# The X-ray Binary-Star Cluster Connection in Late-Type Galaxies

Qiana Hunt Department of Astronomy, University of Michigan, 1085 S University,
Ann Arbor, MI 48109, USA  Rupali Chandar Ritter Astrophysical Research Center,
University of Toledo, Toledo, OH 43606, USA  Elena Gallo Department of
Astronomy, University of Michigan, 1085 S University, Ann Arbor, MI 48109, USA
Matthew Floyd  Ritter Astrophysical Research Center, University of Toledo,
Toledo, OH 43606, USA  Thomas J. Maccarone  Department of Physics & Astronomy,
Texas Tech University, Lubbock, TX 79409, USA  David A. Thilker Department of
Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA

###### Abstract

We conduct one of the largest systematic investigations of bright X-ray
binaries (XRBs) in both young star clusters and ancient globular clusters
(GCs) using a sample of six nearby, star-forming galaxies. Combining complete
CXO X-ray source catalogs with optical PHANGS-HST cluster catalogs, we
identify a population of 33 XRBs within or near their parent clusters. We find
that GCs that host XRBs in spiral galaxies appear to be brighter, more
compact, denser, and more massive than the general GC population. However,
these XRB hosts do not appear preferentially redder or more metal-rich,
pointing to a possible absence of the metallicity-boosted formation of low-
mass X-ray binaries (LMXBs) that is observed in the GCs of older galaxies. We
also find that a smaller fraction of LMXBs is found in spiral GC systems when
compared with those in early-type galaxies: between 8 and 50%, or an average
of 20% across galaxies in our sample. Although there is a non-negligible
probability of a chance superposition between an XRB and an unrelated young
cluster, we find that among clusters younger than 10 Myr, which most likely
host high-mass XRBs, the fraction of clusters associated with an XRB increases
at higher cluster masses and densities. The X-ray luminosity of XRBs appears
to increase with the mass of the cluster host for clusters younger than $\sim
400$ Myr, while the inverse relation is found for XRBs in GCs.

Star clusters (1567) — X-ray binary stars (1811)

††facilities: CXO, HST(STIS), ALMA, VLT††software: Astropy (Astropy
Collaboration et al., 2013), Cigale (Boquien et al., 2019), BAOlab (Larsen,
1999), SciPy (Virtanen et al., 2020)

##  1 Introduction

Over the last $\sim$5 years, the detections of merging compact objects by
LIGO/VIRGO/KAGRA (The LIGO Scientific Collaboration et al., 2021), and the
discovery of a fast radio burst apparently associated with a globular cluster
(GC) in M81 have reignited efforts to self-consistently model the formation
and evolution of exotic binary systems (Sridhar et al., 2021; Kirsten et al.,
2022). A crucial element in these models are X-ray binaries (XRBs) — binaries
consisting of compact object and a luminous, typically main sequence donor
star. XRBs are a means by which we can observe otherwise invisible compact
objects in our galaxy and others. Knowledge of the relative frequency of
bright XRBs serves to inform numerical simulations and population synthesis
models that aim to understand and predict the formation efficiency of rare and
interesting phenomena, such as black hole XRBs, gravitational wave
progenitors, fast radio busts, or intermediate-mass black holes (e.g. Özel et
al., 2010; Farr et al., 2011; Giersz et al., 2015; Romero-Shaw et al., 2023).
Moreover, XRBs themselves are largely important components to the total X-ray
makeup of a galaxy, as they dominate the hard X-ray emissions of a galaxy in
the absence of a bright AGN (e.g. Fabbiano, 2006; Boroson et al., 2011).

Stellar clusters in particular are excellent targets in the search for XRBs in
external galaxies. Not only are clusters optically brighter than the average
extragalactic star, but they are effective factories for XRB formation: they
boost the masses of already massive stars, harden already hard binaries, and
promote the formation of new binaries through increased dynamical interactions
(Fabian et al., 1975; Hills, 1976; Verbunt, 1987; Kundu et al., 2007; Garofali
et al., 2012; Ivanova, 2013). Observations of low-mass XRBs (LMXBs; with donor
stars of mass $\lesssim$ 1 M⊙) indicate they form hundreds of times more
efficiently in GCs than in the field (Clark, 1975; Katz, 1975; Grindlay, 1988;
Hut et al., 1992). Over the past 20 years, observations of nearby, massive
elliptical galaxies with the Chandra X-ray Observatory and the Hubble Space
Telescope (HST) have established that between 20% and 70% of bright
($L_{X}\lower 2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}5\times 10^{37}$ erg
sec-1) LMXBs currently reside in GCs (e.g. Angelini et al., 2001; Kundu et
al., 2002; Jordán et al., 2004; Brassington et al., 2010; Fabbiano et al.,
2010), and that a near constant $4\%-10\%$ of GCs in ellipticals host bright
LMXBs (Maccarone et al., 2003; Jordán et al., 2007; Mineo et al., 2013; Luan
et al., 2018). These high efficiencies suggest that some fraction of field
LMXBs may have formed in GCs that subsequently dissolved, or were dynamically
ejected (e.g. Grindlay, 1984; White et al., 2002), though the impact these
ejected sources have on the total LMXB population is a subject of debate (Piro
& Bildsten, 2002; Kundu et al., 2002, 2007; Peacock & Zepf, 2016; Kremer et
al., 2018; Lehmer et al., 2020).

Studies also show that, in ellipticals, LMXBs are preferentially found in GCs
that are brighter (i.e. more massive; e.g. Sivakoff et al., 2007; Brassington
et al., 2010; Fabbiano et al., 2010; Riccio et al., 2022), more compact (i.e.
smaller effective radii and higher densities; e.g. Kundu et al., 2002; Pooley
et al., 2003; Jordán et al., 2007; Sivakoff et al., 2007; Paolillo et al.,
2011; Riccio et al., 2022), and redder (i.e. metal-rich; e.g Bellazzini et
al., 1995; Kundu et al., 2002, 2003; Jordán et al., 2007; Kundu et al., 2008;
Sivakoff et al., 2007; Luan et al., 2018; Riccio et al., 2022). These surveys
have gone a long way towards helping us understand the mechanism by which XRBs
form within dense stellar environments. However, our understanding of the
connection between XRBs and star clusters in spiral galaxies is comparatively
inferior. The complex morphologies, ongoing star formation, and significant
dust content of spirals make it extremely challenging to identify clusters in
general, let alone to separate ancient GCs — within which LMXBs are produced —
from reddened young clusters, which may host higher-mass XRBs (Chandar et al.,
2004).

It is not obvious whether a comparably high fraction of the LMXB population is
expected to be found in GCs of star-forming galaxies. While ellipticals appear
to host more GCs per unit halo mass than star forming galaxies (Harris et al.,
2015), the LMXB production efficiency likely decreases at higher redshift
(Fragos et al., 2013). Tentatively, a smaller fraction of LMXBs are found in
GCs across late-type galaxies that have been observed so far — between 2% and
13% for extragalactic LMXBs (Chandar et al., 2020; Hunt et al., 2021, 2023),
though as much as 20% of the Galactic LMXB population is located within GCs
(Bahramian & Degenaar, 2022). The number of these galaxies explored remains
too small to definitively prove that this is universally the case, and the
depth of recent X-ray observations of these galaxies compared to prior
observations of early-type galaxies could play a significant role in these
discrepancies.

Furthermore, there is the issue of high-mass X-ray binaries (HMXBs, with donor
star mass in excess of 8 M⊙111The large majority of XRBs with intermediate-
mass donors quickly evolve into low-mass XRBs through a short-lived thermal
mass transfer phase, so this population is often ignored (Podsiadlowski et
al., 2002).). Although HMXBs are believed to form in young clusters, most of
them are expected to be expelled into the general field population by
supernova explosions (Kaaret et al., 2004; Sepinsky et al., 2005; Kalogera et
al., 2007; Zuo & Li, 2010; Poutanen et al., 2013). If the natal kick
velocities from black holes are smaller than those from neutron stars, then
HMXBs with black hole accretors may be more likely to stay within their parent
cluster than neutron star XRBs (Portegies Zwart et al., 2007). However, recent
work posits the kick velocities of black holes can rival those of neutron
stars (Repetto et al., 2017; Atri et al., 2019). Additionally, tight binary
systems may theoretically survive kick velocities well in excess of the
cluster escape velocity (Sippel & Hurley, 2013), and the ejection of black
holes from clusters may not be as efficient as previously predicted, as
indicated by both more recent simulations and the identification of multiple
black holes within compact star clusters (Strader et al., 2012; Morscher et
al., 2015). Therefore, binaries with accreting black holes can be reasonably
expected to be found both outside their birth clusters and within them.

Establishing the ratio of HMXBs in clusters versus HMXBs in the field may help
illuminate the formation mechanisms and evolutionary histories of compact
objects in external spiral galaxies. To date, relatively few papers have
looked at XRBs in young clusters, with only a few late-type galaxies examined
so far. Current observations suggest between 2% and 19% of HMXBs reside within
young clusters (Rangelov et al., 2012; Chandar et al., 2020; Hunt et al.,
2021, 2023). An examination of young cluster populations by Johns Mulia et al.
(2019) find that XRBs in the Antennae galaxies and NGC 4449 prefer more
massive and denser clusters among clusters with ages between 10 and 400 Myr,
but there does not seem to be a similar preference for HMXBs formed in very
young ($<10$ Myr) clusters. Why and whether these patterns hold true for most
late-type galaxies remains unknown.

