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The Occurrence Rate of Small Planets Around
Small Stars
Citation
Dressing, Courtney D., and David Charbonneau. 2013. The Occurrence Rate of Small Planets
Around Small Stars. The Astrophysical Journal 767, no. 1: 95. doi:10.1088/0004-637x/767/1/95.
Published Version
doi:10.1088/0004-637X/767/1/95
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Accepted to ApJ
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
THE OCCURRENCE RATE OF SMALL PLANETS AROUND SMALL STARS
Courtney D. Dressing1,2 and David Charbonneau1
(Dated: February 25, 2013)
Accepted to ApJ
ABSTRACT
Weusetheopticalandnear-infraredphotometryfromtheKeplerInputCatalogtoprovideimproved
3 estimates of the stellar characteristics of the smallest stars in the Keplertarget list. We find 3897
1 dwarfswithtemperaturesbelow4000K,including64planetcandidatehoststarsorbitedby95transit-
0 ingplanetcandidates. WerefitthetransiteventsintheKeplerlightcurvesfortheseplanetcandidates
2 andcombine the revisedplanet/starradiusratioswithourimprovedstellarradiitorevisethe radiiof
the planet candidates orbiting the cooltargetstars. We then comparethe number of observedplanet
b
candidates to the number of stars around which such planets could have been detected in order to
e
F estimate the planet occurrence rate aroundcoolstars. We find that the occurrencerate of0.5 4R⊕
planetswithorbitalperiodsshorterthan50daysis0.90+0.04 planetsperstar. Theoccurrence−rateof
2 −0.03
2 Earth-size (0.5−1.4R⊕) planets is constant across the temperature range of our sample at 0.51+−00..0065
Earth-size planets per star, but the occurrence of 1.4 4R⊕ planets decreases significantly at cooler
−
] temperatures. Our sample includes 2 Earth-size planet candidates in the habitable zone, allowing us
P to estimate that the mean number of Earth-sizeplanets in the habitable zone is 0.15+0.13 planets per
E −0.06
coolstar. Our95%confidencelowerlimitontheoccurrencerateofEarth-sizeplanetsinthehabitable
h. zones of cool stars is 0.04 planets per star. With 95% confidence, the nearest transiting Earth-size
p planetinthehabitablezoneofacoolstariswithin21pc. Moreover,thenearestnon-transitingplanet
- in the habitable zone is within 5 pc with 95% confidence.
o
Subject headings: catalogs – methods: data analysis – planetary systems – stars: low-mass – surveys
r
t – techniques: photometric
s
a
[
1. INTRODUCTION candidates are primarily small objects (196 with Rp <

2 The Keplermission has revolutionized exoplanet 1.25R⊕, 416 with 1.25R⊕ < Rp < 2R⊕, and 421 with
v statistics by increasing the number of known extraso- 2R⊕ < Rp < 6R⊕), but the list also includes 41 larger
7 lar planets and planet candidates by a factor of five candidateswithradii6R⊕ <Rp <15R⊕. Theinclusion
4 oflargercandidatesintheBatalha et al.(2012)sampleis
and discovering systems with longer orbital periods
6 an indication that the original Borucki et al. (2011) list
and smaller planet radii than prior exoplanet surveys
1 wasnotcompleteatlargeplanetradiiandthatcontinued
(Batalha et al. 2011; Borucki et al. 2012; Fressin et al.
. improvements to the detection algorithm may result in
2 2012; Gautier et al. 2012). Kepleris a Discovery-class
furtherannouncementsofplanetcandidateswitharange
0 space-based mission designed to detect transiting exo-
3 planetsbymonitoringthebrightnessofover100,000stars of radii and orbital periods.
1 (Tenenbaum et al. 2012). The majority of Kepler’s tar- In addition to nearly doubling the number of planet
: getstarsaresolar-likeFGK dwarfsandaccordinglymost candidates, Batalha et al. (2012) also improvedthe stel-
v
lar parameters for many target stars by comparing
i of the work on the planet occurrence rate from Ke-
X plerhas been focused on planets orbiting that sample of the estimated temperatures, radii, and surface gravi-
ties in the KeplerInput Catalog (KIC; Batalha et al.
r stars(e.g.,Borucki et al.2011;Catanzarite & Shao2011;
a Youdin 2011; Howard et al. 2012; Traub 2012). Those 2010; Brown et al. 2011) to the values expected from
Yonsei-Yaleevolutionarymodels(Demarque et al.2004).
studiesrevealedthattheplanetoccurrencerateincreases
Rather than refer back to the original photometry,
toward smaller planet radii and longer orbital periods.
Batalha et al. (2012) adopted the stellar parameters of
Howard et al. (2012) also found evidence for an increas-
the closest Yonsei-Yale model to the originalKIC values
ing planet occurrence rate with decreasing stellar effec-
in the three-dimensional space of temperature, radius,
tivetemperature,butthetrendwasnotsignificantbelow
and surface gravity. This approach did not correctly
5100K.
characterizethe coolesttargetstarsbecausethe starting
Howard et al. (2012) conducted their analysis using
points were too far removed from the actual tempera-
the 1235 planet candidates presented in Borucki et al.
tures, radii, and surface gravities of the stars. In addi-
(2011). The subsequent list of candidates published
tion, the Yonsei-Yale models overestimate the observed
in February 2012 (Batalha et al. 2012) includes an ad-
radiiandluminosityofcoolstarsatagiveneffectivetem-
ditional 1091 planet candidates and provides a better
perature (Boyajian et al. 2012).
sample for estimating the occurrence rate. The new
1.1. The Small Star Advantage
1Harvard-Smithsonian Center for Astrophysics, Cambridge,
MA02138 Although early work (Dole 1964; Kasting et al. 1993)
[email protected] suggested that a hypothetical planet in the habitable
2 Dressing & Charbonneau
zone (the range of distances at which liquid water could intrinsicallyfainterthansolar-typestars,75%percentof
exist on the surface of the planet) of an M dwarf would thestarswithin10pcareMdwarfs3 (Henry et al.2006).
be inhospitable because the planet would be tidally- These stars would be among the best targets for fu-
locked and the atmosphere would freeze out on the turespectroscopicinvestigationsofpotentially-habitable
dark side of the planet, more recent studies have been rockyplanetsduetothesmallradiiandapparentbright-
more optimistic. For instance, Haberle et al. (1996) and ness of the stars.
