# oMEGACat III. Multi-band photometry and metallicities reveal spatially well-
mixed populations within $\omega$ Centauri’s half-light radius.

M. S. Nitschai Max Planck Institute for Astronomy, Königstuhl 17, D-69117
Heidelberg, Germany  N. Neumayer Max Planck Institute for Astronomy,
Königstuhl 17, D-69117 Heidelberg, Germany  M. Häberle Max Planck Institute
for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany  C. Clontz Max
Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT
84112, USA  A. C. Seth Department of Physics and Astronomy, University of
Utah, Salt Lake City, UT 84112, USA  A. P. Milone Dipartimento di Fisica e
Astronomia “Galileo Galilei,” Univ. di Padova, Vicolo dell’Osservatorio 3,
Padova, I-35122, Italy  M. Alfaro-Cuello Facultad de Ingeniería y
Arquitectura, Universidad Central de Chile, Av. Francisco de Aguirre 0405, La
Serena, Coquimbo, Chile  A. Bellini Space Telescope Science Institute, 3700
San Martin Drive, Baltimore, MD 21218, USA  S. Dreizler Institut für
Astrophysik und Geophysik, Georg-August-Universität Göttingen, Friedrich-Hund-
Platz 1, 37077 Göttingen, Germany  A. Feldmeier-Krause Max Planck Institute
for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany  Department of
Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180 Wien, Austria
T.-O. Husser Institut für Astrophysik und Geophysik, Georg-August-Universität
Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany  N. Kacharov
Leibniz Institute for Astrophysics, An der Sternwarte 16, 14482 Potsdam,
Germany  S. Kamann Astrophysics Research Institute, Liverpool John Moores
University, 146 Brownlow Hill, Liverpool L3 5RF, UK  M. Latour Institut für
Astrophysik und Geophysik, Georg-August-Universität Göttingen, Friedrich-Hund-
Platz 1, 37077 Göttingen, Germany  M. Libralato INAF, Osservatorio Astronomico
di Padova, Vicolo dell’Osservatorio 5, Padova,I-35122, Italy  G. van de Ven
Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180
Wien, Austria  K. Voggel Université de Strasbourg, CNRS, Observatoire
astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France  Z. Wang
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT
84112, USA

(Received March 28, 2024; Revised May 28, 2024; Accepted May 29, 2024)

###### Abstract

$\omega$ Centauri, the most massive globular cluster in the Milky Way, has
long been suspected to be the stripped nucleus of a dwarf galaxy that fell
into the Galaxy a long time ago. There is considerable evidence for this
scenario including a large spread in metallicity and an unusually large number
of distinct sub-populations seen in photometric studies. In this work, we use
new MUSE spectroscopic and HST photometric catalogs to investigate the
underlying metallicity distributions as well as the spatial variations of the
populations within the cluster up to its half-light radius. Based on 11,050
member stars, the [M/H] distribution has a median of $(-1.614\pm 0.003)$ dex
and a large spread of $\sim$ 1.37 dex reaching from $-0.67$ dex to $-2.04$ dex
for 99.7% of the stars. In addition, we show the chromosome map of the
cluster, which separates the red giant branch stars into different sub-
populations, and analyze the sub-populations of the metal-poorest component.
Finally, we do not find any metallicity gradient within the half-light radius,
and the different sub-populations are well mixed.

Galaxy nuclei(609) — Globular star clusters(656) — Star clusters(1567)

††journal: ApJ††facilities: VLT:Yepun (MUSE), HST (ACS)††software: Astropy
v5.2.1 (Astropy Collaboration et al., 2013, 2018, 2022), Matplotlib v3.7.1
(Hunter, 2007; Caswell et al., 2023), Pandas v1.5.3 (Wes McKinney, 2010;
pandas development team, 2020), Numpy v1.20.3 (Harris et al., 2020), SciPy
v1.10.1 (Virtanen et al., 2020), Scikit-learn v1.2.2 (Pedregosa et al., 2011)

tablenum SIXtablenum

##  1 Introduction

Figure 1: Footprint of data sets. In dark purple with the solid line is the
footprint of the new HST catalog (Paper II) and in yellow with a dashed line
is the MUSE spectroscopic catalog (Paper I) footprint. The light purple dotted
contours show the area where we have a combined catalog with 6 filter
photometry from HST. The black circle is the half-light radius at 5 arcmin
(Harris, 2010) and the red x mark is the Anderson & van der Marel (2010)
center. Background: Cut-out of a wide-field image taken with the ESO/VST
telescope (Image Credit: ESO/INAF-VST/OmegaCAM. Acknowledgement: A. Grado, L.
Limatola/INAF-Capodimonte Observatory,
https://www.eso.org/public/images/eso1119b/)

Nuclear star clusters (NSCs) are massive and compact star clusters in the
innermost region of most galaxies (see e.g. recent review by Neumayer et al.,
2020). They can be found in a wide range of galaxies, with an occupation
fraction peaking at galaxy masses in the range of
${10}^{8}\text{\,}\mathrm{M_{\odot}}{10}^{10}\text{\,}\mathrm{M_{\odot}}$
(e.g. Sánchez-Janssen et al., 2019; Hoyer et al., 2021). Due to their small
sizes (half-light radii of $1\text{\,}\mathrm{pc}10\text{\,}\mathrm{pc}$) and
sizable masses
(${10}^{6}\text{\,}\mathrm{M_{\odot}}{10}^{8}\text{\,}\mathrm{M_{\odot}}$,
Georgiev & Böker 2014; Neumayer et al. 2020), they are the densest stellar
systems in the Universe, $\geq$
${10}^{6}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-3}$ (Walcher et
al., 2005; Neumayer et al., 2020). Moreover, they have extended star formation
histories (Walcher et al., 2006; Seth et al., 2006; Kacharov et al., 2018).

Stripped nuclei are created when smaller galaxies fall into a larger one and
are disrupted by tidal forces. During this process they lose most of their
stellar content, however, similar to globular clusters, NSCs survive these
mergers because they are highly compact and dense. These stripped nuclei will
look like massive globular clusters, and hence be able to hide amongst the
globular clusters in a galaxy. Semi-analytic models predict 2 to 6 stripped
nuclei in our Milky Way halo (Pfeffer et al., 2014; Kruijssen et al., 2019).
One example is M 54, the nucleus of the Sagittarius dwarf galaxy (Alfaro-
Cuello et al., 2019, 2020; Kacharov et al., 2022); it was first discovered as
a globular cluster until the detection of the remains of the galaxy (Ibata et
al., 1994), which is undergoing tidal stripping over the last several
$\mathrm{Gyr}$ (Ibata et al., 1997; Laporte et al., 2018).

The most promising other candidate for a stripped nucleus in the Milky Way
halo is $\omega$ Centauri ($\omega$ Cen, NGC 5139), due to the complexity of
its stellar populations. It shows multiple sequences in its color-magnitude
diagram (e.g. Anderson, 1997; Bedin et al., 2004; Bellini et al., 2010; Milone
et al., 2017a) and has a large spread in metallicity (e.g. Freeman & Rodgers,
1975; Johnson & Pilachowski, 2010). In addition, the internal kinematics of
the cluster show the presence of a central stellar disk and a bias toward
tangential orbits in the outer parts (van de Ven et al., 2006), counter
rotation in the very central region (Pechetti et al., 2024), and fast-moving
stars in the inner 3 arcsec (Häberle et al., 2024b) indicating the presence of
an intermediate-mass black hole. In addition, the orbit of $\omega$ Cen has
been associated with the Gaia-Enceladus merger (Massari et al., 2019; Pfeffer
et al., 2021; Callingham et al., 2022; Limberg et al., 2022) $\sim$ 10 Gyr ago
(Helmi et al., 2018). The star formation and assembly history of the cluster
is still debated and the claims for its populations age span vary from $2-3$
Gyr (Hilker et al., 2004) up to $4-5$ Gyr (Villanova et al., 2007). Other
studies have found models fully consistent with a shorter star formation of
$1\text{\,}\mathrm{Gyr}2\text{\,}\mathrm{Gyr}$ (Joo & Lee, 2013). Finally,
tidal material associated with $\omega$ Cen is directly seen (e.g. Majewski et
al., 2012; Ibata et al., 2019, 2023) which further strengthens the stripped
nucleus scenario.

