Discovery of the first Earth-sized planets orbiting a star other than our Sun in the Kepler-20 system
aa r X i v : . [ a s t r o - ph . E P ] M a y Discovery of the first Earth-sized planets orbiting a starother than our Sun in the Kepler-20 system
Guillermo Torres a , Fran¸cois Fressin a,b a Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA02138, USA, e-mail: [email protected] b CVS Health, 1 CVS Dr., Woonsocket, RI 02895, USA, e-mail: [email protected]
Abstract
Discovering other worlds the size of our own has been a long-held dreamof astronomers. The transiting planets Kepler-20 e and Kepler-20 f, whichbelong to a multi-planet system, hold a very special place among the manygroundbreaking discoveries of the
Kepler mission because they finally realizedthat dream. The radius of Kepler-20 f is essentially identical to that of theEarth, while Kepler-20 e is even smaller (0.87 R ⊕ ), and was the first exoplanetto earn that distinction. Their masses, however, are too light to measurewith current instrumentation, and this has prevented their confirmation bythe usual Doppler technique that has been used so successfully to confirmmany other larger planets. To persuade themselves of the planetary natureof these tiny objects, astronomers employed instead a statistical techniqueto “validate” them, showing that the likelihood they are planets is ordersof magnitude larger than a false positive. Kepler-20 e and 20 f orbit theirSun-like star every 6.1 and 19.6 days, respectively, and are most likely ofrocky composition. Here we review the history of how they were found, andpresent an overview of the methodology that was used to validate them. Keywords:
Kepler mission, transiting planets, false positives, multi-planetsystems, Kepler-20, statistical validation.
1. Introduction
Thanks to the
Kepler mission we now know that small planets similar insize to the Earth are common throughout the Galaxy (Howard et al., 2012;Fressin et al., 2013; Dressing & Charbonneau, 2013, 2015; Petigura et al.,2013; Marcy et al., 2014; Burke et al., 2015). What seems so clear now was
Preprint submitted to New Astronomy Reviews May 14, 2019 ot at all obvious at the time the spacecraft was launched in March of 2009,as no such planets had been found outside the solar system. The ones dis-covered until then by the transit method were all Neptune-size ( ∼ R ⊕ ) orlarger . These had all been confirmed by measuring their dynamical massesthrough high precision radial-velocity observations, to show that they areindeed in the planetary range. Here we recount the developments that led tothe discovery of the first two Earth-sized exoplanets, Kepler-20 e and Kepler-20 f (Fressin et al., 2012), which marked a very important milestone in thefield of exoplanet research. Unlike their larger cousins that are amenableto Doppler studies, the masses of Kepler-20 e and Kepler-20 f have not beenmeasured because the reflex motion they induce on the host star is too smallto detect. For this reason these objects required the use of an entirely differ-ent analysis technique to assess their planetary nature.The importance of careful vetting of candidates and of confirmation bythe Doppler technique became painfully obvious as soon as ground-basedtransit surveys began reporting planetary candidates. It was quickly foundthat the vast majority turned out to be false positives of one kind or another(see, e.g., Brown, 2003), with estimates of the false positive rates reaching ashigh as 90% or 95% in some cases (Konacki et al., 2003; O’Donovan et al.,2006a; Latham, 2007). The most common types of astrophysical false posi-tives, often referred to as “blends”, are eclipsing binaries that happen to bealong the line of sight, whether physically associated with the target or not.When this happens, the otherwise deep eclipses of the binary are greatlyattenuated by the target star and made to look so small that they can beindistinguishable from the transit signals of true planets. As it turns out,however, confirming a planet by measuring the reflex motion of the parentstar is not always feasible. For example, the star may be too faint, it may berotating too rapidly, or it may be too chromospherically active to allow thenecessary precision in the radial velocities. Or even if it does lend itself toDoppler studies, the signal may simply be too small to detect, if the orbitalperiod is long and/or the planetary mass too small relative to the mass ofthe star.This was precisely the case for Kepler-20 e and Kepler-20 f. As their des- The era of smaller planet discoveries began in earnest later that same year with theCoRoT mission (Rouan et al., 1998; Baglin et al., 2006), and the announcement of theplanet CoRoT-7 b (L´eger et al., 2009; Queloz et al., 2009), an object about 1.7 times largerthan the Earth. V = 12 .
