The Hunt for Exomoons with Kepler (HEK): IV. A Search for Moons around Eight M-Dwarfs

1.9 MB·

by  David M. Kipping

Draftversion January28,2014
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
THE HUNT FOR EXOMOONS WITH KEPLER (HEK):
IV. A SEARCH FOR MOONS AROUND EIGHT M-DWARFS †
D. M. Kipping1,2, D. Nesvorny´3, L. A. Buchhave4,5,
J. Hartman6, G. A´. Bakos6,7,8, A. R. Schmitt9
Draft version January 28, 2014
ABSTRACT
4 With their smaller radii and high cosmic abundance, transiting planets around cool stars hold a
1 unique appeal. As part of our on-going project to measure the occurrence rate of extrasolar moons,
0 we here present results from a survey focussing on eight Kepler planetary candidates associatedwith
2
M-dwarfs. Using photodynamical modeling and Bayesian multimodal nested sampling, we find no

n compelling evidence for an exomoon in these eight systems. Upper limits on the presence of such
a bodies probe down to masses of ∼0.4M⊕ in the best case. For KOI-314, we are able to confirm the
J planetary nature of two out of the three known transiting candidates using transit timing variations.
6 Of particular interest is KOI-314c,which is found to have a mass of 1.0+0.4M , making it the lowest
−0.3 ⊕
2 mass transiting planet discovered to date. With a radius of 1.61+0.16R , this Earth-mass world is
−0.15 ⊕
likely enveloped by a significant gaseous envelope comprising ≥ 17+12% of the planet by radius. We
] −13
P also find evidence to support the planetary nature of KOI-784 via transit timing, but we advocate
E furtherobservationsto verifythe signals. Inbothsystems,weinfer thatthe innerplanethasa higher
. density than the outer world, which may be indicative of photo-evaporation. These results highlight
h
boththeabilityofKepler tosearchforsub-Earthmassmoonsandtheexcitingancillarysciencewhich
p
often results from such efforts.
-
o Subjectheadings: techniques: photometric—planetarysystems—planetsandsatellites: detection—
r stars: individual (KIC-8845205, KIC-7603200,KIC-12066335, KIC-6497146, KIC-
t 6425957, KIC-10027323, KIC-3966801, KIC-11187837; KOI-463, KOI-314, KOI-
s
a 784, KOI-3284,KOI-663,KOI-1596,KOI-494,KOI-252)
[

2 1. INTRODUCTION planetarysystems,as wellas providinga potentially fre-
v
quent seat for life in the cosmos (Williams et al. 1997;
0 In recent years, there has been a concerted effort to
Heller 2012; Heller et al. 2013; Forgan & Kipping 2013).
1 determine the occurrence rate of small planets around
Determiningtheoccurrencerateoflargemoonsconsti-
2 main-sequence stars across a broad range of spec-
tutestheprimaryobjectivethe“HuntforExomoonswith
1 tral types and orbital period using various observa-
Kepler” (HEK) project, which seeks evidence of transit-
. tional techniques and strategies (e.g. Howard et al.
1 ing satellites in the Kepler data (Kipping et al. 2012).
2010;Mayor et al.2011;Bonfils et al.2013;Fressin et al.
0 By establishing the occurrence rate as our primary ob-
2013;Dressing & Charbonneau2013;Dong & Zhu2013;
4 jective,thestrategyofoursurveyrequirescarefulconsid-
Petigura et al. 2013). The emerging consensus from
1 eration of upper limits and prior inputs and is therefore
these studies is that small planets are indeed common
:
v with occurrence rates ranging from 5%-90% for differ- substantially more challenging than a simple “fishing-
i ent size and periods ranges. What remains wholly un- trip” style survey. For these reasons, HEK conducts the
X
survey in a rigorous Bayesian framework, which comes
clearis the occurrencerateof“large”(&0.1M )moons
⊕
r attheacceptablecostofhighercomputationaldemands.
around viable planet hosts. Like mini-Neptunes and
a
In previous works, we have surveyed eight planetary
super-Earths, there are no known examples of such ob-
candidates around G and K dwarfs for evidence of exo-
jectsintheSolarSystembuttheirprevalencewouldpro-
moons, with no compelling evidence for such an object
vide deep insights into the formation and evolution of
identified thus far (Kipping et al. 2013a,b). Despite null
1Harvard-Smithsonian Center for Astrophysics, Cambridge, detections, Earth-mass and sub-Earth-mass moons are
MA02138, USA;email: [email protected] excludedinmanycases(Kipping et al.2013a). Although
2NASACarlSaganFellow there have been no transiting exomoon candidates pub-
3Dept. ofSpace Studies, Southwest ResearchInstitute, 1050
lished at this time, recently Bennett et al. (2013) re-
WalnutSt.,Suite300,Boulder,CO80302,USA
4Niels Bohr Institute, University of Copenhagen, DK-2100, portedacandidatefreefloatingplanet-moonpairviami-
Copenhagen, Denmark crolensing for MOA-2011-BLG-262. Unfortunately, this
5CentreforStarandPlanetFormation,NaturalHistoryMu- objectcannotbedistinguishedfromahighvelocityplan-
seumofDenmark,UniversityofCopenhagen, DK-1350,Copen-
etarysysteminthe Galacticbulge(Bennett et al.2013),
hagen, Denmark
6Dept. of Astrophysical Sciences, Princeton University, nor is there much prospect of obtaining a repeat mea-
Princeton,NJ05844, USA surementto confirmthe signal. Atthis time, the sample
7AlfredP.SloanFellow of planets for which statistically robust limits has been
8PackardFellow
9CitizenScience determined is too small to broachthe question of occur-
† BasedonarchivaldataoftheKepler telescope. rence rates. The purpose of this work is to extend the
2 Kipping et al.
sampleofplanetarycandidates whichhavebeen system- duration more appropriately using
atically and thoroughly examined for evidence of exo-
moons.
