SOPHIE velocimetry of Kepler transit candidates XIV. A joint photometric, spectroscopic, and dynamical analysis of the Kepler-117 system
G. Bruno, J.-M. Almenara, S. C. C. Barros, A. Santerne, R. F. Diaz, M. Deleuil, C. Damiani, A. S. Bonomo, I. Boisse, F. Bouchy, G. Hebrard, G. Montagnier
AAstronomy & Astrophysics manuscript no. KOI209 c (cid:13)
ESO 2018March 21, 2018
SOPHIE velocimetry of Kepler transit candidatesXIV. A joint photometric, spectroscopic, and dynamical analysis ofthe Kepler-117 system (cid:63),(cid:63)(cid:63)
G. Bruno , J.-M. Almenara , S. C. C. Barros , A. Santerne , , , R. F. Diaz , M. Deleuil , C. Damiani , A. S. Bonomo ,I. Boisse , F. Bouchy , G. H´ebrard , , and G. Montagnier , Aix Marseille Universit´e, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388, Marseille, France Centro de Astrof´ısica, Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal Instituto de Astrof´ısica e Ciˆencias do Espac¸o, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal Observatoire Astronomique de l’Universit´e de Gen`eve, 51 chemin des Maillettes, 1290 Versoix, Switzerland INAF - Osservatorio Astrofisico di Torino, via Osservatorio 20, 10025 Pino Torinese, Italy Observatoire de Haute Provence, 04670 Saint Michel l’Observatoire, France Institut d’Astrophysique de Paris, UMR7095 CNRS, Universit´e Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, FranceReceived 11 July 2014 / Accepted 8 November 2014
ABSTRACT
As part of our follow-up campaign of Kepler planets, we observed Kepler-117 with the SOPHIE spectrograph at the Observatoire deHaute-Provence. This F8-type star hosts two transiting planets in non-resonant orbits. The planets, Kepler-117 b and c, have orbitalperiods (cid:39) . (cid:39) . ff erent masses (0 . ± .
033 and1 . ± . M J ) and low orbital eccentricities (0 . ± . . ± . Key words.
Stars: planetary systems - Stars: individual: Kepler-117 - Techniques: photometric - Techniques: radial velocities -Techniques: spectroscopic - Methods: statistical
1. Introduction
In the past few years, the number of known multiple planet sys-tems detected by the Kepler space telescope has enormouslyincreased. Multiple transiting planet systems have a low false-positive probability: using conservative hypotheses, Lissaueret al. (2012) estimated a 1.12% probability of observing twofalse positives in the same system and a 2.25% probability for asystem to host a planet and show the features of a false positiveat the same time. Their estimation was based on the assumptionsthat false positives are randomly distributed among the Keplertargets and that there is no correlation between the probability ofa target to host one or more detectable planets and display falsepositives.At the time of writing, Kepler has detected 469 multiple planetsystems. Among them, Kepler-117 (also named KOI-209) hoststhe two transiting planets Kepler-117 b and Kepler-117 c. Theseplanets were presented by Borucki et al. (2011) as candidatesand validated by Rowe et al. (2014) with a confidence level ofmore than 99%, while radial velocity observations were still un- (cid:63)
Appendix A is available in electronic form at . (cid:63)(cid:63) Radial velocity tables are only available at the CDS via anony-mous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/573/A124 . http: // exoplanet.eu / . available. After subjecting Kepler-117 b and c to various false-positive identification criteria, Rowe et al. used the statisticalframework of Lissauer et al. (2012) (further refined in Lissaueret al. 2014) to promote them to bona fide exoplanets. Kepler-117b and c were found to have orbital periods (cid:39) . (cid:39) . (cid:39) .
72 and (cid:39) . R J .Multiple planet systems o ff er insights on their dynamical history(e.g., Batygin & Morbidelli 2013) and can show transit-timingvariations (TTVs; Agol et al. 2005), especially if the planets arein mean motion resonance. TTVs are a powerful tool for detect-ing non-transiting planets, and for the determination of plane-tary masses (Holman et al. 2010; Nesvorn´y et al. 2013; Barroset al. 2014; Dawson et al. 2014) and can be a tracer of stel-lar activity, as well (Barros et al. 2013; Oshagh et al. 2013).Moreover, non-detected TTVs can cause an underestimation ofthe uncertainty on the stellar density derived from the photome-try (Kipping 2014).Using only the first quarter of the Kepler photometric data,Ste ff en et al. (2010) predicted TTVs to be observable for theKepler-117 system. At that time, the photometric time cover-age was not su ffi cient to allow a verification. The TTVs werelater confirmed by Mazeh et al. (2013). According to these au-thors, the ratio between the periodic modulation of the TTVsof the inner planet (b) and the orbital period of the outer one(c), P TTVs , b / P c , is (cid:39) . a r X i v : . [ a s t r o - ph . E P ] J a n . Bruno et al.: Kepler-117 in that paper. The similarity between the two periodicities is astrong indication that the two bodies are in the same system andthus is another argument for the validation of Kepler-117 b andc.In this paper, we included the information from the TTVs in thecombined fit of the system parameters together with the pho-tometry and radial velocities we acquired during our observationcampaigns with the SOPHIE spectrograph at the Observatoire deHaute-Provence. By fully exploiting the data, we obtained a pre-cise measure of the masses and radii of the planets. In Sect. 2,the data acquisition and reduction is discussed. In Sect. 3, wedescribe the treatment of the stellar spectra, and in Sect. 4 wereport on the stellar activity, the measurement of the TTVs, andthe joint Bayesian fit of the system parameters. In Sect. 5, the re-sults are discussed. The implications and conclusions are givenin Sect. 6.
