Constraining Metallicity-dependent Mixing and Extra Mixing Using [C/N] in Alpha-rich Field Giants

Carbon and nitrogen abundances on the surfaces of low-mass stars offer unique probes of their interiors. Hydrogen burning via the CNO bi-cycle occurs near enough to the surface of the star that the convective envelope brings burned material to the surface on the lower giant branch. The dredged-up material affects the star's surface C/N ratio, with the magnitude of the effect depending on the mass of the star and on any interior chemical mixing present. The former effect has been exploited to derive ages for field red giants (Martig et al. 2016; Ness et al. 2016), while the latter has been used to study so-called "extra mixing" in metal-poor stars (e.g., Carbon et al. 1982; Charbonnel & Lagarde 2010). The name "extra mixing" has been used as a placeholder for an as-yet unconfirmed source of mixing in stars that is not included in standard stellar models.

In standard stellar evolution theory, the initial surface abundances of carbon and nitrogen are set by the composition of the birth cloud. While diffusion can affect their surface abundances on the main sequence (Richard et al. 1996), the deepening of the convective envelope as the star expands into a red giant efficiently brings diffused elements to the surface and homogenizes the composition of the envelope at the initial abundance. As the envelope reaches depths where CN cycling has occurred, it mixes these elements into the surface convection zone, increasing the surface nitrogen abundance of these stars. This process is referred to as the first dredge-up and occurs around a surface gravity of log g ∼ 3.5 dex. The [C/N]¹⁵ ratio of stars is then predicted to be constant between the end of the first dredge-up and the tip of the giant branch as well as during the subsequent core-helium-burning phase. The [C/N] ratio after first dredge-up depends on the mass of the star (Iben 1964), both because the depth reached by the convective envelope is larger for higher masses and because these higher-mass stars have hotter cores and thus a larger equilibrium nitrogen abundance.

These simple predictions are in conflict with observations. Analysis of star clusters (e.g., Carbon et al. 1982; Kraft 1994) show that additional mixing is required to match the carbon and nitrogen abundances of stars below log g of ∼2.5. This process is strong in metal-poor stars, but minimal in solar metallicity clusters (Brown 1987). This "non-canonical extra mixing" is present in stars regardless of environment, and has been seen in dwarf spheroidal galaxies (Shetrone et al. 2013; Kirby et al. 2015; Lardo et al. 2016) and in low-metallicity field stars (Lambert & Sneden 1977). In an influential paper, Gratton et al. (2000, hereafter G00) used a sample of 62 low-metallicity field stars to demonstrate that the [Li/H], [C/H], [N/H], and ¹²C/¹³C values changed not only at first dredge-up, but also at the luminosity of the "red bump," when the outwardly moving hydrogen-burning shell reaches the deepest extent of first-dredge-up. The material that participated in first dredge-up has been homogenized throughout the envelope and is therefore more hydrogen-rich than the gas previously encountered by the hydrogen-burning shell. This results in a build-up of red giants at the relevant luminosity and a bump in the luminosity function for the first-ascent red giant branch (RGB) stars. G00 also showed the core-helium-burning "red clump" (RC) stars had C, N, Li, and ¹²C/¹³C values that matched the values at the tip of the RGB. In addition, O and Na did not show any trend with log g, which limits the depth of the extra mixing to above the region of the star where O–N cycling occurs.

A promising theoretical explanation was proposed by Charbonnel & Lagarde (2010) who identified thermohaline mixing as the likely culprit. This mixing occurs because of the "salt-finger instability," when the Ledoux criteria for stability are satisfied, but a decrease in mean molecular weight (μ) with depth causes the sinking of higher density material until it dissolves. In the case of red giant stars, the burning of ³He, via ³He(³He, 2p)⁴He at the outer edge of the hydrogen-burning shell reduces μ. In the region where the convective envelope has never penetrated, the strong μ gradient already present from nuclear burning means that thermohaline mixing cannot begin. However, when the shell reaches the parts of the star where the penetration of the convective envelope has erased any μ gradient present, thermohaline mixing can start. Because the CNO cycle powers the hydrogen-burning shell, the shell must be hotter in metal-poor stars to provide the necessary support. Therefore the inversion of the μ gradient is deeper and hotter in metal-poor stars and the changes in C, N, and the C isotopes are much larger than for metal-rich stars. The excellent agreement between the start of extra mixing and its dependence on metallicity and the properties of thermohaline mixing means that it almost certainly plays an important role in extra mixing.