(R /R )2 T
For our second systematic survey, we focus on plan- SNR = ⊕ ⋆ 14 , (1)
i
etary candidates orbiting M-dwarfs (Teff < 4000K) in CDPP6 r6hours
the Kepler sample. Although considerably rarer than
ΣSNR=SNR N , (2)
i transits
FGK hosts in a magnitude limited survey like Kepler
(Dressing & Charbonneau 2013), M-dwarfs offer several whereCDPP6 is the CombpinedDifferentialPhotomet-
major advantages for seeking exomoons. The most cru- ric Precisionover 6 hours (Christiansen et al. 2012), T14
cial advantage is that since the stars are typically two is the first-to-fourth contact duration, Ntransits is the
to three times smaller than a Sun-like host, an Earth- number of usable transits in the time series, R⊕ is one
sized moon produces a four to nine times greater tran- Earth-radius and R⋆ is the stellar radius. These filters
sit depth. This advantage is somewhat tempered by leaveus with27KOIsfromwhichwe haveselectedeight
the lower intrinsic luminosities of the targets, leading to objects to study in this work. The eight planetary can-
fainter apparent magnitudes (median Kepler magnitude didates are listed along with some basic parameters in
ofoursampleis14.8)andthushigherphotometricnoise. Table 1. Additionally, we list the employed stellar pa-
However,atsuchfaintmagnitudes,Kepler’s noisebudget rameters of each host star in Table 2.
isphoton-dominatedandthusthechanceofinstrumental We noted that KOI-314, being an unusually bright
andtime-correlatednoiseinducing spuriousphotometric Kepler M-dwarf at 12.9 apparent magnitude, has con-
signals masquerading as moons is considerably attenu- siderably more follow-up available than the other seven
ated. Finally, we note that for all things being equal targets. Specifically, Pineda et al. (2013) have derived
except the spectral type of a host star, a planet’s Hill stellar parameters for this object by stacking high-
radius for stable moon orbits is modestly larger (by 25- resolutionspectratakenfromKeck/HIRESandcompar-
45%)inM-dwarfsystems. Thegreatunknownwecannot ingtoempiricallyderivedtemplatesofwell-characterzied
factorintoourchoiceofspectraltypesiswhethertheun- M-dwarfs. We therefore use the Pineda et al. (2013)
derlyingoccurrencerateoflargemoonsisfundamentally stellar parameters rather than those of Muirhead et al.
distinct for M-dwarfs than other spectraltypes, since no (2014) for this unique case. Pineda et al. (2013) do not
confirmed detections of exomoons exist at this time. provide the effective temperature, however, and so we
defer to the measurement of Mann et al. (2013) for this
2. METHODS parameter.
2.1. Target Selection (TS) 2.2. Detrending with CoFiAM
Fromthe thousandsKOIs(KeplerObjectsofInterest) In order to search for the very small expected ampli-
known at this time, we aim to trim the sample down to tudesofexomoonsignalsandaccuratelymodeltheplane-
just eight objects for our survey. This is necessary since tarytransits,itisnecessarytoremovethefluxvariations
eachobjectrequiresdecadesworthofcomputationaltime duetobothinstrumentaleffectsandseveralastrophysical
to process(Kipping et al.2013b). The firstcutwe make effects. Inwhatfollows,weusetheSimpleAperturePho-
was defined earlier in the introduction, where we only tometry (SAP) for quarters 1-15 and perform our own
considerhoststarsforwhichT <4000K.Reliablestel- detrending rather than relying on the Presearch Data
eff
larparametersarechallengingtodetermineforthesecool Conditioning (PDC) data. Short-cadence (SC) data is
faint stars but recent near-infrared spectroscopy cam- used preferentially to long-cadence (LC) data wherever
paigns by Muirhead et al. (2012) and Muirhead et al. available. Short-term variations such as flaring, point-
(2014) provide arguably the most accurate estimates for ing tweaks and safe-mode recoveries (charge trapping)
the cool KOIs. We therefore limit our sample to only are removed manually by simple clipping techniques.
M-dwarfs included in these two catalogs,which contains Long-term variations, such as focus drift and rotational
203 KOIs spread over 134 host stars. modulations, are detrended using the Cosine Filtering
In this work, Target Selection (TS) is conducted with Autocorrelation Minimization (CoFiAM) algorithm.
using the Target Selection Automatic (TSA) algo- CoFiAMwasspecifically developedto aidin searchingfor
rithm described in Kipping et al. (2012) and updated exomoonsandwedirectthereadertoourpreviouspapers
in Kipping et al. (2013a). From the 203 KOIs in the Kipping et al. (2012, 2013a) for a detailed description.
Muirhead et al. (2012) and Muirhead et al. (2014) cat- To summarize the key features of CoFiAM, it is essen-
alogs, we only further consider KOIs which satisfy the tially a Fourier-based method which removes periodici-
following criteria: i) dynamically capable of maintain- tiesoccurringattimescalesgreaterthantheknowntran-
ing a retrograde Earth-mass moon for 5Gyr following sitduration. Thisprocessensuresthatthetransitprofile
the calculationmethod outlined in Kipping et al.(2012) is not distorted and if we assume that a putative exo-
(which uses the expressions of Barnes & O’Brien 2002 moon does not display a transit longer than the known
and Domingos et al. 2006), ii) an Earth-sized transit KOI, then the moon’s signal is also guaranteed to be
with the same duration as that of the KOI would be de- protected. CoFiAM does not directly attempt to remove
tectableto ΣSNR>7.1whenalltransitepochs areused high frequency noise since this process could also easily
andiii)anEarth-sizedtransitwiththesamedurationas end up removing the very small moon signals we seek.
that of the KOI would be detectable to SNR > 1 for However, CoFiAM is able to attempt dozens of differ-
i
a single transit epoch. These three constraints are the entharmonicsandevaluatethe autocorrelationatapre-
same as those in Kipping et al. (2012) except that we selectedtimescale(weuse30minutes)andthenselectthe
slightly modify the SNR equation to account for transit harmonic orderwhich minimizes this autocorrelation,as
IV. The Hunt for Exomoons with Kepler (HEK) 3
TABLE 1
General properties of the planet candidates studied inthis work and used as inputs for our TSA algorithm. SNR (Signal-to-Noise Ratio)
isdefined inEquation 2. Planetary radii were takenfrom the Muirhead et al. (2012), except for KOI-1596.02 for which we use
Dressing & Charbonneau (2013) radius.
