# A Panchromatic Study of the X-ray Binary Population in NGC 300 on Sub-
Galactic Scales

Breanna A. Binder Department of Physics & Astronomy, California State
Polytechnic University, 3801 W. Temple Ave, Pomona, CA 91768, USA  Rosalie
Williams  Department of Physics & Astronomy, California State Polytechnic
University, 3801 W. Temple Ave, Pomona, CA 91768, USA  Jacob Payne  Department
of Physics & Astronomy, California State Polytechnic University, 3801 W.
Temple Ave, Pomona, CA 91768, USA  Michael Eracleous Department of Astronomy &
Astrophysics and Institute for Gravitation and the Cosmos, The Pennsylvania
State University, 525 Davey Lab, University Park, PA 16802, USA  Alexander
Belles Department of Astronomy & Astrophysics and Institute for Gravitation
and the Cosmos, The Pennsylvania State University, 525 Davey Lab, University
Park, PA 16802, USA  Benjamin F. Williams Department of Astronomy, University
of Washington, Box 351580, Seattle, WA 98195, USA

###### Abstract

The population-wide properties and demographics of extragalactic X-ray
binaries (XRBs) correlate with the star formation rates (SFRs), stellar masses
(M⋆), and environmental factors (such as metallicity, $Z$) of their host
galaxy. Although there is evidence that XRB scaling relations ($L_{\rm X}$/SFR
for high mass XRBs [HMXBs] and $L_{\rm X}$/M⋆ for low mass XRBs [LMXBs]) may
depend on metallicity and stellar age across large samples of XRB-hosting
galaxies, disentangling the effects of metallicity and stellar age from
stochastic effects, particularly on subgalactic scales, remains a challenge.
We use archival X-ray through IR observations of the nearby galaxy NGC 300 to
self-consistently model the broadband spectral energy distribution and examine
radial trends in its XRB population. We measure a current ($<$100 Myr) SFR of
0.18$\pm$0.08 M⊙ yr-1 and M⋆= $(2.15^{+0.26}_{-0.14})\times 10^{9}$ M⊙.
Although we measure a metallicity gradient and radially resolved star
formation histories that are consistent with the literature, there is a clear
excess in the number of X-ray sources below $\sim 10^{37}$ erg s-1 that are
likely a mix of variable XRBs and additional background AGN. When we compare
the subgalactic $L_{\rm X}$/SFR ratios as a function of $Z$ to the galaxy-
integrated $L_{\rm X}$-SFR-$Z$ relationships from the literature, we find that
only the regions hosting the youngest ($\lesssim$30 Myr) HMXBs agree with
predictions, hinting at time evolution of the $L_{\rm X}$-SFR-$Z$
relationship.

X-ray binary stars (1811) — High-mass X-ray binary stars (733) — Compact
objects (288) — Spiral galaxies (1560)

††facilities: CXO, Swift (UVOT), GALEX, 2MASS, Spitzer (IRAC and MIPS),
WISE††software: astropy (Astropy Collaboration et al., 2013, 2018, 2022), CIAO
(Fruscione et al., 2006)

##  1 Introduction

Some of the most useful laboratories for studying binary star evolution are
X-ray binaries (XRBs), which contain a compact object (either a neutron star,
NS, or a black hole, BH) accreting mass from a stellar companion. High-mass
XRBs (HMXBs) contain a massive ($\geq 8$M⊙), often post-main sequence,
companion and represent a critical phase in the history of massive star
evolution: their current properties are the result of the orbital and mass-
exchange history of the progenitor binary, and the ultimate fate of some of
these binaries is to form millisecond pulsars, short gamma-ray bursts (GRBs),
or double compact object systems capable of generating detectable
gravitational wave (GW) signals. Feedback from HMXBs is now recognized as an
important contributor to heating the intergalactic medium and re-ionizing the
early universe (Kaur et al., 2022).

It is now well-established that the X-ray luminosity functions (XLFs) and
population-integrated luminosities ($L_{\rm X}$) of HMXBs scale with host
galaxy star formation rate (SFR; Grimm et al., 2003; Ranalli et al., 2003;
Gilfanov et al., 2004; Mineo et al., 2012), while those of low-mass XRBs
(LMXBs, which typically contain companions of $\sim$1-2 M⊙) track host galaxy
stellar mass (M⋆; Boroson et al., 2011; Zhang et al., 2012; Lehmer et al.,
2010; Gilfanov, 2004; Colbert et al., 2004). Previous studies have attempted
to empirically measure the dependence of XRB scaling relations ($L_{\rm
X}^{\rm HMXB}$/SFR or $L_{\rm X}^{\rm LMXB}$/M⋆) on physical parameters such
as age (Kim & Fabbiano, 2010; Lehmer et al., 2014) and metallicity (Basu-Zych
et al., 2013, 2016; Brorby et al., 2016; Tzanavaris et al., 2016; Douna et
al., 2015). Population synthesis models predict that the these scaling
relations in XRB populations can vary by up to an order of magnitude depending
on the metallicity and age of the XRB population (Linden et al., 2010; Fragos
et al., 2008, 2013a, 2013b; Zuo et al., 2014). Nearby galaxies, where XRB
populations can be spatially resolved with X-ray telescopes such as Chandra
and XMM-Newton, span a relatively narrow range in metallicities while hosting
a larger diversity of stellar ages when compared to distant galaxies, and have
experienced a wide variety of star formation histories (SFHs) and
merger/interaction histories. As such, we do not yet precisely understand how
the XRB scaling relations vary with SFH and metallicity.

Studies of the HMXB $L_{\rm X}$/SFR fraction as a function of metallicity
(hereafter referred to as the $L_{\rm X}$-SFR-$Z$ relation; e.g., Brorby et
al., 2014; Basu-Zych et al., 2016) have largely concentrated on high-M⋆
galaxies with high-SFRs where the XRB populations are expected to be dominated
by HMXBs and the XLFs are well-sampled. However, these studies implicitly
assume that the global average SFR and metallicity of galaxies adequately
reflect the sub-kpc scale environments in which HMXBs form. Due to their short
lifetimes, HMXBs do not migrate more than a few hundred parsecs from their
birthplaces (Bodaghee et al., 2012; Binder et al., 2023) and these temporal
and spatial scales are shorter than those over which chemical mixing
efficiently occurs in galaxies (Kreckel et al., 2020). Local variations in
stellar populations and the HMXBs associated with them thus become “smeared
out” on galaxy-wide scales, and may complicate interpretations of the $L_{\rm
X}$/SFR scaling relation (Kouroumpatzakis et al., 2020). Optical integral
field units now make it possible to study thousands of sub-kiloparsec scale
regions across the star-forming disks of nearby galaxies, and 2D metallicity
maps demonstrate that metallicity variations occur on small scales (Williams
et al., 2022), and many nearby galaxies exhibit metallicity gradients (see,
e.g., Belfiore et al., 2019, and references therein).

The nearby (2.0 Mpc, based on measurements of the tip of the red giant branch;
Dalcanton et al., 2009) star-forming galaxy NGC 300 is a particularly suitable
target for studying XRBs in subgalactic environments. Numerous X-ray point
sources are detected within the star-forming disk, which can be imaged in a
single Chandra pointing down to limiting luminosities on the order of
$\sim$1036 erg s-1 (Binder et al., 2012, 2017), it has been observed across
nearly the entire electromagnetic spectrum, and exhibits a metallicity
gradient with metallicity decreasing by $\sim$0.08$Z_{\odot}$ per
kpc;][]Gogarten+10,Kudritzki+08. The SFR of NGC 300 is generally reported to
be $\sim 0.08-0.46$ M⊙ yr-1 by different tracers, such as the HST resolved
stellar populations (Gogarten et al., 2010), UV observations (Mondal et al.,
2019), H$\alpha$ emission (Karachentsev & Kaisina, 2013), and mid-IR
observations (Helou et al., 2004). NGC 300 contains a ULX (e.g., see review by
Binder et al., 2020, and references therein) and the luminous Wolf-Rayet + BH
binary NGC 300 X-1 (Binder et al., 2021), although the majority of XRBs in the
galaxy are of lower luminosity ($\sim 10^{36-37}$ erg s-1; Binder et al.,
2012, 2017), and the specific SFR of NGC 300 suggests that its XRB population
is likely a mix of HMXBs and LMXBs (Lehmer et al., 2019).

We use archival multiwavelength observations to self-consistently model the
stellar mass (M⋆), SFH, metallicity, and XRB population of NGC 300 on
subgalactic scales and search for trends in the $L_{\rm X}$/SFR ratio for
HMXBs as a function of metallicity and stellar age. The paper is organized as
follows: in Section 2 we summarize the archival FUV-IR observations used and
our methodology for performing aperture photometry, and present spectral
energy distributions (SEDs) for NGC 300. We additionally describe how the SED
modeling package Prospector (Leja et al., 2017) is used to extract SFHs,
stellar masses, and metallicites for the subgalactic regions within NGC 300.
In Section 3 we describe the Chandra X-ray observations and compare the
observed X-ray point source population to predicted populations based on prior
studies of the XLFs of HMXBs, LMXBs, and the AGN log$N$-log$S$ distribution.
We discuss our results in Section 4, and conclude with a summary of our main
findings in Section 5. All uncertainties correspond to the 90% confidence
interval unless otherwise specified. Table 1 summarizes the main properties of
NGC 300 that we assume throughout this work, including the distance, central
right ascension and declination coordinates, Galactic absorbing column,
$A_{V}$, and position and inclination angles. The distance to NGC 300 was
measured directly by Dalcanton et al. (2009) using the tip of the red giant
branch, and is thus independent of cosmology.

Table 1: Basic Properties of NGC 300

DistanceaaOnly the first ten entries are shown. A full machine-readable table is available from the journal. | R.A. | Dec. |  $N_{\rm H,Gal}$bbfootnotemark:  |  $A_{V}$ccfootnotemark:  |  $E(B-V)$ccfootnotemark:  | Position | Inclination
---|---|---|---|---|---|---|---
(Mpc) | (J2000) | (cm-2) | (mag) | (mag) | Angleddfootnotemark:  (∘) | Angleddfootnotemark:  (∘)
2.0 | 00:54:53.48 | -37:41:03.8 | 1021 | 0.035 | 0.011 | 111 | 45

References. — aDalcanton et al. (2009), bHI4PI Collaboration et al. (2016),
cSchlafly & Finkbeiner (2011), dde Vaucouleurs et al. (1991)

##  2 Spectral Energy Distributions

###  2.1 The Data

In this section we summarize the publicly-available multiwavelength
observations used to build the SEDs of NGC 300. Foreground stars were
identified using the Gaia DR3 catalog (Gaia Collaboration et al., 2023) and
through visual inspection of multiwavelength images and then masked with
circular apertures of radii of $\sim$15′′. The median flux density per unit
area of the corresponding subgalactic region in which each star was located
was used to approximate the the flux density of the galaxy in the masked
region.

We note that, although NGC 300 has been imaged many times with the HST, the
archival HST observations do not optimally cover the star-forming disk of NGC
300 and were obtained for a variety of filter configurations. For example,
HST/F606W imaging covers only $\sim$50% of the star-forming disk of NGC 300,
while F475W imaging covers less than $\sim$20% of the disk. Inferring galaxy-
integrated optical flux densities from HST would require a significant degree
of extrapolation and carry significant uncertainty, limiting their utility for
SED modeling. We therefore do not include these data in our SED modeling.

####  2.1.1 Spitzer IRAC and MIPS

We first retrieved Spitzer IRAC and MIPS mosaics from the NASA/IPAC Infrared
Science Archive
(IRSA)111https://irsa.ipac.caltech.edu/data/SPITZER/Enhanced/SEIP/overview.html.
The IRAC channels have central wavelengths of to 3.6$\mu$m, 4.5$\mu$m,
5.8$\mu$m, and 8.0$\mu$m, while the MIPS channels have central wavelengths of
24$\mu$m, 70$\mu$m, and 160$\mu$m. The IRAC images have an angular resolution
$\sim 2^{\prime\prime}$, while the angular resolution of the MIPS images are
$\sim 6^{\prime\prime}$, $\sim 18^{\prime\prime}$, and $\sim
40^{\prime\prime}$ at 24$\mu$m, 70$\mu$m, and 160$\mu$m, respectively (Rieke
et al., 2004). All Spitzer images have units of MJy sr-1.

