A Spectroscopic Analysis of a Sample of K2 Planet-host Stars: Stellar Parameters, Metallicities and Planetary Radii

The spectroscopic analysis employed here assumes local thermodynamic equilibrium (LTE) and uses 1D plane parallel model atmospheres from the Kurucz ATLAS9 ODFNEW grid (Castelli & Kurucz 2003). Stellar spectroscopic parameters, namely the effective temperature (T_eff), surface gravity (log g), iron abundance (A(Fe) ¹⁵ ), and microturbulent velocity (ξ) were derived using a standard spectroscopic methodology which is based on measurements of the equivalent widths (EWs) of selected iron lines (Fe i and Fe ii lines).

The adopted line list in this study was taken from Ghezzi et al. (2018), Meléndez et al. (2014), and Friel et al. (2003) and within the Hydra spectral window, a list of 25 Fe i lines and 5 Fe ii lines was selected for analysis. The EWs of the selected lines were measured using the ARES code v2 (Sousa et al. 2015). In Table 2 we present the Fe i and Fe ii lines, the excitation potential energies χ, the oscillator strengths (log gf), and the respective references for the latter.

Three conditions were required for obtaining a consistent solution for T_eff, log g, and ξ for the stars. To obtain T_eff, the excitation equilibrium was required, or, removing trends between A(Fe i) and the excitation potential (χ), of the lines. To obtain log g, the ionization equilibrium was required, or, requiring that the average abundances of the Fe i (A(Fe i)) and Fe ii (A(Fe ii)) lines are equal. By minimizing the slope (<0.005) of the relationship between A(Fe i) and the logarithm of the reduced EWs ($\mathrm{log}(\mathrm{EW}/\lambda$)), the microturbulent velocity, ξ, is obtained. Finally, the iron abundances were consistent with the input model metallicities.

In order to analyze a relatively large number of stars in a homogeneous and efficient way, we used the automated stellar parameter and metallicity code named qoyllur-quipu (or q²). ¹⁶  This is a Python code developed by Ramirez et al. (2014). Briefly, q ² uses an input iron line list and measured EWs, along with the 2019 version of the abundance analysis code MOOG (Sneden 1973), to compute the iron abundances, effective temperatures, and surface gravities. The iterative process starts by interpolating a model atmosphere calculated assuming given values for T_eff, log g, and metallicity and then the values of T_eff, log g, and A(Fe) are increased or decreased iteratively to minimize the slopes of the relationships, and until obtaining a final adjusted value for the spectroscopic parameters of each star. Figure 3 shows an example of the iterated solution for the sample star HIP 81512, obtained for T_eff = 5752 K, log g = 4.34, and ξ = 1.06 km s⁻¹, with the mean metallictity for this star represented by the solid blue line.

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Table 3 presents the derived effective temperatures, surface gravities, metallicities, and microturbulent velocities for all stars in our sample, as well as the stellar radii and masses (see Section 3.3). We note that for a few stars having no measurable Fe ii lines in their spectra, or having only one measurable Fe ii line with a large uncertainty in the EW, the asteroseismic log g, if available, was adopted and used in the derivation of the effective temperature and microturbulence from the Fe i lines. Alternatively, when an asteroseismic log g was also not available, the Fe i lines were used to estimate the parameters by varying log g in steps of 0.05 dex and finding the log g value that produced the most consistent solution in terms of the scatter in the Fe abundances from the individual lines.

As benchmarks to our methodology and analysis techniques, we also analyzed solar-proxy spectra obtained with the Hydra spectrograph for reflected solar light from two asteroids, Astraea and Parthenope, as well as the well-studied solar twin HIP 81512 as a benchmark. Results are presented in Table 4. The parameters and metallicities obtained for the solar-proxy Astraea (T_eff = 5778 K, log g = 4.34, A(Fe) = 7.51, and ξ = 1.16 km s⁻¹) and for Parthenope (T_eff = 5770 K, log g = 4.40, A(Fe) = 7.54, and ξ = 1.06 km s⁻¹) are in excellent agreement with the solar parameters, indicating that our methodology does not harbor strong biases for solar-type stars. We note, however, that the mean metallicity for the solar proxies of A(Fe) =7.52 is slightly more metal-rich than the Asplund et al. (2021) (A(Fe)_⊙ = 7.46) scale, although it is in good agreement with the Magg et al. (2022) (A(Fe)_⊙ = 7.50) scale. We also note excellent agreement with the stellar parameters derived in the high-precision analysis of the solar twin HIP 81512 by Ramirez et al. (2009) and Ramirez et al. (2013); the latter studies are based on the analysis of high-resolution spectra (R = λ/Δλ ≃ 60, 000) obtained with the Robert G. Tull coudé spectrograph on the 2.7 m Harlan J. Smith telescope and measurements of 128 Fe i and 16 Fe ii lines, obtaining T_eff = 5755 ± 32 K, log g = 4.43 ± 0.04 dex, and A(Fe) =7.40 ± 0.04 dex. These comparisons with benchmark spectra can serve as validations for the technique adopted in this study for the analysis of Hydra spectra covering 6050−6350 Å and the Fe i/Fe ii line list from Table 2.

