# Systematically Measuring Ultra-Diffuse Galaxies (SMUDGes). VI. Nuclear Star
Clusters

Mika Lambert Steward Observatory and Department of Astronomy, University of
Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA  Donghyeon J. Khim Steward
Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry
Ave., Tucson, AZ 85721, USA  Dennis Zaritsky Steward Observatory and
Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ
85721, USA  Richard Donnerstein Steward Observatory and Department of
Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

###### Abstract

We present our photometric search for potential nuclear star clusters (NSCs)
in ultra-diffuse galaxies (UDGs) as an extension of the SMUDGes catalog. We
identify 325 SMUDGes galaxies with NSCs and, from the 144 with existing
distance estimates, identify 33 NSC hosts as UDGs ($\mu_{0,g}$ $\geq$ 24 mag
arcsec-2, $r_{e}\geq 1.5$ kpc). The SMUDGes with NSCs lie on the galaxy red
sequence, satisfy the NSC-host galaxy stellar mass relationship, have a mean
NSC stellar mass fraction of 0.02 but reach as high as 0.1, have NSCs that are
displaced from the host center with a standard deviation of 0.10$r_{e}$, and
weakly favor higher density environments. All of these properties are
consistent with previous results from higher surface brightness galaxy
samples, allowing for at most a relatively weak dependence of NSC behavior
with host galaxy surface brightness.

Low surface brightness galaxies (940), Galaxy properties (615), Galaxy
structure (622), Galaxy nuclei (609), Star clusters (1567),

††journal: AJ††software:  Astropy (Astropy Collaboration et al., 2013, 2018),
GALFIT (Peng et al., 2002), Matplotlib (Hunter, 2007), NumPy (van der Walt et
al., 2011), pandas (McKinney, 2010), sep (Barbary, 2016), Source Extractor
(Bertin & Arnouts, 1996), SciPy (Oliphant, 2007; Millman & Aivazis, 2011),
EMCEE (Foreman-Mackey et al., 2013b) skimage (van der Walt et al., 2014)

##  1 Introduction

The origin of massive, compact stellar populations in galaxies, whether those
are globular clusters (GCs), massive black holes, or nuclear star clusters
(NSCs; Caldwell, 1983; Binggeli et al., 1984; Caldwell & Bothun, 1987; Bothun
& Mould, 1988), remains poorly understood. Some models envision their
formation through violent, extreme episodes of star formation (e.g., Mihos &
Hernquist, 1994; Bekki & Couch, 2001; Kravtsov & Gnedin, 2005; Kruijssen et
al., 2012; Renaud, 2018), but such episodes may seem somewhat less likely in
the low surface brightness galaxies that are the focus here. Other models
relate the different population classes to each other, such as those that
posit that NSCs form from the infall and merger of GCs (e.g., Tremaine, 1976;
Gnedin et al., 2014; Sánchez-Salcedo & Lora, 2022; Modak et al., 2023) and
those where central massive black holes form from the dynamical collapse of
NSCs (e.g., Begelman & Rees, 1978; Miller & Hamilton, 2002; Antonini et al.,
2015). Again, processes that may be common and relevant in massive, high
surface brightness galaxies, such as dynamical friction, perhaps play a
diminished role in the low mass, low surface brightness galaxies. Commonality
of features makes it attractive to link these populations into a coherent
scenario (e.g., Wehner & Harris, 2006; Rossa et al., 2006; Ferrarese et al.,
2006; Fahrion et al., 2021).

At least regarding NSCs, basic constraints on any of these scenarios include
the rates at which NSCs are found, the relationship between the NSC and host
stellar masses and stellar populations, any connection to the host galaxy
morphology, and the alignment of the NSC and its host’s dynamical center — all
specified as a function of the relevant host galaxy properties. One key such
property might be the host’s central surface brightness, which presumably
reflects the degree to which dissipation has concentrated matter toward the
galaxy center, where an NSC or a central black hole would reside.

The richness of some of these constraints is already evident, for example, in
the NSC occupation fraction (the fraction of galaxies that host an NSC). The
occupation fraction varies with galaxy mass, rising and then falling as one
proceeds from lower to higher mass galaxies (Neumayer et al., 2020; Hoyer et
al., 2021). This behavior is quite distinct from that of the number of GCs in
a galaxy, which varies proportionally with the host galaxy mass (Blakeslee et
al., 1997; Burkert & Forbes, 2020) even to low masses (Forbes et al., 2020;
Zaritsky, 2022). One might naively have expected a close correspondence
between the rate at which GCs and NSCs are found if NSCs are indeed formed
from merged GCs, but the observed difference may highlight how certain details
of the formation physics, such as the amplitude of dynamical friction, relate
to the host galaxy properties. One can test scenarios along these lines and
attempt to reproduce the NSC occupation fractions (e.g., Lotz et al., 2001;
Capuzzo-Dolcetta & Mastrobuono-Battisti, 2009).

The incidence rate of GCs and the NSC occupation fraction are less well
determined as a function of host galaxy surface brightness, but there are
indications of a dependence of the NSC occupation fraction (Binggeli et al.,
2000; Lim et al., 2018). Of course, the incidence rate is only one of various
properties to explore. One could imagine that the typical mass of an NSC
varies with host mass. In fact, such a trend has been observed (e.g., Balcells
et al., 2003; Ferrarese et al., 2006; Turner et al., 2012; Scott & Graham,
2013; den Brok et al., 2014), and we are tempted to ask whether there is an
analogous relation with the host surface brightness.

Here we begin our exploration of the SMUDGes (Zaritsky et al., 2019, 2021,
2022, 2023) set of ultra-diffuse galaxy (UDG) candidates to explore the nature
of NSCs in low surface brightness galaxies. The value of UDGs to this topic is
that they include the most massive, low surface brightness galaxies known (cf.
van Dokkum et al., 2015) and thus may help us disentangle the roles of mass
vs. surface brightness in shaping NSC properties. The value of the SMUDGes
sample is that it is large and spans all environments, enabling us to also
explore the possibility of a dependence of NSC properties on environment. This
work, in which we focus on the identification of NSCs in SMUDGes galaxies is
followed closely by Khim et al. (2023, in prep.) in which we extend the
analysis to galaxies of somewhat higher surface brightness, but similar
stellar masses, provided by the Sloan Digital Sky Survey (SDSS; Kollmeier et
al., 2017) and images from the DESI Legacy Survey (Dey et al., 2019). That
work will enable us to place the SMUDGes galaxies in a wider context, without
the additional challenge of comparing across studies with disparate image
quality and analysis methodology.

Uniform, well-defined criteria are essential for comparisons of NSC properties
across host mass, surface brightness, or environment. As we will show, the
data characteristics and criteria used to identify NSCs lead to large
variations in sample properties. NSCs are broadly defined to be dense and
massive stellar agglomerations that reside in or near the centers of galaxies
that are brighter than the extrapolated surface brightness profile of the
inner region of that galaxy (Neumayer et al., 2020). However, consistently-
applied, quantitative criteria do not exist for any of these defining
characteristics. The purity and completeness of samples therefore varies among
studies. Such (potential) differences between studies raise questions about
any comparisons one would wish to pursue between, for example, cluster (Lim et
al., 2018) and field samples (Carlsten et al., 2022) of galaxies with NSCs. In
this particular example, both studies find occupation fractions $\sim$ 20%,
but possible systematic differences leave open the question of whether this
agreement is physically meaningful or fortuitous. Other studies have
investigated the prevalence of NSCs in both early and late-type galaxies (Côté
et al., 2006; Georgiev & Böker, 2014) and in dwarf galaxies (Carlsten et al.,
2022), but again comparisons among them remain challenging.

