A Spectroscopic Analysis of the California-Kepler Survey Sample. I. Stellar Parameters, Planetary Radii, and a Slope in the Radius Gap

As previously mentioned, the study by Petigura et al. (2017) was the first to present a spectroscopic analysis of the CKS sample. That study derived spectroscopic parameters using two different spectral synthesis techniques in LTE: SpecMatch and SME@XSEDE. This methodology is very distinct from the one used in our study. SpecMatch was specifically designed for the CKS project. In summary, SpecMatch fits model spectra (computed with Kurucz models by Coelho et al. 2005) individually to five different wavelength segments in the observed spectra and averages the resulting sets of parameters T_eff, log g, metallicity, and v sin i of each segment. The results from the Brewer et al. (2016) catalog were used to calibrate the SpecMatch results.

The spectral synthesis technique SME@XSEDE is an automated version of the spectral synthesis code Spectroscopy Made Easy (SME; Valenti & Piskunov 1996). It uses a line list with atomic parameters taken from the VALD database to interpolate between a grid of plane-parallel MARCS model atmospheres (Gustafsson et al. 2008) until the optimal solution is found, using a χ² minimization. The final results in Petigura et al. (2017) were obtained by applying linear corrections to the raw SME@XSEDE results to put them on the SpecMatch scale (originally calibrated to the Brewer et al. 2016 scale), while for those target stars with consistent results between SpecMatch and SME@XSEDE, the authors adopted the mean value from the two methods.

The top panels of Figure 5 show a comparison of the derived atmospheric parameters T_eff and log g with results from Petigura et al. (2017). In general, there is very good agreement (within the uncertainties) with the stellar parameters derived by Petigura et al. (2017), although there is a small systematic difference of about ∼60 K in the effective temperatures (our T_eff scale being hotter than that of Petigura et al. 2017). When considering mean differences for log g, there is only a small offset with our derived log g values (<Petigura et al. 2017 – this study> = −0.035 dex; rms = 0.14 dex), but it should be noted that 12 stars in our sample have log g > ∼4.7; although their log g errors are within the expected uncertainties (the mean of the log g errors is 0.12 dex), these results are all systematically higher than those of Petigura et al. (2017).

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More recently, Brewer & Fischer (2018) also analyzed the CKS data set. They adopted the SME semi-automated spectral synthesis code (also used in Brewer et al. 2016) to fit the observed spectra and constrain the stellar parameters. The synthesis code uses as input an atomic and molecular line list and a grid of plane-parallel model atmospheres (Castelli & Kurucz 2004) and finds the best solution for global parameters such as T_eff, log g, [M/H], or v_macro. The authors applied an inverse solar analysis to adjust the oscillator strengths, log gfs, of the transitions and performed a combination of spectral synthesis and asteroseismic techniques to obtain their final results.

A comparison of our derived stellar parameters with those obtained by Brewer & Fischer (2018) is shown in the bottom panels of Figure 5 for 847 stars in common. The conclusions are similar to those found with Petigura et al. (2017), which is expected, given that these authors effectively calibrated their results to be on the Brewer & Fischer (2018) and, consequently, the Brewer et al. (2016) scale. For the effective temperatures, the mean difference ($\langle$BF18 – this study$\rangle$) is −69 ± 3 K and the rms = 77 K, again indicating a small systematic offset in the two T_eff scales. For log g, there is a small systematic difference of −0.044 dex, which is not significant.

Buchhave et al. (2012) used multiple observations from high-resolution spectrographs on several telescopes to derive stellar parameters for 152 planet-hosting stars discovered by the Kepler mission. They derived the stellar parameters, T_eff, log g, [m/H], and v_rot, using the spectral synthesis code Stellar Parameter Classification (SPC), which uses a library of model atmospheres (Kurucz 1992) to synthesize the spectrum between 5050 and 5360 Å and measures a cross-correlation function peak that indicates how well the synthetic data reproduce the observed ones.

The top panels of Figure 6 show that there is a small trend in the comparison of our effective temperature scale with that of Buchhave et al. (2012): at higher temperatures (T_eff > 5750 K), our effective temperatures are systematically larger by 82 ± 9 K than those of Buchhave et al. (2012), while for the range between 5200 and 5750 K, our temperatures are systematically lower by 21 ± 10 K; for T_eff values lower than 5200 K, there is no systematic trend but a larger scatter. The agreement with log g is good, with an insignificant systematic difference of −0.02 dex but a higher rms of ∼0.18 dex.

