# Chemical Cartography with APOGEE: Two-process Parameters and Residual
Abundances for 288,789 Stars from Data Release 17

Tawny Sit Department of Astronomy, The Ohio State University, Columbus, OH
43210, USA  David H. Weinberg Department of Astronomy, The Ohio State
University, Columbus, OH 43210, USA  Adam Wheeler Department of Astronomy, The
Ohio State University, Columbus, OH 43210, USA  Christian R. Hayes NRC
Herzberg Astronomy and Astrophysics, 5071 West Saanich Road, Victoria, BC V9E
2E7, Canada  Sten Hasselquist Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, MD 21218, USA  Thomas Masseron Instituto de
Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Jennifer
Sobeck IPAC, MC 314-6, Caltech, 1200 E. California Blvd., Pasadena, CA 91125

###### Abstract

Stellar abundance measurements are subject to systematic errors that induce
extra scatter and artificial correlations in elemental abundance patterns. We
derive empirical calibration offsets to remove systematic trends with surface
gravity $\log(g)$ in 17 elemental abundances of 288,789 evolved stars from the
SDSS APOGEE survey. We fit these corrected abundances as the sum of a prompt
process tracing core-collapse supernovae and a delayed process tracing Type Ia
supernovae, thus recasting each star’s measurements into the amplitudes
$A_{\text{cc}}$ and $A_{\text{Ia}}$ and the element-by-element residuals from
this two-parameter fit. As a first application of this catalog, which is
$8\times$ larger than that of previous analyses that used a restricted
$\log(g)$ range, we examine the median residual abundances of 14 open
clusters, nine globular clusters, and four dwarf satellite galaxies. Relative
to field Milky Way disk stars, the open clusters younger than 2 Gyr show
$\approx 0.1-0.2$ dex enhancements of the neutron-capture element Ce, and the
two clusters younger than 0.5 Gyr also show elevated levels of C+N, Na, S, and
Cu. Globular clusters show elevated median abundances of C+N, Na, Al, and Ce,
and correlated abundance residuals that follow previously known trends. The
four dwarf satellites show similar residual abundance patterns despite their
different star formation histories, with $\approx 0.2-0.3$ dex depletions in
C+N, Na, and Al and $\approx 0.1$ dex depletions in Ni, V, Mn, and Co. We
provide our catalog of corrected APOGEE abundances, two-process amplitudes,
and residual abundances, which will be valuable for future studies of
abundance patterns in different stellar populations and of additional
enrichment processes that affect galactic chemical evolution.

††software:  Matplotlib (Hunter, 2007), astropy (Astropy Collaboration et al.,
2013, 2018, 2022), NumPy (Harris et al., 2020), SciPy (Virtanen et al., 2020),
Korg (Wheeler et al., 2023a, b)

##  1 Introduction

With highly multiplexed, high-resolution spectra, the APOGEE (Majewski et al.,
2017) and GALAH (De Silva et al., 2015) surveys have measured detailed
chemical fingerprints of hundreds of thousands of stars in the Milky Way (MW)
and its closest satellites. Although these surveys measure 15-20 elemental
abundances in each star, the abundance patterns of most MW disk and bulge
stars can be predicted to surprisingly high accuracy by a two-dimensional fit,
based on, e.g., the Mg and Fe abundances or Fe abundance and age (Weinberg et
al., 2019, 2022; Griffith et al., 2019, 2021, 2022, 2023; Ness et al., 2019,
2022; Ting & Weinberg, 2022; Ratcliffe & Ness, 2023). For well measured
elements, the rms intrinsic scatter of residuals from these two-parameter fits
is typically 0.01-0.04 dex. While this scatter is comparable to the
statistical measurement errors for individual stars, the correlation of
residuals demonstrates rich underlying structure that encodes information
about the astrophysical sources of the elements and the processes that govern
chemical enrichment (Ting & Weinberg, 2022; Griffith et al., 2022; Weinberg et
al., 2022).

Abundance measurements in any spectroscopic survey are subject to systematic
errors caused by imperfections in the model atmospheres, spectral synthesis
codes, and analysis pipelines used to infer the abundances from the spectra.
Because the evolved stars targeted by APOGEE span a wide range of $\log(g)$
and $T_{\rm eff}$, differential systematics across the sample can artificially
inflate the scatter of residual abundances and produce spurious correlations
of residuals. Previous studies have addressed this problem by restricting the
$\log(g)$ and $T_{\rm eff}$ range of the sample (Ting & Weinberg, 2022;
Weinberg et al., 2022), by resampling populations to a matched $\log(g)$
distribution before comparing abundance trends (Griffith et al., 2021), or by
applying a local calibration fit to nearest neighbors in a parameter space
that includes $\log(g)$ (Ness et al., 2022). Eilers et al. (2022) fit an
abundance model depending on orbital actions and $\log(g)$, then subtract off
the $\log(g)$ dependencies, in essence making the assumption that all stars on
the same orbit have the same birth abundances. In this paper we use an
empirical approach to calibrate the systematic trends of APOGEE abundances
with $\log(g)$, which enables us to derive corrected abundances that greatly
suppress differential systematics across the range $0\leq\log(g)\leq 3.5$.
Weinberg et al. (2019) and Griffith et al. (2019) show that the median trends
of [X/Mg] vs. [Mg/H] are nearly universal throughout the MW disk and bulge for
all well measured APOGEE elements, provided one separates the low-$\alpha$ and
high-$\alpha$ stellar populations. Given this universality, and the fact that
any individual star evolves through a wide range of $\log(g)$ during its post-
main sequence lifetime, the abundances of most elements in a star remain
relatively constant throughout its life. Therefore, we assume that any
dependence of the median trend on $\log(g)$ is most likely a consequence of
abundance systematics rather than a genuine physical effect. We make this
assumption to derive calibration offsets as a function of $\log(g)$ and [Mg/H]
that force all [X/Mg] median trends to match those of $\log(g)=1.75$ stars.
With a large sample, median abundances can be measured to precision much
higher than that of individual stars, so we can robustly derive offsets at the
0.01-dex level.

After deriving corrected abundances, we follow the approach of Weinberg et al.
(2022, hereafter W22) and fit each sample star with a two-process model
(Weinberg et al., 2019, 2022; Griffith et al., 2019, 2021, 2022, 2023) that
describes the abundances as the sum of a prompt enrichment process associated
with core collapse supernovae (CCSN) and a delayed enrichment process
associated with Type Ia supernovae (SNIa). This procedure recasts each star’s
abundances into two amplitudes $A_{\text{cc}}$ and $A_{\text{Ia}}$, which
capture the main enrichment trends, and element-by-element residuals, which
capture the deviations from these main trends. While we use the same APOGEE
DR17 (Abdurro’uf et al., 2022) data set as W22, our expanded $\log(g)$ range
gives us a final catalog of residual abundances with $\approx 8$ times more
stars. The expanded sample is especially valuable for stellar populations
whose APOGEE targets lie mostly outside the $\log(g)$ and $T_{\rm eff}$ range
of W22, such as luminous low-gravity giants in dwarf satellites and the MW
bulge, and red clump stars in open clusters.

As a first application of our expanded catalog, we examine the median residual
abundances of 14 open clusters, nine globular clusters, and four dwarf
satellites (the Large and Small Magellanic Clouds, the Sagittarius dwarf and
stream, and Gaia Sausage/Enceladus). Taking advantage of medians for large
samples allows us to robustly measure 0.05-dex level abundance differences
between these stellar populations and MW field stars matched in overall CCSN
and SNIa enrichment. The open cluster sample allows us to examine trends of
individual elements with cluster age, similar to the analysis presented by
Griffith et al. (2022) using GALAH data. The most important product of this
paper is our catalog of two-process parameters and residual abundances, which
can also be used to obtain [X/H] values corrected for $\log(g)$ systematics.
We anticipate that this catalog will be useful for many investigations,
including: characterizing additional enrichment sources such as AGB stars or
rare supernovae; quantifying stochastic effects such as incomplete sampling of
the initial mass function (IMF); revealing subtle differences in the abundance
patterns of the bulge, radial and vertical zones of the disk, halo, and
satellites; identifying groups with distinctive abundance patterns that could
indicate a common birth environment; identifying trends with stellar age or
binarity; defining chemically homogeneous samples for reconstructing the
Galactic potential (Price-Whelan et al., 2021); uncovering trends with stellar
age or binarity; and identifying rare outliers that could be signs of exotic
physical processes or unusual measurement errors.

This paper is organized as follows. In Section 2, we describe our data and
sample selection criteria from APOGEE DR17. In section 3, we describe our
derivation of calibration offsets that remove systematic trends with $\log(g)$
in median abundance trends, and apply these offsets to our data. We fit the
two-process model to our sample and derive residual abundances in Section 4.
We then apply our expanded residual abundance catalog to examine residual
abundances in open clusters (Section 5.1), globular clusters (Section 5.2),
and MW satellite galaxies (Section 5.3). Finally, we discuss and summarize our
results, and outline future prospects, in Section 6. Appendix C provides a
user’s guide to the residual abundance catalog. We note that this catalog is
also valuable for “traditional” abundance analyses of APOGEE populations that
span a wide range of $\log(g)$, as the [X/H] abundances themselves are
corrected for $\log(g)$ systematics.

##  2 Data

We use data from DR17 (Abdurro’uf et al., 2022) of the SDSS-IV (Blanton et
al., 2017) APOGEE survey (Majewski et al., 2017). This data release contains
the completed collection of over 650,000 high-resolution ($R\sim 22,500$),
near-infrared (H-band, 1.51–1.70 $\mu$m) spectra from the APOGEE spectrographs
(Wilson et al., 2019) on the 2.5 m Sloan Foundation telescope (Gunn et al.,
2006) at Apache Point Observatory, New Mexico, and the du Pont telescope
(Bowen & Vaughan, 1973) at Las Campanas Observatory, Chile. Targeting for the
APOGEE survey is described in Zasowski et al. (2013, 2017); Beaton et al.
(2021); Santana et al. (2021). Spectral data reduction and calibrations for
APOGEE are performed by a dedicated data processing pipeline described in
Nidever et al. (2015); the reduced spectra then serve as inputs to the APOGEE
Stellar Parameters and Abundances Pipeline (ASPCAP; Holtzman et al. 2015;
García Pérez et al. 2016).

We primarily use the stellar parameters and abundances derived by ASPCAP.
ASPCAP uses FERRE (Allende Prieto et al., 2006) to fit APOGEE spectra to a
grid of synthetic spectra (Mészáros et al., 2012; Zamora et al., 2015)
generated with Synspec (Hubeny & Lanz, 2011) from MARCS model atmospheres
(Gustafsson et al., 2008) and an H-band line list (Shetrone et al., 2015;
Hasselquist et al., 2016; Cunha et al., 2017; Smith et al., 2021), with NLTE
calculations used for Na, Mg, K, and Ca (Osorio et al., 2020). Alternative
analyses using spectral grids synthesized with Turbospectrum (Plez, 2012) are
available, and the major relevant difference between the grids is that
Turbospectrum uses 3D spherical geometry for radiative transfer in
$\log(g)\leq 3.0$ giants but no NLTE calculations. We choose to use the
default DR17 analysis with Synspec for this study. ASPCAP uses a two-step
fitting process: stellar atmospheric parameters are first determined with the
full APOGEE spectrum, and then these parameters are held constant while
individual abundances are fit for in smaller wavelength windows (García Pérez
et al., 2016). We use 17 species measured by ASPCAP in this study: C, N, O,
Na, Mg, Al, Si, S, K, Ca, V, Cr, Mn, Fe, Co, Ni, and Ce. We exclude Ti and Ti
II because they show trends that deviate from those in the literature
(Holtzman et al., in preparation). We use the C abundance measured from
molecular lines because it was shown to be more reliable than the abundance
measured from neutral atomic lines in DR16 (Jönsson et al., 2020).

Additionally, we include abundance measurements from the BACCHUS Analysis of
Weak Lines in APOGEE Spectra (BAWLAS) catalog (Hayes et al., 2022). The BAWLAS
catalog provides abundance measurements of elements with faint or blended
lines (Na, P, S, V, Cu, Ce, and Nd) for a sample of 126,362 high signal-to-
noise (S/N$>$150) spectra of APOGEE red giants using the Brussels Automatic
Code for Characterizing High accUracy Spectra (BACCHUS; Masseron et al. 2016).
It also reports C, N, and O abundances re-derived during the BACCHUS fitting
process (for more details, see Section 4.5.1 of Hayes et al. 2022). We exclude
Nd from this study because less than half of the stars in the sample used for
deriving $\log(g)$ calibrations and two-process vectors (see Section 2.1) had
valid Nd measurements. We also exclude P because its high upper limit in [P/H]
(see Hayes et al. 2022 for more details) resulted in poor coverage over our
sample’s metallicity space. For the common elements between ASPCAP and BAWLAS
(C, N, O, Na, S, V, and Ce), abundance measurements are treated independently,
as if they were separate elements.

Weinberg et al. (2019) advocated for the use of Mg rather than Fe as a
reference element because it has a single enrichment source, CCSN. We follow
their example and also use Mg as the reference element in this study.
Therefore, we first remove all stars whose [Fe/H] and/or [Mg/Fe] abundances
are flagged so that we have a reliable
$\text{[Mg/H]}=\text{[Mg/Fe]}+\text{[Fe/H]}$ value for every star in our
sample. Suspect/flagged abundances are reported as Not a Number (NaN) values
in the BAWLAS catalog. For consistency, we also assign NaN to BAWLAS elements
for stars not in the BAWLAS sample and convert flagged measurements for all
considered ASPCAP elements (other than [Fe/H] and [Mg/Fe], which were removed)
into NaN. The use of NaN allows us to easily omit individual stars from
calculations for specific elements, keep as many stars in the sample as
possible, and simply propagate the invalid element to our final catalog.

The surface abundances of C and N vary for the red giant branch (RGB) stars
that make up the majority of our sample (Section 2.1) due to the CNO cycle and
dredge-up events (Iben, 1965; Shetrone et al., 2019). However, the total
amount of C+N by number remains nearly equal to the birth abundance because
the extra N nuclei in the dredged-up CNO-processed material were created from
C nuclei. Therefore, following the example of W22 and Griffith et al. (2023),
we treat C+N as a single element, calculated as

$\begin{split}\text{[(C+N)/H]}=&\log_{10}\left(10^{\text{[C/H]+8.39}}+10^{\text{[N/H]}+7.78}\right)\\\
&-\log_{10}\left(10^{8.39}+10^{7.78}\right),\end{split}$ (1)

where 8.39 and 7.78 are the logarithmic solar abundances for C and N,
respectively, from Grevesse et al. (2007). C+N is calculated for both the
ASPCAP and BAWLAS measurements of C and N.

