11institutetext:  Instituto de Astronomía y Ciencias Planetarias, Universidad
de Atacama, Copayapu 485, Copiapó, Chile 22institutetext: Instituto de
Astronomía, Universidad Católica del Norte, Av. Angamos 0610, Antofagasta,
Chile 33institutetext: Leibniz-Institut f$\ddot{u}$r Astrophysik Potsdam
(AIP), An der Sternwarte 16, 14482 Potsdam, Germany 44institutetext: Institut
für Physik und Astronomie, Universität Potsdam, Haus 28 Karl-Liebknecht-Str.
24/25, 14476 Golm, Germany 55institutetext: Laboratório Interinstitucional de
e-Astronomia - LIneA, RJ 20921-400, Rio de Janeiro, Brasil 66institutetext:
Universidade de São Paulo, IAG, Rua do Matão 1226, Cidade Universitária, São
Paulo 05508-900, Brazil 77institutetext: Department of Physics and Astronomy
and JINA Center for the Evolution of the Elements, University of Notre Dame,
Notre Dame, IN 46556, USA 88institutetext: Departament de Física Quántica i
Astrofísica (FQA), Universitat de Barcelona (UB), c. Martí i Franquès, 1,
08028 Barcelona, Spain 99institutetext: Institut de Ciències del Cosmos
(ICCUB), Universitat de Barcelona (UB), c. Martí i Franquès, 1, 08028
Barcelona, Spain 1010institutetext: Institut d’Estudis Espacials de Catalunya
(IEEC), c. Gran Capità, 2-4, 08034 Barcelona, Spain 1111institutetext: Depto.
de Cs. Físicas, Facultad de Ciencias Exactas, Universidad Andrés Bello, Av.
Fernández Concha 700, Las Condes, Santiago, Chile 1212institutetext: Vatican
Observatory, V00120 Vatican City State, Italy 1313institutetext: School of
Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China
1414institutetext: Centro de Investigación en Astronomía, Universidad Bernardo
O’Higgins, Avenida Viel 1497, Santiago, Chile

# Galactic ArchaeoLogIcaL ExcavatiOns (GALILEO) II.
t-SNE Portrait of Local Fossil Relics and Structures

Mario Ortigoza-Urdaneta<EMAIL_ADDRESS>Katherine Vieira 11
José G. Fernández-Trincado<EMAIL_ADDRESS>Anna. B. A. Queiroz 334455
Beatriz Barbuy 66 Timothy C. Beers 77 Cristina Chiappini 3355 Friedrich Anders
88991010 Dante Minniti 11111212 Baitian Tang 1313

(Received …; Accepted …)

Based on high-quality APOGEE DR17 and Gaia DR3 data for 1,742 red giants stars
within 5 kpc of the Sun and not rotating with the Galactic disc ($V_{\phi}<$
100 km s-1), we use the nonlinear technique of unsupervised analysis t-SNE to
detect coherent structures in the space of ten chemical-abundance ratios:
[Fe/H], [O/Fe], [Mg/Fe], [Si/Fe], [Ca/Fe], [C/Fe], [N/Fe], [Al/Fe], [Mn/Fe],
and [Ni/Fe]. Additionally, we obtain orbital parameters for each star using
the non-axisymmetric gravitational potential GravPot16. Seven structures are
detected, including the Splash, Gaia-Sausage-Enceladus (GSE), the
high-$\alpha$ heated-disc population, N-C-O peculiar stars, and inner disk-
like stars, plus two other groups that did not match anything previously
reported in the literature, here named Galileo 5 and Galileo 6 (G5 and G6).
These two groups overlap with Splash in [Fe/H], G5 being lower metallicity
than G6, both between GSE and Splash in the [Mg/Mn] versus [Al/Fe] plane, G5
in the $\alpha$-rich in-situ locus, and G6 on the border of the $\alpha$-poor
in-situ one; nonetheless their low [Ni/Fe] hints to a possible ex-situ origin.
Their orbital energy distributions are between the Splash and GSE, with G5
being slightly more energetic than G6. We verified the robustness of all the
obtained groups by exploring a large range of t-SNE parameters, applying it to
various subsets of data, and also measuring the effect of abundance errors
through Monte Carlo tests.

###### Key Words.:

stars: abundances, stars: chemically peculiar, Galaxy: solar neighborhood,
Galaxy: halo, techniques: spectroscopic, methods: statistical

##  1 Introduction

Numerous studies have revealed that the various structures that make up the
Milky Way (MW) have been affected by both intrinsic (e.g., secular evolution,
Vera et al. 2016; Combes 2017) and extrinsic (e.g., merger events, Kruijssen
et al. 2020) processes. As a consequence, our galaxy has mixed populations
comprising both in-situ and ex-situ star formation. The identification of
these structures is fundamental to understanding the formation and
evolutionary history of our galaxy.

In some cases, these structures still retain traces of their original
kinematics, and can be identified by their peculiar orbits as compared to
canonical populations of the MW (Belokurov et al., 2018; Koppelman et al.,
2019a), but in many other cases, their original motion is erased by their
prolonged interaction with the MW. On the other hand, most chemical abundances
in stellar atmospheres are unaffected by the dynamical history of stars. Thus,
a chemical signature becomes a valuable piece of information to determine the
presence and origin of a given population. Taken together, kinematics,
dynamics, and chemical abundances are extremely useful to decode the history
of such population (e.g., Freeman & Bland-Hawthorn, 2002; Hasselquist et al.,
2021; Horta et al., 2023).

Thanks to the data provided by Gaia (Gaia Collaboration et al., 2021) and
spectroscopic surveys (e.g., APOGEE, GALAH Majewski et al., 2017; Buder et
al., 2022), and their unprecedented precision, great advances have been made
in the detection and understanding of structures such as Gaia-Sausage
Enceladus (GSE) (Belokurov et al., 2018; Haywood et al., 2018; Helmi et al.,
2018a; Koppelman et al., 2018; Myeong et al., 2018) or the Helmi stream(s)
(Helmi et al., 1999; Chiba & Beers, 2000; Koppelman et al., 2019b). In both
cases, an ex-situ origin is attributed, coming from past mergers with the MW.
On the other hand, we have the “Splash”, a population thought to have likely
arisen from the heating of the primordial disc by an early merger (Bonaca et
al., 2017, 2020; Haywood et al., 2018; Di Matteo et al., 2019; Belokurov et
al., 2020). The Splash has also been interpreted as an outcome of clumpy star
formation of the early disc (Amarante et al., 2020; Fiteni et al., 2021).

While in the past century astronomical data was often scarce, particularly
regarding chemical abundances, parallaxes, and proper motions, nowadays the
situation is the opposite. This large influx of data allows for a more
detailed analysis of known structures and the discovery of new populations,
although the increased density of data points hampers a clean separation of
the populations, which usually overlap in the many different measurements
available. More data - traditionally analyzed - means it is more difficult to
separate the (probable) members individually.

The tsunami of astronomical data has driven the application of statistical
techniques that allow for automatic searches to identify potential
associations of stars by specific properties. It is necessary to understand
how these methods work and to what extent they can help us analyze the data,
as there are differences in data quality due to both instrumental and natural
factors. Among the widely used techniques is Principal Component analysis
(PCA), which has been successfully applied in many kinds of data with
excellent results (Rebonato & Jäckel, 2011; Ting et al., 2012).

In this paper, we present the use of t-SNE (t-Distributed Stochastic
Stochastic Neighbor Embedding (Hinton & Roweis, 2003; van der Maaten & Hinton,
2008) dimensionality reduction technique for a study of local low-velocity
structures ($V_{\phi}<100$ km s-1, less than 5 kpc from the Sun), from input
data provided by APOGEE+Gaia. t-SNE has been used in several areas of
astronomy, such as pre-main sequence identification (e.g., Rim et al., 2022),
spectral classification (e.g., Verma et al., 2021), and abundance-space
dissection (e.g., Anders et al., 2018), where we note the powerful
applicability for automated searches of populations. We provide an analysis
based on t-SNE detections obtained from ten chemical abundances of our sample
and the dynamics provided by GravPot16111https://gravpot.utinam.cnrs.fr.

In the last few years, several sub-populations have been identified in the MW
halo, e.g., GSE, Splash, Sequoia (Barbá et al., 2019), Kraken (Kruijssen et
al., 2020), and others, due to their peculiar kinematics, dynamics, and/or
chemical abundances. It has been proposed by Naidu et al. (2020) and other
authors that the MW halo is built entirely from accreted dwarfs and heating of
the disc. Those substructures have been generally identified at rather large
vertical distances from the MW plane, but in this study, we look for evidence
of their presence closer to the Sun where they would co-locate with the disc
system.

This paper is organized as follows: data sources and selection are described
in sections 2, 3, and 4, the MW dynamical model in section 5, the t-SNE method
in sections 6 and 7, and our results in Section 8. Section 9 discusses the
accretion origin of some of the structures detected, section 10 is dedicated
to a quick analysis of the effects of the bar pattern speed in the obtained
structures, and our conclusions are in Section 11. Additional figures and
tables are located in the Appendix section.

