Astrophysical Simulations and Data Analyses on the Formation, Detection, and Habitability of Moons Around Extrasolar Planets
AAstrophysical Simulations and Data Analyses onthe Formation, Detection, and Habitability ofMoons Around Extrasolar Planets
Habilitationsschrift eingereicht an der
Fakult¨at f¨ur Physikder Georg-August-Universit¨at G¨ottingen vorgelegt von
Ren´e Heller aus HoyerswerdaG¨ottingen, den 23. September 2019 i ontents Abstract 1
I Scientific Context of this Thesis 3
II Peer-Reviewed Journal Publications 37 v CONTENTS
III Outlook and Appendix 368
ONTENTS v
A Appendix – Non-Peer-Reviewed Conference Proceedings 384
A.1 Constraints on the Habitability of Extrasolar Moons (Heller & Barnes 2014) . . . . . . 384A.2 Hot Moons and Cool Stars (Heller & Barnes 2013b) . . . . . . . . . . . . . . . . . . . 391
B Appendix – Popular Science Publications 396
B.1 Better than Earth (Heller 2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396B.2 Extrasolare Monde – sch¨one neue Welten? (Heller 2013) . . . . . . . . . . . . . . . . . 402B.3 Ist der erste Mond außerhalb des Sonnensystems entdeckt? (Heller 2018b) . . . . . . . 410
Acknowledgements 414 i CONTENTS ist of Figures . − (Io), 3 . − (Eu-ropa), 1 . − (Ganymede), and 1 . − (Callisto) and estimated water contents(see labels) suggest that Io and Europa formed interior to the circumjovian ice line ofJupiter’s early accretion disk, whereas Ganymede and Callisto formed beyond the iceline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 The upper panel shows a computer model of a ring system transiting the star J1407to explain the 70 d of photometric observations from the Super Wide Angle Search forPlanets (SuperWASP) in the lower panel. The thick green line in the background of theupper panel represents the path of the star relative to the rings. Gray annuli indicateregions of the possible ring system that are not constrained by the data. Gradation ofthe red colors symbolizes the transmissivity of each ring. Note that the hypothesizedcentral object of the ring system (possibly a giant planet) is not transiting the star.Image credit: Kenworthy & Mamajek (2015). c (cid:13) AAS. . . . . . . . . . . . . . . . . . . 81.3 The top panel shows the Kepler light curve of the star Kepler-1625. The verticaloffsets are caused by reorientations of the spacecraft every three months (called a Keplerquarter), which changed the position of the target star on the on-board CCD withdifferent pixel sensitivities. The center panel shows the light curve with the flux perquarter normalized to 1. The three transits of the object Kepler-1625 b are now visibleby eye at about 630 d, 1210 d, and 1500 d (Kepler Barycentric Julian Date, KBJD). Thethree panels at the bottom show zooms into these transits. . . . . . . . . . . . . . . . 91.4 This figure is from the discovery paper of the exomoon candidate Kepler-1625 b-i byTeachey et al. (2018). Gray dots with error bars indicate Kepler data (see Figure 1.3),colored curves refer to 100 drawings from a Markov Chain Monte Carlo simulation, andthe the black line refers to the best fit of a dynamical planet-moon model to the threetransits. c (cid:13)
AAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Stellar distances and planetary masses of extrasolar planets listed on . The discovery method of each planet is indicated by the symbol type. The locationsof six planets from the solar system are shown for comparison, indicated by their initials.The cluster of Jovian and super-Jovian planets near 1 AU (along the abscissa) andbetween 1 and 10 M J (along the ordinate) suggests that water-rich moons in their stellarhabitable zones could be highly abundant in the Milky Way. The green-shaded verticalstrip denotes the solar habitable zone defined by the runaway and maximum greenhouse(Kopparapu et al. 2013). This plot is an updated version of Figure 1 from Heller &Pudritz (2015a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Smoothed Particle Hydrodynamic simulations of a late Moon-forming impact using60,000 test particles. The color scale encodes temperatures ranging from 2 000 K (red)to 7 000 K (light blue). Figure adopted from Canup (2004). c (cid:13) Elsevier. . . . . . . . . . 17 iii LIST OF FIGURES (cid:13)
AAS. 203.1 Physical explanation of the barycentric TTV (upper left) and the photocentric TTV(upper right). The two light curves at the bottom illustrate both the TTV b and theTTV p (or PTV) for a moon that is trailing the planet (upper panel, green curve) andleading the planet during the stellar transit (lower panel, red curve). Image credit:Simon et al. (2015). c (cid:13) The Astronomical Society of the Pacific. . . . . . . . . . . . . . 243.2 Orbital sampling effect (OSE) of a simulated transiting exoplanet with moons. Theupper panel shows a model of the phase-folded transit light curve of a Jupiter-sizedplanet around a 0 . R (cid:12) K dwarf star using an arbitrarily large number of transits.The planet is accompanied by three moons of 0 . R ⊕ , 0 . R ⊕ , and 0 . R ⊕ in radialsize, but their contribution to the phase-folded light curve is barely visible with thenaked eye. The lower row of panels shows a sequence of zooms into the prior-to-ingresspart of the planetary transit. The evolution of the OSEs of the three moons is shownfor an increasing number of transits ( N ) used to generate the phase-folded light curves.In each panel, the solid line shows the simulated phase-folded transit and the dashedline shows an analytical model, both curves assuming a star without limb darkening.Image credit: Heller (2014). c (cid:13) AAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Simulated OSE of an m V = 11 K dwarf star (0 . R (cid:12) ) with a Saturn-sized planet andan Earth-sized moin in a Europa-wide orbit as it would be observed with PLATO.The distance between the star and the planet-moon system is 0.3 AU, which yields aperiod of 72 d. Only white noise is assumed, that is to say, the star is assumed to bephotometrically quiet with no spots, flares or the like. . . . . . . . . . . . . . . . . . . 273.4 Transit of Saturn and its ring system in front of the sun as seen by the Cassini spacecraftin September 2016. Note how the rings bend the sun light around the planet, an effectknows as diffraction. Image credit: NASA/JPL/Space Science Institute. . . . . . . . . 284.1 Illumination on an exomoon changes over the course of its orbital revolution aroundthe host planet. In our model that invokes coplanar orbits around the star (of theplanet-moon barycenter) and around the planet (of the moon), the moon’s orbit startsan an orbital phase of 0 and then proceeds in a counter-clockwise fashion to 1. Thecenter of the planetary eclipse occurs at a phase of 1/2. The different contributions oflight falling onto the moon are explained in the legend in the lower left corner. Thisillustration is a modified version of a graphic shown in Heller (2013). . . . . . . . . . . 32 IST OF FIGURES ix
Left : The LongRange Reconnaissance Imager presents details on Io’s sunlit crescent and in the par-tially sunlit plume from the Tvashtar volcano. The source of the plume betrays itslocation as a bright nighttime glow of hot lavas.
Top right : The Multispectral VisibleImaging Camera image shows the contrasting colors of the red lava and blue plume atTvashtar, and the sulfur and sulfur dioxide deposits on Io’s sunlit surface.
Bottom right :The Linear Etalon Imaging Spectral Array image shows that Tvashtar’s glow is evenmore intense at infrared wavelengths and reveals the glows of over ten fainter volcanichot spots on Io’s nightside. Image credit: NASA/Johns Hopkins University AppliedPhysics Laboratory/Southwest Research Institute. . . . . . . . . . . . . . . . . . . . . 344.3 Habitability of circumplanetary orbits for an Earth-sized satellite orbiting a planet often Jupiter masses with an orbital eccentricity of 0.001.
Left : At an age of 100 Myr,the planet is still relatively hot, which makes the moon uninhabitable within a distanceof about 10 R Jup . Right : 900 Myr later, the planet has cooled substantially and thehabitable edge, which defines the radial extent of the the uninhabitable space aroundthe planet, has shrunk to about 7 R Jup . The system is modelled at 1 AU from a sun-like host star. Tidal heating contributes about 10 W m − in both panels. This plotis a modified version of Fig. 4 from Heller & Barnes (2015) that the author of thisthesis presented at the astronomy seminars at Cornell University (11 May 2015) andthe Center for Astrophysics at Harvard University (1 July 2015). . . . . . . . . . . . . 358.1 A sketch of the expected RV curve of Kepler-1625 due to the presence of its gianttransiting planet (plus its hypothetical moon, as the case may be). Symbols on thecurve indicate whether the proposed observations were successful or not (see legend atthe left). The dates shown along the curve refer to the beginning of the night of theproposed observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3728.2 Best fit of a one-planet Keplerian orbit (solid line) to our RV measurements withCARMENES (points with error bars). This solution includes the known transit times(dark red squares) as boundary conditions and it suggest a giant planet with about 5Jupiter masses on an eccentric orbit with e ≈ .
4. Image credit: A. Timmermann. . . 3728.3 Example of a planet-only fit (dashed orange line) to the planet-moon model (solid blackline). The star is sun-like, the planet is Jupiter-sized and in a 30 d orbit around thestar, and the moon is Neptune-sized and in a 3.55 d orbit around the planet. In thisexample, the fitted planet-to-star radius ratio ( r fit ), which goes into the calculationof the transit depth with a power of 2, is slightly overestimated due to the moon’sphotometric signature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3748.4 Some orbital parameters for the planet-moon system. ϕ s is the orbital phase relativeto the line of sight at the time of mid-transit. . . . . . . . . . . . . . . . . . . . . . . . 3758.5 Exomoon indicators for a hypothetical Jupiter-Earth planet-moon system in a 30 d orbitaround a sun-like star as measured from noiseless simulated light curves by fitting aplanet-only model. (a) Transit duration variation (TDV). (b)
Transit timing variation(TTV). (c)
Planetary radius variation (PRV). (d)
TTV vs. TDV diagram. Blue linesrefer to a coplanar planet-moon system with i s = 0 and orange lines depict the effectsfor an inclined planet-moon system with i s = 0 . LIST OF FIGURES
Left:
Without noise.
Right:
With noise contributionsfrom stellar granulation and PLATO-like instrumental noise for an m V = 8 star. Thesubpanels show the autocorrelation. In all panels, blue symbols refer to a coplanarplanet-moon system with i s = 0 and orange symbols depict the effects for an inclinedplanet-moon system with i s = 0 . m V = 8 starobserved with PLATO at a 25 s cadence. . . . . . . . . . . . . . . . . . . . . . . . . . . 3808.8 Transit sequence of the planet radius variation (PRV, left), autocorrelation function ofthe PRV transit sequence (center), and periodogram of the of autocorrelation (right) forvarious planet-moon configurations around an m V = 8 star as observed with PLATO(see Sect. 8.2.5). Blue lines refer to an (cid:15) Eridani-like star, orange lines to a sun-likestar. In the Jupiter-Neptune and Saturn/Super-Earth cases the PRV signal is clearlyvisible in the measured PRV time series. Consequently the signal is also visible in theautocorrelation. Due to only calculating the AC up to a time lag of have the observationlength, a peak in the periodograms is only visible in the P B = 30 d cases and not the P B = 60 d cases. In the Neptune-Earth cases the signal in the PRV time series is barelyvisible, while the autocorrelation and periodogram for the P B = 30 d cases show a clearsignal. There is no visible signal in any of the Earth-Moon cases for in the correspondingtime series, autocorrelation, and periodogram. . . . . . . . . . . . . . . . . . . . . . . . 3818.9 Analysis of the Kepler light curve of Kepler-856 b (KOI-1457.01). (a) PDCSAP Keplerlight curve. (b)
Fractional transit depths variation from Holczer et al. (2016). (c)
Autocorrelation of the transit depth variation. (d)
Periodogram of the transit depthvariation and the autocorrelation function. (e)
Periodograms of the PDCSAP flux of 14Kepler quarters (gray lines) and their mean (black line) after masking out the planetarytransits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 bstract
While the solar system contains as many as about 20 times more moons than planets, no moon hasbeen definitively detected around any of the thousands of extrasolar planets so far. And the questionnaturally arises why an exomoon detection has not yet been achieved. This cumulative habilitationthesis covers three of the key aspects related to the ongoing search for extrasolar moons: 1. thepossible formation scenarios for moons around extrasolar planets; 2. new detection strategies for thesemoons; and 3. the potential of exomoons as hosts to extrasolar life.Maybe one of the most important lessons that can be learned from the history of exoplanet discov-eries is that theories of planet formation in the solar system are poor references for predictions of thekind of planetary systems that can be found around other stars. Or, alternatively, the solar systemhas not proven to be a typical planetary system. As a consequence, one could argue that extrasolarmoon systems might well be very different from the moon systems around the solar system planets. Itwould thus not be very surprising if the first detected exomoon would be extremely different from anyof the solar system moons. And indeed one of the most compelling cases for an exomoon presented inthe literature claims an exomoon candidate, provisionally named Kepler-1625 b-i, which, if confirmed,would be roughly as large as the giant planet Neptune and it would likely have a mass that would beseveral times larger than the combined mass of the planets Mercury, Venus, Earth, and Mars. Thisexomoon candidate is thus a very natural object to be addressed in this thesis and we will look at itboth from a formation and a detection point of view.In fact, the original detection of Kepler-1625 b-i as a candidate was made in search for a phenomenoncalled the orbital sampling effect, a subtle pattern in the transit light curves of extrasolar planets withmoons that was first described in a research paper that is part of this thesis. The proposed exomooncandidate Kepler-1625 b-i is just one example, though a very extreme one, for how future exomoondetections have the potential to fundamentally test our understanding of planet and moon formation.Several studies of moon formation around migrating giant planets in this thesis illustrate a connectionof these models to the solar system, in which the giant planets (and possibly their primordial moonsystems) have supposedly migrated as well. The synthesis of the synchronous formation of giantplanets and their moons is particularly relevant for the availability of water on these moons. In thecase of the four large moons around Jupiter, the so-called Galilean moons, the water contents havepreviously been inferred from a combined analysis of their mean densities, analyses of surface spectra,and their observed internal mass distribution as measured during spacecraft fly-bys. In this thesis,we explain these observations with simulations of the temperature properties in the dusty gas disksaround young, accreting giant planets. We show evidence that the dry Galilean moons Io and Europahave formed interior to the circumjovian water ice line, where water existed in the form of vapourthat could not be accreted, whereas the water-rich Ganymede and Callisto formed beyond the iceline, where they accreted solid water ice. Ultimately, water is observed to be a key ingredient for anyforms of life on Earth. The detections of liquid subsurface water reservoirs on Europa, Enceladus, andpotentially Titan have motivated our studies of exomoon habitability, the possibility of life on moonsaround extrasolar planets. In this thesis, we present the first models of exomoon habitability that takeinto account the irradiation from the star, the reflected and thermal light from the planet, and tidalheating inside the moon.This work is structured as follows. Part I gives a broad introduction to the field of extrasolarmoons with special attention to their formation, detection, and habitability. Part II presents thecumulative part of this thesis with a total of 16 peer-reviewed journal publications listing the authorof this thesis as a lead author, and six publications with the author of this thesis as a co-author.Part III shares some insights into our ongoing research on exomoons in collaboration with masterstudent Anina Timmermann at the Georg-August University of G¨ottingen and former PhD studentKai Rodenbeck at the International Max Planck Research School for Solar System Science and theUniversity of G¨ottingen. The Appendix is a collection of non-peer-reviewed conference proceedingsand popular science publications by the author that further disseminate our research of extrasolarmoons. art I
Scientific Context of this Thesis hapter 1
Introduction
The moons of the solar system planets have become invaluable tracers of the local planet formation,in particular for the Earth-Moon system (Cameron & Ward 1976; Rufu et al. 2017). Crater countson the Moon, for example, serve as a window into the Earth’s bombardment history ( ¨Opik 1960) andgive insights into the number density (or frequency) of interplanetary rocky fragments as a function ofsize (Hartmann 1969). Moreover, the Earth’s Moon has substantially affected the Earth’s spin periodand the direction of its rotational axis (Laskar et al. 1993) from the very beginning. As a consequence,our natural satellite has played a major role in the evolution of the Earth’s climate and therefore inthe origin and evolution of life (Zahnle et al. 2007).Beyond that, the presence of the tiny, asteroid-sized moons Phobos and Deimos around Mars shedslight on the red planet’s bombardment history and early collision frequency (Rosenblatt et al. 2016).The composition of the Galilean moons (Figure 1.1) constrains the temperature distribution in theaccretion disk around Jupiter 4.5 billion years ago (Pollack & Reynolds 1974; Canup & Ward 2002;Heller et al. 2015). And while the major moons of Saturn, Uranus, and Neptune might have formedfrom circumplanetary tidal debris disks (Crida & Charnoz 2012), Neptune’s principal moon Triton hasprobably been captured during an encounter with a minor planet binary (Agnor & Hamilton 2006), aprocessed referred to as tidal disruption. The orbital alignment of the Uranian moon systems suggestsa collisional tilting scenario (Morbidelli et al. 2012) and implies significant bombardment of the youngUranus. The Pluto-Charon system can be considered a planetary binary rather than a planet-moonsystem, since its center of mass is outside the radius of the primary, at about 1.7 Pluto radii. A giantimpact origin of this system delivers important constraints on the characteristic frequency of largeimpacts in the Kuiper belt region (Canup 2005).The presence of the giant planet’s moons has also been used to study the orbital migration history oftheir respective host planets (Deienno et al. 2011; Heller et al. 2015), the properties (such as frequencyand minimum encounter distances) of possible planet-planet scattering events (Deienno et al. 2014),the giant planets’ bombardment histories (Levison et al. 2001), and even the characteristics of the earlyprotoplanetary disk around the sun (Jacobson & Morbidelli 2014). On a smaller scale, planetary ringsconsist of relatively small sub-grain-sized to boulder-sized particles. Their own value is as indicatorsof moon formation and of the tidal or geophysical activity of moons, see Enceladus around Saturn(Spahn et al. 2006).Considering all these beneficial effects of moons for the studies of planet formation in the solarsystem, exomoon and exo-ring discoveries can thus be expected to deliver information on exoplanetformation on a level that is fundamentally new and inaccessible through exoplanet observations alone.Despite the discovery of thousands of planets beyond the solar system within the past two decades,mostly by the space-based Kepler telescope (Borucki et al. 2010; Thompson et al. 2018), however,no moon beyond the solar system (an “exomoon”) has been unambiguously detected and confirmed(Heller 2018a). And so with more than 180 moons known around just eight planets in the solar system,
CHAPTER 1. INTRODUCTION
Io Europa Ganymede Callisto<1% 5% 50% 50%
NASA/JPL/DLR water contents:“rocky” “icy” } Why Are Moons Important?
Constraints on Planet Formation: The Galilean Moon System } Figure 1.1: The Galilean moons around Jupiter The densities of 3 . − (Io), 3 . − (Europa),1 . − (Ganymede), and 1 . − (Callisto) and estimated water contents (see labels) suggestthat Io and Europa formed interior to the circumjovian ice line of Jupiter’s early accretion disk,whereas Ganymede and Callisto formed beyond the ice line.one obvious question to ask about exomoons is: Where are they?Chapter 2 of this thesis gives and overview of the field of moon formation with an emphasis on casestudies for the formation of extrasolar moons. Chapter 5 compiles a range of research papers led bythe author that tackle the question of how large, potentially detectable exomoons could form aroundgas giant planets. So far, most of the searches for exomoons have been executed as piggy-back science on projects with adifferent primary objective. To give just a few examples, Brown et al. (2001) used the exquisite pho-tometry of the
Hubble
Space Telescope (HST) to observe four transits of the hot Jupiter HD 209458 bin front of its host star. As the star has a particularly high apparent brightness and therefore deliversvery high signal-to-noise transit light curves, these observations would have revealed the direct transitsof slightly super-Earth-sized satellites ( (cid:38) . R ⊕ ; R ⊕ being the Earth’s radius) around HD 209458 bif such a moon were present. Alternatively, the gravitational pull from any moon that is more massivethan about 3 M ⊕ ( M ⊕ being the Earth’s mass) could have been detected as well. Yet, no evidencefor such a large moon was found. Brown et al. (2001) also constrained the presence of rings aroundHD 209458 b, which must be either extremely edge-on (so they would barely affect the stellar brightnessduring the transit) or they must be restricted to the inner 1 . .2. AN EXOMOON DETECTION ON THE HORIZON 7 no conclusive evidence.The Hunt for Exomoons with Kepler (HEK) project (Kipping et al. 2012), the first dedicated surveytargeting moons around extrasolar planets, is probably the best bet for a near-future exomoon de-tection. Their analysis combines TTV and TDV measurements of exoplanets with searches for directphotometric transit signatures of exomoons. The most recent summary of their Bayesian photodynam-ical modeling (Kipping 2011) of exomoon transits around a total of 57 Kepler Objects of Interest hasbeen presented by Kipping et al. (2015). Using a different approach that has been developed by theauthor of this thesis, the so-called orbital sampling effect in the transit light curves (see Section 3.2.1Heller 2014), the HEK team recently announced the discovery of an exomoon candidate around thetransiting Jupiter-sized object Kepler-1625 b (Teachey et al. 2018). Section 1.2.2 is devoted to a moredetailed discussion of this object. Other teams found unexplained TTVs in many transiting exoplan-ets from the Kepler mission (Szab´o et al. 2013), but without additional TDVs or direct photometrictransits, a robust exomoon interpretation is impossible.
While a definite exomoon discovery remains to be announced, some tentative claims have already beenpresented in the literature. One of the first exomoon claims was put forward by Bennett et al. (2014)based on the microlensing event MOA-2011-BLG-262. Their statistical analysis of the microlensinglight curve, however, has a degenerate solution with two possible interpretations. It turns out thatan interpretation invoking a 0 . +0 . − . M (cid:12) star with a 17 +28 − M ⊕ planetary companion at 0 . +0 . − . AUis a more reasonable explanation than the hypothetical 3 . M Jup -mass free-floating planet with a0 . M ⊕ -mass moon at a separation of 0.13 AU. Sadly, the sources of microlensing events cannot befollowed-up. As a consequence, no additional data can possibly be collected to confirm or reject theexomoon hypothesis of MOA-2011-BLG-262.In the same year, Ben-Jaffel & Ballester (2014) proposed that the observed asymmetry in the transitlight curve of the hot Jupiter HD 189733 b might be caused by an opaque plasma torus around theplanet, which could be fed by a tidally active natural companion around the planet (which is notvisible in the transit light curve itself, in this scenario). But an independent validation has not beendemonstrated.Using a variation of the exoplanet transit method, Hippke (2015) presented the first evidence ofan exomoon population in the Kepler data. The author used what he refers to as a superstack, acombination of light curves from thousands of transiting exoplanets and candidates, to create kind ofan average transit light curve from Kepler with a very low noise-to-signal level of about 1 part permillion. This superstack of a light curve exibits an additional transit-like signature to both sides ofthe averaged planetary transit, possibly caused by many exomoons that are hidden in the noise ofthe individual light curves of each exoplanet. The depth if this additional transit candidate featurecorresponds to an effective moon radius of 2120 +330 − km, or about 0.8 Ganymede radii. Interestingly,this signal is much more pronounced in the superstack of planets with orbital periods larger than about35 d, whereas more close-in planets do not seem to show this exomoon-like feature. This finding is inagreement with considerations of the Hill stability of moons, which states that stellar gravitationalperturbations may perturb the orbit of a moon around a close-in planet such that the moon will beejected (Domingos et al. 2006).Beyond the exquisite photometric data quality of the Kepler telescope, the COnvection ROtation andplanetary Transits (CoRoT; Auvergne et al. 2009) space mission also delivered highly accurate space-based stellar observations. One particularly interesting candidate object is CoRoT SRc01 E2 1066,which shows a peculiar bump near the center of the transit light curve that might be induced by themutual eclipse of a transiting binary planet system (Lewis et al. 2015), i.e., a giant planet with a verylarge and massive satellite. However, only one single transit of this object (or these two objects) hasbeen observed, and so it is currently impossible to discriminate between a binary planet and a starspot crossing interpretation of the data.There has also been one supposed observation of a transiting ring system, which has been modeled to CHAPTER 1. INTRODUCTION
Figure 1.2: The upper panel shows a computer model of a ring system transiting the star J1407 toexplain the 70 d of photometric observations from the Super Wide Angle Search for Planets (Super-WASP) in the lower panel. The thick green line in the background of the upper panel represents thepath of the star relative to the rings. Gray annuli indicate regions of the possible ring system that arenot constrained by the data. Gradation of the red colors symbolizes the transmissivity of each ring.Note that the hypothesized central object of the ring system (possibly a giant planet) is not transitingthe star. Image credit: Kenworthy & Mamajek (2015). c (cid:13)
AAS.explain the curious brightness fluctuation of the 16 Myr young K5 star 1SWASP J140747.93-394542.6(J1407 for short) observed around 29 April 2007 (Mamajek et al. 2012). The lower panel of Figure 1.2shows the observed stellar brightness variations, and the panel above displays the hypothesized ringsystem that could explain the data. This visualization nicely illustrates the connection between ringsand moons, as the gaps in this proposed ring system could have been cleared by large moons thatwere caught in a stage of ongoing formation (Kenworthy & Mamajek 2015). The most critical aspectof this interpretation though is in the fact that the hypothesized central object has not been observedin transit. Another issue is that the orbital period of this putative ring system around J1407 can onlybe constrained to be between 2.33 yr and 200 yr. In other words, the periodic nature of this proposedtransit event has not actually been established and it could take decades or centuries to re-observethis phenomenon, if the interpretation were valid in the first place.
One particularly interesting case of a tentative exomoon detection is the candidate Kepler-1625 b-i. In July 2017, Teachey et al. (2018) announced the detection of an exomoon-like signature in the .2. AN EXOMOON DETECTION ON THE HORIZON 9 N o r m a li ze d P D C S A P F L UX Time [KBJD] 0.994 0.995 0.996 0.997 0.998 0.999 1 1.001 1.002 1.003 634 635 636 637 638 N o r m a li ze d P D C S A P F L UX Time [KBJD] 0.994 0.995 0.996 0.997 0.998 0.999 1 1.001 1.002 1.003 1209 1210 1211 1212 1213 N o r m a li ze d P D C S A P F L UX Time [KBJD] 0.994 0.995 0.996 0.997 0.998 0.999 1 1.001 1.002 1.003 1497 1498 1499 1500
Figure 1.3: The top panel shows the Kepler light curve of the star Kepler-1625. The vertical offsetsare caused by reorientations of the spacecraft every three months (called a Kepler quarter), whichchanged the position of the target star on the on-board CCD with different pixel sensitivities. Thecenter panel shows the light curve with the flux per quarter normalized to 1. The three transits of theobject Kepler-1625 b are now visible by eye at about 630 d, 1210 d, and 1500 d (Kepler BarycentricJulian Date, KBJD). The three panels at the bottom show zooms into these transits.transit light curve of the exoplanet candidate Kepler-1625 b based on three transits observed withKepler between 2009 and 2013. The candidate moon signal was found using a newly developed andefficient method to search for exomoon signals in large amounts of data that has been previouslydeveloped by the author of this thesis. In brief, this method (referred to as the orbital sampling effect,OSE) is based on a search for an additional dip in both the pre-ingress and post-egress wings of the Formally speaking, the transiting object Kepler-1625 b has not yet been confirmed as an exoplanet, which wouldinvolve observation techniques that are independent of the transit method. Such a confirmation could involve TTVs, e.g.caused by other planets in the system (Agol et al. 2005) or measurements of the stellar radial velocities (RVs), whichwould show periodic variations due the gravitational interaction with Kepler-1625 b. Both TTVs and RVs could be usedto determine the mass of Kepler-1625 b and therefore would enable a classification of this transiting object as either aplanet, or a brown dwarf, or even a low-mass star. The degeneracy of the planetary interpretation of the Kepler transitlight curve alone is treated in Heller (2018c) and in Section 5.4.
Figure 1.4: This figure is from the discovery paper of the exomoon candidate Kepler-1625 b-i byTeachey et al. (2018). Gray dots with error bars indicate Kepler data (see Figure 1.3), colored curvesrefer to 100 drawings from a Markov Chain Monte Carlo simulation, and the the black line refers tothe best fit of a dynamical planet-moon model to the three transits. c (cid:13)
AAS.orbital phase-folded planetary transit light curve, respectively. Geometrically speaking, in the case ofa coplanar moon around a transiting planet, the moon should occasionally start its own passage of thesky-projected stellar disk prior to the planetary transits. At other times then, the planet should enterthe stellar disk first and the moon would lag behind, e.g. by minutes or hours. Averaging over manytransits, one can expect that the moon’s sky-projected orbital position from its host planet shouldsmear out like a ring around the planet, which reveals itself as the above-mentioned additional pre-and post-transit signature around the planetary transit. Details have been developed by Heller (2014)and Heller et al. (2016a) and can be found in Sects. 6.1 and 6.2.Figure 1.3 illustrates the Kepler data of the host star Kepler-1625 (KIC 4760478, KOI-5084). Theupper two panels show the Pre-Search Data Conditioning Simple Aperture Photometry (PDCSAP)flux, which is freely available online at the NASA Exoplanet Archive . The three panels at the bottomof Figure 1.3 give an idea of the size of the planet candidate: with the normalized stellar flux dimmingas much as about d = 1 − .
995 = 0 . R p ) would be about √ d R (cid:63) , or about 7 % the radius of the star.The star, located at about 2181 +332 − pc from the Sun, has been classified as an evolved star witha mass of M (cid:63) = 1 . +0 . − . M (cid:12) , a radius of R (cid:63) = 1 . +0 . − . R (cid:12) , an effective temperature of T eff ,(cid:63) = 5548 +83 − K (Mathur et al. 2017), and a Kepler magnitude of K = 13 .
916 (as per the NASAExoplanet Archive). Hence, R p = 1 . +0 . − . Jupiter radii ( R J ), neglecting any errors from the https://exoplanetarchive.ipac.caltech.edu. .3. THE VALUE OF MOONS FOR ASTROBIOLOGY 11 measurement of d .The transiting object Kepler-1625 b has an orbital period of 287 . ± . ±
00 : 46 : 18) UT. Teachey et al. (2018) have taken observations of this transit with the
HubbleSpace Telescope and the community is eagerly awaiting their improved evaluation of the exomoonhypothesis for Kepler-1625 b-i.Figure 1.4 shows the original light curve presented by Teachey et al. (2018), including the detrendedKepler data (gray dots) and a range of plausible model fits of a transiting planet with a moon. Evena by-eye comparison of the gray data points in Figure 1.4 with the data points shown in the bottompanels of Figure 1.3 reveals significant differences. These differences stem from an additional lightcurve detrending procedure applied by Teachey et al. (2018), which is supposed to remove stellar,instrumental, and other sources of time-correlated flux variability (so-called red noise) from the lightcurve. The aim of this procedure, sometimes referred to as “pre-whitening” of the data (Aigrain &Irwin 2004), is to remove unwanted variations in the data prior to fitting a noiseless model to thedata. As we show in our follow-up study, this approach bears the risk of both removing actual signalfrom the data and of introducing new systematic variability (Rodenbeck et al. 2018; Sect. 6.7). Infact, a moon-like signal can be artificially injected into the data of a planet-only transit, leading to thedetection of a false positive. The simultaneous fitting of the stellar, systematic, and astrophysicallyinduced variability of the light curve, including any possible moon transit features, developed in Helleret al. (2019) is less prone to artificial injection of false positive moon signals (see Sect. 6.8). Butthe Bayesian information criterion (BIC) used by Teachey et al. (2018) to provide evidence for theexistence of Kepler-1625 b-i was shown to be inadequate for this particular data set and given its non-Gaussian noise properties. Even more dramatic is the finding by Kreidberg et al. (2019) of an absenceof the actual moon-like transit feature in the
Hubble data, suggesting that the exomoon candidatearound Kepler-1625 d really is an artifact of the data extraction procedure.Chapter 3 of this book gives a summary of the community’s efforts to search for extrasolar moons.Chapter 6 is dedicated to a number of peer-reviewed studies by the author of this thesis that presentnovel detection methods for exomoons, with a focus on the analysis of space-based transit photometry.
While the interest in the possibility of an independent origin of life beyond Earth has been speculativeor fictional for most of humanity’s history, the advent of modern astronomy has opened the possibilityof the detection and scientific study of extraterrestrial life. Among the several approaches that havebeen pursued in the search for life beyond Earth, the most prominent ones include the search foractive or past life on Mars (McKay et al. 1996), the search for organic (that is carbon-involving) or saltchemistry in the water ice plumes of Saturn’s moon Enceladus (Porco et al. 2006; Hsu et al. 2015), andthe search for extraterrestrial intelligence via interstellar radio communication (Cocconi & Morrison1959; Isaacson et al. 2017). In fact, astrophysical concepts and technological know-how have maturedso rapidly over the past few decades that the in-situ detection of extraterrestrial biological activityin the solar system, the remote detection of intelligent life, or the detection of chemical biomarkersin the atmospheres of transiting exoplanets (Schwieterman et al. 2016) may all be promising on theirown over the coming decades.Most studies and concepts dedicated to the search for life beyond Earth focus on planets, which isa natural approach given the fact the only world known to harbor life is a planet, Earth. This has ledplanetologists and astronomers to develop the concept of the stellar habitable zone, which characterizesan exoplanet’s surface habitability as a function of its distance from the star and its absorbed stellarenergy flux. The oldest record of a description of a circumstellar zone suitable for life traces back toWhewell (1853, Chap. X, Section 4) who, referring to the local stellar system in a qualitative way, The transit epoch of (2 , , . . ± . . ± . called this distance range the “Temperate Zone of the Solar System”. The modern, much more complexversion of this concept was introduced by Kasting et al. (1993). Their one-dimensional atmosphericclimate model includes a parameterization of the CO feedback and of the geological carbon-silicatecycle, both of which are key to the location of the inner and the outer edges of the HZ around the hoststar. The inner edge is defined by the activation of the moist or runaway greenhouse process, whichdesiccates the planet by evaporation of atmospheric hydrogen; the outer edge is defined by CO freezeout from the atmosphere, which breaks down the greenhouse effect whereupon the planet transitionsinto a permanent snowball state.The principal assumption of this astronomer’s concept of habitability is in the presence of liquidsurface water. It has been argued that this pre-requisite is vital since, on Earth, life exists in virtuallyany place where there is liquid water. Some primitive forms of life, such as bacteria, archea, and uni-cellular eukaryotes can grow at temperatures as low as − ◦ C and it has been possible to grow planetsfrom seeds that had been frozen for about 32,000 yr (Yashina et al. 2012). The lower temperaturelimit for higher plants and invertebrates to show ongoing metabolism and to reproduce, however, is atabout − ◦ C (Clarke 2014) and liquid water has been identified as a key ingredient for the completionof their life cycles.While extrasolar planets are natural targets for humans to search for extraterrestrial life, it hasrecently become clear that moons could be equally or even better suited to host life. In fact, mostof the liquid water in the solar system is stored in the sub-surface oceans of Jupiter’s moon Europa,which is supposed to contain as much as two to three Earth oceans worth of water, details dependingon the yet unknown depth of the ocean (Squyres et al. 1983; Carr et al. 1998). Other moons, such asGanymede (around Jupiter) as well as Titan and Enceladus (both around Saturn) also show convincingevidence for liquid subsurface water. The reason for the abundance of liquid water in the outer regionsof the solar system is in the distribution of water during the formation of the planets and moons,almost 4.6 billion years ago. While the rocky planets all formed within the hot and dry regionsof the protoplanetary nebula, where water only existed in the form of vapor that could hardly beincorporated into the newly forming planets , the giant planets and their moons formed beyond theso-called water ice line of the solar system. The ice line, thought to have been located between 2 AUand 3 AU (Hayashi 1981; Lecar et al. 2006), has been identified as a dividing line between the dryand wet (or frozen) regions of the solar system. In fact, water ice was important to rapidly form thegrowing cores of the soon-to-be giant planets (Lissauer et al. 2009). As a consequence, the moons ofthe giant planets had to form beyond the sun’s water ice line as well and so sufficient amounts of waterwere present to also be incorporated into their structures.Now given that water-rich moons are abundant around the giant planets of the solar system, andgiven that most giant planets beyond the solar system are actually found near their stellar HZs (Heller& Pudritz 2015a), one may speculate that the moons of these giant planets have taken a piggybackride on their giant planets from their water-rich formation regions beyond the stellar ice line to theirpresent location similar to that of the Earth around the Sun. Figure 1.5 illustrates the distributionof the knows exoplanets with mass measurements as a function of distance to their respective hoststars. The abundance of Jupiter- and super-Jupiter-sized planets can readily be observed near 1 AU.It is important to keep in mind, however, that Figure 1.5 is heavily biased by the sensitivities of thedifferent detection techniques (see figure legend). As a consequence, the observed planet populationsare by no means representative of the actual exoplanet abundances, or occurrence rates (Howard et al.2012; Dressing & Charbonneau 2015). In fact, most of the validated exoplanets have been found withKepler via the transit method, and so they lack a firm mass estimate. These planets do not evenshow up in Figure 1.5. If, instead, one would plot the distance distribution of all planets with knownradii (rather than masses), then this figure would be dominated by a population of super-Earths andmini-Neptunes near 0.1 AU. The source of water on Earth has, in fact, not been unambiguously determined. Different lines of evidence suggesteither dehydration of water originally stored in Earth’s internal hydrate minerals (Schmandt et al. 2014) or the deliveryof water from asteroids or comets (Sarafian et al. 2014; Chan et al. 2018). .3. THE VALUE OF MOONS FOR ASTROBIOLOGY 13
V E J S U Nsolarhabitablezone p l a n e t a r y m a ss [ J up it e r m a ss e s ] stellar distance [AU] transitingRV or astrometrydirect imagingsolar system -2 -1 -2 -1 Figure 1.5: Stellar distances and planetary masses of extrasolar planets listed on .The discovery method of each planet is indicated by the symbol type. The locations of six planetsfrom the solar system are shown for comparison, indicated by their initials. The cluster of Jovian andsuper-Jovian planets near 1 AU (along the abscissa) and between 1 and 10 M J (along the ordinate)suggests that water-rich moons in their stellar habitable zones could be highly abundant in the MilkyWay. The green-shaded vertical strip denotes the solar habitable zone defined by the runaway andmaximum greenhouse (Kopparapu et al. 2013). This plot is an updated version of Figure 1 from Heller& Pudritz (2015a).The formation, dynamical (orbital) stability, and the resistance of moons against evaporative de-struction during their possible circumstellar migration with their host planets has been addressedrecently in the research literature. As a key outcome, Lehmer et al. (2017) showed that moons moremassive than Ganymede could maintain significant amounts of surface water in their stellar habitablezones, whereas less massive moons (e.g. as light as Europa) would tend to lose their waters ratherquickly through a runaway greenhouse effect and subsequent H O photodissociation in the upper at-mosphere, which would lead to the escape of atmospheric hydrogen and, hence, entire desiccation ofthe moon. The possibility of massive moons around Jovian and super-Jovian planets has been demon-strated through simulations of the circumplanetary accretion disks (Heller & Pudritz 2015a), and theorbital stability of these large satellites has been shown to be possible in many (though not all) cases(Namouni 2010; Spalding et al. 2016; Hong et al. 2018).It is thus possible that moons offer a large part of the extraterrestrial habitable real estate andthat large moons beyond the solar system could be habitable and detectable in the near future, as wediscuss in Chapter 4. This has led the author of this thesis to develop a model for the habitable zonesof moons around giant planets, which is presented in Chapter 7. hapter 2
Formation Mechanisms for Moons
Although no moon has been detected beyond the solar system, there is no reason to believe theydon’t exist. In fact, most astronomers were certain that planets should exist around other stars evenprior to the first detections of planets beyond the solar system (Wolszczan & Frail 1992). And justa few years after the discovery of extrasolar planets around solar-type stars (Mayor & Queloz 1995),exoplanets had soon outnumbered the eight (by then formally nine) solar system planets. In the nearfuture, thanks to the improvements of ground-based radial velocity surveys, such as the High-AccuracyRadial velocity Planetary Searcher (HARPS; Mayor et al. 2003), and thanks to the many discoveriesof transiting extrasolar planets and candidates with the Kepler space telescope (Batalha et al. 2013;Rowe et al. 2014; Thompson et al. 2018), and in anticipation of thousands of new exoplanet discoverieswith NASA’s Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015) and ESA’s PLAnetaryTransits and Oscillations of stars (PLATO; Rauer et al. 2014) satellite, we will soon know as many asa thousand times more exoplanets than planets in the solar system. Will we be able to find any moonsaround them? How many of these planets do actually host moons and what are their properties?The architecture of the solar system has proven inappropriate for most extrasolar planetary systemsin many regards. For one thing, the most abundant population of exoplanets turned out to be apreviously unknown class of super-Earths or mini-Neptunes in relatively tight stellar orbits. Theplanets have radii between about 2 to 4 Earth radii ( R ⊕ ) and orbital distances of about 0.1 AU totheir host stars. For comparison, the solar system has no planet with a radius between 1 R ⊕ and3 . R ⊕ , the radius of Neptune. And the innermost planet, Mercury, orbits the sun at about 0.4 AU.For another thing, the first exoplanets were discovered around pulsars (Wolszczan & Frail 1992), theburnt-out remnants of rather massive progenitor stars with about 10 M (cid:12) to 30 M (cid:12) . The survivalof these planets during the late stages of stellar evolution, including the violent disruption during asupernova, remains unsolved until today. Moreover, the discovery of a hitherto unknown family ofhot Jupiters (Jupiter-mass planets in very tight orbits of typically 0.05 AU) came to a big surprise ofastronomers (Mayor & Queloz 1995), which triggered new theories of planet formation such as planetmigration of various types (Lin et al. 1996; Trilling et al. 1998). With no firm exomoon detectionavailable as of today, any speculation on their formation must necessarily be based on observations ofthe solar system moons. Hence, throughout this chapter we must keep in mind the perils of using thesolar system architecture as a proxy for extrasolar moon systems.The formation of moons in the solar system can broadly be classified into three distinct scenarios, allof which we discuss in the following subsections of this chapter: (1.) re-accretion after giant impacts onterrestrial planets; (2) accretion in the circumplanetary disks of young giant planets; and (3) capturevia the tidal disruption during a close encounter between a massive planet and a planetary binary. The formation of moons around terrestrial (that is, Earth-like, rocky) planets is arguably of highestimportance to us. Several lines of evidence lead to the conclusion that the Moon formed through a giant collision of the proto-Earth with a Mars-sized object (sometimes referred to as Theia; Quarles& Lissauer 2015) about 30 Myr to 50 Myr after the formation of the primordial Earth (Hartmann &Davis 1975). Our home planet is the only rocky planet in the solar system with a substantial moon,which had a major effect on the evolution of the Earth’s spin and therefore on its climate. We shallstart with an overview of the key observations of the Earth-Moon system.1. The Moon has a small iron core, possibly with a radius of about 350 km or less (Wieczorek 2006).The Moon is also depleted in volatile elements relative to the Earth (Khan et al. 2006).2. Isotope dating of lunar rock suggests an age essentially equal to the age of the Earth.3. The Moon’s orbit carries most of the angular momentum in the Earth-Moon system (Cameron &Ward 1976).4. The Moon is receding from Earth at a rate of about 3.8 cm per year (Walker & Zahnle 1986;Dickey et al. 1994).5. The length of the Earth’s day is increasing at a rate of about 1.8 ms per century (Stephensonet al. 2016).The total angular momentum of the Earth-Moon binary ( L tot (cid:36) ⊕ ) is composed of the two contributionsfrom the rotational angular momenta of the Earth ( L rot ⊕ ) and the Moon ( L rot (cid:36) ) and of the system’sorbital angular momentum ( L orb (cid:36) ⊕ ): L rot ⊕ = I ⊕ ω ⊕ = 25 M ⊕ R ⊕ ω ⊕ = 7 . × kg m s − (2.1) L rot (cid:36) = I (cid:36) ω (cid:36) = 25 M (cid:36) R (cid:36) ω (cid:36) = 2 . × kg m s − (2.2) L orb (cid:36) ⊕ = | (cid:42) r (cid:36) ⊕ × (cid:42) p (cid:36) | = r (cid:36) ⊕ M (cid:36) v orb (cid:36) = 2 . × kg m s − , (2.3)where I is the moment of inertia of a sphere, ω is the spin frequency, M is the mass, R is the radius,and v orb is the orbital velocity of the respective object, with indices ⊕ and (cid:36) referring the Earthand Moon, respectively. Note that in Equation (2.3) we have assumed that the Moon is in a circularorbit, which is an adequate approximation given its low orbital eccentricity of about 0.0549. Mostimportant, Equations (2.1) to (2.3) show that most of the angular momentum is currently containedin the Moon’s orbit, that is, about 80.3 %. This is a particular distinction from all other moons in thesolar system, where most of the angular momentum is stored in the rotation of the host planet. Why?The leading theory that embraces all the above observations envisions a collision between a Mars-sized impactor and the proto-Earth. This giant impact generated a circumplanetary halo of ejectedmaterial that collapsed into a disk and then into a moon within only a few years (Canup & Asphaug2001). The satellite supposedly formed at about 3 . R ⊕ , which is near the Earth’s Roche radius,interior to which any rocky moon would be shredded by differential gravitational forces across itsdiameter, that is, by tidal forces. An illustration of this process is shown in Figure 2.1 that is basedon Smoothed Particle Hydrodynamic (SPH) simulations from Canup (2004).This primordial distance of the Moon must be compared to its current orbital separation of about384,000 km, or about 60 R ⊕ . The driver for the Moon’s orbital recession is found in the Earth’s tides.The gravitational pull from the Moon deforms the Earth across its diameter and thereby raises twoprincipal tidal bulges, one roughly facing the Moon, the other one on the opposite side of the Earth.The Earth’s rotation period of about 24 hr is much smaller than the Moon’s orbital period of about27 d. This asynchronicity between the Earth’s rotation and the Moon’s orbit combined with the factthat the tidal bulges cannot instantaneously align with the line connecting the centers of mass of theEarth and the Moon due to friction inside the Earth, yields that the Earth’s Moon-facing tidal bulgeis constantly leading the Moon. This lag corresponds to about 600 s (Lambeck 1977; Neron de Surgy .1. GIANT IMPACTS ON TERRESTRIAL PLANETS 17 Figure 2.1: Smoothed Particle Hydrodynamic simulations of a late Moon-forming impact using 60,000test particles. The color scale encodes temperatures ranging from 2 000 K (red) to 7 000 K (light blue).Figure adopted from Canup (2004). c (cid:13)
Elsevier.& Laskar 1997). Naturally, the Moon-opposing tidal bulge on the Earth is lagging by the same timelag. However, with the Moon-facing tidal bulge being closer to the Moon, there will be a net torqueof the two bulges on the Earth’s rotation, which is dominated by the Moon-facing bulge. This torqueis the origin of both the Moon’s recession from Earth (item 4. in the list above) and the increase ofthe Earth’s length of day (item 5.). The connection of the two processes is the transfer of angularmomentum from the Earth’s rotation to the Moon’s orbit.In fact, observations of the time lag of the Earth’s tidal bulges can be used to estimate the recessionrate and vice versa. Let us consider a tidal model for a two-body system of two deformable bodies(Hut 1981; Leconte et al. 2010), here the Earth and Moon. In this model, the change of the orbitalsemimajor axis d a/ d t can be calculated from first principles as (Heller et al. 2011)d a d t = 2 a GM ⊕ M (cid:36) (cid:88) i (cid:54) = j Z i (cid:18) cos( ψ i ) f ( e ) β ( e ) ω i n − f ( e ) β ( e ) (cid:19) , i ∈ {⊕ , (cid:36) } , (2.4)where G is the gravitational constant, n is the orbital frequency, ω i are the spin frequencies, ψ i arethe spin-orbit angular misalignments, τ i are the time lags, e is the orbital eccentricity Z i = 3 G k ,i M j ( M i + M j ) R i a τ i , (2.5) k ,i is the potential Love number of degree 2 of the i th body, and the extension functions in e are β ( e ) = (cid:112) − e ,f ( e ) = 1 + 312 e + 2558 e + 18516 e + 2564 e ,f ( e ) = 1 + 152 e + 458 e + 516 e following the nomenclature of Hut (1981). Note that these extensions collapse to 1 for zero eccen-tricities. In case of the Earth-Moon system, where the orbit is nearly circular ( e ≈ ψ i ≈ ω (cid:36) = n ), Equation (2.4) simplifies tod a d t ≈ a GM ⊕ M (cid:36) G k , ⊕ M (cid:36) ( M ⊕ + M (cid:36) ) R ⊕ a τ ⊕ (cid:16) ω ⊕ n − (cid:17) . (2.6)With a nominal value for the Earth’s second degree tidal Love number of 0.3, Equation (2.6) yields arecession rate of d a/ d t = 3 . − , in remarkable agreement with observations (Walker & Zahnle1986; Dickey et al. 1994). A similar equation can be derived for the spin-down rate of the Earth,d ω ⊕ / d t , which also yields a value close to the observed increase of the length of the day.One major longstanding riddle of the Earth-Moon system has been in the fact that a numericalintegration of d a/ d t and d ω ⊕ / d t backwards in time predicts a gradual infall of the Moon into theEarth with a merger at only about 1.5 billion years in the past (Bills & Ray 1999). This result is instrong disagreement with the isotopic dating of lunar rock brought back from the Apollo missions,which suggest a lunar age of about 4.5 billion years. The solution to this problem can be found inthe frequency-dependence of the efficiency of tidal dissipation in the Earth’s oceans, which must havebeen much weaker in the past (Webb 1980). In fact, the Earth currently seems to be going through aphase of enhanced tidal forcing.Moving on to the possible formation of Moon-like natural satellites around extrasolar planets, nu-merical N -body simulations of gas-free particle disks around Earth-mass planets were able to reproducemany of the features of the Earth-Luna system (Hyodo et al. 2015). The resulting planet-to-moonmass ratios of these post-impact simulations, for example, are usually between several times 10 − anda few times 10 − , compared to the value of about 1 . × − or roughly 1 /
81 for the Moon-to-Earthmass ratio. As an interesting side note, in about a third of these simulations by Hyodo et al. (2015),the resulting moon system consists of two moons rather than one. These results are in agreement withthe findings of Elser et al. (2011), who also used N -body simulations to study post-impact satelliteformation around Earth-like planets and who found similar mass ratios. Both studies seem to suggest,however, that the Earth’s moon has become relatively large compared to the distribution of the out-comes from these simulations. As a consequence, large and possibly detectable (Luna-like) exomoonsaround terrestrial planets might be rare even if moon formation via giant impacts is common aroundEarth-like exoplanets. The most common process for moon formation in the solar system was the in-situ formation in thecircumplanetary accretion disks that formed around giant planets. It has been suggested that thesedisks of gas ( ≈
99 % of the disk mass) and dust ( ≈
1. Solids-enhanced minimum mass model
In what is referred to as the “solids enhanced minimum mass model” (Mosqueira & Estrada2003a,b; Estrada et al. 2009), the formation of moons commences once sufficient gas has beenremoved from what started out as a massive subnebula, at which point turbulence in this newcircumplanetary disk has subsided. Protomoons form from the solid materials transported to andaccreted by the disk through ablation and capture of planetesimal fragments passing through the .2. ACCRETION DISKS AROUND YOUNG GAS GIANT PLANETS 19 massive disk. A key ingredient to this model is the division of the disk in two density and opacityregimes. The inner, high-density part of the disk is thought to be the leftover of the gas accretedby the young planet, whereas the outer part is the result from longterm gas accretion onto theplanet once it has opened up a gap in the circumstellar accretion disk. In this model, the innerGalilean moons Io, Europa, and Ganymede formed on a short time scale of between 10 and 10 yrin the inner disk around Jupiter, while Ganymede formed in the outer disk and on a time scale of 10 yr.
2. Gas-starved disk model
By contrast, the “actively supplied gaseous accretion disk model” (Canup & Ward 2002, 2006, 2009)suggests that the moons around the giant planets that we observe today are the last of a number ofmoon generations that were constantly formed and destroyed around their respective host planets.This model of an actively supplied gaseous accretion disk assumes a low-mass, viscously evolvingprotosatellite disk with peak surface densities around 100 g cm that is continuously supplied with massfrom the circumstellar protoplanetary disk. In the traditional picture of this model, the temperatureprofile of the circumplanetary disk is dominated by viscous heating of the gas, while the planetaryluminosity has long been assumed a minor role. Protomoons are assumed to build up from dustgrains that are supplied by gas infall. Once a protomoon has grown massive enough, it becomessubject to asymmetric gravitational torques from the protosatellite disk, which cause it to lose angularmomentum and radially move toward the planet through a process known as type I migration (Tanakaet al. 2002). One of the key arguments in favor of the gas-starved disk model over other models is itsability to explain the universal scaling law of the observed total moon mass around the giant planetsto be about 10 − times the mass of the host planet (Canup & Ward 2006) by a balance between themoon loss rate towards the planet (via type I moon migration) and the resupply of material for newmoons (from the circumstellar disk). Further refinements of the model were given by Sasaki et al.(2010), who introduced an inner magnetic cavity to the accretion disk around Jupiter and included theeffect of Jupiter’s gap opening in the protoplanetary disk around the sun, both of effects of which wereirrelevant for Saturn. The resulting N -body simulations with collisions produced satellite architecturesaround the two gas giant planets that were ver distinct and in agreement with observations: systemsof multiple large moons around Jupiter and systems dominated by a single massive moon aroundSaturn. The compositional aspects of the resulting moons, with a focus on their water ice contents,were investigated by Ogihara & Ida (2012).Several papers in this thesis have contributed to the further development of the gas-starved diskmodel. In Heller & Pudritz (2015b), we presented the first 2D computations of the gas temperatureand density distributions around Jupiter-like planets and of super-Jovian planets under the explicitinclusion of all four heating terms: (1) viscous heating in the disk, (2) accretion heating from thematter falling onto the disk, (3) the planetary luminosity, and (4) the ambient temperature of thesolar nebula. An example of the resulting temperature distribution around a Jupiter-like planet at1 AU from its young host star, about 10 yr after the onset of accretion is shown in Fig. 2.2. One keyresult of this study is that super-Jovian planets at about 1 AU around their sun-like stars, severaldozens of which have already been discovered, should regularly form massive moon systems. In fact,we found that the scaling law for the satellite-to-planet mass ratio of M s /M p ≈ − (Canup &Ward 2006) extends from the solar system giant planets to planets as massive as ten Jupiter masses.One crucial ingredient to this mass scaling is the location of the water ice line during the final stagesof planet accretion (see the white arc in Fig. 2.2). Beyond the water ice line, ice crystals can beaccreted and incorporated into the forming moons, whereas closer to the planet water vapor is toovolatile to be accreted. Our results for the continued scaling law are thus the outcome of an interplaybetween the luminosity evolution of the planet, which determines the evolution of the radial positionof the ice line, and the radial extend of the circumplanetary disk: more massive planets have theirwater ice lines farther out, thereby preventing the formation of massive Ganymede-like ice moonsin the inner regions of their disks, but they also have larger disks. In Heller & Pudritz (2015a) we Figure 2.2: Temperature distribution in Jupiter’s accretion disk about 1 Myr after then onset of planetformation. The planet (not shown) is located at the origin of the coordinate system. Distances alongthe axes are measured in units of Jupiter radii. The instantaneous water ice line, where the diskreaches 170 K, is indicated with a white arc. The contemporary orbits of the Galilean moons areshown with black arcs and labeled with initials (I: Io, E: Europa, G: Ganymede, C: Callisto). The twolevels of the disk correspond to the disk mid-plane, where the pressure reaches its vertical maximum,and to the disk photosphere, where the optical depth equals 2/3. Temperatures are encoded in colors(see color bar at the bottom). Figure from Heller & Pudritz (2015b). c (cid:13)
AAS.extended our computations to giant exoplanets at various distances from their host stars to study thebehavior of the water line under different magnitudes of stellar irradiation. We found that the mostlow-mass giant planets in our test sample of simulated planets, which were as massive as Jupiter,struggle to have a water ice line in the first place if they are closer to their star than about 4.5 AU.This finding naturally led a follow-up study (Heller et al. 2015), in which we studied the evolutionof the circumjovian ice line during the phase of Jupiter’s orbital migration in the early solar system,a scenario known as the Grand Tack model (Walsh et al. 2011; Raymond & Morbidelli 2014). TheGrand Tack scenario explains the occurrence and distribution of the present-day S and C types ofasteroids and the low mass of Mars (10 % of the Earth’s mass) as an outcome of Jupiter’s and Saturn’scoupled migration in the early solar system. In particular, Jupiter’s possible formation beyond 3.5 AUand its inward migration to as few as 1.5 AU from the sun have dramatic effects on the water iceline in the accretion disk around the young Jupiter. We found that the icy moons Ganymede andCallisto cannot possibly have accreted their water contents during Jupiter’s supposed Grand Tacksimply because the planetary accretion disk, if still active, was too warm to build water ice in thefirst place. After Jupiter and Saturn got caught up in an orbital mean motion resonance, whichpulled Jupiter outward to its current orbital position at 5.2 AU from the sun, any material in anaccretion disk around Jupiter would have been dry due to the evaporative processes triggered duringits journey to the inner, warm regions of the solar system. The fact that Ganymede and Callisto aremade up of about 50 % of water by mass, respectively, means that they must have acquired largeamounts of water either before or after the possible Grand Tack. If they would have received theirwater reservoirs by late accretion of planetesimals, however, then Io and Europa should be water-rich,too, because computer simulations have shown that planetesimal accretion on these inner two moonsis very efficient (Tanigawa et al. 2014). As a consequence, our results imply that Ganymede andCallisto (and likely also Io and Europa) must have formed prior to the Grand Tack scenario, if thisscenario for giant planet migration in the early solar system is actually correct. This might open thepossibility of tracing the migration history of the Galilean moons through the inner solar system with .3. CAPTURE THROUGH TIDAL DISRUPTION DURING PLANETARY ENCOUNTERS 21 the JUICE mission of ESA (Grasset et al. 2013), which is currently scheduled for launch in 2022 andarrival at Jupiter in 2029.
3. Tidally spreading disk model
Finally, Crida & Charnoz (2012) presented a model of a primordial ring system (a late, gas-freeaccretion disk) around a giant planet, in which satellite seeds initially form near the planet’s Rocheradius interior to which their aggregation is prevented due to the tidal forces of the planet. Theprotosatellites then move outward under the combined effects of the planetary tides and the torquefrom the disk. The moons then grow in mass as they move outward and clear the disk. At the sametime, new material is supplied to the disk from the reservoir of debris interior to the Roche radius.This model produces small moons close to the planet and large moons far from the planet. Thisarchitecture is in agreement with the regular moon systems around Uranian, Neptunian, and – tosome extent – with the Saturnian and Jovian moon systems.
As we have seen, the Earth’s moon has likely formed after a giant collision between the proto-Earth anda Mars-sized protoplanetary impactor (Sect. 2.2) whereas most of the regular massive moons aroundthe giant planets have formed through some sort of in-situ accretion process within the dusty gas disksthat fed their young host planets (Sect. 2.1). There is a third mechanism for moon formation thatsuccessfully explains the odd orbital and geological properties of Neptune’s principal moon, Triton.For one thing, Triton is the only major moon in the solar system that is on a retrograde orbit aroundits host planet. Its orbital plane is inclined to Neptune’s equator by about 157 ◦ and crater countssuggest that its surface has regions of different ages, with all regions being younger than about 50 Myrand some are as young as 6 Myr. This makes it one of the youngest planetary surfaces in the solarsystem (Schenk & Zahnle 2007) and it might be the result of ongoing geological activity includingcryovolcanism.Agnor & Hamilton (2006) have shown that a retrograde Triton can be naturally formed during anencounter between Neptune and a planetary binary, of which one component would be ejected fromNeptune’s well of gravity and of which one would be permanently captured in a stable orbit. Williams(2013) extended this model to the capture of (possibly detectable) moons around giant planets beyondthe solar system. Ultimately, Heller (2018c) showed that a tidal capture scenario is the most viablemechanism to form a giant moon like the proposed Neptune-sized moon around the super-Jovianplanet Kepler-1625 (see Sect. 5.4). hapter 3 Detection Methods for Exomoons About a dozen different theoretical methods have been proposed to search and characterize exomoons.For the purpose of this introduction into the field of exomoon detections, we will group them intothree classes: (1.) dynamical effects of the transiting host planet; (2.) direct photometric transits ofexomoons; and (3) other methods.
The moons of the solar system are small compared to their planet, and so the natural satellites ofexoplanets are expected to be small as well. The depth ( d ) of an exomoon’s photometric transit scaleswith the satellite radius ( R s ) squared: d ∝ R . Consequently, large exomoons could be relatively easyto detect (if they exist), but small satellites would tend to be hidden in the noise of the data.Alternatively, instead of hunting for the tiny brightness fluctuations caused by the moons themselves,it has been suggested that their presence could be derived indirectly by measuring the TTVs and TDVsof their host planets. The amplitudes of both quantities (∆ TTV and ∆
TDV ) are linear in the mass of thesatellite: ∆
TTV ∝ M s ∝ ∆ TTV (Sartoretti & Schneider 1999; Kipping 2009a). Hence, the dynamicaleffect of low-mass moons is less suppressed than the photometric effect of small-radius moons.
In a somewhat simplistic picture, neglecting the orbital motion of a planet and its moon around theircommon center of gravity during their common stellar transit, TTVs are caused by the tangentialoffset of the planet from the planet-moon barycenter (see upper left illustration in Figure 3.1, where“BC” denotes the barycenter). In a sequence of transits, the planet has different offsets during eachindividual event, assuming that it is not locked in a full integer orbital resonances with its circumstellarorbit. Hence, its transits will not be precisely periodic but rather show TTVs, approximately on theorder of seconds to minutes (compared to orbital periods of days to years).Two flavors of observable TTV effects have been discussed in the literature. One is called thebarycentric TTV method (TTV b ; Sartoretti & Schneider 1999; Kipping 2009a), and one is referredto as the photocentric TTV method (TTV p or PTV; Szab´o et al. 2006; Simon et al. 2007, 2015). Agraphical representation of both methods is shown in Figure 3.1.TTV b measurements refer to the position of the planet relative to the planet-moon barycenter.From the perspective of a light curve analysis, this corresponds to measuring the time differences ofthe planetary transit only, e.g. of the ingress, center and/or, egress (Sartoretti & Schneider 1999).PTV measurements, on the other hand, take into account the photometric effects of both the planetand its moon, and so the corresponding amplitudes can actually be significantly larger than TTV b This chapter is based on a review by the author (Heller 2018a) published in the “Handbook of Exoplanets” (Deeg &Belmonte 2018). To avoid any repetition, this paper is not contained in Part II of the peer-reviewed journal publicationof this thesis. like TTV and TDV; Kipping et al. 2012, 2014), and the analysisof orbital sampling effects (Heller 2014). In addition, there hasbeen a plethora of other techniques identified that can play im-portant role in confirming such detections. These include theRossiter – McLaughlin effect (Simon et al. 2009, 2010; Zhuanget al. 2012), timing variations in pulsars ’ signal (Lewis et al.2008), microlensing (Han & Han 2002), excess emission dueto a moon in the spectra of distant Jupiters (Williams & Knacke2004), direct imaging of tidally heated exomoons (Peters &Turner 2013), plasma tori around giant planets by volcanicallyactive moons (Ben-Jaffel & Ballester 2014), and modulation ofplanetary radio emissions (Noyola et al. 2014).Also, there were attempts to identify the potential Kepler candidates that could host a possible exomoon (Szabó et al.2013; Kipping et al. 2012, 2014; the HEK Project). Even though
Kepler is theoretically capable of such a detection, there is nocompelling evidence for an exomoon around the KOI (
Kepler
Objects of Interest) targets so far.In this paper, we turn to the next space telescope,
CHEOPS ,and characterize chances of an exomoon detection by using themethod of the photocentric transit timing variation (PTV, TTV p ;Szabó et al. 2006; Simon et al. 2007). Our aim is to explore thecapabilities of CHEOPS in this field and calculate how manytransit observations are required to a firm detection in the caseof a specific range of exoplanet – exomoon systems. We did thisaccording to the CHEOPS science goals and planets to be ob-served, without investigating the theoretically expected occur-rence rate and probability of exomoons generally. We test adecision algorithm, which gives the efficiency of discoveringexomoons without misspending expensive observing time.We perform simulations and bootstrap analysis to demonstratethe operation of our newly developed decision algorithm andcalculate detection statistics for different planet – moon config-urations. Finally, we introduce a folding technique modifiedby the PTV to increase the possibility of an exomoon detectiondirectly in the phase-folded light curve, which is similar to theidea presented by Heller (2014).
2. TRANSIT TIMING VARIATIONS: THE MEANINGOF TTV AND PTV
First, for the sake of clarity, we have to note that the photo-centric transit timing variation (PTV) differs from the conven-tional transit timing variation (TTV). Because of their blurredmeanings (PTV and TTV were called as TTV p and TTV b , re-spectively, by Simon et al. [2007]), we briefly clarify the differ-ences and show their usability and limitations in the following.Sartoretti & Schneider (1999) suggested that a moon aroundan exoplanet can be detected by measuring the variation in thetransit time of the planet due to gravitational effects. In theirmodel, the barycenter of the system orbits the star with a con-stant velocity, and transits strictly periodically. As the planet re-volves around the planet – moon barycenter, its relative positionto the barycenter is varying, so the transit of the planet starts sometimes earlier, sometimes later (Fig. 1). This time shift isthe conventional TTV:TTV ∼ m s m s þ m p ≈ m s m p ¼ χϑ ; (1)where m s and m p are the masses of the moon and the planet,and χ and ϑ are the ratios (moon/planet) of the densities andradii, respectively. We note that TTV can be caused not onlyby an exomoon itself. Several other processes; e.g., an addi-tional planet in the system (e.g., Agol & Steffen 2005; Nesvornet al. 2014), exotrojans (Ford & Gaudi 2006), and periastronprecession (Pál & Kocsis 2008) can also cause TTVs.The traditional TTV simply measures the timing variationof the planetary transit and does not consider the tiny photo-metric effect of the moon. Szabó et al. (2006) and Simon et al.(2007) argued against this simplification and proposed a newapproach for obtaining the variation of the central time of thetransit and calculated the geometric central line of the lightcurve by integrating the time-weighted occulted flux. They de-rived a formula which showed that there is a fixed point on theplanet – moon line, which is the so called photocenter (PC inFig. 2). In this photocenter, an imaginary celestial body causesthe same photometric timing effect as the planet and the mooncombined together. The motion of this visual body around theplanet – moon barycenter leads to the variation of the transittime, which is the photocentric transit timing variation(PTV in Fig. 2): PTV ∼ j ϑ χϑ j : (2)This model takes into account both the photometric and bar-ycentric effect of the moon.Both methods have advantages and disadvantages. Thetraditional TTV uses parametric model light curve fitting(e.g., Mandel & Agol 2002) to derive the timing variations Planet BC Moon BC Planet
Moon
TTV F IG . 1. — The conventional TTV: the time shift due to the revolution of thecompanion around the common barycenter (Sartoretti & Schneider 1999).Not to scale.
CHEOPS
PERFORMANCE FOR EXOMOONS 10852015 PASP, :1084 – This content downloaded from 23.235.32.0 on Fri, 4 Dec 2015 03:45:57 AMAll use subject to JSTOR Terms and Conditions of the transit. This does not need equally sampled data and theresult does not significantly depend on the number of the meas-urements. Furthermore, it allows characterizing the star-planetpair immediately and gives a physical interpretation of the sys-tem. However, the results depend on the model adopted to fit thedata. For distorted light curves, blind model fitting can lead tononphysical parameter combinations and ultimately misleadingresults; hence, model-independent tests are always very impor-tant. Moreover, mapping large parameter spaces for fitting ex-tensive sets of observations can be extremely time-consuming,making a full, detailed analysis nearly impossible. In contrast,the technique of the PTV is a so-called nonparametric methodwhich can provide fast reduction of the light curve shapes evenfor large sets of data. There is no need to make assumptions onthe analytic light curve model, because PTV uses only a numer-ical summation of time-weighted fluxes in a window with cer-tain width; hence, there is no need to increase the number ofmodel parameters for distorted transit curves. It is true for boththe TTV and the PTV that their usage is limited by the spatialconfiguration of the systems. Independently of their size, forclose-in moons, mutual eclipses occur during the transit; there-fore, due to the opposite motion of the moon, the apparent tim-ing variation will be largely canceled out.Another difference between the techniques is demonstratedin Figure 3, where the green light curve shows a transiting sys-tem with a leading moon, while the red one corresponds to theopposite case, when the moon is in a trailing position. As can beseen, the two effects are opposite in sign and the PTV can pro-duce a larger signal. Interestingly, one can show from the de-rived formula of the TTV and the PTV that the former ismore sensitive to the mass of the moon, while the latter givesbetter constraints on the radius of the moon (Simon et al. 2007).To measure the entire PTV effect, the evaluation windowshould be longer than the transit duration. The size of this win-dow depends on the parameters of the systems (Simon et al.2012), mainly on the semimajor axis of the moon, and therefore on the Hill sphere as well. For our simulation the choice of awindow with triple duration width was enough to measure thetiny drops due to the moon for all cases. Even if this smallbrightness decline on the left or right shoulder of the main lightcurve cannot be seen directly, they can cause measurable timeshift in the transit time (Simon et al. 2007).We note that the magnitude of these effects depends on thedensity ratio of the companions. For example, if the density ofthe moon is double that of the planet, which can be the case for agas giant-rocky (or icy) moon system, the TTV dominates onlyif the size ratio is higher than 0.25. If the moon-planet densityratio is about 0.5 for an Earth-Moon system, the PTV alwayssurpasses the TTV (Fig. 4).Hereafter, we will take the advantages of the PTV method toinvestigate the expected performance of the
CHEOPS spacetelescope. However, to put the results into broader context,we will also compare PTV and TTV in selected cases.
Planet BC Moon PC Planet BC Moon
PC PTV F IG . 2. — The definition of the PTV: the time delay due to the revolution of atheoretical body around the barycenter (Simon et al. 2007). Not to scale. pp m Time [min]leading moon0 100 200 PTVTTVtrailing moon F IG . 3. — The schematic illustration of the differences between PTVand TTV:the PTV effect is larger and appears in the opposite direction than that of theTTV. See the electronic edition of the
PASP for a color version of this figure. t i m i ng v a r i a t i on χ =0.5 PTVTTV0 0.02 0.04 0.06 0.08 0.10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 t i m i ng v a r i a t i on ϑ =r m /r p χ =2.0 PTVTTV F IG . 4. — Comparison between the half-amplitudes of the timing variations(TTV and PTV). The moon-planet density ratio χ is indicated in the upper rightcorners of the panels. The timing variations are expressed in arbitrary units onthe y -axis. See the electronic edition of the PASP for a color version of thisfigure.
SIMON ET AL. :1084 – This content downloaded from 23.235.32.0 on Fri, 4 Dec 2015 03:45:57 AMAll use subject to JSTOR Terms and Conditions of the transit. This does not need equally sampled data and theresult does not significantly depend on the number of the meas-urements. Furthermore, it allows characterizing the star-planetpair immediately and gives a physical interpretation of the sys-tem. However, the results depend on the model adopted to fit thedata. For distorted light curves, blind model fitting can lead tononphysical parameter combinations and ultimately misleadingresults; hence, model-independent tests are always very impor-tant. Moreover, mapping large parameter spaces for fitting ex-tensive sets of observations can be extremely time-consuming,making a full, detailed analysis nearly impossible. In contrast,the technique of the PTV is a so-called nonparametric methodwhich can provide fast reduction of the light curve shapes evenfor large sets of data. There is no need to make assumptions onthe analytic light curve model, because PTV uses only a numer-ical summation of time-weighted fluxes in a window with cer-tain width; hence, there is no need to increase the number ofmodel parameters for distorted transit curves. It is true for boththe TTV and the PTV that their usage is limited by the spatialconfiguration of the systems. Independently of their size, forclose-in moons, mutual eclipses occur during the transit; there-fore, due to the opposite motion of the moon, the apparent tim-ing variation will be largely canceled out.Another difference between the techniques is demonstratedin Figure 3, where the green light curve shows a transiting sys-tem with a leading moon, while the red one corresponds to theopposite case, when the moon is in a trailing position. As can beseen, the two effects are opposite in sign and the PTV can pro-duce a larger signal. Interestingly, one can show from the de-rived formula of the TTV and the PTV that the former ismore sensitive to the mass of the moon, while the latter givesbetter constraints on the radius of the moon (Simon et al. 2007).To measure the entire PTV effect, the evaluation windowshould be longer than the transit duration. The size of this win-dow depends on the parameters of the systems (Simon et al.2012), mainly on the semimajor axis of the moon, and therefore on the Hill sphere as well. For our simulation the choice of awindow with triple duration width was enough to measure thetiny drops due to the moon for all cases. Even if this smallbrightness decline on the left or right shoulder of the main lightcurve cannot be seen directly, they can cause measurable timeshift in the transit time (Simon et al. 2007).We note that the magnitude of these effects depends on thedensity ratio of the companions. For example, if the density ofthe moon is double that of the planet, which can be the case for agas giant-rocky (or icy) moon system, the TTV dominates onlyif the size ratio is higher than 0.25. If the moon-planet densityratio is about 0.5 for an Earth-Moon system, the PTV alwayssurpasses the TTV (Fig. 4).Hereafter, we will take the advantages of the PTV method toinvestigate the expected performance of the
CHEOPS spacetelescope. However, to put the results into broader context,we will also compare PTV and TTV in selected cases.
Planet BC Moon PC Planet BC Moon
PC PTV F IG . 2. — The definition of the PTV: the time delay due to the revolution of atheoretical body around the barycenter (Simon et al. 2007). Not to scale. pp m Time [min]leading moon0 100 200 PTVTTVtrailing moon F IG . 3. — The schematic illustration of the differences between PTV and TTV:the PTV effect is larger and appears in the opposite direction than that of theTTV. See the electronic edition of the
PASP for a color version of this figure. t i m i ng v a r i a t i on χ =0.5 PTVTTV0 0.02 0.04 0.06 0.08 0.10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 t i m i ng v a r i a t i on ϑ =r m /r p χ =2.0 PTVTTV F IG . 4. — Comparison between the half-amplitudes of the timing variations(TTV and PTV). The moon-planet density ratio χ is indicated in the upper rightcorners of the panels. The timing variations are expressed in arbitrary units onthe y -axis. See the electronic edition of the PASP for a color version of thisfigure.
SIMON ET AL. :1084 – This content downloaded from 23.235.32.0 on Fri, 4 Dec 2015 03:45:57 AMAll use subject to JSTOR Terms and Conditions
Figure 3.1: Physical explanation of the barycentric TTV (upper left) and the photocentric TTV (upperright). The two light curves at the bottom illustrate both the TTV b and the TTV p (or PTV) for amoon that is trailing the planet (upper panel, green curve) and leading the planet during the stellartransit (lower panel, red curve). Image credit: Simon et al. (2015). c (cid:13) The Astronomical Society ofthe Pacific.amplitudes, details depending on the actual masses and radii of both objects (Simon et al. 2015).
Planetary TDVs can be caused by several effects. First, they can be produced by the change of theplanet’s tangential velocity component around the planet-moon barycenter between successive transits(referred to as the TDV V component; Kipping 2009a). When the velocity component in the planet-moon system that is tangential to the observer’s line of sight adds to the circumstellar tangentialvelocity during the transit, then the event is relatively short. On the other hand, if the transit catchesthe planet during its reverse motion in the planet-moon system, then the total tangential velocity islower than that of the barycenter, and so the planetary transit takes somewhat longer.TDV effects can also be introduced if the planet-moon orbital plane is inclined with respect tothe circumstellar orbital plane of their mutual center of gravity. In this case, the planet’s apparentminimum distance from the stellar center will be different during successive transits, in more technicalterms: its transit impact parameter will change between transits or, if the moon’s orbital motionaround the planet is fast enough, even during the transits. This can induce a TDV TIP component inthe transit duration measurements of the planet (Kipping 2009b).It is important to realize that the waveforms of the TTV and TDV curves are offset by an angle of π/ .2. DIRECT TRANSIT SIGNATURES OF EXOMOONS 25 the line of sight with the planet-moon barycenter, then the corresponding TDV measurement is eitherlargest (for moons on obverse motion) or smallest (for moons on reverse motion), since the planetwould have the largest/smallest possible tangential velocity in the planet-moon binary system. Thisphase difference is key to breaking the degeneracy of simultaneous M s and a s measurements ( a s beingthe moon’s semi-major axis around the planet). When plotted in a TTV-TDV diagram (Montaltoet al. 2012; Awiphan & Kerins 2013), the resulting ellipse contains predictable dynamical patterns,which can help to discriminate an exomoon interpretation of the data from a planetary perturber,and it may even allow the detection of multiple moons (Heller et al. 2016b). In a related context,where close stars show mutual eclipses (rather than transits), combination of eclipse timing variations(ETVs), eclipse duration variations (EDVs), and radial velocity (RV) measurements of the stars canreveal the presence of S-type extrasolar planets, as we demonstrated in (Oshagh et al. 2017). Like planets, moons could naturally imprint their own photometric transits into the stellar lightcurves, if they were large enough (Tusnski & Valio 2011). The lower two panels of Figure 3.1 show anexomoon’s contribution to the stellar bright variation in case the moon is trailing (upper light curve)or leading (lower light curve) its planet. Note that if the moon is leading, then its transit starts priorto the planetary transit, and so the exomoon transit affects the right part of the planetary transit inthe light curve. As mentioned in Section 3.1, the key challenge is in the actual detection of this tinycontribution, which has hitherto remained hidden in the noise of exoplanet light curves.As a variation of the transit method, it has been suggested that mutual planet-moon eclipses duringtheir common stellar transit might betray the presence of an exomoon or binary planetary companion(Sato & Asada 2009; P´al 2012). This is a particularly interesting method, since the mutual eclipsesof two transiting planets have already been observed (de Wit et al. 2016). Yet, in the latter case,the two planets were known to exist prior to the observation of their common transit, whereas fora detection of an exomoon through mutual eclipses it would be necessary to test the data against apossible origin from star spot crossings of the planet (Lewis et al. 2015) and to use an independentmethod for validation.
One way to generate transit light curves with very high signal-to-noise ratios in order to reveal ex-omoons is by folding the measurements of several transits of the same object into one phase-foldedtransit light curve. Figure 3.2 shows a simulation of this phase-folding technique, which is referred toas the orbital sampling effect (OSE; Heller 2014; Heller et al. 2016a). The derived light curve does noteffectively contain “better” data than the combination of the individual transit light curves (in fact itloses any information about the individual TTV and TDV measurements), but it enables astronomersto effectively search for moons in large data sets, as has been done by Hippke (2015) to generatesuperstack light curves from Kepler (see Section 1.2.1).Ultimately, Teachey et al. (2018) used the OSE to detect the first viable exomoon candidate in theKepler data (see Section 1.2.2).
The minimum possible noise level of photometric light curves is given by the shot noise (or Poissonnoise, white noise, time-uncorrelated noise), which depends on the number of photons collected and,thus, on the apparent brightness of the star. For the amount of photons typically collected with “S-type” planets are in orbit around one single star in a satellite-like fashion. Planets can also orbit a closer stellarbinary in a wide orbit. These planets are often referred to as circumbinary (or “P-type”) planets (Dvorak 1986). A thirdconfiguration is referred to as “L-type” (or sometimes “T-type”, “T” for Trojan), where the planet is located at eitherthe L or L Lagrangian point (Dvorak 1986).
Figure 3.2: Orbital sampling effect (OSE) of a simulated transiting exoplanet with moons. The upperpanel shows a model of the phase-folded transit light curve of a Jupiter-sized planet around a 0 . R (cid:12) K dwarf star using an arbitrarily large number of transits. The planet is accompanied by three moonsof 0 . R ⊕ , 0 . R ⊕ , and 0 . R ⊕ in radial size, but their contribution to the phase-folded light curveis barely visible with the naked eye. The lower row of panels shows a sequence of zooms into theprior-to-ingress part of the planetary transit. The evolution of the OSEs of the three moons is shownfor an increasing number of transits ( N ) used to generate the phase-folded light curves. In each panel,the solid line shows the simulated phase-folded transit and the dashed line shows an analytical model,both curves assuming a star without limb darkening. Image credit: Heller (2014). c (cid:13) AAS.space-based optical telescopes, the minimum possible signal-to-noise ratio (SNR) of a light curve canbe approximated as the square root of the number of photons ( n ): SNR ∝ √ n . And so the SNR ofa phase-folded transit light curve of a given planet goes down with the square root of the numberof phase-folded transits ( N ): SNR OSE ∝ √ N . In other words, for planets transiting photometricallyquiet host stars, the noise-to-signal ratio (1/SNR) of the phase-folded light curve converges to zero foran increasing number of transits.If the planet is accompanied by a moon, however, then the variable position of the moon withrespect to the planet induces an additional noise component. As a consequence, and although theaverage light curve is converging towards analytical models (see Figure 3.2), the noise in the planetarytransit is actually in creasing due to the moon. For large N , once dozens and hundreds of transits canbe phase-folded, the OSE becomes visible together with a peak in the noise, the latter of which hasbeen termed the scatter peak (Simon et al. 2012). As an aside, the superstack OSE candidate signalfound by Hippke (2015) was not accompanied by any evidence of a scatter peak. .3. OTHER METHODS FOR EXOMOON DETECTION 27 After 20 Transits N o r m a li z ed S t e ll a r B r i gh t ne ss [ % ] Time Around Barycenter Mid-Transit [Hours]97.5098.0098.5099.0099.50100.00 -10 0 10Simulated Phase-FoldedLightcurveOrbital Sampling EffectModel
Figure 3.3: Simulated OSE of an m V = 11 K dwarf star (0 . R (cid:12) ) with a Saturn-sized planet and anEarth-sized moin in a Europa-wide orbit as it would be observed with PLATO. The distance betweenthe star and the planet-moon system is 0.3 AU, which yields a period of 72 d. Only white noise isassumed, that is to say, the star is assumed to be photometrically quiet with no spots, flares or thelike. In some cases, where the planet and its moon (or multiple moons) are sufficiently far from their hoststar, it could be possible to optically resolve the planet from the star. This has been achieved morethan a dozen times now through a method known as direct imaging (Marois et al. 2008). Thoughdirect imaging cannot, at the current stage of technology, deliver images of a resolved planet withindividual moons, it might still be possible to detect the satellites. One could either try and detect theshadows and transits of the moons across their host planet in the integrated (i.e. unresolved) infraredlight curve of the planet-moon system (Cabrera & Schneider 2007; Heller 2016), or one could search forvariations in the position of the planet-moon photocenter with respect to some reference object, e.g.another star or nearby exoplanet in the same system (Cabrera & Schneider 2007; Agol et al. 2015).Fluctuations in the infrared light received from the directly imaged planet β Pic b, as an example,could be due to an extremely tidally heated moon (Peters & Turner 2013) that is occasionally seenin transit or (not seen) during the secondary eclipse behind the planet. A related method is in thedetection of a variation of the net polarization of light coming from a directly imaged planet, whichmight be caused by an exomoon transiting a luminous giant planet (Sengupta & Marley 2016).It could also be possible to detect exomoons through spectral analyses, e.g. via excess emissionof giant exoplanets in the spectral region between 1 and 4 µ m (Williams & Knacke 2004); enhancedinfrared emission by airless moons around terrestrial planets (Moskovitz et al. 2009; Robinson 2011);and the stellar Rossiter-McLaughlin effect of a transiting planet with moons (Simon et al. 2010; Zhuanget al. 2012) or the Rossiter-McLaughlin effect of a moon crossing a directly imaged, luminous giantplanet (Heller & Albrecht 2014).Some more exotic exomoon detection methods invoke microlensing (Han & Han 2002; Bennett et al.2014; Skowron et al. 2014), pulsar timing variations (Lewis et al. 2008), modulations of radio emissionfrom giant planets (Noyola et al. 2014, 2016), or the generation of plasma tori around giant planets Figure 3.4: Transit of Saturn and its ring system in front of the sun as seen by the Cassini spacecraftin September 2016. Note how the rings bend the sun light around the planet, an effect knows asdiffraction. Image credit: NASA/JPL/Space Science Institute.by volcanically active moons (Ben-Jaffel & Ballester 2014).
Just like moons are very common around the solar system planets, rings appear to be a commonfeature as well. Naturally, the beautiful ring system around Saturn was the first to be discovered.Less obvious rings have also been detected around all other gas giants in the solar system and evenaround an asteroid (Braga-Ribas et al. 2014). In the advent of exoplanet detections, astronomers havethus started to develop methods for the detection of rings around planets outside the solar system.
The detection of rings around exoplanets is closely related to many of the above-mentioned methods(see Section 3.2) of direct photometric transit observations of exomoons. Like moons, rings can causeadditional dips in the planetary transit light curve (Tusnski & Valio 2011). But rings do not induce anydynamical effects on the planet and, hence, there will be no TTVs or TDVs. In fact, a ring system canbe expected to impose virtually the same pattern on each individual transit of its host planet becauserings should look the same during each transit. Moons, however, would have a different positionrelative to the planet during individual transits (except for the case of full-integer orbital resonancesbetween the circumstellar and the circumplanetary orbits). This static characteristic of the expectedring signals make it susceptible to misinterpretation, e.g. by a standard fit of a planet-only model to ahypothetically observed planet-with-ring light curve: in this case, the planet radius would be slightlyoverestimated, while the ring could remain undetected. However, the O-C diagram (O for observed,C for calculated) could still indicate the ring signature (Barnes & Fortney 2004; Zuluaga et al. 2015)As a consequence, rings could induce a signal into the phase-folded transit light curve, which is verysimilar to the OSE (Section 3.2.1) since the latter is equivalent to a smearing of the moon over itscircumplanetary orbit – very much like a ring. The absence of dynamical effects like TTVs and TDVs,however, means that there would also be no scatter peak for rings (Section 3.2.2). Thus, an OSE-like .5. CONCLUSIONS 29 signal in the phase-folded light curve without an additional scatter peak could indicate a ring ratherthan a moon.One particular effect that has been predicted for light curves of transiting ring systems is diffraction,or forward-scattering (Barnes & Fortney 2004). Diffraction describes the ability of light to effectivelybend around an obstacle, a property that is rooted in the waveform nature of light. In our context,the light of the host star encounters the ring particles along the line of sight, and those of which are µ m- to 10 m-sized will tend to scatter light into the forward direction, that is, toward the observer.In other words, rings cannot only obscure the stellar light during transit, they can also magnify ittemporarily (see Figure 3.4). Beyond those potential ring signals in the photometric transit data, rings might also betray theirpresence in stellar transit spectroscopy. The crucial effect here is similar to the Rossiter-McLaughlineffect of transiting planets: as a transiting ring proceeds over the stellar disk, it modifies the apparent,disk-integrated radial velocity of its rotating host star. This is because the ring covers varying partsof the disk, all of which have a very distinct contribution to the rotational broadening of the stellarspectral lines (Ohta et al. 2009). Qualitatively speaking, if the planet with rings transits the star inthe same direction as the direction of stellar rotation, then the ring (and planet) will first cover theblue-shifted parts of the star. Hence, the stellar radial velocity will occur redshifted during about thefirst half of the transit – and vice versa for the second half.Another more indirect effect can be seen in the Fourier space (i.e. in the frequency domain ratherthan the time domain) of the transit light curve, where the ring can potentially stand out as anadditional feature in the curve of the Fourier components as a function of frequency (Samsing 2015).
In this chapter, we discussed about a dozen methods that various researchers have worked out overlittle more than the past decade to search for moons and rings beyond the solar system. Althoughsome of the original studies, in which these methods have been presented, expected that moons andrings could be detectable with the past CoRoT space mission or with the still active Kepler spacetelescope (Sartoretti & Schneider 1999; Barnes & Fortney 2004; Kipping et al. 2009; Heller 2014), noexomoon or exo-ring has been unequivocally discovered and confirmed as of today.This is likely not because these extrasolar objects and structures do not exist, but because they aretoo small to be distinguished from the noise. Alternatively, and this is a more optimistic interpretationof the situation, those features could actually be detectable and present in the available archival data(maybe even in the HST archival data of transiting exoplanets), but they just haven’t been found yet.The absence of numerous, independent surveys for exomoons and -rings lends some credence to thislatter interpretation: Out of the several thousands of exoplanets and exoplanet candidates discoveredwith the Kepler telescope alone, only a few dozen have been examined for moons and rings withstatistical scrutiny (Heising et al. 2015; Kipping et al. 2015).It can be expected that the Kepler data will be fully analyzed for moons and rings within the nextfew years. Hence, a detection might still be possible. Alternatively, an independent, targeted searchfor moons/rings around planets transiting apparently bright stars – e.g. using the HST, CHEOPS, ora 10 m scale ground-based telescope, might deliver the first discoveries in the next decade. If none ofthese searches would be proposed or proposed but not granted, then it might take more than a decadefor the PLAnetary Transits and Oscillations of stars (PLATO) mission (Rauer et al. 2014) to find anexomoon or exo-ring in its large space-based survey of bright stars. Either way, it can be expectedthat exomoon and exo-ring discoveries will allow us a much deeper understanding of planetary systemsthan is possibly obtainable by planet observations alone. hapter 4
Habitability of Exomoons
As explained in Sect. 1.3, astronomers regularly focus on surface habitability in their search for lifebeyond the solar system because this sort of life has the highest (if any) chances to be remotelydetectable in the foreseeable future. Life on the surface has the observational advantage of interactingwith the atmosphere, which might result in the chemical imprints of life that could be detectablethrough transit spectroscopy, e.g. with the James Webb Space Telescope (Beichman et al. 2014) or itssuccessor missions HabEx or LUVOIR (Wang et al. 2018). The detection of atmospheric biomarkerson exomoons through transit spectroscopy or similar techniques using transits will be more challengingthan it will be for planets, because the moon signal will almost inevitably be contaminated with thesignal from its host planet. The separation of the two signals will be possible in principle by usingobservations of both planet and moon in transit and of only the moon in transit, but this approachwill increase the amount of transit observations required (Kaltenegger 2010). Other means of remotedetection of life on exomoons include observations of surface features in the photometric phase curvesof planet-moon systems (Cowan et al. 2012; Forgan 2017), which also restricts this method to life onthe surface. For life to exist on the surface of an exomoon, surface water is required (see Sect. 1.3). Asa consequence, when referring to habitable exomoons in this thesis, we refer to exomoons that havethe potential of hosting liquid water on the surface.Although the basic concept of habitability is identical for planets and moons, there is a range ofphysical processes that are relevant to exomoon habitability and that might be irrelevant for exoplan-ets.
The effect of planetary illumination onto an exomoon has first been described by Heller (2012)and Heller & Barnes (2013a). In this model, the moon’s orbit around the planet is assumed tobe coplanar with the orbit of the planet-moon barycenter around around the star, and it is assumedthat the spin axis of the moon is also aligned with the orbital plane normal. Planetary oblateness isneglected and all objects involved are treated as spheres. The planet is divided into a day side thatreflects a certain fraction of the incoming star light (the magnitude of which is determined by theplanetary Bond albedo, α p ) and a night side, which does not receive or reflect any stellar illumination.Both hemispheres are also emitters of thermal radiation, the amount of which is determined by theireffective temperatures. A sketch of the model is shown in Fig. 4.1. The vast majority of planets known today, including the eight planets of the solar system and over4000 planets and candidates known beyond, are in orbit around single stars. These planets have oneprincipal light source, neglecting the possibility of additional light coming from their moons. Moons,however, have two principal light sources, namely, the star and the planet (Heller & Barnes 2013a).
Figure 4.1: Illumination on an exomoon changes over the course of its orbital revolution around thehost planet. In our model that invokes coplanar orbits around the star (of the planet-moon barycenter)and around the planet (of the moon), the moon’s orbit starts an an orbital phase of 0 and then proceedsin a counter-clockwise fashion to 1. The center of the planetary eclipse occurs at a phase of 1/2. Thedifferent contributions of light falling onto the moon are explained in the legend in the lower leftcorner. This illustration is a modified version of a graphic shown in Heller (2013).On Earth, the amount of reflected sun light received from the Moon is irrelevant to our climate. The Earth receives about 1365 W m − of photonic energy from the sun at the top of the atmosphere.For comparison, even the full moon only contributes some 0.01 W m − of reflected sun light to theEarth’s top-of-the-atmosphere photon flux, or just about five orders of magnitude less than the solarillumination. Exomoons around giant planets, however, can receive tens or even hundreds of W m − of reflected star light from their planets, provided that the planet-moon system is close to the starand that the moon is close to its planet (Heller & Barnes 2013a). In addition to the star light reflected off of the planetary surface (or atmosphere) back to an exomoon,these satellites can be subject to significant amounts of thermal illumination from their host planets(Heller & Barnes 2015; Heller 2016). This is particularly true for moons around young giant planets,which can be as hot as the most low-mass stars on the main sequence, that is, about 2000 K. Similareffective temperatures have indeed been deduced from direct imaging observations of young giantplanets, e.g. of 1RXS J160929.1 − That said, it is certainly clear that the lunar cycle regulates a vast range of biological processes on Earth (Foster &Roenneberg 2008) .2. TIDAL HEATING AS A SOURCE OF INTERNAL ENERGY 33 luminosity during the first billion years after formation (Mordasini 2013). Taking into account thatmoons around Jovian and super-Jovian planets supposedly form at distances of around ten planetaryradii (Heller & Pudritz 2015b), or hundreds of times closer to their planet than Earth to the sun, thismeans that these young moons may receive up to 100 W m − of thermal infrared radiation from theirplanets. As these exoplanets cool on time scales of millions to billions of years, the amount of infraredirradiation received by their potential moons also vanishes until, at some point, star light will entirelydominate the light budget on the moon (Heller & Barnes 2015).The orbital architecture of planet-moon systems can lead to frequent stellar eclipses on the moon(Scharf 2006; Heller 2012). This may lead to the odd situation in which the subplanetary point on themoon, where the planet is in the zenith above the observer, experiences its darkest time of the dayat noon (during stellar eclipse), while midnight can be quite bright with the planet acting as a giantmirror for the star light in the zenith (Heller & Barnes 2013a). These configurations are illustrated atphases 0 and 1/2, respectively, in Fig. 4.1. The amount of frictional heating that emanates the surface of a tidally heated body with radius R s and at a distance a to its primary (with a mass M p ) scales roughly as R /a × M . For the principalmoons around Jupiter (Ganymede) and Saturn (Titan), this term (in units of kg m − ) is about 10 or 10 , respectively. For the principal planet in the solar system, Jupiter, and taking the sun asthe primary, this terms is only 10 . Even for the innermost planet, Mercury, we only have 10 . ForEarth-sized planets in the habitable zone around M dwarf stars ( a = 0 . . But generally speaking, it can be expected that moons (habitable or not)tend to be subject to stronger tidal heating rates than planets in the stellar habitable zone (Helleret al. 2014).The orbital evolution driven by tidal friction results in a circular orbit of the planet around the star,in which the planetary spin axis is aligned with its orbital plane normal and in which the rotationperiod of the planet equals its orbital period around the planet. As a consequence, tidal heating ceasestypically within much less than the first 100 Myr years after moon formation (Porter & Grundy 2011)in a two-body system and in the absence of continuous gravitational perturbations, e.g. by othermoons or planets, or by the star. That being said, gravitational perturbations of exoplanet-exomoonsystem could be quite common. The most prominent example in the solar system is the 1:2:4 orbitalmean motion resonance of the inner three Galilean moons Io, Europa, and Ganymede, the so-calledLaplace resonance. This resonance prevents Io’s orbit from being circularized by tides and currentlyforces its eccentricity to a value of about 0.0041. The resulting magnitude of tidal heating has beenpredicted to melt parts of the structure of the moon. This prediction was published by Peale et al.(1979) just shortly before the Voyager 1 probe flew by Jupiter and its moons and actually observedactive volcanos on Io in 1979 (Morabito et al. 1979). A more recent view of an active volcanic regionon Io, taken by the New Horizons spacecraft in 2007, is shown in Fig. 4.2.While Io’s tidal heating is maintained by gravitational interaction with its neighboring moons,moons beyond the solar system could regularly be subject to tidal heating driven by the gravitationalperturbations from their host stars. As predicted by Heller (2012) and then shown by Zollinger et al.(2017) using N -body numerical integrations of the combined tidal and secular forces, moons in thestellar habitable zones around M dwarfs could even struggle to be habitable. With the habitable zonebeing so close around M dwarfs (Kasting et al. 1993; Kopparapu et al. 2013), moons around habitablezone planets will inevitably be subject to stellar perturbations. The resulting non-circularity of theirorbits will lead to continuous tidal heating, possibly over billions of years. This additional internalheating would add to the warming effect of the stellar illumination in the habitable zone and thereforedrive a moist or even runaway greenhouse effect on these moons, which would not allow for thepresence of liquid surface water and ultimately lead to desiccation through hydrogen escape into spacevia photolysis in the upper atmosphere. Figure 4.2: This montage shows three images of Io taken almost simultaneously by different instru-ments aboard the New Horizons spacecraft on 1 March 2007 (UT).
Left : The Long Range Recon-naissance Imager presents details on Io’s sunlit crescent and in the partially sunlit plume from theTvashtar volcano. The source of the plume betrays its location as a bright nighttime glow of hot lavas.
Top right : The Multispectral Visible Imaging Camera image shows the contrasting colors of the redlava and blue plume at Tvashtar, and the sulfur and sulfur dioxide deposits on Io’s sunlit surface.
Bottom right : The Linear Etalon Imaging Spectral Array image shows that Tvashtar’s glow is evenmore intense at infrared wavelengths and reveals the glows of over ten fainter volcanic hot spots onIo’s nightside. Image credit: NASA/Johns Hopkins University Applied Physics Laboratory/SouthwestResearch Institute.
All things from Sects. 4.1 and 4.2 combined, we can define the circumplanetary habitable space formoons. As in the scenario for planets, we consider a moon habitable if its globally averaged surfacetemperature is above 0 ◦ and the moon is not subject to a runaway greenhouse effect, which wouldultimately free the moon of any water (Goldblatt & Watson 2012). As a principal threshold for themaximum top-of-the-atmosphere energy flux for the moon to be habitable, we estimate the runawaygreenhouse limit ( F RG ) from the framework of (Pierrehumbert 2010). For an Earth-sized object, thisframework gives a critical flux of 295 W m − .This limit needs to be compared to the total energy budget of the moon under consideration. Let usimagine a satellite with a radius R s , an optical albedo α s , opt , and an infrared albedo α s , IR in an orbitaround a planet with semimajor axis a s and eccentricity e s . Let us further assume that the satellitehas an effective energy redistribution factor f s , which accounts for the fact that the cross section ofthe moon is πR while its surface area is 4 πR . We also consider that the moon spends a fraction1 − x s of its orbit in the shadow of the planet, neglecting the effect of ingress and egress. In otherwords, x s is the fraction of the satellite orbit that is not spent in the shadow of the planet. Movingon to the planet, we consider a radius R p , a Bond albedo α p , an orbital semimajor axis a p , and aneccentricity e p around the star . Finally, the stellar luminosity is referred to as L (cid:63) and the planetary Strictly speaking, the orbit of the planet around the star is defined by the three-body motions of the star-planet-moonsystem, in which the planet (and moon) orbit their common barycenter, which itself is in a Keplerian orbit around the .3. A CONCEPT OF EXOMOON HABITABILITY BASED ON ENERGY BUDGED 35 -2 -1 0 1 2log (X/R Jup )-2-1012 l og ( Y / R J up )
100 Myr -2 -1 0 1 2log (X/R Jup )-2-1012 l og ( Y / R J up ) Tidal VenusTidal-Illumination VenusSuper-IoTidal EarthEarth-like (prograde)Earth-like (retrograde)Roche radius tidal heat → runaway greenhousetidal heat + illumination → runaway greenhousetidal heat > 2 W/m → no runaway greenhouse0.04 W/m ≤ tidal heat ≤ → no runaway greenhousetidal heat ≤ → no runaway greenhousetidal destruction1 Billion years100 Million years Roche radiusTidal Venus Tidal-Illumination VenusSuper-IoTidal Earth Earth-like (prograde) Earth-like (retrograde) } ☠ ☠ ☠ uninhabitable Figure 4.3: Habitability of circumplanetary orbits for an Earth-sized satellite orbiting a planet often Jupiter masses with an orbital eccentricity of 0.001.
Left : At an age of 100 Myr, the planet isstill relatively hot, which makes the moon uninhabitable within a distance of about 10 R Jup . Right :900 Myr later, the planet has cooled substantially and the habitable edge, which defines the radialextent of the the uninhabitable space around the planet, has shrunk to about 7 R Jup . The systemis modelled at 1 AU from a sun-like host star. Tidal heating contributes about 10 W m − in bothpanels. This plot is a modified version of Fig. 4 from Heller & Barnes (2015) that the author of thisthesis presented at the astronomy seminars at Cornell University (11 May 2015) and the Center forAstrophysics at Harvard University (1 July 2015).luminosity is L p . The globally averaged energy flux received by the moon from the star and from theplanet can then be estimated as F globs = L ∗ (1 − α s , opt )4 πa (cid:113) − e (cid:32) x s πR α p f s a (cid:33) + L p (1 − α s , IR )4 πa f s (cid:112) − e + h s + W s . (4.1)The first term defines the direct stellar flux received by the satellite. The terms in braces account foreclipses (e.g. x s = 1 means no eclipses) and for the amount of reflected sun light from the planet (e.g. f s = 4 means perfect redistribution of the planetary light over the moon’s surface). The term invoking L p and 1 − α s , IR relates to the infrared radiation emitted by the planet and reflected by the planet.Finally, h s is the heating of the satellite by tides and W s is any other source of heat, e.g. heat fromradioactive decay.As an example, if we switch the roles of the moon and the planet and apply Eq. (4.1) to the Earth,on which tidal heating and other sources of heat are negligible and where the effects of lunar eclipsesand illumination are entirely negligible, we obtain F globs = 239 W m − using an optical albedo of 70 %and zero orbital eccentricity.A primitive version of Eq. (4.1) was first developed by Heller & Barnes (2013a), who neglectedthe effects of eclipses and the possibility of inefficient energy redistribution. Heller (2012) introducedthe effect of eclipses and in Dobos et al. (2017) we presented the formula as shown above. In Heller& Barnes (2015) we then introduced the effects caused by the evolution of stellar and planetary barycenter of the system. For M p (cid:29) M s , however, our approximation is correct. The total flux of internal heat through the surface of the present-day Earth is about 0.086 W m − on average, ofwhich approximately 0.04 W m − are due to the radiogentic decay of long-lived isotopes K, Th,
U, and
U(Zahnle et al. 2007). The residual heat is of primordial origin from the Earth’s accretion process and from tides raisedby the Moon. luminosities to identify several classes of exomoon habitability around extrasolar planets (see Fig. 4.3).The radial extent of these classes is defined by the a − law for the planetary illumination onto themoon and the a − law for the effect of tidal heating in the moon. Further investigations of exomoonclimates were carried out by Forgan & Kipping (2013) and Forgan & Yotov (2014), who developed a1D latitudinal energy balance model to assess exomoon habitability under the combined stellar andplanetary illumination including a geophysical carbon-silicate as a global thermostat similar to theone observed on Earth. Then Forgan & Dobos (2016) added a more sophisticated tidal model thatincludes the feedback mechanisms from tidal heating on the rheology of the moon (and vice versa). InHaqq-Misra & Heller (2018) we expanded exomoon climate studies into three spatial dimensions byapplying an idealized 3D general circulation model with simplified hydrologic, radiative, and convectiveprocesses. More detailed investigations of the possible processes that regulate atmospheric escape onexomoons were done by Lehmer et al. (2017), so that the field of exomoon habitability and exomoonclimates has evolved to a state in which the basic processes have been identified and in which furtherprogress might only be possible using exomoon observations that will allow us to test these predictions. art II Peer-Reviewed Journal Publications hapter 5
Formation of Moons in the AccretionDisks Around Young Giant Planets
Contribution:RH did the literature research, worked out the mathematical framework, translated the math intocomputer code, performed the simulations, created all figures, led the writing of the manuscript, andserved as a corresponding author for the journal editor and the referees. r X i v : . [ a s t r o - ph . E P ] M a y Draft version May 6, 2015
Preprint typeset using L A TEX style emulateapj v. 04/17/13
WATER ICE LINES AND THE FORMATION OF GIANT MOONS AROUND SUPER-JOVIAN PLANETS
Ren´e Heller and Ralph Pudritz Origins Institute, McMaster University, Hamilton, ON L8S 4M1, [email protected], [email protected]
Draft version May 6, 2015
ABSTRACTMost of the exoplanets with known masses at Earth-like distances to Sun-like stars are heavier thanJupiter, which raises the question of whether such planets are accompanied by detectable, possiblyhabitable moons. Here we simulate the accretion disks around super-Jovian planets and find that giantmoons with masses similar to Mars can form. Our results suggest that the Galilean moons formedduring the final stages of accretion onto Jupiter, when the circumjovian disk was sufficiently cool.In contrast to other studies, with our assumptions, we show that Jupiter was still feeding from thecircumsolar disk and that its principal moons cannot have formed after the complete photoevaporationof the circumsolar nebula. To counteract the steady loss of moons into the planet due to type Imigration, we propose that the water ice line around Jupiter and super-Jovian exoplanets acted as amigration trap for moons. Heat transitions, however, cross the disk during the gap opening within ≈ yr, which makes them inefficient as moon traps and indicates a fundamental difference betweenplanet and moon formation. We find that icy moons larger than the smallest known exoplanet canform at about 15 - 30 Jupiter radii around super-Jovian planets. Their size implies detectabilityby the Kepler and
PLATO space telescopes as well as by the
European Extremely Large Telescope .Observations of such giant exomoons would be a novel gateway to understanding planet formation,as moons carry information about the accretion history of their planets.
Keywords: accretion disks – planets and satellites: formation – planets and satellites: gaseous planets– planets and satellites: physical evolution – planetdisk interactions CONTEXT
While thousands of planets and planet candidates havebeen found outside the solar system, some of which areas small as the Earth’s moon (Barclay et al. 2013), nomoon around an exoplanet has yet been observed. Butif they transit their host stars, large exomoons could bedetectable in the data from the
Kepler space telescopeor from the upcoming
PLATO mission (Kipping et al.2012; Heller 2014). Alternatively, if a large moon tran-sits a self-luminous giant planet, the moon’s planetarytransit might be detectable photometrically or even spec-troscopically, for example with the
European ExtremelyLarge Telescope (Heller & Albrecht 2014). It is thereforetimely to consider models for exomoon formation.Large moons can form in the dusty gas disks aroundyoung, accreting gas giant planets. Several models ofmoon formation posit that proto-satellites can be rapidlylost into the planet by type I migration (Pollack &Reynolds 1974; Canup & Ward 2002, 2006; Sasaki et al.2010). The water (H O) condensation ice line can actas a planet migration trap that halts rapid type I migra-tion in circumstellar disks (Kretke & Lin 2007; Hasegawa& Pudritz 2011, 2012), butthis trap mechanism has notbeen considered in theories of moon formation so far.The position of the H O ice line has sometimes beenmodeled ad hoc (Sasaki et al. 2010) to fit the H O dis-tribution in the Galilean moon system (Mosqueira &Estrada 2003a).An alternative explanation for the formation of the Department of Physics and Astronomy, McMaster University Postdoctoral fellow of the Canadian Astrobiology TrainingProgram
Galilean satellites suggests that the growingIo, Europa,and Ganymede migrated within an optically thick ac-cretion disk the size of about the contemporary orbit ofCallisto and accreted material well outside their instanta-neous feeding zones (Mosqueira & Estrada 2003a,b). Inthis picture, Callisto supposedly formed in an extendedoptically thin disk after Jupiter opened up a gap in thecircumsolar disk. Callisto’s material was initially spreadout over as much as 150 Jupiter radii ( R Jup ), then ag-gregated on a 10 yr timescale, and migrated to Callisto’scurrent orbital location. Yet another possible formationscenario suggests that proto-satellites drifted outwards asthey were fed from a spreading circumplanetary ring ina mostly gas-free environment(Crida & Charnoz 2012).We here focus on the “gas-starved” model of an ac-tively supplied circumplanetary disk (CPD) (Makalkinet al. 1999; Canup & Ward 2002) and determine the time-dependent radial position of the H O ice line. There areseveral reasons why water ice lines could play a funda-mental role in the formation of giant moons. The totalmass of a giant planet’s moon system is sensitive to thelocation of the H O ice line in the CPD, where the surfacedensity of solids (Σ s ) increases by about factor of three(Hayashi 1981), because the mass of the fastest growingobject is proportional to Σ / . This suggests that themost massive moons form at or beyond the ice line. Inthis regard, it is interesting that the two lightest Galileansatellites, Io (at 6 . R Jup from the planetary core) andEuropa (at 9 . R Jup ), are mostly rocky with bulk den-sities > − , while the massive moons Ganymede(at 15 . R Jup ) and Callisto (at 27 . R Jup ) have densitiesbelow 2 g cm − and consist by about 50 % of H O (Show-
Ren´e Heller & Ralph Pudritz man & Malhotra 1999). It has long been hypothesizedthat Jupiter’s CPD dissipated when the ice line was be-tween the orbits of Europa and Ganymede, at about 10to 15 R Jup (Pollack & Reynolds 1974). Moreover, simula-tions of the orbital evolution of accreting proto-satellitesin viscously dominated disks around Jupiter, Saturn, andUranus indicate a universal scaling law for the total massof satellite systems ( M T ) around the giant planets in thesolar system (Canup & Ward 2006; Sasaki et al. 2010),where M T ≈ − times the planetary mass ( M p ). METHODS
In the Canup & Ward (2002) model, the accretionrate onto Jupiter was assumed to be time-independent.Canup & Ward (2006) focussed on the migration andgrowth of proto-moons, but they did not describe theirassumptions for the temperature profile in the plane-tary accretion disk. Others used analytical descriptionsfor the temporal evolution of the accretion rates or forthe movement of the H O ice lines (Makalkin & Doro-feeva 1995; Mousis & Gautier 2004; Canup & Ward 2006;Sasaki et al. 2010; Ogihara & Ida 2012) or they did notconsider all the energy inputs described above (Alibertet al. 2005).We here construct, for the first time, a semi-analyticalmodel for the CPDs of Jovian and super-Jovian planetsthat is linked to pre-computed planet evolution tracksand that contains four principal contributions to the diskheating: (i) viscous heating, (ii) accretion onto the CPD,(iii) planetary irradiation, and (iv) heating from the am-bient circumstellar nebula. Compared to previous stud-ies, this setup allows us to investigate many scenarioswith comparatively low computational demands, and wenaturally track the radial movement of the H O ice lineover time. This approach is necessary, because we donot know any extrasolar moons that could be used tocalibrate analytical descriptions for movement of the iceline around super-Jovian planets. We focus on the largepopulation of Jovian and super-Jovian planets at around1 AU from Sun-like stars, several dozens of which hadtheir masses determined through the radial velocity tech-nique as of today.
Disk Model
The disk is assumed to be axially symmetric and inhydrostatic equilibrium. We adopt a standard viscousaccretion disk model (Canup & Ward 2002, 2006), pa-rameterized by a viscosity parameter α (10 − in our sim-ulations) (Shakura & Sunyaev 1973), that is modified toinclude additional sources of disk heating (Makalkin &Dorofeeva 2014). We consider dusty gas disks aroundyoung ( ≈ yr old), accreting giant planets with finalmasses beyond that of Jupiter ( M Jup ). These planetsaccrete gas and dust from the circumstellar disk. Theiraccretion becomes increasingly efficient, culminating inthe so-called runaway accretion phase when their massesbecome similar to that of Saturn (Lissauer et al. 2009;Mordasini 2013). Once they reach about a Jovian mass Amalthea, although being very close to Jupiter (at 2 . R Jup ),has a very low density of about 0 .
86 g cm − (Anderson et al. 2005),which seems to be at odds with the compositional gradient in theGalilean moons. But Amalthea likely did not form at its currentorbital position as is suggested by the presence of hydrous mineralson its surface (Takato et al. 2004). (depending on their distance to the star, amongst oth-ers), they eventually open up a gap in the circumstel-lar disk and their accretion rates drop rapidly. Hence,the formation of moons, which grow from the accumula-tion of solids in the CPD, effectively stops at this pointor soon thereafter. A critical link between planet andmoon formation is the combined effect of various energysources (see the four heating terms described above) onthe temperature distribution in the CPD and the radialposition of the H O ice line.In our disk model, the radial extent of the inner, op-tically thick part of the CPD, where moon formation issuspected to occur, is set by the disk’s centrifugal ra-dius ( r cf ). At that distance to the planet, centrifugalforces on an object with specific angular momentum j are balanced by the planet’s gravitational force. Using3D hydrodynamical simulations, Machida et al. (2008)calculated the circumplanetary distribution of the angu-lar orbital momentum in the disk and demonstrated theformation of an optically thick disk within about 30 R Jup around the planet. An analytical fit to their simulationsyields (Machida et al. 2008) j ( t ) = . × (cid:18) M p ( t ) M Jup (cid:19) (cid:16) a ⋆ p (cid:17) / m s − for M p < M Jup . × (cid:18) M p ( t ) M Jup (cid:19) / (cid:16) a ⋆ p (cid:17) / m s − for M p ≥ M Jup , (1)where we introduced the variable t to indicate that theplanetary mass ( M p ) evolves in time. The centrifugalradius is then given by r cf = j GM p , (2)with G as Newton’s gravitational constant. For Jupiter,this yields a centrifugal radius of about 22 R Jup , whichis slightly less than the orbital radius of the outermostGalilean satellite, Callisto, at roughly 27 R Jup . Part ofthis discrepancy is due to thermal effects that are ne-glected in the (Machida et al. 2008) disk model. Machida(2009) investigated these thermal effects on the centrifu-gal disk size by comparing isothermal with adiabaticdisk models. They found that adiabatic models typicallyyield larger specific angular momentum at a given plane-tary distance, which then translate into larger centrifugaldisk radii that nicely match the width of Callisto’s orbitaround Jupiter. We thus introduce a thermal correctionfactor of 27/22 to the right-hand side of Equation (2)following Machida (2009), and therefore include Callistoat the outer edge of the optically thick part of our diskmodel. We note, though, that this slight rescaling hardlyaffects the general results of our simulations. In particular, their isothermal model M1I, used to fit ourEq. (1), has a radial specific momentum distribution that is about1.1 times smaller at Callisto’s orbital radius than their adiabaticmodel M1A2. This offset means an (1 . = 1 . / ≈ . ater ice lines and the formation of giant moons around super-Jovian planets The latter authors argue that Callisto formedin the low-density regions of an extended CPD with highspecific angular momentum after Jupiter opened up agap in the circumstellar disk. In their picture, the youngCallisto accreted material from orbital radii as wide as150 R Jup . In our model, however, Callisto forms in thedense, optically thick disk, where we suspect most of thesolid material to pile up. Simulations by Canup & Ward(2006) and Sasaki et al. (2010) show that our assumptioncan well reproduce the masses and orbits of the Galileanmoons.The disk is assumed to be mostly gaseous with an ini-tial dust-to-mass fraction X (set to 0.006 in our simula-tions, Hasegawa & Pudritz 2013). Although we do notsimulate moon formation in detail, we assume that thedust would gradually build planetesimals, either throughstreaming instabilities in the turbulent disk (Johansenet al. 2014) or through accumulation within vortices(Klahr & Bodenheimer 2003), to name just two possibleformation mechanisms. The disk is parameterized with afixed Planck opacity ( κ P ) in any of our simulations, butwe tested various values. The fraction of the planetarylight that contributes to the heating of the disk surfaceis parameterized by a coefficient k s , typically between0.1 and 0.5 (Makalkin & Dorofeeva 2014). This quan-tity must not be confused with the disk albedo, whichcan take values between almost 0 and 0.9, dependingon the wavelength and the grain properties (D’Alessioet al. 2001). The sound velocity in the hydrogen (H)and helium (He) disk gas usually depends on the meanmolecular weight ( µ ) and the temperature of the gas, butin the disk midplane it can be approximated (Keith &Wardle 2014) as c s = 1 . − p T m ( r ) / T m . µ = 2 .
34 kg mol − .Further, the disk viscosity is given by ν = αc / Ω K ( r ),with Ω K ( r ) = p GM p /r as the Keplerian orbital fre-quency.The steady-state gas surface density (Σ g ) in the opti-cally thick part of the disk can be obtained by solvingthe continuity equation for the infalling gas at the disk’scentrifugal radius (Canup & Ward 2006), which yieldsΣ g ( r ) = ˙ M πν × Λ( r ) l (3)where More advanced 3D MHD simulations would need to take intoaccount the actual formation of the planet (assumed to be a sinkparticle by Gressel et al. 2013) and would require resolving theinner parts of the compact disk to test whether this argument holdsin favor of the gas-starved model. Λ( r ) = 1 − r r cf r d − (cid:18) rr c (cid:19) l = 1 − r R p r d (4)is derived from a continuity equation for the infallingmaterial and based on the angular momentum deliveredto the disk (Canup & Ward 2006; Makalkin & Dorofeeva2014), and ˙ M is the mass accretion rate through theCPD, assumed to be equal to the mass dictated by thepre-computed planet evolution models (Mordasini 2013).We set r d = R H /
5, which yields r d ≈ R Jup for Jupiter(Sasaki et al. 2010; Makalkin & Dorofeeva 2014).The effective half-thickness of the homogeneous flareddisk, or its scale height, is derived from the solution ofthe vertical hydrostatic balance equation as h ( r ) = c s ( T m ( r )) r / p GM p . (5)We adopt the standard assumption of vertical hydro-static balance in the disk and assume that the gas den-sity ( ρ g ) in the disk decreases exponentially with distancefrom the midplane as per ρ g ( r ) = ρ e − z s h ( r )2 (6)where z is the vertical coordinate and ρ the gas densityin the disk midplane. The gas surface density is given byvertical integration over ρ ( r, z ), that is,Σ g ( r ) = Z +inf − inf dz ρ ( r, z ) . (7)Inserting Equation (6) into Equation (7), the latter canbe solved for ρ and we obtain ρ ( r ) = r π Σ g ( r )2 h ( r ) , (8)which only depends on the distance r to the planet. Atthe radiative surface level of the disk, or photosphericheight ( z s ), the gas density equals ρ s ( r ) = ρ e − z s h ( r )2 (9)where we calculate z s as z s ( r ) = erf − (cid:16) −
23 2Σ g ( r ) κ P (cid:17) √ h ( r ) . (10)The latter formula is derived using the definition of thedisk’s optical depth τ = Z + inf z s d z κ P ρ ( r, z ) = κ P Z + inf z s d z ρ ( r, z ) (11)and our knowledge of τ = 2 / Ren´e Heller & Ralph Pudritz
23 = κ P Z + inf z s d z ρ g ( r, z ) ⇔ z s ( r ) = erf − (cid:16) − r π h ( r ) ρ ( r ) κ P (cid:17) √ h ( r ) . (12)Using Equation (8) for ρ ( r ) in Equation (12), we obtainEquation (10).Following the semi-analytical disk model of Makalkin& Dorofeeva (2014), the disk surface temperature is givenby the energy inputs of various processes as per T s ( r ) = κ P Σ g ( r )) − σ SB (cid:16) F vis ( r ) + F acc ( r )+ k s F p ( r ) (cid:17) + T ! / , (13)where F vis ( r ) = 38 π Λ( r ) l ˙ M Ω K ( r ) F acc ( r ) = X d χGM p ˙ M πr r e − ( r/r cf ) F p ( r ) = L p sin (cid:16) ζ ( r ) r + η ( r ) (cid:17) π ( r + z ) (14)are the energy fluxes from viscous heating, accretion ontothe disk, and the planetary illumination, and T neb de-notes the background temperature of the circumstellarnebula (100 K in our simulations). The geometry of theflaring disk is expressed by the angles ζ ( r ) = arctan π R p p r + z ! η ( r ) = arctan (cid:18) d z s d r (cid:19) − arctan (cid:16) z s r (cid:17) . (15)Taking into account the radiative transfer within the op-tically thick disk with Planck opacity κ P , the midplanetemperature can be estimated as (Makalkin & Dorofeeva2014) T m ( r ) − T s ( r ) T m ( r ) = 12 µχκ P π σ SB R g γ × ˙ M α Ω K ( r ) × (cid:18) Λ( r ) l (cid:19) q s ( r ) , (16)where σ SB is the Stefan-Boltzmann constant, R Gas theideal gas constant, γ = 1 .
45 the adiabatic exponent (orratio of the heat capacities), and q s ( r ) = 1 − κ P Σ g ( r ) is the vertical mass coordinate at z s . Planet Evolution Tracks
We use a pre-computed set of planet formation mod-els by Mordasini (2013) to feed our planet disk modelwith the fundamental planetary properties such as theplanet’s evolving radius ( R p ), its mass, mass accretionrate ( ˙ M p ), and luminosity ( L p ). Figure 1 shows the evo-lution of these quantities with black solid lines indicat-ing an accreting gas giant that ends up with one Jupitermass or about 318 Earth masses ( M ⊕ ). In total, we haveseven models at our disposal, where the planets have fi-nal masses of 1, 2, 3, 5, 7, 10, and 12 M Jup . These tracksare sensitive to the planet’s core mass, which we assumeto be 33 M ⊕ for all planets. Jupiter’s core mass is actu-ally much lower, probably around 10 M ⊕ (Guillot et al.1997). Lower final core masses in these models translateinto lower planetary luminosities at any given accretionrate. In other words, our results for the H O ice linesaround the Jupiter-mass test planet are actually upper,or outer limits, and a more realistic evaluation of theconditions around Jupiter would shift the ice lines closerto the planet. Over the whole range of available planettracks with core masses between 22 and 130 M ⊕ , we notethat the planetary luminosities at shutdown are 10 − . and 10 − . solar luminosities, respectively. As the dis-tance of the H O ice line in a radiation-dominated diskscales with L / , different planetary core masses wouldthus affect our results by less than ten percent.The pre-computed planetary models cover the first few10 yr after the onset of accretion onto the planet. Weinterpolate all quantities on a discrete time line with astep size of 5,000 yr. At any given time, we feed Equa-tions (14) with the planetary model and solve the coupledEquations (1)-(10) in an iterative framework. With T s provided by Equation (13), we finally solve the 5th orderpolynomial in Equation (16) numerically.Once the planetary evolution models indicate that ˙ M p has dropped below a critical shutdown accretion rate( ˙ M shut ), we assume that the formation of satellites haseffectively stopped. As an example, no Ganymede-sizedmoon can form once ˙ M shut < M Gan
Myr − ( M Gan beingthe mass of Ganymede) and if the disk’s remaining lifetime is < yr (see Figure 1c). As ˙ M p determines thegas surface density through Equation (3), different valuesfor ˙ M shut mean different distributions of Σ g ( r ). In par-ticular, Σ g ( r = 10 R Jup ) equals 7 . × , 9 . × ,and 7 . × kg m − once ˙ M shut reaches 100, 10, and1 M Gan
Myr − , respectively, for the planet that ends upwith one Jupiter mass (see Figure 4 in Heller & Pudritz2015).In any single simulation run, κ P is assumed to be con-stant throughout the disk, and simulations of the plane-tary H O ice lines are terminated once the planet accretesless than a given ˙ M shut . To obtain a realistic picture ofa broad range of hypothetical exoplanetary disk proper-ties, we ran a suite of randomized simulations, where κ P and ˙ M shut were drawn from a lognormal probability den-sity distribution. For log ( κ P / [m kg − ]) we assumed amean value of − ( ˙ M shut / [ M Gan
Myr − ]) and a standardvariation of 1.To get a handle on the plausible surface absorptivi- ater ice lines and the formation of giant moons around super-Jovian planets Figure 1.
Evolution of a Jupiter-like model planet and its cir-cumplanetary disk. Values taken from Mordasini (2013) are la-beled “M13”. (a)
Circumplanetary disk properties. (b)
Growthof the solid core, gaseous envelope, and total mass. (c)
Total massaccretion rate. The dashed horizontal line indicates our fiducialshutdown rate for moon formation of 10 M Gan
Myr − . The dashedvertical line marks the corresponding shutdown for moon formationat about 1 . × yr. (d) Planetary luminosity evolution. ties of various disk, we tested different values of k s be-tween 0.1 and 0.5. For each of the seven test planets, weperformed 120 randomized simulations of the disk evo-lution and then calculated the arithmetic mean distanceof the final water ice line. The resulting distributions areskewed and non-Gaussian. Hence, we compute both thedownside and the upside semi standard deviation (corre-sponding to σ/ . / .
15 % and − .
15 % of the simu-lations around the mean. We also calculate the 2 σ semi-deviation, corresponding to +95 . / .
75 % and − .
75 % around the mean. Downside and upside semideviations combined deliver an impression of the asym-metric deviations from the mean, and their sum equalsthat of the Gaussian standard deviations.The total, instantaneous mass of solids in the disk atthe time of moon formation shutdown is given as M s = 2 πX Z r ice r in d r r Σ g ( r )+3 Z r cf r ice d r r Σ g ( r ) ! , (17)where r in is the inner truncation radius of the disk (as-sumed at Jupiter’s corotation radius of 2.25 R Jup , Canup& Ward 2002), r ice is the distance of the H O ice line, and r cf is the outer, centrifugal radius of the disk (Machidaet al. 2008). Depending on the density of the disk gasand the size of the solid grains, the water sublimationtemperature can vary by several degrees Kelvin (Lewis1972; Lecar et al. 2006), but we adopt 170 K as our fidu-cial value (Hasegawa & Pudritz 2013).We simulate the evolution of the H O ice lines in thedisks around young super-Jovian planets at 5.2 astro-nomical units (AU, the distance between the Sun andthe Earth) from a Sun-like star, facilitating comparisonof our results to the Jovian moon system. These planetsbelong to the observed population of super-Jovian plan-ets at ≈ RESULTS AND PREDICTIONS
Figure 1(a) shows, on the largest radial scales, the Hillradius (black crosses) of the accreting giant planet. Theplanetary radius (black solid line) is well within the Hillsphere, but it is quite extensive for 0.9 Myr, so muchthat the CPD (gray solid line) has not yet formed bythat time. It only appears after 0.9 Myr of evolutionof the system. Within that disk, we follow the timeevolution of two features – the heat transition (orangeopen circles) and the H O ice line (blue dots). Theheat transition denotes the transition from the viscous tothe irradiation heating regime in the disk (see Hasegawa& Pudritz 2011), and it appearsat the outer disk edgeabout 0 . × yr after the onset of accretion. It movesrapidly inwards and within ≈ × yr it reaches theinner disk edge, which sits roughly at the radius of theplanet. At the same time ( ≈ . × yr after the on-set of accretion), the H O ice line appears at the outerdisk radius and then moves slowly inward as the planetcools. The ice line reverses its direction of movement at ≈ . × yr due to the decreasing gas surface densities,while the opacities are assumed to be constant through- Ren´e Heller & Ralph Pudritz
Figure 2.
Disk temperatures around a forming Jupiter-like planet10 yr after the onset of accretion. The two disk levels represent themidplane and the photosphere (at a hight z s above the midplane).At a given radial distance ( r ) to the planet, measured in Jupiterradii, the midplane is usually warmer than the surface (see colorbar). The orbits of Io, Europa, Ganymede, and Callisto (labelledI, E, G, and C, respectively) and the location of the instantaneousH O ice line at ≈ . R Jup are indicated in the disk midplane.The arrow attached to the ice line indicates that it is still movinginward before it reaches its final location at roughly the orbit ofGanymede. This simulation assumes fiducial disk values ( k s =0 . , κ P = 10 − m kg − ), some 10 yr before the shutdown of moonformation. out the disk, see Equation (13).Figure 1(b) displays the mass evolution of the plane-tary core (gray dashed line) and atmosphere (gray solidline). Note that the rapid accumulation of the envelopeand the total mass at around 0 . × yr correspondsto the runaway accretion phase. Panel (c) presents thetotal mass accretion rate onto the planet (black solidline). The dashed horizontal line shows an example for˙ M shut (here 10 M Gan
Myr − ), which corresponds to atime 1.08 Myr after the onset of accretion in that partic-ular model. Note that shutdown accretion rates withinone order of magnitude around this fiducial value occur0.1 - 0.3 Myr after the runaway accretion phase, that is,after the planet has opened up a gap in the circumstellardisk. In panel (d), the planetary luminosity peaks dur-ing the runaway accretion phase and then dies off as theplanet opens up a gap in the circumstellar disk, whichstarves the CPD.Figure 2 shows a snapshot of the temperature struc-ture of the disk surface and midplane around a Jupiter-mass planet, 10 yr after the onset of mass accretion.The location of the instantaneous water ice line is indi-cated with a white dotted line, and the current positionsof the Galilean satellites are shown with black dottedlines. Over the next hundred thousand years, the heatingrates drop and the ice line moves inward to Ganymede’spresent orbit as the planet’s accretion rate decreases dueto its opening of a gap in the circumsolar disk. We arguethat the growing Ganymede moved with the ice line trapand was parked in its present orbit when the circumjo-vian disk dissipated. Due to the rapid decrease of massaccretion onto the planet after gap opening (up to aboutan order of magnitude per 10 yr), this process can bereasonably approximated as an instant shutdown on thetime scales of planet formation (several 10 yr), althoughit is truly a gradual process.Figure 3 shows the radial positions of the ice linesaround super-Jovian planets as a function of time and Figure 3.
Evolution of the H O ice lines in the disks aroundsuper-Jovian gas planets. Black solid lines, labeled (1) - (3), indi-cate the locations of the H O ice lines assuming different massaccretion rates for the shutdown of moon formation ( ˙ M shut ∈{ , , }× M Gan
Myr − ). The shaded area embraces the orbitsof Europa and Ganymede around Jupiter, where Jupiter’s H O iceline must have been at the time when the Galilean satellites com-pleted formation. The most plausible shutdown rate for the Joviansystem (black line with label 2) predicts ice lines between roughly10 and 15 R Jup over the whole range of super-Jovian planetarymasses. Simulations assume k s = 0 . κ P = 10 − m kg − . for a given disk surface absorptivity ( k s ) and disk Planckopacity ( κ P ). More massive planets have larger disksand are also hotter at a given time after the onset ofaccretion, which explains the larger distance and lateroccurrence of water ice around the more massive giants.Solid black lines connect epochs of equal accretion rates(1, 10, and 100 M Gan per Myr). Along any given iceline track, higher accretion rates correspond to earlierphases. The gray shaded region embraces the orbitalradii of Europa and Ganymede, between which we ex-pect the H O ice line to settle. The H O ice line aroundthe 1 Jupiter mass model occurs after ≈ . × yrat the outer edge of the disk, passes through the currentorbit of Ganymede, and then begins to move outwardsaround 1 . × yr due to the decreasing gas surfacedensities (note, the opacities are assumed constant). Inthis graph, ˙ M shut ≈ M Gan
Myr − can well explain thementioned properties in the Galilean system.In Figure 4, we present the locations of the ice lines ina more global picture, obtained by performing 120 ran-domized disk simulations for each planet, where ˙ M shut and κ P were drawn from a lognormal probability distri-bution. We also simulated several plausible surface ab-sorptivities of the disk (D’Alessio et al. 2001; Makalkin& Dorofeeva 2014) (0 . ≤ k s ≤ . k s = 0 .
2. The mean orbital radius of the ice line at thetime of shutdown around the 1 M Jup planet is almostprecisely at Ganymede’s orbit around Jupiter, which weclaim is no mere coincidence. Most importantly, despitea variation of ˙ M shut by two orders of magnitude and con-sidering more than one order of magnitude in planetarymasses, the final distances of the H O ice lines only varybetween about 15 and 30 R Jup . Hence, regardless of theactual value of ˙ M shut , the transition from rocky to icymoons around giant planets at several AU from Sun-likestars should occur at planetary distances similar to theone observed in the Galilean system.We ascribe this result to the fact that the planetary lu- ater ice lines and the formation of giant moons around super-Jovian planets Figure 4.
Distance of the H O ice lines at the shutdown of moonformation around super-Jovian planets. The solid line indicatesthe mean, while shaded areas denote the statistical scatter (darkgray 1 σ , light gray 2 σ ) in our simulations, based on the posteriordistribution of the disk Planck mean opacity ( κ P ) and the shut-down accretion rate for moon formation ( ˙ M shut ). The dashed linerepresents the size of the optically thick part of the circumplan-etary disk, or its centrifugal radius. All planets are assumed toorbit a Sun-like star at a distance of 5.2 AU and k s is set to 0.2.Labeled circles at 1 M Jup denote the orbits of the Galilean satelliteIo, Europa, Ganymede, and Callisto. Orange indicates rocky com-position, blue represents H O-rich composition. Circle sizes scalewith moon radii. Note that Ganymede sits almost exactly on thecircumjovian ice line. minosity is the dominant heat source at the time of moonformation shutdown. Planetary luminosity, in turn, isdetermined by accretion (and gravitational shrinking),hence a given ˙ M shut translates into similar luminositiesand similar ice line radii for all super-Jovian planets.Planets above 1 M Jup have substantially larger parts oftheir disks beyond their water ice lines (note the logarith-mic scale in Figure 4) and thus have much more materialavailable for the formation of giant, water-rich analogs ofGanymede and Callisto.Figure 5 shows the total mass of solids at the time ofmoon formation shutdown around super-Jovian planets.Intriguingly, for any given shutdown accretion rate thetotal mass of solids scales proportionally to the planetarymass. This result is not trivial, as the mass of solidsdepends on the location of the H O ice line at shutdown.Assuming that ˙ M shut is similar among all super-Jovianplanets, we confirm that the M T ∝ − M p scaling lawobserved in the solar system also applies for extrasolarsuper-Jupiters (Canup & Ward 2006; Sasaki et al. 2010).In addition to the evolution of the H O ice lines, wealso tracked the movements of the heat transitions, aspecific location within the disk, where the heating fromplanetary irradiation is superseded by viscous heating.Heat transitions cross the disk within only about 10 yr(see Figure 1a), several 10 yr before the shutdown ofmoon formation, and thereby cannot possibly act asmoon traps. Their rapid movement is owed to the abruptstarving of the planetary disk due to the gap opening ofthe circumstellar disk, whereas the much slower photoe-vaporation of the latter yields a much slower motion ofthe circumstellar heat trap. The ineffectiveness of heattraps for satellites reflects a key distinction between theprocesses of moon formation and terrestrial planet for-mation. DISCUSSION
Figure 5.
Instantaneous mass of solids in the disks around super-Jovian planets at moon formation shutdown. Solid and dashedlines refer to disk absorptivities of k s = 0 . ≈ . M Gan ). For any given shutdownrate, we find a linear increase in the mass of solids at the timeof moon formation shutdown as a function of planetary mass, inagreement with previous simulations for the solar system giants(Canup & Ward 2006; Sasaki et al. 2010). Super-Jovian planetsof 10 M Jup should thus have moon systems with total masses of ≈ − × M Jup , or 3 times the mass of Mars.
Accretion and Migration of both Planets and Moons
While Canup & Ward (2002) stated that accretionrates of 2 × − M Jup yr − (about 2 . × M Gan
Myr − )best reproduced the disk conditions in which the Galileansatellites formed, our calculations predict a shutdownaccretion rate that is considerably lower, closer to10 M Gan
Myr − . The difference in these results is mainlyowed to two facts. First, Canup & Ward (2002) only con-sidered viscous heating. Our additional heating terms(illumination from the planet, accretion onto the disk andstellar illumination) contribute additional heat, whichimply smaller accretion rates to let the H O ice linesmove close enough to the Jupiter-like planet. Second, theparameterization of planetary illumination in the Canup& Ward (2002) model is different from ours. While theyassume an r − / dependence of the midplane tempera-ture from the planet ( r being the planetary radial dis-tance), we do not apply any pre-described r -dependence.In particular, T m ( r ) cannot be described properly by asimple polynomial due to the different slopes of the var-ious heat sources as a function of planetary distance.Previous models assume that type I migration ofthe forming moons leads to a continuous rapid loss ofproto-satellites into the planet (Canup & Ward 2002;Mosqueira & Estrada 2003a,b; Alibert et al. 2005; Sasakiet al. 2010). (Alibert et al. 2005) considered Jupiter’saccretion disk as a closed system after the circumstel-lar accretion disk had been photo-evaporated, whereasSasaki et al. (2010) described accretion onto Jupiterwith an analytical model. In the Mosqueira & Estrada(2003a,b)(ME) model, satellites migrate via type I butperturb the gas as they migrate and eventually stall andopen a gap, ensuring their survival. In opposition to theCanup & Ward (CW) theory, their model does not pos-tulate “generations” of satellites, which are subsequently Canup & Ward (2002) discuss the contribution of planetaryluminosity to the disk’s energy budget, but for their computationsof the gas surface densities they ignore it.
Ren´e Heller & Ralph Pudritz lost into the planet, because satellite formation doesn’tstart until the accretion inflow onto the planet wanes.There are two difficulties with the CW picture. First,type I migration can be drastically slowed down as grow-ing giant moons get trapped by the ice lines or at theinner truncation radius of the disk . Thus it is not obvi-ous that a conveyor belt of moons into their host planetsis ever established. Second, our Figure 5 also contra-dicts this scenario, because the instantaneous mass ofsolids in the disk during the end stages of moon forma-tion (or planetary accretion) is not sufficient to form thelast generation of moons. In other words, whenever theinstantaneous mass of solids contained in the circumjo-vian disk was similar to the total mass of the Galileanmoons, the correspondingly high accretion rates causedthe H O ice line to be far beyond the orbits of Europaand Ganymede.We infer, therefore, that the final moon populationaround Jupiter and other Jovian or super-Jovian exo-planets must, at least to a large extent, have built duringthe ongoing, final accretion process of the planet, whenit was still fed from the circumstellar disk. In orderto counteract the inwards flow due to type I migration,we suggest a new picture in which the circumplanetaryH O ice line and the inner cavity of Jupiter’s accretiondisk have acted as migration traps. This important hy-pothesis needs to be tested in future studies. The effectof an inner cavity will also need to be addressed, as itmight have been essential to prevent Io and Europa fromplunging into Jupiter.In our picture, Io should have formed dry and its mi-gration might have been stopped at the inner truncationradius of Jupiter’s accretion disk, at a few Jupiter radii(Takata & Stevenson 1996). It did not form wet and thenlose its water through tidal heating. Ganymede may haveformed at the water ice line in the circumjovian disk,where it has forced Io and Europa in the 1:2:4 orbitalmean motion resonance (Laplace et al. 1829). From aformation point of view, we suggest that Io and Europabe regarded as moon analogs of the terrestrial planets,whereas Ganymede and Callisto resemble the precursorsof giant planets.Our combination of planet formation tracks and a CPDmodel enables new constraints on planet formation frommoon observations. As just one example, the “GrandTack” (GT) model suggests that Jupiter migrated asclose as about 1.5 AU to the Sun before it reversed itsmigration due to a mutual orbital resonance with Saturn(Walsh et al. 2011). In the proximity of the Sun, however,solar illumination should have depleted the circumjovianaccretion disk from water ices during the end stages ofJupiter’s accretion (Heller et al. 2015, in prep.). Thus,Ganymede and Callisto would have formed in a dry envi-ronment during the GT, which is at odds with their highH O ice contents. They can also hardly have formed overmillions of years (Mosqueira & Estrada 2003a) thereafter,because Jupiter’s CPD (now truncated from its environ-ment by a gap) still would have been dry. Alternatively, An inner cavity can be caused by magnetic coupling betweenthe rotating planet and the disk (Takata & Stevenson 1996), and itcan be an important aspect to explain the formation of the Galileansatellites (Sasaki et al. 2010) This conclusion is similar to that proposed by the ME model,but for reasons that are very different. one might suggest that Callisto and Ganymede formed after the GT from newly accreted planetesimals into astill active, gaseous disk around Jupiter. But then Io andEuropa might have been substantially enriched in water,too. Tanigawa et al. (2014, see their Figure 8) foundthat planetesimal accretion via gas drag is most efficientbetween 0.005 and 0.001 Hill radii ( R H ) or about 4 to8 R Jup where gas densities are relatively high.To come straight to the point, our preliminary stud-ies suggest that in the GT paradigm, the icy Galileansatellites must have formed prior to
Jupiter’s excursionto the inner solar system (Heller et al. 2015, in prep.).This illustrates the great potential of moons to constrainplanet formation, which is particularly interesting for theGT scenario where the timing of migration and planetaryaccretion is yet hardly constrained otherwise (Raymond& Morbidelli 2014).
Parameterization of the Disk
Finally, we must address a technical issue, namely, ourchoice of the α parameter (10 − ). While this is consis-tent with many previous studies, how would a variationof α change our results? Magnetorotational instabili-ties might be restricted to the upper layers of CPDs,where they become sufficiently ionized (mostly by cosmichigh-energy radiation and stellar X-rays). Magnetic tur-bulence and viscous heating in the disk midplane mightthus be substantially lower than in our model (Fujii et al.2014). On the other hand, Gressel et al. (2013) modeledthe magnetic stresses in CPDs with a 3D magnetohydro-dynamic model and inferred α values of 0.01 and larger,which would strongly enhance viscous heating. Obvi-ously, sophisticated numerical simulations of giant planetaccretion do not yet consistently describe the magneticproperties of the disks and the associated α values.Given that circumstellar disks are almost certainlymagnetized, CPD can be expected to have inherited mag-netic fields from this source. This makes it likely thatmagnetized disk winds can be driven off the CPD (Fendt2003; Pudritz et al. 2007) which can carry significantamounts of angular momentum. Even in the limit ofvery low ionization, Bai & Stone (2013) demonstratedthat magnetized disk winds will transport disk angularmomentum at the rates needed to allow accretion ontothe central object. However, independent of these uncer-tainties, the final positions of the H O ice line producedin our simulations turn out to depend mostly on plane-tary illumination, because viscous heating becomes neg-ligible almost immediately following gap opening. Hence,even substantial variations of α by a factor of ten wouldhardly change our results for the ice line locations atmoon formation shutdown since these must develop inradiatively dominated disk structure (but it would al-ter them substantially in the viscous-dominated regimebefore and during runaway accretion).Our assumption of a constant Planck opacity through-out the disk is simplistic and ignores the effects of graingrowth, grain distribution within the disk, as well as theevolution of the disk properties. In a more consistentmodel, κ P depends on both the planetary distance anddistance from the midplane, which might entail signifi-cant modifications in the temperature distribution thatwe predict. ater ice lines and the formation of giant moons around super-Jovian planets CONCLUSIONS
We have demonstrated that ice lines imprint impor-tant structural features on systems of icy moons aroundmassive planets. Given that observations show a strongconcentration of super-Jovian planets at ≈ yr, whichis too fast for it to act as a moon migration trap. Alter-natively, we propose that moon migration can be stalledat the H O ice line, which moves radially on a 10 yrtimescale. For Jupiter’s final accretion phase, when theGalilean moons are supposed to form in the disk, ourcalculations show that the H O line is at about the con-temporary radial distance of Ganymede, suggesting thatthe most massive moon in the solar system formed at acircumplanetary migration trap. Moreover, dead zonesmight be present in the inner CPD regions (Gressel et al.2013) where they act as additional moon migration traps,but this treatment is beyond the scope of this paper.Our model confirms the mass scaling law for the mostmassive planets, which suggests that satellite systemswith total masses several times the mass of Mars awaitdiscovery. Their most massive members will be rich inwater and possibly parked in orbits at their host planet’sH O ice lines at the time of moon formation shutdown,that is, between 15 and 30 R Jup from the planet. A Mars-mass moon composed of 50 % of water would have a ra-dius of ≈ . European Extremely Large Telescope , with poten-tial for follow-up observations of the planetary Rossiter-McLaughlin effect (Heller & Albrecht 2014).More detailed predictions can be obtained by includingthe migration process of the accreting planet, which wewill present in an upcoming paper. Ultimately, we expectthat there will be a competition between the formationof water-rich, initially icy moons beyond the circumplan-etary H O ice line and the gradual heating of the disk(and loss of ices) during the planetary migration towardsthe star. Such simulations have the potential to gener-ate a moon population synthesis with predictions for theabundance and detectability of large, water-rich moonsaround super-Jovian planets.The report of an anonymous referee helped us toclarify several aspects of the manuscript. We thankC. Mordasini for sharing with us his planet evolutiontracks, G. D’Angelo for discussions related to disk ioniza-tion, A. Makalkin for advice on the disk properties, and Y. Hasegawa for discussions of ice lines. R. Heller is sup-ported by the Origins Institute at McMaster Universityand by the Canadian Astrobiology Program, a Collabora-tive Research and Training Experience Program fundedby the Natural Sciences and Engineering Research Coun-cil of Canada (NSERC). R. E. Pudritz is supported by aDiscovery grant from NSERC.REFERENCES
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Contribution:RH did the literature research, worked out the mathematical framework, translated the math intocomputer code, performed the simulations, created all figures, led the writing of the manuscript, andserved as a corresponding author for the journal editor and the referees. r X i v : . [ a s t r o - ph . E P ] A p r Astronomy & Astrophysicsmanuscript no. ms c (cid:13)
ESO 2015April 8, 2015
Conditions for water ice lines and Mars-mass exomoons aroundaccreting super-Jovian planets at 1 - 20 AU from Sun-like stars
R. Heller ⋆ and R. Pudritz Origins Institute, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4M1, Canadae-mail: [email protected] , e-mail: [email protected]
Received December 9, 2014; accepted April 4, 2015
ABSTRACT
Context.
The first detection of a moon around an extrasolar planet (an “exomoon”) might be feasible with NASA’s
Kepler or ESA’supcoming
PLATO space telescopes or with the future ground-based
European Extremely Large Telescope . To guide observers and touse observational resources most e ffi ciently, we need to know where the largest, most easily detected moons can form. Aims.
We explore the possibility of large exomoons by following the movement of water (H O) ice lines in the accretion disks aroundyoung super-Jovian planets. We want to know how the di ff erent heating sources in those disks a ff ect the location of the H O ice linesas a function of stellar and planetary distance.
Methods.
We simulate 2D rotationally symmetric accretion disks in hydrostatic equilibrium around super-Jovian exoplanets. Theenergy terms in our semi-analytical framework – (1) viscous heating, (2) planetary illumination, (3) accretional heating of the disk,and (4) stellar illumination – are fed by precomputed planet evolution models. We consider accreting planets with final masses between1 and 12 Jupiter masses at distances between 1 and 20 AU to a solar type star.
Results.
Accretion disks around Jupiter-mass planets closer than about 4.5 AU to Sun-like stars do not feature H O ice lines, whereasthe most massive super-Jovians can form icy satellites as close as 3 AU to Sun-like stars. We derive an empirical formula for thetotal moon mass as a function of planetary mass and stellar distance and predict that super-Jovian planets forming beyond about5 AU can host Mars-mass moons. Planetary illumination is the major heat source in the final stages of accretion around Jupiter-massplanets, whereas disks around the most massive super-Jovians are similarly heated by planetary illumination and viscous heating. Thisindicates a transition towards circumstellar accretion disks, where viscous heating dominates in the stellar vicinity. We also study abroad range of circumplanetary disk parameters for planets at 5.2 AU and find that the H O ice lines are universally between about 15and 30 Jupiter radii in the final stages of accretion when the last generation of moons is supposed to form.
Conclusions.
If the abundant population of super-Jovian planets around 1 AU formed in situ, then these planets should lack thepreviously predicted population of giant icy moons, because those planets’ disks did not host H O ice in the final stages of accretion.But in the more likely case that these planets migrated to their current locations from beyond about 3 to 4.5 AU they might be orbitedby large, water-rich moons. In this case, Mars-mass ocean moons might be common in the stellar habitable zones. Future exomoondetections and non-detections can provide powerful constraints on the formation and migration history of giant exoplanets.
Key words.
Accretion, accretion disks – Planets and satellites: formation – Planets and satellites: gaseous planets – Planets andsatellites: physical evolution – Astrobiology
1. Introduction
Now that the detection of sub-Earth-sized objects has becomepossible with space-based photometry (Muirhead et al. 2012;Barclay et al. 2013) and almost 2000 extrasolar planets havebeen confirmed (Batalha et al. 2013; Rowe et al. 2014), tech-nological and theoretical advances seem mature enough to findmoons orbiting exoplanets. Natural satellites similar in size toMars (0 .
53 Earth radii, R ⊕ ) or Ganymede (0 . R ⊕ ) could bedetectable in the available Kepler data (Kipping et al. 2012;Heller 2014), with the upcoming
PLATO space mission (Simonet al. 2012; Rauer et al. 2014) or with the
European ExtremelyLarge Telescope ( E-ELT ) (Heller & Albrecht 2014). Exomoonsat about 1 AU or closer to their star might be detected during stel-lar transits (Sartoretti & Schneider 1999; Szabó et al. 2006; Kip-ping et al. 2009). Young, self-luminous planets beyond 10 AUmight reveal their satellites in the infrared through planetarytransits of their moons, if the unresolved planet-moon binary ⋆ Postdoctoral Fellow of the Canadian Astrobiology Training Pro-gram can be directly imaged (Peters & Turner 2013; Heller & Al-brecht 2014). Moon formation theories can guide observers anddata analysts in their searches to streamline e ff orts. In turn, thefirst non-detections of exomoons (Brown et al. 2001; Pont et al.2007; Kipping et al. 2013b,a, 2014) and possible future findingsprovide the first extrasolar observational constraints on satelliteformation.In a recent study, we developed a circumplanetary disk accre-tion model and simulated the radial motion of the water (H O)ice line around super-Jovian exoplanets at 5.2 AU around Sun-like stars (Heller & Pudritz 2014). The H O ice line is criticalfor the formation of large, possibly detectable moons, becausehere the surface density of solids increases sharply by a factor of3 to 4 (Hayashi 1981). The composition and the masses of theGalilean moons are usually considered as records of the locationof the H O ice line and, more generally, of the temperature distri- Throughout the paper, our reference to “super-Jovian” planets in-cludes planets with masses between that of Jupiter ( M Jup ) and 12 M Jup ,where the latter demarcates the transition into the brown dwarf regime.Article number, page 1 of 11 & Aproofs: manuscript no. ms bution in Jupiter’s accretion disk (Pollack & Reynolds 1974; Lu-nine & Stevenson 1982). The inner moons Io and Europa turnedout mostly rocky and comparatively light, whereas Ganymedeand Callisto formed extremely rich in water ices (about 50 %)but became substantially more massive (Showman & Malhotra1999).Our main findings were that (1) super-Jovian planets at5.2 AU around Sun-like stars have their H O ice lines betweenabout 15 and 30 Jupiter radii ( R Jup ) during the late stages of ac-cretion, when the final generation of moons form. This rangeis almost independent of the planetary mass ( M p ). (2) With themost massive planets having the most widely extended disks,these disks host the largest reservoirs of H O ices. In particular,the total instantaneous mass of solids ( M sld ) in these disks scalesroughly proportional with M p . (3) The current orbital positionof Ganymede is very close the mean radial location of the cir-cumjovian H O ice line in our simulations, suggesting a novelpicture in which Ganymede formed at a circumplanetary ice linetrap. (4) Heat transitions, however, transverse the accretion diskson a very short timescale ( ≈ yr) once the planet opens upa gap in the circumstellar accretion disk and thereby drasticallyreduces the supply of material. This timescale is short comparedto the satellite migration time scale (10 - 10 yr) (Tanaka et al.2002; Canup & Ward 2002, 2006; Sasaki et al. 2010; Ogihara& Ida 2012) and the time that is left for the final accretion untilshutdown (10 yr). Hence, heat transitions around super-Jovianplanets cannot act as moons traps, which indicates a di ff erent be-havior of planet and moon formation (Menou & Goodman 2004;Hasegawa & Pudritz 2011; Kretke & Lin 2012).We here extend the parameter space of our previous paperand consider planetary accretion between 1 and 20 AU from thestar. This range is motivated by two facts. First, super-Jovians arethe most abundant type of confirmed planets, that is, objects withaccurate mass estimates at these distances (see Fig. 1) . And sec-ond, this range contains the stellar habitable zone around Sun-like stars, which is of particular interest given the fact that gi-ant, water-rich moons might be frequent habitable environments(Heller & Armstrong 2014; Heller et al. 2014). Our main goalis to locate the H O ice lines in the circumplanetary accretiondisks at the time of moon formation shutdown. Their radial sep-arations from the planet will correspond to the orbital radii ator beyond which we can suspect the most massive, water-richmoons to reside. In particular, stellar illumination at only a fewAU from the star will prevent the accretion disks from havingH O ice lines and therefore from hosting large moons. We deter-mine these critical stellar distances below.
2. Methods
We use the framework developed in Heller & Pudritz (2014),which models a 2D axisymmetric accretion disk in hydrostaticequilibrium around the planet. It considers four heating terms,or energy fluxes, of the disk, namely (1) viscous heating ( F vis ),(2) planetary illumination ( F p ), (3) accretional heating of thedisk ( F acc ), and (4) stellar illumination, all of which determinethe midplane and surface temperature of the disk as a functionof planetary distance ( r ). This model (based on earlier workby Makalkin & Dorofeeva 1995, 2014) considers F vis as a dis-tributed energy source within the disk, while F p , F acc , and stel-lar illumination are considered external heat sources. The model Data from the
Kepler space telescope suggests, however, that terres-trial planets are more abundant at about 1 AU around Sun-like stars thangas giant planets (Dong & Zhu 2013; Mulders et al. 2015).
Fig. 1.
Stellar distances and planetary masses of extrasolar planets listedon as of 5 April 2015. Symbols indicate the dis-covery method of each planet, six solar system planets are shown forcomparison. Note the cluster of red dots around 1 AU along the abscissaand between 1 and 10 M Jup along the ordinate. These super-Jovian plan-ets might be hosts of Mars-mass moons. The shaded region denotes thesolar habitable zone defined by the runaway and maximum greenhouseas per Kopparapu et al. (2013). also includes an analytical treatment for the vertical radiative en-ergy transfer, which depends on the Planck mean opacity ( κ P ). Inreal disks, κ P will depend on the disk temperature and the com-position of the solids. In particular, it will be a function of theradial distance to the planet and it will evolve in time as smallparticles stick together and coagulate. To reduce computationaldemands, we here assume a constant κ P throughout the disk, butwe will test values over two orders of magnitude to explore thee ff ects of changing opacities. The vertical gas density in the diskis approximated with an isothermal profile, which is appropri-ate because we are mostly interested in the very final stages ofaccretion when the disk midplane and the disk surface have simi-lar temperatures. Furthermore, deviations between an isothermaland an adiabatic vertical treatment are significant only in the verydense and hot parts of the disk inside about 10 R Jup . As the H Oice line will always be beyond these distances, inaccuracies inour results arising from an isothermal vertical model are negligi-ble.The heating terms (1)-(3) are derived based on precomputedplanet evolution models (provided by courtesy of C. Mordasini,Mordasini 2013), which give us the planetary mass, planetarymass accretion rate ( ˙ M ), and planetary luminosity ( L p ) as a func-tion of time ( t ) (see Fig. 1 in Heller & Pudritz 2014). L p ( t ) is akey input parameter to our model as it determines F p ( t ), and itis sensitive to the planet’s core mass. Yet, in the final stages ofplanetary accretion, L p di ff ers by less than a factor of two forplanetary cores between 22 to 130 Earth masses. With F p ∝ L p and temperature scaling roughly with ( F p ) / , uncertainties inthe midplane or surface temperatures of the disk are lower than20 %, and uncertainties in the radial distance of the H O ice lineare as large as a few Jovian radii at most. For all our simulationsthe precomputed planetary models assume a final core mass of33 Earth masses, which is about a factor of three larger than themass of Jupiter’s core (Guillot et al. 1997). With higher final coremasses meaning higher values for L p at any given accretion rate, Article number, page 2 of 11. Heller and R. Pudritz: Water ice lines and Mars-mass exomoons around accreting super-Jovian planets at 1 - 20 AU
Table 1.
Parameterization of the circumplanetary disk as described in Heller & Pudritz (2014).
Symbol Meaning Fiducial ValueConstant or parameterized planetary disk parameters r distance to the planet R H planetary Hill radius r cf centrifugal radius r c transition from optically thick to optically thin (viscously spread) outer part 27 . / × r cf ( a ) r d outer disk radius R H / ( b,c ) Λ ( r ) radial scaling of Σ ( r ) T m ( r ) midplane temperature T s ( r ) surface temperature Σ ( r ) gas surface density h ( r ) e ff ective half-thickness ρ ( r ) gas density in the midplane ρ s ( r ) gas density at the radiative surface level z s ( r ) radiative surface level, or photospheric height q s ( r ) vertical mass coordinate at the radiative surface c s ( T m ( r )) speed of sound α Viscosity parameter 0.001 ( d ) ν ( r , T m ) gas viscosity Γ adiabat exponent, or ratio of the heat capacities 1.45 χ dust enrichment relative to the protostellar, cosmic value 10 ( c ) X d dust-to-mass fraction 0 . ( e,f ) µ mean molecular weight of the H / He gas 2 .
34 kg / mol (g) Variable planetary disk parameters κ P Planck mean opacity 10 − − − m kg − h ) k s fraction of the solar radiation flux contributing to disk heating at z ≤ z s . − . ( c ) ˙ M shut shutdown accretion rate for moon formation 100 , , M Gan / Myr
Notes.
See Heller & Pudritz (2014) and additional references for details about the respective parameter values or ranges: ( a ) Machida et al. (2008) ( b ) Sasaki et al. (2010) ( c ) Makalkin & Dorofeeva (2014) ( d ) Keith & Wardle (2014) ( e ) Lunine & Stevenson (1982) ( f ) Hasegawa & Pudritz (2013) ( g ) D’Angelo & Bodenheimer (2013) ( h ) Bell et al. (1997) . we actually derive upper, or outer, limits for the special case ofthe H O ice line around a Jupiter-like planet 5.2 AU around aSun-like star.We interpolate the Mordasini (2013) tracks on a linear scalewith a time step of 1 000 yr and evaluate the four heating terms asa function of r , which extends from Jupiter’s co-rotation radius(2 . R Jup in our simulations, Sasaki et al. 2010) out to the disk’scentrifugal radius ( r cf ). At that distance, the centrifugal force act-ing on a gas parcel equals the gravitational pull of the planet. Wecompute r cf using the analytical expression of Machida et al.(2008), which they derived by fitting a power law expression totheir 3D hydrodynamical simulations of circumplanetary accre-tion disks. In this model, r cf ∝ M / for super-Jovian plan-ets at a given stellar distance. Viscous heating is governed bythe α viscosity parameter (Shakura & Sunyaev 1973), which wefix to a value of 10 − in our simulations (for a discussion seeSect. 4 in Heller & Pudritz 2014). Note that even variations of α by an order of magnitude would only change our results for the circumplanetary H O ice line location during the final stagesof planetary accretion by a few planetary radii at most, becausethen the disk is mostly heated by planetary illumination.The sound velocity ( c s ) in the disk midplane is evaluatedas c s = . − √ T m ( r ) / Σ g ) that is derived analyti-cally by solving the continuity equation of the mass inflow ontoa centrifugally supported, viscous disk with a uniform flux perarea (Canup & Ward 2002). The equations of energy transport(Makalkin & Dorofeeva 2014) then allow us to derive the tem-perature profile both at the disk surface, where the optical depth τ = /
3, and in the disk midplane. This model invokes ab-sorption of planetary illumination in the disk photosphere at aheight z s ( r ) above the midplane, modeled by an absorption co-e ffi cient ( k s ), as well as the transport of energy through an opti-cally thick disk to the surface, modeled by κ P . For the dust-to-gas Article number, page 3 of 11 & Aproofs: manuscript no. ms
Fig. 2.
Temperature structure in the disk around a Jupiter-mass planet 5.2 AU from a Sun-like star. Solid black lines indicate disk midplanetemperatures, solid gray lines disk surface temperatures. The other lines indicate a hypothetical disk surface temperature assuming only one heatsource (see legend). (a) : At 0.992 Myr in this particular simulation, the disk has su ffi ciently cooled to allow the appearance of an H O ice line in thedisk midplane at the outer disk edge at 25 . R Jup . The planetary accretion rate is about 4 × M Gan
Myr − . (b) : At 1.073 Myr, when the planetaryaccretion rate has dropped to 10 M Gan
Myr − , the H O ice line has settles between the orbits of Europa and Ganymede (see colored symbols). Inthese final stages of accretion, disk temperatures are governed by planetary illumination (see dashed line for T p ). Fig. 3.
Same as Fig. 2, but now for a 12 M Jup planet. Note that the accretion disk is substantially larger than in the 1 M Jup case. (a) : The H O iceline appears much later than in the Jovian scenario, here at 1.334 Myr. The planetary accretion rate is about 10 M Gan
Myr − and viscous heating isdominant in this phase. (b) : Once the planetary accretion rate has dropped to 10 M Gan
Myr − at 1 .
43 Myr, the H O ice line is at a similar locationas in the Jupiter-like scenario, that is, between the orbits of Europa and Ganymede. ratio ( X d ) we take a fiducial value of 0.006 (Lunine & Steven-son 1982; Hasegawa & Pudritz 2013) in the inner dry regionsof the disk, and we assume that it jumps by a factor of three atthe H O ice line. The resulting mathematical framework con-tains several implicit functions, which we solve in an iterative,numerical scheme. Table 1 gives a complete overview of the pa-rameters involved in the model. Our model neglects the e ff ectsof planetary migration, which is an adequate approximation be-cause the planets don’t migrate substantially within the ≈ yrrequired for moon formation. We also do not actually simulatethe accretion and buildup of moons (Ogihara & Ida 2012).As an extension of our previous study, where all planets wereconsidered at 5.2 AU around a Sun-like star, we here place hy-pothetical super-Jovian planets at stellar distances between 1 and20 AU to a solar type host. Although the precomputed planet for-mation tracks were calculated at 5.2 AU around a Sun-like star, we may still consider di ff erent stellar distances because the ac-cretion rates through any annulus in the circumstellar disk areroughly constant at any time (and thereby similar to the instan-taneous stellar accretion rate). The surface densities of gas andsolids are naturally lower at larger stellar separations. Hence, theaccretion rates provided by the tracks will actually overestimate˙ M and L p in wider orbits at any given time. However, this willnot a ff ect our procedure as we are not primarily interested in theevolution as a function of absolute times but rather as a func-tion of accretion rates. In particular, we introduced the shutdownaccretion ( ˙ M shut ) in Heller & Pudritz (2014), which we use as ameasure to consider as the final stages of moon formation aroundaccreting giant planets. ˙ M shut is a more convenient quantity (orindependent variable) than time to refer to, because, first, the ini-tial conditions of planet formation as well as the age of the sys-tem are often poorly constrained. And, second, accretion rates Article number, page 4 of 11. Heller and R. Pudritz: Water ice lines and Mars-mass exomoons around accreting super-Jovian planets at 1 - 20 AU
Fig. 4.
Surface densities around the four test planets shown in Figs. 2and 3. Black lines refer to the Jupter-mass planets, gray lines to the12 M Jup -mass super-Jovian. Note that the disk of the Jupiter twin ismuch smaller than that of the super-Jovian and that both surface densitydistributions decrease with time. The radial positions of the Galileanmoons are indicated at the bottom of the figure. can be inferred more easily from observations than age. Hence,planetary accretion rates allow us to consider planet (and moon)formation at comparable stages, independent of the initial con-ditions.Variations of the stellar distance will a ff ect both the size ofthe circumplanetary accretion disk (Machida et al. 2008) and thestellar heating term in our model (Makalkin & Dorofeeva 2014).Disks around close-in planets are usually smaller (though notnecessarily less massive) than disks around planets in wide stel-lar orbits owing to their smaller Hill spheres and their lower av-erage specific angular momentum. Moreover, substantial stellarillumination will prevent these disks from having H O ice linesin the vicinity of the star. The temperature of the circumstellaraccretion disk, in which the planet is embedded, is calculatedunder the assumption that the disk is transparent to stellar irradi-ation (Hayashi 1981) and that the stellar luminosity equals thatof the contemporary Sun. In this model, the circumstellar H Oice line is located at 2.7 AU from the star. We will revisit the faintyoung Sun (Sagan & Mullen 1972) as well as other parameter-izations of the circumstellar disk in an upcoming paper (Helleret al. 2015, in prep.).In Figs. 2 and 3 we show an application of our disk modelto a Jupiter-mass and a 12 M Jup planet at 5.2 AU around a Sun-like star, respectively. All panels present the disk surface tem-peratures ( T s , solid gray lines) and disk midplane temperatures( T m , solid black lines) as well as the contributions to T s fromviscous heating ( T v , black crosses), planetary illumination ( T p ,solid black lines), accretion onto the disk ( T a , gray dashed lines),and heating from the circumstellar accretion disk, or “nebula”( T n , gray dotted lines). Any of these contributions to the disksurface temperature is computed assuming that all other contri-butions are zero. In other words, T v , T p , T a , and T n depict thetemperature of the disk photosphere in the hypothetical case thatviscous heating, planetary illumination, accretional heating ontothe disk, or the stellar illumination were the single energy source,respectively. The di ff erent slopes of these curves, in particu-lar of T v and T p , lead to the appearance of heat transitions (notshown), which traverse the disk on relatively short time scales As an example, T v is calculated setting F acc = F p = T neb = (Heller & Pudritz 2014). Also note that in all the simulationsshown, the disk midplane is warmer than the disk surface. Onlywhen accretion drops to about 10 M Gan
Myr − in panels (b) do T s and T m become comparable throughout the disk, both aroundthe Jupiter- and the super-Jupiter-mass planet.Panels (a) in Figs. 2 and 3 are chosen at the time at whichthe H O ice line first appears at the outer edge of the disk,while panels (b) illustrate the temperature distribution at the timewhen ˙ M = M Gan , with M Gan as Ganymede’s mass. Duringthis epoch, the H O ice line around the Jupiter-mass planet issafely between the orbital radii of Europa and Ganymede (seelabeled arrow in Fig. 2b), which suggests that this correspondsto the shutdown phase of moon formation. The colored symbolsin each panel depict the rocky (orange) or icy (blue) composi-tion of the Galilean moons. Symbol sizes scale with the actualmoon radii. In these simulations, the Planck mean opacity ( κ P )has been fixed at a fiducial value of 10 − m kg − throughout thedisk and the disk absorptivity is assumed to be k s = . r c ≈ R Jup ) is substantially smaller than the disk radius aroundthe super-Jovian test planet ( r c ≈ R Jup ) – note the di ff erentdistance ranges shown in the figures! Moreover, the midplanesand surfaces in the inner regions of the super-Jovian accretiondisks are substantially hotter at any given distance around thesuper-Jupiter. Note also that planetary illumination is the mainenergy source around the Jupiter-mass planet in Figs. 2(a) and(b), while viscous heating plays an important role during the ap-pearance of the H O ice line around the super-Jovian planet inFig. 3(a). In the final stages of accretion onto the 12 M Jup planet,shown in Fig. 3(b), viscous heating and planetary illuminationare comparable throughout the disk.In Fig. 4 we present the radial distributions of the gas sur-face densities around our Jovian and super-Jovian test planets.While solid lines refer to panels (a) in Figs. 2 and 3, dashedlines refer to panels (b), respectively. In particular, solid linesrefer to that instant in time when the H O ice lines first appearat the outer edges of the accretion disks around those two plan-ets, whereas the dashed lines depict the accretion phase when˙ M = M Gan
Myr − , which is the phase when the H O ice linearound our Jupiter-mass test planet is in good agreement with thecompositional gradient observed in the Galilean system (Heller& Pudritz 2014). At any given planetary distance in these partic-ular states, the super-Jovian planet has a gas surface density thatis higher by about a factor of five compared to the Jupiter-massplanet. Moreover, we find that the H O ice lines in both casesneed about 10 yr to move radially from the outer disk edges totheir final positions (see labels). Our values for Σ g are similar tothose presented by Makalkin & Dorofeeva (2014) in their Fig. 4.Similar to the procedure applied in Heller & Pudritz (2014),we performed a suite of simulations for planets with masses be-tween 1 and 12 M Jup at 5.2 AU from a Sun-like star, where ˙ M shut and κ P were randomly drawn from a Gaussian probability dis-tribution. For log ( κ P / [m kg − ]) we took a mean value of − ( ˙ M shut / [ M Gan
Myr − ]) we assumed amean value of 1 with a standard variation of 1, which nicely re-produced the compositional H O gradient in the Galilean moons(Heller & Pudritz 2014). As an extension of our previous simula-tions, we here consider various disk absorptivities ( k s = . O ice line in the vicinity of the star.