Joshi et al. (1997) demonstrated that sufficient quanti- Third, confirming the planetary nature and measur-
tiesofcarbondioxidecouldpreventtheatmospherefrom ing the mass of an Earth-size planet orbiting within the
freezing. In addition, Pierrehumbert (2011) reported habitable zone of an M dwarf is easier than confirming
thatatidally-lockedplanetcouldbe inapartiallyhabit- and measuring the mass of an Earth-size planet orbit-
able “EyeballEarth”state in which the planet is mostly ing within the habitable zone of a G dwarf. The radial
frozen but has a liquid water ocean at the substellar velocity signal induced by a 1M⊕ planet in the middle
point. Moreover, planets orbiting M dwarfs might be- of the habitable zone (a=0.28AU) of a 3800K,0.55M⊙
come trapped in spin-orbit resonances like Mercury in- Mdwarfis23cm/s. Incomparison,theRVsignalcaused
stead of becoming spin-synchronized. by a 1M⊕ planet in the habitable zone of a G dwarf is
A second concern for the habitability of planets or- 9cm/s. TheprospectsforRVconfirmationareevenbet-
biting M dwarfs is the possibility of strong flares and ter for planets around mid-to-late M dwarfs: an Earth-
highUV emissioninquiescence(France et al.2012). Al- size planet in the habitable zone of a 3200K M dwarf
though a planet without a magnetic field could require would produce an RV signal of 1 m/s, which is achiev-
yearsto rebuild its ozone layer after experiencing strong able with the current precision of modern spectrographs
flare, the majority of the UV flux would never reach the (Dumusque et al. 2012). Prior to investing a significant
surface of the planet. Accordingly, flares do not present amount of resources in investigations of the atmosphere
a significant obstacle to the habitability of planets or- ofapotentiallyhabitableplanet,itwouldbewisetofirst
biting M dwarfs (Segura et al. 2010). Furthermore, the guarantee that the candidate object is indeed a high-
specific role of UV radiation in the evolution of life on density planet and not a low-density mini-Neptune.
Earthis uncertain. A baseline level of UV flux might be Finally, upcoming facilities such as JWST and GMT
necessary to spur biogenesis (Buccino et al. 2006), yet willhavethecapabilitytotakespectraofEarth-sizeplan-
UV radiation is also capable of destroying biomolecules. etsinthehabitablezonesofMdwarfs,butnotEarth-size
Having establishedthat planets in the habitable zones planets in the habitable zones of more massive stars. In
of M dwarfs could be habitable despite the initial con- orderto finda sample ofhabitable zone Earth-sizeplan-
cern of the potential hazards of tidal-locking and stel- ets for which astronomers could measure atmospheric
lar flares, the motivation for studying the coolest tar- properties with the next generation of telescopes, as-
get stars is three-fold. First, several more years of Ke- tronomers need to look for planets around small dwarfs.
plerobservations will be required to detect Earth-size
planets in the habitable zones of G dwarfs due to the 1.2. Previous Analyses of the Cool Target Stars
higher-than-expected photometric noise due to stellar In light of the advantages of searching for habit-
variability (Gilliland et al. 2011), but Kepleris already able planets around small stars, several authors have
sensitivetothepresenceofEarth-sizeplanetsinthehab- worked on refining the parameters of the smallest Ke-
itable zones of M dwarfs. Although a transiting planet plertarget stars. Muirhead et al. (2012a) collected
in the habitable zone of a G star transits only once per medium-resolution, K-band spectra of the cool planet
year,atransitingplanetinthehabitablezoneofa3800K candidate host stars listed in Borucki et al. (2011) and
M star transits five times per year. Additionally, the ge- presentedrevisedstellar parametersfor those hoststars.
ometric probability that a planet in the habitable zone Their sample included 69 host stars with KIC tempera-
transits the star is 1.8 times greater. Furthermore, the tures below 4400K as well as an additional 13 host stars
transit signal of an Earth-size planet orbiting a 3800K with higher KIC temperatures but with red colors that
M star is 3.3 times deeper than the transit of an Earth- hint that their KIC temperatures were overestimated.
sizeplanetacrossaGstarbecausethestaris45%smaller Muirhead et al. (2012a) determined effective tempera-
than the Sun. The combination of a shorter orbital pe- ture and metallicity directly from their spectra using
riod,anincreasedtransitprobability,andadeeper tran- the H O-K2 index (Rojas-Ayala et al. 2012) and then
2
sitdepthgreatlyreducesthedifficultyofdetectingahab- constrainstellar radii and masses using Dartmouth stel-
itableplanetandhasmotivatednumerousplanetsurveys larevolutionarymodels(Dotter et al.2008;Feiden et al.
totargetMdwarfs(Delfosse et al.1999;Endl et al.2003; 2011). We adopt the same set of stellar models in this
Nutzman & Charbonneau2008;Zechmeister et al.2009; paper. Muirhead et al. (2012a) found that one of the 82
Apps et al. 2010; Barnes et al. 2012; Berta et al. 2012; targets (Kepler Object of Interest (KOI) 977) is a giant
Bowler et al. 2012; Giacobbe et al. 2012; Law et al. star and that three small KOIs (463.01, 812.03, 854.01)
2012). lie within the habitable zone.
