APOGEE Data and Spectral Analysis from SDSS Data Release 16: Seven Years of Observations Including First Results from APOGEE-South

In DR14, ATLAS-9 atmospheric models (Kurucz 1979, and updates) were used to generate synthetic spectra, but a grid of cooler Model Atmospheres in Radiative and Convective Scheme (MARCS) models (Gustafsson et al. 2008) was used for T_eff < 3500 K; see Mészáros et al. (2012), Zamora et al. (2015), and Holtzman et al. (2018) for details. For DR16, we (B. Edvardsson) computed a new all-MARCS grid of atmospheric models, and these were used exclusively (apart from stars with T_eff > 8000 K, see Section 4.4). The main motivation for this change is to avoid the discontinuity between the two sub-grids seen in DR14-data (compare Figure 11); MARCS models are required to handle the lowest effective temperatures of our targets since the ATLAS grid has a lower limit of 3500 K. Additionally, a transition to MARCS models has made it possible to use spherical models for $\mathrm{log}\,g$ ≤ 3. Figure 2 shows the location of models in the T_eff − $\mathrm{log}\,g$-plane; the grid has finer spacing for 3000 K ≤ T_eff ≤ 4000 K where the model structure changes more with specified T_eff.

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For every grid point shown in the T_eff − $\mathrm{log}\,g$ plane, [M/H] is varied from −2.50 to +1.00 in steps of 0.25 dex (15 steps), [α/Fe] (which includes changes in O, Ne, Mg, Si, S, Ar, Ca, and Ti) is varied between −1.0 and +1.0 in steps of 0.25 dex (nine steps), and [C/Fe] is varied between −1.00 and +1.00 in steps of 0.25 dex (nine steps), meaning that every grid point shown in Figure 2 in fact represents 1215 model atmospheres. This adds up to 300,105 attempted calculated atmospheric models for the warmer grid, and 173,745 models in the cooler, finer spaced grid, and 442,260 models in total (there are some overlapping grid points in the two grids, see Figure 2). However, only 358,123 of these models converged, leading to 84,137 holes in our grid. The fraction of holes in the T_eff − $\mathrm{log}\,g$-plane is shown in the small numbers as well as in the color coding in Figure 2. In general, most of the holes are in regions of the grid where we do not expect many stars, for example with high T_eff and low $\mathrm{log}\,g$, and/or with elemental abundances near the grid edges in the abundance dimension in question. However, of particular interest is that of the models with $\mathrm{log}\,g$ = −0.5, only 1% of all models converged.

The holes in the model atmospheric grid obviously translate to holes in the grid of synthetic spectra, but as described in Section 4.3.1, these holes are filled before the analysis of data using radial basis function (RBF) interpolation (and extrapolation) in flux space.

In DR14 we used the atomic and molecular line lists described in Shetrone et al. (2015; this set of lists is internally labeled as 20150714 based on the date of adoption, in the format YYYYMMDD). In short, these line lists were based on a thorough, up-to-date literature review and evaluation by comparing to observed high-resolution spectra of standard stars (Smith et al. 2013). For the atomic lines, the transition probabilities were adjusted within the quoted uncertainties to match the spectra of the Sun and Arcturus (Livingston & Wallace 1991; Hinkle et al. 1995). For DR16, we decided to launch another literature review to find possibly newer, more accurate line data. This has led to the addition of lines and/or updates of atomic data for almost all atomic species compared to the DR14 line list, and also several updates regarding molecular transitions. Most notably, our line lists now include transitions from Ce ii (Cunha et al. 2017), more transitions from Nd ii (Hasselquist et al. 2016), and the FeH molecule (Hargreaves et al. 2010). The line list and its creation is thoroughly described in V. Smith et al. (2020, in preparation).

For very limited parts of the spectrum, we were not able to fit the Sun and/or Arcturus well in this process. Reasons for this could be missing transitions in our line list, and/or too small uncertainties cited in the atomic data reference, which limited our code from adjusting the transition probability. These regions have been masked out in subsequent analysis, and therefore have not affected our results.

The resulting DR16 line list is internally labeled 20180901.

As in DR14, the synthetic spectra for the main spectral grids were made using Turbospectrum (Alvarez & Plez 1998; Plez 2012). Plane parallel and spherical radiative transfer was used, consistent with the model atmosphere in question.

To ensure regular dimensions in the grid of synthetic spectra (same range in $\mathrm{log}\,g$ for all values of T_eff) and to enable the entire grid to be loaded in memory during the running of FERRE, the grid of synthetic spectra has, as in previous data releases, been divided into sub-grids in ASPCAP (Zamora et al. 2015). The division is somewhat different in DR16 compared to the previous data release and is shown in Figure 2: the solid green and red lines mark what we label the GK- and M-giant grids, respectively, while the dashed blue, green, and red lines indicate the F-, GK-, and M-dwarf grids, respectively.

In the calculation of synthetic spectra, we change some of the dimensionality compared to the dimensions of the grid of the atmospheric models, and also in several instances compared to the grids used for DR14:

• 1.  
  We do not use the models with [α/Fe] = −1.00, limiting the grid of synthetic spectra to eight steps in [α/Fe] between −0.75 and +1.00.
• 2.  
  We add a microturbulent velocity dimension having values of 0.3, 0.6, 1.2, 2.4, and 4.8 km s⁻¹ (five steps). In the calculated model atmospheres, a value of 1.0 km s⁻¹ is used for models with $\mathrm{log}\,g$ > 3 and a value of 2.0 km s⁻¹ for models with $\mathrm{log}\,g$ ≤ 3.
• 3.  
  In the giant sub-grids, we add grid points with [C/Fe] = −1.25 and [C/Fe] = −1.50 using the otherwise appropriate atmospheric model with [C/Fe] = −1.00, for a total of 11 steps in [C/Fe] between −1.50 and +1.00.
• 4.  
  In the dwarf sub-grids, we do not use all of the available models in the [C/Fe] dimension, restricting [C/Fe] from −0.50 to +0.50 in steps of 0.25 (five steps).
• 5.  
  In the giant sub-grids, we add an [N/Fe]-dimension from −0.50 to +2.00 in steps of 0.50 (six steps), while we go from −0.50 to +1.50 in steps of 0.50 (five steps) in the dwarf sub-grids, using the otherwise appropriate atmospheric model. The nitrogen abundance is not expected to affect the model atmosphere structure, so the N abundance was varied in the synthesis only.
• 6.  
  In the dwarf sub-grids, we add a projected rotational velocity ($v\,\sin \,i$) dimension with values of 1.5, 3.0, 6.0, 12.0, 24.0, 48.0, and 96.0 km s⁻¹ (seven steps), using the rotational line broadening from Gray (2005) using a linear limb-darkening coefficient appropriate for the near-IR,  = 0.25.
• 7.  
  In the giant sub-grids, where there is no rotational broadening, we adopt a macroturbulent velocity broadening with the same prescription as that used for DR14: v_mac = 10^{(0.471–0.254[M/H])}.

The final dimensionality of the different sub-grids is listed in Table 2. For the dwarf sub-grids, a solar value of ¹²C/¹³C = 89.9 (Lodders 2003) is used when calculating the synthetic spectra, while for the giant grids, a carbon isotopic ratio such that ¹²C/¹³C = 15 has been adopted. The single value of ¹²C/¹³C = 15 represents a typical isotopic ratio in red giants within a mass range of M ∼ 1–2M_⊙ spanning a moderate range of metallicities, from [Fe/H] ∼ −1.0 to +0.3. Lagarde et al. (2019) present a set of stellar models to probe red giant mixing and compare theoretical values of ¹²C/¹³C with observations from a number of studies of open and globular clusters; the Lagarde et al. (2019) models include additional mixing mechanisms from both stellar rotation and thermohaline mixing. The observed values of ¹²C/¹³C from the various globular and open clusters, which have red giant masses ranging from M ∼ 0.9 to 2.5M_⊙, are between ∼5 and 25, with 15 being a representative value (see Lagarde et al. 2019, their Figure 12 for a summary of the range of ¹²C/¹³C as a function of red giant mass for both the observations of cluster and field red giants, along with predictions from their stellar models).