We measured the surface brightness profile of NGC 300 using the the IRAC1
(3.6$\mu$m) image. We constructed a set of 25 elliptical annuli centered on
the galaxy center and using a position angle of 111∘ for the major axis. The
resulting surface brightness profile, in units of mJy arcsec-2, is shown by
the black dots in Figure 1. We modeled the surface brightness profile with an
fit an exponentially decaying function,

$I(r)=I_{0}e^{-r/R_{d}}+I_{\rm bkg},$ (1)

using SciPy’s curve_fit package, where $I_{0}$ is the central surface
brightness, $R_{d}$ is the disk scale length, and $I_{\rm bkg}$ is the
background level surface brightness. We find best-fit parameters
$I_{0}=(2.79\pm 0.14)\times 10^{-5}$ Jy arcsec-2, $R_{d}=(140.3\pm 4.7)$
arcsec (corresponding to a physical length of $1.36\pm 0.05$ kpc), and $I_{\rm
bkg}=(3.20\pm 0.47)\times 10^{-7}$ Jy arcsec-2 (see the blue line in Figure
1). This scale length is consistent with those measured in previous optical
and IR studies (Gogarten et al., 2010; Jang et al., 2020). The UV disk of NGC
300 is known to be more extended than the optical/IR disk ($\sim$2.7 kpc in
the NUV, see Mondal et al., 2019). For comparison, we also show the GALEX NUV
surface brightness profile in Figure 1, for which we measure a scale length of
$\sim$2.4 kpc (see Section 2.1.5 for a discussion of GALEX data.)

The 3.6$\mu$m scale length was then used to divide the galaxy into four equal-
width radial bins. We opted for the 3.6$\mu$m scale length, rather than the
more extended NUV scale length for three main reasons: (1) the smaller annular
regions reduce the amount of “smearing out” of key environmental factors such
at metallicity and star formation history, both of which are known to vary
across the NGC 300 disk (Kudritzki et al., 2008; Gogarten et al., 2010); (2)
since the expected AGN contribution increases with surface area, utilizing
smaller annular regions reduces the expected contribution from AGN within each
ring; and (3) at many wavelengths the relative uncertainty in the fainter flux
densities at large galactocentric radius results in more uncertain SED models
(see Section 2). The four radial bins are an “inner” bin ($r\leq R_{d}$), an
“inner middle” bin ($R_{d}<r\leq 2R_{d}$), an “outer middle” bin
($2R_{d}<r\leq 3R_{d}$), and an “outer” bin ($3R_{d}<r\leq 4R_{d}$). At a
distance of $4R_{d}$, the surface brightness falls to $e^{-4}\approx 1.8$% of
the central value; the uncertainties in the total flux density beyond $4R_{d}$
are large, diminishing our ability to confidently constrain the metallicity,
stellar mass, and SFR in these outer regions.

Figure 1: The Spitzer/IRAC 3.6$\mu$m surface brightness profile (in mJy
arcsec-2; data in black, best-fit exponential disk model in blue) of NGC 300,
with the four radial bins (each with a width of $R_{d}=1.36$ kpc) indicated by
vertical light blue lines. The GALEX NUV surface brightness (normalized to the
peak value of the 3.6$\mu$m profile) is shown in purple. The purple upward and
downward triangle indicates the scale length of the the NUV disk.

####  2.1.2 WISE

We used the WISE Image Service hosted on the NASA/IPAC Infrared Science
Archive222https://irsa.ipac.caltech.edu/Missions/wise.html to retrieve WISE
images in bands W1 (3.4 $\mu$m), W2 (4.6 $\mu$m), W3 (12 $\mu$m) and W4
(22$\mu$m). The angular resolution is 6.1′′, 6.4′′, 6.5′′ and 12′′ for the W1,
W2, W3, and W4 bands, respectively (Wright et al., 2010). Instrumental
magnitudes are converted to AB magnitudes using the magnitude zero-points for
each filter (in order of increasing central wavelength: 20.5 mag, 19.5 mag,
18.0 mag, and 13.0 mag) and the AB magnitude offset defined for each band (in
order of increasing central wavelength: 2.70 mag, 3.34 mag, 5.17 mag, 6.62
mag; Cutri et al., 2013).

####  2.1.3 2MASS

We retrieved background-subtracted FITS images of NGC 300 from the 2MASS Large
Galaxy Atlas333https://irsa.ipac.caltech.edu/applications/2MASS/LGA/intro.html
(Jarrett et al., 2003). The $J$, $H$, and $K_{s}$ bands used by 2MASS are
particularly sensitive to older stellar populations, which dominate the
stellar mass of NGC 300 (Gogarten et al., 2010). The 2MASS images have an
angular resolution of $\sim 4^{\prime\prime}$ and the photometric zero-point
calibration of the images is accurate to $\sim 2-3$%.

####  2.1.4 Swift Ultraviolet/Optical Telescope

We retrieved all available data from the Ultraviolet/Optical Telescope on the
Neil Gehrels Swift Observatory from the High Energy Astrophysics Science
Archive Research Center (HEASARC)444https://heasarc.gsfc.nasa.gov/cgi-
bin/W3Browse/swift.pl that are within 17 arcminutes of NGC 300 (Gehrels et
al., 2004; Roming et al., 2005). For each of the six filters (UVW2, UVM2,
UVW1, u, b, v), we processed the data by correcting for large scale
sensitivity variations and masking bad pixels. Prior to stacking the data, we
corrected each individual exposure for the time dependent detector sensitivity
loss
555https://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/swift/docs/uvot/uvotcaldb_throughput_06.pdf
and subtracted off the necessary counts to normalize the background count rate
across exposures. Normalizing the background is especially important in the
optical filters due to effects of scattered light from the sun, Earth, and
moon. In total, the final stacked images included 151, 129, 136, 94, 24.3,
18.5 ks for the UVW2, UVM2, UVW1, u, b, and v filters respectively. The
angular resolution of the UVOT filters is $\sim 2.5^{\prime\prime}$.

As Swift was designed to study point sources, photometry must be done with
care. Using the apertures defined in Section 2.1.1, we calculated the
accumulated count rate as a function of radius, following Gil de Paz et al.
(2007). This brightness profile was then used to determine the magnitude in
each radial bin using the zero-point offsets from Breeveld et al. (2011). As
discussed in Decleir et al. (2019) and Belles et al. (2023), the zero-point
offsets are defined based on a 5 arcsecond aperture. Therefore, we divided our
measured fluxes by the asymptote of the curve of growth (Breeveld et al.,
2010) in order to account for the difference in the effective aperture area.
We used the Python package dust_extinction666https://dust-
extinction.readthedocs.io/en/stable/ and the “F99” dust model from Fitzpatrick
(1999), along with $A_{V}$ and $E(B-V)$ provided in Table 1, to correct the
Swift flux densities for dust extinction and attenuation.

####  2.1.5 GALEX

We retrieved GALEX FUV and NUV imaging from the Mikulski Archive for Space
Telescopes (MAST: http://dx.doi.org/10.17909/k7y9-yp18 (catalog
10.17909/k7y9-yp18)). The images were obtained on 2004 October 26, with a
total exposure time of 12987.6 s in both filters. The angular resolution is
$4.3^{\prime\prime}$ in the FUV and $5.3^{\prime\prime}$ in the NUV, and
images are in units of counts per second. Images are converted to flux density
units of erg s-1 cm-2 Å-1 by multiplying by $1.4\times 10^{-15}$ (FUV) or
$2.06\times 10^{-16}$ (NUV; Martin et al., 2005; Morrissey et al., 2007). As
for the Swift/UVOT observations (Section 2.1.4), the GALEX flux densities are
corrected for Galactic dust extinction and attenuation.

###  2.2 Modeling the SED

Figure 2 shows an RGB-rendered image of NGC 300 (using GALEX NUV, Spitzer
3.6$\mu$m, and Spitzer 24$\mu$m imaging in the blue, green, and red channels,
respectively) with the elliptical annuli corresponding to a width of $R_{d}$
superimposed (the locations of Chandra X-ray sources used in our XRB analysis
are shown in yellow, while X-ray sources that are not used are shown in white;
see Section 3). The $D_{25}$ isophotal ellipse (de Vaucouleurs et al., 1991),
shown in cyan for comparison, is slightly larger than the outer annular
region. Table 2 provides the photometry measurements that are used to
construct SEDs for the entire galaxy and the four subgalactic annuli. We use
Prospector (Leja et al., 2017; Johnson et al., 2021)777See also
https://github.com/bd-j/prospector to model the SEDs of NGC 300. Prospector
forward-models the observed photometry given parameters that describe the gas,
dust, and stellar populations present in the galaxy in addition to
instrumental parameters. The code uses noise models and prior distributions to
compute the likelihood and posterior probability for the given model and
performs Monte Carlo sampling of the posterior probability distribution.
Prospector has been extensively tested on SEDs and spectroscopy of nearby
galaxies (Leja et al., 2017, 2018), dwarf galaxies (Greco et al., 2018; Pandya
et al., 2018), and has been used in spatially-resolved studies (Patrício et
al., 2018; Oyarzún et al., 2019).

Figure 2: An RGB-rendered image of NGC 300, with the with four subgalactic
regions shown by white elliptical annuli. The $D_{25}$ isophote (de
Vaucouleurs et al., 1991) is shown in cyan for comparison. The GALEX NUV image
is shown in the blue channel, the Spitzer IRAC 3.6$\mu$m is shown in green,
and the Spitzer MIPS 24$\mu$m is shown in red. Chandra X-ray sources that
heavily absorbed and likely to be AGN are shown inred, sources that are
consistent with being XRBs are shown as yellow crosses, and X-ray sources that
are not used are shown as white crosses (see Section 3).

We use the non-parametric “continuity star formation history” (continuity_sfh)
model (Leja et al., 2019; Johnson et al., 2021) to fit piecewise constant (or
step function) SFHs in seven time bins. Rather than directly fitting for the
SFR in each time bin, this model utilizes parameters that describe the ratio
of SFRs in adjacent temporal bins. The default priors on these ratios are
described by Student-$t$ distributions and a Kroupa (2001) IMF. This model
results in SFHs that tend towards constant SFRs, as dramatic differences in
SFRs between adjacent temporal bins are down-weighted. We selected this SFH
model given that NGC 300 shows no evidence of having undergone dramatic merger
or interaction events that would lead to brief, high-SFR episodes. We adopt
the same time bins as Leja et al. (2019): 0-30 Myr, 30-100 Myr, 100-330 Myr,
330 Myr-1.1 Gyr, 1.1-3.6 Gyr, 3.6-11.7 Gyr, and 11.7-13.7 Gyr. The first two
bins are directly relevant to the current HMXB population, while the last time
bin is selected to permit a maximally old population. The intermediate time
bins spanning from 100 Myr to 11.7 Gyr are equally spaced in logarithmic time
(see, e.g., Ocvirk et al., 2006).