Second, as predicted by Salpeter (1955) and Chabrier Johnson et al. (2012) announced the discovery of
(2003), studies of the solar neighborhood have revealed KOI 254.01, the first short-period gas giant orbiting an
that M dwarfs are twelve times more abundant than Mdwarf. Theplanethasaradiusof0.96R andorbits
Jup
G dwarfs. The abundance of M dwarfs, combined with its host star KIC 5794240 once every 2.455239 days. In
growingevidenceforanincreaseintheplanetoccurrence additiontodiscussing KOI254.01,Johnson et al.(2012)
rate at decreasing stellar temperatures (Howard et al. alsocalibratedarelationfordeterminingthemassesand
2012), implies thatthe majorityofsmallplanets may be
locatedaroundthecooleststars. AlthoughMdwarfsare 3 http://www.recons.org/census.posted.htm
Small Planets Around Small Stars 3
metallicities of M dwarfs from broad-band photometry. providethe mass, luminosity,temperature, surfacegrav-
TheyfoundthatJ K colorisareasonable( 0.15dex) ity, metallicity, helium fraction, and α-element enrich-
− ±
indicator of metallicity for stars with metallicities be- ment at each evolutionary time step. We consider the
tween 0.5and0.5dexandJ K colorswithin0.1mag- full range of Dartmouth model metallicities ( 2.5
− − − ≤
nitudes of the main sequence J K at the V K [Fe/H] 0.5), but we restrict our set of models to stars
− − ≤
color of the star in question. The relationship between with solar α-element enhancement, masses below 1M⊙,
infrared colors and metallicities was first proposed by and temperatures below 7000K. We exclude models of
Mould & Hyland (1976) and subsequently confirmed by more massive stars because solar-like stars are well-fit
Leggett (1992) and L´epine et al. (2007). by the ATLAS9 models used in the construction of the
Mann et al.(2012)took the firststeps towardaglobal KIC and it is unlikely that a star as massive as the Sun
reanalysisofthe coolKeplertargetstars. They acquired would have been assigned a temperature lower than our
medium-resolution,visiblespectraof382targetstarsand selection cut T 5050K.
KIC
≤
classified all of the cool stars in the target list as dwarfs The Dartmouth team supplies synthetic photometry
or giants using “training sets” constructed from their for a range of photometric systems by integrating the
spectraandliteraturespectra. Mann et al. (2012) found spectrum of each star over the relevant bandpass. We
that the majority of bright, cool target stars are giants downloaded the synthetic photometry for the 2MASS
in disguise and that the temperatures of the cool dwarf and Sloan Digital Sky Survey Systems (SDSS) and used
stars are systematically overestimated by 110 K in the relations 1–4 from Pinsonneault et al. (2012) to convert
KIC. Mann et al. (2012) reported that correctly classi- theobservedKICmagnitudesforeachKeplertargetstar
fying and removing giant stars removes the correlation to the equivalent magnitudes in the SDSS system. For
between cool star metallicity and planet occurrence ob- cool stars, the correction due to the filter differences is
served by Schlaufman & Laughlin (2011). After remov- typically much smaller than the assumed errors in the
ing giant stars from the target list, Mann et al. (2012) photometry (0.01 mag in gri and 0.03 mag in zJHK,
calculated a planet occurrence rate of 0.37 0.08 plan- similar to the assumptions in Pinsonneault et al. 2012).
±
ets per cool star with radii between 2 and 32 R⊕ and All stars have full 2MASS photometry, but 21% of the
orbital periods less than 50 days. Their result is higher targetstarsaremissingphotometryinoneormorevisible
than the occurrence rate we report in Section 5.3, most KICbands. Forthosestars,wecorrectforthe linearoff-
likely because of our revisions to the stellar radii. setinallbandsandapplythemediancorrectionfoundfor
Inthispaper,wecharacterizethecoolestKeplertarget the whole sample of stars for the color-dependent term.
stars by revisiting the approach used to create the Ke- Inourfinalcooldwarfsample, 70starslackg-bandpho-
plerInputCatalog(Brown et al.2011)andtailoringthat tometry and 29 stars lack z-band. We exclude all stars
method for applicationto cool stars. Specifically, we ex- with more than one missing band.
tract grizJHK photometry from the KIC for the 51813 Our final sample of model stars is drawnfrom a set of
planet search target stars with KIC temperature esti- isochrones with ages 1–13 Gyr and spans a temperature
mates 5050K and for the 13402 planet search target range2708–6998K.The starshavemasses0.01–1.00M⊙,
≤
stars without KIC temperature estimates and compare radii 0.102–223R⊙, and metallicities 2.5 < [Fe/H] <
−
the observedcolors to the colorsofmodel starsfrom the 0.5. All model stars have solar α/Fe ratios. There is
Dartmouth Stellar EvolutionaryDatabase (Dotter et al. a deficit of Dartmouth model stars with radii 0.32
−
2008; Feiden et al. 2011). We discuss the features of the 0.42R⊙; we cope with this gap by fitting polynomials
Dartmouthstellar models in Section 2.1and explainour to the relationships between temperature, radius, mass,
procedureforassigningrevisedstellarparametersinSec- luminosity, and colors at fixed age and metallicity. We
tion 2.2. We present revised stellar characterizations in theninterpolatethoserelationshipsoveragridwithuni-
Section3andimprovedplanetaryparametersfor the as- form (0.01R⊙) spacing between 0.17R⊙ and 0.8R⊙ to
sociated planet candidates in Section 4. We address the derive the parametersfor stars that would have fallen in
implicationsoftheseresultsontheplanetoccurrencerate the gapin the originalmodel grid. We compute the sur-
in Section 5 and conclude in Section 6. face gravities for the resulting interpolated models from
their masses and radii. When fitting stars, we use the
2. METHODS originalgridofmodelstarssupplementedbytheinterpo-
lated models. Our fitted parameters may be unreliable
2.1. Stellar Models
for stars younger than 0.5 Gyr because those stars are
The Dartmouth models incorporate both an internal
still undergoing Kelvin-Helmholtz contraction.
stellarstructurecodeandamodelatmospherecode. Un-
like the ATLAS9 models (Castelli & Kurucz 2004) used 2.1.1. Distinguishing Dwarfs and Giants
in development of the KeplerInput Catalog, the Dart-
We specifically include giant stars in our model set so
mouth models perform well for low-mass stars because
that we have the capability to identify red giants that
the package uses PHOENIXatmospheres to model stars
have been misclassified as red dwarfs (and vice versa).
coolerthan10,000K.ThePHOENIXmodelsincludelow-
Muirhead et al. (2012a) discoveredone such masquerad-
temperature chemistry and are therefore well-suited for
ing giant (KOI 977) in their spectroscopic analysis of
use with low-mass dwarfs (Hauschildt et al. 1999a,b).
the cool planet candidate host stars and Mann et al.