Fundamental stellar properties, such as stellar radius, age, or mass, can be estimated by comparing the positions of stars in a color–magnitude diagram with theoretical isochrones. Gaia DR3 (Gaia Collaboration et al. 2021) currently provides high-precision parallaxes that can be used to determine the absolute magnitudes of large numbers of stars.

Different codes that are available to the community, such as PARAM (Girardi et al. 2000; da Silva et al. 2006; Rodrigues et al. 2014, 2017), or isochrones (Morton 2015), have been developed as interfaces to find best fits to various isochrones, such as MESA Isochrones & Stellar Tracks (MIST; Choi et al. 2016; Dotter 2016), and several works have adopted similar methodologies to obtain stellar masses and radii (e.g., Johnson et al. 2017; Mayo et al. 2018; Wittenmyer et al. 2020).

In this work, stellar masses and radii were computed using the isochrone method via the q² code (qoyllur-quipu; Ramirez et al. 2014), which determines stellar mass, age, luminosity, and radius, using a grid of Yonsei–Yale isochrones (Yi et al. 2001). Briefly, to determine which isochrones best represent a particular set of observed stellar parameters (spectroscopic T_eff and [Fe/H] from this work, along with the M_V absolute magnitude derived from the parallax), probability distributions of those parameters are determined and matched to the isochrones, assuming that the errors in the observed stellar parameters (δ T_eff, δM_V , and δ[Fe/H]) have Gaussian probability distributions. For more details see Ramirez et al. (2013) and Ramirez et al. (2014). Isochrone-derived masses and radii of the sample stars are shown in Table 3. For five stars, instead of using M_V , we used the log g values to compute the stellar radii and the latter are flagged with an "a" in Table 3. In particular, we opted for using log g when there was a range of V magnitudes reported for a star, which was the case, for example, for two of the eclipsing binaries in our sample. We note that the stellar masses in these cases were estimated in the same way as the other stars using Y² isochrones.

Planetary radii were then obtained using the derived stellar radii and the value of the transit depth (ΔF), which is the fraction of stellar flux lost at the minimum of the planetary transit, given by the equation from Seager & Mallen-Ornelas (2003):

$$\begin{eqnarray}&&{R}_{\mathrm{pl}}=109.1979\times \sqrt{{\rm{\Delta }}F\times {10}^{-6}}\times {R}_{\mathrm{star}},\end{eqnarray} \tag{ 1 }$$

where the radius of the planet is in Earth radii.

We note that in this study, only values of ΔF from confirmed planets were used; we did not consider planets classified as planet candidates and false positives (according to the Kepler 1 and K2 notes in the NASA Exoplanet Archive). Most of the planetary transit depths of the K2 stars were from Kruse et al. (2019) (for 66 confirmed planets); for planets not cataloged by Kruse et al. (2019) we used values from Vanderburg et al. (2016), Pope et al. (2016), Barros et al. (2016), Rizzuto et al. (2017), and Livingston et al. (2018). We note that for K2-100 we noticed that the ΔF from Kruse et al. (2019) was discrepant when compared to other literature sources (e.g., Livingston et al. 2018; Stefansson et al. 2018; Mann et al. 2017; Libralato et al. 2016; Pope et al. 2016) and in this study we adopted the transit depth from Livingston et al. (2018). The transit depth of the planets of Kepler 1 stars were from Thompson et al. (2018) (for 56 confirmed planets). The ΔF values are provided in the Table 6.

The formal errors adopted for the stellar parameters T_eff, log g, and ξ were computed using q², which follows the error analysis discussed by Epstein et al. (2010) and Bensby et al. (2014). Errors in the iron abundances, A(Fe i) and A(Fe ii), were obtained by combining errors estimated from the EW measurements with the stellar parameter uncertainties. The individual errors of these parameters are presented in Tables 3 and 4.

The median errors in the stellar parameters derived in this study are reported in Table 5: δT_eff = 154 K, δlog g = 0.36 dex, δA(Fe) = 0.09 dex, and δξ = 0.24 km s⁻¹. We note that these are somewhat larger than the typical values from analyses of high-resolution spectra (R ∼ 60,000) in the literature (e.g., Ramirez et al. 2014; Martinez et al. 2019; Ghezzi et al. 2021). Larger errors here are to be expected given that the Hydra spectra have lower resolution (R ∼ 18,500) and smaller wavelength coverage, which results in having a smaller number measurable Fe i lines (25) and Fe ii lines (5).

Since the stellar masses and radii and planetary radii of our sample were derived from other parameters, we considered the individual contributions of the errors in each one of the parameters to estimate the error budget (similar to the discussions by Fulton & Petigura 2018; Martinez et al. 2019). The internal precisions (median errors) in the derived effective temperatures and metallicities were discussed above. The error in the V magnitude contributes ∼1% to the stellar radius error, when taking 0.07 mag to be the median error in V magnitude for our stars. The contribution due to errors in the parallaxes corresponds to a median error of 0.02 mas and represents a 0.45% error in the stellar mass and radius.

