An Increase in Small-planet Occurrence with Metallicity for Late-type Dwarf Stars in the Kepler Field and Its Implications for Planet Formation
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An Increase in Small-planet Occurrence with Metallicity for Late-type Dwarf Stars in the KeplerField and Its Implications for Planet Formation
Cicero X. Lu, Kevin C. Schlaufman, and Sihao Cheng ( 程 思 浩 ) Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA (Received May 8, 2020; Revised August 31, 2020; Accepted September 9, 2020)
Submitted to Astronomical JournalABSTRACTWhile it is well established that giant-planet occurrence rises rapidly with host star metallicity, it isnot yet clear if small-planet occurrence around late-type dwarf stars depends on host star metallicity.Using the Kepler Data Release 25 planet candidate list and its completeness data products, we exploreplanet occurrence as a function of metallicity in the Kepler field’s late-type dwarf stellar population.We find that planet occurrence increases with metallicity for all planet radii R p down to at least R p ≈ R ⊕ and that in the range 2 R ⊕ (cid:46) R p (cid:46) R ⊕ planet occurrence scales linearly with metallicity Z . Extrapolating our results, we predict that short-period planets with R p (cid:46) R ⊕ should be rarearound early M dwarf stars with [M / H] (cid:46) − . / H] (cid:46) +0 .
0. Thisdependence of planet occurrence on metallicity observed in the Kepler field emphasizes the need tocontrol for metallicity in estimates of planet occurrence for late-type dwarf stars like those targetedby Kepler’s K2 extension and the Transiting Exoplanet Survey Satellite (TESS). We confirm thetheoretical expectation that the small planet occurrence–host star metallicity relation is stronger forlow-mass stars than for solar-type stars. We establish that the expected solid mass in planets aroundlate-type dwarfs in the Kepler field is comparable to the total amount of planet-making solids intheir protoplanetary disks. We argue that this high efficiency of planet formation favors planetesimalaccretion over pebble accretion as the origin of the small planets observed by Kepler around late-typedwarf stars.
Keywords:
Exoplanet formation (492); Exoplanets (498); Extrasolar rocky planets (511); Extrasolarice giants (2024); Late-type dwarf stars (906); Planet hosting stars (1242) INTRODUCTIONPlanet formation is seeded by the metals present in aprotoplanetary disk. It must be the case that the totalheavy-element content of a protoplanetary disk providesan upper limit on the solid mass of planets formed inthat disk. For that reason, there must be a metallic-ity below which even Earth-mass planets cannot form.Observations have shown that the occurrence of giantplanets around FGKM dwarf stars rises rapidly withhost star metallicity (e.g., Santos et al. 2004; Fischer& Valenti 2005; Johnson & Apps 2009a; Johnson et al.
Corresponding author: Cicero X. [email protected] Z (cid:63) and Z disk should be the same.The mass of a disk during the epoch of planet forma-tion has been found to scale roughly linearly with stel-lar mass with fixed disk-to-star mass ratio M disk /M (cid:63) in a r X i v : . [ a s t r o - ph . E P ] S e p Lu et al. (2020) the range 0 . (cid:46) M disk /M (cid:63) (cid:46) .
006 (e.g., Andrewset al. 2013). Though there is more than an order-of-magnitude scatter in the Andrews et al. (2013) rela-tion, those authors favor an inherently linear relation-ship between M disk and M (cid:63) . The net effect is that theamount of planet-making material available in a proto-planetary disk should scale roughly linearly with bothstellar metallicity Z (cid:63) and mass M (cid:63) .Assuming the solar metallicity Z (cid:12) = 0 .
014 (e.g.,Asplund et al. 2009) and a disk-to-star mass ratio M disk /M (cid:63) = 0 .
01 (Andrews et al. 2013), an early Mdwarf with M (cid:63) = 0 . M (cid:12) and [M / H] = − . M ⊕ of planet-making material in itsdisk. In that case, the available planet-making materialis much less than the amount required to make a singleNeptune-size planet with radius R p = 4 R ⊕ (Neptunehas at least 13 M ⊕ of metals as shown by Podolak et al.2019). On the other hand, 10 M ⊕ of planet-making ma-terial would be enough to make an Earth-compositionsuper-Earth mass planet with R p (cid:46) . R ⊕ (e.g., Zenget al. 2019). If the timescale for growing Earth-mass em-bryos scales with the amount of planet-making materialas suggested by detailed calculations (Movshovitz et al.2010), then the probability of forming a planet with asignificant gaseous envelope in the few Myr available be-fore its parent protoplanetary disk is dissipated shouldalso scale with the amount of planet-making material.There are hints that this effect becomes important at[M / H] ≈ − . g − r of late-typedwarf stars with small-planet candidates was 4 σ redderthan the average color of a control sample of similarstars without identified planet candidates. They arguedthat their observation was evidence for a metallicity dif-ference between late-type dwarf stars with and withoutsmall planets.The Schlaufman & Laughlin (2011) result was criti-cized by Mann et al. (2012, 2013b), who argued that the g − r photometric metallicity indicator used by Schlauf-man & Laughlin (2011) is insensitive to metallicity andthat the possible presence of giant stars mistaken fordwarf stars in the Schlaufman & Laughlin (2011) con-trol sample could produce a similar g − r offset unre-lated to metallicity. Both of these criticisms can nowbe conclusively addressed. Photometric metallicity re-lations for late-type dwarf stars calibrated by reliableAPO Galactic Evolution Experiment (APOGEE) high-resolution H -band spectroscopy are now available (Ma-jewski et al. 2016; Schmidt et al. 2016). Kepler astero-seismology (Hekker et al. 2011; Huber et al. 2011; Stelloet al. 2013; Huber et al. 2014; Mathur et al. 2016; Yuet al. 2016, 2018) and Gaia DR2 parallaxes (Gaia Col-laboration et al. 2016, 2018; Arenou et al. 2018; Hamblyet al. 2018; Lindegren et al. 2018; Luri et al. 2018) enablethe construction of samples of dwarf stars without planetcandidates completely free of subgiant or giant star con-tamination. Advances in the analysis of Kepler data andthe public availability of its completeness data productsnow permit differential planet occurrence calculations.Thanks to these developments, photometric metallici-ties have become a powerful tool for the exploration ofthe small-planet occurrence–metallicity relation.Advances in the theory of planet formation have alsorevealed the possible significance of the accretion of“pebbles”, or material significantly smaller than the km-size planetesimals historically studied (Ormel & Klahr2010; Lambrechts & Johansen 2012). This “pebble ac-cretion” process invokes the accretion by planetary em-bryos of small particles experiencing strong aerodynamicdrag. Since not all of this rapidly migrating materialcan be accreted by a planetary embryo, pebble accre-tion is inherently lossy in the sense that more than 90%of a disk’s initial complement of planet-making material mall Planet Occurrence Increases with Metallicity for Late-type Dwarfs M disk ∼ . M (cid:12) and Z disk ∼ Z (cid:12) = 0 . R p ≈ R ⊕ and that in the range 2 R ⊕ (cid:46) R p (cid:46) R ⊕ planet oc-currence scales linearly with metallicity. In Section 2 wediscuss our sample selection, describe the photometriceffective temperature and metallicity relations we use,and outline the process we use to remove giant stars fromour sample of stars without planet candidates. We splitboth planet candidate-host and non-planet-candidate-host samples into metal-rich and metal-poor subsam-ples and illustrate two different occurrence calculationsin Section 3. We then calculate planet formation effi-ciency in an attempt to infer the relative importance ofplanetesimal accretion and pebble accretion. In Section4 