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Fine Guidance Sensor Data

The Kepler and K2 missions collected Fine Guidance Sensor (FGS) data in addition to the science data, as discussed in the Kepler Instrument Handbook (KIH, Van Cleve and Caldwell 2016). The FGS CCDs areframe transfer devices located in the corners of the Kepler focal plane, which are read out 10 times every second. The FGS data are being made available to the user community for scientific analysis as flux and centroid time series, along with a limited number of FGS full frame images which may be useful for constructing a World Coordinate System (WCS) or otherwise putting the time series data in context. This document will describe the data content and file format, and give example MATLAB scripts to read the time series.

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A Bayesian analysis of HAT-P-7b using the EXONEST algorithm

The study of exoplanets (planets orbiting other stars) is revolutionizing the way we view our universe. High-precision photometric data provided by the Kepler Space Telescope (Kepler) enables not only the detection of such planets, but also their characterization. This presents a unique opportunity to apply Bayesian methods to better characterize the multitude of previously confirmed exoplanets. This paper focuses on applying the EXONEST algorithm to characterize the transiting short-period-hot-Jupiter, HAT-P-7b (also referred to as Kepler-2b). EXONEST evaluates a suite of exoplanet photometric models by applying Bayesian Model Selection, which is implemented with the MultiNest algorithm. These models take into account planetary effects, such as reflected light and thermal emissions, as well as the effect of the planetary motion on the host star, such as Doppler beaming, or boosting, of light from the reflex motion of the host star, and photometric variations due to the planet-induced ellipsoidal shape of the host star. By calculating model evidences, one can determine which model best describes the observed data, thus identifying which effects dominate the planetary system. Presented are parameter estimates and model evidences for HAT-P-7b.

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Overstable librations can account for the paucity of mean motion resonances among exoplanet pairs

We assess the multi-planet systems discovered by the Kepler satellite in terms of current ideas about orbital migration and eccentricity damping due to planet-disk interactions. Our primary focus is on first order mean motion resonances, which we investigate analytically to lowest order in eccentricity. Only a few percent of planet pairs are in close proximity to a resonance. However, predicted migration rates (parameterized by τ{sub n}=n/| n-dot |) imply that during convergent migration most planets would have been captured into first order resonances. Eccentricity damping (parameterized by τ{sub e}=e/| e-dot |) offers a plausible resolution. Estimates suggest τ {sub e}/τ {sub n} ∼ (h/a){sup 2} ∼ 10{sup –2}, where h/a is the ratio of disk thickness to radius. Together, eccentricity damping and orbital migration give rise to an equilibrium eccentricity, e {sub eq} ∼ (τ {sub e}/τ {sub n}){sup 1/2}. Capture is permanent provided e {sub eq} ≲ μ{sup 1/3}, where μ denotes the planet to star mass ratio. But for e {sub eq} ≳ μ{sup 1/3}, capture is only temporary because librations around equilibrium are overstable and lead to passage through resonance on timescale τ {sub e}. Most Kepler planet pairs have e {sub eq} > μ{sup 1/3}. Since τ {sub n} >> τ {sub e} is the timescale for migration between neighboring resonances, only a modest percentage of pairs end up trapped in resonances after the disk disappears. Thus the paucity of resonances among Kepler pairs should not be taken as evidence for in situ planet formation or the disruptive effects of disk turbulence. Planet pairs close to a mean motion resonance typically exhibit period ratios 1%-2% larger than those for exact resonance. The direction of this shift undoubtedly reflects the same asymmetry that requires convergent migration for resonance capture. Permanent resonance capture at these separations from exact resonance would demand μ(τ {sub n}/τ {sub e}){sup 1/2} ≳ 0.01, a value that estimates of μ from transit data and (τ {sub e}/τ {sub n}){sup 1/2} from theory are insufficient to match. Plausible alternatives involve eccentricity damping during or after disk dispersal. The overstability referred to above has applications beyond those considered in this investigation. It was discovered numerically by Meyer and Wisdom in their study of the tidal evolution of Saturn's satellites.

