M dwarf stars are the most promising hosts for detection and characterization of small and potentially habitable planets and provide leverage relative to solartype stars to test models of planet formation and evolution. Using Gaia astrometry adaptive optics imaging and calibrated gyrochronologic relations to estimate stellar properties and filter binaries we refined the radii of 117 Kepler objects of interest confirmed or candidate planets transiting 74 single late Ktype and early Mtype stars and assigned stellar rotationbased ages to 113 of these. We constructed the radius distribution of 115 small inlineformulatexmath idTM0001 notationLaTeXlt 4 rm Rrmoplustexmathinlineformula planets and assessed their evolution. As for solartype stars the inferred distribution contains distinct populations of superEarths at inlineformulatexmath idTM0002 notationLaTeXsim 1.3  rm Rrmoplustexmathinlineformula and subNeptunes at inlineformulatexmath idTM0003 notationLaTeXsim 2.2  rm Rrmoplustexmathinlineformula separated by a gap or valley at inlineformulatexmath idTM0004 notationLaTeXapprox 1.7  rm Rrmoplustexmathinlineformula that has a period dependence that is significantly weaker powerlaw index of 0.03inlineformulatexmath idTM0005 notationLaTeX0.010.03texmathinlineformula than for solartype stars. SubNeptunes are largely absent at short periods inlineformulatexmath idTM0006 notationLaTeXlt 2  rm dtexmathinlineformula and high irradiance a feature analogous to the Neptune desert observed around solartype stars. The relative number of subNeptunes to superEarths declines between the younger and older halves of the sample median age 3.86 Gyr although the formal significance is low inlineformulatexmath idTM0007 notationLaTeXp  0.08texmathinlineformula because of the small sample size. The decline in subNeptunes appears to be more pronounced on wider orbits and low stellar irradiance. This is not due to detection bias and suggests a role for HSUB2SUBO as steam in inflating the radii of subNeptunes andor regulating the escape of HHe from them.
['Astrophysics - Earth and Planetary Astrophysics']
The radius distribution of M dwarfhosted planets and its evolution
{'Eric Gaidos 1 , 2 , ★ Aleezah Ali 3 , Adam L. Kraus 4 , Jason F. Rowe 5': "- 1 Department of Earth Sciences, University of Hawai'i at M¯anoa, 1680 East-West Rd, Honolulu, HI 96822, USA\n- 2 Institute for Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180 Vienna, Austria\n- 3 Institute for Astronomy, University of Hawai'i at M¯anoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\n- 4 Department of Astronomy, University of Texas at Austin, 2515 Speedway Stop C1400, Austin, TX 78712, USA\n- 5 Department of Physics & Astronomy, Bishop's University, 2600 College Street Sherbrooke, QC, Canada, J1M 1Z7", 'ABSTRACT': 'M dwarf stars are the most promising hosts for detection and characterization of small and potentially habitable planets, and provide leverage relative to solar-type stars to test models of planet formation and evolution. Using Gaia astrometry, adaptive optics imaging, and calibrated gyrochronologic relations to estimate stellar properties and filter binaries we refined the radii of 117 Kepler Objects of Interest (confirmed or candidate planets) transiting 74 single late Kand early M-type stars, and assigned stellar rotation-based ages to 113 of these. We constructed the radius distribution of 115 small ( < 4 𝑅 ⊕ ) planets and assessed its evolution. As for solartype stars, the inferred distribution contains distinct populations of \'super-Earths" (at ∼ 1.3 𝑅 ⊕ ) and \'sub-Neptunes" (at ∼ 2.2 𝑅 ⊕ ) separated by a gap or \'valley" at ≈ 1.7 𝑅 ⊕ that has a period dependence that is significantly weaker (power law index of -0.03 + 0 . 01 -0 . 03 ) than for solar-type stars. Sub-Neptunes are largely absent at short periods ( < 2 days) and high irradiance, a feature analogous to the \'Neptune desert" observed around solar-type stars. The relative number of sub-Neptunes to super-Earths declines between the younger and older halves of the sample (median age 3.86 Gyr), although the formal significance is low ( 𝑝 = 0 . 08) because of the small sample size. The decline in sub-Neptunes appears to be more pronounced on wider orbits and low stellar irradiance. This is not due to detection bias and suggests a role for H2O as steam in inflating the radii of sub-Neptunes and/or regulating the escape of H/He from them.  \nKey words: stars: pre-main sequence - planetary systems - planet-star interactions - planets &satellites: protoplanetary disks - open clusters and associations', '1 INTRODUCTION': 'The launch of the Kepler mission in 2009 (Borucki et al. 2010) ushered in a revolution in the study of planetary systems, the occurrence of rocky, Earth-like planets, and the potential for habitable and life-hosting planets in the Galaxy. Kepler revealed a population of small (Earth- to Neptune-size) planets on close-in orbits around solar-type stars (e.g., Burke et al. 2015; Hsu et al. 2019; Bryson et al. 2021). Analyses showed that the occurrence of these planets increases with decreasing stellar mass or later spectral type (Dressing & Charbonneau 2015; Mulders et al. 2015; Gaidos et al. 2016; Hardegree-Ullman et al. 2019; Ment & Charbonneau 2023). This trend is important to the search for Earth-like planets since the low luminosities and compact \'habitable zones" of M dwarfs mean that they have the potential to host many rocky planets with temperate climates; the elevated space density of M dwarfs relative to their solar-mass counterparts (Smart et al. 2021; Ravinet et al. 2022) means that many host stars are nearby and can thus be prime targets for follow-up observations to detect and characterize atmospheres, e.g., by JWST (Greene et al. 2023; Zieba et al. 2023),  \nHowever, M dwarfs differ from their solar-mass counterparts in ways that could impact the properties and evolution of their planets, including an extended pre-main sequence phase (Luger & Barnes 2015), longer rotational spin-down times (e.g., Johnstone et al. 2021), and elevated X-ray and extreme ultraviolet emission relative to bolometric emission (e.g., France et al. 2016). There may also be differences in their protoplanetary disks, i.e. their masses (Gaidos 2017), lifetimes (Silverberg et al. 2020), and chemistry (Tabone et al. 2023). To assess potential variation in planetary evolution, precise measurement of the properties of a large number of planets in systems with robust ages that span a wide age range is needed. Currently, the confirmed and candidate transiting planets among the Kepler Objects of Interest (KOIs) provide the largest, most uniform, and most precisely measured set of planet properties such as radii and periods (e.g., Thompson et al. 2018).  \nThe radius distribution of sub-Jovian planets around solar-type stars detected by the Kepler mission contains two distinct peaks, representing populations of \'super-Earths"- and \'sub-Neptunes" separated by a gap or \'valley" at ∼ 1.7 𝑅 ⊕ (Owen & Wu 2013; Fulton et al. 2017; Berger et al. 2020). Complementary measurements of planet mass (and hence density) indicate that sub-Neptunes and super-Earth have rocky cores with and without envelopes of volatile,  \nlow molecular-weight constituents (Weiss & Marcy 2014; Rogers 2015). These envelopes could consist of hydrogen and helium captured from the primordial gas disk during planet formation (e.g., Béthune & Rafikov 2019), and/or H 2 Oand other volatiles acquired as ices (Turbet et al. 2020; Luque & Pallé 2022). Unfortunately, mass measurements of small planets are observationally expensive, often not precise, and the mass-radius relations are at least partly degenerate with composition (e.g., Rogers & Seager 2010).  \nPursuing an alternative approach, numerous statistical studies have made inferences about the nature of these planets and their manner of their evolution by correlating variation in the radius distribution with parameters such as orbital period (semi-major axis) and host star mass (Zhu & Dong 2021, and references therein). The former sets the stellar irradiation of the planet and the latter - with M dwarfs at one extreme- governs the timescale over which that irradiation changes (Barnes et al. 2009; Johnstone et al. 2021). Due to their faintness, limited inclusion in the Kepler survey, and more poorly understood properties of M dwarf stars, studies of their planets\' radius distribution have lagged those of solar-type stars (e.g., Dressing & Charbonneau 2015; Gaidos et al. 2016; Hirano et al. 2018; Kanodia et al. 2019).  \nThe question of planetary evolution has also been addressed directly by looking for variation with host star age. Among FGK stars, the number of sub-Neptunes (with envelopes) relative to superEarths (without envelopes) planets is found to decrease with age (Berger et al. 2020; Sandoval et al. 2021; David et al. 2021; Chen et al. 2022), consistent with cooling and contraction of the envelopes and/or escape of light elements to space (e.g., Howe & Burrows 2015). The mean size of sub-Neptunes may also decrease with time (Chen et al. 2022) - but see Petigura et al. (2022).  \nReconstructing the planet radius distribution of M dwarf planets and its evolution requires radii and ages that are more precise and accurate than have previously been available. Transiting planet radius depends on host star radius, and calculations of the radius distribution that account for detection efficiency require stellar density (both mass and radius). New empirical calibrations for M dwarfs permit estimation of mass and radius based on the precise absolute luminosities made possible by Gaia parallaxes (Mann et al. 2015, 2019). Moreover, both Gaia astrometry and ground-based adaptive optics (AO) imaging allow more thorough vetting for stellar companions, which, if not properly accounted for, can affect a radius estimate. Binary stars also appear to have a planet population distinct from single stars (Kraus et al. 2016; Sullivan et al. 2023).  \nFor field M dwarfs, rotation (gyrochronology) is the most promising method to assign ages compared to other techniques (i.e., isochrone-fitting, asteroseismology, depletion of lithium, or kinematics), but in the absence of accurate, first-principles spindown models, this approach requires empirical calibration. Dungee et al. (2022) constructed a rotation sequence (vs. effective temperature 𝑇 eff ) for late K-type and early-type M dwarfs in the 4 Gyr-old M67 cluster, extended calibrated age-dating of 𝑇 eff ≈ 3200-4200K stars to nearly half the age of the Galactic disk. Moreover, a comparison with younger clusters suggests these stars are spinning down according to a simple power-law prescription (Skumanich 1972) by ages of a few Gyr, justifying an extrapolation to older ages. Gaidos et al. (2023) constructed 𝑇 eff -dependent age-rotation relations using rotation sequences and assigned ages to the cool host stars of known exoplanets, including Kepler confirmed and candidate planets.  \nIn this work we revise the radius distribution of Kepler Objects of Interest (KOIs, i.e., confirmed, validated or candidate planets) around single late K- and early M-type dwarf stars and examine its variation with age, taking advantage of precise Gaia parallaxes  \nfor luminosity calculations and revised mass/radius/luminosity relations based on nearby stars; identification and screening of binary systems using astrometry and AO imaging; and assignment of robust rotation-based ages based on the Gaidos et al. (2023) gyrochronology. We compare the M dwarf planet radius distribution to that derived for solar-type stars and interpret it and its change with time to different scenarios for planet composition and evolution.', '2 ANALYSIS': 'Wematched the Kepler Input Catalog (KIC, Brown et al. 2011) with Gaia Data Release 3 (DR3) sources (Gaia Collaboration et al. 2023) using an angular separation criterion of < 0".71. Absolute 𝐾 𝑠 -band (2.2 𝜇 m) magnitudes ( 𝑀 𝐾 ) were computed using photometry from the 2MASS Point Source Catalog (Skrutskie et al. 2006; Cutri et al. 2013) and parallaxes from Gaia Data Release 3 (Gaia Collaboration et al. 2016, 2023). We restricted our analysis to stars observed by Kepler during Quarters 1-17 with 𝑀 𝐾 > 4 . 5 (inclusive of spectral types later than K5, Pecaut & Mamajek 2013), and that appear in the Santos et al. (2019) catalog of late K and M-type dwarfs with rotation signals from Kepler lightcurves (about 59% of the 24,162 stars analyzed). We further limited the sample to stars with 𝑇 eff values in Santos et al. (2019) between 3200 and 4200K, the valid limits of the Gaidos et al. (2023) gyrochronology to be applied to these stars , yielding a final sample of 5433 target stars that Kepler observed with the relevant host star parameters. We identified KOIs around these stars from the DR25 catalog, which is based on data from Quarters 1-17 (Thompson et al. 2018).  \nWe excluded KOIs that have false positive (FP) dispositions in the DR25 Supplemental Catalog, which includes a few planets (candidates) erroneously categorized as false positives in DR25. 1 . We also excluded those with a disposition of \'candidate" and a modified astrophysical FP probability > 0 . 01 calculated with vespa (Morton 2015). This threshold was chosen based on the distributions of FP probability; above 𝑝 > 0 . 01, FPs outnumber confirmed or validated planets. The modified probability is the sum of the probabilities of the undiluted eclipsing binary (EB) and hierarchical EB scenarios, plus the background EB scenario if our AO imaging cannot exclude this. The latter entails that imaging was obtained, that there are no additional sources detected within 2" of the KOI (and thus falling on the same Kepler pixel and inducing minimal centroid motion, Adams et al. 2012) and that the contrast limit of the imaging is sufficient to detect an EB sufficiently bright to produce the observed signal. For this we adopt a maximum eclipse depth of 50%(zero impact parameter) and a Kepler -𝐾 𝑠 reddening of 0.24 for background sources. The latter is based on a mean total reddening 𝐸 ( 𝐵 -𝑉 ) = 0 . 12 for the Kepler field (Schlafly & Finkbeiner 2011) and the extinction coefficients of Yuan et al. (2013), substituting Sloan 𝑟 for the Kepler bandpass. Hereafter we refer to all KOIs that pass these tests as \'planets".  \nHost star masses were calculated from an empirical massluminosity relationship (Eqn. 4 in Mann et al. 2019). Radii were calculated from 𝑀 𝐾 using Eqns. 4 or 5 in Mann et al. (2015) for stars with or without estimates of [Fe/H] retrieved from the Exoplanet Archive. 2 Errors were estimated using Monte Carlo calculations assuming normally-distributed errors in 𝐾 𝑠 -band magnitudes and  \nparallaxes. Finally, we identified those KOIs with gyrochronologic age assignments in Gaidos et al. (2023).', '2.1 Filtering of binary stars': 'Stars in binary or multiple systems with separations ≲ 50 au are known to host a planet population different from that around single stars (Kraus et al. 2016). Furthermore, binary stars with separations ( < 100 au) have different rotational histories than their single counterparts (e.g., Messina et al. 2017; Stauffer et al. 2018), meaning that a rotation-age relation developed for single stars cannot be applied. 90% of M dwarfs in the Kepler field are more distant than 135 pc, thus these physical separations correspond to angular separations ≲ 1". Companions or any nearby sources unresolved or partially resolved by Kepler (which had 4" pixels and a 3.1-7.5" PSF) can also dilute transit signals and make identification of the host star itself and thus planet properties - ambiguous. The 𝐾 𝑠 -band photometry used to estimate radius and mass will be inaccurate for binaries that are unresolved by the 2MASS survey (resolution of ∼ 3"). Thus we identified and excluded KOIs around binary stars with separations < 4". Binary/multiple systems or candidate systems were identified and excluded using (1) Gaia astrometry of resolved sources; (2) Gaia astrometric error relative to single-star astrometric solutions; (3) adaptive optics imaging; and (4) over-luminosity relative to a single-star locus in a color-absolute magnitude diagram.  \nGaia resolved companions: Weidentified companions resolved by the Gaia satellite ( ≳ 1 arcsec, Ziegler et al. 2018a) based on similarity of parallax (distance) and proper motion. Rather than fixed criteria for similarity (e.g., El-Badry et al. 2021), we adopted a Bayesian approach. We identified candidate companions with cone searches of the DR3 catalog with a radius of 74" (corresponding to about 10 4 auat the median distance of Kepler Mdwarfs)aroundeach Kepler M dwarf. We computed the probability of companionship as 𝑝 binary = 𝑃 𝑐 /( 𝑃 𝑐 + 𝑃 𝑏 ) , where 𝑃 𝑐 and 𝑃 𝑏 are the likelihoods that the source is a companion or background star, respectively. We evaluated these assuming normally-distributed errors using:  \n𝑃 𝑐 = Δ 𝜇 𝜎 Δ 𝜋 GLYPH<16> 𝜎 2 Δ 𝜇 + 𝜇 2 orb GLYPH<17> exp        -Δ 𝜇 2 2 GLYPH<16> 𝜎 2 Δ 𝜇 + 𝜇 2 orb GLYPH<17> -Δ 𝜋 2 2 𝜎 2 Δ 𝜋        , (1)  \nand  \n𝑃 𝑏 = 1 𝑁 -1 𝑆 ∑︁ 𝑖 = 1 Δ 𝜇 𝑖 𝜎 Δ 𝜋 GLYPH<16> 𝜎 2 Δ 𝜇 + 𝜇 2 orb GLYPH<17> exp        -Δ 𝜇 2 𝑖 2 GLYPH<16> 𝜎 2 Δ 𝜇 + 𝜇 2 orb GLYPH<17> -Δ 𝜋 2 𝑖 2 𝜎 2 Δ 𝜋        (2)  \nwhere Δ 𝜇 and 𝜎 Δ 𝜇 are the absolute scalar difference in proper motion between the star and source and its standard error, Δ 𝜋 and 𝜎 Δ 𝜋 are the difference in parallax and its standard error, and 𝜇 orb is a \'softening" term that accounts for the angular orbital motion of a binary. This last term we set to 5 mas yr -1 based on Eqn. 4 in El-Badry & Rix (2018). Constant factors that divide out are omitted from Eqns. 1 and 2. The difference in the terms involving parallax and proper motion reflects the latter\'s two-dimensionality. To improve the statistical accuracy by averaging over many stars, the summation for 𝑃 𝑏 is performed over the 𝑆 stars in all the 𝑁 -1 other cone searches (a few thousand stars) that cannot be companions. We imposed no weighting with angular separation other than the generous search radius. Based on the observed distribution of 𝑃 binary values relative to those calculated using stars from other fields, we identified a threshold of 𝑃 binary > 0 . 