A major criticism of the idea that thermohaline mixing is solely responsible for extra mixing is the efficiency of the mixing may need to be higher than conceivably possible. Denissenkov (2010) argued, based on 2D simulations, that the efficiency of mixing could be more than 10× smaller than that used by Charbonnel & Lagarde (2010) to reproduce the observations. 3D simulations are far more appropriate (Garaud & Brummell 2015), but have been been hampered by their computational expense and inability to reach the low Prandtl numbers found in stars. Whether 3D simulations show efficient mixing is disputed in the literature (e.g., Denissenkov & Merryfield 2011; Radko & Smith 2012; Brown et al. 2013). Denissenkov et al. (2009) suggested that magnetic buoyancy could be combined with the thermohaline instability to create sufficient mixing, building on the work of Busso et al. (2007). This effect would also start at the red giant bump.

Additional indirect arguments raised by Denissenkov et al. (2009) against thermohaline mixing as the source of extra-mixing are that it appears to be inefficient in stars that have carbon-rich material transfered to them from an asymptotic giant branch (AGB) companion and that, contrary to observations, slowly rotating red giants would be more likely to show enhanced lithium. More stringent tests of thermohaline mixing could be performed if the metallicity dependence and mass dependence could be carefully compared with models.

However, previous results have had wide metallicity bins, small samples, or contamination by other effects present in globular clusters. The stars that G00 studied were limited in number, and were analyzed as a single metallicity bin from −2 ≤ [Fe/H] ≤ −1. While this sample could be expected to have similar mass, the small number of stars made it difficult to understand the trends with metallicity. The other option for samples of stars of very similar mass is RGB stars in clusters, and they have been very important in studies of the extra-mixing phenomenon. However, globular cluster stellar populations also have light-element anomalies, such as two populations in CN band strength (Suntzeff 1981; Carbon et al. 1982) and an anti-correlation between O and Na in the abundances in individual stars (Drake et al. 1992; Sneden et al. 1992). While these anomalies are present at the main-sequence turnoff (Gratton et al. 2001), well before first-dredge-up and are therefore primordial rather than evolutionary in nature, they complicate the interpretation of extra mixing in globular cluster stars. Open clusters do not share these anomalies, but are also not present at [Fe/H] ≤ −1 in the Galaxy and do not have many red giants in an individual cluster. To make progress interpreting extra mixing, we would therefore desire a sample that has good carbon and nitrogen measurements, known masses, a wide range in metallicity, and made up of field stars free from the multiple abundance populations in globular clusters. These requirements are simultaneously met by the stars in the APOGEE-2 (Majewski et al. 2017; Wilson et al. 2019) survey. This analysis provides carbon and nitrogen abundances for >100,000 field red giants throughout the Galaxy.

Masseron et al. (2017) found evidence for extra mixing along the giant branch by the increased [N/Fe] ratio above the red giant bump in thin and thick disk stars throughout the entire Data Release 12 (Alam et al. 2015) APOGEE sample. However, their analysis did not extensively explore the metallicity dependence of extra mixing. In addition, the thin disk covers a wide range of ages and masses, making the interpretation of their [C/N] evolution more complex (Masseron et al. 2017). Around 2000 of the APOGEE DR12 red giants, most close to solar metallicity, have published masses (Pinsonneault et al. 2014) and evolutionary states (Elsworth et al. 2017), as a result of asteroseismic measurements. Masseron et al. (2017) used this sample to show that higher-mass RGB stars have lower [C/N] values after first dredge-up than stars with lower mass, as expected. What was not expected was that the [N/Fe] values for the core-helium-burning RC stars would be ∼0.2 below the [N/Fe] values of the average on the RGB. Because the RC stars are the immediate descendants of stars at the tip of the RGB, they should have similar [N/Fe] values, as indeed found by G00 for more metal-poor stars. Masseron et al. (2017) proposed the dredge-up of high carbon/low nitrogen material from the helium-burning shell near the red giant tip as a possible explanation. Unfortunately, the Pinsonneault et al. (2014) sample extends only down to ∼−0.65 and contains relatively few stars above the RGB bump.

For this analysis, we therefore use the Data Release 14 (Abolfathi et al. 2018) APOGEE-2 spectroscopic sample of red giants, which has ∼100,000 more stars than DR12, to map the metallicity and gravity dependence of [C/N] depletion of the RGB. We choose to analyze only stars with high [α/Fe] values to restrict the sample in mass, guided by our understanding of stellar populations from the APOKASC sample. We divide our sample into eight metallicity bins to examine the effect of extra mixing as a function of metallicity, updating G00 for the age of large spectroscopic surveys.

