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

Cintia F. Martinez Observatório Nacional, Rua General José Cristino, 77,
20921-400 São Cristóvão, Rio de Janeiro, RJ, Brazil  Katia Cunha  Observatório
Nacional, Rua General José Cristino, 77, 20921-400 São Cristóvão, Rio de
Janeiro, RJ, Brazil  Steward Observatory, University of Arizona, 933 North
Cherry Ave., Tucson, AZ 85721, USA  Luan Ghezzi  Observatório Nacional, Rua
General José Cristino, 77, 20921-400 São Cristóvão, Rio de Janeiro, RJ, Brazil
Verne V. Smith  National Optical Astronomy Observatory, 950 North Cherry
Avenue, Tucson, AZ 85719, USA

(Received October, 2018; Revised February, 2019)

###### Abstract

We present results from a quantitative spectroscopic analysis conducted on
archival Keck/HIRES high-resolution spectra from the California-$Kepler$
Survey (CKS) sample of transiting planetary host stars identified from the
$Kepler$ mission. The spectroscopic analysis was based on a carefully selected
set of Fe I and Fe II lines, resulting in precise values for the stellar
parameters of effective temperature (Teff) and surface gravity (log $g$).
Combining the stellar parameters with $Gaia$ DR2 parallaxes and precise
distances, we derived both stellar and planetary radii for our sample, with a
median internal uncertainty of 2.8$\%$ in the stellar radii and 3.7$\%$ in the
planetary radii. An investigation into the distribution of planetary radii
confirmed the bimodal nature of this distribution for the small radius planets
found in previous studies, with peaks at: $\sim$1.47 $\pm$ 0.05 R⊕ and
$\sim$2.72 $\pm$ 0.10 R⊕, with a gap at $\sim$ 1.9R⊕. Previous studies that
modeled planetary formation that is dominated by photo-evaporation predicted
this bimodal radii distribution and the presence of a radius gap, or photo-
evaporation valley. Our results are in overall agreement with these models.
The high internal precision achieved here in the derived planetary radii
clearly reveal the presence of a slope in the photo-evaporation valley for the
CKS sample, indicating that the position of the radius gap decreases with
orbital period; this decrease was fit by a power law of the form Rpl $\propto$
P-0.11, which is consistent with photo-evaporation and Earth-like core
composition models of planet formation.

(stars:) planetary systems — stars: fundamental parameters — techniques:
spectroscopic, parallaxes

††journal: ApJ††software: Python packages: astropy (Astropy Collaboration,
2013), $iso\\-cla\\-ssi\\-fy$ (Huber et al., 2017) and R packages: astro
(Kelvin, 2014), mclust (Scrucca et al., 2016), MASS (Venables & Ripley, 2002).

##  1 Introduction

One well-known axiom in exoplanetary studies connects the properties of
exoplanets with the properties of their host stars through the expression that
“one can only know the planet to the level that the host star is known”. To
know certain exoplanetary physical properties (such as radius, mass, and mean
density) requires the knowledge of those same physical properties for the host
star: for transiting exoplanets, it is possible to determine the planetary
radius relative to the stellar radius (Rp/R⋆) from the analysis of the transit
light curve, while the planetary mass depends on the host stellar mass (Mp
$\propto$ M${}_{\star}^{2/3}$) and is derived from the radial velocity curve.

Although several thousand exoplanet candidates have been discovered by the
$Kepler$ mission (Borucki et al., 2010; Koch et al., 2010; Borucki, 2016), the
initial stellar parameters, as derived from the Kepler Input Catalog (KIC),
were limited in accuracy, as the KIC provided 7-band photometry (g, r, i, and
z, plus J, H, and K from 2MASS), along with a narrow filter centered on the Mg
I b-lines used as a luminosity indicator (Borucki, 2016). The deduced stellar
radii were found to have a scatter of about 30-40$\%$ (Huber et al., 2014),
with the errors for late-type dwarfs being even larger (Dressing &
Charbonneau, 2013). Errors of this size result in large uncertainties in
exoplanetary properties and could mask correlations and trends in exoplanet
properties or the types of systems they inhabit.

Improvements in the derived stellar radii can be accomplished by conducting
precise, quantitative spectroscopic analyses of the host stars, in particular,
using high-quality, high-resolution spectra. In addition to high-resolution
spectroscopic data, $Gaia$ (Gaia Collaboration et al., 2018) has now provided
precise parallaxes for a large number of $Kepler$ exoplanet host-stars. The
$Gaia$ parallaxes, combined with tightly-constrained stellar parameters
derived from spectroscopy, result in stellar and exoplanetary radii with
accuracies of 3-5$\%$ (Stassun et al., 2017).

A recent example of how improved measurements for stellar radii can reveal new
characteristics in exoplanet populations can be found from the
California-$Kepler$ Survey (CKS- Petigura et al., 2017), where Fulton et al.
(2017) discovered a bimodal distribution for small planet radii, with peaks at
$\sim$1.3R⊕ and $\sim$2.4R⊕, and a gap in-between that points to a transition
radius separating super-Earths from sub-Neptunes. An earlier detection of this
small-planet gap was prevented due to the large uncertainties in the
exoplanetary radii, although a small-planet gap was originally predicted by
several formation models (Owen & Wu, 2013; Lopez & Fortney, 2014; Jin et al.,
2014; Chen & Rogers, 2016; Lopez & Rice, 2016), which predicted that gaseous
planets may suffer photo-evaporation of their envelopes by radiation coming
from their host stars. The presence of the small-planet gap has now been
confirmed by other studies (Van Eylen et al., 2018; Fulton & Petigura, 2018;
Berger et al., 2018). In addition to photo-evaporation, Ginzburg et al. (2016)
and more recently, Ginzburg et al. (2018) have shown that the small-planet gap
can also be produced by a young, hot planetary core, whose energy can drive
atmospheric mass loss, with the ability to retain an atmosphere depending on
the mass of the planet. Other processes, such as internal planetary outgassing
(Dorn et al., 2018), or large impacts on young planets (Inamdar & Schlichting,
2016) can both produce a planetary atmosphere or remove it.