Table 2.  Dimensionality and Parameter Ranges of the Final Sub-grids of Synthetic Spectra

	GK Giant	M Giant	F Dwarf	GK Dwarf	M Dwarf
Teff	3500 ⋯ 6000 (250, 11)	3000 ⋯ 4000 (100, 11)	5500 ⋯ 8000 (250, 11)	3500 ⋯ 6000 (250, 11)	3000 ⋯ 4000 (100, 11)
$\mathrm{log}\,g$	+0.0 ⋯ +4.5 (0.5, 10)	−0.5 ⋯ +3.0 (0.5, 8)	+2.5 ⋯ +5.5 (0.5, 7)	+2.5 ⋯ +5.5 (0.5, 7)	+2.5 ⋯ +5.5 (0.5, 7)
[M/H]	−2.50 ⋯ +1.00 (0.25, 15)	−2.50 ⋯ +1.00 (0.25, 15)	−2.50 ⋯ +1.00 (0.25, 15)	−2.50 ⋯ +1.00 (0.25, 15)	−2.50 ⋯ +1.00 (0.25, 15)
[α/Fe]	−0.75 ⋯ 1.00 (0.25, 8)	−0.75 ⋯ +1.00 (0.25, 8)	−0.75 ⋯ 1.00 (0.25, 8)	−0.75 ⋯ 1.00 (0.25, 8)	−0.75 ⋯ 1.00 (0.25, 8)
[C/Fe]	−1.50 ⋯ +1.00 (0.25, 11)	−1.50 ⋯ +1.00 (0.25, 11)	−0.50 ⋯ +0.50 (0.25, 5)	−0.50 ⋯ +0.50 (0.25, 5)	−0.50 ⋯ +0.50 (0.25, 5)
[N/Fe]	−0.50 ⋯ +2.00 (0.50, 6)	−0.50 ⋯ +2.00 (0.50, 6)	−0.50 ⋯ +1.50 (0.50, 5)	−0.50 ⋯ +1.50 (0.50, 5)	−0.50 ⋯ +1.50 (0.50, 5)
vmic	0.3, 0.6, 1.2, 2.4, 4.8 (5)	0.3, 0.6, 1.2, 2.4, 4.8 (5)	0.3, 0.6, 1.2, 2.4, 4.8 (5)	0.3, 0.6, 1.2, 2.4, 4.8 (5)	0.3, 0.6, 1.2, 2.4, 4.8 (5)
$v\,\sin \,i$	1.5 (1)		1.5, 3.0, 6.0, 12.0, 24.0, 48.0, 96.0 (7)
N	4356000	3484800	8085000	8085000	8085000

Note. The step size and number of steps are shown in the parentheses.

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For DR14, four differently smoothed grids were created to roughly match the different LSFs of the different fibers in the APO instrument. For the DR16 grids, we have made a corresponding characterization of the LSFs for the LCO instrument, so each sub-grid has eight different versions; the appropriate one is used when analyzing a particular spectrum taken with a given instrument and mean fiber. This issue and procedure are described in more depth in Holtzman et al. (2018). While the use of four different LSFs for each instrument significantly reduces the dependence of parameters on fiber number, some low-level dependence may still remain; see, e.g., Ness et al. (2018).

4.3.1. Filling of "Holes"

One of the difficulties of computing model atmospheres is the possible lack of convergence of their iteration algorithm. This issue affects both ATLAS and MARCS atmospheres (Mészáros et al. 2012) and is usually solved by interpolating in the atmospheric structure space. However, it may be more accurate to interpolate in the flux space of the synthetic spectra (Mészáros & Allende Prieto 2013).

In DR14, the holes in the grid of synthetic spectra were filled by spectral syntheses using the "closest" neighboring model atmosphere according to a metric specified in Holtzman et al. (2018). This can be extremely inaccurate if the number of holes is significant. For DR16, we instead implemented RBF interpolation to fill the missing synthetic spectra in the grids.

The RBF is a real-valued function whose value depends only on the distance from the known points, and works in any number of D dimensions (D ≥ 1) (Buhmann 2003). The interpolated value is represented as a sum of N RBFs (where N is the number of known points). These functions are strictly positive definite functions, and the most widely used definitions are Gaussian, multi-quadric, polyharmonic spline, or thin plate spline. We chose the multi-quadric form defined below, as it is the most versatile when used with sparse data sets like ours while still achieving the necessary accuracy. Each RBF function is associated with a different known point x_i, weighted by an appropriate coefficient w_i, and scaled by the parameter r₀:

$\begin{eqnarray*}&&y(x)=\displaystyle \sum _{i=1}^{N}{w}_{i}\cdot (| | x-{x}_{i}| {| }^{2}+{{r}_{0}}^{2}{)}^{0.5}\end{eqnarray*}$

The known points in our case are the synthetic spectra calculated with effective temperature, metallicity, surface gravity, etc., of the converged model atmospheres. Determining the w_i weights can be accomplished by solving a system of N linear equations, but round-off errors grow large, and the required computation time becomes impossibly long for high values of N, since the computation complexity scales as O(N³).

Therefore, many iterative methods have been developed to reduce the required computation time. One such method is a Krylov subspace algorithm developed by Faul et al. (2005) for multi-quadric interpolation in multiple dimensions, which scales as O(N²), a significant improvement compared to direct methods. We implemented this algorithm based on a previous implementation in Matlab available from Gumerov & Duraiswami (2007) who also further optimized Faul et al.'s algorithm by reducing its complexity to O(N*logN). Faul et al.'s algorithm includes two main steps:

• 1.  
  A precondition phase that depends only on the distances between the known points and a parameter, q, which is the power of the Lagrange functions of the interpolation, and
• 2.  
  An iteration phase that provides the desired weights for the interpolation.

In the preconditioning phase, Faul et al. (2005) carefully select a set of q points for each known point to construct the preconditioner. This preconditioner is used to build a set of directions in the Krylov space for the iteration phase. Larger q values will result in fewer iterations (of the order of ∼10 depending on the particular problem), but calculating the preconditioner takes significantly longer. In general cases, when q ≪ N, a good compromise is to have q around 30–50 to limit the computation time of the preconditioner.

In APOGEE's case, we need a different approach. While the spectra depend on the atmospheric parameters, in a single spectrum, the flux only depends on the wavelength, so we do not need to compute the preconditioning phase for every single wavelength. This allows us to save significant computation time by using the same preconditioning for every frequency by selecting q = N. While this increases the complexity of the preconditioning phase, the overall time to determine the weights for all spectra is reduced significantly, because the choice of q = N makes the algorithm converge in only two or three iterations.

A given APOGEE spectral sub-grid contains of the order of 1,000,000 spectra (see Table 2). While the Gumerov & Duraiswami (2007) algorithm can handle such a large number of points in a reasonable time, our internal testing showed that the accuracy of how well we can recover missing models degrades significantly when N > 2000. It is our goal to be able to recover spectra with 0.01–0.02 or better in normalized flux, an accuracy that is possible to achieve only if we can select four or five known points in each dimension. For this reason, we chose to implement Faul et al.'s method for simplicity and for the fact that it is faster than the Gumerov & Duraiswami (2007) approach when N < 2000–3000.

To fill each hole, we use a small grid of models around the hole, where the size of this grid depends on the location in parameter space, but generally has three to five points in each dimension. We determine the RBF coefficients for this grid from the filled points and use them to fill the missing point. The shape of the RBF is controlled through the r₀ scale factor, which is recommended to be greater than the minimal distance between points, and significantly less than the maximal distance. It is important to note that no established method exists for determining the best scale factor in terms of accuracy. The best way to evaluate the uncertainties is to temporarily delete known spectra from the grid, recreate them with interpolation, and compare the interpolated spectra with the original ones. After extensive testing of this type, we found that r₀ = 1 provided the best accuracy for all grids, except the F-dwarf sub-grid where we chose r₀ = 0.5. An example of interpolation errors in one of these tests for three different values of r₀ is shown in Figure 3.

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The full grids of synthetic spectra are internally labeled "l33" (DR14 used "l31c"), and are available for download from the Science Archive Server.³⁵ These are available in a series of FITS files, as well as in the FERRE-format described in the code's manual (see also Allende Prieto et al. 2018).

For DR16, we added a sub-grid suitable for hot stars (T_eff > 8000 K), thereby analyzing the more featureless spectra of stars that mainly were targeted for removal of telluric lines in the spectra of main survey target stars. The model atmospheres used for this grid are ATLAS-9, the line list is the atomic DR13/14 line list (20150714), and the spectral synthesis code used is Synspec (Hubeny 1988; Hubeny & Lanz 2017). The final sub-grid only has four grid dimensions; 7000 K ≤ T_eff ≤ 20,000 K in steps of 500 K (27 steps), 3.0 ≤ $\mathrm{log}\,g$ ≤ 5 in steps of 0.5 dex (five steps), −2.5 ≤ [M/H] ≤ 1.0 in steps of 0.5 dex (eight steps), and a projected rotational velocity ($v\,\sin \,i$) dimension with values of 1.5, 3.0, 6.0, 12.0, 24.0, 48.0, 96.0 km s⁻¹ (seven steps).

The analysis of these spectra is extremely challenging; after all, these stars were targeted to show as few spectral features as possible, and often hydrogen lines are the only strong features. Still, at least providing an estimate of the basic stellar parameters for these stars might be useful for some science applications. However, it should be noted that these values are not fully evaluated and should be used with caution, preferably by users familiar with hot stars and their spectra.

Even after dividing the total number of synthetic spectra into sub-grids, these are still too large to hold in memory. Hence we have, as in previous APOGEE data releases, used PCA to reduce the dimensionality of the sub-grids. Previously, this was done by splitting the APOGEE spectra into 30 pieces and using 30 PCA components for every piece, giving 900 PCA components, which provides almost a factor of 10 reduction in grid size. Tests on synthetic data, comparing the reconstructed spectra to originally calculated spectra, have shown that better accuracy is achieved with the same total number of PCA parameters, but by dividing the spectra into 12 pieces and using 75 PCA components for each piece, so this was implemented for the DR16 grids. Interpolation is done in the PCA coefficients, and the resulting values are multiplied by the PCA basis functions to create an interpolated spectrum.