2. Observations and data reduction
Kepler-117 was observed by the Kepler space telescope fromquarter 1 to 17 between May 2009 and May 2013. The firstthree quarters were covered by a sampling of 29.4 minutes (long-cadence data, LC), the following were sampled every 58.5 sec-onds (short-cadence data, SC). We chose to use the LC data onlyfor quarters from 1 to 3, and relied on the SC data for the others.The light curves, already reduced by the Kepler pipeline (Jenkinset al. 2010), are publicly available on the Mikulski Archive forSpace Telescopes (MAST). We made use of the light curvescorrected by the Presearch Data Conditioning (PDC) module,available in the light curve fits file.For all the quarters and for both the LC and the SC data, the dis-persion of the contamination of nearby stars, corrected for by thepipeline, is lower than 1%. The contamination value, then, wasfixed in the following combined analysis (Sect. 4.2.2).We isolated the photometric signal around every transit usinga preliminary estimate of the ephemeris, following Rowe et al.(2014). The transits of the two planets sometimes superpose be-cause of their di ff erent periods ( (cid:39) . (cid:39) . σ clipping. Kepler-117 is part of our follow-up program of Kepler candi-dates (Bouchy et al. 2011; Santerne et al. 2012). We acquired15 spectra of this star during two observing seasons, betweenJuly 2012 and November 2013, using the SOPHIE spectro-graph at the 1.93 m telescope of the Observatoire de Haute-Provence (Perruchot et al. 2008; Bouchy et al. 2013). The in-strument was set in high-e ffi ciency mode, with a spectral reso-lution λ/ ∆ λ ∼ / N per pixel at 550 nm between 9 and 17. The spectrawere reduced using the SOPHIE pipeline (Bouchy et al. 2009).The radial velocities (RVs) and their uncertainties were obtainedthrough a Gaussian fit of the cross-correlation function (CCF) http: // archive.stsci.edu / index.html. with numerical masks corresponding to the F0, G2, and K5 spec-tral types. The final RVs were measured with the G2 mask be-cause the spectral analysis showed Kepler-117 to be close to aG star (i.e., Rowe et al. 2014, verified in Sect. 3). However,using di ff erent masks to compute the CCF did not result in asystematic di ff erence between the RVs. The RV reference starHD185144 (Howard et al. 2010; Bouchy et al. 2013; Santerneet al. 2014) was used to correct the RVs by between ∼ ∼
30 m s − . Three spectra were a ff ected by the moonlight: wecorrected them for the RV of the Moon, as discussed in Baranneet al. (1996), Pollacco et al. (2008), and H´ebrard et al. (2008).The charge transfer ine ffi ciency e ff ect was corrected for usingthe prescription of Santerne et al. (2012). The first three echelleorders at the blue edge of the spectrum were not used to calcu-late of the RVs because their low S / N degrades the precision ofthe measurements.We rejected the point at BJD = . / N (9 at 550 nm, the lowest of all the set).We checked for linear correlations between the bisector spanof the CCF and the RVs, following Queloz et al. (2001) (Fig.1). If linear correlations are observed, the planetary scenario isvery likely to be rejected in favor of a blend. The Spearman-rank-order correlation coe ffi cient between the bisector span andthe RVs, excluding the points contaminated by the moonlight,is − . ± .
32. The p-value for this coe ffi cient, with the nullhypothesis of no correlation, is 0.98. Similarly, the Spearmancorrelation coe ffi cient between the full width at half maximumof the CCFs of the spectra and their respective RV is 0 . ± . (cid:37) in the date are contaminated by moonlight,while the one with † was discarded from the analysis (Sect. 2.2). Date BJD v rad σ v rad S / N at- 2450000 [ km s − ] [ km s − ] 550 nm ∗ (cid:37) (cid:37) , † (cid:37) ∗ Measured by the SOPHIE pipeline.
3. Host star
To analyze the stellar atmosphere we used only spectra that werenot a ff ected by moonlight. The only spectrum at an S / N <
2. Bruno et al.: Kepler-117 ∆ RV [km/s]0.50.40.30.20.10.00.10.20.3 B i s e c t o r s p a n [ k m / s ] Fig. 1: Bisector span of the CCF plotted with respect to the radialvelocity measurements (the mean RV has been subtracted). Thered points indicate contamination by the Moon. The uncertaintyon the bisector span of each point is twice the uncertainty on theRV for that point.normalized. The final S / N in the continuum, at 550 nm, is (cid:39) ff ective temperature T e ff , surfacegravity log g , metallicity [Fe / H], and projected rotationalvelocity v sin i (cid:63) with the VWA software (Bruntt et al. 2010a,b,and references therein). This method is based on the fit of themetal spectral lines, especially the iron lines. The best param-eters are those that minimize the correlation of the elementabundances with the excitation potential and the equivalentwidth of the spectral lines. We obtained T e ff = ±
80 K,log g = . ± .
11, and [Fe / H] = . ± .
13, appropriate of anF8V-type star.The estimate of log g was confirmed with the pressure-sensitivelines of CaI at 612.2 nm and the MgIb triplet. Finally, the couple v sin i (cid:63) and v macro was jointly measured by fitting a rotationalprofile on a set of isolated spectral lines. This measure of v sin i (cid:63) (6 ± − ) agrees with the one obtained with the fit of theCCF (Boisse et al. 2010): 6 . ± . − .A first combined fit of the data sets (Sect. 4.2.2) showed a ∼ σ di ff erence between the spectroscopic log g (4 . ± . . ± . g = . ± .
15 and 4 . ± .
15 (reach-ing, in this case, a parameter limit in their model). Rowe et al.(2014), instead, used the publicly available spectra recordedwith the HIRES spectrograph at the Keck I telescope and foundlog g = . ± . and not in the literature, so that we didnot consider them. The log g is known to be a problematic pa- https: // cfop.ipac.caltech.edu / home / . rameter to measure accurately and is correlated with T e ff and[Fe / H]. In particular, a decrease in log g is usually reflected by adecrease in T e ff and in [Fe / H]. The complete set of the three pa-rameters in the articles we referred to is reported in Table 2. Thestellar densities derived from the SOPHIE spectrum, the HIRESspectrum (both calculated with the Dartmouth tracks), and thosefrom the TTVs are shown in Fig. 2.We were unable to identify a problem in the SOPHIE spectra norin our analysis method. We therefore chose to use the publishedcombination of T e ff , log g and [Fe / H] whose log g is the clos-est to our posterior, which converges to a sharp distribution evenin the tail of the large spectroscopic prior of SOPHIE. The finalvalues we adopted are those of Rowe et al. (2014), which wereused as priors in the Bayesian analysis.Table 2: Published spectroscopic parameters for Kepler-117compared with those of this work. Authors T e ff [K] log g [Fe / H]Everett et al. (2013) (1) 6185 ±
75 4 . ± . − . ± . ∗ ±
75 4 . ± .
15 0 . ± . ±
100 4 . ± . − . ± . ±
80 4 . ± .
