Almost All of Kepler's Multiple Planet Candidates are Planets

2.2 MB·English

by  Jack J. Lissauer

ApJ in press
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
ALMOST ALL OF Kepler’S MULTIPLE PLANET CANDIDATES ARE PLANETS
Jack J. Lissauer1, Geoffrey W. Marcy2, Jason F. Rowe1,3, Stephen T. Bryson1, Elisabeth Adams4, Lars A.
Buchhave5,6, David R. Ciardi7, William D. Cochran8, Daniel C. Fabrycky9,10, Eric B. Ford12, Francois Fressin4,
John Geary4, Ronald L. Gilliland13, Matthew J. Holman4, Steve B. Howell1, Jon M. Jenkins1,3, Karen
Kinemuchi1,14, David G. Koch1, Robert C. Morehead12, Darin Ragozzine4, Shawn E. Seader1,3, Peter G.
Tanenbaum1,3, Guillermo Torres4, Joseph D. Twicken1,3
ApJ in press
2
ABSTRACT
1
0 We present a statistical analysis that demonstrates that the overwhelming majority of Kepler can-
2 didate multiple transiting systems (multis) indeed represent true, physically-associated transiting
planets. Binary stars provide the primary source of false positives among Kepler planet candidates,
n
implying that false positives should be nearly randomly-distributed among Kepler targets. In con-
a
J trast, true transiting planets would appear clustered around a smaller number of Kepler targets if
detectable planets tend to come in systems and/orif the orbitalplanes of planets encircling the same
6
star are correlated. There are more than one hundred times as many Kepler planet candidates in
2
multi-candidate systems as would be predicted from a random distribution of candidates, implying

that the vast majorityaretrue planets. Mostofthese multis aremultiple planet systems orbiting the
]
P Kepler targetstar,but there are likely cases where (a) the planetary system orbits a fainter star,and
E theplanetsarethussignificantlylargerthanhasbeenestimated,or(b)theplanetsorbitdifferentstars
. within abinary/multiple starsystem. We use the lowoverallfalse positive rateamongKepler multis,
h
together with analysis of Kepler spacecraft and ground-based data, to validate the closely-packed
p
Kepler-33 planetary system, which orbits a star that has evolvedsomewhat off of the main sequence.
-
o Kepler-33 hosts five transiting planets with periods ranging from 5.67 to 41 days.
r Subject headings: planetary systems; stars: individual (Kepler-33/KOI-707/KIC9458613); methods:
t
s statistical
a
[
1. INTRODUCTION (2011) estimated the fidelity of Kepler’s planet candi-
1 dates (fraction of the candidates expected to be actual
Roughly one-third of Kepler’s planet candidates an-
v
planets) to be above 90%. Yet the fidelity of multiple
4 nouncedbyBorucki et al.(2011)areassociatedwithtar-
planet candidates is likely to be higher than that for
2 gets that have more than one candidate planet. False
singles (Latham et al. 2011; Lissauer et al. 2011a). We
4 positives(FPs)plagueground-basedtransitsearches,but
show herein that the vast majority of Kepler’s multiple
5 the exquisite quality of Kepler photometry, combined
planet candidates are true multiple planet systems.
. with the ability to measure small deviations in center of
1 The majority of Kepler planet discoveries announced
light during transits (Jenkins et al. 2010; Batalha et al.
0 to date have been confirmed dynamically, using radial
2010),havebeenusedtocleansethesamplepriortopre-
2 velocity variations of the target star or departures of
sentationinBorucki et al.(2011). Accounting forcandi-
1 observed transit times from a linear ephemeris (tran-
dates oneachone’s individualmerit, Morton & Johnson
v: sit timing variations, TTVs). However, three of the
i [email protected] first sixteen Kepler planets do not have dynamical ev-
X 1NASAAmesResearchCenter,MoffettField,CA94035,USA idencesupportingthediscovery,butrathertheseplanets
r 2Astronomy Department, University of California, Berkeley, have been “validated” by showing that the probability
a CA3S9E47T2I0I,nUstSiAtute/NASAAmesResearchCenter,MoffettField, that the observed signal is produced by a planet around
CA94035,USA the host star is at least 100 times as large as that of
4Harvard-Smithsonian Center for Astrophysics, 60 Garden a false positive (Torres et al. 2011; Fressin et al. 2011;
Street, Cambridge,MA02138, USA
Lissauer et al. 2011b). For these three planets, the vali-
5Niels Bohr Institute, University of Copenhagen, DK-2100, dation process was based on the results of the BLENDER
Copenhagen, Denmark
6CentreforStarandPlanetFormation,NaturalHistoryMu- code, which performs detailed comparisons between the
seum of Denmark, University of Copenhagen, DK-1350 Copen- observedKepler lightcurveandthe lightcurvespredicted
hagen, Denmark
foranensembleoftheoreticallyconceivableFPstoreject
7Exoplanet Science Institute/Caltech, Pasadena, CA 91125,
USA FPmodelsthatarenotconsistentwiththeobservations.
8DepartmentofAstronomy,UniversityofTexas,Austin,TX BLENDER then sums the a priori probability for the al-
78712, USA lowed false positives and compares that to the a priori
9DepartmentofAstronomy&Astrophysics,UniversityofCal- probability for the planet model. BLENDER validates a
ifornia,SantaCruz,CA95064, USA
11HubbleFellow planetwhentheresultingoddsratioverystronglyfavors
12University of Florida, 211 Bryant Space Science Center, theplanetmodel. Althoughallthreeoftheseplanetsor-
Gainesville,FL32611, USA bit stars that also have dynamically-confirmed planets,
13Space Telescope Science Institute, Baltimore, MD 21218,
planetarymultiplicity wasnotusedquantitativelytoaid
USA
14BayAreaEnvironmentalInstitute, CA in their validation.
2 Lissauer et al.
Morton & Johnson (2011) do a more cursory analy- ine the target and candidates individually, supplement-
sis, but one that is far easier, enabling them to per- ing the Kepler Input Catalog (KIC, Brown et al. 2011)
form it for all of the candidate planets (“candidates”, with spectroscopic observations of the target and high-
henceforth) that are listed in the Borucki et al. (2011) resolution imaging, including Keck guider camera expo-
paper. The study by Morton & Johnson (2011) as- sures and speckle imaging (Howell et al. 2011) and/or
sumes a flat 20% planet occurrence rate as a prior for adaptive optics (Troy et al. 2000; Hayward et al. 2001),
theirBayesiananalysis(i.e.,theaveragenumberofplan- tosearchforotherstarsinthetargetaperture. Wecheck
ets per star is assumed to be 0.2), and they do not the lightcurve for a shape consistent with a transit. We
consider planets orbiting background stars to be FPs. verifythatthecentroidmotionisnilorconsistentwitha
Morton & Johnson (2011) use the depth and period of transit around the target star and not a transit/ eclipse
the transit, together with the magnitude, stellar prop- of any other stars observed nearby. But we do not per-
erties and galactic latitude of the target star (hence- formdetailedmatchingofobservedlightcurvestoFPsce-
forth “target”) to provide an upper bound on the FP narios. We then use multiplicity to effectively increase
prior; for high SNR transits, this upper bound on the the planet priors and allow for validation in a Bayesian
FP prior can be much higher than the one computed sense. Note that for candidate giant (∼ 1 Jupiter ra-
using detailed lightcurve matching by BLENDER to rule dius) planets, one would need to consider the possibility
out many FP scenarios(Torres et al. 2011; Fressin et al. ofthetransitsignalbeingproducedbyalateMstarpar-
2011). Indeed, Morton & Johnson (2011) do not exam- tiallyeclipsingthetargetstar(perhapswithdilution);for
ine Kepler lightcurves, but rather assume for their cal- smaller candidates, the possibility of transits of fainter
culations that various false-positive vetting procedures stars by larger planets must be taken into account.
