# The Chemical Diversity of the Metal-Poor Milky Way

Nicole Buckley,1 Payel Das,1 Paula Jofré,2 Robert M. Yates3 and Keith Hawkins
4
1Department of Physics, University of Surrey, Stag Hill, Guildford, GU2 7XH,
UK
2Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias,
Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile
3Centre for Astrophysics Research, University of Hertfordshire, Hatfield, AL10
9AB, UK
4Department of Astronomy, The University of Texas at Austin, 2515 Speedway
Boulevard Austin, TX 78712, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a detailed study of the chemical diversity of the metal-poor Milky
Way (MW) using data from the GALAH DR3 survey. Considering $17$ chemical
abundances relative to iron ([X/Fe]) for $9,923$ stars, we employ Principal
Component Analysis (PCA) and Extreme Deconvolution (XD) to identify $10$
distinct stellar groups. This approach, free from chemical or dynamical cuts,
reveals known populations, including the accreted halo, thick disc, thin disc,
and in-situ halo. The thick disc is characterised by multiple substructures,
suggesting it comprises stars formed in diverse environments. Our findings
highlight the limited discriminatory power of magnesium in separating accreted
and disc stars. Elements such as Ba, Al, Cu, and Sc are critical in
distinguishing disc from accreted stars, while Ba, Y, Eu and Zn differentiate
disc and accreted stars from the in-situ halo. This study demonstrates the
potential power of combining a latent space representation of the data (PCA)
with a clustering algorithm (XD) in Galactic archaeology, in providing new
insights into the galaxy’s assembly and evolutionary history.

###### keywords:

Galaxy: abundances – Galaxy: structure – Stars: abundances – Stars: Population
II – Methods: data analysis – Methods: observational

††pubyear: 2023††pagerange: The Chemical Diversity of the Metal-Poor Milky
Way–A

##  1 Introduction

The Milky Way’s (MW) stellar components have distinct star formation histories
(SFHs) that manifest in unique chemical signatures. Chemical diversity, which
measures the spread of elemental abundances, can indicate contributions from
both nucleosynthetic processes and accreted systems. Variations in chemical
diversity are more significant in the stellar halo than in the disc. This is
because the halo has accumulated stars from various smaller galaxies, each
with distinct SFHs, whereas the disc has experienced more thorough chemical
homogenisation due to effective material mixing (Feng & Krumholz, 2014; Reddy
et al., 2006).

The thick disc, first described by Gilmore & Reid (1983), consists of prograde
stars ($L_{z}>0$) with scale heights between $530\pm 32$ pc and $630\pm 29$ pc
above the Galactic plane (Kumar et al., 2021). These stars are older than
those in the thin disc ($\geq 10$ Gyr), more metal-poor ($-2.2\leq$ [Fe/H]
(dex)$\leq-0.5$), and show higher enhancements of $\alpha$ elements relative
to iron ([$\alpha$/Fe] $>0.2$ dex) (Fuhrmann, 1998; Freeman & Bland-Hawthorn,
2002; Navarro et al., 2011; Hayden et al., 2015). The thick disc likely formed
quickly, marked by an early burst of star formation in the MW’s history. Its
formation theories include: (i) secular evolution through heating and
primordial collapse (Prantzos et al., 2023), (ii) a significant merger, e.g.
Gaia Enceladus Sausage (GES), influencing the proto-disc before the thin
disc’s formation (Helmi et al., 2018; Belokurov et al., 2018; Bignone et al.,
2019; Ciucă et al., 2023), and (iii) direct accretion of stars from satellite
galaxies (Reddy et al., 2006).

The thin disc dominates the stellar material in spiral galaxies, with a high
and constant star formation rate (SFR) due to abundant gas, leading to many
young stars ($<8$ Gyr (Yu et al., 2021)). It is the low-$\alpha$
([$\alpha$/Fe] $<0.2$ dex (Navarro et al., 2011)) region of the galactic disc
and is more metal-rich due to later gas accretion. The thin disc extends to
heights of $\approx 230\pm 20$ pc to $330\pm 11$ pc (Kumar et al., 2021).

The MW stellar halo comprises two main parts: the inner and outer halo
(Carollo et al., 2007; Nissen & Schuster, 2010; Beers et al., 2012). Within
the inner halo ($R_{\mathrm{gal}}<15\mathrm{kpc}$), there are high-$\alpha$
(Ishigaki et al., 2012) and low-$\alpha$ components (Nissen & Schuster, 2010;
Hawkins et al., 2014; Nissen & Schuster, 2012; Sheffield et al., 2012; Bensby
et al., 2003; Jackson-Jones et al., 2014). The high-$\alpha$ component
includes in-situ stars, GES, and the heated high-$\alpha$ disc (Zolotov et
al., 2009; Nissen & Schuster, 2010; Tissera et al., 2014; Pillepich et al.,
2015), formed from bursty star formation triggered by mergers (Fernández-Alvar
et al., 2019; Ledinauskas & Zubovas, 2018; Liu et al., 2019; Emami et al.,
2021). The low-$\alpha$ component is comprised of disrupted satellites with
extended, less intense episodes of star formation. The inner halo is primarily
influenced by GES, a significant dwarf galaxy accreted $\approx 8-11$ Gyr ago,
marking the MW’s last major minor merger (Belokurov et al., 2020).

Due to the availability of large stellar spectroscopic surveys, e.g. GALAH
(Buder et al., 2021), APOGEE (Majewski et al., 2017), SDSS/SEGUE (Yanny et
al., 2009), LAMOST (Cui et al., 2012) and H3 (Conroy et al., 2019), many
studies have used integrals of motion and chemical abundances to cluster and
disentangle systems with different histories. Dynamical tagging is one method
used, in particular using quasi-conserved quantities such as orbital actions
($J_{\rm z}$, $J_{\rm R}$ and $L_{\rm z}$) and energy ($E$). Although these
quantities are not conserved over long timescales, the $E-L_{Z}$ space has
been instrumental in identifying distinct accreted structures such as GES
(Belokurov et al., 2018; Helmi et al., 2018; Helmi & de Zeeuw, 2002), Sequoia
(Myeong et al., 2019), and Kraken (a possible pre-disc in-situ population)
(Kruijssen et al., 2020; Forbes, 2020).

Chemical tagging is the process of grouping stars together based on chemical
abundance ratios 111[X/Y] $=\log(N_{X}/N_{Y})_{*}-\log(N_{X}/N_{Y})_{\odot}$,
is the logarithmic ratio between two abundances with respect to solar
abundances, where $N_{X}$ represents the number of ‘X’ atoms per unit volume.
(Freeman & Bland-Hawthorn, 2002). It assumes that stars born in the same
population are chemically similar to each other but chemically distinct from
other stars born at different times and/or locations. This allows us to use
their preserved chemical signatures as snapshots of the galaxy’s accretion
history, facilitating a detailed reconstruction of the MWs formation through
chemical tagging (Robertson et al., 2005; Font et al., 2006; Bedell et al.,
2018). Buder et al. (2022) analyses up to $30$ element abundances, using them
to successfully distinguish GES from the in-situ populations and find that the
chemical signatures of Mg, Si, Na, Al, Mn, Fe, Ni, and Cu, significantly
differ from in-situ MW stars. Through Gaussian mixture models applied to these
chemical abundances, they isolate $1049$ stars associated with GES.

In the literature, two-dimensional hyperplanes, such as the Tinsley-
Wallerstein hyperplane which features [$\alpha$/Fe] and [Fe/H] (Tinsley, 1979)
have been used. The early-type dwarf galaxies formed stars early and have
predominantly old stellar populations. The later-types with ongoing star
formation have shallower [$\alpha$/Fe] - [Fe/H] slopes, which separates it
from the steeper gradients present for MW stars. This is due to a difference
in SFR, which is a result of the relative size difference between the smaller
satellite systems and the MW (Das et al., 2020).

Another informative two-dimensional plane is [Mg/Mn] - [Al/Fe], shown in Das
et al. (2020) to effectively distinguish between accreted populations and the
metal-poor disc (Hawkins et al., 2015; Mackereth et al., 2019; Price-Jones et
al., 2020). Mg and Mn are relatively pristine tracers of core-collapse
supernovae (CCSNe) and type 1a supernovae (SNe1a), respectively (Matteucci,
2016). Similarly to [$\alpha$/Fe], the contrasting timescales of these events
make [Mg/Mn] an invaluable tool for decoding the SFHs of systems. As for
[Al/Fe], accreted populations, such as GES (Hawkins et al., 2015) exhibit sub-
solar values of this ratio due to the metallicity dependence of the light
odd-Z element production (Ting et al., 2012; Kobayashi et al., 2006).

Exploring other nucleosynthesis channels, Manea et al. (2023) emphasised the
role of neutron-capture elements in distinguishing between disc stars,
significantly reducing the number of chemical doppelgangers. The ratio of
s-process to r-process elements, especially [Ba/Eu], is important here.
Analysing [Ba/Eu] ratios within these galaxies reveals sub-solar values at low
metallicities ([Fe/H] $<-1.7$ dex), followed by a marked increase as more
massive stars dominate ISM enrichment (Lanfranchi et al., 2006). The
difference between light s-process (ls) elements, such as yttrium and
lanthanum, and heavy s-process (hs) elements, such as Ba is also crucial.
These elements are primarily synthesised during the AGB phase of stellar
evolution, with the [hs/ls] ratio serving as a gauge for neutron source
efficiency and s-process output. Lower metallicity in AGB stars typically
results in a higher [hs/ls] ratio, indicating stronger neutron-capture
activity and the significant influence of the 13C pocket (Kappeler et al.,
1989; Busso et al., 1999). Due to the obscurity surrounding the sites and
yields of neutron-capture elements, there has been little analysis of these
heavier elements in abundance space.

While two-dimensional abundance hyperplanes effectively highlight enrichment
differences between the halo and disc, the boundary between the inner halo and
metal-poor disc requires more information to break down the degeneracy-
implying that this region is more chemically complex and has a greater
chemical diversity. To uncover the extent of the chemical diversity in the MW,
techniques have been applied to reveal the number of dimensions in chemical
space (hereafter referred to as $\mathcal{C}$-space, drawing from Freeman &
Bland-Hawthorn 2002), and the abundance ratios which are most effective at
breaking the chemical degeneracy. Having a high-dimensional
$\mathcal{C}$-space is heavily recommended in Andrews et al. (2017) for the
next generation of multi-element stellar abundance surveys.

The choice of appropriate abundances is key when trying to recover signatures
of accretion events. In Ting & Weinberg (2022), they determine that for an
observational uncertainty of $\approx 0.01-0.015$ dex, $5-7$ abundances must
be conditioned on. This is shown in Anders et al. (2018), who used t-SNE
(t-distributed Stochastic Neighbor Embedding), a dimensionality-reduction
technique, to dissect $\mathcal{C}$-space revealing insights into the chemical
diversity of the disc and suggesting that it is formed from various progenitor
systems and is chemically complex. In this paper, we investigate this chemical
diversity and assess the chemical degeneracy between the inner halo and the
metal-poor MW (defined by $\mathrm{[Fe/H]}\leq-0.5$ dex) in the GALAH survey.
The focus on the metal-poor MW is due to the high level of overlap in in-situ
disc, halo and accreted populations when viewed in two-dimensional abundance
planes and to also find known without the use of dynamical or chemical
selection cuts- other than a metallicity cut. We aim to reveal nucleosynthesis
contributions in the metal-poor MW and identify reproducible chemical
hyperplanes for clustering chemically similar groups. After applying XD to the
PCA-transformed $\mathcal{C}$-space, we distinguish these groups by their
chemodynamical properties and compare them with known structures. By examining
the metal-poor MW’s chemical diversity, we seek to determine: (i) the most
effective chemical abundances for resolving chemical degeneracy in the disc
and disc-halo region; (ii) the effectiveness of unsupervised clustering
algorithms in identifying chemically coherent groups in the transformed
$\mathcal{C}$-space; (iii) insights into the formation histories of the thick
disc and stellar halo.

