# The Similarity of Abundance Ratio Trends and Nucleosynthetic Patterns in the
Milky Way Disk and Bulge

Emily Griffith The Department of Astronomy and Center of Cosmology and
AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA
David H. Weinberg  The Department of Astronomy and Center of Cosmology and
AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA
Jennifer A. Johnson  The Department of Astronomy and Center of Cosmology and
AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA
Rachael Beaton  The Carnegie Observatories, 813 Santa Barbara Street,
Pasadena, CA 91101, USA  D. A. García-Hernández  Instituto de Astrofísica de
Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Universidad de La Laguna
(ULL), Departamento de Astrofísica, E-38206 La Laguna, Tenerife, Spain  Sten
Hasselquist  NSF Astronomy and Astrophysics Postdoctoral Fellow  Department of
Physics & Astronomy, University of Utah, Salt Lake City, UT, 84112, USA  Jon
Holtzman  Department of Astronomy, New Mexico State University, Las Cruces, NM
88003, USA  James W. Johnson  The Department of Astronomy and Center of
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  Lund Observatory, Department of Astronomy
and Theoretical Physics, Lund University, Box 43, SE-22100 Lund, Sweden
Richard R. Lane  Instituto de Astronomía y Ciencias Planetarias de Atacama,
Universidad de Atacama, Copayapu 485, Copiapó, Chile  David M. Nataf  Center
for Astrophysical Sciences and Department of Physics and Astronomy, The Johns
Hopkins University, Baltimore, MD 21218  Alexandre Roman-Lopes Departamento de
Astronomía, Universidad de La Serena - Av. Juan Cisternas, 1200 North, La
Serena, Chile

###### Abstract

We compare abundance ratio trends in a sample of $\sim 11,000$ Milky Way bulge
stars ($R_{\rm GC}<3\rm\;kpc$) from the Apache Point Observatory Galactic
Evolution Experiment (APOGEE) to those of APOGEE stars in the Galactic disk
($5\rm\;kpc<R_{\rm GC}<11\rm\;kpc$). We divide each sample into low-Ia
(high-$[{\rm Mg/Fe}]$) and high-Ia (low-$[{\rm Mg/Fe}]$) populations, and in
each population we examine the median trends of $[{\rm X}/{\rm Mg}]$ vs.
$[{\rm Mg}/{\rm H}]$ for elements X = Fe, O, Na, Al, Si, P, S, K, Ca, V, Cr,
Mn, Co, Ni, Cu, and Ce. To remove small systematic trends of APOGEE abundances
with stellar $\log(g)$, we resample the disk stars to match the $\log(g)$
distributions of the bulge data. After doing so, we find nearly identical
median trends for low-Ia disk and bulge stars for all elements. High-Ia trends
are similar for most elements, with noticeable (0.05-0.1 dex) differences for
Mn, Na, and Co. The close agreement of abundance trends (with typical
differences $\lesssim 0.03$ dex) implies that similar nucleosynthetic
processes enriched bulge and disk stars despite the different star formation
histories and physical conditions of these regions. For example, we infer that
differences in the high mass slope of the stellar initial mass function (IMF)
between disk and bulge must have been $\lesssim 0.30$. This agreement, and the
generally small scatter about the median sequences, means that one can predict
all of a bulge star’s APOGEE abundances with good accuracy knowing only its
measured $[{\rm Mg/Fe}]$ and $[{\rm Mg}/{\rm H}]$ and the observed trends of
disk stars.

nuclear reactions, nucleosynthesis, abundances – Galaxy: abundances – Galaxy:
bulge – Galaxy: formation – Stars: abundances

††journal: ApJ††software: VICE (Johnson & Weinberg, 2019), Astropy (Astropy
Collaboration et al., 2013, 2018)

##  1 Introduction

To paint a complete picture of our Galactic enrichment history, we need to
study the chemical fingerprints of stars from the outer edges to the inner
depths of the Milky Way. Large scale Galactic surveys like GALAH111GALactic
Archaeology with HERMES (De Silva et al., 2015; Martell et al., 2017), Gaia-
ESO (Gilmore et al., 2012), and APOGEE222APOGEE = Apache Point Observatory
Galactic Evolution Experiment, currently a part of the Sloan Digital Sky
Survey IV (SDSS-IV) (Blanton et al., 2017; Majewski et al., 2017) give us the
power to study stellar populations throughout vast regions of the Milky Way.
Each provides a glimpse of the past and unveils the chemical makeup of the
interstellar medium (ISM) in which the stars were born. Using APOGEE data,
Nidever et al. (2015) and Hayden et al. (2015) showed that stellar populations
follow the same trends in [$\alpha$/Fe] vs. $[{\rm Fe/H}]$ throughout the
Milky Way disk, though the distribution of stars along these tracks depends
strongly on galactocentric radius $R_{\rm GC}$ and midplane distance $|Z|$.
Extending these results, Weinberg et al. (2019, hereafter W19) found that the
median trends of $[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ for all APOGEE
elements are independent of Galactic location, suggesting universality in the
nucleosynthetic processes that determine these abundance ratios. In this work,
we aim to extend their conclusions to the Galactic bulge by leveraging the
extensive bulge coverage of APOGEE DR16 to study its stellar chemical
abundances.

In addition to these large scale programs, many groups have observed smaller
($\lesssim 100$ stars) samples of the inner Galaxy and debated their chemical
similarity to the disk. Works such as McWilliam & Rich (1994) (12 giants),
Cunha & Smith (2006) (7 giants), Fulbright et al. (2007) (27 giants), and
Lecureur et al. (2007) (53 giants) found enhancements in the bulge
$\alpha$-element abundances relative to disk samples333$\alpha$ elements such
as O, Mg, and Si are produced mainly by core collapse supernovae while iron
peak elements in stars with solar [$\alpha$/Fe]=0 have roughly equal
contributions from core collapse and Type Ia supernovae.. However, subsequent
work by Meléndez et al. (2008) (20 giants) found bulge [O/Fe] values in line
with their thick-disk sample. More recently the bulge/disk chemical
similarities/differences have been examined by studies of both giant (e.g.
Johnson et al., 2014; Jönsson et al., 2017; Lomaeva et al., 2019; Forsberg et
al., 2019; Duong et al., 2019b, a) and dwarf (e.g. Bensby et al., 2013, 2017)
stars. These works span $\alpha$-elements to neutron capture elements and
include all those observed by APOGEE. Within these works, small differences
between the bulge and thick disk [X/Fe] vs. [Fe/H] median abundance trends or
abundance distributions are found (see above citations). We note that the
collective studies do not come to a consensus on bulge/thick disk abundance
differences.

Large scale surveys such as APOGEE can provide more substantial bulge coverage
than these smaller studies. While the number of APOGEE bulge observations
drastically increased with the inclusion of DR16 data (Ahumada et al., 2019),
the bulge has been studied in prior data releases as well. Zasowski et al.
(2019) present [X/Fe] vs. [Fe/H] abundance distributions and median trends of
4000 bulge stars ($R_{\rm GC}<4\rm\;kpc$) using DR14. They compare bulge
median trends with a solar radius population, finding good agreement with
their high-$\alpha$ population but not with their low-$\alpha$ stars,
especially at low $[{\rm Fe/H}]$. APOGEE observations of 424 bulge stars
through Baade’s window agree with prior bulge works, suggesting that APOGEE
DR13 data do not suffer systematic biases in the bulge (Schultheis et al.,
2017).

All of the prior studies cited here present bulge abundances in [X/Fe] vs.
[Fe/H] space and generally treat the bulge as a single population, comparable
to the thick disk. It is well known that stellar populations at the solar
annulus show two distinct trends in [$\alpha$/Fe] vs. $[{\rm Fe/H}]$ space
(Fuhrmann, 1998; Prochaska et al., 2000; Bensby et al., 2003). These are
commonly referred to as the high-$\alpha$ and low-$\alpha$ populations, though
in this paper we will refer to them as low-Ia and high-Ia, respectively, since
the pre-physical differences between them arise from the Type-Ia supernova
contribution to their iron abundances. The bimodality of [$\alpha$/Fe] is less
evident in the bulge, so most studies have treated the bulge as a single
evolutionary sequence. However a recent analysis of APOGEE DR14 bulge
($R<3.5\rm\;kpc$) stars by Rojas-Arriagada et al. (2019) found the [Mg/Fe]
distribution to be bimodal for stars near solar $[{\rm Fe/H}]$, with a
distinct low-[Mg/Fe] (high-Ia) sequence that had been previously categorized
as a continuation of the low-Ia population. Other recent works with APOGEE
DR16 data confirm that the bulge exhibits a disk-like double sequence (e.g.
Bovy et al., 2019; Queiroz et al., 2020; Lian et al., 2020).

Stellar populations that span high to low-$[{\rm Mg/Fe}]$ can be decomposed
into high-Ia and low-Ia components, as done in W19. Unlike prior bulge
studies, we can now leverage the large bulge population observed by APOGEE
DR16 to separately study the high-Ia and low-Ia populations of the inner
Galaxy. When viewed in [X/Mg] vs. [Mg/H] space, these two populations inform
us on the relative contribution of core collapse supernovae (CCSN) and Type-Ia
supernovae (SNIa). Elements dominantly produced by prompt CCSN enrichment
(e.g. O, Mg, Si, Ca) will display high-Ia and low-Ia sequences with little
separation, while elements with an increased SNIa contribution (e.g. Fe, Ni,
Mn) will show a larger sequence separation. The two-process decomposition
model developed by W19 allows us to quantify the relative prompt and delayed
contribution to each element. By studying the abundances in Mg space rather
than Fe space, we can conduct a clearer comparison of the nucleosynthetic
contributions to the bulge and disk than has been done before, since Mg comes
almost entirely from CCSN. .

Using APOGEE DR14 disk stars, W19 showed that the median trends of [X/Mg] vs.
[Mg/H] for the low-Ia and high-Ia populations remained constant throughout the
disk for O, Na, Mg, Al, Si, P, S, K, Ca, V, Cr, Mn, Fe, Co, and Ni. Griffith
et al. (2019, hereafter GJW) expanded this work to many new elements in GALAH
and drew similar conclusions. In this paper we use APOGEE DR16 data to carry
out a similar analysis of the bulge. Previous bulge studies have focused
mostly on the distribution of stars in [Fe/H], [$\alpha$/Fe], or other element
ratios. These distributions are sensitive to many aspects of chemical
evolution such as star formation history, star formation efficiency, and
outflows. Here were focus on the median trends of $[{\rm X}/{\rm Mg}]$ vs.
$[{\rm Mg}/{\rm H}]$, which are sensitive mainly to the relative
nucleosynthetic yields of these elements (see W19). Comparing disk and bulge
sequences therefore allows us to ask whether the supernovae that enriched the
bulge are similar to the supernovae that enriched the disk, even if the
relative contributions of CCSN and SNIa are different.

After discussing our data selection in Section 2, we present the APOGEE DR16
bulge abundances in Mg and Fe space in Section 3. Here we also construct a
comparison disk sample and analyze the similarities and differences between
the bulge and disk median high-Ia and low-Ia trends. In Section 4 we review
the two-process model of W19, apply it to the disk trends, and predict the
bulge abundances based on the best fit parameters. Section 5 contains a
discussion of potential constrains on the IMF difference between the bulge and
the disk. We summarize our work in Section 6.

##  2 Data

In this paper, we use data from the sixteenth data release (DR16) of the
SDSS/APOGEE survey (Ahumada et al., 2019; Jönsson et al., 2020, hereafter
J20). DR16 extends APOGEE’s Galactic view to the southern hemisphere, as we
observe with two nearly identical, 300 fiber APOGEE spectographs (Wilson et
al., 2019) on the 2.5 m Sloan Foundation telescope (Gunn et al., 2006) at
Apache Point Observatory (APO) in New Mexico and the 2.5 m du Pont telescope
(Bowen & Vaughan, 1973) at Las Campanas Observatory (LCO) in Chile. The
addition of LCO allows APOGEE to observe a greater number of stars in the
inner Galaxy, crucial for the work done in this paper. Targeting selection for
APOGEE-2 is described by Zasowski et al. (2017) and is updated in Santana et
al. (in prep.). Bulge stars are selected as a part of the main APOGEE red
giant sample. The APOGEE data are reduced as described by Nidever et al.
(2015) and fed into the APOGEE Stellar Parameters and Chemical Abundances
Pipeline (ASPCAP; Holtzman et al., 2015, 2018; García Pérez et al., 2016).
ASPCAP returns the best fit effective temperatures, surface gravities, and
elemental abundances employed in this paper. See J20 and references therein
for a more detailed description of the DR16 data reduction analysis and
validation.

We use data from an internal data release including observations through
November 2019 rather than the public DR16. These additional observations have
been processed with the same DR16 pipeline, so we will call this data set
DR16+444APOGEE allStar file allStar-r13-l33-58814. The 4 months of additional
observations add many more bulge fields, increasing the number of stars in our
bulge population from 6,978 in DR16 to 11,382 in DR16+. The additional data
and their ASPCAP analysis will be included in APOGEE DR17 (planned for
mid-2021). We have repeated our analysis with the smaller DR16 data set and
find very similar results. DR16+ provides more robust median trends for
difficult to observe elements (e.g. Na, K, Mn, Cu, Ce) and for the less
populated, low-metallicity regime.

We apply quality cuts to the DR16+ APOGEE catalog to extract stars with
reliable, calibrated abundance measurements. We cut all stars with flags set
for many bad pixels, bright neighbors, high persistence levels, broad lines,
or radial velocity warnings (STARFLAGs 0, 3, 9, 16, and 17) as well as those
with expected bad or dubious determinations of $T_{\rm eff}$, $\log(g)$,
metallicity, and $\alpha$ element content (ASPCAPFLAGs 0, 3, 16, 17).
Duplicate observations are removed. We require there to be no elemental flags
for [Mg/Fe] and [Fe/H] in our main sample, and we further discard stars with
flagged abundances in their respective elemental analyses. After removing
flagged stars we make a signal-to-noise ratio (SNR) cut of SNR$>100$.

