On the Occurrence Rate of Hot Jupiters in Different Stellar Environments
aa r X i v : . [ a s t r o - ph . E P ] D ec to appear in ApJ Preprint typeset using L A TEX style emulateapj v. 08/22/09
ON THE OCCURRENCE RATE OF HOT JUPITERS IN DIFFERENT STELLAR ENVIRONMENTS
Ji Wang , Debra A. Fischer , Elliott P. Horch and Xu Huang (Received; Accepted) to appear in ApJ ABSTRACTMany Hot Jupiters (HJs) are detected by the Doppler and the transit techniques. Fromsurveys using these two techniques, however, the measured HJ occurrence rates differ by afactor of two or more. Using the California Planet Survey sample and the Kepler sample,we investigate the causes for the difference of HJ occurrence rate. First, we find that12 . ± .
24% of HJs are misidentified in the Kepler mission because of photometric dilutionand subgiant contamination. Second, we explore the differences between the Doppler sampleand the Kepler sample that can account for the different HJ occurrence rate. Third, wediscuss how to measure the fundamental HJ occurrence rates by synthesizing the resultsfrom the Doppler and Kepler surveys. The fundamental HJ occurrence rates are a measureof HJ occurrence rate as a function of stellar multiplicity and evolutionary stage, e.g., the HJoccurrence rate for single and multiple stars or for main sequence and subgiant stars. Whilewe find qualitative evidence that HJs occur less frequently in subgiants and multiple stellarsystems, we conclude that our current knowledge of stellar properties and stellar multiplicityrate is too limited for us to reach any quantitative result for the fundamental HJ occurrencerates. This concern extends to η Earth , the occurrence rate of Earth-like planets.
Subject headings: INTRODUCTION
Hot Jupiters (HJs) are among the most promi-nent astronomical discoveries of the past cen-tury (Mayor & Queloz 1995; Marcy & Butler 1996).Their existence challenged the previously acceptedclassic planet formation model, which is tailored forthe solar system (Lissauer 1993). Our knowledge ofplanet formation has moved beyond the solar sys-tem ever since the first HJ discovery.Among all exoplanets, HJs are the easiest todetect by the Doppler technique because of theirshort periods (P <
10 days) and large planetarymasses ( m > M J ). Doppler planet surveys showthat the occurrence rate of HJs is ∼ . ± . . ± .
1% (Marcy et al. 2005) and 1 . ± .
6% (Cumming et al. 2008). Based on RV data ob-tained from the HARPS and ELODIE, Mayor et al.(2011) measured the HJ occurrence rate at 0 . ± . Electronic address: [email protected] Department of Astronomy, Yale University, New Haven,CT 06511 USA Department of Physics, Southern Connecticut State Uni-versity, 501 Crescent Street, New Haven, CT 06515, USA Department of Astrophysical Sciences, Peyton Hall, 4 IvyLane, Princeton University, Princeton, NJ 08540 USA
Gould et al. (2006) estimated that the HJ occur-rence rate is 0 . +0 . − . %. The number from the Su-perLupus Survey is 0 . +0 . − . % (Bayliss & Sackett2011). A more recent result from the Kepler mis-sion suggested that the HJ occurrence rate is 0 . ± .
1% (8 . R ⊕ ≤ R P ≤ . R ⊕ , P <
10 days,Howard et al. 2012) or 0 . ± .