Article number, page 5 of 11 & Aproofs: manuscript no. ms
Fig. 5.
Distance of the circumplanetary H O ice lines as a function of planetary mass. A distance of 5.2 AU to a Sun-like star is assumed. Theshaded area indicates the 1 σ scatter of our simulations based on the posterior distribution of the disk Planck mean opacity ( κ P ) and the shutdownaccretion rate for moon formation ( ˙ M shut ). The labeled circles at 1 M Jup denote the orbital positions of Jupiter’s moons Io, Europa, Ganymede, andCallisto. Orange indicates rocky composition, blue represents H O-rich composition. Circle sizes scale with the moons’ radii. (a) : Disk reflectivity( k s ) is set to 0 . (b) : Same model parameterization but now with disk reflectivity k s = . Fig. 6.
Similar to Fig. 5, but now in units of fractional disk radius (see ordinate). (a) : Disk reflectivity ( k s ) is set to 0 . (b) : Same model parameter-ization but now with disk reflectivity k s = .
4. The negative slope in both panels indicates that the most massive giant planets have larger fractionsof their disks fed with water ice and, consequently, relatively more space and material to form big moons.
3. H O ice lines around planets of different masses
Figures 2(a) and (b) show that the temperature distribution in thelate accretion disks around Jupiter-mass planets is determinedmostly by the planetary illumination rather than viscous heat-ing. This indicates a key di ff erence between moon formation incircumplanetary accretion disks and planet formation in circum-stellar disks, where the positions of ice lines have been shownto depend mostly on viscous heating rather than stellar illumina-tion (Min et al. 2011; Hasegawa & Pudritz 2011). In this con-text, Figs. 3(a) and (b) depict an interesting intermediate casein terms of the mass of the central object, where both viscousheating and illumination around a 12 M Jup planet show compa-rable contributions towards the final stages of accretion in panel(b). Also note that in Fig. 3(b) the temperature distribution inthe disk outskirts is determined by the background temperatureprovided by stellar illumination. This suggests that the extendeddisks around super-Jovian exoplanets in wide circumstellar or-bits (beyond about 10 AU) might also feature CO and other icelines due to the even weaker stellar illumination.Figures 5(a) and (b) display the radial positions of the cir-cumplanetary H O ice lines around a range of planets withmasses between 1 and 12 M Jup in the final stages of accretion, where we randomized κ P and ˙ M shut as described above, and thestellar distance is 5.2 AU for all planets considered. Panel (a)shows the same data as Fig. 4 in Heller & Pudritz (2014), whilepanel (b) assumes k s = .
4. The larger absorptivity in panel (b)pushes the H O ice lines slightly away from the accreting plan-ets compared to panel (a) for all planets except for the 12 M Jup planet, which we ascribe to an insignificant statistical fluctua-tion. As a key result of our new simulations for k s = .
4, and inagreement with our previous study, the H O ice line is betweenabout 15 and 30 R Jup for all super-Jovian planets and almost in-dependent of M p . The e ff ect of changing k s by a factor of two ismoderate, pushing the H O ice line outward by only a few R Jup on average.Figure 6 presents a di ff erent visualization of these results,again with panel (a) assuming k s = . k s = .
4. Now the ordinate gives the circumplanetary distance inunits of the fractional disk radius, and so the H O ice lines arelocated closer to the planet with increasing M p . This is a conse-quence of the larger disk sizes of the more massive planets. Inother words, larger giant planets have larger fractions of the ac-cretion disks beyond the H O ice lines. Naturally, this means thatthese super-Jovians should form the most massive, icy moons.
Article number, page 6 of 11. Heller and R. Pudritz: Water ice lines and Mars-mass exomoons around accreting super-Jovian planets at 1 - 20 AU
Fig. 7.
Evolution of the total mass of solids ( M sld ) in circumplane-tary accretion disks of several test planets at 5.2 AU from a Sun-likestar. Colored dots along the mass evolution tracks indicate accretionrates ( ˙ M ). If moon formation around all of these planets shuts down atcomparable disk accretion rates ( ˙ M shut ), then M sld ( ˙ M shut ) ∝ M p (see alsoFig. 5 in Heller & Pudritz 2014). Compared to a Jupiter-mass planet, a12 M Jup planet would then have 12 times the amount of solids availablefor moon formation.
In Fig. 7, we therefore analyze the evolution of M sld (in unitsof M Gan ) around these planets as a function of time. Indicatedvalues for ˙ M along these tracks help to visualize M sld as a func-tion of accretion rates (see crosses, squares, and circles), becausetime on the abscissa delivers an incomplete picture of the massevolution due to the di ff erent timescales of these disks. We findthat disks around lower-mass super-Jovians contain less solidmass at any given accretion rate than disks around higher-masssuper-Jovians. Solids also occur earlier in time as the appearanceof the disk itself is regulated by the shrinking of the initiallyvery large planet in our models: lower-mass giant planets con-tract earlier (Mordasini 2013). Assuming that moon formationshuts down at similar accretion rates around any of the simulatedplanets, the example accretion rates (100, 10, 1 M Gan
Myr − ) in-dicate an increase of the mass of solids as a function of M p . Fora given ˙ M shut , this scaling is M sld ∝ M p (see also Fig. 5 in Heller& Pudritz 2014), which is in agreement with the scaling relationfor the total moon mass around giant planets found by Canup &Ward (2006).
4. H O ice lines and total mass of solids aroundplanets at various stellar distances
Having examined the sensitivity of our results to M p and tothe disk’s radiative properties, we now turn to the question ofwhat exomoon systems are like around super-Jovian exoplanetsat very di ff erent locations in their disks than our own Jupiter at5.2 AU. Here, we shall encounter some significant surprises.Figure 8(a) shows the final circumplanetary distance ofthe H O ice line around a Jupiter-mass planet between 2 and20 AU from the star, assuming a shutdown accretion rate of100 M Gan
Myr − . Di ff erent styles of the blue lines correspondto di ff erent disk opacities (see legend), while the solid black We also simulated planets as close as 0.2 AU to the star, but their ac-cretion disks are nominally smaller than the planetary radius, indicatinga departure of our model from reality. Anyways, since the hypotheticaldisks around these planets do not harbor H O ice lines and moon for-mation in the stellar vicinity is hard in the first place (Barnes & O’Brien2002; Namouni 2010), we limit Fig. 10 to 2 AU. line indicates the radius of the circumplanetary accretion disk,following Machida et al. (2008). Circles indicate the radial dis-tances of the Galilean moons around Jupiter, at 5.2 AU from thestar. In this set of simulations, the H O ice lines always endsup between the orbits of Ganymede and Callisto, which is notin agreement with the observed H O compositional gradient inthe Galilean system. As stellar illumination decreases at largerdistances while all other heating terms are constant for the givenaccretion rate, the ice lines move towards the planet at greaterdistances.Most importantly, however, we find that Jovian planets closerthan about 4.8 AU do not have an H O ice line in the first place.Hence, if the large population of Jupiter-mass planets around1 AU (see Fig. 1) formed in situ and without substantial in-ward migration from the outer regions, then these giant plan-ets should not have had the capacity to form giant, icy moons,that is, scaled-up versions of Ganymede or Callisto. These giantmoons with masses up to that of Mars (suggested by Canup &Ward 2006; Heller et al. 2014; Heller & Pudritz 2014), may onlybe present if they have completed their own water-rich forma-tion beyond about 4.8 AU from their star, before they migratedto their current circumstellar orbits together with their host plan-ets. In Fig. 8(b), the planetary accretion rate has droppedby a factor of ten, and the ice lines have shifted. For κ P = − m kg − and κ P = − m kg − they moved towardsthe planet. But for κ P = − m kg − they moved outward.The former two rates actually place the H O ice line aroundJupiter at almost exactly the orbit of Ganymede, which is inbetter agreement with observations. In these simulations, Jovianplanets closer than about 4.5 AU do not have a circumplanetaryH O ice line.In Fig. 9 we vary the stellar distance of a 12 M Jup planet.First, note that the disk (black solid line) is larger at any givenstellar separation than in Fig. 8. Second, note that the H O iceline for a given ˙ M shut in Figs. 9(a) and (b), respectively, is fur-ther out than in the former case of a Jupiter-mass planet. This isdue to both increased viscous heating and illumination from theplanet for the super-Jovian object. Nevertheless, the much largerdisk radius overcompensates for this e ff ect and, hence, all thesesimulations suggests that accretion disks around the most mas-sive planets can have H O ice lines if the planet is not closer thanabout 3.8 AU (panel a) to 3.1 AU (panel b) to a Sun-like star. Thelatter value refers to a shutdown accretion rate of 10 M Gan
Myr − ,which is in good agreement with the water ice distribution in theGalilean moon system. Assuming that moon formation stops atcomparable accretion rates around super-Jovian planets, we con-sider a critical stellar distance of about 3 AU a plausible estimatefor the critical stellar distance of a 12 M Jup accreting planet toshow a circumplanetary H O ice line.Figure 10 presents the total instantaneous mass of solids incircumplanetary accretion disk as a function of stellar distancefor two di ff erent shutdown accretion rates and three di ff erentdisk Planck opacities. The calculation of M sld follows Eq. (17)in Heller & Pudritz (2014), that is, we integrate the surfacedensity of solids between the inner disk truncation radius andthe outer centrifugal radius. Panel (a) for a Jupiter-mass planetdemonstrates that indeed the amount of solids in its late accretiondisk is negligible within about 4.5 AU from a Sun-like star. Thisgives us crucial insights into Jupiter’s migration history withinthe Grand Tack framework (Walsh et al. 2011), which we willpresent in a forthcoming paper (Heller et al. 2015, in prep.).Moreover, both panels show that the e ff ect of di ff erent disk opac-ities on M sld is small for a given M p and ˙ M shut . Article number, page 7 of 11 & Aproofs: manuscript no. ms
Fig. 8.
Distances of the circumplanetary H O ice lines around a Jupiter-like planet (ordinate) as a function of distance to a Sun-like star (abscissa)at the time of moon formation shutdown. Each panel assumes a di ff erent shutdown formation rate (see panel titles). Three Planck opacities throughthe circumplanetary disk are tested in each panel (in units of m kg − , see panel legends). The circumplanetary orbits of the Galilean satellites arerepresented by symbols as in Fig. 2. The black solid line shows the disk’s centrifugal radius (Machida et al. 2008). (a) At ˙ M shut = M Gan / Myr theH O ice lines at ≈ R Jup beyond Ganymede’s current orbit. (b)
At ˙ M shut = M Gan / Myr values of 10 − m kg − ≤ κ ≤ − m kg − place the H O ice line slightly inside the current orbit of Ganymede and thereby seem most plausible.
Fig. 9.
Same as Fig. 8, but now for a 12 M Jup mass planet.
Intriguingly, for any given disk parameterization (see legend)a circumjovian disk at 5.2 AU from a Sun-like star seems to har-bor a relatively small amount of solids compared to disks aroundplanets at larger stellar distances. The red dashed line indicatesa best fit exponential function to the simulations of our fiducialdisk with ˙ M shut = M Gan
Myr − and κ P = − m kg − beyond5.2 AU, as an example. We choose this disk parameterization asit yields the best agreement with the radial location of the icyGalilean satellites (Heller & Pudritz 2014). It scales as M sld = − . M Gan × (cid:18) r ⋆ AU (cid:19) . , (1)where r ⋆ is the stellar distance. Assuming that moon formationstops at similar mass accretion rates around any super-Jovianplanet and taking into account the previously known scaling ofthe total moon masses ( M T ) with M p , we deduce a more generalestimate for the total moon mass: M T = M GM × M p M Jup ! × (cid:18) r ⋆ . (cid:19) . , ( r ⋆ ≥ . , (2) where M GM = . M Gan is the total mass of the Galilean moons.As an example, a 10 M Jup planet forming at 5.2 AU around a Sun-like star should have a moon system with a total mass of about10 M GM or 6 times the mass of Mars. At a Saturn-like stellardistance of 9.6 AU, M T would be doubled. Simulations by Helleret al. (2014) show that this mass will be distributed over three tosix moons in about 90 % of the cases. Hence, if the most massivesuper-Jovian planets formed moon systems before they migratedto about 1 AU, where we observe them today (see Fig. 1), thenMars-mass moons in the stellar habitable zones might be veryabundant.
5. Shutdown accretion rates and loss of moons
Canup & Ward (2002) argued that accretion rates of2 × − M Jup yr − (or about 2 . × M Gan
Myr − ) best repro-duce the disk conditions in which the Galilean system formed.Based on the condition that the H O ice line needs to be be-tween the orbits of Europa and Ganymede at the final stagesof accretion, our calculations predict a shutdown accretion ratethat is considerably lower, closer to 10 M Gan
Myr − (see Fig. 8).The di ff erence in these results is mainly owed to two facts. Article number, page 8 of 11. Heller and R. Pudritz: Water ice lines and Mars-mass exomoons around accreting super-Jovian planets at 1 - 20 AU
Fig. 10.
Total instantaneous mass of solids in circumplanetary accretion disks in units of Ganymede masses and as a function of stellar distance.Labels indicate two di ff erent values for ˙ M shut , and di ff erent line types refer to di ff erent disk opacities (see legend). Black dashed lines in bothpanels indicate our fiducial reference disk. (a) : The 1 M Jup test planet starts to have increasingly large amounts of solids only beyond 4.5 AU,depending on the accretion rate and disk opacity. The red dashed line indicates our fit as per Equation (1). (b) : The 12 M Jup test planet starts tocontain substantial amounts of solids as close as about 3.1 AU to the star, owed to its larger disk.
First, Canup & Ward (2002) only considered viscous heating andplanetary illumination. Our additional heating terms (accretiononto the disk and stellar illumination) contribute additional heat,which imply smaller accretion rates to let the H O ice lines moveclose enough to the Jupiter-like planet. Second, the parameteri-zation of planetary illumination in the Canup & Ward (2002)model is di ff erent from ours. While Canup & Ward (2002) as-sume an r − / dependence of the midplane temperature from theplanet ( r being the planetary radial distance), we do not applyany predescribed r -dependence. In particular, T m ( r ) cannot bedescribed properly by a simple polynomial due to the di ff erentslopes of the various heat sources as a function of planetary dis-tance (see the black solid lines in Figs. 2 and 3).While our estimates of ˙ M shut are about three orders of mag-nitude lower than the values proposed by Canup & Ward (2002),they are also one to two orders of magnitude lower than the val-ues suggested by Alibert et al. (2005). Their moon formationmodel is similar to the so-called “gas-starved” disk model ap-plied by Canup & Ward (2002). Accretion rates in their modelwere not derived from planet formation simulations (as in ourcase) but calculated using an analytical fit to previous simula-tions. Again, the additional energy terms in our model are onereason for the lower final accretion rates that we require in orderto have the H O ice line interior to the orbit of Ganymede.Makalkin & Dorofeeva (2014) also studied Jupiter’s ac-cretion rates and their compatibility with the Galilean moonsystem. They found that values between 10 − M Jup yr − and10 − M Jup yr − (about 10 M Gan
Myr − to 10 M Gan
Myr − ) satisfythese constraints. Obviously, our results are at the lower endof this range, while accretion rates up to three orders of mag-nitude higher should be reasonable according to Makalkin &Dorofeeva (2014). Yet, these authors also claimed that plane-tary illumination be negligible in the final states of accretion,which is not supported by our findings (see Fig. 2). We ascribethis discrepancy to the fact that they estimated the planetary lu-minosity ( L p ) analytically, while in our model L p , R p , M p , and˙ M are coupled in a sophisticated planet evolution model (Mor-dasini 2013). In the Makalkin & Dorofeeva (2014) simulations,10 − L ⊙ . L p . − L ⊙ (depending on assumed values for R p , M p , and ˙ M ), whereas in our case L p remains close to 10 − L ⊙ once a circumjovian H O ice line forms, while ˙ M and R p evolve rapidly due to the planet’s gap opening in the circumsolar disk(see Fig. 1 in Heller & Pudritz 2014).We also calculate the type I migration time scales ( τ I ) ofpotential moons that form in our circumplanetary disks. UsingEq. (1) from Canup & Ward (2006) and assuming a Ganymede-mass moon, we find that 0 . . τ I .
100 Myr in thefinal stages of accretion, with the shortest time scales referringto close-in moons and early stages of accretion when the gas sur-face density is still high. Hence, since the remaining disk lifetime( ≈ yr) is comparable or smaller than the type I migrationtime scale, migration traps might be needed to stop the moonsfrom falling into the planet. As the gas surface density is de-creasing by about an order of magnitude per 10 yr (see Fig. 1(c)in Heller & Pudritz 2014) and since τ I is inversely proportionalto Σ g , type I migration slows down substantially even if the pro-tosatellites still grow. On the other hand, if a moon grows largeenough to open up a gap in the circumplanetary accretion disk,then type II (inwards) migration might kick in, reinforcing theneed for moon migration traps. All these issues call for a de-tailed study of moon migration under the e ff ects of ice lines andother traps.Another issue that could cause moon loss is given by theirpossible tidal migration. If a planet rotates very slowly, its coro-tation radius will be very wide and any moons would be forcedto tidally migrate inwards (ignoring mean motion resonances forthe time being). But if the planet rotates quickly and is not sub-ject to tides raised by the star, such as Jupiter, then moons usuallymigrate outwards due to the planetary tides and at some pointthey might become gravitationally unbound. Barnes & O’Brien(2002, see their Fig. 2) showed that Mars-mass moons around gi-ant planets do not fall into the planet and also remain bound forat least 4.6 Gyr if the planet is at least 0.17 AU away from a Sun-like star. Of course, details depend on the exact initial orbit of themoon and on the tidal parameterization of the system, but as weconsider planets at several AU from the star, we conclude thatloss of moons due to tidal inward or outward migration is not anissue. Yet, it might have an e ff ect on the orbital distances wherewe can expect those giant moons to be found, so additional tidalstudies will be helpful. Article number, page 9 of 11 & Aproofs: manuscript no. ms
6. Discussion
If the abundant population of super-Jovian planets at about 1 AUand closer to Sun-like stars formed in situ, then our results sug-gest that these planets could not form massive, super-Ganymede-style moons in their accretion disks. These disks would havebeen too small to feature H O ice lines and therefore the growthof icy satellites. The accretion disks around these planets mightstill have formed massive, rocky moons, similar in compositionto Io or Europa, which would likely be in close orbits (therebyraising the issue of tidal evolution, Barnes & O’Brien 2002; Cas-sidy et al. 2009; Porter & Grundy 2011; Heller & Barnes 2013;Heller et al. 2014), since circumplanetary accretion disks at 1 AUare relatively small. If these large, close-orbit rocky moons exist,they also might be subject to substantial tidal heating. Alterna-tively, super-Jovians might have captured moons, e.g. via tidaldisruption of binary systems during close encounters (Agnor &Hamilton 2006; Williams 2013), so there might exist indepen-dent formation channels for giant, possibly water-rich moons at1 AU.If, however, these super-Jovian planets formed beyond 3 to4.5 AU and then migrated to their current locations, then theycould be commonly orbited by Mars-mass moons with up to50 % of water, similar in composition to Ganymede and Callisto.Hence, the future detection or non-detection of such moons willhelp to constrain rather strongly the migration history of theirhost planets. What is more, Mars-mass ocean moons at about1 AU from Sun-like stars might be abundant extrasolar habitats(Williams et al. 1997; Heller & Barnes 2013; Heller et al. 2014),Our results raise interesting questions about the formationof giant planets with satellite systems in the solar system andbeyond. The field of moon formation around super-Jovian exo-planets is a new research area, and so many basic questions stillneed to be answered. (1) The “Grand Tack”.
In the “Grand Tack” scenario(Walsh et al. 2011), the fully accreted Jupiter migrated as closeas 1.5 AU to the Sun during the first few million years of thesolar system, then got caught in a mean-motion orbital reso-nance with Saturn and then moved outward to about 5 AU. Ourresults suggest that the icy moons Ganymede and Callisto canhardly have formed during the several 10 yr Jupiter spent inside4.5 AU to the Sun. If they formed before Jupiter’s tack, couldtheir motion through the inner solar system be recorded in thesemoons today? Alternatively, if Callisto required 10 - 10 yr toform (Canup & Ward 2002; Mosqueira & Estrada 2003) andassuming that Jupiter’s accretion disk was intact until after thetack, one might suppose that Ganymede and Callisto might haveformed thereafter. But then how did Jupiter’s accretion disk re-acquire the large amounts of H O that would then be incorpo-rated into Callisto after all water had been sublimated during thetack? We will address these issues in a companion paper (Helleret al. 2015). (2) Migrating planets.
Future work will need to include theactual migration of the host planets, which we neglected in thispaper. Namouni (2010) studied the orbital stability of hypothet-ical moons about migrating giant planets, but the formation ofthese moons was not considered. Yet, the timing of the accretionevolution, the gap opening, the movement within the circumstel-lar disk (and thereby the varying e ff ect of stellar heating), andthe shutdown of moon formation will be crucial to fully assessthe possibility of large exomoons at about 1 AU. These simula-tions should be feasible within the framework of our model, butthe precomputed planet evolution tracks would need to considerplanet migration. Ultimately, magneto-hydrodynamical simula- tions of the circumplanetary accretion disks around migratingsuper-Jovian planets might draw a full picture. (3) Directly imaged planets. Upcoming ground-based ex-tremely large telescopes such as the
E-ELT and the
Thirty Me-ter Telescope , as well as the
James Webb Space Telescope havethe potential to discover large moons transiting directly imagedplanets in the infrared (Peters & Turner 2013; Heller & Albrecht2014). Exomoon hunters aiming at these young giant planets be-yond typically 10 AU from the star will need to know how moonformation takes place in these possibly very extended circum-planetary accretion disks, under negligible stellar heating, and inthe low-density regions of the circumstellar accretion disk. (4) Ice line traps.
If circumplanetary H O ice lines can actas moon migration traps and if solids make up a substantial partof the final masses accreted by the planet, then the accretion ratesonto giant planets computed under the neglect of moons mightbe incorrect in the final stages of accretion. The potential of theH O ice line to act as a moon migration trap is new (Heller & Pu-dritz 2014) and needs to be tested. It will therefore be necessaryto compute the torques acting on the accreting moons within thecircumplanetary accretion disk as well as the possible gap open-ing in the circumplanetary disk by large moons, which mighttrigger type II migration (Canup & Ward 2002, 2006).
7. Conclusions
Planetary illumination is the dominant energy source in the late-stage accretion disks around Jupiter-mass planets at 5.2 AU fromtheir Sun-like host stars (Fig. 2), while viscous heating can becomparable in the final stages of accretion around the most mas-sive planets (Fig. 3).At the time of moon formation shutdown, the H O ice linein accretion disks around super-Jovian planets at 5.2 AU fromSun-like host stars is between roughly 15 and 30 R Jup . This dis-tance range is almost independent of the final planetary mass andweakly dependent on the disk’s absorption properties (Fig. 5).With more massive planets having more extended accretiondisks, this means that more massive planets have larger fractions(up to 70 %) of their disks beyond the circumplanetary H O iceline (compared to about 25 % around Jupiter, see Fig. 6).Jupiter-mass planets forming closer than about 4.5 AU to aSun-like star do not have a circumplanetary H O ice line (Fig. 8),depending on the opacity details of the circumstellar disk. A de-tailed application of this aspect to the formation of the Galileansatellites might help constraining the initial conditions of theGrand Tack paradigm. Due to their larger disks, the most mas-sive super-Jovian planets can host an H O ice line as close asabout 3 AU to Sun-like stars (Fig. 9). With the circumstellar H Oice line at about 2.7 AU in our model of an optically thin cir-cumstellar disk (Hayashi 1981), the relatively small accretiondisks around Jupiter-mass planets at several AU from Sun-likestars thus substantially constrain the formation of icy moons.The extended disks around super-Jovian planets, on the otherhand, might still form icy moons even if the planet is close to thecircumstellar H O ice line.We find an approximation for the total mass available formoon formation ( M T ), which is a function of both M p and r ⋆ (seeEq. 2). The linear dependence of M T ∝ M p has been knownbefore, but the dependence on stellar distance ( M T ∝ r . ⋆ ,for r ⋆ ≥ . M shut ≈ M Gan / yr (Fig. 10) yields the bestresults for the position of the circumjovian H O ice line at theshutdown of the formation of the Galilean moons (Fig. 2b); and
Article number, page 10 of 11. Heller and R. Pudritz: Water ice lines and Mars-mass exomoons around accreting super-Jovian planets at 1 - 20 AU it assumes that this shutdown accretion rate is similar around allsuper-Jovian planets.Our results suggest that the observed large population ofsuper-Jovian planets at about 1 AU to Sun-like stars should notbe orbited by water-rich moons if the planets formed in-situ.However, in the more plausible case that these planets migratedto their current orbits from beyond about 3 to 4.5 AU, theyshould be orbited by large, Mars-sized moons with astrobiolog-ical potential. As a result, future detections or non-detections ofexomoons around giant planets can help to distinguish betweenthe two scenarios because they are tracers of their host planets’migration histories.
Acknowledgements.
We thank Sébastien Charnoz for his referee report whichhelped us to clarify several passages of this manuscript. We thank ChristophMordasini for providing us with the precomputed planetary evolution tracks.René Heller is supported by the Origins Institute at McMaster University andby the Canadian Astrobiology Training Program, a Collaborative Research andTraining Experience Program funded by the Natural Sciences and Engineer-ing Research Council of Canada (NSERC). Ralph E. Pudritz is supported bya Discovery grant from NSERC. This work made use of NASA’s ADS Biblio-graphic Services and of The Extrasolar Planet Encyclopaedia ( ). Computations have been performed with ipython 0.13.2 on python2.7.2 (Pérez & Granger 2007), and all figures were prepared with gnuplot4.6 References
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Article number, page 11 of 11 .3. THE FORMATION OF THE GALILEAN MOONS AND TITAN IN THE GRAND TACKSCENARIO (Heller et al. 2015) 62
Contribution:RH did the literature research, contributed to the mathematical framework, translated the math intocomputer code, created all figures, led the writing of the manuscript, and served as a correspondingauthor for the journal editor and the referees. r X i v : . [ a s t r o - ph . E P ] J un Astronomy & Astrophysicsmanuscript no. ms c (cid:13)
ESO 2015June 29, 2015 L etter to the E ditor The formation of the Galilean moons and Titanin the Grand Tack scenario
R. Heller , ⋆ , G.-D. Marleau ⋆⋆ , and R. E. Pudritz , Origins Institute, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4M1, Canada Department of Physics and Astronomy, McMaster University, [email protected] | [email protected] Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany, [email protected] 17 April 17 2015 / Accepted 2 June 2015
ABSTRACT
Context.
In the Grand Tack (GT) scenario for the young solar system, Jupiter formed beyond 3.5 AU from the Sun and migratedas close as 1.5 AU until it encountered an orbital resonance with Saturn. Both planets then supposedly migrated outward for several10 yr, with Jupiter ending up at ≈ Aims.
We study the formation of Ganymede and Callisto, both of which consist of ≈
50 % H O and rock, in the GT scenario. Weexamine why they lack dense atmospheres, while Titan is surrounded by a thick N envelope. Methods.
We model an axially symmetric circumplanetary disk (CPD) in hydrostatic equilibrium around Jupiter. The CPD is warmedby viscous heating, Jupiter’s luminosity, accretional heating, and the Sun. The position of the H O ice line in the CPD, which iscrucial for the formation of massive moons, is computed at various solar distances. We assess the loss of Galilean atmospheres due tohigh-energy radiation from the young Sun.
Results.
Ganymede and Callisto cannot have accreted their H O during Jupiter’s supposed GT, because its CPD (if still active) wastoo warm to host ices and much smaller than Ganymede’s contemporary orbit. From a thermal perspective, the Galilean moons mighthave had significant atmospheres, but these would probably have been eroded during the GT in < yr by solar XUV radiation. Conclusions.
Jupiter and the Galilean moons formed beyond 4 . ± . ≈ yr after its closest solar approach, ending up at about 7 AU. Key words.
Accretion, accretion disks – Planets and satellites: atmospheres – Planets and satellites: formation – Planets and satellites:physical evolution – Sun: UV radiation
1. Introduction
Recent simulations of the early solar system suggest that the fourgiant planets underwent at least two epochs of rapid orbital evo-lution. In the Grand Tack (GT) model (Walsh et al. 2011), theyformed beyond the solar water (H O) ice line (Ciesla & Cuzzi2006) between 3.5 AU and 8 AU. During the first few 10 yr inthe protostellar disk, when Jupiter had fully formed and Saturnwas still growing, Jupiter migrated as close as 1.5 AU to the Sunwithin some 10 yr, until Saturn gained enough mass to migrateeven more rapidly, catching Jupiter in a 3:2 or a 2:1 mean motionresonance (Pierens et al. 2014). Both planets then reversed theirmigration, with Jupiter ending up at approximately 5 AU. Theother gas planets also underwent rapid orbital evolution due togravitational interaction, pushing Uranus and Neptune to and be-yond about 10 AU, respectively, while Saturn settled at ≈ ≈ × yrlater, according to the Nice model (Tsiganis et al. 2005; Mor-bidelli et al. 2005; Gomes et al. 2005), when Jupiter and Saturncrossed a 2:1 mean motion resonance, thereby rearranging thearchitecture of the gas giants and of the minor bodies. ⋆ Postdoctoral Fellow, Canadian Astrobiology Training Program ⋆⋆ Member of the International Max Planck Research School
While the GT delivers adequate initial conditions for theNice model, the initial conditions of the GT itself are not wellconstrained (Raymond & Morbidelli 2014). Details of the mi-grations of Jupiter and Saturn depend on details of the solar ac-cretion disk and planetary accretion, which are also poorly con-strained (Jacobson & Morbidelli 2014). We note, however, thatthe moons of the giant planets provide additional constraints onthe GT that have scarcely been explored. As an example, using N -body simulations, Deienno et al. (2011) studied the orbitalstability of the Uranian satellites during the Nice instability. Hy-pothetical moons beyond the outermost regular satellite Oberontypically got ejected, while the inner moons including Oberonremained bound to Uranus. Their results thus support the valid-ity of the Nice model. Later, Deienno et al. (2014) found thatone of three Jovian migration paths proposed earlier (Nesvorný& Morbidelli 2012) is incompatible with the orbital stability andalignment of the Galilean moons, yielding new constraints onthe Nice model.To our knowledge, no study has used the Galilean moons orSaturn’s major moon Titan, which is surrounded by a thick ni-trogen (N ) atmosphere, to test the plausibility of the GT model.In this Letter, we focus on the early history of the icy Galileanmoons, Ganymede and Callisto, and on Titan in the GT scenario;assuming that the GT scenario actually took place in one form or Article number, page 1 of 4 & Aproofs: manuscript no. ms another, we identify new constraints on the timing of their for-mation. The novel aspect of our study is the evolution of the H Oice line in Jupiter’s circumplanetary disk (CPD) under the e ff ectof changing solar illumination during the GT. As the positionand evolution of the ice line depends on irradiation from boththe forming Jupiter and the Sun (Heller & Pudritz 2015a), wewant to understand how the observed dichotomy of two mostlyrocky and two very icy Galilean moons can be produced in a GTsetting.
2. The icy Galilean moons in the Grand Tack model
Both Ganymede and Callisto consist of about 50 % rock and50 % H O, while the inner Galilean satellites Io and Europa aremostly rocky. This has been considered a record of the tempera-ture distribution in Jupiter’s CPD at the time these moons formed(Pollack & Reynolds 1974; Sasaki et al. 2010). In particular, theH O ice line, which is the radial distance at which the disk iscool enough for the transition of H O vapor into solid ice, shouldhave been between the orbits of rocky Europa at about 9 . R Jup ) and icy Ganymede at about 15 . R Jup from Jupiter.We simulate a 2D axisymmetric CPD in hydrostatic equilib-rium around Jupiter using a “gas-starved” standard disk model(Canup & Ward 2002, 2006) that has been modified to includevarious heat sources (Makalkin & Dorofeeva 2014), namely, (i)viscous heating, (ii) planetary illumination in the “cold-start sce-nario”, (iii) direct accretion onto the CPD, and (iv) stellar illu-mination. The model is coupled to pre-computed planet evolu-tion tracks (Mordasini 2013), and we here assume a solar lu-minosity 0.7 times its current value to take into account thefaint young Sun (Sagan & Mullen 1972). Details of our semi-analytical model are described in Heller & Pudritz (2015a,b).We evaluate the radial position of the circumjovian H O iceline at di ff erent solar distances of Jupiter and study several diskopacities ( κ P ) as well as di ff erent shutdown accretion rates formoon formation ( ˙ M shut ), all of which determine the radial dis-tributions of the gas surface density and midplane temperaturein the CPD. We follow the CPD evolution after the planet opensup a gap in the circumstellar disk (CSD) until the planetary ac-cretion rate ( ˙ M p ) in the pre-computed track drops to a particu-lar value of ˙ M shut . In Heller & Pudritz (2015a,b) we found that˙ M shut = M Gan
Myr − positions the H O ice line between Eu-ropa and Ganymede for a broad range of κ P values.To assess the e ff ect of heating on the position of the H O iceline in the CPD as a function of solar distance, we compare twoCSD models. First, we use the model of Hayashi (1981, H81, hisEq. 2.3), which assumes that the CSD is mostly transparent in theoptical. Second, we use the model for an optically thick disk ofBitsch et al. (2015, B15, their Eqs. A.3 and A.7), which takesinto account viscous and stellar heating as well as radiative cool-ing and opacity transitions. We consider both a low-metallicity( Z = . Z = .
02) with a stel-lar accretion rate ˙ M ⋆ = . × − M ⊙ yr − corresponding toJupiter’s runaway accretion phase (Mordasini 2013), in whichthe planet opens up a gap and moon formation shuts down.Figure 1 shows the results of our calculations, assuming fidu-cial values κ P = − m − kg − and ˙ M shut = M Gan
Myr − aswell as a centrifugal CPD radius as per Machida et al. (2008).The abscissa denotes distance from the young Sun, the ordinateindicates distance from Jupiter. The vertical dotted line high-lights the critical solar distance ( a crit ) at which the CPD losesits H O ice line in the Z = P l a n e t a r y D i s t a n c e [ R J ] Stellar Distance [AU] a crit IEH O ice line (H81)H O ice line (B15) ( Z = O ice line (B15) ( Z = d i s k r a d i u s icydry GC Fig. 1.
Radial distance of Jupiter’s H O ice line (roughly horizontallines) and centrifugal disk radius (curved solid line) during its final ac-cretion phase. Inside about 4.5 AU from the Sun, Jupiter’s disk does notcontain an H O ice line in any of the CSD models (Hayashi 1981; Bitschet al. 2015, see legend). Locations of the Galilean moons are indicatedby symbols at 5.2 AU. Symbol sizes scale with the physical radii of themoons (orange: rocky, blue: icy composition). to our specific CPD scaling (Heller & Pudritz 2015b), whereasGanymede’s position (circle labeled “G”) near the H O ice lineis not a fit but a result. The CPD radius shrinks substantially to-wards the Sun owing to the increasing solar gravitational force,while the ice line recedes from the planet as a result of enhancedstellar heating. A comparison of the di ff erent CSD models (seelegend) indicates that variations of stellar metallicity or solar il-lumination have significant e ff ects on the circumjovian ice line,but the critical e ff ect is the small CPD radius in the solar vicinity.Most importantly, Fig. 1 suggests that Ganymede and Cal-listo cannot have accreted their icy components as long as Jupiterwas closer than about a crit = . O ice line around Jupiter vanishes for both the H81 and theB15 model. We varied κ P and ˙ M shut by an order of magnitude(not shown), which resulted in changes of this critical solar dis-tance of . . . ± . O because water first would have vaporized and thenbeen photodissociated into hydrogen and oxygen. If Ganymedeor Callisto had acquired their H O from newly accreted plan-etesimals after the GT (e.g. through gas drag within the CDP,Mosqueira & Estrada 2003), then Io (at 0.008 Hill radii, R H )and Europa (at 0 . R H ) would be water-rich, too, because plan-etesimal capture would have been e ffi cient between 0.005 and0 . R H (Tanigawa et al. 2014). Hence, Ganymede and Callistomust have formed prior to the GT, at least to a large extent. Not only must a successful version of the GT model explain theobserved H O ice contents in the Galilean moons, it also must be Heller & Pudritz (2015b) argue that this suggests that Jupiter’s H Oice line acted as a moon migration trap for Ganymede.Article number, page 2 of 4. Heller et al.: The formation of the Galilean moons and Titan in the Grand Tack scenario F r a c t i o n o f t h e A t m o s p h e r e A b o v e ˆ v ˆ v [km s − ] F r a c t i o n o f t h e A t m o s p h e r e A b o v e ˆ v ˆ v [km s − ] H O , 400 KH O , 300 KH O , 200 K − − − − − − − E C I G • N , 400 KN , 300 KN , 200 K − − − − − − − E C I G • Fig. 2.
Integrated Maxwell–Boltzmann velocity distribution of N andH O gas molecules with temperatures similar to Callisto’s surface tem-perature during accretion. The black vertical line denotes Callisto’s es-cape velocity, values for the other Galilean moons are indicated by grayvertical lines. The black circle refers to an example discussed in the text. compatible with both the observed absence of thick atmosphereson the Galilean moons and the presence of a massive N atmo-sphere around Saturn’s moon Titan.Several e ff ects can drive atmospheric escape: (1) thermal (or“Jeans”) escape, (2) direct absorption of high-energy (X-ray andultraviolet, XUV) photons in the upper atmosphere, (3) directabsorption of high-energy particles from the solar wind, (4) im-pacts of large objects, and (5) drag of heavier gaseous compo-nents (such as carbon, oxygen, or nitrogen) by escaping lighterconstituents (such as hydrogen) (Hunten et al. 1987; Pierrehum-bert 2010). Although additional chemical and weather-relatede ff ects can erode certain molecular species (Atreya et al. 2006),we identify the XUV-driven non-thermal escape during the GTas a novel picture that explains both the absence and presence ofGalilean and Titanian atmospheres, respectively. As an example, we assess Callisto’s potential to lose an initialN or H O vapor atmosphere via thermal escape. Our choice ismotivated by Callisto’s relatively low surface temperature duringaccretion, which was about 300 K at most, taking into accountheating within Jupiter’s accretion disk and from the accretionof planetesimals (Lunine & Stevenson 1982). The other moonsmight have been subject to substantial illumination by the youngJupiter, so the e ff ect of thermal escape might be harder to assess.We compute F MB (ˆ v ) = R ∞ ˆ v d v f MB ( v ), with f MB ( v ) as theMaxwell–Boltzmann velocity distribution, for various gas tem-peratures v . Then F normMB (ˆ v ) ≡ F MB (ˆ v ) / F MB (0) is an approximationfor the fraction of the atmosphere that is above a velocity ˆ v .In Figure 2 we compare F normMB (ˆ v ) for N (black lines) andH O (red lines) molecules at temperatures between 200 K and400 K (see legend) with Callisto’s gravitational escape veloc-ity (2 .
44 km s − , black vertical line). As an example, F normMB (ˆ v > .
44 km s − ) . − for an N troposphere at 300 K (see blackcircle), suggesting that a negligible fraction of the atmospherewould be beyond escape velocity. Atmospheric drag (5) can only occur if any of the other e ff ects (1)–(4) is e ffi cient for a lighter gas component. Taking into account collisions of gas particles, their finitefree path lengths, and the extent and temperature of the exo-sphere, Pierrehumbert (2010) estimates the loss time of N fromTitan as ≈ yr. For Callisto, which has about 80 % of Ti-tan’s mass, this timescale would only be reduced by a factor of afew (Pierrehumbert 2010, Eq. 8.37) even if it had been as closeas 1.5 AU from the Sun. At that distance, Callisto would havereceived about 40 times higher irradiation than Titan receivestoday (assuming an optically thin CSD), and its exosphere tem-peratures might have been about 40 / ≈ . atmosphere on Callisto, even during the GT as close as1.5 AU from the Sun. The X-ray and UV luminosities of the young Sun were as highas 10 (Ribas et al. 2005) and 10 (Zahnle & Walker 1982) timestheir current values, respectively, raising the question whetherdirect absorption of high-energy photons or atmospheric dragwould have acted as e ffi cient removal processes of an early Cal-listonian atmosphere. Such a non-thermal escape would haveeroded a hypothetical initial N atmosphere from Earth in only afew 10 yr (Lichtenegger et al. 2010), owing to the high exobasetemperatures (7000-8000 K) and the significant expansion ofthe thermosphere above the magnetopause (Tian et al. 2008).However, the Earth’s primitive atmosphere was likely CO -rich,which cooled the thermosphere and limited the N outflow.If Callisto initially had a substantial CO or H O steam at-mosphere, perhaps provided by outgassing, both gases wouldhave been photo-dissociated in the upper atmosphere, whichthen would have been dominated by escaping H atoms; N andother gases would have been dragged beyond the outer atmo-sphere and lost from the moon forever. Lammer et al. (2014) sim-ulated this gas drag for exomoons at 1 AU from young Sun-likestars and found that the H, O, and C inventories in the initiallythick CO and H O atmosphere around moons 10 % the massof the Earth (four times Ganymede’s mass) would be lost withina few 10 yr depending on the initial conditions and details ofthe XUV irradiation. Increasing the moon mass by a factor offive in their computations, non-thermal escape times increasedby a factor of several tens. Given that strong a dependence ofXUV-driven escape on a moon’s mass, Ganymede and Callistowould most certainly have lost initial N atmospheres during theGT (perhaps within 10 yr), even if they approached the Sun asclose as 1.5 AU rather than 1 AU as in Lammer et al. (2014). If XUV-driven atmospheric loss from the Galilean moons indeedoccurred during the GT, this raises the question why Titan isstill surrounded by a thick N envelope, since Saturn supposedlymigrated as close as 2 AU to the young Sun (Walsh et al. 2011).We propose that the key lies in the di ff erent formation timescalesof the Galilean moons and Titan.In the GT simulations of Walsh et al. (2011), Saturn accretesabout 10 % of its final mass over several 10 yr after its tack,when Jupiter is already fully formed. Hence, while the Galileanmoons must have formed before Jupiter’s GT (Sect. 2.1) andmigrated towards the Sun, thereby losing any primordial at-mospheres, Titan formed after Saturn’s tack and on a longertimescale. Titan actually must have formed several 10 yr afterSaturn’s tack, or it would have plunged into Saturn via its own Article number, page 3 of 4 & Aproofs: manuscript no. ms type I migration within the massive CPD (Canup & Ward 2006;Sasaki et al. 2010) because there would be no ice line to trap itaround Saturn (Heller & Pudritz 2015b).The absence of N atmospheres around the Galilean satellitessupports the GT scenario and is compatible with the presenceof a thick N atmosphere around Titan. The Galilean satelliteslikely lost their primordial envelopes while approaching the Sunas close as 1.5 AU, whereas Titan formed after Saturn’s tack atabout 7 AU under less energetic XUV conditions. Its N atmo-sphere then built up through outgassing of NH accreted fromthe protosolar nebula (Mandt et al. 2014).
3. Discussion
Although the GT paradigm is still controversial, our results pro-vide additional support for it. We show that the outcome ofatmosphere-free icy Galilean satellites and of a thick N atmo-sphere around Titan is possible in the GT scenario. This demon-strates how moon formation can be used to constrain the mi-gration and accretion history of planets – in the solar systemand beyond. The detection of massive exomoons around the ob-served exo-Jupiters at 1 AU from Sun-like stars would indicatethat those gas giants migrated from beyond about 4.5 AU.We tested two CSD models (H81 and B15) to study thee ff ect of changing solar heating on the radial position of theH O ice line in Jupiter’s CPD. Both models yield similar con-straints on the critical solar distance ( a crit ) beyond which theicy Galilean satellites must have formed. We varied opacities,shutdown accretion rates, and stellar metallicities to estimate thedependence of our results on the uncertain properties of the pro-toplanetary and protosatellite disks and found that variations of a crit are . . a crit = . ± . ff erentiation (Kirk & Stevenson 1987).The total heat flux of several TW in both Ganymede and Cal-listo (Mueller & McKinnon 1988) translates into a few tens ofmW m − on the surface. Our results, however, suggest that sun-light might have been a significant external energy source. At1.5 AU from the Sun, illumination might have reached severalW m − , if the CSD was at least partly transparent in the optical.Near-surface temperatures might have been approximately 10 Khigher for & yr (e.g. in Fig. 3a of Nagel et al. 2004). TheGT might have significantly retarded the cooling of the Galileansatellites.Alternatively, Ganymede and Callisto might have becomeH O-rich after the GT, maybe through ablation of newly accretedplanetesimals (Mosqueira et al. 2010), but then a mechanism isrequired that either prevented Io and Europa from accreting sig-nificant amounts of icy planetesimals or that triggered the lossof accreted ice. Tidal heating and the release of large amounts ofkinetic energy from giant impacts might account for that.As H is an e ff ective UV absorber (Glassgold et al. 2004)CSD gas might have shielded early Galilean atmospheres quitee ff ectively. The net UV blocking e ff ect depends on the disk scaleheight, the gas density profile in Jupiter’s gap, and the residualgas flow (Fung et al. 2014). Ultraviolet photons might have beenscattered deep into the gap, maybe even via a back-heating ef- fect from the dusty wall behind the gap (Hasegawa & Pudritz2010). This aspect of our theory needs deeper investigation anddedicated CSD simulations with Jupiter migrating to 1.5 AU.
4. Conclusions
In this Letter we show that the Grand Tack model for the mi-gration of Jupiter and Saturn, if valid, imposes important con-straints on the formation of their massive icy moons, Ganymede,Callisto, and Titan: (1)
Ganymede and Callisto (probably also Ioand Europa) formed prior to the GT (Sect. 2.1), (2) their forma-tion took place beyond 4 . ± . (3) the Galilean moons would have lost any primordial atmospheresduring the GT via non-thermal XUV-driven escape due to the ac-tive young Sun (Sect. 2.2), and (4) Titan’s thick N atmosphereand constraints from moon migration in CPDs suggest that Titanformed after Saturn’s tack (Sect. 2.3).Detailed observations of the Galilean moons by ESA’s up-coming JUpiter ICy moons Explorer (JUICE), scheduled forlaunch in 2022 and arrival at Jupiter in 2030 (Grasset et al. 2013),could deliver fundamentally new insights into the migration his-tory of the giant planets. If Ganymede and Callisto formed priorto Jupiter’s Grand Tack, then JUICE might have the capabilitiesto detect features imprinted during the moons’ journey throughthe inner solar system. Acknowledgements.
We thank C. Mordasini, B. Bitsch, J. Blum, D. N. C Lin, W.Brandner, J. Leconte, C. P. Dullemond, E. Gaidos, and an anonymous referee fortheir helpful comments. RH is supported by the Origins Institute at McMasterU. and by the Canadian Astrobiology Program, a Collaborative Research andTraining Experience Program by the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC). REP is supported by an NSERC Discovery Grant.
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Article number, page 4 of 4 .4. THE NATURE OF THE GIANT EXOMOON CANDIDATE KEPLER-1625 B-I (Heller 2018c)67 stronomy & Astrophysics manuscript no. ms c (cid:13)
ESO 2018March 1, 2018
The nature of the giant exomoon candidate Kepler-1625 b-i
René Heller Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany; [email protected] 11 August 2017; Accepted 21 November 2017
ABSTRACT
The recent announcement of a Neptune-sized exomoon candidate around the transiting Jupiter-sized object Kepler-1625 b couldindicate the presence of a hitherto unknown kind of gas giant moon, if confirmed. Three transits of Kepler-1625 b have been observed,allowing estimates of the radii of both objects. Mass estimates, however, have not been backed up by radial velocity measurements ofthe host star. Here we investigate possible mass regimes of the transiting system that could produce the observed signatures and studythem in the context of moon formation in the solar system, i.e. via impacts, capture, or in-situ accretion. The radius of Kepler-1625 bsuggests it could be anything from a gas giant planet somewhat more massive than Saturn (0 . M Jup ) to a brown dwarf (BD) (upto 75 M Jup ) or even a very-low-mass star (VLMS) (112 M Jup ≈ . M (cid:12) ). The proposed companion would certainly have a planetarymass. Possible extreme scenarios range from a highly inflated Earth-mass gas satellite to an atmosphere-free water-rock companion ofabout 180 M ⊕ . Furthermore, the planet-moon dynamics during the transits suggest a total system mass of 17 . + . − . M Jup . A Neptune-mass exomoon around a giant planet or low-mass BD would not be compatible with the common mass scaling relation of the solarsystem moons about gas giants. The case of a mini-Neptune around a high-mass BD or a VLMS, however, would be located in asimilar region of the satellite-to-host mass ratio diagram as Proxima b, the TRAPPIST-1 system, and LHS 1140 b. The capture of aNeptune-mass object around a 10 M Jup planet during a close binary encounter is possible in principle. The ejected object, however,would have had to be a super-Earth object, raising further questions of how such a system could have formed. In summary, thisexomoon candidate is barely compatible with established moon formation theories. If it can be validated as orbiting a super-Jovianplanet, then it would pose an exquisite riddle for formation theorists to solve.
Key words. accretion, accretion disks – eclipses – planetary systems – planets and satellites: composition – planets and satellites:formation – planets and satellites: individual (Kepler-1625 b)
1. Introduction
The moons in the solar system serve as tracers of their hostplanets’ formation and evolution. For example, the Earth’s spinstate, a key factor to our planet’s habitability, is likely a resultof a giant impact from a Mars-sized object into the proto-Earth(Cameron & Ward 1976), followed by the tidal interaction of theEarth-Moon binary (Touma & Wisdom 1994). The water con-tents and internal structures of the Galilean moons around Jupiterhave been used to reconstruct the conditions in the accretiondisk around Jupiter, in which they supposedly formed (Makalkinet al. 1999; Canup & Ward 2002; Heller et al. 2015). The moonsaround Uranus suggest a collisional tilting scenario for this icygas giant (Morbidelli et al. 2012). It can thus be expected that thediscovery of moons around extrasolar planets could give funda-mentally new insights into the formation and evolution of exo-planets that cannot be obtained by exoplanet observations alone.Another fascinating aspect of large moons beyond the so-lar system is their potential habitability (Reynolds et al. 1987;Williams et al. 1997; Scharf 2006; Heller et al. 2014). In fact,habitable moons could outnumber habitable planets by far, giventheir suspected abundance around gas giant planets in the stellarhabitable zones (Heller & Pudritz 2015a).Thousands of exoplanets have been found (Mayor & Queloz1995; Morton et al. 2016), but no exomoon candidate has un-equivocally been confirmed. Candidates have been presentedbased on a microlensing event (Bennett et al. 2014), asymme-tries detected in the transit light curves of an exoplanet (Ben-Ja ff el & Ballester 2014), and based on a single remarkable ex- oplanet transit in data from CoRoT (Lewis et al. 2015). More-over, hints at an exomoon population have been found in thestacked lightcurves of the
Kepler space telescope (Hippke 2015).The most recent and perhaps the most plausible and testablecandidate has been announced by Teachey et al. (2018). Intheir search for the moon-induced orbital sampling e ff ect (Heller2014; Heller et al. 2016a) in exoplanet transit lightcurves from Kepler , they found the exomoon candidate Kepler-1625 b-i.As the masses of Kepler-1625 b and its proposed compan-ion remain unknown and the radius of the evolved host star ispoorly constrained, the transiting object could indeed be a gi-ant planet. It might also, however, be much larger than Jupiterand, thus, much more massive. Several Jupiter-sized transitingobjects from
Kepler that have been statistically validated but notconfirmed through independent methods, such as stellar radialvelocity (RV) measurements, later turned out to be very-low-mass stars (VLSMs) rather than planets (Shporer et al. 2017).Here, we report on the plausible masses of Kepler-1625 b andits satellite candidate and we show to what extent the possiblescenarios would be compatible with planet and moon formationscenarios in the solar system.
2. Methods
Kepler-1625 (KIC 4760478, KOI-5084), at a distance of2181 + − pc, has been classified as an evolved G-type star Article number, page 1 of 6 a r X i v : . [ a s t r o - ph . E P ] F e b & A proofs: manuscript no. ms with a mass of M ? = . + . − . M (cid:12) , a radius of R ? = . + . − . R (cid:12) , an e ff ective temperature of T e ff ,? = + − K(Mathur et al. 2017), and a Kepler magnitude of K = . The transiting object Kepler-1625 b has an orbital period of287 . ± . ±
00 : 46 : 18) UT , and Teacheyet al. (2018) have secured observations of this transit with the Hubble Space Telescope .The mass of Kepler-1625 b is unknown. Nevertheless, arange of physically plausible masses can be derived fromthe observed transit depth and the corresponding radius ra-tio with respect to the star. The best model fits to the threetransit lightcurves from Teachey et al. (2018) suggest tran-sit depths for Kepler-1625 b (the primary) and its potentialsatellite (the secondary) of about d p = . d s = .
38 ppt, which translate into a pri-mary radius of R p = . + . − . R Jup and a secondary radius of R s = . + . − . R Jup = . + . − . R Nep . The error bars are domi-nated by the uncertainties in R ? .Figure 1 shows a mass-radius curve for non-irradiated sub-stellar objects at an age of 5 Gyr based on “COND” evolutiontracks of Bara ff e et al. (2003). Also indicated on the figure arethe 1 σ confidence range of possible radii of the primary. Thesurjective nature of the mass-radius relationship prevents a di-rect radius-to-mass conversion. As a consequence, as long asthe mass of Kepler-1625 b is unknown, for example from long-term RV observations, the radius estimate is compatible with twomass regimes (see the blue bars at the bottom of Fig. 1). The low-mass regime extends from approximately 0 . M Jup , (roughly 1.3Saturn masses) to about 40 M Jup . The high-mass regime spansfrom 76 M Jup to about 112 M Jup ≈ . M (cid:12) . The entire range cov-ers more than two orders of magnitude. Beyond that, the metal-licity, age, and rotation state of Kepler-1625 b might have sig-nificant e ff ects on its radius and on the curve shown in Fig. 1,allowing for an even wider range of plausible masses for a givenradius.The transition between gas giant planets and brown dwarfs(BDs), that is, between objects forming via core accretion thatare unable to burn deuterium on the one hand and deuterium-burning objects that form through gravitational collapse on theother hand, is somewhere in the range between 10 M Jup and25 M Jup (Bara ff e et al. 2008). Objects more massive than about85 M Jup can ignite hydrogen burning to become VLMSs (Kumar1963; Stevenson 1991). Based on the available radius estimatesalone then, Kepler-1625 b could be anything from a gas planet toa VLMS.Similarly, we may derive mass estimates for the proposedcompanion. The minimum plausible mass for an atmosphere-free object can be estimated by assuming a 50 /
50 water-rockcomposition, giving a mass of about 8 M ⊕ (Fortney et al. 2007).Most objects the size of this candidate do have gas envelopes (atleast for orbital periods .
50 d; Rogers 2015), however, and asubstantial atmosphere around the secondary would be likely. Infact, it could be as light as 1 M ⊕ , considering the discovery ofextremely low-density planets such as those around Kepler-51(Masuda 2014). On the other end of the plausible mass range, ifwe consider the upper radius limit and an object composed of a NASA Exoplanet Archive: https: // exoplanetarchive.ipac.caltech.edu. The transit epoch of (2 , , . + . ± . . ± . BrownDwarfs StarsPlanets
Baraffe et al. (2003)JupiterSaturn upper radius limitbest radius fitlower radius limitallowed mass ranges dynamical transitstransit depth → structure models P r i m a r y R a d i u s [ R J up ] Primary Mass [ M Jup ]0.70.80.91.01.11.21.31.41.5 1 10 100
Fig. 1.
Mass-radius isochrone for substellar objects and VLMSs at anage of 5 Gyr (Bara ff e et al. 2003, black dots). Jupiter’s and Saturn’s po-sitions are indicated with crosses. Horizontal lines indicate the rangeof possible radii for Kepler-1625 b estimated from the Teachey et al.(2018) transit lightcurves and from the uncertainties in the stellar ra-dius. The blue bars at the bottom refer to the mass estimates that arecompatible with the isochrone (lower intervals) and as derived from thedynamical signatures in the transits (upper interval). massive core with a low-mass gas envelope akin to Neptune, weestimate a maximum mass of about 20 M ⊕ (Bara ff e et al. 2008). Given the 287 d orbit of Kepler-1625 b,
Kepler could have ob-served five transits during its four-year primary mission. Yet,only transits number two (T2), four (T4), and five (T5) of thetransit chain were observed. The lightcurves by Teachey et al.(2018) suggest that T2 starts with the ingress of the proposedsatellite, meaning that the satellite would have touched the stel-lar disk prior to its host. The transit T4 shows the opposite con-figuration, starting with the primary and ending with the un-confirmed secondary. Then, T5 indicates that the ingress of themoon candidate precedes the ingress of the planet. In contrast toT2, however, the planet leaves the stellar disk first whereas theproposed satellite would still be in transit for an additional 10 hr.This suggests a transit geometry in which the moon performsabout half an orbit around the planet during the transit, betweenthe two maximum angular deflections as seen from Earth. In fact,the 10 hr moon-only part of the transit suggest a sky-projectedseparation of 17 . R Jup , close to the best fit for the orbital semi-major axis of a ps = . + . − . R p (Teachey et al. 2018). Thus, T5carries important information about the possible planet-moon or-bital period.Assuming that a full orbit would take about twice the timerequired by the proposed satellite to complete T5 (i.e., its orbitalperiod would be roughly P ps =
72 hr), and assuming further anorbital semimajor axis of a ps = . + . − . R p (Teachey et al. 2018)together with the uncertainties in the planetary radius, Kepler’sthird law of motion predicts a barycentric mass of 17 . + . − . M Jup .This value would be compatible with the ten-Jupiter mass objectdescribed by Teachey et al. (2018), although it remains unclearhow the total mass is shared between the primary and secondary.Despite our neglect of uncertainties in the orbital period, our es-timate suggests that proper modeling of the dynamical transit
Article number, page 2 of 6ené Heller: The nature of the giant exomoon candidate Kepler-1625 b-i signature can deliver much tighter constraints on the masses ofthe two bodies than radius estimates alone (see Fig. 1).
We have investigated the plausible masses of Kepler-1625 b andits proposed companion in the context of three moon formationscenarios that have been proposed for the solar system moons.
Impacts . Moon accretion from the gas and debris disk thatforms after giant impacts between planet-sized rocky bodies isthe most plausible scenario for the origin of the Earth-Moonsystem (Canup 2012) and for the Pluto-Charon binary (Canup2005; Walsh & Levison 2015). A peculiar characteristic of thesetwo systems is in their high satellite-to-host mass ratios of about1 . × − for the Earth and 1 . × − for Pluto. In-situ accretion . In comparison, the masses of the moon sys-tems around the giant planets in the solar system are between1 . × − and 2 . × − times the masses of their host planets.This scaling relation is a natural outcome of satellite formationin a “gas-starved” circumplanetary disk model (Canup & Ward2006) and theory predicts that this relation should extend into thesuper-Jovian regime, where moons the mass of Mars would formaround planets as massive as 10 M Jup (Heller & Pudritz 2015b).
Capture . The retrograde orbit and the relative mass of Nep-tune’s principal moon Triton cannot plausibly be explained byeither of the above scenarios. Instead, Agnor & Hamilton (2006)proposed that Triton might be the captured remnant of a formerbinary system that was tidally disrupted during a close encounterwith Neptune. The Martian moons Phobos and Deimos have alsolong been thought to have formed via capture from the aster-oid belt. But recent simulations show that they also could haveformed in a post-impact accretion disk very much like the proto-lunar disk (Rosenblatt et al. 2016).To address the question of the origin of Kepler-1625 b andits potential satellite, we start by calculating the satellite-to-hostmass ratios for a range of nominal scenarios for Kepler-1625 band its proposed companion. These values shall serve as a first-order estimate of the formation regime which the system couldhave emerged from.We then performed numerical calculations of accretion disksaround giant planets as per Heller & Pudritz (2015b) to predictthe satellite-to-host mass ratios in super-Jovian and BD regime,where satellites have not yet been observed. Our disk modelis based on the gas-starved disk theory (Canup & Ward 2006;Makalkin & Dorofeeva 2014). While the conventional scenarioassumes that the planetary luminosity is negligible for the tem-perature structure in the disk, we have included viscous heat-ing, accretion heating, and the illumination of the disk by theyoung giant planet. The evolution of the planet is simulated usingpre-computed planet evolution tracks provided by C. Mordasini(2013). We considered seven di ff erent giant planets with massesof 1 , , , , ,
10, and 12 Jupiter masses. Our model could alsoinclude the relatively weak stellar illumination, but we neglectedit here to avoid unnecessary complexity.We tested a range of reasonable disk surface reflectivi-ties, 0 . ≤ k s ≤ .
3, and assume a constant Planck opacity of10 − m kg − throughout the disk. Models in this range of theparameter space have been shown to reproduce the compositionand orbital radii of the Galilean moons (Heller & Pudritz 2015b).The total amount of solids in the disk is determined by two con-tributions: (1) the initial solids-to-gas ratio of 1 / M cap < M p GM esc π bv enc ∆ v ! / − M esc , (1)where G is the gravitational constant, M esc is the escaping mass, b is the encounter distance, v enc = p v + v ∞ is the encountervelocity, v ∞ is the relative velocity of the planet and the incomingbinary at infinity, v esc = p GM p / b is the escape speed from theplanet at the encounter distance b , ∆ v > q v + v ∞ − s GM p b − a ! , (2)is the velocity change experienced by M cap during capture, a < . a ? p M p M ? ! / + b (3)is the orbital semimajor axis of the captured mass around theplanet, and a ? p is the semimajor axis of the planet around thestar. This set of equations is valid under the assumption that thecaptured mass would be on a stable orbit as long as its apoapsiswere smaller than half the planetary Hill radius.
3. Results
Table 1 summarizes our nominal scenarios for Kepler-1625 b.Scenarios (1aa) to (1bb) refer to the lower stellar radius esti-mate, scenarios (2aa) and (2ab) to the nominal stellar radius, andscenarios (3aa) and (3ab) to the maximum stellar radius. In eachcase, possible values of R p and R s are derived consistently fromthe respective value of R ? . Scenario (TKS) assumes the prelim-inary characterization of the transiting system as provided byTeachey et al. (2018), except that we explicitly adapted a com-panion mass equal to that of Neptune although the authors onlydescribe “a moon roughly the size of Neptune”. The last col-umn in the table shows a list of the corresponding secondary-to-primary mass ratios. Figure 2 illustrates the locations of the saidscenarios in a mass ratio diagram. The positions of the solar sys-tem planets are indicated as are the various formation scenariosof their satellites (in-situ accretion, impact, or capture). The lo-cations of three examples of planetary systems are included forcomparison, namely Proxima b (Anglada-Escudé et al. 2016),the seven planets around TRAPPIST-1 (Gillon et al. 2017), andLHS 1140 b (Dittmann et al. 2017).We find that a Saturn-mass gas planet with either an Earth-mass gas moon (1aa) or a Neptune-mass water-rock moon (1ab) Article number, page 3 of 6 & A proofs: manuscript no. ms
Table 1.
Possible scenarios of the nature of Kepler-1625 b and its proposed companion.
Scenario R ? [ R (cid:12) ] R p [ R Jup ] R s [ R Jup ] M p [ M Jup ] M s [ M ⊕ ] M s / M p (1aa) Saturn-mass gas planet, Earth-mass gas moon (a) (b) (b) . (3) (1) . × − (1ab) Saturn-mass gas planet, Neptune-mass water-rock moon (2) . × − (1ba) brown dwarf, Earth-mass gas moon (3) (1) . × − (1bb) brown dwarf, Neptune-mass water-rock moon (2) . × − (2aa) very-low-mass star, mini-Neptune planet (a) (b) (b) (3) (4) . × − (2ab) very-low-mass star, super-Earth water-rock planet (2) . × − (3aa) very-low-mass star, Neptune-like planet (a) (b) (b) (4) (4) . × − (3ab) very-low-mass star, super-Saturn water-rock planet (2) . × − (TKS) super-Jovian planet, Neptune-like moon – 1 (5) . (5) (5) (5) . × − Notes. ( a ) The stellar radius estimates are based on Mathur et al. (2017). ( b ) The radii of the transiting primary and proposed secondary wereestimated from the lightcurves of Teachey et al. (2018). The corresponding masses of the objects were estimated using structure models andevolution tracks from the following references.
References. (1) Masuda (2014); (2) Fortney et al. (2007); (3) Bara ff e et al. (2003); (4) Bara ff e et al. (2008); (5) Teachey et al. (2018). BrownDwarfs StarsPlanets dynamical transitstransit depth → structure models allowed mass rangesfor Kepler-1625 b ( M a ss o f t h e C o m p a n i on s ) / ( M a ss o f t h e H o s t ) Mass of the Host Object [ M Jup ]simulated impact capture in-situ accretion10 -8 -7 -6 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 Pluto Mars Earth UranusNeptune Saturn Jupiter TRAPP.-1LHS 1140 bProxima b(1aa)(1ab) (1ba)(1bb) (2aa)(2ab) (3aa)(3ab)(TKS)
Kepler-1625 b
Fig. 2.
Mass ratios of companions and hosts, i.e., moons around planets and planets around VLMSs. Host masses are shown along the abscissa,mass ratios on the ordinate. Solar system planets with moons are shown with di ff erent symbols to indicate the respective formation scenariosof their satellites (see legend above the panel). Three VLMSs with roughly Earth-mass planets (TRAPPIST-1, LHS 1140, Proxima Centauri) areplotted as examples of formation via accretion in the stellar regime. The solid black line expanding from Jupiter’s position signifies simulationsof moon formation in the super-Jovian regime, with the gray shaded region referring to uncertainties in the parameterization of the accretion disk.Possible scenarios for the planetary, BD, and VLMS nature of Kepler-1625 b are indicated with blue open circles (see Table 1 for details). Theplausible mass range for Kepler-1625 b is shown with a blue line in the lower right corner and is the same as shown in Fig. 1. would hardly be compatible with the common scaling law of thesatellite masses derived from the gas-starved disk model. Themass ratio would rather be consistent with an impact scenario –just that the mass of the host object would be much higher thanthat of any host to this formation scenario in the solar system.The remaining scenarios, which include a BD orbited by ei-ther an Earth-mass gas moon (1ba) or a Neptune-mass water-rock moon (1bb), as well as a VLMS orbited by either an in-flated mini-Neptune (2aa), or a water-rock super-Earth (2ab), ora Neptune-like binary planet (3aa), or a water-rock super-Saturn-mass object (3ab) could all be compatible with in-situ formationakin to the formation of super-Earths or mini-Neptunes aroundVLMSs.In Fig. 3, we plot the maximum captured mass, that is, themass of what would now orbit Kepler-1625 b, over the mass thatwould have been ejected during a binary encounter with Kepler-1625 b. We adopted a parameterization that represents the (TKS) scenario, where M ? = . M (cid:12) , a ? p = .
87 AU, M p = M jup and b = . R Jup . We tested various plausible values for v ∞ ,ranging from a rather small velocity di ff erence of 1 km / s to amaximum value of v orb =
30 km / s. The latter value reflects theKeplerian orbital speed of Kepler-1625 b, and crossing orbitswould have relative speeds of roughly e × v orb , where e is theheliocentric orbital eccentricity (Agnor & Hamilton 2006).We find that the capture of a Neptune-mass object by Kepler-1625 b is possible at its current orbital location. The escapingobject would have needed to be as massive as 5 M ⊕ to 50 M ⊕ forrelative velocities at infinity between 1 km / s and 30 km / s. Fornearly circular heliocentric orbits of Kepler-1625 b and the for-mer binary, small values of v ∞ would be more likely and preferthe ejection of a super-Earth or mini-Neptune companion froman orbit around the proposed companion to Kepler-1625 b. Article number, page 4 of 6ené Heller: The nature of the giant exomoon candidate Kepler-1625 b-i v ∞ = k m / s k m / s k m / s k m / s NeptuneSaturn M a x i m u m C a p t u r e d S a t e llit e M a ss [ M ⊕ ] Escaping Mass After Capture [ M ⊕ ] 1 10 100 1 10 80 Fig. 3.
Possible masses for the candidate around Kepler-1625 b in acapture scenario. The Keplerian speed at the orbit of Kepler-1625 b isabout 30 km / s and provides a reasonable upper limit on the relative col-lision speed between it and a possible binary approaching on a close-encounter trajectory.
4. Discussion
The mass estimates derived from the analysis of the lightcurvespresented by Teachey et al. (2018) permit an approximation ofthe total mass of Kepler-1625 b and its proposed companion.Photodynamical modeling (Huber et al. 2013) accounting for theorbital motion of the transiting system (Kipping 2011) as well asthe upcoming
Hubble observations will deliver much more ac-curate estimates, provided that the proposed companion aroundKepler-1625 b can be validated. If this exomoon candidate is re-jected, then the dynamical mass estimate of the planet-moon sys-tem (see Sect. 2.1.2) will naturally become meaningless and amass-radius relation akin to Fig. 1 will be the only way to esti-mate the mass of Kepler-1625 b from photometry alone.Consistent modeling of the transiting system based on T2,T4, and T5 could allow predictions of the relative orbital geom-etry of the transiting system during the upcoming transit (Helleret al. 2016b), depending on the as yet unpublished uncertaintiesin P ps / P ? p by Teachey et al. (2018). If these predictions couldbe derived and made publicly available prior to the Hubble ob-servations of T11 on 29 October 2017, then a confirmation of theproposed planet-moon system in the predicted orbital configura-tion would lend further credibility to a discovery claim.The post-capture orbital stability at about 1 AU around Sun-like stars has been demonstrated for Earth-sized moons aroundgiant planets (Porter & Grundy 2011). Most intriguingly, abouthalf of the resulting binaries in these simulations were in a ret-rograde orbit, akin to Triton around Neptune. Hence, if futureobservations of Kepler-1625 b confirm the presence of a com-panion and if it is possible to determine the sense of orbital mo-tion, then this could be a strong argument for a formation throughcapture. Measurements of the sense of orbital motion is impos-sible given even the best available photometric space-based re-sources nowadays available (Lewis & Fujii 2014; Heller & Al-brecht 2014). Nevertheless, if Kepler-1625 b turns out to be a BDor VLMS, then its infrared spectrum could be used to determineits RV variation during the transit. Combined with the variationof the tangential motion of Kepler-1625 and its proposed com-panion that is available from the lightcurve, this would determinethe sense of orbital motion (Oshagh et al. 2017).The (TKS) scenario, suggesting a Neptune-sized moon in or-bit around a 10 M Jup planet, would imply a companion-to-host mass ratio of about 5 . × − , assuming a moon mass equal tothat of Neptune. On the one hand, this is just a factor of a fewsmaller than the relative masses of the Earth-Moon system. Onthe other hand, this value is more than an order of magnitudelarger than the 10 − scaling relation established by the solar sys-tem giant planets. As a consequence, the satellite-to-host massratio does not indicate a preference for either the post-impactformation or in-situ accretion scenarios. In fact, Kepler-1625 band its possible Neptune-sized companion seem to be incompat-ible with both.If the companion around Kepler-1625 b can be confirmedand both objects can be validated as gas giant objects, then itwould be hard to understand how these two gas planets couldpossibly have formed through either a giant impact or in-situaccretion at their current orbits around the star. Instead, theymight have formed simultaneously from a primordial binary ofroughly Earth-mass cores that reached the runaway accretionregime beyond the circumstellar iceline at about 3 AU (Goldre-ich & Tremaine 1980). These cores then may then have startedmigrating to their contemporary orbits at about 0.87 AU as theypulled down their gaseous envelopes from the protoplanetary gasdisk (Lin et al. 1996; Mordasini et al. 2015).The case of a VLMS or massive BD with a super-Earth com-panion would imply a companion mass on the order of 10 − times the host object, which reminds us of the proposed univer-sal scaling relation for satellites around gas giant planets. Hence,this scenario might be compatible with a formation akin to thegas-starved disk model for the formation of the Galilean satel-lites around Jupiter, though in an extremely high mass regime.
5. Conclusions
We derived approximate lower and upper limits on the mass ofKepler-1625 b and its proposed exomoon companion by two in-dependent methods. Firstly, we combined information from theKepler transit lightcurves and evolution tracks of substellar ob-jects. The radius of Kepler-1625 b is compatible with objects aslight as a 0 . M Jup planet (similar to Saturn) or as massive as a0 . M (cid:12) star. The satellite candidate would be securely in theplanetary mass regime with possible masses ranging betweenabout 1 M ⊕ for an extremely low-density planet akin to Kepler-51 b or c and about 20 M ⊕ if it was metal-rich and similar incomposition to Neptune. These uncertainties are dominated bythe uncertainties in the stellar radius. Secondly, we inspected thedynamics of Kepler-1625 b and its potential companion duringthe three transits published by Teachey et al. (2018) and esti-mate a total mass of 17 . + . − . M Jup for the binary. The error barsneglect uncertainties in our knowledge of the planet-moon or-bital period, which we estimate from the third published transitto be about 72 hr.We conclude that if the proposed companion around Kepler-1625 b is real, then the host is most likely a super-Jovian object.In fact, a BD would also be compatible with both the mass-radiusrelationship for substellar objects and with the dynamical transitsignatures shown in the lightcurves by Teachey et al. (2018). Ifthe satellite candidate can be confirmed, then dynamical model-ing of the transits can deliver even better mass estimates of thistransiting planet-moon system irrespective of stellar RVs.Our comparison of the characteristics of proposed exomooncandidate around Kepler-1625 b with those of the moon sys-tems in the solar system reveal that a super-Jovian planet witha Neptune-sized moon would hardly be compatible with con-ventional moon formation models, for example, after a giantimpact or via in-situ formation in the accretion disk around a
Article number, page 5 of 6 & A proofs: manuscript no. ms gas giant primary. We also investigated formation through anencounter between Kepler-1625 b and a planetary binary sys-tem, which would have resulted in the capture of what has pro-visionally been dubbed Kepler-1625 b-i and the ejection of itsformer companion. Although such a capture is indeed possibleat Kepler-1625 b’s orbital distance of 0.87 AU from the star, theejected object would have had a mass of a mini-Neptune itself.And so this raises the question how this ejected mini-Neptunewould have formed around the Neptune-sized object that is nowin orbit around Kepler-1625 b in the first place. If the proposedexomoon can be validated, then measurements of the binary’ssense of orbital motion might give further evidence against or infavor of the capture scenario.The upcoming
Hubble transit observations could potentiallyallow a validation or rejection of the proposed exomoon candi-date. If the moon is real, then dynamical transit modeling willallow precise mass measurements of the planet-moon system. Ifthe moon signature turns out to be a ghost in the detrended
Ke-pler data of Teachey et al. (2018), however, then the transit pho-tometry will not enable a distinction between a single transitinggiant planet and a VLMS. Stellar spectroscopy will then be re-quired to better constrain the stellar radius, and thus the radius ofthe transiting object, and to determine the mass of Kepler-1625 bor Kepler-1625 B, as the case may be.
Acknowledgements.
The author thanks Laurent Gizon, Matthias Ammler-vonEi ff , Michael Hippke, and Vera Dobos for helpful discussions and feedback onthis manuscript. This work was supported in part by the German space agency(Deutsches Zentrum für Luft- und Raumfahrt) under PLATO Data Center grant50OO1501. This work made use of NASA’s ADS Bibliographic Services. Thisresearch has made use of the NASA Exoplanet Archive, which is operated by theCalifornia Institute of Technology, under contract with the National Aeronauticsand Space Administration under the Exoplanet Exploration Program. References
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New Detection and Validation Methodsfor Moons Around Extrasolar Planets .1. DETECTING EXTRASOLAR MOONS AKIN TO SOLAR SYSTEM SATELLITES WITHAN ORBITAL SAMPLING EFFECT (Heller 2014) 75 r X i v : . [ a s t r o - ph . E P ] A p r Received 2014 January 10; accepted 2014 March 23; published 2014 April 29 in The Astrophysical Journal
Preprint typeset using L A TEX style emulateapj v. 04/17/13
DETECTING EXTRASOLAR MOONS AKIN TO SOLAR SYSTEM SATELLITESWITH AN ORBITAL SAMPLING EFFECT
Ren´e Heller
Origins Institute, McMaster University, Hamilton, ON L8S 4M1, [email protected]
Received 2014 January 10; accepted 2014 March 23; published 2014 April 29 in The Astrophysical Journal
ABSTRACTDespite years of high accuracy observations, none of the available theoretical techniques has yet allowedthe confirmation of a moon beyond the solar system. Methods are currently limited to masses about anorder of magnitude higher than the mass of any moon in the solar system. I here present a new methodsensitive to exomoons similar to the known moons. Due to the projection of transiting exomoon orbitsonto the celestial plane, satellites appear more often at larger separations from their planet. Afterabout a dozen randomly sampled observations, a photometric orbital sampling effect (OSE) starts toappear in the phase-folded transit light curve, indicative of the moons’ radii and planetary distances.Two additional outcomes of the OSE emerge in the planet’s transit timing variations (TTV-OSE)and transit duration variations (TDV-OSE), both of which permit measurements of a moon’s mass.The OSE is the first effect that permits characterization of multi-satellite systems. I derive and applyanalytical OSE descriptions to simulated transit observations of the
Kepler space telescope assumingwhite noise only. Moons as small as Ganymede may be detectable in the available data, with Mstars being their most promising hosts. Exomoons with the 10-fold mass of Ganymede and a similarcomposition (about 0.86 Earth radii in radius) can most likely be found in the available
Kepler data ofK stars, including moons in the stellar habitable zone. A future survey with
Kepler -class photometry,such as
Plato 2.0 , and a permanent monitoring of a single field of view over 5 years or more will verylikely discover extrasolar moons via their OSEs.
Keywords: instrumentation: photometers – methods: analytical – methods: data analysis – methods:observational – methods: statistical – planets and satellites: detection CONTEXT AND MOTIVATIONAlthough more than 1000 extrasolar planets have beenfound, no extrasolar moon has been confirmed. Var-ious methods have been proposed to search for exo-moons, such as analyses of the host planet’s transit tim-ing variation (TTV; Sartoretti & Schneider 1999; Simonet al. 2007), its transit duration variation (TDV; Kip-ping 2009a,b), direct photometric observations of exo-moon transits (Tusnski & Valio 2011), scatter analysesof averaged light curves (Simon et al. 2012), a wobbleof the planet-moon photocenter (Cabrera & Schneider2007), mutual eclipses of the planet and its moon ormoons (Cabrera & Schneider 2007; Sato & Asada 2009;P´al 2012), excess emission of transiting giant exoplan-ets in the spectral region between 1 and 4 µ m (Williams& Knacke 2004), infrared emission by airless moonsaround terrestrial planets (Moskovitz et al. 2009; Robin-son 2011), the Rossiter-McLaughlin effect (Simon et al.2010; Zhuang et al. 2012), microlensing (Han & Han2002), pulsar timing variations (Lewis et al. 2008), di-rect imaging of extremely tidally heated exomoons (Pe-ters & Turner 2013), modulations of radio emission fromgiant planets (Noyola et al. 2013), and the generation ofplasma tori around giant planets by volcanically activemoons (Ben-Jaffel & Ballester 2014). Recently, Kippinget al. (2012) started the Hunt for Exomoons with Kepler Department of Physics and Astronomy, McMaster University Postdoctoral fellow of the Canadian Astrobiology TrainingProgram (HEK), the first survey targeting moons around extra-solar planets. Their analysis combines TTV and TDVmeasurements of transiting planets with searches for di-rect photometric transit signatures of exomoons.Exomoon discoveries are supposed to grant fundamen-tally new insights into exoplanet formation. The satellitesystems around Jupiter and Saturn, for example, showdifferent architectures with Jupiter hosting four massivemoons and Saturn hosting only one. Intriguingly, thetotal mass of these major satellites is about 10 − timestheir planet’s mass, which can be explained by their com-mon formation in the circumplanetary gas and debrisdisk (Canup & Ward 2006), and by Jupiter opening upa gap in the heliocentric disk during its own formation(Sasaki et al. 2010). The formation of Earth is inextri-cably linked with the formation of the Moon (Cameron& Ward 1976), and Uranus’ natural satellites indicate asuccessive “collisional tilting scenario”, thereby explain-ing the planet’s unusual spin-orbit misalignment (Mor-bidelli et al. 2012). Further interest in the detection ofextrasolar moons is triggered by their possibility to haveenvironments benign for the formation and evolution ofextrasolar life (Reynolds et al. 1987; Williams et al. 1997;Heller & Barnes 2013). After all, astronomers have founda great number of super-Jovian planets in the habitablezones (HZs) of Sun-like stars (Heller & Barnes 2014).In this paper, I present a new theoretical method thatallows the detection of extrasolar moons. It can be ap-plied to discover and characterize multi-satellite systems Ren´e Heller top viewedge view (a)(b)(c) xy x = - r c o s () r d r d x-r planetmoon Figure 1.
Geometry of a moon’s OSE. Assuming a constant sam-pling frequency over one moon orbit (panel (a), an observer inthe moon’s orbital plane would recognize a non-uniform projecteddensity distribution (panel (b)). All snapshots combined in onesequence frame, the moon is more likely to occur at larger separa-tions x from the planet. The probability distribution P s ( x ) alongthe projected orbit can be constructed as P s ( x ) = r d ϕ/ d x (panel(c)). and to measure the satellites’ radii and orbital semi-major axes around their host planet, assuming roughlycircular orbits. This assumption is justified because ec-centric moon orbits typically circularize on a million yeartime scale due to tidal effects (Porter & Grundy 2011;Heller & Barnes 2013). The method does not dependon a satellite’s direction of orbital motion (retrogradeor prograde), and it relies on high-accuracy averagedphotometric transit light curves. I refer to the physi-cal phenomenon that generates the observable effect asthe Orbital Sampling Effect (OSE). It causes three dif-ferent effects in the phase-folded light curve, namely,(1) the photometric OSE, (2) TTV-OSE, and (3) TDV-OSE. Similarly to the photometric OSE, the scatter peakmethod developed by Simon et al. (2012) makes use oforbit-averaged light curves. But I will not analyze thescatter. While the scatter peak method was describedto be more promising for moons in wide orbits, the OSEworks best for close-in moons. Also, with an orbital semi-major axis spanning 82 % of the planet’s Hill sphere, theexample satellite system studied by Simon et al. (2012)would only be stable if it had a retrograde orbital motion(Domingos et al. 2006). THE ORBITAL SAMPLING EFFECT2.1.
Probability of Apparent Planet-moon Separation
Figure 2.
Normalized sampling frequency (or probability density P s ( x )) for three moons in an Io-wide, Europa-wide, and Ganymede-wide orbit in units of planetary radii. Bars show the results froma randomized numerical simulation, curves show the distributionaccording to Equation (4). The integral, that is, the area undereach curve equals 1. This explains why moons on tighter orbitshave a higher sampling frequency at a given planet-moon distance. Imagine a moon orbiting a planet on a circular orbit.In a satellite system with n moons, this particular moonshall be satellite number s , and its orbital semi-majoraxis around the planet be a ps . Imagine further look-ing at the system from a top view position, such that thesatellite’s apparent path around the planet forms a circle.With a given, but arbitrary sampling frequency you takesnapshots of the orbiting moon, and once the moon hascompleted one revolution, you stack up the frames to ob-tain a frame sequence. This sequence is depicted in panel(a) of Figure 1. An observer in the orbital plane of thesatellite, taking snapshots with the same sampling fre-quency and stacking up a sequence frame from its edgeview post, would see the moon’s positions distributedalong a line. The planet sits in the center of this line,which extends as far as the projected semi-major axis toeither side (panel (b) of Figure 1). Due to the projectioneffect, the moon snapshots pile up toward the edges of theprojected orbit. In other words, if the snapshots wouldbe taken randomly from this edge-on perspective, thenthe moon would most likely be at an apparently wideseparation from the planet. We can assume to observemost transiting exomoon systems in this edge view be-cause the orbital plane of the planet-moon system shouldbe roughly in the same plane as the orbital plane of theplanet-moon barycenter around the star (Heller et al.2011b).The likelihood of a satellite to appear at an appar-ent separation x from the planet can be described by aprobability density P s ( x ), which is proportional to the“amount” of orbital path r × d ϕ , with ϕ as the angularcoordinate and r as the orbital radius, divided by theprojected part of this interval along the x -axis, d x (see Speaking about the orbital geometry of a planet-moon binaryas depicted in Figure 1, I will refer to the moon’s orbital radiusaround the planet as r . In case of a multi-satellite system, I desig-nate the planet-satellite orbital semi-major axis of satellite number s as a ps . etecting Extrasolar Moons akin to Solar System Satellites with an Orbital Sampling Effect x = − r cos( ϕ ), we thus have P s ( x ) ∝ r d ϕ d x = r dd x arccos (cid:18) − xr (cid:19) = 1 r r − (cid:16) xr (cid:17) . (1) P s ( x ) must fulfill the condition Z + r − r d x P s ( x ) = 1 , (2)because it is a probability density, and the moon mustbe somewhere. With Z + r − r d x r − (cid:16) xr (cid:17) = Z + r − r d x r dd x arccos (cid:18) − xr (cid:19) = πr (3)we thus have the normalized sampling frequency P s ( x ) = 1 πr r − (cid:16) xr (cid:17) . (4)In Figure 2, I plot Equation (4) for a three-satellite sys-tem. The innermost moon with probability density P I isin an orbit as wide as Io’s orbit around Jupiter, thatis, it has a semi-major axis of 6.1 planetary radii ( R p ).The central moon with normalized sampling frequency P E follows a circular orbit 9 . R p from the planet, sim-ilar to Europa, and the outermost moon corresponds toGanymede at 15 . R p and with probability density P G .The shaded areas illustrate a normalized suite of random-ized measurements of the projected orbital separation x that I have simulated with a computer. The lines followEquation (4), where r is replaced by the respective or-bital semi-major axis a ps , and they nicely match the ran-domized sampling. As the area under each curve equals1, the innermost moon has a higher probability to ap-pear at a given position within the interval [ − a pI , + a pI ],with a pI as the sky-projected orbital semi-major axis ofthe satellite in an Io-wide orbit, than any of the othermoons. The normalized sampling frequency or probabil-ity density of apparent separation P s ( x ) is independentof the satellite radius.2.2. The Photometric OSE
The Photometric OSE in Averaged Transit LightCurves
From the perspective of a data analyst, it is appealingthat the OSE method does not require modeling of theorbital evolution of the moon or moons during the tran-sit or between transits, that is, during the circumstellarorbit. Assuming that the satellite is not in an orbital res-onance with the circumstellar orbital motion, the OSEwill smear out over many light curves and always yielda probability distribution as per Equation (4). What ismore, this formula allows an analytic description of theactual effect in the light curve. In other words, once thephase-folded light curve is available after potential TTVsor TDVs – induced by the moons or by other planets – planet star P s ( x ) (1) (2) (3)time no r m a li ze d s t e ll a r b r i gh t n e ss } planetarytransit Figure 3.
Photometric OSE during ingress. At epoch (1), a satel-lite’s probability distribution P s ( x ) along its circumplanetary orbittouches the stellar disk, from then on causing a steep (but small)decline in stellar brightness (see lower panel). As circumplanetaryorbits with lower values of P s ( x ) (visualized by lighter colors) enterthe stellar disk, the brightness decrease weakens. At epochs (2),the planet enters the stellar disk and induces a dramatic decrease instellar brightness, depicted by two slanted lines in the lower panel.From that point on, larger values of P s ( x ) enter the stellar disk, sothe slope of the stellar flux decrease as the OSE increases until themoon orbits have completely entered the stellar disk at epochs (3). as well as red noise (Lewis 2013) have been removed, theOSE can be measured with a simple fit to the binned datapoints. I refer to this effect as the photometric OSE.Yet, what does the effect actually look like? Figure 3visualizes the ingress of the planet-moon binary in frontof the stellar disk from a statistical point of view. Thesatellite orbit is assumed to be coplanar with the cir-cumstellar orbit, and the impact parameter (b) of theplanet, corresponding to its minimal distance from thestellar center during the transit, in units of stellar radii,is zero. The thick shaded line drawn through the planetsketches the probability density P s ( x ) and describes asmoothed-out version of the frame shown in panel (b) ofFigure 1. If one were able to repeatedly take snapshotsof a certain moment during subsequent transits (for ex-ample at epochs (1), (2), or (3) in this sketch), then theaveraged positions of the moon would scatter accordingto this shaded distribution. I call the part right of theplanet the right wing of P s ( x ) (or P rw ) and that part of P s ( x ), which is left of the planet, the left wing of P s ( x )(or P lw ).As P rw touches the stellar disk with its highest val-ues, visualized by dark shadings in Figure 3, it causes asteep (though weak) decrease in the averaged transit lightcurve, as depicted in the bottom panel. The decreas-ing probability distribution then proceeds over the stel-lar disk and causes a declining decrease in stellar bright-ness. At epochs (2), the planet enters the stellar disk andtriggers a major brightness decrease, commonly knownfrom a planetary transit light curve. Moving on to point(3), that part of the probability function that actuallymoves into the stellar disk has increasingly higher val-ues toward P lw , thereby inducing an increasing declinein stellar brightness. Note that the lower panel of Fig-ure 3 visualizes the averaged light curve. During eachparticular transit, the moon can be anywhere along itsprojected orbit around the planet, and the decrease in Ren´e Heller stellar brightness follows a completely different curve.2.2.2.
Analytic Description of the Photometric OSE
Equation (4) allows for an analytic description of thephotometric OSE. In Figure 3, the right wing of the prob-ability distribution enters the stellar disk first, which cor-responds to the right side of P s ( x ) shown in Figure 2.Stellar light is blocked between a ps to the right and x ′ ( t )to the left, where the latter variable describes the timedependence of the moving left edge of P s ( x ). To calcu-late the amount of blocked stellar light due to the photo-metric OSE during the ingress of a one-satellite system( F (1)OSE , in ), I integrate P s ( x ) from x ′ ( t ) to a ps and subtractthis area from the normalized, apparent stellar brightness( B OSE ), which equals 1 out of transit. The amplitude ofthe blocked light is given by ( R s /R ⋆ ) , where R ⋆ is thestellar radius, and hence F (1)OSE , in = (cid:18) R s R ⋆ (cid:19) Z a ps x ′ ( t ) d x P s ( x )= 1 π (cid:18) R s R ⋆ (cid:19) (cid:20) π − arccos (cid:18) − x ′ ( t ) a ps (cid:19)(cid:21) n for − ≤ − x ′ ( t ) a ps ≤ o (5)and B (1)OSE , in ( t ) ≡ − F (1)OSE , in ( t ) is the brightness duringingress. For − x ′ ( t ) /a ps < −
1, that is, as long as theprobability distribution of the moon has not yet touchedthe stellar disk, the arccos() term is not defined, so Iset F (1)OSE , in = 0 in that case. To parameterize x ′ ( t ), Ichoose a coordinate system whose origin is in the centerof the stellar disk. Time is 0 at the center of the tran-sit, so that x ′ ( t ) = − R ⋆ − v orb t , with v orb = 2 πa ⋆ b /P ⋆ b as the circumstellar orbital velocity of the planet-moonbarycentric mass M b , assumed to be equal to the cir-cumstellar orbital velocity of both the planet and themoon, P ⋆ b = 2 π p a ⋆ b / ( G ( M ⋆ + M b )) as the circum-stellar orbital period of M b , G as Newton’s gravitationalconstant, and a ⋆ b as the orbital semi-major axis betweenthe star and M b . To consistently parameterize the tran-sit light curve and the OSE, the star-planet system mustthus be well-characterized. Assuming that the planet ismuch more massive than the moon, taking M b ≈ M p isjustified.Once the whole probability density of the moon has en-tered the stellar disk, − x ′ ( t ) /a ps > +1 and the arccos()term is again not defined, so I take F (1)OSE , in = ( R s /R ⋆ ) in that case. During egress of the probability function,the moon, on average, uncovers a fraction F (1)OSE , eg of thestellar disk. Its mathematical description is similar toEquation (5), except for x ′ ( t ) = + R ⋆ − v orb t . Again, F (1)OSE , eg = 0 before the egress of the probability functionand 1 after it has left the disk, so that the normalized,apparent stellar brightness becomes B (1)OSE ( t ) = 1 − F p ( t ) − F (1)OSE , in ( t ) + F (1)OSE , eg ( t ), with F p ( t ) as the stellarflux masked by the transiting planet (see Appendix A).While Equation (5) is valid for one-satellite systems,it can be generalized to a system of n satellites by sub-tracting the stellar flux that is blocked subsequently by the integrated density functions via B ( n )OSE ( t ) = − F p ( t ) − n X s=1 F (s)OSE , in ( t )+ n X s=1 F (s)OSE , eg ( t ) for | x p ( t ) | > R ⋆ − F p ( t ) − n X s=1 F (s)OSE , in ( t )+ n X s=1 F (s)OSE , eg ( t ) + A mask for | x p ( t ) | ≤ R ⋆ (6)where x p ( t ) is the position of the planet, and A mask = 2 π n X s=1 (cid:18) R s R p (cid:19) (cid:20) arccos (cid:18) − R p a ps (cid:19) − arccos (cid:18) + R s a ps (cid:19)(cid:21) (7)compensates for those parts of the probability functionsthat do not contribute to the OSE because of planet-moon eclipses. This masking can only occur during theplanetary transit when | x p ( t ) | ≤ R ⋆ . Note that partialplanet-moon eclipses as well as moon-moon eclipses areignored (but treated by Kipping 2011a).In this model, the ingress and egress of the moons areneglected, which is appropriate because even the largestmoons that can possibly form within the circumplane-tary disk around a 10-Jupiter-mass planet are predictedto have masses around ten times that of Ganymede, or ≈ . M ⊕ (Canup & Ward 2006), and even if they arewater-rich their radii will be < R ⊕ . The effect of amoon’s radial extension on the duration of the OSE willthus be < R ⊕ / (5 R J ) ≈ . R J )and < . R ⊕ / (10 R J ) ≈ .
46 % for a Mars-sized moonwith a semi-major axis of 10 R J .2.2.3. Numerical Simulations of the Photometric OSE
To simulate a light curve that contains a photomet-ric OSE, I construct a hypothetical three-satellite sys-tem that is similar in scale to the three innermost moonsof the Galilean system. The planet is assumed to havethe 10-fold mass of Jupiter and a radius R p = 1 . R J ). According to the Canup & Ward(2006) model, I scale the satellite masses as M = 10 M I , M = 10 M E , and M = 10 M G , with the indices I, E, andG referring to Io, Europa, and Ganymede, respectively.I derive the moon radii of this scaled-up system as perthe Fortney et al. (2007) structure models for icy/rockyplanets by assuming ice-to-mass fractions (imf) similar tothose observed in the Jovian system, that is, imf = 0 . = 0 .
08, and imf = 0 .
45 (Canup & Ward 2009).The model then yields R = 0 . R ⊕ , R = 0 . R ⊕ , and R = 0 . R ⊕ . The star is assumed to be a K star 0.7times the mass of the Sun ( M ⊙ ) with a radius ( R ⋆ ) of etecting Extrasolar Moons akin to Solar System Satellites with an Orbital Sampling Effect Figure 4.
Simulated transit light curve of a hypothetical three-satellite system around a giant planet of 1.05 Jupiter radii and10 Jupiter masses transiting a 0 . R ⊙ K star in the HZ. On thisscale, the photometric OSE is barely visible as a small decreasein stellar brightness just before planetary ingress and as a smalldelay in reaching 100 % of stellar brightness after planetary egress.Stellar limb darkening is neglected. R ⊙ ) (for Sun-like metallicity at an ageof 1 Gyr, following Bressan et al. 2012), and the planet-satellite system is placed into the center of the stellarHZ at 0.56 AU (derived from the model of Kopparapuet al. 2013). The impact parameter is b = 0, the moons’orbits are all in the orbital plane of the planet-moonbarycenter around the star, and the satellites’ orbitalsemi-major axes around the planet are a p1 = a JI R p /R J , a p2 = a JE R p /R J , and a p3 = a JG R p /R J for the inner-most, the central, and the outermost satellite, respec-tively. The values of a JI , a JE , and a JG correspond to thesemi-major axes of Io, Europa, and Ganymede aroundJupiter, respectively, and consequently this system issimilar to the one shown in Figure 2.Figure 4 shows the transit light curve of this system,averaged over an arbitrary number of transits with in-finitely small time resolution, excluding any sources ofnoise, and neglecting effects of stellar limb darkening.Individual transits are modeled numerically by assum-ing a random orbital positioning of the moons duringeach transit. In reality, a moon’s relative position to theplanet during a transit is determined by the initial con-ditions, say at the beginning of some initial transit, aswell as by the orbital periods both around the star andaround the planet-moon barycenter. I here only assumethat the ratio of these periods is not a low value integer.Then from a statistical point of view, positions duringconsecutive transits can be considered randomized.In the simulations shown in Figure 4, the analytic de-scription of Equation (6) is not yet applied – this pseudophase-folded light curve is a randomized, purely numer-ical simulation. If one of the moons turns out to bein front of or behind the planet, as seen by the ob-server, then it does not cause a flux decrease in the lightcurve. On this scale, the combined OSE of the threesatellites is hardly visible against the planetary tran-sit because the depth of the satellites’ features scale as( ≈ . R ⊕ / (0 . R ⊙ )) = 10 − , while the planet causesa depth of about 2 . . −
10 hr), and theinner, or first, moon succeeding at roughly − . − . − . − . P rw3 )of the outermost moon’s probability function leaving thestellar disk even before the planet enters. This indicatesthat the projected orbital separation of the outermostsatellite is larger than the radius of the stellar disk.The upper right panel shows the egress of the planetand the three-satellite system from the stellar disk, whichappears as a mirror-inverted version of the upper leftpanel. In this panel, the left wings of the satellites’ prob-ability functions that leave the stellar disk replace theright wings that enter the disk from the left panel.The wide lower panel of Figure 5 illustrates the moons’photometric OSEs at the bottom of the planetary tran-sit trough. Chronologically, this phase of the transit isbetween the ingress (upper left) and egress (upper right)phases of the planet-satellite system. The downtrendbetween about − − . P rw1 , P rw2 , and P rw3 from the upper left panel.What is more, between about − . P s ( x ) entering and leaving the stellar disk.In particular, the long curved feature between approxi-mately − . P rw2 ) overlaid by the ingressof increasingly high sampling frequencies in the left wingof the central moon ( P lw1 ). The fact that the end of theingress of P lw1 and the beginning of the egress of P rw1 almost coincide at planetary mid-transit means that thewidth of the projected semi-major axis of the innermostsatellite equals almost exactly the stellar radius.2.2.4. Emergence of the Photometric OSE in Transit LightCurves
Assuming the moons do not change their positions rel-ative to the planet during individual transits, then theirapparent separations are discrete in the sense that theyare defined by one value. Only if numerous transits areobserved and averaged will the moons’ photometric OSEsappear, because the sampling frequencies, or the under-lying density distributions P s ( x ), will take shape. Drop-ping the assumption of a fixed planet-satellite separationduring transit, each individual transit will cause a dy-namical imprint in the light curve. However, the pho-tometric OSE will converge to the same analytical ex-pression as given by Equation (6) after a large numberof transits.In Figure 6, I demonstrate the emergence of the photo-metric OSE in the hypothetical three-satellite system foran increasing number of transits N . Each panel shows Ren´e Heller
Figure 5.
Photometric OSEs of a hypothetical three-satellite system around a super-Jovian planet transiting a 0 . R ⊙ K star in the HZ(zoom into Figure 4). The radii of the outermost (the 3rd), central (2nd), and innermost (1st) satellites are R = 0 . R ⊕ , R = 0 . R ⊕ ,and R = 0 . R ⊕ , following planetary structure models (Fortney et al. 2007) and assuming ice-to-mass fractions of the Galilean moons.Due to the smaller range of orbits over which the probability function P ( a p1 ) of the innermost satellite is spread, its dip in this averagedstellar light curve is deeper than the brightness decrease induced by the outermost satellite, although the innermost satellite is smaller( R > R ). An arbitrarily large number of transits has been averaged to obtain this curve, and no noise has been added. Figure 6.
Emergence of the photometric OSE during the transit ingress of a hypothetical three-satellite system for an increasing numberof averaged transits N (zoom into upper left panel of Figure 5). Solid lines show the averaged noiseless light curves, while dashed curvesillustrate the combined photometric OSEs of the three satellites as per Equation (6). the same time interval around planetary mid-transit asthe upper left panel in Figure 5, that is, the ingress ofthe right wings of the probability distributions. The solidline shows the averaged transit light curve coming frommy randomized transit simulations, and the dashed lineshows the predicted OSE signal as per Equation (6). Inorder to draw the dashed line, I make use of the known satellite radii R s (s = 1 , , a ps , the stellar radius, the planetary radius,and the orbital velocity of the transiting planet-moonsystem v orb . The dashed curve is thus no fit to the simu-lations but a prediction for this particular exomoon sys-tem.In the left-most panel, after the first transit of theplanet-satellite system in front of the star ( N = 1), onlyone of the three moons shows a transit, shortly after − × − reveals the second moon as theoriginator, because ( R /R ⋆ ) ≈ × − . As an increas-ing number of transits is collected in the second-to-leftand the center panel, the photometric OSEs of the three-satellite system emerge. Obviously, between N = 10and N = 100, a major improvement of the OSE sig-nal strength occurs, suggesting that at least a few dozentransits are necessary to characterize this system. Af-ter N = 1000 transits, the noiseless averaged OSE curvebecomes indistinguishable from the predicted function.Whatever the precise value of the critical number oftransits ( N OSE ) necessary to recover the satellite systemfrom the solid curve in Figure 6, it is a principled thresh-old imposed by the very nature of the OSE. Noise addedduring real observations increases the number of tran- etecting Extrasolar Moons akin to Solar System Satellites with an Orbital Sampling Effect N obs , and the relation is deemed to be N obs ≥ N OSE .To estimate realistic values for N obs , it is necessary tosimulate noisy pseudo-observed data and try to recoverthe input systems. Section 3 is devoted to this task.2.3. OSE of Exomoon-induced Transit DurationVariations (TDV-OSE)
Consider a single massive exomoon orbiting a planet.As a result of the two bodies’ motion around their com-mon barycenter, the planet’s tangential velocity compo-nent with respect to the observer is different during eachtransit, and hence the duration of the planetary tran-sit shows deviations from the mean duration during eachtransit (Kipping 2009a). While each individual planet-moon transit has its individual TDV offset, all TDVobservations combined will reveal what I refer to as aTDV-OSE. In the reference system shown in Figure 1,the planet’s velocity component is projected onto the x -axis and is given by ˙ x p = ωr p sin( ϕ ), with ω = 2 π/P ps as the planet-satellite orbital frequency and P ps as theorbital period. The probability for a single planetarytransit to show a certain projected velocity around theplanet-satellite barycenter is then given by the samplingfrequency P TDVp ( ˙ x p ) ∝ d ϕ d ˙ x p = dd ˙ x p arcsin (cid:18) ˙ x p ωr p (cid:19) = 1 ωr p s − (cid:18) ˙ x p ωr p (cid:19) . (8)This distribution describes the fraction of randomly sam-pled angles ϕ that lies within an infinitesimal velocityinterval d ˙ x p of the planet projected onto the x -axis,whereas P s ( x ) in Equation (1) measures how much ofan infinitely small orbital path element of a satellite liesin a projected planet-moon distance interval. In analogyto the normalization of the latter position probability inEq. (2), the integral over P TDVp ( ˙ x p ) must equal 1 between − ωr p and + ωr p , which yields P TDVp ( ˙ x p ) = 1 πωr p s − (cid:18) ˙ x p ωr p (cid:19) (9)and has a shape similar to the functions shown in Fig-ure 2. Consequently, as the planet’s velocity wobble in-duced by a single moon distributes within the interval − ωr p ≤ ˙ x p ≤ ωr p as per Equation (9), planetary TDVswill also distribute according to an OSE.Assuming circular orbits, the spread of this TDV dis-tribution is determined by the peak-to-peak TDV ampli-tude∆ TDV = 2 t T × r a ⋆ p a ps s M M b ( M b + M ⋆ ) , (10)where t T is the duration of the transit between first andfourth contact (Kipping 2009a). Then the TDV-OSE distribution is given by P TDVp ( t ) = 1 π ∆ TDV r − (cid:16) t ∆ TDV (cid:17) , (11)where t is time. With M ⋆ ≫ M p and M p ≫ M s theright-most square root in Equation (10) simplifies to p M / ( M p M ⋆ ) . M ⋆ can be determined via spectralclassification, M p may be accessible via radial veloc-ity measurements, t T is readily available from the lightcurve, a ⋆ p can be inferred by Kepler’s third law, and a ps can be measured by the photometric OSE. With M s as the remaining free parameter, a fit of Equation (11)to the moon-induced planetary TDV distribution gives adirect measurement of the satellite mass. But note thelimited applicability of the TTV and TDV methods forshort-period moons (Section 6.3.8 in Kipping 2011b).2.4. OSE of Exomoon-induced Transit TimingVariations (TTV-OSE)
There is, of course, also an OSE for the distribution ofthe TTVs. In circular one-satellite systems, the ampli-tude of this TTV-OSE distribution is given by∆
TTV = 2 × a ps M s M b p a ⋆ p G ( M ⋆ + M b ) (12)(Sartoretti & Schneider 1999; Kipping 2009a), and thedistribution itself is given by P TTVp ( t ) = 1 π ∆ TTV r − (cid:16) t ∆ TTV (cid:17) . (13)With all the parameters on the right-hand side ofEq. (13) known from radial velocity measurements,the photometric OSE and TDV-OSE, this thirdmanifestation of the OSE can be used to furtherimprove the confidence of any exomoon detection, asit needs to be consistent with all three OSE observations. RECOVERING EXOMOON-INDUCEDPHOTOMETRIC OSE WITH
KEPLER
To assess the prospects of measuring the photomet-ric OSE as described in Section 2.2.2, I generate a suiteof pseudo-observed averaged light curves for a range ofgiven star-planet-satellite systems and try to detect theinjected exomoon signal. Following Kipping et al. (2009),I assume that the out-of-transit baseline is known to asufficiently high degree, which is an adequate assumptionfor
Kepler high-quality photometry with large amountsof out-of-transit data. My simulations also imply thatred noise has been corrected for, which can be a time-consuming part of the data reduction. Stellar limb dark-ening is neglected, but this does not alter the photometricOSE substantially (Heller et al., in preparation).3.1.
Noise and Binning of Phase-Folded Light Curves
To simulate a
Kepler -class photometry, I induce stellarbrightness variations σ ⋆ , detector noise σ d , a quarter-to-quarter noise component σ q , and shot noise σ s into the Ren´e Heller
Figure 7.
Simulated
Kepler noise as per Equation (14), used forthe synthesis of pseudo-observed transit light curves. raw light curves that contain a moon-induced photomet-ric OSE. The total noise is then given by σ K = s σ ⋆ + σ + σ + 1Γ ph t int , (14)with σ ⋆ = 19 . σ D = 10 . σ Q = 7 . ph = 6 . × hr − × − . m K − (15)as Kepler ’s photon count rate (Kipping et al. 2009), and t int as the integration time. The stellar noise of 19.5 ppmis typical for a G-type star, while K and M dwarfs tendto show more intrinsic noise. Figure 7 visualizes the de-crease of Kepler noise as a function of integration timeand for five different stellar magnitudes. The
Kepler short cadence and long cadence integration times at 1and 29.4 min are indicated with vertical lines, respec-tively.Figure 8 shows the pseudo phase-folded transit lightcurve of the hypothetical three-satellite system describedin Section 2.2.3 after 100 transits. The K star is chosento have a
Kepler magnitude m K = 12. I simulateeach individual transit and virtually “observe” the stel-lar brightness every 29.4 min, corresponding to the Ke-pler long-cadence mode, and add Gaussian noise follow-ing Equation (14). For each transit, I introduce a randomtiming offset to the onset of observations, so that eachtransit light curve samples different parts of the transit.Each individual light curve is normalized to 1 and addedto the total pseudo phase-folded light curve. This pseudophase-folded light curve is again normalized to 1. Whiledots in Figure 8 visualize my simulated
Kepler measure-ments, the solid line depicts the analytic prediction of theOSE following Equation (6). Even in this noisy data, theOSE is readily visible with the unaided eye shortly be-fore and after the planetary transit (upper two panels),suggesting that less than 100 transits are necessary todiscover – yet maybe not to unambiguously characterize– extrasolar multiple moon systems with the photometricOSE. At the bottom of the pseudo phase-folded transitlight curve, however, the OSE remains hardly noticeable(lower panel). In Figure 9, I show the same simulated data, butnow binned to intervals of 30 min (see Appendix B). Be-fore and after planetary ingress, the OSE now becomesstrongly apparent. Note that the standard deviationsare substantially smaller than the scatter around the an-alytic model. This is not due to observational noise, butdue to the discrete sampling of the transits. This scat-ter from the model decreases for an increasing numberof transits, N (see Figure 6). I also tried binning theshort cadence Kepler data and used binning intervals of10, 30, and 60 min, of which the 30 min binning of longcadence observations showed the most reliable results, atleast for this particular star-planet-moon system. Witha 60 min binning, the individual ingress and egress of thethree moons are poorly sampled, while the 10 min sam-pling shows too much of a sampling scatter around themodel.3.2.
Recovery of Injected Exomoon OSE Signals
Next, I evaluate the odds of characterizing extraso-lar moons with the photometric OSE. Although multi-satellite systems can, in principle, be detected and char-acterized by the analytical OSE model (Equation (6)),I focus on a one-satellite system. I simulate a range oftransits of a single exomoon orbiting a Jupiter- and 10-Jupiter-mass planet. In the former case, the moon isa Ganymede analog of 0 . R ⊕ , in the latter case themoon’s mass is scaled by a factor of ten and its radius of0 . R ⊕ is derived from the Fortney et al. (2007) struc-ture models using imf = 0 .
45. This moon correspondsto the outermost (or third) moon in the system simu-lated in Section 2.2.1. I study the detectability of thesetwo moons around these two planets orbiting three dif-ferent stars at 12th magnitude: a Sun-like star, a 0 . M ⊙ K star as considered in the previous sections, and an0 . M ⊙ -mass M dwarf with a radius of 0 . R ⊙ (for solarmetallicity at an age of 1 Gyr, derived from Bressan et al.2012).I start to simulate the pseudo phase-folded Kepler lightcurve of each of these hypothetical systems after N = 5transits and fit the binned data with a χ minimizationof the analytical model given by Equation (6), where thetwo free parameters are the satellite radius ( R ) and theorbital semi-major axis between the moon and the planet( a p1 ) (see Appendix C). This procedure is performed 100times for a given N , and if both R and a p1 are recoveredwith an error of less than 10 % in at least 68 of the 100runs (corresponding to a 1 σ confidence for the recoveryrate), then the number of transits is stored as N obs . Ifthe satellite cannot be recovered under these boundaryconditions, then the number of transits N is increasedby an amount ∆ N = ⌊ log ( N )+0 . ⌉ , representing theseries N ∈ { , , , ..., , , , ..., , , , ... } , andI repeat my fit of the analytical model to 100 pseudophase-folded light curves.Figure 10 presents an example light curve of theJupiter-mass planet and its Ganymede-like satellite inthe K star toy system after N = 100 transits. Thedata points show the simulated Kepler observations, thesolid line corresponds to the predicted light curve of theknown system, and the dashed line indicates the best fit The notation ⌊ x ⌉ denotes a rounding of the real number x tothe next integer. etecting Extrasolar Moons akin to Solar System Satellites with an Orbital Sampling Effect Figure 8.
Photometric OSEs after 100 transits in simulated
Kepler observations of a hypothetical three-satellite system around a Jupiter-sized planet 10 times the mass of Jupiter, transiting a 0 . R ⊙ K star in the HZ (same as in Figure 5). Noise is simulated after Equation (14).The scale of the ordinate is much wider than in Figure 5. Although the OSE seems invisible at the bottom of the transit light curve (bottompanel) due to the noise, the three moons together mask a measurable amount of stellar light that triggers the fit. model. In this example, the best-fit radius of the moonof 0 . ± . R ⊕ is very close to the input value of0 . R ⊕ , and similarly the fit of the planet-moon orbitaldistance of 15 . ± . R p almost matches the inputvalue of 15 . R p . I consider the formal 1 σ uncertaintiesof the χ fitting as physically unrealistic, which is why Irepeat the fitting procedure 100 times to get more robustestimates of N obs . RESULTS AND PREDICTIONSFigure 11 shows the outcome of the data fitting. Theupper table refers to transits of a Jupiter-sized hostplanet with a Ganymede-like moon, and the lower ta-ble lists the results for the super-Jovian planet with asuper-Ganymede exomoon. Abscissae and ordinates ofthese charts indicate distance to the star and stellarmass, respectively, while the entries show N obs as wellas the equivalent observational time t obs , computed as N obs times the orbital period around the star. Assumingcircular orbits, I use the semi-analytic model of Domin-gos et al. (2006) to test all systems for orbital stability.While a Jupiter-mass planet 0.1 AU from a Sun-like starcannot hold a moon in a Ganymede-wide orbit, a fewother configurations only allow the moon to be stable inretrograde orbital motion. The latter cases are labeledwith an asterisk. Shaded regions indicate the locationsof the respective stellar HZ following Kopparapu et al.(2013). Green cell borders demarcate the observation cy-cle of the Kepler space telescope that has been coveredbefore the satellite’s reaction wheel failure. PhotometricOSEs of exomoons within these boundaries could be de-tectable in the available
Kepler data of photometricallyquiet stars.Inspecting the upper panel for the Jupiter-Ganymededuet, three trends readily appear. First, large values of N obs appear in the top row referring to a Sun-like hoststar, intermediate values for the K star in the center row,and small values for the M dwarf host star in the bot-tom line. This trend is explained by the ratio of thesatellite radius to the stellar radius. The moon’s transitis much deeper in the M dwarf light curve than in thelight curve of the Sun-like star, hence, fewer transits arerequired to detect it. Second, going from short to widestellar distances, N obs decreases. This decline is causedby the decreasing orbital velocity of the planet-moon bi-nary around the star. In wider stellar orbits, transitshave a longer duration, and so the binary’s passage ofthe stellar disk yields more data points. The binned datathen has smaller error bars and allows for more reliable χ fitting. Third, N obs converges to a minimum valuein the widest orbits. For a Sun-like star, this minimumnumber of transits is about 120 at 0.9 AU and beyond,it is 34 for the 0 . M ⊙ star at 0.5 AU and beyond, androughly a dozen transits beyond 0.4 AU around a 0 . M ⊙ star. In these regimes, white noise is negligible and therecovery of the injected moons depends mostly on theratio R s /R ⋆ . N obs is then comparable to the principledthreshold N OSE imposed by the OSE nature (see Sec-tion 2.2.4).Translated into the required duty cycle of a telescope,timescales increase towards wider orbits, simply becausethe circumstellar orbital periods get longer. As an ex-ample, 107 transits were required in my simulations todiscover a Ganymede-like satellite around a Jupiter-likeplanet orbiting a 0 . M ⊙ star at 0.1 AU, corresponding toa monitoring over 4 yr. Only 34 transits were required forthe planet-moon system at 0.5 AU around the same star,but this means an observation time of 14.4 yr. The oddsof finding a Ganymede-sized moon transiting a G star inthe available Kepler data are poor, with t obs > . Ren´e Heller
Figure 9.
Same as Figure 8 but binned to intervals of 30 min. While the OSE of the three-satellite system clearly emerges in the ingressand egress parts of the probability functions P s ( x ) (upper two panels), it remains hardly visible at the bottom of the transit curve. Yet,the combined stellar transits of the moons still influence the depth of this light curve trough. any stellar orbit, and t obs >
100 yr beyond about 0.8 AU.K stars are more promising candidates with t obs as smallas 4 yr at a distance of 0.1 AU. M dwarfs, finally, showthe best prospects for Ganymede-like exomoons, because t obs < . M ⊙ star, this dis-tance encases planet-moon binaries in the stellar HZ.In the lower chart of Figure 11, a planet with the 10-fold mass of Jupiter and an exomoon of 0 . R ⊕ is con-sidered. In most cases, N obs for a given star and stellardistance is smaller than in the left panel, because themoon is larger and causes a transit signal that is betterdistinguishable from the noise. But different from the leftchart, N obs does not strictly decrease towards low-massstars. The 0 . M ⊙ K star shows the best prospects forthis exomoon’s photometric OSE detection in the avail-able
Kepler data. While a Sun-like host star could re-veal the satellite after as few as 30 transits or 7.6 yr at0.4 AU, the K star allows detection after 13 transits or3.9 yr at the same orbital distance, but still 27 transitsor 10.7 yr of observations would be required for the Mstar. The photometric OSE of such a hypothetical super-Ganymede moon could thus be measured in the available
Kepler data for planets as far as 0.4 AU around K stars,thereby encompassing the stellar HZs.Intriguingly, values of N obs are larger in the bottomline of the lower panel than in the bottom line of theupper panel, at a given stellar distance. This is coun-terintuitive, as the larger satellite radius (lower chart)should decrease the number of required transits, in anal-ogy to the Sun-like and K star cases. The discrepancyin the M dwarf scenario, however, is both an artifactof my boundary conditions, which require the fitted val-ues of R s and a p1 to deviate less than 10 % from thegenuine values of the injected test moons, and the verynature of the photometric OSE. For very small stellarradii, such as the M dwarf host star, and relatively large satellite radii, such as the one used in the right chart, theOSE scatter becomes comparatively larger than observa-tional noise effects. Hence, the K star in the center lineof the lower panel of Figure 11 actually yields the mostpromising odds for the detection of super-Ganymede-likeexomoons.My choice of a 10 % deviation in both R s and a ps be-tween the genuine injected exomoon and the best fit isarbitrary. To test its credibility, I ran a suite of random-ized planet-only transits and corresponding χ fits for aJupiter-like planet orbiting the G, K, and M dwarf starsat 0.5 AU, respectively, which is roughly in the center ofthe top panel of Figure 11. I generated 100 white-noisyphase-folded light curves after 151 (for the G star), 34 (Kdwarf), and 13 (M dwarf) transits, corresponding to thenumber of transits required to gather 68 % of the genuinesatellite systems within 10 % of the injected moon param-eters in my previous simulations. After these additional3 ×
100 “no moon” runs, the corresponding best-fit distri-butions turned out to be almost randomly distributed inthe R s - a ps plane with a slight clustering toward smallersatellite radii, and no 10 % ×
10 % bin contained morethan a few of the best fits. In contrast, if a moon werepresent, the best-fit systems would be distributed accord-ing to a Gaussian distribution around the genuine radiusand planetary distance of the moon. I conclude that aby-chance clustering within 10 % of any given locationin the R s - a ps plane is . − . In turn, finding at least68 % of the measurements within any 10 % ×
10 % bin inthe R s - a ps plane makes a genuine moon system a highlyprobable explanation. DISCUSSION5.1.
Methodological Comparison with other ExomoonDetection Techniques etecting Extrasolar Moons akin to Solar System Satellites with an Orbital Sampling Effect Figure 10. χ fit of the analytical OSE model via Equation (6) (dashed lines) to the binned, simulated Kepler data of a one-satellitesystem (data points) after N = 100 transits. The solid line shows the OSE light curve for the known input system. The moon is similarto Ganymede in terms of mass (0 . M ⊕ ), radius (0 . R ⊕ ), and planet-moon distance (15 . R p ). In this simulation, the χ fit yields amoon radius R = 0 . ± . R ⊕ and planet-moon semi-major axis a p1 = 15 . ± . R p . TTV/TDV-based Exomoon Searches
While TTV and TDV refer to variations in the planet’stransit light curve, the photometric OSE directly mea-sures the decrease in stellar brightness caused by oneor multiple moons. Combined TTV and TDV mea-surements allow computations of the planet-satellite or-bital semi-major axis a ps and a moon’s mass M s (Kip-ping 2009b,a). All descriptions of the TTV/TDV-basedsearch for exomoons are restricted to one-satellite sys-tems. The photometric OSE enables measurements of a ps and the satellite radii R s in multiple exomoon sys-tems. In comparison to the TTV/TDV method, theOSE can be measured with analytical expressions (Equa-tion (6) for the photometric OSE; Equation (11) forTDV-OSE; Equation (13) for TTV-OSE). Note thatTTV and TDV still need to be removed prior to analysesof the photometric OSE.A major distinction between TTV and TDV correctionfor the purpose of OSE analysis, compared to the actualdetection of an exomoon via its TTV and TDV imposedon the planet, lies in the irrelevance of the TTV/TDVorigin for the photometric OSE analysis. On the con-trary, for TTV/TDV-based exomoon searches these cor-rections need to be accounted for in a consistent star-planet-satellite model of the system’s orbital dynamics toexclude perturbing planets as the TTV/TDV source. Ifsuch a dynamical model is not applied, then still a ps and M s can be inferred if both TTV and TDV can be mea-sured (Kipping 2009a,b) and a single moon is assumed as the originator of the signal. Exomoon detection viaphotometric OSE, on the other hand, can be achievedwith much less computational power, practically withinminutes, as the transit light curve is phase-folded with-out TTV/TDV corrections. Determination of a ps should be mostly unaffected, because TTV and TDV are sup-posed to be of the order of minutes (Kipping et al. 2009;Heller & Barnes 2013; Awiphan & Kerins 2013), whereasthe photometric OSE extends as far as a few to tensof hours around the planetary transit (see Figures 9 and10), depending on the mass of the host star and the semi-major axis of the planet-satellite barycenter around thestar. The satellite radius is even more robust against un-corrected TTV/TDV, since it is derived from the shapeand depth of the OSE signal, not from its duration (seeFigure 5).Ultimately, the photometric OSE allows for the de-tection and characterization of multi-satellite systems,whereas currently available models of the TTV/TDVstrategy cannot unambiguously disentangle the under-lying satellite architecture of multiple satellite systems.Since the number of satellites around a giant planet issupposed to vary with planetary mass and dependingon the formation scenario (Sasaki et al. 2010), the pho-tometric OSE is a promising alternative to TTV/TDV-based exomoon searches when it comes to understandingthe formation history of extrasolar planets and moons.In Sections 2.3 and 2.4, I examine the distribution ofexomoon-induced TDV and TTV measurements. Com-bined with radial stellar velocity measurements and withthe photometric OSE, they allow for a full parameteriza-tion of a star-planet-moon system. TTV-OSEs or TDV-OSEs alone may not unambiguously yield exomoon de-tections because they can be mimicked by planet-planetinteractions (Mazeh et al. 2013). But if photometricOSEs indicate a satellite system, then TTV-OSE andTDV-OSE can be used to further strengthen the validityof the detection. In particular, TDV-OSE and TTV-OSE both offer the possibility of measuring a satellite’smass, which is unaccessible via the photometric OSE. In2 Ren´e Heller
Figure 11.
Number of transits required to detect the photometric OSE ( N obs , bold numbers) and equivalent observation time of aone-satellite system around a Jupiter-sized planet ( upper panel) and a 10 Jupiter-mass planet (lower panel). The three rows correspondto a 0.4, 0.7, and 1 M ⊙ -mass host star, respectively, while the columns depict the semi-major axis of the planet-moon binary around thestar. Striped areas indicate the stellar HZ, and green cell borders embrace the observation cycle that has been covered with Kepler beforeits reaction wheel failure. In all simulations, the moon is assumed to orbit in a Ganymede-wide orbit around the planet, that is, at about15 R p . the spirit of Occam’s razor, simultaneous observationsof the photometric OSE, TTV-OSE, and TDV-OSE inlongterm observations of a system would make an exo-moon system the most plausible explanation, rather thana tilted transiting ring planet suffering planet-planet per-turbations.5.1.2. Direct Photometric Exomoon Transits
First, in comparison to direct observations of individ-ual transits, the photometric OSE technique does notrequire dynamic modeling of the orbital movements ofthe star-planet-moon system, which drastically reducesthe demands for computational power compared to pho-todynamic modeling (Kipping 2011a). Second, the am-plitude of the photometricOSE signal in the phase-foldedlight curve is similar to the transit depth of a single ex-omoon transit, namely about ( R s /R ⋆ ) . But in contrastto single-transit analyses (a method not applied by Kip-ping 2011a, by the way), OSE comes with a substantialincrease in signal-to-noise by averaging over numeroustransits (see Figures 8 and 9). Third, OSE has a morecomplex imprint in the phase-folded light curve – theingress and egress patterns of the probability functions as well as a contribution to the total depth of the ma-jor transit trough (see Figure 5) – and thereby offersa larger “leverage” to tackle moon detections more se-curely. However, speed is only gained in exchange for lossof information, which is reasonable for the most likelycases considered in this paper (coplanarity, circularity,masses separated by orders of magnitudes, radii and dis-tances differing by at least an order of magnitude).5.1.3. Other Techniques
Besides TTV/TDV measurements and direct photo-metric transit observations, a range of other techniquesto search for exomoons have been proposed (see Sec-tion 1). In comparison to direct imaging of a planet-moon binary’s photocentric wobble, which requires anangular resolution of the order of microarcseconds for aplanet-moon binary similar to Saturn and Titan (Cabr-era & Schneider 2007) , the technical demands for pho-tometric OSE measurements are much less restrictive. In Note that the authors use a mass ratio of 0.01 between Titanand Saturn to yield this threshold. However, the true mass ratiois actually about 2 × − , so the value for the required angularresolution might even be much smaller. etecting Extrasolar Moons akin to Solar System Satellites with an Orbital Sampling Effect Kepler telescope and the upcoming
Plato 2.0 mission. Detec-tions of planet-moon mutual eclipses require modelingof the system’s orbital dynamics, which costs substan-tially more computing time than fitting Equation (6) toa phase-folded transit curve or Equations (11) and (13) tothe distribution of TDV and TTV measurements. Mu-tual eclipses can also be mimicked or blurred by starspots, and they are hardly detectable for moon’s as smallas Ganymede orbiting a Jovian planet. And referring todirect imaging of tidally heated exomoons, a giant planetwith a spot similar to Jupiter’s Giant Red Spot could alsomimic a satellite eclipse.Detections of the Rossiter-McLaughlin effect of aGanymede-sized moon around a Jupiter-like planet re-quires accuracies in radial velocity measurements of theorder of a few centimeters per second (Simon et al. 2010)and will only be feasible for extremely quiet stars andwith future technology (Zhuang et al. 2012).In comparison to detections with microlensing, obser-vations of exomoon-induced OSEs are reproducible. An-nouncements of possible exomoon detections around free-floating giant planets in the Galactic Bulge show theobstacles of this technique and should be treated withparticular skepticism. Not only are microlensing obser-vations irreproducible, but also from a formation point ofview, a moon with about half the mass of Earth cannotpossibly form in the protosatellite disk around a roughlyJupiter-sized planet (Canup & Ward 2006).Exomoons orbiting exoplanets around pulsars consti-tute a bizarre family of hypothetical moons, but as thefirst confirmed exoplanets actually orbit a pulsar (Wol-szczan & Frail 1992), they might exist. Analyses byLewis et al. (2008) suggest that exomoons around pul-sar planet PSR B1620 −
26 b, if they exist, need to be atleast as massive as about 5 % the planetary mass, leavingsatellites akin to solar system moons undetectable. Moregenerally, their time-of-arrival technique can hardly ac-cess moons as massive as Earth even in the most promis-ing cases of planets and moons in wide orbits (KarenLewis, private communication).Direct imaging searches for extremely tidally heatedexomoons also imply a yet unknown family of exomoons(Peters & Turner 2013), where the satellite is at least aslarge as Earth and orbits a giant planet in an extremelyclose, eccentric orbit. Exomoon-induced modulations ofa giant planet’s radio emission require the moon (not theplanet) to be as large as Uranus (Noyola et al. 2013), thatis, quite big. Such a moon does also not exist in the solarsystem. 5.1.4.
Comparison of Detection Thresholds
The detection limit of the combined TTV-TDVmethod using
Kepler data is estimated to be as smallas 0 . M ⊕ for moons around Saturn-like planets tran-siting relatively bright M stars with Kepler magnitudes m K <
11 (Kipping et al. 2009). Note, however, thatthe TTV-TDV method is susceptible to the satellite-to-planet mass ratio M s /M p , not to the satellite mass ingeneral. The HEK team achieves accuracies down to M s /M p ≈ ? ). Us-ing the correlation of exomoon-induced TTV and TDVon planets transiting less bright M dwarfs ( m K = 12 . M ⊕ , and such an exo-moon’s host planet would need to be as light as 25 M ⊕ .Those systems would be considered planet binaries ratherthan planet-moon systems. Lewis (2011) simulatedTTVs caused by the direct photometric transit signa-ture of moons and found that these variations could in-dicate the presence of moons as small as 0 . R ⊕ , withthis limit increasing towards wide orbital separations. Asimilar threshold has been determined by Simon et al.(2012), based on their scatter-peak method applied to Kepler short cadence data.Hence, depending on the actual analysis strategy ofmoon-induced TTV and TDV signals, and dependingon the stellar apparent magnitude, planetary mass, andplanet-moon orbital separation, a wide range of detectionlimits is possible. Most important, detection capabili-ties via TTV/TDV measurements as per Kipping et al.(2009) decrease for increasing planetary mass, whichmakes them most sensitive to very massive moons or-biting relatively light gas planets such as Saturn andNeptune. Yet, the most massive planets are predictedto host the most massive moons (Canup & Ward 2006;Williams 2013).In comparison, the photometric OSE presented in thispaper is not susceptible to planetary mass and can detectGanymede- or Titan-sized moons around even the mostmassive planets. This technique is thus well-suited forthe detection of extrasolar moons akin to solar systemsatellites. What is more, the photometric OSE is thefirst method to enable the detection and classification ofmulti-satellite systems.Rings of giant planets could mimic the OSE of exo-moons. However, planets at distances . N obs of transits requiredto discover the hypothetical one-satellite system arounda giant planet may be overestimations, because I useda fixed data binning of 30 min. This binning deliveredthe most reliable detections for the simulated Jupiter-satellite system in the HZ around a K star, but moonsystems at different stellar orbital distances and aroundother host stars have different orbital velocities and theirapparent trajectories differ in duration. Thus, more suit-able data binning – for example a 10 min binning forshort-period transiting planets – will yield more reliablefittings than derived in this report. My simulations arethought to cover a broad parameter space rather thana best fit for each individual hypothetical star-planet-exomoon system. 5.2. Red Noise Ren´e Heller
While my noise model assumes white noise only (Sec-tion 3.1), detrending real observations will have to dealwith red noise (Lewis 2013). Instrumental effects such asCCD aging as well as red noise induced by stellar granu-lation and spots need to be removed or corrected for be-fore the assumption of a light curve dominated by whitenoise becomes appropriate. In cases where red noise iscomparable to white noise, N obs and t obs as presented inFigure 11 will increase substantially.The results shown in Figure 11 do, however, still applyto a subset of photometrically quiet stars, such as thehost star of transiting planet TrES-2b that has also beenobserved by Kepler (Kipping & Bakos 2011). Basri et al.(2010) have shown that about every second K dwarf andabout 16 % of the M dwarfs in the
Kepler sample are lessactive than the active Sun. Gilliland et al. (2011) foundsimilar activity levels of K and M dwarfs but cautionedthat the small sample of K and M dwarfs in the
Kepler data as well as contamination by giant stars could spoilthese rates.An OSE-based exomoon survey focusing on quiet starswill automatically tend to avoid spotted stars. If never-theless present, clearly visible signatures of big star spotscan be removed from the individual transits before thelight curve is detrended and phase-folded. As long asthese removals are randomized during individual tran-sits, no artificial statistical signal will be induced intothe phase-folded curve. But if the circumstellar orbitalplane of the planet-satellite system were substantiallyinclined against the stellar equator and if the star hadspot belts, then spot crossings would occur at distinctphases during each single transit (see HAT-P-11 for anexample, Sanchis-Ojeda & Winn 2011). Such a geome-try would strongly hamper exomoon detections via theirphotometric OSE.5.3.
OSE detections with Plato 2.0
As the simulations in Section 4 show,
Kepler ’s pho-tometry is sufficiently accurate to detect the photomet-ric OSE of transiting exomoons around relatively quietK and M dwarfs with intrinsic stellar noise below about20 ppm. Hence, from a technological point of view, the
Plato 2.0 telescope with a detector noise similar to thatof
Kepler offers a near-future possibility to observe ex-oplanetary transits with similarly high accuracy. How-ever, even with arbitrarily precise photometry the num-ber of observed transits determines the prospects of OSEdetections. Given that
Plato 2.0 is planned to observetwo star fields for two to three years (Rauer 2013), Fig-ure 11 suggests that this mission could just deliver asmany transits as are required to enable photometric OSEdetections around M and K stars. Yet, the results pre-sented in this paper strongly encourage longterm moni-toring over at least five years of a given star sample toallow exomoons to imprint their OSEs into the transitlight curves. If the survey strategy of
Plato 2.0 can beadjusted to observe one field for about five years or more,then the search for exomoons could become an additionalscience objective of this mission.The
Transiting Exoplanet Survey Satellite (TESS), In 2014 February, the
Plato 2.0 mission has just been selectedby ESA as its third medium-class mission within the Cosmic Visionprogram. Launch is expected around 2024. however, is planned to have an observing duty cycle ofonly two years, and it will observe a given star for 72 daysat most. Hence, TESS cannot possibly discover the pho-tometric OSE of exomoons. CONCLUSIONSThis paper describes a new method for the detectionof extrasolar moons, which I refer to as the Orbital Sam-pling Effect (OSE). It is the first technique that allowsfor reproducible detections of extrasolar multiple satel-lite systems akin to those seen in the solar system. TheOSE appears in three flavors: (1) the photometric OSE(Section 2.2), (2) the TDV-OSE (Section 2.3), and (3)the TTV-OSE (Section 2.4). The photometric OSE canreveal the satellite radii in units of stellar radii as well asthe planet-moon orbital semi-major axes, but it cannotconstrain the satellite masses. TDV-OSE and TTV-OSEcan both constrain the satellite mass. Photometric OSE,TDV-OSE, and TTV-OSE offer important advantagesover other established techniques for exomoon searchesbecause (1) they do not require modeling of the moons’orbital movements around the planet-moon barycenterduring the transit, (2) planet-moon semi-major axes,satellite radii, and satellite masses can be measured or fitwith analytical expressions (Equations (6), (11), (13)),and (3) the photometric OSE is applicable to multi-satellite systems. TDV-OSE and TTV-OSE can also re-veal the masses of moons in multi-satellite systems, butthis parameterization is beyond the scope of this paper.My simulations of photometric OSE detections with
Kepler -class photometry show that Ganymede-sized ex-omoons orbiting Sun-like stars cannot possibly be discov-ered in the available
Kepler data. However, they couldbe found around planets as far as 0.1 AU from a 0 . . M ⊙ M dwarf.The latter case includes planet-moon binaries in the stel-lar HZ. Exomoons with the 10-fold mass of Ganymedeand Ganymede-like composition (implying radii around0 . R ⊕ ) are detectable in the Kepler data around plan-ets orbiting as far as 0.2 AU from a Sun-like host star,0.4 AU from the K dwarf star, or about 0.2 AU from theM dwarf. The latter two cases both comprise the re-spective stellar HZ. What is more, such large moons arepredicted to form locally around super-Jovian host plan-ets (Canup & Ward 2006; Sasaki et al. 2010) and aretherefore promising targets to search for.To model realistic light curves or to fit real observa-tions with a photometric OSE model, stellar limb dark-ening needs to be included into the simulations (Helleret al., in preparation). Effects on N obs are presumablysmall for planet-moon systems with low impact param-eters, because the stellar brightness increases to roughly60 % when the incoming moon has traversed only the first5 % of the stellar radius during a transit (Claret 2004).What is more, effects of red noise have not been treatedin this paper, and so the numbers presented in Figure 11are restricted to systems where either (1) the host staris photometrically quiet at least on a ≈
10 hr timescaleor (2) removal of red noise can be managed thoroughly.The prescriptions of the three OSE flavors delivered inthis paper can be enhanced to yield the sky-projectedangle between the orbital planes of the satellites and thediameter of the star. In principle, the photometric OSEallows measuring the inclinations of each satellite orbit etecting Extrasolar Moons akin to Solar System Satellites with an Orbital Sampling Effect H planet R p A (black area) xy R ✶ star center of thestellar disk Figure 12.
Parameterization of the planetary ingress. separately. The effect of mutual moon eclipses will besmall in most cases but offers further room for improve-ment. Ultimately, when proceeding to real observations,a Bayesian framework will be required for the statisticalassessments of moon detections. As part of frequentistsstatistics, the χ method applied in this paper is onlyappropriate because I do not choose between differentmodels since the injected moon architecture is known apriori.Another application of the OSE technique, which is be-yond the scope of this paper, lies in the parameterizationof transiting binary systems. If not only the secondaryconstituent (in this paper the moon) shows an OSE butalso the primary (in this paper the planet), then bothorbital semi-major axes ( a and a ) around the commoncenter of mass can be determined. If the total binarymass M b = M + M were known from stellar radialvelocity measurements, then it is principally possible tocalculate the individual masses via a /a = M /M andsubstituting, for example, M = M b − M . This proce- dure, however, would be more complicated than in themodel presented in this paper, because the center of theprimary transit could not be used as a reference any-more. Instead, as both the primary and the secondaryorbit their common center of mass, this barycenter wouldneed to be determined in each individual light curve andused as a reference for phase-folding.To sum up, the photometric OSE, the TDV-OSE, andthe TTV-OSE constitute the first techniques capable ofdetecting extrasolar multiple satellite systems akin tothose around the solar system planets, in terms of masses,radii, and orbital distances from the planet, with cur-rently available technology. Their photometric OSE sig-nals should even be measurable in the available data,namely, that of the Kepler telescope. After the recentfailure of the
Kepler telescope, the upcoming
Plato 2.0 mission is a promising survey to yield further data forexomoon detections via OSE. To increase the likelihoodof such detections, it will be useful to monitor a givenfield of view as long as possible, that is, for several years,rather than to visit multiple fields for shorter periods.APPENDIX
A. PARAMETERIZATION OF THE PLANETARY INGRESS
During the ingress of the planet in front of the stellar disk, the planet blocks an increasing area A of the stellar disk(black area in Fig. 12). With the star being substantially larger than the planet, the curvature of the stellar disk canbe neglected and A is determined by the height H of the circular segment by A = R arccos (cid:18) − HR p (cid:19) − q R p H − H ( R p − H ) . (A1)In my simulations, H = H ( t ) is a function of time. The temporary increase (during ingress) and decrease (duringegress) of A is visualized in Figure 4 by the gradual decrease in stellar brightness between about − − B. BINNING OF SIMULATED
KEPLER
DATA
The set of simulated, noisy brightness measurements shown in Figure 8 is given by data points b i . I divide thesimulated observations into time intervals with running index j . The mean value of brightness measurements in the6 Ren´e Heller j th time bin is ¯ b j = 1 K j K j X k =1 b k , (B1)with K j as the number of data points b k in bin j . The variance s j within each bin is given by s j = 1 K j − K j X k =1 ( b k − ¯ b j ) (B2)and the standard deviation of the mean in that bin equals σ j = s p K j = vuut K j ( K j − K j X k =1 ( b k − ¯ b j ) . (B3)By increasing the bin width, say from 30 to 60 min, it is possible to collect more data points per interval and toincrease K j , which in turn decreases σ j and increases accuracy. However, this comes with a loss in time resolution.The best compromise I found, at least for a satellite system on orbits similar to those of the Galilean moons buttransiting in the HZ around a K star, is a 30 min binning of long cadence Kepler data. Transits of systems on widercircumstellar orbits have a longer duration, and thus might yield best results with a binning longer than 30 min. Yet,their transits are less frequent, so this argument is only adequate for a comparable number of transits. C. χ MINIMIZATION
I fit my simulated
Kepler observations of a one-satellite system with a brute force χ minimization technique, thatis, I compute χ R ,a p1 = 1 K K X j =1 ( b j − m j ) σ j (C1)for the whole parameter space and search for the global minimum. The parameter grid I explore spans0 . R ⊕ ≤ R ≤ R ⊕ in increments of 0 . R ⊕ and 2 R p ≤ a p1 ≤ R p in steps of 0 . R p . In Equation (C1), K = P j j is the number of binned data points to fit, b j denotes the binned simulated data points, and m j refers tothe normalized brightness in bin j predicted by the analytical model (Equation (6)) for the satellite’s radius R andsemi-major axis a p1 to be tested (see Figure 10).I thank an anonymous reviewer for her or his valuable report. Karen Lewis’ feedback also helped clarify severalpassages in this paper, and I thank Brian Jackson for his thoughtful comments. This work made use of NASA’sADS Bibliographic Services. Computations have been performed with ipython 0.13.2 on python 2.7.2 (P´erez &Granger 2007), and most figures were prepared with gnuplot 4.4 Awiphan, S., & Kerins, E. 2013, MNRAS, 432, 2549Basri, G., Walkowicz, L. M., Batalha, N., et al. 2010, ApJ, 713,L155Ben-Jaffel, L., & Ballester, G. E. 2014, ApJ, 785, L30Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, 427, 127Cabrera, J., & Schneider, J. 2007, A&A, 464, 1133Cameron, A. G. W., & Ward, W. R. 1976, in Lunar andPlanetary Inst. Technical Report, Vol. 7, Lunar and PlanetaryInstitute Science Conference Abstracts, 120Canup, R. M., & Ward, W. R. 2006, Nature, 441, 834—. 2009, Origin of Europa and the Galilean Satellites, ed. R. T.Pappalardo, W. B. McKinnon, & K. K. Khurana, 59Claret, A. 2004, A&A, 428, 1001Domingos, R. C., Winter, O. C., & Yokoyama, T. 2006, MNRAS,373, 1227Fortney, J. J., Marley, M. S., & Barnes, J. W. 2007, ApJ, 659,1661Gilliland, R. L., Chaplin, W. J., Dunham, E. W., et al. 2011,ApJS, 197, 6Han, C., & Han, W. 2002, ApJ, 580, 490Heller, R., & Barnes, R. 2013, Astrobiology, 13, 18Heller, R., & Barnes, R. 2014, International Journal ofAstrobiology, FirstView, 1Heller, R., Barnes, R., & Leconte, J. 2011a, Origins of Life andEvolution of the Biosphere, 41, 539Heller, R., Leconte, J., & Barnes, R. 2011b, A&A, 528, A27 Kipping, D., & Bakos, G. 2011, ApJ, 733, 36Kipping, D. M. 2009a, MNRAS, 392, 181—. 2009b, MNRAS, 396, 1797—. 2011a, MNRAS, 416, 689—. 2011b, The Transits of Extrasolar Planets with MoonsKipping, D. M., Bakos, G. ´A., Buchhave, L., Nesvorn´y, D., &Schmitt, A. 2012, ApJ, 750, 115Kipping, D. M., Forgan, D., Hartman, J., et al. 2013a, ApJ, 777,134Kipping, D. M., Fossey, S. J., & Campanella, G. 2009, MNRAS,400, 398Kipping, D. M., Hartman, J., Buchhave, L. A., et al. 2013b, ApJ,770, 101Kipping, D. M., Nesvorn´y, D., Buchhave, L. A., et al. 2014, ApJ,784, 28Kopparapu, R. K., Ramirez, R., Kasting, J. F., et al. 2013, ApJ,765, 131Lewis, K. 2011, in European Physical Journal Web ofConferences, Vol. 11, European Physical Journal Web ofConferences, 1009Lewis, K. M. 2013, MNRAS, 430, 1473Lewis, K. M., Sackett, P. D., & Mardling, R. A. 2008, ApJ, 685,L153Mazeh, T., Nachmani, G., Holczer, T., et al. 2013, ApJS, 208, 16Morbidelli, A., Tsiganis, K., Batygin, K., Crida, A., & Gomes, R.2012, Icarus, 219, 737 etecting Extrasolar Moons akin to Solar System Satellites with an Orbital Sampling Effect Moskovitz, N. A., Gaidos, E., & Williams, D. M. 2009,Astrobiology, 9, 269Noyola, J. P., Satyal, S., & Musielak, Z. E. 2013, ArXiv e-printsP´al, A. 2012, MNRAS, 420, 1630P´erez, F., & Granger, B. E. 2007, Comput. Sci. Eng., 9, 21Peters, M. A., & Turner, E. L. 2013, ApJ, 769, 98Porter, S. B., & Grundy, W. M. 2011, ApJ, 736, L14Rauer, H. 2013, European Planetary Science Congress 2013, held8-13 September in London, UK, id.EPSC2013-707, 8, 707, seethe arXiv versionReynolds, R. T., McKay, C. P., & Kasting, J. F. 1987, Advancesin Space Research, 7, 125Robinson, T. D. 2011, ApJ, 741, 51Sanchis-Ojeda, R., & Winn, J. N. 2011, ApJ, 743, 61Sartoretti, P., & Schneider, J. 1999, A&AS, 134, 553 Sasaki, T., Stewart, G. R., & Ida, S. 2010, ApJ, 714, 1052Sato, M., & Asada, H. 2009, PASJ, 61, L29Simon, A., Szatm´ary, K., & Szab´o, G. M. 2007, A&A, 470, 727Simon, A. E., Szab´o, G. M., Kiss, L. L., & Szatm´ary, K. 2012,MNRAS, 419, 164Simon, A. E., Szab´o, G. M., Szatm´ary, K., & Kiss, L. L. 2010,MNRAS, 406, 2038Tusnski, L. R. M., & Valio, A. 2011, ApJ, 743, 97Williams, D. M. 2013, Astrobiology, 13, 315Williams, D. M., Kasting, J. F., & Wade, R. A. 1997, Nature,385, 234Williams, D. M., & Knacke, R. F. 2004, Astrobiology, 4, 400Wolszczan, A., & Frail, D. A. 1992, Nature, 355, 145Zhuang, Q., Gao, X., & Yu, Q. 2012, ApJ, 758, 111 .2. MODELING THE ORBITAL SAMPLING EFFECT OF EXTRASOLAR MOONS (Helleret al. 2016a) 93
Contribution:RH contributed to the literature research, worked out the mathematical framework, translated themath into computer code, performed the simulations shown in Fig. 6, created Figs. 1-4 and 6, ledthe writing of the manuscript, and served as a corresponding author for the journal editor and thereferees. r X i v : . [ a s t r o - ph . E P ] M a r Published in The Astrophysical Journal 820:88 (11pp), 2016 April 1
Preprint typeset using L A TEX style emulateapj v. 04/17/13
MODELING THE ORBITAL SAMPLING EFFECT OF EXTRASOLAR MOONS
Ren´e Heller
Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 G¨ottingen, Germany; [email protected]
Michael Hippke
Luiter Straße 21b, 47506 Neukirchen-Vluyn, Germany; [email protected]
Brian Jackson
Carnegie Department of Terrestrial Magnetism, 5241 Broad Branch Road, NW, Washington, DC 20015, USA andDepartment of Physics, Boise State University, Boise, ID 83725-1570, USA; [email protected]
Published in The Astrophysical Journal 820:88 (11pp), 2016 April 1
ABSTRACTThe orbital sampling effect (OSE) appears in phase-folded transit light curves of extrasolar planetswith moons. Analytical OSE models have hitherto neglected stellar limb darkening and non-zerotransit impact parameters and assumed that the moon is on a circular, co-planar orbit around theplanet. Here, we present an analytical OSE model for eccentric moon orbits, which we implement ina numerical simulator with stellar limb darkening that allows for arbitrary transit impact parameters.We also describe and publicly release a fully numerical OSE simulator (
PyOSE ) that can model arbitraryinclinations of the transiting moon orbit. Both our analytical solution for the OSE and
PyOSE can beused to search for exomoons in long-term stellar light curves such as those by
Kepler and the upcoming
PLATO mission. Our updated OSE model offers an independent method for the verification of possiblefuture exomoon claims via transit timing variations and transit duration variations. Photometricallyquiet K and M dwarf stars are particularly promising targets for an exomoon discovery using the OSE.
Keywords: instrumentation: photometers – methods: data analysis – methods: analytical – methods:observational – methods: statistical – planets and satellites: detection CONTEXT AND MOTIVATIONThe race toward the first detection of an extrasolarmoon picks up pace. While the first attempts to detectexomoons were byproducts of planet-targeted observa-tions (Brown et al. 2001; Pont et al. 2007; Maciejew-ski et al. 2010), high-accuracy space-based observationsof thousands of transiting exoplanets and planet can-didates by
Kepler (Borucki et al. 1997; Batalha et al.2013) now allow for dedicated exomoon searches (Kip-ping et al. 2012; Szab´o et al. 2013; Hippke 2015). Upcom-ing data from the European Space Agency’s
CHEOPS and
PLATO space missions will provide further promis-ing avenues toward an exomoon detection (Hippke &Angerhausen 2015; Simon et al. 2015).Exomoon detections will be highly valuable for our un-derstanding of the origin and fate of planetary systemsbecause they probe the substructures of planet formationthat is not accessible through planet observations alone.As for the solar system, moons provide key insights intothe formation of Earth (by a giant collision; Cameron& Ward 1976; Canup 2012), into the temperature dis-tributions within the circumplanetary accretion disks ofJupiter and Saturn (Pollack & Reynolds 1974; Canup &Ward 2002; Sasaki et al. 2010; Heller & Pudritz 2015a),and into the cause of Uranus’ tilted spin axis (by grad-ual collisional tilting; Morbidelli et al. 2012). As of today,no moon has been confirmed around a planet beyond thesolar system. Hence, exoplanetary science suffers from afundamental lack of knowledge about the fine structureof planetary systems. Heller (2014) recently identified a new exomoon signa-ture in planetary transit light curves, which occurs due tothe additional darkening of the star by a transiting moon.This phenomenon, which we refer to as the photomet-ric orbital sampling effect (OSE), occurs in phase-foldedtransit light curves, because an exomoon’s sky-projectedposition with respect to its host planet is variable in sub-sequent transits but is statistically predictable for a largenumber of transits ( N & The photometric OSE isnot sensitive to the satellite mass ( M s ), but it is verysensitive to its radius ( R s ). This makes the photomet-ric OSE particularly sensitive to low-density, water-richmoons like the three most massive moons in the solarsystem, Ganymede and Callisto around Jupiter and Ti-tan around Saturn. These moons would hardly be de-tectable in the available Kepler data by combined TTVand TDV measurements, which are sensitive to roughlyEarth-mass moons (Szab´o et al. 2013). The most ad-vanced exomoon search as of today, the “Hunt for Exo-moons with Kepler” (HEK; Kipping et al. 2012), using aphotodynamical model, achieves much better detectionlimits down to a few Ganymede masses, depending onthe mass of the host planet, amongst other parameters(Kipping et al. 2015). However, if giant planets were ableto give their fully fledged, icy moon systems a piggybackride to . The OSE appears in three flavors (as per Heller 2014), one ofwhich is the photometric OSE, which we focus on in this paper.The other two manifestations of the OSE appear in the planetarytransit timing variations (TTV-OSE) and transit duration varia-tions (TDV-OSE).
Heller, Hippke & Jackson
Figure 1.
Prospective detection thresholds of the photometricOSE (horizontal dashed line) for moons around a photometricallyquiet star after &
30 transits (as per Heller 2014, Fig. 11 therein).Note that the photometric OSE is sensitive to an exomoon’s ra-dius but not to its mass. Hence, large (potentially low-density)moons are the most promising targets. The open circle denotes aMars-mass moon with a Ganymede-like composition as predictedby Heller & Pudritz (2015b). The mass-radius relationship formoons of various compositions is according to Fortney et al. (2007). dant in radial velocity survey data (Dawson & Murray-Clay 2013; Heller & Pudritz 2015a), then the photomet-ric OSE could be a promising method to find these exo-moons. The mass-radius diagram in Figure 1 illustratesthe detection threshold for moons transiting photometri-cally quiet M stars. Note that Mars-mass moons with awater-rock composition similar to Ganymede, Callisto,and Titan are significantly larger than Mars (see theopen circle). Hence, they could be detectable aroundphotometrically quiet stars. These moons are predictedto form frequently around the most massive super-Jovianplanets (Heller & Pudritz 2015b).Hippke (2015) searched for the photometric OSE inthe archival data of the
Kepler telescope and found in-dications for an OSE-like signal in the combined transitlight curves of hundreds of planets and planet candidateswith orbital periods >
35 d. The original descriptionof the OSE by Heller (2014) was purely analytical andthereby fast, but it made several simplifications: stel-lar limb darkening was neglected; the planet–moon or-bit was assumed to be non-eccentric ( e = 0) and non-inclined ( i = 0 ◦ ) with respect to the circumstellar orbit;the planet–moon system was assumed to transit the staralong the stellar diameter; that is, the transit impactparameter ( b ) was set to 0. Consequently, this modelwas not broadly applicable. Hippke (2015) performeda mostly numerical study that did include non-circularplanet–moon orbits and arbitrary inclinations, but he didnot explore a wide parameter range of the OSE.We here first derive a novel equation for the orbitalsampling frequency P s ( x ) of exomoons on eccentric or- top viewedge view (a)(b)(c) xy d s rdx planetmoon r p - r a Figure 2.
Construction of an exomoon’s orbital sampling fre-quency P s ( x ) for elliptical orbits. (a) The moon’s orbital positionaround the planet is measured with a constant sampling frequency,or frame rate. (b)
The moon’s variable orbital velocity yields anasymmetric probability density with respect to the planet. (c)
The probability density can be derived as P s ( x ) = ds ( x ) /dx . Thespecial case of a circular orbit is given in Figure 1 of (Heller 2014). bits ( e ≥
0; Section 2.1). We then incorporate our for-mula into a numerical simulator that models the phase-folded transits of exoplanets with moons in front ofstars with limb darkening (Section 2.2). Our simula-tions are compared to real
Kepler data. We also performpurely numerical simulations for a wide range of inclinedplanet–moon orbits ( i ≥ b ≥ R s ), and semimajor axes ofthe satellite’s orbit around the planet ( a ; Section 3). A DYNAMICAL MODEL FOR THEPHOTOMETRIC OSE2.1.
Sampling Frequency for Eccentric Exomoon Orbits
In Heller (2014), the OSE was described for circularmoon orbits only. Given that the eccentricities of the 10largest moons in the solar system are all smaller thanthat of the Earth’s Moon ( ≈ . odeling the OSE of Exomoons v ) is higherthan around apoapsis. Thus, the probability of observ-ing the moon, e.g. during a common stellar transit withits host planet, at a given orbital position is asymmetricwith respect to the sky-projected distance to the planet.This fact is visualized in Figure 2(b), where the planet–moon system is sampled from a co-planar perspective.Due to the projection effect, the moon(s) pile(s) up to-ward the edges of the projected major axis, but the prob-ability distribution to the left of the planet is differentfrom what it looks to the right of the planet. Figure 2(c)shows how we construct the satellite’s probability den-sity P s ( x ) as a function of its projected distance ( x ) tothe planet.In its most general form, P s ( x ) describes what frac-tion of its orbital period ( P ps ) the satellite spends in aninfinitely small interval ( dx ) on its sky-projected orbitalong the x -axis (see Figure 2(c)). Hence, it is given as P s ( x ) ∝ P ps dtdx , (1)where dt is an infinitesimal change in time. With ds asan infinitely small interval on the moon’s elliptical orbit,the Keplerian velocity is v ( x ) = ds ( x ) /dt , hence P s ( x ) ∝ P ps v ( x ) ds ( x ) dx , (2)and the challenge is then in finding ds ( x ) /dx . From Fig-ure 2(a) we learn that ds ( x ) = r ( x ) dϕ ( x ). We use theparameterization of a Keplerian orbit r ( ϕ ) = a (1 − e )1 + e cos( ϕ ) (3)and solve it for ϕ ( x ) via x ( ϕ ) = r ( ϕ ) cos( ϕ ) = a (1 − e ) (cid:18) ϕ ) + e (cid:19) ⇔ ϕ ( x ) = arccos (cid:18)h ax (1 − e ) − e i − (cid:19) . (4)Equation (2) then becomes P s ( x ) ∝ r ( x ) P ps v ( x ) dϕ ( x ) dx (5)= r ( x ) P ps v ( x ) ddx arccos (cid:18)h ax (1 − e ) − e i − (cid:19)| {z } = A vuuut − (cid:18) Ax − e (cid:19) (cid:18) Ax − e (cid:19) x − , where we introduced A ≡ a (1 − e ). In Equation (5), r ( x )can be derived by inserting Equation (4) into (3), hence r ( x ) = A − e (cid:16) Ax − e (cid:17) − , (6)and v ( x ) in Equation (5) is given by v ( r ) = s µ (cid:18) r ( x ) − a (cid:19) , (7)where µ = G ( M p + M s ) and G is Newton’s gravitationalconstant.So far, we assumed the line of sight to be parallel to the y -axis in Figure (2), that is, along the orbital semiminoraxis ( b = a (1 − e ) / ). Of course, the moon orbit can berotated in the x - y plane by an angle ω . We can imaginethat once ω = 90 ◦ , the observer samples the moon orbitaccording to P s ( y ). In this case, the sky-projected or-bit appears symmetric with an apparent radius (˜ a ( e, ω ))equal to b . In general, we have˜ a ( e, ω ) = r(cid:16) a cos( ω ) (cid:17) + (cid:16) b sin( ω ) (cid:17) = a p cos( ω ) + (1 − e ) sin( ω ) . (8)We then replace e with ˜ e = e cos( ω ) and A with ˜ A =˜ a (1 − ˜ e ) in Equations (5)–(7) to obtain P s ( x ) for arbi-trary orientations ω .At last, we can swap the proportionality sign in Equa-tion (5) with an equality sign by normalizing Z + r p − r a dx P s ( x ) ≡ , (9)where r a and r p are the moon’s orbital radii at apoapsisand periapsis, respectively (see Figure 2(c)). We evaluatethis integral numerically for arbitrary a , e , and ω and findan approximate solution Heller, Hippke & Jackson P s ( x ) =2(1 + e ) (cid:18) a ˜ a ( e, ω ) (cid:19) ˜ r ( x ) P ps ˜ v ( x ) (10) × ˜ A vuuuut − ˜ Ax − ˜ e ! ˜ Ax − ˜ e ! x − , where ˜ r ( x ) and ˜ v ( x ) refer to Equations (6) and (7), re-spectively, but swapping e for ˜ e and A for ˜ A . The errorin Equation 10 is ≪ e < . R p ). The upperpanel depicts the orbital geometries and the lower panelshows P s ( x ) in each case. Four cases with eccentricitiesbetween 0 and 0.4 assume ω = 0 as presented in Figure 2,and the e = 0 . ω = 90 ◦ . Note thatin the latter scenario, where the semiminor axis b is sub-stantially smaller than a and the line of sight is parallelto a , P s ( x ) is significantly higher at a given planetaryseparation because the moon occupies a smaller circum-planetary region along the x -axis. This means that theOSE becomes particularly prominent in the phase-foldedlight curve of eccentric moon systems with ω close to 90 ◦ or 270 ◦ . On the other hand, it becomes relatively weak(or “smeared”) for ω near 0 ◦ or 180 ◦ .2.2. The Photometric OSE with Limb Darkening
Dynamical Simulations
We built a numerical model to simulate the stellartransit of an exoplanet for arbitrary probability func-tions P s ( x ) , including multiple functions in the case ofmulti-moon systems. As an update to Heller (2014), oursimulator now considers stellar limb darkening and thetransit impact parameter can be varied. In comparison toHeller (2014, Section 2.2.2 therein), where an analyticalsolution for the actual light curve (the stellar brightness B ( n )OSE due to a transiting planet with n exomoons) with-out stellar limb darkening has been derived, we do notderive an analytical solution for the light curve of theOSE with limb darkening. Instead, we simulate the OSEby generating a computer model of a limb-darkened stel-lar disk, approximated as a circle touching the the edgesof a square sized 1 000 × P s ( x ) enters the stellar disk, we multiply thestellar intensity ( I px ⋆ ) in any pixel that is covered withthe probability density P pxs ( x ) in this pixel, so that dx × I px ⋆ ×P pxs ( x ) is the relative amount of stellar bright-ness that is obscured in this pixel. In this notation, dx corresponds to the pixel width. Planet–moon eclipses are For e = 0, we have ˜ e = 0, ˜ a = a , ˜ A = a , ˜ r = a ,and ˜ v = ( µ/a ) / . Hence, setting e = 0 in Equation (10)and using P ps = 2 πa ( a/µ ) / we obtain the circular case P s ( x ) = 1 / (cid:0) πa [1 − ( x/a ) ] / (cid:1) derived in Heller (2014, Equa-tion (4) therein). Figure 3.
Different orbital eccentricities ( e ) in the planet–moon system (upper panel) cause different orbital sampling fre-quencies (lower panel). Six different values are shown: e ∈{ , . , . , . , . , . } . automatically taken into account, because the planet issimulated as a black circle. Consequently, I px ⋆ = 0 insidethe planetary radius and P pxs ( x ) cannot contribute to thephotometric OSE in the planetary shadow. At any givenobservation time t , the sum F ( t ) = (cid:18) R s R ⋆ (cid:19) X px dx × I px ⋆ × P pxs ( x ) (11)over all occulted pixels within the stellar radius ( R ⋆ )gives us the relative loss in stellar brightness B ( t ) =1 − F ( t ) due to the photometric OSE of the first satel- odeling the OSE of Exomoons ✾✾(cid:0)✁✂✾✾(cid:0)✄☎✾✾(cid:0)✄✂✾✾(cid:0)✾☎✾✾(cid:0)✾✂✶☎☎(cid:0)☎☎✶☎☎(cid:0)☎✂ ✲✆ ✲✝ ✲✞ ✲✶ ☎ ✶ ✞ ✝ ✆♥✟✠✡☛☞✌✍✎✏✑✒✎☞☞☛✠✓✠✌✔✕✒♥✎✑✑✖✗✎✠✘✎♥✒✙ t✚✛✜ ✢✣✤✥✦✧ ★✩✢✦✜t✢✣✪ ✛✚✧✲t✣✢✦✫✚t ✬✭✤✥✣✫✮✧✪✦✢✛✚❞✢✩ ✛✤✧✜✩ ✯✚t✭ ✩✚✛✰ ✧✢✣✱✜✦✚✦✳✧✪✦✢✛✚❞✢✩ ✛✤✧✜✩ ✯✚t✭✤✥t ✩✚✛✰ ✧✢✣✱✜✦✚✦✳✢✦✢✩✪t✚❞✢✩ ✜❛★✣✜✫✫✚✤✦ ✴✵✜✩✩✜✣ ✞☎✶✆✷ ✲✝☎✲✞☎✲✶☎☎ ✲✝(cid:0)☎ ✲✞(cid:0)✂ ✲✞(cid:0)☎♥✟✠✡☛☞✌✍✎✏✑✒✎☞☞☛✠✓✠✌✔✕✒♥✎✑✑✖✗✗✡✙ t✚✛✜ ✢✣✤✥✦✧ ★✩✢✦✜t✢✣✪ ✛✚✧✲t✣✢✦✫✚t ✬✭✤✥✣✫✮✧✪✦✢✛✚❞✢✩ ✛✤✧✜✩ ✯✚t✭ ✩✚✛✰ ✧✢✣✱✜✦✚✦✳✧✪✦✢✛✚❞✢✩ ✛✤✧✜✩ ✯✚t✭✤✥t ✩✚✛✰ ✧✢✣✱✜✦✚✦✳✢✦✢✩✪t✚❞✢✩ ✜❛★✣✜✫✫✚✤✦ ✴✵✜✩✩✜✣ ✞☎✶✆✷ Figure 4.
Comparison of the dynamical photometric OSE model with limb darkening (solid line), the dynamic photometric OSE modelwithout limb darkening (dashed line), and the analytical expression for a non-limb-darkened star as per Eq. (6) in Heller (2014).
Left:
TheOSE is hardly visible as a the slight brightness decrease in the wings of the planetary transit.
Right:
A zoom into the ingress of the moon.The dynamic model without limb darkening reproduces the analytical prediction very well. However, only the dynamic limb darkeningmodel will be useful for fitting real observations. This hypothetical star–planet system is similar to the KOI 255.01 system, except that wehere assume b = 0. The toy moon is as large as Ganymede (0 . R ⊕ ) and the planet–moon orbit is 15 . R p wide, equivalent to Ganymede’sorbit around Jupiter. lite. In the more general case of n satellites, F n OSE ( t ) = n X s " (cid:18) R s R ⋆ (cid:19) X px dx × I px ⋆ × P pxs ( x ) (12)and B n OSE ( t ) = 1 − F n OSE ( t ). We apply the nonlinearlimb darkening law of Claret (2000, Equation (7) therein)and use stellar limb darkening coefficients (LDCs) listedin Claret & Bloemen (2011), which depend on stellareffective temperature ( T ⋆, eff ), metallicity ([Fe/H]), andsurface gravity (log( g )).As an example, Figure 4 shows the simulated tran-sit of KOI 255.01 together with the OSE of a hypothet-ical Ganymede-sized moon at a = 15 . R p , equivalentto Ganymede around Jupiter. The 2 . R ⊕ super-Earth( R ⊕ being the Earth’s radius) is an interesting objectas it transits a 0 . M ⊙ -mass M dwarf with a radiusof 0 . ± . R ⊙ every 27 . ± . × − d. Hence, the photometric OSE of even a Ganymede-sizedmoon could be significant. In our simulations, the tran-sit impact parameter is set to b = 0 . b = 0 . . , − . a = 0 . a = 0 . a = 0 = a . In both panels of Figure 4, the solid linerefers to our dynamical OSE model with limb darkening,the dashed line shows the dynamical model with the limbdarkening option switched off, and the dotted line showsthe analytical OSE model by Heller (2014), which alsoneglects limb darkening.The left panel shows that the analytical solution ismuch less accurate than the dynamical OSE model insidethe planetary transit trough because it neglects stellarlimb darkening. In the wings of the transit curve, how-ever, the analytical solution without stellar limb dark- All values taken from http://exoplanetarchive.ipac.caltech.eduas of 2014 May 30. ening and the dynamic OSE model with limb darkeningdiffer by < × − compared to a maximum OSE depthof about 3 × − just before the planetary ingress (rightpanel). The analytic model might thus offer sufficientaccuracy for an initial OSE survey of a large data set.With such an approach, a first and preliminary search forOSE candidates within thousands of phase-folded lightcurves would be a matter of minutes. Even more en-couraging, the dynamic OSE model with limb darkeningalmost resembles a straight line, at least in this config-uration where the moon’s semimajor axis is roughly aswide as the stellar diameter. A straight line fit would, ofcourse, simplify an initial OSE search even further, as itwould mostly depend on the moon’s radius squared (interms of maximum depth) and on the moon’s semimajoraxis (in terms of duration).The right panel of Figure 4 also reveals that the dy-namical OSE model causes a slightly smaller depth in thelight curve for about two-thirds into the OSE ingress.This effect is caused by stellar limb darkening and thestellar brightness in the occulted regions being lowerthan the average brightness on the disk. In the finalthird of the OSE ingress, the dynamical OSE model thenyields a deeper absorption because of the increasing stel-lar brightness towards the disk center. This division intotwo-thirds, in which the OSE model without stellar limbdarkening is deeper than the one with limb darkening,and the one-third where things are reversed, is not a uni-versal relation. Toward the stellar limb, e.g., the modelwithout stellar limb darkening produces a deeper OSEsignal during the entire transit.2.2.2. Comparison with Kepler Data
We now apply our model to observations. This inves-tigation is not an in-depth search for moons around atest planet, but it shall serve as an illustration of theOSE by reference to actual observed data. Our sur-vey for confirmed, super-Neptune-sized
Kepler planetswith orbital periods >
10 d (to ensure Hill stability of an
Heller, Hippke & Jackson
Figure 5.
