# Disentanglement of the chemodynamical assembly: mapping the Milky Way disks

Elvis Cantelli,1 Ramachrisna Teixeira,1
1Instituto de Astronomia, Geofísica e Ciências atmosféricas - Universidade de
São Paulo, Rua do matão, 1226, 05508-090, São Paulo, Brazil
E-mail<EMAIL_ADDRESS>
(Accepted 2024 April 12. Received 2024 April 06; in original form 2023
September 30)

###### Abstract

The formation and structure of the Milky Way has a fundamental role in our
understanding of the universe and its evolution, and thanks to the Gaia
mission and large spectroscopic surveys, we live an exceptional moment of data
availability, allowing us to trace the building blocks of the Galactic disk
and their relations. In this sense, we propose here the exploration of a large
dataset in a top-down fashion, elaborating a similarity map of the local
Galactic volume in order to segregate and characterise its main components,
searching for hints about their relations. We have used the t-SNE algorithm
with chemical, orbital and kinematic properties of the stars to produce 2D
manifolds and dissect their structure by isolating populations to further
analyse their behaviour. The young thin disk could be clearly separated from
the older thick disk, also showing a puzzling transition zone with hints about
the aftermath of the Gaia-Sausage-Enceladus merger. Moving groups and resonant
features also appear prominently in the maps, splitting the disk into inner
and outer portions as consequence of the resonances produced by the Galactic
bar. The dynamical halo appears as an extreme end related to the heated
portion of the thick disk, showing sub-structures corresponding to known
accreted populations. Open and globular clusters also appear in their
chemical/evolutionary context. We present details of the developed strategy,
an overview of the different populations and their relations, as well as a
discussion and insights of our results in the scenario of Galactic evolution.

###### keywords:

Galaxy: structure – Galaxy: abundances – Galaxy: kinematics and dynamics

††pubyear: 2024††pagerange: Disentanglement of the chemodynamical assembly:
mapping the Milky Way disks–B

##  1 Introduction

A richer and more detailed understanding of the history of our Galaxy has,
since the 18th century, been one of the main goals of astronomy, with early
research conducted by William Herschel’s maps of star densities, spiral
structures detected by William Parsons in other galaxies, the flattened disk
shape proposed by Jacobus Kapteyn, followed by the great debate between
Shapley and Curtis that finally culminated in Hubble (1925) confirming the
"island universes" scenario in the 1920’s. Our view about its nature and
structure started to take shape with Lindblad (1927) and Oort (1927) deducing
the rotation of the Galaxy, Baade (1944) separating the stellar populations I
and II, Baade (1951) inferring the existence of the spiral arms, and Vyssotsky
(1951) and Schwarzschild (1952) discovering the Galactic halo. More recently,
a bar (deVaucouleurs, 1964) and an older thicker disk (Gilmore & Reid, 1983)
came to our knowledge, as well as several stream-like structures (eg. Ibata et
al. 1994, Helmi et al. 1999) that trace the hierarchical building of the
$\Lambda$-CDM (Springel et al., 2006) cosmological scenario, and evidences
(eg. Nissen & Schuster 2010, Deason et al. 2013) of a major merger event that
built this configuration, as proposed earlier in Chiappini et al. (1997). As
we expand and refine the databases that supports our understanding of the
Milky Way, new details emerge that lead us on to continue scrutinising our
Galaxy.

With the advent of the Gaia mission (Gaia Collaboration et al., 2016) and
large spectroscopic surveys, the data availability became significantly
larger, encompassing a more isotropic coverage and thus, more homogeneous
sampling of the local volume of the Galaxy, allowing the discovery of many new
components of the Galactic halo, such as the Gaia-Sausage-Enceladus (hereafter
GSE, Belokurov et al. 2018, Haywood et al. 2018a, Helmi et al. 2018), Sequoia
(Myeong et al., 2019) and several others (see Naidu et al. 2020, Forbes 2020,
Myeong et al. 2022).

These halo substructures, especially the GSE, are associated with earlier
dwarf galaxy merger events, which have shown to be the direct cause of the
heating of an early Galactic disk and formation of the "hot" thick disk,
previously assigned as an "inner halo" (Di Matteo et al., 2019) or a "metal-
weak thick disk" (eg. Beers et al. 2002 and Hayes et al. 2018). Still, a
proper characterisation of disk populations is the target of a myriad of
studies (eg. Gilmore et al. 1989, de Souza & Teixeira 2007, Hawkins et al.
2015), employing distinct techniques to segregate the populations based on
their kinematics, chemistry and ages and disentangle these components, tracing
their contribution as Galactic building blocks, as well as dynamically-induced
populations by bar resonances (see Gaia Collaboration et al. (2023b)) such as
the Hercules stream (Pérez-Villegas et al., 2017). The disk heating in plane-
perpendicular velocity beyond $\sim$10 kpc was associated by Das et al. (2024)
with a first encounter with the Sagittarius dSph satellite galaxy. Feuillet et
al. (2022) were able to identify old, accreted populations within the
dynamically cool stellar disk, unveiling a whole new range of disk components
yet to be discovered.

The use of unsupervised learning algorithms is rapidly growing among
astronomers as an efficient method to classify and characterise observed and
simulated data in a vast and diverse range of applications. The
dimensionality-reduction algorithm t-distributed stochastic neighbour
embedding (t-SNE, van der Maaten & Hinton 2008) is one of the most useful
algorithms to visualise and characterise high-dimensional data, already
employed in several areas, including star-forming galaxies (Steinhardt et al.,
2020), stellar spectroscopy (Traven et al., 2017), galaxy morphology (Dai &
Tong, 2018), and even instrumentation (House et al., 2023).

Anders et al. (2018) employed the t-SNE for a sample of 530 HARPS-GTO stars
with reliable chemical abundances to separate different Galactic components in
a multi-dimensional chemistry space, and was able to separate the disc sub-
populations in the solar vicinity with more reliable visualisations than
common 2D chemical diagrams. da Silva & Smiljanic (2023) has recently used
t-SNE to analyse 6618 metal-poor stars with a space composed of orbital
actions and chemical abundances, enabling the characterisation of Galactic
halo groups and their connections to the major accretion events (Sequoia and
GSE). Queiroz et al. (2023) applied the t-SNE to samples of sub-giants of
three large spectroscopic surveys (4638 for APOGEE, 9420 for GALAH, and 12834
for LAMOST) by using a chemical and age parameter space, separating the thin
disk from the thick disk by clustering the manifold via HDBSCAN. In this work
we propose to use both chemical abundances and orbital parameters to feed
t-SNE with stars in the local volume focused on the disk, without relying in
random initialisation, and for a much larger sample of stars.

Our aim is to characterise the different Galactic components with sub-samples
of two high-resolution spectroscopic surveys, namely APOGEE (Apache-POint
Galactic Evolution Experiment, Majewski et al. 2017) and GALAH (GALactic
Archaeology with HERMES, De Silva et al. 2015) using kinematic parameters
calculated from the Gaia catalogue (orbital energies, angular momenta, radii,
eccentricities and velocities, since these properties can summarise the
orbital profiles) and different sets of chemical abundance ratios
(metallicity, alpha-element, odd-Z, iron-peak and s-process, summarising the
most important chemical-tagging features). We assume that this chemodynamic
parameter space is capable of describing the formation and evolution of the
main Milky Way populations and enable their segregation within a large area
around the sun. For this, we apply the t-SNE algorithm to perform the
dimensionality reduction to a 2-D space and produce a structural map, where
each chemo-kinematic population accommodates in separated locuses from each
other, decomposing and bringing into light the several sub-structures that lie
hidden in the Galactic disk to dissect each one and assemble a comprehensive
scenario of the local disk configuration. In this sense, it is expected to
obtain a clear separation between thin and thick disk based on their different
chemical signature and velocity patterns, signatures of resonant features from
the Galactic bar and their associated kinematic populations, clues on older
populations lying in the outermost portion of the disk, as well as the
isolation of dynamical halo stars and their possible connection to the thick
disk. In our strategy, instead of trying to isolate manifold regions only by
their concentration or using clustering algorithms directly, we colour the
maps with the parameter values by using different palettes that can visually
bring out the map behaviour and guide the segregation process more
conceptually.

The paper is outlined as follows. In Section 2 we describe the dataset and
sample selection, and in Section 2.4 we give details on the orbital
integration and show validations of the Galactic potential used. The t-SNE
strategy and parameters for manifold convergence are shown in Section 3, the
results with the component segregation are shown in Section 4, and the
analysis of the different Galactic components and their properties is shown in
Section 5. Finally, in Section 6 we summarise the results and bring new
insights about the use of the technique proposed in this work

##  2 Data

###  2.1 Survey data

We have obtained the data from Gaia DR3 (Gaia Collaboration et al., 2023a)
catalogue by applying the selection criteria described in Gaia Collaboration
et al. (2018a), requiring a determined value for radial velocity and a
magnitude in G band up to 15, where DR3 radial velocity sample is essentially
complete. This primary Gaia DR3 sample had 20 million stars with full 6D phase
space.

The distances were calculated using the formalism of Astraatmadja & Bailer-
Jones (2016) through the code made available by F. Anders at
https://github.com/fjaellet/abj2016, with 0.3 pc for distance PDF resolution.

The spectroscopic survey data were obtained from APOGEE DR17 (Abdurro’uf et
al. 2022, Wilson et al. 2019a, Gunn et al. 2006a) and GALAH DR3 (Buder et al.,
2021) by downloading the full catalogues available in the websites of the
surveys.

APOGEE is a high-resolution ($R\approx 22500$) spectroscopic survey in the
near-infrared operated primarily in the northern hemisphere with the Sloan
telescope (Gunn et al., 2006b) and later in the southern hemisphere with the
duPont telescope (Bowen & Vaughan, 1973), with a dedicated spectrograph
(Wilson et al., 2019b). It was designed to observe red clump stars, red giant
branch stars, RR-Lyrae, among many other stellar objects. The fields are sub-
divided into two types: grid pointings which sample a semi-regular grid in
Galactic longitude and latitude over a component of the Milky Way (such as the
disk and bulge), and non-grid pointings that sample particular objects of
interest (dwarf galaxies, tidal streams, clusters). The stellar parameters and
elemental abundances are obtained via the ASPCAP automated pipeline (García
Pérez et al., 2016), which compares the observed spectra to libraries of
synthetic spectra calculated with model atmospheres (described in detail in
Jönsson et al. 2020) and spectral line lists (Smith et al., 2021).

