The Second APOKASC Catalog: The Empirical Approach

In Pinsonneault et al. (2014), which we will refer to as APOKASC-1, we presented the first major catalog using both asteroseismic and spectroscopic data for a large sample of cool giants. There are two natural applications of this approach: detailed studies of stellar physics and studies of stellar populations. The availability of simultaneous mass and composition data can be used to search for correlations between mass, age, and spectroscopic observables. This is an especially exciting prospect because the set of stars with spectroscopic data from large surveys greatly exceeds the sample with asteroseismic data, which can be used to calibrate such relationships. For example, the surface [C/N] abundance is a product of the first dredge-up on the red giant branch, which is both expected on theoretical grounds to be mass and composition dependent (Salaris et al. 2015) and observed to be so in open cluster data (Tautvaisiene et al. 2015). Data sets prior to APOGEE, however, were sparse, and the samples were small. APOKASC-1 data were used to calibrate mass using both [C/N] (Martig et al. 2016) and the full APOGEE spectra (Ness et al. 2016) using the CANNON methodology. This approach has also been used for stellar population studies (Silva Aguirre et al. 2018).

Another early result from the APOKASC-1 data was the discovery of a significant population of high-mass stars with high [α/Fe] by Martig et al. (2015); this was discovered independently by Chiappini et al. (2015) using a combination of CoRoT and APOGEE data in the related CoRoGEE project. This is a striking result because high-[α/Fe] stars are typically regarded as a purely old, and by extension low-mass, population. Some of these objects are evolved blue stragglers, or merger products (Jofre et al. 2016), but explaining all of them with this channel would require a very high merger rate (Izzard et al. 2018). The alternative is an unusual nucleosynthetic origin; see Chiappini et al. (2015) for a discussion. The discovery and characterization of unusual chemical stellar populations are major prospects for Galactic archeology in general, as is the understanding of the products of binary star interactions. The joint data set has also enabled detailed studies of stellar physics, including tests of models of extra mixing on the red giant branch (Masseron et al. 2017) and of the structure of core He-burning stars (Constantino et al. 2015; Bossini et al.2017).

However, there are recognized drawbacks to the approach used in the initial paper. Important populations, such as members of open clusters, very metal-poor stars, and luminous giants were relatively sparsely sampled. Of more import, the APOKASC-1 effort did not attempt to calibrate the masses, radii, and uncertainties against fundamental data. This is not a priori unreasonable, as initial checks of asteroseismic radii against interferometric values (Huber et al. 2012) and those inferred from Hipparcos parallaxes combined with T_eff (Silva Aguirre et al. 2012) found encouraging agreement at the 5% level. However, even early on there was a recognized tension between masses derived from simple scaling relations and those expected for red giants in the old open cluster NGC 6791 (Brogaard et al. 2012). With the advent of the APOKASC-1 catalog, larger field star samples could be obtained and additional tests were possible. The masses for halo stars derived from scaling relations in APOKASC-1 were found to be well above astrophysically reasonable values for old stellar populations (Epstein et al. 2015). Offsets between fundamental and asteroseismic mass and radius values were also found for eclipsing binary stars (Gaulme et al. 2016). These results highlighted the need for improvements in the overall approach, which we now describe.

The APOKASC-1 catalog contained asteroseismic and spectroscopic data for 1916 stars. Since that time, there has been both a substantial increase in the sample size and a change in the data analysis techniques. The APOKASC-1 approach used spectroscopic data from the 10th data release (hereafter DR10) of the Sloan Digital Sky Survey (Ahn et al. 2010); two different temperature scales were considered to account for scale shifts in spectroscopic data. The asteroseismic analysis was based on standard scaling relations. Measurements and theoretically estimated random uncertainties were taken from a single analysis pipeline with average results close to the mean of the measurements from all methods. Differences between pipelines were then used to infer systematic uncertainties and added in quadrature to the random ones to derive a total error budget. Our final stellar properties were derived by including constraints from both the asteroseismic parameters and stellar interior models (a procedure usually called grid-based modeling). In our revised catalog, we critically examine each of these assumptions.

The spectroscopic pipeline has been extensively tested and modified since DR10 (see Section 2.2 below); the key ingredient for our purposes is T_eff, which enters directly into the formulas for asteroseismic surface gravities, masses, and radii. If grid modeling is being performed, T_eff, [Fe/H], and [α/Fe] are needed to predict stellar parameters from evolutionary tracks. The effective temperature is a defined quantity that can be measured in stars with known radius and total luminosity; such stars define a true fundamental T_eff reference system. Because the revised APOGEE effective temperatures are tied to the Infrared Flux Method (IRFM) fundamental scale (Holtzman et al. 2015), we do not explicitly compare different overall temperature scales in the current effort. We have, however, assessed the impact of systematic changes in the underlying methodology by comparing results from the same stars for different SDSS data releases; the differences in derived masses arising from adopting DR13 as opposed to DR14 parameters are less than 1% in mass with small scatter, which is well below other identified error sources.

We employ multiple methods for measuring the asteroseismic parameters ν_max and Δν. In APOKASC-1, we adopted the solar-scaled hypothesis, which assumes that the measurements themselves are all measured relative to a method-specific solar reference value. Under this assumption, if a given analysis method returns a solar ν_max 10% lower than the norm, all of the ν_max measurements would be expected to be systematically 10% lower than other techniques. In this paper, we replace the solar-scaled hypothesis with a data-driven approach for comparing the measurements; we have also revised our techniques for estimating both random and systematic measurement uncertainties.

Once we have a set of asteroseismic and spectroscopic observables, we then convert them to inferred masses and radii via scaling relations. The Δν scaling relation is theoretically well motivated but not expected to be exact (Stello et al. 2009; White et al. 2011). In a detailed work, Belkacem et al. (2013) studied the physics of the asteroseismic scaling relation for Δν, emphasizing how departures from homology in the structures of evolved stars perturb the scaling relation. We therefore explore theoretically motivated corrections to the Δν scaling relation, which are known to improve agreement between asteroseismic stellar parameters and fundamental ones (Sharma et al. 2016; Handberg et al. 2017; Rodrigues et al. 2017). These corrections are sensitive to the internal structure, so knowledge of the evolutionary state is essential; evolutionary state is also important for ages. We therefore also include asteroseismic and spectroscopic evolutionary state measurements in this paper. This was not done in APOKASC-1, which did not report ages or use corrections.

The empirical ν_max scaling relation has a weaker theoretical basis than the Δν scaling relation, although there have been detailed physical studies of its basis (Belkacem et al. 2011). Despite this concern, it performs well when compared with empirical data. However, adjustments in the zero-point for evolved stars are certainly reasonable, and different methods also yield different values even for the Sun. We therefore treat the absolute zero-point for the ν_max scaling relation as a free parameter that can be calibrated against fundamental mass data.

Finally, we consider the impact of adopting grid-based modeling for evolved giant stars. Grid-based modeling is in principle powerful, because it includes all of the constraints from observables and theory on the derived properties of the star. For stars on or near the main sequence, precisely measured T_eff, log g, and abundances can set stringent constraints on mass and radius that complement asteroseismic measurements; see Serenelli et al. (2017) for our discussion in the dwarf context. Unfortunately, one cannot test the validity of the underlying models if their accuracy is assumed in the solution, and Tayar et al. (2017) found significant offsets between theoretical expectations from solar-calibrated isochrones and APOKASC data. The origin of these differences may be in the treatment of the mixing length, as noted in that paper and by Li et al. (2018), or it may be tied to other choices of input physics as discussed in Salaris et al. (2018). In either case, there is no guarantee that solar-calibrated models agree in the mean with data for evolved stars. A direct consequence is that there will be systematic offsets between stellar properties inferred from the tracks alone and stellar properties inferred from asteroseismology alone, which can inject complex systematic differences in the derived stellar properties unless the models are explicitly calibrated to remove such differences. As a result, there is benefit in choosing to test the asteroseismic scale itself directly against fundamental data, rather than doing so with a hybrid grid-modeling value. In this paper, we therefore do not impose grid-based modeling constraints on our observables. A companion paper (A. Serenelli et al. 2018, in preparation) investigates asteroseismic parameters from our data including grid-based modeling. Finally, for usage in stellar population studies, we have taken our data and used it to infer ages and extinctions.

In summary, the improvements and changes in our APOKASC-2 analysis are:

• 1.  
  Our spectroscopic parameters and uncertainties are taken from the 14th data release (hereafter DR14) of the Sloan Digital Sky Survey (Abolfathi et al. 2017) instead of DR10.
• 2.  
  We have inferred evolutionary states for virtually all of the stars in our sample for APOKASC-2, either from asteroseismology or from spectroscopic diagnostics calibrated on asteroseismic observables.
• 3.  
  The relative zero-points for ν_max and Δν from different pipelines are inferred from the data and not assumed to be strictly defined by their relative solar reference values.
• 4.  
  With zero-point differences accounted for, the scatter of the individual pipeline values about the ensemble mean is used to infer the random uncertainty for each star, rather than relying on formal theoretical error estimates.
• 5.  
  Δν- and ν_max-dependent differences between individual pipeline values and the ensemble mean are treated as systematic error sources.
• 6.  
  The Δν scaling relation is corrected with the same theoretically motivated approach as that in A. Serenelli et al. (2018, in preparation), rather than being treated as exact.
• 7.  
  The absolute zero-point of the ν_max scaling relation is set by requiring agreement with fundamental radii and masses in star clusters with asteroseismic data, as opposed to adopting a solar reference value.
• 8.  
  We do not use grid-based modeling in APOKASC-2.
• 9.  
  We provide ages and extinction estimates.

The outcome of this exercise is tabulated for the full joint sample, and the sample properties are then discussed. The remainder of the paper is organized as follows. We discuss the sample selection in Section 2 and present our basic data there. The relative mean asteroseismic parameters and the absolute calibration from open cluster members are derived in Section 3. The catalog itself is presented in Section 4, and the conclusions are given in Section 5.

