The Orbit and Mass of the Third Planet in the Kepler-56 System
Oderah Justin Otor, Benjamin T. Montet, John Asher Johnson, David Charbonneau, Andrew Collier-Cameron, Andrew W. Howard, Howard Isaacson, David W. Latham, Mercedes Lopez-Morales, Christophe Lovis, Michel Mayor, Giusi Micela, Emilio Molinari, Francesco Pepe, Giampaolo Piotto, David F. Phillips, Didier Queloz, Ken Rice, Dimitar Sasselov, Damien Ségransan, Alessandro Sozzetti, Stéphane Udry, Chris Watson
DDraft version July 3, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THE ORBIT AND MASS OF THE THIRD PLANET IN THE KEPLER-56 SYSTEM
Oderah Justin Otor , Benjamin T. Montet , John Asher Johnson , David Charbonneau ,Andrew Collier-Cameron , Andrew W. Howard , Howard Isaacson , David W. Latham ,Mercedes Lopez-Morales , Christophe Lovis , Michel Mayor , Giusi Micela , Emilio Molinari ,Francesco Pepe , Giampaolo Piotto , David F. Phillips , Didier Queloz , Ken Rice , Dimitar Sasselov ,Damien S´egransan , Alessandro Sozzetti , St´ephane Udry , Chris Watson Draft version July 3, 2018
ABSTRACTWhile the vast majority of multiple-planet systems have orbital angular momentum axes that alignwith the spin axis of their host star, Kepler-56 is an exception: its two transiting planets are coplanaryet misaligned by at least 40 degrees with respect to the rotation axis of their host star. Additionalfollow-up observations of Kepler-56 suggest the presence of a massive, non-transiting companion thatmay help explain this misalignment. We model the transit data along with Keck/HIRES and HARPS-N radial velocity data to update the masses of the two transiting planets and infer the physicalproperties of the third, non-transiting planet. We employ a Markov Chain Monte Carlo sampler tocalculate the best-fitting orbital parameters and their uncertainties for each planet. We find the outerplanet has a period of 1002 ± . ± .
38 M
Jup . We also place a 95%upper limit of 0.80 m s − yr − on long-term trends caused by additional, more distant companions. Subject headings: planets and satellites: fundamental parameters, planets and satellites: individual:Kepler-56, techniques: radial velocities INTRODUCTION
Red giant Kepler-56 (KOI-1241, KIC 6448890) is anatypical star to host transiting planets. While the vastmajority of known transiting planets orbit solar-typeFGK stars (Batalha et al. 2013; Burke et al. 2014; Mul-lally et al. 2015; Rowe et al. 2015; Grunblatt et al. 2016;Van Eylen et al. 2016), Kepler-56 is one of only a fewpost-main sequence stars known to host them (Lillo-Boxet al. 2014; Ciceri et al. 2015; Quinn et al. 2015; Pepperet al. 2016). Detecting transits of these stars is difficultbecause they are much larger than main sequence stars [email protected] Department of Astrophysical Sciences, Princeton University,4 Ivy Lane, Princeton, NJ 08544, USA Cahill Center for Astronomy and Astrophysics, CaliforniaInstitute of Technology, 1200 E. California Blvd., MC 249-17,Pasadena, CA 91106, USA Harvard-Smithsonian Center for Astrophysics, 60 GardenStreet, Cambridge, MA 02138, USA SUPA, School of Physics & Astronomy, University of St. An-drews, North Haugh, St. Andrews Fife, KY16 9SS, UK Department of Astronomy, University of California, Berke-ley CA 94720, USA Observatoire Astronomique de l’Universit´e de Gen`eve, 51 ch.des Maillettes, 1290 Versoix, Switzerland INAF - Osservatorio Astronomico di Palermo, Piazza delParlamento 1, 90134 Palermo, Italy INAF - Fundacion Galileo Galilei, Rambla Jose Ana Fernan-dez Perez 7, 38712 Brena Baja, Spain INAF - IASF Milano, via Bassini 15, 20133, Milano, Italy Dipartimento di Fisica e Astronomia “Galileo Galilei”, Uni-versita’di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy INAF - Osservatorio Astronomico di Padova, Vicolodell’Osservatorio 5, 35122 Padova, Italy Cavendish Laboratory, J. J. Thomson Avenue, CambridgeCB3 0HE, UK SUPA, Institute for Astronomy, University of Edinburgh,Royal Observatory, Blackford Hill, Edinburgh, EH93HJ, UK INAF - Osservatorio Astrofisico di Torino, via Osservatorio20, 10025 Pino Torinese, Italy Astrophysics Research Centre, Queen’s University Belfast,Belfast BT7 1NN, UK and have higher levels of correlated noise (Barclay et al.2015). As such, when selecting targets for
Kepler , mis-sion scientists prioritized capturing main sequence FGKstars over other stellar types (Batalha et al. 2010).Nevertheless, Kepler-56 was targeted in the original
Kepler mission (Borucki et al. 2010), and two transitingplanet candidates with periods of 10 .