In Section 2 we discuss the selection of the $17$ element enhancements
([X/Fe]) from GALAH DR3, and the sample cuts. In Section 3 we utilise PCA to
transform a high-dimensional ($17$) $\mathcal{C}$-space to a lower one ($9$),
on which Extreme Deconvolution (XD) is applied to find chemically coherent
groups. In Section 4, we evaluate the contributions of the $17$ [X/Fe] ratios
to the variance of the $\mathcal{C}$-space, which is examined in Section 5.

##  2 Data

We first present the GALAH+ (GALactic Archaeology with HERMES) DR3 (Buder et
al., 2021) data and then discuss the chemical elements that will be included
to break down degeneracy in $\mathcal{C}$-space. The GALAH survey aims to
explore the chemical and kinematic histories of the MW. Using the HERMES
spectrograph at the Australian Astronomical Observatory’s $3.9$m Anglo-
Australian Telescope, GALAH is a high resolution ($R\approx 28,000$), optical
survey. One of its core strengths is its large amount of data and
determination of chemical abundances for up to $30$ different elements for
five different nucleosynthesis channels. Their abundance determination
workflow uses Spectroscopy Made Easy (SME), which is a spectral synthesis
fitting code (Piskunov & Valenti, 2017).

In this work, we select $17$ [X/Fe] ratios (X = Mg, Si, Ca, Ti, Ba, La, Y, Eu,
Mn, Zn, Co, Cr, Cu, Ni, Al, K and Sc), which were taken from an initial set of
$30$ provided by GALAH. These abundance ratios have been chosen for their
reliability and ability to capture key nucleosynthesis signatures in metal-
poor stars. Abundance ratio measurements that are not measured in the imposed
quality cuts are not included. Some elements were deliberately not included in
this analysis: O and Na (both are light elements that are affected by internal
mixing and depletion (Cohen, 2004)). Additionally [Fe/H], which is the iron
abundance, was not included as [Fe/H] is highly correlated with the abundance
ratios [X/Fe] that are considered in this work. We briefly summarise the
elements included here below.

  * •

$\mathrm{[Mg,Si,Ca,Ti/Fe]}$ \- The [$\alpha$/Fe] ratio is widely used to
differentiate between CCSNe, which enrich the ISM with $\alpha$-elements on
short timescales, and SNe1a, which contribute iron-peak elements across longer
timescales. The play-off between these nucleosynthesis processes leads to a
characteristic ‘knee’ in the [Fe/H]-[$\alpha$/Fe] plane, the location of which
can indicate the mass of the system, as more massive galaxies retain and
accumulate more metals before SNe1a begin to dominate (Hendricks et al., 2014;
De Boer et al., 2014). Previous works highlight the chemical dichotomy of
[$\alpha$/Fe] in both the disc and halo of the MW, acting as key indicators of
SFH and the timing of chemical enrichment (Tinsley, 1979).

  * •

$\mathrm{[Ba/Fe]}$ \- As a hs-element, Ba features super-solar values in
systems with extended SFHs, indicative of enhancement due to AGB
nucleosynthesis around $10$ Myrs after star formation (De Los Reyes et al.,
2022). At low metallicities, Ba is synthesized via the r-process, contributing
to the chemical diversity observed in ancient stellar populations (Busso et
al., 1999; Frebel, 2018).

  * •

$\mathrm{[La,Y/Fe]}$ \- Observations suggest that ls elements exhibit more
scatter than their heavy counterparts and are more overabundant at low
metallicities (Burris et al., 2000; Aoki et al., 2005). This scatter could be
indicative of varied nucleosynthetic sites and processes contributing to their
abundances, reflecting different chemical evolution paths.

  * •

$\mathrm{[Eu/Fe]}$ \- Eu is predominantly produced by the r-process and serves
as an excellent tracer of neutron-capture processes. The sources of the
r-process are currently thought to be CCSNe, which contributes to the elevated
Eu-enhancement observed at low-metallicities, as well as neutron star mergers
(NSMs). NSMs occur on similar delay timescales to SNe1a (Wanajo et al., 2021),
which leads to an overall increase of Eu abundance in time. Although studies
indicate that NSMs are rare events, the observed r-process abundance trends in
dwarf galaxies can predominantly result from stochastic effects (Beniamini et
al., 2016; Ji et al., 2016; Carrillo et al., 2022). This pattern is evidenced
in dwarf galaxies from the Local Group such as Carina and Draco, which exhibit
notable Eu enrichment ([Eu/Fe] $\approx 0.5$ dex) as observed by Lanfranchi et
al. (2006).

  * •

$\mathrm{[Mn/Fe]}$ \- Mn is a more pristine tracer of SNe Ia compared to iron,
which has other formation channels. Studies have shown that [Mn/Fe] is
underabundant in low-metallicity stars and increases towards solar values at
higher metallicities (Feltzing et al., 2007; North et al., 2012).

  * •

$\mathrm{[Zn,Co/Fe]}$ \- Zn and Co are interesting for their roles in
hypernovae (HNe) nucleosynthesis, providing insights into the high-energy end
of stellar end-of-life processes (Kobayashi et al., 2020). High [Zn/Fe] ratios
in very metal-poor stars suggest CCSNe as a major site for Zn production but
the required quantities are more consistent with energetic HNe. In later
phases of galactic evolution, the major Zn-production site likely switches
from HNe to SNe1a. This dual contribution led to the solar isotope composition
of Zn and solar [Zn/Fe] values in the disc (Tsujimoto & Nishimura, 2018). Low
[Zn/Fe] can imply formation in an external environment with a low star
formation efficiency and where the gas has been poorly enriched by massive
stars (Minelli et al., 2021).

  * •

$\mathrm{[Cr,Cu,Ni/Fe]}$ \- Cr is primarily produced in SNe1a and CCSNe, with
its abundance reflecting the integrated yield of these events over time. Cu,
on the other hand, has a more complex nucleosynthetic origin, involving both
SNe1a and CCSNe, but also significant contributions from AGB stars. Ni is
predominantly synthesised in SNe1a, making its abundance a valuable tracer for
these events.

  * •

$\mathrm{[Al,K,Sc/Fe]}$ \- Al and K are included for their potential to reveal
unique nucleosynthetic signatures from massive stars and AGB stars. The in-
situ and accreted halo appear to exhibit different values of [Al/Fe], with
metal-poor populations having a lower efficiency of Al production due to
inefficient C, N and O in the ISM (see Kobayashi et al. 2006 for further
details).

We select stars with a sufficient spectrum signal-to-noise ratio (S/N), with
no identified problems with stellar parameter determination, no identified
problems with abundance determination of any of the elements we consider, and
with a low uncertainty in abundance ratios by imposing the following cuts to
the GALAH data (as recommended in Buder et al. 2021):

  * •

snr$\\_$c3$\\_$iraf $>30$

  * •

flag$\\_$sp $==0$

  * •

flag$\\_$Fe$\\_$H $==0$

  * •

flag$\\_$X$\\_$Fe $==0$

  * •

e$\\_$X$\\_$Fe $<0.2$

Without these cuts, spurious relations can appear and distort the abundance
relations by large residual correlations (Ting & Weinberg, 2022; Jofré et al.,
2019). We only select red giant branch (RGB) stars ($\log{g}<3$ and
$\log{T_{\mathrm{eff}}}<3.73$), due to their high intrinsic luminosity which
allows us to probe more distant stars. By conditioning on the stellar
parameters $T_{\mathrm{eff}}$ and $\log{g}$, we also ensure that our sample is
not impacted by the different measurement systematics between spectral types
(Jofré et al., 2019; de Mijolla et al., 2021). A final additional cut of
[Fe/H] $\leq-0.5$ dex has been made to focus on metal-poor stars. After making
these cuts, our sample size is reduced from $678,423$ to $9,923$ stars.

Figure 1: The logarithmic surface gravity and effective temperature of GALAH
stars. The grey points represent the full sample (without the cuts discussed
above), while overlaid points are of our working sample coloured by their
metallicity. This limits our sample to RGB stars, which are selected due to
their high luminosity.

.

##  3 Methods

In the following section, we describe the details of using PCA to find a
compact latent space transformation of the high-dimensional
$\mathcal{C}$-space. The transformation allows us to explore the importance of
each element in the chemical diversity of the dataset and helps the XD
clustering method by reducing the dimensionality of the dataset (Bovy et al.,
2011; Mukherjee et al., 2024).

###  3.1 Principal Component Analysis (PCA)

Principal Component Analysis (PCA) is a dimensionality reduction technique
that forms linear combinations of the original variables, ordered by their
information content (i.e. contribution to the total variance). By removing
those principal components (PCs) with the least information, the data can be
simplified in an optimal way, making it easier to visualise and allowing
clustering algorithms to find groups more cleanly (Jolliffe & Cadima, 2016).

We refer to our analysis as Principal Component Abundance Analysis (PCAA,
coined by Andrews et al. (2012)). Previous studies have shown PCA’s potential
to reveal latent correlations between chemical elements, providing insights
into nucleosynthetic pathways and the galaxy’s chemical evolution (Ting et
al., 2012; Andrews et al., 2012; Ting et al., 2019). This process yields
eigenvectors (PCs) and eigenvalues representing the variance captured by each
component. A larger eigenvalue indicates that its corresponding PC captures
more variance, holding more ‘information’ about the original data. By
projecting onto a subset of the most significant eigenvectors, the data set
can be dimensionally reduced.

PCA’s linearity, unlike t-SNE, necessitates that our data reflect this
property. The decision to work in [X/Fe] space is based on two assumptions:
(1) [X/Fe] correlations are linear due to their logarithmic nature; and (2)
[X/Fe] errors are Gaussian distributed. [X/H] ratios are highly correlated,
offering less distinctiveness in the $\mathcal{C}$-space. Ting et al. (2012)
analysed the dimensionality of $\mathcal{C}$-space in various metallicity
bands, revealing that neutron-capture abundances significantly contribute to
observed variances in low-metallicity ranges ($-3.5<\mathrm{[Fe/H]}<-1.5$
dex). They demonstrated that the dimensionality of $\mathcal{C}$-space,
defined by the number of PCs required to encapsulate intrinsic variations, is
accurately represented when the cumulative variance of the first $k$ PCs is
around $85\%$. This threshold is influenced by sample size, observed
abundances, and their uncertainties. Unlike two-dimensional chemical
hyperplanes, PCA probes how different elements evolve together, revealing the
chemical diversity of the MW.

###  3.2 Extreme Deconvolution (XD)

In this study we employ XD, a clustering algorithm, designed to recover an
underlying multi-Gaussian distribution, even in the presence of noise. We
start with the assumption that the $N$ observed data points are drawn from a
mixture of $k$ multivariate Gaussian distributions with means
$\mu_{k}\in\mathbb{R}^{d}$ and covariances $\Sigma_{k}\in\mathbb{R}^{d\times
d}$, where $d$ is the number of dimensions. For each observed data point
$x_{i}$, there is an associated observational error, which can be modelled as
a Gaussian. The covariance matrix of this error is represented as $\chi_{i}$.
The initial predictions on the fraction of the total $N$ points that belong to
the $k^{\mathrm{th}}$ Gaussian component are given by $\pi_{k}$, such that
$\sum^{K}_{k=1}\pi_{k}=1$. The initial estimates for $\pi_{k}$, $\mu_{k}$ and
$\Sigma_{k}$ are initialised using the `sklearn.mixture.GaussianMixture`
package, and are generated using K-Means. The XD algorithm employs the
Expectation-Maximisation (EM) technique, which consists of two steps: the
E-step, where the posterior probabilities $\omega_{k,i}$ are computed, and the
M-step, where the Gaussian parameters are updated.

