Chemical Cartography with APOGEE: Mapping Disk Populations with a Two-Process Model and Residual Abundances

Over the past decade, large and systematic spectroscopic surveys have mapped elemental abundance patterns of hundreds of thousands of stars across much of the Galactic disk, bulge, and halo, including RAVE, SEGUE, LAMOST, Gaia-ESO, APOGEE, GALAH, and H3 (Steinmetz et al., 2006; Yanny et al., 2009; Luo et al., 2015; Gilmore et al., 2012; De Silva et al., 2015; Majewski et al., 2017; Conroy et al., 2019). The APOGEE survey of SDSS-III (Eisenstein et al., 2011) and SDSS-IV (Blanton et al., 2017) is especially well suited to mapping the inner disk and bulge because it observes at near-IR wavelengths where dust obscuration is dramatically reduced, because it targets luminous evolved stars that can be observed at large distances, and because its high spectral resolution allows separate determinations of 15 or more elemental abundances per target star.¹¹1SDSS = Sloan Digital Sky Survey. APOGEE = Apache Point Observatory Galactic Evolution Experiment. We use APOGEE to refer to both the SDSS-III program and its SDSS-IV extension (a.k.a. APOGEE-2). In SDSS-V (Kollmeier et al., 2017) the Milky Way Mapper program is using the APOGEE spectrographs to observe a sample ten times larger than that of SDSS-III + IV. These surveys share two primary goals: to understand the astrophysical processes that govern the synthesis of the elements, and to trace the chemical evolution of the Milky Way, which is itself shaped by many processes including gas accretion, star formation, outflows, and radial migration of stars. This paper introduces a novel approach to characterizing and mapping abundance patterns in APOGEE, one that opens new avenues to addressing both of these goals.

Our study builds on a series of investigations that have used APOGEE data to characterize the multi-element abundance distributions of the Galactic disk and bulge (Anders et al., 2014; Hayden et al., 2014, 2015; Nidever et al., 2014; Ness et al., 2016; Ting et al., 2016; Mackereth et al., 2017; Schiavon et al., 2017; Bovy et al., 2019; Fernández-Trincado et al., 2019a, 2020b; Weinberg et al., 2019; Zasowski et al., 2019; Griffith et al., 2021a; Ting & Weinberg, 2021; Vincenzo et al., 2021a). Its most direct predecessors are the papers of Hayden et al. (2015, hereafter H15), who mapped the distribution of stars in $[\alpha/{\rm Fe}]-[{\rm Fe}/{\rm H}]$ as a function of Galactocentric radius $R$ and midplane distance $|Z|$, and Weinberg et al. (2019, hereafter W19), who examined the median trends of other abundance ratios as a function of $R$ and $|Z|$. Because $\alpha$ elements such as O, Mg, and Si are produced mainly by core collapse supernovae (CCSN), while Fe is produced by both CCSN and Type Ia supernovae (SNIa), the $[\alpha/{\rm Fe}]$ ratio is a diagnostic of the relative contribution of these two sources to a star’s chemical enrichment. Many studies have shown that stars in the solar neighborhood have a bimodal distribution of $[\alpha/{\rm Fe}]$, with “thin disk” stars having roughly solar abundance ratios and “thick disk” stars (which have larger vertical velocities and consequently larger excursions from the disk midplane) having elevated $[\alpha/{\rm Fe}]$ (e.g., Fuhrmann 1998; Bensby et al. 2003; Adibekyan et al. 2012; Vincenzo et al. 2021a). H15 showed that the locus of the “high-$\alpha$” sequence in the $[\alpha/{\rm Fe}]-[{\rm Fe}/{\rm H}]$ plane is nearly constant throughout the disk (see also Nidever et al. 2014) but the relative number of high-$\alpha$ and low-$\alpha$ stars and the distribution of those stars in $[{\rm Fe}/{\rm H}]$ changes systematically with $R$ and $|Z|$. W19 advocated the use of Mg rather than Fe as a reference element because it traces a single enrichment source (CCSN), and they showed that the median trends of $[{\rm X}/{\rm Mg}]$ for nearly all of the elements measured by APOGEE are universal throughout the disk, provided that one separates the high-$\alpha$ and low-$\alpha$ populations. Griffith et al. (2021a) showed that this universality of abundance ratio trends extends to the bulge.

Since the star formation and enrichment histories do change across the Galaxy, W19 argued that the universal median sequences must be determined mainly by IMF-averaged nucleosynthetic yields.²²2IMF = Initial Mass Function They interpreted these sequences in terms of a “2-process model,” which describes APOGEE abundances as the sum of a core collapse process representing the IMF-averaged yields of CCSN and a Type Ia process reflecting the IMF-averaged yields of SNIa. The elemental abundances of a given star can be summarized by the two parameters $A_{\rm cc}$ and $A_{\rm Ia}$ that scale the amplitudes of these processes. The success of the 2-process model means that all of a disk or bulge star’s APOGEE abundances can be predicted to surprisingly high accuracy from its Mg and Fe abundances alone. They can be predicted to similar accuracy from the combination of $[{\rm Fe}/{\rm H}]$ and age (Ness et al., 2019). Nonetheless, the residual abundances of other elements at fixed $[{\rm Fe}/{\rm H}]$ and $[{\rm Mg}/{\rm Fe}]$ contain rich information, as demonstrated empirically by Ting & Weinberg (2021, hereafter TW21), who show that one must condition on at least seven APOGEE elements (e.g., Fe, Mg, O, Si, Ni, Ca, Al) before the correlations among the remaining abundances are reduced to a level consistent with observational uncertainties. In this paper, therefore, we turn our attention from median trends to the star-by-star abundance patterns described by 2-process model parameters and the residuals from this description. As argued by TW21, the correlations of residual abundances encode crucial information about nucleosynthetic processes and stochastic effects in chemical evolution.

Although “high-$\alpha$” stars have elevated $[\alpha/{\rm Fe}]$ compared to the Sun, this difference really arises because they have a lower contribution of SNIa to Fe rather than enhanced production of $\alpha$ elements by CCSN (Tinsley, 1980; Matteucci & Greggio, 1986; McWilliam, 1997). Adopting this physical interpretation, we will refer to high-$\alpha$ and low-$\alpha$ stars in this paper as the “low-Ia” and “high-Ia” populations, respectively, following terminology introduced by Griffith et al. (2019).