As with every study, this one too has its weaknesses and strengths. Among the
weaknesses relative to existing NSC studies is that we are working with
shallower, lower resolution images than the state-of-the-art (e.g., Hubble
Space Telescope images have been used that have a $\sim$ 3 mag deeper point
source magnitude than our images; Lim et al., 2018). Thus, we only probe the
bright end of the NSC luminosity function for the majority of our hosts,
potentially suffer greater contamination, and obtain more uncertain
photometric parameters for the NSCs themselves. On the other hand, the
strength of this, and our sister study (Khim et al. 2023, in prep), is that
our galaxy sample spans mass, environment in low luminosity, low surface
brightness galaxies with consistent classification, thereby simplifying
comparisons. In §2, we describe our methodology. In §3, we discuss the results
as follows: 1) an NSC classification catalog for the entire SMUDGes sample; 2)
our constraints on the relative co-centricity of NSCs and their hosts; 3) NSC
properties and their relationship to host galaxy properties and; 4) any
relation to the host galaxy environment. We use a standard WMAP9 cosmology
(Hinshaw et al., 2013), although the results are insensitive to different
choices of cosmological parameters at the level of current uncertainties, and
magnitudes are from SDSS/DESI and are thus on the AB system (Oke, 1964; Oke &
Gunn, 1983).

##  2 Methodology

###  2.1 The Data

We begin with the 6,805 visually-confirmed UDG candidates in the SMUDGes
catalog (Zaritsky et al., 2023). These candidates were selected to have a low
central surface brightness in the $g$-band, $\mu_{0,g}$ $\geq$ 24 mag
arcsec-2, and a large effective radius on the sky, r${}_{e}\geq$ 5.3 arcsec.
In addition, we now impose a stricter color criterion than in the original
work to remove likely background interlopers ($\sim$ 0.2 mag redder than the
red sequence) and candidates with unphysical blue colors (0 $<g-r<0.8$; which
removes 225 galaxies from the sample) and an angular size criterion to remove
nearby galaxies that are unlikely to be UDGs and which corresponds to half our
extracted image size ($r_{e}>26$ arcsec; which removes 38 additional
galaxies). We retain 6,542 galaxies to analyze. The angular size cuts help our
model fitting (see §2.2) by eliminating images that are either too small
relative to our resolution or too large relative to the extracted image. These
cuts do not easily translate to criteria on physical size because our
candidates span a range of distances.

Importantly, for a study of NSCs in low surface brightness galaxies, the
estimate of the central surface brightness of the galaxy used to define the
catalog was calculated using Sérsic model fitting where high surface
brightness objects, including any potential NSC, were masked (Zaritsky et al.,
2022, 2023). This aspect of the parent survey, and also whether a limit on the
value of the Sérsic $n$ value is imposed (SMUDGes imposes $n<2$ in the
fitting), can affect the sample selection and care must be taken if comparing
results drawn from different catalogs of low surface brightness galaxies.

For our photometric analysis, we extract 200$\times$200 pixel (52.4 $\times$
52.4 arcsec) $r$-band images of each of the 6,542 candidates from the 9th data
release (DR9) of the Legacy Survey (Dey et al., 2019). These cutouts provide a
sufficiently large field of view to include adequate background coverage.

###  2.2 Preparing for Model Fitting

To explore the morphology of each UDG candidate further than what was done in
the SMUDGes catalog papers, and to determine if there is evidence of an NSC,
we use the photometric model fitting software package GALFIT (Peng et al.,
2010). We take three preparatory steps before fitting any model.

First, to assess the nature of a possible unresolved source near the center of
each UDG candidate, we need the point spread function (PSF) of each image.
NSCs in our images are unresolved because their half light radii are typically
$<$ 10 pc (Böker et al., 2004; Turner et al., 2012; Georgiev & Böker, 2014),
which corresponds to an angular size $<$ 1 arcsec for distances $>$ 2 Mpc. For
each UDG candidate, we adopt the PSF model provided in the Legacy Survey for
the relevant image.

Second, we generate an image of the pixel-by-pixel uncertainties, a
$\sigma$-image, that is used to assess the likelihoods of the models GALFIT
produces. We calculate the $\sigma$-image using the inverse-variance image
provided by the Legacy Survey that was calculated using the image stack
contributing to each pixel.

Finally, we isolate each UDG candidate from any surrounding bright sources
that could affect the modeling. This is a complicated, iterative process that
we describe in more detail below when discussing our model fitting procedure.

###  2.3 Selecting Among Models

To identify NSCs in UDG candidates, we first assess whether there is evidence
for a concentration of light beyond what can be described by a Sérsic model
with index $n\leq 2$. If there is, then we assess the nature of that excess
component. Because our targets are extremely diffuse and faint, the GALFIT
fitting results are often highly sensitive to the adopted starting parameters
for the model fitting, as well as to the presence of a central component, such
as a bulge or NSC. To mitigate the impact of these factors on our results, we
adopt a two-stage approach that we describe below and repeat the fitting
multiple times with different adopted starting parameter values in each of the
two stages. In the fitting, we use a convolution box set to half the length of
the image, a magnitude zero-point of 22.5, originating from the definition of
nanomaggies, and a plate scale of 0.262 arcsec pix-1 for the Legacy Survey. We
also summarize the procedure in Table 1.

Our goal for the first fitting stage is to obtain a best-fit single Sérsic
model for each candidate UDG that is free from the influence of nearby objects
or other stellar components (e.g., NSC or bulge). The results from this fit
guide the fitting of the more complex models in the second stage.

We fit this single Sérsic model, utilizing distinct image masks for each
galaxy. We create these masks (for an example, see Figure 1) using the Source
Extractor Python library (SEP, see Barbary, 2016, for details) based on
(SExtractor; Bertin & Arnouts, 1996). We start by subtracting the spatially
varying background generated by SEP. We then identify objects defined as
groupings of at least 5 adjacent pixels each with a flux that is 1.5$\sigma$
above the background. We mask these objects except for the galaxy itself. We
remove from further consideration the four UDG candidates whose masked regions
cover more than 50% of the entire image area (leaving 6538 for study at this
point) because these are poorly constrained in the fitting. Furthermore, in
many cases, the fitting results would be strongly affected by an existing
central NSC or bulge in the host galaxy. Thus, we augment the mask to include
any central region of each candidate that contains at least 5 adjacent pixels
that are 1.5 times (0.44 mag) brighter than $\mu=$ 24 mag arcsec-2. This step
introduces a potential bias against low luminosity NSCs that are not masked
because they do not rise to this level and therefore are incorporated into the
base Sérsic model. We address sample completeness further below.

Figure 1: An example UDG candidate with a plausible NSC (SMDG0004118+163159,
the second galaxy with what we will classify as an NSC in our right ascension
ordered catalog) with several nearby contaminating sources is presented in the
left panel. The mask in the right panel highlights most of the visible nearby
sources with the central object shaded in gray indicating that it is unmasked
for the final model fitting, but would masked in the initial model fitting
(see text for details).