05% (6 . R ⊕ ≤ R P ≤ . R ⊕ , 0.8 days ≤ P ≤ . +0 . − . % (Mortier et al.2012). If the stellar metallicity for stars inDoppler surveys is systematically higher thanthose in transit surveys, then the HJ occurrencerate from the the Doppler surveys is higher thanit is for the transit surveys. Given the strongdependence of the HJ occurrence rate on stellarmetallicity (Fischer & Valenti 2005), a metallicitydifference of 0.15 dex may explain the difference ofHJ occurrence rate.Stellar age may also explain the difference ofHJ occurrence rate. If Kepler stars are on av- Wang et al.erage older, then the fraction of evolved stars ishigher. Since there is evidence that evolved starshost fewer HJs (Bowler et al. 2010; Johnson et al.2010; Schlaufman & Winn 2013), a higher fractionof evolved stars may lead to a lower HJ occurrencerate for the Kepler sample.Another possible explanation may be stellar pop-ulation. Transit surveys usually target galacticbulge (e.g., Gould et al. 2006; Bayliss & Sackett2011), where stellar population is dominatedby M dwarfs (Henry et al. 1997). In compari-son, Doppler surveys usually selected solar-typestars (e.g., Valenti & Fischer 2005; Sousa et al.2011). Since HJs occur less frequently around Mdwarfs (Johnson et al. 2010; Bonfils et al. 2013), asample of stars with higher fraction of M dwarfsleads to a lower HJ occurrence rate. However,the stellar population difference is less of a con-cern for the comparison between the Kepler sam-ple and the Doppler sample. Unlike other transitsurveys, Kepler stars mainly consist of solar-typestars (Brown et al. 2011).In this paper, we investigate other possibilitiesthat may reconcile the difference of HJ occurrencerate from the Kepler and Doppler survey. In § §
3, wesummarize the potential explanations for the dis-crepancy of HJ occurrence rate. In § §
5, wesynthesize the Kepler and Doppler results and dis-cuss how to probe the fundamental HJ occurrencerate as a function of stellar multiplicity rate andevolutionary stage. In §
6, we discuss our conclu-sions and their implications to future investigationsof planet occurrence rate. FRACTION OF MISIDENTIFIED HJS
Simulated HJs
We attempt to quantify the fraction of HJs thatare misidentified as smaller planets due to two ef-fects: photometric dilution and subgiant contami-nation. Fig. 1 shows the flow chart of our simu-lation. We start with 100,000 HJs (1 day ≤ P ≤
10 days, 5 . R ⊕ ≤ R P ≤ . R ⊕ ). The lower limitof HJs roughly corresponds to 0.1 M J assumingthe mass-radius relationship used in Lissauer et al.(2011). The radius distribution of these HJs fol-lows a power law of − . ± .
11 (Howard et al.2012). The period distribution between 2 and 10days follows a power law of -1, i.e., uniform distri-bution in logarithmic space. The period distribu-tion between 1 and 2 days follows the same powerlaw distribution but with half the probability of2 day ≤ P ≤
10 days. This assumption accountsfor the lack of HJs within the 2-day orbital pe-riod (Howard et al. 2012).
Kepler Solar-Type Dwarf Stars
We obtain Kepler stellar properties from theNASA Exoplanet Archive (NEA, Huber et al.2014) . We apply a cut in effective temperature http://exoplanetarchive.ipac.caltech.edu/ (4100 K ≤ T eff ≤ K ) and surface grav-ity (4 . ≤ log g ≤ . Photometric Dilution in Multiple StellarSystems
The transit signal of a planet is diluted by theflux of companion stars. The dilution leads totwo consequences: (1), a missing planet or (2), aplanet with an underestimated planet radius in thedetection case. For the Kepler mission, a planetis missed when the detection signal to noise ra-tio (SNR) is lower than 7.1 (Jenkins et al. 2010a).The SNR is calculated based on the following equa-tion (Wang et al. 2014a):
S/N = δCDP P eff p N transits , (1)where δ is transit depth, CDP P eff is the ef-fective combined differential photometric preci-sion (Jenkins et al. 2010b), a measure of photomet-ric noise, and N transits is the number of observedtransits. The transit depth is calculated by the fol-lowing equation: δ = R R ∗ F ∗ F ∗ + F c , (2)where R PL is planet radius, R ∗ is the radius of thestar that the planet is transiting, F denotes flux,and subscripts ∗ and c indicate the planet host starand the contaminating star, respectively. Through-out the paper, we only consider the case in whicha HJ transits the primary star because we focus onHJs around solar-type stars and the secondary starmay not be a solar-type star in most cases.Even if a planet is detected in the presence ofphotometric dilution, the photometric dilution willcause some HJs to be misidentified as smaller plan-ets. These misidentified HJs are not accounted inthe statistics for HJs. Thus, the fraction of misiden-tified HJs needs to be quantified and corrected for.The measured planet radius ( R ′ PL ) can be calculatedby the following equation: R ′ PL = R PL r F ∗ F ∗ + F c . (3) Gravitationally-Bound Multiple Stellar Systems