In the E-step, the algorithm calculates the probability that data point
$x_{i}$ belongs to the $k^{\mathrm{th}}$ Gaussian component using Bayes’
theorem:

$\omega_{k,i}=\frac{p(x_{i}|\mu_{k},\Sigma_{k}+\chi_{i})}{\sum_{j=1}^{k}\pi_{j}p(x_{i}|\mu_{j},\Sigma_{j}+\chi_{i})},$
(1)

where $p(x_{i}|\mu_{k},\Sigma_{k}+\chi_{i})$ represents the probability
density function of the Gaussian distribution with mean $\mu_{k}$ and
covariance $\Sigma_{k}$ evaluated at $x_{i}$. In the M-step, the estimates for
$\mu_{k}$ and $\Sigma_{k}$ are updated. The revised mean $\mu_{k}$ for all $k$
is:

$\mu_{k}=\frac{1}{N_{k}}\sum_{i=1}^{N}\omega_{k,i}x_{i},$ (2)

with $N_{k}$ being the effective count of data points ascribed to the
$k^{\mathrm{th}}$ Gaussian component. The updated covariance estimate for all
$k$ is:

$\Sigma_{k}=\frac{1}{N_{k}}\sum_{i=1}^{N}\omega_{k,i}(x_{i}-\mu_{k})(x_{i}-\mu_{k})^{T}-\chi_{i}$
(3)

This formula accounts for the observational error, refining the true
distribution’s covariance estimate. Lastly, the prior probability $\pi_{k}$ is
adjusted:

$\pi_{k}=\frac{N_{k}}{N}$ (4)

The XD algorithm iterates between the E-step and M-step until convergence is
reached. XD’s robustness against noise and its ability to discern intricate
structures make it particularly valuable in astrophysics. This was show cased
by Bovy et al. (2011) and Belokurov et al. (2018), to break down the complex
velocity distributions of solar neighbourhood stars and the MW stellar halo,
respectively.

###  3.3 Group Number (k) Estimation

After employing XD, the next critical task is to determine the optimal number
of Gaussian groups, hereafter defined as $k$. The Bayesian Information
Criterion (BIC) emerges as a practical tool for this purpose (Fraley &
Raftery, 2002). It provides a trade-off between the fit of the model to the
data and the dimensionality (hereafter defined as $d$) of the model. The
formula for BIC is given by:

$\mathrm{BIC}=-2\ln(L)+m\ln(N),$ (5)

where $L$ is the likelihood of the observed data given the model, $m$ is the
number of free parameters in the model and as before, $N$ is the number of
data points. $L$ is the product of the probabilities of each data point, given
the model parameters and is given by:

$L=\prod^{N}_{i=1}p(\mathbf{x}_{i}|\mathbf{\theta}),$ (6)

where $\mathbf{\theta}$ denotes the set of parameters of the Gaussian mixture
model. The value of $m$ is given by:

$m=d\frac{d+3}{2}k.$ (7)

The BIC tends to favour models with fewer parameters (simpler models) for
smaller datasets and models with more parameters (more complex models) for
larger datasets. As a result of this, the BIC penalises overfitting,
discouraging overly complex models. In our study, under-estimating the number
of groups is more advantageous than over-estimating, as during subsequent
dynamical analyses these merged groups could be teased apart based on their
dynamical properties.

$d$-dimensions |  $k$-groups | $X_{\mathrm{err}}$ |  $k$-predicted
---|---|---|---
3 | 3 | 0.0, 0.05, 0.10, 0.20 | 3, 3, 3, 3
3 | 5 | 0.0, 0.05, 0.10, 0.20 | 5, 4, 5, 6
3 | 10 | 0.0, 0.05, 0.10, 0.20 | 5, 5, 5, 4
5 | 3 | 0.0, 0.05, 0.10, 0.20 | 3, 3, 3, 2
5 | 5 | 0.0, 0.05, 0.10, 0.20 | 6, 6, 6, 6
5 | 10 | 0.0, 0.05, 0.10, 0.20 | 11, 11, 11, 8
7 | 3 | 0.0, 0.05, 0.10, 0.20 | 3, 3, 3, 3
7 | 5 | 0.0, 0.05, 0.10, 0.20 | 8, 5, 5, 5
7 | 10 | 0.0, 0.05, 0.10, 0.20 | 11, 6, 9, 9
Table 1: The compiled results of the XD mock tests, investigating the impact
of the dimensionality ($d$), number of groups ($k$) and the errors
$X_{\mathrm{err}}$.

In order to test the accuracy and limitations of XD, we generated a set of
mock multivariate distributions, adjusting the number of groups ($k$), means
($\mu_{k}$), covariances ($\Sigma_{k}$), weights ($\pi_{k}$), number of
dimensions ($d=2$, $5$ and $7$) and magnitude of the errors
($X_{\mathrm{err}}=0.0,0.05,0.10,0.20$ dex). The summarised results are found
in Table 1. These errors were chosen to span the range of errors that are
encountered in the GALAH data. To keep these tests applicable to the ranges of
our PCA-transformed data, we use the same sample size ($N=9,923$), generating
random samples with the following number of artificially included groups: $3$,
$5$, $7$ and $10$. XD performs well in recovering the correct number of
groups, $k$, but this performance is influenced by both the dimensionality,
$d$, of the space and the magnitude of $k$. In a three-dimensional space with
three groups ($d=3,k=3$), the number of groups predicted by the BIC,
$k_{\mathrm{predicted}}$, is accurate. However, when the number of groups is
increased to $10$ in the same three-dimensional space the prediction becomes
less reliable, resulting in $k_{\mathrm{predicted}}=4$, when the errors are
set to $X_{\mathrm{err}}=0.2$ dex. As the dimensionality increases, the
accuracy of the group recovery seems to improve, for example, in the case of
$d=7$ and $k=10$, the BIC is able to predict a range of values close to the
actual number, $k_{\mathrm{predicted}}=6$ to $11$.

Figure 2: The explained variance of a PC is the eigenvalue of that PC.
Applying a cut-off at $85\%$ cumulative explained variance shows that $9$
components are required to reliably capture the patterns present in the data.
Thus a dimensionality reduction can be made from a $17$ to a $9$ dimensional
space.

Figure 3: The $17$ [X/Fe] abundances are coloured approximately by their
nucleosynthetic families: $\alpha$ (blue), light-odd Z (green), heavy
s-process (red), light s-process (purple), r-process (pink) and iron-peak
(orange). The first PC (left plot) makes up $29\%$ of the total variance of
the $\mathcal{C}$-space. The second PC (right plot) makes up $18\%$ of the
total variance. The weights (positive and negative) show how each [X/Fe]
chemical abundance ratio contributes to the PC.

This suggests that higher-dimensional spaces provide better separation between
groups, allowing for more accurate predictions by the BIC. In summary, when
looking at various dimensions and group numbers, $k_{\mathrm{predicted}}$ does
not consistently indicate a tendency toward overfitting or underfitting our
mock tests. Nevertheless, in cases where $k=10$, which is the number used in
this work as described in Section 4.2, the predicted number of groups falls
short across all dimensions, indicating a tendency towards underfitting.
Consequently, we conclude that when applying this method to our data, the BIC
score’s ‘knee’ (the point where adding more groups does not significantly
improve the model) represents a lower limit of the true number of groups in
our dataset, so is unlikely to subdivide the $\mathcal{C}$-space into too many
groups.

##  4 Results

Here, we describe the results of applying PCA and our clustering analysis,
followed by a presentation of the chemodynamical properties of the revealed
groups- assigning known stellar components to the $10$ derived groups as
follows: the thick disc (G1, G4, G6, G7 and G8), GES (G2), the in-situ halo
(G3 and G5) and the thin disc (G9). G10 is interpreted as an outlier group.

###  4.1 PCAA

PCA is applied to the $17$-dimensional $\mathcal{C}$-space, with the explained
variance shown in Figure 2. The explained variance is the fraction of the
total variance in the original data set that is contained in each PC.
Consistent with the approach of Ting et al. (2012), we have implemented a
cumulative explained variance cutoff of $85\%$, thereby excluding those PCs
dominated by noise or errors. Notably, the cumulative explained variance
increases gradually, requiring a substantial number of PCs to reach the $85\%$
threshold. This pattern could suggest, within the metallicity range of
$-1.7\leq\mathrm{[Fe/H]}\leq-0.5$, the presence of multiple independent
nucleosynthetic processes that influence the observed chemical patterns, as
well as different SFHs contributing to the enrichment of this region. These
aspects are explored further in Section 5. Implementing the $85\%$ cutoff
reduces the $17$-dimensional space to a $9$-dimensional space, highlighted in
Figure 2. We note that changing the number of PCs we consider by a few does
not significantly affect the number of groups that are identified by our
subsequent clustering analysis in Section 4.2.

Figure 4: The ‘core’ is the group centered around PC1 $\approx 0$ and PC2
$\approx-2$. The ‘branch’ is the group centered around PC1 $\approx 5$ and PC2
$\approx 0$. The ‘spray’ is the group centered around PC1 $\approx 0-1$ and
PC2 $\approx 10$. Figure 5: The PC1-PC2 hyperplane coloured by various
abundances ratios ([Fe/H], [Al/Fe], [Zn/Fe], [Cu/Fe], [Eu/Fe], [Ba/Fe],
[Y/Fe], [Mg/Fe] and [Sc/Fe]). The only abundance ratio here that was not
included in the $\mathcal{C}$-space is [Fe/H]. We evaluate the chemical
enrichment in three regions: the ‘core’, the ‘branch’ and the ‘spray’.

To further breakdown the elemental contributions to the variance, Figure 3
shows the weights (i.e. eigenvector coefficients for each element enhancement)
for the first two PCs, which contain $50\%$ of the variance of the data. In
PC1, [Al/Fe], [Cu/Fe] and [Ba/Fe] contribute the most to the variance (in
terms of absolute magnitude). In PC2, there is a greater contribution from the
ls-elements, [Y/Fe] and [La/Fe]. The lack of contribution, positive or
negative, from elements like [Eu/Fe] and [K/Fe] in PC1, and [Mn/Fe], [Al/Fe],
[Co/Fe], [Ni/Fe], and [Cu/Fe] in PC2, is a consequence of the non-linearity of
some nucleosynthesis processes, e.g., Al, where two trends can cancel out. To
see more clearly how these elements contribute to the spread of observed
abundances, we show the PC1 - PC2 plane in Figure 4. We use the following
terms to define the three general regions that are identifiable by eye in this
two-dimensional plane; the ‘core’, the ‘branch’ and the ‘spray’. We argue in
the following that the ‘core’ is made up of disc populations, the ‘branch’ is
made up from a single accreted population (GES) and the ‘spray’ features
super-solar s-process enhanced stars which are in-situ in origin. These
regions are used for reference when analysing the distributions of each [X/Fe]
abundance ratio in the PC1 - PC2 plane in Figure 5. We can also identify that
the positive PC1 direction varies between disc (‘core’) and accreted stars
(‘branch’), with hs-element [Ba/Fe] having a positive contribution and iron-
peak and light odd-Z elements having a negative contribution which points
towards metal-poor populations with extended SFHs. The positive PC2 direction
varies between disc and in-situ populations (‘spray’), pointing towards
s-process and $\alpha$-enhancement. Our main findings are as follows:

  * •

Core: The ‘core’ is characterized by relatively metal-rich stars with [Fe/H]
$\approx-0.6$ dex, noting that the sample is selected to have [Fe/H]
$\leq-0.5$ dex. This region shows significant enhancement in [Al/Fe],
indicating differences in SFHs compared to other regions- with the ‘core’
experiencing recent/ongoing star formation compared with the accreted
populations in the ‘branch’. The majority of the ‘core’ shows enhanced
[Cu/Fe], except for the lower region occupying PC1 $\approx-2$, PC2
$\approx-4$. The ‘core’ has near-solar values of [Ba/Fe], suggesting it is
representative of disc populations. There is a lower [Mg/Fe] value in the
lower region of the ‘core’ compared with the upper ‘core’, with a difference
of $\approx 0.3$ dex, which is greater than the average errors of [Mg/Fe]
($0.07$ dex). This could indicate the presence of thin disc stars.
Additionally, some [Sc/Fe] enhancement is observed in the left region of the
‘core’ (at PC1 $\approx-5$ and PC2 $\approx 0$).

