#  KPM: A flexible and data-driven K-process model for nucleosynthesis

Emily J. Griffith NSF Astronomy and Astrophysics Postdoctoral Fellow  Center
for Astrophysics and Space Astronomy, Department of Astrophysical and
Planetary Sciences, University of Colorado, 389 UCB, Boulder, CO 80309-0389,
USA  David W. Hogg Center for Cosmology and Particle Physics, Department of
Physics, New York University, 726 Broadway, New York, NY 10003, USA  Max-
Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany
Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue,
New York, NY 10010, USA  Julianne J. Dalcanton  Center for Computational
Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
Department of Astronomy, Box 351580, University of Washington, Seattle, WA
98195  Sten Hasselquist Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218  Bridget Ratcilffe Leibniz-Institut für Astrophysik
Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany  Melissa Ness
Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue,
New York, NY 10010, USA  Department of Astronomy, Columbia University, Pupin
Physics Laboratories, New York, NY 10027, USA  David H. Weinberg The
Department of Astronomy and Center of Cosmology and AstroParticle Physics, The
Ohio State University, Columbus, OH 43210, USA

###### Abstract

The element abundance pattern found by APOGEE and GALAH in Milky Way disk
stars is close to two-dimensional, dominated by production from one prompt
process and one delayed process. This simplicity is remarkable, since the
elements are produced by a multitude of nucleosynthesis mechanisms operating
in stars with a wide range of progenitor masses. We fit the abundances of 14
elements for 48,659 red-giant stars from APOGEE DR17 using a flexible, data-
driven $K$-process model—dubbed KPM. In our fiducial model, with $K=2$, each
abundance in each star is described as the sum of a prompt and a delayed
process contribution. We find that KPM with $K=2$ is able to explain the
abundances well, recover the observed abundance bimodality, and detect the
bimodality over a greater range in metallicity than previously has been
possible. We compare to prior work by Weinberg et al. (2022), finding that KPM
produces similar results, but that KPM better predicts stellar abundances,
especially for elements C+N and Mn and for stars at super-solar metallicities.
The model makes assumptions, including especially that it fixes some
parameters to break degeneracies and improve interpretability; we find that
some of the nucleosynthetic implications are dependent upon these detailed
parameter choices. We add a third and fourth process (to make $K=4$), finding
that the additional processes give the model more freedom and improve the
model’s ability to predict the stellar abundances, as expected, but they don’t
qualitatively change the story. The results of KPM have implications for the
formation of the Galaxy disk, the relationship between abundances and ages,
and the physics of nucleosynthesis.

Nucleosynthesis (1131), Stellar abundances (1577), Galaxy chemical evolution
(580), Core-collapse supernovae (304), Type Ia supernovae (1728),

††journal: ApJ††software: matplotlib (Hunter, 2007), NumPy (Harris et al.,
2020), pandas (pandas development team, 2020; Wes McKinney, 2010), astropy
(Astropy Collaboration et al., 2013, 2018, 2022), and jax (Bradbury et al.,
2018).††facilities: Sloan, Kepler

##  1 Introduction

After hydrogen, helium, lithium, and beryllium, all other naturally occurring
elements are made in stars, supernovae, and the collisions of stars. Stellar
surface abundances—the abundances measured by taking a spectrum of a stellar
photosphere—are thought to deliver a relatively unprocessed record of the
element abundances in the gas from which the star formed (though see, e.g.,
Pinsonneault et al., 2001; Oh et al., 2018; Souto et al., 2019; Vincenzo et
al., 2021b). These birth abundances were set by a combination of
nucleosynthetic processes involved in making heavy atomic nuclei, and
astrophysical processes involved in delivering atoms from stellar interiors to
star-formation sites (e.g., Johnson et al., 2020). Thus nuclear physics and a
wide swath of astrophysics are critically intertwined in our understanding of
stellar surface abundances, motivating theoretical, experimental, and
observational work.

At the present day, stellar surface abundances are not very well explained by
purely ab initio, physics-driven models. Theoretical yields vary from data set
to data set, as they are dependent on progenitor properties and explosion
assumptions (e.g., Rybizki et al., 2017; Blancato et al., 2019; Buck et al.,
2021; Griffith et al., 2021b). The wide parameter space of progenitor and
supernova models coupled with uncertainties in reaction rates and explosion
physics hinder the creation of an accurate nucleosynthetic model from theory
alone. In the long run, it is incumbent upon us to understand these issues and
correct the assumptions or calculations underlying our nucleosynthetic and
astrophysical models. In the short run, however, we gather data—tens of
millions of abundance measurements on millions of stars in different
astronomical surveys such as RAVE, SEGUE, LAMOST, Gaia-ESO, APOGEE/MWM, GALAH,
and H3 (Steinmetz et al., 2006; Yanny et al., 2009; Gilmore et al., 2012; De
Silva et al., 2015; Luo et al., 2015; Majewski et al., 2017; Conroy et al.,
2019). This raises the question: _Can we take a data-driven approach to
nucleosynthesis?_

In this Article, we build a purely data-driven model for the surface element
abundances observed in stars. We treat each star as being a linear combination
of nucleosynthetic processes, beginning with one that is primarily responsible
for the $\alpha$-element Mg (prompt enrichment, such as core-collapse
supernovae or CCSN, e.g., Andrews et al., 2017), and one that is primarily
_not_ responsible for Mg (delayed enrichment, such as Type-Ia supernovae or
SNIa). Beyond these up-front assumptions, we try to be agnostic about how the
elements are produced.

We build upon the work of Griffith et al. (2019, 2022, hereafter G22) and
Weinberg et al. (2019, 2022, hereafter W19, W22), who used the bimodality in
[Mg/Fe] vs. [Fe/H]111Where $[{\rm X}/{\rm Y}]=\log({\rm X}/{\rm Y})-\log({\rm
X}/{\rm Y})_{\odot}$ and $({\rm X}/{\rm Y})_{\odot}$ is the solar abundance
ratio. (e.g., Fuhrmann, 1998; Bensby et al., 2003; Adibekyan et al., 2012) to
separate stars into populations with high and low SNIa enrichment. Using the
median [X/Mg] vs. [Mg/H] abundance trends, these works explain data from the
GALAH222GALAH = GALactic Archaeology with HERMES. and SDSS-IV APOGEE333APOGEE
= Apache Point Observatory Galactic Evolution Experiment, part of the Sloan
Digital Sky Survey surveys, respectively, with a two-process model. Because
the median abundance trends in [X/Mg] vs. [Mg/H] space are largely insensitive
to aspects of chemical evolution, such as outflows and variations in star
formation history (W19), the population abundance trends are set by the
nucleosynthetic processes and can be used to empirically constrain Galactic
enrichment.

These works, as well as Ting & Weinberg (2022) and Ratcliffe & Ness (2023),
find that the Milky Way stellar abundances are well fit by two components,
grounded in [Fe/H] and [Mg/Fe], down to residuals of 0.01 to 0.03 dex for the
most precisely measured elements and 0.05 to 0.1 dex for elements (such as Na,
C, and Ce) with large measurement errors. Simultaneously, Frankel et al.
(2018) and Ness et al. (2022) have found that disk abundances are also well
described by a two-component model of birth radius and age. Correlations
between two-process model parameters and stellar ages and kinematics (W22) as
well as the success of a two-component model of [Fe/H] and age in predicting
APOGEE abundances (Ness et al., 2019) suggest that these two 2-dimensional
models are somehow interconnected.

Beyond standard CCSN and SNIa enrichment, many elements have contributions
from additional nucleosynthetic processes, such as the rapid ($r$) and slow
($s$) neutron capture processes (e.g., Arlandini et al., 1999; Bisterzo et
al., 2014) in asymptotic giant branch (AGB) stars (e.g., Simmerer et al.,
2004; Karakas & Lugaro, 2016), merging neutron stars (e.g., Kilpatrick et al.,
2017), or atypical supernova explosions (e.g., Nomoto et al., 2013). After
predicting stellar abundances from $[{\rm Fe/H}]$ and $[{\rm Mg/Fe}]$, Ting &
Weinberg (2022) identify correlated abundance residuals that are unexplained
by observational uncertainties, indicative of additional nucleosynthetic
processes that standard disk CCSN and SNIa enrichment cannot explain. Results
from G22 and W22 support this conclusion, and both works attempt to add
additional processes to their models to account for non-CCSN and non-SNIa
enrichment, though in a restrictive manner. Other sources of abundance
scatter, such as stochastic sampling of the Initial Mass Function (IMF), IMF
variations, and bursty star formation history could also cause deviations away
from a two-process model (Belokurov et al., 2018; Griffith et al., 2023).

To date, survey abundances have not been fully exploited to create a data-
driven model of nucleosynthesis. While works such as Ting et al. (2012), Casey
et al. (2019), and Ratcliffe et al. (2020) effectively use clustering
algorithms to identify elements with like sources and reduce abundance
dimensionality, the results are difficult to translate into a model of
nucleosynthesis. Clustering components can be linked to nucleosynthesis
sources and enrichment history, but have not yet been used to describe the
enrichment of a single star.

In this work, our main innovations are to relax the assumptions made in G22
and W22, to be more agnostic about the nucleosynthetic processes, and to be
more principled with the measurements or inferences from data. In the $K$
Process Model (KPM), we find the intersection between reliable facts about
nucleosynthesis and good abundance measurements to build an edifice of
Galactic enrichment. The model is hierarchical, in that it learns some
parameters (process vectors) that are shared across all stars, but different
for each element, and some parameters (process amplitudes) that are shared
across all elements, but different for each star. The parameters output by our
model can thus be used as de-noised abundance labels; these will sharpen
relationships between abundances and stellar parameters (including birth
location and time). Our main contribution is to construct a data-driven model
for nucleosynthesis that has good statistical properties, only enforcing
constraints to break rotational degeneracies. All other KPM parameters are set
by the data with no fixed normalization.

This paper is organized as follows. In Section 2 we present the assumptions
and the implementation of KPM. In Section 3 we describe the APOGEE data sample
employed in this paper. We apply KPM to the APOGEE data in Section 4 and
compare our results to those of W22 in Section 4.2. In Section 5 we explore
variations from the fiducial model, changing our assumptions about Fe
production as well as the number of model components. Finally, we discuss and
summarize our results in Section 6.

##  2 The $K$-Process Model

As in W22 and G22, we propose that all stellar abundances can be generated by
a combination of $K$ nucleosynthetic processes. In this picture, each element
has $K$ metallicity-dependent process vector components that are shared across
the full stellar sample, while each star individually has $K$ process
amplitudes, which apply across all elements, such that the expected
logarithmic abundance of element $j$ relative to H in star $i$ ($m_{ij}$) is
defined as:

$m_{ij}=\log_{10}\,\sum^{K}_{k=1}A_{i}^{k}\,q_{k,j}^{Z}~{}.$ (1)

Each star $i$ has $K$ process amplitudes ($A^{k}_{i}$) and each element $j$
has $K$ metallicity dependent process vector components ($q_{k,j}^{Z}$). The
$Z$ superscript denotes the dependence of the process vectors on metallicity,
$Z$, taken to be [Mg/H]. The observed abundance can be expressed

$[\text{X}_{j}/\text{H}]_{i}=m_{ij}+\text{noise},$ (2)

where “noise” represents observational noise and/or other sources of intrinsic
abundance scatter that are not included in this model. For detailed examples
of a similar model with $K=2$, see Section 2 and Figures 2 and 3 of W22.

