The First APOKASC Catalog of Kepler Dwarf and Subgiant Stars

The APOGEE stellar parameters and abundances released prior to Data Release 13 (DR13) were derived assuming that stellar rotation was a negligible contribution to line broadening at the resolution of the APOGEE spectra. Therefore, the synthetic spectra grid used for ${\chi }^{2}$ minimization was not convolved with stellar rotation broadening profiles. While this assumption is usually appropriate for giants, which are the majority of the stars observed by APOGEE, warmer dwarfs (${T}_{\mathrm{eff}}\gtrsim$ 6000 K) and younger dwarfs of all temperatures can still show noticeable rotation even at APOGEE resolution. This was realized at an early stage of this project, during the preparation of DR12, so a modified pipeline was developed to correct for this effect by adding rotation velocity as another dimension in the grid of atmosphere models. When rotation is included in the optimization procedure, C and N are fixed to solar values, i.e., $[{\rm{C}}/\mathrm{Fe}]=0$ and $[{\rm{N}}/\mathrm{Fe}]=0$, to keep the computational needs constrained.

The DR12-rot spectroscopic parameters were used in combination with asteroseismic data (Section 2.2) by all the GBM pipelines (see Section 3) to carry out comparisons among the codes. The offsets and different standard deviations found in these comparisons do not depend on whether we adopt the DR12-rot or the DR13 spectroscopic values.

The final ASPCAP spectroscopic parameters used in the catalog correspond to DR13 (SDSS Collaboration et al. 2016). DR13 includes by default a grid of models including rotation for dwarfs, an improved relation between microturbulence and gravity, and a relation for macroturbulence. Some of the improvements brought about by DR13 are discussed in SDSS Collaboration et al. (2016) and in greater extent by J. Holtzman et al. (2017, in preparation). Here we focus on ${T}_{\mathrm{eff}}$ and metallicity [M/H] for the APOKASC sample.

We present results in the catalog using two different sets of temperatures: photometric temperatures from measurements in the SDSS griz bands (Pinsonneault et al. 2012) and ASPCAP spectroscopic temperatures from DR13. A comparison between photometric and spectroscopic temperature scales in the Kepler field is shown in Figure 1. For Kepler targets, there is a systematic offset (defined as spec-phot) of −165 K between 5000 and 6000 K with a 134 K dispersion. At hotter temperatures the offset is −220 K with a 163 K dispersion. The formal ${T}_{\mathrm{eff}}$ uncertainty returned by ASPCAP is 70 K, whereas the median uncertainty for the SDSS ${T}_{\mathrm{eff}}$ scale is 62 K with a dispersion of 27 K. Two concerns with photometric temperatures are (1) the zero-point shifts from using different photometric color–${T}_{\mathrm{eff}}$ relations and (2) the uncertainties caused by reddening corrections. The latter are small, with E(B − V) median and dispersion of 0.023 and 0.015, respectively, for the Kepler dwarfs and subgiants (Huber et al. 2017). Concerning the impact of the KIC extinction values used in Pinsonneault et al. (2012), this has been tested by Huber et al. (2017) who found that using improved reddening estimates over the original KIC values and stellar metallicities (Pinsonneault et al. [2012] assumed a constant $[\mathrm{Fe}/{\rm{H}}]=-0.2$ value for the whole sample, as $[\mathrm{Fe}/{\rm{H}}]$ information was not available at the time for the whole Kepler sample) leads to a $\approx -20$ K zero-point shift of the SDSS ${T}_{\mathrm{eff}}$ scale.

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To test the temperature scale further, we calculated photometric temperatures for stars observed by APOGEE in the open cluster M67, which has a well-known solar metallicity, [Fe/H] = −0.01 (Jacobson et al. 2011), and reddening, E(B − V) = 0.04 mag (Taylor 2007), as well as accurate optical B- and V-band photometry (Sandquist 2004). To investigate the effect of different color–${T}_{\mathrm{eff}}$ relations, we used Casagrande et al. (2010) to convert B − V colors and Boyajian et al. (2013) to convert V − K colors to temperatures. The uncertainties in the ${T}_{\mathrm{eff}}$ from photometric errors are < 80 K. These comparisons are also included in Figure 1. We also compared the ASPCAP temperatures to the temperatures for five Gaia benchmark stars (Heiter et al. 2015), which are based on interferometric radii and parallax measurements. The random uncertainties in these measurements are $\lt$ 2%. Figure 1 also shows that the photometric M67 temperatures and Gaia benchmark star temperatures agree well with the SDSS temperature scale, particularly below 6250 K. This agrees with the conclusion of Pinsonneault et al. (2012) that there was good agreement among photometric scales for stars in this temperature range. Therefore, our preferred temperature scale for the Kepler field stars is the SDSS griz temperature scale. For the spectroscopic temperature scale, we include a systematic uncertainty component in the seismically determined stellar parameters to account for the systematic offset with respect to the photometric scale. This is done by varying the ASPCAP temperature scales by ±100 K and quantifying the impact in the final estimated stellar parameters (g, $\langle \rho \rangle$, R, M, and τ). Details are given in Section 4.3.3.