[X/H] abundances are calculated from the sum of [X/Fe] (reported by either
ASPCAP or BAWLAS) and [Fe/H] (reported by ASPCAP). For the uncertainty on
[X/H], we use the reported [X/Fe] measurement uncertainty from ASPCAP
(X_FE_ERR) or the empirical [X/Fe] uncertainty from BAWLAS (X_FE_ERR_EMP).
ASPCAP estimates uncertainties empirically as a function of S/N, $T_{\rm
eff}$, and metallicity using repeated measurements of stars, and these
uncertainties typically are larger than those reported by spectral model
fitting (Jönsson et al., 2020, Section 5.4). BAWLAS reports two sets of
uncertainties: a measurement uncertainty, determined from the scatter in line-
by-line measurements of a given element, and and an empirical uncertainty,
estimated from repeat observations (Hayes et al., 2022, Section 4.7). We use
the BAWLAS empirical uncertainties because they are most similar in spirit to
the ASPCAP uncertainties, though there are small differences in methodology
(e.g., BAWLAS empirical uncertainties do not depend on SNR or overall
metallicity, but do depend on individual abundances). Since each star uses the
same [Fe/H] measurement in our calculation of [X/H], and we are primarily
interested in the differential scatter between elements not between individual
stars, we do not use the reported ASPCAP [Fe/H] uncertainty except when
considering $\text{X}=\text{Fe}$. For the combined element C+N, we follow W22
and take the [C/Fe] uncertainty as the [(C+N)/H] uncertainty.

We also divide our sample (Section 2.1) into high-Ia (“low-$\alpha$”) and low-
Ia (“high-$\alpha$”) sequences following the dividing line in Weinberg et al.
(2019), where the low-Ia stars are defined as as those with:

$\begin{cases}\text{[Mg/Fe]}>0.12-0.13\text{[Fe/H]},&\text{[Fe/H]}<0\\\
\text{[Mg/Fe]}>0.12,&\text{[Fe/H]}>0.\end{cases}$ (2)

For consistency with Weinberg et al. (2019) and W22, this division is
determined prior to adding the $\log(g)$ calibration and zero-point offsets
(Sections 3 and 4).

Table 1: Sample Sizes | Low-IaaaSample cut defined in Equation 2. | High-IabbAll spectra not in the low-Ia sample. | Total
---|---|---|---
CalibrationccS/N $>$ 100, EXTRATARG = 0, 3 kpc $\leq R\leq$ 13 kpc, $|Z|<2$ kpc, and all full sample [Mg/H], $\log(g)$, and $T_{\rm eff}$ cuts. The calibration sample does not contain duplicate stars. | 40,513 | 111,051 | 151,564
BAWLASddSpectra having at least one valid BAWLAS measurement. Parentheses indicate the number of BAWLAS stars in the calibration sample. | 37,499 (18,406) | 88,662 (48,974) | 126,161 (67,380)
Fullee$-0.75\leq$ [Mg/H] $\leq 0.45$, $0\leq\log(g)\leq 3.5$, $3000\text{ K}\leq T_{\rm eff}\leq 5500\text{ K}$, and S/N $>$ 80. | 91,144 | 219,283 | 310,427

###  2.1 Sample Selection

We first define a “training” sample of DR17 stars to derive the $\log(g)$
calibrations (Section 3) and two-process vectors (Section 4). For this
calibration sample, we impose a S/N threshold of S/N $>$ 100, and we use only
stars targeted as part of the main APOGEE survey by requiring the flag
EXTRATARG=0, to avoid selection biases from special targeting classes.
Weinberg et al. (2019) showed that median trends in [X/Mg]—[Mg/H] are
universal throughout the Galactic disk, so we also restrict this sample to the
disk by selecting stars with 3 kpc $\leq R\leq$ 13 kpc and $|Z|<2$ kpc. $R$
and $Z$ are from the AstroNN value-added catalog for DR17 (Leung & Bovy,
2019a, b), which uses APOGEE spectra and Gaia eDR3 parallaxes to obtain
spectro-photometric distances from deep learning. From W22, we require that
$-0.75\leq$ [Mg/H] $\leq 0.45$; this metallicity range samples a wide range of
enrichment histories in the Galactic disk. Finally, we make additional cuts in
$\log(g)$ and $T_{\rm eff}$: $0\leq\log(g)\leq 3.5$, and $3000\text{ K}\leq
T_{\rm eff}\leq 5500\text{ K}$. These $\log(g)$ and $T_{\rm eff}$ cuts select
nearly the entire RGB and all red clump (RC) stars within our metallicity
range, as illustrated in Figure 1. In total, we use 151,564 stars to derive
$\log(g)$ calibrations and two-process vectors.

Figure 1: $\log(g)$ vs. $T_{\rm eff}$ for all APOGEE DR17 stars with
$-0.75\leq$ [Mg/H] $\leq 0.45$ and SNR$>$80\. Stars with flagged [Fe/H] and/or
[Mg/Fe] are not included. We show our sample selection cuts, $0<\log(g)<3.5$
and $3000\text{ K}<T_{\rm eff}<5500$ K, in the solid red box. For comparison,
the $\log(g)$ and $T_{\rm eff}$ cuts of W22 are shown by the dotted red box.

We use a less stringent set of cuts to define an “application” sample of stars
to which we apply our $\log(g)$ calibration, fit with a two-process model, and
derive residual abundances. We require only that $-0.75\leq$ [Mg/H] $\leq
0.45$, $0\leq\log(g)\leq 3.5$, $3000\text{ K}\leq T_{\rm eff}\leq 5500\text{
K}$, and S/N $>$ 80\. This larger sample includes all stars observed by
APOGEE, regardless of targeting status or location in the Galaxy. These cuts
result in a total of 310,427 entries in our final residual abundance catalog.

There are some stars in APOGEE that have duplicate entries (i.e., were
processed separately) in ASPCAP because they were observed in multiple unique
fields (see Section 2 of Jönsson et al. 2020). In our residual abundance
catalog, each ASPCAP entry is treated as a separate star, so all duplicates
(i.e., same APOGEE_ID) whose ASPCAP parameters meet the sample selection
criteria are included. We find that including the FIELD or LOCATION_ID columns
in our final catalog is sufficient to break degeneracies in APOGEE_ID. There
are no duplicate stars in the smaller calibration (“training”) sample. In the
larger final (“application”) sample, there are 288,789 unique stars in the
total 310,427 spectra/ASPCAP entries; 17,694 stars have two or more spectra.

##  3 log(g) Calibrations

In general, the differential impact of abundance measurement systematics can
be minimized by limiting a sample to a small range of $T_{\rm eff}$ and
$\log(g)$. However, W22 showed that subtle trends in residual abundances with
$T_{\rm eff}$ still exist and can induce artificial correlations between
elements; they correct for these trends by applying an additional, $T_{\rm
eff}$-dependent corrective offset to the ASPCAP abundances. Additionally,
Griffith et al. (2021) showed that $\log(g)$ systematics cause noticeable
changes in median ASPCAP abundance trends. Because of the correlation between
$\log(g)$ and $T_{\rm eff}$ on the giant branch (Figure 1), one can use either
parameter for calibrating systematics, and we choose $\log(g)$ because of its
more intuitive connection to evolutionary state.

In Figure 2, we show median abundance trends with [Mg/H] for an example
element, Al, in different surface gravity bins. The stars in this figure are
all high-Ia (low-$\alpha$) RGB stars in the MW disk from the main APOGEE
survey sample (i.e., no special targeting). Any single star traverses a range
of $\log(g)$ as it evolves off the main sequence, so we do not expect birth
abundances to vary with $\log(g)$. Figure 2 shows that this is not the case
for the measured Al abundance: the median [Al/Mg] vs. [Mg/H] trends are
visibly different for different $\log(g)$. This figure also illustrates the
power of using median abundances to identify subtle trends, since the
$\log(g)$ systematics are comparable in magnitude to the star-to-star scatter
but are measured at high precision in the median curves.

Figure 2: [Al/Mg] as a function of [Mg/H], colored by 0.5 dex bins in
$\log(g)$. Stars are randomly downsampled by a factor of five to reduce
crowding. In each $\log(g)$ bin, median [Al/Mg] values are calculated for
every 0.1 dex [Mg/H] bin and plotted in connected large points. Error bars,
usually smaller than the points, show uncertainties on the medians computed
from 1000 bootstrap resamplings of each ([Mg/H], $\log(g)$) bin. Abundances
are taken directly from ASPCAP with no further calibrations. This figure uses
only high-Ia (low-$\alpha$) stars in the SNR$>$100 main survey disk sample
used for calibration (see Section 2.1), and red clump stars have been removed.

We remove trends with $\log(g)$ similarly to the $T_{\rm eff}$ corrections in
W22 by applying a corrective offset to the ASPCAP or BAWLAS pipeline
abundance. Our key assumption is that the median abundance trends ([X/Mg] vs.
[Mg/H]) of the low-Ia and high-Ia disk populations should be independent of
$\log(g)$ over the range of our sample. As discussed in the introduction,
these trends are nearly universal through the disk and bulge (Weinberg et al.,
2019; Griffith et al., 2021), and any individual star crosses most of our
$\log(g)$ range during its red giant lifetime, so a dependence of the median
trend on $\log(g)$ is much more likely to be an artifact of abundance
systematics than a physical effect.

For each star, we apply offsets to the raw ASPCAP/BAWLAS abundance as follows:

$\text{[X/H]}_{\text{corr}}=\text{[X/H]}_{\text{raw}}+C^{X}_{\text{ZP}}+C^{X}_{\log(g)}(\text{[Mg/H]},\log(g)).$
(3)

The zero-point offset $C_{\text{ZP}}$ is further discussed in Section 4. The
$\log(g)$ calibration correction for each star $C_{\log(g)}$ depends on the
star’s metallicity [Mg/H] and surface gravity $\log(g)$.

Figure 3: Median differences for Al as a function of $\log(g)$ for selected
[Mg/H] bins. In each $\log(g)$ bin at a given [Mg/H], the median [Al/Mg] is
computed and the median [Al/Mg] in the fiducial $\log(g)=1.75$ bin is
subtracted. The median difference at $\log(g)=1.75$ is thus zero by definition
for all [Mg/H]. The $\log(g)=0.25$ bins are not shown because no [Mg/H] bin at
this $\log(g)$ contained more than 25 stars with valid Al measurements. The
top row shows the median differences used to compute $C_{\log(g)}$, and the
bottom row shows the median differences calculated after $C_{\log(g)}$ was
applied. The left and right columns separate the high-Ia (low-$\alpha$) and
low-Ia (high-$\alpha$) sequences, defined in Equation 2; the final
$C_{\log(g)}$ value is computed from the number-weighted average of the two
sequences. Red, green, blue, cyan, and magenta represent
$\text{[Mg/H]}=-0.5,-0.3,-0.1,0.1,\text{and }0.3$ bins. Filled circles
represent the red clump sample. Dashed gray lines indicate the median per star
measurement uncertainty for the high-/low-Ia samples. Uncertainty in the
median differences is shown by the shaded regions or error bars for the red
clump and is estimated from 1000 bootstrap resamplings of each ([Mg/H],
$\log(g)$) bin. This figure uses only the calibration sample (Section 2.1).

Stars in the smaller calibration sample (see Section 2.1) were divided into
high-Ia (low-$\alpha$) and low-Ia (high-$\alpha$) sequences following Equation
2. For each sequence, we calculated the median abundances of each element in
0.1 dex bins in [Mg/H] and 0.5 dex bins in $\log(g)$. The raw ASPCAP/BAWLAS
abundances were used to calculate these medians, and we required a minimum of
25 stars with valid abundance measurements in each bin to calculate a median.
Red clump (RC) stars are He-core burning and thus may have slightly different
systematics from the H-shell burning red giant branch (RGB) stars in the rest
of our sample. Therefore, all stars in the DR17 value-added RC catalog (Bovy
et al., 2014) were considered as their own separate $\log(g)$ bin, and they
were also split into high- and low-Ia sequences following Equation 2. Then,
for each [Mg/H] bin, we calculated the difference between the median [X/H]
abundance in each $\log(g)$ bin from the $\log(g)$ bin centered at a fiducial
value of $\log(g)=1.75$. We plot these median differences for a subset of
[Mg/H] bins for an example element, Al, in the top row of Figure 3. The
differences are similar, though not identical, for high-Ia and low-Ia stars,
supporting our interpretation that they are driven by systematics in the
spectral synthesis modeling that correlate primarily with $\log(g)$ and
overall metallicity.

Figure 4: $\log(g)$ correction values $C^{X}_{\log(g)}$ as a function of
$\log(g)$ for the $\text{[Mg/H]}=0$ bin. A flat offset has been applied so
that each sequence passes through 0 at $\log(g)=1.75$. Blue solid lines with
triangles and orange dashed lines with squares indicate ASPCAP and BAWLAS
abundances, respectively. Filled triangles/squares at $\log(g)=2.5$ indicate
the red clump sample. The gray band represents the $\pm 0.05$ range. This
figure uses the calibration sample (Section 2.1).

Next, we took the average between the high- and low-Ia sequences of the median
differences, weighted by the number of stars in the high/low-Ia bins. In doing
so, we further assume that the $\log(g)$ systematics do not depend on the
[$\alpha$/Fe] ratio at fixed [Mg/H]. Due to this weighted average, the median
differences after calibration are not perfectly flat at $\Delta\text{[X/H]}=0$
for the high- and low-Ia sequences individually, as illustrated by the bottom
row of Figure 3. However, in this example (and most cases) the median
differences in the well-populated $\log(g)$ bins are below 0.02 dex after
calibration. We could have chosen to force these differences to zero for the
(larger) high-Ia sample at the expense of slightly larger deviations for the
low-Ia sample. Finally, we added a flat offset to ensure that the median
$\text{[X/H]}=0$ at ($\text{[Mg/H]}=0$, $\log(g)=1.75$) for the high-Ia
sequence. These values define a correction function on a 2D grid of ([Mg/H],
$\log(g)$) points that is the same across the high and low-Ia sequences. We
plot the $C_{\log(g)}$ values as a function of $\log(g)$ at $\text{[Mg/H]}=0$
for all elements in Figure 4. For each non-RC star, we interpolate along this
grid to the star’s [Mg/H] and $\log(g)$ to obtain the $C_{\log(g)}$ for each
element. For RC stars, we interpolate only along [Mg/H] to obtain
$C_{\log(g)}$.

For most $\log(g)$ bins, the calibration offsets we apply are smaller than the
typical $\approx$0.04-dex observational uncertainty in [Al/H] for individual
stars. Because of the large number of stars per [Mg/H] bin, the median offset
is measured much more precisely, and failing to remove these $\log(g)$
systematics would lead to spurious abundance trends. Therefore, we expect the
most reliable calibrations in the most well-populated bins:
$1.25\geq\log(g)\geq 2.75$ and $-0.4\geq\text{[Mg/H]}\geq 0.2$. Outside of
this range, some elements may have too few stars in bin ($N\leq 25$) to
calculate a median difference; in these cases the final calibration offset is
copied from the nearest $\log(g)$ or [Mg/H] bin with a valid offset.
Furthermore, even in bins with enough stars, the abundance measurements
themselves can become more uncertain at the low $\log(g)$ (e.g.,
$\log(g)=0.25$, 0.75 bins) because the spectra of cooler stars become affected
by line blends. This may contribute to the remaining spread in median
differences for the low-Ia $\log(g)=0.75$ bin in Figure 3. However, strong
trends in individual offsets outside of the overall most reliable ($\log(g)$,
[Mg/H]) bins can still be fairly robust: in Figure 4, V shows the strongest
$\log(g)$ correction of any element (at [Mg/H] = 0) in the $\log(g)=3.25$ bin,
where the median differences used to compute the offset were calculated from
3671 stars (for ASPCAP).

