Radius and Mass Distribution of Ultra-short-period Planets

Ultra-short-period (USP) planets are an intriguing population of exoplanets with orbital periods <1 day. The first confirmed transiting USP planet was CoRoT-7b, with a radius of 1.68 R_⊕ and an orbital period of 0.85 days (Léger et al. 2009).

Since then, many additional USP planets have been discovered by the Kepler spacecraft. During its 4 yr mission, Kepler monitored the brightness of around 200,000 stars. In general, USP planets were not targeted by the Kepler pipeline, and many searches of the Kepler database utilized the box-least-squares method of detecting planetary transits. Sanchis-Ojeda et al. (2014, hereafter referred to as SO14) applied Fourier transforms to the light curves of Kepler target stars to reduce noise and isolate the periodic USP planetary transits. SO14 presented 106 USP planet candidates whose radii fall almost exclusively below 2 R_⊕, and found that USP planets occur around 1.10% of M dwarfs and 0.83% of K dwarfs. The K2 mission has nearly doubled the census of this enigmatic planet population (e.g., Adams et al. 2017, 2016, 2021). Adams et al. (2020) identified 74 candidate USP planets in the first half of K2.

The USP planet population exhibits several characteristics that offer clues toward the planets' formation and evolution processes. The overall occurrence rate of USP planets is similar to that of hot Jupiters, though (in contrast to hot Jupiters) USP planets are more common around lower mass M dwarf stars than G and K dwarfs (SO14). USP planets are typically small (≲1.8 R_⊕). Among the USP planets with bulk density constraints, most are consistent with having an Earth-like compositions (Dai et al. 2019), though a subset are iron enhanced (Price & Rogers 2020). USP planets are typically accompanied by other planets with orbital periods between 1 and 50 days (SO14), though the period ratio between USP planets and their nearest neighbors is generally >4 (Steffen & Farr 2013), compared to the period ratios of 1.3−4 for adjacent planet pairs in the broader sample of Kepler multi-planet systems out to orbital periods of 100 days (Fabrycky et al. 2014). USP planet have larger mutual inclinations than other planets among the Kepler multi-planet systems (Dai et al. 2018). The host stars of USP planets have a similar metallicity distribution to the host stars of hot sub-Neptunes (planets with radii <4 R_⊕ and orbital periods between 1–10 days); unlike hot Jupiters, USP planets do not show an association with metal-rich stars (Winn et al. 2017). Finally, the ages of USP host stars are indistinguishable from the ages of field stars (Hamer & Schlaufman 2020), indicating that USP planets do not experience tidal inspiral while their host stars are on the main sequence.

In this study, we revisit the USP planet radius distribution. We update the radius distribution from SO14 to include the improved stellar (and hence planetary) characterization and planet candidate vetting that has been completed in the intervening years. We focus on the USP planet sample discovered by Kepler to leverage the injection-recovery completeness characterization performed by SO14. From the radius distribution, we derive the USP planet mass distribution, considering various assumed planet composition distributions. When computing model masses for the closest orbiting planets (with P_orb < 10 hr), we apply three-dimensional interior structure models to take into account the tidal distortions of the planets' induced by their host stars. In Section 2, we report the updated radii of USP planets and their host stars. Section 3 addresses the occurrence rate calculations, and in Section 4 we present the mass distribution. We discuss the implications of the inferred radius and mass distributions for the USP formation process in Section 5, and summarize our findings in Section 6.

This work aims to builds upon the USP planet characterizations of SO14 by using updated stellar host parameters. The stellar population considered in this study matches that of SO14, focusing on Kepler-targeted G and K dwarfs, that is, stars with 4100 K < T_eff < 6100 K and 4.0 < log(g) < 4.9. We also limit the population to stars with Kepler magnitudes m_Kep < 16 and with at least one 90 day quarter of data, leaving 97,940 stars. We identify 58 USP planet candidates in the SO14 catalogue with stellar hosts in the outlined population, eliminating 10 candidates that have since been denoted false positives in the NASA Exoplanet Archive. ⁴

We first revise the USP planet stellar host properties to incorporate updated stellar spectra and distance measurements collected since the original SO14 catalog. We expand on the work of Winn et al. (2017) by including recent parallax (π) measurements from Gaia Data Release 2 (DR2), linked to their respective Kepler targets by Megan Bedell's gaia-kepler.fun crossmatch database. Using the isochrones Python package (Morton 2015), we fit K-band magnitude, Gaia parallax (π), and when available, spectroscopic parameters (${T}_{\mathrm{eff}},\mathrm{log}(g),[\mathrm{Fe}/{\rm{H}}]$) from the California Kepler Survey (Winn et al. 2017; Fulton & Petigura 2018) to stellar evolution models. Using MESA Isochrones and Stellar Tracks (MIST) stellar isochrones (Choi et al. 2016), we determine the host star's mass and radius. SO14's median host star radius uncertainties were +28%, −8.3%, while our median uncertainties are +2.5%, −2.4%.