5, midG-type star in the constellation Lyra (Gautier et al., 2012), and one of themany multi-planet systems (“multis”) that
Kepler would find. For a reviewof the discovery and implications of multis, we refer the reader to the articleby Steffen & Lissauer (2019) in this Special Issue. With transit depths eachunder 100 parts per million hinting at objects of both small sizes and smallmasses, the absence of corresponding radial-velocity signatures was not allthat surprising. As an alternative to dynamical confirmation, an attemptwas made by Gautier et al. (2012) to “validate” these two signals in a sta-tistical way using a procedure that had become known as
BLENDER . The ideabehind
BLENDER is to simulate blend configurations all the way through tothe light curves they are expected to produce, and to use the shape of the realtransits to discriminate against as many of those blends as possible. Otherblends may be rejected if additional observations indicate the intruding ob-jects would have been uncovered. The technique then aims to show in aquantitative way that the likelihood of the remaining false positives is muchsmaller than that of a true planet. This approach had been used in a fewother cases before, but it was relatively new at the time and was not sufficientto demonstrate the planetary nature of Kepler-20 e and 20 f to a high enoughdegree of confidence. Further improvements to
BLENDER would be required,as we describe below, and the procedure did eventually succeed in showingbeyond a reasonable doubt that the two objects are indeed Earth-sized orsmaller planets, with radii of R p = 0 . R ⊕ and 1 . R ⊕ , respectively, asmeasured initially by Fressin et al. (2012).The development of the validation methodology represents a significantadvance in our ability to discover small transiting planets. So far it is theonly alternative we have when the mass cannot be measured directly, eitherby the Doppler effect or by modeling transit timing variations (TTVs) inmulti-planet systems. In fact, statistical validation is now the approach thathas verified the largest number of transiting planets from Kepler and itssuccessor mission K2 , and promises to be invaluable for future space-basedtransit searches as well. Because of its considerable impact for small planetsdiscoveries, and in the spirit of this special issue, we chronicle in the nextsection the history of how validation came about, leading up to its applicationto Kepler-20 e and 20 f. The more technical details of the method may befound in the sources cited below. 3 . Statistical validation: a pathway to the discovery of small plan-ets Readers familiar with the early history of photometric searches for transit-ing planets may recall that the very first lists of candidates, following the mo-mentous discovery of HD 209458 b (Charbonneau et al., 2000; Henry et al.,2000), were released by the Optical Gravitational Lensing Experiment (OGLE;Udalski et al., 2002a,b). Out of one of those lists emerged the second knowntransiting planet, OGLE-TR-56 b (Konacki et al., 2003), a Jovian-size, Jovian-mass object with an orbital period of just 1.2 days that was also the first tobe discovered in a photometric survey . Although it was confirmed dynami-cally, the mass determination for OGLE-TR-56 b was based on few and verychallenging radial-velocity observations given the faintness of the host star( V = 16 . did turn out tobe a false positive (OGLE-TR-33).The capability was later added to generate blend light curves simulta-neously in other passbands, and to predict the overall colors of the blend, HD 209458 b, the first known transiting exoplanet, was originally found in a radial-velocity survey and only later discovered to undergo transits.
When the
Kepler mission began finding very small transit-like signalsthat could not be confirmed dynamically, it became clear that the simula-tion approach would come to be crucial. A way was needed not only toimprove the ability to reject false positives as much as possible, but more im-portantly, to quantify the probability that any one of the remaining blendsmight actually be causing the small drops in brightness. The first real testcame with Kepler-9 d (Torres et al., 2011), a super-Earth with a size of about1.6 R ⊕ . This was the third signal found in the multi-planet system Kepler-9 (Holman et al., 2010) featuring two larger Saturn-sized objects ( ∼ R ⊕ ),which were also the first to display unambiguous TTVs (see the review articleby Ragozzine & Holman, 2019, this Special Issue).The simulations for Kepler-9 d were expanded to include a more complete,grid-based exploration of parameter space for false positives. In addition toeclipsing binaries, the numerical experiments now included scenarios involv-ing an intruding single star located anywhere along the line of sight that istransited by a larger planet, rather than by another star. While it may beargued that this type of contaminant is in itself a bona fide planetary system,the unfortunate