At temperatures greater than 6250 K, there are causes for concern for both temperature scales. The near-infrared H-band spectra of hotter stars have fewer absorption features, decreasing their sensitivity to temperature (Holtzman et al. 2017, in preparation). Pinsonneault et al. (2012) noted discrepancies among the photometric and spectroscopic measurements when they compared with their SDSS-color-based temperatures in this temperature range. Possible reasons include (1) rapid rotation producing non-isothermal surface temperatures, which violates an assumption of the infrared flux method, or (2) a lack of calibrators for color–${T}_{\mathrm{eff}}$ relations at these hotter temperatures. Currently ongoing efforts to carefully measure and cross-validate interferometric angular diameters with the CHARA array for a range of spectral types (I. Karovicova et al. 2017, in preparation), as well as planned detailed investigation of individual hot G stars, will contribute to understanding this discrepancy.

The metallicities $[{\rm{M}}/{\rm{H}}]$ we use throughout this work are the ASPCAP metallicities corresponding to DR13. Formal uncertainties returned by ASPCAP have a median value of 0.025 dex with a very small (0.004 dex) dispersion. These estimates of uncertainty are based on the small measurement scatter in members of star clusters. Figure 2 shows the comparison of ASPCAP $[{\rm{M}}/{\rm{H}}]$ values with results from optical spectroscopy, including $[\mathrm{Fe}/{\rm{H}}]$ measurements for 71 stars in common with the Bruntt et al. (2012) sample and $[{\rm{M}}/{\rm{H}}]$ for 400 stars in common with Buchhave & Latham (2015). The overall agreement is very good in both cases. The median and median absolute deviation²⁵ of the $[{\rm{M}}/{\rm{H}}]$ difference between ASPCAP and Bruntt et al. (2012) are −0.019 and 0.052 dex, respectively. With respect to Buchhave & Latham (2015), these values are −0.028 and 0.046 dex, respectively. The figure also shows ASPCAP results for the calibration cluster M67, for which we adopt the mean cluster value $[\mathrm{Fe}/{\rm{H}}]=-0.01\pm 0.05$ dex reported by Jacobson et al. (2011) as our fiducial value. Only stars in the range $\mathrm{log}g=3.3\mbox{--}4.5$ dex appropriate for our sample are shown. The ASPCAP dispersion for stars in this gravity range is 0.08 dex. Finally, two Gaia benchmark subgiants from Heiter et al. (2015) are also shown.

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Based on these comparisons, we do not find significant offsets in the ASPCAP $[{\rm{M}}/{\rm{H}}]$ scale, but typically the dispersion is larger than the formal uncertainties returned by ASPCAP, which highlights the possibility that the true uncertainty is underestimated. In light of the dispersion that we observe in external measurements of subgiant stars relative to APOGEE, we add a 0.1 dex uncertainty in quadrature to the ASPCAP formal uncertainty and adopt these as our final $[{\rm{M}}/{\rm{H}}]$ uncertainties for the catalog. Note that in constructing the catalog we use the same $[{\rm{M}}/{\rm{H}}]$ values in combination with both temperature scales. This is formally inconsistent and might be a systematic source of uncertainty. However, as shown in this section and particularly with the comparison with the results of Buchhave & Latham (2015), ASPCAP $[{\rm{M}}/{\rm{H}}]$ determinations are robust, so combining them with either temperature scale is a safe procedure.

The second major source of uncertainties has its origin in the GBM pipelines. Several aspects need to be considered: the stellar evolution tracks, the statistical method to extract stellar parameters, and the calculation of ${\rm{\Delta }}\nu$ in evolutionary tracks.

As described in Section 3, different pipelines typically employ different grids of stellar models. Stellar evolution theory still has some uncertainty; there is not a unique choice of what different modelers consider is the best physics that should be employed in computation of stellar models. In the realm of low-mass stars, different possibilities are available, among others, for nuclear reaction rates, radiative opacities (atomic and molecular), treatment of microscopic diffusion, and implementation and amount of overshooting, among other physical inputs or processes. Moreover, different numerical implementations of similar physics and numerical aspects of the integration of the equations of stellar structure and evolution can also lead to differences in the computed stellar models. Detailed comparisons of stellar models for dwarf stars using different physics but the same code or different codes but the same underlying physics have been the object of some studies, although of limited scope (Lebreton et al. 2008; Marconi et al. 2008; Stancliffe et al. 2016, among others). Additionally, GBM pipelines can implement differently a given set of stellar tracks (e.g., interpolating stellar tracks to a finer grid) and also employ different statistical approaches to extract stellar parameters and their uncertainties, so even using the same underlying grid of stellar models does not imply that different GBM pipelines will lead to the same results.

C14 has considered this problem and performed several comparisons aimed at disentangling the effects of statistical methods and the use of different grids of stellar models. Here, we have also carried out such comparisons based on results from the 12 GBM sets of results available (Table 1). Extensive GBM one-to-one comparisons are presented in Appendix B. In this section, we use a subset of results to illustrate the typical systematic uncertainties arising from four different sources: different evolutionary tracks, different statistical methods to determine stellar parameters, the use of ${\rm{\Delta }}{\nu }_{0}$ or ${\rm{\Delta }}{\nu }_{\mathrm{scl}}$, and the solar calibration of ${\rm{\Delta }}{\nu }_{0}$ (Section 4.1).

The typical relevance of each uncertainty source is shown, for all stellar parameters, in Figure 8, where each column shows relative differences in the central values of the stellar parameters determined with two different GBM runs that differ from one another in just one aspect. Differences are taken in the sense indicated at the top of each column. In each panel, red solid and blue dashed lines are the statistical uncertainties returned by the first and second pipelines, respectively, binned in 60 K intervals. Note that it is not within the scope of this paper to look into detail at the origin of the differences among GBM results; rather, we present typical cases and point out some possible causes for systematic differences.