Figure 1 compares the stellar host radii calculated in this study to those used in Berger et al. (2020). The deviation between the values is likely created by the our use of host star spectroscopic parameters from Winn et al. (2017). Berger et al. (2020) uses Gaia parallaxes, stellar metallicities with an error of 0.15 dex, and stellar g and K_s photometry to derive T_eff, log(g), radii, and masses for targets. We verify that these stellar radius calculations agree with those of Berger et al. (2020) with a median deviance of 2.4%.

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We calculate the planetary radius, R_p , from the stellar radius, R_⋆, using the same observed transit depths, δ, as SO14. For each planet, we take 1000 samples from both the stellar host radius posterior distribution calculated by isochrones and the transit depth posterior distribution (assumed to be normal), and calculate 1000 samples of planetary radii, using ${R}_{p}=\sqrt{\delta }{R}_{\star }$.

Figure 2 shows the calculated radii of the 58 Kepler USP planets. The majority of USP planets are super-Earth-size, with their radii between 1.25 and 2 R_⊕. Planets are more rare at shorter orbital periods, also noted in previous literature (SO14; Howard et al. 2012). The relative scarcity of USP sub-Neptunes (2–4 R_⊕) and gas giants has also been noted previously by SO14, suggesting that planets larger than 2 R_⊕ cannot survive tidal forces and photoevaporation and remain intact as close to their host stars as smaller planets.

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Occurrence rates quantify how intrinsically common or rare planets are, accounting for both planets that are discovered and those that are missed. Factors such as observation bias, non-transiting geometries, and low signal-to-noise ratios (S/Ns) lead to missed planets. In this section, we use the updated Kepler USP planet sample to calculate revised occurrence rates, accounting for undiscovered planets.

For transit surveys like Kepler, occurrence rates are often calculated within 2D bins in radius–orbital period space (e.g., Howard et al. 2012; Petigura et al. 2013; Dressing & Charbonneau 2015, NASA Exoplanet Exploration Program Analysis Group Study Analysis Group 13 Report). ⁵ We follow an approach similar to that of Howard et al. (2012) to derive USP planet occurrence rates within 2D bins in radius–orbital period space. We deviate from the methods of Howard et al. (2012) in how we (1) account for measurement uncertainties on the planet radii (following an approach similar to Silburt et al. 2015; Dressing & Charbonneau 2015), (2) use the results injection-recovery tests from SO14 to characterize the completeness of the USP planet detection pipeline instead of an S/N cutoff (following an approach similar to, e.g., Petigura et al. 2013), and (3) compute uncertainties on the inferred occurrence rates. Ultimately, Poisson-likelihood-based hierarchical Bayesian approaches or approximate Bayesian computing (e.g., Youdin 2011; Foreman-Mackey et al. 2014; Neil & Rogers 2020; Bryson et al. 2021) could be used to fit a parameterized planet distribution function and eliminate the need for binning when inferring occurrence rates, but these approaches are beyond the scope of this work.

We follow an approach similar to that of Howard et al. (2012) to derive USP planet occurrence rates, considering 2D bins in radius–orbital period space. Howard et al. (2012) calculated the occurrence rate contribution of a planet with a given radius and orbital period as the ratio of the inverse transit probability and the number of stars from which that planet could have been detected. The occurrence rate contribution f_i for a planet i of a specific radius and orbital period is given by

$$\begin{eqnarray}&&{f}_{i}=\displaystyle \frac{{a}_{i}/{R}_{* ,i}}{{N}_{* ,i}},\end{eqnarray} \tag{ 1 }$$

where a is the planet's semimajor axis, a/R_* is the inverse transit probability, and N_*,i is the total number of stars in the survey around which the planet transit would have a sufficient S/N to be detected. The total occurrence rate within a specified planet radius–orbital period bin, f_cell, is the sum of the individual occurrence rate contributions

$$\begin{eqnarray}&&{f}_{\mathrm{cell}}=\sum _{i=1}^{{n}_{\mathrm{pl},\mathrm{cell}}}\displaystyle \frac{{a}_{i}/{R}_{* ,i}}{{N}_{* ,i}}\end{eqnarray} \tag{ 2 }$$

of all n_pl,cell planets detected within the 2D parameter space of the bin.

In order to calculate accurate occurrence rates, we must take into account the likelihood that a particular planet's transit signal would be sufficiently strong to be detected orbiting the stars in the survey. This is encapsulated by the factor N_*,i in Equation (1).

The S/N of a planet-star system measures the strength of the transit signal in comparison to the star's intrinsic noise. A higher S/N leads to a higher probability of a planet being detected. For each planet i, we calculated the S/N of the planet around each star j in the survey, as follows:

$$\begin{eqnarray}&&{\left[{\rm{S}}/{\rm{N}}\right]}_{{ij}}=\frac{{\delta }_{{\rm{ij}}}}{{\sigma }_{{\rm{CDPP}},j}}\sqrt{\frac{{T}_{0,i}{t}_{j}}{(6\,{\rm{hr}}){P}_{{\rm{orb}},i}}},\end{eqnarray} \tag{ 3 }$$

where δ is the transit depth, σ_CDPP is the star's combined differential photometric precision (CDPP, a measure of a its intrinsic noisiness) over 6 hr, T₀ is the transit duration, t is the time spent actively observing the star (here the product of the data span and duty cycle), and P_orb is the planet's orbital period in hours.