KOI PP [days] RP [R⊕] SNRi SN¯R Multiplicity Teq [K] KP
KOI-463.01 18.5 1.50 3.27 30.37 1 268 14.7
KOI-314.02 23.0 1.68 4.07 33.74 3 431 12.9
KOI-784.01 19.3 1.67 1.20 10.86 2 395 15.4
KOI-3284.01 35.2 0.91 2.24 15.02 1 286 14.5
KOI-663.02 20.3 1.58 3.52 31.16 2 534 13.5
KOI-1596.02 105.4 2.35 1.87 7.27 2 308 15.2
KOI-494.01 25.7 1.65 2.23 17.50 1 405 14.9
KOI-252.01 17.6 2.36 1.43 13.59 1 472 15.6
TABLE 2
General properties of the planet candidate host stars studied inthis work. Last column provides the reference from which we lift the
stellar parameters. † implies effectivetemperature comes from Mannet al. (2013) instead of the quoted reference.
KOI Teff[K] logg M∗ [M⊙] R∗ [R⊙] Sp. Type Reference
KOI-463 3389+57 4.96+0.14 0.32±0.05 0.31±0.04 M3V Muirheadetal.(2014)
−41 −0.13
KOI-314 3871+58 † 4.73+0.09 0.57±0.05 0.54±0.05 M1V Pinedaetal.(2013)
−58 −0.09
KOI-784 3767+135 4.78+0.13 0.51±0.06 0.48±0.06 M1V Muirheadetal.(2012)
−51 −0.12
KOI-3284 3748+50 4.75+0.08 0.55±0.04 0.52±0.04 M1V Muirheadetal.(2014)
−100 −0.07
KOI-663 3834+50 4.78+0.13 0.51±0.06 0.48±0.06 M1V Muirheadetal.(2012)
−57 −0.12
KOI-1596 3880+143 4.77+0.14 0.51±0.07 0.49±0.07 M0V Muirheadetal.(2012)
−103 −0.14
KOI-494 3789+219 4.78+0.22 0.50±0.10 0.48±0.10 M1V Muirheadetal.(2012)
−159 −0.20
KOI-252 3745+53 4.76+0.06 0.54±0.03 0.51±0.03 M1V Muirheadetal.(2014)
−71 −0.06
quantifiedusing the Durbin-Watson statistic. This “Au- withatargetefficiencyof0.1andusethesameparameter
tocorrelation Minimization” component of CoFiAM pro- setsandpriorsdescribedinKipping et al.(2013a),giving
videsoptimizeddataforsubsequentanalysis. Incontrast us7freeparametersformodel P and14forS. Theonly
totheinitialsamplestudiedinKipping et al.(2013a),we changefromthe Kipping et al.(2013a)priors,is thatwe
found that none of the eight detrended KOI light curves fitthequadraticlimbdarkeningcoefficientsusingtheq -
1
retainedsignificant(>3σ)autocorrelationafterCoFiAM, q parameter set suggested in Kipping (2013a), which
2
which is consistent with a photon-noise dominated sam- provides more efficient parameter exploration. Contam-
ple, which one might expect for these generally fainter ination factors for each quarter are accounted for us-
targets. ingthe methoddevisedinKipping & Tinetti(2010)and
long-cadencedataisresampledusingN =30follow-
resam
2.3. Light Curve Fits ing the technique devised in Kipping (2010).
In Kipping et al. (2013a), we described four basic de-
The transit light curves for a planet-with-moon sce-
tection criteria (B1-B4) to test whether an exomoon fit
nario (model S) are modeled using the analytic photo-
dynamicalalgorithmLUNA(Kipping 2011a), aswith pre- can be further considered as a candidate or not. The
criteria essentially demand that the moon signal is both
vious HEK papers. Photodynamical modeling accounts
significant and physically reasonable:
for not only the transits of the planet and the moon but
also the transit timing and duration variations (TTVs
B1 Improved evidence of the planet-with-moon fits at
and TDVs) expected (Kipping 2009a,b) in a dynamic
≥4σ confidence.
framework. These fits always assume just a single moon
and thus our search is implicitly limited to such cases. B2 Planet-with-moon evidences indicate a preference
For comparison, we consider a simple planet-only model
for a) a non-zero radius moon b) a non-zero mass
(model P) as well, for which we employ the standard
moon.
Mandel & Agol (2002) routine. Finally, we perform a
planet fit on each individual transit to derive TTVs and B3 Parameterposteriorsarephysical,inparticularρ .
P
TDVs(model I),whichisusefulincomparingourmoon
model against perturbing planet models later on. B4 a) Mass and b) radius of the moon converge away
Light curve fits are performed in a Bayesian frame- from zero.
work with the goal of both deriving parameter posteri-
ors and computing the Bayesian evidence (Z) for each One slight difference to Kipping et al. (2013a) is that
model. The moon fits are particularly challenging re- wehavesplitsomeofthebasicdetectioncriteriaintosub-
quiring 14 free parameters exhibiting a large number of criteria. Thiswillbeusefullatersinceitisofteneasierto
modesandcomplexinter-parametercorrelations. Tothis evaluate just a subset of the sub-criteriaand a failure to
end, we employ the multimodal nested sampling algo- passoneoftheseallowsustoquicklyrejecttheobjectas
rithm MultiNest (Feroz et al. 2008, 2009) as with pre- an exomoon candidate. Note that in practice we reject
vious HEK papers. For all fits, we use 4000 live points anytrialsforwhichthedensityoftheplanetisexcessively
4 Kipping et al.
lowandthiscausestheradiusofthemoontobenon-zero Parameterestimates from the marginalizedposteriorsof
and so criterion B4b is always satisfied by virtue of the the favored light curve model are provided for all KOIs
priorsusedinourmodel. B4aisdefinedasbeingsatisfied inTable3,exceptKOI-314andKOI-784whichareavail-
if there is a less than a 5%false-alarm-probabilityof the able later in Table 6.