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The eclipsing system V404 Lyr: Light-travel times and γ Doradus pulsations

We present the physical properties of V404 Lyr exhibiting eclipse timing variations and multiperiodic pulsations from all historical data including the Kepler and SuperWASP observations. Detailed analyses of 2922 minimum epochs showed that the orbital period has varied through a combination of an upward-opening parabola and two sinusoidal variations, with periods of P {sub 3} = 649 days and P {sub 4} = 2154 days and semi-amplitudes of K {sub 3} = 193 s and K {sub 4} = 49 s, respectively. The secular period increase at a rate of +1.41 × 10{sup –7} days yr{sup –1} could be interpreted as a combination of the secondary to primary mass transfer and angular momentum loss. The most reasonable explanation for both sinusoids is a pair of light-travel-time effects due to two circumbinary objects with projected masses of M {sub 3} = 0.47 M {sub ☉} and M {sub 4} = 0.047 M {sub ☉}. The third-body parameters are consistent with those calculated using the Wilson-Devinney binary code. For the orbital inclinations i {sub 4} ≳ 43°, the fourth component has a mass within the hydrogen-burning limit of ∼0.07 M {sub ☉}, which implies that it is a brown dwarf. A satisfactory model for the Kepler light curves was obtained by applying a cool spot to the secondary component. The results demonstrate that the close eclipsing pair is in a semi-detached, but near-contact, configuration; the primary fills approximately 93% of its limiting lobe and is larger than the lobe-filling secondary. Multiple frequency analyses were applied to the light residuals after subtracting the synthetic eclipsing curve from the Kepler data. This revealed that the primary component of V404 Lyr is a γ Dor type pulsating star, exhibiting seven pulsation frequencies in the range of 1.85-2.11 day{sup –1} with amplitudes of 1.38-5.72 mmag and pulsation constants of 0.24-0.27 days. The seven frequencies were clearly identified as high-order low-degree gravity-mode oscillations which might be excited through tidal interaction. Only eight eclipsing binaries have been known to contain γ Dor pulsating components and, therefore, V404 Lyr will be an important test bed for investigating these rare and interesting objects.

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CALCULATING ASTEROSEISMIC DIAGRAMS FOR SOLAR-LIKE OSCILLATIONS

With the success of the Kepler and CoRoT missions, the number of stars with detected solar-like oscillations has increased by several orders of magnitude; for the first time we are able to perform large-scale ensemble asteroseismology of these stars. In preparation for this golden age of asteroseismology we have computed expected values of various asteroseismic observables from models of varying mass and metallicity. The relationships between these asteroseismic observables, such as the separations between mode frequencies, are able to significantly constrain estimates of the ages and masses of these stars. We investigate the scaling relation between the large frequency separation, {Delta}{nu}, and mean stellar density. Furthermore we present model evolutionary tracks for several asteroseismic diagrams. We have extended the so-called C-D diagram beyond the main sequence to the subgiants and the red giant branch. We also consider another asteroseismic diagram, the {epsilon} diagram, which is more sensitive to variations in stellar properties at the subgiant stages and can aid in determining the correct mode identification. The recent discovery of gravity-mode period spacings in red giants forms the basis for a third asteroseismic diagram. We compare the evolutionary model tracks in these asteroseismic diagrams with results from pre-Kepler studies of solar-like oscillations and early results from Kepler.

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THE EVIL-MC MODEL FOR ELLIPSOIDAL VARIATIONS OF PLANET-HOSTING STARS AND APPLICATIONS TO THE HAT-P-7 SYSTEM

We present a new model for Ellipsoidal Variations Induced by a Low-Mass Companion, the EVIL-MC model. We employ several approximations appropriate for planetary systems to substantially increase the computational efficiency of our model relative to more general ellipsoidal variation models and improve upon the accuracy of simpler models. This new approach gives us a unique ability to rapidly and accurately determine planetary system parameters. We use the EVIL-MC model to analyze Kepler Quarter 0-2 (Q0-2) observations of the HAT-P-7 system, an F-type star orbited by a {approx} Jupiter-mass companion. Our analysis corroborates previous estimates of the planet-star mass ratio q = (1.10 {+-} 0.06) Multiplication-Sign 10{sup -3}, and we have revised the planet's dayside brightness temperature to 2680{sup +10}{sub -20} K. We also find a large difference between the day- and nightside planetary flux, with little nightside emission. Preliminary dynamical+radiative modeling of the atmosphere indicates that this result is qualitatively consistent with high altitude absorption of stellar heating. Similar analyses of Kepler and CoRoT photometry of other planets using EVIL-MC will play a key role in providing constraints on the properties of many extrasolar systems, especially given the limited resources for follow-up and characterization of these systems. However, as we highlight, there are important degeneracies between the contributions from ellipsoidal variations and planetary emission and reflection. Consequently, for many of the hottest and brightest Kepler and CoRoT planets, accurate estimates of the planetary emission and reflection, diagnostic of atmospheric heat budgets, will require accurate modeling of the photometric contribution from the stellar ellipsoidal variation.