11 0 . ± . ∗ Two set of parameters are presented in Everett et al. (2013). Thefit marked with (2) is reported to have reached a parameter limit inthe models. . . . . . . ρ ? /ρ (cid:12) . . . . . . . P D F Fig. 2: Stellar densities derived from the spectroscopic parame-ters of the SOPHIE spectrum (dashed line) and the HIRES spec-trum (Rowe et al. 2014) (continuous line). In red, the posteriordistribution from the PASTIS analysis, shaded according to the1-, 2-, and 3 σ intervals.
4. System analysis
The light curve shows small periodic variations, arguably dueto starspots. To identify the periodicities, we removed the tran-sit features and computed the Lomb-Scargle periodogram (LSP:Press & Rybicki 1989) of the light curve (Fig. 3), finding a peakat 10 . ± .
028 days. The uncertainty is underestimated be-cause it does not take into account the position of the spots on thestellar surface and the di ff erential rotation. The (cid:39)
3. Bruno et al.: Kepler-117 (Fig. 3, bottom panel), precisely, by the main peak at 11 . ± . . ± . . ± . v sin i (cid:63) (Sect. 3) and the stellar radius R (cid:63) (Sect.4.2.2), assuming that the rotation axis is perpendicular to the lineof sight (11 . ± . P (cid:63) = . ± . v sin i (cid:63) shows that the stellar inclination is compatible with 90 ◦ . L o m b - S c a r g l e A u t o c o rr e l a t i o n Fig. 3:
Top panel : Lomb-Scargle periodogram of the light curveafter removing the transits.
Bottom panel : autocorrelation of thelight curve. The green dotted lines indicate the Gaussian fit tothe peaks. The red line corresponds to the maximum rotation pe-riod of the star deduced by the v sin i (cid:63) and the stellar radius. Thered shadowed regions highlight the 1-, 2-, and 3 σ confidenceintervals for the rotation period. We performed a combined Bayesian analysis of photometry, ra-dial velocities, stellar parameters, and TTVs. We begin by de-scribing the measurement and modeling of the TTVs and thenpresent the Bayesian analysis.
After reducing the photometric data (Sect. 2.1), we fitted thetransit times with a procedure similar to the one discussed inBarros et al. (2011). All the transits were fitted simultaneouslyto constrain the shape parameters, that is, the normalized separa-tion of the planet a / R (cid:63) , the ratio of planet-to-star radius R p / R (cid:63) ,and the orbital inclination i . For each transit, the primary transitepoch T and three normalization parameters were also fitted toaccount for a quadratic trend with time. The derived transit timesfor each planet are given in Tables A.1 and A.2. After removinga linear ephemeris, the transit times of the two planets showedsignificant TTVs. The TTVs exhibited by planet b are ∼ (cid:39)
28 min against (cid:39) ffi -cients. To model the TTVs, we performed dynamical simulationswith the mercury code, version 6.2 (Chambers 1999). The inte-grations were executed with a Bulirsch-Stoerwe algorithm. Foreach of them, we identified the transit times by interpolating thepassage of the planets through the line of sight. We computed theTTVs by subtracting a linear fit to the transit times. As a com-promise between execution time and accuracy of the TTVs withrespect to the measured uncertainties, we set the simulations tocover the time span of the Kepler photometry, with a step of 0.4days, that is, 1 / The Kepler photometry, reduced as described in Sect. 2.1 andcorrected for the TTVs as explained in Sect. 4.2.1, and theSOPHIE radial velocities were fitted together using the planetanalysis and small transit investigation software (
PASTIS )described in D´ıaz et al. (2014). This software has been primarilydesigned to calculate the Bayesian odds ratios between compet-ing scenarios in planetary validation problems.
PASTIS allowssimultaneously modeling of several data sets and obtainingsamples of the parameter posteriors with a Markov chain MonteCarlo (MCMC) algorithm.An exploration phase was started at random points drawn fromthe priors listed in Table A.3. From the chains computed inthis phase, we used the one with the highest likelihood forthe starting values of the final MCMC set. The solution of theexploration phase with the highest likelihood has the lowesteccentricities.To take into account the di ff erences between the stellar mod-els, we used four evolutionary tracks as input for the stellarparameters: Dartmouth (Dotter et al. 2008), PARSEC (Bressanet al. 2012), StarEvol (Palacios, priv. com.; Lagarde et al.2012), and Geneva (Mowlavi et al. 2012). However, the intrinsicuncertainties in the models were not taken into account. Weran twenty-five chains of about 10 steps for each of the stellarevolutionary tracks. At each step of the MCMC, the modellight curves were oversampled and then binned by a factor10, to correct for the distortions in the signal due to the finiteintegration time (Kipping 2010). We derived the stellar density ρ (cid:63) from the spectroscopic T e ff , log g , and [Fe / H] (Sect. 3) andset it, together with the spectroscopic T e ff and [Fe / H], as ajump parameter, with normal priors for all three of them. Foreach planet, we used Gaussian priors for the period P and theprimary transit epoch T and non-informative priors for theargument of periastron ω , the inclination i , and the eccentricity e . We stress this last point: without imposing zero eccentricities,we consistently measured these key parameters by taking intoaccount all the available information, TTVs included.We used uniform priors for the coe ffi cients of quadratic limb-darkening, for the planetary-to-stellar radius ratio R p / R (cid:63) , andfor the radial velocity amplitude K . For Kepler-117 b, wefitted the longitude of the ascending node Ω , too, for which weimposed a uniform prior. The Ω of planet c was fixed at 180 ◦ because the symmetry of the problem allows freely choosingone of the two Ω s.We expressed the Kepler normalized flux o ff set, the systemicvelocity, and the RV linear drift with uniform priors (separatingLC and SC data in the photometry). Finally, we modeled theinstrumental and astrophysical systematic sources of error with
4. Bruno et al.: Kepler-117 . . . . . . . . R e l a t i v e fl u x − . − .
005 0 .
000 0 .
005 0 . Orbital phase − − O - C [ pp m ] . . . . . . . . R e l a t i v e fl u x − . − .
005 0 .
000 0 .