alignment with the star one is interested in causes the tran-sits to appear shallower, simulating the presence of a smaller planet orbitingthe target star. As the goal was to prove the existence of a planet of small sizearound the target (rather than a larger one of unknown size around the com-panion), these configurations were considered as false positives. Allowancewas made also for eccentric orbits for all categories of simulated blends, andfor differential interstellar extinction between the intruding star or binaryand the target. Additionally, a more thorough use was made of availablefollow-up observations to help rule out blends. Detection limits from high-resolution imaging were now taken into account, as well as limits on unseenspectroscopic companions in high-resolution spectra, measured color indices5or the target, and other limits on nearby companions based on an analysisof the flux centroids from the Kepler images themselves. All of these helped,but many false positive scenarios still remained viable.The expected numbers of viable false positives of different kinds is ofcourse a function of the number density of stars at the sky location of thetarget, and depends also on how common eclipsing binaries and larger plan-ets are. For Kepler-9 d these blend frequencies were calculated in discretemagnitude bins by counting up the ones that were permitted by all obser-vational constraints, using number densities from Galactic structure modelsalong with estimates of the rates of occurrence of eclipsing binaries and largerplanets from the early
Kepler results. It was also realized that in order toobtain a proper false alarm probability (FAP), or equivalently a confidencelevel that the signal is due to a true super-Earth-sized planet, an estimatewas required of the rate of occurrence of such planets. Expressed in terms ofthe numbers of expected false positives and planets, FAP = N FP / ( N FP + N p ).However, the planet occurrence rate (referred to as the “planet prior”) wasnot well known at the time, so arguments were made drawing on statisticsfrom Doppler surveys, on theoretical considerations, and on preliminary Ke-pler results that were based on candidate detections rather than confirmedplanets. The most conservative of those estimates allowed Kepler-9 d to bevalidated to a sufficiently high level of confidence corresponding to a falsealarm probability of 6 × − (Torres et al., 2011).This framework for simulating blend scenarios and performing statisticalvalidation became known as BLENDER , and over the next year or so it wasapplied in a few other cases with relatively minor changes. The softwareto perform the computationally intensive blend simulations and map outparameter space was ported to the Pleiades supercomputer at the NASAAmes Research Center (California, USA), with the help of Chris Henze.
Kepler-20 e and 20 f were more demanding still than Kepler-9 d because ofthe shallower transits and the associated smaller signal-to-noise ratios. Thismeant they contain less information with which to constrain the detailedshape of the transit and rule out blends. The simple-minded procedure oftabulating blends in discrete magnitude bins to compute their frequencies wasreplaced by a more sophisticated Monte Carlo approach. Background starswere drawn at random from a Galactic structure model, and were assigned6ither a stellar or a planetary companion, depending on the type of false pos-itive. This was done taking into account the known properties of eclipsingbinaries and the size distribution of larger planet candidates, as determinedfrom the
Kepler mission itself. For blends consisting of a planet transitinganother star physically associated with the target (hierarchical triple config-uration) the simulations placed such companions in randomly oriented orbitsaround the host star following the known distributions of periods, mass ra-tios, and eccentricities of binary systems. The frequencies for each type offalse positive were then calculated after removing configurations inconsistentwith constraints from the lightcurve morphology and the follow-up observa-tions. The outcome of this exercise for Kepler-20 e gave a blend frequencyof background eclipsing binaries of 3 . × − , a frequency of backgroundstars transited by larger planets of 2 . × − , and a frequency of hierarchi-cal triple configurations of 5 . × − . These three contributions added upto a total blend frequency of 7 . × − . For Kepler-20 f the numbers were1 . × − + 4 . × − + 3 . × − ≈ . × − . The planet priors, i.e., the apriori chance that the parent star Kepler-20 has a planet of a similar size asimplied by each of the two signals, were estimated again using the catalog of Kepler objects of interest (KOIs), which had expanded by then. As KOIs arestill only candidates, the conservative assumption was made that only 10%of them are real planets, even though other estimates at the time were nearlyan order of magnitude larger (e.g., Morton & Johnson, 2011). The