The details of the Kepler data itself and the light curve reduction procedures used are described in Y. Elsworth et al. (2018, in preparation). We employed five distinct pipelines for asteroseismic analysis of the reduced light curves known in the literature by three-letter acronyms: A2Z, CAN, COR, OCT, and SYD. We briefly reference each method below. For a more detailed discussion of the different approaches, see A. Serenelli et al. (2018, in preparation). The same data preparation method is not used in all cases. Two different methods were used with A2Z preparing their own data sets following Garcia et al. (2011), and CAN, COR, OCT, and SYD all using data prepared using the Handberg & Lund (2014) method. A comparison and review of the methods is given in Hekker et al. (2011) and further discussed in Hekker et al. (2012), where they looked at the impact of data duration on the detectability of the oscillations and the precision of the parameters. For this paper, the precise method used to determine the average asteroseismic parameters is not of major importance because here we seek to show how the differences can be mitigated. Nevertheless, we give basic references to the method of operation of each pipeline. The A2Z pipeline was first described in Mathur et al. (2010b) and, together with their method of data preparation, is updated in Garcia et al. (2014). The CAN pipeline is described in Kallinger et al. (2010), the COR pipeline is described in Mosser & Appourchaux (2009), the OCT pipeline is described in Hekker et al. (2010), and the SYD pipeline is described in Huber et al. (2009).

We collected the spectroscopic data using the 2.5 m Sloan Foundation telescope (Gunn et al. 2006) and the APOGEE near-infrared spectrograph at Apache Point Observatory. These spectra were obtained during SDSS-III, and the target selection criteria for stars in the Kepler field are described in Zasowski et al. (2013). All spectra are re-reduced and re-analyzed for each data release. The procedures used to flat-field, co-add, extract, and calibrate the spectra are described in Nidever et al. (2015). The spectra were then processed through the APOGEE Stellar Parameters and Chemical Abundances Pipeline, or ASPCAP (Garcia Pérez et al. 2016), which derives T_eff, log g, metallicity, and other properties through a ${\chi }^{2}$ minimization of differences with a grid of theoretical spectra as described below.

The APOGEE survey has presented data in four SDSS data releases. The first set of results, in Sloan DR10, was described in Meszaros et al. (2013). The subsequent DR12 data analysis technique was documented in Holtzman et al. (2015), while the data released in DR13 (as well as the subsequent DR14) is discussed in Holtzman et al. (2018) and Jönsson et al. (2018). Each data release contained both "raw" and "calibrated" atmospheric parameters. The "raw" values reflect the output of the automated pipeline analysis, while the "calibrated" values can include corrections to bring the results into agreement with external standards.

As the survey has progressed, the corrections inferred from the calibration process have in general become smaller, because improvements implemented in ASPCAP allowed the APOGEE team to produce more accurate and precise atmospheric parameters. The first APOKASC catalog was compiled using DR10 parameters, while results presented in this paper use DR14 parameters, the latest SDSS-IV release. In this section, we detail the most important improvements to ASPCAP and changes in the calibration of effective temperature, [Fe/H], and [α/Fe] between DR10 and DR14; these ingredients are the ones relevant to the data presented in this paper. There have been important changes made in the reduction techniques, the line list, model atmospheres, and spectrum synthesis. In addition to these changes, abundances of nearly 23 elements are determined in DR13 and DR14, instead of three broad indices being reported, as was the case in DR10.

Data reduction in DR13 and DR14 included improved line spread function characterization, and telluric and persistence correction. ASPCAP pipeline results are benchmarked against the solar spectrum and that of Arcturus, with less secure line strengths empirically adjusted to match specified values, using the line list from Shetrone et al. (2015). A new set of Arcturus abundances has also been adopted for tuning the line strengths, which are not the same as the solar mixture. The solar reference abundances table was changed from Grevesse & Sauval (1998; DR10) to Asplund et al. (2005) in DR12 onwards. APOGEE reports results relative to the Sun, so for some purposes the abundances can be treated as differential. The choice of solar mixture can, however, have subtle effects on the derived abundances. The solar mixture directly impacts the synthetic spectra that were generated; some oscillator strengths were also tweaked to yield the calibrated abundances of some species in the Sun and Arcturus. Molecular bands are especially important for the light CNO elements, however, and their oscillator strengths were not treated as free parameters. The overall metallicity in APOGEE is also an average across lines of many species. As a result, the APOGEE abundances are in practice a mixture of differential and absolute measurements. For the purposes of this paper, we have treated them as purely differential relative to the Sun, and the stellar interiors models used to interpret the oscillation frequencies have employed different solar mixtures from those adopted by APOGEE. We discuss the impact of the adopted mixture on our results in Section 3.2.

New ATLAS model atmospheres (Meszaros et al. 2012) were computed for DR12 and are still in use in DR14 using the solar reference from Asplund et al. (2005). A new set of synthetic spectra covering the range 2500 K < T_eff < 4000 K, based on custom MARCS atmospheres, was included. All synthetic spectra were calculated using Turbospectrum (Alvarez & Plez 1998); previous syntheses were done using ASSeT in DR10. From DR12 onwards, a finer grid spacing was adopted in metallicity ([M/H]), with 0.25 dex steps instead of the 0.5 dex spacing used in DR10. The grid of model atmospheres was also extended to a higher metallicity of [M/H] = +0.75. A macroturbulent velocity relation was determined based on a fit with a subset of data and a macroturbulence dimension, rather than using a fixed value, as illustrated in Figure 7 of Holtzman et al. (2018).

DR13 and DR14 use a multistep analysis through multiple grids to determine the main atmospheric parameters. Initial characterization was carried out using F, G+K, and M coarse grids. Once stars have passed quality control steps, the ASPCAP pipeline is then used to do a full solution in 6D or 7D space depending on the location of the star in the Hertzsprung–Russell (HR) diagram. This high dimensionality is required because the APOGEE spectral region is heavily influenced by CNO molecular features. Therefore, in addition to the 4D ingredients typically considered in model atmosphere fits (T_eff, surface gravity, overall metallicity, and microturbulence), ASPCAP also includes three additional dimensions: alpha-element enhancement (including O), C, and N. The final step is the derivation of individual abundances, which were not included in DR10, and which use spectral windows rather than additional dimensions in the atmospheres grid.

For DR10, effective temperatures were calibrated to be in agreement with color temperatures for stars belonging to open and globular clusters. This comparison sample was improved in subsequent data releases by replacing the limited cluster calibration set with field stars that have low extinction, which have the advantage of providing many more calibrators in a larger metallicity and surface gravity phase space. In DR10, the effective temperature correction was fairly large (around 110–200 K, depending on T_eff and metallicity). As ASPCAP improved, spectroscopic temperatures showed better agreement with photometric ones. This resulted in no correction applied in the DR13 data, as published. However, a modest metallicity-dependent offset was discovered post-release; a similar metallicity-dependent temperature correction was therefore introduced again for DR14. The uncertainty was estimated from the scatter between spectroscopic and photometric temperatures for a subsample of targets.

Metallicities in DR13 and DR14 have been calibrated to remove T_eff trends using members of star clusters; the underlying assumption is that any systematic trends in inferred abundance within a cluster sample are analysis artifacts, as cluster stars share the same true metallicity. This is a significant departure from DR10, where [M/H] was calibrated to mean literature abundances for open and globular clusters as a whole, not star by star. This external calibration for [M/H] has been introduced again for DR14, but was not done in DR12 and DR13. It is important to point out that these calibrations induce changes generally smaller than 0.1 dex and become larger than that only for the most metal-poor stars below [M/H] < −1.0. The DR14 metallicity calibration effects are also smaller than those of DR10.

We illustrate the net impact of these changes in two figures. Figure 1 compares the spectroscopic parameters for stars in APOKASC-1 between DR10 and DR13. Systematic shifts are more important than random scatter, and the differences largely reflect changes in the choice of calibrators for the spectroscopic solution and improvements in the ASPCAP spectroscopic pipeline. By comparison, the differences between DR13 and DR14, illustrated in Figure 2, are milder, although there are still clear zero-point offsets in the metallicity and scatter in the inferred carbon to nitrogen ratio, a diagnostic of the first dredge-up in evolved stars. Uncertainties are discussed in Section 3.1.3.

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The full APOGEE data sample we use was observed in SDSS-III (but analyzed in SDSS-IV) and contains 11,877 stars. Many of these targets were not explicitly observed for asteroseismology, however, and some of the remainder turned out to be subgiants. A total of 8604 of these stars had calibrated spectroscopic log g < 3.5 and were therefore potential cool giant asteroseismic targets. Target selection for this sample was discussed in APOKASC-1. However, not all light curves were sufficiently long to detect asteroseismic signals: some had data artifacts, and a substantial number in the high surface gravity domain (3.3 < log g < 3.5) are technically challenging to analyze because their oscillation frequencies are close to the Kepler 30 minute sampling for long-cadence data.

2.3.1. The Asteroseismic Parameter Calibration Sample

2.3.2. The Catalog Sample

The APOKASC-2 sample analyzed in this paper contains 6676 targets with reduced light curves which were selected for asteroseismic analysis. There are 122 stars for which we were not able to return asteroseismic data or which had bad spectra. We have asteroseismic evolutionary states for 6076 of the remaining objects in Elsworth et al. (2017), including 2453 RC stars and 3623 RGB stars. (The calibration set described above is smaller because we required asteroseismic parameter measurements from all pipelines for calibration, but report catalog values if any pipeline returned measurements.) For the 478 stars without asteroseismic evolutionary state assignments from Elsworth et al. (2017), we infer DR13 spectroscopic evolutionary states as described in Holtzman et al. (2018). This includes 276 RC stars and 152 RGB stars; only 50 stars had ambiguous evolutionary states given their spectroscopic properties. Our data for the stars without asteroseismic state data, and for stars with no seismic parameters, are illustrated in Figure 3. This Kiel diagram is an analog of the classical HR diagram, as surface gravity is related to luminosity. The cluster of targets with log g > 3.1 without results are stars where the asteroseismic frequencies are close to, or exceed, the Nyquist sampling frequency from the Kepler data. The remainder are an admixture of stars close to the boundary between the RGB and the RC, where it is most challenging to distinguish RC from RGB stars spectroscopically. This group of targets also includes a substantial number of higher mass and surface gravity (log g > 2.6) core He-burning stars, and the hotter RGB sample includes a number of very metal-poor targets. For our remaining analysis, we will treat the stars with spectroscopic evolutionary state assignments in a manner similar to the approach taken for targets with asteroseismic states; the sole exception is the group with ambiguous evolutionary states, for which the final mass and radius estimates are more uncertain (see Section 3.2). A more detailed discussion of the evolutionary states of our targets, and a comparison of spectroscopic and asteroseismic methods, can be found in Y. Elsworth et al. (2018, in preparation) and Holtzman et al. (2018).

