DETAILED ABUNDANCES OF STARS WITH SMALL PLANETS DISCOVERED BY KEPLER. I. THE FIRST SAMPLE^*

Stellar parameters for each star have been derived directly from the spectra using excitation and ionization balance of Fe i and Fe ii. Initial parameters for each star were taken from the planet validation papers as listed in the introduction. The parameters were then altered until no correlation existed between the derived [Fe i/H] and lower excitation potential (χ) and [Fe i/H] and reduced EW [log (EW/λ)] for each Fe i line measured, and until the [Fe/H] abundances derived from Fe i and Fe ii lines were equal to within two significant digits. Deriving stellar parameters in this way results in unique solutions only if there is no ab initio correlation between χ and EW of the measured Fe i lines. Such correlations did exist for the initial Fe i line lists for Kepler-20, Kepler-22, and Kepler-37, but the correlations were eliminated by removing two, four, and seven Fe i lines from their respective line lists. Uncertainties in T_eff and ξ are the differences in the adopted parameters and those that result in 1σ correlations in the [Fe/H] versus χ and reduced EW relations, respectively. The derived surface gravities are sensitive to the Fe abundance derived from both Fe i and Fe ii lines, with a greater sensitivity to the latter, and thus the uncertainty in log g is dependent on the uncertainty in the Fe abundances, as more fully described in Bubar & King (2010).

The derived stellar parameters, Fe abundances, and their uncertainties are given in Table 2. Parameters and Fe abundances have been derived for Kepler-21 from each of the three spectra—KFOP, Keck (non-KFOP), and KPNO—obtained for this star. The results from the three different spectra are indistinguishable within the combined uncertainties, with those from the KFOP and Keck observations in particularly good agreement: ΔT_eff = −52 K, Δlog g = −0.09, and Δ[Fe/H] = −0.02. This is reassuring given that the spectra were taken with the same telescope and spectrograph. The log g derived from the KPNO spectroscopy deviates from the KFOP and Keck values by 0.28 and 0.19 dex, respectively. While the differences are approaching statistical significance, they are readily understood given the significantly lower S/N of the KPNO spectrum compared to the Keck and KFOP spectra, as well as the small number of Fe ii lines, and thus higher uncertainty in the mean [Fe ii/H] abundance, measurable in the KPNO spectrum. The adopted stellar parameters and Fe abundances for Kepler-21 are those derived from the Keck (non-KFOP) spectrum and have been used in the derivation of the other elements. All additional references to the parameters of Kepler-21 will refer to these adopted values.

The abundances of individual elements that require special attention are described in the subsections that follow. The stellar and solar EW measurements and the respective absolute abundance for each line of all elements analyzed are given in the appendix. The final adopted abundances, given relative to the solar, and their uncertainties for each star are provided in Table 3.

3.2.1. Lithium

Li abundances have been derived from the λ6707 Li resonance feature using spectral synthesis. The adopted line list is that from King et al. (1997) updated as described in Schaeuble & King (2012). The synthetic spectra were smoothed using Gaussian profiles with FWHMs determined from weak, unblended absorption lines in the same spectral order as the Li feature. Sample spectra and fits of the λ6707 Li region are given in Figure 2.

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3.2.2. Carbon and Oxygen

Abundances of the light elements C and O have been derived from a combination of atomic and molecular lines. For C, we make use of five high-excitation C i lines and two C₂ features. Formation of high-excitation C i lines is susceptible to non-LTE (NLTE) effects, but abundances derived from the lines measured here have been shown to be largely free of deviations from the LTE approximation for solar-type stars (Asplund et al. 2005; Takeda & Honda 2005). The two C₂ features (λ5086 and λ5136) are blends of multiple components of the C₂ Swan system; spectral synthesis was used to derive C abundances from these features. The final adopted C abundance for each star is the mean of the relative abundances derived from the C i and C₂ lines, except for Kepler-21, for which only C i lines were measurable, and for Kepler-37, for which the mean abundance derived from the C i lines is 0.16 dex higher than that derived from the C₂ lines. Abundances derived from high-excitation lines, in particular the λ7775 high-excitation O i triplet (χ = 9.15 eV) and λ6053 S i line (χ = 7.87 eV), using one-dimensional LTE analyses have been shown to increase dramatically with decreasing T_eff in stars with T_eff ≲ 5400 K, contrary to canonical NLTE predictions (Schuler et al. 2004, 2006; Ramírez et al. 2013; Teske et al. 2013). Similar increases in Fe abundances derived from Fe ii lines relative to Fe i lines are seen in the same temperature regime (Schuler et al. 2003; Yong et al. 2004; Schuler et al. 2010). Taken together, the overabundances derived from high-excitation and singly ionized lines in the spectra of cool stars suggest that the electron populations of these energy states are not accurately modeled by our standard LTE analysis. We suspect that the difference in C i (χ = 7.68, 8.65 eV) and C₂ abundances of Kepler-37, with T_eff = 5406 K, is due to the overpopulation of the high-excitation C i states and thus adopt the C₂ based abundances for this star. The C abundances of all the stars are given in Table 4.

Table 4.  Carbon Abundances

Star	[C i/H]	σ	[C2/H]	σ	$\langle [{\rm{C}}/{\rm{H}}]\rangle$ a	N	σμ
Kepler-20	+0.01	0.03	+0.04	0.01	+0.02	5	0.01
Kepler-21	−0.02	0.02	⋯	⋯	−0.02	4	0.01
Kepler-22	−0.23	0.02	−0.26	0.01	−0.24	6	0.01
Kepler-37	−0.21	0.09	−0.37	0.01	−0.37	2	0.01
Kepler-68	+0.08	0.03	+0.10	0.02	+0.08	6	0.01
Kepler-100	+0.14	0.07	+0.05	0.02	+0.11	6	0.03
Kepler-130	−0.24	0.01	−0.28	0.04	−0.25	6	0.01

Note.

^aFinal adopted abundance.