resultingplanet priors were 3 . × − for Kepler-20 e and 7 . × − for Kepler-20 f.By this time it had already been shown that multi-planet systems suchas Kepler-20 tend to be coplanar (Lissauer et al., 2011). Because Kepler-20was already known to have three other transiting planets (Kepler-20 b, 20 c,and 20 d; see below), this made it much more likely that Kepler-20 has atransiting planet at the periods of Kepler-20 e and 20 f than a random Kepler target. With this “multiplicity boost”, the planet priors for 20 d and 20 fincreased to 2 . × − and 7 . × − , respectively. Comparing these to thetotal blend frequencies from above, the false alarm rates became 3 . × − for Kepler-20 e and 7 . × − for Kepler-20 f, which were deemed sufficientlysmall to declare the candidates validated as true Earth-sized planets.Beyond the success in demonstrating the planetary nature of the first twoknown Earth-sized planets, statistical validation using BLENDER has beenapplied to many other planets, including some of the most iconic discov-eries of the
Kepler mission. Examples include
Kepler ’s first rocky planet(Kepler-10 b; Batalha et al., 2011), the first small planets in the habitable7one of their parent stars (Kepler-22 b, Borucki et al. 2012; and Kepler-62 f, Borucki et al. 2013; see also the article by Borucki et al. 2019 in thisSpecial Issue), the first two transiting planets ever discovered in a cluster(Kepler-66 b and Kepler-67 b; Meibom et al., 2013), the discovery of a sub-Mercury-sized planet (Kepler-37 b; Barclay et al., 2013), a transiting planetnear the snow line of its parent star (Kepler-421 b; Kipping et al., 2014),a super-Earth in the habitable zone of a G2 star with an orbital periodnear one year (385 days, Kepler-452 b; Jenkins et al., 2015), the discoveryof a sub-Neptune-sized planet in the open cluster Ruprecht 147 (K2-231 b;Curtis et al., 2018), and others.Statistical validation as an exoplanet discovery tool when Doppler con-firmation is not feasible is now mainstream, and while
BLENDER led the way,several other versions of the same approach with different strengths have nowbeen implemented that were inspired by
BLENDER , such as vespa (Morton,2012) and
PASTIS (D´ıaz et al., 2014). These methods all work by comparingpriors from various scenarios (true planets and false positives) to arrive ata confidence level for planethood. A different technique to validate candi-dates in multi-planet systems was developed by Lissauer et al. (2012), andrefined by Lissauer et al. (2014), which is based on planet multiplicity statis-tics. With reasonable assumptions on the nature and distribution of falsepositives, these authors showed that almost all multi-planet candidates aretrue planets rather than false positives, and that the higher the multiplicity,the more likely the candidates are real planets. This immediately allowedthe validation in bulk of hundreds of
Kepler planets in multis.Interestingly, of the several thousand exoplanets now known, the vastmajority were actually validated (most with vespa , or based on multiplic-ity statistics when in multis) rather than confirmed dynamically (see, e.g.,Rowe et al., 2014; Morton et al., 2016; Crossfield et al., 2016; Mayo et al.,2018). This is partly a reflection of the fact that small planets with unde-tectable Doppler signals far outnumber larger ones, that many of the
Kepler host stars are faint and unsuitable for Doppler studies, and that observingfacilities capable of high-precision (m s − ) radial-velocity measurements arestill very few and far between.
3. Kepler-20 e and Kepler-20 f: two planets the size of the Earth
The discovery of the Kepler-20 multi-planet system was announced tothe community by Gautier et al. (2012). It featured three transiting planets820 b, 20 c, 20 d) with orbital periods ranging from 3.7 to 78 days that wereconfirmed in the traditional way, and that have sizes estimated by thoseauthors of 1.9, 3.1, and 2.8 R ⊕ . The detections were based on eight quarters of Kepler long-cadence (30 min) observations made between 2009 May and 2011March. The same paper gave news of the detection of two additional transit-like signals in the same star that were much shallower and had periods of6.1 and 19.6 days, respectively, but they were left unconfirmed, as mentionedearlier. The validation of these two signals as Kepler-20 e and 20 f, the firsttwo Earth-sized planets, was left to Fressin et al. (2012). The planetary sizesreported by these authors, based on a determination of the properties of thehost star and fits to the light curves, were 0 . +0 . − . R ⊕ and 1 . +0 . − . R ⊕ ,respectively. Fressin et al. (2012) noted also that the first of these planets,Kepler-20 e, is potentially smaller than Venus. Until then the smallest knownexoplanet around a Sun-like star had been Kepler-10 b, with a measuredradius of 1.42 R ⊕ (Batalha et al., 2011). While the difference may not seemall that significant, the validation of Kepler-20 e and 20 f was seen by manyas crossing a