The stellar radii uncertainty and the transit depth (ΔF) errors have a direct impact on the determination of planetary radii errors. The median internal uncertainty in our derived stellar radii distribution is 4.2%; we adopted the transit depth values ΔF and respective errors from Kruse et al. (2019) and Thompson et al. (2018), which, for the planets in our sample, result in a 3.4% internal precision in ΔF. Finally, these uncertainties lead to a 4.4% internal precision for the R_pl error budget. A summary of the contributions to the error budgets in the R_star and R_pl determinations is presented in Table 5. To assess possible differences in the choice of isochrones we adopted the Darmouth isochrones instead of the Yonsei–Yale ones and found no signicant differences in the derived stellar radii and masses. As an example, for the star Kepler-62 we find R_star = 0.659 ± 0.018 R_⊙ and M_star = 0.711 ± 0.024 M_⊙ (using Dartmouth isochrones), while the result in this study is the same but just with a slightly higher uncertainty in the mass: R_star = 0.659 ± 0.018 R_⊙ and M_star = 0.711 ± 0.026 M_⊙ (using Yonsei–Yale isochrones).

In addition to the various sources of uncertainty discussed in the previous subsection, stellar magnetic activity can also affect the determinations of T_eff, ξ, and [Fe/H], as shown by Flores et al. (2016), Yana Galarza et al. (2019), and Spina et al.(2020).

Three stars in our sample had activity indices reported by Brown et al. (2022), who compiled a database of chromospheric activity measurements and surface-averaged large-scale magnetic-field measurements for a sample of FGK main-sequence stars. These stars are K2-229, EPIC 202089657, and Kepler-409, with log R'_HK = −4.73, −4.75, and −4.82, respectively, keeping in mind that the Sun has a value of log R'_HK = −5.02, which varies by about ±0.01 dex over the Solar activity cycle (Lorenzo-Oliveira et al. 2018).

Stellar magnetic activity effects have been quantified by Spina et al. (2020) via the log R_HK index, based on a stellar parameter analysis that relies on relations between the Fe i excitation equilibria and the Fe i reduced EWs (EW/λ), both as functions of the Fe i abundance, in addition to the ionization balance of Fe i and Fe ii. These constraints provide the values of T_eff, log g, ξ, and [Fe/H], and this is the technique used in our study. The impact that stellar activity has on the derivations of the specific stellar parameters using Fe i and Fe ii lines is found to be the greatest for T_eff, ξ, and [Fe/H], with almost no effect on log g (as illustrated in Figure 3 of Spina et al. 2020). In rough numbers (from Figure 4 of Spina et al. 2020), R_HK ∼ –4.25 leads to a lower value of T_eff by ∼100 K, while log R_HK ∼ –4.40 causes a metallicity change of [Fe/H] = −0.05, and a value of log R_HK ∼ –4.35 would lead to a derived microturbulent velocity that is too large by ∼0.3 km s⁻¹. A much lower activity level of log R_HK ∼ −4.7–4.8, as measured, for example, for the three target stars mentioned above, would cause an unmeasurable change in the effective temperature, a metallicity change of −0.02 dex, and would lead to a microturbulent velocity change of ∼0.05 km s⁻¹. All of these variations are well within the uncertainties in our analysis. These are also in line with the results of Lorenzo-Oliveira et al. (2018) who, for the young solar twin HIP 36515 (log R_HK ∼ –4.70), the stellar parameters derived over its six year activity cycle find a scatter in T_eff of ±10 K, ±0.01 in [Fe/H], and ±0.07 km s⁻¹ in microturbulent velocity. It is possible, however, that other stars in our sample may have higher levels of activity, although such high activity levels would suggest ages <1 Gyr (Lorenzo-Oliveira et al. 2018) or that the stars are members of close binary systems (Oláh 2007).

As part of an analysis of the young active solar twin HIP 36515 (age ∼ 0.4 Gyr), Yana Galarza et al. (2019) established a list of Fe i and Fe ii lines that are sensitive to stellar magnetic activity. Among the lines in our line list that were identified as sensitive to magnetic fields are Fe i 6173 Å, 6200 Å, 6213 Å, 6219 Å, 6252 Å, 6265 Å, and 6270 Å (Table 2) and these have both large Landé factors, coupled with large EWs (through a combination of log gf values and excitation potentials). None of our Fe ii lines were identified as sensitive to stellar magnetic activity.

We investigated the impact that the inclusion of "sensitive" Fe i lines has on the derivation of stellar parameters by removing these sensitive lines and rederiving the parameters for a selected sample of 15 stars. This subsample contains the three stars with measured values of log R_HK mentioned above, plus five stars in our sample with approximate ages from q² of <4 Gyr (K2-223, K2-44, Kepler-1339, KOI-6, and KOI-293), and two stars (K2-100 and K2-101) which are members of the young (650 ± 70 Myr; Martin et al. 2018) open cluster M 44. We also added the stars K2-186, K2-34, Kepler-139, Kepler-1445, and Kepler-396, as these have measured flares with energies >10³⁴ erg along with log R_Hα > −4.4 in Su et al. (2022), who determined the R_Hα activity index and flare energy using LAMOST spectra and light curves of stars observed by Kepler 1 and K2.

