Spectral Properties of Cool Stars: Extended Abundance Analysis of Kepler Objects of Interest

The majority of the spectra in Brewer et al. (2016) were observed at a S/N per pixel >100 and the uncertainties for that analysis were derived from that data (Table 1). The authors noted that the scatter in the recovered abundances doubled for the spectra with S/N < 80. Because the targets in the CKS sample are fainter, the mean S/N is ∼ 50, with a significant fraction having S/N < 40. To quantify the impact of decreasing S/N on our ability to reliably recover stellar parameters and abundances across a variety of stellar types, we simulated observations of the Sun and a cool star at a range of S/N ratios and analyzed those spectra as if they were real observations using the procedure above. The increase in the standard deviation in the returned parameters over that at S/N = 100 for each of the S/Ns then gives us a measure of the increased uncertainty for the lower S/N spectra.

To generate the simulated observations of the Sun, we convolved the solar atlas of Wallace et al. (2011) with a Gaussian of the same width as the instrumental profile of the Keck HIRES B5 decker using the gaussbroad function from SME. We then resampled the spectrum to match the pixel scale from HIRES using the resamp function, also from SME. With this “observation” we then generated 25 realizations of the spectrum at S/N ratios of 20, 30, 40, 50, 60, 80, and 100 by adding Gaussian random noise to every pixel. For the cool star, we chose an existing spectrum with a S/N ∼ 380 and added noise to generate 25 new spectra in the same bins as for the simulated solar spectra. After analyzing these simulations, we found the standard deviation in all parameters for the two sets of 25 spectra as a function of S/N to quantify the changing uncertainty (Table 2).

Using the catagories of small planets identified by Fulton et al. (2017), we categorized all planets with R_p < 1.8 R_⊕ as “Super-Earths” and all with 1.8 < R_⊕ < 3.5 as “sub-Neptunes.” Fulton et al. (2017) also made several quality cuts in the data to avoid contaminating their sample with stars with poorly determined parameters. Specifically, they removed false positives, stars fainter than K_p < 14.2, transit impact parameters greater than 0.7, orbital periods greater than 100 days, evolved stars, and stars outside of the range $4700\,{\rm{K}}\lt {T}_{\mathrm{eff}}\lt 6500\,{\rm{K}}$. We performed the same cuts to remain consistent with their analysis, though a check on the full sample without those cuts showed no qualitative difference in our results. This resulted in 433 super-Earth sized planets and 267 sub-Neptunes. Most stars host only planets that are all about the same size (Weiss et al. 2018), so we are tempted to remove from our sample those outlier hosts that contain both super-Earths and sub-Neptunes (∼10%). However, we could not guarantee that our sample would be free of mixed systems, only that we had not detected them. We allow these hosts to count in both samples when comparing hosts instead of planets. Finally, we excluded any planets that were around evolved stars to avoid both evolutionary effects and analysis trends in our abundances. The resulting sample contained 433 Earth-sized planets around 335 dwarf hosts, and 267 sub-Neptunes around 216 dwarf hosts.

We present the stellar parameters determined from fitting observed spectra with synthetic spectra in two tables. Tables 3 and 4 match Tables 8 and 9 in B16 and contain the global stellar parameters and the final trend-corrected abundances, respectively. This layout was chosen to better facilitate combining the two data sets. Table 5 contains masses, radii, and ages from stellar isochrones derived using the spectroscopic parameters combined with parallaxes from Gaia DR2. In Table 6 we provide the per-star uncertainties for both the global stellar parameters and abundances using Equation (2).