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The middle panels of Figure 6 compare our results with those from Everett et al. (2013) for a sample of 268 faint candidate exoplanet-hosting stars discovered by the Kepler mission. Everett et al. (2013) obtained low-resolution spectra (R = 3000) using the RCSpec long-slit spectrograph on the 4 m telescope at Kitt Peak Observatory and determined T_eff, log g, and [Fe/H] by fitting the observed spectra with synthetic ones computed from stellar model atmospheres (Castelli & Kurucz 2003). Effective temperatures from the Everett et al. (2013) results show a more significant mean offset of ∼100 K relative to ours, with our T_eff scale being hotter, with a similar value for the rms scatter. It is also noticeable that there is a negative trend in the log g differences: for log g > ∼4.3 dex, our derived log g values are systematically larger than theirs (with a mean difference of 0.14 ± 0.01 dex), while for log g smaller than ∼4.3 dex, our results are systematically lower (with a mean difference of 0.05 ± 0.04 dex). The bottom panels of Figure 6 compare results for 343 KOIs in common that were observed with the near-infrared (λ 1.5–1.7 μm), high-resolution spectroscopic (R ∼ 22,500) Apache Point Observatory Galactic Evolution Experiment (APOGEE; Majewski et al. 2017), which is a survey in SDSS-IV.

The stellar parameters shown are part of APOGEE Data Release 14 (DR14; Holtzman et al. 2018) and were derived automatically using the APOGEE Stellar Parameters and Chemical Abundances pipeline (ASPCAP; García Pérez et al. 2016), which fits the observed spectra to grids of synthetic spectra using χ² minimization. There is a significant systematic offset in effective temperatures, with our T_eff scale being hotter than that of APOGEE DR14 by 160 K in the mean, with an rms scatter of 126 K. There is also an offset in log g of −0.06 dex (rms = 0.17 dex), with a negative trend in the mean difference <APOGEE – this study> as a function of log g, in particular for log g values larger than ∼4.0. It is well known, however, that the surface gravities from ASPCAP DR14 have systematic offsets for red giants, as well as for dwarfs (Holtzman et al. 2018; Jönsson et al. 2018).

In this study, we use the Fe i and Fe ii lines in order to derive surface gravities for the stars, with the log g derivation being done concomitantly with the determinations of effective temperatures, microturbulent velocities, and iron abundances (Section 3.1). Correlations that exist between these parameters can lead to systematic errors in the derived parameters that can be investigated. In fact, one of the stellar parameters that is typically not very well constrained via spectroscopy is the surface gravity. Asteroseismology, on the other hand, can provide accurate log g values (to 0.05 dex; Pinsonneault et al. 2018) that can serve as valuable benchmarks to investigate the possible systematic offsets in spectroscopic determinations of log g (Chaplin et al. 2014; Silva Aguirre et al. 2015; Huber et al. 2017; Lundkvist et al. 2018 and references therein).

Figure 7 shows the comparison between our derived log g values with those determined via asteroseismology. The asteroseismic data for 40 sample stars were collected from Silva Aguirre et al. (2015), Huber et al. (2017), and Serenelli et al. (2017). It should be noted that both T_eff and [Fe/H] are essential to constrain the asteroseismic log g. Huber et al. (2017) and Serenelli et al. (2017) used the APOGEE results for effective temperatures and metallicities in deriving log g, while Silva Aguirre et al. (2015) used SME (Valenti & Piskunov 1996) and SPC (Buchhave et al. 2012), two spectral synthesis techniques, to fit optical high-resolution spectra to synthetic ones.

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The seismic results presented in Figure 7 indicate good agreement between the log g determinations in the three different asteroseismic studies (Silva Aguirre et al. 2015, Huber et al. 2017, and Serenelli et al. 2017) for the few stars in common. Our derived log g values (based on the Fe i and Fe ii lines) also compare well with the asteroseismic ones, resulting in an insignificant mean offset in log g (<asteroseismic – this study>) of −0.01 dex and a reasonable rms of 0.08 dex. Such an agreement is completely within the expected uncertainties in spectroscopic log g determinations.