50 and 21 .
41 dayswere identified in the first data release (Borucki et al.2011). These candidates interacted dynamically, withobserved, anticorrelated variations in their times of tran-sit (Ford et al. 2011, 2012; Steffen et al. 2012). Steffenet al. (2013) analyzed the times of transit and the orbitalstability of the system to confirm these two candidatesas planets, making Kepler-56 the latest stage star knownat the time to host multiple transiting planets.As a red giant, Kepler-56 exhibits convection-drivenoscillations that vary on timescales long enough to beobservable with
Kepler long-cadence photometry. Huberet al. (2013) analyzed its observed asteroseismic modesto infer a stellar mass of 1 . ± .
13 M (cid:12) and radius of4 . ± .
15 R (cid:12) . Through radial velocity (RV) and transittiming observations of the transiting planets, Huber et al.(2013) then determined their masses to be 22 . +3 . − . M ⊕ and 181 +21 − M ⊕ , respectively. Through a combination ofasteroseismology and dynamical instability simulations,they also detected that the orbits of the planets, whilecoplanar with each other, are tilted with respect to theaxis of stellar rotation by ∼
45 degrees.Huber et al. (2013) also detected the presence of a long-term RV acceleration in the data consistent with at leastone additional massive companion. While the accelera-tion by itself cannot provide a unique orbit for the outercompanion, they proposed that both the planetary obliq-uity and long-term RV trend could be explained by anon-transiting companion with a period of 900 days andmass 3 . Jup . a r X i v : . [ a s t r o - ph . E P ] S e p However, the duration of their RV observations onlycovered a baseline of ∼
100 days. Equipped with fourmore years of RV data, we are now able to measure theorbital parameters of this purported planet, which hasthe third-longest orbital period of any confirmed planetorbiting a Kepler star (Kostov et al. 2015; Kipping et al.2016). We are also able to place upper limits on thepresence of additional planets from the lack of additionallong-term trends in the RV curve.In Section 2 we describe our data collection and reduc-tion. In Section 3, we describe our RV model. In Section4, we present our best estimates for this planet’s orbitalparameters, as well as the likelihood of another compan-ion. We discuss our results in Section 5 and summarizeour findings in Section 6. DATA COLLECTION AND ANALYSIS
Our analysis is based on 43 RV observations of Kepler-56 obtained from 2013 to 2016 with two different spec-trographs: 24 with Keck/HIRES (Vogt et al. 1994) and19 with HARPS-North (Cosentino et al. 2012).