$\omega$ Cen is the brightest, most massive globular cluster in the Milky Way
($\sim$ $3.55\text{\times}{10}^{6}\text{\,}\mathrm{M_{\odot}}$, Baumgardt &
Hilker, 2018). Since it is only at a distance of $\sim$
$5.43\text{\,}\mathrm{kpc}$ (Baumgardt & Vasiliev, 2021), and not heavily
obscured by dust (Schlafly & Finkbeiner, 2011, $E(B-V)=0.12$), it provides us
with an ideal laboratory to study a NSC. Further, because the masses and star
formation histories of NSCs track the galaxies they lived in (e.g. Kacharov et
al., 2018; Sánchez-Janssen et al., 2019) and are much longer lived than
stellar streams, it can also provide us with valuable information about the
host galaxy and its merger with the Milky Way.

Figure 2: Color-magnitude Diagrams (CMDs). On the left is the full CMD of our
combined data set color-coded with the metallicity (with 145,531 stars). On
the right side is a zoom-in on the red giant branch. The gray shaded areas
indicate the regions excluded in this work and gray circle dots on the left of
the RGB are the excluded AGB and (evolved) BSSs from the RGB sample.

In our oMEGACat project, we study $\omega$ Cen in great detail with the aim to
explore its formation history and interactions with the Galaxy. To do that, we
created a spectroscopic catalog (Nitschai et al., 2023, from here on Paper I)
using the Multi-Unit Spectroscopic Explorer (MUSE), and a photometric and
astrometric catalog (Häberle et al., 2024a, from here on Paper II) using the
Hubble Space Telescope (HST) up to the half-light radius of the cluster (see
Fig. 1 for the coverage of each catalog). In this work, we make use of these
data and investigate the metallicity information we have for the stars in
$\omega$ Cen. In Section 2 we summarize the observations used to create the
catalogs and the quality cuts used in this specific work. Afterwards in
Section 3, we present the full metallicity distribution, the chromosome map
and we investigate any possible spatial variation of the metallicity. Finally,
in Section 4 we summarize and discuss our findings.

##  2 Data

###  2.1 Spectroscopic data

A detailed description of the spectroscopic data used can be found in Paper I.
In summary, the spectroscopic data were acquired with MUSE (Bacon et al.,
2010, 2014), a second-generation Very Large Telescope (VLT) instrument mounted
on the UT4 at the Paranal Observatory in Chile observing in the optical domain
($480\text{\,}\mathrm{nm}$– $930\text{\,}\mathrm{nm}$). The observations were
carried out between February 2021 and September 2022 in run 105.20CG.001 (PI:
N. Neumayer) and in addition, we also used the complementary MUSE guaranteed
time observations (hereafter “GTO data”). These data together have a full
coverage out to the half-light radius (4.65’ or $7.04\text{\,}\mathrm{pc}$,
Baumgardt & Hilker, 2018) of the cluster.

For the analysis of the data we first extracted the spectra for the individual
stars using
PampelMuse111https://pampelmuse.readthedocs.io/en/latest/about.html (Kamann et
al., 2013) and the HST catalog from Anderson & van der Marel (2010).
Afterward, we used spexxy222https://github.com/thusser/spexxy (Husser et al.,
2016) to measure the physical parameters of the stars such as effective
temperature, metallicity, and line-of-sight velocity. The observed spectra are
compared to synthetic spectra from the Phoenix library (Husser et al., 2013)
with varying log(g), $\rm T_{\rm eff}$, [M/H] while minimizing the $\chi^{2}$
difference. During the fit with spexxy, log(g) is fixed to the value provided
by the isochrone from the PARSEC database (Marigo et al., 2017) used to
estimate the initial guesses, the $\alpha$-enhancement is kept constant at
[$\alpha$/Fe]=0.3 dex, while $T_{eff}$, [Fe/H] and the line-of-sight velocity
are determined.

In addition, we performed multiple tests to check for the robustness of our
results, which included an error analysis and membership determination, as
well as necessary corrections, like the atomic diffusion correction (ADC). For
more details and the spectroscopic data themself, we refer to Paper I.

###  2.2 Photometric data

The detailed description of the photometric data is available in Paper II and
here we only briefly summarize it. The astro-photometric part of the catalog
is based on observations with the HST spanning over 20 years and using the
Advanced Camera for Surveys Wide Field Channel (ACS/WFC), and the Wide Field
Camera 3 UVIS Channel (WFC3/UVIS). The data was obtained for various general
observing and calibration programs, including a new dedicated program
(GO-16777, PI: A. Seth) aimed at providing complementary filter coverage out
to the half-light radius and additional epochs required for proper motion
measurements. All HST data used for the creation of the astro-photometric
catalog can be found under the following DOI in the Mikulski Archive for Space
Telescopes (MAST): http://dx.doi.org/10.17909/26qj-g090 (catalog
10.17909/26qj-g090).

The catalog contains both high-precision photometry spanning from the UV to
the near-infrared and proper motions for around 1.4 $\times~{}10^{6}$ stars.
The photometric measurements were performed using techniques specifically
designed to analyze crowded environments like the cores of GCs (see Bellini et
al. 2017a for details). Six filters have coverage over the full field
(ACS/WFC: F435W, F625W, F658N; WFC3/UVIS: F275W; F336W, F814W), in addition,
the central region is also covered with multiple epochs of the WFC3/UVIS F606W
filter. The proper motions were determined using the technique established and
improved in Bellini et al. (2014, 2018); Libralato et al. (2018) and Libralato
et al. (2022).

In Paper II an empirical correction for spatially dependent photometry
variations has been calculated for all seven filters, which is needed due to
differential reddening and systematic zero-point variations caused by
instrumental effects. Whenever we use this new HST photometry we add the
correction for differential reddening and instrumental effects to the
magnitudes except where explicitly noted. The coverage for the combined data
catalog with 6-filter HST photometry is shown in Fig. 1.

To combine the photometry from Paper II with the [M/H] values from Paper I, we
perform an astrometric crossmatch between the two catalogs using a matching
radius of 40 mas (equivalent to approximately one WFC3/UVIS pixel). In
addition, we require a mag difference in the F625W and F435W filters of
$\Delta$mag$<$0.1 between the new photometry and the Anderson & van der Marel
(2010) photometry used for the extraction of the spectra. This leaves us with
307,030 matched stars out of the total of 342,797 from the catalog published
in Paper I.

###  2.3 Quality cuts

As default, we assume the quality cuts for the spectroscopic catalog described
in Paper I, which are given as an extra flag in the catalog (‘Flag’). In
summary, we use stars with a minimum SNR of 10, $>$95% membership probability,
and with a relative accuracy of recovered magnitude from spectrum extraction
with PampelMuse $\geq 0.6$. We also exclude stars near the edge of the field
($<$ 5 pixel) and where the cross-correlation for the velocity is not
reliable. These cuts decrease the number of stars from 342,797 to 156,871. The
color-magnitude diagram (CMD) in the left panel of Fig. 2 shows 145,531 stars
that fulfill these criteria and are matched to the HST photometry as described
in the previous section.