In KPM, we put forth the following set of assumptions:

#### 1\. $K$ processes

All elements on the periodic table are produced by a combination of
nucleosynthetic processes such as CCSN, SNIa, AGB stars, and merging neutron
stars (Johnson et al., 2020). The majority of $\alpha$, light odd-$Z$, and Fe-
peak elements (the elements observed by APOGEE) are dominantly produced by
$K=2$ sources, with one being a prompt process or mix of prompt processes, and
one being a delayed process or a mix of delayed processes. This is
substantiated by theoretical yields (e.g., Anderson, 2019; Rybizki et al.,
2017) and past successful data-driven models (e.g., Ness et al., 2019, G22,
W22, Ting & Weinberg, 2022; Ratcliffe & Ness, 2023). In this paper we
therefore assume that $K\geq 2$, though KPM could in principle be implemented
with $K=1$.

#### 2\. Linearity

At every metallicity, the (linear) (X/H) abundances of a star can be expressed
as a linear combination of $K$ processes. These processes themselves will
depend on metallicity, but a linear sum is sufficient to explain all element
abundances at any overall metallicity. Because yields depend on stellar masses
and progenitor structure, and because different stars can get to their
metallicities by different histories, this assumption must be at least
slightly wrong in detail.

#### 3\. Non-negativity

All process vector components for all elements are non-negative and all
process amplitudes are non-negative. This assumption implies that the elements
considered here are only produced, and not ever destroyed, by the $K$
processes (relative to hydrogen). This makes the model similar to a non-
negative matrix factorization (Blanton & Roweis 2007; Tsalmantza & Hogg 2012).
In KPM, this assumption is enforced by requiring that the process vector
components and amplitudes are always greater than or equal to zero, such that

$q_{k,j}^{Z}\geq 0\;\forall\;Z,k,j\quad\text{and}\quad A_{i}^{k}\geq
0\;\forall\;k,i~{}.$ (3)

#### 4\. Mg production

All Mg is produced in a prompt process and no other processes contribute to
its production. This is substantiated by theoretical yields where Mg is purely
produced by prompt CCSN (e.g., Anderson, 2019; Rybizki et al., 2017). This
assumption (along with non-negativity) breaks a set of symmetries in the
process space and makes the processes quasi-interpretable in terms of
nucleosynthesis sources. Because such a prompt process is likely dominated by
CCSN (e.g., Andrews et al., 2017), we label the first process with “CC”.

In KPM, this assumption is enforced by fixing the Mg process vector components
such that

$q_{{\rm CC,Mg}}^{\,Z}=1,\quad q_{k>1,{\rm Mg}}^{\,Z}=0$ (4)

at all metallicities. Equation 4 also imposes that the Mg process is
metallicity independent.

#### 5\. Fe production

Fe is produced through a combination of a prompt and delayed process. Because
the delayed process is likely dominated by SNIa (e.g., Andrews et al., 2017),
we label the delayed process with “Ia”444While other enrichment channels with
similar timescales may be included in the respective processes, the “CC” and
“Ia” naming convention conforms to the choices in W22 and G22, and avoids the
possible confusion of process numbers (1 and 2) with supernova type (II and
Ia).. While the prompt process constraint (Mg) is grounded in nucleosynthesis
theory, there is no equivalent nucleosynthesis fact to constrain the delayed
process. To break model degeneracies, we also fix the Fe process vector
components such that

$q_{{\rm CC,Fe}}^{\,Z}=0.4,\quad q_{{\rm Ia,Fe}}^{\,Z}=1-q_{{\rm
CC,Fe}}^{\,Z}\quad q_{k>2,{\rm Fe}}^{\,Z}=0$ (5)

at all metallicities. This assumption places a star with purely prompt
enrichment on the low metallicity $[{\rm Mg/Fe}]$ plateau near 0.4 dex, in
agreement with APOGEE observations but in contention with recent results from
Conroy et al. (2022) which place the $[{\rm Mg/Fe}]$ plateau near 0.6 dex. We
explore the impact of different $q_{{\rm CC,Fe}}^{Z}$ assumptions in Section
5.1.

#### 6\. Metallicity dependence

We permit the process vector components for all elements other than Mg and Fe
to float as a function of metallicity. The variation is parameterized by a
linear spline in log-process space, attached to a set of variable control
points. We assume that a particular set of hard-coded knots are sufficient to
capture the metallicity dependence.

#### 7\. APOGEE abundances and uncertainties

We assume that the APOGEE abundances and uncertainties can be used for this
project. This is not the same as assuming that they are correct, but rather
that it is possible and useful to build an interpretable model to explain
them. We describe the potential data systematics in Section 3. For our
purposes, we care mainly about the statistical observational errors rather
than systematics that arise from imperfect modeling of the spectra such as
NLTE effects, though differential systematics across the sample can
artificially add abundance scatter. The actual derived values of $q_{k,j}^{Z}$
will be affected by systematic offsets in the abundances. We add a softening
parameter $Q$ (Equation 7) to allow for the possibility that APOGEE
observational errors are underestimated or that there is intrinsic scatter
around the KPM predictions.

#### 8\. Robust likelihood function

The observed value of [X/H] can be described as the $K$ process expected value
plus observational noise and/or other sources of intrinsic abundance scatter,
as described by Equation 2. The expression in this equation can be thought of
as the key assumption underlying our likelihood function. In detail the
(negative two times the) log likelihood function is given by a chi-squared
($\chi^{2}$) objective

$\chi^{2}=\sum_{ij}\frac{1}{\sigma_{ij}^{2}}\,([{\rm X}/{\rm
H}]_{ij}-m_{ij})^{2}~{},$ (6)

where $1/\sigma_{ij}^{2}$ is the (robust; see below) inverse variance on
measurement $ij$.

Because we don’t want to be too drawn or influenced by outlier points, we
don’t use the observed errors $\sigma_{{\rm obs},ij}$ in the likelihood, but
instead we soften them in the spirit of iteratively reweighted least squares
(Holland & Welsch 1977):

$\frac{1}{\sigma_{ij}^{2}}=\frac{Q^{2}/\sigma_{{\rm obs},ij}^{2}}{Q^{2}+([{\rm
X}/{\rm H}]_{ij}-m_{ij})^{2}/\sigma_{{\rm obs},ij}^{2}},$ (7)

where $Q$ is a softening parameter. Our results are largely insensitive to the
choice of $Q$ (e.g., $1<Q<10$), so we choose to set $Q=5$. Very small values
(e.g., $Q=.1$) will erase some of the abundance structure and produce poorer
fits.

#### 9\. Implementation and optimization

With the above assumptions in place, the likelihood function can be optimized
to a set of stellar abundances. The model is initialized at the Mg and Fe
process vector components from Equations 4 and 5. It subsequently optimizes
the process amplitudes (dubbed the $A$-step) using only Mg and Fe at fixed
process vector components, and then optimizes the process vector components
(dubbed the $q$-step) for all elements at fixed process amplitudes. The
$A$-step and $q$-step are alternated, repeating 48 rounds of optimization, in
the $K=2$ case, and updating the best-fit parameters when the objective
function improves. We find few differences in the best-fit parameters when we
decrease the number of iterations to 32, indicating that the model quickly
finds a good solution. In detail the optimizations are performed with a
nonlinear $\chi^{2}$ minimization algorithm (Gauss–Newton nonlinear least-
squares) from jaxopt555https://jaxopt.github.io/.

KPM mirrors the two-proccess model from prior work (G22, W22) but, unless
otherwise noted, the assumptions are weaker, there is a likelihood function in
play, and the implementation is more general. In particular, we don’t assume
anything about the relationships between the process vector components and the
morphologies of observed element-abundance ratio diagrams.

##  3 Data

In this paper, we employ stellar abundances from APOGEE DR17 (Abdurro’uf et
al., 2022), part of the SDSS-IV (Majewski et al., 2017). The APOGEE survey
obtains high-resolution ($R\sim 22,500$) near-infrared (IR) observations
(Wilson et al., 2019) for stars in the Galactic disk, halo, bulge, and nearby
satellites/streams. Observations are taken with two nearly identical
spectrographs on the 2.5m Sloan Foundation telescope (Wilson et al., 2019) at
Apache Point Observatory in New Mexico and the 2.5m du Pont Telescope (Bowen &
Vaughan, 1973) at the Las Campanas Observatory in Chile. Spectral data are
reduced and calibrated with the APOGEE data processing pipeline (Nidever et
al., 2015), after which stellar parameters and abundances are calculated with
ASPCAP (APOGEE Stellar Parameter and Chemical Abundance Pipeline; Holtzman et
al., 2015; García Pérez et al., 2016). See Jönsson et al. (2020, DR16) and
Holtzman et al. (in prep., DR17) for a more detailed description of APOGEE
data reduction and analysis, and Zasowski et al. (2013, 2017), Beaton et al.
(2021), and Santana et al. (2021) for a discussion of survey targeting.

APOGEE DR17 reports stellar parameters, including $T_{\rm eff}$ and $\log(g)$,
as well as 20 elemental abundances: C, C I, N, O, Na, Mg, Al, Si, S, K, Ca, Ti
I, Ti II, V, Cr, Mn, Fe, Co, Ni, and Ce for 657,135 stars. In DR17, new
spectral libraries (Hubeny et al., 2021) are generated using the Synspec code
and incorporate NLTE corrections for Na, Mg, K, and Ca (Osorio et al., 2020).
Among the reported elements and ions, some are measured more precisely than
others. We exclude Ti from our analysis as there are large differences between
the abundances derived from the Ti I and Ti II lines (Jönsson et al., 2020).
We also exclude P and V, as the P abundances are measured from a few very weak
spectra features and V abundances are one of the least precise and least
accurate labels (Jönsson et al., 2020). Both P and V display strong abundance
artifacts and large scatter. Among the remaining elements, we note the
following concerns: weak Na spectral features, large abundance scatter in S,
significant systematic artifacts in Cr abundances at super-solar
metallicities, potentially strong unaccounted-for NLTE effects on Mn
abundances (Bergemann et al., 2019), and large abundance scatter in Co and Ce.
For a more detailed discussion of abundance systematics and their effects on
population trends, see Jönsson et al. (2020) and Griffith et al. (2021a).

