On the Inclination and Habitability of the HD 10180 System
aa r X i v : . [ a s t r o - ph . E P ] A ug Submitted for publication in the Astrophysical Journal
Preprint typeset using L A TEX style emulateapj v. 5/2/11
ON THE INCLINATION AND HABITABILITY OF THE HD 10180 SYSTEM
Stephen R. Kane , Dawn M. Gelino Submitted for publication in the Astrophysical Journal
ABSTRACTThere are numerous multi-planet systems that have now been detected via a variety of techniques.These systems exhibit a range of both planetary properties and orbital configurations. For those sys-tems without detected planetary transits, a significant unknown factor is the orbital inclination. Thisproduces an uncertainty in the mass of the planets and their related properties, such as atmosphericscale height. Here we investigate the HD 10180 system which was discovered using the radial velocitytechnique. We provide a new orbital solution for the system which allows for eccentric orbits for allplanets. We show how the inclination of the system affects the mass/radius properties of the planetsand how the detection of phase signatures may resolve the inclination ambiguity. We finally evaluatethe Habitable Zone properties of the system and show that the g planet spends 100% of an eccentricorbit within the Habitable Zone.
Subject headings: astrobiology – planetary systems – stars: individual (HD 10180) INTRODUCTION
Multi-planet systems discoveries have revealed a diver-sity of system architectures, many of which significantlydiverge from that of our own Solar System. Many of therecent multi-planet system discoveries have been made asa result of data from the
Kepler mission, such as Kepler-62 (Borucki et al. 2013). These systems tend to harbora mixture of terrestrial and Neptune-size planets, someof which are in the Habitable Zone (HZ) of their hoststar. There have also been several discoveries of systemswith more than four planets that have been discoveredby radial velocity (RV) surveys, such as the 55 Cancrisystem (McArthur et al. 2004; Endl et al. 2012). The or-bital inclination of the planets in these cases is generallyunknown, although they can be constrained through ex-amination of the dynamical stability of the system (e.g.,Correia et al. (2010)).A multi-planet system of particular interest is theHD 10180 system, due to the both the relatively largenumber of planets and their relatively low masses. Therehave been various interpretations of the RV data for thissystem with respect to the number of planets present.Lovis et al. (2011) provide a seven planet solution wherethe detection of the inner “b” planet is considered tenta-tive. A further solution by Tuomi (2012) demonstratesthat the system may harbor nine planets. Although inboth cases the planets likely have low masses, this de-pends on the inclination of the system with respect to theplane of the sky. As the inclination decreases the mass ofthe planets increases and thus their physical propertieschange. The inclination ambiguity can be resolved us-ing several techniques, such as astrometry (Tuomi et al.2009) and phase curve analysis Kane & Gelino (2011,2012a). Determining the true masses of the planets isa key factor in determining the significance of their loca-tions within the stellar HZ (Kopparapu et al. 2014). [email protected] Department of Physics & Astronomy, San Francisco StateUniversity, 1600 Holloway Avenue, San Francisco, CA 94132,USA NASA Exoplanet Science Institute, Caltech, MS 100-22, 770South Wilson Avenue, Pasadena, CA 91125, USA
TABLE 1Stellar Parameters (1)
Parameter Value V B − V (2) . ± . T eff (K) 5911 ± g . ± . . ± . M ⋆ ( M ⊙ ) 1 . ± . R ⋆ ( R ⊙ ) (3) . ± . (1) Lovis et al. (2011) (2) van Leeuwen (2007) (3)
Torres et al. (2010)
Here we present the results of a new analysis of theHD 10180 system in which we discuss the orbital param-eters, inclination, predicted phase signatures, and HZstatus of the planets. In Section 2 we provide a newKeplerian orbital solution for the system with eccentricorbits for all planets. Section 3 investigates the effects oforbital inclination on planet masses and possible radii.Section 4 discusses the phase variations of the systemin different inclination scenarios and the detectability ofthose signatures. In Section 5 we present an analysis ofthe system HZ in the context of various inclinations. Weprovide concluding remarks in Section 6. SYSTEM CONFIGURATION
HD 10180 is a star which is quite similar to solar(G1V) in terms of its fundamental properties. Theseare summarized in Table 1, where the majority of pa-rameters are those provided by Lovis et al. (2011). Thedistance is derived from
Hipparcos parallax measure-ments (van Leeuwen 2007) and the stellar radius is calcu-lated from the mass-radius relationships determined byTorres et al. (2010). As noted by Lovis et al. (2011), theactivity index shows that HD 10180 is a relatively inac-tive star, a property that will be of particular relevancewhen discussing the photometry in Section 4.The Keplerian orbital solution provided by Lovis et al.(2011) includes seven planets and forces a circular orbit Stephen R. Kane & Dawn M. Gelino
Fig. 1.—
Top panel: The 190 RV measurements of HD 10180 along with the best-fit 6-planet Keplerian solution. The solution, shown inTable 2, allows the eccentricities of all planets to be free parameters. Bottom panel: The RV residuals (observed minus computed) fromthe best-fit model shown above.