As discussed in Paper I the above cuts are a compromise that allow us to keep
a high number of stars with reliable measurements. Since we want to study the
metallicity distribution, chromosome map, and spatial variations of sub-
populations we want precise and unbiased [M/H] values over our whole field of
view. Hence in addition to the default cuts, we use only stars brighter than a
general magnitude cut at $m_{\rm F625W}\leq 17$. This restriction to bright
stars aids in avoiding completeness issues of the spectroscopic data set at
fainter magnitudes (see Appendix A), getting higher precision on the [M/H]
values with a median SNR of $\sim$ 54, and removes any [M/H] biases between
the different original data sets (GTO, GO, NFM, see Appendix B), that can be
caused by different exposure times. To obtain a clean red giant branch (RGB)
sample, in a further step, we remove by eye (evolved) blue straggler stars
(BSSs) or asymptotic giant branch (AGB) stars. This selection restricts the
sample to 11,050 RGB stars (see the right panel in Fig. 2).

We do not expect a significant fraction of binaries within our sample. In
general, $\omega$ Cen has a low binary fraction, only 5% (Elson et al., 1995;
Mayor et al., 1996) and even lower in recent work at 2.70% $\pm$ 0.08%
(Bellini et al., 2017b) and 2.1 $\pm$ 0.4 % (Wragg et al. submitted).
Therefore, we can safely assume that binaries will not have a significant
effect on our results, especially since Wragg et al. find similar binary
fractions for all stellar evolutionary stages using the GTO subsample of the
oMEGACat catalog (Nitschai et al., 2023). Only the BSSs show an enhanced
binary fraction ($>20\%$). Because of the higher binary fraction among BSSs
and the chemically peculiar atmospheres of hot stars due to diffusion (at
$T_{\rm eff}\gtrsim$ 7,800 K, Lovisi et al., 2013), we exclude all stars bluer
than the main RGB track (see gray areas in Fig. 2.)

$1$ Name Number of stars Description MUSE 342,797 spectroscopy Paper I HST
matched 307,030 spectroscopy with astro-photometry Paper II MUSE QC 156,871 QC
of spectroscopy HST QC 145,531 HST matched with QC MUSE RGB 11,050 HST QC and
$m_{\rm F625W}\leq 17$ RGB phot 10,850 RGB that has a measurement in F625W,
F435W, F275W, F336W and $m_{\rm F814W}\leq 17$

Table 2: Data samples

Note. — QC: quality cuts described in Paper I and beginning of Section 2.3.
The RGB phot sample is needed to have RGB stars in all filters required for
the Chromosome map, see Section 3.2.1.

With these cuts, we get a mean [M/H] error of 0.041 dex. Further, by comparing
our MUSE data to the photometric catalog used for the extraction of the
spectra (Anderson & van der Marel, 2010) we find a completeness $>$80% in all
individual cubes, with a median completeness at 17 mag of 97%. We spatially
bin the sample stars and find no significant variation in the mean magnitude
with radius or at any point within the field of view. This suggests we can
make spatial comparisons with minimal concern about instrumental issues.

Note that all metallicities unless otherwise indicated include the ADC
correction, we always use the scaled errors for the metallicity as described
in Paper I Section 4.2 and the new photometry from Paper II.

In the following section, we will investigate the metallicity information we
have for the cluster in detail using the above quality cuts. All the data
samples described in this section are summarized in Table 2, in Section 3 we
will use either the RGB or RGB phot sample.

Figure 3: Metallicity distribution. The figure on the left shows the
normalized distribution of [M/H] values in the spectroscopic catalog (Paper I)
using the default quality cuts in the black line (MUSE QC), using the extra
condition of a minimum brightness of 17 mag in the gray shaded histogram (RGB)
and the purple line is the KDE estimator for that distribution. The small
horizontal lines on the left side show the bandwidth, 0.030 dex, for the
distributions, which are close to the mean errors of the [M/H] values. On the
right, we show the same distribution (RGB) as probability density but fitted
with a Gaussian mixture model, shown as a black solid line, with each
component (11 in total) shown as a blue dashed line. Note that one component
has a small amplitude and is not visible in this plot.

##  3 Results

###  3.1 Metallicity distribution

First, we analyze the total metallicity distribution of the 156,871 stars
(MUSE QC) passing the default criteria, see the left side of Fig. 3. The
additional magnitude-cutoff ($m_{\rm F625W}\leq$17 mag) leaves us with 11,050
stars (RGB) and decreases the more extended tail to lower values (below
$\sim-2$ dex) but also shifts the peak of the distribution to lower values
since on average the metal-rich stars have a main-sequence turn-off at fainter
F625W magnitudes and hence more metal-rich stars are removed. The cut however
ensures that both GTO and GO data sets have a similar distribution and no bias
is caused due to different data sets, see Appendix B. We also used a Kernel
Density Estimator (KDE) to estimate the probability density of the
distribution, which we normalized to one to plot it onto the same figure. In
this distribution multiple peaks are visible and we will identify them later
in this section.

The mean [M/H] value is at ($-1.550\pm 0.002$) dex and the intrinsic standard
deviation is ($0.255\pm 0.002$) dex, which is close to the mean value of the
distribution with no magnitude cut ($-1.53$ dex). The median is ($-1.614\pm
0.003$) dex with half of the differences between the 16th and 84th percentile
being ($0.221\pm 0.003$) dex. The median of the overall distribution without a
magnitude cut is $-1.54$ dex, which is closer to the mean value since more
metal-rich stars are included. All the values and errors (68%) were calculated
with bootstrapping.

The range in metallicity for stars brighter than 17 mag reaches from $\sim$
$-2$ dex to almost $-0.5$ dex. In detail, 68% of the stars are between $-1.77$
dex and $-1.33$ dex and 99.7% between $-2.04$ dex and $-0.67$ dex. This huge
metallicity spread is, as mentioned previously, an indication of multiple
stellar populations, which indeed are visible as distinct sequences in the
color-magnitude diagram (CMD, see Fig. 2).

All our results are comparable to the values in Husser et al. (2020) where in
a smaller sample of 1,247 stars they found a mean value of $-1.50$ dex, a
median of $-1.65$ dex and a range from $-1.82$ to $-1.31$ dex for the 1st and
3rd quartiles. However, since our sample is 10 times larger we have even more
accurate values with small uncertainties.

We also used a Gaussian Mixture Model (GMM) to describe our distributions and
the result can be seen on the right side of Fig. 3 and in Table 3. We find
that the best number of Gaussians to describe the distribution is eleven using
the Bayes Information Criterion (BIC), but eight components would give an
almost equally good fit. However, one component is not visible in the Figure
(index 11 in the Table) since it has a very small amplitude compared to the
rest and would not count as a separate population but is needed to describe
the distribution in the model. Similarly, component 7 is needed for the
extended tail towards lower metallicities but together with components 1, 2,
and 4 they are not distinguishable by eye in the highest peak of the
distribution. Components 3, 8, 9, and 10 can be seen by eye as smaller peaks,
and components 5, and 6 are more difficult to distinguish but are still
visible.

$2$ index Mean Intrinsic standard deviation Fraction of stars # [dex] [dex] %
1 -1.779 0.030 18.2 2 -1.715 0.030 16.0 3 -1.553 0.036 13.8 4 -1.642 0.032
13.4 5 -1.454 0.043 11.8 6 -1.348 0.049 9.7 7 -1.849 0.048 4.6 8 -1.196 0.063
4.3 9 -0.828 0.082 4.2 10 -1.039 0.058 3.8 11 -2.033 0.210 0.3

Table 3: Multi-Gaussian Components of the metallicity distribution

Note. — The index # is assigned from the highest to the lowest fraction of
stars.