satellite-to-planetmass ratio(M /M ) being zero using
S P
the Lucy & Sweeney (1971) test. 3.2. Null Detections
Another change we implement follows the strategy
The first category of objects we discuss in this sec-
used recently for Kepler-22b (Kipping et al. 2013b),
where we allow LUNA to model negative-radius moons tion are the null detections. We define these objects to
be ones for which the mode of the derived satellite-to-
(which we treat as inverted transits) during model S.
planet mass ratio (M /M ) posterior distribution is at,
By doing so, it is no longer necessary to run a separate S P
or close to, zero plus other detection criteria are also
moon fit to test B2a, where we would have previously
failed. Note that this is distinct from requiring that the
locked the radius to zero (although this fit is sometimes
objects fail criterion B4a, although any object in this
still a useful tool). This trick essentially saves compu-
category will indeed fail B4a too. We find five objects
tation time yet retains our ability to test sub-criterion
in this category: KOI-663.02,KOI-1596.02,KOI-494.01,
B2a. We define that sub-criterion B2a is unsatisfied if
KOI-463.01 and KOI-3284.01. Null detections, such as
the 38.15% quantile (lower 1σ quantile) of the derived
these,allowforasimplecalculationoftheupperlimiton
satellite-to-planet radius ratio (R /R ) is negative.
S P the ratio (M /M ) by posterior marginalization. These
Although we use precisely the same definition for B1 S P
upperlimitsrangefrom0.29to0.92(95%confidenceup-
as that of previous papers, we reiterate here that B1 is
per limits) and are available in Table 3.
computed by evaluating if the Bayesian evidences be-
As well as failing criteria B4a, all of these objects fail
tween models P and S indicates a ≥ 4σ preference for
at least one other detection criteria, as listed in Table 4.
the latter.
Forseveralcases,aperiodogramsearchoftheTTVsand
Ifallofthebasicdetectioncriteriaaresatisfied,wealso
TDVs for these objects appears to indicate possible per-
have three follow-up criteria described in Kipping et al.
turbations, as shown in Figures 1-5. We have listed the
(2013a) to further vet candidates. The concept here is
mostsignificantpeaks from a periodogramsearchin Ta-
to only fit 75%ofthe available data in the originaltran-
ble 5.
sit fits and thus deliberately exclude 25% of time series,
KOI-463.01 and KOI-3284.01 are two single KOI sys-
which serves as “follow-up” data.
tems which show > 4σ significant TTVs (from an F-
F1 All four basic criteria are still satisfied when new test). This may therefore be evidence for additional
planets although we consider the signal-to-noise of the
data is included.
availabledatainsufficienttoreliablydeduceauniqueun-
F2 Thepredictivepowerofthemoonmodelissuperior seen perturber solution. We consider the TTVs/TDVs
to that of a planet-only model. ofthe only othersingle KOIclassedasanexomoonnull-
detection, KOI-494.01,to be insignificant.
F3 Aconsistentandstatisticallyenhancedsignalisre- Twooutofthefivenulldetectioncasesresideinmulti-
covered with the inclusion of more data. transiting planet systems where interactions may be a-
priori expected. However, the period ratios between
The most powerful of these criteria is F2 and this KOI-663.02/KOI-663.01 and KOI-1596.02/KOI-1596.01
test is performed for all KOIs studied in this work. In are7.4and17.8respectivelyi.e. too farto expectsignif-
Kipping et al. (2013a), the 75% original data was con- icantinteractions. Thispointisreinforcedempiricallyby
sidered to be quarters 1-9 and the follow-up data was notingthattheperiodogramsshownostrongoverlapping
quarter 10-12. In this framework, F2 tests the extrapo- peaks and no significant (≥ 4σ) power. We are there-
latedbestmoonmodelfit. Onesignificantchangeinthis foreunabletoconfirmthesetwomulti-planetarysystems
workisthatwenowchoosethe25%follow-updatatolie using TTVs.
somewherearoundthe middle ofourtotalavailabletime
series. This means that F2 now tests an interpolation of 3.3. Spurious Photometric Detections
the model rather than an extrapolation. An advantage
A spurious photometric detection is one where we hy-
of doing this is that parameters such as the orbital pe-
pothesizethataresidualamountoftime-correlatednoise
riod of the planet are best constrained by maximizing
drivesa non-zerosatellite radius solution,which then it-
the baseline of available data and so this new approach
self drives a non-zero satellite mass solution due to the
allows for improved parameter estimates.
factweenforceonlyphysicallyreasonablesatellitedensi-
3. RESULTS tiesinthefits. Weidentifysuchcasesbythefactthat: i)
theposteriordistributionfor(M /M )doesnotpeakat
3.1. Overview S P
zero (and thus it is not a “null” detection), ii) other de-
In this work, all eight KOIs can be placed into one tectioncriteriaarefailed(suggestingsome kindofspuri-
of three categories: i) null detections (§3.2), ii) spurious ousdetection)andiii)performingafitwherethemoon’s
photometricdetections(§3.3)andiii)spuriousdynamical radius is fixed to zero (model S ) yields an (M /M )
R0 S P
detections(§3.4). Eachcategoryisdefinedintherelevant posterior which peaks at zero, as expected for a null de-
subsectionbut it shouldbe clear fromtheir naming that tection. This final point indicates that when we switch
wefindnoconfirmedorcandidateexomoonsinthissam- off the moon’s radius, the planet no longer seems to be
ple. For each KOI system studied, the TTVs and TDVs perturbedandthus the moon’sradiuswaslikely aresult
of all planetary candidates can be seen in Figures 1-8. of the regression fitting out non-astrophysical artifacts
IV. The Hunt for Exomoons with Kepler (HEK) 5
TABLE 3
Final parameter estimatesfrom the favored light curve model for KOIs studied in our sample, except KOI-314 and KOI-784 which are
provided inTable 6. In all cases, the favored model isa simple planet-only model.