##  4 Two-process Model and Residual Abundances

###  4.1 Two-process Vectors, $q^{X}_{\mathrm{cc}}(z)$ and
$q^{X}_{\mathrm{Ia}}(z)$

In this work, we adopt the two-process model as described in W22. In brief,
the two-process model describes chemical abundances as a combination of IMF-
averaged yields from CCSN and SNIa such that an element X produced from only
these two sources will have a solar-scaled abundance described by

$\text{[X/H]}=\log_{10}{\left[A_{\text{cc}}q^{X}_{\text{cc}}(z)+A_{\text{Ia}}q^{X}_{\text{Ia}}(z)\right]},$
(4)

where the metallicity $z=10^{\text{[Mg/H]}}$. $A_{\text{cc}}$ and
$A_{\text{Ia}}$ are amplitudes describing the relative amount of CCSN/SNIa
contribution and are fit to each individual star. They are normalized such
that $A_{\text{cc}}=A_{\text{Ia}}=1$ for a star with solar abundances, [X/H] =
0. The process vectors $q^{X}_{\text{cc}}(z)$ and $q^{X}_{\text{Ia}}(z)$ for
each element X are derived from the observed [X/Mg] vs. [Mg/H] median
sequences of the high-Ia and low-Ia stellar populations in the Galactic disk.
The actual derivation of $q^{X}_{\text{cc}}(z)$ and $q^{X}_{\text{Ia}}(z)$
follows Section 2 of W22 and makes the following key assumptions:

  1. 1.

Mg is a pure CCSN element, so $q^{\text{Mg}}_{\text{Ia}}(z)=0$.

  2. 2.

The Mg and Fe processes are independent of metallicity, so
$q^{\text{Mg}}_{\text{cc}}(z)=q^{\text{Mg}}_{\text{cc}}$,
$q^{\text{Fe}}_{\text{cc}}(z)=q^{\text{Fe}}_{\text{cc}}$, and
$q^{\text{Fe}}_{\text{Ia}}(z)=q^{\text{Fe}}_{\text{Ia}}$.

  3. 3.

The [Mg/Fe] abundance of the low-Ia population plateaus at
$\text{[Mg/Fe]}_{\text{pl}}=0.3$ ( W22, Figure 1, left panel), and stars on
the plateau only have Fe enrichment from CCSN ($A_{\text{Ia}}=0$).

The adopted value of $\text{[Mg/Fe]}_{\text{pl}}$ is a choice, and Griffith et
al. (2023) illustrate the impact of choosing slightly different values or a
slightly tilted plateau.

The APOGEE abundances contain zero-point shifts chosen to make the mean
abundance ratio of solar metallicity stars in the solar neighborhood [X/Fe] =
0 (Jönsson et al. 2020; Holtzman et al. in prep). The BAWLAS abundances are
similarly calibrated to a zero-point derived from a sample of solar
neighborhood stars (Hayes et al., 2022). To ensure that stars on the high-
Ia/low-$\alpha$ sequence with solar metallicity have solar [X/Mg] abundances,
we apply an additional flat zero-point offset $C^{X}_{\text{ZP}}$ that forces
the median [X/Mg] trend of the high-Ia population to pass through
$\text{[X/Mg]}=0$ in the $\text{[Mg/H]}=0$ bin. The spirit of our
$C_{\text{ZP}}$ correction is similar to that of the APOGEE and BAWLAS
analyses, but because our sample is distinct the zero-point offsets are not
identical. We effectively assume that the solar abundance mixture is typical
of high-Ia disk stars with [Mg/H] = 0. $C_{\text{ZP}}$ is derived from the
SNR$>$100 main survey disk sample, and the abundances used to calculate the
median values had the $\log(g)$ calibration $C_{\log(g)}$ previously applied
so that they trace the $\log(g)=1.75$ stars. We provide a table of
$C_{\text{ZP}}$ and $C_{\log(g)}$ values along with sample code for applying
the calibrations in Appendix A.

With both $\log(g)$ and zero-point corrections in hand, we calculate the
median low- and high-Ia trends in 0.1 dex [Mg/H] bins from
[X/H]${}_{\text{corr}}$ (Equation 3) for the SNR $>$ 100 main survey disk
sample ($N=151,564$, Section 2.1). We plot the median trends for the new
BAWLAS-only element Cu in Figure 5. The trend with increasing metallicity for
[Mg/H] $>-0.2$ and the $\approx$0.2 dex separation between the high- and low-
Ia tracks are similar to that of Co and Ni, the other Fe-peak elements heavier
than Fe, in this metallicity range, but the Cu trend is steeper. This suggests
that Cu has a similar level of delayed contributions as Co and Ni, but
potentially a stronger metallicity dependence at higher metallicities. In this
[Mg/H] $>-0.2$ metallicity range, our results also qualitatively match optical
results from GALAH. At [Mg/H] $<-0.2$, the [Cu/Mg] abundance increases with
decreasing [Mg/H], a trend only shared with V (in both the ASPCAP and BAWLAS
measurements of V). However, both elements have weak lines in the APOGEE
wavelengths, and the BAWLAS median Cu trend does not match those from optical
GALAH spectra at the same [Mg/H] (e.g., Griffith et al., 2022), so the
observed decreasing trend may be somewhat biased towards stars with higher Cu
abundances and stronger lines at lower metallicities. Hayes et al. (2022)
compared the overall BAWLAS Cu distribution to literature measurements from
high-resolution spectra and found similar trends, although Fe was used as the
reference element. The median trends of the other elements for both ASPCAP and
BAWLAS are similar to W22 so we do not reproduce them here, except for C+N and
O (Figure 6) which we discuss further in Section 4.3.

Figure 5: [Cu/Mg] vs. [Mg/H] for the SNR$>$100 main survey disk sample. High-
Ia stars are in blue and low-Ia stars in red. Stars are randomly downsampled
by a factor of two to reduce crowding. Connected large points indicate the
median values in bins of [Mg/H]. Abundances in this plot have been corrected
following Equation 3 so that $\log(g)$ trends are removed and the high-Ia
median sequence (large blue points) passes through $\text{[Cu/Mg]}=0$ at
$\text{[Mg/H]}=0$. The separation of low-Ia and high-Ia sequences indicates a
significant delayed contribution to Cu, which could come from SNIa, AGB stars,
or both. Figure 6: [X/Mg] vs. [Mg/H] for ASPCAP (left column) and BAWLAS
(right column) measurements of O (top row) and C+N (bottom row). Stars are
randomly downsampled by 10 for ASPCAP measurements and 5 for BAWLAS
measurements to reduce crowding. Colors and symbols are identical to that of
Figure 5. This figure uses data from the calibration sample.

We then use these medians to derive the process vectors $q^{X}_{\text{cc}}(z)$
and $q^{X}_{\text{Ia}}(z)$ for all elements following Equations 25 and 26 from
W22. We report the values of $q^{X}_{\text{cc}}(z)$ and $q^{X}_{\text{Ia}}(z)$
in Tables 3 and 4 in Appendix B.

Griffith et al. (2023) propose an alternative method of inferring process
vectors that does not involve median sequences. In the disk regime where these
sequences are clearly defined, Griffith et al. (2023) show that their method
gives process vectors nearly identical to those of the W22 method, but their
approach adapts more easily to small samples or to the low metallicity regime
where [$\alpha$/Fe] bimodality is not obvious. Since our calibration sample is
limited to the MW disk, we use the W22 method in this work, but we expect that
applying the Griffith et al. (2023) method to our $\log(g)$-calibrated
abundances should yield similar process vectors.

###  4.2 Two-process Amplitudes, $A_{\textrm{cc}}$ and $A_{\textrm{Ia}}$

Using Equation 4 and our derived $q^{X}_{\text{cc}}(z)$ and
$q^{X}_{\text{Ia}}(z)$ vectors, we can now fit for the process amplitudes
$A_{\text{cc}}$ and $A_{\text{Ia}}$ for all 310,427 observations of 288,789
unique stars in our final sample. First, $C^{X}_{\log(g)}$ is calculated for
each element of each star, and added to the raw reported abundance along with
the appropriate $C^{X}_{\text{ZP}}$ for that element. Following W22, we infer
$A_{\text{cc}}$ and $A_{\text{Ia}}$ from six APOGEE elements (Mg, O, Si, Ca,
Fe, Ni) that have small observational errors, a range of relative
contributions from CCSN/SNIa, and are theoretically unlikely to have
contributions from non-CCSN/SNIa sources. Specifically, we use Equation 4 to
calculate a predicted two-process [X/H] for X = Mg, O, Si, Ca, Fe, Ni, then
use the observed, corrected [X/H] (Equation 3) and reported measurement errors
to calculate $\chi^{2}$. For each star, we minimize the $\chi^{2}$ value of
the fit to these six elements using an iterative process. Starting with an
initial guess of $A_{\text{cc}}$ and $A_{\text{Ia}}$ based only on [Mg/H],
[Fe/Mg], and the plateau
$\text{[Fe/Mg]}_{\text{pl}}=-\text{[Mg/Fe]}_{\text{pl}}=-0.3$ (W22 Eqs. 13,
18), we hold either $A_{\text{cc}}$ or $A_{\text{Ia}}$ fixed while minimizing
the other. To avoid outliers impacting the best-fit $A_{\text{cc}}$ and
$A_{\text{Ia}}$ values, O, Si, Ca, and/or Ni were eliminated from the fitting
process if their abundance deviated by $>$5$\sigma$ from the initial guess.
This six-element method reduces the effect of measurement aberrations in the
residual abundances compared to using only Fe and Mg (e.g., Griffith et al.,
2019; Weinberg et al., 2019), as demonstrated in W22.

Figure 7: Histograms of the $\chi^{2}$ (left) and reduced $\chi^{2}$
($\chi^{2}_{\text{red}}$, right) of the two-process fits to our final sample
($N=\text{310,427}$). $\chi^{2}_{\text{red}}$ is defined by Equation 5. The
solid, dashed, and dotted gray lines mark the median, 95th percentile, and
99th percentile.

Once the best-fit process amplitudes are derived for each star, we use these
amplitudes and the global process vectors $q^{X}_{\text{cc}}(z)$ and
$q^{X}_{\text{Ia}}(z)$ in Equation 4 to predict the abundances for all
elements. The $\chi^{2}$ value reported in our catalog for each star is
calculated from all valid (non-NaN, i.e., existing and unflagged) abundances.
We also report a reduced $\chi^{2}$ for each star in our catalog, defined as

$\chi^{2}_{\text{red}}=\frac{\chi^{2}}{N_{\text{elem}}-2}$ (5)

where $N_{\text{elem}}$ is the number of elements with valid abundance
measurements for that star. The model has two free parameters, $A_{\text{cc}}$
and $A_{\text{Ia}}$, so we subtract 2 to obtain the total degrees of freedom.

The distributions of $\chi^{2}$ and $\chi^{2}_{\text{red}}$ for the final
sample are shown in Figure 7. The median $\chi^{2}$ value is $\approx$44 for a
maximum of 21 degrees of freedom (23 total elements, as we considered ASPCAP
and BAWLAS measurements separately, with two free parameters), and the median
$\chi^{2}_{\text{red}}$ is $\approx$2.7. This suggests that the true
distribution of abundances has intrinsic scatter not captured by the two-
process model and/or that the true abundance uncertainties are larger than
those predicted for Gaussian noise with the reported measurement error. W22
reported a median $\chi^{2}$ of 30 for 14 degrees of freedom, which is a
similar ratio to our results. Based on correlations of residual abundances,
Ting & Weinberg (2022), Griffith et al. (2022), and W22 demonstrate that at
least some of the excess $\chi^{2}$ is a consequence of physical intrinsic
scatter rather than underestimated statistical errors.

Figure 8: The process amplitudes of the full sample ($N=\text{310,427}$),
plotted as $A_{\text{Ia}}/A_{\text{cc}}$ vs. $A_{\text{cc}}$. The high-Ia
stars are in blue and the low-Ia stars in red. The bin size is 0.02 along both
axes.

In Figure 8, we plot the distribution of best-fit
$A_{\text{Ia}}/A_{\text{cc}}$ vs. $A_{\text{cc}}$ values. The high-Ia (in
blue) and low-Ia (in red) populations separate, similar to a typical [Mg/Fe]
vs. [Fe/H] plot but more closely analogous to [Fe/Mg] vs. [Mg/H], with
$A_{\text{cc}}$ providing a measure of metallicity as traced by CCSN elements.
The division of the two populations at $A_{\text{Ia}}/A_{\text{cc}}\approx
0.5$ reflects the definition of the populations (Equation 2). Unlike the flat
plateau at $\text{[Fe/Mg]}=-0.3$ for the low-Ia population in an [Fe/Mg] vs.
[Mg/H] plot, the low-Ia population shows a steady rise in
$A_{\text{Ia}}/A_{\text{cc}}$ as $A_{\text{cc}}$ increases. The high-Ia
population shows a similar but steeper increase in
$A_{\text{Ia}}/A_{\text{cc}}$ before plateauing at
$A_{\text{Ia}}/A_{\text{cc}}\approx 1$. The $A_{\text{Ia}}/A_{\text{cc}}$ rise
in both populations traces an increase in time-delayed SNIa enrichment over
time, and the shallower rise in the high-Ia trend plausibly results from
higher star formation efficiency, which allows the population to reach higher
$A_{\text{cc}}$ within the characteristic SNIa delay time
$\tau_{\text{Ia}}\approx 1-2$ Gyr. The flat plateau at
$A_{\text{Ia}}/A_{\text{cc}}\approx 1$ suggests that these stars form when the
CCSN and SNIa rates are approximately equal to each other, producing near-
solar abundance ratios. These general trends are more obvious in the
calibration sample, which consists only of MW disk stars. The high
$A_{\text{Ia}}/A_{\text{cc}}$ feature at $A_{\text{cc}}<0.5$ is not seen in
the disk and consists primarily of dwarf galaxy members (Section 5.3). In
detail, the appearance of this diagram depends on our key model assumption
that [Mg/Fe] = 0.3 is the ratio produced by pure CCSN enrichment
($A_{\text{Ia}}=0$) (Section 4.1).

###  4.3 Residual Abundances, $\Delta$[X/H]

With the two-process vectors and fitted $A_{\text{cc}}$ and $A_{\text{Ia}}$,
we can predict all of a star’s ASPCAP and BAWLAS abundances, but we do not
expect these predictions to be perfect. Some deviations will arise from
problematic spectra and/or measurement errors. However, other deviations,
particularly those common to certain populations of stars, may be real and
encode information about additional nucleosynthetic sources, chemically
peculiar stars, and abundance trends in stellar populations.

Figure 9: Residual abundances (Section 4.3) of Al as a function of $\log(g)$
before (left) and after (right) applying the $\log(g)$ calibration offsets
$C_{\log(g)}$. The calibration offsets were applied to abundances prior to
calculation of the median [Al/Mg] vs. [Mg/H] tracks, so that the left and
right plots use different sets of two-process vectors. The purple, magenta,
and peach lines indicate the median, 25th/75th percentiles, and 10th/90th
percentiles of each $\log(g)$ bin. Stars falling outside the 10th/90th
percentiles are plotted as points; red clump stars are indicated in blue. For
the $\log(g)$ bins centered at $\log(g)\geq 0.75$, our calibrations clearly
flatten out the overall trend with $\log(g)$ around median
$\Delta\text{[Al/H]}=0$. This figure uses the calibration sample (Section
2.1).