Companions stars may be gravitationally boundor optical doubles or multiples (i.e., unbound). Weconsider these two cases separately. For gravita-tionally bound multiple stellar systems, the stellarmultiplicity rate for planet host stars (Wang et al.2014b,a) is significantly lower than that forthe solar neighborhood (Raghavan et al. 2010).We adopt the stellar multiplicity rate measuredfrom Wang et al. (2014a) for stellar separationssmaller than 1000 AU, i.e., 24% ± Optical doubles and multiples
For photometric dilution caused by optical dou-bles and multiples, we use the TRILEGAL galaxymodel (Girardi et al. 2005) to study the probabilityof such cases. Horch et al. (2014) describes the pro-cess in detail. Ten fields with a field of view of 1square degree are simulated. These fields have dif-ferent galactic latitudes so the combination of theresults from the fields gives a better statistical re-sult of the entire Kepler field of view. In each offield, TRILEGAL model is used to construct a sim-ulated stellar population with effective temperaturedistribution that is similar to Kepler stars. Binaryparameters are turned off in the simulation becausewe focus on optical doubles and multiples, gravita-tionally bound multiple stellar systems have alreadybeen discussed.From the simulations, we calculated the proba-bility of optical doubles or multiples. We find that55.7% of the simulated stars have at least one visualcompanion within 4 ′′ down to Kepler magnitudeof 27.0. The separation of 4 ′′ corresponds to thepixel scale of the Kepler CCD detector. Since stel-lar companions with small differential magnitudescause large adjustment of planet radius (Equation3), we also report the fraction of simulated starswith stellar companions down to a certain differ-ential magnitude. For differential magnitudes of 1,3, and 5 mag, the fractions of simulated stars withstellar companions are 2.8%, 11.3%, and 26.7%, re-spectively. Fig. 2 shows a contour plot of joint prob-ability of Kmag and Kmag , where Kmag is theKepler magnitude of the primary star and Kmag is the Kepler magnitude of the secondary star. Op-tical doubles and multiples with small delta magni-tude are more likely for faint Kepler stars (Kmag ≥ ≤ Contamination of Subgiants
In a transit observation, the ratio of planet ra-dius to star radius is measured. When convertingthe ratio to planet radius, the star radius needs to be multiplied. A systematic error in star radius es-timation can lead to inaccurate planet radius andthus misidentify a planet into an incorrect category.While we use the NEA stellar property catalog toselect solar-type dwarf stars, chances are that somethe selected stars are actually subgiant stars. Re-ported planet radii from the NEA may be systemati-cally lower than they should be with the contamina-tion of subgiants. Bastien et al. (2014) used short-time scale photometric variation of bright Keplerstars (Kmag ≤ ∼
30% of Kepler planets are underesti-mated by 20%-30%. We use a value of 25% for thepercentage by which the stellar radius is underesti-mated. To account for subgiant contamination inthe Kepler dwarf sample, we randomly assign 30%of simulated stars with HJs as subgiants that aremisidentified as ”dwarf” stars. We adopt the ad-justed subgiant radius to calculate SNR based onEquation 1. If the SNR is lower than 7.1, then theHJ is marked as a missing HJ due to insufficientSNR. In the case of detection (SNR ≥ . Result
We repeat the above simulation for 100 times andcalculate the fraction of HJs that are misidentifiedas smaller planets. We find that 12.48% ± ± ± ± DIFFERENCES BETWEEN THE DOPPLER ANDKEPLER SAMPLE
While there are numerous works on the HJ occur-rence rate for the Doppler sample and the Keplersample, we focus on two works that have the mostconsistency between samples, i.e., Wright et al.(2012) and Howard et al. (2012). Howard et al.(2012) used the Kepler sample to measure the oc-currence rate of HJs, f Kepler . Interpolating theirresult, the occurrence rate is 0.60 ± . R ⊕ ≤ R P ≤ . R ⊕ and P < ± Metallicity Difference
The occurrence rate of HJs is a strong functionof stellar metallicity (Gonzalez 1997; Santos et al.2001, 2004; Fischer & Valenti 2005; Johnson et al.2010; Sousa et al. 2011; Wang & Fischer 2013). Aslight difference of stellar metallicity between theDoppler and Kepler sample will result in a signif-icant difference of HJ occurrence rate. The CPSsurvey targets stars in the solar neighborhood. Themetallicity distribution of these nearby stars canbe obtained from the SPOCS (Spectroscopic Prop-erties Of Cool Stars) catalog (Valenti & Fischer2005). We apply the effective temperature andsurface gravity cut (4100 K ≤ T eff ≤ K ,4 . ≤ log g ≤ .