To quantify the completeness of the USP planet search pipeline, SO14 conducted transit signal injection-recovery tests in which they measured the probability of a planet being detected given its S/N. We fit a gamma cumulative distribution function to data from the lower right panel of Figure 7 in SO14, which had χ² = 0.026, as follows:

$$\begin{eqnarray}&&P(x)=\displaystyle \frac{a}{{\rm{\Gamma }}(b)}\gamma \left(b,\displaystyle \frac{x-c}{d}\right),\end{eqnarray} \tag{ 4 }$$

where P is the probability of detection, x is the S/N, a = 0.977, b = 1.833, c = 4.061, d = 0.977, Γ is the gamma function, and γ is the lower incomplete gamma function. Using Equation (4), we can quantify the probability of detection for each planet orbiting each star in the population of Kepler stellar targets with the characteristics described at the beginning of Section 2.

N_*,i is the total number of Kepler target stars around which a particular planet (indexed by i) can be detected. When calculating N_*,i , we use Equation (4) to estimate the probability of a planet being detected around every star j in our sample. We sum the probabilities as fractional stars to find the total number of stars around which that the planet could be detected:

$$\begin{eqnarray}&&{N}_{\ast ,i}=\displaystyle \sum _{j}P\left({[{\rm{S}}/{\rm{N}}]}_{{ij}}\right).\end{eqnarray} \tag{ 5 }$$

After evaluating N_*,i for each planet in our sample, we calculate f_cell by summing the individual occurrence rate contributions of all the planets in the bin (Equation (2)). We use 1000 radius samples for each planet, so each sample is weighted as $\tfrac{1}{1000}$ to take planet radius uncertainties into account.

The finite number of USP planets detected leads to inherent uncertainty in estimates of the occurrence rate, f_cell. In calculating uncertainties on f_cell, we consider the occurrence rate of transiting planets within the radius–orbital period bin, calculated as

$$\begin{eqnarray}&&{f}_{\mathrm{tr},\mathrm{cell}}=\sum _{i=1}^{{n}_{\mathrm{pl},\mathrm{cell}}}\displaystyle \frac{1}{{N}_{* ,i}}.\end{eqnarray} \tag{ 6 }$$

We model the detection of transiting planets within a specified radius-period bin as a binomial process with the occurrence rate, ${f}_{\mathrm{tr},\mathrm{cell}}$, as the probability of success (i.e., the probability that a transiting planet with specified properties is present around the star). We calculate uncertainties on ${f}_{\mathrm{tr},\mathrm{cell}}$ using the Wilson confidence interval to within 1σ (15.85th−84.15th percentiles):

$$\begin{eqnarray}&&\displaystyle \frac{1}{1+\tfrac{{z}^{2}}{n}}\left(\hat{p}+\displaystyle \frac{{z}^{2}}{2n}\right)\pm \displaystyle \frac{z}{1+\tfrac{{z}^{2}}{n}}\sqrt{\displaystyle \frac{\hat{p}(1-\hat{p})}{n}+\displaystyle \frac{{z}^{2}}{4{n}^{2}}},\end{eqnarray} \tag{ 7 }$$

where z is the appropriate z-score (in this case, 1), $\hat{p}$ the observed proportion of successes (here ${f}_{\mathrm{tr},\mathrm{cell}}$), and n the effective number of trials (here ${n}_{* ,\mathrm{eff},\mathrm{cell}}={n}_{\mathrm{pl},\mathrm{cell}}/{f}_{\mathrm{tr},\mathrm{cell}}$). The bounds of the confidence interval on ${f}_{\mathrm{tr},\mathrm{cell}}$ are then scaled by ${f}_{\mathrm{cell}}/{f}_{\mathrm{tr},\mathrm{cell}}$ to set the uncertainties on the total occurrence rate of both transiting and non-transiting planets, f_cell.

This method of calculating the uncertainties takes into account the fact that only transiting USP planets were detectable by Kepler. The total number of planets successfully detected in each bin (n_pl,cell), thus depends most directly on the occurrence rate of transiting planets and not the total number of planets, transiting and non-transiting. In this, we differ from the work of Howard et al. (2012) by calculating the effective number of stars searched as $n={n}_{* ,\mathrm{eff},\mathrm{cell}}={n}_{\mathrm{pl},\mathrm{cell}}/{f}_{\mathrm{tr},\mathrm{cell}}$ instead of n_pl,cell/f_cell, and the probability of success as $\hat{p}={f}_{\mathrm{tr},\mathrm{cell}}$ instead of f_cell.

Figure 3 shows the planetary occurrence rates as a function of radius and orbital period. The general shape of Figure 2 can be seen in the color gradient, with the majority of planets falling under 2 R_⊕ with orbital periods close to the 1 day threshold. The most common USP planets have orbital periods between 15.3 and 24 hr and radii between 1 and 1.4 R_⊕. They occur at 1.5 planets per 1000 stars, a rate similar to that of hot Jupiters. When marginalized over planet radius, the values agree with those presented in SO14 to within 2σ.