Keck/HIRES Observations
Our Keck/HIRES observations were obtained largelyfollowing the standard procedures of the CaliforniaPlanet Survey (CPS) team (Howard et al. 2010), modi-fied slightly for the faint stars of the
Kepler field, follow-ing the approach of Huber et al. (2013). For all obser-vations, we used the C2 decker (14 . (cid:48)(cid:48) × . (cid:48)(cid:48) R ≈ HARPS-North Observations
We also obtained 19 observations of Kepler-56 withHARPS-North, a high-precision echelle spectrograph atthe 3.6 m Telescopio Nazionale Galileo (TNG) at theRoque de los Muchachos Observatory, La Palma, Spain.HARPS-N is a fiber-fed high-resolution ( R = 115 , − apparent shift between the two sets. TABLE 1RV Observations of Kepler-56
Time(BJD-2,400,000) RV (m s − ) RVuncertainty Spectrograph56076.904 -38.30 2.51 HIRES56099.841 -13.18 2.47 HIRES56109.825 57.33 1.74 HIRES56116.089 -4.45 1.56 HIRES56134.000 46.27 1.73 HIRES56144.079 17.81 2.02 HIRES56153.087 88.74 3.29 HIRES56163.981 37.40 1.91 HIRES56166.962 44.33 1.83 HIRES56176.856 88.23 2.29 HIRES56192.844 108.96 1.86 HIRES56450.040 19.85 1.78 HIRES56469.099 3.95 1.86 HIRES56472.114 16.87 1.99 HIRES56476.995 1.60 2.03 HIRES56478.884 -29.12 1.65 HIRES56484.063 -72.43 2.00 HIRES56484.883 -72.22 1.50 HIRES56489.997 -13.62 1.62 HIRES56506.878 -84.42 1.78 HIRES56512.910 -14.65 1.77 HIRES56521.883 -54.61 1.73 HIRES56533.873 -38.28 2.05 HIRES56613.758 -99.60 2.23 HIRES56462.573 -54305.87 4.45 HARPS-N56514.602 -54269.66 2.92 HARPS-N56514.623 -54258.05 3.05 HARPS-N56515.556 -54259.96 4.82 HARPS-N56515.557 -54271.49 5.02 HARPS-N56515.578 -54258.19 4.35 HARPS-N56545.423 -54331.28 2.42 HARPS-N56549.407 -54343.28 3.12 HARPS-N56829.617 -54375.25 3.16 HARPS-N56831.525 -54359.40 2.30 HARPS-N56850.615 -54386.77 4.12 HARPS-N56865.533 -54359.15 2.21 HARPS-N57123.719 -54256.32 4.55 HARPS-N57181.709 -54148.86 2.71 HARPS-N57254.564 -54202.52 6.42 HARPS-N57330.394 -54147.12 3.19 HARPS-N57528.706 -54303.21 4.78 HARPS-N57565.651 -54280.83 5.24 HARPS-N57566.674 -54290.92 3.12 HARPS-N Note . — The Keck/HIRES pipeline returns RVs with the sys-temic RV, γ , removed; this offset is retained in the HARPS-N RVs,leading to an apparent shift of 54.25 km s − . ORBIT FITTING
With the RV data in hand, we can determine the or-bital parameters of the outer planet. We develop codethat, for a given set of orbital parameters, returns theexpected RV contribution from each planet at a listof user-specified times following Lehmann-Filh´es (1894)and Eastman et al. (2013).Our algorithm does not include variations caused bydynamically interacting planets. However, Kepler-56 b’sRV signal is small relative to our RV precision and themagnitude of Kepler-56 c’s perturbation is small relativeto its orbital period, so we do not expect to see anyperturbation signal in the data. The two spectrographpipelines return different RV offsets, so we make an initialguess for the relative offset between the two in our fitting.For each planet, we include the minimum mass( m sin i ), including the unknown inclination of the non-transiting planet, and two vectors which define the ec-centricity and argument of periastron ( √ e cos ω and √ e sin ω ), following Eastman et al. (2013).For the outer planet alone, we include orbital period( P ) and time of transit ( t tr , if it were so aligned); thesevalues are fixed for the inner planets. Stellar mass ( M (cid:63) ),separate instrumental offsets ( γ ), and RV jitter terms( σ jitter ) for HARPS and HIRES complete our list of pa-rameters. Functionally, as the HARPS pipeline returnsa measurement with the systemic RV included (ignor-ing features like the gravitational redshift and convectiveblueshift), the offset associated with that instrument ap-proximates the true systemic velocity of the star whilethe offset for HIRES brings these two sets of observa-tions onto the same scale.We only consider models of three planets plus a long-term RV acceleration. While it is possible that two plan-ets in circular orbits with orbital periods near a 2:1 pe-riod ratio can masquerade in RV observations as a singleplanet with a higher eccentricity (Anglada-Escud´e et al.2010), there is no evidence that such an effect is occurringin our data set. However, we lack the phase coverage tofully rule out this hypothesis. More observations whereour coverage is sparse would be helpful to probe for afourth planet in resonance with the third.After solving Kepler’s equation to obtain the Keplerianorbital elements, the function produces radial velocitiesfollowing:RV = (cid:18) πGP (cid:19) / m sin i ( M (cid:63) + m ) / √ − e × (cid:20) cos ( θ ( e, ω, t obs , t obs ) + ω ) + e cos ω (cid:21) . (1)Here, θ represents the true anomaly, t obs is its specificvalue, and ω is the argument of periastron.With our function’s ability to generate an RV curvefor any specified period, we can test various combina-tions of the outer companion’s orbital parameters. Weexploit this ability in performing successive fits to obtainan initial estimate of our planetary parameters. Maximum Likelihood Estimation
