K2-138 g: Spitzer Spots a Sixth Planet for the Citizen Science System
Kevin K. Hardegree-Ullman, Jessie L. Christiansen, David R. Ciardi, Ian J. M. Crossfield, Courtney D. Dressing, John H. Livingston, Kathryn Volk, Eric Agol, Thomas Barclay, Geert Barentsen, Björn Benneke, Varoujan Gorjian, Martti H. Kristiansen
aa r X i v : . [ a s t r o - ph . E P ] F e b Draft version February 19, 2021
Typeset using L A TEX twocolumn style in AASTeX63
K2-138 g:
Spitzer
Spots a Sixth Planet for the Citizen Science System
Kevin K. Hardegree-Ullman, Jessie L. Christiansen, David R. Ciardi, Ian J. M. Crossfield, Courtney D. Dressing, John H. Livingston, Kathryn Volk, Eric Agol, Thomas Barclay,
Geert Barentsen, Bj¨orn Benneke, Varoujan Gorjian, and Martti H. Kristiansen
12, 13 Caltech/IPAC-NExScI, M/S 100-22, 1200 E. California Blvd, Pasadena, CA 91125, USA Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045 Department of Astronomy, University of California, Berkeley, Berkeley, CA 94720 Department of Astronomy, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Lunar and Planetary Laboratory, The University of Arizona, 1629 E University Blvd, Tucson, AZ 85721 Astronomy Department and Virtual Planetary Laboratory, University of Washington, Seattle, WA 98195 USA NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA University of Maryland, Baltimore County, 1000 Hilltop Cir, Baltimore, MD 21250, USA Bay Area Environmental Research Institute, P.O. Box 25, Moffett Field, CA 94035, USA Department of Physics and Institute for Research on Exoplanets, Universit´e de Montr´eal, Montreal, QC, Canada Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Brorfelde Observatory, Observator Gyldenkernes Vej 7, DK-4340 Tølløse, Denmark DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Lyngby, Denmark
ABSTRACT K2 greatly extended Kepler ’s ability to find new planets, but it was typically limited to identifyingtransiting planets with orbital periods below 40 days. While analyzing K2 data through the ExoplanetExplorers project, citizen scientists helped discover one super-Earth and four sub-Neptune sized planetsin the relatively bright ( V = 12 . K = 10 .
3) K2-138 system, all which orbit near 3:2 mean motionresonances. The K2 light curve showed two additional transit events consistent with a sixth planet.Using Spitzer photometry, we validate the sixth planet’s orbital period of 41 . ± .
006 days andmeasure a radius of 3 . +0 . − . R ⊕ , solidifying K2-138 as the K2 system with the most currently knownplanets. There is a sizeable gap between the outer two planets, since the fifth planet in the system,K2-138 f, orbits at 12.76 days. We explore the possibility of additional non-transiting planets in thegap between f and g. Due to the relative brightness of the K2-138 host star, and the near resonanceof the inner planets, K2-138 could be a key benchmark system for both radial velocity and transittiming variation mass measurements, and indeed radial velocity masses for the inner four planets havealready been obtained. With its five sub-Neptunes and one super-Earth, the K2-138 system providesa unique test bed for comparative atmospheric studies of warm to temperate planets of similar size,dynamical studies of near resonant planets, and models of planet formation and migration. Keywords:
Exoplanet systems; Exoplanets INTRODUCTIONThe NASA K2 mission searched for exoplanets in dif-ferent fields spanning the ecliptic plane, subsequent tothe loss of two reaction wheels, which inhibited the Ke-pler spacecraft’s ability to precisely point at the original
Kepler field for extended durations. Using solar pres-
Corresponding author: Kevin K. [email protected] sure and thrusters,
Kepler was able to point to fieldsalong the ecliptic plane for a period of ∼
83 days eachbefore the spacecraft was rotated to prevent sunlightfrom entering the telescope (Putnam & Wiemer 2014;Howell et al. 2014). K2 has so far enabled the discoveryof 425 new planets and an additional 889 planet candi-dates. http://exoplanetarchive.ipac.caltech.edu/docs/counts detail.html,as of February 2021. Hardegree-Ullman et al.
K2-138 was the first K2 planet system discovered bycitizen scientists through the Exoplanet Explorers pro-gram on the Zooniverse (Christiansen et al. 2018). Thecitizen scientists were able to identify four sub-Neptunesized planets by visual inspection of the light curve. Acloser inspection of the diagnostic plots from the TERRA algorithm (Petigura et al. 2013b,a) elucidated a super-Earth interior to the orbits of the other four planets. Us-ing LcTools (Kipping et al. 2015; Schmitt et al. 2019),Christiansen et al. (2018) also identified two additionaltransits 41.97 days apart, indicating a possible sixthplanet for the system.Lopez et al. (2019) obtained radial velocity (RV) mea-surements of K2-138 with HARPS, yielding mass mea-surements of 3 . ± .