The SPOCS ID is in column (1) of Tables 3 and 4, followed by the star name in column (2). The SPOCS ID numbers are shared with Valenti & Fischer (2005) and B16 and all stars new to this catalog have ID numbers >3247. Columns (3) and (4) of Table 3 contain the R.A. and decl. of the star, columns (5)–(7) contain the ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and [M/H] derived from this analysis. Note that [M/H] is the metallicity of the model atmosphere used to derive the best-fit model spectrum but does not necessarily match the [Fe/H] value for the star. For stars with available broadband colors and $0.4\lt B-V\lt 1.0$, we calculated both of the chromospheric activity indicators S_HK (column (8)) and $\mathrm{log}{R}_{{HK}}^{{\prime} }$ (column (9)) values following the procedure outlined in Isaacson & Fischer (2010). In modeling the spectra, we first derive the total rotational broadening, ${v}_{\mathrm{broad}}$ (column (10)); then, using the final parameters and abundances, fix ${v}_{\mathrm{mac}}$ (column (12)), using the relations in B16 and solve for $v\sin i$ (column (11)). The barycentric radial velocity, ${v}_{\mathrm{rad}}$ (column (13)), is the average over all epochs. The average SNR (column (14)) for the spectra, along with the average rms residuals between model and observation in the continuum and line regions (columns (15) and (16)) give a measure of the fit. The final column of Table 3 (column (17)) contains the number of spectra analyzed that were averaged to give the final parameters and abundances.

The stars in the sample span dwarfs to early giants or late subgiants from 4700 to 6800 K (Figure 1). As a star evolves off the main sequence, abundances in the photosphere can be altered by changes in the extent of the convective zone. When making abundance comparisons, we will limit ourselves to stars on the main sequence, and so divide our sample using the ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ relation developed in J. Myles et al. (2018, in preparation).

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In Table 4, columns (18)–(32) contain the solar relative log number abundances for 15 elements. The abundances are differential with respect to the Sun, using asteroid spectra as a reference, and then are calibrated to minimize trends in abundance with temperature using a sample of high-S/N dwarf spectra (Brewer et al. 2016). The abundances for both main sequence and evolved stars cover the same ranges as those in B16 (Figure 2). For most elements there are no strong trends in abundance with ${T}_{\mathrm{eff}}$ for dwarfs. However, for elements that suffer from NLTE effects, such as oxygen and manganese at temperatures above 6100 K, evolved stars are more strongly affected. This behavior is expected due to the sensitivity of NLTE effects to atmospheric structure (Bergemann et al. 2012) and care should be taken when comparing the abundance ratios of evolved and main sequence stars.

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The CKS Catalog includes parameters for 1305 stars, including some that are hotter or cooler than those included in this analysis. In addition, some spectra analyzed by P17 did not successfully converge in our analysis due to low S/N or did not meet our quality cuts for fit, $4700\,{\rm{K}}\,\lt$ ${T}_{\mathrm{eff}}$ $\lt \,6800\,{\rm{K}}$, 2.5 < $\mathrm{log}g$ < 5.0, or $0.5\,\mathrm{km}\,{{\rm{s}}}^{-1}\,\lt$ ${v}_{\mathrm{broad}}\lt 25\mathrm{km}\,{{\rm{s}}}^{-1}$. In total, there are 1011 stars in common between the two catalogs. We discard one of those (KOI-1060) in the following discussion because P17 report its $v\sin i$ as 456 km s⁻¹, which is likely a failure in their interpolation for this star, as visual inspection shows the spectrum to be a low-S/N but slowly rotating star ($v\sin i$ = 7.7 km s⁻¹) and is well fit by our model.

In general, the parameters in the CKS catalog and this work agree to within our mutual uncertainties, although there is a slightly higher dispersion in ${T}_{\mathrm{eff}}$ (Figure 4). There is a 26 K offset in the mean ${T}_{\mathrm{eff}}$ between the two samples, and the standard deviation about this mean is 79 K. When comparing the difference in temperatures as a function of $\mathrm{log}g$ there is a trend between 3.7 ≲ log g ≲ 4.4 (Figure 3). At lower gravities as measured in this work, the temperatures from P17 are systematically higher, decreasing to be equal at about solar $\mathrm{log}g$, then increasing sharply again at higher gravities. This behavior can be seen in the temperatures alone in the ${T}_{\mathrm{eff}}$ panel of Figure 4 and there is a slight trend in surface gravity itself at log g > 3.7. As these issues were not encountered in comparisons with other catalogs in B16, we assume that these are an artifact of the interpolation process in P17. The temperature average was forced to agree with our scale, but the fluctuations match roughly with the 250 K spacing of their model grid (E. Petigura 2018, private communication).