First, we perform maximum likelihood estimation viaPython’s scipy.optimize.minimize routine. For thepossible companion, we take all values as unknown.Specifically, we fit for √ e cos ω , √ e sin ω , m sin i , P , M (cid:63) , t tr , and ˙ γ , the acceleration of the entire system over time.Since our measurements come from two instruments, weinclude independent offset terms for each, γ HARP S and γ HIRES , where γ is the systemic RV offset term intro-duced in Section 3. There are 17 free parameters in to-tal – these, plus √ e cos ω , √ e cos ω , and m sin i for eachplanet (as mentioned in Section 3).Maximum likelihood estimation is a process in whichwe calculate the logarithm of likelihood ( L ) by comparing our data ( D ) to the sum of our generated RV curvesthrough the standard equation:ln ( L ) = − N − (cid:88) i (cid:18) D i − RV i σ jitter,i (cid:19) + 12 N ln (2 πσ jitter,i ) (2) σ jitter,i = σ i + j (3)We use σ jitter in order to incorporate jitter. Sourcesof jitter include uncertainties in measurements beyondphoton noise that arise from sources like noise in thedetector or stellar activity. For sub-giant stars, typicaljitter values are 3-5 m s − (Johnson 2008). Given thelonger exposures for this star relative to previous studiesof planets around relatively bright subgiants, we mightexpect a lower level of jitter as the integrations will av-erage over the higher-order modes.We initialize the fit with values from Huber et al.(2013). However, we note a typo in Table 1 of the discov-ery paper: the listed times of transit in that paper aretoo large by 20 days. They should be 2454958.2556 and2454958.6560 days for Kepler-56 b and c, respectively,rather than 2454978.2556 and 2454978.6560 days.We reject trials with nonphysical results such as nega-tive masses and periods. For steps that are not rejected,we apply normal priors with expected values and 1 σ un-certainties based on measurements from Huber et al.(2013) for the asteroseismic mass of the host star andthe inner planets’ photodynamical eccentricity vectors,based on the TTV analysis of the Kepler light curve.The sum of the logarithm of each prior term is saved foreach set of parameters that is tested.Then, we calculate the logarithm of the posterior prob-ability for each model, which is the sum of the log-priorand log-likelihood terms (as maximizing the logarithmof a function is equivalent to maximizing the functionitself). Equation 4 illustrates this process:ln (cid:2) p (RV | D ) (cid:3) = ln (cid:2) (cid:89) N ( θ ) (cid:3) + ln (cid:2) L ( D | RV) (cid:3) (4)Equation 4 calculates the logarithm of the posteriorprobability distribution function for any set of model pa-rameters ( θ ) as compared to our RV data ( D ). The com-bination of parameters found by this process to make thedata most probable then becomes the initial guess for ourfinal fitting process. Markov Chain Monte Carlo Analysis
We use the result of maximum-likelihood estima-tion from Section 3.1 as the initialization for emcee (Foreman-Mackey et al. 2013), a Markov Chain MonteCarlo (MCMC) implementation for Python of the affine-invariant ensemble sampler of Goodman & Weare (2010).Our 17 parameter simulation uses 150 walkers and 6000steps, with an observed burn-in of 1500 steps. RESULTS
We detect a massive, non-transiting companion, des-ignated Kepler-56 d, with final best-fit values and un-certainties listed in Table 2. The RV curve generatedby our highest-confidence combination of parameters canbe seen in tandem with its uncertainties and our origi-nal RV data in Figure 1. In the same figure, we alsoshow the maximum likelihood orbits for each individualplanet, as well as the data with the maximum likelihoodsignals from the other two planets removed. These dataare only for visualization purposes; at all times we fit thecontributions from all three planets simultaneously.