we discuss our results and their implications for thetheory of planet formation. We conclude and summarizeour findings in Section 5. DATAWe seek to assemble the sample of late-type dwarfstars with Kepler light curves that have been searchedfor transiting planet candidates. To do so, we selectlate-type stars from the Kepler Input Catalog (KIC -Brown et al. 2011) with effective temperature T eff inthe range 3600 K (cid:46) T eff (cid:46) H -band spectroscopy[M / H] = a + a ( r − z ) + a ( W − W , (1) T eff = b + b ( r − z ) + b [M / H] , (2)with the coefficients a i = ( − . , . , − . b i = (4603 . , − . , . ugriz , Two Micron All SkySurvey (2MASS - Skrutskie et al. 2006) JHK s , andWide-field Infrared Survey Explorer (WISE - Wrightet al. 2010; Mainzer et al. 2011) W W r − z and W − W T eff and [M / H]. The uncertaintiesin individual [M / H] and T eff estimates produced us-ing Equations (1) and (2) are approximately 0.2 dexin [M / H] and 100 K in T eff . We require all late-typestars in our sample to have been observed for at leastone quarter during the Kepler mission.It is well known that the surface gravity log g estimatesin the KIC are imperfect (e.g., Mann et al. 2012; Dress-ing & Charbonneau 2015). To ensure that there are nogiant stars in our sample, we reject stars identified asgiants via either asteroseismic oscillations or Gaia DR2parallaxes. We first select Kepler target stars with KIClog g >
4. We then remove stars identified through as-teroseismology as subgiants or as giants/red clump starsby Hekker et al. (2011), Huber et al. (2011), Stello et al.(2013), Huber et al. (2014), Mathur et al. (2016), or Yuet al. (2016, 2018). We also use Gaia DR2 parallaxesto calculate Gaia G -band absolute magnitudes and thenexclude 11 giant stars that are several magnitudes abovethe Hamer & Schlaufman (2019) empirical Pleiades zero-age main sequence.We cross match this purified sample of late-type dwarfstars with the Kepler DR25 list of KOIs dispositioned asplanet candidates (Thompson et al. 2018). We use thehomogeneous DR25 planet candidate list because it wasgenerated in a fully automated fashion that eliminatedhuman vetting of threshold crossing events. That lackof intervention made its completeness straightforward toalgorithmically assess. Because giant planet host starsare known to be metal rich, we exclude from our analysisstars that host planets with R p > R ⊕ . We also verifiedthat using the updated stellar radii from Berger et al.(2018) did not change any of our subsequent conclusions.Our final planet candidate-host sample consists of the99 late-type dwarfs with at least one planet candidatewith R p ≤ R ⊕ listed in Table 1. We refer to these starsas our planet candidate-host sample from this point on.We also select a sample of 3,395 late-type dwarfs thatwere part of the main transiting exoplanet search pro- Lu et al. (2020) gram, passed all of our selection criteria listed above,and have no detected planet candidate. We list thesestars in Table 2 and refer to them as our non-planet-candidate-host sample from here. We plot r − z ver-sus W − W T eff and [M / H] valuesinferred using Equations (1) and (2) for both our planetcandidate-host and non-planet-candidate-host samplesin Figure 2. ANALYSISWe explore the connection between host star metal-licity and small-planet occurrence in three ways. First,we use logistic regression to estimate the significanceof metallicity and effective temperature for the predic-tion of planet occurrence in our complete sample. Wenext separate our complete sample into metal-rich andmetal-poor subsamples for which we independently cal-culate planet occurrence as a function of metallicity,orbital period P , and planet radius R p using the Ke-pler DR25 completeness data products. We then use amass–radius relation combined with the small-planet oc-currence maps inferred for our complete sample as wellas our metal-rich and metal-poor subsamples to roughlyestimate the planet formation efficiency in the proto-planetary disks that once existed around the stars inour sample. 3.1. Logistic Regression
We use logistic regression—a natural extension of lin-ear regression for probability—to obtain a first look atthe relationship between host star metallicity & effec-tive temperature and the probability of the presence ofa small planet candidate in the system P host . We usethe logistic regression model P host = 11 + e − x , (3) x = β + β T eff + β [M / H] , (4)and the statsmodel.logit (Genz 2004; Seabold &Perktold 2010) implementation of logistic regression.We give the result of our calculation in Table 3.We find that the coefficient for metallicity in the logis-tic regression equation is positive and significantly dif-ferent than zero, while the coefficient for effective tem-perature is consistent with zero (see Table 3). The im-plication is that planet occurrence increases with hoststar metallicity and is insensitive to host star effectivetemperature. The coefficient of a continuous predictorvariable in a logistic regression model gives the expectedchange in the natural logarithm of the odds ratio of the modeled outcome with a one-unit change in that con-tinuous predictor variable. Since we will subsequentlyfind in the next subsection that the metallicity differ-ence between our metal-rich and metal-poor subsamplesis about 0.3 dex, we use our logistic regression model toestimate the effect of a 0.3 dex change in [M / H] on theprobability of finding a planet candidate in a system P host . When the probability of an event is small andtherefore x must be small as well, the logistic regressionfunction is approximately an exponential regression P = e x and the coefficients β i can be interpreted as the frac-tional change of the odds of an event’s occurrence. Wefind that P host ([M / H] + 0 .
3) = 1 . +0 . − . P host ([M / H]).In words, the probability that a late-type dwarf star wasobserved by Kepler to host at least one small planet can-didate increases by a factor of about 1 . +0 . − . for a 0.3dex change in [M / H] (a factor of two in Z (cid:63) ).The logistic regression analysis handles single- andmultiple-planet systems in the same way and thereforedoes not account for multiplicity. It does not controlfor the decrease in transit probability with semimajoraxis or the incompleteness of the Kepler DR25 planetcandidate list. It implicitly assumes that a star withno observed planet candidates is equivalent to a starwithout planets. This last assumption is only valid inthe parts of parameter space where planet occurrenceis low (i.e., P (cid:46)
10 days and 2 R ⊕ (cid:46) R p (cid:46) R ⊕ ).Because planet occurrence increases with both increas-ing orbital period and with decreasing planet radius, itis important to account for transit probability and Ke-pler DR25 completeness to explore the connection be-tween host star metallicity and small-planet occurrencefor the much more common long-period and/or small-radius planets.3.2. Occurrence as a Function of Metallicity, OrbitalPeriod, and Planet Radius
To complement the logistic regression analysis in theprevious subsection, we calculate planet occurrence asa function of metallicity, orbital period, and planet ra-dius using the Kepler DR25 planet candidate list andits completeness data products. This approach takesinto account planet multiplicity and allows us to explorethe connection between host star metallicity and small-planet occurrence at longer periods and smaller radiithan the logistic regression approach.We first separate our complete sample into metal-richand metal-poor subsamples by splitting at the metallic-ity that separates our planet candidate-host sample intotwo nearly equal halves. We split our complete sampleinto two nearly equally sized subsamples to minimizethe effects of sample size differences. We therefore set mall Planet Occurrence Increases with Metallicity for Late-type Dwarfs Table 1.