We note that the coverage in the range log g = 3.4–3.8 is rather limited, and the comparison is based only on three stars, two of them with seismic log g values higher than the ones derived here by 0.13 dex. More stars with seismic log g values would be needed in order to reach a firmer conclusion; based on the current data, there is no indication of significant offsets in our spectroscopic log g determinations.

Stellar radii for the CKS sample were previously derived in Johnson et al. (2017) by converting the spectroscopic T_eff, log g, and [Fe/H] from Petigura et al. (2017) into stellar masses, radii, and ages using the Dartmouth Stellar Evolution Program models (Dotter et al. 2008) interpolated with the isochrones package (Morton 2015). Johnson et al. (2017) also used their derived stellar radii to determine the planetary radii. Fulton et al. (2017) used the planetary radii computed by Johnson et al. (2017) to make completeness corrections and calculate the resulting radius distribution. As Gaia DR2 became available, Fulton & Petigura (2018) computed stellar radii using distances from the Gaia DR2 inverted parallaxes and also derived the planetary radii for the CKS sample.

Stellar radii computed by taking into account the parallaxes are, in principle, more precise. Fulton & Petigura (2018) estimated that the errors in their stellar and planetary radii, using distances from the inversion of the Gaia parallaxes, are at the level of 2% and 5%, respectively. Although Fulton & Petigura (2018) obtained a scatter of 13% in the ratios of stellar radii when compared with those from Johnson et al. (2017), their distribution of planet sizes remained basically the same.

Figure 8 shows the comparison between our derived stellar radii with those from Fulton & Petigura (2018); there is an overall good agreement between the results for the vast majority of the targets. The mean stellar radii ratio between Fulton & Petigura (2018) and this study ("F&P18/Us") is 0.9851 ± 0.0004 with a rms scatter of 0.013. A closer look at the results in Figure 8 indicates that the systematic differences are slightly larger for R_⋆ > ∼2.5 R_⊙; if we compute the mean stellar radii ratio and the rms for those stars, we obtain 0.973 ± 0.002 and 0.010, respectively.

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Figure 9 shows the comparison of the planetary radii. The results show good agreement, but there are some outliers for which Fulton & Petigura (2018) compute unrealistically large planetary radii when compared to ours (R_pl > 23 R_⊕ in Fulton & Petigura 2018). In addition, Fulton & Petigura (2018) have a few planets with R_pl ranging roughly between 1000 and 24,000 R_⊕ (e.g., for K03891.01, R_pl = 23,732 R_⊕; planets with R_pl > 200 R_⊕ are off scale in Figure 9(a)). It is worth noting that the errors associated with these unrealistic radii in Fulton & Petigura (2018) are as large as the planetary radii themselves. We note that the discrepant planetary radii shown in the comparison of the results in Figure 9 are not due to differences in the stellar radii, as the comparison between our respective stellar radii values agrees quite well.

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If we remove the outliers from the comparison or the planets with R_pl > 23 R_⊕ from the Fulton & Petigura (2018) sample, we obtain 0.957 ± 0.007 for the mean ratio "F&P18/Us" and an rms of 3%. It is also found that the differences in planetary radii between this study and those from Fulton & Petigura (2018) increase somewhat for the smallest planets (R_pl < 1 R_⊕, with a mean planetary ratio of 0.925), where Figure 9(b)) shows the comparison of our planetary radii ≤1 R_⊕ with those derived by Fulton & Petigura (2018) for planets in common.

Figure 10 shows comparisons of the stellar radii derived in this study with results from asteroseismology. Such comparisons are particularly valuable because asteroseismology can deliver precise stellar radii, providing the basis for an assessment of both the accuracy and precision of our derived stellar radii. The three panels in Figure 10 show results for samples of stars in common with three asteroseismic studies (these same studies were used for the log g comparisons in Figure 7): Serenelli et al. (2017, panel (a)), Silva Aguirre et al. (2015, panel (b)), and Huber et al. (2017, panel (c)). The mean values for the ratios "other studies/this study," along with the corresponding rms, are presented in each panel of Figure 10; it is clear that our stellar radii compare very well with those from asteroseismology.