1, 6 . +1 . − . , 7 . +1 . − . , and 13 . ± . M ⊕ for planets b, c, d, and e, respectively. Pre-cise masses for K2-138 f and the putative planet K2-138 g were not measured. K2-138 f has an orbital pe-riod of 12.76 days, about half of the 24 . ± . M ⊕ on K2-138 fand g, respectively. Due to the near 3:2 orbital reso-nances, K2-138 is amenable to transit timing variation(TTV) measurements to constrain planet masses. Usingtheir measured masses and assuming zero eccentricity,Lopez et al. (2019) computed TTV amplitudes between2.0 and 7.3 minutes for the inner five planets, similar tothe amplitudes computed by Christiansen et al. (2018).Though Christiansen et al. (2018) were not able to de-tect significant TTVs in the 30 minute cadence K2 data,higher cadence observations with instruments such asCHEOPS, which were scheduled for late 2020 (ProgramID 017 (EP); PI: T. Lopez), should allow TTV massmeasurements of planets c, d, and e, making K2-138 animportant benchmark system for comparing TTV andRV masses. Since RV mass measurements are currentlylimited to host stars brighter than V .
13, TTVs enablemass measurements for a much wider pool of planets(Holczer et al. 2016). However, fewer than 10 systemshave both RV and TTV mass measurements, and detec-tion sensitivity may bias RV measurements for planetswith orbital periods larger than 11 days (Mills & Mazeh2017; Petigura et al. 2018). These reasons highlight theimportance of adding new TTV/RV benchmark systems https://github.com/petigura/terra in order to cross-check masses between measurementtechniques.In this paper we verify the outermost planet K2-138 gwith an orbital period of 41 . +0 . − . days. Thisadds to the nine systems with six or more planets cur-rently known, makes K2-138 the K2 discovered systemwith the most planets, and yields one of the longest pe-riod K2 planets. Using the Spitzer Space Telescope , weobserved a third transit of K2-138 g within one hour ofthe time predicted from the K2 ephemeris. We presentour observations and data reduction in Section 2 anddiscuss our results in Section 3. OBSERVATIONS AND DATA REDUCTION2.1.
Stellar Classification
We obtained a 0.38 to 0.7 µ m spectrum of K2-138 us-ing the Goodman spectrograph (Clemens et al. 2004) onthe Southern Astrophysical Research Telescope (Pro-gram ID 2019A-0364; PI: K. Hardegree-Ullman), anda 0.7 to 2.4 µ m spectrum using the SpeX spectrograph(Rayner et al. 2003) on the NASA Infrared TelescopeFacility (Program ID 2017A-106; PI: K. Hardegree-Ullman). We followed the procedures outlined in § µ m to optical SDSS spectral templates fromKesseli et al. (2017) following the procedures outlined in § µ m spectrum compared to G7 V, G8 V, and G9 Vtemplate spectra.In Table 1 we list stellar parameters for K2-138 compiled from the Ecliptic Plane Input Cat-alog (EPIC; Huber et al. 2016), RAVE DR5(Kunder et al. 2017), Christiansen et al. (2018), Gaia
DR2 (Gaia Collaboration et al. 2018; Bailer-Jones et al.2018), the
TESS
Input Catalog Candidate Target List(TIC CTL; Stassun et al. 2019), Lopez et al. (2019),and Hardegree-Ullman et al. (2020). The measuredparameters are all consistent to within 1 σ , except forone measurement of log( g ) which is within 2 σ . It isreassuring that different data sets and pipelines yieldsimilar results, but when it comes to calculating planetparameters, small differences in stellar parameters canhave a large impact. For large surveys and populationstudies of exoplanets, it is crucial to have a uniformly Kruse et al. (2019) identified six candidate planets in EPIC210965800, five of which have yet to be confirmed. Table 1.
K2-138 stellar parameters.Spectral Type T eff log( g ) [Fe/H] R ⋆ M ⋆ Distance ReferenceK log(cm s − ) dex R ⊙ M ⊙ pc · · · ±
156 4 . ± .
030 0 . ± .
12 0 . ± .
046 0 . ± .
052 182 . ± .
15 1 · · · ±
76 4 . ± .
12 0 . ± . · · · · · · . ± .
869 2K1 V ± ±
60 4 . ± .
07 0 . ± .
04 0 . ± .
08 0 . ± .
06 183 ±
17 3 · · · +209 − · · · · · · . +0 . − . · · · . +2 . − . · · · ±
129 4 . ± . · · · . ± .
049 0 . ± .
11 202 . ± .
009 5G8 5350 ±
80 4 . ± .
15 0 . ± .
10 0 . +0 . − . . ± .
02 201 . ± .