APOGEE-2 is a part of the Sloan Digital Sky Survey IV (Blanton et al. 2017) and is collecting high-resolution (R = 22,500) spectra of ∼300,000 stars across the Milky Way (Zasowski et al. 2017) using the Sloan Foundation Telescope (Gunn et al. 2006) at Apache Point Observatory. The spectra are first run through the data reduction pipeline (Nidever et al. 2015), which includes flat-field correction, extraction of the 1D spectra, wavelength calibration, sky subtraction, and combination of spectra taken at different dither positions and on different nights. The resulting 1D spectra are then compared to a grid of synthetic spectra (Zamora et al. 2015) using the ASPCAP analysis pipeline (García Pérez et al. 2016) to determine the stellar parameters

2.1. Stellar Parameter and Abundance Measurements

2.2. Sample Selection

Because the [C/N] ratio on the giant branch is correlated with stellar mass, we require a sample of stars with similar mass to isolate the effects of mixing. Therefore, we chose only stars that are α-enhanced ([α/M] > 0.14, see Figure 1). Galactic chemical evolution models predict that such stars formed before a substantial number of Type Ia supernova exploded (Tinsley 1979) and must therefore be old and low mass. Haywood et al. (2015), for example, calculated isochrone-based ages for turnoff stars in the solar neighborhood and found that red giants with [α/Fe] > 0.1 dex have ages older than ∼10 Gyr (masses less than ∼1.1 M_⊙).

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We removed all stars that were targeted as hot stars to be used for the removal of telluric absorption features (TARGFLAGS: APOGEE_TELLURIC), all stars flagged as having "bad" or no stellar parameters (ASPCAPFLAGS: STAR_BAD, ATMOS_HOLE_BAD, BAD_PIXELS, BAD_RV_COMBINATION) and all stars targeted as candidate members of globular clusters or dwarf galaxies (TARGFLAGS: APOGEE_SCI_CLUSTER, APOGEE_SGR_DSPH, APOGEE2_DSPH_CANDIDATE). To further remove potential globular cluster contaminates we excluded any stars if they fell within two tidal radii and had velocities within 25 km s⁻¹ of the cluster mean for all clusters listed in the 2003 version of the Harris catalog (Harris 1996). The DR14 APOGEE data release does not contain any upper limits but does include errors that may be so large as to represent non-detection. To avoid this problem, we eliminated points with very large [C/N] errors by propagating the errors in [C/Fe] and [N/Fe] and requiring the total error to be less than 0.3 dex. This cut in [C/N] preferentially removes stars which are some combination of warm and metal-poor. A total of 1085 RGB stars and 101 RC stars were rejected by this cut. The sample spans a range in [Fe/H] from +0.1 to –2.58 and a range in surface gravity from –0.18 to 3.88. The errors on the more metal-poor stars become very significant below a [Fe/H] of –1.7 so we limit our analysis to above this metallicity. We also limit the surface gravity range to be between 3.7 and 0.8 to avoid grid edge effects, issues with non-spherical model atmospheres, and significant AGB star contamination. Our resulting sample of alpha-enhanced stars contains 5724 RC stars and 20,473 RGB stars with measured Fe/H, alpha/Fe, log g, and [C/N]. (see Table 1) We note that the median trends in [N/Fe] are below +0.8 dex even among the most metal-poor stars of our final sample, well contained within the [N/M] grid, which extends from –1.0 dex to +1.0 dex.

Table 1.  Sample Data^a

APOGEE Name	[Fe/H]	[A/M]	log g	[C/Fe]	[C/Fe]err	[N/Fe]	[N/Fe]err	Classification
2M00001653+5540107	−0.089	0.150	1.568	0.101	0.019	0.267	0.028	RGB
2M00004072+5714404	−0.248	0.180	0.966	0.375	0.021	0.697	0.030	RGB
2M00004819−1939595	−1.670	0.335	1.630	−0.518	0.094	0.529	0.152	RGB
2M00010088+1649201	−0.579	0.231	2.530	0.043	0.047	0.067	0.080	RGB
2M00010132+0031530	−0.180	0.172	2.935	0.072	0.044	0.253	0.080	RGB
2M00011390+6228585	−0.447	0.216	1.674	0.092	0.027	0.116	0.041	RGB
2M00011871+0011076	−0.616	0.298	2.821	0.105	0.069	−0.092	0.125	RGB
2M00012134+0106579	−0.164	0.185	3.171	0.054	0.038	0.165	0.067	RGB
2M00012224+1530157	−0.079	0.146	2.772	0.137	0.029	0.195	0.049	RGB
2M00012412+6427175	−0.260	0.202	1.023	0.181	0.017	0.281	0.024	RGB
2M00012523+0012037	−0.672	0.263	2.741	0.006	0.064	0.043	0.115	RGB
2M00012984+7052497	−0.238	0.190	2.359	0.157	0.042	0.129	0.076	RC

Note.