005 0 . Orbital phase − − O - C [ pp m ] − . − . − . − . − . − . − . R ad i a l V e l o c i t y [ k m s − ] . . . . . . Orbital Phase − − O - C [ m s − ] − . − . − . − . − . − . − . R ad i a l V e l o c i t y [ k m s − ] . . . . . . Orbital Phase − − O - C [ m s − ] − − − − − TT V [ m i n ] . . . . . . Phase (P = 50.79 d) − O - C [ m i n ] − − − TT V [ m i n ] . . . . . . Phase (P = 169.9 d) − O - C [ m i n ] Fig. 4:
Top : phase-folded plot of the best transit model of planet b (left) and c (right), over the SC data. In black the model, in redthe data binned every hundredth of orbital phase.
Center : the same for the radial velocities.
Bottom : The TTVs of planet b folded atthe orbital period of planet c (left) and those of planet c folded at the first peak of its Lomb-Scargle periodogram (right, Sect. 5.1).For each plot, the lower panel shows the residuals as observed minus calculated ( O − C ) points.a jitter term for Kepler, two for the TTVs (one for each planet),and one for SOPHIE. A uniform prior was assigned to all thejitter terms.After they were sampled, every posterior distributionwas thinned according to its correlation length. A combinedposterior distribution was derived by taking the same numberof points from each stellar evolutionary track. This combineddistribution gave the most probable values and the confidenceintervals for the system parameters.Finally, the derived log g and the posterior stellar radius R (cid:63) , T e ff , and [Fe / H], together with the magnitudes in Table A.4,were set as priors for another MCMC run to derive the distanceof the system using the spectral energy distribution (SED). Themagnitudes were fitted to sample the posterior distributions ofthe distance of the system, the interstellar extinction E ( B − V ), and the jitter of the SED. The model SED was interpolatedfrom the PHOENIX / BT-Settl synthetic spectral library (Allardet al. 2012), scaled with the distance, the stellar radius, andthe reddening E ( B − V ), expressed through a Fitzpatrick(1999) extinction law. For both the distance and the reddening,non-informative priors were used.In Table 3 we present the mode and the 68.3% equal-tailedconfidence intervals of the system parameters. According toour analysis, Kepler-117 A is a (cid:39) . ± . . ± . ff er widely in theirmass, but less so in their radii: 0 . ± . M J , 0 . ± . R J for planet b and 1 . ± . M J , 1 . ± . R J for planet c. Theplanetary radii, in particular, agree with the estimate of Roweet al. (2014): (cid:39) . ± . R J for planet b and (cid:39) . ± . R J
5. Bruno et al.: Kepler-117 for planet c.We remark that the measured drift of the RVs is compatible with0 km s − ; a non-zero drift would have been an indication of apossible third companion in the system that a ff ected the ampli-tude of the TTVs. − − F l u x [ e r g sc m − s − A − ] Wavelength [ ˚A] − . . . O - C [ m ag ] Fig. 5: Model SED on the photometric bands. The residuals areshown in the lower panel.
5. Discussion
We tested the robustness of our result by inspecting the peri-odic modulation of the measured and the modeled TTVs. To dothis, we compared their Lomb-Scargle periodograms (Fig. 6).The main peaks coincide for both planets and also agree withthe periodicities found by Mazeh et al. 2013 (see also Ofir et al.2014). The periodogram of the modeled TTVs of planet b re-produces that of the measured TTVs well. Some of the peaksof planet c, on the other hand, are due to noise. This can be ex-plained by the di ff erent amplitude of the signal in the two cases. The RV amplitude produced by Kepler-117 b (6 . ± . − )is close to the sensitivity of SOPHIE for a (cid:39) . V star.Indeed, the SOPHIE RVs alone do not have the precisionrequired to measure this planet’s mass and can only providean upper limit. Including the TTVs allowed us to accuratelydetermine this parameter.The combined fit of the TTVs reduces the uncertainties onother parameters, too. This is because the amplitude of theTTVs is mainly determined by orbital separations, periods,and eccentricities of the orbits and masses ratios of the planets(Agol et al. 2005). The strong constraint on the eccentricities,combined with the constraint on the stellar density (which isdetermined with a precision similar to the precisions achievedin asteroseismology), reduces the error bars on the planetaryparameters. Once again, we stress that the derived uncertaintiesdo not include the uncertainties on all the models of stellaratmospheres and evolutionary tracks we used.To check the impact of the TTVs on the combined fit, we ran PASTIS without them. The di ff erent posteriors of the mosta ff ected parameters, with or without the TTVs, are comparedin Fig. 7 and 8. The mass of planet b presents the most evidentdi ff erence because of its poorly constrained RV amplitude: its value reaches from an upper limit (0.28 M J at 68.3% confidencelevel) to a better constrained value (0 . ± . M J ). Thedi ff erence is smaller for planet c because the amplitude ofthe RVs is larger and better fitted. However, its uncertaintyis roughly reduced by 40%. This indicates that, if possible,including the TTVs in the combined fit is more e ff ective thanfitting them a posteriori , using a set of orbital parametersderived without considering them.We remark that the mass of planet c found with the RVs aloneand with the TTVs are fully compatible. Therefore, the observedTTVs are completely explained by the two planets, within thedata error bars. This agrees with the absence of any RV drift(Sect. 4.2.2).The transit signature is degenerate with respect to the stellarhemisphere the planet covers, while the TTVs are not. In atwo-planet scenario, this can lead to strong correlations betweenthe two inclinations. While running PASTIS , we thereforeconstrained one of the transits in one of the hemispheres andleft the other free to vary. As the inclination of planet b is lowerthan that of planet c, the inclination of planet b was limited toone hemisphere (50 ◦ < i < ◦ ) and that of planet c was leftfree to vary between both (89 ◦ < i < ◦ ). In spite of this, ourfit allowed both hemispheres to be transited by planet c becausethe final inclination of its orbit is almost symmetric with respectto the stellar equator (Fig. 7, bottom line). The solutions with i > ◦ are compatible with those without TTVs (all < ◦ )at 1 σ . In particular, for planet c, we found 89 . ± . ◦ withTTVs and i = . ± . ◦ without them.Using the stellar inclination (Sect. 4.1) and the systemparameters, we calculated the expected amplitude of theRossiter-McLaughlin e ff ects following Eq. 11 of Gaudi & Winn(2007). For planet b, we found 10 . ± . − , for planet c79 ±
13 m s − . Measuring the spin-orbit misalignment wouldthen be possible for planet c, but the transit duration ( (cid:39) ff ort from di ff erent locations tocover a whole transit.The di ff erence between the resulting longitude of the ascendingnode Ω for planet b (177 . ± . ◦ ) and that of planet c (fixedto 180 ◦ ) is compatible with 0 ◦ . Combined with the similarinclinations, this implies two almost coplanar orbits. As most ofthe Kepler planetary systems (Fabrycky et al. 2014), Kepler-117clearly has a flat configuration of the orbits.We ran an MCMC set without the RVs to determinethe reliability of the fit. As expected, the posterior distribu-tions are the same as with the RVs. Systems with low-massplanets presenting TTVs, which are challenging for the RVobservations, would benefit from the approach used in this paper.Finally, we verified that the configuration of the most proba-ble solution is dynamically stable. We ran mercury over a timespan of 10 Myr (Fig. 9). The semi-major axes, eccentricities,and orbital inclinations oscillate over a time scale of around 200years, but all the parameters are stable in the long term. The possibility of a third non-transiting companion can beprobed with the RVs and the TTVs. As already mentionedin Sect. 4.2.2, the absence of stellar drift in the RVs bringsno evidence of the possible presence of a third non-transitingplanet in the system. Moreover, the agreement between themass of planet c, found with the RVs and with or without the
6. Bruno et al.: Kepler-117
Table 3: Planetary and stellar parameters with their 68.3% central confidence intervals.