Figure 4 provides comparisons between the stellar parameters derived in this study (without taking into consideration the possible effects of magnetic fields in the measured Fe i lines) and those obtained including only those Fe i lines that were deemed as insensitive to magnetic fields. The median and median absolute deviation (MAD) values of the differences for each stellar parameter are included in each panel of the figure. The upper left panel shows that the differences in the effective temperatures are less than ∼50 K for most stars, with only two stars, K2-44 and Kepler-1445, having larger differences of ΔT_eff ("Non-sensitive"—"This Work") = −112 K and −142 K, respectively. This result suggests that the differences in the effective temperatures are not significant and, in general, fall within the range of our uncertainties (see Table 5). In the case of log g (top right panel), all stars have differences less than 0.09 dex, with the exceptions of K2-44 and Kepler-409 having differences of −0.2 and −0.15 dex, respectively. Given the uncertainties in log g, all differences are well within the estimated median uncertainty of 0.36 dex. A similar result is found for values of [Fe/H] (lower left panel), as the median uncertainty in [Fe/H] in this study is 0.09 dex (Table 5), with the largest metallicity difference found being −0.118 for Kepler-1445. According to Spina et al. (2020), magnetic activity is expected to result in an increase in microturbulent velocities for log R_HK > −5.0, or in active young stars (ages < 4–5 Gyr). The median difference found for the microturbulent velocity parameter is −0.01 and the MAD is 0.06 km s⁻¹ (Figure 4), indicating that there is no significant evidence that magnetic activity is measurably affecting the microturbulent velocities in this analysis.

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In summary, the stellar parameters derived without the use of the magnetically sensitive lines are within the uncertainties when compared to the parameters from this study that include some Fe i lines that are deemed as lines sensitive to the effects of magnetic fields and activity. This result suggests that the final stellar parameters derived for this sample of stars have not been perturbed significantly by strong magnetic activity.

In general, there are more results available for Kepler 1 mission targets than for K2. Several studies in the literature (Petigura et al. 2017; Brewer & Fischer 2018; Martinez et al. 2019; Ghezzi et al. 2021) analyzed the CKS (Petigura et al. 2017) but used different analysis techniques. The stellar parameters obtained by Petigura et al. (2017) were derived using synthetic spectra and the codes SpecMatch and SME@XSEDE; Brewer & Fischer (2018) derived the stellar parameters also using spectral synthesis and the Spectroscopy Made Easy (SME; Piskunov & Valenti 2017) code, while Martinez et al. (2019) and Ghezzi et al. (2021) adopted a methodology that was based on the classical spectroscopic EW method and used the code MOOG (Sneden 1973). Concerning results for K2 targets, in particular, Petigura et al. (2018a), using the same methodology of Petigura et al. (2017), analyzed a sample of 141 K2 candidate planet-host stars, while Wittenmyer et al. (2020) analyzed a sample of 129 K2 planet candidate host stars whose spectra were observed by the K2-HERMES program (Wittenmyer et al. 2018; Sharma et al. 2019; Clark et al. 2022) in the Galactic Archeology with HERMES (GALAH; Buder et al. 2021) survey. In addition, results for a large number of both Kepler 1 and K2 targets were obtained by the low-resolution optical spectroscopic LAMOST survey (Cui et al. 2012; Zong et al. 2018), as well as the high-resolution spectroscopic near-infrared APOGEE survey (Majewski et al. 2017; see also Wilson et al. 2018).

Comparisons of our T_eff values with those from the studies mentioned above are shown in the different panels of Figure 5; for each case, the median differences "Other Work–This Work" are given in the top left of each panel, along with the corresponding MADs; in all panels the gray diamonds correspond to Kepler 1 stars, while the blue circles correspond to K2 stars. From the medians and MADs in Figure 5 we can conclude that there is overall good agreement between our effective temperatures and those from the APOGEE and LAMOST surveys, as well as those available from the literature. The mean of the median differences is small, ΔT_eff ∼ −45 K and the MADs are all below 70 K, except for the comparison with the GALAH results, where MAD = 119 K (Wittenmyer et al. 2020) and 114 K (GALAH). Overall our T_eff scale is just slightly hotter than the other scales, except for the comparison with Petigura et al. (2018a; K2 targets median difference = 8 K) and Martinez et al. (2019; median difference = 0 K).

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Comparisons of the surface gravities derived here—from Fe i and Fe ii ionization balance—with those from the works discussed above are shown in Figure 6 (as in Figure 5, the median differences ("Other Work"—"This Work") ± MAD are given in each panel). The median log g differences for all studies and the surveys are surprisingly small (see Section 4.1.1) given the different analysis methodologies and line lists, all <0.06 dex, except for Petigura et al. (2018a) that has a mean log g difference of 0.12 dex. In all comparisons the MAD values are smaller or equal to 0.14 dex, which is smaller than typical uncetainties in log g values.