^aTable 1 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.

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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There are a few alpha-rich stars that are not low-mass (see, e.g., Chiappini et al. 2015 for massive migrated stars, and Jofré et al. 2016 for binary mergers). However, these higher-mass alpha-enhanced stars are fairly rare. In a DR12 APOGEE sample of RGB stars with asteroseismic masses, Martig et al. (2015) found that 6% of their alpha-rich sample had M > 1.2 M_⊙. The bottom panel of Figure 1 shows stars from the second APOKASC catalog (Pinsonneault et al. 2018). This catalog combines asteroseismology from Kepler light curves with accurate temperatures of APOGEE to determine the masses. There are 6041 stars in the lower panel of which 770 have alpha abundances greater than our 0.14 dex selection criterion. The mean mass of these 770 alpha-enhanced stars is 1.033 ± 0.007 (σ = 0.200) solar masses. There are 83 stars, 10.8% of the sample, with masses greater than 1.233 solar masses, one sigma deviation from the mean, which is greater than one might expect from a single-valued distribution with Gaussian errors. This is more support for the conclusions of Martig et al. (2015). The level of contamination suggested by Martig et al. is small enough that, with some care, it should not bias any conclusions on the bulk properties of normal low-mass, alpha-enhanced giant evolution. To separate out the RGB from the RC stars we use the spectroscopic evolutionary classification discussed above.

In Figure 2, we show the evolution of the [C/N] abundance ratio in our sample stars as a function of surface gravity, similar to Figure 10 of G00. No error bars are shown in this figure because of the high density of points but the sample was chosen to have no errors larger than 0.3 dex in [C/N] and the average error among the RGB sample is 0.09 (σ = 0.05). The RC sample has been plotted separately to the right of the RGB sample. A prominent feature among the RGB sample is the high density of points found near log g = 2.5 in the more metal-rich samples. We attribute this high density of points to the bump in the luminosity function. The log g of the bump in our sample along with the mean log g of the RC are listed in Table 2. To assist in the visual interpretation of Figure 2 we have added several colored lines that represent the median [C/N] value in a small range of log g as shown at the top of the figure. The blue region was selected to be below the first dredge-up region in luminosity. The magenta region was selected to be above the first dredge-up but below the bump in the luminosity function. The green line represents the median [C/N] value selected from the green region shown at the top of Figure 2 and was selected to show the [C/N] abundance after the bump and be clear of any RC contamination. The red line represents the median [C/N] value from the red region shown at the top of Figure 2 and was selected to be the most evolved, luminous RGB stars in our sample.

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To determine the surface gravity at which the first dredge-up occurs, we fit fit a hyperbolic tangent function as a function for each metallicity bin over the range 2.6 ≤ log g ≥ 3.6 dex. The mid-point of the hyperbolic tangent transition, the total [C/N] transition height, and the [C/N] ratio before the drop are reported in Table 2. For the lowest metallicity bins in our sample, we lack the high-gravity stars needed to accurately measure the location of the first dredge-up and so do not report it. The lower metallicity stars, [Fe/H] < −0.3, exhibit an additional drop just above the luminosity bump, usually referred to as extra mixing. We fit this extra mixing with a second hyperbolic tangent function using stars with surface gravities between 1.5 and 2.9 dex and report the results that show a significant change in Table 2. However, the most metal-poor stars, [Fe/H] < −0.9, exhibit further decline in the [C/N] ratios after the bump. Rather than a single rapid drop in [C/N], the evolution is better approximated by a linear change in [C/N] with gravity as the star ascends the giant branch. We fit this change using linear regression for stars at surface gravities between 0.8 and 1.6 and report its slope in Table 2.