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It should be kept in mind, however, that the selected asteroseismic studies use different methodologies (scaling relations versus modeling of individual frequencies), as well as different sets of stellar parameters (effective temperatures and metallicities), which are needed for their determinations of stellar radii. The internal precisions estimated for their derived radii are 2.7% for Serenelli et al. (2017), 1.2% for Silva Aguirre et al. (2015), and 2.7% for Huber et al. (2017). The internal precision inferred from our determinations of stellar radii (Table 3) is also at a similar level: ∼2.8%. Concerning systematic offsets, we find that our derived radii are just slightly higher that the asteroseismic ones, being on average 4.1% larger than those in Serenelli et al. (2017), 1.7% larger than those in Silva Aguirre et al. (2015), and 0.8% larger than those in Huber et al. (2017). The rms values obtained for the three studies are also quite small, indicating a small scatter of 0.02–0.03. All in all, we can conclude that our derived stellar radii from precise parallaxes and spectroscopic determinations of the effective temperatures achieve a comparable precision against stellar radii derived from asteroseismology and do not contain systematic offsets that are much larger than the typical offsets seen between the different asteroseismic studies themselves.

Until recently, the detailed properties for the unprecedented variety of planets discovered by the Kepler mission (Borucki et al. 2010; Koch et al. 2010; Borucki 2016) have been ambiguous due in part to uncertainties in the planetary radii that stem from uncertainties in the stellar radii, which have been for the most part estimated from broadband photometry.

Based on improved measurements for the stellar radii for the CKS sample by Johnson et al. (2017; using stellar and planetary parameters from Petigura et al. 2017 and Johnson et al. 2017), Fulton et al. (2017) found a bimodal distribution for the small-planet radii having peaks at ∼1.3 and ∼2.4 R_⊕, with a gap between 1.5 and 2.0 R_⊕. Van Eylen et al. (2018), using asteroseismic radii for a small subsample of CKS targets with very precise radii, also detected a bimodal distribution with a clear gap around 2 R_⊕. A similar gap in planetary radius was confirmed by Fulton & Petigura (2018) using Gaia DR2 parallaxes. The dearth of planets at R_pl ∼ 1.8 R_⊕ has been predicted by theoretical models and is interpreted as a transition radius separating planets with masses large enough to retain their gas envelope and those that have lost their atmospheres and consist of their remnant cores (Owen & Wu 2013; Lopez & Fortney 2014; Chen & Rogers 2016; Lopez & Rice 2016; Owen & Wu 2017; Ginzburg et al. 2018).

The distribution of planetary radii derived in this study is shown in the different histograms in Figure 11. Although the methodology used in the determination of stellar parameters (needed to derive the stellar and planetary radii) for the CKS sample is completely independent of previous studies in the literature analyzing the same data set, the planetary radii distributions (presented in all panels of Figure 11) are found to be bimodal, showing the clear presence of a valley in the small-sized planet radius distribution. These independent results give support to the fact that the lack of planets around 2.0 R_⊕ is not an analysis artifact but rather represents a real transition between rocky planets and those with extensive atmospheres (Weiss & Marcy 2014; Rogers 2015).

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The full sample analyzed in this study (composed of 1633 planets), without any cuts, is shown in panel (a) of Figure 11. It should be kept in mind, however, that the full sample does not include KOIs with planets deemed "false positives" (see Section 2). The uncorrected-completeness radius distribution shows two clear peaks: one at ∼1.6 R_⊕ and another at ∼2.8 R_⊕. The radius gap is seen roughly between 1.8 and 2.2 R_⊕.

It is of interest to investigate the positions of the peaks and the planetary radius gap using only precise planetary radii, as it is clear that the derived radii have different levels of precision depending, for example, on the errors in the Gaia DR2 parallaxes, which are folded into the distance error estimates by Bailer-Jones et al. (2018). In panel (b), we show a similar histogram as in panel (a), but in this case, we restricted our sample to consider only those planets with uncertainties in the derived radii of less than 8% (corresponding to ∼2× the median uncertainty; Section 3.3). In panel (c), we applied the same selection as in panel (b), but, similarly to Fulton et al. (2017), we removed from the sample any planets with P > 100 days and in a transit configuration corresponding to the impact parameter, b, being larger than 0.7; this is our "clean" sample, which is composed of 965 planets.

The radii histograms shown in Figure 11 panels (a) (for the entire sample), (b) (for planets with precise radii), and (c) (for the "clean" sample) are in general quite similar: in all distributions, there is a dearth of planets at ∼2.0 R_⊕, and the location of the two peaks in the distributions is similar as well. The full sample has a small excess of planets at R_pl ∼ 10 R_⊕, and this is mostly due to the presence of planets with P > 100 days that were removed in the "clean" sample. There is also a large population of small planets with R_pl < 1.0 R_⊕, but, as pointed out by Berger et al. (2018), it is expected that some of these small planets will be classified as "false positives" in the future.