The Dartmouth models include evolutionary tracks
(2012) have argued that giant stars comprise 96% 1%
and isochrones for a range of stellar parameters. The ±
ofthe populationofbright(Kepmag<14)and7% 3%
tracks and isochrones are available electronically4 and ±
of the population of dim (Kepmag > 14) cool target
stars. We are confident in the ability of our photomet-
4 http://stellar.dartmouth.edu/models/grid.html ric analysis to correctly identify the luminosity class of
4 Dressing & Charbonneau
cool stars because the infrared colors of dwarfs and gi- We set the metallicity prior by assuming that the
ant stars are well-separated at low temperatures. For metallicity distribution of the M dwarfs in the Ke-
instance, our photometric analysis classifies KOI 977 plertarget list is similar to the metallicity distribution
(KIC 11192141) as a cool giant with an effective tem- ofthe343nearbyMdwarfsstudiedbyCasagrande et al.
perature of 3894+−5504K, radius R⋆ =36+−32R⊙, luminosity (2008). FollowingBrown et al.(2011),weproduceahis-
L⋆ = 260+−2285L⊙, and surface gravity logg = 1.3+−00..0065. tlooggarraimthmoficthmeeltoagllairciitthymbionfatnhde tnhuemnbfietraofpsotlayrnsoimniaelactho
The reported mass (0.99+−00..0015M⊙) is near the edge of the distribution. We extrapolate the polynomial down
our model grid, so refitting the star with a more mas-
to [Fe/H] = 2.5 and up to [Fe/H] = 0.5 to cover the
sive model grid may yield different results for the stellar −
fullrangeofallowedstellarmodels. Ourfinalmetallicity
parameters.
prior and the histogram of M dwarf metallicities from
Casagrande et al. (2008) are shown in Figure 1. The
2.2. Revising Stellar Parameters distribution peaks at [Fe/H]= 0.1 and has a long tail
−
We assignrevisedstellarparametersbycomparingthe extendingdowntowardlowermetallicities. Weadoptthe
observedopticalandnearinfraredcolorsofall51813cool same height prior as Brown et al. (2011): the number of
(T 5050K)andall13402unclassifiedKeplerplanet stars falls off exponentially with increasing height above
KIC
search≤target stars to the colors of model stars. We ac- theplaneofthegalaxyandthescaleheightofthediskis
count for interstellar reddening by determining the dis- 300 pc (Cox 2000). Our photometric distance estimates
tance at which the apparent J-band magnitude of the for77%ofourcooldwarfsarewithin300pc,soadopting
model star would match the observed apparent J-band thispriorhaslittleeffectonthechosenstellarparameters
magnitude of each target star. We then apply a band- and the resulting planet occurrence rate.
dependent correction assuming 1 magnitude of extinc-
tion per 1000 pc in V-band in the plane of the galaxy
(Koppen & Vergely 1998; Brown et al. 2011). We find
2.0
the best-fit model for each target star by computing the
differenceinthecolorsofagiventargetstarandallofthe
model stars. We then scale the differences by the photo-
1.5
metric errors in each band and add them in quadrature )
s
to determine the χ2 for a match to each model star. t
n
AsexplainedinSection2.2.1,weincorporatepriorson u
o 1.0
the stellar metallicity and the height of stars above the C
planeofthegalaxy. Werescaletheerrorssothatthemin- g(
imum χ2 is equal to the number of colors (generally 6) o
l 0.5
minusthenumberoffittedparameters(3forradius,tem-
perature,andmetallicity). Wethenadoptthestellarpa-
rameterscorrespondingto the best-fitmodelandsetthe 0.0
errorbarstoencompasstheparametersofallmodelstars
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4
falling within the 68.3% confidence interval. For exam-
[Fe/H]
ple, for KOI 2626 (KID 11768142), we find 68.3% confi-
dence intervalsR⋆ =0.35R⊙+−00..1015, T⋆ =3482+−15270K,and
[Fe/H]=−0.1+−00..11. We findabest-fitmass0.36+−00..1026M⊙ Fig.1.—Logarithmicnumberofstarsversuslogarithmicmetal-
andluminosity0.016+−00..00205L⊙,resultinginadistancees- lliicciittyiesbiinn.tThheeCbalsaacgkrahnisdteogertaaml.d(i2s0p0la8y)sstahmepdleistarnibdutthioengorfeemneltianle-
timate of 159+−6237 pc. The corresponding surface gravity isouradopted metallicityprior.
is therefore logg =4.91+0.08.
−0.12
2.2.1. Priors on Stellar Parameters 2.3. Assessing Covariance Between Fitted Parameters
We find that fitting the target stars without assuming Our procedure for estimating stellar parameters ex-
prior knowledge of the metallicity distribution leads to pressly considers the covariance between fitted parame-
an overabundance of low-metallicity stars, so we adopt tersbysimultaneouslydeterminingthelikelihoodofeach
priors on the underlying distributions of metallicity and of the models and determining the range of tempera-
height above the plane. We then determine the best-fit tures, metallicities, and radii that would encompass the
model by minimizing the equation full 68.3% confidence interval. The provided error bars
thereforeaccountforthefactthathigh-metallicitywarm
χ2i =χ2i,color−2lnPmetallicity,i−2lnPheight,i (1) Mdwarfsandlow-metallicitycoolMdwarfshavesimilar
colors.