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Our main sample is shown in Figure 4, and it illustrates the power of asteroseismic evolutionary state classification. As one would expect on stellar populations grounds, the RC stars are, on average, hotter than the RGB ones. Higher mass RC stars had a non-degenerate He flash, however, which produces an interesting population feature. Stars with masses between 2.2 and 3.0 M_☉ can have a smaller He-core mass at ignition than their low- or high-mass counterparts, so they show up at lower luminosity in the secondary RC than the typical old RC star. The excess of red giants around log g = 2.6 is the RGB bump, where the H-burning shells of ascending RGB stars cross the composition discontinuity produced by the first dredge-up at the maximum depth of the surface convection zone. More luminous RGB stars are seen to span a wider range of T_eff than less luminous ones. This is because they include a mixture of metal-poor objects (preferentially seen at greater distances) and double-shell source, or AGB, stars that are asteroseismically similar to first ascent shell H-burning stars. There is also an admixture of higher mass RGB stars in the same Kiel diagram position as RC stars; these can only be distinguished asteroseismically and would be missed in traditional survey methods. Scatter plots with many points have a tendency to emphasize outliers. An alternative visualization (combining RC and secondary clump) is illustrated in Figure 5, where the data are binned to illustrate the density of points. For a further discussion of spectroscopic evolutionary states, we refer to Y. Elsworth et al. (2018, in preparation).

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Inferring masses from asteroseismic scaling relations has rested on two major assumptions: that the measurements from each pipeline can be scaled relative to their solar values and that the uncertainties are the internal values returned by those pipelines. Both assumptions are, in principle, reasonable; but neither is exact. Measuring the frequency of the maximum oscillation power requires us to first model and account for the background, then to make choices about the smoothing and/or fitting of the power excess. The asteroseismic frequency spectrum contains a pattern that is close to uniformly spaced, and the large frequency spacing is a theoretically well-posed quantity. However, there are deviations from uniform spacing, caused, for example, by the way acoustic glitches perturb the comb structure of the pressure modes (Vrard et al. 2015). This measurement must also be normalized and defined in a specific method-dependent frequency domain (Mosser et al. 2013b). Different pipelines can therefore have both random and systematic differences from one another that are non-negligible, and there can also be differential scale factors relative to the individual solar zero-points. In practice, this means that any relative bias in solar measurements from a given pipeline does not necessarily translate into a similar relative bias when it is used for measurements in evolved stars. First principles error estimation is notoriously difficult to perform, and it can be challenging to disentangle systematic and theoretical uncertainties. Fortunately, we can test both assumptions with our data set; we have multiple pipeline results for a large sample of objects. The relative values inferred from different methods for the same targets provide robust constraints on the differential zero-points of the various techniques, and the dispersion in values once systematic differences are accounted for yields guidance on errors. We begin with RGB stars as a calibrating set, and then follow with an analogous study of RC stars. Our asteroseismic values for individual pipelines are given in the Appendix; the values derived from our method are used to derive the stellar observables and are given in the main catalog table (Table 5).

3.1.1. Relative Pipeline zero-points

To motivate the averaging discussion, we will begin with methods used in prior efforts and then generalize to the current one. We use ν_max as an example, but the same considerations apply to Δν. In this discussion, $\langle {\rm{q}}{\rm{u}}{\rm{a}}{\rm{n}}{\rm{t}}{\rm{i}}{\rm{t}}{\rm{y}} \rangle$ refers to a simple average of multiple measurements of that quantity from different pipelines for a given star. The scaling relations require both a measurement (ν_max in this case) and a solar reference value ${\nu }_{\max }^{\mathrm{ref}};$ both are in general different for different methods.

The approach used in Pinsonneault et al. (2014) for evolved stars and in Serenelli et al. (2017) for dwarfs used the solar-scaled hypothesis. In this case, the quantity to be averaged is the ratio of the measurement to the solar value for each pipeline: $\tfrac{{\nu }_{\max }^{\mathrm{star}}}{{\nu }_{\max }^{\mathrm{ref}}}=\langle \tfrac{{\nu }_{\max }^{\mathrm{pipe}}}{{\nu }_{\max ,\odot }^{\mathrm{pipe}}}\rangle$. As we will show below, however, there are significant mean offsets between pipelines using this averaging method. Another logical approach would then be to decouple the averaging of the measurements and the reference values. In this limit, $\tfrac{{\nu }_{\max }^{\mathrm{star}}}{{\nu }_{\max }^{\mathrm{ref}}}=\tfrac{\langle {\nu }_{\max }^{\mathrm{pipe}}\rangle }{\langle {\nu }_{\max ,\odot }\rangle }$. The mean value would then be a simple average of the absolute measurements, and the solar reference could be averaged in the same way. The different pipelines also show significant average differences in the absolute measurements, unfortunately. We therefore treat the relative normalization of the different pipelines, and the choice of reference values, as quantities to be solved for empirically.

We define the ensemble solar reference values as the average of the individual pipeline solar values: ${\nu }_{\max ,\mathrm{ref}}=\langle {\nu }_{\max ,\odot }\rangle$ and ${\rm{\Delta }}{\nu }_{\mathrm{ref}}=\langle {\rm{\Delta }}{\nu }_{\odot }\rangle$, respectively. This choice is not fundamental; if another method were included, the mean would shift. These averaged solar reference values, and the individual values on which they are based, are included in Table 1. We note that the COR pipeline (Mosser et al. 2013a) has a published correction term for asteroseismic scaling relations, implying a different solar normalization; as we are correcting for this physical effect separately (see Section 3.2), we use the solar values instead. However, in our final results, we calibrate the overall zero-point of the ν_max scaling relation to reproduce fundamental data, as described in Section 3.2; because the different Δν methods have very similar solar reference values, and our empirical data constrains only the ratio of the solar reference values, we did not attempt separate empirical adjustments for both solar reference values.

Table 1.  Solar Reference Values

Quantity	A2Z	CAN	CORa	OCT	SYD	Average
${\nu }_{\max ,\odot }$	3097.33	3140	3050	3139	3090	3103.266
Δν☉	135.2	134.88	135.5	135.05	135.1	135.146

Note. Solar reference values for individual pipelines. All measurements are in μHz. Uncertainties are not included when computing the mean, as the zero-point is ultimately inferred empirically.

^aDoes not include Mosser et al. (2013a) scaling relation corrections.

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We are searching for scale factors for each pipeline such that they all, on average, return the same mean values over the full sample. Once these scale factors are defined, we then scale and average the results to obtain our star-by-star measurements, and we use the dispersion between the scaled values to estimate measurement uncertainties. We proceed as follows. For each star (index j), we have measurements from five pipelines (index i) and can define mean values

$$\begin{eqnarray}&&\langle {\nu }_{\max }^{j}\rangle =\displaystyle \frac{1}{{{\rm{N}}}_{\mathrm{pipe}}}\displaystyle \sum _{I=1}^{{{\rm{N}}}_{\mathrm{pipe}}}\displaystyle \frac{{\nu }_{\max }^{i,j}}{{X}_{{\nu }_{\max }}^{i}}\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&\langle {\rm{\Delta }}{\nu }^{j}\rangle =\displaystyle \frac{1}{{N}_{\mathrm{pipe}}}\displaystyle \sum _{I=1}^{{N}_{\mathrm{pipe}}}\displaystyle \frac{{\rm{\Delta }}{\nu }^{i,j}}{{X}_{{\rm{\Delta }}\nu }^{i}},\end{eqnarray} \tag{ 8 }$$

where N_pipe is the total number of pipelines available for that star. We have 2833 targets classified as RGB or AGB/RGB with data returned from all five pipelines; we use this sample to compute the X scale factors. The scale factor is defined in a two-step process. We first determine the factor by which an individual seismic value for a given pipeline differs from the unweighted average over all the returns for that star by defining the normalization factors ${Y}_{{\nu }_{\max }}^{i,j}$ and ${Y}_{{\rm{\Delta }}\nu }^{i,j}$ for each pipeline i and star j by

$$\begin{eqnarray}&&{Y}_{{\nu }_{\max }}^{i,j}={N}_{\mathrm{pipe}}\displaystyle \frac{{\nu }_{\max }^{i,j}}{{\displaystyle \sum }_{i=1}^{{N}_{\mathrm{pipe}}}{\nu }_{\max }^{i,j}}\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&{Y}_{{\rm{\Delta }}\nu }^{i,j}={N}_{\mathrm{pipe}}\displaystyle \frac{{\rm{\Delta }}{\nu }^{i,j}}{{\displaystyle \sum }_{i=1}^{{N}_{\mathrm{pipe}}}{\rm{\Delta }}{\nu }^{i,j}}.\end{eqnarray} \tag{ 10 }$$

For each star j, we can also compute the absolute measurement dispersions ${\sigma }_{{\nu }_{\max }}^{j}$ and ${\sigma }_{{\rm{\Delta }}\nu }^{j}$. The second stage in the determination of the scale factors (Xⁱ) is to use the Y^ij together with the σ^j values to form a weighted average for a given pipeline. The overall normalization factors ${X}_{{\nu }_{\max }}^{i}$ and ${X}_{{\rm{\Delta }}\nu }^{i}$ for each pipeline i are then defined by

$$\begin{eqnarray}&&{X}_{{\nu }_{\max }}^{i}=\displaystyle \frac{{\displaystyle \sum }_{j=1}^{{N}_{\mathrm{star}}}\tfrac{{Y}_{{\nu }_{\max }}^{i,j}}{{({\sigma }_{{\nu }_{\max }}^{j})}^{2}}}{{\displaystyle \sum }_{j=1}^{{N}_{\mathrm{star}}}\tfrac{1}{{({\sigma }_{{\nu }_{\max }}^{j})}^{2}}}\end{eqnarray} \tag{ 11 }$$

and

$$\begin{eqnarray}&&{X}_{{\rm{\Delta }}\nu }^{i}=\displaystyle \frac{{\displaystyle \sum }_{j=1}^{{N}_{\mathrm{star}}}\tfrac{{Y}_{{\rm{\Delta }}\nu }^{i,j}}{{({\sigma }_{{\rm{\Delta }}\nu }^{j})}^{2}}}{{\displaystyle \sum }_{j=1}^{{N}_{\mathrm{star}}}\tfrac{1}{{({\sigma }_{{\rm{\Delta }}\nu }^{j})}^{2}}},\end{eqnarray} \tag{ 12 }$$

with uncertainties defined by standard error propagation. The use of the σ^j terms ensures that stars for which there is a large spread in the determinations are given a lower weight in the formation of the average.