##  2 APOGEE-2 DR 17

The data employed in this study were obtained as part of the second phase of
the Apache Point Observatory Galactic Evolution Experiment (APOGEE-2; Majewski
et al., 2017), which was one of the four Sloan Digital Sky Survey IV surveys
(SDSS-IV; Blanton et al., 2017). APOGEE-2 was a high-resolution ($R\sim
22,500$) near-infrared (NIR) spectroscopic survey containing observations of
$657,135$ stars, whose spectra were obtained using the cryogenic, multi-fiber
(300 fibers) APOGEE spectrograph (Wilson et al., 2019) mounted on the 2.5m
SDSS telescope (Gunn et al., 2006) at Apache Point Observatory to observe the
Northern Hemisphere (APOGEE-2N), and expanded to include a second APOGEE
spectrograph on the 2.5m Irénée du Pont telescope (Bowen & Vaughan, 1973) at
Las Campanas Observatory to observe the Southern Hemisphere (APOGEE-2S). Each
instrument records most of the H-band ($1.51$ $\mu$m – $1.70$ $\mu$m) on three
detectors, with coverage gaps between $\sim$1.58 – 1.59 $\mu$m and $\sim$1.64
– 1.65 $\mu$m, and with each fiber subtending a $\sim$2” diameter on-sky field
of view in the northern instrument and 1.3” in the southern. We refer the
interested reader to Zasowski et al. (2013), Zasowski et al. (2017), Beaton et
al. (2021), and Santana et al. (2021) for further details regarding the
targeting strategy and design of the APOGEE-2 survey.

The final version of the APOGEE-2 catalog was published in December 2021 as
part of the 17th data release of the Sloan Digital Sky Survey (DR17;
Abdurro’uf et al., 2022) and is available publicly online through the SDSS
Science Archive Server and Catalog Archive Server222SDSS DR17 data:
https://www.sdss.org/dr17/irspec/spectrodata/. The APOGEE-2 data reduction
pipeline is described in Nidever et al. (2015), while stellar parameters and
chemical abundances in APOGEE-2 have been obtained within the APOGEE Stellar
Parameters and Chemical Abundances Pipeline (ASPCAP; García Pérez et al.,
2016). ASPCAP derives stellar atmospheric parameters, radial velocities, and
as many as 26 individual elemental abundances for each APOGEE-2 spectrum by
comparing each to a multidimensional grid of theoretical MARCS model
atmosphere grid (Zamora et al., 2015), employing a $\chi^{2}$ minimization
routine with the code FERRE (Allende Prieto et al., 2006) to derive the best-
fit parameters for each spectrum. We used the abundances computed from the
synspec_fix spectral synthesis code.

The accuracy and precision of the atmospheric parameters and chemical
abundances are extensively analyzed in Holtzman et al. (2018), Jönsson et al.
(2018), and (Jönsson et al., 2020), while details regarding the customized
H-band line list are fully described in Shetrone et al. (2015), Hasselquist et
al. (2016), Cunha et al. (2017), and Smith et al. (2021).

##  3 Input parameters

###  3.1 Elemental abundances and radial velocities

We make use of high-quality elemental abundances for ten chemical species,
including the light- (C, N), $\alpha$\- (O, Mg, Si, Ca), odd-Z (Al), and iron-
peak (Mn, Fe, Ni) elements, as well as precise ($<$1 km s-1) APOGEE-2
spectroscopic radial velocity measurements when calculating kinematics and
orbits for giant stars in the APOGEE DR 17 catalog (Abdurro’uf et al., 2022).
The other 16 abundances available from ASPCAP were not used because they are
affected by telluric bands (e.g., Na), blended with other atomic/molecular
lines, or have a lower signal-to-noise ratio (SNR). Also, for some elements a
reduced number of stars have measurements.

###  3.2 StarHorse distances

We also make use of the precise spectro-photo-astrometric distances
($d_{\rm\texttt{StarHorse}}$) estimated with the Bayesian StarHorse code
(Santiago et al., 2016; Queiroz et al., 2018; Anders et al., 2019; Queiroz et
al., 2020), which have been published in the form of a value-added catalog
(VAC)333https://data.sdss.org/datamodel/files/APOGEE_STARHORSE/. StarHorse
combines high-resolution spectroscopic data from APOGEE-2 DR 17 with broad-
band photometric data from several sources (Pan-STARSS1, 2MASS, and AllWISE),
as well as parallaxes from Gaia DR 3 when available, along with their
associated uncertainties, in order to derive distances, extinctions, and
astrophysical parameters for APOGEE-2 stars through a Bayesian isochrone-
fitting procedure. These parameters are robust to changes in the Galactic
priors assumed and corrections for the Gaia parallax zero-point offset.

###  3.3 Proper motions

Astrometric data are taken from Gaia DR 3 (Gaia Collaboration et al., 2021)
that are also available for the APOGEE-2 sample. For this study, when
calculating orbital parameters the RUWE (renormalized unit weight error)
astrometric quality indicator was imposed to be RUWE $<$ 1.40, in order to
have astrometrically well-behaved sources (see, e.g., Lindegren et al., 2018).

##  4 Sample selection

The APOGEE-2 DR 17 catalog contains more than six hundred thousand entries.
Several cuts were applied to refine our sample. For quality control of the
stellar parameters and elemental abundances, we first cleaned the sample from
sources with unreliable parameters by keeping only sources with STARFLAG and
ASPCAPFLAG equal to zero (see e.g., Holtzman et al., 2015, 2018), and stars
with good spectra (SNR $>$ 70 pixel-1) and good stellar abundances, which are
flagged as X_FE_FLAG$=$0 (X$=$ C, N, O, Mg, Al, Si, Ca, Mn, Fe, and Ni). These
cuts ensure that there are not major flagged issues, such as a low SNR, poor
synthetic spectral fit, stellar parameters near grid boundaries, and potential
problematic object spectra.

We also cut in distance precision, by selecting only stars with less than 20%
error distances from $d_{\rm\texttt{StarHorse}}$. Application of these cuts
yields an initial sample of 144,476 stars. Then, we make a cut to select red
giants, by selecting from the initial sample only those stars with surface
gravity values $\log$ g $<$ 3.6, and stellar effective temperatures in the
range T${}_{\rm eff}<$ 5500K, yielding a sample of 142,305 stars. The upper
effective temperature limit was imposed to remove stars at the hot edge of the
main stellar-atmospheric model grids used in the APOGEE-2 analysis. The
following cut is in kinematics, by choosing only stars with Galactocentric
azimuthal velocity $V_{\phi}<100$ km s-1, which minimizes the presence of the
dominant Galactic disc population, yielding a sample of 9544 stars.

We also checked that this sample did not have repeated Gaia DR3 source IDs,
because the main APOGEE catalog does have, in a few cases, pairs of different
APOGEE IDs associated with the same Gaia source ID. This is relevant in the
next cut for RUWE¡1.4, for which a true Gaia match is obviously needed, since
the match is done via the Gaia DR3 source ID listed in the APOGEE catalog. The
cut in RUWE yielded a sample of 9107 stars. We then cut the sample to stars
located within a 5 kpc sphere around the Sun ($d_{\rm\texttt{StarHorse}}<$ 5
kpc), yielding a sample of 1,905 stars that sample a sufficiently large volume
to reach the inner-halo region, but avoiding the Bulge region of influence. In
this paper, we refer to inner halo as the part of the Milky Way halo dominated
by debris from past events of accretion, involving massive objects such as
dwarf galaxies or globular clusters (e.g., Helmi et al., 2018b).

Finally, we discard known globular cluster member stars as listed by Mészáros
et al. (2020), Baumgardt & Vasiliev (2021), and Vasiliev & Baumgardt (2021).
Our final sample for the analysis below contains 1742 red giant stars. It is
also important to notice that we make no correction for the selection biases
within the APOGEE-2 DR 17 survey. Thus, stars close to the Solar Neighborhood
will be over-represented in our sample.

##  5 Dynamical Model

In order to construct a comprehensive orbital study of the local structures
across of the Solar vicinity, we use a state-of-the-art orbital integration
model in a non-axisymmetric gravitational potential that fits the structural
and dynamical parameters based on recent knowledge of our Galaxy.

For the computations in this work, we have employed the rotating “boxy/peanut”
bar of the Galactic potential model called GravPot16, along with other
composite stellar components. The considered structural parameters of our bar
model, e.g., mass, present-day orientation, and pattern speeds, are within
observational estimations that lie in the range of 1.1$\times$1010 M⊙ and 20∘,
in line with Tang et al. (2018) and Fernández-Trincado et al. (2020d), and
31–51 km s-1 kpc-1 (see e.g., Bovy et al., 2019; Sanders et al., 2019) in
increments of 10 km s-1 kpc-1, respectively. The bar scale lengths are
$x_{0}=$1.46 kpc, $y_{0}=$ 0.49 kpc, $z_{0}=$0.39 kpc, and the middle region
ends at the effective major semiaxis of the bar Rc $=3.28$ kpc (Robin et al.,
2012). The density profile of the adopted “boxy/peanut” bar is the same as in
Robin et al. (2012).