In this paper, we undertake a first systematic study of the association
between bright XRBs and dense star clusters in a sample of six nearby face-on
spiral galaxies that were observed by both the Chandra X-ray Observatory and
the HST. We make use of the Physics at High Angular Resolution in Nearby
Galaxies (PHANGS) survey222https://sites.google.com/view/phangs/home, which
produced high-fidelity star cluster catalogs (Maschmann et al. in prep.; Lee
et al., 2022; Thilker et al., 2022). We cross-match these catalogs with
complete compact X-ray source catalogs based on deep Chandra observations, as
compiled by Lehmer et al. (2019, hereafter L19). The paper is organized as
follows: in §2, we describe the X-ray source and optical cluster catalogs,
their astrometric alignment, and the positional uncertainties of the
observations; in §3, we describe the cluster properties we analyze and how
they are calculated; in §4, we discuss the population of clusters that we
determined host XRBs and the properties of those clusters compared to the
general cluster population; finally, in §5, we compare our findings to cluster
XRBs found in previous studies, particularly in early-type galaxies.

Figure 1: Our galaxy sample, as imaged by the Digitized Sky Survey. Small red
circles represent the coordinates and 2$\sigma$ radii of X-ray sources from
Lehmer et al. (2019), while magenta contours show the outermost density
contour of class 1 and 2 clusters in the PHANGS catalog, which we use to
denote the PHANGS cluster distribution. Outlined in black are the isophotal
ellipses that trace the K$\rm{}_{s}~{}\approx~{}20$ mag arcsec-2 galactic
surface brightness of each galaxy (Jarrett et al., 2003; Lehmer et al., 2019).
Cyan stars represent XRBs that we have identified as having at least one
cluster association.

##  2 Catalogs

###  2.1 X-ray source catalog

A comprehensive catalog of more than 4000 X-ray sources across 38 nearby
galaxies was compiled by L19 from 5.8 Ms of Chandra ACIS data. Chandra has an
excellent angular resolution of $0.\arcsec 3$ at its aim-point, which allows
us to view nearby XRBs as point-like sources (Grimm et al., 2008). The
galaxies in the sample are all within 30 Mpc and have inclinations
$\lesssim~{}70^{\circ}$ to our line of sight, which ensures a low degree of
local absorption. Because the galaxy selection was based mainly on those
covered by both Chandra and the Spitzer Infrared Nearby Galaxies Survey
(SINGS; Kennicutt et al., 2003), the sample brackets a diverse range of
properties, including stellar mass and star formation rate. L19 also provides
a careful assessment of the X-ray completeness for each target galaxy: roughly
3500 X-ray sources are above the $90\%$ X-ray completeness limits of their
respective galaxies.

Additionally, L19 presents a model for estimating the expected contributions
from LMXBs, HMXBs and cosmic X-ray background sources (CXBs) to the total
X-ray luminosity function within the footprint of each galaxy. The galactic
footprints are defined by the isophotal ellipse tracing the $K_{s}\approx 20$
mag arcsec-2 galactic surface brightness of each galaxy (L19; Jarrett et al.,
2003). Specifically, L19 provides a functional relationship to estimate the
number of LMXBs and HMXBs in each galaxy based on the total stellar mass and
star formation rates, respectively, within the isophotal ellipses, as mapped
from combined K-band Two Micron All Sky Survey (2MASS) and optical Sloan
Digital Sky Survey (SDSS), and combined FUV GALEX and 24$\mu$m Spitzer
observations. The results of L19’s sub-galactic XRB population modeling yield
a useful framework for comparing the field vs. cluster XRB populations that we
aim to identify with our analysis.

###  2.2 Cluster catalog

The Physics at High Angular Resolution in Nearby Galaxies
(PHANGS)333https://phangs.stsci.edu program is a high-resolution, multi-
wavelength survey that utilizes Atacama Large Millimeter/Submillimeter Array
(ALMA), HST and Very Large Telescope (VLT) observations to probe spiral
galaxies within $\sim 20$ Mpc (Kreckel et al., 2019, 2021; Leroy et al., 2021;
Lee et al., 2022). Of particular interest to this study is the sample of tens
of thousands of compact star clusters identified by the PHANGS-
HST444https://archive.stsci.edu/hlsp/phangs-hst;
https:/archive.stsci.edu/hlsps/phangs-cat/ survey (Deger et al., 2021; Turner
et al., 2021; Whitmore et al., 2021). These span 38 nearby galaxies with five-
band (NUV, U, B, V, and I), high-resolution HST imaging data taken with the
WFC3 camera. The sample is purposefully chosen to ensure that the cluster
radii are broader than the point-spread function (PSF), thereby minimizing the
limitations of instrument resolution. Clusters are identified via a
combination of manual and machine-learning classification techniques using a
Multiple Concentration Index scheme to select cluster candidates (Whitmore et
al., 2021; Lee et al., 2022; Thilker et al., 2022).

In this work, we only use centrally concentrated clusters (class 1 and 2),
which span all ages. A separate effort by Floyd et al. (in prep) has
distinguished ancient GCs from their younger counterparts, based on color
selection followed by a careful visual examination. The PHANGS data were
obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space
Telescope Science Institute and can be accessed via
https://doi.org/10.17909/t9-r08f-dq31 (catalog DOI: 10.17909/t9-r08f-dq31).

Table 1: Galaxy sample

Galaxy | $\alpha\rm_{J2000}$ | $\delta\rm_{J2000}$ | Dist. | Scale | log LX,90% | PHANGS limitsb | Number of sources
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
|  |  | (Mpc) | (pc arcs-1) | (erg s-1) | (mV) | (MV) | X-ray Sourcesa | Clustersb
NGC 0628 | 03 36 41.8 |  $+$15 47 00.5 | 9.84 $\pm$ 0.63 | 35.3 | 36.9 | 24.9 | -5.1 | 155 (47) | 750 (598)
NGC 3351 | 10 43 57.7 |  $+$11 42 13.0 | 9.96 $\pm$ 0.33 | 45.1 | 37.0 | 25.3 | -4.7 | 65 (39) | 335 (313)
NGC 3627 | 11 20 15.0 |  $+$12 59 28.6 | 11.32 $\pm$ 0.48 | 45.4 | 37.5 | 24.7 | -5.6 | 105 (64) | 1251 (1236)
NGC 4321 | 12 22 54.9 |  $+$15 49 20.6 | 15.12 $\pm$ 0.49 | 69.2 | 37.4 | 24.8 | -6.1 | 109 (61) | 868 (866)
NGC 4569 | 12 36 49.8 |  $+$13 09 46.3 | 15.76 $\pm$ 2.36 | 79.7 | 37.8 | 24.0 | -7.0 | 66 (27) | 540 (527)
NGC 4826 | 12 56 43.7 |  $+$21 40 57.6 | 4.41 $\pm$ 0.19 | 36.2 | 36.7 | 23.9 | -4.3 | 49 (33) | 175 (175)

Note. — Galaxy positions are as reported in L19. We adopt the updated
distances used in the PHANGS-HST pipeline (Lee et al., 2022), and the $90\%$
X-ray luminosity completeness limits (LX,90%) are adjusted from the L19 values
to reflect the new distances. The limiting optical magnitude of the PHANGS
(VPHANGS,lim.) is taken as the optical V-band magnitude of the dimmest cluster
identified by PHANGS. The number of X-ray sources and clusters within each
galaxy is given, where numbers in parentheses represent the number of sources
that fall within the Ks-band isophotal ellipses, as described in L19.

aLehmer et al. (2019); bWhitmore et al. (2021); Lee et al. (2022); Thilker et
al. (2022)

###  2.3 Galaxy Selection

This paper considers the six spiral galaxies that constitute the intersection
between the L19 X-ray source catalogs and the PHANGS-HST cluster catalogs: NGC
628, NGC 3351, NGC 3627, NGC 4321, NGC 4569, and NGC 4826. The target
galaxies, which are located between $4-16$ Mpc, are shown in Figure 1. The L19
X-ray sources are plotted in red, where the radii of each red circle
represents the 2$\sigma$ positional uncertainties of the X-ray data. These are
calculated, in part, using the $68\%$ and $90\%$ positional uncertainty
equations of Kim et al. (2007), which incorporates the X-ray counts and off-
axis angles of each source provided by L19. The final 2$\sigma$ positional
uncertainties include astrometric uncertainties, discussed in §2.4 below,
added in quadrature to the positional uncertainties. In addition, the magenta
contours represent the outermost distribution of the star clusters within the
PHANGS-HST field of view (Lee et al., 2022). The basic properties of each
galaxy, along with the number of X-ray point sources and optically selected
clusters, are given in Table 1. For each galaxy, we utilize the updated
distances that were used to identify and estimate cluster properties (see Lee
et al., 2022), and we adjust the X-ray completeness limit and luminosities of
the sample accordingly. The two catalogs capture a total of 549 X-ray sources
and 3919 optical clusters across the six galaxies, 271 and 3715 of which fall
within the $K_{s}$-band isophotal ellipses.