Table 2: UV, Optical, and IR Photometry of NGC 300 |  | Flux Densities (Jy)
---|---|---
Instrument | Wavelength | Whole Galaxy | Inner | Inner Middle | Outer Middle | Outer
GALEX FUV | 1535.08 Å | 0.231$\pm$0.003 | 0.026$\pm$0.001 | 0.084$\pm$0.001 | 0.067$\pm$0.002 | 0.054$\pm$0.003
Swift UVW2 | 2054.61 Å | 0.261$\pm$0.005 | 0.035$\pm$0.001 | 0.096$\pm$0.003 | 0.076$\pm$0.003 | 0.056$\pm$0.002
Swift UVM2 | 2246.43 Å | 0.253$\pm$0.005 | 0.034$\pm$0.001 | 0.093$\pm$0.003 | 0.074$\pm$0.003 | 0.054$\pm$0.002
GALEX NUV | 2300.78 Å | 0.316$\pm$0.003 | 0.039$\pm$0.001 | 0.111$\pm$0.001 | 0.091$\pm$0.003 | 0.074$\pm$0.003
Swift UVW1 | 2580.74 Å | 0.305$\pm$0.007 | 0.047$\pm$0.002 | 0.113$\pm$0.004 | 0.086$\pm$0.004 | 0.059$\pm$0.004
Swift u | 3467.05 Å | 0.621$\pm$0.007 | 0.099$\pm$0.002 | 0.247$\pm$0.005 | 0.158$\pm$0.004 | 0.118$\pm$0.003
Swift b | 4349.56 Å | 1.288$\pm$0.028 | 0.247$\pm$0.006 | 0.421$\pm$0.013 | 0.351$\pm$0.017 | 0.269$\pm$0.018
Swift v | 5425.33 Å | 1.945$\pm$0.018 | 0.421$\pm$0.005 | 0.635$\pm$0.009 | 0.508$\pm$0.010 | 0.382$\pm$0.011
2MASS $J$ | 1.235 $\mu$m | 4.109$\pm$0.069 | 1.065$\pm$0.018 | 1.660$\pm$0.018 | 1.024$\pm$0.062 | 0.360$\pm$0.068
2MASS $H$ | 1.662 $\mu$m | 4.047$\pm$0.064 | 1.076$\pm$0.019 | 1.500$\pm$0.019 | 1.018$\pm$0.057 | 0.453$\pm$0.063
2MASS $K_{s}$ | 2.159 $\mu$m | 3.088$\pm$0.054 | 0.811$\pm$0.014 | 1.208$\pm$0.014 | 0.742$\pm$0.050 | 0.327$\pm$0.053
WISE W1 | 3.4 $\mu$m | 1.948$\pm$0.004 | 0.458$\pm$0.001 | 0.723$\pm$0.003 | 0.494$\pm$0.004 | 0.273$\pm$0.005
Spitzer IRAC1 | 3.6 $\mu$m | 1.857$\pm$0.101 | 0.410$\pm$0.016 | 0.750$\pm$0.038 | 0.454$\pm$0.090 | 0.243$\pm$0.109
Spitzer IRAC2 | 4.5 $\mu$m | 1.251$\pm$0.049 | 0.285$\pm$0.013 | 0.480$\pm$0.045 | 0.304$\pm$0.073 | 0.181$\pm$0.086
WISE W2 | 4.6 $\mu$m | 1.143$\pm$0.004 | 0.266$\pm$0.001 | 0.453$\pm$0.004 | 0.278$\pm$0.006 | 0.151$\pm$0.006
Spitzer IRAC3 | 5.8 $\mu$m | 1.115$\pm$0.464 | 0.271$\pm$0.121 | 0.507$\pm$0.248 | 0.247$\pm$0.109 |  $<$1.299
Spitzer IRAC4 | 8.0 $\mu$m | 2.156$\pm$0.034 | 0.500$\pm$0.012 | 0.999$\pm$0.023 | 0.453$\pm$0.028 | 0.203$\pm$0.031
WISE W3 | 12 $\mu$m | 1.949$\pm$0.097 | 0.434$\pm$0.025 | 0.902$\pm$0.057 | 0.419$\pm$0.068 | 0.195$\pm$0.080
WISE W4 | 22 $\mu$m | 2.017$\pm$0.765 | 0.465$\pm$0.192 | 1.280$\pm$0.413 | 0.510$\pm$0.467 | 0.222$\pm$0.133
Spitzer MIPS1 | 24 $\mu$m | 1.993$\pm$0.479 | 0.389$\pm$0.128 | 1.092$\pm$0.330 | 0.348$\pm$0.186 |  $<$0.423
Spitzer MIPS2 | 70 $\mu$m | 39.242$\pm$2.878 | 7.333$\pm$0.980 | 18.728$\pm$0.134 | 8.680$\pm$0.152 | 4.502$\pm$0.167
Spitzer MIPS3 | 160 $\mu$m | 144.855$\pm$6.201 | 24.264$\pm$2.997 | 57.777$\pm$0.123 | 39.179$\pm$0.139 | 23.635$\pm$0.148

The metallicity, stellar mass formed, and dust attenuation optical depth
($\tau_{\rm dust}$) are additional free parameters in our model. Metallicity
priors are assumed to be Gaussian. Although the slopes of the metallicity
gradients reported by Kudritzki et al. (2008) and Gogarten et al. (2010)
agree, their central metallicity values differ by $\sim$0.3$Z_{\odot}$ (see
Table 3). The central metallicity value depends on the calibration used, and
the value reported by Kudritzki et al. (2008) was derived from a combination
of abundance estimates from mostly Fe, Ti, and Cr spectral lines. The CMD
fitting technique utilized by Gogarten et al. (2010) relies on synthetic
photometry of theoretical stellar isochrones to determine the best metallicity
for a given stellar population. Given the consistency of the slopes, Gogarten
et al. (2010) adopt the Kudritzki et al. (2008) central value when discussing
the implications of the metallicity gradient. We initialized the metallicity
prior by setting the mean value equal to the direct measurement value derived
from Kudritzki et al. (2008) at each ring midpoint. The uncertainty in the
metallicity is computed from the uncertainty in the observed metallicity
gradient and the effect of the finite thickness of the radial bin added in
quadrature; we set the standard deviation of the Gaussian prior to twice this
metallicity uncertainty value (typically $\approx$0.3). This standard
deviation was large enough to be inclusive of the Gogarten et al. (2010)
values across the disk. We note that we obtained similar metallicity values
when we used an uninformative prior on the metallicity, but we opted to retain
the Gaussian prior on the metallicity as it decreased the amount of time
required by Prospector to obtain a good fit.

NGC 300 is estimated to have a total stellar mass of $\sim 2.4\times 10^{9}$
M⊙ (Puche et al., 1990; Sheth et al., 2010; Querejeta et al., 2015). We assume
this mass is distributed in the same manner as the 3.6$\mu$m surface
brightness profile when providing an initial estimate of the stellar mass
contained in each radial bin, and we assume a log uniform prior on the stellar
mass (with minimum and maximum stellar mass values set to $10^{6}$ M⊙ and
$10^{10}$ M⊙, respectively). The dust attenuation parameter assumes a flat
(top-hat) prior with minimum and maximum values of 0.0 and 2.0. We
experimented with changing the maximum and minimum values used by the dust
attenuation and stellar mass priors, and with changing the initial values of
the stellar mass and metallicity. Altering these values had no significant
effect on the resulting fits. Changing the initial value of the stellar mass
by up to a factor of $\sim$5 similarly did not significantly affect our
results. The best-fit model was determined via $\chi^{2}$ minimization, and
parameter values around the best-fit were sampled with an ensemble MCMC
algorithm (Goodman & Weare, 2010) using the emcee Python package (Foreman-
Mackey et al., 2013). We initialize emcee with 128 walkers and 512 iterations
to build up a distribution of SED models. The best-fit parameters and
uncertainties were defined by the 16th, 50th, and 84th percentile quantities
of each distribution.

Table 3: Previous Spatially-Integrated Measurements of NGC 300 Physical
Parameters

Method | Value | Reference
---|---|---
Star Formation Rate (M⊙ yr-1)
Prospector | 0.177 | this work
FUV+NUV | 0.46 | Mondal et al. (2019)
FUV | 0.32 | Lee et al. (2009)
UV | 0.28 | Kaisina et al. (2012)
CMD fitting | 0.15 | Gogarten et al. (2010)
H$\alpha$ | 0.14 | Helou et al. (2004)
H$\alpha$ | 0.17 | Kaisina et al. (2012)
NUV+70$\mu$m | 0.09 | Mineo et al. (2012)
8$\mu$m | 0.08-0.11 | Helou et al. (2004)
Stellar Mass (M⊙)
Prospector | 2.15$\times 10^{9}$ | this work
H I | 4.2$\times 10^{9}$ | Puche et al. (1990)
IR (S4G) | 2.1$\times 10^{9}$ | Querejeta et al. (2015)
Optical+IR (S4G) | 2.0$\times 10^{9}$ | Laine et al. (2016)
Central Metallicity ($Z/Z_{\odot}$) aaOnly the first ten entries are shown. A
full machine-readable table is available from the journal.
Prospector | 0.54 | this work
12+log(O/H) | 0.76 | Bresolin et al. (2009)
12+log(O/H) | 0.78 | Micheva et al. (2022)
$[M/H]$ | 0.59 | Gogarten et al. (2010)
$Z/Z_{\odot}$ | 0.87 | Kudritzki et al. (2008)

Note. — aMetallicites have been converted to $Z/Z_{\odot}$ for ease of
comparison.

###  2.3 Comparison to Previous Studies

Table 3 provides an overview of the spatially-integrated SFR, $M_{\star}$, and
central metallicity888Bresolin et al. (2009) and Micheva et al. (2022) report
12+log(O/H). We convert to $Z/Z_{\odot}$ assuming 10X-8.69 (Asplund et al.,
2009), where $X$ is the reported 12+log(O/H) value. values reported for NGC
300 from the literature and our Prospector modeling; the detailed fit
parameters and SFHs from Prospector are are summarized in Table 4. We
additionally provide the stellar mass formation history, which is the SFR in
each time bin integrated over the duration of the time bin. This gives a
measure of the total mass in stars that formed in each epoch; the current
stellar mass M⋆ represents the surviving stellar mass at the present time and
is lower than the total mass formed in stars. The best-fit SED parameters for
the spatially-integrated SED and the radial subgalactic regions are modeled
independently; i.e., column (3) of Table 4 is not the sum of columns (4)-(7).
We note that the sums of the stellar masses and SFRs over the subgalactic
regions yields values that are generally within $2\sigma$ of the value derived
from the spatially-integrated SED. Figure 3 shows the UV-to-IR galaxy-
integrated SED of NGC 300 (out to 4$R_{d}$) along with the best-fit model from
Prospector, while Figure 4 shows the best SED models and the corresponding UV-
IR photometry for the four subgalactic radial bins. The best-fit model for the
entire galaxy yields a stellar mass, recent ($<$100 Myr) SFR, and metallicity
that are all consistent with values previously reported in the literature (and
summarized in Table 3).