For our stellar sample, we select a subset of APOGEE DR17 stars with the goal
of minimizing statistical errors from poor observations and systematic errors
from abundance trends with $T_{\rm eff}$ and/or $\log(g)$ while preserving a
sufficient number of stars to conduct a meaningful statistical analysis across
the Galactic disk. To remove poor quality data points, we require ASPCAP flags
STAR_BAD and NO_ASPCAP_RESULT equal zero. We only include stars from the main
survey sample (EXTRATARG = 0), and use named abundances (X_FE), as recommended
by Jönsson et al. (2020). In addition to these quality cuts, we apply the
following sample selection:

  * •

$[{\rm Mg}/{\rm H}]>-0.75$

  * •

S/N $\geq 200$

  * •

$\log(g)=1-3$ dex

  * •

$T_{\rm eff}=4000-5200$ K.

To eliminate red clump (RC) stars, which show abundance variations from the
RGB sample (Vincenzo et al., 2021a), we cross-match with and remove stars that
appear in the APOGEE DR17 RC
VAC666https://www.sdss4.org/dr17/data_access/value-added-
catalogs/?vac_id=apogee-red-clump-(rc)-catalog (Bovy et al., 2014).

These cuts result in a sample of 48,659 stars that span the Galactic disk. We
plot their $Z$ vs. $R$ location, as well as the distributions of distances and
eccentricities in Figure 1, taking distances and kinematics from (Queiroz et
al., 2023). While our stellar sample extends from Galactic center, to the
outer disk, to the halo, the majority of our stars ($75\%$) are within 3.5 kpc
of the sun. Further, $94\%$ of our stellar sample has an eccentricity less
than 0.4, indicative of in situ origin. In this paper, we assume that the KPM
fits will be consistent across the Galactic disk, as the median high-Ia and
low-Ia [X/Mg] vs. [Mg/H] abundance tends are insensitive to Galactic location
(W19, Griffith et al., 2021a).

Figure 1: Left: distribution of our stellar sample in $Z$ (kpc) vs. $R$ (kpc)
where (0,0) is the Galactic center. Center: distribution of stellar distances
(kpc). Right: distribution of stellar eccentricity. Our stellar sample spans
the Galactic disk, but the majority of our stars are within 3.5 kpc of the sun
and have kinematics consistent with in situ origin.

We present abundances for Mg, O, Si, S, Ca, C+N, Na, Al, K, Cr, Fe, Ni, Mn,
Co, and Ce. In the analysis of each element, X, we drop stars with X_FE_FLAG.
Ce abundances are flagged in the most stars, resulting in $\sim 700$ Ce labels
being excluded. While the surface abundances of C and N differ from the
stellar birth abundances for RGB stars due to the CNO processes and dredge-up
events (Iben, 1965; Shetrone et al., 2019), the total C+N abundance remains
constant. As in W22, we consider C+N as an element, taking [(C+N)/H] to be

$[\text{C+N}/\text{H}]=\log_{10}(10^{\text{[C/H]}+8.39}+10^{\text{[N/H]}+7.78})-\log_{10}(10^{8.39}+10^{7.78}),$
(8)

using logarithmic solar abundances for C (8.39) and N (7.78) from Grevesse et
al. (2007). We further adopt the error on the [C/Fe] abundance as the error on
[C+N/Fe], since C typically dominates in the abundance ratio.

We plot the distributions of all abundances in [X/Mg] vs. [Mg/H] for our
sample in the first column of Figures 2 and 3.

Figure 2: Abundance distributions and KPM parameters for C+N, $\alpha$, and
light odd-$Z$ elements. First column: observed abundance distributions in
[X/Mg] vs. [Mg/H]. Second column: The predicted [X/Mg] vs. [Mg/H] abundance
distribution of the fiducial model plus estimated noise. By comparing the
first two columns we can evaluate the success of KPM in reproducing the
observed abundance distributions. Third column: process vector components
$q_{{\rm CC,}j}^{Z}$ (thin, purple) and $q_{{\rm Ia,}j}^{Z}$ (thick, orange)
from this work (G23, solid, dark lines) and W22 (light, dashed lines). Overall
offsets between the solid and dashed lines are driven largely by our
normalization that places the [Mg/Fe] plateau at $+0.4$, rather than $+0.3$ in
W22. Fourth column: distribution of fractional contribution from the prompt
process ($f_{ij}^{\rm CC}$) predicted by the fiducial model. We plot the
median $f_{ij}^{\rm CC}$ values of the low-Ia (orange square) and high-Ia
(purple circle) populations in the solar metallicity bin from W22 for
comparison. All density plots are logarithmically scaled. Figure 3: Same as
Figure 2, but for Fe-peak elements and Ce.

##  4 The Fiducial Model

We fit the APOGEE sample with our fiducial model of $K=2$, such that

$m_{ij}=\log_{10}(A_{i}^{\rm CC}\,q_{{\rm CC},j}^{Z}+A_{i}^{\rm Ia}\,q_{{\rm
Ia},j}^{Z})$ (9)

with the assumptions from Section 2. This fit produces process vector
components $q_{{\rm CC,}j}^{Z}$ and $q_{{\rm Ia,}j}^{Z}$ as a function of
$[{\rm Mg}/{\rm H}]$ for each element and process amplitudes $A^{\rm CC}_{i}$
and $A^{\rm Ia}_{i}$ for each star. From the model parameters, we can
calculate fractional contributions from each process as well as a full suite
of predicted $K=2$ process abundances, shown in the second column of Figures 2
and 3.

###  4.1 Process Parameters and Fractional Contributions

We plot the process vector components as a function of $[{\rm Mg}/{\rm H}]$ in
the third column of Figures 2 and 3 and provide the values at the [Mg/H] knots
in Tables 1 and 2. The process vector components inform us about the relative
contribution of prompt and delayed processes to the formation of the elements,
as well as the metallicity dependence of the enrichment. By definition,
$q_{{\rm CC,Fe}}^{Z}=0.4$ at all metallicities. For Mg and Fe, we also require
$q_{{\rm CC,}j}^{Z}+q_{{\rm Ia,}j}^{Z}=1$, implying $q_{{\rm Ia,Fe}}^{Z}=0.6$.
No such constraints are placed on other elements. We note that KPM differs
from the previous two-process models in this regard, as G22 and W22 require
that the process vector components for all elements sum to 1 at solar
metallicity.

Table 1: Fiducial model $q_{{\rm CC,}j}^{Z}$ values at [Mg/H] knot values for each element.  [Mg/H] | C+N | O | Na | Mg | Al | Si | S | K | Ca | Cr | Mn | Fe | Co | Ni | Ce
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
-0.8 | 0.34 | 1.06 | 0.26 | 1.0 | 0.62 | 0.90 | 0.98 | 0.78 | 0.77 | 0.33 | 0.19 | 0.4 | 0.37 | 0.44 | 0.40
-0.5 | 0.44 | 1.04 | 0.36 | 1.0 | 0.77 | 0.82 | 1.02 | 0.88 | 0.70 | 0.34 | 0.22 | 0.4 | 0.48 | 0.48 | 0.27
-0.4 | 0.48 | 1.03 | 0.33 | 1.0 | 0.80 | 0.79 | 0.98 | 0.88 | 0.67 | 0.35 | 0.24 | 0.4 | 0.52 | 0.50 | 0.21
-0.3 | 0.52 | 1.01 | 0.37 | 1.0 | 0.84 | 0.77 | 0.96 | 0.92 | 0.65 | 0.35 | 0.25 | 0.4 | 0.58 | 0.51 | 0.18
-0.2 | 0.55 | 0.98 | 0.42 | 1.0 | 0.86 | 0.75 | 0.93 | 0.93 | 0.64 | 0.38 | 0.27 | 0.4 | 0.62 | 0.53 | 0.17
-0.1 | 0.58 | 0.97 | 0.45 | 1.0 | 0.85 | 0.74 | 0.90 | 0.96 | 0.61 | 0.39 | 0.29 | 0.4 | 0.66 | 0.53 | 0.18
0.0 | 0.62 | 0.95 | 0.46 | 1.0 | 0.84 | 0.72 | 0.87 | 0.99 | 0.60 | 0.39 | 0.29 | 0.4 | 0.67 | 0.52 | 0.22
0.1 | 0.64 | 0.93 | 0.43 | 1.0 | 0.85 | 0.71 | 0.80 | 0.98 | 0.60 | 0.40 | 0.24 | 0.4 | 0.62 | 0.49 | 0.26
0.2 | 0.57 | 0.85 | 0.26 | 1.0 | 0.85 | 0.66 | 0.68 | 0.92 | 0.63 | 0.49 | 0.09 | 0.4 | 0.49 | 0.45 | 0.37
0.3 | 0.61 | 0.78 | 0.18 | 1.0 | 0.78 | 0.63 | 0.62 | 0.76 | 0.66 | 0.64 | 0.01 | 0.4 | 0.48 | 0.44 | 0.47
0.6 | 0.66 | 0.65 | 0.61 | 1.0 | 0.66 | 0.58 | 0.49 | 0.64 | 0.70 | 0.64 | 0.28 | 0.4 | 0.70 | 0.50 | 0.36
Table 2: Fiducial model $q_{{\rm Ia,}j}^{Z}$ values at [Mg/H] knot values for each element.  [Mg/H] | C+N | O | Na | Mg | Al | Si | S | K | Ca | Cr | Mn | Fe | Co | Ni | Ce
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
-0.8 | 0.54 | 0.15 | 0.28 | 0.0 | 0.02 | 0.08 | 0.37 | 0.17 | 0.21 | 0.38 | 0.55 | 0.6 | 0.47 | 0.49 | 0.28
-0.5 | 0.62 | 0.01 | 0.67 | 0.0 | 0.16 | 0.13 | 0.27 | 0.11 | 0.18 | 0.61 | 0.83 | 0.6 | 0.74 | 0.63 | 0.60
-0.4 | 0.54 | 0.01 | 0.70 | 0.0 | 0.11 | 0.15 | 0.18 | 0.10 | 0.24 | 0.63 | 0.81 | 0.6 | 0.67 | 0.56 | 0.78
-0.3 | 0.46 | 0.04 | 0.66 | 0.0 | 0.11 | 0.19 | 0.15 | 0.09 | 0.29 | 0.64 | 0.79 | 0.6 | 0.58 | 0.51 | 0.93
-0.2 | 0.42 | 0.08 | 0.54 | 0.0 | 0.10 | 0.23 | 0.15 | 0.07 | 0.33 | 0.60 | 0.75 | 0.6 | 0.48 | 0.46 | 0.98
-0.1 | 0.39 | 0.09 | 0.50 | 0.0 | 0.13 | 0.25 | 0.15 | 0.04 | 0.36 | 0.58 | 0.74 | 0.6 | 0.43 | 0.45 | 0.88
0.0 | 0.40 | 0.10 | 0.51 | 0.0 | 0.12 | 0.26 | 0.14 | 0.00 | 0.35 | 0.60 | 0.79 | 0.6 | 0.45 | 0.49 | 0.67
0.1 | 0.45 | 0.12 | 0.68 | 0.0 | 0.08 | 0.27 | 0.19 | 0.03 | 0.32 | 0.63 | 0.95 | 0.6 | 0.59 | 0.57 | 0.49
0.2 | 0.61 | 0.21 | 1.05 | 0.0 | 0.06 | 0.31 | 0.31 | 0.15 | 0.25 | 0.53 | 1.26 | 0.6 | 0.83 | 0.65 | 0.28
0.3 | 0.63 | 0.29 | 1.32 | 0.0 | 0.15 | 0.35 | 0.35 | 0.40 | 0.19 | 0.42 | 1.46 | 0.6 | 0.96 | 0.70 | 0.09
0.6 | 0.62 | 0.37 | 1.13 | 0.0 | 0.30 | 0.34 | 0.36 | 0.48 | 0.13 | 0.43 | 1.40 | 0.6 | 0.83 | 0.64 | 0.16

In the fourth column of Figures 2 and 3, we plot the distribution of
fractional contributions from the prompt process ($f_{ij}^{\rm CC}$) to each
element, where

$f_{ij}^{\rm CC}=\frac{A^{\rm CC}_{i}\,q_{{\rm CC,}j}^{Z}}{A^{\rm
CC}_{i}\,q_{{\rm CC,}j}^{Z}+A^{\rm Ia}_{i}\,q_{{\rm Ia,}j}^{Z}}.$ (10)

We generally find that the distributions are bimodal, like the observed
abundance patterns, as the high-Ia and low-Ia populations have differing
fractional contributions from prompt and delayed sources.