In DR14, we did an initial coarse characterization of all stellar spectra to decide which synthetic spectra sub-grid(s) to use when performing the stellar parameter determination. This coarse characterization was made by passing all stars through reduced-size F-dwarf, GK-giant, and M-giant grids with [C/M] = [N/M] = 0. Based on the outcome of these runs, the spectrum was finally analyzed using the sub-grid that yielded the best fit, or the two adjacent sub-grids for cases when the best fit was near a grid edge. After the proper sub-grid(s) to be used was determined, FERRE was run with 12 different starting positions (to avoid being trapped within local minima) distributed evenly in T_eff, $\mathrm{log}\,g$, and [M/H] in the chosen sub-grid(s), and the final stellar parameters were set to the best fitting of these 12 (or 2 × 12 in the case of border line cases) runs. A more thorough description of this process can be found in Holtzman et al. (2018).

In DR16, we instead created one, large "coarse" grid with dimensions 3000 K ≤ T_eff ≤ 8000 K (11 steps of 500 K), 0 ≤ $\mathrm{log}\,g$ ≤ 5 (six steps of 1 dex), −2.5 ≤ [M/H] ≤ 1.0 (eight steps of 0.5 dex), −0.5 ≤ [α/M] ≤ 1.0 (four steps of 0.5 dex), −0.5 ≤ [C/M] ≤ 0.5 (five steps of 0.25 dex), −0.5 ≤ [N/M] ≤ 1.0 (four steps of 0.5 dex), five steps of v_mic; 0.3, 0.6, 1.2, 2.4, and 4.8 km s⁻¹, and seven steps of $v\,\sin \,i$; 1.5, 3.0, 6.0, 12.0, 24.0, 48.0, and 96.0 km s⁻¹, which was used to decide which "fine" sub-grid(s) to use when analyzing the spectrum. Furthermore, the derived values of the stellar parameters from the "coarse" run were adopted as starting values when executing the second "fine" run with FERRE. This means that in the new scheme, we run FERRE significantly fewer times for every star (one coarse and one or two fine), as compared to DR14 (three coarse and 12 or 24 fine). This led to a reduction in analysis time, something that is sorely needed as the data set increases for every release (see Table 1). However, in addition, we changed the choice of minimizing algorithm in FERRE from the default Nelder–Mead algorithm (Nelder & Mead 1965), identified in the code with the option ALGOR = 1, to the Unconstrained Optimization BY Quadratic Approximation (UOBYQA; Powell 2002), ALGOR = 3 in FERRE, and this led to a compensating increase in analysis time. Both algorithms perform numerical optimization without the need for the explicit evaluation of derivatives, but while Nelder–Mead indicates a prescription for the motion of the vertices of a simplex in the search space that on convergence contains the minimum of the objective function (the χ² in our case), UOBYQA builds quadratic models for minimizing the objective function within trust regions. These changes were motivated by tests analyzing synthetic spectra—that then of course have "known" stellar parameters—which showed that the new scheme produces more accurate results.

For DR16, a revised scheme was used to normalize the spectra. First, the reduction process was improved to provide spectra with smoother variations and fewer "wiggles" (see Section 3), helping the normalization of the observed spectra when comparing to the synthetic spectra. In addition, the observed spectra have been slightly continuum-adjusted for the final analysis, based on the fit from the "coarse" fit of stellar parameters. The ratio of the observed spectra to the best-fit "coarse" model spectrum was smoothed with a broad median filter (with a width of 750 pixels), and the observed spectrum was divided by the smoothed residual before being passed to the "fine" run. Manual inspection of spectra and their final, "fine" stellar parameter fits has shown this scheme to greatly improve the continuum fits, and perhaps more importantly, to homogenize the APOGEE-N and APOGEE-S data and decrease the spread in derived stellar parameters/abundances for stars observed with both APOGEE instruments. Finally, both these corrected observed spectra and the synthetic spectra are normalized with a fourth-order polynomial in the wavelength region covered by each of the three APOGEE detectors.

For DR16, this final continuum normalization is now made inside FERRE, allowing for rejection of the same pixels (e.g., those contaminated by night sky emission) in the observed and synthetic spectra, based on pixels flagged in the observed spectrum. In previous data releases, the continuum fit of the observed spectrum was made ignoring flagged pixels, while the continuum normalization of the synthetic spectra used all pixels, leading to possible inconsistencies for some spectra.

After the stellar parameters (and "abundance parameters") have been determined, these are held fixed for additional runs of FERRE to determine the elemental abundances. For these, only windows in the spectra that are sensitive to the element in question are used, and only the most relevant abundance dimension of the grid is varied; [M/H] (for Na, Al, P, K, V, Cr, Mn, Fe, Co, Ni, Cu, Ge, Rb, Ce, Nd, and Yb), [α/M] (for O, Mg, Si, S, Ca, Ti, and Ti ii), [C/M] (for C and C i), or [N/M] (for N). The windows are chosen based on where our synthetic spectra are sensitive to a given element, and at the same time not sensitive to another element in the same abundance group. Based on this, different weights are assigned to pixels in different abundance windows, just as in DR14.

In DR16, however, we performed some test analyses using one window at a time for a subset of spectra for the elements with less than 10 windows, with the aim of weeding out windows that produced deviant results for one reason or another, possibly caused by bad/missing atomic data in the window, unrecognized blends, or 3D/non-LTE (NLTE) effects. These analyses were run on a validation sample, which consists of spectra with high S/Ns, and including stars from across the HR-diagram, stars in the Kepler field, stars with independently determined stellar parameters and abundances, etc.

Based on manual inspection of the derived "window-abundances" as compared to each other, and to expected astrophysical trends in the solar neighborhood, and as a function of T_eff in open clusters, some of the windows used in DR14 were removed for Al, P, S, Ti, V, Cr, Mn, Co, and Yb. The windows and their weights used for DR16 are provided in Table 3.

For the elemental abundance determination for DR16, we have used the new TIE option in FERRE for elements that were fit using the [M/H] dimension of the grid. Using this dimension, abundances of all elements are varied together during the fit. The TIE option allows the [α/M], [C/M], and [N/M] dimensions to be varied oppositely in lockstep, such that the abundances of C, N, and the α elements are not varied as the best-fitting abundance from the [M/H] variation is determined.

We updated FERRE from version 4.7.1 to the latest version at the time of production, 4.8.5. The updates to the code between these releases are rather minor, but include the important TIE option.

The data were all processed on the SDSS cluster at the University of Utah, which is comprised of 27 nodes with 16 cores each. For processing with FERRE, two jobs are run on each node at once to accommodate the significant memory usage required to load a single sub-grid, but the multiprocessing option in FERRE is used to run 16 threads simultaneously for each job. The total processing time is approximately 8–10 hr per field for fields with a single cohort of ∼160 stars.

The RVs are provided in the VHELIO_AVG entry in the allStar file. As in DR14, these velocities are given in the solar system barycentric frame, not the heliocentric frame as the name suggests; the naming convention has been maintained from earlier releases for historical reasons. For stars that have been observed with multiple visits, the scatter of the individually derived RVs is provided in VSCATTER. This can be used, for example, to filter out possible binary systems.

As in previous data releases, and as described in previous sections in this paper, the ASPCAP stellar parameters include the "classic" spectroscopic stellar parameters T_eff, $\mathrm{log}\,g$, [M/H], v_mic, and $v\,\sin \,i$ (for dwarfs; a prescribed v_mac in the case of giants) as well as some initial estimate of abundances; [α/M], [C/M], and [N/M]. The "abundance parameters" are needed for several reasons; for many of our cool, metal-rich targets, CNO-bearing molecular lines cover more or less the entire APOGEE spectral region, and a correct modeling of these is required to fit the classical stellar parameters. Furthermore, since the α-elements are important electron donors, modeling these correctly as the stellar parameters are determined is necessary, and, additionally, some of our targets have carbon abundances far enough from solar that the atmospheric structure is altered. These "abundance parameters" are determined from a global fit of the entire spectrum simultaneously with the other stellar parameters.

In the second stellar abundance measurement stage, these abundances are re-determined using windows in the spectra covering only spectral lines sensitive to the abundance in question (see Section 5.3). For that reason, we recommend the use of these "windowed" abundances in most cases, but even so, the "abundance parameters" are stored and can be found in the FPARAM array as well as in the ALPHA_M tag.

As in previous data releases, some of the spectroscopically determined stellar parameters have been calibrated to match other, independent measurement of the parameters. These calibrations have varied over the data releases, and we include below a description of what has been done in DR16.

The spectroscopic and calibrated abundance parameters are provided in the FPARAM and PARAM arrays; there are nine entries in these arrays for each star, corresponding to T_eff, $\mathrm{log}\,g$, $\mathrm{log}({v}_{\mathrm{mic}})$, [C/M], [N/M], [α/M], log($v\,\sin \,i$), and O (currently unused). Many of these are also split out into appropriately named tags in the allStar file, as described below.

5.2.1. Effective Temperature, T_eff

The spectroscopic T_eff for all stars has been calibrated to the photometric scale of González Hernández & Bonifacio (2009, hereafter GHB) using linear relations as a function of metallicity and effective temperature:

$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{eff},\mathrm{cal}} & = & {T}_{\mathrm{eff}}+610.81-4.275\cdot [{\rm{M}}/{\rm{H}}]^{\prime} \\ & & -\ 0.116\cdot {T}_{\mathrm{eff}}^{{\prime} }\end{array}\end{eqnarray} \tag{ 1 }$

where [M/H] and T_eff are the uncalibrated values of [M/H] and T_eff, and the "primed" values are clipped to lie in the range −2.5 < [M/H]' < 0.75 and $4500\lt {T}_{\mathrm{eff}}^{{\prime} }\lt 7000$. The clipping is applied since the bulk of the stars in GHB fall within these limits, so we prefer not to extrapolate; outside of these ranges, the offsets from the end of the valid range were applied. Figure 4 shows the data from which this relation was derived.