As we alluded to previously, to mitigate sensitivity to the adopted initial
GALFIT fitting parameter values, we repeat the fitting six times using
different values. We use combinations of two different effective radii (30 and
50 pix) and three surface brightness values at $r_{e}$ (25, 28, and 31 mag
arcsec-2). We calculate the reduced chi-squared statistic, $\chi^{2}_{\nu}$,
within a circular region of radius 50 pixels ($\sim 13.1$ arcsec) centered on
the image center and select the model with the smallest $\chi^{2}_{\nu}$
value. The value of 50 pixels ensures that the bulk of the host galaxies falls
within the evaluation region. We find that GALFIT occasionally produces a
model fit with acceptable $\chi^{2}_{\nu}$ but unusually large final parameter
uncertainties, even though other realizations (i.e. different initial
parameters) result in fits with typical uncertainties. Because these solutions
offer no meaningful constraints on the fit parameters, we only consider models
where $r_{e}>2\sigma_{r_{e}}$. We settled on this criterion after exploring a
range of options, but these odd cases are clearly distinct from the remainder
and various criteria could have been adopted without qualitatively affecting
the results. In the rare case where all models for a particular galaxy fail
this criterion (10 galaxies), we consider the galaxy as having failed our
fitting procedure. Additionally, we have 364 galaxies for which all models are
rejected with at least 90% confidence based on their $\chi_{\nu}^{2}$. These
are also categorized as having failed our fitting procedure.

Figure 2: Model and residual images for SMDG0004118+163159. Left to right: the
original image, single Sérsic model, residual image using the single Sérsic
model, Sérsic + PSF model, and residual image using the Sérsic + PSF model.
The need for a central unresolved source is evident when comparing the
residual images. Table 1: Model Summary

1st component | 2nd component | 3rd component
---|---|---
Initial GALFIT model (Stage 1)
S1 | … | …
Final GALFIT model (Stage 2)
S1 | … | …
S1 | S2 | …
S1 | N1,PSF | …
S1 | S2 | N1,PSF
S1 | N1,PSF | N2,PSF

Note. — Columns indicate morphological components used in the six different
model combinations. The initial parameters used for of the second stage are
those of the best-fit model from the first stage. We adopt two- or three-
component models only when they are significantly favored over the one- or
two-component models, respectively. Similarly, we only adopt models with
unresolved sources when they are significantly favored over those with
resolved components. A more comprehensive description can be found in Section
2.3.

In the second stage, we unmask the center of the host galaxy and all other
sources within $0.5r_{e}$ and compare new fits using five independent model
classes for each candidate. These model classes consist of: 1) a single Sérsic
profile, to model UDG candidates with no unresolved nuclear source; 2) a pair
of Sérsic profiles, to model UDG candidates with an additional extended
central source, such as a separate bulge component; 3) a Sérsic profile plus a
PSF profile, to model UDG candidates with an unresolved central source; 4) a
pair of Sérsic profiles plus a PSF profile, to model UDG candidates with an
extended central source, such as a separate bulge component, and an unresolved
central source; and 5) a Sérsic profile plus two PSF profiles, to model UDG
candidates with two unresolved central sources. In the second and fourth model
classes, we refer to the first Sérsic component, intended to model the galaxy
as a whole as S1 and the second, more compact, component, which is intended to
model any nuclear excess, as S2. For models with an unresolved central source,
we refer to that component as N1,PSF. In the fifth model class, we refer to
the PSF component closer to the center of the Sérsic component as N1,PSF and
the further one as N2,PSF. Among the NSC targets, we have exclude those with a
point source magnitude error in either $g$ or $r-$band larger than 0.2 mags.

As done in the first fitting stage, we fit each model class six times using
different initial parameters. The initial parameters we adopt for S1 are those
from the best-fit model in the first stage, except for the central surface
brightness. For the initial brightness of S1 and S2 we take permutations of
the best fit from the first stage and values that are three magnitudes fainter
and brighter. For the size of S2 we adopt a starting size of five pixels. We
found no improvement in fitting when varying $r_{e}$ for S1, so we simply use
the value obtained in the first stage.

We provide GALFIT with the original UDG candidate image, the mask, the PSF,
the $\sigma$-image, and a constraint file that sets the search range for each
of the free parameters. The free parameters are the following: central
positions, Sérsic indices (n), effective radii ($r_{e}$), magnitudes (m), axis
ratios (AR), and position angles (PA) for S1 and S2; and the position and
amplitude for N1,PSF and N2,PSF; and the background level. The constraint file
provides initial parameter values and range limits for the chosen model
parameters (Peng et al., 2002). We constrain the centers of the various
components to lie within a 40 by 40 pixel square centered on the candidate
UDG. In some studies of NSCs, the centrality of the source is a critical
criterion (e.g., Côté et al., 2006; Georgiev & Böker, 2014), but at this stage
of our analysis we allow for significantly offset unresolved components.
Because our sample consists of diffuse galaxies, we set the upper limit of the
Sérsic index to be 2.0 (our UDG candidates have $\langle n\rangle<1$; Zaritsky
et al., 2022). On the other hand, we allow the compact Sérsic component (S2)
to have $n$ as large as 5.0 because such a component could be morphologically
comparable to a bulge. We also set a lower limit on the axis ratio of the
Sérsic components of 0.3 to prevent unrealistically elongated models (SMUDGes
sample has an axis ratio threshold of $0.34<b/a$; Zaritsky et al., 2023).
Finally, we require that the effective radius of each Sérsic component exceed
0.75 times the size of the PSF to ensure differentiation between what we
consider to be a resolved component from an unresolved one.

Before comparing the resulting models, we exclude models that can be rejected
with at least 90% confidence given their $\chi^{2}_{\nu}$. Because we are
primarily concerned with modeling the center of the UDG, the area within which
we evaluate $\chi^{2}_{\nu}$ is now a circular region of radius $0.5r_{e}$,
centered on the UDG candidate, where $r_{e}$ and the central position come
from the single Sérsic model fit from the first fitting stage. Again, we only
consider models where $r_{e}>2\sigma_{r_{e}}$. If no models survive our
$\chi^{2}_{\nu}$ and $r_{e}>2\sigma_{r_{e}}$ criteria, then that galaxy is
considered to have failed our fitting procedure.

Although $\chi^{2}_{\nu}$ values provide a goodness-of-fit measure, they are
inappropriate for selecting among models of differing intrinsic complexity.
Because a model with greater fitting freedom will naturally fit the data
somewhat better, one must account for this additional flexibility when
assessing whether there is statistical evidence in favor of the more complex
model. The Akaike Information Criterion (AIC; Akaike, 1974) is one formulation
that incorporates a penalty for models with higher complexity.