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The first column in Figure 8 compares results obtained with the YB pipeline but with different evolutionary tracks, YREC2 or BaSTI. For g, $\langle \rho \rangle$, R, and M a systematic offset is seen between the sets of results. A possible reason for systematic offset could be that ${T}_{\mathrm{eff}}$ scales or even luminosities in the models are somewhat different. A mismatch in the ${T}_{\mathrm{eff}}$ scale is possible even if both sets of tracks are based on a solar calibration of the mixing length parameter, especially if the solar calibration and the evolutionary tracks have been computed with different physical assumptions. This is the case of the BaSTI tracks, which include microscopic diffusion in the solar calibration but not in the library of stellar models (Pietrinferni et al. 2004), which leads to a ${T}_{\mathrm{eff}}$ offset as discussed in Section 4.1. Another source of differences might be the choice of the critical ${}^{14}{\rm{N}}{({\rm{p}},\gamma )}^{15}{\rm{O}}$ rate, which is about a factor of two smaller in YREC2 than in BaSTI (see Tables 7 and 8 in Appendix A for a description of input physics used in the stellar models used in this work). This rate affects the ${T}_{\mathrm{eff}}$ scale of low-mass stars (typically $M\lt 1.3\,{M}_{\odot }$) close to the turnoff, when the CNO cycle takes over pp-chains as the dominant H-burning process. It also affects the whole main sequence for more massive models and the luminosity of the subgiant branch, where models with a larger rate are less luminous (Magic et al. 2010).

An additional difference is the inclusion (YREC2), or not (BaSTI models used in this work), of overshooting, which will also affect the ${T}_{\mathrm{eff}}$ scale of stellar models that have a convective core during the main sequence. It is also interesting that the systematic offset in stellar mass is not translated into ages, as would be naively expected. This points toward YREC2 predicting longer main-sequence lifetimes than BaSTI, which is consistent (at least for stars with $M\gtrsim 1.2\,{M}_{\odot }$ that have convective cores during the main sequence) with the fact that YREC2 models include convective overshooting whereas the BaSTI models used here do not. It is also possible that, even in the absence of convective cores during the main sequence, BaSTI and YREC2 models do not predict the same evolutionary timescales at equal stellar mass and composition.

Despite all the above differences, it is reassuring that in almost all cases the differences are smaller than or comparable to the 1σ statistical uncertainties returned by the pipelines. The exception is a small number of stars with ${T}_{\mathrm{eff}}\lt 5400$ K for which R and M have larger fractional differences.

The second column compares results obtained by two sets of GBM results that employ the same underlying set of BaSTI tracks but that are implemented differently. These two GBMs, SFP/B and MPS/B, also determine central parameters and uncertainties by different methods as well. Here it is very difficult to single out reasons why results from both GBM pipelines show such dis-persions. A clear difference is the mass resolution because SFP/B uses an interpolated grid with a finer mass resolution than the original BaSTI grid. This may play an important role especially when seismic uncertainties are small, which can lead to statistical uncertainties well below the mass resolution of the grid. For all stellar quantities, the dispersion of results seen among pipelines sharing the same evolutionary tracks is comparable to those seen in the first column, showing the same pipeline with different stellar models. Again, we note that systematic differences are almost in all cases not larger than statistical uncertainties returned by the pipelines.

The third column compares results from the BAS/G pipeline where the only difference is the use of ${\rm{\Delta }}{\nu }_{\mathrm{scl}}$ or ${\rm{\Delta }}{\nu }_{0}$. This case has also been discussed in the literature, as described in Section 3, but with emphasis in the change in $\langle \rho \rangle$, which is most directly affected by the choice of ${\rm{\Delta }}\nu$ in the stellar models. Clearly, and as expected, the systematic effects follow the inverted shape of the ${\rm{\Delta }}{\nu }_{0}/{\rm{\Delta }}{\nu }_{\mathrm{scl}}$ ratio (Figure 4) multiplied by 2. This clear trend with ${T}_{\mathrm{eff}}$ is also reflected in R and M estimates, as well as in $\mathrm{log}g$, which to first order is independent of ${\rm{\Delta }}\nu$ according to the scaling relations. Stellar ages are affected as well, but in this case they reflect an almost one-to-one correspondence with the mass variation, so changes are quite small. Note that ${\rm{\Delta }}{\nu }_{\mathrm{scl}}$ has been scaled such that the reference solar ${\rm{\Delta }}{\nu }_{\odot }=135.1\,\mu \mathrm{Hz}$ is reproduced. On the other hand, the grid using ${\rm{\Delta }}{\nu }_{0}$ has not been calibrated to match ${\rm{\Delta }}{\nu }_{\odot }$.

The fourth column compares results from BeS/G–${\rm{\Delta }}{\nu }_{\mathrm{scl}}$ and BeS/G–${\rm{\Delta }}{\nu }_{0}$, and in this case both ${\rm{\Delta }}{\nu }_{\mathrm{scl}}$ and ${\rm{\Delta }}{\nu }_{0}$ have been rescaled, as explained in Section 4.1, so that a solar model is reproduced properly. The trends seen in the panels of this column are very similar to those in the previous column but with an offset that reflects the rescaling of ${\rm{\Delta }}{\nu }_{0}$ by the factor f_{Δ ν} defined in Section 4.1. It is important to point out that, despite the f_{Δ ν} correction (or calibration) factor, the differences between BeS/G–${\rm{\Delta }}{\nu }_{\mathrm{scl}}$ and BeS/G–${\rm{\Delta }}{\nu }_{0}$ are not zero at the solar ${T}_{\mathrm{eff}}$ because the majority of stars in the APOKASC sample around this ${T}_{\mathrm{eff}}$ range are past the main sequence or do not have a solar mass and composition, so the ${\rm{\Delta }}{\nu }_{0}/{\rm{\Delta }}{\nu }_{\mathrm{scl}}$ is different from solar (see Figure 4).

