# ELVES II: GCs and Nuclear Star Clusters of Dwarf Galaxies; The Importance of
Environment

Scott G. Carlsten Department of Astrophysical Sciences, 4 Ivy Lane, Princeton
University, Princeton, NJ 08544  Jenny E. Greene Department of Astrophysical
Sciences, 4 Ivy Lane, Princeton University, Princeton, NJ 08544  Rachael L.
Beaton Hubble Fellow  Department of Astrophysical Sciences, 4 Ivy Lane,
Princeton University, Princeton, NJ 08544  The Observatories of the Carnegie
Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101
Johnny P. Greco NSF Astronomy & Astrophysics Postdoctoral Fellow  Center for
Cosmology and AstroParticle Physics (CCAPP), The Ohio State University,
Columbus, OH 43210, USA

###### Abstract

We present the properties of the globular clusters (GCs) and nuclear star
clusters (NSCs) of low-mass ($10^{5.5}<M_{\star}<10^{8.5}$ M⊙) early-type
satellites of Milky Way-like and small group hosts in the Local Volume (LV)
using deep, ground-based data from the ongoing Exploration of Local VolumE
Satellites (ELVES) Survey. This sample of 177 dwarfs significantly increases
the statistics for studying the star clusters of dwarfs in low-density
environments, offering an important comparison to samples from nearby galaxy
clusters. The LV dwarfs exhibit significantly lower nucleation fractions at
fixed galaxy mass than dwarfs in nearby clusters. The mass of NSCs of LV
dwarfs show a similar scaling of $M_{\star,\mathrm{NSC}}\propto
M_{\star,\mathrm{gal}}^{0.4}$ as that found in clusters but offset to lower
NSC masses. To deal with foreground/background contamination in the GC
analysis, we employ both a statistical subtraction and Bayesian approach to
infer the average GC system properties from all dwarfs simultaneously. We find
that the GC occupation fraction and average abundance are both increasing
functions of galaxy stellar mass, and the LV dwarfs show significantly lower
average GC abundance at fixed galaxy mass than a comparable sample of Virgo
dwarfs analyzed in the same way, demonstrating that GC prevalence also shows
an important secondary dependence on the dwarf’s environment. This result
strengthens the connection between GCs and NSCs in low-mass galaxies. We
discuss these observations in the context of modern theories of GC and NSC
formation, finding that the environmental dependencies can be well-explained
by these models.

methods: observational – techniques: photometric – galaxies: distances and
redshifts – galaxies: dwarf

††journal: ApJ††software:  astropy (Astropy Collaboration et al., 2018) sep
(Barbary, 2016) imfit (Erwin, 2015) emcee (Foreman-Mackey et al., 2013)
SExtractor (Bertin & Arnouts, 1996)

##  1 Introduction

Globular clusters (GCs) are generally old (age $\sim 10$ Gyr), massive
($M_{\star}\sim 10^{5}$ M⊙) stellar clusters that are ubiquitously found in
nearby galaxies, with essentially all galaxies with $M_{\star}>10^{9}$ M⊙
hosting a system of GCs. GCs are generally thought to form in a similar manner
as young massive clusters (YMCs) observed in nearby merging and/or star-
bursting galaxies (e.g. Holtzman et al., 1992; Whitmore et al., 1999) but
instead at $z\sim 2-3$ (Ashman & Zepf, 1992; Elmegreen & Efremov, 1997;
Kravtsov & Gnedin, 2005; Elmegreen, 2010; Kruijssen, 2015; Pfeffer et al.,
2018). The high densities and pressures in the ISM of galaxies at $z\sim 2-3$
are conducive to the formation of significant populations of massive clusters
(Shapiro et al., 2010; Kruijssen, 2012; Reina-Campos & Kruijssen, 2017).
Understanding the physics of massive cluster formation at high redshift and
the physical processes that connect the high redshift clusters to GCs we
observe today are important open areas of research. Due to their age and
prevalence, GCs have long been noted as important ‘fingerprints’ to decipher
the process of galaxy assembly (for reviews of GCs in the context of galaxy
formation, see Brodie & Strader, 2006; Kruijssen, 2014; Forbes et al., 2018b;
Beasley, 2020).