  * •

Spray: The ‘spray’ region displays a complex mixture of metallicities. It is
notable for hs-enhancement in [Ba/Fe] and super-solar [Y/Fe] stars, indicating
a distinct chemical signature from both the ‘core’ and ‘branch’. PC2 shows a
high positive weighting for s-process elements towards the ‘spray’, suggesting
significant contributions from AGB stars in these populations. This could also
indicate the presence of Ba-rich stars resulting from binary processes. Higher
AGB contributions imply these populations underwent prolonged or multiple-
phase star formation, allowing intermediate-mass stars to evolve into AGB
stars, enriching the ISM with s-process elements. The ‘spray’ has a varied
range of abundance ratios for the other elements considered, showing no clear
pattern like the ‘core’ or ‘branch’, which makes it a more heterogeneous group
in terms of chemical composition.

  * •

Branch: The ‘branch’ contains more metal-poor stars, with [Fe/H] $\approx-1.4$
dex. It exhibits significant variance in [Eu/Fe], pointing to stochasticity in
r-process element production in the metal-poor regime. The ‘branch’ exhibits
lower [Al/Fe] than the ‘core’ or ‘spray’ regions, indicating significant
differences in SFHs, with low [Al/Fe] being a property of accreted
populations. Many of the iron-peak elements have negative weights in PC1,
which can be seen through the sub-solar values of elements such as [Cu/Fe]
towards the ‘branch’. A similar trend is observed in [Cu/Fe], where the lower
[Cu/Fe] agrees with findings for accreted halo stars. The ‘branch’ also has
high [Ba/Fe] values, and a slightly sub-solar [Mg/Fe] present in the lower
region, potentially indicative of low-$\alpha$ halo stars. In PC1, [Ba/Fe],
[Zn/Fe], [Y/Fe], and [La/Fe] have positive weights, indicating these elements
increase towards the accreted population in the ‘branch’.

The PC1-PC2 hyperplane is just a two-dimensional representation of the reduced
$9$-dimensional $\mathcal{C}$-space, which indicates why there are only three
components selected by eye (see Figure 4). This will be explored in more
detail in the following Section 4.2, where we cluster the latent space using
XD to estimate the underlying number of groups.

###  4.2 Clustering the PCA-Transformed Space

Our goal is to cluster the metal-poor MW stars into groups with similar
chemistry to better understand the chemical diversity and formation history of
components such as the thick disc, thin disc, and stellar halo. We apply XD to
our $9$D-transformed $\mathcal{C}$-space for a varying number of groups and
calculate the BIC for each case. The BIC scores shown in Figure 6 suggest the
presence of $\approx 10$ groups, which we label as G1-G10. It is important to
note that the optimal number of groups is not very clear from the BIC
analysis, as a range of about $10$ to $14$ groups returns similar BIC scores.
For simplicity and to facilitate our analysis, we proceed with the minimum
number of groups suggested by the BIC knee, which is $10$, noting also that a
slight under-estimation of the total number of groups is preferable to an
over-estimation, as discussed in Section 3.3. Additionally, we only assign
stars to a group if they have a $\geq 70\%$ probability of belonging to that
Gaussian component.

Figure 6: BIC scores for clustering the $9$D-transformed $\mathcal{C}$-space
of metal-poor MW stars. The minimum BIC, the ‘knee’, suggests $10$ groups. The
inset shows the minimum BIC score at $10$ groups. Although $10$ to $14$ groups
yield similar scores, we proceed with $10$ groups for simplicity and
robustness.

###  4.3 Mapping the Groups to Known Populations

In Figure 7, we show the location of the groups in commonly used
chemodynamical hyperplanes, [Mg/Mn] - [Al/Fe], $L_{z}$ \- $E$ and
$L_{z}/L_{c}$ \- [Fe/H], while also showing where the groups lie in the
reduced PC1 - PC2 space. While [Mg/Fe] has a relatively low weight of $-0.2$
in PC1 (shown in Figure 3), indicating it does not vary significantly, we
still utilise it to explore its associations with the thick disc, thin disc
and halo populations. Combining this with information from the other [X/Fe]
distributions for each group, found in the Appendix A, we can determine the
chemodynamical properties of each of them. We also refer the reader to Table 2
for a complete collection of means and errors of the chemodynamical
distributions.

####  4.3.1 G1, G4, G6, G7 and G8 - Thick Disc Groups

Figure 7: Plots of all groups in varying chemodynamical and PC planes. Top:
the thick disc groups, G1, G4, G6, G7 and G8, in order from left to right: PC2
- PC1, [Mg/Mn] - [Al/Fe], $L_{z}$ \- $E$ and $L_{z}/L_{c}$ \- [Fe/H]. Middle:
GES (G2) and the thin disc (G9), in order from left to right: PC2 - PC1,
[Mg/Mn] - [Al/Fe], $E$ \- $L_{z}$ and [Fe/H] - $L_{z}/L_{c}$. Bottom: the in-
situ halo groups (G3 and G5) in order from left to right: PC2 - PC1, [Mg/Mn] -
[Al/Fe], $E$ \- $L_{z}$ and [Fe/H] - $e$.

G1, G4, G6, G7, and G8 contain $763$, $648$, $2235$, $832$, and $610$ stars,
respectively. In the PC1-PC2 plane these groups form the ‘core’. They occupy
an $\alpha$-rich region with [Mg/Fe]μ values of $\approx 0.3$ dex. Their
[Fe/H] distributions show the most metal-enhanced values among the groups of
around $-0.6$ dex. All have prograde orbits ($L_{z}\approx 1200$ kpc kms-1)
and $e$ distributions peaking at $\approx e=0.30$.

Regarding their chemistry, the [Ba/Fe] values differ: G4 and G8 have super-
solar values ([Ba/Fe]${}_{\mu}=0.28$ dex and [Ba/Fe]${}_{\mu}=0.21$ dex
respectively), while G6 and G7 have slightly sub-solar values
([Ba/Fe]${}_{\mu}=-0.05$ dex and [Ba/Fe]${}_{\mu}=-0.10$ dex respectively).
The [Y/Fe] trend is similar. [Eu/Fe] abundances are consistently super-solar
across these groups ($\approx 0.3$ dex). G4 shows a high Zn enhancement
([Zn/Fe]${}_{\mu}=0.42$ dex), while others are closer to solar ($\approx 0.1$
dex). G7 and G8 show bi-modalities in [Zn/Fe] with solar and sub-solar
components (e.g., G7${}_{\mathrm{subsolar}}=-0.3$ dex and
G7${}_{\mathrm{solar}}=0.1$ dex). The [Al/Fe] distribution is narrow across
all groups, with abundances ranging from [Al/Fe]${}_{\mu}=0.22-0.45$ dex. For
[Cu/Fe], all groups have solar values except G4, which has
[Cu/Fe]${}_{\mu}=-0.08$ dex. In [Ni/Fe], all groups have super-solar Ni-
abundances except G4 ([Ni/Fe]${}_{\mu}=0.00$ dex). Finally, the [Sc/Fe]
distributions are similar across G4, G6, and G7, with tails extending between
[Sc/Fe] $=0.0-0.2$ dex. G1 has an extended distribution with [Sc/Fe] reaching
$\approx 0.4$ dex. G8 is the only group with sub-solar values,
[Sc/Fe]${}_{\mu}=0.04$ dex, and a tail extending down to $\approx-0.2$ dex.
These groups (G1, G4, G6, G7, and G8) lie in the thick disc region of the
[Mg/Mn] - [Al/Fe] plane.

The properties of these groups, including their $\alpha$-rich nature,
relatively high metallicity, prograde orbits, and specific chemical abundance
patterns, align with the expected SFH of the thick disc. This history involves
rapid early star formation from gas enriched by CCSNe, followed by prolonged
or multiple-phase star formation episodes (Bland-Hawthorn & Gerhard, 2016; Yu
et al., 2021). G4 and G8 have some exceptions to expected abundance patterns,
in the case of [Ba/Fe] and [Zn/Fe], this is further discussed in Section 5.1.

####  4.3.2 G2 - GES

G2 contains $671$ stars and dominates the ‘branch’ in the PC2 - PC1 plane. G2
has low-metallicity values ([Fe/H]${}_{\mu}=-1.00$ dex) and
$L_{z,\mu}=145.13\pm 521.97$ kpc kms-1, with an eccentricity $e_{\mu}=0.80\pm
0.21$, extending up to a peak at $e\approx 1$, making it the most dynamically
hot of all groups.

It is $\alpha$-rich, slightly less so than the groups in the ‘core’, with
[Mg/Fe]${}_{\mu}=0.13$ dex, and is the most Mn-depleted group present, with
[Mn/Fe]${}_{\mu}=-0.37$ dex. G2 has a high Ba enhancement,
[Ba/Fe]${}_{\mu}=0.41$ dex, though the ls-process element Y enhancement is
[Y/Fe]${}_{\mu}=0.11$ dex, indicating a lower abundance compared to the hs-
process. G2 shows super-solar [Eu/Fe] values at $\mu=0.44$ dex and a uni-
modal, solar [Zn/Fe] distribution with [Zn/Fe]${}_{\mu}=0.08$ dex. The [Al/Fe]
distribution is sub-solar, the lowest across all groups, with
[Al/Fe]${}_{\mu}=-0.12$ dex. Notably, G2 has a dominant peak at very sub-solar
[Cu/Fe] values, [Cu/Fe]${}_{\mu}=-0.46$ dex, making it the least Cu-enriched
in the sample. The [Ti/Fe] distribution shows that while G2 is enhanced in Mg
(which is also an $\alpha$-element), it extends to [Ti/Fe] abundances as low
as those of G9. G2 has [Ni/Fe]${}_{\mu}=-0.11$ dex, and
[Sc/Fe]${}_{\mu}=-0.01$ dex.

In the [Mg/Mn] - [Al/Fe] plane, G2 overlaps with the ‘blob’ identified in Das
et al. (2020) and also mirrored by other studies (Myeong et al., 2022),
indicating the presence of GES stars. The low metallicity of these stars
position G2 within the more metal-rich extent of GES ([Fe/H]${}_{\mu}=-1.05\pm
0.20$ dex (Ortigoza-Urdaneta et al., 2023), although GES is thought to extend
down to [Fe/H] $\approx-1.5$ dex (Naidu et al., 2022), a metal-poor region not
fully captured in our sample, which extends down to [Fe/H] $\approx-0.6$ dex.
G3 and G10 also show GES-like stars, with G10 reaching lower metallicities.
The high [Ba/Fe] values in G2 present a discrepancy with studies that indicate
GES has negligible s-process contribution and sub-solar [Ba/Fe] values
($-0.02$ dex) (Carrillo et al., 2022). However, our [Ba/Fe]${\mu}=0.41\pm
0.19$ dex closely matches Myeong et al. (2022)’s estimate for metal-rich GES
([Ba/Fe] = $0.38$ dex, [Fe/H] $=-0.96$ dex). G2 also shows r-process
enhancement with [Eu/Fe]${}_{\mu}=0.44$ dex, aligning with the metal-rich GES
cited in the literature ([Eu/Fe] $=0.47$ dex (Myeong et al., 2022)). The sub-
solar [Al/Fe]${}_{\mu}=-0.12$ dex is consistent with an accreted population,
matching Myeong et al. (2022)’s metal-rich GES ([Al/Fe] $=-0.17\pm 0.09$ dex).
G2’s extreme sub-solar [Cu/Fe] values define it distinctly.