Late-type Dwarf Kepler Targets with at Least One DR25 Planet Candidate with R p ≤ R ⊕ KIC Number Kepler Name KOI Name R.A. Decl. r z W σ W W σ W (deg) (deg) (mag) (mag) (mag) (mag) (mag) (mag)10118816 · · · K01085 281.05011 47.188148 15.29 14.28 12.164 0.023 12.103 0 . . · · · K04419 281.11646 43.282440 15.16 14.29 12.145 0.023 12.132 0 . . . . . . . . Note —The typical r - and z -band uncertainties are 0.02 mag (Brown et al. 2011). This table is ordered by right ascensionand is available in its entirety in the machine-readable format. Table 2.
Late-type Dwarf Kepler Targets with No Observed Planet CandidatesKIC Number R.A. Decl. r z W σ W W σ W (deg) (deg) (mag) (mag) (mag) (mag) (mag) (mag)7797376 279.708780 43.53535 15.78 15.14 13.108 0.024 13.106 0.0257867105 279.996150 43.67716 14.32 13.53 11.488 0.023 11.520 0.0217867279 280.120470 43.68729 15.39 14.70 12.614 0.023 12.642 0.0237658133 280.122220 43.35148 15.52 14.80 12.790 0.022 12.796 0.02310382584 280.132140 47.59633 15.85 14.93 12.756 0.023 12.793 0.0247581219 280.137160 43.21006 15.97 14.68 12.352 0.023 12.331 0.02110317398 280.232649 47.45096 15.03 14.39 12.458 0.023 12.436 0.02210251684 280.305889 47.39492 15.09 14.36 12.340 0.022 12.354 0.0227581487 280.315260 43.22374 15.76 14.62 12.377 0.023 12.357 0.0227867585 280.358460 43.61532 16.01 15.01 12.824 0.023 12.821 0.023 Note —The typical r - and z -band uncertainties are 0.02 mag (Brown et al. 2011). This tableis ordered by right ascension and is published in its entirety in the machine-readable format. Table 3.
Logistic Regression ResultsVariable Value Uncertainty t -statistic p -value T eff − . × − . × − − . . / H] 1 . .
65 2 . . the dividing line at [M / H] = − .
15 as shown Figure 2.The resulting metal-rich subsample has 74 planet candi-dates (49 planet candidate hosts) and 1,299 non-planet- candidate hosts while the metal-poor subsample has 76planet candidates (50 planet-candidate hosts) and 2,096non-planet-candidate hosts. We find that the averagemetallicities of our metal-rich and metal-poor samplesare [M / H] = +0 . / H] = − . Lu et al. (2020) r z [mag] W W [ m ag ] [ M / H ] = - . [ M / H ] = - . [ M / H ] = - . [ M / H ] = . [ M / H ] = . Non-KOI hostsKOI hosts r z [mag] W W [ m ag ] K K K K K K K Non-KOI hostsKOI hosts
Figure 1.
Distribution of late-type dwarf stars in the r − z versus W − W / H] according to Schmidt et al. (2016). Right: dashed curves represent lines ofconstant T eff according to Schmidt et al. (2016). T eff [K] M e t a lli c i t y [ M / H ] Non-KOI hostsKOI hosts
Figure 2. [M / H] as a function of T eff for planet candidate-host and non-planet-candidate-host samples inferred using r − z and W1 − W2 colors in Equations (1) and (2). The bluecircles represent the planet candidate-host sample while thegray shading represents the distribution of the non-planet-candidate-host sample after kernel smoothing. The red lineat [M / H] = − .
15 indicates the median host star metallic-ity for the planet candidate sample. While there are fewerstars above the red line than below it, the metal-rich starshost more multiple planet candidate systems. We representthe typical [M / H] and T eff uncertainties resulting from un-certainties in the input photometry with the cross at the topright. in [M/H] and the median metallicity of the metal-richsubsample would remain the same.To go from the observed frequency of planet candi-dates to their underlying occurrence, it is necessary todivide the observed frequency by its completeness. For Period [Days] P l a n e t R a d i u s [ R ] ( 5.0 ) ( 2.4 ) ( 2.4 ) ( 2.4 ) ( 2.4 ) ( 2.4 ) ( 2.4 ) ( 2.4 ) ( 2.8 ) ( 4.8 ) ( 1.6 ) ( 1.6 ) ( 1.6 ) ( 1.6 ) ( 1.6 ) ( 1.6 ) ( 1.6 ) ( 1.9 ) ( 3.5 ) ( 1.1 ) ( 1.1 ) ( 1.1 ) ( 1.1 ) ( 1.1 ) ( 1.1 ) ( 1.1 ) ( 1.2 ) ( 2.1 ) ( 0.8 ) ( 0.7 ) ( 0.7 ) ( 0.7 ) ( 0.7 ) ( 0.7 ) ( 0.7 ) ( 0.8 ) ( 1.0 ) ( 0.6 ) ( 0.5 ) ( 0.5 ) ( 0.5 ) ( 0.5 ) ( 0.5 ) ( 0.5 ) ( 0.6 ) ( 0.3 ) ( 0.6 ) ( 0.3 ) ( 0.3 ) ( 0.3 ) ( 0.3 ) ( 0.3 ) ( 0.3 ) ( 0.4 ) ( 0.0 ) ( 0.6 ) ( 0.2 ) ( 0.2 ) ( 0.2 ) ( 0.2 ) ( 0.2 ) ( 0.2 ) ( 0.3 ) ( 0.0 ) ( 0.4 ) ( 0.2 ) ( 0.2 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.2 ) ( 0.0 ) ( 0.2 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.0 ) ( 0.0 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) ( 0.1 ) C o m p l e t e n e ss [ % ] Figure 3.