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THE PHOTOECCENTRIC EFFECT AND PROTO-HOT JUPITERS. I. MEASURING PHOTOMETRIC ECCENTRICITIES OF INDIVIDUAL TRANSITING PLANETS

Exoplanet orbital eccentricities offer valuable clues about the history of planetary systems. Eccentric, Jupiter-sized planets are particularly interesting: they may link the 'cold' Jupiters beyond the ice line to close-in hot Jupiters, which are unlikely to have formed in situ. To date, eccentricities of individual transiting planets primarily come from radial-velocity measurements. Kepler has discovered hundreds of transiting Jupiters spanning a range of periods, but the faintness of the host stars precludes radial-velocity follow-up of most. Here, we demonstrate a Bayesian method of measuring an individual planet's eccentricity solely from its transit light curve using prior knowledge of its host star's density. We show that eccentric Jupiters are readily identified by their short ingress/egress/total transit durations-part of the 'photoeccentric' light curve signature of a planet's eccentricity-even with long-cadence Kepler photometry and loosely constrained stellar parameters. A Markov Chain Monte Carlo exploration of parameter posteriors naturally marginalizes over the periapse angle and automatically accounts for the transit probability. To demonstrate, we use three published transit light curves of HD 17156 b to measure an eccentricity of e = 0.71{sup +0.16}{sub -0.09}, in good agreement with the discovery value e = 0.67 {+-} 0.08 based on 33 radial-velocity measurements. We present two additional tests using Kepler data. In each case, the technique proves to be a viable method of measuring exoplanet eccentricities and their confidence intervals. Finally, we argue that this method is the most efficient, effective means of identifying the extremely eccentric, proto-hot Jupiters predicted by Socrates et al.

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DECOUPLING PHASE VARIATIONS IN MULTI-PLANET SYSTEMS

Due to the exquisite photometric precision, transiting exoplanet discoveries from the Kepler mission are enabling several new techniques of confirmation and characterization. One of these newly accessible techniques analyzes the phase variations of planets as they orbit their stars. The predicted phase variation for multi-planet systems can become rapidly complicated and depends upon the period, radius, and albedo distributions for planets in the system. Here we describe the confusion that may occur due to short-period terrestrial planets and/or non-transiting planets in a system, which can add high-frequency correlated noise or low-frequency trends to the data stream. We describe these sources of ambiguity with several examples, including that of our solar system. We further show how decoupling of these signals may be achieved with application to the Kepler-20 and Kepler-33 multi-planet systems.

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EMERGING TRENDS IN A PERIOD-RADIUS DISTRIBUTION OF CLOSE-IN PLANETS

We analyze the distribution of extrasolar planets (both confirmed and Kepler candidates) according to their orbital periods P and planetary radii R. Among confirmed planets, we find compelling evidence for a paucity of bodies with 3 R {sub Circled-Plus} < R < 10 R {sub Circled-Plus }, where R {sub Circled-Plus} is Earth's radius and P < 2-3 days. We have christened this region a sub-Jovian Pampas. The same trend is detected in multiplanet Kepler candidates. Although approximately 16 Kepler single-planet candidates inhabit this Pampas, at least 7 are probable false positives (FPs). This last number could be significantly higher if the ratio of FPs is higher than 10%, as suggested by recent studies. In a second part of the paper we analyze the distribution of planets in the (P, R) plane according to stellar metallicities. We find two interesting trends: (1) a lack of small planets (R < 4 R {sub Circled-Plus }) with orbital periods P < 5 days in metal-poor stars and (2) a paucity of sub-Jovian planets (4 R {sub Circled-Plus} < R < 8 R {sub Circled-Plus }) with P < 100 days, also around metal-poor stars. Although all these trends are preliminary, they appear statistically significant and deserve further scrutiny. If confirmed, they could represent important constraints on theories of planetary formation and dynamical evolution.

