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

David H. Weinberg Department of Astronomy and Center for Cosmology and
AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA
Institute for Advanced Study, Princeton, NJ 08540, USA  Jon A. Holtzman
Department of Astronomy, New Mexico State University, Las Cruces, NM 88003,
USA  Jennifer A. Johnson  Department of Astronomy and Center for Cosmology and
AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA
Christian Hayes  Department of Astronomy, University of Washington, Seattle,
WA 98195, USA  Sten Hasselquist  Department of Physics & Astronomy, University
of Utah, Salt Lake City, UT 84112,USA  Matthew Shetrone  University of
California, Santa Cruz, UCO/Lick Observatory, 1156 High St., Santa Cruz, CA
95064, USA  Yuan-Sen Ting (丁源森)  Institute for Advanced Study, Princeton, NJ
08540, USA  Department of Astrophysical Sciences, Princeton University,
Princeton, NJ 08540, USA  Observatories of the Carnegie Institution of
Washington, 813 Santa Barbara Street, Pasadena, CA 91101, USA  Research School
of Astronomy & Astrophysics, Australian National University, Cotter Rd.,
Weston, ACT 2611, Australia  Rachael L. Beaton  The Observatories of the
Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena, CA
91101, USA  Timothy C. Beers  Department of Physics and JINA Center for the
Evolution of the Elements, University of Notre Dame, Notre Dame, IN 46556, USA
Jonathan C. Bird  Department of Physics and Astronomy, Vanderbilt University,
VU Station 1807, Nashville, TN 37235, USA  Dmitry Bizyaev  Apache Point
Observatory, P.O. Box 59, Sunspot, NM 88349  Michael R. Blanton  Center for
Cosmology and Particle Physics, Department of Physics, 726 Broadway, Room
1005, New York University, New York, NY 10003, USA  Katia Cunha  José G.
Fernández-Trincado  Instituto de Astronomía y Ciencias Planetarias,
Universidad de Atacama, Copayapu 485, Copiapó, Chile  Instituto de Astronomía,
Universidad Católica del Norte, Av. Angamos 0610, Antofagasta, Chile  Peter M.
Frinchaboy  Department of Physics & Astronomy, Texas Christian University,
Fort Worth, TX 76129, USA  D. A. García-Hernández Instituto de Astrofísica de
Canarias, 38205 La Laguna, Tenerife, Spain  Universidad de La Laguna (ULL),
Departamento de Astrofísica, E-38206 La Laguna, Tenerife, Spain  Emily
Griffith  Department of Astronomy and Center for Cosmology and AstroParticle
Physics, The Ohio State University, Columbus, OH 43210, USA  James W. Johnson
Department of Astronomy and Center for Cosmology and AstroParticle Physics,
The Ohio State University, Columbus, OH 43210, USA  Henrik Jönsson Materials
Science and Applied Mathematics, Malmö University, SE-205 06 Malmö, Sweden
Richard R. Lane  Centro de Investigación en Astronomía, Universidad Bernardo
O’Higgins, Avenida Viel 1497, Santiago, Chile  Henry W. Leung  Dunlap
Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George
Street, Toronto, ON M5S 3H4, Canada  J. Ted Mackereth  Dunlap Institute for
Astronomy and Astrophysics, University of Toronto, 50 St. George Street,
Toronto, ON M5S 3H4, Canada  David A. Dunlap Department for Astronomy and
Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S
3H4, Canada  Canadian Institute for Theoretical Astrophysics, University of
Toronto, 60 St. George Street, Toronto, ON, M5S 3H8, Canada  Steven R.
Majewski  Department of Astronomy, University of Virginia, Charlottesville, VA
22904, USA  Szabolcz Mészáros  ELTE Eötvös Loránd University, Gothard
Astrophysical Observatory, 9700 Szombathely, Szent Imre H. st. 112, Hungary
MTA-ELTE Lendület Milky Way Research Group, Hungary  MTA-ELTE Exoplanet
Research Group, Hungary  Christian Nitschelm  Centro de Astronomía (CITEVA),
Universidad de Antofagasta, Avenida Angamos 601, Antofagasta 1270300, Chile
Kaike Pan  Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349  Ricardo
P. Schiavon  Astrophysics Research Institute, Liverpool John Moores
University, Liverpool, L3 5RF, UK  Donald P. Schneider  Department of
Astronomy and Astrophysics, The Pennsylvania State University, University
Park, PA 16802, USA  Institute for Gravitation and the Cosmos, The
Pennsylvania State University, University Park, PA 16802, USA  Mathias
Schultheis  Observatoire de la Côte d’Azur, Laboratoire Lagrange, 06304 Nice
Cedex 4, France  Verne Smith  NSF’s National Optical-Infrared Astronomy
Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA  Jennifer
S. Sobeck  Department of Astronomy, University of Washington, Seattle, WA
98195, USA  Keivan G. Stassun  Department of Physics and Astronomy, Vanderbilt
University, VU Station 1807, Nashville, TN 37235, USA  Guy S. Stringfellow
Center for Astrophysics and Space Astronomy, Department of Astrophysical and
Planetary Sciences, University of Colorado, 389 UCB, Boulder, CO 80309-0389,
USA  Fiorenzo Vincenzo  Department of Astronomy and Center for Cosmology and
AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA
John C. Wilson  Department of Astronomy, University of Virginia,
Charlottesville, VA 22904, USA  Gail Zasowski  Department of Physics and
Astronomy, University of Utah, 115 S. 1400 E., Salt Lake City, UT 84112, USA

###### Abstract

We apply a novel statistical analysis to measurements of 16 elemental
abundances in 34,410 Milky Way disk stars from the final data release (DR17)
of APOGEE-2. Building on recent work, we fit median abundance ratio trends
[X/Mg] vs. [Mg/H] with a 2-process model, which decomposes abundance patterns
into a “prompt” component tracing core collapse supernovae and a “delayed”
component tracing Type Ia supernovae. For each sample star, we fit the
amplitudes of these two components, then compute the residuals $\Delta[{\rm
X}/{\rm H}]$ from this two-parameter fit. The rms residuals range from $\sim
0.01-0.03$ dex for the most precisely measured APOGEE abundances to $\sim 0.1$
dex for Na, V, and Ce. The correlations of residuals reveal a complex
underlying structure, including a correlated element group comprised of Ca,
Na, Al, K, Cr, and Ce and a separate group comprised of Ni, V, Mn, and Co.
Selecting stars poorly fit by the 2-process model reveals a rich variety of
physical outliers and sometimes subtle measurement errors. Residual abundances
allow comparison of populations controlled for differences in metallicity and
[$\alpha$/Fe]. Relative to the main disk ($R=3-13\,{\rm kpc}$), we find nearly
identical abundance patterns in the outer disk ($R=15-17\,{\rm kpc}$),
0.05-0.2 dex depressions of multiple elements in LMC and Gaia
Sausage/Enceladus stars, and wild deviations (0.4-1 dex) of multiple elements
in $\omega\,$Cen. Residual abundance analysis opens new opportunities for
discovering chemically distinctive stars and stellar populations, for
empirically constraining nucleosynthetic yields, and for testing chemical
evolution models that include stochasticity in the production and
redistribution of elements.

.

##  1 Introduction

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

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

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

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

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

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

##  2 The 2-process model

Figure 1: (Left) $[{\rm Mg}/{\rm Fe}]$ vs. $[{\rm Mg}/{\rm H}]$ for sample
stars in the low-Ia (red) and high-Ia (blue) populations. Points are randomly
downsampled by a factor of four to reduce crowding. Connected large points
show the median $[{\rm Mg}/{\rm Fe}]$ in bins of $[{\rm Mg}/{\rm H}]$. An
offset of 0.053 dex has been applied to the APOGEE Fe abundances (decreasing
[Mg/Fe] by 0.053) so that the median high-Ia sequence passes through $[{\rm
Mg}/{\rm Fe}]=0$ at $[{\rm Mg}/{\rm H}]=0$. The dotted horizontal line shows
the ratio $[{\rm Mg}/{\rm Fe}]_{\rm pl}=0.30$ that is assumed to correspond to
pure CCSN enrichment in our 2-process modeling. (Right) The fraction of iron
inferred (via the 2-process model) to arise from CCSN at points along the
median low-Ia (red) and high-Ia (blue) sequences (eq. 6). The dashed curve
shows $A_{\rm cc}=10^{[{\rm Mg}/{\rm H}]}$.

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

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

${({\rm X}/{\rm H})_{*}\over({\rm X}/{\rm H})_{\odot}}=A_{\rm cc,*}q^{X}_{\rm
cc}+A_{\rm Ia,*}q^{X}_{\rm Ia}~{}.$ (1)

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

$z\equiv 10^{[{\rm Mg}/{\rm H}]}~{},$ (2)

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

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

Figure 2: Illustration of the 2-process model for the simple case of two
abundances, (Mg/H) and (Fe/H), both linear ratios scaled to solar values. The
(Mg,Fe) components of the two-process vectors are $\vec{q}_{\rm cc}=(1,0.5)$
and $\vec{q}_{\rm Ia}=(0,0.5)$, respectively. Large points show the abundances
of stars with $(A_{\rm cc},A_{\rm Ia})=(1.0,1.0)$ (filled blue circle),
(0.33,0.33) (open blue circle), (1.0,0.2) (filled red square), (0.33,0.02)
(open red square). The inset marks the position of these four stars in the
[Mg/Fe]-[Fe/H] plane, with red and blue curves showing the observed median
sequences of low-Ia and high-Ia stars from Fig. 1. The dotted blue line marks
the locus of $[{\rm Mg}/{\rm Fe}]=0$.