In §2 we describe the 2-process model, which is similar to that of W19 and Griffith et al. (2019) but with adjustments that make the model more flexible and easier to generalize. In §3 we describe our selection of APOGEE stars from SDSS Data Release 17 (DR17): red giants in a restricted range of ${\rm log}\,g$, $T_{\rm eff}$, and $[{\rm Mg}/{\rm H}]$ intended to minimize statistical and differential systematic errors while sampling the disk in the range $3\leq R\leq 13\,{\rm kpc}$ and $|Z|\leq 2\,{\rm kpc}$. Section 4 presents median abundance trends from this sample and uses them to infer the CCSN and SNIa 2-process vectors, i.e., the abundance of each of the APOGEE elements associated with these two processes at a given metallicity. In §5, the heart of the paper, we fit each sample star’s abundances with the 2-process model and examine the distributions and correlations of the abundance residuals. As in TW21, we find a rich correlation structure among these residuals, and we further examine the correlation of these residuals with stellar age and kinematics. In §6 we investigate stars whose abundance patterns deviate unusually far from the 2-process model fits, a group that includes both genuine physical outliers and stars with measurement errors that exceed the reported uncertainties. In §7 we examine the residual abundances of a few special populations, such as likely halo stars that reside within the geometrical boundaries of the disk and members of the rich cluster $\omega\,$Cen, which is thought to be the stripped core of an accreted dwarf spheroidal galaxy. Section 8 discusses ways to go beyond the 2-process model, first with a conceptual $N$-process formulation, then with an empirical approach that fits two additional components to the APOGEE abundance residuals. We review our conclusions and outline prospects for future studies in §9.

This is a long paper covering many interconnected topics. Readers who want to start with a bird’s-eye view can read the introduction to the 2-process model at the start of §2, look through figures with particular attention to the median trends (Figure 4-7), distributions and covariance of residual abundances (Figures 12 and 15), examples of high-$\chi^{2}$ stars (Figure 20), and residual patterns of selected populations (Figure 22), then read the conclusions and loop back to earlier sections as needed. For readers interested in overall abundance trends and their interpretation, §4 is the most relevant. For readers interested in unusual stars and stellar populations, §6 and §7 are the most relevant. Readers interested in the dimensionality of the stellar distribution in abundance space and its connection to the physics of nucleosynthesis and chemical evolution should pay particular attention to §5 and §8. The challenge of determining accurate, precise, robust abundances for many elements in large stellar samples is a running theme throughout the paper.

We begin with a conceptual introduction to the 2-process model. The left panel of Figure 1 shows the distribution of our APOGEE sample (described in §3) in the familiar plane of $[{\rm Mg}/{\rm Fe}]$ vs. $[{\rm Mg}/{\rm H}]$, with the low-Ia and high-Ia populations as red and blue points, respectively. Like W19, we adopt $[{\rm Mg}/{\rm H}]$ as our reference abundance on the $x$-axis because Mg is well measured in APOGEE and, unlike Fe, it is thought to come from a single nucleosynthetic source (CCSN).³³3We choose Mg in preference to O because the observed trends for O are significantly different between optical and near-IR surveys. We choose Mg in preference to Si or Ca because these have non-negligible SNIa contributions. In the conventional interpretation of this diagram, which we adopt in this paper, the “plateau” in the abundances of metal-poor low-Ia stars at $[{\rm Mg}/{\rm Fe}]\approx 0.3$ represents the Mg/Fe ratio of CCSN yields, and stars that lie below this plateau do so primarily because they have additional Fe from SNIa. While the relative number of low-Ia and high-Ia stars depends strongly on Galactic location, the median $[{\rm Mg}/{\rm Fe}]-[{\rm Mg}/{\rm H}]$ tracks of these populations, shown by red and blue lines in Figure 1, are nearly universal throughout the disk (Nidever et al. 2014; H15; W19). The median $[{\rm X}/{\rm Mg}]-[{\rm Mg}/{\rm H}]$ tracks for other APOGEE elements are also universal throughout the disk (W19) and bulge (Griffith et al., 2021a), provided that one separates the low-Ia and high-Ia populations. This universality motivates the hypothesis that these tracks are governed by stellar yields and that differences between the two [X/Mg] tracks reflect the contribution of SNIa enrichment to element X.

In the 2-process model, the position of a star in the $M$-dimensional space of its measured abundances is approximated as the weighted sum of two “vectors” that represent the contributions from CCSN and SNIa:

The 2-process vectors $q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ are taken to be universal for the stellar sample under study, though they may depend on metallicity. The amplitudes $A_{\rm cc}$ and $A_{\rm Ia}$ vary from star to star, and they are normalized such that $A_{\rm cc}=A_{\rm Ia}=1$ for solar abundances. For notational compactness we define metallicity by

i.e., the Mg abundance in solar units. As discussed in §§2.1-2.3 below, we infer $q^{X}_{\rm cc}(z)$ and $q^{X}_{\rm Ia}(z)$ from median abundance ratios of low-Ia and high-Ia stars, then determine $A_{\rm cc}$ and $A_{\rm Ia}$ for all stars in the observational sample by a $\chi^{2}$ fit to a subset of their measured abundances.

Figure 2 illustrates the simple case of $M=2$ dimensions, with points marking the location of four representative stars in (Fe/H) vs. (Mg/H), expressed in solar units. The 2-process (Mg,Fe) vectors are $\vec{q}_{\rm cc}=(1,0.5)$ and $\vec{q}_{\rm Ia}=(0,0.5)$, reflecting our model assumptions that Mg is produced entirely by CCSN and that solar Fe comes equally from CCSN and SNIa. The filled blue circle has solar abundances, with $A_{\rm cc}=A_{\rm Ia}=1$ and $({\rm Mg/H},{\rm Fe/H})=A_{\rm cc}\vec{q}_{\rm cc}+A_{\rm Ia}\vec{q}_{\rm Ia}=(1,1)$. The open blue circle has $A_{\rm cc}=A_{\rm Ia}=1/3$, so its (Mg/H) and (Fe/H) abundances are 1/3 solar but its (Mg/Fe) ratio is solar. Filled and open red squares represent low-Ia stars with $A_{\rm cc}=1$ and $1/3$, respectively. These stars have $A_{\rm Ia}<A_{\rm cc}$, so they have (Mg/Fe) ratios above solar (“$\alpha$-enhanced” or “iron poor”). With $M=2$, the two parameters $A_{\rm cc}$ and $A_{\rm Ia}$ suffice to fit each star’s abundances perfectly, but they recast the information from the space of individual elements to the space of the processes that produce those elements.