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THE OCCURRENCE RATE OF SMALL PLANETS AROUND SMALL STARS

We use the optical and near-infrared photometry from the Kepler Input Catalog to provide improved estimates of the stellar characteristics of the smallest stars in the Kepler target list. We find 3897 dwarfs with temperatures below 4000 K, including 64 planet candidate host stars orbited by 95 transiting planet candidates. We refit the transit events in the Kepler light curves for these planet candidates and combine the revised planet/star radius ratios with our improved stellar radii to revise the radii of the planet candidates orbiting the cool target stars. We then compare the number of observed planet candidates to the number of stars around which such planets could have been detected in order to estimate the planet occurrence rate around cool stars. We find that the occurrence rate of 0.5-4 R{sub Circled-Plus} planets with orbital periods shorter than 50 days is 0.90{sup +0.04}{sub -0.03} planets per star. The occurrence rate of Earth-size (0.5-1.4 R{sub Circled-Plus }) planets is constant across the temperature range of our sample at 0.51{sub -0.05}{sup +0.06} Earth-size planets per star, but the occurrence of 1.4-4 R{sub Circled-Plus} planets decreases significantly at cooler temperatures. Our sample includes two Earth-size planet candidates in the habitable zone, allowing us to estimate that the mean number of Earth-size planets in the habitable zone is 0.15{sup +0.13}{sub -0.06} planets per cool star. Our 95% confidence lower limit on the occurrence rate of Earth-size planets in the habitable zones of cool stars is 0.04 planets per star. With 95% confidence, the nearest transiting Earth-size planet in the habitable zone of a cool star is within 21 pc. Moreover, the nearest non-transiting planet in the habitable zone is within 5 pc with 95% confidence.

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PLANET HUNTERS: A TRANSITING CIRCUMBINARY PLANET IN A QUADRUPLE STAR SYSTEM

We report the discovery and confirmation of a transiting circumbinary planet (PH1b) around KIC 4862625, an eclipsing binary in the Kepler field. The planet was discovered by volunteers searching the first six Quarters of publicly available Kepler data as part of the Planet Hunters citizen science project. Transits of the planet across the larger and brighter of the eclipsing stars are detectable by visual inspection every {approx}137 days, with seven transits identified in Quarters 1-11. The physical and orbital parameters of both the host stars and planet were obtained via a photometric-dynamical model, simultaneously fitting both the measured radial velocities and the Kepler light curve of KIC 4862625. The 6.18 {+-} 0.17 R{sub Circled-Plus} planet orbits outside the 20 day orbit of an eclipsing binary consisting of an F dwarf (1.734 {+-} 0.044 R{sub Sun }, 1.528 {+-} 0.087 M{sub Sun }) and M dwarf (0.378 {+-} 0.023 R{sub Sun }, 0.408 {+-} 0.024 M{sub Sun }). For the planet, we find an upper mass limit of 169 M{sub Circled-Plus} (0.531 Jupiter masses) at the 99.7% confidence level. With a radius and mass less than that of Jupiter, PH1b is well within the planetary regime. Outside the planet's orbit, at {approx}1000 AU, a previously unknown visual binary has been identified that is likely bound to the planetary system, making this the first known case of a quadruple star system with a transiting planet.

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SPITZER OBSERVATIONS OF GJ 3470 b: A VERY LOW-DENSITY NEPTUNE-SIZE PLANET ORBITING A METAL-RICH M DWARF

We present Spitzer/IRAC 4.5 {mu}m transit photometry of GJ 3470 b, a Neptune-size planet orbiting an M1.5 dwarf star with a 3.3 day period recently discovered in the course of the HARPS M-dwarf survey. We refine the stellar parameters by employing purely empirical mass-luminosity and surface brightness relations constrained by our updated value for the mean stellar density, and additional information from new near-infrared spectroscopic observations. We derive a stellar mass of M{sub *}= 0.539{sup +0.047}{sub -0.043} M{sub sun} and a radius of R{sub *}= 0.568{sup +0.037}{sub -0.031} R{sub sun}. We determine the host star of GJ 3470 b to be metal-rich, with a metallicity of [Fe/H] = +0.20 {+-} 0.10 and an effective temperature of T{sub eff} = 3600 {+-} 100 K. The revised stellar parameters yield a planetary radius R{sub p}= 4.83{sub -0.21}{sup +0.22} R{sub Circled-Plus} that is 13% larger than the value previously reported in the literature. We find a planetary mass M{sub p}= 13.9{sup +1.5}{sub -1.4} M{sub Circled-Plus} that translates to a very low planetary density, {rho}{sub p}= 0.72{sup +0.13}{sub -0.12} g cm{sup -3}, which is 33% smaller than the original value. With a mean density half of that of GJ 436 b, GJ 3470 b is an example of a very low-density low-mass planet, similar to Kepler-11 d, Kepler-11 e, and Kepler-18 c, but orbiting a much brighter nearby star that is more conducive to follow-up studies.