We have compared our metallicity findings with previous studies already in
Paper I where we made a one-to-one comparison. Also, the overall distribution
agrees with previous studies (e.g. Johnson & Pilachowski, 2010; Mészáros et
al., 2021; Alvarez Garay et al., 2024), even though we find more peaks in our
distribution. Johnson & Pilachowski (2010) find 5 peaks in the distribution at
$-1.75,-1.50,-1.15,-1.05$ and $-0.75$ dex, while Mészáros et al. (2021) and
Alvarez Garay et al. (2024) find four peaks, $-1.65,-1.35,-1.05$ and $-0.7$
dex, and $-1.85,-1.55,-1.15$, and $-0.80$ dex respectively. All these peaks
are in agreement with at least one of our peaks within the 3 $\sigma$ range.
The difference in the number of peaks can be explained due to the difference
in the data, specifically the amount of data, since it can make it difficult
to identify and separate peaks in the distribution with a lower amount of
stars. All the above-mentioned studies have stars that reach larger radii than
ours, but have fewer than 1,000 stars, while we have more than 10 times their
number which allows us to distinguish different metallicity components more
easily. However, even in our case, one could reduce the number of components
to five if one only looks at the distribution by eye, since peaks 1, 2, 4, and
7 can be taken as one, and components 5 and 6 or 8 and 10 are not that clearly
visible peaks in the overall distribution and component 11 is not at all
visible. Even with the GMM, we can see that eight components would give an
almost as perfect fit to the distribution and that small variations in the
sample of stars can also change the preferred number of Gaussians. This
demonstrates the advantage of our huge data set that allows for much more
detailed studies and analysis.

Figure 4: Chromosome map of $\omega$ Cen. The chromosome map of $\omega$ Cen
is color-coded with metallicity. One can identify different sub-populations in
the map but since we do not aim to separate them precisely in this work we
also do not draw the line separating first and second-generation stars as it
is usually done in globular clusters.

In Husser et al. (2020) they identify nine sub-populations and give their mean
metallicity values. Our multi-Gaussian Mixture Model does not separate the
stars in different sub-populations, since also photometric analysis needs to
be taken into account. However, we can compare their position on the
chromosome map in Section 3.2.1 and identify which component belongs to which
sub-population (P) identified in Husser et al. (2020). P1, P2 and P3 (mean
[Fe/H]= $-1.83,-1.80,-1.74$ dex) would correspond to our components 7, 1 and
2, P5 ($-1.24$ dex) to component 6, P8($-1.50$ dex), P6($-1.47$ dex) and
P4($-1.50$ dex) to components 3 and 5. Our component 6 could be part of P4 or
P3. From the chromosome map, we can identify component 9 as P7 ($-0.18$ dex)
and components 10 and 8 as P9 ($-0.72$ dex) but the metallicity values
disagree the most with our findings. However, they have a lot fewer stars than
we have especially since their P7 and P9 are not well populated. With our
data, they can be much better studied and separated. In the following
sections, we investigate a bit further different sub-populatins. We do not aim
to precisely separate and identify all sub-populations in detail, this will be
done in future work (Clontz et al., in prep). Nevertheless, this analysis
already shows how our huge data set can improve sub-population studies.

Lastly, in Johnson et al. (2020) they find two very metal-poor populations,
below $-2$ and $-2.25$ dex. Our component 11 contains 30 stars at the very low
end of the metallicity distribution reaching as low as $-2.738$ dex and as
high as -1.990 dex but only 4 stars above $-2.25$ dex. However, this component
is just a tiny fraction of the whole sample and its stars do not cover the
whole CMD range (most of them lie close to magnitude 17 mag), indicating that
they are not a separate population and are not the same stars as in Johnson et
al. (2020) where most of these stars lie on the brighter RGB. Hence, our
component 11 stars are individual stars with special [M/H] values either due
to measurement problems or they could be binary stars. Overall, we can
conclude that we do not find the most metal-poor population from Johnson et
al. (2020) in our sample, even though we have a few low metallicity stars.

###  3.2 Photometry and Metallicity

####  3.2.1 Chromosome map

In this subsection, we focus again on RGB stars with $m_{\rm F625W}<17$ mag.
In addition, we require the stars to have accurate photometric measurements in
5 different HST filters (F275W, F336W, F435W, F625W, and F814W) in the
photometric catalog of Paper II. These requirements leave us with a sample of
10,850 stars (RGB phot).

Almost all globular clusters host significant light-element abundance spreads
within them, following a basic pattern, a population with abundances akin to
field stars, and one or more anomalous populations, with varying enrichment in
some elements (e.g., He, N, Na) and depletion in others (e.g., C, O), but the
specifics of each cluster are unique (see the review by Bastian & Lardo,
2018). These abundance spreads are typical for old and massive clusters and
are known as multiple populations. However, abundance variations in Fe and
heavy elements are rare and only have been detected in more complex clusters
such as $\omega$ Cen (Willman & Strader, 2012).

The chromosome map is a photometric diagram used to identify multiple stellar
populations in globular clusters. It uses multi-color photometric information
that is sensitive to element abundance variations to characterize the presence
and diversity of multiple populations (Milone et al., 2017b). Many studies
have provided accurate investigations of the sub-populations of $\omega$ Cen
(e.g. Tailo et al., 2016; Latour et al., 2021; Husser et al., 2020). Here, we
take advantage of an unprecedented dataset that comprises photometry and
spectroscopy of more than 10,000 RGB stars to better investigate them.

To construct the chromosome map, we follow the technique from Milone et al.
(2017a) and use the filters F814W, F336W, F275W, and F435W. In Milone et al.
(2015a, b) it has been shown that the combination of the pseudo-color $C_{\rm
F275W,F336W,F438W}$ (in our case $C_{\rm F275W,F336W,F435W}$) (Piotto et al.,
2015) with the $m_{\rm F275W}-m_{\rm F814W}$ color maximizes the separation
between stellar populations along the main sequence and the RGB. Here we
analyze only the RGB populations, deferring a more comprehensive view of sub-
populations and their exact separation in the cluster to a later paper (Clontz
et al., in prep).

In detail, the pseudo-color $C_{\rm F275W,F336W,F435W}$ is selected since it
is sensitive to the degree of CNO processing in the multiple populations. This
is thanks to the fact that the F275W filter includes the OH molecular band,
the F336W the NH band, and the F435W the CH and CN bands (e.g., Milone et al.,
2015a). Simultaneously the F275W and F814W filters provide a wide color range
being sensitive to the effective temperature and thus to different metallicity
or helium abundances, since helium enhanced stars are hotter at the same
luminosity (e.g., Milone et al., 2015a, 2012).