GravPot16 considers, on a global scale, a 3D steady-state gravitational
potential for the MW, modeled as the superposition of axisymmetric and non-
axisymmetric components. The axisymmetric potential comprises the
superposition of many composite stellar populations belonging to seven thin
discs. For each ith component of the thin disc, we implemented an Einasto
density-profile law Einasto (1979), as described in Robin et al. (2003),
superposed with two thick-disc components, each one following a simple
hyperbolic secant squared decreasing vertically from the Galactic plane plus
an exponential profile decreasing with Galactocentric radius, as described in
Robin et al. (2014). We also implemented the density profile of the
interstellar matter (ISM) component with a density mass as presented in Robin
et al. (2003).

The model also correctly accounts for the underlying stellar halo, modeled by
a Hernquist profile as already described in Robin et al. (2014), and
surrounded by a single spherical Dark Matter halo component Robin et al.
(2003); no time dependence of the density profiles is assumed. The most
important limitations of our model are:

  * (i)

We ignore secular changes in the MW potential over time, which are expected
although our Galaxy has had a quiet recent accretion history.

  * (ii)

We do not consider perturbations due to spiral arms, as an in-depth analysis
is beyond the scope of this paper.

For reference, the Galactic convention adopted by this work is: $X-$axis is
oriented toward $l=$ 0∘ and $b=$ 0∘, the $Y-$axis is oriented toward $l$ = 90∘
and $b=$0∘, and the disc rotates toward $l=$ 90∘; the velocities are also
oriented in these directions. In this convention, the Sun’s orbital velocity
vector is [U⊙,V⊙,W⊙] = [$11.1$, $248.5$, $7.25$] km s-1, in line with
Brunthaler et al. (2011) and Reid & Brunthaler (2020). The model has been
rescaled to the Sun’s Galactocentric distance, $R_{\odot}=$8.178 kpc (GRAVITY
Collaboration et al., 2019) and Z${}_{\odot}=25$ pc (Jurić et al., 2008).

The orbital trajectories for our entire sample were integrated by adopting a
simple Monte Carlo scheme and the Runge-Kutta algorithm of seventh-eight order
elaborated by Fehlberg (1968), in order to construct the initial conditions
for each star, taking into account the uncertainties in the radial velocities
provided by the APOGEE-2 survey, the absolute proper motions provided by the
Gaia DR3 catalog (Gaia Collaboration et al., 2022), and the spectro-photo-
astrometric distances provided by StarHorse (Queiroz et al., 2023). The
uncertainties in the input data (e.g., $\alpha$, $\delta$, distance, proper
motions, and line-of-sight velocity errors), were randomly propagated as
1$\sigma$ variations in a Gaussian Monte Carlo re-sampling. For each star, we
computed one thousand orbits, computed backward in time over 3 Gyr. The
average value of the orbital elements was found for these thousand
realizations, with uncertainty ranges given by the 16th and 84th percentile
values.

For this study, we use the average value of the amplitude of the vertical
oscillation $|Z_{max}|$, the perigalactic distance $R_{peri}$, apogalactic
distance $R_{apo}$, the eccentricity $e$, the orbital Jacobi constant $E_{j}$
computed in the reference frame of the bar and the “characteristic” orbital
energy $E_{char}$ as envisioned by Moreno et al. (2015). The $E_{char}$ vs.
$E_{J}$ plane is used to broadly discriminate orbit populations, the lower the
$E_{char}$ the more radially confined to the Galaxy, the lower the $E_{J}$ the
more vertically confined it is.

We also consider the minimum and maximum of the z-component of the angular
momentum in the inertial frame, which defines their orbital behavior. Under a
non-axisymmetric potential, the angular momentum is not conserved, thus during
the 3 Gyr integration time, a star can be either always prograde (P), always
retrograde (R), or change from prograde to retrograde or vice versa (P-R).
That is, a P orbit has $L_{min}<0$ and $L_{max}<0$, an R orbit has $L_{min}>0$
and $L_{max}>0$, and a P-R orbit has $L_{min}$ and $L_{max}$ with opposite
signs. In general, P-R orbits are extremely eccentric, indicating that their
transition from prograde to retrograde orbit, or vice versa, is achieved by
following a very radial orbit, with their periastron rather close to the
Galactic center. Their dynamics can be subject to chaotic effects with very
sensitive orbits, and/or could be trapped in a resonance. Also, when orbits
are isotropically heated, not just vertically, we suspect the mechanism
keeping them that way, internal or external, is different from the one that
formed the MW thick disc.

##  6 t-SNE

The t-Distributed Stochastic Neighbor Embedding (t-SNE) algorithm is a
nonlinear technique of unsupervised analysis for the study of high-dimensional
ensembles through a dimension reduction that allows its visualization (2D or
3D), as presented by van der Maaten & Hinton (2008). Many areas of study make
use of this technique. due to its utility in finding possibly related
structures automatically and transferring the information to spaces that allow
us to visualize them. In other words, t-SNE is able to detect similarity among
data points in high-dimensional space, by transforming them onto 2D values
(the t-SNE plane), so that nearby points are considered as likely members of
the same population, or at least somehow related.

Astronomy presents an ideal scenario for the use of this technique since we
have now large catalogs with a variety of information for each star. In the
context of our work, t-SNE has proven useful, such as in Anders et al. (2018),
where the interested reader can find a summarized yet comprehensive
description of the operation of t-SNE. In the following paragraphs, we provide
an overall description of how the method works.

The t-SNE algorithm comprises two main stages. First, it constructs a
probability distribution over pairs of high-dimensional objects so that the
closer/farther objects are, the higher/lower probability assigned for being
related. Second, starting from a Gaussian low-dimensional distribution, t-SNE
iteratively constructs a probability distribution of points by minimizing the
Kullback–Leibler divergence between the high- and low-dimensional
distributions using gradient descent. As a result, the final low-dimensional
distribution resembles the probability of the high-dimension one, so that
nearby objects in one are also close in the other.

The method has one main parameter, called the perplexity $p$, which can be
thought of as a guess about the number of close neighbors each point has; the
ideal value for $p$ depends on the sample size. Since a change in perplexity
has in many cases a complex effect on the resulting map, it is recommended to
try different values for $p$ (Wattenberg et al., 2016). According to Linderman
& Steinerberger (2017), two other hyper-parameters of t-SNE can be optimally
chosen: the learning rate ($\sim$ 1) and the early exaggeration parameter (10%
of sample size). We used these recommended values as a starting point but
eventually, other values were set, guided by the best separation of the
detected groups in the t-SNE plane.

We use the Python implementation of t-SNE included in the scikit-learn package
(Pedregosa et al., 2011). A word of caution is given: with the same data and
parameter inputs, we recover the exact same t-SNE planes up to version 1.1.3
of scikit-learn, but when using newer updates, i.e, version 1.2.0 onwards, the
look of the t-SNE plane changes. We checked that our detected groups remain
virtually the same, therefore our conclusions are not altered.

The distribution of points in the t-SNE plane depends not only on the
multidimensional data input, but also on the order each of these vectors is
entered. This means that the same data input sorted in different ways will
yield different-looking t-SNE planes, but in all cases, the same points will
be associated quite similarly. In our case, the stellar data were inserted in
order of increasing [Fe/H], as an additional check on the obtained results.

The t-SNE algorithm does have two major weaknesses: 1) no missing individual
data is allowed, meaning each star must have measured values in all the
dimensions considered; and 2) it does not consider the effect of individual
uncertainties. We deal with these issues by 1) using only stars with all ten
chosen abundances measured and 2) performing a Monte Carlo experiment to show
that our results are robust to abundance uncertainties.

##  7 Strategy adopted with t-SNE

The input entries fed into t-SNE are the abundance ratios [Fe/H], [O/Fe],
[Mg/Fe], [Si/Fe], [Ca/Fe], [C/Fe], [N/Fe], [Al/Fe], [Mn/Fe], and [Ni/Fe]. We
also include [Mg/Mn], which has proven to be useful in separating accreted
from in-situ structures (Das et al., 2020; Naidu et al., 2020), considering
also that from previous work we know of structures that overlap in [Mg/Mn] but
are separate in [Mg/Fe] and [Mn/Fe], or vice-versa. Our final results were
obtained under perplexity = 25, early_exageration = 130, init = pca,
learning_rate = auto, and n_iter = 7000, in which the detected associations of
stars were more visibly separated (See Figure 1, central panel). A total of
seven structures were identified, and are presented for separate analysis in
the coming sections. To choose this result as the final one, we thoroughly
tested the effect of the input parameters and the abundance errors in the
obtained results, as described below. Also, we checked their dynamical
parameters to help confirm if the detected populations also exhibited
additional differences in their orbits.