TABLE 2HD 10180 Planetary Parameters
Parameter c d e f g h P (days) 5 . ± . . ± . . ± .
025 122 . ± .
232 604 . ± .
42 2205 . ± . T p (1) . ± .
426 4022 . ± .
157 4006 . ± .
91 4024 . ± .
03 4002 . ± . . ± . e . ± .
031 0 . ± .
052 0 . ± .
033 0 . ± .
054 0 . ± .
152 0 . ± . ω (deg) 328 ±
24 325 ±
23 147 ±
54 327 ±
27 327 ±
59 142 ± K (m s − ) 4 . ± .
154 2 . ± .
173 4 . ± .
169 2 . ± .
186 1 . ± .
380 3 . ± . M p sin i ( M J ) 0 . ± . . ± . . ± . . ± . . ± . . ± . a (AU) 0 . ± . . ± . . ± . . ± . . ± .
028 3 . ± . (1) BJD – 2,450,000 for several of those planets, including planet g. Since thesemi-amplitude of the RV signal for the b planet is sig-nificantly lower than the others, we performed our ownfit to the RV data to obtain a Keplerian orbital solutionin which all eccentricities were allowed to vary as free pa-rameters. The RV data were extracted from the VizieRCatalog Service . These consist of 190 measurements ob-tained with the HARPS spectrograph at the ESO 3.6mtelescope at La Silla Observatory. We fit the data usingthe partially linearized, least-squares fitting proceduredescribed in Wright et al. (2009) and estimated param-eter uncertainties using the BOOTTRAN bootstrappingroutines described in Wang et al. (2012). Our best solu-tion includes six planets where there is no significant RVtrend in the data. We adopted a slightly larger stellarjitter value than Lovis et al. (2011) of 1.39 m s − which http://vizier.u-strasbg.fr/ forces the reduced χ value to unity. The resulting or-bital solution is shown in Figure 1 and Table 2. Theresiduals shown in the bottom panel of Figure 1 have anRMS scatter of 1.5 m s − . The main differences with thesolution by Lovis et al. (2011) are: (1) no planet b, (2)a significant eccentricity for planet g, and (3) a smallerorbital period and eccentricity for planet h. Note thatLovis et al. (2011) force the eccentricity of the g planetto zero since a non-zero eccentricity produces an almostidentical χ . Here we consider a non-zero eccentricityfor the g planet since it is consistent with the data andis relevant to our subsequent habitability discussion inSection 5.Since this eccentric solution is different from thosepreviously published, it is important to establish if itis dynamically sound. To explore this, we performeddynamical simulations using N-body integrations withthe Mercury Integrator Package, described in more de-he HD 10180 System 3 Fig. 2.—
Dynamical simulations of the HD 10180 system, showing the eccentricity oscillations of the g and h planets over a period of10 years. The g planet primarily dynamically interacts with the h planet with additional minor perturbations caused by the f planet. tail by Chambers (1999). We adopted the hybridsymplectic/Bulirsch-Stoer integrator and used a Jacobicoordinate system. This coordinate system generallyprovides more accurate results for multi-planet systems(Wisdom & Holman 1991; Wisdom 2006) except in casesof close encounters (Chambers 1999). The integrationswere performed for a simulation of 10 years, in steps of100 years, starting at the present epoch.Our simulations indicate that the orbital configurationpresented in Table 2 are stable over the 10 year sim-ulation. The planets do exchange angular momentumthrough secular oscillations of their eccentricities, butthis remains at a relatively low level. The two mainexamples of this are the outermost (g and h) planets.The eccentricity oscillations for both of these planetsare shown in Figure 2 for the complete simulation pe-riod. The secular oscillations complete approximately 15cycles during the 10 year simulation with a period of ∼ INCLINATION AND PLANETARY PROPERTIES
There is a well studied relationship between plan-etary mass and radius. Early work on mass-radiusrelationships for gas giants (Fortney et al. 2007) andsuper-Earths (Seager et al. 2007) paved the way for un-derstanding the results of Kepler discoveries. Keplerplanets have subsequently allowed empirical relations tobe developed for low-mass planets (Weiss et al. 2013;Weiss & Marcy 2014). The nature of radial velocity dis-coveries of exoplanet that lack confirmation from othertechniques is that it is only the minimum mass that isknown. The true planetary masses depend on the incli-nation of the system, from edge-on ( i = 90 ◦ ) to face-on( i = 0 ◦ ) with respect to the plane of the sky.In Figure 3 we show the increase in the HD 10180planet masses as a function of orbital inclination. The fand g planets have similar masses and so their lines inthe plots are almost indistinguishable. The dynamicalanalysis of the HD 10180 system by Lovis et al. (2011)shows that the system is still stable for i = 30 ◦ but notfor i = 10 ◦ , concluding that an instability transition oc- Fig. 3.—