Finally, we remove the cool stars by requiring $T_{\text{eff}}>3500$ K. For
some elements cool stars produce systematic structures in [X/Fe] vs. [Fe/H]
space, likely due to a failure in the APSCAP fit. Our cut at 3500 K does not
remove all observational artifacts, but it significantly cleans up the sample.
A larger discussion of these artifacts can be found in Section 3.1 and in J20.

Stars that pass these cuts span the Milky Way. While the boundaries between
the disk, inner Galaxy, and bulge are debated, we define the Galactic bulge as
$R_{\rm GC}<3\rm\;kpc$. We apply a height cut of $|Z|<5\rm\;kpc$. We use
spectrophotometric distances from Rojas-Arriagada et al. (2020) and take the
Galactic center to be at 8.178 kpc (Gravity Collaboration et al., 2019). To
assure that our stars reside in the inner galaxy we remove those with quoted
distance errors greater than 10% of their distance. While we choose to use the
Rojas-Arriagada et al. (2020) distances, astroNN (Bovy et al., 2019; Leung &
Bovy, 2019) and StarHorse (Queiroz et al., 2020) distances are also publicly
available for DR16 data (see the cited papers for more detailed comparison of
methods and results). When we repeat our analysis with these two alternative
distance sets, we find good agreement between the derived median abundance
trends. Our results are therefore independent of the specific distance
derivation used. The distribution of our bulge sample in $|Z|$ vs. $R_{\rm
GC}$ is shown in the top panel of Figure 1.

Figure 1: Top: The distribution of our bulge stellar sample in the $|Z|$ vs.
$R_{\rm GC}$ plane. Middle: Density plot of stars in the bulge. The dashed
line denotes our division between the high-Ia and low-Ia populations. We
exclude high-Ia stars with $[{\rm Fe/H}]<-0.5$, located in the grey region. We
include a histogram of our bulge sample’s $[{\rm Mg/Fe}]$ distribution in the
right hand panel. Bottom: Median trends for the low-Ia (squares) and high-Ia
(circles) bulge populations are shown in dark purple. Those for the disk (as
defined by W19) are shown in pink.

These quality and spatial cuts leave us with a sample of 11,382 stars in the
bulge, which have a median SNR of 144. The middle panel of Figure 1 shows the
density of these stars in $[{\rm Mg/Fe}]$ vs. $[{\rm Fe/H}]$ space and the
distribution of their $[{\rm Mg/Fe}]$ values. We define the low-Ia population
as stars which satisfy the following set of equations:

$\begin{cases}[{\rm Mg/Fe}]\geq 0.185,&[{\rm Fe/H}]\leq-0.5\cr[{\rm
Mg/Fe}]\geq 0.12-0.13[{\rm Fe/H}],&-0.5<[{\rm Fe/H}]\leq 0\cr[{\rm Mg/Fe}]\geq
0.12,&[{\rm Fe/H}]>0.\cr\end{cases}$ (1)

This dividing line is plotted on Figure 1. We note that our division differs
from that of W19 at $[{\rm Fe/H}]\leq-0.5$, as we add a plateau to
successfully separate the low metallicity stars. We further require high-Ia
stars to have $[{\rm Fe/H}]>-0.5$ to avoid contamination with the stars in the
low-Ia population that scatter below $[{\rm Mg/Fe}]=0.12$ at lower
metallicities.

While previous studies of the bulge have seen a single, continuous sequence in
the [$\alpha$/Fe] vs. [Fe/H]plane (e.g. Ness & Freeman, 2016), the histogram
in Figure 1 strongly suggests the presence of two distinct populations. The
distribution of $[{\rm Mg/Fe}]$ has a minimum at $[{\rm Mg/Fe}]\approx 0.13$
with bimodality particularly evident near $[{\rm Fe/H}]\approx 0$, as seen
previously by Rojas-Arriagada et al. (2019). However, compared to the disk
(see Figure 6), the high-Ia sequence is less evident at $[{\rm Fe/H}]<0$. The
$[{\rm Mg/Fe}]$ distribution is affected by the height distribution of the
sample (e.g. Bovy et al., 2016). Here, we have not corrected for the latitude
sampling bias, so our $[{\rm Mg/Fe}]$ distribution may not be representative
of the full bulge population. In their study of the bulge bimodality, Lian et
al. (2020) do correct this bias. After corrections they find a stronger signal
of bimodality than in the parent population. We likely see more low-Ia stars
than would be representative of this Galactic region due to the higher
latitude observations. For the purposes of this paper, the separation of our
sample into two populations is important mainly because it allows us to make
accurate comparisons of bulge and disk populations with similar levels of SNIa
enrichment. We do not make conclusions about the relative number of stars on
the low-Ia or high-Ia sequence, nor the bulge MDF, so we do not discuss the
sampling bias further.

Compared to the Galactic disk (W19; Figure 1), the low-Ia sequence of the
bulge remains well populated to lower metallicity, though the low-Ia sequences
of both the bulge and the disk reach an $[{\rm Fe/H}]$ of $\sim-1.3$. The
bottom panel of Figure 1 shows the median trends of the high-Ia and low-Ia
populations for the bulge and disk. Here, we use the W19 disk definition
($3\rm\;kpc<R<15\rm\;kpc$, $|Z|<2\rm\;kpc$, $1<\log(g)<2$, and
$3700\rm\;K<T_{\rm eff}<4600\rm\;K$), but recalculate with DR16+ data and the
quality cuts listed above. We will refer to this sample as “W19 disk”
throughout the paper. The $[{\rm Mg/Fe}]$ vs. $[{\rm Fe/H}]$ median trends for
W19 and the bulge agree to within $\sim 0.05$ dex at all metallicities.
Agreement is further improved by matching the $\log(g)$ distributions of the
disk and bulge samples as discussed below (see Section 3.3). We find that the
$[{\rm Mg/Fe}]$ downturn, or knee, of both populations occurs at the same
$[{\rm Fe/H}]$, in agreement with Zasowski et al. (2019).

##  3 Bulge Stellar Abundances

In this section, we discuss the median abundance trends of the APOGEE DR16+
bulge stars (see Tables 1 and 2). Figure 2 plots the APOGEE abundances in
$[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ space for Fe, O, Na, Al, Si, P,
S, K, Ca, V, Cr, Mn, Co, Ni, Cu, and Ce along with the median trends for the
high-Ia and low-Ia populations. Median values are calculated in bins of width
0.1 dex in $[{\rm Mg}/{\rm H}]$ and require $>20$ stars per bin.

Figure 2: Bulge stellar abundance distributions in [X/Mg] vs. [Mg/H] space for APOGEE elements. High-Ia stars are in orange and low-Ia stars are in purple. The medians of the high/low-Ia populations are over-plotted, where we bin by 0.1 dex and require $>20$ stars per bin. The dotted lines in each panel denote solar [X/Mg] and [Mg/H]. The number in the bottom right hand corner of each subplot corresponds to the number of stars for each elemental population which pass our cuts. High-Ia stars with Cr abundances within the dashed box are excluded in the calculation of the high-Ia median. Table 1: Median high-Ia (top) and low-Ia (bottom) sequences for APOGEE DR16+ $\alpha$-elements and light-$Z$ elements. We calculate medians in bins with a width of 0.1 dex in [Mg/H], requiring $>20$ stars per bin. Zero-point shifts discussed in Section 4.1 are included.  [Mg/H] | [O/Mg] | [Na/Mg] | [Al/Mg] | [Si/Mg] | [P/Mg] | [S/Mg] | [K/Mg] | [Ca/Mg]
---|---|---|---|---|---|---|---|---
-0.262 | -0.044 | 0.002 | – | -0.085 | -0.018 | 0.022 | 0.025 | –
-0.149 | -0.017 | 0.019 | -0.041 | -0.036 | -0.004 | 0.022 | 0.022 | 0.048
-0.043 | -0.022 | 0.008 | -0.090 | -0.003 | -0.018 | 0.014 | -0.006 | -0.028
0.056 | -0.012 | 0.005 | -0.044 | 0.013 | -0.017 | 0.002 | -0.029 | -0.017
0.153 | 0.001 | -0.003 | 0.012 | 0.025 | -0.017 | -0.004 | -0.034 | 0.007
0.255 | 0.008 | -0.010 | 0.081 | 0.026 | -0.018 | -0.027 | -0.041 | 0.020
0.354 | 0.013 | -0.020 | 0.170 | 0.023 | -0.014 | -0.056 | -0.051 | 0.065
0.445 | 0.010 | -0.030 | 0.230 | 0.019 | -0.014 | -0.070 | -0.057 | 0.073
0.532 | -0.000 | -0.043 | 0.266 | 0.003 | -0.021 | -0.091 | -0.069 | 0.094
-1.254 | -0.264 | -0.011 | – | -0.527 | -0.015 | -0.190 | 0.231 | -0.056
-1.145 | -0.232 | 0.011 | – | -0.468 | 0.015 | -0.062 | 0.199 | -0.005
-1.042 | -0.244 | 0.017 | – | -0.411 | 0.016 | -0.200 | 0.283 | -0.031
-0.95 | -0.226 | -0.005 | – | -0.377 | 0.011 | -0.201 | 0.234 | -0.028
-0.844 | -0.229 | -0.017 | – | -0.314 | 0.019 | -0.216 | 0.233 | -0.047
-0.746 | -0.246 | -0.043 | – | -0.233 | 0.000 | -0.273 | 0.161 | -0.058
-0.646 | -0.270 | -0.053 | – | -0.211 | -0.007 | -0.281 | 0.153 | -0.079
-0.547 | -0.269 | -0.056 | – | -0.211 | -0.032 | -0.266 | 0.110 | -0.086
-0.444 | -0.270 | -0.056 | – | -0.213 | -0.066 | -0.221 | 0.077 | -0.085
-0.345 | -0.274 | -0.056 | – | -0.165 | -0.060 | -0.173 | 0.078 | -0.069
-0.248 | -0.269 | -0.051 | -0.188 | -0.128 | -0.063 | -0.134 | 0.044 | -0.060
-0.149 | -0.255 | -0.045 | -0.201 | -0.096 | -0.069 | -0.113 | 0.016 | -0.044
-0.049 | -0.230 | -0.040 | -0.185 | -0.062 | -0.064 | -0.084 | -0.004 | -0.036
0.046 | -0.197 | -0.030 | -0.153 | -0.040 | -0.061 | -0.076 | -0.016 | -0.028
0.142 | -0.152 | -0.025 | -0.119 | -0.017 | -0.059 | -0.069 | -0.043 | -0.030
0.241 | -0.126 | -0.029 | -0.094 | 0.001 | -0.060 | -0.070 | -0.072 | -0.030
0.345 | -0.103 | -0.051 | -0.020 | -0.004 | -0.056 | -0.067 | -0.076 | 0.014
0.445 | -0.086 | -0.071 | 0.024 | -0.017 | -0.065 | -0.139 | -0.106 | 0.019
Table 2: Same as Table 1, but for APOGEE DR16+ Fe-peak elements, Cu, and Ce  [Mg/H] | [V/Mg] | [Cr/Mg] | [Mn/Mg] | [Fe/Mg] | [Co/Mg] | [Ni/Mg] | [Cu/Mg] | [Ce/Mg]
---|---|---|---|---|---|---|---|---
-0.262 | 0.015 | 0.001 | – | – | -0.069 | -0.030 | – | –
-0.149 | 0.005 | 0.006 | 0.040 | -0.200 | -0.032 | -0.026 | -0.045 | –
-0.043 | -0.004 | -0.012 | 0.020 | -0.127 | -0.049 | -0.031 | -0.032 | -0.075
0.056 | -0.002 | -0.021 | 0.013 | -0.063 | -0.040 | -0.018 | -0.003 | -0.070
0.153 | -0.001 | 0.005 | 0.045 | -0.030 | -0.021 | 0.005 | -0.013 | -0.140
0.255 | -0.004 | 0.024 | 0.057 | 0.025 | 0.001 | 0.021 | 0.010 | -0.187
0.354 | -0.009 | 0.041 | 0.068 | 0.082 | 0.032 | 0.051 | 0.057 | -0.190
0.445 | -0.026 | 0.058 | 0.066 | 0.133 | 0.074 | 0.071 | 0.065 | -0.198
0.532 | -0.041 | 0.072 | 0.052 | 0.145 | 0.106 | 0.082 | 0.105 | -0.204
-1.254 | -0.049 | -0.001 | -0.406 | -0.620 | -0.443 | -0.247 | 0.169 | -0.237
-1.145 | -0.033 | -0.052 | -0.410 | -0.614 | -0.429 | -0.254 | 0.151 | -0.273
-1.042 | -0.014 | -0.114 | -0.414 | -0.639 | -0.403 | -0.248 | 0.142 | -0.251
-0.95 | -0.015 | -0.171 | -0.361 | -0.622 | -0.309 | -0.230 | 0.097 | -0.259
-0.844 | -0.004 | -0.166 | -0.366 | -0.612 | -0.303 | -0.216 | 0.047 | -0.172
-0.746 | -0.031 | -0.154 | -0.304 | -0.594 | -0.264 | -0.210 | -0.016 | -0.118
-0.646 | -0.056 | -0.187 | -0.328 | -0.590 | -0.268 | -0.223 | -0.069 | -0.123
-0.547 | -0.066 | -0.153 | -0.291 | -0.566 | -0.238 | -0.211 | -0.116 | -0.181
-0.444 | -0.084 | -0.141 | -0.270 | -0.535 | -0.191 | -0.202 | -0.121 | -0.244
-0.345 | -0.104 | -0.168 | -0.280 | -0.510 | -0.194 | -0.203 | -0.140 | -0.291
-0.248 | -0.114 | -0.152 | -0.272 | -0.490 | -0.164 | -0.189 | -0.160 | -0.331
-0.149 | -0.119 | -0.131 | -0.258 | -0.443 | -0.146 | -0.175 | -0.172 | -0.333
-0.049 | -0.120 | -0.130 | -0.242 | -0.383 | -0.126 | -0.160 | -0.166 | -0.316
0.046 | -0.120 | -0.125 | -0.219 | -0.342 | -0.112 | -0.141 | -0.164 | -0.290
0.142 | -0.105 | -0.113 | -0.197 | -0.256 | -0.101 | -0.116 | -0.126 | -0.233
0.241 | -0.088 | -0.080 | -0.183 | -0.157 | -0.078 | -0.097 | -0.087 | -0.206
0.345 | -0.075 | -0.048 | -0.174 | -0.107 | -0.055 | -0.065 | -0.092 | -0.255
0.445 | -0.080 | -0.039 | -0.134 | – | -0.042 | -0.043 | – | –

While APOGEE reports C, N, and Ti for some bulge stars, we do not include them
in this paper. We remove the C and N abundances as mixing of processed
material in giant stars causes the observed atmospheric abundances to differ
from their birth values. We exclude Ti because the Ti I abundances show
inaccurate [Ti/Fe] vs. [Fe/H] trends when compared to optical measurements and
Ti II abundances have excessive scatter (Hawkins et al., 2016, J20).