using Kepler photometry have been done, such as elimi- Theprobabilityofacandidateinamultiplesystembe-
nationofcandidates withV-shapedlightcurvesorwhere ing a false positive is contrasted to that of a single can-
the centroid of light received on the Kepler focal plane didate in Section 2. Other factors involved in validating
is in a different position during a transit than it is out- transiting planet candidates are discussed in Section 3.
side of transit, and that the planet candidates show no We apply our method to the KOI-707 candidate multi-
evidence for being FPs. However, these cuts were not planet system in Section 4. We conclude in Section 5
applied to all of the candidates listed in Borucki et al. with a summary of our results and a discussion of their
(2011). Some candidates were not vetted as extensively implications.
asassumedbyMorton & Johnson(2011),andothercan-
didates did not pass one of the cleanness tests but still 2. STATISTICALVALIDATIONOFMULTI-PLANET
remainviable,ifsomewhatmoresuspect, planetarycan- CANDIDATES
didates (e.g., grazing planetary transits, as well as graz- Kepler has found far more multiple candidate systems
ing stellar eclipses, produce V-shaped lightcurves), and than would be the case if candidates were randomly
thus remain on Borucki et al. (2011)’s candidate list. distributed among target stars (Lissauer et al. 2011a).
We develop herein a quantitative method to esti- False positives are expected to be nearly randomly dis-
mate the true planet fraction of the overall sample of tributed among Kepler targets, whereas true transiting
Kepler candidate multiple planets that is based upon planets could be clustered if planets whose sizes and pe-
the observation that there are far more multiples than riodsareadequatefortransitstobe detectedoftencome
would be expected for a random distribution of can- in multiples, as is the case for planets detected by radial
didates among Kepler targets (Lissauer et al. 2011a). velocity variations (Wright et al. 2009), and/or if plane-
This “overabundance” of multiples is illustrated by the tarysystems tendto be flat, sogeometryleads to higher
fact that fewer than 1% of targets have a candidate on transitprobabilitiesforotherplanetsifoneplanetisseen
Borucki et al.(2011)’scandidatelist,whereasmorethan to transit (Ragozzine & Holman 2010). We quantify the
15% of targets with at least one candidate have multi- excess of multiples and its implications for the overall
ple candidates, and more than 25% of targets with at fidelity of the sample of multiple candidates in this sec-
least two candidates have a third candidate. Such a re- tion.
sult is expected if planets often appear within systems We use the following notation and input parameters:
of planets, especially if such systems tend to be flat in The numberofplanetcandidates is n =1199,the num-
c
the distribution of inclinations. High multiplicity, low beroftargetswithtwoormorecandidates,i.e.,thenum-
inclination, planetary systems are consistent with Ke- ber of candidate multiple planet systems, is n = 170,
m
pler observations (Lissauer et al. 2011a). and the number of targets from which the sample is
We describe an approachto validationof specific mul- drawnisn =160,171. Thesenumericalvaluesaretaken
t
tiple planet system candidates that is intermediate be- fromLissauer et al.(2011a),whoremovedfromthe1235
tween those of Morton & Johnson (2011) and BLENDER, candidates listed in Borucki et al. (2011) those objects
and we apply these techniques to validate the very flat thatwereestimatedtohaveradiigreaterthantwicethat
and dynamically-packed KOI-707 system14. In this ap- of Jupiter, those for which only one transit had been
proach,appliedherein exclusivelyto KOI-707,we exam- observed, and those identified as FPs by Howard et al.
(2011). We denote the number of actual planets among
14 Targets with planetary candidates are given an integer KOI thecandidatesasnp,thenumberoffalsepositivesasnf,
(KeplerObjectofInterest)number,andindividualcandidatesare andthefidelityofthesample(fractionofcandidatesthat
denotedbythetargetKOInumberfollowedbyadecimalpointand are planets) as P, so n =n +n and P =n /n .
a two digit number specifying the order in which they were iden- c p f p c
For our calculations in Equations (1–9), we make the
tified. Forexample,KOI-707.02isthesecondplanetarycandidate
to have been identified by transit-likesignatures inthe lightcurve following two assumptions:
ofthetargetstarKOI-707. i) FPs are randomly distributed among the targets;
High Reliability of Kepler’s Multiple Planet Candidates 3
ii) there is no correlation between the probability of a The number of targets expected to have four or more
targettohostone ormore detectableplanets andtodis- false positives is very small and can be neglected for our
play FPs. purposes. Equation (1) also yields the expected number
Incontrast,wedonotassumethatplanetsarerandomly oftargets with a single FP under the same assumptions,
distributed among the targets. The previous verifica- 597.3 or 119.8.
tions of the two-planet Kepler-10 system (Batalha et al. The assumed lack of correlation between the propen-
2011; Fressin et al. 2011), the three-planet Kepler-9 sity of a target to have false positives and true plan-
(Holman et al. 2010; Torres et al. 2011) and Kepler-18 ets implies that the probabilitythat a giventargethosts
systems (Cochran et al. 2011), and especially the six- both a planet and one or more false positive is equal to
planet Kepler-11system (Lissauer et al. 2011b), provide the product of these individual probabilities. The prob-
strong evidence of the non-random distribution of tran- ability of a target having a detected transiting planet is
siting planets among Kepler target stars. givenbyPn /n ,andtheprobabilityofitshowingafalse
c t
The above two assumptions, together with observed positive is given by Equation (1). Thus, the estimated
values of n , n and n and an assumed lower bound numbersoftargetswithatleastoneplanetaswellasone
c m t
on P, allow us to estimate lower bounds on the fidelity or more false positives are given by:
ofcandidatesinvariousclassesofcandidatemulti-planet
systems using simple algebra. For example, assumption n P n (1−P)
(1) implies that the expected number of targets with k 1 planet +1 FP: c × c ×n (4)
t
n n
falsepositives,E(k),isgivenbyaPoissondistributionof t t
n members whosemean is givenby the averagenumber n2
oft false positives per target, λ = (1 − P)nc/nt. This =P(1−P)nct
expectation value is given by the following formula:
=8.98P(1−P)=2.25 or 0.81;
λke−λ ((1−P)nc)ke−(1−P)nnct
E(k)= k! nt= ntk! nt (1) 1 planet +2 FPs: P(1−P)2 n3c (5)
((1−P)nc)k 2n2t
≈ nt nt, =0.034P(1−P)2
k!
wheretheapproximationinEquation(1)isvalidforλ≪ =0.0042 or 3×10−4.