98 to select companions.  \n,  \nGaia astrometric error: Binaries with separations < 1" cannot be resolved by Gaia (Ziegler et al. 2018b) but can often be identified based on the poor fit of Gaia astrometry to a single-star solution (Belokurov et al. 2020). This is quantified as the Reduced Unit Weight Error (RUWE, Lindegren et al. 2018) and values significantly greater than unity are indicative of a multiple star system. We selected a threshold of 1.4 based on Lindegren et al. (2021).  \nAO imaging: AO imaging of KOIs was obtained of all but 8 systems with the NIRC2 infrared imager and AO system on the Keck-2 telescope (McLean & Chaffee 2000). Observations used the narrow camera (10 mas pixel scale) and either the 𝐾 𝑐𝑜𝑛𝑡 or 𝐾 𝑝 filters. Images were processed, sources extracted, and astrometry performed using the pipeline described in Kraus et al. (2016). We selected those sources with separations < 2" and contrast Δ 𝐾 < 4 as highly likely to be bound companions based on Kraus et al. (2016).  \nPhotometry: Unresolved late K and M dwarf binaries can be identified by their over-luminosity relative to single stars on the main sequence since these stars evolve imperceptibly on the main sequence. Metallicity is the sole confounding factor, and differences in [Fe/H] produce a dispersion in color-absolute magnitude relations. 3 By experimenting with different color-magnitude combinations constructed from ground-based photometry, we found that a color-magnitude locus of Kepler Mdwarfs constructed using PanSTARRS 𝑔 -𝑦 vs. 𝑀 𝑦 (Tonry et al. 2012) has the smallest dispersion and is thus the least sensitive to metallicity. We identified stars that are more luminous than the best fit by more than two standard deviations as probable binaries.', '2.2 Planet radius distribution': 'The posterior distribution in radius of each planet was determined by multiplying each value from the Monte Carlo Markov Chain (MCMC) posterior chains for planet-to-star radius ratio from DR25 Q1-17 (Hoffman & Rowe 2017) with a value from the stellar posterior distribution in radius calculated above, assuming Gaussian errors in magnitude and parallax. To correct for detection efficiency, we normalized each distribution by the summed probability for detection of an equivalent planet around every star observed by Kepler that is a late K- or early M-type dwarf 𝑀 𝐾 > 4 . 5, 3200 < 𝑇 eff < 4200K) and that appears in the Santos et al. (2019) catalog of stars evaluated for rotation periods. We used the KeplerPORTs code (Burke & Catanzarite 2017b) to calculate the probability of detection (including the geometric transit probability). The required window functions, one-sigma detection functions (Burke & Catanzarite 2017a), \'dataspans" (timespans of data), and duty cycles were retrieved from the NASA Exoplanet Archive. Non-linear (four-coefficient) limb-darkening coefficients were obtained from the DR25 Stellar Properties Catalog (Mathur et al. 2017). To mimic the removal of binaries from the KOI list, we excluded all target stars with Gaia RUWE values > 1.4. Finally, KOIs were then matched with entries in the TIME Table of Gaidos et al. (2023) to retrieve gyrochronologic age estimates.  \nThe summed probabilities over all stars represent the expected number of detections if each star in the observed sample had such a planet. Each posterior distribution in radius was divided by this quantity and then summed to give the number of objects per unit radius per star. As a metric of uncertainty due to radius error and finite  \ncounting statistics, we calculated 68% confidence intervals by constructing 1000 Monte Carlo realizations using bootstrap sampling of the planet population with replacement. 4', '2.3 Validation': 'To validate our methodology for reconstructing the planet radius distribution we constructed a sample of synthetic KOIs the same size as the actual one (115 planets, see Sec. 3), drawing radii from the sum of two Gaussian distributions representing super-Earths and sub-Neptunes. The fractional populations in the two distributions were set to 0.4 and 0.6, the centroids were set to 1.3 and 2.1 𝑅 ⊕ , and the standard deviations were set to 0.15 and 0.25 𝑅 ⊕ , respectively. These values produces a reasonable radius distribution that roughly approximates the observed distribution but because this a test of the fidelity of recovery, the precise values do not matter. Orbital periods were selected from a power-law distribution between 1 and 40 days that reproduces the observed distribution (Gaidos et al. 2016). Synthetic planets were randomly assigned to the Kepler late K/early M population with measured rotation periods as defined above. To calculate the planet-star radius ratio, the stellar radii were determined using the metallicity-dependent relation of Mann et al. (2015), with metallicities drawn from the Gaussian distribution found among Kepler Mdwarfs Gaidos et al. (2016). However, for added realism these artificial metallicities were not propagated to the analysis of synthetic KOIs since the metallicities of many KOIs are not well determined; instead, the metallicity-independent relation of Mann et al. (2015) was used. The total probability of detection (including the geometric factor) was calculated for each star-planet pair as described above and the pair accepted conditional to a random uniform draw below this probability. In lieu of actual posterior distributions for the observed radius ratio we adopted Gaussians with fractional standard deviation that are 1/SNR, where SNR is the predicted transit signal-to-noise (Howard et al. 2012):  \nSNR = 𝛿 𝜎 CDPP √︂ 𝑁𝜏 2 hr , (3)  \nwhere 𝛿 is the transit depth (approximately the radius ratio squared), 𝜎 CDPP is the measured combined differential photometric precision of the star over 2 hours (Christiansen et al. 2012, , 2 hr is a typical transit duration), 𝑁 is the number of transits (equal to the dataspan times the duty cycle divided by the orbital period), and 𝜏 the transit duration (after averaging over a uniformly distributed impact parameter). Host star-planet pairs were generated until the accepted number equaled the size of the actual sample We ran the synthetic KOIs through the same analysis pipeline as the actual KOIs to produced a simulated observed radius distribution. This is plotted along with the input distribution in Fig. 1. The bimodal distribution is largely reproduced, but with a noticeable broadening of the super-Earth distribution, probably because of the absence of measured [Fe/H] "measurements" and the effect of noise on the radii of the smallest detectable planets. The input ratio of super-Earths to Neptunes (divided by the vertical grey line) is 0.74 while the retrieved ratio is 0.78, an increase of 5%. The increase is due to the greater scatter of the more poorly determined super-Earths into the sub-Neptune range compared to vice-versa. The location of the  \n4  \nFigure 1. Radius distribution of simulated planets (red dotted curve) and distribution retrieved from 115 simulated KOIs (solid black curve). The vertical grey line at 1.68 𝑅 ⊕ is the approximate divide between super-Earths and sub-Neptunes found among actual KOIs.  \n<!-- image -->  \napparent valley is shifted to a slightly larger radius (from 1.64 to 1.75 𝑅 ⊕ ) for the same reason.', '3.1 Planet Radius-Period Distribution': 'We identified 117 confirmed/validated or candidate planets around 74 single late K and early M dwarfs with matches to Gaia DR3 sources and rotation periods in Santos et al. (2019). Thirteen have NASAExoplanet Archive dispositions of "candidate" while the rest are "confirmed" (i.e., validated). Orbital periods, radii and ages of KOIs are provided in Table 5. Of these, 114 (in 73 systems) have rotation-based ages: nearly all are from Gaidos et al. (2023) but two missing systems (KOIs 2542 and 2705) were assigned ages using the methods of Gaidos et al. (2023). Four planets in three other systems (KOIs 430, 4957, and 6863) have rotation periods in Santos et al. (2019) that are shorter than the calibrated range in Gaidos et al. (2023) and are not age-dated.  \nThe two largest objects in the sample, excluded from further analysis, are the hot Jupiter KOI-254.01 (Kepler-45b, Johnson et al. 2012), and the 5 . 