We further note the exclusion of some Na, K, Mn, and Ce abundances. As in
previous data releases, Na remains one of the most imprecisely measured
elements in DR16 with scatter of $\sim 0.1$ dex (J20). As [Na/Fe] errors
exceed 0.1 dex at low metallicities, we cut all stars with $[{\rm Fe/H}]<-0.5$
in the Na analysis. While our quality cuts remove all stars below 3500 K,
ASPCAP does not report calibrated abundances for Na, K, Mn, and Ce below
$\sim$4000 K. Systematic trends with temperature in all four elements motivate
their exclusion. Radial velocity shifts that move Ce lines into chip gaps or
out of APOGEE’s wavelength range cause additional omission. After our cuts the
[Na/Mg] high-Ia and low-Ia medians still have some temperature dependence. We
find that lower temperature sub-samples have higher median [Na/Mg] values than
higher temperature sub-samples at the same [Mg/H]. Appendix A further explores
this temperature systematic and its influence on the median trends of Na and
other elements.

###  3.1 Systematics and Artifacts

Although we apply extensive quality cuts to our data, artifacts and unexpected
abundance structures still appear. Most obviously, the high-Ia Cr stars split
into two populations, one with higher [Cr/Mg] values, and one with lower.
Clumps and bands can also be seen in [V/Mg], [P/Mg], and [Al/Mg]. Figure 3
plots these same data, but in $[{\rm X}/{\rm Fe}]$ vs. $[{\rm Fe/H}]$ space.
Here the aforementioned artifacts become more apparent and other structures
not seen in $[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ space pop out. We
review the observed features that are caused by two broad problems in the data
reduction process (J20) below.

The $\alpha$ finger: In the [O/Fe], [Ca/Fe], and [Si/Fe] vs. $[{\rm Fe/H}]$
panels of Figure 3, a ‘finger’ can be seen protruding from the abundance
tracks. Zasowski et al. (2019) see a similar feature in their DR14 bulge
abundance distributions. The finger feature affects a small fraction of cool
giants and is caused by an error on the abundance determination (J20). It is
also seen in the [$\alpha$/M] vs. [M/H] trends. Si exhibits a smaller finger
than O and Ca, and Mg is free of the structure. While cutting stars below
$3500$ K removed many of the finger stars from our analysis, the artifact
still remains visible. However, due to the low number of stars in the finger,
their exclusion changes the median high-Ia and low-Ia $[{\rm X}/{\rm Mg}]$
trends by $<0.05$ dex (see Appendix A). As such, we continue in our analysis
with their inclusion. Interestingly, the finger artifact mainly plagues APOGEE
bulge fields.

Cr bimodality: In both Mg and Fe space, the split in high-Ia Cr abundances is
obvious. J20 describe this artifact as the “low abundance trend” seen for some
giant stars with $T_{\rm eff}<4000$ K and suggest that it comes from the TIE-
option in the FERRE interpolation. While this structure affects many elements,
it afflicts Cr the most. The inclusion of both groups would create a high-Ia
median trend that poorly describes either one. We choose to exclude the high-
Ia stars with lower [Cr/Mg] from our analysis. These stars are located within
the box drawn in the Cr panel of Figure 2, having $0<[{\rm Mg}/{\rm H}]<0.75$
and $-0.3<$[Cr/Mg]$<-0.1$. We cut the low-Cr group because Cr trends in higher
$\log(g)$ disk populations from APOGEE DR16 bear greater resemblance to the
high-Cr track. Knowledge of previous works in the solar neighborhood (e.g.
Bensby et al., 2014) also support this decision. The excluded stars have been
colored grey in Figure 3. They have $[{\rm Fe}/{\rm Mg}]$ near solar, so they
shift location relative ot the low-Ia population, which has $[{\rm Fe/H}]<0$.

Al and P banding: In both elemental plots, the high-Ia and low-Ia populations
appear to be divided into multiple sequences or clumps. These are both
instances of the low abundance trend artifacts, similar to that seen in Cr
(J20). We are unable to remove or account for these systematics as they are
blended into the other features, so the abundance trends should be viewed with
caution. We note that the stars within the bands are different for P, Al, and
Cr.

V Clump: A small cluster of high-Ia stars can be seen at higher [V/Mg] and
[V/Fe] than the majority of the population. These stars are not the same stars
in the Al or P banding. As with the $\alpha$ finger, the few stars in this
clump have little effect on the median trend, so we do not eliminate them.

The inclusions/exclusions of stars afflicted by the artifacts discussed in
this section slightly change the high-Ia and low-Ia median sequences with the
only large impact being the high-Ia [Cr/Mg] trend. For a larger discussion of
the effect that these stars have on the median trends, see Appendix A. J20
note that the low abundance trends can also be seen in Ni and Co, but this
structure is not apparent in our stellar selection.

Figure 3: Bulge star distributions in [X/Fe] vs. [Mg/Fe] space for APOGEE
elements. High-Ia stars are in orange and low-Ia stars are in purple. The
dotted lines in each panel denote solar [X/Fe] and [Fe/H]. The number in the
bottom right hand corner of each subplot corresponds to the number of stars
for each elemental population which pass our cuts. Excluded Cr stars have been
colored grey.

###  3.2 Bulge Definition and Bar Influence

If the location and separation of the $[{\rm X}/{\rm Mg}]$ median high-Ia and
low-Ia trends differs between the bulge and the disk, implying some radial
dependence, we might also expect the bulge definition to influence the
observed median trends. In this paper we take the bulge to be all stars within
the cylinder defined by $R_{\rm GC}<3\rm\;kpc$ and $|Z|<5\rm\;kpc$. Previous
studies of the bulge and inner Galaxy have used similar definitions (see
Section 1), though not identical as the bulge/disk division is not clearly
defined. We investigate the radial dependence of the abundance trends in the
inner Galaxy by subdividing the bulge ($R_{\rm GC}<3\rm\;kpc$) into three
groups—$2\rm\;kpc\leq R_{\rm GC}<3\rm\;kpc$, $1\rm\;kpc\leq R_{\rm
GC}<2\rm\;kpc$, and $R_{\rm GC}<1\rm\;kpc$. For each group, we calculate the
median high-Ia and low-Ia trends and compare with the full sample. We find
that the locations of the median trends are identical for all three groups
within the uncertainties, though Mn, K, Cu, and Ce do not have enough stars on
the high-Ia sequence to derive median trends for the innermost radial bin. We
extend this comparison to the vertical bounds as well. Our bulge cut extends
to high $|Z|$, so we divide the stars into two height bins of
$|Z|<0.25\rm\;kpc$ and $1\rm\;kpc$ $\leq|Z|<5\rm\;kpc$ to probe the mid-plane
and extended bulge. We again find that the high-Ia and low-Ia trends remain
constant. We do not see a radial or height dependence of the high-Ia or low-Ia
medians for stars within a radius of $3\rm\;kpc$.

Additionally, the Milky Way bar may have a distinct chemical signature. Work
by Bovy et al. (2019) found that inner galaxy stars ($R_{\rm GC}<5\rm\;kpc$)
inside and outside of the bar followed the same [O/Fe] vs. $[{\rm Fe/H}]$
tracks, but that there were more metal poor stars ($[{\rm Fe/H}]<0$) in the
bar than the outside of the bar. Lian et al. (in prep) find similarly small
differences between the [Fe/H] and [Mg/Fe] distributions on and off the inner
bar. We investigate whether the $[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$
trends differ between on and off bar inner Galaxy regions. We use the bar
definition from Bovy et al. (2019): an ellipse with a semi major axis of 5 kpc
angled at 25∘ clockwise from the Sun-Galaxy line with an axis ratio of 0.4. We
take the 2000 stars in this ellipse with $R_{\rm GC}<3\rm\;kpc$ and find the
median high-Ia and low-Ia abundance trends for each element with sufficient
observations. While the metallicity distribution varies, we find good
agreement between the median tracks of the bar and the full inner galaxy
population. Small differences ( 0.05 dex) are seen at the low $[{\rm Mg}/{\rm
H}]$ end of the low-Ia sequence for Al, Si, V, Cr, and Co. Many of these
elements show the low abundance artifacts noted by J20, so the deviations are
likely due to the small number of stars in the sample and the influence of
these artifacts.

###  3.3 Creating a Comparison Sample

By comparing the median abundance trends of the bulge and disk, we can learn
about the similarities and/or differences in their chemical enrichment
histories. Upon first analysis of the bulge and W19 disk medians, we see
differences of $\sim 0.1-0.2$ dex in half of the elements. The first row of
Figure 4 shows those of Al and Si. The low-Ia [Al/Mg] bulge sequence dips
below that of the W19 disk at sub-solar $[{\rm Mg}/{\rm H}]$. The differences
in Si are subtler, but both the bulge high-Ia and low-Ia medians appear
suppressed relative to the disk.

Figure 4: Median low-Ia trends for Al and Si, comparing different stellar
samples. Top row: Bulge medians (dark purple) compared to W19 disk medians
(light purple). Middle row: Bulge medians (purple) compared to disk stars of
$5\leq R_{G}\leq 11$ whose $\log(g)$ distribution matches that of the bulge
(resampled disk; pink). Bottom row: W19 disk medians (pink) compared to a
subset of stars in the bulge sample that reproduce the $1<\log(g)<2$
distribution of the W19 disk (resampled bulge; orange). After $\log(g)$
systematics are accounted for in the lower two rows, the differences seen in
the top row disappear.

These two populations, however, do not probe similar stellar samples. APOGEE
observes different stellar samples at different locations in the Galaxy. As
the bulge is farther away, APOGEE observes only the most luminous giants. This
means that the $\log(g)$ distribution of stars differs between the bulge and
the W19 disk (Figure 5, panels 1 and 2). Systematic abundance errors that
correlate with $\log(g)$ could cause artificial differences between the bulge
and disk medians (e.g. Santos-Peral et al., 2020). To test this possibility,
we randomly resample the full disk population to reflect the $\log(g)$
distribution of the bulge. To ensure that we sample disk stars and not inner
galaxy stars on the bulge border, we restrict our disk to $5\rm\;kpc<R_{\rm
GC}<11\rm\;kpc$. In each $\log(g)$ bin, we select a number of APOGEE disk
stars equal to the number of bulge stars in the same bin. We will hereafter
refer to this stellar population as the “resampled disk”. Figure 5 plots
histograms of the bulge and resampled disk’s $\log(g)$ distribution in panels
1 and 3 of the top row and their radial position in the bottom row (panel 4).
Stars in both populations are distributed throughout the bulge and disk,
respectively.

After this re-sampling, the median $[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm
H}]$ high-Ia and low-Ia trends come into closer agreement for many elements.
The middle panel of Figure 4 compares [Al/Mg] and [Si/Mg] vs. $[{\rm Mg}/{\rm
H}]$ medians for the bulge and the $\log(g)$ resampled disk. Differences seen
between the W19 disk and bulge almost disappear. To confirm that this is a
result of $\log(g)$ systematics and dissimilar sampling we repeat for the
opposite case–randomly resampling the bulge to reflect the $\log(g)$
distribution of the W19 disk. We again see that the median [Al/Mg] and [Si/Mg]
vs. $[{\rm Mg}/{\rm H}]$ trends agree between the two population with similar
$\log(g)$ distributions (Figure 4, bottom row). We conclude that $\log(g)$
systematics cause small but detectable changes in the elemental median trends.
In order to accurately compare Galactic regions, we must have samples with
similar $\log(g)$ distributions. Our bulge and resampled disk populations
satisfy this requirement.

Figure 5: Top row: $\log(g)$ distributions for stars in the bulge (left; dark
purple), the W19 disk sample (middle; purple), and the resampled disk (right;
pink). Bottom: Radial location of stars with a given $\log(g)$ for the bulge
and the resampled disk populations. We overplot median trends binned radially
by 0.5 kpc. Bulge and disk stars appear evenly radially distributed with no
clear dependence on $\log(g)$.

A density plot of our resampled disk can be found in Figure 6. We use the same
low-Ia definition as in Equation (1). The high-Ia and low-Ia populations are
clearly defined and separated. The resampled disk has more stars on the high-
Ia sequence and a less extended low-Ia sequence than the bulge, reflective of
the distribution found by W19. We apply the same exclusions to the resampled
disk as described above.

Figure 6: Density plot of the stars in the resampled disk that recreate the
$\log(g)$ distribution of the bulge. The dashed line denotes our division
between the high-Ia and low-Ia stars used in the bulge. The two sequences can
clearly be distinguished, as in Figure 1 of W19.