Figure 3: Illustration of the 2-process model for the 16 elements considered
in our analysis (one of which is the element combination C+N). In the top
panel, open circles and filled triangles show the 2-process vectors
$q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ at $[{\rm Mg}/{\rm H}]=0$ inferred from
the APOGEE data in §4. For a star with $A_{\rm cc}=A_{\rm Ia}=1$, the
predicted abundances are the sum of these two vectors, which in this case
yield exactly solar values (blue points) by construction. Red points show the
predicted abundances for a low-Ia star with $A_{\rm cc}=1$, $A_{\rm Ia}=0.2$.
The lower panel shows analogous results at $[{\rm Mg}/{\rm H}]=-0.5$, with
predicted abundances multiplied by a factor of three in this panel to aid
visual comparison with the top panel. In each panel, colored horizontal lines
group elements with similar physical properties. The four combinations of
$(A_{\rm cc},A_{\rm Ia})$ in this Figure are the same cases illustrated in
Figure 2.

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

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

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

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

###  2.1 Model assumptions and basic equations

We can express equation (1) in the alternative form

$\left({\rm X}\over{\rm H}\right)_{*}=A_{\rm cc,*}p_{\rm cc}^{X}(z)+A_{\rm
Ia,*}p_{\rm Ia}^{X}(z)~{},$ (3)

with

$q^{X}_{j}(z)\equiv{p_{j}^{X}(z)\over({\rm X}/{\rm H})_{\odot}}~{}.$ (4)

While it may seem gratuitous to introduce both $p$ and $q$, many of our
equations can be written more compactly in terms of $p$, and these two forms
of the process vectors respond differently to changes in the adopted solar
abundance values. For example, if stellar abundances are inferred by purely ab
initio model-fitting then the $p$ vectors are directly determined while the
$q$ vectors depend on adopted solar abundances. Conversely, if zero-point
offsets are used to calibrate the abundance scale to reproduce solar values,
then the $q$ vectors are directly determined and the conversion to $p$ vectors
depends on the adopted solar abundances. At a given $z$, $p_{\rm cc}^{X}$ is a
set of discrete values, one for each element X being modeled, and likewise for
$p_{\rm Ia}^{X}$. For brevity, we will frequently drop the explicit
$z$-dependence of $p_{\rm cc}^{X}$ and $p_{\rm Ia}^{X}$ in our equations if it
is not needed for clarity, but it is only for Mg and Fe that we assume that
these processes are actually independent of metallicity.

We define $A_{\rm cc}=A_{\rm Ia}$ in the Sun. Therefore, if we ignore the
possible contribution of other processes,

$p_{\rm cc}^{X}(z=1)+p_{\rm Ia}^{X}(z=1)=\left({\rm X}\over{\rm
H}\right)_{\odot}~{}$ (5)

for all X. In a star with metallicity $z$ and amplitudes $A_{\rm cc,*}$ and
$A_{\rm Ia,*}$, the fraction of element $X$ that arises from CCSN is

$\displaystyle f^{X}_{\rm cc}$ $\displaystyle=$ $\displaystyle{A_{\rm
cc,*}p_{\rm cc}^{X}(z)\over A_{\rm cc,*}p_{\rm cc}^{X}(z)+A_{\rm Ia,*}p_{\rm
Ia}^{X}(z)}$ (6) $\displaystyle=$ $\displaystyle\left[1+(A_{\rm Ia,*}/A_{\rm
cc,*})(q^{X}_{\rm Ia}/q^{X}_{\rm cc})\right]^{-1}~{},$ (7)

where we have used the fact that $q^{X}_{\rm Ia}/q^{X}_{\rm cc}=p_{\rm
Ia}^{X}/p_{\rm cc}^{X}$. More generally, the denominator of equation (6)
should be the sum of all processes that contribute to element X (see §8.1).
For the Sun we have $A_{\rm Ia}=A_{\rm cc}=1$ and $q^{X}_{\rm Ia}=1-q^{X}_{\rm
cc}$, which simplifies equation (7) to

$f^{X}_{\rm cc,\odot}=q^{X}_{\rm cc}(z=1).$ (8)

For our implementation of the 2-process model, we assume that the Mg and Fe
processes are independent of metallicity and that Mg is a pure core collapse
element:

$\displaystyle p_{\rm cc}^{\rm Mg}(z)$ $\displaystyle=$ $\displaystyle p_{\rm
cc}^{\rm Mg},\qquad p_{\rm Ia}^{\rm Mg}=0,$ (9) $\displaystyle p_{\rm cc}^{\rm
Fe}(z)$ $\displaystyle=$ $\displaystyle p_{\rm cc}^{\rm Fe},\qquad p_{\rm
Ia}^{\rm Fe}(z)=p_{\rm Ia}^{\rm Fe}~{}.$ (10)

Standard supernova models predict that Mg and Fe yields are approximately
independent of metallicity (see, e.g., Fig. 20 of Andrews et al. 2017) and
that the SNIa contribution to Mg is negligible. At low metallicity, the
high-$\alpha$ population in APOGEE and other surveys exhibits a nearly flat
plateau in $[{\rm Mg}/{\rm Fe}]$ at

$[{\rm Mg}/{\rm Fe}]_{\rm pl}\approx 0.3$ (11)

(see, e.g., Adibekyan et al. 2012; Bensby et al. 2014; Buder et al. 2018;
Griffith et al. 2019; and Figure 1 above). This flatness provides empirical
support for metallicity independence of the Mg and Fe CCSN processes. A flat
plateau could also arise if CCSN yields of these elements have the same
metallicity dependence while keeping the Mg/Fe ratio constant. Our formalism
could be adapted to metallicity-dependent Mg and Fe processes if there were
motivation to do so, but this would introduce some mathematical complication
so we do not consider this generalization here.

Combining equations (9) and (3) implies

$\left({\rm Mg}\over{\rm H}\right)_{*}=A_{\rm cc,*}p_{\rm cc}^{\rm Mg}=A_{\rm
cc,*}\left({\rm Mg}\over{\rm H}\right)_{\odot}$ (12)

and thus

$A_{\rm cc,*}=10^{[{\rm Mg}/{\rm H}]_{*}}~{}.$ (13)

Equation (13) provides a simple way to estimate $A_{\rm cc,*}$, from a star’s
Mg abundance alone, though in practice we will use a multi-element fit as
described below (§2.3).

For iron, equations (10) and (3) imply

${({\rm Fe}/{\rm Mg})_{*}\over({\rm Fe}/{\rm Mg})_{\odot}}={A_{\rm cc,*}p_{\rm
cc}^{\rm Fe}+A_{\rm Ia,*}p_{\rm Ia}^{\rm Fe}\over A_{\rm cc,*}p_{\rm cc}^{\rm
Mg}}\cdot{p_{\rm cc}^{\rm Mg}\over p_{\rm cc}^{\rm Fe}+p_{\rm Ia}^{\rm
Fe}}~{},$ (14)

which can be rearranged to yield

$10^{[{\rm Fe}/{\rm Mg}]_{*}}={p_{\rm cc}^{\rm Fe}+(A_{\rm Ia,*}/A_{\rm
cc,*})p_{\rm Ia}^{\rm Fe}\over p_{\rm cc}^{\rm Fe}+p_{\rm Ia}^{\rm Fe}}~{}.$
(15)

Our third key assumption is that iron in stars on the $[{\rm Mg}/{\rm Fe}]$
plateau comes from CCSN alone, implying $A_{\rm Ia,*}=0$ and thus

$10^{[{\rm Fe}/{\rm Mg}]_{\rm pl}}={p_{\rm cc}^{\rm Fe}\over p_{\rm cc}^{\rm
Fe}+p_{\rm Ia}^{\rm Fe}}=f^{\rm Fe}_{\rm cc,\odot}~{}.$ (16)

By definition, if $A_{\rm Ia,*}=A_{\rm cc,*}=1$ we are at solar abundances for
Mg and Fe (because they are assumed to have no contributions from other
processes), and therefore $[{\rm Fe}/{\rm Mg}]=0$ as implied by equation (15).

With a bit of manipulation one can write

$\displaystyle{A_{\rm Ia,*}\over A_{\rm cc,*}}$ $\displaystyle=$
$\displaystyle{({\rm Fe}/{\rm Mg})_{*}-({\rm Fe}/{\rm Mg})_{\rm pl}\over({\rm
Fe}/{\rm Mg})_{\odot}-({\rm Fe}/{\rm Mg})_{\rm pl}}$ (17) $\displaystyle=$
$\displaystyle{10^{[{\rm Fe}/{\rm Mg}]_{*}}-10^{[{\rm Fe}/{\rm Mg}]_{\rm
pl}}\over 1-10^{[{\rm Fe}/{\rm Mg}]_{\rm pl}}}~{}.$ (18)

In the first equation, the numerator is the amount SNIa Fe in the star
relative to Mg, and the denominator is the amount of SNIa Fe at solar $[{\rm
Fe}/{\rm Mg}]$, for which $A_{\rm Ia,*}/A_{\rm cc,*}=1$. The second equation
relates $A_{\rm Ia,*}/A_{\rm cc,*}$ to the displacement of $[{\rm Fe}/{\rm
Mg}]_{*}$ below the CCSN plateau. We adopt $[{\rm Fe}/{\rm Mg}]_{\rm
pl}=-[{\rm Mg}/{\rm Fe}]_{\rm pl}=-0.3$ as the observed level of the plateau,
and thus $10^{[{\rm Fe}/{\rm Mg}]_{\rm pl}}\approx 0.5$.

Equation (18) provides a simple way to estimate $A_{\rm Ia,*}$ after
estimating $A_{\rm cc,*}$ from equation (13). In the right panel of Figure 1,
red and blue curves show the inferred values of $f^{\rm Fe}_{\rm cc}$
(equation 7) for points along the low-Ia and high-Ia median sequences shown in
the left panel. The values of $A_{\rm cc}$ and $A_{\rm Ia}$ are derived from
the $[{\rm Mg}/{\rm H}]$ and $[{\rm Mg}/{\rm Fe}]$ values along these
sequences as described above. On the high-Ia sequence the inferred core
collapse iron fraction is about 0.5 at all $[{\rm Mg}/{\rm H}]$. On the low-Ia
sequence the fraction declines from nearly 100% at low $[{\rm Mg}/{\rm H}]$ to
about 0.6 at the highest $[{\rm Mg}/{\rm H}]$. The dashed curve shows the
value of $A_{\rm cc}$ corresponding to $[{\rm Mg}/{\rm H}]$ via equation (13).