Using the same four combinations of $(A_{\rm cc},A_{\rm Ia})$ as Figure 2, Figure 3 illustrates the 2-process model for the full set of 16 abundances that we consider in this paper, one of which is the element combination C+N (eq. 27 in §3). In the upper panel, open circles and filled triangles show the components of $q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ that we derive from the APOGEE median abundance trends in §4 below, at metallicity $z=1$. For $\alpha$-elements (on the left), $q^{X}_{\rm cc}$ values are much larger than $q^{X}_{\rm Ia}$ values, while for iron-peak elements (on the right) they are roughly equal. For $A_{\rm cc}=A_{\rm Ia}=1$, the predicted abundances are exactly solar by construction (blue points). For $A_{\rm cc}=1$ and $A_{\rm Ia}=0.2$, the predicted abundances (red points) are only slightly above $q^{X}_{\rm cc}$ (black open circles), with sub-solar (X/H) values for all elements that have a substantial SNIa contribution in the Sun.

The lower panel shows our inferred 2-process vectors for $[{\rm Mg}/{\rm H}]=-0.5$ ($A_{\rm cc}=z=1/3$). These vectors are similar to those found at $z=1$, but they are not identical because some elements have metallicity dependent yields. As a result, the predicted abundances for $A_{\rm Ia}=A_{\rm cc}=1/3$ have element ratios that are approximately but not exactly solar (blue points). For a star on the low-Ia (high-$\alpha$) plateau, the predicted abundances (red points) are just slightly above $A_{\rm cc}q^{X}_{\rm cc}$. The predicted abundances (blue and red points) are multiplied by a factor of three so that the patterns can be visually compared to those of the $[{\rm Mg}/{\rm H}]=0$ stars shown in the upper panel.

With 16 abundances, any given star will not be perfectly reproduced by a 2-parameter $(A_{\rm cc},A_{\rm Ia})$ fit, in part because of measurement errors, but also because the 2-process model is not a complete physical description of stellar abundances. For example, the model does not allow for stochastic variations around IMF-averaged yields or for varying contributions from other sources such as AGB enrichment. The 2-process vectors themselves are useful tests of supernova nucleosynthesis predictions (e.g., Griffith et al. 2019, 2021b), and the distributions of $(A_{\rm cc},A_{\rm Ia})$ and their correlations with stellar age and kinematics are useful diagnostics of Galactic chemical evolution. However, our primary focus in this paper will be the star-by-star departures from the 2-process predictions, and what these departures can tell us about the astrophysical sources of the APOGEE elements, about distinct stellar populations within the geometric boundaries of the Milky Way disk, and about rare stars with distinctive abundance patterns. Disk stars span a range of $>1$ dex in [Fe/H] and typically 0.3 dex or more in [X/Fe]. With two free parameters, the 2-process model fits the measured APOGEE abundances of most disk stars to within $\sim 0.1$ dex, and for the best measured elements to within $\sim 0.01-0.04$ dex, so focusing on these residual abundances allows us to discern subtle patterns that might be lost within the much larger dynamic range of a conventional [X/Fe]-[Fe/H] analysis.

We now proceed to a more formal definition of the 2-process model and its assumptions, and our methods for inferring the 2-process vectors and fitting $A_{\rm cc}$ and $A_{\rm Ia}$ values. While our approach is similar to that of W19, here we define the model in a way that is more general and allows natural extension to include other processes as discussed in §8.

We use data from the 17th data release (DR17) of the SDSS/APOGEE survey (Majewski et al., 2017). The APOGEE disk sample consists primarily of evolved stars with 2MASS (Skrutskie et al., 2006) magnitudes $7<H<13.8$, sampled largely on a grid of sightlines at Galactic latitudes $b=0^{\circ}$, $\pm 4^{\circ}$, and $\pm 8^{\circ}$ and many Galactic longitudes. Targeting for APOGEE is described in detail by Zasowski et al. (2013, 2017), Beaton et al. (2021), and Santana et al. (2021). APOGEE obtains high-resolution ($R\sim 22,500$) H-band spectra (1.51-1.70 $\mu$m) using 300-fiber spectrographs (Wilson et al., 2019) on the 2.5m Sloan Foundation telescope (Gunn et al., 2006) at Apache Point Observatory in New Mexico and the 2.5m du Pont Telescope (Bowen & Vaughan, 1973) at Las Campanas Observatory in Chile. The great majority of spectra in the main APOGEE sample have signal-to-noise ratio per pixel ${\rm SNR}>80$ (with a typical pixel width $\approx 0.22$Å). Spectral reductions and calibrations are performed by the APOGEE data processing pipeline (Nidever et al., 2015), which provides input to the APOGEE Stellar Parameters and Chemical Abundances Pipeline (ASPCAP; Holtzman et al. 2015; Garc´ıa Pérez et al. 2016). ASPCAP uses a grid of synthetic spectral models (Mészáros et al., 2012; Zamora et al., 2015) and H-band linelists (Shetrone et al., 2015; Hasselquist et al., 2016; Cunha et al., 2017; Smith et al., 2021) compiled from a variety of laboratory, theoretical, and astrophysical sources, fitting effective temperatures, surface gravities, and elemental abundances.

A detailed description of the APOGEE DR17 data will be presented by J. Holtzman et al. (in preparation), updating the comparable description of the APOGEE DR16 data by Jönsson et al. (2020). These papers explain the spectral fitting and calibration procedures, the estimation of observational uncertainties, and comparisons to literature values. Notably, the DR17 abundances used here employ a synthetic spectral grid generated by Synspec (Hubeny & Lanz, 2017) with NLTE treatments of Na, Mg, K, and Ca (Osorio et al., 2020). These spectra are based on MARCS atmospheric models (Gustafsson et al., 2008), with spherical geometry in the ${\rm log}\,g$ range used for our analysis. The Synspec synthesis uses these structures but assumes plane parallel geometry. DR17 uses improved H-band wavelength windows for the s-process element Ce (Cunha et al., 2017), providing higher precision measurements than previous APOGEE data releases. We do not distinguish isotopes for any elements, as APOGEE does not have the resolution to clearly separate different isotopic lines.

Stellar abundance measurements are subject to statistical errors arising from photon noise and data reduction and to systematic errors that arise because one is fitting the data with imperfect models. These imperfections include incomplete or inaccurate linelists, astrophysical effects such as departures from local thermodynamic equilibrium (LTE), and observational effects such as inexact spectral linespread functions. The systematic effects will change with stellar parameters such as $T_{\rm eff}$, ${\rm log}\,g$, and metallicity, but if the range of parameters in the sample is small then the differential systematics within the sample will be limited, so the systematics will produce zero-point offsets but will not add much in the way of scatter or correlated abundance deviations for stars of the same $[{\rm Mg}/{\rm H}]$ and $[{\rm Mg}/{\rm Fe}]$.