Example completeness map for KIC 1577265.The values in every cell denote the completeness and thescatter of completeness within the cell in percent. each star in both subsamples, we use the
KeplerPORTS software described in Burke & Catanzarite (2017) to es-timate the completeness of the Kepler Pipeline that pro-duced the DR25 planet candidate list as a function oforbital period and planet radius. We present in Fig-ure 3 an example completeness map for KIC 1577265 (arandomly selected star from our complete sample).We combine individual completeness maps for all starsin each subsample to obtain representative complete-ness maps for both the metal-rich and metal-poor sub-samples. For each point in orbital period–planet ra-dius space, we take the average value of the complete-ness maps produced for all stars in a given subsam-ple. We thereby obtain a representative completenessmap that corresponds to the typical completeness av-eraged over an entire subsample. Completeness maps mall Planet Occurrence Increases with Metallicity for Late-type Dwarfs P and R p . For each cell, we takethe representative completeness value to be the medianof all completeness estimates in that cell. For example,in Figure 3 the value in each cell is the median of allindividual completeness estimates in that cell from theinitial higher-resolution completeness map.We compute the occurrence of planet candidates asa function of orbital period and planet radius in bothmetal-rich and metal-poor subsamples using their therepresentative completeness maps. The occurrence ineach cell depends on the total number of observed planetcandidates N PC and the total number of equivalentstars searched N (cid:63) in that cell. We define N (cid:63) as theproduct of our estimated representative completeness inthat cell and the total number of stars in a subsample.Since there are uncertainties in the measurement of eachplanet candidate’s orbital period and radius , we usea 1,000 iteration Monte Carlo simulation to distributethe impact of an individual planet candidate detectionacross multiple cells using a two-dimensional Gaussiankernel with a diagonal covariance matrix with the 1- σ period and radius uncertainties on the diagonal. Wethen define N PC as the number of counts in each cellaveraged over the Monte Carlo simulation.We adopt a Bayesian framework to estimate planet oc-currence η . We model occurrence with a binomial like-lihood and use a Beta distribution prior Beta ( α, β ). Inthat situation, the Beta distribution is a conjugate priorand the posterior distribution of occurrence will be aBeta distribution that depends on the prior parameters, N PC , and N (cid:63) P ( η | N PC , N (cid:63) ) = (5)Beta ( α + N PC , β + N (cid:63) − N PC ) , where α and β are parameters of the prior. We assumea weak uninformative prior with α = β = 1.We plot the results of our occurrence calculations inFigures 4 and 5 and give them in tabular form in Table 4.Figure 4 shows planet candidate occurrence as a functionof orbital period and planet radius for both our metal- Since the planet radius uncertainties provided in the DR25 planetcandidate list only include the effect of stellar radius uncertain-ties, we calculated our own planet radius uncertainties accountingfor both transit depth and stellar radius uncertainties. rich and metal-poor subsamples, while Figure 5 showsplanet candidate occurrence as a function of orbital pe-riod and planet radius for our complete sample. Thedifferences in planet occurrence between the metal-richand metal-poor subsamples illustrate the effect of metal-licity on small planet formation: small planets are lesscommon around metal-poor stars than around metal-rich stars. We indicate cells with no detected planetcandidates in Table 4 and with black borders in bothFigures 4 and 5. Our metal-rich and metal-poor sub-samples are large enough and Kepler DR25’s complete-ness is high enough that for cells with P (cid:46)
100 days and R p (cid:38) R ⊕ the product of sample size and completenessis larger than 10 (see Table 4). In this case, the signalimplicit in a non-detection is at least an a factor of fivelarger than the signal weakly implied by our prior.Planets with R p (cid:38) R ⊕ are almost certain to possesssignificant H/He envelopes (e.g., Rogers 2015; Chen &Kipping 2017). We therefore separately study the dif-ference in “rocky” (0 . R ⊕ (cid:46) R p (cid:46) R ⊕ ) and “H/Heenvelope” (2 R ⊕ (cid:46) R p (cid:46) R ⊕ ) planet occurrence be-tween our metal-rich and metal-poor subsamples. This R p ≈ R ⊕ boundary also corresponds to the so-called“Fulton Gap” (e.g., Fulton et al. 2017; Fulton & Pe-tigura 2018; Berger et al. 2018). To faithfully accountfor the effect of uncertainty on this calculation, we con-duct a Monte Carlo simulation. For every cell of eachof the metal-rich and metal-poor subsamples, we sampleplanet candidate occurrence from its posterior distribu-tion. For each cell, we then take the planet candidateoccurrence difference between the metal-rich and metal-poor subsamples and sum the difference across all peri-ods (excluding the longest-period cells because of theirsub-percent completeness levels). We obtain a numberthat describes the cumulative differential occurrence be-tween metal-rich and metal-poor subsamples. We repeatthis process 10,000 times to fully sample the differentialoccurrence distribution. We take the median and the16th and 84th percentiles of the distribution as the typ-ical planet candidate occurrence difference and its as-sociated uncertainty. We call this statistic our “planetoccurrence difference” from here. We visualize these re-sults in Figure 6 and present them in tabular form inTable 5.We calculate the enhanced occurrence of planets inthe metal-rich subsample relative to the occurrence ofplanets in the metal-poor sample in one more way. Forevery cell of each of the metal-rich and metal-poor sub-samples, we sample planet