In this study, a homogeneous spectroscopic analysis has been carried out in
order to derive precise stellar parameters (effective temperatures, Teff and
surface gravity) for a sample of $Kepler$ hosts, using a homogeneous set of
high-resolution optical Keck/HIRES spectra made public by the
California-$Kepler$ Survey team (Petigura et al., 2017; Johnson et al., 2017;
Fulton et al., 2017). Stellar radii are derived using the $Gaia$ DR2
parallaxes (Bailer-Jones et al., 2018; Lindegren et al., 2018), with the
improvements in the stellar radii given by the precise parallaxes and better
distances from $Gaia$. Based on our analysis, we independently derive
planetary radii and confirm the presence of a small-planet gap in the
distribution of exoplanetary sizes.

##  2 Observations

The high-resolution spectra analyzed in this study were obtained as part of
the California-$Kepler$ Survey (CKS- Petigura et al., 2017; Johnson et al.,
2017; Fulton et al., 2017), a large observational campaign targeting stars
identified as $Kepler$ Objects of Interest (KOI’s). The CKS campaign was
conducted between 2012 - 2014 using the High Resolution Echelle Spectrometer
(HIRES- Vogt et al., 1994) at the Keck telescope. All CKS spectra analyzed
here were reduced by Petigura et al. (2017) and are publicly available on the
Keck Observatory Archive. The spectra were obtained from https://california-
planet-search.github.io/cks-website/. (Sample HIRES spectra of the CKS are
shown in Figure 2 from Petigura et al. 2017).

From the full CKS sample of 1305 stars, we removed a small sample of 20 stars
that had low quality spectra (all having S/N ratios lower than $\sim$30). The
remaining sample containing 1285 stars was analyzed spectroscopically in this
study; most of these stars have spectra with S/N ratios between 40-70, while
some $\sim$ 8$\%$ of the spectra have excellent quality with S/N higher than
100.

##  3 Analysis

###  3.1 Spectroscopic Stellar Parameters

We derived stellar parameters (Teff, log $g$, and microturbulent velocities),
as well as metallicities (taken to be represented by [Fe/H]) for the studied
stars using standard techniques employed in quantitative stellar spectroscopy,
which relies on equivalent width (EW) measurements of selected samples of Fe I
and Fe II lines.

All abundance calculations were done under the assumption of local
thermodynamic equilibrium (LTE) using 1D model atmospheres. The atmospheric
parameters were obtained by iterating until the line-by-line Fe I abundances,
A(Fe I), exhibit no dependence with the excitation potential of the
transitions ($EP$; excitation equilibrium) and, at the same time, the values
of A(Fe I) show no trend with the reduced equivalent widths
(log(EW/$\lambda$)), while finally, the mean Fe I and Fe II abundances reach
agreement (ionization equilibrium). These 3 conditions define the stellar
Teff, log $g$, and microturbulent velocity ($\xi$). Fig. 1 shows an example of
the iterated solution for the effective temperature, log $g$, and
microturbulent velocity, as well as the abundance of iron for sample star,
KOI-1.

Figure 1: An example of the applied methodology to KOI-1, one of the stars in
our sample. Iron abundances (A(Fe I)) as a function of the excitation
potential (EP) of the Fe I and Fe II transitions (top panel) and the reduced
equivalent width of the lines (EW/$\lambda$; bottom panel). The atmospheric
parameters, iron abundances and microturbulent velocities are obtained once
the correlation coefficients of the dashed lines show no dependence with the
line parameters.

In order to analyze the large number of stars in our sample in an efficient
and homogeneous way, we used the automated stellar parameter and metallicity
pipeline described in detail in Ghezzi et al. (2010, 2018). This code uses the
updated version of the routine ARES (Sousa et al., 2015) to measure EWs of Fe
I and Fe II lines automatically, the abundance analysis code MOOG (Sneden,
1973) to compute the iron abundances, and model atmospheres from the Kurucz
ATLAS9 ODFNEW grid (Castelli & Kurucz, 2004). In summary, the code starts with
a model atmosphere calculated assuming solar values for Teff, log $g$, and
metallicity and then iterates until obtaining a final adjusted value for the
spectroscopic parameters of each star.

The adopted line list in this study was taken from Ghezzi et al. (2018) and
consists of 158 Fe I and 18 Fe II isolated and unblended lines. The log $gf$
values of the Fe I and II lines were obtained in Ghezzi et al. (2018) from an
inverse solar analysis using a Kurucz ATLAS9 ODFNEW model atmosphere for the
Sun (Teff = 5777 K, log $g$ = 4.44, [Fe/H] = 0.00 and $\xi$ = 1.00 km s-1) and
an adopted solar abundance (A(Fe⊙) = 7.50) from Asplund et al. (2009).