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We do not detect any statistically significant dependence of the USP planet radius distribution on orbital period within the Kepler USP planet sample. To evaluate whether USP planet radii differ with orbital period, we perform two-sample Kolmogorov–Smirnov (K-S) tests comparing the radius distributions in each of the three lowest orbital period bins in Figure 3 to that of the outermost bin. The resulting p-values of 0.36, 0.29, and 0.17 for the first, second, and third bins, respectively, show that the null hypothesis (that the planet radii within each orbital period bin were drawn from the same underlying planet radius distribution) cannot be ruled out.

From the USP planet radius distribution, we derive a mass distribution using models of planet interior structure and various assumed composition distributions.

The mapping from planet radius distribution to planet mass distribution depends on the composition distribution of USP planets. To address this, we calculate USP planet masses under a variety of composition assumptions: Earth-like, Mercury-like, and a random assortment of rocky compositions. Herein, we consider only rocky compositions (comprised of iron and silicates) for the USP planets. On USP orbits, primordial hydrogen-dominated envelopes would rapidly be lost to photoevaporation (e.g., Owen & Wu 2013; Lundkvist et al. 2016; Lopez 2017). Further, though planets on USP orbits could retain high-metallicity water-dominated envelopes, water-rich USP planets would commonly be ≳2 R_⊕, which is inconsistent with the observed USP planet radius distribution (Lopez 2017). Dai et al. (2019) found that most of the sample of 11 hot Earths with mass and radius measurements that they analyzed were consistent with an Earth-like composition, so we adopt the scenario wherein all USP planets have Earth-like compositions as our fiducial case. Since some USP are constrained to be iron enhanced (Price & Rogers 2020), we also consider an extreme scenario wherein all USP planets have a Mercury-like composition to set an upper limit on the planet masses, along with a scenario wherein each planet is randomly assigned an iron core mass fraction to quantify the effect of compositional diversity.

We use one-dimensional models of planet interior structure to convert from the USP planet radius distribution to the USP planet mass distribution in Section 4.1. We include the effect of empirical mass measurements (for the subset of USP planets in our sample that have them) in Section 4.2. Finally, we quantify the effect of tidal distortion on the USP planet mass distribution using three-dimensional models of planet interior structure in Section 4.3.

4.1. USP Mass Distribution from Spherical Models

We first use spherical models to serve as a lower limit on the planet masses when orbital periods approach 1 day. The models are created by integrating the equation for the mass in a spherical shell, and the equation of hydrostatic equilibrium:

$$\begin{eqnarray}&&\displaystyle \frac{{dm}}{{dr}}=4\pi {r}^{2}\rho \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{dp}}{{dr}}=-\rho g,\end{eqnarray} \tag{ 9 }$$

where r is the distance to the planet's center, m is mass interior to r, ρ is the density, and p is pressure. We consider planets consisting of an iron core surrounded by a silicate mantle comprised of enstatite at pressures below 23 GPa, and perovskite phase MgSiO₃ at higher pressures. We use the same equations of state as Price & Rogers (2020) to determine the density at a specified pressure (Olinger 1977; Ahrens 1995; Karki et al. 2000; Anderson et al. 2001; Seager et al. 2007). We integrate in terms of ln(p) starting from the planet center using the Livermore solver for ordinary differential equations, with automatic method switching for stiff and nonstiff problems (LSODA; Hindmarsh 1983; Petzold 1983) numerical integration scheme with a maximum step size of 0.1 and an absolute tolerance of 0.5 × 10⁻⁶. Thus, given an input central pressure, p_c , and core-mantle boundary pressure, p_cmb, the total mass and radius of the planet is determined by the values of m and r at the surface, where p = 1 Pa. The core mass and iron core mass fraction (CMF) are determined from the mass interior to the core-mantle boundary.

Using this integration procedure, we create planet interior structure models and compute planet mass, radius, and CMF over a grid of input core-mantle boundary pressures p_cmb and central to core-mantle boundary pressure ratios ${\hat{p}}_{\max }={p}_{\max }/{p}_{\mathrm{cmb}}$. In our grid, we consider 64 equally spaced values of log (p_cmb/1 Pa) ranging between 8 and 14, and 64 values of log(${\hat{p}}_{\max }$) with 10 ranging from 0−0.1 and 54 ranging from 0.11−2. The additional resolution close to zero was necessary in order to interpolate masses for planets with very low CMFs.

We interpolate within the grid of interior structure models to estimate the mass of each USP planet, for a specified CMF. For each of the 1000 radius samples of each planet, we perform a 2D linear interpolation to determine the values of p_cmb and ${\hat{p}}_{\max }$ that correspond to that particular combination of planet radius and CMF. We then use the pressure coordinates to interpolate a sample of the planet's mass for the specified iron core/silicate mantle composition. This distribution of 1000 mass samples can then be used to determine the median mass and uncertainty for each USP planet, listed in Table 1.