We adopt a slight modification of the original AIC formulation that is
referred to as the AICc criterion (Sugiura, 1978),

$AICc=\chi^{2}+2p+\frac{2p(p+1)}{N-p-1}$ (1)

where $p$ is the number of model parameters and $N$ is the number of data
points that are fit, to compare among models. This modification is appropriate
when there are a small number of degrees of freedom and AICc will converge to
the original AIC criterion as the degrees of freedom increase. The model with
the smaller AICc value is the statistically preferred model although the
greater the difference ($\Delta$AICc) the greater the confidence with which
one can discriminate among them. Because AICc values describe likelihoods and
are distributed like $\chi^{2}$, we are able to calculate the confidence level
corresponding to any specific value of $\Delta$AICc. For our situation, a
2$\sigma$ confidence level corresponds to $\Delta$AICc $=11.83$. We adopt this
threshold to assess if the best-fitting two-component models are significantly
preferred over the best-fitting one-component model. If it is, then we use
this same threshold to assess if the model with S1 \+ N1,PSF or S1 \+ N1,PSF
\+ N2,PSF is significantly preferred over that with S1 \+ S2. Figure 2 shows
an example of a model and residual images created by GALFIT for the single
Sérsic model and the S1 \+ N1,PSF model for SMDG0004118+163159. At this stage,
we identify 757 SMUDGes for which the S${}_{1}+$ N1,PSF or S1 \+ N1,PSF \+
N2,PSF model is preferred over other competing models with greater than
$2\sigma$ confidence.

Models where S2 has a large $n$ value, $\geq 4$ and so greater than that
corresponding to a de Vaucouleurs profile (de Vaucouleurs, 1948), often appear
visually indistinguishable from models with N1,PSF. Furthermore, many of these
S2 components have small $r_{e}$, resulting in their observed morphology being
dominated by the convolution with the PSF. For the vast majority of these, the
models with N1,PSF are statistically favored over models with S2, but not at
the $2\sigma$ confidence level. After visual inspection, we decided to
reclassify the subset of systems where two-component models are statistically
favored over a one-component model at greater than 2$\sigma$ confidence and
where that S2 component has $n\geq 4$, $r_{e}<10$ pix (2.62 arcsec), and lies
within 5 pixels of the competing N1,PSF as S1 \+ N1,PSF (see Figure 3). This
increases the total number of SMUDGes within which we identify unresolved
sources to 842. Nevertheless, the exact division, if one even exists, between
a PSF-convolved unresolved source and a steep inner profile is not a simple
issue (Côté et al., 2007).

Figure 3: Distribution of $n$ and $r_{e}$ for S2 components when two
components are statistically favored over a single component, but no
statistically significant difference exists between competing multi-component
models. The box highlights the parameter region where, after visual
examination, we reclassify systems as having unresolved sources.

We present a summary of our classifications in Table 2 and an electronic
version of a SMUDGes (Zaritsky et al., 2023) line-matched catalog of our
classifications as well as an example in Table 3. For each galaxy, we assign
the most specific and complex model that is statistically preferred by the
AICc criterion at the $>$ 2$\sigma$ confidence level, except for the S1
classification which includes all cases where S1 is favored, even if not at
$>$ 2$\sigma$ confidence, and cases where another more complex model was
statistically favored but not at beyond the $2\sigma$ confidence level. To
provide more detail, in the case where we assign the
S${}_{1}+$S${}_{2}+$N1,PSF classification it must be statistically preferred
over both the S${}_{1}+$S2 model and the S${}_{1}+$N1,PSF model at greater
than the $2\sigma$ confidence level. In Table 2 we divide the classifications
into objects with and without unresolved sources. The latter includes galaxies
with indications of possible unresolved sources such as those where models
with N1,PSF are preferred but not at the 2$\sigma$ level and those for which
S${}_{1}+$S${}_{2}+$ N1,PSF is preferred at the $2\sigma$ level. We included
the latter in this category because following visual inspection we decided
that these are generally highly complex systems that are simply difficult to
model and do not necessarily show evidence for NSCs. As a group these are
identified as S${}_{1}+$ N${}_{1,PSF}?$ in the Table.

Table 2: Classification Summary

Class | N | $r_{p}/r_{e}<0.10$ | ND | $r_{p}/r_{e}<0.10$
---|---|---|---|---
No unresolved |  |  |  |
source |  |  |  |
S1 | 2979 | … | 698 | …
S1+S2 | 1558 | … | 263 | …
S1+N${}_{1,PSF}?$ | 795 | … | 176 | …
Total | 5332 | … | 1137 | …
Unresolved |  |  |  |
source |  |  |  |
S1+N1,PSF | 673 | 242 | 221 | 105
S1+N1,PSF | 169 | 83 | 63 | 39
\+ N2,PSF |  |  |  |
Total | 842 | 325 | 284 | 144
Failed fitting | 364 | … | 69 | …
Total | 6538 | 325 | 1490 | 144

Note. — Classifications for the SMUDGes candidates. The second column
describes the classification outcomes for the SMUDGes that satisfy 0
$<g-r<0.8$ and $r_{e}<26$ arcsec based on the original catalog (Zaritsky et
al., 2023) and are not heavily masked. The third column presents the numbers
of those with identified unresolved sources that lie within a normalized
projected separation from the S1 component that is $<0.10$, where the
normalization is done using $r_{e}$ as measured from the initial fitting of
the single Sérsic model. The fourth column refers back to the second column
and identifies the number of galaxies with estimated distances in the original
catalog. The fifth column refers back to the objects in the fourth column and
identifies the number of those that satisfy the projected separation
criterion.

###  2.4 False Positives

Among the 842 candidate UDGs for which the S${}_{1}+$ N1,PSF or S1 \+ N1,PSF
\+ N2,PSF models are preferred with $\geq 2\sigma$ confidence many show large
offsets between $S_{1}$ and N1,PSF (for example, 44 have offsets larger than
20 pixels or 5.2 arcsec). This result raises the question of what constitutes
a nuclear star cluster.

Our candidates span a range of projected offsets from the center of the host
galaxy (Figure 4). The offset distribution, plotted in normalized, distance
independent terms of $r_{p}/r_{e}$, where $r_{p}$ is the projected separation
between S1 and N1,PSF and $r_{e}$ is the effective radius of the host galaxy
(as measured in our initial fitting pass), appears to have two components: a
concentrated central one and a far more extended one. The distribution is
somewhat difficult to interpret because the positional offsets we allowed for
GALFIT translate to different cutoffs in $r_{p}/r_{e}$ for each galaxy.
Nevertheless, it suggests that our sample consists of populations we might
call ‘true NSCs’ and ’contamination’. The contaminating population may be a
combination of sources physically associated with the galaxy, such as non-
nuclear star clusters and star-forming regions, and physically unassociated
sources, such as foreground stars and background galaxies. Critically,
however, the details of how these populations are differentiated will impact
certain questions related to the possibility of off-center nuclear star
clusters, whether that be because they have stalled in their inward migration
or been jostled off-center by a dynamical event.

To better understand the NSC and background populations, so that we can
optimize our selection of NSCs, we explore a set of models for the radial
distribution of all candidate NSCs in Figure 4. In those models, we describe
the NSC population alternatively as an empirically-motivated projected 1-D
Gaussian distribution in normalized projected radial offsets, $r_{p}/r_{e}$, a
2-D Gaussian distribution in $r_{p}/r_{e}$ on the sky, or a 2-D exponential in
$r_{p}/r_{e}$. The contaminating population we describe alternatively as a
Sérsic distribution with the parameters of the host galaxy, assuming the
contaminants come primarily from the galaxy itself, or a uniform distribution
on the sky, assuming the contaminants are principally either foreground or
background sources. Finally, we also account for our radial completeness by
evaluating the fraction of the sample at each radius for which our selection
criteria would have allowed us to find an NSC. The radial completeness is
affected both by our criteria that we only search for unresolved sources
within $0.5r_{e}$ and allow GALFIT to explore positional offsets within a box
of 40 by 40 pixels.