KOI 463.01 3284.01 663.02 1596.02 494.01 252.01
P ............. 18.477637+0.000014 35.23266+0.00027 20.306531+0.000023 105.35794+0.00060 25.695907+0.000070 18.604627+0.00022
−0.000014 −0.00026 −0.000022 −0.00064 −0.000070 −0.000022
τ [BKJDUTC] . 758.07461+−00..0000003333 769.5611+−00..00003364 781.85484+−00..0000004499 665.4733+−00..00002233 779.7936+−00..00001122 874.68224+−00..0000005552
(RP/R⋆) ..... 0.0475+−00..00001110 0.01837+−00..001110 0.02431+−00..0000100555 0.0355+−00..00002108 0.03246+−00..00100975 0.0457+−00..00003252
ρ⋆,obs[gcm−3] 27.0+−38..85 3.41+−01..9485 7.2+−12..14 15.5+−36..76 3.64+−01..6305 2.1+−11..52
b .............. 0.29+0.25 0.30+0.32 0.31+0.26 0.31+0.31 0.30+0.29 0.58+0.21
−0.20 −0.21 −0.21 −0.21 −0.21 −0.38
q1 ............ 0.22+−00..2183 0.43+−00..3268 0.42+−00..2154 0.50+−00..3228 0.27+−00..3137 0.47+−00..3220
q2 ............ 0.29+−00..3281 0.44+−00..3350 0.62+−00..2256 0.51+−00..3311 0.37+−00..3285 0.27+−00..3167
RP[R⊕] ...... 1.61+−00..2211 1.04+−00..1110 1.28+−00..1176 1.90+−00..3209 1.70+−00..3366 2.56+−00..2230
u1 ............ 0.48+−00..1280 0.78+−00..4450 0.99+−00..1134 0.91+−00..3388 0.59+−00..2256 0.74+−00..1275
u2 ............ −0.03+−00..3108 −0.18+−00..3320 −0.36+−00..2101 −0.26+−00..3236 −0.10+−00..3109 −0.03+−00..3219
Seff[S⊕] ...... 0.896+−00..204988 2.23+−00..9398 3.12+−00..9365 0.221+−00..100401 3.54+−10..4810 7.7+−52..43
(ρ⋆,obs/ρ⋆,true) 1.65+−00..9770 0.59+−00..2276 1.01+−00..5472 2.3+−11..72 0.51+−00..5219 0.36+−00..2169
emin .......... 0.18+−00..1142 0.18+−00..1183 0.00+−00..1070 0.00+−00..1070 0.22+−00..2252 0.33+−00..2117
(MS/MP) .... <0.92 <0.29 <0.52 <0.60 <0.66 <0.32
TABLE 4
Detectioncriteriaresults for each of the eight KOIssurveyed inthis work. Only KOI-314.02 and KOI-784.01 pass the five criterialisted
here.
KOI B1 B2a B3a B4a F2
KOI-463.01 X(+2.89σ) X(−0.33) X(4.82%) X(FAP=29.86%) X(∆χ2=27.8forN =1815)
KOI-314.02 X(+14.3σ) X(+0.28) X(0.00%) X(FAP=0.066%) X(∆χ2=−925.0forN =9161)
KOI-784.01 X(+6.50σ) X(+0.32) X(0.00%) X(FAP=0.018%) X(∆χ2=−39.0forN =9431)
KOI-3284.01 X(+3.32σ) X(+0.58) X(94.2%) X(FAP=37.28%) X(∆χ2=15.8forN =597)
KOI-663.02 X(+0.76σ) X(−0.44) X(38.8%) X(FAP=47.61%) X(∆χ2=12.8forN =9676)
KOI-1596.02 X(+1.62σ) X(+0.34) X(2.09%) X(FAP=34.13%) X(∆χ2=−4.1forN =322)
KOI-494.01 X(+1.23σ) X(−0.46) X(39.6%) X(FAP=30.55%) X(∆χ2=16.0forN =836)
KOI-252.01 X(+1.82σ) X(+0.52) X(0.40%) X(FAP=16.13%) X(∆χ2=−4.9forN =33048)
0.015 é 0.04 K
@DTTVd---000000......000000110001505050 èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéééééééééé éèéèéèéèé @DTDVd--0000....00004202 èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèééééééééééé éèéèéèéèé OI0463.01-
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
0.5
wer0.4 wer0.15
po0.3 po0.10
TTV00..12 TDV0.05
0.0 0.00
100 500 1000 3000 100 500 1000 3000
Period@dD Period@dD
Fig.1.—TTVs(left)and TDVs(right)forthe planetary candidate of KOI-463. Pointsused inthe photodynamical moon fitsare filled
circles,whereas thosepointsignoredforsubsequentpredictivetestsareopencircles. Bottompanelsshowtheperiodograms oftheTTVs&
TDVs(which use the entire available data set).
in the light curve. KOI-252.01 is the only object in this we enforce (R /R ) = 0 in the 13-dimensional model
S P
survey which we classify in this category. S . The results from this model are now consistent
R0
As seen in Table 4, B1 and B4a are both failed for with a null detection andthus we may derive upper lim-
KOI-252.01. Thelatterisbecausethederived(M /M ) itson(M /M )asusual(thisissimilartotheprocedure
S P S P
peaks away from zero (at around 0.5) and yet is ex- used for KOI-303.01 in Kipping et al. 2013a). For KOI-
tremelydisperseoverthe entireprior. Theformeressen- 252.01 then, we derive that (M /M ) < 0.33 to 95%
S P
tially implies that whatever is driving the fit is actually confidence.
of relatively low significance.
By definition of being a spurious photometric detec-
tion, the (MS/MP) posterior has a mode at zero when 3.4. Spurious Dynamical Detections
6 Kipping et al.
0.10 0.6 èé K
@DTTVd--0000....10000505èéèé èé èéèé èé é é èéèé èé èéèé @DTDVd---000000......642024èéèé èé èéèé é é èéèé èé èéèé OI3284.01-
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
1.0
TTVpower0000....2468 TDVpower0000....2468
0.0 100 500 1000 3000 0.0 100 500 1000 3000
Period@dD Period@dD
Fig.2.—TTVs(left)andTDVs(right)fortheplanetarycandidate ofKOI-3284. Pointsusedinthephotodynamical moonfitsarefilled
circles,whereas thosepointsignoredforsubsequentpredictivetestsareopencircles. Bottompanelsshowtheperiodograms oftheTTVs&
TDVs(which use the entire available data set).