Stars in all the structures identified were persistently associated in the
test runs despite changing the t-SNE parameters over a range of values:
$20<$perplexity$<40$, and $80<$early_exageration$<130$, although a few of the
smaller groups were more unstable in their location in the t-SNE plane, moving
from one location to other, getting closer or farther from the other groups.
The diagrams generated by t-SNE can strongly vary the shape of the
distribution in the plane depending on the initial parameters imposed (see
Figure 13 in Appendix A), but indeed some of the identified structures
predominate and simply relocate, always staying as a single structure (Groups
G1, G2, G5, and G7 mainly). Therefore, the final selection shown considered
how their location in the t-SNE plane changed with respect to the other
detected associations. Another important test was to verify the separation of
adjacent or nearby structures by running t-SNE on smaller input subsets.
Moreover, we used the obtained t-SNE values as additional inputs for a second
t-SNE run, to check that the selected groups were “squeezed” further in the
new t-SNE plane. These test helped us to trace the boundaries between nearby
groups. Results of this test are shown in Appendix A, Figure 14. We also found
that the most extended structures (G1 and G2) kept each one cohesively in
these runs, without further separation.

To understand the effect of abundance errors, we generated random samples
based on the uncertainty of each abundance and then fed those as input to
t-SNE. For each star, 100 random measurements were drawn from a Gaussian
centered on the measured abundance and dispersion corresponding to its
uncertainty. Samples were then created by randomly taking one out of 100
simultaneously in all input abundances and applying the t-SNE to each
generated sample with the same parameters as the one chosen for our final
results. We found that, despite significant variation in the distribution
shape of the points in the t-SNE plane (see Figure 15 in Appendix A), the same
structures kept together consistently and were detected in the different t-SNE
realizations. Therefore, we conclude that abundance uncertainties do not lead
to significant changes in the group identifications. Moreover, this reinforces
the fact that the structures detected do not correspond to random
associations.

##  8 Structures detected in the t-SNE projection

The large panel shown in Figure 1 is our reference t-SNE projection, on which
we have identified and named seven substructures that clearly emerge from it.
The figure also shows the distribution of these substructures in several
elemental-abundance diagrams (small panels), which have been color-coded
according to the substructures’ names. This t-SNE plane is also shown in
Figure 2, color-coded by the indicated abundances and dynamical parameters
$e$, $|Z_{max}|$ in kpc, and $R_{apo}$ in kpc.

From inspection of the t-SNE map in Figure 1, we can immediately appreciate
two dominant and well-populated substructures, which fit very well with the
descriptions of the Splash-like population (G1, blue points, Fig. 1) and a
significant past merger event (G2, beige points, Fig. 1) established in the
literature. Five other smaller substructures were identified as well. Each
substructure is described individually below, considering not only their
chemical abundances but also their dynamical properties, which were not fed to
the t-SNE but exhibit distinctive ranges. In all corresponding figures,
$V_{\phi}$ is in km s-1, distances are given in kpc, and energies in $10^{5}$
km2 s-2. Eccentricity $e$ and t-SNE X and t-SNE Y values are dimensionless.
Figure 3 shows the distribution of the chemical abundances for some of the
detected groups, to prove they were all indeed different from each other in at
least one of the elements. In the Appendix section B, Table 1 contains the
median and median absolute deviations for all the chemical abundances and
dynamical parameters of each detected group.

Figure 1: The t-SNE plane, shown in the large main panel, with the selected
groups identified by colors and labels. The surrounding panels correspond to
the ten APOGEE abundances used as inputs for t-SNE. Figure 2: The t-SNE plane
color-coded by the indicated abundances and dynamical parameters $e$,
$|Z_{max}|$ in kpc, and $R_{apo}$ in kpc. Figure 3: Comparison of Groups G1
(Splash), G2 (GSE), G5 (Galileo 5), and G6 (Galileo 6), in each of the
chemical abundances used in this work, displayed in the form of a violin plot.
A common trend in the compared abundances for the $\alpha$-elements can be
seen, as well as in the iron-peak elements. The observed distributions
demonstrate that each group is different from the others in at least one
chemical abundance.

###  8.1 Group 1: The Splash

This is the largest and most consistent structure detected by the t-SNE
method, showing higher $\alpha$-element abundances $\gtrsim+0.20$ and slightly
enriched levels of Al ([Al/Fe] $>+0.1$) at metallicities [Fe/H] $>-1$ (Figure
1, blue points) to solar metallicity, chemistry consistent with the thick
disc. Dynamically (see Figure 4), it exhibits a wide range of $|Z_{max}|$,
spanning from in-plane to 14 kpc from the Galactic plane, clearly overlapping
with the inner-halo region of the Galaxy.

$R_{apo}$ spans mostly from 3 to 12 kpc, but a few stars reach up to 25 kpc,
while $R_{peri}$ is mostly below 2.5 kpc. As expected from such values, these
stars follow eccentric orbits, typically with $e\gtrsim 0.6$ (see also the
lower panels of Figure 2, color-coded by $e,|Z_{max}|$, and $R_{apo}$). The
orbital energies of these stars span a large range, from low to mid values.
Much of this group fits very well with the descriptions established for the
so-called Splash by different authors (Bonaca et al., 2017, 2020; Haywood et
al., 2018; Di Matteo et al., 2019; Belokurov et al., 2020).

G1 is mostly (76%) comprised of prograde orbits stars (see Table 2), which
supports the idea of its origin as heated-disc stars. Nonetheless, a non-
negligible portion of G1 stars (see Table 2 in Appendix B) are P-R and
retrograde orbits, which are not only vertically but also radially heated.
Therefore, these stars have the chemical signature of the MW thick disc, but
are dynamically hotter, pointing at some additional mechanism (internal or
external) giving these stars a higher “kick”.

Figure 4: Group 1 (G1), identified as the Splash. Energies units are $10^{5}$
km2 s-2. Details are explained in subsection 8.1.

###  8.2 Group 2: Gaia-Enceladus-Sausage-like

This structure matches very well the group dubbed Gaia-Sausage-Enceladus
(Helmi et al., 2018b; Belokurov et al., 2018; Haywood et al., 2018; Mackereth
et al., 2019), showing its characteristic low concentrations of aluminum and
iron: [Al/Fe] $\lesssim 0$ and [Fe/H]$<-0.6$, in its component stars (see
Figure 3). Our Group 2 (G2) is deficient in carbon, with [C/Fe] $\lesssim 0$,
although some outliers are seen with higher values. For the ratios [Si/Fe],
[Mg/Fe], and [Ca/Fe], G2 follows the known sequence vs [Fe/H] for GSE,
including the so-called “knee” at [Fe/H] $\sim-1$. The elements [N/Fe],
[O/Fe], and [Ni/Fe] are more spread.

The G2 stars (see Fig. 5) exhibit rather eccentric orbits, spanning mainly
between $0.6\leq e\leq 1$ with a visible concentration at $e>0.9$, and heights
exceeding 2 kpc, reaching to 20 kpc. The energy plane $E_{char}$ versus
$E_{J}$ exhibits high values (mainly $E_{J}>-2$ and $E_{char}>-2$), spread in
a way consistent with the dynamics of the so-called inner-halo component,
first identified by (Carollo et al., 2007, 2010; Beers et al., 2012). Most of
the stars in G2 are similarly distributed between prograde and retrograde
orbits with a smaller fraction in the P-R classification (See Table 2), which
means G2 stars are evenly distributed in $L_{Z}$, not showing a preferential
rotation direction. We noticed that G2 stars with the lowest $V_{\phi}$ values
- potentially members of Sequoia - are all also retrograde, but they spread
all over the G2 locus, meaning t-SNE did not detect anything closely in common
among them. Therefore, we keep those as G2 members. The periastron distances
for many G2 stars are within 1 kpc of the Galactic center, with apoastron
distances between 8 and 25 kpc. The highest vertical distance from the
Galactic plane for these stars spans between 2 and 25 kpc. One star (APOGEE
ID: 2M13393889+1836032 ) reaches $R_{apo}\sim$ 80 kpc and $|Z_{max}|\sim$ 80
kpc as well. All these features are in agreement with previous results
(Mackereth & Bovy, 2020; Naidu et al., 2020; Feuillet et al., 2021; Buder et
al., 2022).

Figure 5: Group 2 (G2), identified as Gaia-Sausage-Enceladus. Details are
explained in subsection 8.2. Panels and units are as in Figure 4.

###  8.3 Group 3: High-$\alpha$ heated-disc population

From the t-SNE plane in Figure 1, G3 is next and seemingly connected to G1,
yet we diagnosed G3 as a separate group because in many tests (see Figs. 13,
15) it always appeared at one extreme of G1, looking like an appendage
connected by only a few points, and being always more metal-poor than G1 (see
Fig. 2, row 2, column 5). We also noticed that, in the t-SNE double iteration
(see Fig. 14), G3 is detached from G1 while G1 itself, despite exhibiting some
internal clustering, remains mostly together. Additionally, when applying
t-SNE only to the stars we identified in G1 and G3 alone, under various
conditions, G3 is often separated from G1, while the latter keeps its
integrity.

Although our Group 3 (G3) stellar abundances overlap with either G1 and/or G2
in various elements, it is clear from Figure 1 that this structure sits
between them as a separate population in the [Si/Fe] versus [Fe/H] and [Al/Fe]
versus [Fe/H] planes, being the structure richest in silicon of our whole
dataset. As Hawkins et al. (2015) describe in their Figure 1, the abundance
locus of canonical halo MW stars in the [$\alpha$/Fe] versus [Fe/H] plane is
also fulfilled well by G3 stars. Dynamically though, G3 star’s orbital
parameters spread over ranges similar to those of G1 (Splash) (see Figure 6),
although more dispersed. A population of such features has been found by
Nissen & Schuster (2010): their high-$\alpha$ heated population, and also by
Hayes et al. (2018): their high-Mg thick disc-like population.