9) to the SPOCS catalog to se-lect solar-type stars. The cut is consistent withthe solar-type star definition from Howard et al.(2012) for a proper comparison to the Kepler sam-ple. A total of 694 stars from the SPOCS are se-lected. The mean and median metallicity for theDoppler sample is -0.01 and 0.02 dex. In compar-ison, the mean and median metallicity of the Ke-pler sample -0.04 and -0.03 dex for 12,400 Keplerstars (Dong et al. 2014). The metallicity differencebetween the Doppler sample and the Kepler sam-ple is thus 0.03 dex (mean) or 0.05 dex (median),which results in a factor of 1.15 (mean) or 1.26 (me-dian) difference in the HJ occurrence rate if using apower law of 2.0 (Fischer & Valenti 2005). There-fore, the stellar metallicity alone cannot accountfor the difference of HJ occurrence rate. Giventhe sample size for metallicity determination, i.e.,694 for the Doppler sample and 12,400 for the Ke-pler sample, the standard error of metallicity mea-surement is small. However, the systematic errorof metallicity measurement is estimated at ∼ Evolutionary Stage Difference
HJs occur less frequently around evolved starsthan around main sequence stars (Johnson et al.2010; Bowler et al. 2010). Inclusion of evolved starsmay lower the HJ occurrence rate for a survey. TheCPS uses SPOCS as the input catalog. Followingthe selection criteria of Wright et al. (2012),
V < B − V < .
2, and ∆ M V < . g > . g < .
1. The range 3 . < log g < . Stellar Multiplicity Rate Difference
Doppler planet surveys usually target single starsand stars without nearby (sep < ′′ ) stellar com-panions (Wright et al. 2012), so the stellar multi-plicity rate for the Doppler sample should be lowerthan what is known for stars in the solar neighbor-hood (Duquennoy & Mayor 1991; Raghavan et al.2010). Given the median distance of 30 pc for thestars selected in Wright et al. (2012), a separation of2 ′′ corresponds to 60 AU, which is roughly the peakof stellar separation distribution (Raghavan et al.2010). Therefore, the stellar multiplicity rate for theDoppler sample is at most half of the stellar mul-tiplicity rate in the solar neighborhood, i.e., ∼ ∼ ′′ and photometric aperture is usually at leasttwice as large, so the Kepler mission observes bothsingle stars and multiple stellar systems with stel-lar separations almost extends to the tail of stellarseparation distribution. Therefore, the stellar mul-tiplicity rate of the Kepler sample should be higherthan the Doppler sample. Given that planet for-mation is suppressed in multiple star systems (e.g.,Wang et al. 2014a), including more multiple starsystems in the sample may lead to a lower planetoccurrence rate. False Positive Rate for HJs from Kepler
Not all HJ candidates detected by the Ke-pler are bona fide planets. The false posi-tive rate of HJs is estimated between 10% and35% (Morton & Johnson 2011; Santerne et al. 2012;Fressin et al. 2013). Considering the false positiverate for HJs, the HJ occurrence rate from the Keplermission should be even lower. Fressin et al. (2013)found the HJ occurrence rate to be 0.43% account-ing for the false positive rate. PROBING THE FUNDAMENTAL HJ OCCURRENCERATE
We have covered a variety of potential causes thatmay account for the difference of HJ occurrencerates from the Doppler and Kepler sample. Thefollowing equations summarize how these potentialcauses can be combined to probe the fundamentalHJ occurrence rate as a function of stellar multi-plicity and evolutionary stage. The measured HJoccurrence rate from the Kepler or the Doppler sur-vey is a combined result of fundamental HJ occur-rence rates: f = f MS × (1 − SGR) + f SG × SGR , or (4)J Occurrence Rate 5 f = f single × (1 − MR) + f multiple × MR , (5)Where f is measured HJ occurrence rate, MS rep-resents main sequence, SG represents subgiant, MRis stellar multiplicity rate and SGR is the frac-tion of subgiants in the sample. We have twomeasurements from the Kepler and Doppler sur-veys. However, there are four fundamental HJ oc-currence rates that we wish to solve for, f MS , single , f MS , multiple , f SG , single , and f SG , multiple . In addition,there are correcting factors for the difference be-tween the Kepler and Doppler sample. For exam-ple, we should consider ξ [Fe / H] , the correction fac-tor for the metallicity difference ( § ξ misidentified ,the correction factor for the misidentified HJs dueto photometric dilution and subgiant contamination( § § f Doppler ∼
1% for HJoccurrence rate (Mayor et al. 2011; Cumming et al.2008; Wright et al. 2012). f Kepler measured fromdifferent independent works agree at ∼ . − .