38 6G7 5303 ±
138 4 . ± .
150 0 . ± .
235 0 . +0 . − . . +0 . − . . +2 . − . ± +122 − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +2 . − . References —(1) EPIC (Huber et al. 2016), (2) RAVE DR5 (Kunder et al. 2017), (3) Christiansen et al. (2018), (4)
Gaia
DR2 (Gaia Collaboration et al. 2018; Bailer-Jones et al. 2018), (5) TIC CTL v8.01 (Stassun et al. 2019), (6) Lopez et al.(2019), (7) Hardegree-Ullman et al. (2020), (8) This work.
Figure 1.
K2-138 spectrum (black) compared to G7 V(top), G8 V (middle), and G9 V (bottom) template spec-tra from Kesseli et al. (2017). The G8 V template spectrumis the most similar to K2-138. derived set of stellar parameters (e.g., Fulton et al. 2017;Berger et al. 2020; Hardegree-Ullman et al. 2020). Forindividual systems, however, it is typical to choose asingle set of stellar parameters, which may be suscep-tible to systematic bias. Rather than cherry pickingmeasurements from different references, we combinedall the available measurements for T eff , log( g ), [Fe/H],and M ⋆ . Instead of using a weighted mean, which would produce uncharacteristically small uncertainties, we in-stead employed the following Monte Carlo method. Foreach measurement with symmetric uncertainties, werandomly drew 10 values from a Gaussian distribution,and for asymmetric uncertainties we drew 10 valuesfrom a split normal distribution. The posterior distri-butions were concatenated and we took the median,16th, and 84th percentiles of the resultant distributionas our measurement and errors.We computed a bolometric magnitude ( M bol ) us-ing the Gaia distance of 202 . +2 . − . pc reportedby Bailer-Jones et al. (2018), accounting for interstel-lar reddening with the dustmaps package (Green et al.2018), and applying a bolometric correction found us-ing isoclassify (Huber et al. 2017). We computedthe bolometric luminosity using L bol = L × − . M bol ,where L = 3 . × W is the zero point ra-diative luminosity (Mamajek et al. 2015). From theStefan-Boltzmann law we derived a new stellar radius of0 . +0 . − . R ⊙ with our Monte Carlo averaged effectivetemperature and bolometric luminosity. Stellar param-eters for K2-138 are listed in Table 1.2.2. K2 Photometry
EPIC 245950175 (K2-138) was observed with K2 in30 minute long cadence mode during Campaign 12 be-tween 2016 December 15 and 2017 March 04, with a five A weighted mean (ˆ µ = (Σ x i /σ i ) / (Σ1 /σ i ), σ (ˆ µ ) = 1 / (Σ1 /σ i ))would yield T eff = 5300 ±
35 K, log( g ) = 4 . ± .
03, [Fe/H] =0 . ± .
03, and M ⋆ = 0 . ± .
02. The uncertainties on theweighted mean parameters are ∼ Hardegree-Ullman et al. day gap in the data about two-thirds of the way throughthe campaign due to a spacecraft safe-mode event. K2-138 was included in Campaign 12 Guest Observing Pro-grams 12049, 12071, 12083, and 12122 (PIs E. Quintana,D. Charbonneau, A. Jensen, and A. Howard).For our analysis, we used the instrumentalsystematics-corrected light curve (Figure 2) pro-duced by the k2phot pipeline (Petigura et al. 2015;Aigrain et al. 2016). We compared the k2phot lightcurve to those produced by EVEREST (Luger et al. 2016)and
K2SFF (Vanderburg & Johnson 2014), and foundthat the k2phot light curves had the lowest overallRMS scatter and the fewest outliers. We first maskedout data that was flagged in the k2phot pipeline asa thruster fire event or an outlier in background flux.Periodic transit signals were initially found by flatten-ing the light curve with a Savitsky-Golay filter overa window of 101 points ( ∼
50 hours), then running abox least squares periodogram, iteratively masking outthe higher signal-to-noise transits until there were nomore convincing planet signals in the data. This searchgave estimates of planet periods, transit times, andtransit depths. Next, we made use of the exoplanet toolkit to model stellar variability using a Gaussianprocess with a simple harmonic oscillator kernel, whilesimultaneously fitting planet transits as described inForeman-Mackey et al. (2017). In order to simultane-ously fit the K2 and Spitzer data ( § IRAC Photometry
We observed K2-138 with the Infrared Array Camera(IRAC) on the