To assure that the trend described in the previous section is not an artifact of completeness and affected by the lack of detectability of planets with small radii and/or long orbital periods by Kepler, we reconstruct the planet occurrence rate of the Kepler sample after applying completeness corrections of Christiansen et al. (2016, 2015), Fulton et al. (2017), Fulton & Petigura (2018), and Mulders et al. (2018, 2016).

We used the injection-recovery experiments described in Christiansen et al. (2015, 2016). They measured the Kepler pipeline detection efficiency by injecting simulated transiting planets into the raw pixel data and analyzed the recovery rate after processing them with the Kepler pipeline to reconstruct the planet occurrence rate of the Kepler sample.

Considering injections in the transit signals for the stars included in our sample, we used our derived stellar and planetary radii to calculate the reliability of the transit measure (m) following the procedure described in Fulton et al. (2017),

$$\begin{eqnarray*}&&m={\left(\displaystyle \frac{{R}_{\mathrm{pl}}}{{R}_{\star }}\right)}^{2}\displaystyle \frac{1}{{\mathrm{CDPP}}_{\mathrm{dur}}}\sqrt{\displaystyle \frac{{t}_{\mathrm{obs}}}{P}},\end{eqnarray*}$$

where t_obs is the time that a star of radius R_⋆, harboring a planet of radius R_pl and orbital period P, was observed, while the combined differential photometric precision (CDPP_dur; Koch et al. 2010) is the noise of a transit signal interpolated to the transit duration time. The transit parameters necessary to do the calculations were taken from the Kepler database (DR25; Thompson et al. 2018).

To account for the pipeline efficiency, Fulton et al. (2017) fit a Γ cumulative distribution function of the form

$$\begin{eqnarray*}&&C(m;k,\theta ,l)={\int }_{0}^{\tfrac{m-l}{\theta }}{x}^{k-1}{e}^{-x}\cdot {dx}\end{eqnarray*}$$

to model the distribution of values from the injection-recovery transit signals (m); we use the k, l, and θ values as determined by Fulton et al. (2017).

We determined the detection probability, p_det, obtained by using C(m) values, and the geometric transit probability, p_tr, to check for the survey sensitivity.

The transit probability, p_tr, defined as the geometric probability that a planet with radius R_pl transiting a host star of radius R_⋆ at a distance of a could be detected, is p_tr = 0.7 R_⋆/a, where the factor 0.7 corresponds to the imposed limit in the impact parameter of the planet candidates, according to the work of Fulton et al.

To compensate for incompleteness due to a lack of detection efficiency, p_det, or a low probability of transit detection, p_tr, we weighted each planet by the inverse of these probabilities:

$$\begin{eqnarray*}&&{w}_{i}=\displaystyle \frac{1}{{p}_{\det }\times {p}_{\mathrm{tr}}}.\end{eqnarray*}$$

The true measure of the occurrence rate, f_bin, the number of planets per star in any orbital period or radius bin, is then given by

$$\begin{eqnarray*}&&{f}_{\mathrm{bin}}=\displaystyle \frac{1}{{N}_{\star }}\displaystyle \sum _{i=1}^{{n}_{\mathrm{pl},\mathrm{bin}}}{w}_{i}.\end{eqnarray*}$$

Figure 11(d) presents the same distribution of planetary radii for our "clean sample" (shown in panel (c)) in comparison with the completeness-corrected distribution, shown as the dashed line. It is noticeable that the location of the gap and the peaks in the completeness-corrected distribution are shifted to slightly smaller radii when compared with the uncorrected distribution, similar to what has been found by Fulton & Petigura (2018).

The location of the peaks of the completeness-corrected planetary radius distribution can be estimated from the kernel density estimate for a Gaussian distribution (shown in Figure 11(e)); the peak positions of the distribution are found at 1.47 ± 0.05 and 2.72 ± 0.10 R_⊕, and there is a radius gap at 1.89 ± 0.07 R_⊕ (Figure 11(e)). The completeness-corrected distribution from Fulton & Petigura (2018) is also shown for comparison as the gray line in Figure 11(e). There is a marginal shift in our planetary distribution relative to Fulton & Petigura (2018). Van Eylen et al. (2018) also investigated the planetary radii distributions in their, albeit smaller, sample but with precise radii from asteroseismic parameters; they find the peaks to be at 1.5 and 2.5 R_⊕, respectively, with the radius gap minimum falling at 2.0 R_⊕.