Stellar parameters from the combined analysis
Stellar density ρ (cid:63) [ ρ (cid:12) ] 0.291 + . − . Stellar mass [ M (cid:12) ] 1.129 + . − . Stellar radius [ R (cid:12) ] 1.606 ± T e ff [K] 6150 ± / H] [dex] -0.04 ± g [cgs] 4.102 ± t [Gyr] 5.3 ± ± E ( B − V ) 0.057 ± ffi cient u a ± ffi cient u b ± − yr − ] 12 ± V r [ km s − ] -12.951 ± Kepler-117 b Kepler-117 c
Orbital period P [days] 18.7959228 ± . × − ± . × − Orbital semi-major axis a [AU] 0.1445 + . − . + . − . Primary transit epoch T [BJD-2450000] 4978.82204 ± . × − ± . × − Orbital eccentricity e ± ± ω [deg] 254.3 ± ± i [deg] 88.74 ± ± † Transit impact parameter b prim ± + . − . ‡ Transit duration T [h] 7.258 ± ± a / R (cid:63) ± ± R p / R (cid:63) ± ± K [ m s − ] 6.5 ± ± Ω [deg] 177.9 ± M p [M J ] 0.094 ± ± R p [R J ] 0.719 ± ± ρ p [ g cm − ] 0.30 ± ± g p [cgs] 2.67 + . − . ± T eq [K] ∗ ±
18 704 ± Data-related parameters
Kepler jitter (LC) [ppm] 67 + − Kepler jitter (SC) [ppm] 0 + SOPHIE jitter a [ m s − ] 0 + SED jitter [mags] 0.043 + . − . TTV1 jitter [min] 0.95 ± ± † From the posterior distribution, reflected with respect to i = ◦ . ‡ Reflected as the inclination, with respect to b = ∗ T eq = T e ff (1 − A ) / (cid:113) R (cid:63) a , with A (planet albedo) = a This value is compatible with the low level of activity observed in the photometry (Sect. 4.1) and confirms the estimate of the error bars onthe RVs (Sect. 2.2). M (cid:12) = . × kg, R (cid:12) = . × m, M J = . × kg, R J = TTVs (Sect. 5.2), shows that the TTVs are not a ff ected by anon-transiting body.A more precise constraint can be obtained by subtracting themodeled RVs of the two planets from the RV measurements. Wefolded the residuals for several periods and fitted them with asinusoid. The amplitude of the sinusoid and the mass of the star(Table 3) allow extracting the maximum mass of the possiblecompanion.The result is plotted in Fig. 10 (filled regions) for the 68.27%,95.45%, and 99.73% confidence intervals. The RVs allow thepresence of a Jupiter-mass planet for some orbital periods. TheTTVs, however, impose a stronger constraint, since includinga third body with the combination of mass and period allowedby the RVs (with the simplifying assumptions of a circular orbit and 90 ◦ inclination) would not fit the TTVs. The black line inthe plot represents a 3 σ di ff erence in the residuals between thefit of the TTVs with a third planet and the best solution withtwo.Therefore, under some simplifying assumptions, the presenceof a non-detected third companion above ∼ . M J on an orbitshorter than ∼
100 days, as well as that of a giant companionwith an orbit shorter than ∼
250 days, is very unlikely.
6. Summary and conclusions
We presented the combined analysis of the Kepler photometry,the TTVs, the SOPHIE RVs, and the spectroscopic observations
7. Bruno et al.: Kepler-117
50 100 150 200 250 300Period [days]0.00.20.40.60.81.0 N o r m a li z e d L o m b - S c a r g l e p o w e r
100 150 200 250 300 350 400 450 500Period [days]0.00.20.40.60.81.0 N o r m a li z e d L o m b - S c a r g l e p o w e r Fig. 6: Lomb-Scargle periodogram of the measured (red) and modeled (black) TTVs for planet b (left) and c (right). The bluetriangle in the plot on the left indicates the orbital period of planet c. . . . . . . ρ ? /ρ (cid:12) P D F Without TTVsWith TTVs . . . . . . . e P D F Planet b without TTVsPlanet b with TTVsPlanet c without TTVsPlanet c with TTVs . . . . . . M/M J . . . . . . . P D F . . . . . . . . . . R/R J P D F . . . . . . . i [ ◦ ]0 . . . . . . . . . P D F Fig. 7:
Upper left: probability density function (PDF) of the stellar density including or excluding the TTVs; to make the plot morereadable, the PDF using the TTVs is divided by 2. The prior from spectroscopy (HIRES, Dartmouth) is shown for comparison.
Upper right: the planetary eccentricities from di ff erent sets of data; the PDF using the TTVs is divided by 12. The color code is thesame in the following plots. Central line: planetary masses and radii from the fit with and without TTVs. The PDF of the massesusing the TTVs is divided by 5, that of the radii by 2.