Our approach, by construction, ensures that all pipelines return the same mean values when averaged over the full sample, but this approach does not account for how these differences change as a function of the mean values themselves. To quantify trends in the pipeline means, we rank-ordered our data in ν_max and then broke it up into non-overlapping bins of 100 targets. For each bin and pipeline i, we then computed the average ${X}_{{\nu }_{\max }}^{i}$ that we would have obtained. To test the impact of adopting a solar normalization, we can define an analog of X, ${X}_{{\nu }_{\max }}^{{\prime} ,i}$, where the quantity being averaged is not the absolute measurement ${\nu }_{\max }^{i}$ but $\tfrac{{\nu }_{\max }^{i}}{{\nu }_{\mathrm{ax},\odot }^{i}}$. We would expect ${X}_{{\nu }_{\max }}^{{\prime} ,i}=1$ for all pipelines if the solar-scaled hypothesis were correct. We then repeated the rank-ordering and binning exercises for these alternate values. The results are shown in Figure 4, where the top panel shows the solar-normalized ratio ${X}_{{\nu }_{\max }}^{{\prime} ,i}$ for bins of 100 stars and the bottom panel compares the absolute ratio ${X}_{{\nu }_{\max }}^{i}$ for the same bins as a function of the mean ν_max of the bins. To place both panels on the same scale, we multiplied the average ${X}_{{\nu }_{\max }}^{{\prime} ,i}$ values for the bins by the pipeline mean solar reference value ${\nu }_{\max ,\odot }=3103.266\,\mu \mathrm{Hz}$. Because we are defining the value relative to the mean for all five pipelines, the important feature here is the spread between the pipelines, not the absolute position of any one pipeline. However, the usage of solar-normalized values increases the dispersion between them in Figure 6 rather than decreasing it, disfavoring the scaling of each pipeline output relative to the results it would have obtained for the Sun. To quantify this effect, we compute the dispersion in the ratio of individual values to the total; we obtain σ = 0.0131 for the solar-normalized values and σ = 0.0088 for the absolute values.

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We repeat this exercise for Δν, and the results are illustrated in Figure 7. In contrast to the ν_max case, there is a slight improvement in the agreement between pipelines in the solar-normalized case (σ = 0.0041 for solar-normalized ratios, as opposed to σ = 0.0047 for absolute ones); however, there are still significant systematic trends between pipeline values as a function of Δν.

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Our final mean values are taken using the scale factors presented in Table 2, which also contains a statistical description of our data. In this normalized framework, trends in the mean values for different pipelines as a function of ν_max and Δν are shown in Figure 8. For intermediate values of the asteroseismic parameters, the normalized values of the pipelines agree well; for the most luminous targets (with small frequency spacings and low frequency of maximum power), pipeline-to-pipeline systematics are apparent at the ±1% percent level for ν_max and the ±0.5% percent level for Δν. Pipeline-to-pipeline differences for ν_max are also somewhat larger than the norm in the least luminous targets as well. These trends are consistent with well-understood data analysis challenges: for high-luminosity giants, the limited frequency resolution yields less reliable measurements of asteroseismic parameters, while oscillations near the long-cadence Nyquist frequency cause systematic differences in estimating the background noise level.

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Table 2.  Relative Pipeline zero-points

Quantity	A2Z	CAN	COR	OCT	SYD
${X}_{{\nu }_{\max },\odot }$	0.9981	1.0118	0.9828	1.0115	0.9957
${X}_{{\nu }_{\max },\mathrm{RGB}}$	1.0023(2)	1.0082(2)	0.9989(2)	0.9900(2)	1.0006(2)
${X}_{{\nu }_{\max },\mathrm{RC}}$	1.0035(3)	1.0067(2)	0.9909(2)	0.9979(4)	1.0010(3)
${\sigma }_{{\nu }_{\max },\mathrm{RGB}}$	0.010	0.006	0.006	0.012	0.009
${X}_{{\rm{\Delta }}\nu ,\odot }$	1.0004	0.9980	1.0026	0.9993	0.9997
${X}_{{\rm{\Delta }}\nu ,\mathrm{RGB}}$	0.9993(1)	1.0007(1)	1.0051(1)	0.9955(1)	0.9995(1)
${X}_{{\rm{\Delta }}\nu ,\mathrm{RC}}$	0.9965(3)	1.0108(2)	0.9960(1)	0.9935(2)	1.0032(2)
${\sigma }_{{\rm{\Delta }}\nu ,\mathrm{RGB}}$	0.006	0.004	0.004	0.005	0.003

Note. Error-weighted mean ratios of values from individual pipelines to the ensemble average.

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These differences illustrated here are a systematic error source. For example, if we had adopted the OCT pipeline as our reference, similar to APOKASC-1, then the purple dashed lines in this figure would be straight lines at 1 on the Y axis, and the entire bundle of lines would have been shifted up or down accordingly as a function of Δν and ν_max. By construction, the mean value would be unchanged; but, especially for luminous giants, the relative values would be different, and so would the derived masses. This figure also illustrates an important point about the nature of our empirical normalization procedure: it is specific to this sample and this data set, rather than being an absolute and universal calculation. For example, if the fraction of intrinsically luminous targets in our sample were larger, the systematic differences between methods at low ν_max and Δν would have had more weight in our solution, and we would have inferred different absolute scale factors.

We can also use our large sample of RC stars to test whether the data for them have the same overall behavior as that of RGB stars. As the observed frequency spectra are quite different, evolutionary-state dependent differences are certainly possible. We present the RC results in Figure 9. If we use the same scale factors as those derived for the RGB, it is apparent that there are real pipeline-to-pipeline differences in the relative RC zero-points. Although the scatter would be reduced if we were to adopt an RC-specific normalization, there are significant trends with the seismic parameter (ν_max or Δν), which would still yield significant method-dependent scatter. As our masses and radii are ultimately calibrated on RGB stars, we choose to adopt the RGB normalization for the relative pipeline values. We will use the systematic differences in Figure 9 as a guide to systematic uncertainties in the relative derived RC masses, which are significantly higher than the corresponding trends on the RGB (as reflected in Figure 8).

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3.1.2. Random Uncertainties

Our treatment of the random uncertainties represents another significant change in the model. Prior work has used the formal values returned by the pipelines as a measure of random errors and combined these values with pipeline-to-pipeline differences in results to infer total error budgets. To test the pipeline error models, we compare the dispersion of measurements from each pipeline around the normalized mean to the dispersion that would have been predicted by the formal pipeline uncertainties. In effect, this is testing whether the pipeline uncertainties behave as one would expect for random error sources, i.e., that they reflect how well the given method predicts the average measurements returned by all analysis techniques. The results are presented in Figure 10 (for ν_max on the left and for Δν on the right). Two features become immediately obvious: (1) the pipelines usually (but not always) predict uncertainties much larger than the observed method-to-method differences; (2) there is no clear mapping between the formal predicted uncertainties for the different pipelines and their true scatter around the ensemble mean. To take a concrete example, if we were to use the pipeline-predicted uncertainties in an error-weighted mean, we would have assigned very high weight to the CAN measurements of Δν and lower weight to the COR measurements. However, the two pipelines are very similar in terms of how well they predict the ensemble mean values; there is no evidence that COR values have larger random measurement scatter.

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Some caution is in order, as the dispersion that we are testing here is not necessarily the random error in the underlying data. Rather, we are measuring the dispersion between methods being used to infer the mean asteroseismic parameters. In the case of the Kepler light curves for evolved stars, however, the targets are bright and the time series are extremely long. As a result, it is at least a plausible hypothesis that our ability to interpret the light curves, rather than signal to noise, is the primary contributor to the error budget. This would not in general be true for shorter time series or lower signal-to-noise data, for example in K2 or TESS. We therefore adopt an empirical random uncertainty based on the concordance between different methods and test this error model against star cluster data in the next section.

From the results presented here, it is clear that weighting the various pipelines by their formal errors, or comparing method to method based on the formal errors, does not accurately capture how well the different techniques can predict the ensemble mean of any given star. The data in Figure 10 also collapse the distribution of uncertainties down to a single figure of merit, σ, which assumes a normal error distribution. We would ideally like to determine whether we can justify a single overall uncertainty estimate or whether we need a star-by-star measurement.

We therefore proceed as follows. For each star, we compute an unweighted mean across pipelines of the asteroseismic properties using the relative normalizations illustrated in Figure 8, and we compute the dispersion about the mean. The results are displayed in Figures 11 and 12 (for ν_max and Δν respectively). There are four distinct groups represented on each figure: (1) the calibrating sample (RGB stars with results from all five pipelines, in red); (2) RGB stars with results from fewer than five pipelines, in pink; (3) RC stars with results from all five pipelines, in blue; and (4) RC stars with results from some pipelines, in cyan. There are some striking trends in the data. For the RGB sample with results from all pipelines, the distribution of dispersions matches well the expectations from a normally distributed distribution with small uncertainties. RGB targets where one or more pipelines failed, however, had substantially larger scatter, and the distribution of dispersions is clearly not drawn from a normal distribution. The dispersions for RC stars are larger than those for RGB stars; this is a combination of systematic differences between pipelines and truly larger random differences. The RC stars with partial detections have a larger dispersion than those with detections from all methods, but the differences with RC stars that were measured by all methods are not as stark as they are for the RGB case. One possible factor is that the most problematic RC stars lacked an evolutionary state classification (see Figure 1), which removed them from our sample. For Δν, the peak in the dispersion for RC stars around 0.01 in the third panel from the top reflects systematic zero-point differences between pipelines relative to the means for the RGB targets.