Spitzer Space Telescope (DDT 13253; PIJ. Christiansen) at the predicted transit time of the pu-tative sixth planet. We used Channel 2 (4.5 µ m) since itis less affected by intrapixel sensitivity variations thanChannel 1 (3.6 µ m). The observation began with a 30minute pre-observation stare which was discarded in theanalysis, but included in the Astronomical ObservationRequest to allow the telescope and instruments to set-tle after slewing (Grillmair et al. 2012). To minimizethe pixel-phase effect and achieve a pointing accuracyto within ∼ https://github.com/petigura/k2phot https://exoplanet.dfm.io/en/stable/ hours centered near the predicted time of mid-transitfrom the K2 ephemeris of the sixth planetary signalfound by Christiansen et al. (2018). Individual frameexposure times were set to two seconds to stay in thelinear regime of the detector for this bright target. Thesubarray mode was used to minimize readout times anddata volume. In total, 19,840 individual frames weretaken.We performed centroiding and aperture photometryusing photutils (Bradley et al. 2019), fitting a 2DGaussian to each image. To select the optimal aper-ture radius, we computed photometry from the centroidpositions using fixed radii between 1.5 and 3.0 pixels in0.1 pixel increments. Background levels were found bythe method described by Knutson et al. (2011). Thisprocess entails masking out the regions within a radiusof 12 pixels from the centroid along with the central tworows and columns, then finding the median backgroundvalue of the pixels after clipping 3 σ outliers.We modeled systematics in the Spitzer light curvesusing pixel-level decorrelation (PLD; Deming et al.2015). PLD has become a premier techniquefor correcting
Spitzer systematics in planet transitanalyses (e.g., Beichman et al. 2016; Benneke et al.2017; Dressing et al. 2018; Feinstein et al. 2019;Livingston et al. 2019; Berardo et al. 2019), and wasdeveloped to account for intra-pixel sensitivity varia-tions which produce intensity fluctuations in the pho-tometry. In our analysis, we used PLD to model the
Spitzer systematics simultaneously with the exoplanetsystem parameters. The full model is described by: S ( t ) = n P i =1 w i D i ( t ) n P i =1 D i ( t ) + m · t + M tr ( θ, t ) , (1)where w i are individual time-independent pixel weightsin the n selected pixels in the region centered on the star, D i ( t ) is the observed flux (or counts) in the individualpixels of the selected region for each time step t , m isthe slope of a linear temporal ramp, and M tr ( θ, t ) is thetransit model with model parameters θ . The first part ofthis equation normalizes the individual pixel intensitiesso their sum at each time step is unity. In our analysiswe used a 3 × Transit Fitting
Using emcee (Foreman-Mackey et al. 2013), we simul-taneously model the K2 and Spitzer data, computingthe posterior probability distributions for six transitingplanets and the
Spitzer systematics. To model the tran-sits we used batman (Kreidberg 2015), which solves the Figure 2. (Upper) The raw K2 light curve from the k2phot pipeline (gray points) with the Gaussian process fit (black) andsix planet fits (colors). (Middle) The Gaussian process flattened light curve with planet fits. (Lower) Residuals after removal ofall planet transit signals. analytic equations for an exoplanet transit as derived inMandel & Agol (2002). We computed posterior proba-bility distributions for the mid-transit times T , orbitalperiods P , the ratios of planet to star radii R p /R ⋆ , thescaled semi-major axes a/R ⋆ , impact parameters b , twosets of quadratic limb darkening coefficients q and q (one set for K2 and another for Spitzer ), and the ninepixel weights w i and linear slope m from Equation 1. Weperformed an autocorrelation analysis to ensure chainconvergence. Due to our large set of parameters, weused 500 walkers and 250,000 steps.The resultant K2 light curve fit using the median val-ues of the posteriors is shown in Figure 2, with the phasefolded light curves shown in Figure 3. Figure 4 shows a https://emcee.readthedocs.io/en/latest/tutorials/autocorr/ clear transit event in the Spitzer data for K2-138 g, con-firming the existence of a sixth planet in the K2-138 sys-tem. We note that Christiansen et al. (2018) obtainedhigh resolution AO imaging of the K2-138 system, rul-ing out nearby stellar companions that could contami-nate or mimic a planet signal. Further, since K2-138 isa multi-planet system, it is more likely that additionaltransit-like signals come from another planet (validationby multiplicity, e.g., Lissauer et al. 2014; Sinukoff et al.2016). Table 2 lists all the derived planet parametersfor the K2 and Spitzer data. We compare the K2 and Spitzer light curves for K2-138 g in Figure 5. The tran-sit durations for the light curves are nearly identical, butthe transit depth posterior distributions show a slightlylarger radius in the
Spitzer data, although the differenceis < σ . Hardegree-Ullman et al.
Figure 3.
Phase folded K2 light curves for the six transiting K2-138 planets with best-fit transit models overlaid. The twotransits for K2-138 g are shown in different colors and shapes in the lower right panel. Figure 4. (Upper) Raw
Spitzer flux (gray points) with transit and systematics model for K2-138 g. (Middle) Systematicscorrected flux (gray points) with transit model and ∼
20 minute binned data (red) to highlight the drop in flux. (Lower) Residualsfrom the transit model fit.