Left:
Raw PDCSAP FLUX for Kepler-229 c – individual quarters come offset from one another, and the approximately 90transits are visible as large dips.
Right:
Detrended data. moons) around a & . R ⊙ star (allowing detection of ex-omoons akin to the largest solar system moons) revealedKepler-229 c (KOI 757.01) as a promising object. It is a4 . R ⊕ planet transiting a 0 . R ⊙ star every 16 . b = 0 .
25 at a distance of about0 .
117 AU (Rowe et al. 2014). The stellar mass can thenbe estimated via Kepler’s third law as 0 . M ⊙ , and theplanetary Hill radius ( R H ) is about 78 R ⊕ ≈ R p , as-suming that Kepler-229 c’s mass is similar to that of Nep-tune. Stellar LDCs are interpolated from Claret & Bloe-men (2011) tables using the stellar effective temperatureand surface gravity of Kepler-229 (Rowe et al. 2014),yielding a = 0 . a = − . a = 1 . a = − . (30 minutes) pub-licly available Kepler data for Kepler-229 c. We ana-lyzed the PDCSAP FLUX data, from which the
Kepler mission has attempted to remove instrumental variabil-ity. Nevertheless, these data still exhibit significant vari-ability unrelated to transits, as seen in Figure 5 (leftpanel). The creation of the PDCSAP fluxes by the
Kepler mission involves the removal of common modevariability from the light curves attributable to instru-mental effects, which can distort real astrophysical (suchas stellar) variability but primarily at medium to longtimescales. Since we consider relatively short-periodplanets, these possible distortions are unlikely to affectour analysis.To condition each quarter’s observations, we sub-tracted the quarter’s mean value from all data pointsand then normalized by that mean. To these mean-subtracted, mean-normalized data, we applied a meanboxcar filter with a width equal to twice the transit du-ration plus 10 hr. This window is chosen to maximallyremove non-transit variations while preserving the shapeof the transit and OSE signals. The right panel of Fig- The OSE itself is an averaging effect, as it appears in the phase-folded light curve and only after several transits. Hence, for theOSE curve it does not make a difference if the data is taken inshort cadence and then binned into 30 minute intervals or if it istaken in 30 minute intervals in the first place. http://archive.stsci.edu/kepler/data search/search.php ure 5 shows the resulting detrended data. Finally, westitched together all quarters and masked out 10 σ out-liers. Figure 6 (right panel) shows the final result forKepler-229 c, after we folded the detrended data on a pe-riod of 16.968618 d. Gray dots present the detrended,phase-folded data, and black dots with error bars showthe binned data. The solid black line shows the directoutput of our dynamical OSE simulator but for a planetwithout a moon, and the dashed red line shows the tran-sit including the photometric OSE of an injected moon.A red cross on that curve at about 0.5 hr highlights theorbital configuration that is depicted in the left panel.The left panel of Figure 6 illustrates our dynamicalOSE model at work for Kepler-229 c. The planet alongwith the probability distribution of one hypothetical ex-omoon can be seen in transit. The injected moon has aradius of 0 . R ⊕ , the vertical width of the moons P s ( x )is to scale to both the planetary and the stellar radius.The moon orbit is set to 8 R p , which is R H / . R ⊕ -sized moon), we also show a blue dottedline indicating the OSE of a hypothetical Earth-sizedmoon. Note that the width of individual error bars ofthe binned data is ≈ − , which corresponds to thedepth of an Earth-sized moon’s photometric OSE. Withabout 5 of such binned data points (or about 500 un-binned data points during the OSE ingress), a searchfor the exomoon-induced photometric OSE around thisplanet could yield statistical constraints on the presenceof moons the size of Earth and smaller around this par-ticular exoplanet. A more elaborate statistical analysisof exomoon effects in transit light curves is deferred to afuture study (R. Heller et al. 2016, in preparation). NUMERICAL SIMULATIONS OF THEPHOTOMETRIC OSE We define σ to be the standard deviation estimated as 1.4826 × the median absolute deviation (Bevington & Robinson 2003). odeling the OSE of Exomoons Numerical OSE Simulator Applied to Kepler-229 c P s ( x ) × left panel no moon 0.7 R ! R ! Comparison of OSE Simulations with
Kepler
Data
Figure 6.
Visualization of our dynamical photometric OSE transit model. This example shows the transit of Kepler-229 c, a 4 . R ⊕ planetorbiting a 0 . R ⊙ star every ≈
17 d with a transit impact parameter b = 0 . Left:
The planet (black circle) and the probability function(shaded horizontal strip) of a hypothetical 0 . R ⊕ moon with an orbital semimajor axis of 8 R p transit the limb-darkened star (large brightcircle). Right:
The red cross on the OSE curve at 0.5 hr refers to the moment shown in the left panel. The inset zooms into the wingsof the transit ingress. Three models are shown: no moon (black solid), a 0 . R ⊕ moon (red dashed), and a 1 R ⊕ moon (blue dotted), allmoons with a semimajor axis of 8 R p around the planet. To simulate the OSE in more complex configurationswith inclined moon orbits, for which an analytical solu-tion is not available, we wrote a numerical OSE simulatorin python , which we call “
PyOSE ”. The code and exam-ples are publicly available under the MIT license. Allof the following figures were generated with this code.3.1.
Parameterization in
PyOSE In PyOSE , the moon’s orbital ellipse is defined by itseccentricity ( e ), circumplanetary semimajor axis ( a ), itsorbital inclination with respect to the circumstellar orbit( i s ), the longitude of the ascending node (Ω), and theargument of the periapsis ( ω ). Figure 7 (top) shows ahypothetical 0 . R ⊕ exomoon around Kepler-229 c on acircular, inclined orbit ( i s = 83 ◦ , Ω = 30 ◦ , a = 8 R p ) atthe time of the planetary mid-transit. In our numericalimplementation, the planet–moon ensemble transits thestar from left to right, which is an arbitrary choice. Themotion of the planet and the moon around their commonbarycenter during transit can be simulated with PyOSE .The center panel in Figure 7 shows a river plot (Carteret al. 2012) representation of 250 of these transit lightcurves, with one shown in each row. Each of these lightcurves corresponds to a different fixed position of themoon during the stellar transit. The orbital phase of themoon (along the ordinate in Figure 7) corresponds to themean anomaly. Two horizontal gray regions at phases ≈ . ≈ . https://github.com/hippke/pyose http://opensource.org/licenses/MIT black arrow at phase ≈ . PyOSE can simulate transits of planets withmoons, we will focus on the moon’s OSE signature in thefollowing and study a range of moon transits for variousorbital parameters. The transit model is the one pre-sented by Mandel & Agol (2002). For each OSE curve,we chose to sample at least 100 moon transits equallyspaced in time (not necessarily in space for e = 0; seeFigure 2) to achieve convergence (Heller 2014). The limb-darkened stellar disk is represented by a numerical grid of1 000 × & < PyOSE allows for arbitrarily large pixelgrids at the cost of CPU time, which is proportional tothe total number of pixels in the grid or to the square of R s (in units of pixels).3.2. Mutual Planet–Moon Eclipses
Heller, Hippke & Jackson − . − . . . . distance [stellar radii] − . − . . . . d i s t a n c e [ s t e ll a rr a d ii ] b P -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20time around planetary mid-transit [days]0.00.20.40.60.81.0 p h a s e -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20time around planetary mid-transit [days]-100-80-60-40-200 n o r m a li z e d s t e ll a r b r i g h t n e ss [ pp m ] Figure 7.
Top:
Sky-projected view generated with
PyOSE of bothKepler-229 c and a hypothetical exomoon in transit.
Center:
Riverplot of the moon transit only. The black arrow at phase 0.2 showsthe moon’s position chosen in the top panel.
Bottom:
Average(phase-folded) transit light curve of the system after an arbitrarilylarge number of transits, where the moon orbit has been equallysampled in time.
We treat both the planet and the moon as black disks.Thus, it is irrelevant whether the satellite eclipses behindor in front of the planet. For each of the simulatedplanet–moon transit configurations,
PyOSE compares thedistance between the planet and moon disk centers ( d ps )to the sum of their radii. If d ps < R p + R s , a mutualeclipse occurs and PyOSE calculates the area ( A ) of theasymmetric lens defined by the intersection of the twocircles as A = R arccos d + R − R d ps R s ! + R arccos d + R − R d ps R p ! − q ( − d ps + R s + R p )( d ps + R s − R p ) × q ( d − R s + R p )( d ps + R s + R p ) ! (13)This area does not contribute to the stellar blocking bythe moon, as it is covered by the silhouette of the tran-siting planet. 3.3. Parameter Study
In the following, we study variations of the OSE signaldue to variations in the parameterization of the star–planet–moon orbital and physical configuration. We useKepler-229 c as a reference case and modify one param-eter at a time, as specified below.3.3.1.
The Moon’s Semimajor Axis ( a ) For circular moon orbits, changes in a modify the shapeof the OSE, while the area under the integral remainsunaffected (see Equation 9). For e = 0 or i s = 0, how-ever, both the shape and the integral will change. Theleft panel of Figure 8 visualizes this effect for two cases;one in which the moon is barely stable from a dynamicalpoint of view ( a = 0 . R H ), and one for a close-in moonnear the Roche limit ( a ≈ . R H ≈ R p ). The du-ration of the OSE signal is longer for moons with largersemimajor axes. Planet–moon eclipses are visualized bybumps in the OSE curve of the moon at a = 0 . R H ,near ± .
07 days. The moon in the wider orbit is notsubject to planet–moon eclipses.The right panel of Figure 8 shows the integral underthe OSE curve as a function of a . This is an importantquantity as it serves as a measure for the significance ofthe moon-induced OSE imprint on the light curve. Thedecline of the OSE integral up to a ≈ . R H is mostlydue to stellar limb darkening: the larger a , the more fluxwill be blocked closer to the stellar center for this spe-cific orbital configuration. Beyond 0 . R H , moon transitswill occasionally be missed during planetary transits andthe OSE signal decreases. The dashed part of the curve, This fact makes the OSE insensitive to the satellite’s sense oforbital motion around the planet (Heller & Albrecht 2014; Lewis& Fujii 2014). odeling the OSE of Exomoons -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20time around planetary mid-transit [days]-100-80-60-40-200 n o r m a li z e d s t e ll a r b r i g h t n e ss [ pp m ] a s = 0 . R H a s = 0 . R H -13-12-11-10-9-8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 O S E i n t e g r a t e d f l u x [ pp m d a y s ] planet-moon semi-major axis [ R Hill ] Figure 8.
Variation of the OSE for different semimajor axes of a hypothetical exomoon around Kepler-229c.
Left:
Photometric OSEfor two cases where the satellite is at 0 .
128 and 0 . R H around the planet. Right:
Integral under the OSE curve as a function of theplanet–moon orbital semimajor axis. The solid line shows the limiting cases at 0 .
128 and 0 . R H , corresponding to the two scenarios shownin the left panel. The dashed line represents moon orbits that are unphysically wide for prograde moons. -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20time around planetary mid-transit [days]-100-80-60-40-200 n o r m a li z e d s t e ll a r b r i g h t n e ss [ pp m ] R S = 0 . R ⊕ R S = 0 . R ⊕ R S = 0 . R ⊕ Figure 9.
Variation of the OSE for different satellite radii. Theinset zooms into the ingress of the phase-folded satellite transit,showing that the time of first contact between the satellite’s sil-houette and the stellar disk depends on the satellite radius. corresponding to moons beyond 0 . R H , is physically im-plausible for prograde moons, and valid only for progrademoons (Domingos et al. 2006).3.3.2. The Moon’s Radius ( R s ) Larger moons naturally cause deeper transits. Keepingeverything else fixed, variations in R s cause variationsin the OSE amplitude roughly proportional to R , asillustrated in Figure 9 (see also the term ( R s /R ⋆ ) inEquation 11). Comparing the upper and lower curves,we see a signal increase by a factor of four (-25 ppm vs.-100 ppm) for a change in R s by a factor of two (from0.35 to 0 . R ⊕ ). Changes in transit duration occur dueto the different timings of the first and last contact ofthe planetary silhouette with the stellar disk (see insetin Figure 9).3.3.3. The Planetary Impact Parameter ( b ) andthe Inclination of the Moon’s Orbit ( i s ) -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20time around planetary mid-transit [days]-100-80-60-40-200 n o r m a li z e d s t e ll a r b r i g h t n e ss [ pp m ] b = 0 , i s = 90 ◦ b = 0 , i s = 0 ◦ b = 0 . , i s = 0 ◦ b = 0 . , i s = 90 ◦ Figure 10.
Variation of the OSE for different planetary transitimpact parameters and different inclinations of the satellite orbit.Solid lines refer to b = 0 and dashed lines relate to b = 0 .
95. Forboth cases, we show i s = 0 ◦ and 90 ◦ . The planetary impact parameter and the inclinationof the satellite orbit determine the fraction of planetarytransits without moon transits, that is, planetary tran-sits with the moon passing beyond the stellar disk. InFigure 10, we show four different scenarios of a hypo-thetical moon around Kepler-229 c. Solid lines refer to b = 0, dashed lines to b = 0 .
95, and we examine in-clinations i s = 0 ◦ (face-on view) and i s = 90 ◦ (edge-onview). In the b = 0 . i s = 90 ◦ case (upper dashed line),a bump around planetary mid-transit gives evidence ofplanet–moon eclipses. In the b = 0 . i s = 0 ◦ and b = 0, i s = 0 ◦ cases, planet–moon eclipses do not occur. In the b = 0, i s = 90 ◦ case (lower solid line), the bump fromplanet–moon eclipses is very broad and deformed intotwo minor bumps at ± .
07 d in the moon’s OSE.3.4.
Multiple Exomoons
Multiple moons are common in our solar system. Al-though photodynamical modeling (Kipping 2011) cantackle multi-satellite systems in principle (Kipping et al.0
Heller, Hippke & Jackson -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20time around planetary mid-transit [days]-100-80-60-40-200 n o r m a li z e d s t e ll a r b r i g h t n e ss [ pp m ] Figure 11.
OSE of a multi-moon system. The two solid linesshow the OSE of one 0 . R ⊕ moon at 0 . R H and a second 0 . R ⊕ moon at 0 . R H . The dashed line shows the combined OSE. Kepler datato single-moon systems. Our Equation (12) is an analyt-ical description of the additive OSE in multi-moon sys-tems and describes how the dynamical OSE model han-dles multi-satellite systems. Our purely numerical simu-lator
PyOSE can handle multi-satellite systems as well.That being said, with real observational data, distin-guishing between an OSE signal of a single large moonand a signal from several smaller moons may prove dif-ficult, if not impossible. This is demonstrated in Fig-ure 11, where we show the individual OSE signals fromtwo moons (solid lines) and their combined OSE signa-ture (dashed line). This dashed curve demonstrates thatthe photometric OSE is additive as long as moon–moonoccultations can be neglected. The dashed curve isalso what would be observed in reality, noise effects aside.With noise taking into account, there would be substan-tial degeneracies among the model parameters and thenumber of moons. DISCUSSIONIn addition to the orbital and physical parameterstreated above, there are several minor effects on the OSE,the most important being stellar limb darkening. Weran a series of simulations using different LDCs for FGKmain-sequence stars. For a typical uncertainty in stellartemperature ≈ PyOSE currently neglects moon–moon occultations. Theywould only occur in < Some aspects of our OSE model might hardly be acces-sible with near-future technology. For moon orbits thatare co-planar with the circumstellar orbit ( i s = 90 ◦ ),moderate moon eccentricities will not cause an OSE sig-nal much different from a circular moon orbit. Never-theless, there are configurations in which these parame-ters make all the difference in determining the presenceof an exomoon (see Fig. 10). Moreover, the discoveryof the formerly unpredicted hot Jupiter population, theunsuspected dominance of super-Earth-sized exoplanetsin orbits as short as that of Mercury, and the puzzlingabundance of close-in planets with highly misaligned or-bits suggests that the solar system does not present a re-liable reference for extrasolar planetary systems. Hence,any model used to explore yet undiscovered exomoonswill need to be able to search a large parameter spacebeyond the margins suggested by the solar system.Both numerical simulations and observations will al-ways be undersampled and only converge to the ana-lytical solution. In terms of observations, this is be-cause of the limited number of observed transits, usu-ally <
100 for an observational campaign over a fewyears, and because of telescope downtime and observa-tional windows. Beyond that, the orbital periods of boththe planet–moon barycenter (around the star) and themoon(s) (around the planet–moon barycenter) determine a non-randomized sampling of the moon orbit in succes-sive transits. If an exoplanet’s orbital period were – bywhatsoever reason – an integer multiple of its moon’speriod, then the moon were to appear at the same po-sition relative to the planet in successive transits. Theresulting phase-folded light curve would not display theOSE and therefore not converge to our solutions, but itwould show two transits: one caused by the planet andone caused by the moon.
PyOSE can simulate large numbers of OSE curves forarbitrary star–planet–multi-moon configurations to testreal observations. Beyond the functionalities demon-strated in this paper,
PyOSE can add noise (parameter-ized or injected real noise).Both our dynamical OSE model (Sect. 2.2.1) and ournumerical OSE simulator (
PyOSE , Sect. 3) are computa-tionally inexpensive and easy to implement in computercode, which is crucial for the independent verification orrejection of possible exomoon signals. More advancedmethods may suffer from a large parameter space to beexplored, resulting in a huge number of simulations (10 ;Kipping et al. 2013). These could be difficult to verify.For an independent verification of an exomoon search,either the original code needs to be released for review,or an independent implementation is required. To ex-clude processing, runtime, and hardware errors (bit errorrates are typically 10 − ), a search based on TBs of dataultimately needs to be repeated on different hardware.The average computational burden for this is large, withan average of roughly 33,000 CPU hours required percandidate using photodynamical modeling. The mone-tary equivalent, e.g. using Amazon’s EC2 on-demandfacilities , is about $50,000 U.S. dollars (2015 Decem-ber prices) for a single candidate check.The dynamical OSE simulation in Figure 4 contains48 data points and was computed within 10 to 14 s on https://aws.amazon.com/ec2 odeling the OSE of Exomoons Hence, this setup can generatea grid of about 10 such OSE light curves per day. OurOSE model involves 11 independent parameters: M ⋆ , R ⋆ ,the planet’s orbital period around the star ( P ⋆ p ), R p , b , a , a , a , a , R s , and a . For a well parameterized star–planet system, M ⋆ , R ⋆ , P ⋆ p , and b can be observed andfit without considerations of any potential moon, assum-ing that moon-induced variations of the transit impactparameter (Kipping 2009) are negligible. If one were tocarry out a search for the photometric OSE in the Kepler data, a limb darkening law with two LDCs should stilldo a good job in a first broad survey, leaving us with R p , R s , and a plus the two LDCs to be fit per phase-foldedlight curve. Testing 10 values per parameter would thenimply a grid of 10 OSE light curves per planet or planetcandidate. With 10 LCs simulated per day, one
Kepler planet or planet candidate could be checked for an OSEsignature within a week, given a standard desktop com-puter. If dedicated high-speed computational resourcescould be used, all
Kepler planets and candidates (about4000 as of today) could be checked for a photometric OSEwithin maybe a month. A simple straight line fit of theOSE, as suggested above, would dramatically reduce thistime frame to much less than one day. A detailed sta-tistical analysis, e.g. within a Bayesian framework (Kip-ping et al. 2012) and using an injection-retrieval method(Hippke 2015), could then be used to infer the signifi-cance of the best-fit model for each object. CONCLUSIONWe present a new formula to describe the orbital sam-pling frequency of a moon on an eccentric orbit arounda planet, that is, the probability of a moon residing ata specific sky-projected distance from the planet (Equa-tion 10). This formula assumes co-planar circumstellarand circumplanetary orbits. We implemented it in a dy-namical OSE simulator with stellar limb darkening thatcan be applied to arbitrary transit impact parameters.In contrast to a previously derived framework, in whichstellar limb darkening was neglected and b was requiredto be zero (Heller 2014), our new dynamical OSE simu-lator can now be applied to observations (Figure 6).Using an independent numerical OSE simulatordubbed PyOSE , we examined the moon’s part of the OSEparameter space, spanned by its orbital semimajor axis,its physical radius, the inclination of its orbit with re-spect to the line of sight, the orbit’s longitude of theascending node, and the transit impact parameter of theplanet (and therefore of the moon).The OSE might give evidence of a multi-moon configu-ration, but the precise characterization of multi-satellitesystem will be extremely challenging using OSE only.Therefore, the OSE method could be used for prelimi-nary analyses of a large number of systems, while morecostly methods (Kipping et al. 2012; Heller & Albrecht2014; Agol et al. 2015) could be used to focus on themost promising subset of targets. Beyond that, the OSEmethod can generally be used as an independent meansto verify an exomoon claim via planetary TTV and TDV. Computations took 14 s on a MacBook Pro Retina 8-core,2.8 GHz Intel Core i7 processor, 16 GB of total memory, 1600 MHzDDR3 and 10 s on a desktop computer with an 8-core, 3.6 GHzIntel Core i7-3820 CPU, 32 GB of total memory.
We thank the referee for diligent reports. Ren´e Hellerhas been supported by the Origins Institute at McMas-ter University, by the Canadian Astrobiology Program(a Collaborative Research and Training Experience Pro-gram funded by the Natural Sciences and EngineeringResearch Council of Canada), by the Institute for Astro-physics G¨ottingen, and by a Fellowship of the GermanAcademic Exchange Service (DAAD). This work madeuse of NASA’s ADS Bibliographic Services. Compu-tations were performed with ipython 0.13 on python2.7.2 (P´erez & Granger 2007), and figures were pre-pared with gnuplot 4.6 ( ).REFERENCES Agol, E., Jansen, T., Lacy, B., Robinson, T. D., & Meadows, V.2015, ApJ, 812, 5Batalha, N. M., Rowe, J. F., Bryson, S. T., et al. 2013, ApJS,204, 24Bevington, P. R., & Robinson, D. K. 2003, Data reduction anderror analysis for the physical sciencesBorucki, W. J., Koch, D. G., Dunham, E. W., & Jenkins, J. M.1997, in Astronomical Society of the Pacific Conference Series,Vol. 119, Planets Beyond the Solar System and the NextGeneration of Space Missions, ed. D. Soderblom, 153Brown, T. M., Charbonneau, D., Gilliland, R. L., Noyes, R. W.,& Burrows, A. 2001, ApJ, 552, 699Cameron, A. G. W., & Ward, W. R. 1976, in Lunar andPlanetary Institute Science Conference Abstracts, Vol. 7, Lunarand Planetary Institute Science Conference Abstracts, 120Canup, R. M. 2012, Science, 338, 1052Canup, R. M., & Ward, W. R. 2002, AJ, 124, 3404Carter, J., Agol, E., Chaplin, W. J., et al. 2012, Science, 337, 556Claret, A. 2000, A&A, 363, 1081Claret, A., & Bloemen, S. 2011, A&A, 529, A75´Cuk, M. 2007, Science, 318, 244Dawson, R. I., & Murray-Clay, R. A. 2013, ApJ, 767, L24Domingos, R. C., Winter, O. C., & Yokoyama, T. 2006, MNRAS,373, 1227Doyle, L. R., Carter, J. A., Fabrycky, D. C., et al. 2011, Science,333, 1602Fortney, J., Marley, M. S., & Barnes, J. W. 2007, ApJ, 659, 1661Goldreich, P. 1963, MNRAS, 126, 257Gong, Y.-X., Zhou, J.-L., Xie, J.-W., & Wu, X.-M. 2013, ApJ,769, L14Heller, R. 2012, A&A, 545, L8—. 2014, ApJ, 787, 14Heller, R., & Albrecht, S. 2014, ApJ, 796, L1Heller, R., & Pudritz, R. 2015a, A&A, 578, A19—. 2015b, ApJ, 806, 181Hippke, M. 2015, ApJ, 806, 51Hippke, M., & Angerhausen, D. 2015, ApJ, 810, 29Kipping, D. M. 2009, MNRAS, 396, 1797—. 2011, MNRAS, 416, 689Kipping, D. M., Bakos, G. ´A., Buchhave, L., Nesvorn´y, D., &Schmitt, A. 2012, ApJ, 750, 115Kipping, D. M., Forgan, D., Hartman, J., et al. 2013, ApJ, 777,134Kipping, D. M., Nesvorn´y, D., Buchhave, L. A., et al. 2014, ApJ,784, 28Kipping, D. M., Schmitt, A. R., Huang, X., et al. 2015, ApJ, 813,14Lewis, K. M., & Fujii, Y. 2014, ApJ, 791, L26Maciejewski, G., Dimitrov, D., Neuh¨auser, R., et al. 2010,MNRAS, 407, 2625Mandel, K., & Agol, E. 2002, ApJ, 580, L171Mayor, M., & Queloz, D. 1995, Nature, 378, 355Morbidelli, A., Tsiganis, K., Batygin, K., Crida, A., & Gomes, R.2012, Icarus, 219, 737Payne, M. J., Deck, K. M., Holman, M. J., & Perets, H. B. 2013,ApJ, 775, L44P´erez, F., & Granger, B. E. 2007, Comput. Sci. Eng., 9, 21Pollack, J. B., & Reynolds, R. T. 1974, Icarus, 21, 248Pont, F., Gilliland, R. L., Moutou, C., et al. 2007, A&A, 476, 1347Rowe, J. F., Bryson, S. T., Marcy, G. W., et al. 2014, ApJ, 784,45Sasaki, T., Stewart, G. R., & Ida, S. 2010, ApJ, 714, 1052Simon, A. E., Szab´o, G. M., Kiss, L. L., Fortier, A., & Benz, W.2015, PASP, 127, 1084 Heller, Hippke & Jackson
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Contribution:RH did the literature research, worked out the mathematical framework, translated the math intocomputer code, created Figs. 1-4, led the writing of the manuscript, and served as a correspondingauthor for the journal editor and the referees. r X i v : . [ a s t r o - ph . E P ] S e p Draft version September 26, 2014
Preprint typeset using L A TEX style emulateapj v. 04/17/13
HOW TO DETERMINE AN EXOMOON’S SENSE OF ORBITAL MOTION
Ren´e Heller
Origins Institute, McMaster University, Hamilton, ON L8S 4M1, Canada; [email protected] andSimon Albrecht
Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,Denmark; [email protected]
Draft version September 26, 2014
ABSTRACTWe present two methods to determine an exomoon’s sense of orbital motion (SOM), one with respectto the planet’s circumstellar orbit and one with respect to the planetary rotation. Our simulationsshow that the required measurements will be possible with the European Extremely Large Telescope(E-ELT). The first method relies on mutual planet-moon events during stellar transits. Eclipses withthe moon passing behind (in front of) the planet will be late (early) with regard to the moon’s meanorbital period due to the finite speed of light. This “transit timing dichotomy” (TTD) determinesan exomoon’s SOM with respect to the circumstellar motion. For the ten largest moons in the solarsystem, TTDs range between 2 and 12 s. The E-ELT will enable such measurements for Earth-sizedmoons around nearby stars. The second method measures distortions in the IR spectrum of the ro-tating giant planet when it is transited by its moon. This Rossiter-McLaughlin effect (RME) in theplanetary spectrum reveals the angle between the planetary equator and the moon’s circumplanetaryorbital plane, and therefore unveils the moon’s SOM with respect to the planet’s rotation. A reason-ably large moon transiting a directly imaged planet like β Pic b causes an RME amplitude of almost100 m s − , about twice the stellar RME amplitude of the transiting exoplanet HD209458 b. Both newmethods can be used to probe the origin of exomoons, that is, whether they are regular or irregularin nature. Keywords: eclipses — methods: data analysis — methods: observational — planets and satellites:individual ( β Pic b) — techniques: photometric — techniques: radial velocities CONTEXT
Although thousands of extrasolar planets and candi-dates have been found, some as small as the Earth’sMoon (Barclay et al. 2013), no extrasolar moon has beendetected. The first dedicated hunts for exomoons havenow been initiated (Pont et al. 2007; Kipping et al. 2012;Szab´o et al. 2013), and it has been shown that the Kepleror PLATO space telescopes may find large exomoons inthe stellar light curves (Kipping et al. 2009; Heller 2014),if such worlds exist.The detection of exomoons would be precious from aplanet formation perspective, as giant planet satellitescarry information about the thermal and compositionalproperties in the early circumplanetary accretion disks(Canup & Ward 2006; Heller & Pudritz 2014). Moonscan also constrain the system’s collision history (see theEarth-Moon binary, Hartmann & Davis 1975) and bom-bardment record (see the misaligned Uranian system,Morbidelli et al. 2012), they can trace planet-planet en-counters (see Triton’s capture around Neptune, Agnor &Hamilton 2006), and even the migration history of close-in giant planets (Namouni 2010). Under suitable condi-tions, an exomoon observation could reveal the absolutemasses and radii in a star-planet-moon system (Kipping2010). What is more, moons may outnumber rocky plan-ets in the stellar habitable zones (Heller & Barnes 2014) Department of Physics and Astronomy, McMaster University Postdoctoral fellow of the Canadian Astrobiology TrainingProgram and therefore could be the most abundant species of hab-itable worlds (Williams et al. 1997; Heller et al. 2014).A moon’s sense of orbital motion (SOM) is crucialto determine its origin and orbital history. About adozen techniques have been proposed to find an extra-solar moon (Heller 2014), but none of them can deter-mine an exomoon’s SOM with current technical equip-ment (Lewis & Fujii 2014). We here identify two meansto determine an exomoon’s SOM relative to the circum-stellar orbit and with respect to the planet’s direction ofrotation. In our simulations, we use the European Ex-tremely Large Telescope (E-ELT) as an example for oneof several ELTs now being built. METHODS
An Exomoon’s Transit Timing Dichotomy (TTD)
For our first new method to work, the moon needs tobe large enough (and the star’s photometric variabilitysufficiently low) to cause a direct transit signature in thestellar light curve (Sartoretti & Schneider 1999). De-pending on the moon’s orbital semi-major axis aroundthe planet ( a ps ) and on the orbital alignment, some stel-lar transits of the planet-moon pair will then show mu-tual planet-moon eclipses. These events have been sim-ulated (Cabrera & Schneider 2007; Sato & Asada 2009;Kipping 2011a; P´al 2012), and a planet-planet eclipse(Hirano et al. 2012) as well as mutual events in a stel- Construction of the E-ELT near the Paranal Observatory inChile began in June 2014, with first light anticipated in 2024.
Ren´e Heller & Simon Albrecht ! ! (a) prograde orbit ! ! (b) retrograde orbit t o E a r t h III IIIIV (c) top view (d) edge view could beI - II - III - IV (pro)orIV - III - II - I (retro)star ! ! planetmoon orbital motion II IIII IV ! t =late eclipsesearly eclipses t o E a r t h c a ps " ! ! t I ( z oo m ) (e) Figure 1.
Orbital geometry of a star-planet-moon system in circu-lar orbits. (a)
Top view of the system’s orbital motion. The moon’scircumplanetary orbit is prograde with respect to the circumstellarorbit. (b)
Similar to (a) , but now the moon is retrograde. (c)
Topview of the moon’s orbit around the planet. Roman numbers I toIV denote the ingress and egress of mutual planet-moon eclipses. (d)
Edge view (as seen from Earth) of a circumplanetary moon or-bit. (e)
Transit timing dichotomy of mutual planet-moon events.Due to the finite speed of light, an Earth-bound observer witnessesevents I and IV with a positive time delay ∆ t compared to eventsII and III, respectively. lar triple system (Carter et al. 2011) have already beenfound in the Kepler data.Figure 1 illustrates the difficulty in determining amoon’s SOM. Panels (a) and (b) visualize the two possi-ble scenarios of a prograde and a retrograde SOM withrespect to the circumstellar motion. Panel (c) presentsthe four possible ingress and egress locations (arbitrarilylabelled I, II, III, and IV) for a mutual planet-moon eventduring a stellar transit. Panel (d) shows the projection ofthe three-dimensional moon orbit on the two-dimensionalcelestial plane. If the moon transit is directly visible inthe stellar light curve, then events I and II can be distin- guished from events III and IV, simply by determiningwhether the moon enters the stellar disk first and thenperforms a mutual event with the planet (III and IV) orthe planet enters the stellar disk first before a mutualevent (I and II). However, this inspection cannot discernevent I from II or event III from IV. Therefore, progradeand retrograde orbits cannot be distinguished from eachother.Figure 1 (e) illustrates how the I/II and III/IV ambi-guities can be solved. Due to the finite speed of light,there will be a time delay between events I and II, andbetween events III and IV. It will show up as a transittiming dichotomy (TTD) between mutual events wherethe moon moves in front of the planetary disk (early mu-tual events) or behind it (late mutual events). Events Iand IV will be late by ∆ t = 2 a ps /c compared to eventsII and III, respectively. Imagine that two mutual events,either I and [II or III] or IV and [III or II]), are observedduring two different stellar transits and that the moonhas completed n circumplanetary orbits between the twomutual events. Then it is possible to determine the se-quence of late and early eclipses, and therefore the SOMwith respect to the circumstellar movement, if (1) theorbital period of the planetary satellite ( P ps ) can be de-termined independently with an accuracy δP ps < P ps /n ,and if (2) the event mid-times can be measured with aprecision < ∆ t . As a byproduct, measurements of ∆ t yield an estimate for a ps = ∆ t × c .Concerning (1), the planet’s transit timing variation(TTV) and transit duration variation (TDV) due to themoon combined may constrain the satellite mass ( M s )and P ps (Kipping 2011b). As an example, if two mutualevents of a Jupiter-Ganymede system at 0.5 AU arounda Sun-like star were observed after 10 stellar transits (or3.5 yr), the moon ( P ps ≈ .
02 yr) would have completed n ≈
175 circumplanetary orbits. Hence, δP ps . t , we can safely approximatethat a moon enters a mutual event at a radial distance a ps to the planet, because δ ∆ t = a ps c (cid:16) − cos( α ) (cid:17) = a ps c − cos n arcsin (cid:16) R p a ps (cid:17)o! ≪ ∆ t , (1)with c being the speed of light, R p the planetary ra-dius, and α defined by sin( α ) = R p /a ps as shown in Fig-ure 1(e). For a moon at 15 R p from its planet, such asGanymede around Jupiter, α ≈ . ◦ and δ ∆ t ≈ .
005 s,which is completely negligible. Orbital eccentricitiescould also cause light travel times different from the oneshown in Figure 1. But even for eccentricities comparableto Titan’s value around Saturn, with 0.0288 the largestamong the major moons in the solar system, TTDs wouldbe affected by < ow to determine an exomoon’s sense of orbital motion Figure 2.
Theoretical emission spectra of a hot, Jupiter-sizedplanet (orange, upper) and a Sun-like star as reflected by the planetat 10 AU (blue, middle) and 100 AU (light blue, lower). For theplanet, a Jupiter-like bond albedo of 0.3 is assumed.
The Rossiter-McLaughlin effect (RME) in thePlanetary Emission Spectrum
In the solar system, all planets except Venus (Gold &Soter 1969) and Uranus rotate in the same direction asthey orbit the Sun. One would expect that the orbitalmotion of a moon is aligned with the rotation of theplanet. However, collisions, gravitational perturbations,capture scenarios etc. can substantially alter a satel-lite’s orbital plane (Heller et al. 2014). Hence, knowl-edge about the spin-orbit misalignment, or obliquity, ina planet-exomoon system would be helpful in inferringits formation and evolution.One such method to constrain obliquities is theRossiter-McLaughlin effect (RME), a distortion in the ro-tationally broadened absorption lines caused by the par-tial occultation of the rotating sphere (typically a star)by a transiting body (usually another star or a planet).This distortion can either be measured directly (Albrechtet al. 2007; Collier Cameron et al. 2010) or be picked upas an RV shift during transit. The shape of this anoma-lous RV curve reveals the projection of the angle betweenthe orbital normal of the occulting body (in our casethe moon) and the rotation axis of the occulted body(here the planet). Originally observed in stellar binaries(Rossiter 1924; McLaughlin 1924), this technique experi-enced a renaissance in the age of extrasolar planets, withthe first measurement taken by Queloz et al. (2000) forthe transiting hot Jupiter HD209458 b. Numerous mea-surements revealed planets in aligned, misaligned, andeven retrograde orbits – strongly contrasting the archi-tecture of the solar system (e.g. Albrecht et al. 2012).The stellar RME of exomoons has been studied before(Simon et al. 2010; Zhuang et al. 2012), but here we referto the RME in the planetary infrared spectrum causedby the moon transiting a hot, young giant planet.For this method to be effective, a giant planet’s lightneeds to be measured directly. Starting with the plan-etary system around HR 8799 (Marois et al. 2008) andthe planet candidate Fomalhaut b (Kalas et al. 2008), 18giant exoplanets have now been directly imaged, most of ∼ rheller). Figure 3.
Transit timing dichotomies of the ten largest moonsin the solar system. Moon radii are symbolized by circle sizes.The host planets Jupiter, Saturn, Neptune, Uranus, and Earth areindicated with their initials. Note that the largest moons, causingthe deepest solar transits, induce the highest TTDs. which are hot ( > <
100 Myr). Up-coming instruments like SPHERE (Beuzit et al. 2006)and GPI (McBride et al. 2011) promise a rapid increaseof this number. Stellar and planetary spectra can alsobe separated in velocity space without the need of spatialseparation (Brogi et al. 2012; de Kok et al. 2013; Birkbyet al. 2013), but for β Pic-like systems, instruments likeCRIRES can also separate stellar and planetary spectraspatially. Snellen et al. (2014) determined the rotationperiod of β Pic b to be 8 . ± . ≈
25 km s − ) favors a large RME amplitude,but makes RV measurements more difficult.Contamination of the planetary spectrum by the starvia direct stellar light on the detector and stellar reflec-tions from the planet might pose a challenge. Conse-quently, observations need to be carried out in the near-IR, where the planet is relatively bright and presentsa rich forest of spectral absorption lines. Figure 2shows that contamination of the planetary spectrum by reflected star light becomes negligible beyond several10 AU, with no need for additional cleaning. RESULTS AND PREDICTIONS
Transit Timing Dichotomies
We computed the TTDs of the ten largest moons inthe solar system, yielding values between about 2 and12 s (Figure 3). Most intriguingly, the largest moons(Ganymede, Titan, and Callisto), which have the deep-est solar transit signatures, also have the largest TTDs.This is owed to the location of the water ice line in theaccretion disks around Jovian planets, which causes themost massive icy satellites to form beyond about 15 R Jup (Heller & Pudritz 2014). Figure 3 indicates that timingprecisions of 1 - 6 s need to be achieved on transit eventswith depths of only about 10 − , corresponding to thetransit depth of an Earth-sized moon transiting a Sun-like star.Precisions of 6 s in exoplanet transit mid-times havebeen achieved from the ground using the Baade 6.5 mtelescope at Las Campanas Observatory in Chile (Winn Models were provided by T.-O. Husser (priv. comm.), basedon the spectral library by Husser et al. (2013) available athttp://phoenix.astro.physik.uni-goettingen.de.
Ren´e Heller & Simon Albrecht
Figure 4.
Probability of mutual planet-moon events as a func-tion of the moon’s planetary distance around a Jupiter-like planet,which is assumed to orbit a Sun-like star at 1 AU. The five curvesindicate the frequency of 0, 1, 2, 3, or 4 mutual events during stellarstellar transits as measured in our transit simulations. The orbitsof Io, Europa, Ganymede, and Callisto are indicated with symbolsalong each curve. et al. 2009). On the one hand, the planet in these obser-vations (WASP-4b) was comparatively large to its hoststar with a transit depth of about 2.4 %. On the otherhand, the star was not particularly bright, with an appar-ent visual magnitude m V ≈ .
6. The photon collectingpower of the E-ELT will be (39 . / . ≈
37 timesthat of the Baade telescope. And with improved datareduction methods, timing precisions of the order of sec-onds should be obtainable for transit depths of 10 − withthe E-ELT for very nearby transiting systems.We calculate the probabilities of one to four mutualevents ( P i , i ∈ { , , , } ) during a stellar transit of acoplanar planet-moon system. The distance of the moontravelled on its circumplanetary orbit during the stel-lar transit s = 2 R ⋆ P ⋆ b a ps / ( P ps a ⋆ b ), with a ⋆ b denotingthe circumstellar orbital semi-major axis of the planet-moon barycenter and P ⋆ b being its circumstellar orbitalperiod. We analyze a Jupiter-like planet at 1 AU froma Sun-like star and simulate 10 transits for a range ofpossible moon semi-major axes, respectively, where themoon’s initial orbital position during the stellar transitis randomized. As the stellar transit occurs, we followthe moon’s circumplanetary orbit and measure the num-ber of mutual events (of type I, II, III, or IV) during thetransit, which can be 0, 1, 2, 3, or even 4.For siblings of Io, Europa, Ganymede, and Callisto wefind P = 21, 13, 8, and 5 % as well as P = 50, 24,11, and 5 %, respectively (Figure 4). Notably, the prob-abilities for two mutual events (purple long-dashed line)are higher than the likelihoods for only one (red solidline) if a ps . . × km. For moons inside about half-way between Io and Europa, the probability of having noevent (black solid line) is <
50 %, so it is more likely tohave at least one mutual event during any transit thanhaving none. Moons inside 200,000 km (half the semi-major axis of Io) can even have three or more events,allowing the TTD method to work with only one stellartransit. Deviations from well-aligned orbits due to hightransit impact parameters or tilted moon orbits wouldnaturally reduce the shown probabilities. Nevertheless,mutual events could obviously be common during tran-sits. ✲(cid:0) ✲✁ ✵ ✁ (cid:0)❚✂✄☎ ✆✝✞✄ ✄✂✟✲✠✝✡☛☞✂✠ ✌✍✝✎✲✁✵✵✵✁✵✵❘✏✑✒✒✑✓✔✕✖✗✘✙ ✮ ♣✚✛✜✚✢✣✤ ✥✛✣✤✦✚✤r✚✛✜✚✢✣✤ ✥✛✣✤✦ ♣✚✛✜✚✢✣✤ ✧★✥✩✦✢r✤✣ ✣✢r✢✚✤r✚✛✜✚✢✣✤ ✧★✥✩✦✢r✤✣ ✣✢r✢
Figure 5.
Simulated Rossiter-McLaughlin effect of a giant moon(0 . R ⊕ ) transiting a hot, Jupiter-sized planet similar to β Pic b.Solid and dashed lines correspond to a prograde and a retrogradecoplanar orbit, respectively. Full and open circles indicate simu-lated E-ELT observations.
A single stellar transit with ≥ P is zero beyondabout 10 km. In close orbits, the fraction of two-eventcases with TTD information is P , TTD ≈ R p ( a ps π ) − . ForIo, as an example, P , TTD = (6 . × π ) − ≈ The Planetary Rossiter-McLaughlin Effect due toan Exomoon
We simulated the RME in the near-IR spectrum ofa planet similar to β Pic b assuming a planetary rota-tion speed of 25 km s − (Snellen et al. 2014) and a Jo-vian planetary radius ( R Jup ). The moon was placed ina Ganymede-like orbit ( a ps = 15 R Jup ) and assumed tobelong to the population of giant moons that form atthe water ice lines around super-Jovian planets, witha radius of up to about 0 . R ⊕ ) (Heller& Pudritz 2014). Using the code of Albrecht et al.(2007, 2013), we simulated absorption lines of the ro-tating planet as distorted during the moon’s transit witha cadence of 15 min. Focusing on the same spectral win-dow (2 . − . µ m) as Snellen et al. (2014), we thenconvolved the planetary spectrum (Figure 2) with thesedistorted absorption lines. Employing the CRIRES Ex-posure Time Calculator and incorporating the increase of(39 . / . ≈
23 in collecting area for the E-ELT, weobtain a S/N of 75 per pixel for a 15 min exposure – thesame values as obtained by Snellen et al. (2014) for a sim-ilar calculation. The resulting pseudo-observed spectraare finally cross-correlated with the template spectrum,and a Gaussian is fitted to the cross-correlation functionsto obtain RVs.Pro- and retrograde coplanar orbits are clearly dis-tinguishable in the resulting RME curve (Figure 5).In particular, the RME amplitude of ≈
100 m s − isquite substantial. In comparison, the RME amplitude ofHD209458 b is ≈
40 m s − (Queloz et al. 2000), and re- ow to determine an exomoon’s sense of orbital motion − (Winn et al.2010). DISCUSSION AND CONCLUSION
We present two new methods to determine an exo-moon’s sense of orbital motion. One method, which werefer to as the transit timing dichotomy, is based on alight travelling effect that occurs in subsequent mutualplanet-moon eclipses during stellar transits. For the tenlargest moons in the solar system, TTDs range between2 and 12 s. If the planet-moon orbital period can bedetermined independently (e.g. via TTV and TDV mea-surements) and with an accuracy of . − need to be obtainedalong with mid-event precisions . R ⊙ / . ≈ . R Jup , implyingtransit probabilities of up to about 16%. With orbitalperiods of a few days, moon transits occur also muchmore frequently than for a common Kepler planet. Per- manent, highly-accurate IR photometric monitoring of afew dozen directly imaged giant exoplanets thus has ahigh probability of finding an extrasolar moon.The report of an anonymous referee was very helpful inclarifying several passages in this letter. We thank Tim-Oliver Husser for providing us with the PHOENIX mod-els. Ren´e Heller is supported by the Origins Institute atMcMaster University and by the Canadian AstrobiologyProgram, a Collaborative Research and Training Experi-ence Program funded by the Natural Sciences and Engi-neering Research Council of Canada (NSERC). Fundingfor the Stellar Astrophysics Centre is provided by TheDanish National Research Foundation (Grant agreementno.: DNRF106). REFERENCES
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Ren´e Heller & Simon Albrecht
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Contribution:RH contributed to the literature research, contributed to the mathematical framework, created Fig. 1,led the writing of the manuscript, and served as a corresponding author for the journal editor and thereferees. r X i v : . [ a s t r o - ph . E P ] J un Astronomy & Astrophysicsmanuscript no. ms c (cid:13)
ESO 2016June 21, 2016
Predictable patterns in planetary transit timing variations andtransit duration variations due to exomoons
René Heller , Michael Hippke , Ben Placek , Daniel Angerhausen , , and Eric Agol , Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany; [email protected] Luiter Straße 21b, 47506 Neukirchen-Vluyn, Germany; [email protected] Center for Science and Technology, Schenectady County Community College, Schenectady, NY 12305, USA;[email protected] NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA; [email protected] USRA NASA Postdoctoral Program Fellow, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771,USA Astronomy Department, University of Washington, Seattle, WA 98195, USA; [email protected] NASA Astrobiology Institute’s Virtual Planetary Laboratory, Seattle, WA 98195, USAReceived 22 March 2016; Accepted 12 April 2016
ABSTRACT
We present new ways to identify single and multiple moons around extrasolar planets using planetary transit timing variations (TTVs)and transit duration variations (TDVs). For planets with one moon, measurements from successive transits exhibit a hitherto unde-scribed pattern in the TTV-TDV diagram, originating from the stroboscopic sampling of the planet’s orbit around the planet–moonbarycenter. This pattern is fully determined and analytically predictable after three consecutive transits. The more measurementsbecome available, the more the TTV-TDV diagram approaches an ellipse. For planets with multi-moons in orbital mean motion reso-nance (MMR), like the Galilean moon system, the pattern is much more complex and addressed numerically in this report. Exomoonsin MMR can also form closed, predictable TTV-TDV figures, as long as the drift of the moons’ pericenters is su ffi ciently slow. Wefind that MMR exomoons produce loops in the TTV-TDV diagram and that the number of these loops is equal to the order of theMMR, or the largest integer in the MMR ratio. We use a Bayesian model and Monte Carlo simulations to test the discoverability ofexomoons using TTV-TDV diagrams with current and near-future technology. In a blind test, two of us (BP, DA) successfully retrieveda large moon from simulated TTV-TDV by co-authors MH and RH, which resembled data from a known Kepler planet candidate.Single exomoons with a 10 % moon-to-planet mass ratio, like to Pluto-Charon binary, can be detectable in the archival data of the
Kepler primary mission. Multi-exomoon systems, however, require either larger telescopes or brighter target stars. Complementarydetection methods invoking a moon’s own photometric transit or its orbital sampling e ff ect can be used for validation or falsification.A combination of TESS , CHEOPS , and
PLATO data would o ff er a compelling opportunity for an exomoon discovery around a brightstar. Key words. eclipses – methods: numerical – planets and satellites: detection – planets and satellites: dynamical evolution andstability – planets and satellites: terrestrial planets – techniques: photometric
1. Introduction
The search for moons around planets beyond the solar systemis entering a critical phase. The first dedicated exomoon sur-veys have now been implemented using space-based highly ac-curate
Kepler photometry (Kipping et al. 2012; Szabó et al.2013; Hippke 2015) and more will follow in the near futureusing
CHEOPS (Simon et al. 2015) and
PLATO (Hippke &Angerhausen 2015). An important outcome of the first exomoonsearches is that moons at least twice as massive as Ganymede,the most massive local moon, are rare around super-Earths (Kip-ping et al. 2015).More than a dozen techniques have been proposed to searchfor exomoons: (i.) transit timing variations (TTVs; Sartoretti &Schneider 1999; Simon et al. 2007) and transit duration vari-ations (TDVs; Kipping 2009a,b) of exoplanets; (ii.) the directphotometric transit signature of exomoons (Sartoretti & Schnei-der 1999; Brown et al. 2001; Szabó et al. 2006; Charbonneauet al. 2006; Pont et al. 2007; Tusnski & Valio 2011; Kip-ping 2011a); (iii.) microlensing (Han & Han 2002; Liebig & Wambsganss 2010; Bennett et al. 2014); (iv.) mutual eclipsesof directly imaged, unresolved planet–moon binaries (Cabrera& Schneider 2007); (v.) the wobble of the photometric centerof unresolved, directly imaged planet–moon systems (Cabrera& Schneider 2007; Agol et al. 2015) (vi.) time-arrival analy-ses of planet–moon systems around pulsars (Lewis et al. 2008);(vii.) planet–moon mutual eclipses during stellar transits (Sato& Asada 2009; Pál 2012); (viii.) the Rossiter-McLaughlin ef-fect (Simon et al. 2010; Zhuang et al. 2012); (ix.) scatter peakanalyses (Simon et al. 2012) and the orbital sampling e ff ect ofphase-folded light curves (Heller 2014; Heller et al. 2016); (x.)modulated radio emission from giant planets with moons (Noy-ola et al. 2014, 2016); (xi.) the photometric detection of moon-induced plasma torii around exoplanets (Ben-Ja ff el & Ballester2014); (xii.) and several other spectral (Williams & Knacke2004; Robinson 2011; Heller & Albrecht 2014) and photometric Article number, page 1 of 19 & Aproofs: manuscript no. ms (Moskovitz et al. 2009; Peters & Turner 2013) analyses of theinfrared light in exoplanet-exomoon systems. None of these techniques has delivered a secure exomoon de-tection as of today, which is partly because most of these meth-ods are reserved for future observational technologies. Somemethods are applicable to the available data, for example, fromthe
Kepler primary mission, but they are extremely computer in-tense (Kipping et al. 2012). We present a novel method to findand characterize exomoons that can be used with current tech-nologies and even the publicly available
Kepler data. We identifya new pattern in the TTV-TDV diagram of exoplanet-exomoonsystems that allows us to distinguish between single and mul-tiple exomoons. Detection of multiple moons is naturally morechallenging than the detection of single exomoons owing to thehigher complexity of models that involve multiple moons (Heller2014; Hippke & Angerhausen 2015; Kipping et al. 2015; Helleret al. 2016; Noyola et al. 2016).
2. Patterns in the TTV-TDV diagram
An exoplanet transiting a star can show TTVs (Sartoretti &Schneider 1999) and TDVs (Kipping 2009a,b) if accompaniedby a moon. Various ways exist to measure TTVs, for example,with respect to the planet-moon barycenter or photocenter (Sz-abó et al. 2006; Simon et al. 2007, 2015). We utilize the barycen-tric TTV. On their own, TTVs and TDVs yield degenerate solu-tions for the satellite mass ( M s ) and the orbital semimajor axis ofthe satellite around the planet ( a s ). If both TTV and TDV can bemeasured repeatedly, however, and sources other than moons beexcluded, then the M s and a s root mean square (RMS) values canbe estimated and the degeneracy be solved (Kipping 2009a,b).This method, however, works only for planets (of mass M p )with a single moon. Moreover, observations will always under-sample the orbit of the moon and P s cannot be directly measured(Kipping 2009a). This is because the orbital period of the planetaround the star ( P p ) is & P s ) to ensure Hill stability. Strictly speaking,it is the circumstellar orbital period of the planet–moon barycen-ter ( P B ) that needs to be & P s , but for M s / M p → P p → P B . Finally, a predictable pattern in TTV-TDV measure-ments has not been published to date.We present new means to determine (1) the remainder of thedivision of the orbital periods of the moon and the planet forone-moon systems; (2) the TTV and TDV of the planet duringthe next transit for one-moon systems; and (3) the number ofmoons in multiple moon systems.Our approach makes use of the fact that TTVs and TDVsare phase-shifted by π/ Imagine a transiting exoplanet with a single moon. We let P B be f = n + r times P s , where n is an integer and 0 ≤ r ≤ P B ≈ .
25 d, P s ≈ . P B ≈ . P s . Hence, n =
13 and r ≈ . Naturally, these methods are not fully independent, and our enumer-ation is somewhat arbitrary. liver any TTV or TDV, because there is no reference value yetto compare our measurements to yet. After the second transit,we may obtain a TDV because the duration of the transit mightbe di ff erent from the first transit. We are still not able to ob-tain a TTV because determining the transit period requires twoconsecutive transits to be observed. Only after the third transitare we able to measure a variation of the transit period, i.e., ourfirst TTV (and our second TDV). While we observe more andmore transits, our average transit period and transit duration val-ues change and converge to a value that is unknown a priori butthat could be calculated analytically for one-moon systems if thesystem properties were known.Figure 1 qualitatively shows such an evolution of a TTV-TDV diagram. Di ff erent from the above-mentioned procedure,measurements in each panel are arranged in a way to graduallyform the same figure as shown in panel (e), as if the average TTVand TDV for N obs = ∞ were known a priori. In each panel, N obs denotes the number of consecutive transit observations requiredto present those hypothetical data points. In panels (b)-(d), thefirst data point of the data series is indicated by a solid line. Gen-erally, TTV and TDV amplitudes are very di ff erent from eachother, so the final figure of the TTV-TDV diagram in panel (e) isan ellipse rather than a circle. However, if the figure is normal-ized to those amplitudes and if the orbits are circular, then thefinal figure is a circle as shown. The angle ρ corresponds to theremainder of the planet-to-moon orbital period ratio, which canbe deduced via r = ρ/ (2 π ). Knowledge of ρ or r makes it pos-sible to predict the planet-moon orbital geometry during stellartransits, which enables dedicated observations of the maximumplanet-moon apparent separation (at TTV maximum values) orpossible planet-moon mutual eclipses (at TDV maximum val-ues). Yet, n remains unknown and it is not possible to determinehow many orbits the moon has completed around the planet dur-ing one circumstellar orbit of their common barycenter. As anaside, ρ cannot be used to determine the sense of orbital motionof the planet or moon (Lewis & Fujii 2014; Heller & Albrecht2014).If the moon’s orbit is in an n : 1 orbital mean motion reso-nance (MMR) with the circumstellar orbit, then the moon alwaysappears at the same position relative to the planet during subse-quent transits. Hence, if ρ = = r , there is e ff ectively no TTV orTDV and the moon remains undiscovered. In an ( n + ) : 1 MMR,we obtain r = and subsequent measurements in the TTV-TDVdiagram jump between two points. Again, the full TTV-TDV fig-ure is not sampled and the moon cannot be characterized. In gen-eral, in an ( n + x ) : 1 MMR, the diagram jumps between x pointsand r = x .Orbital eccentricities of the satellite ( e s ) complicate this pic-ture. Figure 2 shows the TTV-TDV diagram for a planet–moonsystem akin to the Earth-Moon binary around a Sun-like star.The only di ff erence that we introduced is an e s value of 0 . For single moons on circular orbits, the TTV-TDV diagram canbe calculated analytically if the TTV and TDV amplitudes areknown (Sartoretti & Schneider 1999; Kipping 2009a,b). For sys-tems with more than N = Article number, page 2 of 19ené Heller et al.: Predictable patterns in planetary TTVs and TDVs due to exomoons do not exist. Hence, for systems with more than one moon (andone planet), we resort to numerical simulations. N -body simulations to TTVs and TDVs In the following, we generate TTV-TDV diagrams of exoplanet-exomoons systems using a self-made, standard N -body integra-tor that calculates the Newtonian gravitational accelerations act-ing on N point masses.The average orbital speeds of the major solar system moonsare well known. The initial planetary velocities in our simula-tions, however, are unknown and need to be calculated. We take v B = m B . With index p referring tothe planet, and indices s1, s2, etc. referring to the satellites, wehave v B = v p m p + v s1 m s1 + v s2 m s2 + . . . m B = ⇔ v p = − v s1 m s1 − v s2 m s2 − . . . m p (1)Our simulations run in fixed time steps. We find that 10 steps per orbit are su ffi cient to keep errors in TTV and TDVbelow 0 . steps to obtain < . / TDV error of 0.1 s would generally be consideredto be too large for studies dedicated to the long-term orbital evo-lution and stability of multiplanet systems. For our purpose ofsimulating merely a single orbit to generate the correspondingTTV-TDV diagram, however, this error is su ffi cient because theresulting errors in TTV and TDV are smaller than the linewidthin our plots. The processing runtime for 10 steps in a five-moonsystem is < Under the assumption that the orbits of the planet–moonbarycenter around the star and of the planet with moons aroundtheir local barycenter are coplanar, TTVs depend solely on thevariation of the sky-projected position of the planet relative tothe barycenter. On the other hand, TDVs depend solely on thevariation of the orbital velocity component of the planet that istangential to the line of sight. For a given semimajor axis ofthe planet–moon barycenter ( a B ) and P B around the star, we firstcompute the orbital velocity of the planet on a circular circum-stellar orbit. We then take the stellar radius ( R ⋆ ) and convert vari-ations in the relative barycentric position of the planet into TTVsand variations in its orbital velocity into TDVs. Each of these ≈ orbital configurations corresponds to one stellar transit http: // nssdc.gsfc.nasa.gov / planetary / factsheet / joviansatfact.html https: // github.com / hippke / TTV-TDV-exomoons In case of nonaligned orbits, the planet shows an additional TDVcomponent (Kipping 2009b). of the planet–moon system. It is assumed that the planet–moonsystem transits across the stellar diameter, that is, with a transitimpact parameter of zero, but our general conclusions would notbe a ff ected if this condition were lifted. In the Sun–Earth–Moonsystem, for example, Earth’s maximum tangential displacementof 4 763 km from the Earth–Moon barycenter corresponds to aTTV of 159 s compared to a transit duration of about 13 hr. Aswe are interested in individual, consecutive TTV and TDV mea-surements, we use amplitudes rather than RMS values.Our assumption of coplanar orbits is mainly for visualiza-tion purposes, but more complex configurations with inclined or-bits should be revisited in future studies to investigate the e ff ectsof variations in the planetary transit impact parameter (Kipping2009b) and the like. For this study, coplanar cases can be jus-tified by the rareness of high moon inclinations ( i s ) in the solarsystem; among the 16 largest solar system moons, only four haveinclinations > ◦ ; Moon (5.5 ◦ ), Iapetus (17 ◦ ), Charon (120 ◦ ),and Triton (130 ◦ ). In Fig. 3, we show the dynamics of a two-moon system arounda Jupiter-like planet, where the moons are analogs of Io and Eu-ropa. The left panel shows the actual orbital setup in our N -bodycode, the right panel shows the outcome of the planetary TTV-TDV curve over one orbit of the outermost moon in a Europa-wide orbit. Numbers along this track refer to the orbital phaseof the outer moon in units of percent, with 100 correspondingto a full orbit. In the example shown, both moons start in a con-junction that is perpendicular and “to the left” with respect tothe line of sight of the observer (left panel), thereby causing amaximum barycentric displacement of the planet “to the right”from the perspective of the observer. Hence, if the planet were totransit the star in this initial configuration of our setup, its transitwould occur too early with respect to the average transit periodand with a most negative TTV (phase 0 in Fig. 3).As an analytical check, we calculate the orbital phases of theouter moon at which the planetary motion reverses. The plan-etary deflection due to the Io-like moon is of functional form f I ( t ) ∝ − k I cos( n I t ), where n I is the orbital mean motion of Io, k I is the amplitude of the planetary reflex motion due to the moon,and t is time. The planetary displacement due to the Europa-likemoon is f E ( t ) ∝ − k E cos( n E t ), where n E is the orbital meanmotion of Europa and k E is the amplitude. The total planetarydisplacement then is f tot ( t ) ∝ f I ( t ) + f E ( t ) and the extrema canbe found, where0 ! = ddt f tot ( t ) , (2)that is, where the planetary tangential motion reverses and theplanet “swings back”. We replace n I with 2 n E and k I with k E (since k I = . k E ) and find that Eq. (2) is roughly equivalent to0 ! = n E t ) + sin( n E t ) , (3)which is true at n E t =
0, 1 . π , 4 . π , 8 . n E t / (2 π ) = ,
29 %, 50 %, 71 %,100 %, 129 %. These values agree with the numerically derivedvalues at the maximum displacements (Fig. 3, right panel) andthey are independent of the spatial dimensions of the system.Next, we explore di ff erent orbital period ratios in two-moonMMR systems. In Fig. 4, we show the TTV-TDV diagrams for Article number, page 3 of 19 & Aproofs: manuscript no. ms a Jupiter-like planet (5.2 au from a Sun-like star) with an Io-likeplus a Europa-like moon, but in a 3:1 MMR (top panel) and a4:1 MMR (bottom panel). Intriguingly, we find that the order ofthe MMR, or the largest integer in the MMR ratio, determinesthe number of loops in the diagram. Physically speaking, a loopdescribes the reverse motion of the planet due to the reversal ofone of its moons. This behavior is observed in all our numericalsimulations of MMRs of up to five moons (see Appendix A).The sizes of any of these loops depends on the moon-to-planet mass ratios and on the semimajor axes of the moons. Thisdependency is illustrated in Fig. 5, where we varied the massesof the moons. In the upper (lower) panels, a 2:1 (4:1) MMR isassumed. In the left panels, the mass of the outer moon ( M s2 )is fixed at the mass of Ganymede ( M Gan ), while the mass of theinner moon ( M s1 ) is successively increased from 1 M Gan (blacksolid line) over 2 M Gan (blue dotted line) to 3 M Gan (red dashedline). In the right panels, the mass of the inner moon is fixed at1 M Gan , while the mass of the outer moon is varied accordingly.As an important observation of these simulations, we find thatmassive outer moons can make the loops extremely small andessentially undetectable.
Orbital MMRs can involve librations of the point of conjunctionas well as drifts of the pericenters of the moons. In fact, the peri-joves (closest approaches to Jupiter) in the Io-Europa 2:1 MMRshows a drift of about 0 ◦ . − . Hence, the MMR is only validin a coordinate system that rotates with a rate equal to the drift ofthe pericenters of the respective system. The libration amplitudeof the pericenters on top of this drift has been determined obser-vationally to be 0 ◦ . ± ◦ . ff ects would smear the TTV-TDV fig-ures obtained with our simulations, if the drift is significant onthe timescale on which the measurements are taken.If the moons are not in a MMR in the first place, then there isan additional smearing e ff ect of the TTV-TDV figures because ofthe di ff erent loci and velocities of all bodies after one revolutionof the outermost moon. We explore this e ff ect using an arbitraryexample, in which we add a second moon to the Earth-Moonsystem. We choose an arbitrary semimajor axis (50 % lunar) andmass (0.475% lunar), and reduced the Moon’s mass to 70 % lu-nar. The upper panel in Fig. 6 shows the TTV-TDV diagram af-ter a single orbit of the outermost moon. As this system does notinclude a low-integer MMR, the TTV-TDV figure is very com-plex, involving an hourglass shape main figure with two minuteloops around the origin (Fig. 6, center panel). As expected, theresulting TTV-TDV diagram after multiple moon orbits exhibitsa smearing e ff ect (Fig. 6, bottom panel). In this particular ex-ample system, the overall shape of the TTV-TDV figure actuallyremains intact, but much more substantial smearing may occurin other systems.
3. Blind retrieval of single and multiple moonsystems
Next, we want to know whether the above-mentioned TTV-TDVpatterns can actually be detected in a realistic dataset. Above all,observations only deliver a limited amount of TTV-TDV mea-surements per candidate system and white noise and read noiseintroduce uncertainties. To which extent does real TTV-TDV data enable the detection of exomoons, and permit us to discernsingle from multiple exomoon systems?A χ test can determine the best-fitting model if the modelparameters are known. In a realistic dataset, however, the numberof parameters is generally unknown since the number of moonsis unknown. Hence, we performed a Bayesian test, in which twoof us (MH, RH) prepared datasets that were then passed to theother coauthors (BP, DA) for analyses. The preparation teamkept the number of moons in the data secret but constrained itto be either 0, 1, or 2. It was agreed that any moons would bein circular, prograde, and stable orbits, that is, beyond the Rocheradius but within 0 . R H (Domingos et al. 2006). Our code al-low us to simulate and retrieve eccentric moons as well, but fordemonstration purpose we restrict ourselves to circular orbits inthis study. Yet, eccentric and inclined orbits would need to beconsidered in a dedicated exomoon survey. We chose to use an example loosely based on KOI-868, a sys-tem searched for exomoons by Kipping et al. (2015). We kept P p =
236 d, M ⋆ = . ± . M ⊙ , and R ⋆ = . ± . R ⊙ ,and the measured timing errors of ∼ . . M Jup planet would be too small.Hence, we assumed an Earth-mass planet ( M p = M ⊕ ± . − (Fressin et al. 2012). Our hypothesized planetwould have a much smaller radius than KOI-868 b, potentiallyresulting in less accurate timing measurements than we assumed.This neglect of an additional source of noise is still reasonable,since there is a range of small Kepler planets with very precisetiming measurements, for example, Kepler-80 d with a radius of R p = . ± . R ⊕ and timing uncertainties ∼ . In our first example, we assumed a heavy moon (0 . M ⊕ ) in astable, circular Moon-wide orbit ( a s = . × km or 34 %the Hill radius of this planet, R H ). This high a mass yields amoon-to-planet mass ratio that is still slightly smaller than thatof the Pluto-Charon system. Its sidereal period is 26.2 d, slightlyshorter than the 27.3 d period of the Moon. The resulting TTVand TDV amplitudes are 23 . ± . Kepler primary observations. The data were simulated using our N -body integrator, then stroboscopically spread over the TTV-TDV diagram (see Sect. 2.1) and randomly moved in the TTV-TDV plane assuming Gaussian noise. The retrieval team treated M ⋆ , R ⋆ , M p , and a B as fixed and only propagated the errors of R ⋆ and M p , which is reasonable for a system that has been char-acterized spectroscopically. Our preliminary simulations showed that multiple exomoon re-trieval based on
Kepler -style data quality only works in extreme
Article number, page 4 of 19ené Heller et al.: Predictable patterns in planetary TTVs and TDVs due to exomoons cases with moons much larger than those known from the so-lar system or predicted by moon formation theories. Hence, weassumed an up-scaled space telescope with a theoretical instru-ment achieving ten times the photon count rate of
Kepler , cor-responding to a mirror diameter of ≈ . Hubble Space Telescope (2.5 m),but smaller than the
James Webb Space telescope (6.5 m). Ne-glecting other noise sources such as stellar jitter, our hypothet-ical telescope reaches ten times higher cadence than
Kepler atthe same noise level. We also found that seven data points donot sample the TTV-TDV figure of the planet su ffi ciently to re-veal the second moon. Rather more than thrice this amount isnecessary. We thus simulated 25 data points, corresponding to15 years of observations. This setup is beyond the technologicalcapacities that will be available within the next decade or so, andour investigations of two-moon systems are meant to yield in-sights into the principal methodology of multiple moon retrievalusing TTV-TDV diagrams.Our N -body simulations suggested that the second mooncannot occupy a stable, inner orbit if the more massive, outermoon has a mass & . M p , partly owing to the fact that bothmoons must orbit within 0 . R H . We neglect exotic stable config-urations such as the Klemperer rosette (Klemperer 1962) as theyare very sensitive to perturbations. Instead, we chose masses of0 . M ⊕ for the inner moon and 0 . M ⊕ for the outer moon,which is about the mass of the Moon. We set the semimajor axisof the outer moon ( a s2 ) to 1 . × km, half the value of theEarth’s moon. The inner moon was placed in a 1:2 MMR witha semimajor axis ( a s1 ) of 1 . × km. The masses of the innerand outer satellites are referred to as M s1 and M s2 , respectively. In order to robustly select the model that best describes the givendata, we employ Bayesian model selection (Sivia & Skilling2006; Knuth et al. 2015), which relies on the ability to computethe Bayesian evidence Z = Z θ π ( θ ) × L ( θ ) d θ. (4)Here, π ( θ ) represents the prior probabilities for model parame-ters θ and quantifies any knowledge of a system prior to analyz-ing data. L ( θ ) represents the likelihood function, which dependson the sum of the square di ff erences between the recorded dataand the forward model.The evidence is an ideal measure of comparison betweencompeting models as it is a marginalization over all model pa-rameters. As such, it naturally weighs the favorability of a modelto describe the data against the volume of the parameter space ofthat model and thus aims to avoid the overfitting of data. It canbe shown that the ratio of the posterior probabilities for two com-peting models with equal prior probabilities is equal to the ratioof the Bayesian evidence for each model. Therefore the modelwith the largest evidence value is considered to be more favor-able to explain the data (Knuth et al. 2015).We utilize the MultiNest algorithm (Feroz & Hobson 2008;Feroz et al. 2009, 2011, 2013) to compute log-evidences used inthe model selection process, and posterior samples used for ob-taining summary statistics for all model parameters. MultiNestis a variant on the Nested Sampling algorithm (Skilling 2006)and is e ffi cient for sampling within many dimensional spacesthat may or may not contain degeneracies. Nested sampling al-gorithms are becoming increasingly useful in exoplanet science, Table 1.
Normalized Bayesian evidence for our simulated datasets asper Eq. (6). no. of moons in model 1-moon dataset 2-moon dataset0 0 % 0 %1 57.2 % 0.00004 %2 42.8 % 99.99996 %
Table 2.
Specification and blind-fitting results for the one-moon system.
Parameter True Fitted M s1 . M ⊕ . ± . M ⊕ a s1 ,
399 km 413 , ± ,
450 km
Table 3.
Specification and blind-fitting results for the two-moon system.
Parameter True Fitted M s1 . M ⊕ . M ⊕ ± . M ⊕ M s2 . M ⊕ . M ⊕ ± . M ⊕ a s1 ,
000 km 131 , ± ,
560 km a s2 ,
000 km 183 , ± ,
000 kme.g. for the analysis of transit photometry (Kipping et al. 2012;Placek et al. 2014, 2015) or for the retrieval of exoplanetary at-mospheres from transit spectroscopy (Benneke & Seager 2013;Waldmann et al. 2015).As inputs, MultiNest requires prior probabilities for allmodel parameters as well as the log-likelihood function. Priorsfor all model parameters were chosen to be uniform between rea-sonable ranges. For the single moon case, we explored a range ofpossible moon masses and semimajor axes with 0 ≥ M s ≥ . M ⊕ and 10 km ≥ a s ≥ . × km. The lower and upper limits for a s correspond to the Roche lobe and to 0 . R H , respectively. Forthe two-moon scenario, the prior probabilities for the orbital dis-tances were kept the same but the moon masses were taken torange within 0 ≥ M s ≥ . M ⊕ . The upper mass limit has beendetermined by an N -body stability analysis.Since both TDV and TTV signals must be fit simultaneously,a nearest neighbor approach was adopted for the log-likelihoodfunction. For each data point, the nearest neighbor model pointwas selected, and the log-likelihood computed for that pair. Thisapproach neglects the temporal information contained in theTTV-TDV measurements or, in our case, simulations. A fit ofthe data in the TTV-TDV plane is similar to a phase-folding tech-nique as frequently used in radial velocity or transit searches forexoplanets. A fully comprehensive data fit would test all the pos-sible numbers of moon orbits during each circumstellar orbit ( n ,see Sect. 2.1), which would dramatically increase the CPU de-mands. Assuming Gaussian noise for both TTV and TDV sig-nals, the form of the log-likelihood was taken to belog L = − σ N X i = (cid:0) M TTV , i − D TTV , i (cid:1) − σ N X i = (cid:0) M TDV , i − D TDV , i (cid:1) (5)where M TTV , i and M TDV , i are the TTV and TDV coordinates ofthe nearest model points to the i th data points, D TTV , i and D TDV , i , Article number, page 5 of 19 & Aproofs: manuscript no. ms N is the number of data points, and σ is the signal variance.Ultimately, we normalize the evidence for the i th model as per(normalized evidence) i = Z i Z + Z + Z (6)to estimate the probability that a model correctly describes thedata. The left-hand side of Eq. (6) would change if we were toinvestigate models with more than two moons.
4. Results
The results of our log-evidence calculations for the blind exo-moon retrieval are shwon in Table 1. In the case where a one-moon system had been prepared for retrieval, the Bayesian log-evidences are log Z = − . ± .
02, log Z = − . ± .
10, andlog Z = − . ± .
15, indicating a slight preference of the one-moon model over both the two-moon and the zero-moon cases.The best fits to the data are shown in Fig. 7. While the di ff erencein log-evidence between the one- and two-moon models is small,our retrieval shows that an interpretation with moon, be it a one-or a multiple system, is strongly favored over the planet-only hy-pothesis. A moon with zero mass is excluded at high confidence.In the two-moon case, the log-evidences for the zero-, one-,and two-moon models are log Z = − . ± .
02, log Z = − . ± .
37, and log Z = − . ± .
47, respectively. The fitsto the data are shwon in Fig. 8. In this case, the two-moon modelis highly favored over both the zero- and one-moon models.Once the most likely number of moons in the system hasbeen determined, we were interested in the parameter estimatesfor the moons. For the one-moon case, our estimates are listedin Table 2. The one-moon model, which has the highest log-evidence, predicts M s1 = . ± . M ⊕ and a s1 = , ± ,
450 km. The relatively large uncertainties indicate an M – a s1 degeneracy. Figure 9 shows the log-likelihood contours ofthe one-moon model applied to this dataset. Indeed, the curvedprobability plateau in the lower right corner of the plot suggests adegeneracy between the moon mass and orbital distance, whichmay be exacerbated by the small amplitude of the TDV signal.The estimates for the two-moon case are shown in Table 3.The favored model in terms of log-evidence is indeed the two-moon model, predicting M s1 = . ± . M ⊕ and a s1 = , ± ,
560 km. The outer moon is predicted to have M s2 = . ± . M ⊕ and a s2 = , ± ,
000 km.Hence, both pairs of parameters are in good agreement with thetrue values. The log-likelihood landscape is plotted in Fig. 10.The landscape referring to the inner moon (left panel) shows apeak around the true value ( M s1 , a s1 ) = (0 . M ⊕ , ,
000 km)rather than the above-mentioned plateau in the one-moon case,implying that the parameters of the inner moon are well con-strained. The mass of the outer moon is tightly constrained aswell, but a s2 has large uncertainties visualized by the high-log-likelihood ridge in the right panel.
5. Discussion
Our exomoon search algorithm uses an N -body simulator to gen-erate TTV-TDV diagrams based on a s , R s , and optionally e s and i s ). In search of one-moon systems, however, analytic solutionsexist. In particular, the TTV-TDV ellipse of planets hosting oneexomoon in a circular orbit can be analytically calculated usingthe TTV and TDV amplitudes via Eqs. (3) in Kipping (2009a) and (C7) in Kipping (2009b) and multiplying those RMS valuesby a factor √
2. This would dramatically decrease the computa-tion times, but computing times are short for our limited param-eter range in this example study anyway.TTVs can also be caused by additional planets (Agolet al. 2005), but the additional presence or absence of TDVsputs strong constraints on the planet versus moon hypothesis(Nesvorný et al. 2012). Planet-induced TDVs have only beenseen as a result of an apsidal precession of eccentric planetscaused by perturbations from another planet and in circumbi-nary planets (Doyle et al. 2011). Moreover, an outer planetaryperturber causes TTV on a timescale that is longer than the or-bital period of the planet under consideration (thereby causingsine-like TTV signals), whereas perturbations from a moon acton a timescale much shorter than the orbital period of the planet(thereby causing noise-like signals).An important limitation of our method is in the knowledgeof M p , which is often poorly constrained for transiting planets. Itcan be measured using TTVs caused by other planets, but a TTV-TDV search aiming at exomoons would try to avoid TTV signalsfrom other planets. Alternatively, stellar radial velocity measure-ments can reveal the total mass of the planet–moon system. Ifthe moon(s) were su ffi ciently lightweight it would be possibleto approximate the mass of the planet with the mass of the sys-tem. Joint mass-radius measurements have now been obtainedfor several hundred exoplanets. If the mass of the planet remainsunknown, then TTV-TDV diagrams can only give an estimate of M s / M p in one-moon systems. As an additional caveat, exomoonsearches might need to consider the drift of the pericenters ofthe moons (Sect. 2.2.3). This would involve two more parame-ters beyond those used in our procedure: the initial orientationof the arguments of periapses and the drift rate. This might evenresult in longterm transit shape variations for su ffi ciently largemoons on eccentric and / or inclined orbits. Finally, another im-portant constraint on the applicability of our method is the im-plicit assumption that the transit duration is substantially shorterthan the orbital period of the moon. For an analytical descriptionsee constraints α
3) (regarding TTV) and α
7) (regarding TDV) inKipping (2011b).TTV and TDV e ff ects in the planetary transits of a star–planet–moon(s) system are caused by Newtonian dynamics. Thisdynamical origin enables measurements of the satellite massesand semimajor axes but not of their physical radii. The lattercan be obtained from the direct photometric transit signature ofthe moons. In these cases, the satellite densities can be derived.Direct moon transits could potentially be observed in individ-ual planet-moon transits or in phase-folded transit light curves.A combination of TTV-TDV diagrams with any of these tech-niques thus o ff ers the possibility of deriving density estimatesfor exomoons. TTV-TDV diagrams and the orbital sampling ef-fect are sensitive to multiple moon systems, so they would be anatural combination for multiple moon candidate systems.We also point out an exciting, though extremely challenging,opportunity of studying exomoons on timescales shorter than theorbital period of a moon around a planet. From Earth, it is onlypossible to measure the angle ρ (or the numerical remainder r of P B / P s ) in the TTV-TDV diagram (see Fig. 1). However, KatjaPoppenhäger (private communication 2015) pointed out that asecond telescope at a su ffi ciently di ff erent angle could observetransits of a given planet-moon system at di ff erent orbital phasesof the moon. Hence, the orbit of the moon could be sampledon timescales smaller than P s . Then n could be determined, en-abling a complete measurement of ρ , P s , and n . Article number, page 6 of 19ené Heller et al.: Predictable patterns in planetary TTVs and TDVs due to exomoons
Beyond the star–planet–moon systems investigated in thispaper, our concept is applicable to eclipsing binary stars withplanets in satellite-type (S-type) orbits (Rabl & Dvorak 1988).Here, the TTV-TDV diagram of the exoplanet host star wouldprovide evidence of the planetary system around it.
6. Conclusions
We identify new predictable features in the TTVs and TDVs ofexoplanets with moons. First, in exoplanet systems with a sin-gle moon on a circular orbit, the remainder of the planet–moonorbital period (0 ≤ r ≤
1) appears as a constant angle ( ρ ) inthe TTV-TDV diagram between consecutive transits. The pre-dictability of ρ determines the relative position of the moon tothe planet during transits. This is helpful for targeted transit ob-servations of the system to measure the largest possible planet-moon deflections (at maximum TTV values) or observe planet-moon eclipses (at maximum TDV values).Second, exoplanets with multiple moons in an orbital MMRexhibit loops in their TTV-TDV diagrams. These loops corre-spond the reversal of the tangential motion of the planet duringits orbital motion in the planet–moon system. We find that thelargest integer in the MMR of an exomoon system determinesthe number of loops in the corresponding TTV-TDV diagram,for example, five loops in a 5:3 MMR. The lowest number isequal to the number of orbits that need to be completed by theoutermost moon to produce a closed TTV-TDV figure, for exam-ple, three in a 5:3 MMR.Planetary TTV-TDV figures caused by exomoons are cre-ated by dynamical e ff ects. As such, they are methodologicallyindependent from purely photometric methods such as the di-rect transit signature of moons in individual transits or in phase-folded light curves. Our novel approach can thus be used to inde-pendently confirm exomoon candidates detected by their own di-rect transits, by planet–moon mutual eclipses during stellar tran-sits, the scatter peak method, or the orbital sampling e ff ect.We performed blind retrievals of two hypothetical exomoonsystems from simulated planetary TTV-TDV. We find that anEarth-sized planet with a large moon (10 % of the planetarymass, akin to the Pluto-Charon system) around an M dwarf starexhibits TTVs / TDVs that could be detectable in the four yearsof archival data from the
Kepler primary mission. The oddsof detecting exomoon-induced planetary TTVs and TDVs with
PLATO are comparable.
PLATO will observe about ten times asmany stars as
Kepler in total and its targets will be significantlybrighter (4 ≤ m V ≤ Kepler might, however, be compensated by
PLATO ’s shorter ob-servations of its two long-monitoring fields (two to three yearscompared to four years of the
Kepler primary mission; Raueret al. 2014). Our blind retrieval of a multiple moon test sys-tem shows that the TTV-TDV diagram method works in prin-ciple, from a technical perspective. In reality, however, multiplemoons are much harder to detect, requiring transit observationsover several years by a space-based photometer with a collectingarea slightly larger than that of the
Hubble Space Telescope .As modern space-based exoplanet missions have duty cy-cles of a few years at most, exomoon detections via planetaryTTVs and TDVs will be most promising if data from di ff erentfacilities can be combined into long-term datasets. Follow-upobservations of planets detected with the Kepler primary mis-sion might be attractive for the short term with four years ofdata being readily available. However,
Kepler stars are usuallyfaint. Long-term datasets should be obtained for exoplanets tran-siting bright stars to maximize the odds of an exomoon detec- tion. A compelling opportunity will be a combination of
TESS , CHEOPS , and
PLATO data.
TESS (mid-2017 to mid-2019) willbe an all-sky survey focusing on exoplanets transiting brightstars.
CHEOPS (late 2017 to mid-2021) will observe stars knownto host planets or planet candidates. Twenty percent of its sci-ence observation time will be available for open-time scienceprograms, thereby o ff ering a unique bridge between TESS and
PLATO (2024 to 2030). The key challenge will be in the precisesynchronization of those datasets over decades while the timinge ff ects occur on a timescale of minutes. Note added in proof . After acceptance of this paper, the au-thors learned that Montalto et al. (2012) used a TTV-TDV di-agram to search for exomoons around WASP-3b. Awiphan &Kerins (2013) studied the correlation between the squares of theTTV and the TDV amplitudes of exoplanets with one moon.
Article number, page 7 of 19 & Aproofs: manuscript no. ms
TDV TTVTDV TTV ! TDV TTV !! TDV TTVTDV TTV N obs = 3 N obs = 4 N obs = 5 N obs = 13 N obs = ! (a)(b)(c)(d)(e) Fig. 1.
Evolution of a TTV-TDV diagram of an exoplanet with 1 moon.TTV and TDV amplitudes are normalized to yield a circular figure. -3 -2 -1 0 1 2 3transit timing variation [minutes]-0.4-0.3-0.2-0.10.00.10.20.30.4 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] Fig. 2.
TTV-TDV diagram for the Earth-Moon system transiting theSun, but with the orbital eccentricity of the Moon increased to 0.25.Article number, page 8 of 19ené Heller et al.: Predictable patterns in planetary TTVs and TDVs due to exomoons ✁ ✂ ✄ ☎ ✆ ✝ (cid:0)✟ ✠✡ ✂ (cid:0) ✠ ✄ ☛ ☞ ✌ ☞ ✌✍ ✎ ✍ ✎ ✏ ✑✏ ✑ ✎ ✑✎ ✑✑ ✑ ✒ ✑✓ ☞ ✲ (cid:0) ✁ (cid:0) ✂ ✲ (cid:0) ✁ (cid:0) ✄ (cid:0) ✁ (cid:0) (cid:0) (cid:0) ✁ (cid:0) ✄ (cid:0) ✁ (cid:0) ✂t ☎ ✆ ✝ ✞ ✟ t t ✟ ✠ ✟ ✝ ✡ ✈ ✆ ☎ ✟ ✆ t ✟ ❛ ✝ ❬ ✠ ✟ ✝ ✉ t ☛ ✞ ☞✲ (cid:0) ✁ ✌ ✍✲ (cid:0) ✁ ✌ (cid:0)✲ (cid:0) ✁ (cid:0) ✍(cid:0) ✁ (cid:0) (cid:0)(cid:0) ✁ (cid:0) ✍(cid:0) ✁ ✌ (cid:0)(cid:0) ✁ ✌ ✍ ✎✏ ✑ ✒ ✓ ✔ ✎ ❞✕ ✏ ✑ ✎ ✔ ✖ ✒ ✗ ✑ ✏ ✔ ✑ ✎ ✔ ✖ ✒ ✘ ✙ ✔ ✒✕ ✎ ✚ ✓ ✛ ✍ (cid:0) ✼ ✌✂ (cid:0)✻ (cid:0) ✄ ✍(cid:0) ✵ ✌ (cid:0) (cid:0) ✌ ✶ Fig. 3.
Jupiter-Io-Europa system in a 2:1 MMR. Numbers denote the percental orbital phase of the outer moon, Europa.
Left : Top-down perspectiveon our two-dimensional N -body simulation. The senses of orbital motion of the moons are indicated with curved arrows along their orbits. Right :TDV-TTV diagram for the same system. The progression of the numerical TTV-TDV measurements is clockwise. Article number, page 9 of 19 & Aproofs: manuscript no. ms -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06transit timing variation [minutes]-0.20-0.15-0.10-0.050.000.050.100.150.20 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] Fig. 4.
TTV-TDV diagrams for a Jupiter-like planet with 2 moons akinto Io and Europa, but in di ff erent MMRs. A 2:1 MMR (top) produces 2ellipses, a 3:1 MMR (center) 3, and a 4:1 MMR (bottom) 4.Article number, page 10 of 19ené Heller et al.: Predictable patterns in planetary TTVs and TDVs due to exomoons -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4transit timing variation [minutes]-1.0-0.50.00.51.0 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4transit timing variation [minutes]-1.0-0.50.00.51.0 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4transit timing variation [minutes]-1.0-0.50.00.51.0 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4transit timing variation [minutes]-1.0-0.50.00.51.0 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] Fig. 5.
TTV-TDV diagrams of a Jupiter-like planet with 2 moons. Top panels assume a 2:1 MMR akin to the Io-Europa resonance; bottom panelsassume a 4:1 MMR akin to the Io-Ganymede resonance. Models represented by black solid lines assume that both moons are as massive asGanymede. In the left panels, blue dotted lines assume the inner moon is twice as massive ( M s1 = M s2 = M Gan ), and red dashed lines assumethe inner moon is thrice as massive. In the right panels, blue assumes the outer moon is twice as massive ( M s2 = M s1 ) and red assumes the outermoon is thrice as massive. The inner loop becomes invisibly small for low M s1 / M s2 ratios in the top right panel. Article number, page 11 of 19 & Aproofs: manuscript no. ms -4 -2 0 2 4transit timing variation [minutes]-0.6-0.4-0.20.00.20.40.6 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] -0.04 -0.02 0.00 0.02 0.04transit timing variation [minutes]-0.04-0.020.000.020.04 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] -4 -2 0 2 4transit timing variation [minutes]-0.6-0.4-0.20.00.20.40.6 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] Fig. 6.
Top : TTV-TDV diagram the Earth-Moon-like system involvinga second satellite at half the distance of the Moon to Earth with 70 % ofthe lunar mass.
Center : Zoom into the center region showing the turningpoints.
Bottom : After sampling 10 orbits, a smearing e ff ect becomesvisible as the 2 moons are not in a MMR. Exomoon retrieval from such amore complex TTV-TDV figure needs to take into account the evolutionof the system. (cid:0) !" (cid:0) (cid:0) $" " $" (cid:0) ! (cid:0) (cid:0) $"$ & ( ) * + ,- . / (cid:0) *55,(*53/6.75 (cid:0) *55,(*53/6 Fig. 7.
One- and two-moon fits to the model generated data of onemoon. (cid:0) ! (cid:0) " (cid:0) (cid:0) (cid:0) $%&$$%& ’ ) * + , -. / (cid:0) +66-)+6407/86 (cid:0) +66-)+6407 Fig. 8.
One- and two-moon fits to the model generated data of twomoons.Article number, page 12 of 19ené Heller et al.: Predictable patterns in planetary TTVs and TDVs due to exomoons M !" M (cid:0) % a ! " $ & ’ % ( ()(* ()" ()"* ()+()*"")*++)*,,)*--)** ✁ *(( ✁ -*( ✁ -(( ✁ ,*( ✁ ,(( ✁ +*( ✁ +(( ✁ "*( ✁ "(( ✁ *(."( * /01 L Fig. 9.
Log-likelihood landscape of the one-moon model applied to thesimulated one-moon data. Lighter areas indicate regions of high prob-ability and the red cross indicates the true parameter values. One canclearly see a degeneracy between a s1 and M s1 in the form of a curvedplateau in the bottom right portion of the plot. Appendix A: Higher-order MMRs
In the following, we present a gallery of TTV-TDV diagrams forplanets with up to five moons in a chain of MMRs. This collec-tion is complete in terms of the possible MMRs. In each case, aJupiter-mass planet around a Sun-like star is assumed. All moonmasses are equal to that of Ganymede. The innermost satellite isplaced in a circular orbit at an Io-like semimajor axis ( a s1 ) withan orbital mean motion n s1 around the planet. The outer moons,with orbital mean motions n s i (where 2 ≤ i ≤ a s i = a s1 ( n s1 / n s i ) / corresponding tothe respective MMR. Acknowledgements.
References
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Log-likelihood landscape of the two-moon model applied to simulated two-moon data. Lighter areas indicate regions of better fits. Thered cross indicates the location of the simulated system to be retrieved. The parameters corresponding to the inner moon are shown in the leftpanel and those of the outer moon are shown in the right panel. The vertical ridge of high log-likelihood in the right panel indicates that the orbitaldistance of the outer moon ( a s2 ) is not as well constrained as that of the inner moon.Article number, page 14 of 19ené Heller et al.: Predictable patterns in planetary TTVs and TDVs due to exomoons -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20transit timing variation [minutes]-0.4-0.20.00.20.4 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] Fig. A.1.
TTV-TDV diagrams of planets with two moons in MMRs. Article number, page 15 of 19 & Aproofs: manuscript no. ms -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15transit timing variation [minutes]-0.6-0.4-0.20.00.20.40.6 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] Fig. A.2.
TTV-TDV diagrams of planets with three moons in MMRs.Article number, page 16 of 19ené Heller et al.: Predictable patterns in planetary TTVs and TDVs due to exomoons -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15transit timing variation [minutes]-0.6-0.4-0.20.00.20.40.6 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] Fig. A.2. (continued) Article number, page 17 of 19 & Aproofs: manuscript no. ms -0.10 -0.05 0.00 0.05 0.10transit timing variation [minutes]-0.8-0.6-0.4-0.20.00.20.40.60.8 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] Fig. A.3.
TTV-TDV diagrams of planets with four moons in MMRs.Article number, page 18 of 19ené Heller et al.: Predictable patterns in planetary TTVs and TDVs due to exomoons -0.10 -0.05 0.00 0.05 0.10transit timing variation [minutes]-0.8-0.6-0.4-0.20.00.20.40.60.8 tr a n s i t du r a t i o n v a r i a t i o n [ m i nu t e s ] Fig. A.4.
TTV-TDV diagram of a planets with five moons in MMRs. Article number, page 19 of 19 .5. TRANSITS OF EXTRASOLAR MOONS AROUND LUMINOUS GIANT PLANETS (Heller2016) 133 r X i v : . [ a s t r o - ph . E P ] M a r Astronomy & Astrophysicsmanuscript no. ms c (cid:13)
ESO 2016March 2, 2016
Transits of extrasolar moons around luminous giant planets(Research Note)
R. Heller Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany; [email protected] October 2, 2015; Accepted February 1, 2016
ABSTRACT
Beyond Earth-like planets, moons can be habitable, too. No exomoons have been securely detected, but they could be extremelyabundant. Young Jovian planets can be as hot as late M stars, with e ff ective temperatures of up to 2000 K. Transits of their moonsmight be detectable in their infrared photometric light curves if the planets are su ffi ciently separated ( &
10 AU) from the stars to bedirectly imaged. The moons will be heated by radiation from their young planets and potentially by tidal friction. Although stellarillumination will be weak beyond 5 AU, these alternative energy sources could liquify surface water on exomoons for hundreds ofMyr. A Mars-mass H O-rich moon around β Pic b would have a transit depth of 1 . × − , in reach of near-future technology. Key words.
Astrobiology – Methods: observational – Techniques: photometric – Eclipses – Planets and satellites: detection –Infrared: planetary systems
1. Introduction
Since the discovery of a planet transiting its host star (Charbon-neau et al. 2000), thousands of exoplanets and candidates havebeen detected, mostly by NASA’s
Kepler space telescope (Roweet al. 2014). Planets are natural places to look for extrasolar life,but moons around exoplanets are now coming more and moreinto focus as potential habitats (Reynolds et al. 1987; Williamset al. 1997; Scharf 2006; Heller et al. 2014; Lammer et al. 2014).Key challenges in determining whether an exomoon is habitableor even inhabited are in the extreme observational accuraciesrequired for both a detection, e.g., via planetary transit timingvariations plus transit duration variations (Sartoretti & Schneider1999; Kipping 2009), and follow-up characterization, e.g., bytransit spectroscopy (Kaltenegger 2010) or infrared (IR) spectralanalyses of resolved planet-moon systems (Heller & Albrecht2014; Agol et al. 2015). New extremely large ground-based tele-scopes with unprecedented IR capacities (
GTM , 1st light 2021;
TMT , 1st light 2022;
E-ELT , 1st light 2024) could achieve dataqualities required for exomoon detections (Quanz et al. 2015).Here I investigate the possibility of detecting exomoons tran-siting their young, luminous host planets. These planets need tobe su ffi ciently far away from their stars ( &
10 AU) to be directlyimaged. About two dozen of them have been discovered aroundany stellar spectral type from A to M stars, most of them attens or hundreds of AU from their star. The young super-Jovianplanet β Pic b (11 ± ff ec-tive temperature is about 1 700 K (Baudino et al. 2014), and con-tamination from its host star is su ffi ciently low to allow for directIR spectroscopy with CRIRES at the
VLT (Snellen et al. 2014).Exomoons as heavy as a few Mars masses have been predictedto form around such super-Jovian planets (Canup & Ward 2006;Heller et al. 2014), and they might be as large as 0.7 Earth radiifor wet / icy composition (Heller & Pudritz 2015b). The transitdepth of such a moon (10 − ) would be more than an order of magnitude larger than that of an Earth-sized planet around a Sun-like star. Photometric accuracies down to 1 % have now beenachieved in the IR using HST (Zhou et al. 2015). Hence, transitsof large exomoons around young giant planets are a compellingnew possibility for detecting extrasolar moons.
2. Star-planet versus planet-moon transits
Planets and moons form on di ff erent spatiotemporal scales. Wethus expect that geometric transit probabilities, transit frequen-cies, and transit depths di ff er between planets (transiting stars)and moons (transiting planets). The H O ice line, beyond whichrunaway accretion triggers formation of giant planets in the pro-toplanetary disk (Lissauer 1987; Kretke & Lin 2007), was atabout 2.7 AU from the Sun during the formation of the local gi-ant planets (Hayashi 1981). In comparison, the circum-JovianH O ice line, beyond which some of the most massive moonsin the solar system formed, was anywhere between the orbitsof rocky Europa and icy Ganymede (Pollack & Reynolds 1974)at 10 and 15 Jupiter radii ( R J ), respectively. The ice line radius( r ice ), which is normalized to the physical radius of the host ob-ject ( R ), was 2 . / . R J ≈
480 times larger in the solar ac-cretion disk than in the Jovian disk. With r ice depending on R andthe e ff ective temperature ( T e ff ) of the host object as per r ice ∝ q R T ff , (1)we understand this relation by comparing the properties of ayoung Sun-like star ( R = R ⊙ [solar radius], T e ff = R = R J , T e ff = √ × =
250 times wider H O ice line forthe star, neglecting the complex opacity variations in both cir-cumstellar and circumplanetary disks (Bitsch et al. 2015).
Article number, page 1 of 4 & Aproofs: manuscript no. ms
The mean geometric transit probabilities ( ¯ P ) of the mostmassive moons should thus be larger than ¯ P of the most massiveplanets. Because of their shorter orbital periods, planet-moontransits of big moons should also occur more often than stel-lar transits of giant planets. In other words, big moons shouldexhibit higher transit frequencies ( ¯ f ) around planets than giantplanets around stars, on average. However, Eq. (1) does not giveus a direct clue as to the mean relative transit depths ( ¯ D ). I exclude moons around the local terrestrial planets (most no-tably the Earth’s moon) and focus on large moons around the so-lar system giant planets to construct an empirical moon samplerepresentative of moons forming in the accretion disks aroundgiant planets. This family of natural satellites has been suggestedto follow a universal formation law (Canup & Ward 2006). Weneed to keep in mind, though, that these planets orbit the outer re-gions of the solar system, where stellar illumination is negligiblefor moon formation (Heller & Pudritz 2015a). However, manygiant exoplanets are found in extremely short-period orbits (the“hot Jupiters”); and stellar radial velocity (RV) measurementssuggest that giant planets around Sun-like stars can migrate to1 AU, where we observe them today. Moreover, at least one largemoon, Triton, has probably been formed through a capture ratherthan in-situ accretion (Agnor & Hamilton 2006).For the planet sample, I first use all RV planets confirmed asof the day of writing. I exclude transiting exoplanets from myanalysis as these objects are subject to detection biases. RV ob-servations are also heavily biased (Cumming 2004), but we knowthat they are most sensitive to close-in planets because of the de-creasing RV amplitude and longer orbital periods in wider orbits.Thus, planets in wide orbits are statistically underrepresented inthe RV planet sample, but these planets are not equally underrep-resented as planets in transit surveys. Transit surveys also preferclose-in planets, as the photometric signal-to-noise ratio scalesas ∝ n tr ( n tr the number of transits, Howard et al. 2012).Mean values of ¯ P (similarly of ¯ f and ¯ D ) are calculated as¯ P = ( P N p i P i ) / N p , where P i is the geometric transit probability ofeach individual RV planet and N p is the total number of RV plan-ets. Standard deviations are measured by identifying two binsaround ¯ P : one in negative and one in positive direction, each ofwhich contains
68 % =
34 % of all the planets in the distribu-tion. The widths of these two bins are equivalent to asymmetric1 σ intervals of a skewed normal distribution.The geometric transit probability P = R / a , with a as the or-bital semimajor axis of the companion. For the RV planets, ¯ P = .
028 ( + . , − . The ¯ P value of an unbiased exoplanetpopulation would be smaller as it would contain more long-period planets. On the contrary, no additional detection of a largesolar system moon is expected, and ¯ P = .
114 ( + . , − . f ≈ / (2 π ) p ( GM ) / a , where G isNewton’s gravitational constant and M the mass of the centralobject, assuming that M is much larger than the mass of the com-panion. For the RV planets, ¯ f = .
040 ( + . , − . / day. The value of ¯ f , which would be corrected for detection biases,would be smaller with long-period planets having lower frequen- Information for both R and a was given for 398 RV planets listed on as of 1 October 2015. The orbital period was given for 612 RV planets
Fig. 1.
Dependence of (a) the mean geometric transit probability, (b)mean transit frequency, and (c) mean transit depth of the largest solarsystem moons on the number of moons considered. The abscissa refersto the ranking ( N s ) of the moon among the largest solar system moons. cies. For comparison, ¯ f of the 20 largest moons of the solar sys-tem giant planets is 0 .
170 ( + . , − . / day.Stellar limb darkening, star spots, partial transits, etc. aside,the maximum transit depth depends only on R and on the ra-dius of the transiting object ( r ) as per D = ( r / R ) . In my cal-culations of D , I resort to the transiting exoplanet data because r is not known for most RV planets. For the transiting plan-ets, ¯ D = .
00 ( + . , − . × − . It is not clear whethera debiased ¯ D value would be larger or smaller than that. Thisdepends on whether long-period planets usually have larger orsmaller radii than those used for this analysis. For comparison,¯ D of the 20 largest moons of the solar system giant planets is6 .
319 ( + . , − . × − .The choice of the 20 largest local moons for comparison withthe exoplanet data is arbitrary and motivated by the number ofdigiti manus. Figure 1(a)-(c) shows ¯ P , ¯ f , and ¯ D as functions ofthe number of natural satellites ( N s ) taken into account. Solidlines denote mean values, shaded fields standard deviations. Thefirst moon is the largest moon, Ganymede, the second moon Ti-tan, etc., and the 20th moon Hyperion. The trends toward highertransit probabilities (a), higher transit frequencies (b) and lowertransit depths (c) are due to the increasing amount of smallermoons in short-period orbits. The negative slope at moons 19and 20 in (a) and (b) is due to Nereid and Hyperion, which arein wide orbits around Neptune and Saturn, respectively.The key message of this plot is that ¯ P ( N s = f ( N s = D ( N s =
20) used above for comparison with the exoplanetdata serve as adequate approximations for any sample of solarsystem moons that would have been smaller because variationsin those quantities are limited to a factor of a few.
An exomoon hosting planet needs to be su ffi ciently far from itsstar to enable direct imaging and to reduce contamination fromIR stellar reflection. This planet needs to orbit the star beyondseveral AU, where alternative energy sources are required onits putative moons to keep their surfaces habitable, i.e., to pre-vent freezing of H O. This heat could be generated by (1) tides(Reynolds et al. 1987; Scharf 2006; Cassidy et al. 2009; Heller& Barnes 2013), (2) planetary illumination (Heller & Barnes2015), (3) release of primordial heat from the moon’s accretion Information for both r and R was given for 1202 transiting planets.Article number, page 2 of 4. Heller: Transits of extrasolar moons around luminous giant planets (RN) Fig. 2.
Habitable zone (HZ) of a Mars-sized moon around a Jupiter-mass (light green) and a β Pic b-like 11 M J -mass (dark green) planet at & Left:
Both HZs refer to a planet at an age of 10 Myr, where planetary illumination is significant.
Right:
BothHZs refer to a planet at an age of 1 Gyr, where planetary illumination is weak and tidal heating is the dominant energy source on the moon. (Kirk & Stevenson 1987), or (4) radiogenic decay in its mantleand / or core (Mueller & McKinnon 1988). All of these sourcestend to subside on a Myr timescale, but (1) and (2) can competewith stellar illumination over hundreds of Myr in extreme, yetplausible, cases. (3) and (4) usually contribute ≪ − at thesurface even in very early stages. Earth’s globally averaged in-ternal heat flux, for example, is 86 mW m − (Zahnle et al. 2007),which is mostly fed by radiogenic decay in the Earth’s interior.The globally averaged absorption of sunlight on Earth is239 W m − . If an Earth-like object (planet or moon) absorbsmore than 295 W m − , it will enter a runaway greenhouse ef-fect (Kasting 1988). In this state, the atmosphere is opaque inthe IR because of its high water vapor content and high IR opac-ity. The surface of the object heats up beyond 1 000 K until theglobe starts to radiate in the visible. Water cannot be liquid un-der these conditions and the object is uninhabitable by definition.For a Mars-sized moon, which I consider as a reference case, therunaway greenhouse limit is at 266 W m − , following a semian-alytic model (Eq. 4.94 in Pierrehumbert 2010). If the combinedillumination plus tidal heating (or any alternative energy source)exceed this limit, then the moon is uninhabitable.On the other hand, there is a minim energy flux for a moon toprevent a global snowball state that is estimated to be 0.35 timesthe solar illumination absorbed by Earth (83 W m − , Kopparapuet al. 2013), which is only weakly dependent on the object’smass. I use the terms global snowball and runaway greenhouselimits to identify orbits in which a Mars-sized moon would behabitable, which is similar to an approach presented by Heller &Armstrong (2014). A snowball moon might still have subsurfaceoceans, but because of the challenges of even detecting life onthe surface of an exomoon, I here neglect subsurface habitabil-ity. I calculate the total energy flux by adding the stellar visualillumination and the planetary thermal IR radiation absorbed bythe moon to the tidal heating within the moon. Illumination iscomputed as by Heller & Barnes (2015), and I neglect both stel-lar reflected light from the planet and the release of primordialand radiogenic heat. Assuming an albedo of 0.3, similar to theMartian and terrestrial values, the absorbed stellar illuminationby the moon at 5.2 AU from a Sun-like star is 9 W m − . I con-sider a Jupiter-mass planet and a β Pic b-like 11 M J -mass planet,both in two states of planetary evolution; one, in which the sys-tem is 10 Myr old (similar to β Pic b) and the planet is still very hot and inflated; and one, in which the system has evolved to anage of 1 Gyr and the planet is hardly releasing any thermal heat.Planetary luminosities are taken from evolution models(Mordasini 2013), which suggest that the Jupiter-mass planetevolves from R = . R J and T e ff =
536 K to R = . R J and T e ff =
162 K over the said period. For the young 11 M J -massplanet, I take R = . R J (Snellen et al. 2014) and T e ff = R = . R J and T e ff =
480 K.Tidal heating is computed as by Heller & Barnes (2013), follow-ing an earlier work on tidal theory by Hut (1981).Figure 2 shows the HZ for exomoons around a Jupiter-like(dark green areas) and a β Pic b-like planet (light green areas)at ages of 10 Myr (left) and 1 Gyr (right). Planet-moon orbitaleccentricities ( e , abscissa) are plotted versus planet-moon dis-tances (ordinate). In both panels, the HZ around the more mas-sive planet is farther out for any given e , which is mostly dueto the strong dependence of tidal heating on M p . Most notably,the planetary luminosity evolution is almost negligible in the HZaround the 1 M J planet. In both panels, the corresponding darkgreen strip has a width of just 1 R J , details depending on e .The HZ around the young β Pic b, on the other hand, spansfrom 18 R J to 33 R J for e . . ∝ a − , causing a muchsmoother transition from a runaway greenhouse (at 18 R J ) to asnowball state (at 33 R J ) than on a moon that is fed by tidal heat-ing alone, which scales as ∝ a − . In the right panel, planetary il-lumination has vanished and tides have become the principal en-ergy source around an evolved β Pic b. However, the HZ aroundthe evolved β Pic b is still a few times wider (for any given e )than that around a Jupiter-like planet; note the logarithmic scale.
3. Discussion
A major challenge for exomoon transit observations around lu-minous giant planets is in the required photometric accuracy. Al-though
E-ELT will have a collecting area 1 600 times the size of
Kepler ’s, it will have to deal with scintillation. The IR flux ofan extrasolar Jupiter-sized planet is intrinsically low and comeswith substantial white and red noise components. Photometric
Article number, page 3 of 4 & Aproofs: manuscript no. ms exomoon detections will thus depend on whether
E-ELT canachieve photometric accuracies of 10 − with exposures of a fewminutes. Adaptive optics and the availability of a close-by ref-erence object, e.g., a low-mass star with an apparent IR bright-ness similar to the target, will be essential. Alternatively, sys-tems with multiple directly imaged giant planets would provideparticularly advantageous opportunities, both in terms of reli-able flux calibrations and increased transit detection probabili-ties. Systems akin to the HR 8799 four-planet system (Maroiset al. 2008, 2010) will be ideal targets.Transiting moons could also impose RV anomalies on theplanetary IR spectrum, known as the Rossiter-McLaughlin (RM)e ff ect (McLaughlin 1924; Rossiter 1924). The RM reveals thesky-projected angle between the orbital plane of transiting objectand the rotational axis of its host, which has now been measuredfor 87 extrasolar transiting planets . E-ELT might be capable ofRM measurements for large exomoons transiting giant exoplan-ets that can be directly imaged (Heller & Albrecht 2014).Even nontransiting exomoons might be detectable in the IRRV data of young giant planets. The estimated RV 1 σ confi-dence achievable on a β Pic b-like giant exoplanet with a high-resolution ( λ/ ∆ λ ≈
100 000, λ being the wavelength of the ob-served light) near-IR spectrograph mounted to the E-ELT couldbe as low as ≈
70 m s − in reasonable cases (Heller & Al-brecht 2014). The RV amplitude of an Earth-mass exomoon ina Europa-wide ( a ≈ R J ) orbit around a Jupiter-mass planetwould be 43 m s − with a period of 3.7 d. RV detections of super-massive moons, if they exist, would thus barely be possible evenwith E-ELT
IR spectroscopy. We should nevertheless recall that“hot Jupiters” had not been predicted prior to their detection(Mayor & Queloz 1995). In analogy, observational constraintscould still allow for detections of a so-far unpredicted class ofhot super-Ganymedes; they might indeed be hot due to enhancedtidal heating (Peters & Turner 2013) and / or illumination from theyoung planet (Heller & Barnes 2015).Moon eccentricities tend to be tidally eroded to zero in a fewMyr. Perturbations from other moons or from the star can main-tain e > e varying in time, exomoonhabitability could be episodical. Uncertainties in the tidal qualityfactor ( Q ) and the 2nd order Love number ( k ), which scale thetidal heating as per ∝ k / Q , can be up to an order of magnitude.However, because of the strong dependence on a , the HZ limitsin Fig. 2 would be a ff ected by < R J .
4. Conclusions
Large moons around the local giant planets transit their planetsmuch more likely from a randomly chosen geometrical perspec-tive and significantly more often than the RV exoplanets transittheir stars. This is likely a fingerprint of planet and moon forma-tion acting on di ff erent spatiotemporal scales. If the occurrencerate of planets per star is similar to the occurrence rate of largemoons around giant planets (1-10 per system), the probability ofobserving a moon transiting a randomly chosen giant planet is atleast four times higher than the probability of observing a planettransiting a randomly chosen star; planet-moon transits are atleast four times more frequent than star-planets transits; the av-erage transit depth of the transiting exoplanets is five times largerthan the average transit depth of the twenty largest moons around Transits of a moon that formed at the H O ice line around a β Pic b-like planet (20 R J , Heller & Pudritz 2015a) take about 1 hr:13 min. Holt-Rossiter-McLaughlin Encyclopaedia at the local giant planets. However, each solar system giant planethas at least one moon with a transit depth of 10 − . Following thegas-starved accretion disk model for moon formation (Canup &Ward 2006; Heller & Pudritz 2015b), a super-Ganymede aroundthe young super-Jovian exoplanet β Pic b could have a transitdepth of ≈ . × − , which is 18 times as deep as the tran-sit of the Earth around the Sun.Beyond the stellar HZ, more massive planets have widercircumplanetary HZs. The HZ for a Mars-sized moon around β Pic b is between 18 R J and 33 R J for moon eccentricities e . .
1. As the planet ages, the HZ narrows and moves in. After afew hundred Myr, tidal heating may become the moon’s majorenergy source. Owing to the strong dependence of tidal heatingon the planet-moon distance, the HZ around β Pic b narrows to afew R J within 1 Gyr from now, details depending on e . An exo-moon would have to reside in a very specific part of the e - a spaceto be continuously habitable over a Gyr.The high geometric transit probabilities of moons around gi-ant planets, their higher transit frequencies, and the possibility oftransit signals that are one order of magnitude deeper than thatof the Earth around the Sun, make transit observations of moonsaround young giant planets a compelling science case for theupcoming GMT , TMT , and
E-ELT extremely large telescopes.Most intriguingly, these exomoons could orbit their young plan-ets in the circumplanetary HZ and therefore be cradles of life.
Acknowledgements.
This study has been inspired by discussions with RoryBarnes, to whom I express my sincere gratitude. The reports of an anonymousreferee helped to clarify several passages of this paper. I have been supported bythe German Academic Exchange Service, the Institute for Astrophysics Göttin-gen, and the Origins Institute at McMaster University, Canada. This work madeuse of NASA’s ADS Bibliographic Services.
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Article number, page 4 of 4 .6. HOW ECLIPSE TIME VARIATIONS, ECLIPSE DURATION VARIATIONS, AND RADIALVELOCITIES CAN REVEAL S-TYPE PLANETS IN CLOSE ECLIPSING BINARIES (Oshaghet al. 2017) 138
Contribution:RH contributed to the literature research, computed the expected eclipse timing variations (ETVs)and eclipse duration variations (EDVs), wrote the equations into computer code, created Figs. 1, 3,5, 8, and A.1, and contributed to the writing of the manuscript.
NRAS , 1–9 (2016) Preprint 20 December 2016 Compiled using MNRAS L A TEX style file v3.0
How eclipse time variations, eclipse duration variations,and radial velocities can reveal S-type planets in closeeclipsing binaries
M. Oshagh ? , R. Heller † , and S. Dreizler Institut f¨ur Astrophysik, Georg-August-Universit¨at, Friedrich-Hund-Platz 1, 37077 G¨ottingen, Germany Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 G¨ottingen, Germany
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
While about a dozen transiting planets have been found in wide orbits around aninner, close stellar binary (so-called “P-type planets”), no planet has yet been detectedorbiting only one star (a so-called “S-type planet”) in an eclipsing binary. This is de-spite a large number of eclipsing binary systems discovered with the
Kepler telescope.Here we propose a new detection method for these S-type planets, which uses a cor-relation between the stellar radial velocities (RVs), eclipse timing variations (ETVs),and eclipse duration variations (EDVs). We test the capability of this technique bysimulating a realistic benchmark system and demonstrate its detectability with exist-ing high-accuracy RV and photometry instruments. We illustrate that, with a smallnumber of RV observations, the RV-ETV diagrams allows us to distinguish betweenprograde and retrograde planetary orbits and also the planetary mass can be esti-mated if the stellar cross-correlation functions can be disentangled. We also identifya new (though minimal) contribution of S-type planets to the Rossiter-McLaughlineffect in eclipsing stellar binaries. We finally explore possible detection of exomoonsaround transiting luminous giant planets and find that the precision required to detectmoons in the RV curves of their host planets is of the order of cm s − and thereforenot accessible with current instruments. Key words: methods: numerical, observational– planetary system–stars: binaries–techniques: photometric, radial velocities, timing
Among the detection of thousands of extrasolar planets andcandidates around single stars, the
Kepler telescope has alsodelivered the first transit observations of planets in stellarmultiple systems, eleven in total as of today (Doyle et al.2011; Welsh et al. 2012; Orosz et al. 2012a,b; Schwamb et al.2013; Kostov et al. 2014; Welsh et al. 2015; Kostov et al.2016). Most of these binaries show mutual eclipses, thereforeallowing precise radius estimates of both stars and planets ina given system. All of these planets are classical “circumbi-nary planets”, or P-type planets (Dvorak 1982), orbiting onthe circumbinary orbit around an inner stellar binary (ortheir common center of mass) on a wide orbit. In other sys-tems, the planetary transit cannot be observed directly, butstellar eclipse timing variations (ETVs) still signify the pres-ence of one or more P-type planets (Beuermann et al. 2010; ? E-mail: [email protected] † E-mail: [email protected]
Qian et al. 2010; Beuermann et al. 2011; Schwarz et al. 2011;Qian et al. 2012; Beuermann et al. 2012, 2013; Marsh et al.2014; Schwarz et al. 2016), some of which are still beingdisputed (Horner et al. 2012, 2013; Bours et al. 2016).Planets also exist in S-type configurations (“S” for satel-lite), where the planet orbits a star in a close orbit, whileboth the planet and its host star are gravitationally boundto an additional star. Many of the known exoplanets are in-deed S-type planets, mostly in wide-orbit binaries. A handfulof S-type planets are orbiting stellar binaries with separa-tions .
20 AU (the tightest known binary which hosts as-type planet is KOI-1257 system with the semi-major axisof 5.3 AU (Santerne et al. 2014)) , but none of these systems A third configuration is referred to as ‘L-type” (or sometimes“T-type”, “T” for Trojan), where the planet is located at eitherthe L or L Lagrangian point (Dvorak 1986). Such planets havebeen suggested, and could possibly be detectable via eclipse timevariations (Schneider & Doyle 1995; Schwarz et al. 2015), butremain undiscovered as of today.c (cid:13) a r X i v : . [ a s t r o - ph . E P ] D ec M. Oshagh, R. Heller, and S. Dreizler are eclipsing. In other words, there has been no detection ofan S-type planet in a stellar eclipsing binary.Formation of S-type planets, particularly of giant plan-ets, in close binaries may be hampered by gravitational anddynamical perturbations from the stars (Whitmire et al.1998; Nelson 2000; Th´ebault et al. 2004; Haghighipour 2009;Thebault & Haghighipour 2014). However, to explain thediscovered S-type planets in close (non-eclipsing) binaries,several solutions have been proposed. For instance, a binaryorbit might be initially wide and tighten only after planetformation and migration have completed (Malmberg et al.2007; Th´ebault et al. 2009; Thebault & Haghighipour 2014).Alternatively, S-type planets, giant or terrestrial, in closeeclipsing binaries could build up through other formationscenarios, such as gravitational instabilities (Boss 1997) orin-situ core-accretion (Boss 2006; Mayer et al. 2007; Duchˆene2010).Therefore, S-type planets in close eclipsing binariesmight be expected to be rare from both a formation andan orbital stability point of view (Dvorak 1986; Holman& Wiegert 1999; Horner et al. 2012). Yet, the discoveriesof planets around pulsars (Wolszczan & Frail 1992), of thepreviously unpredicted families of hot Jupiters (Mayor &Queloz 1995), and of circumbinary planets (Doyle et al.2011) remind us that formation theories do not necessarilydeliver ultimate constraints on the actual presence of planetpopulations. As a consequence, the discovery of an S-typeplanet in close eclipsing binaries would have major impacton our understanding of planet formation and evolution. Thedetection of almost 3,000 close eclipsing binaries with
Kepler (Kirk et al. 2016) suggests that a dedicated search could in-deed be successful. In Figure 1, we show the orbital period( P ) distribution of these systems. Note that binaries with
P <
10 d have Hill spheres of about ten solar radii or smaller,so any planets might be subject to removal over billions ofyears. In systems with periods beyond 10 d, however, S-typeplanets might be long-lived and present today.Here we report on a new method to detect S-type plan-ets in close eclipsing binaries using radial velocities (RVs),eclipse timing variations (ETVs), and eclipse duration varia-tion (EDVs). It is an extension of recently developed meth-ods to find extrasolar moons around transiting exoplanetsvia the planet’s transit timing variations (TTVs; Sartoretti& Schneider 1999) and transit duration variations (TDVs;Kipping 2009a) independent of the moon’s direct photomet-ric signature. In comparison to S-type exoplanet searchesbased on ETVs only, the combination of ETVs and EDVsprovides a non-degenerate solution for the planet’s mass( M p ) and its semi-major axis ( a ) around the secondary star(Kipping 2009a). The RV method, using proper treatmentof the stellar cross-correlation functions (CCFs) as describedin this paper, offers additional measurements of M p , whichmust be consistent with the value derived from the ETV-EDV approach. What is more, the RV-ETV diagram offersa means to determine the sense of orbital motion of theplanet (prograde or retrograde).This paper is organised as follows: In Section 2 we de-scribe the basic idea behind our method and we describe thesimulations which we perform to evaluate the feasibility of http://keplerebs.villanova.edu Figure 1.
Period distribution of almost 3,000 stellareclipsing binaries observed by
Kepler space telescope fromhttp://keplerebs.villanova.edu (Kirk et al. 2016). The horizontalbar represents a density distribution of the histogram with darkblue corresponding to the highest densities and white correspond-ing to zero. our method. In Section 3. we present the results of our tests.We summarise our conclusions in Section 4.
Consider an eclipsing binary system consisting of a primarystar, which shall be the more massive one, and a secondarystar. Without loss of generality, the secondary star shall beorbited by a close-in planet. Then the planet will cause thesecondary star to perform a curly, wobbly orbit around thecenter of mass of the stellar binary. Hence, the eclipses willnot be exactly periodic: sometimes they will be too early, atother times they will be too late compared to the averageeclipse period. The resulting ETVs are essentially equiva-lent to the previously predicted TTVs of transiting plan-ets due to the gravitational perturbations of an extrasolarmoon (Sartoretti & Schneider 1999). Additionally, the tan-gential orbital velocity component of the secondary star inthe star-planet system causes EDVs, which are analogous tothe transit duration variations of a transiting planet with amoon (Kipping 2009a). Both ETVs and EDVs (or TTVs andTDVs) are phase-shifted by π/
2, that is, they form an el-lipse in the ETV-EDV (or TTV-TDV) diagram (Heller et al.2016). By using high-precision photometric time series, e.g.with the
Kepler space telescope, very high-precision ETV-EDV measurements can be obtained.
Kepler ’s median tim-ing precision turned out to be .
30 s (Welsh et al. 2013)and ETV accuracies obtained by Borkovits et al. (2016) aretypically below this level.Beyond ETVs and EDVs, the planet imposes a periodicRV variation onto its host star. RV observations of the sec-ondary star, however, would be challenging because, first,
MNRAS , 1–9 (2016)
TVs, EDVs, and RVs can reveal S-type planets − − RV ( m / s ) a) RV of primary due to secondary − − RV ( m / s ) b) RV of secondary due to planet − − RV ( m / s ) c) RV of secondary diluted
Time (days) − − RV ( m / s ) d) RV of the whole system
Figure 2.
Radial velocity components of a binary system with an S-type planet around the secondary star. Panel a) shows the RV ofthe primary due to the gravitational pull of the secondary. Red dots denote times of eclipses, when the secondary star is in front of theprimary (as seen from Earth) and the RV signal is zero. Panel b) illustrates the true RV variation of the secondary star due to its S-typeplanet and panel c) depicts the secondary’s RV, which is diluted by the brightness contrast between the primary and the secondary.Panel d) shows the total RV of the system, i.e. the sum of a) and c) . the spectrum of the secondary star would be diluted by theprimary star; and second, the RV signal of the secondarystar will be on top of a much more pronounced RV signalof the stellar binary. To tackle the latter aspect, we proposeto obtain the RV measurements exactly just before or aftereclipse. Thus, as the orbital geometry of the stellar binarywill be consistent during each such RV measurement, theRV component of the stellar binary will be equal for eachmeasurement as well. In particular, the binary’s RV compo-nent will be zero (see Figure 2a) and the observed RV will bedue to the secondary’s motion around the secondary-planetcenter of mass only (see Figure 2d). By limiting ourselvesto measurements just before or after eclipse, we also avoidcontributions of light travel time effects to our ETV mea-surements (Woltjer 1922; Irwin 1952).In Figure 3, we illustrate the correlation of ETVs withRVs and of EDVs with RVs acquired around eclipse. Theupper panel refers to a prograde planetary orbit, where theplanet’s sense of orbital motion is the same as the stellarbinary’s sense of orbital motion; the lower panel depicts aretrograde planetary orbit. The key feature of the RV-ETVcorrelation is in the slope of the observed curve. A negativeslope (upper panel) indicates a prograde planetary orbit,whereas a positive slope (lower panel) is suggestive of a ret-rograde orbit. We arbitrarily normalized both the RV andthe EDV amplitudes to obtain a circle. In general, however,the RV-EDV figure will be an ellipse with its semi-majorand semi-minor axis given by the RV and EDV amplitudes(or vice versa). To assess the feasibility of our method, we simulate obser-vations of two toy systems.
Our first test case consists of a Sun-like G-dwarf primary starand K-dwarf secondary star hosting an S-type Jupiter-styleplanet. The orbital period of the binary is chosen to be 50 dand the planet is put in a 4 d orbit. This choice is dominantlymotivated by constraints on the long-term orbital stability ofthe planet. Numerical simulations by Domingos et al. (2006)showed that the outermost stable orbit of a prograde satelliteis at about half the Hill radius around the secondary, whichtranslates into a maximum orbital period for the planet ofabout 1/9 the orbital period of the binary (Kipping 2009a).We consider the CCFs of the publicly available HARPSspectra of 51 Peg (our primary) and HD 40307 (our sec-ondary) as reference stars. In Table 1 we list their phys-ical parameters. As mentioned above, our hypothetical RVobservations are supposed to be taken near eclipse whenthe primary’s CCF will have zero RV. We therefore cor-rect the CCF of 51 Peg to be centered around zero RVthroughout our computations. The RV amplitude ( K ) im-posed by a 4 d Jupiter-sized planet on an HD 40307 is about150 m s − . Hence, we periodically shift the secondary’s CCFaccording to the planet’s orbital phase with an amplitudeof K = 150 m s − . We normalize the secondary CCF to the http://archive.eso.org/wdb/wdb/adp/phase3 spectral/form?collection name=HARPSMNRAS , 1–9 (2016) M. Oshagh, R. Heller, and S. Dreizler
RV RVETV EDV
Prograde Orbit of the Secondary with Companion
RV RV
Retrograde Orbit of the Secondary with Companion
ETV EDV
Figure 3. Top : Illustration of the RV-ETV and RV-EDV corre-lations for a prograde S-type planet (small black circle) around asecondary star (intermediate circle) eclipsing a primary star (bigyellow circle). The colors of the secondary indicate its instanta-neous RV. Blue means a negative RV-shift (toward the observer),white means zero RV, red means positive contribution (away fromthe observer). Arrows indicate the planet’s direction of motionaround the secondary. Images 1-4 depict four specific scenariosfor the secondary-planet orbital geometry during eclipse, whichcan be found in the respective diagrams.
Bottom : Same as topbut for a retrograde planet. total flux of the system to properly model the brightnesscontrast between the two stars (see Figure 4). We then addthe shifted CCF of the secondary and the zero-RV CCF ofthe primary to obtain the combined CCF (CCF tot ). For dif-ferent orbital phases of the planet around the secondary, wethen fit a Gaussian function to CCF tot to approximate anobserved RV measurement of the unresolved system. A sim-ilar approach was taken by Santos et al. (2002) to study theRV signal of HD 41004 AB.To estimate the ETV and EDV amplitudes of the sec-ondary star due its planet, we utilize the analytical frame-work of Kipping (2009a) and Kipping (2009b), which hasoriginally been developed to estimate the TTVs and TDVsof transiting planets with moons. Each individual ETV andEDV measurement in our simulations is calculated by weigh-
Table 1.
Physical parameters of 51 Peg and HD 40307.Star Mass Radius T eff v sin( i ) m V M (cid:12) R (cid:12) K km s −
51 Peg 1.11 1.266 5793 2.8 5.49HD 40307 0.77 0.68 4977 < ing the respective amplitude with a sine function of the plan-etary orbital phase, ETV and EDV being shifted by π/ Our second test case consists of an M-dwarf and a luminousJupiter-like transiting planet with a Neptune-mass invisiblemoon. The orbital periods are the same as we used for thefirst toy system and Hill stability is ensured. The Jupiter-like transiting planet cannot be considered as a canonical“hot Jupiter” because the illumination from the M-dwarfwould be too weak to heat it up significantly. Hence, in orderto make the planet visible in thermal emission, its intrinsicluminosity would need to be fed by primordial heat, whichcould be sufficient for detection during the first ∼
100 Myrafter the formation of the system.We model the CCF of the M-dwarf using a HARPS ob-served CCF of GJ 846, an M0.5 star (Henry et al. 2002) witha mass of approximately 0.63 solar masses ( M (cid:12) ), assumingsolar age and metallicity (Chabrier & Baraffe 2000). Sincethere is no direct observation of any hot Jupiter emissionspectrum, there also is no observed CCF available for oursimulations. Instead, we use the CCF of 51 Peg b observedby Martins et al. (2015) in reflected light. We consider aGaussian function with same depth and width as the CCFof 51 Peg b reported by Martins et al. (2015) for the transit-ing planet. Our preliminary tests showed that no moon, which canreasonably form through in-situ accretion within the gasand dust accretion disk around a young giant planet (a fewMars masses at most; Canup & Ward 2006; Heller & Pu-dritz 2015), could induce a detectable RV or ETV-EDV sig-nature. Hence, to explore the detection limit, we assumedan extreme version of a Neptune-sized satellite around theJovian-style transiting planet, which essentially results in aplanetary binary orbiting the M-dwarf host star. The K am-plitude of this planetary binary, which we use to shift theplanetary CCF as a function of the planet’s orbital phasearound the common center of mass, is around 500 m s − . As the primary star rotates, one hemisphere is moving to-ward the observer (and therefore subject to an RV blue-shift), while the other hemisphere is receding (and therefore Note that 51 Peg b is orbiting on a shorter period orbit, there-fore it is much brighter in the reflected light than our test planetin a 50 d orbit. MNRAS , 1–9 (2016)
TVs, EDVs, and RVs can reveal S-type planets red-shifted). During eclipse of a prograde binary, the sec-ondary star first occults part of the blue-shifted hemisphereand, thus, induces an RV red-shift. During the second half ofan eclipse, assuming zero transit impact parameter, the RVshift is blue. The resulting RV signal, known as the Rossiter-McLaughlin effect (RM) (Rossiter 1924; McLaughlin 1924),has been used to determine the projected stellar rotation ve-locity ( v sin i ) and the angle between the sky-projections ofthe stellar spin axis and the orbital plane of eclipsing bina-ries, and recently has been used for the transiting exoplanetsin almost 100 cases as of today. We suppose that the presence of an S-type planetaround the secondary star could create an additional RVcomponent in the RM signal because the light coming fromthe secondary star will show different RV shifts during sub-sequent eclipses (see Figure 3). To test our hypothesis, weuse the publicly available SOAP-T software (Oshagh et al.2013a) , which can simulate a rotating star being occultedby a transiting foreground object. SOAP-T generates boththe photometric lightcurve and the RV signal and it can alsodeliver the time-resolved CCFs of the system.We assume that the primary star rotates with the ro-tation period of 24 days, which is close to the Sun’s value.We simulate the eclipse of the secondary and generate theCCFs of the combined system for three orbital phases ofthe secondary star around its common center of mass withits S-type planet. One orbital phase simply assumes a zeroRV shift of the secondary, corresponding to an orbital ge-ometry where the secondary and its planet are both on theobserver’s line of sight. The second scenario assumes that thesecondary is moving toward the observer with its K ampli-tude (see Section 2.2) during eclipse and so the secondary’sCCF is maximally blue-shifted in this eclipse simulation.We then estimate the observed RV by fitting a Gaussian toall those CCFs, and obtain the RM signal that is contami-nated by a redshifted secondary. Finally, the third scenarioassumes a maximum redshift of the secondary during eclipse. In the following, we focus on our first test case of a Sun-likehost with a K-dwarf companion and a Jovian S-type planet(Section 2.2.1). Our second test case of an M-dwarf with aplanetary binary is considered in a separate Section 3.4.
We first explore the correlation between the secondary’sRVs, as measured directly from the combined dG-dK CCF(CCF tot ), with their corresponding ETV and EDV observa-tions. Figure 5 shows the RV-ETV and RV-EDV patternsfor both a prograde (upper panels) and a retrograde (lowerpanels) S-type planet. Nominal error bars of 30 s (
Kepler ’smedian timing precision; Welsh et al. 2013) for ETVs andEDVs as well as 1 m s − for the RV simulations (obtainablewith HARPS) are shown in each panel. The sequence of the RV (km/s) . . . . . . R e l a t i v e c o n t r a s t Gaussian fit51 Peg’s CCFHD 40307’s CCFHD 40307’s CCF - NormalizedHD 40307’s CCF - Normalized- Shiftted
Figure 4.
The solid green line represents the CCF of 51 Pegas observed with HARPS and normalized to its continuum. Theblack dashed line (used in Section 3.3) shows the best Gaussianfit to the CCF of 51 Peg. The dotted blue line shows the CCF ofHD 40307 obtained with HARPS, and normalized to its contin-uum. The dashed dotted cyan line shows the CCF of HD 40307 atzero RV and normalized to the total flux of system. The dashedred line illustrates the same CCF of HD 40307 but shifted by themaximum RV induced by our S-type test planet ( K = 150 m s − ). first five measurements is indicated with red symbols andnumbers. As a reading example, consider measurement “1”in the upper row. A maximum ETV observation of ∼ +3min in panel (a) means that the passage of the secondarystar in front of the primary is maximally delayed. With re-gards to Figure 3, the secondary has its maximum deflec-tion to the left (see configuration labelled “1”) so that theprograde planet is maximally deflected to the right. Theprograde motion of the planet then implies that the planetis receding from the observer while the star is approach-ing and, hence, the latter is subject to a maximal RV blueshift of about −
40 m s − in Figure 5a. Regarding the EDVcomponent of “1” in Figure 5b, it must be zero as the sec-ondary’s motion around the local secondary-planet center ofmass has no tangential velocity component with respect tothe observer’s line of sight.One intriguing result of these simulations is that theslopes of the RV-ETV figures for the prograde (panel a) andthe retrograde planet (panel c) indeed exhibit different alge-braic signs. The prograde scenario shows a negative slope,the retrograde case a positive slope, as qualitatively pre-dicted in Section 2.1. Also note that the sequence of simul-taneous RV-EDV measurements in the prograde case (panelb) is a horizontally mirrored version of the retrograde case(panel d). The main advantage of our method is that by ob-taining a small number of RV observations we will be able to Sequences of both ETVs and EDVs always relate to an averageeclipse period (for ETVs) or average eclipse duration (for EDVs).Hence, the first eclipse observation would, by definition, have noETV or EDV. Only once multiple eclipses have been observedwill the ETV-EDV measurements settle along the figures shownin Figure 5 and measurement “1” will turn out to have an ETVof about +3 min as in this example.MNRAS , 1–9 (2016)
M. Oshagh, R. Heller, and S. Dreizler
12 345 Nominal Error (a) R a d i a l V e l o c it y [ m / s ] Eclipse Timing Variation [min]dG + dK + GP (prograde)-50-40-30-20-10 0 10 20 30 40 50 -3 -2 -1 0 1 2 3 (b) R a d i a l V e l o c it y [ m / s ] Eclipse Duration Variation [min]dG + dK + GP (prograde)-50-40-30-20-10 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 1.5
12 345Nominal Error (c) R a d i a l V e l o c it y [ m / s ] Eclipse Timing Variation [min]dG + dK + GP (retrograde)-50-40-30-20-10 0 10 20 30 40 50 -3 -2 -1 0 1 2 3 (d) R a d i a l V e l o c it y [ m / s ] Eclipse Duration Variation [min]dG + dK + GP (retrograde)-50-40-30-20-10 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 1.5
Figure 5.
Correlation of the RVs, ETVs, and EDVs of the secondary K-dwarf star due to a giant planet (GP) companion, both orbitinga dG primary star. In each panel, the sequence of the first five measurements is labeled with red numbers. Nominal error bars of ± − in RV and 30 s in both ETV and EDV are shown in each panel. Panels (a) - (b) assume a prograde sense of orbital motion of the dK-GPsystem. Panels (c) - (d) assume a retrograde sense of orbital motion. detect the S-type planet in the system, distinguish betweenthe prograde and retrograde orbit.In Appendix A, we also present the correlation betweenthe Bisector Inverse Slope (BIS) (Queloz et al. 2001) andthe FWHM of the CCFs and the ETV. By fitting the Gaussian profile to CCF tot , we find that theRV variation of the secondary due to its Jupiter-sized S-typeplanet is strongly diluted by the primary star. We detect anRV amplitude of about 40 m s − . Hence, the correspondingmass of the planet would be strongly underestimated. Oursimulated RV observations are shown Figure 6.To derive a more accurate measurement of planet’smass, we remove the contribution of primary light fromCCF tot . We first fit a Gaussian function to the CCF of theprimary’s star and then use this fit as a template represent-ing the CCF of primary (as shown in Figure 4 in dashedblack line). We then remove this template Gaussian fromall simulated CCF tot measurements, which in reality wouldrepresent the observed CCF of the unresolved binary system.We then fit a Gaussian function to all measurements in our simulated CCF time series, from which the primary’scontribution has been removed, to estimate the RVs of thesecondary due its S-type planet. We retrieve the RV vari-ation amplitude induced by the S-type planet on the sec-ondary star that is 85 % ± ± While RV-ETV and RV-EDV correlations require the RVmeasurements to be taken either slightly before or aftereclipse in order to minimize effects of the eclipsing star onthe spectrum of the primary in the background (Section 2.1),we now consider the actual in-eclipse RVs of the system asdescribed in Section 2.3.In Figure 7 we show the impact of the S-type planet onthe observed RM as derived from the total CCF of the dG-dK binary. Although the additional effect is only a few per-cent ( ∼ . − ) of the total RM amplitude ( ∼
75 m s − ),it could be detectable with current high-precision spectro-graphs. If not taken into account properly, the asymmetry inthe RM contributions could lead to inaccurate estimations MNRAS , 1–9 (2016)
TVs, EDVs, and RVs can reveal S-type planets Time (days) RV ( m / s ) RV of secondaryObserved RV at eclipseRV of secondary (primary removed)Observed RV at eclipse (primary removed)Expected amplitude
Figure 6.
The red solid line represents the RV signal of the unresolved system derived by fitting a Gaussian profile to CCF tot . ObservedRVs near eclipses are represented with the magenta triangles. The dashed blue line displays the RV of the system after removing thecontribution of the primary. Green circles show observed RVs near eclipses. The black dotted line at K = 150 m s − presents the actualamplitude of RV signal due to S-type planet. A nominal error bar of ± − is indicated in the upper left corner. .
45 0 .
50 0 .
55 0 . Orbital phase RV ( m / s ) zero shiftpositive shiftnegative shift Figure 7.
The green solid line shows the RM effect in the com-bined spectrum of a dG-dK binary with the K-dwarf star passingin front of the dG star. The dashed blue line refers to the same sys-tem, but now the K-dwarf star is subject to a maximum blue shiftdue to a Jupiter-sized S-type planet in a 4 d orbit. The dashed-dotted red line refers to the reverse motion, where the secondaryis moving away from the observer. of the spin-orbit angle of the eclipsing binary. But if the vari-ation in the RM amplitude during subsequent eclipses canbe correlated with the presence of an S-type planet, then itcould serve as an additional means of verifying the S-typeplanet. That being said, note that there are several otherphenomena that can affect the RM signal, such as gravi-tational microlensing (Oshagh et al. 2013c), the convectiveblueshift (Cegla et al. 2016), and occultations active regionon the background star (Oshagh et al. 2016).
Our simulations of an extremely massive moon around atransiting, luminous giant planet in orbit about an M-dwarfstar are shown in Figure 8. The predicted RV amplitude ofthe observations is of order of cm s − . Note that the ampli-tude of the planet’s actual RV motion of 500 m s − is muchlarger than the 150 m s − that we estimated for the K-dwarfdue its S-type planet in our first test case. However, the fluxratio between a Jovian planet and its M-dwarf host star ismuch lower.Hence, we find that the precision required to detecteven a ridiculously massive exomoon, or essentially a bi-nary planet, is unachievable with current RV facilities. Nearfuture facilities such as ESPRESSO at the Very Large Tele-scope might be able to go down to the cm s − accuracy levelin favorable cases (Pepe et al. 2010), so future detections ofhighly massive exomoons or binary planets might be possiblein principle, but will certainly be challenging. We present a novel theoretical method to detect and ver-ify S-type planets in stellar eclipsing binaries by correlatingthe RVs of the secondary star with its ETVs and EDVs.We test the applicability of our method by performing re-alistic simulations and find that it can be used to detecta short-period S-type planet (e.g. a hot Jupiter) around aK-dwarf or lower-mass secondary star in a moderately wideorbit ( ∼
50 d) around a Sun-like primary star, e.g. using
Kepler photometry and HARPS RV measurements. We also
MNRAS , 1–9 (2016)
M. Oshagh, R. Heller, and S. Dreizler
12 345 Nominal Error (a) R a d i a l V e l o c it y [ m / s ] Transit Timing Variation [min]dM + GP + moon (prograde)-0.01-0.005 0 0.005 0.01 -3 -2 -1 0 1 2 3 (b) R a d i a l V e l o c it y [ m / s ] Transit Duration Variation [min]dM + GP + moon (prograde)-0.01-0.005 0 0.005 0.01 -1.5 -1 -0.5 0 0.5 1 1.5
Figure 8.
RVs, TTVs, TDVs of the transing planet due to a Neptune-sized moon. The host star is of spectral type M-dwarf. We assumea prograde sense of orbital motion of the exomoon system. Nominal error bars of 30 s in both TTV and TDV are shown in each panel.Nominal error bars of RV measurements are assumed to be in the order of 10 cm s − . find that the RV-ETV diagram can be used to distinguishbetween prograde and retrograde S-type orbits. The senseof orbital motion is a key tracer of planet formation and mi-gration, so the RV-ETV correlation identified in this papercould be very useful in studying the origin of close-in planetsin binaries.We show that the removal of the primary’s CCF fromthe combined CCF of the unresolved stellar binary can yieldrealistic estimates of the planetary mass through RV mea-surements. With the ETV-EDV relation offering a method-ologically independent measurement, we find that combinedRV-ETV-EDV observations offer a means to both detectand confirm/validate S-type planets at the same time. ETV-EDV measurements also deliver the planet’s orbital semi-major axis around the secondary star. In this paper, we pro-pose that RV observations be taken near eclipse in order tocorrelate them with ETVs and EDVs. After about a dozeneclipses, or if additional RV measurements could be takenfar from eclipse, it could be possible to securely identify theplanet’s orbital period around the secondary star. And ifthe secondary’s mass can be estimated from its spectrumand using stellar classification schemes (e.g. stellar evolutionmodels), then RVs could also yield an independent measure-ment of the planet’s semi-major axis, which needs to be inagreement with the value derived from the ETV-EDV data.Various physical phenomena can mimic RV-ETV andRV-EDV correlations, such as the stellar activity. Transit-ing planets crossing stellar active regions, for instance, cancause TTVs and TDVs (Oshagh et al. 2013b). It is also well-known that active regions on a rotating star affect the CCF,and thus produce RV variations, even if the planet does nottransit (Queloz et al. 2001). We therefore presume that it isplausible that stellar activity produces some kind of corre-lation between RVs and ETVs or EDVs, although it mightbe very different from the patterns we predict for S-typeplanets.Eclipsing binaries from Kepler are usually faint withtypical
Kepler magnitudes 11 . m K .
15 (Borkovits et al.2016). RV accuracy of ∼ − will be hard to achieve formany of these systems. The PLATO mission, scheduled forlaunch in 2025, will observe ten thousands of bright stars with m V <
11 (Rauer et al. 2014), many of which will turnout to be eclipsing binaries. PLATO will therefore discovertargets that allow both more accurate ETV-EDV measure-ments and high-accuracy ground-based RV follow-up.
APPENDIX A: VARIATIONS OF THEBISECTOR SPAN AND THE FULL WIDTH ATHALF MAXIMUM
As an extension of the data shown in Figure 5, we appendfigures of the bisector inverse slope (BIS) (Queloz et al. 2001)and of the full width at half maximum (FWHM) of the CCFsas a function of the stellar ETVs. The data refers to theprograde scenario of a Jupiter-sized S-type planet in a 4 dorbit around a K-dwarf star, both of which orbit a Sun-likeprimary star every 50 d (see Section 2.2.1). Figure A1 clearlyreveals additional BIS-ETV and FWHM-ETV correlations.However, our follow-up simulations did not indicate a uniquecorrelation between the planet’s sense of orbital motion andeither the BIS or the FWHM.
ACKNOWLEDGEMENTS
MO acknowledges research funding from the Deutsche Forschungsgemein-schaft (DFG , German Research Foundation) - OS 508/1-1. This work madeuse of NASA’s ADS Bibliographic Service. We would like to thank theanonymous referee for insightful suggestions.
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TVs, EDVs, and RVs can reveal S-type planets
12 345Nominal Error (a) B i s ec t o r S p a n ( B I S ) [ m / s ] Eclipse Timing Variation [min]dG + dK + GP (prograde)-100-90-80-70-60-50-40-30-20-10 0 10 -3 -2 -1 0 1 2 3
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Contribution:RH contributed to the writing of the text, guided the direction of research, and contributed to theinterpretation of the results. stronomy & Astrophysics manuscript no. ms c (cid:13)
ESO 2018June 27, 2018
Revisiting the exomoon candidate signal around Kepler-1625 b
Kai Rodenbeck , , René Heller , Michael Hippke , and Laurent Gizon , Institute for Astrophysics, Georg August University Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germanye-mail: (rodenbeck/heller/gizon)@mps.mpg.de Sonneberg Observatory, Sternwartestr. 32, 96515 Sonneberg, Germany e-mail: [email protected]
Accepted 26.6.2018
ABSTRACT
Context.
Transit photometry of the Jupiter-sized exoplanet candidate Kepler-1625 b has recently been interpreted to show hints of amoon. This exomoon, the first of its kind, would be as large as Neptune and unlike any moon we know from the solar system.
Aims.
We aim to clarify whether the exomoon-like signal is indeed caused by a large object in orbit around Kepler-1625 b, or whetherit is caused by stellar or instrumental noise or by the data detrending procedure.
Methods.
To prepare the transit data for model fitting, we explore several detrending procedures using second-, third-, and fourth-order polynomials and an implementation of the Cosine Filtering with Autocorrelation Minimization (CoFiAM). We then supply alight curve simulator with the co-planar orbital dynamics of the system and fit the resulting planet-moon transit light curves to theKepler data. We employ the Bayesian Information Criterion (BIC) to assess whether a single planet or a planet-moon system is amore likely interpretation of the light curve variations. We carry out a blind hare-and-hounds exercise using many noise realizationsby injecting simulated transits into di ff erent out-of-transit parts of the original Kepler-1625 light curve: (1) 100 sequences with threesynthetic transits of a Kepler-1625 b-like Jupiter-size planet and (2) 100 sequences with three synthetic transits of a Kepler-1625 b-likeplanet with a Neptune-sized moon. Results.
The statistical significance and characteristics of the exomoon-like signal strongly depend on the detrending method (poly-nomials versus cosines), the data chosen for detrending, and on the treatment of gaps in the light curve. Our injection-retrievalexperiment shows evidence of moons in about 10 % of those light curves that do not contain an injected moon. Strikingly, many ofthese false-positive moons resemble the exomoon candidate, i.e. a Neptune-sized moon at about 20 Jupiter radii from the planet. Werecover between about a third and half of the injected moons, depending on the detrending method, with radii and orbital distancesbroadly corresponding to the injected values.
Conclusions. A ∆ BIC of − . ff erent solutions depending on the details of the detrending method.We find it concerning that the detrending is so clearly key to the exomoon interpretation of the available data of Kepler-1625 b. Furtherhigh-accuracy transit observations may overcome the e ff ects of red noise but the required amount of additional data might be large. Key words.
Planets and satellites: detection – Eclipses – Techniques: photometric – Methods: data analysis
1. Introduction
Where are they? – With about 180 moons discovered around theeight solar system planets and over 3 ,
500 planets confirmed be-yond the solar system, an exomoon detection could be imminent.While many methods have indeed been proposed to search formoons around extrasolar planets (Sartoretti & Schneider 1999;Han & Han 2002; Cabrera, J. & Schneider, J. 2007; Moskovitzet al. 2009; Kipping 2009; Simon et al. 2010; Peters & Turner2013; Heller 2014; Ben-Ja ff el & Ballester 2014; Agol et al.2015; Forgan 2017; Vanderburg et al. 2018) , only a few ded-icated surveys have actually been carried out (Szabó et al. 2013;Kipping et al. 2013b,a, 2014; Hippke 2015; Kipping et al. 2015;Lecavelier des Etangs et al. 2017; Teachey et al. 2018), one ofwhich is the “Hunt for Exomoons with Kepler” (HEK for short;Kipping et al. 2012).In the latest report of the HEK team, Teachey et al.(2018) find evidence for an exomoon candidate aroundthe roughly Jupiter-sized exoplanet candidate Kepler-1625 b,which they provisionally refer to as Kepler-1625 b-i. Kepler- For reviews see Heller et al. (2014) and Heller (2017). M ? = . + . − . M (cid:12) ( M (cid:12) being the solar mass), a radius of R ? = . + . − . R (cid:12) (with R (cid:12) as the solar radius), and an ef-fective temperature of T e ff ,? = + − K (Mathur et al. 2017).Its Kepler magnitude of 15.756 makes it a relatively dim Ke-pler target. The challenge of this tentative detection is in thenoise properties of the data, which are a ff ected by the system-atic noise of the Kepler space telescope and by the astrophysicalvariability of the star. Although the exomoon signal did show upboth around the ingress / egress regions of the phase-folded tran-sits (known as the orbital sampling e ff ect; Heller 2014; Helleret al. 2016a) generated by Teachey et al. (2018) and in the se-quence of the three individual transits, it could easily have beenproduced by systematics or stellar variability, as pointed out inthe discovery paper.The noise properties also dictate a minimum size for an ex-omoon to be detected around a given star and with a giveninstrument. In the case of Kepler-1625 we calculate the root-mean-square of the noise level to be roughly 700 ppm. Asa consequence, any moon would have to be at least about NASA Exoplanet Archive: https: // exoplanetarchive.ipac.caltech.eduArticle number, page 1 of 16 a r X i v : . [ a s t r o - ph . E P ] J un & A proofs: manuscript no. ms
Fig. 1.
Kepler light curve of Kepler-1625.
Left:
Simple Aperture Photometry (SAP) flux.
Right:
Pre-search Data Conditioning Simple AperturePhotometry (PDCSAP) flux. The top panels show the entire light curves, respectively. The second to fourth rows illustrate zooms into transits 2,4, and 5 of Kepler-1625 b, respectively. These transits were shifted to the panel center and ±
10 d of data are shown around the transit mid-points.Some examples of jumps and gaps in the light curve are shown. Time is in Barycentric Kepler Julian Date. p
700 ppm × . R (cid:12) ≈ . R ⊕ ( R ⊕ being the Earth’s radius)in size, about 30% larger than Neptune, in order to significantlyovercome the noise floor in a single transit. The three observedtransits lower this threshold by a factor of √
3, suggesting a min-imum moon radius of ≈ R ⊕ . In fact, the proposed moon candi-date is as large as Neptune, making this system distinct from anyplanet-moon system known in the solar system (Heller 2018).Here we present a detailed study of the three publicly avail-able transits of Kepler-1625 b. Our aim is to test whether theplanet-with-moon hypothesis is favored over the planet-only hy-pothesis. In brief, we 1. develop a model to simulate photometric transits of a planetwith a moon (see Sect. 2.2.2).2. implement a detrending method following Teachey et al.(2018) and explore alternative detrending functions.3. detrend the original Kepler-1625 light curve, determine themost likely moon parameters, and assess if the planet-with-moon hypothesis is favored over the planet-only hypothesis.4. perform a blind injection-retrieval test. To preserve the noiseproperties of the Kepler-1625 light curve, we inject planet-with-moon and planet-only transits into out-of-transit partsof the Kepler-1625 light curve. Article number, page 2 of 16ai Rodenbeck et al.: Revisiting the exomoon candidate signal around Kepler-1625 b
2. Methods
The main challenge in fitting a parameterized, noise-less modelto observed data is in removing noise on time scales similar orlarger than the time scales of the e ff ect to be searched; at thesame time, the structure of the e ff ect shall be untouched, anapproach sometimes referred to as “pre-whitening” of the data(Aigrain & Irwin 2004). The aim of this approach is to removeunwanted variations in the data, e.g. from stellar activity, sys-tematics, or instrumental e ff ects. This approach bears the riskof both removing actual signal from the data and of introducingnew systematic variability. The discovery and refutal of the exo-planet interpretation of variability in the stellar radial velocitiesof α Centauri B serves as a warning example (Dumusque et al.2012; Rajpaul et al. 2016). Recently developed Gaussian pro-cess frameworks, in which the systematics are modeled simulta-neously with stellar variability, would be an alternative method(Gibson et al. 2012). This has become particularly important forthe extended Kepler mission (K2) that is now working with de-graded pointing accuracy (Aigrain et al. 2015).That being said, Teachey et al. (2018) applied a pre-whitening technique to both the Simple Aperture Photometry(SAP) flux and the Pre-search Data Conditioning (PDCSAP)flux of Kepler-1625 to determine whether a planet-only or aplanet-moon model is more likely to have caused the observedKepler data. In the following, we develop a detrending andmodel fitting procedure that is based on the method applied byTeachey et al. (2018), and then we test alternative detrendingmethods.During Kepler’s primary mission, the star Kepler-1625 hasbeen monitored for 3.5 years in total, and five transits could havebeen observed. This sequence of transits can be labeled as tran-sits 1, 2, 3, 4, and 5. Due to gaps in the data, however, only threetransits have been covered, which correspond to transits 2, 4,and 5 in this sequence. Figure 1 shows the actual data set that wediscuss. The entire SAP (left) and PDCSAP (right) light curvesare shown in the top panels, and close-up inspections of the ob-served transit 2, 4, and 5 are shown in the remaining panels. Thetime system used throughout the article is the Barycentric Ke-pler Julian Date (BKJD), unless marked as relative to a transitmidpoint.
A key pitfall of any pre-whitening or detrending method is theunwanted removal of signal or injection of systematic noise, thelatter of which could mimic signal. In our case of an exomoonsearch, we know that the putative signal would be restricted toa time-window around the planetary mid-transit, which is com-patible with the orbital Hill stability of the moon. This criteriondefines a possible window length that we should exclude fromour detrending procedures. For a nominal 10 Jupiter-mass planetin a 287 d orbit around a 1 . M (cid:12) star (as per Teachey et al. 2018),this window is about 3.25 days to both sides of the transit mid-point (see Appendix A).Although this window length is astrophysically plausible toprotect possible exomoon signals, many other choices are simi-larly plausible but they result in significantly di ff erent detrend-ings. Figure 2 illustrates the e ff ect on the detrended light curveif two di ff erent windows around the midpoint of the planetarytransit (here transit 5) are excluded from the fitting. We chosea fourth-order polynomial detrending function and a 7 . r e l a t i v e f l u x [ pp t ] Transit 51497 1498 1499time [BKJD]5.02.50.0 Transit 5
Fig. 2.
Example of how the detrending procedure alone can produce anexomoon-like transit signal around a planetary transit. We use transit 5of Kepler-1625 b as an example.
Top:
Gray dots indicate the Kepler PD-CSAP flux. The lines show a 4th-order polynomial fit for which we ex-clude 7 . Center:
Dots show the detrended light curve de-rived from the blue polynomial fit in the top panel. The blue line illus-trates a planet-only transit model.
Bottom:
Dots visualize the detrendedlight curve using the orange polynomial fit from the top panel. Note theadditional moon-like transit feature caused by the overshooting of theorange polynomial in the top panel. The orange line shows a planet-moon transit model with moon parameters as in Table 1 (see Fig. 4 fortransit dynamics). As an alternative interpretation, the blue detrendingfunction filters out an actually existing moon signature while the orangedetrending fit preserves the moon signal. poses. In particular, with the latter choice we produce a moon-like signal around the planetary transit similar to the moon sig-nal that appears in transit 5 in Teachey et al. (2018). For the for-mer choice, however, this signal does not appear in the detrendedlight curve.Teachey et al. (2018) use the Cosine Filtering with Auto-correlation Minimization (CoFiAM) detrending algorithm to de-trend both the SAP and PDCSAP flux around the three transitsof Kepler-1625 b. CoFiAM fits a series of cosines to the lightcurve, excluding a specific region around the transit. CoFiAMpreserves the signal of interest by using only cosines with a pe-riod longer than a given threshold and therefore avoids the injec-tion of artificial signals with periods shorter than this threshold.Teachey et al. (2018) also test polynomial detrending functionsbut report that this removes the possible exomoon signal. Wechoose to reimplement the CoFiAM algorithm as our primarydetrending algorithm to remain as close as possible in our analy-sis to the work in Teachey et al. (2018). In our injection-retrievaltest we also use polynomials of second, third, and fourth orderfor detrending. While low-order polynomials cannot generally
Article number, page 3 of 16 & A proofs: manuscript no. ms fit the light curve as well as the series of cosines, the risk ofinjecting artificial signals may be reduced.
We implement the CoFiAM detrending algorithm as per the de-scriptions given by Kipping et al. (2013b) and Teachey et al.(2018). In the following, we refer to this reimplementation astrigonometric detrending as opposed to polynomial functionsthat we test as well (see Sect. 2.4.4).The light curves around each transit are detrended indepen-dently. For each transit, we start by using the entire SAP flux ofthe corresponding quarter. We use the SAP flux instead of thePDCSAP flux to reproduce the methodology of Teachey et al.(2018) as closely as possible. The authors argue that the use ofSAP flux avoids the injection of additional signals into the lightcurve that might have the shape of a moon signal. First, we re-move outliers using a running median with a window length of20 h and a threshold of 3 times the local standard deviation withthe same window length. In order to achieve a fast convergenceof our detrending and transit fitting procedures, we initially esti-mate the transit midpoints and durations by eye and identify dataanomalies, e.g. gaps and jumps (e.g. the jump 2 d prior to transit2 and the gap 4 d after transit 4, see Fig. 1).Jumps in the light curve can have multiple reasons. Thejumps highlighted around transit 2 in Fig. 1 are caused by a re-action wheel zero crossing event; the jump 5 d after transit 4 iscaused by a change in temperature after a break in the data col-lection. Following Teachey et al. (2018), who ignore data pointsbeyond gaps and other anomalous events for detrending, we cutthe light curve around any of the transits as soon as it encountersthe first anomaly, leaving us with a light curve of a total duration D around each transit (see top left panel in Fig. 3). In Sect. 2.4.4,we investigate the e ff ect of including data beyond gaps. The de-trending is then applied in two passes, using the first pass to getaccurate transit parameters. In particular, we determine the dura-tion ( t T ) between the start of the planetary transit ingress and theend of the transit egress (Seager & Mallén-Ornelas 2003) andthe second pass to generate the detrended light curve. First pass:
Using the estimated transit midpoints and dura-tions, we calculate the time window ( t c , see top left panel inFig. 3) around a given transit midpoint to be cut from the de-trending fit as t c = f t c t T , where the factor f t c , relating the timecut around the transit to the transit duration, is an input parame-ter for the detrending algorithm. Specifically, t c denotes the totallength of time around the transit excluded from the detrending.We fit the detrending function G k ( t , * a , * b ) = a + k X l = a l cos l π D t ! + b l sin l π D t ! (1)to the light curve (excluding the region t c around the transit) byminimizing the χ between the light curve and G k ( t , * a , * b ), where * a = ( a , a , ..., a k ) and * b = ( b , b , ..., b k ) are the free modelparameters to be fitted. The parameter k is a number between1 and k max = round(2 D / t p ), where t p = f t p t T is the time scalebelow which we want to preserve possible signals. f t p is an inputparameter to the detrending algorithm. For each k we divide thelight curve by G k , giving us the detrended light curves F k . Wecalculate the first-order autocorrelation according to the Durbin& Watson (1950) test statistic for each F k (excluding again theregion around the transit). For each transit we select the F k withthe lowest autocorrelation F k min and combine these F k min around each transit into our detrended light curve F . We fit the planet-only transit model to the detrended light curve F and computethe updated transit midpoints and duration t T . Second pass:
The second pass repeats the steps of the firstpass, but using the updated transit midpoints and durations as in-put. The resulting detrended light curve F is then used for ourmodel fits with the ultimate goal of assessing whether an ex-omoon is a likely interpretation of the light curve signatures ornot. We estimate the noise around each transit by taking the vari-ance of F , excluding the transit region.Figure 3 shows the detrending function as well as the de-trended light curve for f t c = . f t p = .
4, corresponding to t c = . t p = . We construct two transit models, one of which contains a planetonly and one of which contains a planet with one moon. Wedenote the planet-only model as M (the index referring to thenumber of moons) and the planet-moon model as M . We do notconsider models with more than one moon. M assumes a circular orbit of the planet around its star. Giventhe period of that orbit ( P ) and the ratio between stellar radiusand the orbital semimajor axis ( R ? / a ), the sky-projected appar-ent distance to the star center relative to the stellar radius can becalculated as z = s" aR ? sin π ( t − t ) P ! + " b cos π ( t − t ) P ! , (2)where b is the transit impact parameter and t is the time of thetransit midpoint. We use the python implementation of the Man-del & Agol (2002) analytic transit model by Ian Crossfield tocalculate the transit light curve based on the planet-to-star ra-dius ratio ( r p = R p / R ? ) and based on a quadratic limb-darkeninglaw paramterized by the limb-darkening parameters q and q as given in Kipping (2013). We call this model light curve withzero moons F . In our planet-moon model, we assume a circular orbit of the lo-cal planet-moon barycenter around the star with an orbital pe-riod P B , a semimajor axis a B , and a barycentric transit midpointtime t , B . The projected distance of the barycenter to the star cen-ter relative to the stellar radius is calculated the same way as ineq. 2. The planet and moon are assumed to be on circular orbitsaround their common center of mass with their relative distancesto the barycenter determined the ratio of their masses M p and M s to the total mass M p + M s . The individual orbits of both theplanet and the moon are defined by the total distance betweenthe two objects a ps , the planet mass M p , the moon mass M s andby the time of the planet-moon conjunction t , s , that is, the timeat which the moon is directly in front of the planet as seen froman observer on Earth.This model is degenerate in terms of the sense of orbital mo-tion of the moon. In other words, a given planet-moon transitlight curve can be produced by both a prograde and a retrograde Available at http: // / ~ianc / files as python.py.Article number, page 4 of 16ai Rodenbeck et al.: Revisiting the exomoon candidate signal around Kepler-1625 b
635 636 637 638 63935803600 Transit 2 Dt c t T
635 636 6370-2.5-5 Transit 21206 1208 1210 1212 1214568057005720 f l u x [ e / s ] Transit 4 1210 1211 12120-2.5-5 r e l a t i v e f l u x [ pp t ] Transit 41496 1497 1498 1499 1500 1501time [BKJD]50405060 Transit 5 1497 1498 1499time [BKJD]0-2.5-5 Transit 5
Fig. 3.
Left:
Kepler SAP flux around the transits used for the trigonometric detrending, our reimplementation of the CoFiAM algorithm. The datapoints denoted by open circles around the transits are excluded from the detrending fit. The black line shows the resulting light curve trend withoutthe transit.
Right:
Detrended transit light curves as calculated by the trigonometric detrending. moon (Lewis & Fujii 2014; Heller & Albrecht 2014). We restrictourselves to prograde moons. The planet mass is set to a nominal10 Jupiter masses, as suggested by Teachey et al. (2018) and inagreement with the estimates of Heller (2018). This constraintsimplifies the interpretation of the results substantially since themoon parameters are then una ff ected by the planetary parame-ters. The moon mass is assumed to be much smaller than that ofthe planet. In fact, for a roughly Neptune-mass moon around a 10Jupiter-mass planet, we expect a TTV amplitude of 3 to 4 min-utes and a TDV amplitude of 6 to 7 minutes, roughly speaking.Hence, we simplify our model and set M s =
0, which means that a ps is equal to the distance between the moon and the planet-moon barycenter, a s . The moon is assumed to have a coplanarorbit around the planet and, thus, to have the same transit impactparameter as the planet.With these assumptions the projected distance of the planetcenter to the star center relative to the stellar radius z p is equalto that of the barycenter z B . The projected distance of the mooncenter to the star center relative to the stellar radius z s is given by z = " a B R ? sin π ( t − t , B ) P B ! + a ps R ? sin π ( t − t , s ) P s ! + " b cos π ( t − t , B ) P B ! , (3)where P s is the orbital period of the moon calculated from thefixed masses and a ps . We calculate the transit light curves of both bodies and com-bine them into the total model light curve, which we call F .We use the limb-darkening parameter transformation from Kip-ping (2013). For computational e ffi ciency, we do not considerplanet-moon occultations. For the planet-moon system of inter-est, occultations would only occur only about half of the transits(assuming a random moon phase) even if the moon orbital planewould be perfectly parallel to the line of sight. Such an occulta-tion would take about 1.5 h and would only a ff ect 5-10 % of thetotal moon signal duration.In Table 1 we give an overview of our nominal parameteri-zation of the planet-moon model. In Fig. 4 we show the orbitaldynamics of the planet and moon during transits 2, 4, and 5 usingthe nominal parameters in Table 1. This nominal parameteriza-tion was chosen to generate a model light curve that is reason-ably close to the preferred model light curve found in Teacheyet al. (2018), but it does not represent our most likely model fitto the data. We use the Markov Chain Monte Carlo (MCMC) implementa-tion
Emcee (Foreman-Mackey et al. 2013) to estimate the pos-terior probability distribution of the parameters for model M ( i ) ( M or M ). For this purpose, we need to formulate the proba- Article number, page 5 of 16 & A proofs: manuscript no. ms
Transit 2 r e l a t i v e f l u x [ pp t ] Transit 4
24 12 0 12 24time from transit midpoint [h]43210
Transit 5
Transit 2
Transit 4 R ] Transit 5
Fig. 4.
Left : Example of a simulated planet-moon transit light curve for transits 2, 4, and 5 using the nominal parameterization given in Table 1.The relative flux is the di ff erence to the out-of-transit model flux and is given in parts per thousand (ppt). Right : Visualization of the orbitalconfigurations during transits 2 (left column), 4 (center column), and 5 (right column). Labels 1-5 in the light curves refer to configurations 1-5(see labels along the vertical axis). An animation of this figure is available online.
Table 1.