Following the convention of W22 and Griffith et al. (2022), we define residual
abundances in the rest of this work as observed$-$predicted, specifically:

$\Delta\text{[X/H]}=\text{[X/H]}_{\text{corr}}-\text{[X/H]}_{\text{2proc}},$
(6)

where [X/H]${}_{\text{corr}}$ is defined in Equation 3 and
[X/H]${}_{\text{2proc}}$ is the abundance predicted by the two-process model
(Equation 4). The catalog of $A_{\text{cc}}$, $A_{\text{Ia}}$, and residual
abundances for our sample of 288,789 stars is a major product of this paper.
We describe how to access and use it in Appendix C.

In Figure 9, we plot the residual abundances of Al, $\Delta$[Al/H], with and
without applying the $\log(g)$ calibrations offsets $C_{\log(g)}$ (Section 3).
The figure illustrates that our calibrations are effective at removing overall
trends with $\log(g)$ in the residual abundances, as expected. We have also
confirmed that they remove trends with $T_{\rm eff}$. No $\log(g)=0.25$ bin at
any metallicity contained enough stars in the calibration sample to calculate
$\log(g)$ offsets, resulting in a small number of stars with $\log(g)=0-0.5$
using offsets from the $\log(g)=0.75$ bin at matching [Mg/H].

Figure 10: Scatter in two-process residuals as a function of $\log(g)$,
quantified as half of the difference between 84th and 16th percentiles ($\pm
1\sigma$ of a Gaussian distribution) to reduce the effect of outliers. ASPCAP
abundances are shown in solid lines and triangles. BAWLAS abundances are shown
in dashed lines and squares. Filled triangles/squares indicate the red clump
sample, which has median $\log(g)\approx 2.5$. Filled diamonds at
$\log(g)=2.0$ show $\sigma_{68}$ of the two-process residuals from the smaller
sample in W22, with $\log(g)=1-2.5$, for comparison. The points marked around
$\log(g)=0$ represent the median measurement error for each element; the
colors and symbols of these points match those of the two-process residual
scatter. Data in this figure are from the full sample.

In Figure 10, we plot the dispersion of two-process residuals for each element
as a function of $\log(g)$. To eliminate sensitivity to outliers, we compute
the dispersion $\sigma_{68}$ as half the difference between the
84${}^{\text{th}}$ and 16${}^{\text{th}}$ percentile values of $\Delta$[X/H]
in a given $\log(g)$ bin. The scatter in the residuals is fairly flat over our
sample’s $\log(g)$ range overall, although there is a slight increase in the
lowest $\log(g)$ bins, which have the fewest stars. Compared to W22, who used
a more limited range in surface gravity and effective temperature
($\log(g)=1-2.5$ and $T_{\rm eff}=4000-4600$) and thus had a sample $\approx$8
times smaller than in this work, the dispersion in our sample is only slightly
higher on average. If we did not introduce $\log(g)$ calibration offsets, the
residual abundance scatter would be substantially larger because it would be
inflated by systematic errors over the large $\log(g)$ range of our sample.
This is best illustrated by six ASPCAP elements that show a $>$25% decrease in
$\sigma_{68}$ after applying the $\log(g)$ calibration offsets: O (0.007 dex,
26.4%), Al (0.014 dex, 27.3%), Ca (0.010 dex, 28.7%), V (0.055 dex, 31.6%),
and Mn (0.020 dex, 40.5%). The flatness of the scatter trends in Figure 10 and
the similarity to the scatter found in the narrower sample of W22 demonstrate
the effectiveness of the calibration offsets derived from the median abundance
trends. We also see generally higher $\sigma_{68}$ at low [Mg/H], with the
most notable increases at [Mg/H] $<-0.4$. This effect is most likely due to
lower measurement precision at lower metallicities, but we did not distinguish
measurement uncertainty vs. intrinsic scatter in our calculation of
$\sigma_{68}$.

For elements shared between ASPCAP and BAWLAS, the BAWLAS measurements
generally show a similar amount of scatter. The dispersion in V shows the
largest reduction from ASPCAP to BAWLAS, especially at higher $\log(g)$,
suggesting that a significant amount of the total dispersion in V is
observational rather than intrinsic. On the other hand, C+N and O show notably
higher two-process residual scatter in BAWLAS than ASPCAP. This increased
dispersion can be seen when comparing [X/Mg] values around the median for
these elements in Figure 6, and BAWLAS reports higher median measurement
errors than ASPCAP for both elements.

Furthermore, the top row in Figure 6 shows that O has a dramatically different
median trend with [Mg/H] in BAWLAS than ASPCAP: the ASPCAP median trend in
[O/Mg] is nearly flat for $\text{[Mg/H]}\gtrsim-0.4$ while BAWLAS [O/Mg]
steeply declines with increasing [Mg/H]. This difference has long been
observed between O abundances measured from near-IR and optical spectra (e.g.,
Bensby et al., 2014; Griffith et al., 2022), but BAWLAS abundances are
measured from the same near-IR APOGEE spectra as ASPCAP. We note that the
optical O measurements use atomic lines, while the near-IR ASPCAP and BAWLAS
measurements use molecular OH lines. Hayes et al. (2022) attribute the
difference between the ASPCAP and BAWLAS O trends to the BAWLAS analysis using
calibrated stellar parameters for $T_{\rm eff}$ and $\log(g)$, whereas ASPCAP
used the uncalibrated parameters. In essence, ASPCAP adopts the $T_{\rm eff}$
and $\log(g)$ that gives the best fit to the whole APOGEE spectrum
simultaneously with several other stellar parameters (including overall
metallicity [M/H] and overall $\alpha$-element abundance [$\alpha$/H], which
includes O; see García Pérez et al. (2016) for details). BAWLAS adopts
adjusted stellar parameters that have been calibrated using external
references (e.g., astroseismic surface gravities, photometric temperatures).
It is not obvious which approach is preferable for deriving abundances, or why
systematics associated with the use of calibrated vs. uncalibrated parameters
would reproduce the optical abundance trends.

The closeness of the high- and low-Ia sequences indicates that O production is
dominated by prompt CCSN for both ASPCAP and BAWLAS determinations, but
ASPCAP’s mostly-flat slope implies metallicity-independent yields while the
downward slope of BAWLAS implies that O yields decrease with increasing
metallicity. The crossing of the high-Ia sequence above the low-Ia sequence in
BAWLAS O at $\text{[Mg/H]}=0$ leads to negative $q_{\text{Ia}}$ values at sub-
solar metallicities. Negative $q_{\text{Ia}}$ values are technically
unphysical, according to the two-process model assumption that
$q^{X}_{\text{Ia}}=0$ represents pure CCSN enrichment for element $X$, but the
high- and low-Ia sequences are close enough to one another that the negative
values are most likely the result of scatter in the BAWLAS O measurements. A
smaller effect seems to be present for C+N, where the median [(C+N)/Mg] trend
more steeply increases with [Mg/H] in ASPCAP than BAWLAS.

For some elements the scatter of the two-process residual is similar in
magnitude to the typical reported measurement error. However, some elements
show scatter significantly above the measurement errors (e.g., Ca, Na, Mn),
suggesting a substantial contribution of intrinsic scatter. The estimation of
intrinsic scatter and correlation of residuals is discussed at length by W22
(see their Figures 13 and 15), so we will not repeat this analysis here. Other
discussion of the intrinsic scatter of APOGEE abundances around a 2-parameter
fit can be found in Ting & Weinberg (2022) and Ness et al. (2022).

##  5 Residual Abundances of Stellar Populations

One application of the two-process model and residual abundances is to examine
differences in the enrichment patterns of different stellar populations, which
may be defined by spatial, kinematic, and age cuts or membership in star
clusters or satellite galaxies. These populations often span a range in
overall metallicity and [$\alpha$/Fe] ratio, producing correlated variations
across many elements, so a traditional “chemical tagging” approach risks
counting the same physical effect many times in different elements. Two-
process modeling encapsulates these correlated variations in the
$A_{\text{cc}}$ and $A_{\text{Ia}}$ parameters, and residual abundances can
reveal subtle deviations that could reflect different levels of AGB
enrichment, differences in supernova populations, or other effects. Because we
“train” the two-process vector on the disk median sequences, residuals of
stellar populations reveal systematic differences from typical disk stars of
matched [Mg/H] and [$\alpha$/Fe]. W22 applied this approach to find 0.05-0.2
dex differences for multiple elements in stellar populations from the Large
Magellanic Cloud (LMC) and the Gaia-Sausage/Enceladus (GSE) remnant, and much
larger deviations for $\omega$ Cen. Conversely, they found no differences (at
the 0.01-0.02 dex level) between stars at $R=15-17$ kpc and stars in the main
body of the disk.

Our stellar sample is $\approx$8 times larger than that of W22 overall, and
the gain is even larger for star clusters and dwarf satellites, where many of
the APOGEE targets lie outside the $\log(g)$ and $T_{\rm eff}$ cuts used by
W22. As a first application of our expanded catalog, therefore, we investigate
residual abundances of a selection of open clusters, globular clusters, and MW
satellites. To ensure robust residual abundance measurements, we apply a
S/N$>$100 cut for the following sections, but utilizing the lower S/N$>$80
limit of the overall catalog does not meaningfully change any results.

###  5.1 Open Clusters

Open clusters are groups of stars that formed from the same molecular cloud at
the same time in Galactic history, and are thus expected to have nearly
uniform chemical abundances. Previous work has shown that members of open
clusters generally show chemical homogeneity within 0.02-0.04 dex (e.g., Silva
et al., 2006; Liu et al., 2016; Bovy, 2016; Casamiquela et al., 2020; Poovelil
et al., 2020; Ness et al., 2022). Within the context of the two-process model,
we would expect that any enhancements or depletions seen in the residual
abundances will be shared by the member stars of an open cluster. Dating of
open clusters via color-magnitude diagrams is likely to be more accurate than
isochrone dating of individual stars. Combining open cluster ages with the use
of median deviations to reduce the impact of potential measurement errors and
statistical uncertainties within a chemically homogeneous population can
provide interesting and robust insights into changes in disk abundance
patterns with age.

Griffith et al. (2022) analyzed the residual abundances of 14 open clusters
using GALAH. They found that younger clusters have larger residual abundances
than older clusters as a general trend and noted significant deviations in the
elements O, Ca, K, Cu, Y, and Ba. The positive Y and Ba residuals were
consistent with previous literature, including super-solar abundances of these
elements in Spina et al. (2021) and enhancement of Y and Ba in young open
clusters (e.g., Baratella et al., 2021; Casamiquela et al., 2021), which
suggest an enhanced s-process (D’Orazi et al., 2009). O, Ca, and K had
positive residuals (i.e. enhancement compared to field stars) that were
stronger in the younger clusters, while Cu had the opposite trend, showing
more negative residuals (i.e. depletion compared to field stars) in the
younger clusters. Based on these results, Griffith et al. (2022) argue that
open clusters’ unique enrichment patterns and age-correlated residuals make
them promising candidates for testing the feasibility of chemical tagging.

Here, we perform a similar analysis using our APOGEE sample. The differences
in surveys used and sample cuts result in several important differences
between our analysis and Griffith et al. (2022):

  1. 1.

GALAH is an optical survey while APOGEE is a near-IR survey, so the lines
available in each spectral range are different, and the resulting analyses
from the different surveys will result in different systematics. This is well-
illustrated by the notable negative trend in [O/Mg] vs. [Mg/H] in GALAH
measurements not present in ASPCAP abundances. Interestingly, this trend is
also seen in the BAWLAS measurements of APOGEE spectra. In theory, the true
abundances of elements shared by both surveys should be the same.

  2. 2.

Griffith et al. (2022) selected dwarf and subgiant stars with
$\log(g)=3.5-4.5$ and $T_{\rm eff}=4200-6200$ K. In contrast, we select
evolved stars with $\log(g)=0-3.5$ and $T_{\rm eff}=3000-5000$ K.

  3. 3.

The Griffith et al. (2022) open cluster sample contains many more young $<$1
Gyr clusters. The youngest cluster in our sample, NGC 6705, would be the sixth
oldest cluster in the Griffith et al. (2022) sample. We also have two clusters
older than the oldest cluster in Griffith et al. (2022). The difference in age
distributions may reflect the evolutionary state of the stars in our
respective samples.

To obtain our open cluster sample, we cross-match our two-process residual
catalog with the Open Cluster Chemical Abundances and Mapping (OCCAM) survey
for DR17 (Myers et al., 2022). We consider only clusters with at least 8
member stars in OCCAM and designated as “high-quality” based on their color-
magnitude diagrams (Table 1, Myers et al. 2022). We also only consider stars
in those clusters within 3$\sigma$ of the cluster mean in proper motion,
radial velocity, and [Fe/H] (following the recommendations in Myers et al.
2022). We find 26 open clusters in OCCAM represented in our sample with these
constraints, and ultimately select 14 of the 15 clusters that have $N>10$
stars with two-process residuals. Dolidze 41 was excluded because it contained
exactly the same stars as Berkeley 85. The ages of the clusters are also taken
from the OCCAM catalog. The sample of stars within each OC generally spans a
reasonably wide range of $T_{\rm eff}$-$\log(g)$ space, with the exception of
NGC 1245 and ESO 211-03, whose stars are both concentrated around $T_{\rm
eff}\approx 5100-5200$ K and $\log(g)\approx 2.75-3$. However, these two
clusters show different residual abundance patterns from each other (Figure
11), so there do not appear to be remaining systematics from $\log(g)$ or
$T_{\rm eff}$.

Figure 11: Left: [Fe/Mg] vs. [Mg/H] abundances for the 8 youngest ($\leq$2
Gyr) open clusters in our sample. The median low-Ia (red) and high-Ia (blue)
sequences for the calibration (disk) sample are shown for reference.
Individual stars in each cluster are shown by black stars. Right: residual
abundances of the 8 youngest open clusters in our sample. Light gray lines
show the residuals for each individual member star for the ASPCAP elements and
BAWLAS Cu. Individual residuals for BAWLAS O, S, C+N, Na, V, and Ce are shown
in small orange points. Large blue points indicate the median deviation of
ASPCAP measurements for each element and the corresponding ASPCAP sample size
is shown in blue text. Large orange points indicate the median of BAWLAS
measurements and the BAWLAS sample size is shown in orange text. We note that
the sample size for a specific element may be less than the listed number if
any individual star’s measurement was flagged (Section 2). A blue line has
been plotted connecting different elements’ median residuals, with preference
to the ASPCAP measurement due to larger sample size, to guide the eye. Error
bars show the standard deviation on the median computed from 1000 bootstrap
resamplings of each cluster. When not visible, these error bars are smaller
than the plotted points. Note that the panel for ESO 211-03 has a different
vertical range. Figure 12: Same as Figure 12 for the 6 oldest ($\geq$2 Gyr)
open clusters in our sample.

In the left panels of Figures 11 and 12, we plot [Fe/Mg] vs. [Mg/H] for all
stars in each of the 14 clusters in our sample. We notice immediately that the
stars in all of the clusters lie on or above the high-Ia median sequence, with
the older clusters having lower [Fe/Mg] values and lying closer to the median
sequence than the younger clusters. This generally makes sense because the gas
from which the youngest clusters formed would likely have had more time to be
enriched by delayed SNIa contributions. The member stars in each cluster are
generally well-concentrated in [Fe/Mg]-[Mg/H] space as expected. However, the
observed spread within each cluster is $\approx$0.1-0.2 dex, larger than the
inhomogeneity reported in higher SNR studies. Furthermore, the measurement
errors on ASPCAP [Fe/H] and [Mg/Fe], when added in quadrature, yield median
[Mg/H] uncertainties of $\approx$0.012-0.020 dex across our open cluster
sample and are insufficient to fully explain the observed spread.