###  3.4 Comparison to APOGEE Disk

Now that we have similar stellar samples, we can compare the high-Ia and low-
Ia median abundance trends of the bulge and the resampled disk. Figure 7 plots
the median bulge trends (same as Figure 2) and the median trends of the
resampled disk. Both are binned by 0.1 dex in $[{\rm Mg}/{\rm H}]$, requiring
$>20$ stars per bin. Overall, the medians agree well. The resampled disk and
bulge trends are nearly identical for Fe, O, Al, Si, S, K, Ca, V, Cr, Ni, and
Cu. This similarity suggests that the bulge and the disk experienced similar
chemical enrichment and that nucleosynthesis pathways are identical throughout
the Galaxy–extending the conclusions of W19 from the disk to the inner Galaxy.
We do see minor differences between the Na, P, Mn, Co, and Ce trends of bulge
and resampled disk. All of these difference, though, are on the scale of 0.1
dex or smaller (see Figure 12). In the following subsections we will more
closely examine the median trends, discussing APOGEE’s elemental precision,
nucleosynthetic origins, and differences/similarities between the bulge and
the resampled disk. We will reference the chemical evolution models of Andrews
et al. (2017), hereafter AWSJ17, for theoretical yield predictions.

Figure 7: APOGEE bulge star median abundances trends of high-Ia (orange
circles) and low-Ia (purple squares) populations, as in Figure 2. Median high-
Ia and low-Ia abundances of resampled disk stars are plotted as light grey
circles and dark grey squares, respectively, for comparison. Dotted lines
correspond to solar values of [X/Mg] and [Mg/H].

####  3.4.1 $\alpha$-elements

CCSN dominate the production of $\alpha$-elements such as O, Si, S, and Ca. O
is predicted to come almost entirely from CCSN, as is Mg, while Si, S, and Ca
have some SNIa contribution (AWSJ17). As such, we see that O traces Mg and has
little to no separation between the high-Ia and low-Ia [O/Mg] medians.
APOGEE’s [O/Mg] trends are flat, supporting metallicity independent CCSN
yields. This agrees with other IR studies (J20) but is inconsistent with
metallicity dependent [O/Mg] trends derived from optical abundances (GJW).

Si and Ca show larger median high-Ia and low-Ia separations than O, in
agreement with theoretical predictions that they have a non-dominant SNIa
contribution. We note that Si is one of the most precisely measured elements
in APOGEE with scatter of $\lesssim 0.02$ dex (J20). While AWSJ17 find that S
has a larger SNIa contribution than Si, we see no separation between the S
high-Ia and low-Ia median trends, indicative of little SNIa contribution. S is
the only $\alpha$-element for which we infer metallicity dependent yields.

The high-Ia and low-Ia median trends of the $\alpha$-elements agree very well
between the bulge and the resampled disk. The median absolute differences
between the sequences are all $\leq 0.02$ dex and within observational errors.
We do not see the enhancement in the bulge $\alpha$ abundances relative to the
local thick disk at a give metallicity, as was found in previous works (e.g.
Bensby et al., 2013; Johnson et al., 2014)

####  3.4.2 Light odd-$Z$ elements

Like the $\alpha$-elements, we expect CCSN to be the predominant source of the
light odd-$Z$ elements (Na, Al, P, and K) and expect SNIa to contribute in a
non-dominant way. The abundance tracks of these elements, however, should be
distinguished from $\alpha$-elements by stronger metallicity dependence
(AWSJ17). While [Na/Mg] high-Ia and low-Ia median trends do show a positive
metallicity gradient, they also show a significant separation. This suggests a
large SNIa or other delayed production source. The median trend separation
agrees with results for the APOGEE DR14 disk (W19) and GALAH DR2 sample (GJW).
Both papers include a larger discussion of the nucleosynthetic implications of
the empirical results. We see a relatively similar sequence separation in the
Na trends of the resampled disk and the bulge, but the two samples have
different high-Ia metallicity dependencies. While the low-Ia tracks are in
good agreement, the high-Ia trends have a median absolute difference of 0.06
dex, one of the largest differences between the bulge and the resampled disk.
We reiterate that Na is not precisely determined by APOGEE (with scatter of
$\sim 0.1$ dex), and that temperature systematics have not been accounted for
(J20). The difference between bulge and disk trends for the high-Ia population
could be a result of different chemical enrichment histories, but it could
also be a result of residual systematic errors on Na abundance measurements.

Al and P are also plagued by systematics and reduction artifacts. Both show
some separation in their median trends, though not to the extent of Na. This
separation is unexpected for Al, which is predicted to be a pure CCSN element
(AWSJ17). W19 find no separation in their analysis of DR14 data. Al trends do
show a positive metallicity dependence, as expected. The bulge and resampled
disk [Al/Mg] trends agree well after the disk resampling. Al abundances have
high precision ($\lesssim 0.04$ dex) in APOGEE DR16 (J20).

The [P/Mg] trends show a 0.05-0.1 dex differences between the bulge and the
resampled disk in some $[{\rm Mg}/{\rm H}]$ bins. A closer examination of
stars in the high-Ia and low $[{\rm Mg}/{\rm H}]$ bins–most discrepant between
the bulge and the disk–reveals that the $\log(g)$ distribution of stars in the
respective populations differ significantly. While the resampled disk stars
span the parent sample’s $\log(g)$ range of 0-2.5, the less populated bulge
bin only includes stars near $\log(g)=0$, perhaps because only the most
luminous stars yield successful P measurements at low $[{\rm Mg}/{\rm H}]$ at
these distances. The $\log(g)$ systematics could be responsible for some of
the median bin differences. We reiterate that P is the least precisely
determined element in APOGEE (with scatter of 0.15 dex or greater) and is
subject to systematic errors (J20). Masseron et al. (2020), for example, find
that low-metallicity P-rich stars are missed by ASPCAP entirely. We do not
trust the P abundances and do not draw conclusions from these median trends.

We see the weakest metallicity dependence of the light odd-$Z$ elements in the
[K/Mg] median trends. The high-Ia and low-Ia medians show little separation
and suggest CCSN dominated production. This agrees with theoretical yields
(AWSJ17). We see good agreement between the bulge and the resampled disk,
though the resampled disk’s low-Ia median trend sits at a higher [K/Mg] than
the bulge low-Ia median.

####  3.4.3 Fe-peak elements

Nucleosynthetic yields and chemical evolution models predict comparable CCSN
and SNIa contribution to Fe-peak elements at solar abundances (AWSJ17). Our
observations agree, as we see significant separation between the high-Ia and
low-Ia median trends for all APOGEE Fe-peak elements. AWSJ17 further predict
that the odd-$Z$ elements (V, Mn, Co) will have a positive metallicity
dependence while the even-$Z$ elements (Cr, Fe, Ni) will have flatter trends.
We observe Mn and Co to have a stronger metallicity dependence than the
even-$Z$ elements. V exhibits a shallower trend than Co and Mn. [V/Mg] and
[Co/Mg] have similar, small separation between the high-Ia and low-Ia median
trends, indicative of significant but non-dominant SNIa contribution. As in
W19 and GJW, Mn has the largest separation of all included elements, and thus
the largest SNIa contribution. Among the even-$Z$ elements, [Cr/Mg] trends
show larger separation than [Fe/Mg] (with the caveat that we cut out
low-[Cr/Mg], high-Ia stars as discussed in Section 3.1), while [Ni/Mg] trends
show less separation. This suggests that SNIa dominate Cr production, while
CCSN make a larger contribution to Ni. Both elements strongly resemble Fe.

The V, Cr, Fe, and Ni median trends agree well between the bulge and resampled
disk. All have absolute median differences $\leq 0.02$ dex. Both populations’
$[{\rm Fe}/{\rm Mg}]$ low-Ia median sequences plateau around $-0.3$. Mn and Co
show larger differences. Both the high-Ia and low-Ia resampled disk [Mn/Mg]
median trends sit at slightly higher [Mn/Mg] than those of the bulge and have
a flatter metallicity dependence. The bulge and resampled disk’s high-Ia and
low-Ia [Mn/Mg] trends have median absolute differences of 0.07 and 0.05 dex
respectively. We reiterate that APOGEE DR16 Mn abundances are not populated
for cool stars ($T\lesssim 4000$ K) due to temperature systematics, so we have
a much smaller sample than the rest of the bulge with few high-Ia stars. While
cluster calibrations were applied to DR14 data to remove temperature effects
(Holtzman et al., 2018), these calibrations were not done in DR16. Mn shows
some of the largest differences between the bulge and the resampled disk,
particularly for high-Ia stars, but due to the small bulge sample and
potential temperature systematics we interpret these results cautiously. The
Co high-Ia and low-Ia resampled disk medians are slightly higher in [Co/Mg],
with flatter metallicity dependence than the bulge. The deviations in the Co
low-Ia trends seen at sub-solar $[{\rm Mg}/{\rm H}]$ can also be seen, to a
much smaller degree, in Ni, and V.

####  3.4.4 Cu and Ce

In GJW, we classify Cu as a “Fe-cliff” element, as it resides on the steeply
falling edge of the Fe abundance peak. APOGEE DR16+ Cu abundances are not the
most reliable, having a low precision and low accuracy (J20). The upturned
tail at low $[{\rm Mg}/{\rm H}]$ and $[{\rm Fe/H}]$ seen in Figures 2 and 3 is
unexpected and probably not trustworthy. We will limit our interpretation to
Cu abundances above solar $[{\rm Mg}/{\rm H}]$. Here we see separation between
the high-Ia and low-Ia sequences and a positive metallicity dependence. Cu
production is thought to be dominated by CCSN, with yields suggesting a strong
metallicity dependence (W19). The observed separation in median trends,
however, implies a non-zero SNIa or other delayed component, in agreement with
optical trends from GALAH (GJW). At all metallicities, we see good agreement
between the bulge and resampled disk median trends.

While all of the above elements are likely produced through CCSN and SNIa
nucleosynthesis, Ce is produced by a combination of the rapid and slow neutron
capture processes ($r$-process and $s$-process, respectively) 555The
$r$-process and $s$-process do not automatically translate into prompt and
delayed, since, for example, some $s$-process production occurs in massive
CCSN progenitors. In practice, most $s$-process enrichment arises in AGB stars
with intermediate lifetimes, and the main sources of $r$-process enrichment
appear to be prompt.. Separation in the Ce high-Ia and low-Ia median trends
indicates that Ce is dominated by a delayed enrichment source, in agreement
with measurements by Arlandini et al. (1999) and Bisterzo et al. (2014) who
find Ce to have $\sim 80\%$ $s$-process contribution. We see similar sequence
separation between the bulge and the resampled disk, though the resampled
disk’s low-Ia trend has slightly higher [Ce/Mg] than that of the bulge. This
difference should be viewed with some caution, as APOGEE Ce abundances have
low precision (scatter of 0.15 dex or greater) due to its single line analysis
(J20). We do, however, think that Ce abundances are should be accurate
(excluding low metallicity stars). Neutron capture elements are not well
studied in the bulge, so APOGEE Ce observations can give insight into
nucleosyntheic processes beyond CCSN and SNIa.

DR16 is the first APOGEE data release to include Cu and Ce. A larger
discussion of their abundance trends, nucleosynthetic implications, and
comparison to GALAH data can be found in Appendix B.

##  4 Two-process Model

With the exception of Ce, the elements studied in this paper are theoretically
expected to originate almost entirely from CCSN or SNIa (Johnson, 2019). While
individual CCSN produce elements in different ratios depending on progenitor
mass, the IMF-averaged production should yield fixed ratios for supernovae of
a given metallicity. W19 interpreted the median abundance-ratio sequences from
APOGEE DR14 in terms of a two-process model, which represents the elemental
abundances of any given star (or median sequence point) as the sum of a CCSN
process with amplitude $A_{\rm cc}$ and a SNIa process with amplitude $A_{\rm
Ia}$:

$\bigg{(}\frac{\rm X}{\rm H}\bigg{)}=A_{\rm cc}p_{\rm cc}^{\rm X}(Z)+A_{\rm
Ia}p_{\rm Ia}^{\rm X}(Z).$ (2)

To account for metallicity-dependent yields, the process vectors $p_{\rm
cc}^{\rm X}$ and $p_{\rm Ia}^{\rm X}$ are allowed to depend on the star’s
metallicity. GJW extend the analysis to GALAH DR2 abundance ratios, including
elements not measured by APOGEE. In W19, GJW, and here, the metallicity
dependencies are modeled as power laws in (Mg/H), with index $\alpha_{\rm cc}$
and $\alpha_{\rm Ia}$ for $p_{\rm cc}^{\rm X}$ and $p_{\rm Ia}^{\rm X}$,
respectively.

Here we fit the DR16 abundance-ratio sequences of the resampled disk to infer
the two-process model parameters $\alpha_{\rm cc}$, $\alpha_{\rm Ia}$, and

$R_{\rm Ia}^{X}=\frac{p_{\rm Ia,\odot}^{\rm X}}{p_{\rm cc,\odot}^{\rm X}}\,,$
(3)

which is the ratio of the two processes for element X in a star with solar
abundances $[{\rm Mg}/{\rm H}]=[{\rm Fe}/{\rm Mg}]=0$. Differences from W19
arise partly from the differences between DR14 and DR16 abundance
determinations and partly from the different $\log(g)$ distributions of the
samples. The values of $R_{\rm Ia}^{X}$, $\alpha_{\rm cc}$, and $\alpha_{\rm
Ia}$ derived here are not necessarily more reliable than those of W19 but they
allow us to predict bulge stars abundances using a model “trained”
independently on the disk. In Section 5, we use these comparisons to place
rough limits on the possible difference between the high mass IMF slope of
bulge and disk populations.