In terms of the solar-scaled process vectors, our model assumptions and $[{\rm
Fe}/{\rm Mg}]_{\rm pl}$ value correspond to $q_{\rm cc}^{\rm Mg}=1$ and
$q_{\rm cc}^{\rm Fe}=q_{\rm Ia}^{\rm Fe}=0.5$. For an element X that is
produced entirely by CCSN and SNIa, the solar-scaled abundances are

$[{\rm X}/{\rm H}]_{*}=\log_{10}\left[A_{\rm cc,*}q^{X}_{\rm cc}(z)+A_{\rm
Ia,*}q^{X}_{\rm Ia}(z)\right]~{}.$ (19)

Subtracting $[{\rm Mg}/{\rm H}]_{*}=\log_{10}A_{\rm cc,*}$ gives

$[{\rm X}/{\rm Mg}]_{*}=\log_{10}\left[q^{X}_{\rm cc}(z)+q^{X}_{\rm
Ia}(z)A_{\rm Ia,*}/A_{\rm cc,*}\right]~{}.$ (20)

More generally, an element may have contributions from CCSN or other “prompt”
enrichment sources (e.g., massive star winds) that rapidly follow star
formation, and additional contributions from enrichment sources with a
distribution of delay times (e.g., SNIa, AGB stars). When modeled with the
2-process formalism, $q^{X}_{\rm cc}$ represents the prompt contributions and
$q^{X}_{\rm Ia}$ represents the contributions that follow SNIa iron
enrichment, with the implicit assumption that the ISM is sufficiently well
mixed to average out the diverse properties of individual supernovae or other
sources.

To apply the 2-process model to APOGEE data, we must first determine the
values of $q^{X}_{\rm cc}(z)$ and $q^{X}_{\rm Ia}(z)$ from the ensemble of
measurements, then determine the amplitudes $A_{\rm cc,*}$ and $A_{\rm Ia,*}$
for each star. We can then predict each star’s abundances and measure the
residuals, i.e., the difference between the observed abundances and the
2-process predictions.

###  2.2 Inferring the 2-process vectors from median sequences

Similar to W19, we infer the process vectors $q^{X}_{\rm cc}(z)$ and
$q^{X}_{\rm Ia}(z)$ from the observed median sequences of $[{\rm X}/{\rm Mg}]$
vs. $[{\rm Mg}/{\rm H}]$ for the low-Ia and high-Ia stellar populations. We do
this separately in each bin of $[{\rm Mg}/{\rm H}]$, and we will henceforth
drop the $z$-dependence from our notation with the understanding that
$q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ can change from bin to bin. In principle
we could perform a global $\chi^{2}$ fit to the abundances of stars in each
$[{\rm Mg}/{\rm H}]$ bin, but inferring the process vectors from the median
sequences is much easier, and it is also more robust because outliers (whether
physical or observational) have minimal impact on median values in a large
data set. The median values of $[{\rm Fe}/{\rm Mg}]$ are significantly
different between the two populations even at high $[{\rm Mg}/{\rm H}]$, so
there is sufficient leverage to separate the CCSN and SNIa contributions.
Statistical errors on the median abundance ratios are very small because there
are many stars in each bin, though we are still sensitive to systematic errors
in the abundance measurements.

W19 assumed a power-law $z$-dependence of the process vectors, but here we
allow a general metallicity dependence. For each element and each $[{\rm
Mg}/{\rm H}]$ bin, there are two measurements, $[{\rm X}/{\rm Mg}]_{\rm med}$
of the low-Ia and high-Ia populations, to fit with two parameters, $q^{X}_{\rm
cc}$ and $q^{X}_{\rm Ia}$, so the 2-process model can exactly reproduce the
observed median sequences by construction. We adopt a general (bin-by-bin)
$z$-dependence in part to capture possibly complex trends, but for purposes of
this paper our primary motivation is to ensure that the mean star-by-star
residuals from the 2-process predictions are close to zero at all $[{\rm
Mg}/{\rm H}]$. Although the more restrictive power-law formulation usually
allows a good fit to the observed median sequences, there are departures for
some elements in some $[{\rm Mg}/{\rm H}]$ ranges, and residuals could easily
be dominated by these global differences rather than star-to-star variations.

Using the observed values of $[{\rm Fe}/{\rm Mg}]_{\rm pl}$ and of the median
values of $[{\rm Fe}/{\rm Mg}]$ on the high-Ia and low-Ia sequences in the
$[{\rm Mg}/{\rm H}]$ bin under consideration, we define

$R_{\rm high}\equiv\left({A_{\rm Ia}\over A_{\rm cc}}\right)_{\rm
high}={10^{[{\rm Fe}/{\rm Mg}]_{\rm high}}-10^{[{\rm Fe}/{\rm Mg}]_{\rm
pl}}\over 1-10^{[{\rm Fe}/{\rm Mg}]_{\rm pl}}}$ (21)

and

$R_{\rm low}\equiv\left({A_{\rm Ia}\over A_{\rm cc}}\right)_{\rm
low}={10^{[{\rm Fe}/{\rm Mg}]_{\rm low}}-10^{[{\rm Fe}/{\rm Mg}]_{\rm
pl}}\over 1-10^{[{\rm Fe}/{\rm Mg}]_{\rm pl}}}~{}.$ (22)

From equation (20) we have

$\displaystyle 10^{[{\rm X}/{\rm Mg}]_{\rm high}}$ $\displaystyle=$
$\displaystyle q^{X}_{\rm cc}+R_{\rm high}q^{X}_{\rm Ia}$ (23) $\displaystyle
10^{[{\rm X}/{\rm Mg}]_{\rm low}}$ $\displaystyle=$ $\displaystyle q^{X}_{\rm
cc}+R_{\rm low}q^{X}_{\rm Ia}~{}.$ (24)

Solving these equations yields

$q^{X}_{\rm Ia}={10^{[{\rm X}/{\rm Mg}]_{\rm high}}-10^{[{\rm X}/{\rm
Mg}]_{\rm low}}\over R_{\rm high}-R_{\rm low}}$ (25)

and

$q^{X}_{\rm cc}=10^{[{\rm X}/{\rm Mg}]_{\rm low}}-{10^{[{\rm X}/{\rm Mg}]_{\rm
high}}-10^{[{\rm X}/{\rm Mg}]_{\rm low}}\over R_{\rm high}/R_{\rm low}-1}~{}.$
(26)

If there is no difference between $[{\rm X}/{\rm Mg}]$ on the two sequences we
get $q^{X}_{\rm cc}=10^{[{\rm X}/{\rm Mg}]}$ and $q^{X}_{\rm Ia}=0$. For a
point with $[{\rm Fe}/{\rm Mg}]_{\rm low}=[{\rm Fe}/{\rm Mg}]_{\rm pl}$, we
get $R_{\rm low}=0$ and $q^{X}_{\rm cc}=10^{[{\rm X}/{\rm Mg}]_{\rm low}}$,
which is as expected because such a point has no SNIa contribution. We use
equations (25) and (26) to infer $q^{X}_{\rm Ia}$ and $q^{X}_{\rm cc}$ for
each element X in each bin of $[{\rm Mg}/{\rm H}]$ (see Figures 4-7 below).

###  2.3 Fitting stellar values of $A_{\rm cc}$ and $A_{\rm Ia}$

Equations (13) and (18) provide a simple way to estimate a star’s 2-process
amplitudes $A_{\rm cc}$ and $A_{\rm Ia}$ from its Mg and Fe abundances. This
is the method used by W19, and because Mg and Fe are well measured by APOGEE
it is accurate enough for many purposes. However, for our goal of studying the
correlations of residual abundances it has an important disadvantage: random
measurement errors in $[{\rm Mg}/{\rm H}]$ and $[{\rm Fe}/{\rm Mg}]$ will
induce spurious apparent correlations in the residuals of other elements. For
example, if a star’s measured $[{\rm Mg}/{\rm H}]$ fluctuates low, its $A_{\rm
cc}$ will be underestimated, and all of the star’s other $\alpha$-elements
will tend to lie above the 2-process prediction. TW21 examined the closely
connected question of residual abundances after conditioning on $[{\rm
Fe}/{\rm H}]$ and $[{\rm Mg}/{\rm Fe}]$. They described the spurious
correlations that arise from random Mg and Fe abundance errors as “measurement
aberration,” caused by defining the residual abundances relative to a
(randomly) incorrect reference point.

We can mitigate the effects of measurement aberration by estimating a star’s
$A_{\rm cc}$ and $A_{\rm Ia}$ from multiple abundances, since the random
errors in these abundances tend to average out. As discussed in §5 below, we
choose to infer $A_{\rm cc}$ and $A_{\rm Ia}$ from the abundances of six
elements (Mg, O, Si, Ca, Fe, Ni) that have small statistical errors in APOGEE
and that collectively provide good leverage on the 2-process amplitudes
because they have a range of relative contributions from SNIa vs. CCSN. These
elements are not expected to have significant contributions from sources other
than CCSN and SNIa. We fit each star’s $A_{\rm cc}$ and $A_{\rm Ia}$ by
$\chi^{2}$ minimization using the observational measurement uncertainties
reported by APOGEE.