As discussed in Section 5.1.1, one mechanism suggested to explain the bimodal small-planet radius distribution is photoevaporation: X-ray and UV fluxes from the young planet-hosting star evaporate the envelopes of the H–He rich sub-Neptunes, exposing their stripped rocky cores. In addition to photoevaporation, Ginzburg et al. (2018) modeled the radius valley as being caused by the energy from young, hot planetary cores driving planetary-mass-dependent atmospheric mass loss (or "core-powered mass loss"). Several theoretical models also predict the shape and slope of the evaporation valley (e.g., Owen & Wu 2013, 2017; Jin & Mordasini 2018); the planetary radii for the CKS derived in this study are precise enough to investigate the presence of such signatures in the planetary radius as a function of orbital period or incident flux.

Figures 12 and 13 present the derived exoplanetary radii as a function of their orbital periods (with Kepler DR25 periods taken from Thompson et al. 2018) and insolation fluxes, respectively. The stellar fluxes were calculated from the derived T_eff and R_⋆, while the semimajor axes of the planetary orbits, assuming an orbital eccentricity equal to zero, were used to compute the incident flux at the planet.

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In both figures, panel (a) shows the full sample with 1633 planets, without any cuts, and panel (b) shows a subset of the "clean" sample for the region containing the smaller exoplanets (radii less than ∼4 R_⊕). Visual inspection of these figures clearly indicates the presence of two clouds representing the population density of the distributions of super-Earths and sub-Neptunes in the log radius – log period/incident flux planes, with a low density of exoplanets at radii between the two clouds, with R_pl ∼ 2 R_⊕. This "evaporation valley" corresponds to the separation between the super-Earth and sub-Neptune regimes.

The exoplanet population plotted in Figures 12 and 13 also shows an overall lack of sub-Neptunes with short orbital periods (P < ∼3 days) and incident fluxes (relative to Earth) > ∼700, which is likely related to the photoevaporation of the atmospheres of sub-Neptune-sized exoplanets that are very close to their parent stars and suffer high stellar incident flux levels (Ikoma & Hori 2012; Lopez et al. 2012; Ciardi et al. 2013; Owen & Wu 2013; Wu & Lithwick 2013). It should be pointed out, however, that some exoplanets have been detected in the sub-Neptune "desert" region (see West et al. 2018).

Close inspection of Figure 12 gives some hints that the value of R_pl in the evaporation valley minimum overall decreases with increasing orbital period. The change in the radius gap minimum as a function of orbital period can be quantified in a simple way by dividing our "clean" sample of planets (with radii errors ≤8%, P ≤ 100 days, and b ≤ 0.7) into 10 orbital period bins containing an equal number of planets. Within each orbital period bin, the minimum value in the radius gap was measured, resulting in a linear relation defined by log(R_pl) = (−0.11 ± 0.02)*log(P) + (0.39 ± 0.01), with R_pl in units of Earth radii and P in days, or R_pl scaling as P^−0.11; the slope, along with the corresponding prediction interval, is shown in Figure 12(b)). A similar fit was done for the valley observed between the planetary radii and the incident fluxes (Figure 13(b)), finding a linear relation defined as log(R_pl) = (+0.12 ± 0.02)*log(F) + (0.04 ± 0.03), with F in units of incident flux at the Earth, or R_pl scaling as F^+0.12.

Van Eylen et al. (2018) also detected a slope in the planetary radius valley as a function of the planetary orbital period from a small sample of 75 stars, well characterized by asteroseismology and having precise radii for their 117 associated planets. Their derived slope of ${P}^{-{0.09}_{-0.04}^{+0.02}}$ for the small-planet radius gap, although from a much smaller sample, is in good agreement with ours within the uncertainties. They also model the valley slope after restricting their sample to periods <25 days and find P^−0.10; if we follow the same cut for the much larger CKS sample, we obtain a similar slope of P^{−0.11 ± 0.03}.