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The expected outcome from adding increasing amounts of noise to our model is increasing uncertainty in our fit and should follow a 1/(S/N) relation. The scatter in the fits to simulated solar and cool star spectra showed slightly different responses to the decreasing S/N ratio (see Figures 5, 6, and 7). In general, the result followed a 1/(S/N) relation, with some parameters having a slightly flatter and others a slightly steeper relation. Since the differences were small and inconsistent between the solar and cool star case, we chose the single functional form proportional to 1/(S/N) for all of the parameters.

The increase in the scatter in returned parameters was approximately proportional to $\tfrac{1}{{\rm{S}}/{\rm{N}}}$, with some variations between the solar case and the cool star case. To calculate the additional uncertainties for spectra with S/N < 100, we chose this form and began at the mean standard deviation at S/N = 100 of the Sun and cool star for each of the parameters (Equation (1)). The exception was $v\sin i$, which was negligible in the cool star and had no variation in recovered $v\sin i$ (Figure 5). We simply subtracted 0.05 km s⁻¹ from the solar case to obtain a reasonable approximation to the $v\sin i$ values.

$$\begin{eqnarray}&&{\rm{\Delta }}\sigma ={\sigma }_{100}\ast \displaystyle \frac{100}{{\rm{S}}/{\rm{N}}}.\end{eqnarray} \tag{ 1 }$$

To calculate the additional uncertainties for spectra with S/N < 100, we then calculated this additional σ for all of the parameters and abundances using Equation (1), where σ₁₀₀ is the mean standard deviation between the solar and cool star case at S/N = 100, and added them in quadrature to the uncertainties provided in B16 (Equation (2)). The uncertainties in that work were calculated by determining the distribution of differences between all stars with multiple observations and S/N > 100, then multiplying by two to account for possible systematic trends. Table 1 provides the means for Equation (1) to calculate new uncertainties (Equation (2)) for the lower S/N spectra from Brewer et al. (2016).

Two parameters that differed significantly between the solar and cool star simulations were $v\sin i$ and [N/H]. In the case of $v\sin i$, the cool star showed negligible rotational broadening and our model fitting consistently returned the minimum $v\sin i$,

$$\begin{eqnarray}&&{\sigma }_{\mathrm{new}}=\sqrt{{\sigma }_{b16}^{2}+{\rm{\Delta }}{\sigma }^{2}}.\end{eqnarray} \tag{ 2 }$$

The uncertainties listed in Table 6 are the statistical uncertainties from fitting models to observations. They provide a measure of the precision, but with the exception of $\mathrm{log}g$, do not incorporate any estimate of the accuracy of the parameters. At solar temperature, gravity, and metallicity, B16 demonstrated that the parameters and abundances were accurate, consistently returning the solar values.

We also looked at the change in the mean itself to see if any of the parameters or abundances had systematic trends induced by the decreasing signal-to-noise. In general, there are no appreciable or consistent shifts in the mean as a function of the decreasing S/N. However, the mean returned nitrogen abundance decreased to −0.04 dex at S/N = 30 and −0.09 dex at S/N = 20. We have added these offsets in quadrature to the uncertainties in the nitrogen abundance at these S/N ratios. The means in the oxygen and manganese abundances also show trends, though they are much smaller, reaching positive and negative 0.024 dex, respectively, at S/N = 20, and they do not appreciably affect the uncertainties.