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The abundances of O have been derived from the λ6300 forbidden [O i] line and the λ7775 high-excitation O i triplet. The λ6300 line is blended with two isotopic components of a Ni i transition at 6300.34 Å which must be taken into account when deriving O abundances from this feature. We use the blends driver within the MOOG software package for this purpose. The necessary input for the abundance derivation includes the EW of the λ6300 feature, atomic parameters (χ and log gf) for both the O and Ni transitions, and a Ni abundance. The atomic parameters for the [O i] line are taken from the fine analysis of Allende Prieto et al. (2001); those for the Ni blend are from the laboratory analysis of Johansson et al. (2003). The adopted Ni abundances are those derived here and given below. Unlike the λ6300 forbidden [O i] line which has been shown to be well described by LTE (e.g., Takeda 2003), formation of the high-excitation O i triplet is sensitive to NLTE effects (e.g., Kiselman 1991). NLTE corrections for the O i triplet based abundances have been estimated using the analytical formula provided by (Takeda 2003). We also tested the NLTE corrections of Fabbian et al. (2009), which cover a smaller parameter space compared to Takeda et al. The corrections are found to be similar for stars falling within the parameter space of both studies.

The derived O abundances are provided in Table 5, where it can be seen that the [O i] and the NLTE-corrected O i triplet abundances are in good agreement for Kepler-20 and Kepler-21. Because we consider the [O i]-based abundances to be more reliable, they are adopted as the final O abundances for these two stars. The [O i] line was not measurable in the Kepler-22, Kepler-68, Kepler-100, and Kepler-130 spectra, so the NLTE-corrected triplet-based abundances are adopted for these stars. We note, however, that the NTLE corrections for Kepler-22 and Kepler-68 are expected to be of the same order as for the Sun (Takeda 2003), so the LTE and NLTE abundances should not differ significantly. This is indeed observed, as the LTE and NLTE-based abundances differ by only a few dex in each case (Table 5). Both the [O i] and O i triplet features were measurable for Kepler-37, but as described above, the triplet based abundance is likely unreliable due to the overpopulation of high-excitation states in this cool star. We therefore adopt the [O i] based abundance as the final O abundance for Kepler-37.

Table 5.  Oxygen Abundances

Star	[O i]	O i Triplet				[O/H]a
	[O/H]	[O/H]LTE	σ	[O/H]NLTE	σ
Kepler-20	−0.04	−0.03	0.02	−0.01	0.02	−0.04
Kepler-21	−0.05	+0.15	0.03	0.00	0.00	−0.05
Kepler-22	⋯	−0.21	0.02	−0.19	0.02	−0.19
Kepler-37	−0.21	−0.28	0.02	⋯	⋯	−0.21
Kepler-68	⋯	+0.12	0.01	+0.09	0.00	+0.09
Kepler-100	⋯	+0.29	0.11	+0.19	0.08	+0.19
Kepler-130	⋯	+0.14	0.08	+0.10	0.07	+0.10

Note.

^aFinal adopted abundance.

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3.2.3. Odd-Z Elements: Sc, V, Mn, and Co

3.2.4. Abundance Uncertainties

Uncertainties in the derived stellar parameters (T_eff, log g, and ξ) and the uncertainty in the mean abundance (σ_μ) of a given element contribute to the total internal uncertainty in the derived abundance of that element. For elements whose abundances are based on two or less lines, the total uncertainty is based on the parameter uncertainties alone. Abundance sensitivities to the stellar parameters were determined for each star by calculating changes in the final abundances due to parameter changes of ±150 K in T_eff, ±0.25 dex in log g, and ±0.30 km s⁻¹ in ξ; representative sensitivities are given in Table 6. Abundance uncertainties to each parameter are then calculated by scaling the abundance sensitivities by the uncertainty in the respective stellar parameter. The final total internal uncertainty is the quadratic sum of the individual uncertainties due to the stellar parameters and σ_μ, for those elements with abundances derived from more than two lines.

Table 6.  Abundance Sensitivities

Species	Kepler-20			Kepler-21			Kepler-22
	ΔTeff	Δlog g	Δξ	ΔTeff	Δlog g	Δξ	ΔTeff	Δlog g	Δξ
	(±150 K)	(±0.25 dex)	(±0.30 km s−1)	(±150 K)	(±0.25 dex)	(±0.30 km s−1)	(±150 K)	(±0.25 dex)	(±0.30 km s−1)
Fe i	±0.09	±0.01	∓0.05	±0.10	±0.01	∓0.02	±0.01	±0.09	∓0.02
Fe ii	${}_{+0.10}^{-0.07}$	${}_{-0.10}^{+0.13}$	∓0.05	∓0.02	±0.10	∓0.05	±0.11	∓0.06	∓0.03
C i	${}_{+0.13}^{-0.09}$	${}_{-0.07}^{+0.10}$	∓0.01	∓0.08	±0.09	∓0.01	∓0.10	±0.09	∓0.01
O i	${}_{+0.19}^{-0.15}$	${}_{-0.05}^{+0.09}$	∓0.02	∓0.11	±0.07	∓0.05	∓0.15	±0.08	∓0.02
Na i	±0.10	∓0.05	∓0.03	±0.07	∓0.02	∓0.02	±0.09	∓0.03	∓0.02
Mg i	±0.07	∓0.03	${}_{+0.05}^{-0.09}$	±0.07	∓0.02	∓0.03	±0.08	∓0.03	∓0.02
Al i	±0.08	∓0.02	∓0.02	⋯	⋯	⋯	±0.08	∓0.01	∓0.01
Si i	±0.02	±0.02	∓0.02	±0.05	∓0.01	∓0.02	±0.01	±0.02	∓0.01
S i	${}_{+0.09}^{-0.06}$	${}_{-0.07}^{+0.10}$	∓0.01	∓0.04	±0.08	∓0.01	⋯	⋯	⋯
K i	±0.14	${}_{+0.09}^{-0.12}$	${}_{+0.04}^{-0.07}$	±0.13	∓0.06	∓0.12	±0.15	∓0.10	∓0.07
Ca i	±0.13	∓0.06	∓0.06	±0.10	∓0.03	∓0.05	±0.12	∓0.04	∓0.04
Sc ii	∓0.01	±0.11	∓0.02	±0.03	±0.10	∓0.02	0.00	±0.11	∓0.01
Ti i	±0.18	∓0.03	∓0.10	±0.14	∓0.02	∓0.03	±0.17	∓0.02	∓0.06
Ti ii	∓0.01	±0.09	∓0.10	±0.02	±0.10	∓0.10	∓0.01	±0.10	∓0.07
V i	±0.18	∓0.01	∓0.04	±0.14	∓0.02	∓0.01	±0.18	∓0.01	∓0.01
Cr i	±0.11	∓0.03	∓0.05	±0.10	∓0.01	∓0.02	±0.11	∓0.02	∓0.03
Mn i	±0.16	∓0.03	∓0.11	±0.14	∓0.02	∓0.01	±0.15	∓0.01	∓0.03
Co i	±0.11	±0.03	∓0.02	±0.13	∓0.02	∓0.01	±0.11	±0.03	∓0.01
Ni i	±0.07	±0.02	∓0.05	±0.10	∓0.01	∓0.02	±0.08	±0.02	∓0.03
Zn i	±0.01	${}_{+0.03}^{+0.03}$	∓0.11	±0.08	±0.02	∓0.10	0.00	±0.05	∓0.07