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In light of these results, we choose to treat the fractional standard deviation of our sample measurements for the RGB calibrators (0.009 in ν_max and 0.004 in Δν from the top panels of Figures 11 and 12) as a minimum fractional random uncertainty for the asteroseismic parameters. If the fractional dispersions of the normalized measurements around the mean are larger than these minimum values, we adopt them instead for our error analysis. This conservative approach assigns larger uncertainties to targets where different analysis methods disagree by more than the norm, while avoiding unphysical small formal error estimates for targets with small formal dispersions. It is not uncommon to have multiple measurements agree much better than their formal dispersion would predict if the sample size is small; our approach avoids this pitfall.

3.1.3. Systematic Uncertainties

Our systematic errors include the systematic differences that could arise from the choice of pipeline, the ${f}_{{\nu }_{\max }}$ scale factor, and the ${f}_{{\rm{\Delta }}\nu }=1$ individual correction factors. There is also a global uncertainty in the absolute temperature scale, which also enters into our mass and radius formulae; systematic errors in the abundances may also be significant and are of order 0.1 dex between DR13 and DR14. We do not include these in our systematic error analysis, however, because they are implicitly accounted for in our empirical calibration process.

The effective temperatures that we adopt are tied to the IRFM fundamental scale in low extinction fields, which should be reliable across the T_eff domain of our data. APOGEE uses the González Hernández & Bonifacio (2009) scale. There are modest differences, at the 30 K level, between their system and the more recent Casagrande et al. (2010) scale for dwarfs. Unfortunately, the latter is not currently available for evolved stars, but the differences there should be comparable to those for main-sequence stars. In uncalibrated models, fractional systematic errors of order 0.005 in T_eff would propagate to 0.0025 in R and 0.0075 in M. However, in our empirical calibration approach, we derive the ${f}_{{\nu }_{\max }}$ factor by requiring our open cluster sample to match empirical data, which would affect the absolute value of ${f}_{{\nu }_{\max }}$ but not the derived masses or radii.

As illustrated in Figures 8 and 9, there are systematic offsets in both asteroseismic observables between our mean scale and the one that would have been obtained for each of our five analysis methods. These effects dominate our systematic error budget. In Figure 13, we show the implied shifts in the mass, radius, and surface gravity scales that we would have observed if we had chosen each of the pipelines as the preferred scale, rather than an ensemble mean. Our RC results have substantially more variance than the RGB ones. Masses have larger systematic differences than radius, which in turn has more uncertainty than surface gravities do.

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To infer errors, we take the conservative approach of fitting a straight line to the upper envelope of the family of curves in Figure 13 for M and R. This yields fractional systematic errors in mass of 0.09–5 × 10⁻⁴ × ν_max for RC stars and 0.04–1 × 10⁻⁴ × ν_max for RGB stars, and fractional systematic errors in radius of 0.03–2.5 × 10⁻⁴ × ν_max for RC stars and 0.015–5 × 10⁻⁵ × ν_max for RGB stars. For log g and $\langle \rho \rangle$, the systematic errors can be inferred directly from Figures 8 and 9, as they depend solely on ν_max and Δν respectively, rather than a combination of the two. A corresponding fit to their upper envelopes for log g is 0.01–5 × 10⁻⁵∗ν_max for RC stars and 0.005 for RGB stars; for $\langle \rho \rangle$, we adopt fractional uncertainties of 0.01 for RC stars and 0.005–1.5 × 10⁻⁵∗ν_max for RGB stars.

The uncertainty in our mean calibrator mass scale of  2% induces a systematic error for our ${f}_{{\nu }_{\max }}$ of 0.007, which we treat as a systematic error source. Our derived ${f}_{{\rm{\Delta }}\nu }=1$ factors depend on the stellar parameters and are larger for RGB stars than RC ones. As a result, systematic errors in the differential values for the two groups exist. We discuss this systematic error source, which is larger for RC stars than RGB ones, at the end of Section 3.2.

As an illustration of the potential systematic effects that could be present, we examined all stars with data from all five pipelines. We used the scale factors in Table 2 to place the Δν and ν_max measurements from all pipelines on the same average scale. We then constructed mean asteroseismic parameters $\langle {\rm{\Delta }}\nu \rangle$ and $\langle {\nu }_{\max }\rangle$ by averaging these corrected values. For each star, we then used Equations (1) and (2) to infer the masses that we would have obtained using each pipeline alone, as well as the mass implied by the ensemble average. The ratios of the masses that we would have obtained from each pipeline to the ensemble mean are shown for RGB and RC stars in Figures 14 and 15, respectively. Note that the pipeline values for Δν and ν_max were placed on the same mean system before computing masses, so zero-point shifts have been suppressed. This exercise uses simple scaling relations. We see well-behaved errors in the RGB case, consistent with method-dependent systematics being well controlled there and larger offsets in the RC case. For a fuller discussion of these systematics, we refer the interested reader to A. Serenelli et al. (2018, in preparation).

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In a departure from the APOKASC-1 approach, we apply theoretically motivated corrections to the Δν scaling relation. These corrections can be computed with knowledge of the stellar mass, composition, evolutionary state, surface gravity, and effective temperature. We do this by implementing stellar models for which one derives Δν from adiabatically derived radial modes on the one hand and from the Δν scaling relation on the other; see White et al. (2011) for an example. For our open cluster calibrators, we assume a known true mass and do a table lookup to infer the correction using the procedure below. For the general case, we must iteratively solve for the correction factor and the mass (as our final mass estimate requires knowledge of the correction term itself.) This approach is similar in spirit to that employed by Rodrigues et al. (2017).

The determination of the correction to the Δν scaling relation is done using a modified version of grid-based modeling as implemented in the BeSPP code (A. Serenelli et al. 2018, in preparation) in which the input data are stellar mass, surface gravity, [Fe/H], and [α/Fe]. The procedure is implemented as follows. We use the average solar reference values defined in Table 1. For the asteroseismic measurements, we take the average of the normalized measurements (the raw values divided by the factors given in Table 2). The uncertainties are the larger of our minimum uncertainty for all stars (0.009 and 0.004 for ν_max and Δν, respectively) and the fractional dispersion in normalized measurements.

For star cluster members, we used the asteroseismic ν_max and the spectroscopic T_eff to infer log g. We then perform a table lookup at that surface gravity using the mean cluster mass, [Fe/H], and [α/Fe] in Table 3 to infer f_Δν factors for our open cluster calibration exercise.

Table 3.  Open Cluster Global Properties for Mass Calibration

Cluster	${M}_{\mathrm{fund}}^{\mathrm{cluster}}$	[Fe/H]	[α/Fe]	${M}_{{sc}}^{\mathrm{cluster}}$	${M}_{\mathrm{trial}}^{\mathrm{cluster}}$	${f}_{{\nu }_{\max }}$
NGC 6791	1.15(2) M☉	0.42	0.04	1.294(17)	1.103(15)	1.014(8)
NGC 6819	$1.55(4){M}_{\odot }$	0.11	0.00	1.771(22)	1.582(20)	0.993(14)
Mean	⋯	⋯	⋯	⋯	⋯	1.009(7)

Note. The benchmark masses are derived from the sources described in the text, and the cluster abundances are the means of the DR14 values for the targets with asteroseismic measurements. ${M}_{{sc}}^{\mathrm{cluster}}$ is the mean mass for RGB cluster members without any corrections to the scaling relations (${f}_{{\nu }_{\max }}={f}_{{\rm{\Delta }}\nu }=1$). ${M}_{\mathrm{trial}}^{\mathrm{cluster}}$ is the mean mass for RGB cluster members including a correction to the Δν scaling relation only (${f}_{{\nu }_{\max }}=1$). The final column is the ${f}_{{\nu }_{\max }}$ factor needed to reproduce the calibrating mass for RGB members relative to an assumed solar ν_max reference value of 3103.266 μHz. The bottom row gives the adopted mean ${f}_{{\nu }_{\max }}$ factor, corresponding to an effective solar ν_max reference value of 3076 ± 21μHz.

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For the remainder of the sample, we had to adopt an iterative approach. Initial guesses M₀ and g₀ for the stellar mass and the surface gravity, respectively, are obtained using the scaling relations in Equations (3) and (4) with T_eff, Δν, and ν_max as inputs. The uncertainties ${\sigma }_{{M}_{0}}$ and ${\sigma }_{{g}_{0}}$ of these quantities are also determined from the scaling relations by propagating the errors in the input quantities. Then, an iterative procedure is run by feeding BeSPP with these quantities at step i (${M}_{i},{g}_{i},[{Fe}/H],[\alpha /{Fe}],{\sigma }_{{M}_{i}},{\sigma }_{{g}_{0}},{\sigma }_{[{Fe}/H]},{\sigma }_{[\alpha /{Fe}]}$) to compute a new value ${f}_{{\rm{\Delta }}\nu ,i}$ defined as

$$\begin{eqnarray}&&{f}_{{\rm{\Delta }}\nu }=\displaystyle \frac{{\rm{\Delta }}{\nu }_{{sc}}}{{\rm{\Delta }}{\nu }_{l=0}}.\end{eqnarray} \tag{ 13 }$$

Note that, because of the notation used in this paper, this ratio is the inverse of the value frequently used in the literature. The mean value is determined from the probability distribution of f_Δν given the quoted uncertainties. For subsequent iterations, g₀, [Fe/H], [α/Fe], and their uncertainties are held fixed, while the mass estimate is updated based upon applying the f_Δν factor. The iteration procedure continues until the mean value of f_Δν converges to one part in 10⁵. The uncertainty of f_Δν is defined as the standard deviation of the final probability distribution, and we include it as a random error source.

As a result of this procedure, T_eff is only used to determine the initial values M₀ and g₀. Moreover, T_eff and [Fe/H] are never used simultaneously. This has the positive effect that the GBM scheme is not directly sensitive to the T_eff scale in the stellar tracks, and it is thus robust with respect to its calibration. If we had done a table lookup in correction as a function of T_eff instead of surface gravity, by contrast, systematic errors in the input stellar tracks would have a much larger impact on the derived corrections because of the steep dependence of log g on T_eff in stellar evolution tracks.