Table 2: Astrometric corrections

| L19 X-ray sources | PHANGS clusters
---|---|---
|  |
| $\Delta_{RA}$ | $\Delta_{Dec}$ | $\sigma_{RA}$ | $\sigma_{Dec}$ | $\Delta_{RA}$ | $\Delta_{Dec}$
Galaxy | (arcsec) | (arcsec) | (arcsec) | (arcsec) | (arcsec) | (arcsec)
NGC 0628 | -0.03 | -0.18 | 0.04 | 0.11 | -0.06 | -0.13
NGC 3351 | -0.21 | 0.29 | 0.48 | 0.45 | 0.01 | -0.01
NGC 3627 | -0.18 | -0.08 | 0.05 | 0.04 | -0.33 | -0.18
NGC 4321 | -0.40 | -0.19 | 0.06 | 0.08 | -0.65 | -0.65
NGC 4569 | 0.23 | -0.07 | 0.06 | 0.05 | 0.01 | -0.02
NGC 4826 | -0.27 | 0.06 | 0.16 | 0.29 | 0.01 | 0.02

Note. — The astrometric corrections on the L19 (Chandra) X-ray sources and
PHANGS (HST) clusters are given as translational shifts $\Delta$ along right
ascension and declination. For the X-ray sources, the standard deviation of
the shifts between the L19 reference source coordinates and their optical
counterparts on the HST image, $\sigma$, are used to calculate the positional
uncertainties of the X-ray sources, which are included in the 2$\sigma$
positional uncertainty calculations.

###  2.4 Astrometric corrections

To identify candidate PHANGS-HST cluster counterparts to X-ray sources, we
first correct the coordinate offsets between the two catalogs, since
observations made by Chandra and HST inherently contain a degree of positional
uncertainty that yields an imperfect alignment between the two. HST images
obtained from the Hubble Legacy Archives (HLA555http://hla.stsci.edu/) provide
a good reference against which to adjust the coordinates of the L19 X-ray
sources, since X-ray catalogs are likely to contain objects with optically
visible counterparts such as bright background AGN (e.g., see the methods of
Hunt et al., 2021). The PHANGS-HST catalogs are generated using large mosaics
of HST observations, so the catalog positions of PHANGS clusters are shifted
slightly when compared to individual HST fields.

For each galaxy in our sample, we select one HST field from the HLA that was
imaged in at least three filters (BVI) and that overlaps with as much of the
area enclosed by the $K_{s}$-band ellipses as possible. A handful of sources
with visibly identifiable counterparts in the HST field are selected as
reference sources for the astrometric calculations of each catalog: background
AGN, foreground stars, and isolated GCs in clear proximity to X-ray sources
are used for the X-ray catalog, while isolated GCs are used for PHANGS.
Ideally, at least 15 reference sources are used per catalog per galaxy, but in
some cases (particularly for the L19 catalog), there are fewer than 10
reference sources suitable for use.

The sources in each catalog are then shifted by the median offsets in right
ascension and declination ($\Delta_{\rm{RA}}$ and $\Delta_{\rm{Dec}}$) between
the HST image and the catalog coordinates of the select reference sources. For
PHANGS, a simple translational shift is sufficient to properly align cluster
coordinates onto the HST image. For the X-ray sources, there is an additional
positional uncertainty introduced when combining Chandra and HST observations
(e.g. by rotation, instrumental resolution, etc.). These are reflected in the
standard deviations of the right ascension and declination correction
($\sigma_{\rm{RA}}$ and $\sigma_{\rm{Dec}}$) and are factored into the
2$\sigma$ positional uncertainty of each X-ray source as described in §2.3.
The final astrometric corrections are listed in Table 2. After corrections,
$95\%$ of the new 2$\sigma$ positional uncertainties of the L19 sources fall
between $0.\arcsec 21-1.\arcsec 96$, with a median of 1.″07 across all
sources. Within the $K_{s}$-band ellipses — a region that inherently limits
the off-axis angle of each X-ray source and thus the degradation of the
Chandra PSF — the maximum 2$\sigma$ uncertainty is $2.\arcsec 15$.

In the following sections, we use the L19 X-ray source catalog as the basis of
our investigation. We search for clusters associated with each XRB, where a
cluster is determined to be associated with an X-ray source if it falls within
a radius equal to the 2$\sigma$ positional uncertainty of said source. We
explore the properties of these XRB clusters compared to both the total
cluster population and the field XRBs within these spiral galaxies in §4. We
compare our results to previous studies of XRBs within elliptical and spiral
galaxies in §5.

##  3 Star Cluster Properties

The basic properties of the stellar clusters were determined from multi-band
HST images of our sample galaxies taken as part of PHANGS-HST (Turner et al.,
2021; Lee et al., 2022, Floyd et al. in prep). The cluster selection
methodology is briefly summarized in Section §2; here we describe estimates of
the age, mass, and effective radius of each cluster within the galaxy sample.

###  3.1 Age

The PHANGS-HST survey estimates the age and reddening of each cluster by
comparing the measured photometry with predictions from population synthesis
models. Because PHANGS galaxies have formed stars continuously over their
lifetimes, they contain clusters spanning a wide age range, from very recently
formed clusters with ages $\approx 1$ Myr to ancient GCs with ages of $\approx
12$ Gyr. The significant interstellar medium (ISM) in PHANGS galaxies means
that breaking the well-known age-reddening degeneracy, where red colors for
clusters are either due to older ages or because they are affected by dust, is
critical for correctly age-dating clusters.

Details of the method used to estimate the age and reddening for each cluster
are described in Turner et al. (2021), and we only provide a brief summary
here. The fluxes measured in five broad-band filters (equivalent to NUV, U, B,
V, and I bands) are compared with predictions from the solar-metallicity model
for cluster ages between 1 Myr and 12 Gyr (Bruzual & Charlot, 2003). Age and
reddening are the two free parameters in the fit, and reddening is allowed to
vary from an E(B-V) of 0.0 to 1.5 mag. The software Cigale (Boquien et al.,
2019) fits the photometry for a cluster to each age-reddening combination in
the grid and finds the combination that minimizes $\chi^{2}$.

While the PHANGS-HST study estimates ages correctly for many clusters,
incorrect determinations have been found to disproportionately affect the
ancient GC populations. Whitmore et al. (2023) and Floyd et al. (in prep) have
found that $\sim 80$% of ancient globular clusters, which comprise $\approx
5-10$% of the PHANGS cluster catalogs, have been incorrectly age-dated, with
both too young ages and too high reddening. For our sample, we assign an age
of 10 Gyr to any GC that is coincident with an XRB, regardless of the age
estimated in the PHANGS-HST pipeline. We similarly correct the ages of any
intermediate-age ($10-400$ Myr) clusters which have incorrectly been assigned
too young of an age from the PHANGS-HST pipeline if a visual inspection
indicates they are not heavily affected by dust, using the best-fit age where
a low maximum extinction of E(B-V)$<0.1$ mag is allowed. For the rest of the
clusters, we adopt the age (and reddening) determined by the default PHANGS-
HST pipeline.

###  3.2 Mass

To estimate cluster mass, PHANGS-HST adopts a mass-to-light ratio predicted by
the evolutionary model of solar metallicity clusters by Bruzual & Charlot
(2003) at the best-fit age obtained in §3.1. The mass is then calculated using
the extinction-corrected luminosity.

We adopt these default pipeline masses for all clusters where the PHANGS-HST
pipeline gives good age estimates, which includes most non-GC clusters. For
non-GCs which have incorrect age-dating, a revised mass is calculated by using
the M$/$LV ratio predicted for the new best fit age. To estimate the masses of
GCs, we adopt a mass-luminosity ratio of 1.5 $M_{\odot}$ /$L_{\odot}$, as
given in Chandar et al. (2007), where the V-band luminosity of each GC is
calculated from the HST V-band magnitude: $L_{V}\equiv
10^{0.4(M_{V,\odot}-M_{V})}$. The absolute V-band Vega magnitude of the Sun
($M_{V,\odot}$) is given by Willmer (2018) as 4.91 for the WFC3 F555W filter.
The mass estimates are typically uncertain to within a factor of 2 (e.g. see
Chandar et al., 2010).

###  3.3 Effective Radius and Density

Floyd et al. (in prep.) measures the effective radius $R\rm{{}_{eff}}$ of each
cluster in the PHANGS-HST catalog using the Ishape routine in BAOlab (Larsen,
1999), a standard software package used to estimate the half-light radii of
clusters. Ishape convolves a series of model clusters having King profiles
(with an assumed tidal-to-core radius ratio of 30) with a user-provided PSF.
These convolved models are then fitted to the V-band image of each cluster,
and the fits are converted to effective radius using the appropriate pixel
scale for the WFC3 or ACS camera. A number of checks were performed to verify
the size results, including reproducing the size measurements of clusters
published in M51 by Chandar et al. (2016) and verifying that the Ishape
measurements track other size estimates, such as the concentration index (the
magnitude difference measured in aperture radii of 1 and 3 pixels). The
distribution of effective (and half-light) radii for ancient globular clusters
is consistent with those reported in prior studies (e.g. Jordán, 2004; Jordán
et al., 2007; Puzia et al., 2014).

With the sizes and masses of the clusters, we estimate the density of each
cluster by defining the internal half-mass density as $\rho_{h}\equiv 3M/8\pi
R^{3}_{h}$, where $M$ is the mass of the cluster, and the 3-dimensional half-
mass radius $R_{h}$ is calculated from the 2-dimensional effective radius
using $R_{h}=(4/3)R_{\rm eff}$ (e.g. see Chandar et al., 2007; McLaughlin &
Fall, 2008).