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Accurately derived stellar parameters are essential for the characterization of exoplanets and as asteroseismic inputs. Kepler's incredible success at finding planetary candidates during its primary mission has dramatically increased the number of stars for which effective temperatures, surface gravities, and metallicities are needed in short order, so the size and equilibrium temperature of the planet(s) can be accurately determined. Automated (spectrum fitting) and semi-automated (spectral line synthesis) routines, such as Spectroscopy Made Easy (SME; Valenti & Piskunov 1996), SPC (Buchhave et al. 2012), Automatic Routine for line Equivalent widths in stellar Spectra (ARES; Sousa et al. 2007), and Versatile Wavelength Analysis (VWA; Bruntt et al. 2002) to name a few, are generally used to meet this demand. In Table 7 we compare results in the existing literature to ours derived by "by hand" to provide a check of those determined using automated or semi-automated routines, and to ensure that the most accurate stellar parameters are being used in planetary characterization and asteroseismic studies. Here we discuss the comparisons for each star.

Table 7.  Comparison of Stellar Parameters

Star	Teff	log g	[Fe/H]	Method	Reference	Comment
Kepler-20	5514 ± 25	4.44 ± 0.12	+0.06 ± 0.06	MOOG	This study	⋯
	5455 ± 100	4.4 ± 0.1	+0.01 ± 0.04	SME	Gautier et al. (2012)	⋯
	5466 ± 93	4.443 ± 0.075	+0.02 ± 0.04	SME	Fressin et al. (2012)	⋯
	5547 ± 50	4.55 ± 0.10	+0.04 ± 0.08	SPC	Buchhave et al. (2014)	⋯
Kepler-21	6177 ± 42	3.99 ± 0.15	−0.08 ± 0.07	MOOG	This study	⋯
	6131 ± 44	4.0 ± 0.1	−0.15 ± 0.06	Multiple	Howell et al. (2012)	a
	6148 ± 111	3.94 ± 0.21	−0.19 ± 0.21	ROTFIT	Molenda-Żakowicz et al. (2013)	b
	6409 ± 74	4.43 ± 0.12	−0.03 ± 0.06	ARES/MOOG	Molenda-Żakowicz et al. (2013)	⋯
	5928 ± 75	3.25 ± 0.15	−0.20 ± 0.10	spec fit	Everett et al. (2013)	c
	6190 ± 60	4.00 ± 0.03	−0.16 ± 0.06	VWA	Bruntt et al. (2012)	⋯
	6123 ± 50	3.86 ± 0.10	−0.20 ± 0.08	SPC	Buchhave et al. (2014)	g
	6286 ± 70	${4.018}_{-0.004}^{+0.003}$	−0.01	Astero	Silva Aguirre et al. (2012)	d
	6305 ± 50	4.026 ± 0.04	−0.03 ± 0.10	Astero	Silva Aguirre et al. (2015)	e
	6120 ± 70	4.01 ± 0.01	−0.17 ± 0.07	Astero	Mathur et al. (2012)	f
	5838 ± 178	${4.008}_{-0.031}^{+0.011}$	−0.20 ± 0.30	Astero	Chaplin et al. (2014)	g
	6190 ± 84	${4.010}_{-0.009}^{+0.010}$	−0.16 ± 0.09	Astero	Chaplin et al. (2014)	h
Kepler-22	5622 ± 30	4.57 ± 0.08	−0.25 ± 0.04	MOOG	This study
	5518 ± 44	4.44 ± 0.06	−0.29 ± 0.06	SME	Borucki et al. (2012)
	5642 ± 50	4.49 ± 0.10	−0.27 ± 0.08	Multiple	Borucki et al. (2012)	i
	5626 ± 50	4.54 ± 0.10	−0.23 ± 0.08	SPC	Buchhave et al. (2014)
	5642 ± 50	4.443 ± 0.027	−0.27 ± 0.08	Astero	Huber et al. (2014)	j
Kepler-37	5406 ± 28	4.49 ± 0.13	−0.32 ± 0.08	MOOG	This study	⋯
	5417 ± 75	4.5667 ± 0.0065	−0.32 ± 0.07	Mulitple	Barclay et al. (2013)	k
	5442 ± 50	4.52 ± 0.10	−0.33 ± 0.08	SPC	Buchhave et al. (2012)	⋯
	5430 ± 50	4.61 ± 0.10	−0.34 ± 0.08	SPC	Buchhave et al. (2014)
	5241 ± 75	4.33 ± 0.15	−0.55 ± 0.10	spec fit	Everett et al. (2013)	c
Kepler-68	5887 ± 28	4.45 ± 0.10	+0.13 ± 0.05	MOOG	This study
	5793 ± 74	4.281 ± 0.06	+0.12 ± 0.074	Multiple	Gilliland et al. (2013)	l
	5758 ± 50	4.21 ± 0.10	+0.07 ± 0.08	SPC	Buchhave et al. (2012)	⋯
	5777 ± 50	4.21 ± 0.10	+0.04 ± 0.08	SPC	Buchhave et al. (2014)
	5793 ± 74	4.280 ± 0.003	+0.12 ± 0.07	Astero	Silva Aguirre et al. (2015)	m
Kepler-100	5855 ± 40	3.98 ± 0.15	+0.06 ± 0.08	MOOG	This study	⋯
	5909	4.30	⋯	SME	Batalha et al. (2013)	n
	5825 ± 75	4.125 ± 0.03	+0.02 ± 0.10	Multiple	Marcy et al. (2014a)	o
	5798 ± 49	4.09 ± 0.10	+0.03 ± 0.08	SPC	Buchhave et al. (2012)	⋯
	5824 ± 49	4.13 ± 0.10	+0.07 ± 0.08	SPC	Buchhave et al. (2014)
	5825 ± 75	4.125 ± 0.004	+0.02 ± 0.10	Astero	Silva Aguirre et al. (2015)	p
Kepler-130	5958 ± 37	4.41 ± 0.15	−0.20 ± 0.07	MOOG	This study	⋯
	5884 ± 75	4.304 ± 0.053	−0.22 ± 0.010	Multiple	Huber et al. (2013)	q
	5873 ± 50	4.29 ± 0.10	−0.22 ± 0.08	SPC	Buchhave et al. (2012)	⋯
	5861 ± 50	4.29 ± 0.10	−0.15 ± 0.08	SPC	Buchhave et al. (2014)

Notes.