The BeSPP models that we used adopt the Grevesse & Noels (1993) mixture of heavy elements, which differs from the Asplund et al. (2005) abundance scale adopted by APOGEE. This could impact the predicted mechanical structure of the models. However, the corrections themselves are insensitive even to large change in the overall metallicity scale at fixed mean density. This is because the core mass is strongly correlated with L and the envelope is nearly adiabatic, so the impact of composition on the structure of evolved stars is not large.

As a cross-check on our system, we independently computed f_Δν using the method of Sharma et al. (2016).³⁸ We then performed the open cluster calibration exercise (described below, in Section 3.3) with this alternate approach, and propagated the masses through for the full sample. The mass differences between our base approach (BeSPP) and this alternate approach (Sharma) are illustrated for the full sample in Figure 16. For RGB stars and RC stars, the BeSPP values are higher on average by a scale factor of 1.006 and 1.002, respectively. The dispersion in the correction factor between methods is modest for the majority of cases (0.003 and 0.007 for RGB and RC stars respectively, for objects with f_Δν between 0.99 and 1.04). By comparison, in our reference BeSPP method, the formal uncertainty for RGB stars is 0.001 and that for RC stars is 0.012. We use the scale shifts to estimate the magnitude of the induced systematic uncertainties, which we include in our error model. In our mass calibration, which scales as ${f}_{{\nu }_{max}}^{3}/{f}_{{\rm{\Delta }}\nu }^{4}$, a scale shift of 1.006 for RGB stars would have implied a compensating scale shift of 1.008 in ${f}_{{\nu }_{\max }}$. The net impact would be systematic shifts in M, R, log g, and $\langle \rho \rangle$ of 0.016, 0.004, 0.004, and 0.002 for RC stars. RGB stars would have systematic shifts in M, R, log g, and $\langle \rho \rangle$ of 0 (by construction), 0.004, 0.004, and 0.006.

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Our calibration for the scale factor ${f}_{{\nu }_{\max }}$ comes from requiring that the masses derived for RGB members of star clusters be in agreement with fundamental measurements. For this purpose, we treat the zero-point of the Δν scaling relation as being fixed. In a technical sense, that there is a degeneracy between the ratio of the adopted solar reference values $({\nu }_{\max ,\odot }^{3}/{\rm{\Delta }}{\nu }_{\odot }^{4})$ and the ${f}_{{\nu }_{\max }}$ factor that we define here, so the latter factor can also be thought of as defining an effective or calibrated solar reference value.

Mass loss complicates the expected initial masses of RC stars, so our mass calibration uses only stars asteroseismically classified as RGB. In both NGC 6791 and NGC 6819, there are eclipsing binary stars near the turnoff with well-measured masses. The relative masses expected on the RGB are only weakly model dependent. For NGC 6791, Brogaard et al. (2012) did a comprehensive study of the expected masses of lower RGB stars, and we adopt their value of 1.15 ± 0.02 M_☉. NGC 6819 has a higher uncertainty, largely because of complications in interpreting the eclipsing binary system closest to the turnoff there; see Brewer et al. (2016) for a discussion. For NGC 6819, we used the isochrones and ages in Brewer et al. (2016) to infer a mean predicted RGB mass of 1.55 ± 0.04 M_☉.

For each such star j, we can compute $\langle {\rm{\Delta }}\nu {\rangle }^{j}$, $\langle {\nu }_{\max }{\rangle }^{j}$, M^j, and their associated errors, using the procedure outlined in Section 3.1. We define a mean cluster mass derived from scaling relations by

$$\begin{eqnarray}&&{M}_{{sc}}^{\mathrm{cluster}}=\displaystyle \frac{{\displaystyle \sum }_{j=1}^{{N}_{\mathrm{rgb}}}\tfrac{{M}_{{sc}}^{j}}{{({\sigma }_{M}^{j})}^{2}}}{{\displaystyle \sum }_{j=1}^{{N}_{\mathrm{rgb}}}\tfrac{1}{{({\sigma }_{M}^{j})}^{2}}}.\end{eqnarray} \tag{ 14 }$$

We can then define a trial mean cluster mass that we would have obtained after applying the star-by-star corrections to the Δν scaling relation, but setting ${f}_{{\nu }_{\max }}=1$, by

$$\begin{eqnarray}&&{M}_{\mathrm{trial}}^{\mathrm{cluster}}=\displaystyle \frac{{\displaystyle \sum }_{j=1}^{{N}_{\mathrm{rgb}}}\tfrac{{M}_{{sc}}^{j}}{{({f}_{{\rm{\Delta }}\nu }^{j})}^{4}{({\sigma }_{M}^{j})}^{2}}}{{\displaystyle \sum }_{j=1}^{{N}_{\mathrm{rgb}}}\tfrac{1}{{({\sigma }_{M}^{j})}^{2}}}.\end{eqnarray} \tag{ 15 }$$

If we require that the corrected scaling relation mass M_cor from Equation (5) equal the fundamental mass, we can then solve for ${f}_{{\nu }_{\max }}$ by

$$\begin{eqnarray}&&{f}_{{\nu }_{\max }}^{3}=\left(\displaystyle \frac{{M}_{\mathrm{fund}}^{\mathrm{cluster}}}{{M}_{\mathrm{trial}}^{\mathrm{cluster}}}\right).\end{eqnarray} \tag{ 16 }$$

We have 17 RGB stars in NCG 6791 and 23 in NGC 6819; of these, 10 stars in each cluster have asteroseismic parameters from all pipelines, while the remainder have measurements from a minimum of two pipelines. Our cluster data in a color–magnitude diagram are shown in Figure 17, with the RGB and RC stars used in this analysis illustrated as large colored solid and open symbols respectively. For reference, we compare both data sets to isochrones.

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As we are using ensemble averages to infer mean masses, we have to be alert to biases induced by stars with an unusual evolutionary origin—for example, evolved overmassive blue stragglers. None of our NGC 6791 targets have statistically unusual masses. However, five stars in NGC 6819 were flagged as problematic and not used in computing mean sample properties, and all were flagged as outliers in Handberg et al. (2017). The RGB star KIC 5113061, visible in the upper-left corner of Figure 17, is in an unusual part of the CMD, even though its mass is in a reasonable range. The RGB star KIC 5024272 is a high-mass outlier at 2.65 ± 0.41. The RC star KIC 4937011 is highly undermassive (below 0.6 solar masses formally), while KIC 5023953 and KIC 5024476 are statistical outliers on the high-mass side (at 2.05 ± 0.11 and 2.47 ± 0.16 solar masses, respectively.) Two other overmassive stars in Handberg et al. (2017), KIC 5112880 and KIC 5112361, are statistically consistent with the mean cluster trend and were retained in our analysis. We caution that our uncertainties for some of these stars are large, and their unusual masses may reflect measurement difficulties rather than an anomalous origin; because of large uncertainties, they also would not have significantly impacted our mean properties had we included them.

In Figure 18, we show the scaling relation masses prior to any corrections for NGC 6791 and NGC 6819. As known from prior work, the classical scaling relation masses are overestimated, with formal mean RGB masses of 1.294 ± 0.017 M_☉ for NGC 6791 and 1.772 ± 0.034 M_☉ for NGC 6819, both ruled out at high statistical significance.

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We then apply corrections to the Δν scaling relation and solve for the best-fit ${f}_{{\nu }_{\max }}$ values for both clusters. We illustrate our resulting fits for the cluster stars in Figure 19. The mean masses for both systems that we would have derived using Equations (14) and (15), and the associated random uncertainties, are given in Table 3, as are the ${f}_{{\nu }_{\max }}$ factors derived from Equation (16). There are several features here worth discussing. We can achieve concordance between the fundamental and asteroseismic mass scale with small adjustments to the ν_max zero-point. If we combine the two clusters, we obtain a final ${f}_{{\nu }_{\max }}$ factor of 1.009 ± 0.007, corresponding to a reference "effective solar" ν_max value of 3076 ± 21 μHz.

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We can also use our data to check on trends with mass, metallicity, and surface gravity over a limited domain. The NGC 6819 and NGC 6791 mass ranges (1.55 and 1.15 solar masses, respectively) roughly bracket the mass range for the majority of RGB stars in our field sample; there is no statistically significant evidence for a mass trend in the ν_max relationship once the f_Δν corrections have been applied to the Δν scaling relationship. Across most of the surface gravity domain (2.0 < log g < 3.2), we see no strong trends in the asteroseismic masses relative to expectations from isochrones. This is significant because the predicted corrections are themselves functions of surface gravity, and if there was an error in the adopted functional form, we could have seen strong residual trends. However, as is visually apparent in Figures 18 and 19, the most luminous RGB stars (low surface gravity) have mass estimates systematically lower than the cluster mean, albeit with large random uncertainties. This represents a possible caution about the scaling relations at low surface gravities, but it is only of marginal statistical significance. A similar result was obtained by Mosser et al. (2013b) in M giants.

The dispersion of the points around the mean predicted by our error model for both clusters is close to that seen in the real data (with a reduced χ² per degree of freedom of 0.79 and 0.60 for NGC 6819 and NGC 6791, respectively). We take this as confirmation that our uncertainty estimates are conservative but reasonable relative to astrophysical expectations. In NGC 6819, in particular, the higher mass scatter has been noted before and a potential explanation has been the presence of a large number of stars experiencing binary mass interactions, and we did have to remove several stars from our sample because of either highly discrepant masses or anomalous HR diagram position. However, the excess mass scatter in NGC 6819 is present for the majority of targets, which may indicate that some outlier values may simply reflect the difficulty in inferring asteroseismic properties in this mass domain. A more detailed examination of the possibilities will be covered in J. Zinn et al. (2018).