Table 1. USP Planet Radii and Masses

KIC	KOI	Star Radius (R⊙)	Planet Radius (R⊕)	Planet Mass (M⊕)	Distorted Planet Mass (M⊕)
2718885		${1.454}_{-0.05}^{+0.053}$	${1.551}_{-0.124}^{+0.131}$	${5.169}_{-1.417}^{+1.912}$	${8.116}_{-2.501}^{+3.468}$
3112129	4144	${1.012}_{-0.027}^{+0.028}$	${1.142}_{-0.041}^{+0.045}$	${1.67}_{-0.212}^{+0.244}$
4665571	2393	${0.741}_{-0.017}^{+0.016}$	${1.299}_{-0.044}^{+0.041}$	${2.661}_{-0.321}^{+0.315}$
8435766		${0.757}_{-0.015}^{+0.015}$	${1.25}_{-0.025}^{+0.026}$	${2.312}_{-0.155}^{+0.181}$	${2.624}_{-0.179}^{+0.201}$
9642018	4430	${0.827}_{-0.02}^{+0.021}$	${1.349}_{-0.082}^{+0.073}$	${3.052}_{-0.636}^{+0.653}$	${3.884}_{-0.768}^{+0.803}$
11187332		${0.983}_{-0.03}^{+0.03}$	${1.203}_{-0.071}^{+0.072}$	${2.012}_{-0.4}^{+0.468}$	${2.393}_{-0.459}^{+0.544}$
11550689		${0.647}_{-0.012}^{+0.013}$	${1.435}_{-0.029}^{+0.031}$	${3.839}_{-0.272}^{+0.338}$	${4.513}_{-0.318}^{+0.383}$
1717722	3145	${0.669}_{-0.015}^{+0.015}$	${1.193}_{-0.051}^{+0.053}$	${1.952}_{-0.281}^{+0.332}$
3444588	1202	${0.667}_{-0.015}^{+0.015}$	${1.453}_{-0.051}^{+0.05}$	${4.044}_{-0.508}^{+0.529}$
4055304	2119	${0.812}_{-0.016}^{+0.017}$	${1.383}_{-0.031}^{+0.032}$	${3.362}_{-0.281}^{+0.289}$
4144576	2202	${0.856}_{-0.019}^{+0.017}$	${1.188}_{-0.032}^{+0.032}$	${1.922}_{-0.177}^{+0.201}$
5040077	3065	${0.956}_{-0.026}^{+0.025}$	${1.203}_{-0.076}^{+0.076}$	${2.014}_{-0.427}^{+0.498}$
5095635	2607	${1.003}_{-0.027}^{+0.028}$	${1.69}_{-0.073}^{+0.073}$	${7.213}_{-1.151}^{+1.329}$
5175986	2708	${0.654}_{-0.015}^{+0.016}$	${1.612}_{-0.047}^{+0.047}$	${5.99}_{-0.647}^{+0.735}$
5513012	2668	${0.894}_{-0.021}^{+0.021}$	${1.53}_{-0.037}^{+0.042}$	${4.895}_{-0.432}^{+0.531}$
5942808	2250	${0.788}_{-0.018}^{+0.018}$	${1.753}_{-0.048}^{+0.054}$	${8.364}_{-0.867}^{+1.123}$
5972334	191	${0.951}_{-0.028}^{+0.028}$	${1.394}_{-0.052}^{+0.046}$	${3.459}_{-0.461}^{+0.436}$
6129524	2886	${0.847}_{-0.03}^{+0.033}$	${1.517}_{-0.073}^{+0.071}$	${4.733}_{-0.79}^{+0.905}$
6265792	2753	${1.322}_{-0.041}^{+0.041}$	${1.158}_{-0.052}^{+0.055}$	${1.755}_{-0.272}^{+0.32}$
6294819	2852	${1.027}_{-0.058}^{+0.065}$	${1.56}_{-0.099}^{+0.116}$	${5.276}_{-1.153}^{+1.692}$
6310636	1688	${1.61}_{-0.066}^{+0.067}$	${1.701}_{-0.086}^{+0.086}$	${7.407}_{-1.37}^{+1.611}$
6362874	1128	${0.86}_{-0.019}^{+0.019}$	${1.303}_{-0.028}^{+0.03}$	${2.695}_{-0.217}^{+0.226}$
6867588	2571	${1.236}_{-0.039}^{+0.04}$	${1.541}_{-0.062}^{+0.064}$	${5.035}_{-0.715}^{+0.845}$
6934291	1367	${0.747}_{-0.015}^{+0.015}$	${1.498}_{-0.032}^{+0.034}$	${4.52}_{-0.342}^{+0.409}$
6964929	2756	${1.053}_{-0.031}^{+0.034}$	${1.141}_{-0.042}^{+0.044}$	${1.665}_{-0.213}^{+0.237}$
6974658	2925	${0.886}_{-0.019}^{+0.019}$	${0.919}_{-0.092}^{+0.087}$	${0.778}_{-0.237}^{+0.292}$
7102227	1360	${0.736}_{-0.016}^{+0.017}$	${0.918}_{-0.053}^{+0.053}$	${0.773}_{-0.14}^{+0.169}$
7605093	2817	${0.773}_{-0.021}^{+0.021}$	${1.55}_{-0.065}^{+0.071}$	${5.15}_{-0.773}^{+0.969}$