The detailed steps to calculate the $\Delta$ values can be found in Appendix
C. In general, the following relations are used:

$\displaystyle\Delta_{\rm F275W,F814W}=W_{\rm F275W,F814W}\cdot$
$\displaystyle\frac{X-X_{fiducialR}}{X_{fiducialR}-X_{fiducialB}}$ (1)

and

$\displaystyle\Delta_{\rm F275W,F336W,F435W}=W_{\rm F275W,F336W,F435W}\cdot$
$\displaystyle\frac{Y_{fiducialR}-Y}{Y_{fiducialR}-Y_{fiducialB}}$ (2)

with fiducialR the redder fiducial line and fiducialB the bluer line in color
space. Further, $X=m_{\rm F275W}-m_{\rm F814W}$, $Y=C_{\rm F275W,F336W,F435W}$
and the pseudo-color $C_{\rm F275W,F336W,F435W}=(m_{\rm F275W}-m_{\rm
F336W})-(m_{\rm F336W}-m_{\rm F435W})$. The final chromosome map is shown in
Fig. 4. The few stars lying on the very blue side of the chromosome map are
most likely remaining evolved BSSs or/and binaries (see Marino et al., 2019a;
Martins et al., 2020; Kamann et al., 2020) that have not been removed from our
RGB sample because they lie close to the bluest RGB stars. Including the
bright (evolved) BSSs (gray shaded points in Fig. 2) would create a more
extended tail to bluer $\Delta_{\rm F275W,F814W}$ values in the chromosome
map.

Figure 5: $\Delta$ values as a function of [M/H]. The chromosome $\Delta$
values, left metallicity against $\Delta_{\rm F275W,F814W}$ and right
$\Delta_{\rm F275W,F336W,F435W}$ plotted against metallicity in a density
histogram. On the left, the red line separates the two streams visible and the
black line indicates the metal-rich stars that are excluded from the analysis
in Section 3.4.3.

The x-axis of this chromosome map is mostly sensitive to stellar populations
with different helium (He) content and metallicity (Fe), whereas the y-axis is
mostly efficient in identifying stellar groups of stars with different
nitrogen (N) abundance (see e.g. Figures 25 and 27 in Marino et al. 2019b and
Figure 5 in Milone et al. 2020, for details). In a typical GC, what is known
as first (1G) and second-generation (2G) stars would lie around $\Delta_{\rm
F275W,F814W}\sim 0$, with 1G being centered at the origin (0,0) and the 2G
stars found above ($\Delta_{\rm F275W,F336W,F435W}>0$). In $\omega$ Cen the
most metal-poor stars follow this pattern, while higher metallicity stars have
different tracks at $\Delta_{\rm F275W,F814W}>0$. Further abundance studies
will help us distinguish the different sub-populations even better.

####  3.2.2 Metallicity dependency

Since we have the metallicity information for all these stars we can
investigate the relationship between the $\Delta$-values and [M/H], see Fig.
5. $\Delta_{\rm F275W,F336W,F435W}$ vs [M/H] shows as expected a similar
structure to the chromosome map since $\Delta_{\rm F275W,F814W}$ tracks the
[M/H] variations. Specifically [He/H] and [Fe/H] variations, are the main
cause for the horizontal displacement in the chromosome map (see Figure 5 in
Milone et al. 2020 and Figure 27 in Marino et al. 2019b).

Interestingly, we notice two distinct sequences or streams in the $\Delta_{\rm
F275W,F814W}$ vs [M/H] diagram, which correspond to stars with different
light-element abundances (see also Fig. 11 in Section 3.4.3). Specifically,
the upper stream is composed of the most extreme stars in chemical properties,
enhanced in [Na/Fe], [Al/Fe] and depleted in [O/Fe], [C/Fe] with respect to
the lower stream stars with the same metallicity (Marino et al., 2019b; Milone
et al., 2020). It is also known that the vertical displacement in the
chromosome map is mainly caused by [N/Fe] (e.g., Marino et al., 2019b; Milone
et al., 2020), hence stars located in the upper stream have higher [N/Fe]
abundance than stars in the lower stream. In addition, He-enhanced and Mg-
depleted stars are expected in the upper stream (Milone et al., 2015b, 2018;
Marino et al., 2019b; Milone et al., 2020). Further, Fe enrichment occurs in
both streams, the upper has a slightly higher peak (Marino et al., 2019b),
which we will also show in Section 3.2.3 and Section 3.4.3.

In the future we are aiming to perform a detailed study on the abundances of
individual stars (Wang et al. in prep.), using DD-Payne (Ting et al., 2017;
Xiang et al., 2019) and further investigate the effect the abundance
variations have on the chromosome map. That will allow us to identify the main
cause for the two sequences seen in the left panel of Fig. 5. We are also
working on combining stellar spectra of the same subpopulations to identify
the overall abundance differences between subpopulations (Di Stefano et al. in
prep.), which will also help us tackle the cause of the two sequences.

Figure 6: Multiple populations in the metal-poor stars. On the left is the
chromosome map only for the metal-poor stars with the probability density
histograms of the $\Delta$ values. There we can identify 3 peaks in the
$\Delta_{\rm F275W,F336W,F435W}$ histogram and plot their metallicity
distribution on the right. The metallicity distributions for the populations
indicated with the red and black lines are very close, while the population
plotted in yellow exhibits a higher metallicity.  Figure 7: Metallicity maps.
(a.) The LOESS smoothed metallicity map for visualization only, (b.) the
median map from the 200 nearest neighbors, and (c.) the 68% dispersion map
from the 200 nearest neighbors. In (b.) the black dashed circle in the left
corner shows the mean radius of the 200 neighbors. The pointing structure from
MUSE is shown with faint squares, the red elliptical annuli are the bin edges
used in Section 3.4, and the minor axis of the elliptical annuli corresponding
to the rotation axis is shown as a black dashed line.

####  3.2.3 The metal-poor component and its multiple populations

The most metal-poor stars ([M/H] $\lesssim-1.60$ dex) of the cluster follow a
similar trend as the stars of mono-metallic globular clusters that separate in
the so-called 1G and 2G stars along the $\Delta_{\rm F275W,F336W,F435W}$ axis
due to light-element variations.

To identify these metal-poor stars in the chromosome map, we used stars in the
metal poorest components of the GMM (Index 1, 2, 4, 7, 11). Each star is
assigned to the component that has the highest contribution to the GMM at the
star’s metallicity. This creates sharp metallicity cutoffs for the GMM
populations. Then we remove blue or red outliers in the chromosome map (see
the left panel in Fig. 6) for the remaining stars.

Figure 8: Overall metallicity gradient. Black are all the stars used for this
analysis red is the median value for each bin and blue is the mean value. The
dashed lines show the total median or mean value for the whole sample. The
left panel shows the whole [M/H] range while the right side is a zoom into
metallicities around the mean and median.

If we only look at those stars, we can see that they also have a metallicity
spread reaching from $-1.82$ to $-1.68$ dex for 68% of the stars, see Fig. 6.
With a simple Gaussian Mixture Model fit we identify 3 peaks (using BIC) in
the $\Delta_{\rm F275W,F336W,F435W}$ distribution, separating them into three
different populations. We see that the third peak (highest $\Delta_{\rm
F275W,F336W,F435W}$ values) has the highest metallicity (similar to the
findings in Marino et al., 2019b) and that all have a significant width. We
investigate that further by looking at their metallicity distribution. The
yellow peak has the highest mean metallicity at $-1.721$ dex and an intrinsic
standard deviation of $\sigma$ = 0.065 dex, the red is at $-1.762$ dex and
$\sigma$ = 0.071 dex and the black one is at $-1.762$ dex and $\sigma$ = 0.067
dex. They have similar standard deviations [M/H] but the yellow peak is offset
to higher [M/H], with a p-value below 1% for the Anderson-Darling and
Kolmogorov-Smirnov tests. The red and black distributions have higher p-values
$>$25% meaning that they are consistent to be drawn from the same
distribution. Overall, even in this one metallicity component, there is a
spread in [M/H] and it rises to higher [M/H] for higher $\Delta_{\rm
F275W,F336W,F435W}$ values as seen also in Marino et al. (2019b). However,
since the difference is small, even though it seems to be significant (see
also Fig. 9), we do not have pure iron abundance but [M/H] which could be
increased with enrichment in Na and Al which is expected for the black and
even stronger for the yellow peak.