Table 3: Properties of XRBs and associated clusters

Galaxy | L19 ID | log LX | PHANGS ID | V | U-B | V-I | Age | Mass | Reff | Density
---|---|---|---|---|---|---|---|---|---|---
|  | (erg s-1) |  | (Mag) | (mag) | (mag) | (log yrs) | (log M⊙) | (pc) | (log M⊙ pc-3)
Host clusters of singly-associated XRBs
NGC 0628 | L19X038 | 36.8 | 481 | -6.30 | -1.38 | 0.51 | 7.28 | 3.47 | 4.80 | 0.13
| L19X039 | 37.9 | 7257 | -9.75 | 0.20 | 1.22 | 10.00* | 6.04 | 4.65 | 2.74
| L19X048 | 38.4 | 7375 | -6.60 | -0.01 | 1.75 | 10.00* | 4.78 | 3.38 | 1.90
| L19X059 | 37.0 | 5558 | -6.21 | -0.67 | 0.43 | 7.88 | 3.75 | 4.27 | 0.56
| L19X060 | 36.9 | 4602 | -8.13 | 0.49 | 1.57 | 10.00* | 5.39 | 4.16 | 2.24
| L19X086 | 36.7 | 4841 | -7.94 | -0.01 | 1.27 | 10.00* | 5.31 | 3.38 | 2.43
| L19X087 | 37.5 | 7497 | -11.30 | — | 0.23 | 6.70 | 5.11 | 4.56 | 1.84
NGC 3351 | L19X014 | 37.1 | 151 | -7.88 | -0.81 | -0.20 | 6.70 | 3.31 | 7.49 | -0.61
| L19X026 | 38.6 | 3845 | -5.91 | — | 1.50 | 10.00* | 4.5 | 1.37 | 2.79
| L19X040 | 37.3 | 2881 | -7.85 | 0.24 | 1.26 | 10.00* | 5.28 | 2.20 | 2.96
NGC 3627 | L19X032 | 38.2 | 9731 | -12.32 | -1.10 | 0.49 | 6.00 | 6.17 | 1.95 | 4.00
| L19X040 | 38.8 | 7367 | -8.04 | -0.05 | 1.13 | 10.00* | 5.36 | 1.26 | 3.75
| L19X063 | 37.1 | 1421 | -6.78 | -0.50 | 0.99 | 10.00* | 4.85 | 4.28 | 1.66
| L19X074 | 37.5 | 5530 | -8.65 | -1.04 | 0.62 | 7.51 | 4.41 | 3.20 | 1.60
| L19X078 | 39.4 | 5876 | -7.51 | 0.44 | 1.20 | 10.00* | 5.15 | 6.94 | 1.32
| L19X083 | 38.7 | 10673 | -6.69 | 0.39 | 1.01 | 10.00* | 4.81 | 5.03 | 1.41
NGC 4321 | L19X055 | 38.1 | 2466 | -8.21 | 0.13 | 1.14 | 10.00* | 5.42 | 1.70 | 3.43
| L19X056 | 37.0 | 831 | -9.70 | -0.94 | 0.83 | 6.90 | 4.43 | 1.98 | 2.24
| L19X070 | 37.1 | 368 | -10.58 | -1.19 | 0.48 | 6.00 | 5.28 | 2.05 | 3.05
| L19X071 | 37.3 | 878 | -7.88 | -0.77 | 0.72 | 7.66 | 4.13 | 2.12 | 1.85
NGC 4826 | L19X011 | 36.5 | 1885 | -8.31 | 0.07 | 0.76 | 10.00* | 5.46 | 3.16 | 2.67
| L19X021 | 37.7 | 876 | -8.26 | 0.53 | 1.18 | 10.00* | 5.44 | 1.49 | 3.62
| L19X023 | 36.8 | 10 | -11.19 | -0.12 | 1.09 | 10.00* | 6.62 | 2.70 | 4.02
| L19X028 | 36.4 | 33 | -9.98 | -0.18 | 1.09 | 10.00* | 6.13 | 1.49 | 4.31
| L19X034 | 36.9 | 287 | -9.53 | -0.10 | 1.12 | 10.00* | 5.95 | 1.21 | 4.40
Candidate host clusters of multiply-associated XRBs
NGC 0628 | L19X033 | 36.5 | 3546 | -8.15 | -1.48 | -0.07 | 6.00 | 4.07 | 2.64 | 1.51
|  | 36.5 | 3557 | -8.12 | -1.20 | 0.17 | 6.00 | 4.33 | 2.21 | 2.00
NGC 3351 | L19X032 | 38.2 | 3382 | -8.12 | -0.99 | 0.08 | 6.70 | 3.57 | 2.71 | 0.97
|  | 38.2 | 3399 | -9.54 | -0.51 | 1.06 | 8.21** | 5.15 | 4.15 | 2.00
|  | 38.2 | 3499 | -8.50 | -0.64 | 0.58 | 8.06** | 4.67 | 7.17 | 0.80
| L19X033 | 38.4 | 3627 | -10.96 | -0.85 | 0.74 | 7.96** | 5.59 | 2.35 | 3.18
|  | 38.4 | 3660 | -9.90 | -0.98 | 0.46 | 6.60 | 4.5 | 6.08 | 0.85
|  | 38.4 | 3673 | -9.98 | -1.39 | 0.36 | 6.00 | 4.89 | 6.12 | 1.24
|  | 38.4 | 3705 | -9.36 | -1.09 | 0.37 | 6.48 | 4.61 | 3.28 | 1.76
| L19X034 | 37.7 | 3000 | -12.72 | -1.31 | 0.17 | 6.30 | 5.85 | 4.70 | 2.54
|  | 37.7 | 3009 | -11.92 | -1.16 | 0.36 | 6.48 | 5.6 | 4.60 | 2.31
|  | 37.7 | 3017 | -10.81 | -1.29 | 0.19 | 6.00 | 5.22 | 6.80 | 1.43
|  | 37.7 | 3028 | -9.77 | -1.07 | 0.85 | 6.90 | 4.42 | 4.58 | 1.14
| L19X037 | 38.0 | 3081 | -9.54 | -0.96 | 0.77 | 6.90 | 4.29 | 4.45 | 1.05
|  | 38.0 | 3159 | -9.27 | -1.07 | 0.67 | 6.90 | 4.11 | 13.09 | -0.54
| L19X039 | 38.0 | 3624 | -11.60 | -1.35 | 0.23 | 6.00 | 5.49 | 3.03 | 2.75
|  | 38.0 | 3631 | -9.40 | -1.13 | 0.15 | 6.60 | 4.14 | 1.72 | 2.14
|  | 38.0 | 3650 | -10.36 | -1.31 | 0.27 | 6.00 | 5.0 | 4.72 | 1.68
| L19X044 | 37.1 | 3630 | -7.41 | 0.43 | 1.33 | 10.00* | 5.11 | 1.37 | 3.40
|  | 37.1 | 3712 | -8.53 | 0.01 | 1.15 | 10.00* | 5.55 | 1.37 | 3.84
NGC 3627 | L19X020 | 37.2 | 7021 | -9.01 | -1.31 | 0.12 | 6.48 | 4.33 | 2.62 | 1.78
|  | 37.2 | 7031 | -9.19 | -1.00 | 0.71 | 6.90 | 4.26 | 3.88 | 1.19

Note. — Singly-associated XRBs refer to those with a single cluster
association, while multiply-associated XRBs are those with multiple clusters
within 2$\sigma$. The U-B and V-I colors are calculated given the magnitudes
reported in the PHANGS, while age, mass, effective radius, and density are
obtained as described in §3.

* XRBs associated with GCs are assigned an age of 10 log yrs due to complications in estimating the ages of GCs (see §3.1).

** Some intermediate-age clusters are assigned manually-corrected ages, as the
PHANGS-HST pipeline provides inaccurate age estimates for certain young
clusters.

##  4 X-ray binary Host Clusters

Within the fields of view captured by the PHANGS cluster catalog (see Figure
1, magenta contours), there are 33 X-ray point sources that are spatially
coincident with at least one cluster. Depending on the galaxy distance and
physical size, a non-negligible fraction of the X-ray point source population
may be background AGN, even within the $K_{s}$-band ellipses (see Figure 10 in
L19). The contamination by CXBs to the total X-ray emissions of the galaxy is
estimated by L19; it is likely that unseen CXBs are embedded within the field
X-ray source population, which are described and corrected in §5.2.

Observation-based simulations show that the likelihood of a randomly-
distributed X-ray source coinciding with an unassociated cluster is small
(Rangelov et al., 2012; Hunt et al., 2023). We further demonstrate the
unlikelihood of chance superpositions by performing a similar simulation: for
each galaxy in our sample, we randomly scatter $N$ simulated X-ray sources,
where $N$ is equivalent to the number of XRBs expected to fall within some
area based on the enclosed SFR and M⋆ (see §2.1, §5.2 and L19 for details).
The chosen area in each galaxy is the region covered by both the $K_{s}$-band
isophotal ellipse and the PHANGS cluster distribution (the intersection of the
black ellipses and magenta contours in Figure 1). Each simulated source is
restricted to this area and assigned a positional uncertainty equal to the
median 2$\sigma$ of the L19 sources above the 90% X-ray completeness limit and
within the region. The sources are scattered 1000 times per galaxy, yielding a
statistical measure of the number of chance superpositions with a cluster. We
find that across all galaxies, the mean and median number of chance
superpositions between an XRB and cluster per simulation run is approximately
zero, except in the case of NGC 3351, which averaged a single superposition
per simulation run (equivalent to 5% of the simulated XRB population). Thus,
we conclude that the probability of a randomly distributed X-ray source
intersecting with a cluster is negligible, consistent with previous estimates
(Rangelov et al., 2012; Riccio et al., 2022; Hunt et al., 2023).