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The spectroscopically determined T_eff is given in a new TEFF_SPEC tag while the calibrated T_eff, as in previous data releases, can be found in the TEFF tag.

5.2.2. Surface Gravity, $\mathrm{log}\,g$

As in DR14, the spectroscopic $\mathrm{log}\,g$ for giant stars has been calibrated using relations determined from stars in the Kepler field for which asteroseismic surface gravities are available (Pinsonneault et al. 2018). As with previous data releases, we find that the relationship between the spectroscopic and asteroseismic values is complex; in particular, we find different offsets for red clump and red giant stars that occur in similar locations in a T_eff − $\mathrm{log}\,g$ diagram.

New for DR16 is that we also provide calibrated surface gravities for dwarfs, for which we use a combination of techniques: for warmer dwarfs, we have asteroseismic values that we use, while for cooler dwarfs, we derive an approximate calibration using isochrones.

The classification of stars into these different "calibration-categories" was done according to the following criteria:

• 1.  
  All stars with uncalibrated $\mathrm{log}\,g$ > 4 or T_eff > 6000 K are considered dwarf stars.
• 2.  
  Stars with uncalibrated 2.38 < $\mathrm{log}\,g$ < 3.5 and

  $\begin{eqnarray*}&&[{\rm{C}}/{\rm{N}}]\gt 0.04\mbox{--}0.46\cdot [{\rm{M}}/{\rm{H}}]-0.0028\cdot {dT}\end{eqnarray*}$

  are considered red clump stars. Here dT is defined as

  $\begin{eqnarray*}\begin{array}{rcl}{dT} & = & {T}_{\mathrm{eff},\mathrm{spec}}-(4400+552.6\cdot (\mathrm{log}\,{g}_{\mathrm{spec}}-2.5)\\ & & -324.6\cdot [{\rm{M}}/{\rm{H}}].\end{array}\end{eqnarray*}$
• 3.  
  RGB stars are defined as the stars with uncalibrated log g < 3.5 and T_eff < 6000 K that do not fall in the red clump category, as defined above.
• 4.  
  For stars with uncalibrated 3.5 < $\mathrm{log}\,g$ < 4.0 and T_eff < 6000 K, a correction is determined using both the RGB and dwarf calibrations, and a weighted correction is adopted based on $\mathrm{log}\,g$.

These classifications are shown in Figure 5 in the T_eff–$\mathrm{log}\,g$ plane, although this does not show the dependence of the RC/RGB-classification on [M/H] and [C/N].

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The calibration relations for dwarf, RC, and RGB stars are, respectively,

Dwarf stars:

$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}\,{g}_{\mathrm{cal}} & = & \mathrm{log}\,g-(-0.947+1.886\cdot {10}^{-4}\cdot {T}_{\mathrm{eff},\mathrm{spec}}\\ & & +\ 0.410\cdot [{\rm{M}}/{\rm{H}}]).\end{array}\end{eqnarray} \tag{ 2 }$

Red clump stars:

$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}\,{g}_{\mathrm{cal}} & = & \mathrm{log}\,g-(-4.532+3.222\cdot \mathrm{log}\,g\\ & & -\ 0.528\cdot {(\mathrm{log}g)}^{2}).\end{array}\end{eqnarray} \tag{ 3 }$

Red giant stars:

$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}\,{g}_{\mathrm{cal}} & = & \mathrm{log}\,g-(-0.441+0.7588\cdot \mathrm{log}\,g^{\prime} \\ & & -\ 0.2667\cdot {(\mathrm{log}g^{\prime} )}^{2}+0.02819\cdot {(\mathrm{log}g^{\prime} )}^{3}\\ & & +\ 0.1346\cdot [{\rm{M}}/{\rm{H}}]^{\prime} )\end{array}\end{eqnarray} \tag{ 4 }$

where

$\begin{eqnarray*}\begin{array}{rcl}\mathrm{log}\,g^{\prime} & = & \mathrm{log}\,g\ \mathrm{for}\ \mathrm{log}\,g\geqslant 1.2795\\ \mathrm{log}\,g^{\prime} & = & 1.2795\ \mathrm{for}\ \mathrm{log}\,g\lt 1.2795\\ \left[{\rm{M}}/{\rm{H}}\right]^{\prime} & = & [M/H]\ \mathrm{for}\ [{\rm{M}}/{\rm{H}}]\leqslant 0.5\\ \left[{\rm{M}}/{\rm{H}}\right]^{\prime} & = & 0.5\ \mathrm{for}\ [{\rm{M}}/{\rm{H}}]\gt 0.5\end{array}\end{eqnarray*}$

where the fixed value of $\mathrm{log}\,g^{\prime}$ at low surface gravity and [M/H]' at high metallicity avoids extrapolation into a region where there are few calibrators.

The functional forms for these calibrations were determined from inspection of the relations between spectroscopic, asteroseismic, and isochrone surface gravities. While these capture a significant portion of the relationships, small trends with other parameters may certainly exist, and the calibrated surface gravities cannot be assumed to be more accurate than ∼0.05 dex.

We note that no smooth transition is implemented between the RGB and RC calibrations, resulting in a small discontinuity in $\mathrm{log}\,g$ at the transition value. Based on the asteroseimsic results, we find that 93% of the RGB stars and 96% of the RC stars are classified correctly by our procedure. For the incorrectly classified stars, the calibrated surface gravities will be systematically off. However, since we perform the abundance analysis using the uncalibrated parameters, the abundances are unaffected.

The spectroscopic $\mathrm{log}\,g$ is given in the LOGG_SPEC tag in the allStar file, while the calibrated $\mathrm{log}\,g$, as in previous data releases, can be found in the LOGG tag.

5.2.3. The Abundance Parameters: [M/H], [α/M], [C/M], and [N/M]

In DR16, the abundance determination of 26 species is attempted: C, C i, N, O, Na, Mg, Al, Si, P, S, K, Ca, Ti, Ti ii, V, Cr, Mn, Fe, Co, Ni, Cu, Ge, Rb, Ce, Nd, and Yb. Note that, as in previous data releases, the uncalibrated spectroscopic stellar parameters were used when determining the stellar abundances. The reason for this is that the spectroscopic parameters give the best general fit to the stellar spectrum, and thereby give the best description of possible blends when determining the abundances from the abundance windows.

All of the "raw" abundance measurements for all stars are presented in the FELEM array, in which the order of the array elements for each star is by atomic number, with entries as listed above. Note that, in this array, the abundances for different elements are given with respect to either the total metals or to hydrogen, depending on which grid dimension was used during the fit.

In previous data releases, a T_eff-dependent calibration was applied to each individual elemental abundance to remove apparent trends in the uncalibrated abundances, based on observations of star clusters. For DR16, no such calibration is applied because, with the modification to the abundance pipeline, the trends with effective temperature for most elements have reduced amplitude in the cluster sample as compared with previous data processing. That being said, inspection of the full data set suggests that some trends of abundances with stellar parameters can exist for some elements, such that users need to exercise caution when comparing abundances across different regions of stellar parameters space (see Section 6.5).

The only calibration applied to the DR16 abundances is a zero-point shift to force stars with solar [M/H] in the solar neighborhood to have a mean [X/M] = 0. This is done separately for giants and dwarfs, where "giants" in this case are defined as stars with $\mathrm{log}g\lt 2+({T}_{\mathrm{eff}}-3500)/650$ and $\mathrm{log}\,g$ < 4 and T_eff < 7000 K, and all others are defined as dwarfs. More specifically, we average the "raw" abundances of all stars within 0.5 kpc of the Sun, based on Gaia DR2 parallaxes (Gaia Collaboration et al. 2016, 2018; Lindegren et al. 2018), and with −0.05 < [M/H] < 0.05, and subtract this value from the "raw" [X/M] of all stars. The applied shifts are tabulated in Table 4 (compare Table 5 in Holtzman et al. 2018 for the shifts applied in DR13 and DR14); they are generally small (of the order of hundredths of a dex), but are substantial for a handful of elements such as Al, P, V, and Mn. Note that this calibration is a zero-point offset only. Formally, using bracket notation ([X/Fe]) suggests that the abundances are relative to those of the Sun; we did not choose this procedure because many of the lines/elements that we measure in cooler stars are very weak in the solar spectrum, so an APOGEE-based solar abundance measurement has significant uncertainties. Instead, we build upon many results reported in the literature that suggest that the mean [X/Fe] in solar neighborhood stars is close to solar at solar abundance (Reddy et al. 2003; Adibekyan et al. 2012; Bensby et al. 2014, among others). Small intrinsic spread in [X/Fe] at solar abundance as found by Bedell et al. (2018) will still be reflected in the calibrated abundances, as we only apply a single mean offset to all stars.

The calibrated abundances are provided in the X_H and X_M arrays in the allStar file, where the difference between these is just the value of M_H. For further discussion about the APOGEE abundance scale, see Section 5.3.2.