whereχ2 isthetotalcolordifferencebetweenatarget We confirm that the quoted errors on the stellar pa-
i,color
starandmodelstari, P is the probabilitythat rameters are large enough to account for the errors in
metallicity,i
a star has the metallicity of model star i, and P is thephotometrybyconductingaperturbationanalysisin
height,i
the probability that a star would be found at the height whichwecreate100copiesofeachoftheKeplerMdwarfs
at which model star i would have the same apparent J- and add Gaussian distributed noise to the photometry
bandmagnitudeasthetargetstar. Weweightthe priors based on the reported uncertainty in each band. We
so that each prior has the same weight as a single color. then run our stellar parameter determination pipeline
Small Planets Around Small Stars 5
and compare the distribution of best-fit parameters for
each star to our original estimates. We find that there
0.7 Fractional Parallax Error
is a correlation between higher temperatures and higher
metallicities, but that our reported error bars are larger es) 0.00 0.01 0.02 0.03 0.03 0.04 0.05
s 0.6
than the standard deviation of the best-fit parameters. s
a
M
2.4. Validating Methodology ar 0.5
ol
S
We confirm that we are able to recover accurate pa- s ( 0.4
rametersforlowmassstarsfromphotometrybyrunning s
a
our stellar parameter determination pipeline on a sam- d M 0.3
ple of stars with known distances. We obtained a list of e
at
438Mdwarfswithmeasuredparallaxes,JHK photome- m 0.2
tryfrom2MASS,andg′r′i′photometryfromtheAAVSO sti
E 1:1
Photometric All-Sky Survey5 (APASS) from Jonathan 0.1 Fit to the Data
Irwin (personal communication, January 2, 2013) and
performed a series of quality cuts on the sample. We 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Mass from Parallax & Delfosse Relation (Solar Masses)
removed stars with parallax errors above 5% and and
stars with fewer than two measurements in the APASS
database. We then visually inspected the 2MASS pho-
Fig.2.— Mass estimated by our photometric stellar parameter
tometry of the remaining 230 stars to ensure that none determinationpipelineversusmasspredictedbytheDelfosserela-
of them belonged to multiple systems that could have tion. Thedashedredlineindicatesa1:1relationandthesolidblue
been unresolved in APASS and resolved in 2MASS. We lineisfitto the data. Thepoints arecolor-coded bythe reported
fractionalerrorintheparallaxmeasurement.
removed 203 stars with other stars or quasars within 1’,
resulting in a final sample of 26 stars.
We estimate the masses ofthe 26 starsby running our stars classified as “ambiguous” were not assigned tem-
stellar parameter determination pipeline to match the peratures in the KIC. We find that 96 98% of cool
−
observed colors to the colors of Dartmouth model stars. bright(Teff <4000K,Kepmag<14)starsand5 6%of
The APASS g′r′i′ photometry was acquired using fil- coolfaint (Teff <4000K,Kepmag >14)starsare−giants,
tersmatching the originalSDSSg′r′i′ bands; we convert whichisconsistentwithMann et al.(2012). (Theprecise
the APASS photometry to the unprimed SDSS 2.5mgri fractions of giant stars depend on whether the ambigu-
bands using the transformation equations provided on ousstarsarecountedasgiantstars.) Oneoftheexcluded
the SDSS Photometry White Paper.6 We then compare ambiguous stars is KID 8561063 (KOI 961), which was
the masses assigned by our pipeline to the masses pre- confirmedbyMuirhead et al.(2012b)asa0.17 0.04R⊙,
±
dicted fromthe empiricalrelationbetweenmass andab- 3200 65Kstarhostingsub-Earth-sizethree planetcan-
±
soluteKsmagnitude(Delfosse et al.2000). Asshownin didates. The KIC does not include z-band photometry
Figure2,ourmassestimatesareconsistentwiththemass forKOI961andwewereunabletoruleoutmatcheswith
predictedbytheDelfosserelation. Themassespredicted giant stars using only griJHK photometry.
bythepipelinearetypically5%lowerthanthemasspre- The distributions of temperature, radius, metallicity,
dicted by the Delfosse relation, but none of these stars andsurfacegravityforthestarsinoursampleareshown
have reported z-band photometry whereas 96% of our in Figure 3. For comparison, we display both fits made
finalsampleofKeplerMdwarfshavefullgrizJHK pho- without using priors (left panels) and fits including pri-
tometry. Accordingly,wedonotfitforacorrectionterm ors on the stellar metallicity distribution and the height
because the uncertainty introduced by adding a scaling of stars above the plane of the galaxy (right panels). In
term based on fits made to stars with only five colors both cases the radii of the majority of stars are signif-
wouldbecomparabletotheoffsetbetweenourpredicted icantly smaller than the values given in the KIC and
massesandthemassespredictedbytheDelfosserelation. the surface gravities are much higher. As discussed in
Section 2.2.1, the primary difference between the two
3. REVISEDSTELLARPROPERTIES modelfitsisthatsettingapriorontheunderlyingmetal-
Our final sample of cool Keplertarget stars includes licity distribution reduces the number of stars with re-
3897 stars with temperatures below 4000K and surface vised metallicities below [Fe/H]= 0.6. Since such stars
−
gravities above logg = 3.6. The sample consists pri- should be relatively uncommon, we choose to adopt the
marily of late-K and early-M dwarfs, but 201 stars have stellarparametersgivenbyfittingthestarsassumingpri-
revisedtemperaturesbetween3122 3300K.Therevised ors on metallicity and height above the plane.
parameters for all of the cool dw−arfs are provided in Incorporatingpriors,themediantemperatureofastar
the Appendix in Table 4. We exclude 4420 stars from inthesampleis3723Kandthemedianradiusis0.45R⊙.
the final sample because their photometry is consistent Most of the stars in the sample are slightly less metal-
with classification as evolved stars (logg <3.6) and 608 rich than the Sun (median [Fe/H]= 0.1), but 21% have
−
starsbecause their photometry is insufficient to discrim- metallicities 0.0 [Fe/H]< 0.5. Although nearly all of
≤
inate between dwarf and giant models. We refer to the the stars in the sample (96%) had KIC surface gravities
stars that could be fit by either dwarf or giant mod- below logg = 4.7, our reanalysis indicates that 95% ac-
els as “ambiguous” stars. The majority (80%) of the tuallyhavesurfacegravitiesabovelogg =4.7. Asshown
by the purple histograms in each of the panels, the dis-
5 http://www.aavso.org/apass tribution of stellar parameters for the planet candidate
6http://www.sdss.org/dr5/algorithms/jeg_photometric_eq_dr1.hthmlost stars matches the overall distribution of stellar pa-
6 Dressing & Charbonneau
rameters for the cool star sample. (2012a) at higher temperatures, we choose to plot only
The two-dimensionaldistribution of radiiand temper- the 32 stars with revised temperatures below 4000K
aturesforourchosenmodelfitisshowninFigure4. The and spectroscopic metallicities from Muirhead et al.