The shape and, in particular, the value of the slope in the evaporation valley can constrain planetary formation models. Lopez & Rice (2016) used different models to point out that the transition radius between rocky super-Earths and sub-Neptunes with volatile envelopes scales differently with the orbital period depending on the planet formation scenario; the transition radius should decrease for longer orbital periods in the case of a photoevaporation scenario (R_pl scaling as P^−0.15). Contrarily, it should increase if the primordial rocky planets formed after the protoplanetary disks dissipated in a gas-poor formation model with a positive slope for the radius gap (R_pl scaling as ${P}^{+0.07\pm 0.10}$).

According to models by Owen & Wu (2017), the slope of the transition radius (or the upper envelope of the super-Earth radius) with period derived through evaporation models can change when considering different evaporation efficiencies in these models, ranging from P^−0.25, when a constant evaporation efficiency is considered for all of the planets, to P^−0.16, when evaporation efficiency depends on the planet density. Owen & Wu (2017), as well as Jin & Mordasini (2018), also investigate how the bulk composition affects the planetary radii of super-Earths and sub-Neptunes as a function of period (or orbital semimajor axis). Depending on the mixture of iron, silicates, or ices, for example, the maximum radius of super-Earths changes as a function of orbital period. Using Figure 10 from Owen & Wu (2017) as an example, their model, which mimics the composition of the Earth (with ∼one-third iron) and considers a variable efficiency for the evaporation, provides the closest match to the upper envelope defining super-Earth radii versus period for periods less than ∼8 days. The model then declines more steeply than the observed distribution toward increasing periods, with the result that the observed maximum radius for super-Earths falls well above the Owen & Wu (2017) model core with one-third iron. Since their "icy" models with one-third ice and two-thirds silicates have larger radii than the iron cores, the observed super-Earths with larger radii at longer periods may signal a shift in the overall compositions of super-Earths with short periods compared to those with long periods.

The location of the radius gap in Figure 12 and its negative slope agree with what is expected from photoevaporation models, as well as core-powered mass-loss models, which predict a slope of −0.11 (Gupta & Schlichting 2018). The change in the maximum radius versus orbital period observed for super-Earths in Figure 12 agrees qualitatively with model cores from Owen & Wu (2017); a quantitative comparison with model predictions can constrain the core compositions and may even be able to map compositions at different orbital periods around different types of host stars.

The shape and location of the planet radius versus orbital period valley is also in qualitative agreement with the core-powered mass-loss mechanism from Ginzburg et al. (2018). In this model, atmospheric mass loss is more effective if the equilibrium temperature of the planet, T_eq, is high, which is, in general, the case for more closely orbiting planets, or more massive main-sequence stars, which have larger luminosities. Ginzburg et al. (2018) evolved a model planetary distribution and found that the position of the valley minimum shifts from ∼2 R_⊕ for P < 10 days to ∼1.5 R_⊕ for longer orbital periods. Recently, Fulton & Petigura (2018) established a relation between the cumulative distribution of planets versus the incident flux as a function of stellar mass, with the distribution shifted to higher incident fluxes for larger stellar masses.

Figure 13, where planetary radii are plotted versus incident flux, tells effectively the same story as Figure 12. The incident flux shown is the current flux, although within the photoevaporation model, Owen & Wu (2013, 2017) pointed out that it is the flux from the young, presumably active host star that is most important in sculpting the distributions of planetary radii in Figures 12 and 13. Photoevaporation is most effectively driven by the stellar integrated X-ray and extreme UV (EUV) fluxes during the first 100 Myr of the life of the star (Owen & Wu 2013). The planetary composition also plays a role, as H and He are most affected by the EUV flux, while the metals are more easily evaporated by the X-ray flux (Owen & Wu 2017). The planetary radii distributions as functions of orbital period and incident flux shown in Figures 12 and 13 are thus the result of their early X-ray and EUV radiation environments and distance from their young host star.

The CKS planetary sample studied here can be divided into four exoplanet size regimes as follows.

• 1.  
  Jupiters with 8 R_⊕ < R_pl ≤ 20 R_⊕.
• 2.  
  Sub-Saturns with 4 R_⊕ < R_pl ≤ 8 R_⊕.
• 3.  
  Sub-Neptunes with 2 R_⊕ < R_pl ≤ 4 R_⊕.
• 4.  
  Super-Earths with R_pl ≤ 2 R_⊕.