In practice, we take the parameter estimates from Mg and Fe as an initial
guess, then iterate between optimizing $A_{\rm cc}$ and $A_{\rm Ia}$, an
approach that is computationally cheap and quickly converges to a 2-d
$\chi^{2}$ minimum. To avoid fit parameters being affected by outlier
abundances (which could well be observational errors), we eliminate O, Si, Ca,
or Ni from the fit if their abundance differs by more than $5\sigma$ from the
value predicted based on the initial guess. This criterion leads to the
elimination of 206 O measurements, 118 Si measurements, 279 Ca measurements,
and 625 Ni measurements from our sample of 34,410 stars. Fitting six
abundances with two parameters does not add any more freedom to the model, but
instead of fitting Mg and Fe exactly it chooses compromise values that give
the best overall fit to the selected elements. We demonstrate the reduced
measurement aberration from six-element fitting in Fig. 15 below.

##  3 APOGEE data sample

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

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

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

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

  1. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

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

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

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

  1. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

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

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

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

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

$\displaystyle[{\rm(C+N)}/{\rm H}]=$ $\displaystyle\log_{10}\left(10^{[{\rm
C}/{\rm H}]+8.39}+10^{[{\rm N}/{\rm H}]+7.78}\right)-$ (27)
$\displaystyle\log_{10}\left(10^{8.39}+10^{7.78}\right)~{},$

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

Table 1: Zero-point offsets and $T_{\rm eff}$ trend slopes  Elem | Offset | $10^{3}\alpha_{T}$ |  | Elem | Offset | $10^{3}\alpha_{T}$
---|---|---|---|---|---|---
Mg | $0.000$ | $0.94$ |  | K | $0.002$ | $1.68$
O | $-0.016$ | $2.28$ |  | Cr | $0.048$ | $4.35$
Si | $0.038$ | $-3.22$ |  | Fe | $0.053$ | $0.76$
S | $0.008$ | $5.29$ |  | Ni | $0.030$ | $1.33$
Ca | $0.071$ | $-6.01$ |  | V | $0.222$ | $14.9$
C+N | $0.022$ | $4.12$ |  | Mn | $0.002$ | $16.3$
Na | $0.043$ | $8.89$ |  | Co | $-0.032$ | $8.86$
Al | $0.050$ | $-12.3$ |  | Ce | $0.125$ | $-2.64$

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

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

##  4 Median sequences and 2-process vectors

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

$\begin{cases}[{\rm Mg}/{\rm Fe}]>0.12-0.13[{\rm Fe}/{\rm H}],&[{\rm Fe}/{\rm
H}]<0\cr[{\rm Mg}/{\rm Fe}]>0.12,&[{\rm Fe}/{\rm H}]>0.\cr\end{cases}$ (28)

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

Figure 4: (Left) $[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ for the
$\alpha$ elements of sample stars, color-coded as low-Ia (red) or high-Ia
(blue). Stars are randomly downsampled by a factor of four to reduce crowding.
Connected large points show the median values in bins of $[{\rm Mg}/{\rm H}]$,
which the 2-process model fits exactly by construction. (Right) Solar-scaled
values of the CCSN and SNIa process vector components for each element,
$q^{X}_{\rm cc}$ (circles) and $q^{X}_{\rm Ia}$ (triangles), inferred by
fitting the observed median sequences. Red and blue curves show the CCSN
fractions $f^{X}_{\rm cc}$, which depend on the values of $q^{X}_{\rm cc}$ and
$q^{X}_{\rm Ia}$ and on the amplitude ratio $A_{\rm Ia}/A_{\rm cc}$ at the
corresponding point on the median sequence (eq. 7). Our abundances include
zero-point calibrations (Table 1) that force the high-Ia sequence to pass
through $[{\rm X}/{\rm Mg}]=0$ at $[{\rm Mg}/{\rm H}]=0$. For solar
abundances, $f^{X}_{\rm cc}=q^{X}_{\rm cc}$, so blue curves in the right
panels always pass through the open circle at $[{\rm Mg}/{\rm H}]=0$.

Figure 5: Same as Fig. 4 but showing the light odd-$Z$ elements Na, Al, and K
and the element combination C+N.

Figure 6: Same as Fig. 4 but for even-$Z$ iron-peak elements.

Figure 7: Same as Fig. 4 but for odd-$Z$ iron-peak elements and the s-process
element Ce.

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

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

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

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

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

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

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

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

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

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

##  5 Residual abundances and their correlations

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

Figure 8: Distribution of stars in the 2-process parameters $A_{\rm Ia}/A_{\rm
cc}$ vs. $A_{\rm cc}$, in zones of Galactocentric radius (columns) and
midplane distance (rows) as labeled. $A_{\rm cc}$ measures the abundance of
CCSN elements (e.g., Mg) relative to solar, and $A_{\rm Ia}/A_{\rm cc}$
measures the ratio of SNIa to CCSN enrichment; $A_{\rm Ia}=A_{\rm cc}=1$ for
solar abundances. Red and blue points show stars in the low-Ia and high-Ia
population, respectively. To improve coverage of the inner Galaxy, this plot
uses the SN100 sample, which has a SNR threshold of 100 at all $[{\rm Mg}/{\rm
H}]$ values. Although we use six elements to fit stellar values of $A_{\rm
cc}$ and $A_{\rm Ia}$, they are generally close to the values implied by Mg
and Fe, so this plot resembles a plot of $[{\rm Fe}/{\rm Mg}]$ vs. $[{\rm
Mg}/{\rm H}]$ but with transformed variables that are linearly proportional to
the inferred CCSN and SNIa content.

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

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

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

Figure 9: Examples of 2-process fits to element abundance ratios, for two low-
Ia stars (rows one and two) with sub-solar $[{\rm Mg}/{\rm H}]$ and two high-
Ia stars (rows three and four) with near-solar $[{\rm Mg}/{\rm H}]$ and $[{\rm
Mg}/{\rm Fe}]$. The first and third stars have $\chi^{2}$ values near the
median of the distribution for sample stars, while the second and fourth stars
have $\chi^{2}$ values near the 98th-percentile of this distribution. The
model has two free parameters for each star and is fit to the six elements
listed in blue on the horizontal axis (Mg, O, Si, Ca, Fe, Ni). In each panel,
filled points with error bars show the measured value of [X/H] and the ASPCAP
error. Open circles show the abundances predicted by the 2-process fit. Solid
and dotted lines are present to guide the eye. Colored bars along the bottom
of each panel group $\alpha$ elements (blue), light odd-$Z$ elements (green),
even-$Z$ iron-peak elements (red), and odd-$Z$ iron-peak elements (magenta).

###  5.1 Removing trends with $T_{\rm eff}$

We have limited the range of ${\rm log}\,g$ and $T_{\rm eff}$ in our sample in
order to minimize the differential impact of abundance measurement systematics
on our results. Nonetheless, there are subtle trends of residual abundances
with $T_{\rm eff}$ over this range, as illustrated for four elements in the
top row of Figure 10. In this and all subsequent plots we adopt the sign
convention

$\Delta[{\rm X}/{\rm H}]\equiv[{\rm X}/{\rm H}]_{\rm data}-[{\rm X}/{\rm
H}]_{\rm model}~{}.$ (29)

Mn residuals have the strongest trend, with the median abundance residual
changing from $0.047$ to $-0.033$ as $T_{\rm eff}$ increases from $4050\,{\rm
K}$ to $4550\,{\rm K}$. Co residuals have a weaker trend of the same sign, Ca
residuals have a trend of similar magnitude but opposite sign, and Ni
residuals have almost no trend with $T_{\rm eff}$.

Figure 10: (Top) Correlation of abundance residuals (measured abundance -
2-process prediction) with $T_{\rm eff}$ for Ca, Ni, Mn, and Co. Magenta
points and lines show the median residuals in $100\,{\rm K}$ bins of $T_{\rm
eff}$ and the linear trends fit to these medians. (Middle) Correlation of
abundance residuals for the element pairs Ca-Ni, Ca-Mn, Co-Ni, and Co-Mn (left
to right). Correlation coefficients are listed in each panel. (Bottom) Same as
middle using abundances that have been corrected for the temperature trends as
described in the text. In all panels, points have been downsampled by a factor
of ten to reduce crowding, and abundance residuals are shown in dex.

To avoid artificial correlations induced by these trends, we fit them with
linear relations and apply corrections to the DR17 APOGEE abundance values:

$[{\rm X}/{\rm H}]_{\rm corr}=[{\rm X}/{\rm H}]_{\rm APO}+{\rm
Offset}+\alpha_{T}(T_{\rm eff}-4300)/100~{}.$ (30)

The values of the zero-point offsets (discussed in §3) and the slopes
$\alpha_{T}$ are listed in Table 1, with values of the latter ranging from
$|\alpha_{T}|\sim 0.001$ (Mg, O, K, Fe, Ni) to $|\alpha_{T}|\sim 0.01$ (Ca,
Na, Al, V, Mn, Co). For Mn, for example, all abundances are increased by 0.002
dex, and the abundances of stars with $T_{\rm eff}=4600\,{\rm K}$ are further
increased by $0.0163\times 3=0.0489$ dex, thus increasing the median residual
abundance to near zero. We have confirmed that residual trends with $T_{\rm
eff}$ and ${\rm log}\,g$ are negligible for all elements after applying these
corrections. Note that the abundances plotted in Figures 4-7 include the zero-
point offsets but do not have the $T_{\rm eff}$ corrections applied. Median
sequences and derived $q^{X}_{\rm cc}$ and $q^{X}_{\rm Ia}$ values are
negligibly affected by these $T_{\rm eff}$ trends. They matter for our
subsequent analysis (and are used in all subsequent calculations and plots)
because they can affect correlations of residual abundances.