45 ± 0 . 27 𝑅 ⊕ KOI-2715.01 (Kepler-1321b, Morton et al. 2016) orbiting a late K dwarf (neither shown in Fig. 5). The next largest planet is the 3 . 20 ± 0 . 14 𝑅 ⊕ sub-Neptune KOI-253.01 (Kepler-2000b, Valizadegan et al. 2023), highlighting how Neptunesize and larger planets are very rare around M dwarfs. Figure 2 plots the radius vs. orbital period of the 115 small planets, along with a Gaussian kernel density estimator (KDE) to highlight structure. Planets with assigned ages are plotted as black points; while those lacking ages are plotted in grey.  \nWecalculated the detection efficiency (i.e. fraction of detected planets with a given radius and period if distributed uniformly among the late K/early M type stars), in among 3200-4200K stars in Santos et al. (2019) using the procedures described in Sec. 2. This is represented as the grey contours in Fig. 2. As expected, detection efficiency is lowest for smaller planets on wider orbits, which explains the absence of detections in the lower right corner of the plot. There is a single object, KOI-571.05, which is in a region of  \nFigure 2. Orbital periods and radii of 115 transiting planets and planet candidates around late K and early M dwarfs detected by Kepler . Black points represent those planets with rotation-based ages in Gaidos et al. (2023). The shading is a Gaussian KDE with a bandwidth of 0.25 and major features of the distribution are labeled. The grey contours demarcate the Kepler detection efficiency (including the geometric factor) for the DR25 catalog with late K and early M dwarf host stars of all ages, i.e. the fraction of planets detected if every star in the sample hosted a planet with a given radius and orbital period.  \n<!-- image -->  \nparticularly low detection efficiency ( < 0.002) and would normally warrant scrutiny as a potential false alarm, but this was validated as a member of the multi-planet Kepler-186 system by Quintana et al. (2014).  \nAs was shown for solar-type host stars (Fulton et al. 2017) and subsequently for late K- and M-type dwarfs (Hirano et al. 2018; Cloutier & Menou 2020; Van Eylen et al. 2021), the planet distribution of this sample contains two distinct clusters at about 1.3 𝑅 ⊕ (\'super-Earths") and 2.2 𝑅 ⊕ (\'sub-Neptunes"), with an intervening gap or \'valley" at 1.6-1.7 𝑅 ⊕ . Around solar-type hosts there is a deficit of (sub-)Neptunes on short-period (or high-irradiance) orbits, the so-called \'Neptune desert" (Szabó & Kiss 2011; Mazeh et al. 2016). There are no universally accepted boundaries for this desert but it is expected to depend on stellar mass (McDonald et al. 2019). Adopting the conservative criterion of a period of < 2 days (corresponding to irradiances 𝑆 ≳ 200 𝑆 ⊕ ), the \'desert" has a single unambiguous denizen in our sample: KOI-952.05 (Kepler-32f), a member of a multi-transiting planet system (Swift et al. 2013), has an orbital period of 0.74 days and radius of 2 . 04 ± 0 . 11 𝑅 ⊕ , consistent with but more precise than that determined by Morton et al. (2016) and Berger et al. (2018) but significantly larger than previous estimates (e.g., Swift et al. 2013; Mann et al. 2013). There are five other Earth- and super-Earth-size Ultra Short Period (USP, 𝑃 < 1 d) planets in the sample. The smallest object in the sample is the 0 . 52 0 . 04 0 . 05 𝑅 ⊕ candidate KOI-7793.01.  \nCloutier & Menou (2020) found that the planet radius distribution hosted by stars with masses 0.08-0.42M ⊙ (corresponding to spectral types M3-M8) differs from that of more massive stars in lacking a significant population of (sub)Neptunes. Ment & Charbonneau (2023) also found a very pronounced dearth of sub-Neptunes around mid-to-late type M dwarfs. Our sample does not substantially overlap in stellar mass with these analyses and falls off below 0.5 𝑀 ⊕ , although there is at least one KOI hosted by a mid-M dwarf just above the radius valley (Fig. 3).  \nFigure 3. Planet radius vs. stellar luminosity (absolute 𝐾 𝑠 magnitude) and stellar mass. Black point indicate KOIs in systems with ages assigned by gyrochronology. Multi-planet systems appear as vertical chains of points. The shading is a Gaussian KDE with a bandwidth of 0.25.  \n<!-- image -->  \nThe location of the radius valley and its dependence on orbital period (a proxy for irradiance) is used to test models of escape of planetary atmospheres/envelopes (see Sec. 4). Following past work (e.g., Ho & Van Eylen 2023, and references therein), we performed a regression on the full sample with support vector machines (SVM) as implemented in the Python sci-kit learn package, with a choice of regularization parameter of 𝐶 = 10, following Van Eylen et al. (2019). Super-Earth and sub-Neptune classification was initially assigned initially based on radii smaller or larger than 1.68 𝑅 ⊕ (the value predicted for this sample from the relation in Ho et al. (2024)), then classification was iteratively redone (10 times) using the SVM. Confidence intervals were calculated with 10 4 Monte Carlo simulations where planet radii were randomized according to their uncertainties. We plot the final assignments and converged regression (log 𝑅 𝑝 = -0 . 03 + 0 . 01 -0 . 03 log 𝑃 + 0 . 26 + 0 . 04 -0 . 01 ) as the black line in Fig. 4, along with previously published regressions.  \nOur regression is only weakly period-independent and contrasts with the negative slopes for M dwarf host stars derived by Van Eylen et al. (2021, blue dashed line in Fig. 4), from combined solar- and late-type Kepler host stars derived by Ho et al. (2024) and evaluated at the 0.59M ⊙ median sample mass of our sample (dashed magenta line), and for a sample of TESS -detected planets around M dwarfs with mass measurements (Bonfanti et al. 2024, dashed orange line). It is also inconsistent with the positive slope found by Cloutier & Menou (2020) for planets detected around latetype hosts by Kepler / K2 (dashed cyan line). However, the slope is consistent with that derived from a small sample of M dwarf planets with RV-determined masses by Luque & Pallé (2022). The cause of the disagreements between these estimates is not known, but could arise from sensitivity of the slope to objects with the shortest or longest orbital periods combined with refinement of radii by exclusion of binaries and improved stellar properties.', '3.2 Corrected Radius Distribution': 'The detection efficiency-corrected radius distribution constructed for the entire small-planet sample is shown in the top panel of  \nFigure 4. Linear boundary (black line) between \'super-Earths\' and \'subNeptunes\' in log period-log radius space derived by support vector machine regression, using the full sample. Planets are color-coded according to the self-consistent (iterated) SVM classification and the grey region represents the 68% confidence range based on 10 4 Monte Carlo bootstrap representations of the data. Regressions of the valley derived for M dwarf host stars by other investigations are represented by the colored dashed lines. The Ho et al. (2024) slope is the mass-and period-dependent equation log 𝑅 𝑝 = -0 . 10 log 𝑃 + 0 . 17 log 𝑀 ∗ + 0 . 36 evaluated at the median stellar mass of our sample of 0.59M ⊙ .  \n<!-- image -->  \nFig. 5. The grey region spans the 68% confidence interval, and the individual planet radii are marked by the ticks at the bottom. The total occurrence is the integral over the distribution and is 1.84 planets per star. This presentation also shows the super-Earths and sub-Neptunes sa peaks separated by an intervening gap or \'valley" around 1.6-1.7 𝑅 ⊕ . We computed a valley depth, i.e. the ratio of the mean of the adjacent maximum to the minima, of 3.85. This is similar to that found for solar-type stars by Ho & Van Eylen (2023), and inconsistent with the trend of shallower valley with lower stellar mass found by Ho et al. (2024).  \nWhile Fig. 5 appears to resolve the super-Earth population into two overlapping peaks, this structure is not significant compared to the uncertainties in the reconstruction (represented by the shaded area). To verify this, for each of the 1000 bootstrap reconstructions we assessed whether the radius interval 1-1.7 𝑅 ⊕ contained one or two maxima and, in the latter case, calculated the ratio of the intervening minimum to the lowest of the two maxima. 75% of the reconstructions contain two maxima (at ≈ 1.2 and 1.4 𝑅 ⊕ ), but among these, the ratio of the minimum to lowest maximum is continuously distributed between about 0.6 to unity, with most values near unity. This indicates that the two observed peaks are a random occurrence from a continuous distribution of possible outcomes.  \nThere are other, more ambiguous features in the radius distribution, including a \'shoulder" at 3 𝑅 ⊕ , representing four planets of similar size (Fig. 2). The radius distribution also hints at the existence of a population of Earth-size objects distinct from superEarths (second peak from left in Fig. 5). This is not an artefact of detection bias, since the efficiency only decreases smoothly with radius in this regime (Fig. 2). The prominent peak at 0.64 𝑅 ⊕ in Fig. 5 is produced by KOI-314.03 (Kepler-138b), which has a low detection probability due to its comparatively wide (10.3 day) orbit. It is a member of a three-planet system and was discovered only after analysis with optimized, automated routines following the discov-  \nFigure 5. Empirical, non-parametric radius distribution of planets hosted by late K- and early M-type dwarfs observed by Kepler and assigned ages based on rotation, corrected for detection efficiency. The top panel shows the distribution of the entire sample, with radii of individual objects denoted by tick marks. The bottom panel show the distributions for objects with assigned ages partitioned into equal younger ( < 3 . 86 Gyr) and older halves. Shaded regions are 68% confidence interval determined from Monte Carlo sampling with replacement.  \n<!-- image -->  \nother, larger members (Batalha et al. 2013). Its presence in the sample could be just a statistical fluke, or it could represent a large population of \'sub-Earths" (Sinukoff et al. 2017).', '3.3 Radius Evolution': 'We divided the planets with assigned ages into halves that are younger and older than the median gyrochronologic age of 3.86 Gyr (mean ages of 2.6 and 5.4 Gyr, respectively). These are plotted red and blue curves in the bottom panel of Fig. 5. The sub-Neptune peak is much less prominent in the older sample while the super-Earth is larger, with the ratio between the two (integrated over 1.68-4 𝑅 ⊕ and 0.7-1.68 𝑅 ⊕ , respectively) declining from 2.38 to 0.58 from the younger to older moieties. This is consistent with the evolution of planets by the loss and/or contraction of low molecular-weight envelopes over Gyr. Indeed, the integrated probabilities over miniNeptunes (1.7-4 𝑅 ⊕ ) fall from 0.59 to 0.31 and is roughly equivalent to the increase in the Earth- and super-Earth (0.7-1.7 𝑅 ⊕ ) from 0.25 to 0.53. There is no evidence for a decrease in the mean radius of the sub-Neptunes with age nor a deepening of the radius valley as has been seen in other analyses of solar-type stars (Petigura et al. 2022; Chen et al. 2022) but the valley is indistinct in the older sample due to the small number of sub-Neptunes. A two-sided KolmogorovSmirnov test provide weak statistical support for the young and old samples to be drawn from different populations ( 𝑝 = 0 . 08).  \nWe investigated four systematic effects that could potentially mimic radius evolution: (1) an artefact due to a sensitivity of the radius distribution to the choice of divisor between younger and older sub-samples; (2) a variation in stellar properties with age that influences detection efficiency; (3) a difference in host star properties brought about by selecting for stars with detectable rotation period that in turn affects the detected planets; and (4) a variation in stellar  \nFigure 6. Radius vs. orbital period of planets in systems with ages younger (red points) or older (blue points) than the median age of the sample (3.86 Gyr). The grey contours map the Kepler detection efficiency (including the geometric factor) for the DR25 catalog with late K and early M dwarf host stars of all ages, i.e. the fraction of planets detected if every star in the sample hosted a planet of a given radius and orbital period.  \n<!-- image -->  \nproperties (i.e. 𝑇 eff or mass) with age combined with a correlation between the planet population and stellar properties. We find no evidence that any of these is responsible for the observed difference in radius distribution between the younger and old sub-samples, and details are provided in Appendix A.  \nFigure 6 compares the period-radius distribution of the younger (red points) and older (blue points) sub-samples and hints at nonuniformity in this evolution. While there are both younger and older sub-Neptunes ( > 1 . 7 𝑅 ⊕ , above the horizontal line) at periods < 25 days, nearly all those at longer periods ( > 25 days) are younger. This is discussed further in Sec. 4.3.', '4.1 The Radius Valley': 'Models that explain the dichotomy between super-Earths and subNeptunes predict a dependence of the location of the radius valley on orbital period or stellar irradiation similar that can be compared to observations (e.g., see Table 6 in Ho & Van Eylen 2023). Both photoevaporation and core-powered mass loss models predict negative slopes with period, in the former due to stellar X-ray and/or EUV radiation as the driver (e.g., Owen & Jackson 2012), and in the latter a consequence of the role of stellar bolometric irradiation (Gupta &Schlichting 2021). For example, Gupta & Schlichting (2019) derived an index of -0.11 for core-powered loss, independent of stellar mass, while Owen & Kollmeier (2017) derive -0.16 for photoevaporative loss. Such models are consistent with the slopes reported by Van Eylen et al. (2021), Ho et al. (2024), and Bonfanti et al. (2024) (Fig. 4). In contrast, a model of planet formation in a gas-depleted inner disk predicts a positive slope of +0.11 (Lopez & Rice 2018), consistent with the findings of Cloutier & Menou (2020). None of these are consistent with the near-zero slopes found by Luque & Pallé (2022) and our analysis ( -0 . 03 + 0 . 01 -0 . 03 ; Fig. 4). Weak dependence on period as well as on stellar luminosity (Fig. 3), suggests a mechanism largely independent of the host star, e.g. differences  \nin water abundance (see Burn et al. 2024, for a discussion of some possible scenarios).  \nThe depth of the radius valley relative to the adjacent superEarth and sub-Neptune peaks has also been compared to model predictions. In particular, Ho et al. (2024) found that the valley became shallower (filled in) at lower host star masses. A trend could appear due to error in the radius ratio 𝑅 𝑝 / 𝑅 ∗ from fitting the transit lightcurve, which, all else being equal, is expected to be higher for shorter duration transits around smaller M dwarf host stars (Petigura 2020). Astrophysical interpretations of a shallower radius valley include: (a) scatter in the XUV-emission luminosity of young stars at a given mass and age; (b) heterogeneity in the composition of planetary cores; and (c) collisions (Ho et al. 2024). Our failure to reproduce the trend motivates a deeper dive into the systematics that could affect reconstructions of the planet radius distribution.', '4.2 Evolution of Sub-Neptunes into Super-Earths': 'The decline of the sub-Neptune population and commensurate rise in super-Earths between our younger and older sub-samples (Fig. 5) indicates that some sub-Neptunes have evolved into super-Earths over a timescale of ∼ 3 Gyr. On orbits with 𝑃 > 20 day this radius evolution has presumably caused some planets to leave the sample due to the low detection efficiency of super-Earths on wider orbits (grey contours in Fig. 2). The observed change is analogous to that seen around solar-type Kepler host stars (Berger et al. 2020; Sandoval et al. 2021; David et al. 2021; Chen et al. 2022), which has been explained by a scenario where sub-Neptunes are superEarths with thick hydrogen-dominated envelopes which can escape by XUV-driven photoevaporation or core-powered mass-loss from rocky/icy cores (Owen 2019; Gupta & Schlichting 2021; Owen & Schlichting 2024).  \nHigh-energy emission from solar-type host stars is most intense during the first few hundred Myr (Johnstone et al. 2021), so a Gyrtimescale for planet evolution has been previously interpreted as supporting core-powered (Berger et al. 2020). However, more quantitative estimates of photoevaporative timescales are longer (King & Wheatley 2021) and both mechanisms may operate at different times in different regimes (Owen & Schlichting 2024). Moreover, our sample cannot be used to investigate shorter timescales because it does not include systems with ages much younger than 1 Gyr. This is because low mass stars take longer to converge to an initial condition-independent rotation sequence, where gyrochronology can be applied.  \nIn a scenario where sub-Neptunes are rocky cores with thick envelopes of high molecular weight volatiles, specifically H 2 O (Turbet et al. 2020), radius evolution could be at least partly explained by contraction due to cooling (Howe & Burrows 2015; Vazan et al. 2018), and/or chemical reaction between the envelope and the rocky core (Kite et al. 2020; Kim et al. 2023). These processes have timescales different from that of atmospheric escape mechanisms.', '4.3 The Cool Sub-Neptune Collapse': 'The near-complete absence of sub-Neptunes in the \'desert" very close to the host stars is consistent with the canonical picture of envelope loss driven by stellar irradiation. However, the disappearance of sub-Neptunes elsewhere appears to be non-uniform (Fig. 6). There is a striking paucity of older sub-Neptunes on the widest orbits  \n( > 20 days), where irradiation is lowest and escape should be slower than at intermediate orbits (2-20 days) where older sub-Neptunes are found (Fig. 6).  \nThe top panel of Figure 7 shows planet radius versus stellar irradiance, comparing the younger and older sub-samples. Irradiance is based on bolometric luminosities calculated using interpolations of the corrections to 𝐾 -band luminosities tabulated in Pecaut & Mamajek (2013), and assuming near-circular orbits. Constructed irradiance distributions for younger and older subNeptunes ( 𝑅 𝑝 > 1 . 7 𝑅 ⊕ , dotted grey line) are shown in the bottom panel. Both Gaussian kernel density estimators (solid curves) and histograms with Poisson statistics errors (shaded bars) suggest a deficit of older sub-Neptunes at irradiances below several times the present terrestrial value ( 𝑆 ⊕ ). The fraction of sub-Neptunes in the older-sub-sample falls from 53% at irradiances > 5 𝑆 ⊕ to 17% at lower irradiance (see red and blue histograms). This cannot be due to a differences in detection efficiency since planets on wider orbits should be more readily detected around older, photometrically quieter stars. Nor is there evidence for a combination of correlations between age with host star mass (or luminosity), and stellar mass with planet period distribution (Figs. A5 and A6 in Appendix A) that could mimic this effect. There are no sub-Neptunes in the sample below an irradiance of 1 𝑆 ⊕ , although this could be due in part to low detection efficiency at long orbital periods. The two planets that are marginally that are least irradiated are KOIs 812.03 (Kepler-235e) and 952.03 (Kepler-32d), with ages of 7 . 8 ± 1 . 43 and 5 . 60 ± 0 . 96 Gyr, respectively (Gaidos et al. 2023).  \nThe persistence of \'warm" sub-Neptunes and disappearance of cool sub-Neptunes at older ages could be due to an inflation mechanism that depends on stellar irradiation or planet equilibrium temperature and hence proximity to the host star. Turbet et al. (2020) showed that primarily rocky planets with water-dominated atmospheres could have sub-Neptune-like radii at irradiances exceeding a \'runaway" threshold where H 2 O cannot condense into a liquid ocean.  \nKopparapu et al. (2013) and Turbet et al. (2019) found the irradiance threshold for a runaway greenhouse on Earth-like planets to be ≈ 1.05 𝑆 ⊕ . Around M dwarfs stars the threshold is slightly lower because of reduced Rayleigh scattering and increased absorption by atmospheric H 2 Oat the longer wavelengths where cooler stars emit more radiation; the value is predicted to be 0.9 𝑆 ⊕ for 𝑇 eff =3900K or M0 spectral type (dashed vertical green line in Fig. 7. Threedimensional general circulation models find a somewhat higher threshold that depends on planet rotation rate (Leconte et al. 2013; Kodama et al. 2018, ; vertical orange dashed line). The higher Bond albedo of a cloudy steam atmosphere (that of Venus is around 0.76 (Moroz et al. 1985) but measurements of the phase function suggest a value as high as 0.9 (Mallama et al. 2006) 5 ), might move the runaway threshold to ∼ 5 𝑆 ⊕ . This is the approximate boundary where sub-Neptunes become persistent (vertical dashed magenta line). Finally, atmospheres containing significant H/He will have a lower threshold (Innes et al. 2023).  \nWhile a decline in sub-Neptune occurrence across the water condensation boundary could, in principle, be explained by the collapse of steam atmospheres, it is unclear if H 2 O alone can explain the observed distributions with radius and age. In Fig. 7 we plot predictions of Turbet et al. (2020) for H 2 O mass fractions be-  \n5 The albedo depends would depend strongly on cloud droplet size and the high reflectivity of Venus is largely due to sulfuric acid aerosols and a similar scenario might not apply to steam atmospheres.  \nFigure 7. Top: Radius vs. total stellar irradiance or \'instellation" (in units of the current terrestrial value) for planets in systems with ages younger (red) or older (blue) than the median age of the sample (3.86 Gyr). Horizontal error bars are smaller than the points. Vertical lines mark different irradiance thresholds for a runaway water vapor greenhouse atmosphere above which a planet\'s atmosphere is predicted to be inflated. Cyan lines mark predicted radii of planets with a 3 𝑀 ⊕ rocky core and (bottom to top) 0.1, 1, 10, and 25% H2O mass fractions (Turbet et al. 2020). Bottom: Relative distribution of younger (red) and older (blue) sub-Neptunes (1.7-4 𝑅 ⊕ ) with irradiance. Shaded bars present histograms spanning the 68% probability range based on Poisson statistics. The solid curves were generated by Gaussian kernel density estimators with Scott\'s bandwidth (Scott 1992).  \n<!-- image -->  \ntween 0.1 and 25% (cyan lines), assuming a 3 𝑀 ⊕ rocky core mass chosen to reproduce super-Earth radii. This comparison suggests that H 2 O mass fractions exceeding 25% are necessary to explain some sub-Neptunes. Models of H 2 O-rich planets produce radii up to 3 𝑅 ⊕ (Burn et al. 2024), but Vivien et al. (2022) found that pure water/steam atmospheres on sub-Earths become unstable if the atmospheric scale height exceeds 0.1 times the planet radius. Moreover, some internal governing mechanism (e.g., reaction with a magma ocean, Schaefer et al. 2016) is needed to explain the radius evolution over Gyr since the change in luminosity of these very lowmass host stars is small and the response of an atmosphere nearly instantaneous.  \nAlternatively, the inflated radii of sub-Neptunes and the differential evolution of cool vs. warm objects could be explained by atmospheres of mixed H/He and H 2 O. H/He lowers the mean molecular weight and decreases the irradiance threshold for a runaway water-vapor greenhouse (Innes et al. 2023). Slow escape of H/He would eventually collapse sub-Neptunes to smaller objects. At an elevated irradiances above the H 2 O condensation threshold, H 2 O could be persist in the upper atmosphere where its radiative cooling the lowers temperature and inhibits H/He escape (Yoshida et al. 2022). In a cooler sub-Neptune, water condenses into clouds deeper in the atmosphere and cannot play that role. The upper atmosphere is therefore hotter, H/He escape faster, and the sub-Neptunes eventually collapses to a rocky core with a geometrically thin atmosphere.  \nFinally, we searched for irradiance-dependent evolution of sub-  \nNeptunes among Kepler transiting planets around solar-type host stars with isochrone ages determined by Berger et al. (2018), restricting the sample to those with ages more precise than 30%. Although the median age of the sample is 5 Gyr, we use the same division of 3.86 Gyr between young and old populations for comparison. No such effect is obvious in the irradiance distributions (Fig. 8). The ratio of young to old planets is statistically indistinguishable ( ≈ 0 . 44) for planets experiencing 𝑆 > 5 𝑆 ⊕ and 𝑆 < 5 𝑆 ⊕ . Three explanations for the difference between the two samples are (1) the late K/early M dwarf sample is affected by small-number statistics and is a statistical fluke; (2) the ages of the Berger et al. (2018) sub-sample with relatively cool planets are more uncertain because the host stars are likely to be late G and early K dwarfs which evolve very slowly in a color-magnitude diagram; and (3) there is a fundamental difference in the evolution of the two planet populations. At a given bolometric irradiance, planets around M dwarfs will experience far more X-ray and UV irradiation than their solar-type counterparts at (McDonald et al. 2019) and it is this radiation that drives photoevaporation of H/He atmospheres (Owen 2019). Thus temperate sub-Neptunes around solar-type stars will have retained H/He atmospheres while their counterparts around M dwarfs have lost most or all their H/He and have radii controlled by the behavior of water-dominated atmospheres.  \nThese different pathways are illustrated in Fig. 9, where the Kepler M dwarf sample spans Zones I, II, and III (but not Zone V due to small sample size), while the Berger et al. (2018) sample spans Zones I, II, and IV. In Zone I, rapid stellar-driven escape of H/He is not suppressed by atmospheric H 2 O and leads to planets with extended steam-rich atmospheres or rocky super-Earths (Fig. 9). In Zone II, water-rich (but not water-poor) sub-Neptunes retain thick H/He envelopes. In Zone III (only around very low-mass stars), H/He envelopes are lost, leaving \'ocean worlds" and rocky superEarths, and in contrast to Zones 4 and 5, where these envelopes are retained. Measurements of planet mass by RVs and observations of atmospheres by JWST for examples around much closer, brighter host stars could distinguish between these scenarios.', '4.4 Caveats and Limitations of this Analysis': "Biases and selection effects that could possibly mimic the observed radius evolution are summarized in Sec. 3.3 and evaluated in Appendix A. There are other effects that could quantitatively impact our results. We restricted our sample to planet host stars with detected rotational signals in Kepler lightcurves and these are a biased subset of the overall population. Younger stars will be rapidly rotating, magnetically active, and highly spotted and their rotational periods can be readily measured. However, due that rotational variability as well as intense flaring can inhibit the detection of transiting planets (Gaidos et al. 2017). On the other hand, the oldest stars will be slowly rotating, less spotted, and exhibit less rotational variability, and the rotation periods can be similar to the Kepler quarterly roll (90 days) and thus more difficult to retrieve (Santos et al. 2019). This is particularly true for the coolest stars in the sample (mid M-dwarfs) because these stars have the longest rotation periods for a given age. These selection effects can manifest themselves in the overall distribution of M dwarf planet host ages (e.g., Gaidos et al. 2023). However, we see no trend of planet properties (i.e., radius) with host star properties (i.e., luminosity, Fig. 3).  \nThe overall occurrence rate estimated from KOIs with rotation periods could be biased upwards to the extent that stellar rotation and planetary orbits are aligned (low stellar obliquity Albrecht et al. 2022). Stars viewed at a high inclination (equator on) will have the  \nFigure 8. Top: Distributions of planet radius vs. stellar irradiation for systems around solar-type stars with ages that are younger (red points) or older (blue points) than 3.86 Gyr, from Berger et al. (2018). Bottom: Relative distribution of younger (red) and older (blue) sub-Neptunes (1.7-4 𝑅 ⊕ , above the horizontal grey dotted line in the top panel) with irradiance. Shaded bars present histograms spanning the 68% probability range based on Poisson statistics. The solid curves were generated by Gaussian kernel density estimators with Scott's bandwidth. Compare to Fig. 7 for late K and early M-type stars. Here, systematic errors in irradiance are likely larger than counting statistics.  \n<!-- image -->  \nstrongest rotational signal and their planets will be more likely to transit compared to the overall population. We quantified this effect by determining the fractional numbers of KOIs around K/M dwarfs observed by Kepler with rotation periods in Santos et al. (2019) vs. K/M dwarfs lacking rotation periods. We applied the same cuts ( 𝑀 𝐾 > 4 . 5, RUWE < 1 . 4, and not otherwise identified as binaries) to both samples, but we did not apply a 𝑇 eff cut since temperatures are only available for the Santos et al. (2019) sample. The KOI fractions are 2 . 70 ± 0 . 24% and 2 . 50 ± 0 . 39%, respectively, with uncertainties from counting statistics. These are thus indistinguishable and we conclude the effect is negligible among these types of stars, perhaps because of the high (%73) success rate of detecting rotation among these stars.  \nTransit parameters and stellar parameters were not computed self-consistently: the former adopted DR25 values and the latter were based on revised luminosities and new empirical relations. Stellar density is covariant with transit impact parameter which, along with 𝑇 eff (which controls limb darkening coefficients) is covariant with the planet-to-star radius ratio 𝑅 𝑝 / 𝑅 ∗ . Lastly, our filtering of binaries from the Kepler late K- and early M-dwarf target sample was less thorough because AO imaging is not available for the overall Kepler late K/early M dwarf population as it is for the KOIs.", '5 SUMMARY AND FUTURE DIRECTIONS': 'We refined the radii of 117 transiting planets detected by Kepler around 74 cool (3200-4200K) single dwarf stars and assigned ages to 113 based on their rotation and a revised gyrochronology. Based  \nFigure 9. A pictorial schematic of zones in the orbit- and stellar-massdependent evolution of sub-Neptunes. Boundaries between the zones are merely illustrative and not to scale. Within each zone, the upper and lower pathways represent water-rich and water-poor planets, respectively. Red represents hydrogen/helium, blue represents H2O, and black represents rocky cores. Interior to the water condensation limit (blue boundary), H2O is present in the upper atmospheres of water-rich planets. Exterior to the strong atmospheric escape limit, atmospheric water inhibits escape of H/He. Exterior to the weak atmospheric escape limit, escape of H/He is negligible. The difference in slopes represent the greater sensitivity of bolometric luminosity to stellar mass compared to XUV luminosity (e.g., McDonald et al. 2019).  \n<!-- image -->  \non these values, we conclude that: (1) analogous to the distribution found for solar-type host stars, the detection efficiency-corrected radius distribution of M dwarf planets contains two peaks representing super-Earths and sub-Neptunes, with the sub-Neptunes largely absent in a \'desert" at periods < 2 days; (2) in contrast to some previous investigations of both solar-type and cooler dwarfs, the planet radius valley separating these two populations is only weakly dependent on orbital period (dlog 𝑅 𝑝 /dlog 𝑃 = -0 . 03 + 0 . 01 -0 . 03 ); (3) there is a decline in sub-Neptunes and rise in super-Earths over the ∼ 3 Gyr separating the younger and older halves of the sample; and (4) the decline in sub-Neptunes is mostly at lower stellar irradiances where water can condense as clouds deep in the atmosphere. The last effect is not seen among Kepler planets around solar-type stars where the water condensation limit is much more distant. Taken together these suggest a governing role for intrinsic compositional differences in the radius dichotomy of small planets e.g., planets having water-rich and water-poor envelopes lying above and below the valley, respectively. The relative decline in cooler sub-Neptunes relative to their warmer counterparts with time could be due to cooling of the upper atmosphere by water and inhibition of escape of H/He.  \nOur findings, while intriguing, are limited in their statistical robustness by sample size rather than the precision with which stellar and planet parameters can be determined. The small number of late K and M dwarf KOIs is the result of the selection criteria for targets of the Kepler prime mission and the telescope\'s single fixed pointing. Both the K2 and TESS mission have identified many more transiting planet systems around such stars, some of which have ages assigned by gyrochronology (Gaidos et al. 2023). However, determining the rotation periods of late-type field stars from K2  \nTable 1. Kepler MDwarf Planet Radii and Ages 1  \n1 The full table is available as a machine-readable table (http://zenodo.org).  \nand especially TESS photometry alone is challenging (Reinhold & Hekker 2020; Claytor et al. 2022). In the Gaidos et al. (2023) catalog of late K- and early M-type stars with established rotation periods and ages there are only 18 TESS Objects of Interest (TOIs). This situation will improve with the discovery of more TOIs hosted by very cool stars, plus long-period planets near the water-condensation limit detected with only one or two transits by TESS and confirmed by other observatories (Gill et al. 2020; Cañas et al. 2020; Yao et al. 2021; Magliano et al. 2024). However, obtaining rotation periods of middle-aged cool host stars for gyrochronology will in most cases require sources of photometry other than TESS or analysis of activity indicators in the spectra obtained for radial velocity monitoring. The PLATO mission and its long-duration (two-year) fields could yield additional systems with gyrochronologic ages, depending on target selection (Nascimbeni et al. 2022).  \nThe water condensation limit should be marked by a transition in the abundance of H 2 O in the upper atmospheres of cool subNeptunes, a boundary that is beginning to be explored by infrared transmission spectroscopy by JWST , e.g. TOI-270d (Holmberg & Madhusudhan 2024). There are also additional investigations that could be performed with larger samples, for example, the expected dynamical evolution of multi-planet systems due to mutual perturbations (Teixeira & Ballard 2023). 6', 'ACKNOWLEDGEMENTS': 'EG and AA acknowledge support from the NASA Exoplanet Research Program (Award 80NSSC20K0957). ALK acknowledges support from the NASA Exoplanet Research Program (Award 80NSSC22K0781). Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. This research has made use of the NASA Exoplanet Archive,  \nwhich is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https:// www.cosmos.esa.int/web/gaia/dpac/consortium ). Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement.', 'DATA AVAILABILITY': 'The KIC and DR25 KOI catalogs are available from the, Gaia DR3 astrometry and photometry are available from the CDS, and AO imaging is publicly available from the Keck Observatory Archive.', 'Evolution of M Dwarf Planet Radii': 'Nascimbeni V., et al., 2022, A&A, 658, A31  \nOwen J. 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B., Liu X. W., Xiang M. S., 2013, MNRAS, 430, 2188  \nZhu W., Dong S., 2021, ARA&A, 59, 291  \nZieba S., et al., 2023, Nature, 620, 746  \nZiegler C., et al., 2018a, AJ, 156, 83  \nZiegler C., et al., 2018b, AJ, 156, 259  \nFigure A1. Sensitivity of the change in radius distribution to the divisor between young and old samples. Top: Planet radius vs. gychronologic system age. Three possible age cuts between young and old stars are shown; the median age (3.86 Gyr) is represented by the solid line, and 40 and 60 percentile ages (3.65 and 4.59 Gyr) bound the grey region. Bottom: Radius distributions of young (red) and old (blue) planets using the three divisor ages, with those using the median represented by the solid line, and those constructed using the 40 and 60 percentile values spanning the shaded regions.  \n<!-- image -->', 'APPENDIX A: BIASES AND SELECTION EFFECTS': 'We evaluated potential biases and selection effects that could cause the reconstruction of the planet radius distribution to mimic the observed evolution.  \nChoice of age divisor: We investigated the sensitivity of the difference in radius distribution between young and older systems to the choice of age to divide the sub-samples. The choice of median (3.86 Gyr, horizontal line in the top panel of Fig. A1) is a statistical convenience and gyrochronologic ages have significant uncertainties which can \'blur" age dependence. To assess this, we generated alternative radius distributions, dividing the sample a the 40 and 60 percentile values of the age distribution (3.65 and 4.59 Gyr). These bound the grey shaded region in the top panel of Fig. A1. In the bottom panel, the radius distributions for the younger and older sub-samples are shown as the red and blue curves, respectively, with the solid curves those constructed using the median age and the shaded regions bounded by the results from the 40 and 60 percentile values. All younger distributions exhibit an elevated number of sub-Neptunes to super-Earths whereas for older distributions the reverse is true. Thus the evidence for radius evolution is robust to the exact choice of sample divisor.  \nPlanet detection bias: We also investigated whether the observed radius evolution could be an artefact of differences in planet detection efficiency between young and old stars 7 . Rotation-driven magnetic activity (spots and flaring) on cool stars introduces noise into lightcurves. As stars age and spin down, the noise decreases (e.g., Davenport 2016), meaning that, all else being equal, smaller  \nFigure A2. Ratio of mean planet detection efficiency between younger ( < 4 Gyr, based on rotation) and older sub-samples of the late K and early M dwarfs observed by Kepler with rotation periods reported in Santos et al. (2019). Each point represents a planet in our KOI sample, and the red points are binned averages of 20 planets with error bars representing the scatter within the bin divided by √ 20. The approximate radius domains of superEarths and sub-Neptunes are delineated.  \n<!-- image -->  \nand/or more distant planets can be detected around older stars. To evaluate the magnitude of this systematic, we divided the Kepler late K/early M dwarf sample with rotation periods in Santos et al. (2019) into 773 (older) stars more slowly rotating than the rotation sequence of the 4 Gyr-old cluster M67 (Dungee et al. 2022), and 1936 (younger) stars that are more rapidly rotating than that sequence. The age of M67 is within error of our median sample age (3.86 Gyr). We calculated overall detection efficiency among the two samples for each planet in our KOI list and plot the ratio vs. planet radius in Fig. A2. All planets in the radius range of interest are less detectable by about 10% around the more rapidly rotating (\'younger") stars than around the more slowly rotating (\'older") stars and the difference is largest among the smallest planets. However, the relative decline in detection efficiency between super-Earths and sub-Neptunes is < 10% and cannot explain the factor of ∼ 2 differences between the younger and older radius distributions (Fig. 5). Because we cannot adequately filter the overall sample for binary stars, which tend to be more rapidly rotating than single stars and also dilute any transit signals, we do not explicitly correct for this difference in detection efficiency.  \nRotation period selection bias: This observed change in planet radii could possibly arise from a difference in the properties of younger and older stars brought about by selection for detectable rotation periods. Cooler dwarf stars tend to rotate more slowly at a given age (e.g., Curtis et al. 2019) and their rotation is more difficult to detect due to the longer time baseline needed to capture variability as well as the lower amplitude of variability from decreased magnetic activity. Thus, cooler stars are more likely to be included in the younger sub-sample, an effect which is apparent in the 𝑇 eff distribution of Kepler target stars (not just KOIs) with rotation-based ages younger or older than 3.86 Gyr (Fig. A3). This bias with 𝑇 eff should also bias the distributions with stellar radius, and potentially detection of transiting planets. However, we find that this does not produce any significant difference in planet detectability. The distribution of predicted transit signal-to-noise ratio (SNR) of the two subsets of target stars are nearly indistinguishable  \nFigure A3. Effective temperature distribution of younger ( < 4 Gyr, based on rotation) and older subsets of Kepler late K- and early M-type target stars, showing the difference due to selection effects (e.g., detectable rotation).  \n<!-- image -->  \nFigure A4. Distribution of the relative detection SNR at a given planet radius and orbital period based on a scaling with stellar parameters (see text), showing no significant difference between younger and old stars.  \n<!-- image -->  \n(Fig. A4). The median values are 0.35 and 0.36 and a two-sided Kolmogorov-Smirnov (K-S) returns a probability of 0.13 that the two samples, excluding the tails with relative SNR > 1.3, are drawn from the same distribution. Here, we calculated the relative SNR for a given 𝑅 𝑝 and 𝑃 𝐾 as the inverse of the stellar radius times the square root of the brightness times the transit duration, which scales as 𝑅 -3 / 2 ∗ 𝑀 -1 / 2 ∗ 10 -𝑚 / 5 , where Gaia 𝐺 magnitude was used as a proxy for Kepler magnitude 𝑚 . Cooler dwarf stars are smaller but also intrinsically fainter, and these two opposing trends apparently cancel out in this field.  \nPlanet population variation: A dependence of the intrinsic planet population on stellar mass, combined with a difference in the mass distribution between younger and older sub-samples, could mimic evolution. The occurrence ratio of sub-Neptunes to superEarths to sub-Neptunes modestly decreases with stellar mass between 0.5 and 1.4M ⊙ (about 0.04 per 0.1 dex in mass, top panels of Fig. 12 in Petigura et al. 2022), but seems to markedly increase with increasing stellar mass in the M dwarf range (Hardegree-Ullman et al. 2019; Ment & Charbonneau 2023). Thus, if there was a bias towards more higher-mass stars in the older sample due to a detec-', '14 Gaidos et al.': 'Figure A5. Distribution of gyrochronologic host star age vs. stellar luminosity (absolute 𝐾 𝑠 -band magnitude), an age-independent proxy for mass for these main sequence stars. Corresponding stellar masses and spectral types from Pecaut & Mamajek (2013) are at the top and bottom axes, respectively.  \n<!-- image -->  \nFigure A6. Orbital period vs. host star luminosity ( 𝑀 𝐾 ) for younger (red) and older (blue) sub-Neptunes (planets with radii between 1.7 and 4 𝑅 ⊕ ). There is no correlation that could explain the absence of older sub-Neptunes at larger radii.  \n<!-- image -->  \ntion bias for shorter rotation periods (see above), the older sample should contain more sub-Neptunes, opposite to what is observed. Moreover, there is no apparent trend between host star luminosity (here 𝑀 𝐾 , a proxy for mass) and age in our sample (Fig. A5). A two-sided K-S test returns a probability of 0.29 that the luminosities of younger and older stars are drawn from the same distribution.  \nThis paper has been typeset from a T E X/L A T E X file prepared by the author.'}