From the discussion of the previous section it becomes clear that a complete determination of systematic uncertainties is not possible. First, it would require surveying all physical inputs in stellar models (some of which are not even properly modeled, such as overshooting or semiconvection) and quantifying the resulting model uncertainties. Second, it would be necessary to account for differences in stellar properties due to the statistical methods that can be employed in GBMs. Instead, we follow the pragmatic approach initiated in C14 for defining and computing uncertainties associated with systematic differences arising from the use of various GBMs. The method is based on using all available sets of GBM results described in Section 3 and relies on results returned by GBMs being robust and that a good measure of the systematic uncertainty due to stellar models and of the statistical inference is given by the dispersion among all GBM results.

A measure of the robustness of the results returned by the GBM can be obtained by computing the difference of the central values returned by each pipeline with respect to reference values, which we take here as those returned by the combination BeSPP/GARSTEC–${\rm{\Delta }}{\nu }_{0}$, and then normalizing these differences by the median of the statistical uncertainty across all GBM pipelines. These normalized residuals give an overall picture of the scatter across different sets of GBM results. The resulting distributions for g, $\langle \rho \rangle$, R, M, and τ are shown in Figure 9, including the 68.3% and 95.4% confidence levels, which, in all cases except for $\langle \rho \rangle$, correspond to narrower than normal distributions of standard deviation equal to 1, i.e., when systematic and statistical uncertainties are equal.

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The distribution of $\langle \rho \rangle$ has a slight shift introduced by the choice of reference values. The reason for the shift is related to the fact that the reference values have been computed with a grid of stellar models using ${\rm{\Delta }}{\nu }_{0}$, as opposed to ${\rm{\Delta }}{\nu }_{\mathrm{scl}}$, and 9 out of the 12 GBM sets of results employ ${\rm{\Delta }}{\nu }_{\mathrm{scl}}$. As discussed by White et al. (2011), in the ${T}_{\mathrm{eff}}$ range between 5000 and 6300 K, composing most of our sample, ${({\rm{\Delta }}{\nu }_{0}/{\rm{\Delta }}{\nu }_{\odot })\gt (\langle \rho \rangle /\langle {\rho }_{\odot }\rangle )}^{1/2}$, whereas, by definition, ${({\rm{\Delta }}{\nu }_{\mathrm{scl}}/{\rm{\Delta }}{\nu }_{\odot })=(\langle \rho \rangle /\langle {\rho }_{\odot }\rangle )}^{1/2}$. This implies that GBM relying on ${\rm{\Delta }}{\nu }_{\mathrm{scl}}$ will introduce a (small) systematic shift toward higher $\langle \rho \rangle$ values than GBMs relying on the more physically correct ${\rm{\Delta }}{\nu }_{0}$, as is reflected in the distribution shown in Figure 9. The second feature to notice are the extended tails of the distribution toward both positive and negative values that are formed by results corresponding, in all cases, to stars with very small ${\rm{\Delta }}\nu$ uncertainties, below 0.4%. For all cases where the absolute value of the normalized residuals is larger than 2, the statistical uncertainty of $\langle \rho \rangle$ is smaller than 1.5%, a value driven by the high precision of ${\rm{\Delta }}\nu$ for most of our sample.

The same arguments that explain the shift of peak of the $\langle \rho \rangle$ distribution are behind the shifts of the central peaks in the R and M distributions. The shifts here take the opposite sign with respect to the change in $\langle \rho \rangle$ because both quantities depend inversely on ${\rm{\Delta }}\nu$ (Equations (3)–(4)). Finally, the age distribution simply mirrors differences in masses, as is apparent from the bottom two panels in Figure 9. The similarity between these two distributions is, once again, reassuring in that evolutionary timescales obtained from different stellar evolution codes are in reasonably good agreement, so that a given fractional difference in mass translates into a very similar fractional difference in age.

Results discussed show that differences among GBM sets of results are typically consistent with each other when measured against the statistical uncertainties, i.e., given the current limitations imposed by uncertainties in the seismic and spectroscopic data.

The systematic GBM uncertainty for each star and quantity is defined as the standard deviation σ of the central values returned by all 12 GBM sets after applying outlier rejection. Initially, results lying beyond 3σ of the median of the results are removed. Then, the median of the remaining sample is recalculated and the process is iterated until no values beyond the 3σ level remain. The standard deviation of the remaining values determines the systematic GBM uncertainty.

We have considered the presence of a systematic effect in the calibration of the ${T}_{\mathrm{eff}}$ ASPCAP scale. This is included in the present catalog as an additional systematic uncertainty in the derived stellar parameters. To obtain this uncertainty, we have performed two additional GBM runs with BeSPP/GARSTEC–${\rm{\Delta }}{\nu }_{0}$ where the ASPCAP ${T}_{\mathrm{eff}}$ scale has been modified by ±100 K. Then, for each stellar parameter, the systematic uncertainty has been defined as the absolute value of half the difference between those two sets of GBM results. The median fractional uncertainties for the whole sample are 1.0% ($\mathrm{log}g$), 0.3% ($\langle \rho \rangle$), 1% (R), 3% (M), and 13% (τ). In the case of M and τ, these are smaller than but comparable to median statistical uncertainties as discussed in Section 4.2, whereas they are substantially smaller for the other stellar parameters.