1. The number of targets expected to have both a planet
We next compute the expected number of Kepler tar- andthreeormorefalsepositivesisverysmallandcanbe
gets with multiple candidates at least one of which is a neglectedforourpurposes. Somewhatlowerestimatesof
FP. In some cases, we make approximations that yield the number of multi- candidate systems with one planet
conservative estimates of the expected numbers of true andatleastoneFP thangivenbyEquations(4)and(5)
planets (i.e., overestimatethe expected number of FPs). can be derived by noting that the observed multiplicity
Ourresults,presentedinEquations(2–7),aregivenfirst implies fewer (than n ) targets with a nonzero number
c
as general formulae, then as a function of the fraction of planet candidates.
of candidates that are true planets, P, for the observed To derive an estimate of the expected number of tar-
values of 160,171targets,1199 candidates, and 170mul- gets with multiple true planets as well as at least one
tiple candidate systems stated above, and finally these FP, replace the term in Equations (4) and (5) that rep-
value if P = 0.5 or 0.915. These final numbers are to be resents the probability of a given target having a true
comparedtotheobservationthatthereare115candidate planet,Pn /n ,withatermrepresentingtheprobability
c t
2-planet systems, with a total of 230 planet candidates, of a given target having a multi-planet system, n /n .
m t
and55candidatesystemsof3ormoretransitingplanets, (Assuming all multis are multi-planet systems to esti-
with a total of 178 planet candidates. mateofthistermisconservativeinthesensethatitmay
Equation(1) yields estimates of the number oftargets lead to an overestimate in the number of false positives
with two or three false positives: in multi-candidate systems.)
2 FPs: ((1−P)nc)2 =4.49(1−P)2 (2) 2 or more planets +1 FP: nm × nc(1−P) ×n (6)
2nt nt nt t
=1.12 or 0.045; =(1−P)nmnc =1.27(1−P)
n
t
=0.64 or 0.13;
((1−P)n )3
3 FPs: c =0.011(1−P)3 (3)
6n2
t
=0.0014 or 1.1×10−5. 2 or more planets +2 FPs: nm ×(1−P)2× n2c(7)
n 2n
t t
15WechoosetopresentnumericalresultsforP =0.9becausewe =0.0048(1−P)2
considerittobeareasonableestimateoftheactualplanetfidelity
rate, and P = 0.5 to show that even for an overall FP rate far =0.0012 or 5×10−5.
higher than studies suggest, the expected number of FPs among
themultisisremarkablylow. The number of targets expected to have both multiple
4 Lissauer et al.
planets and three or more false positives is extremely prior unchanged. Thus the probability of the planet in-
small and can be neglected for our purposes. terpretationaccountingformultiplicityisrelatedtothat
neglectingmultiplicityaccordingtothefollowingapprox-
2.1. Numerical Estimates for Kepler Candidates
imate formulae:
AddinguptheaboveestimatesofFPsofdifferenttypes
25P
yields expected totals of 4.5 or 0.9 FPs (for assumedFP P ≈ 1 (8)
rates of 50% and 10%, respectively) in doubles (com- 2 25P1+(1−P1)
pared to a total of 230 candidates) and 0.65 or 0.13 FPs 1−P
(≈1− 1 for 1−P ≪1, and ≈25P for P ≪1)
in systems with three or more candidates (compared to 1 1 1
25
a total of 178 candidates). These numbers suggest that
validation of doubles at the 99% level by this method
alone is marginal. Triples and higher multiplicity candi- 50P
P ≈ 1 (9)
dates are more strongly validatable, enough so that the 3+ 50P +(1−P )
1 1
one dynamically-unstable candidate triple system noted
1−P
by Lissauer et al. (2011a), KOI-284, which has a pair of ( ≈1− 1 for 1−P ≪1, and ≈50P for P ≪1),
1 1 1
candidates with a period ratio 1.0383, should be viewed 50
as somewhat surprising (but see Section 2.4).
where in each case we use the smaller value computed
Itisofinteresttocomparethe(low)falsepositiverates
in the previous paragraph. Rough estimates for the
for multis estimated in the previous paragraph with the
planetprobabilitiesofindividualKeplercandidatemultis
presumed FP rates for singles upon which their calcula-
can be obtained by applying Expressions (8) and (9) to
tions are based. For 50% FPs in singles, our estimates
theestimatesofplanetprobabilitybyMorton & Johnson
abovecorrespondto2%FPsindoubles(comparethefirst
(2011); the Morton & Johnson (2011) planet probabili-
numbersattheendofequations2and4withthe230can-
ties are given by P in the above formulae since their
didates in Kepler candidate 2-planet systems) and 0.4% 1
numbers do not account for multiplicity. Note that this
FPs for targets with three or more candidates (compare
multiplicityboostincorporatesself-consistentlyboththe
thefirstnumbersattheendequations3and5–7withthe
coplanarityboostdiscussedinCochran et al.(2011)and
178candidatesinKepler candidate3ormoreplanetsys-
thesystemsboost(thatplanetstendtocomeinsystems)
tems); for 10% FPs in singles, we estimate 0.4% FPs in
without need to disentangle these effects.
doubles (compare the final numbers at the end of equa-
The above calculations of the multiplicity boost do
tions 2 and 4 with the 230 candidates in Kepler candi-
not assume that any of the candidates of the target un-
date 2-planet systems) and 0.1% for targets with three
der consideration have been confirmed or validated as
or more candidates (compare the final numbers at the
planets. Does validating one or more candidate as a
end equations 3 and 5–7 with the 178 candidates in Ke-
true planet(s)substantially increasethe probability that
pler candidate 3 or more planet systems). These values
other candidates of the same target are also planets?
areconsistentwitha“multiplicityboost”intheBayesian
For realistic estimates of FP rates (e.g., 10%), the equa-
priorforplanetsof∼25forasystemwithoneadditional
tions above show that most FPs in multis are likely to
candidateandof∼ 100for asystemwithmorethan one
be combinations of 1 FP with one or more true plan-
additionalcandidate. Theprobabilitythatatargethosts
ets, so validating one candidate does not substantially
aplanetcandidate,1/150,canbe comparedtothe prob-
increase confidence in the others from a statistical view-
abilityofatargetthathostsonecandidatehostsanother,
point. Nonetheless, validating one candidate does elim-
1/6,andthatanadditionalcandidatehasbeenfoundfor
inate some of the non-statistical issues raised in the fol-
highermultiplicity systems,1/3. Theselatter values,to-
lowing three subsections. In particular, if that validated
gether with the assumption that FPs and true planets
candidate is shown to not only be a planet but also to
are uncorrelated, lead to an estimate of the multiplicity
orbit the Kepler target star, then the probability that
boostforcandidatedoublesof30,slightlylargerthanthe
the other candidates are physically-associated with the
valueof25computedabove,andtoa somewhatreduced
target increases.
estimateof∼50forcandidatesinsystemsofhighermul-
tiplicity.