threshold of sorts, advancing the frontier of discovery into therealm of planets the size of our own, and smaller.The Kepler-20 multi-planet system has received additional attention morerecently. Buchhave et al. (2016) reported new radial-velocity measurements,and revisited both the stellar parameter determination and the photometricsolutions, now using the full complement of short-cadence (1 min) observa-tions of the star obtained in Quarters 3 through 17, rather than the smallernumber of long-cadence data used previously. This is important becausethe very brief ingress and egress phases of the transit that are so criticalfor constraining the planetary sizes are much better resolved with the 1 minintegrations than the 30 min integrations. The new stellar properties alsobenefited from asteroseismic constraints the authors were able to extractfrom the short-cadence observations.The updated properties obtained by Buchhave et al. (2016) for Kepler-20 e and 20 f are listed in Table 1. They include the orbital semimajor axesand the equilibrium temperatures, on the assumption of full energy redistri-bution and a Bond albedo of 0.3. The orbits were assumed to be circular. Theplanetary sizes are considerably better determined than before, although theactual values differ little from those of Fressin et al. (2012). This is the resultof a trade-off in the recent work between a small increase in the stellar radiusand a small reduction in the radius ratios R p /R ⋆ . Further improvements inthe stellar properties of Kepler-20 are now possible thanks to the availability9 able 1: Main properties of Kepler-20 e and 20 f (Buchhave et al., 2016) Parameter Kepler-20 e Kepler-20 f P (days) 6 . +0 . − . . +0 . − . T c * 968 . +0 . − . . +0 . − . R p ( R ⊕ ) 0 . +0 . − . . +0 . − . i (deg) 87 . +1 . − . . +0 . − . a (au) 0 . +0 . − . . +0 . − . T eq (K)** 1040 ±
22 705 ±
16* Time of mid transit expressed as BJD − , , Gaia cat-alog (Gaia Collaboration et al., 2018), which places the star at a distanceof 282 . ± . R ⋆ = 0 . +0 . − . R ⊙ ,which is about 8% smaller than the determination by Buchhave et al. (2016)of 0 . ± . R ⊙ (a 1.9 σ difference). The new value would reduce thesizes of Kepler-20 e and 20 f even further to about 0.80 and 0.92 R ⊕ , respec-tively, making them both nominally smaller than the Earth, though withuncertainties increased by a factor of two.
4. Kepler-20 architecture, formation, and planetary composition
With their new radial-velocity measurements Buchhave et al. (2016) wereable to improve the mass determinations for the three larger planets Kepler-20 b, 20 c, and 20 d, but the two smaller ones remain below the detectionthreshold. Even assuming a rocky composition, which would maximize theirradial-velocity signal, they are expected to induce reflex motions on the starwith semi-amplitudes of only ∼
20 cm s − . Interestingly, however, the newDoppler measurements have revealed a sixth planet (Kepler-20 g) in this al-ready extraordinary system, which does not undergo transits. It is nestledbetween the outer two previously known planets 20 f ( P = 19 . P = 77 . M p sin i = 20 . +3 . − . M ⊕ , which is larger than that10f Neptune. For completeness, we summarize in Table 2 the main propertiesof this new planet and the remaining ones in the system. We include themeasured velocity semi-amplitudes, K , the estimated orbital eccentricities,the planetary masses, and the mean densities, ρ p . A schematic view of thearchitecture of the Kepler-20 system is shown in Figure 1.Kepler-20 is quite remarkable in that it is very compact: its six knownplanets, five of which transit, are all packed within the orbital distanceof Mercury in our own solar system. Compactness has now been foundto be a feature of many other multi-planet systems as well. Furthermore,Gautier et al. (2012) pointed out a striking feature of Kepler-20 that is thepresence of small and likely rocky Earth-size planets (20 e and 20 f) inter-spersed between larger sub-Neptune planets at smaller and larger orbitalsemimajor axes. This is quite different from our own solar system, in whichthe terrestrial planets, gas giants, and ice giants are neatly segregated inregions with increasing distance from the Sun. Recent studies of samples ofmulti-planet systems have found that the patterns in planet sizes, masses,and spacing are linked through formation and subsequent orbital dynamics,although the full complexity of planetary system architectures is not yet wellunderstood (see, e.g., Millholland et al., 2017; Weiss et al., 2018).The long-term stability of the Kepler-20 system was investigated numer- Table 2: Main properties of the larger Kepler-20 planets (Buchhave et al., 2016)
Parameter Kepler-20 b Kepler-20 c Kepler-20 g Kepler-20 d P (days) 3.696 10.854 34.940 77.611 T c * 967.50201 971.60796 967.50027 997.7303 R p ( R ⊕ ) 1.868 3.047 . . . 2.744 i (deg) 87.4 89.8 < . a (au) 0.0463 0.0949 0.2055 0.3506 e K (m s − ) 4.20 3.84 4.10 1.57 M p ( M ⊕ )*** 9.7 12.8 20.0 10.1 ρ p (g cm − ) 8.2 2.5 . . . 2.7 T eq (K) 1105 772 524 401* Time of mid transit expressed as BJD − , , M p sin i .11 igure 1: The Kepler-20 system:
Orbital configuration of the Kepler-20 system, re-produced from Figure 1 of Buchhave et al. (2016). All six planets are contained withinthe orbital distance of Mercury in our Solar System. Orbital distances are drawn to scale,and planet sizes are rendered in correct proportion to each other, though not on the samescale as the orbits. The size of Kepler-20 g was estimated using its mass and assuminga composition similar to Kepler-20 c. Orbits drawn in blue represent planets with massmeasurements. ically by Gautier et al. (2012) prior to the discovery of the massive non-transiting planet Kepler-20 g, and by Buchhave et al. (2016) afterward. Bothstudies assumed masses for Kepler-20 e and 20 f of about 0.65 and 1.0 M ⊕ ,respectively, and neither found any indication of instability over a 10 millionyear period, provided the eccentricities (which are still poorly determined)are small. Buchhave et al. (2016) concluded that the Kepler-20 system isconsistent with being dynamically cold, with relatively small eccentricitiesand inclination angles, and that it may have formed during the transitionphase when the circumstellar disk has a high solid surface density but alow or moderate gas surface density, according to theoretical modeling byLee & Chiang (2016). 12epler-20 e and 20 f are so small that they most likely have a rocky com-position like the Earth. Based on the properties of the parent star and theorbital semimajor axes of these two bodies, we estimate they now receive,respectively, about 187 and 39 times the incident radiation that the Earthreceives from the Sun. Any primordial gaseous envelopes would have beencompletely lost to atmospheric photoevaporation (e.g., Lopez et al., 2012) orperhaps other processes such as impact erosion (e.g., Inamdar & Schlichting,2015).Thanks to the improved mass determinations by Buchhave et al. (2016)for the other transiting planets in the system, their nature is now also cominginto better focus. Despite the large radius of Kepler-20 b, the innermostplanet, interior structure models by Zeng & Sasselov (2013) indicate it has arocky composition that is consistent with an iron-to-silicate ratio similar tothat of the Earth. We are likely seeing the bare core of a planet that lost itsprimordial atmosphere due to strong irradiation from the star, equivalent tomore than 350 times the flux of the Sun impinging on the Earth. The massesand radii of Kepler-20 c and 20 d, on the other hand, lead to densities thatindicate the presence of volatiles and/or a hydrogen/helium envelope.
5. Final words
Kepler-20 e and 20 f marked the first time astronomers were able to verifythe presence of another world the size of our own around a Sun-like star.Since then, efforts have continued to push toward ever smaller planets, notonly from space but also from the ground. As of this writing there are some150 transiting planets known that are about the size of the Earth or smaller ,all but a handful being Kepler discoveries. The exceptions are some of theplanets in the fascinating multi-planet system TRAPPIST-1 (Gillon et al.,2017), detected recently from the ground around a nearby late-type M dwarf,and observed also by K2 as well as Spitzer . The record-holder in terms ofthe smallest measured size is still Kepler-37 b (Barclay et al., 2013), a bodythat is smaller than the planet Mercury in our solar system.In this growing collection of small planets only a few that reside inmulti-planet systems have been confirmed dynamically, not by the Dopplertechnique but by taking advantage of their TTVs to measure their masses. Count based on the tabulation at http://exoplanets.org/table , consulted onNovember 15, 2018.
13n example is the TTV measurement of the mass of the Mars-sized planetKepler-138 b by Jontof-Hutter et al. (2015). The rest have all been validatedstatistically. The use of the validation approach that has been so successfulis likely to continue into the future in support of missions such as the Tran-siting Exoplanet Survey Satellite (TESS; Ricker et al., 2015), which has nowbegun to scrutinize the sky looking for small planets like Kepler-20 e and 20 faround bright nearby stars.
Acknowledgments
We thank Jack Lissauer for the invitation to write this review, and forhelpful comments. Our two referees also provided helpful suggestions thatimproved the manuscript. We wish to acknowledge the invaluable help ofChris Henze (NASA/Ames), who implemented important modifications tothe
BLENDER program to improve the mapping of the range of possible blends,and managed the processing on the Pleiades supercomputer. Finally, we aredeeply grateful to the engineers, managers, and scientists who were respon-sible for the
Kepler mission that has led to so many important discoveriesincluding Kepler-20 e and 20 f.Declarations of interest: none.
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