Nissen & Schuster (2010) suggest that these stars may be ancient disc or bulge
stars “heated” to halo kinematics by merging satellite galaxies, or they could
have formed as the first stars during the collapse of a proto-Galactic gas
cloud, while Hayes et al. (2018) claims that the HMg population is likely
associated with in-situ formation. In Figure 11, G3 sits on the high-$\alpha$
in-situ locus. In the chemical-evolution model for a Milky Way galaxy proposed
by Horta et al. (2021) (their Fig. 2), this location roughly corresponds to an
in-situ early disk population. We conclude that G3 corresponds to this
structure, which we tag as the High-$\alpha$ population.

Figure 6: Group 3 (G3), identified as the High-$\alpha$ heated-disc
population. Details are explained in subsection 8.3. Panels and units are as
in Figure 4.

###  8.4 Group 4: N-C-O peculiar stars

This is a small substructure in the t-SNE space, whose abundances relative to
Fe are shown in small panels in Figure 1. A large fraction of the stars in
this substructure is typified by near-solar and anomalously high levels of
[N/Fe] that are well above ($\gtrsim+0.5$) typical Galactic levels over a
range of metallicities, accompanied by decreased abundances of [C/Fe]
$<+0.15$, as seen in Figures 1 and 2. A portion of its stars also exhibit high
[Al/Fe]. This substructure also matches very well with the atypical nitrogen-
enriched population, as envisioned by a series of works (see, e.g., Schiavon
et al., 2017; Fernández-Trincado et al., 2016, 2017; Fernández-Trincado et
al., 2019a, b, 2020c, 2020b, 2020a, 2021a, 2021b, 2021c; Fernández-Trincado et
al., 2022) that have attributed these stars to exhibit Globular Cluster (GC)
second-generation-like chemical patterns, and are likely GC debris (Fernández-
Trincado et al., 2021b, a) that have now become part of the Galactic field
population. The dynamics of these stars have inner-halo-like orbits, with
values spreading with no particular concentration in any parameter (see Figure
7).

A closer examination of their abundances reveals that G4 contains three
subgroups of stars:

  * •

N-rich and C-poor stars: [N/Fe] $>$ +0.5 and [C/Fe] $<$ +0.15, amounting to 20
stars falling into this category, including 17 already reported by Fernández-
Trincado et al. (2022), plus 3 others (APOGEE IDs: 2M06572697$+$5543115,
2M21372238$+$1244305 and 2M16254515-2620462), located within the abundance
limits defined by Fernández-Trincado et al. (2022) for this kind of stars, and
proposed by these authors to GC second-generation or chemically anomalous
debris. These three stars are genuine newly identified N-rich stars, the last
one (2M16254515-2620462) being a potential extra-tidal star of M 4.

  * •

N-poor and C-poor stars: [N/Fe] $<+0.55$ and [C/Fe] $<-0.1$, which amount to 5
stars that also exhibit sub-solar [Al/Fe] abundance ratios. When examined in
the [Mg/Mn]–[Al/Fe] plane these stars clearly fall in the accreted halo region
(see Figure 11). We found that three of the five stars (APOGEE IDs
2M12211605-2310262, 2M23364836$-$1135404 and 2M21420647$-$3019385) are within
GSE’s t-SNE footprint in all the tests shown in Fig. 15. We postulate that
they are part of a merger remnants (and/or likely associated with GSE, or to a
number of unknown events contributing to this sub-population) present in the
Milky Way.

  * •

N-rich, C-rich, and O-rich stars: [N/Fe] $>$ +0.4, [C/Fe] $>$ +0.15 and [O/Fe]
$>$ +0.2, which amount to 3 stars. These stars also have super-solar values in
Mg and Si. Stars with such enhancements have been reported by, e.g., Beers &
Christlieb (2005). It is likely that the atmosphere of these mildly carbon-
enhanced stars has been contaminated either by an intrinsic process, such as
self-enrichment (likely a TP-AGB), or by a past extrinsic event, that is, the
mass-transfer hypothesis (binary mass transfer systems). However, with the
available radial velocity scatter (VSCATTER $<1$ km s-1) from three APOGEE-2
visits, it is not possible to support or reject either possibility.

In our entire sample of 1742 stars, we have 24 matches with the 412 N-rich
stars listed by Fernández-Trincado et al. (2022), of which the 7 not in G4
were assigned by t-SNE to G2 (GSE). Three of these latter stars have [Al/Fe]
$>$ 0, which is not consistent with the main feature that distinguishes GSE,
but the other four have [Al/Fe] $<$ 0, and we postulate these as second-
generation or chemically anomalous debris from GSE’s own GCs. This is
corroborated by their locus in Figure 11.

Some of the differences between the results of our work and that of Fernández-
Trincado et al. (2022) could also be explained by two factors: (i) these
authors based their analysis on a proprietary catalog internally distributed
to the collaboration, which contained the resulting ASPCAP solutions from the
“$\dots$l33$\dots$” runs, while our present study is based on the published
ASPCAP run “$\dots$synspec_fix$\dots$”, and (ii) t-SNE does an un-supervised
search for stars associated in the used abundances space, while Fernández-
Trincado et al. (2022) does a supervised one.

Figure 7: Group 4 (G4), identified as N-C-O peculiar stars. Details are
explained in subsection 8.4. Panels and units as in Figure 4.

###  8.5 Group 5: New structure

Stars in Group 5 (G5) are located next to the metal-rich border of G2 (GSE) in
many abundances versus [Fe/H] in the small panels of Figure 1. This small
structure appears frequently between G2 and G7 in the many t-SNE tests
performed in this work. The G5 structure sits in a distinctive separate locus
in the [Mg/Fe], [Si/Fe], and [Al/Fe] versus [Fe/H] planes. Compared to G1
(Splash), G5 is lower in oxygen, magnesium, silicon, carbon, and aluminum,
although it has G1-like metallicity. Dynamically, the values of
$e,R_{peri},R_{apo}$, and $|Z_{max}|$ in G5 stars spread over the values
spanned by the full dataset, but its energy and eccentricity value are
slightly higher than G1 (see Figure 2, lowest row 5th panel). The G5 structure
has mostly $-2.2\lesssim E_{J}\lesssim-1.5$ and $-2.1\lesssim
E_{char}\lesssim-1.6$ in the energy plot of Figure 8. We notice one star,
APOGEE ID 2M15113246+4813218, which is an outlier of G5 in [Mg/Fe, [Si/Fe],
[Ca/Fe], and [Al/Fe], despite its location on the t-SNE plane being well
within the footprint of this group.

Chemically, there is an overlap between G5 and the metal-poor side of the
structure Eos described by Myeong et al. (2022). Unlike these authors’ Eos,
our G5 group does not overlap much with either Splash (G1) or GSE (G2). Our
detection also extends the eccentricity range towards slightly lower values,
as we do not cut on this parameter. Like the metal-poor half of Eos, G5 is
lower in nickel than the MW disc, and low values of [Ni/Fe] have been linked
to accretion from present-day dwarf spheroidal galaxies (Nissen & Schuster,
2010), although the latter are more metal poor ([Fe/H] $<-0.8$) than G5. The
investigation of Montalbán et al. (2021) carefully dated ex-situ stars with
$e>0.7$ that fall below the line [Mg/Fe] $=-0.2$ [Fe/H] $+$0.05, and found
that they also exhibit sub-solar values of [Ni/Fe] and are slightly richer
than GSE in [(C$+$N)/O], as seen in Figure 12. These stars are found to be
slightly younger than the high-Mg in-situ halo stars. Most of our G5 stars
fall below the above-mentioned line and occupy the same locus in the plot
[Ni/Fe] versus [(C+N)/O] as the confirmed ex-situ low-Mg stars.

We did a cross-match between the 177 stars labeled as members of Eos by Myeong
et al. (2022) (private communication) with our full sample. Only 49 were in
common between both studies: 11 (22%) in G1, 7 (14%) in G2, 1 (2%) in G3, 0
(0%) in G4, 19 (39%) in G5, 8 (16%) in G6 and 3 (6%) in G7. Given their
chemical similarity, it is not unexpected that a good portion of the common
stars (55%) belongs to G5 and G6, yet a non-negligible 22% portion was
assigned by t-SNE to G1 (Splash). Only 169 listed members of Eos have a
$d_{\rm\texttt{StarHorse}}$, of which 65 are beyond 5 kpc, and therefore
outside the volume sampled by our study.

While Myeong et al. (2022) propose that Eos originated from the gas polluted
by the GSE and evolved to resemble the (outer) thin disc of the MW, we instead
speculate that what they detect as the single structure Eos, was separated by
t-SNE in our work into two portions, the more metal-poor one being our G5 and
of ex-situ origin, and the more metal-rich one which may correspond to our
detected group G6, described next. We postulate G5 as a new structure, which
we name Galileo 5, after the name of the project “Galactic ArchaeoLogIcaL
ExcavatiOns” (GALILEO), thus retaining the G5 acronym.