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4.1.1. Asteroseismic versus Spectroscopic Surface Gravities

The stellar parameters derived here (T_eff, log g, ξ, and [Fe/H]) are based on an analysis of a sample of Fe i and Fe ii lines, where correlations between parameters in such a spectroscopic analysis can lead to systematic errors, especially in the derived values for log g; for example, Fe i lines are typically stronger than Fe ii lines, leading to potential correlations between the microturbulent velocity and log g. Accurate surface gravities can thus be one of the more difficult parameters to constrain via spectroscopy, especially if the Fe ii lines are few in number and weak. In order to investigate possible systematic offsets in the log g values derived in this study, we compare our results with those computed via asteroseismology, where the surface gravity can be derived with quite good precision (Pinsonneault et al. 2018).

For such a comparison, we collected in Figure 7 asteroseismology results for 83 stars from Huber et al. (2016), Aguirre et al. (2017), and Serenelli et al. (2017). In order to compute surface gravities via asteroseismology, Huber et al. (2016) and Serenelli et al. (2017) used the T_eff and [Fe/H] from the APOGEE survey (DR13), while Aguirre et al. (2017) adopted T_eff and [Fe/H] from several literature sources. Most of the stars in common with this study are from Huber et al. (2016); the log g values from that study show no significant offsets and are in good agreement when compared to ours, with a median log g difference ("Huber et al. 2016–This Work") = 0.033 ± 0.164 dex. However, there are four stars for which the log g differences are >1 dex. These four discrepent stars (which are labeled in Figure 7) have asteroseismic gravities that would indicate that they are giant or subgiant stars, whereas other studies have derived surface gravities that suggest that these stars are dwarfs as discussed below:

• 1.  
  K2-36: this star was analyzed individually by Damasso et al. (2019) using high-resolution spectra from HARPS-N, with stellar parameters derived using the spectroscopic technique and finding log g values of 4.73, 4.60, and 4.57 with EWs, Atmospheric Stellar Parameters from Cross-Correlation Functions (CCFpams), and Stellar Parameter Classification (SPC; Buchhave et al. 2014), respectively. Using SME and SpecMatch with HIRES spectra, Sinukoff et al. (2016), Brewer et al. (2016), and Crossfield et al. (2016) derived log g = 4.65, 4.55, and 4.60, respectively. Finally, using SPC in Tillinghast Reflector Echelle Spectrograph (TRES) spectra, Vanderburg et al. (2016) derived log g = 4.70. Our result (log g = 4.45) is ∼0.1–0.3 dex lower than these studies, although all of these results indicate that K2-36 is a dwarf.
• 2.  
  K2-58: using SpecMatch and SME with HIRES spectra, Crossfield et al. (2016) and Brewer & Fischer (2018) derived log g = 4.52 and 4.50, respectively. Using SPC with TRES spectra, Vanderburg et al. (2016) derived log g = 4.54, with these values being consistent with our log g = 4.50 dex, pointing to K2-58 as being a dwarf.
• 3.  
  K2-128: using SPC with TRES spectra, Crossfield et al. (2018) and Mayo et al. (2018) derived log g = 4.70; while this result is 0.20 dex larger than our log g, taken together, these studies point to K2-128 as being a dwarf.
• 4.  
  EPIC 210423938: for this star, the log g value obtained by Mayo et al. (2018) is 4.69. Stassun et al. (2019) determined that log g = 4.55 from its stellar radius and mass. Considering our result (log g = 4.55 dex), we identify EPIC 210423938 as a dwarf.

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In summary, spectroscopic studies have found the four stars discussed above to have dwarf-star surface gravities. In addition, this status is confirmed from their DR3 Gaia parallaxes and distances, with distances to K2-36 of 109 pc, K2-58 of 182 pc, K2-128 of 114 pc, and EPIC 210423938 of 149 pc, resulting in absolute Gaia or V magnitudes of M_V = 6.5, M_G = 5.8, M_V = 7.3, and M_V = 7.1, respectively. These absolute magnitudes confirm them as K dwarfs. Without considering the four discrepant stars discussed above, we find median log g differences with Huber et al. (2016) = −0.049 ± 0.136. We note that for the few stars in common with Aguirre et al. (2017) and Serenelli et al. (2017), there is excellent agreement.

Figure 8 summarizes a comparison of the metallicities for K2 stars (blue circles) and Kepler 1 stars (gray diamonds) derived in this work with those derived in other studies and surveys. The metallicities from the literature were obtained either via spectrum synthesis methods (Petigura et al. 2017, 2018a; Brewer & Fischer 2018; Wittenmyer et al. 2020), (Buder et al. 2021; GALAH), (Jonsson et al. 2020; APOGEE), (Zong et al. 2018; LAMOST) or were based on EW measurements of Fe i and Fe ii lines (Ghezzi et al. 2021). For Kepler 1 stars in common with Petigura et al. (2017), the median metallicity difference is 0.040 ± 0.034 dex, while with Brewer & Fischer (2018) it is 0.024 ± 0.064 dex. There are 22 stars in common with Ghezzi et al. (2021), with a median difference of 0.017 ± 0.037 dex. The K2 stars in common with Petigura et al. (2018a) have a median difference of 0.021 ± 0.052 dex, and for those in common with Wittenmyer et al. (2020) the median [Fe/H] difference is −0.015 ± 0.073 dex. Comparison of the metallicities with the spectroscopic surveys also finds excellent agreement: 0.001 ± 0.041 dex for APOGEE DR17, while with GALAH DR3 it is −0.014 ± 0.069 dex. Finally, the median difference for stars in common with LAMOST DR5 is 0.010 ± 0.053 dex.