GALAH is a high-resolution ($R\approx 28000$) spectroscopic survey in the
optical region operated in the southern hemisphere with the Anglo-Australian
Telescope’s HERMES spectrograph (Sheinis et al., 2015). The target selection
had two phases: the first one being a magnitude-limited survey covering stars
with magnitude between $12<V<15$, declination $\delta<10^{\circ}$ and angular
Galactic plane distance $|b|>10^{\circ}$; and the second one more focused on
age estimation, sampling the main-sequence turnoff, and the Galactic plane
previously not covered. In its third data release, the stellar parameters and
abundances were obtained with Spectroscopy Made Easy (Piskunov & Valenti,
2016), where the main stellar parameters are estimated by fitting synthetic
spectra to 46 segments containing lines with reliable data for this analysis,
improving the result further in a second iteration. The elemental abundances
are obtained separately for each element, using the same segment-driven
strategy for the parameters. A full description is given in Buder et al.
(2021).

The cross-match of the surveys with the primary Gaia sample by their source_id
resulted in a reduction of 733901 to 342263 stars for APOGEE and 588571 to
491790 for GALAH. The sharp reduction was due to the selection function of
Gaia, that required a measured radial velocity and $G_{mag}$ < 15 (giving a
little extra from the completeness drop of the radial velocity around G =
14.5, Katz et al. 2023). After this, we applied a selection filter to the
[Fe/H] and [Mg/Fe] abundances, requiring the errors to be < 0.2 dex with no
pipeline flags, resulting in a reduction to 307871 stars for APOGEE and 391710
for GALAH (see Table 1). Although the radial velocities from the spectroscopic
surveys have smaller errors due to the higher resolution, we chose to use the
Gaia RVS to keep the homogeneity of the sample phase space.

Table 1: Median, 5th and 95th percentiles of the error distributions of parallaxes, proper motions (pm), radial velocities (RV), effectivce temperatures (Teff), gravities, abundances and signal-to-noise ratios in the main-sequence (MS) and red giant branch (RGB) subsamples for APOGEE and GALAH. | APOGEE | GALAH
---|---|---
Parameter | p05 | p50 | p95 | p05 | p50 | p95
| Main sequence
parallax [mas] | 0.01 | 0.018 | 0.039 | 0.01 | 0.015 | 0.03
pmα [mas/yr] | 0.01 | 0.018 | 0.042 | 0.01 | 0.016 | 0.032
pmδ [mas/yr] | 0.011 | 0.016 | 0.037 | 0.01 | 0.014 | 0.029
RV [km/s] | 0.3 | 1.53 | 5.74 | 0.5 | 1.94 | 5.15
Teff [K] | 9 | 20 | 54 | 75 | 94 | 123
log G [dex] | 0.017 | 0.025 | 0.045 | 0.178 | 0.183 | 0.195
[Fe/H] | 0.005 | 0.009 | 0.016 | 0.052 | 0.084 | 0.132
[Mg/Fe] | 0.011 | 0.016 | 0.025 | 0.037 | 0.094 | 0.181
SNR | 42 | 126 | 397 | 22 | 42 | 101
| Red giant branch
parallax [mas] | 0.01 | 0.015 | 0.025 | 0.01 | 0.015 | 0.024
pmα [mas/yr] | 0.009 | 0.015 | 0.028 | 0.009 | 0.015 | 0.026
pmδ [mas/yr] | 0.01 | 0.014 | 0.023 | 0.01 | 0.013 | 0.022
RV [km/s] | 0.16 | 0.61 | 2.93 | 0.18 | 0.72 | 2.81
Teff [K] | 6 | 9 | 18 | 74 | 93 | 130
log G [dex] | 0.02 | 0.025 | 0.038 | 0.185 | 0.206 | 0.281
[Fe/H] | 0.007 | 0.008 | 0.013 | 0.046 | 0.071 | 0.114
[Mg/Fe] | 0.009 | 0.012 | 0.021 | 0.034 | 0.075 | 0.172
SNR | 77 | 219 | 703 | 26 | 55 | 135

We separated the red giant branch (RGB) based on the criteria from Gaia
Collaboration et al. (2018b) for stars with absolute magnitude in G band
$M_{G}<3.9$ and intrinsic colour $(G_{BP}-G_{RP})_{0}>0.95$, but we chose to
perform a visual cut in the colour-magnitude (CM) diagram to compensate for
the reddening of sub-giants above $G_{M}>4$, and also because there were
spurious main-sequence (MS) stars in the upper portion of the Kiel diagrams of
the samples, preventing a direct log G selection cut. Furthermore, reddening
data were not available for the full sample and, even if applied for the
available stars, produced a glitchy CM diagram. Figure 1 displays the
selection cuts for MS and RGB stars from the APOGEE (left) and GALAH (right)
datasets. The dashed polygons represent the cuts, with colour-coded points to
distinguish between the successive subsets produced by filters and cuts. The
coordinates for the polygons are shown in the Appendix A. In the upper panels
we show the CM diagram of Gaia cross-correlation (gray) with the Fe and Mg
filter (green), and the subsequent selection cuts for RGB (red sub-samples)
and MS (blue sub-samples). This CM selection was projected into the Kiel
diagram to perform further cuts avoiding the turn-off and contamination from
the MS as shown in the middle panels for the RGB (brown sub-samples), and
lower panels for the MS (light blue sub-samples), resulting in 155773 and
132468 stars on the RGB sample, and 99037 and 161668 stars on the MS sample
for APOGEE and GALAH respectively. All selections were made respecting the
same boundary polygon on both surveys, except for GALAH MS kiel cut with two
vertexes modified to admit more stars. The last step was to impose that the
RGB samples had determined abundances for the elements used for t-SNE inputs,
as discussed in Section 2.2, where we achieved a final RGB catalogue (yellow
sub-samples) of 151534 stars for APOGEE and 124686 for GALAH, shown in the
middle panels.

Figure 1: Selection cuts of the main-sequence and giants sub-samples from the
APOGEE (left) and GALAH (right) samples. Upper panels show the Gaia cross-
correlation in gray, filtered stars by Fe and Mg quality in green, and the
selected RGB (red) and MS stars (blue) as subsets of quality-filtered stars.
Middle panels show the Kiel diagram with the RGB selected by CM (red), the
selection by the Kiel diagram cut (brown), and the stars that have determined
abundances for the elements discussed in Section 2.2 in yellow. Same for the
lower panels, but for the MS selected by CM (blue) and Kiel diagram cut (light
blue). All polygons used for selection cuts are displayed by dashed lines.

The main analysis and t-SNE maps were made with the RGB sample, since these
stars have more reliable radial velocities, proper motions and chemical
abundances, where the typical errors are smaller than the MS with larger
signal-to-noise ratios, as shown in Table 1. We display the median errors for
the selected samples of each survey together with the 16th and 84th
percentiles, since the error distributions are mostly asymmetric. Also, APOGEE
has roughly $\sim$300 stars in the MS with all the required elements in the
filtered sample, enabling only the use of the RGB for both surveys to keep
sampling consistency. The GALAH main-sequence sample was used in the Galactic
potential validation for the orbital parameters described in Section 2.4,
since this sample encompasses around $\sim$1.2 kpc around the sun, in contrast
with about $\sim$0.6 kpc from the APOGEE MS sample. Moreover, GALAH MS sample
is much larger and includes hotter and more massive younger MS stars, as seen
in the lower panels of Figure 1, which captures more of the thin disk
kinematic behaviour.

The distribution of the final sample RGB stars is shown in the projection of
Figure 2, with the sampling space mostly dependent on APOGEE DR17 and GALAH
DR3. In this visualisation we can notice the southern celestial hemisphere
sampling of GALAH in contrast with the northern-dominated sampling of APOGEE,
and also a lack of stars in the Galactic plane spanning roughly $\sim$180
degrees towards the Galactic centre.

Figure 2: Coverage of the APOGEE (ocre) and GALAH (cyan) RGB final samples in
galactic coordinates.

###  2.2 Elemental abundances

If we are segregating different populations in the Galaxy by tracing
astrophysical events such as a major merger or accretion of dwarf galaxies,
the inclusion of more chemical data (in addition to metallicity and
$\alpha$-elements) is very useful since mass inflows can cause starburst
events in-situ, and the own satellite abundances give hints about their mass
and evolutionary history. In particular the odd-Z elements, the iron-peak and
the heavy elements, since associations of light elements like Na and Al
together with Mn and Mg are able to segregate the halo/accreted populations
(Hawkins et al. 2015, Das et al. 2020, Feuillet et al. 2021), while the s- and
r-process elements can trace different astrophysical scenarios, older
populations and halo stars (Aguado et al. 2021, Carrillo et al. 2022, and da
Silva & Smiljanic 2023), as well as serving as good chemical clocks (Spina et
al. 2017, Magrini et al. 2018, see Appendix B).

Nandakumar et al. (2022) has produced calibration catalogues to scale between
APOGEE and GALAH abundances, but since our elemental selection spans beyond
[Fe/H] and [$\alpha$/Fe] with different species representing each elemental
group (Al as light and Ce as heavy for APOGEE, and Na as light and Ba as heavy
for GALAH), we chose to use the original abundances. Moreover, the use of
different chemical catalogues in the same segregation methodology can be a
good validation to the results produced by each one.

The main challenge in this abundance scenario is to properly select the
available elements within each survey, as each one has a different set of
elements beyond [Fe/H] and [Mg/Fe]. We show in Figure 3 the odd-Z and Mn
elements from both surveys in order to compare their quality and usability to
detect accreted populations.

Figure 3: Elemental abundance patterns in the RGB sub-samples of spectroscopic
surveys. Top panels: odd-Z elements vs [Fe/H], where the sparse "tail"
represents the dynamical halo with accreted populations, for APOGEE on the
left and GALAH on the right. Lower panels: [Mg/Mn] vs odd-Z, showing the
separability of the accreted populations in different cases for the surveys.

In APOGEE, the Al is well determined and shows the low [Al/Fe], [Mg/Mn]
abundances that characterise low star-formation rates typical of low-mass
galaxies as the indicative of accreted stars, where in GALAH this population
is over-estimated for Al and can be seen better with Na and Cu (see eg.
Giribaldi & Smiljanic 2023). Kobayashi et al. (2020) shows that these odd-Z
elements have similar trends with metallicity, and for the GALAH sample, Cu
seems to produce a better distinction of the accreted population than Na
(lower right panel of Figure 3). However, even though Cu has a similar
nucleosynthetic origin, it is more related to the iron-peak than to light
elements like Al and Na, and this use for Cu is shown in the Figure 27 of
Buder et al. (2021) in the plane [Mg/Cu] vs [Na/Fe] as an alternative to
[Mg/Mn] vs [Al/Fe] to segregate the accreted populations. Moreover, APOGEE
lists Cu for the abundance list, but the whole column is nulled, preventing
its use for the analysis. Even though this plane is able to segregate the
accreted halo, the disk collapses to a central concentration at [Na/Fe] $\sim$
0.1 and [Mg/Cu] $\sim$ 0, whereas in the [Mg/Mn] vs [odd-Z/Fe] planes of
Figure 3 there is a split of two main concentrations, one around solar value
for both ratios that is associated with the metal-rich thin disk, and the
other at [Al/Fe] $\sim$ 0.25 and [Mg/Mn] $\sim$ 0.5 that is associated with
the thick disk (discussed furthermore and shown in Figure 24). Therefore, we
chose the use of Mn as the iron-peak element and Na as the light odd-Z for the
GALAH analysis and preserve some consistency with APOGGE.