Figure 8: Group 5, a new structure identified in this work and named Galileo 5
(G5). Details are explained in subsection 8.5. Panels and units are as in
Figure 4.

###  8.6 Group 6: New structure or Eos?

Stars in Group 6 (G6) are more metal-rich than G5 but more metal-poor than G7,
and are slightly lower than G5 in all the $\alpha$ elements, as seen in Figure
1. In several t-SNE tests, G6 stars were very close to or attached to G7.
Similar to the previous group, G6 also occupies a distinctive separate locus
in the [Mg/Fe], [Si/Fe], and [Al/Fe] versus [Fe/H] planes, and is more
$\alpha$-poor than G1 (Splash), overlapping partially with the abundance locus
of thin-disc stars. Dynamically, like G5, G6 also exhibits high
eccentricities, but is slightly less energetic than G5.

Figure 3 shows an element-by-element comparison between groups G1 (Splash), G2
(GSE), G5 (Galileo 5), and G6. For all the $\alpha$-elements, G1, G2, G5, and
G6 go from richest to poorest in their mean values; for the iron-peak
elements, aluminum, and carbon, G2, G5, and G6 follow an ascending trend, with
G1 being either the richest or the second richest. G1 stands out as the
richest structure in aluminum; G2 is easily separated from the other
structures by its extremely low values in carbon, aluminum, and iron; G5 and
G6 separate from each other in iron. We notice one star in G6, APOGEE ID
2M16045805-2418515, which sits a bit far from the locus of G6 in the t-SNE
plane, and also looks like an outlier in several abundances. This star was
associated with G6 when running t-SNE only on the G5 and G6 data.

As commented in the previous subsection, G6 overlaps with the more metal-rich
part of the reported structure Eos. Our data selection is more stringent than
Myeong et al. (2022), and this may have helped t-SNE in separating G6 from G5
in [Fe/H]. In our data, G5 and G6 have distinctive values and distribution of
metallicity, as seen in Figure 3, therefore we trust they are separate
structures. G6 abundances overlap with Aleph, as reported by Horta et al.
(2023), but Aleph clearly follows a disc-like orbit, which is not the case for
G6. When studying various halo substructures in the MW halo, Horta et al.
(2023) explains that halo substructures with low nickel due to a low star-
formation rate can be identified by their having disc-like values of [Mn/Fe],
which is the case for both G5 and G6. On the other hand, unlike G5, G6
overlaps in many abundances with the thick-disc population, although G6 stars’
orbits are clearly more energetic. If we assume G6 stars were formed in the
disc, a mechanism different or additional to the one that formed the thick
disc is needed to keep G6 stars in those more energetic orbits. With G6 it is
harder to ascertain if it has an ex-situ origin or if it matches Eos - which
is proposed to be formed in situ -. We postulate G6 as a likely new ex-situ
structure formed by debris from a past merger event and we name it Galileo 6,
keeping the G6 acronym. Nonetheless, we recognize there is also a chance of it
being the Eos structure.

Figure 9: Group 6, a new structure identified in this work and named Galileo 6
(G6). Details are explained in subsection 8.6. Panels and units as in Figure
4.

###  8.7 Group 7: Inner disk-like stars

Our Group 7 (G7) contains the richest stars of the whole dataset in the iron-
peak elements: [Fe/H] $\gtrsim-0.2$ and [Mn/Fe] $\gtrsim-0.2$, and also poorer
in all the $\alpha$-elements than all the other groups. G7 has the chemical
signature of inner disk-like stars. About half of the G7 stars have low energy
values, and this subset is also concentrated at $V_{\phi}\sim$ 80 km s-1,
eccentricity at $e\sim 0.82,Z_{max}\lesssim 2$ kpc, $R_{apo}<5$ kpc, and
$R_{peri}<0.5$ kpc. Nonetheless, this subset does not cluster in a particular
locus within the G7 footprint in the t-SNE plane, and not all the stars around
$V_{\phi}\sim$ 80 km s-1 belong to this subset. A non-negligible portion of
the $V_{\phi}\sim$ 80 km s-1 stars have higher energies and halo-like orbits.
The energies of G7 stars in the $E_{char}$ versus $E_{J}$ plane fall within
the footprint of MW Bulge GCs shown in Fernández-Trincado et al. (2021d),
their Figure 4 panel d. In the t-SNE runs, we noticed that G7 kept close to G1
in several tests but it did separate sometimes, while G1 - despite some
internal clustering - always retained its integrity. This was a hint of G7
being different from G1, which was independently confirmed by its dynamics.

Figure 10: Group 7 (G7), identified as inner disk-like stars. Details are
explained in subsection 8.7. Panels and units as in Figure 4.

##  9 Accretion origin for some structures

The plot [Mg/Mn] versus [Al/Fe], shown in Figure 11, has been used by several
authors (e.g., Das et al., 2020; Naidu et al., 2020) as an empirical
diagnostic for the origin of structures as either in-situ or accretion.
Moreover, Horta et al. (2021) proves the validity of this plot with their
chemical-evolution model. From the structures detected in this work, only G2
(which we identify as GSE) falls right on the locus of accreted structures in
this plot, confirming its nature. Also, there are five G4 N-poor and C-poor
stars that fall on this locus, which we postulate as GSE’s GC first-generation
debris. We also notice that, among the G2 stars, there are four N-rich and
probable GSE’s GC second-generation debris.

As discussed in Sections 8.5 and 8.6, the plot of [Ni/Fe] versus [(C$+$N)/O]
has been used by Montalbán et al. (2021) to pinpoint the loci of identified
in-situ and ex-situ stars, for which also precise seismic ages were
calculated. The ex-situ stars are slightly younger than the in-situ halo
stars. Low values of [Ni/Fe] with disc-like values of [Mn/Fe] have been linked
to satellite galaxies of the MW with low star-formation rates, and such an
abundance pattern has been observed in both G5 and G6 (see Figure 12). Both
reasons motivate us to postulate G5 and G6 as newly identified accreted
structures in the MW inner halo.

Figure 11: [Mg/Mn] versus [Al/Fe] plot, color-coded for each detected group. A
kernel density estimation (KDE) plot of MW disc stars is shown in grey curves.
These were selected from APOGEE DR17 following the same quality criteria as
our halo stars, and $V_{\phi}\geq 100$ km s-1. Straight lines mark the
boundaries of the accretion and in-situ high/low-$\alpha$ loci used in Das et
al. (2020). The G2 (GSE) locus is mostly limited in this plot by [Al/Fe] $<$ 0
and [Mg/Mn] $>$ +0.2. Figure 12: [Ni/Fe] versus [(C$+$N)/O] plot, color-coded
for each detected group. A kernel density estimation (KDE) plot of MW disc
stars is shown in grey curves. These were selected from APOGEE DR17 following
the same quality criteria as our halo stars, and $V_{\phi}\geq 100$ km s-1.
[Ni/Fe] $<$ 0 and [(C+N)/0] $<-0.2$ has been linked to ex-situ stars. In this
work, G2 (Splash) and G5 (Galileo 5) fall in the locus of accreted stars. G6
(Galileo 6) overlaps with the thick-disc population.

##  10 The Galactic bar pattern speed $\Omega_{bar}$ role in the detected
inner halo groups

In previous works studying peculiar chemical and/or kinematical structures in
the solar vicinity (e.g., Naidu et al., 2020, 2022; Horta et al., 2023), the
dynamical analysis was performed using axisymmetric gravitational potentials.
This work is the first to analyze the role of a non-axisymmetric potential, in
this case, the one induced by the bar on the dynamics of the inner-halo groups
detected – a relevant issue as they spend a good portion of their orbits in
the volume occupied by the MW disc, therefore some effect is expected. Our
analysis considered $\Omega_{bar}=31,41,$ and $51$ km s-1 kpc-1. The results
obtained for each of these values were very similar; we did not detect a
visible or significant difference in the distribution of each of the dynamical
parameters examined, $|Z_{max}|,R_{peri},R_{apo},e,E_{j}$, and $E_{char}$.
Therefore, the bar pattern velocity does not have an effect on the dynamical
behavior of the structures detected, and we conclude that its role seems to be
absent or undetectable at the error level of the data.

##  11 Conclusions

From this investigation, we conclude that:

  * •

We have used high-quality APOGEE abundances and Gaia astrometric data for 1742
red giant stars located within 5 kpc of the Sun, in order to detect coherent
structures associated in the chemical-kinematic-dynamical space. We limited
our study to stars not rotating with the disc, i.e, $V_{\phi}<100$ km s-1.

  * •

We determined orbital parameters using the non-axisymmetric galactic potential
model GravPot16, which, together with the stellar components for the disc and
halo of the MW, also includes a rotating bar it boxy/peanut. The bar pattern
velocity adopted was 41 km s-1 kpc. They were also determined for velocities
31 and 51 km s-1 kpc in order to measure the impact on the orbital parameters,
since these stars may spend a significant portion of their orbits close to the
Galactic plane. We did not detect significant variations in this regard.