As in W19, we assume that Mg is purely produced by CCSN with metallicity
independent yields ($R_{\text{Ia}}^{\text{Mg}}=0$ and
$\alpha_{\text{cc}}^{\text{Mg}}=\alpha_{\text{Ia}}^{\text{Mg}}=0$), that Fe is
produced by both SNIa and CCSN with metallicity independent yields
($\alpha_{\text{cc}}^{\text{Fe}}=\alpha_{\text{Ia}}^{\text{Fe}}=0$), and that
stars on the low-Ia plateau, at $[{\rm Mg/Fe}]\approx 0.3$, have pure CCSN
enrichment (implying $R_{\text{Ia}}^{\text{Fe}}=1$). Therefore, given any
star’s $[{\rm Mg/Fe}]$, the ratio of its SNIa to CCSN enrichment is

$\frac{A_{\rm Ia}}{A_{\rm cc}}=10^{0.3-[{\rm Mg/Fe}]}-1\,.$ (4)

From the global elemental parameters ($R_{\rm Ia}^{X}$, $\alpha_{\rm cc}$,
$\alpha_{\rm Ia}$) and stellar abundances ($[{\rm Mg}/{\rm H}]$ $[{\rm
Mg/Fe}]$), W19 derive an expression to calculate the expected abundance of
element $X$ in a star as

$[{\rm X}/{\rm Mg}]=\alpha_{\rm cc}[{\rm Mg}/{\rm H}]+\\\
\log\bigg{[}\frac{1+R_{\rm Ia}^{X}(A_{\rm Ia}/A_{\rm cc})10^{(\alpha_{\rm
Ia}-\alpha_{\rm cc})[{\rm Mg}/{\rm H}]}}{1+R_{\rm Ia}^{X}}\bigg{]},$ (5)

with $A_{\rm Ia}/A_{\rm cc}$ from Equation (4). As in W19 and GJW, we will use
the median $[{\rm X}/{\rm Mg}]$ and $[{\rm Mg}/{\rm H}]$ trends to find the
best $R_{\rm Ia}^{X}$, $\alpha_{\rm cc}$, and $\alpha_{\rm Ia}$ values for
each element. For elements that have substantial production by AGB stars (Ce
and possibly Na, P, Cu), the model parameters are only qualitatively
meaningful since the AGB enrichment delay is different from that of SNIa.

###  4.1 Fitting the Resampled Disk

We perform an unweighted, least-squares fit to the resampled disk’s high-Ia
and low-Ia $[{\rm Mg}/{\rm H}]$ vs. $[{\rm X}/{\rm Mg}]$ median trends for
each element with the two-process model to derive the best fit $R_{\rm
Ia}^{X}$, $\alpha_{\rm cc}$, and $\alpha_{\rm Ia}$ values. We allow all three
parameters to vary and conduct a grid search with a step size of 0.01 for
each. Due to the low number of stars at low metallicity, we only fit median
points with $[{\rm Mg}/{\rm H}]>-0.7$. We choose to fit the resampled disk
rather than bulge because the resampled disk medians track the bulge medians
and its high-Ia sequence is more well populated.

As pointed out by GJW (see their Figure 10), the two-process model necessarily
predicts [X/Mg] = 0 for a star, or median sequence point, with $[{\rm Fe}/{\rm
Mg}]=[{\rm Mg}/{\rm H}]=0$. Figure 7 shows that this is not the case for all
elements in our sample, but the small departures from solar $[{\rm X}/{\rm
Mg}]$ on the high-Ia sequence are plausibly a consequence of small calibration
errors in the abundance determinations. Following GJW, therefore we apply
zero-point offsets to all elements in order to force the high-Ia sequence to
pass through $[{\rm X}/{\rm Mg}]=0$ at $[{\rm Mg}/{\rm H}]=0$ prior to
performing the fits. These offsets are listed in Table 3 and are applied to
the bulge and resampled disk sequences in all further analysis. We note that
APOGEE does already apply zero point offsets to force stars with solar [M/H]
in the solar neighborhood to have a mean [X/M]=0 (J20), but the $\log(g)$
distribution of that sample is different from ours. (See GJW Figure 10 for an
example of two-process fits with and without zero-point offsets.)

Under the two process model, elements solely produced by CCSN should have no
separation between their high-Ia and low-Ia median sequences and should be
best fit by $R_{\rm Ia}^{X}\approx 0$. As the SNIa contribution to an element
increases, the sequence separation should also increase, driving the $R_{\rm
Ia}^{X}$ value up. Elements with $R_{\rm Ia}^{X}\approx 1$ are produced
equally by CCSN and SNIa in stars of solar abundance, and SNIa dominate the
production of those with $R_{\rm Ia}^{X}>1$. The $R_{\rm Ia}^{X}$ value can be
converted to a CCSN fraction by the equation

$f_{\rm cc}=\frac{1}{(1+R_{\rm Ia}^{X})}\,.$ (6)

Elements with $\alpha_{\rm cc}\approx\alpha_{\rm Ia}\approx 0$ and $R_{\rm
Ia}^{X}\approx 1$ will have median $[{\rm X}/{\rm Mg}]$ trends that follow the
corresponding $[{\rm Fe}/{\rm Mg}]$ sequences. As $\alpha_{\rm cc}$ increases,
the slope of both sequences should also increase, with positive/negative
$\alpha_{\rm cc}$ representing a positive/negative metallicity dependence. The
$\alpha_{\rm Ia}$ parameter will further change the slope of the high-Ia
sequence relative to the low-Ia sequence.

Figure 8 shows the two-process model fits to the resampled disk’s high-Ia and
low-Ia medians (including an offset). The $R_{\rm Ia}^{X}$, $\alpha_{\rm cc}$,
and $\alpha_{\rm Ia}$ values for each element as well as the zero-point offset
applied prior to the fit are listed in the respective panels and in Table 3.
The two-process model well describes the high-Ia and low-Ia trends of the
resampled disk down to $[{\rm Mg}/{\rm H}]=-0.7$ for most elements. However,
we see that the Al, P, and Cr high-Ia medians are poorly fit at the high
$[{\rm Mg}/{\rm H}]$ end and that the two-process model is unable to reproduce
the non-linear metallicity dependence of the [Cu/Mg] and [Ce/Mg] trends. If we
allowed arbitrary metallicity dependencies, then the two-process model would
be able to fit the observed median sequences by construction, so imperfect
fits are a consequence of the power-law restriction. Comparisons like Figure 8
are not in themselves a strong test of the two-process model’s underlying
assumptions; rather, these fits allow us to convert the observed sequences
into physically meaningful quantities given those assumptions. Better tests of
the model’s validity come from predicting the abundance trends of other
stellar populations or from predicting star-by-star deviations from median
trends, which we will examine briefly in Section 4.2 and in more detail in
future work.

Table 3: Zero-point offsets and the best fit two-process model parameters for each element as fit to the median high-Ia and low-Ia points of the resampled disk. $f_{\rm cc}$ denotes the fractional CCSN contribution, as defined in Equation (6). [X/Mg] | Offset | $R_{\rm Ia}^{X}$ | $\alpha_{\rm cc}$ | $\alpha_{\rm Ia}$ | $f_{\rm cc}$
---|---|---|---|---|---
Fe | $0.060$ | $0.99$ | $0.00$ | $0.00$ | $0.503$
O | $0.003$ | $0.14$ | $0.00$ | $-0.02$ | $0.877$
Na | $0.044$ | $0.81$ | $-0.30$ | $0.70$ | $0.552$
Al | $0.049$ | $0.22$ | $0.22$ | $0.06$ | $0.820$
Si | $0.046$ | $0.24$ | $-0.02$ | $0.06$ | $0.806$
P | $-0.023$ | $0.45$ | $0.10$ | $-0.38$ | $0.690$
S | $0.034$ | $0.07$ | $-0.17$ | $0.42$ | $0.935$
K | $0.014$ | $0.08$ | $0.03$ | $0.70$ | $0.926$
Ca | $0.057$ | $0.42$ | $-0.10$ | $0.03$ | $0.704$
V | $0.034$ | $0.49$ | $-0.15$ | $0.52$ | $0.671$
Cr | $0.039$ | $1.30$ | $-0.15$ | $0.12$ | $0.435$
Mn | $-0.033$ | $2.10$ | $0.17$ | $0.23$ | $0.323$
Co | $0.023$ | $0.46$ | $0.01$ | $0.13$ | $0.685$
Ni | $0.059$ | $0.64$ | $-0.02$ | $0.10$ | $0.610$
Cu | $-0.035$ | $0.61$ | $-0.26$ | $0.49$ | $0.621$
Ce | $0.128$ | $1.83$ | $-0.18$ | $-0.40$ | $0.353$
Figure 8: Two-process model fits to the resampled disk median trends.
Orange/pink circles/squares represent the high/low-Ia population medians
including offsets. Solid lines are the best fit two-process sequences. Best
fit sequences only extend to a lower bound of $[{\rm Mg}/{\rm H}]=-0.7$ as
fits were computed on median points with $[{\rm Mg}/{\rm H}]\geq-0.7$.
Elemental best fit two-process values and zero-point offsets are given in each
cell.

W19 report two-process fits to the APOGEE DR14 disk for all elements included
here except Cu and Ce (new to DR16). They discuss each element, its two-
process model fit and the nucleosynthetic implications, and they compare the
derived $f_{\rm cc}$ value to theoretical results from Rybizki et al. (2017).
We find very similar $f_{\rm cc}$ values (a difference of $\leq 10\%$) for Fe
(by construction), and for O, Na, Si, S, Ca, Cr, Mn, and Ni, so we do not go
into detail about the two-process model implications in this paper. We discuss
the two-process model implications for Cu and Ce in Appendix B.

We do, however, see differences between our derived two-process fits and those
of W19 for Al, P, K, V, and Co . In all of these cases, we have repeated our
fits using the DR16 abundances but the disk $\log(g)$ and geometry cuts used
by W19. The differences persist, indicating that they arise primarily from
changes in the ASPCAP abundance pipeline, not from differences in the
$\log(g)$ distribution or location of the sample. For Al, the higher $f_{\rm
cc}=0.97$ found by W19 (vs. 0.82 here) is in better agreement with theoretical
expectations that Al is an almost pure CCSN element. Conversely, for K the
higher $f_{\rm cc}=0.94$ (vs. 0.80 in W19) agrees better with the expectation
that CCSN dominate its production. For Co we find $f_{\rm cc}=0.69$ vs.
$f_{\rm cc}=0.80$ in W19, in better agreement with the GJW findings from GALAH
but lower than predicted by the yield models of Rybizki et al. (2017). For P
and V, smaller separation of $[{\rm X}/{\rm Mg}]$ values in DR16 leads to
higher inferred values of $f_{\rm cc}$. J20 stress that P and V are two of the
least precise and potentially least accurate abundances in DR16. We do not
have simple explanations for these differences or a clear indication of which
abundances are more accurate for a given element, DR14 or DR16. The enormous
sample size and generally high abundance precision in APOGEE allow us to see
artifacts that would be hidden in smaller samples or less precise data. The
artifacts discussed in Section 3.1 and the two-process parameter differences
discussed here imply that systematic uncertainties remain in the APOGEE
measurements at a level that is physically interesting.

###  4.2 Predicting Bulge Abundances

Figure 9: Predicted abundances for the bulge population given the resampled
disk’s two-process model parameters. For each bulge star, $[{\rm X}/{\rm Mg}]$
values are predicted based on the star’s measured $[{\rm Mg}/{\rm H}]$ and
$[{\rm Fe}/{\rm Mg}]$. Low-Ia/high-Ia stars are colored pink/orange
respectively. Lighter shades indicate extrapolated abundance predictions
($[{\rm Mg}/{\rm H}]<-0.7$). The bulge median observed high-Ia and low-Ia
trends, including offsets listed in the top left corner, are over-plotted in
dark orange circles and purple squares, respectively.

Using Equations (4) and (5) as well as the best-fit two-process model
parameters from Table 3, we can predict the full set of abundance ratios for
any star given only its $[{\rm Mg}/{\rm H}]$ and $[{\rm Fe}/{\rm Mg}]$ values.
If the disk and the bulge are enriched by the same nucleosynthetic processes,
then the two-process model parameters used to describe the disk should
accurately predict the bulge abundances. To test this, we calculate the APOGEE
abundance suite of every bulge star using the model parameters from the
resampled disk. These predicted distributions in $[{\rm X}/{\rm Mg}]$ vs.
$[{\rm Mg}/{\rm H}]$ space are shown in Figure 9. Median trends of the high-Ia
and low-Ia bulge populations (Figure 2) are overplotted for comparison,
including the zero-point offsets discussed above. As we only fit the two-
process model above $[{\rm Mg}/{\rm H}]=-0.7$, trends below this metallicity
are not included in our subsequent comparison and analysis but are included in
the plot for completeness.

The $[{\rm Fe}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ predicted abundances are
identical to those observed, by construction. All elements with low $R_{\rm
Ia}^{X}$ values have ‘tight’ predicted abundance distributions and little
spread. This is a direct result of the two-process model assumptions: elements
whose production is dominated by CCSN should trace Mg and have little to no
separation between the high-Ia and low-Ia sequences. We do not add spread
induced by observational errors to our predictions, so the width of our
predicted abundance distributions is smaller than that of the observed
distributions (Figure 2). The only source of scatter in the two-process
predictions comes from the scatter of $[{\rm Fe}/{\rm Mg}]$ at fixed $[{\rm
Mg}/{\rm H}]$, which is interpreted as star-to-star variations in the SNIa-to-
CCSN enrichment ratio $A_{\rm Ia}/A_{\rm cc}$. For elements with $R_{\rm
Ia}^{X}\approx 0$ ($f_{\rm cc}\approx 1$), the induced scatter is minimal
because SNIa do not contribute to this element in any case.

Figure 9 shows good agreement between the predicted abundances and observed
bulge median trends of most elements. The most noticeable exceptions are for
Na, Al, P, and Ce, particularly at super-solar metallicities. For Na, this
difference reflects the difference in the observed median sequences of the
resampled disk and bulge (Figure 7). For Al, P, and Ce, the power-law form of
the two-process model leads to over-predicting the disk trends themselves at
high $[{\rm Mg}/{\rm H}]$ (Figure 8), leading to over-prediction of the bulge
trends. The trends for many elements diverge from predictions below $[{\rm
Mg}/{\rm H}]=-0.7$ because the power-law extrapolation of the two-process
model becomes inaccurate.

Interestingly, for [Mn/Mg] the agreement between predicted and observed trends
in Figure 9 is better than the agreement of the observed medians in Figure 7.
This improvement arises at least partly because the median $[{\rm Fe}/{\rm
Mg}]$ values are slightly lower for the bulge than for the resampled disk.
Within the two-process model, this difference implies that bulge stars have
slightly lower SNIa contributions at a given $[{\rm Mg}/{\rm H}]$. Because Mn
has a large inferred $R_{\rm Ia}^{X}=2.1$, this difference translates into
lower predicted [Mn/Mg] for the bulge, producing better agreement. Given the
observational uncertainties, these improvements could be a fluke, but it could
be a sign that the two-process description is correctly capturing the impact
of subtle differences in the relative SNIa/CCSN contributions between the disk
and the bulge.