Table 4: Best-Fit SED Models from Prospector |  | Galaxy Inte- | Radial Bin | Sum of All
---|---|---|---|---
Parameter | Units | grated ($\leq$4$R_{d}$) | $\leq R_{d}$ | $R_{d}<r\leq 2R_{d}$ | $2R_{d}<r\leq 3R_{d}$ | $3R_{d}<r\leq 4R_{d}$ | Radial Bins
M⋆ | 109 M⊙ | 2.15${}^{+0.26}_{-0.14}$ | 0.64${}^{+0.03}_{-0.02}$ | 0.94${}^{+0.08}_{-0.07}$ | 0.38${}^{+0.06}_{-0.04}$ | 0.27${}^{+0.03}_{-0.02}$ | 2.23${}^{+0.20}_{-0.15}$
SFR($<$100 Myr) | M⊙ yr-1 | 0.177$\pm$0.070 | 0.016${}^{+0.009}_{-0.007}$ | 0.094${}^{+0.026}_{-0.033}$ | 0.045${}^{+0.035}_{-0.026}$ | 0.014${}^{+0.011}_{-0.006}$ | 0.169${}^{+0.081}_{-0.072}$
log($Z$) | log[$Z_{\odot}$] | -0.239${}^{+0.014}_{-0.033}$ | -0.266${}^{+0.017}_{-0.013}$ | -0.360$\pm$0.013 | -0.459${}^{+0.065}_{-0.046}$ | -0.628${}^{+0.011}_{-0.012}$ |  $\cdots$
$\tau_{\rm dust}$ |  | 0.122${}^{+0.008}_{-0.005}$ | 0.155${}^{+0.007}_{-0.009}$ | 0.170$\pm$0.007 | 0.145${}^{+0.009}_{-0.008}$ | 0.122$\pm$0.09 |  $\cdots$
Star Formation History |
0-30 Myr | M⊙ yr-1 | 0.048$\pm$0.018 | 0.005${}^{+0.003}_{-0.002}$ | 0.016${}^{+0.007}_{-0.005}$ | 0.017${}^{+0.007}_{-0.006}$ | 0.013${}^{+0.005}_{-0.004}$ | 0.051${}^{+0.022}_{-0.017}$
30-100 Myr | M⊙ yr-1 | 0.234${}^{+0.094}_{-0.093}$ | 0.021${}^{+0.012}_{-0.009}$ | 0.128${}^{+0.034}_{-0.046}$ | 0.057${}^{+0.034}_{-0.048}$ | 0.014${}^{+0.013}_{-0.007}$ | 0.220${}^{+0.093}_{-0.110}$
100-330 Myr | M⊙ yr-1 | 0.462${}^{+0.231}_{-0.086}$ | 0.070${}^{+0.015}_{-0.029}$ | 0.161${}^{+0.082}_{-0.045}$ | 0.105${}^{+0.035}_{-0.048}$ | 0.068${}^{+0.032}_{-0.045}$ | 0.404${}^{+0.164}_{-0.167}$
330 Myr-1.1 Gyr | M⊙ yr-1 | 0.263${}^{+0.056}_{-0.085}$ | 0.065${}^{+0.015}_{-0.012}$ | 0.053${}^{+0.023}_{-0.027}$ | 0.044${}^{+0.023}_{-0.024}$ | 0.093${}^{+0.033}_{-0.024}$ | 0.255${}^{+0.094}_{-0.087}$
1.1-3.6 Gyr | M⊙ yr-1 | 0.239${}^{+0.064}_{-0.096}$ | 0.017${}^{+0.015}_{-0.010}$ | 0.066${}^{+0.027}_{-0.023}$ | 0.126${}^{+0.025}_{-0.029}$ | 0.030${}^{+0.013}_{-0.010}$ | 0.239${}^{+0.080}_{-0.072}$
3.6-11.7 Gyr | M⊙ yr-1 | 0.239${}^{+0.041}_{-0.034}$ | 0.100${}^{+0.009}_{-0.010}$ | 0.087${}^{+0.024}_{-0.023}$ | 0.023${}^{+0.013}_{-0.018}$ | 0.022${}^{+0.007}_{-0.005}$ | 0.232${}^{+0.053}_{-0.056}$
11.7-13.7 Gyr | M⊙ yr-1 | 0.222${}^{+0.276}_{-0.078}$ | 0.055${}^{+0.053}_{-0.043}$ | 0.261${}^{+0.163}_{-0.116}$ | 0.007${}^{+0.005}_{-0.009}$ | 0.032${}^{+0.032}_{-0.015}$ | 0.355${}^{+0.253}_{-0.183}$
Stellar Mass Formation History |
0-30 Myr | log[M⊙] | 6.14${}^{+0.14}_{-0.20}$ | 5.18$\pm$0.19 | 5.67$\pm$0.15 | 5.69${}^{+0.13}_{-0.23}$ | 5.58${}^{+0.14}_{-0.17}$ | 6.17${}^{+0.14}_{-0.23}$
30-100 Myr | log[M⊙] | 7.21${}^{+0.15}_{-0.22}$ | 6.16${}^{+0.20}_{-0.25}$ | 6.95${}^{+0.10}_{-0.19}$ | 6.60${}^{+0.26}_{-0.39}$ | 6.00${}^{+0.29}_{-0.27}$ | 7.19${}^{+0.24}_{-0.30}$
100-330 Myr | log[M⊙] | 8.03${}^{+0.18}_{-0.09}$ | 7.21${}^{+0.09}_{-0.24}$ | 7.57${}^{+0.18}_{-0.14}$ | 7.39${}^{+0.16}_{-0.17}$ | 7.19${}^{+0.17}_{-0.47}$ | 7.97${}^{+0.16}_{-0.21}$
330 Myr-1.1 Gyr | log[M⊙] | 8.30${}^{+0.08}_{-0.17}$ | 7.70$\pm$0.09 | 7.61${}^{+0.16}_{-0.31}$ | 7.53${}^{+0.19}_{-0.32}$ | 7.85$\pm$0.13 | 8.29${}^{+0.14}_{-0.18}$
1.1-3.6 Gyr | log[M⊙] | 8.78${}^{+0.10}_{-0.22}$ | 7.64${}^{+0.28}_{-0.37}$ | 8.22${}^{+0.15}_{-0.19}$ | 8.51${}^{+0.09}_{-0.10}$ | 7.88${}^{+0.15}_{-0.17}$ | 8.78$\pm$0.14
3.6-11.7 Gyr | log[M⊙] | 9.29$\pm$0.07 | 8.91${}^{+0.04}_{-0.05}$ | 8.85${}^{+0.10}_{-0.13}$ | 8.28${}^{+0.25}_{-0.36}$ | 8.26${}^{+0.09}_{-0.16}$ | 9.07${}^{+0.09}_{-0.10}$
11.7-13.7 Gyr | log[M⊙] | 8.66${}_{-0.35}^{+0.19}$ | 8.05${}^{+0.29}_{-0.68}$ | 8.73${}^{+0.21}_{-0.26}$ | 7.17${}_{-0.35}^{+0.61}$ | 7.82${}^{+0.30}_{-0.26}$ | 8.86${}^{+0.25}_{-0.30}$
Figure 3: The best-fit, whole-galaxy (out to $4R_{d}$) SED for NGC 300. The
observed photometry (from Table 2) is shown as blue dots. The best-fit model
spectrum is shown in black, and black open squares indicate the model-
predicted photometry for each filter used. The filter transmission curves are
shown in multiple colors across the lower horizontal axis. Figure 4: Best-fit
radial SEDs from Prospector. All four SEDs are shown in each plot in gray,
with the relevant sub-region highlighted in bold color. The galaxy-integrated
best-fit SED model from Figure 3 is shown by the thick, dark gray line. The
white circles show the photometry measurements from Table 2; in most cases,
the uncertainties are smaller than the plotting symbol. Black squares show the
predicted flux density in each filter.

The metallicity gradient recovered from the Prospector is shown in the top
panel of Figure 5 compared to the gradients of Kudritzki et al. (2008, light
blue) and Gogarten et al. (2010, dark blue). Our SED modeling of NGC 300
yields a metallicity gradient of

$\text{log}\left(Z/Z_{\odot}\right)=\left(-0.193\pm
0.016\right)-\left(0.092\pm 0.005\right)r,$ (2)

where $r$ is the galactocentric radius in kpc. This gradient shows good
agreement with those taken from the literature and predicts a central
metallicity value that is intermediate between the Gogarten et al. (2010) and
Kudritzki et al. (2008) values. Our utilization of Prospector is similar to
CMD fitting in that we rely on similar theoretical stellar models to construct
galaxy SEDs for comparison to flux densities, rather than directly fitting for
abundances from spectra. We note that the radial gradient obtained by Gogarten
et al. (2010) was derived using a narrow strip of three HST fields that
spanned the disk of NGC 300 (which also prohibited measurement of integrated
galaxy mass and SFR), while our SED modeling uses observations extending over
the full annular rings shown in Figure 2. The metallicity gradient of NGC 300
from Kudritzki et al. (2008) was derived from a sample of 24 supergiants (with
spectral types ranging from B8 to A4), and Bresolin et al. (2009) found that
these direct stellar metallicity measurements were in excellent agreement with
those derived from a sample of 28 H II regions for which auroral lines were
detected. The stellar and auroral nebular metallicities did not, however,
agree with abundances derived from strong-line methods: the metallicity
gradient based on strong oxygen lines in Zaritsky et al. (1994) and Deharveng
et al. (1988) $\sim$50% steeper than that found in Kudritzki et al. (2008) and
Bresolin et al. (2009). The metallicities of H II regions in two other well-
studied spiral galaxies – the Milky Way and M33 – show significant
discrepancies depending on the lines and methods used, and these differences
cannot be explained by measurement errors. Using mid- and far-IR lines largely
resolves the discrepancies in the Milky Way (Martín-Hernández et al., 2002;
Rudolph et al., 2006), bringing the nebular metallicity gradient into
agreement with that of direct stellar measurements, but these lines are rarely
used to derive metallicities or abundance gradients in external spiral
galaxies. Additional discussion of metallicity and its role in HMXB evolution
is discussed in Section 4.

Figure 5: Top: The metallicity gradient of NGC 300. The black points show the
results of our SED modeling with Prospector, and the dashed black line shows
the best-fit metallicity gradient. The metallicity gradients taken from the
literature (and their corresponding uncertainties) are shown in light blue
(Kudritzki et al., 2008) and dark blue (Gogarten et al., 2010). Bottom: The
SFH of NGC 300 (black; gray shaded region indicates uncertainties).

The overall SFH for the entire galaxy (Figure 5, bottom panel) is in broad
agreement with previous studies, showing an enhancement in the SFR at early
times (roughly 90% the stars ever formed in NGC 300 were formed more than a
few Gyr ago, as found by Gogarten et al., 2010). The average SFR on timescales
relevant for HMXBs ($<$100 Myr) agrees with estimates derived by other SFR
indicators, our SED modeling suggests that most of the star formation activity
occurred $\sim$30-100 Myr ago, with the most recent SFR ($<$30 Myr) occurring
at only $\sim$20% the rate found for the older time bin. Williams et al.
(2013) found that most HMXBs in NGC 300 are $\sim$40-60 Myr old through CMD
analysis of the resolved stellar populations in the vicinity of HMXB
candidates, with NGC 300 ULX-1 being the only notable exception of a younger
HMXB (the massive HMXB NGC 300 X-1 is likely $<$30 Myr as well, given the
large BH mass and evolved state of the Wolf-Rayet companion; Binder et al.,
2020, 2021). We next compared the SFHs derived by Prospector to the radially-
resolved SFHs of Gogarten et al. (2010), which assumed a Salpeter (1955) IMF
during CMD fitting but report results scaled to a Kroupa (2001) IMF. Although
the outermost radii ($\sim$5.4 kpc) are similar in both studies, we use four
bins each with a width of 1.36 kpc while Gogarten et al. (2010) used six bins
of width 0.9 kpc, and the time bins used for in constructing the SFHs differs.
Furthermore, the Gogarten et al. (2010) analysis was limited to the small area
covered by the HST fields, whereas our SED modeling is performed for annular
rings across the entire disk. This introduces some additional uncertainty in
the comparison of the two radially-resolved SFHs. The SFHs show broad
agreement (uncertainties in the Gogarten et al. (2010) SFRs are typically on
the order of $\sim$20%; Figure 6), with the largest discrepancies appearing in
the oldest time bin at $r<2R_{d}$, where Gogarten et al. (2010) predict a
higher SFR at very early times compared to our Prospector results,
particularly at $r\lesssim$4 kpc. These differences will not affect our
interpretation of the HMXB population properties, which depend only on the
most recent SFRs.

Different SFR indicators are sensitive to different population ages, and
Kouroumpatzakis et al. (2020) demonstrated that the relationship between
$L_{\rm X}$ and SFR of HMXB populations depends on choice of SFR indicator. As
shown in Table 3, different SFR indicators can yield up to a factor of $\sim$5
difference in inferred SFR in NGC 300 on a galaxy-integrated scale.
Metallicity, too, can be measured through a variety of means; nearly all
studies of the $L_{\rm X}$-SFR-$Z$ relationship assume the the overall
metallicity of a galaxy is well traced by the nebular oxygen abundance, 12 +
log(O/H). While emission lines from H II region spectra provide the bulk of
direct present-day abundance measurements, many studies utilize metallicities
derived from strong-line indices such as [O III] $\lambda$5007\. Auroral
emission lines that directly probe both the composition and the electron
temperatures of H II regions (e.g., [O III] $\lambda$4363) become fainter with
both increasing distance and metallicity (as the electron temperature
decreases due to cooling via metal lines; (Berg et al., 2015; Croxall et al.,
2015, 2016), eventually becoming to faint to detect. Strong-line metallicity
indices therefore depend either on empirical calibrations tied to more local
auroral line measurements (Pettini & Pagel, 2004; Pilyugin & Thuan, 2005;
Liang et al., 2007; Florido et al., 2022) or theoretical calibrations derived
from grids of photoionization models (McGaugh, 1991; Kewley & Dopita, 2002;
Tremonti et al., 2004) These different calibrations provide large differences
($\sim$0.7 dex) in the inferred metallicities (Kewley & Ellison, 2008), and
may be significantly contributing to the observed scatter in the $L_{\rm
X}$-SFR-$Z$ relationships.

Figure 6: Radially-resolved SFHs derived from Prospector modeling (color-coded
as in Figure 4) with increasing galactocentric radius (from left to right)
compared to the SFHs derived by Gogarten et al. (2010, solid and dashed black
lines) at similar galactocentric distances. The SFHs from Gogarten et al.
(2010) carry a typical uncertainty of $\sim$10-20% and were computed over a
much smaller surface area of NGC 300 than the Prospector-derived SFHs. The
galaxy-integrated SFH from Figure 5 is shown by the thick gray line.