The dependence of the HD 10180 planetary propertiesof mass (top panel) and radius (bottom panel) on the system in-clination, where 90 ◦ is an edge-on orientation and 0 ◦ is face-on.The vertical lines represent the likely lower inclination limit of thesystem as estimated by Lovis et al. (2011). curs around i ∼ ◦ . We tested this instability transitionby repeating our stability analysis described in Section 2for a variety of inclinations. We confirm that the systembecomes unstable at i ∼ ◦ (shown as a vertical dashedline in Figure 3) with the ejection of the d planet, but theg planet remains stable despite the higher eccentricity.The range of g planet eccentricities described in Section Stephen R. Kane & Dawn M. Gelino Fig. 4.—
Photometry of HD 10180 from the
Hipparcos missionwhich shows photometric stability at the 1% level. PHASE VARIATIONS
A further means through which to constrain the in-clination of the system is by the detection of phasevariations. This technique was described in detail byKane & Gelino (2012a) where the amplitude of the phasevariations depend on the planetary properties which varywith inclination (see Section 3). One aspect of the sys-tem that affects the ability to detect such phase signa-tures is the activity of the star. HD 10180 is known tobe a relatively inactive star with a mean activity indexof log R ′ HK = − .
00 (Lovis et al. 2011). We used pub-licly available data from the
Hipparcos satellite to searchfor low-frequency photometric variations of HD 10180.
Hipparcos acquired a total of 125 measurements span-ning a period of 1184 days during the course of its three-year mission (Perryman et al. 1997; van Leeuwen 2007).These data are shown in Figure 4. The 1 σ RMS scatter ofthe 125 HD 10180 measurements is 0.013 mag, while themean of the measurement uncertainties is 0.011. Thusit is consistent with a photometrically stable star at the1% level. A Fourier analysis of the
Hipparcos data do notreveal any significant periodic signatures and indeed thedata Nyquist frequency of 0.0528 days − is slightly abovethe predicted period of the stellar rotation ( ∼
24 days).The
Hipparcos data sampling is therefore unlikely to de-tected stellar rotation variability.Although it is advantageous that the star is relativelyquiet, the phase variations occur on a much lower level.We calculate the predicted phase variations of the sys-tem by adopting the formalism of Kane & Gelino (2010).This formalism accounts for planetary size, orbital eccen-tricity, and the variation of geometric albedo with sepa-ration from the host star. The flux ratio of the planet tothe host star is given by ǫ ( α, λ ) ≡ f p ( α, λ ) f ⋆ ( λ ) = A g ( λ ) g ( α, λ ) R p r (1)where A g is the geometric albedo, g ( α, λ ) is the phasefunction, R p is the planetary radius, and r is the star–planet separation. The resulting flux variations of thesystem are shown in the top three panels of Figure 5where we have calculated the variations for system in-clinations of 90 ◦ (edge-on), 30 ◦ , and 10 ◦ . In each panelthe phase flux variations due to the individual planetsare shown as solid lines and the combined variations are indicated by the dotted line. These are shown for onecomplete orbital phase of the outer planet. As describedin Section 3, the inclination must be larger than 10 ◦ inorder to retain a stable orbital configuration for the sys-tem.The phase variations of the c and d planets are alsolabeled on the right of each panel in Figure 5. The peakflux variations are dominated by the inner (c) planet foreach inclination. The planet has a calculated radius of0.37 and 0.92 Jupiter radii for inclinations of 90 ◦ and 10 ◦ respectively. As shown by Kane & Gelino (2011), the ef-fect of decreasing the inclination is to remove the timevariability of the phase function resulting in flux vari-ations caused exclusively by orbital eccentricity. Thiscan be particularly seen for the d planet in the bottompanel which retains a photometrically variable signaturedue to its eccentricity of 0.131 (see Table 2). The am-plitude of the total variations remains very similar withdecreasing inclination due to the compensation of the in-creased planetary radii. However, the constantly