^aHowell et al. (2012) used multiple sources to estimate the adopted parameters. ^bThe ROTFIT package is described in Frasca et al. (2003). ^cThe authors developed a spectrum fitting routine catered to low-resolution spectroscopy. ^dThe [Fe/H] for the asteroseismic analysis is taken from Casagrande et al. (2011). The T_eff is determined in an iterative fashion using the infrared flux method (IRFM). See Silva Aguirre et al. (2012) for details. ^e[Fe/H] and T_eff are derived using a high-resolution spectrum obtained from the Kepler Community Follow-up Observing Program (CFOP). See Silva Aguirre et al. (2015) for details. ^f[Fe/H] and T_eff are those of the optimal model from the Asteroseismic Modeling Portal, while the corresponding uncertainties are those reported for the spectroscopically derived values. The uncertainty in log g is the quadratic sum of the uncertainties in the derived radius and mass. ^g[Fe/H] and T_eff are field-average values and determined using the IRFM, respectively. See Chaplin et al. (2014) for details. ^h[Fe/H] and T_eff adopted from Bruntt et al. (2012). ⁱStellar parameters are derived from five separate spectra using various spectrum fitting techniques. See Borucki et al. (2012) for details. ^j[Fe/H] and T_eff are taken from Borucki et al. (2012). ^kSME and SPC were used to analyze two different spectra obtained with two different telescopes. The initial set of parameters and metallicity were used as input for an asteroseismic analysis to determine log g. [Fe/H] and T_eff were then rederived while holding the log g fixed at the asteroseismic value. See Barclay et al. (2013) for details. ^lAn initial set of parameters and metallicity were derived using SME and then used as input for an asteroseismic analysis to determine log g. [Fe/H] and T_eff were then rederived while holding the log g fixed at the asteroseismic value. See Gilliland et al. (2013) for details. ^m[Fe/H] and T_eff are taken from Gilliland et al. (2013). ⁿUncertainties in T_eff and log g are not provided, and no [Fe/H] is reported. ^oAn initial set of parameters and metallicity were derived using SME and then used as input for an asteroseismic analysis to determine log g. [Fe/H] and T_eff were then rederived while holding the log g fixed at the asteroseismic value. See for Marcy et al. (2014a) details. ^p[Fe/H] and T_eff are taken from Marcy et al. (2014a). ^qAn initial set of parameters and metallicity were derived using SME or SPC (which is not specified) and then used as input for an asteroseismic analysis to determine log g. [Fe/H] and T_eff were then rederived while holding the log g fixed at the asteroseismic value. See for Huber et al. (2013) details.

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Kepler-20: stellar parameters and metallicities have been derived for Kepler-20 by Gautier et al. (2012) and Fressin et al. (2012), both of which employed SME using the same Keck/HIRES spectrum obtained as part of the KFOP. The two studies find nearly identical parameters and metallicity for this star. The T_eff and [Fe/H] of both studies are in agreement with ours within uncertainties, and the surface gravity values are the same at log g = 4.44.

Kepler-21: numerous spectroscopic and asteroseismic studies have determined the stellar parameters and metallicity of Kepler-21. Howell et al. (2012) carried out an extensive spectroscopic study of the star, analyzing three separate spectra obtained with three different telescopes. Stellar parameters and metallicities were determined from each spectrum using automated spectrum-fitting routines and compared to existing literature values. The final adopted parameters and metallicity were chosen from the accumulated data set. Molenda-Żakowicz et al. (2013) derived two sets of parameters and metallicities using two different routines: the spectrum fitting package ROTFIT (Frasca et al. 2003) and the semi-automated routine ARES along with MOOG. The two sets of results, with the exception of the metallicities, differ at greater than the 1σ level, with the ARES/MOOG results being particularly disparate; it is the only analysis suggesting that Kepler-21 has the surface gravity of an MS dwarf rather than of a subgiant. Everett et al. (2013) developed a spectrum-fitting routine to analyze low-resolution (R ∼ 3000) spectra obtained with the KPNO 4 m Mayall telescope and RCSpec long-slit spectrograph. Bruntt et al. (2012) used the semi-automated routine VWA to derive T_eff, metallicity, and the abundances of 11 other elements, all of which are included in our analysis. The stellar parameters and metallicities of Howell et al. (2012), Bruntt et al. (2012), and the ROTFIT analysis of Molenda-Żakowicz et al. (2013) are in agreement with ours within uncertainties, while the stellar parameters of Everett et al. (2013), based on low-resolution spectroscopy, are lower by more than the combined uncertainties. Also, the abundances of the eleven additional elements derived by Bruntt et al. (2012) are in good agreement with our results, with all of them agreeing within uncertainties except for the abundance of carbon, our abundance (−0.02 ± 0.05) for which is +0.18 dex higher than theirs (−0.20 ± −0.07).

Asteroseismic analyses generally require T_eff and [Fe/H] as input, and they determine the fundamental stellar properties of mass, radius, and density. With the mass and radius, the surface gravity of a star can be accurately determined (e.g., Chaplin & Miglio 2013). No less than five different papers report fundamental properties of Kepler-21 based on asteroseismology, and three of the analyses include deriving the input T_eff and [Fe/H] parameters themselves, either from high-resolution spectroscopy or photometry (Howell et al. 2012; Silva Aguirre et al. 2012, 2015), one adopts the parameters from the literature (Mathur et al. 2012), and one does both (Chaplin et al. 2014). The derived T_eff values from these studies range from 5838 to 6305 K, and the [Fe/H] values from −0.20 to −0.01. Our adopted T_eff (6177 ± 42 K) and [Fe/H] (−0.08 ± 0.07) fall nicely in the middle of these ranges. More impressive is the tight agreement in the asteroseismically derived surface gravities, which range from 4.0 to 4.03 dex, despite the large ranges in T_eff and [Fe/H]. Our adopted surface gravity, log g = 3.99 ± 0.15, is in near-perfect agreement with these values.