0.02 0.06 K
@DTTVd-000...000101èéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èééééééé éèé èéèéèé èéèéèéééé èéèéèéèéèéèéèéèé @DTDVd--00000.....0000042024èéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èééééééé éèé èéèéèé èéèéèéééé èéèéèéèéèéèéèéèéOI0663.0-
-0.02 -0.06 2
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
@DTTVd--000000......000000210123èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé @DTDVd-000...000505èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéKOI0663.0-
-0.0355000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200 1
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
0.15 0.30
power0.10 power000...122505
V0.05 V0.10
TT TD0.05
0.00 0.00
100 500 1000 3000 100 500 1000 3000
Period@dD Period@dD
Fig. 3.— TTVs (left) and TDVs (right) for the planetary candidates of KOI-663. Each row shows the observations (circles) with the
best-fit sinusoid from a periodogram using the same coloring as that for which the name of the planet is highlighted in on the RHS. For
KOI-663.02, pointsusedinthephotodynamical moon fitsarefilledcircles, whereas thosepointsignored forsubsequent predictivetestsare
opencircles. Bottompanelsshowtheperiodograms oftheTTVs&TDVs(whichusetheentireavailable dataset),withverticalgridlines
marking the locations where one might expect power from planet-planet interactions.
TABLE 5
Summary ofthe highestpowerpeaks foundintheperiodograms oftheTTVsandTDVsoftheplanetary candidates analyzed inthiswork.
KOI TTVPeriod[d] TTVAmp. [mins] TTVSignif. TDVPeriod[d] TDVAmp. [mins] TDVSignif.
KOI-463.01 125.1 6.5 4.8σ 123.6 7.1 2.3σ
KOI-314.02 112.9 21.7 9.3σ 307.3 6.9 3.9σ
KOI-314.01 913.8 2.5 3.8σ 38.7 3.1 2.4σ
KOI-314.03 434.0 16.4 3.2σ - - -
KOI-784.01 541.9 25.6 4.9σ 53.9 54.1 4.8σ
KOI-784.02 29.0 7.2 3.3σ - - -
KOI-3284.01 176.9 84.8 5.3σ 97.9 5.0 3.3σ
KOI-663.02 133.1 4.2 2.5σ 116.6 15.0 2.9σ
KOI-663.01 11.1 2.0 2.9σ 50.1 7.1 5.4σ
KOI-1596.02 1050.0 22.1 2.7σ 802.7 48.5 1.7σ
KOI-1596.01 36.1 24.1 2.7σ - - -
KOI-494.01 116.1 13.1 2.6σ 116.1 33.9 2.7σ
KOI-252.01 128.4 3.3 3.4σ 81.1 12.9 2.9σ
IV. The Hunt for Exomoons with Kepler (HEK) 7
@DTTVd--00000.....0000021012 èé èé èé é é èé èé èé èé @DTDVd-000...000505 èé èé èé é é èé èé èé èé KOI1596.0-
2
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Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
0.10 èé K
@DTTVd-000...000505 èé èéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èé èéèéèéèéèé @DTDVd OI1596-
.0
-0.10 èé 1
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
power0000....00116802 power0000....12235050
TTV00..0024 TTV00..0150
0.00 0.00
100 500 1000 3000 500 1000 3000
Period@dD Period@dD
Fig.4.—TTVs (left) and TDVs (right) for the planetary candidates of KOI-1596. Each row shows the observations (circles) with the
best-fit sinusoid from a periodogram using the same coloring as that for which the name of the planet is highlighted in on the RHS. For
KOI-1596.02, points used in the photodynamical moon fits are filled circles, whereas those points ignored for subsequent predictive tests
are open circles. Bottom panels show the periodograms of the TTVs & TDVs (which use the entire available data set), with vertical grid
linesmarking the locations where one might expect power from planet-planet interactions.
@DTTVd---0000000.......00000006420246550èé0èé0èéèéèé5èé5èé2èé0èé0èé èéé5é5400ééé55é6é00ééèé5580èé0èéèéèé5èé6èé0èé00èéèé5620èé0èéèé @DTDVd--00000.....1000105050550èé0èé0èéèéèé5èé5èé2èé0èé0èé èéé5é5400ééé55é6é00ééèé5580èé0èéèéèé5èé6èé0èé00èéèé5620èé0èéèé KOI0494.01-
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
0.30
er0.20 er0.25
w w
o0.15 o0.20
Vp0.10 Vp00..1105
TT0.05 TD0.05
0.00 0.00
100 500 1000 3000 100 500 1000 3000
Period@dD Period@dD
Fig.5.—TTVs(left)and TDVs(right)forthe planetary candidate of KOI-494. Pointsused inthe photodynamical moon fitsare filled
circles,whereas thosepointsignoredforsubsequentpredictivetestsareopencircles. Bottompanelsshowtheperiodograms oftheTTVs&
TDVs(which use the entire available data set).
8 Kipping et al.
0.02 0.04 K
@DTTVd-000...000101èéèéèéèéèéèéèéééééèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèééééééééééééééèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé @DTDVd-000...000202èéèéèéèéèéèéèéééééèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèééééééééééééééèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéOI0252-
-0.02 -0.04 èé .01
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
0.14
wer00..1102 wer00..1102
TTVpo0000....00002468 TDVpo0000....00002468
0.00 0.00
100 500 1000 3000 100 500 1000 3000
Period@dD Period@dD
Fig.6.—TTVs(left)and TDVs(right)forthe planetary candidate of KOI-252. Pointsused inthe photodynamical moon fitsare filled
circles,whereas thosepointsignoredforsubsequentpredictivetestsareopencircles. Bottompanelsshowtheperiodograms oftheTTVs&
TDVs(which use the entire available data set).