5.3.1. "Named" Abundance Tags, X_FE

5.3.2. The Abundance Scale

As in DR14, we find that the uncertainties for parameters and abundances returned by the fitting routine in FERRE are unrealistically low in most cases. As a result, we take an alternate approach to derive empirical uncertainties, and adopt for the final uncertainties the larger of the FERRE and empirical uncertainty estimates.

For T_eff and $\mathrm{log}\,g$, we estimate uncertainties from the scatter around the calibration relations, parameterized as a function of T_eff, [M/H], and S/N, which captures the main dependencies of the scatter. The form of the adopted uncertainty parameterization is:

$\begin{eqnarray}&&\mathrm{ln}\,\sigma =A+B\cdot {T}_{\mathrm{eff}}^{{\prime\prime} }+C\cdot {\rm{S}}/{\rm{N}}^{\prime\prime} +D\cdot [{\rm{M}}/{\rm{H}}]\end{eqnarray} \tag{ 5 }$

where ${T}_{\mathrm{eff}}^{{\prime\prime} }$ = TEFF_SPEC-4500 and S/N'' =  SNREV-100 for SNREV ≤ 200; otherwise it is capped at a value of 100. The coefficients are presented in Table 6. The final uncertainties are presented in the TEFF_ERR and LOGG_ERR tags in the allStar file.

Table 6.  Coefficients Describing the Supplied Uncertainties in Stellar Parameters, Compare Equation (5)

Parameter	A	B	C	D	eA
Teff	4.583	2.965 × 10−4	−2.177 × 10−3	−0.117	98
$\mathrm{log}\,g$ (dwarfs)	−2.327	−1.349 × 10−4	2.269 × 10−4	−0.306	0.10
$\mathrm{log}\,g$ (RC)	−3.444	9.584 × 10−4	−5.617 × 10−4	−0.181	0.03
$\mathrm{log}\,g$ (RGB)	−2.923	2.296 × 10−4	6.900 × 10−4	−0.277	0.05

Note. Due to the parameterization, e^A can be taken as a measure of a typical uncertainty for a star with T_eff = 4500 K, [M/H] = 0.0, and S/N = 100.

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For the uncertainties for the derived stellar abundances, we adopted a new scheme in DR16 using repeat observations of the same star. As mentioned in Section 2, there are a moderate number of stars that were observed in multiple overlapping fields with different field centers, and since the reduction and analysis pipeline is built on processing field-by-field, these stars are completely independently analyzed more than once by ASPCAP. The differences between the derived abundances from the different visits provide some information about the uncertainties. These repeat observations include stars covering a large region in stellar parameter space. To supplement the coverage in S/N, several individual visit spectra of cluster stars were processed using ASPCAP. The differences are larger when the S/N is lower and also in regions of parameter space where lines are generally weaker (lower [M/H] and higher T_eff). These "repeat abundances" and their deviations as a function of T_eff, [M/H], and S/N were used to estimate uncertainties for the entire sample of stars. Specifically, the differences between pairs of measurements were tabulated for all of the repeats, along with the mean T_eff, [M/H], and S/N (only pairs with S/N the same within 20% were considered). A fit was then performed to these differences (multiplied by $\sqrt{\pi }/2$ to provide an unbiased estimator of the standard deviation, σ) using the form:

$\begin{eqnarray}&&\mathrm{ln}\sigma =A+B\cdot {T}_{\mathrm{eff}}^{{\prime} }+C\cdot S/N^{\prime} +D\cdot [{\rm{M}}/{\rm{H}}]+E\cdot {({T}_{\mathrm{eff}}^{{\prime} })}^{2}\end{eqnarray} \tag{ 6 }$

where ${T}_{\mathrm{eff}}^{{\prime} }$ = TEFF_SPEC-4500 and S/N' = SNREV-100 for SNREV ≤ 200; otherwise it is capped at a value of 100. The functional form was chosen to provide a reasonable match to the observed distribution of measured differences, but it is still just an approximation. We adopt the same uncertainties for both X_H and X_M in the X_H_ERR and X_M_ERR arrays.

Figure 6 shows an example of the methodology for Mg. Each panel in the plot shows data for different bins in T_eff and [M/H], where individual points are the differences for different pairs of observation/measurements; note that these individual differences are noisy by definition. The curves show the function fits to the data as described by Equation (6).

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Figure 7 shows a graphical summary of the derived uncertainties for all elements as a function of T_eff and [M/H], for an S/N of 125. The coefficients for all elements are presented for dwarfs in Table 7 and giants in Table 8.

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It is important to note that the uncertainties estimated in this way depend only on T_eff, [M/H], and S/N. Individual stars may have larger uncertainties if pixels with important information have larger uncertainties, e.g., if they happen to fall near a sky line in all of the visit spectra for the star. However, this is not reflected in the tabulated uncertainties unless the uncertainty returned by the fitting routine is larger than that estimated by Equation (6).

The uncertainties are included in the summary files in the X_H_ERR and X_M_ERR arrays, and in the individual X_FE_ERR tags.

Several changes have been made to the summary allStar file since DR14:

• 1.  
  The uncalibrated, spectroscopically determined T_eff and $\mathrm{log}\,g$ are now presented in "named tags" called TEFF_SPEC and LOGG_SPEC.
• 2.  
  Data from Gaia DR2 (Gaia Collaboration et al. 2016, 2018) have been added as a service for the user, and are provided in the tagsGAIA_SOURCE_ID, GAIA_PARALLAX,GAIA_PARALLAX_ERROR, GAIA_PMRA,GAIA_PMRA_ERROR, GAIA_PMDEC,GAIA_PMDEC_ERROR,GAIA_PHOT_G_MEAN_MAG,GAIA_PHOT_BP_MEAN_MAG,GAIA_PHOT_RP_MEAN_MAG,GAIA_RADIAL_VELOCITY, andGAIA_RADIAL_VELOCITY_ERROR.To perform the cross-match between APOGEE and Gaia, the 2MASS cross-match provided by the Gaia collaboration was used when available; otherwise, a positional match, taking the brightest star within 3'', was made.
• 3.  
  Distances from Bailer-Jones et al. (2018) have been added as a service for the user, and are provided in the tags GAIA_R_EST, GAIA_R_LO, and GAIA_R_HI.
• 4.  
  New bits have been added in the PARAMFLAG for OTHER_WARN and ERR_WARN.
• 5.  
  New bits have been added to PARAMFLAG[1] that provide information about which $\mathrm{log}\,g$ calibration was applied: LOGG_CAL_RC (red clump, RC), LOGG_CAL_RGB (red giant branch, RGB), LOGG_CAL_MS (main sequence, MS), or LOGG_CAL_RGB_MS (RGB-MS transition).
• 6.  
  New bits have been added in APOGEE2_TARGET2 and APOGEE2_TARGET3 targeting flags.

In addition, a new allStarLite file is provided that eliminates some of the information in the allStar file, but contains much of the information likely to be of interest for most users. This file is about a third of the size of the full allStar file. It eliminates information about the individual visits that went into the combined frame, as well as several of the arrays that contain uncalibrated stellar parameters and abundances.

The full data model for the two allStar file versions is described (see footnote 30).

5.6.1. Incorrect IDs in Some Value Added Catalogs

Users of the data should pay attention to the bitmask flagging to sort out spectra/stars that are likely inaccurately analyzed by the pipeline. This flagging system is the same as in DR14, and is described in Holtzman et al. (2018, Tables 10–12) and in the online documentation.³⁹

While there are multiple flags for different quantities, two particularly useful ones are the EXTRATARG bitmask, which allows the user to select only the main survey sample (those objects with EXTRATARG == 0) and the ASPCAPFLAG STAR_BAD bit, which is a "collection"-bit that flags stars for which the derived quantities are unreliable, both from the data point of view, as well as from the determined stellar parameters point of view. For users of the spectra, the STARFLAG bit may be of utility to identify potentially problematic object spectra.

We note that there is a small bug in the DR16 summary allStar file ASPCAPFLAG bit related to stars falling near a grid edge. For such stars, the appropriate bit (GRIDEDGE_BAD or GRIDEDGE_WARN) is set in the relevant PARAMFLAG, but this is not propagated, as it should be, into triggering parameter bits in ASPCAPFLAG (e.g., TEFF_BAD or TEFF_WARN if a GRIDEDGE bit is set in the T_eff PARAMFLAG). However, if GRIDEDGE_BAD is set in any parameter, the STAR_BAD bit is correctly triggered in ASPCAPFLAG.

The precision of the RVs is a function of S/N, T_eff, and [M/H], with higher precision for brighter, cooler, and more metal-rich stars. This is demonstrated in Figure 8, which shows histograms of VSCATTER for stars with more than five visits. Judging from the peak of the histograms, the RV precision is of the order of 100 m s⁻¹ for the best-measured stars and is better than 500 m s⁻¹ for almost all stars with T_eff < 8000 K.

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To assess the accuracy of the RVs, we compare the APOGEE RVs to a set of high-quality literature values accurate to ∼30 m s⁻¹ (Nidever et al. 2002; Chubak et al. 2012) as well as to RVs from Gaia (Gaia Collaboration et al. 2016, 2018). We compare stars with spectra meeting the criteria APOGEE S/N > 30, APOGEE VSCATTER < 1 km s⁻¹, APOGEE VERR < 1 km s⁻¹, and Gaia RV error < 1 km s⁻¹. From Table 9, we find that the APOGEE DR16 RVs are offset from the values of Nidever et al. (2002) and Chubak et al. (2012) by ∼0.4 km s⁻¹, as was the case for the DR12 RVs (Nidever et al. 2015). The DR14 RVs showed a smaller offset of ∼0.2 km s⁻¹ to these two comparison studies (Holtzman et al. 2018).