TABLE 2Orbital Parameters for the Kepler-56 System
Parameters Maximum-likelihoodbest-fits emcee median fits& 1 σ uncertaintiesKepler-56 b √ e cos ω ± √ e sin ω -0.04 -0.04 ± e ± ω (Radians) a -0.20 -0.19 ± M sin i (M ⊕ ) 29.4 30.0 ± √ e cos ω -0.00 -0.01 ± √ e sin ω -0.12 -0.05 ± e ± ω (Radians) a -1.61 -1.70 ± M sin i (M ⊕ ) 191 195 ± √ e cos ω ± √ e sin ω -0.12 -0.12 ± e ± ω (Radians) a -0.27 -0.26 ± M sin i (M ⊕ ) 1767 1784 ± M sin i (M Jup ) 5.55 5.61 ± P (days) 1002 1002 ± t tr , (BJD-2,400,000) 56449 56450 ± γ (m s − yr − ) -0.26 -0.25 ± γ HARPS (m s − ) -54276.1 -54276.2 ± γ HIRES (m s − ) -27.7 -27.7 ± σ jitter, HARPS (m s − ) 0.72 1.23 ± σ jitter, HIRES (m s − ) 1.68 1.80 ± a Derived quantity
For Kepler-56 d itself, we return a Doppler semiampli-tude of 95.21 ± − , corresponding to a mini-mum mass of 5 . ± .
38 M
Jup (1784 ±
120 M ⊕ ). Wealso measure a period of 1002 ± . ± .
01, and a semimajor axis of 2 . ± .
08 AU.
Limits on a Fourth Planet
A fourth planet beyond the orbit of Kepler-56 d, if itexists, could be observable through the detection of along-term trend in the data. Given our three-year base-line of observations, we can place limits on the presenceof such an outer companion. From our emcee results, wefind a long-term RV acceleration of − . ± .
32 m s − yr − . The 95 th percentile value of the emcee posteriorprobability distribution for ˙ γ provides an upper limit onacceleration from a fourth planet of 0.80 m s − yr − .From Montet et al. (2014), we know the maximumtrend caused by a planetary companion on a circular or-bit is˙ γ = (6 .
57m s − yr − ) (cid:18) m p sin iM J (cid:19)(cid:18) a (cid:19) − , (5)where m p is the mass of the planet, M J the mass of Jupiter, and a the orbital semimajor axis. From this,we can place limits on the presence of outer companionswith m sin i larger than 0 . M J at 10 AU and 1 . M J at 20 AU; such companions must be at particular pointsin their orbits or at low inclination in order to evade RVdetection.At V (cid:39)
13 mag, Kepler-56 falls just within
Gaia ’sbright-star limit (Perryman et al. 2014). A fourthplanet’s acceleration on Kepler-56 in
Gaia astrometrymight be detectable at the level of 10-20 µ as/yr overthe course of the mission. Averaging over flat priors fororbital angles and eccentricity, at the nominal distanceof Kepler-56 ( d ∼
850 pc),
Gaia could in principle detectcurvature due to orbital motion of a companion of (cid:38) J at 10 AU or (cid:38)
80 M J at 20 AU. These values in Equa-tion 5 return, at the lowest, an acceleration of 32.85 ms − yr − . This is much higher than the limits returned byour fit, suggesting that, save for face-on orbits, Gaia willbe less helpful than continued RV observation in placingfurther limits on a fourth planet.A fourth planet in a near-resonant orbit with Kepler-56 d could masquerade as a single eccentric planet, asdescribed by Anglada-Escud´e et al. (2010). However,we find the probability of this scenario to be low. Re-running our emcee fit with the outer planet’s eccentricityfixed at 0 leads to decreased likelihoods for the fit as awhole, and we do not detect any long-term structure inthe residuals. However, our observations do not have thetime resolution necessary to make a definitive assertionon this effect. Complete phase coverage of Kepler-56 d isneeded to answer this question. DISCUSSION
Comparison to Previous Work
Our research supports that of Huber et al. (2013) infinding strong evidence for a massive, non-transiting ex-oplanet in the Kepler-56 system. Now that our observa-tions span a full Kepler-56 d orbit, we can compare ourresults with the projections from Huber et al. (2013), whopredicted that both the planetary obliquity and long-term RV trend could both be broadly explained by anon-transiting companion with a period of 900 days andmass 3 . Jup .Both our minimum mass and period are similar tothe representative values listed by Huber et al. (2013).Kepler-56 d’s minimum mass could be commensuratewith that of a giant planet or a brown dwarf (for inclina-tions below 30 degrees). This could have implications forthe near 2:1 resonance of the inner planets’ orbits, as wellas for the misalignment of their orbital plane with thatof Kepler-56 ’s rotation. Indeed, Li et al. (2014) simu-lated several scenarios and found a higher probability ofthe observed misalignment being of a dynamical origin(e.g. Fabrycky & Tremaine 2007) than from migrationof the bodies in a tilted protoplanetary disk (e.g. Bateet al. 2010) or through angular momentum transport inthe star itself that led to an apparent misalignment, evenif the system was originally aligned (Rogers et al. 2012).While Kepler-56 d is a possible source of dynamicalperturbation, Gratia & Fabrycky (2016; submitted) sim-ulate the scattering of two giant outer planets and findscattering between a system of three outer planets is re-quired to excite the two inner planets of the system to