We find that the $\alpha$-elements (O, Si, S, Ca) are best fit with $q_{{\rm
CC,}j}^{Z}$ and $f_{ij}^{\rm CC}>0.5$ at all metallicities. This is in
agreement with theoretical prediction that $\alpha$-elements are dominated by
prompt CCSN production (e.g. Andrews et al., 2017). O, a Mg-like element
theoretically purely produced in prompt CCSN, shows $f_{i,{\rm O}}^{\rm CC}$
near 1 from $[{\rm Mg}/{\rm H}]=-0.75$ to solar. At supersolar metallicity,
the delayed process contributes to O production, driving the $f_{i,{\rm
O}}^{\rm CC}$ value down to $\sim 0.8$ at $[{\rm Mg}/{\rm H}]=0.4$. S behaves
like O, with almost entirely prompt production up to solar metallicity, after
which delayed enrichment contributes more significantly. Conversely, we find
that Si and Ca are best fit with prompt and delayed enrichment at all
metallicities, though the prompt process always dominates. For Si, the delayed
process appears to increase linearly with $[{\rm Mg}/{\rm H}]$, while the Ca
delayed enrichment increases from $[{\rm Mg}/{\rm H}]$ of $-0.75$ to $-0.1$
and then decreases from $[{\rm Mg}/{\rm H}]$ of $-0.1$ to $0.5$.

The process vector components of light odd-$Z$ elements Al and K resemble
those of the $\alpha$-elements, such as S. Both exhibit $q_{{\rm CC,}j}^{Z}$
and $f_{ij}^{\rm CC}$ near 1 through solar metallicity, with an increase in
$q_{{\rm Ia,}j}^{Z}$ and downturn in $f_{ij}^{\rm CC}$ at supersolar
metallicities (especially for K). The behavior of the Na process vector
components is more complex, with peaks and troughs in $q_{{\rm Ia,Na}}^{Z}$.
We find that Na has the strongest contributions from the delayed process of
all $\alpha$ and light odd-$Z$ elements, with $q_{{\rm Ia,Na}}^{Z}\gtrsim 0.5$
at almost all values of $[{\rm Mg}/{\rm H}]$ and $f_{i,{\rm Na}}^{\rm CC}<0.3$
at $[{\rm Mg}/{\rm H}]>0$. The strong delayed contribution to Na is in
agreement with findings of W22 and G22, and in tension with theoretical yields
(e.g. Andrews et al., 2017; Rybizki et al., 2017).

Unlike $\alpha$ and light odd-$Z$ elements whose delayed production is
dominated by SNIa, C and N are thought to be promptly produced in CCSN with
additional delayed enrichment from AGB stars (e.g. Andrews et al., 2017). We
find that the prompt and delayed processes both contribute significantly, and
nearly equally, across our stellar sample. Though theoretical N yields from
AGB stars have a strong metallicity dependence (Karakas, 2010; Ventura et al.,
2013; Cristallo et al., 2015; Johnson et al., 2022), we observe only a slight
positive metallicity dependence in $q_{{\rm CC,C+N}}^{Z}$ and a shallow dip in
$q_{{\rm Ia,C+N}}^{Z}$. We find a population of stars with $f_{i,{\rm
C+N}}^{\rm CC}$ near 0.9 and a population near 0.4.

The Fe-peak elements (Cr, Mn, Fe, Co, Ni) are thought to be produced through
prompt CCSN production and delayed SNIa production (e.g. Andrews et al.,
2017). By construction, $q_{{\rm CC,Fe}}^{Z}=0.4$ and $q_{{\rm
Ia,Fe}}^{Z}=0.6$ at all metallicities. This produces a bimodal distribution in
$f_{i,{\rm Fe}}^{\rm CC}$ similar to that observed in abundance space. Because
of our choice of $q_{{\rm CC,Fe}}^{Z}$, only a few stars have $f_{i,{\rm
Fe}}^{\rm CC}=1$ (see Section 5.1). We instead observe a population with
$f_{i,{\rm Fe}}^{\rm CC}$ near 0.8 and a population near 0.4. The process
vector components and $f_{i,{\rm Fe}}^{\rm CC}$ distribution for Cr and Ni
strongly resemble those of Fe. All three elements have even atomic numbers. At
supersolar metallicity, we find that the prompt process dominates Cr
production, resulting in an upturn in $f_{ij}^{\rm CC}$. Conversely, Ni
displays a dominant, and increasing, delayed process vector component at
supersolar metallicities. The process vector components for Mn and Co (odd
atomic numbers) show a complex metallicity dependence, more resembling that of
Na. Both elements display a strong delayed process, with the $q_{{\rm
Ia,Mn}}^{Z}>0.5$ at all metallicities and $>1$ for $[{\rm Mg}/{\rm H}]>0.1$.
Mn is the only element for which $f_{i,{\rm Mn}}^{\rm CC}$ decreases to 0 for
$[{\rm Mg}/{\rm H}]\gtrsim 0.2$.

Finally, we find that the delayed process dominates Ce production at
intermediate metallicity, with $q_{{\rm Ia,Ce}}^{Z}$ increasing up to $[{\rm
Mg}/{\rm H}]\approx-0.2$ and then decreasing to nearly 0 at $[{\rm Mg}/{\rm
H}]\approx 0.3$. The $f_{i,{\rm Ce}}^{\rm CC}$ values are clustered near 0.25
around $[{\rm Mg}/{\rm H}]$ of 0.2, then increase such that the abundances are
almost entirely dominated by prompt enrichment at high metallicity.

In addition to process vector components, each star is fit with a prompt and
delayed process amplitude, $A^{\rm CC}_{i}$ and $A^{\rm Ia}_{i}$ respectively
(Table 3). All elemental abundances are used in the calculation of these
amplitudes, so they can be interpreted as “de-noised” abundance labels that
suppress observational scatter by averaging over elements via the data-driven
model. The value of $A^{\rm CC}_{i}$ traces the metallicity (specifically
$[{\rm Mg}/{\rm H}]$) while the ratio of $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$
traces the $[{\rm Fe}/{\rm Mg}]$ abundance. In the left panel of Figure 4 we
plot the distribution of $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ vs. $A^{\rm CC}_{i}$.
We find a bimodal distribution, similar to the Tinsley-Wallerstein diagram
($[{\rm Mg/Fe}]$ vs. $[{\rm Fe/H}]$, Wallerstein, 1962; Tinsley, 1979, 1980),
as was found in W22 and G22. We stress that the presence of the abundance
bimodality was not fed into our model, and yet it is recovered in the best-fit
process amplitudes. The stars with larger $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$
values correspond to the high-Ia population, and those with low $A^{\rm
Ia}_{i}/A^{\rm CC}_{i}$ correspond to the low-Ia population. While in the
Tinsley-Wallerstein diagram the two populations blend together at high
metallicity, they are more distinguishable in our amplitude space. We plot
$A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ vs. $[{\rm Mg}/{\rm H}]$ in the center panel
of Figure 4. The high-Ia and low-Ia populations are clearly separable through
$[{\rm Mg}/{\rm H}]$ of 0.4. This is further shown through the $A^{\rm
Ia}_{i}/A^{\rm CC}_{i}$ distributions in the right panel of Figure 4 for
[Mg/H] bins of $-0.75$ to $-0.425$, $-0.425$ to $-0.1$, $-0.1$ to 0.225, and
0.225 to 0.55. The three lowest metallicity bins display a bimodal
distribution and the highest metallicity bin is dominated by high-Ia stars.

Table 3: Fiducial model $A^{\rm CC}_{i}$ and $A^{\rm Ia}_{i}$ values for our
stellar sample.

APOGEE ID | [Mg/H] | $A^{\rm CC}_{i}$ | $A^{\rm Ia}_{i}$
---|---|---|---
2M00000546+6152107 | -0.20 | 0.61 | 0.48
2M00000866+7122144 | -0.10 | 0.82 | 0.69
2M00001328+5725563 | 0.04 | 1.09 | 1.02
2M00001653+5540107 | 0.06 | 1.13 | 0.36
2M00001717+6147500 | -0.23 | 0.60 | 0.39
… | … | … | …

Note. — Full table available online.

Figure 4: Left: distribution of $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ vs. $A^{\rm
CC}_{i}$ for the fiducial model. This plot is similar to a [Fe/Mg] vs. [Mg/H]
distribution where $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ is a proxy for [Fe/Mg] and
$A^{\rm CC}_{i}$ is a proxy for [Mg/H]. Note that $A^{\rm CC}_{i}=1$
corresponds to $[{\rm Mg}/{\rm H}]=0$. Center: distribution of $A^{\rm
Ia}_{i}/A^{\rm CC}_{i}$ vs. $[{\rm Mg}/{\rm H}]$. In the left and center
panels, we can clearly see the bimodality to high values of $A^{\rm CC}_{i}$
and $[{\rm Mg}/{\rm H}]$. Both density plots are logarithmically scaled.
Right: distribution of $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ for ranges of [Mg/H]
with metallicity bin increasing from top to bottom. [Mg/H] bins are of width
0.325 dex and span $-0.75$ to 0.55.

With the optimized process parameters in hand, we can use Equation 2 to
calculate predicted abundances for the fiducial model—the abundances our
stellar population would have if the model assumptions are correct and only
one prompt and one delayed process contribute. To simulate observational
noise, we add an error drawn from a Gaussian distribution with $\sigma$ equal
to the reported error on each abundance for each star. In Figure 2 and 3 we
plot the predicted abundances plus estimated noise in the second columns.
These distributions can be compared to the observed abundance distribution in
the first columns.