The sample (in this case, we are considering the full sample of planets without any cuts) has a majority of exoplanets with small sizes, split roughly in equal numbers between the classes of super-Earths (736 exoplanets) and sub-Neptunes (706 exoplanets). It has a much smaller number of sub-Saturns (96 exoplanets) and 93 exoplanetary systems containing at least one Jupiter-like planet. About 20% of these systems have a single hot Jupiter without any detectable inner or outer companions, while almost half of the warm Jupiter sample (∼30 out of ∼70) is found around a variety of exoplanetary system architectures. We define hot Jupiters as those with periods less than 10 days, while warm Jupiters have periods larger than 10 days.

Figure 12(a) also shows, as red symbols, the weighted median values and rms scatter of the planetary radii and orbital period distributions for each planet size domain: super-Earths (filled square), sub-Neptunes (circle), sub-Saturns (triangle), hot Jupiters (open square), and warm Jupiters (diamond). It is clear that the super-Earths have a median value for planetary radii and orbital periods that is lower than that of the sub-Neptunes, suggesting a possible correlation in the sense that the sizes of the exoplanets generally increase with orbital periods; some of this correlation is due to incompleteness in transit measurements given, for example, the difficulty in detecting small planets at larger distances from the parent star (with larger orbital periods). The correlation between planetary radii and orbital periods extends, however, toward the larger exoplanet groups containing the sub-Saturns and warm Jupiters. For these larger planets, the observational biases should not be significant, in particular for systems with orbital periods less than ∼500 days (taking completeness values from Silburt et al. 2015).

The hot Jupiters do not fit into the trend of increasing orbital period with increasing exoplanet size. The median value of R_pl–P for the warm Jupiters (diamond) generally follows the trend delineated by the smaller planets, while the hot Jupiters occupy a different locus in the R_pl–orbital period plane (median represented by the open square), having much shorter orbital periods, on average, and being much closer to their parent stars. Hot Jupiters are believed to have formed several astronomical units away from their parent stars and undergone extreme migration into the exoplanetary system's inner region, destabilizing any small exoplanets, scattering them out of the system, or destroying them (Latham et al. 2011; Morbidelli 2014; Mustill et al. 2015). We also note that Huang et al. (2016) analyzed a sample of 45 hot Jupiters and 27 warm Jupiters from the Kepler catalog and proposed that warm Jupiters with low-mass companions would be formed in situ, not affecting their small neighbors, and that warm Jupiters with no detectable companions may be a distinct population.

With the position of the hot Jupiters in the R_pl–P plane dominated by migration, the other size groups (super-Earths, sub-Neptunes, sub-Saturns, and warm Jupiters) may define a seemingly tight correlation of median radius with median orbital period. This correlation is influenced by the dearth of sub-Neptunes with short orbital periods.

To evaluate how much the correlation depends on the limits adopted for cutting the planet sample in terms of radii and orbital periods, we analyze three different samples that will correspond to different levels of completeness. (1) We consider the entire sample of warm Jupiters, sub-Saturns, sub-Neptunes, and super-Earths at any orbital period, without any considerations about observational biases. A fit to the trend in log–log space results in a well-defined power-law relation, such that R_pl ∼ P^0.84±0.11, with R_pl in Earth radii and orbital period in days. (2) If we limit the sample to include only those planets with R_pl ≥ 2 R_⊕ (equivalent to assuming that roughly all sub-Neptunes and larger planets can be detected at any orbital period), we obtain a steeper power law but with smaller errors in the fit: R_pl ∼ P^1.1±0.3. (3) If we now add a cut in orbital period, considering only those planets with R_pl ≥ 2 R_⊕ and P < 500 days, we obtain the same power law as before but with an increase in the uncertainties in the fit (R_pl ∼ P^1.09±0.13). We expect that the Kepler completion levels would be significant for this regime and verify from these tests that the correlation of planetary radii with orbital period does not change significantly in the three samples analyzed.

Helled et al. (2016) also suggested the existence of a correlation between planetary radius and orbital period for exoplanets with radii smaller than 4 R_⊕ using Kepler data. They did a statistical analysis that took into consideration completeness values for the detection of planets by Kepler (Silburt et al. 2015) to conclude that this correlation was not the result of a selection bias. Helled et al. (2016) obtained a power-law relation between planet radius and orbital period of R_pl ∼ P^0.5–0.6, which is similar, within the uncertainties, to the value of ∼0.8 obtained here. If true, the correlation between radii and orbital periods found for the smaller planets in the CKS sample may imply that larger planets would also more likely form at larger distances from the host star.