As a consistency check, we also measured manually the equivalent widths of a
total of 540 Fe I and Fe II in four sample stars (KOI’s 64, 268, 280 and 5782
with a mean S/N of 70 in their spectra) using the IRAF package $splot$. A
comparison of our manual EW measurements with the automatic ones using the
ARES code is presented in Figure 2. Despite the fact that there were some
lines with discrepant equivalent width measurements (these are not an issue in
the final solution because the pipeline performs two rounds of sigma-clipping
to remove lines with abundances that are too discrepant from the average
values) the equivalent widths compared well, showing, on average, a small
offset of 1.25 mÅ (in the sense of ARES EW being larger than ours) and a RMS
scatter of 3.18 mÅ.

Figure 2: Comparison between automatic (using ARES code) and manual equivalent
width measurements (using $splot$ task from IRAF) for a sample of 540 Fe I and
Fe II lines in four target stars: KOI-64 (open squares), KOI-268 (filled
circles), KOI-280 (open diamonds) and KOI-5782 (open triangles). The red
dashed line represents perfect agreement.

Table 1 presents the resulting effective temperatures, surface gravities,
microturbulent velocities, and stellar radii (these will be discussed in
Section 3.2), and their respective uncertainties, for all stars in our sample.
The metallicities obtained for the CKS sample will be presented and discussed
in a forthcoming paper (Martinez et al. 2019 - Paper II).

Table 1: Stellar Parameters and Radii

$KOI$ | $T_{\rm eff}$ | $\delta T_{\rm eff}$ | log $g$ |  $\delta$log $g$ |  |  | $\xi$ | $\delta\xi$ | $R_{\star}$ | $\delta R_{\star}$
---|---|---|---|---|---|---|---|---|---|---
No. | (K) | (K) | (dex) | (dex) |  |  | (km s-1) | (km s-1) | ($R_{\odot}$) | ($R_{\odot}$)
K00245 | 5410 | 11 | 4.55 | 0.02 |  |  | 0.83 | 0.04 | 0.78 | 0.01
K01925 | 5418 | 18 | 4.48 | 0.06 |  |  | 0.82 | 0.04 | 0.90 | 0.02
K01612 | 6167 | 35 | 4.32 | 0.09 |  |  | 1.43 | 0.06 | 1.24 | 0.03
K00069 | 5646 | 21 | 4.50 | 0.04 |  |  | 0.91 | 0.03 | 0.94 | 0.02
K03167 | 5467 | 29 | 4.48 | 0.07 |  |  | 1.03 | 0.06 | 0.88 | 0.02
K00082 | 4849 | 49 | 4.51 | 0.08 |  |  | 0.62 | 0.12 | 0.73 | 0.01
K00975 | 6227 | 42 | 4.00 | 0.03 |  |  | 1.50 | 0.06 | 1.95 | 0.08
K02687 | 5800 | 22 | 4.55 | 0.06 |  |  | 1.02 | 0.03 | 0.94 | 0.02
K01924 | 5973 | 28 | 3.86 | 0.06 |  |  | 1.53 | 0.03 | 2.57 | 0.14
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  |  |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$

Note. — This table is published in its entirety in the machine readable
format. A portion is shown here for guidance regarding its form and content.

###  3.2 Stellar and Planetary Radii

Precise stellar radii result in precise planetary radii - a crucial parameter
necessary to unveil planetary composition and to ultimately define the
transition from rocky to gaseous planets.

To calculate the stellar radii (R⋆) we used the Stefan-Boltzmann law, which
depends on the Stefan-Boltzmann constant ($\sigma_{sb}$), stellar effective
temperature (Teff) and luminosity (L⋆),

$R_{\star}=(\frac{L_{\star}}{4\pi\sigma_{sb}T_{\rm eff}^{4}})^{1/2}$

where L⋆

$L_{\star}=L_{0}10^{-0.4M_{bol}}$

L0 is the zero point of bolometric magnitude scale (Mamajek et al., 2015) and
Mbol the bolometric magnitude related with photometric apparent magnitude
(mk), extinction (Ak) and bolometric correction (BC) in the same band, and
distance modulus ($\mu$) via

$M_{bol}=m_{k}-A_{k}-\mu+BC_{k}$

For each target star we used the distances estimated by Bailer-Jones et al.
(2018), when available. Bailer-Jones et al. (2018) adopted a bayesian approach
and geometric priors to obtain the distances from the $Gaia$ DR2 parallaxes
and considered the systematic parallax offsets determined from $Gaia$’s
observations of quasars (Lindegren et al., 2018; Zinn et al., 2018). We used
the 2MASS K-band, combined with the reddening E(B-V) derived from the 3D dust
map of Green et al. (2018) and transformed into Ak extinction using the
relations in Bilir et al. (2008); and the $isoclassify$ package (Huber et al.,
2017) in its “direct mode” using Teff, log $g$, [Fe/H] and Av extinction as
inputs to interpolate bolometric corrections from MIST grids (Choi et al.,
2016) and calculate absolute magnitudes and stellar luminosities.

In order to calculate the stellar radii, we combined all these parameters with
our derived Teff; the results are presented in Table 1.

Finally, we used the stellar radii to determine planet radii using the values
of transit depth, $\Delta$F, which are the fraction of stellar flux lost at
the minimum of the planetary transit, cataloged in Thompson et al. (2018); and
the equation from Seager & Mallén-Ornelas (2003):

$R_{pl}=109.1979\times(\Delta F\times 10^{-6})^{1/2}\times R_{\star}$ (1)

To assure a higher level of reliability in the computed planetary radii, we
only included in our sample those planets whose host stars have $\leq$10$\%$
error in their derived stellar radius (see Figure 4). In addition, we removed
KOI’s that have been classified as “false positives” (adopting the same
dispositions as in Thompson et al. 2018). The derived radii for the final
planetary sample in this study are presented in Table 2.