The lower half of Figure 10 shows scatterplots of residual abundances for four
pairs of these four abundances before (middle row) and after (bottom row)
removing the $T_{\rm eff}$ trends. The Ca-Ni correlation is minimally affected
because Ni residuals have almost no trend with $T_{\rm eff}$. The correlation
coefficient changes from $-0.30$ before correction to $-0.28$ after
correction. However, Ca and Mn residuals have significant and opposite trends
with $T_{\rm eff}$ before correction, causing an artificial anti-correlation
with coefficient $-0.32$ that is almost entirely removed by the $T_{\rm eff}$
correction. The Co-Ni correlation, like Ca-Ni, is minimally affected by
$T_{\rm eff}$ trends. However, the Mn-Co correlation is artificially boosted
because residuals of both elements are anti-correlated with $T_{\rm eff}$, and
correcting the $T_{\rm eff}$ trend lowers the correlation coefficient from
0.38 to 0.19.

In sum, we apply small ($\lesssim 0.05$-dex) detrending corrections to the
ASPCAP abundances that remove weak correlations between residual abundances
and $T_{\rm eff}$.

###  5.2 Simulating the impact of observational errors

If all stars were perfectly described by the 2-process model, the measured
abundances would still depart from the predicted abundances because of
statistical errors induced by observational noise. It is tricky to predict the
distribution and correlation of these noise-induced residuals because the
observational uncertainties span a significant range from star-to-star and
element-to-element and because some of the measured values are used to fit the
2-process amplitudes $A_{\rm cc}$ and $A_{\rm Ia}$. We have therefore created
a simulated data set in which we take each star from the sample, set its true
abundances exactly equal to the 2-process predictions given its measured
$A_{\rm cc}$ and $A_{\rm Ia}$, then add an “observational” error to each
abundance, drawn from a Gaussian distribution with the star’s ASPCAP
uncertainty for that element. If an abundance measurement is flagged in the
APOGEE data, then we flag it in the simulation as well. We can then apply the
same analysis to the simulated data that we apply to our observed sample to
understand the results that would be expected if all stars followed the
2-process model and all measurement errors were described by Gaussian noise
with the reported observational uncertainties.

###  5.3 Distribution of residuals

The black curve in Figure 11 shows the cumulative distribution of $\chi^{2}$
values for the 34,410 sample stars, computed using all of the elements shown
in Figure 9 but omitting flagged element values. The median, 95th, and 99th-
percentiles of this distribution are 30, 134, and 438, respectively. Only 13%
of stars have $\chi^{2}$ values below 14, the number of degrees of freedom for
16 elements and two free parameters, so either the true distribution has
intrinsic element scatter relative to the 2-process model or the observational
abundance errors exceed those predicted for Gaussian noise with the ASPCAP
uncertainties, or both. The red curve shows the $\chi^{2}$ distribution for
the simulated data set described in §5.2, and in this case 45% of stars have
$\chi^{2}<14$.

The green curve shows the $\chi^{2}$ distribution if $A_{\rm Ia}$ and $A_{\rm
cc}$ are inferred from [Mg/H] and [Fe/Mg] alone instead of the six-element
2-process fit. This leads to only a small increase in $\chi^{2}$ values. The
blue curve shows the distribution of $\chi^{2}$ if we compute element
residuals from the observed median sequences instead of the 2-process fits,
interpolating the sequences in $[{\rm Mg}/{\rm H}]$ to avoid any effects of
metallicity variation across our 0.1-dex $[{\rm Mg}/{\rm H}]$ bins. These
$\chi^{2}$ values are substantially higher (e.g., a median value of 78, vs. 30
for the 2-process residuals), a first demonstration that the 2-process model
is removing physical scatter present in the stellar abundance distribution. In
other words, a star’s APOGEE abundances are typically closer to those
predicted by the 2-process model than they are to the median abundances of
stars of the same $[{\rm Mg}/{\rm H}]$ and the same population (low-Ia or
high-Ia), by an amount that is highly statistically significant.

Figure 11: Cumulative distribution of $\chi^{2}$ values for model fits to the
sample of 34,110 stars. Values of $\chi^{2}$ are computed using all 16
elements shown in Figure 9, omitting flagged values. The black solid curve
shows results from applying the full 2-process fit, with points marking the
median, 95th-percentile, and 99th-percentile values of the distribution
(listed in the lower right). The green dotted curve shows results when $A_{\rm
Ia}$ and $A_{\rm cc}$ values are inferred from [Mg/H] and [Fe/Mg] alone. The
blue dashed curve shows results when abundances are predicted from the
observed median trends (blue and red curves in the left panels of Figures
4-7), independent of the 2-process model. The red-dashed curve shows results
for a simulated data sample in which all stars lie on the 2-process model and
arise only from Gaussian noise with the reported abundance errors.

Figure 12: Distribution of [X/H] deviations from model predictions, normalized
by the reported abundance error. Black histograms show results using the full
2-process fit to six abundances. Blue histograms show deviations from the
median sequences. Red curves show a unit Gaussian for reference. The mean
ASPCAP error is listed in each panel. For the median trend deviations, the
$[{\rm Mg}/{\rm H}]$ deviation is zero by definition, giving a zero-width blue
histogram.

Figure 12 examines the deviation distributions element-by-element. For those
elements that have a mean ASPCAP abundance error smaller than 0.015 dex, the
deviations from the 2-process model predictions (black histograms) are
significantly narrower than the deviations from the median sequence
predictions (blue). Some of this improvement arises because many of these
elements are used in the 2-process fit, but even if we use only Mg and Fe to
determine the 2-process parameters the deviation distributions for the other
fit elements are narrower than the distribution of deviations from the median
sequences. Mn and C+N, which are not used in the 2-process fit, both show
narrower deviations from the 2-process predictions than from the median
sequences. For other elements, with larger mean errors, there are only small
differences between the 2-process deviation distribution and the median-
sequence deviation distribution, presumably because the deviations in both
cases are dominated by observational errors. However, the extended tails of
the distributions are still noticeably reduced for Al, Cr, Co, and Ce.

Red curves show a unit Gaussian, which is narrower than the 2-process
deviation distributions for all elements except Mg and Fe. The simulated star
sample drawn from the 2-process model yields deviation distributions (not
shown) very close to these unit Gaussians, as expected. For many elements the
extended tails of the deviation distribution appear more nearly exponential
than Gaussian, and for some elements there is a significant asymmetry between
positive and negative deviations in these extended tails. Extended tails and
asymmetries could arise from either genuine physical deviations or measurement
errors. We investigate this question to a limited degree in §6, though it is
difficult to quantify the relative contribution of physical and observational
outliers without detailed investigation of the abundance determinations for a
large number of stars.

Figure 13: Top: Dispersion of [X/H] values relative to 2-process predictions,
in dex. Filled circles show results for the data. Open squares show results
from the simulation in which stars lie on the 2-process model except for
Gaussian errors. Red crosses show the mean observational error. Elements used
in the 2-process fit are denoted in blue on the horizontal axis. Bottom:
Filled circles, repeated from the top panel, show the total dispersion. Open
circles show the intrinsic dispersion estimated by subtracting in quadrature
the dispersion of the simulated data set (i.e., the squares in the top panel).
Open triangles show an alternative estimate of the intrinsic dispersion
obtained by subtracting half of the 16-84% percentile range ($\pm 1\sigma$ for
a Gaussian distribution), in quadrature, for the simulated data from the same
quantity for the observed data.

In both panels of Figure 13, filled circles show the rms difference between
ASPCAP abundances and 2-process model predictions. In the upper panel, red
crosses show the mean abundance error reported by ASPCAP for that element.
Open squares show the rms deviations for the simulated data set. These
simulated dispersions are generally very similar to the mean abundance errors,
though they are significantly lower for Mg and Fe, which carry high weight in
the 2-process fit. In the lower panel, we estimate the intrinsic dispersion by
subtracting the dispersion of the simulated data from the dispersion of the
observational data. If intrinsic and observational scatter contributed equally
to the variance, then the intrinsic dispersion would be $1/\sqrt{2}=70.7\%$ of
the total dispersion. Half of the elements have an intrinsic/total dispersion
ratio near or below this value (Mg, O, Si, S, Cr, Fe, V, Co), and the other
half have higher ratios that imply intrinsic dispersion dominating over the
observational scatter. However, the observational contribution could be
underestimated, and the intrinsic dispersion overestimated, if the reported
measurement uncertainties are systematically low or if non-Gaussian tails of
the measurement errors inflate the dispersion. As already noted, the residual
distributions for many elements exhibit exponential tails, which could
represent real deviations or non-Gaussian measurement errors. As an
alternative estimate of dispersion, we have taken half of the 16-84%
percentile range ($\pm 1\sigma$ for a Gaussian distribution) for the observed
residuals, then computed the same quantity for the simulated data and
subtracted in quadrature, obtaining the open triangles in Figure 13. These
alternative estimates characterize scatter in the core of the residual
abundance distribution, with less sensitivity to large deviations (whether
physical or observational).

For Mg, O, and Fe we estimate rms intrinsic scatter of only 0.003-0.005 dex,
but given the weight of these elements in the 2-process fit a low scatter is
expected. For other elements the rms intrinsic scatter ranges from $\approx
0.01-0.02$ dex (Si, Ca, C+N, Ni, Mn, Co) to $\approx 0.05-0.08$ dex (Na, K, V,
Ce). The percentile-based intinsic scatter estimate is lower for all elements,
with Mg, O, Si, S, Ca, C+N, Al, Cr, Fe, Ni, Mn, and Co having values $\lesssim
0.02$ dex and the Na, K, V, and Ce scatter reduced to 0.04-0.07 dex. Our
estimates of intrinsic scatter, including the relative values of different
elements, are similar to those inferred by TW21 for scatter in APOGEE
abundances conditioned on [Mg/H] and [Mg/Fe]. We originally performed our
analysis for the APOGEE DR16 data set, and while the relative ranking of
elements was nearly the same, the total scatter and estimated intrinsic
scatter were consistently higher. The lower estimates of intrinsic scatter in
DR17 likely reflect improvements in the abundance analysis that reduce the
number of large measurement errors.