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PLANETS NEAR MEAN-MOTION RESONANCES

The multiple-planet systems discovered by the Kepler mission exhibit the following feature: planet pairs near first-order mean-motion resonances prefer orbits just outside the nominal resonance, while avoiding those just inside the resonance. We explore an extremely simple dynamical model for planet formation, in which planets grow in mass at a prescribed rate without orbital migration or dissipation. We develop an analytic version of this model for two-planet systems in two limiting cases: the planet mass grows quickly or slowly relative to the characteristic resonant libration time. In both cases, the distribution of systems in period ratio develops a characteristic asymmetric peak-trough structure around the resonance, qualitatively similar to that observed in the Kepler sample. We verify this result with numerical integrations of the three-body problem. We show that for the 3 : 2 resonance, where the observed peak-trough structure is strongest, our simple model is consistent with the observations for a range of mean planet masses 20-100 M{sub Circled-Plus }. This predicted mass range is higher-by at least a factor of three-than the range expected from the few Kepler planets with measured masses, but part of this discrepancy could be due to oversimplifications in the dynamical model or uncertainties in the planetary mass-radius relation.

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AN UNDERSTANDING OF THE SHOULDER OF GIANTS: JOVIAN PLANETS AROUND LATE K DWARF STARS AND THE TREND WITH STELLAR MASS

Analyses of exoplanet statistics suggest a trend of giant planet occurrence with host star mass, a clue to how planets like Jupiter form. One missing piece of the puzzle is the occurrence around late K dwarf stars (masses of 0.5-0.75 M{sub Sun} and effective temperatures of 3900-4800 K). We analyzed four years of Doppler radial velocity (RVs) data for 110 late K dwarfs, one of which hosts two previously reported giant planets. We estimate that 4.0% {+-} 2.3% of these stars have Saturn-mass or larger planets with orbital periods <245 days, depending on the planet mass distribution and RV variability of stars without giant planets. We also estimate that 0.7% {+-} 0.5% of similar stars observed by Kepler have giant planets. This Kepler rate is significantly (99% confidence) lower than that derived from our Doppler survey, but the difference vanishes if only the single Doppler system (HIP 57274) with completely resolved orbits is considered. The difference could also be explained by the exclusion of close binaries (without giant planets) from the Doppler but not Kepler surveys, the effect of long-period companions and stellar noise on the Doppler data, or an intrinsic difference between the two populations. Our estimates for late K dwarfs bridge those for solar-type stars and M dwarfs, and support a positive trend with stellar mass. Small sample size precludes statements about finer structure, e.g., a ''shoulder'' in the distribution of giant planets with stellar mass. Future surveys such as the Next Generation Transit Survey and the Transiting Exoplanet Satellite Survey will ameliorate this deficiency.

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UNDERSTANDING TRENDS ASSOCIATED WITH CLOUDS IN IRRADIATED EXOPLANETS

Unlike previously explored relationships between the properties of hot Jovian atmospheres, the geometric albedo and the incident stellar flux do not exhibit a clear correlation, as revealed by our re-analysis of Q0-Q14 Kepler data. If the albedo is primarily associated with the presence of clouds in these irradiated atmospheres, a holistic modeling approach needs to relate the following properties: the strength of stellar irradiation (and hence the strength and depth of atmospheric circulation), the geometric albedo (which controls both the fraction of starlight absorbed and the pressure level at which it is predominantly absorbed), and the properties of the embedded cloud particles (which determine the albedo). The anticipated diversity in cloud properties renders any correlation between the geometric albedo and the stellar flux weak and characterized by considerable scatter. In the limit of vertically uniform populations of scatterers and absorbers, we use an analytical model and scaling relations to relate the temperature-pressure profile of an irradiated atmosphere and the photon deposition layer and to estimate whether a cloud particle will be lofted by atmospheric circulation. We derive an analytical formula for computing the albedo spectrum in terms of the cloud properties, which we compare to the measured albedo spectrum of HD 189733b by Evans et al. Furthermore, we show that whether an optical phase curve is flat or sinusoidal depends on whether the particles are small or large as defined by the Knudsen number. This may be an explanation for why Kepler-7b exhibits evidence for the longitudinal variation in abundance of condensates, while Kepler-12b shows no evidence for the presence of condensates despite the incident stellar flux being similar for both exoplanets. We include an 'observer's cookbook' for deciphering various scenarios associated with the optical phase curve, the peak offset of the infrared phase curve, and the geometric albedo.