We evaluate the parameters for each of these model combinations using a
Bayesian approach and the Markov Chain Monte Carlo (MCMC) ensemble sampler
called EMCEE (Foreman-Mackey et al., 2013a). We model the distribution using
different combinations of either a 1-D Gaussian or exponential primary
distribution and a Sérsic profile or uniform background secondary
distribution. We adopt uniform priors for the amplitude and standard deviation
of the central population and for the amplitude of the second component. We
find a slight preference for the models that describe the distribution as a
1-D Gaussian central component plus Sérsic-distributed contamination, but the
exponential-distributed central component plus uniform background is nearly as
good a fit (Figure 4). As such, we conclude that we cannot distinguish whether
the contaminating population is primarily within the host galaxy or
unassociated on the basis of this data and fitting. This question will be
reexamined in future work where we explore the extended population in greater
detail. The contaminating population is potentially physically interesting
because it might include clusters that are otherwise similar to NSCs but found
at large radii.

Figure 4: The distribution of normalized radial offsets between S1 and N1,PSF,
$r_{p}/r_{e}$. Solid blue lines represent the data, the blue dotted lines show
the models for the central Gaussian and background that together, modulated by
the radial completeness, combine to produce the red dashed line. In the left
panel, the background is assumed to follow the S1 profile, while in the right
panel, it is assumed to be randomly distributed on the sky. The models fit the
data indistinguishably well, demonstrating that we cannot differentiate
between the two background scenarios at the current time and that our
conclusions are insensitive to this choice.

To distinguish true NSCs from contamination, the choice of model becomes
irrelevant because using either model we conclude that the contamination in
our recovered NSC sample is 15% (the percentage that we have set as our
target) when we reject candidates with $r_{p}/r_{e}>0.10$. Setting 0.10 as an
upper limit on the normalized radial offset, we retain 325 UDG candidates with
NSCs, and reject 517. Examples of our final NSC sample are presented in Figure
5 and this is the sample we present as NSC hosting SMUDGes. This severe
selection demonstrates directly the impact of any imposed radial selection on
the overall population. Consider that a slightly more permissive criterion of
$r_{p}/r_{e}<0.2$ results in $\sim$50% more candidates (463), although a
larger fraction of these will come from the contaminating population.

To highlight again how comparing among studies is fraught, we note that
Poulain et al. (2021) accept the brightest unresolved source out to a radial
offset of $0.5r_{e}$ as an NSC. After an initial reading of our results one
would conclude that their sample is dominated by contaminants. However, their
use of superior imaging data (both deeper and of higher angular resolution)
might allow them to reject contamination far better than we can in our data.
Without carefully examining both data sets and redoing parts of the analysis
similarly, it is not possible to reach definitive conclusions regarding a
comparison between our two studies.

Our strict radial selection cut ensures relatively high purity (defined to be
85%) but excludes true NSCs that lie beyond this radial cut. We use our best-
fit model for this component (a 1-D Gaussian with $\sigma=0.10$) to calculate
that 35% of this component lies outside of $r_{p}/r_{e}=0.10$. From the 325
candidate UDGs with NSCs at projected offsets less than $r_{p}/r_{e}=0.10$, we
calculate that 276 are true NSCs after correcting for the 15% contamination
and that this corresponds to a total population of UDG candidates with NSCs of
340 after correcting for systems that lie at $r_{p}/r_{e}>0.10$.

Figure 5: Mosaic of the first 21 SMUDGes sources in Right Ascension that have
an identified NSC. Images span slightly over an arcmin on a side with North at
the top and East to the left. These are drawn from the Legacy Surveys on-line
viewer (https://www.legacysurvey.org/viewer). Object labels are included at
the top of each panel.

###  2.5 False Negatives

We classify as single component 2979 candidate UDGs. Among this set, there may
be some for which the NSCs fall below our detection limit and these are, as
such, false negatives. To estimate our limiting magnitude, we randomly select
100 galaxies that are preferred by a single Sérsic model and insert artificial
point sources of varying brightness ranging from 18 to 30 magnitudes, in
intervals of 0.04 magnitudes, at the centers of the images. We model these
point sources using a Gaussian of 1.5 pixel width that extends out to 20
pixels and run our pipeline. When we do not recover the point source, there
are two failure modes: (1) the best-fit model is not the Sérsic+PSF model; (2)
the best-fit model is the Sérsic+PSF model but has a confidence level of less
than 2$\sigma$ ($\Delta$AICc $<$11.83). For each of the 100 galaxies, we set
the magnitude of the brightest inserted point source that we fail to recover
as the detection limit for that galaxy. In Figure 6 we present the set of
these detection limits as a function of the host galaxy central surface
brightness. The detection limits lie mostly between magnitudes of 23 to 25. We
find that the limits correlate weakly with central surface brightness
(confidence level 99.2% and Spearman Rank correlation coefficient of 0.26).
This result matches our intuition that it should be more difficult to detect
an NSC in a galaxy that is itself intrinsically brighter in its center.
However, given the large scatter in detection limits about this mean trend
(Figure 6), there must be other factors at play as well and we neglect the
subtle, but real, surface brightness dependence in our subsequent qualitative
discussion of completeness.

Figure 6: NSC detection limits in the $r$ band determined for 100 Single
Sérsic SMUDGes images as a function of $r-$band central surface brightness,
$\mu_{0,r}$. Points represent the brightest simulated point source that was
not recovered as an NSC by our procedure. Horizontal lines mark where we reach
the corresponding incompleteness percentage. The three labeled values
correspond to limiting magnitudes of $r_{PSF}$ of 23.2, 24.1, and 24.9 mag.
Figure 7: Absolute magnitude of NSCs, for SMUDGes with estimated distances,
across redshift. Systems that are UDGs ($r_{e}\geq 1.5$ kpc; filled blue
circles) and non-UDGs ($r_{e}<1.5$ kpc; open red circles) are plotted. The
curves designate completeness limits corresponding to the three incompleteness
fractions shown in Figure 6. We also include a vertical line that indicates
the redshift beyond which any object in the SMUDGes catalog would satisfy the
UDGs size criterion, and the redshifts corresponding to the Virgo and Coma
clusters.

We now examine the consequences of using shallower, lower resolution data by
comparing our classification results to those of Lim et al. (2018) for Coma
galaxies using Hubble Space Telescope images. Lim et al. (2018) presented 44
UDGs in their Table 1, 26 of which match SMUDGes. The majority of the
unmatched galaxies are either of smaller angular extent than the SMUDGes limit
or at a surface brightness where SMUDGes is incomplete. From the matched 26,
Lim et al. (2018) concluded from visual inspection that five contain NSCs.
Among these, we find an NSC in only one of those, but the normalized
separation ($r_{p}/r_{e}=0.226$) is greater than our separation criterion
($r_{p}/r_{e}<0.10$). We attribute our failure to identify NSCs in the other
four as the result of the difference between our magnitude limit of $\sim$24
mag and theirs of $\sim$ 27.4 mag (Lim et al., 2018). Lim et al. (2018) do not
present photometry for their NSCs, so we cannot confirm that these are indeed
fainter than our detection limit, but as we show in Figure 7, our detection
limit is likely to exclude nearly all NSCs by the time we are considering
galaxies at the distance of the Coma cluster. This incompleteness also helps
to explain why our occupation fraction (the fraction of candidates that we
identify to host NSCs), which is globally $\sim 0.05$ (340/6538) for our
sample, is so much lower than the $\gtrsim$ 0.20 that is commonly found (e.g.,
den Brok et al., 2014; Lim et al., 2018; Eigenthaler et al., 2018; Carlsten et
al., 2022), although it may also reflect that our sample is not as constrained
to high density environments and the occupation fraction is measured to be
higher in high density environments (Lim et al., 2018; Sánchez-Janssen et al.,
2019; Poulain et al., 2021; Carlsten et al., 2022).