Dwarf galaxies ($M_{\star}\lesssim 10^{9}$ M⊙) offer an interesting window
into the physics that sculpt the $z=0$ properties of GCs. Their relatively
simple formation pathways (compared to the Milky Way, for instance) allows one
to better isolate different physical mechanisms relevant to GCs. Knowledge of
the GC systems of dwarf galaxies can then, in turn, inform our understanding
of the build-up of the GC systems of more massive galaxies, like the Milky Way
(MW), which consist of a large fraction of accreted, ex situ GCs (e.g. Leaman
et al., 2013; Kruijssen et al., 2019). The GC systems of large samples of
nearby dwarf galaxies have been studied in numerous past works, primarily
using _HST_ observations (e.g. Miller et al., 1998; Lotz et al., 2004; Sharina
et al., 2005; Miller & Lotz, 2007; Georgiev et al., 2009b, 2010). These works
have studied how the properties of the GC systems depend on various properties
of the host galaxy, including mass/luminosity, color, and morphological type.

Another, less explored, property of the host galaxy that appears to have an
effect on the GC system is the larger scale environment of the host. There is
moderate evidence that galaxies in denser environments have richer GC systems
at fixed stellar mass. Peng et al. (2008) found that dwarf galaxies in Virgo
had higher GC specific frequencies when closer to the center of Virgo. While
Liu et al. (2019) did not find any difference in GC specific frequency between
Fornax and Virgo dwarfs (even though the two clusters are almost an order of
magnitude different in mass), they too found that dwarfs in denser regions of
the clusters have somewhat higher specific frequencies. However, Miller & Lotz
(2007) did not find a trend in specific frequency with clustercentric distance
amongst lower mass dwarf galaxies in Virgo. These studies have been limited to
just the environments of Fornax and Virgo, and a study encompassing a much
wider dynamic range in environment is needed. Investigating this is one of the
key goals of the current work.

There have been some further early indications that the environment of a dwarf
galaxy does indeed have an effect on its GC properties from recent work on
‘ultra-diffuse galaxies’ (UDGs). UDGs are large $r_{e}\gtrsim 1.5$ kpc dwarf
galaxies ($10^{7}\lesssim M_{\star}\lesssim 10^{9}$ M⊙) that have seen
significant interest in recent years, at least partly because of unexpectedly
rich GC systems (van Dokkum et al., 2016; Peng & Lim, 2016; van Dokkum et al.,
2017)111However, see also Saifollahi et al. (2020) for different results
regarding a Coma UDG.. These early results were for UDGs in the Coma cluster,
and results for UDGs in other, less dense environments suggest that these
ultra-rich GC systems might be unique to the Coma environment and are
generally not found in lower mass clusters (like Virgo or Fornax) or groups
(Prole et al., 2019; Lim et al., 2020; Somalwar et al., 2020), although with a
few exceptions (e.g. Müller et al., 2021). As UDGs are outliers in the mass-
size relation for dwarf galaxies, it is unclear how this environmental
dependence translates for the whole dwarf galaxy population.

In addition to GCs, another dense, bound stellar component that many galaxies,
including dwarf galaxies, possess are nuclear star clusters (NSCs). NSCs are
central stellar clusters with sizes comparable to GCs ($\sim$ few pc) but with
a broad range in stellar mass ranging from roughly that of a GC to several
orders of magnitude more ($M_{\star}\sim 10^{5}-10^{8}$M⊙; Walcher et al.,
2005; Côté et al., 2006; Turner et al., 2012; Georgiev & Böker, 2014). For a
recent review on the properties of NSCs see Neumayer et al. (2020). NSCs are
found in a wide variety of galaxies, spanning different masses, morphological
types, and environment (Binggeli et al., 1987; Carollo et al., 1997, 1998;
Lisker et al., 2007; Georgiev et al., 2009a; Sánchez-Janssen et al., 2019).
For early-type galaxies, studies of Virgo and Fornax cluster galaxies and
nearby groups show NSC occurrence is an increasing function of galaxy stellar
mass up to $\sim 10^{9}$ M⊙ where it reaches $\sim 90$%, beyond which it
starts to decline (Turner et al., 2012; den Brok et al., 2014; Ordenes-Briceño
et al., 2018a; Sánchez-Janssen et al., 2019; Habas et al., 2020).