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Detecting extrasolar moons akin to solar system satellites with an orbital sampling effect

Despite years of high accuracy observations, none of the available theoretical techniques has yet allowed the confirmation of a moon beyond the solar system. Methods are currently limited to masses about an order of magnitude higher than the mass of any moon in the solar system. I here present a new method sensitive to exomoons similar to the known moons. Due to the projection of transiting exomoon orbits onto the celestial plane, satellites appear more often at larger separations from their planet. After about a dozen randomly sampled observations, a photometric orbital sampling effect (OSE) starts to appear in the phase-folded transit light curve, indicative of the moons' radii and planetary distances. Two additional outcomes of the OSE emerge in the planet's transit timing variations (TTV-OSE) and transit duration variations (TDV-OSE), both of which permit measurements of a moon's mass. The OSE is the first effect that permits characterization of multi-satellite systems. I derive and apply analytical OSE descriptions to simulated transit observations of the Kepler space telescope assuming white noise only. Moons as small as Ganymede may be detectable in the available data, with M stars being their most promising hosts. Exomoons with the ten-fold mass of Ganymede and a similar composition (about 0.86 Earth radii in radius) can most likely be found in the available Kepler data of K stars, including moons in the stellar habitable zone. A future survey with Kepler-class photometry, such as Plato 2.0, and a permanent monitoring of a single field of view over five years or more will very likely discover extrasolar moons via their OSEs.

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A STELLAR-MASS-DEPENDENT DROP IN PLANET OCCURRENCE RATES

The Kepler spacecraft has discovered a large number of planets with up to one-year periods and down to terrestrial sizes. While the majority of the target stars are main-sequence dwarfs of spectral type F, G, and K, Kepler covers stars with effective temperatures as low as 2500 K, which corresponds to M stars. These cooler stars allow characterization of small planets near the habitable zone, yet it is not clear if this population is representative of that around FGK stars. In this paper, we calculate the occurrence of planets around stars of different spectral types as a function of planet radius and distance from the star and show that they are significantly different from each other. We further identify two trends. First, the occurrence of Earth- to Neptune-sized planets (1-4 R {sub ⊕}) is successively higher toward later spectral types at all orbital periods probed by Kepler; planets around M stars occur twice as frequently as around G stars, and thrice as frequently as around F stars. Second, a drop in planet occurrence is evident at all spectral types inward of a ∼10 day orbital period, with a plateau further out. By assigning to each spectral type a median stellar mass, we show that the distance from the star where this drop occurs is stellar mass dependent, and scales with semi-major axis as the cube root of stellar mass. By comparing different mechanisms of planet formation, trapping, and destruction, we find that this scaling best matches the location of the pre-main-sequence co-rotation radius, indicating efficient trapping of migrating planets or planetary building blocks close to the star. These results demonstrate the stellar-mass dependence of the planet population, both in terms of occurrence rate and of orbital distribution. The prominent stellar-mass dependence of the inner boundary of the planet population shows that the formation or migration of planets is sensitive to the stellar parameters.

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EVOLUTIONARY MODELS OF SUPER-EARTHS AND MINI-NEPTUNES INCORPORATING COOLING AND MASS LOSS

We construct models of the structural evolution of super-Earth- and mini-Neptune-type exoplanets with H{sub 2}–He envelopes, incorporating radiative cooling and XUV-driven mass loss. We conduct a parameter study of these models, focusing on initial mass, radius, and envelope mass fractions, as well as orbital distance, metallicity, and the specific prescription for mass loss. From these calculations, we investigate how the observed masses and radii of exoplanets today relate to the distribution of their initial conditions. Orbital distance and the initial envelope mass fraction are the most important factors determining planetary evolution, particularly radius evolution. Initial mass also becomes important below a “turnoff mass,” which varies with orbital distance, with mass–radius curves being approximately flat for higher masses. Initial radius is the least important parameter we study, with very little difference between the hot start and cold start limits after an age of 100 Myr. Model sets with no mass loss fail to produce results consistent with observations, but a plausible range of mass-loss scenarios is allowed. In addition, we present scenarios for the formation of the Kepler-11 planets. Our best fit to observations of Kepler-11b and Kepler-11c involves formation beyond the snow line, after which they moved inward, circularized, and underwent a reduced degree of mass loss.

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