Nominal parameterization of the planet-moon model to repro-duce the transit shape suggested by Teachey et al. (2018). The no-moonmodel uses the same parameter set (excluding the moon parameters),except that R ? and a B are combined into a single parameter R ? / a B . Parameter Nominal Value Description r p a B b t , B P B R ? R (cid:12) stellar radius q ffi -cient q ffi -cient r s a s R J orbital semimajor axis of theplanet-moon binary t , s N de-trended flux measurements (see Sect. 2.1.1). Given a set of pa- rameters * θ , model M i produces a model light curve F i ( t , * θ ).We assume that the noise is uncorrelated (see Appendix B) andGaussian with a standard deviation σ j at time t j . This simplifiesthe joint probability density to a product of the individual prob-abilities. The joint probability density function of the detrendedflux F ( t ) is given by p ( F | * θ , M i ) = N Y j = q πσ j exp − (cid:16) F ( t j ) − F i ( t j , * θ ) (cid:17) σ j . (4)The noise dispersion σ j has a fixed value for each transit.Table 2 shows the parameter ranges that we explore. Aprior is placed on the stellar mass according to the mass of1 . + . − . M (cid:12) determined by Mathur et al. (2017). The stellarmass for a given parameter set is determined from the system’stotal mass using P B and a B and subtracting the fixed planet massof 10 Jupiter masses.A total of 100 walkers are initiated with randomly chosen pa-rameters close to the estimated transit parameters. For the sakeof fast computational performance, the walkers are initially sep-arated into groups of 16 for the planet-only model and 24 for theplanet-moon model (twice the number of parameters plus 2, re-spectively), temporarily adding walkers to fill the last group. Totransform the initially flat distribution of walkers into a distri-bution according to the likelihood function, the walkers have togo through a so-called burn-in phase, the resulting model fits ofwhich are discarded. We chose a burn-in phase for the walkers of500 steps in both groups. Afterwards, we discard the temporarilyadded walkers, merge the walkers back together, and perform asecond burn-in phase of 2 200 steps with a length determined byvisual inspection. Finally, we initiate the main MCMC run witha total of 8 000 steps.We run the MCMC code on the detrended light curve usingboth the planet-only and the planet-moon model. Article number, page 6 of 16ai Rodenbeck et al.: Revisiting the exomoon candidate signal around Kepler-1625 b
Table 2.
Parameter ranges explored with our planet-moon model. Theranges of the no-moon model parameters are the same for the sharedparameter and is propagated to the derived parameter R ? / a . Min. Value Parameter Max. Value0 ≤ r p ≤ ≤ a B ≤ ≤ b ≤ − P B / ≤ t , B ≤ P B / ≤ P B ≤
300 d0 ≤ R ? ≤ . R (cid:12) ≤ q ≤ ≤ q ≤ ≤ r s ≤ r p ≤ a s ≤ R Hill / − P s / ≤ t , s ≤ P s / We use the Bayesian Information Criterion (BIC) to evaluatehow well a model describes the observations in relation to thenumber of model parameters and data points. The BIC of a givenmodel M i with m i parameters is defined by Schwarz (1978) asBIC( M i | F ) = m i ln N − p ( F | * θ max , M i ) , (5)where * θ max is the set of parameters that maximizes the probabil-ity density function p ( F | * θ , M i ) for a given the light curve F andmodel M i .The di ff erence of the BICs between two models gives anindication as to which model is more likely. In particular, ∆ BIC( M , M ) ≡ BIC( M ) − BIC( M ) < M ismore likely. We consider ∆ BIC < ∆ BIC >
6) as strongevidence in favor of (or against) model M (see, e.g., Kass &Raftery 1995).The best-fitting set of parameters derived from our MCMCruns ( * θ max ) is then used to calculate ∆ BIC( M , M ). For ourcalculations, we only use those parts of the light curve aroundthe transits that could potentially be a ff ected by a moon (3 .
25 don each side of the transits, determined by the Hill radius R Hill and the orbital velocity of the planet-moon barycenter, see Ap-pendix A).
In order to estimate the likelihood of an exomoon feature tobe due to either a real moon or due to noise, we perform sev-eral injection-retrieval experiments. One of us (MH) injectedtwo cases of transits into the out-of-transit parts of the originalPDCSAP Kepler flux. In one case, a sequence of three planet-only transits (similar to the sequence of real transits 2, 4, and5) was injected, where the planet was chosen to have a radiusof 11 Earth radii. In another case, a sequence of three transitsof a planet with moon with properties similar to the proposedJupiter-Neptune system was injected. Author KR then appliedthe Baysian framework described above in order to evaluate theplanet-only vs. the planet-with-moon hypotheses and in order tocharacterize the planet and (if present) its moon.As an important trait of our experiment, KR did not knowwhich of the light curves contained only a planet and which con-tained also a moon.
For the injection part, we use
PyOSE (Heller et al. 2016a,b) tocreate synthetic planet and moon ensemble transits. This codenumerically integrates the non-occulted areas of the stellar diskto calculate the instantaneous flux of the star, which makes ita computationally slow procedure. Hence we use the analyticalmodel described Sect. 2.2 for the retrieval part. In our model,the moon’s orbit is defined by its eccentricity ( e s , fixed at 0), a s ,its orbital inclination with respect to the circumstellar orbit ( i s ,fixed at 90 ◦ ), the longitude of the ascending node, the argumentof the periapsis, and the planetary impact parameter ( b , fixedat zero). Due to the small TTV and TDV amplitudes comparedto the 29.4 min exposure of the Kepler long cadence data, weneglect the planet’s motion around the planet / moon barycenter,although PyOSE can model this dynamical e ff ect as well, andassume that the moon orbits the center of the planet.Our numerical code creates a spherical limb-darkened star ona 2-dimensional grid of floating-point values. The sky-projectedshapes of both the planet and the moon are modeled as blackcircles. The spatial resolution of the simulation is chosen to be afew million pixels so that the resulting light curve has a numeri-cal error < ≈
700 ppmnoise level of the Kepler light curve. The initial temporal resolu-tion of our model is equivalent to 1 000 steps per planetary transitduration, which we then downsample to the observed 29.4 mincadence. The creation of one such light curve of a planet with amoon takes about one minute on a modern desktop computer.We create a set of 100 such transit simulations of the planet-moon ensemble, where the two bodies move consistently dur-ing and between transits. All orbits are modeled to be co-planarand mutual planet-moon occultations are also included. For eachtransit sequence, the initial orbital phase of the planet-moon sys-tem is chosen randomly.With P B = . P s = . ≈ .
13 in phase between each subsequent planetarytransit ( P B / P s ≈ . ≈ .
36 rad in phase (the planetary transit durationis 0 . ± . As a first validation of our injection-retrieval experiment andour implementation of the Bayesian statistical framework, wegenerate a new set of white noise light curves to test only themodel comparison part of our pipeline without any e ff ects thatcould possibly arise from imperfect detrending. Any e ff ects thatwe would see in our experiments with the real Kepler-1625light curve but not in the synthetic light curves with noise onlycould then be attributed to the imperfect detrending of the time-correlated (red) noise.MH generated 200 synthetic light curves with ten di ff er-ent levels of white noise, respectively, ranging from root meansquares of 250 ppm to 700 ppm in steps of 50 ppm. This resultsin a total of 2 000 synthetic light curves. MH used the methoddescribed in Sect. 2.4.1 to inject three transits of a planet onlyinto 100 light curves per noise level and three transits of a planetwith a moon into the remaining 100 light curves per noise level.The initial orbital phases were randomly chosen and are di ff er- Article number, page 7 of 16 & A proofs: manuscript no. ms
Fig. 5. Di ff erence between the BIC of the planet-moon model and the no-moon model for 2 000 artificial white noise light curves at di ff erent noiselevels, injected with simulated transits. On the left (100 ×
10 light curves) a planet and moon transit was injected, on the right (100 ×
10 lightcurves) only the planet. Each light curve consists of three consecutive transits. Each column is sorted by the ∆ BIC. The ∆ BIC threshold, overwhich a planet-moon or planet-only system is clearly preferred is ± ∆ BIC between those values considered to beambiguous. ent from the ones used to generate the light curves in Sect. 2.4.3.MH delivered these light curves to KR without revealing theirspecific contents. KR then ran our model selection algorithm tofind the ∆ BIC for each of the 2 000 systems. After the ∆ BICswere found, MH revealed the planet-only or the planet-moon na-ture of each light curve.Fig. 5 shows the resulting ∆ BICs for each of the 2 000 lightcurves, separated into the planet-only (left panel) and planet-moon injected systems (right panel) and sorted by the respectivewhite noise level (along the abscissa). Each vertical column con-tains 100 light curves, respectively. For a noise level of 250 ppm,as an example, our algorithm finds no false positive moons in theplanet-only data, that is, no system with a ∆ BIC < −
6, while 1case remains ambiguous ( − < ∆ BIC <
6) and the other 99cases are correctly identified as containing no moons. In the caseof an injected planet-moon system instead, the algorithm cor-rectly retrieves the moon in 100 % of the synthetic light curves,that is, ∆ BIC < − a s and R s for each of the maximum-likelihood fits shown in Fig. 5. Each panel in Fig. 6 refers toone white noise level, that is, to one column in Fig. 5 of eitherthe planet-only or the planet-moon injected system. In the caseof an injected planet only (upper panels), the most likely valuesof a s are distributed almost randomly over the range of valuesthat we explored. On the other hand, R s is constrained to a smallrange from about 1 . R ⊕ at 250 ppm to roughly 3 R ⊕ at 700 ppmwith the standard variation naturally increasing with the noiselevel. The lower part of Fig. 6 shows the outcome of our planet-moon injection-retrievals from the synthetic light curves withwhite noise only. The correct parameters are generally recov-ered at all noise levels. In fact, we either recover the moon witha similar radius and orbital separation as the injection values(symbolized by blue points) or we find the moon to have verydi ff erent radius and orbit while also rejection the hypothesis ofits presence in the first place (symbolized by red points). The dis-tribution of these false negatives in the a s - R s plane resembles thedistribution of the true negatives in the corresponding no-mooncases. The ambiguous runs with a ∆ BIC around 0 still mostlyrecover the injected moon parameters. This is especially clearfor the 700 ppm level, with 50 % more ambiguous runs than truepositives, where most of the runs still recover the injected pa-rameters.
We inject synthetic transits into the Kepler-1625 PDCSAP dataprior to our own detrending (see Sect. 2.4.1). We use the PDC-SAP flux instead of SAP flux because (1) it was easier for us toautomate the anomaly detection and (2) PDCSAP flux has beencleaned from common systematics. Since the PDC pipeline re-moves many of the jumps in the data, we can focus on a singletype of anomaly, that is gaps. Gaps are relatively easy to detectin an automated way, removing the requirement of visual inspec-tion of each light curve. For the injection, we select out-of-transitparts of the Kepler-1625 light curve that have at least 50 d ofmostly uninterrupted data (25 d to both sides of the designatedtime of transit injection), but accept the presence of occasionalgaps with durations of up to several days during the injectionprocess.The set of 200 synthetic light curves was provided by MH toKR for blind retrieval without any disclosure as to which of thesequences have a moon. The time of mid-transit was communi-
Article number, page 8 of 16ai Rodenbeck et al.: Revisiting the exomoon candidate signal around Kepler-1625 b B I C m oo n un li k e l y m oo n li k e l y planet-only system injected median moon semi-major axis [ R J ] m e d i a n m oo n r a d i u s [ R ] B I C m oo n un li k e l y m oo n li k e l y planet+moon system injected median moon semi-major axis [ R J ] m e d i a n m oo n r a d i u s [ R ] Fig. 6.
Distribution of the median likelihood R s and a s for all the runs for the di ff erent noise levels, with the runs injecting planet and moon on thetop and runs injecting only a planet in the bottom. The ∆ BIC of the planet-moon model compared to the no-moon model for all runs is indicatedby the color. Generally runs with a low ∆ BIC (indicating the presence of a moon) also are in the vicinity of the injected parameter. cated with a precision of 0.1 days to avoid the requirement of apre-stage transit search. This is justified because (i.) the originaltransits of Kepler-1625 b have already been detected and (ii.) thetransit are visible by-eye and do not necessarily need computer-based searches. We provide the 200 datasets to the communityfor reproducibility and encourage further blind retrievals. The detrending procedure for our injection-retrieval experimentdi ff ers from the one used to detrend the original light curvearound the Kepler-1625 b transits (see Sect. 2.1.1) in two re-gards.First, we test the e ff ect of the detrending function. In addi-tion to the trigonometric function, we detrend the light curve bypolynomials of second, third and fourth order.In addition, we test if the inclusion or neglect of data beyondany gaps in the light curve a ff ects the detrending. In one vari-ation of our detrending procedure, we use the entire ±
25 d ofdata (excluding any data within t c ) around a transit midpoint. Inanother variation, we restrict the detrending to the data up to thenearest gap (if present) on both sides of the transit. Available on Zenodo, [10.5281 / zenodo.1202034], Hippke (2018) To avoid the requirement of time-consuming visual inspec-tions of each light curve, we construct an automatic rule to de-termine the presence of gaps, which are the most disruptive kindof artifact to our detrending procedure. We define a gap as aninterruption of the data of more than half a day. Whenever we dodetect a gap, we cut another 12 h at both the beginning and theend of the gap, since our visual inspection of the data showedthat many gaps are preceded or followed by anomalous trends(see e.g. the gap 4 d after transit 4 in Fig. 1).We ignore any data points within t c around the transitmidpoint (see Sect. 2.1.1). If a gap starts within an interval[ t c / , t c / +
12 h] around the transit midpoint, then we lift ourconstraint of dismissing a 12 h interval around gaps and use allthe data within [ t c / , t c / +
12 h] plus any data up to 12 h aroundthe next gap.If all these cuts result in no data points for the detrending pro-cedure to one side of one of the three transits in a sequence, thenwe ignore the entire sequence for our injection-retrieval experi-ment. This is the case for 40 out of the 200 artificially injectedlight curves. This high loss rate of our experimental data is a nat-ural outcome of the gap distribution in the original Kepler-1625light curve. We exclude these 40 light curves for all variationsof the detrending procedure that we investigate. All things com-bined, these constraints produce synthetic light curves with gap
Article number, page 9 of 16 & A proofs: manuscript no. ms
Table 3.
Definition of the detrending identifiers in relation to the re-spective detrending functions that we explored in our transit injection-retrieval experiment of the Kepler-1625 data. We define a gap as anyempty parts in the light curve that show more than 12 h between con-secutive data points. The trigonometric function refers to our reimple-mentation of the CoFiAM algorithm. P2 to P4 refer to polynomials ofsecond to fourth order. T refers to our trigonometric detrending. G standsfor the inclusion of data beyond gaps, N stands for the exclusion of databeyond gaps. Identifier Detrending Function Reject Data Beyond Gap?
P2/G
P2/N
P3/G
P3/N
P4/G
P4/N
T/G trigonometric yes
T/N trigonometric nocharacteristics similar to the original Kepler-1625 b transits (seeFig. 3), that is, we allow the simulation of light curves with gapsclose to but not ranging into the transits. The four detrendingfunctions and our two ways of treating gaps yield a total of eightdi ff erent detrending methods that we investigate (see Table 3).
3. Results
Our first result is a reproduction of a detrended transit light curveof Kepler-1625 b that has the same morphology and moon char-acterization as the one proposed by Teachey et al. (2018) and thathas a negative ∆ BIC. We explore the variation of the free param-eters of our trigonometric detrending procedure, f t p and f t c , andidentify such a detrended light curve for f t p = . f t c = . a s = . + . − . R J and R s = . + . − . R ⊕ . While both the moon radius and semima-jor axis are well constrained, the distribution of the initial planet-moon orbital conjunction ( t , s ) fills out almost the entire allowedparameter range from − / P s to + / P s . The planetary radiusis 0 . + . − . R J , the stellar radius is R ? = . + . − . R (cid:12) , and thedensity is ρ ? = . + . − . ρ (cid:12) .The point of maximum likelihood in the resulting MCMCdistribution is at a s = . R J , R s = . R ⊕ , R ? = . R (cid:12) , ρ ? = . ρ (cid:12) and R p = . R J . The ∆ BIC( M , M ) we foundis -4.954, indicating moderate evidence in favor of an exomoonbeing in the light curve. In Fig. 9 we show the ∆ BIC for the 160 simulated Kepler lightcurves that were not rejected by our detrending method due togaps very close to a transit. The left panel shows our results forthe analysis of planet-only injections and the right panel refersto planet-moon injections. The tables in the panel headers listthe true negative, false positive, true positive, and false positiverates as well as the rates of ambiguous cases. With “positive”(“negative”), we here refer to the detection (non-detection) of amoon. − − Transit 2 t , B = 636 . d MCMC Fitted ModelDetrended SAP Flux -36 -24 -12 t , B
12 24 36 − − − Transit 4 t , B = 1211 . d-36 -24 -12 t , B
12 24 36 − r e l a t i v e fl u x [ pp t ] − − Transit 5 t , B = 1498 . d-36 -24 -12 t , B
12 24 36time from transit midpoint t , B [h] − Light Curve Detrending and Fitting
Fig. 7.
The observed 2nd, 4th, and 5th transits of Kepler 1625 b. Blackdots refer to our detrended light curve from the trigonometric detrend-ing procedure, and orange curves are the model light curves generatedusing the 100 best fitting parameter sets of the MCMC run. The ∆ BIC,calculated from the most likely parameters, is − . In particular, we find the true negative rate (left panel, ∆ BIC ≥
6) to be between 65 % and 87.5 % and the true posi-tive rate (right panel, ∆ BIC ≤ −
6) to be between 31.25 % and46.25 % depending on the detrending method, respectively.The rates of false classifications is between 8.75 % and17.5 % for the injected planet-only systems with a falsely de-tected moon (false positives) and between 30 % and 41.25 % forthe injected planet-moon systems with a failed moon recovery(false negatives).The rates of classification as a planet-moon system dependssignificantly on the treatment of gaps during the detrending pro-cedure. Whenever the light curve is cut at a gap, the detectionrates for a moon increase – both for the false positives and forthe true positives. Among all the detrending methods, this e ff ectis especially strong for the trigonometric detrending. The falsepositive rate increases by almost a factor of two from 8.75 %(T / N) to 16.25 % (T / G) and the true positive rate increases by15 % to 46.25 %. The e ff ect on the true negative rate is strongestfor the trigonometric detrending, decreasing from 87.5 % whenthe light curve is not cut at gaps (T / N) to 72.5 % if the light curveis cut (T / G). The false negative rate for the second order polyno-mial detrending decreases from 41.25 % (P2 / N) to 30 % (P2 / G)when gaps are cut, while the false negative rates of the other de-trending methods remain almost una ff ected.Of all the light curves with an injected planet only, 21.25 %have an ambiguous classification with at least one of the detrend-ing methods showing a negative and a di ff erent method showing Article number, page 10 of 16ai Rodenbeck et al.: Revisiting the exomoon candidate signal around Kepler-1625 b p o s t e r i o r d e n s i t y -3-2-10123 t , s [ d ] p o s t e r i o r d e n s i t y a s [ R J ]012345 R s [ R ] -2.5 0.0 2.5 t
0, s [d] 0 1 2 3 4 5 R s [ R ] 0.00.20.40.6 p o s t e r i o r d e n s i t y p o s t e r i o r d e n s i t y Fig. 8.
Posterior probability distribution of the moon parameters gen-erated by the MCMC algorithm for the light curve detrended by thetrigonometric detrending. The black vertical lines show the median ofthe posterior distribution, the black horizontal lines indicate the 1 σ range around the median. The red vertical lines show the point of max-imum likelihood. The locations of the Galilean moons are included inthe lower left panel for comparison. a positive ∆ BIC above the threshold. For the light curves with aninjected planet-moon system, there are 18.75 % with ambiguousclassification and another 18.75 % of the injected planet-moonsystems are classified unanimously as true positives by all de-trending methods.Fig. 10 shows the distribution of the retrieved moon param-eters a s and R s as well as the corresponding ∆ BIC (see colorscale) for each of the detrending methods.For the light curves with an injected planet-moon system(lower set of panels), the maximum likelihood values of a s and R s of the true positives (blue) generally cluster around the in-jected parameters. In particular, we find that the moon turns outto be more likely (deeper-blue dots) when it is fitted to have alarger radius. The parameters of the false positives (blue dots inthe upper set of panels) are more widely spread out, with moonradii ranging between 2 and 5 R ⊕ and the moon semimajor axesspread out through essentially the entire parameter range that weexplored. The clustering of median a s at around 100 R J is an ar-tifact of taking the median over a very unlocalized distributionalong a s . For the polynomial detrending methods there are a cer-tain number of what one could refer to as mischaracterized truepositives. In these cases the ∆ BIC-based planet-only vs. planet-moon classification is correct but the maximum likelihood valuesare very di ff erent from the injected ones.The correctly identified planet-only systems show a similardistribution of a s and R s as in our experiment with white noiseonly and a 700 ppm amplitude (Fig. 5).Most surprisingly, and potentially most worryingly, the falsepositives (blue dots in the upper set of panels in Fig. 10) clus-ter around the values of the moon parameters found by Teacheyet al. (2018), in particular if the light curve is cut at the first gap.
4. Discussion
In this article we compare several detrending methods of thelight curve of Kepler-1625, some of which were used by Teacheyet al. (2018) in their characterization of the exomoon candidatearound Kepler-1625 b. However, we do not perform an exhaus-tive survey of all available detrending methods, such as Gaussianprocesses (Aigrain et al. 2016).We show that the sequential detrending and fitting procedureof transit light curves is prone to introducing features that can bemisinterpreted as signal, in our case as an exomoon. This “pre-whitening” method of the data has thus to be used with cau-tion. Our investigations of a polynomial-based fitting and of atrigonometric detrending procedure show that the resulting best-fit model depends strongly on the specific detrending function,e.g. on the order of the polynomial or on the minimum time scale(or wavelength) of a cosine. This is crucial for any search of sec-ondary e ff ects in the transit light curves – moons, rings, evapo-rating atmospheres etc. – and is in stark contrast to a claim byAizawa et al. (2017), who stated that neither the choice of thedetrending function nor the choice of the detrending window ofthe light curve would have a significant e ff ect on the result. Wefind that this might be true on a by-eye level but not on a levelof 100 ppm or below. Part of the di ff erence between our findingsand those of Aizawa et al. (2017) could be in the di ff erent timescales we investigate. While they considered the e ff ect of stellarflairs on time scales of less than a day, much less than the ∼ ff erent choicesfor this protected time scale around the transit yield di ff erentconfidences and di ff erent solutions for a planet-moon system.We find that the previously announced solution by Teachey et al.(2018) is only one of many possibilities with similar likelihoods(specifically: Bayesian Information Criteria). This suggests, butby no means proves, that all of these solutions could, in fact, bedue to red noise artifacts (e.g. stellar or instrumental) rather thanindicative of a moon signal.Our finding of higher true positive rate compared to a falsepositive rate from injection-retrieval experiments could be inter-preted as slight evidence in favor of a genuine exomoon. This in-terpretation, however, depends on the number of transiting plan-ets and planet candidates around stars with similar noise charac-teristics that were included in the Teachey et al. (2018) search.Broadly speaking, if more than a handful of similar targets werestudied, the probability of at least one false positive detectionbecomes quite likely.
5. Conclusions
We investigated the detrending of the transit light curve ofKepler-1625 b with a method very similar to the one used byTeachey et al. (2018) and then applied a Bayesian frameworkwith MCMC modeling to search for a moon. Our finding of a ∆ BIC of − .
954 favors the planet-moon over the planet-onlyhypothesis. Although significant, this tentative detection fails tocross the threshold of −
6, which we would consider strong evi-
Article number, page 11 of 16 & A proofs: manuscript no. ms
Fig. 9. Di ff erence between the BIC of the planet-moon model and the no-moon model using di ff erent detrending methods for 160 light curves,generated using the PDCSAP flux of Kepler 1625, injected with three simulated transits. On the left (80 light curves) a planet and moon transit wasinjected, on the right (80 light curves) only the planet. Each light curve consists of three consecutive transits. Each row of 8 detrending methodsuses the same light curve. The rows are sorted by their mean ∆ BIC, with black lines indicating the ∆ BIC = {− , , } positions for the mean ∆ BICper row. dence of a moon. Our ∆ BIC value would certainly change if wecould include the additional data from the high-precision transitobservations executed in October 2017 with the Hubble SpaceTelescope (Teachey et al. 2018) in our analysis. Moreover, byvarying the free parameters of our detrending procedure, we alsofind completely di ff erent solutions for a planet-moon system, i.e.di ff erent planet-moon orbital configurations during transits anddi ff erent moon radii or planet-moon orbital semimajor axes.As an extension to this validation of the previously publishedwork, we performed 200 injection-retrieval experiments into theoriginal out-of-transit parts of the Kepler light curve. We alsoextended the previous work by exploring di ff erent detrendingmethods, such as second-, third-, and fourth-order polynomialsas well as trigonometric methods and find false-positive rates be-tween 8.75 % and 16.25 %, depending on the method. Surpris-ingly, we find that the moon radius and planet-moon distancesof these false positives are very similar to the ones measured byTeachey et al. (2018). In other words, in 8.75 % to 16.25 % of thelight curves that contained an artificially injected planet only, wefind a moon that is about as large as Neptune and orbits Kepler-1625 b at about 20 R J .To sum up, we find tentative statistical evidence for a moonin this particular Kepler light curve of Kepler-1625, but we alsoshow that the significant fraction of similar light curves, whichcontained a planet only, would nevertheless indicate a moon withproperties similar to the candidate Kepler-1625 b-i. Clearly, stel-lar and systematic red noise components are the ultimate barrierto an unambiguous exomoon detection around Kepler-1625 band follow-up observations have the potential of solving this rid-dle based on the framework that we present.Of all the detrending methods we investigated, the trigono-metric method, which is very similar to the CoFiAM methodof Teachey et al. (2018), can produce the highest true positiverate. At the same time, however, this method also ranks amongthe ones producing the highest false positive rates as well. Toconclude, we recommend that any future exomoon candidate be detrended with as many di ff erent detrending methods as possibleto evaluate the robustness of the classification. Acknowledgements.
We thank James Kuszlewicz and Jesper Schou for usefuldiscussions. This work was supported in part by the German Aerospace Center(DLR) under PLATO Data Center grant 50OL1701. This paper includes data col-lected by the Kepler mission. Funding for the Kepler mission is provided by theNASA Science Mission directorate. This work has made use of data provided byNASA and the Space Telescope Science Institute. K.R. is a member of the Inter-national Max Planck Research School for Solar System Science at the Universityof Göttingen. K.R. contributed to the analysis of the simulated light curves, tothe interpretation of the results, and to the writing of the article.
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Distribution of the median likelihood R s and a s for the transits injected into di ff erent parts of the Kepler-1625 light curve, using di ff erentdetrending methods. The ∆ BIC of the planet-only model compared to the planet-moon model is indicated by the symbol color. The values of themoon semimajor axis (abscissa) and radius (ordinate) suggested by Teachey et al. (2018) are indicated with thin, gray lines in each sub-panel.
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Appendix A: Effect of the window length on theBayesian Information Criterion
Given the constraint of orbital stability, a moon can only possi-bly orbit its planet within the planet’s Hill sphere. Hence, transitsmay only occur within a certain time interval around the mid-point of the planetary transit. This time t Hill can be calculatedas t Hill = η R Hill v orbit = η P π s M p M ? , (A.1)where v orbit is the orbital velocity of the planet-moon systemaround the star, M p and M ? the planet and star mass, P the orbitalperiod of the planet-moon system, and R Hill is the Hill radius ofthe planet. η is a factor between 0 and 1, which has been nu-merically determined for prograde moons ( η ≈ .
5) and forretrograde moons ( η ≈ η = .
5. For a 10 M J planet in a 287 d orbit around a1 . M (cid:12) star the Hill time is t Hill = .
25 d.As shown in Fig. 2, the length of the light curve, which is ne-glected for the polynomial fit has a strong e ff ect on the resultingdetrended light curve. Figure A.1 shows the e ff ect that di ff erentcutout times t c and detrending base lines D can have on whethera moon is detected or not. Article number, page 14 of 16ai Rodenbeck et al.: Revisiting the exomoon candidate signal around Kepler-1625 b
630 635 640 6454202 r e l a t i v e f l u x [ pp t ] B I C B I C B I C
630 635 640 6454202 r e l a t i v e f l u x [ pp t ] B I C B I C B I C Fig. A.1.
Detrending for di ff erent cutout times t c and base length D , color coded by the resulting ∆ BIC using a 2nd- and 4th-order polynomialfunction. While some of the detrending models corresponding to a large negative ∆ BIC are clearly results of wrong detrending, it is much lessclear for many other detrending models. Article number, page 15 of 16 & A proofs: manuscript no. ms
Appendix B: Autocorrelation of Detrended LightCurves
The autocorrelations of the detrended light curves are shown inFig. B.1. For all three transits, the autocorrelation is close to zero,except for the zero-lag component. This suggests that it is rea-sonable to model the noise covariance matrix as a diagonal ma-trix (see Sect. 2.2.3). a u t o c o rr e l a t i o n Transit 40 5 10 15 20time lag [h]0.00.51.0 Transit 5
Fig. B.1.
The autocorrelation of the di ff erence between the detrendedlight curve and the best fitting model.Article number, page 16 of 16 .8. AN ALTERNATIVE INTERPRETATION OF THE EXOMOON CANDIDATE SIGNAL INTHE COMBINED KEPLER AND HUBBLE DATA OF KEPLER-1625 (Heller et al. 2019) 165 Contribution:RH guided the work, did the literature research, created Fig. 5, led the writing of the manuscript, andserved as a corresponding author for the journal editor and the referees. stronomy & Astrophysics submitted c (cid:13)
ESO 2019April 24, 2019
An alternative interpretation of the exomoon candidate signalin the combined
Kepler and
Hubble data of Kepler-1625
René Heller , Kai Rodenbeck , , and Giovanni Bruno Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, [email protected], [email protected] Institute for Astrophysics, Georg August University Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany INAF, Astrophysical Observatory of Catania, Via S. Sofia 78, 95123 Catania, Italy, [email protected] 18 December 2018; Accepted 25 February 2019
ABSTRACT
Context.
Kepler and
Hubble photometry of a total of four transits by the Jupiter-sized exoplanet Kepler-1625 b have recently beeninterpreted to show evidence of a Neptune-sized exomoon. The key arguments were an apparent drop in stellar brightness after theplanet’s October 2017 transit seen with
Hubble and its 77.8 min early arrival compared to a strictly periodic orbit.
Aims.
The profound implications of this first possible exomoon detection and the physical oddity of the proposed moon, i.e., its giantradius prompt us to examine the planet-only hypothesis for the data and to investigate the reliability of the Bayesian informationcriterion (BIC) used for detection.
Methods.
We combined
Kepler ’s Pre-search Data Conditioning Simple Aperture Photometry (PDCSAP) with the previously pub-lished
Hubble light curve. In an alternative approach, we performed a synchronous polynomial detrending and fitting of the
Kepler data combined with our own extraction of the
Hubble photometry. We generated five million parallel-tempering Markov chain MonteCarlo (PTMCMC) realizations of the data with both a planet-only model and a planet-moon model, and compute the BIC di ff erence( ∆ BIC) between the most likely models, respectively.
Results.
The ∆ BIC values of − . Hubble data) and − . ff erent from the best-fit solutions,suggesting that the likelihood function that best describes the data is non-Gaussian. We measure a 73.7 min early arrival of Kepler-1625 b for its Hubble transit at the 3 σ level. This deviation could be caused by a 1 d data gap near the first Kepler transit, stellaractivity, or unknown systematics, all of which a ff ect the detrending. The radial velocity amplitude of a possible unseen hot Jupitercausing the Kepler-1625 b transit timing variation could be approximately 100 m s − . Conclusions.
Although we find a similar solution to the planet-moon model to that previously proposed, careful consideration of itsstatistical evidence leads us to believe that this is not a secure exomoon detection. Unknown systematic errors in the
Kepler / Hubble data make the ∆ BIC an unreliable metric for an exomoon search around Kepler-1625 b, allowing for alternative interpretations of thesignal.
Key words. eclipses – methods: data analysis – planets and satellites: detection – planets and satellites: dynamical evolution andstability – planets and satellites: individual: Kepler-1625 b – techniques: photometric
1. Introduction
The recent discovery of an exomoon candidate around thetransiting Jupiter-sized object Kepler-1625 b orbiting a slightlyevolved solar mass star (Teachey et al. 2018) came as a sur-prise to the exoplanet community. This Neptune-sized exomoon,if confirmed, would be unlike any moon in the solar system, itwould have an estimated mass that exceeds the total mass of allmoons and rocky planets of the solar system combined. It is cur-rently unclear how such a giant moon could have formed (Heller2018).Rodenbeck et al. (2018) revisited the three transits obtainedwith the
Kepler space telescope between 2009 and 2013 andfound marginal statistical evidence for the proposed exomoon.Their transit injection-retrieval tests into the out-of-transit
Ke-pler data of the host star also suggested that the exomoon couldwell be a false positive. A solution to the exomoon question wassupposed to arrive with the new
Hubble data of an October 2017transit of Kepler-1625 b (Teachey & Kipping 2018). The new evidence for the large exomoon by Teachey & Kip-ping (2018), however, remains controversial. On the one hand,the
Hubble transit light curve indeed shows a significant de-crease in stellar brightness that can be attributed to the previ-ously suggested moon. Perhaps more importantly, the transit ofKepler-1625 b occurred 77.8 min earlier than expected from a se-quence of strictly periodic transits, which is in very good agree-ment with the proposed transit of the exomoon candidate, whichoccurred before the planetary transit. On the other hand, an up-grade of
Kepler ’s Science Operations Center pipeline from ver-sion 9.0 to version 9.3 caused the exomoon signal that was pre-sented in the Simple Aperture Photometry (SAP) measurementsin the discovery paper (Teachey et al. 2018) to essentially van-ish in the SAP flux used in the new study of Teachey & Kip-ping (2018). This inconsistency, combined with the findings ofRodenbeck et al. (2018) that demonstrate that the characteriza-tion and statistical evidence for this exomoon candidate dependstrongly on the methods used for data detrending, led us to revisitthe exomoon interpretation in light of the new
Hubble data.
Article number, page 1 of 8 a r X i v : . [ a s t r o - ph . E P ] A p r & A submitted
Here we address two questions. How unique is the pro-posed orbital solution of the planet-moon system derived withthe Bayesian information criterion (BIC)? What could be the rea-son for the observed 77.8 min di ff erence in the planetary transittiming other than an exomoon?
2. Methods
Our first goal was to fit the combined
Kepler and
Hubble datawith our planet-moon transit model (Rodenbeck et al. 2018)and to derive the statistical likelihood for the data to representthe model. In brief, we first model the orbital dynamics of thestar-planet-moon system using a nested approach, in which theplanet-moon orbit is Keplerian and unperturbed by the stellargravity. The transit model consists of two black circles, onefor the planet and one for the moon, that pass in front of thelimb-darkened stellar disk. The resulting variations in the stellarbrightness are computed using Ian Crossfield’s python code ofthe Mandel & Agol (2002) analytic transit model. The entiremodel contains 16 free parameters and it features three majorupdates compared to Rodenbeck et al. (2018): (1) Planet-moonoccultations are now correctly simulated, (2) the planet’s motionaround the local planet-moon barycenter is taken into account,and (3) inclinations between the circumstellar orbit of the planet-moon barycenter and the planet-moon orbit are now included.We used the emcee code of Foreman-Mackey et al. (2013)to generate Markov chain Monte Carlo (MCMC) realizations ofour planet-only model ( M ) and planet-moon model ( M ) andto derive posterior probability distributions of the set of modelparameters ( θ ). We tested both a standard MCMC samplingwith 100 walkers and a parallel-tempering ensemble MCMC(PTMCMC) with five temperatures, each of which has 100 walk-ers. As we find a better convergence rate for the PTMCMC sam-pling, we use it in the following. Moreover, PTMCMC can sam-ple both the parameter space at large and in regions with tightpeaks of the likelihood function. The PTMCMC sampling is al-lowed to walk five million steps.The resulting model light curves are referred to as F i ( t , θ ),where t are the time stamps of the data points from Kepler and
Hubble ( N measurements in total), for which time-uncorrelatedstandard deviations σ j at times t j are assumed, following thesuggestion of Teachey & Kipping (2018). This simplifies thejoint probability density of the observed (and detrended) fluxmeasurements ( F ( t )) to the product of the individual probabil-ities for each data point, p ( F | θ , M i ) = N Y j = q πσ j exp − (cid:16) F ( t j ) − F i ( t j , θ ) (cid:17) σ j . (1)We then determined the set of parameters ( θ max ) that maxi-mizes the joint probability density function ( p ( F | θ max , M i )) fora given light curve F ( t j ) and model M i and calculated the BIC(Schwarz 1978)BIC( M i | F ) = m i ln N − p ( F | θ max , M i ) . (2)The advantage of the BIC in comparison to χ minimization, forexample, is in its relation to the number of model parameters / ∼ ianc / files Available at http: // dfm.io / emcee / current / user / pt ( m i ) and data points. The more free parameters in the model,the stronger the weight of the first penalty term in Eq. (2),thereby mitigating the e ff ects of overfitting. Details of the ac-tual computer code implementation or transit simulations aside,this Bayesian framework is essentially what the Hunt for Exo-moons with Kepler survey used to identify and rank exomooncandidates (Kipping et al. 2012), which ultimately led to the de-tection of the exomoon candidate around Kepler-1625 b after itsfirst detection via the orbital sampling e ff ect (Heller 2014; Helleret al. 2016a). In a first step, we used
Kepler ’s Pre-search Data ConditioningSimple Aperture Photometry (PDCSAP) and the
Hubble
WideField Camera 3 (WFC3) light curve as published by Teachey &Kipping (2018) based on their quadratic detrending. Then weexecuted our PTMCMC fitting and derived the ∆ BIC values andthe posterior parameter distributions.In a second step, we did our own extraction of the
Hubble light curve including an exponential ramp correction for each
Hubble orbit. Then we performed the systematic trend correc-tion together with the transit fit of a planet-moon model. Ourown detrending of the light curves is not a separate step, but it isintegral to the fitting procedure. For each calculation of the like-lihood, we find the best fitting detrending curve by dividing theobserved light curve by the transit model and by fitting a third-order polynomial to the resulting light curve. Then we removethe trend from the original light curve by dividing it through thebest-fit detrending polynomial and evaluate the likelihood. Wealso performed a test in which the detrending parameters werefree PTMCMC model parameters and found similar results forthe parameter distributions but at a much higher computationalcost. We note that the resulting maximum likelihood is (and mustbe) the same by definition if the PTMCMC sampling converges.Kepler-1625 was observed by
Hubble under the GO program15149 (PI Teachey). The observations were secured from Oc-tober 28 to 29, 2017, to cover the ∼
20 hr transit plus severalhours of out-of-transit stellar flux (Teachey & Kipping 2018).The F130N filter of WFC3 was used to obtain a single directimage of the target, while 232 spectra were acquired with theG141 grism spanning a wavelength range from 1.1 to 1 . µ m.Due to the faintness of the target, it was observed in staring mode(e.g. Berta et al. 2012; Wilkins et al. 2014) unlike the most re-cent observations of brighter exoplanet host stars, which weremonitored in spatial scanning mode (McCullough & MacKenty2012). Hence, instead of using the IMA files as an intermediateproduct, we analyzed the FLT files, which are the final outputof the calwfc3 pipeline of Hubble and allow a finer manipula-tion of the exposures during consecutive nondestructive reads.Each FLT file contains measurements between about 100 and300 electrons per second, with exposure times of about 291 sec-onds.We used the centroid of the stellar image to calculate thewavelength calibration, adopting the relations of Pirzkal et al.(2016). For each spectroscopic frame, we first rejected the pix-els flagged by calwfc3 as “bad detector pixels”, pixels with un-stable response, and those with uncertain flux value (Data Qual-ity condition 4, 32, or 512). Then we corrected each frame withthe flat field file available on the Space Telescope Science Insti-tute (STScI) website by following the prescription of the WFC3 / hst / wfc3 / analysis / grism_obs / calibrations / wfc3_g141.htmlArticle number, page 2 of 8ené Heller et al.: Alternative explanations for the exomoon interpretation for Kepler-1625 s p a ti a l d i r ec ti on e l ec t r on s / s e l ec t r on s / s Fig. 1.
Top : Example of a WFC3 exposure of Kepler-1625. The abscissashows the column pixel prior to wavelength calibration. The yellow boxindicates the region used for background estimation. The spectrum ofKepler-1625 is at the center of the frame, around row 150 in the spatialdirection, while several contaminant sources are evident in other regionsof the detector. The color bar illustrates the measured charge values.
Bottom : Background value measured across the rows of the same frame. online manual. We performed the background subtraction on acolumn-by-column basis. Due to a number of contaminant starsin the observation field (Fig. 1, top panel), we carefully selecteda region on the detector that was as close as possible to the spec-trum of Kepler-1625, close to row 150 in spatial dimension, andfar from any contaminant. For each column on the detector, weapplied a 5 σ clipping to reject the outliers and then calculatedthe median background flux value in that column. FollowingSTScI prescriptions, we also removed pixels with an electron-per-second count larger than 5. An example for the backgroundbehavior is shown in the bottom panel of Fig. 1.We inspected each frame with the image registration package (Baker & Matthews 2001) to search drifts in both axesof the detector with respect to the very last frame, and then ex-tracted the spectrum of Kepler-1625 by performing optimal ex-traction (Horne 1986) on the detector rows containing the stel-lar flux. This procedure automatically removes bad pixels andcosmic rays from the frames by correcting them with a smooth-ing function. We started the extraction with an aperture of a fewpixels centered on the peak of the stellar trace and gradually in-creased its extension by one pixel per side on the spatial directionuntil the flux dispersion reached a minimum.We performed another outlier rejection by stacking all theone-dimensional spectra along the time axis. We computed amedian-filtered version of the stellar flux at each wavelengthbin and performed a 3 σ clipping between the computed fluxand the median filter. Finally, we summed the stellar flux acrossall wavelength bins from 1.115 to 1.645 µ m to obtain the band-integrated stellar flux corresponding to each exposure. Before performing the PTMCMC optimization, we removedthe first Hubble orbit from the data set and the first data pointof each
Hubble orbit, as they are a ff ected by stronger instrumen-tal e ff ects than the other observations (Deming et al. 2013) andcannot be corrected with the same systematics model. We alsoremoved the last point of the 12th, 13th, and 14th Hubble or-bit since they were a ff ected by the passage of the South AtlanticAnomaly (as highlighted in the proposal file, available on theSTScI website). According to Teachey et al. (2018), the 2017
Hubble transit ofKepler-1625 b occurred about 77.8 min earlier than predicted, ane ff ect that could be astrophysical in nature and is referred to asa transit timing variation (TTV). As proposed by Teachey et al.(2018), this TTV could either be interpreted as evidence for anexomoon or it could indicate the presence of a hitherto unseenadditional planet. Various planetary configurations can cause theobserved TTV e ff ect such as an inner planet or an outer planet.At this point, no stellar radial velocity measurements of Kepler-1625 exist that could be used to search for additional nontransit-ing planets in this system.In the following, we focus on the possibility of an innerplanet with a much smaller orbital period than Kepler-1625 bsimply because it would have interesting observational conse-quences. We use the approximation of Agol et al. (2005) for theTTV amplitude ( δ t ) due to a close inner planet, which wouldimpose a periodic variation on the position of the star, and solvetheir expression for the mass of the inner planet ( M p , in ) as a func-tion of its orbital semimajor axis ( a p , in ), M p , in = δ t M ? a p , out a p , in P p , out , (3)where a out = .
87 AU is the semimajor axis of Kepler-1625 b.The validity of this expression is restricted to coplanar systemswithout significant planet-planet interaction and with a out (cid:29) a in ,so that TTVs are only caused by the reflex motion of the stararound its barycenter with the inner planet.As we show in Sect. 3.2, the proposed inner planet could be ahot Jupiter. The transits of a Jupiter-sized planet, however, wouldbe visible in the Kepler data. As a consequence, we can esti-mate the minimum orbital inclination ( i ) between Kepler-1625 band the suspected planet to prevent the latter from showing tran-sits. This angle is given as per i = arctan( R ? / a p , in ) and we use R ? = . + . − . R (cid:12) (Mathur et al. 2017). We can exclude certain masses and orbital semimajor axes foran unseen inner planet based on the criterion of mutual Hill sta-bility. This instability region depends to some extent on the un-known mass of Kepler-1625 b. Mass estimates can be derivedfrom a star-planet-moon model, but these estimates are irrelevantif the observed TTVs are due to an unseen planetary perturber.Hence, we assume a nominal Jupiter mass ( M Jup ) for Kepler-1625 b.The Hill sphere of a planet with an orbital semimajoraxis a p around a star with mass M ? can be estimated as R H = a p ( M p / [3 M ? ]) / , which suggests R H = R Jup for
Article number, page 3 of 8 & A submitted
636 6380 . . . . r e l a t i v e flu x a . . b . . . . r e l a t i v e flu x c d
636 6380 . . . . r e l a t i v e flu x e . . f . . . . r e l a t i v e flu x g h moon ingressno moon egress Fig. 2.
Orbital solutions for Kepler-1625 b and its suspected exomoon based on the combined
Kepler and
Hubble data. ( a,b,c ) Kepler
PDCSAP fluxand ( d ) the quadratic detrending of the Hubble data from Teachey & Kipping (2018). The blue curves show 1 000 realizations of our PTMCMCfitting of a planet-moon model. Our most likely solution (red line) is very similar to the one found by Teachey & Kipping (2018), but di ff erssignificantly from the one initially found by Teachey et al. (2018). ( e,f,g ) Kepler
PDCSAP flux and ( h ) our own detrending of the Hubble lightcurve (in parallel to the fitting). The ingress and egress of the model moon are denoted with arrows and labels in panel h as an example. Kepler-1625 b. We calculate the Hill radius of the proposed in-ner planet accordingly, and identify the region in the mass-semimajor axis diagram of the inner planet that would lead toan overlap of the Hill spheres and therefore to orbital instability.
3. Results ∆ BIC
Regarding the combined data set of the
Kepler and
Hubble dataas detrended by Teachey & Kipping (2018), we find a ∆ BIC of − . Kepler and
Hubble light curves based on our own extraction of the WFC3 data yieldsa ∆ BIC of − .
0. Formally speaking, both of these two valuescan be interpreted as strong statistical evidence for an exomooninterpretation. The two values are very di ff erent, however, whichsuggests that the detrending of the Hubble data has a significante ff ect on the exomoon interpretation. In other words, this illus-trates that the systematics are not well-modeled and poorly un-derstood.In Fig. 2a-d, we show our results for the PTMCMC fit-ting of our planet-moon model to the four transits of Kepler-1625 b including the Hubble data as extracted and detrended byTeachey & Kipping (2018) using a quadratic fit. Although ourmost likely solution shows some resemblance to the one pro-posed by Teachey & Kipping (2018), we find that several aspectsare di ff erent. As an example, the second Kepler transit (Fig. 2b)is fitted best without a significant photometric moon signature,that is to say, the moon does not pass in front of the stellar disk ,whereas the corresponding best-fit model of Teachey & Kipping(2018) shows a clear dip prior to the planetary transit (see theirFig. 4). What is more, most of our orbital solutions (blue lines)di ff er substantially from the most likely solution (red line). Inother words, the orbital solutions do not converge and various Martin et al. (2019) estimate that failed exomoon transits should ac-tually be quite common for misaligned planet-moon systems, such asthe one proposed by Teachey & Kipping (2018). planet-moon orbital configuration are compatible with the data,though with lower likelihood.In Fig. 2e-h, we illustrate our results for the PTMCMC fittingof our planet-moon model to the four transits of Kepler-1625 bincluding our own extraction and detrending of the
Hubble tran-sit. Again, the orbital solutions (blue lines) do not converge. Acomparison of panels d and h shows that the di ff erent extractionand detrending methods do have a significant e ff ect on the indi-vidual flux measurements, in line with the findings of Rodenbecket al. (2018). Although the time of the proposed exomoon transitis roughly the same in both panels, we find that the best-fit so-lution for the data detrended with our own reduction proceduredoes not contain the moon egress (panel h), whereas the best-fit solution of the data detrended by Teachey & Kipping (2018)does contain the moon egress (panel d). A similar fragility ofthis particular moon egress has been noted by Teachey & Kip-ping (2018) as they explored di ff erent detrending functions (seetheir Fig. 3).Our Fig. 3 illustrates the distribution of the di ff erential likeli-hood for the planet-moon model between the most likely modelparameter set ( θ max ) and the parameter sets ( θ ) found after fivemillion steps of our PTMCMC fitting procedure, p ( θ | F , M ) − p ( θ max | F , M ). For the combined Kepler and
Hubble data de-trended by Teachey & Kipping (2018) (left panel) and for ourown
Hubble data extraction and detrending (right panel), we findthat most model solutions cluster around a di ff erential likelihoodthat is very di ff erent from the most likely solution, suggestingthat the most likely model is, in some sense, a statistical out-lier. We initially detected this feature after approximately thefirst one hundred thousand PTMCMC fits. Hence, we increasedthe number of PTMCMC samplings to half a million and finallyto five million to make sure that we sample any potentially nar-row peaks of the likelihood function near the best-fit model at p ( θ | F , M ) − p ( θ max | F , M ) = ffi cient accuracy. Wefind, however, that this behavior of the di ff erential likelihooddistribution clustering far from the best-fit solution persists, ir-respective of the available computing power devoted to the sam-pling. Article number, page 4 of 8ené Heller et al.: Alternative explanations for the exomoon interpretation for Kepler-1625 p ( ~✓ | F, M ) ln p ( ~✓ max | F, M )0 . . . . . . p r o b a b ili t y d e n s i t y p ( ~✓ | F, M ) ln p ( ~✓ max | F, M )0 . . . . . . . p r o b a b ili t y d e n s i t y Fig. 3. Di ff erential likelihood distribution between the most likely planet-moon model and the other solutions using 10 steps of our PTMCMCfitting procedure. Left : Results from fitting our planet-moon transit model to the original data from Teachey & Kipping (2018).
Right : Results fromfitting our planet-moon transit model to our own detrending of the
Kepler and WFC3 data. In both panels the most likely model is located at 0along the abscissa by definition. In both cases the models do not converge to the best-fit solution, suggesting that the best-fit solution could in factbe an outlier.
Table 1.
Results of our PTMCMC fitting procedure to the combined
Ke-pler and
Hubble data. The
Hubble data was either based on the photom-etry extracted by Teachey & Kipping (2018, TK18b, central column) orbased on our own extraction (right column).TK18b
HST photometry Our
HST photometry r s [%] 1 . + . − . . + . − . a ps [ R ? ] 2 . + . − . . + . − . P s [d] 27 + − + − ϕ [rad] 3 . + . − . . + . − . f M [%] 1 . + . − . . + . − . i s [rad] − . + . − . − . + . − . Ω s [rad] − . + . − . − . + . − . Notes.
Figure 4 illustrates quite clearly that the posterior distributionsare not normally distributed and often not even representative of skewednormal distributions. The confidence intervals stated in this table havethus to be taken with care.
Figure 4 shows the posterior distributions of the moon pa-rameters of our planet-moon model. The top panel refers toour PTMCMC fitting of the combined
Kepler and
Hubble data(
Hubble data as detrended and published by Teachey & Kipping2018), and the bottom panel shows our PTMCMC fitting of the
Kepler data combined with our own extraction and detrending ofthe
Hubble light curve. The respective median values and stan-dard deviations are noted in the upper right corners of each sub-panel and summarized in Table 1.A comparison between the upper and lower corner plots inFig. 4 reveals that the di ff erent detrending and fitting techniqueshave a significant e ff ect on the resulting posterior distributions,in particular for i s and Ω s , the two angles that parameterize theorientation of the moon orbit. At the same time, however, themost likely values (red dots above the plot diagonal) and me-dian values (blue crosses below the plot diagonal) of the sevenparameters shown are well within the 1 σ tolerance. The following features can be observed in both panels ofFig. 4. The moon-to-star radius ratio (Col. 1, leftmost) showsan approximately normal distribution, whereas the scaled planet-moon orbital semimajor axis (Col. 2) shows a more complicated,skewed distribution. The solutions for the orbital period of theexomoon candidate (Col. 3) show a comb-like structure owingto the discrete number of completed moon orbits that would fit agiven value of the moon’s initial orbital phase (Col. 4), which isessentially unconstrained. The moon-to-planet mass ratio (Col.5) then shows a skewed normal distribution with a tail of largemoon masses. Our results for the inclination i s between the satel-lite orbit (around the planet) and the line of sight, and for thelongitude of the ascending node of the moon orbit are shownin Cols. 6 and 7. The preference of i s being either near 0 ornear ± π (the latter is equivalent to a near-coplanar retrogrademoon orbit) illustrates the well-known degeneracy of the pro-grade / retrograde solutions available from light curve analyses(Lewis & Fujii 2014; Heller & Albrecht 2014). Next we consider the possibility of the transits being caused by aplanet only. Neglecting the
Hubble transit, our PTMCMC sam-pling of the three
Kepler transits with our planet-only transitmodel gives an orbital period of P = . ± . t = . ± . − , , . Hubble transit is 3222 . ± . Hubble transitgives a transit midpoint at 3222 . ± . . ± . Hubble transit occurred 77.8 minearlier than predicted. This observed early transit of Kepler-1625 b has a formal ∼ σ significance. We note, however, thatthis 3 σ deviation is mostly dictated by the first transit observedwith Kepler (see Fig. S12 in Teachey et al. 2018). We also notethat this transit was preceded by a ∼ Article number, page 5 of 8 & A submitted r s =1 . +0 . . % a p s / R ? a ps R ? =2 . +1 . . P s [ d ] P s =27 +15 d ' ' =3 . +2 . . f M [ % ] f M =1 . +1 . . % i s i s = . +2 . . r s [%] ⌦ s a ps /R ?
15 30 45 P s [d] 0 2.5 5 ' f M [%] i s . . . s ⌦ s = . +2 . . r s =1 . +0 . − . % a p s / R ? a ps R ? =2 . +1 . − . P s [ d ] P s =29 +17 − d ϕ ϕ =3 . +2 . − . f M [ % ] f M =2 . +1 . − . % − i s i s = − . +2 . − . r s [%] − Ω s a ps /R ? P s [d] 0 2.5 5 ϕ f M [%] − i s − . . . s Ω s = − . +2 . − . Fig. 4.
Posterior distributions of a parallel tempering ensemble MCMC sampling of the combined
Kepler and WFC3 data with our planet-moonmodel.
Top : Results for the original data from Teachey & Kipping (2018).
Bottom : Results for our own detrending of the
Kepler and WFC3 data. Inboth figures, scatter plots are shown with black dots above the diagonal, and projected histograms are shown as colored pixels below the diagonal.The most likely parameters are denoted with an orange point in the scatter plots. Histograms of the moon-to-star radius ratio r s , scaled semimajoraxis of the planet-moon system ( a ps / R ? ), satellite orbital period ( P s ), satellite orbital phase ( ϕ ), moon-to-planet mass ratio ( f M ), orbital inclinationof the satellite with respect to our line of sight ( i s ), and the orientation of the ascending node of the satellite orbit ( Ω s ) are shown on the diagonal.Median values and standard deviations are indicated with error bars in the histograms.Article number, page 6 of 8ené Heller et al.: Alternative explanations for the exomoon interpretation for Kepler-1625 m a ss o f s u s p ec t e d p l a n e t [ M J up ] semi-major axis [AU]exoplanet.eu 0.1 1 10 0.02 0.1 1 un s t a b l e
151 m/s923 m/s7.8 o o Fig. 5.
Mass estimate for the potential inner planet around Kepler-1625based on the observed TTV of 73.728 min. The thin pale blue fan aroundthe solid curve shows the 1 σ tolerance fan of ± .
016 min. Values forsemimajor axes > . . M (cid:12) and 1 . M (cid:12) . A conservative estimate of a dy-namically unstable region for the suspected inner planet, where its Hillsphere would touch the Hill sphere of Kepler-1625 b with an assumedmass of 1 M Jup , is shaded in pale red. RV amplitudes and minimum or-bital inclination with respect to Kepler-1625 b are noted along the curvefor the planetary mass estimate. of the TTV e ff ect being due to the large deviation from the lin-ear ephemeris of the first transit, stellar (or any other systematic)variability could have a large (but unknown) e ff ect on the errorbars that go into the calculations.In Fig. 5 we show the mass of an unseen inner planet thatis required to cause the observed 73 . . M Jup at 0.03 AU to 1 . M Jup at0.1 AU. Values beyond 0.1 AU cannot be assumed to fulfill theapproximations made for Eq. (3) and are therefore shown with adashed line. The actual TTV amplitude of Kepler-1625 b couldeven be higher than the ∼
73 min that we determined for the
Hubble transit, and thus the mass estimates shown for a possibleunknown inner planet serve as lower boundaries.The resulting radial velocity amplitudes of the star of923 m s − (at 0.03 AU) and 151 m s − (at 0.1 AU), respectively,are indicated along the curve. Even if the approximations fora coplanar, close-in planet were not entirely fulfilled, our re-sults suggest that RV observations of Kepler-1625 with a high-resolution spectrograph attached to a very large (8 m class)ground-based telescope could potentially reveal an unseen planetcausing the observed TTV of Kepler-1625 b. Also shown alongthe curve in Fig. 5 are the respective minimum orbital inclina-tions (rounded mean values shown) between Kepler-1625 b andthe suspected close-in planet required to prevent Kepler-1625 bfrom transiting the star. The exact values are i = . + . − . degreesat 0.03 AU and i = . + . − . degrees at 0.1 AU.The pale red shaded region is excluded from a dynamicalpoint of view since this is where the planetary Hill spheres wouldoverlap. The extent of this region is a conservative estimate be-cause it assumes a mass of 1 M Jup for Kepler-1625 b and neglectsany chaotic e ff ects induced by additional planets in the systemor planet-planet cross tides etc. The true range of unstable orbitsis probably larger. The black dots show all available exoplanetmasses and semimajor axes from the Exoplanet Encyclopaedia, which illustrates that the suspected planet could be more massivethan most of the known hot Jupiters.
4. Conclusions
With a ∆ BIC of − . Hubble data of Teachey& Kipping 2018) or − . Hubble extractionand detrending) between the most likely planet-only model andthe most likely planet-moon model, we find strong statisticalevidence for a roughly Neptune-sized exomoon. In both casesof the data detrending, the most likely orbital solution of theplanet-moon system, however, is very di ff erent from most ofthe other orbital realizations of our PTMCMC modeling and themost likely solutions do not seem to converge. In other words,the most likely solution appears to be an outlier in the distri-bution of possible solutions and small changes to the data canhave great e ff ects on the most likely orbital solution found forthe planet-moon system. As an example, we find that the twodi ff erent detrending methods that we explored produce di ff erentinterpretations of the transit observed with Hubble : in one caseour PTMCMC sampling finds the egress of the moon in the lightcurve, in the other case it does not (Fig. 2).Moreover, the likelihood of this best-fit orbital solution isvery di ff erent from the likelihoods of most other solutions fromour PTMCMC modeling. We tested both a standard MCMCsampling and a parallel-tempering MCMC (Foreman-Mackeyet al. 2013); the latter is supposed to explore both the parame-ter space at large and the tight peaks of the likelihood function indetail. Our finding of the nonconvergence could imply that thelikelihood function that best describes the data is non-Gaussian.Alternatively, with the BIC being an asymptotic criterion that re-quires a large sample size by definition (Stevenson et al. 2012),our findings suggest that the available data volume is simply toosmall for the BIC to be formally applicable. We conclude thatthe ∆ BIC is an unreliable metric for an exomoon detection forthis data set of only four transits and possibly for other data setsof
Kepler as well.One solution to evaluating whether the BIC or an alternativeinformation criterion such as the Akaike information criterion(AIC; Akaike 1974) or the deviance information criterion (DIC;Spiegelhalter et al. 2002) is more suitable for assessing the likeli-hoods of a planet-only model and of a planet-moon model couldbe injection-retrieval experiments of synthetic transits (Helleret al. 2016b; Rodenbeck et al. 2018). Such an analysis, however,goes beyond the scope of this paper.We also observe the TTV e ff ect discovered by Teachey &Kipping (2018). If the early arrival of Kepler-1625 b for its late-2017 transit was caused by an inner planet rather than by an exo-moon, then the planet would be a super-Jovian mass hot Jupiter,the exact mass limit depending on the assumed orbital semimajoraxis. For example, the resulting stellar radial velocity amplitudewould be about 900 m s − for a 5 . M Jup planet at 0.03 AU andabout 150 m s − for a 1 . M Jup planet at 0.1 AU. From the ab-sence of a transit signature of this hypothetical planet in the fouryears of
Kepler data, we conclude that it would need to have anorbital inclination of at least i = . + . − . (if it were at 0.03 AU)or i = . + . − . degrees (if it were at 0.1 AU). If its inclination isnot close to 90 ◦ , at which point its e ff ect on the stellar RV ampli-tude would vanish, then the hypothesis of an unseen inner planetcausing the Kepler-1625 TTV could be observationally testable.Ground-based photometric observations are hardly practica-ble to answer the question of this exomoon candidate becausecontinuous in- and near-transit monitoring of the target is re- Article number, page 7 of 8 & A submitted quired over at least two days. Current and near-future space-based exoplanet missions, on the other hand, will likely not beable to deliver the signal-to-noise ratios required to validate orreject the exomoon hypothesis. With a
Gaia
G-band magnitudeof m G = .
76 (Gaia Collaboration et al. 2016, 2018) the star israther faint in the visible regime of the electromagnetic spectrumand the possible moon transits are therefore beyond the sensitiv-ity limits of the
TESS , CHEOPS , and
PLATO missions. observations suggest that Kepler-1625 is somewhat brighter inthe near-infrared (Cutri et al. 2003), such that the
James WebbSpace Telescope (launch currently scheduled for early 2021)should be able to detect the transit of the proposed Neptune-sized moon, for example via photometric time series obtainedwith the NIRCam imaging instrument.All things combined, the fragility of the proposed photomet-ric exomoon signature with respect to the detrending methods,the unknown systematics in both the
Kepler and the
Hubble data,the absence of a proper assessment of the stellar variability ofKepler-1625, the faintness of the star (and the resulting photo-metric noise floor), the previously stated coincidence of the pro-posed moon’s properties with those of false positives (Roden-beck et al. 2018), the existence of at least one plausible alterna-tive explanation for the observed TTV e ff ect of Kepler-1625 b,and the serious doubts that we have about the ∆ BIC as a reliablemetric at least for this particular data set lead us to conclude thatthe proposed moon around Kepler-1625 b might not be real. Wefind that the exomoon hypothesis heavily relies on a chain of del-icate assumptions, all of which need to be further investigated.A similar point was raised by Teachey & Kipping (2018),and our analysis is an independent attempt to shed some lighton the “unknown unknowns” referred to by the authors. For thetime being, we take the position that the first exomoon has yetto be detected as the likelihood of an exomoon around Kepler-1625 b cannot be assessed with the methods used and data cur-rently available.
Acknowledgements.
The authors thank Kevin Stevenson, Hannah Wakeford andMegan Sosey for their help with the data analysis, and Nikole Lewis for thefeedback on the manuscript. The authors would also like to thank the refereefor a challenging and constructive report. This work was supported in part bythe German space agency (Deutsches Zentrum für Luft- und Raumfahrt) underPLATO Data Center grant 50OO1501. This work made use of NASA’s ADSBibliographic Services and of the Exoplanet Encyclopaedia (http: // exoplanet.eu).RH wrote the manuscript, proposed Figs. 2 - 4, generated Fig. 5, and guided thework. KR derived the star-planet-moon orbital simulations and the respectivestatistics and generated Figs. 2 - 4. GB performed the light curve extraction fromthe WFC3 Hubble data and generated Fig. 1. All authors contributed equally tothe interpretation of the data.
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Astrophysical Constraints on theHabitability of Extrasolar Moons xomoon habitability constrained by illumination and tidal heating
René Heller I , Rory Barnes II,III I Leibniz-Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany, [email protected] II Astronomy Department, University of Washington, Box 951580, Seattle, WA 98195, [email protected]
III
NASA Astrobiology Institute – Virtual Planetary Laboratory Lead Team, USA
Abstract
The detection of moons orbiting extrasolar planets (“exomoons”) has now become feasible. Once they are discovered in the circumstellar habitable zone, questions about their habitability will emerge. Exomoons are likely to be tidally locked to their planet and hence experience days much shorter than their orbital period around the star and have seasons, all of which works in favor of habitability. These satellites can receive more illumination per area than their host planets, as the planet reflects stellar light and emits thermal photons. On the contrary, eclipses can significantly alter local climates on exomoons by reducing stellar illumination. In addition to radiative heating, tidal heating can be very large on exomoons, possibly even large enough for sterilization. We identify combinations of physical and orbital parameters for which radiative and tidal heating are strong enough to trigger a runaway greenhouse. By analogy with the circumstellar habitable zone, these constraints define a circumplanetary “habitable edge”. We apply our model to hypothetical moons around the recently discovered exoplanet Kepler-22b and the giant planet candidate KOI211.01 and describe, for the first time, the orbits of habitable exomoons. If either planet hosted a satellite at a distance greater than 10 planetary radii, then this could indicate the presence of a habitable moon.Key Words: Astrobiology – Extrasolar Planets – Habitability – Habitable Zone – Tides
1. Introduction
The question whether life has evolved outside Earth has prompted scientists to consider habitability of the terrestrial planets in the Solar System, their moons, and planets outside the Solar System, that is, extrasolar planets. Since the discovery of the first exoplanet almost two decades ago (Mayor & Queloz 1995), roughly 800 more have been found, and research on exoplanet habitability has culminated in the targeted space mission
Kepler , specifically designed to detect Earth-sized planets in the circumstellar irradiation habitable zones (IHZs, Huang 1959; Kasting et al. 1993; Selsis et al. 2007; Barnes et al. 2009) around Sun-like stars. No such Earth analog has been confirmed so far. Among the 2312 exoplanet candidates detected with Kepler (Batalha et al. 2012), more than 50 are indeed in the IHZ (Borucki et al. 2011; Kaltenegger & Sasselov 2011; Batalha et al. 2012), yet most of them are significantly larger and likely more massive than Earth. Habitability of the moons around these planets has received little attention. We argue here that it will be possible to constrain their habitability on the data available at the time they will be discovered. ! Various astrophysical effects distinguish investigations of exomoon habitability from studies on exoplanet habitability. On a moon, there will be eclipses of the star by the planet (Dole 1964); the planet’s stellar reflected light, as well the planet’s thermal emission, might affect the moon’s climate; and tidal heating can provide an additional energy source, which must be considered for evaluations of exomoon habitability (Reynolds et al. 1987; Scharf 2006; Debes & Sigurdsson 2007; Cassidy et al. 2009; Henning et al. 2009). Moreover, tidal processes between the moon and its parent planet will determine the orbit and spin evolution of the moon. Earth-sized planets in the IHZ around low-mass stars tend to become tidally locked, that is, one hemisphere permanently faces the star (Dole 1964; Goldreich 1966; Kasting et al. 1993), and they will not have seasons because their obliquities are eroded (Heller et al. 2011a,b). On moons, however, tides from the star are mostly negligible compared to the tidal drag from the planet. Thus, in most cases exomoons will be tidally locked to their host planet rather than to the star (Dole 1964; Gonzalez 2005; Henning et al. 2009; Kaltenegger 2010; Kipping et al. 2010) so that ( i .) a satellite’s rotation period will equal its orbital period about the planet, ( ii .) a moon will orbit the planet in its submitted to Astrobiology: April 6, 2012 accepted by Astrobiology: September 8, 2012 published in Astrobiology: January 24, 2013this updated draft: October 30, 2013 doi:10.1089/ast.2012.0859 A related but more anthropocentric circumstellar zone, termed “ecosphere”, has been defined by Dole (1964, p. 64 therein). Whewell (1853, Chapter X, Section 4 therein) presented a more qualitative discussion of the so-called “Temperate Zone of the Solar System”. quatorial plane (due to the Kozai mechanism and tidal evolution, Porter & Grundy 2011), and ( iii .) a moon’s rotation axis will be perpendicular to its orbit about the planet. A combination of ( ii .) and ( iii .) will cause the satellite to have the same obliquity with respect to the circumstellar orbit as the planet. ! More massive planets are more resistive against the tidal erosion of their obliquities (Heller et al. 2011b); thus massive host planets of exomoons can maintain significant obliquities on timescales much larger than those for single terrestrial planets. Consequently, satellites of massive exoplanets could be located in the IHZ of low-mass stars while, firstly, their rotation would not be locked to their orbit around the star (but to the planet) and, secondly, they could experience seasons if the equator of their host planet is tilted against the orbital plane. Both aspects tend to reduce seasonal amplitudes of stellar irradiation (Cowan et al. 2012) and thereby stabilize exomoon climates. ! An example is given by a potentially habitable moon in the Solar System, Titan. It is the only moon known to have a substantial atmosphere. Tides exerted by the Sun on Titan’s host planet, Saturn, are relatively weak, which is why the planet could maintain its spin-orbit misalignment, or obliquity, ψ p of 26.7° (Norman 2011). Titan orbits Saturn in the planet’s equatorial plane with no significant tilt of its rotation axis with respect to its circumplanetary orbit. Thus, the satellite shows a similar obliquity with respect to the Sun as Saturn and experiences strong seasonal modulations of insolation as a function of latitude, which leads to an alternation in the extents and localizations of its lakes and potential habitats (Wall et al. 2010). While tides from the Sun are negligible, Titan is tidally synchronized with Saturn (Lemmon et al. 1993) and has a rotation and an orbital period of ≈ ψ p ≈ ! No exomoon has been detected so far, but it has been shown that exomoons with masses down to 20% the mass of Earth ( M ⊕ ) are detectable with the space-based Kepler telescope (Kipping et al. 2009). Combined measurements of a planet’s transit timing variation (TTV) and transit duration variation (TDV) can provide information about the satellite’s mass ( M s ), its semi-major axis around the planet ( a ps ) (Sartoretti & Schneider 1999; Simon et al. 2007; Kipping 2009a), and possibly about the inclination ( i ) of the satellite’s orbit with respect to the orbit around the star (Kipping 2009b). Photometric transits of a moon in front of the star (Szabó et al. 2006; Lewis 2011; Kipping 2011a; Tusnski & Valio 2011), as well as mutual eclipses of a planet and its moon (Cabrera & Schneider 2007; Pál 2012), can provide information about its radius ( R s ), and the photometric scatter peak analysis (Simon et al. 2012) can offer further evidence for the exomoon nature of candidate objects. Finally, spectroscopic investigations of the Rossiter-McLaughlin effect can yield information about the satellite’s orbital geometry (Simon et al. 2010; Zhuang et al. 2012), although relevant effects require accuracies in stellar radial velocity of the order of a few centimeters per second (see also Kipping 2011a). Beyond, Peters & Turner (2013) suggest that direct imaging of extremely tidally heated exomoons will be possible with next-generation space telescopes. It was only recently that Kipping et al. (2012) initiated the first dedicated hunt for exomoons, based on Kepler observations. While we are waiting for their first discoveries, hints to exomoon-forming regions around planets have already been found (Mamajek et al. 2012). ! In Section 2 of this paper, we consider general aspects of exomoon habitability to provide a basis for our work, while Section 3 is devoted to the description of the exomoon illumination plus tidal heating model. Section 4 presents a derivation of the critical orbit-averaged global flux and the description of habitable exomoon orbits, ultimately leading to the concept of the “habitable edge”. In Section 5, we apply our model to putative exomoons around the first Neptune-sized planet in the IHZ of a Sun-like star, Kepler-22b, and a much more massive, still to be confirmed planet candidate, the “Kepler Object of Interest” (KOI) 211.01 , also supposed to orbit in the IHZ. We summarize our results with a discussion in Section 6. Detailed illustrations on how we derive our model are placed into the appendices.
2. Habitability of exomoons
So far, there have been only a few published investigations on exomoon habitability (Reynolds et al. 1987; Williams et al. 1997; Kaltenegger 2000; Scharf 2006; Porter & Grundy 2011). Other studies were mainly concerned with the observational aspects of exomoons (for a review see Kipping et al. 2012), their orbital stability (Barnes & O’Brien 2002; Domingos et al. 2006; Donnison 2010; Weidner & Horne 2010; Quarles et al. 2012; Sasaki et al. 2012), and eventually with the detection of biosignatures (Kaltenegger 2010). Thus, we provide here a brief overview of some important physical and biological aspects that must be accounted for when considering exomoon habitability.
Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating Planets with radii between 2 R ⊕ and 6 R ⊕ are designated Neptune-sized planets by the Kepler team (Batalha et al. 2012). Although KOI211.01 is merely a planet candidate we talk of it as a planet, for simplicity, but keep in mind its unconfirmed status.
Williams et al. (1997) were some of the first who proposed that habitable exomoons could be orbiting giant planets. At the time of their writing, only nine extrasolar planets, all of which are giant gaseous objects, were known. Although these bodies are not supposed to be habitable, Williams et al. argued that possible satellites of the jovian planets 16 Cygni B and 47 Ursae Majoris could offer habitats, because they orbit their respective host star at the outer edge of the habitable zone (Kasting et al. 1993). The main counter arguments against habitable exomoons were ( i .) tidal locking of the moon with respect to the planet, ( ii .) a volatile endowment of those moons, which would have formed in a circumplanetary disk, that is different from the abundances available for planets forming in a circumstellar disk, and ( iii .) bombardment of high-energy ions and electrons within the magnetic fields of the jovian host planet and subsequent loss of the satellite’s atmosphere. Moreover, ( iv. ) stellar forcing of a moon’s upper atmosphere will constrain its habitability. ! Point ( i .), in fact, turns out as an advantage for Earth-sized satellites of giant planets over terrestrial planets in terms of habitability, by the following reasoning: Application of tidal theories shows that the rotation of extrasolar planets in the IHZ around low-mass stars will be synchronized on timescales ≪ rather than to the star (Porter & Grundy 2011). This configuration would not only prevent a primordial atmosphere from evaporating on the illuminated side or freezing out on the dark side ( i .) but might also sustain its internal dynamo ( iii .). The synchronized rotation periods of putative Earth-mass exomoons around giant planets could be in the same range as the orbital periods of the Galilean moons around Jupiter (1.7d − ≈ . The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about P ∗ p /9, P ∗ p being the planet’s orbital period about the star (Kipping 2009a). Since the satellite’s rotation period also depends on its orbital eccentricity around the planet and since the gravitational drag of further moons or a close host star could pump the satellite’s eccentricity (Cassidy et al. 2009; Porter & Grundy 2011), exomoons might rotate even faster than their orbital period. ! Finally, from what we know about the moons of the giant planets in the Solar System, the satellite’s enrichment with volatiles ( ii .) should not be a problem. Cometary bombardment has been proposed as a source for the dense atmosphere of Saturn’s moon Titan, and it has been shown that even the currently atmosphere-free jovian moons Ganymede and Callisto should initially have been supplied with enough volatiles for an atmosphere (Griffith & Zahnle 1995). Moreover, as giant planets in the IHZ likely formed farther away from their star, that is, outside the snow line (Kennedy & Kenyon 2008), their satellites will be rich in liquid water and eventually be surrounded by substantial atmospheres. ! The stability of a satellite’s atmosphere ( iv .) will critically depend on its composition, the intensity of stellar extreme ultraviolet radiation (EUV), and the moon’s surface gravity. Nitrogen-dominated atmospheres may be stripped away by ionizing EUV radiation, which is a critical issue to consider for young (Lichtenberger et al. 2010) and late-type (Lammer et al. 2009) stars. Intense EUV flux could heat and expand a moon’s upper atmosphere so that it can thermally escape due to highly energetic radiation ( iii .), and if the atmosphere is thermally expanded beyond the satellite’s magnetosphere, then the surrounding plasma may strip away the atmosphere nonthermally. If Titan were to be moved from its roughly 10AU orbit around the Sun to a distance of 1AU (AU being an astronomical unit, i.e., the average distance between the Sun and Earth), then it would receive about 100 times more EUV radiation, leading to a rapid loss of its atmosphere due to the moon’s smaller mass, compared to Earth. For an Earth-mass moon at 1AU from the Sun, EUV radiation would need to be less than 7 times the Sun’s present-day EUV emission to allow for a long-term stability of a nitrogen atmosphere. CO provides substantial cooling of an atmosphere by infrared radiation, thereby counteracting thermal expansion and protecting an atmosphere’s nitrogen inventory (Tian 2009). ! A minimum mass of an exomoon is required to drive a magnetic shield on a billion-year timescale ( M s ≳ M ⊕ , Tachinami et al. 2011); to sustain a substantial, long-lived atmosphere ( M s ≳ M ⊕ , Williams et al. 1997; Kaltenegger 2000); and to drive tectonic activity ( M s ≳ M ⊕ , Williams et al. 1997), which is necessary to maintain plate tectonics and to support the carbon-silicate cycle. Weak internal dynamos have been detected in Mercury and Ganymede (Kivelson et al. 1996; Gurnett et al. 1996), suggesting that satellite masses > 0.25 M ⊕ will be adequate for considerations of exomoon habitability. This lower limit, however, is not a fixed number. Further sources of energy – such as radiogenic and tidal Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating Maintained by Robert Jacobson, http://ssd.jpl.nasa.gov. eating, and the effect of a moon’s composition and structure – can alter our limit in either direction. An upper mass limit is given by the fact that increasing mass leads to high pressures in the moon’s interior, which will increase the mantle viscosity and depress heat transfer throughout the mantle as well as in the core. Above a critical mass, the dynamo is strongly suppressed and becomes too weak to generate a magnetic field or sustain plate tectonics. This maximum mass can be placed around 2 M ⊕ (Gaidos et al. 2010; Noack & Breuer 2011; Stamenkovi ć et al. 2011). Summing up these conditions, we expect approximately Earth-mass moons to be habitable, and these objects could be detectable with the newly started Hunt for Exomoons with Kepler (HEK) project (Kipping et al. 2012).
The largest and most massive moon in the Solar System, Ganymede, has a radius of only ≈ R ⊕ ( R ⊕ being the radius of Earth) and a mass of ≈ M ⊕ . The question as to whether much more massive moons could have formed around extrasolar planets is an active area of research. Canup & Ward (2006) have shown that moons formed in the circum-planetary disk of giant planets have masses ≲ -4 times that of the planet’s mass. Assuming satellites formed around Kepler-22b, their masses will thus be 2.5 × -3 M ⊕ at most, and around KOI211.01 they will still weigh less than Earth’s Moon. Mass-constrained in situ formation becomes critical for exomoons around planets in the IHZ of low-mass stars because of the observational lack of such giant planets. An excellent study on the formation of the Jupiter and the Saturn satellite systems is given by Sasaki et al. (2010), who showed that moons of sizes similar to Io, Europa, Ganymede, Callisto, and Titan should build up around most gas giants. What is more, according to their Fig. 5 and private communication with Takanori Sasaki, formation of Mars- or even Earth-mass moons around giant planets is possible. Depending on whether or not a planet accretes enough mass to open up a gap in the protostellar disk, these satellite systems will likely be multiple and resonant (as in the case of Jupiter), or contain only one major moon (see Saturn). Ogihara & Ida (2012) extended these studies to explain the compositional gradient of the jovian satellites. Their results explain why moons rich in water are farther away from their giant host planet and imply that capture in 2:1 orbital resonances should be common. ! Ways to circumvent the impasse of insufficient satellite mass are the gravitational capture of massive moons (Debes & Sigurdsson 2007; Porter & Grundy 2011; Quarles et al. 2012), which seems to have worked for Triton around Neptune (Goldreich et al. 1989; Agnor & Hamilton 2006); the capture of Trojans (Eberle et al. 2011); gas drag in primordial circum-planetary envelopes (Pollack et al. 1979); pull-down capture trapping temporary satellites or bodies near the Lagrangian points into stable orbits (Heppenheimer & Porco 1977; Jewitt & Haghighipour 2007); the coalescence of moons (Mosqueira & Estrada 2003); and impacts on terrestrial planets (Canup 2004; Withers & Barnes 2010; Elser et al. 2011). Such moons would correspond to the irregular satellites in the Solar System, as opposed to regular satellites that form in situ. Irregular satellites often follow distant, inclined, and often eccentric or even retrograde orbits about their planet (Carruba et al. 2002). For now, we assume that Earth-mass extrasolar moons – be they regular or irregular – exist.
A prominent argument against the habitability of moons involves high-energy particles, which a satellite meets in the planet’s radiation belt. Firstly, this ionizing radiation could strip away a moon’s atmosphere, and secondly it could avoid the buildup of complex molecules on its surface. In general, the process in which incident particles lose part of their energy to a planetary atmosphere or surface to excite the target atoms and molecules is called sputtering. The main sources for sputtering on Jupiter’s satellites are the energetic, heavy ions O + and S + , as well as H + , which give rise to a steady flux of H O, OH, O , H , O, and H from Ganymede’s surface (Marconi 2007). A moon therefore requires a substantial magnetic field that is strong enough to embed the satellite in a protective bubble inside the planet’s powerful magnetosphere. The only satellite in the Solar System with a substantial magnetic shield of roughly 750nT is Ganymede (Kivelson et al. 1996). The origin of this field is still subject to debate because it can only be explained by a very specific set of initial and compositional configurations (Bland et al. 2008), assuming that it is generated in the moon’s core. ! For terrestrial planets, various models for the strength of global dipolar magnetic fields B dip as a function of planetary mass and rotation rate exist, but none has proven exclusively valid. Simulations of planetary thermal evolution have shown that B dip increases with mass (Tachinami et al. 2011; Zuluaga & Cuartas 2012) and rotation frequency (Lopez-Morales et al. 2012). The spin of exomoons will be determined by tides from the planet, and rotation of an Earth-sized exomoon in the IHZ can be much faster than rotation of an Earth-sized planet orbiting a star. Thus, an exomoon could be prevented from Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating idal synchronization with the host star – in support of an internal dynamo and thus magnetic shielding against energetic irradiation from the planet and the star. Some studies suggest that even extremely slow rotation would allow for substantial magnetic shielding, provided convection in the planet’s or moon’s mantle is strong enough (Olson & Christensen 2006). In this case, tidal locking would not be an issue for magnetic shielding. ! The picture of magnetic shielding gets even more complicated when tidal heating is considered, which again depends on the orbital parameters. In the Moon, tidal heating, mostly induced by the Moon’s obliquity of 6.68° against its orbit around Earth, occurs dominantly in the core (Kaula 1964; Peale & Cassen 1978). On Io, however, where tidal heating stems from Jupiter’s effect on the satellite’s eccentricity, dissipation occurs mostly in the mantle (Segatz et al. 1988). In the former case, tidal heating might enhance the temperature gradient between the core and the mantle and thereby also enhance convection and finally the strength of the magnetic shielding; in the latter case, tidal heating might decrease convection. Of course, the magnetic properties of terrestrial worlds will evolve and, when combined with the evolution of EUV radiation and stellar wind from the host star, define a time-dependent magnetically restricted habitable zone (Khodachenko et al. 2007; Zuluaga et al. 2012). ! We conclude that radiation of highly energetic particles does not ultimately preclude exomoon habitability. In view of possible deflection due to magnetic fields on a massive satellites, it is still reasonable to consider the habitability of exomoons.
On Earth, the thermal equilibrium temperature of incoming and outgoing radiation is 255K. However, the mean surface temperature is 289K. The additional heating is driven by the greenhouse effect (Kasting 1988), which is a crucial phenomenon to the habitability of terrestrial bodies. The strength of the greenhouse effect depends on numerous variables – most importantly on the inventory of greenhouse gases, the albedo effect of clouds, the amount of liquid surface water, and the spectral energy distribution of the host star. ! Simulations have shown that, as the globally absorbed irradiation on a water-rich planetary body increases, the atmosphere gets enriched in water vapor until it gets opaque. For an Earth-like body, this imposes a limit of about 300W/m to the thermal radiation that can be emitted to space. If the global flux exceeds this limit, the body is said to be a runaway greenhouse. Water vapor can then leave the troposphere through the tropopause and reach the stratosphere, where photodissociation by stellar UV radiation allows the hydrogen to escape to space, thereby desiccating the planetary body. While boiling oceans, high surface temperatures, or high pressures can make a satellite uninhabitable, water loss does by definition. Hence, we will use the criterion of a runaway greenhouse to define an exomoon’s habitability. ! Surface temperatures strongly depend on the inventory of greenhouse gases, for example, CO . The critical energy flux F RG for a runaway greenhouse, however, does not (Kasting 1988; Goldblatt & Watson 2012). As in Barnes et al. (2013), who discussed how the interplay of stellar irradiation and tidal heating can trigger a runaway greenhouse on exoplanets, we will use the semi-analytical approach of Pierrehumbert (2010) for the computation of F RG : F RG = o SB lR ln ✓ P / r P g s ( M s , R s ) k ◆ 1CCCA (1) with P = P ref exp ⇢ lR T ref , (2) P ref = 610.616Pa, l is the latent heat capacity of water, R is the universal gas constant, T ref = 273.13K, o = 0.7344 is a constant designed to match radiative transfer simulations, σ SB is the Stefan-Boltzmann constant, P = 104Pa is the pressureat which the absorption line strengths of water vapor are evaluated, g s = GM s /R is the gravitational acceleration at thesatellite’s surface, and k = 0.055 is the gray absorption coefficient at standard temperature and pressure. Recall that the runaway greenhouse does not depend on the composition of the atmosphere, other than it contains water. As habitability requires water and Eq. (1) defines a limit above which the satellite will lose it, the formula provides a conservative limit to Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating abitability. ! In addition to the maximum flux F RG to allow for a moon to be habitable, one may think of a minimum flux required to prevent the surface water from freezing. On terrestrial exoplanets, this freezing defines the outer limit of the stellar IHZ. On exomoons, the extra light from the planetary reflection and thermal emission as well as tidal heating in the moon will move the circumstellar habitable zone away from the star, whereas eclipses will somewhat counterbalance this effect. While it is clear that a moon under strong tidal heating will not be habitable, it is not clear to what extent it might actually support habitability (Jackson et al. 2008). Even a relatively small tidal heating flux of a few watts per square meter could render an exomoon inhospitable; see Io’s global volcanism, where tidal heating is a mere 2W/m (Spencer et al. 2000). Without applying sophisticated models for the moon’s tidal heating, we must stick to the irradiation aspect to define an exomoon’s circumstellar habitable zone. At the outer edge of the stellar IHZ, the host planet will be cool and reflected little stellar flux. Neglecting tidal heating as well thermal emission and reflection from the planet, the minimum flux for an Earth-like moon to be habitable will thus be similar to that of an Earth-like planet at the same orbital distance to the star. Below, we will only use the upper flux limit from Eq. (1) to constrain the orbits of habitable exomoons. This will lead us to the concept of the circumplanetary “habitable edge”.
3. Energy reservoirs on exomoons
Life needs liquid water and energy, but an oversupply of energy can push a planet or an exomoon into a runaway greenhouse and thereby make it uninhabitable. The critical, orbit-averaged energy flux for an exomoon to turn into a runaway greenhouse is around 300W/m , depending on the moon’s surface gravity and atmospheric composition (Kasting 1988; Kasting et al. 1993; Selsis et al. 2007; Pierrehumbert 2010; Goldblatt & Watson 2012). An exomoon will thus only be habitable in a certain range of summed irradiation and tidal heat flux (Barnes et al. 2013). ! We consider four energy reservoirs and set them into context with the IHZ: ( i .) stellar illumination, ( ii .) stellar reflected light from the planet, ( iii .) thermal radiation from the planet, and ( iv .) tidal heating on the moon. Here, primordial heat from the moon’s formation and radiogenic decay is neglected, and it is assumed that the moon’s rotation is tidally locked to its host planet, as is the case for almost all the moons in the Solar System. Our irradiation model includes arbitrary orbital eccentricities e ∗ p of the planet around the star . While we compute tidal heating on the satellite as a function of its orbital eccentricity e ps around the planet, we assume e ps = 0 in the parametrization of the moon’s irradiation. This is appropriate because typically e ps ≪ e ps ≠ e ps ≪ i of the moon’s orbit with respect to the orbit of the planet-moon barycenter around the star. If one assumed that the moon always orbits above the planet’s equator, that would imply that i is equal to the planetary obliquity ψ p , which is measured with respect to the planet’s orbit around the star. We do not need this assumption for the derivation of our equations, but since ψ p ≈ i for all the large moons in the Solar System, except Triton, observations or numerical predictions of ψ p (Heller et al. 2011b) can provide reasonable assumptions for i . ! In our simulations, we consider two prototype moons: one rocky Earth-mass satellite with a rock-to-mass fraction of 68% (similar to Earth) and one water-rich satellite with the tenfold mass of Ganymede and an ice-to-mass fraction of 25% (Fortney et al. 2007). The remaining constituents are assumed to be iron for the Earth-mass moon and silicates for the Super-Ganymede. The more massive and relatively dry moon represents what we guess a captured, Earth-like exomoon could be like, while the latter one corresponds to a satellite that has formed in situ. Note that a mass of 10 M G ( M G being the mass of Ganymede) corresponds to roughly 0.25 M ⊕ , which is slightly above the detection limit for combined TTV and TDV with Kepler (Kipping et al. 2009). Our assumptions for the Super-Ganymede composition are backed up by observations of the Jupiter and Saturn satellite systems (Showman & Malhotra 1999; Grasset et al. 2000) as well as
Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating In the following, a parameter index “ ∗ ” will refer to the star, “p” to the planet, and “s” to the satellite. The combinations “ ∗ p” and “ps”, e.g., for the orbital eccentricities e ∗ p and e ps , refer to systems of a star plus a planet and a planet plus a satellite, respectively. For a vector, e.g. r p ⇤ , the first letter indicates the starting point (in this case the planet) and the second index locates the endpoint (here the star). errestrial planet and satellite formation studies (Kuchner 2003; Ogihara & Ida 2012). These papers show that in situ formation naturally generates water-rich moons and that such objects can retain their water reservoir for billions of years against steady hydrodynamic escape. Concerning the habitability of the water-rich Super-Ganymede, we do not rely on any assumptions concerning possible life forms in such water worlds. Except for the possible strong heating in a water-rich atmosphere (Matsui & Abe 1986; Kuchner 2003), we see no reason why ocean moons should not be hospitable, in particular against the background that life on Earth arose in (possibly hot) oceans or freshwater seas. ! For the sake of consistency, we derive the satellites’ radii R s from planetary structure models (Fortney et al. 2007). In the case of the Earth-mass moon, we obtain R s = 1 R ⊕ , and for the much lighter but water-dominated Super-Ganymede R s = 0.807 R ⊕ . Equation (1) yields a critical flux of 295W/m for the Earth-mass moon, and 266W/m for the Super-Ganymede satellite. The bond albedo of both moons is assumed to be 0.3, similar to the mean albedo of Earth and of the Galilean satellites (Clark 1980). In the following, we call our Earth-like and Super-Ganymede satellites our “prototype moons”. Based on the summary of observations and the model for giant planet atmospheres provided by Madhusudhan & Burrows (2012), we also use a bond albedo of 0.3 for the host planet, although higher values might be reasonable due to the formation of water clouds at distances 1AU from the host star (Burrows et al. 2006a). Mass and radius of the planet are not fixed in our model, but we will mostly refer to Jupiter-sized host planets. The total bolometric illumination on a moon is given by the stellar flux ( f ∗ ), the reflection of the stellar light from the planet ( f r ), and the planetary thermal emission ( f t ). Their variation will be a function of the satellite’s orbital phase 0 ≤ φ ps ( t ) = ( t − t )/ P ps ≤ t being time, t as the starting time [0 in our simulations], and P ps as the period of the planet-moonorbit), the orbital phase of the planet-moon duet around the star ( φ ∗ p , which is equivalent to the mean anomaly M ⇤ p dividedby 2 π ), and will depend on the eccentricity of the planet around the star ( e ∗ p ), on the inclination ( i ) of the two orbits, on the orientation of the periapses ( η ), as well as on longitude and latitude on the moon’s surface ( φ and θ ). ! In Fig. 1, we show the variation of the satellite’s illumination as a function of the satellite’s orbital phase φ ps . For this plot, the orbital phase of the planet-moon pair around the star φ ∗ p = 0 and i = 0. Projection effects due to latitudinal variation have been neglected, starlight is assumed to be plane-parallel, and radii and distances are not to scale. ! In our irradiation model of a tidally locked satellite, we neglect clouds, radiative transfer, atmospheric circulation, geothermal flux , thermal inertia, and so on, and we make use of four simplifications: ! ! !! ! !! ( i .) ! We assume the planet casts no penumbra on the moon. There is either total illumination from the star or none. This ! ! !! ! !! !! ! ! assumption is appropriate since we are primarily interested in the key contributions to the moon’s climate. ! ! !! ! ! ( ii .) ! The planet is assumed to be much more massive than the moon, and the barycenter of the planet-moon binary is ! ! !! ! !! !! ! ! placed at the center of the planet. Even if the planet and the moon had equal masses, corrections would be small ! ! !! ! !! !! ! ! since the range between the planet-moon barycenter and the star ≫ a ps . ! ! !! ! ( iii .) !! For the computation of the irradiation, we treat the moon’s orbit around the planet as a circle. The small eccentrici- ! ! !! ! !! !! ! ! ties which we will consider later for tidal heating will not modify our results significantly. ! ! !! ! ( iv .) !! The distance between the planet-satellite binary and the star does not change significantly over one satellite orbit, ! ! !! ! !! !! ! ! which is granted when either e ∗ p is small or P ps ≪ P ∗ p . ! In the following, we present the general results of our mathematical derivation. For a more thorough description and discussions of some simple cases, see Appendices A and B.
The stellar flux on the substellar point on the moon’s surface will have a magnitude L ∗ /(4 π r s ∗ ( t ) ), where L ∗ is stellar luminosityand r s ⇤ is the vector from the satellite to the star. We multiply this quantity with the surface normal n ⇥, /n ⇥, on the moonand r s ⇤ /r s ⇤ to include projection effects on a location ( φ , θ ). This yields
Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating Tidal heating will be included below, but we will neglect geothermal feedback between tidal heating and irradiation. f ⇤ ( t ) = L ⇤ r s ⇤ ( t ) r s ⇤ ( t ) r s ⇤ ( t ) n ⇥, ( t ) n ⇥, ( t ) . (3) f r s ⇤ and n have an antiparallel part, then f ∗ < 0, which is meaningless in our context, and we set f ∗ to zero. The task is now tofind r s ⇤ ( t ) and n ⇥, ( t ) . Therefore, we introduce the surface vector from the subplanetary point on the satellite to the planet, s ⌘ n , , and the vector from the planet to the star, r p ⇤ ( t ) , which gives r s ⇤ ( t ) = r p ⇤ ( t ) + s ( t ) (see Fig. 1). ApplyingKepler’s equations of motion, we deduce r p ⇤ ( t ) ; and with a few geometric operations (see Appendix A) we obtain n ⇥, ( t ) : r p ⇤ ( t ) = a ⇤ p ˜ c e ⇤ p q e ⇤ p ˜ s (4) n ⇥, ( t ) = a ps ¯ sS ˜ C + ¯ c ( ˜ CcC ˜ Ss ) ¯ sS ˜ S + ¯ c ( ˜ ScC ˜ Cs )¯ sC + ¯ ccS (5) withwhere i , η , 0 ≤ φ ≤ ≤ θ ≤
90° are provided in degrees, and φ and θ are measured from the subplanetary point (see Fig. 1). E ⇤ p ( t ) e ⇤ p sin ⇣ E ⇤ p ( t ) ⌘ = M ⇤ p ( t ) (7) defines the eccentric anomaly E ⇤ p and Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 1.
Geometry of the triple system of a star, a planet, and a moon with illuminations indicated by different shadings (pole view). For ease of visualization, the moon’s orbit is coplanar with the planet’s orbit about the star and the planet’s orbital position with respect to the star is fixed. Combined stellar and planetary irradiation on the moon is shown for four orbital phases. Projection effects as a function of longitude φ and latitude θ are ignored, and we neglect effects of a penumbra. Radii and distances are not to scale, and starlight is assumed to be plane-parallel. In the right panel, the surfacenormal on the subplanetary point is indicated by an arrow. For a tidally locked moon this spot is a fixed point on the moon’ssurface. For φ ps = 0 four longitudes are indicated. c = cos ✓ ⇤ ( ⇧ ps ( t ) + ⌅ ) ◆ s = sin ✓ ⇤ ( ⇧ ps ( t ) + ⌅ ) ◆ ¯ c = cos ⇣ ⇥ ⇤ ) ⌘ ¯ s = sin ⇣ ⇥ ⇤ ) ⌘ ˜ c = cos ( E ⇤ p ( t )) ˜ s = sin ( E ⇤ p ( t )) C = cos ⇣ i ⇤ ⌘ S = sin ⇣ i ⇤ ⌘ ˜ C = cos ⇣ ⇤ ⌘ ˜ S = sin ⇣ ⇤ ⌘ (6) M ⇤ p ( t ) = 2 ( t t ) P ⇤ p (8) s the mean anomaly. The angle η is the orientation of the lowest point of the moon's inclined orbit with respect to the star at periastron (see Appendix A). Kepler's equation (Eq. 7) is a transcendental function which we solve numerically. ! To compute the stellar flux over one revolution of the moon around the planet, we put the planet-moon duet at numerous orbital phases around the star (using a fixed time step d t ), thus ˜ c and ˜ s will be given. At each of these positions, we then evolve φ ps from 0 to 1. With this parametrization, the moon’s orbit around the planet will always start at the left, corresponding to Fig. 1, and it will be more facile to interpret the phase functions. If we were to evolve the moon’s orbitconsistently, ⇡ M ⇤ p would have to be added to the arguments of c and s . Our simplification is appropriate as long as r ⇤ p does not change significantly over one satellite orbit. Depending on the orientation of an eventual inclination between the two orbits and depending on the orbital position of the planet-moon system around the star, the can be eclipsed by the planet for a certain fraction of φ ps as seen from the moon. This phenomenon might have significant impacts on exomoon climates. Eclipses occur if the perpendicular part r ? = sin arccos ✓ r s ⇤ r p ⇤ | r s ⇤ || r p ⇤ | ◆ ! | r s ⇤ | (9) of r s ⇤ with respect to r p ⇤ is smaller than the radius of the planet and if | r s ⇤ | > | r p ⇤ | , that is, if the moon is behind the planetas seen from the star and not in front of it. The angular diameters of the star and the planet, β ∗ and β p , respectively, are given by ⇤ = 2 arctan ✓ R ⇤ a ⇤ p + a ps ◆ p = 2 arctan ✓ R p a ps ◆ . (10) If β p > β ∗ then the eclipse will be total. Otherwise the stellar flux will be diminished by a factor [1 – ( β p / β ∗ ) ]. We now consider two contributions to exomoon illumination from the planet, namely, reflection of stellar light ( f r ) and thermal radiation ( f t ). If the planet's rotation period is ≲ T be , p , and the dark back sidehas a temperature T de , p (see Appendix B). With d T = T be , p T de , p as the temperature difference between the hemispheresand α p as the planet's bond albedo, that is, the fraction of power at all wavelengths scattered back into space, thermal equilibrium yields p ( T be , p ) ⌘ ( T be , p ) + ( T be , p d T ) T , ⇤ (1 p ) R ⇤ r ⇤ p = 0 . (11) For a given d T , we search for the zero points of the polynomial p ( T be , p ) numerically. In our prototype system at 1AU from aSun-like star and choosing d T = 100K, Eq. (11) yields T be , p = 291 K and T de , p = 191 K . Finally, the thermal flux receivedby the moon from the planet turns out aswhere ⇥ ( t ) = 12 ( ⇣ ⇤ ( t ) ⌘ cos ⇣ ( t ) ⇤ p ( t ) ⌘) (13) weighs the contributions from the two hemispheres, Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating f t ( t ) = R ⌅ SB a cos ✓ ⇧⇤ ◆ cos ✓ ⇤ ◆ ⇥ " ( T be , p ) ⇥ ( t )+( T de , p ) (1 ⇥ ( t )) , (12) ⇤ p ( t ) = arccos ✓ cos( E ⇤ p ( t )) e ⇤ p e ⇤ p cos( E ⇤ p ( t )) ◆ (14) is the true anomaly, ( t ) = 2 arctan s y ( t ) q s ( t ) + s ( t ) + s x ( t ) ⇥ ( t ) = arccos s y ( t ) q s ( t ) + s ( t ) + s ( t ) , (15) and s x,y,z are the components of s = ( s x , s y , s z ) (see Appendix B). ! Additionally, the planet reflects a portion ⇥R p of the incoming stellar light. Neglecting that the moon blocks a smallfraction of the starlight when it passes between the planet and the star (< 1% for an Earth-sized satellite around a Jupiter-sized planet), we find that the moon receives a stellar flux f r ( t ) = R ⇤ ⇧ SB T , ⇤ r ⇤ ⌅R p a cos ✓ ⌃⌅ ◆ cos ✓ ⇥⌅ ◆ ⇤ ( t ) (16) from the planet. ! In Fig. 2, we show how the amplitudes of f t ( t ) and f r ( t ) compare. Therefore, we neglect the time dependence and compute simply the maximum possible irradiation on the moon’s subplanetary point as a function of the moon’s orbit around the planet, which occurs in our model when the moon is over the substellar point of the planet. Then it receives maximum reflection and thermal flux at the same time. For our prototype system, it turns out that f t > f r at a given planet-moon distance only if the planet has an albedo ≲ α p = 0.093 in this case, can be obtained by comparing Eqs. (12) and (16) (see Appendix B). For increasing α p , stellar reflected flux dominates more and more; and for α p ≳ f ∗ is over a magnitude stronger than f t . ! The shapes of the curves can be understood intuitively, if one imagines that at a fixed semi-major axis (abscissa) the reflected flux received on the moon increases with increasing albedo (ordinate), whereas the planet’s thermal flux increases when it absorbs more stellar light, which happens for decreasing albedo. ! The shaded area in the upper left corner of the figure indicates where the sum of maximum f t and f r exceeds the limit of 295W/m for a runaway greenhouse on an Earth-sized moon. Yet a satellite in this part of the parameter space would not necessarily be uninhabitable, because firstly it would only be subject to intense planetary radiation for less than about half its orbit, and secondly eclipses could cool the satellite half an orbit later. Moons at a ps ≲ R p are very likely to experience eclipses. Note that a moon’s orbital eccentricity e ps will have to be almost perfectly zero to avoid intense tidal heating in such close orbits (see Section 3.2). ! Since we use a ps in units of planetary radii, f t and f r are independent of R p . We also show a few examples of Solar System moons, where we adopted 0.343 for Jupiter’s bond albedo (Hanel et al. 1981), 0.342 for Saturn (Hanel et al. 1983), 0.32 for Uranus (Neff et al. 1985; Pollack et al. 1986; Pearl et al. 1990), 0.29 for Neptune (Neff et al. 1985; Pollack et al. 1986; Pearl & Conrath 1991), and 0.3 for Earth. Flux contours are not directly applicable to the indicated moons because the host planets Jupiter, Saturn, Uranus, and Neptune do not orbit the Sun at 1AU, as assumed for our prototype exomoon system . Only the position of Earth’s moon, which receives a maximum of 0.35W/m reflected light from Earth, reproduces the true Solar System values. The Roche radius for a fluid-like body (Weidner & Horne 2010, and references therein) is indicated with a gray line at 2.07 R p . Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating At a distance of 5.2AU from the Sun, Europa receives roughly 0.5W/m reflected light from Jupiter, when it passes the planet’s subsolar point. Jupiter’s thermal flux on Europa is negligible. .1.3 The circumstellar habitable zone of exomoons We next transform the combined stellar and planetary flux into a correction for the IHZ, for which the boundaries are proportional to L ∗ (Selsis et al. 2007). This correction is easily derived if we restrict the problem to just the direct and the reflected starlight. Then, we can define an “effective luminosity” L eff that is the sum of the direct starlight plus the orbit-averaged reflected light. We ignore the thermal contribution as its spectral energy distribution will be much different from the star and, as shown below, the thermal component is the smallest for most cases. Our IHZ corrections are therefore only lower limits. From Eqs. (3), (13), and (16) one can show that L e = L ⇤ ⇣ p R a ⌘ , (17) where we have averaged over the moon's orbital period. For realistic moon orbits, this correction amounts to 1% at most for high α p and small a ps . For planets orbiting F dwarfs near the outer edge of the IHZ, a moon could be habitable about 0.05AU farther out due to the reflected planetary light. In Fig. 3, we show the correction factor for the inner and outer boundaries of the IHZ due to reflected light as a function of α p and a ps . With Eqs. (3), (12), and (16), we have derived the stellar and planetary contributions to the irradiation of a tidally locked moon in an inclined, circular orbit around the planet, where the orbit of the planet-moon duet around the star is eccentric. Now, we consider a satellite’s total illuminationFor an illustration of Eq. (18), we choose a moon that orbits its Jupiter-sized host planet at the same distance as Europa orbits Jupiter. The planet-moon duet is in a 1AU orbit around a Sun-like star, and we arbitrarily choose a temperature difference of d T = 100K between the two planetary hemispheres. Equation (18) does not depend on M s or R s , so our irradiation model is not restricted to either the Earth-sized or the Super-Ganymede prototype moon. ! In Fig. 4 we show f s ( t ) as well as the stellar and planetary contributions for four different locations on the moon’s surface. For all panels, the planet-moon duet is at the beginning of its revolution around the star, i.e. M ⇤ p = 0 , and we set i = 0.Although M ⇤ p would slightly increase during one orbit of the moon around the planet, we fix it to zero, so the moon startsand finishes over the illuminated hemisphere of the planet (similar to Fig. 1). The upper left panel depicts the subplanetary point, with a pronounced eclipse around φ ps = 0.5. At a position 45° counterclockwise along the equator (upper right panel), the stellar contribution is shifted in phase, and f t as well as f r are diminished in magnitude (note the logarithmic scale!) by a factor cos(45°). In the lower row, where φ = 90°, θ = 0° (lower left panel) and φ = 180°, θ = 80° (lower right panel), there are no planetary contributions. The eclipse trough has also disappeared because the star’s occultation by the planet cannot be seen from the antiplanetary hemisphere. ! For Fig. 5, we assume a similar system, but now the planet-moon binary is at an orbital phase φ ∗ p = 0.5, corresponding to M ⇤ p = , around the star. We introduce an eccentricity e ∗ p = 0.3 as well as an inclination of 45° between the two orbital Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 2.
Contours of constant planetary flux on an exomoon as a function of the planet-satellite semi-major axis a ps and the planet’s bond albedo α p . The planet-moon binary orbits at 1AU from a Sun-like host star. Values depict the maximum possible irradiation in terms of orbital alignment, i.e., on the subplanetary point on the moon, and when the moon is over the substellar point of the planet. For α p ≈ f r and f t intersect, i.e., both contributions are equal. An additional contour is added at 295W/m , where the sum of f r and f t induces a runaway greenhouse on an Earth-sized moon. Some examples from the Solar System are given: Miranda (Mi), Io, Rhea (Rh), Europa (Eu), Triton (Tr), Ganymede (Ga), Titan (Ti), Callisto (Ca), and Earth’s moon (Moon). f s ( t ) = f ⇤ ( t ) + f t ( t ) + f r ( t ) . (18) lanes. The first aspect shifts the stellar and planetary contributions by half an orbital phase with respect to Fig. 4. Considering the top view of the system in Fig. 1, this means eclipses should now occur when the moon is to the left of the planet because the star is to the right, “left” here meaning φ ps = 0. However, the non-zero inclination lifts the moon out of the planet’s shadow (at least for this particular orbital phase around the star), which is why the eclipse trough disappears. Due to the eccentricity, stellar irradiation is now lower because the planet-moon binary is at apastron. An illustration of the corresponding star-planet orbital configuration is shown in the pole view (left panel) of Fig. 6. ! The eccentricity-driven cooling of the moon is enhanced on its northern hemisphere, where the inclination induces a winter. Besides, our assumption that the moon is in the planet’s equatorial plane is equivalent to i = ψ p ; thus the planet also experiences northern winter. The lower right panel of Fig. 5, where φ = 180° and θ = 80°, demonstrates a novel phenomenon, which we call an “antiplanetary winter on the moon”. On the satellite’s antiplanetary side there is no illumination from the planet (as in the lower two panels of Fig. 4); and being close enough to the pole, at θ > 90° – i for this occasion of northern winter, there will be no irradiation from the star either, during the whole orbit of the moon around the planet. In Fig. 6, we depict this constellation in the edge view (right panel). Note that antiplanetary locations close to the moon’s northern pole receive no irradiation at all, as indicated by an example at φ = 180°, θ = 80° (see arrow). Of course there is also a “proplanetary winter” on the moon, which takes place just at the same epoch but on the proplanetary hemisphere on the moon. Theopposite effects are the “proplanetary summer”, which occurs on the proplanetary side of the moon at M ⇤ p = 0 , at least forthis specific configuration in Fig. 5, and the “antiplanetary summer”. ! Finally, we compute the average surface flux on the moon during one stellar orbit. Therefore, we first integrate d φ ps f s ( φ ps ) over 0 ≤ φ ps ≤ φ ∗ p = 0), which yields the area under the solid lines in Fig. 4. We then step through ≈
50 values for φ ∗ p and again integrate the total flux. Finally, we average the flux over one orbit of the planet around the star, which gives the orbit-averaged flux F s ( φ , θ ) on the moon. In Fig. 7, we plot these values as surface maps of a moon in four scenarios. The two narrow panels to the right of each of the four major panels show the averaged flux for − ≤ φ ∗ p ≤ +1/4 and +1/4 ≤ φ ∗ p ≤ +3/4, corresponding to northern summer (ns) and southern summer (ss) on the moon, respectively. ! In the upper left panel, the two orbits are coplanar. Interestingly, the subplanetary point at φ = 0 = θ is the “coldest” spot along the equator (if we convert the flux into a temperature) because the moon passes into the shadow of the planet when the star would be at zenith over the subplanetary point. Thus, the stellar irradiation maximum is reduced (see the upper left panel in Fig. 4). The contrast between polar and equatorial irradiation, reaching from 0 to ≈ , is strongest in this panel. In the upper right panel, the subplanetary point has turned into the “warmest” location along the equator. On the one hand, this is due to the inclination of 22.5°, which is why the moon does not transit behind the planet for most of the orbital phase around the star. On the other hand, this location gets slightly more irradiation from the planet than any other place on the moon. In the lower left panel, the average flux contrast between equatorial and polar illumination has decreased further. Again, the subplanetary point is slightly warmer than the rest of the surface. In the lower right panel, finally, where the moon’s orbital inclination is set to 90°, the equator has become the coldest region of the moon, with the subplanetary point still being the warmest location along 0° latitude. ! While the major panels show that the orbit-averaged flux contrast decreases with increasing inclination, the side panels indicate an increasing irradiation contrast between seasons. Exomoons around host planets with obliquities similar to that of Jupiter with respect to the Sun ( ψ p ≈
0) are subject to an irradiation pattern corresponding to the upper left panel of Fig. 7. The upper right panel depicts an irradiation pattern of exomoons around planets with obliquities similar to Saturn (26.7°) and
Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 3.
Contours of the correction factor for the limits of the IHZ for exomoons, induced by the star’s reflected light from the planet. Since we neglect the thermal component, values are lower limits. The left-most contour signifies 1.01. The dotted vertical line denotes the Roche lobe. eller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 4.
Stellar and planetary contributions to the illumination of our prototype moon as a function of orbital phase φ ps . Tiny dots label the thermal flux from the planet ( f t ), normal dots the reflected stellar light from the planet ( f r ), dashes the stellar light ( f ∗ ), and the solid line is their sum. The panels depict different longitudes and latitudes on the moon’s surface. The upper left panel is for the subplanetary point, the upper right 45° counterclockwise along the equator, the lower left panel shows a position 90° counterclockwise from the subplanetary point, and the lower right is the antiplanetary point. FIG. 5.
Stellar and planetary contributions to the illumination of our prototype moon as in Fig. 4 but at a stellar orbital phase φ ∗ p = 0.5 in an eccentric orbit ( e ∗ p = 0.3) and with an inclination i = π /4 ≘
45° of the moon’s orbit against the circumstellar orbit.s qualitatively in good agreement with the yearly illumination pattern of Titan as simulated by Mitchell (2012, see his Fig. 1c). Exomoons around a planet with a Uranus-like obliquity (97.9°) will have an irradiation similar to the lower right panel. ! The typical orbit-averaged flux between 300 and 400W/m in Fig. 7 is about a quarter of the solar constant. This is equivalent to an energy redistribution factor of 4 over the moon’s surface (Selsis et al. 2007), indicating that climates on exomoons with orbital periods of a few days (in this case 3.55d, corresponding to Europa’s orbit about Jupiter) may be more similar to those of freely rotating planets rather than to those of planets that are tidally locked to their host star. Tidal heating is an additional source of energy on moons. Various approaches for the description of tidal processes have been established. Two of the most prominent tidal theories are the “constant-time-lag” (CTL) and the “constant-phase-lag” (CPL) models. Their merits and perils have been treated extensively in the literature (Ferraz-Mello et al. 2008; Greenberg 2009; Efroimsky & Williams 2009; Hansen 2010; Heller et al. 2010,2011b; Lai 2012) and it turns out that they agree for low eccentricities. To begin with, we arbitrarily choose the CTL model developed by Hut (1981) and Leconte et al. (2010) for the computation of the moon’s instantaneous tidal heating, but we will compare predictions of both CPL and CTL theory below. We consider a tidal time lag τ s = 638s, similar to that of Earth (Lambeck 1977; Neron de Surgy & Laskar 1997), and an appropriate second-order potential Love number of k = 0.3 (Henning et al. 2009). ! In our two-body system of the planet and the moon, tidal heating on the satellite, which is assumed to be in equilibrium rotation and to have zero obliquity against the orbit around the planet, is given bywhere
Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 6.
Illustration of the antiplanetary winter on the moon with the same orbital elements as in Fig. 5. The arrow in the edge view panel indicates the surface normal at φ =180°, θ =80°, i.e., close to pole and on the antiplanetary side of the moon. For all orbital constellations of the moon around the planet ( φ ps going from 0 to 1), this location on the moon receives neither irradiation from the star nor from the planet (see lower right panel in Fig. 5). Shadings correspond to the same irradiation patterns as in Fig. 1. ˙ E eqtid , s = Z s ( e ps ) f ( e ps ) f ( e ps ) f ( e ps ) , (19) Z s ⌘ G k , s M ( M p + M s ) R a ⇥ s (20) and ( e ps ) = q e ,f ( e ps ) = 1 + 312 e + 2558 e + 18516 e + 2564 e ,f ( e ps ) = 1 + 152 e + 458 e + 516 e ,f ( e ps ) = 1 + 3 e + 38 e (21) . ere, G is Newton's gravitational constant, and M p is the planet's mass. The contribution of tidal heating to the moon'senergy flux can be compared to the incoming irradiation when we divide ˙ E eqtid , s by the surface of the moon and define itssurface tidal heating h s ⌘ ˙ E eqtid , s / (4 R ) . We assume that h s is emitted uniformly through the satellite’s surface. ! To stress the importance of tidal heating on exomoons, we show the sum of h s and the absorbed stellar flux for four star-planet-moon constellations as a function of e ps and a ps in Fig. 8. In the upper row, we consider our Earth-like prototype moon, in the lower row the Super-Ganymede. The left column corresponds to a Jupiter-like host planet, the right column to a Neptune-like planet, both at 1AU from a Sun-like star. Contours indicate regions of constant energy flux as a function of a ps and e ps . Far from the planet, illustrated by a white area at the right in each plot, tidal heating is negligible, and the total heat flux on the moon corresponds to an absorbed stellar flux of 239W/m , which is equal to Earth’s absorbed flux. The right-most contour in each panel depicts a contribution by tidal heating of 2W/m , which corresponds to Io’s tidal heat flux (Spencer et al. 2000). Satellites left of this line will not necessarily experience enhanced volcanic activity, since most of the dissipated energy would go into water oceans of our prototype moons, rather than into the crust as on Io. The blue contour corresponds to a tidal heating of 10W/m , and the red line demarcates the transition into a runaway greenhouse, which occurs at 295W/m for the Earth-mass satellite and at 266W/m for the Super-Ganymede. For comparison, we show the positions of some prominent moons in the Solar System, where a ps is measured in radii of the host planet. Intriguingly, both the Earth-like exomoon and the Super-Ganymede, orbiting either a Jupiter- or a Neptune-mass planet, would be habitable in a Europa- or Miranda-like orbit (in terms of fractional planetary radius and eccentricity), while they would enter a runaway greenhouse state in an Io-like orbit. Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 7.
Illumination of our prototype exomoon (in W/m ) averaged over the orbit of the planet-moon duet around their host star. Major panels present four different orbital inclinations: i = 0° (upper left), i = 22.5° (upper right), i = 45° (lower left), and i = 90° (lower right). The two bars beside each major panel indicate averaged flux for the northern summer (ns) and southern summer (ss) on the moon. Contours of constant irradiation are symmetric about the equator; some values are given. The Roche radii are ≈ R p for an Earth-type moon about a Jupiter-like planet (upper left panel in Fig. 8), ≈ R p for an Earth-type moon about a Neptune-mass planet (upper right), ≈ R p for the Super-Ganymede about a Jupiter-class host (lower left), and ≈ R p for the Super-Ganymede orbiting a Neptune-like planet (lower right). The extreme tidal heating rates in the red areas may not be realistic because we assume a constant time lag τ s of the satellite’s tidal bulge and ignore its dependence on the driving frequency as well as its variation due to the geological processes that should appear at such enormous heat fluxes. ! In Fig. 8, irradiation from the planet is neglected; thus decreasing distance between planet and moon goes along with increasing tidal heating only. An Earth-like exomoon (upper panels) could orbit as close as ≈ R p , and tidal heating would not induce a runaway greenhouse if e ps ≲ e ps ≈ ≳ R p to prevent a runaway greenhouse. Comparison of the left and right panels shows that for a more massive host planet (left) satellites can be slightly closer and still be habitable. This is because we draw contours over the fractional orbital separation a ps / R p on the abscissa. In the left panels 10 R p ≈ × km, whereas in the right panel 10 R p ≈ × km. When plotting over a ps / R p , this discrepancy is somewhat balanced by the much higher mass of the host planet in the left panel and the strong dependence of tidal heating on M p (see Eq. 20). Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 8.
Contours of summed absorbed stellar irradiation and tidal heating (in logarithmic units of W/m ) as a function of semi-major axis a ps and eccentricity e ps on an Earth-like (upper row) and a Super-Ganymede (lower row) exomoon. In the left panels, the satellite orbits a Jupiter-like planet, in the right panels a Neptune-mass planet, in both cases at 1AU from a Sun-like host star. In the white area at the right, tidal heating is negligible and absorbed stellar flux is 239W/m . The right-most contours in each panel indicate Io’s tidal heat flux of 2W/m , a tidal heating of 10W/m , and the critical flux for the runaway greenhouse (295W/m for the Earth-like moon and 266W/m for the Super-Ganymede). Positions of some massive satellites in the Solar System are shown for comparison. .2.1 Planet-moon orbital eccentricity Tidal heating on the moon will only be significant, if e ps ≠
0. If the eccentricity is free , that is, the moon is not significantly perturbed by other bodies, then e ps will change with time due to tidal damping, and orbit-averaged equations can be applied to simulate the tidal evolution. For those cases considered here, d e ps /d t < 0. Similar to the approach used by Heller et al. (2011b), we apply a CTL model (Leconte et al. 2010) and a CPL model (Ferraz-Mello et al. 2008) to compute the tidal evolution, albeit here with a focus on exomoons rather than on exoplanets. ! In Fig. 9, we show the evolution of our Earth-sized prototype moon in orbit around a Jupiter-like planet. The upper two panels show the change of eccentricity e ps ( t ), the lower two panels of tidal surface heating h s ( t ). In the left panels, evolution is backward until − yr; in the right panels, evolution is forwards until +10 yr. The initial eccentricity is set to 0.1, and for the distance between the planet and the moon we choose the semi-major axis of Europa around Jupiter. Going backward, eccentricity increases as does tidal heating. When e ps → ≈ − to 1300W/m . The right panels show that free eccentricities will be damped to zero within <10Myr and that tidal heating becomes negligible after ≈ ! Although e ps is eroded in <10Myr in these two-body simulations, eccentricities can persists much longer, as the cases of Io around Jupiter and Titan around Saturn show. Their eccentricities are not free, but they are forced , because they are excited by interaction with other bodies. The origin of Titan’s eccentricity e ps = 0.0288 is still subject to debate. As shown Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating For the CPL model, e ps converges to 0.285 for − t ≳ ε and ε (for details see Ferraz-Mello et al. 2008; Heller et al. 2011b). We have tried other initial eccentricities, which all led to this convergence. FIG. 9.
Evolution of the orbital eccentricity (upper row) and the moon’s tidal heating (lower row) following the two-body tidal models of Leconte et al. (2010) (a “constant-time-lag” model, solid line) and Ferraz-Mello et al. (2008) (a “constant-phase-lag” model, dashed line). Initially, an Earth-sized moon is set in an eccentric orbit ( e ps = 0.1) around a Jupiter-mass planet at the distance in which Europa orbits Jupiter. In the left panels evolution is backward, in the right panels into the future. Both tidal models predict that free eccentricities are eroded and tidal heating ceases after <10Myr.y Sohl et al. (1995), tidal dissipation would damp it on timescales shorter than the age of the Solar System. It could only be primordial if the moon had a methane or hydrocarbon ocean that is deeper than a few kilometers. However, surface observations by the Cassini Huygens lander negated this assumption. Various other possibilities have been discussed, such as collisions or close encounters with massive bodies (Smith et al. 1982; Farinella et al. 1990) and a capture of Titan by Saturn ≪ e ps = 0.0041 lies in the moon’s resonances with Ganymede and Callisto (Yoder 1979). Such resonances may also appear among exomoons. ! Using the publicly available N -body code Mercury (Chambers 1999) , we performed N -body experiments of a hypothetical satellite system to find out whether forced eccentricities can drive a tidal greenhouse in a manner analogous to the volcanic activity of Io. We chose a Jupiter-mass planet with an Earth-mass exomoon at the same distance as Europa orbits Jupiter, placed a second exomoon at the 2:1 external resonance, and integrated the resulting orbital evolution. In one case, the second moon had a mass equal to that of Earth, in the other case a mass equal to that of Mars. For the former case, we find the inner satellite could have its eccentricity pumped to 0.09 with a typical value of 0.05. For the latter, the maximum eccentricity is 0.05 with a typical value near 0.03. Although these studies are preliminary, they suggest that massive exomoons in multiple configurations could trigger a runaway greenhouse, especially if the moons are of Earth-mass, and that their circumstellar IHZ could lie further away from the star. A comprehensive study of configurations that should also include Cassini states and damping to the fixed-point solution is beyond the scope of this study but could provide insight into the likelihood that exomoons are susceptible to a tidal greenhouse.
4. Orbits of habitable exomoons
By analogy with the circumstellar habitable zone for planets, we can imagine a minimum orbital separation between a planet and a moon to let the satellite be habitable. The range of orbits for habitable moons has no outer edge, except that Hill stability must be ensured. Consequently, habitability of moons is only constrained by the inner edge of a circum-planetary habitable zone, which we call the “habitable edge”. Moons inside the habitable edge are in danger of running into a greenhouse by stellar and planetary illumination and/or tidal heating. Satellites outside the habitable edge with their host planet in the circumstellar IHZ are habitable by definition. ! Combining the limit for the runaway greenhouse from Section 2.2 with our model for the energy flux budget of extra-solar moons from Section 3, we compute the orbit-averaged global flux ¯ F globs received by a satellite, which is the sum of the averaged stellar ( ¯ f ⇤ ) , reflected ( ¯ f r ) , thermal ( ¯ f t ) , and tidal heat flux ( h s ). Thus, in order for the moon to be habitable F RG > ¯ F globs = ¯ f ⇤ + ¯ f r + ¯ f t + h s = L ⇤ (1 s )16 ⇥a ⇤ p q e ⇤ p ⇥R p a ! + R ⇤ SB ( T eqp ) a (1 s )4 + h s , (22) where the critical flux for a runaway greenhouse F RG is given by Eq. (1), the tidal heating rate h s ⌘ ˙ E eqtid , s / (4 R ) by Eq. (19),and the planet’s thermal equilibrium temperature T eqp can be determined with Eq. (11) and using d T = 0. Note that the addend“1” in brackets implies that we do not consider reduction of stellar illumination due to eclipses. This effect is treated in a companion paper (Heller 2012). ! In Fig. 10, we show the F RG = ¯ F globs orbits of both the Earth-like (blue lines) and the Super-Ganymede (black lines)prototype moon as a function of the planet-moon semi-major axis and the mass of the giant host planet, which orbits the Sun-like star at a distance of 1AU. Contours are plotted for various values of the orbital eccentricity, which means that orbits to the left of a line induce a runaway greenhouse for the respective eccentricity of the actual moon. These innermost, limiting orbits constitute the circumplanetary habitable edges. ! When the moon is virtually shifted toward the planet, then illumination from the planet and tidal heating increase, reaching F RG at some point. With increasing eccentricity, tidal heating also increases; thus F RG will be reached farther away from the planet. Blue lines appear closer to the planet than black contours for the same eccentricity, showing that more massive moons can orbit more closely to the planet and be prevented from becoming a runaway greenhouse. This is a purely atmospheric Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating For consistency, we computed the planetary radius (used to scale the abscissa) as a function of the planet’s mass (ordinate) by fitting a high-order polynomial to the Fortney et al. (2007) models for a giant planet at 1AU from a Sun-like star (see line 17 in their Table 4). ffect, determined only by the moon’s surface gravity g s ( M s , R s ) in Eq. (1). We also see that, for a fixed eccentricity, moons can orbit closer to the planet – both in terms of fractional and absolute units – if the planet’s mass is smaller. ! Estimating an exomoon’s habitability with this model requires a well-parametrized system. With the current state of technology, stellar luminosity L ∗ and mass M ∗ can be estimated by spectral analysis and by using stellar evolution models. In combination with the planet’s orbital period P ∗ p , this yields a ∗ p by means of Kepler’s third law. The planetary mass M p could be measured with the radial-velocity (RV) method and assuming it is sufficiently larger than the moon’s mass. Alternatively, photometry could determine the mass ratios M p / M ∗ and M s / M p . Thus, if M s were known from spectroscopy and stellar evolution models, then all masses were accessible (Kipping 2010). Combining RV and photometry, the star-planet orbital eccentricity e ∗ p can be deduced (Mislis et al. 2012); and by combination of TTV and TDV, it is possible to determine M s as well as a ps (Kipping 2009a). Just like the planetary radius, the moon’s radius R s can be determined from photometric transit observations if its transit can directly be observed. The satellite’s second-order tidal Love number k and its tidal time lag τ s , however, would have to be assumed. For this purpose, Earth or Solar System moons could serve as reference bodies. If the age of the system (i.e., of the star) was known or if the evolution of the satellite’s orbit due to tides could be observed, then tidal theory could give a constraint on the product of k and τ s (or k / Q s in CPL theory) (Lainey et al. 2009; Ferraz-Mello 2012). Then the remaining free parameters would be the satellite’s albedo α s and the orbital eccentricity of the planet-moon orbit e ps . N -body simulations of the system would allow for an average value of e ps . ! We conclude that combination of all currently available observational and theoretical techniques can, in principle, yieldan estimation of an exomoon’s habitability. To that end, the satellite’s global average energy flux ¯ F globs (Eq. 22) needs to becompared to the critical flux for a runaway greenhouse F RG (Eq. 1).
5. Application to Kepler-22b and KOI211.01
We now apply our stellar-planetary irradiation plus tidal heating model to putative exomoons around Kepler-22b and KOI211.01, both in the habitable zone around their host stars. The former is a confirmed transiting Neptune-sized planet (Borucki et al. 2012), while the latter is a much more massive planet candidate (Borucki et al. 2011). We choose these two planets that likely have very different masses to study the dependence of exomoon habitability on M p . ! For the moon, we take our prototype Earth-sized moon and place it in various orbits around the two test planets to investigate a parameter space as broad as possible. We consider two planet-moon semi-major axes: a ps = 5 R p , which is similar to Miranda’s orbit around Uranus, and a ps = 20 R p , which is similar to Titan’s orbit around Saturn . We also choose two eccentricities, namely, e ps = 0.001 (similar to Miranda) and e ps = 0.05 (somewhat larger than Titan’s value). With this parametrization, we cover a parameter space of which the diagonal is spanned by Miranda’s close, low-eccentricity orbit around Uranus and Titan’s far but significantly eccentric orbit around Saturn (see Fig. 8). We must keep in mind, however, that strong additional forces, such as the interaction with further moons, are required to maintain eccentricities of 0.05 in the Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating Note that in planet-moon binaries close to the star Hill stability requires that the satellite’s orbit is a few planetary radii at most. Hence, if they exist, moons about planets in the IHZ around M dwarfs will be close to the planet (Barnes & O’Brien 2002; Domingos et al. 2006; Donnison 2010; Weidner & Horne 2010).
FIG. 10.
Innermost orbits to prevent a runaway greenhouse, i.e., the “habitable edges” of an Earth-like (blue lines) and a Super-Ganymede (black lines) exomoon. Their host planet is at 1AU from a Sun-like star. Flux contours for four eccentricities of the moons’ orbits from e ps = 10 − to e ps = 10 − are indicated. The larger e ps , the stronger tidal heating and the more distant from the planet will the critical flux be reached.ssumed close orbits for a long time, because tidal dissipation will damp e ps . Finally, we consider two orbital inclinations for the moon, i = 0° and i = 45°. Orbiting its solar-mass, solar-luminosity ( T eff, ∗ = 5518K, R ∗ = 0.979 R ⊙ , R ⊙ being the radius of the Sun) host star at a distance of ≈ M ⊕ , consistent with their photometric and radial-velocity follow-up measurements but not yet well constrained by observations. This ambiguity leaves open the question whether Kepler-22b is a terrestrial, gaseous, or transitional object. ! In Fig. 11, we show the photon flux, coming both from the absorbed and re-emitted illumination as well as from tidal heating, at the upper atmosphere for an Earth-sized exomoon in various orbital configurations around Kepler-22b. Illumination is averaged over one orbit of the planet-moon binary around the star. In the upper four panels a ps = 5 R p (similar to the Uranus-Miranda semi-major axis), while in the lower four panels a ps = 20 R p (similar to the Saturn-Titan semi-major axis). The left column shows co-orbital simulations ( i = 0°); in the right column i = 45°. In the first and the third line e ps = 0.001; in the second and fourth line e ps = 0.05. ! In orbits closer than ≈ R p even very small eccentricities induce strong tidal heating of exomoons around Kepler-22b. For e ps = 0.001 (upper row), a surface heating flux of roughly 6250W/m should induce surface temperatures well above the surface temperatures of Venus, and for e ps = 0.05 (second row), tidal heating is beyond 10 W/m , probably melting the whole hypothetical moon (Léger et al. 2011). For orbital distances of 20 R p , tidal heating is 0.017W/m for the low-eccentricity scenario; thus the total flux is determined by stellar irradiation (third line). However, tidal heating is significant for the e ps = 0.05 case (lower line), namely, roughly 42W/m . ! Non-inclined orbits induce strong variations of irradiation over latitude (left column), while for high inclinations, seasons smooth the distribution (right column). As explained in Section 3.1.4, the subplanetary point for co-planar orbits is slightly cooler than the maximum temperature due the eclipses behind the planet once per orbit. But for tilted orbits, the subplanetary point becomes the warmest spot. ! We apply the tidal model presented in Heller et al. (2011b) to compute the planet’s tilt erosion time t ero and assess whether its primordial obliquity ψ p could still persist today. Due to its weakly constrained mass, the value of the planet’s tidal quality factor Q p is subject to huge uncertainties. Using a stellar mass of 1 M ⊙ and trying three values Q p = 10 , 10 and 10 , we find t ero = 0.5Gyr, 5Gyr, and 50Gyr, respectively. The lowest Q p value is similar to that of Earth, while the highest value corresponds approximately to that of Neptune. Thus, if Kepler-22b turns out mostly gaseous and provided that it had a significant primordial obliquity, the planet and its satellites can experience seasons today. But if Kepler-22b is terrestrial and planet-planet perturbations in this system can be neglected, it will have no seasons; and if its moons orbit above the planet’s equator, they would share this tilt erosion.
KOI211.01 is a Saturn- to Jupiter-class planet candidate (Borucki et al. 2011). In the following, we consider it as a planet. Its radius corresponds to 0.88 R J , and it has an orbital period of 372.11d around a 6072K main-sequence host star, whichyields an estimate for the stellar mass and then a semi-major axis of 1.05AU. The stellar radius is 1.09 R ⊙ , and using modelsfor planet evolution (Fortney et al. 2007), we estimate the planet’s mass to be 0.3 M J . This value is subject to various uncertainties because little is known about the planet’s composition, the mass of a putative core, the planet’s atmospheric opacity and structure, and the age of the system. Thus, our investigations will serve as a case study rather than a detailed prediction of exomoon scenarios around this particular planet. Besides KOI211.01, some ten gas giants have been confirmed in the IHZ of their host stars, all of which are not transiting. Thus, detection of their putative moons will not be feasible in the foreseeable future. ! In Fig. 12, we present the flux distribution on our prototype Earth-like exomoon around KOI211.01 in the same orbital configurations as in the previous subsection. Tidal heating in orbits with a ps ≲ R p can be strong, depending on eccentricity, Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating See Gavrilov & Zharkov (1977) and Heller et al. (2010) for discussions of Q values and Love numbers for gaseous substellar objects. eller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 11.
Orbit-averaged flux (in units of W/m ) at the top of an Earth-sized exomoon’s atmosphere around Kepler-22b for eight different orbital configurations. Computations include irradiation from the star and the planet as well as tidal heating. The color bar refers only to the lower two rows with moderate flux. eller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 12.
Orbit-averaged flux (in units of W/m ) at the top of an Earth-sized exomoon’s atmosphere around KOI211.01for eight different orbital configurations. Computations include irradiation from the star and the planet as well as tidalheating. The color bar refers only to the first and the lower two rows with moderate flux.nd make the moon uninhabitable. For e ps = 0.001, tidal heating is of the order of 1W/m and thereby almost negligible for the moderate flux distribution (first line). However, it increases to over 3200W/m for e ps = 0.05 and makes the moon uninhabitable (second line). Tidal heating is negligible at semi-major axes ≳ R p even for a significant eccentricity of 0.05. ! By comparison of Fig. 11 with Fig. 12, we see that in those cases where tidal heating can be neglected and stellar irradiation dominates the energy flux, namely, in the third line of both figures, irradiation is higher on comparable moons around KOI211.01. But in the lower panel line, the total flux on our Kepler-22b prototype satellite becomes comparable to the one around KOI211.01, which is because of the extra heat from tidal dissipation. Surface flux distributions in the low-eccentricity and the moderate-eccentricity state on our KOI211.01 exomoons are virtually the same at a ps = 20 R p . ! Since KOI211.01 is a gaseous object, we apply a tidal quality factor of Q p = 10 , which is similar to, but still lower than, the tidal response of Jupiter. We find that t ero of KOI211.01 is much higher than the age of the Universe. Thus, satellites of KOI211.01 will experience seasons, provided the planet had a primordial obliquity and planet-planet perturbations can be neglected. ! The radius of KOI211.01 is about 4 times greater than the radius of Kepler-22b; thus moons at a certain multitude of planetary radii distance from the planet, say 5 R p or 20 R p as we considered, will be effectively much farther away from KOI211.01 than from Kepler-22b. As tidal heating strongly depends on a ps and on the planet’s mass, a quick comparison between tidal heating on exomoons around Kepler-22b and KOI211.01 can be helpful. For equal eccentricities, the fraction of tidal heating in two moons about Kepler-22b and KOI211.01 will be equal to a fraction of Z s from Eq. (20). By taking the planet masses assumed above, thus M KOI ≈ M Ke , and assuming that the satellite mass is much smaller than the planets’ masses, respectively, we deduce Z Kes Z KOIs = M M ( M Ke + M s )( M KIO + M s ) a KOIps a Keps ! M s negligible ⇡ . ⇡ , (23) where indices ‘Ke’ and ‘KOI’ refer to Kepler-22b and KOI211.01, respectively. The translation of tidal heating from any panel in Fig. 11 to the corresponding panel in Fig. 12 can be done with a division by this factor. We now set our results in context with observables to obtain a first estimate for the magnitude of the TTV amplitude induced by habitable moons around Kepler-22b and KOI211.01. For the time being, we will neglect the actual detectability of such signals with
Kepler but discuss it in Section 6 (see also Kipping et al. 2009). ! To begin with, we apply our method from Section 4 and add the orbit-averaged stellar irradiation on the moon to the averaged stellar-reflected light, the averaged thermal irradiation from the planet, and the tidal heating (in the CTL theory).We then compare their sum ¯ F globs (see Eq. 22) to the critical flux for a runaway greenhouse F RG on the respective moon. InFig. 13, we show the limiting orbits ( ¯ F globs = F RG ) for a runaway greenhouse of an Earth-like (upper row) and a Super-Ganymede (lower row) exomoon around Kepler-22b (left column) and KOI211.01 (right column). For both moons, we consider two albedos ( α s = 0.3, gray solid lines; and α s = 0.4, black solid lines) and three orbital eccentricities ( e ps = 0.001, 0.01, 0.1). Solid lines define the habitable edge as described in Section 4. Moons in orbits to the left of a habitable edge are uninhabitable, respectively, because the sum of stellar and planetary irradiation plus tidal heating exceeds the runaway greenhouse limit. Dashed lines correspond to TTV amplitudes and will be discussed below. Let us first consider the three black solid lines in the upper left panel, corresponding to an Earth-like satellite with α s = 0.4 about Kepler-22b. Each contour represents a habitable edge for a certain orbital eccentricity of the moon; that is, in these orbits the average energy flux on the moon is equal to F RG = 295W/m . Assuming a circular orbit around the star ( e ∗ p = 0), the orbit-averaged stellar flux absorbed by a moon with α s = 0.4 isusing the parametrization presented in Section 5.1. Thus, black contours indicate an additional heating of ≈ . The indicated eccentricities increase from left to right. This is mainly due to tidal heating, which increases for larger eccentrici- Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating L ⇤ (1 s )16 ⇥a ⇤ p ⇡
227 W / m , (29) ies but decreases at larger separations a ps . Thus, the larger e ps , the farther away from the planet the moon needs to be to avoidbecoming a greenhouse. If a moon is close to the planet and shall be habitable, then its eccentricity must be small enough. ! Next, we compare the black lines to the gray lines, which assume an albedo of 0.3 for an Earth-like satellite. With this albedo, it absorbs ≈ of stellar irradiation. Consequently, a smaller amount of additional heat is required to push it into a runaway greenhouse, so the corresponding habitable edges are farther away from the planet, where both tidal heating and irradiation from the planet are lower. ! In the lower left panel, we consider a Super-Ganymede around Kepler-22b. Its critical flux of 266W/m is smaller than for an Earth-like moon, so for an albedo of 0.4 only 39W/m of additional heating is required for the moon to turn into a runaway greenhouse. Yet, compared to the Earth-like satellite in the upper left panel, the habitable edges (black lines) are still slightly closer to the planet, because tidal heating in the smaller Super-Ganymede is much weaker at a given orbital-semi major axis. More important, gray lines are at much larger distances in this panel because with an albedo of 0.3 the satellite’s absorbed stellar flux (265W/m ) is almost as high as the critical flux (266W/m ). This is irrespective of whether our Ganymede-like object is orbiting a planet. ! Next, we consider moons in orbit about KOI211.01, shown in the right column of Fig. 13. First note the absence of gray
Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. 13.
Habitable edges for an Earth-like (upper row) and a Super-Ganymede (lower row) exomoon in orbit around Kepler-22b (left column) and KOI211.01 (right column). Masses of both host planets are not well constrained; thus abscissae run over several decades (in units of M ⊕ for Kepler-22b and M J for KOI211.01). We consider two albedos α s = 0.3 (gray solid lines) and α s = 0.4 (black solid lines) and three eccentricities ( e ps = 0.001, 0.01, 0.1) for both moons. No gray solid lines are in the right column because both prototype moons would not be habitable with α s = 0.3 around KOI211.01. TTV amplitudes (in units of seconds) for coplanar orbits are plotted with dashed lines.olid lines. For moons with an albedo of α s = 0.3, the orbit-averaged stellar irradiation will be 315W/m , with the parametrization presented in Sect 5.2. This means that stellar irradiation alone is larger than the critical flux of both our Earth-like (295W/m , upper panel) and Super-Ganymede (266W/m , lower panel) prototype moons. Thus, both moons with α s = 0.3 around KOI211.01 are not habitable, irrespective of their distance to the planet. ! Moons with α s = 0.4 around KOI211.01 absorb 270W/m , which is less than the critical flux for the Earth-like moon (upper right panel) but more than the Super-Ganymede moon could bear (lower right panel). Thus, the black lines are absent from the latter plot but show up in the upper right. Contours for given eccentricities are closer to the planet than their counterparts in the case of Kepler-22b. This is because we plot the semi-major axis in units of planetary radii – while KOI211.01 has a much larger radius than Kepler-22b – and tidal heating, which strongly depends on the absolute distance between the planet and the moon. Now we compare the habitable edges as defined by Eq. (22) to the amplitude of the TTV induced by the moon on the planet. TTV amplitudes Δ in units of seconds are plotted with dashed lines in Fig. 13. Recall, however, that Δ would not directly be measured by transit observations. Rather the root-mean-square of the TTV wave would be observed (Kipping 2009a). We apply the Kipping et al. (2012) equations, assuming that the moon’s orbit is circular and that both the circumstellar and the circumplanetary orbit are seen edge-on from Earth. ! Each panel of Fig. 13 shows how the TTV amplitude decreases with decreasing semi-major axis of the satellite (from right to left) and with increasing planetary mass (from bottom to top). Thus, for a given planetary mass, a moon could be habitable if its TTV signal is sufficiently large. Comparison of the upper and the lower panels in each column shows how much larger the TTV amplitude of an Earth-like moon is with respect to our Super-Ganymede moon. ! Corrections due to an inclination of the moon’s orbit and an accidental alignment of the moon’s longitude of the periapses are not included but would be small in most cases. For non-zero inclinations, the TTV amplitude Δ will bedecreased. This decrease is proportional to p cos( i s ) sin( s ) , where i s is measured from the circumstellar orbitalplane normal to the orbital plane of the planet-moon orbit, and $ s denotes the orientation of the longitude of the periapses(see Eq. (6.46) in Kipping 2011b). Thus, corrections to our picture will only be relevant if both the moon’s orbit is significantly tilted and $ s ≈ $ s ≈ Δ by <10%, while for $ s ≈ i s could be as large as 40° to produce a similar correction. Simulations of the planet’s and the satellite’s tilt erosion can help assess whether substantial misalignments are likely (Heller et al. 2011b). While such geometric blurring should be small for most systems, prediction of the individual TTV signal for a satellite in an exoplanet system – as we suggest in Fig. 13 – is hard because perturbations of other planets, moons, or Trojans could affect the TTV.
6. Summary and discussion
Our work yields the first translation from observables to exomoon habitability. Using a scaling relation for the onset of the runaway greenhouse effect, we have deduced constraints on exomoon habitability from stellar and planetary irradiation aswell as from tidal heating. We determined the orbit-averaged global energy budget ¯ F globs for exomoons to avoid a runawaygreenhouse and found that for a well parametrized system of a star, a host planet, and a moon, Eq. (22) can be used to evaluate the habitability of a moon. By analogy with the circumstellar habitable zone (Kasting et al. 1993), these rules define a circumplanetary “habitable edge”. To be habitable, moons must orbit their planets outside the habitable edge. ! Application of our illumination plus tidal heating model shows that an Earth-sized exomoon about Kepler-22b with a bond albedo of 0.3 or higher would be habitable if ( i .) the planet’s mass is ≈ M ⊕ , ( ii .) the satellite would orbit Kepler-22b with a semi-major axis ≳ R p , and ( iii .) the moon’s orbital eccentricity e ps would be <0.01 (see Fig. 13). If Kepler-22b turns out more massive, then such a putative moon would need to be farther away to be habitable. Super-Ganymede satellites of Kepler-22b meet similar requirements, but beyond that their bond albedo needs to be ≳ ! Super-Ganymede or smaller moons around KOI211.01 will not be habitable, since their critical flux for a runaway greenhouse effect ( ≤ ) is less than the orbit-averaged irradiation received by the star (270W/m ). But Earth-like or more massive moons can be habitable if ( i .) the planet’s mass ≈ M J , ( ii .) the satellite’s bond albedo ≳ iii .) the satellite orbits KOI211.01 with a semi-major axis ≳ R p , and ( iv .) the moon’s orbital eccentricity e ps < 0.01. If the planet turns out less massive, then its moons could be closer and have higher eccentricities and still be habitable. Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating Kepler-22 and KOI211 both are mid-G-type stars, with
Kepler magnitudes 11.664 (Borucki et al. 2012) and 14.99 (Borucki et al. 2011), respectively. As shown by Kipping et al. (2009), the maximum
Kepler magnitude to allow for the detection of an Earth-like moon is about 12.5. We conclude that such moons around Kepler-22b are detectable if they exist, and their habitability could be evaluated with the methods provided in this communication. Moons around KOI211.01 will not be detectable within the 7 year duty cycle of
Kepler . Nevertheless, our investigation of this giant planet’s putative moons serves as a case study for comparably massive planets in the IHZ of their parent stars. ! Stellar flux on potentially habitable exomoons is much stronger than the contribution from the planet. Nevertheless, the sum of thermal emission and stellar reflected light from the planet can have a significant impact on exomoon climates. Stellar reflection dominates over thermal emission as long as the planet’s bond albedo α p ≳ once the moon is in a close orbit ( a ps ≲ R p ) and the planet has a high albedo ( α p ≳ ! Due to the weak tides from its host star, KOI211.01 can still have a significant obliquity, and if its moons orbit the planet in the equatorial plane they could have seasons and are more likely to be discovered by transit duration variations of the transit impact parameter (TDV-TIP, Kipping 2009b). For Kepler-22b, the issue of tilt erosion cannot be answered unambiguously until more about the planet’s mass and composition is known. ! If a moon’s orbital inclination is small enough, then it will be in the shadow of the planet for a certain time once per planet-moon orbit (see also Heller 2012). For low inclinations, eclipses can occur about once every revolution of the moon around the planet, preferentially when the subplanetary hemisphere on the moon would experience stellar irradiation maximum (depending on e ∗ p ). Eclipses have a profound impact on the surface distribution of the moon’s irradiation. For low inclinations, the subplanetary point on the moon will be the “coldest” location along the equator, whereas for moderate inclinations it will be the “warmest” spot on the moon due to the additional irradiation from the planet. Future investigations will clarify whether this may result in enhanced sub planetary weathering instabilities, that is, runaway CO drawdown rates eventually leading to very strong greenhouse forcing, or sub planetary dissolution feedbacks of volatiles in sub planetary oceans, as has been proposed for exoplanets that are tidally locked to their host stars and thus experience such effects at the fixed sub stellar point (Kite et al. 2011). ! We predict seasonal illumination phenomena on the moon, which emerge from the circumstellar season and planetary illumination. They depend on the location on the satellite and appear in four versions, which we call the “proplanetary summer”, “proplanetary winter”, “antiplanetary summer”, and “antiplanetary winter”. The former two describe seasons due to the moon’s obliquity with respect to the star with an additional illumination from the planet; the latter two depict the permanent absence of planetary illumination during the seasons. ! For massive exomoons with a ps ≲ R p around Kepler-22b and around KOI211.01, tidal heating can be immense, presumably making them uninhabitable if the orbits are substantially non-circular. On the one hand, tidal heating can be a threat to life on exomoons, in particular when they are in close orbits with significant eccentricities around their planets. If the planet-moon duet is at the inner edge of the circumstellar IHZ, small contributions of tidal heat can render an exomoon uninhabitable. Tidal heating can also induce a thermal runaway, producing intense magmatism and rapid resurfacing on the moon (B ě hounková et al. 2011). On the other hand, we can imagine scenarios where a moon becomes habitable only because of tidal heating. If the host planet has an obliquity similar to Uranus, then one polar region will not be illuminated for half the orbit around the star. Moderate tidal heating of some tens of watts per square meter might be just adequate to prevent the atmosphere from freezing out. Or if the planet and its moon orbit their host star somewhat beyond the outer edge of the IHZ, then tidal heating might be necessary to make the moon habitable in the first place. Tidal heating could also drive long-lived plate tectonics, thereby enhancing the moon’s habitability (Jackson et al. 2008). An example is given by Jupiter’s moon Europa, where insolation is weak but tides provide enough heat to sustain a subsurface ocean of liquid water (Greenberg et al. 1998; Schmidt et al. 2011). On the downside, too much tidal heating can render the body uninhabitable due to enhanced volcanic activity, as it is observed on Io. ! Tidal heating has a strong dependence on the moon’s eccentricity. Eccentricities of exomoons will hardly be measurable even with telescopes available in the next decade, but it will be possible to constrain e ps by simulations. Therefore, once exomoon systems are discovered, it will be necessary to search for further moons around the same planet to consistently simulate the N -body ( N > 2) evolution with multiple-moon interaction, gravitational perturbations from other planets, and the gravitational effects of the star. Such simulations will also be necessary to simulate the long-term evolution of the orientation η of the moon’s inclination i with respect to the periastron of the star-related orbit, because for signifiant Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating ccentricities e ∗ p it will make a big difference whether the summer of either the northern or southern hemisphere coincides with minimum or maximum distance to the star. Technically, these variations refer to the apsidal precession (the orientation of the star-related eccentricity, i.e., of a ∗ p ) and the precession of the planet’s rotation axis, both of which determine η . For Saturn, and thus Titan, the corresponding time scale is of order 1Myr (French et al. 1993), mainly induced by solar torques on both Saturn’s oblate figure and the equatorial satellites. Habitable exomoons might preferably be irregular satellites (see Section 2) for which Carruba et al. (2002) showed that their orbital parameters are subject to particularly rapid changes, driven by stellar perturbations. ! We find that more massive moons can orbit more closely to the planet and be prevented from becoming a runaway greenhouse (Section 2.2). This purely atmospheric effect is shared by all terrestrial bodies. Similar to the circumstellar habitable zone of extrasolar planets (Kasting et al. 1993), we conclude that more massive exomoons may have somewhat wider habitable zones around their host planets – of which the inner boundary is defined by the habitable edge and the outer boundary by Hill stability – than do less massive satellites. In future investigations, it will be necessary to include simulations of the moons’ putative atmospheres and their responses to irradiation and tidal heating. Thus, our irradiation plus tidal heating model should be coupled to an energy balance or global climate model to allow for more realistic descriptions of exomoon habitability. As indicated by our basal considerations, the impact of eclipses and planetary irradiation on exomoon climates can be substantial. In addition to the orbital parameters which we have simulated here, the moons’ climates will depend on a myriad bodily characteristics ! Spectroscopic signatures of life, so-called “biosignatures”, in the atmospheres of inhabited exomoons will only be detectable with next-generation, several-meter-class space telescopes (Kaltenegger 2010; Kipping et al. 2010). Until then, we may primarily use our knowledge about the orbital configurations and composition of those worlds when assessing their habitability. Our method allows for an evaluation of exomoon habitability based on the data available at the time they will be discovered. The recent detection of an Earth-sized and a sub-Earth-sized planet around a G-type star (Fressin et al. 2012) suggests that not only the moons’ masses and semi-major axes around their planets can be measured (e.g., by combined TTV and TDV, Kipping 2009a) but also their radii by direct photometry. A combination of these techniques might finally pin down the moon’s inclination (Kipping 2009b) and thus allow for precise modeling of its habitability based on the model presented here. ! Results of ESA’s
Jupiter Icy Moons Explorer (“ JUICE ”) will be of great value for characterization of exomoons. With launch in 2022 and arrival at Jupiter in 2030, one of the mission’s two key goals will be to explore Ganymede, Europa, and Callisto as possible habitats. Therefore, the probe will acquire precise measurements of their topographic distortions due to tides on a centimeter level; determine their dynamical rotation states (i.e., forced libation and nutation); characterize their surface chemistry; and study their cores, rocky mantles, and icy shells. The search for water reservoirs on Europa, exploration of Ganymede’s magnetic field, and monitoring of Io’s volcanic activity will deliver fundamentally new insights into the planetology of massive moons. ! Although our assumptions about the moons’ orbital characteristics are moderate, that is, they are taken from the parameter space mainly occupied by the most massive satellites in the Solar System, our results imply that exomoons might exist in various habitable or extremely tidally heated configurations. We conclude that the advent of exomoon observations and characterization will permit new insights into planetary physics and reveal so far unknown phenomena, analogous to the staggering impact of the first exoplanet observations 17 years ago. If observers feel animated to use the available
Kepler data, the
Hubble Space Telescope, or meter-sized ground-based instruments to search for evidence of exomoons, then one aim of this communication has been achieved.
AppendixAppendix A: Stellar irradiation
We include here a thorough explanation for the stellar flux f ∗ ( t ) presented in Section 3.1.1. To begin with, we assume that the irradiation on the moon at a longitude φ and latitude θ will be f ⇤ ( t ) = L ⇤ r s ⇤ ( t ) r s ⇤ ( t ) r s ⇤ ( t ) n ⇥, ( t ) n ⇥, ( t ) , (A.1) Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating ith L ∗ as the stellar luminosity and r s ⇤ ( t ) as the vector from the moon to the star (see Fig. A.1). The product of the surface normal n ⇥, /n ⇥, on the moon and r s ⇤ ( t ) /r s ⇤ ( t ) accounts for projection effects on the location ( φ , θ ), and we set f ∗ to zeroin those cases where the star shines from the back. Figure A.1 shows that r s ⇤ ( t ) = r p ⇤ ( t ) + s ( t ) . While the Keplerian motion,encapsulated in r p ⇤ ( t ) , is deduced in Section 3.1.1, we focus here on the moon’s surface normal n ⇥, . ! In the right panel of Fig. A.1, we show a close-up of the star-planet-moon geometry at t = 0, which corresponds to the initial configuration of the orbit evolution. The planet is at periastron ( φ ∗ p = 0) and for the case where η = 0 the vector s = n , ( t ) from the subplanetary point on the moon to the planet would have the form a ps cos ⇣ ( t ⇥ ) P ps ⌘ cos ⇣ i ⌘ sin ⇣ ( t ⇥ ) P ps ⌘ cos ⇣ ( t ⇥ ) P ps ⌘ sin ⇣ i ⌘1CCCCCCCCA , (A.2) where all angles are provided in degrees. The angle η depicts the orientation of the lowest point of the moon’s orbit withrespect to the projection of r p ⇤ on the moon’s orbital plane at t = 0. At summer solstice on the moon’s nothern hemisphere,the true anomaly ν ∗ p equals η (see Eq. B.8). This makes η a critical parameter for the seasonal variation of stellar irradiation on the moon because it determines how seasons, induced by orbital inclination i , relate to the changing distance to the star, induced by star-planet eccentricity e ∗ p . In particular, if η = 0 then northern summer coincides with the periastron passage about the star and nothern winter occurs at apastron, inducing distinctly hot summers and cold winters. It relates to the conventional orientation of the ascending node Ω as η = Ω + 270° and can be considered as the climate-precession parameter. , For η ≠
0, we have to apply a rotation M ( η ): (cid:1) → (cid:1) of s around the z -axis (0,0,1), which is performed by the rotation matrix M ( ) = cos( ) sin( ) 0sin( ) cos( ) 00 0 1 . (A.3) With the abbreviations introduced in Eq. (6) (with φ = 0 for the time being), we obtain Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating FIG. A.1.
Geometry of the triple system of a star, a planet, and a moon. In the left panel the planet-moon duet has advanced by an angle ν ∗ p around the star, and the moon has progressed by an angle 2 π φ ps . The right panel shows a zoom-in to the planet-moon binary. As in the left panel, time has proceeded, and a projection of r p ⇤ ( t ) at time t = 0 has been included to explain the orientation of the moon’s orbit, which is inclined by an angle i and rotated against the star-moon periapses the by an angle η . s ( t ) ⌘ n , ( t ) = a ps ˜ CcC ˜ Ss ˜ ScC ˜ CscS . (A.4) his allows us to parametrize the surface normal s ( t ) /s ( t ) of the subplanetary point on the moon for arbitrary i and η . ! Finally, we want to find the surface normal n ⇥, ( t ) /n ⇥, ( t ) for any location ( φ , θ ) on the moon’s surface. Therefore, wehave to go two steps on the moon’s surface, each one parametrized by one angle: one in longitudinal direction φ and one in latitudinal direction θ . The former one can be walked easily, just by adding 2 πφ /360° in the sine and cosine arguments inEq. (A.2) (see first line in Eq. 6), thus n , = s ( ⇤ ps + 2 ⇥/ ) . This is equivalent to a shift along the moon’s equator. For the second step, we know that if θ = 90°, then the surface normal will be along the rotation axis N = a ps S ˜ C S ˜ SC (A.5) of the satellite; that is, we are standing on the north pole. The vector n ⇥, can then be obtained by tilting s ( ⇤ ps + 2 ⇥/ ) by an angle θ toward N . With N = a ps = s , we then derive n ⇥, ( t ) = a ps sin( ) NN + a ps cos( ) s ( ⌅ ps + 2 ⇥⇤/ ) s ( ⌅ ps + 2 ⇥⇤/ ) = sin( ) N + cos( ) s ( ⌅ ps + 2 ⇥⇤/ )= a ps ¯ sS ˜ C + ¯ c ( ˜ CcC ˜ Ss ) ¯ sS ˜ S + ¯ c ( ˜ ScC ˜ Cs )¯ sC + ¯ ccS , (A.6) which solves Eq. (A.1). Appendix B: Planetary irradiation
In addition to stellar light, the moon will receive thermal and stellar-reflected irradiation from the planet, f t ( t ) and f r ( t ), respec-tively. One hemisphere on the planet will be illuminated by the star and the other one will be dark. On the bright side of the planet, there will be both thermal emission from the planet as well as starlight reflection, while on the dark side the planet will only emit thermal radiation, though with a lower intensity than on the bright side due to the lower temperature. ! We begin with the thermal part. The planet’s total thermal luminosity L th,p will be the sum of the radiation from the brightside and from the dark side. On the bright side, the planet shall have a uniform temperature T be , p , and on the dark side itstemperature shall be T de , p . Then thermal equilibrium between outgoing thermal radiation and incoming stellar radiation yieldswhere the first term in the second line describes the absorbed radiation from the star and the second term ( W p ) can be any additional heat source, for example, the energy released by the gravitation-induced shrinking of the gaseous planet. For Jupiter, Saturn, and Neptune, which orbit the Sun at distances >5AU, W p is greater than the incoming radiation. However, for gaseous planets in the IHZ it will be negligible once the planet has reached an age ≳ Owed to the negligibility for our purpose and for simplicity we set W p = 0. ! In our model, we want to parametrize the two hemispheres by a temperature difference d T ⌘ T be , p T de , p . We define
Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating Note that Barnes et al. (2013) identified 100Myr as the time required for a runaway greenhouse to sterilize a planet. L th , p = 2 ⇥R ⇤ SB ⇣ ( T be , p ) + ( T de , p ) ⌘ = ⇥R (1 p ) 4 ⇥R ⇤ ⇤ SB T , ⇤ ⇥ r ⇤ + W p , (B.1) p ( T be , p ) ⌘ ( T be , p ) + ( T be , p d T ) T , ⇤ (1 p ) R ⇤ r ⇤ p = 0 (B.2) nd, once d T is given, search for the first zero point of p ( T be , p ) above T eqe , p = T e , ⇤ ✓ (1 p ) R ⇤ r ⇤ p ◆ / , (B.3) which yields T be , p and T de , p . In our prototype exomoon system,we assume d T = 100K, which is equivalent to fixing the efficien-cy of heat redistribution from the bright to the dark hemisphere on the planet (Burrows et al. 2006b; Budaj et al. 2012). ! From Eq. (B.1), we can deduce the thermal flux on the subplanetary point on the moon. With increasing angular distance from the subplanetary point, f t will decrease. We can parametrize this distance on the moon’s surface by longitude φ and latitude θ , which givesThe time dependence of the irradiation is now packed into ξ ( t ), which serves as a weighting function for the two contributions from the bright and the dark side. It is given by ( t ) = 12 ⇣ l ( t ) ⌘! , (B.5) where l ( t ) = arccos ( cos ⇣ ⇥ ( t ) ⌘ cos ⇣ ( t ) ⇤ p ( t ) ⌘) (B.6) is the angular distance between the moon's projection on the planetary surface and the substellar point on the planet. In other words, l ( t ) is an orthodrome on the planet’s surface, determined by Φ ( t ) and ϑ ( t ) (see Fig. B.1). This yields ⇥ ( t ) = 12 ( ⇣ ⇤ ( t ) ⌘ cos ⇣ ( t ) ⇤ p ( t ) ⌘) . (B . Since the subplanetary point lies in the orbital plane of the planet, it will be at a position ( ν ∗ p ( t ),0) on the planetary surface, where ⇤ p ( t ) = arccos ✓ cos( E ⇤ p ( t )) e ⇤ p e ⇤ p cos( E ⇤ p ( t )) ◆ (B . is the true anomaly. Moreover, with s x , s y , and s z as the components of s = ( s x , s y , s z ) we have Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating planet illuminatedhemisphere darkhemisphere orbital plane substellarpoint( ! p, l (0,0) respect to the projection of r ! p ! on the moon’s orbital planeat t =
0. At summer solstice on the moon’s northern hemi-sphere, the true anomaly m ) p equals g (see Eq. B8). This makes g a critical parameter for the seasonal variation of stellarirradiation on the moon because it determines how seasons,induced by orbital inclination i, relate to the changing dis-tance to the star, induced by star-planet eccentricity e ) p . Inparticular, if g =
0, then northern summer coincides with theperiastron passage about the star and northern winter occursat apastron, inducing distinctly hot summers and cold win-ters. It relates to the conventional orientation of the ascend-ing node U as g = U + ! and can be considered as theclimate-precession parameter.For g s
0, we have to apply a rotation M ( g ): R / R ofaround the z axis (0,0,1), which is performed by the rotationmatrix M ( g ) ¼ cos ( g ) sin ( g ) 0sin ( g ) cos ( g ) 00 0 1 (A3)With the abbreviations introduced in Eq. 6 (with / = s ! $ n !
0, 0 ( t ) ¼ a ps ~ CcC ~ Ss ~ ScC ~ CscS (A4)This allows us to parametrize the surface normal s ! ( t ) = s ( t ) ofthe subplanetary point on the moon for arbitrary i and g .Finally, we want to find the surface normal n ! / , h ( t ) = n / , h ( t )for any location ( / , h ) on the moon’s surface. Therefore, wehave to go two steps on the moon’s surface, each one pa-rametrized by one angle: one in longitudinal direction / andone in latitudinal direction h . The former one can be walkedeasily, just by adding 2 p/ /360 ! in the sine and cosine ar-guments in Eq. A2 (see first line in Eq. 6), thus n ! / , 0 ¼ s ! ( u þ p/ = & ). This is equivalent to a shift alongthe moon’s equator. For the second step, we know that if h = ! , then the surface normal will be along the rotation axis N ! ¼ a ps S ~ C S ~ SC (A5)of the satellite; that is, we are standing on the northpole. The vector n ! / , h can then be obtained by tilting s ! ( u þ p/ = & ) by an angle h toward N ! . With N = a ps = s, we then derive n ! / , h ( t ) ¼ a ps sin ( h ) N ! N þ a ps cos ( h ) s ! ( u þ p/ = & ) s ( u þ p/ = & ) ¼ sin ( h ) N ! þ cos ( h ) s ! ( u þ p/ = & ) ¼ a ps ! sS ~ C þ ! c ( ~ CcC ~ Ss ) ! sS ~ S þ ! c ( ~ ScC ~ Cs ) ! sC þ ! ccS (A6)which solves Eq. A1. Appendix B: Planetary Irradiation
In addition to stellar light, the moon will receive thermaland stellar-reflected irradiation from the planet, f t ( t ) and f r ( t ),respectively. One hemisphere on the planet will be illumi-nated by the star, and the other one will be dark. On thebright side of the planet, there will be both thermal emissionfrom the planet as well as starlight reflection, while on thedark side the planet will only emit thermal radiation, though FIG. B1.
Geometry of the planetary illumination. Thesubsatellite point on the planetary surface is at ( F , w ). Theangular distance l of the moon from the substellar point(0, m *p ) determines the amount of light received by the moonfrom the two different hemispheres. FIG. A1.
Geometry of the triple system of a star, a planet, and a moon. In the left panel the planet-moon duet has advanced byan angle m *p around the star, and the moon has progressed by an angle 2 pu ps . The right panel shows a zoom-in to the planet-moon binary. As in the left panel, time has proceeded, and a projection of r ! ps ( t ) at time t = i and rotated against the star-moon periapses by an angle g . EXOMOON HABITABILITY 23
AST-2012-0859-ver9-Heller_1P.3d 01/02/13 6:25pm Page 23 ! "
FIG. B.1.
Geometry of the planetary illumination. The sub-satellite point on the planetary surface is at ( Φ , ϑ ). The angular distance l of the moon from the substellar point ( ν ∗ p ,0) determines the amount of light received by the moon from the two different hemispheres. f t ( t ) = R ⌅ SB a cos ✓ ⇧⇤ ◆ cos ✓ ⇤ ◆ ⇥ " ( T be , p ) ⇥ ( t ) + ( T de , p ) ⇣ ⇥ ( t ) ⌘ . (B . ( t ) = 2 arctan s y ( t ) q s ( t ) + s ( t ) + s x ( t ) ⇥ ( t ) = arccos s y ( t ) q s ( t ) + s ( t ) + s ( t ) (B . nd thus determined f t ( t ). ! We now consider the reflected stellar light from the planet. For the derivation of f t ( t ), we assumed that the planet is divided in two hemispheres, one of which is the bright side and one the dark side. Now the bright part coincides with the hemisphere from which the planet receives stellar reflected light, while there is no contribution to the reflectance from the dark side. Thus, the deduction of the geometrical part of f r ( t ), represented by ξ ( t ), goes analogously to f t ( t ). We only have to multiply the stellar flux received by the planet R ⇤ BS T , ⇤ /r ⇤ with the amount ⇥R p that isreflected from the planet and weigh it with thesquared distance decrease a between the pla-net and its satellite. Consideration of projec-tional effects of longitude and latitude yields ! In Section 3.1.2, we have compared thermal radiation and stellar reflection from the planet as a function of the planet-moon distance a ps and the planet’s albedo α p (see Fig. 2). To compute the amplitudes of f t and f r over the moon’s orbit around the planet, we assume that the moon is over the substellar point on the planet where the satellite receives both maximum reflected and thermal radiation from the planet. Then ξ ( t ) = 1 and equating Eq. (B.4) with (B.10) allows us to compute that value of α p for which the two contributions will be similar. We neglect projectional effects of longitude and latitude and derive For our prototype system this yields α p = 0.093. For higher planetary albedo, stellar reflected light will dominate irradiation on the moon. Appendix C: Computer code of our model: exomoon.py
Finally, we make the computer code exomoon.py python and optimized for use with ipython (Pérez & Granger 2007). Care has been taken to make it easily human-readable, and a downloadable manual is available, so it can be modified by non-expert users. The output format are ascii tables, which can be accessed with gnuplot and other plotting software. ! In brief, the program has four operation modes, which allow the user to compute ( i .) phase curves of f ∗ ( φ ps ), f r ( φ ps ), and f t ( φ ps ), ( ii .) orbit-averaged flux maps of exomoon surfaces, (iii .) the orbit of the planet-moon duet around their common host star, and (iv .) the runaway greenhouse flux F RG . In Fig. C.1, we show an example for such an orbit calculation. The moon’s orbit is inclined by 45° against the plane spanned by the planet and the star, and the stellar orbit has an eccentricity e ∗ p = 0.3. Near periastron (at the right of the plot), where M ⇤ p = 0 , the orbital velocity of the planet-moon duet is greater than at apastron.This is why the moon’s path is more curly at the left, where M ⇤ p = ⇡ . Note that the starting point and the final point of themoon’s orbit at the very right of the figure do not coincide! This effect induces TTVs of the planet. Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating star s ! r p ! ! r s ! ! planetsatellite startend FIG. C.1.