Table 2: Planetary Radii

Planet | Rpl |  $\delta$ Rpl
---|---|---
| (R⊕) | (R⊕)
K00001.01 | 13.62 | 0.30
K00002.01 | 17.93 | 1.07
K00007.01 | 4.49 | 0.13
K00010.01 | 16.62 | 1.14
K00017.01 | 14.31 | 0.35
K00018.01 | 16.78 | 0.80
K00020.01 | 21.19 | 0.63
K00022.01 | 14.06 | 0.34
K00041.01 | 2.48 | 0.07
K00041.02 | 1.38 | 0.04
$\cdots$ |  $\cdots$ |  $\cdots$

Note. — This table is published in its entirety in the machine readable
format. A portion is shown here for guidance regarding its form and content.

###  3.3 Uncertainties in the Derived Parameters

The internal errors in the derived effective temperatures and microturbulent
velocities were calculated by changing these parameters until the slopes of
both the A(Fe I) versus $EP$, and A(Fe I) versus log(EW/$\lambda$),
respectively, reach the same values as the errors in the slopes from the
converged solution. The uncertainties in log $g$ were estimated by varying
this parameter until the Fe I and Fe II mean abundances differed by one
standard deviation of the mean of the A(Fe I).

Table 3: Error Budget  Parameter | Median Uncertainty
---|---
mk | 0.022 mag
Ak | 0.009 mag
BCk | 0.03 mag
$\mu$ | 0.006 mag
Teff | 40 K
Rstar | 2.8$\%$
$\Delta F$ | 4$\%$
Rpl | 3.7$\%$

In order to estimate the total error budget in the derived stellar and
planetary radii for our sample, we consider the individual contributions of
the errors in each one of the parameters used in the computation of the radii.
(See also the discussion in Fulton & Petigura 2018).

The error in the $K$-band stellar magnitudes (mk) contributes with 1$\%$ to
the stellar radius error if we consider 0.022 mag to be the median error in mk
from 2MASS (Skrutskie et al., 2006) for our target stars. The contribution due
to errors in the extinction Ak is even smaller, given that the Ak values for
the target stars are quite small, ranging between 0.001 to 0.052 mag, with a
median Ak of 0.009 mag. If we were to completely neglect extinction, this
would result in an error of 0.6$\%$ in the stellar radii (using the median Ak
in the estimate). The errors in the $K$-band bolometric corrections (BCk) are
mainly dominated by uncertainties in the effective temperatures. We estimate
an error in BCk in the same manner as in Fulton & Petigura (2018), by changing
the effective temperature of a solar-type star by the median error in our
Teff’s (40 K) and investigating the corresponding change in BCk. (We also
investigated the effect of the errors in log $g$ and metallicities, but these
were found to be negligible). Taking the error obtained in BCk for a test
solar type star as typical, we estimate a change of 0.03 mag in BCk (Huber et
al. 2017 also estimate the error in BCk to be 0.03 mag) and an error of 0.3
$\%$ for the stellar radius.

The distances and the respective errors for the target stars were taken
directly from Bailer-Jones et al. (2018); and correspond to a median error of
0.006 mag for the “distance modulus” ($\mu$) and 0.08$\%$ error in R⋆. The
internal precision (median error) in the effective temperatures in this study
is 40 K, corresponding to 2$\%$ in the R⋆ error.

Combining all of the errors in the parameters discussed above in quadrature
and propagating the errors, we obtain a median internal uncertainty in our
derived stellar radii distribution of $\sim$ 2.8$\%$. This uncertainty in the
stellar radii has a direct impact, along with the transit depth ($\Delta$F)
errors, on the determination of the planetary radii errors. We adopted the
transit depth values $\Delta$F and respective errors from Thompson et al.
(2018), which, for the planets in our sample, result in 4$\%$ internal
precision in $\Delta$F and corresponds to a 3.7$\%$ internal precision for the
Rpl error budget. A summary of the contributions to the error budgets in the
R⋆ and Rpl determinations are presented in Table 3.

##  4 Results

The atmospheric parameter distributions for the studied sample are shown in
the different histograms of Figure 3; most of the target stars have effective
temperatures roughly between 4800 - 6500 K, having a peak around the solar
effective temperature (Teff = 5777 K) and a smaller peak corresponding to
cooler stars at Teff $\sim$ 5000 K. The log $g$ distribution for the studied
sample corresponds mostly to unevolved stars, with the distribution having a
peak between roughly log $g$ = 4.3 – 4.5 dex, but it also contains a tail with
more evolved stars having log $g$ $<$ $\sim$4.2. Most of the target stars
(about 80$\%$) are from the solar neighborhood having distances (from Bailer-
Jones et al., 2018) generally within $\sim$1 Kpc of the Sun (Fig. 3, panel c).

As discussed in the previous section, the stellar radii in this study were
calculated using distances based on $Gaia$ DR2 parallaxes. In Figure 4 we
present a HR diagram showing the derived effective temperatures and stellar
radii, color coded by the errors in the stellar radii. The median internal
uncertainty in the radii distribution in this study is $\sim$ 2.8$\%$ (Section
3.3) and most target stars follow the expected sequences in the HR diagram,
showing a densely populated main sequence and the presence of some evolved
stars (as expected from the Teff and log $g$ distributions obtained for the
targets and shown in Fig. 3). There are, however, clear outliers that occupy
unexpected loci in the diagram corresponding to at least 10$\%$ error in the
stellar radii (represented by green, yellow and red points in Figure 4).