Time (JD-2,400,000) R a d i a l V e l o c i t y ( m / s ) Kepler-56 b
Period = 10.501 days
Kepler-56 c
Period = 21.405 days
Kepler-56 d
Period = 1001.709 days
Phase (days)
Phase (days)
Time (JD-2,400,000)
Fig. 1.—
A suite of results from our MCMC fit. (Top) Kepler-56 RV data – with HIRES points in red and those of HARPS in navy –together with a curve whose gradient represents the differing confidence levels of MCMC’s results, with the darkest navy representing themedian fit and lighter shades corresponding to the 1, 2, and 3 σ uncertainties on the RV of the star as a function of time. (Middle) Identicalto Panel 1, save that each data point and confidence curve has been subtracted by the median fit RV in order to show the data residualsand uncertainty as a function of time. (Lower left) Individual, phase folded RV contribution of Kepler-56 b to the system’s total RV. Thecontributions of the other two planets are subtracted from the HARPS and HIRES data displayed on the plot for visualization purposes.(Lower middle) The phase folded version of the lower left plot for Kepler-56 c with the signals from the other two planets removed forvisualization purposes. (Lower right) Kepler-56 d’s individual, non-phase folded RV contribution, again with the other two planets removed. inclinations similar to those observed in the data whilepreserving coplanarity. These additional planets, if real,must be scattered to large orbital separations or ejectedentirely to evade detection by our RV observations. The Effect of Kepler-56 d on Transits of the InnerPlanets
Huber et al. (2013) inferred masses of the system’sinner planets by dynamically modeling their transits,ignoring possible perturbations from the third, outerplanet. We verify that this is a reasonable assumptionby checking two effects that may be significant: a tidalterm corresponding to the change in the gravitational po-tential as Kepler-56 d completes its orbit, and a Roemerdelay as the distance to the inner planets and host starvary over the orbit of the outer planet.Following Equations 25-27 of Agol et al. (2005), thetidal effects would cause, over a long time baseline, thetransits of an inner planet with mass m and period P to be perturbed with a standard deviation σ = 3 βe √ − e ) / (cid:20) − e − e − e (cid:21) / , (6)where e is the eccentricity of the outer planet and β = m π ( m + m ) P P . (7)Here, m is the mass of the outer planet with orbitalperiod P , and m the mass of the host star.For the values in Table 2 for our system, we find per-turbations in the time of transit on the order of four sec- onds for Kepler-56 b and sixteen seconds for Kepler-56 c.Given that the precision in the measurement of times oftransit of these planets is typically tens of minutes, we donot expect these perturbations to affect, or be noticeablein, the measured times of transit.The light travel time, or Roemer, delay is the resultof changes in the physical distance between the observerand the host star due to the orbit of the outer body.Following Equations 6 and 7 of Rappaport et al. (2013),its magnitude is bounded such that A R ≤ G / c (2 π ) / P / (cid:20) m sin i ( m + m + m ) / (cid:21) , (8)where G is Newton’s constant, c the speed of light, andall other terms retain their meaning from the previousequation. Inserting values from Table 2 again, we find anexpected light travel time signal not to exceed 5 seconds,significantly smaller than the observed uncertainties, sowe do not expect Kepler-56 d to affect the orbits of theinner planets in any observable way. Alternative Methods of Measuring Kepler-56 d
From our model, we measure a time of transit forKepler-56 d of BJD-2 , ,
000 = 56 , ± Kepler dataset that is visible byeye, but do not observe one in this window. As we knowthe posterior distribution of allowed times of transit, wecan determine the probability the planet transited dur-ing a data gap. In Quarters 6 and 7, there are four datagaps larger than 12 hours in which a transit could re-side. Together, these gaps represent 1.7% of the massof the posterior distribution of the time of central tran-sit. The transit duration allows us to place even tighterconstraints. If Kepler-56 d transited with an impact pa-rameter b = 0, the transit would have a duration of 3.1days. As none of the gaps are longer than 20 hours induration, we can additionally rule out any transits with b < .