Orbits of an exomoon and an exoplanet around their commonhost star as computed with our exomoon.py software. f r ( t ) = R ⇤ ⇧ SB T , ⇤ r ⇤ ⌅R p a (B . ⇥ cos ✓ ⌃⌅ ◆ cos ✓ ⇥⌅ ◆ ⇤ ( t ) . p = 1 ⇥ ✓ r p R ◆ T be , p T e , ! | f t = f r . (B.11) cknowledgements ipython 0.13 on python 2.7.2 and figures have been prepared with gnuplot 4.4 gimp 2.6 References
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Heller & Barnes (2013) – Exomoon habitability constrained by illumination and tidal heating .2. EXOMOON HABITABILITY CONSTRAINED BY ENERGY FLUX AND ORBITALSTABILITY (Heller 2012) 210 r X i v : . [ a s t r o - ph . E P ] S e p Astronomy&Astrophysicsmanuscript no. 2012-4˙Exomoons˙LMS c ! ESO 2012September 4, 2012
Exomoon habitability constrained byenergy flux and orbital stability
R. Heller Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany, e-mail: [email protected]
Received 13 July 2012 / Accepted 30 August 2012
ABSTRACT
Context.
Detecting massive satellites that orbit extrasolar planets has now become feasible, which led naturally to questions aboutthe habitability of exomoons. In a previous study we presented constraints on the habitability of moons from stellar and planetaryillumination as well as from tidal heating.
Aims.
Here I refine our model by including the e ff ect of eclipses on the orbit-averaged illumination. I then apply an analytic approxi-mation for the Hill stability of a satellite to identify the range of stellar and planetary masses in which moons can be habitable. Moonsin low-mass stellar systems must orbit their planet very closely to remain bounded, which puts them at risk of strong tidal heating. Methods.
I first describe the e ff ect of eclipses on the stellar illumination of satellites. Then I calculate the orbit-averaged energyflux, which includes illumination from the planet and tidal heating to parametrize exomoon habitability as a function of stellar mass,planetary mass, and planet-moon orbital eccentricity. The habitability limit is defined by a scaling relation at which a moon loses itswater by the runaway greenhouse process. As a working hypothesis, orbital stability is assumed if the moon’s orbital period is lessthan 1 / Results.
Due to eclipses, a satellite in a close orbit can experience a reduction in orbit-averaged stellar flux by up to about 6 %. Thesmaller the semi-major axis and the lower the inclination of the moon’s orbit, the stronger the reduction. I find a lower mass limit of ≈ . M for exomoon host stars that allows a moon to receive an orbit-averaged stellar flux comparable to the Earth’s, with whichit can also avoid the runaway greenhouse e ff ect. Precise estimates depend on the satellite’s orbital eccentricity. Deleterious e ff ects onexomoon habitability may occur up to ≈ . M if the satellite’s eccentricity is ! . Conclusions.
Although the traditional habitable zone lies close to low-mass stars, which allows for many transits of planet-moonbinaries within a given observation cycle, resources should not be spent to trace habitable satellites around them. Gravitational pertur-bations by the close star, another planet, or another satellite induce eccentricities that likely make any moon uninhabitable. Estimatesfor individual systems require dynamical simulations that include perturbations among all bodies and tidal heating in the satellite.
Key words.
Astrobiology – Planets and satellites: general – Eclipses – Celestial mechanics – Stars: low-mass
1. Introduction
The detection of dozens of Super-Earths and Jupiter-mass plan-ets in the stellar habitable zone naturally makes us wonderabout the habitability of their moons (Williams et al. 1997;Kaltenegger 2010). So far, no extrasolar moon has been con-firmed, but dedicated surveys are underway (Kipping et al.2012). Several studies have addressed orbital stability of extra-solar satellite systems (Donnison 2010; Weidner & Horne 2010)and tidal heating has been shown to be an important energysource in satellites (Reynolds et al. 1987; Scharf 2006; Cassidyet al. 2009). In a recent study (Heller & Barnes 2012, HB12in the following), we have extended these concepts to the illu-mination from the planet, i.e. stellar reflected light and thermalemission, and presented a model that invokes the runaway green-house e ff ect to constrain exomoon habitability.Earth-mass moons about Jupiter-mass planets have beenshown to be dynamically stable for the lifetime of the solar sys-tem in systems where the stellar mass is greater than > . M (Barnes & O’Brien 2002). I consider here whether such moonscould actually be habitable. Therefore, I provide the first studyof exomoon habitability that combines e ff ects of stellar illumi-nation, reflected stellar light from the planet, the planet’s thermalemission, eclipses, and tidal heating with constraints by orbitalstability and from the runaway greenhouse e ff ect.
2. Methods
To assess the habitability of a satellite, I estimated the globalaverage ¯ F globs = L ∗ (1 − α s )16 π a ∗ p ! − e ∗ p " x s + π R α p a + R σ SB ( T eqp ) a (1 − α s )4 + h s (1)of its energy flux over one stellar orbit, where L ∗ is stellar lumi-nosity, a ∗ p the semi-major axis of the planet’s orbit about the star, a ps the semi-major axis of the satellite’s orbit about the planet, e ∗ p the circumstellar orbital eccentricity, R p the planetary radius, α p and α s are the albedos of the planet and the satellite, respec-tively, T eqp is the planet’s thermal equilibrium temperature, h s the satellite’s surface-averaged tidal heating flux, σ SB the Stefan-Boltzmann constant, and x s is the fraction of the satellite’s orbitthat is not spent in the shadow of the planet. Tidal heating h s depends on the satellite’s eccentricity e ps , on its radius R s , andstrongly on a ps . I avoid repeating the various approaches avail-able for a parametrization of h s and refer the reader to HB12 and Heller et al. (2011), where we discussed the constant-time-lag model by Leconte et al. (2010) and the constant-phase-lagmodel from Ferraz-Mello et al. (2008). For this study, I arbitrar-ily chose the constant-time-lag model with a tidal time lag ofthe model satellites similar to that of the Earth, i.e. τ s =
638 s(Neron de Surgy & Laskar 1997).In HB12 we derived Eq. (1) for x s =
1, thereby neglect-ing the e ff ect of eclipses on the average flux. Here, I explorethe decrease of the average stellar flux on the satellite due toeclipses. Eclipses occur most frequently – and will thus havethe strongest e ff ect on the moon’s climate – when the satellite’scircum-planetary orbit is coplanar with the circumstellar orbit. Ifthe two orbits are also circular and eclipses are total x s = − R p / ( π a ps ) . (2)Applying the Roche criterion for a fluid-like body (Weidner &Horne 2010), I derive1 − . π R p R s " M s M p / < x s ≤ , (3)where x s = x s >
79 %.Due to tidal e ff ects, the moon’s semi-major axis will typicallybe > R p , then x s > − / (5 π ) ≈ . ≈ . ff ects of the same order of magnitude.Eclipses will be most relevant for exomoons in low-massstellar systems for two reasons. Firstly, tides raised by the staron the planet will cause the planet’s obliquity to be eroded in ’ ff ect of an eclipse-induced decrease of stellar irra-diation for some examples. The irradiation habitable zone (IHZ) is defined as the circum-stellar distance range in which liquid surface water can persiston a terrestrial planet (Dole 1964). On the one hand, if a planetis too close to the star, its atmosphere will become saturated withH O. Photodissociation then drives an escape of hydrogen intospace, which desiccates the planet by turning it into a runawaygreenhouse (Kasting et al. 1993). On the other hand, if the planetis too far away from the star, condensation of the greenhouse gasCO will let any liquid surface water freeze (Kasting et al. 1993).This picture can be applied to exomoons as well. However, in-stead of direct stellar illumination only, the star’s reflected lightfrom the planet, the planet’s thermal emission, and tidal heatingin the moon need to be taken into account for the global energyflux (HB12).Analytic expressions exist that parametrize the width of theIHZ as a function of stellar luminosity. I used the set of equa-tions from Selsis et al. (2007) to compute the Sun-Earth equiva-lent distance l between the host star and the planet – or in thiscase: the moon. To a certain extent, the constraints on exomoonhabitability will therefore still allow for moons with massive car-bon dioxide atmospheres and strong greenhouse e ff ect to exist in the outermost regions of the IHZ but I will not consider thesespecial cases.Exomoon habitability can be constrained by an upper limitfor the orbit-averaged global flux. Above a certain value, typ-ically around 300 W / m for Earth-like bodies, a satellite willbe subject to a runaway greenhouse e ff ect. This value does notdepend on the atmospheric composition, except that it containswater (Kasting 1988). As in HB12, I used the semi-analytic ex-pression from Pierrehumbert (2010) to compute this limit, F RG .For computations of the satellite’s radius, I used the model ofFortney et al. (2007) with a 68 % rock-to-mass fraction, similarto the Earth. A body’s Hill radius R H is the distance range out to which thebody’s gravity dominates the e ff ect on a test particle. In realis-tic scenarios the critical semi-major axis for a satellite to remainbound to its host planet is merely a fraction of the Hill radius,i.e. a ps < f R H (Holman & Wiegert 1999), with a conservativechoice for prograde satellites being f = / P ps / P ∗ p " /
9, where P ∗ p is the circumstellar period of the planet-moon pair and P ps is the satellite’s orbital period about the planet (Kipping 2009).With M ∗ ) M p ) M s and the planet-moon barycenter orbit-ing at an orbital distance l to the star, Kepler’s third law gives a ps < l $
181 ( M p + M s )( M ∗ + M p ) % / . (4)With this dynamical restriction in mind, we can think of sce-narios in which the mass of the star is very small, so that l liesclose to it, and a moon about a planet in the IHZ must be veryclose to the planet to remain gravitationally bound. Then therewill exist a limit (in terms of minimum stellar mass) at whichthe satellite needs to be so close to the planet that its tidal heat-ing becomes so strong that it will initiate a greenhouse e ff ect andwill not be habitable. Below, I determine this minimum mass forstars to host potentially habitable exomoons.The P ps / P ∗ p " /
3. Results
In Fig. 1 I show two examples for x s < The ordinate givesthe perpendicular displacement r ⊥ of the satellite from the planetin units of planetary radii and as a function of circumstellar or-bital phase ϕ ∗ p . Thus, if r ⊥ < Computations were performed with my
Moon’s perpendicular displacement [ R p ] Circumstellar orbital phase ϕ *p Fig. 1.
Perpendicular displacement of a moon as seen from the star for two scenarios. The thin and highly oscillating curve corre-sponds to the orbit of a satellite in a Europa-wide orbit (in units of R p ) about a Jupiter-sized planet at 1 AU from a Sun-like star. Thesatellite’s orbit about the Jovian planet is inclined by i = ◦ . The thicker gray line shows the Miranda-wide orbit of a satelliteorbiting a Neptune-mass object at ≈ .
16 AU from a 0 . M star, i.e. in the center of the IHZ. Here, the moon orbits in the sameplane as the planet orbits the low-mass host star, i.e. i = ◦ . Time spent in eclipse is emphasized with a thick, black line. Bothorbits are normalized to the circumstellar orbital phase ϕ ∗ p .planet (and not in front of it), an eclipse occurs. E ff ects of theplanet’s penumbra are neglected.In one scenario (thin, highly oscillating line) I placed a satel-lite in a Europa-wide orbit, i.e. at 9 . R p , about a Jupiter-sizedplanet in 1 AU distance from a Sun-like star. I applied a stel-lar orbital eccentricity of 0 . ◦ against the circumstellar orbit. In this specific scenario,the eccentricity causes the summer (0 ≤ ϕ ∗ p " . . " ϕ ∗ p <
1) to be shorter than the winter(0 . " ϕ ∗ p " . ϕ ∗ p ≈ .
18 and ϕ ∗ p ≈ .
82. Owingto the relatively wide satellite orbit, occultations are short com-pared to the moon’s orbital period. Thus, e ff ects of eclipses onthe satellite’s climate are small, in this case I find x s ≈ . . R p , comparable to Miranda’s orbit about Uranus,about a Neptune-mass planet, which orbits a 0 . M star in thecenter of the IHZ at ≈ .
16 AU. The planet’s circumstellar orbitwith a period of roughly 36 days is circular and the moon’s three-day orbit is coplanar with the stellar orbit. In this case, transits(thick black line) occur periodically once per satellite orbit ande ff ects on its climate will be significant with x s ≈ . In Fig. 2 contours present limiting stellar masses (abscissae)and host planetary masses (ordinate) for satellites to be habit-able. I considered two moons: one with a mass ten times thatof Ganymede (10 M Ga , left panel) and one with the mass of theEarth ( M ⊕ , right panel), respectively. Variation of the satellite’smass changes the critical orbit-averaged flux F RG (see title of each panel). Contours of ¯ F globs = F RG are drawn for five dif-ferent eccentricities of the satellite, four of which are taken fromthe solar system: e Io = . e Ca = . e Eu = . e Ti = . .
05. Moons in star-planet systems with masses M ∗ and M p leftto a contour for a given satellite eccentricity are uninhabitable.As a confidence estimate of the tidal model, dashed contoursprovide the limiting mass combinations for strongly reduced( τ s = . τ s = e ps = .
05 example.Satellites with the lowest eccentricities can be habitable evenin the lowest-mass stellar systems, i.e. down to 0 . M and be-low for extremely circular planet-moon orbits. With increasingeccentricity, however, the satellite needs to be farther away fromthe planet to avoid a greenhouse e ff ect, consequently it needs tobe in a wider orbit about its planet, which in turn means that theplanet needs to be farther away from the star for the moon tosatisfy Eq. (4). Thus, the star needs to be more massive to have ahigher luminosity and to reassure that the planet-moon binary iswithin the IHZ. This trend explains the increasing minimum stel-lar mass for increasing eccentricities in Fig. 2. For e ps = . . M for a 10 M Gan -massmoon (left panel) and 0 . M for an Earth-mass satellite (rightpanel). Uncertainties in the parametrization of tidal dissipationincrease this limit to almost 0 . M .
4. Conclusions
Eclipses of moons that are in tight orbits about their planet cansubstantially decrease the satellite’s orbit-averaged stellar illu-mination. Equation (2) can be used to compute the reduction by
Fig. 2.
Contours of maximum stellar masses (abscissa) for two possibly habitable moons with given planetary host masses (ordinate).Solid lines correspond to the satellite’s time lag τ s of 638 s and one out of five eccentricities, namely those of Io, Callisto, Europa,Titan, and 0.05, from left to right. Dashed lines refer to e ps = .
05 and variation of τ s by a factor of ten. A satellite with host starand host planet masses located left to its respective eccentricity contour is uninhabitable.total eclipses for circular and coplanar orbits. In orbits similar tothe closest found in the solar system, this formula yields a reduc-tion of about 6 . < " . M can hardly host habit-able exomoons. These moons’ orbits about their planet wouldneed to be almost perfectly circularized, which means that theywould need to be the only massive moon in that satellite system.Moreover, the nearby star will excite substantial eccentricitiesin the moon’s orbit. This increases the minimum mass of hoststars for habitable moons. To further constrain exomoon habit-ability, it will be necessary to simulate the eccentricity evolutionof satellites with a model that considers both N -body interactionand tidal evolution. Such simulations are beyond the scope ofthis communication and will be conducted in a later study.The full Kozai cycle and tidal friction model of Porter &Grundy (2011) demonstrates very fast orbital evolution of satel-lites that orbit M stars in their IHZ. The star forces the satel-lite orbit to be eccentric, thus the moon’s semi-major axis willshrink due to the tides on timescales much shorter than one Myr.Eventually, its orbit will be circular and very tight. Indeed, amoon’s circum-planetary period in the IHZ about an M0 starwill be even shorter than the P ∗ p / ’ ff ect. This con-tradiction makes my estimates for the minimum stellar mass fora habitable moon star more robust.Additionally, moons about low-mass stars will most likelynot experience seasons because they would orbit their planetin its equatorial plane (Porter & Grundy 2011). The paucityof Jovian planets in the IHZ of low-mass stars (Borucki et al.2011) further decreases the chances for habitable Earth-sized ex-omoons in low-mass stellar systems.I conclude that stars with masses " . M cannot host hab-itable moons, and stars with masses up to 0 . M can be a ff ected by the energy flux and orbital stability criteria combined in thispaper. A model that couples gravitational scattering with tidalevolution is required to further constrain exomoon habitabilityabout low-mass stars. Acknowledgements.
I thank the anonymous referee and the editor TristanGuillot for their valuable comments. I am deeply grateful to Rory Barnes forour various discussions on the subject of this paper. This work has made use ofNASA’s Astrophysics Data System Bibliographic Services. Computations wereperformed with ipython 0.13 (P´erez & Granger 2007) on python 2.7.2 andfigures were prepared with gnuplot 4.4 ( ) as well as with gimp 2.6 ( ). References
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A dynamically-packed planetary system around GJ 667C with threesuper-Earths in its habitable zone ⋆ ⋆⋆
Guillem Anglada-Escud´e , Mikko Tuomi , , Enrico Gerlach , Rory Barnes , Ren´e Heller , James S. Jenkins ,Sebastian Wende , Steven S. Vogt , R. Paul Butler , Ansgar Reiners , and Hugh R. A. Jones Universit¨at G¨ottingen, Institut f¨ur Astrophysik, Friedrich-Hund-Platz 1, 37077 G¨ottingen, Germany Centre for Astrophysics, University of Hertfordshire, College Lane, Hatfield, Hertfordshire AL10 9AB, UK University of Turku, Tuorla Observatory, Department of Physics and Astronomy, V¨ais¨al¨antie 20, FI-21500, Piikki¨o, Finland Technical University of Dresden, Institute for Planetary Geodesy, Lohrmann-Observatory, 01062 Dresden, Germany Astronomy Department, University of Washington, Box 951580, WA 98195, Seattle, USA Leibniz Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany Departamento de Astronom´ıa, Universidad de Chile, Camino El Observatorio 1515, Las Condes, Casilla 36-D Santiago, Chile UCO / Lick Observatory, University of California, Santa Cruz, CA 95064, USA Carnegie Institution of Washington, Department of Terrestrial Magnetism, 5241 Broad Branch Rd. NW, 20015 Washington D.C.,USAsubmitted Feb 2013
ABSTRACT
Context.
Since low-mass stars have low luminosities, orbits at which liquid water can exist on Earth-sized planets are relativelyclose-in, which produces Doppler signals that are detectable using state-of-the-art Doppler spectroscopy.
Aims.
GJ 667C is already known to be orbited by two super-Earth candidates. We have applied recently developed data analysismethods to investigate whether the data supports the presence of additional companions.
Methods.
We obtain new Doppler measurements from HARPS extracted spectra and combined them with those obtained fromthe PFS and HIRES spectrographs. We used Bayesian and periodogram-based methods to re-assess the number of candidates andevaluated the confidence of each detection. Among other tests, we validated the planet candidates by analyzing correlations of eachDoppler signal with measurements of several activity indices and investigated the possible quasi-periodic nature of signals.
Results.
Doppler measurements of GJ 667C are described better by six (even seven) Keplerian-like signals: the two known candidates(b and c); three additional few-Earth mass candidates with periods of 92, 62 and 39 days (d, e and f); a cold super-Earth in a 260-dayorbit (g) and tantalizing evidence of a ∼ ⊕ object in a close-in orbit of 17 days (h). We explore whether long-term stable orbits arecompatible with the data by integrating 8 × solutions derived from the Bayesian samplings. We assess their stability using secularfrequency analysis. Conclusions.
The system consisting of six planets is compatible with dynamically stable configurations. As for the solar system,the most stable solutions do not contain mean-motion resonances and are described well by analytic Laplace-Lagrange solutions.Preliminary analysis also indicates that masses of the planets cannot be higher than twice the minimum masses obtained from Dopplermeasurements. The presence of a seventh planet (h) is supported by the fact that it appears squarely centered on the only island ofstability left in the six-planet solution. Habitability assessments accounting for the stellar flux, as well as tidal dissipation e ff ects,indicate that three (maybe four) planets are potentially habitable. Doppler and space-based transit surveys indicate that 1) dynamicallypacked systems of super-Earths are relatively abundant and 2) M-dwarfs have more small planets than earlier-type stars. These twotrends together suggest that GJ 667C is one of the first members of an emerging population of M-stars with multiple low-mass planetsin their habitable zones. Key words.
Techniques : radial velocities – Methods : data analysis – Planets and satellites : dynamical evolution and stability –Astrobiology – Stars: individual : GJ 667C
Send o ff print requests to : G. Anglada-Escud´e e-mail: [email protected] - Mikko Tuomi, e-mail: [email protected] ⋆ Based on data obtained from the ESO Science Archive Facility un-der request number ANGLADA36104. Such data had been previouslyobtained with the HARPS instrument on the ESO 3.6 m telescope un-der the programs 183.C-0437, 072.C-0488 and 088.C-0662, and withthe UVES spectrograph at the Very Large Telescopes under the pro-gram 087.D-0069. This study also contains observations obtained at theW.M. Keck Observatory- which is operated jointly by the Universityof California and the California Institute of Technology- and observa-tions obtained with the Magellan Telescopes, operated by the Carnegie
1. Introduction
Since the discovery of the first planets around other stars,Doppler precision has been steadily increasing to the pointwhere objects as small as a few Earth masses can currently bedetected around nearby stars. Of special importance to the ex-oplanet searches are low-mass stars (or M-dwarfs) nearest tothe Sun. Since low-mass stars are intrinsically faint, the orbits
Institution, Harvard University, University of Michigan, University ofArizona, and the Massachusetts Institute of Technology. ⋆⋆ Time-series are available in electronic format at CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strabg.fr/cgi-bin/qcat?J/A+A/ at which a rocky planet could sustain liquid water on its surface(the so-called habitable zone, Kasting et al. 1993) are typicallycloser to the star, increasing their Doppler signatures even more.For this reason, the first super-Earth mass candidates in the hab-itable zones of nearby stars have been detected around M-dwarfs(e.g. GJ 581, Mayor et al. 2009; Vogt et al. 2010)).Concerning the exoplanet population detected to date, it isbecoming clear that objects between 2 M ⊕ and the mass ofNeptune (also called super-Earths) are very common around allG, K, and M dwarfs. Moreover, such planets tend to appear inclose in / packed systems around G and K dwarfs (historicallypreferred targets for Doppler and transit surveys) with orbitscloser in than the orbit of Mercury around our Sun. These fea-tures combined with a habitable zone closer to the star, point tothe existence of a vast population of habitable worlds in multi-planet systems around M-dwarfs, especially around old / metal-depleted stars (Jenkins et al. 2013).GJ 667C has been reported to host two (possibly three)super-Earths. GJ 667Cb is a hot super-Earth mass object in anorbit of 7.2 days and was first announced by Bonfils (2009). Thesecond companion has an orbital period of 28 days, a minimummass of about 4.5 M ⊕ and, in principle, orbits well within the liq-uid water habitable zone of the star (Anglada-Escud´e et al. 2012;Delfosse et al. 2012). The third candidate was considered tenta-tive at the time owing to a strong periodic signal identified in twoactivity indices. This third candidate (GJ 667Cd) would have anorbital period between 74 and 105 days and a minimum mass ofabout 7 M ⊕ . Although there was tentative evidence for more pe-riodic signals in the data, the data analysis methods used by bothAnglada-Escud´e et al. (2012) and Delfosse et al. (2012) studieswere not adequate to properly deal with such high multiplicityplanet detection. Recently, Gregory (2012) presented a Bayesiananalysis of the data in Delfosse et al. (2012) and concluded thatseveral additional periodic signals were likely present. The pro-posed solution, however, contained candidates with overlappingorbits and no check against activity or dynamics was done, cast-ing serious doubts on the interpretation of the signals as planetcandidates.E ffi cient / confident detection of small amplitude signals re-quires more specialized techniques than those necessary to de-tect single ones. This was made explicitly obvious in, for ex-ample, the re-analysis of public HARPS data on the M0V starGJ 676A. In addition to the two signals of gas giant planetsreported by Forveille et al. (2011), Anglada-Escud´e & Tuomi(2012) (AT12 hereafter) identified the presence of two very sig-nificant low-amplitude signals in closer-in orbits. One of themain conclusions of AT12 was that correlations of signals al-ready included in the model prevent detection of additional low-amplitude using techniques based on the analysis of the residu-als only. In AT12, it was also pointed out that the two additionalcandidates (GJ 676A d and e) could be confidently detected with30% less measurements using Bayesian based methods.In this work, we assess the number of Keplerian-like sig-nals around GJ 667C using the same analysis methods asin Anglada-Escud´e & Tuomi (2012). The basic data consistsof 1) new Doppler measurements obtained with the HARPS-TERRA software on public HARPS spectra of GJ 667C(see Delfosse et al. 2012, for a more detailed descrip-tion of the dataset), and 2) Doppler measurements fromPFS / Magellan and HIRES / Keck spectrometers (available inAnglada-Escud´e & Butler 2012). We give an overview ofGJ 667C as a star and provide updated parameters in Section2. The observations and data-products used in later analysesare described in Section 3. Section 4 describes our statistical tools, models and the criteria used to quantify the significanceof each detection (Bayesian evidence ratios and log–likelihoodperiodograms). The sequence and confidences of the signals inthe Doppler data are given in section 5 where up to seven planet-like signals are spotted in the data. To promote Doppler signalsto planets, such signals must be validated against possible corre-lations with stellar activity. In section 6, we discuss the impactof stellar activity on the significance of the signals (especiallyon the GJ 667Cd candidate) and we conclude that none of theseven candidates is likely to be spurious. In section 7, we in-vestigate if all signals were detectable in subsets of the HARPSdataset to rule out spurious detections from quasi-periodic vari-ability caused by stellar activity cycles. We find that all signalsexcept the least significant one are robustly present in both thefirst and second-halves of the HARPS observing campaign inde-pendently. A dynamical analysis of the Bayesian posterior sam-ples finds that a subset of the allowed solutions leads to long-term stable orbits. This verification steps allows us promotingthe first six signals to the status of planet candidates. In Section8 we also investigate possible mean-motion resonances (MMR)and mechanisms that guarantee long-term stability of the system.Given that the proposed system seems physically viable, we dis-cusses potential habitability of each candidate in the context ofup-to-date climatic models, possible formation history, and thee ff ect of tides in Section 9. Concluding remarks and a summaryare given in Section 10. The appendices describe additional testsperformed on the data to double-check the significance of theplanet candidates.
2. Properties of GJ 667C
GJ 667C (HR 6426 C), has been classified as an M1.5V star(Geballe et al. 2002) and is a member of a triple system, sinceit is a common proper motion companion to the K3V + K5Vbinary pair, GJ 667AB. Assuming the HIPPARCOS distanceto the GJ 667AB binary ( ∼ ∼
230 AU. Spectroscopic measurements of the binary have re-vealed a metallicity significantly lower than the Sun (Fe / H = -0.59 ± >
100 days, see Section 6) also support an age of > / VLT spectro-graph (program 87.D-0069). Both the HARPS and the UVESspectra show no H α emission. The value of the mean S-indexmeasurement (based on the intensity of the CaII H + K emissionlines) is 0 . ± .
02, which puts the star among the most inac-tive objects in the HARPS M-dwarf sample (Bonfils et al. 2013).By comparison, GJ 581(S = .
45) and GJ 876 (S = = . Table 1.
Parameter space covered by the grid of synthetic mod-els.
Range Step size T e ff [K] 2,300 – 5,900 100log( g ) 0.0 – + Fe / H ] -4.0 – -2.0 1.0-2.0 – + and molecular lines for the spectral fitting without accountingfor magnetic and / or rotational broadening. UVES observationsof GJ 667C were taken in service mode in three exposures dur-ing the night on August 4th 2011. The high resolution mode witha slit width of 0 . ′′ was used to achieve a resolving power of R ∼
100 000. The observations cover a wavelength range from640 nm to 1020 nm on the two red CCDs of UVES.The spectral extraction and reduction were done using theESOREX pipeline for UVES. The wavelength solution is based,to first order, on the Th-Ar calibration provided by ESO. Allorders were corrected for the blaze function and also normalizedto unity continuum level. Afterwards, all orders were mergedtogether. For overlapping orders the redder ends were used dueto their better quality. In a last step, an interactive removal of badpixels and cosmic ray hits was performed.The adjustment consists of matching the observed spectrumto a grid of synthetic spectra from the newest PHOENIX / ACESgrid (see Husser et al. 2013)). The updated codes use a newequation of state that accounts for the molecular formation ratesat low temperatures. Hence, it is optimally suited for simula-tion of spectra of cool stars. The 1D models are computed inplane parallel geometry and consist of 64 layers. Convection istreated in mixing-length geometry and from the convective ve-locity a micro-turbulence velocity (Edmunds 1978) is deducedvia v mic = . · v conv . The latter is used in the generation of thesynthetic high resolution spectra. An overview of the model gridparameters is shown in Table 1. Local thermal equilibrium is as-sumed in all models.First comparisons of these models with observations showthat the quality of computed spectra is greatly improved in com-parison to older versions. The problem in previous versions ofthe PHOENIX models was that observed spectra in the ϵ - and γ -TiO bands could not be reproduced assuming the same e ff ectivetemperature parameter (Reiners 2005). The introduction of thenew equation of state apparently resolved this problem. The newmodels can consistently reproduce both TiO absorption bandstogether with large parts of the visual spectrum at very high fi-delity (see Fig. 1).As for the observed spectra, the models in our grid are alsonormalized to the local continuum. The regions selected for thefit were chosen as una ff ected by telluric contamination and aremarked in green in Fig. 1. The molecular TiO bands in the regionbetween 705 nm to 718 nm ( ϵ -TiO) and 840 nm to 848 nm ( γ -TiO) are very sensitive to T e ff but almost insensitive to log g . Thealkali lines in the regions between 764 nm to 772 nm and 816 nmto 822 nm (K- and Na-atomic lines, respectively) are sensitive tolog g and T e ff . All regions are sensitive to metallicity. The simul-taneous fit of all the regions to all three parameters breaks thedegeneracies, leading to a unique solution of e ff ective tempera-ture, surface gravity and metallicity.As the first step, a three dimensional χ -map is produced todetermine start values for the fitting algorithm. Since the modelgrid for the χ -map is discrete, the real global minimum is likelyto lie between grid points. We use the parameter sets of the three Table 2.
Stellar parameters of GJ 667C
Parameters Value Ref.R.A. 17 18 57.16 1Dec -34 59 23.14 1 µ R . A . [mas yr − ] 1129.7(9.7) 1 µ Dec . [mas yr − ] -77.0(4.6) 1Parallax [mas] 146.3(9.0) 1Hel. RV [km s − ] 6.5(1.0) 2V [mag] 10.22(10) 3J [mag] 6.848(21) 4H [mag] 6.322(44) 4K [mag] 6.036(20) 4 T ef f [K] 3350(50) 5[Fe / H] -0.55(10) 5log g [g in cm s − ] 4.69(10) 5Derived quantitiesUVW LSR [km s − ] (19.5, 29.4,-27.2) 2Age estimate > ⊙ ] 0.33(2) 5 L ∗ / L ⊙ References. (1) van Leeuwen (2007); (2)Anglada-Escud´e et al. (2012);(3) Mermilliod (1986); (4) Skrutskie et al. (2006); (5) This work (seetext) smallest χ -values as starting points for the adjustment proce-dure. We use the IDL curvefit -function as the fitting algorithm.Since this function requires continuous parameters, we use threedimensional interpolation in the model spectra. As a fourth freeparameter, we allow the resolution of the spectra to vary in or-der to account for possible additional broadening (astrophysicalor instrumental). For this star, the relative broadening is alwaysfound to be <
3% of the assumed resolution of UVES, and isstatistically indistinguishable from 0. More information on themethod and first results on a more representative sample of starswill be given in a forthcoming publication.As already mentioned, the distance to the GJ 667 sys-tem comes from the HIPPARCOS parallax of the GJ 667ABpair and is rather uncertain (see Table 2). This, combinedwith the luminosity-mass calibrations in Delfosse et al. (2000),propagates into a rather uncertain mass (0.33 ± M ⊙ )and luminosity estimates (0 . ± . L ⊙ ) for GJ 667C(Anglada-Escud´e et al. 2012). A good trigonometric parallaxmeasurement and the direct measurement of the size of GJ 667Cusing interferometry (e.g. von Braun et al. 2011) are mostlyneeded to refine its fundamental parameters. The updated valuesof the spectroscopic parameters are slightly changed from pre-vious estimates. For example, the e ff ective temperature used inAnglada-Escud´e et al. (2012) was based on evolutionary modelsusing the stellar mass as the input which, in turn, is derived fromthe rather uncertain parallax measurement of the GJ 667 sys-tem. If the spectral type were to be understood as a temperaturescale, the star should be classified as an M3V-M4V instead ofthe M1.5V type assigned in previous works (e.g. Geballe et al.2002). This mismatch is a well known e ff ect on low metallic-ity M dwarfs (less absorption in the optical makes them appearof earlier type than solar metallicity stars with the same e ff ec-tive temperature). The spectroscopically derived parameters andother basic properties collected from the literature are listed inTable 2. Fig. 1.
Snapshots of the wavelength regions used in the spectral fit to the UVES spectrum of GJ 667C. The observed spectrum isrepresented in black, the green curves are the parts of the synthetic spectrum used in the fit. The red lines are also from the syntheticspectrum that were not used to avoid contamination by telluric features or because they did not contain relevant spectroscopicinformation. Unfitted deep sharp lines- especially on panels four and five from the top of the page- are non-removed telluric features.
3. Observations and Doppler measurements
A total of 173 spectra obtained using the HARPS spectrograph(Pepe et al. 2002) have been re-analyzed using the HARPS-TERRA software (Anglada-Escud´e & Butler 2012). HARPS-TERRA implements a least-squares template matching algo- rithm to obtain the final Doppler measurement. This methodand is especially well suited to deal with the highly blendedspectra of low mass stars. It only replaces the last step of acomplex spectral reduction procedure as implemented by theHARPS Data Reduction Software (DRS). Such extraction is au-tomatically done by the HARPS-ESO services and includes all the necessary steps from 2D extraction of the echelle orders, flatand dark corrections, and accurate wavelength calibration usingseveral hundreds of Th Ar lines accross the HARPS wavelengthrange (Lovis & Pepe 2007). Most of the spectra (171) were ex-tracted from the ESO archives and have been obtained by othergroups over the years (e.g., Bonfils et al. 2013; Delfosse et al.2012) covering from June 2004 to June 2010. To increase thetime baseline and constrain long period trends, two additionalHARPS observations were obtained between March 5th and 8thof 2012. In addition to this, three activity indices were also ex-tracted from the HARPS spectra. These are: the S-index (pro-portional to the chromospheric emission of the star), the full-width-at-half-maximum of the mean line profile (or FWHM, ameasure of the width of the mean stellar line) and the line bi-sector (or BIS, a measure of asymmetry of the mean stellarline). Both the FWHM and BIS are measured by the HARPS-DRS and were taken from the headers of the corresponding files.All these quantities might correlate with spurious Doppler o ff -sets caused by stellar activity. In this sense, any Doppler signalwith a periodicity compatible with any of these signals will beconsidered suspicious and will require a more detailed analy-sis. The choice of these indices is not arbitrary. Each of themis thought to be related to an underlying physical process thatcan cause spurious Doppler o ff sets. For example, S-index vari-ability typically maps the presence of active regions on the stel-lar surface and variability of the stellar magnetic field (e.g.,solar-like cycles). The line bisector and FWHM should havethe same period as spurious Doppler signals induced by spotscorotating with the star (contrast e ff ects combined with stel-lar rotation, suppression of convection due to magnetic fieldsand / or Zeeman splitting in magnetic spots). Some physical pro-cesses induce spurious signals at some particular spectral re-gions (e.g., spots should cause stronger o ff sets at blue wave-lengths). The Doppler signature of a planet candidate is con-stant over all wavelengths and, therefore, a signal that is onlypresent at some wavelengths cannot be considered a crediblecandidate. This feature will be explored below to validate thereality of some of the proposed signals. A more comprehen-sive description of each index and their general behavior in re-sponse to stellar activity can be found elsewhere (Baliunas et al.1995; Lovis et al. 2011). In addition to the data products derivedfrom HARPS observations, we also include 23 Doppler mea-surements obtained using the PFS / Magellan spectrograph be-tween June 2011 and October 2011 using the Iodine cell tech-nique, and 22 HIRES / Keck Doppler measurements (both RVsets are provided in Anglada-Escud´e & Butler 2012) that havelower precision but allow extending the time baseline of the ob-servations. The HARPS-DRS also produces Doppler measure-ments using the so–called cross correlation method (or CCF). Inthe Appendices, we show that the CCF-Doppler measurementsactually contain the same seven signals providing indirect con-firmation and lending further confidence to the detections.
4. Statistical and physical models
The basic model of a radial velocity data set from a singletelescope-instrument combination is a sum of k Keplerian sig-nals, with k =
0, 1, ..., a random variable describing the instru-ment noise, and another describing all the excess noise in thedata. The latter noise term, sometimes referred to as stellar RVjitter (Ford 2005), includes the noise originating from the stel-lar surface due to inhomogeneities, activity-related phenomena,and can also include instrumental systematic e ff ects. FollowingTuomi (2011), we model these noise components as Gaussian random variables with zero mean and variances of σ i and σ l ,where the former is the formal uncertainty in each measurementand the latter is the jitter that is treated as a free parameter of themodel (one for each instrument l ).Since radial velocity variations have to be calculated withrespect to some reference velocity, we use a parameter γ l thatdescribes this reference velocity with respect to the data meanof a given instrument. For several telescope / instrument combi-nations, the Keplerian signals must necessarily be the same butthe parameters γ l (reference velocity) and σ l (jitter) cannot beexpected to have the same values. Finally, the model also in-cludes a linear trend ˙ γ to account for very long period compan-ions (e.g., the acceleration caused by the nearby GJ 667AB bi-nary). This model does not include mutual interactions betweenplanets, which are known to be significant in some cases (e.g.GJ 876, Laughlin & Chambers 2001). In this case, the relativelylow masses of the companions combined with the relativelyshort time-span of the observations makes these e ff ects too smallto have noticeable impact on the measured orbits. Long-term dy-namical stability information is incorporated and discussed later(see Section 8). Explicitly, the baseline model for the RV obser-vations is v l ( t i ) = γ l + ˙ γ ( t i − t ) + k ! j = f ( t i , β j ) + g l " ψ ; t i , z i , t i − , r i − , (1)where t is some reference epoch (which we arbitrarily chooseas t = g is a function describing the specific noiseproperties (instrumental and stellar) of the l -th instrument on topof the estimated Gaussian uncertainties. We model this functionusing first order moving average (MA) terms Tuomi et al. (2013,2012) that and on the residual r i − to the previous measurementat t i − , and using linear correlation terms with activity indices(denoted as z i ). This component of the model is typically pa-rameterized using one or more “nuisance parameters” ψ that arealso treated as free parameters of the model. Function f repre-sents the Doppler model of a planet candidate with parameters β j (Period P j , Doppler semi-amplitude K j , mean anomaly at refer-ence epoch M , j , eccentricity e j , and argument of the periastron ω j ).The Gaussian white noise component of each measurementand the Gaussian jitter component associated to each instrumententer the model in the definition of the likelihood function L as L ( m | θ ) = N $ i = % π ( σ i + σ l ) exp ⎧⎪⎪⎨⎪⎪⎩ − * m i − v l ( t i ) + σ i + σ l ) ⎫⎪⎪⎬⎪⎪⎭ , (2)where m stands for data and N is the number of individual mea-surements. With these definitions, the posterior probability den-sity π ( θ | m ) of parameters θ given the data m ( θ includes the or-bital elements β j , the slope term ˙ γ , the instrument dependentconstant o ff sets γ l , the instrument dependent jitter terms σ l , anda number of nuisance parameters ψ ), is derived from the Bayes’theorem as π ( θ | m ) = L ( m | θ ) π ( θ ) / L ( m | θ ) π ( θ ) d θ . (3)This equation is where the prior information enters the modelthrough the choice of the prior density functions π ( θ ). This way,the posterior density π ( θ | m ) combines the new information pro-vided by the new data m with our prior assumptions for the pa-rameters. In a Bayesian sense, finding the most favored model and allowed confidence intervals consists of the identificationand exploration of the higher probability regions of the poste-rior density. Unless the model of the observations is very sim-ple (e.g., linear models), a closed form of π ( θ | m ) cannot be de-rived analytically and numerical methods are needed to exploreits properties. The description of the adopted numerical methodsare the topic of the next subsection. Given a model with k Keplerian signals, we draw statisti-cally representative samples from the posterior density of themodel parameters (Eq. 3) using the adaptive Metropolis al-gorithm Haario et al. (2001). This algorithm has been usedsuccessfully in e.g. Tuomi (2011), Tuomi et al. (2011) andAnglada-Escud´e & Tuomi (2012). The algorithm appears to bea well suited to the fitting of Doppler data in terms of its rela-tively fast convergence – even when the posterior is not unimodal(Tuomi 2012) – and it provides samples that represent well theposterior densities. We use these samples to locate the regionsof maximum a posteriori probability in the parameter space andto estimate each parameter confidence interval allowed by thedata. We describe the parameter densities briefly by using themaximum a posteriori probability (MAP) estimates as the mostprobable values, i.e. our preferred solution, and by calculatingthe 99% Bayesian credibility sets (BCSs) surrounding these esti-mates. Because of the caveats of point estimates (e.g., inability todescribe the shapes of posterior densities in cases of multimodal-ity and / or non-negligible skewness), we also plot marginalizeddistributions of the parameters that are more important from adetection and characterization point of view, namely, velocitysemi-amplitudes K j , and eccentricities e j .The availability of samples from the posterior densities ofour statistical models also enables us to take advantage of thesignal detection criteria given in Tuomi (2012). To claim thatany signal is significant, we require that 1) its period is well-constrained from above and below, 2) its RV amplitude has adensity that di ff ers from zero significantly (excluded from the99% credibility intervals), and 3) the posterior probability of themodel containing k + k signals.The threshold of 150 on condition (3) might seem arbitrary,and although posterior probabilities also have associated un-certainties (Jenkins & Peacock 2011), we consider that such athreshold is a conservative one. As made explicit in the def-inition of the posterior density function π ( θ | m ), the likelihoodfunction is not the only source of information. We take into ac-count the fact that all parameter values are not equally probableprior to making the measurements via prior probability densities.Essentially, our priors are chosen as in Tuomi (2012). Of spe-cial relevance in the detection process is the prior choice for theeccentricities. Our functional choice for it (Gaussian with zeromean and σ e = ff erent prior choices, seethe dedicated discussion in Tuomi & Anglada-Escude (2013). Because the orbital period (or frequency) is an extremely non-linear parameter, the orbital solution of a model with k + Δ l og L Δ l og L Δ l og L Δ l og L Δ l og L Δ l og L
10 100 1000
Period [days] Δ l og L b c def g ? (h) (all circular) Fig. 2.
Log–likelihood periodograms for the seven candidate sig-nals sorted by significance. While the first six signals are easilyspotted, the seventh is only detected with log–L periodograms ifall orbits are assumed to be circular.method based on stochastic processes, there is always a chancethat the global maxima of the target function is missed. Ourlog–likelihood periodogram (or log–L periodogram) is a toolto systematically identify periods for new candidate planets ofhigher probability and ensure that these areas have been wellexplored by the Bayesian samplings (e.g., we always attempt tostart the chains close to the five most significant periodicities leftin the data). A log–L periodogram consists of computing the im-provement of the logarithm of the likelihood (new model with k + k planets) at each test period. Log–L peri-odograms are represented as period versus ∆ log L plots, wherelog is always the natural logarithm. The definition of the likeli-hood function we use is shown in Eq. 2 and typically assumesGaussian noise sources only (that is, di ff erent jitter parametersare included for each instrument and g = ∆ log L can also be used for estimating the frequentist falsealarm probability (FAP) of a solution using the likelihood-ratiotest for nested models. This FAP estimates what fraction oftimes one would recover such a significant solution by an un-fortunate arrangement of Gaussian noise. To compute this FAPfrom ∆ log L we used the up-to-date recipes provided by Baluev(2009). We note that that maximization of the likelihood involvessolving for many parameters simultaneously: orbital parame-ters of the new candidate, all orbital parameters of the alreadydetected signals, a secular acceleration term ˙ γ , a zero-point γ l for each instrument, and jitter terms σ l of each instrument (seeEq. 1). It is, therefore, a computationally intensive task, espe- cially when several planets are included and several thousand oftest periods for the new candidate must be explored.As discussed in the appendices (see Section A.1), allow-ing for full Keplerian solutions at the period search level makesthe method very prone to false positives. Therefore while a fullKeplerian solution is typically assumed for all the previously de-tected k -candidates, the orbital model for the k + ff erent from uniform. As discussed later,the sixth candidate is only confidently spotted using log–L peri-odograms (our detection criteria is FAP < red line beyond which our detection criteria becomes strongly dependenton our choice of prior on the eccentricity. The same applies tothe seventh tentative candidate signal.
5. Signal detection and confidences
As opposed to other systems analyzed with the same techniques(e.g. Tau Ceti or HD 40307, Tuomi et al. 2012, 2013), we foundthat for GJ 667C the simplest model ( g = ffi cient description of the data. For brevity,we omit here all the tests done with more sophisticated param-eterizations of the noise (see Appendix C) that essentially leadto unconstrained models for the correlated noise terms and thesame final results. In parallel with the Bayesian detection se-quence, we also illustrate the search using log–L periodograms.In all that follows we use the three datasets available at this time :HARPS-TERRA, HIRES and PFS. We use the HARPS-TERRADoppler measurements instead of CCF ones because TERRAvelocities have been proven to be more precise on stable M-dwarfs (Anglada-Escud´e & Butler 2012).The first three periodicities (7.2 days, 28.1 days and 91days) were trivially spotted using Bayesian posterior samplingsand the corresponding log–L periodograms. These three sig-nals were already reported by Anglada-Escud´e et al. (2012) andDelfosse et al. (2012), although the last one (signal d, at 91 days)remained uncertain due to the proximity of a characteristic time-scale of the star’s activity. This signal is discussed in the contextof stellar activity in Section 6. Signal d has a MAP period of 91days and would correspond to a candidate planet with a mini-mum mass of ∼ ⊕ .After that, the log–L periodogram search for a fourth sig-nal indicates a double-peaked likelihood maximum at 53 and Table 3.
Relative posterior probabilities and log-Bayes factorsof models M k with k Keplerian signals given the combinedHARPS-TERRA, HIRES, and PFS RV data of GJ 667C. Factor ∆ indicates how much the probability increases with respect tothe best model with one less Keplerian and P denotes the MAPperiod estimate of the signal added to the solution when increas-ing k . Only the highest probability sequence is shown here (ref-erence solution). A complete table with alternative solutions cor-responding to local probability maxima is given in Appendix B.2 k P ( M k | d ) ∆ log P ( d |M k ) P [days] ID0 2.7 × − – -602.1 –1 3.4 × − × -516.0 7.22 1.3 × − × -486.3 913 8.9 × − × -444.5 284 1.9 × − × -436.2 534 1.2 × − × -436.7 625 1.0 × − × -420.0 39, 535 1.0 × − × -422.3 39, 626 4.1 × − × -408.7 39, 53, 2566 4.1 × − × -411.0 39, 62, 2567 0.057 14 -405.4 17, 39, 53, 2567 0.939 230 -402.6 17, 39, 62, 256
62 days -both candidate periods receiving extremely low false-alarm probability estimates (see Figure 2). Using the recipes inDawson & Fabrycky (2010), it is easy to show that the two peaksare the yearly aliases of each other. Accordingly, our Bayesiansamplings converged to either period equally well giving slightlyhigher probability to the 53-day orbit ( × > . In appendix B.2 we provide a detailedanalysis and derived orbital properties of both solutions andshow that the precise choice of this fourth period does not sub-stantially a ff ect the confidence of the rest of the signals. As willbe shown at the end of the detection sequence, the most likelysolution for this candidate corresponds to a minimum mass of2 . ⊕ and a period of 62 days.After including the fourth signal, a fifth signal at 39.0 daysshows up conspicuously in the log–L periodograms. In this case,the posterior samplings always converged to the same period of39.0 days without di ffi culty (signal f). Such a planet would havea minimum mass of ∼ ⊕ . Given that the model probabilityimproved by a factor of 5.3 × and that the FAP derived fromthe log-L periodogram is 0 . × . The log–L periodograms did spot the same signal as themost significant one but assigned a FAP of ∼
20% to it. This ap-parent contradiction is due to the prior on the eccentricity. Thatis, the maximum likelihood solution favors a very eccentric orbitfor the Keplerian orbit at 62 days ( e e ∼ . .
5% which would then qual-ify as a significant detection. Given that the Bayesian detectioncriteria are well satisfied and that the log–L periodograms alsoprovide substantial support for the signal, we also include it in
Fig. 3.
Marginalized posterior densities for the Doppler semi-amplitudes of the seven reported signals.
Phase [days] -9-6-30369 R V [ m s - ] b Phase [days] -9-6-30369 R V [ m s - ] c Phase [days] -9-6-30369 R V [ m s - ] d Phase [days] -9-6-30369 R V [ m s - ] e Phase [days] -9-6-30369 R V [ m s - ] f Phase [days] -9-6-30369 R V [ m s - ] g Phase [days] -9-6-30369 R V [ m s - ] (h) Fig. 4.
RV measurements phase-folded to the best period for each planet. Brown circles are HARPS-TERRA velocities, PFS ve-locities are depicted as blue triangles, and HIRES velocities are green triangles. Red squares are averages on 20 phase bins of theHARPS-TERRA velocities. The black line corresponds to the best circular orbital fit (visualization purposes only).
Table 4.
Reference orbital parameters and their corresponding 99% credibility intervals. While the angles ω and M are uncon-strained due to strong degeneracies at small eccentricities, their sum λ = M + ω is better behaved and is also provided here forreference. b (h) c f e ∗ P [days] 7.2004 [7.1987, 7.2021] 16.946 [16.872, 16.997] 28.140 [28.075, 28.193] 39.026 [38.815, 39.220] 62.24 [61.69, 62.79]e 0.13 [0.02, 0.23] 0.06 [0, 0.38] 0.02 [0, 0.17] 0.03 [0, 0.19] 0.02 [0, 0.24]K [m s − ] 3.93 [3.55, 4.35] 0.61 [0.12, 1.05] 1.71 [1.24, 2.18] 1.08 [0.62, 1.55] 0.92 [0.50, 1.40] ω [rad] 0.10 [5.63, 0.85] 2.0 [0, 2 π ] 5.1 [0, 2 π ] 1.8 [0, 2 π ] 0.5 [0, 2 π ]M [rad] 3.42 [2.32, 4.60] 5.1 [0, 2 π ] 0.3 [0, 2 π ] 5.1 [0, 2 π ] 4.1 [0, 2 π ] λ [deg] 201[168, 250] 45(180) † †
34 (170) † † M sin i [M ⊕ ] 5.6 [4.3, 7.0] 1.1 [0.2, 2.1] 3.8 [2.6, 5.3] 2.7 [1.5, 4.1] 2.7 [1.3, 4.3]a [AU] 0.0505 [0.0452, 0.0549] 0.0893 [0.0800, 0.0977] 0.125 [0.112, 0.137] 0.156 [0.139, 0.170] 0.213 [0.191, 0.232]d g Other model parametersP [days] 91.61 [90.72, 92.42] 256.2 [248.3, 270.0] ˙ γ [m s − yr − ] 2.07 [1.79, 2.33]e 0.03 [0, 0.23] 0.08 [0, 0.49] γ HARPS [m s − ] -30.6 [-34.8, -26.8]K [m s − ] 1.52 [1.09, 1.95] 0.95 [0.51, 1.43] γ HIRES [m s − ] -31.9 [-37.0,, -26.9] ω [rad] 0.7 [0, 2 π ] 0.9 [0, 2 π ] γ PFS [m s − ] -25.8 [-28.9, -22.5]M [rad] 3.7 [0, 2 π ] 4.1 [0, 2 π ] σ HARPS [m s − ] 0.92 [0.63, 1.22] σ HIRES [m s − ] 2.56 [0.93, 5.15] λ [deg] 251(126) † † σ PFS [m s − ] 1.31 [0.00, 3.85]M sin i [M ⊕ ] 5.1 [3.4, 6.9] 4.6 [2.3, 7.2]a [AU] 0.276 [0.246, 0.300] 0.549 [0.491, 0.601] Notes. † Values allowed in the full range of λ . Full-width-at-half-maximum of the marginalized posterior is provided to illustrate the most likelyrange (see Figure 10). ∗ Due to the presence of a strong alias, the orbital period of this candidate could be 53 days instead. Such an alternativeorbital solution for planet e is given in Table B.2.8uillem Anglada-Escud´e et al.: Three HZ super-Earths in a seven-planet system the model (signal g). Its amplitude would correspond to a planetwith a minimum mass of 4.6 M ⊕ .When performing a search for a seventh signal, the posteriorsamplings converged consistently to a global probability maxi-mum at 17 days (M sin i ∼ . ⊕ ) which improves the modelprobability by a factor of 230. The global probability maximumcontaining seven signals corresponds to a solution with a pe-riod of 62 days for planet e. This solution has a total probability ∼
16 times larger than the one with P e =
53 days. Althoughsuch a di ff erence is not large enough to make a final decision onwhich period is preferred, from now on we will assume that ourreference solution is the one with P e = . ∼ . ⊕ , it would be among the least massive exoplanets dis-covered to date.As a final comment we note that, as in Anglada-Escud´e et al.(2012) and Delfosse et al. (2012), a linear trend was always in-cluded in the model. The most likely origin of such a trend isgravitational acceleration caused by the central GJ 667AB bi-nary. Assuming a minimum separation of 230 AU, the acceler-ation in the line-of-sight of the observer can be as large as 3.7m s − , which is of the same order of magnitude as the observedtrend of ∼ − yr − . We remark that the trend (or part ofit) could also be caused by the presence of a very long periodplanet or brown dwarf. Further Doppler follow-up, astrometricmeasurements, or direct imaging attempts of faint companionsmight help addressing this question.In summary, the first five signals are easily spotted usingBayesian criteria and log–L periodograms. The global solutioncontaining seven-Keplerian signals prefers a period of 62.2 daysfor signal e, which we adopt as our reference solution. Still, aperiod of 53 days for the same signal cannot be ruled out at themoment. The statistical significance of a 6th periodicity dependson the prior choice for the eccentricity, but the Bayesian oddsratio is high enough to propose it as a genuine Keplerian signal.The statistical significance of the seventh candidate (h) is closeto our detection limit and more observations are needed to fullyconfirm it.
6. Activity
In addition to random noise (white or correlated), stellar activitycan also generate spurious Doppler periodicities that can mimicplanetary signals (e.g., Lovis et al. 2011; Reiners et al. 2013).In this section we investigate whether there are periodic varia-tions in the three activity indices of GJ 667C (S-index, BIS andFWHM are described in Section 3). Our general strategy is thefollowing : if a significant periodicity is detected in any of the in-dices and such periodicity can be related to any of the candidatesignals (same period or aliases), we add a linear correlation termto the model and compute log–L periodograms and new sam-plings of the parameter space. If the data were better describedby the correlation term rather than a genuine Doppler signal, theoverall model probability would increase and the planet signalin question should decrease its significance (even disappear).Log–L periodogram analysis of two activity indices (S-indexbut specially the FWHM) show a strong periodic variabilityat 105 days. As discussed in Anglada-Escud´e et al. (2012) and
10 100 1000051015 Δ l og L
10 100 1000
Period [days] Δ l og L dd c ef g(h) 105 days29 days S-index (1)S-index (2)
10 100 100005101520 Δ l og L
10 100 1000
Period [days] Δ l og L dd c ef g(h) FWHM (1)FWHM (2)
105 days110 days
Fig. 5. Top two panels
Log–L periodograms for up to 2 signalsin the S-index. The most likely periods of the proposed planetcandidates are marked as vertical lines.
Bottom two panels.
Log–L periodograms for up to 2 signals in the FWHM. Giventhe proximity of these two signals, it is possible that both ofthem originate from the same feature (active region corotatingwith the star) that is slowly evolving with time.Delfosse et al. (2012), this cast some doubt on the candidate at91 days (d). Despite the fact that the 91-day and 105-day peri-ods are not connected by first order aliases, the phase samplingis sparse in this period domain (see phase–folded diagrams ofthe RV data for the planet d candidate in Fig. 4) and the log–Lperiodogram for this candidate also shows substantial power at105 days. From the log–L periodograms in Figure 2, one can di-rectly obtain the probability ratio of a putative solution at 91 daysversus one with a period of 105 days when no correlation termsare included. This ratio is 6 . × , meaning that the Dopplerperiod at 91 days is largely favoured over the 105-day one. AllBayesian samplings starting close to the 105-day peak ended-upconverging on the signal at 91 day. We then applied our valida-tion procedure by inserting linear correlation terms in the model(g = C F × FWHM i or g = C S × S i ), in eq. 1) and computed bothlog–L periodograms and Bayesian samplings with C F and C S as free parameters. In all cases the ∆ log L between 91 and 105days slightly increased, thus supporting the conclusion that thesignal at 91 days is of planetary origin. For example, the ratio oflikelihoods between the 91 and 105 day signals increased from6.8 × to 3.7 × when the correlation term with the FWHMwas included (see Figure 6). The Bayesian samplings includingthe correlation term did not improve the model probability (seeAppendix C) and still preferred the 91-day signal without anydoubt. We conclude that this signal is not directly related to thestellar activity and therefore is a valid planet candidate.
10 100 1000
Period [days] Δ l og L No correlation termIncluding correlation term
105 days91 days
Fig. 6.
Log–likelihood periodograms for planet d (91 days) in-cluding the correlation term (red dots) compared to the originalperiodogram without this term (black line). The inclusion of thecorrelation term increases the contrast between the peaks at 91and 105 days, favoring the interpretation of the 91 days signalsas a genuine planet candidate.Given that activity might induce higher order harmonics inthe time-series, all seven candidates have been analyzed anddouble-checked using the same approach. Some more details onthe results from the samplings are given in the Appendix C.2. Allcandidates -including the tentative planet candidate h- passed allthese validation tests without di ffi culties.
7. Tests for quasi-periodic signals
Activity induced signals and superposition of several indepen-dent signals can be a source of confusion and result in detec-tions of “apparent” false positives. In an attempt to quantify howmuch data is necessary to support our preferred global solution(with seven planets) we applied the log-L periodogram analysismethod to find the solution as a function of the number of datapoints. For each dataset, we stopped searching when no peakabove FAP 1% was found. The process was fully automated sono human-biased intervention could alter the detection sequence.The resulting detection sequences are show in Table 5. In addi-tion to observing how the complete seven-planet solution slowlyemerges from the data one can notice that for N obs <
100 the k = k = ∼ N obs =
75 case in moredetail and made a supervised / visual analysis of that subset.The first 7.2 days candidate could be easily extracted. Wethen computed a periodogram of the residuals to figure out ifthere were additional signals left in the data. In agreement withthe automatic search, the periodogram of the residuals (bottomof Figure 7) show a very strong peak at ∼
100 days. The peak wasso strong that we went ahead and assessed its significance. It hada very low FAP ( < .
16 32 64 128 256
Period [days] P o w e r Real set +28d+91d+63d+39d+260d+17d
Fig. 7.
Sequence of periodograms obtained from synthetic noise-less data generated on the first 75 epochs. The signals in Table 4were sequentially injected from top to bottom. The bottom panelis the periodogram to the real dataset after removing the first 7.2days planet candidate.planet solution and generated the exact signal we would expectif we only had planet c (28 days) in the first 75 HARPS-TERRAmeasurements (without noise). The periodogram of such a noise-less time-series (top panel in Fig. 7) was very di ff erent from thereal one. Then, we sequentially added the rest of the signals.As more planets were added, the periodogram looked closer tothe one from the real data. This illustrates that we would havereached the same wrong conclusion even with data that had neg-ligible noise. How well the general structure of the periodogramwas recovered after adding all of the signals is rather remarkable(comparing the bottom two panels in Fig. 7). While this is not astatistical proof of significance, it shows that the periodogram ofthe residuals from the 1-planet fit (even with only 75 RVs mea-surements) is indeed consistent with the proposed seven-planetsolution without invoking the presence of quasi-periodic signals.This experiment also shows that, until all stronger signals couldbe well-decoupled (more detailed investigation showed this hap-pened at about N obs ∼ ffi cult. We repeated the same ex-ercise with N obs = ff ect is not new and the literature con-tains several examples that cannot be easily explained by sim-plistic aliasing arguments- e.g., see GJ 581d (Udry et al. 2007;Mayor et al. 2009) and HD 125612b / c (Anglada-Escud´e et al.2010; Lo Curto et al. 2010). The fact that all signals detected inthe velocity data of GJ 667C have similar amplitudes – exceptperhaps candidate b which has a considerably higher amplitude– made this problem especially severe. In this sense, the cur-rently available set of observations are a sub–sample of the manymore that might be obtained in the future, so it might happen thatone of the signals we report “bifurcates” into other periodicities.This experiment also suggests that spectral information beyondthe most trivial aliases can be used to verify and assess the sig-nificance of future detections (under investigation). Table 5.
Most significant periods as extracted using log–L periodograms on subsamples of the first N o bs measurements. Boldfacedvalues indicate coincidence with a signal of our seven-planet solution (or a first order yearly alias thereof). A parenthesis in the lastperiod indicates a preferred period that did not satisfy the frequentist 1% FAP threshold but did satisfy the Bayesian detectabilitycriteria. N obs a – – –120 – – – –143 a a (260) –160 a
275 (16.9) a − alias of the preferred period in Table 4.
16 32 64 128 256
Period [days] P o w e r Real set +28d+91d+63d+39d+260d+17d
16 32 64 128 256
Period [days] P o w e r Real set +28d+91d+63d+39d+260d+17d
16 32 64 128 256
Period [days] P o w e r Real set +28d+91d+63d+39d+260d+17d
Fig. 8.
Same as 7 but using the first 100 epochs (left), first 120 (center) and all of them (right).
As an additional verification against quasi-periodicity, we inves-tigated if the signals were present in the data when it was di-vided into two halves. The first half corresponds to the first 86HARPS observations and the second half covers the remainingdata. The data from PFS and HIRES were not used for this test.The experiment consists of removing all signals except for one,and then computing the periodogram on those residuals (firstand second halfs separately). If a signal is strongly present inboth halfs, it should, at least, emerge as substantially significant.All signals except for the seventh one passed this test nicely.That is, in all cases except for h, the periodograms prominentlydisplay the non-removed signal unambiguously. Besides demon-strating that all signals are strongly present in the two halves, italso illustrates that any of the candidates would have been triv-ially spotted using periodograms if it had been orbiting alonearound the star. The fact that each signal was not spotted be-fore (Anglada-Escud´e et al. 2012; Delfosse et al. 2012) is a con-sequence of an inadequate treatment of signal correlations whendealing with periodograms of the residuals only. Both the de-scribed Bayesian method and the log-likelihood periodogramtechnique are able to deal with such correlations by identify-ing the combined global solution at the period search level.As for other multiplanet systems detected using similar tech-niques (Tuomi et al. 2013; Anglada-Escud´e & Tuomi 2012), op-timal exploration of the global probability maxima at the signalsearch level is essential to properly detect and assess the signif-icance of low mass multiplanet solutions, especially when sev- eral signals have similar amplitudes close to the noise level ofthe measurements.Summarizing these investigations and validation of the sig-nals against activity, we conclude that – Up to seven periodic signals are detected in the Doppler mea-surements of GJ 667C data, with the last (seventh) signalvery close to our detection threshold. – The significance of the signals are not a ff ected by corre-lations with activity indices and we could not identify anystrong wavelength dependence with any of them. – The first six signals are strongly present in subsamples ofthe data. Only the seventh signal is unconfirmed using halfof the data alone. Our analysis indicates that any of the sixstronger signals would had been robustly spotted with halfthe available data if each had been orbiting alone around thehost star. – Signal correlations in unevenly sampled data are the reasonwhy Anglada-Escud´e et al. (2012) and Delfosse et al. (2012)only reported three of them. This is a known problem whenassessing the significance of signals using periodograms ofresiduals only (see Anglada-Escud´e & Tuomi 2012, as an-other example).Given the results of these validation tests, we promote six of thesignals (b, c, d, e, f, g) to planet candidates. For economy oflanguage, we will refer to them as planets but the reader mustremember that, unless complementary and independent verifica-tion of a Doppler signal is obtained (e.g., transits), they should be Δ l og L Δ l og L Δ l og L Δ l og L Δ l og L Δ l og L Period [days] Δ l og L Period [days] bcdefg(h)
First half Second half
Fig. 9.
Periodograms on first and second half of the time series as obtained when all signals except one were removed from the data.Except for signal h, all signals are significantly present in both halves and could have been recovered using either half if they hadbeen in single planet systems.called planet candidates . Verifying the proposed planetary sys-tem against dynamical stability is the purpose of the next section.
8. Dynamical analysis
One of the by-products of the Bayesian analysis described in theprevious sections, are numerical samples of statistically allowedparameter combinations. The most likely orbital elements andcorresponding confidence levels can be deduced from these sam-ples. In Table 4 we give the orbital configuration for a planetarysystem with seven planets around GJ 667C, which is preferredfrom a statistical point of view. To be sure that the proposedplanetary system is physically realistic, it is necessary to confirmthat these parameters not only correspond to the solution favoredby the data, but also constitute a dynamically stable configura-tion. Due to the compactness of the orbits, abundant resonancesand therefore complex interplanetary interactions are expectedwithin the credibility intervals. To slightly reduce this complex-ity and since evidence for planet h is weak, we split our analysisand present- in the first part of this section- the results for thesix-planet solution with planets b to g. The dynamical feasibilityof the seven-planet solution is then assessed by investigating thesemi-major axis that would allow introducing a seventh planetwith the characteristics of planet h.
A first thing to do is to extract from the Bayesian samplingsthose orbital configurations that allow stable planetary motionover long time scales, if any. Therefore we tested the stability of each configuration by a separate numerical integration using thesymplectic integrator SABA of Laskar & Robutel (2001) with astep size τ = . ff ects were neglected for these runs. Possible e ff ects of tides arediscussed separately in Section 9.4. The integration was stoppedif any of the planets went beyond 5 AU or planets approachedeach other closer than 10 − AU.The stability of those configurations that survived the inte-gration time span of 10 orbital periods of planet g (i. e. ≈ D k for eachplanet k as the relative change in its mean motion n k over twoconsecutive time intervals as was done in Tuomi et al. (2013).For stable orbits the computed mean motion in both intervalswill be almost the same and therefore D k will be very small.We note that this also holds true for planets captured inside amean-motion resonance (MMR), as long as this resonance helpsto stabilize the system. As an index for the total stability of a con-figuration we used D = max( | D k | ). The results are summarizedin Figure 10. To generate Figure 10, we extracted a sub-sampleof 80,000 initial conditions from the Bayesian samplings. Thoseconfigurations that did not reach the final integration time arerepresented as gray dots. By direct numerical integration of theremaining initial conditions, we found that almost all configu-rations with D < − survive a time span of 1 Myr. This cor-responds to ∼ D < − ) are depicted as black crosses.In Figure 10 one can see that the initial conditions takenfrom the integrated 80,000 solutions are already confined to a λ [ d e g ] a [au] λ [ d e g ] a [au] a [au] b c de f g Fig. 10.
Result of the stability analysis of 80,000 six-planet solutions in the plane of initial semi-major axis a vs. initial meanlongitude λ obtained from a numerical integration over T ≈ < − ). Black crosses represent the most stable solutions (D < − ), and can last overmany Myr. Table 6.
Astrocentric orbital elements of solution S . Planet P (d) a (AU) e ω ( ◦ ) M ( ◦ ) M sin i (M ⊕ )b 7.2006 0.05043 0.112 4.97 209.18 5.94c 28.1231 0.12507 0.001 101.38 154.86 3.86f 39.0819 0.15575 0.001 77.73 339.39 1.94e 62.2657 0.21246 0.001 317.43 11.32 2.68d 92.0926 0.27580 0.019 126.05 243.43 5.21g 251.519 0.53888 0.107 339.48 196.53 4.41 very narrow range in the parameter space of all possible orbits.This means that the allowed combinations of initial a and λ arealready quite restricted by the statistics. By examining Figure10 one can also notice that those initial conditions that turnedout to be long-term stable are quite spread out along the areaswhere the density of Bayesian states is higher. Also, for someof the candidates (d, f and g), there are regions were no orbitwas found with D < − . The paucity of stable orbits at certainregions indicate areas of strong chaos within the statistically al-lowed ranges (likely disruptive mean-motion resonances) and il-lustrate that the dynamics of the system are far from trivial.The distributions of eccentricities are also strongly a ff ectedby the condition of dynamical stability. In Figure 11 we showthe marginalized distributions of eccentricities for the sampleof all the integrated orbits (gray histograms) and the distribu-tion restricted to relatively stable orbits (with D < − , red his-tograms). We see that, as expected, stable motion is only pos-sible with eccentricities smaller than the mean values allowedby the statistical samples. The only exceptions are planets band g. These two planet candidates are well separated from the other candidates. As a consequence, their probability densitiesare rather una ff ected by the condition of long-term stability. Wenote here that the information about the dynamical stability hasbeen used only a posteriori . If we had used long-term dynamicsas a prior (e.g., assign 0 probability to orbits with D > − ), mod-erately eccentric orbits would have been much more stronglysuppressed than with our choice of prior function (Gaussian dis-tribution of zero mean and σ = .
3, see Appendix A.1). In thissense, our prior density choice provides a much softer and unin-formative constraint than the dynamical viability of the system.In the following we will use the set of initial conditions thatgave the smallest D for a detailed analysis and will refer to itas S . In Table 6, we present the masses and orbital parametersof S , and propose it as the favored configuration. To doublecheck our dynamical stability results, we also integrated S for10 years using the HNBody package (Rauch & Hamilton 2002)including general relativistic corrections and a time step of τ = − years. Publicly available at http: // janus.astro.umd.edu / HNBody / e b F r equen cy e c F r equen cy e d F r equen cy e e F r equen cy e f F r equen cy e g F r equen cy Fig. 11.
Marginalized posterior densities for the orbital eccentricities of the six planet solution (b, c, d, in the first row; e, f, g in thesecond) before (gray histogram) and after (red histogram) ruling out dynamically unstable configurations.
Although the dynamical analysis of such a complex system withdi ff erent, interacting resonances could be treated in a separatepaper, we present here a basic analysis of the dynamical archi-tecture of the system. From studies of the Solar System, we knowthat, in the absence of mean motion resonances, the variations inthe orbital elements of the planets are governed by the so-calledsecular equations. These equations are obtained after averagingover the mean longitudes of the planets. Since the involved ec-centricities for GJ 667C are small, the secular system can be lim-ited here to its linear version, which is usually called a Laplace-Lagrange solution (for details see Laskar (1990)). Basically, thesolution is obtained from a transformation of the complex vari-ables z k = e k e ıϖ k into the proper modes u k . Here, e k are theeccentricities and ϖ k the longitudes of the periastron of planet k = b , c , . . . , g . The proper modes u k determine the secular vari-ation of the eccentricities and are given by u k ≈ e i ( g k t + φ k ) .Since the transformation into the proper modes depends onlyon the masses and semi-major axes of the planets, the secularfrequencies will not change much for the di ff erent stable config-urations found in Figure 10. Here we use solution S to obtainnumerically the parameters of the linear transformation by a fre-quency analysis of the numerically integrated orbit. The secularfrequencies g k and the phases φ k are given in Table 7. How wellthe secular solution describes the long-term evolution of the ec-centricities can be readily seen in Figure 12. Table 7.
Fundamental secular frequencies g k , phases φ k and cor-responding periods of the six-planet solution. k g k φ k Period[deg / yr] [deg] [yr]1 0.071683 356.41 5022.092 0.184816 224.04 1947.883 0.991167 116.46 363.214 0.050200 33.63 7171.375 0.656733 135.52 548.176 0.012230 340.44 29435.80 From Figure 12, it is easy to see that there exists a strong sec-ular coupling between all the inner planets. From the Laplace- e cc en t r i c i t y t [yr]b cfed g Fig. 12.
Evolution of the eccentricities of solution S . Coloredlines give the eccentricity as obtained from a numerical integra-tion. The thin black lines show the eccentricity of the respectiveplanet as given by the linear, secular approximation. Close toeach line we give the name of the corresponding planet.Lagrange solution, we find that the long-term variation of theeccentricities of these planets is determined by the secular fre-quency g − g with a period of ≈ ≈
600 years ( g − g ), whilethe eccentricities of planets c and f vary with a period of almost3000 years ( g + g ). Such couplings are already known to pre-vent close approaches between the planets (Ferraz-Mello et al.2006). As a result, the periastron of the planets are locked andthe di ff erence ∆ ϖ between any of their ϖ librates around zero.Although the eccentricities show strong variations, thesechanges are very regular and their maximum values remainbounded. From the facts that 1) the secular solution agrees sowell with numerically integrated orbits, and 2) at the same timethe semi-major axes remain nearly constant (Table 8), we canconclude that S is not a ff ected by strong MMRs.Nevertheless, MMRs that can destabilize the whole systemare within the credibility intervals allowed by the samplings andnot far away from the most stable orbits. Integrating some of the initial conditions marked as chaotic in Figure 10 one finds that,for example, planets d and g are in some of these cases temporar-ily trapped inside a 3:1 MMR, causing subsequent disintegrationof the system. Table 8.
Minimum and maximum values of the semi-major axesand eccentricities during a run of S over 10 Myr. k a min a max e min e max b 0.050431 0.050433 0.035 0.114c 0.125012 0.125135 0.000 0.060f 0.155582 0.155922 0.000 0.061e 0.212219 0.212927 0.000 0.106d 0.275494 0.276104 0.000 0.087g 0.538401 0.539456 0.098 0.116 After finding a non-negligible set of stable six-planet solutions, itis tempting to look for more planets in the system. From the dataanalysis, one even knows the preferred location of such a planet.We first considered doing an analysis similar to the one for thesix-planet case using the Bayesian samples for the seven-planetsolution. As shown in previous sections, the subset of stable so-lutions found by this approach is already small compared to thestatistical samples in the six-planet case ( ∼ . ffi cient.As a first approximation to the problem, we checked whetherthe distances between neighboring planets are still large enoughto allow stable motion. In Chambers et al. (1996) the mean life-time for coplanar systems with small eccentricities is estimatedas a function of the mutual distance between the planets, theirmasses and the number of planets in the system. From their re-sults, we can estimate the expected lifetime for the seven-planetsolution to be at least 10 years.Motivated by this result, we explored the phase space aroundthe proposed orbit for the seventh planet. To do this, we use solu-tion S and placed a fictitious planet with 1.1 M ⊕ (the estimatedmass of planet h as given in Table 4) in the semi-major axis rangebetween 0.035 and 0.2 AU (step size of 0.001 AU) varying theeccentricity between 0 and 0.2 (step size of 0.01). The orbital an-gles ω and M were set to the values of the statistically preferredsolution for h (see Table 4). For each of these initial configu-rations, we integrated the system for 10 orbits of planet g andanalyzed stability of the orbits using the same secular frequencyanalysis. As a result, we obtained a value of the chaos index D at each grid point. Figure 13 shows that the putative orbit of happears right in the middle of the only island of stability left inthe inner region of the system. By direct numerical integrationof solution S together with planet h at its nominal position, wefound that such a solution is also stable on Myr timescales. Withthis we conclude that the seventh signal detected by the Bayesiananalysis also belongs to a physically viable planet that might beconfirmed with a few more observations. Due to the lack of reported transit, only the minimum massesare known for the planet candidates. The true masses depend on e cc en t r i c i t y -6-5-4-3 m a x | D k | b h c f Fig. 13.
Stability plot of the possible location of a 7th planet inthe stable S solution (Table 5). We investigate the stability ofan additional planet with 1.1 Earth masses around the locationfound by the Bayesian analysis. For these integrations, we variedthe semi-major axis and eccentricity of the putative planet on aregular grid. The orbital angles ω and M were set to the valuesof the statistically preferred solution, while the inclination wasfixed to zero. The nominal positions of the planets as given inTable 6 are marked with white crosses.the unknown inclination i of the system to the line-of-sight. Inall the analysis presented above, we implicitly assume that theGJ 667C system is observed edge-on ( i = ◦ ) and that all truemasses are equal to the minimum mass Msin i . As shown in thediscussion on the dynamics, the stability of the system is frag-ile in the sense that dynamically unstable configurations can befound close in the parameter space to the stable ones. Therefore,it is likely that a more complete analysis could set strong lim-itations on the maximum masses allowed to each companion.An exploration of the total phase space including mutual incli-nations would require too much computational resources and isbeyond the scope of this paper. To obtain a basic understandingof the situation, we only search for a constraint on the maximummasses of the S solution assuming co-planarity. Decreasing theinclination of the orbital plane in steps of 10 ◦ , we applied the fre-quency analysis to the resulting system. By making the planetsmore massive, the interactions between them become stronger,with a corresponding shrinking of the areas of stability. In thisexperiment, we found that the system remained stable for at leastone Myr for an inclination down to i = ◦ . If this result can bevalidated by a much more extensive exploration of the dynamicsand longer integration times (in prep.), it would confirm that themasses of all the candidates are within a factor of 2 of the min-imum masses derived from Doppler data. Accordingly, c,f ande would be the first dynamically confirmed super-Earths (truemasses below 10 M ⊕ ) in the habitable zone of a nearby star.
9. Habitability
Planets h–d receive 20–200% of the Earth’s current insolation,and hence should be evaluated in terms of potential habitability.Traditionally, analyses of planetary habitability begin with deter-mining if a planet is in the habitable zone (Dole 1964; Hart 1979;Kasting et al. 1993; Selsis et al. 2007; Kopparapu et al. 2013),but many factors are relevant. Unfortunately, many aspects can- not presently be determined due to the limited characterizationderivable from RV observations. However, we can explore theissues quantitatively and identify those situations in which hab-itability is precluded, and hence determine which of these plan-ets could support life. In this section we provide a preliminaryanalysis of each potentially habitable planet in the context of pre-vious results, bearing in mind that theoretical predictions of themost relevant processes cannot be constrained by existing data.
The HZ is defined at the inner edge by the onset of a “moistgreenhouse,” and at the outer edge by the “maximum green-house” (Kasting et al. 1993). Both of these definitions assumethat liquid surface water is maintained under an Earth-like at-mosphere. At the inner edge, the temperature near the surfacebecomes large enough that water cannot be confined to the sur-face and water vapor saturates the stratosphere. From there, stel-lar radiation can dissociate the water and hydrogen can escape.Moreover, as water vapor is a greenhouse gas, large quantities inthe atmosphere can heat the surface to temperatures that forbidthe liquid phase, rendering the planet uninhabitable. At the outeredge, the danger is global ice coverage. While greenhouse gaseslike CO can warm the surface and mitigate the risk of globalglaciation, CO also scatters starlight via Rayleigh scattering.There is therefore a limit to the amount of CO that can warma planet as more CO actually cools the planet by increasing itsalbedo, assuming a moist or runaway greenhouse was never trig-gered.We use the most recent calculations of the HZ(Kopparapu et al. 2013) and find, for a 1 Earth-mass planet,that the inner and outer boundaries of the habitable zone forGJ 667C lie between 0.095–0.126 AU and 0.241–0.251 AUrespectively. We will adopt the average of these limits as aworking definition of the HZ: 0.111 – 0.246 AU. At the inneredge, larger mass planets resist the moist greenhouse and theHZ edge is closer in, but the outer edge is almost independentof mass. Kopparapu et al. (2013) find that a 10 M ⊕ planet canbe habitable 5% closer to the star than a 1 M ⊕ planet. However,we must bear in mind that the HZ calculations are based on1-dimensional photochemical models that may not apply toslowly rotating planets, a situation likely for planets c, d, e, fand h (see below).From these definitions, we find that planet candidate h ( a = . a = .
125 AU) is close to the inner edge but is likely tobe in the HZ, especially since it has a large mass. Planets f ande are firmly in the HZ. Planet d is likely beyond the outer edgeof the HZ, but the uncertainty in its orbit prevents a definitiveassessment. Thus, we conclude that planets c, f, and e are inthe HZ, and planet d might be, i.e. there up to four potentiallyhabitable planets orbiting GJ 667C.Recently, Abe et al. (2011) pointed out that planets withsmall, but non-negligible, amounts of water have a larger HZthan Earth-like planets. From their definition, both h and d arefirmly in the HZ. However, as we discuss below, these planets arelikely water-rich, and hence we do not use the Abe et al. (2011)HZ here.
Planet formation is a messy process characterized by scat-tering, migration, and giant impacts. Hence precise calcula-tions of planetary composition are presently impossible, but seeBond et al. (2010); Carter-Bond et al. (2012) for some generaltrends. For habitability, our first concern is discerning if a planetis rocky (and potentially habitable) or gaseous (and uninhab-itable). Unfortunately, we cannot even make this rudimentaryevaluation based on available data and theory. Without radiimeasurements, we cannot determine bulk density, which coulddiscriminate between the two. The least massive planet knownto be gaseous is GJ 1214 b at 6.55 M ⊕ (Charbonneau et al.2009), and the largest planet known to be rocky is Kepler-10 bat 4.5 M ⊕ (Batalha et al. 2011). Modeling of gas accretion hasfound that planets smaller than 1 M ⊕ can accrete hydrogen incertain circumstances (Ikoma et al. 2001), but the critical massis likely larger (Lissauer et al. 2009). The planets in this systemlie near these masses, and hence we cannot definitively say ifany of these planets are gaseous.Models of rocky planet formation around M dwarfs havefound that those that accrete from primordial material arelikely to be sub-Earth mass (Raymond et al. 2007) and volatile-poor (Lissauer 2007). In contrast, the planets orbiting GJ 667Care super-Earths in a very packed configuration summing up to > M ⊕ inside 0.5 AU. Therefore, the planets either formed atlarger orbital distances and migrated in ( e.g. Lin et al. 1996), oradditional dust and ice flowed inward during the protoplanetarydisk phase and accumulated into the planets Hansen & Murray(2012, 2013). The large masses disfavor the first scenario, andwe therefore assume that the planets formed from material thatcondensed beyond the snow-line and are volatile rich. If notgaseous, these planets contain substantial water content, whichis a primary requirement for life (and negates the dry-worldHZ discussed above). In conclusion, these planets could beterrestrial-like with significant water content and hence are po-tentially habitable.
Stellar activity can be detrimental to life as the planets can bebathed in high energy photons and protons that could strip theatmosphere or destroy ozone layers. In quiescence, M dwarfsemit very little UV light, so the latter is only dangerous if flaresoccur frequently enough that ozone does not have time to bereplenished (Segura et al. 2010). As already discussed in Section2, GJ 667C appears to be relatively inactive (indeed, we wouldnot have been able to detect planetary signals otherwise), andso the threat to life is small today. If the star was very activein its youth- with mega-flares like those on the equal mass starAD Leo (Hawley & Pettersen 1991)- any life on the surface ofplanets might have been di ffi cult during those days (Segura et al.2010). While M dwarfs are likely to be active for much longertime than the Sun (West et al. 2008; Reiners & Mohanty 2012),GJ 667C is not active today and there is no reason to assume thatlife could not form after an early phase of strong stellar activity. Planets in the HZ of low mass stars may be subject to strongtidal distortion, which can lead to long-term changes in orbitaland spin properties (Dole 1964; Kasting et al. 1993; Barnes et al.2008; Heller et al. 2011), and tidal heating (Jackson et al. 2008;Barnes et al. 2009, 2013). Both of these processes can a ff ect hab- t lock t ero t lock t ero t lock t ero t lock t ero h 0.07 0.08 18.2 20.4 0.55 0.77 66.9 103c 0.62 0.69 177 190 4.7 8.1 704 1062f 2.2 2.3 626 660 18.5 30.1 2670 3902e 14.2 15.0 4082 4226 129 210 > > d 70.4 73 > >
692 1094 > > Table 9.
Timescales for the planets’ tidal despinning in units of Myr. “CPL” denotes the constant-phase-lag model ofFerraz-Mello et al. (2008), “CTL” labels the constant-time-lag model of Leconte et al. (2010).
Fig. 14.
Tidal evolution of the spin properties of planet GJ 667C f. Solid lines depict predictions from constant-time-lag theory(“CTL”), while dashed lines illustrate those from a constant-phase-lag model (“CPL”). All tracks assume a scenario similar to the“base” configuration (see text and Table 9).
Left : Despinning for an assumed initial rotation period of one day. The CPL modelyields tidal locking in less than 5 Myr, and CTL theory predicts about 20 Myr for tidal locking.
Right : Tilt erosion of an assumedinitial Earth-like obliquity of 23 . ◦ . Time scales for both CPL and CTL models are similar to the locking time scales.itability, so we now consider tidal e ff ects on planets c, d, e, f andh. Tides will first spin-lock the planetary spin and drivethe obliquity to either 0 or π . The timescale for these pro-cesses is a complex function of orbits, masses, radii and spins,(see e.g. Darwin 1880; Hut 1981; Ferraz-Mello et al. 2008;Leconte et al. 2010) but for super-Earths in the HZ of a ∼ . M ⊙ star, Heller et al. (2011) found that tidal locking should oc-cur in 10 –10 years. We have used both the constant-time-lagand constant-phase-lag tidal models described in Heller et al.(2011) and Barnes et al. (2013) (see also Ferraz-Mello et al.2008; Leconte et al. 2010), to calculate how long tidal lockingcould take for these planets. We consider two possibilities. Ourbaseline case is very similar to that of Heller et al. (2011) inwhich the planets initially have Earth-like properties: a 1-dayrotation period, an obliquity of 23.5 ◦ and the current tidal dissi-pation of the Earth (a tidal Q of 12-Yoder (1995) or time lag of638 s-Lambeck (1977); Neron de Surgy & Laskar (1997)). Wealso consider an extreme, but plausible, case that maximizes thetimescale for tidal locking: 8-hour rotation period, obliquity of89.9 ◦ and a tidal Q of 1000 or time lag of 6.5 s. In Table 9 weshow the time for the obliquity to erode to 1 ◦ , t ero , and the timeto reach the pseudo-synchronous rotation period, t lock .In Figure 14, we depict the tidal evolution of the rotation pe-riod (left panel) and obliquity (right panel) for planet f as an ex-ample. The assumed initial configuration is similar to the “base” scenario. Time scales for rotational locking and tilt erosion aresimilar to those shown in Table 9. As these planets are on nearly circular orbits, we ex-pect tidally-locked planets to be synchronously rotating, al-though perhaps when the eccentricity is relatively largepseudo-synchronous rotation could occur (Goldreich 1966;Murray & Dermott 1999; Barnes et al. 2008; Correia et al. 2008;Ferraz-Mello et al. 2008; Makarov & Efroimsky 2013). FromTable 9 we see that all the planets h–f are very likely syn-chronous rotators, planet e is likely to be, but planet d is uncer-tain. Should these planets have tenuous atmospheres ( < . Note that evolution for the CPL model is faster with our parameter-ization. In the case of GJ 581 d, shown in Heller et al. (2011), the planetwas assumed to be less dissipative in the CPL model ( Q p = Leconte et al. (2010) model and the Earth’s dissipation, we findthat tidal heating of the HZ planets will be negligible for mostcases. Consider an extreme version of planet h, which is justinterior to the HZ. Suppose it has the same tidal dissipation asthe Earth (which is the most dissipative body known), a rota-tion period of 10 hr, an eccentricity of 0.1, and an obliquityof 80 ◦ . The Leconte et al. (2010) model predicts such a planetwould have a tidal heat flux of nearly 4000 W m − . However, thatmodel also predicts the flux would drop to only 0.16 W m − injust 10 years. The timescale for a runaway greenhouse to ster-ilize a planet is on the order of 10 years (Watson et al. 1981;Barnes et al. 2013), so this burst of tidal heating does not forbidhabitability.After tidal locking, the planet would still have about 0.14 Wm − of tidal heating due to the eccentricity (which, as for theother candidates, can oscillate between 0 and 0.1 due to dynam-ical interactions). If we assume an Earth-like planet, then about90% of that heat is generated in the oceans, and 10% in the rockyinterior. Such a planet would have churning oceans, and about0.01 W m − of tidal heat flux from the rocky interior. This num-ber should be compared to 0.08 W m − , the heat flux on the Earthdue entirely to other sources. As e = . §
8, the actual heat flux is probably muchlower. We conclude that tidal heating on planet h is likely to benegligible, with the possibility it could be a minor contributor tothe internal energy budget. As the other planets are more distant,the tidal heating of those planets is negligible. The CPL predictshigher heating rates and planet c could receive ∼ .
01 W m − ofinternal heating, but otherwise tidal heating does not a ff ect theHZ planets. Assuming planets c, f and e have habitable surfaces (see Figure15), what might their climates be like? To first order we ex-pect a planet’s surface temperature to be cooler as semi-majoraxis increases because the incident flux falls o ff with distancesquared. However, albedo variations can supersede this trend, e.g. a closer, high-albedo planet could absorb less energy than amore distant low-albedo planet. Furthermore, molecules in theatmosphere can trap photons near the surface via the greenhousee ff ect, or scatter stellar light via Rayleigh scattering, an anti-greenhouse e ff ect. For example, the equilibrium temperature ofVenus is actually lower than the Earth’s due to the former’s largealbedo, yet the surface temperature of Venus is much larger thanthe Earth’s due to the greenhouse e ff ect. Here, we speculate onthe climates of each HZ planet based on the our current under-standing of the range of possible climates that HZ planets mighthave.Certain aspects of each planet will be determined by the red-der spectral energy distribution of the host star. For example, the“stratosphere” is expected to be isothermal as there is negligi-ble UV radiation (Segura et al. 2003). On Earth, the UV lightabsorbed by ozone creates a temperature inversion that delin-eates the stratosphere. HZ calculations also assume the albedo ofplanets orbiting cooler stars are lower than the Earth’s becauseRayleigh scattering is less e ff ective for longer wavelengths, andbecause the peak emission in the stellar spectrum is close toseveral H O and CO absorption bands in the near infrared.Therefore, relative to the Earth’s insolation, the HZ is fartherfrom the star. In other words, if we placed the Earth in orbitaround an M dwarf such that it received the same incident radi-ation as the modern Earth, the M dwarf planet would be hotteras it would have a lower albedo. The di ff erent character of the light can also impact plant life, and we might expect less pro-ductive photosynthesis (Kiang et al. 2007), perhaps relying onpigments such as chlorophyll d (Mielke et al. 2013) or chloro-phyll f (Chen et al. 2010).Planet c is slightly closer to the inner edge of the HZ thanthe Earth, and so we expect it to be warmer than the Earth, too.It receives 1230 W m − of stellar radiation, which is actuallyless than the Earth’s solar constant of 1360 W m − . Assumingsynchronous rotation and no obliquity, then the global climatedepends strongly on the properties of the atmosphere. If the at-mosphere is thin, then the heat absorbed at the sub-stellar pointcannot be easily transported to the dark side or the poles. Thesurface temperature would be a strong function of the zenith an-gle of the host star GJ 667C. For thicker atmospheres, heat re-distribution becomes more significant. With a rotation period of ∼
28 days, the planet is likely to have Hadley cells that extendto the poles (at least if Titan, with a similar rotation period, isa guide), and hence jet streams and deserts would be unlikely.The location of land masses is also important. Should land beconcentrated near the sub-stellar point, then silicate weatheringis more e ff ective, and cools the planet by drawing down CO (Edson et al. 2012).Planet f is a prime candidate for habitability and receives788 W m − of radiation. It likely absorbs less energy than theEarth, and hence habitability requires more greenhouse gases,like CO or CH . Therefore a habitable version of this planethas to have a thicker atmosphere than the Earth, and we can as-sume a relatively uniform surface temperature. Another possi-bility is an “eyeball” world in which the planet is synchronouslyrotating and ice-covered except for open ocean at the sub-stellarpoint (Pierrehumbert 2011). On the other hand, the lower albedoof ice in the IR may make near-global ice coverage di ffi cult(Joshi & Haberle 2012; Shields et al. 2013).Planet e receives only a third the radiation the Earth does, andlies close to the maximum greenhouse limit. We therefore expecta habitable version of this planet to have > . Theplanet might not be tidally locked, and may have an obliquitythat evolves significantly due to perturbations from other plan-ets. From this perspective planet e might be the most Earth-like,experiencing a day-night cycle and seasons.Finally planet d is unlikely to be in the habitable zone, but itcould potentially support sub-surface life. Internal energy gen-erated by, e.g. , radiogenic heat could support liquid water be-low an ice layer, similar to Europa. Presumably the biologicallygenerated gases could find their way through the ice and be-come detectable bio-signatures, but they might be a very smallconstituent of a large atmosphere, hampering remote detection.While its transit probability is rather low ( ∼ ∼
40 milliarcseconds. Thisvalue is the baseline inner working angle for the Darwin / ESAhigh-contrast mission being considered by ESA (Cockell et al.2009) so planet d could be a primary target for such a mission.
In addition to planets, extrasolar moons have been suggestedas hosts for life (Reynolds et al. 1987; Williams et al. 1997;Tinney et al. 2011; Heller & Barnes 2013). In order to sustaina substantial, long-lived atmosphere under an Earth-like amountof stellar irradiation (Williams et al. 1997; Kaltenegger 2000),to drive a magnetic field over billions of years (Tachinami et al.2011), and to drive tectonic activity, Heller & Barnes (2013)concluded that a satellite in the stellar HZ needs a mass ! . M ⊕ in order to maintain liquid surface water. Fig. 15.
Liquid water habitable zone of GJ 667C with the proposed seven candidates as estimated using the updated relations inKopparapu et al. (2013). Three of the reported planets lie within the HZ. The newly reported planets f and e are the most comfortablylocated within it. The inner edge is set by the moist greenhouse runaway limit and the outer edge is set by the snow ball runawaylimit. The empirical limits set by a recent uninhabitable Venus and an early habitable Mars are marked in brown solid lines. Thepresence of clouds of water (inner edge) or CO (outer edge) might allow habitable conditions on planets slightly outside thenominal definition of the habitable zone (Selsis et al. 2007).If potential moons around planets GJ 667C c, f, or e formedin the circumplanetary disk, then they will be much less mas-sive than the most massive satellites in the Solar System(Canup & Ward 2006) and thus not be habitable. However, if oneof those planets is indeed terrestrial then impacts could have cre-ated a massive moon as happened on Earth (Cameron & Ward1976). Further possibilities for the formation of massive satel-lites are summarized in Heller & Barnes (2013, Sect. 2.1).As the stellar HZ of GJ 667C is very close to this M dwarfstar, moons of planets in the habitable zone would have slightlyeccentric orbits due to stellar perturbations. These perturbationsinduce tidal heating and they could be strong enough to preventany moon from being habitable (Heller 2012). Moons aroundplanet d, which orbits slightly outside the stellar HZ, could o ff era more benign environment to life than the planet itself, if theyexperience weak tidal heating of, say, a few watts per square me-ter (see Jupiter’s moon Io, Reynolds et al. 1987; Spencer et al.2000).Unless some of these planets are found to transit, there isno currently available technique to identify satellites (Kipping2009; Kipping et al. 2012). The RV technique is only sensitiveto the combined mass of a planet plus its satellites so it might bepossible that some of the planets could be somewhat lighter– buthost a massive moon.
10. Conclusions
We describe and report the statistical methods and tests usedto detect up to seven planet candidates around GJ 667C us-ing Doppler spectroscopy. The detection of the first five plan-ets is very robust and independent of any prior choice. Inaddition to the first two already reported ones (b and cAnglada-Escud´e & Butler 2012; Delfosse et al. 2012) we showthat the third planet also proposed in those papers (planet d)is much better explained by a Keplerian orbit rather than anactivity-induced periodicity. The next two confidently detectedsignals (e and f) both correspond to small super-Earth mass ob-jects with most likely periods of 62 and 39 days. The detectionof the 6th planet is weakly dependent on the prior choice of the orbital eccentricity. The statistical evidence for the 7th candidate(planet h) is tentative and requires further Doppler follow-up forconfirmation. Gregory (2012) proposed a solution for the sys-tem with similar characteristics to the one we present here buthad fundamental di ff erences. In particular, he also identified thefirst five stronger signals but his six-planet solution also includeda candidate periodicity at 30 days- which would be dynamicallyunstable- and activity was associated to the signal at 53 dayswithout further discussion or verification. The di ff erence in ourconclusions are due to a slightly di ff erent choice of priors (es-pecially on the eccentricity), more data was used in our analysis-only HARPS-CCF data was used by Gregory (2012)-, and weperformed a more thorough investigation of possible activity-related periodicities.Numerical integration of orbits compatible with the posteriordensity distributions show that there is a subset of configurationsthat allow long-term stable configurations. Except for planetsb and g, the condition of dynamical stability dramatically af-fects the distribution of allowed eccentricities indicating that thelower mass planet candidates (c, e, f) must have quasi-circularorbits. A system of six planets is rather complex in terms ofstabilizing mechanisms and possible mean-motion resonances.Nonetheless, we identified that the physically allowed configu-rations are those that avoid transient 3:1 MMR between planetsd and g. We also found that the most stable orbital solutions arewell described by the theory of secular frequencies (Laplace-Lagrange solution). We investigated if the inclusion of a seventhplanet system was dynamically feasible in the region disclosedby the Bayesian samplings. It is notable that this preliminarycandidate appears around the center of the region of stability.Additional data should be able to confirm this signal and pro-vide detectability for longer period signals.The closely packed dynamics keeps the eccentricities smallbut non-negligible for the lifetime of the system. As a result, po-tential habitability of the candidates must account for tidal dis-sipation e ff ects among others. Dynamics essentially a ff ect 1) thetotal energy budget at the surface of the planet (tidal heating),2) synchronization of the rotation with the orbit (tidal locking),and 3) the timescales for the erosion of their obliquities. These dynamical constraints, as well as predictions for potentially hab-itable super-Earths around M dwarf stars, suggest that at leastthree planet candidates (planets c, e and f) could have remainedhabitable for the current life-span of the star. Assuming a rockycomposition, planet d lies slightly outside the cold edge of thestellar HZ. Still, given the uncertainties in the planet parametersand in the assumptions in the climatic models, its potential hab-itability cannot be ruled out (e.g., ocean of liquid water undera thick ice crust, or presence of some strong green-house e ff ectgas).One of the main results of the Kepler mission is that high-multiplicity systems of dynamically-packed super-Earths arequite common around G and K dwarfs (Fabrycky et al. 2012).The putative existence of these kinds of compact systems aroundM-dwarfs, combined with a closer-in habitable zone, suggeststhe existence of a numerous population of planetary systemswith several potentially-habitable worlds each. GJ 667C is likelyto be among first of many of such systems that may be discov-ered in the coming years. Acknowledgements.
We acknowledge the constructive discussions with the ref-erees of this manuscript. The robustness and confidence of the result greatlyimproved thanks to such discussions. G. Anglada-Escud´e is supported bythe German Federal Ministry of Education and Research under 05A11MG3.M. Tuomi acknowledges D. Pinfield and RoPACS (Rocky Planets AroundCool Stars), a Marie Curie Initial Training Network funded by the EuropeanCommission’s Seventh Framework Programme. E. Gerlach would like to ac-knowledge the financial support from the DFG research unit FOR584. R. Barnesis supported by NASA’s Virtual Planetary Laboratory under CooperativeAgreement Number NNH05ZDA001C and NSF grant AST-1108882. R. Hellerreceives funding from the Deutsche Forschungsgemeinschaft (reference num-ber scho394 / / ff orts of thePFS / Magellan team in obtaining Doppler measurements. We thank Sandy Keiserfor her e ffi cient setup of the computer network at Carnegie / DTM. We thank DanFabrycky, Aviv Ofir, Mathias Zechmeister and Denis Shulyak for useful and con-structive discussions. This research made use of the Magny Cours Cluster hostedby the GWDG, which is managed by Georg August University G¨ottingen andthe Max Planck Society. This research has made extensive use of the SIMBADdatabase, operated at CDS, Strasbourg, France; and NASA’s Astrophysics DataSystem. The authors acknowledge the significant e ff orts of the HARPS-ESOteam in improving the instrument and its data reduction software that made thiswork possible. We also acknowledge the e ff orts of the teams and individual ob-servers that have been involved in observing the target star with HARPS / ESO,HIRES / Keck, PFS / Magellan and UVES / ESO.
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Table A.1.
Reference prior probability densities and ranges ofthe model parameters.
Parameter π ( θ ) Interval Hyper-parametervalues K Uniform [0 , K max ] K max = − ω Uniform [0, 2 π ] – e ∝ N (0 , σ e ) [0,1] σ e = . M Uniform [0, 2 π ] – σ J Uniform [0 , K max ] (*) γ Uniform [ − K max , K max ] (*) φ Uniform [-1, 1]log P Uniform [log P , log P max ] P = . P max = Notes. * Same K max as for the K parameter in first row. Appendix A: Priors
The choice of uninformative and adequate priors plays a cen-tral role in Bayesian statistics. More classic methods, such asweighted least-squares solvers, can be derived from Bayes theo-rem by assuming uniform prior distributions for all free parame-ters. Using the definition of Eq. 3, one can note that, under coor-dinate transformations in the parameter space (e.g., change from P to P − as the free parameter) the posterior probability distri-bution will change its shape through the Jacobian determinant ofthis transformation. This means that the posterior distributionsare substantially di ff erent under changes of parameterizationsand, even in the case of least-square statistics, one must be veryaware of the prior choices made (explicit, or implicit throughthe choice of parameterization). This discussion is addressedin more detail in Tuomi & Anglada-Escude (2013). For theDoppler data of GJ 667C, our reference prior choices are sum-marized in Table A.1. The basic rationale on each prior choicecan also be found in Tuomi (2012), Anglada-Escud´e & Tuomi(2012) and Tuomi & Anglada-Escude (2013). The prior choicefor the eccentricity can be decisive in detection of weak signals.Our choice for this prior ( N (0 , σ e )) is justified using statistical,dynamical and population arguments. A.1. Eccentricity prior : statistical argument
Our first argument is based on statistical considerations to min-imize the risk of false positives. That is, since e is a stronglynon-linear parameter in the Keplerian model of Doppler signals(especially if e > − jitter level), and search for the maximum likeli-hood solution using the log–L periodograms approach (assum-ing a fully Keplerian solution at the period search level, seeSection 4.2). Although no signal is injected, solutions with aformal false-alarm probability (FAP) smaller than 1% are foundin 20% of the sample. On the contrary, our log–L periodogramsearch for circular orbits found 1.2% of false positives, matchingthe expectations given the imposed 1% threshold. We performedan additional test to assess the impact of the eccentricity prior onthe detection completeness. That is, we injected one Kepleriansignal ( e = .
8) at the same observing epochs with amplitudes of 1.0 m s − and white Gaussian noise of 1 m s − . We then per-formed the log–L periodogram search on a large number of thesedatasets (10 ). When the search model was allowed to be a fullyKeplerian orbit, the correct solution was only recovered 2.5% ofthe time, and no signals at the right period were spotted assuminga circular orbit. With a K = . − , the situation improvedand 60% of the orbits were identified in the full Keplerian case,against 40% of them in the purely circular one. More tests areillustrated in the left panel of Fig. A.1. When an eccentricity of0.4 and K = − signal was injected, the completeness of thefully Keplerian search increased to 91% and the completenessof the circular orbit search increased to 80%. When a K >
2m s − signal was injected, the orbits were identified in >
99% ofthe cases. We also obtained a histogram of the semi-amplitudesof the false positive detections obtained when no signal was in-jected. The histogram shows that these amplitudes were smallerthan 1.0 m s − with a 99% confidence level (see right panel ofFig. A.1). This illustrates that statistical tests based on pointestimates below the noise level do not produce reliable assess-ments on the significance of a fully Keplerian signal. For a givendataset, information based on simulations (e.g., Fig. A.1) or aphysically motivated prior is necessary to correct such detectionbias (Zakamska et al. 2011).In summary, while a uniform prior on eccentricity onlylooses a few very eccentric solutions in the low amplituderegime, the rate of false positive detections ( ∼ missed if the search is done assuming strictlycircular orbits. This implies that, for log–likelihood periodogramsearches, circular orbits should always be assumed when search-ing for a new low-amplitude signals and that, when approach-ing amplitudes comparable to the measurement uncertainties,assuming circular orbits is a reasonable strategy to avoid falsepositives. In a Bayesian sense, this means that we have to bevery skeptic of highly eccentric orbits when searching for sig-nals close to the noise level. The natural way to address thisself-consistently is by imposing a prior on the eccentricity thatsuppresses the likelihood of highly eccentric orbits. The log–Likelihood periodograms indicate that the strongest possibleprior (force e = π ( e ) = δ (0))and a bit more constraining than a uniform prior ( π ( e ) =
1) canprovide a more optimal compromise between sensitivity and ro-bustness. Our particular functional choice of the prior (Gaussiandistribution with zero-mean and σ = .
3) is based on dynamicaland population analysis considerations.
A.2. Eccentricity prior : dynamical argument
From a physical point of view, we expect a priori that eccentric-ities closer to zero are more probable than those close to unitywhen multiple planets are involved. That is, when one or twoplanets are clearly present (e.g. GJ 667Cb and GJ 667Cc aresolidly detected even with a flat prior in e ), high eccentricities inthe remaining lower amplitude candidates would correspond tounstable and therefore physically impossible systems.Our prior for e takes this feature into account (reducing thelikelihood of highly eccentric solutions) but still allows high ec-centricities if the data insists so (Tuomi 2012). At the samplinglevel, the only orbital configurations that we explicitly forbidis that we do not allow solutions with orbital crossings. Whilea rather weak limitation, this requirement essentially removesall extremely eccentric multiplanet solutions ( e > Injected amplitude [m s -1 ] C o m p l e t e n ess ( i n % ) Injected e=0.8, Keplerian searchInjected e=0.8, circular searchInjected e=0.4, Keplerian searchInjected e=0.4, circular search
Recovered K from false positives [m s -1 ] C u m u l a t i ve p r ob a b ili t y e>0.0e=0 positiveFalseratio Fig. A.1.
Left panel. Detection completeness as a function of the injected signal using a fully Keplerian search versus a circularorbit search. Red and blue lines are for an injected eccentricity of 0.8, and the brown and purple ones are for injected signals witheccentricity of 0.4. Black horizontal lines on the left show the fraction of false-positive detections satisfying the FAP threshold of1% using both methods (Keplerian versus circular). While the completeness is slightly enhanced in the low K regime, the fractionof false positives is unacceptable and, therefore, the implicit assumptions of the method (e.g., uniform prior for e ) are not correct. Right panel . Distribution of semi-amplitudes K for these false positive detections. Given that the injected noise level is 1.4 m s − (1m s − nominal uncertainty, 1 m s − jitter), it is clear that signals detected with fully Keplerian log–L periodograms with K belowthe noise level cannot be trusted.move orbital configurations with close encounters between plan-ets, and the solutions we receive still have to be analyzed by nu-merical integration to make sure that they correspond to stablesystems. As shown in Section 8, posterior numerical integrationof the samplings illustrate that our prior function was, after all,rather conservative. A.3. Eccentricity prior : populationargument
To investigate how realistic our prior choice is compared to thestatistical properties of the known exoplanet populations, weobtained the parameters of all planet candidates with M sin i smaller than 0.1 M jup as listed in The Extrasolar PlanetsEncyclopaedia as at 2012 December 1. We then produceda histogram in eccentricity bins of 0 .
1. The obtained distribu-tion follows very nicely a Gaussian function with zero meanand σ e = .
2, meaning that our prior choice is more unin-formative (and therefore, more conservative) than the currentdistribution of detections. This issue is the central topic ofTuomi & Anglada-Escude (2013), and a more detailed discus-sion (plus some illustrative plots) can be found in there.
Appendix B: Detailed Bayesian detectionsequences
In this section, we provide a more detailed description of detec-tions of seven signals in the combined HARPS-TERRA, PFS,and HIRES data. We also show that the same seven signals (withsome small di ff erences due to aliases) are also detected indepen-dently when using HARPS-CCF velocities instead of HARPSTERRA ones. The PFS and HIRES measurements used are againthose provided in Anglada-Escud´e & Butler (2012). B.1. HARPS-CCF,PFS andHIRES
First, we perform a detailed analysis with the CCF values pro-vided by Delfosse et al. (2012) in combination with the PFS and http:exoplanet.eu HIRES velocities. When increasing the number of planets, pa-rameter k , in our statistical model, we were able to determinethe relative probabilities of models with k = , , , ... rather eas-ily. The parameter spaces of all models could be sampled withour Markov chains relatively rapidly and the parameters of thesignals converged to the same periodicities and RV amplitudesregardless of the exact initial states of the parameter vectors.Unlike in Anglada-Escud´e et al. (2012) and Delfosse et al.(2012), we observed immediately that k = ff ect its significance (seealso Sec. 6).We could identify three more signals in the data at 39, 53,and 260 days with low RV amplitudes of 1.31, 0.96, and 0.97ms − , respectively. The 53-day signal had a strong alias at 62days and so we treated these as alternative models and calculatedtheir probabilities as well. The inclusion of k = k = k = ff er significantly from Table B.1.
Relative posterior probabilities and log-Bayes factors of models M k with k Keplerian signals derived from the combinedHARPS-CCF, HIRES, and PFS RV data on the left and HARPS-TERRA HIRES, PFS on the right. Factor ∆ indicates how muchthe probability increases with respect to the best model with one less Keplerian signal. P s denotes the MAP period estimate of thesignal added to the solution when increasing the number k . For k =
4, 5, 6, and 7, we denote all the signals on top of the three mostsignificant ones at 7.2, 28, and 91 days because the 53 and 62 day periods are each other’s yearly aliases and the relative significanceof these two and the signal with a period of 39 days are rather similar.
HARPS-CCF, PFS, HIRES HARPS-TERRA, PFS, HIRES k P ( M k | d ) ∆ log P ( d |M k ) P s [days] P ( M k | d ) ∆ log P ( d |M k ) P s [days]0 2.2 × − – -629.1 – 2.7 × − – -602.1 –1 2.4 × − × -550.0 7.2 3.4 × − × -516.0 7.22 1.3 × − × -526.9 28 1.3 × − × -486.3 913 8.7 × − × -503.6 91 8.9 × − × -444.5 284 5.1 × − × -494.2 39 1.5 × − × -436.4 394 1.0 × − × -488.9 53 1.9 × − × -436.2 534 2.0 × − × -495.2 62 1.2 × − × -436.7 625 8.0 × − × -474.7 39, 53 1.0 × − × -420.0 39, 535 5.4 × − × -482.0 39, 62 1.0 × − × -422.3 39, 626 3.4 × − × -463.3 39, 53, 256 4.1 × − × -408.7 39, 53, 2566 1.3 × −
16 -471.2 39, 62, 256 4.1 × − × -411.0 39, 62, 2567 0.998 2.9 × -454.6 17, 39, 53, 256 0.057 14 -405.4 17, 39, 53, 2567 1.5 × − Table B.2.
Seven-Keplerian solution of the combined RVs of GJ 667C with HARPS-CCF data. MAP estimates of the parametersand their 99% BCSs. The corresponding solution derived from HARPS-TERRA data is given in Table 4. Note that each datasetprefers a di ff erent alias for planet f (53 versus 62 days). Parameter b h c f P [days] 7.1998 [7.1977, 7.2015] 16.955 [16.903, 17.011] 28.147 [28.084, 28.204] 39.083 [38.892, 39.293] e K [ms − ] 3.90 [3.39, 4.37] 0.80 [0.20, 1.34] 1.60 [1.09, 2.17] 1.31 [0.78, 1.85] ω [rad] 0.2 [0, 2 π ] 2.3 [0, 2 π ] 2.3 [0, 2 π ] 3.6 [0, 2 π ] M [rad] 3.2 [0, 2 π ] 6.0 [0, 2 π ] 2.9 [0, 2 π ] 2.8 [0, 2 π ]e d g P [days] 53.19 [52.73, 53.64] 91.45 [90.81, 92.23] 256.4 [248.6, 265.8] e K [ms − ] 0.96 [0.48, 1.49] 1.56 [1.11, 2.06] 0.97 [0.41, 1.53] ω [rad] 0.8 [0, 2 π ] 3.0 [0, 2 π ] 6.2 [0, 2 π ] M [rad] 5.9 [0, 2 π ] 5.4 [0, 2 π ] 1.0 [0, 2 π ] γ [ms − ] (HARPS) -32.6 [-37.3, -28.2] γ [ms − ] (HIRES) -33.3 [-38.9,, -28.2] γ [ms − ] (PFS) -27.7 [-31.0, -24.0]˙ γ [ms − a − ] 2.19 [1.90, 2.48] σ J , [ms − ] (HARPS) 0.80 [0.20, 1.29] σ J , [ms − ] (HIRES) 2.08 [0.54, 4.15] σ J , [ms − ] (PFS) 1.96 [0.00, 4.96] zero. The probability of the model with k = k = k =
7. The correspondingorbital parameters of these seven Keplerian signals are shown inTable B.2. Whether there is a weak signal at roughly 1200-2000days or not remains to be seen when future data become avail-able. The naming of the seven candidate planets (b to h) followsthe significance of detection.
B.2. HARPS-TERRA,PFS andHIRES(reference solution)
The HARPS-TERRA velocities combined with PFS and HIRESvelocities contained the signals of GJ 667C b, c, and d veryclearly. We could extract these signals from the data with easeand obtained estimates for orbital periods that were consistentwith the estimates obtained using the CCF data in the previ- ous subsection. Unlike for the HARPS-CCF data, however, the91 day signal was more significantly present in the HARPS-TERRA data and it corresponded to the second most signifi-cant periodicity instead of the 28 day one. Also, increasing k improved the statistical model significantly and we could againdetect all the additional signals in the RVs.As for the CCF data, we branched the best fit solution intothe two preferred periods for the planet e (53 /
62 days). The solu-tion and model probabilities are listed on the right-side of TableB.1. The only significant di ff erence compared to the HARPS-CCF one is that the 62-day period for planet e is now preferred.Still, the model with 53 days is only seventeen times less proba-ble than the one with a 62 days signal, so we cannot confidentlyrule out that the 53 days one is the real one. For simplicity andto avoid confusion, we use the 62-day signal in our referencesolution and is the one used to analyze dynamical stability andhabitability for the system. As an additional check, preliminary dynamical analysis of solutions with a period of 53 days forplanet e showed very similar behaviour to the reference solutionin terms of long-term stability (similar fraction of dynamicallystable orbits and similar distribution for the D chaos indices).Finally, we made several e ff orts at sampling the eight-Keplerianmodel with di ff erent initial states. As for the CCF data, there arehints of a signal in the ∼ Appendix C: Further activity-related tests
C.1. Correlatednoise
The possible e ff ect of correlated noise must be investigated to-gether with the significance of the reported detections (e.g. GJ581; Baluev 2012; Tuomi & Jenkins 2012). We studied the im-pact of correlated noise by adding a first order Moving Averageterm (MA(1), see Tuomi et al. 2012) to the model in Eq. 1 andrepeated the Bayesian search for all the signals. The MA(1)model proposed in (Tuomi et al. 2012) contains two free pa-rameters: a characteristic time-scale τ and a correlation coe ffi -cient c . Even for the HARPS data set with the greatest num-ber of measurements, the characteristic time-scale could notbe constrained. Similarly, the correlation coe ffi cient (see e.g.Tuomi et al. 2013, 2012) had a posteriori density that was notdi ff erent from its prior, i.e. it was found to be roughly uniformin the interval [-1,1], which is a natural range for this parameterbecause it makes the corresponding MA model stationary. Whilethe k = C.2. Includingactivity correlationin themodel
Because the HARPS activity-indices (S-index, FWHM, andBIS) were available, we analyzed the data by taking into accountpossible correlations with any of these indices. We added an ex-tra component into the statistical model taking into account lin-ear correlation with each of these parameters as F ( t i , C ) = C z i (see Eq. 1), where z i is the any of the three indices at epoch t i .In Section 6, we found that the log–L of the solution forplanet d slightly improved when adding the correlation term.The slight improvement of the likelihood is compatible with aconsistently positive estimate of C for both FWHM and the S-index obtained with the MC samplings (see Fig. C.1, for an ex-ample distribution of C S − index as obtained with a k = ffi cient was always within the95% credibility interval. Moreover, we found that the integratedmodel probabilities decreased when compared to the model withthe same number of planets but no correlation term included.This means that such models are over-parameterized and, there-fore, they are penalized by the principle of parsimony. C.3. Wavelength dependenceofthe signals
Stellar activity might cause spurious Doppler variability thatis wavelength dependent (e.g., cool spots). Using the HARPS-TERRA software on the HARPS spectra only, we extracted one
Fig. C.1.
Value of the linear correlation parameter of the S-index ( C S ) with the radial velocity data for a model with k = N obs = × = ff ects (unknown at this level of pre-cision), each echelle aperture can be treated as an independentinstrument. Therefore, the reference velocities γ l and jitter terms σ l of each aperture were considered as free parameters. To as-sess the wavelength dependence of each signal, the Keplerianmodel of the i-th planet (one planet at a time) also containedone semi-amplitude K i , l per echelle aperture and all the other pa-rameters ( e i , ω i , M , i and P i ) were common to all apertures. Theresulting statistical model has 72 × + × k − k Keplerian signals are included and one is investigated (250free parameters for k = K of each signal measuredas a function of wavelength. Plotting this K against central wave-length of each echelle order enabled us to visually assess ifsignals had strong wavelength dependencies (i.e. whether therewere signals only present in a few echelle orders) and check ifthe obtained solution was consistent with the one derived fromthe combined Doppler measurements. By visual inspection andapplying basic statistical tests, we observed that the scatter inthe amplitudes for all the candidates was consistent within theerror bars and no significant systematic trend (e.g. increasing K towards the blue or the red) was found in any case. Also, theweighted means of the derived amplitudes were fully consistentwith the values in Table 4. We are developing a more quantitativeversion of these tests by studying reported activity-induced sig-nals on a larger sample of stars. As examples of low–amplitudewavelength-dependent signals ruled out using similar tests inthe HARPS wavelength range see : Tuomi et al. (2013) on HD40307 (K3V), Anglada-Escud´e & Butler (2012) on HD 69830(G8V) and Reiners et al. (2013) on the very active M dwarf ADLeo (M3V). Table C.2.
HARPS-TERRA Doppler measurements of GJ 667C. Measurements are in the barycenter of the Solar System and corrected forperspective acceleration. The median has been velocity has been subtracted for cosmetic purposes. Nominal uncertainties in FWHM and BIS are2.0 and 2.35 times the corresponding σ ccf (see Section 4.5 in Zechmeister et al. 2013). No consistent CCF measurement could be obtained forthe last two spectra because of conflicting HARPS-DRS software versions (di ff erent binary masks) used to produce them. Except for those twospectra and according to the ESO archive documentation, all CCF measurements were generated using the default M2 binary mask.BJD RV TERRA σ TERRA RV ccf σ ccf FWHM BIS S-index σ S (days) (m s − ) (m s − ) (m s − ) (m s − ) (km s − ) (m s − ) (–) (–)2453158.764366 -3.10 0.95 -3.11 1.05 3.0514 -7.93 0.4667 0.00952453201.586794 -11.88 1.25 -11.8 1.09 3.0666 -9.61 0.4119 0.00742453511.798846 -7.61 0.89 -9.22 1.07 3.0742 -7.42 0.5915 0.00882453520.781048 -3.92 1.17 -0.37 1.23 3.0701 -11.99 0.4547 0.00822453783.863348 0.25 0.61 0.34 0.65 3.0743 -13.31 0.4245 0.00532453810.852282 -3.48 0.55 -3.00 0.54 3.0689 -10.62 0.4233 0.00442453811.891816 2.20 1.08 0.24 1.02 3.0700 -9.37 0.4221 0.00662453812.865858 -0.34 0.71 -0.56 0.72 3.0716 -9.78 0.4125 0.00542453814.849082 -10.16 0.49 -10.06 0.47 3.0697 -10.63 0.4848 0.00422453816.857459 -9.15 0.52 -9.89 0.65 3.0698 -12.20 0.4205 0.00512453830.860468 -6.96 0.56 -7.29 0.59 3.0694 -11.59 0.4729 0.00522453832.903068 -0.49 0.64 -0.35 0.68 3.0706 -13.33 0.4930 0.00582453834.884977 -1.50 0.72 -1.68 0.57 3.0734 -8.20 0.4456 0.00492453836.887788 -6.99 0.48 -6.24 0.48 3.0723 -8.27 0.4864 0.00442453861.796371 6.38 0.59 7.84 0.59 3.0780 -11.47 0.6347 0.00602453862.772051 6.69 0.76 8.00 0.74 3.0768 -12.54 0.5534 0.00652453863.797178 4.57 0.59 4.58 0.56 3.0759 -10.71 0.4891 0.00512453864.753954 1.21 0.68 2.52 0.65 3.0783 -9.21 0.4854 0.00552453865.785606 -1.85 0.61 -2.55 0.55 3.0752 -7.73 0.4815 0.00502453866.743120 -1.36 0.58 -2.32 0.49 3.0770 -7.49 0.5277 0.00452453867.835652 -0.48 0.66 -0.05 0.64 3.0816 -10.55 0.4708 0.00552453868.813512 2.34 0.56 0.62 0.61 3.0754 -10.01 0.4641 0.00532453869.789495 3.85 0.63 4.73 0.65 3.0795 -12.71 0.4837 0.00552453870.810097 2.37 0.88 2.82 0.81 3.0813 -10.48 0.4567 0.00622453871.815952 -1.11 0.61 -3.03 0.81 3.0790 -9.16 0.5244 0.00682453882.732970 -2.96 0.52 -4.17 0.51 3.0795 -8.09 0.5121 0.00472453886.703550 -4.54 0.58 -3.78 0.48 3.0757 -10.11 0.4607 0.00422453887.773514 -5.97 0.48 -3.98 0.44 3.0700 -10.94 0.4490 0.00412453917.737524 -4.12 0.88 -2.44 1.14 3.0666 -10.91 0.5176 0.00842453919.712544 0.98 0.99 0.69 1.17 3.0774 -8.01 0.4324 0.00732453921.615825 -1.67 0.49 -1.24 0.51 3.0671 -9.87 0.4305 0.00432453944.566259 -2.02 0.98 -2.16 1.00 3.0776 -9.25 0.6143 0.00792453947.578821 3.89 1.68 5.83 2.43 3.0806 -8.54 0.7079 0.01342453950.601834 -1.01 0.89 1.65 0.92 3.0780 -11.80 0.5612 0.00712453976.497106 2.40 0.61 3.52 0.60 3.0791 -12.74 0.5365 0.00542453979.594316 -2.67 0.95 -0.48 1.19 3.0776 -9.20 0.5517 0.00912453981.555311 -4.77 0.64 -4.29 0.57 3.0749 -13.12 0.5339 0.00552453982.526504 -4.36 0.81 -2.88 0.69 3.0717 -11.84 0.4953 0.00612454167.866839 -1.87 0.62 -2.51 0.61 3.0798 -10.14 0.5141 0.00532454169.864835 -0.10 0.59 -0.04 0.63 3.0793 -11.94 0.4729 0.00522454171.876906 5.17 0.71 6.08 0.58 3.0744 -7.24 0.4893 0.00502454173.856452 -1.18 0.83 -1.44 0.61 3.0746 -10.33 0.4809 0.00522454194.847290 1.27 0.59 0.85 0.69 3.0756 -8.43 0.4586 0.00542454196.819157 -3.57 0.79 -3.06 0.79 3.0759 -12.33 0.4809 0.00612454197.797125 -3.83 0.86 -4.71 0.97 3.0726 -9.12 0.4584 0.00692454198.803823 -4.06 0.76 -4.99 0.79 3.0708 -9.33 0.5685 0.00682454199.854238 0.18 0.55 0.97 0.51 3.0714 -10.66 0.4652 0.00442454200.815699 1.30 0.60 2.55 0.57 3.0708 -10.26 0.4468 0.00472454201.918397 0.54 0.79 2.31 0.63 3.0681 -11.27 0.4690 0.00562454202.802697 -2.96 0.69 -3.23 0.66 3.0696 -8.49 0.4954 0.00562454227.831743 -1.26 0.84 0.47 0.95 3.0619 -9.96 0.4819 0.00712454228.805860 3.35 0.68 5.19 0.65 3.0651 -15.03 0.4603 0.00552454229.773888 7.44 1.29 7.23 1.28 3.0708 -6.34 0.5213 0.00822454230.845843 1.51 0.58 1.97 0.62 3.0631 -8.92 0.4409 0.00532454231.801726 -0.57 0.62 -1.15 0.55 3.0704 -8.86 0.5993 0.00552454232.721251 -0.63 1.15 -2.17 1.41 3.0719 -9.70 0.3737 0.00792454233.910349 -1.27 1.29 -2.10 1.68 3.0687 -12.12 0.5629 0.01122454234.790981 -1.89 0.74 -1.48 0.66 3.0672 -8.39 1.2169 0.00932454253.728334 0.99 0.79 1.65 0.84 3.0773 -10.30 0.4509 0.00622454254.755898 -2.64 0.54 -3.25 0.52 3.0779 -7.99 0.4426 0.00462454255.709350 -2.92 0.74 -2.83 0.72 3.0775 -7.36 0.4829 0.005926uillem Anglada-Escud´e et al.: Three HZ super-Earths in a seven-planet system Table C.2. continued.BJD RV
TERRA σ TERRA RV ccf σ ccf FWHM BIS S-index σ S (days) (m s − ) (m s − ) (m s − ) (m s − ) (km s − ) (m s − ) (–) (–)2454256.697674 -0.21 0.97 -0.45 0.84 3.0775 -9.19 0.4608 0.00632454257.704446 2.93 0.66 2.39 0.70 3.0766 -11.09 0.4549 0.00552454258.698322 4.19 0.83 5.19 0.63 3.0799 -9.57 0.4760 0.00522454291.675565 -5.58 1.16 -4.45 1.35 3.0802 -9.95 0.4298 0.00862454292.655662 -4.37 0.75 -1.25 0.76 3.0820 -11.83 0.4487 0.00562454293.708786 0.89 0.63 2.84 0.59 3.0732 -11.52 0.5344 0.00562454295.628628 3.05 0.92 3.67 1.03 3.0786 -6.85 0.4975 0.00722454296.670395 -4.68 0.75 -3.99 0.74 3.0703 -7.79 0.5453 0.00672454297.631678 -5.53 0.63 -4.81 0.55 3.0725 -10.38 0.5212 0.00532454298.654206 -5.39 0.67 -6.73 0.71 3.0743 -5.18 0.5718 0.00662454299.678909 -1.46 0.85 -2.26 0.92 3.0785 -6.46 0.5299 0.00702454300.764649 0.14 0.74 -0.07 0.63 3.0693 -12.07 0.4803 0.00572454314.691809 -0.53 1.88 -2.89 2.22 3.0756 -12.48 0.3823 0.01022454315.637551 3.41 1.12 2.31 1.47 3.0701 -10.42 0.4835 0.00912454316.554926 5.78 0.96 6.61 1.12 3.0746 -6.02 0.4402 0.00692454319.604048 -6.64 0.79 -7.01 0.59 3.0694 -7.54 0.4643 0.00522454320.616852 -5.58 0.65 -6.49 0.69 3.0698 -3.94 0.4611 0.00572454340.596942 -1.52 0.60 -0.55 0.55 3.0691 -10.11 0.4480 0.00482454342.531820 -2.39 0.66 -1.74 0.54 3.0667 -9.95 0.4573 0.00482454343.530662 0.55 0.64 1.39 0.61 3.0669 -7.25 0.4900 0.00552454346.551084 -0.17 1.01 -0.82 1.14 3.0677 -5.48 0.5628 0.00862454349.569500 -5.24 0.65 -4.02 0.77 3.0658 -11.12 0.3809 0.00582454522.886464 -1.68 0.70 -1.11 0.61 3.0688 -9.85 0.5582 0.00562454524.883089 4.38 0.69 3.05 0.69 3.0668 -8.66 0.4779 0.00572454525.892144 1.96 0.72 0.69 0.58 3.0692 -8.77 0.4202 0.00472454526.871196 -1.08 0.54 0.30 0.52 3.0717 -9.58 0.4898 0.00462454527.897962 -2.69 0.64 -3.31 0.65 3.0689 -8.46 0.4406 0.00522454528.903672 -2.80 0.71 -5.04 0.74 3.0679 -8.40 0.4666 0.00582454529.869217 0.48 0.63 -0.10 0.62 3.0664 -8.23 0.4255 0.00502454530.878876 1.40 0.68 1.38 0.53 3.0667 -7.31 0.4331 0.00442454550.901932 -6.95 0.70 -6.46 0.58 3.0680 -5.44 0.4330 0.00472454551.868783 -3.73 0.65 -3.44 0.53 3.0654 -7.36 0.4287 0.00452454552.880221 0.24 0.59 -0.25 0.50 3.0665 -9.12 0.4342 0.00422454554.846366 2.14 0.57 1.73 0.68 3.0699 -5.49 0.4116 0.00522454555.870790 -2.84 0.58 -2.26 0.58 3.0663 -7.66 0.4704 0.00502454556.838936 -4.14 0.59 -3.47 0.51 3.0686 -11.20 0.4261 0.00432454557.804592 -4.56 0.66 -4.37 0.60 3.0650 -8.87 0.4306 0.00492454562.905075 0.67 0.70 0.80 0.57 3.0668 -7.11 0.4709 0.00512454563.898808 -1.37 0.71 -1.39 0.53 3.0656 -11.93 0.4127 0.00462454564.895759 -2.63 0.85 -2.82 0.71 3.0680 -8.67 0.5068 0.00612454568.891702 3.27 0.87 4.85 1.02 3.0735 -11.20 0.4682 0.00692454569.881078 -0.46 0.83 0.46 0.78 3.0720 -16.02 0.4939 0.00612454570.870766 -1.70 0.72 -1.21 0.88 3.0715 -10.68 0.4606 0.00632454583.933324 0.44 1.00 1.56 1.11 3.0711 -17.51 0.5177 0.00872454587.919825 -0.50 0.90 -1.42 1.10 3.0824 -7.86 0.4602 0.00782454588.909632 4.05 0.98 3.18 1.05 3.0828 -6.95 0.5501 0.00802454590.901964 4.22 0.93 4.19 0.93 3.0758 -9.05 0.4707 0.00732454591.900611 1.69 0.91 -1.27 0.96 3.0753 -7.39 0.5139 0.00752454592.897751 -2.50 0.68 -2.50 0.63 3.0757 -8.84 0.4741 0.00572454593.919961 -2.30 0.74 -2.58 0.65 3.0680 -12.41 0.5039 0.00632454610.878230 9.08 0.88 10.36 0.95 3.0671 -9.46 0.4037 0.00692454611.856581 5.49 0.56 6.40 0.54 3.0650 -8.37 0.4296 0.00502454616.841719 4.81 0.91 5.15 0.88 3.0713 -8.09 0.3999 0.00652454617.806576 8.12 0.93 7.30 1.33 3.0753 -14.38 0.4948 0.00862454618.664475 10.67 1.76 7.01 2.51 3.0854 -7.21 0.6755 0.01352454639.867730 3.14 1.06 4.26 1.10 3.0588 -8.27 0.4083 0.00832454640.723804 5.06 0.64 7.07 0.66 3.0705 -13.61 0.4387 0.00552454642.676950 -0.81 0.47 1.56 0.61 3.0704 -10.27 0.4720 0.00532454643.686130 -2.06 0.72 -4.52 0.76 3.0709 -9.26 0.4809 0.00642454644.732044 -1.19 0.46 -1.85 0.56 3.0680 -8.64 0.5097 0.00542454646.639658 5.74 1.11 5.01 0.95 3.0737 -10.14 0.4316 0.00662454647.630210 5.37 0.68 3.28 0.72 3.0693 -6.35 0.4938 0.00622454648.657090 2.58 0.92 0.96 0.94 3.0720 -8.85 0.4597 0.00682454658.650838 -4.20 0.97 -3.30 0.88 3.0714 -13.06 0.4193 0.00652454660.650214 -0.82 1.13 -0.40 1.06 3.0728 -10.20 0.4224 0.00742454661.760056 1.72 0.73 1.76 0.84 3.0737 -11.56 0.4238 0.0065 27uillem Anglada-Escud´e et al.: Three HZ super-Earths in a seven-planet system Table C.2. continued.BJD RV
TERRA σ TERRA RV ccf σ ccf FWHM BIS S-index σ S (days) (m s − ) (m s − ) (m s − ) (m s − ) (km s − ) (m s − ) (–) (–)2454662.664144 3.30 0.72 2.58 0.97 3.0713 -8.43 0.4675 0.00702454663.784376 -1.92 0.93 -1.14 0.78 3.0643 -11.52 0.3811 0.00612454664.766558 -1.00 1.51 0.0 1.85 3.0765 -9.85 0.4702 0.01062454665.774513 -1.88 0.87 -2.51 0.85 3.0695 -9.89 0.4183 0.00652454666.683607 -0.37 0.87 0.36 0.79 3.0717 -9.64 0.4098 0.00602454674.576462 4.82 1.01 6.41 1.39 3.0901 -6.38 0.4226 0.00832454677.663487 7.37 1.78 8.63 3.11 3.1226 -4.66 0.4452 0.01172454679.572671 2.94 1.26 -1.23 1.48 3.0822 -5.41 0.5622 0.01032454681.573996 2.51 0.89 2.86 1.10 3.0780 -6.26 0.4443 0.00752454701.523392 -0.50 0.68 -0.28 0.67 3.0719 -4.48 0.5141 0.00582454708.564794 -0.12 0.86 -0.67 0.79 3.0803 -12.80 0.5160 0.00622454733.487290 8.06 3.51 10.75 3.89 3.0734 -3.79 0.5017 0.01462454735.499425 0.00 1.04 -2.22 1.19 3.0720 -11.44 0.4337 0.00722454736.550865 -3.28 0.91 -4.99 1.05 3.0671 -8.70 0.4647 0.00752454746.485935 -4.49 0.58 -5.00 0.53 3.0611 -13.01 0.4259 0.00452454992.721062 6.84 0.79 7.80 0.65 3.0748 -10.71 0.4826 0.00532455053.694541 -3.20 0.84 -3.32 1.09 3.0741 -11.73 0.4427 0.00782455276.882590 0.27 0.74 1.86 0.77 3.0732 -11.03 0.4699 0.00612455278.827303 1.84 0.92 1.02 0.85 3.0760 -8.41 0.5883 0.00742455280.854800 5.26 0.76 4.41 0.87 3.0793 -12.50 0.4817 0.00652455283.868014 -0.69 0.68 -0.09 0.61 3.0793 -8.23 0.5411 0.00542455287.860052 4.99 0.72 5.42 0.64 3.0779 -11.76 0.5366 0.00562455294.882720 8.56 0.69 6.81 0.58 3.0775 -8.71 0.5201 0.00512455295.754277 10.15 1.06 8.21 0.98 3.0743 -8.94 0.5805 0.00762455297.805750 4.95 0.64 3.95 0.70 3.0779 -10.19 0.4614 0.00572455298.813775 2.52 0.75 2.95 0.67 3.0807 -7.72 0.5828 0.00612455299.785905 3.74 1.60 4.62 2.19 3.0793 -10.36 0.4187 0.01062455300.876852 5.07 0.60 5.78 0.75 3.0792 -9.53 0.5104 0.00602455301.896438 9.54 0.99 8.40 1.32 3.0774 -12.07 0.4395 0.00852455323.705436 8.56 0.86 8.82 0.78 3.0702 -7.78 0.4349 0.00672455326.717047 2.17 1.05 0.67 1.27 3.0649 -9.48 0.5955 0.01032455328.702599 1.56 1.02 1.83 1.01 3.0658 -8.48 0.5077 0.00892455335.651717 1.01 0.92 3.02 1.22 3.0593 -10.79 0.4685 0.00922455337.704618 7.58 1.03 6.13 1.24 3.0725 -11.55 0.4859 0.00902455338.649293 13.01 1.97 12.32 2.54 3.0687 -5.74 0.4969 0.01342455339.713716 6.70 1.03 5.13 1.57 3.0700 -11.07 0.4760 0.00972455341.789626 -0.40 0.63 0.01 0.71 3.0812 -11.27 0.4916 0.00612455342.720036 4.80 0.91 5.68 1.13 3.0718 -7.25 0.4674 0.00762455349.682257 6.55 0.78 4.00 0.97 3.0685 -4.609 0.4787 0.00732455352.601155 12.92 1.11 14.24 1.35 3.0700 -7.48 0.4530 0.00932455354.642822 7.52 0.60 8.38 0.65 3.0663 -10.63 0.4347 0.00572455355.576777 6.41 0.97 4.76 0.96 3.0681 -9.41 0.4278 0.00742455358.754723 8.54 1.17 10.00 1.30 3.0619 -11.04 0.3527 0.00952455359.599377 6.89 0.90 6.90 0.97 3.0724 -11.39 0.3649 0.00702455993.879754 12.28 1.04 – – – – 0.5179 0.00862455994.848576 15.43 1.30 – – – – 0.6712 0.012028 .4. RUNAWAY GREENHOUSE EFFECT ON EXOMOONS DUE TO IRRADIATION FROMHOT, YOUNG GIANT PLANETS (Heller & Barnes 2015) 244 Contribution:RH did the literature research, contributed to the mathematical framework, created Figs. 1, 4, and 5,led the writing of the manuscript, and served as a corresponding author for the journal editor and thereferees. r X i v : . [ a s t r o - ph . E P ] N ov Runaway greenhouse effect on exomoons due toirradiation from hot, young giant planets
R. Heller and R. Barnes , McMaster University, Department of Physics and Astronomy, 1280 Main Street West, Hamilton (ON) L8S 4M1, [email protected] University of Washington, Department of Astronomy, Seattle, WA 98195, [email protected] Virtual Planetary Laboratory, USAsubmitted June 15, 2013 – revision
ABSTRACT
Context.
The
Kepler space telescope has proven capable of detecting transits of objects almost as small as the Earth’s Moon. Somestudies suggest that moons as small as 0 . Kepler data by transit timing variations and transitduration variations of their host planets. If such massive moons exist around giant planets in the stellar habitable zone (HZ), then theycould serve as habitats for extraterrestrial life.
Aims.
While earlier studies on exomoon habitability assumed the host planet to be in thermal equilibrium with the absorbed stellarflux, we here extend this concept by including the planetary luminosity from evolutionary shrinking. Our aim is to assess the dangerof exomoons to be in a runaway greenhouse state due to extensive heating from the planet.
Methods.
We apply pre-computed evolution tracks for giant planets to calculate the incident planetary radiation on the moon as afunction of time. Added to the stellar flux, the total illumination yields constraints on a moon’s habitability. Ultimately, we includetidal heating to evaluate a moon’s energy budget. We use a semi-analytical formula to parametrize the critical flux for the moon toexperience a runaway greenhouse e ff ect. Results.
Planetary illumination from a 13-Jupiter-mass planet onto an Earth-sized moon at a distance of ten Jupiter radii can drivea runaway greenhouse state on the moon for about 200 Myr. When stellar illumination equivalent to that received by Earth from theSun is added, then the runaway greenhouse holds for about 500 Myr. After 1000 Myr, the planet’s habitable edge has moved inward toabout 6 Jupiter radii. Exomoons in orbits with eccentricities of 0.1 experience strong tidal heating; they must orbit a 13-Jupiter-masshost beyond 29 or 18 Jupiter radii after 100 Myr (at the inner and outer boundaries of the stellar HZ, respectively), and beyond 13Jupiter radii (in both cases) after 1000 Myr to be habitable.
Conclusions.
If a roughly Earth-sized, Earth-mass moon would be detected in orbit around a giant planet, and if the planet-moon duetwould orbit in the stellar HZ, then it will be crucial to recover the orbital history of the moon. If, for example, such a moon around a13-Jupiter-mass planet has been closer than 20 Jupiter radii to its host during the first few hundred million years at least, then it mighthave lost substantial amounts of its initial water reservoir and be uninhabitable today.
Key words.
Astrobiology – Celestial mechanics – Planets and satellites: general – Radiation mechanisms: general
1. Introduction
The advent of exoplanet science in the last two decades has ledto the compelling idea that it could be possible to detect a moonorbiting a planet outside the solar system. By observational se-lection e ff ects, such a finding would reveal a massive moon, be-cause its signature in the data would be most apparent. Whilethe most massive moon in the solar system – Jupiter’s satelliteGanymede – has a mass roughly 1 /
40 the mass of Earth ( M ⊕ ), adetectable exomoon would have at least twice the mass of Mars,that is, 1 / M ⊕ (Kipping et al. 2009). Should these relativelymassive moons exist, then they could be habitats for extrasolarlife.One possible detection method relies on measurements ofthe transit timing variations (TTV) of the host planet as it pe-riodically crosses the stellar disk (Sartoretti & Schneider 1999;Simon et al. 2007; Kipping 2009a; Lewis 2013). To ultimatelypin down a satellite’s mass and its orbital semi-major axis aroundits host planet ( a ps ), it would also be necessary to measure thetransit duration variation (TDV, Kipping 2009a,b). As shown byAwiphan & Kerins (2013), Kepler ’s ability to find an exomoon is crucially determined by its ability to discern the TDV signal,as it is typically weaker than the TTV signature. Using TTV andTDV observations together, it should be possible to detect moonsas small as 0 . M ⊕ (Kipping et al. 2009).Alternatively, it could even be possible to observe the di-rect transits of large moons (Szab´o et al. 2006; Tusnski & Valio2011; Kipping 2011), as the discovery of the sub-Mercury-sizedplanet Kepler-37b by Barclay et al. (2013) recently demon-strated. Now that targeted searches for extrasolar moons are un-derway (Kipping et al. 2012; Kipping et al. 2013; Kipping et al.2013) and the detection of a roughly Earth-mass moon in thestellar habitable zone (Dole 1964; Kasting et al. 1993, HZ in thefollowing) has become possible, we naturally wonder about theconditions that determine their habitability. Indeed, the searchfor spectroscopic biosignatures in the atmospheres of exomoonswill hardly be possible in the near future, because the moon’stransmission spectrum would need to be separated from that ofthe planet (Kaltenegger 2010). But the possible detection of ra-dio emission from intelligent species on exomoons still allowsthe hypothesis of life on exomoons to be tested. ff ect on exomoons due to planetary irradiation The idea of habitable moons has been put forward byReynolds et al. (1987) and Williams et al. (1997). Both stud-ies concluded that tidal heating can be a key energy source if amoon orbits its planet in a close, eccentric orbit (for tidal heatingin exomoons see also Scharf 2006; Debes & Sigurdsson 2007;Cassidy et al. 2009; Henning et al. 2009; Heller 2012; Heller& Barnes 2013). Reflected stellar light from the planet and theplanet’s own thermal emission can play an additional role ina moon’s energy flux budget (Heller & Barnes 2013; Hinkel& Kane 2013). Having said that, eclipses occur frequently inclose satellite orbits which are coplanar to the circumstellarorbit. These occultations can significantly reduce the averagestellar flux on a moon (Heller 2012), thereby a ff ecting its cli-mate (Forgan & Kipping 2013). Beyond that, the magnetic en-vironment of exomoons will a ff ect their habitability (Heller &Zuluaga 2013).Here, we investigate another e ff ect on a moon’s global en-ergy flux. So far, no study focused on the impact of radialshrinking of a gaseous giant planet and the accompanying ther-mal illumination of its potentially habitable moons. As a gi-ant planet converts gravitational energy into heat (Bara ff e et al.2003; Leconte & Chabrier 2013), it may irradiate a putativeEarth-like moon to an extent that makes the satellite subjectto a runaway greenhouse e ff ect. Atmospheric mass loss mod-els suggest that desiccation of an Earth-sized planet (or, in ourcase, of a moon) in a runaway greenhouse state occurs as fastas within 100 million years (Myr). Depending on the planet’ssurface gravitation, initial water content, and stellar XUV irradi-ation in the high atmosphere (Barnes et al. 2013, see Sect. 2 andAppendix B therein), this duration can vary substantially. But aswater loss is a complex process – not to forget the possible stor-age of substantial amounts of water in the silicate mantle, thehistory of volcanic outgassing, and possible redelivery of waterby late impacts (Lammer 2013) – we here consider moons in arunaway greenhouse state to be temporarily uninhabitable, ratherthan desiccated forever.
2. Methods
In the following, we consider a range of hypothetical planet-moon binaries orbiting a Sun-like star during di ff erent epochsof the system’s life time. As models for in-situ formation ofexomoons predict that more massive host planets will developmore massive satellites (Canup & Ward 2006; Sasaki et al. 2010;Ogihara & Ida 2012), we focus on the most massive host plan-ets that can possibly exist, that is, Jovian planets of roughly 13Jupiter masses. Following Canup & Ward (2006) and Heller etal. (2013), such a massive planet can grow Mars- to Earth-sizedmoons in its circumplanetary disk.In Fig. 1, we show the detections of stellar companions withmasses between that of Uranus and 13 M J , where M J denotesJupiter’s mass. While the ordinate measures planetary mass( M p ), the abscissa depicts orbit-averaged stellar illumination re-ceived by the planet, which we compute via We include a term W p for an additional source of illumination fromthe planet onto the moon in our orbit-averaged Eq. (B1) in Heller &Barnes (2013). This term can be attributed to heating form the planet.In Heller & Zuluaga (2013), we use methods developed here, but illu-mination from Neptune-, Saturn-, and Jupiter-like planets is weak. Given only the mass, such an object might well be a brown dwarfand not a planet. We here assume that this hypothetical object is a giantplanet and that it can be described by the Bara ff e et al. (2003) evolutionmodels (see below). Data from as of Sep. 25, 2013.
Fig. 1.
Planetary masses as a function of orbit-averaged stellarillumination for objects with masses between that of Uranus and13 times that of Jupiter. Non-transiting objects are shown as opencircles; red circles correspond to transiting objects, which couldallow for the detection of exomoons via TTV or TDV tech-niques. HD 217786 b is labeled because it could host Mars- toEarth-sized moons in the stellar HZ. HAT-P-44c is the most mas-sive transiting planet in the recent Venus / early Mars HZ, whosewidth independent of the host star is denoted by the solid verti-cal lines. The dashed vertical lines denote the inner and outer HZborders of the solar system (Kopparapu et al. 2013). Obviously,some ten super-Jovian planets reside in or close to the stellarHZs of their stars, but most of which are not known to transittheir stars. F ⋆ = σ SB T ⋆ q − e ⋆ p R ⋆ a ⋆ p ! . (1)Here, σ SB is the Stefan-Boltzmann constant, T ⋆ the stellar e ff ec-tive temperature, R ⋆ the stellar radius, e ⋆ p the orbital eccentricityof the star-planet system, and a ⋆ p the planet’s orbital semi-majoraxis.Dashed vertical lines illustrate the inner and outer bound-aries of the solar HZ, which Kopparapu et al. (2013) locateat 1 . × S e ff , ⊙ for the runaway greenhouse e ff ect and at0 . × S e ff , ⊙ for the maximum possible greenhouse e ff ect( S e ff , ⊙ = − being the solar constant). For stars otherthan the Sun or planetary atmospheres other than that of Earth,the HZ limits can be located at di ff erent flux levels. Planetsshown in this plot orbit a range of stars, most of which are on themain-sequence. The two solid vertical lines denote a flux intervalwhich is between the recent Venus and early Mars HZ bound-aries independent of stellar type, namely between 1 . × S e ff , ⊙ (recent Venus) and 0 . × S e ff , ⊙ (early Mars) (Kopparapu et al.2013). The purpose of Fig. 1 is to confirm that super-Jovianplanets (and possibly their massive moons) exist in or near theHZ of main-sequence stars in general, rather than to explic- ff ect on exomoons due to planetary irradiation itly identify giant planets in the HZ of their respective hoststars. Yet, the position of HD 217786 b is indicated in Fig. 1,because it roughly corresponds to our hypothetical test planetin terms of orbit-averaged stellar illumination at the outer HZboundary (483 W m − = . × S e ff , ⊙ ) and planetary mass( M p = ± . M J , Moutou et al. 2011). We also highlightHAT-P-44c because it is the most massive transiting planet inthe recent Venus / early Mars HZ (Hartman et al. 2013) and couldthus allow for TTV and TDV detections of its massive moons ifthey exist. The electromagnetic spectrum of a main-sequence star is verydi ff erent from that of a giant planet. M, K, and G dwarf starshave e ff ective temperatures between ≈ ≈ A s , opt = .
3, while anEarth-like planet orbiting a cool star with an e ff ective tempera-ture of 2500 K has an infrared albedo A s , IR = .
05 (Kasting et al.1993; Selsis et al. 2007; Kopparapu et al. 2013) . We thereforeuse a dualpass band to calculate the total illumination absorbedby the satellite F i = L ∗ ( t )(1 − A s , opt )16 π a ∗ p + L p ( t )(1 − A s , IR )16 π a , (2)where the first term on the right side of the equation describes thestellar flux absorbed by the moon, and the second term denotesthe absorbed illumination from the planet. L ∗ ( t ) and L p ( t ) arethe stellar and planetary bolometric luminosities, respectively,while a ∗ p is the orbital semi-major axes of the planet aroundthe star. The factor 16 in the denominators indicates that weassume e ff ective redistribution of both stellar and planetary ir-radiation over the moon’s surface (see Sect. 2.1 in Selsis et al.2007). Stellar reflected light from the planet is neglected (for adescription see Heller & Barnes 2013).To parametrize the luminosities of the star and the coolingplanet, we use the cooling tracks from Bara ff e et al. (1998) andBara ff e et al. (2003), respectively. Figure 2 shows the evolutionof the luminosity for a Sun-like star and the luminosity of threegiant planets with 13, 5, and 1 M J . Note the logarithmic scale:at an age of 0 . M J planet. Thesegiant planets models assume arbitrarily large initial temperaturesand radii. Yet, to actually assess the luminosity evolution of agiant exoplanet during the first ≈
500 Myr, its age, mass, andluminosity would need to be known (Mordasini 2013). Our study Strong infrared absorption of gaseous H O and CO in the atmo-spheres of terrestrial worlds and the absence of Rayleigh scatteringcould further decrease A s , IR . These e ff ects, however, would tend to warmthe stratosphere rather than the surface. Stratospheric warming couldthen be re-radiated to space without contributing much to warming thesurface. Equation (2) implies that the planet is much more massive than themoon, so that the planet-moon barycenter coincides with the planetarycenter of mass. It is also assumed that a ps ≪ a ∗ p and that both orbits arecircular. Fig. 2.
Luminosity evolution of a Sun-like star (according toBara ff e et al. 1998) and of three giant planets with 13, 5, and1 M J (following Bara ff e et al. 2003).is by necessity illustrative. Once actual exomoons are discoveredmore realistic giant planet models may be more appropriate. Tidal heating has been identified as a possible key source for amoon’s energy flux budget (see Sect. 1). Hence, we will also ex-plore the combined e ff ects of stellar and planetary illuminationon extrasolar satellites plus the contribution of tidal heating.While tidal theories were initially developed to describe thetidal flexing of rigid bodies in the solar system (such as theMoon and Jupiter’s satellite Io, see Darwin 1879, 1880; Pealeet al. 1979; Segatz et al. 1988), the detection of bloated gi-ant planets in close circumstellar orbits has triggered new ef-forts on realistic tidal models for gaseous objects. Nowadays,various approaches exist to parametrize tidal heating, and twomain realms for bodies with an equilibrium tide (Zahn 1977)have emerged: constant-phase-lag (CPL) models (typically ap-plied to rigid bodies, see Ferraz-Mello et al. 2008; Greenberg2009), and the constant-time-lag (CTL) models (typically ap-plied to gaseous bodies, see Hut 1981; Leconte et al. 2010).We here make use of the CPL model of Ferraz-Mello et al.(2008) to estimate the tidal surface heating F t on our hypotheti-cal exomoons. This model includes tidal heating from both cir-cularization (up to second order in eccentricity) and tilt ero-sion (that is, tidally-induced changes in obliquity ψ s , Helleret al. 2011). For simplicity, we assume that F t distributes evenlyover the satellite’s surface, although observations of Io, Titan,and Enceladus suggest that tidal heat can be episodic and heatwould leave a planetary body through hot spots (Ojakangas &Stevenson 1986; Spencer et al. 2000; Sotin et al. 2005; Spenceret al. 2006; Porco et al. 2006; Tobie et al. 2008; Bˇehounkov´aet al. 2012), that is, volcanoes or cryovolcanoes. The obliquityof a satellite in an orbit similar to Jupiter’s Galilean moons andSaturn’s moon Titan, with orbital periods .
16 d, is eroded inmuch less than 1 Gyr. As in Heller & Barnes (2013), we thus as-sume ψ s =
0. We treat the moon’s orbit to have an instantaneouseccentricity e ps . Tidal heating from circularization implies that e ps approaches zero, but it can be forced by stellar, planetary, oreven satellite perturbations to remain non-zero. Alternatively, itcan be the remainder of an extremely large initial eccentricity,or be caused by a recent impact. Note also that Titan’s orbital ff ect on exomoons due to planetary irradiation eccentricity of roughly 0.0288 is still not understood by any ofthese processes (Sohl et al. 1995). To assess whether our hypothetical satellites would be habitable,we compare the sum of total illumination F i (Eq. 2) and theglobally averaged tidal heat F t to the critical flux for a runawaygreenhouse, F crit . The latter value is given by the semi-analyticalapproach of Pierrehumbert (2010, Eq. 4.94 therein) (see alsoEq. 1 in Heller & Barnes 2013), which predicts the initiationof the runaway greenhouse e ff ect based on the maximum possi-ble outgoing longwave radiation from a planetary body with anatmosphere saturated in water vapor.We study two hypothetical satellites. In the first case of arocky Earth-mass, Earth-sized moon, we obtain a critical flux of F crit =
295 W / m for the onset of the runaway greenhouse e ff ect.In the second case, we consider an icy moon of 0 . M ⊕ andan ice-to-mass fraction of 25 %. Using the structure models ofFortney et al. (2007), we deduce the satellite radius (0.805 timesthe radius of Earth) and surface gravity (3 .
75 m / s ) and finallyobtain F crit =
266 W / m for this Super-Ganymede. If one of ourtest moons undergoes a total flux that is beyond its critical limit,then it can be considered temporarily uninhabitable. With decreasing distance from the planet, irradiation from theplanet and tidal forces on the moon will increase. While anEarth-sized satellite in a wide circumplanetary orbit may essen-tially behave like a freely rotating planet with only the star as arelevant light source, moons in close orbits will receive substan-tial irradiation from the planet – at the same time be subject toeclipses – and eventually undergo enormous tidal heating.To illustrate combined e ff ects of illumination from the planetand tidal heat as a function of distance to the planet, we introducean exomoon menagerie (Barnes & Heller 2013). It consists of thefollowing specimen (colors in brackets refer to Figs. 4 and 5): • Tidal Venus (red): F t ≥ F crit (Barnes et al. 2013) • Tidal-Illumination Venus (orange): F i < F crit ∧ F t < F crit ∧ F i + F t ≥ F crit • Super-Io (hypothesized by Jackson et al. 2008, yellow): F t > / m ∧ F i + F t < F crit • Tidal Earth (blue):0 .
04 W / m < F t < / m ∧ F i + F t < F crit • Earth-like (green): F t < .
04 W / m and within the stellar HZAmong these states, a Tidal Venus and a Tidal-InsolationVenus are uninhabitable, while a Super-Io, a Tidal Earth, andan Earth-like moon could be habitable. The 2 and 0 .
04 W / m limits are taken from examples in the solar system, where Io’sextensive volcanism coincides with an endogenic surface flux of This mass corresponds to about ten times the mass of Ganymedeand constitutes roughly the detection limit of
Kepler (Kipping et al.2009). roughly 2 W / m (Spencer et al. 2000). Williams et al. (1997) es-timated that tectonic activity on Mars came to an end when itsoutgoing energy flux through the surface fell below 0 .
04 W / m .For our menageries, we consider the rocky Earth-type moonorbiting a giant planet with a mass 13 times that of Jupiter.We investigate planet-moon binaries at two di ff erent distancesto a Sun-like star. In one configuration, we will assume that theplanet-moon duet orbits a Sun-like host at 1 AU. In this scenario,the planet-moon system is close to the inner edge of the stellarHZ (Kopparapu et al. 2013). In a second setup, the binary is as-sumed to orbit the star at a distance equivalent to 0 . S e ff , ⊙ ,which is the average of the maximum greenhouse and the earlyMars limits computed by Kopparapu et al. (2013). In this sec-ond configuration, hence, the binary is assumed at a distance of1 .
738 AU from a Sun-like host star, that is, at the outer edge ofthe stellar HZ.To explore the e ff ect of tidal heating, we consider fourdi ff erent orbital eccentricities of the planet-moon orbit: e ps ∈{ − , − , − , − } . For the Tidal Venus and the Tidal-Illumination Venus satellites, we assume a tidal quality factor Q s = Q s =
10 for the others. Our choiceof larger Q s , corresponding to lower dissipation rates, for theTidal Venus and the Tidal-Illumination Venus moons is moti-vated by the e ff ect we are interested in, namely volcanism, whichwe assume independent of tidal dissipation in a possible ocean.Estimates for the tidal dissipation in dry solar system objectsyields Q s ≈
100 (Goldreich & Soter 1966). On the other moons,tidal heating is relatively weak, and we are mostly concernedwith the runaway greenhouse e ff ect, which depends on the sur-face energy flux. Hence, dissipation in the ocean is crucial, andbecause Earth’s dissipation constant is near 10, we choose thesame value for moons with moderate and weak tidal heating.Ultimately, we consider all these constellations at three di ff erentepochs, namely, at ages of 100, 500, and 1000 Myr.In total, two stellar distances of the planet-moon duet,four orbital eccentricities of the planet-moon system, and threeepochs yield 24 combinations, that is, our circumplanetary exo-moon menageries.
3. Results
In Fig. 3, we show how absorbed stellar flux (dashed black line),illumination from the planet (dashed red line), and total illumi-nation (solid black line) evolve for a moon at a ps = R J from a13 M J planet. At that distance, irradiation from the planet alonecan drive a runaway greenhouse for about 200 Myr on both theEarth-type moon and the Super-Ganymede. What is more, whenwe include stellar irradiation from a Sun-like star at a distanceof 1 AU, then the total illumination at 10 R J from the planet isabove the runaway greenhouse limit of an Earth-like satellite forabout 500 Myr. Our hypothetical Super-Ganymede would be ina runaway greenhouse state for roughly 600 Myr.At a distance of 15 R J , flux from the planet would be(10 / = . ¯4 times the red dashed line shown in Fig. 3.After 200 Myr, illumination from the planet would still be 0 . ¯4 ×
310 W / m , and with the additional 190 W / m absorbed from the For comparison, Io, Europa, Ganymede, and Callisto orbit Jupiterat approximately 6.1, 9.7, 15.5, and 27.2 Jupiter radii. In-situ formationof moons occurs mostly between roughly 5 and 30 R p , from planets themass of Saturn up to planets with the 10-fold mass of Jupiter (Sasakiet al. 2010; Heller et al. 2013).4eller & Barnes: Runaway greenhouse e ff ect on exomoons due to planetary irradiation Fig. 3.
The total illumination F i absorbed by a moon (thick blackline) is composed of the absorbed flux from the star (blackdashed line) and from its host planet (red dashed line). The crit-ical values for an Earth-type and a Super-Ganymede moon toenter the runaway greenhouse e ff ect ( F crit ) are indicated by dot-ted lines at 295 and 266 W / m , respectively. Both moons orbit a13 M J planet at a distance of 10 R J , at 1 AU from a Sun-like hoststar. Illumination from the planet alone can trigger a runawaygreenhouse e ff ect for the first ≈
200 Myr.star, the total irradiation would still sum up to about 328 W / m at an age of 200 Myr. Consequently, even at a distance similar tothat of Ganymede from Jupiter, an Earth-sized moon could un-dergo a runaway greenhouse e ff ect around a 13 M J planet overseveral hundred million years. At a distance of 20 R J , total il-lumination would be 268 W / m after 200 Myr, and our Super-Ganymede test moon would still be uninhabitable. Clearly, ther-mal irradiation from a super-Jupiter host planet can have a majore ff ect on the habitability of its moons. In Fig. 4, we show four examples for our circumplanetary ex-omoon menageries. Abscissae and ordinates denote the dis-tance to the planet, which is chosen to be located at the centerat (0,0) [note the logarithmic scale!]. Colors illustrate a TidalVenus (red), Tidal-Illumination Venus (orange), Super-Io (yel-low), Tidal Earth (blue), and an Earth-like (green) state of anEarth-sized exomoon (for details see Sect. 2.4). Light green de-picts the Hill radius for retrograde moons, dark green for a pro-grade moons. In all panels, the planet-moon binary has an age of500 Myr. The two panels in the left column show planet-moonsystems at the inner edge of the stellar HZ; the two panels to the right show the same systems at the outer edge of the HZ. In thetop panels, e ps = − and tidal heating is strong; in the bottompanels e ps = − and tidal heating is weak.The outermost stable satellite orbit is only a fraction of theplanet’s Hill radius and depends, amongst others, on the distanceof the planet to the star (Domingos et al. 2006): the closer thestar, the smaller the planet’s Hill radius. This is why the greencircles are smaller at the inner edge of the HZ (left panels). Inthis particular constellation, the boundary between the dark andthe light green circles, that is, the Hill radii for moons in pro-grade orbits, is at 70 R J , while the outermost stable orbit for ret-rograde moons is at 132 R J . At the outer edge of the HZ (rightpanels), these boundaries are 121 at and 231 R J , respectively.A comparison between the top and bottom panels showsthat tidal heating, triggered by the substantial eccentricity inthe upper plots, can have a dramatic e ff ect on the circumplan-etary, astrophysical conditions. The Tidal Earth state (blue) in ahighly eccentric orbit (upper panels) can be maintained betweenroughly 21 and 36 R J both at the inner and the outer edge of thestellar HZ. But this state does not exist in the extremely low ec-centricity configuration at the inner HZ edge (lower left panel).In the latter constellation, the region of moderate tidal heatingis inside the circumplanetary sphere in which stellar illumina-tion plus illumination from the planet are strong enough to trig-ger a runaway greenhouse e ff ect on the moon (red and orangecircles for a Tidal Venus and Tidal-Illumination Venus state, re-spectively). In the lower right panel, a very thin rim of a TidalEarth state exists at roughly 8 R J .Each of these four exomoon menageries has its own circum-planetary death zone, that is, a range of orbits in which an Earth-like moon would be in a Tidal Venus or Tidal-Illumination Venusstate. In Heller & Barnes (2013), we termed the outermost orbit,which would just result in an uninhabitable satellite, the “habit-able edge”. Inspection of Fig. 4 yields that the habitable edgesfor e ps = − (top) and e ps = − (bottom) are located ataround 20 and 12 R J with the planet-moon binary at the inneredge of the stellar HZ (left column), and at 15 and 8 R J withthe planet-moon pair orbiting at the outer edge of the HZ (rightcolumn, top and bottom panels), respectively.Note that for moons with surface gravities lower than that ofEarth, the critical energy flux for a runaway greenhouse e ff ectwould be smaller. Although tidal surface heating in the satellitewould also be smaller, as it is proportional to the satellite’s ra-dius cubed, the habitable edge would be even farther away fromthe planet (see Fig. 10 in Heller & Barnes 2013). For Mars- toEarth-sized moons, and in particular for our Super-Ganymedehypothetical satellite, the habitable edges would be even fartheraway from the planet than depicted in Fig. 4.Figure 5, finally, shows our whole model grid of 24 exo-moon menageries. The four examples from Fig. 4 can be foundat the very top and the very bottom of the diagram in the cen-ter – though now on a linear rather than on a logarithmic scale.The left, center, and right graphics show a range of exomoonmenageries at ages of 100, 500, and 1000 Myr, respectively. Inall three epochs, highly eccentric exomoon orbits are shownat the top ( e ps = − ), almost circular orbits at the bottom( e ps = − ). The four circumplanetary circles at the left illus-trate menageries with the planet-moon system assumed at theinner edge of the HZ, while the four menageries at the right vi-sualize the planet-moon duet at the outer edge of the HZ.In the highly eccentric cases at the top, tidal heating domi-nates the circumplanetary orbital conditions for exomoon hab- Tidal heating in the moon does not depend on the stellar distance.5eller & Barnes: Runaway greenhouse e ff ect on exomoons due to planetary irradiation Fig. 4.
Circumplanetary exomoon menageries for Earth-sized satellites around a 13 Jupiter-mass host planet at an age of 500 Myr. Ineach panel, the planet’s position is at (0,0), and distances are shown on a logarithmic scale. In the left panels, the planet-moon binaryorbits at a distance of 1 AU from a Sun-like star; in the right panels, the binary is at the outer edge of the stellar HZ at 1 .
738 AU.In the upper two panels, e ps = − ; in the lower two panels, e ps = − . Starting from the planet in the center, the white circlevisualizes the Roche radius, and the exomoon types correspond to Tidal Venus (red), Tidal-Illumination Venus (orange), Super-Io(yellow), Tidal Earth (blue), and Earth-like (green) states (see Sect. 2.4 for details). Dark green depicts the Hill sphere of progradeEarth-like moons, light green for retrograde Earth-like moons. Note the larger Hill radii at the outer edge of the HZ (right panels)!itability. In the system’s youth at 100 Myr, the circumplanetaryhabitable edge is as far as 29 R J (upper left panel, at the inneredge of the HZ) or 18 R J (right panel in the leftmost diagram,outer edge of the HZ) around the planet. The region for TidalEarth moons spans from roughly 29 to 36 R J at the inner edge ofthe HZ and from 21 to 36 R J at the outer edge of the stellar HZ.At an age of 1000 Myr (diagram at the right), when illumination from the planet has decreased by more than one order of magni-tude (see red line in Fig. 2), the habitable edge has moved inwardto roughly 19 R J at the inner edge and 16 R J at the outer edge ofthe HZ, while the Tidal Earth state is between 21 to 36 R J in boththe inner and outer HZ edge cases.In the low-eccentricity scenarios at the bottom, tidal heat-ing has a negligible e ff ect, and thermal flux from the planet ff ect on exomoons due to planetary irradiation dominates the evolution of the circumplanetary conditions. At100 Myr, the Tidal-Illumination Venus state, which is just in-side the planetary habitable edge, ranges out to 28 R J at theinner edge of the HZ (lower left) and to 14 R J at the outeredge of the HZ (lower right menagerie in the left-most dia-gram). At an age of 1000 Myr, those values have decreased to5 and 2.5 R J for the planet-moon duet at the inner and outer HZboundaries, respectively (bottom panels in the rightmost sketch).Moreover, at 1000 Myr illumination from the planet has becomeweak enough that moons with substantial tidal heating in theselow-eccentricity scenarios could exist between 5 and 6 R J (TidalEarth) at the inner HZ edge or between 2.5 and 6 R J (Super-Ioand Tidal Earth) at the outer HZ boundary.