spreadintheradiiofthemodelpointsatagiventemper- (2012a). The top panel of Figure 7 compares our re-
ature is due to the range of metallicities allowed in the visedmetallicities to the spectroscopicmetallicities from
modelsuite. Atagiventemperature,themajorityofthe Muirhead et al. (2012a). We observe a systematic off-
originalradiifrom the KIC lie abovethe modelgridin a set in metallicity with our values typically 0.17 dex
region of radius–temperature space unoccupied by low- lower than the metallicities reported in Muirhead et al.
massstars. Thediscrepancybetweenthemodelradiiand (2012a).
theKICradiiispartiallyduetotheerrorsintheassumed The metallicity difference is dependent on the spec-
surfacegravities. AsshowninFigure3,thesurfacegrav- troscopic metallicity of the star, as depicted in the
itiesassumedintheKICpeakatlog(g)=4.5withalong lower panel of Figure 7, which shows the metallicity
tailextendingtolowersurfacegravitieswhereasthemin- difference as a function of the metallicity reported in
imum expected surface gravity for cool stars is closer to Muirhead et al. (2012a). For stars with Muirhead et al.
log(g)=4.7. (2012a) metallicities between -0.2 and -0.1 dex, our re-
Foratypicalcoolstar,wefindthattherevisedradiusis vised metallicities are 0.05 dex lower, but for stars with
only69%oftheoriginalradiuslistedintheKICandthat Muirhead et al. (2012a) metallicities above 0.1 dex, our
the revised temperature is 130Kcooler than the original revised metallicities are 0.3 dex lower.
temperature estimate. The majority (96%) of the stars
haverevisedradiismallerthantheradiilistedintheKIC 4. REVISEDPLANETCANDIDATEPROPERTIES
and 98% of the stars are cooler than their KIC temper- Our sample of cool stars includes 64 host stars with
atures. The revised radius and temperature distribution 95 planet candidates. As part of our analysis, we down-
ofplanetcandidatehoststarsissimilartotheunderlying loaded the Keplerphotometry for the 95 planet candi-
distribution of cool target stars. The median changes in dates and inspected the agreement between the planet
radius and temperature for a cool planet candidate host candidate parameters provided by Batalha et al. (2012)
star are 0.19R⊙ ( 29%) and 102K, respectively. and the Keplerdata. We used long cadence data from
− − −
We compare the revised and initial parameters for the Quarters1 6forallKOIsexceptKOI531.01,forwhich
−
host stars in more detail in Figure 5. For all host stars we utilized short cadence data from Quarters 9 and 10
except for KOI 1078 (KID 10166274), the revised radii due to the range of apparent transit depths observed in
are smaller than the radii listed in the KIC and the re- the long cadence data. The long cadence data provide
vised temperatures for all of the stars are cooler than measurements of the brightness of the target stars every
the KIC temperatures. Unlike the original values given 29.4 minutes and the short cadence data provide mea-
in the KIC, the revised temperatures and radii of the surements every 58.9 seconds.
cool stars align to trace out a main sequence in which We detrended the data by dividing each data point
smaller stars have cooler temperatures by construction. by the median value of the data points within the sur-
rounding 1000 minute interval and masked transits of
3.1. Comparison to Previous Work
additional planets in multi-planet systems. We found
We validate our revised parameters by comparing that the distribution of impact parameters reported by
our photometric effective temperatures for a subset Batalha et al. (2012) for these planet candidates was
of the cool target stars to the spectroscopic effec- biased towards high values (median b = 0.75) and
tive temperatures from Muirhead et al. (2012a) and that the published parameters for several candidates
Mann et al. (2012). We exclude the stars KIC 5855851 did not match the observed depth or shape. Accord-
and KIC 8149616 from the comparison due to concerns ingly, we used the IDL AMOEBA minimization algo-
that their spectra may have been contaminated by light rithmbasedonPress et al.(2002)todeterminethebest-
from another star (Andrew Mann, personal communi- fit period and ephemeris for each planet candidate. We
cation, January 15, 2013). As shown in Figure 6, our then ran a Markov Chain Monte Carlo analysis using
revised temperatures are consistent with the literature Mandel & Agol(2002)transitmodelstorevisetheplanet
resultsfor starswithrevisedtemperatures below4000K, radius/star radius ratio, stellar radius/semimajor axis
which is the temperature limit for our final sample. ratio, and inclination for each of the candidates. For
Athighertemperatures,wefindthatourtemperatures each star, we determined the limb darkening coefficients
are systematically hotter than the literature values re- by interpolating the quadratic coefficients provided by
ported by Muirhead et al. (2012a). The temperatures Claret & Bloemen(2011)fortheKeplerbandpassatthe
given in Muirhead et al. (2012a) are determined from effective temperature and surface gravity found in Sec-
theH O-K2index(Rojas-Ayala et al.2012),whichmea- tion 2.2. We adopt the median values of the resulting
2
surestheshapeofthespectruminK-band. Althoughthe parameter distributions as our best-fit values and pro-
H O-K2 index is an excellent temperature indicator for vide the resulting planet candidate parameters in the
2
coolstars,theindexsaturatesaround4000K,accounting AppendixinTable5. Figures8-11displaydetrendedand
for the disagreementbetweenour temperatureestimates fittedlightcurvesforthethreehabitablezoneplanetcan-
andthe Muirhead et al. (2012a) estimates for the hotter didates in our sample and for one additional candidate
stars in our sample. at short cadence.