Figure 13: Residual abundances of each open cluster in our sample as a
function of cluster age for selected elements. The division into two panels is
only to improve visibility. Blue circles, yellow diamonds, green triangles,
orange squares, and pink circles represent the median residual abundance of
C+N, Na, S, Cu, and Ce respectively. The error bars show the standard
deviation on the median computed from 1000 bootstrap resamplings of each
cluster. ASPCAP measurements are used for all elements except the BAWLAS-only
element Cu. The median residual abundance of Ca is also shown in open black
circles on both plots as an example of an element with little residual
abundance variation with age in our sample, and the bootstrapped uncertainties
are smaller than the marker size.

The right panels of Figures 11 and 12 show the residual abundances of our 14
open clusters. We include error bars on median residuals based on bootstrap
resampling, but these are usually smaller than the plotted points. This plot
demonstrates the value of investigating median residuals, as differences at
the 0.1 dex level can be detected at high significance even when individual
stars may have 0.05-dex level observational uncertainties (which one can judge
from the grey lines showing individual stars). We generally find a similar
trend to Griffith et al. (2022), where the youngest clusters show the
strongest residuals while the older clusters show much smaller residuals, if
any. In our sample, the abundances become close to that of field stars by
$\approx$1.7-1.8 Gyr. The two youngest clusters in our sample, NGC 6705 and
Berkeley 85, show enhancements in S, C+N, Na, Cu, and Ce. We plot these
elements as a function of cluster age in Figure 13.

We observe Ce enhancement of 0.05-0.2 dex in all clusters with ages up to
$\approx$2 Gyr (all clusters in Figure 11, and NGC 2204), and in Berkeley 18
(age = 4.3 Gyr). This age trend is different from other elements shown in
Figure 13, which only show clear enhancement in the two youngest clusters. The
enhancement in Ce is consistent with Sales-Silva et al. (2022), who found
higher [Ce/Fe] and [Ce/$\alpha$] in open clusters younger than 4 Gyr, as well
as enhancement in other s-process elements like Y and Ba in young open
clusters (e.g., Baratella et al., 2021; Casamiquela et al., 2021).
Furthermore, the dispersion in Ce enhancement in the $\approx$2 Gyr old OCs is
resembles a similar trend in Ba observed by Mishenina et al. (2015), which
they argue is a signature of an intermediate neutron-capture process
(“i-process”). We would likely need to compare our Ce residuals to other
neutron-capture elements to confirm or rebut an i-process nucleosynthesis
signature.

Ce enhancement sometimes arises in concert with C+N enhancement as a
consequence of binary mass transfer (e.g., Smith et al. 2000, and discussed in
the context of residual abundances in W22). However, we expect this phenomenon
to affect only a small fraction of stars, so the open cluster Ce enhancement
is likely present in birth abundances. Outside clusters, Casali et al. (2023)
studied low-$\alpha$ stars with astroseismic ages and found that younger stars
have higher [Ce/Fe] at fixed [Fe/H] and higher [Ce/$\alpha$] at fixed radial
position. In residual abundances, W22 find a trend of enhanced Ce in young
APOGEE field stars, and Griffith et al. (2022) find a similar trend for Y and
Ba in GALAH, so it seems probable that both cluster and field stars reflect a
general increase of AGB enrichment in the ISM at late times, relative to SNIa
enrichment. The C+N enhancement in our two youngest clusters might be a
consequence of such extra AGB enrichment, and it is tempting to ascribe the
elevated levels of Na, S, and Cu in these clusters to the same source.
However, the theoretically predicted AGB contribution to these three elements
is small, in contrast to Ce and C+N (see, e.g., Fig. 10 of Andrews et al. 2017
and Fig. 13 of Rybizki et al. 2017).

Na overabundance in evolved stars can be caused by mixing during the first
dredge-up in low- to intermediate-mass red giants (Smiljanic et al., 2016, and
references therein). Our observed Na enhancement in only the youngest clusters
could reflect this phenomenon since their evolved stars (which make up our
sample) are more massive compared to older clusters. In contrast to Griffith
et al. (2022), we do not see any notable Ca or K enhancement in any clusters;
they also did not see any enhancement in Na. We also observe mild Cu
enhancement in our youngest clusters where they observed depletion. The other
elements showing enhancement (S and C+N) were not considered in Griffith et
al. (2022). An advantage of APOGEE for investigating young clusters is that
the abundance determinations for red giants are less likely to be affected by
rotational line broadening than determinations from main sequence stars that
dominate GALAH cluster samples.

ESO 211-03 and NGC 2158 show significant ($>$0.2 dex) depletion in Co. To
investigate this depletion, we compare the strongest Co line in the observed
APOGEE spectra to synthetic spectra. Using the 1D spectral synthesis code Korg
(Wheeler et al., 2023a, b) with the APOGEE DR17 line list (most similar to
that of the DR16 line list, described in Smith et al. 2021), we generate two
synthetic spectra for every star in each cluster: one spectrum with reported
ASPCAP abundances and stellar parameters, and one spectrum where only the Co
abundance is changed to that predicted by the two-process model. We note that
we use the uncalibrated, best-fit ASPCAP spectroscopic $T_{\rm eff}$ and
$\log(g)$ as Korg inputs to synthesize the spectra. The synthetic spectra are
downsampled to APOGEE resolution, then normalized by dividing out a moving
mean, where at each pixel the mean is calculated using a bandwidth of 25Å. For
the observed spectra, we use the combined, but not normalized, APOGEE spectra
in the apStar/asStar files and normalize using the same moving mean procedure
with a weighted mean to account for flux uncertainties. Finally, we calculate
the pixel-by-pixel median of the normalized spectra over every star in the
cluster. As illustrated in Figure 14, we find that the median observed
spectrum is a closer match to the median synthetic spectrum with reported
ASPCAP Co abundances in both ESO 211-03 and NGC 2158. This indicates that the
Co prediction of the two-process model results in a poorer fit, and thus that
the observed depletion is most likely real. However, the physical origin of
this Co depletion is unclear. An alternative explanation for the significant
mismatch between the observed spectra and two-process prediction is that the
spectra are affected by small errors in sky subtraction: the 16762Å Co line
falls on a telluric feature in both ESO 211-03 and NGC 2158. In either case,
we show that our median residual abundance analysis can reveal real
differences in spectra of stellar populations. A caveat of this analysis is
that while we use identical line lists and model atmospheres, Korg is not the
same spectral synthesis code used to generate the ASPCAP spectral grids for
DR17, so we are in effect generating new spectra to compare to APOGEE.

Figure 14: Spectra around the strongest Co absorption line for all member
stars in our sample of the open clusters ESO 211-03 and NGC 2158, which showed
significant depletion in their median Co residual abundances. This line was
identified from the wavelength range where the synthetic spectra, generated
with identical abundances except Co, showed the largest difference. We note
that three other Co lines were identified through this method; we only plot
the strongest one here. For each star in the clusters, synthetic spectra were
generated with Korg (Wheeler et al., 2023a, b) using the reported ASPCAP
abundances only (blue) and two-process predicted Co abundance with ASPCAP
abundances for all other elements (orange). Observed APOGEE spectra are in
gray. Thin, lighter lines represent individual cluster stars while the thick,
dark lines represent the median spectrum. All individual spectra were
normalized using a moving mean prior to calculation of the median spectrum;
synthetic spectra were downsampled to APOGEE resolution prior to
normalization. Better agreement of the observed spectra with the ASPCAP Co
abundance indicates that the cluster stars are genuinely depleted in Co
relative to the two-process prediction.

NGC 1245, NGC 7789, and NGC 2420 show a $\approx$0.1-0.2 dex enhancement in
median V. We performed a similar analysis to that of the Co depletion for
ASPCAP V (chosen over BAWLAS due to a higher number of measurements) and found
that the median observed APOGEE spectra more closely matched that of the
median synthetic spectrum with predicted V abundances for all three clusters.
From this result we conclude that the V enhancements seen in the panels of
Figure 11 are likely not real, as the median line depth is consistent with the
two-process prediction (i.e., the residual should be close to 0). The V
residuals show some of the largest variations between individual stars in the
majority of the clusters, so there are likely large uncertainties in the
measurements as well.

We repeated the median spectrum analysis for the Na enhancement, again using
ASPCAP over BAWLAS due to a larger number of measurements, with the two
youngest clusters, NGC 6705 and Berkeley 85. However, the continuum level of
the median observed spectra did not match that of the synthetic spectra
despite identical moving-mean normalization procedures prior to the median
calculation, making it difficult to accurately judge the relative depth of the
strongest Na lines in each median spectrum. It is therefore unclear whether
the median Na enhancement in NGC 6705 and Berkeley 85 is real from this
analysis. We note that V and Na are both elements whose lines can become weak
at warmer temperatures, and it is possible that some of the V and Na
measurements are fitting noise in the spectra in young clusters. This is more
likely for V than Na given that the median BAWLAS Na is also elevated in NGC
6705 and Berkeley 85, and significant Na enhancement has been independently
reported for NGC 6705 (Loaiza-Tacuri et al., 2023).

Finally, we check the enhancements for the remaining single elements showing
age trends, S, Ce, and Cu, using the median spectrum analysis in Korg. For Ce,
we performed this analysis on five clusters showing $>$0.1 dex median
enhancement (NGC 6705, NGC 1245, ESO 211-03, NGC 2420, and NGC 2204) and used
the ASPCAP abundances to increase the number of measurements in each cluster.
The median spectra show clear enhancements in Ce, similarly to the Co case
(Figure 14) but to a smaller degree, in all clusters except NGC 6705. NGC 6705
suffered similar continuum matching issues as in the Na case. In S, we found
that the median observed spectra were actually shallower than even the two-
process prediction i.e., the clusters have less S than predicted. Upon further
investigation, it is likely that several line blends in S in combination with
the initial fitting of S with the other alpha elements resulted in ASPCAP
overestimating the S abundance. However, in NGC 6705 and Berkeley 85, using
the smaller number of BAWLAS measurements of S, whose methods are intended to
better measure abundances of elements suffering from blended lines, yields
very similar results to using ASPCAP. For Cu, we examined NGC 6705 and
Berkeley 85, and we find a similar result to S, where the median observed
spectra were shallower than the median synthetic spectra. However, the
$\approx$0.1 dex median Cu enhancements seen in these clusters is insufficient
to generate a clear difference between the median synthetic spectra generated
with measured or predicted (two-process) Cu at APOGEE resolution, so it would
have been difficult to determine which is closer to the data regardless.

Overall, we find that open clusters appear increasingly similar to the MW disk
as they get older. We find age trends in the elements C+N, Na, S, Cu, and Ce
where the two youngest clusters show the substantial enhancements, and all
clusters younger than 2 Gyr show enhanced Ce. Detailed examination of spectral
fits shows that the Na, S, and Cu enhancements are not fully convincing, as
all of them arise from weak features. However, the strong Co depletions in ESO
211-03 and NGC 2158 and the enhancements in Ce in NGC 1245, ESO 211-03, NGC
2420, and NGC 2204 do appear to be real.

###  5.2 Globular Clusters

Once thought to be simple stellar populations, the last several decades have
shown that globular clusters (GCs) are complex systems showing evidence of
multiple generations of stars. Almost all GCs show star-to-star variation in
their light element abundances, an observation that is commonly interpreted as
a signature of self-enrichment within the cluster (e.g., Gratton et al., 2004,
2012, and references therein). The most notable chemical signatures include
the Na-O anticorrelation and the Al-Mg anticorrelation/Al-Si correlation,
which originate from high-temperature proton-capture reactions (i.e.,
H-burning) only possible in the cores of the first-generation, low-metallicity
stars that go on to enrich later generations (e.g., Carretta et al., 2009a,
2010, 2013; Mészáros et al., 2020).

In this section, we examine the stellar populations of GCs through residual
abundance analysis. By using residual abundances, we are able to control for
the bulk abundance variations resulting from CCSN and SNIa contributions,
which are largely shared by field stars in the MW. Thus, this method
highlights element variations and correlations potentially unique to GCs,
which in turn could be a signature of their distinctive evolutionary histories
or interesting nucleosynthetic pathways more common in these systems. Many
proposed early-generation polluters of GC stars, such as AGB stars (Ventura et
al., 2001), mass loss from fast-rotating massive stars (Decressin et al.,
2007), and binary massive stars (de Mink et al., 2009), are not necessarily
accounted for by our two-process model.

We select GC member stars by cross-matching our residual abundance catalog
with the DR17 value-added catalog of Galactic GCs (Schiavon et al., 2023),
then impose a stricter set of membership cuts. We require that each star have
a membership probability $>$0.8 using either set of independent membership
probabilities reported in the catalog. Furthermore, the GC star sample is
divided into Likely and Outlier groups; we use only stars selected using a set
of metallicity-independent angular distance, proper motion, and radial
velocity criteria in the Likely group (iFlag=0) (see Schiavon et al. 2023 for
more details). After these membership cuts, a total of 23 GCs are represented
in our sample, and we select eight with $N>8$ stars. M54 was dropped from our
GC sample due to heavy contamination from stars in the Sagittarius Dwarf
Galaxy (Sgr). We found significant overlap between the Schiavon et al. (2023)
M54 sample ($N=313$) and our $N=595$ Sgr sample in Section 5.3, and it is
difficult to separate the M54 and Sgr populations at the metallicity range
examined in this work. We choose to examine Sgr as a whole, without separating
out any M54 stars, in Section 5.3. The stars selected in each GC typically
span the full giant branch in $T_{\rm eff}$ and $\log(g)$, except for NGC
6388, NGC 6380, and NGC 6760, whose member stars are all upper RGB
($\log(g)\lesssim 1.5$).

Figure 15: Same as Figure 11 for the 8 globular clusters with more than 8
stars in our sample ordered by sample size, except for $\omega$ Cen, which has
a different vertical range and is placed last.

In Figure 15 we plot the residual abundances of our sample of 8 GCs as well as
their member stars’ location in [Fe/Mg] vs. [Mg/H] space. Because GCs are
typically metal-poor but we impose a cutoff of $\text{[Mg/H]}\geq-0.75$ in our
sample selection (Section 2.1), we have ultimately selected relatively metal-
rich GCs and/or sampled the metal-rich end of their metallicity distributions.
The most metal-poor cluster in our sample is $\omega$ Cen, with
$\text{[Fe/H]}=-1.53$ (Harris, 1996, 2010 edition), whose stars have lower
SNIa contributions than the median low-Ia plateau, as seen in the left panel
of Figure 15. Our GCs all populate the low [Mg/H] bins, and most also lie on
the low-Ia sequence, except for NGC 6388 (a relatively metal-rich bulge
cluster).

Figure 16: Na-O anticorrelation plot showing residual abundances in ASPCAP
[O/H] on the x-axis and ASPCAP [Na/H] on the y-axis for six globular clusters
in our sample with statistically significant ($p<0.05$) Spearman correlation
coefficients, reported in the upper right of each plot for each cluster.
One-$\sigma$ uncertainties are estimated from the standard deviation of the
correlation coefficients from 1000 bootstrap resamplings of each cluster.