4. Conclusions
Young and hot giant planets can illuminate their potentially hab-itable, Earth-sized moons strong enough to make them uninhab-itable for several hundred million years. Based on the planetaryevolution models of Bara ff e et al. (2003), thermal irradiationfrom a 13 M J planet on an Earth-sized moon at a distance of10 R J can trigger a runaway greenhouse e ff ect for about 200 Myr.The total flux of Sun-like irradiation at 1 AU plus thermal fluxfrom a 13 M J planet will force Earth-sized moons at 10 or 15 R J into a runaway greenhouse for 500 or more than 200 Myr, re-spectively. A Super-Ganymede moon 0 .
25 times the mass ofEarth and with an ice-to-mass fraction of 25 % would undergoa runaway greenhouse e ff ect for longer periods at the same or-bital distances or, equivalently, for the same periods at largerseparations from the planet. Even at a distance of 20 R J from a13 M J host giant, it would be subject to a runaway greenhousee ff ect for about 200 Myr, if it receives an early-Earth-like illumi-nation from a young Sun-like star. In all these cases, the moonscould lose substantial amounts of hydrogen and, consequently,of water. Such exomoons would be temporarily uninhabitableand, perhaps, uninhabitable forever.If tidal heating is included, the danger for an exomoon to un-dergo a runaway greenhouse e ff ect increases. The habitable edgearound young giant planets, at an age of roughly 100 Myr andwith a mass 13 times that of Jupiter, can extend out to about 30 R J for moons in highly eccentric orbits ( e ps ≈ . ff ects we considered here, that is,stellar irradiation, thermal irradiation from the planet, and tidalheating, other heat sources in moons may exist. Considerationof primordial thermal energy (or “sensible heat”), radioactive de-cay, and latent heat from solidification inside an exomoon wouldincrease the radii of the circumplanetary menageries and pushthe habitable edge even further away from the planet.Our estimates for the instantaneous habitability of exomoonsshould be regarded as conservative because ( i. ) the minimumseparation for an Earth-like moon from its giant host planet tobe habitable could be even larger than we predict. This is be-cause a giant planet’s luminosity at 1 AU from a Sun-like starcan decrease more slowly than in the Bara ff e et al. (2003) mod-els used in our study (Fortney et al. 2007). ( ii. ) The runawaygreenhouse limit, which we used here to assess instantaneoushabitability, is a conservative approach itself. A moon with a to-tal energy flux well below the runaway greenhouse limit may beuninhabitable as its surface is simply too hot. It could be caughtin a moist greenhouse state with surface temperatures up to thecritical point of water, that is, 647 K if an Earth-like inventoryof water and surface H O pressure are assumed (Kasting 1988).( iii. ) Moons with surface gravities smaller than that of Earth will have a critical energy flux for the runaway greenhouse e ff ect thatis also smaller than Earth’s critical flux. Hence, their correspond-ing Tidal Venus and Tidal-Illumination Venus states would reachout to wider orbits than those depicted in Figs. 4 and 5. ( iv. ) Tidalheating could be stronger than the values we derived with theFerraz-Mello et al. (2008) CPL tidal model. As shown in Helleret al. (2011), the CTL theory of Leconte et al. (2010) includesterms of higher orders in eccentricity and yields stronger tidalheating than the CPL of Ferraz-Mello et al. (2008). Yet, suchmathematical extensions may not be physically valid (Greenberg2009), and parametrization of a planet’s or satellite’s tidal re-sponse with a constant tidal quality factor Q or a fixed tidal timelag τ remains uncertain (Leconte et al. 2010; Heller et al. 2011;Efroimsky & Makarov 2013).Massive moons can bypass an early runaway greenhousestate if they form after the planet has cooled su ffi ciently. Possiblescenarios for such a delayed formation include gravitational cap-ture of one component of a binary planet system, the captureof Trojans, gas drag or pull-down mechanisms, moon mergers,and impacts on terrestrial planets (for a review, see Sect. 2.1 inHeller & Barnes 2013). Alternatively, a desiccated moon in thestellar HZ could be re-supplied by cometary bombardment later.Reconstruction of any such event would naturally be di ffi cult.Once extrasolar moons will be discovered, assessments oftheir habitability will depend not only on their current orbitalconfiguration and irradiation, but also on the history of stellarand planetary luminosities. Even if a Mars- to Earth-sized moonwould be found about a Jupiter- or super-Jupiter-like planet at,say, 1 AU from a Sun-like star, the moon could have lost sub-stantial amounts of its initial water reservoir and be uninhabit-able today. In the most extreme cases, strong thermal irradiationfrom the young, hot giant host planet could have desiccated themoon long ago by the runaway greenhouse e ff ect. Acknowledgements.
The referee reports of Jim Kasting, Nader Haghighipour,and an anonymous reviewer significantly improved the quality of this study.We thank Jorge I. Zuluaga for additional comments on the manuscript andJean Schneider for technical support. Ren´e Heller is funded by the CanadianAstrobiology Training Program and a member of the Origins Institute atMcMaster University. Rory Barnes acknowledges support from NSF grant AST-1108882 and the NASA Astrobiology Institute’s Virtual Planetary Laboratorylead team under cooperative agreement No. NNH05ZDA001C. This work hasmade use of NASA’s Astrophysics Data System Bibliographic Services and ofJean Schneiders exoplanet database . References
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Exomoon menageries at 100 (left), 500 (center), and 1000 Myr (right) for a hypothetical Earth-like satellite around a 13 M J planet. For each epoch, one suite of menageries is at the inner edge of the HZ (left) and another one at the outer edge of the HZaround a Sun-like star. Four di ff erent orbital eccentricities e ps of the satellite are indicated. Distances from the central planet in eachmenagerie are on a linear scale, and absolute values can be estimated by comparison with the four examples shown on a logarithmicscale in Fig. 4. The Hill sphere for prograde moons (boundary between dark and light green) is at 70 R J at the inner HZ boundaryand at 121 R J at the outer HZ edge. .5. MAGNETIC SHIELDING OF EXOMOONS BEYOND THE CIRCUMPLANETARYHABITABLE EDGE (Heller & Zuluaga 2013) 254 Contribution:RH contributed to the literature research, contributed to the mathematical investigations, contributedto the writing of the computer code and to the generation of the figures, led the writing of themanuscript, and served as a corresponding author for the journal editor and the referees. raft version September 5, 2013
Preprint typeset using L A TEX style emulateapj v. 04/17/13
MAGNETIC SHIELDING OF EXOMOONS BEYOND THE CIRCUMPLANETARY HABITABLE EDGE
Ren´e Heller McMaster University, Department of Physics and Astronomy, Hamilton, ON L8S 4M1, Canada; [email protected] andJorge I. Zuluaga FACom - Instituto de F´ısica - FCEN, Universidad de Antioquia, Calle 70 No. 52-21, Medell´ın, Colombia; jzuluaga@fisica.udea.edu.co
Draft version September 5, 2013
ABSTRACTWith most planets and planetary candidates detected in the stellar habitable zone (HZ) being super-Earths and gas giants, rather than Earth-like planets, we naturally wonder if their moons could behabitable. The first detection of such an exomoon has now become feasible, and due to observationalbiases it will be at least twice as massive as Mars. But formation models predict moons can hardlybe as massive as Earth. Hence, a giant planet’s magnetosphere could be the only possibility for sucha moon to be shielded from cosmic and stellar high-energy radiation. Yet, the planetary radiationbelt could also have detrimental effects on exomoon habitability. We here synthesize models for theevolution of the magnetic environment of giant planets with thresholds from the runaway greenhouse(RG) effect to assess the habitability of exomoons. For modest eccentricities, we find that satellitesaround Neptune-sized planets in the center of the HZ around K dwarf stars will either be in an RGstate and not be habitable, or they will be in wide orbits where they will not be affected by theplanetary magnetosphere. Saturn-like planets have stronger fields, and Jupiter-like planets could coatclose-in habitable moons soon after formation. Moons at distances between about 5 and 20 planetaryradii from a giant planet can be habitable from an illumination and tidal heating point of view, butstill the planetary magnetosphere would critically influence their habitability.
Keywords: astrobiology – celestial mechanics – methods: analytical – planets and satellites: magneticfields – planets and satellites: physical evolution – planet-star interactions INTRODUCTION
The search for life on worlds outside the solar systemhas experienced a substantial boost with the launch ofthe
Kepler space telescope in 2009 March (Borucki et al.2010). Since then, thousands of planet candidates havebeen detected (Batalha et al. 2013), several tens of whichcould be terrestrial and have orbits that would allow forliquid surface water (Kaltenegger & Sasselov 2011). Theconcept used to describe this potential for liquid water,which is tied to the search for life, is called the “habitablezone” (HZ) (Kasting et al. 1993). Yet, most of the
Kepler candidates, as well as most of the planets in the HZ de-tected by radial velocity measurements, are giant planetsand not Earth-like. This leads us to the question whetherthese giants can host terrestrial exomoons, which mayserve as habitats (Reynolds et al. 1987; Williams et al.1997).Recent searches for exomoons in the
Kepler data (Kip-ping et al. 2013a,b) have fueled the debate about theexistence and habitability of exomoons and incentivizedothers to develop models for the surface conditions onthese worlds. While these studies considered illuminationeffects from the star and the planet, as well as eclipses,tidal heating (Heller 2012; Heller & Barnes 2013b,a), andthe transport of energy in the moon’s atmosphere (For-gan & Kipping 2013), the magnetic environment of exo-moons has hitherto been unexplored.Moons around giant planets are subject to high-energyradiation from (1) cosmic particles, (2) the stellar wind,and (3) particles trapped in the planet’s magnetosphere(Baumstark-Khan & Facius 2002). Contributions (1) and (2) are much weaker inside the planet’s magneto-sphere than outside, but effect (3) can still have detri-mental consequences. The net effect (beneficial or detri-mental) on a moon’s habitability depends on the actualorbit, the extent of the magnetosphere, the intrinsic mag-netosphere of the moon, the stellar wind, etc.Stellar mass-loss and X-ray and extreme UV (XUV)radiation can cause the atmosphere of a terrestrial worldto be stripped off. Light bodies are in particular dan-ger, as their surface gravity is weak and volatiles canescape easily (Lammer et al. 2013). Mars, for example,is supposed to have lost vast amounts of CO , N, O,and H (the latter two formerly bound as water) (McEl-roy 1972; Pepin 1994; Valeille et al. 2010). Intrinsic andextrinsic magnetic fields can help moons sustain their at-mospheres and they are mandatory to shield life on thesurface against galactic cosmic rays (Grießmeier et al.2009). High-energy radiation can also affect the atmo-spheric chemistry, thereby spoiling signatures of spectralbiomarkers, especially of ozone (Segura et al. 2010).Understanding the evolution of the magnetic environ-ments of exomoons is thus crucial to asses their habit-ability. We here extend models recently applied to theevolution of magnetospheres around terrestrial planetsunder an evolving stellar wind (Zuluaga et al. 2013). Em-ployed on giant planets, they allow us a first approachtoward parameterizing the potential of a giant planet’smagnetosphere to affect potentially habitable moons. METHODS
In the following, we model a range of hypothetical sys-tems to explore the potential of a giant planet’s mag-
Ren´e Heller & Jorge Zuluaga netosphere to embrace moons that are habitable from atidal and energy budget point of view.
Bodily Characteristics of the Star
M dwarf stars are known to show strong magneticbursts, eventually coupled with the emission of XUV ra-diation as well as other high-energy particles (Gurzadian1970; Silvestri et al. 2005). Moreover, stars with massesbelow 0 . . M (cid:12) ) cannot possibly hosthabitable moons, since stellar perturbations excite haz-ardous tidal heating in the satellites (Heller 2012). Wethus concentrate on stars more similar to the Sun. Gdwarfs, however, are likely too bright and too massive toallow for exomoon detections in the near future. As acompromise, we choose a 0 . M (cid:12) K dwarf star with solarmetallicity Z = 0 . R ? ) and ef-fective temperature ( T eff ,? ) at an age of 100 Myr (Bressanet al. 2012): R ? = 0 . R (cid:12) ( R (cid:12) being the solar radius), T eff ,? = 4270 K. These values are nearly constant overthe next couple of Gyr. Bodily Characteristics of the Planet
We use planetary evolution models of Fortney et al.(2007) to explore two extreme scenarios, between whichwe expect most giant planets: (1) mostly gaseous with acore mass M c = 10 Earth masses ( M ⊕ ) and (2) planetswith comparatively massive cores. Class (1) correspondsto larger planets for given planetary mass ( M p ). Mod-els for suite (2) are constructed in the following way.For M p < . M Jup , we interpolate between radii ofplanets with core masses M c = 10, 25, 50 and 100 M ⊕ to construct Neptune-like worlds with a total amountof 10 % hydrogen (H) and helium (He) by mass. For M p > . M Jup , we take the precomputed M c = 100 M ⊕ grid of models. These massive-core planets (2) yield anestimate of the minimum radius for given M p . To ac-count for irradiation effects on planetary evolution, weapply the Fortney et al. (2007) models for planets at1 AU from the Sun.As it is desirable to compare our scaling laws for themagnetic properties of giant exoplanets with known mag-netic dipole moments of solar system worlds, we start outby considering a Neptune-, a Saturn-, and a Jupiter-classhost planet. For the sake of consistency, we attribute to-tal masses of 0.05, 0.3, and 1 M Jup , as well as M c = 10,25, and again 10 M ⊕ , respectively. Bodily Characteristics of the MoonKepler has been shown capable of detecting moons assmall as 0 . M ⊕ combining measurements of the planet’stransit timing variation and transit duration variation(Kipping et al. 2009). The detection of a planet as smallas 0 . R ⊕ ), almost half the radius of Mars(Barclay et al. 2013), around a K star suggests that directtransit measurements of Mars-sized moons may be pos-sible with current or near-future technology (Sartoretti& Schneider 1999; Szab´o et al. 2006; Kipping 2011). Yet,in-situ formation of satellites is restricted to a few times10 − M p at most (Canup & Ward 2006; Sasaki et al.2010; Ogihara & Ida 2012). For a Jupiter-mass planet,this estimate yields a satellite of about 0 . M ⊕ ≈ . .
94 Mars radii or 0 . R ⊕ for a Mars-mass exomoon.Tidal heating in the moon is calculated using the modelof Leconte et al. (2010) and assuming an Earth-like timelag of the moon’s tidal bulge τ s = 638 s as well as a sec-ond degree tidal Love number k , s = 0 . The Stellar Habitable Zone
We investigate a range of planet-moon binaries locatedin the center of the stellar HZ. Therefore, we computethe arithmetic mean of the inner HZ edge (given by themoist greenhouse effect) and the outer HZ edge (given bythe maximum greenhouse) around a K dwarf star (Sec-tion 2.1) using the model of Kopparapu et al. (2013). Forthis particular star, we localize the center of the HZ at0 .
56 AU.
The Runaway Greenhouse and Io Limits
The tighter a moon’s orbit around its planet, themore intense the illumination it receives from the planetand the stronger tidal heating. Ultimately, there existsa minimum circumplanetary orbital distance, at whichthe moon becomes uninhabitable, called the “habitableedge” (Heller & Barnes 2013b). As tidal heating dependsstrongly on the orbital eccentricity e ps , amongst others,the radius of the HE also depends on e ps .We consider two thresholds for a transition into anuninhabitable state: (1) When the moon’s tidal heat-ing reaches a surface flux similar to that observed onJupiter’s moon Io, that is 2 W m − , then enhanced tec-tonic activity as well as hazardous volcanism may occur.Such a scenario could still allow for a substantial area ofthe moon to be habitable, as tidal heat leaves the surfacethrough hot spots, (see Io and Enceladus, Ojakangas &Stevenson 1986; Spencer et al. 2006; Tobie et al. 2008),and there may still exist habitable regions on the surface.Hence, we consider this “Io-limit HE” (Io HE) as a pes-simistic approach. (2) When the moon’s global energyflux exceeds the critical flux to become a runaway green-house (RG), then any liquid surface water reservoirs canbe lost due to photodissociation into hydrogen and oxy-gen in the high atmosphere (Kasting 1988). Eventually,hydrogen escapes into space and the moon will be desic-cated forever. We apply the semi-analytic RG model ofPierrehumbert (2010) (see Equation 1 in Heller & Barnes2013b) to constrain the innermost circumplanetary orbitat which a moon with given eccentricity would just behabitable. For our prototype moon, this model predictsa limit of 269 W m − above which the moon would tran-sition to an RG state. We call the corresponding criticalsemi-major axis the “runaway greenhouse HE” (RG HE)and consider it as an optimistic approach.We calculate the Io and RG HEs for e ps ∈{ . , . , . } using Equation (22) from Heller &Barnes (2013b) and introducing two modifications. First,the planetary surface temperature ( T p ) depends on theequilibrium temperature ( T eqp ) due to absorbed stellarlight and on an additional component ( T int ) from inter-nal heating: T p = ([ T eqp ] + T ) / . Second, we use a AGNETIC SHIELDING OF EXOMOONS R ✌ – R R ✌ R M a gn e t o s ph e r e M a g n e t o p a u s e incomingstellarwind Figure 1.
Sketch of the planetary magnetosphere. R S denotesthe standoff distance, R M labels the radius of the magnetopause.A range of satellite orbits illustrates how a moon can periodicallydive into and out of the planetary magnetosphere. Conceptually,the dashed orbit resembles that of Titan around Saturn, with oc-casional shielding and exposition to the solar wind (Bertucci et al.2008). At orbital distances & R S , the fraction of the moon’s orbitspent outside the planet’s magnetospheric cavity reaches ≈ Bond albedo α opt = 0 . α IR = 0 .
05 for the light absorbedfrom the relatively cool planet (Heller & Barnes 2013a).
Planetary Dynamos
We apply scaling laws for the magnetic field strength(Olson & Christensen 2006) that consider convection ina spherical conducting shell inside a giant planet and aconvective power Q conv (Equation (28) in Zuluaga et al.2013), provided by the Fortney et al. (2007) models.The ratio between inertial forces to Coriolis forces is cru-cial in determining the field regime — be it dipolar- ormultipolar-dominated — for the dipole field strength onthe planetary surface. We scale the ratio between dipo-lar and total field strengths following Zuluaga & Cuartas(2012).The core density ( ρ c ) is estimated by solving a poly-tropic model of index 1 (Hubbard 1984). Radius andextent of the convective region are estimated by apply-ing a semi-empirical scaling relationship for the dynamoregion (Grießmeier 2006). Thermal diffusivity κ is as-sumed equal to 10 − for all planets (Guillot 2005). Elec-trical conductivity σ is assumed to be 6 × for planetsrich in H and He, and σ = 1 . × for the ice-rich giants(Olson & Christensen 2006).We have verified that our model predicts dynamo re-gions that are similar to results obtained by more so-phisticated analyses. For Neptune, our model predictsa dynamo radius R c = 0 .
77 planetary radii ( R p ) and ρ c = 3000 kg m − , in good agreement to Kaspi et al.(2013). We also ascertained the dynamo scaling laws toreasonably reproduce the planetary dipole moments ofGanymede, Earth, Uranus, Neptune, Saturn, and Jupiter(J. I. Zuluaga et al., in preparation). For the mass rangeconsidered here, the predicted dipole moments agreewithin a factor of two to six. Discrepancies of this magni-tude are sufficient for our estimation of magnetospheric properties, because they scale with M / . Significant un-derestimations arise for Neptune and Uranus, which havestrongly non-dipolar surface fields. Evolution of the Magnetic Standoff Distance
The shape of the planet’s magnetosphere can be ap-proximated as a combination of a semi-sphere with ra-dius R M and a cylinder representing the tail region (Fig-ure 1). The planet is at a distance R M − R S off thesphere’s center, with R S = (cid:18) µ f π (cid:19) / M / P − / (1)being the standoff distance, µ = 4 π × − N A − thevacuum magnetic permeability, f = 1 . M planetary magnetic dipole moment, and P sw ∝ n sw v sw the dynamical pressure of the stellar wind. Num-ber densities n sw and velocities v sw of the stellar wind arecalculated using a hydrodynamical model (Parker 1958).Both quantities evolve as the star ages. Thus, we use em-pirical formulae (Grießmeier et al. 2007) to parameterizetheir time dependence.Observations of stellar winds from young stars are chal-lenging, and thus the models only cover stars older than700 Myr. We extrapolate these models back to 100 Myr,although there are indications that stellar winds “satu-rate” when going back in time (J. Linsky 2012, privatecommunication). RESULTS
Evolution of Magnetic Standoff Distance versusRunaway Greenhouse and Io Habitable Edges
Figure 2 visualizes the evolution of R S (thick blue line)as a snail curling around the planet, indicated by a darkcircle in the center. Stellar age ( t ? ) is denoted in unitsof Gyr along the snail, starting at 0 . . R S , as well as theRG and Io HEs, are given in units of R p . At t ? = 0 . t ? = 4 . R p = 0 . . . R Jup (0 . . . R Jup ) for the Neptune-, Saturn-, andJupiter-like planets, respectively.In panel (a) for the Neptune-like host, R S starts veryclose to the planet and even inside the RG HE for the e ps = 0 .
001 case (thin black snail), at roughly 2 R p fromthe planetary center. This means, at an age of 0 . e ps = 0 .
001 would need toorbit beyond 2 R p to avoid transition into a RG state,where it would not be affected by the planet’s magneto-sphere. As the system ages, wider orbits are enshroudedby the planetary magnetic field, until after 4 . R S reaches as far as the Io HE for e ps = 0 . In summary, moons in low-eccentricity orbitsaround Neptune-like planets can be close the planet andbe habitable from an illumination and tidal heating pointof view, but it will take at least 300 Myr in our specificcase until they get coated by the planet’s magnetosphere.Exomoons on more eccentric orbits will either be unin-habitable or affected by the planet’s magnetosphere aftermore than 300 Myr. Note that as the planet shrinks, the RG and Io HEs move out-ward, too. This is because of our visualization in units of planetaryradii, while the values of the HEs remain constant in secular units.
Ren´e Heller & Jorge Zuluaga
RG e ps =0.1RG e ps =0.01RG e ps =0.001R S IoIoIo
Figure 2.
Evolution of the magnetic shielding (blue curves) compared to the RG HEs (solid black lines) and Io-like HEs (dashed greenlines). Thick lines correspond to HEs for e ps = 0 .
1, intermediate thickness to e ps = 0 .
01, and thin circles to e ps = 0 . Moving on to Figure 2(b) and the Saturn-like host,we find that R S reaches the e ps = 0 .
001 RG HE afterroughly 200 Myr, but it transitions all the other HEs sub-stantially later than in the case of a Neptune-like host.After ≈ e ps = 0 .
01 RG HE and the Io HE for e ps = 0 .
01 are transversed. After 4 . . e ps = 0 .
001 will be bathed inthe planetary magnetosphere at stellar ages as young as100 Myr. After roughly 1 . R S transitions the RGHE for e ps = 0 . e ps = 0 .
01. Eventhe Io HE with an eccentricity of e ps = 0 . . . R p . Clearly, exomoons aboutJupiter-like planets face the greatest prospects of inter-ference with the planetary magnetosphere, even in orbitsthat are sufficiently wide to ensure negligible tidal heat-ing. Minimum Magnetic Dipole Moment
AGNETIC SHIELDING OF EXOMOONS Figure 3.
Magnetic dipole moments (ordinate) for a range ofplanetary masses (abscissa). The straight lines depict M dip as itwould be required at an age of 0 . Looking at the standoff radii for the three cases in Fig-ure 2, we wonder how strong the magnetic dipole moment M dip would need to be after 0.5 Gyr, when the atmo-spheric buildup should have mostly ceased, in order tomagnetically enwrap the moon at a given HE. This ques-tion is answered in Figure 3.While planets with relatively massive cores (brownsolid line) shield a wider range of orbits for given M p be-low roughly 1 M Jup , the low-mass core model (blue solidline) catches up for more massive giants. What is more,our model tracks for the predicted M dip , which we expectto be located between the thick brown and thick blue line,“overtake” the RG and Io-like HEs for a range of eccen-tricities. HE contours that fall within the shaded area,are magnetically protected, while moons above a certainHE are habitable from a tidal and illumination point ofview.We finally examine temporal aspects of magneticshielding in Figure 4. The question answered in this plotis: “How long would it take a planet to magnetically coatits moon beyond the habitable edges?”. Again, planetarymass is along the abscissa, but now stellar life time t ? isalong the ordinate. Clearly, the magnetic standoff radiusof lower-mass planets requires more time to expand outto the respective HEs. While low-mass giants with low-mass cores (blue lines) require up to 3 . e ps = 0 .
01, equally mass planets but with ahigh-mass core (thin brown line) could require as few as650 Myr.Planets more massive than roughly 0 . M Jup , will coattheir moons at the RG HE for e ps = 0 .
01 as early as 1 Gyrafter formation. For lower eccentricities, time scales de-crease. As the RG HE is more inward to the planet thanthe Io HE, it is coated earlier than the Io limit for given e ps . RG and Io HEs for e ps = 0 .
001 are not shown asthey are covered earlier than ≈
300 Myr in all cases. CONCLUSION
Figure 4.
Time required by the planet’s magnetic standoff radius R S to envelop the Io (dashed) and RG (solid) HEs. Shaded regionsillustrate uncertainties coming from planetary structure models,with the boundaries corresponding to a high- and a low-core massplanetary model, respectively (brown and blue lines). In general,the magnetic standoff distance around more massive planets re-quires less time to enshroud moons at a given HE. Mars-sized exomoons of Neptune-sized exoplanets inthe stellar HZ of K stars will hardly be affected by plan-etary magnetospheres if these moons are habitable froman illumination and tidal heating point of view. Whilethe magnetic standoff distance expands for higher-massplanets, ultimately Jovian hosts can enshroud their mas-sive moons beyond the HE, depending on orbital eccen-tricity. In any case, exomoons beyond about 20 R p willbe habitable in terms of illumination and tidal heating,and they will not be coated by the planetary magneto-sphere within about 4.5 Gyr. Moons between 5 and 20 R p can be habitable, depending on orbital eccentricity, andbe affected by the planetary magnetosphere at the sametime.Uncertainties in the parameterization of tidal heatingcause uncertainties in the extent of both the RG andIo HEs. Once a potentially habitable exomoon wouldbe discovered, detailed interior models for the satellite’sbehavior under tidal stresses would need to be explored.In a forthcoming study, we will examine the evolutionof planetary dipole fields, and we will apply our methodsto planets and candidates from the Kepler sample. Ob-viously, a range of giant planets resides in their stellarHZs, and these planets need to be prioritized for follow-up search on the potential of their moons to be habitable.The referee report of Jonathan Fortney substantiallyimproved the quality of this study. We have made useof NASA’s ADS Bibliographic Services. Computationshave been performed with ipython 0.13 on python2.7.2 (P´erez & Granger 2007). RH receives fundingfrom the Canadian Astrobiology Training Program. JIZis supported by CODI-UdeA and Colciencias.REFERENCES Barclay, T., Rowe, J. F., Lissauer, J. J., et al. 2013, Nature, 494,452Batalha, N. M., Rowe, J. F., Bryson, S. T., et al. 2013, ApJS,204, 24
Ren´e Heller & Jorge Zuluaga
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Contribution:RH did the literature research, worked out the mathematical framework for the concept of the exomoonmenagerie, translated the math into computer code, performed all computations, created all figures,led the writing of the manuscript, and served as a corresponding author for the journal editor and thereferees. uperhabitable Worlds
René Heller , John Armstrong McMaster University, Department of Physics and Astronomy, Hamilton, ON L8S 4M1, Canada ([email protected]) Department of Physics, Weber State University, 2508 University Circle, Ogden, UT 84408-2508 ([email protected])To be habitable, a world (planet or moon) does not need to be located in the stellar habitable zone (HZ), and worlds in the HZ are not necessarily habitable. Here, we illustrate how tidal heating can render terrestrial or icy worlds habitable beyond the stellar HZ. Scientists have developed a language that neglects the possible existence of worlds that offer more benign environments to life than Earth does. We call these objects “superhabitable” and discuss in which contexts this term could be used, that is to say, which worlds tend to be more habitable than Earth. In an appendix, we show why the principle of mediocracy cannot be used to logically explain why Earth should be a particularly habitable planet or why other inhabited worlds should be Earth-like. Superhabitable worlds must be considered for future follow-up observations of signs of extraterrestrial life. Considering a range of physical effects, we conclude that they will tend to be slightly older and more massive than Earth and that their host stars will likely be K dwarfs. This makes Alpha Centauri B, member of the closest stellar system to the Sun that is supposed to host an Earth-mass planet, an ideal target for searches of a superhabitable world.
Key Words:
Extrasolar terrestrial Planets –
Extraterrestrial Life –
Habitability –
Planetary environments – Tides
1. Introduction
A substantial amount of research is conducted and resources are spent to search for planets that could be habitats for extrasolar life. Engineers and astronomers have developed expensive instruments and large ground-based telescopes, such as the
High Accuracy Radial Velocity Spectrograph (HARPS) at the 3.6m ESO telescope and the
Ultraviolet-Visual Echelle Spectrograph (UVES) at the
Very Large Telescope (VLT), and launched the
CoRoT and
Kepler space telescopes with the explicit aim to detect and characterize Earth-sized planets. Even larger facilities are being planned or constructed, such as the
European Extremely Large Telescope (E-ELT) and the
James Webb Space Telescope (JWST), and an ever growing community of scientists is working to solve not only the observational but also the theoretical and laboratory challenges.
At the theoretical front, the concept of the stellar “habitable zone” (HZ) has been widely used to identify potentially habitable planets (Huang, 1959; Dole, 1964; Kasting et al., 1993). To the confusion of some, planets that reside within a star’s HZ are often called “habitable planets.” However, a planet in the HZ need not be habitable in the sense that it has at least some niches that allow for the existence of liquid surface water. Naturally, as Earth is the only inhabited world we know, this object usually serves as a reference for studies on habitability. Instruments are being designed in a way to detect and characterize Earth-like planets and spectroscopic signatures of life in Earth-like atmospheres (Des Marais et al., 2002; Kaltenegger and Traub, 2009; Kaltenegger et al., 2010; Rauer et al., 2011; Kawahara et al., 2012). However, other worlds can offer conditions that are even more suitable for life to emerge and to evolve. Besides planets, moons could be habitable, too (Reynolds et al., 1987; Williams et al., 1997; Kaltenegger, 2010; Porter and Grundy, 2011; Heller and Barnes, 2013a). To find a habitable and ultimately an inhabited world, a characterization concept is required that is biocentric rather than geo- or anthropocentric.
In Section 2 of this paper, we illustrate how tidal heating can make planets inside the stellar HZ uninhabitable, and how it can render exoplanets and exomoons beyond the HZ habitable. Section 3 is devoted to conditions that could make a world more hospitable for life than Earth is. We call these objects “superhabitable worlds.” Though our considerations are anticipatory, they still rely on the assumption that life needs liquid water. Our conclusions on the nature and prospects for finding superhabitable worlds are presented in Section 4. In Appendix A, we disentangle confusions between planets in the HZ and habitable planets, and we address related disorder that emerges from language issues. Appendix B is dedicated to the principle of mediocracy – in particular why it cannot explain that Earth is a typical, inhabited world.
2. Habitability in and Beyond the Stellar Habitable Zone
A natural starting point towards the characterization of a world’s habitability is computing its absorbed stellar energy flux. This approach has led to what is called the “stellar habitable zone.” The oldest record of a description of a circumstellar zone suitable for life traces back to Whewell (1853, Chap. X, Section 4), who, referring to the local stellar system in a qualitative way, called this distance range the “Temperate Zone of the Solar System.” More than a century later, Huang (1959) presented a more general discussion of the “Habitable Zone of a Star,” which considers time scales of stellar evolution, dynamical constraints in stellar multiple systems, and the stellar galactic orbit. A much broader, less anthropocentric elaboration on submitted to
Astrobiology : August 6, 2013 accepted by
Astrobiology : December 5, 2013 published in
Astrobiology : January, 2014this draft: January 9, 2014 doi:10.1089/ast.2013.1088 abitability has then been given by Dole (1964), who termed the circumstellar habitable zone “ecosphere.” The most widely used concept as of today is the one presented by Kasting et al. (1993), who applied a one-dimensional climate model and identified the CO feedback to ensure the inner and the outer edges of the stellar HZs. The inner edge is defined by the activation of the moist or runaway greenhouse process, which desiccates the planet by evaporation of atmospheric hydrogen; the outer edge is defined by CO freeze out, which breaks down the greenhouse effect whereupon the planet transitions into a permanent snowball state. Extensions of this concept to include orbital eccentricities have been given by Selsis et al. (2007) and Barnes et al. (2008), and a recent revision of the input model used by Kasting et al. (1993) has been presented by Kopparapu et al. (2013). Considering the aging of the star, which involves a steady increase of stellar luminosity as long as the star is on the main sequence, the distance range within the HZ that is habitable for a certain period (say over the last 4.6 Gyr in the case of the Solar System) has been termed the continuous habitable zone (CHZ) (Kasting et al., 1993; Rushby et al., 2013). From an observational point of view, the CHZ provides a more useful tool because life needs time to evolve to a certain level such that it modifies its atmosphere on a global scale. Life seems to have appeared relatively early after the formation of Earth. Chemical and fossil indicators for early life can be found in sediments that date back to about 3.5 – 3.8 Gyr ago (Schopf, 1993; Mojzsis et al., 1996; Schopf, 2006; Basier et al., 2006), that is, less than about 1 Gyr after Earth had formed. If correct, then life would have recovered within 100 Myr or so after the Late Heavy Bombardment (LHB) on Earth (Gomes et al., 2005). However, life required billions of years before it modified Earth’s atmosphere substantially and imprinted substantial amounts of bio-relevant signatures in the atmospheric transmission spectrum. Stars more massive than the Sun have shorter lifetimes. Thus, although the lifetime of a 1.4 solar-mass ( M ⊙ ) star is still about 4.5 Gyr, superhabitable planets will tend to orbit stars that are as massive as the Sun at most. Further modifications of the circumstellar HZ include effects of tidal heating (Jackson et al., 2008a; Barnes et al., 2009), orbital evolution due to tides (Barnes et al., 2008), tidal locking of planetary rotation (Dole, 1964; Kasting et al., 1993), planetary obliquity (Spiegel et al., 2009), loss of seasons due to tilt erosion (Heller et al., 2011), land-to-ocean fractional coverage on planets (Spiegel et al., 2008), stellar irradiation in eccentric orbits (Dressing et al., 2010; Spiegel et al., 2010), the formation of water clouds on tidally locked planets (Yang et al., 2013), and the dependence of the ice-albedo feedback on the stellar spectrum and the planetary atmosphere (Joshi and Haberle, 2012; von Paris et al., 2013). These studies show that planets in the HZ of stars with masses M ∗ ≲ M ⊙ can be subject to enormous tidal heating, substantial variations in their semi-major axis, loss of seasons, and tidal locking. Above all, they demonstrate that the circumstellar HZ, though a helpful working concept, does not define a planet’s habitability. For one and the same star, two different planets can have a different HZ, depending on a myriad of bodily and orbital characteristics. Accounting for some of these effects, we can imagine a menagerie of terrestrial worlds. In Fig. 1, we show these planets, all of which are assumed to have a mass 1.5 times that of Earth ( M p = 1.5 M ⊕ ), a radius 1.12 that of Earth , and a host star similar to Gl581 (Mayor et al., 2009). Rather than discussing the exact orbital limits for any of these hypothetical worlds, we shall illustrate here the range of possible scenarios. Irradiation from the star is given by F i , tidal heat flux by F t (computed with the Leconte et al., 2010 tidal equilibrium model), and the critical flux for the planet to initiate a runaway greenhouse effect by F RG (Goldblatt and Watson, 2012). Using the analytical expression given in Pierrehumbert (2010), we estimate F RG = 301 W/m for our test planet, and we compute the HZ boundaries with the model of Kopparapu et al. (2013). Following the approach of Barnes et al. (2013) and Heller and Barnes (2013b), we identify the following members of the menagerie: • Tidal Venus: F t ≥ F RG (Barnes et al., 2013) • Insolation Venus: F i ≥ F RG • Tidal-Insolation Venus: F t < F RG , F i < F RG , F t + F i ≥ F RG • Super-Io: F t > 2 W/m , F t + F i < F RG (hypothesized by Jackson et al., 2008b) • Tidal Earth: < F t < 2 W/m , F t + F i < F RG and within the HZ • Super-Europa: < F t < 2 W/m and beyond the HZ • Earth twin: F t < 0.04 W/m and within the HZ • Snowball Earth: F t < 0.04 W/m and beyond the HZAmong these worlds, a Tidal Venus, an Insolation Venus, and a Tidal-Insolation Venus are uninhabitable by definition, while a Heller and Armstrong – Superhabitable Worlds 2 Dole notes that the term “ecosphere” goes back to Strughold (1955). The radius is derived with an assumed Earth-like rock-to-mass fraction of 0.68 and using the analytical expression provided by Fortney et al. (2007). uper-Io, Tidal Earth, Super-Europa, and an Earth Twin could be habitable. The surface of a Snowball Earth is also uninhabitable because it is so cold that even atmospheric CO would condense, and the warming greenhouse effect could not operate to maintain liquid surface water. Note that the 2 and 0.04 W/m thresholds are taken from the Solar System, where it has been observed that Io’s global volcanism coincides with a surface flux of 2 W/m (Spencer et al., 2000). Moreover, Williams et al. (1997) suggested that tectonic activity on Mars ceased when its endogenic surface flux fell below 0.04 W/m . Concerning the Super-Europa class, note that O’Brien et al. (2002) estimated Europa’s tidal heat flux to about 0.8 W/m . This menagerie illustrates that terrestrial planets can be located in the HZ and yet be uninhabitable. Tidal heating during the planet’s orbital circularization can be an additional heat source that causes a planet to enter a runaway greenhouse state. What is more, tidal heating could make a world habitable beyond the HZ, possibly the Super-Europa planets in our menagerie. Elevated orbital eccentricities would induce tidal friction in these planets, which would transform orbital energy into heat. Such highly eccentric orbits would tend to be circularized, and hence perturbations from other planets or stars in the system would be required to maintain substantial eccentricities. Then tidal heating could partly compensate for the reduced stellar illumination beyond the stellar HZ and potentially maintain liquid water reservoirs.
In exomoons beyond the stellar HZ, tidal heat could even become the major source of energy to allow for liquid water – be it on the surface or below (Reynolds et al., 1987; Scharf, 2006; Debes and Sigurdsson, 2007; Cassidy et al., 2009; Henning et al., 2009; Heller and Barnes, 2013a,b). Imagine a moon the size and mass of Earth in orbit around a planet the size and mass of Jupiter, and assume that this binary orbits a star of solar luminosity at a distance of 1 AU. If the moon is in a wide orbit, say beyond 20 planetary radii from its host, then it will hardly receive stellar reflected light or thermal emission from the planet (Heller and Barnes, 2013a,b), its orbit-averaged stellar illumination will not be substantially reduced by eclipses behind the planet (Heller, 2012), tidal heating will be insignificant , and the moon will essentially be heated by illumination absorbed from Heller and Armstrong – Superhabitable Worlds 3 Only if the moon’s rotation is fast after formation, then it can experience tidal heating in a wide orbit due to the deceleration towards synchronous rotation. In this particular constellation of an Earth-like moon around a Jupiter-like planet at 1 AU from a Sun-like star, this tidal locking takes less than 4.5 Gyr, even in the widest possible orbits (Hinkel and Kane , , , star-planet orbital eccentricity star-planet orbital eccentricity TidalVenus 0.1 0.2 0.3star-planet semi-major axis [AU]0.00.10.20.30.40.50.60.70.8 star-planet orbital eccentricity
TidalVenusInsolation Venus 0.1 0.2 0.3star-planet semi-major axis [AU]0.00.10.20.30.40.50.60.70.8 star-planet orbital eccentricity
TidalVenusInsolation Venus
T I V star-planet orbital eccentricity
TidalVenusInsolation Venus
T I V
S u p e r I o star-planet orbital eccentricity
TidalVenusInsolation Venus
T I V
S u p e r I o
TidalEarthEarthTwin0.1 0.2 0.3star-planet semi-major axis [AU]0.00.10.20.30.40.50.60.70.8 star-planet orbital eccentricity
TidalVenusInsolation Venus
T I V
S u p e r I o
TidalEarthEarthTwin
S u p e rE u r o p a star-planet orbital eccentricity
TidalVenusInsolation Venus
T I V
S u p e r I o
TidalEarthEarthTwin
S u p e rE u r o p a
HZ HZ Snowball Earth
Fig. 1 : Menagerie of terrestrial planets based on stellar irradiation and tidal heating. A planet with a mass of 1.5 M ⊕ in orbit around a star similar to Gl581 is assumed. Dashed lines indicate the boarders of the HZ following Kopparapu et al. (2013). The inner edge is constituted by the runaway greenhouse effect, the outer limit by the maximum greenhouse effect. Note that tidal heating can potentially heat planets beyond the HZ and open a class of Super Europas.he star. But as the moon is virtually shifted into a closer orbit around the planet, illumination from the planet and tidal heating increase, and the total energy flux can become large enough to render the moon uninhabitable. The critical orbit, in which the total energy flux equals the critical flux for the moon to enter the runaway greenhouse effect, has been termed the circumplanetary “habitable edge” (Heller and Barnes, 2013a). Moons inside the habitable edge are uninhabitable. Imagine further that the planet-moon binary is virtually shifted away from the star. Due to the reduced stellar illumination, the habitable edge moves inward towards the planet because tidal heat and illumination from the planet can outbalance the loss of stellar illumination. When the planet-moon system is shifted even beyond the stellar HZ, then the moon will need to be close enough to the planet such that it will prevent transition into a snowball state. In this sense, giant planets beyond the stellar HZ have their own circumplanetary HZ, defined by illumination from the planet and tidal heating in the moon. In Fig. 2, we illustrate this scenario for two different orbital eccentricities of the planet-moon binary, 0.01 and 0.001. The abscissa denotes stellar distance of the planet-moon system, and the ordinate shows the distance between the planet and its satellite. Green areas denote orbits, in which the total flux – composed of stellar plus planetary illumination and tidal heating – varies between the minimum and maximum energy flux ( S eff,MaxGr and S eff,MoiGr , respectively) identified by Kopparapu et al. (2013) to define the solar HZ. To compute the total energy flux, we chose the same model as in Heller and Barnes (2013a) . As the planet-moon system is assumed at increasing stellar distances, the habitable edge (red line) moves closer to the planet. Ultimately, beyond the stellar HZ, the satellite must be closer to its planet than a certain maximum distance such that it receives enough tidal heating. Moving to the outer regions of the star system, stellar irradiation vanishes and tidal heat becomes the dominant source of energy. Comparison of the two stripes in Fig. 2 indicates that moons in orbits with only small orbital eccentricities would need to be closer to the planet to experience substantial tidal heating. Note that the orbital eccentricities of the Galilean satellites around Jupiter are all larger than 0.001, and that Titan’s eccentricity around Saturn is 0.0288. While the reason for Titan’s enhanced eccentricity remains unclear (Sohl et al., 1995), the eccentricities of the major Jovian moons are not free but forced, that is, they are excited by the satellites’ gravitational
Heller and Armstrong – Superhabitable Worlds 4 This model, which includes a tidal theory presented by Leconte et al. (2010), neglects the feedback between tidal heating and the rheology of the moon. Yet, it has been shown that increasing tidal heat can melt a terrestrial body, thereby shutting down tidal heating itself (Zahnle et al. , , , Fig. 2:
Habitable orbits for an Earth-like exomoon around a Jupiter-like planet around a solar luminosity star. Green areas ilustrate orbits, in which the total energy flux of absorbed illumination and tidal heating is above the maximum greenhouse limit S eff,MaxGr and below the moist greenhouse threshold S eff,MoiGr . Within these stripes, orbits with tidal heating rates above 100 W/m are highlighted in orange. The circumplanetary habitable edge, here defined by the moist greenhouse, is indicated with a red line.nteraction (Yoder, 1979). Thus, as rocky and icy exomoons are predicted to exist around extrasolar Jovian planets (Sasaki et al., 2010; Ogihara and Ida, 2012), and if these moons encounter substantial orbital perturbations by other moons, then possibly many habitable exomoons in and beyond the stellar HZ await their discovery.
3. Physical Characteristics of Superhabitable Worlds
All exoplanets detected so far are either subject to stellar irradiation that is very different from the amount or spectral distribution currently received by Earth, or they have masses larger than a few Earth masses. This has led astrobiologists to speculate about extremophile life forms that could cope with more bizarre conditions and maybe survive on a planet that is more hostile than Earth (for a brief review see Dartnell, 2011). The word “bizarre” is here to be understood from an anthropocentric point of view. From a potpourri of habitable worlds that may exist, Earth might well turn out as one that is marginally habitable , eventually bizarre from a biocentric standpoint. In other words, it is not clear why Earth should offer the most suitable regions in the physicochemical parameter space that can be tolerated by living organisms. Such an anthropocentric assumption could mislead research for extrasolar habitable planets because planets could be non-Earth-like but yet offer more suitable conditions for the emergence and evolution of life than Earth did or does, that is, they could be superhabitable. As to how superhabitable planets could look like or under which conditions a world could occupy a more benign zone within the physicochemical volume, we now discuss planetary characteristics that are relevant to planetary habitability (for a broader review see Gaidos et al., 2005; Lammer et al., 2009). These considerations will allow us to deduce quantitative estimates for superhabitable worlds. Instead of elaborating on extremophile or even completely different forms of life, we will still stick to liquid water as a pre-requisite for life and explore more comfortable environments as those found on Earth. Thus, our extensions of habitability towards superhabitability are incremental and still carry a geocentric flavor. What could we understand under a superhabitable world? So far, the term has not been in use, and thus its meaning remains obscure. We propose a context family in which it might be used with reason. • Habitable surface area : An Earth-sized planet on which the surface area that permits liquid water is larger than that of Earth (Spiegel et al., 2009; Pierrehumbert, 2010, § 1.9.1) could be regarded as superhabitable. • Total surface area : A more uneven surface, or simply a larger planet with more space for living forms, could make a planet superhabitable. Due to the higher surface gravity of a more massive planet, however, both characteristics tend to exclude one another. To increase a planet’s habitability, the body cannot be arbitrarily large. As mass typically increases with increasing radius for terrestrial planets, plate tectonics will cease to operate at a certain mass (see below). Moreover, a terrestrial planet that is much heavier than Earth might not get rid of its primordial hydrogen atmosphere, which could hamper the emergence of life (Huang, 1960). A planet slightly larger than Earth can, however, still be regarded superhabitable. Note that Earth, the only inhabited planet known so far, is the largest terrestrial planet in the Solar System. • Land-to-ocean-fraction and distribution : The amount of surface water compared to the amount of land is not only crucial for planetary climate but also for the emergence and diversification of life. Giant continents, as Earth’s Gondwana about 500 Myr ago, may have vast deserts in their interiors, as they are not subject to the moderating effect of oceans. In contrast, planets with more fractionate continents and archipelagos should favor superhabitable environments due to their enhanced richness in habitats. Earth’s shallow waters have a higher biodiversity than the deep oceans (Gray, 1997). Hence, we expect that planets with shallow waters rather than those with deep extended oceans tend to be superhabitable.
What is more, Abe et al. (2011) found that planets dryer than Earth should have wider stellar HZs. At the inner HZ boundary, these “Dune” planets are more tolerant against transition into the runaway greenhouse effect, because their low-humidity equatorial regions can emit above the critical flux, assumed for an atmosphere saturated in water. But still the atmosphere is somewhat opaque in the infrared and thus exerts a global greenhouse effect, which prevents water at poles from freezing. On dry planets at the outer HZ boundary, the low humidity in the tropics hampers formation of clouds and thus snowfall. Dry planets will thus tend to have lower albedos than frozen aqua planets (such as Earth), and they will effectively absorb more stellar illumination and be less susceptible to transitioning into a snowball state. In addition, daytime temperatures will be higher on dryer planets at the outer HZ regions due to their smaller thermal inertia. Combined with the shallow-waters argument, considerations of dry planets thus suggest that planets with a lower fractional surface coverage of water, and with bodies of liquid water that are distributed over many reservoirs rather than combined in
Heller and Armstrong – Superhabitable Worlds 5 Note that Earth is located at the very inner margin of the solar habitable zone (Kopparapu et al. , We here understand superhabitability as a state in which a terrestrial world is generally more habitable than Earth. Conventionally, habitability is considered a binary condition, an “on/off” or “1/0” state, just as a sow is in pig or not. In this sense, we discuss the prospects of a sow being pregnant with several farrows, a state more fertile than only “on” or “1”. ne big ocean, can be considered superhabitable. • Plate tectonics : On Earth, plate tectonics drive the carbon-silicate cycle. In this planet-wide geochemical reaction, near-surface weathering of calcium silicate (CaSiO ) rocks leads to the formation of quartz-like minerals, that is, silicon dioxide (SiO ). At the same time, carbon dioxide (CO , for example from the atmosphere) combines with the residual carbon atoms to form calcium carbonate (CaCO ). When subducted to deeper sediments, elevated pressures and temperatures reverse this reaction, ultimately leading to volcanic outgassing of CO . If this cycle stopped or if it never started on a hypothetical terrestrial, water-rich planet, then silicate weathering would draw down atmospheric CO , which could lead to a global snowball state. On a planet that receives more stellar illumination or has other internal heat sources (for example tidal or radiogenic heating), this collapse could be avoided. The period over which radiogenic heating is strong enough to maintain plate tectonics increases with increasing planetary mass (Walker et al., 1981). To a certain degree, more massive terrestrial planets should thus tend to be superhabitable. However, planets with masses several times that of Earth develop high pressures in their mantle, and the resulting enhanced viscosities make plate tectonics less likely (Noack and Breuer, 2011). Moreover, a stagnant lid forms at the core-mantle-boundary that allows only a reduced heat flow from the core and thereby also frustrates tectonics (Stamenkovi ć et al., 2011). Too high a mass thus impedes plate tectonics and therefore also subduction that is required for the carbon-silicate. Yet, “propensity of plate tectonics seems to have a peak between 1 and 5 Earth masses” (Noack and Breuer, 2011), which, of course, depends on composition and primordial heat reservoir. We conclude that planets with masses up to about 2 M ⊕ tend to be superhabitable from the tectonic point of view. • Magnetic shielding : To allow for surface life, a world must be shielded against high-energy radiation from interstellar space (termed “cosmic radiation”) and from the host star (Baumstark-Khan and Facius, 2002). Too strong an irradiation could destroy molecules relevant for life, or it could strip off the world’s atmosphere, an effect to which low-mass terrestrial worlds are particularly prone (Luhmann et al., 1992). Protection can be achieved by a global magnetic field, whether it is intrinsic as on Earth or extrinsic as may be the case on moons (Heller and Zuluaga, 2013), and by the atmosphere. While a giant planet’s magnetosphere can shield a moon against cosmic rays and stellar radiation, it may itself induce a bombardment of the moon with ionized particles that are trapped in the planet’s radiation belt (see Jupiter; Fischer et al., 1996).
To sustain an intrinsic magnetic field strong enough for protection over billions of years, a terrestrial world needs to have a liquid, rotating, and convecting core. Within Earth, this liquid is composed of molten iron alloys in the outer core, that is, between 800 and 3000 km from its center. Less massive planets or moons will have weaker, short-lived magnetic shields. Williams et al. (1997) estimated a minimum mass of 0.07 M ⊕ for a world under solar irradiation to retain atmospheric oxygen and nitrogen over 4.5 Gyr. Beyond that, the dipole component of the magnetic moment ℳ depends on the core radius r o , rotation frequency Ω , and the thickness D of the core rotating shell where convection occurs via ℳ ∝ r o3 D Ω (Olson and Christensen, 2006; López-Morales et al., 2011), which implies that tidally locked planets and moons in wide orbits may have weak magnetic shielding. • Climatic thermostat : A more reliable global thermostat that impedes ice ages and snowball states would prevent an existing ecosystem from experiencing mass extinctions, which would decelerate or even frustrate evolution. There should exist atmospheric and geological processes whose interplay constitutes a thermostat that makes a planet superhabitable.
Triggered by the recent discoveries of super-Earth planets in or near the stellar HZ, recycling mechanisms of atmospheric CO and CH have been proposed for potentially water-rich planets (Kaltenegger et al., 2013). These planets are predicted to be completely covered by a deep liquid water ocean on top of high-pressure ices and without direct contact to the rocky interior. On such worlds, an Earth-like carbon-silicate cycle cannot possibly operate as there would be no CO weathering. Alternatively, lattices of high-pressure water molecules could trap CO as guest molecules, a chemical substance known as carbon clathrate, and provide an effective climatic thermostat by moderating the H O and CO levels in water-rich super-Earths. A similar clathrate mediation has been shown possible for CH instead of CO (Levi et al., 2013), that is, methane clathrate. Clathrate convection could be an effective mechanism to transport CH and/or CO from a water-rich planet’s silicate-iron core through a high-pressure ice-mantle into the ocean and, ultimately, into the atmosphere (Fu et al., 2010). • Surface temperature : On worlds with substantial atmospheres, in other words with surface pressures P at least as high as those on Mars (where 1 mb ≲ P ≲
10 mb), surface temperatures will generally be different from the thermal equilibrium temperature given by stellar irradiation and planetary albedo alone (Selsis et al., 2007; Leconte et al., 2013). The biodiversity, or the richness of families and genera, seems to have multiplied during warmer epochs on Earth (Mayhew et al., 2012), indicating that worlds warmer than Earth could be more habitable.
A slightly warmer version of Earth might have extended
Heller and Armstrong – Superhabitable Worlds 6 As candidates for such water-rich planets, Levi et al. (2013) propose Kepler-11b, Kepler-18, and Kepler-20b. Kaltenegger et al. (2013) suggest Kepler-62e and f. ropical zones that would allow for more biological variance. This is suggested by both the “cradle model” and the “museum model” used in evolutionary biology. The former approach suggests that rapid diversification occurred recently and rapidly in the tropics, while the latter theory claims that the tropics provide particularly favorable circumstances for slow accumulation and preservation of diversity over time (McKenna and Farrell, 2006; Moreau and Bell, 2013). However, warming Earth does not necessarily yield increased biodiversity. Warming on short timescales causes mass extinction, which can currently be witnessed on Earth. Only a planet that is warm compared to Earth on a Gyr timescale or a world that warms gently over millions and billions of years could have more extended surface regions suitable for liquid water and biodiversity.
On the downside, with fewer temperate zones and no arctic regions, an enormous range of life forms known from Earth could not exist. Above all, a world that is substantially warmer than Earth might have anoxic oceans. On Earth, Oceanic Anoxic Events occurred in periods of warm climate, with average surface temperatures above 25°C compared to pre-industrial 14°C (IPCC, 1995), and resulted in extensive extinctions like the Permian/Triassic around 250 Myr ago (Wignall and Twitchett, 1996). While the concatenation of circumstances that led to extinctions during hot periods is complicated and may reflect problems of Earth’s ecosystem, it cannot be excluded that a world moderately warmer than Earth could be superhabitable. A colder planet, however, can be assumed to be less habitable as less energy input would slow down chemical reactions and metabolism on a global scale. • Biological diversification : An inhabited planet whose flora and fauna are more diverse than they are on Earth could reasonably be termed superhabitable as it empirically shows that its environment is more benign to life. An evolutionary explosion, such as the Cambrian one on Earth, could occur earlier in a planet’s history than it did on Earth – or simply long enough ago to make the respective planet more diversely inhabited than Earth is today. Alternatively, evolution could have progressed faster on other planets. Jumps in diversification or accelerated evolution can be triggered by nearby supernovae and by enhanced radiogenic or ultraviolet radiation. • Multihabitability and panspermia : Stellar systems could be more habitable than the Solar System if there were more than one terrestrial planet or moon in the HZ (Borucki et al., 2013; Anglada-Escudé et al., 2013 ). If, for example, the Moon-forming impact had distributed the mass more evenly between Earth and the Moon, then both objects might have been habitable. Alternatively, in a hypothetical Solar System analog in which only the orbits of Mars and Venus would be exchanged, there could exist three habitable planets. With the possibility of massive moons about giant planets, there might also exist satellite systems with several habitable exomoons. Such stellar systems could be called “multihabitable.” Impacts of comets, asteroids, or other interplanetary debris might trigger exchange of material between those worlds. This exchange could then induce mutual fertilization among multiple habitable worlds, a process known as panspermia (Hoyle and Wickramasinghe, 1981; Weber and Greenberg, 1985). Worlds in multihabitable systems, whether they are planets or moons, could thus be regarded as superhabitable because they have a higher probability to be inhabited. • Localization in the stellar habitable zone : Recent work emphasized that Earth is scraping at the very inner edge of the Sun’s HZ (Kopparapu et al., 2013; Worsworth and Pierrehumbert, 2013). Terrestrial worlds that are located more towards the center of the stellar HZ could be considered superhabitable. These objects would be more resistant against transitioning into a moist or runaway greenhouse state (at the inner edge of the HZ) than Earth is. • Age : From a biological point of view, older worlds can be assumed to be more habitable because Earth experienced a steady increase in biodiversity as it aged (Mayhew et al., 2012). This diversification indicates that non-intelligent life itself is able to modify an environment so as to make it more suitable for its ancestors. A stronger claim has been put forward by what is now known as the Gaia hypothesis, which suggests that the global biosphere as a whole can be regarded as a creature controlling “the global environment to suit its needs” (Lovelock, 1972). Whether considered as a global entity or not, Earth’s ecosystem obviously influences global geochemical processes, which has perpetually led to an increase in biodiversity over billions of years. As an example, note that after the Great Oxygen Event about 2.5 Gyr ago (Anbar et al., 2007) , which was likely induced by oceanic algae, Earth’s surface became more habitable, allowing life to conquer the continents about 360 to Heller and Armstrong – Superhabitable Worlds 7 In the case of GL 667C, it is entertaining to imagine how the structure of that system might influence the development of an intelligent species’ astronomy and human space flight activities, with three potentially habitable worlds and three complete stellar systems to study up-close. Intriguingly, now that the first form of life on Earth able to call itself intelligent it causes a drastic decrease in biodiversity. But even in case evolution typically leads to intelligent life, then if an intelligence destroyed itself, it can be assumed that the respective ecosystem would be able to recover on a Myr or Gyr timescale, of course depending on the magnitude of the caused extinction and the environmental effects left behind by the intelligence. Analyses of chromium isotopes and redox-sensitive metals of drill cores from South Africa by Crowe et al. (2013) indicate a first slight increase in atmospheric oxygen about 3 Gyr ago, which could be related to the emergence of oxygenic photosynthesis.
80 Myr ago (Kenrick and Crane, 1997). Therefore, older planets should tend to be more habitable, or superhabitable if inhabited. • Stellar mass : The mass of a star on the main sequence determines its luminosity, its spectral energy distribution, and its lifetime. The Sun emits most of its light between 400 nm and 700 nm, which is the part of the spectrum visible to the human eye. This is also the spectral range in which plants and other organisms perform oxygenic photosynthesis. On worlds orbiting stars with masses ≲ M ⊙ (known as M dwarfs, Baraffe and Chabrier, 1996), these forms of life might not have the capacity to properly harvest energy for their survival because their stars have their radiation maxima in the infrared. However, Miller et al. (2005) found a free-living cyanobacterium that is able to use near-infrared photons at wavelengths > 700 nm. This discovery, as well as the ability of the oxygenic photosynthetic cyanobacterium Acaryochloris marina to use chlorophyll d for harvesting photons at 750 nm (Chen and Blankenship, 2011), suggests that – of course provided that many other conditions are met – oxygenic photosynthesis on planets orbiting cool stars is possible. Discussing the results of Kiang et al. (2007a,b) and Stomp et al. (2007), Raven (2007) also concluded that photosynthesis can occur on exoplanets in the HZ of M dwarfs. Ultimately, the transmissivity of the planet's atmosphere needs to be appropriate to allow an adequate amount of spectral energy to arrive at the planet’s surface. We will not go deeper in possible extremophile life – extremophile from the standpoint of an Earthling – and, for the time being, consider M stars as less likely hosts for superhabitable planets. However, these reflections show that stars slightly less massive than the Sun could still provide the appropriate spectral energy distribution for photosynthesis. • Stellar UV irradiation : Stellar UV radiation can damage deoxyribonucleic acid (DNA) and thus impede the emergence of life. Today, Earth has a substantial stratospheric ozone column that absorbs solar irradiation almost completely between 200 and 285 nm (UVC) and most of the radiation between 280 and 315 nm (UVB). During the Archean (3.8 to 2.5 Gyr ago), this ozone shield did not exist, but yet life managed to form. We can assume that terrestrial planets with anoxic primordial atmospheres would be more habitable than early Earth if they received less hazardous UV irradiation. M stars remain very active and emit a lot of X-ray and UV radiation during about the first Gyr of their lifetime (Scalo et al., 2007). The activity-driven XUV flux of G stars, such as the Sun, falls off much more rapidly, but their quiescent UV flux is enhanced with respect to K and M dwarfs. What is more, while the UV flux of young M stars is generally much stronger than that of young Sun-like stars, quiescent UV radiation from evolved M dwarfs may be too weak for some essential biochemical compounds to be synthesized (Guo et al., 2010). Thus, they do not seem to offer superhabitable primordial environments. K stars offer a convenient compromise between moderate initial and long-term high-energy radiation. This is supported by considerations of the weighted irradiance spectrum of complex carbon-based molecules, indicating that planets in the HZs of K main sequence stars experience particularly favorable UV environments (Cockell, 1999). This indicates that K dwarf stars are favorable host stars for superhabitable planets. • Stellar lifetime : With a planet’s tendency to be superhabitable increasing with age, the star must burn long enough for existing life forms to evolve. Stars less massive than the Sun have longer lifetimes, and planets or moons can spend more time within the HZ before they transition inside the expanding inner edge (Rushby et al., 2013). Against the background from the two previous items and accounting for the relatively stable spectral radiance once they have settled on the main-sequence, we propose that K dwarfs are more likely to host superhabitable planets than the Sun or M dwarfs.•
Early planetary bombardment : The nature of Earth is closely coupled to its bombardment history. From the lunar forming impact (Cameron and Ward, 1976) to the Late Heavy Bombardment (Gomes et al., 2005), the impact history influenced the surface environment, delivery of organic molecules and volatiles (Chyba and Sagan, 1992; Raymond et al., 2009), and spin/orbital evolution of Earth. This means that the history of Earth's evolution is closely coupled to the orbital dynamics of the planetary system. It is possible that the LHB itself is responsible for Earth's habitability, since it helped deliver water and other volatiles to Earth's surface from farther out in the Solar System.
While the exact cause of the LHB is uncertain, the debate has focused on effects of a continuous, though gradually tapering, history of impacts versus a spiked delivery of material caused by changes in orbital dynamics (Ryder, 2002). Either way, the system architecture played a large role in determining the extent of these impacts (Raymond et al., 2004). Is it possible that a system with more dynamical instability early in a planet’s history would result in a longer, more extensive LHB, or – in the case of a stochastic LHB – a sequence of LHB-type events? Such a history could have little effect on the ongoing evolution of marine or subterranean microbes, yet result in a richer volatile inventory for the host planet or moon, and even encourage multihabitability by enhancing transfer of material between planets in the same system.•
Planetary spin : The initial spin-orbit misalignment, or obliquity, and rotation rate of a planet are largely due to the random events that lead to a planet's formation (Miguel and Brunini, 2010), but the subsequent evolution is tightly coupled to orbital dynamics. Conventional wisdom suggests that Earth is an "ideal" habitable world, since it has a large – and presumably rare
Heller and Armstrong – Superhabitable Worlds 8 moon to stabilize its tilt relative against the orbital forcing from the Sun and other planets (Laskar et al., 1993). However, there are a couple of assumptions in this: 1) that a stable spin is required or even desired for a habitable planet, and 2) that this effect is not mitigated by the crucial role the Moon has had on the evolution of Earth’s spin rate. For example, studies have indicated that Earth's rotation axis could be stable without the presence of a massive satellite (Lissauer et al., 2012), and that such stability is perhaps not desirable (Spiegel et al., 2009; Armstrong et al., 2013). In the latter case, planets with a large tilt can break the ice-albedo feedback at locations further from the star, keeping the planet from entering the "snowball Earth" stage (Williams and Kasting, 1997), and systems with varying tilts could provide slow but steady changes in ecosystems that encourage evolution of life.
It is uncertain whether any given spin rate is desirable for life, as long as it helps keep the surface uniformly habitable, while radical changes in such a spin rate might be detrimental. Did the existence of the Moon encourage life to evolve by changing the diurnal and tidal cycles, or was this an impediment to evolution?
Could moderate changes of a world’s obliquity or rotation rate even force life to adapt to a broader range of environmental conditions, thereby triggering more diverse evolution? Ultimately, is it possible that a terrestrial planet without a massive moon, or a planet more subject to changes in spin, could be superhabitable?•
Orbital dynamics : It is occasionally claimed that Earth is habitable largely owing to its stable, circular orbit. However, climate studies indicate a range of dramatic shifts in climate due to subtle changes in Earth's orbit. These oscillations of obliquity, precession, orbital eccentricity, and rotation period – mainly driven by gravitational interaction with the Sun, the Moon, Jupiter, and Saturn – are known as Milankovitch cycles (Hays et al., 1976; Berger, 1976). The very stability of our orbit, in these cases, makes such events treacherous, as Earth may experience only subtle changes again to help rectify the problem. In fact, it is entirely possible that such stability might put the brakes on biological evolution.
Planets with eccentric orbits would still provide a range of seasonally viable habitats while perhaps acting as a “vaccine” against life-threatening snowball events. Tidal heating in planets or moons on eccentric orbits may even act as a buffer against transition into a global snowball state (Reynolds et al., 1987; Scharf, 2006; Barnes et al., 2009). Planets with large swings in eccentricity can also influence the planetary tilt, which has its own, perhaps positive, impacts on the habitability of a planet. We thus claim that moderate variations in the orbital elements of a terrestrial world need not necessarily hamper the evolution or inhibit the formation of life. Consequently, we see no terminating argument that Earth’s configuration in an almost circular orbit with mild changes in its orbital elements should be considered the most benign situation. Planets that undergo soft variations in their orbital configurations may still be superhabitable. • Atmosphere : The atmosphere of an exoplanet or exomoon is essential to its surface life, as it serves as a mediator of transport for water and, to a lesser degree, nutrients. Atmospheric composition and the gases’ partial pressures will determine surface temperatures and, hence, have a key role in shaping the environment and providing the preconditions for formation and evolution of life.
Just as an example of how an atmosphere different from that of Earth could make an otherwise similar world superhabitable, note that (i.) enhanced atmospheric oxygen concentration allows a larger range of metabolic networks (Berner et al., 2007); (ii.) variations in the atmospheric oxygen concentration seem to constrain the maximum possible body size of living forms (Harrison et al., 2010; Payne et al., 2011); and (iii.) there are no known multicellular organisms that are strictly anaerobic. Today, Earth’s atmosphere contains about 21% oxygen by volume or partial pressure (pO ). Limited by runaway wildfires for pO > 35% and lack of fire at pO < 15% (Belcher and McElwain, 2008), a range of oxygen partial pressures is compatible with an ecosystem broadly similar to Earth’s. Obviously, atmospheric oxygen contents can be much greater than on Earth today, and worlds with oxygen-rich atmospheres could be entitled superhabitable, because of items (i.) – (iii.). While atmospheres less massive than that of Earth would offer weaker shielding against high-energy irradiation from space, weaker balancing of day-night temperature contrasts, retarded global distribution of water, etc., somewhat more massive atmospheres could induce positive effects for habitability. Again, this indicates that planets slightly more massive than Earth should tend to be superhabitable because, first, they acquire thicker atmospheres and, second, their initially extended hydrogen atmospheres can envelope gaseous nitrogen and thereby prevent its loss due to non-thermal ion puck up under an initially strong stellar UV irradiation (Lammer, 2013). Some of the conditions listed in this Section are already, or will soon be, accessible remotely (namely, orbital and bodily characteristics of extrasolar planets or moons), some will be modeled and thereby constrained (such as orbital evolution and composition), and others will remain hidden and induce random effects on habitability (climate history, radiogenic heating, ocean salinity, former presence of meanwhile ejected planets or satellites, etc.) from the viewpoint of an observer. This list is by far not complete, and it is not our goal to provide such a complete list. However, it is supposed to illustrate that a range of physical characteristics and processes can make a world exhibit more benign environments than Earth does. Given the amount of planets that exist in the Galaxy, it is therefore reasonable to predicate that worlds with more comfortable settings for life than Heller and Armstrong – Superhabitable Worlds 9 arth exist. Earth might still be rare, but this does not make the emergence and existence of extraterrestrial life impossible or even very unlikely because superhabitable worlds exist.
4. Conclusions
Utilization of any flavor of the HZ concept implies that a planet is either in the HZ and habitable or outside it and uninhabitable. Resuming our considerations from Section 2, our results are threefold: (i.) Extensions of the HZ concept which include tidal heating, show that planets (“Super-Europas” in our terminology) can exist beyond the HZ and still be habitable. (ii.) Fed by tidal heating, moons of planets beyond the HZ can be habitable. (iii.) Intriguingly, none of all the discussed concepts for the HZ describes a circumstellar distance range that would make a planet a more suitable place for life than Earth currently is.
Terrestrial planets that are slightly more massive than Earth, that is, up to 2 or 3 M ⊕ , are preferably superhabitable due to the longer tectonic activity, a carbon-silicate cycle that is active on a longer timescale, enhanced magnetic shielding against cosmic and stellar high-energy radiation, their larger surface area, a smoother surface allowing for more shallow seas, their potential to retain atmospheres thicker than that of Earth, and the positive effects of non-intelligent life on a planet’s habitability, which can be observed on Earth. Higher biodiversity made Earth more habitable in the long term. If this is a general feature of inhabited planets, that is to say, that planets tend to become more habitable once they are inhabited, a host star slightly less massive than the Sun should be favorable for superhabitability. These so-called K dwarf stars have lifetimes that are longer than the age of the Universe. Consequently, if they are much older than the Sun, then life has had more time to emerge on their potentially habitable planets and moons, and – once occurred – it would have had more time to “tune” its ecosystem to make it even more habitable. The K1V star Alpha Centauri B ( α CenB), member of the closest stellar system to the Sun and supposedly hosting an Earth-mass planet in a 3.235-day orbit (Dumusque et al., 2012), provides an ideal target for searches of planets in the HZ and, ultimately, for superhabitable worlds. Age estimates for α CenB, derived via asteroseismology, chromospheric activity, and gyrochronology (Thévenin et al., 2002; Eggenberger et al., 2004; Thoul et al., 2008; Miglio and Montalbán, 2008; Bazot et al., 2012), show the star to be slightly evolved compared to the Sun, with estimates being 4.85 ±0.5 Gyr, 6.52 ±0.3 Gyr, 6.41 Gyr, 5.2 – α CenB (Forgan, 2012). If life on a planet or moon in the HZ of α CenB evolved similarly as it did on Earth and if this planet had the chance to collect water from comets and planetesimals beyond the snowline (Wiegert and Holman, 1997; Haghighipour and Raymond, 2007), then primitive forms of life could already have flourished in its waters or on its surface when the proto-Earth collided with a Mars-sized object, thereby forming the Moon. Eventually, just as the Solar System turned out to be everything but typical for planetary systems, Earth could turn out everything but typical for a habitable or, ultimately, an inhabited world. Our argumentation can be understood as a refutation of the Rare Earth hypothesis. Ward and Brownlee (2000) claimed that the emergence of life required an extremely unlikely interplay of conditions on Earth, and they concluded that complex life would be a very unlikely phenomenon in the Universe. While we agree that the occurrence of another truly Earth-like planet is trivially impossible, we hold that this argument does not constrain the emergence of other inhabited planets. We argue here in the opposite direction and claim that Earth could turn out to be a marginally habitable world. In our view, a variety of processes exists that can make environmental conditions on a planet or moon more benign to life than is the case on Earth.
Appendix A: Usage and Meaning of Terms Related to Habitability
Discussions about habitability suffer from diverging understanding of the terms “habitability,” “habitable,” etc. Recall that a planet in the stellar illumination HZ, as it is defined by physicists and astronomers (see Section 2), need not necessarily be habitable. It is thus precipitate, if not simply false, to state that the planet Gl581d is “habitable, but not much like home” (Schilling, 2007). Analogously, a world such as a tidally heated moon outside the HZ need not necessarily be uninhabitable. Claiming that “Being inside the habitable zone is a necessary but not sufficient condition for habitability” (Selsis et al., 2007) can be wrong, depending on the meaning of the word “habitable.” If that statement means that habitable planets are in the HZ by definition, then the sentence is tautological. If, however, it means that a planet needs to be in the HZ to provide liquid surface water, then it can be proven wrong.
Confusions from blurred pictures are not restricted to the qualitative. As an example, quantitative problems occur in discussions about the occurrence rate of planets similar to Earth that orbit Sun-like stars. The parameter η ⨁ has been introduced to quantify their abundance. Unfortunately, different understandings of η ⨁ occur in the literature. It has been used as “fraction of stars with Earth-mass planets in the habitable zone” (Howard et al., 2009), “the fraction of Sun-like stars that have planets like Earth” (Catanzarite and Shao, 2011), “the fraction of Sun-like stars with Earth-like planets in their habitable zones” (O’Malley-James et al., 2012), “the fraction of habitable planets for all Sun-like stars” (Catanzarite and Shao, 2011), “the Heller and Armstrong – Superhabitable Worlds 10 raction of Sun-like stars that have at least one planet in the habitable zone” (Lunine et al., 2008), the “frequency of Earth-mass planets in the habitable zone” (Wittenmyer et al., 2011), the fraction of “Earth-like planets with M sin i = 0.5-2M Earth and P < 50 days” (Howard et al., 2010), “the frequency of habitable planets orbiting M dwarfs” (Bonfils et al., 2013b), “the frequency of 1 < m sin i < 10 M ⊕ planets in the habitable zone of M dwarfs” (Bonfils et al., 2013a), “the frequency of terrestrial planets in the habitable zone [...] of solar-like stars in our galaxy” (Jenkins, 2012), and “the number of planets with 0.1 M ⨁ < M p < 10 M ⨁ in the 3 Gyr CHZ (a < 0.02AU)” (Agol, 2011). The latter two definitions stand out because Jenkins (2012) restricts η ⨁ to the Milky Way, and Agol (2011) introduces η ⨁ as a total count, and yet, he uses it as a frequency. Intriguingly, (i . ) as it is not clear whether a planet must be similar to Earth to be habitable, (ii.) as the definitions diverge in their reference to the stellar type, and (iii.) as it sometimes remains obscure what “Earth-like planets” are in the respective context, none of these understandings is equivalent to at least one of the others, except for the Howard et al. (2009) and Wittenmyer et al. (2011) explanations. As a consequence, different estimates for η ⨁ must occur. Although physical, observational, and systematic effects play a role, a quantitative divergence of estimates for η ⨁ will remain as long as there is no consensus about the meaning, that is, the usage, of this word or variable. This problem is not physical, but it is a logical consequence of the diverging understanding of η ⨁ . Imagine a situation in which all the authors of the mentioned studies sit around a desk to discuss their values for η ⨁ and the implications! If they were not aware of the meaning/usage drift of “their” respective η ⨁ , then their dialogue would founder on a language problem. The crux of the matter lies in the meaning of any of these terms, which again depends on the context in which any term is used. Following the Austrian philosopher Ludwig Wittgenstein and his
Philosophische Untersuchungen (Wittgenstein, 1953), many logical problems occur when terms are alienated from their ancestral use and then unreasonably applied in other contexts. Ultimately, as astrobiology is an interdisciplinary science, it is exposed to those dangers of confusion and contradiction to a special degree. In this communication, we shall not infringe the use of language and terminology but unravel possible perils. In other words, we ought to be descriptive rather than normative (Wittgenstein, 1953, §124). To answer the question of whether a planet is habitable, it must be clear what we understand a habitable planet to be. And following semantic holism, a doctrine in the philosophy of language, the term “habitable” then is defined by its usage in the language. Appendix B: An Algebraic Approach to Superhabitable Planets
Astronomers have developed an inclination to evaluate habitability in terms of geocentric conditions. Expressions such as “Earth-like,” “Earth analog,” “Earth twin,” “Earth-sized,” and “Earth-mass” are often used to evaluate a planet’s habitability. Although being a natural body of reference, if other inhabited worlds exist – and obviously some scientists assume that and look for them – then it would be presumptuous to claim that they need to be Earth-like or that Earth offers the most favorable conditions. We can use set algebra to discern and display planet families. This somewhat unconventional approach would allow us to identify Earth as one sort of an habitable and inhabited world and to become acquainted with superhabitable worlds.
Appendix
B.1: Set theory
Consider a set T of terrestrial planets. We assume that any solar or extrasolar planet will either be an element of T or not. Planets have been detected with masses of about 5 to 10 Earth masses, and they likely constitute a transitional regime between terrestrial and, as the case may be, icy or gaseous. They may still have their bulk mass in solid form but also have a substantial atmosphere. Nevertheless, we use a sharp classification here for simplicity. We concentrate here on the genuine terrestrial planets. As an example, Earth (e ⊕ ) is a terrestrial planet (e ⊕ ∈ T), whereas Jupiter is not.
The elements of T are the terrestrial planets: T = {t ∈ T | t terrestrial} (see Fig. A). Some of these planets will be habitable and thus be an element of the set of habitable, terrestrial planets H = {h ∈ T | h habitable} (dotted area). The complement of this set is the set of uninhabitable, terrestrial planets U = ¯H = {u ∈ T | u uninhabitable} (blank area). There are no planets that are both habitable and uninhabitable. Hence, the union of H and U is equal to the terrestrial planets: H ∩ U = T. Beyond, there will be a set of Earth-like planets E = {e ∈ T | e Earth-like} (vertically striped area). Our intuition, trained by the usage of the term “Earth-like” in literature, by talks, and conversations, suggests that Earth-like planets are habitable. For the time being, we prefer to take a more general point of view and allow Earth-like planets also to be uninhabitable. E thus overlaps with U in Fig.
Heller and Armstrong – Superhabitable Worlds 11 In this context, M is planetary mass, i is the inclination of the planet’s orbital plane with respect to an Earth-based observer’s line of sight, M
Earth is an Earth-mass, and P is the planet’s orbital period about the star. Here, m is planetary mass and i the inclination of the planet’s orbital plane with respect to an Earth-based observer’s line of sight. In this context, M p is planetary mass, “CHZ” is an abbreviation for the “continuous habitable zone”, and a is the planet’s orbital semi-major axis. (Wittgenstein , . Yet, to be inhabited, a terrestrial planet must also be habitable. Thus, the set I = {i ∈ T | i inhabited} (green area) of inhabited planets is a subset of H: I ⊆ H. Note that the equality is only valid if all the habitable planets were indeed inhabited. It is reasonable to assume that there exists at least one terrestrial planet that is habitable but yet uninhabited. Thus, we can securely state I ⊂ H ⇔ (t ∈ I ⇒ t ∈ H). With Earth being Earth-like, habitable, and inhabited, we have e ⊕ ∈ (H ∩ I ∩ E). Finally, we propose that there exists a set S = {s ∈ T | s superhabitable} (horizontally striped area) of terrestrial planets, whose elements (that is, superhabitable planets) offer more comfortable environments to life than Earth does. From a statistical perspective, this statement reads:A randomly chosen element s ∈ S is more likely to be inhabited than a randomly chosen element e ∈ E. (1)Alternatively, with p being the probability of a planet to be inhabited: p (s) > p (e) (s ∈ S, e ∈ E) (2)In Fig. A, we insinuate sentences (1) and (2) by plotting the relative area of (S ∩ I) to S larger than the relation of (E ∩ I) to E. An equivalent sentence to (2) is |S ∩ I| / |S| = p (s) > |E ∩ I| / |E| = p (e) (s ∈ S, e ∈ E), (3)where |X| is the number of elements, or “cardinality,” of X.
Sentences (1) – (3) say nothing about the absolute number of inhabited worlds from sets E and S, which corresponds to the size of the areas of E and S in Fig. A. Perhaps there are only two superhabitable planets in our galactic neighborhood, both of which are inhabited, and it may be that there are one hundred Earth-like planets in a similar volume, of which, say, ten are inhabited. Then still (2) is true because p(s) = 2/2 = 1 > p (e) = 10/100 = 0.1. But there would be five times as many Earth-like planets with life than there are superhabitable inhabited planets. In debates about habitable planets, it is subliminally assumed that there are more Earth-like inhabited planets than there are non-Earth-like inhabited planets: |E ∩ I| > | Ē ∩ I|. However, the numbers |E ∩ I|, |E ∩ Ī |, |( Ē ∩ T) ∩ I|, and | ( Ē ∩ T) ∩ Ī | are truly not known, say for a local volume of 100 pc about the Sun. There are only the following constraints: |E ∩ I| ≥ Ē ∩ T) ∩ Ī | ≳
30 = |{CoRoT-7b, Kepler-10b, 55Cnc e, Kepler-18b, Kepler-20e, Kepler-20f, Kepler-36b, Kepler-42b, Kepler-42c, Kepler-42d,
Heller and Armstrong – Superhabitable Worlds 12 This question of equality is related to the question how long it took life to occur on Earth after the planet became habitable. In fact, planets may generally become inhabited very shortly after becoming habitable. This would allow one to advocate the I ⊆ H relation or even I = H.
Fig. A:
Set of terrestrial worlds T and subsets. The set membership of Earth e ⊕ ∈ (E ∩ I) is indicated with a symbol. This graphic visualizes our claim that habitable planets (H) need not be Earth-like (E), and that there may well exist a set of superhabitable worlds (S). The cardinality of S may be greater than that of E, and the fraction of planets inside S that are actually inhabited (I, green) may be greater than the fraction of Earth-like, inhabited planets. For this purpose, S is depicted to be larger than E, and (E ∩ I) is chosen to be smaller with respect to E than (S ∩ I) with respect to S.epler-62c, and others}| (Léger et al., 2009; Batalha et al., 2011; Winn et al., 2011; Cochran et al., 2011; Fressin et al., 2012; Carter et al., 2012; Muirhead et al., 2012; Borucki et al., 2013). More terrestrial planet candidates are known, but they lack either radius or mass determinations (for example Gl581d, GJ667Cb to h, GJ1214b, HD 88512, and Alpha Centauri Bb). The possible existence of S has fundamental observational implications. Were it possible to describe S and predict the characteristics of its elements s, as we attempt in this communication, then the search for extraterrestrial life could be made more efficient. Assume two planets were found; one (ê) being Earth-like and another one ( ŝ ) being member of S. Then it would be more reasonable to spend research resources on ŝ rather than on ê in order to find extrasolar life. And intriguingly, ŝ could be less Earth-like than ê. Ultimately, a superhabitable world may already have been detected but not yet noticed as such. Appendix B.2: The principle of mediocracy
The principle of mediocracy claims that, if an item is drawn at random from one of several categories, it is likelier to come from the most numerous category than from any of the other less numerous categories (Section 1 in Kukla, 2010). As an example, consider the cardinalities of two sets A and B were known; |A| < |B| and A ∩ B = ∅ , where ∅ is the empty set. Further, A ∪ B = M = {m | m ∈ A ∨ m ∈ B} is the set of all elements. Then if an arbitrary element ḿ ∈ M were drawn, it would be more likely to come from B than from A. This is all the principle of mediocracy states. In this reading, it comes as a truism. Note that the proportion of A and B, that is, the prior |A| < |B|, is known and it is the probability for the drawing that is inferred: p ( ḿ ∈ B) > p ( ḿ ∈ A).
In a second reading of the principle of mediocracy, and this is the one subliminally applied in modern searches for inhabited planets, the functions of the prior and the drawing are reversed. Here, ḿ (which in our example from Section B.1 is Earth, e ⊕ ) has already been drawn. It is recognized as an element of a certain set (E ∩ I), and it is claimed that this set is more abundant than the other one. In the terrestrial worlds scenario (Section B.1), this ventured conclusion reads “e ⊕ ∈ (E ∩ I) ⇒ |E ∩ I| > | Ē ∩ I|”. We paraphrase it because it is not justified. Given that we have almost no antecedent knowledge of E, Ē , and I, this claim is not logic. What is more, it is not logical to state that the choice of e ⊕ has been random; humans have not chosen the Earth by random from a set T (Mash, 1993). To make things worse, even if we could have chosen e ⊕ randomly from T and if our assumption were correct, then what could we conclude from only one drawing? Numerous drawings, in other words observations and knowledge about inhabitance of many Earth-like and non-Earth-like planets, would be required to reconstruct the prior with statistical significance. Hence, Earth cannot be justified as a reference for astrobiological investigations with the principle of mediocracy. The claim “e ⊕ ∈ (E ∩ I) ⇒ |E ∩ I| > | Ē ∩ I|” remains arbitrary and current searches for life might not be designed optimally.
To conclude, the principle of mediocracy cannot explain why Earth should be considered a particularly benign, inhabited world. When applied to our set of terrestrial worlds, the principle simply states that a randomly chosen world most likely comes from the most numerous subset of worlds. In this understanding, the cardinality of the subsets of terrestrial worlds is the prior – it is know before the drawing – and the probability of affiliation with any subset can be predicted. Yet, concluding that inhabited worlds are most likely Earth-like is not logical, because, first, the roles of the prior (here: the inhabited worlds) and the drawing (here: Earth) are reversed and, second, Earth has not been drawn (by whom?) at random.
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Heller and Armstrong – Superhabitable Worlds 18 .7. FORMATION, HABITABILITY, AND DETECTION OF EXTRASOLAR MOONS (Helleret al. 2014) 280
Contribution:RH invited all co-authors to this review, structured the content of this manuscript, and moderated thecorrespondence among the co-authors. RH guided the literature research, contributed to the calcula-tions illustrated in Figs. 7, 8, and 13, created Figs. 1, 7, 8, and 13, led the writing of the manuscript,and served as a corresponding author for the journal editor and the referees. RH also created an illus-tration of Europa, Enceladus, Ganymede, and Titan (similar to Fig. 1 in the paper) that was chosen asa journal cover for the September 2014 issue of
Astrobiology ormation, Habitability, and Detection of Extrasolar Moons
René Heller , Darren Williams , David Kipping , Mary Anne Limbach , Edwin Turner , Richard Greenberg , Takanori Sasaki , Émeline Bolmont , Olivier Grasset , Karen Lewis , Rory Barnes , Jorge I. Zuluaga Abstract
The diversity and quantity of moons in the Solar System suggest a manifold population of natural satellites to exist around extrasolar planets. Of peculiar interest from an astrobiological perspective, the number of sizable moons in the stellar habitable zones may outnumber planets in these circumstellar regions. With technological and theoretical methods now allowing for the detection of sub-Earth-sized extrasolar planets, the first detection of an extrasolar moon appears feasible. In this review, we summarize formation channels of massive exomoons that are potentially detectable with current or near-future instruments. We discuss the orbital effects that govern exomoon evolution, we present a framework to characterize an exomoon’s stellar plus planetary illumination as well as its tidal heating, and we address the techniques that have been proposed to search for exomoons. Most notably, we show that natural satellites in the range of 0.1 to 0.5 Earth mass (i.) are potentially habitable, (ii) can form within the circumplanetary debris and gas disk or via capture from a binary, and (iii.) are detectable with current technology.
Key Words : Astrobiology – Extrasolar Planets – Habitability – Planetary Science – Tides
1. Introduction
Driven by the first detection of an extrasolar planet orbiting a Sun-like star almost 20 years ago (Mayor and Queloz, 1995), the search for these so-called exoplanets has nowadays achieved the detection of over one thousand such objects and several thousand additional exoplanet candidates (Batalha et al., 2013). Beyond revolutionizing mankind’s understanding of the formation and evolution of planetary systems, these discoveries allowed scientists a first approach towards detecting habitats outside the Solar System. Ever smaller and lighter exoplanets were found around Sun-like stars, the record holder now smaller than Mercury (Barclay et al., 2013). Furthermore, ever longer orbital periods can be traced, now encompassing Earth-sized planets in the stellar habitable zones (Quintana et al. 2014) and beyond. While the realm of extrasolar planets is being explored in ever more detail, a new class of objects may soon become accessible to observations: extrasolar moons. These are the natural satellites of exoplanets, and based on our knowledge from the Solar System planets, they may be even more abundant.
Moons are tracers of planet formation, and as such their discovery around extrasolar planets would fundamentally reshape our understanding of the formation of planetary systems. As an example, the most massive planet in the Solar System, Jupiter, has four massive moons – named Io, Europa, Ganymede, and Callisto – whereas the second-most massive planet around the Sun, Saturn, hosts only one major moon, Titan. These different architectures were likely caused by different termination time scales of gas infall onto the circumplanetary disks, and they show evidence that Jupiter was massive enough to open up a gap in the circumsolar primordial gas and debris disk, while Saturn was not (Sasaki et al., 2010).
Besides clues to planet formation, exomoons excite the imagination of scientists and the public related to their possibility of being habitats for extrasolar life (Reynolds et al., 1987; Williams et al., 1997; Heller and Barnes, 2013). This idea has its roots submitted: January 20, 2014 accepted: July 3, 2014 published: August 22, 2014 this draft: August 26, 2014 doi:10.1089/ast.2014.1147 Origins Institute, McMaster University, Department of Physics and Astronomy, Hamilton, ON L8S 4M1, Canada; [email protected] Penn State Erie, The Behrend College School of Science, 4205 College Drive, Erie, PA 16563-0203; [email protected] Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA; [email protected] Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA; [email protected] Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA The Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa 227-8568, Japan; [email protected] Lunar and Planetary Laboratory, University of Arizona, 1629 East University Blvd, Tucson, AZ 85721-0092, USA; [email protected] Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan; [email protected] Université de Bordeaux, LAB, UMR 5804, 33270 Floirac, France; [email protected] CNRS, LAB, UMR 5804, F-33270, Floirac, France Planetology ad Geodynamics, University of Nantes, CNRS, France; [email protected] Earth and Planetary Sciences, Tokyo Institute of Technology, Japan; [email protected] Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195, USA; [email protected] NASA Astrobiology Institute – Virtual Planetary Laboratory Lead Team, USA FACom - Instituto de Física - FCEN, Universidad de Antioquia, Calle 70 No. 52-21, Medellín, Colombia; [email protected] n certain Solar System moons, which may – at least temporarily and locally – provide environments benign for certain organisms found on Earth. Could those niches on the icy moons in the Solar System be inhabited? And in particular, shouldn’t there be many more moons outside the Solar System, some of which are not only habitable beyond a frozen surface but have had globally habitable surfaces for billions of years? While science on extrasolar moons remains theoretical as long as no such world has been found, predictions can be made about their abundance, orbital evolution, habitability, and ultimately their detectability. With the first detection of a moon outside the Solar System on the horizon (Kipping et al., 2012; Heller 2014), this paper summarizes the state of research on this fascinating, upcoming frontier of astronomy and its related fields. To begin with, we dedicate Section 2 of this paper to the potentially habitable icy moons in the Solar System. This section shall provide the reader with a more haptic understanding of the possibility of moons being habitats. In Sect ion
3, we tackle the formation of natural satellites, thereby focussing on the origin of comparatively massive moons roughly the size of Mars. These moons are suspected to be a bridge between worlds that can be habitable in terms of atmospheric stability and magnetic activity on the one hand, and that can be detectable in the not-too-far future on the other hand. Section 4 is devoted to the orbital evolution of moons, with a focus on the basics of tidal and secular evolution in one or two-satellite systems. This will automatically lead us to tidal heating and its effects on exomoon habitability, which we examine in Section 5, together with aspects of planetary evolution, irradiation effects on moons, and magnetic protection. In Section 6, we outline those techniques that are currently available to search for and characterize exomoons. Section 7 presents a summary and Section 8 an outlook.
2. Habitable Niches on Moons in the Solar System
Examination of life on Earth suggests that ecosystems require liquid water, a stable source of energy, and a supply of nutrients. Remarkably, no other planet in the Solar System beyond Earth presently shows niches that combine all of these basics. On Mars, permanent reservoirs of surface water may have existed billions of years ago, but today they are episodic and rare. Yet, we know of at least three moons that contain liquids, heat, and nutrients. These are the Jovian companion Europa, and the Saturnian satellites Enceladus and Titan. Ganymede’s intrinsic and induced magnetic dipole fields, suggestive of an internal heat reservoir and a liquid water ocean, make this moon a fourth candidate satellite to host a subsurface habitat. What is more, only four worlds in the Solar System other than Earth are known to show present tectonic or volcanic activity. These four objects are not planets but moons: Jupiter’s Io and Europa, Saturn’s Enceladus, and Neptune’s Triton. Naturally, when discussing the potential of yet unknown exomoons to host life, we shall begin with an inspection of the Solar System moons and their prospects of being habitats. While the exomoon part of this review is dedicated to the surface habitability of relatively large natural satellites, the following Solar Systems moons are icy worlds that could only be habitable below their frozen surfaces, where liquid water may exist.
Europa is completely surrounded by a global ocean that contains over twice the liquid water of Earth on a body about the size of Earth’s Moon. Its alternative heat source to the weak solar irradiation is tidal friction. Tidal heating tends to turn itself off by circularizing orbits and synchronizing spins. However, Europa’s orbit is coupled to the satellites Io and Ganymede through the Laplace resonance (Peale et al., 1979; Greenberg, 1982). The orbital periods are locked in a ratio of 1:2:4, so their mutual interactions maintain eccentricities. As a result, enough heat is generated within Europa to maintain a liquid subsurface ocean about 10 to 100 km deep, which appears to be linked to the surface. Its surface is marked by dark lines and splotches, and the low rate of impact craters suggests that the surface is younger than 50 Myr (Zahnle et al., 2003). Tectonics produce linear features (cracks, ridges, and bands), and thermal effects produce splotches (chaotic terrain) (Fig. 1). These global scale lineaments roughly correlate with the patterns of expected tidal stress on the ice shell (Helfenstein and Parmentier, 1983; Greenberg et al., 1998) and may record past deviations from uniform synchronous rotation. Double ridges likely form on opposite sides of cracks due to the periodic tidal working over each 80 hr orbit, thereby squeezing up material from the crack and out over the surface. This process would be especially effective if the crack extended from the ocean (Greenberg et al., 1998). Although tidal tension is adequate to crack ice, at depth the overburden pressure counteracts this stress. Therefore, it is unlikely that the ice is thicker than 10 km if ridges form in this way.
The most distinctive crack patterns associated with tidal stress are the cycloids, chains of arcs, each about 100 km long, connected at cusps, and extending often over 1000 km or more. Cycloids are ubiquitous on Europa, and they provided the first observational evidence for a liquid water ocean, because they fit the tidal stress model so well and because adequate tidal distortion of Europa would be impossible if all the water were frozen (Hoppa et al., 1999; Groenleer and Kattenhorn, 2008). Given that Europa’s surface age is less than 1% of the age of the Solar System, and that many cycloids are among the
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 2 reshest features, it seemed evident that the ocean must still exist today. Confirmation came with measurements of Jupiter’s magnetic field near the satellite, which showed distortions consistent with the effect of a near-surface electrical conducting salty ocean (Kivelson et al., 2000). Active plumes observed at the moon’s south pole deliver further evidence of a liquid water reservoir, but it remains unclear whether these waters that feed these geysers are connected to the global subsurface ocean or they are local (Roth et al., 2014).
Large plates of Europa’s surface ice have moved relative to one another. Along some cracks, often hundreds of kilometers long, rifts (called dilation bands) have opened up and filled with new striated ice (Tufts et al., 2000). The dilation can be demonstrated by reconstructing the surface, matching opposite sides, like pieces of a picture puzzle. The apparent mobility of large plates of surface ice shows that the cracks must penetrate to a fluid or low-viscosity layer, again indicating that the ice is less than 10 km thick. Chaotic terrain covers nearly half of Europa's surface (Riley et al., 2000) and appears to have been thermally disrupted, leaving a lumpy matrix between displaced rafts, on whose surfaces fragments of the previous surface are clearly visible. The crust appears to have melted, allowing blocks of surface ice to float, move, and tilt before refreezing back into place (Carr et al., 1998). Only modest, temporary, local or regional concentrations of tidal heat are required for substantial melt-through. If 1% of the total internal heat flux were concentrated over an area about 100 km across, the center of such a region would melt through in only a few thousand years and cause broad exposure over tens of km wide in 10 yr (O'Brien et al., 2000). If the ice were thick, the best prospects for life would be near the ocean floor, where volcanic vents similar to those on the Earth’s ocean floor could support life by the endogenic substances and tidal heat. Without oxygen, organisms would require alternative metabolisms whose abundance would be limited (Gaidos et al., 1999). For a human exploration mission, Europan
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 3
Figure 1 : Europa, Enceladus, Ganymede, and Titan are regarded as potentially habitable moons. Global lineaments on Europa’s surface and ridges on Enceladus indicate liquid water as close as a few kilometers below their frozen surfaces. Ganymede’s surface is much older with two predominant terrains, bright, grooved areas and older, heavily cratered, dark regions. Titan has a dense nitrogen atmosphere and liquid methane/ethane seas on its surface. While the atmosphere is intransparent to the human eye, the lower right image contains information taken in the infrared. Note the different scales! Moon diameters are indicated below each satellite. [Image credits: NASA/JPL/Space Science Institute/Ted Stryk]ife would only be accessible after landing on the surface, penetrating about 10 km of viscous ice, and diving 100 km or more to the ocean floor, where the search would begin for a hypothetical volcanic vent. If the ice is thin enough to be permeable, the odds that life is present and detectable increase. At the surface, oxidants – especially oxygen and hydrogen peroxide – are produced by impacts of energetic charged particles trapped in the Jovian magnetic field (Hand et al., 2007). Although this radiation must also sterilize the upper 10 cm of ice, organisms might be safe below that level. Photosynthesis would be possible down to a depth of a few meters, and the oxidants, along with organic compounds from cometary debris, are mixed to that depth by micrometeoroid impacts. Any active crack, periodically opening and closing with the tide, might allow the vertical flow of liquid water from the ocean and back down. Organisms living in the ice or the crack might take advantage of the access to near-surface oxidants and organics as well as oceanic substances and the flow of warm (0°C) water (Greenberg et al., 2002). Based on the size of the double ridges, cracks probably remain active for tens of thousands of years, and so organisms would have time to thrive. But once a crack freezes, they would need to hibernate in the ice or migrate into the ocean or to another active site. Exposure of water at the surface would allow some oxygen to enter the ocean directly. The gradual build-up of frozen ocean water over the surface exposes fresh ice to the production of oxidants and also buries ever deeper the previously oxygenated ice. Based on resurfacing rates of the various geological processes, oxygen may enter the ocean at a rate of about 3 × moles/yr (Greenberg, 2010), equivalent to the respiration requirements of 3 million tons of terrestrial fish. Hence, this delivery rate could allow a concentration of oxygen adequate to support complex life, which is intrinsically less efficient than microorganisms, due to their extra operational overhead. Moreover, with this oxygen source, an ecosystem could be independent of photosynthesis. In the ocean, interaction of the oxygen with the rocky or clay seafloor could be significant and eventually result in a drastic decrease in the pH of the water (Pasek and Greenberg, 2012). How much this process could affect life depends on the efficiency of the contact between the water and the rock. Moreover, if organisms consume the oxygen fast enough, they could ameliorate the acidification. With its 2634 km in radius, the most massive moon in the Solar System, Ganymede, features old, densely cratered terrain and widespread regions that may have been subject to tectonic resurfacing. The great variety in geologic and geomorphic units has been dated over a range of several billions of years, and shows evidence of past internal heat release. Besides Mercury and Earth, Ganymede is one of only three solid bodies in the Solar System that generate a magnetic dipole field. It also possesses a small moment of inertia factor of 0.3115 (Schubert et al., 2004) , which is indicative of a highly differentiated body. This moon is thought to have (i) an iron-rich core, in which a liquid part must be present to generate the magnetic field, (ii) a silicate shell, (iii) a hydrosphere at least 500 km thick, and (iv) a tenuous atmosphere (Anderson et al., 1996; Kivelson et al., 2002; Sohl et al., 2002; Spohn and Schubert, 2003). There is no evidence of any present geologic activity on Ganymede. Locally restricted depressions, called “paterae”, may have formed through cryovolcanic processes, and at least one of them is interpreted as an icy flow that produced smooth bright lanes within the grooved terrain (Giese et al., 1998; Schenk et al., 2004). This suggests that cryovolcanism and tectonic processes played a role in the formation of bright terrain. There is no geologic evidence on Ganymede that supports the existence of shallow liquid reservoirs. And analyses of magnetic field measurements collected by the Galileo space probe are still inconclusive regarding an interpretation of a global subsurface ocean (Kivelson et al., 1996). The challenge is in the complex interaction among four field components, namely, a possibly induced field, the moon’s intrinsic field, Jupiter’s magnetosphere, and the plasma environment. But still, the presence of an ocean is in agreement with geophysical models, which predict that tidal dissipation and radiogenic energy keep the water liquid (Spohn and Schubert, 2003; Hussmann et al., 2006; Schubert et al. 2010).
While Europa with its relatively thin upper ice crust and a global ocean, which is likely in contact with the rocky ocean floor, has been referred to as a class III habitat (Lammer et al., 2009) , Ganymede may be a class IV habitat. In other words, its hydrosphere is split into (i) a high-pressure ice layer at the bottom that consists of various water-rich ices denser than liquid water, (ii) a subsurface water ocean that is likely not in direct contact with the underlying silicate floor, and (iii) an ice-I layer forming the outer crust of the satellite. In this model, Ganymede serves as an archetype for the recently suggested class of water world extrasolar planets (Kuchner, 2003; Léger et al., 2004; Kaltenegger et al., 2013; Levi et al., 2013). Ganymede’s liquid layer could be up to 100 km thick (Sohl et al., 2010), and it is not clear whether these deep liquid oceans can be habitable. Chemical and energy exchanges between the rocky layer and the ocean, which are crucial for habitability, cannot be ruled out, but they require efficient transport processes through the thick high-pressure icy layer. Such processes are indeed possible (Sohl et al., 2010) but not as clear-cut as the exchanges envisaged for Europa, where they probably prevailed Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 4 This dimensionless measure of the moment of inertia equals I Cal /( M Cal R Cal2 ), where I Cal is Callisto’s moment of inertia, M Cal its mass, and R Cal its radius. This factor is 0.4 for a homogeneous spherical body but less if density increases with depth. See cases 3 and 4 in their Fig. 11. ntil recent times.
Another potential moon habitat is Enceladus. With an average radius of merely 250 km, Saturn’s sixth-largest moon should be a cooled, dead body, if it were not subject to intense endogenic heating due to tidal friction. Contrary to the case of Europa, Enceladus’ orbital eccentricity of roughly 0.0047 cannot be fully explained by gravitational interactions with its companion moons. While perturbations from Dione may play a role, they cannot explain the current thermal flux observed on Enceladus (Meyer and Wisdom, 2007). The current heat flux may actually be a remainder from enhanced heating in the past, and it has been shown that variations in the strength of this heat source likely lead to episodic melting and resurfacing events (B ě hounková et al., 2012). Given the low surface temperatures on Enceladus of roughly 70 K, accompanied by its low surface gravity of 0.114 m/s, viscous relaxation of its craters is strongly inhibited. The existence of a large number of shallow craters, however, suggests that subsurface temperatures are around 120 K or higher and that there should have been short periods of intense heating with rates up to 0.15 W/m (Bland et al., 2012). These conditions agree well with those derived from studies of its tectonically active regions, which yield temporary heating of 0.11 to 0.27 W/m (Bland et al., 2007; Giese et al., 2008). Enceladus’ internal heat source results in active cryovolcanic geysers on the moon’s southern hemisphere (Porco et al., 2006; Hansen et al., 2006). The precise location coincides with a region warm enough to make Enceladus the third body, after Earth and Io, whose geological energy flow is accessible by remote sensing. Observations of the
Cassini space probe during close flybys of Enceladus revealed temperatures in excess of 180 K along geological formations that have now become famous as “tiger stripes” (Porco et al, 2006; Spencer et al., 2008) (Fig. 1). Its geological heat source is likely strong enough to sustain a permanent subsurface ocean of liquid saltwater (Postberg et al., 2011). In addition to water (H O), traces of carbon dioxide (CO ), methane (CH ), ammonia (NH ), salt (NaCl), and Ar have been detected in material ejected from Enceladus (Waite et al., 2006; Waite et al., 2009; Hansen et al., 2011). The solid ejecta, about 90% of which fall back onto the moon (Hedman et al, 2009; Postberg et al., 2011), cover its surface with a highly reflective blanket of µm-sized water ice grains. With a reflectivity of about 0.9, this gives Enceladus the highest bond albedo of any body in the Solar System (Howett et al., 2010).
While Europa’s subsurface ocean is likely global and may well be in contact with the moon’s silicate floor, Enceladus’ liquid water reservoir is likely restricted to the thermally active south polar region. It is not clear if these waters are in contact with the rocky core (Tobie et al., 2008; Zolotov, 2007) or if they only form pockets in the satellite’s icy shell (Lammer et al., 2009).
All of these geological activities, and in particular their interference with liquid water, naturally open up the question of whether Enceladus may be habitable. Any ecosystem below the moon’s frozen ice shield would have to be independent from photosynthesis. It could also not be based on the oxygen (O ) or the organic compounds produced by surface photosynthesis (McKay et al., 2008). On Earth, indeed, such subsurface ecosystems exist. Two of them rely on methane-producing microorganisms (methanogens), which themselves feed on molecular hydrogen (H ) released by chemical reactions between water and olivine rock (Stevens and McKinley, 1995; Chapelle et al., 2002). A third such anaerobic ecosystem is based on sulfur-reducing bacteria (Lin et al, 2006). The H required by these communities is ultimately produced by the radioactive decay of long-lived uranium (U), thorium (Th), and potassium (K). It is uncertain whether such ecosystems could thrive on Enceladus, in particular due to the possible lack of redox pairs under a sealed ocean (Gaidos et al., 1999). For further discussion of possible habitats on Enceladus, see the work of McKay et al. (2008). What makes Enceladus a particularly interesting object for the in-situ search for life is the possibility of a sample return mission that would not have to land on the moon’s surface (Reh et al, 2007; Razzaghi et al., 2007; Tsou et al., 2012). Instead, a spacecraft could repeatedly dive through the Enceladian plumes and collect material that has been ejected from the subsurface liquid water reservoir. This icy moon may thus offer a much more convenient and cheap option than Mars from which to obtain biorelevant, extraterrestrial material. What is more, once arrived in the Saturnian satellite system, a spacecraft could even take samples of the upper atmosphere of Titan (Tsou et al., 2012).
With its nitrogen-dominated atmosphere and surface pressures of roughly 1.5 bar, Titan is the only world in the Solar System to maintain a gaseous envelope at least roughly similar to that of Earth in terms of composition and pressure. It is also the only moon beyond Earth’s Moon and the only object farther away from the Sun than Mars, from which a spacecraft has returned in-situ surface images. Footage sent back from the surface by the Huygens lander in January 2005 show pebble-sized, rock-like objects some ten centimeters across (mostly made of water and hydrocarbon ices) on a frozen ground with compression properties similar to wet clay or dry sand (Zarnecki et al., 2005). Surface temperatures around 94 K imply that water is frozen and cannot possibly play a key role in the weather cycle as it does on Earth. Titan’s major atmospheric constituents are molecular nitrogen (N , 98.4 %), CH (1.4 %), molecular hydrogen (0.1 %), and smaller traces of acetylene Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 5 C H ) as well as ethane (C H ) (Coustenis et al., 2007). The surface is hidden to the human eye under an optically thick photochemical haze (Fig. 1), which is composed of various hydrocarbons. While the surface illumination is extremely low, atmospheric C H could act as a mediator and transport the energy of solar ultra-violet radiation and high-energy particles to the surface, where it could undergo exothermic reactions (Lunine, 2010). Intriguingly, methane should be irreversibly destroyed by photochemical processes on a timescale of 10 to 100 million years (Yung et al., 1984; Atreya et al., 2006). Hence, its abundance suggests that it is continuously resupplied. Although possibilities of methanogenic life on Titan have been hypothesized (McKay and Smith, 2005; Schulze-Makuch and Grinspoon, 2005), a biological origin of methane seems unlikely because its C/ C ratio is not enhanced with respect to the Pee Dee Belemnite inorganic standard value (Niemann et al., 2005). Instead, episodic cryovolcanic activity could release substantial amounts of methane from Titan’s crust (LeCorre et al., 2008), possibly driven by outgassing from internal reservoirs of clathrate hydrates (Tobie et al., 2006).
In addition to its thick atmosphere, Titan’s substantial reservoirs of liquids on its surface make it attractive from an astrobiological perspective (Stofan et al., 2007; Hayes et al, 2011). These ponds are mostly made of liquid ethane (Brown et al., 2008) and they feed a weather cycle with evaporation of surface liquids, condensation into clouds, and precipitation (Griffith et al., 2000). Ultimately, the moon’s non-synchronous rotation period with respect to Saturn as well as its substantial orbital eccentricity of 0.0288 (Sohl et al., 1995; Tobie et al., 2005) point towards the presence of an internal ocean, the composition and depth of which is unknown (Lorenz et al., 2008; Norman, 2011). If surface life on Titan were to use non-aqueous solvents, where chemical reactions typically occur with much higher rates than in solid or gas phases, it would have to rely mostly on ethane. However, laboratory tests revealed a low solubility of organic material in ethane and other non-polar solvents (McKay, 1996), suggesting also a low solubility in liquid methane. While others have argued that life in the extremely cold hydrocarbon seas on Titan might still be possible (Benner et al., 2004), the characterization of such ecosystems on extrasolar moons will not be possible for the foreseeable future. In what follows, we thus exclusively refer to habitats based on liquid water.
3. Formation of Moons
Ganymede, the most massive moon in the Solar System, has a mass roughly 1/40 that of Earth or 1/4 that of Mars. Supposing Jupiter and its satellite system would orbit the Sun at a distance of one astronomical unit (AU), Ganymede’s ices would melt but it would not be habitable since its gravity is too small to sustain a substantial atmosphere. Terrestrial sized objects must be larger than 1 to 2 Mars masses, or 0.1 to 0.2 Earth masses ( M ⨁ ), to have a long-lived atmosphere and moist surface conditions in the stellar habitable zones (Williams et al., 1997). Hence, to assess the potential of extrasolar moons to have habitable surfaces, we shall consult satellite formation theories and test their predictions for the formation of Mars- to Earth-sized moons. Ganymede as well as Titan presumably formed in accretion disks surrounding a young Jupiter and Saturn (Canup and Ward, 2002), and apparently there wasn’t enough material or accretion efficiency in the disks to form anything larger. Disks around similar giant planets can form much heavier moons if the disk has a lower gas-to-solid ratio or a high viscosity parameter α (Canup and Ward, 2006). Assuming disks similar to those that existed around Jupiter and Saturn, giant exoplanets with masses five to ten times that of Jupiter might accrete moons as heavy as Mars. The orbit of Triton, the principal moon of Neptune, is strongly suggestive of a formation via capture rather than agglomeration of solids and ices in the early circumplanetary debris disk. The satellite’s orbital motion is tilted by about 156° against the equator of its host planet, and its almost perfectly circular orbit is embraced by various smaller moons, some of which orbit Neptune in a prograde sense (that is, in the same direction as the planet rotates) and others which have retrograde orbits. With Triton being the seventh-largest moon in the Solar System and with Earth- to Neptune-sized planets making up for the bulk part of extrasolar planet discoveries (Batalha et al. 2013; Rowe et al. 2014), the capture of substantial objects into stable satellites provides another reasonable formation channel for habitable exomoons.
Moving out from the Solar System towards moons orbiting extrasolar planets, and eventually those orbiting planets in the stellar habitable zones, we ask ourselves: “How common are massive exomoon systems around extrasolar planets?” Ultimately, we want to know the frequency of massive moons the size of Mars or even Earth around those planets. In the following, we discuss recent progress towards addressing these questions from the formation theory point of view.
Various models for satellite formation in circumplanetary protosatellite disks have been proposed recently. One is called the solids enhanced minimum mass model (Mosqueira and Estrada, 2003a, 2003b; Estrada et al., 2009), another one is an actively supplied gaseous accretion disk model (Canup and Ward, 2002, 2006, 2009), and a third one relies on the viscous spreading
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 6 f a massive disk inside the planet’s Roche limit (Crida and Charnoz, 2012). All these models have been applied to study the formation of the Jovian and Saturnian satellite systems. In the Mosqueira and Estrada model, satellite formation occurs once sufficient gas has been removed from an initially massive subnebula and turbulence in the circumplanetary disk subsides. Satellites form from solid materials supplied by ablation and capture of planetesimal fragments passing through the massive disk. By contrast, in the Canup and Ward picture satellites form in the accretion disk at the very end of the host planet’s own accretion, which should reflect the final stages of growth of the host planets. Finally, the Crida and Charnoz theory assumes that satellites originate at the planet’s Roche radius and then move outward due to gravitational torques experienced within the tidal disk. While the latter model may explain some aspects of satellite formation around a wide range of central bodies, that is, from bodies as light as Pluto to giant planets as heavy as Saturn, it has problems reproducing the Jovian moon system.
The actively supplied gaseous accretion disk model postulates a low-mass, viscously evolving protosatellite disk with a peak surface density near 100 g/cm that is continuously supplied by mass infall from the circumstellar protoplanetary disk. The temperature profile of the circumplanetary disk is dominated by viscous heating in the disk, while the luminosity of the giant planet plays a minor role. “Satellitesimals”, that is, proto-moons are assumed to form immediately from dust grains that are supplied by gas infall. Once a satellite embryo has grown massive enough, it may be pushed into the planet through type I migration driven by to satellite-disk interaction (Tanaka et al., 2002). The average total mass of all satellites resulting from a balance between disposal by type I migration and repeated satellitesimal accretion is universally of the order of 10 -4 M p , where M p is the host planet mass (Canup and Ward, 2006). Beyond a pure mass scaling, Sasaki et al. (2010) were able to explain the different architectures of the Jovian and the Saturnian satellite systems by including an inner cavity in the circum-Jovian disk, while Saturn’s disk was assumed to open no cavity, and by considering Jupiter’s gap opening within the solar disk. Ogihara and Ida (2012) improved this model by tracing the orbital evolution of moonlets embedded in the circumplanetary disk with an N -body method, thereby gaining deeper insights into the compositional evolution of moons. As a gas giant grows, its gravitational perturbations oppose viscous diffusion and pressure gradients in the gas disk and thereby open up a gap in the circumstellar protoplanetary disk (Lin and Papaloizou, 1985). Since the critical planetary mass for the gap opening at 5 - 10 AU is comparable to Jupiter’s mass, it is proposed that Jupiter’s final mass is actually determined by the gap opening (Ida and Lin, 2004). While the theoretically predicted critical mass for gap opening generally increases with orbital radius, Saturn has a mass less than one-third of Jupiter’s mass in spite of its larger orbital radius. It is conventionally thought that Saturn did not open up a clear gap and the infall rate of mass onto Saturn and its protosatellite disk decayed according to the global depletion of the protoplanetary disk. Since core accretion is generally slower at larger orbital radii in the protoplanetary disk (Lin and Papaloizou, 1985), it is reasonable to assume that gas accretion onto Saturn proceeded in the dissipating protoplanetary disk, while Jupiter’s growth was truncated by the formation of a clear gap before the disk lost most of its gas.
Although gap opening may have failed to truncate the incident gas completely, it would certainly reduce the infall rate by orders of magnitude (D’Angelo et al., 2003). After this truncation, the protosatellite disk would be quickly depleted on its own viscous diffusion timescale of about (10 -3 / α )10 years, where α represents the strength of turbulence (Shakura and Sunyaev, 1973) and has typical values of 10 -3 to 10 -2 for turbulence induced by magneto-rotational instability (Sano et al., 2004). On the other hand, if Saturn did not open up a gap, the Saturnian protosatellite disk was gradually depleted on the much longer viscous diffusion timescales of the protoplanetary disk, which are observationally inferred to be 10 to 10 years (Meyer et al., 2007). The different timescales come from the fact that the protosatellite disk is many orders of magnitude smaller than the protoplanetary disk. The difference would significantly affect the final configuration of satellite systems, because type I migration timescales of protosatellites (approximately 10 years) are in between the two viscous diffusion timescales (Tanaka et al., 2002). Jovian satellites may retain their orbital configuration frozen in a phase of the protosatellite disk with relatively high mass at the time of abrupt disk depletion, while Saturnian satellites must be survivors against type I migration in the final less massive disk. The different disk masses reflected by the two satellite systems indicate that their disks also had different inner edges. By analogy with the observationally inferred evolution of the inner edges of disks around young stars (Herbst and Mundt, 2005), Sasaki et al. (2010) assumed (i.) a truncated boundary with inner cavity for the Jovian system and (ii.) a non-truncated boundary without cavity for the Saturnian system. Condition (i.) may be appropriate in early stages of disk evolution with high disk mass and accretion rate and strong magnetic field, while in late stages with a reduced disk mass, condition (ii.) may be more adequate. The Jovian satellite system may have been frozen in a phase with condition (i.), since the depletion timescale Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 7 The recently proposed Grand Tack model (Walsh et al., 2011), however, suggests that Jupiter and Saturn actually opened up a common gap (Pierens and Raymond, 2011). f its protosatellite disk is much shorter than typical type I migration timescales of protosatellites. In this system, type I migration of satellites is terminated by a local pressure maximum near the disk edge. On the other hand, the Saturnian system that formed in a gradually dissipating protosatellite disk may reflect a later evolution phase with condition II. Sasaki et al. (2010) explained how these different final formation phases between Jupiter and Saturn produce the significantly different architectures of their satellite systems, based on the actively supplied gaseous accretion protosatellite disk model (Canup and Ward, 2002). They applied the population synthesis planet formation model (Ida and Lin, 2004, 2008) to simulate growth of protosatellites through accretion of satellitesimals and their inward orbital migration caused by tidal interactions with the gas disk. The evolution of the gas surface density of the protosatellite disk is analytically given by a balance between the infall and the disk accretion. The surface density of satellitesimals is consistently calculated with accretion by protosatellites and supply from solid components in the incident gas. They assumed that the solid component was distributed over many very small dust grains so that the solids can be delivered to the protosatellite disk according to the gas accretion to the central planet. The model includes type I migration and regeneration of protosatellites in the regions out of which preceding runaway bodies have migrated leaving many satellitesimals. Resonant trapping of migrating protosatellites is also taken into account. If the inner disk edge is set, the migration is halted there and the migrated protosatellites are lined up in resonances from the inner edge to the outer regions. When the total mass of the trapped satellites exceeds the disk mass, the halting mechanism is not effective, such that they release the innermost satellite into the planet. Sasaki et al. (2010) showed that in the case of the Jovian system, a few satellites of similar masses were likely to remain in mean motion resonances. These configurations form by type I migration, temporal stopping of the migration near the disk inner edge, and quick truncation of gas infall by gap opening in the solar nebula. On the other hand, the Saturnian system tended to end up with one dominant body in its outer regions caused by the slower decay of gas infall associated with global depletion of the solar nebula. Beyond that, the compositional variations among the satellites was consistent with observations with more rocky moons close to the planet and more water-rich moons in wider orbits.
Ultimately, we want to know whether much more massive satellites, for example the size of Mars, can form around extrasolar giant planets. If the Canup and Ward (2006) mass scaling law is universal, these massive satellites could exist around super-Jovian gas giants. To test this hypothesis, we applied the population synthesis satellite formation model of Sasaki et al. (2010) to a range of hypothetical “super-Jupiters” with tenfold the mass of Jupiter. The results from 100 runs are depicted in Fig. 2, showing that in about 80% of all cases, four to six large bodies are formed. In the right panel, we show the averaged semi-major axes (abscissa) and masses (ordinate), along with the 1 σ standard deviations, of those 36 systems that contain four large satellites. In these four-satellite-systems, the objects reach roughly the mass of Mars, and they are composed of rocky materials as they form in massive protosatellite disks with high viscous heating. We conclude that massive satellites around extrasolar gas giants can form in the circumplanetary disk and that they can be habitable if they orbit a giant planet in the Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 8
Figure 2 : Results after 100 simulations of moon formation around a 10 M Jup planet. Left panel: Multiplicity distribution of the produced satellite systems for satellite masses M s > 10 -2 M ⨁ . Right panel: Averaged M s and semi-major axis ( a ) are shown as filled circles, their standard deviations are indicated by error bars for the 36 four-satellite-systems.tellar habitable zone. Other ways of forming giant moons include collision and capture, both of which require strong dynamical mixing of giant planets and terrestrial objects in a protoplanetary disk. A gas giant planet can migrate both inward and outward through a protoplanetary disk once it grows large enough to clear a gap in the disk. The migrating gas giant is then able to sweep through terrestrial material, as is thought to have occurred in the Solar System (Gomes et. al, 2005), possibly enabling close encounters that might result in the capture of an over-sized moon. In all capture scenarios, the approaching mass must be decelerated below the planet’s escape velocity and inserted into a bound, highly elliptical orbit that can later be circularized by tides (Porter and Grundy, 2011). The deceleration may result from the impactor striking the planet, a circumplanetary disk (McKinnon and Leith, 1995), or a fully formed satellite, but in each of these cases, the intercepting mass – gas or solid – must be comparable to the mass of the impactor to make a significant change in its encounter trajectory. For an impact with the planet to work, the impactor must tunnel deep enough through a planet’s outer atmosphere to release enough energy but still shallow enough to avoid disintegration. An impact with a massive satellite already in orbit around the planet might work if the collision is head-on and the impactor-to-satellite mass ratio is smaller than a few, which makes this formation scenario unlikely. Irrespective of the actual physical reason or history of a possible capture, Porter and Grundy (2011) followed the orbital evolution of captured Earth-mass moons around a range of giant planets orbiting in the HZ of M, G, and F stars. Stellar perturbations on the planet-satellite orbit were treated with a Kozai Cycle and Tidal Friction model, and they assumed their satellites to start the planetary entourage in highly elliptical orbits with eccentricities e ps > 0.85 and apoapses beyond 0.8 times the planetary Hill radius R Hill , p = ✓ M p M ? ◆ / a ? p , (1)where M (cid:1) is the stellar mass, and a (cid:1) p is the orbital semi-major axis of the star-planet system. Initial orbital inclinations and moon spin states were randomized, which allowed investigations of a potential preference for prograde or retrograde orbital stability. Most importantly, their results showed that captured exomoons in stellar HZs tend to be more stable the higher the stellar mass. While about 23% of their captured satellites around Neptune- and Jupiter-sized planets in the HZs of an M0 star remained stable, roughly 45% of such planet-moon binaries in the HZ of a Sun-like star survived and about 65% of similar scenarios in the HZ of an F0 star stabilized over one billion years. This effect is related to the extent of the planet’s Hill sphere, which scales inversely proportional to stellar distance while stellar illuminations goes with one over distance squared. No preference for either pro- or retrograde orbits was found. Typical orbital periods of the surviving moons were 0.9, 2.1, and 3.6 days for planet-moon binaries in the HZs of M, G, and F stars, respectively. Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 9
Figure 3 : Maximum captured mass (ordinate) as a function of escaping mass (abscissa) and encounter distance b = 5 R p , 10 R p , and 15 R p (contours). The curves are calculated from Eq. (3) with the planetary mass set to 0.3 M Jup and the distance from the K star a ✭ = 0.8 AU in both panels. The encounter speed at infinity v ∞ = 0.5 km/s in the left panel and v ∞ = 5 km/s in the right panel. A plausible scenario for the origin of these captures was recently discussed by Williams (2013, W13 hereafter), who showed that a massive moon could be captured if it originally belonged to a binary-terrestrial object (BTO) that was tidally disrupted from a close-encounter with a gas-giant planet. During the binary-exchange interaction, one of the BTO members is ejected while the other is captured as a moon. In fact, it is this mechanism that gives the most reasonable formation scenario for Neptune’s odd principal moon Triton (Agnor and Hamilton, 2006). The first requirement for a successful capture is for the BTO to actually form in the first place. The second requirement is to pass near enough to the planet (inside five to ten planet radii) to be tidally disrupted. This critical distance is where the planet’s gravity exceeds the self-gravity of the binary, and is expressed in W13 as ✓ bR p ◆ . ✓ a B R ◆ ✓ ⇢ p ⇢ ◆ / , (2)where the subscript “1“ refers to the more massive component of the binary, “p“ refers to the planet, and a B is the binary separation. Setting a B / R = 10, and using densities ρ p and ρ appropriate for gaseous and solid planets, respectively, yields b ≈ R p , which shows that the encounter must occur deeper than the orbits of many of the major satellites in the Solar System. A second requirement for a successful binary-exchange is for the BTO to rotate approximately in the encounter plane as it passes the planet. The binary spin thereby opposes the encounter velocity of the retrograde-moving binary mass so that its final velocity may drop below the escape velocity from the planet if the encounter velocity of the BTO at infinity is small, say ≲ m as a function of the escaping mass m , in addition to other parameters such as planet mass M p , impact parameter b , encounter velocity v enc , and the periapsis velocity v peri of the newly captured mass (W13). Expressed in compact form, this relation is m < M p ✓ G m ⇡ b v enc ( v enc v peri ) ◆ / m , (3)with G as Newton’s gravitational constant and secondary expressions for v enc and v peri given in Eqs. (4) and (5) of W13. It is apparent from this expression that the heaviest moons (large m ) will form from the closest encounters (small b ) occurring at low velocity (small v enc ). The dependence on planet mass is not as straightforward. It appears from Eq. (3) that larger planets should capture heavier masses, but the encounter velocity and periapsis velocity in the denominator both increase with M p , making the dependence on planet mass inverted. This is borne out in Fig. 7 of W13, which shows the size of the captured mass to decrease as planet mass increases with the impact parameter b held constant. Therefore, in general, it is easier to capture a moon around a Saturn- or Neptune-class planet than around a Jupiter or a super-Jupiter because the encounter speeds tend to be smaller. As detailed in W13, (see his Figs. 3 to 7), moons the size of Mars or even Earth are possibly formed if the ratio of captured mass to escaping mass is not too large. The limiting ratio depends on encounter details such as the size and proximity of both the planet and the star, as well as the encounter velocity. Yet, a ratio > 10:1 could yield a moon as big as Earth around a Jupiter at 2 AU from the Sun (W13, right side of Fig. 5 therein). Mars-sized moons can possibly form by ejecting a companion the size of Mercury or smaller. Expanding the cases considered in W13, we assume here a Saturn-mass planet in the habitable zone (0.8 AU) around a K-star and vary the encounter distance b as well as the encounter velocity to examine whether velocities as large as 5 km/s make a binary-exchange capture impossible. According to the left panel of Fig. 3, losing an object the size of Mercury ( ≈ M ⨁ ) in a 0.5 km/s encounter would result in the capture of a Mars-sized moon, with tidal disruption occurring as far as 15 R p from the planet. The right panel of Fig. 3 shows that such an exchange is also possible at ten times this encounter velocity, provided the approach distance is reduced by a factor of three. This is a promising result given the broad range of encounter speeds and approach distances expected between dynamically interacting planets in developing planetary systems. To sum up, close encounters of binary-terrestrial objects can reasonably provide a second formation channel for moons roughly the mass of Mars.
4. Orbital Dynamics
Once the supply of the circumplanetary disk with incident gas and dust has ceased several million years after the formation of the planetary system or once a satellite has been captured by a giant planet, orbital evolution will be determined by the
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 10 idal interaction between the planet and the moons, gravitational perturbations among multiple satellites, the gravitational pull from the star, and perturbations from other planets. These effects give rise to a range of phenomena, such as spin-orbit resonances, mean motion resonances among multiple satellites, chaos, ejections, and planet-satellite mergers.
As an example, a giant planet orbiting its host star closer than about 0.5 AU will have its rotation frequency Ω p braked and ultimately synchronized with its orbital motion n ✶ p around the star. This rotational evolution, Ω p = Ω p ( t ) ( t being time), implies an outward migration of the planet’s corotation radius, at which Ω p = n ps , with n ps as the orbital mean motion of a satellite around the planet. Due to the exchange of orbital and rotational momentum via the planet’s tidal bulge raised by a moon, satellites inside the corotation radius will spiral towards the planet and perhaps end in a collision (see Phobos’ fall to Mars; Efroimsky and Lainey, 2007), while satellites beyond the corotation radius will recede from their planets and eventually be ejected (Sasaki et al., 2012). Hence, the evolution of a planet’s corotation radius due to stellar-induced tidal friction in the planet affects the stability of moon systems. Barnes and O’Brien (2002) considered Earth-mass moons subject to an incoming stellar irradiation similar to the solar flux received by Earth and found that these satellites can follow stable orbits around Jupiter-like planets if the host star’s mass is greater than 0.15 M ⨀ , with M ⨀ as one solar mass. From another perspective, satellite systems around giant planets that orbit their stars beyond 0.6 AU should still be intact after 5 Gyr. Cassidy et al. (2009), however, claimed that an Earth-sized moon could even follow a stable orbit around a hot Jupiter if the planet is rotationally synchronized to its orbit around the star. Then the tidal forcing frequencies raised by the moon on the planet would be in a weakly dissipative regime, allowing for a slow orbital evolution of the moon. Combining the results of Barnes and O’Brien (2002) with those of Domingos et al. (2006), Weidner and Horne (2010) concluded that 92% of the transiting exoplanets known at that time (almost all of which have masses between that of Saturn and a few times that of Jupiter) could not have prograde satellites akin to Earth’s Moon. Further limitations on the orbital stability of exomoons are imposed by the migration-ejection instability, causing giant planets in roughly 1-day orbits to lose their moons during migration within the circumstellar disk (Namouni, 2010). Investigations on the dynamical stability of exomoon systems have recently been extended to the effects of planet-planet scattering. Gong et al. (2013) found that planetary systems whose architectures are the result of planet-planet scattering and mergers should have lost their initial satellite systems. Destruction of moon systems would be particularly effective for scattered hot Jupiters and giant planets on eccentric orbits. Most intriguingly, in their simulations, the most massive giant planets were not hosts of satellite systems, if these planets were the product of former planet-planet mergers. In a complimentary study, Payne et al. (2013) explored moons in tightly packed giant planet systems, albeit during less destructive planet-planet encounters. For initially stable planet configurations, that is, planet architectures that avoid planet-planet mergers or ejections, they found that giant exoplanets in closely packed systems can very well harbor exomoon systems. Orbital effects constrain the habitability of moons. Heller (2012) concluded that stellar perturbations would force exomoons into elliptical orbits, thereby generating substantial amounts of tidal heat in the moons. Giant planets in the HZs around low-mass stars have small Hill radii, and so their moons would necessarily have small semi-major axes. As a consequence of both the requirement for a close orbit and the stellar-induced orbital eccentricities, Mars- to Earth-sized exomoons in the HZs of stars with masses below 0.2 to 0.5 M ⨀ would inevitably be in a runaway greenhouse state due to their intense tidal heating. To illustrate the evolution in satellite systems in the following, we simulate the orbital evolution of hypothetical exomoon systems and discuss secular and tidal processes in more detail. Stellar perturbations are neglected. At first, we consider single satellite systems and then address multiple satellite systems. Imagine a hypothetical Earth-mass satellite orbiting a Jupiter-mass planet. Both the tides raised on the planet by the satellite (the planetary tide) and the tides raised on the satellite by the planet (the satellite tide) can dissipate energy from this two-body system. For our computations, we use the constant-time-lag model, which is a standard equilibrium tidal model (Hut, 1981). As in the work of Bolmont et al. (2011,2012), we solve the secular equations for this system assuming tidal dissipation in the moon to be similar to that on Earth, where Δ τ ⨁ = 638s is the time lag between Earth’s tidal bulge and the line connecting the two centers of mass (Neron de Surgy and Laskar, 1997), and k ⨁ × Δ τ ⨁ = 213 s, with k ⨁ as the Earth’s second-degree tidal Love number. For Jupiter, we use the value given by Leconte et al. (2010) ( k × Δ τ Jup = 2.5 × -2 s) and include the evolution of the planetary radius R Jup , of k , and of the radius of gyration squared ( r g,Jup ) = I Jup /( M Jup R Jup2 ), where I Jup is Jupiter’s moment of inertia, following the evolutionary model of Leconte and Chabrier (2012, 2013). This model was scaled so that the radius of Jupiter at the age 4.5 Gyr is equal to its present value.
The left panel of Fig. 4 shows the evolution of the semi-major axes (top panel) and spin-orbit misalignments, or “obliquities” (bottom panel), of various hypothetical Earth-mass satellites around a Jupiter-mass planet for different initial semi-major axes. The orbital eccentricity e ps is zero and the moon is in synchronous rotation, so once the obliquity is zero only the planetary tide influences the evolution of the system. Not shown in these diagrams is the rotation of the Jupiter, Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 11 hich spins up due to contraction and which is modified by the angular momentum transfer from the satellite’s orbit (Bolmont et al., 2011). Figure 4 shows that the corotation radius (ensemble of dashed blueish lines) – defined as the distance at which the satellite’s orbital frequency equals the planetary rotation frequency – shrinks due to the spin-up of the planet. A satellite initially interior to the corotation radius migrates inward, because the bulge of the Jupiter is lagging behind the position of the satellite, and eventually falls onto the planet. A satellite initially exterior to the corotation radius migrates outward. Most notably, an Earth-mass satellite can undergo substantial tidal evolution even over timescales that span the age of present Jupiter. The bottom left panel of Fig. 4 shows the evolution of the obliquities of the satellites, whose initial value is set to 11.5° for all tracks. In all cases, the obliquities rapidly eroded to zero, in less than a year for a very close-in satellite and in a few 10 yr for a satellite as far as 7 × − AU. Thus, in single satellite systems, satellite obliquities are likely to be zero. The timescale of evolution of the satellite’s rotation period is similar to the obliquity evolution timescale, so a satellite gets synchronized very quickly (for e ps ≈
0) or pseudo-synchronized (if e ps is substantially non-zero). The right panel of Fig. 4 shows the evolution of semi-major axes (top panel) and eccentricities (bottom panel) of the same Earth-Jupiter binaries as in the left panel, but now for different initial eccentricities. In the beginning of the evolution, the eccentricity is damped by both the planetary and the satellite tides in all simulations. What is more, all solid tracks start beyond their respective dashed corotation radius, so the satellites should move outward. However, the satellite corresponding to the light blue curve undergoes slight inward migration during the first 10 years, which is due to the satellite tide that causes eccentricity damping. There is a competition between both the planetary and the satellite tides, and when the eccentricity is > 0.08 then the satellite tide determines the evolution causing inward migration. But if e ps < 0.08, then the planetary tide dominates and drives outward migration (Bolmont et al., 2011). The planet with initial e ps = 0.5 undergoes rapid inward migration such that it is interior to the corotation radius when its eccentricity approaches zero. Hence, it then transfers orbital angular momentum to the planet until it merges with its host. We consider now a system of two satellites orbiting the same planet. One moon is a Mars-mass satellite in a close-in orbit at 5 R Jup , the other one is an Earth-mass satellite orbiting farther out, between 10 and 50 R Jup . All simulations include the structural evolution of the Jupiter-mass planet as in the previous subsection. We use a newly developed code, based on the
Mercury code (Chambers, 1999), which takes into account tidal forces between both satellites and the planet as well as
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 12
Figure 4 : Tidal evolution of an Earth-sized moon orbiting a Jupiter-sized planet. Left panels: Semi-major axis (top) and obliquity (bottom) evolution for different initial semi-major axes, while all other initial parameters are equal. The black dashed dotted line in the top panel represents the planetary radius; four overlapping dashed lines indicate the corotation radii. A red dashed line represents the Roche limit. Right panels: Semi-major axis (top) and eccentricity (bottom) evolution for the same system but for different initial eccentricities.ecular interaction. We also consider tides raised by each of the satellites on the planet, assuming these planetary bulges are independent, and the tide raised by the planet in each of the two satellites. The code consistently computes the orbital evolution as well as the evolution of the rotation period of the three bodies and their obliquities (Bolmont et al., 2014). The inner satellite is assumed to have an initial eccentricity of 0.05, an orbital inclination of 2° with respect to the planet’s equatorial plane, and an obliquity of 11.5°. The outer satellite has an initial eccentricity of 0.1, an inclination of 5° to the planet’s equator, and an obliquity of 7°. Tidal dissipation parameters for the satellite and the planet are assumed as above.
Figure 5 shows the results of these simulations, where each panel depicts a different initial semi-major axis of the outer moon. In all runs, the outer massive satellite excites the eccentricity of the inner moon. Excitations result in higher eccentricities if the outer satellite is closer in. Both eccentricities decrease at a similar pace, which is dictated by the distance of the outer and more massive satellite: the farther away the outer satellite, the more slowly the circularization. At the same time, the inner smaller satellite undergoes inward migration due to the satellite tide. When its eccentricity reaches values below a few 10 − , the planetary tide causes outward migration because the moon is beyond the corotation radius (panels 1 to 3). In panels 4 to 9, e ps of the outer moon decreases on timescales > 10 Myr, so the system does not reach a state in which the planetary tide determines the evolution to push the inner satellite outward. Panel 1 of Fig. 5 shows an instability between 10 and 10 yr. The inner satellite eccentricity is excited to values close to the outer satellite eccentricity. At a few 10 yr, e ps of the inner satellite is excited to even higher values, but the satellite tide Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 13
Figure 5 : Evolution of the semi-major axes ( a ps ) and eccentricities ( e ps ) of the two satellites orbiting a Jupiter-mass planet. Red lines correspond to the inner Mars-mass satellite, blue lines to the outer Earth-mass satellite, and black lines represent the corotation distance. Each panel depicts a different initial distance for the outer satellite, increasing from panels 1 to 9.uickly damps it. This rapid circularization is accompanied by a substantial decrease of the inner moon’s semi-major axis. Once the eccentricity reaches values of 10 -2 , evolution becomes smoother. Both eccentricities then decrease and dance around a kind of “equilibrium” at e ps ≈ − . On even longer timescales, both eccentricities are expected to decrease further towards zero (Mardling, 2007). Figure 6 shows the evolution of the obliquities, inclinations, rotation periods, and tidal surface heat flux for both satellites in the scenario of panel 2 of Fig. 5. The inner satellite reaches pseudo-synchronization in < 100 yr, during that time its obliquity decreases from initially 11.5° to about 2°. The outer satellite reaches pseudo-synchronization in a few 1000 yr, while its obliquity decreases to a few 10 − degrees within some 10 yr. Once pseudo-synchronization is reached on both satellites, their obliquities slowly decrease to very low values between 10 and 10 yr in Fig. 6. The inclination of the small inner satellite oscillates around the value of the more massive outer one, whose inclination does not vary significantly during the first 10 yr because the planetary tides are not efficient enough to adjust the orbital planes to the planet’s equatorial plane. Moving on to the tidal heat flux in the right panel of Fig. 6, note that the internal heat flux through the Earth’s surface is around 0.08 W/m (Pollack et al., 1993) and Io’s value is between 2.4 and 4.8 W/m (Spencer et al., 2000). Initially, the tidal heat flux is extremely large in both satellites. While being pseudo-synchronized during the first ≈ yr, the tidal heat flux of the outer satellite decreases from some 10 to about 10 W/m . After 10 yr, the decay of tidal heating in both satellites is due to the circularization and tilt erosion (Heller et al., 2011b) on a 10 Myr timescale. The tidal surface heat flux of the outer satellite is about an order of magnitude lower than Earth’s internal heat flux through its surface, but the tidal heat going through a surface unit on the inner moon is several times the tidal surface heat flux on Io, probably causing our Mars-mass test moon to show strong tectonic activity for as long as its first 100 Myr after formation. The long-term evolution of this system might lead to interesting configurations. For example, the outward migration of the inner satellite decreases the distance between the two satellites, eventually resulting in an orbital resonance (Bolmont et al., 2014). In such a resonance, the eccentricities of both satellites would be excited and thereby reignite substantial tidal heating. Not only would such a resurgence affect the satellites’ potentials to be, remain, or become habitable, but their thermal emission could even prevail against the host planet’s thermal emission and make the moon system detectable by near-future technology (Section 6.4).
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 14
Figure 6 : Evolution of the obliquity (top left), inclination (center left), rotation period (bottom left), and internal heat flux (right) of a Mars- (red lines) and an Earth-like (blue lines) satellite orbiting a Jupiter-mass planet (configuration of panel 2 in Fig. 9). In the bottom left panel, the black line represents the rotation period of the planet, and the dashed lines correspond to the pseudo-synchronization period of the satellites. In the right panel, the dotted black horizontal line corresponds to the internal heat flux of Earth (0.08 W/m ), the dashed black lines correspond to tidal surface heating on Io (2.4 - 4.8 W/m ). . Effects of Orbital and Planetary Evolution on Exomoon Habitability Predictions of exomoons the size of Mars orbiting giant planets (Section 3) and the possible detection of exomoons roughly that size with the
Kepler space telescope or near-future devices (Section 6) naturally make us wonder about the habitability of these worlds. Tachinami et al. (2011) argued that a terrestrial world needs a mass ≳ M ⨁ to sustain a magnetic shield on a billion-year timescale, which is necessary to protect life on the surface from high-energy stellar and interstellar radiation. Further constraints come from the necessity to hold a substantial, long-lived atmosphere, which requires satellite masses M s ≳ M ⨁ (Williams et al., 1997; Kaltenegger, 2000). Tectonic activity over billions of years, which is mandatory to entertain plate tectonics and to promote the carbon-silicate cycle, requires M s ≳ M ⨁ (Williams et al., 1997). Hence, formation theory, current technology, and constraints from terrestrial world habitability point towards a preferred mass regime for habitable moons that can be detected with current or near-future technology, which is between 0.1 and 0.5 M ⨁ . While these mass limits for a habitable world were all derived from assuming a cooling, terrestrial body such as Mars, exomoons have an alternative internal heat source that can retard their cooling and thus maintain the afore-mentioned processes over longer epochs than in planets. This energy source is tidal heating, and its effects on the habitability of exomoons have been addressed several times in the literature (Reynolds et al, 1987; Scharf, 2006; Henning et al, 2009; Heller, 2012; Heller and Barnes, 2013, 2014; Heller and Zuluaga, 2013; Heller and Armstrong, 2014). In their pioneering study, Reynolds et al. (1987) point out the remarkable possibility that water-rich extrasolar moons beyond the stellar HZ could be maintained habitable mainly due to tidal heating rather than stellar illumination. Their claim was supported by findings of plankton in Antarctica lakes, which require an amount of solar illumination that corresponds to the flux received at the orbit of Neptune. Heller and Armstrong (2014) advocated the idea that different tidal heating rates allow exomoons to be habitable in different circumplanetary orbits, depending on the actual distance of the planet-moon system from their common host star.
Assuming an exomoon were discovered around a giant planet in the stellar HZ, then a first step towards assessing its habitability would be to estimate its global energy flux ¯ F globs . Were the moon to orbit its planet too closely, then it might be undergoing a moist or runaway greenhouse effect (Kasting, 1988; Kasting et al., 1993) and be temporarily uninhabitable or even desiccated and uninhabitable forever (Heller and Barnes, 2013, 2014). In addition to the orbit-averaged flux absorbed from the star ( ¯ f ? ), the moon absorbs the star’s reflected light from the planet ( ¯ f r ), the planet’s thermal energy flux ( ¯ f t ), and it can undergo substantial tidal heating ( h s , the amount of tidal heating going through a unit area on the satellite surface). The total global average flux at the top of a moon’s surface is then given by ¯ F globs = ¯ f ? + ¯ f r + ¯ f t + h s + W s = L ⇤ (1 ↵ s , opt )4 ⇡a ⇤ p q e ⇤ p x s ⇡R ↵ p f s a ! + L p (1 ↵ s , IR )4 ⇡a f s q e + h s + W s = R ⇤ T ,? (1 ↵ s , opt ) a ⇤ p q e ⇤ p x s ⇡R ↵ p f s a ! + R SB T , e↵ (1 ↵ s , IR ) a f s q e + h s + W s , (4)where L ✶ and L p are the stellar and planetary luminosities, respectively, α s,opt and α s,IR the satellite’s optical and infrared albedo, respectively (Heller and Barnes, 2014), α p is the planetary bond albedo, a ✶ ,p and a p,s are the star-planet and planet-satellite semi-major axes, respectively, e ✶ ,p and e p,s the star-planet and planet-satellite orbital eccentricities, respectively, R p is the planetary radius, x s the fraction of the satellite orbit that is not spent in the planetary shadow (Heller, 2012), σ SB the Stefan-Boltzmann constant, T p,eff the planet’s effective temperature, h s is the tidal surface heating of the satellite, and W s denotes contributions from other global heat sources such as primordial thermal energy (or “sensible heat”), radioactive decay, and latent heat from solidification. The planet’s effective temperature T p,eff is a function of long timescales as young, hot giant planets cool over millions of years; and it depends on the planetary surface temperature T p,eq , triggered by thermal equilibrium between absorbed and re-emitted stellar light, as well as on the internal heating contributing a surface component T p,int . Hence, T p,eff = ([ T p,eq ] + [ T p,int ] ) ¼ (Heller and Zuluaga, 2013). The factor f s in the denominator of the terms describing the reflected and thermal irradiation from the planet accounts for the efficiency of the flux distribution over the Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 15 It is assumed that the planet is much more massive than its moons and that the barycenter of the planet and its satellite system is close to the planetary center. atellite surface. If the planet rotates freely with respect to the planet, then f = 4, and if the satellite is tidally locked with one hemisphere pointing at the planet permanently, then f = 2 (Selsis et al, 2007). Another factor of 2 in the denominator of the term related to the stellar light reflected from the planet onto the satellite accounts for the fact that only half of the planet is actually starlit. The split-up of absorbed stellar and planetary illumination in Eq. (4), moderated by the different optical and infrared albedos, has been suggested by Heller and Barnes (2014) and is owed to the different spectral regimes, in which the energy is emitted from the sources and absorbed by the moon. A progenitor version of this model has been applied to examine the maximum variations in illumination that moons in wide circumplanetary orbits can undergo, indicating that these fluctuations can be up to several tens of W/m (Hinkel and Kane, 2013). Taking into account stellar illumination and tidal heating, Forgan and Kipping (2013) applied a one-dimensional atmosphere model to assess the redistribution of heat in the Earth-like atmospheres of Earth-like exomoons. In accordance with the results of Hinkel and Kane, they found that global mean temperatures would vary by a few Kelvin at most during a circumplanetary orbit. While global flux or temperature variations might be small on moons with substantial atmospheres, the local distribution of stellar and planetary light on a moon can vary dramatically due to eclipses and the moon’s tidal locking (Heller, 2012; Heller and Barnes, 2013; Forgan and Yotov, 2014). Equation (4) can be used to assess an exomoon’s potential to maintain liquid reservoirs of water, if ¯ F globs is compared to the critical flux F RG for a water-rich world with an Earth-like atmosphere to enter a runaway greenhouse state. We use Eq. (8) of Fortney et al. (2007b) and an Earth-like rock-to-mass fraction of 68% to calculate the two satellites’ radii R s = 0.5 R ⨁ and 1 R ⨁ , respectively, where R ⨁ symbolizes the radius of Earth. Then, using the semi-analytic atmosphere model of Pierrehumbert (2010, see also Eq. 1 in Heller and Barnes, 2013), we compute F RG = 269 W/m and 295 W/m for a Mars-mass and and Earth-mass exomoon, respectively. The closer an exomoon is to its host planet, the stronger tidal heating and the stronger illumination from the planet. Ultimately, there exists a minimum circumplanetary orbit at which the moon transitions into a runaway greenhouse state, thereby becoming uninhabitable. This critical orbit has been termed the circumplanetary “habitable edge”, or HE for short (Heller and Barnes, 2013). Giant planets in the stellar HZ will not have an outer habitable edge for their moons, because moons in wide orbits will essentially behave like free, terrestrial planets. Beyond about 15 R jup , tidal heating will be weak or negligible; illumination from the planet will be very low; eclipses will be infrequent and short compared to the moon’s circumplanetary orbital period; and magnetic effects from the planet will be weak. Thus, by definition of the stellar HZ, exomoons could be habitable even at the planetary Hill radius. As an application of Eq. (4), we assume a planet-moon binary at 1 AU from a Sun-like star. Following Heller and Barnes
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 16
Figure 7 : Circumplanetary habitable edges (HEs) for a Mars-mass (orange lines) and an Earth-mass (blue lines) exomoon orbiting a range of host planets (masses indicated along the ordinate). All system are assumed to be at 1 AU from a Sun-like star. HEs are indicated for four different orbital eccentricities: e ps ∈ {0.0001, 0.001, 0.01, 0.1}. The larger e ps , the further away the moons need to be around a given planet to avoid transition into a runaway greenhouse due to extensive tidal heating. Examples for the orbital distances found in the Jovian and Saturnian satellite systems are indicated with labeled dots.2014), we assume that the exomoon’s albedo to starlight is α s,opt = 0.3, while that due to the planet’s thermal emission is α s,IR = 0.05. This difference is due to the expectation that the planet will emit primarily in the infrared, where both Rayleigh scattering is less effective and molecular absorption bands are more prominent, decreasing the albedo (Kopparapu et al., 2013). Figure 7 shows the HEs for the Mars- and Earth-like satellites described above (orange and blue lines, respectively), assuming four different orbital eccentricities, e ps ∈ {0.0001, 0.001, 0.01, 0.1} from left to right. Along the abscissa, semi-major axis a ps is given in units of planetary radii R p , which depends on planetary mass M p , which is depicted along the ordinate. To intertwine abscissa and ordinate with one another, we fit a polynomial to values predicted for giant planets with a core mass of 10 M ⨁ at 1 AU from a Sun-like star (Table 4 in Fortney et al., 2007a). Examples for Solar System moons in this mass-distance diagram are indicated with black dots. For e ps ≳ h s ∝ R s3 . Thus, although the Earth-like moon is more resistant against a runaway greenhouse state than the Mars-like satellite, in the sense that 295 W/m > 269 W/m , it is more susceptible to enhanced tidal heating in a given orbit. We also see a trend of both suites of HEs moving outward from the planet as M p increases. This is because, firstly, h s ∝ M p3 for M p ≫ M s and, secondly, we measure the abscissa in units of planetary radii and radii of super-Jovian planets remain approximately constant around R Jup . For e ps ≲ M p . Again, this is owed to the strong dependence of h s on M p and explains why the more resistive Earth-like moon, in terms of a transition to a runaway greenhouse state, can be closer to less massive host planets than the Mars-like moon: in low-eccentricity orbits around low-mass planets, tidal heating is negligible and the HEs depend mostly on additional illumination from the planet. Orbital eccentricities of exomoons will typically be small due to tidal circularization of their orbits (Porter and Grundy, 2011; Heller and Barnes, 2013). And given that the highest eccentricity among the most massive satellites in the Solar System is 0.0288 for Titan, Fig. 7 suggests that exomoons the mass of Mars or Earth orbiting a giant planet at about 1 AU around a Sun-like star can be habitable if their orbital semi-major axis around the planet-moon barycenter ≳ R p , with closer orbits still being habitable for lower eccentricities and lower-mass host planets. In this section, we examine the evolution of incident planetary radiation (“inplanation”) on the habitability of exomoons, encapsulated in L p in Eq. (4). Following the prescriptions of Heller and Barnes (2014), we approximate Eq. (4) by considering stellar light, the evolution of inplanation due to the radial shrinking of young gaseous giant planets (Baraffe et al., 2003; Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 17
Figure 8 : Circumplanetary exomoon menageries for Mars-sized satellites around a 10 M Jup host planet at ages of 100 Myr (left panel) and 1 Gyr (right panel). The planet is assumed to orbit in the middle of the HZ of a 0.7 M ⨀ star, and the moon orbits the planet with an eccentricity of 10 − . In each panel, the planet’s position is at (0,0), and distances are shown on logarithmic scales. Note that exomoons in a Tidal Venus or a Tidal-Illumination Venus state are in a runaway greenhouse state and thereby uninhabitable.econte and Chabrier, 2013), and assuming constant tidal heating. While Heller and Barnes (2014) considered several configurations of star-planet-moon systems, we limit our example to that of a Mars-sized moon (see Section 5.1) around a 10 M Jup planet that orbits a K dwarf in the middle of the HZ. Further, we arbitrarily consider a moon with e ps = 10 − , a small value that is similar to those of the Galilean satellites. Heller and Barnes (2014) found that Earth-sized moons orbiting planets less massive than 5 M Jup are safe from desiccation by inplanation, so our example exomoon may be at risk of losing its water from inplanation.
As in the previous subsection, our test exomoon may be considered habitable if the total flux is < 269 W/m . What is more, the duration of the runaway greenhouse is important. Barnes et al. (2013) argued that ≈
100 Myr are sufficient to remove the surface water of a terrestrial body, based on the pioneering work of Watson et al. (1981). While it is possible to recover habitability after a runaway greenhouse, we are aware of no research that has examined how a planet evolves as energy sources drop below the limit. However, we conjecture that the flux must drop well below the runaway greenhouse limit to sufficiently weaken a CO greenhouse to permit stable surface water. In Fig. 8, we classify exomoons with different surface fluxes as a function of semi-major axis of the planet-moon orbit. We assume satellites of Earth-like properties in terms of overall tidal response, and that 90% of the energy is dissipated in a putative ocean, with the remainder in the solid interior. For this example, we use the “constant-phase-lag” tidal model, as described by Heller et al. (2011a), and assume a tidal quality factor Q s of 10, along with k = 0.3. We classify planets according to the scheme presented by Barnes and Heller (2013) and Heller and Barnes (2014), which the authors refer to as the “exomoon menagerie”. If tidal heating alone is strong enough to reach the runaway greenhouse limit, the moon is a “Tidal Venus” (Barnes et al., 2013), and the orbit is colored red. If the tidal heating flux is less, but the total flux is still sufficient to trigger the runaway greenhouse, the moon is a “Tidal-Illumination Venus,” and the orbit is orange. These two types of moons are uninhabitable. If the surface flux from the rocky interior is between the runaway limit and that observed on Io ( ≈ ; Veeder et al., 1994; Spencer et al., 2000), then we label the moon a “Super-Io” (see Jackson et al., 2008; Barnes et al., 2009a, 2009b), and the orbit is yellow. If the tidal heating of the interior is less than Io’s but larger than a theoretical limit for tectonic activity on Mars (0.04 W/m ; Williams et al., 1997), then we label the moon a “Tidal Earth,” and the orbit is blue. For lower tidal heat fluxes, the moon is considered Earth-like, and the orbit is green. The outer rim of this sphere containing Earth-like moons is determined by stability criteria found by Domingos et al. (2006). Satellites orbiting their planet in the same direction as the planet orbits the star, that is, in a prograde sense, are bound to the planet as far as about 0.49 R Hill,p . Moons orbiting their planet in the opposite direction, in a retrograde sense, can follow stable orbits out to 0.93 R Hill,p , depending on eccentricities.
The left panel of Fig. 8 shows these classes for a 100 Myr old planet, the right after 1 Gyr. As expected, all boundaries are the same except for the Tidal-Illumination Venus limit, which has moved considerably inward (note the logarithmic scale!). This shrinkage is due to the decrease in inplanation, as all other heat sources are constant . In this case, moons at distances ≲ R Jup are in a runaway greenhouse for at least 100 Myr, assuming they formed contemporaneously with the planet, and thus are perilously close to permanent desiccation. Larger values of planet-moon eccentricity, moon mass, and/or moon radius increase the threat of sterilization. Thus, should our prototype moon be discovered with a ps ≲ R Jup , then it may be uninhabitable due to an initial epoch of high inplanation.
Beyond irradiation and tidal heating, the magnetic environments of moons determine their surface conditions. It is now widely recognized that magnetic fields play a role in the habitability of exoplanetary environments (Lammer et al., 2010). Strong intrinsic magnetic fields can serve as an effective shield against harmful effects of cosmic rays and stellar high energy particles (Grießmeier et al., 2005). Most importantly, an intrinsic magnetic shield can help to protect the atmosphere of a terrestrial planet or moon against non-thermal atmospheric mass loss, which can obliterate or desiccate a planetary atmosphere, see Venus and Mars (Zuluaga et al., 2013). To evaluate an exomoon’s habitability, assessing their magnetic and plasmatic environments is thus crucial (Heller and Zuluaga, 2013).
To determine the interaction of a moon with the magnetic and plasmatic environment of its host planet, we need to estimate the size and shape of the planetary magnetosphere. Magnetospheres are cavities within the stellar wind, created by the intrinsic magnetic field of the planet (Fig. 9). The scale of a magnetosphere is given by the distance between the planetary center and the planetary magnetopause. This so-called “standoff radius” R S has been measured for giant planets in the Solar System (see Table 1). For extrasolar planets, however, R S can only be estimated from theoretical and semi-empirical models, which predict that R S scales with the dynamic pressure of the stellar wind P sw ∝ n sw v sw2 , with n sw as the particle density and v sw as the particle velocity, and with the planetary dipole moment (cid:1) as Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 18 Without external perturbations, tidal heating should also dissipate as the eccentricity is damped, but here we do not consider orbital evolution. S / M / ⇥ P /↵ mag sw . (5)Here, α mag depends on the magnetospheric compressibility, that is, the contribution to internal stress provided by magnetospheric plasma. Values for α mag range from 6 in the case of relatively empty magnetospheres such as that of Earth, Uranus, and Neptune (Zuluaga et al., 2013; Heller and Zuluaga, 2013), to 4 if important magnetospheric plasma sources are present as is the case for Jupiter and Saturn (Huddleston et al., 1998; Arridge et al., 2006). Predictions of R S for the Solar System giant planets are listed in Table 1, together their actually observed dipole moments and present values of the solar wind properties. As a giant planet ages, its internal heat source recedes and thus the planet’s internal dynamo as well as its magnetic field weaken . Yet, the stellar wind also weakens as the stellar activity decreases. The combination of both effects induces a net expansion of the magnetosphere over billions of years. Figure 10 shows the evolution of R S for a Jupiter-analog in the center of the Sun’s HZ, following methods presented by Heller and Zuluaga (2013): M p = M Jup , the planetary core mass M core = 10 M ⨁ , and the planet’s rotation P rot = 10 h. Magnetospheres of HZ giant planets could be very compressed during the first billion years, thereby leaving their moons unshielded. As an example, compare the blue spiral in Figure 10, which denotes R S , with the orbits of the Galilean moons. While an Io-analog satellite would be coated by the planetary magnetic field as early as 500 Myr after formation, a moon in a Europa-wide orbit would be protected after about 2 Gyr, and a satellite in a Ganymede-wide orbit would only be shielded after 3.5 Gyr. This evolutionary effect of an exomoon’s magnetic environment can thus impose considerable constraints on its habitability. Heller and Zuluaga (2013), while focussing on moons around giant planets in the HZ of K stars, concluded that exomoons beyond 20 planetary radii from their host will hardly ever be shielded by their planets’ magnetospheres. Moons in orbits between 5 and 20 R p could be shielded after several hundred Myr or some Gyr, and they could also be habitable from an illumination and tidal point of view. Moons closer to their planets than 5 R p could be shielded during the most hazardous phase of the stellar wind, that is, during the first ≈
100 Myr, but they would likely not be habitable in the first place, due to enormous tidal heating and strong planetary illumination (Heller and Barnes, 2014).
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 19 The evolution of the magnetic field strength could be much more complex if the interplay between the energy available energy for the dynamo and the Coriolis forces are taken into account (Zuluaga and Cuartas, 2012).
Figure 9 : Schematic representation of a planetary magnetosphere. Its roughly spherical dayside region has a standoff radius R S to the planetary center. Between the bow shock and the magnetopause lies the magnetosheat, a region of shocked stellar wind plasma and piled up interplanetary magnetic fields. Inside the magnetosphere, plasma is dragged by the co-rotating magnetic field, thereby creating a particle stream called the “co-rotating magnetoplasma”. Moons at orbital distances < R S (completely shielded, CS) can be subject to this magnetospheric wind and, hence, experience similar effects as unshielded (US) moons exposed to stellar wind. Partially shielded (PS) moons that spend most of their orbital path inside the magnetosphere will experience both effects of interplanetary magnetic fields and stellar wind periodically. .3.1 Unshielded exomoons The fraction of a moon’s orbit spent inside the planet’s magnetospheric cavity defines three different shielding conditions: (i.) unshielded (US), (ii.) partially shielded (PS), and (iii.) completely shielded (CS) (Fig. 9). Depending on its membership to any of these three classes, different phenomena will affect an exomoon, some of which will be conducive to habitability and others of which will threaten life. Assuming that the dayside magnetosphere is approximately spherical and that the magnetotail is cylindrical, a moon orbiting its host planet at a distance of about 2 R S will spend more than 50% of its orbit outside the planetary magnetosphere. To develop habitable surface conditions, such a partially shielded exomoon would need to have an intrinsic magnetic field similar in strength to that required by a terrestrial planet at the same stellar distance (Zuluaga et al., 2013; Vidotto et al., 2013). For moons in the HZ of Sun-like stars, intrinsic magnetic dipole moments larger than about that of Earth would allow for habitable surface conditions. Moons near the HZs around less massive stars, however, which show enhanced magnetic activity, need intrinsic dipole moments that are several times larger to prevent an initial satellite atmosphere from being exposed to the strong stellar wind (Vidotto et al., 2013). As formation models predict that moons can hardly be as massive as Earth (see Section 3), the maximum magnetic field attainable by an hypothetical exomoon may be insufficient to prevent atmospheric erosion (Williams et al., 1997). Perhaps moons with alternative internal heat sources could drive a long-lived, sufficiently strong internal dynamo. Ganymede may serve as an example to illustrate this peril. It is the only moon in the Solar System with a strong intrinsic magnetic field (Ness et al., 1979). With a dipole moment about 2 × − times that of Earth , a Ganymede-like, unshielded moon (see “US” zone in Fig. 9) orbiting a giant planet in the HZ of a Sun-like star would only have a standoff radius of about 1.5 Ganymede radii at an age of 1 Gyr. This would expose a potentially thin and extended atmosphere of such a moon to the high-energy particles from the stellar wind, eventually stripping off the whole atmosphere. Inside the planetary magnetosphere, the main threat to a partially shielded exomoon is that of the so-called co-rotating magnetoplasma (Neubauer et al., 1984). Magnetospheric plasma, which is dragged by the planetary magnetic field to obtain the same rotational angular velocity as the planet, will produce a “magnetospheric wind”. Around gas giants akin to those in the Solar System, the circumplanetary angular velocity of the co-rotating plasma is typically much higher than the Keplerian
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 20
Table 1 : Measured magnetic properties of giant planets in the Solar System. (cid:1) : Magnetic dipole moment (Bagenal, 1992; Guillot, 2005). R Sobs : observed range of magnetospheric standoff radii (Arridge et al., 2006; Huddleston et al., 1998). Note that standoff distances vary with solar activity. R Sth : range of predicted average standoff distances. Predicted standoff distances depend on magnetospheric compressibility (see text). R Sth,HZ : extrapolation of R Sth assuming that the planet would be located at 1 AU from the Sun. Semi-major axes a ps of selected major moons are shown for comparison. The final two columns depict the shielding status (see Fig. 13) of the moons for their actual position in the Solar System and assuming a distance of 1 AU to the Sun.elocity of the moons. Hence, the magnetospheric wind will hit exomoon atmospheres and may erode them in orbits as wide as a significant fraction of the standoff distance. This effect could be as intense as effects of the direct stellar wind. For instance, it has been estimated that Titan, which is partially shielded by Saturn’s magnetosphere, could have lost 10% of its present atmospheric content just because of charged particles from the planet’s co-rotating plasma (Lammer and Bauer, 1993). Towards the inner edge of the HZ around low-mass stars, the stellar wind flux is extremely strong. What is more, the planetary magnetosphere will be small, thereby increasing the density of the magnetospheric wind. Other sources of plasma such as stellar wind particles, planetary ionosphere gasses, and ions stripped off from other moons could also be greatly enhanced. Moons that are completely shielded by the planetary magnetic shield are neither exposed to stellar wind nor to significant amounts of cosmic high-energy particles. What is more, being well inside the magnetosphere, the co-rotating plasma will flow at velocities comparable to the orbital velocity of the moon. Hence, no strong interactions with the satellite atmosphere are expected.
On the downside, planetary magnetic fields can trap stellar energetic particles as well as cosmic rays with extremely high energies. Around Saturn and Jupiter, electrons, protons, and heavy ions with energies up to hundreds of GeV populate the inner planetary magnetospheres within about ten planetary radii, thereby creating what is known as radiation belts. Around Jupiter, as an example, the flux of multi-MeV electrons and protons at a distance of about 15 R Jup to the planet is between 10 and 10 cm − s − (Divine and Garrett, 1983), that is four to six orders of magnitude larger than that of solar high-energy particles received by Earth. Exposed to these levels of ionizing radiation, moon surfaces could absorb about 1 to 10 J kg -1 Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 21
Figure 10 : Evolution of the standoff radius R S (thick blue line) around a Jupiter-like planet in the center of the stellar HZ of a Sun-like star. For comparison, the orbits of the Galilean moons are shown (see legend). Angles with respect the vertical line encode time in Gyr. Time starts at the “12:00” position in 100 Myr and advances in steps of 0.3 Gyr up to the present age of the Solar System. Thin gray circles denote distances in intervals of 5 planetary radii. The filled circle in the center denotes the planetary radius.in -1 (National Research Council, 2000) , which is many orders of magnitude above the maximum tolerable radiation levels for most microorganisms found on Earth (Baumstark-Khan and Facius, 2002). Magnetic field intensities inside compressed magnetospheres will be larger for a given dipole moment of a Solar System giant planet. Moreover, stellar high-energy particle fluxes, feeding radiation belts, will be also larger. As a result, the flux of high energy particles accelerated inside the magnetosphere of giant planets in the HZ could be strongly enhanced with respect to the already deadly levels expected on Jupiter’s moon Europa.