Models for the formation of NSCs generally fall into two main classes. First,
many works have shown they could be the result of dynamical friction-induced
inspiral and subsequent merging of GCs (e.g. Tremaine et al., 1975; Capuzzo-
Dolcetta, 1993; Oh & Lin, 2000; Lotz et al., 2001; Antonini, 2013; Gnedin et
al., 2014). The second main proposed formation mechanism is that of in situ
formation of the NSC. In this model, dissipational gas transfer into the
center of galaxies drives star formation and a buildup of a dense stellar
nucleus. The gas can be sent to the center either via wet-mergers or other
secular processes (e.g. Mihos & Hernquist, 1994; Milosavljević, 2004). It is
almost certain that both mechanisms are at work (e.g. Hartmann et al., 2011;
Leigh et al., 2012). However, there is growing consensus that both models are
not equally important at all host masses, but the relative importance of each
pathway changes as a function of host mass. In particular, GC-inspiral is
likely the dominant mechanism for dwarf ($M_{\star}<10^{9}$ M⊙) galaxies while
the in situ star formation is dominant at higher masses (Turner et al., 2012;
Sánchez-Janssen et al., 2019; Neumayer et al., 2020).

The works referenced above have elucidated how various properties of an NSC
depend on the properties of its host, including stellar mass and type.
However, as with GCs, there is some evidence that the properties of NSCs
(particularly their frequency) depend also on the larger scale environment in
which the galaxy resides. Starting with Ferguson & Sandage (1989), there has
been growing evidence that the nucleation fraction of dwarf early-type
galaxies depends on the local density. The spatial distribution of nucleated
dE’s in Fornax and Virgo are more centrally concentrated than non-nucleated
dE’s (Lisker et al., 2007; Ordenes-Briceño et al., 2018b). Additionally, dEs
in higher mass clusters, like Coma, are nucleated at a higher frequency than
in Virgo or Fornax (den Brok et al., 2014; Sánchez-Janssen et al., 2019;
Zanatta et al., 2021). Sánchez-Janssen et al. (2019) tentatively showed that
the nucleation fraction is even lower in the Local Group (LG); however, the
statistics afforded by just the LG are not enough to make a strong conclusion.
The cause of this environmental dependence is not clear.

With a few notable exceptions, the above-cited observational works that
present the properties of GCs and NSCs in dwarf galaxies do so for dwarfs in
nearby clusters. The primary goal of the current work is to systematically
present the GC and NSC properties of dwarfs in a lower density environment,
particularly in halos of nearby roughly Milky Way (MW)-sized galaxies in the
Local Volume (LV; $D\lesssim 12$ Mpc). As discussed more below, we focus only
on early-type dwarfs, directly comparing to early-type dwarfs in nearby
clusters. We find that adding the axis of environment opens a valuable new
perspective on the formation of GCs and NSCs.

Due to the low density of these environments, achieving the statistics
necessary to meaningfully compare with samples of cluster dwarfs has been a
major hurdle. It is not until recently that enough of the LV has been surveyed
to build up a significant sample of dwarf satellites of MW-like galaxies with
confirmed distances and well-quantified completeness. In particular, we use
the dwarf catalogs from the ongoing Exploration of Local VolumE Satellites
(ELVES) Survey. In Section 2, we describe this sample, in Section 3 we
describe our analysis of the NSCs and GCs of these dwarfs, in Sections 4 and 5
we present our results regarding NSCs and GCs, respectively, in Section 6 we
discuss these results in the context of modern theories of how GCs and NSCs
form, and we conclude in Section 7.