The GCs in our sample all show unique residual abundance patterns. While there
are some common trends, none are universal among all clusters in our sample
except the very small residuals in Mg, Si, and Fe, which is to be expected
since these elements are well-measured in ASPCAP and were used to fit
$A_{\text{cc}}$ and $A_{\text{Ia}}$. As for the other elements that the two-
process model was conditioned on, NGC 6380 and $\omega$ Cen show a
$\approx$0.1 dex enhancement in Ca, and $\omega$ Cen shows a $\approx$0.2 dex
depletion in Ni. S, K, and Mn also generally show very small median residuals
among our GC sample. The only notable K median residuals are $\approx$0.1 dex
enhancements in NGC 6380 and NGC 6760 and an $\approx$0.1 dex depletion in
$\omega$ Cen; similarly, the only notable Mn residuals are a $\approx$0.1 dex
enhancement in M4 and $\approx$0.1 dex depletion in $\omega$ Cen.

####  5.2.1 Oxygen and Sodium

The median O value is close to the two-process prediction for all clusters but
shows notable star-by-star scatter in most (this effect is most visible in GCs
with large $N$). The clusters with the most obvious star-to-star scatter in O
also have large star-to-star scatter in Na, which has a known anticorrelation
with O in GCs (e.g., Carretta et al., 2009a, b, 2010). We plot the O and Na
residual abundances in Figure 16 for six GCs in our sample with statistically
significant ($p<0.05$) Spearman correlation coefficients. We choose to use the
Spearman test because it is more robust to outliers and makes no assumptions
about the linearity of the correlation, and we also use residuals from the
ASPCAP measurements to maintain a sufficiently large sample size. All six GCs
in Figure 16 have negative Spearman correlation coefficients, indicating that
the O and Na residual abundances are indeed anticorrelated. The two clusters
with statistically insignificant ($p>0.05$) correlation coefficients, NGC 6388
and NGC 6760, show smaller spreads in $\Delta$[O/H] and/or $\Delta$[Na/H]
values compared to the other GCs in Figure 15. We note that Na has fairly weak
lines in the H-band, and previous studies of APOGEE GCs have opted out of
studying the Na-O anticorrelation for this reason (e.g., Mészáros et al.
2020).

In 47 Tuc, M4, M71, M107, and (to a lesser extent) $\omega$ Cen, we observe a
bifurcation in the star-by-star residuals for Na in Figure 15; this may be a
signature of at least two distinct populations within those GCs. The median Na
residual would then reflect the relative size of those multiple populations.
It is not clear whether these multiple populations exist in NGC 6338, NGC
6380, or NGC 6760 based on their star-to-star spread in Na residuals, although
NGC 6380 does show two fairly distinct “clumps” of stars in
$\Delta$[O/H]-$\Delta$[Na/H] space. NGC 6380 and NGC 6760 are also notably the
only two GCs with only positive Na residuals.

Figure 17: Correlations between residual abundances of ASPCAP [Na/H] (top
row), ASPCAP [Al/H] (middle row), and ASPCAP [Ce/H] (bottom row) with ASPCAP
[(C+N)/H] for the globular clusters 47 Tuc (left column), M4 (middle column),
and $\omega$ Cen (right column). Points are colored by the [C/N] ratio
calculated from raw (not adjusted for $C_{\log(g)}$ and $C_{ZP}$) ASPCAP
[C/Fe] and [N/Fe] values. Spearman correlation coefficients are reported in
the upper left for each cluster, with their 1$\sigma$ uncertainties estimated
from the standard deviation of the correlation coefficients from 1000
bootstrap resamplings of each cluster-element combination. All correlation
coefficients are statistically significant ($p<0.05$) except in the
$\Delta$[Ce/H]-$\Delta$[(C+N)/H] panel for M4.

####  5.2.2 Carbon, Nitrogen, Aluminum, and Cerium

All the clusters in our sample show elevated C+N except for M71, which is
unusually similar to the MW disk stars compared to the rest of the sample and
not significantly enhanced in any element except Ce. The clusters all also
show some amount of enhancement in Ce ranging from slight ($\approx$0.1 dex,
47 Tuc and M71) to extreme ($\approx$1 dex, $\omega$ Cen). 47 Tuc and $\omega$
Cen show statistically significant ($p<0.05$) positive correlations between
ASPCAP $\Delta$[(C+N)/H] and $\Delta$[Ce/H], plotted in the bottom row of
Figure 17. NGC 6388 and NGC 6380 also show statistically significant
correlations for this pair of elements, but are not plotted. The elevated C+N
and Ce abundances could be a signature of second-generation AGB enrichment in
these clusters, especially $\omega$ Cen (e.g., Smith et al., 2000).

M4, M107, $\omega$ Cen, and NGC 6380 show enhancement in Al. The star-by-star
Al residual abundances in $\omega$ Cen (as well as NGC 6388 and NGC 6380,
though limited by the much smaller sample sizes) show large, somewhat bimodal
scatter around the median, similar to the pattern seen in the Na residuals,
and could be a signature of multiple populations within the cluster. However,
the star-by-star residual scatter around Al in M4, M71, and M107 is much
tighter and appears mostly centered around the median residual value,
consistent with the small Al spreads found in Mészáros et al. (2020) for these
GCs. An Al-Mg anticorrelation and an Al-Si correlation originating from the
Al-Mg cycle in the high temperature ($>$70 million K) cores of early-
generation low-metallicity stars has been noted in the literature for many GCs
(e.g., Carretta et al., 2009a, 2010, 2013; Mészáros et al., 2020), but we do
not see evidence of the Al-Mg relation in our residual abundance data. Five of
our GC sample (47 Tuc, M4, NGC 6388, $\omega$ Cen, and M54) have some
correlation between Al and Si residuals, but we caution that the spread in
$\Delta$[Si/H] is relatively small, only $\approx$0.1-0.2 dex. Mg and Si are
two of the six elements used to fit $A_{\text{cc}}$ and $A_{\text{Ia}}$ in the
two-process model, which leads us to expect little scatter in the Mg and Si
residuals.

In the first two rows of Figure 17, we plot the residual abundance
correlations of Na and Al with C+N for 47 Tuc, M4, and $\omega$ Cen, with
points colored by [C/N] ratio. These three clusters are both well represented
in our residual abundance catalog ($N>81$) and show notable residuals in the
light odd-Z elements and Ce in Figure 15. While not plotted, all GCs in our
sample show similar statistically significant positive correlations between
the C+N and Al residuals; NGC 6388 and NGC 6380 show the same between the C+N
and Na residuals.

The strong color gradient along the positive correlations in Figure 17 shows
that [C/N] values decrease as $\Delta$[(C+N)/H] increases, so the C+N
enhancement in these GCs is most likely driven by enhancement in N.
Considering this, the correlation we observe between Al and C+N residuals is
likely a reflection of an Al-N correlation (e.g., Fernández-Trincado et al.,
2019b; Mészáros et al., 2020); similarly, our $\Delta$[Na/H]-$\Delta$[(C+N)/H]
correlation likely reflects a Na-N correlation (e.g., Carretta et al., 2009b).
These correlations are, like other known elemental correlations in GCs,
believed to be related to pollution from early-generation stars hot enough to
burn hydrogen through the Ne-Na and/or Mg-Al cycles. The strong N dependence
on the C+N enhancement in these GCs and the correlations with Al and Na are
common with many N-enhanced stars found in the MW and have been suggested as
evidence for these N-rich stars having GC origin (e.g., Martell et al., 2016;
Schiavon et al., 2017; Fernández-Trincado et al., 2019a).

####  5.2.3 Iron-peak Elements

The two Fe-peak elements (Fe and Ni) used to fit the two-process amplitudes
generally have a very small, if any, median residual in our GC sample except
for $\omega$ Cen, which shows a $\approx$0.2 dex depletion in Ni. $\omega$ Cen
also has $\approx$0.1-0.3 dex depletions (depending on the cluster/element) in
the other Fe-peak elements V, Mn, and Co in addition to the aforementioned Ni;
this chemical pattern appears more similar to our sample of massive dwarf
satellite galaxies (Figure 18) than the other GCs so we discuss its potential
origins in Section 5.3.4. M4 has an $\approx$0.2 dex median residual in Cr,
while NGC 6388 and NGC 6380 show smaller $\approx$0.1 dex Cr residuals. NGC
6388 and NGC 6380 also have $\approx$0.1-0.2 dex enhancements in Co. NGC 6388,
NGC 6380, and NGC 6760 show $\approx$0.1-0.2 dex enhancement in V. V also
shows large star-by-star scatter in its residuals; however, because the star-
by-star deviations are present and large in other stellar populations (e.g.
Figures 11, 12, 18), this is unlikely to be a signature of multiple
populations, unlike in the case of Na. Furthermore, M107 shows a V enhancement
based on the BAWLAS measurement and a V depletion based on the ASPCAP
measurements, which suggests that the variation in V is likely due to large
measurement errors. This effect is present but smaller in $\omega$ Cen, where
the median BAWLAS residual is just barely positive while the median ASPCAP
residual shows a small depletion of $\approx$0.1 dex. Only four GCs in our
sample (47 Tuc, M4, M71, and $\omega$ Cen) have any Cu measurements. Of these
four GCs, only 47 Tuc has more than six measurements, and it has
$\Delta$[Cu/H]$\approx$0.

####  5.2.4 Omega Centauri

The chemical properties of $\omega$ Cen have long been known to be extreme
relative to other GCs (e.g., Johnson & Pilachowski, 2010; Mészáros et al.,
2021). From Figure 15, its residual abundances are much higher than other
clusters, but the overall residual abundance pattern—particularly the enhanced
light elements C+N, Na, and Al—is actually quite similar to those of the other
GCs in our sample (most notably 47 Tuc, M4, and NGC 6380). This greater
similarity could be due to sampling only the high metallicity population of
our GCs, or that controlling for CCSN and SNIa contributions removes one
source of overall abundance variation. Because the residual abundance pattern
is similar in form, despite being larger in magnitude, the underlying
nucleosynthetic processes creating the abundance patterns (at least for the
high-metallicity stars) may be similar.

###  5.3 Milky Way Satellite Galaxies

Dwarf galaxies in the Local Group are interesting laboratories for probing
star formation over a range of masses, metallicities, and environments on a
detailed, star-by-star level. The chemical abundance trends of dwarf galaxies
provide further insight on the more subtle details affecting star formation,
such as variations in the initial mass function (e.g., McWilliam et al., 2013;
Hasselquist et al., 2017) and AGB enrichment (e.g., Skúladóttir et al., 2019;
Hansen et al., 2018; Fernández-Trincado et al., 2020). The low-metallicity
environments (compared to the MW disk) of dwarf galaxies could also help
identify and constrain metallicity-dependent yields of certain elements.

Our two-process model is calibrated using median abundance trends in the
Galactic disk. Therefore, by construction, we can consider the residual
abundances in external galaxies as deviations/differences from the chemical
pattern of the Galactic disk. Strong residuals (or lack thereof) from the
disk-trained two-process model would provide insight into which elements are
(or are not) affected by the differences in star formation and chemical
enrichment history between the dwarf galaxies and the Galactic disk.

We select our sample of Milky Way satellite galaxies by cross-matching the
APOGEE IDs of our sample with Table 2 of Hasselquist et al. (2021), which
examined the chemical abundance trends of the five most massive MW satellites.
We find four satellite galaxies represented in our sample: the Large
Magellanic Cloud (LMC), Small Magellanic Cloud (SMC), Sagittarius Dwarf Galaxy
(Sgr), and Gaia Sausage/Enceladus (GSE). For the LMC and SMC, we also checked
for selection effects on our residual abundances by comparing with a sample
chosen using a different set of selection criteria, a combination of APOGEE
flags and observing fields with proper motion and radial velocity cuts from
Nidever et al. (2020), and found little difference in the residual abundance
pattern.

Sgr is currently in the process of merging with the MW. The Hasselquist et al.
(2021) Sgr sample contains about two-thirds main body stars. They also
demonstrate that despite different metallicity distribution functions between
the Sgr core and tidal tails, the [Mg/Fe] vs. [Fe/H] tracks are not
significantly different where they overlap in [Fe/H]. Our
$\text{[Mg/H]}>-0.75$ metallicity cut (Section 2.1) selects the highest
metallicity stars in Sgr, where the main body stars dominate. Therefore, we do
not distinguish between Sgr core and tail stars in this analysis, and we
expect the majority of Sgr stars in our sample to be main body stars.

Figure 18: Same as Figure 11 for four massive MW satellite galaxies: the LMC,
SMC, Sgr, and GSE.

We plot [Fe/Mg] vs. [Mg/H] for the member stars in each of the satellite
galaxies in our sample in the left panels of Figure 18. All four galaxies lie
on or above the median high-Ia sequence, indicating extended star formation
histories that allowed the most metal-rich stars in these systems to be
enriched by many generations of delayed SNIa contributions. GSE shows the
greatest number of stars below the high-Ia median sequence, which makes sense
given that its star formation was most likely cut off $\sim$10 Gyr ago during
its infall into the MW (e.g., Helmi et al., 2018; Gallart et al., 2019;
Vincenzo et al., 2019). A “wall” of stars exists at the leftmost edge of the
panels, right at the $\text{[Mg/H]}\geq-0.75$ sample metallicity cutoff. When
compared to the [Fe/Mg] vs. [Mg/H] plots in Hasselquist et al. (2021) (Figures
6 and 7), it is clear that our work samples only the most metal-rich end of
these satellite galaxies’ metallicity distributions.

On the right panels of Figure 18, we plot the residual abundances of the stars
in our dwarf galaxy sample. Overall, the most striking result are the
significantly lower abundances ($\approx$0.2-0.3 dex below the two-process
predictions, i.e., the median MW disk trends at matched Fe and Mg), in all
light, odd-Z elements except K (which is consistent with the two-process
prediction within $\leq$0.1 dex), shared by all four satellite galaxies. The
low C+N and Al abundances are consistent with their [X/Mg] tracks in
Hasselquist et al. (2021), which both fall significantly below the MW trends
where they overlap in [Mg/H]. We see a similar, though less extreme,
$\approx$0.1-0.2 dex depletion in all four satellites for the odd-Z Fe-peak
elements, excepting Cu. In both of these element groups, the one exception to
the trend is the most massive element, K in the case of the light odd-Z
elements and Cu in the case of the odd-Z Fe-peak elements. The
$\alpha$-elements O, Mg, and Si show close agreement with median disk trends
along with Fe ($\Delta$[X/H]$\approx$0). While this could be a result of using
these elements to fit for $A_{\text{Ia}}$ and $A_{\text{cc}}$ in the two-
process model, we have checked that the median residuals change by $\leq 0.02$
dex if $A_{\text{Ia}}$ and $A_{\text{cc}}$ are estimated from Mg and Fe alone.