####  4.3.3 G9 - Thin Disc

G9 contains $581$ stars and is part of a distinct lower cluster in the
segmented ‘core’ shown in Figure 4. G9 has the highest $L_{z}$ among the
groups, with $L_{z}=2020.10\pm 619.82$ kpc kms-1. There is a slight bi-
modality in $L_{z}$, with a small group of stars having $L_{z}\sim 0$, and it
has the most circular orbits of all the groups, with a distinct peak at
$e\approx 0.1$. G9 has a tail towards $L_{z}\leq 0$, with $9$ stars exhibiting
retrograde kinematics, representing $\approx 1.72\%$ contamination from G2. G9
is the most $\alpha$-poor group in the sample, with [Mg/Fe]${}_{\mu}=0.10$
dex, supported by low [Ti/Fe] values ([Ti/Fe]${}_{\mu}=0.09$ dex). It has the
most overall Mn-rich distribution of the groups, with [Mn/Fe]${}_{\mu}=-0.08$
dex), with only G1 and G10 showing extended tails to higher Mn-enhancement.
The [Fe/H] distribution is the narrowest and also among the most metal-rich
([Fe/H]${}_{\mu}=-0.56$ dex). The s-process enhancement shows slight bi-
modality, more evident in the ls-process element Y, with [Y/Fe] featuring
peaks at solar ($\approx 0.1$ dex) and sub-solar ($\approx-0.2$ dex) values.
G9 has the lowest r-process enhancement with [Eu/Fe]${}_{\mu}=0.15$ dex). The
[Zn/Fe] values are solar, [Zn/Fe]${}_{\mu}=-0.01$ dex, and feature a tail
extending to low [Zn/Fe] values ($<-0.5$ dex). [Al/Fe] aligns with the ‘core’
groups, and the [Cu/Fe] distribution closely resembles that of G1, G6, and G8.
G9 has the lowest [Ti/Fe] of all groups, [Ti/Fe]${}_{\mu}=0.09$ dex, with a
tail extending to sub-solar values, $\approx-0.2$ dex, similar to G2. The
[Ni/Fe] distribution is similar to G1, G4, G6, and G7 ([Ni/Fe]${}_{\mu}=0.04$
dex), and [Sc/Fe] values closely match G8, at [Sc/Fe]${}_{\mu}=0.06$ dex. In
the [Mg/Mn] - [Al/Fe] plane, G9 lies in a region characteristic of the thin
disc. This is corroborated by its predominantly prograde and circular orbits.
The $\alpha$-poor nature and narrow metallicity distribution at [Fe/H] $>-0.8$
dex are consistent with the thin disc population studied by Hawkins et al.
(2015). The retrograde contamination ($1.72\%$) likely originates from GES, as
they possess a similar $\alpha$ enhancement.

####  4.3.4 G3 and G5 - In-situ Halo

G3 and G5 contain $651$ and $141$ stars, respectively. They both occupy the
upper region of the ‘core’ (at PC1 $\approx 0$ and PC2 $\approx 5$), with G3
extending into the ‘branch’ and G5 extending into the ‘spray’. It is evident
that these two groups overlap across multiple hyperplanes, capturing stars on
retrograde and more circular orbits. G3 features a bi-modality at
$L_{z}\approx 1000$ and $0$ kpc kms-1. The prograde peak aligns with G1, G4,
G6, G7, and G8 (i.e. the thick-disc populations), while the $L_{z}\approx 0$
kpc kms-1 peak matches G2 (GES). G5 also shows a bi-modal distribution in
$L_{z}$, with stars at $L_{z}=0$ and retrograde values. The $e$ distribution
in both groups is widely spread between $e=0$ and $1$, with a bi-modality of
thick disc-like and higher eccentricity orbits.

The [Mg/Fe] distribution of G3 is centered at [Mg/Fe]${}_{\mu}=0.32$ dex,
similar to the $\alpha$-enhanced G1, G4, G6, G7, and G8 (thick-disc
populations). G5 has the most $\alpha$-rich stars of all groups, with
[Mg/Fe]${}_{\mu}=0.36$ dex, and a distribution tail extending to $\approx
0.75$ dex. Both groups have wide metallicity distributions, with G5 being more
metal-rich ([Fe/H]${}_{\mu}=-0.75$ dex), and G3 having a lower metallicity
([Fe/H]${}_{\mu}=-0.87$ dex). The Ba-enrichment of G3 is similar to G2
([Ba/Fe]${}_{\mu}=0.44\pm 0.24$ dex), while G5 has the highest [Ba/Fe]
enrichment among all groups ([Ba/Fe]${}_{\mu}=0.97\pm 0.37$ dex). G3 has a
narrow [Y/Fe] distribution ([Y/Fe]${}_{\mu}=0.33\pm 0.17$ dex), unique among
the groups, while G5 has a higher enhancement, [Y/Fe]${}_{\mu}=0.68\pm 0.26$
dex, with a complex distribution featuring about four peaks. The high
s-process enhancements present in G3 are found in the proto-galaxy population
Aurora (Myeong et al., 2022), and are attributed to the prolonged star
formation of in-situ populations. Similarly, Heracles (Horta et al., 2020;
Naidu et al., 2022), an ancient merger event prior to the GES merger has
similar s-process properties to G3. The Eu-enhancement of G3 closely resembles
G1 and G6 (thick-disc groups), featuring a tail of Eu-enhanced stars trailing
up to [Eu/Fe] $>0.7$ dex. G5 has a similar distribution but with a slightly
higher mean enhancement ([Eu/Fe]${}_{\mu}=0.39$ dex), and a tail extending up
to [Eu/Fe] $\approx 0.8$ dex.

The [Al/Fe] distribution are $0.25$ and $0.31$ dex for G3 and G5 respectively,
and are too high to be considered from Heracles, which is limited to [Al/Fe]
$<0$ (Horta et al., 2020). This range however matches well with the in-situ
halo population Aurora, with [Al/Fe] $=0.10\pm 0.16$ dex (Myeong et al.,
2022). For [Zn/Fe], G3 has [Zn/Fe]${}_{\mu}=0.19$ dex and G5 has
[Zn/Fe]${}_{\mu}=0.25$ dex, similar to G1, G6, and G8. Both G3 and G5 extend
to the lowest [Cu/Fe] values ($\approx-0.5$ dex), with G3 having the lowest
mean ([Cu/Fe]${}_{\mu}=-0.14$ dex). The [Ti/Fe] distributions for G3 and G5
are similar to G1, G6, and G7 (thick-disc groups), with [Ti/Fe]${}_{\mu}=0.26$
dex and [Ti/Fe]${}_{\mu}=0.28$ dex respectively, which is also seen in the
case of [Ni/Fe]. The [Sc/Fe] distributions of both G3 and G5 closely match G9
and G8, with [Sc/Fe]${}_{\mu}=0.08$ dex and [Sc/Fe]${}_{\mu}=0.11$ dex
respectively. Retrograde contamination in these groups is $99$ stars for G3
($15.2\%$ of the group) and $15$ stars for G5 ($10.6\%$ of the group). Located
in the [Mg/Mn] - [Al/Fe] plane where in-situ halo stars are found, G3 and G5
are distinguished by their s-process enhancement in [Ba/Fe] and [Y/Fe].

####  4.3.5 G10 - Outlier Group

G10 contains $365$ stars and is an outlier group, overlapping into the
‘branch’, ‘spray’, and ‘core’. G10 appears as a background component in all
hyperplanes. In $L_{z}$, G10 has two peaks with a greater spread towards
prograde values ($L_{z}>2000$ kpc kms-1), and a wide $e$ distribution. G10
shows a broad range of [Mg/Fe] values from $-0.75$ to $0.75$ dex, grouping
together some of the most $\alpha$-rich and $\alpha$-poor stars in the data
set. It also has a spread of extreme [Ba/Fe] values, likely captured in the
‘spray’, with hs-process enhancement ([Ba/Fe] $>2.0$ dex). [Y/Fe] shows a
similar spread. The [Eu/Fe] distribution has wide dispersion and tails to very
high Eu-enhancements ([Eu/Fe] $\approx 1.0$ dex). In contrast, [Ni/Fe] and
[Sc/Fe] distributions are narrower than other chemical abundances, with
[Ni/Fe]${}_{\mu}=0.00\pm 0.16$ dex and [Sc/Fe]${}_{\mu}=0.12\pm 0.17$ dex,
showing similarities to G5 in both abundances. Kinematically, G10 includes
approximately two distinct groups that align with the thick disc kinematics of
G1, G4, G6, G7, and G8, as well as the accreted, $L_{z}\approx 0$, G2 group.
The wide distribution of chemical and kinematic features contrasts with its
tight distributions in both [Ni/Fe] and [Sc/Fe].

##  5 Discussion

In this section, we assess the robustness of our method and compare our
findings with previous studies. We discuss the chemical and kinematic
properties of different groups, emphasising the distinct populations within
the thick disc and their implications for the SFH of the metal-poor MW.
Additionally, we examine the in-situ halo groups and GES and explore the
chemical diversity present in our sample.

Figure 8: Four hyperplanes: (left to right, top to bottom) [Zn/Cu] - [Ba/Eu],
[Sc/Y] - [Al/Zn], [Ba/Zn] - [Sc/Cu] and [Ba/Al] - [Cu/Y]. The five groups are
selected to belong to the thick disc as defined by their high [$\alpha$/Fe]
and thick disc-like kinematics. In these chemical hyperplanes, the thick-disc
populations show distinction in chemistry. G4 is easier to classify separately
in each plane, whereas G7 is most clearly distinguishable in [Al/Zn] and
[Zn/Cu].

###  5.1 Multiple Populations in the Thick Disc

The magnitude of the weights for Ba (PC1 and PC2), Y (PC2), Zn (PC2), Al
(PC1), Cu (PC1), La (PC2), Sc (PC2) and Eu (PC2) are among the highest, and
the distinguishing power of these chemical abundances become a useful tool to
decompose the thick disc. Figure 8 shows the chemical composition of groups we
associate with the thick disc. We select the following hyperplanes: [Zn/Cu] -
[Ba/Eu], [Sc/Y] - [Al/Zn], [Ba/Zn] - [Sc/Cu] and [Ba/Al] - [Cu/Y]. [Zn/Cu] is
selected as they both have different timescales: Cu from SNe1a and AGB (longer
delay) and Zn from CCSNe and HNe (shorter delay). Hence, regions with early
star formation might have higher [Zn/Cu], while prolonged star formation shows
lower [Zn/Cu]. [Ba/Eu] traces s- vs r-process nucleosynthesis. Higher [Ba/Eu]
indicates a stronger AGB star influence and prolonged star formation; lower
[Ba/Eu] suggests significant r-process events linked to early, rapid star
formation. Thus in Figure 8, G4 in the top right (high [Zn/Cu], high [Ba/Eu])
could represent an older population with rapid star formation and early
enrichment. We note that G4 lies in the region of the chemical hyperplane, and
suspect it could have experienced a high number of HNe. [Sc/Y] indicates the
play-off between CCSNe and the ls-process in AGB stars- Y shows a very high
weighting in PC2 and this variance is seen in [Sc/Y]. [Al/Zn] is another probe
for HNe, as well as for lower mass (low Al) populations. This means G4 and G8
both have low [Al/Zn], indicating lower mass systems with evidence of early
intense star formation. The bottom two chemical hyperplanes also demonstrate
how the thick disc populations lie, with G4 and G8 being the most distinctly
separate.

Overall, G1 and G6 are chemically similar, representing the classical thick
disc. G7 differs due to its Mn and Zn enhancement, while G4 is the most
chemically distinct among the thick disc groups captured here. G8 alongside G4
has a distinctive hs-element enhancement which is captured in [Ba/Fe]. This
enhancement in Ba, but not Y, could indicate that there was an environment
with higher neutron density (Bisterzo et al., 2010). Additionally, G4 features
a very high [Zn/Fe] enhancement which could be the contribution from HNe
(Kobayashi et al., 2020). This suggests while some studies have argued that
the thick disc is mostly chemically homogeneous (Haywood et al., 2019; Chandra
et al., 2023), we find evidence that the thick disc’s formation could have
resulted from three separate events or nuanced nucleosynthesis channel which
results in the classical thick disc (G1 + G6 + G7), G4 (high Zn, high Ba) and
G8 (low Sc, high Ba) (Navarro et al., 2011; Belokurov & Kravtsov, 2022).