Overall, the fiducial model successfully reproduces the observed abundance
distributions. It is capable of capturing metalicity dependences and
bimodality. The predicted abundances plus estimated noise are not, however,
able to reproduce the observed abundance scatter. This is especially
noticeable for O, C+N, Na, Al, K, Co, and Ce. For these elements the scatter
in the observed abundance distribution is larger than in the predicted
distribution, suggesting that the APOGEE observational scatter is
underestimated, that there are $T_{\rm eff}$ or $\log(g)$ dependent abundance
trends (e.g., Griffith et al., 2021a, W22), or that the $K=2$ model is
insufficient–a likely case for elements produced by AGB stars, such as C+N and
Ce.

###  4.2 Comparing to W22

As discussed in Sections 1 and 2, KPM is based upon the two-process model
developed in W19 and W22, but with increased flexibility, minimal
normalization, and no forced dependence upon the [Fe/Mg] vs. [Fe/H] bimodality
or population abundance trends. Further, the KPM utilizes all stellar
abundances in the optimization of $A^{\rm CC}_{i}$ and $A^{\rm Ia}_{i}$,
whereas only Mg and Fe are used in W19 and only Mg, O, Si, Ca, Fe, and Ni in
W22.

In the fiducial model, we adopt $K=2$, as in W22, but assume $q_{{\rm
CC,Fe}}^{Z}=0.4$, 0.1 dex lower than the $q_{{\rm CC,Fe}}^{Z}$ value assumed
in W22. In practice, this moves the implied “pure” CCSN enrichment plateau
from $[{\rm Fe}/{\rm Mg}]=-0.3$ to $[{\rm Fe}/{\rm Mg}]=-0.4$ (though the W22
plateau value is determined after they apply a global offset of +0.05 to all
$[{\rm Fe}/{\rm Mg}]$ abundances). Because our model is non-negative, it
requires a lower $q_{{\rm CC,Fe}}^{Z}$ to correctly model the stars on the
$[{\rm Fe}/{\rm Mg}]$ plateau, whereas W22 assigns stars with $[{\rm Fe}/{\rm
Mg}]<-0.3$ negative $A^{\rm Ia}_{i}$ values.

While our stellar samples and model assumptions differ, we plot the W22
$q_{{\rm CC,}j}^{Z}$ and $q_{{\rm Ia,}j}^{Z}$ vector components as well as the
W22 solar metallicity $f_{ij}^{\rm CC}$ values of Figures 2 and 3 for
comparison with our fiducial model. We generally observe similar behavior
between KPM and W22. Our $q_{{\rm Ia,}j}^{Z}$ vector components tend to be
$\sim 0.1$ greater than those of W22 for elements with significant delayed
contributions because of our differing $q_{{\rm CC,Fe}}^{Z}$ assumptions. The
metallicity dependencies agree for most elements, with small variations at the
high-$[{\rm Mg}/{\rm H}]$ end for O, Al, K, and Ce. We also see good agreement
between the KPM and W22 solar metallicity $f_{ij}^{\rm CC}$ values, with the
W22 points slightly offset to larger values for elements with significant
delayed contributions.

To compare the accuracy of the models’ abilities to reproduce the observed
abundances, we identify a subset of $\sim 23,000$ stars in both our sample and
the W22 sample. We calculate the predicted abundances for each star under KPM
and the two-process model, then determine the $\chi^{2}$ value of the fits for
each star (summing over the elements) and for each element (summing over the
stars). We plot the cumulative stellar $\log(\chi^{2})$ distribution and the
total $\chi^{2}$ for each element in the right and left panels, respectively,
of Figure 5. It is important to note that in the calculation of the W22 model
residuals, we do not apply the temperature corrections discussed in Section
5.1 of W22.

We find that, overall, the $\chi^{2}$ decreases between the W22 two-process
model and our $K$-process model, an indication that we better predict all of a
star’s abundances. When looking at each element individually, we find that we
better predict C+N, Na, K, Ni, Mn, Co, and Ce, with major improvements to C+N
and Mn. Our fiducial model is significantly worse at predicting Mg, Ca, and Fe
than the W22 model, three of the six elements that W22 employ to fit the
process amplitudes. Because KPM uses all elements in its optimization, Mg, Ca,
and Fe are effectively de-weighted relative to the W22 model, while C+N and Mn
influence the model parameters. If we re-fit KPM using only the Mg, O, Si, Ca,
Fe, and Ni abundances in the $A$-step (as in W22), we find that KPM and the
two-process model predict the abundances of all elements but Mn with similar
accuracy, and that the two-process model better predicts Mn. Our fiducial
model’s success in predicting C+N and Mn is likely attributable to the
inclusion of the elements in the $A$-step. The choice to include all elements
or a subset of elements in the fits should be considered when implementing
KPM. If searching for stars with anomalous abundances of element X relative to
the expected abundances of others, one may want to exclude X from the
$A$-step.

Figure 5: Left: cumulative distribution of $\log_{10}(\chi^{2})$ for W22
(dashed orange line) and our fiducial model (G23, solid purple line). Right:
$\chi^{2}$ per element for the same model fits with elements ordered by atomic
number. Overall, our model has a smaller cumulative $\chi^{2}$ than the
previous two-process model and better predicts the abundances of C+N, Na, K,
Mn, Co, Ni, and Ce.

We note that our fiducial model is fit to a stellar sample that spans a wider
range of $T_{\rm eff}$ and $\log(g)$ than the W22 sample. If we repeat our
analysis on the W22 stellar sample with $q_{{\rm CC,Fe}}^{Z}=0.5$ we almost
perfectly recover the W22 process vector components, with small deviations at
$[{\rm Mg}/{\rm H}]>0.1$, and more substantially improve upon the stellar and
elemental $\chi^{2}$ values. Most notably, KPM is better able to predict the
abundances of stars with $[{\rm Mg}/{\rm H}]>0$, where the high-Ia and low-Ia
sequences blend together and the W22 categorization of high-Ia and low-Ia
stars may be incorrect.

##  5 Variations Away from the Fiducial Model

###  5.1 Impact of Fe Process Assumptions

While KPM is flexible, we still include some quantitative assumptions, which
we make to break exact model degeneracies (see Section 2). Specifically, for
each process, we choose one element to assign a “known” process vector
component at all metallicities. These are Mg and Fe in the $K=2$ case.
Specifically, for the prompt process, we ground our assumption in the
nucleosynthetic theory that Mg is a pure CCSN element (e.g., Andrews et al.,
2017). Unfortunately, there is no comparable pure SNIa element, nor is there
an element for which we know the relative CCSN/SNIa ratio. In order to break
an exact degeneracy between the prompt and delayed processes, we choose to fix
the Fe process vector components, which effectively makes assumptions about
the exact fractional contribution of CCSN and SNIa (or prompt and delayed
processes) to Fe enrichment.

In the fiducial model, we choose $q_{{\rm CC,Fe}}^{Z}=0.4$ as this parameter
choice is able to reproduce the observed [Fe/Mg] vs. [Mg/H] abundance
distribution, as discussed in Section 4.1. This choice impacts the predicted
abundances as well as the implied $f_{ij}^{\rm CC}$ values of each star.
Because our model is non-negative, the choice of $q_{{\rm CC,Fe}}^{Z}$ sets
the minimum [Fe/Mg] value attainable by our model ($\log_{10}(q_{{\rm
CC,Fe}}^{Z})$). In this Section, we explore the implications of different
$q_{{\rm CC,Fe}}^{Z}$ assumptions, varying the zero point and metallicity
dependence ($dq_{{\rm CC,Fe}}^{Z}/dZ$). In Figure 6, we plot the minimum
[Fe/Mg] as a function of $[{\rm Mg}/{\rm H}]$ for the $q_{{\rm CC,Fe}}^{Z}$
and $dq_{{\rm CC,Fe}}^{Z}/dZ$ parameters we explore. With a $dq_{{\rm
CC,Fe}}^{Z}/dZ=0.0$, we choose $q_{{\rm CC,Fe}}^{Z}=0.5$ (the W22 value), 0.45
(approximate plateau value at $[{\rm Mg}/{\rm H}]=-0.75$), 0.4 (the fiducial
value that skirts the edge of the distribution), and 0.35 (captures almost all
stars). With a $dq_{{\rm CC,Fe}}^{Z}/dZ=0.15$, which roughly matches the slope
of the low-Ia sequence at intermediate metallicity, we choose $q_{{\rm
CC,Fe}}^{Z}=0.5$ (passes through the center of the low-Ia density), 0.4
(skirts the edge of the distribution), and 0.35 (captures almost all stars).

Figure 6: Minimum [Fe/Mg] value attainable for $q_{{\rm CC,Fe}}^{Z}$ and
$dq_{{\rm CC,Fe}}^{Z}/dZ$ assumptions. Left: for $dq_{{\rm CC,Fe}}^{Z}/dZ=0$
with $q_{{\rm CC,Fe}}^{Z}=0.5$, 0.45, 0.4, and 0.35 (light to dark purple).
Right: for $dq_{{\rm CC,Fe}}^{Z}/dZ=0.15$ with $q_{{\rm CC,Fe}}^{Z}=0.5$, 0.4,
0.35 (light to dark orange).

We repeat the optimization of the KPM with $K=2$ and these $q_{{\rm
CC,Fe}}^{Z}$ assumptions. In Figure 7, we plot the resulting predicted
abundance distributions plus estimated noise alongside the observed
distribution. We do not plot the prediction for the fiducial model ($q_{{\rm
CC,Fe}}^{Z}=0.4$, $dq_{{\rm CC,Fe}}^{Z}/dZ=0.0$), as this is shown in Figure
3. Both models with $q_{{\rm CC,Fe}}^{Z}=0.5$ fail to reproduce the shape and
width of the low-Ia abundance distribution. They instead predict a much
thinner sequence that is flat for $dq_{{\rm CC,Fe}}^{Z}/dZ=0.0$ or slightly
inclined for $dq_{{\rm CC,Fe}}^{Z}/dZ=0.15$. The model with $q_{{\rm
CC,Fe}}^{Z}=0.45$ and $dq_{{\rm CC,Fe}}^{Z}/dZ=0.0$ is better, but still
predicts a low-Ia abundance distribution that is too thin, flat, and dense.
The other four models ($q_{{\rm CC,Fe}}^{Z}=0.35$ and $q_{{\rm
CC,Fe}}^{Z}=0.4$ with $dq_{{\rm CC,Fe}}^{Z}/dZ=0.0$ and $dq_{{\rm
CC,Fe}}^{Z}/dZ=0.15$) predict an abundance distribution that strongly
resembles the observed. There are minor differences in the low [Fe/Mg] and low
[Mg/H] region, but it is difficult to tell by eye which model is best.

Figure 7: Top: observed [Fe/Mg] vs. [Mg/H] abundance distribution. Bottom:
predicted [Fe/Mg] vs. [Mg/H] abundance distributions plus estimated noise for
models with $dq_{{\rm CC,Fe}}^{Z}/dZ=0.0$ (middle row) and $dq_{{\rm
CC,Fe}}^{Z}/dZ=0.15$ (bottom row). The $q_{{\rm CC,Fe}}^{Z}$ value increase
from left to right from 0.35 to 0.5. All density plots are logarithmically
scaled. Note that the center panels have different $q_{{\rm CC,Fe}}^{Z}$
values (middle: 0.45, bottom: 0.4) and that the predicted abundance
distribution for $q_{{\rm CC,Fe}}^{Z}=0.4$ and $dq_{{\rm CC,Fe}}^{Z}/dZ=0.0$
can be found in Figure 3. The $q_{{\rm CC,Fe}}^{Z}$ and $dq_{{\rm
CC,Fe}}^{Z}/dZ$ parameters are listed in each panel. By comparing the lower
panels to the observed distribution (top) we can evaluate the success or
failure of each model; the low-Ia population is most sensitive to the Fe
assumptions.