6. Detection Methods for Extrasolar Moons
Within the arena of transiting planets, there are two basic effects which may betray the presence of an exomoon: (i.) dynamical effects which reveal the mass ratio between the satellite and the planet ( M s / M p ), and (ii.) eclipse effects which reveal the radius ratio between the satellite and the star ( R s / R ✶ ). Detecting both effects allows for a measurement of the bulk density and thus allows one to distinguish between, say, icy moons versus rocky moons. Beyond the possibility to detect extrasolar satellites when using the stellar transits of a planet-moon system, several other methods have been proposed over the past decade . Cabrera and Schneider (2007) suggested that an exomoon might induce a wobble of a directly imaged planet’s photocenter and that planet-moon eclipses might be detectable for directly imaged planets (see also Sato and Asada 2010; Pál 2012). Excess emission of transiting giant exoplanets in the spectrum between 1 and 4 µm (Williams and Knacke 2004) or enhanced infrared of terrestrial planets (Moskovitz et al. 2009; Robinson 2011) might also indicate the presence of a satellite. Further approaches consider the Rossiter-McLaughlin effect (Simon et al. 2010; Zhuang et al. 2012), pulsar timing variations (Lewis et al. 2008), microlensing (Han and Han 2002), modulations of a giant planet’s radio emission (Noyola et al. 2014), and the the generation of plasma tori around giant planets caused by the tidal activity of a moon (Ben-Jaffel and Ballester 2014). In particular, the upcoming launch of the James Webb Space Telescope (
JWST ) inspired Peters and Turner (2013) to propose the possibility of detecting an exomoon’s thermal emission. In the following, we discuss the dynamical effects that may reveal an exomoon as well as the prospects of direct photometric transit observations and constraints imposed by white and red noise. Possibilities of detecting extremely tidally heated exomoons via their thermal emission will also be illustrated.
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 22 Figure 11 : Period-ratio versus mass-ratio scatter plot of the Solar System moons. Transit timing and duration variations (TTV and TDV) exhibit complementary sensitivities with the period-ratio. Using the
Kepler timing measurements from Ford et al. (2012), one can see that the tip of the observed distribution is detectable. The above assumes a planetary period of 100 days and a baseline of 4.35 years of
Kepler data.he first technique ever proposed to detect an exomoon comes from Sartoretti and Schneider (1999), which falls into the dynamical category and is now often referred to as transit timing variations, or simply TTV, although it was not referred to as this in the original paper. For a single moon system, the host planet and the companion orbit a common barycenter, which itself orbits the host star on a Keplerian orbit. It therefore follows that the planet does not orbit the star on a Keplerian orbit and will sometimes transit slightly earlier or later depending upon the phase of the moon. These deviations scale proportional to P P × ( M s / M p ) × ( a s / a p ) and can range from a few seconds to up to a few hours for terrestrial moons. The original model of Sartoretti and Schneider (1999) deals with circular, coplanar moons but extensions to non-coplanar and eccentric satellites have been proposed since (Kipping, 2009a, 2011a). Three difficulties with employing TTV in isolation are that, firstly, other effects can be responsible (notably perturbing planets); secondly, the TTV waveform is guaranteed to be below the Nyquist frequency and thus undersampled, making a unique determination of M s untenable; and, thirdly, moons in close orbits blur their planet’s TTV signature as the planet receives a substantial tangential acceleration during each transit (Kipping 2011a). TTV can be thought of as being conceptually analogous to the astrometric method of finding planets, since it concerns changes in a primary’s position due to the gravitational interaction of a secondary. Astrometry, however, is not the only dynamical method of detecting planets. Notably, Doppler spectroscopy of the host star to measure radial velocities has emerged as one of the work horses of exoplanet detection in the past two decades. Just as with astrometry, an exomoon analogy can be devised by measuring changes in a planet’s transit duration, as a proxy for its velocity (Kipping, 2009a). Technically though, one is observing tangential velocity variations rather than those in the radial direction. Transit duration variations due to velocity variations, dubbed TDV-V, scale as ( P P / P s ) × ( M s / M p ) × ( a s / a p ) and vary from seconds to tens of minutes in amplitude for terrestrial satellites. Just as radial velocity is complementary to astrometry, TDV-V and TTV are complementary since TTV is more sensitive to wide-orbit moons (sensitivity scales as ∝ a s ) and TDV is more sensitive to close-orbit moons (sensitivity scales as ∝ a s-1/2 ). Furthermore, should one detect both signals, the ratio of their root-mean-square amplitudes (that is, their statistical scatter) yields a direct measurement of P s , which can be seen via inspection of the aforementioned scalings. This provides a powerful way of measuring P s despite the fact that the signals are undersampled. Finally, TDV-V leads TTV by a π /2 phase shift in amplitude (Kipping 2009a) providing a unique exomoon signature, which even for undersampled data can be detected with cross-correlation techniques (Awiphan and Kerins, 2013). Just as with TTV though, a TDV-V signal in isolation suffers from both model and parameter degeneracies. An additional source of transit duration variations comes from non-coplanar satellite systems, where the planet’s reflex motion causes position changes not only parallel to the transit chord (giving TTV) but also perpendicular to it. These
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 23
Figure 12 : (
Left ) Six simulated transits using LUNA (Kipping, 2011b) of a HZ Neptune around an M2 star with an Earth-like moon on a wide orbit (90% of the Hill radius). The moon can be seen to exhibit “auxiliary transits” and induce TTVs. (
Right ) Same as left, except the moon is now on a close-in orbit (5% of the Hill radius), causing “mutual events”. Both plots show typical
Kepler noise properties for a 12th-magnitude star observed in short-cadence.ariations cause the apparent impact parameter of the transit to vary leading to transit impact parameter induced transit duration variations, known as TDV-TIP (Kipping, 2009b). This is generally a small effect of order of seconds but can become much larger for grazing or near-grazing geometries. Should the effect be detected, it can be shown to induce a symmetry breaking that reveals whether a moon is prograde or retrograde in its orbit, thereby providing insights into the history and formation of the satellite. For all of the aforementioned techniques, multiple moons act to cancel out the overall timing deviations, except for resonant cases. In any case, the limited number of observables (two or three) make the detection of multiple moons untenable with timing data alone. Despite this, there are reasons to be optimistic for the detection of a large quasi-binary type exomoon. Kipping et al. (2009) and Awiphan and Kerins (2013) estimated that the
Kepler space telescope is sensitive to telluric moons. Recent statistics on the timing sensitivity of
Kepler by Ford et al. (2012) reveal that the tip of the known population of moons in our own Solar System would be detectable with
Kepler too, as shown in Fig. 11.
In terms of detectable habitable exomoons, Kipping et al. (2009) took into account orbital stability criteria from Barnes and O’Brien (2002) and found that moons more massive than about 0.2 M ⨁ and orbiting a Saturn-mass planet in the center of the stellar HZ can be detectable with Kepler -class photometry if the star is more massive than about 0.3 M ⨀ and has a Kepler magnitude ≲
12 (Fig. 6 of Kipping et al, 2009). More massive stars would need to be brighter because the planetary transit and its TTV and TDV signals become relatively dimmer. Simulations by Awiphan and Kerins (2013) suggest that moons in the habitable zone of a 12.5
Kepler magnitude M dwarf with 0.5 times the mass of the Sun would need to be as massive as 10 M ⨁ and to orbit a planet less massive than about 1/4 the mass of Saturn to be detectable via their planet’s TTV and TDV. Such a system would commonly be referred to as a planetary binary system rather than a planet-moon binary because the common center of gravity would be outside the primary’s radius. From this result and similar results (Lewis, 2011b), the detection of a HZ moon in the ≈ Kepler data is highly challenging.
While dynamical effects of the planet can yield the system’s orbital configuration, a transit of the moon itself not only provides a measurement of its radius but also has the potential to distort the transit profile shape leading to the erroneous derivation of TTVs and TDVs. Modeling the moon’s own transit signal is therefore both a critical and inextricable component of hunting for exomoons.
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 24
Figure 13 : Transit of the almost Jupiter-sized planet candidate KOI189.01 around a star of about 0.7 solar radii. Gray dots indicate the original phase-folded
Kepler light curve, the black dashed line indicates a model for the transit assuming a planet only, and the red line assumes an Earth-sized moon in an orbit that is 15 planetary radii wide. Black dots in the right panel indicate data that is binned to 60 minutes. The photometric orbital sampling effect appears in the right panel as a deviation between the red solid and the black dashed line about 6.5 hours before the planetary mid-transit.
Exomoons imprint their signal on the stellar light curve via two possible effects. The first is that the moon, with a wide sky-projected separation from the planet, transits the star and causes a familiar transit shape. These “auxiliary transits” can occur anywhere from approximately 93% of the Hill radius away from the planet (Domingos et al., 2006) to being ostensibly on-top of the planet’s own transit (see Fig. 12 for examples). Since the phase of the moon will be unique for each transit epoch, the position and duration of these auxiliary transits will vary, which can be thought of as TTV and TDV of the moon itself, magnified by a factor of M p / M s . The position and duration variations of the moon will also be in perfect anti-phase with the TTVs and TDVs of the planet, providing a powerful confirmation tool. The second eclipse effect caused by exomoons are so-called “mutual events”. This is where the moon and planet appear separated at the start of a transit but then the moon eclipses (either in front or behind) the planet at some point during the full transit duration. This triple-overlap means that the amount of light being blocked from the star actually decreases during the mutual event, leading to what appears to be an inverted-transit signal (Fig. 12). Mutual events can occur between two planets too (Ragozzine and Holman, 2010), but the probability is considerably higher for a moon. The probability of a mutual event scales as ∝ R p / a s and thus is highest for close-in moons. One major source of false-positives here are starspots crossings (Rabus et al., 2009), which appear almost identical, but of course do not follow Keplerian motion. Accounting for all of these eclipse effects plus all of the timing effects is arguably required to mount a cogent and expansive search for exomoons. Self-consistent modeling of these phenomena may be achieved with full “photodynamic” modeling, where disc-integrated fluxes are computed at each time stamp for positions computed from a three-body (or more) dynamical model. Varying approaches to moon-centric photodynamic modeling exist in the literature including a re-defined TTV for which the photocenter (Simon et al., 2007), circular/coplanar photodynamical modeling (Sato and Asada, 2009; Tusnski and Valio, 2011), and full three-dimensional photodynamic modeling (Kipping, 2011b) have been used.
Another way of detecting an exomoon’s direct transit signature is by folding all available transits of a given system with the circumstellar orbital period into a single, phase-folded light curve. If more than a few dozen transits are available, then the satellites will show their individual transit imprints (Heller 2014). Each moon causes an additional transit dip before and after the planetary transit and thereby allows measurements of its radius and planetary distance. This so-called orbital sampling effect loses the temporal information content exploited in rigorous photodynamical modeling, meaning that the satellite's period cannot be determined directly. And without the satellite period, one cannot compute the planetary density via the trick described by Kipping (2010). Despite this, the computational efficiency of the OSE method makes it attractive as a quick method for identifying exomoon candidate systems, perhaps in the readily available
Kepler data or the upcoming
Plato space mission. But for such a detection, dozens of transits in front of a photometrically quiet M or K dwarf are required (Heller 2014). What is more, to ultimately confirm the presence of an exomoon, additional evidence would be required to exclude photometric variations due other phenomena, both astrophysical (e.g., stellar activity, rings) and instrumental (e.g., red noise). A confirmation could be achieved with photodynamical modeling as applied by the HEK team. As an example for the photometric OSE, Figure 13 presents the phase-folded Kepler light curve (gray dots) of the almost Jupiter-sized planetary candidate KOI189.01, transiting a ≈ R ⨀ K star. Note how the “no moon” model (black dashed line) and the model for an Earth-sized moon (red solid line) diverge in the right panel! The latter is based on an improved model of Heller (2014), now including stellar limb darkening, amongst others, and is supposed to serve as a qualitative illustration but not as a statistical fit to the Kepler data for the purpose of this paper.
One major factor constraining the detection of moons of transiting planets is the type of photometric noise contaminating the light curve. A first detailed analysis of noise effects on the detection of exomoons with
Kepler was performed by Kipping et al. (2009), showing that photon noise and instrument noise strongly increase for stellar apparent magnitudes ≳
13. Combing shot noise,
Kepler ’s instrumental noise, and stellar variability with arguments from orbital stability, they obtained a lower limit of about 0.2 M ⨁ for the detection of moons orbiting Saturn-like planets in the stellar HZ of bright M stars (with Kepler magnitudes < 11) by using TTV and TDV. This mass detection limit increases for Neptune- and Jupiter-like planets because of these planets’ higher densities (see Fig. 3 in Kipping et al., 2009). What is more, absolute mass detection limits are hard to generalize because the TTV-TDV combined method constrains the planet-to-satellite mass ratio M s / M p . In their targeted search for exomoons, Kipping et al. (2013a) achieved accuracies as good as M s / M p ≈ Lewis (2011a) investigated the effects of filtering on Sun-like stellar noise for a variant of TTV, photometric transit timing, and found that realistic photometric noise suppresses the detection of moons on wide circumplanetary orbits. In another work, Lewis (2013) confirmed that these noise sources also hamper the detection of moons around planets that follow distant circumstellar orbits. As a result, exomoons hidden in the noise of the
Kepler data will need to have radii ≳ R ⨁ to be Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 25 In the case of moon detection through perturbation of the Rossiter-McLaughlin effect (Simon et al., 2010; Zhuang et al., 2012) one needs to consider types of noise that contaminate radial velocity measurements. etectable by direct eclipse effects. This value is comparable to the results of Simon et al. (2012), who found that moons of close-in planets, that is, with circumstellar orbital periods ≲
10 days, could be detected by using their scatter peak method on
Kepler short cadence data if the moons have radii ≳ R ⨁ . Uncorrelated noise that follows a Gaussian probability distribution f ( x ) = 1 p ⇡ e ( x µ )22 , (6)where σ is the standard deviation, x is the perturbation to the light curve due to noise, and μ is the mean of the noise (usually zero), is called “white noise”, and it makes moon detection relatively straightforward. At a given time, f ( t ) is assumed to be uncorrelated with the past. As a result, white noise contains equal power at all spectral frequencies. White noise or noise that is nearly white is produced by many instrumental or physical effects, such as shot noise. Processes in stars, in Earth’s atmosphere, and in telescopes can lead to long-term correlated trends in photometry that show an overabundance of long-wavelength components. In astrophysics, these effects are collectively referred to as red noise (Carter and Winn, 2009). Stars pulsate, convect, show spots and rotate, each of which leaves a distinctive red noise signature on the light curve (Aigrain et al., 2004). Solar oscillations have a typical amplitude of a few parts per million and period on the order of five minutes, while for other stars the specific details depend on the stellar size and structure (Christensen-Dalsgaard, 2004). In Sun-like stars, stellar convection leads to granulation, which produces red noise with characteristic timescales between minutes and hours. Finally, starspots can cause red noise in transit light curves, first through spot crossing events and second through spot evolution or rotational modulation. If a planet passes in front of a starspot group during transit, a temporary short-term increase in brightness results. This effect was predicted (Silva, 2003) and has finally been observed in many transiting systems (Pont et al., 2007; Sanchis-Ojeda and Winn, 2011; Kundurthy et al., 2011). Evolution and rotational modulation of starspots can cause long-term photometric variation over tens of days. This variation has been observed and modeled for a range of stars including CoRoT-2a (Lanza et al., 2009a) and CoRoT-4a (Lanza et al.,
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 26
Terrestrial planets ( R p < 1.5 R ⨁ ) Gas giant planets (5 R ⨁ < R p < 15 R ⨁ ) Figure 14 : Calculated timing errors of the transit mid-time (filled circles) and duration (open circles) for
Keple r planet candidates around stars with magnitudes < 13.5 (Mazeh et al., 2013). Planetary radii are estimated by assuming an Earth-like composition (left panel) or a mostly gaseous composition (right panel). Point size is proportional to the stellar radius, with color indicating if the radius is smaller than 0.7 solar radii ( R ⨀ ) (red), between 0.7 R ⨀ and 1.3 R ⨀ (gold), or larger than 1.3 R ⨀ (green). Predictions of the errors in transit mid-time (solid line) and duration (dotted line) following Carter et al. (2008) assume a central transit and white noise for the case of a 1 R ⨁ or 10 R ⨁ radius planet orbiting a 0.7 R ⨀ (red), R ⨀ (gold), or 1.3 R ⨀ (green) star. Photometric transit timing errors, dominated by white noise (dashed line) and realistic solar-type noise (crosses) assume a 12th magnitude G dwarf and relative photometric precision of 2 × − in a 6.5 hour exposure (Lewis, 2013).009b). Ground-based observations can be substantially contaminated with red noise induced by Earth’s atmosphere (Pont et al., 2006), for example, by the steadily changing value of the air mass over the course of a night. The operation of the telescope, as well as the telescope surroundings, can also introduce red noise, as has been noted by the ground-based WASP (Street et al., 2004) and HATNet surveys (Bakos et al., 2002, 2009). For the space-based
Kepler and
CoRoT missions, a number of different processes can lead to trends and jumps in the data, including effects from spacecraft motion, motion of the image on the CCD, and cosmic rays (Auvergne et al., 2009;
Kepler Characteristics Handbook ). When a spacecraft rotates to reposition solar panels or to downlink data, or when it passes into Earth’s shadow , the varying solar irradiation on various parts of the spacecraft’s surface can cause transient photometric perturbations. In addition, the target’s image can move on the CCD, and, finally, damage due to cosmic rays can lead to degradation of the CCDs. Cosmic rays can be isolated as well as certain specific events such as the passage of the south atlantic anomaly in the case of CoRoT , or the three large coronal mass ejections in
Kepler’s quarter 12. While processing pipelines work to reduce effects from cosmic rays (
Kepler Data Processing Handbook ), they are never completely removed in all cases. While uncertainties in the transit mid-time and transit duration are dominated by errors in the ingress and egress parts of the light curve, relatively short lengths of time, red noise acts over longer periods. Hence, these techniques should be quite robust to red noise. However, transit mid-time (filled circles in Fig. 14) and duration (open circles) errors (Mazeh et al., 2013) of
Kepler
Objects of Interest are up to many factors above the values predicted by assuming white noise only. While inclination, stellar luminosity, stellar radius, and planetary radius can explain some of this discrepancy, some of it is undoubtedly due to red noise.
Carter and Winn (2009) investigated the effect of red noise on both mid-time and duration errors with a power spectrum following a power law, showing that parameters can be effectively recovered through use of a wavelet transform. Alternatively, Kipping et al. (2009) and later Awiphan and Kerins (2013) modeled red noise as white noise plus a set of longer period sinusoids. While Kipping et al. (2009) ended up neglecting the difference between white and red noise, their simulations show that the distribution of transit mid-times became slightly non-gaussian, in particular, the tails of the distribution became fatter (Fig. 2 of Kipping et al., 2009). Awiphan and Kerins (2013) confirmed this effect and concluded that this variety of red noise decreases moon detectability. Other recent studies suggest that the presence of starspots, especially near the stellar limb, can alter transit durations and timings (Silva-Válio, 2010; Barros et al., 2013), reducing sensitivity for spotted stars. Mazeh et al. (2013) confirmed this trend in transit timing errors for numerous Kepler Objects of Interest. In particular, they found that TTVs correlate with stellar rotation in active stars, indicating again that noise components due to starspots could alter timing results.
In addition to the standard transit timing technique, photometric transit timing has been proposed (Szabó et al., 2006), a technique that measures timing perturbations with respect to the average transit time weighted by the dip depth. Using solar photometric noise from the Solar and Heliosphere Observatory as a proxy, Lewis (2013) found that realistic red stellar photometric noise dramatically degraded moon detectability for this technique (see crosses in Fig. 14), compared to white noise for the case where the transit duration was longer than three hours. What is more, this degradation was not completely reversed by filtering, independent of the filtering method. This result is unfortunate, as the photometric transit timing statistic is proportional to the moon radius squared (Simon et al., 2007) as opposed to the moon’s mass (Sartoretti and Schneider, 1999), but unsurprising as photometric transit timing uses data from both within and outside the planetary transit, compared to transit timing variation, which only uses shorter sections of data.
While Tusnski and Valio (2011) assumed white photometric noise, other projects have started to consider the effect of red noise in their analyses. The “Hunt for Exomoons with Kepler” (HEK) (Kipping et al., 2012) addresses the effect of red noise in a number of ways. First, they reject candidates with high levels of correlated noise (Kipping et al., 2013a). Second, a high-pass cosine filter is used to remove long-term trends from the data. The possibility of modeling the effects of starspots has also been suggested (Kipping, 2012). To help distinguish between star spot crossings and mutual events, rules have been proposed, for example, it is required that a mutual event have a flat top (Kipping et al., 2012). Also, to allow for secure detection, it is required that a given moon is detected by several detection methods. In an attempt to quantify the true error on detected moon properties, fake moon transits were injected into real data for the case of Kepler-22 b (Kipping et al., 2013b), indicating that an Earth-like moon, if present, would have been detected.
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 27 http://archive.stsci.edu/kepler/manuals/Data_Characteristics.pdf Kepler orbits the Sun in an Earth-trailing orbit, where it avoids transits in the shadow of Earth. At an altitude of 827 km,
CoRoT is in a polar orbit around the Earth, and occasionally passes the Earth’s shadow. http://archive.stsci.edu/kepler/manuals/KSCI-19081-001_Data_Processing_Handbook.pdf .4 Direct imaging of extrasolar moons6.4.1 Principles of direct imaging Direct imaging of exoplanets, especially those in the stellar HZ, is extremely difficult because of the very small star-planet angular separation and the high contrast ratio between the star and planet. In fact, all exoplanets that have been directly imaged are well separated from their host star, and are young systems that are still hot from formation, rather than being heated by stellar irradiation, with effective temperatures around 1000 K. Examples of directly imaged exoplanets include the HR8799 planets, β Pic b, LkCa15b, κ And b, and GJ 504b (Marois et al., 2008; Lagrange et al., 2008; Kraus and Ireland, 2011; Carson et al., 2013; Kuzuhara et al., 2013). Intuitively, one would expect exomoons to be even more difficult to be imaged directly. For exomoons similar to those found in the Solar System, this is likely the case. However, satellites that are heated by sources other than stellar irradiation, such as tidal heating, can behave completely differently. There has been considerable discussion in the literature of the existence of tidally heated exomoons (THEMs) that are extrasolar analogs to Solar System objects such as Io, Europa, and Enceladus (Peale et al., 1979; Yoder and Peale, 1981; Ross and Schurbert, 1987; Ross and Schubert, 1989; Nimmo et al., 2007; Heller and Armstrong 2014). Yet, the possibility of imaging the thermal emission from unresolved THEMs was proposed only recently by Peters and Turner (2013) . From an observational point of view, direct imaging of exomoons has several advantages over exoplanet direct imaging. THEMs may remain hot and luminous for timescales of order the stellar main sequence lifetime and thus could be visible around both young and old stars. Additionally, THEMs can be quite hot even if they receive negligible stellar irradiation, and therefore they may be luminous even at large separations from the system primary. This will reduce or even eliminate the inner-working angle requirement associated with exoplanet high contrast imaging. Furthermore, tidal heating depends so strongly on orbital and physical parameters of the THEM that plausible systems with properties not very different from those occurring in the Solar System will result in terrestrial planet sized objects with effective temperatures up to 1000 K or even higher in extreme but physically permissible cases. The total luminosity of a THEM due to tidal heating is given by Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 28 For exoplanets, direct imaging refers to the imaging of a planet that is spatially resolved from its host star. In the model proposed by Peters and Turner (2013), the tidally heated moon is not spatially resolved from its host planet, but it rather adds to the thermal flux detected at the circumstellar orbital position of the planet. Despite this unresolved imaging, we here refer to this exomoon detection method as direct imaging.
Figure 15 : 5 σ detection limits of JWST’s MIRI for tidally heated exomoons with radii along the abscissa and effective temperatures along the ordinate (following Turner and Peters, 2013). 10,000s of integration are assumed for a star 3 pc from the Sun. MIRI’s nine imaging bands are indicated with different line colors, their names encoding the wavelength in units of microns times 100. Dashed vertical black lines denote the radii of Io, Earth, and Jupiter, which is roughly equal to that of a typical brown dwarf. tid = ✓ ⇡ G ◆ / R ⇢ / µ s Q s ! e / ! , (7)where μ s is the moon’s elastic rigidity, Q s is the moon’s tidal dissipation function (or quality factor), e ps is the eccentricity of the planet-satellite orbit, β ps is the satellite’s orbital semi-major axis in units of Roche radii, ρ s is the satellite density, and R s is the satellite radius (Scharf, 2006; Peters and Turner, 2013). Equation (7) is based on the tidal heating equations originally derived by Reynolds et al. (1987) and Segatz et al. (1988) and assumes zero obliquity. The terms that depend on the exomoon’s physical properties and those that describe its orbit are grouped separately. Although β ps is grouped with the orbital terms, in addition to its linear dependence on the moon’s semi-major axis a ps , it also scales with the planetary mass and the satellite density as ( M p / ρ s ) ⅓ for a fixed a ps . Following Peters and Turner (2013), the scaling relation for Eq. (7) relative to the luminosity of Earth ( L ⨁ = 1.75 × ergs/s) is L s ⇡ L "✓ R s R ◆ ✓ ⇢ s ⇢ ◆ Q s ·
11 dynescm µ s ! ⇥ "⇣ e ps . ⌘ ✓ ps ◆ . (8)Note that Eq. (8) adopts Q s = 36, μ s = 10 dynes/cm as for Io (Peale et al., 1979; Segatz et al., 1988), and Earth’s radius and density as reference values. The reference values of β ps and e ps were then chosen to give L ⨁ . If we assume that THEMs are blackbodies, we can use the exomoon’s luminosity to approximate its effective temperature. The blackbody assumption is a reasonable approximation, but in general we would expect THEMs to emit excess light at bluer wavelengths due to hotspots on the surface. Additionally, exomoons with atmospheres are likely to have absorption lines in their spectrum. Assuming a blackbody, we can define a satellite’s effective temperature ( T s ) from the luminosity via the Stefan-Boltzmann law as a scaling relation relative to a 279 K exomoon, this temperature corresponding to the equilibrium temperature of Earth: T s ⇡ K R s R ◆ ✓ ⇢ s ⇢ ◆ Q s ·
11 dynescm µ s ! / ⇥ "⇣ e ps . ⌘ ✓ ps ◆ (9)(Peters and Turner, 2013). As an example, Eq. (9) yields ≈
60 K for Io, corresponding to the effective temperature of Io if it were not irradiated by the Sun. Note that Eq. (9) adopts the same reference values as Eq. (8) and that it assumes only tidal heating with no additional energy sources, such as stellar irradiation or interior radiogenic heat.
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 29
Figure 16 : Minimum detectable eccentricity (abscissa) and semi-major axis in units of Roche radii ( β ps , ordinate) for tidally-heated exomoons between 0.1 and 1 R ⨁ in size (Turner and Peters, 2013). The assumed bodily characteristics are described around Eqs. (5) - (6). The orange dot indicates Io’s eccentricity and β ps , but note that its size is 0.066 R ⨁ . Satellites below their respective curve will be detectable by MIRI as far as 3 pc from Earth, provided they are sufficiently separated from their star. An exomoon’s brightness is expected to be variable for multiple reasons, including eclipses of the exomoon behind its host planet (Heller, 2012), phase curve variations due to temperature variation across an object’s surface (Heller and Barnes, 2013), and volcanism.
If THEMs exist and are common, are they detectable with current and future instrumentation? As shown by Peters and Turner (2013), Spitzer’s IRAC could detect an exomoon the size of Earth with a surface temperature of 850 K and at a distance of five parsecs (pc) from Earth. Future instruments, such as JWST’s Mid-Infrared Instrument (MIRI), have even more potential for direct imaging of exomoons. Figure 15 shows minimum temperatures and radii of exomoons detectable with MIRI at the 5 σ confidence level in 10,000 seconds for a star 3 pc from the Sun (Glasse et al., 2010). Note that some exomoons shown here have temperatures and radii similar to Earth's! Thus, it is plausible that some of these hypothetical exomoons are not only directly visible with JWST, but they could potentially be habitable, in the sense of having surface temperatures that would allow liquid water to be present. A 300 K, Earth-radius THEM is only 3 × ⁵ fainter at a wavelength of λ ≈ μ m than a Sun-like star, but at the same time it can be at a large distance from it's host star. For example, at 30 AU projected separation, it would be 15” from the star at a distance of 2 pc. At λ ≈ μ m, λ /D = 0.44” for a D = 6.5-m telescope, which means this moon would be at 30 × λ /D. This far away from the star, the airy rings are about 3 × ⁵ times fainter than the core of the star and about the same intensity as the 300 K, Earth-radius moon, indicating that the detection should be possible. This example is not at the limit of JWST's sensitivity and inner working angle. The most challenging THEM detection JWST will be capable of making is a 300 K THEM as far as 4 pc from the Sun. If there is such an Earth-sized 300 K moon orbiting α Cen, MIRI will be able to detect it in 8 of its 9 spectral bands with better than 15 σ signal-to-noise in a 10 s integration. Thus, directly imaging a 300 K, Earth-radius moon that is tidally heated is potentially much easier than resolving an Earth-like exoplanet orbiting in the HZ of its primary! We can ask the question of what orbital parameters would give rise to the temperatures of moons shown in Fig. 15. The minimum detectable temperatures by MIRI for four different radii satellite ( R s = 0.1 R ⨁ , 0.25 R ⨁ , 0.5 R ⨁ , and R ⨁ ) at 3 pc were calculated to be T s = 542 K, 355 K, 282 K, and 229 K, respectively (see Fig. 15). We can use Eq. (9) to calculate the expected orbital parameters β ps and e ps for these four objects. Figure 16 shows β ps and e ps corresponding to these four cases, indicating that a roughly Earth-sized exomoon (black line) at a distance of ten Roche radii, with an eccentricity similar to that of Titan ( e ps = 0.0288), and an effective temperature equal to the reader’s room temperature could be detectable with JWST. Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 30
Figure 17 : Moon-to-planet mass ratio constraints derived so far by the HEK project (Nesvorn ý et al., 2012; Kipping et al., 2013a). Kepler-22b has also been studied and yields M s < 0.5 M ⨁ but is not shown here as the paper is still under review at the time of writing.ompared to their planetary cousins, relatively few searches for exomoons have been conducted to date. The great enabler in this field has been the Kepler mission offering continuous, precise photometric data for years of the same targets. The downside of
Kepler data is that the field is deep (and thus faint) and dominated by Sun-like stars, whereas K- and M-dwarfs provide a significant boost to sensitivity thanks to their smaller sizes.
Within the current literature, one can find claims that various observations are consistent with an exomoon without specifically claiming an unambiguous exomoon detection. For example, Szabo et al. (2013) suggested that many of
Kepler ‘s hot-Jupiters show high TTVs that could be due to exomoons or equally other planets in the system. Others used observations of transiting exoplanets to perform serendipitous searches for exomoons, such as Brown et al. (2001) around HD 209458b, Charbonneau et al. (2006) around HD 149026b, Maciejewski et al. (2010) around WASP-3b, and Montalto et al. (2012) around WASP-3b, none of which revealed evidence for an extrasolar satellite.
At the time of writing, the only known systematic survey is the “Hunt for Exomoons with Kepler” (HEK) (Kipping et al., 2012). Using photodynamic modeling, the HEK team has surveyed seventeen Keple r planetary candidates for evidence of an extrasolar moon so far. Due to the large number of free parameters, the very complex and multimodal parameter space, the demands of full photodynamic modeling, and the need for careful Bayesian model selection, the computational requirements reported by the HEK project have been staggering compared to the a typical planet-only analysis. For example, Kepler-22b required 50 years of modern CPU time (Kipping et al., 2013b). So far, the HEK team reports no compelling evidence for an exomoon, but they have derived strong constraints of M s / M p < 4% for five cases (Kipping et al., 2013a; Nesvorn ý et al., 2012), translating into satellite masses as small as 0.07 M ⨁ (Fig. 17 ) , and M s < 0.5 M ⨁ for the additional case of the habitable-zone planet Kepler-22b (Kipping et al., 2013b). In the latter example, an absolute mass constraint is possible thanks to the availability of radial velocities. In their latest release, Kipping et al. (2014) added another eight null detections of moons around planets transiting M dwarfs, with sensitivities to satellites as tiny as 0.36 M ⨁ in the case of KOI3284.01. Despite the recent malfunction with the Kepler spacecraft, there is a vast volume of unanalyzed planetary candidates for moon hunting and still relatively few conducted so far. In the next one to two years then, we should see about 100 systems analyzed, which will provide a statistically meaningful constraint on the occurrence rate of large moons, η (cid:2) .
7. Summary and Conclusions
This review highlights a remarkable new frontier of human exploration of space: the detection and characterization of moons orbiting extrasolar planets. Starting from the potentially habitable icy satellites of the Solar System (Section 2), we show that the formation of very massive satellites the mass of Mars is possible by in-situ formation in the circumplanetary disk and how a capture during binary exchange can result in a massive moon, too (Section 3). After discussing the complex orbital evolution of single and multiple moon systems, which are governed by secular perturbations and tidal evolution (Section 4), we explore illumination, tidal heating, and magnetic effects on the potential of extrasolar moons to host liquid surface water (Section 5). Finally, we explain the currently available techniques for searching for exomoons and summarize a recently initiated survey for exomoons in the data of the
Kepler space telescope (Section 6).
From a formation point of view, habitable exomoons can exist. Mars-sized exomoons can form by either in-situ formation in the circumplanetary disks of super-Jovian planets (Section 3.1) or by gravitational capture from a former planet-moon or planet-planet binary (Section 3.2). This mass of about 0.1 M ⨁ is required to let any terrestrial world be habitable by atmospheric and geological considerations (Section 5). Stellar perturbations on these moons’ orbits, their tidal orbital evolution, tidal heating in the moons, and satellite-satellite interactions prevent moons in the HZs of low-mass stars from being habitable (Section 4). Moreover, while more massive giant planets should tend to produce more massive moons, strong inplanation from young, hot planets onto their moons can initiate a runaway greenhouse effect on the moons and make them at least temporarily uninhabitable (Section 5.2). With this effect becoming increasingly severe for the most massive giant planets, we identify here the formation and habitability of moons as competing processes. Although growing moons can accumulate more mass from the disks around more massive giant planets, the danger of a planet-induced runaway greenhouse effect on the moons increases, too. Finally, the host planet’s intrinsic magnetic field and the stellar wind affect the moon’s habitability by regulating the flux of high-energy particles (Section 5.3.2). Exomoons in the
Kepler data may be detectable if they belong to a class of natural satellites that does not exist in the Solar System. While the most massive known moon, Ganymede, has a mass roughly 1/40 M ⨁ , exomoons would need to have about ten times this mass to be traceable via their planet’s TTV and TDV (Section 6.1). Another detection channel would be through a moon’s direct photometric transit, which could be measurable in the Kepler data for moons as small as Mars or even Ganymede (Sections 6.2 and 6.3).
Combining the key predictions from formation, detection, and habitability sections, we find a favorable mass regime for the first extrasolar moons to be detected. Notably, there is a mass overlap of the most massive satellites that (i.) can possibly
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 31 orm in protosatellite disks around super-Jovian planets, (ii.) can be detected with current and near-future technology, and (iii.) can be habitable in terms of atmospheric stability. This regime is roughly between one and five times the mass of Mars or 0.1 – 0.5 M ⨁ . Heavier moons would have better odds to be habitable (Heller and Armstrong, 2014), and they would be easier to detect, though it is uncertain whether they exist. Although the observational challenges of exomoon detection and characterization are huge, our gain in understanding planet formation and evolution would be enormous. The moons of Earth, Jupiter, Saturn, and Neptune have proven to be fundamentally important tracers of the formation of the Solar System – most notably of the formation of Earth and life – and so the characterization of exomoon systems engenders the power of probing the origin of individual extrasolar planets (Withers and Barnes, 2010). Just as the different architectures of the Jovian and Saturnian satellite systems are records of Jupiter’s gap opening in the early gas disk around the Sun, of an inner cavity in Jupiter’s very own circumplanetary disk, and of the evolution of the H O snow line in the disks, the architectures of exomoon systems could trace their planets’ histories, too. We conclude that any near-future detection of an exomoon, be it in the
Kepler data or by observations with similar accuracy, could fundamentally challenge formation and evolution theories of planets and satellites. Another avail of exomoon detections could lie in a drastic increase of potentially habitable worlds. With most known planets in the stellar HZ being gas giants between the sizes of Neptune and Jupiter rather than terrestrial planets, the moons of giant planets could actually be the most numerous population of habitable worlds.
8. Outlook
On the theoretical front, improvements of our understanding of planet and satellite formation might help to focus exomoon searches on the most promising host planets. The formation and movement of water ice lines in the disks around young giant planets, as an example, has a major effect on the total mass of solids available for moon formation. Preliminary studies show that super-Jovian planets might host water-rich giant moons the size of Mars or even larger (Heller and Pudritz, in prep.), in agreement with the scaling relation suggested by Canup and Ward (2006). Hence, even future null detections of moons around the biggest planets could help to assess the conditions in circumplanetary disks. As an example, the absence of giant moons could indicate hotter planetary environments, in which the formation of ices is prevented, than are currently assumed. Earth-mass moons, however, are very hard to form with any of the viable formation theories, and consequently the suggested low abundance or absence (Kipping et al., 2014) of such moons may not help to constrain models immediately. However, in the most favorable conditions, where a somewhat super-Jovian-mass planet with an entourage of an oversized Galilean-style moon system transits a photometrically quiet M or K dwarf star, a handful of detections might be feasible with the available
Kepler data (Kipping et al., 2012; Heller 2014) and therefore might confirm the propriety of moon formation theories for extrasolar planets.
Another largely unexplored aspect of habitable moon formation concerns the capture and orbital stability of moons during planet migration. A few dozen giant planets near, or in, their host stars’ HZs have been detected by radial velocity techniques (see Fig. 1 in Heller and Barnes, 2014), most of which cannot possibly have formed at their current orbital locations, as giant planets’ cores are assumed to form beyond the circumstellar snow lines (Pollack et al., 1996; Ward, 1997). This prompts the compelling question of whether migrating giant planets, which end up in their host star’s HZ, can capture terrestrial planets into stable satellite orbits during migration. A range of studies have addressed the post-capture evolution or stability of moons (Barnes and O'Brien, 2002; Domingo et al., 2006; Donnison et al., 2006; Porter and Grundy, 2011; Sasaki et al., 2012; Williams, 2013) and the loss of moons during planet migration towards the hot Jupiter regime (Namouni, 2010). But dynamical simulations of the capture scenarios around migrating giant planets near the HZ are still required in order to guide searches for habitable exomoons.
NASA’s all-sky survey with the
Transiting Exoplanet Survey Satellite ( TESS ), to be launched in 2017, will use the transit method to search for planets orbiting bright stars with orbital periods ≲
72 days (Deming et al., 2009). As stellar perturbations play a key role in the orbital stability of moons orbiting giant planets (see Section 4),
TESS will hardly find such systems. The proposed ESA space mission
PLAnetary Transits and Oscillations of stars ( PLATO 2.0 ) (Rauer et al., 2013), with launch envisioned for 2022-2024, however, has been shown capable of finding exomoons as small as 0.5 R ⨁ , that is, the size of Mars (Simon et al., 2012; Heller 2014). Since the target stars will be mostly K and M dwarfs in the solar neighborhood, precise radius and mass constraints on potentially discovered exomoons will be possible (Kipping, 2010). What is more, PLATO ’s long pointings of two to three years would allow for exomoon detections in the HZs of K and M dwarfs.
Future exomoon surveys may benefit by expanding their field of view while keeping a continuous staring mode and using the redder part of the spectrum to look at K and M dwarfs. Targeting these low-mass stars increases the relative dip in the light curves due to an exomoon as well as the transit frequency, and it decreases the minimum transit duration of a planet capable of hosting long-lived stable moons. Searches for exomoons around specific transiting planets will need to target relatively inactive stars or stars with noise spectra that are amenable to filtering (Carter and Winn, 2009) in order to reduce the effect of starspot modulation and errors due to spot crossing events. While recently developed procedures of white and
Heller et al. (2013) – Formation, Habitability, and Detection of Extrasolar Moons 32 ed noise filtering in stellar light curves represent major steps towards secure moon detections, they do not directly address additional timing noise due to spots on the stellar limb (Silva-Válio, 2010; Barros et al., 2013) and the empirical correlation between the timing error and the slope of the out of transit light curve (Mazeh et al., 2013). Further progress, either through the modeling-based approach used by the HEK project, the filtering approach suggested by Carter and Winn (2009), or the observational approach used by Mazeh et al. (2013) is required to securely detect and constrain moons of
Kepler planets. Finally, telescopes capable of multi-color measurements will help discriminate between starspot occupations on the one hand and planet/moon transits on the other hand.
JWST will be sensitive to a hypothetical family of tidally heated exomoons (THEMs) at distances as far as 4 pc from the Sun and as a cool as 300 K. These moons would need to be roughly Earth-sized and, together with the host planets, separated from their host star by at least 0.5”. Future surveys that aim to detect smaller and colder THEMs would need to operate in the mid- to far-infrared with sub-microjansky sensitivity. Although THEMs need not be close to their host star to be detectable, instruments able to probe smaller inner working angles are likely to detect more THEMs simply because there will presumably be more planets closer into the star that can host THEMs. A preliminary imaging search for THEM around the nearest stars with the use of archival
Spitzer IRAC data has recently been completed and will deliver the first constraints on the occurrence of THEMs (Limbach and Turner in prep).
Once an exomoon has been detected around a giant planet in the stellar HZ, scientists will try to discern whether it is actually inhabited. Kaltenegger (2010) showed that
JWST would be able to spectroscopically characterize the atmospheres of transiting exomoons in nearby M dwarf systems, provided their circumplanetary orbits are wide enough to allow for separate transits. In her simulations, the spectroscopic signatures of the biologically relevant molecules H O, CO , and O could be observable for HZ exomoons transiting M5 to M9 stars as far as 10 pc from the Sun. Although stellar perturbations on those moon’s orbits could force the latter to be eccentric and thereby generate substantial tidal heating (Heller, 2012), the threat of a runaway greenhouse effect would be weak for moons in wide orbits because tidal heating scales inversely to a high power in planet-moon distance. Exomoon science will benefit from results of the
Jupiter Icy Moons Explorer ( JUICE ) (
JUICE
Assessment Study Report , 2011; Grasset et al., 2013), the first space mission designated to explore the emergence of habitable worlds around giant planets. Scheduled to launch in 2022 and to arrive at Jupiter in 2030, JUICE will constrain tidal processes and gravitational interaction in the Galilean system and thereby help to calibrate secular-tidal models for the orbital evolution of moons around extrasolar planets. As an example, the tidal response of Ganymede’s icy shell will be measured by laser altimetry and radio science experiments. The amplitudes of periodic surface deformations are suspected to be about 7 to 8 m in case of a shallow subsurface ocean, but only some 10 cm if the ocean is deeper than roughly 100 km or not present at all. With Ganymede being one of the three solid bodies in the Solar System – besides Mercury and Earth – that currently generate a magnetic dipole field (Kivelson et al., 2002),
JUICE could constrain models for the generation of intrinsic magnetic fields on extrasolar moons, which could be crucial for their habitability (Baumstark-Khan and Facius, 2002; Heller and Zuluaga, 2013). Measuring the bodily and structural properties of the Galilean moons, the mission will also constrain formation models for moon systems in general. In view of the unanticipated discoveries of planets around pulsars (Wolszczan and Frail, 1992), Jupiter-mass planets in orbits extremely close to their stars (Mayor and Queloz, 1995), planets orbiting binary stars (Doyle et al., 2011), and small-scale planetary systems that resemble the satellite system of Jupiter (Muirhead et al., 2012), the discovery of the first exomoon beckons, and promises yet another revolution in our understanding of the universe.
Acknowledgements
The helpful comments of two referees are very much appreciated. We thank Alexis Carlotti, Jill Knapp, Matt Mountain, George Rieke, Dave Spiegel and Scott Tremaine for useful conversations and Ted Stryk for granting permission to use a reprocessed image of Europa. René Heller is supported by the Origins Institute at McMaster University and by the Canadian Astrobiology Training Program, a Collaborative Research and Training Experience Program funded by the Natural Sciences and Engineering Research Council of Canada (NSERC). Darren Williams is a member of the Center for Exoplanets and Habitable Worlds, which is supported by the Pennsylvania State University, the Eberly College of Science, and the Pennsylvania Space Grant Consortium. Takanori Sasaki was supported by a grant for the Global COE Program, “From the Earth to ‘Earths’”, MEXT, Japan, and Grant-in-Aid for Young Scientists (B), JSPS KAKENHI Grant Number 24740120. Rory Barnes acknowledges support from NSF grant AST-1108882 and the NASA Astrobiology Institute’s Virtual Planetary Laboratory lead team under cooperative agreement no. NNH05ZDA001C. Jorge I. Zuluaga is supported by CODI/UdeA. This research has been supported in part by World Premier International Research Center Initiative, MEXT, Japan. This work has made use of NASA’s Astrophysics Data System Bibliographic Services.
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Astrophysical Journal .8. EXOMOON HABITABILITY AND TIDAL EVOLUTION IN LOW-MASS STAR SYSTEMS(Zollinger et al. 2017) 322
Contribution:RH contributed to the literature research and to the mathematical investigations, assisted in the gener-ation of the figures, contributed to the writing of the manuscript, and contributed to the interpretationof the results.
NRAS , 1–20 (2017) Preprint 25 July 2017 Compiled using MNRAS L A TEX style file v3.0
Exomoon Habitability and Tidal Evolution in Low-MassStar Systems
Rhett R. Zollinger, ? John C. Armstrong, † Ren ´e Heller ‡ Southern Utah University, Cedar City, Utah 84720, USA Weber State University, Ogden, Utah 84408, USA Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 G¨ottingen, Germany
Accepted 2017 July 20. Received 2017 July 14; in original form 2016 August 9
ABSTRACT
Discoveries of extrasolar planets in the habitable zone (HZ) of their parent star lead toquestions about the habitability of massive moons orbiting planets in the HZ. Aroundlow-mass stars, the HZ is much closer to the star than for Sun-like stars. For a planet-moon binary in such a HZ, the proximity of the star forces a close orbit for the moon toremain gravitationally bound to the planet. Under these conditions the effects of tidalheating, distortion torques, and stellar perturbations become important considerationsfor exomoon habitability.Utilizing a model that considers both dynamical and tidal interactions simulta-neously, we performed a computational investigation into exomoon evolution for sys-tems in the HZ of low-mass stars ( . . M (cid:12) ). We show that dwarf stars with masses . . M (cid:12) cannot host habitable exomoons within the stellar HZ due to extreme tidalheating in the moon. Perturbations from a central star may continue to have delete-rious effects in the HZ up to ≈ . M (cid:12) , depending on the host planet’s mass and itslocation in the HZ, amongst others. In addition to heating concerns, torques due totidal and spin distortion can lead to the relatively rapid inward spiraling of a moon.Therefore, moons of giant planets in HZs around the most abundant type of star areunlikely to have habitable surfaces. In cases with lower intensity tidal heating the stel-lar perturbations may have a positive influence on exomoon habitability by promotinglong-term heating and possibly extending the HZ for exomoons. Key words: planets and satellites: dynamical evolution and stability – planets andsatellites: physical evolution
The exploration of the moons of Jupiter and Saturn has pro-vided immense understanding of otherworldly environments.Some of these moons have reservoirs of liquids, nutrients,and internal heat (Squyres et al. 1983; Hansen et al. 2006;Brown et al. 2008; Saur et al. 2015) – three basic componentsthat are essential for life on Earth. Recent technological andtheoretical advances in astronomy and biology now raise thequestion of whether life might exist on any of the moons be-yond the solar system (“exomoons”). Given the abundantpopulation of moons in our system, exomoons may be evenmore numerous than exoplanets (Heller & Pudritz 2015).No moon outside the Solar System has been detected,but the first detection of an extrasolar moon appears to be ? E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] on the horizon (Kipping et al. 2009, 2012; Heller et al. 2014)now that modern techniques enable detections of sub-Earthsized extrasolar planets (Muirhead et al. 2012; Barclay et al.2013). A recent review of current theories on the formation,detection, and habitability of exomoons suggests that nat-ural satellites in the range of 0.1-0.5 Earth masses (i) arepotentially habitable, (ii) can form within the circumplane-tary debris and gas disk or via capture from a binary, and(iii) are detectable with current technology (Heller et al.2014). Considering the potential for current observation, weexplore the expected properties of such exomoons in thispaper.When investigating planet habitability, the primaryheat source is typically the radiated energy from a centralstar. Considerations of stellar radiation and planet surfacetemperatures have led to the adoption of a concept referredto as the stellar “habitable zone” (HZ) that is the regionaround a star in which an Earth-like planet with an Earth-like atmosphere can sustain liquid surface water (Kasting © a r X i v : . [ a s t r o - ph . E P ] J u l R. R. Zollinger et al. et al. 1993). Tidal heating is typically less important as anenergy source for planets in the HZ, though it might be rel-evant for eccentric planets in the HZs around low-mass starsof spectral type M (Barnes et al. 2008). For the moon of alarge planet, tidal heating can work as an alternative inter-nal heat source as well. Several studies have addressed theimportance of tidal heating and its effects on the habitabilityof exomoons (Reynolds et al. 1987a; Scharf 2006; Henninget al. 2009; Porter & Grundy 2011a; Heller 2012; Forgan &Kipping 2013; Heller & Barnes 2013; Heller & Zuluaga 2013;Heller & Barnes 2015; Dobos & Turner 2015; Dobos et al.2017). Tidal heat can potentially maintain internal heat-ing over several Gyr (see Jupiter’s moon Io; Spencer et al.2000a), which contributes to surface heating and potentiallydrives important internal processes such as plate tectonics.Tidal interactions between a massive moon and its hostplanet become particularly interesting for planets in the HZsof M dwarfs, which are smaller, cooler, fainter, and less mas-sive than the Sun. They are the predominant stellar popula-tion of our Galaxy (Chabrier & Baraffe 2000), and as such,these low-mass stars may be the most abundant planet hostsin our Galaxy (Petigura et al. 2013; Dressing & Charbon-neau 2015). Their lower core temperatures and decreasedenergy output result in a HZ that is much closer to thestar in comparison to Sun-like stars. If a planet in the HZaround an M dwarf star has a massive moon, the close-inorbital distances could potentially influence the orbit of themoon around the planet (Heller 2012). The relatively closecentral star can serve as a continual source of gravitationalperturbation to the moon’s orbit, which has important im-plications to tidal heating and orbital evolution of moons inthese systems.With the possible abundance of planetary systemsaround low-mass stars and their unique characteristics forhabitability, we are interested in exploring their potentialfor exomoon habitability. To accomplish this we employed adistinctive approach to simultaneously consider both orbitaland tidal influences. In this paper, we review some theorieson tidal interactions between two massive bodies and thenconduct a computational investigation into the importanceof these interactions on the long-term evolution of hypo-thetical exomoons. For our study we focus on massive moonsaround large planets in the HZ of low-mass stars ( . . M (cid:12) )which demonstrates the long-term effect of gravitational per-turbations from a central star. Tidal bulges raised in both a planet and satellite will dissi-pate energy and apply torques between the two bodies. Therate of dissipation strongly depends on the distance betweenthe two objects. As a result of the tidal drag from the planet,a massive satellite will be subject to essentially four effectson its spin-orbital configuration:(i)
Semi-major axis:
Tidal torques can cause a moon toeither spiral in or out (Barnes & O’Brien 2002; Sasaki et al.2012). The direction of the spiral depends on the alignmentbetween the planet’s tidal bulge (raised by the moon) andthe line connecting the two centers of mass. If the planet’srotation period is shorter than the orbital period of the satel-lite, the bulge will lead (assuming prograde orbits) and the moon will slowly spiral outward. This action could eventu-ally destabilize the moon’s orbit, leading to its ejection. Onthe other hand, if the planet’s rotation period is longer, thebulge will lag and the moon will slowly spiral inward. As thishappens, the tidal forces on the moon become increasinglygreater. If the inward migration continues past the Rochelimit, the satellite can be disintegrated.(ii)
Eccentricity:
Non-circular orbits will be circularizedover the longterm. The timescale for which the eccentricityis damped can be estimated as (Goldreich & Soter 1966) τ e ≈ M s M p (cid:18) aR s (cid:19) Q s n , (1)where n is the orbital mean motion frequency, Q s is the tidaldissipation function, a is the semi-major axis, M s and R s arethe mass and radius of the satellite, and M p the mass of thehost planet.(iii) Rotation frequency:
The moon will have its ro-tation frequency braked and ultimately synchronized withits orbital motion around the planet (Dole 1964; Porter &Grundy 2011b; Sasaki et al. 2012), a state that is commonlyknown as tidal locking.(iv)
Rotation axis:
Any initial spin-orbit misalignment(or obliquity) will be eroded, causing the moon’s rotationaxis to be perpendicular to its orbit about the planet. In ad-dition, a moon will inevitably orbit in the equatorial plane ofthe planet due to both the Kozai mechanism and tidal evolu-tion (Porter & Grundy 2011b). The combination of all theseeffects will result in the satellite having the same obliquityas the planet with respect to the circumstellar orbit. As forthe host planet, massive planets are more likely to maintaintheir primordial spin-orbit misalignment than small planets(Heller et al. 2011b). Therefore, satellites of giant planetsare more likely to maintain an orbital tilt relative to thestar than even a single terrestrial planet at the same dis-tance from a star.The issue of tidal equilibrium in three-body hierarchicalstar-planet-moon systems in the Keplerian limit has recentlybeen treated by Adams & Bloch (2016), who demonstratedthat a moon in tidal equilibrium will have its circumplane-tary orbit widened until it ultimately gets ejected by stellarperturbations. N -body gravitational perturbations from thestar set an upper limit on the maximum orbital radius of themoon around its planet at about half the planetary Hill ra-dius for prograde moons (Domingos et al. 2006a). However,moons can be ejected from their planet even from close or-bits around their planets, e.g. when the planet-moon systemmigrates towards the star and stellar perturbations increasethe moon’s eccentricity fatally (Spalding et al. 2016). Per-turbations from other planets can also destruct exomoonsystems during planet-planet encounters (Gong et al. 2013;Payne et al. 2013; Hong et al. 2015). As a result of the tidal orbital evolution, orbital energy istransformed into heat in either the planet or the moon orboth. Several quantitative models for tidal dissipation havebeen proposed, the most widely used family of which is com-monly referred to as equilibrium tide models (Darwin 1879;Hut 1981; Efroimsky & Lainey 2007; Ferraz-Mello et al.
MNRAS , 1–20 (2017) xomoon Habitability and Tidal Evolution H ) of a satellite as H = ( GM p ) / M p R Q s a − / e , (2)where G is the gravitational constant and e is the satel-lite’s eccentricity around the planet (Peale & Cassen 1978;Jackson et al. 2008b). Equation (2) implies that tidal heat-ing drops off quickly with increasing distance and altogetherceases for circular orbits ( e = 0). In Eqs. (1) and (2), Q parameterizes the physical re-sponse of a body to tides (Peale et al. 1979). Its specific use-fulness is that it encapsulates all the uncertainties about thetidal dissipation mechanisms. For solid bodies, this functioncan be related to the rigidity µ and the standard Q -value, Q = Q ( + µ / g ρ R ) , where g is the gravitational acceler-ation at the surface of the body and ρ is its mean density.The dissipation function can also be defined in terms of thetidal Love number ( k L ) as Q = Q / k L .The tidal heating in Jupiter’s moons Io and Europaworks to circularize their orbits relative to Jupiter. Interest-ingly, estimates for their eccentricity damping timescales areconsiderably less than the age of the Solar System (Murray& Dermott 1999, p. 173). Yet, their eccentricities are notice-ably not zero (0.0043 and 0.0101, respectively). This incon-sistency has been explained by the observed Laplace reso-nance between them and the satellite Ganymede (Peale et al.1979; Yoder & Peale 1981). The orbital periods of the threebodies are locked in a ratio of 1:2:4, so their gravitationalinteractions continually excite the orbits and maintain theirnon-zero eccentricities. This example demonstrates the needto include external gravitational influences when consideringthe long-term tidal evolution of non-isolated planet-moonsystems. For systems in the HZ of low-mass star systems,external perturbations from the relatively close star couldserve to maintain non-zero satellite eccentricity in much thesame way as the orbital resonances in the Galilean moonsystem (Heller 2012).Equation (2) represents the energy being tidally dissi-pated by the whole of a satellite. However, to assess thesurface effects of tidal heating on a potential biosphere itis necessary to consider the heat flux through the satellite’ssurface. Assuming the energy eventually makes its way tothe moon’s surface, the tidal surface heat flux ( h ) can berepresented as h conv = H / π R . (3)Note in the definition that a subscript is used to identifyEq. (3) as the surface heat flux based on the conventionaltidal model for H . Below, we will define another tidal heatflux based on a different tidal model. This model breaks down for large e (Greenberg 2009) and itneglects effects of tidal heating from obliquity erosion (Heller et al.2011b) and from rotational synchronization. In terms of the effects of tidal heating on a moon’s sur-face habitability, Barnes et al. (2009) proposed a conserva-tive limit based on observations of Io’s surface heat flux ofabout h = − (Spencer et al. 2000b; McEwen et al.2004), which results in intense global volcanism and a litho-sphere recycling timescale on order of 10 yr (Blaney et al.1995; McEwen et al. 2004). Such rapid resurfacing could pre-clude the development of a biosphere, and thus − canbe considered as a pessimistic maximum rate of tidal heatingto still allow habitable environments.Tidal heating is not the only source of surface heat flowin a terrestrial body. Radiogenic heating, which comes fromthe radioactive decay of U, Th, and K, is an additional sourcefor surface heat flow. Barnes et al. (2009) used this combi-nation to also set a lower limit of h min ≡ − for thetotal surface heat flux of a terrestrial body by consideringthat internal heating must be sufficient to drive the vitallyimportant plate tectonics. This value was based on theoreti-cal studies of Martian geophysics which suggest that tectonicactivity ceased when the radiogenic heat flux dropped be-low this value (Williams et al. 1997). Even though the pro-cesses which drive plate tectonics on Earth are not fully un-derstood (Walker et al. 1981; Regenauer-Lieb et al. 2001),it is accepted that an adequate heat source is essential.The phenomenon of plate tectonic is considered importantfor habitability because it drives the carbon-silicate cyclethereby stabilizing atmospheric temperatures and CO lev-els on timescales of ∼ yr. For reference, the Earth’s com-bined outward heat flow (which includes both tidal and ra-diogenic heat) is 0.065 W m − through the continents and0.1 W m − through the ocean crust (Zahnle et al. 2007),mostly driven by radiogenic heating in the Earth.Radiogenic heating scales as the ratio of volume to area(Barnes et al. 2009). Consequently, for most cases involvingclosely orbiting bodies that are significantly smaller than theEarth, it is believed that tidal heating probably dominates(Jackson et al. 2008b,a). The theoretical moons consideredin this study meet these conditions. Therefore, we assumethat radiogenic heating is negligible and that the total sur-face heat flux is equal to the tidal flux. We also adopt theheating flux limits for habitability presented above, so that h min < h < h max could allow exomoon surface habitability. Investigations of exomoon habitability can be distinguishedfrom studies on exoplanet habitability in that a variety ofastrophysical effects can be considered in addition to the il-lumination received by a parent star. For example, a moon’sclimate can be affected by the planet’s stellar reflected lightand its thermal emission. Moons also experience eclipses ofthe star by the planet, and tidal heating can provide an ad-ditional energy source that is typically less substantial forplanets. Heller & Barnes (2013) considered these effects in-dividually, and then combined them to compute the orbit-averaged global flux ( F glob ) received by a satellite. Morespecifically, this computation summed the averaged stellar, At the orbital distance of Mars, any contribution from tidalheating would be very low. Therefore, the total internal heatingis essential equal to the radiogenic heating in this case.MNRAS , 1–20 (2017)
R. R. Zollinger et al. reflected, thermal, and tidal heat flux for a satellite. In theirstudy, they provided a convenient definition for the globalflux as F glob = L ? ( − α s ) π a ? p q − e ? p + π R α p a ! + R σ SB ( T eq p ) a − α s + h s (4)where L ? is the luminosity of the star, a ? p is the semi-major axis of the planet about the star and a ps is the satel-lite’s semimajor axis about the planet, α is the Bond albedo, e is eccentricity, σ SB is the Stefan-Boltzmann constant, h s is the tidal heat flux in the satellite, and T eqp is the planet’sthermal equilibrium temperature.As an analogy with the circumstellar HZ for planets,there is a minimum orbital separation between a planet andmoon that will allow the satellite to be habitable. Moons in-side this minimum distance are in danger of runaway green-house effects by stellar and planetary illumination and/ortidal heating. There is not a corresponding maximum sepa-ration distance (other than stability limits) because satelliteswith host planets in the stellar HZ are habitable by defini-tion. The benefit of Eq. (4) is that it can be used to explorethe minimum distance. This is accomplished by comparingthe global flux to estimates of the critical flux for a run-away greenhouse ( F RG ). Heller & Barnes (2013) discusseda useful definition for F RG originally derived by Pierrehum-bert (2010). Applying that definition to an Earth-mass ex-omoon gives a critical flux of
295 W m − for a water-richworld with an Earth-like atmosphere to enter a runawaygreenhouse state. In comparison to the conservative limit of − for the tidal heating discussed above, the runawaygreenhouse limit defines the ultimate limit on habitability.Moons with a higher top-of-the-atmosphere energy flux can-not be habitable by definition, since all surface water will bevaporized. Heller (2012) suggested that low-mass stars cannot possi-bly host habitable moons in the stellar habitable zones be-cause these moons must orbit their planets in close orbitsto ensure Hill stability. In these close orbits they would besubject to devastating tidal heating which would trigger arunaway greenhouse effect and make any initially water-richmoon uninhabitable. This tidal heating was supposed to beexcited, partly, by stellar perturbations. While tidal pro-cesses in the planet-moon system would work to circularizethe satellite orbit, the stellar gravitational interaction wouldforce the moon’s orbital eccentricity around the planet toremain non-zero. However, Heller (2012) acknowledged thathis model did not couple the tidal evolution with the grav-itational scattering of a hypothetical satellite system so theextent of the gravitational influence of the star was surmised,but not tested. The need therefore remains to simulate theeccentricity evolution of satellites about low-mass stars witha model that considers both N -body gravitational accelera-tion and tidal interactions.We can estimate the stellar mass below which no habit-able moons can exist for a given system age and planet mass. Neglecting all atmospheric effects on a water-rich, Earth-likeobject, we have a HZ ≈ (cid:18) L ? L (cid:12) (cid:19) / AU ≈ (cid:18) M ? M (cid:12) (cid:19) / AU , (5)where L (cid:12) and M (cid:12) are the luminosity and mass of the Sun,respectively. The planetary Hill radius can be approximatedas R Hill ≈ M / ? (cid:18) M p (cid:19) / AU M / (cid:12) . (6)Using the timescale for tidal orbital decay as used in Barnes& O’Brien (2002; Eq. (7) therein), we have a crit = τ dec k , p M s R Q p s GM p + R / ! / ! < R Hill , (7)for tidal love number k , p for the planet, where the latterrelation must be met to allow for tidal survival over a time τ dec . Hence M ? ! > a crit (cid:18) M p (cid:19) / M / (cid:12) AU ! / . (8)For a Jupiter-like planet ( Q p = , k , p = . ) with anEarth-like moon, and assuming tidal survival for 4.5 Gyr,we estimate a minimum stellar mass of . M (cid:12) .For a computational study, many popular and welltested computer codes are available to simulate the grav-itational (dynamical) evolution of many bodied systems(Chambers & Migliorini 1997; Rauch & Hamilton 2002).Such codes are particularly useful for studying the long-termstability of planetary systems. These codes, however, do notinclude tidal interactions in their calculations. A modifica-tion of the Mercury N -body code (Chambers & Migliorini1997) to include tidal effects (Bolmont et al. 2015) has re-cently been used to simulate, for the first time, the evolu-tion of multiple moons around giant exoplanets (Heller et al.2014). Yet, these orbital calculations neglected stellar effectsin the planet-moons system.Useful derivations for a tide model that provides the ac-celerations from tidal interactions at any point in a satellitesorbit derivations were presented by Eggleton et al. (1998),whose work was based on the equilibrium tide model by Hut(1981). Their particular interest was to consider tidal inter-actions between binary stars. Eggleton et al. (1998) derivedfrom first principles equations governing the quadrupole ten-sor of a star distorted by both rotation and the presence ofa companion in a possibly eccentric orbit. The quadrupoledistortion produces a non-dissipative acceleration f QD . Theyalso found a functional form for the dissipative force of tidalfriction which can then be expressed as the acceleration dueto tidal fiction f TF . These acceleration terms are useful be-cause they can be added directly to the orbital equation ofmotion for the binary: (cid:220) r = − GM r r + f QD + f TF , (9)where r is the distance vector between the two bodiesand M is the combined mass. This enables the evaluationof both the dynamical and tidal evolution of a binary starsystem. MNRAS , 1–20 (2017) xomoon Habitability and Tidal Evolution Mardling & Lin (2002) were the first to recognize that theformulations by Eggleton et al. (1998) provided a powerfulmethod for calculating the complex evolution of not just bi-nary stars, but planetary systems as well. Based on theirformulations, Mardling & Lin (2002) presented an efficientmethod for calculating self-consistently the tidal plus N -body evolution of a many-bodied system. Their work hada particular focus on planets, yet they emphasized that themethod did not assume any specific mass ratio and thattheir schemes were entirely general. As such, they could beapplied to any system of bodies.The Mardling & Lin (2002) method lends itself bestto a hierarchical (Jacobi) coordinate system. Since our pri-mary interest involves the evolution of moons around a largeplanet, we use the planet’s position as the origin of a givensystem. The orbit of the next closest body will be a moon,whose position is referred to the planet. The orbit of a thirdbody is then referred to the center of mass of the planet andinnermost moon, while the orbit of a fourth body is referredto the center of mass of the other three bodies. This systemhas the advantage that the relative orbits are simply per-turbed Keplerian orbits so the osculating orbital elementsare easy to calculate (Murray & Dermott 1999).Mardling & Lin (2002) parameterized the accelerationterms to create equations of motion for systems containingup to four bodies. Let the masses of the four objects be m , m , m , and m . For our study, m always represented aplanet and m represented a moon. The third mass, m , rep-resented a central star for 3-body systems, which are cre-ated by simply setting m equal to zero. The planet andmoon (being the closest pair of bodies in the system) areendowed with structure that is specified by their radii S and S , moments of inertia I and I , spin vectors Ω and Ω , their quadrupole apsidal motion constants (or half theappropriate Love numbers for planets with some rigidity) k and k , and their Q -values Q and Q . Body 3 is assumed tobe structureless, meaning it is treated as a point mass. Weshould note one necessary correction to Eq. (7) in Mardling& Lin (2002), the m β term in the first set of bracketsshould be replaced with m β .The total angular momentum and total energy for eachsystem were calculated as well as the evolution of the spinvectors for bodies 1 and 2 (Eqs. 9, 10, and 15 in Mardling& Lin 2002). We should also note one minor correction toEq. (14) in Mardling & Lin (2002), the factor S should bereplaced by S .The acceleration from quadrupole distortion f QD wasderived from a potential and under prevailing circumstanceswould conserve total energy. On the other hand, the ac-celeration due to tidal friction f TF cannot be derived froma potential and represents the effects of a slow dissipationof orbital energy. The dissipation within the moon causeschange in its orbital semi-major axis and spin vector. Wedefined the rate of energy loss from tidal heating (cid:219) E tide usingEq. (71) in Mardling & Lin (2002). While total orbital en-ergy is not conserved, we still expect conservation of totalangular momentum. Since (cid:219) E tide represents the tidal heatingin a satellite according to this tidal model, the surface heat flux in the satellite can be defined as h = (cid:219) E tide / π S . (10) Using the equations of motion defined in the previous sub-section, we designed a computer program that has the abilityto simultaneously consider both dynamical and tidal effects,and with it, we simulated exomoon tidal evolution in low-mass star systems. The program code was written in C++and a Bulirsch-Stoer integrator with an adaptive timestep(Press et al. 2002) was used to integrate the equations ofmotion.The robustness of the code was evaluated by trackingthe relative error in total angular momentum, defined as ( L out − L in )/ L in , where L in is the system angular momentumat the start of a simulation and L out is the angular momen-tum at a later point. The size of the relative error couldbe controlled by adjusting an absolute tolerance parameter.However, as is often the case with direct orbit integrators,the problem of systematic errors in the semi-major axis ex-isted. The simulations were monitored to ensure a maximumallowed error of 10 − for the total angular momentum. Thealgorithm efficiency required about seven integration stepsper orbit, for the smallest orbit. Relative errors in total en-ergy ( ( E out − E in )/ E in ) were also tracked, although, conserva-tion of mechanical energy was not expected for our systemsand, hence, not included as a performance constraint in oursimulations.A primary drawback to directly integrating orbital mo-tion is the extensive computational processing times re-quired to simulate long-term behavior. With our particularevolution model the situation is compounded by extra calcu-lations for tidal interactions. Early tests for code efficiencyindicated a processing time of roughly two days per Myrof simulated time for 3-body systems (two extended objectsand a third point mass object). Integration time significantlyincreases when additional bodies are considered, so a maxi-mum of 3 bodies was chosen for this initial study. Since theintegration timestep is effectively controlled by the objectwith the shortest orbital period, the exact processing timesvaried significantly between wide orbit and short orbit satel-lites. For our study the size of the orbit was determined fromthe theoretical habitable zone around a low-mass star (seesubsection 3.1.2). From these results we predicted an abil-ity to simulate satellite systems with timescales on order of10 yr with our limited computational resources. As part of our investigation into the evolution of exomoonsaround giant planets in the HZ of low-mass stars, we eval-uated two different system architectures. The first involveda minimal 2-body system consisting of only a planet and amoon; the second was a 3-body system of one planet, onemoon, and a central star. For each system, the planet andmoon were given structure while a star was treated as apoint mass (which significantly reduce the required number
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Table 1.
Physical properties for a hypothetical Mars-like exo-moon. The parameters A , B and C are the principal moments ofinertia. Parameter Value
Mass ( M ) 0.107 M ⊕ Mean Radius ( R ) 0.532 R ⊕ Love Number ( k L ) 0.16Bond Albedo ( α ) 0.250Dissipation Factor ( Q ) 80 C / M R C / A C / B of calculations per timestep). For each 3-body simulation,a corresponding 2-body simulation was performed for thesame planet-moon binary. Comparing the two simulationswould provide a baseline for determining the stellar contri-bution to a moon’s long-term evolution.Although thousands of extrasolar planets have been de-tected, very little information is known about their internalstructure and composition. Notwithstanding, work is beingconducted to offer a better understanding (for example, seeUnterborn et al. (2016) and Dorn et al. (2017)). Recogniz-ing that limits for tidal dissipation depend critically on theseproperties, without this information, objects in the solar sys-tem provide the best guide for hypothesizing the internalstructure and dynamics of extrasolar bodies. For this reason,we used known examples from our Solar System to modelthe hypothetical extrasolar satellite systems. Formation models for massive exomoons show reasonablesupport for the formation of moons with roughly the massof Mars around super-Jovian planets (Canup & Ward 2006;Heller & Pudritz 2015), which is near the current detectionlimit of ∼ M ⊕ (Heller 2014; Kipping et al. 2015) and alsolies in the preferred mass regime for habitable exomoons (seeSect. 1). For these reasons, we chose to model the physicalstructure of our hypothetical exomoons after planet Mars.The specific physical properties used in our moon modelare shown in Table 1. The Bond albedo represents currentestimates for Mars, although one could argue that an Earth-like value of 0.3 might be equally appropriate since we con-sider habitable exomoons. Bond albedos are not actuallyused in the evolution simulations, they are utilized after-wards to estimate the global flux F glob received by the moon,see Eq. (4). Habitability considerations from the global fluxare made in comparison to the critical flux for a runawaygreenhouse ( F RG , see subsection 2.2). For a Mars-mass ex-omoon, the critical flux is F RG =
269 W m − . We decided touse the lower Bond albedo of Mars, keeping in mind that itwould produce slightly higher estimates of F glob . If the sim-ulation results showed a global flux >
269 W m − we wouldthen also consider an Earth-like Bond albedo.The Love number and dissipation factor are also basedon recent estimates for Mars (Yoder et al. 2003; Bills et al.2005; Lainey et al. 2007; Konopliv et al. 2011; Nimmo &Faul 2013). Their specific values represent the higher dissi-pation range of the estimates. This choice produces a slightly faster tidal evolution and also tests the extent of the gravi-tational influence of the star against a slightly higher rate ofenergy loss that continually works to circularize the orbit ofthe moon. The principal moments of inertia also reflect esti-mates for Mars (Bouquillon & Souchay 1999) and representa tri-axial ellipsoid for the overall shape of the body. Werealize that a Mars-like exomoon orbiting close to a giantplanet would undoubtedly develop a different bodily shapethan the current shape of Mars. If we assume a constantmoment of inertia, then the principal moments only becomeimportant in calculating the evolution of the moon’s spinvector (see Eq. 15 in Mardling & Lin 2002). To minimizethis importance, we set the moon’s initial obliquity to zeroand started each simulation with the moon in synchronousrotation. Under these conditions there is minimal change tothe moon’s spin vector and the given triaxial shape is effec-tive at keeping the moon tidally locked to the planet as itsorbit slowly evolves. In that sense, another choice in shapecould have equally served the same purpose.We only considered prograde motion for the moon rel-ative to the spin of the planet. In keeping with our use oflocal examples, we modeled the moon’s orbital distance afterknown satellite orbits around giant planets in the Solar Sys-tem. The large solar system moons Io, Europa, Ganymede,Titan, and Callisto happen to posses roughly evenly spacedintervals for orbital distance in terms of their host planet’sradius R p (5.9, 9.6, 15.3, 21.0, and 26.9 R p , respectively).In reference to this natural spacing, when discussing orbitaldistances of a moon we will often refer to them as Io-like orEuropa-like orbits, etc.Considering stability constraints, and assuming oursatellite systems are not newly formed, we would not ex-pect high eccentricities for stable exomoon orbits. For thisreason we use an initial eccentricity of 0.1 for our 2-bodyand 3-body models. We then monitor the moon’s evolutionas the tidal dissipation works to circularize the orbit. Moon formation theories suggest that massive terrestrialmoons will most likely be found around giant planets.With this consideration, we chose to model our hypothet-ical planet after the two most massive planets in our SolarSystem, specifically, planets Jupiter and Saturn. The physi-cal properties used in our planet models are shown in Table2. The Bond albedos, Love numbers, and dissipation factorsare based on current estimates (Gavrilov & Zharkov 1977;Hanel et al. 1981, 1983; Meyer & Wisdom 2007; Lainey et al.2009). We chose values that represent more substantial heat-ing in the planets, similar to our choice for the Mars-likemoons.The normalized moments of inertia were also based onrecent estimates (Helled 2011; Helled et al. 2011) and thethree equal principal moments imply a spherical shape forthe planet. This shape may be unrealistic in that a planetorbiting close to a star with a massive moon is unlikely tomaintain a truly spherical shape. With spherical bodies wealso do not match the exact shapes of the solar system plan-ets, i.e., we ignore their oblateness that results from theirshort rotation periods (about 10 hrs for Jupiter, 11 hrs forSaturn). However, the planet’s exact shape is not particu-larly import for the purposes of our simulations. We start
MNRAS , 1–20 (2017) xomoon Habitability and Tidal Evolution Table 2.
Physical properties for hypothetical giant exoplanets.The planet shape is assumed spherical with principal moments ofinertia A = B = C . Jupiter-likeParameter Value
Mass ( M ) 318 M ⊕ Mean Radius ( R ) 11.0 R ⊕ Bond Albedo ( α ) 0.343Love Number ( k L ) 0.38Dissipation Factor ( Q ) 35000 C / M R Saturn-likeParameter Value
Mass ( M ) 95.2 M ⊕ Mean Radius ( R ) 9.14 R ⊕ Bond Albedo ( α ) 0.342Love Number ( k L ) 0.341Dissipation Factor ( Q ) 18000 C / M R each planet with zero obliquity and synchronous rotationrelative to its orbit around the central star, which resultsin a rotation period that ranges from about 15 to 120 days.This setup is consistent with the prediction that planets inthe HZ of low-mass stars will most likely be tidally lockedto the star.To define the HZ for a given star we followed the workby Kopparapu et al. (2013), who estimated a variety of lim-its for the inner and outer edges around stars with effectivetemperatures ( T eff ) in the range 2600 K ≤ T eff ≤ log R ? R (cid:12) = .
03 log M ? M (cid:12) + . . (11)Note that Eq. (11) is valid only when M ? . M (cid:12) , which isvalid for the dwarf stars we consider. The stellar luminositycan be estimated from the mass by λ = . µ + . µ + . µ + . , (12)where λ = log ( L / L (cid:12) ) and µ = log ( M ? / M (cid:12) ) (Scaloet al. 2007). With luminosity and radius, the stellar effec-tive temperature can be calculated using the familiar rela-tionship L = πσ SB R ? T . (13)An online database of confirmed exoplanet detections lists almost 3,500 confirmed planets in total. Of those, onlyabout 8 percent have host stars with mass . . M (cid:12) . This NASA Exoplanet Archive at exoplanetarchive.ipac.caltech.edu/ relatively small sample is most likely due to selection bias asdetection techniques have evolved. The majority of these de-tections occurred in recent years and the number is expectedto grow. While several large planets can be found orbitinglow-mass stars, only a small number orbit inside the HZ. Thepercentage of these systems is questionable since a signifi-cant number of confirmed planets in the database are miss-ing key parameters such as planet mass or orbit distance.One known planet in particular corresponds nicely with ourSaturn model. The planet HIP 57050 b has a mass of 0.995 (± . ) M Sat × sin ( i ) (Haghighipour et al. 2010), where M Sat is the mass of Saturn and i is the unknown inclination be-tween the normal of the planetary orbital plane and our lineof sight. Since no other information is available about thephysical properties of these giant planets, only direct com-parisons relating to their masses can be made. None of thelow-mass HZ candidates matches directly with Jupiter, buttwo have masses about double that of Jupiter (GJ 876 b andHIP 79431 b). This at least supports the plausibility for theexistence of Jupiter mass planets in the HZ of dwarf stars.Each simulated system consisted of only one planet.Each planet started with a circular orbit relative to the starat a distance inside the stellar HZ. To conserve computa-tional processing time, we limited exploration of the HZ tojust two specific orbital distances per planetary system. Thefirst location was in the center of the HZ for a given starmass, otherwise defined as a center = ( inner edge + outer edge )/ . (14)This particular location was to serve as a referencepoint for the next round of simulations. If most of our hy-pothetical satellites already experienced intense tidal heat-ing at the center, then the next location should be furtherout in the zone. On the other hand, if the surface heatingrates were below the proposed maximum for habitability( h max = − ), then we would move inward for the nextround. As we will show, after simulating satellite systems inthe center of the HZ it became clear that the second roundof simulations should involve the inner HZ.While the innermost edge was a reasonable option toexplore, an Earth-equivalent distance had obvious attrac-tion. By ‘Earth-equivalent’ we refer to the Earth’s relativeposition in the Sun’s HZ as compared to the total width ofthe zone. Conservative estimates by Kopparapu et al. (2013)place the inner edge of the Sun’s HZ at 0.99 AU and the outeredge at 1.67 AU. With these boundaries we define Earth’srelative location within the solar HZ as d rel = − ( .
67 AU − )/( .
67 AU − .
99 AU ) = . . (15)For a given star, we use this relative location and thewidth of its HZ to define a planet’s Earth-equivalent orbitaldistance as a eq = inner edge + ( outer edge − inner edge ) × d rel . (16) Information relating to a star’s mass, radius, and effectivetemperature is required to calculate its circumstellar HZ and
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Table 3.
Eccentricity damping timescale estimates for a Mars-like moon at different orbital distances.
Jupiter-like Host PlanetOrbital Distance τ e (years) Io-like ( . R Jup ) 4 x Europa-like ( . R Jup ) 9 x Ganymede-like ( . R Jup ) 2 x Titan-like ( . R Jup ) 1 x Callisto-like ( . R Jup ) 7 x Saturn-like Host PlanetOrbital Distance τ e (years) Io-like ( . R Sat ) 6 x Europa-like ( . R Sat ) 1 x Ganymede-like ( . R Sat ) 3 x Titan-like ( . R Sat ) 2 x Callisto-like ( . R Sat ) 1 x to estimate the global flux received by a moon. In our sim-ulations the stars are treated as point masses, and so thestellar mass is the only physical characteristic used in theactual simulation of a system. For our study we considereda star mass range from 0.075 M (cid:12) to 0.6 M (cid:12) . These wouldconsist of mostly M spectral type stars and possibly somelate K type stars, with typical surface temperatures less than4,000 K. This range of stars is sometimes referred to as reddwarf stars. We can estimate the eccentricity damping timescales ( τ e )for our hypothetical satellite systems following Eq. (1). Thetimescales are summarized in Table 3 for the two planetmodels and for the various moon orbital distances consid-ered. It was fortunate to see estimates around 1 Myr sinceintegration times of order yr represent achievable compu-tational processing times. Results showed that we were ableto simulate complete tidal evolution for moons with Io-likeand Europa-like orbital distances. However, it was unlikelythat we could show any significant evolution for the widerorbits on these timescales. It should be noted that Eq. (1)was derived from a two body calculation. As such, it doesnot take into account any perturbing effects from additionalbodies. Since the degree to which the central star would in-fluence a moon’s orbit was unknown, we decided to includethe wider orbits in our considerations. Figure 1 demonstrates some typical evolutions for Mars-likemoons around a Jupiter-like planet. Solid curves are sim-ulations that included tidal interactions. The lower, mid-dle, and upper red curves represent Io-like, Europa-like, andGanymede-like moon orbital distances, respectively. As ex-pected, the rate of change depended strongly on the moon’sorbital distance. When repeated without tidal interactions,all simulations had the same result, with no significantchange to the orbits (see dashed lines). Simulations for Titan-like and Callisto-like orbital distances showed very lit-tle change over 10 Myr whether tides were considered or not.Their results are nearly identical to the dashed curves in eachplot so they were not included in Fig. 1.Simulations for a Saturn-like host planet were also con-ducted. The satellite evolutions are similar to those in Fig. 1for a Jupiter-like host planet, with one exception. An Io-like moon orbit around Saturn experienced significantlygreater change in semi-major axis than the Io-like moon or-bit around Jupiter. In just 7 Myr the semi-major axis de-cayed to 75% of its initial value with a Saturn-like hostplanet, compared to about 96% with a Jupiter-like host.The explanation is related to our approach of normalizingplanet-moon distances by the radius of the respective hostplanet. A moon’s initial orbital distance is defined by R p , theplanet radius (an ‘Io-like’ orbit is 5.9 R p ). Because of Sat-urn’s smaller radius a moon with an Io-like orbital distanceis actually closer to the planet than an Io-like orbit aroundJupiter, since 5.9 R Sat < R Jup . Following this formulationfor the orbital distances, identical moons around each planetwill experience different heating rates.While different tidal heating rates explain part of thediscrepancy in semi-major axis evolution between Saturn-like and Jupiter-like host planets, it does not account for allof it. Figure 1(a) shows the Io-like orbit around the Jupiter-like host was circularized after ≈ . Myr, at which pointthe tidal heating ceases in our synchronized moon with zerospin-orbit alignment. Therefore, any change in the semi-major axis after ≈ . Myr is no longer attributed to tidaldissipation in the moon. Instead, tidal dissipation must oc-cur in the planet, which itself is synchronized to the star,causing the planet’s tidal bulge to lag behind the moon inour systems. This continued evolution makes the moon spiraltowards the planet. Ultimately, the moon will be destroyednear the planetary Roche radius (Barnes & O’Brien 2002).Tidal evolution of moons in Europa-wide orbits are muchslower, with a fractional decrease of the semi-major axis of ≈ . per 10 Myr.Using the corresponding slope in Fig. 1(b) for theJupiter system, we roughly estimate a total lifetime of ≈ Myr for the moon with an Io-like orbit. As a compari-son of this timescale to theoretical predictions, Goldreich &Soter (1966) derived a relation for the change in semi-majoraxis of a satellite as dadt = (cid:18) GM p (cid:19) / R p Q p M s a / . (17)This model assumes that the planet’s Q -value is frequency-independent, which is only a fair approximation over a verynarrow range of frequencies and therefore implicitly limitsthe model to low eccentricities and inclinations (Greenberg2009).As an alternative, Mardling (2011) suggested to use theconstant time lag ( τ ) model to study the tidal evolution ofhot Jupiters based on the knowledge of tidal dissipation inJupiter. This is done by assuming that τ (rather than Q ) iscommon to Jupiter-mass planets. Mardling (2011) proposeto approximate the Q -value of a planet with orbital period P via Q / Q J = P / P Io , where Q J and P Io = . d are the tidaldissipation constant of Jupiter and the orbital period of Io,respectively. The shortest orbital period for our Jupiter-likeplanets was 13 d, which gives us an estimated Q -value 7.3 MNRAS , 1–20 (2017) xomoon Habitability and Tidal Evolution (a)(b) Figure 1.
Orbital evolution of a single Mars-like moon around aJupiter-like planet in four different cases. The dashed line assumesno tides. The solid lines all assume an initial eccentricity of 0.1but with the moon starting at different semi-major axes from theplanet, i.e. in an Io-wide, a Europa-wide, and in a Ganymede-wideorbit around the planet (see labels). times that for Jupiter. Applying this result and using thevalues listed in Table 2, Eq. (17) estimates a timescale of ≈ Myr for the orbital decay. While a few times longerthan the rough estimate from our simulations, they agreeon the order of magnitude.A graphical representation of the spin evolution for theIo-like systems is provided in Fig. 2. Comparing the planetspin rates (dashed lines) in Fig. 2(a), the Saturn-like planet(black dashed line) had a noticeable increase relative theJupiter-like host (red dashed line), note the log scaling forthe ordinate axis. This difference is explained by the unequaldistances for the otherwise identical Mars-like moons as wellas the unequal moments of inertia for the planets. The shortsatellite orbit around Saturn combined with its lower mo-ment of inertia leads to the small, but noticeable increase inspin. That small increase causes the moon to almost doubleits initial spin rate. On the other hand, the wider orbit and (a)(b)
Figure 2.
Spin evolution of a moon and its host planet, assum-ing an Io-like moon orbit. ( a ) Spin frequencies of the planets(dashed lines) and moons (solid lines) assuming host planets akinto Jupiter (red) and Saturn (black). ( b ) Ratio of spin magnitude( Ω ) and mean motion ( n ) for the moons represented in plot a. more massive Jupiter host planet caused very little changefor both the moon and planet in this system.The tidally locked state of the moons throughout thesimulated time are demonstrated in Fig. 2(b). A perfectlysynchronized spin rate would produce a value of 1 for theratio between spin magnitude and mean orbital motion. No-tice that the ratio is not exactly 1 for the first few Myr,although the difference is small. This discrepancy can beexplained by the shapes of the orbit during that time. Re-ferring to the eccentricity plotted in Fig. 1(a), the spin can-not completely synchronize with the orbital period until theeccentricity approaches zero.As a final observation, one similarity between theJupiter and Saturn-like systems is that there is littlechange in the orbital elements over a period of 10 Myr forGanymede-like orbits and greater. This was not particularly MNRAS , 1–20 (2017) R. R. Zollinger et al.
Table 4. R Hill .Gray shaded cells represent unstable moon orbits.
Earth-Equivalent Planet Orbit
Star Mass Moon Semi-major Axis5.9 R Jup R Jup R Jup R Jup R Jup M (cid:12) M (cid:12) M (cid:12) M (cid:12) M (cid:12) M (cid:12) Planet Orbit in Center of HZ
Star Mass Moon Semi-major Axis5.9 R Jup R Jup R Jup R Jup R Jup M (cid:12) M (cid:12) M (cid:12) M (cid:12) M (cid:12) M (cid:12) surprising as the result was predicted from the dampingtimescales. At these wider orbits the dissipation rate wouldbe much lower for a given eccentricity due to the a − / de-pendence of tidal heating. On the other hand, a central star’spotential for exciting the eccentricity is still untested, so weincluded the wider orbits in our 3-body analysis. We continue with systems consisting of one star, one planet,and one moon. The Mars-like moon was given an initial ec-centricity of 0.1, measured relative to the planet. We usedthis high value for eccentricity to cover a wide range of for-mation possibilities and allowed the eccentricity to decay asa result of tidal dissipation in the moon and planet. We mon-itored the simulations to see if the moon orbits would settleto steady, non-zero eccentricities long after the orbits shouldhave circularized due to the tidal heating. For each satellitesystem we also ran a nearly identical simulation with themoon instead starting with a circular orbit. In this way wecould test if the stellar perturbation raised the eccentricityto the same steady state value as was achieved following thedecay from a higher initial value. For each 3-body simula-tion, we ran a corresponding 2-body simulation with justthe planet and moon to show what the evolution would bewithout the influence of the star.
With a central star ranging in mass from . M (cid:12) to . M (cid:12) and our confined planetary orbits within the relatively closestellar HZs, the exomoon Hill stability became very limited. Table 5. R Hill .Gray shaded cells represent unstable moon orbits.
Earth-Equivalent Planet Orbit
Star Mass Moon Semi-major Axis5.9 R Sat R Sat R Sat R Sat R Sat M (cid:12) M (cid:12) M (cid:12) M (cid:12) M (cid:12) M (cid:12) Planet Orbit in Center of HZ
Star Mass Moon Semi-major Axis5.9 R Sat R Sat R Sat R Sat R Sat M (cid:12) M (cid:12) M (cid:12) M (cid:12) M (cid:12) M (cid:12) We defined the stability region by the Hill radius ( R Hill ),given by the relation R Hill = a P (cid:18) M p M ? (cid:19) / , (18)where a P is the semimajor axis of the planet’s orbitaround the star, M p and M ? are the masses of the planet andstar, respectively. An extended study of moon orbital stabil-ity was conducted by Domingos et al. (2006b), who showedthat the actual stability region depends upon the eccentric-ity and orientation of the moon’s orbit. For prograde moonsthe stable region is: a s , max = . R Hill ( . − . e p − . e s ) , (19)where e p and e s are the orbital eccentricities of theplanet and satellite, respectively. Following Eqs. (18) and(19), we expected a stability limit of ∼ . R Hill for our cho-sen moon orbits, and so we only simulated systems for whichthe moon’s semi-major axis about the planet was < . R Hill .Our initial tests showed that for even the tightest exomoonorbit, which was an Io-like orbit at a = . R p , no stable sys-tems were found around stars less than 0.1 M (cid:12) . We thereforelimited extended simulations to stellar masses ≥ . M (cid:12) , us-ing intervals of . M (cid:12) for our systems (i.e., the consideredstar masses were 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6 M (cid:12) ).The stable region around a planet slowly increased withstellar mass (i.e. with increasingly larger HZ distances). Asexplained in Sect. 3.1.1, we considered a discrete range ofmoon orbital distances for each star-planet pair. Not sur-prising, many of the wider moon orbits were unstable forthe lowest star masses. This instability resulted in the moonbecoming unbound from the planet. We found that the 3-body simulations were only able to maintain long-term sta-bility for a satellite orbit of a s . . R Hill , which was thepredicted value described above. A summary of the stable
MNRAS , 1–20 (2017) xomoon Habitability and Tidal Evolution After completing our investigations into system stability andshort-term behavior, we were left with a set of 3-body sys-tems for which we could expect stability and reasonably pre-dict the long-term behavior. In other words, our systemswere feasable from a strictly dynamical perspective. The in-tegrated times for long-term simulations were extended un-til one of three results was achieved: (1) the average surfaceheat flux fell well below the proposed maximum limit forhabitability, h max = − ; (2) the average surface heatflux settled to a reasonably constant value; or (3) the tidalevolution was slow enough that it was not practical to con-tinue the simulation further.Examples of tidal evolution plots for Io-like and Europa-like moons with a Jupiter-like host planet are shown inFigs. 3 and 4. Similar plots for a Saturn-like host planet areincluded in Figs. 5 and 6. Plots for Ganymede-like, Callisto-like, and Titan-like moon simulations are not included sincethere was little to no change in the surface heat flux of thosemoons during the integrated time periods. Displayed in thesefigures is the average surface heat flux ( h ) in the moon as afunction of time, with h defined by Eq. (10). The individualplots in each figure represent low and high limit mass valuesconsidered for the central star. Note that a greater star masssignifies a larger orbital distance for the planet. Each plotincludes four curves with two solid ones referring to 3-bodysystems and two dashed ones referring to 2-body systems.Both pair of curves contains one case with zero initial ec-centricity and another scenario with an initial eccentricityof 0.1. Comparing the 2-body and 3-body curves is usefulin demonstrating the star’s influence on the long-term tidalevolution of the moon. The focus of this study involves the long-term behavior ofthe solid red curves in Figs. 3 through 6. With them wetest the extent to which a low-mass star has influence overa moon’s evolution in the HZ, as a consequence of the shortdistance of the HZ. This is done by comparing the solidred curve to the dashed red curve, which represents an iso-lated planet-moon binary that began the simulation withthe same initial conditions, minus the star. The dashed redcurve shows what the solid red curve would look like withoutthe influence of the central star. Since tidal heating works to circularize the orbits, ini-tially circular orbits (black curves) should continue with lit-tle to no heating, unless there are significant perturbationsto the orbit. The purpose of these simulations was to test ifthe central star would excite the heating to the same steadystate value that was achieved through the slow decay of aninitially higher value. If that result was observed, then thereis confidence that a non-zero, steady-state value for the sur-face heat flux was the product of perturbations from the star,and not simply based on our choice of initial conditions oran external artifact of our computer code. This anticipatedresult was indeed observed for Io-like and Europa-like or-bital distances, but not for the wider orbits (Ganymede-like,Titan-like, and Callisto-like orbits). This difference betweenthe eccentric and circular simulations for wide orbit systemsremained constant throughout a test period of 10 Myr.Our specific interest relates to the difference in thesolid red and dashed red curves by the end of each simu-lation. When the slope of the solid red line deviates fromthe dashed red line and eventually levels off, it suggests asustainable value for the surface heat flux. This indicatesthat the star has significant influence on the moon, and thecontinual perturbations are restricting the moon’s eccentric-ity from reaching zero. This behavior is observed for all theIo-like and Europa-like orbital distances. For our systemsthe timescales necessary to achieve the constant state weremore than double the estimates listed in Table 3. For con-venience, we will refer to Io-like and Europa like orbits as“short” orbits. At these distances the initially circular sim-ulations eventually reached the same approximate steadystate values as the non-circular systems. This result helpsto confirm the degree of excitation to which the central starcan influence the tidal evolution of a moon.For the shorter orbits, our results show that pertur-bations from a low-mass star can prevent circularizationand are able to maintain tidal heating in the moon forextended periods of time. While this effect was predicted(Heller 2012), this is the first instance in which it has ac-tually been tested with an evolution model that simultane-ously and self-consistently considered gravitational and tidaleffects.At orbital distances a s & R p , little change to the sur-face heat flux occurred over 10 Myr. For convenience, ref-erence to these distances as “wide” orbits will correspondto Ganymede, Titan, and Callisto-like moon orbits. In wideorbits, we found little deviation from the 2-body models,which was expected based on estimates of eccentricity damp-ing timescales and early 2-body test runs. Tidal heat fluxesfor these orbits are . . − , even with the influence ofthe star. This is an order of magnitude less than the shorterorbit moons and well below h max . Such low heating rateswould not be expected to noticeably affect the orbit overthe timescales considered.An examination of Fig. 5 shows unique behavior thatis not observed in the other figures. In these simulationsthe surface heat flux rapidly increases near the end of thesimulated time. The cause for this behavior was already dis-cussed in Sect. 4. Using 2-body models, we found that Mars-like moons with Io-like orbits and Saturn-like host planetsexperience significant evolution in their semi-major axis asa result of tidal torques and exchanges of angular momen-tum. Our 3-body results indicate that the star’s influence MNRAS , 1–20 (2017) R. R. Zollinger et al. (a) (b)
Figure 3.
Examples of the orbital evolution of a Mars-mass moon in an Io-like orbit around a Jupiter-like planet.(a) (b)
Figure 4.
Examples of the orbital evolution of a Mars-mass moon in a Europa-like orbit around a Jupiter-like planet. does little to change this long-term behavior. In this case,the rapid rise in tidal heating is due to the equally rapid de-cay of the moon’s semi-major axis as the moon spirals evercloser to the planet. The sudden end to the plots in Fig. 5reflects the early termination of the simulations when themoon approached the Roche radius at R p , where it wouldexperience tidal disruption. Note that the complete inwardspiral occurred in < Myr.
Our results for the 3-body simulations of Io-like and Europa-like satellite systems in the center of the HZ are summarizedin Table 6, while the results for systems at Earth-equivalentdistances are summarized in Table 7. Our interest is in heat-ing values which are developed after noticeable tidal evolu-tion and may be maintained for extended periods of time bystellar perturbations. The tables do not include results forwider orbits since, as discussed previously, there was no sig-nificant orbital evolution in these systems during the consid-ered time periods. Therefore, surface heating values for wideorbit moons would simply reflect our choice for the initial
MNRAS , 1–20 (2017) xomoon Habitability and Tidal Evolution (a) (b) Figure 5.
Examples of the orbital evolution of a Mars-mass moon in an Io-like orbit around a Saturn-like planet.(a) (b)
Figure 6.
Examples of the orbital evolution of a Mars-mass moon in a Europa-like orbit around a Saturn-like planet. orbital parameters of those systems and would not allow forany conclusive statements as to their ultimate tidal evolu-tion. It is worth noting that very few wide orbit moons areeven stable in our low-mass star range, and of those, all wideorbit systems had surface heat flux values below − formoon eccentricities at or below 0.1.Tables 6 and 7 show the average values at the end ofeach simulation and only include simulations that startedwith e = . for the moon. Included is the moon’s aver-age surface heat flux, the average eccentricity, and the av-erage semi-major axis (in comparison to its initial value).The initial value for the semi-major axis is listed as the moon distance. We also included the ‘conventional’ surfaceheat flux h conv . This value was calculated with Eq. (3) usingthe simulation results for the final eccentricity and semi-major axis and serves as a comparison between our chosenmodel and the conventional model for tidal heating. In re-gards to overall exomoon habitability, we included the orbit-averaged global flux ( F glob ) received by a satellite, as definedby Eq. (4). The total integration time considered for eachsimulation is also included.Results from a 2-body simulation for each moon orbitaldistance were also included in Tables 6 and 7 to demonstratethe influence of the low-mass star in comparison to an iso- MNRAS , 1–20 (2017) R. R. Zollinger et al.
Table 6. h max . Values of ** indicate themoon spiraled into the planet. Jupiter-like Host PlanetMoon Distance Star Mass Planet h h conv e a / a F glob Sim. Time( R Jup ) ( M (cid:12) ) Distance ( AU ) (W/m ) (W/m ) (W/m ) (10 Years)Io-like5.9 − − − − − − h h conv e a / a F glob Sim. Time( R Sat ) ( M (cid:12) ) Distance ( AU ) (W/m ) (W/m ) (W/m ) (10 Years)Io-like5.9 − − ** ** ** ** ** 100.2 0.11 ** ** ** ** ** 100.3 0.16 ** ** ** ** ** 100.4 0.22 ** ** ** ** ** 100.5 0.30 ** ** ** ** ** 100.6 0.41 ** ** ** ** ** 10Europa-like9.6 − − − lated planet-moon system. The planet’s spin rate in thesesimulations reflected a tidally locked rotation around a 0.6solar mass star, which is the slowest of all considered planetspins.As discussed in the previous subsection, moons withinitial Io-like orbits and Saturn-like host planets spiraledinto the planet after ≈ Myr. Over that same time period,distortion torques were much less effective at evolving thesemi-major axis for Io-like orbits around Jupiter-like hosts.Note that for a given star mass, the two planets were atequal distances from the star and they both started witha tidally locked rotation relative to the star. This resultsin approximately equal spin rates for the two planets. Therelative effectiveness of the torques is then due to the dif-ference in orbital distances for the satellites. While moonswith a Jupiter host evolved much slower in comparison toa Saturn host, their inward migration was still noticeable.If we assume the inward migration rate will only increase,we estimate a total lifetime < Myr for a Mars-like moonwith an initial Io-like orbit and Jupiter-like host planet inthe HZ of a low-mass star.Lifetime estimates for the inward spiral of all the Io-like and Europa-like moon orbit simulations are shown inFig. 7, which shows results from simulations that startedwith circular moon orbits and thereby limits orbital effectsof tidal heating and emphasizes evolution due to distortion torques. Simulations that started with eccentric orbits haveshorter estimates for moon lifetimes. Estimated lifetimes forEuropa-like orbits are significantly larger than those for Io-like orbits, around a maximum of 1 Gyr for Saturn-like hostplanets and several Gyr for a Jupiter-like host. Figure 7 in-dicates that the lifetime of moons at larger orbital distancesare more sensitive to the planet’s position in the HZ sincethey are more weakly bound to the planet and can experi-ence greater influence from the central star. The sensitivity,however, becomes negligible for HZ distances around 0.6 so-lar mass stars.One result that stands out in Tables 6 and 7 is the dif-ference between the conventional model for surface heat fluxand our chosen model. Slightly lower conventional values ap-ply to systems involving higher eccentricities. As mentionedin Sect. 2.1, the conventional model can break down for higheccentricities, while our chosen model is still appropriate atlarge values. For systems involving lower eccentricities, h conv is consistently higher than h . The difference in the two val-ues scales down for the lowest eccentricities until there islittle difference between them. The greatest total differencewas less than a factor of 2. Since the exact mechanisms oftidal dissipation are still poorly understood, some differencebetween two separate tidal models is not surprising.A similarity between the two tidal models is a directdependence on the tidal Love number k L and an inverse MNRAS , 1–20 (2017) xomoon Habitability and Tidal Evolution Table 7. h max . Values of ** indicates themoon spiraled into the planet. Jupiter-like Host PlanetMoon Distance Star Mass Planet h h conv e a / a F glob Sim. Time( R Jup ) ( M (cid:12) ) Distance ( AU ) (W/m ) (W/m ) (W/m ) (10 Years)Io-like5.9 − − − − − − h h conv e a / a F glob Sim. Time( R Sat ) ( M (cid:12) ) Distance ( AU ) (W/m ) (W/m ) (W/m ) (10 Years)Io-like5.9 − − ** ** ** ** ** 100.2 0.076 ** ** ** ** ** 100.3 0.11 ** ** ** ** ** 100.4 0.15 ** ** ** ** ** 100.5 0.21 ** ** ** ** ** 100.6 0.29 ** ** ** ** ** 10Europa-like9.6 − − − dependence on the dissipation factor Q . Higher values for Q decrease the heating estimates and represent lower dis-sipation rates in the moon. Higher values for k L producehigher estimates for tidal heating and represent an increasedsusceptibility of the moon’s shape to change in response toa tidal potential. This would also produce a larger accel-eration due to the quadrupole moment of the moon, re-sulting in a shorter inward spiral towards the planet. Re-cent measurements of these two parameters for the so-lar system planet Mars contain significant uncertainty (e.g. k L = . ± . Q = ± ; Nimmo & Faul 2013). Whilethese uncertainties can be considered in the simulation re-sults for a single orbit, the long term effects of varying k L and Q would require each simulation to be repeated withthe different values.It is important to compare the 3-body results with theprovided 2-body result (the shaded rows) for each moon dis-tance in Tables 6 and 7. In each case, the difference in tidalheating between the 3-body and 2-body models decreaseswith increasing star mass. In other words, the influence ofthe star can be seen to decrease as the HZ moves outwardfor higher mass stars. Other than these general similarities,we will discuss each table separately. A graphical represen-tation of the surface heat flux in each table, as a function ofstellar mass, is also provided in Fig. 8.Satellites Systems at the Center of the HZ (Table 6)In the tables, surface heat flux values highlighted in redare above h max = − and represent a tidally evolved, steady state value from the 3-body simulations. Continuingwith the assumption that exomoon habitability might be injeopardy above h max , the table indicates that most of themoons would have the potential for habitability.The highest surface heat flux involves an Io-like orbitaround a Jupiter-like planet with a 0.1 solar mass centralstar. This was the only stable satellite orbit for a 0.1 M (cid:12) starand represents the tightest 3-body simulation (smaller starsystems were considered, but none were stable). The effectsof such a tight system can be seen by the long-term excita-tion of the moon in comparison to an isolated planet-moonbinary whose orbit was completely circularized during thesame time period. It is interesting that the final eccentricityis not unreasonably large ( e = 0.035), yet it is more thanenough to generate extreme heating when combined withthe short orbital distance. Heating rates for star masses of0.2 M (cid:12) are significantly lower in comparison. However, thesurface fluxes are still about 3 times more than the mostgeologically active surface in our solar system, so we will ex-clude them from habitability considerations as well. Noticethat all systems with a 0.2 M (cid:12) star had heating rates wellabove our maximum for habitability.There were mixed results for a star mass of 0.3 M (cid:12) .Both planet models had stable Europa-like moon orbits andthe moons were comfortably below h max . For Io-like moonorbits the surface heat flux was below h max for our model,but just above that for the conventional model. Given theinherent uncertainty in the tidal heating estimates, ratherthan discussing which model gives the better estimate, it is MNRAS , 1–20 (2017) R. R. Zollinger et al.
Figure 7.
Estimated lifetimes from each 3-body simulation for the moon to completely spiral into the host planet. more relevant at this point to discuss the global flux ( F glob )observed. Excluding the previously mentioned case of ex-treme tidal heating, the global flux for all the satellites is ≈
110 W m − . Although the global flux does not lead to adirect estimate for the exomoon’s surface temperature, auseful comparison can be made with the critical flux ( F RG )estimate of
269 W m − for a runaway greenhouse in a Mars-mass exomoon. Clearly, the habitability of exomoons at thislocation is not at risk based on runaway greenhouse condi-tions from the global energy flux. In this case, tidal heatingmay actually be beneficial in warming large bodies of surfacewater that would otherwise freeze, which might be helpfulalso beyond the stellar HZ (Reynolds et al. 1987b; Heller &Armstrong 2014).Central star masses of 0.4 M (cid:12) mark a cutoff for all ex-omoon habitability concerns based on intense tidal heatingalone, since all the hypothetical exomoons are comfortablybelow h max at this point. The star does influence the shortorbit moons, but the influence may be seen as a benefit inpromoting surface activity in the exomoons rather than arestriction for habitability.For the tidally evolved short orbits there is not a largedifference in the final satellite heating rates between the twoplanet models and a given star mass. For example, with astar mass of 0.3 M (cid:12) and an Io-like moon orbit, the moonaround Jupiter had h = .
49 W m − . In comparison, the moonaround Saturn had h = .
56 W m − . Some examples matcheven closer. The same cannot be said for the orbital ele-ments, yet, these parameters would have to vary in orderto compensate for the unique physical characteristics of theplanet. Given the physical differences and the correspond-ing difference in orbital distances for the moons, we did notexpect the tidally evolved heating rates to be so similar.As explained earlier, the 3-body simulation resultsinvolving the center of the HZ helped determine our secondlocation for exploration inside the zone. Considering thatthe majority of the satellites had heating rates below h max and that the global flux was well below the critical flux fora runaway greenhouse, it was clear that the next round ofsimulations should involve the inner HZ.Satellites Systems at Earth-equivalent Distances (Table 7)For the second round of 3-body simulations we essen-tially moved the planet-moon binary closer to the star inthe prior 3-body systems. By decreasing the orbital distancefrom the star the Hill radius of the planet also decreased.This led to a reduction in stable satellite systems, especiallyfor wider orbits.All stable satellite systems involving a central star ofmass 0.4 M (cid:12) and below experience surface heating greaterthan h max . Since the surface heat fluxes represent tidallyevolved, steady state values, the heating has the potentialto be maintained for extended periods of time. There is onlysmall differences in final heating rates between specific moonorbits around Jupiter or Saturn-like host planets.In regards to global flux, extreme tidal heating causedone satellite to be above the critical flux of
269 W m − . Inthat situation the extreme heating was already enough torule it out for habitability. The rest of the satellites arecomfortably below the critical flux, leaving the surface heatflux as our primary consideration for exomoon habitabilityin these systems. At least, that is the case for the chosenphysical parameters. An interesting future study would beto vary parameters in regards to the global flux and theninvestigate changes to the tidal evolution of each system. We performed a computational exploration into specificcharacteristics of putative exomoon systems. Our study fo-cused on satellite systems in the habitable zone (HZ) of low-mass stars, which contain significantly reduced HZ distances
MNRAS , 1–20 (2017) xomoon Habitability and Tidal Evolution Figure 8.
Graphical representations of the surface heat fluxes listed in Tables 6 and 7. The red shaded regions represent exomoon tidalheating rates above h max ≡ − or below h min ≡ .
04 W m − . The dashed vertical lines at 0.4 and 0.5 M (cid:12) are where Ganymede-likeorbits (wider orbits) begin to be gravitationally stable towards higher stellar masses. in comparison to Sun-like stars. For our planet-moon bina-ries the close proximity of the star presents dynamical re-strictions to the stability of the moon, forcing it to orbit closeto the planet to remain gravitationally bound. At short or-bital distances, tidal heating and tidal torques between theplanet and moon become more substantial. The relativelyclose star can also influence the long-term tidal evolutionof the moon by continually exciting its orbit through gravi-tational perturbations. Key results from the computationalsimulations are highlighted as follows: • M dwarfs with masses ... . M (cid:12) cannot host hab-itable exomoons within the stellar habitable zone. Considering planets up to one Jupiter mass, stars withmasses . . M (cid:12) can support a Mars-like satellite out todistances of ∼ R p (Europa-like orbits). Though these sys-tems are gravitationally stable, the close proximity to thestar continually excites non-zero eccentricities in the moon’sorbit. With limited orbital distances, tidal heating in themoon was significant, resulting in surface heating rates wellabove the proposed limit of h max = − . Considering theexcessively high heating estimates these systems are unlikelyto host habitable exomoons. This conclusion is consistentwith our theoretical estimate based on Eqs. (5) - (8).Our results confirm that exomoon habitability is morecomplex than traditional definitions for planet habitability,which are based primarily on irradiation from a host star.Massive moons in the stellar HZ are not necessarily hab-itable by definition. Since the intense heating rates in thehypothetical exomoons are maintained by stellar perturba-tions, moving the host planet slightly outside the HZ shouldreduce the stellar influence. In this case, tidal heating in exo-moons beyond the stellar HZ could make up for the reducedstellar illumination so that adequate surface temperaturesfor liquid surface water could be maintained. • In the mass range . M (cid:12) < M ? < . M (cid:12) , pertur-bations from the central star may continue to influ- ence the long-term tidal evolution of exomoons inthe stellar habitable zone. For stellar masses & . M (cid:12) , the distinction for habit-able exomoons became less defined when based primarily ontidal evolution. Our results suggest that the host planet’slocation in the HZ has to be taken into consideration. Re-sults from simulations involving Earth-equivalent distances(the inner HZ) show that M dwarfs with masses . . M (cid:12) promote surface heating beyond our accepted limit for hab-itability. In comparison, planetary orbits in the center of theHZ are within the established limits for habitability. Since astar’s influence on a moon decreases with distance, so doesits ability to excite higher heating rates. Therefore, this re-sult for the center of HZs can be applied, by extension, toouter HZs.Our adopted maximum limit for habitability ( h max = − ) is based on a single example – Jupiter’s volcani-cally active moon Io. Although there is little doubt that Iodoes not support a habitable environment, there is no ev-idence that a Mars-sized or even an Earth-sized exomooncould not remain habitable given the same internal heatingrate. This is especially true considering the uncertainty inthe exact mechanisms of tidal dissipation and the efficiencyof plate tectonics in terrestrial bodies. For these reasons, h max should be considered more as a unit of comparisonthan a hard cutoff for habitable exomoons. The ultimateconstraint on surface habitability will be given by the run-away greenhouse limit. With this consideration and treatingthe HZ as a whole, perturbations from a central star maycontinue to have deleterious effect on exomoon habitabilityup to ≈ . M (cid:12) .In contrast to the perils of intense tidal heating, pertur-bations from the star may actually have a positive influenceon the habitability of exomoons. Figure 8 shows many sur-face heating rates below h max , yet above the proposed mini-mum for habitability ( h min = .
04 W m − ). These results sug- MNRAS , 1–20 (2017) R. R. Zollinger et al. gest that satellite systems around stars in this mass rangewould need to be considered on a case by case basis de-pending on the planet mass and the specific location in theHZ. Moderate tidal heating might actually help to sustaintectonic activity (thus possibly a carbon-silicate cycle) andan internal magnetic dynamo that might help to shield themoon from high-energy radiation (Heller & Zuluaga 2013)on a Gyr timescale. • Considerations of global energy flux do not restricthabitability of exomoons in the HZs around starswith masses above . M (cid:12) . The conclusions thus far are in agreement with predic-tions made by Heller (2012) who followed considerations ofenergy flux and gravitational stability. This study, which fo-cused on tidal evolution, has verified those predictions usinga tidal model that considered both N -body interaction andtidal evolution. Similar considerations for energy flux wereincorporated within this study with the global averaged flux( F glob ) listed in Tables 6 and 7. Compared to the criticalflux of
269 W m − for a runaway greenhouse on a Mars-masssatellite, the 3-body global flux results support the conclu-sion that star masses . . M (cid:12) are unlikely to host habitableexomoons. Above that mass, exomoon habitability was notconstrained by global energy flux. • Torques due to spin and tidal distortion betweenthe planet and moon can result in rapid inward spi-raling of a moon for orbital distances ... R p . In specific simulations involving a Saturn-like hostplanet and Io-like ( a ∼ R p ) moon orbits, distortion torquesresulted in the complete inward spiral of a moon in < yr.The inward migration was connected to the assumption thatthe giant host planet was tidally locked to the star. Whilethe orbital decay rate was slower for more massive Jupiter-like host planets, a conservative estimate for the maximumlifetime of Io-like moon orbits was only 200 Myr. Comparedto the geological age of the Earth, this is a short lifespan.Assuming a Mars-like moon in an Io-like orbit was initiallyhabitable, implications for the development of life may beconsiderable. For the sake of minimizing computational demands we as-sumed coplanar orbits and no spin-orbit misalignments ofthe moons. However, we expect new effects to arise in morecomplex configurations, such as spin-orbit resonances be-tween the circumstellar and circumplanetary orbits as wellas substantial effects on the longterm evolution of tidal heat-ing. Important observational predictions can be obtainedby studying the obliquity evolution of initially misalignedplanet spins due to tidal interaction with the star. Willthis “tilt erosion” (Heller et al. 2011a) tend to align themoons’ orbits with the circumstellar orbit? These investi-gations could help predicting and interpreting possible vari-ations of the planetary transit impact parameter due to thepresence of an exomoon (Kipping 2009).Of specific interest are retrograde satellites, which weneglected in this study. These satellites can form throughdirect capture or tidal disruption during planet-planet en-counters (Agnor & Hamilton 2006; Williams 2013), and theycan be orbitally stable as far as the Hill radius (Domingos et al. 2006b). Hence, we expect that retrograde moons couldstill be present or habitable in some cases that we identifiedas unstable or even uninhabitable for prograde moons. Whatis more, the detection of an exomoon in a very wide circum-planetary orbit near or beyond about 0.5 Hill radii mightonly be explained by a retrograde moon.Owing to computational restrictions, the number ofbodies considered was limited to three. A study with ad-ditional bodies would be of interest. The effects of orbitalresonances for mutliple moons could be considered as wellas the influence of additional planets in the system just out-side the HZ. It was also necessary to limit many physicalparameters such as the moments of inertia, tidal dissipationfactors ( Q ), and tidal Love numbers ( k L ). More fundamen-tally, these parameters are treated as constants in our ap-plied tidal model. However, under the effect of tidal heating,the rheology and tidal response of a moon (or planet) canchange substantially (Henning & Hurford 2014; Dobos &Turner 2015). Hence, more advanced tidal theory needs tobe incorporated in our mathematical treatment of tidal plus N -body interaction to realistically model tidal effects in theregime of enhanced tidal heating. Modifying these values cancoincide with variations in other physical parameters suchas mass and radius for the extended bodies, which shouldnoticeably affect the dynamical stability and tidal evolutionof the systems.One result from this study is the hypothetical exis-tence of extremely tidally heated moons. Peters & Turner(2013) proposed the direct imaging of tidally heated exo-moons. Closer examination is warranted to see if our com-puter model would be useful in providing orbital constraintson directly detectable exomoons. When considering extremeheating in a massive body, the issue of inflation may becomeimportant. Inflation is a physical response that was not in-corporated into our model. Planetary inflation was consid-ered by Mardling & Lin (2002) and future plans include theintegration of this effect into our tidal evolution code. Rhett Zollinger was supported by funding from the UtahSpace Grant Consortium. Ren´e Heller has been supportedby the Origins Institute at McMaster University, by theCanadian Astrobiology Program (a Collaborative Researchand Training Experience Program funded by the NaturalSciences and Engineering Research Council of Canada), bythe Institute for Astrophysics G¨ottingen, by a Fellowshipof the German Academic Exchange Service (DAAD), andby the German space agency (Deutsches Zentrum f¨ur Luft-und Raumfahrt) under PLATO grant 50OO1501. This workmade use of NASA’s ADS Bibliographic Services. We wouldlike to thank Benjamin C. Bromley, from the University ofUtah, for his valuable insight and advice regarding this work.Finally, we would like to thank the reviewers for their helpfulsuggestions and useful insights.
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Contribution:RH assisted in the literature research, translated parts of the mathematical framework into com-puter code, generated Figs. 1-2, helped to structure Figs. 3-4, and contributed to the writing of themanuscript. r X i v : . [ a s t r o - ph . E P ] M a r Astronomy & Astrophysics manuscript no. tidal + illum_aa_corr2 c (cid:13) ESO 2017March 8, 2017
The Effect of Multiple Heat Sources on Exomoon Habitable Zones
Vera Dobos , , René Heller , and Edwin L. Turner , Konkoly Thege Miklós Astronomical Institute, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sci-ences, H–1121 Konkoly Thege Miklós út 15-17, Budapest, Hungary Geodetic and Geophysical Institute, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences, H–9400Csatkai Endre u. 6-8., Sopron, Hungary Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany Department of Astrophysical Sciences, Princeton University, 08544, 4 Ivy Lane, Peyton Hall, Princeton, NJ, USA The Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, 227-8583, 5-1-5 Kashiwanoha,Kashiwa, JapanReceived 1 June 2015 / Accepted 1 June 2015
ABSTRACT
With dozens of Jovian and super-Jovian exoplanets known to orbit their host stars in or near the stellar habitable zones, it hasrecently been suggested that moons the size of Mars could o ff er abundant surface habitats beyond the solar system. Several searchesfor such exomoons are now underway, and the exquisite astronomical data quality of upcoming space missions and ground-basedextremely large telescopes could make the detection and characterization of exomoons possible in the near future. Here we explorethe e ff ects of tidal heating on the potential of Mars- to Earth-sized satellites to host liquid surface water, and we compare the tidalheating rates predicted by tidal equilibrium model and a viscoelastic model. In addition to tidal heating, we consider stellar radiation,planetary illumination and thermal heat from the planet. However, the e ff ects of a possible moon atmosphere are neglected. We mapthe circumplanetary habitable zone for di ff erent stellar distances in specific star-planet-satellite configurations, and determine thoseregions where tidal heating dominates over stellar radiation. We find that the ‘thermostat e ff ect’ of the viscoelastic model is significantnot just at large distances from the star, but also in the stellar habitable zone, where stellar radiation is prevalent. We also find thattidal heating of Mars-sized moons with eccentricities between 0.001 and 0.01 is the dominant energy source beyond 3–5 AU froma Sun-like star and beyond 0.4–0.6 AU from an M3 dwarf star. The latter would be easier to detect (if they exist), but their orbitalstability might be under jeopardy due to the gravitational perturbations from the star. Key words. astrobiology – methods: numerical – planets and satellites: general – planets and satellites: interiors
1. Introduction
Although no exomoon has been discovered as of today, ithas been shown that Mars-mass exomoons could exist aroundsuper-Jovian planets at Sun-like stars, from a formation pointof view (Heller et al. 2014; Heller and Pudritz 2015a,b). Ifthey exist, these moons could be detectable with currentor near-future technology (Kipping et al. 2009; Heller 2014;Hippke and Angerhausen 2015) and they would be potentiallyhabitable (Williams et al. 1997; Heller 2014; Lammer et al.2014).For both planets and moons, it is neither su ffi cient noreven necessary to orbit their star in the habitable zone (HZ)– the circumstellar region where the climate on an Earth-like planet would allow the presence of liquid surface wa-ter (Kasting et al. 1993). Instead, tidal heating can result ina suitable surface temperature for liquid water at large stel-lar distances (Reynolds et al. 1987; Peters and Turner 2013;Heller and Armstrong 2014; Dobos and Turner 2015). More-over, most of the water – and even most of the liquid water –in the solar system can be found beyond the location of the pastsnowline around the young Sun (Kereszturi 2010), which is sup-posed to have been at about 2.7 AU during the final stages of thesolar nebula (Hayashi 1981). Hence, these large water reservoirs Send o ff print requests to : V. Dobos, e-mail: [email protected] could be available for life on the surfaces of large exomoons atwide stellar separations.However, observations of moons orbiting at several AU fromtheir stars cannot be achieved with the conventional transitmethod, because transit surveys can only detect exomoons withsignificant statistical certainty after multiple transits in front oftheir star. This implies relatively small orbital separations fromthe star, typically < Q ) and with a uniform rigidity of the rockymaterial of the moon ( µ ), both of which are considered con-stants. This family of the tidal equilibrium models is called the“constant-phase-lag” (CPL) models because they assume a con-stant lag of the tidal phases between the tidal bulge of the moonand the line connecting the moon and its planetary perturber(Efroimsky and Makarov 2013). Alternatively, equilibrium tidescan be described by a ”constant-time-lag“ (CTL) model, where it Article number, page 1 of 8 & Aproofs: manuscript no. tidal + illum_aa_corr2 is assumed that the tidal bulge of the moon lags the line betweenthe two centers of mass by a constant time.In reality, however, all these parameters depend on tempera-ture. Another problem with equilibrium tide models is that thevalues of the parameters (e.g. of Q ) are di ffi cult to calculate(Remus et al. 2012) or constrain observationally even for solarsystem bodies: Q can vary several orders of magnitude for dif-ferent bodies: from ≈
10 for rocky planets to > ff ects of two kinds of tidalheating models, a CTL model and a viscoelastic model, on thehabitable edge for moons within the circumstellar HZ. We alsocalculate the contribution of viscoelastic model to the total en-ergy flux of hypothetical exomoons with an emphasis on moonsfar beyond the stellar habitable zone.
2. Method
We neglect any greenhouse or cloud feedbacks by a possiblemoon atmosphere and rather focus on the global energy budget.The following energy sources are included in our model: stellarirradiation, planetary reflectance, thermal radiation of the planet,and tidal heating. The globally averaged energy flux on the moonis estimated as per (Heller et al. 2014, Eq. 4) F globs = L ∗ (cid:16) − α s , opt (cid:17) π a q − e x s + π R α p f s a + L p (cid:0) − α s , IR (cid:1) π a f s p − e + h s + W s , (1)where L ∗ and L p are the stellar and planetary luminosities, re-spectively, α p is the Bond albedo of the planet, α s , opt and α s , IR are the optical and infrared albedos of the moon, a p and a s arethe semi-major axes of the planet’s orbit around the star and ofthe moon’s orbit around the planet , e p and e s are the eccentric-ities of the planet’s and the satellite’s orbit, R p is the radius ofthe planet, x s is the fraction of the satellite’s orbit that is notspent in the shadow of the host planet, and f s describes the ef-ficiency of the flux distribution on the surface of the satellite.The first and second terms of this equation account for illumi-nation e ff ects from the star and the planet, whereas h s indicatesthe tidal heat flux through the satellite’s surface. W s denotes ar-bitrary additional energy sources such as residual internal heatfrom the moon’s accretion or heat from radiogenic decays, butwe use W s =
0. We also neglect eclipses, thus x s =
1, and weassume that the satellite is not tidally locked to its host planet,hence f s =
4. The planetary radius is calculated from the mass, We assume that the moon’s mass is negligible compared to the plan-etary mass and that the barycenter of the planet-moon system is in thecenter of the planet. using a polynomial fit to the data given by Fortney et al. (2007,Table 4, line 17). The albedos of both the planet and the moon(in the optical and also in the infrared) were set to 0.3, that is, toEarth-like values.We estimate the runaway-greenhouse limit of the globallyaveraged energy flux ( F RG ), which defines the circumplanetaryhabitable edge interior to which a moon becomes uninhabit-able, using a semi-analytic model of Pierrehumbert (2010) asdescribed in Heller and Barnes (2013). In this model, the outgo-ing radiation on top of a water-rich atmosphere is calculated us-ing an approximation for the wavelength-dependent absorptionspectrum of water. The runaway greenhouse limit then dependsexclusively on the moon’s surface gravity (i.e. its mass and ra-dius) and on the fact that there is enough water to saturate theatmosphere with steam. Our test moons are assumed to be rockybodies with masses between the mass of Mars (0.1 Earth masses, M ⊕ ) and 1 M ⊕ . Our test host planets are gas giants around eithera sun-like star or an M dwarf star.We apply the model of Kopparapu et al. (2014) to calcu-late the borders of the circumstellar HZ, which are di ff erent forthe 0.1 and the 1 Earth-mass moons. The corresponding energyfluxes are then used to define and evaluate the habitability of ourtest moons, both inside and beyond the stellar HZ. Tidal heat flux is calculated using two di ff erent models: a vis-coelastic one with temperature-dependent tidal Q and a CTLone, the latter of which converges to a CPL model for small ec-centricities as the time-lag of the principal tide becomes 1 / Q (Heller et al. 2011). For the CTL model we use the same frame-work and parameterization as in the work of Heller and Barnes(2013), which goes back to Leconte et al. (2010) and Hut (1981).In particular, we use the following parameters: k = . τ = τ ∼ / ( nQ ) and k . Hence, changes in these parameters usu-ally do not have dramatic e ff ects on the tidal heating. In contrast,changes in the radius or mass of the tidally distorted object, orin the orbital eccentricity or semi-major axis result in significantchanges of the tidal heating rates.For the viscoelastic tidal heating calculations, we use themodel described by Dobos and Turner (2015), which was origi-nally developed by Henning et al. (2009). Tidal heat flux is cal-culated from h s , visc = − Im ( k ) R n e G , (2)where Im ( k ) is the complex Love number, which describes thestructure and rheology in the satellite (Segatz et al. 1988). In theMaxwell model, the value of Im ( k ) is given by (Henning et al.2009) − Im ( k ) = ηω ρg R m + + µ ρg R m ! η ω µ , (3)where η is the viscosity, ω is the orbital frequency, and µ is the shear modulus of the satellite. The temperature depen-dency of the viscosity and the shear modulus is described by Article number, page 2 of 8era Dobos et al.: The E ff ect of Multiple Heat Sources on Exomoon Habitable Zones Fischer and Spohn (1990) and Moore (2003); and the adaptedvalues and equations are listed by Dobos and Turner (2015).Since only rocky bodies like the Earth are considered as satel-lites in this work, the solidus and liquidus temperatures, at whichthe material of the rocky body starts melting and becomes com-pletely liquid, were chosen to be 1600 K and 2000 K, respec-tively. We assume that disaggregation occurs at 50% melt frac-tion, which implies a breakdown temperature of 1800 K.The viscoelastic tidal heating model also describes the con-vective cooling of the body. The iterative method described byHenning et al. (2009) was used for our calculations of the con-vective heat loss: q BL = k therm T mantle − T surf δ ( T ) , (4)where k therm is the thermal conductivity ( ∼ / mK), T mantle and T surf are the temperatures in the mantle and on the surface, re-spectively, and δ ( T ) is the thickness of the conductive layer. Weuse δ ( T ) =
30 km as a first approximation, and then for the iter-ation δ ( T ) = d a RaRa c ! − / (5)is used, where d is the mantle thickness ( ∼ a is theflow geometry constant ( ∼ Ra c is the critical Rayleigh num-ber ( ∼ Ra is the Rayleigh number which can be ex-pressed as Ra = α g ρ d q BL η ( T ) κ k therm . (6)with α ( ∼ − ) as the thermal expansivity, κ = k therm / ( ρ C p ) asthe thermal di ff usivity, and C p = / (kg K). The iterationof the convective heat flux ends when the di ff erence of the lasttwo values becomes smaller than 10 − W / m . Once the stableequilibrium temperature is found, we compute the tidal heat flux.This viscoelastic model was already used together with a cli-mate model by Forgan and Dobos (2016) with the aim of deter-mining the location and width of the circumplanetary habitablezone for exomoons. The 1D latitudinal climate model includedeclipses, the carbonate-silicate cycle and the ice-albedo feedbackof the moon (in addition to tidal heating, stellar and planetary ra-diation). The ice-albedo positive loop in the climate model alongwith eclipses result in a relatively close-in outer limit for circum-planetary habitability, if the orbit of the satellite is not inclined.The climate model, however, can only be used for bodies of sim-ilar sizes to the Earth. In this work we investigate the e ff ect ofthe viscoelastic model on smaller exomoons, as well, and for thisreason, instead of a climate model, we apply an orbit-averagedillumination model. Beside solar-like host stars, we made calcu-lations also for M dwarfs.Both the CPL and the viscoelastic models are valid only forsmall orbital eccentricities, that is, for e . .
1. For larger eccen-tricities, the instantaneous tidal heating in the deformed bodycan di ff er strongly from the orbit-averaged tidal heating rate andthe frequency spectrum of the decomposed tidal potential couldinvolve a wide range of frequecies (Greenberg 2009). Both as-pects go beyond the approximations involved in the models, andhence our results for e & . We consider various reference systems of a star, a planet, and amoon.
First, we examine the di ff erent e ff ects of tidal heating in eithera CTL or in a viscoelastic tidal model on the location of thecircumplanetary habitable edge (see Section 3.1). We consider astar with a sun-like radius ( R ⋆ = R ⊙ ) and e ff ective temperature( T e ff = . M ⊕ moon. Theorbital period and eccentricity of the moon are varied between 1and 20 days and between 0.01 and 1, respectively. Second, we map the circumplanetary habitable zone over a widerange of circumstellar orbits (see Section 3.2) using four di ff er-ent star-planet-moon configurations. The star is either sun-like(see Section 2.3.1) or an M3 class main sequence star ( M ∗ = . M ⊙ , R ∗ = . R ⊙ , T e ff = L ∗ = . L ⊙ ,Kaltenegger and Traub 2009, Table 1). The planet is a Jupiter-mass gas giant, and the satellite’s mass is either 0 . M ⊕ or 1 M ⊕ (i.e. a Mars or Earth analogue). The density of the moon is thatof the Earth. The stellar luminosity and the temperature valuesare used for our calculations of the HZ boundaries, and the stel-lar radius and temperature values are required for the incidentstellar flux calculation.
3. Results
We calculated the total energy flux (using Eq. 1) at the top ofthe moon’s atmosphere as a function of distance to the planet inboth the CTL (Fig. 1) and the viscoelastic (Fig. 2) frameworks.In both Figs. 1 and 2, colours show the amount of the total fluxreceived by the moon, and the white contours at 288 W / m indi-cate the habitable edge defined by the runaway-greenhouse limit(Heller and Barnes 2013). Interestingly, in the CTL model thehabitable edge is located closer in to the planet than in the vis-coelastic model.Black contour curves show the tidal heat flux alone at288, 100 and 2 W / m , the latter is the global mean heat flowfrom tides on Io as measured with the Galileo spacecraft(Spencer et al. 2000). This curve in Fig. 2 has a di ff erent slopethan the one labeled ‘tidal heat at 100 W / m ’. The di ff erentslope is caused by the changing equations in the viscoelasticmodel. The tidal flux (and also the convective cooling flux) isdescribed by di ff erent formulae below or above certain temper-atures (namely the solidus, the breakdown and the liquidus tem-peratures). In other words, if the tidal heating is low, di ff erentequations will be used, than in the case of higher tidal forces.Figs. 1 and 2 show substantial di ff erences. The viscoelasticmodel predicts moderate ( ≤ / m ) heating beyond 5 . e = .
01, while the CTL model needs themoon to be as close as 3 . = Article number, page 3 of 8 & Aproofs: manuscript no. tidal + illum_aa_corr2 M s = 0.5 M ¯ , M p = 5 M Jup (CTL model) [W/m ]1 10 orbital period [days] m oon ’ s o r b it a l ecce n t r i c it y W / m = R un a w a y G r ee nhou s e L i m it ti d a l h ea t a t W / m ti d a l h ea t a t W / m ti d a l h ea t a t W / m I G C M s = 0.5 M ¯ , M p = 5 M Jup (CTL model) [W/m ]1 10 orbital period [days] m oon ’ s o r b it a l ecce n t r i c it y W / m = R un a w a y G r ee nhou s e L i m it ti d a l h ea t a t W / m ti d a l h ea t a t W / m ti d a l h ea t a t W / m I G C M s = 0.5 M ¯ , M p = 5 M Jup (CTL model) [W/m ]1 10 orbital period [days] m oon ’ s o r b it a l ecce n t r i c it y W / m = R un a w a y G r ee nhou s e L i m it ti d a l h ea t a t W / m ti d a l h ea t a t W / m ti d a l h ea t a t W / m I G C M s = 0.5 M ¯ , M p = 5 M Jup (CTL model) [W/m ]1 10 orbital period [days] m oon ’ s o r b it a l ecce n t r i c it y W / m = R un a w a y G r ee nhou s e L i m it ti d a l h ea t a t W / m ti d a l h ea t a t W / m ti d a l h ea t a t W / m I G C
Fig. 1.
Energy flux at the top of the atmosphere of a 0.5 Earth-massmoon orbiting a 5 Jupiter-mass planet at 1 AU distance from a Sun-likestar; tidal heating is calculated with a CTL model. The planetary or-bit has 0.1 eccentricity. The stellar radiation, the planetary reflectance,thermal radiation of the planet and tidal heating were considered as en-ergy sources. White contour curve indicates the runaway greenhouselimit considering all energy sources, while the black curves show thetidal heating flux alone. The arrows at the top of the figure indicate theorbital periods of Io (I), Ganymede (G) and Callisto (C). Note that theCTL model is valid only for small orbital eccentricities. Heating con-tours in the upper half of the panel (above 0.1 along the ordinate) couldbe o ff by orders of magnitude in real cases. M M (cid:0)
M MM M (cid:0)
M MM M (cid:0)
M MM M (cid:0)
M M
Fig. 2.
Energy flux at the top of the atmosphere of a 0.5 Earth-massmoon orbiting a 5 Jupiter-mass planet at 1 AU distance from a Sun-like star; tidal heating is calculated with a viscoelastic model. The samecolours, contours and signatures were applied as in Fig. 1. Note that theviscoelastic model is valid only for small orbital eccentricities. Heatingcontours in the upper half of the panel (above 0.1 along the ordinate)could be o ff by orders of magnitude in real cases. flux that is su ffi cient to trigger a runaway greenhouse e ff ect. Inthis case, stellar illumination is not even required to make such amoon uninhabitable. In contrast, with the viscoelastic model themoon would need to have a 1 day orbital period to trigger thesame e ff ect.A comparison of these two plots shows the ‘thermostat ef-fect’ of the viscoelastic model described by Dobos and Turner(2015). The CTL model yields lower tidal heating rates thanthe viscoelastic model in the weak-to-moderate heating regime (tidal heat ≤
100 W / m ), while above 100 W / m the viscoelas-tic model produces lower heating rates, where the CTL modelruns away. As a consequence, in this specific example of a 0.5Earth-mass moon around a 5 Jupiter-mass planet, the habitableedge (red contour curve in the figures) is located at a larger dis-tance from the planet than for the CTL model. It is caused by thestronger tidal heating rate below 100 W / m .Stellar and planetary illuminations can also have importante ff ects. They can break the feedback loop between tidal heat-ing and convective cooling: in an extreme case the system willnot even find a stable convective heat transport rate. However,the most relevant cases of tidally heated exomoons will involvesystems orbiting so far from the star that illumination heating isirrelevant and around planets so old that planetary illuminationis also small. In any cases, such systems will be the ones that canbe most easily understood, if they exist. In addition to the inner habitable edge, we also want to locate theouter boundary of the circumplanetary habitable zone for dif-ferent satellite sizes and stellar classes. Figs. 3 and 4 show thecircumplanetary habitable zones for our Jupiter-like test planetwith a moon around either a sun-like or an M dwarf star, respec-tively. The left and right panels of the figures show the cases of0.1 and 1 Earth-mass satellites, respectively. The eccentricity ofthe moon’s orbit is 0.001 in the top panels and 0.01 in the bottompanels.Grey horizontal lines in each panel indicate zero tidal heatflux. Above these lines, only stellar radiation is relevant sincethe reflected and thermal radiation from the planet are alsonegligibly small. Green areas illustrate habitable surface condi-tions with tidal heating rates below 100 W m − , whereas diag-onally striped orange areas visualize exomoon habitability withtidal heating rates above 100 W m − (color coding adopted fromHeller and Armstrong 2014).As expected, for smaller satellite masses the HZ is locatedcloser to the planet. At zero tidal flux, small plateaus are presentwhich are consequences of the viscoelastic tidal heating model.At the horizontal line tidal forces are turned on, hence there isa change in the position of the green area. At the plateau thetidal heat flux elevates from zero to about 10 W / m or more. Thesudden change is caused by the fact that the tidal heating modelgives result only if the tidal heat flux and the convective coolingflux has stable equilibrium. It means that if tidal forces are weak,then the convective cooling will be weak too, or not present in thebody at all, hence there is no equilibrium. In other words, tidalheating is insu ffi cient to drive convection in the body. There willstill be some heating, but a di ff erent heat transport mechanism(probably conduction) would be in play. A more accurate modelwould consider all heat transport mechanisms at all temperaturesand would probably not exhibit such discontinuities.In the inspected cases the two dominating energy sourcesare the stellar radiation and the tidal heating. Since these twoe ff ects are physically independent, one would not expect themto be of comparable importance except in a small fraction ofcases. In general, however, one e ff ect will dominate the other.To explore this interplay between stellar illumination and tidalheating in more detail, we calculated the distance from the starat which tidal heating (for a given hypothetical moon and or-bit) equals stellar heating. These critical distances are indicatedwith blue dashed curves in Figs. 3 and 4. We find that beyond3-5 AU around a G2 star and beyond 0.4-0.6 AU around an M3star, tidal heating for moons with eccentricities between 0.001 Article number, page 4 of 8era Dobos et al.: The E ff ect of Multiple Heat Sources on Exomoon Habitable Zones s e m i - m a j o r a x i s o f t h e m oo n [ R J ] F tidal = 0 W/m F tidal >100 W/m stellar HZ s t e ll a r r a d i a t i o n d o m i n a t e s t i d a l h ea t i ng d o m i n a t e s e =0.001, M m =0.1M Earth tidal = 0 W/m F tidal >100 W/m stellar HZ s t e ll a r r a d i a t i o n d o m i n a t e s t i d a l h e a t i n g d o m i n a t e s e =0.001, M m =1.0M Earth tidal = 0 W/m F tidal >100 W/m stellar HZ s t e ll a r r a d i a t i o n d o m i n a t e s t i d a l h ea t i ng d o m i n a t e s e =0.01, M m =0.1M Earth semi-major axis of the planet [AU] tidal = 0 W/m F tidal >100 W/m stellar HZ s t e ll a r r a d i a t i o n d o m i n a t e s t i d a l h e a t i n g d o m i n a t e s e =0.01, M m =1.0M Earth semi-major axis of the planet [AU] s e m i - m a j o r a x i s o f t h e m oo n [ R J ] Fig. 3.
Tidal heating habitable zone for 0.1 and 1 Earth-mass exomoons (left and right panels, respectively) around a Jupiter-mass planet hostedby a G2 star as functions of the semi-major axes of the planet and the moon. The red curve indicates the runaway greenhouse limit (habitableedge). The green and the diagonally striped orange areas cover the habitable region, where the striped orange colour indicates that the tidal heatingflux is larger than 100 W / m . The orbital eccentricity of the moon is 0.001 in the top panels, and 0.01 in the bottom panels. Vertical dashed linesindicate the inner and outer boundaries of the circumstellar habitable zone. At the dashed blue contour curve the tidal heating flux equals to thestellar radiation flux, hence it separates the tidal heating dominated and the stellar radiation dominated regime. At the grey horizontal line the tidalflux is zero. and 0.01 will be the dominant source of energy rather than stel-lar radiation (see the overlapping of the grey and dashed bluecurves). This is the case for both satellite masses investigated.For smaller stellar distances, the dominant heat source dependson the circumplanetary distance of the moon.Based on the fact that Io is a global volcano world, one mighttake the 2 W / m limit as a conservative limit for Earth-like sur-face habitability. But other worlds might still be habitable evenin a state of extreme tidal heating near 100 W / m . In fact, thesurface habitability of rocky, water-rich planets or moons hasnot been studied in this regime of extreme tidal heating to ourknowledge. Nevertheless, although 100 W / m seems a lot of in-ternal heating compared to the internal heat flux on Earth, whichis less than 0 . / m (Zahnle et al. 2007), it could still allow thepresence of liquid surface water as long as the total energy flux isbelow the runaway greenhouse limit. The striped orange colourindicates that the body might become volcanic or tectonicallyactive in this regime.It is supposed that the gravitational pull of an M dwarf canforce an exomoon whose orbit is within the stellar HZ into avery eccentric circumplanetary orbit (Heller 2012). The resultingtidal heating might ultimately prevent such moons from beinghabitable. Since the planetary Hill sphere in the stellar HZ ofM dwarfs is not so large, any moon needs to be in a tight orbit.Prograde (regular) moons are only stable out to about 0.5 Hillradius (Domingos et al. 2006). In Fig. 4, the area beyond this line is shown as a shaded region in grey and labeled as ‘Hillunstable’. As a consequence of the closeness of this Hill unstablelimit, the circumplanetary volume for habitable satellites is muchsmaller in systems of M dwarfs than in systems with sun-likestars, in particular if the moons have substantial eccentricities.Note how the circumplanetary space between the red (runawaygreenhouse) curve and the Hill unstable region becomes smalleras the moon’s eccentricity or its mass increases.
4. Discussion
From the results shown in Section 3, the following general find-ings can be concluded.The circumplanetary habitable edge calculated with the vis-coelastic model (Fig. 2) is located at a larger distance than in theCTL model (Fig. 1). If the outer boundary of the circumplane-tary habitable zone is defined by Hill stability, than it means thatthe viscoelastic model reduces the habitable environment. How-ever, Forgan and Yotov (2014) showed that the moon can enterinto a snowball phase, if the ice-albedo feedback and eclipsesare also taken into account. It means that the outer boundary willbe significantly closer to the planet, than in the case when it isdefined by Hill stability. Since the viscoelastic model resulted inhigher tidal fluxes below 100 W / m , than the CTL model, weexpect that the outer boundary defined by the snowball state willbe farther from the planet. Altogether, the circumplanetary HZ is Article number, page 5 of 8 & Aproofs: manuscript no. tidal + illum_aa_corr2 s e m i - m a j o r a x i s o f t h e m oo n [ R J ] H ill u n s t a b l e F tidal = 0 W/m F tidal >100 W/m stellar HZ s t e ll a r r a d i a t i o n d o m i n a t e s t i d a l h e a t i n g d o m i n a t e s e =0.001, M m =0.1M Earth H ill u n s t a b l e F tidal = 0 W/m F tidal >100 W/m stellar HZ s t e ll a r r a d i a t i o n d o m i n a t e s t i d a l h e a t i n g d o m i n a t e s e =0.001, M m =1.0M Earth H ill u n s t a b l e F tidal = 0 W/m F tidal >100 W/m stellar HZ s t e ll a r r a d i a t i o n d o m i n a t e s t i d a l h e a t i n g d o m i n a t e s e =0.01, M m =0.1M Earth semi-major axis of the planet [AU] H ill u n s t a b l e F tidal = 0 W/m F tidal >100 W/m stellar HZ s t e l l a r r a d i a t i o n d o m i n a t e s t i d a l h e a t i n g d o m i n a t e s e =0.01, M m =1.0M Earth semi-major axis of the planet [AU] s e m i - m a j o r a x i s o f t h e m oo n [ R J ] Fig. 4.
Tidal heating habitable zone for 0.1 and 1 Earth-mass exomoons (left and right panels, respectively) around a Jupiter-mass planet hostedby an M3 dwarf as functions of the semi-major axes of the planet and the moon. The orbital eccentricity of the moon is 0.001 in the top panels,and 0.01 in the bottom panels. The same colours and contours were applied as in Fig. 3. The grey area in the upper left corner covers those caseswhere the moon’s orbit is considered unstable. not likely to be thinner in the viscoelastic case, but it is locatedin a larger distance from the planet.Tidal heating in close-in orbits is more moderate with theviscoelastic model and does not generate extremely high rates aspredicted by the CTL model.Moons that are primarily heated by tides, that is, moons be-yond the stellar habitable zone, have a wider circumplanetaryrange of orbits for habitability with the viscoelastic model thanwith the CTL model. This is in agreement with the findingsof Forgan and Dobos (2016). In Heller and Armstrong (2014,Fig. 2), it was shown that the circumplanetary habitable zone cal-culated with the CTL model thins out dramatically as the planet-moon binary is virtually shifted away from their common hoststar. This is due to the very strong dependence of tidal heating inthe CTL model on the moon’s semi-major axis.In the calculations we considered 0.1, 0.5 and 1 Earth-massmoons. Lammer et al. (2014) found that about 0.25 Earth masses(or 2.5 Mars masses) are required for a moon to hold an atmo-sphere during the first 100 Myrs of high stellar XUV activity,assuming a moon in the habitable zone around a Sun-like star.Considering this constraint, a 0.1 Earth-mass moon may not behabitable in the stellar HZ because of atmospheric loss, how-ever, at larger stellar distances the stellar wind and strong stellaractivity do not present such severe danger to habitability.From a formation point of view, smaller moons are moreprobable to form around super-Jovian planets than Earth-masssatellites, according to Canup and Ward (2006), who gave an up-per limit of 10 − to the mass ratio of the satellite system and theplanet. However, moons can also originate from collisions rather than the circumplanetary disk, as in the case of the Moon. Insuch cases, larger mass ratios are reasonable, too.We have considered a large range of orbital eccentrici-ties for the moon with values up to 1. Beyond the fact thatour models are physically plausible only for moderate val-ues ( ≤ . ff ect onthe moon’s orbit (Heller 2012), if other planets can act as or-bital perturbers (Gong et al. 2013; Payne et al. 2013; Hong et al.2015), if other massive moons are present around the sameplanet, or if the planet-moon system has migrated through or-bital resonances with the circumstellar orbit (Namouni 2010;Spalding et al. 2016).Hot spots (hot surface areas generated by geothermical heat,that could generate volcanoes) might be important on both ob-servational and astrobiological grounds. Regarding observations,hot spots produce variability in the brightness and spectral en-ergy distribution of thermal emission from the moon. As Io il-lustrates, hot spots can potentially be big and prominently hot(see Spencer et al. 2000, Fig. 4). This could let one determinethe moon’s spin period and may indicate a geologically ‘active’body. Hot spots also allow liquid water to locally exist and per-sist over long periods of time even if the mean surface temper-ature is far below the freezing point. Enceladus may provide anexample for such phenomenon in the Solar System, where tidalheating maintains a liquid (and probably global) ocean below Article number, page 6 of 8era Dobos et al.: The E ff ect of Multiple Heat Sources on Exomoon Habitable Zones the ice cover, and contributes to the eruption of plumes at thesouthern region of the satellite (Thomas et al. 2016).On the other hand, hot spots can be e ff ective in conductingthe internal heat. The calculated fluxes in this paper are averagesurface fluxes, but in reality hot spots are probable to form. Theseareas are much higher in temperature, meaning that other areason the surface must be somewhat colder than the average. As aresult, the temperature of the surface (excluding the vicinity ofthe hot spots) can be lower than that is calculated.Two end-member models exist for spatial distribution of tidalheat dissipation. In model A the dissipation mostly occurs in thedeep-mantle of the body, and in model B it occurs in the astheno-sphere, which is a thin layer in the upper mantle (Segatz et al.1988). In model A the surface heat flux is higher at the polar re-gions, while in model B the flux is mostly distributed between − ◦ and + ◦ latitudes. From volcano measurements of Io itseems that model B is more consistent with the location of hotspots (Hamilton et al. 2011, 2013; Rathbun and Spencer 2015).The global distribution of volcanoes on the surface is random(Poisson distribution), but closer to the equator they are morewidely spaced (uniform distribution, Hamilton et al. 2013).On Io the total power output of volcanoes and paterae isabout 5 · W (Veeder et al. 2012a,b). About 62% of the fluxis going through volcanoes in Io (Veeder et al. 2012a), whichmeans that there is a large temperature di ff erence in the volcanicareas and lowlands. For moons of larger sizes than Io, or withmuch stronger tidal force, it is probable that even higher percent-age of energy will leave through volcanoes. A geological modelis needed to estimate the connection between the tidal heatingflux and the energy output of volcanoes and hot spots, which isbeyond the scope of this paper.Detectibility of Mars-to-Earth size satellites is not investi-gated in this work, but detections could be possible in the Ke-pler data (Kipping et al. 2009; Heller 2014) or with the PLATO(Hippke and Angerhausen 2015) or CHEOPS (Simon et al.2015) missions, or with the E-ELT (Heller and Albrecht 2014;Heller 2016; Sengupta and Marley 2016). Naturally, largermoons are easier to detect than smaller moons, and so we expectan observational selection bias for the first known exomoons tobe large and potentially habitable.
5. Conclusion
In this work we improved the viscoelastic tidal heating modelfor exomoons (Dobos and Turner 2015) by adding the stellar ra-diation, the planetary reflectance and the planet’s thermal radia-tion to the energy budget. We found that the ‘thermostat e ff ect’of the viscoelastic model is robust even with the inclusion ofthese additional energy sources. This temperature regulation inthe viscoelastic tidal heating model is caused by the melting ofthe satellite’s inner material, since the phase transition preventsthe temperature from rising to extreme heights.We investigated the circumplanetary tidal heating HZ for afew representative configurations. We showed that the extentof the tidal HZ is considerably wide even at large distancesfrom the stellar HZ, predicting more habitable satellite orbitsthan the CTL models. In a previous study with a CTL model,Heller and Armstrong (2014) found that the tidal HZ thinnedout for large stellar distances. We also showed that if tidal heat-ing is present in the moon, then beyond 3–5 AU distance fromsolar-like stars this will be the dominating energy source, andfor M3 main sequence dwarfs tidal heating dominates beyond0.4–0.6 AU distance. At smaller stellar distances the semi-major axes of the moon defines whether stellar radiation or tidal heat-ing dominates.From an observational point of view, M dwarfs are bettercandidates to detect habitable exomoons, since both the stellarand the circumplanetary tidal HZs and closer to the star, mean-ing that the orbital period of the planet is shorter, hence moretransits can be observed. However, in the stellar HZ, the possiblehabitable orbits for exomoons are constrained by Hill stability.At larger stellar distances the circumplanetary tidal HZ and theHill unstable region do not overlap, so there is no such constraint. Acknowledgements
We thank Duncan Forgan for a very helpful referee report. VDthanks László L. Kiss for the helpful discussion. This work wassupported in part by the German space agency (Deutsches Zen-trum für Luft- und Raumfahrt) under PLATO grant 50OO1501.This research has been supported in part by the World Pre-mier International Research Center Initiative, MEXT, Japan. VDhas been supported by the Hungarian OTKA Grants K104607,K119993, and the Hungarian National Research, Developmentand Innovation O ffi ce (NKFIH) grant K-115709. References
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Article number, page 8 of 8 .10. EXPLORING EXOMOON SURFACE HABITABILITY WITH AN IDEALIZEDTHREE-DIMENSIONAL CLIMATE MODEL (Haqq-Misra & Heller 2018) 352
Contribution:RH did the literature research to embed this study in the context of exomoon science, assisted in theconception of the figures and in their interpretation, and contributed to the writing of the manuscript.
NRAS , 1–14 (2018) Preprint 19 June 2018 Compiled using MNRAS L A TEX style file v3.0
Exploring exomoon atmospheres with an idealized generalcirculation model
Jacob Haqq-Misra ? and Ren´e Heller Blue Marble Space Institute of Science, 1001 4th Ave Suite 3201, Seattle, WA 98154, USA Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 G¨ottingen, Germany
Accepted for publication by
MNRAS on 18 June 2018
ABSTRACT
Recent studies have shown that large exomoons can form in the accretion disks aroundsuper-Jovian extrasolar planets. These planets are abundant at about 1 AU from Sun-like stars, which makes their putative moons interesting for studies of habitability.Technological advances could soon make an exomoon discovery with
Kepler or theupcoming
CHEOPS and
PLATO space missions possible. Exomoon climates mightbe substantially different from exoplanet climates because the day-night cycles onmoons are determined by the moon’s synchronous rotation with its host planet. More-over, planetary illumination at the top of the moon’s atmosphere and tidal heating atthe moon’s surface can be substantial, which can affect the redistribution of energyon exomoons. Using an idealized general circulation model with simplified hydrologic,radiative, and convective processes, we calculate surface temperature, wind speed,mean meridional circulation, and energy transport on a 2.5 Mars-mass moon orbit-ing a 10-Jupiter-mass at 1 AU from a Sun-like star. The strong thermal irradiationfrom a young giant planet causes the satellite’s polar regions to warm, which remainsconsistent with the dynamically-driven polar amplification seen in Earth models thatlack ice-albedo feedback. Thermal irradiation from young, luminous giant planets ontowater-rich exomoons can be strong enough to induce water loss on a planet, whichcould lead to a runaway greenhouse. Moons that are in synchronous rotation withtheir host planet and do not experience a runaway greenhouse could experience sub-stantial polar melting induced by the polar amplification of planetary illumination andgeothermal heating from tidal effects.
Key words: planets and satellites: terrestrial planets – planets and satellites: atmo-spheres – hydrodynamics – astrobiology
Low-mass stars are conventionally thought to exhibit themost promising odds for the detection of terrestrial plan-ets, at least from an observational point of view. Theirlow masses enable detections of low-mass companions likeEarth-mass planets using radial velocity measurements, andtheir small radii allow findings of small transiting objectsin photometric time series. The recent detections of sub-Earth-sized planets around the M dwarf stars TRAPPIST-1(Gillon et al. 2016; Gillon et al. 2017), Proxima Centauri(Anglada-Escud´e et al. 2016), and LHS 1140 (Dittmannet al. 2017) serve as impressive benchmark discoveries.Even cooler dwarfs exist. Brown dwarfs (BDs), withmasses between about 13 and 75 Jupiter masses ( M J ) cannotfuse hydrogen, but their eternal shrinking nevertheless con- ? E-mail: [email protected] verts significant amounts of gravitational energy into heatfor billions of years. From the perspective of BD formation,one can expect that satellites of BDs should commonly formin the dusty, gaseous disks around accreting BDs (Payne& Lodato 2007). At the transitional mass regime to giantplanets, models of moon formation have shown that Mars-sized moons should commonly form around the most massivesuper-Jovian gas giant planets (Heller & Pudritz 2015b,a).Photometric accuracies of − have now been achieved onBDs using the Hubble Space Telescope (Zhou et al. 2016),and an improvement of about one order of magnitude shouldallow the detection of moons transiting luminous giant plan-ets that can be directly imaged around their host star (Cabr-era & Schneider 2007; Heller & Albrecht 2014; Heller 2016).The search for exomoons has recently become an activearea of research that several groups are now competing in,mostly using space-based stellar photometry of exoplanet-exomoons transits (Kipping et al. 2012; Szab´o et al. 2013; © a r X i v : . [ a s t r o - ph . E P ] J un Haqq-Misra and Heller
Simon et al. 2015; Hippke 2015) but also using alterna-tives such as radio emission from giant planets with mag-netic fields that are perturbed by moons (Luki´c 2017) orspace-based coronographic methods such as spectroastrom-etry (Agol et al. 2015). With the first tentative detectionof exomoons or exorings recently claimed in the literature(Mamajek et al. 2012; Bennett et al. 2014; Udalski et al.2015; Aizawa et al. 2017; Hill et al. 2017; Teachey et al.2018; Rodenbeck et al. 2018), a first unambiguous discoverycould thus be imminent. Observational biases as well as dy-namical constraints will preferentially reveal large, massivemoons beyond 0.1 AU around their star (Szab´o et al. 2006;Heller 2014), similar to the moon system that we investigatein this study.The first step in estimating the climate conditions on amoon is in the identification of the relevant energy sources.Different from planets, moons receive stellar reflected lightfrom their planetary host (Heller & Barnes 2013) as well asthe planet’s own thermal emission (Heller & Barnes 2015).In particular, a super-Jovian planet’s own luminosity candesiccate its initially water-rich moons over several 100 Myrand make it ultimately uninhabitable. But exomoons aroundgiant planets can also be subject to significant tidal heating(Reynolds et al. 1987; Scharf 2006; Cassidy et al. 2009; Heller& Barnes 2013); see Io around Jupiter for a prominent ex-ample in the solar system (Peale et al. 1979).For moons around giant planets in the stellar habit-able zone (HZ), the reflected plus thermal planetary lightonto the moon is a significant source of energy ( &
10 W m − )if the moon is closer than about 10 Jupiter radii ( R jup ) toits host planet. Global top-of-atmosphere flux maps showedthat eclipses can decrease the average energy flux on themoon’s subplanetary point by tens of W m − relative to themoon’s antiplanetary hemisphere (Heller & Barnes 2013).Eclipses can cause a maximum decrease of the globally av-eraged stellar flux of ≈ . at most (Heller 2012).In analogy to the stellar HZ, Heller & Barnes (2013)defined a circumplanetary “habitable edge” (HE) that is acritical distance to the planet interior to which a moon withan Earth-like atmosphere (mostly N , small amounts CO ,see Kasting et al. 1993) and a substantial water reservoirexperiences a runaway greenhouse effect and is therefore atleast temporarily uninhabitable. In top-of-atmosphere en-ergy flux calculations, moons orbiting planets in the stel-lar HZ encounter an inner HE but no outer HE. Only be-yond the stellar HZ does an outer HE appear around theplanet (Reynolds et al. 1987; Heller & Armstrong 2014).One-dimensional latitudinal energy balance models suggestthat moons near the outer edge of the stellar HZ can face anouter HE owing to eclipses and an ice-albedo effect if theyorbit their star near the outer edge of the HZ (Forgan &Yotov 2014; Forgan & Dobos 2016).Hinkel & Kane (2013) used the Heller & Barnes (2013)model to study the effect of global energy flux variationsfor hypothetical exomoons orbiting four confirmed giant ex-oplanets in or near the stellar HZ ( µ Ara b, HD 28185 b,BD +
14 4559 b, and HD 73534). Their focus was on the or-bital eccentricity of the planet-moon barycenter around the With an eclipse we here refer to a moon moving through thestellar shadow cast by the planet. star with the result that fluctuations of tens of
W m − canoccur on moons orbiting on highly eccentric stellar orbits.Significant improvements in exomoon climate simula-tions were presented by Forgan & Kipping (2013), whoused a 1D latitudinal energy balance model to assess exo-moon surface temperatures under the effect of tidal heat-ing and eclipses. Forgan & Yotov (2014) advanced thismodel by also including planetary illumination, and For-gan & Dobos (2016) showed yet another update including aglobal carbonate-silicate cycle and a viscoelastic tidal heat-ing model. In any of the previous studies that estimatedsurface temperatures on exomoons (Hinkel & Kane 2013;Forgan & Kipping 2013; Forgan & Yotov 2014; Forgan & Do-bos 2016), maximum and minimum surface temperatures onexomoons varied by several degrees Kelvin (K) over the cir-cumplanetary orbit at most, while variations due to a moon’schanging stellar distance on its circumplanetary orbit or dueto eclipses were a mere ≈ . K.General circulation models (GCMs) have been ap-plied to model the effects of eclipses on the atmosphereand surface conditions on Titan, which shows up to 6 hrlong eclipses during ≈ consecutive orbits around Saturnaround equinox. Tokano (2016) showed that eclipse-inducedcooling of Titan’s surface, averaged over one orbit aroundSaturn, is usually < K on the pro-Saturnian hemisphere.Here we present the first simulations of exomoon cli-mates using an idealized GCM (Haqq-Misra & Kopparapu2015). This GCM improves upon previous 1D studies byallowing us to calculate the energy transfer not only as afunction of latitude but also of longitude and height in theexomoon atmosphere. Our main objective is to determinewhether exomoon climates are principally different from ex-oplanet climates, that is: how do planetary illumination andtidal heating contribute to surface habitability in an exo-moon atmosphere? And ultimately, could these climatic ef-fects of the different heat sources possibly be observed withnear-future technology?
Most of the major moons in the solar system are in syn-chronous rotation with their host planet; that is, one andthe same hemisphere faces the planet permanently (exceptmaybe for libration effects). The star, however, does not ahave fixed position in the reference frame of such a moon, asthe satellite rotates with its circumplanetary orbital period( P ps ). Hence, while stellar illumination can be averaged overthe day and night side of the moon (longitudinally but notlatitudinally), the planet will always shine on the moon’ssubplanetary point. This is a principal difference betweenthe illumination effects experienced by a planet and a moon.Furthermore, there will be planet-moon eclipses. Butthey will only be relevant to the global climate if the moonis in a very close orbit that is nearly coplanar with the cir-cumstellar orbit. Beyond Io’s orbit around Jupiter, the orbit-averaged flux decrease will be a few percent at most, and be-yond ten planetary radii around a Jovian planet eclipses willbe completely negligible (Heller 2012). We therefore neglectplanet-moon eclipses in our simulations. MNRAS , 1–14 (2018) xploring exomoon atmospheres with a GCM As an additional heat source, we consider geothermalenergy, which can be produced via tidal heating, radiogenicdecays in the rocky part of the moon, or through release ofprimordial heat from the moon’s accretion. We do not sim-ulate the production of these heat sources in our model, butuse interesting and reasonable fiducial values in our GCMsimulations.The moon’s mass is chosen as . Earth masses ( M ⊕ ),which we consider as an optimal value in terms of exomoonformation around accreting giant planets (Canup & Ward2006; Heller et al. 2014; Heller & Pudritz 2015b), exomoondetectability (Kipping et al. 2009; Lewis 2011; Heller 2014;Heller & Albrecht 2014; Kipping et al. 2015), and exomoonhabitability (Lammer et al. 2014). As these moons shouldbe water-rich, their radii should be about 0.7 Earth radii( R ⊕ ).We consider two moon orbits, one as wide as Europa’sorbit around Jupiter ( R Jup ) and one twice that value. Theformer choice is supported by simulations of moon formationaround Jupiter-mass planets, which suggest that icy moonscan migrate to ∼ R Jup within the circumplanetary disks(Sasaki et al. 2010; Ogihara & Ida 2012). The latter choiceis motivated by the recent prediction that the most massive,water-rich moons around super-Jovian planets beyond 1 AUshould form near the circumplanetary ice line at about 20to R Jup (Heller & Pudritz 2015b,a). The orbital periodsrelated to these two semi-major axes of the satellite orbit( a ps ) are . and . d. We refer to these orbits as our“short-period” and “long-period” cases, respectively. In allcases, the moon’s spin-orbit misalignment with respect to itscircumplanetary orbit is assumed to be zero and variations ofplanetary illumination due to the moon being on an eccentricorbit are neglected.As for the geothermal heat budget of the satellite ( F g ),we consider three cases of , , and
100 W m − . In thosecases where tidal heating is a major contribution, the high-est values will only be reached in very close-in orbits within . R Jup around our test planet. If the moon under con-sideration is the only major satellite, tidal processes willusually act to circularize its orbit (Goldreich 1963), to erodeits obliquity (Heller et al. 2011), and to lock its rotationrate with the orbital mean motion (Makarov et al. 2016).These processes generate tidal heating in the moon, whichwill gradually decay over time. Tidal surface heating rateson moons could be >
100 W m − for about yr for a singlemoon on an initially eccentric orbit Heller & Barnes (2013).In this sense, the physical conditions that we model couldpreferably correspond to young systems rather than evolvedsystems. The system age, however, is not an input parame-ter in our models and our results are not restricted to youngsystems to begin with. In fact, large extrasolar moons havenot been detected unambiguously so far, and so it remainsunclear whether they are only subject to significant tidalheating when they are young. If the moon is member of amulti-satellite system, for example, then mutual interactionand resonances could maintain significant orbital eccentrici- However, Awiphan & Kerins (2013) showed that photometricred noise from stellar variability might make it difficult to detecteven Earth-mass moons in the HZ around low-mass stars with
Kepler . ties and extend this timescale by orders of magnitude (Helleret al. 2014; Zollinger et al. 2017).As for the planetary illumination absorbed by the moon,planet evolution tracks show that the luminosities of youngsuper-Jovian planets 10 times the mass of Jupiter can beas high as − to − solar luminosities ( L (cid:12) ), dependingon the planet’s core mass, amongst others (Mordasini 2013).Even at the lower end of this range, a moon in a Europa-wide orbit at about R Jup would absorb about
500 W m − (maybe for some ten Myr), thereby easily triggering a run-away greenhouse effect on the moon. Hence, we considerfour cases of planetary illumination onto the satellite’s sub-planetary point, namely, , , , and
500 W m − (the lat-ter one only in the short-period moon orbit). All these casesare summarized in Table 1. We use an atmospheric GCM to simulate the climate of anEarth-like moon in orbit around a super-Jovian planet. ThisGCM was developed by the Geophysical Fluid DynamicsLaboratory (GFDL), based upon their ‘Flexible ModelingSystem’ (FMS), with idealized physical components (Frier-son et al. 2006, 2007a; Haqq-Misra et al. 2011; Haqq-Misra& Kopparapu 2015). The dynamical core uses a spectral de-composition method with T42 resolution to solve the Navier-Stokes (or ‘primitive’) equations of motion. We use a shal-low penetrative adjustment scheme to perform convectiveadjustment in the model (Frierson et al. 2007a), which pro-vides a computationally efficient method for restoring verti-cal balance in lieu of a more explicit representation of con-vective processes. The GCM surface is bounded with a diffu-sive boundary layer scheme and a 50 m thick thermodynamicocean layer with a fixed heat capacity. This is analogous toassuming that the moons are fully covered with a static, uni-form ocean ; hence, topography is neglected. Our assump-tion of aquamoon conditions with no topography or ice alsomeans that we neglect ice-albedo feedback. These simplifi-cations allow for computational efficiency, and they allow usto examine fundamental changes in climate structure with-out any positive feedback processes causing the model tobecome numerically unstable.The GCM includes two-stream gray radiative transfer,which uses a gray-gas radiative absorber with a specifiedvertical profile to mimic a greenhouse effect (Frierson et al.2006). The model atmosphere is transparent to incomingstellar ( i.e., shortwave) radiation, so that incoming starlightpenetrates the atmosphere and warms the surface (with afixed surface albedo of 0.31 for shortwave radiation). Stellarradiation is averaged across the surface (so there is no diur-nal cycle). Infrared ( i.e., longwave) radiation is absorbed by This case is particularly interesting in view of the possible detec-tion of moons around young, self-luminous giant planets via directimaging with the
European Extremely Large Telescope (Heller &Albrecht 2014). This is motivated by our assumption that our test moonsformed in the icy parts of the accretion disks around their gianthost planets at several AU from their Sun-like star. The initialH O ice content of the moons would then have liquefied as theplanets and moons migrated to about 1 AU, where ∼ super-Jovian exoplanets are known today.MNRAS , 1–14 (2018) Haqq-Misra and Heller
Table 1.