2.2. Caveats
Themultiplicityboostintroducedinthepreviouspara-
graph can be used to quantify the increased probability To examine the validity of the three assumptions
that aset oftransit-likesignaturesrepresenta realtran- statedabove,we divide FPs into two classes,(I) chance-
siting planet if one or more additional set of transit-like alignment blends such as background eclipsing binaries
signatures are detected for the same Kepler target; i.e., (BGEBs)and(II)physicaltriplestars(PTs),wheregraz-
to compute an estimate the probability of a particular ing eclipses of the targetstar by a stellar companion are
candidate in a multiple candidate system being a true considered together with PTs for this purpose. We note
planetfromanestimateofthatcandidatebeingaplanet thatitisamatterofchoicehowtocountcaseswherethe
that does not account for multiplicity. We denote the transit signal is produced by a planet transiting a star
probability of planethood of the candidate in question other than the Kepler target. Larger planets transiting
computed without factoring in multiplicity by P and starsthatarenotphysicallyassociatedwiththestarpro-
1
the probabilityof it being a planet accountingfor multi- viding most of the target’s flux can be considered to fall
plicitybyP ifitisinatwocandidatesystemandbyP within the chance-alignment blends/BGEB class of FPs
2 3+
if it is a member of a three or more candidate system. (as they must be for Kepler’s statistical census of plan-
In our formulation, multiplicity effectively increases the ets,becausebackgroundstarsarenotinthedenominator
Bayesian estimate of the planet prior but leaves the FP of the ratio taken to calculate the fraction of stars with
High Reliability of Kepler’s Multiple Planet Candidates 5
planets, so their planets must not be included in the nu-
merator), or alternatively not be regarded as FPs when
assessing individual candidates and the reliability of the 1.0
sample. Planets transiting physical companions to the
star providing most of the flux can either be viewed as
members of the PT class ofFPs or not classifiedas FPs;
how these planets are categorizedfor Kepler’s statistical 0.8
census of planets is also a matter of choice.
Assumptions (i) and (ii) lead to the following caveats: n
(i): The expected number of BGEBs (chance- o
ti
alignment blends) that can cause FPs varies with the c
a0.6
magnitude and galactic latitude of the target star, r
f

whereas those of PTs do not (except to the extent that e
v
the distribution of stellar types among Kepler targets ti
a
depends somewhatonmagnitude). Thevariationsinex- ul0.4
pectedBGEBsacttoincreasetheexpectedtotalnumber m
of systems with 2 or more FPs compared to the value u
C
given by the above formulae. When looking at indi-
vidual candidate systems, the FP probability of those (Bkg stars) x (targets)
0.2
aroundfainttargetsandatlowgalacticlatitudeislarger,
Targets (163986)
whereastheplanetaryinterpretationbecomesmorelikely
KOIs (961)
forbrightertargetsandthoseathighergalacticlatitude.
Multis (170)
However, the distributions in galactic latitude of the
0.0
planet candidates and of the multiple candidate systems 6 8 10 12 14 16 18 20
both track that of all Kepler targets, whereas the distri- Galactic latitude (deg)
bution of expected BGEB FPs is quite different (Figure
1);thissuggeststhatthefractionofcandidates(bothsin- Fig.1.— Distributions in galactic latitude of all 163,986 Ke-
gles and multis) that are produced by BGEBs is small. pler Quarter 2 exoplanet targets (solid turquoise curve), the 961
targetswithplanetcandidatesmeetingourcriteria(soliddarkred
The statistical analysis that we presented above does
curve)andthe170multiplecandidatesystems(solidblackcurve).
not discriminate between multiple candidates resulting Thedashed curve shows the distribution of expected BGEB FPs,
from multiple planets orbiting a background star and a whichwecompute byweighting each target by the skydensity of
single candidate resulting from a single planet orbiting starswithmagnitudesbetween15.0and20.0accordingtotheBe-
sancon model of the galaxy (Robin et al. 2003, 2004). All curves
a background star. However, the correlations and lack
are normalized to give a total of unity. Error bars represent the
thereof between the curves shown in Figure 1 also imply statisticaluncertainty(1σbinomialvariation)ofeachoftheplanet
that the fraction of KOIs caused by single or multiple candidate curvesatquartilepoints intheirdistributions. Theba-
planets transiting a background star is small. In some sic shape of the three solid curves is governed by the orientation
of the Kepler field on the sky, where the amount of area covered
cases, it is especially unlikely that a multiple candidate increasesfromapointatgalacticlatitude5◦ toaregionspanning
system is produced by several planets orbiting a back- 16◦ in longitude at 13◦ in galactic latitude and then decreases
ground star, either because its largest member would to a point at 21◦ latitude; the second most important factor is a
be greater than planetary in size, or because the re- weightingtowardsthegalacticplanebecauseofthehigherspatial
density of target stars in that portion of the field. The dashed
quirement of dynamical stability would imply implau-
curveismoreheavilyweightedtowardsthegalacticplanebecause,
sibly small densities for the larger planets orbiting the to first approximation, it is weighted quadratically in the spatial
fainter star. Additionally, as the fraction of giant plan- density of stars (more precisely, the product of the spatial densi-
ets among the multis is smaller than it is for the singles tiesoftargetstarsandbackgroundstars),whereastheothercurves
areonlyweightedlinearly. Thespatialdensityofbackgroundstars
(Latham et al. 2011), the probability of a multi being a dropsbyafactorof17fromthelowestlatitudeKeplerobservesto
systemoflargerplanetsorbitingafainterstarislessthan the highest latitude. The similarity of curves for targets, planets
thatofasinglecandidateorbitingastarfainterthanthe and multis, and the difference of these from the curve estimating
BGEBFPs,suggests that thefractionofcandidates that arepro-
target.
ducedbyBGEBs(aswellasthatresultingfromplanetstransiting
AsecondclassofmultipleFPscanbeproducedfroma backgroundstars)issmall.
single physical system, either a triple star with two sets
of eclipses or a background star with multiple transit-
ing planets. Triple star systems are unstable for small
periodratios andproduce largeeclipse timing variations
for moderate period ratios, so if large timing variations WenotealsothatFPsareinherentlymorelikelyamong
are not detected, we do not need to consider triple star both small (Earth-size and smaller) and large (Jupiter-
system FPs (nor grazing eclipsing binary stars with a size and larger) candidates than among intermediate
transiting planet) unless period ratios are large. (Neptune-size) candidates, independent of multiplicity.