95. By again integrating over the posterior dis-tribution but accounting for the nonzero transit dura-tion, assuming a flat distribution in impact parameter,we find that only 0.07% of allowed transits fall fully in-side a data gap. If Kepler-56 d were to transit, there isa 99.93% proabability it would be observable in the
Ke-pler data. Given this low probability and the a priori small transit probability for a companion on a ∼ m sin i ) and or-bital semimajor axis ( a ) of Kepler-56 d, we can considerthe possibility that the Gaia astrometric mission wouldbe able to constrain its inclination. For lower (more face-on) inclinations, the planet will have a higher mass andthe center of mass of the system will move closer to theplanet. Additionally, the astrometric orbit will changeshape on the sky, with more face-on inclinations appear-ing more circular throughout an orbit.Given the distance to the system ( d ∼
850 pc) andthe inferred semimajor axis 2 . ± .
07 AU, the orbit ofKepler-56 d has a projected semimajor axis on the skyof ∼ . − µ as. Perryman et al. (2014)determined that Gaia will detect planets with astromet-ric signatures larger than 68 µ as for stars as bright asKepler-56, meaning this planet would evade detection atall except the lowest inclinations. However, given thatthe Gaia data can be combined with the prior informa-tion about the orbit of Kepler-56 d from RVs, it may bepossible that the planet will be detected at slightly lowerinclinations. Regardless, the prospects of a robust de-termination of the outer planet’s complete set of orbitalparameters from
Gaia appear unlikely.
Facilities:
Keck:I (HIRES), TNG (HARPS-N) SUMMARY
Kepler-56, a red giant targeted in the telescope’s pri-mary mission, has a massive, non-transiting companiondetected through radial velocities. This star is one ofonly a few red giants known to have transiting planets,and these planets orbit with a nearly 2:1 period ratioon a plane misaligned relative to the spin of their hoststar. The presence of another body in the system wasfirst detected by Huber et al. (2013) with observationsfrom Keck/HIRES; we follow them up with subsequentobservations from HIRES and HARPS-North at TNG.Incorporating these new data, we model the RV curvefor a three-planet system. Our results confirm the exis-tence of Kepler-56 d, with a period of 1002 ± . ± .