Simulated exomoon systems: initialization parameters and global average quantitiesInitialization parameters Average quantitiesCase a ps P ps F g F t T surf ∆ T pole F OLR q strat a R Jup . d − − . . . − . × − R Jup . d −
10 W m − . . . − . × − R Jup . d
10 W m − − . . . − . × − R Jup . d
10 W m −
10 W m − . . . − . × − R Jup . d
10 W m −
100 W m − . . . − . × − b R Jup . d − − . . . − . × − R Jup . d −
10 W m − . . . − . × − R Jup . d
10 W m − − . . . − . × − R Jup . d
10 W m −
10 W m − . . . − . × − R Jup . d
10 W m −
100 W m − . . . − . × − R Jup . d
10 W m −
500 W m − . . . − . × − R Jup . d
100 W m −
500 W m − . . . − . × − Notes.
In all cases the planet-moon binary orbits at 1 AU from a Sun-like star, M p = M Jup , M s = . M ⊕ , and R s = . M ⊕ . Theparameter F g is uniform geothermal heating at all latitudes. The parameter F t is absorbed planetary illumination, with a latitudinaldistribution of F t | cos λ | when ◦ < λ < ◦ and zero otherwise. Mean values from the set of simulations are shown for global surfacetemperature T surf , change in polar surface temperature ∆ T pole , global outgoing longwave radiative flux at the top of the atmosphere F OLR ,and global specific humidity at the 50 hPa level q strat . ( a , b ) We also refer to 1 and 4 as our “slow rotator control” and “rapid rotator control” cases, respectively. the gray-gas atmosphere in proportion to the optical depthat each model layer, with the surface values of optical depthtuned to reproduce an Earth-like value when the GCM isconfigured with Earth-like parameters. Furthermore, watervapor is decoupled from the gray radiative transfer scheme,so that water vapor feedback is neglected. The GCM is alsocloud-free, although we remove excess moisture and energyfrom the atmosphere through large-scale condensation to thesurface.Even with these idealized assumptions, this GCM re-mains capable of representing surface and tropospheric tem-perature profiles, as well as the large-scale circulation fea-tures, of Earth today (Frierson et al. 2006, 2007a; Haqq-Misra et al. 2011). The model maintains symmetry aboutthe equator due to the lack of a seasonal cycle, which can beinterpreted as a mean annual climate state. The model wasoriginally developed to explore the role of moisture on tropo-spheric static stability (Frierson et al. 2006) and the trans-port of static energy in moist climates (Frierson et al. 2007a).We also note that this model has been used to demonstratethat a realistic tropospheric profile can be maintained byeddy fluxes alone, even in the absence of stratospheric ozonewarming (Haqq-Misra et al. 2011). Even so, our applicationof this model to exomoons implies that our results shouldbe interpreted qualitatively, as a conservative estimate withregard to surface temperature values, runaway greenhousethresholds, and large-scale dynamics.The use of a gray-gas absorber allows us to avoid theproblem of choosing a specific atmospheric composition, asany particular choice of greenhouse gases (such as carbondioxide or methane) will yield a unique GCM solution. Al-though many GCM studies of exoplanet atmospheres useband-dependent ( e.g., non-gray) radiation with cloud pa-rameterizations (Yang et al. 2014; Kopparapu et al. 2016;Wolf et al. 2017; Leconte et al. 2013b; Popp et al. 2016;Godolt et al. 2015; Haqq-Misra et al. 2018), such an ap-proach introduces a new set of free parameters for deter- mining the appropriate mix of greenhouse and inert gases.We instead choose to use an idealized gray-gas GCM forour study of exomoon habitability, as others have done forqualitatively exploring the runaway greenhouse threshold(Ishiwatari et al. 2002). Likewise, the assumption that wa-ter vapor is radiatively neutral allows our model to remainstable at high stellar flux levels without initiating a run-away greenhouse state; thus, our assumption of a cloud-freeatmosphere and a simplified convection scheme limits theuse of our GCM for quantitatively determining the runawaygreenhouse threshold. For example, clouds beneath the sub-planetary point could help to delay the onset of a runwaygreenhouse (Yang et al. 2014), although rapid rotation mayweaken this effect (Kopparapu et al. 2016). We emphasizethat our radiation limits, particular for identifying a run-away greenhouse, should be interpreted in a qualitative sensein order to guide more sophisticated investigations with less-idealized GCMs.We fix the moon into synchronous rotation with itsplanet so that the subplanetary point is centered at themoon’s equator, and we set the moon to have a circular orbitand an obliquity of zero. We include an additional source ofinfrared heating at the top of the atmosphere, centered onthe subplanetary point, which we use to represent heatingby the super-Jovian planet in our simulations. Specifically,we set the downward infrared flux at the top of the atmo-sphere equal to F t | cos λ | when ◦ < λ < ◦ , and zerootherwise (where λ is longitude). Because the atmosphere isabsorbing to infared radiation (both in upward and down-ward directions), this planetary infrared flux is absorbed bythe uppermost atmospheric layers, with none of this infraredradiation penetrating through to the surface in any of oursimulations. This upper atmosphere absorption of infraredradiation from the host planet is the primary feature thatdistinguishes an exomoon climate from an Earth-like exo-planet climate.We also add geothermal heating uniformly at the bot- MNRAS , 1–14 (2018) xploring exomoon atmospheres with a GCM tom of the atmosphere, as a representation of tidal heat-ing. The infrared flux from the surface follows the Stefan-Boltzman law, with the geothermal heating term, F g , addedas a secondary source of surface warming. Geothermal heat-ing due to tidal heating may also appear on synchronouslyrotating exoplanets (Haqq-Misra & Kopparapu 2015), par-ticularly those with highly eccentric orbits. Geothermalheating can contribute to habitability by increasing surfacetemperature, while it can also alter atmospheric circulationpatterns by driving stronger poleward transport. Geother-mal heating provides a secondary mechanisms that may con-tribut to climatic features on exomoons.Each case in Table 1 was initialized from a state of restand run for a period of , d in total. The first , dwere discarded, and the average of the subsequent , d ofruntime were used to analyze our cases. The model reaches astatistically steady state within about 500 d of initialization,which takes approximately 4 h to complete using a Linuxworkstation (6 cores at 2.0 GHz). In our presentation of re-sults, we refer to our cases with a 1.175 d rotation rate as‘short-peroid’ and our cases with a 3.324 d rotation rate as‘long-period.’ We also refer to our two cases with geother-mal heating and absorbed planetary illumination set to zeroas our ‘control’ cases. Our set of twelve cases provide anoverview of the dependence of an exomoon’s climate on theproperties of its host planet. For all of our control and experimental cases, we calculateglobal average values of surface temperature, outgoing long-wave flux at the top of the atmosphere, and stratosphericspecific humidity (Table 1). Our control cases (1 and 6) bothshow global average surface temperatures similar to that ofEarth today. Our experimental cases (2-5 and 7-12) show anincrease in temperature as geothermal and planetary fluxesincrease, with a corresponding increase in stratospheric wa-ter vapor and outgoing infrared radiation.
Surface temperature and winds are shown for the two con-trol cases in Figure 1, with the slow rotator on the toprow, the rapid rotator on the middle row, and the differ-ence between the two on the bottom row. The change inrotation rate from the slow to rapid rotator results in anincrease in the strength of the easterly equatorial jet, whichalso corresponds to an increase in the equator-to-pole tem-perature contrast. This expected behavior corroborates theclassic results of Williams & Halloway (1982), who find thateven slower rotation rates will result in a global meridionalcirculation cell that spans the entire hemisphere. Lackinggeothermal or planetary heating, these control cases repre-sent Earth-like climate states for a smaller planet at differentrotation rates.The addition of infrared planetary illumination to thetop of the atmosphere and geothermal heating to the sur-face can lead to departures from an Earth-like climate state.Figure 2 shows surface temperature and winds for the rapidrotator experiments with increasing contributions of infraredheating. The top row shows cases with modest geothermal
Figure 1.
Time average of surface temperature deviation fromthe freezing point of water (shading) and horizontal wind (vec-tors) for the P = . d (top panel) and P = . d (middlepanel) control cases. The bottom row shows the difference of thefirst row minus the second. The subplanetary point falls on thecenter of each panel. The lengths of the vectors are proportionalto the local wind speeds, and a reference vector with a length of5 m s − is shown on the last panel. (The color scale is chosen tomatch the maximum temperature range shown in Figures 2 and3 to ease comparison.) and planetary heating, which leads to both warming of thepoles and destabilization of the easterly equatorial jet. Asadditional heating is applied, shown in the bottom rows ofFigure 2, the predominantly zonal flow of winds becomesdisturbed in favor of a pattern dominated by large-scale vor-tices. Figure 2 also includes dark contours showing the ‘ani-mal habitable zone’ limits of 0 and 50 degrees Celsius, whichrepresents the temperature range where complex animal lifecould survive (Ward & Brownlee 2000; Edson et al. 2012).The two cases in the top row of Figure 2 both show thatthe freezing line of 0 degrees Celsius sits near the boundarybetween midlatitude and polar regions, around 60 degreeslatitude, so that the planet’s habitable real estate is con-fined to the midlatitude and equatorial regions. The bottomleft panel of Figure 2 shows a case where the entire sur-face is within the animal habitability limits, as a result fromstrong thermal heating from the host planet. The most ex-treme case, shown in the bottom right panel of Figure 2, hasthe 50 degree Celsius contour at about 30 degrees latitude, MNRAS , 1–14 (2018)
Haqq-Misra and Heller
Figure 2.
Time average of surface temperature deviation from the freezing point of water (shading) and horizontal surface wind (vectors)for the P = . d experiments. The geothermal ( F g ) and top-of-the atmosphere heat flux from the planet ( F t ) are shown on top of eachpanel. Dark coutours indicate the ‘animal habitability’ temperature bounds of 0 and 50 degrees Celsius where complex life could survive.The lengths of the vectors are proportional to the local wind speeds, and a reference vector with a length of 5 m s − is shown on the lastpanel. with regions closer to the equator being too warm to supportcomplex life.Our set of GCM cases also illustrate the propensity foran exomoon atmosphere to enter a runaway greenhouse asa result of strong thermal heating aloft. As we discussedabove, our idealized GCM neglects water vapor feedback andrelies upon gray-gas radiative transfer, so our considerationof greenhouse states should be interpreted as a qualitativeand conservative estimate. Further GCM development withband-dependent radiative transfer, cloud paramterization,and more realistic convective processes will be required toidentify quantitative thresholds for when we should expectto observe a runaway greenhouse state on an exomoon. Ad-ditionally, a dry moon with a desert surface and little stand-ing water will remain stable past these radiation limits (Abeet al. 2011; Leconte et al. 2013b), so our moist GCM calcu-lations also serve as a conservative estimate of the runawaygreenhouse threshold. Given these caveats, it still remains in-structive to consider how our exomoon simulations comparewith theoretical limits for expecting a runaway greenhouse.The long-period cases (1-5) show a maximum outgo-ing infrared flux of 276.3 W m − , which falls within the sta-ble radiation limits calculated from one-dimensional (Kast-ing 1988; Goldblatt & Watson 2012; Ramirez et al. 2014)and three-dimensional (Leconte et al. 2013a; Wolf & Toon2014) climate models. By contrast, the short-period cases(6-12) include experiments with an outgoing infrared flux of400 W m or larger (cases 11 and 12), which exceeds stableradiation limits and indicates the climate should be in a run-away greenhouse state (Kasting 1988; Goldblatt & Watson 2012; Ramirez et al. 2014; Leconte et al. 2013a). The twocases (11 and 12) with F t = W m − are also both withinthe runaway greenhouse regime, which suggests that theirsurface temperatures, as shown in the bottom row of Fig-ure 2, would continue to increase in a GCM with raditively-coupled water vapor.Water loss can also occur prior to the runaway green-house, due to the photodissociation of water vapor as thestratosphere becomes wet, in a process sometimes known asa ‘moist greenhouse’ (Kasting 1988; Kopparapu et al. 2017).The moist greenhouse state can be inferred by amount ofwater vapor that crosses the tropopause and reaches thestratosphere, which we indicate using specific humidity (theratio of moist to total air), q strat , near the model top (seeTable 1). Calculations with other models indicate that at-mospheres enter a moist greenhouse and begin to rapidly losewater to space when stratospheric specific humidity exceedsa threshold of q strat ≈ − (Kasting 1988; Kopparapu et al.2013; Wolf & Toon 2014; Kopparapu et al. 2016). As a qual-itative approach to this problem, our results illustrate thatthermal heating from the host planet, as well as geothermalheating from tides, are both plausible mechanisms for heat-ing an exomoon atmosphere to the point of initiating waterloss. We compare our potentially habitable long-peroid and short-peroid cases with F g = W m − and F t = W m − withthe corresponding control cases in Figure 3. These cases are MNRAS , 1–14 (2018) xploring exomoon atmospheres with a GCM within stable radiation limits and are not at risk of enter-ing a runaway greenhouse or otherwise losing water to spacedue to a wet stratosphere. Both panels in Figure 3 show thatwarming is concentrated toward the poles, with a more mod-est degree of warming at the tropics and midlatitudes. This‘polar amplification’ is a well-known process that occurs inclimate models, particularly in simulations of global warm-ing, and is an expected consequence of imposing additionalheating on an atmosphere.Polar amplification is commonly attributed to thewarming that results from ice-albedo feedback in polar re-gions, which accelerates the loss of ice, thereby reducingalbedo and continuing to accelerate the rate of warming(Polyakov et al. 2002; Holland & Bitz 2003). However, po-lar amplification is also present in idealized models thatlack sea ice feedback entirely (Alexeev et al. 2005; Langen& Alexeev 2007; Alexeev & Jackson 2013), which suggeststhat atmospheric heat transport alone can provide an ex-planatory mechanism. Alexeev et al. (2005) demonstratedthat polar amplification should occur when a GCM is forcedwith an additional source of uniform surface warming, whichin principle could be casued by both infrared and visiblesources. Alexeev & Jackson (2013) argued that both ice-albedo feedback and meridional energy transport contributeto polar amplification, with the effects of energy transportbeing masked when ice-albedo feedback is present.Our exomoon simulations further illustrate the capabil-ity of GCMs to show polar amplification in the absence ofice-albedo feedback. Previous idealized GCMs have shownpolar amplification from uniform thermal heating sources,and our results demonstrate that similar polar amplifica-tion can be obtained from the non-uniform thermal heat-ing of planetary illumination. Polar amplification on ice-freeplanets occurs as a response to enhancement of the merid-ional circulation on a warmer planet (Lu et al. 2007), whichleads to increased poleward transport of energy and mois-ture (Alexeev et al. 2005).We calculate the polar amplification, ∆ T pole as the dif-ference in the mean temperature at the north pole betweeneach experiment and the corresponding control case. Thevalues of ∆ T pole are shown in Table 1. A constant uniformgeothermal heating of F g = W m − with no planetary il-lumination (cases 3 and 8) yields a polar amplification ofabout K , with the outgoing longwave radiation also about W m − greater than the control case. This indicates thatgeothermal heating is entirely absorbed and re-radiated bythe lower, thicker layers of the atmosphere. Conversely, anon-uniform planetary illumination of F t = W m − withno geothermal heating (cases 2 and 7) shows a smallerpolar amplification of about . K . Cases 2 and 7 showan increase in outgoing longwave radiation of only about W m − compared to the control; however, this is expectedbecause the non-uniform distribution of planetary heating( F t | cos λ | when ◦ < λ < ◦ ) results in a net warming of F t / π ≈ . W m − . Even so, the total polar amplification of . K in cases 2 and 7 is nearly seven times less than the po-lar amplification of K in cases 3 and 8. This indicates thatsome planetary illumination is absorbed by the uppermostlayers of the atmosphere, with the remainder of this energycontributing to surface warming. However, these cases (2and 7) still demonstrate that polar amplification can occurfrom a non-uniform upper-atmospheric heating source. The experiments with equal geothermal and planetary heating of F g = W m − and F t = W m − (cases 4 and 9) show thatthe value of ∆ T pole is equal to the sum of the polar amplifi-cation when each heating source is considered in isolation.Likewise, the value of F OLR for case 4 (and 9) equals thesum of the outgoing longwave radiation terms from cases 2and 3 (7 and 8). Polar amplification continues to increasewhen F t = W m − and greater (cases 5, 10-12), with cor-responding increases in F OLR that indicate a further pen-etration depth for incoming planetary illumination. Theseresults emphasize that polar amplification in GCMs thatlack ice albedo feedback can still occur with both uniformsurface warming and non-uniform stratospheric warming.The expansion of the meridional overturining (i.e.,Hadley) circulation is shown in Fig. 4 for the P = . dexperiments. The top row of Fig. 4 shows the control case,while the bottom row shows the experiment with F g = W m − and F t = W m − . The left column of Fig. 4shows the global average, while the middle and right columnsseparate the atmosphere into the hemispheres east and west,respectively, of the subplanetary point. The purpose of thisdecomposition is to show that the atmosphere responds toa fixed heating source by altering both the direction andwidth of the Hadley circulation in each hemisphere rela-tive. This prediction originates from the shallow water modelof Geisler (1981), which demonstrated that the Hadley cir-culation should change directions on either side of a fixedheating source. Although the Hadley circulation appears toweaken and maintain its width when planetary and geother-mal heating is applied (left column), the hemispheric decom-position shows that the Hadley circulation in eastern hemi-sphere reaches fully to the poles while the western hemi-sphere shows a Hadley circulation with the opposite direc-tion. The zonal wind pattern in these atmospheres remainsrelatively consistent between the two cases, with prominentupper-level jets associated with the descending branch of theglobal Hadley cell that appear identical in the hemisphericdecomposition.We also demonstrate this Hadley cell expansion or the P = . d experiments in Fig. 5, which shows the con-trol experiment along with three other cases of increas-ing planetary illumination and geothermal heating. Fig. 5shows that the global average Hadley circulation tends todiminish as planetary illumination increases, but the hemi-spheric decomposition shows that the Hadley circulation isactually expanding poleward. The two hemispheres showcirculations with opposite direction and approximately thesame strength. Geothermal heating does not significantlyalter the circulation strength, although it does change themorphology of the ascending branch of the Hadley circula-tion. Two upper-level midlatitude jets are present when F t ≤ W m − , while a third equatorial jet emerges when F t = W m − . Strong geothermal heating of F g = W m − tends to sharpen the equatorial jet and raise the altitude ofthe midlatitude jets.We further demonstrate this behavior in our resultsby examining the vertically integrated flux of moist staticenergy as a function of latitude, following Frierson et al.(2007b) and Kaspi & Showman (2015). Moist static energy, m , represents the combination of dry static energy and latent MNRAS , 1–14 (2018)
Haqq-Misra and Heller
Figure 3.
Difference from control cases of the time average of surface temperature (shading) and horizontal wind (vectors) for the P = . d (left panel) and P = . d (right panel) experiments with F g =
10 W m − and F t =
100 W m − . The lengths of the vectors areproportional to the local wind speeds, and a reference vector with a length of 5 m s − is shown on the last panel. Figure 4.
The mean meridional circulation (line contours) and zonal mean zonal wind (shading) are averaged across the entire planet (firstcolumn), eastern hemisphere (second column), and western hemisphere (third column) from the subplanetary point for the P = . dexperiments with F g = F t = W m − (top row) and F g = W m − and F t = W m − (bottom row). Contours are drawn at an interval of ±{ , , } × kg s − . Solid contours indicate positive (northward) circulation, and dashed contours indicate negative (southward)circulation. energy as m = c p T + Φ + L v q , (1)where c p is the specific heat capacity of air, Φ is geopotentialheight, L v is the enthalpy of vaporization, and q is specifichumidity. We decompose moist static energy m into a sumof time mean m and eddy m contributions as m = m + m .This allows us to write the meridional moist static energyflux as v m = ¯ v ¯ m + v m , (2) where v m represents meridional mean energy transport and v m represents meridional eddy energy transport. The ver-tically integrated flux of m is defined as M = π a cos φ ∫ p s v m g dp , (3)where a is planetary radius, φ is latitude, and the overbardenotes a zonal and time mean. Equation (3) gives the totalvalue of M from all dynamical contributions. We can likewiseseparate the mean and eddy contributions to M by replacing MNRAS , 1–14 (2018) xploring exomoon atmospheres with a GCM Figure 5.
The mean meridional circulation (line contours) and zonal mean zonal wind (shading) are averaged across the entire planet (firstcolumn), eastern hemisphere (second column), and western hemisphere (third column) from the subplanetary point for the P = . dexperiments with F g = F t = W m − (first row), F g = W m − and F t = W m − (second row), F g = W m − and F t = W m − (third row), and F g = W m − and F t = W m − (last row). Contours are drawn at an interval of ±{ , , } × kg s − . Solidcontours indicate positive (northward) circulation, and dashed contours indicate negative (southward) circulation. the meridional static energy flux v m in Eq. (3) with ¯ v ¯ m or v m , respectively.We present the total, mean, and eddy fluxes of M in Fig-ure 6 to show the difference between our control cases andour experiments with F g = W m − and F t = W m − .Both panels show a poleward increase in M when geother-mal and planetary heating are added, with all of this con-tribution due to increases in the mean component of M at latitudes φ > ◦ . By contrast, the eddy component of M de-creases in the range ◦ < φ < ◦ when heating is induced.Polar amplification, and an associated increase in moisture,occurs as a result of intensification of the mean polewardtransport of static and latent energy fluxes from both sur-face geothermal and top-of-atmosphere planetary heating.The moist static energy flux also provides an explana-tion for the equatorial band of warming in our differenceplots shown in Figure 3, particularly in the rapid rotating MNRAS , 1–14 (2018) Haqq-Misra and Heller
Figure 6.
Vertically integrated moist static energy flux ( M ) for the P = . d (left panel) and P = . d (right panel) experiments.The total M (black curves), mean contribution to M (green curves), and eddy contribution to M (blue curves) are shown for controlcases with F g = F t = (solid) and experiments with F g =
10 W m − and F t =
100 W m − (dashed). case. Figure 6 also shows a decrease in M at tropical latitudesin the range ◦ < φ < ◦ , which leads to warming and anaccumulation of moisture beneath the subplanetary point.This point corresponds to a maximum in the rising motionof the mean meridional circulation (not shown), as a resultof planetary illumination. Although our GCM does not in-clude cloud processes, convective processes at the subplane-tary point should lead to cloud formation, which could con-tribute to an expansion of the inner habitable region aroundthe host planet (Yang et al. 2014). In general, the circulationpatterns of climates with a fixed source of heating are noteasily characterized by longitudinally-averaged mean merid-ional circulation functions, due to hemispheric reversals inthe direction of these circulation patterns (Haqq-Misra &Kopparapu 2015). Nevertheless, we can still expect strongrising motion beneath the subplanetary point, with bothzonal and meridional transport toward the opposing hemi-sphere. The redistribution of energy from planetary illuminationcauses warming of both the surface as well as the po-lar stratosphere. Figure 7 shows the vertical structure ofmean zonal temperature for the short-peroid cases with F g = F t = W m − (left panel) and F g = W m − and F t = W m − (right panel). The height of the tropopauseis also shown as a dark curve in both panels of Figure 7,which follows the World Meteorological Organization defini-tion of the tropopause as the altitude at which the lapse rateequals − K km − . For the control case (left panel), the trop-ical tropopause extends up to about hPa due to convec-tive heating by both moist and dry processes (Haqq-Misraet al. 2011). The height of the extratropical tropopause isdetermined by the balance between warming from latent andsensible heating in the troposphere below with warming in the stratosphere from the poleward transport of the Brewer-Dobson circulation (Haqq-Misra et al. 2011).When planetary and geothermal heating are included(Figure 7, right panel), the boundaries of the tropicaltropopause sharpen and act to widen the extratropical zonewhile narrowing the tropics. Increased warming beneaththe subplanetary point drives stronger convection, whichcauses the tropopause to extend higher. This increase inconvective heating also increases the poleward flux of moiststatic energy, which causes the extratropical tropopause tosteepen. Warming in the stratosphere occurs primarily inthe polar regions, which is driven by poleward energy trans-port processes such as the Brewer-Dobson circulation (notshown). This stratospheric warming competes with tropo-spheric warming in the tropics, which results in the polarheight of the tropopause remaining relatively constant be-tween the two cases shown. One interpretation of this be-havior is that the expanded extratropics are analogous toan increase in efficiency of the moon’s ‘radiator fins,’ whichprovide a means of transporting and dissipating energy fromthe warming tropics (Pierrehumbert 1995). Planetary illumi-nation thus serves to sharpen the distinction between tropi-cal and extratropical climate zones by redistributing energypoleward both along the surface and aloft in the strato-sphere.The effect of planetary illumination on the atmosphericstructure is also evident from examining vertical tempera-ture profiles along the equator (Fig. 8, left panel). For allprofiles, the long-period cases show a colder stratospherebut a warmer troposphere than the short-period cases withthe same planetary and geothermal heating. These differ-ences correspond to the enhanced meridional transport ofenergy and moisture in the long-period cases, which alsodrives stronger and narrower jets in the short-period cases(Williams & Halloway 1982). The effect of planetary illumi-nation causes a stratospheric temperature inversion, anal-ogous to the ozone-driven stratospheric inversion on Earth MNRAS , 1–14 (2018) xploring exomoon atmospheres with a GCM Figure 7.
Time average of mean zonal temperature (shading and contours) for the P = . d experiments with F g = F t = W m − (left panel) and F g = W m − and F t = W m − (right panel). The dark curve shows the height of the tropopause. (The color scale ischosen to match previous figures, and contours are drawn every ◦ C.) today. The warmest cases with F t = W m − show a profilewith increasing slope that begins approaching an isothermalatmosphere from the additional planetary and geothermalheating.Some planetary illumination is absorbed in the upper-most layers of the model atmosphere, while the rest con-tributes to surface warming. Fig. 8 (right panel) shows ver-tical profiles of the temperature tendency due to radiation(or radiative heating) along the equator. This reflects the di-rect change in temperature due to radiation alone, neglect-ing physical processes such as convection, boundary layerdiffusion, and dynamical heating. The green curves with F t = W m − show modest warming in the upper atmo-sphere of about . K day − , with strong cooling in the mid-dle troposphere and surface heating of about . K day − .The blue curves show increased upper atmosphere warmingto ∼ . K day − with stronger cooling in the middle tropo-sphere. The blue curves also show a lower radiative heatingrate at the surface, even though these cases show a highersurface temperature. The decrease in direct surface radiativeheating is accounted for by increased diffusion of the bound-ary layer, which in turn leads to increased surface warming.The structure of the boundary layer is also evident from theleft panel of Fig. 8 as a change in lapse rate near ∼ hPa.The middle troposphere is characterized by strong convec-tion, which acts to restore the radiative cooling.The more extreme cases with F t = W m − also con-tinue this trend, with radiative cooling of ∼ . K day − in-dicating a transfer of energy to the vertical diffusion of theatmosphere’s boundary layer—which thereby leads to sur-face warming. Note also that case 12 shows reduced upper-atmosphere absorption due to the large geothermal flux of F g = W m − along with stronger cooling (and thus con-vection) in the middle troposphere. The gray-gas radiativeabsorber in this GCM assumes a specified vertical profile asa function of optical depth, which represents a greenhouseeffect in the troposphere and stratosphere. The upper at-mosphere absorption of planetary illumination on an actualexomoon would depend upon the atmospheric compositionand pressure, among other factors, which could be explored with other GCMs that use band-dependent radiative trans-fer. Comparison of several different GCMs in a similar exo-moon configuration would provide more robust constraintson the expected heating profile from planetary illumination. In general, these simulations illustrate that the potentialhabitability of an exomoon depends upon the thermal en-ergy emitted by its host planet. We find stable climate statesfor both slow and rapid rotators with F g ≤ W m − and F t ≤ W m − , which indicates that both geothermal heat-ing and planetary illumination could provide an additionalsource of warming for an exomoon. However, strong ther-mal illumination by the host planet ( F t = W m − in ourexperiments) would likely lead to an onset of a runawaygreenhouse and the loss of all standing water.On the one hand, the habitability of some exomoon sys-tems may therefore be precluded based upon the presenceof a luminous host planet, although planetary illuminationitself does not necessarily limit an exomoon’s habitability(Heller 2016). On the other hand, we expect polar ampli-fication of warming in all cases, which may suggest thatexomoons in orbit around a luminous host planet may beless likely to develop polar ice caps. This tendency for anexomoon to have warmer poles due to planetary illumina-tion suggests that such bodies may have a greater fractionalhabitable area than Earth today (Spiegel et al. 2008), po-tentially improving the prospects of an origin of life (Heller& Armstrong 2014). This prediction of polar warming onexomoons could eventually translate into observables fromthe circumstellar phase curves of an exomoon, if the planet’scontribution to the combined planet-moon phase curve canbe filtered out (Cowan et al. 2012; Forgan 2017).To a lesser extent, the rotational period of the exomoonalso contributes to differences in surface habitability. Theshort-period cases tend to show a greater amount of warm-ing along the equator, near the subplanetary point (Figure 3,right panel), which could serve as an additional source of en- MNRAS , 1–14 (2018) Haqq-Misra and Heller
Figure 8.
Vertical profiles of the zonal mean temperature (left) and radiative heating (right) at the equator for all simulations listedin Table 1. Solid curves indicate short-period cases ( . d) and dashed curves indicate long-period cases ( . d). Black curves showcontrol experiments, while green indicates all simulations with F t ≤ W m − . Blue curves show simulations with F g = W m − and F t = W m − , while red curves show cases with F t = W m − . Case numbers as per Table 1 are given for some of the lines. ergy to maintain regional habitable conditions. Dynamicalchanges in atmospheric jet structure that result from differ-ences in rotation rate also contribute to changes in surfacewind patterns, as well as the general circulation, which willlikely correspond to significant contrasts in resulting cloudpatterns. The large-scale circulation also shows an oppositedirectional sense in the hemispheres east and west of thesubplanetary illumination point, which could also impactthe probable location of clouds.The hemispheric differences in the Hadley circulation(Figs. 4 and 5) show similarities to simulations of terres-trial planets in synchronous rotation around low mass stars,where the host star is fixed upon a substellar point on theplanet. Haqq-Misra et al. (2011) used the idealized FMSGCM to demonstrate that the Hadley circulation shows di-rection in the opposite direction when comparing the hemi-spheres east and west of the substellar point for plan-ets with 1 d and 230 d rotation periods. Haqq-Misra et al.(2018) also find the same hemispheric circulation patterns inan analysis the Community Earth System Model (CESM),which includes band-dependent radiation, cloud processes,and other physical processes. Synchronously rotating plan-ets drive such a circulation when their stellar energy sourceis fixed to a single location; however, our exomoon calcu-lations demonstrate that such a circulation can also be ob-tained when planetary illumination is fixed, even if the moonotherwise experiences variations in incoming starlight.These idealized calculations provide a qualitative de-scription of surface temperature and winds on an exomoon,but the application of more sophisticated GCMs will help toidentify particular threshold where a runaway greenhouseand other water loss processes occur. Non-gray radiativetransfer will allow for particular atmospheric compositionsto be examined, such as a mixture of nitrogen, carbon diox-ide, and water vapor that is characteristic of Earth-likeatmospheres. Implementation of a cloud scheme into theGCM will also provide important insights into habitabil- ity, as clouds could help to delay the onset of a runawaygreenhouse state (Yang et al. 2014).In terms of the odds of an actual detection of the cli-matic effects described in this paper, this could in principlebe possible if the moon’s electromagnetic spectrum (eitherreflection or emission) could be separated from that of theplanet. This might be possible in very fortunate cases wherea large moon is transiting its luminous giant planet (Heller &Albrecht 2014; Heller 2016) or where both the planet and itsmoon transit their common low-mass host star (Kalteneg-ger 2010). Alternatively, if the moon is subject to extremetidal heating, it could even outshine its host planet in theinfrared and therefore directly present its emission spectrum(Peters & Turner 2013) while the planet would still domi-nate the visible part of the spectrum, where it reflects muchmore light than the moon. The technological requirements,however, will go beyond the ones offered by the James WebbSpace Telescope (Kaltenegger 2010) and might not be acces-sible within the next decade. We present the first GCM simulations of the atmospheresof moons with potentially Earth-like surface conditions. Oursimulations illustrate the effects of tidal heating and of plan-etary illumination on the atmospheres of large exomoonsthat could be abundant around super-Jovian planets in thestellar habitable zones.Most of the energy from planetary thermal heating andgeothermal heating is transported toward the poles as a re-sult of enhanced meridional transport of moisture and en-ergy. This suggests that polar ice melt may be less prevalenton exomoons that are in synchronous rotation with theirhost planet. In general, these calculations further illustratethat the poleward expansion of the Hadley circulation en-hances meridional energy transport and can lead to polar
MNRAS , 1–14 (2018) xploring exomoon atmospheres with a GCM amplification of warming, even in the absence of ice albedofeedback.This polar heat transport could increase the fraction ofthe surface that allows the presence of liquid surface waterby compensating for the lower stellar flux per area receivedat the poles of a moon. In other words, illumination fromthe planet might be beneficial for the development of lifeon exomoons. Future observations that are able to distin-guish exomoons from their host planet may be able to de-tect the absence of polar ice caps due to polar amplificationof planetary illumination, such as analysis of photometricphase-curves. ACKNOWLEDGMENTS
The authors thank Ravi Kopparapu for helpful feedbackon a previous version of the manuscript. J.H. acknowl-edges funding from the NASA Astrobiology Institute’s Vir-tual Planetary Laboratory under awards NNX11AC95G andNNA13AA93A, as well as the NASA Habitable Worldsprogram under award NNX16AB61G. R.H. has been sup-ported by the German space agency (Deutsches Zentrumf¨ur Luft- und Raumfahrt) under PLATO Data Center grant50OO1501, by the Origins Institute at McMaster University,and by the Canadian Astrobiology Program, a Collabora-tive Research and Training Experience Program funded bythe Natural Sciences and Engineering Research Council ofCanada (NSERC). Any opinions, findings, and conclusionsor recommendations expressed in this material are those ofthe authors and do not necessarily reflect the views of NASAor NSERC.
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Outlook and Appendix hapter 8
Towards an Exomoon Detection
Of all the exomoon cases put forward in the literature so far, the proposed Neptune-sized exomoonaround the Jupiter-sized transiting planet candidate Kepler-1625 b is currently the most actively de-bated object (see Sect. 1.2). And it will most likely remain a source of debate for some time to comedue to the star’s faintness, which makes conclusive photometric follow-up observations extremely chal-lenging. So far, the scientific debate has focussed on the three transits observed in the four yearsof data from Kepler and one additional transit observed with
Hubble . On 25 October 2017, we havestarted looking at Kepler-1625 and its proposed planet-moon system from a different perspective andthis survey is still ongoing. Our preliminary results are the subject of this section.
To give a little bit of a historical context, the discovery of an exomoon candidate around Kepler-1625 b was announced on 26 July 2017 by Teachey et al. Soon thereafter, we (Principal Investigator:R. Heller) requested Director’s Discretionary Time (DDT) at the 3.5 m telescope at the Calar AltoObservatory to take spectra of the star just before and just after the transit on 29 October 2017. Atthat time, Kepler-1625 b was known only as a transit planet candidate, that is to say, it was estimatedto have a very low false positive probability (FPP) but it had not been confirmed as a planet by anindependent method. The FPP of giant planets in wide orbits has been shown to be particularlyhigh (Morton et al. 2016) and Kepler-1625 b is a giant planet on a wide orbit, with an orbital periodof about 287 d. Hence, there was a non-negligible probability of Kepler-1625 b being a false positivecaused by, e.g., an unrelated eclipsing stellar binary in the background. Alternatively, the objectmight in fact be transiting in front of Kepler-1625 but the relatively large error bars on the physicalradius of the star propagate to the radius of the transiting object and, in this case, permit it to beas large as a very-low-mass star or a brown dwarf (Heller 2018c). Moreover, only a few transitingplanets with similar orbital periods had been confirmed by stellar radial velocity (RV) measurementsat the time. As a consequence, the announcement by Teachey et al. made it a highly conclusive caseto determine the mass of Kepler-1625 b and its moon using RVs and to confirm the planetary natureof the transiting object irrespective of the moon candidate.The purpose of our initial observations in late 2017 was to determine the star’s proper radial velocitycomponent with respect to the solar system since its RV variation due to the planet (and potentiallyits moon) is very close to zero at the time of the transit. Our first service mode observations weresuccessfully taken on 25 and 31 October 2017. Further observations were proposed and granted through The characterization of the star by the Gaia Data Release 2 (Gaia Collaboration et al. 2016, 2018) derives a parallaxof 0 . ± . d = 2460 +238 − pc, and the photometric g mean magnitude is 15 . The original pre-print is freely available at https://arxiv.org/abs/1707.08563v1. .1. NEW CONSTRAINTS ON THE EXOMOON CANDIDATE HOST PLANETKEPLER-1625 B 371
Table 8.1: CARMENES observations of Kepler-1625. observation date observation time BJD S/N25.10.17 18h 14min 2458052.26775 5.372118h 37min 2458052.28352 4.26618h 59min 2458052.29859 5.138331.10.17 18h 31min 2458058.28172 6.924219h 04min 2458058.30206 4.877919h 27min 2458058.31763 5.367828.04.18 03h 41min 2458236.66051 3.533804h 03min 2458236.67569 3.404404h 24min 2458236.69211 7.056504.06.18 02h 14min 2458273.60239 3.489102h 36min 2458273.61722 3.592202h 58min 2458273.63306 3.506430.06.18 01h 10min 2458299.54861 0.829601h 39min 2458299.57736 3.882102h 01min 2458299.59385 4.323102h 24min 2458299.60959 3.916309.08.18 21h 56min 2458340.42432 3.238822h 18min 2458340.43902 4.268522h 38min 2458340.45349 4.374423.10.18 18h 21min 2458415.27194 2.908418h 49min 2458415.29215 2.918819h 12min 2458415.30818 2.8022
Notes.
The first spectrum on 30.06.18 was not analyzed due to poor data quality.This table was prepared by courtesy of A. Timmermann. an Open Time proposal (PI: R. Heller) and another DDT proposal (PI: R. Heller) and the data wereacquired through one orbital cycle of Kepler-1625 b. We took three spectra with 20 min exposuresper night, which we then combined into one spectrum per night that we used for our subsequent RVanalysis. An overview of these observations is shown in Table 8.1 and their expected contribution tothe RV curve over one cycle is illustrated in Fig. 8.1.
The data reduction of the CARMENES spectra was performed using the SERVAL software (Zechmeis-ter et al. 2018) by courtesy of A. Timmermann and in cooperation with M. Zechmeister, the results ofwhich will soon be submitted for peer review. Figure 8.2 shows our best fit of a one-planet Keplerianmodel with eccentricity ( e ), which takes into account the boundary condition of the known transittime. The orbital period and the stellar mass were fixed at 287.38 d and 1.1 solar masses, respectively.The resulting planetary mass is roughly 5 Jupiter masses and the orbital eccentricity is about 0.4.These values are preliminary and we are currently working to deduce robust error bars using Markovchain Monte Carlo simulations (Foreman-Mackey et al. 2013). That said, all things combined we arevery certain that Kepler-1625 b is indeed a transiting planet, whether it has a moon or not. At thetime of writing, our results suggest that among the more than 3100 confirmed transiting exoplanets,2345 of which have been discovered with Kepler , there are only nine with orbital periods longer thanthat of Kepler-1625 b. And so while our RV measurements cannot be used to validate or reject theexomoon hypothesis directly, we find new constraints on the mass of Kepler-1625 b or the combinedplanet-moon mass, as the case may be. Any future characterization of this system that would take intoaccount additional photometric or RV observations will need to be consistent with our new results. Available at https://github.com/mzechmeister/serval. NASA Exoplanet Archive (https://exoplanetarchive.ipac.caltech.edu/docs/counts detail.html) as of 28 August 2019. .1. NEW CONSTRAINTS ON THE EXOMOON CANDIDATE HOST PLANETKEPLER-1625 B 372 ×× transit CARMENES Observations of Kepler-1625 transit O c t Period = 287 days A ug M a r J un not observable × ××
27 Aprno nightly airmass < 1.4 no elevation > 45° D e c F eb J an O c t × Figure 8.1: A sketch of the expected RV curve of Kepler-1625 due to the presence of its giant transitingplanet (plus its hypothetical moon, as the case may be). Symbols on the curve indicate whether theproposed observations were successful or not (see legend at the left). The dates shown along the curverefer to the beginning of the night of the proposed observations. − − R V [ m s − ] ModelTransitsSERVAL Results
Figure 8.2: Best fit of a one-planet Keplerian orbit (solid line) to our RV measurements withCARMENES (points with error bars). This solution includes the known transit times (dark redsquares) as boundary conditions and it suggest a giant planet with about 5 Jupiter masses on aneccentric orbit with e ≈ .
4. Image credit: A. Timmermann. .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 373
In this section we present our ongoing study of three indicators of an exomoon that emerge if well-established planet-only models are fitted to a light curve that contains a planet-moon system: transittiming variations (TTVs), transit duration variations (TDVs), and planetary radius variations (PRVs).We re-evaluate under realistic conditions the previously proposed exomoon signatures in the TTV andTDV series (Sartoretti & Schneider 1999; Kipping 2009a). We simulate light curves of a transitingexoplanet with a single moon, taking into account stellar limb darkening, orbital inclinations, planet-moon occultations, and noise from both stellar granulation and instrumental effects. These modellight curves are then fitted with a planet-only transit model, pretending we wouldn’t know that thereis a moon, and we explore the resulting TTV, TDV, and PTV series for evidence of the moon. Ourresults are preliminary and will soon be submitted to a peer-reviewed journal. This work was done incollaboration with K. Rodenbeck in the context of this PhD thesis (Rodenbeck 2019).As we have seen in Sect. 6, and in particular in Sects. 6.7 (Rodenbeck et al. 2018) and 6.8 (Heller et al.2019), the dynamical modeling of planet-moon systems is computationally demanding. Consideringthe large computational load required for the statistical vetting of exomoon candidates in a star-planet-moon framework and taking into account the large numbers of transiting planets that needto be examined, tools for an uncomplicated identification of the most promising exomoon candidatescould be beneficial to streamline more detailed follow-up studies. Many ways have been proposed tolook for moons in transit light curves (for a review see Heller 2018a). Sartoretti & Schneider (1999)suggested that a massive moon can distract the transits of its host planet from strict periodicity andinduce transit timing variations (TTVs). Kipping (2009a) identified an additional effect of the moonon the planet’s sky-projected velocity component tangential to the line of sight that results in transitduration variations (TDVs). These effects relate to the planet’s position with respect the planet-moonbarycenter, which is modeled on a Keplerian orbit around the star, and they depend on the planet-satellite orbital semi-major axis ( a ps ) and on the satellite’s mass ( M s ) but they do not depend on thesatellite’s radius ( R s ). Heller et al. (2016b) proposed that the combined planetary TTV and TDVeffects can produce distinct ellipsoidal patterns in the TTV-TDV diagram if the moon is sufficientlysmall to avoid any effects on the photometric transit center.Szab´o et al. (2006), however, noted that the stellar dimming caused by a moon with a suf-ficiently large radius can affect the photometric center of the transit light curve, defined as τ = (cid:80) i t i ∆ m i / (cid:80) i ∆ m i with t i as the times when the stellar differential magnitudes (∆ m i ) aremeasured. For sufficiently small (but arbitrarily massive) moons, τ would coincide with the transitmidpoint of the planet but for large moons τ can be substantially offset from the position of themidpoint of the planetary transit. Simon et al. (2007) derived an analytical estimate of the photomet-ric TTV effect (dubbed “TTV p ”) and compared it to the magnitude of the barycentric TTV effect(dubbed “TTV b ”). They define the TTV p as the difference between TTV b and τ .Here we simulate many different transit light curves of a planet with a moon of non-negligible radiusand then fit the resulting data with a planet-only transit model to obtain the TTVs and TDVs ofa hypothetical sequence of transits. Our aim is to provide the exoplanet community with a tool toinspect their TTV-TDV distributions for possible exomoon candidates without the need of developing afully consistent, photodynamical transit modelling of a planet with a moon (Kipping 2011; Rodenbecket al. 2018). Our most recent investigations, which we present below, predict that the previouslydescribed ellipse in the TTV-TDV diagram of an exoplanet with a moon (Heller et al. 2016b; seeSect. 6.4) emerges only for high-density moons. Low-density moons, however, distort the sinusoidalshapes of the TTV and the TDV series due to their photometric contribution to the combined planet-moon transit. Sufficiently large moons can distort the previously proposed ellipse in the TTV-TDV intovery complicated patterns, which are much harder to discriminate from the TTV-TDV distributionof a planet without a moon. After all, we identify a new effect that appears in the sequence of .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 374 − −
100 0 100 200 300time from barycenter midtransit [min]0 . . . . . . . . r e l a t i v e flu x ∝ r t T, fit modelfit Figure 8.3: Example of a planet-only fit (dashed orange line) to the planet-moon model (solid blackline). The star is sun-like, the planet is Jupiter-sized and in a 30 d orbit around the star, and the moonis Neptune-sized and in a 3.55 d orbit around the planet. In this example, the fitted planet-to-starradius ratio ( r fit ), which goes into the calculation of the transit depth with a power of 2, is slightlyoverestimated due to the moon’s photometric signature.planetary radius ( R p ) measurements, which could be used to identify exomoon candidates in largeexoplanet surveys. Sufficiently large moons can nevertheless produce periodic apparent PRVs of theirhost planets that could be observable in the archival Kepler data or with the PLATO mission. Wedemonstrate how the periodogram of the sequence of planetary radius measurements can indicate thepresence of a moon and propose that PRVs could be a more promising means to identify exomoonsin large exoplanet surveys. An inspection of a limited amount of exoplanets from Kepler revealssubstantial PRVs of the Saturn-sized planet Kepler-856 b although an exomoon could only ensure Hillstability in a very narrow orbital range. Figure 8.3 demonstrates how the transit light curve of an exoplanet with a moon (solid black line)differs ever so slightly from the light curve expected for a single planet (orange dashed line). For onething, the midpoint of the planetary transit changes between transits with respect to the planet-moonbarycenter, thereby causing a TTV b effect (Sartoretti & Schneider 1999). Moreover, the additionaldimming of the star by a sufficiently large moon could act to shift the photometric center of thecombined transit, an effect referred to as TTV p (Szab´o et al. 2006; Simon et al. 2007). The sameprinciple of the moon’s photometric (rather than the barycentric) effect should be applicable to theTDV, though it has not been explored in the literature so far. And finally, Fig. 8.3 illustrates how theplanet-only fit to the planet-moon model overestimates the radius of the planet. For single planets,the transit depth is roughly proportional to R , although details depend on the stellar limb darkening(Heller 2019). But a moon can affect the radius estimate for the planet and therefore cause planetary .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 375 star line of sight top-down view line of sightplanet moon ϕ s bary-center a ps side view star line of sight i s b Figure 8.4: Some orbital parameters for the planet-moon system. ϕ s is the orbital phase relative tothe line of sight at the time of mid-transit.radius variations (PRVs) in a sequence of transit measurements.We investigate both the barycentric and the photometric contributions to the measured TTVs andTDVs of a planet-with-moon transit light curve that is fitted with a planet-only model. To separatethe TTV b and TTV p effects, we simulate transits of a hypothetical planet-moon system with a zero-radius or zero-mass moon, respectively, and then fit the simulated light curves with a planet-onlytransit model. Finally, we assume a realistic moon with both mass and radius and redo our fit to theresulting light curve.Synthetic exoplanet-exomoon transits are generated using our implementation of the Mandel &Agol (2002) analytic transit model in a fashion similar to the one presented in Rodenbeck et al.(2018). As an improvement to Rodenbeck et al. (2018), however, our model now includes planet-moonoccultations and the possibility of inclining the planet-moon orbit with respect to the line-of-sight.The planet-only transit model is fitted to the simulated light curve using the planet-to-star radiusratio ( r p ), transit duration ( t T ), and the orbital phase ( ϕ b ), while the orbital period ( P b ) and thelimb darkening coefficients ( q and q for the quadratic limb darkening model as per Kipping 2013)are kept constant at the respective values (for details see Table A.1 in Rodenbeck 2019). Figure 8.3shows an example of a fitted planet-only light curve (orange dashed line) to the simulated noiselesstransit of a planet-moon system. We also investigate the possibility of detecting PRVs. Variations ofthe planetary transit depth have been observed before (Holczer et al. 2016) but to our knowledge theyhave not been treated in the context of exomoons so far.We compare the effect of photon noise between two space missions: Kepler (Borucki et al. 2010)and PLATO (Rauer et al. 2014), the latter of which is currently scheduled for launch in 2026. Forboth missions we test stars of two different apparent magnitudes, namely 8 and 11. The two stellarmagnitudes provide us with two possible amplitudes of the white noise level per data point, which iscomposed of photon noise and telescope noise. For Kepler long cadence (29.4 min) observations, weobtain a white noise level of 9 parts per million (ppm) at magnitude 8 and 36 ppm at magnitude 11(Koch et al. 2010). For PLATO we estimate 113 ppm at magnitude 8 and 448 ppm at magnitude 11,both at a cadence of 25 s. These estimates for PLATO assume an optimal target coverage by all 24normal cameras.We also simulate a time-correlated granulation noise component according to Gilliland et al. (2011),where the granulation power spectrum is modeled as a Lorentzian distribution in the frequency domainand then transformed into the time domain. The granulation amplitude and time scale depend on thestar’s surface gravity and temperature. We use two stellar types, a sun-like star and a red dwarf withsimilar properties as (cid:15) Eridani ( (cid:15)
Eri).We investigate several hypothetical star-planet-moon systems to explore the observability of theresulting TTV, TDV, and PRV effects. We ensure orbital stability of the moons by arranging themsufficiently deep in the gravitational potential of their host planet, that is, at < . .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 376 Hill radius. This distance has been shown to guarantee orbital stability of both pro- and retrogrademoons based on numerical N -body simulations (Domingos et al. 2006). We choose stellar limb dark-ening coefficients ( q , q ) for both the sun and a K2V star akin to (cid:15) Eri from Claret & Bloemen(2011).For case 1, and for all of its sub cases 1a - 1p, we choose the planet to be a Jupiter-sized and themoon to be Earth-sized. The orbit period of the moon can take two values, either the orbital periodof Europa (3.55 d) or that of Io (1.77 d). The barycenter of the planet-moon system is in a P b = 30 dorbit around its Sun-like star. In test cases 2a - 2d, we simulate Neptune-sized moons and vary bothorbital periods involved to test the effect of the number and duration of the transits. We set P b to beeither 30 d orbit or close to twice that value. In fact, however, we shorten the 60 d orbit by 1.69 d to58.31 d for the Io-like orbit and by 1.59 d to 58.41 d for the Europa-wide orbit in order for the moonto show the same orbital advancement between transits, that is, for the remainder of P b /P s to be thesame (Heller et al. 2016b). Cases 3a - 3d refer to a Saturn-sized planet and a super-Earth moon, cases4a - 4d to a Neptune-sized planet and an Earth-sized moon, cases 5a - 5d to an Earth-Moon analog(though at either a 30 d or a 58.31 d orbital period around their star). In cases 6 - 9 we essentiallyredo all these cases except for the star to resemble (cid:15) Eri-like.In cases 1a - 1p, we study the effect of the moon’s orbital phase on the resulting transit shape andgenerate transits of planet-moon systems for orbital moon phases ranging from 0 to 1. We allow fordifferent orbital inclinations of the moon orbit ( i s ) in these test cases (see Fig. 8.4), which may preventoccultations if the planet-moon orbit is sufficiently wide for a given inclination. As we show below, theinclination has an effect on the measured transit parameters if the line connecting the planet and themoon is parallel to the line of sight, that is, if the planet and the moon have a conjunction during thetransit. For test cases 2a - 9d we choose the lowest inclination possible without causing a planet-moonoccultation.As a first application of our search for exomoon indicators, we explore the time series of the fittedtransit depth from Holczer et al. (2016). We then do a by-eye vetting of the transit depth time series,of the autocorrelation functions (ACF), and of the periodograms for each of the 2598 Kepler Objects ofInterest (KOIs) listed in Holczer et al. (2016) and identify KOI-1457.01 (Kepler-856 b) as an interestingcandidate. In brief, this is a roughly Saturn-sized validated planet with a false positive probability of2 . × − (Morton et al. 2016) and an orbital period of about 8 d around an (cid:15) Eri-like host star.
Figure 8.5 shows the TDV (panel a), TTV (panel b), PRV (panel c), and combined TTV-TDV (paneld) effects. Each of these four panels is divided into three subplots showing the contributions of thephotometric distortion of the light curve due to the moon (top subplots, moon mass set to zero), of thebarycentric motion of the planet due to its moon (center subplots, moon radius set to zero), and thecombined effect as derived from the planet-only fit to the simulated planet-moon transit light curve(bottom subplots). All panels assume a Jupiter-sized planet at an orbital period of P b = 30 d, anEarth-like moon with an orbital period of P s = 3 .
55 d (= P Eu ), and a sun-like host star. Blue linesrefer to an orbital inclination of i s = 0, that is to say, to co-planar orbital configurations. This resultsin occultations near moon phases of 0, when the moon is in front of the planet as seen from Earth (seeFig. 8.4), and 0.5, when the planet is behind the moon. Orange lines refer to i s = 0 . .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 377 − R s = 1 R ⊕ , M s = 0 M ⊕ photometric − T D V [ s ] R s = 0 R ⊕ , M s = 1 M ⊕ barycentric i s = 0 . i s = 0 . .
00 0 .
25 0 .
50 0 .
75 1 . − R s = 1 R ⊕ , M s = 1 M ⊕ combined a ) − R s = 1 R ⊕ , M s = 0 M ⊕ photometric − TT V [ s ] R s = 0 R ⊕ , M s = 1 M ⊕ barycentric i s = 0 . i s = 0 . .
00 0 .
25 0 .
50 0 .
75 1 . − R s = 1 R ⊕ , M s = 1 M ⊕ combined b ) R s = 1 R ⊕ , M s = 0 M ⊕ photometric − P R V [ pp m ] R s = 0 R ⊕ , M s = 1 M ⊕ barycentric i s = 0 . i s = 0 . .
00 0 .
25 0 .
50 0 .
75 1 . R s = 1 R ⊕ , M s = 1 M ⊕ combined c ) − R s = 1 R ⊕ , M s = 0 M ⊕ photometric − T D V [ s ] R s = 0 R ⊕ , M s = 1 M ⊕ barycentric i s = 0 . i s = 0 . − −
20 0 20 40TTV [s] − R s = 1 R ⊕ , M s = 1 M ⊕ combined d ) Figure 8.5: Exomoon indicators for a hypothetical Jupiter-Earth planet-moon system in a 30 d orbitaround a sun-like star as measured from noiseless simulated light curves by fitting a planet-only model. (a)
Transit duration variation (TDV). (b)
Transit timing variation (TTV). (c)
Planetary radiusvariation (PRV). (d)
TTV vs. TDV diagram. Blue lines refer to a coplanar planet-moon system with i s = 0 and orange lines depict the effects for an inclined planet-moon system with i s = 0 . ≤ ϕ s ≤ .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 378 Figure 8.6: The PRV, TDV and TTV effects fitted for the transit sequence of an Earth-sized moonin a Europa-like orbit (3.55 d) around a Jupiter-sized planet. The planet-moon barycenter has a 30 dorbit around a sun-like star and the transits (not shown) were sampled with a PLATO-like cadence of25 s.
Left:
Without noise.
Right:
With noise contributions from stellar granulation and PLATO-likeinstrumental noise for an m V = 8 star. The subpanels show the autocorrelation. In all panels, bluesymbols refer to a coplanar planet-moon system with i s = 0 and orange symbols depict the effects foran inclined planet-moon system with i s = 0 . ϕ s = π ) relative to the barycentric effect (panel a), though with roughlythe same amplitude. The resulting TTV signal in the fitted model light curves then turns out to beextremely small (panel c).The photometric PRV depends strongly on the projected separation of planet and moon, with ϕ s = 0 corresponding to the moon being precisely in front of the planet as seen from Earth, ϕ s = π/ ≡ ◦ ≡ “moon phase of 0 . (cid:48)(cid:48) corresponding to the moon being separated as much aspossible from the planet and as “late” as possible for the transit etc. (see Fig. 8.4) The larger thesky-projected apparent separation between the planet and the moon, that is, the better the moon’sphotometric dip in the light curve is separated from the planet’s photometric signature, the smaller thePRV. On the other hand, if the two bodies are sufficiently close to cause a planet-moon occultation,then the PRV drops significantly (see the dip at moon phase 0.5 in Fig. 8.5c).Interestingly, there is a difference in amplitude depending on whether the moon passes behind orin front the planet. This is due to the slightly different transit shapes of the combined planet-moonsystem. If the moon’s effective (tangential) speed across the stellar disk is lower than that of theplanet, the overall transit shape resembles that of a single planet except for an additional signaturein the wings of the transit light curve. Note that this asymmetry of the PRV effect cannot helpto discriminate between prograde and retrograde moon orbits, which is both a theoretical and anobservational challenge for exomoon characterization (Lewis & Fujii 2014; Heller & Albrecht 2014).Moreover, in practice this effect will hardly be observable when noise is present. Note that thebarycentric PRV shows no variation as a function of the moon phase in the case of i s = 0 but somevariation in the case of i s = 0 . .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 379 or lower impact parameter due to its movement around the planet-moon barycenter. At differenttransit impact parameters then, the resulting transit depth will be different due to the stellar limbdarkening effect (Heller 2019). The panels at the left of Fig. 8.6 show the PRV (top), TDV (center), and TTV (bottom) effects,respectively, as measured from a sequence of simulated light curves without noise. Just like in Fig. 8.3,we modeled the planet moon transits as such but fitted a planet-only model. The number of the transitin this sequence is shown along the abscissa. Also shown are the respective autocorrelations beloweach of these three time series. The model systems used for these sequences are our test cases 1b( i s = 0) and 1j ( i s = 0 . P b ) and around the planet-moon barycenter ( P s ) happen to make the moon jump by a phase of 0.448 circumplanetary (or circum-barycentric) orbits between successive transits. This well-known undersampling results in an observedTTV signal period of under 2 P b (Kipping 2009a). The TDV series is also subject to this aliasingeffect, but the autocorrelation suggests some periodicity. Finally, the PRV signal, whose period is justa half of P s , is not affected by the undersampling. Hence, the periodicity can be observed even by eyein the top panel and, of course, also in the corresponding autocorrelation.Moving on to the addition of noise in our simulated light curves, the panels at the right of Fig. 8.6illustrate the same PRV (top), TDV (center), and TTV (bottom) series as in the panels at the left,but now as if the host star were an m V = 8 solar type star observed with PLATO. In both the TDVand the TTV panels we see that the magnitude of the error caused by the noise in the light curves iscomparable to the TTV and TDV amplitudes. As a consequence, both the two time series and theirautocorrelations show only weak hints of periodic variations. For comparison, the PRV signal of theslightly inclined system ( i = 0 . i = 0 system is somewhat weaker.To get a more quantitative handle on the possible periodicities in the PRV, TTV, and TDV se-quences, we investigate the periodograms of the data. In Fig. 8.7 we present the periodograms ofthe time series and of the autocorrelations of the PRV (top), TTV (center), and TDV (bottom) se-quences. Each panel is based on 20 randomized noise realizations of 24 transits of the same system asin Fig. 8.6. The PRVs of each realization show a clear peak at around 9.5 times the orbital period ofthe barycenter around the star. This peak occurs in the periodograms of both the time series and ofthe autocorrelation of the time series. The periodograms of the TDV series and TDV autocorrelation,however, do not show any significant signals. The TTVs show a peak in the periodogram close to theperiod corresponding to the Nyquist frequency (for details see Rodenbeck 2019). The results presented in Figs. 8.6 and 8.7 refer to only two specific test cases, that is, 1b and 1f,respectively. Next, we extend our investigations to all other test cases, while Fig. 8.8 shows the resultsfor a subsection of these cases as a cut through the parameter space. All results shown in Fig. 8.8assume synchronous observations of an m V = 8 star with all 24 normal cameras of PLATO. Wefocus on the PRV because our investigations to this point have indicated that the PRV effect is morepronounced than the TTV and TDV effects.The four systems in the upper row in Fig. 8.8 refer to cases 2b (solid orange), 2d (dashed orange),6b (blue solid), and 6d (blue dashed). The second line of panels shows cases 3b (solid orange), 3d(dashed orange), 7b (solid blue) and 7d (dashed blue). The third line of panels shows cases 4b (solid A complete description of all test cases is given in Rodenbeck (2019). .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 380 f r o m t i m e s . P R V f r o m a u t o c o rr . f r o m t i m e s . T D V f r o m a u t o c o rr . f r o m t i m e s . TT V P b ]01020 f r o m a u t o c o rr . PLATO, magn. 8.0, Sun-Jupiter-Earth, P s : P Eu Figure 8.7: Periodograms of the PRV (top), TDV (center), and TTV (bottom) effects and theirautocorrelations measured from a sequence of 20 simulated light curves for a Jupiter-Earth planetmoon system similar to the one used in Fig. 8.6. The 20 randomized noise realizations contain bothwhite noise and granulation equivalent to an m V = 8 star observed with PLATO at a 25 s cadence.orange), 4d (dashed orange), 8b (blue solid), and 8d (blue dashed). The bottom row of panels refersto cases 5b (orange solid), 5d (orange dashed), 9b (blue solid), and 9d (blue dashed).In cases 2, 3, 6, and 7 involving the large Neptune and super-Earth-sized moons, the PRV signal isalways visible in the time series (left column). The autocorrelation function of the PRV time series(center column) for systems with P b = 60 d, however, is truncated at 6 times the orbital period due tothe low number of transits available at that period. This truncation is caused by our our restrictionof the autocorrelation function to a lag time of at least half the observed number of transits. Forcomparison, in those cases that assume P b = 30 d and, thus, twice the number of transits available forthe analysis, the signal in the corresponding periodogram of the autocorrelation is very clear.For cases 4 and 8 involving an Earth-sized moon, the detectability of the PRV signal depends onthe moon’s orbital period and the size of its star. For the case of a Europa-like orbital period of themoon and the (cid:15) Eridani-like star, a marginally significant peak in the periodogram is present, but nosignificant peak can be seen if the star is sun-like or the moon’s orbital period is that of Io. In none ofthe cases with a Moon-sized moon (cases 5 and 9) we were able to produce a significant PRV signal.In the right column of panels in Fig. 8.8, we observe that our test moons with a Europa-like orbitalperiod all cause a peak in the periodograms at around 9 . P b. For all the other cases involving amoon with an Io-like period (all subcases a and c), the position of the peak in the periodgrams of theautocorrelation is near 12 P b , which is at the edge of the number of orbital periods we deem reasonablefor analysis.The positions of the peak in the periodogram of the PRV autocorrelation function corresponds tothe remainder of the fraction between the circumstellar period of the planet and the circumplanetaryperiod of the moon. If the nominal value of this resulting peak position is smaller than two planetaryorbital periods, then the actual position of the peak is shifted to a position longer than two times P b . .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 381 − −
101 02040 (cid:15)
EriSun P b = 30d P b = 60d − P R V [ pp m ] Saturn - super-Earth − A u t o c o rr e l a t i o n o f P R V P e r i o d og r a m p o w e r − −
101 05100 5 10 15 20transit number − P b ] −
101 2 4 6 8 10 12Period [ P b ]0510Telescope: PLATO, Magnitude: 8, P s : P Eu Figure 8.8: Transit sequence of the planet radius variation (PRV, left), autocorrelation function of thePRV transit sequence (center), and periodogram of the of autocorrelation (right) for various planet-moon configurations around an m V = 8 star as observed with PLATO (see Sect. 8.2.5). Blue lines referto an (cid:15) Eridani-like star, orange lines to a sun-like star. In the Jupiter-Neptune and Saturn/Super-Earth cases the PRV signal is clearly visible in the measured PRV time series. Consequently thesignal is also visible in the autocorrelation. Due to only calculating the AC up to a time lag of havethe observation length, a peak in the periodograms is only visible in the P B = 30 d cases and notthe P B = 60 d cases. In the Neptune-Earth cases the signal in the PRV time series is barely visible,while the autocorrelation and periodogram for the P B = 30 d cases show a clear signal. There is novisible signal in any of the Earth-Moon cases for in the corresponding time series, autocorrelation, andperiodogram. As a first application of our novel PRV method, we searched for PRVs in the published values of thetransit depth sequences of 2598 Kepler Objects of Interest (KOIs), or transiting exoplanet candidates,from Holczer et al. (2016). As a general result of our by-eye inspection of those KOIs with the highestperiodogram power, we noticed that stellar rotation is the major cause of false positives. In the case ofKepler-856 b (KOI-1457.01), however, the interpretation of a false positive is less evident based purelyon the inspection of the light curve. Figure 8.9 shows our vetting sheet constructed for our PRV surveywith Kepler. Kepler-856 b is a Saturn-sized planet with unknown mass in a 8 d orbit around a sun-likestar. The transit sequence has been used to statistically validate Kepler-856 b as a planet with lessthan 1 % probability of being a false positive (Morton et al. 2016).Figure 8.9(a) shows the Pre-Search Conditioning Simple Aperture Photometry (PDCSAP) Keplerflux (Jenkins et al. 2010), which has been corrected for the systematic effects of the telescope, nor-malized by the mean of each Kepler quarter. We note that the apparent periodic variation of themaximum transit depth is probably not the PRV effect that we describe. The apparent variation ofthe maximum transit depth visible in Fig. 8.9(a) is likely an aliasing effect due to the finite exposuretime of the telescope.Figure 8.9(b) shows the series of the transit depths of Saturn-sized object Kepler-856 b, panel (c) .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 382 the autocorrelation function, panel (d) the periodogram power of the transit depth series (blue) and ofthe autocorrelation function (orange), and panel (e) shows the periodograms of the 14 Kepler quartersof the light curve (gray) together with their mean values (black). We used panel (e) as a means toquickly identify stellar variability and compare it to any possible periodicity in the PRV periodogramof panel (d).Using the stellar and planetary parameters derived by Morton et al. (2016), we estimate that theplanetary Hill radius ( R H ) is about 540,000 km wide. Given that prograde moons can only be stableout to about 0 . × R H (Domingos et al. 2006), the range of prograde and gravitationally stable moonorbits is limited to about 270,000 km, corresponding to about 9 R p for Kepler-856 b. For comparison,Io, the innermost of the Galilean moons, orbits Jupiter at about 6.1 planetary radii, corresponding toabout 8 % of Jupiter’s Hill radius. So the space for orbital stability is quite narrow around Kepler-856 bcompared to the orbits of the solar system moons, though a close-in sufficiently large moon could stillbe physically plausible.
400 600 800 1000 1200 1400time [KBJD]0.9900.9951.0001.005 r e l a t i v e f l u x a 500 1000 1500time [KBJD]0.10.00.1 d e p t h v a r i a t i o n b 0 500 1000time lag [d]0.50.00.51.0 A C F c 0 200 400 600time [d]01020 p e r i o d o g r a m p o w e r d depth variationACF 10 time [d]10 p e r i o d o g r a m p o w e r e meanquarters KOI-1457.01
Figure 8.9: Analysis of the Kepler light curve of Kepler-856 b (KOI-1457.01). (a)
PDCSAP Keplerlight curve. (b)
Fractional transit depths variation from Holczer et al. (2016). (c)
Autocorrelation ofthe transit depth variation. (d)
Periodogram of the transit depth variation and the autocorrelationfunction. (e)
Periodograms of the PDCSAP flux of 14 Kepler quarters (gray lines) and their mean(black line) after masking out the planetary transits.
Comparing the TTV, TDV, and PRV distributions as a function of the moon’s orbital phase forsystems of various orbital inclinations, we noticed that systems without occultations (and significantinclinations) exhibit more sinusoidal patterns. A comparison of the blue (with occultations) and orange(without occultations) lines in any of the upper panels of Fig. 8.5(a) - (c) serves as an illustrativeexample for this. Configurations without occultations thus naturally produce higher peaks in in thecorresponding periodograms. An improvement to our PRV indicator, in particular for systems withoccultations, could be achieved if the PRV signal could be modeled and fitted to the PRV time series,or if it could be used as a fitting function in the periodogram. This approach, however, could becomeas mathematically and computationally involved as implementing a planet-moon transit model to thelight curve to begin with.The detectability of moon-induced PRV signals increases with the number of transits observed andit is desirable to at least observe a full period of the PRV signal. Hence, this new exomoon indicatorcan preferably be detected with longterm transit surveys such as Kepler or PLATO, which deliver (orare planned to deliver) continuous stellar high-precision photometry for several years.Localized features on a star’s surface such as spots or faculae can also cause variations in themeasured exoplanet radius. Hence, any search for moon-induced PRVs needs to include the analysisof stellar variations and verify that any candidate signals are not caused by the star.nents to the .2. EXOMOON INDICATORS IN HIGH-PRECISION TRANSIT LIGHT CURVES 383 observed – or in our case the fitted – TTV signal, we noticed that both components tend to canceleach other, see Fig. 8.5(b). This means that for low-density moons the TTV signal will be dominatedby the photometric component, while for high-density moons the barycentric component will be morerelevant. For moons with densities similar to the densities of the terrestrial or planets and moons of thesolar system, which range between about 3 g cm − and 5 g cm − , the overall TTV effect might actuallybe very hard to observe as the barycentric and photometric and contributions essentially cancel eachother out.The amplitude of the PRV signal increases with increasing orbital separation between the planetand the moon. The PRV amplitude is maximized when the orbital semimajor axis around the planetis larger than two stellar radii, at which point the two transits of the planet and the moon can occurwithout any overlap in the light curve. Transits without an overlap happen when the moon’s orbitalphase is near ϕ s = 0 .
25 or ϕ s = 0 .
75, that is, when the planet and the moon show maximum tangentialdeflection on the celestial plane. In these cases of a transit geometry, there is no “contamination” of theplanetary transit by the moon and the lowest possible planetary radius is fitted, thereby maximizingthe variation with respect to those transits of a transit sequence in which the planet and the moontransit along or near the line of sight, that is, near ϕ s = 0 or ϕ s = 0 .
5, at which point the maximumplanetary radius (with maximum contamination by the moon) is fitted.
We use standard tools for the analysis of exoplanet transits to study the usefulness of TTVs, TDVs,and PRVs as exomoon indicators. Our aim is to identify patterns in the data that is now regularlybeing obtained by exoplanet searches, which could betray the presence of a moon around a givenexoplanet without the need of dedicated star-planet-moon simulations.We first test these indicators on simulated light curves of different stellar systems and find that thePRVs observed over multiple transits show periodic patterns that indicate the presence of a companionaround the planet. The TTV and TDV signals, however, may be less useful indicators for the presenceof an exomoon, depending on whether the density of the moon. The moon density determines whetherthe barycentric or photometric component of the respective effect dominates. The PRV signal, forcomparison, is always dominated by the photometric contribution and, as a consequence, has morepredictable, understandable variation.Our search for indications of an exomoon in the 2598 series of transit mid-points, transit durations,and transit depths by Holczer et al. (2016) reveals the hot Saturn Kepler-856 b as an interesting objectwith strong PRVs that are likely not caused by stellar variability. While an exomoon interpretationwould be a natural thing to put forward in this study, we caution that the range of dynamically stablemoon orbits around Kepler-856 b is rather narrow due to its proximity to the star. The planetaryHill sphere is about 18 R p wide, restricting prograde moons to obits less than about 9 R p wide. Atthis proximity to the planet, however, tides may become important and ultimately present additionalobstacles to the longterm survival of a massive moon (Barnes & O’Brien 2002; Heller 2012). Gen-erally speaking, our investigations of the fitted transit depths from Holczer et al. (2016) and of thecorresponding Kepler light curves show that rotation of spotted stars or an undersampling effect ofthe transit light curve can cause false positives.The previously suggested method of finding exomoons in the TTV-TDV diagram (Heller et al.2016b) might not be as efficient a tool to search for moons as thought. The TTV-TDV diagram onlyforms an ellipse in the barycentric regime, that is, when the moon has a very high density. For rockyor icy moons, however, the figure of the TTV-TDV diagram derived from fitting a planet-only modelto the combined planet-moon transit light curve is much more complex than an ellipse.We find that the threshold for a PRV-based exomoon detection around an m V = 8 solar type starwith the Kepler or PLATO missions lies at a minimum radius of about the size of the Earth. ppendix A Appendix – Non-Peer-ReviewedConference Proceedings
A.1 Constraints on the Habitability of Extrasolar Moons (Heller &Barnes 2014) r X i v : . [ a s t r o - ph . E P ] O c t Formation, detection, and characterization of extrasolar habitableplanetsProceedings IAU Symposium No. 293, 2012Nader Haghighipour c ! Constraints on the habitability ofextrasolar moons
Ren´e Heller and Rory Barnes , Leibniz Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdamemail: [email protected] University of Washington, Dept. of Astronomy, Seattle, WA 98195, USA Virtual Planetary Laboratory, NASA, USAemail: [email protected]
Abstract.
Detections of massive extrasolar moons are shown feasible with the
Kepler spacetelescope.
Kepler ’s findings of about 50 exoplanets in the stellar habitable zone naturally makeus wonder about the habitability of their hypothetical moons. Illumination from the planet,eclipses, tidal heating, and tidal locking distinguish remote characterization of exomoons fromthat of exoplanets. We show how evaluation of an exomoon’s habitability is possible based onthe parameters accessible by current and near-future technology.
Keywords. celestial mechanics – planets and satellites: general – astrobiology – eclipses
1. Introduction
The possible discovery of inhabited exoplanets has motivated considerable efforts towardsestimating planetary habitability. Effects of stellar radiation (Kasting et al. 1993; Selsiset al. 2007), planetary spin (Williams & Kasting 1997; Spiegel et al. 2009), tidal evolution(Jackson et al. 2008; Barnes et al. 2009; Heller et al. 2011), and composition (Raymondet al. 2006; Bond et al. 2010) have been studied.Meanwhile,
Kepler ’s high precision has opened the possibility of detecting extrasolarmoons (Kipping et al. 2009; Tusnski & Valio 2011) and the first dedicated searchesfor moons in the
Kepler data are underway (Kipping et al. 2012). With the detectionof an exomoon in the stellar irradiation habitable zone (IHZ) at the horizon, exomoonhabitability is now drawing scientific and public attention. Yet, investigations on exomoonhabitability are rare (Reynolds et al. 1987; Williams et al. 1997; Scharf 2006; Heller &Barnes 2012; Heller 2012). These studies have shown that illumination from the planet,satellite eclipses, tidal heating, and constraints from orbital stability have fundamentaleffects on the habitability of moons – at least as important as irradiation from the star. Inthis communication we review our recent findings of constraints on exomoon habitability.
2. Why bother about exomoon habitability?
The number of confirmed exoplanets will soon run into the thousands with only a handfulbeing located in the IHZ. Why should we bother about the habitability of their moonswhen it is yet so hard to characterize even the planets? We adduce four reasons: ( i. ) If they exist, then the first detected exomoons will be roughly Earth-sized, i.e. havemasses ! . M ⊕ (Kipping et al. 2009). ( ii. ) Expected to be tidally locked to their planets, exomoons in the IHZ have days muchshorter than their stellar year. This is an advantage for their habitability compared toterrestrial planets in the IHZ of M dwarfs, which become tidally locked to the star.1 Ren´e Heller & Rory Barnes
Figure 1.
A hypothetical Earth-sized moon orbiting the recently discovered Neptune-sizedplanet Kepler-22 b in the irradiation habitable zone of a Sun-like host star. The satellite’s orbitis equal to Europa’s distance from Jupiter. A second moon with the size of Europa is in thebackground. ( iii. ) Massive host planets of satellites are more likely to maintain their primordial spin-orbit misalignment than small planets (Heller et al. 2011). Thus, an extrasolar moon inthe stellar IHZ which will likely orbit any massive planet in its equatorial plane (Porter& Grundy 2011) is much more likely to experience seasons than a single terrestrial planetat the same distance from the star. ( iv. ) Extrasolar habitable moons could be much more numerous than planets. In her IAUtalk on Aug. 28, Natalie Batalha has shown the “Periodic Table of Exoplanets”, indicatingmany more Warm Neptunes and Warm Jovians (i.e. potential hosts to habitable moons)than Warm Earths in the
Kepler data ( http://phl.upr.edu ).The confirmation of the Neptune-sized planet Kepler-22 b in the IHZ of a Sun-like star(Borucki et al. 2012) and the detection of Kepler-47 c in the IHZ of a stellar binary system(Orosz et al. 2012) have shown that, firstly, adequate host planets exist and, secondly,their characterization is possible. Figure 1 displays a hypothetical, inhabited Earth-sizedmoon about Kepler-22 b in an orbit as wide as Europa’s semi-major axis about Jupiter.
3. Constraints on exomoon habitability
Illumination
Similar to the case of planets, where the possibility of liquid surface water defines hab-itability (Kasting et al. 1993), we can approach a satellite’s habitability by estimating onstraints on the habitability of extrasolar moons F globs . If this top-of-the-atmosphere quantity is lessthan the critical flux to induce a runaway greenhouse process F RG and if the planet-moonduet is in the IHZ, then the moon can be considered habitable. Of course, a planet-moonsystem can also orbit a star outside the IHZ and tidal heating could prevent the moonfrom becoming a snowball (Scharf 2006). However, the geophysical and atmospheric prop-erties of extremely tidally-heated bodies are unknown, making habitability assessmentschallenging. With Io’s surface tidal heating of about 2 W / m (Spencer et al. 2000) inmind, which leads to rapid reshaping of the moon’s surface and global volcanism, wethus focus on moons in the IHZ for the time being.Computation of ¯ F globs includes phenomena that are mostly irrelevant for planets. Wemust consider the planet’s stellar reflection and thermal emission as well as tidal heatingin the moon. Only in wide circular orbits these effects will be negligible. Let us assume ahypothetical moon about Kepler-22 b, which is tidally locked to the planet. In Fig. 2 weshow surface maps of its flux averaged over one stellar orbit and for two different orbitalconfigurations. In both scenarios the satellite’s semi-major axis is 20 planetary radiiand eccentricity is 0 .
05. Tidal surface heating, assumed to be distributed uniformly overthe surface, is 0 .
017 W / m in both panels. For reference, the Earth’s outward heat flowis 0 .
065 W / m through the continents and 0 . / m through the ocean crust (Zahnleet al. 2007). Parametrization of the star-planet system follows Borucki et al. (2012). Inthe left panel, the moon’s orbit about the planet is assumed to be coplanar with thecircumstellar orbit, i.e. inclination i = 0 ◦ . The satellite is subject to eclipses almost onceper orbit about the planet. An observer on the moon could only watch eclipses from thehemisphere which is permanently faced towards the planet, i.e. the moon’s pro-planetaryhemisphere. Eclipses are most prominent on the sub-planetary point and make it thecoldest point on the moon in terms of average illumination (Heller & Barnes 2012). Inthe right panel, the moon’s orbit is tilted by 45 ◦ against the circumstellar orbit andeclipses occur rarely (for satellite eclipses see Fig. 1 in Heller 2012). Illumination fromthe planet overcompensates for the small reduction of stellar illumination and makes thesub-planetary point the warmest spot on the moon.To quantify a moon’s habitability we need to know its average energy flux. In Heller& Barnes (2012) and Heller (2012) we show that Figure 2.
Orbit-averaged surface illumination of a hypothetical exomoon orbiting Kepler-22 b.Stellar reflection and thermal emission from the planet as well as tidal heating are included.
Left :Orbital inclination is 0 ◦ , i.e. the moon is subject to periodic eclipses. The subplanetary pointat ( φ = 0 ◦ = θ ) is the coldest location. Right : Same configuration as in the left panel, exceptfor an inclination of 45 ◦ . Eclipses are rare and the planet’s illumination makes the subplanetarypoint the warmest location on the moon. Ren´e Heller & Rory Barnes¯ F globs = L ∗ (1 − α s )16 πa ∗ p ! − e ∗ p " x s + πR α p a + R σ SB ( T eqp ) a (1 − α s )4 + h s ( e ps , a ps , R s ) , (1)where L ∗ is stellar luminosity, a ∗ p the semi-major axis of the planet’s orbit about thestar, a ps the semi-major axis of the satellite’s orbit about the planet, e ∗ p the circumstellarorbital eccentricity, R p the planetary radius, α p and α s are the albedos of the planet andthe satellite, respectively, T eqp is the planet’s thermal equilibrium temperature, h s thesatellite’s surface-averaged tidal heating flux, σ SB the Stefan-Boltzmann constant, and x s is the fraction of the satellite’s orbit that is not spent in the shadow of the planet. Notethat tidal heating h s depends on the satellite’s orbital eccentricity e ps , its semi-major axis a ps , and on its radius R s .This formula is valid for any planetary eccentricity; it includes decrease of average stel-lar illumination due to eclipses; it considers stellar reflection from the planet; it accountsfor the planet’s thermal radiation; and it adds tidal heating. Analyses of a planet’s tran-sit timing (Sartoretti & Schneider 1999; Szab´o et al. 2006) & transit duration (Kipping2009a,b) variations in combination with direct observations of the satellite transit (Szab´oet al. 2006; Simon et al. 2007; Tusnski & Valio 2011) can give reasonable constraints onEq. (1) and thus on a moon’s habitability. In principle, these data could be obtainedwith Kepler observations alone but N -body simulations including tidal dissipation willgive stronger constraints on the satellite’s eccentricity than observations.3.2. The habitable edge and Hill stability
The range of habitable orbits about a planet in the IHZ is limited by an outer and aninner orbit. The widest possible orbit is given by the planet’s sphere of gravitationaldominance, i.e. Hill stability, the innermost orbit is defined by the runaway greenhouselimit ¯ F globs = F RG and is called the “habitable edge” (Heller & Barnes 2012).Space for habitable orbits about a planet decreases when the planet’s Hill radius movesinward and when the habitable edge moves outward. This is what happens when wevirtually move a given planet-moon binary from the IHZ of G a star to that of a K starand finally into the IHZ of an M star (Heller 2012). Shrinking the planet’s Hill spheremeans that any moon must orbit the planet ever closer to remain gravitationally bound.Additionally, perturbations from the star on the moon’s orbit become substantial anddue to the enhanced eccentricity and the accompanying tidal heating the habitable edgemoves outward. Hence, the range of habitable orbits vanishes. As the planetary Hill spherein the IHZ about an M dwarf is small and moons follow eccentric orbits, satellites in Mdwarf systems become subject to catastrophic tidal heating, and this energy dissipationinduces rapid evolution of their orbits.Let us take an example: Imagine a planet-moon binary composed of a Neptune-sizedhost and a satellite 10 times the mass of Ganymede (10 M Gan ≈ . M ⊕ , M ⊕ being themass of the Earth). This duet shall orbit in the center of the IHZ of a 0 . M $ star ( M $ being the solar mass), i.e. at a stellar distance of roughly 0 . / ,
000 km.This means that the moon’s circum-planetary orbit must be at least as tight as Io’s orbitabout Jupiter! Recall that Io is subject to enormous tidal heating. Yet, the true tidalheating of this hypothetical moon will depend on its eccentricity (potentially forced by onstraints on the habitability of extrasolar moons stellar mass p l a n e t a r y m ass M dwarfs K dwarfs N ep t une J up i t e r Earth TwinTidal EarthSuper-IoTidalVenus S upe r- E a r t h increasing forced eccentricity e ps (r ap i d o r b i t a l e v o l u t i on ,t i da l d i ss i pa t i on un k no w n ) Figure 3.
Schematic classification of hypothetical 10 M Gan -mass ( ≈ . M ⊕ ) moons in thewidest Hill stable orbits about planets in the stellar IHZ. In the IHZ about M dwarfs a satellite’seccentricity e ps is strongly forced by the close star, which induces strong tidal heating in themoon. A Tidal Venus moon is uninhabitable. the close star and/or by further satellites), and with masses and radii different from thoseof Jupiter and Io, tidal dissipation in that system will be different.Moving on to a 0 . M $ star, the IHZ is now at ≈ .
125 AU and the satellite’s orbitmust be within 255 ,
000 km about the planet. For a 0 . M $ star with its IHZ at about0 .
05 AU the moon’s orbital semi-major axis must be < ,
000 km, i.e. almost as closeas Miranda’s orbit about Uranus. As we virtually decrease the stellar mass and as wemove our planet-moon binary towards the star to remain the IHZ, the star also forces thesatellite’s orbit to become more and more eccentric. We expect that for stellar massesbelow about 0 . M $ no habitable Super-Ganymede exomoon can exist in the stellar IHZdue strong tidal dissipation (Heller 2012).Figure 3 shall illustrate our gedankenexperiment. The abscissae indicates stellar mass,the ordinate denotes planetary mass. For each star-planet system the planet-moon binaryis assumed to orbit in the middle of the IHZ and the moon shall orbit at the widest possi-ble orbit from the planet. With this conservative assumption, tidal heating is minimized.Colored areas indicate the type of planet according to our classification scheme proposedin Barnes & Heller (2012). Exomoons in K dwarf systems will hardly be subject to thedynamical constraints illustrated above (green area), thus Earth twin satellites could ex-ist. However, if roughly Earth-mass exomoons exist in lower-mass stellar systems, thenthey can only occur as Tidal Earths with small but significant tidal heating (yellow area);as Super-Ios with heating > / m but not enough to induce a runaway greenhouseprocess (orange area); or Tidal Venuses, i.e. with powerful tides and ¯ F globs > F RG (redareas). A Tidal Venus is uninhabitable by definition. Tidal dissipation in the upper-leftcorner of Fig. 3 will be enormous and will lead to so far unexplored geological and orbitalevolution on short timescales. Ren´e Heller & Rory Barnes
4. Prospects for habitable extrasolar moons
The quest of habitable moons seeks objects unknown from the solar system. Even themost massive moon, Ganymede, has a mass of only ≈ . M ⊕ . It is not clear whethermoons as massive as Mars ( ≈ . M ⊕ ) or 10 times as massive as Ganymede ( ≈ . M ⊕ )exist (Sasaki et al. 2010; Ogihara & Ida 2012). But given the unexpected presence of giantplanets orbiting their stars in only a few days and given transiting planetary systemsabout binary stellar systems, clearly a Mars-sized moon about a Neptune-mass planetdoes not sound absurd.Although Fig. 3 is schematic and urgently requires deeper investigations, it indicatesthat habitable exomoons cannot exist in the IHZ of stars with masses " . M $ (Heller2012). Orbital simulations, eventually coupled with atmosphere or climate models, haveyet to be done to quantify these constraints. With NASA’s James Webb Space Telescope and ESO’s
European Extremely Large Telescope facilities capable of tracking spectralsignatures from inhabited exomoon are being built (Kaltenegger 2010) and future ob-servers will need a priority list of the most promising targets to host extraterrestrial life.Exomoons have the potential to score high if their habitability can be constrained fromboth high-quality observations and orbital simulations.
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A.2 Hot Moons and Cool Stars (Heller & Barnes 2013b) r X i v : . [ a s t r o - ph . E P ] J a n EPJ Web of Conferences will be set by the publisherDOI: will be set by the publisherc ! Owned by the authors, published by EDP Sciences, 2013
Hot Moons and Cool Stars
René Heller , a and Rory Barnes , , b Leibniz-Institut fürAstrophysik Potsdam (AIP),An der Sternwarte16, 14482 Potsdam, Germany University of Washington, Dept. of Astronomy, Seattle, WA98195 Virtual Planetary Laboratory, USA
Abstract.
The exquisite photometric precision of the
Kepler space telescope now putsthe detection of extrasolar moons at the horizon. Here, we firstly review observationaland analytical techniques that have recently been proposed to find exomoons. Secondly,we discuss the prospects of characterizing potentially habitable extrasolar satellites. Withmoons being much more numerous than planets in the solar system and with most exo-planets found in the stellar habitable zone being gas giants, habitable moons could be asabundant as habitable planets. However, satellites orbiting planets in the habitable zonesof cool stars will encounter strong tidal heating and likely appear as hot moons.
The advent of high-precision photometry from space with the
CoRoT and
Kepler telescopes has dra-matically increased the number of confirmed and putative extrasolar planets. Beyond the sheer numberof detections, smaller and smaller exoplanets were found with the today record being about 0 . Kepler data, most of which are much bigger than Earth.These planets are likely to be gaseous and to resemble Neptune or Jupiter, rather than Earth. Whilethey are not likely to be habitable, their moons might be.Referred to exomoons, Martin Still, Director of the Kepler Guest Observer O ffi ce, said in his talkon the extended Kepler project during this meeting: “They are gonna come.” So how can massiveextrasolar moons be detected, provided they exist in the first place? And to which extent will theypossibly be characterized? a e-mail: [email protected] b e-mail: [email protected] PJ Web of Conferences
Figure 1.
The tectonically most active body in the solar system is Jupiter’s moon Io. Its enhanced volcanism isdriven by tidal dissipation inside the satellite. The two inlets in the left image, taken by the Galileo Orbiter, showa sulfuric plume over a volcanic depression named Pillan Patera (upper photograph) and another eruption calledthe Prometheus plume (lower photograph). The right image shows the strong orange infrared emission of flowinglava in Tvashtar Catena, a chain of calderas on Io. In extrasolar moons, tidal heating may be much stronger andeven make them detectable via direct imaging. (Image credits: NASA / JPL)
Two of the most promising techniques proposed for finding exomoons are transit timing variations(TTVs) and transit duration variations (TDVs) of the host planet. Combination of TTV and TDVmeasurements can provide information about a satellite’s mass, its semi-major axis around the planet[4–6], and possibly about the inclination of the satellite’s orbit with respect to the orbit around the star[7]. The first dedicated hunt for exomoons in the
Kepler data is now underway [8] and could possiblydetect exomoons with masses down to 20% the mass of Earth [9]. This corresponds to roughly 10times the mass of the two most massive moons in the solar system, Ganymede and Titan.
Observations of an exomoon transit itself [10–13] as well as planet-satellite mutual eclipses [14, 15]can provide information about the satellite’s radius. Spectroscopic investigations of a moon’s Rossiter-McLaughlin e ff ect can yield information about its orbital geometry [16, 17], although relevant e ff ectsrequire accuracies in stellar radial velocity of the order of a few cm / s.Moons in the stellar habitable zone of low-mass stars must orbit their host planet very closely toremain gravitationally bound [18, 19]. This will trigger enhanced tidal heating on those hypotheticalmoons and could make them uninhabitable. While a threat to life, enormous tidal heating in terrestrialmoons about giant planets could be strong enough to make them detectable by direct imaging [20].Tidal heating in moons has been observed in the solar system, with Jupiter’s moon Io serving asthe most prominent example (see Fig. 1). As tidal heating in a satellite is proportional to the host ot Planets and Cool Stars runaway greenhouse Earths’s absorbed flux e ps = e ps = e ps = Figure 2.
Left:
Total top-of-the-atmosphere flux (in logarithmic units of W / m ) of a Mars-sized moon about aNeptune-sized planet in the habitable zone of a 0 . M " star. Tidal heating increases with decreasing semi-majoraxis a ps (abscissa) and increasing eccentricity e ps (ordinate). Some examples for orbital elements of solar systemmoons are indicated. Right:
Amplitude of the transit timing variation (dashed lines) for a Mass-sized moon abouta range of host planets. Planetary masses are shown in Earth masses on the ordinate. The habitable edge [21] isindicated for three di ff erent orbital eccentricities e ps of the satellite: 0 .
1, 0 .
01, and 0 . planet’s mass cubed, massive planets provide the most promising targets for direct imaging detectionsof tidally heated exomoons. A massive moon in the stellar habitable zone can be subject to strong tidal heating and hence beuninhabitable. To estimate its habitability, we have set up a model that includes stellar and planetaryillumination as well as tidal heating [21]. If their sum is greater than the critical flux for the moon to besubject to a runaway greenhouse e ff ect [22], the moon will lose all its water and become uninhabitable.In the left panel of Fig. 2 we show contours of the orbit-averaged illumination plus tidal heat flux ofa hypothetical Mars-sized moon orbiting a Neptune-sized planet in the habitable zone of a 0 . M " star. Abscissa indicates the planet-satellite semi-major axis in planetary radii, ordinate shows orbitaleccentricity. In the reddish regions this prototype satellite will be desiccated and uninhabitable.Given su ffi ciently long observational coverage and high-accuracy data, the techniques describedin Sect. 2 make it possible to detect and considerably characterize a sub-Earth-sized moon orbiting agiant planet. In the right panel of Fig. 2 we show the TTV amplitudes of a Mars-sized moon orbitinga range of host planets. It is assumed that the moon’s orbit is circular and that both the circumstellarand the circum-planetary orbit are seen edge-on from Earth [8].To determine a satellite’s mass and orbit via TTV and TDV, many transits of the host planet needto be observed. Thus, to find moons about planets in the stellar habitable zone within the Kepler dutycycle of 7 years, one might be tempted to conclude that they can preferably be detected in cool star
PJ Web of Conferences systems (i.e. around M dwarfs), because their habitable zones are close-by where a planet performspotentially many transits in a given time span. However, the amplitude of a planet’s TTV is smaller incool star systems, given a fixed semi-major axis [10], and the lack of M dwarfs in the
Kepler sample[23] further decreases the chance of finding habitable moons in cool star systems.Beyond that, moons of planets in the habitable zones of cool stars might not be habitable in the firstplace [18]. The planet’s sphere of gravitational dominance, i.e. its Hill sphere, is relatively small dueto the close star. Hence, any moon would have to follow a very tight orbit about the planet. Moreover,the close star will force the satellite’s orbit to be non-circular. Both aspects, small orbital distanceand an eccentric orbit, will cause any Earth-sized moon of a massive gaseous planet to experienceenormous tidal heating. Ultimately, stellar irradiation and tidal heating will sum up to a top-of-the-atmosphere energy flux that exceeds the critical flux for the initiation of the runaway greenhouse e ff ect[21]. Moons of planets in the habitable zones of cool stars will thus be hot rather than habitable.To characterize potentially habitable moons, that is to say, moons in the habitable zones of K andG type stars, using the TTV and TDV techniques will take about a decade of observations, at least.With the Kepler mission being scheduled for a total mission cycle of 7 years, such a detection mightjust be at the edge of what is possible [8–10].
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Appendix – Popular SciencePublications
B.1 Better than Earth (Heller 2015)
This popular science article was published as a cover story of the January 2015 issue of
Scientific Amer-ican
O WE INHABIT THE BEST O ALL POSSIBLE WORLDS? German mathematician Gottfried Leibniz thought so, writing in 1710 that our planet, warts and all, must be the most optimal one imaginable. Leibniz’s idea was roundly scorned as unscientific wishful thinking, most notably by French author Voltaire in his magnum opus,
Candide.
Yet Leibniz might find sympathy from at least one group of scientists – the astronomers who have for decades treated Earth as a golden standard as they search for worlds beyond our own solar system. Because earthlings still know of just one living world – our own – it makes some sense to use Earth as a template in the search for life elsewhere, such as in the most Earth-like regions of Mars or Jupiter’s watery moon Europa. Now, however, discoveries of potentially habitable planets orbiting stars other than our sun – exoplanets, that is – are challenging that geocentric approach. Over the past two decades astronomers have found more than 1,800 exoplanets, and statistics suggest that our galaxy harbors at least 100 billion more. Of the worlds found to date, few closely resemble Earth. Instead they exhibit a truly enormous diversity, varying immensely in their orbits, sizes and compositions and circling a wide variety of stars, including ones significantly smaller and fainter than our sun. Diverse features of these exoplanets suggest to me and to others that Earth may not be anywhere close to the pinnacle of habitability. In fact, some exoplanets, quite different from our own, could have much higher chances of forming and maintaining stable biospheres. These “superhabitable worlds” may be the optimal targets in the search for extraterrestrial, extrasolar life.
AN IMPERFECT PLANET
OF COURSE, our planet does possess a number of properties that, at first glance, seem ideal for life. Earth revolves around a sedate, middle-aged star that has shone steadily for billions of years, giving life plenty of time to arise and evolve. It has oceans of life-giving water, largely because it orbits within the sun’s “habitable zone,” a slender region where our star’s light is neither too intense nor too weak. Inward of the zone, a planet’s water would boil into steam; outward of the area, it would freeze into ice. Earth also has a life-friendly size: big enough to hold on to a substantial atmosphere with its gravitational field but small enough to ensure gravity does not pull a smothering, opaque shroud of gas over the planet. Earth’s size and its rocky composition also give rise to other boosters of habitability, such as climate-regulating plate tectonics and a magnetic field that protects the biosphere from harmful cosmic radiation. Yet the more closely we scientists study our own planet’s habitability, the less ideal our world appears to be. These days habitability varies widely across Earth, so that large portions of its surface are relatively devoid of life – think of arid deserts, the nutrient-poor open ocean and frigid polar regions. Earth’s habitability also varies over time. Consider, for instance, that during much of the Carboniferous period, from roughly 350 million to 300 million years ago, the planet’s atmosphere was warmer, wetter and far more oxygen-rich than it is now. Crustaceans, fish and reef-building corals flourished in the seas, great forests blanketed the continents, and insects and other terrestrial creatures grew to gigantic sizes. The Carboniferous Earth may have supported significantly more biomass than our present-day planet, meaning that Earth today could be considered less habitable than it was at times in its ancient past. Further, we know that Earth will become far less life-friendly in the future. About five billion years from now, our sun will have largely exhausted its hydrogen fuel and begun fusing more energetic helium in its core, causing it to swell to become a “red giant” star that will scorch Earth to a cinder. Long before that, however, life on Earth should already have come to an end. As the sun burns through its hydrogen, the temperature at its core will gradually rise, causing our star’s total luminosity to slowly increase, brightening by about 10 percent every billion years. Such change means that the sun’s habitable zone is not static but dynamic, so that over time, as it sweeps farther out from our brightening star, it will eventually leave Earth behind. To make matters worse, recent calculations suggest that Earth is not in the middle of the habitable This version: 2 March 2015 DOI:10.1038/scientificamerican0115-32
Better Than Earth
Planets quite different from our own may be the best homes for life in the Universe.
By René Heller
This is the author’s version of a cover story published in
Scientific American
SEEKING A SUPERHABITABLE WORLD
IN 2012 I FIRST BEGAN THINKING about what worlds more suitable to life might look like while I was researching the possible habitability of massive moons orbiting gas- giant planets. In our solar system, the biggest moon is Jupiter’s Ganymede, which has a mass only 2.5 percent that of Earth – too small to easily hang on to an Earth-like atmosphere. But I realized that there are plausible ways for moons approaching the mass of Earth to form in other planetary systems, potentially around giant planets within their stars’ habitable zones, where such moons could have atmospheres similar to our own planet. Such massive “exomoons” could be superhabitable because they offer a rich diversity of energy sources to a potential biosphere. Unlike life on Earth, which is powered primarily by the sun’s light, the biosphere of a super-habitable exomoon might also draw energy from the reflected light and emitted heat of its nearby giant planet or even from the giant planet’s gravitational field. As a moon orbits around a giant planet, tidal forces can cause its crust to flex back and forth, creating friction that heats René Heller – Better Than Earth IN BRIEFAstronomers are searching for twins of Earth orbiting other sunlike stars.
Detecting Earth-like twins remains at the edge of our technical capabilities.
Larger “super-Earths” orbiting smaller stars are easier to detect and may be the most common type of planet.
New thinking suggests that these systems, along with massive moons orbiting gas-giant planets, may also be superhabitable – more conducive to life than our own familiar planet.
The Evolution of the Solar Habitable Zone solarhabitable zone1 Astronomical Unit 1 Astronomical Unitdead Earth
Figure 1:
This graphic shows the Earth’s distance to the Sun and its location in the solar habitable zone (nothing is to scale here) during three different epochs of stellar evolution. In general, the habitable zone of a star is the distance range in which an Earth-like planet would have the ability to sustain liquid surface water – the essential ingredient to make a world habitable. As time goes by (see panel titles), the solar luminosity and radius increase, and so the solar habitable zone moves away from the Sun. In about 1.75 billion years, the Earth will leave the habitable zone and become a desiccated giant rock. While Earth is a nice place to live on today, it can be regarded “marginally habitable” from a cosmological point of view – both in spatial and in temporal dimensions. he moon from within. This phenomenon of tidal heating is probably what creates the subsurface oceans thought to exist on Jupiter’s Europa and Saturn’s moon Enceladus. That said, this energetic diversity would be a double-edged sword for a massive exomoon because slight imbalances among the overlapping energy sources could easily tip a world into an uninhabitable state. No exomoons, habitable or otherwise, have yet been detected with certainty, although some may sooner or later be revealed by archival data from observatories such as NASA’s Kepler space telescope. For the time being, the existence and possible habitability of these objects remain quite speculative. Superhabitable planets, on the other hand, may already exist within our catalogue of confirmed and candidate exoplanets. The first exoplanets found in the mid-1990s were all gas giants similar in mass to Jupiter and orbiting far too close to their stars to harbor any life. Yet as planet-hunting techniques have improved over time, astronomers have begun finding progressively smaller planets in wider, more clement orbits. Most of the planets discovered over the past few years are so-called super-Earths, planets larger than Earth by up to 10 Earth masses, with radii between that of Earth and Neptune. These planets have proved to be extremely common around other stars, yet we have nothing like them orbiting the sun, making our own solar system appear to be a somewhat atypical outlier. Many of the bigger, more massive super-Earths have radii suggestive of thick, puffy atmospheres, making them more likely to be “mini Neptunes” than super-sized versions of Earth. But some of the smaller ones, worlds perhaps up to twice the size of Earth, probably do have Earth-like compositions of iron and rock and could have abundant liquid water on their surfaces if they orbit within their stars’ habitable zones. A number of the potentially rocky super-Earths, we now know, orbit stars called M dwarfs and K dwarfs, which are smaller, dimmer and much longer-lived than our sun. In part because of the extended lives of their diminutive stars, these super-sized Earths are currently the most compelling candidates for super-habitable worlds, as I have shown in recent modeling work with my collaborator John Armstrong, a physicist at Weber State University.
THE BENEFITS OF LONGEVITY
WE BEGAN OUR WORK with the understanding that a truly longlived host star is the most fundamental ingredient for superhabitability; after all, a planetary biosphere is unlikely to survive its sun’s demise. Our sun is 4.6 billion years old, approximately halfway through its estimated 10-billion-year lifetime. If it were slightly smaller, however, it would be a much longerlived K dwarf star. K dwarfs have less total nuclear fuel to burn than more massive stars, but they use their fuel more efficiently, increasing their longevity. The middle-aged K dwarfs we observe today are billions of years older than the sun and will still be shining billions of years after our star has expired. Any potential biospheres on their planets would have much more time in which to evolve and diversify. A K dwarf’s light would appear somewhat ruddier than the sun’s, as it would be shifted more toward the infrared, but its spectral range could nonetheless support photosynthesis on a planet’s surface. M dwarf stars are smaller and more parsimonious still and can steadily shine for hundreds of billions of years, but they shine so dimly that their habitable zones are very close-in, potentially subjecting planets there to powerful stellar flares and other dangerous effects. Being longer-lived than our sun yet not treacherously dim, K dwarfs appear to reside in the sweet spot of stellar superhabitability. Today some of these long-living stars may harbor potentially rocky super-Earths that are already several billion years older than our own solar system. Life could have had its genesis in these planetary systems long before our sun was born, flourishing and evolving for billions of years before even the first biomolecule emerged from the primordial soup on the young Earth. I am particularly fascinated by the possibility that a biosphere on these ancient worlds might be able to modify its global environment to further enhance habitability, as life on Earth has done. One prominent example is the Great Oxygenation Event of about 2.4 billion years ago, when substantial amounts of oxygen first began to accumulate in Earth’s atmosphere. The oxygen probably came from oceanic algae and eventually led to the evolution of more energy-intensive metabolisms, allowing creatures to have bigger, more durable and active bodies. This advancement was a crucial step toward life’s gradual emergence from Earth’s oceans to colonize the continents. If alien biospheres exhibit similar trends toward environmental enhancement, we might expect planets around long-lived stars to become somewhat more habitable as they age. To be superhabitable, exoplanets around small, long-lived stars would need to be more massive than Earth. That extra bulk would forestall two disasters most likely to befall rocky planets as they age. If our own Earth were located in the habitable zone of a small K dwarf, the planet’s interior would have grown cold long before the star expired, inhibiting habitability. For example, a planet’s internal heat drives volcanic eruptions and plate tectonics, processes that replenish and recycle atmospheric levels of the greenhouse gas carbon dioxide. Without those processes, a planet’s atmospheric CO ₂ would steadily decrease as rainfall washed the gas out of the air and into rocks. Ultimately the CO ₂ -dependent global greenhouse effect would grind to a halt, increasing the likelihood that an Earth-like planet would enter an uninhabitable “snowball” state in which all of its surface water freezes. Beyond the potential breakdown of a planet-warming greenhouse effect, the cooling interior of an aging rocky world could also cause the collapse of any protective René Heller – Better Than Earth lanetary magnetic field. Earth is shielded by a magnetic field generated by a spinning, convecting core of molten iron, which acts like a dynamo. The core remains liquefied because of leftover heat from the planet’s formation, as well as from the decay of radioactive isotopes. Once a rocky planet’s internal heat reservoir became exhausted, its core would solidify, the dynamo would cease, and the magnetic shield would fall, allowing cosmic radiation and stellar flares to erode the upper atmosphere and impinge on the surface. Consequently, old Earth-like planets would be expected to lose substantial portions of their atmospheres to space, and higher levels of damaging radiation could harm surface life. Rocky super-Earths as much as twice our planet’s size should age more gracefully than Earth, retaining their inner heat for much longer because of their significantly greater bulks. But planets larger than about three to five Earth masses may actually be too bulky for plate tectonics because the pressures and viscosities in their mantles become so high that they inhibit the required outward flow of heat. A rocky planet only two times the mass of Earth should still possess plate tectonics and could sustain its geologic cycles and magnetic field for several billion years longer than Earth could. Such a planet would also be about 25 percent larger in diameter than Earth, giving any organisms about 56 percent more surface area than our world on which to live. LIFE ON A SUPERHABITABLE SUPER-EARTH
WHAT WOULD A SUPERHABITABLE PLANET look like? Higher surface gravity would tend to give a middling super-Earth planet a slightly more substantial atmosphere than Earth’s, and its mountains would erode at a faster rate. In other words, such a planet would have relatively thicker air and a flatter surface. If oceans were present, the flattened planetary landscape could cause the water to pool in large numbers of shallow seas dotted with island chains rather than in great abyssal basins broken up by a few very large continents (see Figure 2). Just as biodiversity in Earth’s oceans is richest in shallow waters near coastlines, such an “archipelago world” might be enormously advantageous to life. Evolution might also proceed more quickly in isolated island ecosystems, potentially boosting biodiversity. Of course, lacking large continents, an archipelago world would potentially offer less total area than a René Heller – Better Than Earth Figure 2:
This example of a hypothetical superhabitable planet visualizes a range of effects that we expect these class of planets to show. First, it appears illuminated by an orange light source, which is owed to its host star being a so-called K dwarf star, a star about 80 percent the mass of the Sun. Second, the planet shows a somewhat opaque atmosphere, which is due to its increased surface gravity compared to Earth and its ability to draw down more gas). Third, while this planet has a large amount of water, its land surface is somewhat more dispersed than on Earth. It resembles an archipelagos world rather than a planet dominated by continents, like Earth. ontinental world for land-based life, which might reduce overall habitability. But not necessarily, especially given that a continent’s central regions could easily become a barren desert as a result of being far from temperate, humid ocean air. Furthermore, a planet’s habitable surface area can be dramatically influenced by the orientation of its spin axis with respect to its orbital plane around its star. Earth, as an example, has a spin-orbit axial tilt of about 23.4 degrees, giving rise to the seasons and smoothing out what would otherwise be extreme temperature differences between the warmer equatorial and colder polar regions. Compared with Earth, an archipelago world with a favorable spin-orbit alignment could have a warm equator as well as warm, ice-free poles and, by virtue of its larger size and larger surface area on its globe, would potentially boast even more life-suitable land than if it had large continents. Taken together, all these thoughts about the features important to habitability suggest that superhabitable worlds are slightly larger than Earth and have host stars somewhat smaller and dimmer than the sun. If correct, this conclusion is tremendously exciting for astronomers because across interstellar distances super-Earths orbiting small stars are much easier to detect and study than twins of our own Earth-sun system. So far statistics from exoplanet surveys suggest that super-Earths around small stars are substantially more abundant throughout our galaxy than Earth-sun analogues. Astronomers seem to have many more tantalizing places to hunt for life than previously believed. One of Kepler’s prize finds, the planet Kepler-186f, comes to mind. Announced in April 2014, this world is 11 percent larger in diameter than Earth and probably rocky, orbiting in the habitable zone of its M dwarf star. It is probably several billion years old, perhaps even older than Earth. It is about 500 lightyears away, placing it beyond the reach of current and near-future observations that could better constrain predictions of its habitability, but for all we know, it could be a superhabitable archipelago world. Closer superhabitable candidates orbiting nearby small stars could soon be discovered by various projects, most notably the European Space Agency’s PLATO mission, slated to launch by 2024. Such nearby systems could become prime targets for the James Webb Space Telescope, an observatory scheduled to launch in 2018, which will seek signs of life within the atmospheres of a small number of potentially superhabitable worlds. With considerable luck, we may all soon be able to point to a place in the sky where a more perfect world exists.
The author : René Heller is a postdoctoral fellow at the Origins Institute at McMaster University in Ontario and a member of the Canadian Astrobiology Training Program. His research focuses on the formation, orbital evolution, detection and habitability of extrasolar moons. He is informally known as the best German rice pudding cook in the world.René Heller – Better Than Earth MORE TO EXPLOREHabitable Climates: The Influence of Obliquity.
David S. Spiegel, Kristen Menou and Caleb A. Scharf in
Astrophysical Journal,
Vol. 691, No. 1, pages 596–610; January 20, 2009. DOI:10.1088/0004-637X/691/1/596
Exomoon Habitability Constrained by Illumination and Tidal Heating.
René Heller and Rory Barnes in
Astrobiology , Vol. 13, No. 1, pages 18–46; 2013. http://arxiv.org/abs/1209.5323
Habitable Zone Lifetimes of Exoplanets around Main Sequence Stars.
Andrew J. Rushby, Mark W. Claire, Hugh Osborn and Andrew J. Watson in
Astrobiology , Vol. 13, No. 9, pages 833–849; September 18, 2013. DOI:10.1089/ast.2012.0938.
Superhabitable Worlds.
René Heller and John Armstrong in
Astrobiology , Vol. 14, No. 1, pages 50-66; January 16, 2014. http://arxiv.org/abs/1401.2392 .2. EXTRASOLARE MONDE – SCH ¨ONE NEUE WELTEN? (Heller 2013) 402
B.2 Extrasolare Monde – sch¨one neue Welten? (Heller 2013) er Nachweis von Monden um extrasolare Planeten, sogenannter Exomonde, ist nunmehr möglich geworden, vor allem durch das Weltraumteleskop
Kepler . Zahlreiche Studien der vergangenen Jahre belegen, dass die Detektion von Exomonden unmittelbar bevorsteht, vorausgesetzt, Satelliten von ungefähr der Größe des Mars existieren außerhalb des Sonnensystems. Angesichts der Schwierigkeit des Unterfangens jedoch stellen sich berechtigte Fragen danach, was wir überhaupt über Exomonde lernen können. Welche Beobachtungsgrößen gibt es und auf welche Eigenschaften von Exomonden werden sie uns schließen lassen?
Noch bevor auch nur eines dieser Objekte nachgewiesen wurde, wissen wir, dass sich diese Welten grundsätzlich verschieden darstellen werden von den zahlreichen bisher erforschten Exoplaneten. Ihr Tag-und-Nacht-Muster unterscheidet sich von dem auf Planeten durch das Wechselspiel von planetarer und stellar Bestrahlung sowie durch
Bedeckungen des
Sterns durch den Planeten. Sowohl planetares Licht als auch Eklipsen haben Auswirkungen auf das Mondklima. Darüber hinaus wirken
Gezeiten auf eine Kopplung der Rotation von Monden an die
Orbitperiode um den Planeten hin. Das heißt, in der habitablen Zone um massearme Sterne, in der erdähnliche Planeten dem Stern stets die gleiche
Seite zuwenden, können erdähnliche Monde durch Gezeiteneffekte verhältnismäßig kurze Tage erfahren. Ähnlich wie Titan im Orbit um Saturn kann eine signifikante Schräglage des Planeten außerdem Jahreszeiten auf Monden hervorrufen, solange diese den Planeten am Äquator umrunden. Diese Effekte könnten sich positiv auf die Stabilität eventueller Atmosphären von Exomonden auswirken und ihnen sogar Vorteile gegenüber terrestrischen Exoplaneten bringen. In engen Orbits eines Satelliten um seinen Planeten spielt darüber hinaus die Gezeitenheizung eine entscheidende Rolle für das Mondklima, eventuelle Plattentektonik und Vulkanismus. Außerdem können je nach dem, ob ein Satellit einen Gasplaneten oder terrestrischen Planeten umrundet, bestimmte Entstehungs-Szenarien und somit materielle Zusammensetzungen angenommen bzw. verworfen werden.
Angesichts der über 2000 Kandidaten für
Planeten in den
Kepler -Daten, von denen sich mehr als 50 meist Neptun- bis Jupiter-große Objekte in der HZ befinden, erleben wir mit Studien zur Physik von Exomonden gerade die Geburt eines neuen Forschungszweiges. Schauen wir uns also an, was er uns bietet!
Entstehung von Monden
Zunächst sollten wir uns darüber im Klaren sein, dass selbst die massereichsten Monde im Sonnensystem, Ganymed und Titan, Leichtgewichte sind im Vergleich zur Erde. Ihre Massen reichen nur an den vierzigsten Teil der Erdmasse. Monde, die mit
Kepler in absehbarer Zeit nachgewiesen werden können, müssten jedoch mindestens ein Viertel der Masse der Erde haben, also um mindestens den Faktor zehn schwerer sein als Ganymed. Forscher suchen somit nach etwas, das wir aus unserem Sonnensystem nicht kennen, was kaum eine Demotivation bedeuten kann angesichts der zuvor unvorstellbaren Vielfalt an Planeten, die man mittlerweile fand.
Basierend auf den Bahnparametern, gruppieren Astronomen die Monde im Sonnensystem in zwei Klassen: reguläre und irreguläre Monde. Erstere weisen fast kreisrunde, meist enge Orbits auf und umrunden den Planeten in seiner Äquatorebene und zwar in der gleichen Richtung wie der Planet sich dreht. Man geht davon aus, dass sie sich in-situ aus einer zirkumplanetaren Scheibe aus Eis und Gesteinen geformt haben. Die meisten der irregulären Monde hingegen sind wahrscheinlich zum größten Teil eingefangen worden. Prominentes Beispiel hierfür ist Neptuns größter Mond Triton. Sein Orbit ist um 157° gegen Neptuns Äquator geneigt, er umläuft den Gasplaneten also retrograd. Wahrscheinlich hat Neptun Triton dereinst aus einem passierenden planetaren Binärsystem herausgerissen, während Tritons ehemaliger Begleiter aus dem Sonnensystem katapultiert wurde [1].
Kritisch für das Vorkommen von massiven, regulären Exomonden sind Ergebnisse zweier Forschergruppen aus Boulder (USA) und Tokio (Japan) aus den vergangenen Jahren. Erstere fand in ihren theoretischen Untersuchungen, dass die Masse des für die Bildung von Monden zur Verfügung stehenden Materials um Gasplaneten ungefähr ein Fünftausendstel der Planetenmasse ausmacht [2]. Diesem Anteil entsprechen z.B. die Summe der Massen der Galileischen Monde im Vergleich zur Masse Jupiters und Titans Masse im Vergleich zu Saturn. Auf der Suche nach einem Mond mit der Masse des Mars müsste man also Planeten untersuchen, die mindestens doppelt so schwer sind wir Jupiter. Für Monde mit der Masse der Erde müsste der Planet bereits die Masse eines Braunen Zwerges haben.
Verantwortlich für diese Massenbarriere während der Entstehung sei die Konkurrenz zweier Prozesse. Während aus der zirkumstellaren Scheibe Material auf den Gasplaneten einfällt, welches das Wachstum Entwurf für Sterne und Weltraum (Potsdam, 12. Juli 2012)
Extrasolare Monde – Schöne Neue Welten? von René Heller
Während mittlerweile knapp 800 Planeten außerhalb des Sonnensystems gefunden wurden, steht der Nachweis von extrasolaren Monden noch aus. Aktuelle Studien zeigen, dass ihre Detektion mit der heutigen Technologie zum ersten Mal möglich ist. Für diese Exomonde sagen Wissenschaftler nun bisher unbekannte astrophysikalische Phänomene voraus.
Monde um extrasolare Planeten könnten lebensfreundliche Bedingungen aufweisen. Ihre Klimaeigenschaften werden u.a. von der Bestrahlung des Sterns und des Planeten sowie von der Gezeitenheizung abhängen. er Trabanten fördert, werden die massivsten Monde von der Reibung mit dem zirkumplanetaren Gas in immer engere Bahnen um den Planeten getriebenen, bis sie schließlich unter enormen Gezeitenkräften zerrissen werden und eventuell auf den Planeten herabbröseln.
In einer Erweiterung dieser Theorie konnte die Tokioter Gruppe allerdings nachweisen, dass auch marsähnliche Monde um jupiterähnliche Planeten entstehen können [3]. Sogar Satelliten mit der Masse der Erde könnten vorkommen, seien aber äußerst selten.
Über das Einfangen von Objekten der Masse des Mars oder der Erde in einen stabilen Orbit um einen Gasplaneten gibt es zahlreiche Untersuchungen, die belegen, dass solche Prozesse unter dynamischen Aspekten durchaus möglich sind – allein wie wahrscheinlich und üblich solche Vorgänge sind, vermögen sie nicht aufzuzeigen. Die Frage nach solchen Wahrscheinlichkeiten werden wohl erst die Detektionen oder gegebenenfalls die Nicht-Detektionen belegen.
Detektion von Exomonden
Bisher gibt es keine bestätigte Beobachtung eines extrasolaren Mondes. Zum einen liegt das an der zu erwartenden Rarität dieser Objekte, zum anderen fehlte bis vor kurzem die dafür erforderlichen Instrumentierung. Denn die zu erwartenden beobachterischen Effekte sind nicht nur extrem selten, sondern auch so winzig, dass eine erdgebundene Suche nach Exomonden um zufällig ausgewählte Planeten aussichtslos wäre. Durch den erfolgreichen Betrieb des
Kepler -Teleskops seit 2009 wurde diese Hürde just genommen und Astronomen begeben sich nun auf die Suche nach den möglicherweise in den bereits gesammelten Daten versteckten Hinweisen auf Exomonde.
Erst Anfang dieses Jahres initiierte ein Team um den Astrophysiker David Kipping vom Harvard-Smithsonian Center for Astrophysics die erste dezidierte Suche nach Exomonden in den
Kepler -Daten. Ihr Programm trägt den Namen “Hunt for Exomoons with Kepler”, kurz HEK [4]. In einigen Artikeln hatten Kipping und seine Kollegen zuvor die theoretischen Grundlagen für den Nachweis von Exomonden gelegt. In diesen Arbeiten und in Studien anderer Forscher konnten mittlerweile mehrere Effekte bestimmt werden, welche die Detektion von Exomonden möglich machen. Einige von ihnen werden direkt durch einen Mond hervorgerufen, andere bestehen aus kleinen Abweichungen des Planeten von seiner Bahn. Beide Kategorien jedoch vereint, dass sie nur für eine bestimmte Klasse von Planeten auftreten: die Transitplaneten. Wagen wir einen kurzen Exkurs zu diesen Objekten, auf dass wir die Effekte ihrer Monde besser verstehen können!
Transitplaneten ziehen, von der Erde aus gesehen, einmal im Laufe ihres Orbits um den Stern vor diesem vorbei und verdunkeln ihn dabei geringfügig. Da ein Planet während des Transits, also während wir seine unbeleuchtete Seite sehen, in guter Näherung schwarz ist im Vergleich zu seinem Stern, kann man die Stärke des Helligkeitsverlusts gut dadurch abschätzen, dass man die Fläche der Planetenscheibe πR durch die Fläche der Sternscheibe πR teilt, also R /R berechnet. Könnte man die Verdunklung der Sonne durch Jupiter betrachten, befände man sich also außerhalb der Bahn Jupiters um das Zentralgestirn, so würde man eine Helligkeitseinbuße der Sonne von ungefähr 0.988% beobachten. Die Verdunklung durch die Erde betrüge 0.0084%, also weniger als ein Zehntausendstel. Mittlerweile kennt man 230 bestätigte Transitplaneten in 196 Sternsystemen, nicht zuletzt Dank der bis vor kurzem unerreichten Anzahl simultan überwachter Sterne des
Kepler -Teleskops, verbunden mit seiner enormen Präzision. Über 2000 weitere
Kepler -Kandidaten harren übrigens ihrer Bestätigung durch weiterführende Analysen der Lichtkurven oder unabhängige Beobachtungen.
Das Haupterkennungsmerkmal für die automatische Detektions-Software der Planetensucher ist die präzise Periodizität der Bedeckungen. Sie entspricht der Orbit-Periode des Transitplaneten um den Stern. Für den Fall, dass der Planet seinen Stern ohne Mond umkreist und vorausgesetzt, dass die Bahnstörungen durch etwaige andere Planeten ausreichend gering sind, ist die Transitperiode konstant. Wird der Planet jedoch von einem Mond begleitet, so verursacht die gravitative Wechselwirkung ein Torkeln des Planeten, denn beide Körper umrunden dann ihren gemeinsamen Schwerpunkt. Die Auslenkung von diesem Massenzentrum erfolgt für die beiden Körper in entgegengesetzter Richtung und wird durch das Hebelgesetzt alsRené Heller: Extrasolare Monde –
Schöne Neue Welten? 2
Box 1: Transit Timing Variation (TTV)
Auf dem nebenstehenden Bild zeigt die blaue, lockige Kurve die Bahn eines Mondes, während der Orbit des Planeten durch die schwarze Ellipse gekennzeichnet ist. Das Duett aus Planet und Mond umrundet gemeinsam den orangefarbenen Stern. Die Länge des Vektors s vom Mond zum Planeten ist dabei stark vergrößert dargestellt, damit das Torkeln des Satelliten sichtbar wird. Der vergrößerte Ausschnitt zeigt einen Zoom in den Bereich um den Zeitpunkt, zu dem das Planet-Mond-System von der Erde aus gesehen vor dem Stern vorbeizieht. Offensichtlich befindet sich der Mond am Ende seines gemeinsamen Orbits mit dem Planeten um den Stern nicht an der gleichen Position wie zum Start der Simulation. Einen entgegengesetzten räumlichen Versatz erfährt dabei auch der Planet. Allerdings ist dieser noch um einiges kleiner als der des Mondes und so ist er auf dieser Abbildung nicht erkennbar. Wir können dennoch aus der Abbildung erahnen (oder kompliziert genau berechnen), dass die Periode des Transitzeitpunkts des Planeten von Orbit zu Orbit kleinen Schwankungen unterworfen sein wird. Diese Abweichung bezeichnet man als Transit Timing Variation (TTV). p M m = d m d p beschrieben, wobei M p und M m jeweils die Masse des Planeten und des Mondes bezeichnen und d p und d m die Abstände der beiden Objekte vom Massenzentrum. Die Auslenkung des Planeten wird also typischerweise viel kleiner sein als die des Mondes. Je nach dem, in welcher Konstellation das Planet-Mond-Paar von der Erde aus gesehen vor dem Stern entlangzieht, wird der Planet mal in Richtung seiner Bewegung um den Stern ausgelenkt sein, mal in die entgegengesetzte Richtung. Im erste Fall passiert der Transit etwas früher als im Durchschnitt, im zweiten Fall etwas später. Diese Variationen sind je nach den Massen- und Abstandsverhältnissen in dem Dreikörper-System aus Stern, Planet und Mond in der Größenordnung von Sekundenbruchteilen bis zu wenigen Minuten. Der englische Ausdruck für dieses Phänomen lautet “transit timing variation” (TTV, siehe Box 1). Bereits Ende der 1990er Jahre wurde vorhergesagt, dass für die Amplitude der zeitlichen Variation der Transitperiode ∆ TTV ∼ M m × a pm gilt, wobei a pm die große Halbachse der Mondbahn um den Planeten ist. Diese Proportionalität allein lässt also nicht eindeutig auf jeweils den Abstand zwischen Planet und Mond die Masse des Mondes schließen. Eine weitere Beobachtungsgröße muss her. Im Rahmen seiner Dissertation am University College London konnte David Kipping einen neuen Effekt ausfindig machen, den der sogenannten “transit duration variation” (TDV). Es handelt sich dabei also um die Variation der Dauer des Planetentransits. Diese Schwankung kann zweierlei Ursprung haben. Zum einen variiert neben der Auslenkung auch die tangentiale Geschwindigkeitskomponente des Planeten. Je nach dem, in welche Richtung der Mond sich während des Transits gerade um den Planeten bewegt, wird der Planet eine zusätzliche Geschwindigkeit in Richtung seines Orbits um den Stern erfahren oder ein wenig langsamer vor der stellaren Scheiben entlang ziehen. Dadurch dauert der Transit jeweils etwas kürzer oder länger als im Durchschnitt. Da die Variation der Geschwindigkeitsrichtung der Auslöser für diese Sorte von TDV verantwortlich ist, wird diese “TDV-V” abgekürzt, wobei das letzte “V” für “velocity”, also die Geschwindigkeit steht.
Zum anderen kann der Orbit des Mondes um seinen Planeten relativ zum Orbit des Planeten um den Stern gekippt sein. Dadurch erfährt der Planet während seiner Transits eine Auslenkung aus der mittleren Bahnebene und zieht mal näher zur Mitte der Sternscheibe, mal eher am Rand der Sternscheibe entlang. Näher zur Mitte ist der Weg über die Sternscheibe länger, in der Mitte selbst entspricht er einfach ihrem Winkeldurchmesser. Somit entsteht die sogenannte “TDV-TIP”, wobei der Suffix “TIP” für das englische “transit impact parameter” steht, also den Abstand des Planetentransits von der Sternmitte.
Die Schwankungen von beiden TDV-Effekten sind in der gleichen Größenordnung wie die der TTV. Phänomenalerweise besteht für die Amplitude der TDV jedoch die Proportionalität ∆ TDV ∼ M m √ a pm ,so dass durch simultane Messung von Δ TTV und Δ TDV die mathematische Entartung von M m und a pm gelöst werden kann. In den kompletten mathematischen Ausdrücken für die beiden Observablen stecken die Masse des Planeten und des Sterns sowie die Orbitperiode des Planet-Mond-Duetts um den Stern. Diese Parameter sind durch zeitaufgelöste Spektroskopie allesamt bestimmbar, so dass schließlich tatsächlich die Masse des Mondes sowie sein Abstand zum Planeten bestimmt werden kann. Die bisher beschriebenen Effekte sind allesamt indirekter Natur, insofern als nicht der Mond, sondern der Planet beobachtet und auf die Existenz des Mondes geschlossen wird. Natürlich kann man sich auch vorstellen, dass der Transit eines Mondes direkt beobachtet wird.
Kepler wurde schließlich zum Zwecke der Detektion von erdgroßen Planeten gestartet und konnte bereits deutlich kleinere Objekte mit der Größe des Mars nachweisen. Ein marsgroßer Mond, wie er um einen jupiterähnlichen Planeten durchaus vorkommen mag, wäre somit direkt nachweisbar. Der entscheidende Gewinn einer solchen Beobachtung läge in der Messung des Mondradius. Zusammen mit der Masse des Mondes ließe sich dann auf seine Dichte und somit die Zusammensetzung schließen.
Mehrere Untersuchungen der Heidelberger Forscherin Lisa Kaltenegger und Kollegen zeigen zudem, dass die spektralen Signaturen von Leben auf Exomonden, sogenannter Biomarker bzw. Bioindikatoren (letztere können auch abiotisch produziert werden), mit Weltraum-Teleskopen der nächsten Generation nachgewiesen werden können. Hierzu zählen molekularer Sauerstoff (O ), Ozon (O ,) Methan (CH ) und Distickstoffmonoxid (auch bekannt als Lachgas, N O) bzw. Kohlendioxid (CO ) und Wasserdampf (H O). Die dafür notwendigen Messungen jedoch wären auf Grund des notwendigen Signal-zu-Rausch-Verhältnisses nur für Monde um M-Sterne in unserer kosmischen Nachbarschaft nachweisbar. Die Nähe zur Sonne ist dabei wichtig für eine ausreichende Lichtausbeute dieser leuchtschwachen Objekte. Da sich ihre habitable Zone zudem sehr nah am Stern befindet, hat ein Planet-Mond-Duett eine kurze Umlaufperiode um den Stern und kann innerhalb weniger Erdjahre ausreichend viele Transits zeigen, deren Signal sich aufsummieren lässt.
Während direkte spektroskopische Beobachtungen von Exomonden noch die ein oder andere Dekade unzugänglich sein werden, könnten wir aus den oben aufgezählten Bahnparametern des Systems aus einem Stern, einem Planeten und einem Mond bereits eine Menge über die Bedingungen auf der Mondoberfläche erschließen. Dabei gilt es verschiedene Effekte zu berücksichtigen, welche für die Charakterisierung von Exoplaneten keine Rolle spielen.
Einstrahlung von Stern und Planet
Ein wesentlicher Unterschied zwischen einem frei rotierenden, erdgroßen Planeten und einem Mond gleicher Größe besteht darin, dass der Mond von zwei bedeutenden Lichtquellen beschienen wird. Auf der Erde, also auf einem Planeten stehend, kennen wir das umgekehrte Phänomen, dass wir in einer klaren Nacht bei Vollmond sogar lesen können. Man stelle sich vor wie hell eine Nacht auf Jupiters Mond Europa sein mag, wenn um Mitternacht der gleißende helle Gasriese im Zenit steht!
Mein Kollege Rory Barnes vom Astrobiology Institute der University of Washington, Seattle, und ich haben uns daran gemacht, die Einstrahlungseffekte eines Planeten auf seine Monde zu ermitteln, mit Fokus auf potenzielle Monde um Exoplaneten. Dabei haben wir sowohl das vom Planeten auf den Mond reflektierte Sternenlicht als auch die thermische Strahlung des Planeten berücksichtigt. Durch Beobachtungen der Monde im Sonnensystem und durch die Theorie der Gezeiten gerechtfertigte Prämisse unseres Modells ist, dass der Mond sich zum Planeten im sogenannten “tidal locking” befindet. Er wendet seinem Planeten also stets die gleiche Hemisphäre zu.
Unternehmen wir nun im Geiste eine Reise auf solche einen Exomond, der einen Gasplaneten umrundet! Wir stellen uns vor, dass auf dem Mond gerade Mitternacht herrscht und dass wir am subplanetaren Punkt auf dem Mond stehen (siehe Abbildung auf Seite 4). Der Planet steht also genau im Zenit, während der Stern sich gerade unter unseren Füßen befindet, auf der Rückseite des Mondes. Stern, Mond und Planet bilden eine Linie. Zwar ist nach den Begriffen, wie wir sie auf der Erde verwenden, gerade Mitternacht auf dem Mond, doch schauen wir hoch in den Zenit, so sehen wir die voll beleuchtete René Heller: Extrasolare Monde –
Schöne Neue Welten? 3cheibe des Planeten. Je nach dem, wie weit unser Mond vom Planeten entfernt ist, welchen Radius der Planet hat und welchen Anteil α p des einfallenden Sternenlichts der Planet reflektiert, wird die einfallende Strahlung eine Leistung zwischen ungefähr einem und hundert Watt pro Quadratmeter haben. Zum Vergleich: Die Sonne strahlt mit einer Leistung von ca. 1400 W/m auf die Erde, während der Vollmond ungefähr 0.01 W/m auf die Erde reflektiert. Das reflektierte Licht eines Exoplaneten mit einer Leistung von einigen zehn Watt pro Quadratmeter kann also durchaus die Nacht buchstäblich zum Tag machen! Zusätzlich zu der Spiegelung des Sternenlichts gibt der Planet thermische Strahlung an den Mond ab. Für realistische Albedos des Planeten fanden wir, dass ihr Beitrag um den Faktor zehn kleiner als das reflektierte Licht und damit meist vernachlässigbar ist. Die Abbildung oben veranschaulicht die interessante Überlagerung von stellar und planetarer Bestrahlung mit zwei Momentaufnahmen aus dem Sonnensystem. Auf dem subplanetaren Punkt des Enceladus im linken Bild – markiert durch ein Kreuz – herrscht gerade die beschriebene Mitternacht. Noch verquerer wird es, wenn wir nun versuchen, uns den Tagesablauf dem Mond vorzustellen. Die Einstrahlung vom Planeten hängt nämlich von dessen Phase ab. Um Mitternacht scheint der Planet über dem subplanetaren Punkt auf dem Mond am stärksten. Danach nehmen seine Phase und Einstrahlungsintensität ab, bis bei Sonnenaufgang nur noch die dem Stern zugewandte Hälfte scheint, sozusagen “Halbplanet” in Anlehnung an den von der Erde aus betrachteten Halbmond. Nun nähern wir uns in Gedanken einem Spektakel. Kurz vor der Mittagszeit nämlich wird es auf einmal stockdunkel. Der Planet schiebt sich täglich zur gleichen Uhrzeit vor den Stern und da wir uns nun über der unbestrahlten Hemisphäre des Planeten befinden, ist es tatsächlich duster, denn die thermische Strahlung des Planeten kann, wie erwähnt, vernachlässigt werden. Die Rückseite des Planeten schneidet derweil einen schwarzen Kreis aus dem Himmel aus – und das zur Mittagszeit! Während dieser Minuten bis wenige Stunden dauernden Bedeckung dürften die Temperaturen auf dem Mond spürbar sinken. Kurz danach geht der gleißend helle Stern hinter dem Planeten wieder auf und bei Sonnenuntergang erscheint nun die andere Hälfte des Planeten beleuchtet und nimmt weiter zu bis Mitternacht. Es sei bemerkt, dass die Taglänge dabei genau der Orbitperiode des Mondes um seinen Planeten entspricht. Die Tage auf Jupiters Galileischen Monden entsprechen daher ungefähr 1,8 (Io), 3,6 (Europa), 7,2 (Ganymed) und 16,7 (Callisto) Erdtagen und Titans Tag dauert 15,9 Erdtage.
Während dieses hypothetischen Vorgangs haben wir angenommen, dass sich das Planet-Mond-Paar nicht nennenswert um den Stern bewegt. Das Bestrahlungsverhalten wird jedoch komplexer, wenn das Massenzentrum von Planet und Mond einen exzentrischen Orbit um den Stern beschreibt (siehe Box 1). Dann hängt die Einstrahlung von einem im Laufe eines Jahres variierenden Abstand zum Stern ab.
In unserem Artikel haben Rory Barnes und ich über das hier geschilderte Szenario hinaus Fälle erwogen, in denen der Orbit des Mondes gegen den Orbit, welchen das Planet-Mond-System um den Stern beschreibt, um den Winkel i geneigt ist. Im Sonnensystem gibt das Saturn-Titan-System hierfür ein schönes Beispiel. Die Rotationsachse des Planeten ist um 26.7° gegen den Orbit um die Sonne geneigt. Saturn erfährt also über einen Zeitraum von 29 ½ Erdjahren Jahreszeiten. Titan umrundet Saturn in dessen Äquatorebene und erfährt somit auch Jahreszeiten. Durch diese starke Inklination kommt es von Titan aus gesehen nicht wie in unserem oben betrachteten Falle, einmal pro Planet-Mond-Orbit zu Eklipsen der Sonne hinter Saturn. Über die meiste Zeit des Jahres geht die Sonne nämlich zur Mittagszeit unter oder über Saturn René Heller: Extrasolare Monde –
Schöne Neue Welten? 4
Beispiele für das kniffelige Wechselspiel der Mond-Bestrahlung durch Stern und Planet finden wir im Sonnensystem. Im linken Bild erscheint Saturns Mond Enceladus von links beleuchtet durch die Sonne und von rechts durch das reflektierte Sonnenlicht seines Planeten. Man achte auf die verschiedenen Farben und Intensitäten der beiden Beleuchtungen! Das Kreuz deutet auf den subplanetaren Punkt des Mondes. Das rechte Bild zeigt Jupiters Monde Europa und Io. Auf Ios nördlicher Hemisphäre speit ein gewaltiger Vulkan. Beide Monde erhalten von links direkte Sonneneinstrahlung, während Io von rechts auch von Jupiters Reflexion erhellt wird. Die Monde sind ca. 800.000 km voneinander entfernt und erscheinen nur in dieser Projektion ihrer momentanen Orbitgeometrie nah beieinander. (Bildnachweis: Courtesy NASA/JPL-Caltech) inweg. Lediglich um den Frühlings- und um den Herbstpunkt des Saturns, also wenn die Sonne seine Äquatorebene durchquert, wird sie einmal in Titans Orbit um Saturn bedeckt.
Gezeitenheizung
Stockdunkle Mittagszeit, hell-erleuchtete Nacht, Tage mit Längen von mehreren Erdtagen – was für bizarre Welten wir uns da vorstellen! Natürlich sind wir noch nicht am Ende. Betrachten wir noch einen Aspekt, nämlich den der Gezeitenheizung! Für Monde in engen Orbits, d.h. mit Abständen von weniger als ungefähr zehn Planetenradien, wird diese Energiequelle signifikant. Prominentestes Beispiel aus dem Sonnensystem ist hierfür Io. In der Abbildung auf Seite 4 ist einer seiner zahlreichen aktiven Vulkane zu sehen. Io gilt als der geologisch aktivste Körper des Sonnensystems. Während auf der Erde ein Wärmefluss von ca. 0.08 W/m aus dem Inneren herrscht, hauptsächlich gespeist durch radioaktive Zerfälle im Kern und nur zum kleinen Teil durch Gezeitenheizung vom Mond, so emittiert Io satte 2 W/m aus seiner Gezeitenheizung. Die Folgen sind globaler Vulkanismus, das Ausströmen von Gasen mit einem durchschnittlichen Massenverlust von 1 Tonne pro Sekunde und wahrscheinlich ein unterirdischer, über 1200°C heißer Magma-Ozean aus diversen Schwefel- und Eisenverbindungen. Auf Europa, der seinen Orbit um Jupiter weiter außen zieht, herrscht verhältnismäßig geringe Gezeitenheizung. Beobachtungen der NASA-Sonde
Galileo aus den 1990er Jahren deuten jedoch darauf hin, dass ihre Leistung ausreicht, unter der gefrorenen Oberfläche des Mondes einen gewaltigen Ozean aus Wasser flüssig zu halten. In Gedanken an die sogenannten Schwarzen Raucher (englisch “black smoker”) am Grunde der Tiefsee auf der Erde, in deren Umgebung man komplexe, von der Erdoberfläche und dem Sonnenlicht unabhängige Ökosysteme gefunden hat, lassen uns
Galileos
Befunde an die Möglichkeit von Leben auf Europa denken.
Extrasolare Monde mit solch einer moderaten Gezeitenheizung können bewohnbar sein oder nicht, wir würden das beim besten Willen nicht voraussagen können. Mit Sicherheit können wir jedoch sagen, dass Exomonde mit Gezeitenheizung vergleichbar der des Io nicht bewohnbar sind. Selbst wenn ein Exomond mit seinem Planeten in der habitablen Zone um den Stern zieht, würde das durch Vulkanismus freigesetzte Kohlendioxid den Mond in ein Treibhaus verwandeln, siehe Venus. Das von Rory Barnes und mir entwickelte Modell, welches die Einflüsse von stellarer sowie planetarer Einstrahlung mit der Gezeitenheizung koppelt, soll helfen, kostbare Beobachtungszeit an den Großteleskopen sinnvoll einzuteilen und nur den aussichtsreichen Objekten Priorität zu geben. Schließlich werden Monde, die eventuell bewohnbare oder gar bewohnte Oberflächen haben und von einer erdähnlichen Atmosphäre umgeben sein könnten, die attraktivsten Ziele darbieten.
Synthese von Einstrahlung und Gezeitenheizung
Unser Modell, das Bestrahlung und Gezeitenheizung auf Monden berücksichtigt, haben wir in ein Computerprogramm geschrieben und konnten damit den Energiefluss auf Monden simulieren. Betrachten wir als Beispiel einen hypothetischen Exomond um den neulich entdeckten Exoplaneten Kepler-22b, der seinen sonnenähnlichen Stern in der habitablen Zone umrundet! Der Planet weist einen Radius R p von ungefähr 2,38 Erdradien auf und ist daher in Masse und Struktur vermutlich ähnlich Uranus oder Neptun. Für unsere Simulation haben wir angenommen, dass der Mond Kepler-22b in 20 R p Entfernung auf einer leicht exzentrischen Bahn umrundet, nämlich mit e = 0,05. Dies sollte zu starker Gezeitenheizung führen. Außerdem legten wir den Orbit des Mondes in die gleiche Ebene wie den Orbit um den Stern, so dass Eklipsen auftreten. Die Abbildung oben zeigt den über einen Orbit (also ein “Jahr” von Kepler-22b) gemittelte Energiefluss auf der Oberfläche des Mondes. Die Ordinate vermittelt geographische Breite ϕ , die Abszisse geographische Länge θ , jeweils gemessen vom subplanetaren Punkt in der Mitte. Über ϕ = 0° = θ steht also Kepler-22b im Zenit. Die Grafik ähnelt somit einer Weltkarte, nur dass hier die Kontinente, Meere und andere Oberflächeneigenschaften unterlassen sind und stattdessen der Energiefluss aufgetragen ist. Dabei beobachten wir ein spannendes Phänomen: der subplanetare Punkt ist der “kühlste” Ort entlang des Äquators. Das rührt von den Eklipsen her, denn auf der dem Planeten abgewandten Seite des Mondes sind Eklipsen nie beobachtbar und verursachen somit auch keine Reduktion der stellaren Einstrahlung. Für Orbits mit Inklinationen von wenigen Grad fanden wir übrigens das Gegenteil. Dort wurde auf Grund der zusätzlichen Einstrahlung vom Planeten der subplanetare Punkt der “heißeste” Ort der gesamten Oberfläche, denn Eklipsen traten nun sehr selten im Laufe des Jahres auf. Auch wenn der Effekt der Gezeiten in dieser Abbildung nicht ohne Weiteres abzulesen ist, so sei erwähnt, dass Gezeitenheizung hier gleichmäßig verteilt 42W/m beiträgt. Ohne Gezeitenheizung würden die Farben bei gleichbleibender Farbskale am Äquator also nur bis ins Orange gehen und nicht ins Rot, während sie an den Polen den Wert 0W/m erreichen würden, was sie hier – zugegeben: schwer erkennbar – nicht tun. Atmosphären und die habitable Zone
Die Atmosphäre ist von grundlegender Bedeutung für die Oberflächeneigenschaften René Heller: Extrasolare Monde –
Schöne Neue Welten? 5
Durchschnittlicher Energiefluss über der Atmosphäre eines hypothetischen Mondes um Kepler-22b. Die Konturen geben konstante Flüsse in Einheiten von W/m an (siehe Farbleiste). Die Einbuchtung um den subplanetaren Punkt, also bei ϕ = 0° = θ resultiert aus den Eklipsen. es Körpers, nicht zuletzt für seine eventuelle Bewohnbarkeit. Ihrer Wärmekapazität führt zu einer zeitlichen Latenz in der Abkühlung der Oberfläche bei Nacht und der Erwärmung bei Tag. Auch bestimmt sie den Wärmetransport von der bestrahlten auf die unbestrahlte Seite des Körpers, bietet ggf. Schutz gegen schädigende Strahlung aus dem All – so schützt auf der Erde die Ozon-Schicht gegen UV-Strahlung der Sonne – und ist schließlich das Medium für den globalen Gas- und Wasseraustausch. Darüber hinaus wirkt sie wie ein Treibhaus. Auf der Erde z.B. beträgt die durchschnittliche Oberflächentemperatur +16°C, obwohl sich aus der Berechnung des thermischen Gleichgewichts lediglich –18°C ergeben (siehe Box 2). Ausschlaggebend dafür ist der Treibhauseffekt, der auch zur Definition der habitablen Zone berücksichtigt wird. Dieser Abstandsbereich um einen Stern definiert einen Orbit-Gürtel, genauer eine Sphäre, innerhalb der ein terrestrischer Planet mit einer Atmosphäre ähnlich der der Erde flüssiges Wasser auf seiner Oberfläche führen würde. Da Wasser als notwendige Bedingung für Leben angesehen wird, nennt man diesen Bereich eben die bewohnbare Zone. Ihr innerer Rand beschreibt den Abstand zum Stern, in dem eine erdähnliche Atmosphäre heiß und mit Wasserdampf gesättigt wäre und außerdem ihre Tropopause verlöre, so dass das Wasser hoch in die Stratosphäre steigen könnte, wo es dann durch die energiereiche Strahlung in Wasserstoff und Sauerstoff zerlegt würde. Der Wasserstoff würde ins All entweichen, der Sauerstoff zurückbleiben und der Planet würde sein Wasservorrat verlieren, also austrocknen und unbewohnbar werden. Diesen Effekt nennen wir ein “runaway greenhouse”, also einen irreversibel zunehmenden Treibhauseffekt. Im Sonnensystem gibt es nur einen Mond innerhalb der habitablen Zone, nämlich den Erdmond. Dieser allerdings ist zu massearm, als dass seine Gravitation im Stande wäre eine signifikante Atmosphäre zu binden. Überhaupt gibt es anscheinend nur zwei Monde im Sonnensystem, die eine nennenswerte Gashülle tragen.
Auf Saturns Mond Titan, ca. 9.5 AU von der Sonne entfernt, könnte ein Mensch dank der dichten Stickstoff-Atmosphäre, die auf der Oberfläche einen Druck von ca. 1.5 bar erzeugt, mit Flügeln ähnlich denen von Vögeln fliegen, auch dank Titans niedriger Oberflächengravitation. Auf Spaziergängen wäre ein Druckausgleich nicht nötig, man benötigte lediglich einen Temperaturanzug und eine Atemmaske. Zur Bespaßung aller Beteiligten könnte man noch kurz die Atemmaske absetzen und ins Feuerzeug ausatmen. Der expirierte Sauerstoff ergäbe mit dem Methan der Atmosphäre ein explosives Gemisch.
Am Rande des Sonnensystems wird Neptuns Mond Triton von einer hauchdünnen Atmosphäre aus Stickstoff, Kohlenmonoxid und Methan umgeben. Wie die Sonde Voyager 2 herausfand, bilden sich in ihr bei -230°C zarte Wolken aus Stickstoff.
Was lernen wir uns zukünftigen Beobachtungen?
Der Nachweis von Wolken auf Exomonden ist allerdings noch eine Strophe in der Zukunftsmusik. Vorerst müssen wir uns damit begnügen, die grundlegenden Parameter von Exomonden zu bestimmen. Diese werden sich jedoch nur im Verhältnis zu anderen Parametern, nämlich denen des Sterns und des Planeten, ermitteln lassen, so die Aufgabe letztlich darin besteht, mindestens drei Körper zu charakterisieren – und eventuell weitere Planeten oder Monde in dem System.
Die stellare Leuchtkraft L s ließe sich entweder aus der Oberflächentemperatur T s des Sterns und seinem Radius R s bestimmen oder aus der Parallaxe und Magnitude des Sterns. Für die erste Methode müssten hochaufgelöste Spektren verfügbar sein und seismische Daten, z.B. aus der präzisen Kepler -Photometrie, welche Rückschlüsse auf den Radius erlauben. Die zweite Methode wäre nur auf Sterne in der Sonnenumgebung anwendbar und benötigte astrometrische Messungen. Aus Modellen für Sternevolution ließe sich dann die Sternmasse M s bestimmen. Unter der Annahme, dass die Masse M p des Planeten viel größer ist als die des Mondes, wäre M p aus Messungen der Radialgeschwindigkeit des Sterns berechenbar und sowohl die Orbitperiode P sp als auch die große Halbachse und Exzentrizität e sp des Planet-Mond-Systems um den Stern würden sich automatisch ergeben. Wie oben ausgeführt, ließen sich für Transitplaneten dann TTV und TDV nutzen, um die Mondmasse M m , die große Halbachse des Mondorbits um seinen Planeten a pm , und eventuell die Bahnneigung i zwischen den beiden Orbits zu erlangen. Die für die Gezeitenheizung essentiell wichtige Exzentrizität e pm des Mondorbits um den Planeten kann lediglich simuliert werden, insbesondere ist hier die Anwesenheit weiterer Monde zu prüfen (s. Io). Die Albedos α p und α m des Planeten und des Mondes müssten voraussichtlich geschätzt werden. Ähnliches gilt für die Materialeigenschaften des Mondes, welche seine Gezeitenkopplung beschreiben, typischerweise ausgedrückt durch Dissipationskonstanten Q oder τ , und k . Zwar wird die direkte Charakterisierung von Exomond-Atmosphären mittelfristig nicht möglich sein, doch können uns auch die durch die derzeitige Technologie und Theorie zugänglichen Parameter bereits viel über die zu erwartenden Oberflächenbedingungen verraten. Aus Masse und Radius lässt sich die Dichte bestimmen, welche wiederum Schlüsse auf die Zusammensetzung zulässt. Und mit der abschätzbaren Einstrahlung, die sich aus unserem Modell ergibt, zusammen mit der aus Masse und Radius ebenfalls berechenbaren Oberflächengravitation können sich gewisse atmosphärische Kompositionen als wahrscheinlich herausstellen.
Der neue Beitrag von Rory Barnes und mir bestand in der Synthese von stellarer und planetarer Einstrahlung mit der Gezeitenheizung auf potentiellen Exomonden zur Beschreibung des Klimas auf solchen Welten. Als Pointe haben wir die Bewohnbarkeit von Exomonden dadurch René Heller: Extrasolare Monde –
Schöne Neue Welten? 6
Box 2: Der Treibhauseffekt
Das thermische Gleichgewicht auf einem Planeten oder Mond ergibt sich aus der Annahme, dass er dieselbe Menge an Strahlung abgibt wie er absorbiert. Mit T ⊙ als der Oberflächentemperatur (genauer der “effektiven Temperatur”) der Sonne, R ⊙ als ihrem Radius, und α ⊕ als der Bond-Albedo, also der Oberflächenreflektivität, der Erde, ergibt sich ihre Oberflächentemperatur im thermischen Äquilibrium zu T eq = T ⊙ R ⊙ (1 − α ⊕ )4 / = 255 K ≡ − ◦ C ’wobei eine Astronomische Einheit (AU) dem Abstand zwischen Sonne und Erde entspricht. Die durchschnittlich gemessen Temperatur auf der Erde beträgt jedoch +16°C. Der Unterschied von 34°C wird auf den Treibhauseffekt zurückgeführt. Dieses Phänomen tritt auf, weil Kohlendioxid, Wasserdampf und Methan in der Erdatmosphäre teilweise intransparent sind für die thermische Abstrahlung der Erde. Ein Teil der absorbierten Sonneneinstrahlung kann also nicht re-emittiert werden und erwärmt die Erde. Für die Venus in 0.723 AU Abstand zur Sonne und mit einer Albedo von 0.75 ergibt die Gleichung eine Gleichgewichtstemperatur von 232 K ≡ –41°C. Jedoch bewirkt der Treibhauseffekt in der von Kohlendioxid dominierten Atmosphäre eine mittlere Oberflächentemperatur von beeindruckenden +464°C. Zwar besteht auch die Atmosphäre des Mars vor allem aus Kohlendioxid, doch ist sie so dünn, dass ihr Treibhauseffekt verschwindend gering bleibt.efiniert, dass die Summe aller beteiligter Energieflüsse gering genug sein muss, so dass ein Mond mit erdähnlicher Masse und Atmosphäre nicht einen irreversiblen Treibhauseffekt erfährt. So konnten wir nachweisen, dass etwaige erdgroße Monde um den neulich entdeckten Planeten Kepler-22b, welcher seinen sonnenähnlichen Stern in der HZ umrundet, bewohnbar sein könnten [5], vorausgesetzt, diese Monde hätten ein TTV-Signal Δ TDV ≲
10 Sekunden.
Noch gibt es auch auf der theoretischen Seite einiges zu tun, z.B. bei der Simulation und Auswertung der Transit-Effekte für den Fall von Mehrfach-Mond-Systemen. Eine analytisch geschlossene Theorie für die TTV und TDV in Systemen mit mehr als einem Mond existiert noch nicht. Numerische Simulationen können derweil eine Stütze bieten. Eine grundlegende Arbeit hierzu könnte aus N -Körper-Simulationen erstellt werden, wobei N die Anzahl aller beteiligten Körper und in diesem Fall größer als drei ist. N -Körper-Simulationen sind auch nötig um die langfristige Evolution der Geometrie des Mondsystems kennenzulernen, welche das Bestrahlungsmuster bestimmt. Untersuchungen der Stabilität solcher Systeme sind darüber hinaus notwendig um etwaige Funde von Exomonden auf Konsistenz zu prüfen und Exomond-Interpretationen von beobachteten TTVs und TDVs zu bekräftigen oder auszuschließen. Die Anfang Mai 2012 von der ESA ernannte große Mission “
Jupiter Icy Moons Explorer ” (
JUICE ) mit Start 2022, wird ab 2030 Jupiters große Monde untersuchen. Eines der Hauptziele des Projekts ist es, das Potenzial der Monde Europa, Ganymed und Callisto als Hort von Leben zu erkunden. Dazu wird
JUICE deren Topographie zentimetergenau vermessen und somit Rückschlüsse auf Verformungen von Gezeiten zulassen. Die präzise Vermessung von Nutation und Libration sowie der Rotationsschieflagen der Monde gegen den Orbit werden weitere interessante Verbesserungen der Modellierung dynamischer Eigenschaften von Satelliten-Systemen erlauben. Die Erkundung der Oberflächenchemie, die Vermessung der strukturellen Zusammensetzung, die Suche nach Wasservorkommen, die Vermessung von Ganymeds Magnetfeld und das Überwachen von Ios vulkanischer Aktivität lassen fundamental neue Einsichten in die Planetologie dieser Monde erwarten. Schließlich wird
JUICE
Schlüsse auf die Physik von Exomonden zulassen, deren Detektion mit
Kepler uns bald bevorstehen mag.
Literaturhinweise [1] Agnor, C. B. & Hamilton, D. P : Neptune’s capture of its moon Triton in a binary-planet gravitational encounter. In: Nature (2006), 441, 192 [2] Canup, R. M. & Ward, W. R : A common mass scaling for satellite systems of gaseous planets. In: Nature (2006), 441, 834 [3] Sasaki, T., Stewart, G. R, Ida, S : Origin of the different architectures of the Jovian and Saturnian satellite systems. In: The Astrophysical Journal (2010), 714, 1052, http://arxiv.org/abs/1003.5737 [4] Kipping, D. M et al. [5] Heller, R. & Barnes, R. : Exomoon habitability constrained by illumination and tidal heating. In: Astrobiology, (2012).
Dr. René Heller befasst sich am Leibniz-Institut für Astrophysik Potsdam (AIP) sowohl mit der Spektralanalyse von Weißen Zwergen sowie mit der Bewohnbarkeit von extrasolaren Planeten und Monden.
René Heller: Extrasolare Monde –
Schöne Neue Welten? 7 .3. IST DER ERSTE MOND AUSSERHALB DES SONNENSYSTEMS ENTDECKT? (Heller2018b) 410
B.3 Ist der erste Mond außerhalb des Sonnensystems entdeckt?(Heller 2018b) ntwurf für
Sterne und Weltraum (Göttingen, 1. Oktober 2018)Wie heißt es doch in einem populären Popsong der deutschen Hip-Hop-Band Die Fantastischen Vier? "Es könnte so einfach sein, isses aber nicht."Seit beinahe 20 Jahren nun machen sich Astrophysiker ernsthafte Gedanken über die Möglichkeiten, Monde außerhalb des Sonnensystems zu entdecken. Die einen favorisieren die Transit-Methode als den verheißungsvollsten Ansatz (siehe Box 1), andere bevorzugen die Suche mittels Microlensing, wieder andere Astronomen haben vorgeschlagen, durch Gezeiten extrem aufgeheizte Monde zu suchen, die im infraroten Licht sichtbar sein könnten –sozusagen, super-Ios. Und so schön die theoretischen Ansätze zur Exomond-Detektion auch in mathematischer Form oder in Computersimulationen auch aussehen, eines haben sie fast alle gemeinsam: sie basierend entweder auf stark vereinfachten Annahmen über das Rauschen in den Daten echter Beobachtungen oder sie vernachlässigen das Rauschen gleich ganz.Und so verwundert es nicht, dass um keinen der beinahe 4000 mittlerweile bekannten Exoplaneten bisher eindeutig ein Exomond gefunden werden konnte.Neue Beobachtungen des Hubble-Weltraumteleskops vom Stern Kepler-1625 jedoch, welcher von einem jupitergroßen Exoplaneten begleitet wird, liefern starke Hinweise auf den ersten Fund eines solchen Exomondes [1]. Die ersten Hinweise hatten sich bereits in den Daten des Kepler-Teleskops gefunden [2]. In drei Transits, die Kepler zwischen 2009 und 2013 beobachtete, zeigte der Stern nicht nur die charakteristische Verdunklung während des Transits seines Planeten Kepler-1625b, sondern es zeigten sich in der gemittelten Lichtkurve aus den drei Transits zusätzliche Hinweise auf eine leichte Verdunklung sowohl vor als auch nach dem Transit des Planeten. Dieser Effekt wurde 2014 vom Autor dieses Artikels vorhergesagt [3] und ist darauf zurückzuführen, dass der Mond im statistischen Mittel über mehrere Transits mal vor und mal nach dem Planeten die Sternscheibe von der Erde aus gesehen passiert. Die detaillierte Untersuchung der drei Transits von Kepler-1625b ergab dann Hinweise auf einen Mond, der ungleich allen Monden des Sonnensystems wäre. Dieser Kandidat wäre so groß wie Neptun und würde seinen jupitergroßen Planeten in einer Entfernung von 5 bis 10 Planetenradien umrunden. Zum Vergleich, die Galileischen Monde um Jupiter befinden sich in Entfernungen von ca. 6 (Io) bis 27 (Kallisto) Jupiterradien. Der schwerste Mond des Sonnensystems jedoch, Ganymed, ist nur halb so schwer wie der leichteste Planet des Sonnensystems, Merkur. Die Exomond-Kandidat um Kepler-1625b aber wäre ca. zehnmal so schwer wie alle Gesteinsplaneten und alle Monde des Sonnensystems zusammengenommen.
Entstehung von Riesenmonden
Wie ein solcher Riesenmond entstehen kann, ist den Astrophysikern derzeit ein Rätsel [4]. Für die Monde des Sonnensystems haben sich drei Entstehungsszenarien als möglich erwiesen. Erstens können Monde nach E i n s c h l ä g e n a u f e r d ä h n l i c h e Gesteinsplaneten entstehen. Dieses Szenario erklärt z.B. am besten die beobachteten Eigenschaften des Erde-Mond-Systems. Zweitens können Monde in den Akkretionsscheiben um junge Gasplaneten entstehen. Dieser Mechanismus wird
Hinweise auf die erste Entdeckung eines Mondes außerhalb des Sonnensystems von René Heller
Während um die acht Planeten des Sonnensystems bisher insgesamt fast 200 Monde entdeckt wurden, konnte noch kein einziger Mond um die annähernd 4000 bekannten Planeten außerhalb des Sonnensystems nachgewiesen werden. Doch neue Beobachtungen des Hubble Weltraumteleskops weisen nun auf die erste Entdeckung eines Exomondes hin.
Zieht ein Exoplanet von der Erde aus gesehen vor seinem Heimatstern entlang, so kommt es zu einer charakteristischen Verdunklung des Sterns. Solch ein Transit dauert typischerweise mehrere Stunden. Wird der Exoplanet zusätzlich von einem Exomond umkreist, so kann auch der Begleiter einen Transiteffekt verursachen und den Stern zusätzlich verdunkeln. Die drei links gezeigten Transit-Lichtkurven sind das Ergebnis aus Simulationen der drei Transits von Kepler-1625 b mit seinem Exomond-Kandidaten, die mit dem Kepler-Teleskop von 2009 bis 2013 beobachtet wurden.Eine Animation zu dieser Abbildung ist online verfügbar: https://goo.gl/kucjUw. ené Heller: Extrasolare Monde –
Schöne Neue Welten? verantwortlich gemacht für die Existenz der beeindruckenden Mondsysteme um Jupiter, Saturn, Uranus und Neptun. Und drittens können Monde eingefangen werden. Prominentestes Beispiel hierfür ist sicherlich Neptuns größter Mond Triton, aber auch viele der kleineren natürlichen Satelliten um die Gasplaneten sind wahrscheinlich eingefangene Asteroiden. Für Kepler-1625b-i, wie der Exomond-Kandidat vorläufig provisorisch genannte wird, kommt wahrscheinlich keiner dieser Prozesse in Frage.
Stochern im Datennebel
Die entscheidende Frage, die die neuen Hubble-Daten liefern sollten, nämlich ob das zusätzliche Exomond-Signal in den Kepler-Daten durch einen Mond oder vielleicht durch stellare Variabilität oder gar instrumentelle Effekte verursacht wird, wurde auch im Angesicht der neuen Beobachtungen nicht eindeutig beantwortet. Zwar sind die Hubble-Daten ungefähr viermal genauer als die Kepler-Daten, allerdings beinhalten sie nur einen Transit, während in den Kepler-Daten drei verfügbar sind. Auch findet sich das vermutete Exomond-Signal erneut in den Daten, allerdings ist das Signal nach einem Update der Kepler-Datenreduktions-Software kurioserweise aus den Kepler-Daten fast verschwunden. In der Gesamtschau der Daten ist damit die Signifikanz des Signals immerhin leicht erhöht gegenüber der vorherigen Analyse der Kepler-Daten.In der Gesamtschau bleibt zu sagen, dass dieses System der bei weitem aussichtsreichste Kandidat für einen Exomond-Fund ist, den es derzeit gibt. Und nicht nur das – würde sich die Existenz dieses Systems mit zukünftigen Beobachtungen bestätigen, dann würde Kepler-1625b und sein Riesenmond ein neues Kapitel in der Forschung der Planetenentstehung öffnen.
Literaturhinweise [1] Teachey, A., Kipping, D.M. : Evidence for a large exomoon orbiting Kepler-1625b. In: Science Advances, 4, eaav1784, 2018 [1] Teachey, A., Kipping, D.M., Schmitt, A.R. : HEK. VI. On the dearth of Galilean analogs in Kepler, and the exomoon candidate Kepler-1625b I. In: The Astronomical Journal, 155, id 36, 2018 https://arxiv.org/abs/1707.08563 [3] Heller, R. : Detecting extrasolar moons akin to solar system satellites with an orbital sampling effect. In: The Astrophysical Journal, 787, id 14, 2014 https://arxiv.org/abs/1403.5839[4] Heller, R. : The nature of the giant exomoon candidate Kepler-1625b-i. In: Astronomy & Astrophysics, 610, A39, 2018 https://arxiv.org/abs/1710.06209[5] Rodenbeck, K., Heller, R., Hippke, M., Gizon, L. : Revisiting the exomoon candidate signal around Kepler-1625b. In: Astronomy & Astrophysics, 617, A49, 2018 https://arxiv.org/abs/1806.04672
Dr. René Heller ist Projektwissenschaftler für die PLATO-Mission der ESA, die ab 2026 erdähnliche Planeten mittels der Transitmethode finden soll. Er hat mehr als ein Dutzend Forschungsartikel zu neuen Detektionsmethoden von Exomonden, zu ihren möglichen Entstehungsmechanismen und zu möglichem Leben auf Exomonden in Fachzeitschriften publiziert. Box 1: Datenaufbereitung von Transit-Lichtkurven
Die Crux in der Interpretation des Exomond-Signals in der Transit-Lichtkurve des Sterns Kepler-1625 liegt in dem Wissen um die Effekte der verschiedenen Zwischenschritte in der Datenaufbereitung. Beispielhaft ist hier die Aufbereitung der Kepler-Daten gezeigt [5]. Das obere Panel zeigt die unbehandelten Rohdaten des Kepler-Teleskops. Jeder Schnipsel der Lichtkurve ist knapp 100 Tage lang. Die starken Helligkeitsschwankungen sind hervorgerufen durch den Drift des Kepler-Teleskops und die damit verbundene Bewegung des Objekts Kepler-1625 über CCD-Pixel verschiedener Sensitivität. Im zweiten Panel wurden diese instrumentellen Effekte korrigiert. Das mittlere Panel zeigt den letzten der drei Transits aus den Kepler-Daten bei ca. 1498 Tagen. Er trägt die Nummer 5, da zwei weitere Transits in Beobachtungslücken unglücklicherweise nicht aufgenommen wurden. In diesem mittleren Panel kann die stellare Variabilität auf einer Zeitskala von mehreren Tagen erkannt werden. Im vorletzten Panel wurden Variationen in der Lichtkurve auf einer Zeitskala größer als die Dauer des Transits entfernt. Im unteren Panel schließlich sehen wir die Beobachtungsdaten nach dem Detrending zusammen mit einer Schar von 100 Transit-Modellen eines Planet-Mond-Systems, das aus einer Sequenz von Markov-Chain-Monte-Carlo-Simulationen ermittelt wurde [5]. Acknowledgements
Firstly, I express my sincere gratitude to my most longterm collaborator and reference for astrobiologyand exoplanet research, Rory Barnes of the Department of Astronomy at the University of Washington,Seattle. Our discussions about the textual and graphical presentations of our work were crucial sourcesof inspiration for me and they still serve as a guidelines for me today. Ralph Pudritz of the Departmentof Physics and Astronomy at McMaster University and founding director of the Origins Institute atMcMaster made key contributions to my theoretical work on moon formation. Interaction with himas my supervisor during my two years as a postdoctoral fellow of the Canadian Astrobiology TrainingProgram was a major stimulation for me to grow not only as a scientist. I am particularly gratefulto professors Stefan Dreizler, Laurent Gizon, and Ansgar Reiners for giving me the opportunities toteach at the Georg-August-University of G¨ottingen. My special thanks go to computer scientist anddata analyst Michael Hippke for many disruptive conversations and for our highly efficient collaborationon several topics related to the science of extrasolar planets and moons.I thank Kai Rodenbeck, former PhD student at the Institute for Astrophysics at the Universityof G¨ottingen and the Max Planck Institute for Solar System Research, for his permission to presentpreliminary results from our ongoing collaboration and for providing me with Figs. 8.3 to 8.9. I thankAnina Timmermann, master student at the Institute for Astrophysics of the University of G¨ottingen,for her cooperation on the analysis of the CARMENES spectra of Kepler-1625 and for providingTable 8.1 and Fig. 8.2.I also wish to thank the Hans and Clara Lenze Foundation in Aerzen for their financial support ofthis thesis with a “Habilitations-Stipendium”.The original source of inspiration for me to work in the field of exomoon research is in the 2009movie Avatar that was written, directed, and produced by James Cameron. ibliography
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