Eclipsing binary systems with asteroseismic detections and well-characterized masses can also be used to test the mass scale. There are interesting samples of such stars (Gaulme et al. 2016). Unfortunately, the overlap between the EB sample and our data set is small, with only KIC 9970396 and KIC 10001167 being stars in common with precise measurements (KIC 4663613 has partial data with large parameter scatter and is not a useful calibration point). The fundamental radii are in good agreement with the radii derived using our method (R_cor/R_EB = 1.005 ± 0.038 and 1.013 ± 0.031, respectively, for KIC 9970396 and KIC 10001167. The masses are in tension, M_cor/M_EB = 1.162 ± 0.106 and 1.092 ± 0.053, respectively, but not with high statistical confidence. This finding is consistent with recent comparisons using other analysis methods by Brogaard et al. (2018) and indicates either a caution on the overall mass scale or complications in the analysis of the eclipsing binary stars.

A more significant concern is the validity of our asteroseismic stellar parameters for RC stars. We cannot assume a calibrating mass for core He-burning stars because of the possibility of significant mass loss on the first ascent red giant branch; in fact, the derived masses of such stars are a science result, not a calibration point (Miglio et al. 2012). Our methodology, when applied to RC stars in open clusters, yields masses equal to their RGB precursors (but with systematic errors large enough that we cannot rule out the possibility of either no mass loss or significant mass loss.)

An alternate approach, using comparisons to theoretical models of the core He-burning phase, can also be employed to infer RC masses. This approach when applied to NGC 6791 suggests moderate mass loss on the RGB, but there are significant model dependencies in the results (D. An et al. 2018, in preparation). We can, however, look at radii, which can be tested for both RGB and RC stars independent of the assumed mass scales.

To test the radius scale, we take advantage of the fact that members of these star clusters have a small intrinsic range in relative distance to us. As a result, we can infer the total luminosity of stars by a combination of their apparent brightnesses, the cluster extinction and distance modulus, and their bolometric corrections. When combined with spectroscopic T_eff, we can then infer a fundamental radius. The weakness of this approach is that changes in the cluster distance and extinction cause correlated zero-point shifts in the radius scale, making this method uncompetitive for calibrating the scaling relations relative to using the (relatively precise) masses. However, we can use the cluster radii to test the relative corrections for RC and RGB stars as described below.

Our V-band photometry is from Brogaard et al. (2012) for NGC 6791 and from Kalirai et al. (2001) for NGC 6819. All of our targets have 2MASS JHK photometry. To constrain the extinction, we take advantage of the large number of APOGEE T_eff measurements that have been calibrated on the González Hernández & Bonifacio (2009) IRFM system. We derive a mean photometric T_eff value as the average of the value derived from the V – J, V – H, and V – Ks reddening-corrected colors using the González Hernández & Bonifacio (2009) color–temperature relationships. Our extinction model uses Equations (5)–(7) from An et al. (2007) to derive color-dependent extinction terms, following the model of Bessell et al. (1998). We then solve for the E(B – V) value required to bring the average difference between the two temperature scales to zero, and treat the dispersion between the values as a representative temperature error. As indicated in Table 4, our derived extinctions (0.123 and 0.144 for NGC 6791 and NGC 6819, respectively) are in good agreement with literature values. There is some discussion of differential extinction in the literature, and we did check the impact of the proposed differences to our answer; they reduced the dispersion in temperature differences somewhat but did not impact our mean values. We adopt V-band bolometric corrections from Flower (1996) with the erratum from Torres (2010), using the spectroscopic T_eff values from DR13. In combination with an adopted cluster distance, we can then predict stellar luminosities and, from knowledge of the effective temperatures, radii.

Table 4.  Open Cluster Global Properties for Radius Calibration

Cluster	${(m-M)}_{o,\mathrm{cmd}}$	${(m-M)}_{o,{seis}}$	$E{(B-V)}_{\mathrm{cmd}}$	$E{(B-V)}_{{sp}+{ph}}$	$\tfrac{{R}_{\mathrm{RC}}}{{R}_{\mathrm{RGB}},{sc}}$	$\tfrac{{R}_{\mathrm{RC}}}{{R}_{\mathrm{RGB}},\mathrm{cor}}$
6791	13.06(8)	13.03(1)	0.122(17)	0.153(5)	0.954(23)	1.011(24)
6819	11.91(4)	11.88(2)	0.160(7)	0.165(5)	0.917(22)	0.9741(23)

Note. NGC 6791 distance and extinction are the average of those from An et al. (2015) and Brogaard et al. (2012). NGC 6819 distance modulus is the apparent distance modulus of 12.38 ± 0.04 from Brewer et al. (2016) modified by the extinction, taken from Anthony-Twarog et al. (2014) and R_V = 3.26. The seismic distance is derived from requiring concordance between the asteroseismic radius and that derived from the combination of L, extinction, and spectroscopic T_eff. The seismic extinction is derived from requiring concordance between the spectroscopic and photometric T_eff. The final two columns are the ratios of RC to RGB radii with scaling relations alone (fifth column) and after corrections to the Δν scaling relation (sixth column).

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Because our method is calibrated to reproduce known masses, concordance between these radii and those predicted by asteroseismology can be achieved with a suitable choice of the cluster distance. This technique for inferring "asteroseismic distances" has been successfully used in the literature, for example, by Stello et al. (2016). For our purposes, we adopt the ν_max zero-point derived from the mass constraints and adjust the cluster distance modulus (m – M) such that the average radii inferred for RGB stars from the calibrated asteroseismic scaling relations were in agreement with the fundamental average radii inferred from our combination of photometric and spectroscopic constraints. The inferred cluster distance moduli (13.03 and 11.88 for NGC 6791 and NGC 6819, respectively) are given in Table 4 and are well within the observational uncertainties. We can then examine the concordance between the radii inferred for RC stars and those inferred for RGB ones; in the absence of corrections to the Δν scaling relation, for example, Miglio et al. (2012) found that the relative radii of the two populations were discordant at the 5% level. The results are shown in Figure 20. In panel a, we show the relative radii that we would have obtained without corrections to the Δν scaling relation. The RC radii are too small relative to the RGB radii, and the radius system overall is inflated relative to expectations. Once Δν corrections are applied, the RGB properties are significantly affected, while the RC properties change only minimally. As a result, the RC radii are inflated relative to the RGB radii. This is tentative evidence that the Δν corrections adopted here may be somewhat too large, although we caution that this result is not highly statistically significant.

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An independent check on this result can be obtained using T_eff values and extinctions derived from SED fitting (Huber et al. 2017; Stassun et al. 2017, 2018). The results are consistent with those derived from spectroscopy, but with somewhat larger uncertainties (RC radii relative to RGB of 1.022 and 1.021 in NGC 6791 and 6819, respectively.) This represents welcome confirmation of the overall results.

After this manuscript was submitted, the Gaia collaboration released their second data release (Lindegren et al. 2018, hereafter DR2). Gaia parallaxes add powerful additional tests of asteroseismic results, but taking advantage of them also requires substantial additional information. To briefly summarize, Gaia parallaxes in combination with photometry, model-dependent bolometric corrections, and extinction estimates can be used to infer the absolute luminosity of stars. Given a luminosity and an effective temperature, one can infer absolute stellar radii, which can serve as a test of asteroseismic radii. Gaia parallaxes can also be used to infer distances to star clusters, which we are using as fundamental calibrators in our method. For NGC 6819 and 6791, however, the Gaia parallaxes have larger systematic uncertainties than the eclipsing binary solutions that we use.

There are two additional effects that complicate the interpretation of Gaia parallaxes for our purposes. For distant objects, such as the evolved stars in the Kepler fields that we are studying, there are significant systematics arising from zero-point uncertainties in the Gaia parallaxes (Lindegren et al. 2018). Parallaxes are also correlated locally, which complicates the error analysis. As a result, a proper treatment of Gaia data is out of scope for this paper. However, Zinn et al. (2018) used asteroseismic data as a reference frame in the Kepler fields to test the parallax zero-point and found a value within the range expected by the Gaia team. Their analysis found no evidence for a significant radius zero-point offset between Gaia and the asteroseismic scale employed in this paper (at the one percent level or less.) A more detailed comparison of Gaia with our results is underway in a separate paper in preparation.

We generate our final stellar parameters as follows.

• 1.  
  We use the DR14 spectroscopic T_eff, as described in Abolfathi et al. (2017) and Holtzman et al. (2018).
• 2.  
  We use measured asteroseismic evolutionary states where available. If there is no consensus evolutionary state, we used spectroscopic evolutionary states from DR13. In cases where the spectroscopic states were ambiguous, we derived asteroseismic parameters assuming that the target had both states and treated the range as an additional systematic error source.
• 3.  
  The raw asteroseismic parameters from each pipeline with data are divided by the scale factors in Table 2, and these normalized values are then averaged (see Section 3.1.)
• 4.  
  We adopt solar reference values of 135.146 μHz and 3076 ± 21 μHz for the Δν and ν_max scaling relations respectively, as discussed in Section 3. This choice incorporates the ${f}_{{\nu }_{\max }}$ factor as a correction to the average solar reference value for ν_max in Table 1.
• 5.  
  Fractional random errors for asteroseismic observables are taken as the larger of the normalized measurement dispersion σ(parameter)/<parameter> and our "best case" uncertainties (0.009 and 0.004 for ν_max and Δν, respectively). Random uncertainties in these observables and f_Δν are propagated in quadrature to obtain the tabulated values.
• 6.  
  Systematic fractional uncertainties are taken from Sections 3.1, 3.2, and 3.3. These include systematic shifts in derived quantities that would have resulted from adopting single pipeline values rather than the ensemble mean, discussed in Section 3.1.3; the measurement technique for f_Δν, discussed in Section 3.2; and the zero-point of the open cluster calibrating mass scale, discussed in Section 3.3, which induces a global uncertainty in ${f}_{{\nu }_{\max }}$.
• 7.  
  We compute the star-by-star correction factors f_Δν iteratively as discussed in Section 3.2. In addition to the T_eff and asteroseismic parameters discussed above, this procedure also uses evolutionary state and the DR14 [Fe/H] and [α/Fe] values.
• 8.  
  We then infer asteroseismic mass, radius, mean density, and surface gravity measurements using Equations (1)–(6) combined with standard error propagation.
• 9.  
  Ages are estimated using our derived mass, surface gravity, [Fe/H], and [α/Fe], and uncertainties, while extinctions are computed using our data and additional photometry.