  * •

The search for structures was performed with the nonlinear algorithm of
dimensional reduction known as t-SNE, using ten chemical-abundance ratios as
input data: [Fe/H], [O/Fe], [Mg/Fe], [Si/Fe], [Ca/Fe], [C/Fe], [N/Fe],
[Al/Fe], [Mn/Fe], and [Ni/Fe].

  * •

Seven structures were detected, of which the following were traced to
previously identified populations in the literature: G1 $\sim$ Splash, G2
$\sim$ GSE, G3 $\sim$ high-$\alpha$ heated-disc population, G4 $\sim$ N-C-O
peculiar stars, and G7 $\sim$ inner disk-like stars. In the many tests
performed, t-SNE occasionally separated the metal-poor high-$\alpha$ plateau
portion of GSE from the metal-rich lower-$\alpha$ declining section at the so-
called knee.

  * •

No known structures were detected to be unambiguously similar to groups
Galileo 5 (G5) and Galileo 6 (G6), therefore we posit them as new structures
found in this work. These groups have low values of [Ni/Fe] but [Mn/Fe] with
thin disc-like chemical abundances, which hints at an ex-situ origin, i.e, an
accreted satellite galaxy. Both G5 and G6 have Splash-like dynamics. G6 may
also correspond to Eos, which was proposed by Myeong et al. (2022) to be
formed in-situ from gas contaminated by GSE.

  * •

All groups except G4 (N-C-O peculiar stars) as detected by t-SNE in this work
occupy distinctive loci in the [Mg/Fe], [Si/Fe], and [Al/Fe] versus [Fe/H]
planes, with little overlap between the groups. We recommend using these
abundances in future searches for these structures.

  * •

We report three new N-rich stars and probable GC debris: APOGEE IDs
2M06572697$+$5543115, 2M21372238$+$1244305, and 2M16254515-2620462.

  * •

The t-SNE algorithm was thoroughly tested on several issues. 1) A wide range
of parameters were tested for the parameters perplexity, early exaggeration,
and number of iterations to find those that produced the greatest visible
separation in the t-SNE plane. 2) In those cases where the groups separation
was not completely clean in the t-SNE plane, several procedures were applied:
(i) t-SNE was applied iteratively, i.e., the outputs t-SNE X and t-SNE Y were
introduced as two additional inputs for a second run of the method; (ii) t-SNE
was applied to sub-samples in which two or more groups were suspected to
coexist, to check if t-SNE separated them or not; and (iii) t-SNE was applied
to the larger individual groups to determine whether or not they remained as a
single entity. 3) Through Monte Carlo realizations considering the
uncertainties in the abundances used, it was determined that the errors in
these do not appreciably affect the groups detected by t-SNE.

  * •

The t-SNE algorithm proved to be very useful for exploring high-dimensional
data sets, objectively separating real structures at various scales present in
a large number of data, e.g., small hidden groups with a very dominant
feature, or large ensembles with more dispersed properties that overlap in
some variables with other structures and whose boundaries are less evident. In
any case, it is recommended to cross-check t-SNE results with independent data
(e.g., dynamics) or previous findings, and also gauge the stability of the
structures detected, as described in the previous item.

  * •

A table with all the relevant data for the full dataset, including a tag for
each of the seven groups detected, is available online. A sample is shown in
Table 3 in Appendix C.

###### Acknowledgements.

This work has received funding from the grant support provided by Agencia
Nacional de Investigación y Desarrollo de Chile (ANID) under the Proyecto
Fondecyt Iniciación 2022 Agreement No. 11220340 (PI: José G. Fernández-
Trincado) and from ANID under the Concurso de Fomento a la Vinculación
Internacional para Instituciones de Investigación Regionales (Modalidad corta
duración) Agreement No. FOVI210020 (PI: José G. Fernández-Trincado) and from
the Joint Committee ESO-Government of Chile 2021 under the Agreement No. ORP
023/2021 (PI: José G. Fernández-Trincado) and from Becas Santander Movilidad
Internacional Profesores 2022, Banco Santander Chile (PI: José G. Fernández-
Trincado).
T.C.B. acknowledges partial support for this work from grant PHY 14-30152;
Physics Frontier Center/JINA Center for the Evolution of the Elements (JINA-
CEE), and from OISE-1927130: The International Research Network for Nuclear
Astrophysics (IReNA), awarded by the U.S. National Science Foundation.
B.T. gratefully acknowledges support from the Natural Science Foundation of
Guangdong Province under grant No. 2022A1515010732, the National Natural
Science Foundation of China through grants No. 12233013, and the China Manned
Space Project No. CMS-CSST-2021-B03.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
website is www.sdss.org. SDSS-IV is managed by the Astrophysical Research
Consortium for the Participating Institutions of the SDSS Collaboration
including the Brazilian Participation Group, the Carnegie Institution for
Science, Carnegie Mellon University, the Chilean Participation Group, the
French Participation Group, Harvard-Smithsonian Center for Astrophysics,
Instituto de Astrofìsica de Canarias, The Johns Hopkins University, Kavli
Institute for the Physics and Mathematics of the Universe (IPMU) / University
of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für
Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatory of China, New Mexico State University, New York University,
University of Notre Dame, Observatório Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
http://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.

##  Appendix A t-SNE tests

The following figures show the various tests performed with t-SNE to assess
its performance, and to make the final selection of adjacent groups that were
not clearly separated in the t-SNE plane chosen, as described in Section 7.
Figure 13 shows the effect of varying the input parameters:
$25<$perplexity$<40$, and $80<$early_exageration$<130$. Figure 14 shows the
iterated t-SNE plane, and Figure 15 shows the effect of abundance errors.
These tests helped to identify truly stable groups that t-SNE kept finding
regardless of the changes introduced.

Figure 13: The t-SNE plane for different input parameter values, as indicated
in each panel. Horizontal and vertical axes correspond, respectively, to the
t-SNE X and Y dimensions.

The panel marked in red is the one chosen as final – i.e. Figure 1 – and best
to select the structures found in this investigation.

Figure 14: The iterated t-SNE plane. Values of t-SNE X and t-SNE Y from Figure
1-central panel were fed as additional data to another run of t-SNE, to check
if adjacent groups in the original t-SNE plane were separated or not, and
where the borders occur. Notice how the overall shape is kept, but in fact in
this case t-SNE flipped the orientation of the points in the plane.

Figure 15: Effect of abundance errors on the t-SNE plane. Horizontal and
vertical axes correspond, respectively, to the t-SNE X and Y dimensions.

Monte-Carlo realizations of the abundances used were fed into the t-SNE
algorithm to check if the groups as selected in Figure 1 were still associated
and kept as coherent structures.

##  Appendix B Summary of chemical abundances and dynamical parameters of each
detected group

Table 1 contains the median value and median absolute deviation
Median$(X-\mbox{Median}(X))$ of each the abundances and dynamical parameters
studied in this work. Table 2 shows the percentage of prograde, P-R, and
retrograde orbits per group. See Section 8 and subsections within for details.