38 M
Jup . We also return an upper limit of acceleration from a possible fourth planetof 0.80 m s − yr − at 95% confidence, severely restrict-ing the possibility of the existence of other giant planetswithin ∼
20 AU. We find that Kepler-56 d should notbe detectable through its dynamical effect on the tran-sits of the two inner planets, but for sufficiently face-on(more massive) orbits could be detectable through
Gaia observations of its astrometric wobble.We thank Eric Agol, Daniel Fabrycky, and Daniel Hu-ber for comments and conversations which improved thequality of this manuscript.O.J.O. thanks the members and friends of the Ban-neker Institute, who made the summer in which thisproject began a fruitful time. He also thanks Neta Bah-call for allowing him to continue this research as his se-nior thesis. He gratefully acknowledges support from theBanneker Institute and Princeton’s astrophysics depart-ment, Class of 1984, and Office of the Dean of Under-graduate Students in facilitating travel to AAS 227 topresent this research. He would be remiss to forget theother members of the Party of Three and their associates.B.T.M. is supported by the National Science Foun-dation Graduate Research Fellowship under Grant No.DGE1144469.J.A.J. is supported by generous grants from the Davidand Lucile Packard Foundation and the Alfred P. SloanFoundation.C.A.W. acknowledges support from STFC grantST/L000709/1.This publication was made possible through the sup-port of a grant from the John Templeton Foundation.The opinions expressed in this publication are those ofthe authors and do not necessarily reflect the views ofthe John Templeton Foundation. This material is basedupon work supported by the National Aeronautics andSpace Administration under grant No. NNX15AC90Gissued through the Exoplanets Research Program.The research leading to these results has received fund-ing from the European Union Seventh Framework Pro-gramme (FP7/2007-2013) under Grant Agreement No.313014 (ETAEARTH.)Some of the data presented herein were obtained at theW.M. Keck Observatory, which is operated as a scientificpartnership among the California Institute of Technol-ogy, the University of California and the National Aero-nautics and Space Administration. The Observatory wasmade possible by the generous financial support of theW.M. Keck Foundation. The authors wish to recognizeand acknowledge the very significant cultural role andreverence that the summit of Maunakea has always hadwithin the indigenous Hawaiian community. We are mostfortunate to have the opportunity to conduct observa-tions from this mountain.The HARPS-N project was funded by the Prodexprogram of the Swiss Space Office (SSO), the HarvardUniversity Origin of Life Initiative (HUOLI), the Scot-tish Universities Physics Alliance (SUPA), the Universityof Geneva, the Smithsonian Astrophysical Observatory(SAO), and the Italian National Astrophysical Institute(INAF), University of St. Andrews, Queens UniversityBelfast, and University of Edinburgh.observations of its astrometric wobble.We thank Eric Agol, Daniel Fabrycky, and Daniel Hu-ber for comments and conversations which improved thequality of this manuscript.O.J.O. thanks the members and friends of the Ban-neker Institute, who made the summer in which thisproject began a fruitful time. He also thanks Neta Bah-call for allowing him to continue this research as his se-nior thesis. He gratefully acknowledges support from theBanneker Institute and Princeton’s astrophysics depart-ment, Class of 1984, and Office of the Dean of Under-graduate Students in facilitating travel to AAS 227 topresent this research. He would be remiss to forget theother members of the Party of Three and their associates.B.T.M. is supported by the National Science Foun-dation Graduate Research Fellowship under Grant No.DGE1144469.J.A.J. is supported by generous grants from the Davidand Lucile Packard Foundation and the Alfred P. SloanFoundation.C.A.W. acknowledges support from STFC grantST/L000709/1.This publication was made possible through the sup-port of a grant from the John Templeton Foundation.The opinions expressed in this publication are those ofthe authors and do not necessarily reflect the views ofthe John Templeton Foundation. This material is basedupon work supported by the National Aeronautics andSpace Administration under grant No. NNX15AC90Gissued through the Exoplanets Research Program.The research leading to these results has received fund-ing from the European Union Seventh Framework Pro-gramme (FP7/2007-2013) under Grant Agreement No.313014 (ETAEARTH.)Some of the data presented herein were obtained at theW.M. Keck Observatory, which is operated as a scientificpartnership among the California Institute of Technol-ogy, the University of California and the National Aero-nautics and Space Administration. The Observatory wasmade possible by the generous financial support of theW.M. Keck Foundation. The authors wish to recognizeand acknowledge the very significant cultural role andreverence that the summit of Maunakea has always hadwithin the indigenous Hawaiian community. We are mostfortunate to have the opportunity to conduct observa-tions from this mountain.The HARPS-N project was funded by the Prodexprogram of the Swiss Space Office (SSO), the HarvardUniversity Origin of Life Initiative (HUOLI), the Scot-tish Universities Physics Alliance (SUPA), the Universityof Geneva, the Smithsonian Astrophysical Observatory(SAO), and the Italian National Astrophysical Institute(INAF), University of St. Andrews, Queens UniversityBelfast, and University of Edinburgh.