candidate occurrence fromits posterior distribution and sum over all cells in a sub-sample. We divide the summed occurrence calculatedfor the metal-rich subsample by the summed occurrence Lu et al. (2020) Period [Days] P l a n e t R a d i u s [ R ] ( +0.3 )( -0.2 ) ( +0.4 )( -0.3 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.5 )( -0.2 ) ( +0.5 )( -0.2 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +1.4 )( -0.9 ) ( +1.1 )( -0.8 ) ( +0.7 )( -0.4 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +3.0 )( -2.3 ) ( +2.1 )( -1.6 ) ( +1.9 )( -1.4 ) ( +1.4 )( -0.9 ) ( +1.0 )( -0.4 ) ( +1.0 )( -0.4 ) ( +1.0 )( -0.4 ) ( +1.0 )( -0.4 ) ( +1.0 )( -0.4 ) ( +5.7 )( -4.6 ) ( +4.1 )( -3.5 ) ( +3.2 )( -2.4 ) ( +2.3 )( -1.5 ) ( +1.5 )( -0.7 ) ( +1.4 )( -0.7 ) ( +1.4 )( -0.7 ) ( +1.4 )( -0.7 ) ( +1.4 )( -0.7 ) ( +7.4 )( -5.1 ) ( +5.3 )( -4.3 ) ( +5.0 )( -4.0 ) ( +3.9 )( -2.9 ) ( +2.2 )( -1.0 ) ( +2.1 )( -1.0 ) ( +2.1 )( -1.0 ) ( +2.1 )( -1.0 ) ( +2.1 )( -1.0 ) ( +10.0 )( -5.3 ) ( +6.9 )( -5.3 ) ( +7.0 )( -5.8 ) ( +6.0 )( -4.6 ) ( +3.3 )( -1.6 ) ( +3.1 )( -1.5 ) ( +3.1 )( -1.5 ) ( +3.1 )( -1.4 ) ( +3.1 )( -1.4 ) ( +19.0 )( -12.1 ) ( +10.3 )( -8.1 ) ( +9.3 )( -7.7 ) ( +5.9 )( -3.5 ) ( +4.5 )( -2.1 ) ( +4.5 )( -2.1 ) ( +4.5 )( -2.1 ) ( +4.5 )( -2.1 ) ( +4.5 )( -2.1 ) ( +27.5 )( -18.7 ) ( +13.5 )( -9.8 ) ( +11.7 )( -9.2 ) ( +6.8 )( -3.2 ) ( +6.6 )( -3.1 ) ( +6.5 )( -3.1 ) ( +6.5 )( -3.1 ) ( +6.4 )( -3.1 ) ( +6.4 )( -3.1 ) ( +32.9 )( -26.3 ) ( +19.3 )( -14.6 ) ( +10.8 )( -5.4 ) ( +9.7 )( -4.8 ) ( +9.3 )( -4.6 ) ( +9.2 )( -4.5 ) ( +9.1 )( -4.5 ) ( +9.1 )( -4.4 ) ( +9.0 )( -4.4 ) O cc u rr e n c e [ % ] Period [Days] P l a n e t R a d i u s [ R ] ( +0.5 )( -0.2 ) ( +0.5 )( -0.2 ) ( +0.5 )( -0.2 ) ( +0.5 )( -0.2 ) ( +0.5 )( -0.2 ) ( +0.5 )( -0.2 ) ( +0.5 )( -0.2 ) ( +0.5 )( -0.2 ) ( +0.5 )( -0.2 ) ( +1.3 )( -0.9 ) ( +1.2 )( -0.9 ) ( +0.8 )( -0.4 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.3 ) ( +1.8 )( -1.1 ) ( +2.1 )( -1.6 ) ( +1.2 )( -0.6 ) ( +1.3 )( -0.8 ) ( +1.0 )( -0.5 ) ( +1.0 )( -0.5 ) ( +1.0 )( -0.5 ) ( +1.0 )( -0.5 ) ( +1.0 )( -0.5 ) ( +4.5 )( -3.4 ) ( +2.9 )( -2.1 ) ( +2.4 )( -1.6 ) ( +1.5 )( -0.7 ) ( +1.5 )( -0.7 ) ( +1.5 )( -0.7 ) ( +1.5 )( -0.7 ) ( +1.6 )( -0.8 ) ( +1.8 )( -1.0 ) ( +7.8 )( -6.1 ) ( +4.6 )( -3.5 ) ( +4.5 )( -3.4 ) ( +4.0 )( -2.9 ) ( +2.2 )( -1.1 ) ( +2.2 )( -1.0 ) ( +2.2 )( -1.0 ) ( +2.2 )( -1.0 ) ( +2.2 )( -1.0 ) ( +12.4 )( -9.5 ) ( +4.8 )( -3.0 ) ( +6.5 )( -5.2 ) ( +6.9 )( -5.7 ) ( +5.2 )( -3.6 ) ( +3.4 )( -1.7 ) ( +3.2 )( -1.5 ) ( +3.2 )( -1.5 ) ( +3.2 )( -1.5 ) ( +19.3 )( -18.4 ) ( +8.6 )( -6.2 ) ( +8.3 )( -6.4 ) ( +7.8 )( -5.9 ) ( +7.6 )( -5.6 ) ( +6.0 )( -3.7 ) ( +4.9 )( -2.5 ) ( +4.6 )( -2.2 ) ( +4.6 )( -2.2 ) ( +26.1 )( -23.7 ) ( +12.2 )( -8.7 ) ( +10.5 )( -7.7 ) ( +10.2 )( -7.5 ) ( +6.7 )( -3.3 ) ( +6.5 )( -3.1 ) ( +6.5 )( -3.1 ) ( +6.5 )( -3.1 ) ( +6.5 )( -3.1 ) ( +33.0 )( -24.8 ) ( +17.3 )( -12.5 ) ( +13.3 )( -9.4 ) ( +12.0 )( -7.9 ) ( +9.2 )( -4.6 ) ( +11.0 )( -6.7 ) ( +10.6 )( -6.2 ) ( +9.1 )( -4.5 ) ( +9.1 )( -4.5 ) ( +34.4 )( -30.8 ) ( +21.8 )( -12.4 ) ( +14.4 )( -7.4 ) ( +12.9 )( -6.6 ) ( +12.6 )( -6.4 ) ( +12.5 )( -6.3 ) ( +12.5 )( -6.3 ) ( +12.5 )( -6.3 ) ( +12.5 )( -6.3 ) O cc u rr e n c e [ % ] Figure 4.
Planet candidate occurrence in metallicity–period–planet radius space with an uninformative prior. The values ineach cell are the occurrence of planet candidates in that cell and its uncertainty. All values are expressed as percents. Cellswith heavy borders have no detected planet candidates. Left: small planet candidate occurrence in our metal-poor subsample.Right: small planet candidate occurrence in our metal-rich subsample. Planet candidates are significantly more common in themetal-rich subsample than in the metal-poor subsample. The product of our samples’ sizes and Kepler DR25’s completenessindicate that the amount of information implicit in a non-detection is at least a factor of five larger than the signal weaklyimplied by our prior for cells with P (cid:46)
100 days and R p (cid:38) R ⊕ . Table 4.
Occurrence of Small Planet Candidates in the Kepler Field with Late-type Dwarf Primaries as a Function ofMetallicityPlanet Radius Period Occurrence [M / H] Description PC Detection Flag Completeness Equivalent Numberof Stars Searched( R ⊕ ) (days) (%) (%)0.5-1.0 0.3-0.5 0 . +0 . − . MP 0 16.41 2260.5-1.0 0.5-1.0 0 . +0 . − . MP 0 10.22 1410.5-1.0 1.0-1.9 1 . +1 . − . MP 1 6.21 850.5-1.0 1.9-3.8 6 . +3 . − . MP 1 3.64 500.5-1.0 3.8-7.3 13 . +5 . − . MP 1 1.97 270.5-1.0 7.3-14.3 9 . +7 . − . MP 1 0.99 140.5-1.0 14.3-27.9 7 . +10 . − . MP 1 0.46 60.5-1.0 27.9-54.5 18 . +19 . − . MP 1 0.20 30.5-1.0 54.5-106.2 27 . +27 . − . MP 1 0.08 10.5-1.0 106.2-207.2 38 . +32 . − . MP 0 0.03 0
Note —In the column “[M/H] Description” the strings “MP”, “MR”, and “All”, correspond to our metal-poor, metal-rich,and complete samples. This table is published in its entirety in the machine-readable format. calculated for the metal-poor subsample to calculate astatistic we define as the occurrence “factor of enhance-ment”. We repeat this process 10,000 times to fullysample the factor of enhancement distribution. We re-port the median and the 16th and 84th percentiles ofthe factor of enhancement distribution in Table 5 andpresent it visually in Figure 6.For planet candidates over the complete range inplanet radius we study 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ , the occur- rence of planet candidates in the metal-rich subsampleis a factor of 1 . +0 . − . higher than in the metal-poor sub-sample. For H/He envelope planet candidates (2 R ⊕ (cid:46) R p (cid:46) R ⊕ ), the occurrence of planet candidates in themetal-rich subsample is a factor of 1 . ± . / H] between our metal-rich and metal-poorsubsamples is about 0.3 dex or a factor of two in Z (cid:63) , sothe occurrence of planets with R p (cid:38) R ⊕ grows roughly mall Planet Occurrence Increases with Metallicity for Late-type Dwarfs Table 5.