Although the differences between the parameters derived in the CKS catalog and this work were small, there were some systematic differences in both $\mathrm{log}g$ and ${T}_{\mathrm{eff}}$. Because the CKS values were then used to derive the stellar radius, and from that the planet radius (Johnson et al. 2017), these differences could have an effect on any interpretations related to the size of the planets. We followed the same procedure as Johnson et al. (2017), using the Python package isochrones (Morton 2015) to fit our parameters and 2MASS (Skrutskie et al. 2006) K magnitudes from SIMBAD (Wenger et al. 2000) to isochrone grids from the Dartmouth Stellar Evolution Database (Dotter et al. 2008). However, we also included Gaia DR2 parallaxes (Collaboration et al. 2016), which we obtained using the CDS cross-match service with a 15 radius, for all but 28 stars that were not found (5) or had no reported parallax (23). More than one Gaia source was found for about 30 of the stars, so we chose the closest match in these cases. Additionally, five had negative parallaxes (Luri et al. 2018), but this was still a usable constraint for the isochrones package. Seven stars without K magnitudes were not fit. The isochrone fitting gives us radii, masses, and ages for the stars and we compile all of those along with their 1-σ uncertainty ranges in Table 5. Including the parallax in the isochrone fits gave us slightly tighter constraints on the radius than we would have, giving us a median uncertainty in radius of only 4.4%, with 90% of the stars having an uncertainty of less than 20%.

We found generally good agreement in the radii for most stars (Figure 8), with an offset of 7.7% and a weighted standard deviation of 11%, though there was a small systematic, with larger stars having slightly larger radii in our analysis. The small-planet comparison looked only at stars with smaller radii, where there is generally better agreement and smaller scatter. We also re-derived the planet radii using our updated stellar radii and verified that the distribution of small-planet radii was relatively unchanged with a small shift to larger radii (Figure 9). We compared the full distributions of all matching planets between Johnson et al. (2017) and this work. To estimate the uncertainties in the empirical PDFs, we generated 200 bootstrap realizations of the radii and found that they are consistent, though there may be a preference for slightly smaller radii of the sub-Neptunes with our new radii.

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Our hypothesis is that chemical differences in the protoplanetary disk, specifically in the Mg/Si ratio, could lead to differences in the mass of the planetary core. This in turn could lead to a bimodal population of small planets. To test this, we used a Gaussian kernel density estimator to calculate the probability density function (PDF) of the host Mg/Si distribution for the super-Earth-sized and sub-Neptune-sized planets.

The Mg/Si distribution of planets with R_p < 1.5R_⊕ is statistically indistinguishable from that of planets with 1.5R_⊕ <R_p < 3.5R_⊕ (Figure 10). To estimate the uncertainties for the PDFs we used bootstrap resampling to generate 200 realizations of each sample and generated PDFs using a Gaussian kernel density estimator. The PDFs appear very similar, but to quantify any differences between them we also performed a two sample Kolmogorov–Smirnov (KS) test on the cumulative distributions. We found that the KS test could not reject the null hypothesis that both samples were drawn from the same distribution and returned a two sided p-value of 0.33. Because the multiplicity of smaller planets is slightly higher than that of larger planets in our sample (∼1.3/star versus ∼1.17/star), we may be slightly shifting the distributions through weighting. However, when we looked at just the distributions of hosts and not planets we saw a qualitatively identical distribution. In fact, the host samples appear even more similar than when looking at the planets. The KS test for the hosts returned a two sided p-value of 0.73.

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We also looked at the distribution of planet radius as a function of the Mg/Si ratio of their host stars by calculating a two-dimensional PDF in radius and Mg/Si space (Figure 11). We found again that there was no appreciable difference in the distributions for super-Earth-sized planets and sub-Neptunes.