Bottom: orbital inclinations from di ff erent sets of data.of Kepler-117. This allowed us to measure the stellar, planetary, and orbital parameters of the system. According to our analysis,
8. Bruno et al.: Kepler-117 − . − . − . − . . . . . . ∆ S e m i - m a j o r a x i s [ au ] . . . . . . . . E cc en t r i c i t y Time [yr] I n c li na t i on [ deg r ee s ] Time [Myr]
Fig. 9: Evolution of semi-major axes (top), eccentricities (center), and orbital inclinations (bottom) over a 10 Myr simulation of themost probable solution. The respective mean has been subtracted from the two semi-major axes. On the left, a zoom on the first 300yr; on the right, the variation intervals of the parameters. In blue planet b, in red planet c. The shaded regions in the left panel of theinclinations correspond to the values resulting in a transit (see Eq. 7 of Winn 2010). . . . . . . . M/M J . . . . . . . . . R / R J JS JS
Fig. 8: Mass-radius diagram for the solutions without (cyan forplanet b, magenta for planet c) and with (blue for planet b, redfor planet c) TTVs. The blue and red solutions are those indi-cated in Table 3. The colors, from the center to the edge of theregions, correspond to the 39.3%, 86.5%, and 98.9% joint confi-dence intervals. Jupiter and Saturn (labeled J and S) are markedfor comparison. The other planet parameters were taken fromWright et al. (2011). Kepler-117 A is an F8-type main-sequence star that is about 5Gyr old. The system is composed of two planets. Kepler-117 c,the outer one, has a period of (cid:39)
51 days and a ∼ M J mass.Kepler-117 b, the inner one, has a period of (cid:39)
19 days and a ∼ M Earth mass. The latter produces a RV semi-amplitude of6 . ± . − , close to the limit of sensitivity of SOPHIE forfaint magnitudes. Therefore, its mass and eccentricity cannot beobtained from the RVs alone, even though the derived upper lim-its confirm that the transiting body belongs to the planet realm.Our analysis shows that the inclusion of the TTVs in the com-bined fit allows tightly constraining the mass of the lighterplanet. Taking into account the TTVs in the fit also allows abetter determination of the other system parameters. The stel-lar density was tightly constrained despite a loose prior onthe spectroscopic log g . The planetary radii were strongly con-strained, as were the orbital eccentricities, even if small (around ∼ . − .
05 for both orbits). Measuring the eccentricity ac-curately is important for testing the dynamical models of youngsystems with giant planets. Simulations show that the complexevolution of systems with two planets can be the result of ejec-tion or merging in systems with three planets and can lead to sta-ble, resonant, and low-eccentricity orbits (e.g., Lega et al. 2013).While the RV- and TTV-measured mass agree for planet c, thesame comparison for planet b would benefit from a spectrographwith higher sensitivity than SOPHIE. Only a small part of theplanets known to date have their mass measured with both RVs
9. Bruno et al.: Kepler-117
Fig. 10: Maximum mass of a third possible companion as a func-tion of the orbital period, based on the RV (orange filled ar-eas) and TTV (blue line) observations. The color, from darkerto lighter, corresponds to the 68.27%, 95.45%, and 99.73% con-fidence intervals from the RV data. The blue line represents a 3 σ di ff erence in the residuals on the TTVs from the best solutionwith two planets.and TTVs (e.g., Barros et al. 2014). As observed by Weiss &Marcy (2014), planets smaller that four Earth radii with a massmeasured with TTVs are systematically lower in the mass-radiusdiagram than those discovered by RV surveys. This could be dueto non-detected companions that might dampen the TTVs, caus-ing a systematic underestimation of the masses, or to a lowerdensity of the planets that show TTVs. Indeed, in the com-pact multiplanetary systems that are likely to produce observ-able TTVs, planets with lower masses for a given size are morelikely to reach stable orbits (Jontof-Hutter et al. 2014). In thiscontext, it is remarkable that the RV- and the TTV-measuredmass of Kepler-117 c agree. In addition, under some simplify-ing assumptions, the TTVs almost exclude a non-detected ∼ . M J companion with an orbit shorter than ∼
100 days, as well asa giant companion with an orbit shorter than ∼
250 days.We cannot exclude that while in this particular case the condi-tions are fulfilled for the combined fit to be e ff ective, this is notthe case in general. In fact, TTVs with su ffi ciently high ampli-tude are necessary. This system exhibits significant TTVs eventhough the orbital period ratio of the planets is far from an ex-act low-order mean motion resonance, for which strong TTVsare expected (e.g., Lithwick & Wu 2012). The orbital period ra-tio between planet c and b ( (cid:39) .
7) places this system on thewide side of the 5:2 mean motion resonance. The overabun-dance of systems with period ratios some percent higher thanresonant values than those with a ratio slightly lower than theseones has been well established for the Kepler systems in the caseof first-order resonances, that is, 2:1 or 3:2 (see Lissauer et al.2011, e.g.). The explanation is given in terms of tidal dissipa-tion related to disk-planet or star-planet interactions for close-in orbits (Lithwick & Wu 2012; Batygin & Morbidelli 2013;Delisle et al. 2014), causing the orbital periods to diverge. Todate, this piling-up appears only for some first-order resonancesin the Kepler systems. Instead, the rest of the period-ratio dis-tribution, including higher order resonances, remains flat (e.g.,Batygin & Morbidelli 2013; Fabrycky et al. 2014).We verified that a system with the most probable solution of ouranalysis is dynamically stable. However, we noted the eccen- tricities and the inclinations show small oscillations, that do nota ff ect the stability of the system. We found the planetary orbitsto be almost coplanar. This places Kepler-117 in the most com-mon population of the Kepler multiplanetary systems with a flatconfiguration, as highlighted by Fabrycky et al. (2014).In conclusion, a deeper understanding of the dynamics of orbitalresonances is needed to better reconstruct the history of Kepler-117, which adds valuable information to our knowledge of mul-tiplanetary systems. Acknowledgements.