To demonstrate more clearly the log g and metallicity dependence of the mixing, we show in Figures 3 and 4 fits to the [C/N] ratio as a function of log g and metallicity for stars in a number of metallicity and gravity bins. The fits are piecewise spline and power series polynomials, of varying order depending on the degree of structure. The coefficients of these fits are listed in Tables 3 and 4. We also show the dispersion about the fits and the errors on the points, on the top and bottom of each figure, for guidance. Because the ends of the fits are poorly constrained, we have removed the first and last 20 points to avoid providing a misleading impression of the substructure. The fits in Figure 3 are directly related to the panels in Figure 2; for example the solid purple line which starts at log g = 2.0 and drops steeply can be seen to come from the middle panel on the right side of Figure 2 which has very few data points above log g > +2.5 but does exhibit the sharp decline in [C/N] to lower log g values. Figure 4 is a different cut through the same data set and shows fits to the same data but at fixed log g bins. To give the reader a feel for the quality of the fits we provide an error of the mean for bins 0.2 dex wide in both figures. In Figure 3 the top red error bars are errors in the mean for the −0.1 < [Fe/H] < +0.1 sample while the bottom black error bars are errors in the mean for the −1.7 < [Fe/H] < −1.5 sample. In Figure 4 the error bars represent errors in the mean for the highest and lowest surface gravity fits in bins of 0.2 dex in metallicity. Generally the error in the mean is larger at high surface gravities and at lower metallicities.

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Our final sample has four stars in common with G00. This small number of stars in common combined with a lack of errors from the G00 sample make a direct star-by-star comparison nearly useless. However, the G00 sample does have eight main-sequence stars, 11 RGB, and three RC stars in the metallicity range −1.95 < = [Fe/H] < = −0.92; see the red points in Figure 2. The G00 RGB stars' CN values are consistent with our RGB CN values. The G00 RC stars' CN values are systematically lower than our RC CN values and are much more in-line with our upper RGB median CN values.

4.1. Mapping Extra Mixing

4.2. Galactic Chemical Evolution of Carbon and Nitrogen

4.3. The RC

4.4. Consequences for Ages from [C/N]

4.5. Consequences for Modeling First Dredge-up in Standard Models

To investigate the magnitude of extra mixing relative to standard expectations, we computed 1D evolutionary models of the first ascent RGB. We use the Yale Rotating Evolution Code, adopting the fiducial physical inputs and parameters from the models of Tayar et al. (2017). We computed models of 0.9 M_⊙, consistent with the mean value of our APOGEE sample, and at three [Fe/H] values: 0.0, −0.6, and −1.2. These alpha-enhanced models have initial [C/N] abundances of 0.0, +0.2, and +0.4, respectively, to be consistent with the Galactic chemical evolution trends we may be seeing in our data as discussed in 4.2.

We compare these models to our data in Figure 5. In the left panel, the gray lines show the [Fe/H] versus [C/N] fits from Figure 4—the labels indicate the mid-range log g values of each fitted sample—and the colored lines show the results of our evolutionary models at several different log g intervals. In the right panel, the solid lines are the results of the model predictions for the surface [C/N] abundance ratios for the three [Fe/H] values. The dashed line and dots represent the APOGEE [C/N] values in 0.2 dex bins around these same metallicities, comparable to the fitted data in Figure 3.

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4.5.1. Agreement with Standard Models

At solar metallicity, the models predict a dredge-up Δ[C/N] of ∼−0.1 dex beginning around $\mathrm{log}{\text{}}g=3.6$ and no additional depletion below log g = 3.0, in reasonable agreement with the APOGEE data. At [Fe/H] = −0.6, the modeled first dredge-up begins at a slightly lower log g and results in a drop in the [C/N] ratio of ∼−0.2 dex. The APOGEE data at this metallicity show a drop of approximately the same magnitude at the predicted log g, although further drops in the [C/N] ratio are seen in the APOGEE data and discussed in the next section. For the lowest [Fe/H] model the onset of the first dredge-up, at log g ∼ 3.2, is consistent with the APOGEE data but beyond this single point of agreement the model is a poor match for the data.

4.5.2. Deviations from Standard Models

Beyond the onset of first dredge-up, the standard models increasingly deviate from the observations, over-predicting the ultimate [C/N] value by 0.1 dex at [Fe/H] = −0.6, and by ∼0.7 dex at [Fe/H] = −1.2. In the right panel of Figure 5 it is notable that the two metal-poor models make reasonable predictions until log g ∼ 2.25, around the RGB bump, but show no additional depletion thereafter while the APOGEE [C/N] ratio in the metal-poor stars continues to decrease.