Kepler-22: two sets of stellar parameters and metallicities have been derived for Kepler-22 by Borucki et al. (2012). One analysis utilized SME and a Keck/HIRES KFOP spectrum, and the other combined the results of spectrum-fitting analyses of five separate spectra obtained with four different telescopes. Borucki et al. (2012) also carried out an asteroseismic analysis, and Huber et al. (2014) calculate a surface gravity based on former's derived mass and radius. Both sets of spectroscopic results are in good agreement with ours, with only T_eff from the SME analysis deviating from our value by more than the combined uncertainties. Also, our derived surface gravity is 0.13 dex higher, just larger than the combined uncertainty, than the asteroseismically derived value.

Kepler-37: stellar parameters and metallicity of Kepler-37 are reported in the Kepler-37b discovery paper, Barclay et al. (2013). They used SME and SPC to analyze two sets of spectra obtained from two different telescopes as part of the KFOP. After deriving initial parameters and metallicity, asteroseismology was used to determine the star's surface gravity, among other things, and then the surface gravity was held fixed while T_eff and [Fe/H] were rederived. The final adopted values are the averages of those from the SME and SPC analyses. The Buchhave et al. (2012) values are in excellent agreement with those of Barclay et al., and both sets of results agree with our derived values within the uncertainties. Everett et al. (2013) also analyzed a low-resolution Mayall/RCSpec spectrum of this star. While their surface gravity agrees with ours within the uncertainties, their T_eff and [Fe/H] are lower than ours by more than the combined uncertainties. Finally, our surface gravity, as well as those of Barclay et al. (2013) and Buchhave et al. (2012), is in good agreement with the asteroseismic value of Silva Aguirre et al. (2015).

Kepler-68: Gilliland et al. (2013) determined the stellar parameters and metallicity of Kepler-68 using SME and a KFOP spectrum with the surface gravity constrained by an asteroseismic analysis, similar to the analysis of Barclay et al. (2013). These results are in good agreement with the Buchhave et al. (2012) values, all of which agree within the uncertainties. The surface gravities are also in agreement with the asteroseismic value determined by Silva Aguirre et al. (2015). While the results of Gilliland et al. (2013) agree with ours within the combined uncertainties, the stellar parameters of Buchhave et al. (2012) and the surface gravity of Silva Aguirre et al. (2015) differ from ours by slightly more than the combined uncertainties, with our values larger than those of these two studies. Despite these differences, the metallicities are in very good agreement.

Kepler-100: initial stellar parameters without uncertainties for Kepler-100 are provided by Batalha et al. (2013) based on an SME analysis of a KFOP spectrum. Refined stellar parameters and a metallicity were derived by Marcy et al. (2014a) using SME and a KFOP spectrum with the surface gravity constrained by an asteroseismic analysis, similar to the analysis of Barclay et al. (2013). These updated results are in good agreement with those of Buchhave et al. (2012). The surface gravities also match well the asteroseismic value of Silva Aguirre et al. (2015). The results of the latter three studies are in agreement with ours within the combined uncertainties, but those of Batalha et al. (2013) differ by more than the uncertainties of our analysis.

Kepler-130: stellar parameters and metallicity of Kepler-130 have been derived by Huber et al. (2013) using either SME or SPC and a KFOP spectrum (which routine and the source of the spectrum is not specified for individual stars) with the surface gravity constrained by an asteroseismic analysis, similar to the analysis of Barclay et al. (2013). These results are in near perfect agreement with those of Buchhave et al. (2012), and both sets of results agree with ours within the combined uncertainties.

Overall, the statistical agreement between our derived stellar parameters and metallicities of these Kepler host stars and those derived by numerous other groups using various automated and semi-auotmated routines with high-resolution spectroscopy is quite good. Only the ARES/MOOG stellar parameters of Kepler-21 (Molenda-Żakowicz et al. 2013), the SME T_eff of Kepler-22 (Borucki et al. 2012), the SPC stellar parameters of Kepler-68 (Buchhave et al. 2012), and the initial SME stellar parameters of Kepler-100 (Batalha et al. 2013) differ from our values by more than the combined uncertainties. In each case, the metallicity of each analysis agrees with ours. In contrast to the high-resolution spectroscopic analyses, the low-resolution spectroscopic analysis of Everett et al. (2013) produces more disparate results for the T_eff and log g of Kepler-21, and the T_eff and [Fe/H] of Kepler-37, all of which are underestimated at greater than the 2σ level. Nonetheless, the low-resolution spectroscopic results are considered to be an improvement over the photometrically derived stellar parameters and metallicities given in the Kepler Input Catalog (Brown et al. 2011).

It is also encouraging that our surface gravities statistically agree with the asteroseismic values for four of the six stars in our sample. For the two stars, Kepler-22 and Kepler-68, whose surface gravities are not in agreement, our values are larger at about the 1.7σ level in each case. It is generally accepted that asteroseismology provides the most accurate and precise surface gravities, because they are largely independent of the input stellar physics (Chaplin & Miglio 2013). It has been shown that, in general, log g determined spectroscopically by forcing ionization balance are larger than asteroseismically determined values, with the difference increasing with increasing T_eff (e.g., Mortier et al. 2014). While not all of the stars in our sample follow this general trend, it does agree with what we find for Kepler-22 and Kepler-68. The source of the discrepancy is not yet fully understood and merits further investigation. Nonetheless, our derived T_eff and [Fe/H] remain robust, because they do not correlate with changes in log g for standard curve-of-growth analyses (Torres et al. 2012; Mortier et al. 2014). Furthermore, based on the derived sensitivities (Table 6), the abundances of the other elements should also not be greatly affected by our slightly overestimated log g for Kepler-22 and Kepler-68.