We also compare our photometric metallicity esti- Ten of the planet candidates in our sample have re-
mates to the spectroscopic metallicity estimates from ported transit timing variations (TTVs), but our fit-
Muirhead et al. (2012a). Given the disagreement be- ting procedure assumed a linear ephemeris. Due to the
tween our temperature estimates and Muirhead et al. smearingof ingressand egress caused by fitting a planet
Small Planets Around Small Stars 7
Without Priors With Priors
1200 Revised Host Stars (x40) 1200
ars 1000 RKIeCv iVseadlu Vesalues ars 1000
St 800 St 800
of of
er 600 er 600
mb 400 mb 400
u u
N 200 N 200
0 0
2950 3338 3725 4112 4500 2950 3338 3725 4112 4500
Effective Temperature (K) Effective Temperature (K)
1200 1200
s 1000 s 1000
ar ar
St 800 St 800
er of 600 er of 600
mb 400 mb 400
u u
N 200 N 200
0 0
0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.4 0.6 0.8 1.0 1.2
Radius (Solar Radii) Radius (Solar Radii)
1500 1500
Revised Host Stars (x40)
of Stars 1000 RKIeCv iVseadlu Vesalues of Stars 1000
er er
b b
m 500 m 500
u u
N N
0 0
-2.0 -1.5 -1.0 -0.5 0.0 0.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
Metallicity Metallicity
1000 1000
s s
ar 800 ar 800
St St
of 600 of 600
er er
b 400 b 400
m m
Nu 200 Nu 200
0 0
4.0 4.3 4.6 4.9 5.2 5.5 4.0 4.3 4.6 4.9 5.2 5.5
log(g) log(g)
Fig. 3.— Histograms of the resulting temperature (top), radius (second from top), metallicity (third from top), and surface gravity
(bottom)distributionsforthetargetstarswithrevisedtemperaturesbelow4000K.Thepanelsontheleftshowthedistributionsresulting
from fitting the stars without setting priors while the stellar parameters in the right panels were fit assuming priors on metallicity and
height above the plane. In all panels, a histogram of the original KIC values is shown in blue and a histogram of the revised values is
plottedinred. Thedistributionofcoolhoststars(multipliedbyforty)isshowninpurpleinallplots.
candidate exhibiting TTVs with a linear ephemeris, our pleted extensive modeling of their light curves. We
simple fitting routine experienced difficulty determin- cannot adopt values from Steffen et al. (2012b) for
ing the transit parameters for those candidates. Rather KOI 250.04 because that planet candidate was an-
than use our poorly constrained fits for the candi- nounced after publication of Steffen et al. (2012b).
dates with TTVs, we choose instead to adopt the lit- Fabrycky et al. (2012a) also present transit parameters
erature values for KOIs 248.01, 248.02, 886.01, and for a fifth planet candidate in the KOI 952 system, but
886.02(Kepler-49b,49c,54b,and54c)fromSteffen et al. we choose not to add KOI 952.05to our sample because
(2012a), KOIs 250.01 and 250.02 (Kepler-26b and 26c) that planet candidate was not included in the February
from Steffen et al. (2012b), KOIs 952.01 and 952.02 2012 planet candidate list (Batalha et al. 2012) and
(Kepler-32b and 32c) from Fabrycky et al. (2012a), and including KOI 952.05 would necessitate including any
KOIs 898.01 and 898.03 from Xie (2012). other planet candidates that were not included in the
We alsoadoptthe transitparametersforKOIs248.03, February 2012 KOI list.
248.04, and 886.03 from Steffen et al. (2012a), For 31 of the remaining 78 planet candidates without
KOI250.03fromSteffen et al.(2012b),KOIs952.03and revisedfitsfromtheliterature,theplanetradius/starra-
952.04 from Fabrycky et al. (2012a), and KOI 254.01 dius ratios from Batalha et al. (2012) lie within the 1σ
from Johnson et al. (2012) because the authors com- error bars of our revised values. The median changes
8 Dressing & Charbonneau
dii) 1.5 KRRMIeeoCvvd iiVesseela Gddlu rVHeidsaolsute Sstars Median Errors ure (K) 44020000 MMuainrhne+a 2d0+1 22012
a at
R r
ar 1.0 pe 3800
ol m
S e
us ( e T 3600
di 0.5 ur
Ra at 3400
r
e
Lit 3200
0.0
3200 3440 3680 3920 4160 4400
3000 3500 4000 4500
Temperature (K) Revised Temperature (K)
Fig. 4.— Revised (red) and original (blue) temperatures and Fig.6.— Spectroscopic effective temperatures from
radiiof the cool target stars. Therevisedvalues weredetermined Muirheadetal. (2012a) (red circles) and Mannetal. (2012)
by comparing the observed colors of stars to the expected colors (blue squares) versus our revised photometric effective temper-
of Dartmouth model stars (gray) and incorporating priors on the ature estimates. The dashed black line indicates a 1:1 relation.
metallicityandheightabovethegalacticplane. Therevisedstellar The disagreement for the hotter stars is attributed to the
parameters for cool planet candidate host stars arehighlighted in saturation of the H2O-K2 index used by Muirheadetal. (2012a)
purple. The position of the KIC radii well above the model grid attemperatures above4000K.
indicatesthatmanyofthecombinationsofradiusandtemperature
foundintheKICarenonphysical.
Temperature Difference (K)
-156 -73 11 95 178 262
1.2
H] 0.0
e/
1.0 d [F -0.2
R)Sun evise --00..64
s ( 0.8 R
adiu nce 0.0
Stellar R 0.6 city Differe --00..42
0.4 KIC etalli -0.6
This Work M
-0.5 -0.3 -0.1 0.1 0.3 0.5
0.2
Spectroscopically Determined [M/H] from Muirhead et al. (2012)
3200 3400 3600 3800 4000 4200
Stellar Effective Temperature (K)
Fig.7.— Comparison of our photometric metallicity estimates
tothespectroscopicmetallicitiesfromMuirheadetal.(2012a)for
Fig.5.— Revised (red circles) and original (blue squares) radii stars with revised T < 4000K. The color-coding indicates our
andtemperatures fortheplanet candidate hoststarswithrevised revised stellar temperatures and the dashed red lines mark a
temperatures below 4000K. Thegraylinesconnect the initialand 1:1 relation between photometric and spectroscopic metallicities.
finalvaluesforeachhoststar. Top: Revised photometric metallicity estimates versus spectro-
scopic metallicity. Bottom: Metallicitydifference (photometric -
to the transit parameters for the refit planet candidates spectroscopic)versusspectroscopicmetallicity.