This systematic component of the total error budget is not included in the catalog based on the SDSS griz ${T}_{\mathrm{eff}}$ scale. The reason, discussed in Section 2.1.3, is that this ${T}_{\mathrm{eff}}$ scale is more accurate than the ASPCAP ${T}_{\mathrm{eff}}$ scale.

The vast majority of stars in the catalog presented in this work were also part of the C14 catalog. The most important difference with respect to C14 is that a single value $[\mathrm{Fe}/{\rm{H}}]=-0.2\pm 0.3$, taken as representative of the solar neighborhood (Silva Aguirre et al. 2011), was adopted for the whole sample in that work. The large uncertainty effectively implied that $[\mathrm{Fe}/{\rm{H}}]$ actually had a relatively minor influence on the derived stellar properties. The SDSS ${T}_{\mathrm{eff}}$ scale used in this work is the same as that used in C14.

We can highlight the impact of having $[{\rm{M}}/{\rm{H}}]$ measurements by directly comparing the stellar properties derived in this catalog with respect to those in C14 when the SDSS ${T}_{\mathrm{eff}}$ scale is used in both cases. This is shown in Figure 15, where fractional differences are shown as a function of the ASPCAP $[{\rm{M}}/{\rm{H}}]$ value. For reference, the $[\mathrm{Fe}/{\rm{H}}]$ adopted in C14 is shown as a vertical dashed line. For all seismically determined stellar quantities there are clear correlations in the differences between the present and the C14 catalogs with the actual $[{\rm{M}}/{\rm{H}}]$ of stars. A simple linear fit $\delta =a\cdot ([{\rm{M}}/{\rm{H}}]+0.2)+b$, shown in each plot together with the 1σ variation in solid and dashed lines, respectively, is indicative of the dependence of each stellar parameter on $[{\rm{M}}/{\rm{H}}]$. The fits have slopes equal to $a\,=0.062(g),-0.044(\langle \rho \rangle ),0.103(R),0.267(M),-0.493(\tau )$. The offsets at $[{\rm{M}}/{\rm{H}}]=-0.2$ for the different quantities are $b=0.004,0.020,-0.016,-0.026,-0.062$ in the same order as above. Stellar parameters in C14 were also determined using BeSPP, models computed with GARSTEC, and ${\rm{\Delta }}{\nu }_{0}$. The reason for the offset in $\langle \rho \rangle$ arises from the fact that contrary to the procedure applied in the present work, in C14 ${\rm{\Delta }}{\nu }_{0}$ in the stellar models were not rescaled according to the factor f_{Δ ν} described in Section 4.1. This leads to a 1% offset in ${\rm{\Delta }}{\nu }_{0}$ between the grids of stellar models that propagates as about twice that value to the $\langle \rho \rangle$ determination, as expected, with a comparable offset for R and a slightly larger one for M, which, in turn, propagates into a τ offset. The scatter seen in the relations can have at least three sources: the asteroseismic data used here are determined in many cases from longer time series and improved seismic analysis, the grids of models have not been computed with the exact same assumptions, and the current version of BeSPP is an evolution of that used in C14, when it was still based on a frequentist approach akin to that used by the YB pipeline.

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As a whole, results in this catalog are consistent with those previously published by C14, but they represent a qualitative and quantitative step forward because metallicity information is now included individually for all stars. The C14 catalog, however, includes about 100 more stars that were not observed with APOGEE (see Section 2).

The LEGACY sample is formed by 66 main-sequence stars of highest quality of asteroseismic data from the Kepler mission (Lund et al. 2017). Asteroseismic analysis of these stars is based on stellar model fitting that relies on using specific combinations of individual frequencies that allow a more precise determination of stellar parameters. Detailed results have been presented in Silva Aguirre et al. (2017).

From the total common sample between APOKASC and LEGACY, we have removed two stars because in one case the ASPCAP ${T}_{\mathrm{eff}}$ and in the other the SDSS ${T}_{\mathrm{eff}}$ values differ from the LEGACY ${T}_{\mathrm{eff}}$ value (based on SPC spectroscopic analysis by Buchhave & Latham 2015) by more than 600 K. In total, we are left with a common sample of 45 stars that can be used as a benchmark. Our comparison of stellar parameters makes use of LEGACY results computed with the BASTA pipeline, which are based on GARSTEC. By using LEGACY results based on the same evolutionary code used to determine the central values in the APOKASC catalogs, we minimize the systematic differences arising from stellar models.

Note that, in addition to the different type of input seismic data used in this catalog (${\rm{\Delta }}\nu$ and ${\nu }_{\max }$) and in the LEGACY work (individual frequencies or frequency separation ratios), another source of deviation between the two sets of results is likely due to the fact that all GBMs presented in this work are based on libraries of stellar models where the mixing length parameter is fixed, whereas in the LEGACY work this is an additional free parameter in the stellar models. This precludes a detailed discussion of star-by-star comparisons. Instead, we focus on a global comparison of mean differences and dispersions.

Figure 16 shows the comparison for all stars and both ${T}_{\mathrm{eff}}$ scales, identifying in blue (circles) results for the SDSS ${T}_{\mathrm{eff}}$ scale and in red (squares) those for the ASPCAP ${T}_{\mathrm{eff}}$ scale. Horizontal lines indicate the weighted mean difference with respect to the LEGACY results. For three stars only ASPCAP ${T}_{\mathrm{eff}}$ values are available.