Quadruple star systems consisting of two pairs of Most small candidates produce low-amplitude transits
eclipsing binaries (e.g., Graczyk et al. 2011) could pro- thatreduceourabilitytoremovefalsepositivesfromthe
duceasmallexcessprobabilityofapairofFPs. Suchsys- candidate list on the basis of the shape of the lightcurve
tems,aswellaslineofsightpairsofphysically-unrelated or from differences in the centroid of the location of the
EBs,alsomightbemorelikelytomimicplanetary-depth dip relative to that of the target. Small candidates, in-
transitsbecauseofmutualdilutionofthefractionoflight cluding multis, are more likely to be planets orbiting
in the Kepler aperture. starsfainterthanthetargetsimplybecausetheNeptune-
6 Lissauer et al.
sizeplanets thatcouldproducesuchsignalsarefarmore tion2.1,oneofthecandidatesystems,KOI-284,ishighly
commonaccordingtoKepler observationsthanaregiant suspect because two of the candidates have periods that
planets(Borucki et al.2011),andthisdifferenceinabun- differ from one another by < 4%, which would rapidly
dance is even larger for the multis than for the singles leadtoadynamicalinstabilityonaveryshorttimescale
(Latham et al.2011). Attheotherextremeoftheplanet (but see the discussion of this system in Section 2.4).
sizedistribution,giantcandidatesappeartobemorecon- Do non-resonant periods reduce the confidence in indi-
taminated by stellar false positives (Demory & Seager vidual multis and do stable systems produce an increase
2011) of the type which plague ground-based surveys in our confidence? The answers to both parts of this
than do Neptune-size candidates. question are yes, but in most cases these effects are mi-
(ii): PlanetsandFPsofbothtypesaremoreeasilyde- nor. While resonances are overpopulated, a substantial
tectableforquietertargets;planetsandPTsaremorede- majority of multis are not in nor near such resonances
tectable for brighter targets,but BGEBs that are bright (Lissauer et al. 2011a; Wright et al. 2011). And while
enough to mimic observable transits of the target star drawing planetary periods randomly from the observed
are more likely for fainter targets. A grazing eclipsing ensemble of transiting planet periods gives many more
binary with a circumstellar or circumbinary transiting nominally unstable systems than are observed, the frac-
planet analogous to Kepler-16 (Doyle et al. 2011) would tionofunstableconfigurationsfromsucharandomdraw
violate our assumption of a lack of correlation between is still small. Nonetheless, in high-multiplicity densely-
planetsandFPs,butsuchaconfigurationrequiresalarge packed systems, wherein most configurations with the
period ratio for stability and an even larger period ratio same number of planets randomly spread over the same
for the planet to have modest TTVs. range in period would likely be unstable, the boost in
The assumptionof no correlationbetweenplanets and confidence from stability can be significant. Note that
FPswouldbeviolatediftheorientationsoforbitalplanes specialperiodratioboostsandstabilityboostsareinad-
of PTs with one star having a planet are not random. dition to the standard multiplicity boost, i.e., they can
Alignments of planetary orbital planes for binary stars be applied together as two multiplicative factors in the
for which both stars possess planets also violate this as- planet prior for appropriate systems.
sumption, although members of this class are true plan-
2.4. KOI-284 & Multis Composed of Planets Transiting
etsthatareallphysicallyassociated(albeitlessdirectly)
Differing Stars in a Binary
with the target star and thus would for many purposes
not be classified as FPs. Averyroughestimate ofthe expectednumber ofmul-
The numbers quoted in Equations (2–9) do not incor- tiple candidate systems that orbit different members of
poratethecaveatsnotedinthissubsection. Correlations a binary/multiple star system (rather than a single sys-
between the propensity of a target to have one or more tem of planets orbiting the same star) can be made by
FPs and to host one or more identified true planets (ii) assuming that half of Kepler’s targets are binaries, that
are likely to be at most a factor of a few. False positives each star within a binary is as likely to host both single
of the BGEB class are likely to be concentrated among and multiple transiting planets as are single stars, and
fainttargetstarsthatarelocatedatlowgalacticlatitude that the probability of one star within a binary hosting
(Morton & Johnson 2011); this concentration increases transiting planets is uncorrelated with that of its binary
the expected number of targets with two (or more) FPs, companion. These assumptions imply nc planets spread
butalsopointstoasubstantialportionofKepler’starget around 3nt/2 stars, and thus (2/3)(nc/nt) planets per
list whose members have a priori FP likelihood that is star and 2n2/(9n ) ≈ 2 cases in which multis are com-
c t
lowerthanorcomparableto those givenby the numbers posedofplanets transitingdiffering members ofa stellar
above. binary. Positive correlations between the propensity of
stars in a binary to have transiting planets (resulting
2.3. Period Ratios: Resonances & Stability
from the abundance of planets and/or alignment of or-
The distribution of candidate period ratios exhibits bital planes) would increase this number, but the fact
spikes near strong mean motion resonances, which im- that a significant majority of binaries are composed of
plies that candidates with such period ratios are more stars of substantially differing surface brightness com-
likely to be true planets than those candidates not in bined with the paucity of giant planets transiting low
or very near such resonances (Lissauer et al. 2011a). luminositystarsshouldreducethenumberofmultiscom-
This clustering can be used to increase confidence in prised of two planetary systems because of the difficulty
candidates with periods near resonances; such a “near- of detecting planets around secondary stars.
resonance boost” can be quantified using the measured As mentioned above, KOI-284, with three planetary
values of the normalized distance from resonance com- candidates having periods of 6.18, 6.42, 18.0 days and
puted from the distribution of planet period ratios in nominalsizes of ∼ 2 R⊕ (Earthradii), is the only one of
Lissauer et al. (2011a). Candidate resonantplanets can, 170 multi-candidate system identified in Borucki et al.
however, arise from period aliases in analysis of light (2011) that would be clearly dynamically unstable if
curves;suchaliasesarenotoftheformoftheastrophysi- all of the planets orbited the same star (Lissauer et al.
calFPsconsideredinthispaper,andwenotethatspecial 2011a)16. Analysis of a McDonald observatory spec-
care to search for them is an important aspect of vali- trumofthisbright(Kepler magnitudeKp=11.8)target
dating candidate resonant planets (see the discussion of
KOI-730and the Note Added in Proof in Lissauer et al. 16 The other multi that was unstable according to the integra-
tionsreportedinLissaueretal.(2011a)wasthe4candidatesystem
(2011a)).