###  5.2 In-situ halo and GES

G3 and G5 are likely the in-situ halo. We look to Heracles, postulated to be
an ancient accreted population. The stars are proposed to have formed prior to
the spin-up of the MW disc. It was discovered chemodynamically by Horta et al.
(2020) and is thought to have an estimated progenitor stellar mass of
$M_{*}\approx 5\times 10^{8}M_{\odot}$. The [Al/Fe] distribution presented in
this paper is on average higher than the expectation for Heracles quoted in
Horta et al. (2023). This is because the authors selected Heracles according
to the following chemical criteria: [Al/Fe] $<-0.07\&$ [Mg/Mn] $\geq 0.25$,
and [Al/Fe] $\geq-0.07\&$ [Mg/Mn] $\geq 4.25\times$ [Al/Fe] $+0.5475$. Other
connections have been made relating to this population, e.g. Kraken (Kruijssen
et al., 2020) and Koala (Forbes, 2020).

Heracles has been found to have very similar properties to the proto-galaxy
population, Aurora, as identified in Myeong et al. (2022). Aurora is
associated with having a higher [$\alpha$/Fe] ratio compared with Heracles,
indicating that Aurora had a higher SFR than Heracles (Horta et al., 2023).
Additionally, as G3 occupies a lower orbital energy region,
$E_{\mu}=-1.82\times 10^{5}$ km2s-2, this appears to kinematically match with
Aurora which has orbital energies of $-1.76\pm 0.15\times 10^{5}$ km2s-2.
Additionally, as noted, G3 is incredibly enhanced in the [Ba/Fe] and [Y/Fe]
ratio, indicating the contribution of metallicity-dependent AGB stars. Our
values of s-process enhancement, [Ba/Fe] $=0.44\pm 0.24$ dex, align with those
presented for Aurora at [Ba/Fe] $=0.48\pm 0.25$ dex (Myeong et al., 2022).

G2’s chemodynamical properties point to it being representative of the
accreted population GES. This structure has been extensively studied in terms
of its chemodynamical properties. In this work, we find that G2 has lower
$\alpha$-enrichment than the thick disc, in agreement with previous works
(e.g. (Haywood et al., 2018; Helmi et al., 2018; Mackereth et al., 2019; Das
et al., 2020; Buder et al., 2022; Horta et al., 2023)). Additionally, as GES
is an accreted population we find that it has a sub-solar [Al/Fe] ratio which
resembles the abundance pattern of stars from satellites of the MW (Hawkins et
al., 2015; Horta et al., 2020). Looking to the neutron-capture elements, we
see that G2 has a high [Eu/Fe] ratio of $0.44\pm 0.15$ dex, this aligns with
many other sources (see e.g. (Matsuno et al., 2021; Aguado et al., 2021; Naidu
et al., 2022; Myeong et al., 2022; Carrillo et al., 2022)). On the other hand,
looking to the s-process abundances, such as Ba and Y, we see that our sample
of GES candidates have an equivalent enhancement to Eu ([Ba/Fe] $=0.41\pm
0.19$ dex). This is in contradiction with some studies who indicate that GES
has solar levels of Ba-enhancement (e.g. (Carrillo et al., 2022)), while
others measure a strong enhancement of up to [Ba/Fe] $=0.38\pm 0.18$ dex
(Myeong et al., 2022). GES is thought to have an extended SFH of $\approx 3.6$
Gyr (Bonaca et al., 2021), meaning theoretically there would be an expected
source of enrichment via AGB stars.

###  5.3 Method Robustness and Comparison

We find in this study of observational data that $9$ PCs are required to make
up $85\%$ of the cumulative variance, reflecting the findings of Ting et al.
(2012) and Price-Jones & Bovy (2019), who argued for the necessity of $\approx
6-10$ abundance dimensions. The results contrast however with those obtained
from applying PCA to a set of chemical abundances predicted by a one-zone
chemical evolution model, Andrews et al. (2017). Here, the authors found that
$\alpha$-elements and elements such as Fe, Mn, and Al dominate the encoded
information, with two PCs accounting for $99\%$ of the variance in their
model. This suggests that chemical evolution models may be underestimating the
true diversity of contributions to the high-dimensional $\mathcal{C}$-space.
While we find that Ba, Zn, and Cu are all important elements in chemically
distinguishing stars, we do not see a significant variance in [Mg/Fe]. Ting et
al. (2012) found that low-mass AGB stars are significant contributors to the
abundance of hs and ls-elements at higher metallicity, which could reflect the
Ba-enhancement we see in G4 and G8 among the thick-disc groups. The s-process
elements are also shown to be effective for chemical tagging in Blanco-
Cuaresma et al. (2015). Weinberg et al. (2022) emphasises that to effectively
chemically tag groups, it is recommended to minimise group overlap and reduce
residual correlations between abundances to below the level of observational
uncertainties.

Figure 9: These box plots show the spread and distribution of absolute errors
in PC1 (upper) and PC2 (lower) for all the groups. The whiskers extend to 1.5
$\times$ the interquartile range. Data points beyond the whiskers are
considered outliers and are marked as individual dots. It’s unlikely that
G10’s wide distribution of chemodynamical properties are due to large errors,
as shown.

Hogg et al. (2016) assessed the potential of chemical tagging in light of
challenges such as noise and incompleteness in chemical abundance measurements
and determined that clustering in an abundance space can be effective. We also
find that by using the PCA latent space as the clustering target we are able
to cleanly recover different histories in the metal-poor MW for the majority
of groups. In Naidu et al. (2020), they conclude that clustering algorithms
(such as DBSCAN, HBDSCAN and k-means) either fracture the space into too many
clusters, or assign all stars to a dominant structure (e.g. GES or the thick
disc). Furthermore, Blanco-Cuaresma et al. (2015) indicate that chemical
homogeneity across differing groups becomes a challenge, with there being a
non-negligible overlap between structures dependent on the
$\mathcal{C}$-space. In light of this, we will discuss the outlier group
(G10). The errors of the PC1 and PC2 variables for each group are shown in
Figure 9, indicating that high errors are not the reason for this group
capturing an extreme range of chemodynamical properties.

Also, we look at the Renormalised Unit Weight Error (RUWE) values of the
s-process enhanced in-situ halo groups (G3 and G5) to evaluate whether they
are unresolved AGB binaries which may possess enhanced s-process values due to
mass transfer and other effects. After examining the RUWE values across all
groups, we quantified the number with RUWE $\geq 1.4$, based on the
recommended cut-off value from Buder et al. (2021). The number of stars with
RUWE $\geq 1.4$ is very low across all groups, indicating that unresolved
binaries are not a significant concern. Additionally, we cross-matched APOGEE
(Prieto et al., 2008) and GALAH surveys to obtain [C/Fe] values for $329$
stars, as high C along with Ba is another indicator of binary AGB stars. The
data suggest that Ba-rich stars are not a concern in our analysis. This is
shown in Figure 10.

Figure 10: G1 to G10 plotted in the [Y/Fe] - [Ba/Fe] chemical space, and
colour-mapped by the RUWE values. At RUWE $\geq 1.4$ (Buder et al., 2021), the
single-star astrometric fit becomes poor and this could indicate the presence
of an unresolved binary. The number of stars with RUWE $\geq 1.4$ is very low
across all groups. Black points represent stars with [C/Fe] values obtained
from cross-matching APOGEE and GALAH data.

##  6 Conclusions

In this study, we explored the chemical diversity of the MW’s metal-poor
populations by combining PCA and XD on $17$ abundances measured for $9,923$
stars in the GALAH DR3 dataset. This methodology allowed for an unbiased and
comprehensive examination of the Galaxy’s chemical landscape, leading to the
identification of $10$ distinct stellar groups. Our approach provides a clear
advantage over traditional techniques by avoiding arbitrary cuts based on
chemistry or kinematics, offering a less biased representation of the Galaxy’s
chemical complexity. Our key findings include:

  * •

Identification of Known Populations: By using chemistry alone, we successfully
identified known stellar populations through a combination of latent space
representation and a detailed examination of chemical abundances beyond a
traditional 2D analysis. This includes GES, the thick disc, the thin disc and
in-situ halo populations.

  * •

Limited Discriminatory Power of Mg: Mg has less impact in separating accreted
and disc stars, with the greatest $\alpha$-separation being between the thick
and thin discs. Other elements like Ba, Al, Cu, and Sc are important for
distinguishing disc from accreted stars, and Ba, Y, Eu, and Zn for
distinguishing disc and accreted stars from in situ halo.

  * •

Multiple Chemically Distinct Populations in the Thick Disc: The thick disc is
revealed to be a composite of multiple populations with distinct chemical
histories, underscoring its complexity and the consideration of it being a
diverse assembly rather than a homogeneous entity.

These findings highlight the intricate chemical structure of the thick disc
and the broad range of influential elements, reflecting the diverse enrichment
environments specific to the metal-poor MW.

## Acknowledgements

PD is supported by a UKRI Future Leaders Fellowship (grant reference
MR/S032223/1). NB would like to thank the GLEAM and ERIS collaborations for
supporting this research. The research for this publication was coded in
`python` and included its packages: `numpy`, `astropy`, `pandas`, `matplotlib`
and `sklearn`. This work made use of the `Extreme Deconvolution` (XD) package
(Bovy et al., 2011), available at: https://github.com/jobovy/extreme-
deconvolution. This work made use of the Third Data Release of the GALAH
Survey (Buder et al., 2021). The GALAH Survey is based on data acquired
through the Australian Astronomical Observatory, under programs: A/2013B/13
(The GALAH pilot survey); A/2014A/25, A/2015A/19, A2017A/18 (The GALAH survey
phase 1); A2018A/18 (Open clusters with HERMES); A2019A/1 (Hierarchical star
formation in Ori OB1); A2019A/15 (The GALAH survey phase 2); A/2015B/19,
A/2016A/22, A/2016B/10, A/2017B/16, A/2018B/15 (The HERMES-TESS program); and
A/2015A/3, A/2015B/1, A/2015B/19, A/2016A/22, A/2016B/12, A/2017A/14 (The
HERMES K2-follow-up program). We acknowledge the traditional owners of the
land on which the AAT stands, the Gamilaraay people, and pay our respects to
elders past and present. This paper includes data that has been provided by
AAO Data Central (datacentral.org.au). This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

## Data Availability

The code and data generated to produce the figures will be shared on
reasonable request to NB.