For the analyses of this paper, we have several goals that affect the choice of sample selection criteria:

1. 1.

   Minimize statistical errors to improve measurements of residuals from 2-process predictions.

2. 2.

   Minimize differential systematic errors across the sample so that scatter and correlated residuals are minimally affected by systematics.

3. 3.

   Cover a substantial fraction of the disk to probe populations with a range of enrichment histories.

4. 4.

   Retain a large enough sample to enable accurate measurements of median trends, scatter, and correlations.

Plots of $[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ show that DR17 ASPCAP abundances still have systematic trends with ${\rm log}\,g$ (see Griffith et al. 2021a). However, to get a large sample one cannot afford to take too narrow a range of ${\rm log}\,g$. Luminous giants provide the best coverage of a wide range of the disk.

From the DR17 data set, we remove stars with the ASPCAP STAR_BAD or NO_ASPCAP_RESULT flags set, and we remove stars with flagged $[{\rm Fe}/{\rm H}]$ or $[{\rm Mg}/{\rm Fe}]$ measurements. We use only stars targeted as part of the main APOGEE survey (flag EXTRATARG=0) to avoid any selection biases associated with special target classes. We use the DR17 “named” abundance tags X_FE, which apply additional reliability cuts for each element (see §5.3.1 of Jönsson et al. 2020). As a compromise among the considerations above, we have adopted the following sample selection cuts:

1. 1.

   $R=3-13\,{\rm kpc}$, $|Z|\leq 2\,{\rm kpc}$ (399,573 stars)

2. 2.

   $-0.75\leq[{\rm Mg}/{\rm H}]\leq 0.45$ (387,218 stars)

3. 3.

   ${\rm SNR}\geq 200$ for $[{\rm Mg}/{\rm H}]>-0.5$; ${\rm SNR}\geq 100$ for $[{\rm Mg}/{\rm H}]<-0.5$ (160,133 stars)

4. 4.

   ${\rm log}\,g=1-2.5$ (65,611 stars)

5. 5.

   $T_{\rm eff}=4000-4600$ (34,410 stars) .

Numbers in parentheses indicate the number of sample stars remaining after each cut. Spectroscopic distances for computing $R$ and $Z$ are taken from the DR17 version of the AstroNN catalog (see Leung & Bovy 2019a); at distances of many kpc, these spectroscopic estimates are more precise than those from Gaia parallaxes. We use a lower ${\rm SNR}$ threshold below $[{\rm Mg}/{\rm H}]=-0.5$ to retain a sufficient number of low metallicity stars. The combination of cuts 4 and 5 eliminates red clump (core helium burning) stars (see Vincenzo et al. 2021a), which might have different measurement systematics from red giant branch (RGB) stars and could thus artificially add scatter or correlated deviations. The APOGEE red clump stars are themselves a well controlled and powerful sample (Bovy et al., 2014), and it would be useful to repeat some of our analyses below for the red clump sample and to understand the origin of any differences.

We compute $[{\rm X}/{\rm H}]$ values as the sum of the ASPCAP quantities X_FE and FE_H. We take the quantity X_FE_ERR as the statistical measurement uncertainty in $[{\rm X}/{\rm H}]$. Although FE_H has its own statistical uncertainty, we are primarily interested in differential scatter among elements, and all abundances in a given star use the same value of FE_H. ASPCAP abundance uncertainties are estimated empirically as a function of SNR, $T_{\rm eff}$, and metallicity using repeat observations of a subset of stars (see §5.4 of Jönsson et al. 2020). These empirical errors are usually larger (by a factor of several) than the $\chi^{2}$ model-fitting uncertainty. This procedure means that the adopted observational uncertainty for a given element is representative of that for stars with the same global properties and SNR but does not reflect the specifics of the individual star’s spectrum near the element’s spectral features. In the rare cases where the $\chi^{2}$ model-fitting uncertainty exceeds the empirical uncertainty, the fitting uncertainty is reported instead. Some stars have flagged values of individual elements, in which case we keep the star in the sample but omit the star from any calculations involving those elements. These cuts eliminate 562 Ce values but no more than two values for other elements.

The C and N surface abundances of RGB stars differ from their birth abundances because the CNO cycle preferentially converts ¹²C to ¹⁴N and some processed material is dredged up to the convective envelope (e.g., Iben 1965; Shetrone et al. 2019). However, because the extra N nuclei come almost entirely from C nuclei, leaving the O abundance much less perturbed, the number-weighted C+N abundance is nearly equal to the birth abundance, with theoretically predicted differences $\sim 0.01$ dex over most of the ${\rm log}\,g$ range considered here (Vincenzo et al., 2021b). We therefore take C+N as an “element” in our analysis, computing

where 8.39 and 7.78 are our adopted logarithmic values of the solar C and N abundances (Grevesse et al., 2007) on the usual scale where the hydrogen number density is 12.0. We somewhat arbitrarily set the uncertainty in $[{\rm(C+N)}/{\rm H}]$ equal to the ASPCAP uncertainty in $[{\rm C}/{\rm Fe}]$, i.e., to C_FE_ERR. While the fractional error in N may exceed the fractional error in C, N contributes only 20% to C+N for a solar C/N ratio.

As discussed by Jönsson et al. (2020) and Holtzman et al. (in prep.), the APOGEE abundances include zero-point shifts of up to 0.2 dex (though below 0.05 dex for most elements) chosen to make the mean abundance ratios of solar metallicity stars in the solar neighborhood satisfy $[{\rm X}/{\rm Fe}]=0$. These zero-point shifts are computed separately for giant and dwarf stars. Here we use a particular set of ${\rm log}\,g$ and $T_{\rm eff}$ cuts and a sample that spans the Galactic disk. We have therefore chosen to apply additional zero-point offsets that force the median abundance ratio trends of the high-Ia population in our sample to run through $[{\rm X}/{\rm Mg}]=0$ at $[{\rm Mg}/{\rm H}]=0$. These offsets are reported in Table 1; the Mg offset is zero by definition. The order of elements in the Table follows that used in plots below, based on dividing elements into related physical groups. The V and Ce offsets are 0.222 dex and 0.125 dex, while others are below 0.1 dex and mostly below 0.05 dex. Since ${\rm log}\,g$ trends are also present in APOGEE at this level, we regard it as reasonable to treat these as calibration offsets rather than assume that the Sun is atypical of stars with similar $[{\rm Mg}/{\rm H}]$ and $[{\rm Mg}/{\rm Fe}]$. However, this is a debatable choice. The most important offset is the one applied to Fe because the $[{\rm Fe}/{\rm Mg}]$ abundances determine the values of $A_{\rm cc}$ and $A_{\rm Ia}$, though we note that our choice of $[{\rm Fe}/{\rm Mg}]_{\rm pl}$ has a similar impact and is uncertain at a similar level. Furthermore, we identify $[{\rm Fe}/{\rm Mg}]_{\rm pl}$ from data that have the Fe offset applied (Figure 1), and much of the impact of a different offset would be absorbed by the associated change in $[{\rm Fe}/{\rm Mg}]_{\rm pl}$. The zero-point offsets for other elements have a small but not negligible impact on our derived values of $q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$. They should have minimal impact on residual abundances, since the 2-process model is calibrated to reproduce the observed median sequences. Table 1 also lists slopes of trends with $T_{\rm eff}$ that are discussed in §5.1 below (see equation 30).