However, this might not necessarily be the case for non-random distributions.
In a secondary test, we find that shifting the positions of our real X-ray
sources by 4$\arcsec$ (a separation sufficiently greater than the 2$\sigma$
positional uncertainty of any source but smaller than the typical distance
over which the properties of the target galaxies change significantly) in any
direction yields a fairly high probability of the X-ray sources coinciding
with at least one cluster: the number of XRB-cluster associations when
shifting in any direction decreases by an average of 57%, and in some cases
(e.g. NGC 4321) there is no change. The chance superpositions are mostly
driven by younger clusters, largely due to the dense distribution of these
sources and HMXBs within active star-forming regions, such as the spiral arms
of their host galaxy. Because LMXBs and GCs tend to follow the underlying,
radial stellar mass distributions and are generally more isolated and more
uniformly distributed than higher-mass XRBs and younger clusters, we consider
any X-ray source that has a GC within 2$\sigma$ to be truly associated with a
cluster, while we approach the population of XRBs associated with clusters
younger that 400 Myr with more caution. This approach to the GC population is
independently supported by prior works (e.g. see Riccio et al., 2022).

XRBs associated with GCs are almost certainly LMXBs, since only low-mass stars
remain in these ancient stellar systems. The nature of the donor star is less
certain for clusters younger than a few hundred million years. Because the
most massive stars tend to sink towards the centers of clusters due to
dynamical friction and have been found to be the most active in forming
binaries (e.g. Trenti, 2006; Kaczmarek et al., 2011; Wu & Zhao, 2021), very
young clusters ($<~{}10$ Myr) are quite likely to host HMXBs. Clusters with
ages between 10 and a few 100 Myr contain both intermediate and low-mass
stars. The masses of XRB donor stars in these intermediate-age clusters are
currently not well understood. For neutron star accretors (i.e., the majority
of the population), intermediate-mass donors quickly evolve into low-mass
stars through a short-lived thermal mass transfer phase (Podsiadlowski et al.,
2002; Pfahl et al., 2003; Chen & Podsiadlowski, 2016). This likely results in
a population of abnormally luminous LMXBs with intermediate-mass donor
progenitors. For these reasons, we analyze intermediate-age clusters as a
separate population from both GCs and very young clusters.

In total, 25 of the 33 point-like X-ray sources are coincident with a single
cluster. Of these, 16 are associated with an ancient GC, 5 are associated with
a very young cluster, and 4 are associated with an intermediate-age cluster.
Eight additional X-ray sources are coincident with more than one cluster
within 2$\sigma$. This is particularly common in bright, crowded, actively
star-forming regions, as stated above. In these cases, it can be challenging
to determine the specific cluster host of an XRB. An inspection of X-ray
sources with multiple cluster candidates indicates that clusters within
2$\sigma$ of an X-ray source generally have similar ages: 1 X-ray source is
associated with 2 GCs, while 7 are each associated with more than 1 cluster
younger than $\sim 400$ Myr. There are 2 XRBs that are associated with both
young and intermediate-age clusters: L19X032 and L19X033 in NGC 3351. We also
find that clusters associated with the same X-ray source can have drastically
different masses, effective radii, and densities. Through our analysis, we
explicitly indicate which clusters are confirmed XRB hosts (i.e. the only
cluster within 2$\sigma$ of a singly-associated XRB) and which are candidate
hosts (i.e. one of several clusters within 2$\sigma$ of a multiply-associated
XRB). All singly-associated and multiply-associated XRBs and the properties of
their (candidate) host clusters are listed in Table 3.

One of the galaxies in our sample, NGC 4569, has zero clusters that host XRBs.
This may, in part, be due to the fact that NGC 4569 has the brightest X-ray
completeness limit of all galaxies in our sample, at 37.8 log LX. Indeed, an
inspection of X-ray sources with log L$\rm{{}_{X}}\geq 37.8$ yields far fewer
sources that are spatially coincident with a cluster: only 12 of the 33 XRB
clusters have X-ray luminosities brighter than the completeness limit of NGC
4569, with NGC 4826 having none. For this reason, we exclude the clusters from
NGC 4569 from the remainder of our analysis, particularly where we calculate
statistics using the full star cluster population.

Figure 2: (a) Color-magnitude diagram of all clusters (grey, excluding
clusters from NGC 4569) compared to the X-ray emitting GCs, very young
clusters, and intermediate-age clusters. The 25 filled symbols represent
clusters that are the sole candidate host of an XRB, whereas the 22 open
symbols represent clusters that are associated with XRBs with multiple cluster
counterparts (1 LMXB and 7 non-GC XRBs). The density contours of the very
young cluster and GC populations are shown in blue and red, respectively. (b)
Color-color diagram of the GC population, where the evolutionary model for
solar-metallicity clusters (Bruzual & Charlot, 2003) is shown in black, and
the arrow represents the direction of reddening. (c) Histograms of the $B-V$
and $V-I$ colors of GCs. The black dashed line and horizontal bar represent
the mean and standard deviation of a Gaussian fit to the total GC population.

###  4.1 Magnitude and color distribution

The PHANGS cluster catalog includes measurements of cluster magnitudes in 5
HST bands, allowing us to compare the colors and brightness of XRB-host
clusters to the general cluster population. Figure 2a shows the V vs. B-V
color-magnitude diagram (CMD) of all clusters, in grey, where the V-band
magnitudes were calculated using the apparent magnitudes reported by PHANGS at
the galactic distances from Lee et al. (2022). The distribution of the GC and
young cluster populations are respectively mapped by red and blue contours.
The CMD shows that XRBs are more common in brighter GCs and very young
clusters, compared to the general cluster population in each age group.
However, no such preference is apparent among intermediate-age clusters.

For ancient GCs, we are interested in establishing whether XRBs are more
likely to be found in redder (and thus presumably more metal-rich) GCs over
blue (metal-poor) ones, as has consistently been found in elliptical galaxies
(Kundu et al., 2003, 2008; Sivakoff et al., 2007; Luan et al., 2018). Figure
2b shows the $B-V$ vs. $V-I$ colors of XRB-hosting GCs, compared to the full
GC population. For reference, the evolutionary model for solar-metallicity
clusters (Bruzual & Charlot, 2003) is plotted in black, and the direction that
the clusters may shift due to reddening is indicated by the arrow. To
determine whether the XRB GCs are statistically redder than the general GC
population, we perform a two-sample independent Wilcoxon test on the $B-V$ and
$V-I$ distributions of the two populations using ranksum from the SciPy
statistical Python package (see Jordán et al., 2007). The $B-V$ and $V-I$
distributions are plotted in Figure 2c. We find that XRB GCs show no color
preference in any way that is inconsistent with the underlying GC population.
The same test performed on the younger clusters indicates a possible tendency
towards bluer colors (and thus presumably younger, more metal-poor systems)
for the intermediate-age XRB clusters, although our sample size is too small
to determine at this time, and there is a non-negligible chance that at least
some of these clusters are spurious associations (as discussed at the
beginning of §4).

###  4.2 XRB cluster masses

Figure 3: Histogram of cluster masses, separated into (a) GCs, (b)
intermediate-age clusters, and (c) young clusters, excluding clusters from NGC
4569. Dashed lines indicate the bins containing the 22 candidate hosts with
XRBs that have multiple cluster associations. In panel (d), the fraction of
clusters that hosts at least one XRB is given as a percentage of the total
clusters per age population per mass bin. The horizontal lines show the
boundaries of each bin.

While the connection between star cluster mass and XRBs has been explored
extensively for GCs in elliptical galaxies, little work has been done to date
to characterize XRB-hosting clusters in late-type galaxies. The top panel of
Figure 3 shows histograms of the masses of PHANGS clusters, separated by (a)
GC, (b) intermediate-age cluster, and (c) very young cluster populations,
where the XRB hosts are highlighted. The bottom panel (Figure 3d) illustrates
the percentage of clusters that host an XRB as a function of cluster mass.

It is clear from Figure 3a that XRB-hosting clusters occupy the more massive
end of the GC mass distribution; that is, bright LMXBs are preferentially
found in more massive GCs. In addition, in Figure 3d we see that the fraction
of GCs that host XRBs increases significantly at higher mass bins. This is
true globally and on a galaxy-by-galaxy basis. The fraction of very young
clusters that host XRBs also increases with increasing mass, with a nearly 2
order of magnitude difference over the range plotted in Figure 3d. The
statistics for intermediate-age XRB host clusters are quite poor, and these
XRBs do not show any evidence of preferentially forming in higher-mass
clusters.

Figure 4: The X-ray luminosities of XRB-hosting clusters as a function of cluster mass. Open symbols represent the 22 candidate clusters of the 8 XRBs with multiple cluster associations. Clusters associated with the same XRB are connected by a dotted line. Panels (b) and (c) represent population sets (i) and (ii) respectively, for which the assumed parent cluster of multiply-associated XRBs are indicated in magenta. The best-fit line for the populations with a statistically significant M⋆-$L_{\rm X}$ correlation — i.e. the intermediate-age XRB cluster and the total non-GC cluster populations — is given (green dashed and grey dash-dotted lines, respectively), where $x=\rm{log}$ M⋆ in solar units. The linear fits to set (i) for GCs (not shown) is $-0.82x+42.01$.  Table 4: Most significant correlations per population set | Set i | Set ii | Set iii
---|---|---|---
Very Young | 0.59a (0.07) | 0.49a (0.12) | 0.17a (0.59)
Intermediate-age | 0.99a ($<$ 0.01) | 0.95a (0.01) | 0.99a ($<$0.01)
Total non-GCs | 0.69a ($<$ 0.01) | 0.57a (0.02) | 0.34b (0.19)
GCs | -0.53b (0.03) | — | -0.49b (0.05)

###  4.3 X-ray luminosity vs. cluster mass

The environment within dense star clusters is conducive to efficient XRB
formation, but whether more massive clusters should be expected to retain
multiple XRBs at a time is difficult to determine. Observational studies
suggest that the brightest metal-rich GCs likely host multiple LMXBs (Angelini
et al., 2001; Kundu et al., 2007); also, multiple LMXBs have been confirmed
within Galactic GCs, such as the metal-poor M15 (White & Angelini, 2001;
Arnason et al., 2015). Johns Mulia et al. (2019) approached the question of
XRBs in younger clusters by comparing the X-ray luminosities to cluster mass,
but found no evidence of a correlation for HMXBs. We adopt a similar analysis
here.