Our results are presented in Table 5, available in machine-readable format; the header describing the contents of the table is reproduced here. For the remainder of this section, we present some comparisons of the current work to the prior catalog and briefly illustrate some of the main properties of the catalog itself. For some stars, we were not able to obtain full results, and the reason is given in the Notes column. As noted in Section 2, there were a number of stars for which we were unable to get asteroseismic parameters; they are flagged as No Seis. Our mass corrections require interpolation in a model grid, and for 25 targets we could not obtain internally consistent parameters; these are flagged as No Fdnu. We also flag stars with unusually large measurement uncertainties in their asteroseismic parameters (as SeisUnc) and those with formal mean ages greater than 14 Gyr (as AgeOld), although we do provide data for them.

Table 5.  The APOKASC-2 Catalog of Stellar Properties

Label	Source	Description
KIC	Kepler Input Catalog
2MASS	2MASS Catalog
T eff	DR14	Teff in K
S Teff	DR14	$\sigma ({{\rm{T}}}_{\mathrm{eff}})$ in K
FeH	DR14	[M/H]
S FeH	DR14	$\sigma ([\mathrm{Fe}/{\rm{H}}])$
AFe	DR14	[α/Fe]
S AFe	DR14	σ([α/Fe])
Nmax	This paper	$\langle {\nu }_{\max }\rangle$, μHz
S Nmax	This paper	$\sigma \tfrac{\langle {\nu }_{\max }\rangle }{\langle {\nu }_{\max }\rangle }$
Dnu	This paper	$\langle {\rm{\Delta }}\nu \rangle$, μHz
S Dnu	This paper	$\sigma \tfrac{\langle {\rm{\Delta }}\nu \rangle }{\langle {\rm{\Delta }}\nu \rangle }$
ES	Elsworth et al. (2017), DR13	Evolutionary Statea
F dnu	This paper	fΔν, Δν correction factor
S Fdnu	This paper	$\sigma ({f}_{{\rm{\Delta }}\nu })$
M(cor)	This paper	Corrected Mass, M☉
S Mran	This paper	$\tfrac{{\sigma }_{\mathrm{ran}}(M)}{M}$
S Msys	This paper	$\tfrac{{\sigma }_{\mathrm{sys}}(M)}{M}$
R(cor)	This paper	Corrected Radius, R☉
S Rran	This paper	$\tfrac{{\sigma }_{\mathrm{ran}}(R)}{R}$
S Rsys	This paper	$\tfrac{{\sigma }_{\mathrm{sys}}(R)}{R}$
logg(seis)	This paper	Log asteroseismic surface gravity
S Gran	This paper	$\tfrac{{\sigma }_{\mathrm{ran}}(\mathrm{log}g)}{\mathrm{log}\ {\rm{g}}}$
S Gsys	This paper	$\tfrac{{\sigma }_{\mathrm{sys}}(\mathrm{log}g)}{\mathrm{log}\ {\rm{g}}}$
$\langle \mathrm{Rho}\rangle$	This paper	$\langle \rho \rangle$ in g cm−3
S Rhoran	This paper	$\tfrac{{\sigma }_{\mathrm{ran}}(\langle \rho \rangle )}{\langle \rho \rangle }$
S Rhosys	This paper	$\tfrac{{\sigma }_{\mathrm{sys}}(\langle \rho \rangle )}{\langle \rho \rangle }$
Log Age	This paper	Log age in Myr
S LogageP	This paper	${\sigma }_{{\rm{r}}{\rm{a}}{\rm{n}}}({\rm{l}}{\rm{o}}{\rm{g}}\,{\rm{a}}{\rm{g}}{\rm{e}})+$
S LogageM	This paper	${\sigma }_{{\rm{r}}{\rm{a}}{\rm{n}}}({\rm{l}}{\rm{o}}{\rm{g}}\,{\rm{a}}{\rm{g}}{\rm{e}})-$
Av	This paper	Extinction AV(RV = 3.1)
S Av	This paper	${\sigma }_{\mathrm{ran}}{A}_{V}$
Notes	This paper	a

Notes. Contents of the main APOKASC-2 data table. Stars without parameters requiring asteroseismic inputs were cases where the time series data were analyzed but no asteroseismic parameters were returned. The asteroseismic observables are derived as described in the text from the raw pipeline values. The Δν correction factor f_Δν for each stat was derived iteratively from the combination of the spectroscopic, evolutionary state, and asteroseismic data with the models of A. Serenelli et al. (2018, in preparation). Masses, radii, and uncertainties were derived from the combined asteroseismic and spectroscopic data as described in the text. Ages were derived from the A. Serenelli et al. (2018, in preparation) models using the catalog mass, radius, [Fe/H], and [α/Fe]. The procedure for the extinction map is described in the text.

^aDerived from asteroseismology if available. Otherwise, uses DR13 spectroscopic evolutionary states, noted by (s) after class. RGB is a shell-burning source only (first ascent red giant or asymptotic giant); RC is a star that has a core He-burning source; Amb(s) is a star where the spectroscopic state was ambiguous. Reject stars are ones with suspect or absent asteroseismic data, while Bad Teff stars did not have reliable DR14 effective temperatures. ^bThe notes in the final column are included for stars whose reported properties are either incomplete or uncertain. "No Seis" flags all stars without reported reliable asteroseismic values (and, by extension, reported masses, radii, log g, or mean density.) "No Fdnu" flags stars where we could not obtain f_dnu correction factor; the only derived quantity is the asteroseismic surface gravity. "SeisUnc" flags all stars with fractional uncertainties in ν_max > 0.05 or Δν > 0.025. "AgeOld" flags all stars with formal mean ages greater than 14 Gyr.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In Figure 21 we compare the current asteroseismic results with values from our 2014 catalog for stars in common. For the asteroseismic parameters, surface gravities are very well behaved, and for the large majority of cases, the changes in the inferred Δν and ν_max are small. We see some excess scatter for RC stars and a handful of cases where the mean parameters changed significantly between the two efforts.

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Our distributions of uncertainties are illustrated in Figure 22. The error distribution is skewed, with the high scatter cases corresponding to the high-uncertainty tails in the measurements of the asteroseismic parameters. As a result, median errors are a better guide to the overall performance in the catalog of our data. The median random and systematic mass uncertainties for RGB stars are at the 4% level, while the corresponding median uncertainties for RC stars are higher (at 9% and 8%, respectively.)

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In Figure 23, we also present two slices of the full data set to illustrate the mass and abundance trends within our sample in a manner analogous to the APOKASC-1 presentation. We selected all RGB stars with −0.1 < [Fe/H] < +0.1 and color-coded their position in a Kiel diagram as a function of corrected asteroseismic mass on the left panel. In the right panel, we selected all RGB stars with 1.1 M_☉ < M < 1.3 M_☉ and color-coded their position in a Kiel diagram as a function of metallicity. Relative to APOKASC-1, we can see that these diagrams are now much more richly sampled.

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Our data also show distinct mass patterns for different populations and strong correlations between the surface [C/N] and masses. To illustrate these points, we have subdivided our sample into two cohorts: stars with [α/Fe] > +0.12 and those with [α/Fe] < +0.12. We show the density of points in RGB stars for both cohorts as a function of mass and metallicity in Figure 24. The α-rich stars are generally low mass and relatively metal-poor, but there is a clear population of high-mass stars seen in our sample. The α-poor stars, by contrast, show a broad distribution in mass and metallicity, with weak (if any) mean mass–metallicity relationship seen.

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The surface [C/N] is correlated with mass, and by extension, age. In metal-poor stars, there is extra mixing on the RGB which complicates the interpretation of the [C/N] ratio, so we plot the [C/N]–mass relationship for α-poor stars in our sample in Figure 25. Interestingly, there is a strong trend in both the RC and the RGB sample, with a flattening of the slope at high mass (or young age.)

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As a by-product of our catalog, we also present ages and extinction values for our targets, as described below. The central values for ages were inferred by interpolation in the BeSPP grid at the input mass, surface gravity, [Fe/H], and [α/Fe] presented in Table 5, and uncertainties around the mean were computed using the tabulated uncertainties. We stress that the age uncertainties are random only and reflect neither systematic errors in the inputs nor systematic errors in the theoretical age inferences. For details on the adopted input physics, see A. Serenelli et al. (2018, in preparation).

Extinctions are derived using the direct method of stellar parameter estimation as implemented in the PARAM code (da Silva et al. 2006; Rodrigues et al. 2014). As a first step, the code estimates the intrinsic luminosity from knowledge of R_sc and T_eff. The luminosity is then transformed into absolute magnitudes in several filters using bolometric corrections inferred from the library of ATLAS 9 synthetic spectra from Castelli & Kurucz (2003) at the spectroscopic T_eff and metallicity and the asteroseismic surface gravity. Probability density functions (PDFs) of the absolute magnitudes are then generated.

As a second step, the absolute magnitude PDFs are combined with apparent magnitude PDFs, resulting in a joint PDF of the apparent distance modulus. We then assume a single interstellar extinction curve (Cardelli et al. 1989), with R_V = 3.1, and infer the PDF for the true distance modulus as a function of A_V for all available passbands. Our best-fitting A_V and its uncertainty are inferred from requiring consistency between values for different filters. For a more detailed description, see Rodrigues et al. (2014).

The apparent magnitudes adopted are SDSS griz as measured by the KIC team (Brown et al. 2011) and corrected by Pinsonneault et al. (2012), JHKs from 2MASS (Skrutskie et al. 2006), and WISE photometry from Wright et al. (2010).

The median uncertainty in A_V is 0.08 mag. Only 3% of the stars have uncertainties greater than 0.2 mag, typically because they have data from fewer filters. Since this is a statistical method, a small fraction of negative extinction values exist and are expected (Rodrigues et al. 2014). In our sample, approximately 2% of the stars have their PDFs better matched with slightly negative values of extinctions. For these stars, we adopt A_V = 0.0. Figure 26 shows our extinction map. After this paper was submitted, extinctions from the Gaia team (Lindegren et al. 2018) were reported. Because their typical uncertainties are much larger than those that we compute here, we do not employ them as a constraint for our data.

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