The iron-peak elements are mainly produced during the thermonuclear explosion
of type-Ia supernovae, and in this case, Mn is produced more than Fe
(Kobayashi & Nomoto, 2009), where Hawkins et al. (2015) showed that the Mn
abundance can be used to distinguish between thin and thick disk stars by
tracing the type-Ia supernovae enriched population.

For the heavy elements, the only one available in APOGEE is Ce, which is an
s-process (second peak) element. In the GALAH sample, the [Ce/Fe] - [Fe/H]
distribution is very poor in comparison to the expected trends of Kobayashi et
al. (2020) and the APOGEE counterpart, as shown in Figure 4. The closest
correspondent element in the GALAH sample is Ba, which has a more similar
distribution and has the same nucleosynthetic origin.

Figure 4: Abundances of heavy s-process elements vs [Fe/H] for the RGB sub-
samples of the surveys. The monotonic pattern of GALAH [Ce/Fe] does not match
with APOGEE [Ce/Fe], where GALAH [Ba/Fe] ratio shows a much better agreement.

###  2.3 Ages

In the process of segregating and characterising populations, the stellar ages
can play a major role in helping to determine the sequence of events that took
place in an astrophysical scenario, and with these parameters, it is possible
to separate populations that would be indistinguishable for sharing similar
properties in other variables and establish a timeline of the assembly of the
Galaxy.

The GALAH sample has ages estimated from the bayesian BSTEP (Sharma et al.,
2017) code, based on stellar observed parameters and the PARSEC-COLIBRI
isochrones (Marigo et al., 2017). In the other hand, APOGEE ages were
estimated using the astroNN (Leung & Bovy, 2018) trained with the
asteroseismic ages from Montalbán et al. (2021) and Mackereth et al. (2019).
However, there are some inconsistencies between the two age estimations if we
compare them by using the common stars between the two surveys, as discussed
in Appendix B. In face of these large inhomogeneities between the default age
determinations of APOGEE and GALAH (astroNN and BSTEP), and the lack of a
large enough matching sample from other sources such as StarHorse (Queiroz et
al., 2023), we chose not to use ages as a main input parameter for t-SNE, as
it would introduce more errors and unwanted effects in an already large
dimensional parameter space, leaving the age as a validation parameter.
Instead, we chose to use the [Ce/H] and [Ba/H] ratios that, as discussed in
Appendix B, may also work as an age estimator better than [Fe/H].

###  2.4 Orbital parameters

As already stated by Antoja et al. (2018), the Gaia era showed that assuming a
time-independent, axisymmetric potential in dynamical equilibrium for the
Milky Way is definitively incorrect, requiring a more rich approach to extract
orbital parameters in order to better characterise stellar kinematics. To
calculate the orbit trajectories and orbital parameters, we employed the galpy
code (Bovy, 2015) using the modified MWPotential2014 with halo mass
$M_{h}=1.4\times 10^{12}M_{\odot}$, twice the original value, together with a
bar using the DehnenBarPotential parametrised with $\Omega_{b}=$ 39 km/s/kpc,
$R_{\Omega}=$ 3.5 kpc, and azimuth angle of 28$\deg$ from the Sun-Galatic
centre line, with the sun at $R=8.3$ kpc, following the results of Lucey et
al. (2023) and solar motion from Schönrich (2012). The integration was made
backwards in time with 1000 steps down to -800 Myr from present, which is
roughly equivalent to 5 bar periods. The parameters obtained were the
apocentric and pericentric radii $R_{ap}$ and $R_{per}$, eccentricity $e$,
maximum Galactic plane distance $Z_{max}$, orbital energy $E$, total and
Z-axis angular momenta $L$ and $L_{z}$, galactocentric radial, tangential and
vertical velocities $V_{r}$, $V_{\phi}$, $V_{z}$, as well as mean orbital
radius $\bar{R}_{o}$.

The use of a consistent bar potential is necessary in order to reproduce and
better constrain kinematic populations that originate from the bar resonances,
such as Hercules, Hyades and Sirius (Khoperskov & Gerhard 2022, Pérez-Villegas
et al. 2017) which are well-established moving groups. The constraining of a
connection between the resonances and their associated moving groups could
validate the possible kinematic effects that may arise from this barred
structure and its influence on the outer portions and dispersion-dominated
components such as the thick disk.

In order to verify the Galactic radius of the bar resonances that arise from
the calculated orbits and match their expected positions, we used the GALAH MS
sub-sample (with a much larger size than the APOGEE counterpart) to inspect
the ridges produced by the accumulation of $\bar{R}_{o}$ (Figure 5), since the
MS sample contains the younger and bluer stars, thus it is more suitable to
investigate the rotation-dominated thin disk with the dynamical effects
produced by the presence of a bar.

Figure 5: Radius and velocity planes for the GALAH sub-sample. Galactocentric
radial velocity ridges are shown in the upper left panel, where the
corresponding star density ridges (R1 to R5) induced by $\bar{R}_{o}$
accumulation are shown with guide lines in the upper right panel. The ridges
selected from the upper panels are projected in the bottom left panel as the
guide lines, and the corresponding stars of the ridge fiducial selection are
shown in the U-V plane together with open clusters in the bottom right panel.

We made fiducial selection cuts around the density ridges (R1 to R5) that
appear along the $\bar{R}_{o}$ which are also visible as $V_{r}$ ridges in the
upper panels of Figure 5, and by fitting linear trends to these ridges it is
possible to marginally constrain the location of these features in other
planes, such as the V-R plane colour-coded by -U (Chen et al., 2022). In this
visualisation, we projected the fiducial lines corresponding to each selected
density ridge group, enabling the comparison of the orbital effects of the bar
potential (mainly $\bar{R}_{o}$) with directly observable parameters (R, U,
V), where the density ridges of $\bar{R}_{o}$ match the location of -U
velocity ridges. This qualitative agreement is an indication that the bar
parameters employed in the potential are a good representation of the true
Galactic bar potential, since higher pattern speeds shifts the Lindblad
resonances (and consequently the position of $\bar{R}_{o}$ ridges) towards the
Galactic centre, dissociating the -U velocity ridges from the $\bar{R}_{o}$
density ridges. Moreover, these fiducial selections are shown in Figure 5 in
the U-V plane together with stars from open clusters available in our sample
in positions, matching the locations of the moving groups of Hercules, Hyades
and Sirius, with the ridge positions in a good agreement with those detected
by Bernet et al. (2022). The red and royal blue lines, once a single density
ridge (R1) at $\bar{R}_{o}\sim 6.5$ kpc (R1 Her1), were splitted with the help
of V-R plane, where the high $-U$ value around -40 km/s in $V$ associated with
this ridge shows an additional small ridge (R1 Her2) with a more pronounced
slope. This duplicity may be first two components (Hercules 1 and 2) of the
multimodal structure of the Hercules stream shown by Asano et al. (2020).

##  3 t-SNE

The t-distributed stochastic neighbour embedding (t-SNE) algorithm (van der
Maaten & Hinton, 2008) is a non-linear dimensionality reduction technique for
visualising high-dimensional data in a low-dimensional space, modelling each
object in a two- or three-dimensional point where similar groups are reduced
to nearby clusters, thus creating islands and gaps in such a way that, for our
purpose, results in a 2-D space where distinct Galactic components are
expected to appear clustered in distinct regions. A good tutorial to
understand the mechanics on how t-SNE works is outlined in Erdem (2020), but
in a nutshell, it calculates the conditional probabilities with gaussian
distribution (that can be understood as similarities) for the data points in
the large-dimensional space, performs a first guess of the low-dimensional
space by using PCA (this is for our case, but it could be set to random) and
calculates the conditional probabilities with a Student-t distribution. After
this, it iterates over the low dimensional space to minimise the Kullback-
Leibler divergence as the cost-function via gradient descent and stops at a
defined number of iterations or when the divergence reaches no significant
change in the last iterations. As a result, it returns the points arranged in
the low dimensional space that can be seen as a similarity space, where
similar groups clump together and relate to neighbouring groups.

Anders et al. (2018) showed a default parametrisation for the sklearn
implementation of t-SNE where two of the main three parameters (learning_rate,
hereafter lr and early_exaggeration, hereafter ee) should be set to 1 and
$0.1\cdot N_{points}$, respectively, that worked for a sample of 530 stars.
The learning_rate governs how fast the gradient descent changes the positions
of the samples in the 2D space across iterations, and the ee governs how fast
the clusters of data move across the space in the initial phase of the
convergence, or as defined in the openTSNE (Poličar et al., 2019)
documentation, it increases the "attractive forces" between points and make
clusters more compressed. The perplexity (hereafter pp) is the parameter that
adjust the bandwidth of the gaussian kernels centered in each sample which
will admit the points to calculate similarities, and can be thought as an
analogous to the k nearest neighbours. In practice, it governs the
preservation of structures, where large values preserve global structures and
smaller values preserve local structures, since it regulates the amount of
neighbours "seen" by a point to be comprehended in what the distribution
admits as a local cluster.

In preliminary tests, having our sample with significantly more stars, the ee
parameter showed to perform better around $0.005\cdot N_{points}$, and lr with
values ranging between 600 and 800, as the maps should resemble scattered
clouds instead of entangled tapes or uniform balls of points, following the
examples of sklearn documentation, where values in the range 100-150 were
appropriate to keep both global and local structures. Another good tutorial
with several examples on how the convergence behave based on the perplexity
are available in Wattenberg et al. (2016). Belkina et al. (2019) showed that
raising the number of iterations in the early exaggeration phase led to better
accommodations of global structures, where clumps belonging to the same group
tended not to scatter around the final manifold. We also experimented with
modifying the sklearn implementation to allow the input of early exaggeration
n_iter to perform more tests, achieving slightly better results, but as
discussed further on, it contributed to our choice of using another
implementation of t-SNE.