This paper includes data collected by the Kepler mission.Funding for the Kepler mission is provided by the NASA Science Missiondirectorate. We made use of the Mikulski Archive for Space Telescopes(MAST). Support for MAST for non-HST data is provided by the NASA O ffi ceof Space Science via grant NNX09AF08G and by other grants and contracts.We thank the technical team at the Observatoire de Haute-Provence for theirsupport with the SOPHIE instrument and the 1.93 m telescope and in particularfor the essential work of the night assistants. Financial support for the SOPHIEobservations comes from the Programme National de Planetologie (PNP) ofCNRS / INSU, France is gratefully acknowledged. We also acknowledge supportfrom the French National Research Agency (ANR-08- JCJC-0102-01).The team at LAM acknowledges support by CNES grants 98761 (SCCB),426808 (CD), and 251091 (JMA). AS acknowledge the support from theEuropean Research Council / European Community under the FP7 throughStarting Grant agreement number 239953. A.S. is supported by the EuropeanUnion under a Marie Curie Intra-European Fellowship for Career Developmentwith reference FP7-PEOPLE-2013-IEF, number 627202. ASB acknowledgesfunding from the European Union Seventh Framework Programme (FP7 / mercury andRosemary Mardling for the fruitful discussions about the dynamic of three-bodysystems.This research was made possible through the use of data from di ff erent surveys:the AAVSO Photometric All-Sky Survey (APASS), funded by the RobertMartin Ayers Sciences Fund; the Two Micron All Sky Survey, which is a jointproject of the University of Massachusetts and the Infrared Processing andAnalysis Center / California Institute of Technology, funded by the NationalAeronautics and Space Administration and the National Science Foundation; theWide-field Infrared Survey Explorer, which is a joint project of the University ofCalifornia, Los Angeles, and the Jet Propulsion Laboratory / California Instituteof Technology, funded by the National Aeronautics and Space Administration.This research has made reference to the Exoplanet Orbit Database and theExoplanet Data Explorer at exoplanets.org.
References
Agol, E., Ste ff en, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567Allard, F., Homeier, D., & Freytag, B. 2012, Royal Society of LondonPhilosophical Transactions Series A, 370, 2765Baranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373Barros, S. C. C., Bou´e, G., Gibson, N. P., et al. 2013, MNRAS, 430, 3032Barros, S. C. C., D´ıaz, R. F., Santerne, A., et al. 2014, A&A, 561, L1Barros, S. C. C., Pollacco, D. L., Gibson, N. P., et al. 2011, MNRAS, 416, 2593Batygin, K. & Morbidelli, A. 2013, AJ, 145, 1Boisse, I., Eggenberger, A., Santos, N. C., et al. 2010, A&A, 523, A88Borucki, W. J., Koch, D. G., Basri, G., et al. 2011, ApJ, 728, 117Bouchy, F., Bonomo, A. S., Santerne, A., et al. 2011, A&A, 533, A83Bouchy, F., D´ıaz, R. F., H´ebrard, G., et al. 2013, A&A, 549, A49Bouchy, F., H´ebrard, G., Udry, S., et al. 2009, A&A, 505, 853Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, 427, 127Bruntt, H., Bedding, T. R., Quirion, P.-O., et al. 2010a, MNRAS, 405, 1907Bruntt, H., Deleuil, M., Fridlund, M., et al. 2010b, A&A, 519, A51Chambers, J. E. 1999, MNRAS, 304, 793Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online DataCatalog, 2246, 0Dawson, R. I., Johnson, J. A., Fabrycky, D. C., et al. 2014, ApJ, 791, 89Delisle, J.-B., Laskar, J., & Correia, A. C. M. 2014, A&A, 566, A137D´ıaz, R. F., Almenara, J. M., Santerne, A., et al. 2014, MNRAS, 441, 983Dotter, A., Chaboyer, B., Jevremovi´c, D., et al. 2008, ApJS, 178, 89Everett, M. E., Howell, S. B., Silva, D. R., & Szkody, P. 2013, ApJ, 771, 107Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2014, ApJ, 790, 146Fitzpatrick, E. L. 1999, PASP, 111, 63Gaudi, B. S. & Winn, J. N. 2007, ApJ, 655, 550H´ebrard, G., Bouchy, F., Pont, F., et al. 2008, A&A, 488, 763
10. Bruno et al.: Kepler-117
Holman, M. J., Fabrycky, D. C., Ragozzine, D., et al. 2010, Science, 330, 51Holman, M. J. & Murray, N. W. 2005, Science, 307, 1288Howard, A. W., Marcy, G. W., Johnson, J. A., et al. 2010, Science, 330, 653Jenkins, J. M., Caldwell, D. A., Chandrasekaran, H., et al. 2010, ApJ, 713, L87Jontof-Hutter, D., Lissauer, J. J., Rowe, J. F., & Fabrycky, D. C. 2014, ApJ, 785,15Kipping, D. M. 2010, MNRAS, 408, 1758Kipping, D. M. 2014, MNRAS, 440, 2164Lagarde, N., Decressin, T., Charbonnel, C., et al. 2012, A&A, 543, A108Lega, E., Morbidelli, A., & Nesvorn´y, D. 2013, MNRAS, 431, 3494Lissauer, J. J., Marcy, G. W., Bryson, S. T., et al. 2014, ApJ, 784, 44Lissauer, J. J., Marcy, G. W., Rowe, J. F., et al. 2012, ApJ, 750, 112Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8Lithwick, Y. & Wu, Y. 2012, ApJ, 756, L11Mazeh, T., Nachmani, G., Holczer, T., et al. 2013, ApJS, 208, 16Mowlavi, N., Eggenberger, P., Meynet, G., et al. 2012, A&A, 541, A41Nesvorn´y, D., Kipping, D., Terrell, D., et al. 2013, ApJ, 777, 3Ofir, A., Dreizler, S., Von Essen, C., & Aharonson, O. 2014, in CoRoT3-KASC7Symposium: The Space Photometry Revolution, ed. J. Ballot, & R. A. Garcia,EPJ Web of Conferences, in pressOshagh, M., Santos, N. C., Boisse, I., et al. 2013, A&A, 556, A19Perruchot, S., Kohler, D., Bouchy, F., et al. 2008, in Society of Photo-OpticalInstrumentation Engineers (SPIE) Conference Series, Vol. 7014, Society ofPhoto-Optical Instrumentation Engineers (SPIE) Conference SeriesPollacco, D., Skillen, I., Collier Cameron, A., et al. 2008, MNRAS, 385, 1576Press, W. H. & Rybicki, G. B. 1989, ApJ, 338, 277Queloz, D., Henry, G. W., Sivan, J. P., et al. 2001, A&A, 379, 279Rowe, J. F., Bryson, S. T., Marcy, G. W., et al. 2014, ApJ, 784, 45Santerne, A., D´ıaz, R. F., Moutou, C., et al. 2012, A&A, 545, A76Santerne, A., H´ebrard, G., Deleuil, M., et al. 2014, A&A, 571, A37Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163Ste ff en, J. H., Batalha, N. M., Borucki, W. J., et al. 2010, ApJ, 725, 1226Weiss, L. M. & Marcy, G. W. 2014, ApJ, 783, L6Winn, J. N. 2010, arXiv:1001.2010Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868Wright, J. T., Fakhouri, O., Marcy, G. W., et al. 2011, PASP, 123, 412
11. Bruno et al.: Kepler-117 , Online Material p 1
Appendix A: Additional figures and tables
Table A.1: Transit-timing variations for planet b.