In sum, the 2-process model predicts a star’s APOGEE abundances better than
the median abundances of similar stars, demonstrating that much of the
abundance scatter within the low-Ia and high-Ia populations arises from
scatter in SNIa/CCSN ratios at fixed $[{\rm Mg}/{\rm H}]$. RMS residuals about
2-process predictions range from $\sim 0.01$ dex for the most precisely
measured elements to $\sim 0.1$ dex for Na, V, and Ce. These dispersions
exceed those expected from observational errors alone, implying intrinsic
scatter at the $\sim 0.01-0.05$ dex level, depending on element.

###  5.4 Covariance of residuals

As emphasized by TW21, correlations are a more robust measure of residual
structure in elemental abundance patterns than dispersion, because estimating
the intrinsic dispersion requires accurate knowledge of the observational
error distribution. The correlations also provide richer information about the
sources of residual structure, which could include additional enrichment
processes, stochastic sampling of the IMF, binary mass transfer, or even
effects like variable depletion of refractory elements in proto-planetary
disks or abundances boosted by giant planet engulfment. We compute elements of
the covariance matrix of element pairs Xi, Xj as

$C_{ij}=\langle(\Delta[{\rm X}_{i}/{\rm H}])(\Delta[{\rm X}_{j}/{\rm
H}])\rangle$ (31)

with $\Delta[{\rm X}/{\rm H}]$ defined as the difference between the observed
abundance and the 2-process model prediction (equation 29).

Figure 14: Bivariate distribution of 2-process residuals in [(C+N)/H] vs.
[Ce/H]. Small dots show a random 10% subset of the full sample. Crosses show
all sample stars that have $|\Delta[{\rm(C+N)}/{\rm H}]|>0.1$. The inset
provides an expanded view of the core of the distribution, from -0.2 to 0.2 in
$\Delta[{\rm Ce}/{\rm H}]$ and -0.1 to 0.1 in $\Delta[{\rm(C+N)}/{\rm H}]$.
Although residuals in the core of the distribution are anti-correlated, there
is a population of rare stars with large positive deviations in (C+N) and Ce.
Two examples of such stars are shown in Figure 20 below and discussed in §6.

Pairwise scatterplots of element residuals generally resemble the examples
shown in Figure 10. In particular, for element pairs with significant
correlation or anti-correlation, the scatterplot shows a consistent slope
between the core of the distribution and the stars with large residuals. The
one exception is (C+N) vs. Ce: the core of the distribution shows a clear
anti-correlation of the residual abundances, but there is a population of rare
outlier stars with large positive residuals in both (C+N) and Ce, as
illustrated in Figure 14. We discuss this population further in §6. To avoid
covariance estimates being driven by rare outliers, we eliminate stars with
element deviations $>10\sigma_{\rm obs}$ before computing covariances
involving that element. This censoring reverses the sign of the (C+N)-Ce
covariance, which would be positive instead of negative if we retained the
extreme outliers. It has a moderate impact on some other matrix elements
involving Ce or (C+N) and minimal effect on other element pairs.

Figure 15a shows the residual abundance covariance matrix of our APOGEE
sample. Diagonal elements are the squares of the rms deviations shown by the
filled circles in Figure 13. This covariance matrix would look qualitatively
similar if we did not remove the temperature trends discussed in §5.1, but the
covariances involving pairs of elements with the strongest trends (largest
values of $|\alpha_{T}|$ in Table 1) would be noticeably affected. Figure 15b
shows the residual covariance if we determine $A_{\rm cc}$ and $A_{\rm Ia}$
values from [Mg/H] and [Mg/Fe] alone, instead of fitting six elements. Here
the correlations are stronger and almost all positive, except those involving
Mg and Fe, which have vanishing residuals by definition. These artificial
correlations span many elements because random Fe and Mg measurement errors
lead to errors in $A_{\rm Ia}$ and $A_{\rm cc}$ and thus to correlated
deviations from the 2-process predictions, the effect that TW21 describe as
“measurement aberration.” Fitting six elements mitigates this effect, though
it does not entirely remove it. Figure 15c shows the covariance of the
simulated data, which has no intrinsic residual correlations by construction.
However, because $A_{\rm cc}$ and $A_{\rm Ia}$ must be fit to abundances with
random statistical errors, measurement aberration still produces off-diagonal
covariances.

The most important conclusion from comparing the simulation covariance matrix
to the data covariance matrix is that measurement aberration is much too small
to explain the observed covariances. Our conclusion agrees with that of TW21,
who investigated the correlation of residual abundances in the conditional
probability distribution $p([{\rm X}/{\rm H}])$ at fixed $[{\rm Fe}/{\rm H}]$
and $[{\rm Mg}/{\rm Fe}]$, using an APOGEE sample nearly identical to ours.
TW21 also find that the measured residual correlations are much larger than
those arising from measurement aberration. Artificial correlations could also
arise from the abundance determination itself, e.g., from random errors in
$T_{\rm eff}$ leading to correlated deviations in the abundances of multiple
elements. TW21 examine this issue by approximately modeling the ASPCAP
measurement procedure and conclude that artificial correlations from the
measurement method are also much smaller than the observed correlations (see
their figure 9).

To estimate the covariance matrix of intrinsic deviations from the 2-process
model, we subtract the simulated covariance matrix (c) from the data
covariance matrix (a). This subtraction produces little change because the
$C_{ij}$ elements for the simulation are generally much smaller than those of
the data. The result is shown in Figure 15(d). The diagonal elements of this
covariance matrix correspond to the open circles in Figure 13. As discussed in
§5.3, the estimates of the intrinsic variance are sensitive to knowledge of
the observational error distribution, so their magnitudes are uncertain.
However, the prominent off-diagonal structures in §5.3 likely represent
genuine physical correlations among abundance residuals. The most obvious of
these structures are the block of correlations among the iron-peak elements
Ni, V, Mn, and Co, and another block of correlations among the elements Ca,
Na, Al, K, and Cr. Ce is also positively correlated with all members of this
latter group except Al. There is also a noticeable positive correlation of Na
with V, Mn, and Co.

Figure 16a converts this intrinsic covariance matrix to the corresponding
correlation matrix,

$c_{ij}=C_{ij}/(C_{ii}C_{jj})^{1/2},$ (32)

thus normalizing all diagonal elements to unity. This transformation depends
on our estimate of the intrinsic variance, and if we used the percentile-based
estimates (triangles in Figure 13) then the off-diagonal correlations would
all be larger, though the structure would be similar. Relative to the
covariance matrix, this conversion highlights the substantial correlations
among elements that have small observational and total scatter. The intrinsic
correlation matrix is similar in its main features to that found by TW21 (see
their figure 10), including the previously noted correlations among the iron-
peak element residuals, positive correlations among O, Si, and S, and positive
correlations among Ca, Na, and Al (extending here to include K and Cr). Since
conditioning on Mg and Fe has much in common with fitting the 2-process model,
we would expect residual correlations to be similar, but details of our
analysis are entirely different and independent, so we consider this agreement
an encouraging indication that the correlations are qualitatively robust to
these details. Many of these correlations are quite strong, e.g. 0.15 or
larger, and would be stronger still if we used the percentile-based intrinsic
variance estimate.

Figure 16b shows the correlations for the simulated data set. This shows that
measurement aberration can induce substantial spurious correlations even with
six-element fitting. The primary effects are a positive correlation between Mg
and Fe and anti-correlations between these elements and the other fit elements
(O, Si, Ca, Ni), and to a lesser degree among those elements themselves. In
principle our subtraction of the simulated covariance matrix from the data
covariance matrix should have removed these artificial correlations from our
estimate of the intrinsic correlation matrix. However, given the uncertainties
in this procedure (primarily our imperfect knowledge of the observational
error distributions), the inferred correlations involving the fit elements
should be treated with some caution. If we used only Mg and Fe to infer
$A_{\rm cc}$ and $A_{\rm Ia}$, then the off-diagonal correlations for the
simulated data set would be comparable in typical magnitude but nearly all
positive.

In sum, the measured covariance of residual abundances is larger than expected
from observational errors alone, demonstrating the existence of intrinsic
physical correlations in the residual abundance patterns. The most prominent
of these are correlations among Ca, Na, Al, K, Cr, and Ce and correlations
among Ni, V, Mn, and Co.

Figure 15: Covariance matrix of deviations between measured [X/H] values and
2-process model predictions. In each panel, filled and open circles denote
positive and negative values, respectively, with area proportional to the
magnitude of the matrix element and consistent scaling used across all panels.
(a) Residual covariance for the APOGEE sample, with $A_{\rm cc}$ and $A_{\rm
Ia}$ fit using the six elements denoted in blue on the axes. The diagonal
elements are the squares of the rms deviations shown by filled circles in
Figure 13. For visual scaling, note that the diagonal elements (Na,Na) and
(Ce,Ce) have values of about $(0.09)^{2}$, (S,S) has a value of about
$(0.05)^{2}$, and (O,O) has a value of about $(0.01)^{2}$. (b) Residual
covariance when $A_{\rm cc}$ and $A_{\rm Ia}$ are fit using Mg and Fe only,
which increases the artificial correlations induced by measurement aberration.
(c) Residual covariance for the simulated data set in which all stars lie on
the 2-process model prediction and have Gaussian observational errors at the
level of the reported ASPCAP uncertainties. Off-diagonal elements arise from
measurement aberration, but they are small compared to the observed
covariances. (d) Intrinsic covariance estimated by subtracting (c) from (a).

Figure 16: (a) Correlation matrix (eq. 32) corresponding to the intrinsic
covariance matrix in Fig. 15d. For visual scaling, note that diagonal elements
have a magnitude of 1.0 by definition, the O-S coefficient is 0.37, the Ni-Mn
coefficient is 0.26, and the O-V coefficient is 0.10. (b) Correlation matrix
corresponding to the simulated data covariance matrix (Fig. 15c). Off-diagonal
coefficients in this matrix are caused by measurement aberration.