The sklearn implementation has several limitations in comparison to the more
recent implementation of openTSNE, including speed, early_exaggeration
iterations (natively, instead of modifying the source code), different
exaggeration settings for each phase, the possibility to obtain snapshots
along the iterations via callback functions (enabling the inspection of the
convergence), as well as novel features, such as perplexity annealing (the
modification of perplexity along the run) and the additiion of new data to an
existing embedding. A first Galactic component classification can be used as a
control along t-SNE iterations and different parametrisations (as discussed
further in Section 3.1), using different variable spaces and subsample
selections, enabling consistency checks of the manifolds produced. Having the
power to fine-tune the parameters and customise the embedding, it is possible
to primarily separate large structures (such as thin disk, thick disk and
halo) with a large perplexity, then sub-classify them by reducing the
perplexity to continue the iterations, resulting in an embedding that makes a
better representation of the Galactic components based on the hierarchy of
parameters. A description of the parameters and procedures used to fine-tune
the t-SNE and the convergence maps will be presented in Section 3.1.

###  3.1 Manifold convergence

The process of convergence for the t-SNE implied in exhaustive parametrisation
tests in order to obtain a satisfactory iteration process in which we could
observe the gradual separation of our pre-labeled structures by their
relational hierarchy. For example, one could expect that the alpha-rich,
dynamically hot component (namely the canonical thick disk) would separate
early from the young metal-rich thin disk, and subsequently show some other
structures such as the dynamical halo. With the openTSNE ability to change and
tune hyperparameters in a continuous iteration process and to communicate with
the iterator object via a callback function, we were able to build a parameter
cascade for each one of the RGB datasets (APOGEE and GALAH) and obtain our
final manifolds, detailed in the following sections.

The kinematic and chemical parameters to use as t-SNE features were $E$,
$Z_{max}$, $L$, $L_{z}/L$, $V_{\phi}$, $e$, $R_{ap}$, $R_{per}$, [Fe/H],
[Mg/Fe], a light odd-Z element ([Na/Fe] for GALAH and [Al/Fe] for APOGEE),
[Mn/Fe] and a heavy s-element ([Ba/Fe] for GALAH and [Ce/Fe] for APOGEE). The
maximum Galactic plane distance $Z_{max}$ is extremely useful to dynamically
trace the thin and thick disks, as well as the halo. We introduce $L_{z}/L$ as
a direct measure of orbital inclination, based on how much the Z component
(associated to the Galactic plane orbital motion) contributes to the total
angular momentum, where smaller values approaches perpendicular orbits and
negative values easily represents retrograde motion. $E$ and $L_{z}$ are
typically used in the Lindblad diagram, and are able to separate accreted and
dispersion-dominated populations, and even though we include only $L$, $L_{z}$
is recovered by the use of $L_{z}/L$, with $V_{\phi}$ not only reinforcing its
character, but also being able to segregate the well-known moving groups in
the U-V plane (as shown in Figure 11 furthermore). Here we use the instant
angular momenta and eccentricities, since these variables are not conservative
in a non-axisymmetric potential. The apocentric and pericentric radii place
the constraints on how deep a star may dive to the centre or elongate to the
outermost parts of the Galactic disk, and the eccentricity $e$ is fundamental
to trace circular or elongated orbits. One could argue that $R_{ap}$,
$R_{per}$ would suffice to map this orbital feature, but as the euclidean
space is linear, $e$ cannot be obtained as a linear combination of the radii,
which is not the case for $L_{z}$, that it is linearly dependent of $L_{z}/L$
and $L$. We performed a min-max normalisation on all parameters (except
$L_{z}/L$ and $e$ that are already in the 0-1 range) before executing the
t-SNE runs, a very important step not only for t-SNE, but also for machine
learning algorithms in general.

The initialisation was set to "pca", where the convergence is not affected by
a random state, producing a deterministic result that sustains itself for any
run, depending only on the convergence parameters. The metric used was the
cosine distance, as recommended by openTSNE documentation. The cascade used
for APOGEE consisted in a primary t-SNE parametrisation with early
exaggeration (corresponding to the first 5 and subsequent 10 tiles of Figure
6), followed by two annealing steps ($anneal_{1}$ and $anneal_{2}$ on Figure
6), and a final annealing ($anneal_{3}$), both without early exaggeration. For
GALAH, we made two steps both with early exaggeration, with the first one
corresponding to the same pattern as APOGEE in Figure 6, and the second
labeled as $anneal_{1}$ for the early exaggeration phase and $anneal_{2}$ for
the normal iterations phase. The hyperparameters used in each convergence step
is summarised in Table 2.

Figure 6: t-SNE convergence iterations with colour-coded reference mapping for the APOGEE sample in the left panel and for GALAH sample in the right panel. The named regions were chosen in an iterative process while converging the manifold. We show the final reference classification made by visually comparing, identifying and cross-matching both APOGEE and GALAH data. The last annealing iterations are shown only up to 600 since the manifold achieves a state that is very close to its final shape. The iteration stages are annotated in the subplots, representing one at every 100 iterations, according to the scheme of Table 2, and they follow a left-to-right and top-to-down order. Table 2: Hyperparameters early exaggeration (ee), exaggeration (ex), learning rate (lr), perplexity (pp), and iterations used in each convergence step for APOGEE and GALAH. Step | ee | ex | lr | pp |  ee iter. | normal iter.
---|---|---|---|---|---|---
| APOGEE
primary | 80 | 2 | auto | 250 | 500 | 1000
annealing 1 | - | 2 | 90 | 170 | - | 200
annealing 2 | - | 5 | 70 | 130 | - | 300
annealing 3 | - | 1.6 | auto | 50 | - | 2000
| GALAH
primary | 90 | 1.7 | auto | 150 | 500 | 1000
annealing | 9 | 1.8 | auto | 25 | 300 | 2000

Following this relational hierarchy idea, we traced the manifold convergence
by colouring and labelling reference groups for some well-known Galactic
components (thick disk, metal-rich thin disk and dynamical halo for example),
monitoring the displacement behaviour of the structures in the manifold in an
iterative process, as shown in Figure 6 for the two survey samples. The
identification of the reference groups was done by visually inspecting
parameter-coloured maps (as shown for example in Figures 8 and 9, discussed in
detail on Section 4), and matching these groups between APOGEE and GALAH in
such a way that both manifolds showed the same structures with very similar
chemical and kinematic patterns.

Figure 7: Final manifolds achieved For APOGEE (left panel) and GALAH (right
panel). The $\odot$ mark represents the sun location in the manifold. The axis
ticks were omitted since the manifold coordinates have no physical meaning.
The labelled "inner" groups are for $R_{avg}<7$ kpc and "outer" are for
$R_{avg}>7$ kpc.

The splitting of the blobs change at the start of each annealing, especially
for GALAH, and we see that the identified groups tend to accommodate and
define themselves more when the new early exaggeration starts in the annealing
stage. The light orange blob representing the outer thick disk transits
between the orange portion (inner part of the thick disk) and the lower left
portion that represents the outer populations, and show some relation to the
scattered metal-poor outer disk, discussed in more detail at Section 5.1.

##  4 Results

The main goal of this work is to produce a map where we can visualise the
Galactic structure in an intuitive fashion, enabling us to spot sub-structures
and relations that would not be easily accessible otherwise. The definition of
regions were done visually, and can be extremely useful to dissect some
specific populations, their peculiarities, and how they fit in the Galactic
assembly, as already shown in the previous section.

The final manifolds are shown in Figure 7 with some of the main identified
structures labelled in different colours for both APOGEE and GALAH with their
mapped positions coinciding between the surveys. The legend is grouped by the
large clusters to guide the eye.

The parameter-coloured manifolds are the most important figures that enable
the distinction of characteristic groups by a careful analysis and comparison
of each panel, where the colour maps of the variables were chosen to highlight
different ranges of values and bring out features on the manifolds to
segregate the populations. These are shown in Figure 8 for APOGEE and Figure 9
for GALAH.

Figure 8: APOGEE manifold colour-coded by the chemical and dynamical
parameters selected for the analysis. Figure 9: GALAH manifold colour-coded by
the chemical and dynamical parameters selected for the analysis.

Also, an important measure for the kinematics of the populations is the
velocity dispersion in the $V_{r}$, $V_{\Phi}$ and $V_{Z}$ components, and
this was made by binning the manifold into hexagons and taking the standard
deviation of the velocities inside each bin. The bin size was chosen in a way
that structures were preserved while keeping at least 5 stars into each bin.
The velocity dispersion manifolds with hexagonal binning are shown in Figure
10 for APOGEE and GALAH. It is possible to verify the match between
populations with higher eccentricities and larger $\sigma V_{r}$ in the
manifolds, and also the match of those with higher $Z_{max}$ and larger
$\sigma V_{z}$ as a good sanity check for the kinematics consistency.

Figure 10: APOGEE (left panel) and GALAH (right panel) manifolds colour-coded
by total velocity dispersion and its radial ($V_{r}$), tangential ($V_{\Phi}$)
and disk plane transversal ($V_{Z}$) components.

The selected groups were drawn by hand, carefully inspecting the parameter-
coloured and the velocity dispersion manifolds to produce consistent
populations for both surveys at the same time, searching for gaps in the
clusters and leaps in the values of the colour-coded variables that could
delineate the region boundaries. This was possible due to the final
configuration of the manifolds, showing the main Galactic features roughly on
the same locations for both APOGEE and GALAH, which would be extremely
difficult to achieve in other implementations of t-SNE, where the fine-tuning
of the convergence steps plays a major role in the desired result.

By analysing the Figures 8 and 9, the main contrast between the survey samples
is the scattering and visual separability. Although the orbital parameters are
derived using the same source, the chemical abundances have a much larger
spread in GALAH than APOGEE, which translates into more concentrated and
distinct regions for the latter. The ages, as stated before, were not used as
t-SNE input features, but are valuable markers for these older populations and
in-survey comparison. Many of the labelled groups carry an inner- or outer-
designation that refers mainly to their $R_{avg}$ location, appearing in
contrast at the upper-right panels of Figures 8 and 9, and further in Figure
13 as a bimodal distribution of the whole sample of APOGEE. In general,
"inner" groups are for $R_{avg}<7$ kpc and "outer" are for $R_{avg}>7$ kpc.