Higher $\Delta_{\rm F275W,F336W,F435W}$ also mean more enhanced Na and Al
element abundances and depleted in O and Mg, as explained in Section 3.2.2 and
hence more He-rich stars (Milone et al., 2015b, 2018; Marino et al., 2019b).
Therefore we will call stars belonging to the red peak Na-poor metal-poor (Na-
poor MP) population, the yellow peak Na-rich metal-poor (Na-rich MP)
population, and in the black peak are stars with intermediate Na enrichment.

There have been studies identifying sub-populations in $\omega$ Cen on the RGB
(e.g Lee et al., 1999; Pancino et al., 2000; Sollima et al., 2005; Husser et
al., 2020) and the main sequence (MS) (e.g. Bedin et al., 2004; Bellini et
al., 2010; Milone et al., 2017a; Bellini et al., 2017b). However, connecting
the RGB and main-sequence stars into the same subpopulations is a complex task
that has not been done so far since the subpopulations overlap and cross each
other below the turnoff point in the CMD. This will be the focus of future
studies (Clontz et al. in prep).

###  3.3 Spatial 2D Metallicity distribution

We analyze the 2D spatial distribution of the metallicity since 2D variations
might average each other out in radial bins.

We create smooth maps that still show details using the median of the 200
nearest neighbors (see Fig. 7) and for visualization only the locally weighted
regression (LOESS) technique (with a regularization factor f=0.1, Cappellari
et al., 2013). We choose 200 nearest neighbors since that gives a smooth map
without patches also in the velocity (similar to Pechetti et al., 2024). In
Fig. 7 we can see that indeed there is some structure, almost ring-like,
consistently visible in both the LOESS and median map. In addition, we also
plot half the difference of 68% of the 200 nearest neighbors as an indication
of the dispersion of the [M/H] values. Regions with lower median metallicity
in the outskirts of the field have a relatively narrow metallicity
distribution, while some of the structures with higher median metallicity also
have a higher dispersion. However, there are also regions with moderately high
median [M/H] and a narrow [M/H] distribution. In general, the variation in the
dispersion map does not perfectly match the ring-like structure of the median
metallicity map. Further, the pattern does not follow the MUSE pointing
structure, reassuring us that the structure is not caused by calibration
differences. We also verified that the structure does not appear if we take
the ratio between the redmost RGB branch (metal-rich stars) and the blue RGB
(metal-poor stars), which suggests that the structure is not caused by biases
in our observations or selection of the sample in specific regions. However,
this also indicates that the structure is not caused by the most metal-rich
population but by some other intermediate [M/H] populations that might have
been accreted later in its evolution and not yet well mixed.

###  3.4 Metallicity gradients

####  3.4.1 Overall cluster gradient

Since the cluster is elongated in the plane of the sky, we do not use circular
radial bins for further analysis but elliptical bins. We use $100^{\circ}$ for
the positional angle (PA) (van de Ven et al., 2006; Kamann et al., 2018), 0.07
for the ellipticity (median value from Geyer et al., 1983; Pancino et al.,
2003; Calamida et al., 2020, inside 5 arcmin) and the equivalent radii for the
bins are (0.5, 1, 1.5, 2, 3, 4, 7) arcmin. These bin edges are overplotted in
red in Fig. 7. In all the following analyses we calculate the median circular
radius in each of the elliptical bins and show that in the figures.

Figure 9: Chromosome map showing the four different subgroups considered in
Section 3.4.2. Red are the metal-rich stars, green the intermediate
population, cyan the metal-poor but Na-rich, and blue the metal- and Na-poor
stars.

Figure 10: Spatial Distribution of the subgroups. The left side shows the
cumulative distribution of the four subgroups, the labels show the median
[M/H] value. On the right is their metallicity versus radius.

In Fig. 8 we plot the mean and median metallicity value for each bin with the
99.7% as 3$\sigma$ error bars, derived with bootstrapping. We can conclude
that there is no significant gradient within the half-light radius of the
cluster. However, there is an indication of the ring we see in the 2D
distribution (Section 3.3), in the bins between 2-4 arcmin, where the median
[M/H] is almost 3$\sigma$ (99.7%) above the global median [M/H] of the
cluster. Also, the last bin lies just outside the 3 $\sigma$ range below the
overall median value. We see these same trends in the mean metallicity
profile. The trend is not as strong as in the 2D maps since it is not a
perfect ring at some lower and higher [M/H] areas cancel each other out when
looking at the radial profile, but the signature is still visible.

Previous studies (e.g. Sollima et al., 2007; Bellini et al., 2009; Calamida et
al., 2017, 2020) find spatial differences in sub-populations of the MS and the
RGB. The bluer MS (associated with the intermediate metallicity RGB by Piotto
et al. (2005)), which is thought to be He-enriched (Bedin et al., 2004;
Norris, 2004; Piotto et al., 2005) is more centrally concentrated than the red
MS (associated to the metal-poor RGB), but increasing towards the outer parts,
above 25 arcminutes. While for the RGB the rich and intermediate follow the
same distribution and are more centrally concentrated with a decline towards 8
to 10 arcminutes (Bellini et al., 2009). The last bin in our metallicity
gradient and lower metallicity towards the outer parts of the 2D map (see
Section 3.3) indicate a similar trend with the more metal-rich or intermediate
populations declining at the edge of the half-light radius. In Calamida et al.
(2017, 2020) they show that the more metal-poor stars follow the cluster
elongation and are more numerous in the northern half, while the more metal-
rich are elongated in the northeast-southwest direction and are more numerous
in the eastern half. This trend is not visible in our data in the 2D
distributions.

In general, for all these previous studies the spatial differences are
prominent at the edge or after the half-light radius which is outside of our
data range. From our analysis, inside the half-light radius, the sub-
populations seem to be well mixed in radial bins.

####  3.4.2 Spatial Differences between sub-populations

Further, we want to investigate if there is a difference in the spatial
distribution of different sub-populations. To this aim, we use four different
subgroups, identified from the GMM and the chromosome map as follows. Na-rich
but metal-poor (Na-rich MP), Na-poor, and metal-poor (Na-poor MP) populations
were identified in Section 3.2.3, then we further select one intermediate
metallicity (Int) population, index 7 in the GMM, and constrain it to the
stars belonging to the clear overdensity in the chromosome map. We emphasize
however that this is not the only intermediate metallicity population but one
of many and we only selected this one because it is easily distinguishable in
the upper sequence of the Chromosome map. Finally, the metal-rich population
(MR) includes the most metal rich stars with a visible separate sequence in
the chromosome map. See Fig. 9 for their location on the chromosome map.

This identification of four sub-populations is a first selection of some
easily detectable sub-populations in the chromosome map that span a wide
spread in [M/H] and are far apart in the chromosome map. We do not aim to find
all subpopulations but to select a few to check their distribution over the
field of view. We plan to use all available information and machine learning
techniques to classify the subpopulations in an upcoming paper (Clontz et al.
in prep.).

The cumulative radial distribution and metallicity gradient of each sub-
population are shown in Fig. 10. The Anderson-Darling test gives a p-value
higher than 25%, so the null hypothesis is not rejected and the sub-
populations could have the same underlying distribution. The only exception is
between the Na-poor MP or Na-rich MP and the Int populations, the p-values are
lower ($\sim$ 8%), however still too high to reject the null hypothesis.
Future more detailed separation between sub-populations might reveal stronger
differences. The populations do not show any significant metallicity gradient,
see Fig. 10 right panel. The right panel also confirms our finding in Section
3.2.3 that the metallicity difference between Na-rich and Na-poor metal-poor
populations is significant (outside the 3$\sigma$ range) for all radii, except
for the first bin, where due to the low number of stars the error bars are
larger. The sub-populations, depending on metallicity or Na-enrichment, we
selected are all well-mixed.