7749002	4325	${0.824}_{-0.023}^{+0.024}$	${1.074}_{-0.055}^{+0.053}$	${1.345}_{-0.228}^{+0.245}$
8278371	1150	${1.837}_{-0.303}^{+0.514}$	${1.684}_{-0.276}^{+0.487}$	${7.11}_{-3.53}^{+13.742}$
8558011	577	${0.846}_{-0.018}^{+0.018}$	${0.973}_{-0.04}^{+0.043}$	${0.95}_{-0.13}^{+0.155}$
8947520	2517	${0.978}_{-0.031}^{+0.031}$	${1.156}_{-0.05}^{+0.056}$	${1.742}_{-0.261}^{+0.322}$
9092504	2716	${0.997}_{-0.041}^{+0.046}$	${1.92}_{-0.091}^{+0.094}$	${12.156}_{-2.202}^{+2.74}$
9149789	2874	${0.831}_{-0.019}^{+0.02}$	${1.158}_{-0.039}^{+0.034}$	${1.755}_{-0.206}^{+0.19}$	${2.005}_{-0.224}^{+0.215}$
9221517	2281	${0.81}_{-0.016}^{+0.016}$	${0.907}_{-0.024}^{+0.024}$	${0.742}_{-0.064}^{+0.074}$
9456281	4207	${0.639}_{-0.013}^{+0.012}$	${1.027}_{-0.084}^{+0.074}$	${1.146}_{-0.294}^{+0.314}$
9472074	2735	${0.807}_{-0.022}^{+0.023}$	${1.487}_{-0.057}^{+0.06}$	${4.399}_{-0.617}^{+0.714}$
9473078	2079	${1.003}_{-0.025}^{+0.025}$	${0.742}_{-0.025}^{+0.025}$	${0.372}_{-0.043}^{+0.044}$
9580167	2548	${0.762}_{-0.017}^{+0.017}$	${1.368}_{-0.092}^{+0.091}$	${3.225}_{-0.734}^{+0.875}$
10024051	2409	${0.697}_{-0.014}^{+0.013}$	${1.512}_{-0.031}^{+0.034}$	${4.667}_{-0.334}^{+0.426}$
10028535	2493	${0.878}_{-0.022}^{+0.023}$	${1.719}_{-0.054}^{+0.053}$	${7.736}_{-0.933}^{+0.971}$
10319385	1169	${1.072}_{-0.027}^{+0.028}$	${1.604}_{-0.044}^{+0.039}$	${5.856}_{-0.583}^{+0.616}$
10468885	2589	${0.857}_{-0.023}^{+0.024}$	${1.392}_{-0.07}^{+0.068}$	${3.443}_{-0.6}^{+0.679}$
10647452	4366	${0.828}_{-0.023}^{+0.023}$	${1.267}_{-0.074}^{+0.065}$	${2.419}_{-0.468}^{+0.493}$
10975146	1300	${0.745}_{-0.023}^{+0.025}$	${1.674}_{-0.054}^{+0.057}$	${6.942}_{-0.833}^{+1.014}$
11401182	1428	${0.708}_{-0.013}^{+0.014}$	${1.716}_{-0.032}^{+0.033}$	${7.679}_{-0.58}^{+0.594}$
11547505	1655	${0.933}_{-0.021}^{+0.022}$	${1.463}_{-0.039}^{+0.04}$	${4.143}_{-0.424}^{+0.423}$
11600889	1442	${1.067}_{-0.025}^{+0.026}$	${1.288}_{-0.033}^{+0.036}$	${2.577}_{-0.236}^{+0.272}$
11752632	2492	${1.089}_{-0.028}^{+0.03}$	${1.025}_{-0.037}^{+0.043}$	${1.139}_{-0.134}^{+0.18}$
11870545	1510	${0.678}_{-0.016}^{+0.017}$	${1.69}_{-0.05}^{+0.053}$	${7.222}_{-0.789}^{+0.963}$
11904151	72	${1.07}_{-0.026}^{+0.027}$	${1.553}_{-0.037}^{+0.042}$	${5.19}_{-0.475}^{+0.543}$
12265786	4595	${0.793}_{-0.019}^{+0.019}$	${1.416}_{-0.115}^{+0.115}$	${3.651}_{-0.974}^{+1.26}$
12405333	3009	${0.804}_{-0.017}^{+0.018}$	${1.086}_{-0.052}^{+0.054}$	${1.396}_{-0.223}^{+0.267}$
5773121	4002	${0.846}_{-0.02}^{+0.02}$	${1.431}_{-0.048}^{+0.042}$	${3.798}_{-0.441}^{+0.457}$
5980208	2742	${0.619}_{-0.012}^{+0.011}$	${1.012}_{-0.034}^{+0.035}$	${1.093}_{-0.125}^{+0.135}$
8416523	4441	${0.767}_{-0.015}^{+0.015}$	${1.399}_{-0.031}^{+0.033}$	${3.499}_{-0.276}^{+0.307}$	${3.965}_{-0.309}^{+0.356}$
9353742	3867	${0.971}_{-0.027}^{+0.027}$	${1.746}_{-0.054}^{+0.053}$	${8.234}_{-0.982}^{+1.085}$
9475552	2694	${0.751}_{-0.016}^{+0.014}$	${1.504}_{-0.035}^{+0.031}$	${4.586}_{-0.371}^{+0.378}$