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The weighted mean and the dispersion (rms) of the fractional differences of the two APOKASC sets of results with respect to the LEGACY results are summarized in Table 6. For all stellar parameters, ASPCAP results show a smaller mean difference than SDSS. Overall, it is reassuring that the systematic offsets between APOKASC and LEGACY results are in all cases substantially smaller than the median error of each quantity in the APOKASC catalog (Figures 10 and 11).

Table 6.  APOKASC−LEGACY

	ASPCAP			SDSS
	Weig. Mean	Weig. rms	Unw. Mean	Weig. Mean	Weig. rms	Unw. Mean
$\delta g/g$	−0.007	0.028	−0.002	0.014	0.021	0.016
$\delta \langle \rho \rangle /\langle \rho \rangle$	−0.004	0.013	−5 × 10−4	0.001	0.014	0.007
$\delta R/R$	−0.004	0.018	−0.001	0.007	0.016	0.009
$\delta M/M$	−0.015	0.061	−0.003	0.027	0.047	0.036
$\delta \tau /\tau$	0.078	0.300	0.213	−0.167	0.183	−0.116

Note. Comparison of APOKASC and LEGACY results shown as APOKASC–LEGACY. Weighted mean and rms fractional differences are shown for both ${T}_{\mathrm{eff}}$ scales used in this work. Additionally, the unweighted fractional mean differences are also given.

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In the following we discuss the comparison in more detail, in particular for stellar mass. The mean fractional difference $\delta M/M$ between APOKASC/ASPCAP and LEGACY is $-1.5 \%$ and between APOKASC/SDSS and LEGACY is 2.7%. The mean ${T}_{\mathrm{eff}}$ in ASPCAP is 1.4% cooler than in the LEGACY sample, and the SDSS ${T}_{\mathrm{eff}}$ scale is 1.3% hotter. By propagating these differences through the scaling relation Equation (3), we infer that asteroseismic masses from global seismology show a systematic mean deviation of about 2% with respect to the LEGACY sample. We have confirmed this by performing an additional GBM run with BeS/G–${\rm{\Delta }}{\nu }_{0}$ using as inputs the combination of our global seismic parameters and the ${T}_{\mathrm{eff}}$ and $[\mathrm{Fe}/{\rm{H}}]$ values used from LEGACY sample (Silva Aguirre et al. 2017). In this case, we find that the mean $\delta M/M=0.6 \%$, in agreement with the above estimate, and an rms $\delta M/M=6 \%$. In this case, the mean and rms values for $\delta \tau /\tau$ are −4% and 26%, respectively. These numbers can be taken as a first-order estimation of the systematic differences in mass and age results between global seismology and more detailed analysis based on modeling many individual mode frequencies. However, a more careful analysis required to be more conclusive is beyond the scope of this paper. For example, global seismic parameters reported for the LEGACY sample (Lund et al. 2017) are not the same as the ones in our work (${\nu }_{\max }$ in particular is different by up to a few percent for some stars), and we have not tested whether this introduces systematic effects. Also, despite that stellar models are computed with GARSTEC in both cases, they are not exactly the same.

From Table 6 we also see that the mass and age rms are larger for the ASPCAP results than for SDSS. For the ASPCAP results, the larger rms is driven by three stars with ${T}_{\mathrm{eff}}$ values different from the LEGACY values by more than 300 K. By removing these stars, the rms decreases to 6.3%, around the 6% found between global and detailed seismic analysis. Mass differences (global-detailed seismology) do not show clear correlation with the ${T}_{\mathrm{eff}}$ differences in the analysis. However, age differences do correlate with the ${T}_{\mathrm{eff}}$ scale. This explains in part the variation in $\delta \tau /\tau$ from almost 8% (ASPCAP) down to −17% (SDSS), which is larger than expected from the differences found for $\delta M/M$. A detailed comparison between global and detailed seismic analysis is worth further investigation.

A final and useful comparison between the APOKASC catalog and the LEGACY results is that of the typical errors with which the different stellar quantities can be determined based on global seismic parameters (APOKASC) or on using individual frequencies. The error distributions presented in Silva Aguirre et al. (2017; see their Figure 2) are the equivalent of the statistical errors in the APOKASC catalog. Following the publication of the original LEGACY work (Lund et al. 2017), an error was identified in the preparation of the frequency ratios (Roxburgh 2017; Lund et al. 2017), which caused an overestimation of uncertainties in the LEGACY results determined with some of the pipelines used in Silva Aguirre et al. (2017). Taking this factor into account (Silva Aguirre, private comm.), typical errors embracing LEGACY results obtained with all pipelines are in the range of 0.5%–1% for $\langle \rho \rangle$, 0.8%–1.8% for R, 2%–4% for M, and 5%–10% for τ. The statistical errors in the catalog presented in this paper (see Table 2 and Figure 6) are about 3% ($\langle \rho \rangle$), 2.3% (R), 4% (M), and 15% (τ). The better precision of the detailed seismic analysis is most evident for $\langle \rho \rangle$, while for R and M results are closer. For τ, the precision of global seismology is on average a factor of two larger than detailed seismic analysis. We note that the final errors in the present catalog include the systematic component to the total uncertainty, whereas in Silva Aguirre et al. (2017) this has not been included in the error estimate because results for all seismic pipelines were reported.

According to our results, the precision of global seismology is typically larger than that of detailed seismology by factors from about 1.5 to 3, depending on the stellar property considered. Note that in this discussion the APOKASC results include stars with time series of all available lengths. On the other hand, the LEGACY sample is formed by the best available Kepler data.