Figure 11: The two streams in [M/H] vs $\Delta_{\rm F275W,F814W}$. The top
panel shows the chromosome map and the bottom panel the $\Delta_{\rm
F275W,F814W}$ vs [M/H]. We color-coded the stars depending on whether we
identified them as belonging to the upper or lower stream in the original Fig.
5.

####  3.4.3 Differences in the two streams of [M/H] vs $\Delta_{\rm
F275W,F814W}$

In the previous section, we chose four sub-populations with different
metallicities and Na-enhancement and found no significant spatial difference.
To further check the difference between sub-populations with different
enhancement in light elements, we use the two streams seen in [M/H] vs
$\Delta_{\rm F275W,F814W}$ (see Section 3.2.2 and Fig. 11). We exclude the
metal-rich stars (black line in Section 3.2.2) since they lie in between the
sequences. The cumulative distribution is shown in the left panel of Fig. 12
and they do not differ significantly, Kolmogorov-Smirnov and Anderson-Darling
tests show 10% and 8% p-values confirming that the null hypothesis is not
rejected for their spatial distribution. However, the p-values are lower than
for the subpopulations indicating that there might be differences that could
get stronger towards larger radii (similar to the difference of the blue and
red MS Sollima et al. (2007); Bellini et al. (2009)).

Their metallicity, however, is different with the lower sequence having a
median metallicity higher than the upper sequence. That is not unexpected
since most of the metal-richer stars lie on the upper stream and slightly
higher iron values are expected in that stream as discussed in Section 3.2.2.
However, again no significant metallicity gradient is visible (see right panel
Fig. 12).

Figure 12: Spatial Distribution of the two streams in [M/H] vs $\Delta_{\rm
F275W,F814W}$. The left side shows the cumulative distribution of the two
sequences. On the right is their metallicity versus radius.

##  4 Conclusion

We present a detailed study of the metallicity distribution of 11,050 RGB
stars in $\omega$ Cen with a mean error of 0.041 dex and a median SNR$\sim$
54\. We find a mean value of $-1.550$ dex and a median value of $-1.614$ dex
spanning 0.44 dex for 68% of the stars with $m_{\rm F625W}<17$ mag. Further,
we identify multiple peaks in the metallicity distribution indicating
different metallicity subgroups.

Additionally, we investigate the chromosome map and its dependencies with
metallicity. We looked in detail at the most metal-poor group and identified
three populations that show a similar spread in [M/H] $\sim$ 0.07 dex while
the [M/H] increases for higher $\Delta_{\rm F275W,F336W,F435W}$.

We also studied the 2D spatial distribution of the [M/H] values and created
metallicity maps showing a ring-like structure with higher values, while
inside and outside the [M/H] values drop. Finally, we check for metallicity
gradients and spatial differences in radial bins for different subgroups of
stars. There we find no gradient in [M/H] over our field of view.

Overall, the different [M/H] populations seem to be well mixed and no strong
gradient is visible in their radial profiles. However, more information on the
abundance might help better separate them and further investigate their
differences. In addition, the different sub-populations may be well mixed
spatially, but their kinematics may be used to separate them, as it takes
longer to erase the stars’ memory of their original orbits.

We plan to determine elemental abundances (Di Stefano et al., in prep; Wang et
al. in prep), and identify sub-populations (Clontz et al., in prep.) and ages
(Clontz et al., in prep). This will bring us closer to uncovering the
formation history of $\omega$ Cen, the nearest nuclear star cluster.

## Acknowledgment

We thank the anonymous referee for the helpful and constructive comments. The
work was based on observations collected at the European Southern Observatory
under ESO program 105.20CG.001. Further, based on archival and new
observations with the NASA/ESA Hubble Space Telescope, obtained at the Space
Telescope Science Institute, which is operated by AURA, Inc., under NASA
contract NAS 5-26555. M.A.C. acknowledges the support from FONDECYT
Postdoctorado project No. 3230727. SK acknowledges funding from UKRI in the
form of a Future Leaders Fellowship (grant no. MR/T022868/1). A.B.
acknowledges support from HST grants GO-15857 and GO-16777. M.L. acknowledges
funding from the Deutsche Forschungsgemeinschaft (grant LA 4383/4-1). A.F.K.
acknowledges funding from the Austrian Science Fund (FWF) [grant DOI
10.55776/ESP542].

Figure 13: Completeness and magnitude. Left plot: Each line shows the
completeness fraction of one pointing compared to the HST catalog from
Anderson & van der Marel (2010) and the thick black line shows the median
completeness fraction. The vertical solid line is at 17 mag and the dotted one
at 18.86 mag (for GO 50% completeness on average). The dashed horizontal lines
are at 80% and 90%. Right plot: the magnitude of the stars above 17 mag versus
radius, which shows that for our quality cuts, there is no bias or gradient
for specific magnitudes.

##  Appendix A Completeness

In Paper I we provided the completeness compared to the Anderson & van der
Marel (2010) catalog and found that at a magnitude of 18.86 mag in the F625W
filter, we have 50% completeness. For our study, we require higher
completeness to avoid biases that could cause gradients or spatial variation
in metallicity to vary. A magnitude cut of 17 mag gives us over 80%
completeness for all fields and even reaches 90% for the majority, see Fig. 13
left plot.

To further ensure that we do not bias our sample we check the radial
dependency of the magnitude, which should be uniform. Indeed Fig. 13 (right)
shows that the mean and median values remain constant over our whole field of
view, not showing any crowding effects, for the RGB stars.

##  Appendix B [M/H] bias

We noticed a metallicity bias in the different data sets, GO, GTO, NFM, when
using all available magnitudes. In Fig. 14 we can see that the [M/H] has
always a trend to go to higher metallicities for lower SNR, but it differs
between data sets. In general, the GTO stars with fainter magnitudes have
higher [M/H] and lower SNR than the GO, which is likely caused by differences
in the observing runs (e.g. different exposure times and number of exposures).
However, the trend that stars with higher [M/H] have lower SNR is true for all
stars and could be because they are redder, on average fainter (in general
lower SNR), and have more absorption lines that make them look noisier to our
spectral fitting routine compared to more metal-poor stars. When excluding
stars fainter than 17 mag in the F625W magnitude we remove most of the stars
with SNR $\leq 20$ where the bias is visible, making our sample consistent
between GTO and GO observations.

Figure 14: Metallicity bias. We plot the metallicity versus the SNR for the
different data sets underlying our spectroscopic catalog. Red dots are the
mean values for each bin and the black solid line is the mean value between 30
and 40 SNR (indicated with blue dashed lines).

Additionally, we check if the distribution of the stars in GO and GTO are
different, see Fig. 15. When using all magnitudes the GO distribution is
broader reaching lower metallicities which would cause the GTO fields always
to be on average more metal-rich. However, when applying the magnitude cut the
distributions agree well not having any systematic difference.

Figure 15: Metallicty distribution GO and GTO. Using all magnitudes the two
different data sets have important differences in their distributions.
However, with a magnitude cut these disappear and the distributions look
almost identical.  Figure 16: The three subsamples used to calculate the
chromosome map. a. The chromosome map of $\omega$ Cen, color-coded with [M/H]
only for the stars in the three subsamples. b. The CMD of the subsamples with
the fiducial lines for each. Blue and red are the 96th and 4th percentile for
the poor population (blue crosses), green is the median line for the
intermediate (green crosses), and yellow is the median line for the rich
population (red crosses). c. The pseudo-CMD of the subsamples with blue
crosses being the poor and intermediate population and red the rich
population. Blue and red lines are the 96% and 4% enclosing lines for the blue
crosses and black and cyan for the red crosses.