Hardegree-Ullman et al.
Figure 5.
Transit comparison between K2 (upper left) and Spitzer (upper right) on the same scale. The lower panel shows thecomputed radius posteriors in Earth radii for both K2 and Spitzer . The
Spitzer radius is larger, but it is still consistent withthe K2 radius within 1 σ . It is also possible that systematics could bias the radius measurements. For example, having only twotransits in the 30 minute cadence K2 data means that any outliers could skew the measured transit depth. Additional transitsat these and other wavelengths will be necessary to constrain atmospheric properties of this planet. - g Table 2 . K2-138 planet parameters.
Planet Period T T R p R p a a b i F T † eq TSMd BJD-2457700 hr R ⋆ R ⊕ R ⋆ AU ◦ F ⊕ Kb 2 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +1 . − . . +126 . − . +78 − . +1 . − . c 3 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +70 . − . +65 − . +10 . − . d 5 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +41 . − . +57 − . +8 . − . e 8 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +23 . − . +52 − . +8 . − . f 12 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +13 . − . +45 − . +32 . − . g K . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +4 . − . . +0 . − . . +0 . − . . +0 . − . . +2 . − . +31 − . +29 . − . g ‡ Spitzer · · · . +0 . − . . +0 . − . . +0 . − . . +0 . − . · · · · · · · · · · · · · · · · · · · · ·† Equilibrium temperatures were computed assuming a Bond albedo of 0.3. ‡ Period, a , b , and i were computed jointly with the K2 data. Hardegree-Ullman et al. DISCUSSION3.1.
Near Resonances and Gap Planets
The ratio of orbital periods between successive K2-138 planets are: c:b = 1.513, d:c = 1.518, e:d = 1.529,f:e = 1.544, g:f = 3.290. In order to determine howclose to 3:2 resonance these planets are, we estimatedthe mean motion resonance widths using the programfrom Volk & Malhotra (2020) , which is based on theanalytical derivations of resonance widths in the single-planet limit from Murray & Dermott (2000). For thiscalculation, we used the masses of planets K2-138 b, c,d, and e from Lopez et al. (2019), and estimated massesof K2-138 f and g (6 . +8 . − . and 8 . +12 . − . M ⊕ ) frommass–radius relationships (Ning et al. 2018). The re-sults of this calculation, out to fourth order mean motionresonances, are shown in Figure 6. Within the upper andlower planet mass limits, K2-138 b, c, d, and e are near(within a few half-widths) their mutual 3:2 resonancesat low eccentricity, but the outer pair of planets are notnear any low-order resonances.The sizeable gap between K2-138 f and g leads to spec-ulation that there could be additional non-transitingplanets in the system. Indeed, Gilbert & Fabrycky(2020) suggest ∼
20% of high multiplicity planet systemshost additional planets in the gaps between detectedplanets. Each consecutive planet pair of K2-138 has pe-riod ratios that slip further away from 3:2, and assumingthe orbital period ratios continued at 1.544 (f:e), planetscould be expected with orbital periods near 19.70, 30.42,and 46.98 days. However, if the K2-138 planets were allin perfect 3:2 resonance with planet b, there would or-bits at 17.87, 26.80, and 40.21 days. Without additionaldata, we are unable to conclude whether or not K2-138 gwould be near a 3:2 resonance with a planet in the gap.Multi-planet systems have been found to be highlycoplanar (e.g., Fabrycky et al. 2014; Zhu et al. 2018;Gilbert & Fabrycky 2020), however, the more distant aplanet orbits, the closer to 90 ◦ inclination it must be tobe in a transiting geometry. Assuming orbital periodsof 19.70 and 30.42 days, planets around K2-138 wouldneed to be at inclinations above 88 . ◦ and 89 . ◦ , respec-tively, for us to observe them in transit. Even withinthe Solar System, the planets are nearly coplanar, yetthey still have mutual inclinations between 0 . ◦
33 and 6 . ◦ (Winn & Fabrycky 2015).We further explore the possibility of planets withinthe gap between planets f and g using DYNAMITE ,which uses population statistics to predict previously https://github.com/katvolk/analytical-resonance-widths https://github.com/JeremyDietrich/dynamite Figure 6.
The location and analytically estimated widthsof mean-motion resonances for the K2-138 system. Eachplanet is plotted in relative size to the other planets along thediscontinuous y-axis indicating orbital period. Eccentricityis given along the x-axis, and extending from each planetis a line out to the eccentricity at which the planet wouldcross another planet’s orbit. Horizontal dashed lines indicatethe locations of interior (e.g., 3-b:2) and exterior (e.g., 3:2-c) resonances up to fourth order, color coded to match thelabel. The shaded regions surrounding each resonance lineare the resonance widths corresponding to the lower (darker)and upper (lighter) planet mass limits. Planets b, c, d, ande are sufficiently near their mutual 3:2 resonances at loweccentricity for their dynamics to be affected, likely inducingTTVs (see Section 3.2). (a) (b)(c) (d) Figure 7.