####  5.3.1 Large and Small Magellanic Clouds

Our median LMC residual abundances are broadly in agreement with those of the
much smaller sample in W22. There is some enhancement in the ASPCAP
measurements of the more massive $\alpha$ elements S ($\approx$0.2 dex) and Ca
($\approx$0.1 dex), which is slightly higher than was seen in W22. However,
ASPCAP S shows signficant star-to-star scatter and the median BAWLAS S
residual is close to 0. The Ca enhancement is consistent with the [Ca/Mg]
track for the LMC in Hasselquist et al. (2021), which is slightly elevated
above the MW trend at higher [Mg/H]. K has a median residual of $\approx$0
dex, indicating a high degree of similarity with the MW disk. This strongly
contrasts its lighter odd-Z element counterparts C+N, Na, and Al, which show
significant depletion compared to the disk. The contrast between the more
mass-dependent Na and Al yields and K has been suggested by Hasselquist et al.
(2017) to be a signature of a top-light IMF in Sgr; the same argument may hold
for the LMC given the similar abundance pattern. However, Hasselquist et al.
(2021) show that a top-light IMF may not be required to fit observed abundance
patterns, and there is little evidence of such a top-light IMF in the $\alpha$
abundances at metallicities lower than those included in our sample (e.g.,
Hansen et al., 2018). Ni shows $\approx$0.1 dex depletion in the LMC, in
contrast with lighter even-Z Fe-peak elements Cr and Fe, which are both well-
predicted by the two-process model. We also see enhancement in Ce of
$\approx$0.1 dex for the ASPCAP measurements and $\approx$0.3 dex for the
BAWLAS measurement; for comparison, W22 observed a $\approx$0.2 dex Ce
enhancement in the LMC. The Ce enhancement is consistent with known
enhancements in its fellow second-peak s-process elements Ba and La, which may
be a signature of increased (compared to the MW) metal-poor AGB enrichment in
the LMC’s history (Van der Swaelmen et al., 2013). The Ni depletion and Ce
enhancement are also consistent with the tracks in Hasselquist et al. (2021).
The median Cu in the LMC is slightly depleted, similar to the other odd-Z Fe-
peak elements.

Despite only having 8 stars overlap with our sample, the SMC still shows a
distinct residual abundance pattern. It shows strong depletion in the same
light odd-Z elements as the other galaxies, and notably is the only galaxy in
our sample to show depletion in K ($\approx$0.1 dex). Its depletions in V and
Co are also somewhat stronger ($\approx$0.2-0.3 dex) than in the other
galaxies. The SMC also shows a very strong enhancement in Ce ($\approx$0.3
dex). We caution that only two stars out of the total eight had an ASPCAP Ce
measurement, but this strong enhancement would be roughly consistent if we
extrapolated the [Ce/Mg] trend for the SMC in Hasselquist et al. (2021) to
higher [Mg/H], where it would fall above the distribution in the MW disk.

####  5.3.2 Sagittarius Dwarf Galaxy

The Sgr dwarf galaxy shows a similar residual abundance pattern to the other
galaxies in our sample. It has large star-to-star scatter in S (similar to the
LMC), but its median ASPCAP S is $\approx$0.1 dex lower than that of the LMC.
The Ce enhancement is also similar to the LMC in both its magnitude and the
difference between median ASPCAP and BAWLAS measurements. The Sgr Ni track is
$\approx$0.1 dex higher than the LMC at high [Mg/H] in Hasselquist et al.
(2021), but the two galaxies show very similar median $\Delta$[Ni/H] in Figure
18. This suggests that the difference in Ni tracks between Sgr and the LMC in
Hasselquist et al. (2021) can be largely explained by differences in CCSN vs.
SNIa contribution, as illustrated by the difference in their Hasselquist et
al. (2021) [Fe/Mg] tracks and position of member stars in the left panels of
Figure 18 relative to the high-Ia median sequence. Sgr shows the strongest Na
and Al depletion ($\approx$0.4 and $\approx$0.3 dex, respectively) in our
sample, consistent with previous work noting strong deficiencies in these
elements in Sgr stars (e.g., Bonifacio et al., 2000; Sbordone et al., 2007;
McWilliam et al., 2013; Hasselquist et al., 2017). Additionally, the lack of K
depletion is consistent with a top-light IMF scenario in Sgr proposed by
McWilliam et al. (2013) and Hasselquist et al. (2017), though we again note
that this scenario has dubious evidence at metallicities lower than our sample
(Hansen et al., 2018). The $\approx$0.2 dex C+N depletion is about the same as
the other galaxies while the $\approx$0.3 dex Co depletion is more similar to
that of the SMC than the LMC or GSE. The most notable feature of Sgr’s
residual abundance pattern is its high ($\approx$0.3 dex) median Cu
enhancement, derived from residual abundances of 44 stars, which is in
disagreement with previously observed Cu deficiencies (e.g., Sbordone et al.,
2007; McWilliam et al., 2013). This could be an artifact of a more general
disagreement where the Cu abundances in the entire BAWLAS sample are
significantly higher than literature abundances at low metallicities
($\text{[Fe/H]}\lesssim-0.8$). The difference is similar in magnitude to our
observed $\approx$0.3 dex median Cu residual abundance. Hayes et al. (2022)
attribute the discrepancy to a combination of weak, blended lines at APOGEE
wavelengths leading to Cu measurements that are biased high at lower
metallicity, and NLTE effects that are stronger in optical wavelengths than
infrared and should push the optical abundances to larger values.

####  5.3.3 Gaia Sausage/Enceladus

Like the other dwarf galaxies in our sample, GSE is also depleted in the light
odd-Z elements C+N, Na, and Al. It has the smallest Na depletion and largest
ASPCAP/BAWLAS discrepancy in our sample, $\approx$0.3 dex using ASPCAP
measurements and only $\approx$0.15 dex using BAWLAS measurements. GSE also
shows no notable deviations in median Ce, unlike the other galaxies in our
sample which show Ce enhancements of $\geq$0.1 dex compared to the disk stars;
this is generally consistent with the [Ce/Mg] tracks in Hasselquist et al.
(2021), which show GSE having $\approx$0.1 dex lower [Ce/Mg] than the other
dwarfs at the same [Mg/H]. Hasselquist et al. (2021) suggest that the MCs and
Sgr had more AGB contributions that led to their Ce enhancements compared to
the MW disk, but GSE’s shorter star formation time (due to its infall into the
MW $\sim$10 Gyr ago) reduced the AGB contribution in its stars. Cu is slightly
depleted in GSE, though to a lesser extent than the other odd-Z Fe-peak
elements; this pattern is closest to that of the LMC.

####  5.3.4 Common Trends

Hasselquist et al. (2021) note and discuss the abundances of Al and Ni
compared to the MW in all satellite galaxies in our sample; they invoke
metallicity-dependent yields and a greater number of metal-poor SNe in these
systems than the MW to explain the observed depletions. Since C+N and Na show
a similar residual pattern to Al and are all theorized to be ejected from
CCSN, it is possible that these elements share similar metallicity-dependent
CCSN yields. The lack of depletion in K compared to Na and Al has been cited
as evidence for a top-light IMF that produces fewer massive CCSN progenitors
in Sgr (McWilliam et al., 2013; Hasselquist et al., 2017). Since this chemical
signature is present in all four dwarf galaxies we examine (especially the
LMC, Sgr, and GSE), this top-light IMF scenario is not unique. However, the
top-light IMF is not necessarily required to reproduce overall abundance
patterns (Hasselquist et al., 2021) and lacks supporting evidence in very
metal-poor Sgr stars Hansen et al. (2018). We also observe that the odd-Z Fe-
peak elements V, Mn, and Co share similar depletion magnitudes to Ni; all of
these Fe-peak elements are known to have metallicity-dependent yields (e.g.,
Woosley & Weaver, 1995; Chieffi & Limongi, 2004; Nomoto et al., 2006).
Furthermore, Ni and Mn depletions have been suggested as potential markers of
enrichment from sub-Chandrasekhar SNIa in dwarf galaxies (e.g., McWilliam et
al., 2018; Kirby et al., 2019; de los Reyes et al., 2020), but these previous
studies have focused on dwarfs of lower mass and metallicity than we examine
here.

In general, low C+N, Na, and Al relative to MW field stars are the strongest
chemical signatures of dwarf galaxy populations, and Ni, Mn, and perhaps Co
are weaker but also potentially useful tracers. On the other hand, near-zero
median $\alpha$-element residuals indicate that dwarf galaxy stars are
chemically similar to MW field stars in these elements at fixed metallicity,
and thus $\alpha$ abundances alone may not be a robust identifier of dwarf
populations.

The residual abundance diagram (Figure 18) provides a compact way to compare
and contrast multi-dimensional enrichment patterns among satellites and
between satellites and the disk, controlling for differences associated with
the overall metallicity and SNIa/CCSN ratio, and leveraging the ability of
median statistics to reveal systematic differences below the level of
individual star measurement errors. Overall, we find that the four massive MW
satellite galaxies represented in our data (LMC, SMC, Sgr, and GSE) show
surprisingly similar residual abundance patterns despite their different star
formation environments and enrichment histories, as well as only sampling from
the most metal-rich stars in these systems due to our sample selection
constraints. Combining the KPM methodology of Griffith et al. (2023) with the
$\log(g)$ calibration strategy developed here would allow these comparisons to
be extended to the lower metallicity populations of dwarf satellites, tidal
streams, and the in-situ halo.

In forthcoming work, Hasselquist et al. (in prep.) examine two-process fits
and residual abundances in dwarf satellites using APOGEE DR17. Although there
are many differences of methodology—including the use of matched-$\log(g)$
samples rather than the $\log(g)$ abundance corrections employed here—their
results for stars with $\text{[Mg/H]}>-0.75$ are in excellent agreement with
those shown in Figure 18, demonstrating the robustness of these conclusions to
analysis details.

##  6 Summary

We present a method of correcting systematic abundance trends with surface
gravity $\log(g)$ by deriving corrective offsets from median [X/Mg] vs. [Mg/H]
abundance trends, which are universal throughout the MW disk and bulge and
thus assumed to be independent of stellar evolutionary state. We apply these
offsets to 310,427 spectra (288,789 unique stars) from APOGEE DR17. We derive
two-process residual abundances for these stars following the method described
in W22. Our sample shares a metallicity range with W22, $-0.75\geq$ [Mg/H]
$\geq 0.45$, but it spans the entire red giant branch and red clump sample in
APOGEE: $0\geq\log(g)\geq 3.5$ and 3000 K $\geq T_{\rm eff}\geq$ 5500 K. We
consider the $\alpha$-elements O, Mg, Si, S, and Ca; the light odd-Z elements
Na, Al, K, and the combined element C+N; Fe-peak elements V, Cr, Mn, Fe, Co,
Ni, and Cu; and the s-process element Ce. Abundances of all elements except Cu
are from the APOGEE abundance measurement pipeline ASPCAP (Holtzman et al.,
2015; García Pérez et al., 2016). The Cu abundances, analyzed in context of a
two-process model and residual abundances for the first time in this work, are
taken from the BAWLAS catalog (Hayes et al., 2022), along with a set of
independent measurements of C+N, O, Na, S, V, and Ce. Our calibration offsets
and two-process residual catalog are available online (see Appendices).

As a first application of our expanded residual abundance catalog, we examine
the median residual abundance trends of 14 open clusters, nine globular
clusters, and four MW dwarf satellites. Many APOGEE targets in these
populations are only accessible through our wide $\log(g)$ and $T_{\rm eff}$
cuts. Using median abundances reduces the impact of an individual star’s
measurement errors and reveals systematic differences below the level of those
measurement errors. We summarize our key findings below:

  1. 1.

Overall, older open clusters show smaller median residual abundances, i.e.,
they are more chemically similar to MW disk stars. The two youngest clusters
show noticeable enhancement in median C+N, Na, S, and Cu compared to older
clusters. Ce enhancement is generally seen in clusters with ages up to
$\approx$2 Gyr.

  2. 2.

We investigate enhancements or depletions in open clusters through spectral
synthesis with Korg (Wheeler et al., 2023a, b), which has not previously been
performed with APOGEE spectra. We find spectral evidence of Co depletion in
ESO 211-03 ($\approx$0.6 dex) and NGC 2158 ($\approx$0.25 dex) through this
method, although the physical origin of this depletion is unclear.

  3. 3.

We recover the well-known Na-O anticorrelation in 6 out of 9 globular clusters
in our sample in $\Delta$[Na/H] and $\Delta$[O/H] residual abundance space.
Our result indicates that the Na-O anticorrelation is not explained by changes
in bulk CCSN/SNIa contributions, and it also highlights the power of residual
abundance correlations in identifying unique enrichment pathways.

  4. 4.

We find $>$0.2 dex C+N enhancements, and statistically significant
correlations in the residual abundances between C+N and Ce, Al, and Na, for
several globular clusters in our sample. The (C+N)-Ce correlation is likely a
signature of AGB enrichment, while the (C+N)-Al and (C+N)-Na correlations are
likely reflections of known Al-N and Na-N correlations. Because our two-
process residuals essentially compare populations to median MW disk trends,
this result supports claims that N-enhanced stars are of globular cluster
origin.

  5. 5.

The globular cluster $\omega$ Cen, which is known to have extremely complex
and unusual abundance patterns, shows enhanced median residual abundances in
C+N, Na, and Al similar to that of 47 Tuc, M4, and NGC 6380. While the
magnitude of the residual abundances in $\omega$ Cen is larger, the similar
chemical abundance pattern suggests similar enrichment mechanisms in the
metal-rich populations of these four clusters.

  6. 6.

The individual abundance patterns of four of the most massive MW satellites
are broadly consistent with previous results comparing these systems to the MW
disk population (e.g., Hasselquist et al., 2021). Our analysis finds
surprisingly similar residual abundance patterns among all four satellites
once bulk changes in metallicity and CCSN/SNIa contributions are accounted for
by the two-process model, despite their different star formation environments
and enrichment histories.

  7. 7.

Strong Al and Na depletions are observed across the median residual abundances
of all massive satellite galaxies in our sample. We also observe a milder
depletion in Ni of similar strength across our sample. Low Al and Ni
abundances are potential signatures of metallicity-dependent SN yields/more
enrichment from metal-poor SNe (Hasselquist et al., 2021).

Our expanded two-process residual abundance catalog, which contains $\approx$8
times more stars than previous work, is only possible with the derivation of
$\log(g)$ calibration offsets that remove artificial abundance trends and
correlations between elements. While we provide values of the $\log(g)$
calibration offsets for our specific sample constraints, our method of
empirically calibrating $\log(g)$ trends is generally applicable to any data
set, provided one has sufficient stars in each ($\log(g)$, [Mg/H]) bin to
derive a robust median value. An immediate usage example that follows from the
analysis presented in this work is to use our $\log(g)$ calibration method in
the derivation of residual abundances at low metallicities using the KPM
method (Griffith et al., 2023), which will in turn enable further comparative
studies of populations in dwarf satellites, tidal streams, and the MW halo at
lower metallicities than our sample here.

There are many further applications of our expanded residual abundance catalog
beyond the comparative study of stellar populations presented in this work. A
systematic exploration of outlier stars with large two-process $\chi^{2}$ fit
values should reveal both stars with unusual abundance patterns, which could
be signatures of rare physical processes or other enrichment sources, and
stars with rare measurement errors. The method of comparing observed spectra
to synthetic spectra with the two-process prediction, used in our open cluster
analysis here, can help distinguish which deviations are real. Unresolved
binaries may be poorly fit by the two-process model and stand out as high
$\chi^{2}$ stars, and trends in residual abundances with, for example, radial
velocity variations, could point to chemical signatures of binarity.
Furthermore, our expanded sample allows for much greater overlap with external
catalogs such as APOKASC (Pinsonneault et al., 2018) and Gaia DR3 (Gaia
Collaboration et al., 2023), enabling analyses of residual abundance trends
with stellar age, kinematics, position in the Galaxy, and more. Furthermore,
we can quantitatively test aspects of galactic chemical evolution models: the
intrinsic scatter in residual abundances provides constraints on stochastic
effects, and the $q_{\text{cc}}$ and $q_{\text{Ia}}$ process vectors provide
information about population-averaged yields and their metallicity dependence.