To quantify any possible correlation between X-ray luminosity and host cluster
mass, we run two correlation analyses from the SciPy statistical library: the
Pearson correlation, and the Spearman correlation (Zwillinger & Kokoska, 2000;
Virtanen et al., 2020). The Pearson correlation measures the degree to which
two variables are linearly related, while the Spearman correlation measures
for nonlinear, monotonic dependence or association. A correlation is
considered statistically significant if it has a p-value $\leq 0.05$, and the
strength of the correlation is measured on a scale from 0 (uncorrelated) to 1
(perfect correlation). We analyze the XRB-hosting clusters in each age
population (GCs, intermediate-age, and young). We also analyze the combined
non-GC population, since the young and intermediate-age populations are
spatially intertwined and the physical distinction between them is unclear.

The total cluster sample is shown in Figure 4a, including all candidate
cluster hosts of multiply-associated XRBs. Because 8 of the 33 XRBs have
multiple cluster associations spanning a wide range of masses, the results of
our correlation analysis depends heavily on the true parent clusters of
multiply-associated XRBs. In total, there are 2304 unique combinations of
cluster counterparts on which the analyses are run.

Since there is only 1 multiply-associated GC XRB, which has only 2 candidate
GC counterparts, the M⋆-$L_{\rm X}$ analysis of the GC population is simple:
there are only 2 unique combinations of GC counterparts, and both show, for
the first time, a statistically significant M⋆-$L_{\rm X}$ anticorrelation.
This is in contrast to the analysis of GCs within M81, which demonstrated the
lack of a clear relation (Hunt et al., 2023), and within NGC 4278, which
indicated a weak positive correlation between $L_{\rm X}$ and brighter
magnitudes (i.e. higher cluster mass; Fabbiano et al., 2010). More complicated
are the non-GC cluster populations, which include 7 multiply-associated XRBs
with 20 cluster candidates. Young and intermediate-age clusters tend to share
similar environments, leading to 2 of the multiply-associated XRBs having both
young and intermediate-age candidate counterparts. Between young and
intermediate-age clusters, there is a total of 1152 unique counterpart
combinations. Of these, 62% yield statistically significant M⋆-$L_{\rm X}$
correlations among the combined (total) young and intermediate-age XRB cluster
population. In analyzing these populations independently, we find that only 6
cluster combinations (0.5%) yield a statistically significant M⋆-$L_{\rm X}$
correlation for young clusters alone. All combinations of intermediate-age XRB
clusters show a nearly perfect linear M⋆-$L_{\rm X}$ correlation, though it
must be noted that there are relatively few intermediate-age XRB clusters, and
the 1152 combinations yield only 6 unique intermediate-age configurations. A
larger cluster analysis is required to determine the robustness of this trend.

To better assess how the combination of cluster candidates impacts the
correlation models, we highlight the analysis of 3 sets of clusters, for which
the parent cluster of each multiply-associated XRB is assumed to be: (i) the
most massive candidate cluster; (ii) the median mass candidate cluster, with
preference given to the more massive of two clusters if no single median is
available; and (iii) the least massive candidate cluster. The results of the
correlation analysis on each of these sets is summarized in Table
LABEL:tab:m-lx_stats. Interestingly, the trials that include either the
highest mass clusters or the median mass clusters (sets i and ii) show strong
positive M⋆-$L_{\rm X}$ correlations at a statistically significant level for
both the intermediate-age cluster and the total non-GC populations. The linear
fits to the non-GC populations of sets (i) and (ii) are shown in Figure 4b and
4c, respectively, with the assumed hosts highlighted in magenta. It is clear
that, of the three cluster sets, the high-mass trial provides the strongest
correlation to the highest statistical significance, followed by the median-
mass set. The set which assumes the least massive cluster is the XRB host, set
(iii), gives the lowest statistical significance nearly across the board.

There is no definitive way of knowing which of the 2304 cluster combinations
describe the true parent cluster population with the data used in this study.
Since we find in §3.2 that the fraction of clusters that host XRBs appears to
increase with increasing mass bins, and because clusters with a greater number
of stars (i.e. greater mass) provide more opportunities for the formation of
an XRB, it is reasonable to presume that the true host clusters for at least
some of the 8 multiply-associated XRBs are likely higher mass. If this is the
case, then based on the results from the correlation analyses on set (i) and
(ii) in Figure 4, there is likely a strong correlation between X-ray
luminosity and mass among the total non-GC X-ray clusters population. However,
this correlation does not appear to hold among the very young host cluster
subsample. For GCs, the M⋆-$L_{\rm X}$ anticorrelation is strongest for the
higher-mass candidate host, but is statistically significant either way.

While, for non-GC XRB hosts, these correlations may indicate the presence of
multiple XRBs within more massive clusters, there is no indication that these
clusters have higher luminosity than the general field XRB population.
Furthermore, the correlation for GCs is negative, despite being more massive
than the non-GC population. Therefore, we conclude that the M⋆-$L_{\rm X}$
correlation for clusters younger than 400 Myr may not be due to the presence
of multiple XRBs within X-ray emitting clusters, though the possibility cannot
be precluded. This result assumes all XRBs included are indeed found within or
near their parent clusters, and the presence of spurious cluster associations
may further complicate this relation.

###  4.4 XRB cluster radii

Figure 5: Histogram of cluster effective radii, separated into (a) GCs, (b)
intermediate-age clusters, and (c) very young clusters, excluding clusters
from NGC 4569. Dashed lines represent candidate hosts for multiply-associated
XRBs. In panel (d), the fraction of clusters that hosts at least one XRB is
given as a percentage of the total clusters per age population per bin. The
horizontal lines show the boundaries of each bin.

The size distribution of clusters in late-type galaxies remains poorly
understood. It has been proposed that star clusters undergo rapid expansion
within their first 10 Myr (Bastian et al., 2012; Chandar et al., 2016; Johns
Mulia et al., 2019). Indeed, there appears to be a preference for XRBs to form
in young clusters with smaller radii, a correlation that flattens from
intermediate-age clusters to GCs, although this might not necessarily be a
consequence of rapid expansion. Generally, young clusters tend to be more
compact than older ones (Johns Mulia et al., 2019). Nevertheless, the size
distribution of XRB hosts appears dependent on cluster age.

Figure 5 shows the size distribution of PHANGS clusters, binned by radius and
divided into (a) GC, (b) intermediate-age cluster, and (c) very young cluster
populations. As with Figure 3, panel (d) shows the percentage of clusters that
host an XRB per age group per radius bin. In all cases, XRBs are
preferentially seen in clusters with smaller radii — although a Wilcoxon test
performed on these populations imply only XRB-hosting GCs are statistically
smaller than their general cluster population. While the percentage of GCs
that host XRBs decreases with increasing radius bin (affirming a strong
dependence on cluster size), the same is not seen for intermediate-age and
very young clusters. In particular, the fraction of very young clusters that
host an XRB appears to increase, albeit slightly, with increasing radius bin.
If these sources are indeed XRB-hosting clusters and not spurious
associations, then this result, in conjunction with the results from §4.2,
suggests that cluster mass is more important for XRB formation in young
clusters than radius, whereas both cluster mass and radius appear to play a
strong role in XRB formation in GCs, as seen in early-type galaxies (e.g.
Paolillo et al., 2011).

###  4.5 XRB cluster densities

Figure 6: Histogram of cluster densities, separated into (a) GCs, (b)
intermediate-age clusters, and (c) very young clusters, excluding clusters
from NGC 4569. Dashed lines represent candidate hosts for multiply-associated
XRBs. In panel (d), the fraction of clusters that hosts at least one XRB is
given as a percentage of the total clusters per age population per bin. The
horizontal lines show the boundaries of each bin.

Figure 6 is a histogram of the densities of (a) GC, (b) intermediate-age
cluster, and (c) very young cluster populations, where, as before, panel (d)
shows the percentage of clusters that host an XRB per age group per density
bin.

The number of XRB-hosting clusters remains relatively flat over increasing
density bins for GCs and young clusters, with all of the singly-associated
XRBs appearing in the lowest-density bin for intermediate-age clusters (panels
a, b, and c). When observed as a percentage of the total population, however,
we see that the fraction of clusters that host XRBs increases drastically with
increasing density for both GCs and very young clusters (panel d). The
fraction of very young clusters that host XRBs in the highest density bin is a
factor of 22 times higher than that of the lowest density bin. For GCs, the
difference between the two bins is a factor of 15.

There are a couple of interesting conclusions that can be drawn from Figure
6d. First, the number of GCs and very young clusters that host XRBs is
strongly dependent on cluster densities. This may be due to more frequent
dynamical interactions in denser clusters, which could enhance the number of
tight binaries that can form in a system (Jordán et al., 2004; Peacock et al.,
2009). Second, across the total galaxy sample, the fraction of GCs that host
XRBs is at least twice that of young clusters for a given density bin. On a
galaxy-by-galaxy basis, the difference between age populations is much less
dramatic, but is generally consistent with GCs being more likely to host XRBs
per density bin. This difference becomes greater if some of the younger
clusters are determined to be spurious associations due to the chance
superpositon on an XRB with a young cluster in star-forming regions.