Nevertheless, it is easy to identify the thick disk due to its alpha-enhanced
character, the lower overall metallicity, higher $Z_{max}$ and tilted orbits
in relation to the Galactic plane (lower $L_{z}/L$), and especially higher
velocity dispersions in the lower-right portion of the manifolds in Figures 8,
9 and 10. The thick disk appears splitted into two components: the inner thick
disk attached to the corotation, and the outer thick disk associated with the
$\bar{R}_{o}$ concentration at R$\sim$8-9 kpc, and this outer component has a
contrasting high $\sigma V_{\phi}$ in comparison with the rest of the
manifold. In both APOGEE and GALAH it is possible to spot a slightly older
concentration, which shows larger radial velocity dispersions and
eccentricities, and will be discussed further in Section 5.3. At the bottomost
region of this population we can also notice a dynamically hotter portion with
even higher values of $Z_{max}$ and highly tilted orbits, achieving negative
values (retrograde orbits) in its right end. This portion is the so-called
heated or splashed thick disk that is associated with the merger with GSE,
where its right end is a piece of the dynamical halo including the accreted
populations with lower Mn, Na and Al abundances.

The upper-central region is mostly comprised of dynamically colder thin disk
stars with higher metallicities, mostly contained inside the solar
galactocentric radius within an inner disk. There are many interesting
features in this structure, and one of them is the higher eccentricity portion
in the right with higher $V_{\phi}$ dispersion (labelled metal-rich eccentric
disk), which according to Figure 11 could be associated to the Hercules moving
group, where Pérez-Villegas et al. (2017) pointed that this bar-induced
dynamical population is composed of stars that move between the solar radius
and the innermost regions of the Galaxy, showing itself as an overdensity in
the U-V plane (Figure 2 of their paper) peaking roughly at -30 km/s in U and
-50 km/s in V, which translates into $(V_{r},V_{\phi})=(18.9,200.24)$ km/s
considering their LSR at (U, V) = (-11.1, -12.24) km/s, and a local galactic
rotation of 238 km/s, whereas our metal-rich eccentric disk populations peaks
in density at $(V_{r},V_{\phi})=(24.34,200.28)$ km/s for APOGEE and
$(V_{r},V_{\phi})=(16.4,203.63)$ km/s for GALAH. Also, the labeled cold metal-
rich thin disk and cold metal-rich local disk could be associated with the
moving groups of Hyades and Sirius, respectively, according to their locations
in the $V_{\phi}-V_{r}$ plane (analogous to U-V plane) as shown by Bernet et
al. (2022). At the lower end of this portion, we can spot a small low-
metallicity tail, being the inner counterpart of the metal-poor populations of
the outer portion (bottom-left region). The proportion of metal-poor stars
between inner and outer populations reproduces the metallicity gradient that
increases towards the Galactic centre (e.g., Sun et al. 2024), where the outer
portion carries a noticeably larger metal-poor fraction than the inner portion
for the thin disk, and no significant radial gradient when considering the
outer thick disk and inner thick disk.

Figure 11: $V_{\phi}-V_{r}$ plane (left panels) and respective manifolds
(right panels) for APOGEE and GALAH metal rich populations, showing metal-rich
eccentric disk (green), cold metal-rich thin disk (orange) and cold metal-rich
local disk (blue) regions.

The bottom-left region is mostly composed of an outer thin disk (with the
exception of the high alpha blob in the GALAH manifolds, that is associated
with the thick disk) reaching the solar Galactic radius and some metal-poor
structures, consistent with the negative metallicity gradient towards the
outer parts of the Milky Way. This outer thick disk portion, as shown in
salmon in Figure 6, stays associated with the thick disk cluster during the
convergence, but after the final annealing with lower perplexities, it tends
to associate with the outer disk region in GALAH, and as a separated cluster
in APOGGE, closer to the bulk of the thick disk.

Also, a transition zone appears between the cold, thin, metal-rich disk and
the alpha-enhanced thick disk, with moderate chemical features for virtually
all of the cases including the ages, where this region is clearly older when
compared to the thin disk. Similar to some of the selected regions, this zone
shows a few sub-structures that will be discussed further in Section 5.4.

One remarkable difference between the two surveys is the size proportions of
the clusters, which are explained by the sky sampling of each one. APOGEE
scans mostly the northern sky, which points towards the Galactic anticentre,
and thus there are more outer disk stars. In the other hand, GALAH scans the
southern sky in the direction of the Galactic centre, so more inner (with
respect to the solar radius) disk stars are expected in the sample. Moreover,
GALAH basically avoids the Galactic plane (with the exception of the Galactic
centre) while APOGEE samples its available sky more uniformly.

##  5 Discussion

It is important to remember that, as a technique that minimises the similarity
distribution divergence, the t-SNE manifolds can be seen as similarity maps of
the Galaxy. In this context, the relative distances between the selected
groups can give hints on how they are related in a Galactic assembly scenario.
In the following topics, we will discuss each of the main components, new
insights and some caveats that are intrinsic to the problem.

###  5.1 Thin disk

First, we need to stress that due to the poor sampling of the Galactic plane
in the direction of its centre (see Figure 2), we miss the bulk of thin disk
stars confined to the lowermost vertical elongations, but still within a
reasonable range considering stellar distances, proper motions and radial
velocity determination errors, which can broadly affect the orbital
parameters. Our thin disk sample, due to this, shows some dispersion of
$Z_{max}$, but still more concentrated within Z$\sim$1 kpc as shown for a
selection of metal-rich portions of the manifolds in Figure 12, together with
thick disk populations for a comparison. These thin disk populations can be
easily spotted through low $Z_{max}$, young ages and their typical [Fe/H]
ranging from -0.25 to +0.3 dex on the colour-coded manifolds of Figures 8 and
9, as well as the low values of velocity dispersion in Figure 10. We show
mostly APOGEE in the figures of this section, mainly for its larger sampling
of the outer disk.

Figure 12: Distribution functions of $Z_{max}$ for the dynamically cold metal-
rich (upper panel) and hot metal-poor (lower panel) populations of the APOGEE
sample.

One of the main differences between the two major regions containing the thin
disk is their mean orbital radius $\bar{R}_{o}$, where the central manifold
portion is bound to a concentration centered at R$\sim$6.4 kpc with a long
tail towards the Galactic centre, and the lower-left portion is associated to
another concentration distributed between R$\sim$8-9 kpc, as illustrated in
Figure 13 for the case of APOGEE, which better covers the mean orbital radius
space. The outer disk was split into the more metal-rich upper portion (outer
disk 1 in magenta, including the cold metal-rich local disk and cold metal-
rich outer disk) and a more metal-poor with larger $R_{min}$ at the bottom
portion (outer disk 2 in salmon, including the metal-poor outer disk and
scattered metal-poor outer disk) to show the origin of its bimodality. We show
the mean orbital radii of the total APOGEE sample calculated with the barred
galactic potential aforementioned in comparison with the MW2014 axisymmetric
potential (also with twice the original halo mass).

Figure 13: Distribution functions of mean orbital radius for the APOGEE
sample. In light gray we show the total sample for the orbits calculated with
the axisymmetric potential, and in dark gray for the barred potential used in
this study. Blue line shows the central manifold portion (inner thin disk).
Inner thick disk (red), outer thick disk (yellow) and transition zone are
shown for comparison.

Our assumed Galactic potential for orbital integration with a bar pattern
speed of $\Omega_{b}$=39 km/s/kpc places the corotation radius around $\sim$6
kpc, but the locus of the resulting radial density caused by the corotation is
subject to other parameters and effects such as bar length and the
aforementioned phase space measurement accuracy, therefore we associate the
R$\sim$6.4 kpc overdensity as the corotation effect on mean orbital radius of
the sample stars. As early stated by Athanassoula (1980), growing galactic
bars can drive the spiral arm structure, and Garma-Oehmichen et al. (2021)
have shown through simulations that the spiral arm pattern is the collective
result of the rotating bar perturbing the stellar orbits, thus forming the
arms through a density-wave mechanism. Dias et al. (2019) studied the spiral
arm structure and placed its corotation radius at R=8.5 kpc, which is tightly
reproduced in our mean orbital radius density distribution (where the outer
populations lie), leading us to suggest that the Gaia observed phase space
combined with the Galactic potential used here could be able to reproduce by
itself the spiral arm structure of the Galactic disk.

Figure 14: Radial distributions for the metal-rich (upper panel) and metal-
poor (lower panel) populations of the APOGEE sample with the mean orbital
radius plotted in thick lines. The thin lines represent the apocentric and
pericentric radii of the metal-rich populations, and the apocentric radii of
the metal-poor populations.

Moreover, the density waves are the main star formation locations, as their
potential favours converging gas inflows (Pettitt et al., 2019). The
dynamically cold metal-rich populations show this mean orbital radius binding
to the corotation of the bar and the spiral arms, and when we inspect the
apocentric and pericentric radius of these populations (upper panel of Figure
14), it is possible to notice that most of their extreme orbital radii are
concentrated in the bar and spiral arm corotations. Lucey et al. (2023)
constrains a dynamical bar length (maximum extent of trapped orbits) of 3.5
kpc with an overdensity spanning up to 4.8 kpc, representing an upper limit
for the direct bar churning influence. The metal-rich eccentric disk,
associated with the Hercules moving group, dives more deeply inside the the
churning influence zone of the bar, and consequently the inner Galaxy, which
might explain its slightly older right-end in the age manifold of Figure 8.
Moreover, by closely inspecting the orbits that belong to Hercules in Figure 3
of Pérez-Villegas et al. (2017) ((d), (g), (h) and (i) panels), their
apocentric and pericentric radii seem to coincide with the ones at Figure 14,
providing more evidences that the metal-rich eccentric disk is the isolated
Hercules stream. The cold metal-rich local disk population, where the sun
lies, even if associated with the outer portion, has a lower apocentric radius
coincident with the spiral arm corotation radius, like most of the metal-rich
populations, feature also visible in the GALAH manifolds on Figure 9.

Figure 15: Tinsley-Wallerstein diagrams (left panels) with contours
representing the 50th percentile of the metal-rich sequence(upper left), 16th,
50th and 84th percentiles of metal-poor sequence (lower left), and respective
manifold locations (right) for selected thin disk populations in APOGEE.
metal-rich and metal-poor are abbreviated to MR and MP, respectively.

Digging more deeply into the chemical behaviour of these populations, we show
in Figure 15 the locations in the Tinsley-Wallerstein diagram of the selected
inner and outer populations for APOGEE (due to the better [Mg/Fe] resolution),
where the outermost ones lie along a metal-poor sequence on the left portion
of the lower [Mg/Fe], and the innermost ones populate its metal-rich end,
commonly associated as the younger population. A further discussion of the
inner metal-rich population is presented in Section 5.2.