visibleillumination of the c and d planets for lower inclinationssignificantly raises the baseline of the planetary reflectedlight received. The effect of this is to raise to signal-to-noise of the variations which makes their detection moreaccessible. A possible method to discriminate betweenthe baseline flux from the planet and the stellar flux isthrough polarized light. The motion of the planet(s) willproduce a polarization signature distinct from the stellarflux due to the scattering of light from the planetary at-mospheres (Berdyugina et al. 2011; Wiktorowicz 2009).However, for Keplerian orbits an additional distinction isavailable via the difference in phase between the times ofperiastron and maximum phase variations (phase anglezero). The bottom panel of Figure 5 shows the differ-ence between the combined flux variations of i = 10 ◦ and i = 90 ◦ after the minimum flux (baseline) has beenremoved. The 58 ◦ separation of the periastron passageof the c planet from the zero phase angle produces a dif-ference in phase signature of amplitude similar to theindividual inclination scenarios. In practice, most sys-tems will have an even larger separation which will aidin this distinction, the amplitude of which will dependon the orbital eccentricities. Resolving this degeneracywill greatly aid in disentangling the components of thephase signature and thus the inclination of the system.The total flux variations are of amplitude several partsper million (ppm) and thus close to the photometric pre-cision achieved by the Kepler mission for the brighteststars monitored. This technique could therefore be usedto rule out low system inclinations and/or high albedosfor multi-planet systems, particularly with data from fu-ture missions such as the James Webb Space telescope(JWST) and the Transiting Exoplanet Survey Satellite(TESS). HABITABILITY OF THE g PLANET
With the revised orbital solution of Section 2 and in-clination effects described in Sections 3 and 4, we finallyinvestigate the HZ of the system. The empirically derivedHZ boundaries of Kasting et al. (1993) have recentlybeen replaced by new calculations by Kopparapu et al.(2013, 2014). We use these calculations to provide newestimates for the HZ in the HD 10180 by adopting thedefinitions of “conservative” and “optimistic” HZ mod-he HD 10180 System 5
Fig. 5.—
The flux variations of the HD 10180 system due to the phase variations of the planets as they orbit the host star. We considerthree inclinations of the plane of the system with respect to the plane of the sky; i = 90 ◦ or edge-on (top panel), i = 30 ◦ (second panel),and i = 10 ◦ (third panel). In each panel the phase variations of individual planets are shown as solid lines phased on the orbital period ofthe outer planet, the combined signature of all six planets is shown as a dotted line, and the phase variations of c and d are labeled. Thebottom panel shows the difference in flux variations between the i = 10 ◦ and i = 90 ◦ scenarios after the baseline flux has been removed. els described by Kane et al. (2013). These definitionsuse different boundaries for the HZ based on assump-tions regarding the amount of time that Venus andMars were able to retain liquid water on their surfaces(Kopparapu et al. 2013). Kane (2014) showed the ex-tent to which HZ boundaries depend on stellar param-eter uncertainties, although the parameters in Table 1are sufficiently well known that the HZ boundary uncer-tainties are negligible. HZ calculations for all known exo- planetary systems are available using the same methodol-ogy through the Habitable Zone Gallery (Kane & Gelino2012b).Figure 6 shows two top-down views of the HD 10180system, one zoomed out to include the outer planet (leftpanel) and the other zoomed in to show the inner planets(right panel). In each panel, the HZ is depicted by theshaded region where the light gray represents the con-servative HZ and the dark gray is the extension to the Stephen R. Kane & Dawn M. Gelino Fig. 6.—
A top-down view of the HD 10180 system showing the extent of the HZ calculated using the stellar parameters of Table 1.The conservative HZ is shown as light-gray and optimistic extension to the HZ is shown as dark-gray. The revised Keplerian orbits of theplanets from Table 2 are overlaid. Left Panel: The full HZ of the system with the outer 5 planets. Right panel: A zoom-in of the systemshowing the orbits of the inner 5 planets and the orbital path of the g planet into the optimistic HZ.