KOI-314.02 and KOI-784.01 are the only two remain-
ingKOIsandweclassifythese bothasspuriousdynami-
P′ j−k
caldetections. A-priori,thesetwosystemshavethehigh-
∆≡ −1, (3)
est chance of exhibiting significant planet-planet inter- P ! j !
actions since KOI-314.02 and KOI-314.01 reside near a
P′
5:3 period commensurability and KOI-784.01 and KOI- P = . (4)
TTV
784.02 are close to 2:1. In the case of KOI-314, there is j|∆|
athirdplanetarycandidateinteriorto the othertwo,for
For the KOI-314 and KOI-784 systems, we have
whichthe closestcommensurabilities are9:4 and5:4 rel-
marked P on the periodograms shown in Figures 7
ative to KOI-314.02and KOI-314.01respectively. Given TTV
& 8. The pair KOI-314.01/.02 indeed shows power at
that the 9:4 commensurabily does not usually exhibit
this frequency and performing a full dynamical fit of
strong perturbations and the fact KOI-314.03 is three
the TTVs (described inthe Appendix) retrievesa quasi-
times smaller than the other two KOIs, the lack of any
sinusoidalsignalconformingwiththis frequency. Incon-
observed interactions induced by this candidate is not
trast,ourdynamicalfitsofKOI-784.01/.02revealanon-
surprising.
sinusoidal waveform, which explains why the expected
ForbothKOI-314.02andKOI-784.01,themoonmodel
super-period has no accompanying power in the peri-
S yieldsan(M /M )posteriorpeakingatunity(i.e. the
S P odogram.
solutionsfavorbinaryplanets). Formally,allofthedetec-
In both systems, the TTVs fitted by our moon model
tion criteriaare satisfied despite this, as seen in Table 4.
S produceanessentiallyequallygoodfittothedataasa
FromtheTTVsshowninFigures7&8andthelistofpe-
planet-planet interaction model. This is clearly evident
riodogramfrequenciesdetailedinTable5,itisclearthat
by comparing the fits shown in Figures 7 & 8. However,
significant power exists at long-periods for both objects.
the moon model requires an entirely new object to be
Since exomoons always have an orbital period less than
introduced into the system to explain the observations,
that of the planet they are bound to (Kipping 2009a),
whereas the planet-planet models naturally explain the
the moon interpretation requires these to be aliases of
TTVs simply using the known transiting planets. This
the true short period. However, producing such a long
fact, combined with the previously discussed concerns
period requires fine tuning of the moon’s period. This is
withthemoonhypothesis,leadsustoconcludethatboth
easily visualized in Figure 9, where one can see that for
KOI-314andKOI-784arespuriousdynamicaldetections
both cases we require a moon’s period to be nearly 1:2
i.e. a planet-planet perturbation is the most likely un-
commensurable to the planet’s period, with a small fre-
derlying reason via Occam’s Razor.
quency splitting which induces the longer super-period.
ForbothKOI-314andKOI-784,weconductedtwosets
Suchfinetuningisastrongblowtothemoonhypothesis
of dynamical TTV fits where we turned on/off the mass
sincethereis noknownexampleofaSolarSystemmoon
ofthe planets. Since ourfits areconductedwithMulti-
being in such a near-commensurability.
Nest, we may compare the Bayesian evidences to eval-
A crude but useful tool in studying TTVs of multi-
uate the statistical significance of the claimed interac-
planet systems is to estimate the “super-period” of the
tions, for which we find healthy confidences of 24.2σ
TTVs assuming the variationsaresinusoidal(Xie 2013).
and 6.7σ for KOI-314.01/.02 and KOI-784.01/.02 re-
Consider two planets with orbital periods P′ and P
spectively. The maximum a-posteriori fit through the
whereP′ >P. For twoplanets in akth orderMMR, the TTVs & TDVs (where available) reduces the χ2 from
ratioofthe orbitalperiodsis simply (P′/P)≃j/(j−k),
1440.1→335.2with218observationsforKOI-314.01/.02
where k is an integer satisfying 0 < k < j and j de-
and 487.7 → 340.0 with 166 observations for KOI-
fines the mutual proximity of the MMR. Following the 784.01/.02. A simple F-test based on these χ2 changes
approximate theory of Xie (2013), the expected period-
finds confidences of 17.1σ and 7.1σ for KOI-314.01/.02
icityoftheobservedTTVs,theso-called“super-period”,
and KOI-784.01/.02 respectively, showing good consis-
is given by:
tency with the Bayesian evidence calculation.
IV. The Hunt for Exomoons with Kepler (HEK) 9
@DTTVd--00000.....0000021012 èé èéèéèéèé èéèé èéèééé éééé ééèé èéèéèéèéèéèéèéèé èéèéèé èé èé èéèéèéèéèéèéèé @DTDVd---0000000.......00000003210123 èé èéèéèéèé èéèé èéèééé éééé ééèé èéèéèéèéèéèéèéèé èéèéèé èé èé èéèéèéèéèéèéèé KOI0314.02-
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
èé
0.010 0.02 K
@DTTVd--0000....000010000505èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèèééèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé @DTDVd--0000....00002101èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèèééèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé OI0314.01-
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
0.10 èé K
O
@DTTVd-000...000505èé èéèéèé èéèéèéèéèéèéèéèé èéèé èéèéèéèéèéèéèéèéèéèé èé èéèéèé @DTDVd I0314-
-0.10 .03
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
TTVpower00001.....24680 ««.02.03.02.03 ««.01.02.01.02 ««.01.03.01.03 TDVpower0000....1234 ««.02.03.02.03 ««.01.02.01.02 ««.01.03.01.03
0.0 100 500 1000 3000 0.0 100 500 1000 3000
Period@dD Period@dD
Fig.7.— TTVs (left) and TDVs (right) for the planetary candidates of KOI-314. Each row shows the observations (circles) with
the best-fit sinusoid from a periodogram using the same coloring as that for which the name of the planet is highlighted in on the RHS.
Additionally, we show the best-fit moon model in blue and best-fit planet-planet model in orange. For KOI-314.02, points used in the
photodynamical moon fits are filled circles, whereas those points ignored for subsequent predictive tests are open circles. Bottom panels
showtheperiodograms oftheTTVs&TDVs(whichusetheentireavailable dataset),withverticalgridlinesmarkingthelocations where
one might expect power from planet-planet interactions.