Figure 3: Effective temperature (panel a) and surface gravity (panel b)
distributions for the sample stars. Panel (c) has the distribution of the
stellar distances for the sample; with distances obtained from $Gaia$ DR2
parallaxes and taken from Bailer-Jones et al. (2018). The median uncertainties
in the parameters are plotted in the upper corner of each panel. Figure 4: The
stellar radii versus the effective temperatures for the target stars. The
stellar radii are based on $Gaia$ DR2 parallaxes and are shown color-coded by
the stellar radii errors: from $\sim$1$\%$ error in blue to 20$\%$ error or
larger (up to $\sim$200$\%$) in red.

###  4.1 Previous Results from the Literature

####  4.1.1 Stellar Parameters Comparisons for the CKS Sample

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 Teff, 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.

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 $\chi^{2}$ 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, Teff 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 $\sim$60 K in the effective temperatures
(our Teff scale being hotter than the 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 twelve stars in our sample have log $g$
$>$ $\sim$4.7, although their log $g$ errors are within the expected
uncertainties (mean of the log $g$ errors is 0.12 dex) these results are all
systematically higher than Petigura et al. (2017).

More recently, Brewer & Fischer (2018) also analyzed the CKS dataset. 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, a grid of
plane-parallel model atmospheres (Castelli & Kurucz, 2004) and finds the best
solution for global parameters, such as Teff, log $g$, [M/H], or vmacro. The
authors applied an inverse solar analysis to adjust the oscillator strengths,
log $gf$s, 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 on the
Brewer et al. (2016) scale. For the effective temperatures, the mean
difference ($<$BF18 - This study$>$) is -69 $\pm$ 3 K and the RMS = 77 K,
again indicating a small systematic offset in the two Teff scales. For log
$g$, there is a small systematic difference of -0.044 dex, which is not
significant.

Figure 5: Comparison of the effective temperatures (panels a) and log $g$
values (panels b) derived in this work and in Petigura et al. (2017) for 1013
stars in common (top panels) and for 847 stars in common with Brewer & Fischer
(2018) (bottom panels). The mean differences between the parameters and the
corresponding RMS scatters are indicated in each case. The red dashed lines
represent the equality.

####  4.1.2 Stellar Parameter Comparisons for Other Samples of Kepler Stars

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, Teff, log $g$, [m/H] and vrot using the spectral synthesis
code Stellar Parameter Classification (SPC), which uses a library of model
atmospheres (Kurucz, 1992) to synthesize the spectrum between 5050 - 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 (Teff $>$ 5750 K), our effective temperatures are
systematically larger by 82 K $\pm$ 9 K than Buchhave et al. (2012), while for
the range between 5200 K and 5750 K our temperatures are systematically lower
by 21 K $\pm$ 10 K; for Teff’s 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
$\sim$0.18 dex.

Figure 6: Comparison of the effective temperatures (panels a) and log $g$
values (panels b) derived in this work and in Buchhave et al. (2012) for 135
stars in common (top panels), for 119 stars in common with Everett et al.
(2013) (middle panels) and for 343 stars in common in the APOGEE DR14 (bottom
panels). The mean differences between the parameters and the corresponding RMS
scatters are indicated in each case. The red dashed lines represent the
equality.

The middle panels of Figure 6 compares 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 = 3,000) using the RCSpec long-slit spectrograph on the
4-m telescope at Kitt Peak Observatory and determined Teff, log $g$ and [Fe/H]
by fitting the observed spectra with synthetic ones computed from stellar
model atmospheres (Castelli & Kurucz, 2003). Effective temperatures from
Everett et al. (2013) results show a more significant mean offset of $\sim$100
K relative to ours, with our Teff 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 $g$ $>$ $\sim$4.3 dex our derived log $g$’s are
systematically larger than theirs (with a mean difference of 0.14 $\pm$ 0.01
dex), while for log $g$ smaller than $\sim$4.3 dex our results are
systematically lower (with a mean difference of 0.05 $\pm$ 0.04 dex). The
bottom panels of Figure 6 compare results for 343 KOI’s in common that were
observed with the near-infrared ($\lambda$ 1.5 - 1.7 $\mu$m), high-resolution
spectroscopic (R $\sim$ 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, Garcia-Perez et al.,
2016), which fits the observed spectra to grids of synthetic spectra using
$\chi^{2}$ minimization. There is a significant systematic offset in effective
temperatures, with our Teff scale being hotter than that of APOGEE DR14 by 160
K in the mean, with a 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 function of log $g$, in particular for log $g$
values larger than $\sim$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).

####  4.1.3 Asteroseismic vs. Spectroscopic Surface Gravities

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 $g$’s (to 0.05
dex; Pinsonneault et al., 2018) that can serve as valuable benchmarks to
investigate 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 Teff 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
Spectroscopic Made Easy (SME- Valenti & Piskunov, 1996) and Stellar Parameter
Classification (SPC- Buchhave et al., 2012), two spectral synthesis
techniques, to fit optical high-resolution spectra to synthetic ones.

The seismic results presented in Figure 7 indicate good agreement between the
log $g$ determinations in the 3 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 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 between 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 $g$’s 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.

Figure 7: Our derived spectroscopic surface gravities in comparison with
asteroseismic surface gravities from Silva Aguirre et al. (2015), Huber et al.
(2017) and Serenelli et al. (2017) for 40 stars in common.