6% (Howard et al. 2012; Fressin et al. 2013). Thecorrecting factor ξ [Fe / H] for the metallicity differ-ence is still uncertain to ∼ ξ misidentified in this work, but its value depends onassumptions of other parameters, such as MR andSGR. FPR for the Kepler sample has been studiedextensively, but its value is still not well constrained,ranging from 10% to 35% (Morton & Johnson 2011;Santerne et al. 2012; Fressin et al. 2013).Understanding the stellar properties and statis-tics is crucial in solving Equation 4 and 5. De-termination of SGR requires measurement of log g . While the stars in the Doppler sample haverelatively well-determined log g (Valenti & Fischer2005; Sousa et al. 2011), log g distribution for theKepler sample is still uncertain especially for thefaint end (Bastien et al. 2014). MR MS for theDoppler sample is not well constrained because ofselection bias. While there is no strong selectionbias against multiple star systems for the Keplersample, but it is not known whether MR MS is thesame for the Kepler sample as for the solar nei-borhood (Raghavan et al. 2010). The knowledge ofMR SG is more uncertain for both the Kepler andDoppler sample. HJ OCCURRENCE RATE IN THE SOLARNEIGHBORHOOD
It is very challenging, if not impossible, to probethe fundamental HJ occurrence rates given the un-certainties of the parameters required for the calcu-lation. In addition, we need some additional con-straints to solve for the four variables, f MS , single , f MS , multiple , f SG , single , and f SG , multiple , with onlytwo measurements from the Doppler and Keplersample. However, the fundamental HJ occurrencerates can be calculated under two extreme and yetunlikely circumstances where the four variables arereduced to two. While the following two extremecases do not necessarily reflect the reality, but theyhelp us to understand how we can use Equation4 and 5 to probe the fundamental HJ occurrencerates. Extreme Case I: f single = f multiple It is still under debate how the HJ occurrencerate in single stars compares to that for multi-ple stellar systems. On one hand, evidence ofsuppressed planet formation is found in multiplestellar systems (e.g., Wang et al. 2014b,a). Onthe other hand, a stellar companion may facilitatethe formation of HJs via Kozai perturbation (e.g.,Wu & Murray 2003), although a recent study showsthat HJ formation via this channel may have an up-per limit of 44% (Dawson et al. 2012). Therefore, itis not an unreasonable assumption that HJ occur-rence rate is the same in single stars as in multiplestellar systems.With this assumption, Equation 4 and 5 arereduced to Equation 4 with two variables, f MS and f SG . Two measurements of f are available.One is from the Kepler sample and the other oneis from the Doppler sample. In this calculation,we adopt f Kepler = 0 . ± .
1% (Howard et al.2012), f Doppler = 1 . ± .
38% (Wright et al.2012). We apply correction factors ξ [Fe / H] = 1 . ξ misidentified = 1 .
14, and FAP = 17% (Fressin et al.2013). We adopt SGR for the Kepler sample tobe 48% (Bastien et al. 2014), and 12.6% for theDoppler sample (Valenti & Fischer 2005). Substi-tuting all the adopted values into Equation 4 forthe Kepler and Doppler sample, we infer that f MS =1 . ± .
52% and f SG = 0 . ± . f MS and f SG are based only on theuncertainties of f Kepler and f Doppler . The result in-dicates that HJs are much less common around sub-giant stars than around main sequence stars, whichis consistent with observations (Bowler et al. 2010;Johnson et al. 2010).
Extreme Case II: f MS = f SG On the other hand, Doppler observations of sub-giants have not completely ruled out the possibil-ity that HJs may be as common around subgiantsas around main sequence stars. The sample inBowler et al. (2010) contains 31 subgiants. It is dif-ficult to rule out that f MS = f SG with the smallsample size. If we assume that f MS = f SG , Equa-tion 4 and 5 are reduced to Equation 5 with twovariables, f single and f multiple . In this case, weadopt MR = 46% for the Kepler sample, whichis the same as the stellar multiplicity rate in thesolar neighborhood (Raghavan et al. 2010). Weadopt MR = 5%, which is the stellar multiplic-ity rate for all the known planets detected by theDoppler technique (Wright et al. 2011). Adopting Wang et al.the same values for other parameters as the previ-ous case, we infer that f single = 1 . ± .