Item | G1 | G2 | G3 | G4 | G5 | G6 | G7
---|---|---|---|---|---|---|---
[C/Fe] | +0.087 $\pm$0.035 | -0.366 $\pm$0.077 | -0.124 $\pm$0.095 | -0.239 $\pm$0.115 | -0.097 $\pm$0.037 | -0.019 $\pm$0.037 | +0.034 $\pm$0.034
$[$N/Fe$]$ | +0.120 $\pm$0.044 | +0.155 $\pm$0.099 | +0.195 $\pm$0.082 | +0.772 $\pm$0.153 | +0.134 $\pm$0.066 | +0.114 $\pm$0.027 | +0.235 $\pm$0.063
$[$O/Fe$]$ | +0.349 $\pm$0.040 | +0.305 $\pm$0.067 | +0.429 $\pm$0.071 | +0.212 $\pm$0.122 | +0.241 $\pm$0.046 | +0.159 $\pm$0.028 | +0.068 $\pm$0.043
$[$Mg/Fe$]$ | +0.339 $\pm$0.026 | +0.195 $\pm$0.053 | +0.327 $\pm$0.028 | +0.176 $\pm$0.065 | +0.162 $\pm$0.040 | +0.129 $\pm$0.018 | +0.052 $\pm$0.049
$[$Al/Fe$]$ | +0.265 $\pm$0.039 | -0.187 $\pm$0.062 | +0.170 $\pm$0.088 | +0.106 $\pm$0.215 | +0.031 $\pm$0.041 | +0.052 $\pm$0.044 | +0.007 $\pm$0.056
$[$Si/Fe$]$ | +0.240 $\pm$0.027 | +0.201 $\pm$0.041 | +0.334 $\pm$0.045 | +0.178 $\pm$0.056 | +0.109 $\pm$0.020 | +0.069 $\pm$0.014 | +0.031 $\pm$0.031
$[$Ca/Fe$]$ | +0.172 $\pm$0.034 | +0.186 $\pm$0.049 | +0.258 $\pm$0.029 | +0.229 $\pm$0.058 | +0.098 $\pm$0.033 | +0.041 $\pm$0.031 | -0.025 $\pm$0.035
$[$Mn/Fe$]$ | -0.136 $\pm$0.034 | -0.297 $\pm$0.055 | -0.279 $\pm$0.054 | -0.321 $\pm$0.081 | -0.166 $\pm$0.025 | -0.093 $\pm$0.015 | +0.026 $\pm$0.079
$[$Fe/H$]$ | -0.570 $\pm$0.103 | -1.230 $\pm$0.197 | -1.105 $\pm$0.105 | -1.027 $\pm$0.136 | -0.692 $\pm$0.042 | -0.455 $\pm$0.031 | +0.058 $\pm$0.170
$[$Ni/Fe$]$ | +0.079 $\pm$0.017 | -0.040 $\pm$0.033 | +0.028 $\pm$0.030 | -0.007 $\pm$0.039 | -0.016 $\pm$0.020 | -0.009 $\pm$0.019 | +0.039 $\pm$0.034
$[$Mg/Mn$]$ | +0.479 $\pm$0.050 | +0.497 $\pm$0.087 | +0.619 $\pm$0.055 | +0.541 $\pm$0.085 | +0.311 $\pm$0.048 | +0.218 $\pm$0.022 | -0.007 $\pm$0.157
$R_{peri}$ | 0.808 $\pm$0.529 | 0.457 $\pm$0.360 | 0.688 $\pm$0.542 | 0.471 $\pm$0.406 | 0.414 $\pm$0.315 | 0.372 $\pm$0.211 | 0.391 $\pm$0.228
$R_{apo}$ | 8.673 $\pm$1.268 | 11.759 $\pm$2.724 | 8.890 $\pm$1.153 | 9.889 $\pm$2.340 | 10.212 $\pm$1.354 | 9.116 $\pm$0.974 | 5.955 $\pm$2.256
$e$ | 0.811 $\pm$0.093 | 0.933 $\pm$0.053 | 0.830 $\pm$0.089 | 0.920 $\pm$0.070 | 0.923 $\pm$0.061 | 0.922 $\pm$0.052 | 0.835 $\pm$0.059
$|Z_{max}|$ | 3.303 $\pm$0.914 | 5.065 $\pm$2.075 | 3.857 $\pm$1.202 | 3.681 $\pm$1.046 | 4.183 $\pm$1.183 | 3.534 $\pm$0.998 | 1.894 $\pm$1.431
$E_{j}$ | -2.115 $\pm$0.100 | -1.789 $\pm$0.166 | -2.003 $\pm$0.154 | -1.934 $\pm$0.207 | -1.867 $\pm$0.116 | -1.877 $\pm$0.077 | -2.284 $\pm$0.218
$E_{char}$ | -1.927 $\pm$0.097 | -1.747 $\pm$0.125 | -1.887 $\pm$0.092 | -1.838 $\pm$0.132 | -1.813 $\pm$0.077 | -1.898 $\pm$0.071 | -2.117 $\pm$0.211

Table 1: Median and median absolute deviation of each of the chemical abundances and dynamical parameters investigated in this work, and plotted in Figs. 1 to 12, for each detected group. $R_{peri}$, $R_{apo}$, and $|Z_{max}|$ are measured in kpc; $E_{j}$ and $E_{char}$ in $10^{5}$ km2 s-2. The first column (Item) indicates the variable evaluated. G | Name | # of P | # of P-R | # of R | Total
---|---|---|---|---|---
1 | Splash | 851 (76%) | 175 (16%) | 99 (8%) | 1125
2 | GSE | 136 (49%) | 55 (20%) | 89 (31%) | 280
3 | High-$\alpha$ | 61 (54%) | 19 (17%) | 34 (29%) | 114
4 | N-rich | 13 (46%) | 9 (32%) | 6 (22%) | 28
5 | Galileo 5 | 39 (49%) | 17 (21%) | 24 (30%) | 80
6 | Galileo 6 | 14 (44%) | 6 (19%) | 12 (37%) | 32
7 | Inner Disk | 64 (77%) | 10 (12%) | 9 (11%) | 83
Table 2: Number and percentage of stars in each group detected according to
their orbit orientation. G$=$group number assigned in this work, P$=$prograde,
P-R$=$prograde-retrograde, R$=$retrograde.

##  Appendix C Supplementary online data

A catalog with all the relevant information, including a tag for each detected
group, is available online. A sample table showing the column labels is shown
in Table 3.

Table 3: Relevant information for each star used in this investigation,
including a tag for each detected group.

ID # | Column name | Units | Column Description
---|---|---|---
1 | APOGEE_ID |  | APOGEE id
2 | RA | deg | $\alpha(J2000)$
3 | DEC | deg | $\delta(J2000)$
4 | SNR |  pixel-1 | Spectral signal-to-noise
5 | RV |  km s-1 | APOGEE-2 radial velocity
6 | VSCATTER |  km s-1 | APOGEE-2 radial velocity scatter
7 | LOGG | [cgs] | Surface gravity
8 | ERROR_LOGG | [cgs] | Uncertainty in LOGG
9 | C_FE |  | [C/Fe] from ASPCAP
10 | ERROR_C_FE |  | Uncertainty in [C/Fe]
11 | N_FE |  | [N/Fe] from ASPCAP
12 | ERROR_N_FE |  | Uncertainty in [N/Fe]
13 | O_FE |  | [O/Fe] from ASPCAP
14 | ERROR_O_FE |  | Uncertainty in [O/Fe]
15 | MG_FE |  | [Mg/Fe] from ASPCAP
16 | ERROR_MG_FE |  | Uncertainty in [Mg/Fe]
17 | AL_FE |  | [Al/Fe] from ASPCAP
18 | ERROR_AL_FE |  | Uncertainty in [Al/Fe]
19 | SI_FE |  | [Si/Fe] from ASPCAP
20 | ERROR_SI_FE |  | Uncertainty in [Si/Fe]
21 | CA_FE |  | [Ca/Fe] from ASPCAP
22 | ERROR_CA_FE |  | Uncertainty in [Ca/Fe]
23 | MN_FE |  | [Mn/Fe] from ASPCAP
24 | ERROR_MN_FE |  | Uncertainty in [Mn/Fe]
25 | FE_H |  | [Fe/H] from ASPCAP
26 | ERROR_FE_H |  | Uncertainty in [Fe/H]
27 | NI_FE |  | [Ni/Fe] from ASPCAP
28 | ERROR_NI_FE |  | Uncertainty in [Ni/Fe]
29 | VR |  km s-1 | Galactocentric radial velocity
30 | VPHI |  km s-1 | Galactocentric azimuthal velocity
31 | PERIGALACTICON | kpc | Perigalactocentric distance
32 | ERROR_PERIGALACTICON | kpc | Uncertainty in PERIGALACTICON
33 | APOGALACTICON | kpc | Apogalactocentric distance
34 | ERROR_APOGALACTICON | kpc | Uncertainty in APOGALACTICON
35 | Eccentricity |  | Orbital eccentricity
36 | ERROR_Eccentricity |  | Uncertainty in eccentricity
37 | $|Zmax|$ | kpc | Maximum vertical excursion from the Galactic plane
38 | ERROR_Zmax | kpc | Uncertainty in Zmax
39 | $E_{j}$ |  $10^{5}$ km2 s-2 | Jacobi energy
40 |  ERROR_$E_{j}$ |  $10^{5}$ km2 s-2 | Uncertainty in Jacobi energy
41 | $E_{char}$ |  $10^{5}$ km2 s-2 | Characteristic orbital energy
42 |  ERROR_$E_{char}$ |  $10^{5}$ km2 s-2 | Uncertainty in characteristic orbital energy
43 | $L_{zmin}$ |  $10^{1}$ km2 s-2 kpc |  Minimum $z$-component of angular momentum
44 |  ERROR_$L_{zmin}$ |  $10^{1}$ km2 s-2 kpc |  Uncertainty in minimum $z$-component of angular momentum
45 | $L_{zmax}$ |  $10^{1}$ km2 s-2 kpc |  Maximum $z$-component of angular momentum
46 |  ERROR_$L_{zmax}$ |  $10^{1}$ km2 s-2 kpc |  Uncertainty in maximum $z$-component of angular momentum
47 | GAIADR3_SOURCE_ID |  | GAIA DR3 source id
48 | ruwe |  | Gaia DR3 renormalized unit weight error
49 | d_STARHORSE | kpc | Bayesian StarHorse distance, 50th percentile
50 | ERROR_d_STARHORSE | kpc | Uncertainty in d_STARHORSE, ERROR = (84th-16th )/2 percentile
51 | pmRA_GaiaDR3 |  mas yr-1 |  $\mu_{\alpha}$ cos ($\delta$) from Gaia DR3
52 | ERROR_pmRA_GaiaDR3 |  mas yr-1 |  Uncertainty in $\mu\alpha$ cos ($\delta$) from Gaia EDR3
53 | pmDEC_GaiaDR3 |  mas yr-1 |  $\mu_{\alpha}$ from Gaia EDR3
54 | ERROR_pmDEC_GaiaDR3 |  mas yr-1 |  Uncertainty in $\mu\alpha$ from Gaia EDR3
55 | Group |  | Group as determined by t-SNE

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