Figure 10: Median absolute difference of the of the observed bulge stars’
[X/Mg] (including offsets) and the predicted [X/Mg] values for high-Ia (light
orange/top) and low-Ia (pink/bottom) populations. Alongside, we plot the
median absolute difference of observed bulge stars’ $[{\rm X}/{\rm Mg}]$ and
the median $[{\rm X}/{\rm Mg}]$ value binned by $[{\rm Mg}/{\rm H}]$ on the
high-Ia (dark orange/top) or low-Ia (purple/bottom) populations. Only stars
with Mg/H $>$ -0.7 are included in the calculations.

To better quantify the similarities and differences between the observed and
predicted populations, we calculate the median of the absolute differences,

$\text{median}(|[{\rm X}/{\rm Mg}]_{\rm(obs)}-[{\rm X}/{\rm
Mg}]_{\rm{pred}}|),$ (7)

for each element, including the corrective offsets. In principle the two-
process model should be a better predictor of the elemental abundances than
the median value of stars on the same sequence with the same $[{\rm Mg}/{\rm
H}]$ because the model accounts for star-by-star scatter in $A_{\rm Ia}/A_{\rm
cc}$. To test this expectation, we compare the median absolute difference of
the observed and the predicted abundances to the median absolute difference of
the observed abundance and the median $[{\rm X}/{\rm Mg}]$ value for a given
$[{\rm Mg}/{\rm H}]$ bin,

$\text{median}(|[{\rm X}/{\rm Mg}]_{\rm{obs}}-[{\rm X}/{\rm
Mg}]_{\rm{med}}|).$ (8)

We plot both statistics for the high-Ia (top) and low-Ia (bottom) populations
of all elements in Figure 10. We only consider stars with $[{\rm Mg}/{\rm
H}]>-0.7$, as the two-process model was only fit for such stars.

To first approximation, Figure 10 shows similar median absolute difference
statistics for all elements regardless of whether the two-process model or the
observed median sequence is used to predict a star’s $[{\rm X}/{\rm Mg}]$.
This similarity suggests that most of the scatter in $[{\rm X}/{\rm Mg}]$
arises from observational errors, and the median absolute differences are
indeed largest for elements measured with relatively low precision, such as
Na, P, and K. For the high-Ia population, the two-process model predicts
[Al/Mg] and [Ca/Mg] worse than the corresponding median trend because it also
predicts the median trend itself poorly (see Figure 9). For the low-Ia
population, the two-process prediction is more accurate than the observed
median trend for Fe-peak elements, particularly for [Mn/Mg] and [Ni/Mg]. This
admittedly subtle difference indicates that star-by-star deviations for Fe-
peak elements track the SNIa contribution to $[{\rm Fe}/{\rm Mg}]$, as the
two-process model predicts. Stronger tests of the two-process model can be
obtained by focusing on subsets of disk stars with the highest SNR spectra, so
that observational contributions to scatter all minimized. We reserve such an
investigation for future work.

##  5 IMF constraints

Recent works by Ballero et al. (2007), Grieco et al. (2012) and Grieco et al.
(2015) find that the bulge (and galactic center) is better fit by a chemical
evolution model with a more top heavy IMF than that which fits the solar
neighborhood. Grieco et al. (2012) employ chemical evolution modeling to
reproduce the metal rich and metal poor bulge populations. They find that a
model with a flatter IMF, such as the Salpeter (1955), fits the MDF and $[{\rm
Mg/Fe}]$ vs. $[{\rm Fe/H}]$ abundance distribution of blue red clump stars
from Hill et al. (2011).

Our measurements allow an entirely different test of IMF differences between
the bulge and the disk because the relative amounts of different elements
produced in a CCSN depend on the mass of the progenitor. Changing the high
mass slope of the IMF will increase or decrease the number of massive stars
and thus the ratio of nucleosynthetic products from the IMF-integrated CCSN
population. If a steeper or shallower IMF induces larger abundance differences
than those observed, we can exclude that possible IMF in the bulge.

We adopt a Kroupa IMF (Kroupa, 2001) as our standard for the disk. This three
slope IMF is shown in Equation (9), with $a_{3}=-2.3$. To produce a top heavy
or light IMF, we change the high mass slope ($a_{3}$) to -2.0 or -2.6,
respectively. Our IMFs are all of the form

$\frac{dN}{dM}=\begin{cases}A_{\rm K}M^{-0.3},&M<0.08\\\ B_{\rm
K}M^{-1.3},&0.08\leq M<0.5\\\ C_{\rm K}M^{a_{3}},&M\geq 0.5\end{cases}$ (9)

with $A_{\rm K},B_{\rm K},$ and $C_{\rm K}$ being the appropriate
multiplicative constants.

Figure 11: Kroupa IMF (black) and edited Kroupa IMFs with high mass slope of
-2.0 (cyan) and -2.6 (pink). We compare the resulting yields of populations
integrated under these three IMFs.

To derive the CCSN abundance yields from our three IMFs, we employ VICE, the
Versatile Integrator for Chemical Evolution, and its yield integration
function (Johnson & Weinberg, 2019). In our analysis we take net CCSN yields
from Chieffi & Limongi (2013, hereafter CL13) at solar metallicity. While CL13
explode all stars to $120\rm M_{\odot}$, we assume for our modeling that
progenitors above 30$\rm M_{\odot}$ collapse to black holes instead of
exploding as CCSN. This sharp cutoff is a simplification of an “explodability
landscape” that is probably much more complex (Pejcha & Thompson, 2015;
Sukhbold et al., 2016). The VICE integrator returns yields in $\rm M_{\odot}$
of element X per $\rm M_{\odot}$ of stars formed. We convert to bracket
notation using solar abundances from Asplund et al. (2009). We note that
APOGEE take solar abundance from Grevesse et al. (2007), but we do not expect
this to impact our results.

The resulting yields and [X/Mg] integrated with all three IMFs can be found in
Table 4 of Appendix C. This Appendix includes a more detailed exploration of
the CL13 yields such as their sensitivity to metallicity and to the choice of
upper mass cutoff for explosion. We also include a similar calculation for the
Limongi & Chieffi (2018, hereafter LC18) yields, which explode stars to 25$\rm
M_{\odot}$ with different explosion criteria. We note that the CL13 yields
force an ejection of $0.1\rm M_{\odot}$ of 56Ni from all stars.

Unfortunately, while predicted yields are mass-dependent, that dependence is
fairly similar for all of the elements examined here. As a result, the
theoretical [X/Mg] abundances for the Kroupa and $a_{3}=-2.0$ or $a_{3}=-2.6$
IMFs differ by small amounts, 0.05 dex for [Ni/Mg] and 0.01-0.02 dex for other
elements. These IMF-induced abundance differences are plotted in Figure 12.
Taking Kroupa as the standard, we show the $\Delta[{\rm X}/{\rm Mg}]$ values
for 12 elements, where

$\Delta[{\rm X}/{\rm Mg}]=[{\rm X}/{\rm Mg}]_{\rm Kroupa}-[{\rm X}/{\rm
Mg}]_{a_{3}=-2.0}$ (10)

or

$\Delta[{\rm X}/{\rm Mg}]=[{\rm X}/{\rm Mg}]_{\rm Kroupa}-[{\rm X}/{\rm
Mg}]_{a_{3}=-2.6}.$ (11)

Although these values are plotted as horizontal lines, they are calculated at
solar $[{\rm Fe/H}]$. We stress that this calculation only investigates star
formation changes that would affect the CCSN contribution to each element. We
do not account for any differences in SNIa nucleosynthesis between the bulge
and disk at fixed $[{\rm Mg}/{\rm H}]$ and $[{\rm Fe}/{\rm Mg}]$. A difference
in SNIa-produced $[{\rm X}/{\rm Fe}]$ ratios could compensate for, or
masquerade as, an IMF-induced change to CCSN contributions.

For comparison, Figure12 also shows the observed differences in the bulge and
resampled disk low-Ia median points,

$\Delta[{\rm X}/{\rm Mg}]=[{\rm X}/{\rm Mg}]_{\rm Disk}-[{\rm X}/{\rm
Mg}]_{\rm Bulge}.$ (12)

We compare only the low-Ia median abundances as the bulge low-Ia sequence is
more well populated than the high-Ia sequence, and we wish to minimize the
SNIa contribution for this test in any case. We have already seen in Figure 7
that disk and bulge median sequences agree well, and in Figure 12 we see that
the differences near solar metallicity are typically 0.02 dex or below. These
are comparable in magnitude to the predicted IMF effects, but some elements
agree better with the steeper IMF and some with the shallower IMF, with no
clear pattern to separate these groups. We are therefore inclined to ascribe
these differences to residual systematics in matching disk and bulge abundance
scales at the 0.02-dex level.

Figure 12: Differences between the bulge and resampled disk low-Ia median
sequences (purple squares) and theoretical abundance changes between the
standard Kroupa IMF ($a_{3}=-2.3$) and a Kroupa IMF with an altered high mass
slope. The inducted elemental differences between the Kroupa and $a_{3}=-2.0$
IMFs are shown in the cyan dashed lines and those for $a_{3}=-2.6$ in the
magenta dash-dot lines. The theoretical and observed abundances should be
compared near solar $[{\rm Mg}/{\rm H}]$, as indicated by the grey band.

Two elements deserve special mention. [Mn/Mg] ratios show the largest
differences between disk and bulge, $\sim$0.05 dex, much larger than predicted
by an IMF change. If real, this difference likely arises from different SNIa
enrichment in the bulge, perhaps connected to the strong apparent metallicity
dependence of Mn yields. [Ni/Mg] ratios, on the other hand, show 0.01-dex
agreement near solar metallicity, much closer than the 0.04-dex changes
predicted by the IMF models. At face value this comparison implies a stringent
limit on IMF differences, with $|\Delta a_{3}|\lesssim 0.1$. However, Ni
predictions may be sensitive to the criteria used to define the mass cut in
supernova models. Ideally we would like to base an IMF test on relative
abundances of Mg, O, Si, and Ca, which are well measured and are predicted
(and empirically inferred) to come predominantly from CCSN with little
metallicity dependence. If abundance scales of different populations can be
reliably cross-calibrated at the 0.01 dex level, then tests of IMF variations
at the $|\Delta a_{3}|\approx 0.2-0.3$ level can be achieved.

##  6 Summary

Using APOGEE DR16+ data, we present the $[{\rm X}/{\rm Mg}]$ vs. $[{\rm
Mg}/{\rm H}]$ and $[{\rm X}/{\rm Fe}]$ vs. $[{\rm Fe/H}]$ abundance trends of
the Milky Way bulge ($R<3\rm\;kpc$) for Fe, O, Na, Al, Si, P, S, K, Ca, V, Cr,
Mn, Co, Ni, Cu, and Ce, the latter two being new to DR16. The addition of the
LCO instrument in APOGEE DR16 provides us with observations of stars in the
southern hemisphere and thus a more complete look at the bulge than previous
APOGEE data releases. After a series of quality cuts, we are left with 11,382
bulge stars with median SNR of 144.

Like the disk, the bulge population shows a bimodal $[{\rm Mg/Fe}]$ vs. $[{\rm
Fe/H}]$ distribution. The DR16+ data reveal that the two populations are
distinct and do not form a continuous sequence. As in W19, we divide the
sample into high-Ia and low-Ia populations in this plane. We see median $[{\rm
Mg/Fe}]$ vs. $[{\rm Fe/H}]$ trends that match those of the W19 disk
definition, though slightly offset. As in Hayden et al. (2015) and W19, we
find that the inner Galaxy contains more low-Ia stars than high-Ia stars, with
the stellar distribution extending to $[{\rm Fe/H}]$ of -1.5.

In Section 3 we present the $[{\rm X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ and
the $[{\rm X}/{\rm Fe}]$ vs. $[{\rm Fe/H}]$ abundance distributions and $[{\rm
X}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$ median trends for the APOGEE elements.
Some data systematics can be seen, including a finger-like feature in low-Ia
O, Ca, and Si in $[{\rm X}/{\rm Fe}]$ space; a bimodal distribution of high-Ia
Cr stars; and banding/clumping in the Al, P, and V abundance distributions. We
only exclude the lower [Cr/Mg] high-Ia stars in our analysis, but we explore
the impact of these systematics in Appendix A.

The main goal of this paper is to determine the similarity or dissimilarity
between the median high-Ia and low-Ia $[{\rm X}/{\rm Mg}]$ trends of the
Galactic bulge and disk. However, upon a comparison of our bulge sample with
the disk as defined by W19, we find systematic abundance differences which
correlate with the stellar $\log(g)$ distribution. We therefore resample the
APOGEE DR16+ disk to select a subset of stars that reproduce the $\log(g)$
distribution of the bulge. We divide the resampled disk stars into high-Ia and
low-Ia populations.

The similarities between the median high-Ia and low-Ia trends of the bulge and
disk shown in Figure 7 are striking. All elemental trends agree to within 0.1
dex and most within 0.05 dex; near solar $[{\rm Mg}/{\rm H}]$ the agreement of
the low-Ia median trends is typically 0.01-0.02 dex (Figure 12). The [Na/Mg]
and [Mn/Mg] vs. [Mg/H] bulge high-Ia median trends rise more steeply than
those of the disk. The low-Ia [Mn/Mg] median trend is about 0.05-dex higher in
the bulge than the disk. The Co bulge medians are offset to lower [Co/Mg] than
the resampled disk at low $[{\rm Mg}/{\rm H}]$ These differences could reflect
differences in the chemical evolution of the disk and bulge, but they are
small enough that they could still reflect imperfect relative calibration of
bulge and disk abundance scales.