As future spectroscopic surveys, notably Milky Way Mapper (Kollmeier et al.,
2017), continue to map out increasingly more multi-element abundances across
the MW and its dwarf satellite neighbors, it becomes imperative to effectively
interpret and analyze the influx of data. As discussed in this paper and
previous works (e.g., W22; Griffith et al. 2022, 2023) a two-process or
K-process model and residual abundances provide a way to reduce/recast high-
dimensional abundance information and disentangle the enrichment histories of
the Galaxy, its satellites, or special stellar populations from the effects of
nucleosynthetic yields. While survey abundance measurement pipelines will
continue to improve, the $\log(g)$ calibration method outlined in this work
can account for remaining systematics and thus maximize the number of stars on
which we can apply these analyses. In turn, we will be rewarded by increasing
our understanding of the astrophysical origins of the elements and the
processes governing the chemical enrichment of our Galaxy and nearest
neighbors.

##  7 Acknowledgements

We thank Emily Griffith and Yuan-Sen Ting for helpful comments on an earlier
draft of this paper.

This work is supported by NSF grants AST-1909841 and AST-2307621. TM
acknowledges financial support from the Spanish Ministry of Science and
Innovation (MICINN) through the Spanish State Research Agency, under the
Severo Ochoa Program 2020-2023 (CEX2019-000920-S).

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.

SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
www.sdss4.org.

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

##  Appendix A Calibration Offsets

All calibration offsets, including the grid of $C_{\log(g)}$ values and values
of $C_{\text{ZP}}$ are provided in a table, available online at
https://zenodo.org/records/10659205. The first five rows are shown in Table 2.
We also include a Jupyter notebook with sample code to load the table, extract
the offset grid for a specific element, and obtain calibrated abundances for
ten example stars. Also included in the notebook is code to reproduce Figure
4. Note that the reported calibration offsets are derived for and thus
applicable to only the data used in this work (described in detail in Section
2.1).

Table 2: Table of Calibration Offsets

Element | Source | C_ZP | MG_H | C_LOGG_0.25 | C_LOGG_0.75 | C_LOGG_1.25 | C_LOGG_1.75 | C_LOGG_2.25 | C_LOGG_2.75 | C_LOGG_3.25 | C_LOGG_RC
---|---|---|---|---|---|---|---|---|---|---|---
O | ASPCAP | -0.025 | -0.7 |  | 0.110 | 0.077 | 0.009 | -0.079 | -0.081 | -0.081 | -0.058
O | ASPCAP | -0.025 | -0.6 |  | 0.072 | 0.062 | 0.009 | -0.069 | -0.090 | -0.090 | -0.058
O | ASPCAP | -0.025 | -0.5 |  | 0.044 | 0.042 | 0.009 | -0.060 | -0.074 | -0.062 | -0.058
O | ASPCAP | -0.025 | -0.4 |  | 0.024 | 0.030 | 0.009 | -0.037 | -0.050 | -0.056 | -0.034
O | ASPCAP | -0.025 | -0.3 |  | 0.004 | 0.019 | 0.009 | -0.020 | -0.023 | -0.032 | -0.019

Note. — Full table available online at https://zenodo.org/records/10659205.

##  Appendix B Process Vector Tables

In Table 3 we report values of the CCSN process vector $q^{X}_{\text{cc}}(z)$
for all elements considered in this work. The same for the SNIa process vector
$q^{X}_{\text{Ia}}(z)$ is reported in Table 4. Values are calculated from the
median sequences of $\log(g)$-calibrated abundances following Equations 25 and
26 in W22. For more details, see Section 4.1.

##  Appendix C Catalog User Guide

The residual abundance catalog is available online at
https://zenodo.org/records/10659205. The table columns are reported in Table
5. We also include a Jupyter notebook with sample code to cross-match with
external catalogs, using AstroNN (Leung & Bovy, 2019a, b) and the DR17
globular cluster catalog (Schiavon et al., 2023) as examples. The notebook
also includes code to make some plots, including reproducing the residual
abundances (right side) of Figure 15 and plotting
$A_{\text{Ia}}/A_{\text{cc}}$ as function of position.

Table 3: Values of $q_{\text{cc}}^{X}$ Element | Source | [Mg/H]
---|---|---
|  | -0.7 | -0.6 | -0.5 | -0.4 | -0.3 | -0.2 | -0.1 | 0.0 | 0.1 | 0.2 | 0.3 | 0.4
O | ASPCAP | 1.129 | 1.047 | 1.014 | 0.992 | 0.974 | 0.957 | 0.952 | 0.953 | 0.943 | 0.916 | 0.877 | 0.862
O | BAWLAS | 1.522 | 1.491 | 1.363 | 1.278 | 1.201 | 1.142 | 1.076 | 1.005 | 0.919 | 0.800 | 0.760 | 0.660
Mg | ASPCAP | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
Si | ASPCAP | 1.036 | 0.976 | 0.925 | 0.892 | 0.868 | 0.844 | 0.831 | 0.819 | 0.792 | 0.747 | 0.697 | 0.682
S | ASPCAP | 1.153 | 1.152 | 1.099 | 1.066 | 1.047 | 1.003 | 0.961 | 0.929 | 0.871 | 0.791 | 0.726 | 0.674
S | BAWLAS | 1.032 | 1.061 | 0.991 | 0.974 | 0.956 | 0.921 | 0.902 | 0.863 | 0.786 | 0.677 | 0.568 | 0.585
Ca | ASPCAP | 0.945 | 0.865 | 0.839 | 0.814 | 0.797 | 0.775 | 0.752 | 0.732 | 0.708 | 0.675 | 0.655 | 0.652
C+N | ASPCAP | 0.441 | 0.483 | 0.528 | 0.554 | 0.585 | 0.617 | 0.659 | 0.706 | 0.729 | 0.694 | 0.649 | 0.605
C+N | BAWLAS | 0.673 | 0.699 | 0.687 | 0.681 | 0.689 | 0.700 | 0.722 | 0.737 | 0.732 | 0.684 | 0.558 | 0.649
Na | ASPCAP | 0.394 | 0.393 | 0.441 | 0.462 | 0.489 | 0.527 | 0.560 | 0.595 | 0.568 | 0.423 | 0.228 | 0.107
Na | BAWLAS | 0.732 | 0.681 | 0.579 | 0.612 | 0.601 | 0.601 | 0.586 | 0.564 | 0.532 | 0.385 | 0.144 | 0.159
Al | ASPCAP | 0.893 | 0.875 | 0.902 | 0.906 | 0.925 | 0.932 | 0.929 | 0.925 | 0.917 | 0.901 | 0.851 | 0.821
K | ASPCAP | 0.913 | 0.899 | 0.904 | 0.905 | 0.92 | 0.941 | 0.969 | 1.004 | 1.047 | 1.084 | 1.065 | 1.086
Cr | ASPCAP | 0.405 | 0.422 | 0.428 | 0.445 | 0.455 | 0.466 | 0.482 | 0.487 | 0.466 | 0.495 | 0.549 | 0.518
Fe | ASPCAP | 0.501 | 0.501 | 0.501 | 0.501 | 0.501 | 0.501 | 0.501 | 0.501 | 0.501 | 0.501 | 0.501 | 0.501
Ni | ASPCAP | 0.537 | 0.561 | 0.574 | 0.582 | 0.590 | 0.597 | 0.604 | 0.606 | 0.567 | 0.517 | 0.493 | 0.448
V | ASPCAP | 0.896 | 0.830 | 0.677 | 0.573 | 0.567 | 0.586 | 0.643 | 0.684 | 0.655 | 0.571 | 0.529 | 0.524
V | BAWLAS | 0.996 | 0.940 | 0.914 | 0.779 | 0.719 | 0.691 | 0.697 | 0.688 | 0.628 | 0.557 | 0.527 | 0.609
Mn | ASPCAP | 0.238 | 0.255 | 0.275 | 0.293 | 0.312 | 0.329 | 0.344 | 0.354 | 0.317 | 0.232 | 0.148 | 0.146
Co | ASPCAP | 0.371 | 0.453 | 0.507 | 0.538 | 0.577 | 0.611 | 0.657 | 0.688 | 0.655 | 0.568 | 0.505 | 0.511
Cu | BAWLAS | – | 0.978 | 0.770 | 0.655 | 0.623 | 0.580 | 0.586 | 0.610 | 0.614 | 0.554 | 0.494 | 0.295
Ce | ASPCAP | 0.500 | 0.412 | 0.396 | 0.386 | 0.374 | 0.350 | 0.351 | 0.393 | 0.431 | 0.480 | 0.562 | 0.725
Ce | BAWLAS | 0.717 | 0.632 | 0.555 | 0.522 | 0.494 | 0.483 | 0.481 | 0.500 | 0.504 | 0.476 | 0.540 | –
Table 4: Values of $q_{\text{Ia}}^{X}$ Element | Source | [Mg/H]
---|---|---
|  | -0.7 | -0.6 | -0.5 | -0.4 | -0.3 | -0.2 | -0.1 | 0.0 | 0.1 | 0.2 | 0.3 | 0.4
O | ASPCAP | 0.094 | 0.086 | 0.042 | 0.027 | 0.02 | 0.033 | 0.040 | 0.047 | 0.063 | 0.091 | 0.122 | 0.131
O | BAWLAS | -0.034 | -0.200 | -0.122 | -0.111 | -0.079 | -0.058 | -0.032 | -0.005 | 0.023 | 0.084 | 0.058 | 0.095
Mg | ASPCAP | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
Si | ASPCAP | 0.065 | 0.102 | 0.143 | 0.164 | 0.174 | 0.184 | 0.180 | 0.181 | 0.200 | 0.241 | 0.282 | 0.288
S | ASPCAP | 0.207 | 0.038 | 0.012 | 0.019 | 0.022 | 0.052 | 0.065 | 0.071 | 0.096 | 0.156 | 0.192 | 0.225
S | BAWLAS | 0.230 | 0.169 | 0.100 | 0.085 | 0.099 | 0.147 | 0.152 | 0.137 | 0.130 | 0.136 | 0.105 | 0.025
Ca | ASPCAP | 0.199 | 0.200 | 0.23 | 0.250 | 0.267 | 0.286 | 0.285 | 0.268 | 0.264 | 0.271 | 0.272 | 0.264
C+N | ASPCAP | 0.309 | 0.441 | 0.395 | 0.373 | 0.335 | 0.303 | 0.285 | 0.294 | 0.337 | 0.429 | 0.511 | 0.582
C+N | BAWLAS | 0.301 | 0.380 | 0.371 | 0.342 | 0.304 | 0.271 | 0.249 | 0.263 | 0.300 | 0.389 | 0.520 | 0.437
Na | ASPCAP | 0.223 | 0.508 | 0.499 | 0.483 | 0.457 | 0.407 | 0.386 | 0.405 | 0.527 | 0.808 | 1.107 | 1.329
Na | BAWLAS | 0.260 | 0.700 | 0.748 | 0.589 | 0.516 | 0.432 | 0.414 | 0.436 | 0.523 | 0.768 | 1.084 | 1.185
Al | ASPCAP | -0.231 | 0.225 | 0.267 | 0.249 | 0.197 | 0.135 | 0.099 | 0.075 | 0.066 | 0.073 | 0.113 | 0.141
K | ASPCAP | 0.100 | 0.117 | 0.104 | 0.096 | 0.083 | 0.053 | 0.024 | -0.004 | -0.025 | 0.010 | 0.088 | 0.117
Cr | ASPCAP | 0.439 | 0.459 | 0.494 | 0.503 | 0.514 | 0.511 | 0.502 | 0.513 | 0.560 | 0.562 | 0.535 | 0.578
Fe | ASPCAP | 0.499 | 0.499 | 0.499 | 0.499 | 0.499 | 0.499 | 0.499 | 0.499 | 0.499 | 0.499 | 0.499 | 0.499
Ni | ASPCAP | 0.363 | 0.490 | 0.459 | 0.437 | 0.412 | 0.381 | 0.374 | 0.394 | 0.466 | 0.545 | 0.590 | 0.660
V | ASPCAP | 0.461 | 0.275 | 0.320 | 0.296 | 0.289 | 0.294 | 0.295 | 0.316 | 0.403 | 0.539 | 0.611 | 0.647
V | BAWLAS | 0.283 | 0.440 | 0.361 | 0.367 | 0.345 | 0.325 | 0.313 | 0.312 | 0.370 | 0.448 | 0.469 | 0.381
Mn | ASPCAP | 0.330 | 0.546 | 0.539 | 0.532 | 0.532 | 0.544 | 0.574 | 0.646 | 0.786 | 0.977 | 1.141 | 1.244
Co | ASPCAP | 0.347 | 0.388 | 0.371 | 0.375 | 0.358 | 0.321 | 0.293 | 0.312 | 0.394 | 0.539 | 0.643 | 0.693
Cu | BAWLAS | – | 0.159 | 0.330 | 0.357 | 0.323 | 0.330 | 0.344 | 0.390 | 0.493 | 0.687 | 0.928 | 1.300
Ce | ASPCAP | 0.271 | 0.544 | 0.652 | 0.738 | 0.863 | 0.955 | 0.835 | 0.607 | 0.444 | 0.252 | 0.104 | -0.073
Ce | BAWLAS | 0.452 | 0.561 | 0.716 | 0.777 | 0.888 | 0.940 | 0.749 | 0.500 | 0.361 | 0.284 | 0.116 | –
Table 5: Columns Included in Catalog Table Label | Description
---|---
APOGEE_IDaaTaken directly from APOGEE DR17 allStar file. | APOGEE object name
LOCATION_IDaaTaken directly from APOGEE DR17 allStar file. | APOGEE field location ID
TEFFaaTaken directly from APOGEE DR17 allStar file. | Calibrated effective temperature
LOGGaaTaken directly from APOGEE DR17 allStar file. | Calibrated surface gravity
SNRaaTaken directly from APOGEE DR17 allStar file. | Estimated signal-to-noise per pixel
A_CC | Best-fit $A_{\text{cc}}$
A_IA | Best-fit $A_{\text{Ia}}$
CHISQ |  $\chi^{2}$ value
RED_CHISQ | Reduced $\chi^{2}$ (Eq. 5)
DFLAGbbHighest SNR spectrum indicated by DFLAG=0; higher numbers rank additional spectra by descending SNR. | Primary/Secondary Spectrum Flag
X_H_SOURCE_RAWccTaken directly from APOGEE DR17 (if SOURCE=ASPCAP) or BAWLAS (if SOURCE=BAWLAS). | Raw [X/H] value
X_H_SOURCE_ADJ | [X/H]${}_{\text{corr}}$ value (Eq. 3)
X_H_SOURCE_ERRccTaken directly from APOGEE DR17 (if SOURCE=ASPCAP) or BAWLAS (if SOURCE=BAWLAS). | [X/Fe] uncertaintyddSee Section 2 for details. [Fe/H] uncertainty used for X=Fe.
X_H_SOURCE_DEV | Two-process residual $\Delta$[X/H]ee$\Delta\text{[X/H]}=\text{[X/H]}_{\text{corr}}-\text{[X/H]}_{\text{2proc}}$ (see Section 4).

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