DYNAMITE predictions of undetected planets. (a) Inputs of known K2-138 planets yields a prediction of a planet orplanets with high relative likelihood in the gap between planets f and g, and additional planet(s) beyond g. (b) When planetsc and e are removed,
DYNAMITE predicts planets at their respective locations, indicating the predictive model yields the resultswe expect. (c) Injection of a planet at 19.70 days results in a planet prediction near 30 days, though with smaller relativelikelihood than in the previous two scenarios. (d) Injection of a planet at 30.42 days yields a planet prediction near 20 dayswith a moderate relative likelihood. undetected planets (Dietrich & Apai 2020). This modeltakes inputs of stellar parameters (radius, mass, tem-perature) and known planet parameters (inclination, ra-dius, period), and yields probability distributions wherethe population models predict a planet or planets mightexist. We considered four different scenarios as inputsto
DYNAMITE , which are shown in Figure 7: (a) all cur-rently known/detected K2-138 planets, (b) removal ofK2-138 c and e, (c) all K2-138 planets with a planet in-jected at 19.70 days, and (d) all K2-138 planets with aplanet injected at 30.42 days. In scenario (a),
DYNAMITE predicted a planet or planets to be within the gap be-tween K2-138 f and g, and beyond K2-138 g. The modelaccurately predicted the locations of K2-138 c and e inscenario (b). From our planet injection tests in scenar- ios (c) and (d), the models predicted a planet near 30and 20 days, respectively.3.2.
Masses and TTVs
Due to its distance from first order resonance, TTVmeasurements for K2-138 g would be difficult. Us-ing
TTVFaster (Agol & Deck 2016) we computed TTVamplitudes of 2 . +0 . − . , 4 . +1 . − . , 8 . +1 . − . , 6 . +3 . − . ,7 . +1 . − . , and 0 . +0 . − . minutes for K2-138 b, c, d,e, f, and g, respectively. Our inputs to TTVFaster were the masses of planets K2-138 b, c, d, and e fromLopez et al. (2019) and the aforementioned estimatedmasses of K2-138 f and g. We also assumed zero eccen-tricity. Our average six minute (1 σ ) K2 timing preci-sion was insufficient to measure TTVs for this system,however, higher cadence (one minute) observations with2 Hardegree-Ullman et al.
CHEOPS should improve the timing precision enoughto allow detection of TTVs of the inner five planets.In measuring the masses of the inner four K2-138 plan-ets, Lopez et al. (2019) did not identify additional plan-ets, though additional signals might have been absorbedby their Gaussian process to fit out stellar activity at the5.6 m s − level. The mass measurement of K2-138 f washindered by its orbital period of 12.8 days, near half the24.7 day stellar rotation period. The stellar rotation pe-riod might also hinder detection of a planet in orbit nearthe next 3:2 resonance beyond planet f around 20 days.Future planet searches and mass measurements for thissystem would likely benefit from simultaneous photo-metric and RV observations, as Kosiarek & Crossfield(2020) suggest this could enhance the precision of RVmeasurements. Lopez et al. (2019) were unable to reli-ably measure a mass of K2-138 g either, but assuming amass of 8 . +12 . − . M ⊕ , we predict an RV semi-amplitudeof 1 . +2 . − . m s − . If there were planets between f and gof similar masses to the other planets in the system, wewould expect them to have RV semi-amplitudes between1.5 and 2.5 m s − , which would make them similarly dif-ficult to detect due to stellar activity levels.We note that the outer five planets of K2-138are all sub-Neptunes similar in size, and planetb is likely a rocky super-Earth with a density of5 . +2 . − . g cm − . Common sizing of multi-planet sys-tems has previously been found for Kepler systems(e.g. Millholland et al. 2017; Wang 2017; Weiss et al.2018; Gilbert & Fabrycky 2020). From a planet for-mation standpoint, Adams et al. (2020) found that en-ergy optimization occurs when planets are nearly equalin mass for low-mass (super-Earth/sub-Neptune) planetsystems, which is consistent with what we see withK2-138. Though, we note that the outer planetsof K2-138 have larger radii than the inner planets,a trend consistent with the findings of Ciardi et al.(2013), Millholland et al. (2017), Kipping (2018), andWeiss et al. (2018), possibly the result of enhanced pho-toevaporation closer to the star. We plot the planet radiiwith respect to incident stellar flux for the K2-138 plan-ets, compared to the population of K2 planets (shown asdensity contours) from Hardegree-Ullman et al. (2020)in Figure 8. K2-138 b has incident flux ( F ⊕ ) over 400times higher than Earth and is the only planet in thesystem below the planet radius valley. The other plan-ets in the system receive less than 250 F ⊕ , apparentlylow enough to retain an atmosphere.From Figure 8, it appears that many of the K2-138planets are inflated relative to their counterparts withsimilar incident stellar flux. If the system was relativelyyoung, we would expect the planets to still be undergo- Figure 8.