The scattered metal-poor outer disk shows an alpha enhancement and larger
velocity dispersions, and in Figure 16 we show the behaviour of this
population in APOGEE (as it samples more of the outer disk) in function of
galactocentric radius and $Z_{max}$ compared to the other thin disk
components. Das et al. (2024) presented results of the heating effect caused
in the outer disk by the interaction of a satellite galaxy (probably
Sagittarius dSph), and in their Figure 4 a clear bump in $\sigma_{z}$ is
visible between $R\sim 9-10$ kpc, comprehended in the same place where the
scattered metal-poor outer disk is located, showing a behaviour more related
to the thick disk in comparison with the other thin disk selected populations.
Besides the location, the age distributions at the lower panel of Figure 16
shows the scattered metal-poor outer disk near $\sim$6 Gyr, and also the
[Fe/H]$\sim$-0.5 together with [$\alpha$/Fe]$\sim$+0.14 (see Figure 15) are in
agreement with Das et al. (2024) results. This suggests that this population
may have a connection to the passage of Sgr dSph. In this panel we also show
populations belonging to thin and thick disk for comparison. In Section 5.4,
we give some insights about the transition zone and its possible relation to
this disk perturbation event.

Figure 16: $Z_{max}$ vs R plane for APOGEE sample (grey scattered points) with
contour plots of the scattered metal-poor outer disk compared with other
selected components in the in upper panel, and age distributions in the lower
panel.

###  5.2 Inner metal-rich population

As stated before, we kept the kinematic homogeneity by using only Gaia DR3
measurements for orbital constraints, but some artefacts could arise from the
use of different chemical data when comparing the manifold results, and also
from the different spatial sampling of APOGEE and GALAH. As an example, the
assigned cold metal-rich inner disk appears as an isolated island in APOGEE,
but for GALAH, the definition of an analogous structure is not trivial, since
the sampling of the surveys is not uniform in a specific region of the
$R_{ap}$-$R_{per}$ plane as shown in Figure 17. Moreover, the light element
used for GALAH was Na, while for APOGEE it was Al, making it difficult to make
a direct comparison for this specific metal-rich and Na-rich population. This
cold metal-rich inner disk structure has a lower $R_{ap}$, confined to the
inner parts of the Galaxy, and was found to be splitted on the GALAH manifold
in 3 sub-structures, also shown in Figure 17. These 3 sub-structures show a
reasonable coincidence in the $R_{ap}$-$R_{per}$ plane with the APOGEE
counterpart, but it is difficult to tell if they represent the exact same
elements. Moreover, still stressing the incompatibility caveats between
surveys, this inner-metal-rich population does not appear alpha-enhanced in
APOGEE, and one possible reason is the lower sampling of the Galactic central
directions in comparison with GALAH.

Figure 17: $R_{ap}$-$R_{per}$ plane for APOGEE and GALAH, with details of the
cold metal-rich inner disk cluster sub-divided into 3 pieces. The blue
substructure is an innermost and more eccentric portion of the region that
appears apart from the other 2 in the GALAH manifold.

An interesting feature of this selected portion is the high Na and Mn together
with moderate Mg content (Figure 9), where its distribution function is
centered at $\sim$0.1 dex, with enhanced Mn and Ba/Eu shown in Figure 18.

Figure 18: Abundance and age distribution functions of GALAH for the metal-
rich populations ([Fe/H] $\gtrsim$ -0.2) cold metal-rich inner disk, cold
metal-rich thin disk and cold metal-rich outer disk structures, together with
a representative of the thick disk and the local disk for comparison. Metal-
rich is abbreviated to MR in the legend.

The high Mn and moderate Mg points towards a fast enrichment in an already
metal-rich environment, since the yield of Na starts to increase above 2 solar
masses (Smiljanic et al., 2016), but the higher [Ba/Eu] indicates that the
s-process operated significantly in the enrichment of this population. This
same behaviour of [Ba/Eu] is observed in the cold metal-rich thin disk and
even stronger in the cold metal-rich outer disk structures, but with lower Na,
Mg and Mn distributions. Karakas et al. (2018) shows that the most significant
production of the heavy s-process elements takes place in stars from
$\sim$1.15 to 4 solar masses, which covers an evolutionary timescale range
between $\sim$0.3 to 7 Gyr. This result favours the aforementioned secular
enrichment of Ba, taking into account the Galactic scenario, which for inner
parts of the Galaxy the more active star formation rapidly recycles the Ba
made available by the constantly evolving stars, while in the outer portions,
the lower star formation rate and thus lower r-process supernovae yields
raises the [Ba/Eu] ratio at a lower pace.

If we consider that the GSE merger took place between 10-8 Gyr ago, a large
infall of gas should have taken place (Deason et al., 2013), and if we take
the peak age of the thick disk (around $\sim$9 Gyr in the last panel of Figure
18) as a proxy to the merger event, the peak distance between the thick disk
and the cold metal-rich inner disk is roughly $\sim$1 Gyr. The peak abundances
of Na for the cold metal-rich inner disk around $\sim$0.4 dex translates
roughly into $\sim$3 solar masses of progenitor stars via the relation
presented in Smiljanic et al. (2016), representing a stellar lifetime around
$\sim$650 Myr, which is reasonable enough to consider the fast recycling
scenario that led to the observed Na and Ba abundances of this population.
Wrapping these pieces together, it is not preposterous to suggest that this
population may be the direct cause of the GSE merger reflected onto the inner
disk of the Galaxy. Of course, the BSTEP stellar age estimations have large
associated errors of around $\sim$3 Gyr for these populations, but working
with the age distributions and its offsets are the best tools available to
this kind of analysis.

As for the cold metal-rich local disk, encompassing the solar neighbourhood,
the [Ba/Eu] ratio shows a broad distribution around the solar value, perhaps
signalling that the sun lies in a region where the Ba recycling is not so
intense and the Eu production rates are enough to settle down the [Ba/Eu]
ratio.

###  5.3 Thick disk

Until now, most studies worked by dividing 2D diagrams (eg. Tinsley-
Wallerstein, Toomre, etc) to distinguish between thin and thick disk and
conduct their analyses, but of course, some contamination is expected from
this methodologies. Here we have used a selection of kinematic and chemical
properties to segregate these older components with a clearer boundary, being
able to visualise at which limit we admit stars to a specific sampling by
making use of the colour-coded manifolds of Figures 8 and 9.

Still from the APOGEE sample (due to its better [Mg/Fe] separation), we trace
the $Z_{max}$ distributions of the metal-poor populations in the lower panel
of Figure 12, evidencing the dynamically hot character of the canonical thick
disk, and the evident presence of the heated (or splashed) thick disk with a
long tail towards larger Z.

We can also see the scattered metal-poor outer disk exhibiting larger Z
elongations, where the mean orbital radius and apocentric radius of the metal-
poor populations are shown in the lower panel of Figure 14. The maximum Z
elongation is achieved near the apocentre of the orbit, which can be up to
R$\sim$12 kpc for this local volume sampling. The mean orbital radius and
apocentric accumulation near the corotation radii also rise for the thick
disk, indicating that the bar and spiral arm structure also affects the
dynamically hot populations.

The so-called heated thick disk was interpreted by Di Matteo et al. (2019) as
the imprint left by a great merging event (GSE) heating the old Galactic disk.
Here we have isolated this population that is significantly hotter and with
much larger velocity dispersions than the canonical thick disk, as shown in
Figures 10 and 12. This population can be distinguished not only by its
kinematics, but also for the chemical properties, where its [Mg/Fe]
distribution is clearly shifted towards higher values, and for [Fe/H], shifted
towards lower values (shown in Figure 19), which corroborates with the
proposition that this heated thick disk was an older disk being splashed by a
merger event. Here we can see the similarity space brought into action when it
comes to the relations between populations, showing the heated thick disk as a
bridge between the thick disk and the dynamical halo.

By a closer inspection of the colour-coded manifolds in the thick disk region
(also in Figure 19), we could identify 2 sub-structures of the inner thick
disk, which are the circular thick disk with low eccentricities, and the old
thick disk with slightly older ages.

Figure 19: Tinsley-Wallerstein diagrams (left panels) with contours
representing the 50th percentile, and manifolds (right panels) for selected
thick disk populations of APOGEE (upper panels) and GALAH (lower panels). The
circular thick disk and the old thick disk are superposed to the inner thick
disk.

The separability of the populations is clearer for APOGEE, and we can see that
the old thick disk concentrates at the most metal-poor and alpha-rich end,
followed by the heated thick disk. The old thick disk together with the more
general inner thick disk and outer thick disk occupy roughly the same
distribution along the alpha-rich population, while the bulge extends up to
the more metal-rich end as expected for the inner Galaxy.

Apparently, the more circular orbits are the only feature that distinguish
this circular thick disk population, and we did not find any other unique
chemical or kinematic signs other than the apocentric and pericentric radii
coinciding with the cold metal-rich thin disk population and the [Mg/Fe] and
[Fe/H] distributions matching those of the outer thick disk.

As for the old thick disk, it can be spotted as a lower age region in the
heart of the thick disk portion of the parameter-coloured manifolds (Figures 8
and 9), and we find the [Mg/Fe] and [Fe/H] distributions in Figure 19 even
more shifted than the heated thick disk. Moreover, if we inspect the age
distributions of the thick disk (Figure 20), this population appears slightly,
but not negligibly shifted towards older ages for both survey samples, effect
that is also observed in the [s-process/H], as proposed here as a possible
chemical clock. This population could be either just an "older tail" of the
thick disk or a fossil record of a primordial Galactic configuration, where we
favour this scenario in the following section.

A small portion of the outermost extents of the bulge population also appears,
but only clear and isolated in the APOGEE. Our GALAH selection even samples
the central Galactic region, but the bulge may be too attached to the thick
disk to reliably isolate it. The bulge population can be spotted as a
stretched clump above the thick disk region in the APOGEE manifold, and as
shown in Figure 8, it is among the lowest apocentric and pericentric radii,
and share the same chemical pattern as the thick disk, with Zmax contained
inside $\sim$1.5 kpc.

Figure 20: Age (left panels) and heavy elements (right panels) distributions
for the thick disk sub-populations in the GALAH (lower panels) and APOGEE
(upper panels). The bump at $\sim$6 Gyr on APOGEE’s astroNN appears as a
slight asymmetry in the GALAH distributions, and the bimodality can be seen in
the [Ba/Eu] distribution.