43% and f multiple = 0 . ± . f single and f multiple are based only on the uncertain-ties of f Kepler and f Doppler .We emphasize that the two extreme cases do notnecessarily reflect the reality. In fact, one extremeassumption leads to a result that is significantly dif-ferent from the other extreme assumption. There-fore, a meaningful solution of these fundamental HJoccurrence rate must lie in between these two ex-tremes. To better understand the fundamental HJoccurrence rate, we need to have better knowledgeof the stellar properties of the Kepler and Dopplersample and some other external constraints. SUMMARY AND DISCUSSION
Summary
We conduct simulation to investigate the frac-tion of HJs missed or misidentified by the Keplermission due to the effects of photometric dilutionand subgiant contamination. Despite these two ef-fects, the Kepler mission rarely missed any HJs be-cause of their large photometric signal. However,12.48% ± ∼ Future Work on the Fundamental HJOccurrence Rates
To quantitatively calculate the fundamental HJoccurrence rate, f MS , single , f MS , multiple , f SG , single ,and f SG , multiple , we must need external constraintsin addition to the Doppler and Kepler results. Forexample, one can calculate f MS , single by carefullyselecting main sequence single stars in the Dopplersurvey. Alternatively, one can select main sequencestars and sub giants in the Kepler sample basedon the ”Flicker” method (Bastien et al. 2014) orspectroscopic methods. From these two sample ofKepler stars, one can calculate the ratio of f MS and f SG assuming similar stellar multiplicity ratefor main sequence and sub giant stars. In addi-tion, the ratio of f MS , single to f MS , multiple is provided by Wang et al. (2014a), although the caveat is thatthe ratio is for a sample of planet candidates thatare mostly small planets. With more constraints,the fundamental planet occurrence may be calcu-lated quantitatively. However, much more effort isneeded to make sure the calculation is meaningful.Future missions such as K2 (Howell et al. 2014) andTESS (Ricker et al. 2014) may provide independentmeasurement of HJ occurrence rate, i.e., f K2 and f TESS , these results can be incorporated to calcu-late the fundamental HJ occurrence rate given thatstellar properties and stellar multiplicity rate areknown to a certain accuracy.
Implications to η Earth
The findings in this paper can be used for thedetermination of η Earth , the occurrence rate of anEarth-like planet in the habitable zone. This issuehas already been complicated by the definition ofthe habitable zone (Seager 2013; Kopparapu et al.2013). Furthermore, the determination of η Earth should take into account the photometric dilutionand subgiant contamination. Finally, the measure-ment of η Earth from either the Kepler or the Dopplersurvey is a combination of the fundamental η Earth asa function of stellar multiplicity rate and evolution-ary stage (Equation 4 and 5). These fundamental η Earth can be precisely determined only if we havea good understanding of the stellar properties andstellar multiplicity rate of the sample.
Acknowledgements
We thank the referee RonaldGilliland for his insightful comments and sugges-tions which substantially improve the paper. Thisresearch has made use of the NASA ExoplanetArchive, which is operated by the California In-stitute of Technology, under contract with the Na-tional Aeronautics and Space Administration underthe Exoplanet Exploration Program.
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Figure 1.
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Figure 2.
Contours of joint probability distribution of Kmag and Kmag for optical doubles and multiples in the Keplerfield of view. Kmag is the Kepler magnitude of the primary star and Kmag is the Kepler magnitude of the secondary star.55.7% of Kepler stars have at least one visual unbound stellar companion. Fainter Kepler stars are more likely to have visualunbound stellar companions with small delta magnitudes. Therefore, planets around faint Kepler stars are more likely to bemisidentified as smaller planets. J Occurrence Rate 9
Figure 3.
Result of simulation to quantify the fraction of HJs that misidentified because of photometric dilution and subgiantcontamination. Color contours show the joint distribution of SNR and measured planet radius of simulated HJs. The marginal-ized distributions of SNR and measured planet radius are shown on the side of each axis. We find that 12.48% ± Table 1