KOI-191. Thisinstabilityresultedfromthestrongperturbationsof
Resonant period ratios increase confidence in various giant planet candidate KOI-191.01 on the nearby outermost can-
candidates, and as mentioned at the beginning of Sec- didate KOI-191.04. As the period ratio of these two candidates
High Reliability of Kepler’s Multiple Planet Candidates 7
yields a temperature of ∼ 6250 K and logg =4.5, which process can be very involved, especially for candidates
suggesta F7V star that is twice as luminous as the Sun. withorbitalperiods longenoughthatastrophysicalfalse
Speckle imagesofKOI-284obtainedon24June 2010re- positivescouldplausiblybeproducedbyobjectswithec-
vealed two stars differing in brightness by less than one centric (not tidally-damped) orbits (Fressin et al. 2011;
magnitude and separated from one another by slightly Lissauer et al. 2011b).
less than 1′′ (Howell et al. 2011). Spectroscopic obser- True planetary transits of the star that is the dom-
vations of each individual star obtained at Keck in 2011 inant source of light in the aperture are wavelength-
show the two stars to have nearly identical spectra and independent (neglecting the small effects of stellar limb-
have a difference in radial velocity of 0.94 ± 0.1 km/s. darkening and contamination by flux from other stars
The nearly identical velocities are consistent with their in the aperture), whereas false positives are not (except
being gravitationallybound. Their separationof 0.9′′ at in the unlikely case that the effective temperatures of
a distance of roughly 500 pc implies a projected separa- the contributing stars are similar). By providing in-
tion of 450 AU; the relative orbital velocities for two 1 frared time series spanning times of transit, the Warm
M⊙ starsonacircularorbitwithsemimajoraxisa=450 SpitzerMissioncanassistinthevalidationofmanytran-
AUis2km/s. Theseobservationssuggestthatthethree siting planet systems. Several candidates in multiple
candidatesmaywellallbetransitingplanetsinthesame systems have been observed, and they all exhibit achro-
stellar system, with one of the stars hosting one of the matic transit depths, consistent with planetary signals
six day period planets and the other two planets orbit- (Desert et al. 2012).
ingtheotherstarinthebinary. Thus,thissystemwould The durations of transit-likesignaturescanbe used to
notcallintoquestionthestatisticalargumentspresented assesstheprobabilityoftheplanetaryinterpretation,es-
in Section 2.1. A detailed analysis of both Kepler and pecially when more than one planet candidate is present
ground-based data to test this hypothesis and elucidate for a given target. Transit and eclipse durations depend
particulars is currently underway (Bryson et al. 2012). on several factors, including the masses and radii of the
twoobjects,theimpactparameter,b,whichisdefinedas
3. TRANSITCHARACTERISTICS:PLANETSVS.FALSE
theminimumprojectedseparationbetweenthecenterof
POSITIVES
the smallerbody (planet in the case ofa planetary tran-
ThenumbersderivedinSection2suggestthatthesam-
sit) and that of the larger body (star) measuredin units
pleofmulti-planetcandidatesidentifiedbyBorucki et al.
of the radius of the larger body, and the period and ec-
(2011) is likely to include only a small fraction of false
centricity of the orbit. Central transits (those for which
positives, of order 1%. But because of the uncertain-
thecenterofthetransitingbodypassesoverthecenterof
ties associated with the caveats discussed in Section 2,
the transited body) by a planet on a circular orbit have
as well as any factors that we may have omitted, we
durationsthatvaryasthecuberootoftheorbitalperiod
do not consider it appropriate to view all of Kepler’s
(Kepler 1619) and inversely to the cube root of the stel-
multi-planet candidates to be validated planets at the
lar density (Seager & Mall´en-Ornelas 2003); these rela-
99% level. Nonetheless, it is possible to identify candi-
tionships hold for FPs as long as the bright star is sub-
dates with various favorable characteristics as detailed
stantially larger and more massive than the other body.
below, andto examine the availabledata for systems in-
When the system’s mass is dominated by the primary
dividually for signs of potential inconsistencies with the
but the radius of the secondary cannot be ignored, then
planethypothesis. SeeHaswell(2011)foramoredetailed the duration, T ,(1st contact–4th contact,whichcor-
discussion of transit characteristics. dur
responds to the values reported in Borucki et al. (2011)
The shape and duration of a candidate transit can be
and in Table 3 of this paper), of a central transit for a
usedtodistinguisheventsproducedbyplanetsfromfalse
circular orbit satisfies the following proportionality rela-
positives produced by known astrophysicalsources. The
tionship:
BLENDERcode(Torres et al.2011)comparestheobserved
R +R
lightcurveto synthetic lightcurvesof both a planettran- T ∝ ⋆ pP1/3. (10)
dur R orb
siting the target star and a vast array of possible astro- ⋆
physical false positives involving binary stars and larger Ifthe transitdurationismeasuredoverthe timeinterval
planets transiting fainter stars. If the planet hypothe- whenthe centerof the planet coversthe disk ofthe star,
sis does not provide an acceptable fit to the data, the the first factor on the right-handside of Expression(10)
candidate is not viable; candidates for which the planet isnolongerrelevantandtheproportionalityissimply to
fit is marginal are called into question, and such can- P1/3. Note that the orbital period, P , is very accu-
didates are not considered further herein. Astrophysi- raotreblymeasuredby Kepler, as is the ratoirobofthe planet’s
cal false positives are considered credible if they provide radius, R , to that of the star, R , provided dilution by
p ⋆
an acceptable fit (3σ) to the data. Then the a priori nearby stars is small.
likelihoods of the planet hypothesis and that of a false Planetary transits with an impact parameter smaller
positive are compared, and a probability that the ap- than half the star’s radius have a duration that exceeds
parent transit is caused by a blend is computed. The 31/2/2 ≈ 86.67% that of a central transit. Such transits
accountforalmosthalfofalltransitingplanets,andover
was 1.258, Lissaueretal. (2011a) speculated that the system was
libratingina 5:4 mean motion resonance and that this resonance halfofthetransitsobservedtoaspecifiedSNRthreshold.
protectedtheplanetsfromcloseapproaches. Subsequentdataanal- The fraction of transits with durations greater than half
ysis shows that the period of KOI-191.04 was underestimated by thatofacentraltransitis31/2/2;againspecifyingaSNR
afactoroftwo. Theupdatedvalueoftheperiodofthiscandidate
is 38.6516 ± 0.0012 days. The revised period places the planets thresholdincreasesthe observedfraction. Orbitaleccen-
much farther apart, and the system is stable with the updated tricities also influence the durations of transits and FPs;
parameters. in most cases, durations are reduced because speeds are
8 Lissauer et al.
highest near periastron (Kepler 1609), where geometric out-of-transit images, so common biases such as PRF
considerationsimplythattransitprobabilitiesarelarger. error should approximately cancel. There is, however,
a quarter-dependent residual bias. Statistical analysis
4. VALIDATIONOFTHE5-PLANETSYSTEMKEPLER-33
acrossmanyuncrowdedKepler targetsindicatesthatthe
There are two primary purposes of this work: demon- residual bias due to PRF error is zero-mean, so we av-
strating the very high credibility of the ensemble of Ke- erage the quarterly centroid positions. Specifically, we
pler multi-planet candidates and providing a framework compute a robust χ2 minimizing best-fit position to the
to validate particular systems of candidates as planets. quarterlypositiondata. Theuncertaintiesoftheaverage
Wenowturntoaspecificsystemthatisbothintrinsically are estimated by taking the uncertainties of the fit and
interesting because it is the closest analog to Kepler-11 performing a bootstrap analysis. This average for each
observed to date (Lissauer et al. 2011a) and has charac- planet candidate is shown on Figure 3.