Epoch Mid-transit time TTV Uncertainty[BJD - 2450000] [min] [min]0 4978.82211 -2.9 2.71 4997.62519 7.5 3.03 5035.19749 -20.6 2.54 5054.00955 2.6 3.55 5072.80069 -4.2 2.97 5110.40214 9.6 2.88 5129.19445 4.5 3.09 5147.98545 -2.6 2.610 5166.78207 -1.6 2.511 5185.57076 -12.0 2.412 5204.37575 1.1 2.214 5241.95112 -22.6 2.215 5260.76796 7.6 2.216 5279.56177 4.6 2.417 5298.35371 -1.2 2.418 5317.15385 4.9 2.419 5335.94633 0.0 2.320 5354.74575 5.1 2.321 5373.52714 -15.8 2.322 5392.32105 -18.7 2.323 5411.13384 5.6 2.224 5429.92650 0.9 2.225 5448.71835 -4.9 2.226 5467.52116 5.0 2.527 5486.31647 4.2 2.428 5505.10995 0.7 2.430 5542.68973 -16.7 2.432 5580.29000 -4.5 2.233 5599.08156 -10.8 2.334 5617.89305 11.7 2.236 5655.47836 2.3 2.437 5674.27106 -2.3 2.438 5693.06165 -9.9 2.539 5711.86685 3.4 2.341 5749.44394 -17.8 2.242 5768.25766 7.9 2.243 5787.05285 6.8 2.244 5805.84231 -2.4 2.245 5824.64242 3.6 2.246 5843.43402 -2.6 2.447 5862.23795 9.0 2.548 5881.02176 -8.5 2.449 5899.80911 -20.8 2.450 5918.62126 2.6 2.451 5937.41712 2.5 2.252 5956.20759 -5.3 2.953 5975.01304 8.4 2.155 6012.59970 1.0 2.256 6031.38973 -7.4 2.457 6050.18187 -12.8 2.358 6068.99024 5.1 2.459 6087.77696 -8.1 2.462 6144.17234 2.9 2.363 6162.96539 -1.2 2.364 6181.75998 -3.1 2.265 6200.55215 -8.5 2.366 6219.35264 -1.9 2.568 6256.93344 -17.8 2.469 6275.74812 9.3 2.470 6294.54117 5.2 2.572 6332.13746 11.6 2.273 6350.92542 0.2 2.274 6369.72444 4.6 2.275 6388.51087 -9.0 2.276 6407.30314 -14.2 2.4. Bruno et al.: Kepler-117 , Online Material p 2
Table A.2: Transit-timing variations for planet c.
Epoch Mid-transit time TTV Uncertainty[BJD - 2450000] [min] [min]0 4968.63008 -3.5 1.31 5019.42134 -2.2 1.32 5070.21489 2.3 1.33 5121.00355 -0.1 1.34 5171.79086 -4.5 1.46 5273.37520 0.6 1.27 5324.16706 2.8 1.38 5374.95617 0.9 1.29 5425.74606 0.3 1.210 5476.53664 0.6 1.311 5527.32559 -1.5 1.312 5578.11883 2.6 1.213 5628.90845 1.6 1.214 5679.69704 -1.0 1.316 5781.27846 0.0 1.217 5832.06706 -2.6 1.218 5882.85959 0.5 1.319 5933.65090 1.9 1.220 5984.43899 -1.4 1.221 6035.23056 0.3 1.322 6086.02001 -1.0 1.323 6136.81231 1.7 1.224 6187.60050 -1.4 1.226 6289.18225 0.0 1.327 6339.97047 -3.1 1.228 6390.76238 -0.8 1.6. Bruno et al.: Kepler-117 , Online Material p 3
Table A.3: Prior distributions used in the combined fit with
PASTIS . U ( a , b ) stands for a uniform distribution between a and b ; N ( µ, σ ) indicates a normal distribution with mean µ and standard deviation σ ; N A ( µ, σ − , σ + ), stands for an asymmetric normalwith mean µ , right width σ + and left width σ − ; S ( a , b ) represents a sine distribution between a and b ; finally, J ( a , b ) means aJe ff reys distribution between a and b . Stellar parameters E ff ective temperature T e ff [K] N (6169 , / H] N ( − . , . ρ (cid:63) [ ρ (cid:12) ] (Dartmouth) N A (0 . , . , . ρ (cid:63) [ ρ (cid:12) ] (PARSEC) N A (0 . , . , . ρ (cid:63) [ ρ (cid:12) ] (StarEvol) N A (0 . , . , . ρ (cid:63) [ ρ (cid:12) ] (Geneva) N A (0 . , . , . ffi cient u a U ( − . , . ffi cient u b U ( − . , . − yr − ] U ( − . , . Planet parameters Kepler-117 b Kepler-117 c
Orbital period P [days] N (18 . , . × − ) N (50 . , . × − )Primary transit epoch T [BJD-2450000] N (4978 . , . × − ) N (4968 . , . × − )Orbital eccentricity e U (0 , U (0 , ω [deg] U (0 , U (0 , i [deg] S (50 , S (89 , Ω [deg] U (135 , R p / R (cid:63) J (0 . , . J (0 . , . K [ m s − ] U (0 . , . U (0 . , . System parameters
Distance [pc] U (0 , × )Interstellar extinction E ( B − V ) U (0 , V r [ km s − ] U ( − . , − . Instrumental parameters
Kepler jitter (LC) [ppm] U (0 , . ff set (LC) [ppm] U (0 . , . U (0 , . ff set (SC) [ppm] U (0 . , . U (0 , U (0 , − ] U (0 , U (0 , Table A.4: Magnitudes of the Kepler-117 system.