Note. The masses reported in the "Planet Mass" column in Table 1 are calculated under an assumption of spherical shape and an Earth-like composition (CMF = 0.33). Distorted masses are reported only for planets with orbital periods less than 10 hr.

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Figure 4 shows the calculated masses for 58 USP planets, assuming they all have Earth-like bulk compositions (CMF = 0.33). Qualitatively, the distribution of modeled planet masses closely resembles the patter of planet radii shown in Figure 2 as a function of orbital period, which is reasonable since the masses were calculated from the radii.

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We calculate the USP planet mass distribution in much the same way as the radius distribution in Section 2. For each planet, 1000 mass samples were used to calculate the occurrence rate in logarithmic mass-period bins, weighted by the occurrence rate contributions from Equation 1. Marginalizing over orbital period, we obtain a distribution of USP planet masses.

Figure 5 shows the distribution of spherical USP planet masses given several different composition assumptions—Earth-like (CMF = 0.33), Mercury-like (CMF = 0.7), and random (normal distribution of CMFs with μ = 0.33 and σ = 0.1). As is expected, the Mercury-like distribution yields more massive planets due to its higher iron content compared with the Earth-like distribution.

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To explore the effects of compositional diversity on the USP planet mass distribution, we randomly draw a core mass fraction for each planet from a Gaussian distribution with μ = 0.33 and σ = 0.1 (truncating at zero). We repeat the random CMF assignments 50 times and plot the derived USP mass distributions for each (as green points in Figure 5). The plotting of multiple points illustrates the uncertainties in the USP mass distribution induced by the as-yet-unknown iron mass fraction distribution of the planets. The cumulative trend of the green points show a variety of possibilities for the true mass distribution of USP planets. Ultimately, the spread introduced by an intrinsic CMF scatter at the level of σ = 0.1 is comparable to (at small masses) or smaller than (at large masses) the uncertainties in the USP mass distribution induced by the small number of USP planets in the Kepler data set.

4.2. Effect of Measured USP Planet Masses

4.3. Effect of Tidal Distortion

Given that USP planets orbit very close to their host stars, tidal forces can have significant impacts on the planets' morphology and thus the observed transit depths. Here, we turn to exploring the effect of tidal distortion on the inferred mass distribution of USP planet, using planet interior structure models that take the 3D tidally distorted shapes of the planets into account.

To model the 3D structures of USP planets, we use an updated version of the code presented in Price & Rogers (2020), which uses the method of Hachisu (1986a, 1986b) to self-consistently compute the shapes of ultra-short-period exoplanets. As described in Price & Rogers (2020), this iterative method takes as input the aspect ratio of the planet (measured at the origin of the coordinate system), the core-mantle boundary pressure p_cmb, the ratio of central to core-mantle boundary pressure ${\hat{p}}_{\max }$, and the scaled distance from the planet to the star $\hat{a};$ it returns as output the parameters of interest such as planet mass, radii, and core mass fraction. Since the parameters of interest are output parameters, we cannot simply simulate a planet based on its mass and radii; instead, we interpolate within the model grid to find the parameters associated with a particular configuration.

For a specified transit depth, we expect tidally distorted masses to be larger than spherical masses because the planet will be stretched toward its host start and thus have a larger volume than a sphere with the transit radius. Figure 6 compares the masses inferred from the spherical and tidally distorted models for the seven USP planets with orbital periods below 10 hr. In each case, the median tidally distorted mass is larger than the median spherical mass, as was predicted. Additionally, the discrepancy between the distorted and spherical masses decreases as orbital period increases. This is also expected, as tidal forces are strongest for planets with shorter orbital periods.

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The USP mass distribution incorporating the effect of tidal distortion (computed by replacing the masses derived from spherical models with the masses derived from 3D tidally distorted models) assuming all the planets have an Earth-like composition is shown by the pink points in Figure 4. The planet mass distribution shifts toward higher masses relative to the distribution based on spherical masses, while staying within the uncertainty on the spherical Earth-like distribution. The effect of tidal distortion on the inferred USP mass distribution is similar in magnitude to the effect of including the measured planet masses.

In summary, tidal distortion can have a significant effect on the inferred masses of individual planets, increasing the modeled mass of the closest orbiting planets in the sample by up to 57%. The most extreme USP planets (P_orb ≲ 10 hr) that are significantly affected by tidal distortion are so rare, however, that the effect of tidal distortion on the inferred USP planet mass distribution is diluted.

Our updated USP planet radius distribution, and inferred USP planet mass distribution provide useful empirical anchors to which the predicted outcomes of USP period planet formation mechanisms can be compared. Figure 7 presents the cumulative mass distributions functions (CMDFs) inferred for the Kepler USP planets, along with constraints from various planet formation scenarios.

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There are various formation mechanisms and evolutionary processes that may contribute to the observed population of USP planets. These mechanism include inward migration of rocky and sub-Neptune planets (e.g., Mardling & Lin 2004; Lee & Chiang 2017; Petrovich et al. 2019; Pu & Lai 2019; Millholland & Spalding 2020), photoevaporation of Neptune and sub-Neptune planets (e.g., Owen & Wu 2013; Lundkvist et al. 2016; Lopez 2017), and Roche-lobe overflow (RLO) of gas giant planets (e.g., Valsecchi et al. 2014, 2015; Jackson et al. 2017; Königl et al. 2017).