To better assess the goodness of fit of each model, we calculate the average
$\chi^{2}$ value per star. The models with $q_{{\rm CC,Fe}}^{Z}$ of 0.5 and
0.45, regardless of $dq_{{\rm CC,Fe}}^{Z}/dZ$, have an average $\chi^{2}$ per
star $>90$ while the models with $q_{{\rm CC,Fe}}^{Z}$ of 0.4 and 0.35 have an
average $\chi^{2}$ per star $<55$. In both the metallicity independent and
metalicity dependent cases, the models with $q_{{\rm CC,Fe}}^{Z}=0.4$ have the
lowest average $\chi^{2}$ per star, at 54.54 and 54.47, respectively, though
the models with $q_{{\rm CC,Fe}}^{Z}=0.35$ have a $\chi^{2}$ that is only
greater by $\sim 0.1$. Of the seven models explored here, the case with
$q_{{\rm CC,Fe}}^{Z}=0.4$ and $dq_{{\rm CC,Fe}}^{Z}/dZ=0.15$ has the lowest
average $\chi^{2}$ per star, indicating that the Fe abundances are best fit by
a metallicity dependent prompt process. Introducing this metallicity
dependence subtly changes the shape of the predicted low-Ia distribution in a
way that achieves better agreement with APOGEE observations.

Though the $q_{{\rm CC,Fe}}^{Z}=0.4$ and 0.35 models are similar in terms of
their goodness of fit, their nucleosynthesis implications are different. In
Figure 8, we plot the median value of $f_{ij}^{\rm CC}$ (Equation 10) for the
low-Ia population at solar metallicity ($-0.05<[{\rm Mg}/{\rm H}]<0.05$),
where low-Ia stars are defined by

$\begin{cases}[{\rm Mg/Fe}]>0.12-0.13\,[{\rm Fe/H}],&[{\rm Fe/H}]<0\cr[{\rm
Mg/Fe}]>0.12,&[{\rm Fe/H}]>0,\cr\end{cases}$ (11)

as in W19, W22, and G22. We only show the median $f_{ij}^{\rm CC}$ values for
the models with $dq_{{\rm CC,Fe}}^{Z}/dZ=0.0$ as the solar metallicity median
$f_{ij}^{\rm CC}$ values for the metallicity dependent models are almost
identical for matching values of $q_{{\rm CC,Fe}}^{Z}$. We find that the
choice of $q_{{\rm CC,Fe}}^{Z}$ has little impact on the median $f_{ij}^{\rm
CC}$ values of elements dominated by CCSN enrichment (e.g., O, Al, S, K). As
the delayed contribution increases, the median elemental $f_{ij}^{\rm CC}$
values decrease more significantly with decreasing $q_{{\rm CC,Fe}}^{Z}$. The
choice of $q_{{\rm CC,Fe}}^{Z}$ most impacts the median $f_{ij}^{\rm CC}$
values for Na, Cr, Fe, Mn, and Ce, with the median $f_{ij}^{\rm CC}$ for Mn
decreasing from 0.42 for $q_{{\rm CC,Fe}}^{Z}=0.5$ to 0.22 for $q_{{\rm
CC,Fe}}^{Z}=0.35$. Because the $q_{{\rm CC,Fe}}^{Z}$ value sets the prompt
enrichment plateau, a lower $q_{{\rm CC,Fe}}^{Z}$ model implies a lower
$f_{ij}^{\rm CC}$ value.

While the high $q_{{\rm CC,Fe}}^{Z}$ model can likely be ruled out due to
poorness of fit, the true $q_{{\rm CC,Fe}}^{Z}$ value and its metallicity
dependence are unknown. It is therefore important not to over-interpret the
specific $f_{ij}^{\rm CC}$ values of a given model. The $f_{ij}^{\rm CC}$
parameter can provide qualitative descriptions of which elements have more or
less prompt/delayed enrichment, but the exact values are uncertain.

Figure 8: Elemental median values of $f_{ij}^{\rm CC}$ at solar metallicity
for the low-Ia population for KPM with $q_{{\rm CC,Fe}}^{Z}=0.35$ (darkest
purple) to 0.5 (lightest purple) and $dq_{{\rm CC,Fe}}^{Z}/dZ=0.0$. Elements
are ordered by atomic number. The median $f_{ij}^{\rm CC}$ changes most
dramatically for elements with strong delayed contributions.

###  5.2 Increasing the Number of Processes

In our fiducial model, we adopt $K=2$ with the two processes representing
prompt, CCSN-like enrichment and delayed, SNIa-like enrichment. While a $K=2$
model can well describe the stellar abundances (e.g., Figures 2 and 3), the
abundance residuals cannot be explained by observational noise alone and hold
information about the intrinsic variations from a $K=2$ model (Ness et al.,
2019 G22, W22, Ting & Weinberg, 2022; Ratcliffe & Ness, 2023). Potential
sources of such scatter include metallicity-dependent SN yields with a bursty
star formation history, environmental variations in the IMF, stochastic
sampling of the IMF, and more than two distinct processes (e.g., AGB stars,
merging neutron stars, and unique classes of SNIa) with different time delays
for enrichment (e.g. Belokurov & Kravtsov, 2022; Griffith et al., 2023). Note
that the existence of many enrichment channels is not in itself sufficient for
producing scatter around a $K=2$ model (or even a $K=1$ model); one needs
star-to-star variation in the relative amplitude of these channels. For
example, in a fully mixed one-zone model, all abundances depend only on time,
even if many enrichment channels contribute.

In this Section, we will explore the impact of adding additional processes to
our model, increasing from $K=2$ to $K=4$. Because KPM is sensitive to
enrichment with different time delays, adding components could be interpreted
as adding sources with distinct enrichment time scales. For example, if AGB
stars and SNIa enrich with the same time delay, the model would fit both
sources in one delayed component. If AGB and SNIa enrich with different delay
times, a third component could pick up delayed AGB enrichment not captured by
the original delayed process. Indeed, evidence of a distinct AGB-like process
is identified in G22 and W22, where correlated residuals are used to expand
the two-process model. However, both works add components in a restrictive
manner that requires choosing which elements to assign to 3rd and/or 4th
processes and that does not allow for the original two processes to vary.

Our goal is to demonstrate the potential for using KPM to flexibly model more
than two enrichment channels and improve the accuracy of the abundance
predictions. We allow the model to identify the elements best-fit with
additional components and modify the $K=2$ process parameters. Ultimately,
such a method could be used to identify elements with more than two enrichment
channels, but our data set may not be capable of doing so robustly. In the
$K=4$ case, our model becomes

$m_{ij}=\log_{10}(A^{\rm CC}_{i}\,q_{{\rm CC,}j}^{Z}+A^{\rm Ia}_{i}\,q_{{\rm
Ia,}j}^{Z}+A_{i}^{3}\,q_{3,j}^{Z}+A_{i}^{4}\,q_{4,j}^{Z}),$ (12)

where $q_{3,j}^{Z}$ and $q_{4,j}^{Z}$ are the third and fourth process vector
components and $A_{i}^{3}$ and $A_{i}^{4}$ are the third and fourth process
amplitudes. The model, however, does require some regularization to converge.
As in the $K=2$ case where we assume that Mg is a pure CCSN element and fix
the $q_{{\rm CC,Fe}}^{Z}$ and $q_{{\rm Ia,Fe}}^{Z}$ values, we need elements
to regulate our 3rd and 4th processes. We choose Ce and Mn, two elements with
larger residuals that likely have additional nucleosynthetic sources—Ce from
AGB stars and Mn from distinct classes of SNIa (e.g. Gallino et al., 1998; de
los Reyes et al., 2020; Gronow et al., 2021). To test the impact of our choice
of representative elements we also fit the $K=4$ model with the third and
fourth processes fixed to C+N and Cr. We find that similar groups of elements
are better fit with additional components. We initialize the $K=4$ model at
the $K=2$ model values of $q_{{\rm CC,}j}^{Z}$, $q_{{\rm Ia,}j}^{Z}$, $A^{\rm
CC}_{i}$, and $A^{\rm Ia}_{i}$ with the added constraints that

$q_{{\rm 3,Mg}}^{\,Z}=0,\quad q_{{\rm 3,Fe}}^{\,Z}=0,\quad q_{{\rm
3,Mn}}^{\,Z}=0,\quad q_{{\rm 3,Ce}}^{\,Z}=1\quad$ (13)

and

$q_{{\rm 4,Mg}}^{\,Z}=0,\quad q_{{\rm 4,Fe}}^{\,Z}=0,\quad q_{{\rm
4,Mn}}^{\,Z}=1,\quad q_{{\rm 4,Ce}}^{\,Z}=0\quad$ (14)

at all metallicities. We first fit the $A$-step to only Mg, Fe, Mn, and Ce,
and then conduct 32 iterations of the $q$-step and $A$-step, as described in
Section 2. We again inflate the scatter according to Equation 7 with $Q=5$.

The model converges upon a set of process vector components and amplitudes
that can be combined with Equation 12 to predict the stellar abundances and
calculate the fractional contribution from each process. In Figure 9, we plot
$A^{k}_{i}/A^{\rm CC}_{i}$ vs. $A^{\rm CC}_{i}$ for the SNIa, third, and
fourth processes. We find that processes 3 and 4 are most prominent in stars
at low metallicity and that there is a large population of stars with
$A^{3}_{i}$ and/or $A^{4}_{i}\approx 0$. In Figure 10 we plot the observed and
predicted abundance distributions as well as the process vector components and
fractional contribution from each component for a subset of elements. We note
that the model parameters $q_{3,j}^{Z}$ and $q_{4,j}^{Z}$ should be
interpreted in conjunction with the amplitudes, as we set the third and fourth
process vector components for Ce and Mn to an arbitrary value with no
metallicity dependence.

Figure 9: $A^{k}_{i}/A^{\rm CC}_{i}$ vs. $A^{\rm CC}_{i}$ distribution for
$A^{\rm Ia}_{i}$ (left), $A^{3}_{i}$ (center), and $A^{4}_{i}$ (right). All
density plots are logarithmically scaled. The third and fourth processes
contribute most to low metallicity stars. Figure 10: Left: [X/Mg] vs. [Mg/H]
abundance distributions for the observed sample (first column), $K=2$ model
(second column) and $K=4$ model (third column) for Ca, C+N, Na, Cr, Ni, Mn,
Co, and Ce. Observational errors are not added to the model predictions. All
density plots are logarithmically scaled. Right: process vector components
(fourth column) and median low-Ia fractional contribution from each process
(fifth column) for the $K=4$ (solid lines) and $K=2$ (dashed lines) models to
the low-Ia population as a function of $[{\rm Mg}/{\rm H}]$. We plot the
median $q_{{\rm CC,}j}^{Z}$ and $f_{ij}^{\rm CC}$ in light purple, $q_{{\rm
Ia,}j}^{Z}$ and $f^{\rm Ia}_{ij}$ in dark purple, $q^{Z}_{3,i}$ and
$f^{3}_{ij}$ in light orange, and $q^{Z}_{4,i}$ and $f^{4}_{ij}$ in dark
orange. In the final column we plot a dotted grey line at 0 for reference.