This close agreement of median $[{\rm X}/{\rm Mg}]$ trends between the bulge
and disk extends the universality of these trends as a function of $R$ and
$|Z|$ within the disk found by W19. It demonstrates that although the
distribution of stars in $[{\rm Mg}/{\rm H}]$ and $[{\rm Fe}/{\rm Mg}]$
depends strongly on location in the Galaxy, the physics that determines $[{\rm
X}/{\rm Mg}]$ ratios for a given $[{\rm Mg}/{\rm H}]$ and $[{\rm Fe}/{\rm
Mg}]$ does not. To quantify this point we compare the observed bulge abundance
ratios with those predicted by W19’s empirical two-process model fit to the
resampled disk. The two-process model describes the abundances of a star or
population of stars as the sum of prompt (CCSN) and delayed (SNIa) components,
with each of these contributions to (X/Mg) having a power law dependence.
Using the parameters derived as the best fit to the resampled disk’s median
abundances for each element ($R_{\rm Ia}^{X}$, $\alpha_{\rm cc}$, and
$\alpha_{\rm Ia}$), we predict the bulge elemental abundances suite based only
on the observed [Fe/Mg] and $[{\rm Mg}/{\rm H}]$ of the bulge stars. Figure 9
shows that the predicted abundances accurately trace the observed [X/Mg]
median trends. Elements for which the predicted and observed trends diverge
show poor two-process model fits (e.g. Al) and/or real differences in the
median trends of the bulge and resampled disk (e.g. Na, Mn, Co). Star-by-star
deviations from median trends appear to be dominated by observational errors,
but the two-process model explains a small fraction of the scatter for some
iron peak elements.

As prior works have supported a bulge IMF with a shallower high mass slope
than the solar neighbourhood (Grieco et al., 2012), we test if the bulge and
sampled disk’s similar abundance tracks can constrain the Galactic IMF
variability. Using the yield integration function of VICE (Johnson & Weinberg,
2019), we calculate the theoretical [X/Mg] abundance at solar $[{\rm Fe/H}]$
for the CL13 yields with three IMFs: Kroupa with high mass slopes of -2.0,
-2.3, or -2.6. Overall, we find small predicted [X/Mg] changes with a varying
IMF, $<0.05$ dex for all elements. If the bulge and disk have different IMFs,
we would expect the median abundance differences of the bulge and resampled
disk to correlate with the theoretical abundance changes induced by an altered
IMF. However, we do not see consistent agreement between the observed and
theoretical $\Delta[{\rm X}/{\rm Mg}]$ trends for either IMF variation (Figure
12). The observed agreement of median trends argues against IMF slope
differences larger than about 0.3. At face value the agreement for [Ni/Mg]
implies a more stringent limit, but the predicted sensitivity of Ni yields to
the IMF may not be robust. The 0.05 dex difference in [Mn/Mg] trends could be
a sign of differences in SNIa enrichment between bulge and disk.

Our principal finding is that the bulge and resampled disk have similar [X/Mg]
vs. $[{\rm Mg}/{\rm H}]$ median abundance tracks, extending the conclusions of
Hayden et al. (2015) and W19 to the inner Galaxy. The universality of the
median [X/Mg] abundance trends suggests that they are not sensitive to most
aspects of chemical evolution and instead depend on the IMF-averaged
nucleosynthetic yields, which appear consistent throughout the Galaxy. As we
obtain additional high SNR observations of the Milky Way bulge, our
understanding of the Galactic nucleosynthetic processes will grow. In future
works with APOGEE and other data sets, we aim to set more stringent empirical
constraints on the astrophysics that governs the creation of the elements
throughout the Galaxy and its closest neighbors.

## Acknowledgements

We thank APOGEE collaborators Julie Imig, Jianhui Lian, Steve Majewski,
Matthew Shetrone, and Gail Zasowski, who were in attendance at the Autumn 2019
OSU workshop and Autumn 2019 bulge workshop and provided many insights on
APOGEE abundances and findings for the bulge. We thank the Center for
Cosmology and AstroParticle Physics for helping to organize the Autumn 2019
OSU workshop.

This work was supported by NSF grant AST-1909841. D.A.G.H. acknowledges
support from the State Research Agency (AEI) of the Spanish Ministry of
Science, Innovation and Universities (MCIU), and the European Regional
Development Fund (FEDER) under grant AYA2017-88254-P. S.H. is supported by an
NSF Astronomy and Astrophysics Postdoctoral Fellowship under award
AST-1801940. H.J. acknowledges support from the Crafoord Foundation,
Stiftelsen Olle Engkvist Byggmästare, and Ruth och Nils-Erik Stenbäcks
stiftelse. D.M.N. acknowledges support from NASA under award Number
80NSSC19K0589. A.R-L acknowledges financial support provided in Chile by
Comisión Nacional de Investigación Científica y Tecnológica (CONICYT) through
the FONDECYT project 1170476 and by the QUIMAL project 130001.

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

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

This research made use of Astropy,666http://www.astropy.org a community-
developed core Python package for Astronomy (Astropy Collaboration et al.,
2013, 2018).

##  Appendix A The effect of data systematics on median trends

In Section 3, we present the median high-Ia and low-Ia abundance trends for
APOGEE elements and discuss some of the systematics and artifacts afflicting
the data in Section 3.1. In the $[{\rm X}/{\rm Fe}]$ vs. $[{\rm Fe/H}]$ plots
(Figure 3) we see a finger-like feature in O, Ca, and Si; bimodality in the Cr
high-Ia stars; banding in Al and P; and a clump in V. In our data analysis we
exclude some stars to mitigate the Cr feature and proceed with the other
artifacts included. Here, we take a closer look these anomalies to see how the
median trends change with their inclusion and exclusion.

Figure 13 presents three versions of the [Cr/Mg] median high-Ia trend. The
first panel shows the observed distribution with no cuts. The high-Ia median
trend follows the Cr-poor stars at low $[{\rm Mg}/{\rm H}]$ but jumps to solar
[Cr/Mg] around $[{\rm Mg}/{\rm H}]\sim 0.3$. The bimodality in the Cr
abundance obviously produces a skewed median trend. In our analysis, we remove
1421 stars with $0<[{\rm Mg}/{\rm H}]<0.75$ and $-0.3<[\rm Cr/\rm Mg]<-0.1$.
This exclusion and the resulting median high-Ia trend are shown in the middle
panel of Figure 13. Alternatively, we could have excluded the 1544 Cr-rich
stars with $0<[{\rm Mg}/{\rm H}]<0.75$ and $-0.1<[\rm Cr/\rm Mg]<0.2$, as
shown in the right panel. As expected, the high-Ia median trend changes
dramatically between the two exclusions. The higher $\log(g)$ disk better
resembles the the Cr-rich high-Ia stars and previous works, so we use this
subset in our analysis.

Figure 13: Bulge [Cr/Mg] vs. [Mg/H] distribution with high-Ia stars in orange
and low-Ia stars in purple. The medians of the high/low-Ia populations are
over-plotted in dark orange/purple circles/squares, where we bin by 0.1 dex
and require $>20$ stars per bin. Left: Median trends for the full population.
Middle: Median trends excluding the Cr poor high-Ia stars inside the dashed
box (exclusion employed in the main body of the paper). Right: Median trends
excluding the Cr rich high-Ia stars inside the dashed box.

While we mitigate the Cr systematic with exclusions in our data analysis, the
$\alpha$ finger and other banding/clumping are left in. Figure 14 and 15 show
examples of how the [Ca/Mg] and [Al/Mg] vs. [Mg/H] trends (respectively) would
change if we removed these features. For Ca, we isolate the finger stars in
[Ca/Fe] vs. [Fe/H] space, where the finger is much easier to identify. We
define the finger as low-Ia stars with $-0.3<[{\rm Fe/H}]<0.5$ and $0.15<[\rm
Cr/\rm Fe]<0.3$. Median trends including and excluding these stars are plotted
in the right panel of Figure 14. The low-Ia median trend changes
insignificantly after the exclusion of 368 finger stars. We find that the
inclusion of the bulge finger stars in our O, Si, and Ca analysis does not
affect our resulting median trends or conclusions.

Figure 14: Left: Bulge [Ca/Fe] vs. [Fe/H] distribution with high-Ia stars in
orange and low-Ia stars in purple. The over-plotted dashed box identifies the
finger stars. Right: [Ca/Mg] vs. [Mg/H] distribution. The medians of the full
high/low-Ia populations are over-plotted in dark orange/purple
circles/squares, where we bin by 0.1 dex and require $>$20 stars per bin. The
black squares show the resulting low-Ia median trend when we exclude the
finger stars (shown in grey).

The low abundance trends (J20) seen in Al and P are much harder to isolate as
they blend into the main abundance tracks. The left panel of Figure 15 plots
[Al/Fe] vs. [Fe/H], where the banding is more pronounced than in Mg space. We
define two exclusion regions. The first (1) is [Al/Fe]$<0.05$ and [Fe/H]$<0.1$
for low-Ia stars and the second (2) is $-0.1<[\rm Al/\rm Fe]<-0.3$ and
$-0.2<[{\rm Fe/H}]<0.7$ for high-Ia stars. Both boxes are drawn on the figure
and include 2700 and 206 stars, respectively. Median trends including and
excluding these stars are shown in the right panel. The high-Ia trend is not
noticeably affected by the small exclusion, but the low-Ia trend deviates from
the full population median at low $[{\rm Mg}/{\rm H}]$. This deviation should
be expected, as we cut out many of the low $[{\rm Mg}/{\rm H}]$ stars. While
the Al ‘tail’ may be a systematic, the median trends above $[{\rm Mg}/{\rm
H}]=-0.25$ are robust. As the bulge and disk median trends agree at this
metallicity, our conclusions about the similarity of the bulge and disk
abundance ratios hold.

Figure 15: Same as Figure 14, but for Al. In the left panel, exclusion region
1 (2) applies to the low-Ia (high-Ia) stars. All excluded stars are shown in
grey in the right panel.

We also explore the temperature correlation of median abundance trends. As
noted in Section 3, APOGEE flags low temperature Na, K, and Mn stars due to
reduction systematics that correlate with temperature. To see if this affects
the median high-Ia and low-Ia trends we compare the full sample medians to
those of smaller temperature divisions. Figure 16 plots [Na/Mg] vs. [Mg/H]
distributions and medians for a sample with no temperature cuts, stars with
$3800\rm\;K\leq T_{\rm eff}<4200\rm\;K$ (2800), and stars with $4200\rm\;K\leq
T_{\rm eff}<4600\rm\;K$ (931). The majority of stars lie within the lower
temperature range, so the medians of this sample track the full population.
Both median sequences of the higher temperature sample lie below those of the
full population, though the differences are $<-0.05$ dex. The median high-Ia
and low-Ia Na trends appear to have some small correlation with temperature.

Figure 16: Bulge [Na/Mg] vs. [Mg/H] distribution with high-Ia stars in light
orange and low-Ia stars in light purple. The medians of the full high/low-Ia
populations are over-plotted in dark orange/purple circles/squares, where we
bin by 0.1 dex and require $>20$ stars per bin. Left: Full sample (3851 stars)
These medians remain the same in each panel. Center: Medians of stars with
$3800\rm\;K\leq T_{\rm eff}<4200\rm\;K$ (2800 stars). Medians of the full
population are in orange/purple and of the temperature cut sample are in
black. The background light orange/purple points show the sub-sample of stars
in the temperature range. Right: Same as middle panel, but for stars with
$4200\rm\;K\leq T_{\rm eff}<4600\rm\;K$ (931 stars) in black.

We repeat this analysis with all other elements using the same temperature
divisions. In most cases, the lower temperature bin dominates the sample and
agrees well with the full population medians. We see no temperature dependent
changes in the $[{\rm X}/{\rm Mg}]$ vs $[{\rm Mg}/{\rm H}]$ median trend
location for Ca, K, Mn, Ni, Cu, and Ce. In the higher temperature bin, we find
that the low-Ia trends of Al, O, and Si sit slightly above the full population
medians and those of Co and Cr sit below (in many cases there are too few
high-Ia median points to draw strong conclusions). Si displays one of the
largest low-Ia median trend changes in the higher temperature bin ($\sim 0.1$
dex), shown in Figure 17. Some elemental abundance variations with
temperature, such as Si, are on the scale of or larger than the difference
between the bulge and disk medians.

Figure 17: Same as Figure 16, but for Si. There are 5254 stars in the lower
temperature range and 2208 in the higher temperature range.

While we match $\log(g)$ distributions between the bulge and disk samples, we
do not match $T_{\rm eff}$ distributions. To verify that our conclusions hold,
we check the median high-Ia and low-Ia disk trends’ variation with
temperature. We find similar deviations in the disk as in the bulge, e.g. the
median Si high-Ia and low-Ia trends of stars with $4200\rm\;K\leq T_{\rm
eff}<4600\rm\;K$ in the disk lie above those of the full population by a
similar amount to that seen in Figure 17. While there is uncertainty in the
exact location of the abundance trends, we have confidence that the bulge and
the disk trends agree.

##  Appendix B Cu and Ce abundances

While the main body of this paper focuses on the abundance trends in the Milky
Way bulge, in this appendix we will take a closer look at the disk Cu and Ce
abundances, continuing the work of W19 and GJW for the two new APOGEE
elements. In Figure 18 we plot the Cu and Ce abundances for all stars using
the W19 disk sample cuts ($1<\log(g)<2$, $3700\rm\;K<T_{\rm eff}<4600\rm\;K$,
$3\rm\;kpc<R<15\rm\;kpc$, $|Z|<2\rm\;kpc$) with median high-Ia and low-Ia
trends (calculated as in Section 3). Over plotted are the GALAH median high-Ia
and low-Ia trends for Cu and La from GJW.

APOGEE and GALAH Cu abundances agree reasonably well above solar $[{\rm
Mg}/{\rm H}]$. Here, both surveys’ [Cu/Mg] median trends show large sequence
separation and positive slopes. The GALAH medians are significantly more
inclined, especially when comparing the high-Ia sequences. As noted in Section
3.4, the APOGEE Cu abundances turn upwards towards higher [Cu/Mg] at low
[Mg/H]. We do not trust the measurements for low $[{\rm Mg}/{\rm H}]$ stars.