The defining characteristic of USP planets is their short orbital period. Because orbital periods of ≤1 day fall interior to the dust-sublimation radius in their nascent protoplanetary disks, USP planets must have experienced inward migration either during formation or later dynamical evolution. USP planet formation scenarios involving low-mass (rocky or sub-Neptune) progenitors entail inward migration driven by tidal dissipation. Possible sources of tidal dissipation include tides raised on the host star (Lee & Chiang 2017), planet eccentricity tides driven by secular planet–planet interactions (Petrovich et al. 2019; Pu & Lai 2019), and planetary obliquity tides driven by Cassini states with forced nonzero obliquities (Millholland & Spalding 2020). In each of these formation pathways, the mass distribution of observed USP planets that we have derived would reflect the mass distribution of the planets (and/or the cores of the planets) at the initial orbital separations from which USP planets are sourced (Millholland & Spalding 2020): near the inner-edge of the magnetospherically truncated protoplanetary disk in the case of Lee & Chiang (2017), orbital periods of ∼5–10 days in the high-eccentricity migration scenario of Petrovich et al. (2019), orbital periods of ∼1–3 days in the moderate-eccentricity migration scenario of Pu & Lai (2019), and orbital periods of ∼1–5 days in the obliquity-driven tidal runaway scenario of Millholland & Spalding (2020).

5.1. Low-mass USP Planet Progenitors

5.2. Gas Giant USP Planet Progenitors

We present an updated radius distribution and a completeness-corrected mass distribution of the Kepler USP planets. Expanding on the work of SO14, we take into account new parallax (Gaia Collaboration et al. 2018) and spectroscopic (Winn et al. 2017) measurements to calculate the stellar host and USP planet radii, and present the occurrence rates of USP planets over logarithmic radius-period bins. We then calculate USP mass distributions using various assumptions for the planet compositions, taking into account the tidal distortion of planets with orbital periods less than 10 hr and USP planet mass measurements (when available).

We explore the implications of our results for USP planet formation mechanisms and evolutionary processes. Comparisons to models from Neil & Rogers (2020), Mulders et al. (2020), and Venturini et al. (2020) suggest that USP planets could be formed by a variety of mechanisms, including photoevaporation of sub-Neptunes, accretion in the presence of orbital damping from gas, and dry pebble accretion. In contrast, formation scenarios that predict planet masses that are significantly higher or lower than our inferred USP planet mass distribution are disfavored. If a significant subpopulation (≳10%) of USPs are the remnant cores of tidally stripped hot Jupiters (as suggested by Königl et al. 2017), the inferred USP mass distribution implies that gas giant cores are typically smaller than 7.6 M_⊕ (the 90th percentile of the Earth-like USP planet CMDF). Typical USP planet masses are significantly lower than the total heavy-element masses of gas giants exoplanets (Thorngren et al. 2016), but are comparable to recent constraints on the mass of Jupiter's central compact core (Bolton et al. 2017; Y. Miguel 2021, private communication) accounting for a gradient of heavy elements dissolved in the envelope.

In this work, we have focused on the USP planet sample discovered by Kepler, leveraging the injection-recovery completeness characterization performed by SO14. With only 58 planet candidates in the sample, however, small number statistics are a dominant source of uncertainty in our derived USP planet mass distribution (swamping the effect of planet radial velocity measurements, tidal distortions, and CMF variance of the order of 0.1). Photometry from K2 and the Transiting Exoplanet Survey Satellite (TESS) is enabling new USP planet discoveries around brighter stars (on average) than the SO14 Kepler sample that are more amenable to follow-up characterization, e.g., K2-141b (Barragán et al. 2018; Malavolta et al. 2018), HD 213885 b (Espinoza et al. 2020), HD 80653 b (Frustagli et al. 2020), and TOI-561 b (Lacedelli et al. 2021; Weiss et al. 2021). Once the completeness and reliability of of the K2 and TESS transit surveys have been fully characterized, the USP mass distribution derived here could be further improved by aggregating USP planet samples and occurrence rates from Kepler, TESS, and K2. Additional USP planet mass measurements will further help to benchmark our assumed CMFs and M–R relations used to map from the radius distribution to the mass distribution. Overall, continued refinement of the radius and mass distribution of USP planets may enable further disambiguation of the formation and evolutionary processes sculpting this intriguing class of exoplanets.

This work made use of the gaia-kepler.fun crossmatch database created by Megan Bedell.

Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.

The authors would like to thank B.J. Fulton for providing data from Fulton & Petigura (2018), Yamila Miguel for providing data from Y. Miguel (2021, private communication), Travis Berger for providing details regarding Berger et al. (2020), and Andrew Neil for helpful discussion and providing data from Neil & Rogers (2020).

L.A.R. acknowledges support from NSF grant AST-1615089, and the Research Corporation for Science Advancement through a Cottrell Scholar Award. This work was completed in part with resources provided by the University of Chicago's Research Computing Center. A.S.M.U. acknowledges support by a NC State Park Scholarship and a Goldwater Scholarship.