An immediate conclusion that can be drawn from the abundances given in Table 3 is that small planets (in this case, 0.30 R_⊕ ≲ R ≲ 3.00 R_⊕) form in environments with a range of metallicities, from metal-poor ([Fe/H] ≈ −0.30) to metal-rich ([Fe/H] ≈ +0.15). This is in concordance with previous suggestions based on stars with planets classified as sub-Neptunes (Udry et al. 2006; Sousa et al. 2008; Ghezzi et al. 2010a) and, in particular, small planet candidates discovered by Kepler (Buchhave et al. 2012; Everett et al. 2013; Buchhave & Latham 2015). We point out that at the time of this writing at least 145 of the 226 planet candidates in the Buchhave et al. (2012) sample, including those discussed herein, have since been confirmed as bona fide planets (according to Kepler's Table of Confirmed Planets¹⁸ ), making more robust their conclusions that small planets form around stars with a wide range of metallicities and that the average metallicity of stars with small planets is approximately solar. Interestingly, the star with the highest metallicity, Kepler-68, is the one star in our sample with a known Jupiter-type giant planet.

Beyond the bulk metallicity of the host stars, the abundances of 15 of the remaining 18 other elements derived for each star show no deviations from general Galactic abundance distributions. In Figure 3 we compare the abundances of each element relative to Fe for the stars in our sample to those from studies that have analyzed a large number of stars with and without detected planets in the Galactic disk. Carbon and oxygen abundances are compared to abundances from Delgado Mena et al. (2010). Additional C abundances are taken from Bensby & Feltzing (2006) and additonal O abundances from Bensby et al. (2014). The abundances of the remaining elements are compared to those from Bensby et al. (2014) and Adibekyan et al. (2012b); the abundances of the latter are for stars with T_eff between 4900 and 6150 K. As can be seen in the figure, the abundances of each element of the Kepler planet host stars fall along the Galactic trends, demonstrating that these abundance ratios for this small subset of stars with small planets are typical of the general Galactic disk population. Generally speaking, this implies that the abundances of elements other than Fe scale with Fe in a way that is typical for similar stars in the Galactic disk.

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To further demonstrate this, we show in Figures 4 and 5 the distribution of the abundances relative to solar of 13 elements for our sample of Kepler host stars excluding Kepler-68 and the sample of stars with stars with no detected planets, stars with Neptune- and super-Earth-size planets (no Jupiter-type giants), and Jupiter-type giant planets from Adibekyan et al. (2012b). We do not include Kepler-68 in these histograms, because our main interest here is to compare the compositions of stars with small planets to a general stellar population. Kepler-68 is a giant planet (Kepler-68d) host, and given that it has the highest metallicity in our sample, its inclusion would certainly skew the comparisons. Our sample of only six stars prevents us from making firm statistical conclusions based on these comparisons, but they are suggestive of possible correlations. Indeed, the histograms for the Kepler host stars, stars without known planets, and stars with Neptune- and super-Earth-size planets largely overlap, whereas the histograms for the Kepler host stars and stars with Jupiter-type giant planets appear to be offset from each other, with the latter shifted toward higher abundances.

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To investigate these comparisons further, two-sample Kolmogorov–Smirnov (KS) tests were carried out for each element in Figures 4 and 5 for the Kepler host star sample and each of the samples of stars without known planets, stars with Neptune- and super-Earth-size planets, and stars with Jupiter-type giant planets. Again, our small sample size limits the sufficiency of the KS test to provide statistically significant probabilities of the relationships between the different samples of stars, and indeed, none of the tests yielded p-values that rejected the null hypothesis that the samples are drawn from the same parent populations. A much larger sample of detailed abundances of stars with small planets is clearly needed to determine if those abundances are in fact typical of the general Galactic disk population, and lower than those of stars with giant planets, as hinted at in Figures 4 and 5. If confirmed, this result would complement studies finding that the occurrence rate of small planets around solar-type stars and M dwarfs could be as high as 50% (e.g., Howard et al. 2012; Bonfils et al. 2013; Fressin et al. 2013; Marcy et al. 2014b; Dressing & Charbonneau 2015; Winn & Fabrycky 2015), whereas the occurrence rate of Jupiter-type giant planets around these stars is below 10% (e.g., Cumming et al. 2008; Winn & Fabrycky 2015), by suggesting that stars do not need enhanced or otherwise distinguishable chemical compositions to form small planets.

We conclude this section with a word on Li. The abundance of Li in stars with planets has been a topic of interest for some time, and there has been a debate on whether planetary hosts exhibit depleted Li abundances relative to stars without known planets for just as long. Gonzalez & Laws (2000) first suggested that stars with RV-detected planets tend to have Li abundances that are lower than field stars with detected Li based on a sample of seven planet host stars. Soon thereafter, Ryan (2000) compared the Li abundances of 16 stars with RV-detected planets to a sample of open cluster and field stars with similar ages, evolutionary states, and T_eff as the planet host stars and found that the Li abundances of the host stars are not different than those of the control sample. Subsequent to these two papers, numerous studies (e.g., Israelian et al. 2004; Takeda & Kawanomoto 2005; Sousa et al. 2010; Delgado Mena et al. 2014; Figueira et al. 2014) argue that stars with RV-detected planets do have Li abundances that are lower on average than stars without detected planets, and numerous studies (e.g., Luck & Heiter 2006; Ghezzi et al. 2010b; Meléndez et al. 2010; Ramírez et al. 2012) argue that the Li abundances of the two groups are indistinguishable. It is not clear how stars with small rocky (transiting) planets fit into this debate, because the Li abundances of a large enough sample of stars with small planets have yet been determined. Among our sample there are three stars with well-determined Li abundances and four with upper limits (Table 3). For the three stars with firm Li detections, their Li abundances are in good agreement with the general Li–T_eff trends of Ghezzi et al. (2010b) and Delgado Mena et al. (2015), and thus the Li abundances of stars with small rocky planets do not appear to be blatantly anomalous. Li abundances of a much larger sample of stars with rocky planets are needed, though, to determine if this is actually the case.