We find that the third process, regularized to Ce, contributes at a low level
to O, Si, S, Al, and K and more significantly to Ca, Na, Cr, and Ce. The
fourth process, regularized to Mn, contributes at a low level to K and more
significantly to S, C+N, Na, Cr, Ni, Mn, and Co. These best-fit element
groupings resemble, but are not identical to, the elements selected for
additional components in W22, where the third process included Ca, Na, Al, K,
Cr, and Ce and the fourth process included Ni, V, Mn, and Co.

In the left-most columns of Figure 10, we plot the $[{\rm X}/{\rm Mg}]$ v.s.
$[{\rm Mg}/{\rm H}]$ distributions for the observed stellar sample, the $K=2$
model predictions, and the $K=4$ model predictions for Ca, C+N, Na, Cr, Ni,
Mn, Co, and Ce. We note that the predicted abundances do not have noise added
(unlike Figures 2 and 3) to highlight the differences between the $K=2$ and
$K=4$ predictions. Comparing the predicted abundances from the $K=2$ and $K=4$
process models, we see that the $K=4$ process model is better able to capture
the abundance scatter than the $K=2$ model, especially at the low metallicity
end of the low-Ia population. This result is expected, as adding more model
components will increase the abundance space that KPM is able to reproduce.

In the fourth and fifth columns of Figure 10, we plot the process vector
components and median $f^{k}_{ij}$ as a function of $[{\rm Mg}/{\rm H}]$ of
the low-Ia population (Equation 11), respectively, where

$f^{k}_{ij}=\frac{A_{i}^{k}\,q_{k,j}^{Z}}{A^{\rm CC}_{i}\,q_{{\rm
CC,}j}^{Z}+A^{\rm Ia}_{i}\,q_{{\rm
Ia,}j}^{Z}+A_{i}^{3}\,q_{3,j}^{Z}+A_{i}^{4}\,q_{4,j}^{Z}}.$ (15)

We include $q_{{\rm CC,}j}^{Z}$ and $q_{{\rm Ia,}j}^{Z}$ as well as median
$f_{ij}^{\rm CC}$ and $f^{\rm Ia}_{ij}$ from the $K=2$ model in respective
columns for comparison. We see that the third process contributes
significantly to Ca, Na, Cr, and Ce at low metallicity, with decreasing
contribution up to $[{\rm Mg}/{\rm H}]\approx 0.1$. The fractional
contribution from the $K=2$ prompt and delayed processes to these elements
decreases under the $K=4$ model. The fourth process behaves in a similar
manner but with elements C+N, Na, Cr, Ni, Mn, and Co. The fractional
contribution from the third and fourth process is nearly identical in the
high-Ia population.

=The statistical improvement in the KPM between the $K=2$ and $K=4$ models is
evident in the $\chi^{2}$ values. In Figure 11, we plot the cumulative
$\log_{10}(\chi^{2})$ distributions for the fits to each star and the total
$\chi^{2}$ for each element for the fiducial $K=2$ model and the $K=4$ model.
We find that the cumulative $\log_{10}(\chi^{2})$ distribution decreases with
the increase in model components by greater than two for most stars, expected
for the addition of two degrees of freedom. We also find that the $\chi^{2}$
per element is lower for all elements in the $K=4$ model. Significant
improvements to Ca, C+N, Mn, and Ce are likely due to the additional third and
fourth components capturing abundance scatter that the original two processes
could not. Notably, we also see a significant improvement in the Fe fit, even
though we require $q_{\rm 3,Fe}^{Z}=q_{\rm 4,Fe}^{Z}=0$. Because all elements
influence the $K=2$ model fit, the fiducial model was likely pulled away from
the best solution for Fe to accommodate another element, like Mn. With the
additional components able to account for the non-Fe-like enrichment, the
original two processes are better able to capture the Fe enrichment.

Figure 11: Left: cumulative distribution of $\log_{10}(\chi^{2})$ for the
$K=2$ model (dotted light purple line) and the $K=4$ model (solid dark purple
line). Right: $\chi^{2}$ per element for the same model fits. Elements are
ordered by atomic number.

Through this investigation, we find that KPM is extendable to $K>2$ processes.
The additional processes improve the model quantitatively, but additional work
is needed to improve the nucleosynthetic interpretability. We provide a
discussion of the future science that KPM and the $K=4$ model enable below.

##  6 Discussion

In this paper, we present KPM, a flexible and data-driven model for inferring
nucleosynthesis yields. KPM describes stellar abundances as the sum of $K$
components, where each component is the product of a metallicity-dependent
process vector component (fit to each element) and a process amplitude (fit to
each star). Combined with a likelihood function and a set of assumptions
(Section 2) that make the processes interpretable in terms of nucleosynthetic
sources, the best-fit KPM parameters can be used to calculate fractional
contributions from each process as well as a full suite of predicted $K$
process abundances.

We fit KPM with $K=2$ to abundances labels for 15 elements and 48,659 RGB
stars in APOGEE DR17, selecting a population that minimizes statistical and
systematic errors while spanning an [Mg/H] of -0.8 to 0.5. In the $K=2$ model,
the first process, fixed to Mg, represents prompt CCSN-like enrichment, and
the second process, fixed to Fe, represents delayed SNIa-like enrichment—but
other nucleosynthetic sources with similar time delays may be mixed into each.
Under our adopted assumptions, the prompt process also contributes to Fe, but
the delayed process does not contribute to Mg, in accordance with theoretical
expectations for CCSN and SNIa. Overall, we find that $K=2$ is a good fit to
the data and that the model successfully recovers the global abundance
patterns in the Milky Way. While KPM does not rely on $[{\rm Fe}/{\rm Mg}]$
vs. $[{\rm Mg}/{\rm H}]$ bimodality or median abundance trends, it is able to
recover the observed bimodal abundance distribution. Further, the fit
parameters $A^{\rm CC}_{i}$ and $A^{\rm Ia}_{i}$ act as combined individual-
process abundance labels, revealing a clearer signature of bimodality at high
metallicity in $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ vs. $[{\rm Mg}/{\rm H}]$ space
than in $[{\rm Fe}/{\rm Mg}]$ vs. $[{\rm Mg}/{\rm H}]$. This suggests that the
KPM fit parameters and predicted abundances could be used as a higher signal-
to-noise tracer of nucleosynthesis, as they are using a justified likelihood
to condense information from 15 elements into two variables.

To test the assumptions of the fiducial model, we explore the impact of
varying the fixed value of $q_{{\rm CC,Fe}}^{Z}$. We find that high values of
$q_{{\rm CC,Fe}}^{Z}$ (0.5 and 0.45) are not able to reproduce the observed
[Fe/Mg] vs [Mg/H] abundance distribution, regardless of the process vector
component’s metallicity dependence. Our requirement that $A^{\rm CC}_{i}$ and
$A^{\rm Ia}_{i}$ are non-negative makes it impossible for the model to
reproduce the lowest $[{\rm Fe}/{\rm Mg}]$ values in the APOGEE data. Values
$q_{{\rm CC,Fe}}^{Z}=0.4$ and 0.35 with $dq_{{\rm CC,Fe}}^{Z}/dZ=0$ or 0.15
produces similarly successful fits. While predicted abundance distributions
appear similar for these models, the implied fractional contribution from the
prompt process is dependent upon the Fe assumption for elements with
substantial delayed enrichment. Through this exploration, we conclude that the
quantitative nucleosynthetic interpretation of KPM is dependent upon the input
assumptions, and that there is inherent uncertainty in the $f_{ij}^{\rm CC}$
values.

Finally, we expand KPM from $K=2$ to $K=4$, regularizing the third and fourth
processes to Ce and Mn. KPM builds off of the original model, such that the
$K=4$ model starts at the $K=2$ solution and then finds the best-fit
parameters for $K=4$, altering the original solution and allowing all elements
(except Mg and Fe) to have contributions from additional processes. We find
that S, Ca, C+N, Na, Cr, Mn, Co, Ni, and Ce are best fit with a third and/or
fourth component, with such processes contributing most significantly at low
metallicity. The information for constraining the $q^{Z}_{k,j}$ values for the
third (fourth) process come from the star-by-star deviations of Ce (Mn) from
the $K=2$ model predictions and their correlation with deviations for other
elements $\text{X}_{i}$. Relative to the approach taken in W22 (Section 8),
our $K=4$ model requires non-negative $q^{Z}_{k,j}$ and $A_{i}^{k}$ for all
elements and stars, and it starts by tying the third and fourth processes to
individual elements rather than groups of elements. The $K=4$ model improves
the ability of KPM to fit the abundances of all elements but especially
improves predictions of Ca, C+N, Fe, Mn, and Ce. This successful
implementation of a $K=4$ model shows that KPM can be extended to $K>2$, and
it has potential future use in constraining enrichment beyond a single prompt
and delayed process—critical to understanding enrichment from AGB stars,
merging neutron stars, and rarer novae.

KPM is based upon the two-process model developed in W19 and W22. While the
two models are identical in format for the $K=2$ case, the model assumptions,
parameter derivations, and implementations differ. The W22 two-process model
derives process vector components from median abundance trends, reliant upon
$[{\rm Mg/Fe}]$ vs. $[{\rm Mg}/{\rm H}]$ bimodality, and fits process
amplitudes to a subset of $2-6$ $\alpha$ and Fe-peak elements. KPM, on the
other hand, employs a likelihood function fit to all stars and all elements to
derive both process amplitudes and vector components. Our more data-driven
implementation results in the improved ability of the $K=2$ model fit to
predict all of a star’s abundances. Notably, KPM can better predict C+N and Mn
abundances than the W22 two-process model, since all elements are used to
constrain the fits. The most significant improvement to the original two-
process model, though, is in KPM’s flexibility. The flexible implementation of
the model allows us to easily vary the assumptions, such as $q_{{\rm
CC,Fe}}^{Z}$, and increase the number of model components to study the impact
of our assumptions on the results and push the interpretation of KPM beyond
standard CCSN and SNIa nucleosynthesis in a less restrictive manner than W22
and G22.