HZ with optimistic calculations. Of greatest interest inthis figure from a HZ perspective is the g planet. Theplanet remains in the conservative HZ for 89% of the du-ration of one orbit with the remaining 11% within theoptimistic HZ. The orbital path through the optimisticHZ occurs during the periastron passage and is clearlyshown in the right panel of Figure 6. The orbital stabil-ity analysis from Section 2 showed that the eccentricityof the planet can oscillate to a value as high as ∼ ∼ σ uncertainty shown in Table 2 ( e = 0 . ◦ , the mass of the g planetis 0.0732 Jupiter mass or 23.3 Earth masses. The esti-mated radius according to Figure 3 is 0.5 Jupiter radii.An inclination of 30 ◦ raises these values to 0.1464 M J ,46.5 M ⊕ , and 0.71 R J respectively. Adjusting the in-clination to the stability limit of 10 ◦ further raises thevalues to 0.422 M J , 134.0 M ⊕ , and 1.0 R J respectively.For any value of inclination, the mass of the g planet is farabove the threshold where the planet is likely to have apurely rocky composition (Marcy et al. 2014). Thus the prospects for habitability for the g planet lies within amoon system which the planet may harbor. Searches forexomoons around such HZ giant planets have been un-dertaken (Kipping et al. 2013) but have not yet yieldedpositive results. Recent studies of exomoon habitabilityhave shown that there are a variety of factors which addto the total energy budget including flux from the planetand tidal effects (Heller 2012; Hinkel & Kane 2013). Al-though these additional factors are usually negligiblecompared with the flux from the host star, they maybe sufficient in this case to render the g planet moonsdevoid of surface liquid water considering the planet al-ready moves interior to the conservative HZ. CONCLUSIONS
Multi-planet exoplanetary systems offer exceptionalopportunities for system characterization, such as con-straining the orbital inclinations and eccentricities basedon stability simulations. These systems are particularlyinteresting when one or more of the planets occupy theHZ of their host star. The HD 10180 is just such asystem, with a range of planetary masses and at leastone planet within the HZ. We have presented a new or-bital solution which allows the g planet to have signif-icant orbital eccentricity whilst preserving the stabilityof the system. We have quantitatively shown how theproperties of the planets alter depending on the inclina-tion of the system. These properties in turn change thepredicted phase signatures of the system and provides amethod through which future observations could resolvethe inclination ambiguity.An important consideration for HZ planets is the or-bital eccentricity since the variable stellar insolation cangreatly effect the habitability. The revised orbital solu-tion presented here allows for an eccentric orbit for theonly HZ planet in the system. The HZ calculations de-he HD 10180 System 7scribed here show that the g planet spends most of acomplete orbital period within the conservative HZ andmoves into the interior optimistic HZ for the remainingtime. The mass of the planet is high enough that surfaceliquid water is only possible on any moons the planetmay possess. The topic of exomoons is one of increasingstudy and detection efforts and is thus likely to yield pos-itive detections in the near future. As next-generation in-strumentation is developed for exoplanetary studies, it isimportant to identify the best HZ targets orbiting bright host stars such as the one described here.
ACKNOWLEDGEMENTS