##  5 XRB-hosting star clusters in late vs. early type galaxies

The majority of studies regarding XRBs within star clusters focus on the GCs
of elliptical galaxies (e.g. Angelini et al., 2001; Jordán et al., 2004; Smits
et al., 2006; Peacock & Zepf, 2016; Lehmer et al., 2020; Ferrell et al., 2021;
Riccio et al., 2022). A few, more recent papers report on XRB associations
with GCs in late-type galaxies (e.g. Pfahl et al., 2003; Peacock et al., 2009;
Generozov et al., 2018; Hailey et al., 2018; Hixenbaugh et al., 2022), and
fewer still include XRBs in younger clusters (e.g. Rangelov et al., 2012;
Johns Mulia et al., 2019; Hunt et al., 2023; Avdan et al., 2022) In this
paper, we take the first step towards (i) assessing the fraction of star
clusters — both old and young — that host XRBs in late-type galaxies, and (ii)
comparing the fraction of field vs. cluster XRBs in late-type vs. early-type
galaxies.

For each galaxy in the sample, the sky area over which the full X-ray source
catalog extends is always larger than the sky area encompassing the PHANGS
cluster catalog (i.e., the magenta contours in Figure 1). However, the number
of XRBs expected to be observed above the completeness limit, corrected for
the CXB, is estimated using models for the XRB X-ray luminosity functions of
late-type galaxies generated by L19. These models rely on measurements of M⋆
and SFR, which are confined to the $K_{s}$-band isophotal ellipses. To compare
the field-to-cluster XRB ratios of each galaxy, we must restrict our analysis
to the region covered by both the PHANGS-HST catalog and the corrected XRB
estimates — i.e. the intersection of the isophotal ellipses with the PHANGS
contours in Figure 1. For the sake of consistency, we also use this region for
our analysis of XRB clusters vs. the total cluster population. The
$K_{s}$-band isophotal ellipses are reasonably well-covered by the cluster
catalogs for most of the galaxies; in total, only 2 of the 33 XRB-hosting
clusters fall outside of the $K_{s}$-band ellipses. Out of the remaining XRB
clusters, 22 exceed the X-ray completeness limit of their respective host
galaxies.

Table 5: Statistics of clusters inside the $K_{s}$-band isophotal ellipse of each galaxy | GCs (XRB%) |  $\geq 10$ Myrs (XRB%) |  $<10$ Myrs (XRB%) | Total (XRB%)
---|---|---|---|---
NGC 0628 | 202 (1.5%) | 196 (0.5%) | 200 (0.5%) | 598 (0.7%)
NGC 3351 | 83 (3.6%) | 79 (0+2.5%) | 151 (0+1.3%) | 313 (1.6%)
NGC 3627 | 536 (0.6%) | 494 (0.2%) | 206 (0.5%) | 1236 (0.4%)
NGC 4321 | 216 (0.5%) | 327 (0%) | 323 (0%) | 866 (1.2%)
NGC 4569 | 110 (0%) | 371 (0%) | 46 (0%) | 527 (0%)
NGC 4826 | 43 (7%) | 73 (0%) | 59 (0%) | 175 (1.7%)
Total* | 1080 (1.2%) | 1169 (0.1+0.1%) | 939 (0.5+0.2%) | 3188 (0.7%)
$\rm{}^{(excl.~{}NGC4569)}$ |  |  |  |

###  5.1 Properties of XRB-hosting clusters

In elliptical galaxies, between $4$ and $10\%$ of GCs host LMXBs (e.g.
Maccarone et al., 2003; Jordán et al., 2007; Brassington et al., 2010; Mineo
et al., 2013; Luan et al., 2018). Analyses of these XRB hosts suggest they are
more massive, more luminous, radially smaller, denser, and redder than the
general GC population (e.g. Kundu et al., 2002; Sivakoff et al., 2007; Peacock
et al., 2009; Brassington et al., 2010; Paolillo et al., 2011; Riccio et al.,
2019). Further investigations suggest that metallicity — more so than cluster
age — plays a role in the production of LMXBs (Kundu et al., 2003, 2008;
Sivakoff et al., 2007; Fabbiano, 2006; Peacock et al., 2010; D´Ago, G. et al.,
2014; Luan et al., 2018; Riccio et al., 2022).

In late-type galaxies, the percentage of GCs that host LMXBs is still an
unsettled matter, though a recent study of M81 suggest a fraction consistent
with that of elliptical galaxies — around $4\%$ (Hunt et al., 2023).
Similarly, XRB-hosting GCs in M81 tend to be more compact, denser, and more
massive than the rest of the GC population (Hunt et al., 2023, see also
Bregman et al. 2006 in the context of the Milky Way galaxy). As far as young
clusters go, a study of XRB-hosting clusters younger than 400 Myr in the
Antennae galaxies and NGC 4449 suggests that more massive and denser
intermediate-age clusters may be more likely to host XRBs, but that this
preference is not reflected in the the population of clusters younger than 10
Myr (Johns Mulia et al., 2019).

The total number of clusters and the percent that host XRBs in each galaxy is
given in Table 5. Among these late-type galaxies, the percentage of clusters
that host bright XRBs above the completeness limits of their host galaxy
varies between $0.5-7\%$ for GCs, $0.2-2.5\%$ for intermediate-age clusters,
and $0.5-1.3\%$ for young clusters, excluding NGC 4569 (for which no XRB-
hosting clusters were identified). Combining all clusters within our galaxy
sample (excluding NGC 4569), the percentage of XRB hosts among GCs,
intermediate-age clusters, and young clusters is 1.2%, and 0.1+0.1%, 0.5+0.2%,
respectively. Tentatively, a total of 0.7% of all compact star clusters within
the PHANGS catalog, irrespective of age, host XRBs. As a word of caution, we
note again that the statistics for clusters younger than 400 Myr may be
inflated due to the high probability of a chance superposition with an
unrelated cluster in densely populated, active regions, as determined in §4.

XRBs in late-type galaxies appear to prefer smaller, denser, brighter, and
more massive GCs, just as they do in elliptical galaxies (see Figures 3, 5,
and 6d). This is consistent with the notion that clusters with more stars
(that is, more massive and brighter) that are in closer proximity to each
other (denser and smaller) provide ample opportunities for the formation of a
close binary (Smits et al., 2006; Jordán et al., 2007; Peacock et al., 2009,
2010).

In contrast to elliptical galaxies, we find that XRB-hosting GCs in the star-
forming galaxies under consideration are not redder than the general GC
population — rather, their colors are consistent with those of the full
population (see Figure 2c). For early-type galaxies, it is believed that
metallicity plays a large role in this correlation, possibly due to enhanced
tidal capture rates and core concentrations (smaller radii) in higher-
metallicity GCs, which in turn facilitates the formation of LMXBs (Jordán,
2004; Schulman et al., 2012; Ivanova, 2013; Vulic et al., 2018; Hixenbaugh et
al., 2022). It is interesting to note that, while GCs typically display
bimodal colors corresponding to metal-rich and metal-poor populations (e.g.
Peacock & Zepf, 2016; Hixenbaugh et al., 2022), the GCs in our sample lack a
clear distinction between red and blue GCs (Figure 2b and 2c). This could be
due to the fact that (i) the metal poor “mode” becomes very broad when
different late type galaxies (each with different star forming histories) are
added together, and (ii) the metal rich mode may be triggered by mergers,
which are rarer in spirals (Muratov & Gnedin, 2010; Pfeffer et al., 2023).

Regarding younger clusters, we find strong evidence that brighter, more
massive, and denser clusters younger than 10 Myr are more likely to host XRBs
than other clusters of similar ages (see Figures 2, 3d, and 6d). The sample of
intermediate-age XRB-hosting clusters (younger than 400 Myr) is too small to
draw a solid conclusion, but their properties appear consistent with the
broader intermediate-age population. This is possibly in contrast to NGC 4449
and the Antennae galaxies, which show a tentative correlation between XRB
formation and mass and density in intermediate-age clusters, but no such
correlation in young clusters (Johns Mulia et al., 2019). Larger statistics
are needed to definitively confirm whether a trend exists, as only 11
intermediate-age and 17 young XRB-hosting clusters were observed in the NGC
4449 and the Antennae galaxies, compared to the 17 non-GC XRBs we analyze
here. Furthermore, the likelihood of chance superpositions with unrelated
clusters causing spurious associations to be included in our analysis may
further complicate matters. It is interesting that such strong correlations
between the presence of an XRB and cluster mass and density are identified;
however, this could point to the associations of singly-associated XRBs, at
the very least, being real, and that the background contamination is minimal.