###  5.5 Correlations with age and kinematics

Figure 17 plots the amplitude ratio $A_{\rm Ia}/A_{\rm cc}$ inferred from our
2-process fits against the stellar age inferred from the APOGEE spectra by
AstroNN (Leung & Bovy, 2019b; Mackereth et al., 2019), a Bayesian neural
network model trained on a subset of APOGEE stars with asteroseismic ages. We
use the DR17 AstroNN Value Added Catalog, which will be made available with
the DR17 public release. Specifically we use the age_lowess_correct ages,
which correct the raw neural network ages for biases at low and high age using
a lowess smoothing regression (see Mackereth et al. 2017). We adopt the same
Galactic zones shown previously in Figure 8 and again use the SN100 sample to
improve coverage of the inner Galaxy. In the solar neighborhood ($R=7-9\,{\rm
kpc}$, $|Z|<0.5\,{\rm kpc}$) we compute the median age in narrow bins of
$A_{\rm Ia}/A_{\rm cc}$, and we repeat this median sequence in other panels
for visual reference. We use this order of binning because $A_{\rm Ia}/A_{\rm
cc}$ is measured much more precisely than age, so the median $A_{\rm
Ia}/A_{\rm cc}$ in bins of age cannot be determined as reliably.

Figure 17:  Correlation of the 2-process amplitude ratio $A_{\rm Ia}/A_{\rm
cc}$ with stellar age estimated from the APOGEE spectra by AstroNN, in zones
of $R$ (columns) and $|Z|$ (rows) as labeled. Red and blue points show stars
in the low-Ia and high-Ia population, respectively. In the $R=7-9\,{\rm kpc}$,
$|Z|<0.5\,{\rm kpc}$ panel, black circles show the median age in bins of width
0.1 in $A_{\rm Ia}/A_{\rm cc}$. Black lines repeat this median sequence and
are the same in all panels. To improve coverage of the inner Galaxy, this plot
uses the SN100 sample.

Not surprisingly, the low-Ia stars are systematically older than the high-Ia
stars. There is, furthermore, a continuous trend of age with $A_{\rm
Ia}/A_{\rm cc}$ within both the low-Ia and high-Ia populations, and although
these two populations are separated in $[\alpha/{\rm Fe}]$ the age trend is
continuous across them. The trend is similar in different Galactic zones,
though in the high-Ia population at $|Z|<1\,{\rm kpc}$ the stars are
systematically older in the inner Galaxy and younger in the outer Galaxy at
fixed $A_{\rm Ia}/A_{\rm cc}$, by roughly 1-2 Gyr. The correlation of age with
$[\alpha/{\rm Fe}]$ within the high-Ia population is visible in previous
studies (Martig et al., 2016; Miglio et al., 2021), though it is perhaps more
obvious in this representation.

The primary spectroscopic diagnostic of age in APOGEE spectra comes from
features that trace the C/N ratio (Masseron & Gilmore, 2015; Martig et al.,
2016), because the surface C and N abundances are changed by dredge-up on the
giant branch in a way that depends on stellar mass, and for red giants the age
(slightly longer than the main sequence lifetime) is tightly correlated with
mass. Although AstroNN works directly on spectra, it responds primarily to C
and N features and returns large age uncertainties when these features are
masked. It is therefore unlikely that it is “learning” a correlation between
age and other abundance ratios from its asteroseismic training set and then
applying that to other stars. However, the birth [C/N] ratio likely depends on
stellar metallicity and [$\alpha$/Fe] (Vincenzo et al., 2021b), and these
birth abundance trends could induce systematic age errors that correlate with
abundances.

Figure 18:  Correlation of the abundance residual $\Delta[{\rm X}/{\rm H}]$
(in dex) with stellar age for eight selected elements as labeled. In the first
and third columns, residuals are computed with respect to the median
sequences, while in the second and fourth columns residuals are computed with
respect to the 2-process model predictions. Red and blue points show stars in
the low-Ia and high-Ia population, respectively. To reduce crowding, we plot
only 25% of the stars.

Figure 18 plots residual abundances vs. AstroNN age. We return to using the
high SNR-threshold sample to reduce observational contributions to the
residual scatter, and we have selected a subset of elements that illustrate a
range of behaviors. In the first and third columns, $\Delta[{\rm X}/{\rm H}]$
is computed relative to the median sequence of the low-Ia or high-Ia
population. We see a clear correlation between $\Delta[{\rm Fe}/{\rm H}]$ and
age in the high-Ia population and a weaker correlation in the low-Ia
population. This correlation indicates that even though the scatter about the
median sequence at fixed $[{\rm Mg}/{\rm H}]$ is small (about 0.04 dex in
$[{\rm Fe}/{\rm Mg}]$, see Vincenzo et al. 2021a), it is correlated with age
in the expected sense, with younger stars showing greater Fe enrichment. Mn,
which we infer to have the largest SNIa contribution of all APOGEE elements
(Figure 7), shows a similarly strong correlation, and Na and Ce show similar
correlations in the high-Ia population despite the larger scatter from
observational errors. O residuals show no correlation with age, but Si and Ca
residuals do, consistent with our inference of a significant SNIa contribution
to these two $\alpha$-elements (Figure 4). Residual correlations for C+N are
weak, though there is a population of older high-Ia stars that have below-
median C+N.

In the second and fourth columns, $\Delta[{\rm X}/{\rm H}]$ is computed
relative to the predictions of the 2-process model. The residual scatter is
smaller for the well measured elements, as seen previously in Figures 12 and
13. The age trends seen previously for Fe, Mn, and Si are removed, and the
trend for Ca is reduced though not entirely eliminated. We regard this
flattening of age trends as evidence for the physical validity of the
2-process model, which is constructed with no knowledge of the stellar ages.
However, residuals from the 2-process model could correlate with age if they
are caused by other enrichment processes that have a different time dependence
than SNIa. We see a slight correlation of C+N residual with age in the high-Ia
population, though there is some risk of a systematic effect because of the
central role of these elements in the age determinations.

More strikingly, we see a clear trend of Ce residuals and, to a lesser extent,
Na residuals with age in the high-Ia population. Stars with $\Delta[{\rm
Ce}/{\rm Mg}]>0.1$ have typical AstroNN ages of 2-3 Gyr, while stars with
$\Delta[{\rm Ce}/{\rm Mg}]\approx 0$ have typical ages of 4-6 Gyr. Within the
high-Ia population, the trend continues to negative $\Delta[{\rm Ce}/{\rm
Mg}]$ and older ages, though there is no clear trend within the low-Ia
population. These results are qualitatively consistent with the findings of
Sales-Silva et al. (2021) that APOGEE’s [Ce/Fe] and [Ce/$\alpha$] ratios for
open clusters increase with decreasing cluster age over at least the past
$\sim 4\,{\rm Gyr}$. Previous studies (e.g., da Silva et al. 2012; Nissen
2015; Feltzing et al. 2017) have also shown that abundances of s-process
elements are well correlated with age in thin-disk stars. We have already
noted (§4) that the separation of median sequences for [Na/Mg] is larger than
expected based on the yield models used by Andrews et al. (2017) and Rybizki
et al. (2017). The qualitative similarity of age trends for Na and Ce
residuals suggests that a common source, presumably AGB stars, makes important
contributions to both elements.

In future work we will directly examine trends with asteroseismic ages,
instead of the spectroscopic estimates trained on them. However, the current
sample is not ideally suited for this purpose because our ${\rm log}\,g$ and
$T_{\rm eff}$ cuts eliminate many of the APOGEE stars for which asteroseismic
parameters from Kepler are available (Pinsonneault et al., 2018).

Figure 19:  Correlation of the $\Delta[{\rm Ca}/{\rm H}]$ and $\Delta[{\rm
Mn}/{\rm H}]$ residual abundances with orbital eccentricity (left) and maximum
midplane distance $|Z_{\rm max}|$ (right). As in Fig. 18, residuals in the the
first and third columns are computed with respect to the median sequences,
while those in the second and fourth columns are computed with respect to the
2-process model predictions. Red and blue points show stars in the low-Ia and
high-Ia population, respectively. To reduce crowding, only 25% of the stars
are plotted.

Figure 19 plots residual abundances for Ca and Mn against orbital parameters
derived by AstroNN from APOGEE and Gaia data, computed using the fast method
of Mackereth et al. (2018) implemented in galpy (Bovy, 2015), assuming the
MWPotential2014 gravitational potential from Bovy (2015). The two left columns
show residuals vs. eccentricity. Not surprisingly, low-Ia (“thick disk”) stars
are more likely to have high orbital eccentricity than high-Ia stars. Within
each population, there is a slight tendency for the highest eccentricity stars
to have lower [Ca/H] and [Mn/H] relative to the median sequence, but this
trend is weak, and it vanishes when examining residuals from the 2-process
fits instead of from the median sequence. The right two columns show residuals
vs. $|Z_{\rm max}|$, the maximum distance a star’s orbit reaches from the
midplane. Low-Ia “thick disk” stars are more likely to have high $|Z_{\rm
max}|$, though a number of stars in the high-Ia population have inferred
$|Z_{\rm max}|>2\,{\rm kpc}$. Trends (or lack thereof) are similar to those
seen for eccentricity but somewhat more pronounced. In particular, high-Ia
stars with $|Z_{\rm max}|<0.5\,{\rm kpc}$ tend to have higher [Ca/H] and
[Mn/H] relative to the median sequences, an effect that is subtle (0.02-0.05
dex) but discernible with a large sample. The coldest “thin disk” stars tend
to have higher fractions of elements with SNIa contributions, as expected from
the trend of $A_{\rm Ia}/A_{\rm cc}$ with age (Fig. 17) and the age-velocity
relation.