Relative Planet Candidate Occurrence Statistics Observed Between the Metal-richand Metal-poor SubsamplesCategory Occurrence Difference Factor of Enhancement(%)Rocky (0 . R ⊕ (cid:46) R p (cid:46) R ⊕ ) 80 +62 − . +0 . − . H/He Envelope (2 R ⊕ (cid:46) R p (cid:46) R ⊕ ) 108 +38 − . +0 . − . All (0 . R ⊕ (cid:46) R p (cid:46) R ⊕ ) 188 +72 − . +0 . − . Period [Days] P l a n e t R a d i u s [ R ] ( +0.2 )( -0.1 ) ( +0.3 )( -0.2 ) ( +0.2 )( -0.1 ) ( +0.2 )( -0.1 ) ( +0.2 )( -0.1 ) ( +0.2 )( -0.1 ) ( +0.2 )( -0.1 ) ( +0.2 )( -0.1 ) ( +0.2 )( -0.1 ) ( +0.6 )( -0.4 ) ( +0.5 )( -0.4 ) ( +0.3 )( -0.2 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +0.3 )( -0.1 ) ( +1.0 )( -0.7 ) ( +1.0 )( -0.8 ) ( +0.5 )( -0.3 ) ( +0.6 )( -0.3 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +0.4 )( -0.2 ) ( +2.5 )( -2.0 ) ( +1.7 )( -1.3 ) ( +1.4 )( -1.1 ) ( +0.9 )( -0.5 ) ( +0.6 )( -0.3 ) ( +0.6 )( -0.3 ) ( +0.6 )( -0.3 ) ( +0.7 )( -0.3 ) ( +0.7 )( -0.4 ) ( +4.4 )( -3.7 ) ( +3.1 )( -2.6 ) ( +2.5 )( -2.1 ) ( +2.0 )( -1.5 ) ( +0.9 )( -0.4 ) ( +0.9 )( -0.4 ) ( +0.9 )( -0.4 ) ( +0.9 )( -0.4 ) ( +0.9 )( -0.4 ) ( +6.1 )( -4.7 ) ( +3.7 )( -3.0 ) ( +3.9 )( -3.4 ) ( +3.6 )( -3.0 ) ( +2.3 )( -1.5 ) ( +1.5 )( -0.8 ) ( +1.3 )( -0.6 ) ( +1.3 )( -0.6 ) ( +1.3 )( -0.6 ) ( +10.2 )( -7.8 ) ( +5.3 )( -4.2 ) ( +5.4 )( -4.6 ) ( +4.7 )( -3.8 ) ( +3.5 )( -2.5 ) ( +2.6 )( -1.6 ) ( +2.1 )( -1.1 ) ( +2.0 )( -0.9 ) ( +2.0 )( -0.9 ) ( +15.9 )( -11.4 ) ( +7.8 )( -6.3 ) ( +7.0 )( -5.9 ) ( +5.3 )( -3.9 ) ( +3.0 )( -1.4 ) ( +2.9 )( -1.3 ) ( +2.9 )( -1.3 ) ( +2.9 )( -1.3 ) ( +2.9 )( -1.3 ) ( +22.9 )( -14.0 ) ( +10.7 )( -8.2 ) ( +8.9 )( -7.1 ) ( +5.9 )( -3.7 ) ( +4.3 )( -2.0 ) ( +5.2 )( -3.0 ) ( +5.0 )( -2.8 ) ( +4.2 )( -2.0 ) ( +4.2 )( -2.0 ) ( +31.2 )( -22.5 ) ( +14.5 )( -10.1 ) ( +7.3 )( -3.5 ) ( +6.5 )( -3.1 ) ( +6.2 )( -3.0 ) ( +6.1 )( -2.9 ) ( +6.1 )( -2.9 ) ( +6.0 )( -2.9 ) ( +6.0 )( -2.9 ) O cc u rr e n c e [ % ] Figure 5.
Planet candidate occurrence as a function of pe-riod and planet radius in our complete sample. The valuesin each cell are the occurrence of planet candidates in thatcell and its uncertainty. All values are expressed as percents.Cells with heavy borders have no detected planet candidates.The product of our samples’ sizes and Kepler DR25’s com-pleteness indicate that the amount of information implicit ina non-detection is at least an order-of-magnitude larger thanthe signal weakly implied by our prior for cells with P (cid:46) R p (cid:38) R ⊕ . linearly with Z (cid:63) in our sample. We also note that noneof the planet candidates with 3 R ⊕ (cid:46) R p (cid:46) R ⊕ inour complete sample were found in the metal-poor sub-sample. We therefore conclude that for late-type dwarfstars metallicity is an important parameter in planetoccurrence calculations that should not be neglected forplanets with R p (cid:38) R ⊕ . Studies of small planet occur-rence around late-type dwarfs using K2 or TESS datashould therefore be sure to control for metallicity.For rocky planet candidates with 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ , the occurrence of planet candidates in the metal-rich subsample is a factor of 1 . +0 . − . higher than in themetal-poor subsample. Given the 1- σ significance of thisobservation, we cannot confirm or reject a relationshipbetween host star metallicity and planet candidate oc-currence.To compare with previous estimates of the occur-rence of small planets around late-type dwarfs in the Kepler field, we calculated the occurrence of planetcandidates in the ranges 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ and1 R ⊕ (cid:46) R p (cid:46) R ⊕ with orbital period P <
207 daysin our complete sample. We find that in these radiusranges a late-type dwarf in our complete sample hosts3 ± . . +0 . − . planets respectively. These results areconsistent with those reported in other studies of small-planet occurrence in the late-type dwarf stellar popula-tion in the Kepler field (e.g., Dressing & Charbonneau2015; Hsu et al. 2020). This consistency supports theaccuracy of our occurrence calculations.3.3. Formation Efficiency of Small Planets
The fraction of planet-making material present in aprotoplanetary disk during the epoch of planet forma-tion that ends up sequestered in planets can be thoughtof as that disk’s planet formation efficiency. As we de-scribed in Section 1, pebble accretion is expected to beinefficient with planet formation efficiencies below 10%.On the other hand, the apparent planet formation effi-ciency in the solar system was much higher. We there-fore estimate the planet formation efficiency in the Ke-pler field’s late-type dwarf stellar population in an at-tempt to observationally constrain the planet formationprocess.To estimate planet formation efficiency, we need boththe expectation value for the mass in planets today aswell as the total amount of planet-making material thatwas available in the young disk. To calculate the for-mer, we use the small-planet occurrence we estimatedabove combined with the mass–radius relation presentedin Ning et al. (2018) and implemented in the
MRExo package (Kanodia et al. 2019). We note that the Ninget al. (2018) mass–radius relation does not distinguishbetween a planet’s mass in metals and its mass in hy-drogen and helium. Since the masses of planets smallerthan Neptune are dominated by their metal mass, thisshould only bias our results by about 10% (e.g., Podolaket al. 2019, Schlaufman & Halpern 2020 submitted). Weuse a Monte Carlo simulation in which we sample theoccurrence in each cell of the maps presented in Fig-0
Lu et al. (2020) . - - . - Planet Size [ R ]2001000100200300400 P l a n e t O cc u rr e n c e D i ff e r e n c e ( M R - M P )[ % ] Rocky H/HeEnvelope All . - - . - Planet Size [ R ]01234 F a c t o r o f E nh a n c e m e n t Rocky H/HeEnvelope All
Figure 6.
Violin plots with differences in planet candidate occurrence between metal-rich and metal-poor subsamples as afunction of planet radius. The dark blue bars denote the 16th and 84th percentiles (i.e., the 1- σ region) while the light bluebars represents the 0.13th and 99.7th percentiles (i.e., the 3- σ region). Left: planet occurrence difference as a function of planetradius. Right: factor of enhancement as a function of planet radius. The occurrence of planet candidates is significantly higherin our metal-rich subsample both for the entire range of radii we study 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ and for H/He envelope planetswith 2 R ⊕ (cid:46) R p (cid:46) R ⊕ . Our results are inconclusive for rocky planets with 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ . Table 6.