The overall comparison of the APOKASC catalog, with both ${T}_{\mathrm{eff}}$ scales, with the LEGACY sample shows that there are no strong systematic differences present at levels larger than the mean uncertainties in the catalog. Moreover, the typical precision with which GBM pipelines estimate stellar parameters using only ${\rm{\Delta }}\nu$ and ${\nu }_{\max }$ is comparable to those obtained using individual oscillation frequencies, at least when stars without mixed modes are considered. More detailed work is, however, desirable to understand in full the limitations of global seismology of dwarf and subgiant stars.

Timescales and amplitudes of granulation are tightly related to surface gravities (Kjeldsen & Bedding 2011), which was first observed for red giants (Mathur et al. 2012) and later applied to measure surface gravities of dwarfs and giants from long-cadence data. The so-called flicker method (Bastien et al. 2013) is based on a correlation between the rms variations of the stellar brightness on timescales shorter than 8 hr, the so-called 8 hr flicker F₈, and the surface gravity of stars. The relation between F₈ and $\mathrm{log}g$ was further calibrated and analyzed in Bastien et al. (2016), using in part seismic $\mathrm{log}g$ from C14.

As discussed in Section 5.3.1, the availability of [M/H] from APOGEE leads to changes in the determined seismic $\mathrm{log}g$ values with respect to C14, although these corrections are small in comparison with the typical 0.10 dex uncertainties in $\mathrm{log}g$ obtained from F₈. The left panel in Figure 17 compares the newly determined seismic $\mathrm{log}g$ with the F₈ $\mathrm{log}g$ from Bastien et al. (2016), with symbols coded according to stellar metallicity. The agreement of the flickering gravities with the newly determined seismic gravity is excellent overall, with a mean difference of 0.017 dex (flickering−seismic) and a dispersion of 0.068 dex for the whole APOKASC sample. If the C14 $\mathrm{log}g$ values are used instead for the same sample, the mean and dispersion of the difference are 0.023 and 0.071 dex, respectively. Including the [M/H] information brings both $\mathrm{log}g$ scales (slightly) closer.

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More interesting is perhaps the difference between the flicker and the seismic gravities (${\rm{\Delta }}\mathrm{log}g$) as a function of [M/H], which is shown in the right panel of Figure 17, where colors are the same as in the left panel. Here, stars showing $| {\rm{\Delta }}\mathrm{log}g| \gt 0.2$ dex ($\sim 2{\sigma }_{{{\rm{F}}}_{8}}$) are plotted with squares. The larger negative differences are related to some of the hotter stars in the sample, above $\approx 6500$ K; as pointed out in Bastien et al. (2016), the tight relation between F₈ and $\mathrm{log}g$ seems to break down around this or slightly higher ${T}_{\mathrm{eff}}$. We have performed a linear fit to ${\rm{\Delta }}\mathrm{log}g$ as a function of [M/H], excluding from the fit all cases where $| {\rm{\Delta }}\mathrm{log}g| \gt 0.2$ dex. The metal-poor star KIC 7341231, shown with a black square, is also excluded from the fit. The result is plotted as a blue solid line. We have also carried out linear fits to binned data. This was done by sorting stars by increasing [M/H] computing the mean [M/H] and ${\rm{\Delta }}\mathrm{log}g$ in batches of 10 stars and then performing the linear fit. The result is virtually unchanged compared to the one reported in Figure 17. We have checked that results are not significantly affected by the bin size. Interestingly, the linear fit reproduces well the result for KIC 7341231, although the typical scatter in flicker gravities prevents us from drawing firm conclusions.

The significance of the correlation between ${\rm{\Delta }}\mathrm{log}g$ and $[{\rm{M}}/{\rm{H}}]$ has been tested by computing the Kendall's ${\tau }_{K}$ nonparametric correlation test. Applying the cut $| {\rm{\Delta }}\mathrm{log}g| \gt 0.2$ dex, we obtain for 381 stars ${\tau }_{K}=0.177$, leading to z_K = 5.16 and a p-value of 1.2 × 10⁻⁷. If stars are grouped in 12 bins of 30 stars each, then ${\tau }_{K}=0.522$, i.e., z_K = 2.47 or a p-value of $6.7\times {10}^{-3}$). A highly significant correlation between ${\rm{\Delta }}\mathrm{log}g$ and $[{\rm{M}}/{\rm{H}}]$ exists in both cases.

The result above seems to suggest that there is some degree of correlation between the level of flickering activity F₈ and the metallicity of the star for dwarfs and subgiants. As [M/H] decreases, the tendency is for F₈ to lead to larger estimates of $\mathrm{log}g$ than the seismic values. Based on the ${F}_{8}\mbox{--}\mathrm{log}g$ relation in Bastien et al. (2016), this implies lower levels of activity associated with lower [M/H]. This is in qualitative agreement with theoretical models (Samadi 2011) linking F8 and mode amplitudes with $[{\rm{M}}/{\rm{H}}]$ and recent detailed analysis of red giant stars carried out by Corsaro et al. (2017) based on open star clusters. The relation between stellar activity, in particular as measured by F₈, metallicity, and $\mathrm{log}g$, deserves further and detailed study.