0.06 0.15 èé K
@DTTVd--00000.....0000042024 èéèéèéèéèéèé èéèé èéèéèé èéèéèéèéèéèéèéèéèé èéèéèéèé éééééééééééééééééèéèéèéèéèéèéèéèéèéèé @DTDVd--00000.....1000105050 èéèéèéèéèé èé èéèéèé èéèéèéèéèéèéèéèéèé èéèéèéèé ééééééééééééééééèéèéèéèéèéèéèéèéèéèéOI0784.0-
-0.06 -0.15 1
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
@DTTVd-0000....00002024èéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé èéèéèéèéèéèéèéèéèéèéèéèéèéèé èé èé èéèéèéèéèéèéèéèéèéèéèéèéèéèéèéèé @DTDVd KOI0784-
-0.04 .0
2
55000 55200 55400 55600 55800 56000 56200 55000 55200 55400 55600 55800 56000 56200
Time@BJDUTC-2,400,000D Time@BJDUTC-2,400,000D
0.5 0.5
power00..34 ««.02.02 power00..34 ««.02.02
TTV00..12 .01.01 TDV00..12 .01.01
0.0 0.0
100 500 1000 3000 100 500 1000 3000
Period@dD Period@dD
Fig.8.— TTVs (left) and TDVs (right) for the planetary candidates of KOI-784. Each row shows the observations (circles) with
the best-fit sinusoid from a periodogram using the same coloring as that for which the name of the planet is highlighted in on the RHS.
Additionally, we show the best-fit planet-planet model in orange. For KOI-784.01, points used in the photodynamical moon fits are filled
circles, whereas those points ignored for subsequent predictive tests are open circles. Bottom panels show the periodograms of the TTVs
& TDVs (which use the entire available data set), with vertical grid lines marking the locations where one might expect power from
planet-planet interactions.
10 Kipping et al.
thintailofhighereccentricitiesandhighermasses. These
solutions have much lower sample probability since they
require the fine tuning of ∆̟ ≃ 0 to explain the data.
Given our assumption of uniform priors in ̟, the likeli-
hoodmultipliedbythepriormass,whichdefinesthesam-
pleprobability,ismuchloweralongthistailandthushas
2
(cid:144) amuchlowerBayesianevidence. Forthisreason,thistail
P
P doesnotappearwhenMultiNestoutputstheweighted
posteriors. We decided to explore this tail in a second
fit by enforcing a lower limit of (M /M )>1.5×10−5.
P ⋆
The mode which is picked up by MultiNest down this
tail is easily identified as unphysical, since it requires
that77.4%ofKOI-314.01’sposteriorsamplesexceedthe
mass-strippinglimitofMarcus et al.(2009)(i.e. theden-
sity is unphysically high). The fact the solution is both
unphysical and has a much lower sample probability al-
lows us to discard it.
We therefore consider the original solution to be the
only plausible explanation for the TTVs. We subse-
11.3 11.4 11.5 11.6 11.7 11.8 quently consider these planetary candidates to be con-
P @dD firmed as exhibiting interactions and thus are real plan-
S etsgiventhelowderivedmasses. WerefertoKOI-314.01
and .02 as KOI-314b and KOI-314c respectively from
here on. Note that KOI-314.03 exhibits no detectable
signature in any of the TTVs and thus this object re-
mains a planetary candidate at this time.
Formally, our dynamical model for KOI-784.01/.02 is
favored over a non-interacting model at ∼7σ. Despite
2 this, we prefer to consider these two objects a tentative
(cid:144)
P detection at this time. Our caution is based on the fact
P
thatitis difficulttoactuallyseeanystructureinthe ob-
served TTVs (see Figure 8). Further, there is an appar-
entlackofpowerinthe .01periodogramatthe expected
super-period and yet considerable power at ∼500d (see
Table5). OurbestfittingTTVmodeldoesnotreproduce
anypowerat500d,butdoeshaveitsdominantpowerat
thesuper-period. Forthesereasons,itisunclearwhether
our model is just fitting out some non-white noise com-
ponent remaining in the data. Despite this, we note
that the derived solution is apparently unique, dynam-
ically stable and again yields physically plausible inter-
9.4 9.5 9.6 9.7 9.8 9.9 nal compositions for both objects (see Figure 11). Our
favored solution yields significant eccentricity for KOI-
PS @dD 784.01 of e = 0.182+−00..001241, which is also noteworthy. We
attempted to repeat the fits enforcing low eccentricities
but this yielded an implausibly dense composition for
Fig.9.—Posteriordistributionforaputativemoon’sorbitalpe-
KOI-784.01 with 99.1% of the trials exceeding the mass
riodforKOI-314.02(upperpanel) andKOI-784.01 (lowerpanel).
Inbothcases,theobservedlong-periodTTVrequiresafinelytuned stripping limit of Marcus et al. (2009). We argue that
moonperiodclosetoa2:1commensurabilitywiththeplanet,which confirmation of this signal could be made by collecting
we deem quite improbable. furtherTTVsoridentifyinganobjectresponsibleforthe
∼500d periodogram peak, either through transits or ra-
dial velocities.
Statistics aside, Figure 7 shows that our fits for KOI- Following the strategy of Nesvorny´ et al. (2012) for
314.01/02 clearly reproduce the dominant pattern in KOI-872b, we adjusted the time stamps of the de-
the data. The two sets of TTVs exhibit visible anti- trended photometry for these two systems by offsetting
correlation, which is a well-known signature of planet- the best-fitting dynamical planet-planet model transit
planet interactions (Steffen et al. 2012) and so further times. Since the interacting planets must orbitthe same
improvesour confidence in this fit. Finally, we note that star, this allows us to re-fit the photometry with a com-
the derivedsolution is dynamically stable for ≥Gyr and mon mean stellar density (ρ ) and common limb dark-
⋆
yieldsphysicallyplausibleinternalcompositionsforboth ening coefficients (q and q ) using a simple planet-only
1 2
objects (see Table 6). model. Combining this information yields improvedand
Although MultiNest only identifies one mode and self-consistent parameters for each planet. In these fits,
thus the solution appears unique, the unweighted pos- the eccentricity and argument of periastron passage for
teriors show high likelihood solutions extending down a