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Overall, the metallicity comparisons in Figure 8 indicate good consistency between our results and the other metallicity scales, with median metallicity differences in all cases being smaller than ∼0.04 dex and MAD less than 0.07 dex. Save for a few outliers, the scatter in the [Fe/H] differences are well within the expected uncertainties.

A number of studies in the literature obtained stellar radii and, in some instances, also masses, for Kepler 1 targets using different combinations of codes, models, and parameters. Johnson et al. (2017) used the isochrone method, stellar parameters from Petigura et al. (2017), and Dartmouth Stellar Evolution Program (DSEP) models (Dotter et al. 2008); Fulton & Petigura (2018), computed stellar radii using Gaia DR2 inverted parallaxes, and Martinez et al. (2019) used their stellar parameters, combined with the Gaia DR2 distances from Bailer-Jones et al. (2018). The large Gaia–Kepler stellar properties catalog by Berger et al. (2020b) combined isochrones, Gaia DR2 parallaxes, and spectroscopic metallicities using the isoclassify (Huber et al. 2017) package, while Hardegree-Ullman et al. (2020) used stellar parameters from LAMOST and derived K2 stellar radii from the Stefan–Boltzmann law and stellar masses from stellar radii. Via asteroseismology, Huber et al. (2016) also derived the stellar masses and radii of K2 stars using T_eff and [Fe/H] from APOGEE DR14, while Mayo et al. (2018) used the isochrone package (Morton 2015), which requires the effective temperature, surface gravity, and metallicity as input parameters (these parameters were derived using the spectral synthesis method described by Mayo et al. 2018).

Comparisons of our derived radii (Section 3.3) with those in the six studies mentioned above are shown in the left and right panels of Figure 9, respectively for K2 and Kepler 1 targets. Please note the different scales for the x- and y- axes in the different panels. First, when considering the median radius differences "Other works–This work," overall, there is not a clear bias in any direction. Overall, all results from the literature present similar levels of consistency relative to ours. We note, however, the presence of the four significant outliers of Huber et al. (2016) for which their radii are much larger than ours (the log g values for these cases are also very discrepant as discussed in Section 4.1.1 and shown in Figure 7), which when removed improve the consistency relative to our results (0.029 ± 0.082 R_⊙).

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For example, for the most discrepant case (K2-128), the stellar radius of Huber et al. (2016) would be roughly 15 times larger than ours, which would imply a correspondingly larger planet radius. For the comparisons with Berger et al. (2020b), Mayo et al. (2018), and Martinez et al. (2019), the median differences are ΔR_star < −0.01 R_⊙, noting again the presence of outliers. In the comparison with Mayo et al. (2018), in particular, the outliers are mostly for radii larger than ∼1.5 R_⊙, with a tendency that our radii are larger. For the comparisons with Hardegree-Ullman et al. (2020) and Johnson et al. (2017), the systematics go in the opposite direction, with median ΔR_star ∼ 0.03 R_⊙. In all comparisons, the MADs are less than 0.05 R_⊙, except for Johnson et al. (2017) that is slightly larger (0.07 R_⊙), while Huber et al. (2016) has a much larger MAD of 0.08 R_⊙.

Moving on to discuss the stellar masses derived in this study, comparisons with other results are shown in Figure 10. The mass scale from Mayo et al. (2018) has a median difference (and MAD) that are very small when compared to ours, indicating overall good agreement. We note, however, that for stars more massive than M_star ∼ 1.1 M_⊙, there is significantly more scatter, while for stars with M < 1.1 M_⊙ we find a median mass difference of 0.026 ± 0.020 M_⊙. The mass comparisons for the Kepler 1 stars (Johnson et al. 2017; Berger et al. 2020a; right panels of Figure 10), indicate even smaller offsets, where, the median mass differences are less than 3% and the MADs are ΔM_star ∼ 0.03 M_⊙, although there are also outliers in these comparisons. The median difference for stars with less than 1 M_⊙ are −0.009 ± 0.025 M_⊙ and 0.003 ± 0.021 M_⊙ for Johnson et al. (2017) and Berger et al. (2020a), respectively. The mass scale of Hardegree-Ullman et al. (2020), on the other hand, has a larger systematic offset when compared to ours, being overall more massive than our scale by ∼8.2%; the latter comparison also shows some scatter with an rms value of 0.22 and a MAD of 0.111 M_⊙, which is the largest in Figure 10.