K2-138 planet radii vs. incident stellar flux. Thecontours represent the population of 816 confirmed and can-didate K2 planets from Hardegree-Ullman et al. (2020). Thehigh incident stellar flux on K2-138 b likely stripped awayits atmosphere, placing it below the planet radius valley,whereas the other planets were able to maintain their at-mospheres. ing mass loss. Lopez et al. (2019) computed an age of2 . +0 . − . Gyr for K2-138 based on chromospheric emis-sion, and 2 . +3 . − . Gyr from their joint radial velocity,light curve, and spectral energy distribution analysis.Similarly, we input photometry, stellar parameters, anda rotation period of 24.7 days into the isochrone fittingwith gyrochronology package stardate and computean age of 2 . ± . ∼ ∼ Kepler mission unveiled a statistical overabundance ofplanet pairs just outside of first-order mean motion res-onances, specifically 2:1 and 3:2 (Lissauer et al. 2011;Millholland & Laughlin 2019). As noted in Section 3.1,most of the planet pairs of K2-138 fall just outside a3:2 resonance. Tidal forces from the host star can pushplanets into near-resonant configurations, but host-startides alone cannot explain how all the energy from thisprocess is dissipated to keep planets in this configura-tion. Millholland & Laughlin (2019) showed that obliq- https://github.com/RuthAngus/stardate Figure 9.
The orbits and planets of K2-138 and HD 158259, highlighting the similarities of the two systems. The orbits andthe star sizes are to scale, but the planets are enlarged by 50 × to show detail. Orbital distances of 0.034, 0.046, 0.060, 0.080,0.105, and 0.135 AU for HD 158259 b, c, d, e, f, and tentative planet (g), respectively, were computed from Kepler’s third lawusing stellar mass and planet orbital periods from Hara et al. (2020). uity tides may be the source of energy dissipation thathelps sculpt these near-resonant systems. Consequently,these tidal forces heat the planet interiors, leading toatmospheric inflation (Millholland 2019). We posit thatK2-138 is a strong candidate for planet radius inflationdue to obliquity tides.3.3. Comparison to Other Multi-planet Systems
To date, there have only been nine other exoplanetsystems with six or more confirmed planets , includingradial velocity discovered systems HD 10180 (6 plan-ets), HD 219134 (6 planets), and HD 34445 (6 planets),and transiting systems Kepler-11 (6 planets), Kepler-20 (6 planets), Kepler-80 (6 planets), Kepler-90/KOI-351 (8 planets), TRAPPIST-1 (7 planets), and TOI-178(6 planets). Perhaps most similar to K2-138, however,is the HD 158259 system, with five confirmed planetsand a sixth candidate outer planet (Hara et al. 2020),all near 3:2 orbital mean motion resonances. Four ofthe confirmed planets were detected in radial velocitydata with the SOPHIE spectrograph, and the inner-most planet was found to be transiting in TESS data.The outermost candidate planet orbits every 17.4 days, https://exoplanetarchive.ipac.caltech.edu/cgi-bin/TblView/nph-tblView?app=ExoTbls&config=PS, as of February 2021. close to the stellar rotation period, complicating con-firmation of this planet. The five innermost planetsof K2-138 and HD 158259 are each located at nearlyidentical distances to their host stars, as shown in Fig-ure 9. We estimated HD 158259 planet radii for thefive non-transiting planets using the planet masses fromHara et al. (2020) and the mass-radius relationships ofChen & Kipping (2017). These non-transiting planetsare also all similar-sized sub-Neptunes, with estimatedradii larger than 2 R ⊕ –again consistent with the afore-mentioned common sizing of multi-planet systems. Eachrespective planet in HD 158259 is slightly smaller thanits counterpart K2-138 planet, which could be the resultof HD 158259 being a larger host star (1 . ± . M ⊙ )that is more efficient at stripping away planetary atmo-spheres by intense irradiation (Ehrenreich et al. 2015).We note, however, that there are significant uncertain-ties in planetary mass-radius relationships. Withouttransit data, it is difficult to test whether or not thissystem undergoes tidal radius inflation as mentioned inSection 3.2.We qualitatively compared the orbital spacing ( a/R ⋆ )of these high-multiplicity systems with transiting plan-ets (Figure 10). In addition to the K2-138 system, thereis a sizeable gap between the outermost detected transit-ing planets of the Kepler-11, Kepler-20, and Kepler-804 Hardegree-Ullman et al.