However, the Gaia DR2 RVs (Cropper et al. 2018; Sartoretti et al. 2018; Katz et al. 2019) are in better agreement with the DR16 APOGEE values than the DR14 APOGEE values. We show the distributions of RV differences between DR16 and Gaia in the left three panels of Figure 9. The bright stars (H < 11) agree to 0.13 km s⁻¹ or better for both the northern and southern instruments. This offset is larger for the fainter stars (11 < H < 14), with the northern instrument RVs offset by −0.12 km s⁻¹ and the southern instrument RVs offset by −0.18 km s⁻¹. The standard deviations of the differences plotted in Figure 9 are driven by the individual Gaia RV uncertainties, which are typically 0.35 km s⁻¹, 0.62 km s⁻¹, and 0.84 km s⁻¹ from the bright bin to the faint bin, respectively.

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The differences DR14-DR16 are likely because of the updated wavelength calibration (see Section 3). Despite the better agreement with Gaia, we note that the RVs from RAVE DR5 were found to be offset from the Gaia RVs by a similar magnitude and direction as the APOGEE DR14 RVs (Deepak & Reddy 2018; Steinmetz et al. 2018), and this offset persists in RAVE DR6 (Steinmetz et al. 2020). Note that RV offsets at this level, if real, are unlikely to have a significant effect on stellar parameters and abundances because they are a small fraction of the instrumental resolution of APOGEE, which is Δv = c/R ∼ 13 km s⁻¹.

As another measure of precision and accuracy, we also compare the RVs derived for stars observed from both the northern and southern instruments. The results of this comparison are shown in the right panel of Figure 9 and in Table 9. We find a sample of 1204 stars observed from both hemispheres that meet the criteria in both observations of S/N > 30, VSCATTER < 1 km s⁻¹, and VERR < 1 km s⁻¹. From this sample, we find a RV offset of 0.06 km s⁻¹, such that the stars observed from the LCO 2.5 m have slightly lower RVs than the same stars observed using the APO 2.5 m.

For the faintest stars in DR16, the individual visit spectra can have low S/N, and, as a result, the RV determination can fail. In many, but not all, cases, such objects are flagged as having a bad or suspect RV combination. Users who are working with data for stars with H > 14.5 need to be very careful, as incorrect RVs leads to incorrect spectral combination, which invalidates any subsequent analysis. To minimize inadvertent usage of invalid data, the named abundance tags have not been populated for stars with H > 14.6.

Figure 10 shows the Kiel diagrams (T_eff versus $\mathrm{log}\,g$) of the main survey sample collected from the APO 2.5 m and LCO 2.5 m systems, plotted using the spectroscopic (top panels) and calibrated stellar parameters (bottom panels). Overall, as with previous APOGEE data releases, the general trends are consistent with expectations from isochrones.

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One of the most significant improvements of DR16 compared to DR14 is that of the accuracy and consistency of determined stellar parameters for the coolest giants (and in turn, abundances derived for these stars). Figure 11 shows the Kiel diagrams using the spectroscopic stellar parameters for these stars. The DR16 Kiel diagram looks much better than that from DR14 in several aspects. First, the sub-grid edge at T_eff = 3500 K seen in DR14 has disappeared. This change is due to DR16 using only MARCS models, while DR14 used ATLAS-9 models above T_eff = 3500 K, and MARCS below. Second, the unexpected spread of $\mathrm{log}\,g$ values for the most metal-rich giants with T_eff just above 3500 K in DR14 has disappeared in DR16. It is possible that this improvement can be attributed to the fact that the MARCS models are spherical, while the ATLAS-9 models are plane parallel. And finally—and perhaps most strikingly—the "clumpiness" of the Kiel diagram for giants with T_eff < 3500 K in DR14 is gone. This is likely due to the new treatment of holes in the grid, as discussed in Section 4.3.1.

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All in all, these improvements of the stellar parameters make us more confident in the analysis of the coolest giants in the DR16 data, something that should be welcomed by many users, especially because the coolest giants are, in many cases, the most distant stars. However, note that we do not have any "external" abundances of such cool stars to use for the estimation of accuracy, so users interested in the very coolest giants should still be cautious.

However, comparison of the uncalibrated and calibrated diagrams, along with consideration of the calibration relation, demonstrates several systematic issues with the spectroscopic measurements, which we attempt to compensate for with calibration:

• 1.  
  The spectroscopic surface gravities for cooler dwarfs are systematically low with respect to isochrones.
• 2.  
  The spectroscopic surface gravities for red giant stars are systematically high, and this difference is larger for red clump stars than for stars on the red giant branch.

While we have provided calibrated surface gravities that correct for these effects, we caution that abundances that are sensitive to surface gravity, in particular, could have systematic offsets, especially since we use the spectroscopic surface gravities in the abundance analysis, as described in Section 5.3. Such elements include, but might not be limited to, Ti ii and Ce ii.

As described in Section 4.6, we have taken great care in choosing which final sub-grid to use for analysis of a particular star, and flagged stars that, in the final analysis, end up close to a grid edge. Even so, there are some over-densities of stars corresponding to the locations that are just outside the flagged regions that are plausibly identified with interpolation issues or, possibly, with unusual spectra. Users should be cautious regarding stars with parameters close to grid edges (compare Figure 2).

Figure 12 presents the relations between [X/Fe] and [Fe/H] for main survey stars that do not have the STAR_BAD flag set, which yields 272,120 stars. Furthermore, in each panel, only stars with the relevant element flag and iron flag equal to zero (X_FE_FLAG == 0 and FE_H_FLAG == 0) are plotted. Since different stars have flags set for different elements, the individual panels in the plot have different numbers of stars. Note that no general cuts on stellar parameters nor position on the sky have been made for this plot, which therefore includes giant and dwarf stars of all effective temperatures and in all parts of the Galaxy.

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Numerous studies have presented diagrams like this, largely for stars in the solar neighborhood, so there are some expectations for what we might see in the APOGEE data. Based on those expectations, we highlight several areas of difference/concern:

• 1.  
  Most of the α elements show a plateau at lower metallicities, as expected, but the value of the plateau—which, e.g., in [O/Fe] and [Mg/Fe], is at about 0.3 dex—is lower than that found in, for example, the optical studies in Section 6.8.
• 2.  
  The [O/Fe] versus [Fe/H] panel shows a thin "finger" of stars sticking out of the high [O/Fe] versus [Fe/H] trend with [O/Fe] ∼ 0.25 and [Fe/H] > 0. The same peculiarity is seen in Ca and also in the [α/M] versus [M/H] parameter trend, but not as clearly in the [Si/Fe] versus [Fe/H] trend, and not at all in the [Mg/Fe] versus [Fe/H] trend. The same stars showing this behavior in some of the α elements have a similar feature in C, C i, and [C/M], but oddly enough, no discernible matching peculiarity in N. This feature was present in DR14 (Zasowski et al. 2019), and seems to affect a small fraction of cool giants. Our estimation is that about 4% of the "main" stellar sample of giants with supersolar metallicity and T_eff < 4000 K—or of the order of 400 stars (i.e., on the order of 0.1% of the DR16 stars)—show these unexpectedly high oxygen abundances. The tightness of the feature in abundance space makes it likely that this is caused by an error in the abundance determination, and that this does not represent a peculiar stellar population. However, despite thorough investigation of the abundance analysis, we could not find the origin of this artifact; thus it is still an open question whether the feature arises from something real in the observed spectra that is not discernible by eye (perhaps leading to inaccurately derived parameters/abundances, but still with some underlying physical origin), or if it simply is the result of some intricate interplay between the different ingredients of the analysis that is leading to incorrect results. Therefore, we recommend caution when using the data for this set of stars.
• 3.  
  There is a group of stars with lower-than-expected [Ni/Fe] values creating a thin horizontal sequence in the [Ni/Fe] versus [Fe/H] plot at [Ni/Fe] ∼ −0.25, which leads to an apparent bimodality. The same type of feature can be seen in Cr, to some extent in Al, and possibly in Co. These kinds of features were not present in DR14, and (post-release of DR16) have been traced to be related to the implementation of the TIE option in FERRE. While there is little doubt that using the TIE option is more correct, it seems—together with the PCA, interpolation in the grid, the minimization, etc.—to make the χ² surface of some solutions shallow enough that there may be multiple minima, and measurements seem to cluster around these values. This problem—similarly to the [O/Fe]-[Fe/H] "finger" discussed above—seems to affect giants with T_eff < 4000 K exclusively. However, the two issues do not affect the same stars, which indicates that the two peculiarities have different causes.

The primary targets of the APOGEE main survey are red giant stars. For a single population of stars of a given age and metallicity, one expects that stars should have the same [X/Fe] along the giant branch, with the exception of elements that are modified by mixing with material from the stellar interior (see discussion of N below). For a population of mixed age, there might be some astrophysical spread in [X/Fe] at any location on the giant branch, but still no mean trends of abundance along the red giant branch.