##  3 X-ray Observations from Chandra

We retrieved four Chandra ACIS-I observations (ObsIDs 12238, 16028, 16029, and
22375) from the Chandra archive999https://cda.harvard.edu/chaser/. The
observations ranged in exposure duration from 47.4 ks (ObsID 22375) to 64.2 ks
(ObsID 16028). We reprocessed all evt1 data using the CIAO v4.15 (Fruscione et
al., 2006) task chandra_repro and standard reduction procedures. We ran the
point source detection algorithm wavdetect on each individual exposure to
generate a preliminary list of X-ray source positions and error ellipses. The
major and minor axes of these error ellipses were increased by a factor of
five, and all X-ray sources were masked so that a background light curve could
be extracted for each observation. We inspected these background light curves
but found no evidence for strong, prolonged background flaring events in any
of the four observations. We created good time intervals (GTIs) using lc_clean
and filtered all event data on these GTIs. Finally, we restricted the energy
range of our resulting “clean” evt2 files to 0.5-7 keV.

We updated the absolute astrometry of all four images by using the tasks
wcs_match and wcs_match to align each Chandra observation to the longest
observation (ObsID 16028). Images were finally reprojected to a common tangent
point and combined using the CIAO task reproject_obs and flux_obs. The total
usable Chandra exposure time was 235.9 ks. We next re-ran wavdetect on the
merged observation, using smoothing scales of “1 2 4 8” and a significance
threshold of $10^{-6}$. A total of 154 X-ray sources were detected in the
merged Chandra image. An RGB-rendered image of NGC 300 with the locations of
all Chandra X-ray sources is shown in Figure 2. A summary of the properties of
all the detected X-ray sources is provided in Table 5 and provided as a
machine-readable table from the journal.

Table 5: Properties of All 154 X-ray Sources Detected in Merged Chandra ImageaaOnly the first ten entries are shown. A full machine-readable table is available from the journal. RA | Dec. | Galactocentric | Galactocentric | Source | Net Counts | 12238 |  | 16028 |  | 16029 |  | 22375
---|---|---|---|---|---|---|---|---|---|---|---|---
(J2000) | Radius (kpc) | Radius/$R_{d}$ | Signif. ($\sigma$) | (0.5-7 keV) | HR1 | HR2 |  | HR1 | HR2 |  | HR1 | HR2 |  | HR1 | HR2
13.6980206 | -37.8927248 | 10.10 | 7.43 | 20.5 | 549$\pm$35 |  $\cdots$ |  $\cdots$ |  |  $\cdots$ |  $\cdots$ |  |  $\cdots$ |  $\cdots$ |  | 0.42${}^{+0.05}_{-0.05}$ | -0.20${}^{+0.06}_{-0.06}$
13.7722944 | -37.8583461 | 8.07 | 5.93 | 4.2 | 35$\pm$9 |  $\cdots$ |  $\cdots$ |  |  $\cdots$ |  $\cdots$ |  | 0.75${}^{+0.12}_{-0.12}$ | -0.60${}^{+0.21}_{-0.22}$ |  |  $\cdots$ |  $\cdots$
13.6859116 | -37.8505873 | 8.22 | 6.04 | 13.9 | 179$\pm$18 |  $\cdots$ |  $\cdots$ |  | 0.47${}^{+0.10}_{-0.10}$ | -0.33${}^{+0.14}_{-0.15}$ |  | 0.44${}^{+0.08}_{-0.08}$ | -0.33${}^{+0.10}_{-0.11}$ |  |  $\cdots$ |  $\cdots$
13.6999274 | -37.8355104 | 7.37 | 5.42 | 11.3 | 106$\pm$13 |  $\cdots$ |  $\cdots$ |  | 0.43${}^{+0.15}_{-0.14}$ | -0.31${}^{+0.19}_{-0.22}$ |  | 0.40${}^{+0.10}_{-0.10}$ | -0.15${}^{+0.13}_{-0.15}$ |  |  $\cdots$ |  $\cdots$
13.7920276 | -37.8333377 | 6.91 | 5.08 | 7.2 | 68$\pm$12 |  $\cdots$ |  $\cdots$ |  |  $\cdots$ |  $\cdots$ |  | 0.76${}^{+0.15}_{-0.19}$ | -0.78${}^{+0.23}_{-0.14}$ |  | 0.28${}^{+0.21}_{-0.21}$ | -0.24${}^{+0.27}_{-0.30}$
13.8005427 | -37.8284841 | 6.71 | 4.93 | 16.1 | 177$\pm$17 |  $\cdots$ |  $\cdots$ |  |  $\cdots$ |  $\cdots$ |  | 0.53${}^{+0.09}_{-0.09}$ | -0.30${}^{+0.13}_{-0.14}$ |  | 0.47${}^{+0.08}_{-0.08}$ | -0.08${}^{+0.13}_{-0.14}$
13.7174112 | -37.8202939 | 6.50 | 4.78 | 5.6 | 49$\pm$10 |  $\cdots$ |  $\cdots$ |  | 0.41${}^{+0.22}_{-0.21}$ | -0.18${}^{+0.30}_{-0.36}$ |  | 0.56${}^{+0.22}_{-0.22}$ | -0.44${}^{+0.35}_{-0.32}$ |  | 0.22${}^{+0.49}_{-0.48}$ | -0.08${}^{+0.65}_{-0.63}$
13.6970791 | -37.8204055 | 6.68 | 4.91 | 4.4 | 35$\pm$9 |  $\cdots$ |  $\cdots$ |  | 0.45${}^{+0.19}_{-0.16}$ | -0.01${}^{+0.31}_{-0.37}$ |  | 0.70${}^{+0.17}_{-0.18}$ | -0.58${}^{+0.29}_{-0.26}$ |  |  $\cdots$ |  $\cdots$
13.7205749 | -37.8120096 | 6.09 | 4.48 | 4.6 | 53$\pm$13 |  $\cdots$ |  $\cdots$ |  | 0.52${}^{+0.20}_{-0.19}$ | -0.53${}^{+0.26}_{-0.23}$ |  | 0.44${}^{+0.26}_{-0.28}$ | -0.29${}^{+0.40}_{-0.40}$ |  | -0.01${}^{+0.53}_{-0.54}$ | -0.34${}^{+0.34}_{-0.31}$
13.7953472 | -37.8095754 | 5.85 | 4.30 | 34.4 | 416$\pm$23 |  $\cdots$ |  $\cdots$ |  | 0.42${}^{+0.11}_{-0.11}$ | -0.53${}^{+0.13}_{-0.13}$ |  | 0.36${}^{+0.06}_{-0.07}$ | -0.36${}^{+0.07}_{-0.07}$ |  | 0.51${}^{+0.06}_{-0.06}$ | -0.33${}^{+0.08}_{-0.08}$

We additionally calculate hardness ratios for each X-ray source using the
Bayesian estimation of hardness ratios (BEHR) code (Park et al., 2006)
following the same methodology as in Binder et al. (2012). We define two
hardness ratios,

$\text{HR1}=\frac{M-S}{H+M+S},$ (3)

and

$\text{HR2}=\frac{H-M}{H+M+S},$ (4)

where $S$ is the “soft” band (0.35-1.1 keV), $M$ is the “medium” band (1.1-2.6
keV), and $H$ is the “hard” band (2.6-8.0 keV). We use BEHR to generate 50,000
samples of the source flux probability distributions (in all four
observations) for each of the three bands, which we use to compute 50,000
values of HR1 and HR2 for each source. In Table 5 we report the median value
for each hardness ratio, and the uncertainties are defined using the 16th and
84th percentile values. We use the hardness ratio categories defined in Binder
et al. (2012) to roughly classify sources as likely XRBs, SNRs, highly
absorbed sources (“ABS”), indeterminate soft or hard sources (“SOFT” and
“HARD,” respectively), and sources with indeterminate spectral types
(“INDET”). The hardness ratio diagram in Figure 7 shows the hardness ratios
for each source and the six categories defined in Binder et al. (2012).

Figure 7: Hardness ratio diagram for all 154 sources detected in the merged
Chandra image of NGC 300, with the six hardness ratio categories defined in
Binder et al. (2012) shown in blue. Red lines indicate the physically
meaningful region of the diagram. The typical uncertainty on the hardness
ratios is shown by the black cross in the lower left corner.

###  3.1 Sensitivity Curves and Limiting Fluxes

In addition to generating a merged observation image, we also created
background images using the CIAO blank_sky task. “Blank sky” images were
generated to estimate the source-free background across the detector for each
observation and merged in the same manner as described above for the actual
observation images. These blank sky images can be used to derive sensitivity
curves, which provide an estimate of the probability that an X-ray source with
a given flux will be detected at a given location on the detector. These are
essential components for accurately modeling XLFs and for determining the flux
levels below which a source catalog is highly incomplete.

We follow the methodology of Georgakakis et al. (2008) to construct a
sensitivity curve for the merged Chandra image of NGC 300. We present a
summary of the sensitivity curve construction procedure here; the reader is
referred to Georgakakis et al. (2008) for additional details and an example
application to AGN observations. The merged blank sky image provides us with
an estimate of the mean expected source-free background counts $B$ across the
entire image. The cumulative probability that the observed number of counts
detected in a source (which is a sum of the counts intrinsic to the source and
the background) will exceed some limit $L$ is given by

$P_{B}(\geq L)=\frac{1}{\Gamma(L)}\int_{0}^{B}e^{-t}t^{L-1}dt$ (5)

assuming $B$ and $L$ are integers. If one adopts a value for the threshold
probability $P_{B}$, Equation 5 can be inverted to find the detection limit
$L$ for a given background. We adopt a threshold probability of $P_{B}=4\times
10^{-6}$ (the values assumed by, e.g., Georgakakis et al., 2008; Laird et al.,
2009, which corresponds to approximately 0.5 false sources per Chandra image)
and perform this calculation across the entire merged image area to construct
a map of $L$ values as a function of ($x,y$), given the background counts $B$
predicted by the blank sky map. We can convert $L$ to an energy flux by
assuming a power law spectral model (with photon index $\Gamma=1.7$ and
subject to the Galactic absorbing column) and using the total exposure time at
each ($x,y$) position across the image. From the resulting 2D fluxed image, we
measure the total area over which a source with a given flux could be
detected.

We perform this calculation for the whole galaxy out to a maximum distance of
$4R_{d}$ (we note that beyond this radius the X-ray point source population is
likely to be both highly incomplete due to the lower sensitivity at large off-
axis angle and dominated by background AGN, rather than sources intrinsic to
NGC 300). The 90% limiting 0.5-7 keV flux is 3.38$\times 10^{-15}$ erg s-1
cm-2 (corresponding to a luminosity of 1.6$\times 10^{36}$ erg s-1 at the
distance of NGC 300, in good agreement with Binder et al., 2012, 2017, who
used a subset of the observations analyzed in this work but a somewhat less
restrictive threshold for detection). We hereafter refer to the 90% limiting
luminosity as $L_{\rm X,90}$ and the 90% limiting flux as $S_{\rm X,90}$.

For the remainder of our analysis, we only keep X-ray sources that meet all of
the following three criteria: (1) detected at $r\leq 4R_{d}$, (2) detected at
$\geq 3\sigma$ significance by wavdetect, and (3) with fluxes $\geq S_{\rm
X,90}$. Within $4R_{d}$, there is one supernova remnant (Gross et al., 2019)
and one foreground star (Binder et al., 2012) that we remove from the X-ray
source list. A total of 70 X-ray sources meet these criteria, with 6 located
in the inner region, 16 in the inner middle bin, 22 in the outer middle bin,
and 26 in the outer bin. These sources are indicated in yellow on Figure 2.

###  3.2 The X-ray Binary Population

To estimate the number and total $L_{\rm X}$ produced by XRBs in NGC 300, we
first used the AGN log$N$-log$S$ distribution from the Chandra Multiwavelength
Project (ChaMP, Kim et al., 2004) to estimate the AGN contribution in each
annular region. We estimate 43$\pm$2 AGN above $S_{\rm X,90}$ are found within
the full survey area ($\leq 4R_{d}$), and $\sim$7 of these are AGN with fluxes
above $\sim 2.1\times 10^{-14}$ erg s-1 cm-2 (which corresponds to a
luminosity of $10^{37}$ erg s-1 at the distance of NGC 300). The remaining
$\sim$27 X-ray sources above $S_{\rm X,90}$ ($\sim$6 above $10^{37}$ erg s-1)
are therefore likely intrinsic to NGC 300.