We adopt a high, ${\rm SNR}\geq 200$ threshold for most of our analyses because we want to minimize the impact of observational errors on our results, especially the statistics and correlations of residual abundances. However, for some purposes we want to improve our coverage of the inner Galaxy, where distance and extinction leave fewer stars bright enough to pass this high threshold. For these analyses we lower the SNR threshold to 100 at all $[{\rm Mg}/{\rm H}]$ values, which increases the sample to 55,438 (a factor of 1.6) and increases the number of stars at $R=3-5\,{\rm kpc}$ by a factor of 4.3. We refer to this as the SN100 sample, but our calculations and plots use the higher threshold sample unless explicitly noted otherwise.

We separate our sample into low-Ia and high-Ia populations (conventionally referred to as “high-$\alpha$” and “low-$\alpha$,” respectively), using the same dividing line as W19:

For consistency with W19, we apply this separation to the APOGEE abundances before adding the zero-point offsets in Table 1, but we include the offsets in all of our subsequent calculations and plots. The distribution of our sample stars in $[{\rm Mg}/{\rm Fe}]$ vs. $[{\rm Mg}/{\rm H}]$, together with the median sequences and inferred CCSN iron fractions, have been shown previously in Figure 1.

The left panels of Figure 4 show $[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ distributions for other $\alpha$-elements (O, Si, S, Ca) in the low-Ia and high-Ia populations, with median sequences shown by connected large points. These can be compared to corresponding distributions in Figure 8 of W19, based on DR14 data; not surprisingly, the results are similar. The median [O/Mg] trends are nearly flat, with only a small separation between the low-Ia and high-Ia median sequences, as expected if the production of both O and Mg is dominated by CCSN with metallicity-independent yields. If real, the $\approx 0.05$-dex separation between the sequences suggests a small contribution to O abundances from a delayed source, perhaps AGB stars rather than SNIa. For Si and Ca the sequence separations are progressively larger, implying larger SNIa contributions to these elements as predicted by nucleosynthesis models (e.g., Andrews et al. 2017; Rybizki et al. 2017). For S the two median sequences are very close, implying minimal SNIa contribution to S production, and they are sloped, implying S yields from CCSN that decrease with increasing metallicity. They diverge slightly at high $[{\rm Mg}/{\rm H}]$, implying a growing $q^{X}_{\rm Ia}$.

The right panels of Figure 4 show the values of $q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ derived from these median sequences via equations (26) and (25), using the ratios $A_{\rm Ia}/A_{\rm cc}$ along the two sequences shown in Figure 1. As discussed in §2.2, with a general metallicity dependence the 2-process model fits the observed median sequences exactly; empirical evidence for the qualitative validity of the model will come from the reduction of scatter and correlations in residual abundances shown below in §5. For O the inferred $q^{X}_{\rm cc}\approx 0.9$ at all $[{\rm Mg}/{\rm H}]$, declining slightly at the highest metallicity. For Si and Ca the inferred values of $q^{X}_{\rm Ia}$ at solar $[{\rm Mg}/{\rm H}]$ are 0.19 and 0.26, respectively. For S the low-Ia median sequence crosses above the high-Ia median sequence at low metallicity, a violation of the 2-process model assumptions that leads to slightly negative values of $q^{X}_{\rm Ia}$. Given the large observational scatter in S abundances, this sequence-crossing appears compatible with observational fluctuations, so we do not regard it as a serious problem. The inferred $q^{X}_{\rm cc}$ for S declines continuously with increasing $[{\rm Mg}/{\rm H}]$, tracking the sloped median sequences. Because our zero-point offsets are chosen to give $[{\rm X}/{\rm Mg}]=0$ on the high-Ia sequence at $[{\rm Mg}/{\rm H}]=0$, and because stars with $[{\rm Mg}/{\rm Fe}]=[{\rm Mg}/{\rm H}]=0$ have $A_{\rm Ia}=A_{\rm cc}=1$ by definition, our fits always yield $q^{X}_{\rm cc}+q^{X}_{\rm Ia}=1$ at $[{\rm Mg}/{\rm H}]=0$ (see equation 20). However, this constraint does not apply at other metallicities. Red and blue curves in these panels show the fraction of each element that is inferred to come from CCSN in stars on the low-Ia or high-Ia sequence (equation 6). Even if $q^{X}_{\rm Ia}>0$, the low-Ia population has $f^{X}_{\rm cc}\approx 1$ at low $[{\rm Mg}/{\rm H}]$ because these stars have $[{\rm Mg}/{\rm Fe}]\approx[{\rm Mg}/{\rm Fe}]_{\rm pl}$, implying (at least according to the 2-process model assumptions) that nearly all enrichment is from CCSN.

Figure 5 shows $[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ distributions, median sequences, and inferred 2-process vectors for Na, Al, K, and the element combination C+N. Because they have odd atomic numbers, the nucleosynthesis of Na, Al, and K is fundamentally different from that of $\alpha$-elements even within massive stars. C and N are both expected to have significant contributions from AGB stars in addition to prompt contributions from CCSN and massive star winds, and the AGB yields of N are predicted to have substantial metallicity dependence (e.g., Karakas 2010; Ventura et al. 2013; Cristallo et al. 2015). Similar to W19, the median sequences for K and Al show little separation between the low-Ia and high-Ia populations, indicating a dominant contribution from CCSN; in detail, the DR17 data show a slightly larger sequence separation for Al and slightly smaller for K. More significantly, the DR17 data show [Al/Mg] trends that are essentially flat over this $[{\rm Mg}/{\rm H}]$ range while the DR14 data showed an increasing trend that implied CCSN yields increasing with metallicity.