Figure 10.
Orbital spacing of systems with six or more planets, and at least one transiting planet. Systems are arranged fromlargest (top) to smallest (bottom) stellar host, and the regions are colored according to the host temperatures (Harre & Heller2021). The width of the colored regions are scaled to the stellar radii. Planets are to scale with the stellar radii but enlargedby 10 × for clarity, and are placed at their respective transiting inclination angles (randomly distributed above and below thestellar mid-point; gray line). Non-transiting planet locations are labeled in blue. For clarity, we did not plot HD 219134 h onthis scale, but note that it is located at a/R ⋆ = 857. In addition to K2-138, the systems Kepler-11, Kepler-20, HD 219134, andKepler-80 have notable gaps between their outermost planets. systems. HD 219134 has two transiting planets and fournon-transiting planets detected via RV measurements,again with a large gap between the two outermost plan-ets. This large outermost planet gap is also present inthe RV system HD 34445. Notably, a non-transitingplanet was identified in the gap between outer planetsKepler-20 f and d with RV data (Buchhave et al. 2016).As noted in Section 3.1, planets orbiting further outmust be closer to i = 90 ◦ to be in a transiting geome-try, but RV and TTV data may uncover unseen planets.We encourage further investigations of this outer planetgap feature in high-multiplicity planet systems in orderto disambiguate whether it is caused by observationalbiases or planet formation processes.3.4. JWST, ARIEL, and Future Prospects
We computed the transmission spectroscopy metric(TSM) for the K2-138 planets as defined by Equations1, 2, and 3 of Kempton et al. (2018). The TSM is theexpected signal-to-noise for a 10 hour observing programwith JWST/NIRISS. For planets b, c, d, and e, we usedthe planet masses measured by Lopez et al. (2019), andfor planets f and g, we used our estimated masses. Theequilibrium temperature was calculated assuming zeroalbedo and full day-night heat redistribution. The re-sultant TSM values are listed in Table 2, and are ∼ >
90 for high quality atmosphericcharacterization of sub-Neptune sized planets. For now,the K2-138 planets are unlikely to be selected as high-priority targets for JWST observations.The European Space Agency Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL) spacemission aims to gather transmission spectra of 1000 exo-planets during its four year mission (expected to launchin 2028) in order to study their composition, formation,and evolution. Edwards et al. (2019) compiled a list ofpotential targets for ARIEL, taking into account cur-rently known stellar and planet parameters. K2-138 fallsvery near the average star system considered for this tar-get list. The inner five planets of K2-138 also fall withinthe range of planets considered for the potential targetlist, but very few planets with orbital periods beyond ∼
20 days will likely be considered, all but ruling outobservations of K2-138 g. However, since K2-138 con-tains five similarly-sized sub-Neptunes with a ∼
500 Krange of equilibrium temperatures from warm to tem-perate, these planets might provide a unique test bedfor comparative sub-Neptune atmosphere studies.We have confirmed the existence of K2-138 g, solid-ifying K2-138 as the largest K2 multi-planet system.K2-138 g breaks the continuous near 3:2 mean motionresonance of the inner five planets, but the sizeable gap Spitzer Space Telescope , which was operated by theJet Propulsion Laboratory, California Institute of Tech-nology under a contract with NASA.This paper includes data collected by the K2 mission.Funding for the K2 mission is provided by the NASAScience Mission directorate.This research has made use of the NASA ExoplanetArchive, which is operated by the California Institute of Technology, under contract with the National Aero-nautics and Space Administration under the ExoplanetExploration Program.This research made use of Astropy, a community-developed core Python package for Astronomy(Astropy Collaboration et al. 2013a, 2018a).This research made use of exoplanet (Foreman-Mackey et al. 2020) and its dependen-cies (Agol et al. 2020; Astropy Collaboration et al.2013b, 2018b; Foreman-Mackey et al. 2020;Foreman-Mackey et al. 2017; Foreman-Mackey 2018;Kipping 2013; Luger et al. 2019; Salvatier et al. 2016;Theano Development Team 2016; Van Eylen et al.2019).KV acknowledges funding from NASA (grant80NSSC18K0397). Facilities:
Spitzer, Kepler, Exoplanet Archive
Software:
Astropy (Astropy Collaboration et al.2013a, 2018a), batman (Kreidberg 2015), dustmaps (Green et al. 2018),
DYNAMITE (Dietrich & Apai2020), emcee (Foreman-Mackey et al. 2013), exoplanet (Foreman-Mackey et al. 2020), isoclassify (Huber et al. 2017),
NumPy (Harris et al. 2020), photutils (Bradley et al. 2019),
SciPy (Virtanen et al.2020),
TTVFaster (Agol & Deck 2016)REFERENCES
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