####  4.1.4 Stellar and Planetary Radii Comparisons for the CKS Sample

Stellar radii for the CKS sample were previously derived in Johnson et al.
(2017) by converting the spectroscopic Teff, log $g$, and [Fe/H] from Petigura
et al. (2017) into stellar masses, radii, and ages using the Darmouth 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 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 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
$\pm$ 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⋆
$>$ $\sim$2.5 R⊙; if we compute the mean stellar radii ratio and the RMS for
those stars, we obtain: 0.973 $\pm$ 0.002 and 0.010, respectively.

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 (Rpl $>$
23R⊕ in Fulton & Petigura (2018)). In addition, Fulton & Petigura (2018) have
a few planets with Rpl ranging roughly between 1,000 R⊕ and 24,000 R⊕ (e.g.,
for K03891.01; Rpl = 23,732 R⊕; planets with R${}_{pl}>$200R⊕ are off scale in
Figure 9a). 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.

If we remove the outliers from the comparison, or, remove from the Fulton &
Petigura (2018) sample those planets with Rpl $>$ 23R⊕, we obtain 0.957 $\pm$
0.007 for the mean ratio “F$\&$P18/Us” and a RMS of 3$\%$. It is also found
that the differences in planetary radii between this study and those from
Fulton & Petigura (2018) increases somewhat for the smallest planets
(R${}_{pl}<$1R⊕, with a mean planetary ratio of 0.925, where Figure 9b) shows
the comparison of our planetary radii $\leq$ 1R⊕ with those derived by Fulton
& Petigura (2018) for planets in common.

Figure 8: A comparison of the stellar radii derived from Fulton & Petigura
(2018) with those derived in this work. Our stellar radii are just slightly
larger than Fulton & Petigura (2018). The mean ratio between the radii in the
studies and the corresponding RMS scatter is shown. There is a tendency to
find larger differences for R⋆ $>$ 2.5 R⊙. Figure 9: Panel (a) presents the
comparison of our planetary radii with those derived by Fulton & Petigura
(2018) for planets in common having planetary radii less than 200 R⊕ in Fulton
& Petigura (2018). Panel (b) shows the same comparison shown in panel (a), but
for planetary radii $\leq$ 1R⊕. The offset between the results is clear. The
mean ratio between the radii in the studies and the corresponding RMS scatter
are also shown. The red dashed lines represent the equality line. Figure 10: A
comparison of the derived stellar radii in this work against the precise
stellar radii obtained from asteroseismology from (a) Serenelli et al. (2017),
(b) Silva Aguirre et al. (2015) and (c) Huber et al. (2017).

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.

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:
$\sim$2.6$\%$. Concerning systematic offsets, we find that our derived radii
are just slightly higher that the asteroseismic ones being on average 4.1$\%$
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 precise 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 typical offsets seen between the
different asteroseismic studies themselves.

##  5 Discussion

###  5.1 Exoplanetary Radii, Orbital Periods $\&$ Incident Fluxes

####  5.1.1 The Small Planet Radius Gap

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 broad-band 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 $\sim$1.3R⊕ and
$\sim$2.4R⊕, with a gap between 1.5R⊕ \- 2.0R⊕. 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 2R⊕. A
similar gap in planetary radius was confirmed by Fulton & Petigura (2018)
using $Gaia$ DR2 parallaxes. The dearth of planets at Rpl $\sim$ 1.8R⊕ 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 planet 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 2.0R⊕ is not an analysis artifact, but represents a real
transition between rocky planets and those with extensive atmospheres (Weiss &
Marcy, 2014; Rogers, 2015).

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 KOI’s with planets deemed as “false
positives” (see Section 2). The uncorrected-completeness radius distribution
shows two clear peaks: one at $\sim$1.6R⊕ and another at $\sim$2.8R⊕; the
radius gap is seen roughly between 1.8R⊕ and 2.2R⊕.

It is of interest to investigate the position 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 $\sim$2$\times$ 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
$\sim$2.0R⊕ and the location of the two peaks in the distributions is similar
as well. The full sample has a small excess of planets at Rpl $\sim$ 10R⊕ 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 Rpl $<$ 1.0R⊕, 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.

####  5.1.2 Completeness Corrections

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
Fulton & Petigura (2018); Fulton et al. (2017); Mulders et al. (2018, 2016);
Christiansen et al. (2016, 2015).

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):

$m=(\frac{R_{pl}}{R_{\star}})^{2}\frac{1}{CDPP_{dur}}\sqrt{\frac{t_{obs}}{P}}$

where tobs is the time that a star of radius $R_{\star}$, 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 $\Gamma$
cumulative distribution function of the form

$C(m;k,\theta,l)=\int_{0}^{\frac{m-l}{\theta}}x^{k-1}e^{-x}\cdot dx$

to model the distribution of values from the injection-recovery transit
signals ($m$); we use the $k$, $l$ and $\theta$ 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 of a
planet with radius Rpl transiting a host star of radius R⋆ at a distance of
$a$ could be detected, is $p_{tr}$ = 0.7R⋆/$a$; 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:

$w_{i}=\frac{1}{p_{det}\times p_{tr}}$

The true measure of the occurrence rate, $f_{bin}$, is the number of planets
per star in any orbital period or radius bin, is then given by:

$f_{bin}=\frac{1}{N_{\star}}\sum_{i=1}^{n_{pl,bin}}w_{i}$

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 histogram. 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, similarly 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 $\pm$ 0.05 R⊕ and 2.72 $\pm$ 0.10 R⊕, and a radius gap at
1.89 $\pm$ 0.07 R⊕ (Figure 11(e)). The completeness-corrected distribution
from Fulton & Petigura (2018) is also shown for comparison as the grey line
histogram 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 1.5R⊕ and 2.5R⊕, respectively, with the radius gap minimum
falling at 2.0R⊕.