As in W19, [Na/Mg] trends show a large separation between low-Ia and high-Ia populations implying a substantial delayed contribution to Na enrichment. Standard nucleosynthesis models predict that SNIa and AGB contributions are small compared to CCSN (Andrews et al., 2017; Rybizki et al., 2017), so this evidence for delayed enrichment comes as a surprise. The Na features in APOGEE spectra are weak, making the abundance measurements noisy and susceptible to systematic errors, but there is no obvious effect that would cause artificially boosted Na measurements at this level for high-Ia stars relative to low-Ia stars of the same $[{\rm Mg}/{\rm H}]$. A comparable sequence separation is also found in GALAH DR2 (Griffith et al., 2019). The $q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ values inferred from the 2-process fit are comparable in magnitude over most of the $[{\rm Mg}/{\rm H}]$ range. The inferred metallicity-dependence is complex, but given the scatter and uncertainties of the Na measurements it should be treated with caution.

While W19 considered P as an additional odd-$Z$ elements, the analyses in Jönsson et al. (2020) suggest that APOGEE’s P measurements in DR16 are not robust. The P abundances are improved in DR17, but they remain subject to significant systematics, and we have elected to omit P from this paper.

The two [(C+N)/Mg] sequences show a separation nearly as large as the two [Na/Mg] sequences, again implying a substantial delayed contribution. Nucleosynthesis models predict a moderate AGB contribution to C and a dominant AGB contribution to N (Andrews et al., 2017; Rybizki et al., 2017), so this result is qualitatively expected. The inferred metallicity dependence is complex, with $q^{X}_{\rm cc}$ rising with $[{\rm Mg}/{\rm H}]$ before leveling out and dropping at high metallicity, and the opposite trend for $q^{X}_{\rm Ia}$. We caution, however, that the separation into prompt and delayed contributions is not quantitatively accurate for elements that have large AGB contributions because it is based on tracking Fe from SNIa, and the delay time distributions for AGB enrichment and SNIa enrichment are different (see, e.g., Figure 5 of Johnson & Weinberg 2020). Our present analysis does not allow us to separate the roles of C and N in these trends, though for stars with asteroseismic mass measurements one can apply corrections from stellar evolution models to infer birth abundances of the two elements (see Vincenzo et al. 2021b). The observed [(C+N)/Mg] sequences can themselves provide a quantitative test of chemical evolution models that track both elements. The high-Ia medians of [(C+N)/Mg], [Na/Mg], and [Al/Mg] all show drops in the lowest metallicity bin $-0.75\leq[{\rm Mg}/{\rm H}]<-0.65$. This bin contains just 53 stars, so this drop could simply be a statistical fluctuation, though it might also be affected by accreted halo stars becoming a significant fraction of the high-Ia population at this low metallicity (see §7).

Figures 6 and 7 show sequences and 2-process parameters for iron peak elements with even atomic number (Cr, Fe, Ni) and odd atomic number (V, Mn, Co), respectively. Trends are similar to those shown for DR14 data by W19, though in W19 the $[{\rm Cr}/{\rm Mg}]$ trends are flatter with $[{\rm Mg}/{\rm H}]$ and the $[{\rm V}/{\rm Mg}]$ trends are steeper and with somewhat larger separation between the low-Ia and high-Ia sequences. For Fe, $q^{X}_{\rm Ia}=q^{X}_{\rm cc}=0.5$ at all $[{\rm Mg}/{\rm H}]$ as a consequence of adopting $[{\rm Mg}/{\rm Fe}]_{\rm pl}=0.3\approx\log_{10}2$ and assuming metallicity independence. For Cr we also infer $q^{X}_{\rm Ia}\approx q^{X}_{\rm cc}\approx 0.5$ at $[{\rm Mg}/{\rm H}]=0$. APOGEE Cr abundances exhibit apparent systematics for a significant fraction of stars above solar metallicity (Griffith et al., 2021a), so we regard the median sequences and inferred 2-process parameters as unreliable in the super-solar regime. While $[{\rm Ni}/{\rm Mg}]$ trends are similar to $[{\rm Fe}/{\rm Mg}]$ trends, the separation of low-Ia and high-Ia sequences is smaller, implying that CCSN contribute 60% of the Ni at solar abundances vs. 50% for iron. We find somewhat higher CCSN fractions at solar abundances for Co and V, 67% and 74%, respectively. (To phrase things still more precisely, in a star with [Fe/Mg]=[Ni/Mg]=[Co/Mg]=[V/Mg]=[Mg/H]=0, we infer that 50/60/67/74% of the star’s Fe/Ni/Co/V atoms were produced in CCSN.)

As in W19 we find that Mn has the largest SNIa contribution of any APOGEE element. Note that Bergemann et al. (2019) find a 0.15-dex difference between 1D LTE and 3D NLTE abundances from H-band Mn I lines, with little dependence on $[{\rm Fe}/{\rm H}]$, $T_{\rm eff}$, or ${\rm log}\,g$; here we take the ASPCAP Mn determinations at face value. Although the low-Ia median sequence in $[{\rm Mn}/{\rm Mg}]$ is steeply rising, this slope can be largely explained by the increasing SNIa enrichment fraction along the sequence, so that the inferred metallicity dependence of $q^{X}_{\rm cc}$ is weak. We infer a sharp rise in $q^{X}_{\rm Ia}$ at super-solar metallicity, needed to explain the rising $[{\rm Mn}/{\rm Mg}]$ trend on the high-Ia sequence. Given the spectral modeling and calibration uncertainties in the super-solar regime, the rising trend of $q^{X}_{\rm Ia}$ should be viewed with some caution, but it could suggest a change in the physical properties of SNIa progenitors or explosion mechanisms in super-solar stellar populations. A similar pattern of rising $q^{X}_{\rm Ia}$ at $[{\rm Mg}/{\rm H}]>0$ is seen for C+N, Na, V, Co, and (more weakly) Ni. These common trends could indicate a delayed source (SNIa or AGB) that becomes important for all of these elements at high metallicity, though they could also be a sign that our assumptions for separating prompt and delayed components are breaking down in this regime. Like Na, Al, and C+N, the median trend of the high-Ia population drops sharply in the lowest $[{\rm Mg}/{\rm H}]$ bin for [Mn/Mg] and [Co/Mg], and to a lesser extent for [Ni/Mg].