## References

  * Aguado et al. (2021) Aguado D. S., et al., 2021, The Astrophysical Journal Letters, 908, L8
  * Anders et al. (2018) Anders F., Chiappini C., Santiago B. X., Matijevič G., Queiroz A. B., Steinmetz M., Guiglion G., 2018, Astronomy & Astrophysics, 619, A125
  * Andrews et al. (2012) Andrews B. H., Weinberg D. H., Johnson J. A., Bensby T., Feltzing S., 2012, Principal Component Abundance Analysis of Microlensed Bulge Dwarf and Subgiant Stars (arXiv:1205.4715)
  * Andrews et al. (2017) Andrews B. H., Weinberg D. H., Schönrich R., Johnson J. A., 2017, The Astrophysical Journal, 835, 224
  * Aoki et al. (2005) Aoki W., et al., 2005, The Astrophysical Journal, 632, 611
  * Bedell et al. (2018) Bedell M., et al., 2018, The Astrophysical Journal, 865, 68
  * Beers et al. (2012) Beers T. C., et al., 2012, The Astrophysical Journal, 746, 34
  * Belokurov & Kravtsov (2022) Belokurov V., Kravtsov A., 2022, Monthly Notices of the Royal Astronomical Society, 514, 689
  * Belokurov et al. (2018) Belokurov V., Erkal D., Evans N. W., Koposov S. E., Deason A. J., 2018, Monthly Notices of the Royal Astronomical Society, 478, 611
  * Belokurov et al. (2020) Belokurov V., Sanders J. L., Fattahi A., Smith M. C., Deason A. J., Evans N. W., Grand R. J. J., 2020, Monthly Notices of the Royal Astronomical Society, 494, 3880
  * Beniamini et al. (2016) Beniamini P., Hotokezaka K., Piran T., 2016, The Astrophysical Journal, 832, 149
  * Bensby et al. (2003) Bensby T., Feltzing S., Lundström I., 2003, Astronomy & Astrophysics, 410, 527
  * Bignone et al. (2019) Bignone L. A., Helmi A., Tissera P. B., 2019, The Astrophysical Journal, 883, L5
  * Bisterzo et al. (2010) Bisterzo S., Gallino R., Straniero O., Cristallo S., Käppeler F., 2010, Monthly Notices of the Royal Astronomical Society
  * Blanco-Cuaresma et al. (2015) Blanco-Cuaresma S., et al., 2015, Astronomy & Astrophysics, 577, A47
  * Bland-Hawthorn & Gerhard (2016) Bland-Hawthorn J., Gerhard O., 2016, Annual Review of Astronomy and Astrophysics, 54, 529
  * Bonaca et al. (2021) Bonaca A., et al., 2021, The Astrophysical Journal Letters, 909, L26
  * Bovy et al. (2011) Bovy J., Hogg D. W., Roweis S. T., 2011, The Annals of Applied Statistics, 5
  * Buder et al. (2021) Buder S., et al., 2021, Monthly Notices of the Royal Astronomical Society, 506, 150
  * Buder et al. (2022) Buder S., et al., 2022, Monthly Notices of the Royal Astronomical Society, 510, 2407
  * Burris et al. (2000) Burris D. L., Pilachowski C. A., Armandroff T. E., Sneden C., Cowan J. J., Roe H., 2000, The Astrophysical Journal, 544, 302
  * Busso et al. (1999) Busso M., Gallino R., Wasserburg G. J., 1999, Annual Review of Astronomy and Astrophysics, 37, 239
  * Carollo et al. (2007) Carollo D., et al., 2007, Nature, 450, 1020
  * Carrillo et al. (2022) Carrillo A., Hawkins K., Jofré P., de Brito Silva D., Das P., Lucey M., 2022, Monthly Notices of the Royal Astronomical Society, 513, 1557
  * Chandra et al. (2023) Chandra V., et al., 2023, The Three-Phase Evolution of the Milky Way (arXiv:2310.13050)
  * Ciucă et al. (2023) Ciucă I., et al., 2023, Monthly Notices of the Royal Astronomical Society: Letters, 528, L122
  * Cohen (2004) Cohen J. G., 2004, Chemical Abundance Inhomogeneities in Globular Cluster Stars (arXiv:astro-ph/0412584)
  * Conroy et al. (2019) Conroy C., et al., 2019, The Astrophysical Journal, 883, 107
  * Cui et al. (2012) Cui X.-Q., et al., 2012, Research in Astronomy and Astrophysics, 12, 1197
  * Das et al. (2020) Das P., Hawkins K., Jofre P., 2020, Monthly Notices of the Royal Astronomical Society, 493, 5195
  * De Boer et al. (2014) De Boer T. J. L., Tolstoy E., Lemasle B., Saha A., Olszewski E. W., Mateo M., Irwin M. J., Battaglia G., 2014, Astronomy & Astrophysics, 572, A10
  * De Los Reyes et al. (2022) De Los Reyes M. A. C., Kirby E. N., Ji A. P., Nuñez E. H., 2022, The Astrophysical Journal, 925, 66
  * Emami et al. (2021) Emami N., et al., 2021, The Astrophysical Journal, 922, 217
  * Feltzing et al. (2007) Feltzing S., Fohlman M., Bensby T., 2007, Astronomy & Astrophysics, 467, 665
  * Feng & Krumholz (2014) Feng Y., Krumholz M. R., 2014, Nature, 513, 523
  * Fernández-Alvar et al. (2019) Fernández-Alvar E., Tissera P. B., Carigi L., Schuster W. J., Beers T. C., Belokurov V. A., 2019, Monthly Notices of the Royal Astronomical Society, 485, 1745
  * Font et al. (2006) Font A. S., Johnston K. V., Bullock J. S., Robertson B., 2006, The Astrophysical Journal, 638, 585
  * Forbes (2020) Forbes D. A., 2020, Monthly Notices of the Royal Astronomical Society, 493, 847
  * Fraley & Raftery (2002) Fraley C., Raftery A. E., 2002, Journal of the American Statistical Association, 97, 611
  * Frebel (2018) Frebel A., 2018, Annual Review of Nuclear and Particle Science, 68, 237
  * Freeman & Bland-Hawthorn (2002) Freeman K., Bland-Hawthorn J., 2002, Annual Review of Astronomy and Astrophysics, 40, 487
  * Fuhrmann (1998) Fuhrmann K., 1998, A&A, 338, 161
  * Gilmore & Reid (1983) Gilmore G., Reid N., 1983, Monthly Notices of the Royal Astronomical Society, 202, 1025
  * Hawkins et al. (2014) Hawkins K., Jofré P., Gilmore G., Masseron T., 2014, Monthly Notices of the Royal Astronomical Society, 445, 2575
  * Hawkins et al. (2015) Hawkins K., Jofre P., Masseron T., Gilmore G., 2015, Monthly Notices of the Royal Astronomical Society, 453, 758
  * Hayden et al. (2015) Hayden M. R., et al., 2015, The Astrophysical Journal, 808, 132
  * Haywood et al. (2018) Haywood M., Di Matteo P., Lehnert M., Snaith O., Khoperskov S., Gómez A., 2018, The Astrophysical Journal, 863, 113
  * Haywood et al. (2019) Haywood M., Snaith O., Lehnert M. D., Di Matteo P., Khoperskov S., 2019, Astronomy & Astrophysics, 625, A105
  * Helmi & de Zeeuw (2002) Helmi A., de Zeeuw P. T., 2002, Monthly Notices of the Royal Astronomical Society, 319, 657
  * Helmi et al. (2018) Helmi A., Babusiaux C., Koppelman H. H., Massari D., Veljanoski J., Brown A. G. A., 2018, Nature, 563, 85
  * Hendricks et al. (2014) Hendricks B., Koch A., Lanfranchi G. A., Boeche C., Walker M., Johnson C. I., Peñarrubia J., Gilmore G., 2014, The Astrophysical Journal, 785, 102
  * Hogg et al. (2016) Hogg D. W., et al., 2016, The Astrophysical Journal, 833, 262
  * Horta et al. (2020) Horta D., et al., 2020, Monthly Notices of the Royal Astronomical Society, 500, 1385
  * Horta et al. (2023) Horta D., et al., 2023, Monthly Notices of the Royal Astronomical Society, 520, 5671
  * Ishigaki et al. (2012) Ishigaki M. N., Chiba M., Aoki W., 2012, The Astrophysical Journal, 753, 64
  * Jackson-Jones et al. (2014) Jackson-Jones R., et al., 2014, Astronomy & Astrophysics, 571, L5
  * Ji et al. (2016) Ji A. P., Frebel A., Chiti A., Simon J. D., 2016, Nature, 531, 610
  * Jofré et al. (2019) Jofré P., Heiter U., Soubiran C., 2019, Annual Review of Astronomy and Astrophysics, 57, 571
  * Jolliffe & Cadima (2016) Jolliffe I. T., Cadima J., 2016, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374, 20150202
  * Kappeler et al. (1989) Kappeler F., Beer H., Wisshak K., 1989, Reports on Progress in Physics, 52, 945
  * Kobayashi et al. (2006) Kobayashi C., Umeda H., Nomoto K., Tominaga N., Ohkubo T., 2006, The Astrophysical Journal, 653, 1145
  * Kobayashi et al. (2020) Kobayashi C., Karakas A. I., Lugaro M., 2020, The Origin of Elements from Carbon to Uranium (arXiv:2008.04660)
  * Kruijssen et al. (2020) Kruijssen J. M. D., et al., 2020, Monthly Notices of the Royal Astronomical Society, 498, 2472
  * Kumar et al. (2021) Kumar R., Pradhan A. C., Ojha D. K., Piridi S., Baug T., Ghosh S. K., 2021, Journal of Astrophysics and Astronomy, 42, 42
  * Lanfranchi et al. (2006) Lanfranchi G. A., Matteucci F., Cescutti G., 2006, Monthly Notices of the Royal Astronomical Society, 365, 477
  * Ledinauskas & Zubovas (2018) Ledinauskas E., Zubovas K., 2018, Astronomy & Astrophysics, 615, A64
  * Liu et al. (2019) Liu F., Asplund M., Yong D., Feltzing S., Dotter A., Meléndez J., Ramírez I., 2019, Astronomy & Astrophysics, 627, A117
  * Mackereth et al. (2019) Mackereth J. T., et al., 2019, Monthly Notices of the Royal Astronomical Society, 482, 3426
  * Majewski et al. (2017) Majewski S. R., et al., 2017, The Astronomical Journal, 154, 94
  * Manea et al. (2023) Manea C., Hawkins K., Ness M. K., Buder S., Martell S. L., Zucker D. B., 2023, Chemical Doppelgangers in GALAH DR3: The Distinguishing Power of Neutron-Capture Elements Among Milky Way Disk Stars (arXiv:2310.15257)
  * Matsuno et al. (2021) Matsuno T., Hirai Y., Tarumi Y., Hotokezaka K., Tanaka M., Helmi A., 2021, Astronomy & Astrophysics, 650, A110
  * Matteucci (2016) Matteucci F., 2016, Journal of Physics: Conference Series, 703, 012004
  * Minelli et al. (2021) Minelli A., Mucciarelli A., Massari D., Bellazzini M., Romano D., Ferraro F. R., 2021, The Astrophysical Journal Letters, 918, L32
  * Mukherjee et al. (2024) Mukherjee C. S., Doerkar N., Zhang J., 2024, Capturing the Denoising Effect of PCA via Compression Ratio (arXiv:2204.10888)
  * Myeong et al. (2019) Myeong G. C., Vasiliev E., Iorio G., Evans N. W., Belokurov V., 2019, Monthly Notices of the Royal Astronomical Society, 488, 1235
  * Myeong et al. (2022) Myeong G. C., Belokurov V., Aguado D. S., Evans N. W., Caldwell N., Bradley J., 2022, The Astrophysical Journal, 938, 21
  * Naidu et al. (2020) Naidu R. P., Conroy C., Bonaca A., Johnson B. D., Ting Y.-S., Caldwell N., Zaritsky D., Cargile P. A., 2020, The Astrophysical Journal, 901, 48
  * Naidu et al. (2022) Naidu R. P., et al., 2022, The Astrophysical Journal Letters, 926, L36
  * Navarro et al. (2011) Navarro J. F., Abadi M. G., Venn K. A., Freeman K. C., Anguiano B., 2011, Monthly Notices of the Royal Astronomical Society, pp no–no
  * Nissen & Schuster (2010) Nissen P. E., Schuster W. J., 2010, Astronomy and Astrophysics, 511, L10
  * Nissen & Schuster (2012) Nissen P. E., Schuster W. J., 2012, Astronomy & Astrophysics, 543, A28
  * North et al. (2012) North P., et al., 2012, Astronomy & Astrophysics, 541, A45
  * Ortigoza-Urdaneta et al. (2023) Ortigoza-Urdaneta M., et al., 2023, Astronomy & Astrophysics, 676, A140
  * Pillepich et al. (2015) Pillepich A., Madau P., Mayer L., 2015, The Astrophysical Journal, 799, 184
  * Piskunov & Valenti (2017) Piskunov N., Valenti J. A., 2017, Astronomy & Astrophysics, 597, A16
  * Prantzos et al. (2023) Prantzos N., et al., 2023, Monthly Notices of the Royal Astronomical Society, 523, 2126
  * Price-Jones & Bovy (2019) Price-Jones N., Bovy J., 2019, Monthly Notices of the Royal Astronomical Society, 487, 871
  * Price-Jones et al. (2020) Price-Jones N., et al., 2020, Monthly Notices of the Royal Astronomical Society, 496, 5101
  * Prieto et al. (2008) Prieto C. A., et al., 2008, Astronomische Nachrichten, 329, 1018
  * Reddy et al. (2006) Reddy B. E., Lambert D. L., Prieto C. A., 2006, Monthly Notices of the Royal Astronomical Society, 367, 1329
  * Robertson et al. (2005) Robertson B., Bullock J. S., Font A. S., Johnston K. V., Hernquist L., 2005, The Astrophysical Journal, 632, 872
  * Sheffield et al. (2012) Sheffield A. A., et al., 2012, The Astrophysical Journal, 761, 161
  * Ting & Weinberg (2022) Ting Y.-S., Weinberg D. H., 2022, The Astrophysical Journal, 927, 209
  * Ting et al. (2012) Ting Y. S., Freeman K. C., Kobayashi C., De Silva G. M., Bland-Hawthorn J., 2012, Monthly Notices of the Royal Astronomical Society, 421, 1231
  * Ting et al. (2019) Ting Y.-S., Conroy C., Rix H.-W., Cargile P., 2019, The Astrophysical Journal, 879, 69
  * Tinsley (1979) Tinsley B. M., 1979, The Astrophysical Journal, 229, 1046
  * Tissera et al. (2014) Tissera P., Beers T., Carollo D., Scannapieco C., 2014, Monthly Notices of the Royal Astronomical Society, 439, 3128
  * Tsujimoto & Nishimura (2018) Tsujimoto T., Nishimura N., 2018, The Astrophysical Journal Letters, 863, L27
  * Wanajo et al. (2021) Wanajo S., Hirai Y., Prantzos N., 2021, Monthly Notices of the Royal Astronomical Society, 505, 5862
  * Weinberg et al. (2022) Weinberg D. H., et al., 2022, The Astrophysical Journal Supplement Series, 260, 32
  * Yanny et al. (2009) Yanny B., et al., 2009, The Astronomical Journal, 137, 4377
  * Yu et al. (2021) Yu S., et al., 2021, Monthly Notices of the Royal Astronomical Society, 505, 889
  * Zolotov et al. (2009) Zolotov A., Willman B., Brooks A. M., Governato F., Brook C. B., Hogg D. W., Quinn T., Stinson G., 2009, The Astrophysical Journal, 702, 1058
  * de Mijolla et al. (2021) de Mijolla D., Ness M., Viti S., Wheeler A., 2021, The Astrophysical Journal, 913, 12