In summary, the standard models appear to predict the [Fe/H]-dependent log g value when dredge-up begins and roughly predicts the magnitude of the [C/N] drop down to log g ∼ 2.25. The standard models do not anticipate any additional mixing after log g ∼ 2.25 seen among more metal-poor stars, [Fe/H] < −0.4.

4.5.3. Beyond the Standard Model

As an additional exercise, we considered the impact that arbitrarily deep additional mixing in our models would have on the surface abundance. We found that nuclear burning in the central regions produces enough nitrogen that the observed mixing signals could be achieved if stars were fully mixed to a few pressure scale-heights below the hydrogen-burning shells. However, these calculations do not include additional [C/N] processing that would occur in envelope material mixed down into the hydrogen-burning shell, which could serve as a non-conservative source of additional [C/N] depletion. Finally, we suggest that the morphology of the continuous reduction in [C/N] at low metallicity is qualitatively consistent with more sophisticated models of diffusive mixing (e.g., Placco et al. 2014), but we postpone quantitative comparisons to specific prescriptions of extra mixing to future work.

4.6. Comparisons with Dwarf Galaxies

4.7. Implications for Later Stages of Stellar Evolution

We use the alpha-rich first ascent red giants from the APOGEE field sample to map the evolution of the [C/N] ratio as a function of gravity and metallicity in a restricted mass range. We find a discrepancy between the APOGEE [C/N] ratios for the RC stars in comparison to the tip of the giant branch [C/N] ratios. A comparison with the few RC [C/N] values in G00 suggests that the APOGEE RC C and N abundances may be in error for RC stars. The few RGB [C/N] values in G00 are in good agreement with the APOGEE [C/N] ratios so the discrepancy appears to be limited to the RC stars.

We find that the [C/N] ratio before the first dredge-up is metallicity dependent, with more metal-poor stars having higher initial [C/N] values, possibly as the result of Galactic chemical evolution. We show that standard stellar evolution models can qualitatively match the first dredge-up of the stars in our sample as a function of metallicity if an appropriate, non-solar, mixture is used.

We confirm the existence of extra mixing above the RGB bump in metal-poor stars, [Fe/H] < −0.4 dex, with the mid-point of this extra mixing being 0.14 dex above the bump in the luminosity function. We show that the amount of mixing is a smooth function of metallicity, with increasing mixing with decreasing metallicity up to Δ[C/N] = 0.58 dex at [Fe/H] = −1.4. This extra mixing is not present in standard stellar evolution models, and so these models fail to reproduce the measured [C/N] in low-gravity, metal-poor stars.

We find that either this extra mixing continues down to very low surface gravities at the metal-poor end of our sample or that a second source of extra mixing plays a roll in the upper giant branch in stars with [Fe/H] < −1.0. We provide tables of the measured [C/N] changes as well as fits to our sample of [C/N] measurements as a function of metallicity and gravity which can be compared to more sophisticated models including processes like thermohaline mixing in order to better constrain the mechanism responsible for this extra mixing.

Finally, because of this extra mixing, we caution that calibrations between [C/N] and age should not be used in the low-gravity (above the bump in the luminosity function), metal-poor ([Fe/H] < −0.4) regime unless extra mixing has been explicitly taken into account.

We thank John Beacom for the idea of using hyperbolic tangent functions to fit the drops in [C/N]. The hospitality of the Center for Cosmology and Astroparticle Physics at The Ohio State University during the paper-writing workshop was very much appreciated. We acknowledge support from NSF grant AST-1211853. G.S. acknowledges the support of the Vanderbilt Office of the Provost through the Vanderbilt Initiative in Data-intensive Astrophysics (VIDA) fellowship. Sz.M. has been supported by the Premium Postdoctoral Research Program of the Hungarian Academy of Sciences, and by the Hungarian NKFI Grants K-119517 of the Hungarian National Research, Development and Innovation Office. D.A.G.H., T.M., and O.Z. acknowledge support provided by the Spanish Ministry of Economy and Competitiveness (MINECO) under grant AYA-2017-88254-P. Support for this work was provided by NASA through the NASA Hubble Fellowship grant #51424 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. H.J. acknowledges support from the Crafoord Foundation and Stiftelsen Olle Engkvist Byggmästare. The authors thank Ohio State's Center for Cosmology and AstroParticle Physics for hosting an APOGEE workshop where substantial progress was made on this project. This research made use of Astropy,¹⁶ a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013; Price-Whelan et al. 2018).

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS website is www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatory of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.