Figure 13 shows the abundances [X/Fe] as a function of surface gravity for stars with near-solar (−0.1 < [Fe/H] < 0.1) metallicity within 0.5 kpc of the Sun (based on Gaia parallaxes). These show that, while there is general consistency along the red giant branch, there are some variations. For several elements, the variations are sufficiently large at some surface gravities that they are suspect, and we have chosen in these cases not to populate the named X_FE tags, although the calibrated abundances are available in the X_M and X_H arrays. The unpopulated values are shown in Figure 13 as red points, while the black points show the populated values. Note that the choices of which regions were chosen to be populated was made by visual inspection, and there is some qualitative judgement involved in the choices that were made.

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While the unpopulated X_FE values remove many of the regions of concern for the abundances, several residual effects remain:

• 1.  
  As noted above, a few elements (C, O, Si, Al, P, Cr, Ni) show bimodalities at low $\mathrm{log}\,g$/low T_eff.
• 2.  
  While most elements show relatively little change with $\mathrm{log}\,g$, there is still some low-level variation. As a result, if looking for subtle changes in elemental abundances between different data sets, it may be important to ensure that stars of similar $\mathrm{log}\,g$ are being compared.

The source of variation of abundance along the giant branch is not fully understood. Possible causes include the use of uncalibrated surface gravities in the abundance determination or changing NLTE or 3D effects along the giant branch.

Since ASPCAP determines the stellar parameters by fitting the entire spectrum, stars with abundance patterns deviating from those used in the calculation of the synthetic spectra might be inaccurately analyzed by the pipeline. For example, second-generation globular cluster stars are believed to be oxygen poor, and since all of the α elements are varied together, ASPCAP will struggle to fit a spectrum of a relatively oxygen-poor/calcium-rich star, leading to inaccuracies in the determined stellar parameters and, in turn, in all of the subsequently determined abundances. This is a problem intrinsic to our analysis method that has been there from the initial APOGEE DR10 release, and is described in more detail in Jönsson et al. (2018).

As already mentioned, many stars have been observed multiple times using both the APO 2.5 m as well as LCO 2.5 m telescope/instrument combinations, and, as described in Section 5.4, the independent ASPCAP results from these different observations of the same stars have been used to estimate the uncertainty in the derived stellar abundances. However, these independent measurements can also be used as a way of assessing and validating the precision of the analysis pipeline, as is done in this subsection. First we chose pairs of spectra for all stars observed more than once with the APO 2.5 m and where both spectra have S/N > 100, STAR_BAD == 0, and calibrated values for all stellar parameters (if more than two such spectra were analyzed, we chose the two with the highest S/Ns); this resulted in 15,920 pairs of analyzed spectra of the same 15,920 stars (7651 pairs of giants). The same was done for stars observed more than once with the LCO 2.5 m (529 pairs of spectra, 239 pairs of giants), as well as for stars where one spectrum was observed with the APO 2.5 m and the other with the LCO 2.5 m (341 pairs of spectra, 191 pairs of giants). The results of the comparison of the stellar parameters can be seen in Table 10.

Table 10.  Differences in Calibrated T_eff, $\mathrm{log}\,g$, and [Fe/H], as well as the "Abundance Parameters" [C/M], [N/M], and [α/M] for Pairs of Independently Processed High-quality Spectra of the Same Stars for Different 2.5 m Telescope Combinations

		ΔTeff	Δ$\mathrm{log}\,g$	Δ[Fe/H]	Δ[C/M]	Δ[N/M]	Δ[α/M]
Giants and dwarfs:
APO–APO	15920 pairs	0 ± 21	0.00 ± 0.05	0.00 ± 0.02	0.00 ± 0.03	0.00 ± 0.05	0.00 ± 0.01
LCO–LCO	529 pairs	4 ± 21	0.01 ± 0.06	0.00 ± 0.02	0.00 ± 0.03	0.01 ± 0.06	0.00 ± 0.01
APO–LCO	341 pairs	−4 ± 21	−0.01 ± 0.06	0.02 ± 0.02	0.01 ± 0.03	−0.02 ± 0.05	0.00 ± 0.02
Only giants, $\mathrm{log}\,g$ < 3.5:
APO–APO	7651 pairs	−2 ± 14	0.00 ± 0.04	0.00 ± 0.02	0.00 ± 0.02	0.00 ± 0.03	0.00 ± 0.02
LCO–LCO	239 pairs	−2 ± 14	0.00 ± 0.06	0.00 ± 0.02	0.00 ± 0.03	0.00 ± 0.03	0.00 ± 0.02
APO–LCO	191 pairs	2 ± 13	0.00 ± 0.05	0.01 ± 0.02	0.01 ± 0.02	−0.02 ± 0.03	0.00 ± 0.02

Note. The listed numbers denote the mean and the robust standard deviation (the median absolute deviation divided by 0.67449).

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Comparing the ASPCAP results from these independently processed pairs of spectra leads to the following observations. First, the precision is fairly high, especially if only the giants are considered. Second, the precision is about the same for all three telescope combinations, despite the different numbers of stars utilized in each set. Finally, the precision in [N/M] is somewhat lower when dwarfs are included in the comparison. This is expected, however, since the CN molecular lines—which are the only N-indicators—become very weak in FGK-dwarf stars.

Table 11 lists the comparisons for the derived abundances for the same 7651, 239, and 191 pairs of giant spectra. The elements with the highest [X/Fe] precision in this table are C, Mg, Si, Ca, and Ni, which show scatters as low as 0.02 dex. The elements C i, N, O, Al, Ti, and Mn also yield a relatively high precision, in this table with a scatter less than 0.04 dex. The elements S, K, Cr, and Co could be considered to be of medium precision, based on their scatter of 0.06–0.08 dex, along with Na, V, and Cu, which yield scatters of ∼0.10 dex. Not surprisingly, this exercise shows the elemental species with the least precise [X/Fe] to be P, Ti ii, and Ce, which show scatters above 0.15 dex.

Precision is one thing, but it is also desirable for the stellar parameters and abundances to be accurate. However, this is much harder to evaluate. As an example, the "flat main sequence"—the lack of a correlation between $\mathrm{log}\,g$ and T_eff—seen in the uncalibrated data in Figure 10 as well as in previous APOGEE DRs—is a sign that we have a problem accurately determining $\mathrm{log}\,g$ for dwarf stars. The fact that the main sequence is thin, however, is a sign that the determination of $\mathrm{log}\,g$ is precise, something that is also seen in Table 10. The calibration of T_eff and $\mathrm{log}\,g$ is an attempt to remove systematic uncertainties from our supplied stellar parameters by making them compliant with other, objectively more accurate values (in the case of $\mathrm{log}\,g$), or more widely trusted values (in the case of T_eff).

When it comes to abundances, accuracy is even more difficult to assess, because there are few "objectively accurate" and/or "widely trusted" abundance measurements, especially for giant stars. In Jönsson et al. (2018), we made an attempt to evaluate the accuracy of the abundances in DR13 and DR14 by a comparison to an optical analysis of a subsample of APOGEE giants, and to already published, independent abundance studies having significant overlap with the APOGEE stellar sample (da Silva et al. 2015; Brewer et al. 2016; Jönsson et al. 2017). Table 12 presents the same evaluation for DR16 (compare to Tables 5–6 in Jönsson et al. 2018). Note that the comparison sample of Jönsson et al. (2017) now is expanded with more elements from Lomaeva et al. (2019) and Forsberg et al. (2019). The details of the different samples are given in detail in Jönsson et al. (2018), but in general, the compared stars are giants with 4000 K < T_eff < 5500 K, 1.5 < $\mathrm{log}\,g$ < 3.8, and −1 < [Fe/H] < 0.5. It is important to reiterate, however, that the optical studies are not necessarily guaranteed to present the "true" abundances.

In general, the DR16 abundances are rather similar to those in DR14, but with a smaller scatter. Below we highlight the main differences between the DR16-optical comparison and the DR14-optical comparison in Jönsson et al. (2018):

• 1.  
  In DR14, [N/Fe] had a systematic shift of 0.08 dex compared to the optical references, while Table 12 shows that this value now is −0.15 dex. The trend of derived [N/Fe] with [Fe/H] that was found based on the comparisons to optical measurements in Jönsson et al. (2018) is still there in DR16. However, determining nitrogen abundances from optical spectra is challenging, and the trend might very well come from the comparison samples.
• 2.  
  The possible slight trend of derived [Mg/Fe] with metallicity compared to the optical reference abundances found for DR14 in Jönsson et al. (2018) is essentially gone in DR16.
• 3.  
  Regarding Al, there is a systematic difference as compared to the optical abundances that was not present in DR14 (see Table 12). The spread, in Δ[Al/H], however, is lower in DR16.
• 4.  
  The K abundances derived for both DR14 and DR16 are consistent with one another, but very different than those derived optically. The optical measurements follow a much more scattered [K/Fe] versus [Fe/H] trend, indicating that the discrepancy might lay with the reference abundances and not APOGEE.
• 5.  
  The Ca abundances are very similar in DR14 and DR16, but the possible trend of derived Ca abundance with metallicity seen in DR14 when comparing to optical data seems to have disappeared in DR16.
• 6.  
  The possible trend of determined V abundance with metallicity seen in DR14 as compared to the optical studies has vanished in DR16.
• 7.  
  Based on a smaller scatter and offset compared to the optical references, the DR16 abundances of Co are improved compared to those in DR14.