As mentioned in the Introduction, stellar abundances that correlate with T_c may indicate the presence of rocky planets. All of the Kepler stars in our sample are host to at least one sub-Neptune size or smaller planet, and four have at least one Earth-size planet. RV follow-up observations have, in general, only been able to place upper limits on masses of small planets except in a handful of cases, e.g., Kepler-20b, Kepler-20c, and Kepler-68b, so the densities and refractory element abundances of the majority of small planets have yet to be reliably estimated. Even with the known densities of Kepler-20b ($\rho ={6.5}_{-2.7}^{+2.0}\;{\rm{g}}\;{\mathrm{cm}}^{-3}$), Kepler-20c ($\rho ={2.91}_{-1.08}^{+0.85}\;{\rm{g}}\;{\mathrm{cm}}^{-3};$ Gautier et al. 2012), and Kepler-68b ($\rho ={3.32}_{-0.98}^{+0.86}\;{\rm{g}}\;{\mathrm{cm}}^{-3};$ Gilliland et al. 2013), the compositions are still uncertain, although densities greater than 5 g cm⁻³ are consistent with rocky and iron–nickel composition (Gautier et al. 2012; Marcy et al. 2014b, 2014a). The upper limits on the masses for all the small planets in our sample except for Kepler-20c, Kepler-68b, and Kepler-100c allow for densities greater than 5 g cm⁻³. Moreover, Marcy et al. (2014b) show that for the known exoplanets with 2σ mass determinations (33 planets at the time of this writing), those with radii less than 2 R_⊕ have densities ≳5 g cm⁻³. More recently, Rogers (2015) and Dressing et al. (2015) find that the transition from rocky to primarily non-rocky planets actually occurs at the smaller radius of 1.6 R_⊕. Each of the stars in our sample, save Kepler-22, has at least one planet with a radius less than or approximately equal to 1.6 R_⊕. It is therefore reasonable to assume that with the exception of Kepler-22, each star in our sample is host to at least one planet with a rocky or primarily rocky composition, and it provides a meaningful test of the T_c slope–rocky planet correlation hypothesis.

We quantify the relationship between the stellar abundances and T_c by measuring the slope of a standard least-squares fit to the refractory element (T_c > 900 K) abundances relative to Fe and the 50% condensation temperatures from Lodders (2003). Both unweighted fits and fits weighted by the inverse square of the total uncertainties in the [x/Fe] abundances have been made. The slopes and their 1σ uncertainties are provided in Table 8, and the weighted fits are shown in Figure 6. The difference in the slopes from the unweighted and weighted fits are statistically insignificant. Initial inspection of the fits to the data revealed that the Mn abundance falls below the fit to the [x/Fe]–T_c relation for each star, by more than the 1σ total uncertainty in the Mn abundances for three of the stars, despite the fact that all of the Mn abundances are in agreement within uncertainties with the general Galactic population (see Figure 3). Because the T_c of Mn is 1158 K and falls near the cool end of the T_c range, these low abundances could be biasing the slopes of the fits toward more positive values. We therefore have made unweighted and weighted fits to the data with Mn removed from the list of elements; the slopes of these fits are also included in Table 8. The difference in the slopes from these unweighted and weighted fits are once again found to be statistically insignificant. Moreover, all four slope measurements for each star are consistent within the 1σ uncertainties.

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The slopes for the five MS dwarfs in our sample are all positive at greater than the 2σ level, whereas the slopes for the two subgiants are negative at greater than the 1.5σ level, except for the unweighted and weighted slopes for Kepler-21 with the Mn abundance included in the calculation. In these two cases, the slopes are statistically consistent with positive values. We remind the reader that a positive slope in the [x/Fe]–T_c plane signifies an overabundance of refractory elements relative to the Sun, and in the interpretation of Meléndez et al. (2009) and Ramírez et al. (2010), suggests that the star is not a rocky planet host. On the other hand, a flat or negative slope signifies that the star has a similar refractory element distribution as the Sun or is deficient in refractories relative to the Sun, and thus the star would be a candidate for hosting rocky planets. Therefore, the slopes of the five MS dwarf stars are not consistent with the T_c slope–rocky planet correlation hypothesis. While the slopes for the two subgiants would appear to be consistent with the hypothesis, these stars' changing internal structure, in particular their deepening convective envelopes, have certainly mixed internal material to their surfaces, potentially changing their photospheric compositions. The T_c-planet signature is expected to be sensitive to the depth of the convective envelope during the planet formation process (e.g., Chambers 2010), so the increase in the depth of the convection zone of an evolving star could alter the signature. We therefore omit the two subgiants, Kepler-21 and Kepler-100, from the remaining discussion in this subsection.

Much of the preceding work investigating the possible correlation between T_c–abundance trends and rocky planets have focused on solar twins and analogs, and late F-type dwarfs. The former because of the similar structure to the Sun, and the latter because of the thinner convective envelopes compared to the Sun. In each case, it is thought that a planet formation signature will persist in the photospheres of these stars (e.g., Ramírez et al. 2014), as opposed to less massive stars with larger convective envelopes, in which any signature would be lost to convective mixing on a relatively short timescale. Two of the stars in our sample have convection zones that are estimated to be approximately equal to or smaller than that of the Sun, as discussed below.

The mass of Kepler-68 has been measured to be 1.079 ± 0.051 M_⊕ (Gilliland et al. 2013), and thus this star would be expected to have a convective envelope that is at least similar to or thinner than that of the Sun. It is host to an Earth-size planet, a sub-Neptune, and Jupiter-type giant planet. The mass and density of the Earth-size planet, Kepler-68c, have yet to be firmly determined, but Marcy et al. (2014a) calculate upper limits of M_c ≤ 2. 18 ± 3.5 M_⊕ and ρ_c ≤ 10.77 ± 17.29 g cm⁻³ based on four years of RV observations. Given these upper limits and its size, it is reasonable to assume that Kepler-68c is a rocky planet. The mass and density of the sub-Neptune planet, Kepler-68b, have been determined, and at M_b = 5.97 ± 1.7 M_⊕ and ρ_b = 2.60 ± 0.74 g cm⁻³, the planet's density falls in between those of gas giant and rocky planets. Lopez & Fortney (2014) suggest that planets like Kepler-68b represent a transition between rocky planets and gas giant planets, and may be composed of large quantities of (ice/liquid) water while lacking a large H/He gas envelope. The planetary models of Lopez & Fortney (2014) and Howe et al. (2014) suggest that if Kepler-68b has a H/He envelope at all, it makes up less than 2% of the planet mass. Using the model of Fortney et al. (2007), the composition of Kepler-68b is predicted to be close to 75% water ice and 25% rock. Assuming a 25% rocky composition for Kepler-68b and a rocky composition for Kepler-68c, a total of 3.7 M_⊕ of rocky material is sequestered in these two planets. We do not include the giant planet Kepler-68d in this calculation, because it is unclear if large, gas giant planets contribute to the putative depletion of refractory elements in the host star (Schuler et al. 2011; Ramírez et al. 2014). The amount of refractory material contained in Kepler-68b and Kepler-68c is more than what has been suggested to be missing in the photosphere of the Sun, so if rocky planet formation depletes the refractory elements in host stars, we would expect to observe such a depletion in Kepler-68.