However, KPM is not without its own faults. The assumptions listed in Section
2 may incorrectly skew our results, and the model could benefit from
improvements in implementation. While assumptions (4) and (5) on Mg and Fe
production are flexible, KPM requires that both elements have fixed process
vector components. If our assumptions are incorrect and, for instance, Mg is
not a pure prompt element or (in the $K=4$ case) Fe has contributions from
multiple delayed sources, our nucleosynthetic interpretation of KPM may be
wrong. This becomes more challenging as $K$ increases and we have to make more
assumptions to break rotational symmetries. Additionally, assumption (7)
states that the APOGEE data products can be used for this project, but we
inflate outlying-star abundance errors with a softening parameter, $Q$ to
account for their likely underestimation. It is also possible that $Q$ is
accounting for some of the real intrinsic scatter in the data and inflating
the observational error on true outlier stars. In future KPM implementation,
the development of a more robust method to justifiably down-weight outlier
stars from the global fits would be beneficial. This method should account for
both non-Gaussian observational errors (e.g., from bad telluric subtraction or
unlucky line blends) and physically interesting outliers (e.g., from binary
mass transfer). Finally, KPM fits process vector components along a spline
with 11 knots (assumption 6), and those knots have fixed locations in
metallicity. As this method fits a polynomial between each knot, it can result
in sharp features at the knot locations in metallicity regions with few points
or large scatter. Fitting process vector components with a differentiable
function might be more reasonable, though results shown in Appendix C.1 in G22
suggests that this change might have minimal impact on the results. In general
this model for the metallicity-dependence of the yields is very rigid; a
better model could both have more flexibility and be smoother.

Beyond improvements to the underlying model assumptions and implementation,
KPM needs to include parameter uncertainty. While the model delivers process
vector components and amplitudes, which can be used to calculate $f^{k}_{ij}$
and $K$ process predicted abundances, the current implementation does not
return errors on any variable. The best method to derive such errors has not
been explored, but one could use the likelihood function or bootstrapping.
These methods will encapsulate the uncertainty on process parameters from the
APOGEE abundance errors but will not capture the uncertainty due to model
assumptions, such as $q_{{\rm CC,Fe}}^{Z}$ (Section 5.1).

While such future changes will improve the model, the current form of KPM and
its data products can support ongoing research and will enable new science.
Most immediately, KPM provides high signal-to-noise abundance labels, $A^{\rm
CC}_{i}$ and $A^{\rm Ia}_{i}$, as well as de-noised stellar abundances
($m_{ij}$). The best-fit values of $A^{\rm CC}_{i}$ and $A^{\rm Ia}_{i}$, in
particular, are powerful tracers of nucleosynthesis. They show a bimodality at
all metallicities, as do some of the de-noised abundances. And—because the
model is a maximum-likelihood model—they represent information-theory optimal
combined measures of $\alpha$ and Fe-peak abundances. That is, these data-
driven amplitudes could replace more theory-driven measures of the relative
contributions of CCSN and SNIa enrichment channels.

In Section 4.1, we showed that the high-Ia and low-Ia populations are more
clearly defined in $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ vs. [Mg/H] than in [Fe/Mg]
vs/ [Mg/H]. In amplitude space, the low-Ia population can be re-defined as

$\begin{cases}A^{\rm Ia}_{i}/A^{\rm CC}_{i}<0.35,&[{\rm Mg}/{\rm H}]<-0.2\cr
A^{\rm Ia}_{i}/A^{\rm CC}_{i}<0.5+0.7\,[{\rm Mg}/{\rm H}],&-0.2\geq[{\rm
Mg}/{\rm H}]<0.1,\cr A^{\rm Ia}_{i}/A^{\rm CC}_{i}<0.57,&[{\rm Mg}/{\rm
H}]\geq 0.1\,.\cr\end{cases}$ (16)

Compared to Equation 11 (W19, W22), this new definition re-classifies 647
stars as high-Ia and 224 stars as low-Ia. We show the location of these stars
in $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ vs. [Mg/H] and [Mg/Fe] vs. [Fe/H] in Figure
12. Many of the re-classified stars are at $[{\rm Fe/H}]>-0.1$. When dividing
in [Mg/Fe], it is difficult to correctly separate the populations at high
metallicity, as they are blended together. Our new definition also re-
classifies many stars near [Fe/H] of -0.3 as high-Ia, suggesting that the W19
and W22 high-Ia definition has too shallow a slope. While only $\sim 2\%$ of
stars are re-classified under the new definition, we suggest that Equation 16
be used to chemically define the low-Ia and high-Ia populations if KPM fits
are available, especially if studying stars with $[{\rm Fe/H}]>-0.1$. Beyond
improving the definition of high-Ia and low-Ia populations, the KPM parameters
and predicted abundances could be used in any current analysis that strives to
show trends with abundance labels. We predict that trends of stellar
parameters with [X/H] will be clearer when comparing to $m_{ij}$, $A^{\rm
CC}_{i}$, or $A^{\rm Ia}_{i}$.

Figure 12: Left: distribution of stars in $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ vs.
[Mg/H] where the black dashed line is the dividing line between the high-Ia
and low-Ia populations (Equation 16). Stars that are re-classified as high-Ia
are shown in purple (647 stars) and stars that are re-classified as low-Ia are
shown in orange (224 stars). Right: Same as left, but in [Mg/Fe] vs. [Fe/H]
space. The high-Ia and low-Ia populations have been labeled in both panels for
clarity. The symbols emphasize the stars at the edge of the populations but
the re-classified stars make up only $\sim 2\%$ of the total population.

Such analysis with KPM parameters will be useful in studying nucleosynthesis,
dynamics, disk formation, stellar ages, and much more. However, in this paper
we only present fits for a small population with restricted stellar
parameters, relative to the full APOGEE sample. While KPM could be fit to the
full APOGEE sample, systematic abundance effects with $T_{\rm eff}$ and
$\log(g)$, as well as other abundance artifacts (e.g., Jönsson et al., 2020;
Griffith et al., 2021a), cause the abundance trends to differ across the
Hertzsprung-Russell diagram. The best-fit KPM parameters for the giants would
differ from those for the dwarfs. If such systematics could be accounted for
(see Sit et al. in prep) we could fit the full APOGEE stellar sample with KPM,
or train KPM on a subset of high signal-to-noise stars and apply the fits to
the full sample. This potential future analysis could reveal additional
information about the nucleosynthetic history of our Galaxy and would provide
higher signal-to-noise abundance labels for the full sample.

The success of the two-process model (W19, W22) and KPM with $K=2$ suggests
that the distribution of disk stars in APGOEE abundance space is largely two-
dimensional (2D), though more dimensions are required to fully explain the
data (Ting & Weinberg, 2022, W22). In this paper, we have focused on a 2D
nucleosynthetic model, with the two dimensions representing prompt CCSN-like
enrichment and delayed SNIa-like enrichment. However, another 2D class of
theoretical models for the Milky Way exists, describing stars in terms of
birth radius and birth date (e.g., Frankel et al., 2018; Ness et al., 2022).
Are these two 2D models related? If they are, then the nucleosynthetic
parameters from KPM ($A^{\rm Ia}_{i}$ and $A^{\rm CC}_{i}$) should predict
asteroseismic ages (or masses), up to unpredictable aspects of mass transfer,
as well as the guiding center radius, up to unpredictable aspects of radial
migration. While a deeper study of the implications of the disk’s two-
dimensionality is outside the scope of this work, we show the relationship
between asteroseismic age from the APOKASC sample (Pinsonneault et al., 2018)
and process amplitudes in Figure 13. Here we see a clear gradient in age with
$A^{\rm CC}_{i}$ and $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ (as in G22 and W22),
though outlier stars are scattered throughout. We predict that the KPM
parameters will be better age diagnostics than APOGEE abundances, and that age
outliers may be mass transfer objects.

Figure 13: $A^{\rm Ia}_{i}/A^{\rm CC}_{i}$ vs. $A^{\rm CC}_{i}$ distribution
for stars in the APOKASC sample (Pinsonneault et al., 2018) Each point is
colored by the star’s asteroseismic $\log_{10}$(age), with younger stars in
black and older stars in yellow. A clear gradient with $\log_{10}$(age) and
process parameters exists with young outliers scattered throughout. This shows
that the process amplitudes constitute a good-quality age indicator.

Finally, because of the flexibility of KPM, new scientific applications are
enabled that were not feasible before. Because KPM does not rely on [Mg/Fe]
vs. [Fe/H] bimodality, non-bimodal populations can now be fit with a multi-
component nucleosynthetic model. KPM could be applied to the low metallicity
disk, halo, Gaia Enceladus Sausage, Nubecula Major, Nubecula
Minor777Historically referred to as the LMC and SMC., other Milky Way
satellites, and more. KPM can also be easily extended to $K>2$ in a much less
restricted way than the two-process model. While a $K=2$ model well describes
the global abundance patterns, intrinsic residual scatter on the scale of 0.01
to 0.02 dex remains (Ting & Weinberg, 2022, W22, G22). This scatter could be
signatures of enrichment from non-CCSN/SNIa sources, stochastic sampling of
the IMF, environmental IMF variations, or metallicity-dependent SN yields with
a bursty star formation history (e.g. Belokurov & Kravtsov, 2022; Griffith et
al., 2023). While it is difficult to identify non-CCSN or SNIa enrichment in
the APOGEE data alone, where only C+N and Ce are expected to have significant
contributions from other sources, there may be signatures in other surveys
with better coverage of heavier elements. Applying a $K>2$ model to GALAH
(Buder et al., 2021), or an overlapping sample of APOGEE and GALAH stars
(Nandakumar et al., 2022), could prove more successful.

In the $K=2$ and $K>2$ cases, results from KPM will help us disentangle our
Galactic formation and enrichment history. This data-driven model opens doors
to many new research projects and exciting future scientific results. To use
KPM yourself, please reference the KPM GitHub
repository888https://github.com/13emilygriffith/KProcessModel or contact the
corresponding author.

##  7 Acknowledgements

It is a pleasure to thank Polly Frazer (NYU), Adrian Price-Whelan (Flatiron),
Tawny Sit (OSU), Soledad Villar (JHU), the Darling research group at CU
Boulder, CU Boulder Research Computing services and staff, and the
Astronomical Data group at the Flatiron Institute for valuable discussions and
help.

E.J.G. is supported by an NSF Astronomy and Astrophysics Postdoctoral
Fellowship under award AST-2202135. B.R. acknowledges support by the Deutsche
Forschungsgemeinschaft under the grant MI 2009/2-1.

Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P.
Sloan Foundation, the Heising-Simons Foundation, the National Science
Foundation, and the Participating Institutions. SDSS 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 is managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS Collaboration, including the Carnegie Institution for
Science, Chilean National Time Allocation Committee (CNTAC) ratified
researchers, the Gotham Participation Group, Harvard University, Heidelberg
University, The Johns Hopkins University, L’Ecole polytechnique fédérale de
Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-Planck-
Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für
Extraterrestrische Physik (MPE), Nanjing University, National Astronomical
Observatories of China (NAOC), New Mexico State University, The Ohio State
University, Pennsylvania State University, Smithsonian Astrophysical
Observatory, Space Telescope Science Institute (STScI), the Stellar
Astrophysics Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Illinois
at Urbana-Champaign, University of Toronto, University of Utah, University of
Virginia, Yale University, and Yunnan University.

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