There were no calibrated abundances supplied in DR14 for Cu, and regarding Ce, there were no such abundances in DR14 because these lines were not fully understood until later (Cunha et al. 2017). Hence, there were no comparisons made for these elements in Jönsson et al. (2018) and so a slightly longer discussion is warranted here.

Table 12 and the top panels in Figure 14 indicate a reasonable accuracy for [Cu/Fe] for stars with [Fe/H] > −1. However, the trend of [Cu/Fe] with [Fe/H], presented in Figure 12 below, shows a fair amount of scatter, and the rising trend at lower metallicities does not follow the expectations for a weak s-process element to be subsolar for lower metallicities.

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The [Ce/Fe] versus [Fe/H] trend in Figure 12 exhibits significant scatter, but does in principle show a reasonable (banana-like) shape for an s-process dominated element, something that is less obvious in the smaller sample plotted in Figure 14.

In addition to the comparison works above with large overlap with APOGEE, five stars from the high-quality sample of Gaia FGK benchmark stars (Jofre et al. 2018) are in APOGEE DR16, and the comparison for these is given in Table 13. While this is a small sample of stars, the parameters and abundances in this table highlight some of the issues described in Holtzman et al. (2018), Jönsson et al. (2018), and above:

• 1.  
  The values of APOGEE uncalibrated T_eff are generally too high for metal-rich stars and too low for metal-poor stars.
• 2.  
  The values of APOGEE uncalibrated $\mathrm{log}\,g$ are generally too high for giants, while the calibration makes them lower.
• 3.  
  The APOGEE α abundances are generally low compared to the optical measurements, except Ti i, which has an unexpected rising trend with metallicity in APOGEE.
• 4.  
  The mismatch is particularly striking for HD122563, which is not unexpected, since this star has a benchmark metallicity lower than the ASPCAP grid boundary of [M/H] = −2.5. However, it is worth noting that the uncalibrated parameters and some of the uncalibrated abundances are quite close to the benchmark values, suggesting that users interested in metal-poor giants might find values of use in some of the arrays with uncalibrated values (FPARAM and FELEM).
• 5.  
  The most metal-rich star in the sample, μLeo, shows several deviations. However, in the case of Mg and Si, the values from APOGEE seem more like one would expect from other optical studies of dwarf stars (Reddy et al. 2003; Adibekyan et al. 2012; Bensby et al. 2014, among others).

Table 13.  Comparison to the Giants among the Gaia FGK Benchmark Stars (Jofre et al. 2018) Also within APOGEE DR16, Sorted by Benchmark $\mathrm{log}\,g$

Star	Teff	$\mathrm{log}\,g$	[Fe/H]	[Mg/Fe]	[Si/Fe]	[Ca/Fe]	[Ti/Fe]	[V/Fe]	[Cr/Fe]	[Mn/Fe]	[Co/Fe]	[Ni/Fe]
αTau	3927	1.11	−0.37	0.20	0.19	0.03	0.03	0.17	0.08	−0.12	0.12	−0.03
	3889	1.50	−0.10	0.04	0.04	−0.04	0.15	−0.10	−0.08	0.01	0.14	0.04
	3982	1.33	−0.13	0.06	0.03	−0.01	⋯	0.12	−0.03	⋯	0.14	0.06
Arcturus	4286	1.60	−0.52	0.36	0.27	0.11	0.21	0.08	−0.06	−0.37	0.11	0.03
	4180	1.91	−0.53	0.24	0.22	0.09	0.09	−0.28	−0.07	−0.22	0.16	0.10
	4291	1.75	−0.55	0.25	0.20	0.10	⋯	−0.07	−0.03	−0.09	0.15	0.10
HD122563	4587	1.61	−2.64	0.29	0.31	0.21	0.14	−0.06	−0.44	−0.46	−0.00	−0.05
	4855	1.76	−2.41	0.32	0.35	0.04	−0.64	0.35	0.03	−0.00	0.00	0.02
	5005	1.86	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
HD107328	4496	2.09	−0.33	0.25	0.21	0.01	0.20	0.13	−0.08	−0.35	0.15	0.02
	4299	2.04	−0.41	0.25	0.25	0.04	0.06	−0.03	−0.14	−0.24	0.20	0.12
	4405	1.86	−0.42	0.25	0.22	0.05	0.06	0.16	−0.11	−0.11	0.18	0.11
μLeo	4474	2.51	0.25	0.22	0.27	0.03	0.07	0.05	0.09	−0.16	0.20	0.07
	4519	2.82	0.34	−0.05	0.01	−0.06	0.11	−0.09	−0.09	0.00	0.19	0.06
	4590	2.57	0.31	−0.02	0.01	−0.03	0.14	0.14	−0.03	0.16	0.19	0.08

Note. For every star, there are three lines in the table: the benchmark values, the uncalibrated DR16 values, and the calibrated DR16 values, respectively.

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Star clusters offer another opportunity to check the DR16 results. Two recently published studies have undertaken comparisons of APOGEE results to published cluster data.

As part of the latest contribution from the APOGEE-based Open Cluster Chemical Analysis and Mapping Survey⁴⁰ (OCCAM; Donor et al. 2020), the DR16-version APOGEE [Fe/H] were compared to literature metallicity values for six well-studied open clusters, and a mean difference of 0.004 dex was found, in the sense that the APOGEE metallicities are very slightly higher than the literature values.

For more metal-poor stars, Nidever et al. (2019) made a comparison of DR16 APOGEE [Fe/H] and literature values for first-generation globular cluster stars, and found the APOGEE values to be 0.06 dex higher in the mean, and with a spread of 0.09 dex (see their Figure 8).

While absolute abundance scales are challenging to establish, these results suggest that APOGEE metallicities have few systematic errors for metal-rich stars, but may have systematic offsets for metal-poor stars at the 0.05–0.1 dex level.

In this section we provide some summary notes about the DR16 abundances for each APOGEE-measured element, including the estimated quality of the abundances, any peculiarities about which the user should be aware, and the population of the named X_FE tag for the element.

6.10.1. Carbon, C

6.10.2. Nitrogen, N

The nitrogen abundances are determined from CN lines, in conjunction with the carbon abundance (which is determined from CO lines, as described above). While the CN line list was not modified for DR16, the CO and OH line lists were, and therefore the N abundances in DR16 differ from those in DR14.

When compared to optical measurements, the precision of APOGEE nitrogen abundances appears good (i.e., there is little scatter), but there is an unexplained trend of [N/Fe] with T_eff when compared with the optical abundances. However, this might very well come from the optical measurements. Moreover, the CN lines become weak or nonexistent for warmer stars, which makes the associated uncertainties very large.

Carbon and nitrogen in red giant stars are of particular interest because [C/N] has been found to be an indicator of stellar mass, which corresponds to stellar age for red giant stars (Masseron & Gilmore 2015; Ness et al. 2016). This is because material that has been processed through the CNO cycle is brought to the surface and leads to enhanced N abundances, with the level of enhancement depending on stellar mass and metallicity. Such canonical mixing appears near the base of the giant branch, but metal-poor stars may exhibit extra mixing as the stars ascend the red giant branch (Shetrone et al. 2019). At solar metallicity, [C/N] is expected to be relatively constant above the base of the giant branch. Figure 15 shows [C/N] as a function of $\mathrm{log}\,g$ for stars with −0.1 < [M/H] < 0.1, and demonstrates the expected drop of [C/N] at the base of the giant branch ($\mathrm{log}\,g$ ∼ 3.5).

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The range of [C/N] abundances along the giant branch is plausible if there is a range of ages at solar metallicity. However, the rise of [C/N] at $\mathrm{log}\,g$ < 1 is unexpected, and occurs because [N/Fe] is dropping (see Figure 13) at low surface gravity, which perhaps makes the measurements of N (and corresponding [C/N]) suspicious in this regime. Also worth noting is the locus at $\mathrm{log}\,g$ ∼ 2.5, which corresponds to the RC stars, and the fact that some of them show unexpectedly high [C/N] values above 0.1 (Masseron et al. 2017; Shetrone et al. 2019).

6.10.3. Oxygen, O

6.10.4. Sodium, Na

6.10.5. Magnesium, Mg

6.10.6. Aluminium, Al

6.10.7. Silicon, Si

6.10.8. Phosphorous, P

6.10.9. Sulfur, S

6.10.10. Potassium, K

6.10.11. Calcium, Ca

6.10.12. Titanium, Ti

6.10.13. Vanadium, V

6.10.14. Chromium, Cr

6.10.15. Manganese, Mn

Iron abundances are determined using windows as for the other elements. In general, the resulting [Fe/H] is very close to the [M/H] determined at the atmospheric parameter-determination stage. However, there are a small number of cooler stars that seem to show [Fe/H] offsets from [M/H], with a bimodality similar to that seen in some other elements (see Section 6.4); as with those elements, this is likely some artifact of the analysis.

Stars with [Fe/H] that differ from [M/H] by more than 0.1 dex have the PARAM_MISMATCH_WARN bit in the iron ELEMFLAG set, and those that differ by more than 0.25 dex have the PARAM_MISMATCH_BAD bit set. The FE_H tag is not populated for stars that have either of these bits set. In turn, this will imply that such a star has none of the "named" tags (C_FE, N_FE, O_FE, etc.) populated.

6.11.1. Cobalt, Co

6.11.2. Nickel, Ni

6.11.3. Copper, Cu

6.11.4. Cerium, Ce