We have tested this expectation by calculating the changes in the photospheric abundances of Kepler-68 due to the extraction of 3.7 M_⊕ of material from its convection zone. Using the accretion prescription of Mack et al. (2014), we calculated the abundance changes in the star assuming the accretion of 3.7 M_⊕ of material and then subtracted the changes from the observed abundances to estimate the effects on the slope of the T_c–abundance relation if this amount of material had indeed been sequestered in the planets during the formation of Kepler-68b and Kepler-68c. In addition to the mass of material, the accretion model inputs include the composition of the material, the mass of the star, and the mass of the convection zone. We adopted two different compositions of the extracted material, one that matches the Earth's composition (Earth model) based on McDonough (2001, pp. 3–23) and, following Chambers (2010), one composed of 50% Earth's composition and 50% carbonaceous (CM) chondrite material (50/50 model), with the CM chondrite compositions taken from Wasson & Kallemeyn (1988). Using the mass of Kepler-68 given above, the mass of its convection zone was estimated from Figure 1 of Pinsonneault et al. (2001) using our derived T_eff. The results of the modeled abundance changes are shown in Figure 7. Both the Earth model and the 50/50 model predict abundance changes that result in slopes of the fits to the T_c–abundance data that are consistent with zero. According to the T_c slope–rocky planet correlation hypothesis, a zero slope would be indicative of rocky planet formation. This result reinforces our assertion that if rocky planet formation does in fact deplete the refractory elements in host stars, we would expect to detect the signature in Kepler-68.

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The second star with a convective envelope that is estimated to be roughly equal to the Sun's is Kepler-130. Density (ρ = 0.927 ± 0.053 g cm⁻³) and radius (R = 1.127 ± 0.033 R_⊙) estimates for Kepler-130 based on posterior distributions of stellar evolution models are presented by Rowe et al. (2014); however, these estimates are based on the stellar parameters of Huber et al. (2013), which are about 75 K and 0.11 dex in T_eff and log g, respectively, lower than ours (although, as mentioned above, these are in agreement with ours within the combined uncertainty). Using their radius, density, and log g, as well as our derived log g, we calculate the mass of Kepler-130 to range from 0.932 M_⊙ to 1.19 M_⊙. Thus, assuming a convective envelope on par with that of the Sun's is not unreasonable. Kepler-130 is host to an Earth-size planet, a super-Earth, and a sub-Neptune; unfortunately, definitive masses have not been determined for any of them. However, the small sizes of the planets, particularly Kepler-130b (R = 1.02 ± 0.04 R_⊕) and Kepler-130d (R = 1.64 ± 0.16 R_⊕), are suggestive of rocky compositions. Using the empirical mass–radius relation of Weiss & Marcy (2014), we estimate the masses of the three planets to be M_b = 1.1 M_⊕, M_c = 7.0 M_⊕, and M_d = 4.3 M_⊕ for Kepler-130b, Kepler-130c, and Kepler-130d, respectively. We then use the model of Fortney et al. (2007) to predict the compositions of the planets and find an Earth-like composition for Kepler-130b, a pure ice composition for Kepler-130c, and a pure rock composition for Kepler-130d. Considering Kepler-130b and Kepler-130d only, as much as 5.4 M_⊕ of refractory material could be sequestered in these planets. Adopting a stellar mass of 1.061 M_⊕ (average of the range given above) and estimating a convection zone mass using Pinsonneault et al. (2001), we use the Earth and 50/50 composition models described above to determine the effect on the slope of the T_c–abundance relation if this amount of material had been sequestered in Kepler-130b and Kepler-130d. The results are shown in Figure 8. As found for Kepler-68, both the Earth model and 50/50 model predict abundance changes that result in T_c–abundance slopes that are consistent with zero, suggesting that if this star's refractory elements were depleted due to small planet formation, we should be able to detect it.

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One more system, Kepler-20, also warrants additional scrutiny, because masses and densities have been well determined for two of its planets (Gautier et al. 2012). The density of Kepler-20b is ${\rho }_{{\rm{b}}}={6.5}_{-2.7}^{+2.0}\;{\rm{g}}\;{\mathrm{cm}}^{-3},$ placing this planet squarely in the rocky planet category, and that of Kepler-20c is ${\rho }_{{\rm{c}}}={2.91}_{-1.08}^{+0.85}\;{\rm{g}}\;{\mathrm{cm}}^{-3},$ placing it in the transitional water-world regime. Two more planets in this system, Kepler-20e and Kepler-20f, have radii near that of Earth, so it is reasonable to assume they too are rocky planets. The fifth planet in the system, Kepler-20d, has a radius equal to that of Kepler-20c within uncertainties and is probably of the water-world type planet, as well. Once again we use the empirical mass–radius relation of Weiss & Marcy (2014) to estimate the masses of Kepler-20d (M_d = 6.9 M_⊕), Kepler-20e (M_e = 0.64 M_⊕), and Kepler-20f (M_f = 1.2 M_⊕) and the model of Fortney et al. (2007) to predict the compositions of all five planets. Compositions of the planets are found to be pure rocky for Kepler-20b, 75% water ice and 25% rocky for Kepler-20c, pure ice for Kepler-20d, and Earth-like compositions (∼25% rocky and ∼75% iron) for both Kepler-20e and Kepler-20f. If we include only the lower limits of the masses of Kepler-20b and Kepler-20c, a total of 9.6 M_⊕ of refractory material is contained in these two planets, whereas if we use the upper limits of the masses for these two planets and include the estimated masses of Kepler-20e and Kepler-20f, as much as 17.5 M_⊕ of refractory material could be sequestered in these planets. The mass and radius of the host star have been estimated at M = 0.912 ± 0.034 M_⊙ and $R={0.944}_{-0.095}^{+0.060}\;{R}_{\odot }$ (Fressin et al. 2012; Gautier et al. 2012), and thus it has a larger convective envelope than the Sun. Nonetheless, using Pinsonneault et al. (2001) to estimate the mass of the star's convection zone, both the Earth and 50/50 composition models predict T_c–abundance slopes that are negative or consistent with zero if either 9.6 M_⊕ or 17.5 M_⊕ of refractory material has been sequestered in the Kepler-20 planets (Figure 9). Once again we find that the putative rocky planet signature, if real, should be detectable in this star.

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