Table 3: SMUDGes NSC Extension Catalog

SMDG Designation | mUDG,r | mNSC,r | mNSC,r,err | mNSC,g | mNSC,g,err | $(g-r)_{UDG}$ | $r_{e}$ | $r_{p}/r_{e}$ | Class
---|---|---|---|---|---|---|---|---|---
| (mag) | (mag) | (mag) | (mag) | (mag) | (mag) | (arcsec) |  |
SMDG0000017+325141 | 20.53 |  $-$99.00 |  $-$99.00 |  $-$99.00 |  $-$99.00 | 0.27 | 5.32 |  $-$99.000 | 1
SMDG0000202$-$402435 | 20.95 |  $-$99.00 |  $-$99.00 |  $-$99.00 |  $-$99.00 | 0.30 | 6.66 |  $-$99.000 | 6
SMDG0000334+165424 | 18.24 |  $-$99.00 |  $-$99.00 |  $-$99.00 |  $-$99.00 | 0.55 | 8.56 |  $-$99.000 | 2
SMDG0000453+305356 | 19.90 |  $-$99.00 |  $-$99.00 |  $-$99.00 |  $-$99.00 | 0.18 | 5.50 |  $-$99.000 | 2
SMDG0000473$-$040432 | 23.93 |  $-$99.00 |  $-$99.00 |  $-$99.00 |  $-$99.00 | 0.72 | 12.90 |  $-$99.000 | 7

Note. — We present here the first five lines of the full catalog, which is
available in electronic form. Magnitudes refer to the S1 and N1,PSF components
of the best fit model, while colors and effective radii refer to those derived
from the initial single Sérsic fit, which are found to be more stable. Entries
of $-99.00$ signal invalid values corresponding to SMUDGes galaxies for which
an unresolved source is not identified. We retain these SMUDGes to maintain
line-by-line matching with the Zaritsky et al. (2023) catalog. The numerical
values in the Class column correspond, in order, to the classification
categories in Table 2 ( S${}_{1}\equiv 1$, S${}_{1}+$S${}_{2}\equiv 2$,
S${}_{1}+$N${}_{1,PSF}?\equiv 3$, S${}_{1}+$N${}_{1,PSF}\equiv 4$,
S${}_{1}+$S${}_{2}+$N${}_{1,PSF}\equiv 5$, failed fitting$\equiv 6$) and
galaxies that were excluded prior to fitting ($\equiv 7$). Note that some
S${}_{1}+$N1,PSF and S${}_{1}+$S${}_{2}+$N1,PSF targets have photometric
errors larger than 0.2 mags. These were classified as S${}_{1}\equiv 1$ in
Table 2 and subsequently excluded from our analysis.

##  3 Results

A principal product of this study is an NSC-related classification for each
galaxy in the SMUDGes catalog and a measurement of the NSC properties when
there is one. We present in Table 3 the first five lines of the full catalog,
available electronically, where the objects are row matched with the Zaritsky
et al. (2023) catalog. We provide a summary of the classifications in Table 2
by listing the number of objects in each of our classification classes, the
number of which have an inferred unresolved source that meets our
$r_{p}/r_{e}<0.10$ criterion, the number in each class with distance estimates
from the original SMUDGes catalog (Zaritsky et al., 2023), and the number of
those with distance estimates that have an inferred unresolved source
component that meets our $r_{p}/r_{e}$ criterion.

###  3.1 NSC Positional Offsets

We have defined NSCs as the centrally located sub-population of unresolved
sources coincident with our UDG candidates. The degree to which NSCs are truly
found at the dynamical center of their host galaxy is somewhat difficult to
address because NSCs are often required, by definition, to be at the galaxy’s
center (e.g., Böker et al., 2002; Côté et al., 2006; Neumayer et al., 2011).
For specific examples, we cite Côté et al. (2007) which sets an offset upper
limit of 0.02$r_{e}$ (about 20 pc for the typical distance in their sample)
and Neumayer et al. (2020) who propose an offset limit of 50 pc. In contrast,
Poulain et al. (2021), who allow for larger offsets, find NSCs out to
0.58$r_{e}$. Binggeli et al. (2000) identify some with even larger offsets.
Offsets are potentially interesting to measure because they may be caused by
dynamical interactions (Bellovary et al., 2018) and could help us measure the
shape of the gravitational potential (Miller & Smith, 1992; Taga & Iye, 1998).

Although we too have imposed such a requirement, based on the distribution of
$r_{p}/r_{e}$, we do measure differences in the degree of alignment between S1
and N1,PSF. Careful examination of Figure 4 shows that the observed
$r_{p}/r_{e}$ distribution does not peak at zero separation and we have
measured a dispersion of 0.10. To assess whether these findings reflect
physical scatter in the co-centricity of the NSC and host or are simply the
results of measurement errors always leading to positive offsets, we compare
the measured distribution to the expected scatter arising simply from our
observational uncertainties. When we adopt the GALFIT positional uncertainties
in S1 and N1,PSF for the individual observed systems and use those to randomly
generate S1 and N1,PSF pairs, we find a distribution of $r_{p}/r_{e}$ that
also does not peak at zero but which has a dispersion of only 0.02. From this
result, we conclude that the observed scatter does constitute evidence of
physical offsets. Our measured dispersion is in excellent agreement with the
median offset measured in the MATLAS dwarf galaxy sample of 0.10$r_{e}$
(Poulain et al., 2021). Because the Poulain et al. (2021) sample has galaxies
with a somewhat brighter central surface brightness than SMUDGes111The MATLAS
sample (Poulain et al., 2021) has a mean central surface in the $g-$band of
24.0 mag arcsec-2, whereas the SMUDGes sample is defined to have a central
surface brightness $\geq$ 24 mag arcsec-2. Nevertheless, the MATLAS sample
contains 53 satellites that qualify as UDGs and is an interesting
complementary sample to ours., this agreement may be in conflict with findings
of increasing offsets in lower surface brightness hosts (Binggeli et al.,
2000; Barazza et al., 2003), although, as noted by Poulain et al. (2021) this
measurement is complicated by the likely larger uncertainties in the
measurements of the centers of lower surface brightness galaxies.

A different complication is that it is not clear which component is the better
tracer of the dynamical center. It is possible that the photometric center of
S1 is precisely measured but that it does not reflect the dynamical center.
All we can conclude is that the photometric center of S1 and the position of
N1,PSF show greater scatter than accounted for by the observational
centroiding uncertainties. In fact, the entire dark matter halo may be offset
leading to strong lopsidedness in the center of the galaxy (Prasad & Jog,
2017).