##  2 Data

###  2.1 ELVES Overview

For our primary observational sample, we use results from the ongoing
Exploration of Local VolumE Satellites (ELVES) survey. The overall goal of the
survey is to survey the classical satellites down to luminosities of
$M_{V}\sim-9$ and within $300$ projected kpc of all massive, $M_{K}<-22.4$
mag, hosts within 12 Mpc with full or nearly full distance information for the
satellites. There are 29 hosts that meet this qualification, and further
details on host selection and the list of hosts will be presented in a future
paper. Satellite candidates are detected using deep, wide-field imaging
combined with the detection algorithm specialized for finding low-surface
brightness, diffuse dwarf galaxies of Carlsten et al. (2020a) and Greco et al.
(2018). Through extensive tests with injected artificial galaxies, we ensure
that ELVES will be complete down to $M_{V}\sim-9$ and surface brightness of
$\mu_{0,V}\lesssim 26.5$ mag arcsec-2 (see Carlsten et al., 2020b, for more
detailed information on completeness).

The difficulty in this endeavor is in determining the distance to candidate
satellites to confirm their association with a host. Since these groups are
quite sparse (compared to Virgo or Fornax, for instance), contamination is
often quite high, and candidates selected on their low-surface brightness,
diffuse morphology alone can often consist of a majority ($>80\%$, e.g.
Carlsten et al., 2019a, 2020a) of background contaminants. In studying the
star clusters of dwarfs, it is crucial we only consider satellites that have
distance confirmation. Background, interloping galaxies likely would not have
detectable nuclei or GCs and could artificially bring down the average
inferred abundance of these clusters.

The candidate satellites are confirmed with a variety of distance measurements
including archival redshifts and TRGB distances, but the majority of distances
are measured via surface brightness fluctuations (SBF). SBF (e.g. Tonry &
Schneider, 1988; Carlsten et al., 2019b, a; Greco et al., 2021) can produce
relatively precise distance errors ($\lesssim 15\%$) using modest ground based
data. This is enough to confirm the association of candidate satellites or not
in most cases.

The survey uses a mixture of archival CFHT/MegaCam imaging (Carlsten et al.,
2020a, b, c) and the DESI Legacy Imaging Surveys (Dey et al.,
2019)222https://www.legacysurvey.org/ which includes both the Beijing-Arizona
Sky Survey (BASS Zou et al., 2017, 2018) and the DECam Legacy Survey
(henceforth these surveys are collectively referred to as DECaLS). We find
that we can readily detect dwarfs down to $M_{V}\sim-9$ even using the
relatively shallow DECaLS imaging. While the CFHT/MegaCam data are high enough
quality to apply SBF to even $M_{V}\sim-9$ mag dwarfs, the depth and PSF size
of DECaLS are not adequate for SBF of dwarfs throughout the mass range of the
survey. Thus, we are using deeper Gemini (program IDs: FT-2020A-060 and
US-2020B-037) or Subaru/HSC data, where available, to measure the dwarf
distances via SBF.

As discussed in Carlsten et al. (2020b), for some of the faintest and/or
lowest surface brightness dwarfs, the SBF distances were inconclusive, leaving
some candidates as ‘unconfirmed’ or ‘possible’ satellites. In this work, we
only consider the confirmed satellites, e.g., those measured to be at the
distance of their host. We are actively pursuing even deeper data for SBF or
_HST_ data for TRGB measurements for the remaining candidates and expect that
some will turn out to be real satellites. Thus, the satellite sample used in
the current work is not actually complete to $M_{V}\sim-9$ mag. However,
because the unconfirmed candidate satellites are generally the lowest
luminosity ($M_{V}\gtrsim-10$ mag) candidates, their exclusion will not have a
significant effect on the current work as such faint satellites are unlikely
to host either NSCs or GCs.

A handful of hosts have been previously surveyed in the literature to at least
the level of ELVES (or deeper), with distance confirmation for the satellites.
We take the properties of the MW and M31 satellites from McConnachie (2012).
For the other four (M81, CenA, M94, M101) previously surveyed LV hosts, we
only use the distance confirmation from the previous surveys. We derive all
photometry ourselves (either from DECaLS or other archival data). The relevant
references are: Müller et al. (2019) and Crnojević et al. (2019) (CenA),
Chiboucas et al. (2013) (M81), Smercina et al. (2018) (M94), and Bennet et al.
(2019) (M101). Note that the satellite samples of M81 and CenA are subsamples
(including roughly 80%) of their entire, known satellite systems for which we
could find high-quality multi-band imaging.