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Asteroseismic masses are, in principle, the most accurate masses presented in Figure 10, having the smallest expected uncertainties (Pinsonneault et al. 2018). When compared to our results, there is a clear systematic difference between the mean mass of Huber et al. (2016) and ours, with our masses being smaller than the asteroseismic ones in the median by ∼3%. We note for example, the significant outlier that appears as much more massive (M_star = 1.2 M_⊙) when compared to our mass value (M_star ∼ 0.7 M_⊙) and again refer to the discussion in Section 4.1.1. As in the comparison with Mayo et al. (2018), there is also more scatter for stars with masses larger than 1.1 M_⊙.

The radii of exoplanets orbiting Kepler 1 and K2 host stars have been derived in several studies in the literature and these are compared with our results in the different panels of Figure 11. Here, the planetary radii were computed using Equation (1), which combines our derived stellar radii with transit depth values available in the literature (Section 3.3), and these are presented in Table 6, which contains the star identifications, the planet names, transit depths, and planetary radii along with the error in R_pl.

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For K2, we compare planetary radii with those of Petigura et al. (2018a), Hardegree-Ullman et al. (2020), Kruse et al. (2019), Mayo et al. (2018), Vanderburg et al. (2016), and Crossfield et al. (2016). Overall, the results in Figure 11 indicate that our planetary radii for K2 hosts are, on the median, larger than in the other K2 works, except for Crossfield et al. (2016); however, the median differences in the planetary radii ("Other Work–This Work") are small, with median systematic differences (("Other Work–This Work")/"This Work") of 0.2% for Crossfield et al. (2016), ∼2% for Kruse et al. (2019), ∼3% for Hardegree-Ullman et al. (2020) and Vanderburg et al. (2016), and ∼4% for Mayo et al. (2018). Although the median differences are, in some cases insignificant (at the level of 0%–2%), or small (at the level of 3%–4%), there are some outliers with larger discrepancies in some regimes, as is the case of the comparison with Vanderburg et al. (2016) that reveals larger offsets, in particular for planets with radii larger than ∼2.5 R_⊕. The comparison with Petigura et al. (2018a) finds larger median differences with our derived planetary radii of ∼6% larger.

In this comparison (top left panel), there is one significant outlier in the small planet regime, planet K2-183d, for which we find R_pl = 3.1 R_⊕, while Petigura et al. (2018a) find R_pl = 17.4 R_⊕. Our result is better agreement with the planet radius reported by Livingston et al. (2018) (R_pl = 2.9R_⊕) and Mayo et al. (2018) (R_pl = 2.5 R_⊕), as well as by Hardegree-Ullman et al. (2020) (R_pl = 5.1 R_⊕) and Kruse et al. 2019) (R_pl = 5.2 R_⊕). We also note that the difference in the effective temperature between Petigura et al. (2018a) and our result is ΔT_eff = −81 K and that if we used Petigura et al. (2018a) R_pl/R_star value we would obtain R_pl = 17.56 R_⊕, which is in much closer agreement with Petigura et al. (2018a), indicating that the difference in transit depth is responsible for a large part of the difference in R_pl.

For the Kepler 1 planets studied here, the radii of Petigura et al. (2022) and Fulton & Petigura (2018) show the largest offsets relative to our results, with a median systematic difference of 8%–9%, again with our planetary radius scale being larger. For Martinez et al. (2019), there is a much smaller systematic shift in the planetary radii relative to ours of ∼1%, and also having the smallest MAD of all comparisons, indicating that the scales for planetary radii are very consistent in both studies.

One aspect to keep in mind is the fact that the various studies discussed here may have employed different transit depths. Some studies, such as ours, use literature values, while others derive their own. For planets in common, we should note that we used the transit depths from Kruse et al. (2019) and these were also used by Hardegree-Ullman et al. (2020), while Vanderburg et al. (2016), Crossfield et al. (2016), Mayo et al. (2018), and Petigura et al. (2018a) derived their own transit depths. For the Kepler 1 planets, we used the transit depths from Thompson et al. (2018), which are also used by Martinez et al. (2019) and Petigura et al. (2022), while the transit depths of Fulton & Petigura (2018) come from Mullally et al. (2015).

The final distribution of planetary radii derived from the Hydra spectra of K2 host stars contains 85 confirmed planets orbiting 69 stars. Although this sample is relatively small, the results presented here are derived using an independent methodology that relies on an analysis of a carefully selected set of Fe i and Fe ii lines, which are used to determine fundamental host-star parameters (T_eff, log g, and [Fe/H]), and thus represent a useful addition to the growing number of independently derived K2 planetary radii. Figure 15 (top panel) presents a histogram of K2 planetary radii that result from the derived stellar radii combined with available transit depth catalogs. The 85 planets plotted in Figure 15 reveal a distinct and well-defined radius valley that spans R_pl ∼ 1.6−2.2 R_⊕, with a minimun near 1.9 R_⊕. Figure 15 includes all planetary orbital periods which, for this sample, range from P ∼ 0.5 days up to 53 days and it should be noted that, due to the differing observing techniques between Kepler 1 and K2, the sample studied here is biased toward shorter orbital periods when compared to Kepler 1 periods, e.g., the CKS sample. Nevertheless, the K2 radius valley observed in this sample, with a minimum at R_pl ∼ 1.9 R_⊕, is very similar to the results derived from the CKS sample from several studies (e.g., Fulton et al. 2017; Berger et al. 2018; Fulton & Petigura 2018; Van Eylen et al. 2018; Martinez et al. 2019).