Figure 7 also presents results for the s-process element Ce. Like Na, V, and K, APOGEE Ce abundances have relatively large statistical uncertainties (mean of 0.043 dex in our sample), and the large scatter about the median trends is likely dominated by observational errors. DR14 did not include Ce abundances, so we cannot compare to W19. However, the large separation between the low-Ia and high-Ia sequences and the non-monotonic metallicity dependence of the high-Ia sequence are qualitatively similar to results from GALAH DR2 for the neutron capture elements Y and Ba (Griffith et al., 2019). A rising-then-falling metallicity dependence is expected for AGB nucleosynthesis of heavy s-process elements: at low [Fe/H] the number of seeds available for neutron capture increases with increasing metallicity, but at high [Fe/H] the number of neutrons per seed becomes too low to produce the heavier s-process elements (Gallino et al., 1998). As with C+N, the decomposition into prompt and delayed components implied by the $q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ values should be regarded as qualitative because the delay time distribution for AGB Ce production should differ from that of SNIa Fe production.

We report the values of $q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ for all elements in Tables 4 and 5 in the Appendix , along with the values of $A_{\rm Ia}/A_{\rm cc}$ along the two median sequences (Table 6). The value of $A_{\rm cc}$ follows from equation (13). These quantities can be used in equation (20) to exactly reproduce the median sequences shown in Figures 4-7. The values of $f^{X}_{\rm cc,\odot}=q^{X}_{\rm cc}(z=1)$ (equation 8) can be used to correct observed solar abundances to the abundances produced by CCSN, which can then be used to test the predictions of supernova models, as done by W19 (their fig. 20), by Griffith et al. (2019) (their fig. 17), and most comprehensively by Griffith et al. (2021b), who carefully investigate the interplay between IMF-averaged supernova yields and black hole formation scenarios. The solar values of $q^{X}_{\rm Ia}=1-q^{X}_{\rm cc}$ can be similarly used as a test of SNIa yield models, a task we defer to future work.

With the 2-process vectors determined by fitting the median $[{\rm X}/{\rm Mg}]$ sequences of the low-Ia and high-Ia populations, we proceed to fit the values of $A_{\rm cc}$ and $A_{\rm Ia}$ for all individual sample stars as described in §2.3. We perform a $\chi^{2}$-minimization fit to the abundances $[{\rm Mg}/{\rm H}]$, [O/H], [Si/H], [Ca/H], [Fe/H], and [Ni/H], using the reported ASPCAP observational uncertainties for each measurement. We choose these six elements because they have small mean observational uncertainties, ranging from 0.0084 dex (Fe) to 0.0136 dex (Ca), because they have no major known systematic uncertainties in APOGEE data, because the production of these elements is theoretically expected to be dominated by CCSN and SNIa, and because their $q^{X}_{\rm Ia}/q^{X}_{\rm cc}$ ratios span a wide range, giving strong collective leverage on $A_{\rm Ia}$ and $A_{\rm cc}$. The only other abundance with a mean observational error in this range is [(C+N)/Mg], but we do not use this quantity in our fits because it does not represent a single element and because the production of C and (especially) N is expected to have significant AGB contributions. The [Mn/H] abundance also has a small mean observational uncertainty (0.0144 dex), but the strong and unusual metallicity dependence of $q^{X}_{\rm Ia}$ above $[{\rm Mg}/{\rm H}]=0$ could distort inferred $A_{\rm Ia}/A_{\rm cc}$ ratios in this regime if it is incorrect. The 2-process amplitudes inferred from this six-element fit are close to those inferred from $[{\rm Mg}/{\rm H}]$ and $[{\rm Fe}/{\rm Mg}]$ alone via equations (13) and (18), but they have smaller statistical uncertainties, are robust to observational errors in these two abundances alone, and mitigate artificial correlations among residual abundances from measurement aberration (see Fig. 15 below).

Figure 8 plots the distribution of stars in the plane of $A_{\rm Ia}/A_{\rm cc}$ vs. $A_{\rm cc}$ in zones of Galactic $R$ and $|Z|$, with red and blue points denoting stars in the low-Ia and high-Ia populations, respectively. For this plot we have used our SN100 sample to improve coverage of the inner Galaxy. Although we use our six-element fits for $A_{\rm Ia}$ and $A_{\rm cc}$, this map does not look noticeably different if we use the values inferred from $[{\rm Mg}/{\rm H}]$ and $[{\rm Fe}/{\rm Mg}]$ alone. The $x$-axis quantity $A_{\rm cc}$ is simply a linear measure of metallicity as traced by CCSN elements, in solar units. This plot is analogous to Figure 4 of H15, showing $[\alpha/{\rm Fe}]$ vs. $[{\rm Fe}/{\rm H}]$, and still more closely analogous to Figure 3 of W19, showing $[{\rm Fe}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$. Similar to those element-ratio maps, the low-Ia population is more prominent at small $R$ and large $|Z|$, and the metallicity ($A_{\rm cc}$) distribution of the high-Ia population shifts towards lower values at larger $R$. However, the non-linear relations between $[{\rm Mg}/{\rm H}]$ and $A_{\rm cc}$ (equation 13) and between $[{\rm Fe}/{\rm Mg}]$ and $A_{\rm Ia}/A_{\rm cc}$ (equation 18) highlight three features that are less obvious in these earlier maps. First, the median trend of $A_{\rm Ia}/A_{\rm cc}$ in the low-Ia population rises continuously and approximately linearly with $A_{\rm cc}$ up to $A_{\rm cc}\approx 1.5$, reaching $A_{\rm Ia}/A_{\rm cc}\approx 0.5$. Second, for $A_{\rm cc}<1$ the high-Ia stars also show a clear trend of increasing $A_{\rm Ia}/A_{\rm cc}$ with $A_{\rm cc}$, especially evident in the $R\geq 7\,{\rm kpc}$ annuli. Both of these trends follow from the fact that the low-Ia and high-Ia sequences in $[{\rm Mg}/{\rm Fe}]$ are sloped below $[{\rm Mg}/{\rm H}]=0$ (see Fig. 1), and their persistence in a plot based on six-element fits implies that these slopes are not caused by vagaries of $[{\rm Mg}/{\rm Fe}]$ abundance ratio measurements. Third, the $\approx 0.04$-dex scatter of $[\alpha/{\rm Fe}]$ along the low-Ia and high-Ia sequences, which is dominated by intrinsic scatter rather than measurement noise (Bertran de Lis et al., 2016; Vincenzo et al., 2021a), translates to substantial scatter in $A_{\rm Ia}/A_{\rm cc}$ at a given metallicity within each population. In the solar annulus ($R=7-9\,{\rm kpc}$) the distribution of $A_{\rm Ia}/A_{\rm cc}$ is clearly bimodal at sub-solar metallicity, with typical values of $0-0.3$ for the low-Ia population and $0.6-1.2$ for the high-Ia population. This bimodality reflects the bimodality of $[{\rm Fe}/{\rm Mg}]$ values, which Vincenzo et al. (2021a) demonstrate is an intrinsic feature of the underlying stellar populations that is robust to $|Z|$-dependent and age-dependent selection effects in the APOGEE sample.