##  Appendix C Chromosome map

Since $\omega$ Cen is more complex than typical globular clusters, a more
elaborate analysis is needed to create a useful chromosome map. Our method is
adapted from the procedure described in Milone et al. (2017a).

In summary, one needs more than just two, red and blue, fiducial lines since
there are multiple different populations in $\omega$ Cen. Hence, one
identifies three subsamples, the metal-poor population that also on its own
would look like a chromosome map for typical clusters, an intermediate
population, and a metal-rich population (a. Fig. 16). The process is iterative
and we start by identifying the subsamples in [M/H] first. Metal-poor stars
are those belonging to peaks 1, 2, 4, 7 and 11 of the Gaussian Mixture model
in Section 3.1, metal-rich stars belong to peak 9, and the intermediate
population are stars in the 6 components. We then use slightly adapted
fiducial lines from (Milone et al., 2017a) to get an initial chromosome map.
Next, we identify the subsamples using the different components identified for
the [M/H] distribution and also the photometry as in Milone et al. (2017a).
Including the photometry information for the subsamples narrows CMD tracks and
we calculate new fiducial lines.

The red and blue fiducial lines correspond to the 96th and 4th percentiles in
the CMD and pseudo-CMD. In the $\rm m_{\rm F275W}-m_{\rm F814W}$ CMD (b. Fig.
16) the red and blue fiducial lines correspond to the metal-poor population.
For the intermediate and rich population, one needs to find the median
fiducial line. In the CMD with the pseudo-color, $\rm C_{\rm
F275W,F336W,F435W}$, the populations are more mixed and the intermediate and
poor populations can use the same red and blue fiducial lines, while the
metal-rich have different 96th and 4th percentiles indicated with the black
and cyan lines. For all fiducial lines, we smoothed and corrected by hand for
bright magnitudes.

We repeated this process twice refining our subsample selection and fiducial
lines. Our final fiducial lines are shown in Fig. 17 together with the
verticalization of the colors for all bright stars.

Figure 17: Final Fiducial lines. Left is the CMD for all stars and on the
right is the pseudo-CMD both color-coded with [M/H] and showing the fiducial
lines for the RGB. On the right of each subfigure is the verticalized CMD
proving that the subsamples do follow these straight lines.

To get the chromosome map values we calculate the following weights:

$\displaystyle W_{\rm
F275W,F814W_{1}}=X_{fiducialRed}(15.5~{}mag)-X_{fiducialBlue}(15.5~{}mag)$
(C1) $\displaystyle W_{\rm
F275W,F814W_{2}}=X_{fiducialOrange}(15.5~{}mag)-X_{fiducialRed}(15.5~{}mag)$
(C2) $\displaystyle W_{\rm
F275W,F814W_{3}}=X_{fiducialGreen}(15.5~{}mag)-X_{fiducialOrange}(15.5~{}mag)$
(C3)

where e.g. $X_{fiducialRed}(15.5~{}mag)$ is the color of the fiducial line
interpolated at 15.5 magnitude in F814W. Then we can find the respective
$\Delta_{\rm F275W,F814W_{i}}$ values:

$\displaystyle\Delta_{\rm F275W,F814W_{1}}=W_{\rm
F275W,F814W_{1}}\frac{X-X_{fiducialRed}}{X_{fiducialRed}-X_{fiducialBlue}}$
(C4) $\displaystyle\Delta_{\rm F275W,F814W_{2}}=W_{\rm F275W,F814W_{2}}+W_{\rm
F275W,F814W_{2}}\frac{X-X_{fiducialOrange}}{X_{fiducialOrange}-X_{fiducialRed}}$
(C5) $\displaystyle\Delta_{\rm F275W,F814W_{3}}=W_{\rm F275W,F814W_{2}}+W_{\rm
F275W,F814W_{3}}+W_{\rm
F275W,F814W_{3}}\frac{X-X_{fiducialGreen}}{X_{fiducialGreen}-X_{fiducialOrange}}$
(C6)

Which of these three $\Delta_{\rm F275W,F814W_{i}}$ is used for each star
depends on its relative position to the different fiducial lines, see Table 4
for details. For the $\Delta_{\rm F275W,F336W,F435W}$ value we calculate:

$\displaystyle W_{\rm
F275W,F336W,F435W_{1}}=Y_{fiducialRed}(15.5~{}mag)-Y_{fiducialBlue}(15.5~{}mag)$
(C7) $\displaystyle W_{\rm
F275W,F336W,F435W_{2}}=Y_{fiducialRed}(15.5~{}mag)-Y_{fiducialCyan}(15.5~{}mag)$
(C8) $\displaystyle W_{\rm
F275W,F336W,F435W_{3}}=Y_{fiducialCyan}(15.5~{}mag)-Y_{fiducialBlack}(15.5~{}mag)$
(C9)

and then

$\displaystyle\Delta_{\rm F275W,F336W,F435W_{1}}=W_{\rm
F275W,F336W,F435W_{1}}\frac{Y_{fiducialRed}-Y}{Y_{fiducialRed}-Y_{fiducialBlue}}$
(C10) $\displaystyle\Delta_{\rm F275W,F336W,F435W_{2}}=W_{\rm
F275W,F336W,F435W_{2}}+W_{\rm
F275W,F336W,F435W_{3}}\frac{Y_{fiducialCyan}-Y}{Y_{fiducialCyan}-Y_{fiducialBlack}}.$
(C11)

Depending on each metallicity each star gets assigned either $\Delta_{\rm
F275W,F336W,F435W_{1}}$ or $\Delta_{\rm F275W,F336W,F435W_{2}}$. The condition
is described in Table 4.

$3$ Final chromosome $\Delta$ value $\Delta_{i}$ value Condition $\Delta_{\rm
F275W,F814W_{1}}$, if $\Delta_{\rm F275W,F814W_{1}}\leq 0$, right of the red
fiducial line $\Delta_{\rm F275W,F814W}=$ $\Delta_{\rm F275W,F814W_{2}}$, if
$\Delta_{\rm F275W,F814W_{1}}>0$ & $\Delta_{\rm F275W,F814W_{2}}\leq W_{\rm
F275W,F814W_{2}}$, between red and orange fiducial line $\Delta_{\rm
F275W,F814W_{3}}$, if $\Delta_{\rm F275W,F814W_{3}}>W_{\rm F275W,F814W_{2}}$,
right of the orange fiducial line $\Delta_{\rm F275W,F336W,F435W}=$
$\Delta_{\rm F275W,F336W,F435W_{1}}$, if star does not belong to the most
metal-rich peak ($\neq$ index 9) $\Delta_{\rm F275W,F336W,F435W_{2}}$, if star
belongs to metal-rich peak (= index 9)

Table 4: Final chromosome map values

Note. — The metal-rich peak (index 9) is taken from the Gaussian Mixture model
Table 3 in Section 3.1.

Finally, we color-coded the final chromosome map with all the components of
the Gaussian Mixture model of the [M/H] distribution (Section 3.1) in Fig. 18.
All the components are at specific locations on the map depending on their
[M/H], separating different sub-populations and showing the complexity of the
clusters.

Figure 18: Chromosome map color-coded with the subgroups identified in the
metallicity distribution. We use the components of the Multi Gaussian Mixture
Model used in Section 3.1 and color-coded the stars in the chromosome map with
the component (index) they most likely belong to.

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