When we use the area spanned by each elliptical annulus, the predicted number
of AGN above $S_{\rm X,90}$ in each subgalactic radial bin (from the innermost
region outward) is approximately 3, 8, 14, and 18. Given their far distance,
background AGN are more likely to be highly absorbed than XRBs and other X-ray
sources intrinsic to NGC 300; we therefore examine the number of sources with
hardness ratios placing them in the “ABS” category of Figure 7 in at least one
observation. We find 3, 13, 16, 20 sources show evidence of high absorption in
each of the four radial bins. This is only slightly higher than the predicted
number of background AGN, and consistent with expectations given that some
XRBs can experience significant intrinsic absorption. We therefore expect
approximately 3, 8, 8, and 8 X-ray sources above $S_{X,90}$ in the inner,
inner middle, outer middle, and outer bin, respectively, are XRBs in NGC 300.
We additionally measure the total flux we expect to originate from AGN and
subtract this value from the flux of the total X-ray source population; the
remainder is the approximate flux of XRBs in NGC 300. We use the reported
uncertainties on the ChaMP log$N$-log$S$ distribution to assess the
uncertainty in the number of AGN predicted in our observations; the
uncertainty is $\sim$2-3 AGN for the whole galaxy. These calculations are
summarized in Table 3.2.

Given the relatively low number of likely XRBs observed in each radial bin, we
do not attempt to directly fit the XRB X-ray luminosity functions. Instead, we
predict the expected number of HMXBs and LMXBs and their total $L_{\rm X}$
using the (metallicity-independent) HMXB and LMXB scaling relations of Lehmer
et al. (2020, 2021):

$\displaystyle\text{log}\left(\frac{L_{\rm X}^{\rm HMXB}}{\text{SFR}}\right)$
$\displaystyle=39.71^{+0.14}_{-0.09}\left[\frac{\text{erg
s}^{-1}}{M_{\odot}\text{ yr}^{-1}}\right]$ (6)
$\displaystyle\text{log}\left(\frac{L_{\rm X}^{\rm LMXB}}{M_{\star}}\right)$
$\displaystyle=29.25^{+0.07}_{-0.06}\left[\frac{\text{erg
s}^{-1}}{M_{\odot}}\right]$

These scaling relations were derived from a sample of 24 elliptical galaxies
(Lehmer et al., 2020) and 33 star-forming (average SFR$\sim$1.4 M⊙), HMXB-
dominated galaxies (Lehmer et al., 2021). The XLFs of some of these galaxies
extend below 1037 erg s-1 (4/22 elliptical galaxies had 90% limiting
luminosities below $10^{37}$ erg s-1 and 9/33 star-forming galaxies had 50%
limiting luminosities comparable to $L_{\rm X,90}$ in this work; Lehmer et
al., 2020, 2021), however the majority of the X-ray sources used to derive
these scaling relations were significantly brighter than most of the X-ray
sources coincident with the disk of NGC 300. Given the relatively small number
of galaxies with X-ray source populations imaged to faint fluxes, the
applicability of these scaling relations to faint XRB populations may be
limited (see Section 4 for further discussion).

Since the SFRs and stellar masses considered here are low and XRB formation is
a stochastic process, we use the following approach to estimate the XRB
population properties expected for NGC 300. First, we randomly draw values of
the SFR and M⋆ from within the uncertainties derived from our SED modeling for
each subgalactic region in NGC 300 assuming a normal distribution with the
mean set equal to the best-fit value of each parameter. Since our best-fit
values of the SFR and M⋆ have asymmetric errors, we set the standard deviation
of this normal distribution to the larger uncertainty. We then randomly select
a value from within the uncertainties of the above scaling relations and
compute the expected $L_{\rm X}$ for each population. To convert the predicted
total $L_{\rm X}$ into a number of XRBs, we assume that both HMXBs and LMXBs
follow the broken power law models described in Lehmer et al. (2019, 2020).

To randomly generate a sample of synthetic XRBs that follow a broken power law
distribution, we follow the approach outlined in Clauset et al. (2009,
Appendix D). We first generate a random $L_{\rm X}$ value for an XRB according
to $L_{\rm X,r}=L_{\rm X,90}(1-r)^{-1/(\gamma_{1}-1)}$, where $\gamma_{1}$ is
the faint-end power law slope and $r$ is a number that is randomly drawn from
a uniform distribution in the range $0\leq r<1$. If $L_{\rm X,r}$ is brighter
than the 1039 erg s-1 cutoff luminosity assumed in Lehmer et al. (2019, 2020)
it is rejected. If $L_{\rm X,r}$ is within the observed range of $L_{\rm X}$
values as the true sample, we draw a second uniform random variate $r_{2}$
(also in the range $0\leq r_{2}<1$) and accept $L_{\rm X,r}$ if $(L_{\rm
X,r}/L_{\rm X,90})^{-\gamma_{1}}\geq r_{2}$. If $L_{\rm X,r}$ is rejected, we
use a similar functional form as above to generate a new value of $L_{\rm
X,r}$ that follows the bright end of the XLF: $L_{\rm X,r}=L_{\rm
X,break}(1-r)^{-1/(\gamma_{2}-1)}$, where $\gamma_{2}$ is the slope of the
bright end of the XLF and $L_{\rm X,break}$ is the break luminosity ($3\times
10^{37}$ erg s-1 for LMXBs and $10^{38}$ erg s-1 for HMXBs; Lehmer et al.,
2019, 2020). Synthetic XRBs are generated one at a time and their total
luminosity is tracked until the value predicted using the scaling relations
and relevant SFR and M⋆ values is obtained. This process is repeated 10 000
times to build up a distribution of synthetic HMXB and LMXB populations that
satisfy the scaling relations in Equation 6 for the subgalactic regions of NGC
300. We used the median and 90% confidence interval to define the “predicted”
number of XRBs and their collective $L_{\rm X}$ and report these values in
Table 3.2. For clarity, we use use $N$ and $L_{\rm X}$ to refer to numbers of
X-ray sources and their luminosity measured or inferred from the data,
respectively, and $\mathcal{N}$ and $\mathcal{L}_{X}$ are used to indicate
numbers and luminosities, respectively, that are predicted purely from the XRB
scaling relations as described above.

Table 6: The Predicted and Inferred X-ray Source Population in NGC 300

| Galaxy Inte- | Radial Bin | Sum of All
---|---|---|---
| grated ($r\leq 4R_{d}$) | $r\leq R_{d}$ | $R_{d}<r\leq 2R_{d}$ | $2R_{d}<r\leq 3R_{d}$ | $3R_{d}<r\leq 4R_{d}$ | Radial Bins
Observed X-ray Point Sources
$N_{\rm X}$ | 70 | 6 | 16 | 22 | 26 |  $\cdots$
$N_{\rm AGN}$aafootnotemark:  ($>S_{\rm X,90}$) | 42.8$\pm$2.1 | 2.8$\pm$0.1 | 8.0$\pm$0.4 | 14.1$\pm$0.8 | 17.9$\pm$0.9 |  $\cdots$
$N_{\rm AGN}^{\rm HR}$bbfootnotemark:  | 52 | 3 | 13 | 16 | 20 |  $\cdots$
$N_{\rm XRB}^{\rm HR}$bbfootnotemark:  | 18 | 3 | 3 | 6 | 6 |  $\cdots$
$N_{\rm int}$ ($>L_{\rm X,90}$) | 27.2$\pm$2.1 | 3.2$\pm$0.1 | 8.0$\pm$0.4 | 7.9$\pm$0.8 | 8.1$\pm$0.9 |  $\cdots$
$N_{\rm int}$ ($>10^{37}$ erg s-1) | 6.4$\pm$0.4 | 0.6$\pm$0.1 | 2.8$\pm$0.1 | 3.0$\pm$0.1 |  $<$0.2 |  $\cdots$
$L_{\rm X,int}$ddfootnotemark:  (1038 erg s-1) | 4.9$\pm$0.4 | 0.16$\pm$0.03 | 3.9$\pm$0.3 | 0.4$\pm$0.1 | 0.4$\pm$0.2 |  $\cdots$
Predicted X-ray Point Sourcesccfootnotemark:
$\mathcal{N}_{\rm HMXB}$ ($>L_{\rm X,90}$) | 4${}^{+2}_{-3}$ | 1$\pm$1 | 3${}^{+2}_{-1}$ | 2${}^{+2}_{-1}$ | 1$\pm$1 | 7${}^{+6}_{-4}$
$\mathcal{N}_{\rm HMXB}$ ($>10^{37}$ erg s-1) | 4${}^{+2}_{-1}$ | 1$\pm$1 | 2${}^{+2}_{-1}$ | 2$\pm$1 | 1$\pm$1 | 6${}^{+5}_{-4}$
$\mathcal{N}_{\rm LMXB}$ ($>L_{\rm X,90}$) | 3$\pm$2 | 1${}^{+2}_{-1}$ | 2$\pm$1 | 2${}^{+2}_{-1}$ | 1$\pm$1 | 6${}^{+6}_{-4}$
$\mathcal{N}_{\rm LMXB}$ ($>10^{37}$ erg s-1) | 2${}^{+2}_{-1}$ | 1$\pm$1 | 1${}^{+2}_{-1}$ | 1$\pm$1 | 1$\pm$1 | 4${}^{+5}_{-4}$
$\mathcal{N}_{\rm HMXB+LMXB}$ ($>L_{\rm X,90}$) | 7${}^{+4}_{-5}$ | 2${}^{+3}_{-2}$ | 5${}^{+3}_{-2}$ | 4${}^{+4}_{-2}$ | 2$\pm$2 | 13${}^{+12}_{-8}$
$\mathcal{L}_{X}$(HMXB) ($10^{38}$ erg s-1) | 7.6${}^{+2.6}_{-2.2}$ | 0.6$\pm$0.3 | 4.3${}^{+1.3}_{-1.1}$ | 2.6${}^{+1.2}_{-1.1}$ | 0.6${}^{+0.3}_{-0.2}$ | 8.1${}^{+3.1}_{-2.7}$
$\mathcal{L}_{X}$(LMXB) ($10^{38}$ erg s-1) | 4.5${}^{+0.5}_{-0.4}$ | 1.0$\pm$0.2 | 1.6$\pm$0.2 | 0.7$\pm$0.1 | 0.5$\pm$0.1 | 3.8$\pm$0.6
$\mathcal{L}_{X}^{\rm HMXB+LMXB}$ ($10^{38}$ erg s-1) | 12.1${}^{+3.1}_{-2.6}$ | 1.6$\pm$0.5 | 5.9${}^{+1.5}_{-1.3}$ | 3.3${}^{+1.3}_{-1.2}$ | 1.1${}^{+0.4}_{-0.3}$ | 11.9${}^{+3.7}_{-3.3}$

References. — aInferred from the AGN log$N$-log$S$ distribution (Kim et al.,
2004). bNumber of AGN and XRB candidates inferred from hardness ratio
analysis. cValues are not corrected for the true $L_{\rm X}$ of NGC 300 ULX-1
(see Section 3.3 for details); when corrected for NGC 300 ULX-1, the galaxy-
integrated $L_{\rm X,int}$ is 4.43$\times 10^{39}$ erg s-1 and the
corresponding radial bin ($R_{d}<r\leq 2R_{d}$) $L_{\rm X,int}$ becomes
4.36$\times 10^{39}$ erg s-1.

Subgalactic regions are modeled independently; the sums in columns (3)-(6)
agree (within the uncertainties) with the spatially-integrated results in
column (2).

###  3.3 Observed vs. Predicted X-ray Source Populations

We observe more X-ray sources above $S_{\rm X,90}$ than predicted for the
field of NGC 300 based on the ChaMP AGN log$N$-log$S$ distribution and the
scaling relations of Lehmer et al. (2020, 2021). Figure 8 shows the observed
radial source distribution (black), the AGN contribution inferred from the
ChaMP log$N$-log$S$ distribution (red), the observed X-ray source population
expected to be intrinsic to NGC 300 (blue), and the combined HMXB and LMXB
population expected from the XRB scaling relations (dashed purple) above
$S_{\rm X,90}$. Only in the central region ($r\leq R_{d}$) of NGC 300 is there
near-consistency between observations and predictions (owing to the large
uncertainties in the expected number of XRBs), while the observed number of
X-ray sources in the outer three regions exceeds predictions by $\geq
3\sigma$.

Figure 8: Radial distribution of all X-ray sources (black), the AGN
contribution inferred from the ChaMP log$N$-log$S$ distribution (red), the
observed X-ray source population expected to be intrinsic to NGC 300 (blue),
and the combined HMXB and LMXB population expected from the XRB scaling
relations (dashed purple).