Table 6: Statistics of X-ray sources inside the isophotal ellipse and PHANGS footprint of each galaxy  |  |  |  | Estimated XRBs | Observed XRB-hosting clusters
---|---|---|---|---|---
| M⋆,encl | SFRencl | X-ray |  |
| ($10^{9}$ M⊙) | (M⊙ yr-1) | Sources | LMXBs | HMXBs | GCs |  $\geq 10$ Myrs |  $<10$ Myrs
NGC 0628 | 2.64 | 0.24 | 33 | 6 | 7 | 3 (50%) | 1 (14.3%) | 1 (14.3%)
NGC 3351 | 7.07 | 0.50 | 30 | 9 | 8 | 3 (33.3%) | 0+2 (0+25%) |  $3^{+2}$ (37.5+25%)
NGC 3627 | 19.4 | 1.81 | 48 | 15 | 14 | 3 (20%) | 1 (7.1%) | 1 (7.1%)
NGC 4321 | 14.3 | 1.65 | 45 | 12 | 15 | 1 (8.3%) | 0 | 0
NGC 4569 | 26.4 | 1.02 | 17 | 10 | 4 | 0 | 0 | 0
NGC 4826 | 19.2 | 0.39 | 20 | 15 | 4 | 3 (20%) | 0 | 0
|  | Total* | 176 | 67 | 52 | 13 (19.4%) | 2+2 (3.8+3.8%) | 5+2 (9.6+3.8%)
|  | $\rm{}^{(excl.~{}NGC4569)}$ |  |  |  |  |  |

###  5.2 Cluster XRBs vs. field XRBs

Studies of elliptical galaxies have shown that roughly $25-70\%$ of LMXBs are
found within GCs, while the rest are found in the field (e.g. Sarazin et al.,
2000; Angelini et al., 2001; Kundu et al., 2002, 2003; Maccarone et al., 2003;
Fabbiano et al., 2010; Luan et al., 2018). A growing body of work identifying
LMXBs in late-type galaxies suggests that the ratio of GC LMXBs to field LMXBs
may be smaller in spiral galaxies: an analysis of XRBs in M101 revealed only
$2\%$ of LMXBs are found in GCs (Chandar et al., 2020), while $12\%$ of LMXBs
in M81 are found in GCs (Hunt et al., 2023). On the other hand, around $10\%$
of XRBs appear in clusters younger than 400 Myr in the handful of late-type
galaxies studied so far (Chandar et al., 2020; Palmore et al., 2022), with a
fraction of at least $25\%$ observed in NGC 4449 and the Antennae galaxies
(Johns Mulia et al., 2019). Tentatively, this may suggest the fraction of
HMXBs in young clusters is higher than the fraction of LMXBs in GCs in late-
type galaxies.

While we do not directly classify the XRBs in this study on the basis of their
donor mass, we can estimate the relative number of LMXBs and HMXBs in the
target galaxies using the best-fit X-ray luminosity function of L19. Table 6
shows the total number of X-ray sources within the region over which the
$K_{s}$-band isophotal ellipses intersects the PHANGS region, the number of
XRBs above the $90\%$ X-ray completeness limits in that region, and the
estimated number of LMXBs and HMXBs. The XRB estimates take into account
possible CXB contamination, so the sum of expected LMXBs and HMXBs is always
lower than the total number of X-ray sources. We compare the expected number
of LMXBs and HMXBs to the number that we observe within GCs and clusters
younger than $\sim 400$ Myr over the same region and above the $90\%$
completeness limit of their respective galaxies. We also calculate the
percentages (in parentheses) of LMXBs found in GCs and HMXBs found in non-GCs.

There are a few conclusions that may be drawn from the results presented in
Table 6. First, the fraction of LMXBs within GCs is on the low end of the
estimates from elliptical galaxies; while in early-type galaxies anywhere
between $25-70\%$ of LMXBs are found in GCs, the number is between $\sim
8-50\%$ within our sample of late-type galaxies, or 19.4% over all galaxies,
excluding NGC4569. This is tentatively consistent with previous studies of
cluster XRBs in spiral galaxies (Chandar et al., 2020; Hunt et al., 2023), but
spans a broader range of values.

One possible cause for this apparent suppression of GC-LMXBs could be an
inherent difference in the metallicity distribution of GCs in certain late-
type galaxies, since the presence of XRBs in GCs is correlated with higher
metallicities (e.g. Kundu et al., 2008; Luan et al., 2018). The (often
bimodal) metallicity distribution of GCs is thought to arise from galaxy
mergers and/or GC disruption, the former of which is unlikely in spiral
galaxies (Muratov & Gnedin, 2010; Pfeffer et al., 2023). Another explanation
may be related to the difference in star formation rate. Since new star
formation is essentially negligible in early-type galaxies, the growth of the
XRB field population may slow significantly — particularly because
intermediate-mass donor stars are at least a factor of 5 times more likely to
form an XRB than low-mass stars in the disk of a galaxy (Pfahl et al., 2003),
and elliptical galaxies are dominated exclusively by low-mass stars.
Meanwhile, LMXBs can continue to efficiently form within GCs through dynamical
interactions. This ‘stagnation’ effect may be mitigated if GCs are responsible
for seeding a significant fraction of the field LMXB population (e.g. Lehmer
et al., 2020). The difference in field-to-cluster LMXB ratios between early-
type and late-type galaxies may provide further evidence that ejection from
GCs has limited impact on the total LMXB population (see, e.g., Piro &
Bildsten, 2002; Kundu et al., 2007; Peacock & Zepf, 2016; Kremer et al.,
2018).

The second conclusion that may be drawn from Table 6 is that the fraction of
(presumably high-mass) XRBs that appear in non-GCs is almost consistently
smaller than the fraction of LMXBs that appear in GCs across nearly all
galaxies (except NGC 3351). These fractions do not appear directly correlated
to the total M⋆ or SFR of the host galaxy. This is inconsistent with results
from M101 — for which $\sim 2\%$ of LMXBs are in GCs and $\sim 5\%$ of HMXBs
are in very young clusters — but does match findings from M83 ($\sim 13\%$ vs.
$\sim 3\%$; Hunt et al., 2021) and M81 ($\sim 12\%$ vs. $\sim 2\%$; Hunt et
al., 2023). Furthermore, while we do not measure metallicity directly in our
cluster sample, it is interesting to note that there is no indication of a
color (ergo metallicity) bias among GC LMXBs in our sample of late-type
galaxies. These results are, of course, contingent on our population of
clusters that host singly-associated XRBs being positively identified. As
there is a non-negligible probability of an XRB within an active star-forming
region being associated with an unrelated cluster (see §4), our results must
be broached with caution, though previous studies must also suffer from this
issue. Ultimately, there appear to be variations between galaxies that
requires more exploration to fully understand.

##  6 Summary

This study represents one of the largest systematic investigation of XRBs in
both GCs and young clusters in late-type galaxies to date. We analyze the
properties of clusters that host XRBs, broken into three distinct populations:
ancient GCs, intermediate-age clusters between $\sim 10$ and $400$ Myr old,
and very young clusters younger than 10 Myr. Our analysis of the clusters in
this sample of late-type galaxies yields a number of interesting results:

  * •

While we confirm that the chance of an XRB being wrongly associated with an
unrelated, nearby GC is low (in agreement with the simulations from Hunt et
al. 2023), we find there is a non-negligible probability of a chance
superposition between an XRB and an unrelated cluster in densely populated,
actively star-forming regions such as the spiral arms, which may inflate the
statistics for clusters younger than $\sim 400$ Myr. Therefore, the following
results for XRBs with non-GC counterparts should be approached with caution.

  * •

Clusters that host X-ray sources tend to be brighter than the general
population for both the very young clusters and GCs. This may not necessarily
be the case for intermediate-age clusters.

  * •

GCs in these late-type galaxies do not appear to prefer redder clusters,
contrary to what has been repeatedly found for elliptical galaxies. On the
other hand, there is marginal evidence that XRBs preferentially form in bluer
intermediate-age clusters, though better statistics are required.

  * •

As found in previous studies, X-ray emitting GCs tend to be more massive,
denser, and have smaller effective radii than the general GC population. Among
young cluster populations, there is a higher fraction of very young clusters
that host XRBs in denser, more massive populations.

  * •

There may be a strong positive correlation between cluster mass and X-ray
luminosity in both intermediate-age X-ray emitting clusters and the cumulative
non-GC cluster population, though a greater sample of XRBs with known cluster
hosts is needed. This could indicate the presence of multiple XRBs, though
these clusters are not more X-ray luminous than the general field XRB
population. On the other hand, there is a statistically significant M⋆-$L_{\rm
X}$ anticorrelation for GCs.

  * •

The percentage of GCs that host XRBs in late-type galaxies varies widely
between galaxies ($0.5-7\%$). This is compared to the $4-10\%$ seen in early-
type galaxies. On the other hand, these percentages are $0.2-2.5\%$ for
intermediate-age clusters and $0.5-1.3\%$ for young clusters, depending on the
true parent clusters.

  * •

Whereas up to $70\%$ of LMXBs in early-type galaxies are found within their
parent GCs, we find that a potentially smaller percentage ($\sim 8-50\%$) are
found in GCs in late-type galaxies, in agreement with statistics from previous
late-type galaxy studies.

Throughout our analysis, we note that our ability to gain insight into the
population of XRBs within intermediate-age clusters is hindered by the small
number of such sources in our sample. Whether this is an inherent property of
cluster XRBs or a matter of insufficient data can only be addressed by a
larger study of XRBs in nearby, late-type galaxies. Such a study would benefit
from deeper X-ray observations (including a revisit of NGC 4569), optical
observations covering a larger sky area, X-ray and optical spectroscopy, and
metallicity measurements of compact star clusters. This type of data would
give us a clearer picture of these sources by allowing us to identify black
hole vs. neutron star XRBs within and around clusters, constrain the cluster-
to-field ratios, and study the cluster metallicity dependence of XRBs in late-
type galaxies.

Q.H. is partially funded by a Rackham Merit Fellowship, awarded by the
University of Michigan Rackham Graduate School. Q.H. thanks the LSSTC Data
Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant
no. 1829740, the Brinson Foundation, and the Moore Foundation; her
participation in the program has benefited this work. The Digitized Sky
Surveys were produced at the Space Telescope Science Institute under U.S.
Government grant NAG W-2166. The images of these surveys are based on
photographic data obtained using the Oschin Schmidt Telescope on Palomar
Mountain and the UK Schmidt Telescope. The plates were processed into the
present compressed digital form with the permission of these institutions.

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