As with eccentricity, shifting from median residuals to 2-process residuals
removes these weak correlations. We have examined other elements and find no
convincing correlations of individual 2-process residuals with kinematics or
with Galactic position. As discussed in §8 below, grouping correlated elements
sharpens sensitivity and reveals weak correlations that are difficult to
discern in individual element plots like Figures 18 and 19. In §7 we give
examples of stellar populations whose median residual abundances clearly
depart from those of the main disk sample.

In sum, deviations from median sequences show weak but expected correlations
with age and kinematics, with the stars that have higher SNIa/CCSN ratios
within each population also having younger ages and colder kinematics.
Changing to 2-process residuals removes most of these correlations, but within
the high-Ia population the Ce and Na residuals show significant age
correlations, with younger stars exhibiting higher abundances of both elements
relative to stars with similar $A_{\rm cc}$ and $A_{\rm Ia}$.

##  6 High-$\chi^{2}$ Stars

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

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

Figure 20: Examples of stars that are poorly fit by the 2-process model, in a
format similar to Fig. 9. In each panel, filled circles with error bars show
the APOGEE abundance measurements and open circles show the abundances
predicted by the best 2-process model fit. Each panel lists the star’s best-
fit $A_{\rm cc}$ and $A_{\rm Ia}$, the $\chi^{2}$ of this fit, and the
(usually lower) modified $\chi^{2}$ described in the text.

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

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

2M19531095+4635518 is a metal-rich, high-Ia star ($[{\rm Mg}/{\rm H}]=0.32$,
$[{\rm Fe}/{\rm Mg}]=-0.01$) that is an extreme outlier in both its standard
$\chi^{2}=5575$ and its modified $\chi^{2}=225$. This star has an extremely
high C+N residual, with ASPCAP values of [C/Fe]=0.32 and [N/Fe]=1.04. The
[O/Fe] from ASPCAP is 0.045, implying a C/O number ratio of 1.01 that puts
this star just over the boundary into the carbon star regime, where the
stellar spectral features become very different from those of typical, ${\rm
C/O}<1$ stars. The large deviations in Na, K, V, and Mn may be physical, but
they could also be affected by the very strong blends that occur in the carbon
star regime, such that any inaccuracies in the ASPCAP synthesis could lead to
poor fits or incorrect abundance values.

2M16464736-4430101 displays a pattern that we have found in multiple examples
of high-$\chi^{2}$ stars of solar or super-solar metallicity. Consistently in
these stars, the Na abundance is far above the 2-process prediction (by 0.37
dex in this case), the C+N abundance is moderately elevated ($\sim 0.1-0.2$
dex), the Al, K, and Cr abundances are moderately depressed, and the V, Mn,
and Co abundances are slightly enhanced. Although this could be a distinctive
class of chemically peculiar stars, we suspect that this pattern arises from
an artifact of ASPCAP abundance determinations, in part because we find
anomalous structures in [X/Fe]-[Fe/H] diagrams for Na, Al, and Mn that do not
appear physical. We do not understand the origin of this artifact, though the
elevated C+N hints that it could be related to inaccurate interpolation across
the model grid near the carbon-star regime, where spectral syntheses change
rapidly with stellar parameters.

2M19143234+5954000 has near-solar $[{\rm Mg}/{\rm H}]=-0.03$ and an $[{\rm
Fe}/{\rm Mg}]=-0.26$ that places it below the median of the low-Ia population
at this metallicity (Figure 6). Its high $\chi^{2}$ value is driven by
extremely low K and low V, and to a lesser extent by elevated Ce. Low K values
and to some degree low V values are fairly common among high-$\chi^{2}$ stars,
and the residual distributions for both elements (especially K) are asymmetric
towards negative values (Figure 12). K and V both have weak, sometimes blended
features in APOGEE spectra, so the abundances are more subject to statistical
and systematic errors. Furthermore, follow-up of the low K population shows
that many of them (including this star) have a heliocentric velocity of
$-70\,{\rm km}\,{\rm s}^{-1}$ that happens to place two of APOGEE’s K lines on
top of stronger telluric features. It therefore seems likely that the low K
abundances are a consequence of imperfect telluric subtraction.

2M07384226+2131021 is a low metallicity,444We refer to stars with $[{\rm
Mg}/{\rm H}]<-0.5$ as low metallicity because they lie at the metal poor end
of our sample and of disk populations in general, though of course halo
populations reach to much lower metallicity. high-Ia star ($[{\rm Mg}/{\rm
H}]=-0.55$, $[{\rm Fe}/{\rm Mg}]=-0.04$) with highly elevated C+N (0.41 dex)
and extremely elevated Ce (1.59 dex), as well as moderate enhancements ($\sim
0.2$ dex) in Na and Cr. This star is a member of the old open cluster NGC
2420, and it was identified by Smith & Suntzeff (1987) as an extreme example
of a “barium star” based on its strong excess abundances of s-process
elements. The extreme Ce enhancement is in line with these previous findings.
ASPCAP’s individual C and N abundances are [C/Fe]=0.27 and [N/Fe]=0.73. We
find numerous examples of stars with large enhancements of both C+N and Ce, as
discussed further below. These enhancements may arise from binary mass
transfer from an AGB companion, or from internal AGB enrichment in star
clusters, or both. These s-process enhanced stars are frequently referred to
as barium stars at high metallicity and CH or CEMP-s stars at low metallicity
(e.g., McClure 1985; Lucatello et al. 2005).

2M13264723-4734121 is a low metallicity, low-Ia star ($[{\rm Mg}/{\rm
H}]=-0.65$, $[{\rm Fe}/{\rm Mg}]=-0.26$) with 0.7-1.1 dex enhancements in C+N,
Na, Al, and Ce. The ASPCAP values of [C/Fe] and [N/Fe] are -0.17 and 1.78,
respectively, so the elevated C+N is driven entirely by the extreme N
enhancement. This star is a member of $\omega\,$Cen, a globular cluster that
is often hypothesized to be the stripped core of a dwarf galaxy because of its
large internal $[{\rm Fe}/{\rm H}]$ spread (e.g., Smith et al. 2000). The well
established and distinctive pattern of enhanced N, Na, Al, and s-process
elements is thought to be a signature of self-enrichment by the cluster’s
evolved AGB stars (Smith et al., 2000; Johnson & Pilachowski, 2010; Mészáros
et al., 2020, 2021). A significant number of the most extreme $\chi^{2}$ stars
in our sample are members of $\omega\,$Cen, and we discuss the abundance
pattern of these stars further in §7.

2M18120031-1350169 is a low metallicity star ($[{\rm Mg}/{\rm H}]=-0.55$) with
unusual abundances for many elements. Its $[{\rm Fe}/{\rm Mg}]=-0.50$ lies
well below the low-Ia plateau at $-0.3$, so the 2-process fit assigns it a
negative value of $A_{\rm Ia}$. However, even if we set $A_{\rm Ia}=0$ its
abundances would depart strongly from the 2-process prediction, especially the
high Al, high Ce, low Na, and unusual (0.31-dex) enhancement of Si. The C+N of
of this star is only moderately (0.25 dex) above the 2-process prediction, but
this enhancement is dominated by an unusual N abundance, with ASPCAP values of
[C/Fe]=0.01 and [N/Fe]=0.72. Schiavon et al. (2017) and Fernández-Trincado et
al. (2020c) have previously highlighted 2M18120031-1350169 as a N-rich star
that is a likely escapee from a globular cluster, part of an extensive
population of such stars identified in APOGEE (Schiavon et al., 2017;
Fernández-Trincado et al., 2016, 2017; Fernández-Trincado et al., 2019a,
2020b, 2020c). The pattern of high N, Al, and Ce resembles that found for
$\omega\,$Cen and could reflect a similar self-enrichment process. However,
this star does not show Na enhancement, and the Si enhancement seen here does
not appear in our $\omega\,$Cen stars (see Figure 22 below), though enhanced
Si is found in a population of field stars in the inner halo (dubbed
“Jurassic”; Fernández-Trincado et al. 2019b, 2020a), which may arise from
tidally disrupted globular clusters. Intriguingly, Masseron et al. (2020a)
identified 2M18120031-1350169 as one of 15 APOGEE stars with extreme P
enhancement, finding [P/Fe]=1.65 using a custom analysis of the APOGEE
spectrum (rather than the ASPCAP abundance). The unusually high [X/Fe] values
for Mg, O, Si, and Al are also found in the other members of this P-rich
population (see Figure 9 of Masseron et al. 2020a), and the high [Ce/Fe]
accords with the enhanced s-process abundances found in follow-up optical
spectroscopy by Masseron et al. (2020b). From the overall abundance patterns,
Masseron et al. (2020a, b) argue that the chemical peculiarities of these
stars do not originate in globular clusters or binary mass transfer, and they
are a challenge to explain with existing stellar nucleosynthesis models. The
source of 2M18120031-1350169’s unusual enrichment is unclear, but it is
encouraging that a simple $\chi^{2}$ analysis readily turns up some of the
most interesting stars found in independent studies.

Instead of selecting stars based on $\chi^{2}$ values, one can look for
specific abundance anomaly patterns. For example, motivated by examples like
2M07384226+2131021, we have searched for stars that have unusual enhancements
of both C+N and Ce. With high thresholds of 0.2 dex in C+N and 0.8 dex in Ce,
we find 24 such stars in our disk sample, six of which are members of
$\omega\,$Cen. Further investigation of several of these cases shows evidence
of velocity variations among the multiple APOGEE visits, supporting the idea
that some of these anomalous patterns arise in binary systems. If we lower the
thresholds to 0.15 dex and 0.5 dex, the number of high-(C+N)/high-Ce stars
rises to 87, and to 127 if we select from the larger SN100 sample. We have not
yet carried out a systematic census to assess the frequency of likely binaries
or of cluster members other than $\omega\,$Cen.

Stars with anomalous abundances of single elements may also be interesting,
but these require careful individual vetting. For example, we found two stars
with unusually high Ca abundances that arose because a particular combination
of radial velocity and APOGEE fiber placed one of the Ca lines on previously