teristics very favorable for validation by multiplicity. Bias due to crowding (stars close enough to the tar-
KOI-707 (KIC 9458613) is a Kp = 14 star for which get that they contribute light observedin the aperture),
four planet candidates, ranging in period from 13 to 41 however,doesnotcancelbecause,totheextentthatvari-
days, were announced by Borucki et al. (2011); a fifth ationsinotherfieldstarsarenotcorrelatedwithtransits,
candidateplanet,withshorterorbitalperiodandsmaller fieldstarsdonotcontributetothedifferenceimage. This
size, has subsequently been identified. The lightcurve of introduces a systematic bias component that remains in
this target is shown in Figure 2. the average over quarters. To investigate the possibility
We performed an analysis of pixel-level Kepler space- that the observed offsets are due to crowding bias, we
craft data that shows that the transit signals come from model the local scene using stars from the KIC and the
a location on the sky that is coincident with that of the measuredPRF,inducedthetransitonKOI-707withthe
target star within small uncertainty, and we obtained ephemeris of each planet candidate, and performed the
high-resolution images of the target that did not show same PRF fit analysis on the model difference and out-
anynearbystarscapableofproducingthetransitsignals; of-transit images as that performed on the pixel data.
these studies are presented in Section 4.1. Our analyses If the resulting modeled average offset matches the ob-
ofthe transitlightcurvesand spectra takenofthe target served average offset, then the observed offset can be
star are presented in Section 4.2. In Section 4.3, we re- explained by crowding bias.
view the characteristics of this system that allow for its The locations of the transit signatures (difference im-
validation by multiplicity. We discuss the properties of agecentroids)ofeachofKOI-707’sfiveplanetcandidates
the Kepler-33 planetary system in Section 4.4. lie within1′′ ofthe predictedposition,andgiventhe un-
certainties listed in Table 1, none of the signatures are
4.1. Location of Transit Signature on the Plane of the
significantly offset and all are fully consistent with tran-
Sky
sits on the target star. Furthermore, the lack of near
A seeing-limited image taken at Lick shows a star 7′′ neighbor stars seen in the speckle image rules out some
to the NW that is 4.5 magnitudes fainter than KOI-707, of the possible false positive scenarios that could pro-
butnostarsclosertothe target. On12June2011,KOI- duce a centroid shift of the magnitude suggested by the
707 was imaged in the 880 nm filter (∆λ = 50 nm) of marginally significant offset observed.
ourspecklecameraattheWIYN3.5-mtelescopelocated
4.2. Lightcurves, Spectra, Derived Properties
on Kitt Peak. This star showed no companions within a
2.8 × 2.8 arcsec region down to 3.1 magnitudes fainter We obtained a spectrum of the target star KOI-
thanthe targetstar;starswithin0.05′′ ofthe targetstar 707/Kepler-33 using the HiRES spectrometer at Keck
would not have been detected. Details of this speckle I 10-m telescope. Classification of this spectrum us-
imagingdatacollectionandreductionprecessesarefully ingSME(SpectroscopyMadeEasy,(Valenti & Piskunov
described in Howell et al. (2011). 1996; Valenti & Fischer 2005)) yields the stellar param-
We check to see whether the transit signal is due to eter values shown in Table 2 (the final two columns in
a source other than KOI-707 using the difference image this table show results from constrained fits described
techniquedescribedinTorres et al.(2011). Thismethod at the end of this subsection). The mean stellar den-
fits the measured Kepler pixel response function (PRF) sity based on spectroscopic information was estimated
to a difference image formed from the average in-transit by matching the determined distributions of T ,logg
eff
andaverageout-of-transitpixelimagesinordertocalcu- and [Fe/H] to the Yonsei-Yale evolution tracks. This al-
late the position of the difference image centroid. This lowed the determination of stellar mass and radius and
difference image centroid position is compared with the meanstellardensityposteriordistributionbasedonspec-
position of the PRF-fit centroid of the average out-of- troscopy. The spectrum of KOI-707 is consistent with
transit image, as well as to the predicted deviations in two sets of stellar evolution tracks (Figure 4), leading to
centroid locations due to scene crowding. The results double-peaked probability distributions and asymmetric
for all 5 planet candidates are presented in Table 1 and uncertainties in several stellar parameters. We plot the
shown in Figure 3. probability distributions of the fundamental parameters
Theuncertaintyinthequarterlycentroidlocationsdis- ofstellarmassandageinFigure5andthoseofthestellar
played in Figure 3 is based upon propagating pixel-level density,whichisakeyinputtotransitfitting,inFigure6,
uncertainty and does not include possible PRF fit bias. andweincludeinTable2variousmeasuresofuncertainty
Sources of PRF fit bias include scene crowding, because beyond the simple standard deviation for the values of
the fit is of a single PRF that assumes a single star, and various derived stellar parameters that are either criti-
PRF error. The centroid offset is the difference between cal to the validation and properties of KOI-707 and/or
the positionsofthecentroidsofthe differenceimageand distributed highly asymmetrically. Note that both solu-
High Reliability of Kepler’s Multiple Planet Candidates 9
Fig. 2.— Kepler photometric time series data for KOI-707 (KIC 9458613; Kepler-33) in29.426 minute intervals (standard Kepler long
cadence observations). Calibrated data from the spacecraft with each quarter normalizedto its median are shown in the top panel. The
bottom panel displays the lightcurve after detrending with a polynomial filter (Roweetal. 2010). The entire 8 Quarters of Kepler data
analyzedhereinaredisplayed. Notethedifferenceinverticalscalesbetweenthetwopanels. Themidpointtimesofthefivesetsofperiodic
transits (including those that were not observed because they occurred during data gaps) are indicated by triangles of differing colors.
Close-upsoffoldedlightcurves neartransittimesofeachoftheplanetsareshowninFigures7and8.
tions are somewhat above the main sequence, implying measuring the Doppler shift of the stellar spectrum ob-
an evolved star. tained with the Keck I telescope and HiRES spectrom-
As an independent check of the values of the eter on BJD = 2455782.9441 relative to the solar spec-
SME parameters, we also derived values by matching trumbyaχ2 fitofthetwospectra. Tosetthezeropoint
the spectrum to synthetic spectra (Torres et al. 2002; of the wavelength scale accurately, we measure the dis-
Buchhave et al. 2010), and in addition we employ a new placement of telluric lines from the Earth’s atmosphere,
fitting scheme (Stellar Parameter Classification, SPC) serving as the “iodine cell”. The resulting radial veloc-
that is currently under development and being read- ities are accurate to 0.1 km/s, based on measurements
ied for publication, allowing us to extract precise stel- of 2000 stars (Chubak & Marcy 2012) and comparison
lar parameters from the spectra. The SPC analysis with IAU standard stars. For KOI-707,the radialveloc-
yielded the following parameters: T = 5849±50 K, ityisv =14.09km/s,indicatingmotionofthestaraway
eff ⋆
logg =4.07±0.10,[m/H]=0.12±0.08,vsini=3.4±0.5 from the Solar System at a speed typical of the velocity
km/s, RV = 13.779±0.020 km/s. The SME and SPC dispersion of stars in the Galactic disk. Orbital periods
parameters are fully consistent, and the nominal value of the planets quoted in Table 3 are as perceived from
of the key parameter logg from SPC is slightly (< 1σ) the barycenter of our Solar System. As the Kepler-33
largerthanthe SME value. Incontrast,the derivedstel- systemis recedingfromusat4.7×10−5c,where crepre-
lar radius from SME is 1.82 ± 0.16 R⊙, substantially sents the speed of light, the actual orbitalperiods in the
larger(more than 3σ formal error,see Table 2) than the restframeofthe Kepler-33systemare0.999953times as
radiusestimateof1.29R⊙ basedupontheKICthatwas large as the tabulated values.