Recently, Huber et al. (2017) have used the Gaia DR1 (TGAS) parallaxes in combination with the Two Micron All Sky SurveyKband and ${T}_{\mathrm{eff}}$ from Buchhave & Latham (2015) to determine stellar radii. Almost all stars in the catalog in this paper have been analyzed in Huber et al. (2017) and form their dwarf and subgiant sample. Figure 18 shows histograms comparing stellar radii for our two ${T}_{\mathrm{eff}}$ scales with radii determined from TGAS by Huber et al. (2017) using the same ${T}_{\mathrm{eff}}$ scales. The median differences, indicated by vertical lines, are 3.3% and 7.0% for the SDSS and the ASPCAP ${T}_{\mathrm{eff}}$-based results, respectively, with a dispersion in both cases of 10%. This offset was already highlighted by Huber et al. (2017), who concluded that TGAS results favor a hotter ${T}_{\mathrm{eff}}$ scale such as the SDSS scale rather than the cooler one provided by ASPCAP. The right panel shows differences normalized with respect to the combined APOKASC+TGAS uncertainty, where the offsets are also visible. The dispersion of uncertainties is 0.9σ and 1σ for the ASPCAP and SDSS scales, respectively, and it is dominated by errors in the TGAS results.

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Our results confirm those in Huber et al. (2017), that asteroseismic radii agree with those determined from Gaia parallaxes. However, there is a systematic offset present, and its nature is not fully understood. It might be tempting to present this as a limitation of global seismology (i.e., based on ${\rm{\Delta }}\nu$ and ${\nu }_{\max }$), stellar models (i.e., shortcomings in determination of ${\rm{\Delta }}{\nu }_{0}$), or the scaling relation of ${\nu }_{\max }$. However, our radii compare very well with those from the LEGACY sample (Section 5.3.2), determined from a seismic analysis based on oscillation frequencies. In fact, seismic parallaxes of the LEGACY sample computed by Silva Aguirre et al. (2017; see their Figure 13) show a systematic offset of ∼0.25 mas toward larger values than those from TGAS. Analogous results were also found by Davies et al. (2017) using asteroseismic parallaxes of red clump stars and by Stassun & Torres (2016) using binaries. Because seismic parallaxes are based on computing the total stellar flux, results in Silva Aguirre et al. (2017) are equivalent to having seismic radii smaller than TGAS radii, consistent with our findings. About 2% of the offset in radius can be accounted for by a systematic offset in TGAS parallaxes. Still, further study is required to have a better understanding of whether the reasons for the offset lie with asteroseismic analysis, ${T}_{\mathrm{eff}}$ scale, or a systematic underestimation of TGAS parallaxes. Gaia DR2 will certainly help shed light on this.

The synthetic grid of spectra in ASPCAP incorporates rotational broadening as an additional dimension in the analysis since DR13. The possibility of having a large sample of stars with $v\sin i$ determinations and seismic parameters, even if the inclination i remains unknown, opens up interesting possibilities for studies of gyrochronology, at least in a statistical sense. In order to assess the quality of ASPCAP $v\sin i$ determinations, we use stars in common with Bruntt et al. (2012) for which $v\sin i$ has been determined from optical spectroscopy taken at resolution R ∼ 80,000 and S/N > 200. Results are shown in the left panel of Figure 19, where stars are color-coded according to their mass. There is a very good agreement between both data sets, with a dispersion of $1.8\ \mathrm{km}\,{{\rm{s}}}^{-1}$ between the two data sets and a mean offset of $-0.5\ \mathrm{km}\,{{\rm{s}}}^{-1}$ (in the sense ASPCAP−Bruntt).

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The comparison above lends strong support to $v\sin i$ determinations with ASPCAP. There are 331 stars in the catalog with ASPCAP $v\sin i$ determinations, and they are shown in the right panel of Figure 19, where $v\sin i$ is shown as a function of ${T}_{\mathrm{eff}}$ (SDSS). We have excluded from the plot stars that have been flagged by ASPCAP with problematic $v\sin i$. These are mostly stars with low ($\sim 2\,\mathrm{km}\,{{\rm{s}}}^{-1}$) $v\sin i$. Main-sequence stars are shown as large filled circles, color-coded as in the left panel of the figure. Stars that are close to the end of the main sequence or have already evolved off it are shown as small open circles. This is an approximate classification based on whether the central hydrogen mass fraction X_c, estimated from the BeS/G–${\rm{\Delta }}{\nu }_{\mathrm{scl}}$ GBM results, is larger or smaller than 0.1. The plot includes ASPCAP values of $v\sin i$ for M67 stars. Hyades and Praesepe stars from van Saders & Pinsonneault (2013) are also included. For all three clusters, ${T}_{\mathrm{eff}}$ have been determined using $(B-V)$ colors and the Casagrande et al. (2010) transformation. The general trend in the APOKASC sample is similar to that seen in cluster stars. Cool stars (${T}_{\mathrm{eff}}\lt 6200$ K) show slow rotation in all but a few cases. Above this temperature, as surface convective envelopes thin, magnetic braking becomes less effective and higher rotation rates are present, as in cluster stars (van Saders & Pinsonneault 2013). At ${T}_{\mathrm{eff}}\gt 6200$ K there seems to be an excess of low $v\sin i$ in comparison to cluster stars. This may be indicative that there is a relative lack of fast rotators in seismic samples (Tayar et al. 2015) and the particular exclusion of close binary stars, which, at least for red giant stars, tend to have rapid rotation but suppressed oscillations (Gaulme et al. 2014). Ongoing work includes the comparison of estimated $v\sin i$ distributions obtained for stars with known rotational periods to further confirm the reliability of $v\sin i$ determinations by ASPCAP (G. Simonian et al. 2017, in preparation).