The excess number of observed X-ray sources is also apparent in the XLF, as
shown in Figure 9. The unabsorbed luminosity of an X-ray source is dependent
on the spectral model assumed; oftentimes, a fiducial spectral model
appropriate for both XRBs and AGN is assumed (usually an absorbed power law)
and used to convert count rates to unabsorbed luminosities. We first use
WebPIMMS101010https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl
to implement this approach, assuming a power law model with a photon index
$\Gamma=1.7$ that is subject to the Galactic absorbing column ($N_{\rm
H}=10^{21}$ cm-2 towards NGC 300; HI4PI Collaboration et al., 2016) to convert
the measured count rates to fluxes (and then into luminosities using the
distance to NGC 300). The cumulative XLF can then be computed using

$N(>S)=\sum_{i,S_{i}>S}\frac{1}{A(S_{i})},$ (7)

where $A(S)$ is the area over which a source with flux $S$ could be detected
as measured from the sensitivity curves in Section 3.1. The resulting
cumulative XLF is shown in black in Figure 9. There is one striking example of
a source in NGC 300 where the assumption of a fiducial spectral model fails to
yield a realistic luminosity approximation: NGC 300 ULX-1. Detailed
observations of this source by XMM-Newton and NuSTAR indicate a relatively
constant luminosity of $\sim 4\times 10^{39}$ erg s-1, with the modest count
rates observed by Chandra likely the result of significant local obscuration
by a precessing accretion disk (see, e.g. Binder et al., 2020, and references
therein). We thus replace the fiducial model luminosity for NGC 300 ULX-1 with
the “true” luminosity of the source, and recalculate a “corrected” XLF (dot-
dashed gray line in Figure 9). We additionally show the ChaMP AGN
log$N$-log$S$ distribution (red Kim et al., 2004) and the synthetic HMXB
(dashed blue) and LMXB (dot-dashed dark red) XLFs described in the previous
section. The total predicted XLF (AGN + HMXB + LMXB) and the corresponding
uncertainties are shown in purple. When we replace the incorrect inferred
$L_{\rm X}$ of NGC 300 ULX-1 (from the fiducial model) with the more accurate
$L_{\rm X}$ we find that galaxy-integrated $L_{\rm X,int}$ in Table 3.2
becomes ($4.43\pm 0.04$)$\times 10^{39}$ erg s-1. The value of $L_{\rm X,int}$
becomes (4.36$\pm$0.03) $\times 10^{39}$ erg s-1 in the radial bin containing
NGC 300 ULX-1 ($R_{d}<r\leq 2R_{d}$).

There is excellent agreement between the “corrected” observed XLF and the
predicted XLF above $\sim 10^{37}$ erg s-1. Below this value, we observe
$\sim$20 more X-ray sources than predicted based on the survey area, stellar
mass, and SFR. If we attempt to describe the low-flux end of the XLF by
increasing the AGN contribution, the assumed survey area must be increased by
$\sim$60%, which is well in excess of any area corrections derived from the
uncertainties in $R_{d}$. Furthermore, increasing the AGN contribution leads
to a clear excess of sources at $\sim(1-4)\times 10^{37}$ erg s-1, where our
survey is expected to be highly complete.

We subtract the observed XLF from the predicted XLF and show the resulting
“excess” XLF in the bottom panel of Figure 9. Both a broken power law model
and an exponential cutoff model describe the shape of this excess XLF well; in
either case, the faint-end slope is $\sim$0.7 with a break at $\sim 10^{37}$
erg s-1. In the broken power law model, the bright-end slope is very steep
($\gg$5) but largely unconstrained. There are two plausible explanations for
the observed excess: (1) missing AGN, as the faint-end slope is broadly
consistent with the faint end of the AGN log$N$-log$S$ distribution, and (2)
variable sources, as the broken power law shape of the excess XLF matches the
predicted shape of the XLF of simulated variable XRBs (Binder et al., 2017).
The XRB scaling relations from Equation 6 were derived from on-average
brighter XRB populations (largely above $10^{37}$ erg s-1) than those observed
in the merged NGC 300 image and, as shown in Figure 9 (top panel) the
uncertainties in the HMXB XLF at luminosities below $10^{37}$ erg s-1 are
significant. If we assume that all of the excess sources below $10^{37}$ erg
s-1 are XRBs, their total $L_{\rm X}$ would be $\sim 2.5\times 10^{37}$ erg
s-1, which is comparable to the uncertainties on the predicted
$\mathcal{L}_{X}$ values from the XRB scaling relations (see Table 3.2).
Although the HMXB XLF (and corresponding $L_{\rm X}$/SFR scaling relation)
adequately predicts the integrated $L_{\rm X}$ of NGC 300, it does not predict
the total number of objects in the galaxy below $\sim$1037 erg s-1.

Figure 9: Top: The spatially-integrated XLF of NGC 300. The solid black line
shows the observed XLF when a fiducial spectral model is used to directly
convert count rates to luminosities, while the dot-dashed gray line shows the
XLF after correcting for the “true” X-ray luminosity of NGC 300 ULX-1. The
predicted contributions of AGN, HMXBs, and LMXBs are shown as solid red,
dashed blue, and dot-dashed dark red lines, respectively (the shaded region
indicates uncertainties), and the sum of these three components is shown in
purple. Bottom: The “excess” XLF (the difference between the gray and purple
distributions in the top panel) with a broken power law model (blue)
superimposed. Only the horizontal axis uses a logarithmic scale, which makes
the power law model appear curved. The gray shaded region indicates the 90%
limiting luminosity cutoff.

##  4 Discussion

Numerous works have investigated the relationship between the number and
collective luminosity of HMXBs and host galaxy SFR and metallicity. However,
the majority of these studies have utilized galaxy-integrated measurements of
SFR and $Z$, and have largely targeted vigorously star-forming nearby galaxies
to ensure a well-populated high-luminosity HMXB XLF. We now compare the
results of our subgalactic analysis of NGC 300 to two key prior results: the
$N_{\rm HMXB}$-SFR relationship (Mineo et al., 2012) and the $L_{\rm
X}$-SFR-$Z$ relationship. We seek to emphasize (1) that SFHs of more localized
regions within a galaxy can vary, for example as a result of inside-out disk
growth, which leads to measurable differences in the XRB populations that
form; (2) that metallicity gradients provide a means to test metallicity
dependence of observable XRB properties within a single galaxy; and (3) that
HMXB-SFR scaling relations may not be reliable tools for predicting the XRB
population below $\sim$1037 erg s-1.

There is evidence that HMXBs, in particular, are related to even more local
conditions than considered here: for example, Williams et al. (2013) used CMD
fitting of the HST-resolved stellar populations in the very local ($\sim$50
pc) vicinity of HMXB candidates in NGC 300 and found that the SFRs of HMXB-
hosting regions was a factor of $\sim$2.5 higher than that of randomly
selected regions roughly $\sim$40-60 Myr ago. HMXB formation, however, is an
intrinsically rare and stochastic process – not all highly localized regions
that experienced a similar increase in SFR $\sim$50 Myr produce an HMXB – so
additional assumptions about, e.g., the HMXB formation efficiency, are
required to compared these results to the galaxy-integrated literature. We
additionally examine whether highly localized ($\sim$100 pc) environments
around XRB candidates exhibit significant differences in SFH or metallicity
than the sub-galactic annular regions discussed above.

###  4.1 The $N_{\rm HMXB}$-SFR Relationship

Since our observations are not able to definitively separate HMXBs from LMXBs
for a majority of sources in NGC 300, we assume that the relative number and
luminosities of HMXBs to LMXBs follow the predictions of the XRB scaling
relations for each subgalactic bin to estimate the number of HMXBs (and the
integrated $L_{\rm X}$ of these HMXBs) in each radial bin. In other words,

$N_{\rm HMXB}=N_{\rm int}\times\left(\frac{\mathcal{N}_{\rm
HMXB}}{\mathcal{N}_{\rm HMXB}+\mathcal{N}_{\rm LMXB}}\right)$ (8)

and

$L_{\rm HMXB}=L_{\rm int}\times\left(\frac{\mathcal{L}_{\rm
HMXB}}{\mathcal{L}_{\rm HMXB}+\mathcal{L}_{\rm LMXB}}\right).$ (9)

Mineo et al. (2012) observed a linear relationship between the number of HMXBs
observed in a galaxy and the host galaxy SFR. In that work, a single power law
XLF was fit to the bright ($>10^{37}$ erg s-1) HMXB population and used to
predict the total number of HMXBs above $10^{35}$ erg s-1. To compare with
those results, in Figure 10 we plot the predicted number of HMXBs above
$L_{X,90}$ as a function of SFR for the four subgalactic regions within NGC
300. A simple linear regression fit yields

$N_{\rm HMXB}(>10^{36}~{}\text{erg s}^{-1})=\left(95\pm
19\right)\left(\frac{\text{SFR}}{M_{\odot}\text{ yr}^{-1}}\right).$ (10)

The slope of this relationship is in good agreement (within $\sim 2\sigma$)
with Mineo et al. (2012, slope of 135), which is remarkable given the very
different populations of sources considered and different assumptions invoked
in the two studies. Assuming the predicted number of HMXBs, $\mathcal{N}_{\rm
HMXB}$, derived from the $L_{\rm X}$/SFR scaling relation yields a
significantly shallower slope: 43$\pm$5\. We can additionally estimate the
HMXB production rate by dividing the expected number of HMXBs in each
subgalactic region by the total stellar mass formed in the last 100 Myr within
each region. Across all four subgalactic regions within NGC 300, we estimate
an HMXB production rate of $\sim$(1-2)$\times 10^{-6}$ M⊙-1. These values are
lower than observed in M33 (Lazzarini et al., 2023), but are comparable to the
production rates of HMXBs in M31 $\lesssim$50 Myr ago (Lazzarini et al.,
2021). The agreement between expected number of HMXBs and the HMXB production
rate to other dedicated HMXB studies suggests that the extra X-ray sources
observed below $10^{37}$ erg s-1 are dominated by HMXBs, and not AGN.

Figure 10: The number of HMXBs above $L_{\rm X,90}$ (given by Equation 8) as a
function of SFR in NGC 300. Black points show our data, the best-fit linear
relationship (and the 3$\sigma$ uncertainty) is shown in gray. The solid blue
line shows the relationship derived by Mineo et al. (2012). Figure 11: The
ratio of $L_{\rm X}$/SFR as a function of metallicity for the four subgalactic
regions in NGC 300 (gray) and the whole galaxy within $4R_{d}$ (black)
compared to recent $L_{\rm X}$-SFR-$Z$ relationships from the literature.
Uncertainties in the literature relationships are typically on the order of
$\sim$0.5 dex. The point at $Z\approx 0.43Z_{\odot}$ indicates the subgalactic
region containing the two brightest (and youngest) sources in NGC 300: NGC 300
X-1 and NGC 300 ULX-1. The vertical bright, dashed red lines show the effect
of removing NGC 300 ULX-1 from our analysis, while the faint, dashed red line
shows the effect of additionally removing NGC 300 X-1. The only subgalactic
region with a constant SFR between 0-30 Myr and 30-100 Myr is indicated with
an open circle.

###  4.2 The $L_{\rm X}$-SFR-$Z$ Relationship

In Figure 11 we compare our subgalactic (and galaxy-integrated) observations
of NGC 300 to several $L_{\rm X}$-SFR-$Z$ relationships from the literature
(uncertainties in the literature relationships are typically on the order of
$\sim$0.5 dex). To estimate the $L_{\rm X}$ originating from HMXBs only, we
take the observed $L_{\rm X,int}$ from Table 3.2 and multiply it by the
percentage of the predicted $\mathcal{L}_{\rm X}$ originating in HMXBs from
the XRB scaling relations. The predicted $\mathcal{L}_{\rm X}$ was determined
from simulating XRB populations from within the stellar mass and SFRs for NGC
300, so the uncertainties on $\mathcal{L}_{\rm X}$ are inclusive of both
measurement uncertainties and additional uncertainties that result from
stochastic formation of very bright sources (Section 3.2). These uncertainties
are comparable to the stochastic uncertainties in $L_{\rm X}$/SFR reported by
Lehmer et al. (2021). We find that the galaxy-integrated $L_{\rm X}$/SFR ratio
of NGC 300 agrees with predictions for the value of $Z$ we measure from our
SED modeling if the corrected $L_{\rm X}$ of NGC 300 ULX-1 is not used; using
the higher $L_{\rm X}$ for this one source puts the galaxy-integrated $L_{\rm
X}$/SFR ratio $\sim 1\sigma$ above the literature relationships. Two of the
four subgalactic regions fall below the literature $L_{\rm X}$-SFR-$Z$
relationships, while the outermost region (at the lowest $Z$) is consistent
(within $\sim 1\sigma$) with the literature relationships. The only radial bin
that lies above the literature results is the bin from $R_{d}<r\leq 2R_{d}$,
which has the highest recent SFR and contains the two youngest and brightest
HMXBs in NGC 300: ULX-1 and X-1. If we exclude NGC 300 ULX-1 from our
analysis, the predicted $\mathcal{L}_{\rm X}$ for HMXBs from the XRB scaling
relations in Equation 6 is consistent with the observed $L_{\rm X}$ of NGC 300
X-1 and the $L_{\rm X}$/SFR ratio for this radial bin becomes consistent with
literature $L_{\rm X}$-SFR-$Z$ relationships (bright red, dashed line in
Figure 11).