Turning to residual abundances, Figure 9 shows examples of measured vs. predicted abundances for four stars. Recall that for each sample star we fit the two free parameters $A_{\rm cc}$ and $A_{\rm Ia}$ using the measured abundances of six elements, then predict all of the abundances using these two process amplitudes and the global values of $q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ that have been inferred from the median trends of the full sample. The first two stars in Figure 9 are low metallicity members ($A_{\rm cc}=0.584$ and 0.307) of the low-Ia population ($A_{\rm Ia}=0.054$ and -0.005), one with a $\chi^{2}$ value near the median for all sample stars and one with a high $\chi^{2}$ value near the 98th-percentile of the $\chi^{2}$ distribution. (Negative $A_{\rm Ia}$ values can arise for stars with $[\alpha/{\rm Fe}]$ values above the low metallicity plateau.) The third and fourth stars are solar metallicity ($A_{\rm cc}=1.036$, 1.131) stars from the high-Ia population, again one that is near the median of the $\chi^{2}$ distribution and one near the 98th-percentile.

For the first star, the 2-process model reproduces the observed abundance pattern quite well, though the $\chi^{2}$ value of 30.8 for 14 degrees of freedom (16 elements fit with two free parameters) is inconsistent with a purely statistical fluctuation for Gaussian measurement errors with the reported observational uncertainties. The largest residuals are for S (0.13 dex), V (0.09 dex), Na (0.07 dex), and Cr (0.05 dex), elements with relatively large observational uncertainties in APOGEE. The second star shows $\sim 0.2$-dex residuals for several elements, including C+N, Al, V, and Ce, some overpredicted and some underpredicted. The predicted abundances of the third star are all nearly solar, since $A_{\rm Ia}\approx A_{\rm cc}\approx 1$, and the observed abundances are also near solar, with the largest residual being 0.08 dex for Na. The fourth star shows large residuals ($\sim 0.3$ dex) for Na and Ce and smaller ($\sim 0.1$ dex) but statistically significant residuals for C+N and Mn. We will discuss other examples of high-$\chi^{2}$ outliers in §6.

The 2-process model fits the APOGEE abundances of most disk stars to an accuracy that is comparable to the reported observational uncertainties. However, the estimated intrinsic scatter about the 2-process predictions exceeds 0.01 dex for most elements (Figure 13), and the off-diagonal covariance of abundance residuals demonstrates the physical reality of intrinsic deviations even among stars that appear individually well described by the model (Figures 10 and 15). In this section we examine a selection of stars whose measured abundances are poorly described by the 2-process model, i.e., with high values of $\chi^{2}$. We refer to these stars as outliers, but we note that with sufficiently precise measurements it is likely that most stars would show statistically robust deviations from the 2-process fit. Some of these outlier stars may simply be extreme examples of the same correlated deviations present in the main stellar population, offering clues to the physical drivers of these deviations. In other cases, unusual abundances may arise from rare physical processes that do not affect most stars. Yet other high-$\chi^{2}$ cases arise from measurement errors that are much larger than the reported observational uncertainty, for reasons that may be simple (e.g., a poorly deblended line) or subtle (e.g., inaccurate interpolations in a grid of synthetic spectra at an unusual location in abundance space).

High $\chi^{2}$ values can arise from single deviant measurements, which may have a variety of mundane observational causes. To preferentially select genuine physical outliers, we have used a modified $\chi^{2}$ criterion in which (a) we use the total scatter (filled circles in Figure 13) rather than the observational uncertainty, and (b) for each star, we omit the element that makes the single largest contribution to $\chi^{2}$. This criterion thus requires at least two anomalous abundances, and it downweights the impact of elements that more frequently have observational errors much larger than the reported uncertainties. Figure 20 shows a selection of eight stars drawn from the top 2% of this modified $\chi^{2}$ distribution. We list both the original $\chi^{2}$ and the modified $\chi^{2}$ for each star. For reference, the 98%, 99%, and 99.5% highest values of the modified $\chi^{2}$ are 59.9, 97.0, and 154.7, respectively. We selected these eight stars after examining $\sim 40$ examples in the top 2%, illustrating a few of the common themes that we find within this high-$\chi^{2}$ population.

2M09431719-5350178 has nearly solar values of $[{\rm Mg}/{\rm H}]$ and $[{\rm Fe}/{\rm Mg}]$, but it has low values (relative to the 2-process predictions) of Ca, Na, Al, K, and Ce, by $\sim 0.25$ dex for Ce and 0.1-0.2 dex for the other elements. This star individually exemplifies the pattern shown by the block of observed correlations in Figure 15d, and we find similar behavior in some other high-$\chi^{2}$ stars. These examples and the residual correlations themselves hint at a common physical source that contributes to these elements and is deficient in some stars. However, we do not have a clear physical interpretation of this pattern. We have not noticed comparably clean examples in which all of these elements are high, though Figure 23 (discussed in §8) shows two examples with high average deviations among these elements.

2M05551243+2447549 is a lower metallicity, high-Ia star ($[{\rm Mg}/{\rm H}]=-0.40$, $[{\rm Fe}/{\rm Mg}]=0.01$) that shows a similar deficiency of Na, Al, and K but a Ce abundance that is enhanced by 0.34 dex, demonstrating that large deviations among these elements do not necessarily move in lockstep. This star also shows a large (0.36-dex) enhancement of S relative to the predicted, near-solar [S/Mg]. This star has broader lines than the synthetic spectral fit, suggesting high rotation, and it has a radial velocity spread of $\sim 40\,{\rm km}\,{\rm s}^{-1}$ over the 100 days that it was observed, implying a binary companion. While these properties could be connected to unusual abundances, it is also possible that high rotation is causing systematic errors in the ASPCAP abundance measurements, and in the spectroscopic ${\rm log}\,g$, which is low (by about 0.4 dex) relative to most stars of similar metallicity and $T_{\rm eff}$.