The imaging data that we use in the current work to analyze NSCs and GCs is
roughly about one-third CFHT/Megacam, one-half DECaLS, and the remainder
Subaru/HSC. We prioritize the deeper CFHT and Subaru data, where available.
For each dwarf we always have at least two imaging bands, either $g$ and $r$
or $g$ and $i$. The dwarfs are listed, along with their photometry (see
Section 3.1 below), in a table in Appendix A. In this table, we also list the
data source for each dwarf (e.g. CFHT/Megacam, DECaLS, or other) and filter
combination used. There is the concern that we are dealing with several
different filter systems (DECam, HSC, BASS, MegaCam). All are Sloan-like
filters but will have some differences. We make no attempt to bring the
different measurements onto the same filter system. Instead, in a companion
paper focused on the structural measurements of these galaxies (Carlsten et
al, submitted), we show that the filter systems do not differ by enough to
alter any conclusions. The different filters differ generally by $\lesssim
0.1$ mag.

We use slightly different samples of dwarfs for the NSC analysis and GC
analysis. For the NSC analysis, we include satellites of 27 LV hosts
(including the MW and M31), meaning that nearly all massive LV hosts have been
surveyed to some extent. The list of hosts can be found in Appendix A. This is
a sample of 177 early-type (see §2.4) satellites. However, due to the
proximity of the MW and M31 satellites, we cannot analyze their GCs in the
same way as the other LV hosts, and they are not included in the GC analysis.
However, we do use the known GC properties of MW/M31 satellites as a check on
our results. Additionally, due to their low galactic latitude (and high
density of foreground stars), we do not analyze the satellites of CenA and
NGC5236 for GCs. Thus, we only include the satellites of 23 LV hosts in the GC
analysis, leading to a sample size of 140 early-type satellites.

We note that this is a larger host sample than used in the companion paper
(Carlsten et al., submitted) which focuses on the structural scaling relations
for ELVES dwarfs. In that work, we restricted the sample to hosts which have
distance constraints on a majority of their candidate satellites due to the
importance for that work to use a satellite sample that is as close to
complete to $M_{V}\sim-9$ and $\mu_{0,V}\lesssim 26.5$ mag arcsec-2 as
possible. In this work, we instead maximize the sample statistics by including
the confirmed satellites for a handful of other hosts who lack distance
confirmation for the majority of their candidate satellites. Along this vein,
we also include a handful of confirmed satellites that are a little outside of
the fiducial 300 projected kpc radial coverage of the ELVES Survey. The
conclusions we find are unchanged if we do not include these extra satellites,
although the statistics are lowered.

###  2.2 Virgo Comparison Sample

As mentioned in the introduction, many contemporary samples of dwarfs in
denser, cluster environments exist. Of particular relevance, given their
similar completeness levels to ELVES, are the Next Generation Virgo Survey
(NGVS) (Ferrarese et al., 2012; Sánchez-Janssen et al., 2019), the Next
Generation Fornax Survey (NGFS) (Muñoz et al., 2015; Eigenthaler et al., 2018;
Ordenes-Briceño et al., 2018a), and the Fornax Deep Survey (FDS) (Venhola et
al., 2018; Prole et al., 2019). All three surveys have published analyses of
the NSC properties of their dwarf samples. However, at the time of writing,
only the FDS has published an analysis of the GCs for their entire dwarf
sample (Prole et al., 2019). This is clearly a valuable comparison for the LV
sample, but given the vagaries involved in determining stellar masses and
characterizing the GC systems of dwarfs (cf. Section 3.3 below), it is
important to have a cluster reference sample analyzed in as close to the same
way as the LV sample as possible. Due to the archival availability of the
data, we use the NGVS dwarf sample as the primary comparison sample. We use
the NGVS galaxy catalog from Ferrarese et al. (2020) that considers only the
core 4 deg2 region of Virgo (roughly out to $r\lesssim R_{\mathrm{vir}}/5$).
We acquire the raw CFHT/Megacam data from the Canadian Astronomy Data
Center333http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/, and reduce the data in
the same way as the other CFHT/Megacam data used here (see Carlsten et al.,
2019b, 2020a, for details on the sky subtraction, etc.). Since our focus is
primarily on low-mass dwarf galaxies, we only reduce data for the NGVS dwarfs
with $M_{\star}<10^{9}$M⊙, as given by the list of stellar masses from
Sánchez-Janssen et al. (2019). This sample of NGVS dwarfs is analyzed in the
same way as the main ELVES sample. In Appendix B, we show that the Virgo and
LV dwarf samples have quite similar stellar mass distributions.