Figure 11 thus demonstrates the effect that stochastic formation of bright
HMXBs in modestly star forming galaxies can have on the observed $L_{\rm
X}$/SFR ratio. Many ULXs of comparable $L_{\rm X}$ to NGC 300 ULX-1 are
located within galaxies with ongoing and significantly higher SFRs (often by
an order of magnitude or more) than that observed in NGC 300, which is past
the peak of its SFH. When NGC 300 ULX-1 is excluded, the subgalactic region at
$R_{d}<r\leq 2R_{d}$ ($Z\approx 0.4Z_{\odot}$) is the only radial bin
consistent with literature $L_{\rm X}$-SFR-$Z$ relationships (bright, dashed
red line in Figure 11). The remaining $L_{\rm X}$ from this region is
dominated by the Wolf-Rayet + black hole binary NGC 300 X-1 (Binder et al.,
2021), which is assumed to be extremely young based on the high masses of both
the black hole and donor star. The outermost radial bin ($Z\approx
0.25Z_{\odot}$) falls below the literature $L_{\rm X}$-SFR-$Z$ relationships
only by $\sim 1-2\sigma$. This is the only subgalactic region where the SFR
increased (by $\sim$30%) in the last 30 Myr compared to 30-100 Myr ago, and so
likely contains a higher proportion of younger, more massive HMXBs compared to
other regions of NGC 300.

These differences in the $L_{\rm X}$/SFR ratios of the subgalactic regions
within NGC 300 may be hinting at an HMXB stellar age effect on the $L_{\rm
X}$-SFR-$Z$ relationship, although with a limited number of data points it is
difficult to draw any firm conclusions. That X-ray emission from XRB
populations evolves as a function of time is not new (Fragos et al., 2013b;
Lehmer et al., 2019; Antoniou et al., 2019), but this effect is difficult to
directly measure on the short timescales over which HMXBs evolve
($\lesssim$100 Myr). The XLFs of LMXBs reveal that older galaxies tend to host
a larger number of faint LMXBs ($\lesssim 10^{37}$ erg s-1) compared to
younger galaxies, and that frequency of bright- and ULX-class LMXBs is much
higher in younger galaxies (Zhang et al., 2012). The duration of the HMXB
phase for a single binary is expected to be short ($\sim$104 yr, Mineo et al.,
2012). If more massive HMXBs are systematically more luminous than lower-mass
HMXBs, one would expect the the X-ray emission from an HMXB population to
shift from younger/brighter systems to older/fainter systems over time, and in
the absence of renewed star formation episodes the collective $L_{\rm X}$
would diminish as brighter systems begin to “turn off.” The evolution of
$L_{\rm X}$/M⋆ in star-forming galaxies over cosmic time has been examined
both observationally (Gilbertson et al., 2022) and theoretically (Linden et
al., 2010; Fragos et al., 2013a, b), and this study of NGC 300 which
approximately controls for stellar age on timescales $<$100 Myr provides
another framework by which HMXB age evolution could be studied further.

Prior studies have systematically selected higher-SFR galaxies with ongoing
star formation for analysis, which results in a proportionally larger
population of young and luminous HMXBs (akin to NGC 300 ULX-1 and NGC 300 X-1)
compared to the more “typical,” moderate-luminosity $\sim$40-60 Myr HMXBs that
are readily detectable in nearby galaxies (in NGC 300, but also the Magellanic
Clouds, M31, M33, and NGC 2403; Williams et al., 2013; Lazzarini et al., 2021,
2023; Antoniou & Zezas, 2016; Antoniou et al., 2010). We note that if NGC 300
X-1 were to evolve out of its current X-ray bright phase, the $L_{\rm X}$/SFR
ratio of its corresponding subgalactic region would follow the general
downward trend towards higher metallicities seen in the other three radial
bins (faint red, dashed line in Figure 11). While the stochastic formation of
very bright and short-lived HMXBs is expected to be influenced by metallicity,
the current understanding of the $L_{\rm X}$-SFR-$Z$ relationship and XRB
scaling relations are likely biased towards this small subset of younger,
brighter HMXBs. Whether increasing stellar ages result in an $L_{\rm
X}$-SFR-$Z$ relationship with a different normalization or slope (or both)
compared to the one derived from presumably younger HMXBs is uncertain based
on the present study; a more rigorous test of the effect of stellar age on the
HMXB $L_{\rm X}$-SFR-$Z$ relationship would require a comprehensive study of
the fainter HMXB populations in modestly star-forming galaxies with spatially
resolved SFHs and metallicity gradients.

Evidence of variation and evolution in the $L_{\rm X}$-SFR-$Z$ relationship
may be hidden within the significant scatter seen in previous studies. The
literature $L_{\rm X}$-SFR-$Z$ relationships were derived using a variety of
SFR indicators and stellar IMF assumptions, and some of the scatter in these
relationships may originate from a heterogeneous mix of SFR tracers in
addition to different assumptions about metallicity calibrations. Self-
consistently modeling the modern SFRs, SFHs, and metallicities in subgalactic
regions concurrently with the HMXBs population is required to firmly establish
the extent to which stochastic and stellar age effects contribute to the
$L_{\rm X}$-SFR-$Z$ relationship, and ultimately our understanding of massive
binary star evolution.

###  4.3 Comparison to XRB-Hosting Local Regions

The short lifetimes of HMXBs ($\lesssim$100 Myr) and relatively low natal kick
velocities they experience during initial compact object formation (Giacobbo &
Mapelli, 2020) prevent HMXBs from travelling more than $\sim$100-200 pc from
their birthplaces. We thus explored whether the environments directly
associated with XRB candidate sources showed any significant differences in
terms of their SFHs or metallicities compared to the larger annular regions
described above.

We identified XRB candidate X-ray sources by excluding highly absorbed sources
(e.g., those with HR1 $\geq$ 0.4 which are likely background AGN) and
extracted photometry from circular apertures centered on the X-ray sources
with radii of 10.3′′ ($\sim$100 pc). We verified that none of the individual
circular apertures overlapped; thus the total area covered is 0.03 kpc$\times$
the number of XRB candidates in each annular region (from innermost to
outermost region, we identify 4, 11, 7, and 10 XRB candidates). The SED
resulting from the combined photometry from all XRB candidates within a given
annular region is then modeled with Prospector in a manner identical to that
described in Section 2.

The resulting SFHs and the metallicity gradients from both the XRB-hosting
local regions and the larger annular regions are shown in Figure 12. To
account for the significant differences in areas covered, we show the SFR
surface density $\Sigma_{\rm SFR}$. The SFHs of the XRB-hosting local regions
generally show an enhancement in $\Sigma_{\rm SFR}$ the most recent ($<$30
Myr) time bin by factors of $\sim$3-10 while the SFRs observed in the 30-100
Myr time bins are consistent (within the large uncertainties) with those found
from the annular analysis presented. This leads to an overall increase in the
inferred SFR within the last 100 Myr in the locally-derived SFR compared to
the annuli-averaged SFR. The metallicity gradient derived from the XRB-hosting
regions is nearly identical to the one derived from the larger annular
regions. If the SFRs associated with XRBs is indeed higher than the annuli-
averaged value, the $L_{\rm X}$/SFR ratios shown on Figure 11 would be shifted
down. The deviation between the NGC 300 subgalactic analysis and the
literature $L_{\rm X}$-SFR-$Z$ relationships becomes more pronounced; the NGC
300 sources are underluminous for their metallicity and SFR. This is
consistent with the finding in Section 3.3 (see also Figure 9), where we find
evidence for an excess of possibly variable, low-luminosity XRB population in
excess of the predictions from HMXB and LMXB scaling relations.

|
---|---
Figure 12: Left: The radially-resolved SFHs of the annular regions (solid
lines) compared to the stacked localized regions around XRB candidates (dashed
lines; the number of XRB candidates used to derive the SFHs is indicated in
the legend). Colors are as in Figure 6. Right: The metallicity gradient
derived from the stacked localized regions containing XRBs (green) compared to
the annular regions (black), Kudritzki et al. (2008, light blue), and Gogarten
et al. (2010, dark blue).

##  5 Summary

We have conducted a panchromatic analysis of the nearby galaxy NGC 300 on
subgalactic scales to illustrate how spatially-resolved studies of nearby
galaxies can provide useful insights into the XRB-host galaxy connection. The
3.6$\mu$m scale length ($R_{d}$) is used to divide the galaxy into four equal-
width annular regions, and we perform aperture photometry to measure the UV-
optical-IR flux densities within each subgalactic region. We then construct
SEDs and use Prospector to measure the SFHs and present-day stellar masses,
metallicities, and recent ($<$100 Myr) SFRs across the star-forming disk. The
current galaxy-integrated stellar mass is M⋆=$(2.15^{+0.26}_{-0.14})\times
10^{9}$ M⊙ and the current SFR is $\sim$0.18$\pm$0.07 M⊙ yr-1, in agreement
with previous measurements. Our measured metallicity gradient and radially-
resolved SFH are also consistent with other gradients reported in the
literature.

We used four archival Chandra observations to identify 70 X-ray sources within
$4R_{d}$ that were detected at high significance. We used the ChaMP AGN
log$N$-log$S$ distribution (Kim et al., 2004) and the XRB scaling relations of
Lehmer et al. (2020, 2021) to compare the observed X-ray point source
population to the predicted X-ray point source population. On a galaxy-
integrated scale, the radial source distribution and XLF show a clear excess
in the number of observed X-ray sources at luminosities $<10^{37}$ erg s-1.
The X-ray source excess is consistent with a faint and variable XRB population
that is not well described by the assumed XRB scaling relations, with the
possibility of additional AGN contamination. Our subgalactic analysis reveals
the linear relationship between the number of HMXBs formed and SFR that is
consistent with prior expectations that derived from a large, galaxy-
integrated sample of more vigorously star forming galaxies.

Most intriguingly, we find that the $L_{\rm X}$/SFR ratios as a function of
$Z$ across the four subgalactic regions are only consistent with $L_{\rm
X}$-SFR-$Z$ relationships from the literature for the bins that are likely to
host the youngest HMXBs. These results hint at a possible trend in the $L_{\rm
X}$-SFR-$Z$ relationship with stellar age, as lower-mass HMXBs begin to
dominate the total $L_{\rm X}$ of a galaxy after the star formation episode
that formed those systems ends. We derive SFHs for the local ($\sim$100 pc)
regions immediately surrounding XRB candidate sources and find that their
recent ($<$30 Myr) SFR surface densities are generally higher than those of
the integrated annular regions, which decreases the $L_{\rm X}$/SFR ratio and
makes the underluminous nature of the NGC 300 XRB population even more
pronounced. Expanding this analysis to a larger sample of nearby galaxies that
are just past their peak of star formation, and where the fainter X-ray source
population can be studied on subgalactic scales (and the spatially-resolved
SFH can be directly modeled), is needed to further investigate possible trends
in the $L_{\rm X}$/SFR ratio with both metallicity and stellar age.

B.A.B acknowledges support from the National Science Foundation Launching
Early-Career Academic Pathways in the Mathematical and Physical Sciences
(LEAPS-MPS) award #2213230. This paper employs a list of Chandra datasets,
obtained by the Chandra X-ray Observatory, contained in
https://doi.org/10.25574/cdc.204 (catalog DOI: 10.25574/cdc.204), and software
provided by the Chandra X-ray Center (CXC) in the application package CIAO.
This research made use of Astropy, a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013, 2018, 2022).

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