used by used by Borucki et al. (2011). WeusedQ1-Q8long-cadenceKepler aperturephotom-
WemeasuredtheradialvelocityofKOI-707relativeto etry (i.e., only pixel-level corrections such as bias and
thebarycenterofourSolarSystem. Themethodinvolves dark smear are applied to the data and there is no co-
10 Lissauer et al.
KOI−707.01 KOI−707.02
1.5 1.5
1 1
0.5 0.5
→N (arcsec) 0 9458613, 13.988 →N (arcsec) 0 9458613, 13.988
−0.5 −0.5
−1 −1
−1−.51.5 −1 −0.5 ← E (a0rcsec) 0.5 1 1.5 −1−.51.5 −1 −0.5 ← E (a0rcsec) 0.5 1 1.5
KOI−707.03 KOI−707.04
2.5
1.5
2
1 1.5
1
0.5
→N (arcsec) 0 9458613, 13.988 →N (arcsec) −00..055 9458613, 13.988
−0.5
−1
−1 −1.5
−2
−1.5
−2.5
−1.5 −1 −0.5← E (a0rcsec)0.5 1 1.5 −2.5 −2 −1.5 −1 −0.5← E (a0rcsec0).5 1 1.5 2 2.5
KOI−707.05
3
Fig.4.—PlottedaretheYonsei-Yaleevolutionarytracksfor1.1
2 M⊙ (red),1.2M⊙ (green), 1.3M⊙ (blue),1.4M⊙ (cyan)and1.5
M⊙(magenta)starswith[Fe/H]=0.15. Ages,inGyr,aremarked
1 along the tracks. Note the uneven rates of evolution along these
→N (arcsec) −01 9458613, 13.988 tmsrtaaacrsskssoa.nnTdthh0ee.0gl2orgaGygyd−roitTnteeffdtimptrleaanctkeo;sitalhlrueestasrppaaptceaerdtehnbetygda0isp.t0sr1/ibdMuist⊙cioonnintionsfutefiitleliaeldsr
in these tracks are caused by rapid changes at various stages of
stellarevolution. Thethreeboxesmarkthe1,2and3σconfidence
−2
regionsofthespectroscopicSMEanalysisofthespectrumofKOI-
−−33 −2 −1 ← E (a0rcsec) 1 2 3 7st0e7lltaarkmenasastaKnedckr.adTihuesrefoarrespdeecgifienederavcaileusesinodfeTteerffmainnidngloagugn.iqFuoer
Fig.3.—Centroidfittingresultsareshownforall5planetcandi- example, a 4.4 Gyr, 1.3 M⊙ model and a 5.5 Gyr, 1.2 M⊙ model
datesofKOI-707. Theoriginisthecenteroflightreceivedoutside havenearlyidenticalTeff andlogg. Thelow-mass/oldpeaksinthe
oftransittimes. Theredasteriskisthe expected centroid oflight probability distribution functions shown in Figure 5 result from
duringtransitassumingthe transitis onthe target star; itis dis- the pile-up of stellar evolution tracks as core H is exhausted in
placedfromtheoriginbecausesomeofthelightenteringtheaper- the transition regime from radiative cores of 1.1 M⊙ and smaller
turecomes fromknown background stars whose brightness would stars to convective cores in 1.2 M⊙ and larger stars. The high-
notbeaffectedbytransitsofthetargetstar. (Thenumbersfollow- mass/young peaks result from the overlap between tracks for the
ingtheredasteriskaretheKICnumberandKeplermagnitudeof early shell H burning phase of stars with convective cores with
KOI-707.) The fitted centroids of the transit signatures observed tracksforthelatecoreHburningphaseofstarsthatare∼0.12M⊙
inQuarters1through8areshownasthethinblackcrosses,where moremassive.
the arms of these crosses show the uncertainty in RA and Dec.
Somequartersasmissingforsomecandidatesduetofailureofthe
PRF fit. The robust average centroid across quarters is shown
by the thick black cross, with the black solid circle giving the 3σ
uncertaintyintheoffsetdistance. Therobustaverageofeachmod-
median.
eledquarterlyoffsetsisshownasathickgraycross,withthegray
dashedcircleshowingthe average model3σ offsetdistance uncer- We adopt a simple model to fit the data. We fit for
tainty. Whenthecircleshowingtheobserved3σuncertaintyinthe the mean stellar density (ρ ), and each planet’s orbital
⋆
differenceimagehassignificantintersectionwiththedashedcircle period (P ), transit epoch (T ), scaled planetary ra-
showing the modeled 3σ uncertainty in the position of the target orb 0
dius(R /R )andimpactparameter(b). Theplanetsare
star,thedataareconsistentwiththeobservedtransitsignalbeing p ⋆
onKOI-707. assumed to be on circular orbits; planet-planet interac-
tions are implicitly accounted for by allowing the center
of transit time to vary in order to best-fit transit shape
(minimization of scatter on residuals). The transit was
described with the analytic formulae of Mandel & Agol
detrending). At this level of data reduction, there are (2002), and we adopted a non-linear limb darkening law
still instrumental artifacts on timescales similar to the with coefficients fixed based on the Claret & Bloemen
planetary orbits. To detrend the data (remove long- (2011)tabulationfortheKeplerbandwithstellarparam-
period variations), a second-order polynomial was fitted etersfromSMEanalysis. Abestfitmodelwascomputed
to segments of sequential Kepler photometry data out- usingaLevenberg-Marquardtalgorithmto minimize the
side of transits (i.e., in-transit observations were given χ2 statistic and initial uncertainties were estimated via
zero weight), and then this polynomial was subtracted the construction of a co-variance matrix.
from the data. A segment is defined as a length of data Thebestfitmodelandco-varianceswereusedtoseeda
that is uninterrupted for 5 or more consecutive long ca- hybrid-MCMC (Markov Chain Monte-Carlo) algorithm
dences (∼2.5 hours). Each segment is then combined to todetermineposteriordistributionsofourmodelparam-
produceafinaltimeseriesthatisthennormalizedbythe eters. The model is considered hybrid, as we randomly