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The bottom panel of Figure 15 provides a comparison of radii derived for the CKS sample (all Kepler 1 planets) from Martinez et al. (2019; their Figure 11(d)). Comparing the top and bottom panels of Figure 15 illustrates the similarity in the location of the radius valley between the K2 and Kepler 1 samples. Although the Kepler 1 field focused on a single pointing, encompassing a limited Galactic longitude and latitude (l ∼70°–8°, b ∼10°–20°), the K2 fields were constrained by the ecliptic plane and thus ranged over a broader range of Galactic longitudes and latitudes. The similarity in the position of the radius valley suggests it to be a ubiquitous phenomenon among short-period planets across a broad range of Galactocentric distances in the thin and thick disk populations; Zink et al. (2021) arrived at this conclusion after their analysis of planetary radii in the K2 Campaign fields 1−8 and 10−18.

With the similarity in the position of the radius valley between the K2 and Kepler 1 samples (Mayo et al. 2018; Hardegree-Ullman et al. 2020; Zink et al. 2021), we next investigate planet radius as a function of orbital period, with Figure 16 plotting our planet sample in a period–radius plane. The top panel includes all planets (our full sample) in the different orbital periods, with the K2 planets plotted as black filled circles and the Kepler 1 planets plotted as open diamonds; blue symbols represent the median and MADs of the planetary radii and orbital period distributions for each planet size domain: super-Earths (R_pl ≤ 2 R_⊕; square), sub-Neptunes (2 R_⊕ < R_pl ≤ 4.4 R_⊕; circle), sub-Saturns (4.4 R_⊕ < R_pl ≤ 8 R_⊕; triangle), and Jupiters (8 R_⊕ < R_pl ≤ 20 R_⊕; star). These median values are overall similar to those for the CKS sample of Martinez et al. (2019), but here we adopt the limit of 4.4 R_⊕ for the transition between sub-Neptunes and sub-Saturns, as discussed by Ghezzi et al. (2021).

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Planetary radii <4 R_⊕ are shown in the bottom panels, where the bottom left panel plots only the sample of K2 planets, while in the bottom right panel includes all K2 planets and those Kepler 1 planets having orbital periods less than 100 days. In both plots the slope of the radius valley is shown as a blue line (we used the slope of −0.11 from Martinez et al. 2019). Although we do not fit a trend of the position of the radius gap, R_gap, as a function of orbital period (P) to the 139 planets in our sample, the planetary radii that result from the stellar parameters derived here are consistent with the relation of the radius valley following a power law of the form R_gap ∝ P^−0.11, as shown in the bottom panels of Figure 16.

The distribution of host-star metallicity as a function of planetary radius is shown in the left panel of Figure 17. The 85 K2 planets in this study are the solid black circles and the Kepler 1 planets are the gray open diamonds. The blue symbols represent the median (±MAD) metallicities of the K2 planet hosts dividing the sample in the same planet size domains as in Figure 16: super-Earths, sub-Neptunes, sub-Saturns, and Jupiters. Although the K2 planet sample size is small, we find that, in general, the metallicities of the K2 planet hosts increase with planetary radius, but the increase in metallicity is seen in particular for the transition between the small (super-Earth and sub-Neptune) and large (sub-Neptune and sub-Saturn) planet regimes, a result that is similar to what has been found in previous studies of Kepler planets in the literature (Petigura et al. 2018b; Narang et al. 2018; Ghezzi et al. 2021; see also Beauge & Nesvorny 2013).

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As mentioned in the introduction, previous works have investigated correlations between host-star metallicity and planet orbital period distributions, concluding that small and hot planets with orbital periods P ≲ 8–10 days appear preferentially around metal-rich stars (e.g., Mulders et al. 2016; Wilson et al. 2018). The right panel of Figure 17 shows the host-star metallicity as a function of the planet orbital period for our studied sample (symbols are the same as in the left panel). The horizontal dashed lines represent the mean metallicities for those K2 planets having orbital periods below and above 10 days, and the 10 day boundary is marked as a dashed vertical line. We find that the median host-star metallicity for planets with P < 10 days is slightly metal-rich, 0.059 ± 0.122 dex, while the median host-star metallicity for planets with P > 10 days is slightly metal-poor, −0.060 ± 0.106 dex. Our K2 sample has mostly small planets and only some large planets. If we restrict the sample to that having only planets with R_pl < 4.4 R_⊕, we obtain a similar behavior, with a median metallicity of 0.051 ± 0.124 for P < 10 days and −0.051 ± 0.090 for planets with P > 10 days. We note that adopting a boundary at R_pl = 8.3 R_⊕, as found by Wilson et al. (2018), gives similar metallicity differences between the two orbital period regimes. Summarizing the results obtained here for the K2 host-star metallicities and orbital planetary periods are in line with what was found previously for Kepler 1 systems.