Expected Mass in Planets as aFunction of MetallicitySample Expected Mass in Planets( M ⊕ )Metal-poor 16 . +0 . − . Metal-rich 24 . +0 . − . Complete 13 . +0 . − . ures 4 and 5 from the occurrence posterior in each cell.We next multiply that occurrence by the mass predictedby the Ning et al. (2018) mass–radius relation at the ra-dius of the cell’s midpoint. We then sum the product ofoccurrence and mass for each cell over an entire occur-rence map (excluding the longest-period cells becauseof their sub-percent completeness levels). We save theresulting estimate of the expectation value for the totalmass in planets and repeat the process 10,000 times. Weperform a similar simulation for the complete sample aswell as the metal-rich and metal-poor subsamples. Wereport the expected mass in planets for all three samplesin Table 6.To calculate the total amount of planet-making ma-terial that was available in the protoplanetary disksonce present around the late-type dwarfs in the Ke-pler field, we use the same back-of-the-envelope calcu- lation described in Section 1. We assume Z disk = Z (cid:63) =(0 . , . , . M disk /M (cid:63) = 0 .
01 for an early M dwarfwith M (cid:63) = 0 . M (cid:12) . We therefore estimate the amountof planet-making material available in the protoplane-tary disks around the stars in our metal-poor, complete,and metal-rich samples as 14 M ⊕ , 20 M ⊕ , and 28 M ⊕ .We find planet formation efficiencies in excess of 50%.The implication is that either planet formation is veryefficient or that the small planet candidates observedaround the Kepler field’s late-type dwarf stellar popu-lation formed in disks more massive than the averagedisks observed by Andrews et al. (2013). This could bebecause their parent protoplanetary disks were prefer-entially drawn from the high-mass side of the Andrewset al. (2013) distribution or because these planet can-didates formed in younger and therefore more massivedisks than those observed by Andrews et al. (2013). Wealso ignore the uncertainty in the Ning et al. (2018)mass–radius relation. Nevertheless, our observation’spreference for massive disks is similar to that suggestedin the minimum-mass extrasolar nebula scenario pro-posed by Chiang & Laughlin (2013) and expanded byDai et al. (2020). It is important to note that our esti-mated planet formation efficiency is limited to planetsfalling within our occurrence maps, or R p (cid:46) R ⊕ and P (cid:46)
200 days. mall Planet Occurrence Increases with Metallicity for Late-type Dwarfs DISCUSSIONThe occurrence of small planet candidates in the unionof our metal-poor and metal-rich subsamples is consis-tent with the results of previous studies. Dressing &Charbonneau (2015) found that M dwarfs host on aver-age 2 . ± . R ⊕ (cid:46) R p (cid:46) R ⊕ and orbital period P <
200 days. Our planet candi-date occurrence of 3 . ± . . R ⊕ (cid:46) R p (cid:46) R ⊕ and orbital period0 . < P <
256 days. Their planet occurrenceranges from 4 . +0 . − . to 8 . +1 . − . planets per M dwarf de-pending on the choice of prior. Our estimated occur-rence is 4 . +0 . − . for planets with radii 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ and orbital period 0 . < P <
207 days. Ourresults are consistent with those of Hsu et al. (2020),though we use a slightly smaller maximum period dueto the large and uncertain completeness corrections re-quired for P (cid:38)
200 days.We find significant increases in planet occurrence withmetallicity over both the entire range of planet radii westudy (0 . R ⊕ (cid:46) R p (cid:46) R ⊕ ) and over the range of radiiindicative of planets with significant H/He envelopes(2 R ⊕ (cid:46) R p (cid:46) R ⊕ ). We find period-averaged oc-currences in the metal-rich samples higher than the oc-currences observed in the metal-poor samples by a factor1 . +0 . − . for 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ and a factor of 1 . ± . R ⊕ (cid:46) R p (cid:46) R ⊕ . These factor-of-two enhance-ments are significant at more than the 2- σ level. Sincethe average photometric metallicities of the metal-richand metal-poor subsamples differ by about 0.3 dex in[M / H] (or a factor of two in Z (cid:63) ), the occurrence of smallplanets overall and gas-rich planets specifically scaleslinearly with metallicity. This linear scaling applies atleast in the thin disk metallicity range probed by Keplerduring its prime mission ( − . (cid:46) [M / H] (cid:46) +0 . . R ⊕ (cid:46) R p (cid:46) R ⊕ . We find a period-averaged oc-currence in the metal-rich sample higher than the oc-currence observed in the metal-poor sample by a factor1 . +0 . − . . This hint of an enhancement is only significantat the 1- σ level. We are therefore unable to confirm arelationship between metallicity and planet occurrencefor rocky planets. The lack of a statistically significant relationship be-tween metallicity and occurrence in the range 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ could be due to Kepler’s low completenessfor small planets. It could also be the case that thereis no relationship between metallicity and planet occur-rence for the smallest planets. We assert that the formeris the best explanation. Since the relationship betweenplanet occurrence and metallicity is set during the eraof planet formation, the subsequent atmospheric evo-lution of a planetary system cannot alter the relation.If planets with 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ are the leftovercores of larger planets that were stripped of their H/Heenvelopes, then the dependence of occurrence on metal-licity should be the same for both rocky and gas-richplanets. In other words, the lack of a relationship be-tween metallicity and occurrence for the smallest planetsthat cannot be attributed to low completeness would re-quire that the small planets observed by Kepler aroundlate-type dwarfs formed like terrestrial planets withoutsignificant H/He envelopes. We argue that a more pre-cise quantification of the relationship between metallic-ity and small planet occurrence should be a priority forK2 and TESS planet occurrence studies.We confirm the reality of the relation between metal-licity and small-planet occurrence for late-type dwarfstars first noted by Schlaufman & Laughlin (2010, 2011).Our use of a vetted photometric metallicity relationand removal of all giant stars from our non-planet-candidate-host sample using both Kepler asteroseismol-ogy and Gaia DR2 parallaxes answers the criticisms ofthe Schlaufman & Laughlin (2011) result made by Mannet al. (2012). In accord with Schlaufman & Laughlin(2010), we find that the average metallicity of late-typedwarfs in the Kepler field is [M / H] ≈ − . − . (cid:46) [M / H] (cid:46) +0 . Lu et al. (2020) reaffirmed by analyses making use of large samples ofspectroscopic stellar parameters for Kepler-field starsbased on low-resolution optical spectra from the LargeSky Area Multi-Object Fibre Spectroscopic Telescope(LAMOST) and its massive sky survey (e.g., Zhu et al.2016; Dong et al. 2018).None of the studies listed in the paragraph above havetaken into account Kepler’s completeness and thereforecould not fully explore the connection between host starmetallicity and small planet occurrence. Petigura et al.(2018) was the first to account for completeness andfound for solar-type host stars that the occurrence ofplanets with 1 . R ⊕ (cid:46) R p (cid:46) R ⊕ doubles as stellarmetallicity increases over the range − . (cid:46) [M / H] (cid:46) +0 . Z (cid:63) ). We find a factor of twochange the occurrence of 2 R ⊕ (cid:46) R p (cid:46) R ⊕ planetsover a smaller range of metallicity − . (cid:46) [M / H] (cid:46) +0 . Z (cid:63) ). Our study therefore verifiesthe theoretical expectation that the connection betweenhost star metallicity and small-planet occurrence shouldbe stronger for late-type dwarfs than for solar-type stars.With the connection between small-planet occurrenceand late-type dwarf host star metallicity now firmly es-tablished, it is possible to predict the occurrence andproperties of small planets around late-type dwarf starsas a function of stellar metallicity and mass. Assum-ing a planet-formation efficiency of 50% and that theamount of planet-making material available in a proto-planetary disk with M disk /M (cid:63) = 0 .