are that the planet radius/star radius ratio decreases by
3%, the star radius/semimajor axis ratio increases by
18%, and the inclination increases by 0.7◦. Combining notably KOIs 531.01 and 1843.02.
our improved stellar radii with the revised planet ra- We computed error bars on the planet candidate radii
dius/star radius ratios for all of the planet candidates, by computing the fractional error in the planet ra-
we find that the radius of a typical planet candidate is dius/star radius ratio and the stellar radius and adding
29% smaller than the value found by computing the ra- thosedifferencesinquadraturetodetermineseparateup-
diusfromthetransitdepthgiveninBatalha et al.(2012) per and lower 1σ error bounds for each candidate. For
and the stellar radii listed in the KIC as shown in Fig- a typical candidate in the sample, the 68% confidence
ure 12. The improvements in the stellar radii account region extends from 86 112% of the best-fit planet ra-
−
for most of the changes in the planet candidate radii, dius. Thebest-fitradiiand1σerrorbarsforthesmallest
but the contributions from the revised transit parame- planet candidates are plotted in Figure 13 as a function
ters are non-negligible for a few planet candidates, most of orbital period.
Small Planets Around Small Stars 9
KOI 854.01 KOI 2626.01
Detrended Flux000011......999900999900246802 Detrended Flux0011....990099006802
200 300 400 500 600 200 300 400 500 600
rrstar: 0.040, arstar: 73.140, Inc: 89.483 Time (Days) rrstar: 0.039, arstar: 90.045, Inc: 89.759 rrstar: 0.025, arstar: 91.029, Inc: 89.815 Time (Days) rrstar: 0.036, arstar: 36.283, Inc: 88.538
1.002 1.002
1.001 1.001
Flux01..909090 Flux01..909090
0.998 0.998
0.997 Batalha Revised 0.997 Batalha Revised
-4 -2 0 2 4 -3 -2 -1 0 1 2 3
Time (Hours Since Transit) Time (Hours Since Transit)
0.004
0.003
Residuals 00..000002 Residuals-0000....000000000121
-0.002 -0.002
-4 -2 0 2 4 -3 -2 -1 0 1 2 3
MAST Period: 56.056286 Days Time (Hours Since Transit) MAST t0: 33.000000 Days MAST Period: 38.098240 Days Time (Hours Since Transit) MAST t0: 25.700000 Days
AMOEBA Period: 56.054762 Days AMOEBA t0: 33.001157 Days AMOEBA Period: 38.098428 Days AMOEBA t0: 25.702688 Days
Fig.8.— Light curve for KOI 854.01. Top: Detrended light Fig. 10.— Light curve for KOI 2626.01 in the same format as
curvewithtransittimesmarkedbyreddots. Middle: Lightcurve Figure8.
phasedtothebest-fitperiod. Thebluecurveindicatestheoriginal
transit model and the red curve marks our revised fit. The pa-
rameters for the fit are indicated above the middle panel and the
periodandephemerisaremarkedatthebottomofthefigure. The
“MAST” values indicate the original period and ephemeris listed
intheplanetcandidate listatMASTandthe“AMOEBA”values
indicatetherevisedperiodandephemeris. Bottom: Residualsfor
theoriginaltransitmodel(blue)andourrevisedmodel(red).
KOI 1422.02
Flux1.002
Detrended 001...990990680
200 300 400 500 600
rrstar: 0.040, arstar: 34.830, Inc: 88.700 Time (Days) rrstar: 0.038, arstar: 51.985, Inc: 89.553
1.002
1.001
Flux01..909090
0.998
0.997 Batalha Revised
-2 -1 0 1 2
Time (Hours Since Transit)
0.002
Residuals-000...000000101 tm5o0Fo%ibdgee.olstf1a-1fitn.htd—eptdheLraeiitgoarhdet.dacruecTurhpvreevloeftbotmlreudaKe.rOkcsuTIrh5ov3uee1rg.i0nrr1eadv.yiicTsapeotodepisnfi:tttL.hiniegFhotothrriecgcuillnraovawreilteyprt,rharaoingsnsehlidytt
-2 -1 0 1 2 indicatesrepresentativeerrorbars. Theparametersforthefitsare
MAST Period: 19.850214 Days Time (Hours Since Transit) MAST t0: 14.560000 Days indicatedbetweenthepanels. Bottom: Residualsfortheoriginal
AMOEBA Period: 19.849853 Days AMOEBA t0: 14.559054 Days
transitmodel(blue)andourrevisedmodel(red).
Fig.9.— Light curve for KOI 1422.02 in the same format as systems,6triplesystems,and4quadruple7systems. The
Figure8. fraction of single planet systems (73%) is slightly lower
thanthe79%singlesystemfractionfortheplanetcandi-
4.1. Multiplicity dates around all stars (Fabrycky et al. 2012b), but this
difference is not significant.
Half (48 out of 95) of our cool planet candidates are
located in multi-candidate systems. We mark the multi- 5. PLANETOCCURRENCEAROUNDSMALLSTARS
plicity of each system in Figure 14. As shown in the fig-
We estimate the planet occurrence rate around small
ure, the largest planet candidates (KOIs 254.01, 256.01,
stars by comparing the number of detected planet can-
531.01,and2156.01)areinsystemswithonlyoneknown
didates with the number of stars searched. Our anal-
planet and 93% of the 14 candidates with orbital peri-
ysis assumes that all 64 of the planet candidates are
ods shorter than 2 days belong to single-candidate sys-
bona fide planet candidates and not false positives.
tems. The one exception is KOI 936.02, which has an
This assumption is reasonable because previous stud-
orbital period of 0.89 days and shares the system with
KOI 936.01, a 1.8R⊕ planet in a 9.47 day orbit. At or- 7Fabryckyetal.(2012a)reportthattheKOI952systemhasfive
bital periods longer than 2 days, 59% of the candidates
planetcandidates,butwecountthissystemasaquadrupleplanet
belongtosystemswithatleastoneadditionalplanetcan- systembecauseKOI952.05wasnotincludedintheFebruary2012
didate. Our sample contains 47 single systems, 7 double planetcandidate list.