Figure 11: Radius distribution (a) considering the entire planet sample (1633
planets); (b) taking into account only those planets (1526 planets) having
radii with errors less than 8$\%$; (c) the same as panel (b) but also
discarding planets with $P$ $>$ 100 days and planetary systems with $b$ $>$
0.7 (this defines the “clean” sample having 965 planets); (d) completeness-
corrected histogram of planet radii for our “clean” sample (dashed line) in
comparison with the distribution of planet radii uncorrected for completeness
(solid line); (e) completeness-corrected planet radii for our “clean” sample
(red line) but compared with the one from Fulton & Petigura (2018) (grey
line). We confirm the presence of a gap in the occurrence distribution of
close-in planets with orbital periods less than 100 days at $\sim$1.9R⊕.

####  5.1.3 A Slope in the Planetary Radius Gap

As discussed in Section 5.1.1, one mechanism suggested to explain the bimodal
small planet radius distribution is photo-evaporation: 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 photo-
evaporation, Ginzburg et al. (2018) model 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 the 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 Teff, and R⋆, while the semi-major axis of the planetary
orbits, assuming an orbital eccentricity equal to zero, were used to compute
the incident flux at the planet.

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 $\sim$ 4R⊕). Visual
inspection of these figures indicates clearly 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 in-between the two clouds, with Rpl $\sim$ 2R⊕.
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 $<$ $\sim$ 3 days) and
incident fluxes (relative to Earth) $>$ $\sim$ 700, which is likely related to
the photo-evaporation of the atmospheres of sub-Neptune-size exoplanets that
are very close to their parent stars and suffer high stellar incident flux
levels (Ikoma & Hori, 2012; Lopez et al., 2013; 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 Rpl 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 $\leq$ 8$\%$, $P$ $\leq$ 100 days and $b$ $\leq$ 0.7) into ten
orbital period bins containing 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(Rpl) = (-0.11 $\pm$ 0.02)*log($P$) + (0.39
$\pm$ 0.01), with Rpl in units of Earth radii and P in days, or Rpl 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(Rpl) = (+0.12 $\pm$ 0.02)*log($F$) + (0.04 $\pm$
0.03), with F in units of incident flux at the Earth, or Rpl 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 $P^{-0.09^{+0.02}_{-0.04}}$ 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 $P^{-0.11\pm 0.03}$.

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 photo-evaporation
scenario (Rpl scaling as $P^{-0.15}$). Contrarily, it should increase if the
primordial rocky planets formed after the proto-planetary disks dissipated in
a gas-poor formation model with a positive slope for the radius gap (Rpl
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 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 semi-major 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 $\sim$1/3
iron) and considers variable efficiency for the evaporation, provides the
closest match to the upper envelope defining super-Earth radii versus period,
for periods less than $\sim$8 days. The model then declines more steeply than
the observed distribution towards increasing periods, with the result that the
observed maximum radius for super-Earths falls well above the Owen & Wu (2017)
model core with 1/3 iron. Since their “icy” models with 1/3 ice and 2/3
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 photo-evaporation models. 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, Teq, is high, which is, in
general the case for more closely orbiting planets. Ginzburg et al. (2018)
evolve a model planetary distribution and find that the position of the valley
minimum shifts from $\sim$2R⊕ for P$<$10 days, to $\sim$1.5R⊕ for the longer
orbital periods. Recently, Fulton & Petigura (2018) have 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. The core-powered mass-loss
mechanism is dependent on the properties of the planet itself, with no
expected dependence on the stellar mass; the host-star mass correlation with
the planetary incident flux distribution may favor the photo-evaporation
model.

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 photo-evaporation model, Owen & Wu (2013,
2017) point 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. Photo-evaporation 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.

####  5.1.4 A Possible Correlation between Planetary Radii and Orbital
Periods?

The CKS planetary sample studied here can be divided into four exoplanet size
regimes:

  * •

Jupiters with 8R⊕ $<$ Rpl $\leq$ 20R⊕.

  * •

Sub-Saturns with 4R⊕ $<$ Rpl $\leq$ 8R⊕

  * •

Sub-Neptunes with 2R⊕ $<$ Rpl $\leq$ 4R⊕

  * •

Super-Earths with Rpl $\leq$ 2R⊕

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 ($\sim$30 out of $\sim$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
weighted RMS scatter of the planetary radii and orbital periods 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 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, towards 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 $\sim$ 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 Rpl \- $P$ for the Warm
Jupiters (diamond) generally follows the trend delineated by the smaller
planets, while the Hot Jupiters occupy a different locus in the Rpl \- 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 AUs 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.

Figure 12: (a) Planetary radius as a function of planetary period for the
entire sample. Red symbols and bars indicate the weighted median values and
their weighted uncertainties, respectively, for planetary radii and orbital
periods for each planet size domain: super-Earths (filled square), sub-
Neptunes (circle), sub-Saturns (triangle), Hot Jupiters (open square) and Warm
Jupiters (diamond). (b) Same as (a) but only considering the small-sized
planet regime (radii less than $\sim$ 4R⊕) of our “clean” planet sample. The
best fit slope (red line) and prediction interval (shaded region) to the
radius gap is shown. Figure 13: Panel (a): Planetary radius as a function of
stellar incident flux (relative to Earth) for the entire planet sample. Panel
(b): Same as (a) but only considering the small-sized planet regime (radii
less than $\sim$ 4R⊕) of our “clean” planet sample. The best fit slope (red
line) and prediction interval (shaded region), to the photo-evaporation valley
is also shown.