##  Appendix A Table of Group Properties and Histograms

Group |  Sample Size |  Chemistry (dex) ($\mu\pm\sigma$) |  Kinematics ($L_{z}$ kpc kms-1) ($\mu\pm\sigma$) |  PC1 - PC2 Location |  Notes
---|---|---|---|---|---
G1 |  763 |  [Ba/Fe]$=-0.32\pm 0.21$, [Mg/Fe]$=0.32\pm 0.11$, [Mn/Fe]$=-0.13\pm 0.14$, [Y/Fe]$=-0.02\pm 0.23$, [Eu/Fe]$=0.30\pm 0.10$, [Zn/Fe]$=0.12\pm 0.22$, [Al/Fe]$=0.37\pm 0.11$, [Fe/H]$=-0.62\pm 0.10$, [Cu/Fe]$=0.09\pm 0.10$, [Ti/Fe]$=0.28\pm 0.10$, [Ni/Fe]$=0.07\pm 0.07$, [Sc/Fe]$=0.17\pm 0.07$ |  $L_{\rm{z}}=1175.89\pm 450.05$, $e=0.29\pm 0.19$ |  Core |  Thick disc chemistry and kinematics
G2 |  671 |  [Ba/Fe]$=0.41\pm 0.19$, [Mg/Fe]$=0.13\pm 0.10$, [Mn/Fe]$=-0.37\pm 0.12$, [Y/Fe]$=0.11\pm 0.17$, [Eu/Fe]$=0.44\pm 0.15$, [Zn/Fe]$=0.08\pm 0.15$, [Al/Fe]$=-0.12\pm 0.14$, [Fe/H]$=-1.00\pm 0.20$,[Cu/Fe]$=-0.46\pm 0.12$, [Ti/Fe]$=0.14\pm 0.10$, [Ni/Fe]$=-0.11\pm 0.07$, [Sc/Fe]$=-0.01\pm 0.07$ |  $L_{\rm{z}}=145.13\pm 521.97$, $e=0.80\pm 0.21$ |  branch |  GES
G3 |  651 |  [Ba/Fe]$=0.44\pm 0.24$, [Mg/Fe]$=0.32\pm 0.08$, [Mn/Fe]$=-0.27\pm 0.10$, [Y/Fe]$=0.33\pm 0.17$, [Eu/Fe]$=0.30\pm 0.10$, [Zn/Fe]$=0.19\pm 0.15$, [Al/Fe]$=0.25\pm 0.15$, [Fe/H]$=-0.87\pm 0.18$,[Cu/Fe]$=-0.14\pm 0.14$, [Ti/Fe]$=0.28\pm 0.08$, [Ni/Fe]$=0.00\pm 0.08$, [Sc/Fe]$=0.08\pm 0.08$ |  $L_{\rm{z}}=660.54\pm 599.88$, $e=0.49\pm 0.26$ |  Core/branch |  Bi-modal $L_{\rm{z}}$, in-situ population
G4 |  648 |  [Ba/Fe]$=0.28\pm 0.12$, [Mg/Fe]$=0.31\pm 0.06$, [Mn/Fe]$=-0.17\pm 0.06$, [Y/Fe]$=0.12\pm 0.13$, [Eu/Fe]$=0.24\pm 0.09$, [Zn/Fe]$=0.42\pm 0.08$, [Al/Fe]$=0.22\pm 0.08$, [Fe/H]$=-0.58\pm 0.07$,[Cu/Fe]$=-0.08\pm 0.07$, [Ti/Fe]$=0.20\pm 0.06$, [Ni/Fe]$=0.00\pm 0.06$, [Sc/Fe]$=0.13\pm 0.04$ |  $L_{\rm{z}}=1218.60\pm 441.24$, $e=0.32\pm 0.20$ |  Core |  Thick disc kinematics, high [Zn/Fe] distinguished from other thick disc groups
G5 |  141 |  [Ba/Fe]$=0.97\pm 0.37$, [Mg/Fe]$=0.34\pm 0.12$, [Mn/Fe]$=-0.21\pm 0.11$, [Y/Fe]$=0.68\pm 0.26$, [Eu/Fe]$=0.39\pm 0.13$, [Zn/Fe]$=0.25\pm 0.17$, [Al/Fe]$=0.31\pm 0.14$, [Fe/H]$=-0.75\pm 0.18$,[Cu/Fe]$=-0.04\pm 0.15$, [Ti/Fe]$=0.28\pm 0.08$, [Ni/Fe]$=0.02\pm 0.08$, [Sc/Fe]$=0.11\pm 0.06$ |  $L_{\rm{z}}=840.21\pm 609.00$, $e=0.45\pm 0.25$ |  Core/spray |  High s-process enhancement
G6 |  2235 |  [Ba/Fe]$=-0.05\pm 0.12$, [Mg/Fe]$=0.29\pm 0.07$, [Mn/Fe]$=-0.14\pm 0.06$, [Y/Fe]$=-0.06\pm 0.13$, [Eu/Fe]$=0.31\pm 0.09$, [Zn/Fe]$=0.13\pm 0.08$, [Al/Fe]$=0.37\pm 0.06$, [Fe/H]$=-0.62\pm 0.08$, [Cu/Fe]$=0.08\pm 0.06$, [Ti/Fe]$=0.27\pm 0.06$, [Ni/Fe]$=0.07\pm 0.05$, [Sc/Fe]$=0.14\pm 0.04$ |  $L_{\rm{z}}=1245.04\pm 430.14$, $e=0.31\pm 0.19$ |  Core |  Thick disc, chemically similar to G1 and G8
G7 |  832 |  [Ba/Fe]$=-0.10\pm 0.13$, [Mg/Fe]$=0.27\pm 0.08$, [Mn/Fe]$=-0.23\pm 0.07$, [Y/Fe]$=0.09\pm 0.20$, [Eu/Fe]$=0.34\pm 0.06$, [Zn/Fe]$=-0.08\pm 0.19$, [Al/Fe]$=0.43\pm 0.06$, [Fe/H]$=-0.63\pm 0.09$, [Cu/Fe]$=0.12\pm 0.06$, [Ti/Fe]$=0.24\pm 0.06$, [Ni/Fe]$=0.06\pm 0.05$, [Sc/Fe]$=0.18\pm 0.05$ |  $L_{\rm{z}}=1153.59\pm 416.14$, $e=0.31\pm 0.18$ |  Core |  Thick disc kinematics with bi-modal [Zn/Fe]
G8 |  610 |  [Ba/Fe]$=0.21\pm 0.15$, [Mg/Fe]$=0.36\pm 0.08$, [Mn/Fe]$=-0.18\pm 0.08$, [Y/Fe]$=-0.05\pm 0.18$, [Eu/Fe]$=0.22\pm 0.07$, [Zn/Fe]$=0.15\pm 0.15$, [Al/Fe]$=0.27\pm 0.07$, [Fe/H]$=-0.61\pm 0.08$, [Cu/Fe]$=0.06\pm 0.08$, [Ti/Fe]$=0.19\pm 0.06$, [Ni/Fe]$=0.10\pm 0.05$, [Sc/Fe]$=0.04\pm 0.05$ |  $L_{\rm{z}}=1073.85\pm 462.95$, $e=0.35\pm 0.21$ |  Core |  Thick disc, chemically similar to G1 and G6
G9 |  581 |  [Ba/Fe]$=0.24\pm 0.19$, [Mg/Fe]$=0.10\pm 0.07$, [Mn/Fe]$=-0.08\pm 0.08$, [Y/Fe]$=-0.06\pm 0.18$, [Eu/Fe]$=0.15\pm 0.09$, [Zn/Fe]$=-0.03\pm 0.20$, [Al/Fe]$=0.21\pm 0.09$, [Fe/H]$=-0.56\pm 0.06$, [Cu/Fe]$=0.05\pm 0.09$, [Ti/Fe]$=0.09\pm 0.07$, [Ni/Fe]$=0.04\pm 0.07$, [Sc/Fe]$=0.06\pm 0.06$ |  $L_{\rm{z}}=2020.10\pm 649.82$, $e=0.19\pm 0.19$ |  Core |  Most $\alpha$-poor, thin disc
G10 |  365 |  [Ba/Fe]$=0.63\pm 0.65$, [Mg/Fe]$=0.28\pm 0.23$, [Mn/Fe]$=-0.19\pm 0.28$, [Y/Fe]$=0.37\pm 0.53$, [Eu/Fe]$=0.39\pm 0.24$, [Zn/Fe]$=0.26\pm 0.34$, [Al/Fe]$=0.24\pm 0.28$, [Fe/H]$=-0.78\pm 0.22$, [Cu/Fe]$=-0.11\pm 0.27$, [Ti/Fe]$=0.22\pm 0.21$, [Ni/Fe]$=0.00\pm 0.16$, [Sc/Fe]$=0.12\pm 0.17$ |  $L_{\rm{z}}=982.75\pm 831.54$, $e=0.44\pm 0.31$ |  Core/branch/spray |  Significant variance, overlaps with multiple groups
Table 2: Group characteristics based on sample size, chemistry, kinematics,
PC1 - PC2 location, and notable information.