###  5.4 Transition zone

The so-called transition zone is an intermediate population with properties
that stay between the thin and thick disks, and was interpreted by Recio-
Blanco et al. (2014) as a settling process that the thick disk experienced,
increasing its rotation and lowering the tangential velocity dispersion, what
can be indeed observed in Figure 10. We will refer to the APOGEE sample for
this analysis, since the transition zone appears more clear and defined in the
manifolds. The $Z_{max}$ distribution is compared to other populations in
Figure 12, and follows a pattern that stands between the metal-poor
populations and the thick disk, showing a longer tail towards larger values.
Its chemical and age patterns are rather puzzling, as can be explored in
Figure 8, showing a moderate metallicity around [Fe/H]$\sim$-0.25, moderate
[$\alpha$/Fe]$\sim$0.15, but enriched [Ce/Fe] (comparable to the metal-rich
populations) and older ages (comparable to the thin disk) as seen in the
manifolds. We have splitted the transition zone and its surroundings in the
manifold to better dissect its properties, shown in Figure 21.

Figure 21: Transition zone splitting showing Zmax distributions in the upper
left panel, manifold with the selected regions in the upper right panel, age
distributions in the lower left panel, and colour-coded regions in the lower
right legend, with transition zone abbreviated to TZ and metal-rich to MR. For
comparison, we also show the cold metal-rich thin disk and the old thick disk
in the extremes.

Besides its main portion (shown in yellow at Figure 7, connecting the inner
thin disk and the bulk of the thick disk), the outer portion of the manifold
also has a counterpart (labelled outer heated transition zone) that shares
some characteristics of the transition zone, already disconnected from the
outer thick disk blob. By inspecting the APOGEE convergence map of Figure 6,
we can see in the 100th iteration of annealing 3 (lowest rightmost panel) that
the outer heated transition zone and the outer thick disk were once connected,
representing their similarity relation which can be seen as the outer
counterpart of the transition zone. Moreover, the aforementioned scattered
metal-poor outer disk appears also very close and even stretched upon the
outer thick disk, which also favours its connection with this transition zone.

In the Zmax distributions of Figure 21, the "heated" character of the inner
heated transition zone and outer heated transition zone becomes clear and
contrasts with the remaining populations of the transition zone. These two
populations also have the strongest bimodality among the age distributions,
whose peaks are located at $\sim$6 Gyr and $\sim$8-9 Gyr, which coincides with
the Saggitarius dSph disturbance (Das et al., 2024) and the GSE merger,
respectively. The outer heated transition zone shows the largest ratio of the
6 Gyr peak compared to the 8 Gyr peak, corroborating with Das et al. (2024)
results of a heating effect in the outer regions of the disk. In the other
hand, this ratio gradually decreases for inner heated transition zone, main
transition zone, old transition zone, until it reaches the thick disk
interface and a very weak character in the old thick disk.

If we inspect the Zmax behaviour of some of these populations in the Tinsley-
Wallerstein diagram of Figure 22, an even more puzzling effect appears: for
all the cases shown, the Zmax value increases for decreasing [Mg/Fe], the
opposite of what would be expected in the evolutionary sense.

Figure 22: Tinsley-Wallerstein diagrams with transition zone populations
colour-coded by Zmax, showing its gradients for the case of APOGEE.

This effect partly contradicts the settling process proposed by Recio-Blanco
et al. (2014), which leads to a scenario where low-alpha thin-disk stars were
scattered to higher-Zmax orbits, and the older, dynamically colder stars in
the moderate-alpha sequence are the ones that were probably formed in a post-
merger environment (after GSE), as seen in the age distributions of Figure 21.
The metal-poor transition zone may give some hints about this formation
scenario, and was named mainly due to its bluer hue in the [Fe/H] panel of
Figure 8, in contrast to the rest of the transition zone. While its age
profile seems more similar to the thin disk interface, its tri-modal structure
has peaks at ages that coincide with would be an "aftermath" of each disk
perturbation, possibly due to metal-poor gas inflow.

To better understand the Zmax heating mechanism, we inspected the pericentric
radii of the transition zone in Figure 23 together with dynamically hot and
cold populations.

Figure 23: Pericentric radius distribution functions of dynamically cold and
dynamically hot populations, with transition zone abbreviated to TZ and metal-
rich to MR.

The outer heated transition zone avoids the stronger churning effects (cited
previously in Section 5.1) with pericentric radii located mostly at $\sim$6
kpc, which could explain the preservation of the $\sim$6 Gyr peak ratio. In
general, it seems mostly bound to the bar corotation for pericentric radii,
spiral arm corotation for mean orbital radius, and outer Lindblad resonance at
$\sim$10.5 kpc for apocentric radii. Moreover, Das et al. (2024) shows
chemical and age distributions with a bump at [Fe/H]= -0.5, [$\alpha$/Fe] =
0.1 and age = 7.2 Gyr, where outer heated transition zone has one of the age
peaks exactly at 7.2 Gyr and [$\alpha$/Fe] = 0.1, but at [Fe/H] = -0.3.

Putting these pieces together, we suggest that the transition zone could have
a formation history connected to the disk perturbation from satellite mergers
followed by a subsequent starburst from the infall of metal-poor gas.

###  5.5 Dynamical halo

At the tip of the rightmost end of the heated thick disk structures in the
manifolds shown in Figure 7, there are stars with highly tilted orbits,
together with very low [Fe/H] abundances. This portion represents the stars
belonging to the dynamical halo of the Galaxy, where they approach the solar
neighbourhood during their dynamically hot orbits, containing populations with
high eccentricities, retrograde motions and large velocity dispersions. For
APOGEE, the piece with very low [Al/Fe] abundances represents the accreted
population, as shown by Das et al. (2020), where part of it was interpreted by
Haywood et al. (2018b), Belokurov et al. (2018) and Helmi et al. (2018) as the
remnant of a major merger with a dwarf galaxy between 8 and 10 Gyr ago. In
Figure 24 we present the manifold locations and chemical diagrams for the
dynamical halo and the accreted populations, represented together with the
other main Galactic components.

Figure 24: Manifold (upper left), Tinsley-Wallerstein (lower left) and [Mg/Mn]
vs [Al/Fe] (lower right) diagrams for the dynamical halo and accreted
populations for APOGEE. The accreted population is superposed to the dynamical
halo. Thin and thick disk representatives are also plotted for comparison.

If we continue to further split this dynamical population in the manifold, it
is possible to identify more structures such as Splash (e.g., Myeong et al.
2022) and Sequoia (e.g., Feuillet et al. 2021), as shown in Figure 25.

Figure 25: Manifold (upper right), Tinsley-Wallerstein diagram (upper left),
[Mg/Mn] vs [Al/Fe] plane (lower left) and eccentricity vs mean orbital radius
(lower right) for the segregated dynamical halo populations.

For the case of Sequoia, it is typically associated with retrograde orbits,
but the chemical pattern closely represents the one found by Feuillet et al.
(2021) and the moderate eccentricities reported by da Silva & Smiljanic
(2023), which is another good example of multidimensional segregation of halo
populations using t-SNE. We have checked the $L_{z}$ behaviour of our Sequoia-
assigned population with orbits integrated with axisymmetric potentials, and
they show the same patterns presented in Figure 25, so the bar potential
cannot be the culprit. Also, our restrictive selection could have eliminated
the retrograde portion, as streams are sensitive to the spatial sampling. It
is possible that Sequoia is not exclusively a retrograde population, as result
of the selection cuts of previous studies, but instead distributed along a
broad range of positive and negative $L_{z}$.

It is clear that the manifold clumps represent the real structures that were
identified in the previous works by using bivariate chemodynamical planes, but
without relying on sharp cuts and several planes together to further filter
the sample.

###  5.6 Open and globular clusters

Both APOGEE and GALAH samples contain stars belonging to clusters, where the
open clusters are located together with the thin disk population, while the
globular clusters stand within the thick disk region, or even isolated, as for
the case of NGC 6121 (Messier 4). In Figure 26 we show the cluster locations
in the manifolds for APOGEE and GALAH.

Figure 26: Manifolds of APOGEE and GALAH with the open and globular clusters
of each sample.

The common globular clusters NGC 6397 and NGC 104 appear on the same regions
of the manifolds, which are the two extremes of the heated thick disk, while
the common open cluster NGC 2682 appears at the upper portion of the outer
disk. The other open clusters NGC 6819, NGC 7789, NGC 6253, NGC 1817 and
Collinder 261 are also located in the cold disk portions.

NGC 6121 appears isolated, and in the colour-coded manifolds of Figure 8, it
is possible to see its the yellowish hue, indicating a high eccentricity, with
apocentric radii up to 7 kpc and pericentric radii approaching the Galactic
center at $\sim$0.5 kpc. Nordlander et al. (2024) measured new abundances for
this cluster, with [Fe/H] = -1.13$\pm$0.07 and [Mg/Fe] = +0.35$\pm$0.06 in
agreement within uncertainties with our APOGEE-based values of [Fe/H] =
-1.06$\pm$0.05 and [Mg/Fe] = +0.34$\pm$0.05. Song et al. (2018) reported an
isochrone-based age between 12 and 13.3 Gyr for NGC 6121, characterising it as
a very old cluster. Vitral et al. (2023) reported structural parameters for
NGC 6121 through mass-anisotropy Jeans modelling, and found a central mass
excess of $\sim$800 M⊙, pointing towards the existence of an intermediate-mass
black hole or a supercompact black hole population. Due to its highly
eccentric orbit, metal-poor and alpha-enhanced chemistry, very old age, the
exotic central content, together with its isolated position in the Galactic
similarity context of the manifold, we suggest that NGC 6121 may have an
extragalactic origin, possibly being the remnant of an accreted satellite.

##  6 Summary and conclusions

In this paper we have presented a multivariate approach to embed a similarity
map of the local volume of the Galaxy, making possible to describe it in terms
of orbital, kinematic, chemical and age characteristics, enabling the
distinction and isolation of important structures to further study them. By
means of orbit integration of the observed Gaia data, we could validate the
Galactic bar as an indispensable feature in the study of the milky way disk,
and as an important agent in driving the structure of the spiral arms. We have
shown that the main moving groups arise from orbital resonances and
overdensities produced by the bar and spiral arms, as these structures play an
important role in star formation. The radial gradient of metallicity and
velocity dispersion could also be mapped with the help of the segragation of
populations, showing the heating of an outer disk population as a probable
effect of the perturbation by the passage of Sagittarius dSph. The older
populations of the disk, namely the canonical and heated thick disks were
clearly defined and isolated as very distinct regions from the thin disk, also
showing concentrations in the radial locations of orbital resonances. Also,
the accreted populations that compose the local dynamical halo could be easily
mapped due to their highly tilted orbits and chemical features, showing a
close connection with the heated thick disk. The spatial manifold
distribution, as a consequence of similarities, enables the visualisation of
hierarchical relationships between the disk components and their sub-
structures, giving hints about the process of formation of the milky way in a
general picture and can even locate star clusters. We also have proposed that
the Na rich population in the inner portion of the Galaxy may be the thin disk
consequence of the GSE merger event due to its timescale and chemical
patterns. Moreover, our analysis suggest that the transition zone may have
formed from the interactions with merging satellites and their gas infall.