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One issue that can still affect abundance comparisons using this catalog is the varying S/N of the spectra, especially for fainter stars that are generally observed at lower S/Ns. By simulating more than 700 observations at varying S/Ns, we showed that our analysis method was robust to the increasing noise. At S/Ns above 50, the rms scatter in the returned parameters (Figures 5, 6, and 7) remained relatively small and there were no trends in the mean of the returned parameters and abundances. Below an S/N of 50 the scatter in all parameters increased rapidly and in a few cases, particularly nitrogen abundance, the mean of the abundance also shifted away from the true value. To facilitate comparisons for stars within the catalog, we quantified the increase in our uncertainty due to decreasing S/N and provided a table of individual uncertainties for the parameters of the 1149 stars in this catalog.

The uncertainty corrections that we derived also apply to the previously analyzed stars in B16. Though parameters for most of the stars in that catalog were derived using spectra with S/N > 100, almost 25% used lower quality spectra. The uncertainty equations here will allow their inclusion in abundance studies without the risk of skewing the interpretation.

The analysis procedure used in this work has already been shown to produce accurate gravities in agreement with asteroseismic values to within 0.05 dex (Brewer et al. 2015), and other stellar parameters along with precise abundances that, for most elements, do not have strong trends with temperature (Brewer et al. 2016). In Figure 2, we can see that the stars added in this catalog follow the same general distributions, for both dwarfs and evolved stars, as those from the previous catalog. One interesting feature that can be seen in the abundances is a slight difference in the distribution of dwarfs and evolved stars. This could be due to either flaws in the analysis, which may have trouble at the lower gravities of evolved stars, or actual astrophysical differences at a given ${T}_{\mathrm{eff}}$, or both.

An evolved star at a given effective temperature will be a more massive star that has migrated to its current position on the HR diagram after a comparatively short life compared to dwarf stars of that temperature. This, somewhat counterintuitively, will generally mean that the evolved star will be younger than the dwarfs at similar temperatures and so will have formed at a metallicity the same or higher than its main sequence counterparts. However, during a star’s main sequence lifetime the surface abundances will tend to decrease due to diffusion and gravitational settling out of the convective layer (Michaud et al. 1984; Paquette et al. 1986; Turcotte & Wimmer Schweingruber 2002; Dotter et al. 2017). Once the star begins to evolve off of the main sequence and its convective region begins to grow, some of this material can be re-mixed into the convective region and the surface abundance will be closer to or exceed its initial value (Dotter et al. 2017; Souto et al. 2018). Ratios between most elements remain relatively constant.

In both the B16 sample and this work, the dwarf stars are very slightly more metal-poor than the evolved stars (Figure 12), which agrees qualitatively with our expectations. When looking at the [Si/Fe] ratios, which should match more closely, we instead see that the B16 evolved stars show a significant shift to lower Si/Fe (Figure 13). Closer examination shows that all of the stars with lower Si/Fe have $\mathrm{log}g$ < 3.7, placing them as older subgiants, or at the base of the giant branch. There are comparatively few of these stars in the current work, but we do see that all evolved stars are reporting a systematically lower Si/Fe than dwarfs. This points to a systematic trends in abundance with $\mathrm{log}g$ for the evolved stars with lower surface gravities. We can also see some indications of these effects by eye when looking at the [X/Fe] ratios as plotted in Figures 14 and 15. We only used dwarf stars in our planet radius comparisons, which do not show these trends.

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There is no difference in the Mg/Si ratios between hosts of super-Earth-sized planets and hosts of sub-Neptune planets (Figure 10), nor is there any difference in the Mg/Si ratio as a function of planet radius between the two groups of planets (Figure 11). This contradicts our hypothesis that varying oxygen uptake in planetary cores, leading to different planet masses, could explain the bimodal small-planet populations found in Fulton et al. (2017). This eliminates metallicity as a driver for planet size and shows that planet radius evolution is driven by post-formation sculpting of the planetary atmospheres (Owen & Wu 2013; Fulton et al. 2017; Owen & Wu 2017). There do seem to be differences in planet multiplicity between the two populations, and there are hints of differences in other abundance ratios, but these are beyond the scope of the current paper and will be explored in future studies.