As GALAH does not report Ce abundances, we choose to compare Ce to La, the
closest observed element on the periodic table with similar neutron capture
origins. Both elements show the expected non-linear trends. In Ce and La we
see a clear peak in the high-Ia median trends near solar. The same is seen in
the low-Ia medians, though the Ce peak is smaller and offset to higher $[{\rm
Mg}/{\rm H}]$ (we ignore the upturned low-Ia tail at low $[{\rm Mg}/{\rm H}]$
as there are fewer stars here). As explained in GJW, the trends rise with
$[{\rm Mg}/{\rm H}]$ in the low metallicity regime because of the increasing
numbers of Fe seeds and then decline with $[{\rm Mg}/{\rm H}]$ in the high
metallicity regime because of the decreasing neutron-to-seed ratio (Gallino et
al., 1998).

Figure 18: APOGEE [Cu/Mg] (left) and [Ce/Mg] (right) vs. [Mg/H] abundances for
the W19 disk, with high-Ia stars in light orange and low-Ia stars in purple.
The medians of the high-Ia and low-Ia populations are over plotted in dark
orange circles and dark purple squares, respectively. GALAH high-Ia (black
circles) and low-Ia (black squares) medians from GJW are also included for
[Cu/Mg] (left) and [La/Mg] (right) (GJW).

We fit both the [Cu/Mg] and [Ce/Mg] disk median trends with the two-process
model, as outlined in Section 4 (Figure 19). As both elemental trends are non-
linear, the two-process model does not reproduce observations at the low
metallicity end. However, it does give us a sense of the sequence separation
at $[{\rm Mg}/{\rm H}]=0$. GJW find Cu to have an $R_{\rm Ia}^{X}=0.71$,
$\alpha_{\rm cc}=0.56$, and $\alpha_{\rm Ia}=-0.40$. The $\alpha_{\rm cc}$ and
$\alpha_{\rm Ia}$ values found for APOGEE differ drastically (likely due to
the upturn at low metallicity) but the $R_{\rm Ia}^{X}$ value of 0.66 is in
good agreement with GALAH. APOGEE’s $R_{\rm Ia}^{X}$ for Ce and GALAH’s La
differ more, at 1.59 and 2.31, respectively. The $\alpha_{\rm cc}$ and
$\alpha_{\rm Ia}$ terms hold less meaning, as the abundance tracks do not
follow a power law dependence.

In the right hand panel of Figure 19, we plot the $f_{\rm cc}$ values
corresponding to all four $R_{\rm Ia}^{X}$ measurements. The GALAH and APOGEE
points for Cu overlap around $f_{\rm cc}=60\%$, showing strong agreement that
Cu has a large delayed component. As detailed in GJW, this component may be
due to AGB stars, as SNIa models do not produce substantial amounts of Cu. For
Ce and La, the $f_{\rm cc}$ value is better interpreted as a fractional
$r$-process component ($f_{r}$). APOGEE and GALAH find Ce and La,
respectively, to have around 30-40% $r$-process contribution. We note that our
two-process models are scaled to CCSN and SNIa, so this estimated percentage
could change if the $s$-process and SNIa enrichment have different delay
times. The $f_{r}$ values for our neutron capture elements agree reasonably
well with theory, which predicts both to be $s$-process dominated. Arlandini
et al. (1999) find Ce to have 77% $s$-process contribution ($f_{s}$) and
Bisterzo et al. (2014) find 83%. See GJW for a broader discussion of La.
Figure 19 plots these theoretical $f_{r}$ value for both elements, taking
$f_{r}=1-f_{s}$.

Figure 19: Left: Same as Figure 8 but for Cu abundances in a sample with the
W19 disk cuts. Center: Same for Ce abundances. Right: Fractional CCSN
contribution to Cu, Ce, and La as calculated from $R_{\rm Ia}^{X}$ values for
fits to APOGEE (dark purple squares) and GALAH (light purple circles).
Theoretical $r$-process contributions from Arlandini et al. (1999)(A99, pink
diamonds) and Bisterzo et al. (2014)(B14, thin orange diamonds) are included
for Ce and La.

##  Appendix C CCSN yield exploration

In Section 5, we present the theoretical [X/Mg] abundances at solar [Fe/H], as
calculated from the net CL13 yields integrated with three different IMF high
mass slopes. In this appendix, we provide additional information about the net
CL13 and LC18 CCSN yields and their resulting [X/Mg] abundances. While LC18
cite multiple improvements to the CL13 approach, we note differences in their
explosion criterion: CL13 impose a mass cut that ejects 0.1 $\rm M_{\odot}$ of
56Ni for all stars while LC18 set interior [Ni/Fe] values and choose a mass
cut that ejects 0.07 $\rm M_{\odot}$ of 56Ni for stars with $M\leq 25\rm
M_{\odot}$. Both compute yield calculations for rotating stars, but we do not
explore rotation effects here.

The foundation of our earlier IMF investigation rests on the fact that CCSN
yields have a mass dependence. If the high mass IMF slope steepens/flattens,
then the Galaxy sees fewer/more high mass stars and their nucleosynthetic
products. The IMF integrated abundances shown above are thus dependent upon
the elemental yield’s mass dependence. In our case, we need to know how the
yields vary relative to Mg. Figure 20 shows the net Mg yield and the net X/Mg
yield for O, Al, Si, Ca, and Ni as a function of mass for CL13 and LC18. In
the CL13 yields we see that while Mg has an obvious mass dependence, the alpha
elements (O, Si, Ca) and light odd-$Z$ element (Al) track Mg and thus have
little mass dependence in X/Mg. This explains our findings in Figure 12, where
all four elements show small $\Delta[{\rm X}/{\rm Mg}]$ for a changing IMF.
The Ni/Mg yields steeply decrease with stellar mass, causing larger changes in
the [Ni/Mg] values when the IMF is varied.

Figure 20: Elemental net yields from CL13 (solid) and LC18 (dashed) as a
function of mass for $[{\rm Fe/H}]=0$ with no rotation. Top: Net explosive and
wind Mg yields in $\rm M_{\odot}$ (cyan circles) for published CL13 (solid
marker and line) and unpublished LC18 (empty circle and dashed line). Both are
reported after a full decay of unstable isotopes. Bottom: Net X/Mg yields for
O (light blue squares), Al (dark blue upwards triangle), Si (indigo downwards
triangles), Ca (purple stars), and Ni (pink diamonds) for both CL13 and LC18.

While CL13 report yields for all massive stars, LC18 publish their set R
yields, where stars above $25\rm M_{\odot}$ are assumed to collapse to a black
hole. Figure 20 plots their M set, which imposes the same mass cut as set R,
but explodes stars to 120 $\rm M_{\odot}$. These yields show very similar mass
dependence to CL13. While O/Mg and Al/Mg don’t vary much, Si, Ca, and Ni
yields decrease with respect to Mg between 8 and 30 $\rm M_{\odot}$. Ni/Mg
yields drop the most but are shallower than CL13, suggesting a smaller IMF
induced abundance change than that seen in Figure 12. We have only chosen to
plot a few select elements in these figures, but integrated net yields for all
APOGEE elements can be found in Tables 4 (CL13) and 5 (LC18).

The predicted abundance variability with changing IMF discussed in Section 5
employs net yields for $[{\rm Fe/H}]=0$ and explodes stars up to a birth mass
of $30\rm M_{\odot}$. These choices impact the scale of the theoretical
$\Delta[{\rm X}/{\rm Mg}]$ values. To explore the full range of possible
$\Delta[{\rm X}/{\rm Mg}]$, we repeat our calculations for different
explodability cutoffs and metallicities. Figure 21 plots the $\Delta[{\rm
X}/{\rm Mg}]$ values for the $a_{3}=-2.0$ and $a_{3}=-2.6$ cases as a we
change the cutoff mass for O, Al, and Ni for the CL13 yields. A given mass on
the $x$-axis indicates that all stars with progenitor masses $\leq M$ explode
and contribute to nucleosynthesis while those above do not. The $\Delta$
[O/Mg] and $\Delta$ [Al/Mg] values remain small ($<0.03$), with the $\Delta$
[O/Mg] increasing to a plateau around $\pm 0.025$ and $\Delta$ [Al/Mg] peaking
near $\pm 0.02$ for a cutoff mass of $\sim 60\rm M_{\odot}$. The $\Delta$
[Ni/Mg] values are much more dependent on the cutoff mass, growing to $\sim\pm
0.1$ as we include stars approaching $120\rm M_{\odot}$. This limit gives an
upper bound on the possible observable abundance changes induced by a changing
IMF high mass slope. We also investigate how changing the metallicity affects
the $\Delta[{\rm X}/{\rm Mg}]$ values for LC18 yield, which are calculated for
$[{\rm Fe/H}]=-3.0,-2.0,-1.0,0.0$. We find that the differences as a function
of metallicity are smaller than those caused by the explosion mass cutoff, so
we do not report them here. We will explore the impact of more complex
explodability landscapes (Pejcha & Thompson, 2015; Sukhbold et al., 2016) in
future work.

Figure 21: Theoretical $[{\rm X}/{\rm Mg}]$ abundance changes between the standard Kroupa IMF ($a_{3}=-2.3$) and a Kroupa IMF with an altered high mass slope as a function of changing explodability cutoff mass for O (left, squares), Al (center, triangles), and Ni (right, diamonds). Abundance changes for $a_{3}=-2.0$ are shown in cyan and for $a_{3}=-2.6$ are shown in magenta. Net yields from CL13 are used.. Table 4: VICE integrated yields (left) and [X/Mg] abundances (right) for net CL13 yields with [Fe/H] = 0.0, exploding stars up to 30 M⊙. All APOGEE elements but Ce are reported for all three IMFs investigated above.  | Yield ($\rm M_{\odot}/\rm M_{\odot formed}$) | [X/Mg]
---|---|---
| $a_{3}=-2.0$ | Kroupa | $a_{3}=-2.6$ | $a_{3}=-2.0$ | Kroupa | $a_{3}=-2.6$
Fe | 1.09E-03 | 6.85E-04 | 3.48E-04 | -0.015 | 0.017 | 0.049
O | 8.22E-03 | 4.66E-03 | 2.15E-03 | 0.214 | 0.204 | 0.193
Na | 5.47E-05 | 3.10E-05 | 1.43E-05 | 0.329 | 0.318 | 0.307
Al | 7.86E-05 | 4.50E-05 | 2.09E-05 | 0.208 | 0.201 | 0.195
Si | 1.09E-03 | 6.51E-04 | 3.17E-04 | 0.273 | 0.285 | 0.297
P | 5.59E-06 | 3.29E-06 | 1.58E-06 | 0.039 | 0.045 | 0.052
S | 4.12E-04 | 2.47E-04 | 1.20E-04 | 0.181 | 0.194 | 0.208
K | 4.78E-07 | 2.81E-07 | 1.35E-07 | -0.750 | -0.744 | -0.738
Ca | 5.30E-05 | 3.19E-05 | 1.56E-05 | -0.025 | -0.010 | 0.004
V | 1.94E-07 | 1.18E-07 | 5.89E-08 | -0.157 | -0.136 | -0.113
Cr | 1.47E-05 | 8.93E-06 | 4.42E-06 | 0.005 | 0.023 | 0.044
Mn | 8.66E-06 | 5.29E-06 | 2.65E-06 | -0.039 | -0.018 | 0.007
Co | 3.65E-06 | 2.18E-06 | 1.08E-06 | -0.004 | 0.008 | 0.029
Ni | 1.09E-04 | 7.08E-05 | 3.73E-05 | 0.243 | 0.291 | 0.338
Cu | 3.12E-07 | 1.75E-07 | 8.02E-08 | -0.306 | -0.320 | -0.334
Table 5: Same as Table 4, but for LC18 with net explosive yields for stars up to $25\rm M_{\odot}$. All APOGEE elements are reported.  | Yield ($\rm M_{\odot}/\rm M_{\odot formed}$) | [X/Mg]
---|---|---
| $a_{3}=-2.0$ | Kroupa | $a_{3}=-2.6$ | $a_{3}=-2.0$ | Kroupa | $a_{3}=-2.6$
Fe | 7.24E-04 | 4.57E-04 | 2.34E-04 | -0.103 | -0.077 | -0.050
O | 5.96E-03 | 3.54E-03 | 1.71E-03 | 0.166 | 0.166 | 0.166
Na | 3.65E-05 | 2.13E-05 | 1.01E-05 | 0.245 | 0.237 | 0.229
Al | 4.94E-05 | 2.93E-05 | 1.41E-05 | 0.098 | 0.097 | 0.097
Si | 8.51E-04 | 5.27E-04 | 2.65E-04 | 0.257 | 0.275 | 0.293
P | 6.73E-06 | 4.11E-06 | 2.04E-06 | 0.212 | 0.224 | 0.237
S | 3.14E-04 | 1.95E-04 | 9.89E-05 | 0.155 | 0.175 | 0.196
K | 7.36E-07 | 4.48E-07 | 2.22E-07 | -0.470 | -0.460 | -0.449
Ca | 4.20E-05 | 2.63E-05 | 1.34E-05 | -0.034 | -0.011 | 0.012
V | 1.20E-07 | 7.50E-08 | 3.80E-08 | -0.273 | -0.252 | -0.230
Cr | 8.11E-06 | 5.09E-06 | 2.60E-06 | -0.163 | -0.139 | -0.115
Mn | 4.50E-06 | 2.81E-06 | 1.43E-06 | -0.232 | -0.210 | -0.188
Co | 3.14E-06 | 1.92E-06 | 9.52E-07 | 0.022 | 0.034 | 0.046
Ni | 6.67E-05 | 4.21E-05 | 2.16E-05 | 0.120 | 0.147 | 0.173
Cu | 7.45E-07 | 4.36E-07 | 2.07E-07 | 0.164 | 0.157 | 0.150
Ce | 1.71E-10 | 1.00E-10 | 4.76E-11 | -1.209 | -1.215 | -1.222

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