With the position of the Hot Jupiters in the Rpl \- $P$ 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 Rpl $\sim$ $P^{0.84\pm 0.11}$, with Rpl
in Earth-radii and orbital period in days. 2) If we limit the sample to
include only those planets with Rpl $\geq$ 2R⊕ (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: Rpl
$\sim$ $P^{1.1\pm 0.3}$. 3) If we add now a cut in orbital period, considering
only those planets with Rpl $\geq$ 2R⊕ and with $P$ $<$ 500 days, we obtain he
same power-law as before but with an increase in the uncertainties in the fit
(Rpl $\sim$ $P^{1.09\pm 0.13}$). 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 periods does not change
significantly in the 3 samples analyzed.

Helled et al. (2016) also suggested the existence of a correlation between
planetary radius and orbital periods for exoplanets with radii smaller than
4R⊕ 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: Rpl $\sim$ $P^{0.5-0.6}$, which is
similar, within the uncertainties, to the value of $\sim$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.

##  6 Conclusions

$\bullet$ We have conducted a homogeneous, quantitative spectroscopic analysis
of 1232 exoplanet host stars using the high-resolution Keck/HIRES spectra made
publicly available by the California-$Kepler$ Survey (CKS) team (Petigura et
al., 2017; Johnson et al., 2017). Stellar parameters (Teff and log $g$) were
derived from an equivalent width measurement analysis of a sample of 158 Fe I
and 18 Fe II lines and using the automated pipeline described in Ghezzi et al.
(2010, 2018).

$\bullet$ $Gaia$ DR2 parallaxes (Gaia Collaboration et al., 2018) for the CKS
stars were used to determine precise distances in Bailer-Jones et al. (2018),
which were then used as the foundation for determining stellar luminosities in
this study. With tightly constrained effective temperatures and luminosities,
stellar radii were computed for the entire sample, with a median internal
uncertainty of 2.8$\%$.

$\bullet$ Our derived stellar radii from precise parallaxes and precise
spectroscopic determinations of the effective temperatures achieve a
comparable precision against stellar radii obtained from asteroseismology,
with no significant systematic offsets. Precise stellar radii are important to
constrain the planetary radii - a crucial parameter necessary to unveil
planetary composition.

$\bullet$ Considering the sample of those stars with R⋆ uncertainties less
than 10$\%$ and planetary transit depths from Thompson et al. (2018), we
derive planetary radii for 1633 planets, with a median uncertainty of 3.7$\%$.
Comparisons of our derived planetary radii with those from Fulton & Petigura
(2018) indicate that our planetary radii are systematically larger than theirs
by $\sim$ 4.3$\%$.

$\bullet$ The derived planetary radii clearly exhibit two peaks in the
completeness-corrected planetary radii distributions. In particular, for our
“clean” planet sample (planets with radii errors $<$ 8$\%$, $P$ $<$ 100 days
and $b$ $<$ 0.7), we obtain peaks in the radius distributions corresponding to
1.47R⊕ (super-Earths) and 2.72R⊕ (sub-Neptunes), with a clean minimum (the
“gap”) at 1.9R⊕.

$\bullet$ Given the good internal precision in the derived radii, it was
possible to evaluate not only the location but the shape of the radius gap.
Our results indicate that the radius gap for the CKS sample does not fall at a
constant value of radius, but changes as functions of both planetary orbital
period and incident stellar flux. The position of the radius gap decreases
with orbital period and this decrease can be fit by a power law of the form
Rpl $\propto$ P-0.11; this agrees well with the recent value of -0.09 from Van
Eylen et al. (2018) for a much reduced sample of 117 planets having precise
radii from asteroseismology.

$\bullet$ According to Owen & Wu (2017), the value of the slope in the
evaporation valley can constrain the planet core compositions at different
orbital periods around different types of host stars. In our case, Rpl
$\propto$ P-0.11 matches a terrestrial-like composition model for the planets
in the transition radius.

$\bullet$ The value of the radius gap increases with increasing incident
stellar flux, such that Rpl $\propto$ F+0.12 provides an excellent fit to our
results. The trend of radius gap position with incident flux (and orbital
period) agrees with models of photo-evaporation (e.g., Owen & Wu, 2013, 2017;
Jin & Mordasini, 2018).

$\bullet$ If we divide our planetary sample into Warm Jupiters (Jupiters with
orbital periods larger than 10 days), Sub-Saturns, Sub-Neptunes and Super-
Earths, we find that the mean values for planetary radii and orbital periods
seem to suggest a possible correlation: larger planets seem to form more
distant from their parent stars. Considering only those planets with radii
$\geq$ 2R⊕ and with orbital periods $<$ 500 days (for which $Kepler$
completeness levels should be high), we obtain Rpl $\propto$ $P^{0.8}$; this
slope is similar to what has been found previously for small planets by Helled
et al. (2016).

We thank the referee for giving suggestions that improved the paper. We warmly
thank the California-$Kepler$ Survey team for making their reduced data
publicly available. We thank the European Space Agency (ESA) mission $Gaia$
and NASA Exoplanet Archive for the data used in this work. CM acknowledges the
financial support from Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq). CM also thanks D.F. Morell for his kindly assistance in R
language. KC and VS acknowledge that their work is supported, in part, by the
National Aeronautics and Space Administration under Grant 16-XRP16 2-0004,
issued through the Astrophysics Division of the Science Mission Directorate.
LG would like to thank the financial support from Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES).

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