###  3.2 Nuclear and Non-nuclear Stellar Clusters: Host Galaxy Colors

A related question is whether the unresolved sources associated with the more
extended population include a substantial number of stellar clusters that are
otherwise indistinguishable from those in the NSC population. Such a
population would have important repercussions on models of NSC formation.

To work with a set of unresolved sources that are mostly independent of what
we have identified as the NSC population, we define a non-nuclear class in an
analogous way as we did for the nuclear class, again focusing on purity. We
use the model of the $r_{p}/r_{e}$ distribution but this time search for a
lower limit on $r_{p}/r_{e}$ that ensures $\leq 15$% contamination of the non-
nuclear population by the nuclear population. By setting that lower limit to
be 0.38, we find 134 UDG candidates with unresolved sources that we define to
be a ‘clean’ sample of non-nuclear unresolved sources.

Figure 8: The distribution of host galaxy colors, $(g-r)_{UDG}$ from Table 3,
and apparent magnitudes for systems with unresolved sources with
$r_{p}/r_{e}<0.10$ (NSCs; filled circles) and $r_{p}/r_{e}>0.38$ (non-NSCs;
open circles). The right panel shows the normalized distributions in color of
these two populations (filled histogram represents NSCs; unfilled blue, solid
line represents non-NSCs), as well as that for the full SMUDGes sample
(represented by the dashed red line).

We find that the distributions in color differ markedly (Figure 8). NSC hosts
have $g-r\sim$ 0.6, with a modest dispersion $\sim 0.05$, indicating that
these hosts are predominantly on the red sequence. The number of NSC hosts
that are much bluer ($g-r<0.5$) is only 17 (out of the total of 325) and is
smaller than the anticipated level of contamination (i.e. 15% or 30 galaxies),
suggesting that NSCs, as we have defined them, might only be found in red
sequence hosts galaxies in our sample. In contrast, the hosts of the non-
nuclear unresolved-source population are distributed broadly in color. This
flatter color distribution, and its similarity to the color distribution of
the entire SMUDGes sample, further suggests that a significant fraction of
this population is indeed contamination and not physically associated with the
hosts.

The relative deficit of NSCs in our blue hosts raises the question of whether
NSCs are exclusively a red galaxy phenomenon or reflect a selection bias that
is working against us. Previous studies of late-type galaxies have found large
occupation fractions (Georgiev & Böker, 2014), albeit not in hosts of this low
surface brightness. We identify various challenges in identifying NSCs in blue
SMUDGes. The hosts are more likely to be irregular and difficult to model,
leading both to failures in the fitting and to greater statistical noise in
the centroiding. The latter can both complicate our procedure for assessing
the significance of an additional unresolved source and corrupt our
measurement of the offset. Nevertheless, SMUDGes with color $0.4<g-r<0.5$ are
not highly irregular and yet these already host relatively few NSCs, as can be
seen by comparing the various distributions presented in Figure 8.

###  3.3 NSC properties

We now focus on the internal properties of NSCs in SMUDGes. We have two
distinct populations to discuss. First, we draw inferences from our sample of
325 NSCs for which we do not have distance estimates, to discuss distance-
independent aspects of the population. Their hosts are all galaxies of low
central surface brightness, although it is likely that many will not satisfy
the UDG physical size criterion. Second, we select only those hosts for which
we do have distance estimates (see Zaritsky et al., 2022, 2023, for a
discussion of distance estimation), and then either focus on the physical
properties of the 144 such systems or select only the 33 of those that meet
the UDG size criterion ($r_{e}>1.5$ kpc).

The luminosity (or corresponding stellar mass for similar mass-to-light
ratios) of an NSC scales with that of the host galaxy (Neumayer et al., 2020).
We find this broadly holds for our sample (Figure 9). We adopt a stellar M/L
of 1.8 M⊙/L⊙ for our transformation to stellar masses. The masses we derive,
typically between $10^{5}$ and $10^{7}$ M⊙ are within the previously measured
range for NSCs (Neumayer et al., 2020), although on the lower end as expected
given that the stellar masses of our host sample is also lower than average.
The data are mostly consistent with the previously published relationship
(Neumayer et al., 2020), although a steeper proportionality relation appears
to fit the data better. However, recall that we are incomplete at lower NSC
masses for more distant, hence typically larger and more massive hosts, which
could help fill in the distribution in the lower right portion of the diagram.
Furthermore, distance errors, and we expect that a significant fraction of the
estimated redshifts ($\sim$ 30 %) are incorrect (Zaritsky et al., 2023), will
tend to have a preference for scattering objects to larger distances and
proportionally higher masses along both axes. We conclude that our findings
are consistent with the previous relation obtained with higher surface
brightness galaxies. As such, we conclude that we find no systemic difference
in the relationship between NSC and host galaxy stellar masses for low surface
brightness galaxies relative to what was previously found for the more general
NSC host population.

We find that the stellar mass fraction in NSCs can reach close to 0.1 in the
most extreme objects, and is typically about 1/50th (Figure 9). This
measurement is distance independent. At the upper end of the mass fractions,
our findings are consistent with those of Binggeli et al. (2000) for Virgo
dwarfs and mostly consistent with those of Poulain et al. (2021) for the
MATLAS dwarfs. The latter do find objects with mass fractions $>$0.1, with one
reaching 0.45, although they find only 6 in their sample of 508 with mass
fraction $>$ 0.2. For the lower mass fractions our sample is incomplete, so we
do not discuss that side of the distribution. Our typical value of 0.02 is in
excellent agreement with their median value of 0.017, again suggesting no
strong dependence on host surface brightness.

The objects with the largest mass ratios appear to pose interesting
constraints on formation models as it is difficult to envision how nearly 10%
of a galaxy’s stellar mass could end up as an NSC either in the globular
cluster infall model or in the central star formation model. In Figure 10 we
present the five galaxies and the five UDGs with the most extreme NSC mass
fractions in our sample. In our entire sample, where we can use galaxies
without estimated distances because we are considering only the ratio of
stellar masses, we do find that the most extreme galaxies, such as those shown
in the upper row of the Figure, have slightly more than 10% of their stellar
mass in their NSCs. For the most extreme UDGs that percentage drops to less
than 3%. Given the small number of confirmed UDGs in the sample, we do not yet
know if this represents a real difference between UDGs and other galaxies or
if we have simply not yet sampled the high fraction tail sufficiently well. In
closing, we note that the NSC in one of the UDGs (SMDG0430339-052909) has a
measured color that is anomalously red and so we suspect that this is an
unresolved background galaxy rather than an NSC. Having one contaminant among
the ten galaxies shown in the Figure is consistent with our anticipated 15%
contamination rate.

Figure 9: NSC stellar mass vs. host galaxy stellar mass. Stellar masses
calculated assuming a fixed stellar mass-to-light ratio of 1.8
$M_{\odot}/L_{\odot}$. Solid line and shaded region represent the relationship
and its uncertainty presented by Neumayer et al. (2020). The heavy dotted line
represents a normalized 1:1 relation where M${}_{NSC}=0.019M_{HOST}$. The
thinner dotted line corresponds to M${}_{NSC}=0.1M_{HOST}$ and represents the
upper limit on the NSC mass fraction. Red stars represent UDGs. Blue dots
represent the galaxies that do not meet the UDG physical size criterion.
Figure 10: Mosaic of galaxies with extreme NSC mass fractions. The top row
presents the five galaxies in our sample with the largest NSC mass fraction.
The bottom row presents the five UDGs ($r_{e}\geq 1.5$ kpc) with the largest
NSC mass fraction. Images span slightly over an arcmin on a side with North at
the top and East to the left. These are drawn from the Legacy Surveys on-line
viewer. Object labels are included at the top of each panel.