The t-SNE technique has a myriad of applications for the most variate
purposes, and its use here can be seen as a proof-of-concept to a whole new
possibilities of big data analyses with the upcoming spectroscopic surveys
such as 4MOST, as well as the already available LAMOST and RAVE. Also, for a
more detailed analysis, one could isolate the main regions of interest and
apply t-SNE individually to each one and obtain even better separations. The
sample can be further expanded to encompass a more complete Galactic volume in
exchange for a narrower context of a single chemical survey and a slight loss
of radial velocity homogeneity, but still of great value to shed light into
the big scenario of galactic evolution.

## Acknowledgements

EC acknowledges the initial support of CAPES scholarship and the
infrastructure of IAG-USP. This study made extensive use of TOPCAT (Taylor,
2005) for acquisition, manipulation, cross-match, visualisation and selection
of the catalogues. Data was also manipulated besides orbit integration and
distance calculation with Python3 (Van Rossum & Drake, 2009) and Jupyter
notebooks (Kluyver et al., 2016) for testing with t-SNE, and figures
elaborated with Matplotlib (Hunter, 2007). We thank Rodrigo Silva, Guilherme
Limberg, Amancio Friaça and Angeles Perez-Villegas for the valuable
discussions, and the referee that heavily contributed in making this work
stronger.

## Data Availability

The data underlying this article are available in Zenodo, at
https://dx.doi.org/10.5281/zenodo.10968535. The datasets were derived from
sources in the public domain:

\- Gaia DR3: https://www.cosmos.esa.int/web/gaia/dr3

\- APOGEE DR17: https://www.sdss4.org/dr17/irspec/

\- GALAH DR3: https://www.galah-survey.org/dr3/overview/

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##  Appendix A Selection polygons of CM and Kiel diagrams

The polygons used for the RGB and MS subsample selection are shown in this
appendix as a list of vertex coordinate pairs for the RGB cuts in Table 3, and
for the MS cuts in Table 4.

Table 3: Vertex list of the polygons drawn for the selection cuts of the red giant branch in the CM and Kiel diagrams for both APOGEE and GALAH. BP-RP and G are the Gaia magnitudes, and units of Teff and log G are in K and dex, respectively. CM cut | Kiel cut
---|---
(BP-RP, G) | (Teff, log G)
(0.92, 3.90) | (3449, 0.01)
(1.11, 4.30) | (3431, 0.84)
(1.85, 5.20) | (3758, 1.52)
(3.08, 5.75) | (3839, 1.63)
(4.12, 5.39) | (4311, 2.57)
(3.66, -3.04) | (4511, 2.91)
(1.39, -4.37) | (4792, 3.64)
(0.97, -3.59) | (5037, 4.18)
(0.81, 0.00) | (5436, 4.21)
(0.88, 2.90) | (5653, 4.18)
| (5726, 3.89)
| (5681, 3.48)
| (5490, 2.88)
| (5472, 2.36)
| (5508, 2.13)
| (5381, 1.96)
| (5109, 1.66)
| (4674, 0.76)
| (3885, 0.01)
Table 4: Vertex list of the polygons drawn for the selection cuts of the main sequence in the CM and Kiel diagrams for APOGEE and GALAH separately. Only the second and third points have differences in the Kiel polygon. BP-RP and G are the Gaia magnitudes, and units of Teff and log G are in K and dex, respectively. CM cut | Kiel cut APOGEE | Kiel cut GALAH
---|---|---
(BP-RP, G) | (Teff, log G) | (Teff, log G)
(0.62, -0.15) | (3537, 4.45) | (3537, 4.45)
(0.73, 1.08) | (5700, 4.36) | (5500, 4.23)
(0.88, 3.28) | (6300, 4.20) | (6280, 4.10)
(0.89, 3.85) | (6360, 4.00) | (6360, 4.00)
(1.05, 4.88) | (5762, 3.35) | (5762, 3.35)
(1.64, 6.36) | (5989, 3.20) | (5989, 3.20)
(3.11, 9.67) | (6443, 3.30) | (6443, 3.30)
(2.22, 10.75) | (6972, 3.30) | (6972, 3.30)
(1.48, 10.21) | (8131, 3.50) | (8131, 3.50)
(0.67, 6.96) | (8131, 4.68) | (8131, 4.68)
(0.12, 3.55) | (6972, 4.68) | (6972, 4.68)
(-0.07, -0.57) | (6421, 4.83) | (6421, 4.83)
(0.46, -1.18) | (5924, 4.68) | (5924, 4.68)
| (3494, 4.85) | (3494, 4.85)

##  Appendix B Age analysis

Here we show comparisons between age estimations of APOGEE and GALAH together
with other related catalogues of ages with the same determination methods.
Also, we present comparisons and some insights about the use of chemical
clocks in age estimation, their relation with astroNN and BSTEP, and their
manifold behaviour compared to BSTEP ages.

Spina et al. (2017) found a chemical clock based on [Y/Mg] ratios to determine
solar-twin ages, which can be used together with [Fe/H] as a guide to inspect
the behaviour of the BSTEP x astroNN age relation, and more generally
bayesian-isochrone vs asteroseismology estimators. Figure 27 shows the ages of
common stars between APOGEE and GALAH.

Figure 27: BSTEP ages vs astroNN ages, colour coded by [Fe/H] in the left
panel and by [Y/Mg] (Spina et al., 2017) in the right panel.

Although there is some correlation between the ages ($r_{Pearson}=0.4$), there
is a knee-like trend with scattered metal-rich stars at older ages, probably
originating from the different methods (isochrone-based and asteroseismology)
and two different samples (Montalbán et al. 2021 for low-metallicity stars and
APOKASC-2 from Pinsonneault et al. 2018) in the APOGEE estimation. This large
deviation of metal-rich stars from the 1:1 ratio of astroNN x BSTEP, signals
that one must be cautious when using the ages at this metallicities.

We have compared our GALAH and APOGEE samples (without RGB and MS cuts) with
StarHorse ages, which uses a isochrone-based bayesian estimation similar to
BSTEP (Figure 28), hence there is a much higher correlation between StarHorse
x BSTEP ($r_{Pearson}=0.99$) ages than StarHorse x astroNN ages
($r_{Pearson}=0.67$). In both APOGEE and GALAH, the StarHorse stars with
available ages are concentrated in the turnoff region of the HR diagram, as
shown in Figure 30.

Figure 28: StarHorse vs astroNN ages in the left panel and Starhorse vs BSTEP
ages colour coded by [Y/Mg] in the right panel.

In the same line, we also show a comparison of the ages obtained from
asteroseismology in the K2 survey for a subsample of GALAH (Zinn et al., 2022)
with our common sample (Figure 29), where it shows a higher correlation with
astroNN ($r_{Pearson}=0.57$) than with BSTEP ($r_{Pearson}=0.35$). The K2
sample is concentrated in the red clump region of the HR diagram, also on
Figure 30. The knee pattern seems to arise from all the asteroseismologic age
to isochrone-based age relations present in this analysis.

Figure 29: K2 vs astroNN ages for the common stars in the left panel and BSTEP
vs K2 ages in the right panel, both colour-coded by [Y/Mg].

Figure 30: Sub-samples of stars with external age estimations and their
locuses in the HR diagram. StarHorse stars with available ages are shown in
the left panel and K2 stars are shown in the right panel.

It is clear that the different age estimation methods can lead to large biases
and deviations if directly compared, and this is one of the main reasons that
the age was not an input parameter for the t-SNE convergence. Therefore, we
only recommend comparisons between ages from the same estimation methods.

During the convergence of t-SNE manifolds, we used an intermediate GALAH
convergence to inspect several elemental ratios to compare their patterns with
the BSTEP age estimation, as shown in Figure 31. The colours along different
ratios were matched using TOPCAT’s histogram-scaling to enable a fair and
comprehensible pattern comparison and how the ratios behaved along the
manifold.

Figure 31: Preliminary t-SNE manifolds for the GALAH sample colour-coded by
several abundance ratios and BSTEP ages. In these visualisations, the outer
disk (mostly R > 8 kpc) appears as the bottom-left blob, the inner metal-rich
disk is the large central concentration, and the dynamically hot alpha-
enhanced population appears as the bottom-right concentration.

In these manifolds, by qualitative comparison, we can see that the [Y/Mg]
ratio can closely reproduce most of the features of the metal-rich moderate
and younger age (< 8 Gyr) stars, but fails for the older metal-poor alpha-
enhanced populations, underestimating the ages. The [Fe/H] fails to reproduce
the age relation in contrast with [Ba/H], which shows a very resemblant
pattern to the age in the manifold. We also found an improvement in the
agreement for metal-rich stars by subtracting [Na/Fe] from [Ba/H]. The [Ba/Eu]
relation showed the closest representation of the age pattern for the alpha-
enhanced population, but not so well for the metal-rich portion. Also, APOGEE
lacks a r-process element to normalise Ce, precluding a direct comparison with
GALAH.

As cited by Kobayashi et al. (2020), light s-process elements are also
produced by intermediate-mass asymptotic giant branch (AGB) stars, while heavy
s-process elements are mainly from low-mass AGB stars, so a possible
explanation for this direct [Ba/H]-age correlation is a continuous enrichment
of Ba from low-mass stars along the evolution of the Galaxy, as the effects of
type-Ia (main responsible for the Fe-peak enrichment, needing binary
interacting systems) and type-II supernovae are not the main players in the Ba
enrichment, with exception of the r-process contribution coming from massive
stars, where the [Eu/Fe] can be used to account for this effect and filter the
sample. Alternatively, [Na/Fe] can also be useful as a proxy to reduce this
effect since Na can be produced either by hot-bottom-burning and carbon
burning, both in massive stars (Boeltzig et al., 2016). Unfortunately, without
any r-process elements in the APOGEE sample and the [Na/Fe] not properly
measured for the metal-poor end (Figure 3), it leaves us with only Ce as an
s-process element. Since both Ba and Ce are heavy s-process elements, this
enables the use of [Ce/H] as a possible auxiliary age-tracer in the APOGEE
sample by comparing its behaviour with GALAH’s [Ba/H] and age in the resulting
manifolds presented.

It is good to remind that even though the chemical clocks may, in general, be
good age estimators in the absence of more direct measurements such as
asteroseismology, populations with anomalous abundances of heavy elements can
lead to large mistakes when estimating their ages chemically.