01 scales with stellarmass and metallicity, during the epoch of planet for-mation there will be less than 9 M ⊕ of planet-makingmaterial in the disk around a M (cid:63) ≈ . M (cid:12) early-typeM dwarf with [M / H] (cid:46) − .
5. This meager amount ofplanet-making material is barely sufficient to make evenan Earth-composition 1 . R ⊕ planet (e.g., Zeng et al.2019). Using the same assumptions for a late-type Mdwarf like 2MASS J23062928-0502285 (TRAPPIST-1)with M (cid:63) ≈ . M (cid:12) and [M / H] ≈
0, there will beabout 4 M ⊕ of planet-making material available. As-suming an Earth-like composition for the seven knownTRAPPIST-1 planets implies a total mass of about7 M ⊕ (Gillon et al. 2016, 2017). We therefore predictthat TRAPPIST-1 is metal-rich and/or that its plane-tary system formed early in a massive protoplanetarydisk. In either case, TRAPPIST-1 like systems shouldbe very uncommon in future planet occurrence studiesof late-type M dwarfs like Sestovic & Demory (2020).For early M dwarfs in the Kepler field, we estimatethat more than 50% of the planet-making materialinitially present in their protoplanetary disks was se-questered in planets. Even if we assume disks an order ofmagnitude more massive than the typical disk observed by Andrews et al. (2013), this is still larger than theexpected (cid:46)
10% of planet-making material locked up inplanets as a result of pebble accretion. While both ourinability to differentiate between solid and gas massesfor the small planets in our sample and our exclusionof giant planets may bias our planet formation efficien-cies, we argue that these effects are small. Neptune-sizeor smaller planets have less than 10% of their mass inH/He envelope, while giant planets occur around onlya few percent of early M dwarfs (e.g., Podolak et al.2019; Johnson et al. 2010). We therefore argue that thehigh planet formation efficiencies observed by Dai et al.(2020) and ourselves hint at planetesimal accretion asthe main formation channel for the small planets aroundearly M dwarfs in the Kepler field. While our planet for-mation efficiency calculation has large uncertainties andmay be systematically biased, we hope that future anal-yses of the occurrence of small planets around low-massstars may be able to improve the estimation of planetformation efficiencies and thereby more confidently dif-ferentiate between pebble and planetesimal accretion.Even though our logistic regression analysis cannotaccount for the important issues of completeness andmultiple-planet systems, it still provides an estimate ofthe strength of the small-planet occurrence–host starmetallicity relation that is consistent with our more care-ful occurrence analysis. Specifically, our logistic regres-sion analysis indicates that a 0.3 dex increase in [M / H]increases planet occurrence in the range 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ by a factor of 1 . ± .
3. The more robust occur-rence calculation accounting for completeness and mul-tiple planet systems suggests a factor of 1 . +0 . − . increasefor the same change in metallicity. These two estimatesare consistent at the 1- σ level. The reason for this agree-ment is that the assumptions of the logistic regressionanalysis are reasonable in regions of parameter spacewhere planet occurrence is low. In other words, a lo-gistic regression analysis is an easy way to explore thedependence of planet occurrence on other system param-eters at short orbital periods and/or at relatively largeplanet masses or sizes where planets are intrinsically un-common (see Figure 4). As most planets discovered byK2 and TESS are on short-period orbits because of theirlimited observation durations, we suggest that a logisticregression analysis could easily be used to explore thedependence of planet occurrence on metallicity or othersystem parameters among K2 or TESS discoveries evenwithout accounting for completeness. CONCLUSIONWe find that the occurrence of small planets aroundearly M dwarfs in the Kepler field increases linearly mall Planet Occurrence Increases with Metallicity for Late-type Dwarfs Z (cid:63) for planets with H/He en-velopes in the radius range 2 R ⊕ (cid:46) R p (cid:46) R ⊕ and − . (cid:46) [M / H] (cid:46) +0 .
0. We are unable to confirm or re-ject a relationship between planet occurrence and hoststar metallicity for rocky planets with 0 . R ⊕ (cid:46) R p (cid:46) R ⊕ . Similar analyses have shown an analogous butweaker increase in planet occurrence with metallicity forsolar-type stars in a similar range of host star metallic-ity and period. These observations confirm the theoret-ical expectation that the small-planet occurrence–hoststar metallicity relation should be stronger for low-massstars. Our results provide a hint that planetesimal ac-cretion should be preferred to pebble accretion as thedriving process for the formation of 2 R ⊕ (cid:46) R p (cid:46) R ⊕ planets around early M dwarfs in the Kepler field. Wepredict that even rocky planets with R p (cid:38) . R ⊕ or R p (cid:38) . R ⊕ should be rare around early M dwarfs with[M / H] (cid:46) − . / H] (cid:46) +0 .
0. Weargue that future small planet occurrence calculationsfor M dwarfs targeted by K2 and/or TESS should con-trol for metallicity. ACKNOWLEDGMENTSWe thank Hsiang-Chih Hwang, Bin Ren, Josh Winn,and Winston Wu for useful discussions. This paper in-cludes data collected by the Kepler mission. Fundingfor the Kepler mission is provided by the NASA Sci-ence Mission directorate. This research has made use ofthe NASA Exoplanet Archive, which is operated by theCalifornia Institute of Technology, under contract withthe National Aeronautics and Space Administration un-der the Exoplanet Exploration Program. Some/all ofthe data presented in this paper were obtained from theMikulski Archive for Space Telescopes (MAST). STScI isoperated by the Association of Universities for Researchin Astronomy, Inc., under NASA contract NAS5-26555.This research has made use of the NASA/IPAC In-frared Science Archive, which is funded by the NationalAeronautics and Space Administration and operated bythe California Institute of Technology. This publicationmakes use of data products from the Wide-field InfraredSurvey Explorer, which is a joint project of the Uni-versity of California, Los Angeles, and the Jet Propul-sion Laboratory/California Institute of Technology, andNEOWISE, which is a project of the Jet PropulsionLaboratory/California Institute of Technology. WISEand NEOWISE are funded by the National Aeronau-tics and Space Administration. This work has madeuse of data from the European Space Agency (ESA)mission
Gaia
Gaia
Gaia
MultilateralAgreement. This research has made use of NASA’s As-trophysics Data System. This research has made use ofthe SIMBAD database, operated at CDS, Strasbourg,France (Wenger et al. 2000). This research has madeuse of the VizieR catalogue access tool, CDS, Stras-bourg, France (DOI: 10.26093/cds/vizier). The originaldescription of the VizieR service was published in 2000,A&AS 143, 23 (Ochsenbein et al. 2000). This projectwas developed in part at the 2018 NYC Gaia Sprint,hosted by the Center for Computational Astrophysicsof the Flatiron Institute in New York City, New York.
Facilities:
Exoplanet Archive, Gaia, Kepler, IRSA,MAST, NEOWISE, WISE
Software:
Astropy (Astropy Collaboration et al.2013, 2018),
KeplerPORTS (Burke & Catanzarite 2017),
MRExo (Kanodia et al. 2019), pandas (McKinney et al.4
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