###  3.4 Environment

The hosts of NSCs in the SMUDGes sample are almost exclusively on the red
sequence (Figure 8). Given the connection between UDG color and environment
(Prole et al., 2018; Kadowaki et al., 2021), this finding may point to an
environmental dependence of the occupation fraction, in the sense as others
have found (Lim et al., 2018; Sánchez-Janssen et al., 2019; Poulain et al.,
2021; Carlsten et al., 2022).

Figure 11: Environmental properties of UDGs with and without NSCs. We compare
the local measures of environment, $\sigma_{ENV}$ and $N_{ENV}$, measures of
the local velocity dispersion and richness provided by Zaritsky et al. (2023),
for UDGs with NSCs (red stars) and UDGs without NSCs (smoothed blue
distribution). The dotted lines mark the regions set by Zaritsky et al. (2023)
to differentiate low ($N_{ENV}<15$ and $\sigma_{ENV}<500$) and high
($N_{ENV}>15$ and $\sigma_{ENV}>500$) density environments.

In Figure 11 we compare the distribution of UDGs with NSCs to those without
NSCs in the environmental parameter space defined by the estimated velocity
dispersion, $\sigma_{ENV}$, and the richness, $N_{ENV}$, of the hosting
overdensity that are provided by Zaritsky et al. (2023). Those parameters are
known to be highly uncertain measures of environment, so Zaritsky et al.
(2023) suggest using them in combination to define poor and rich environments
(the lower left and upper right quadrants in Figure 11, respectively). We are
limited in this comparison by the small number of UDGs with NSCs (33), but the
visual impression from the Figure is that UDGs with NSCs tend to higher values
of $\sigma_{ENV}$ than the overall sample. Although the sense of this behavior
is as previously observed (e.g., Lim et al., 2018; Sánchez-Janssen et al.,
2019; Poulain et al., 2021), statistical tests (comparison of means, medians,
KS test) do not yield a statistically significant detection of a difference in
the two populations presented here.

Confirming this preliminary result is important because it would extend the
relation between NSC occupation fraction and environment to that of groups and
the field and to UDGs. This will require spectroscopic redshifts of many more
SMUDGes galaxies to provide distances and enable us to convert angular
measurements to physical ones.

##  4 Summary

We present the results of our photometric search for potential nuclear star
clusters (NSCs) hosted by the ultra-diffuse galaxy candidates in the SMUDGes
catalog (Zaritsky et al., 2023). Using $r$-band images from DR9 of the Legacy
Survey (Dey et al., 2019), we develop an algorithm to statistically determine
if additional photometric components beyond a single Sérsic model within
0.5$r_{e}$ are needed, and then if among those components there is one that is
best modeled as an unresolved object. We find that slightly over half of the
SMUDGes sample does show evidence for additional components. Among those, we
identify 1059 for which we find with 90% confidence the need for an unresolved
source.

The distribution in projected radius of these unresolved sources shows a peak
near zero separation and a second more extended component. We explore models
and quantify the nature of the two components, attributing only the central
component to NSCs. We use our models of the radial distribution to define a
maximum projected separation for our defined NSC sample that ensures an 85%
pure sample of candidate NSCs (0.10$r_{e}$).

We explore our incompleteness using simulations and establish that we are
significantly incomplete due to our relatively shallow imaging (NSC magnitude
limits ranging from $\sim 23$ to 25 mag depending on the specifics of the host
galaxy and surroundings). Nevertheless, we are able to confirm with confidence
325 SMUDGes galaxies with NSCs, 144 of which also have estimated distances
provided by Zaritsky et al. (2023). Among those 144, we confirm 33 as UDGs
with NSCs.

Despite our identification of NSCs as a population that is closely aligned
with the center of their host galaxy, we find that the observed scatter of
positional offsets between NSCs and their hosts is greater than expected from
measurement errors alone. Our estimate of the dispersion in offsets
(0.10$r_{e}$) is in good agreement with the median offset measured from MATLAS
dwarf galaxies (Poulain et al., 2021). Such offsets could be used to constrain
formation scenarios and models of the host’s gravitational potential (Miller &
Smith, 1992; Taga & Iye, 1998; Bellovary et al., 2018).

We find that our sample of NSCs is hosted almost exclusively by galaxies on
the red sequence. The number of NSCs found in bluer hosts is consistent with
the expected level of contamination. This result may reflect the color-
environment trend identified for such galaxies (Prole et al., 2018; Kadowaki
et al., 2021) and the greater NSC occupation fraction in denser environments
(Lim et al., 2018; Sánchez-Janssen et al., 2019; Poulain et al., 2021;
Carlsten et al., 2022). Unresolved sources away from the nucleus are found in
hosts whose color distribution matches that of the SMUDGes sources overall,
suggesting no physical connection. We discuss some possible selection biases
against our finding NSCs in blue hosts.

Despite our focus on low surface brightness galaxies, and UDGs in particular,
we find that the NSCs in our sample fall on the NSC-host galaxy stellar mass
relationship found previously (Neumayer et al., 2020) from higher surface
brightness objects. There is potentially a deviation for the largest objects
in our sample (the UDGs) but our strong selection effects and the currently
necessity of comparing across studies rather than within a single study limits
our conclusions in this regard. The typical NSC in our sample contains about
0.02 of the total stellar mass of the host galaxy, although the most extreme
objects reach a fraction of nearly 0.1. These results are in agreement with
previous studies of NSCs in somewhat different galaxy samples (Binggeli et
al., 2000; Poulain et al., 2021).

Finally, we search for possible environmental effects in the NSC population.
Despite this being principally a field sample, we find a suggestion that NSCs
are more likely in UDGs in higher density environments, in agreement with
previous results (Lim et al., 2018; Sánchez-Janssen et al., 2019; Poulain et
al., 2021; Carlsten et al., 2022). However, quantitative analysis of this
trend in our data does not yet yield statistically significant results.
Increasing the sample of UDGs with NSCs, by having spectroscopic redshifts of
a larger fraction of the SMUDGes NSC sample, will allow us to further assess
this possibility in the future.

The SMUDGes catalog and the NSC extension catalog provided here enable ongoing
work on the nature of NSCs and their hosts, extending the latter to low
surface brightness, physically large galaxies. This sample will benefit
greatly from complementary high angular resolution imaging to come from
surveys carried out with the Euclid and Nancy Grace Roman telescopes,
spectroscopy from highly multiplexed surveys such as DESI, and from continued
dedicated follow-up observations.

The authors acknowledge financial support from NSF AST-1713841 and
AST-2006785. An allocation of computer time from the UA Research Computing
High Performance Computing (HPC) at the University of Arizona and the prompt
assistance of the associated computer support group is gratefully
acknowledged. The authors also thank Min-Su Shin for their on-line description
on scaling fits files to png images using their img_scale.py module.

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