While the NGVS Virgo dwarf sample is our primary point of comparison,
particularly for the GC analysis, we do make additional comparisons to other
galaxy cluster samples, including from Fornax and Coma. These samples are
described more in the Results sections where the comparisons are made.

###  2.3 Definition of ‘Environment’

Throughout this work we investigate the large-scale environmental dependence
of the star cluster properties of dwarf galaxies, so it is important to
clarify what is meant by ‘environment’. In our analysis, we use ‘environment’
to denote the mass of the parent halo in which the dwarf satellites reside as
subhalos. As we describe in more detail in Carlsten et al. (submitted), the LV
hosts are significantly lower in halo mass than either cluster. Various
estimates put the LV hosts in the halo mass range of $10^{12}\lesssim
M_{\mathrm{halo}}/M_{\odot}\lesssim 10^{13}$. On the other hand, the Fornax
cluster has a dynamical mass of $M_{200}\sim 7\times 10^{13}$ M⊙ (Kourkchi &
Tully, 2017), the Virgo cluster has an estimated mass closer to $5-6\times
10^{14}$ M⊙ (Ferrarese et al., 2012; Kourkchi & Tully, 2017; Kashibadze et
al., 2020), and Coma is even more massive at $\sim 10^{15}$M⊙ (Łokas & Mamon,
2003).

The environmental difference may be exacerbated by the fact that the cluster
samples are from the very central $r_{\mathrm{proj}}\lesssim
R_{\mathrm{vir}}/4$ (see §2.2 above) regions while the LV host satellite
samples are generally complete to at least $r_{\mathrm{proj}}\gtrsim
R_{\mathrm{vir}}/2$.

###  2.4 Galaxy Type

In this work, we only consider the GC and NSC properties of early-type dwarf
galaxies in the ELVES sample. This is partly an observational necessity as
late-type dwarfs contain a plethora of young star clusters, star-forming
clumps, and HII regions that would greatly confuse any attempt to identify
NSCs or count GCs. Late-type dwarfs require the resolution of _HST_ to be able
to robustly identify these star clusters (see e.g. Seth et al., 2004; Georgiev
et al., 2009a, b, 2010). Additionally, considering only the early-type dwarfs
makes the comparison with the cluster samples (which are almost exclusively
early-type) more direct.

To split the dwarfs into early-type and late-type groups, we use a visual
inspection of the dwarf morphology. Given the generally quite deep imaging
data available (deep enough to apply SBF) and proximity of these dwarfs, we
believe this is the most robust way to split dwarfs with the data
available444These two classes also separate clearly in color-magnitude space.
A simple cut in this space splits the dwarfs into two classes with the same
result as the visual inspection about 90% of the time.. Dwarfs with smooth,
featureless morphology are classed as early-types while dwarfs with clear
star-forming regions, blue clumps, dust-lanes, or any other kinks in their
surface brightness profile are classed as late-type.

Measurements of H$\alpha$ and/or HI are available for about a third of the
ELVES sample. We find that these measurements closely corroborate the visual
morphology classification. The vast majority of early-type dwarfs with spectra
do not have significant H$\alpha$ emission (equivalent widths $<2$Å) while
$\sim 85$% of the late-type dwarfs with spectra do have H$\alpha$ above this
level (with more likely having H$\alpha$ emission that is simply below the
sensitivity of the archival spectra, which is generally from SDSS). Based on
this, we assert that the visual classification into late/early-type is
essentially a physical distinction into star-forming and quenched dwarfs,
respectively.

Throughout the paper, we focus only on dwarfs in the stellar mass range of
$5.5<\log(M_{\star}/{\rm M}_{\odot})<8.5$, and there are 177 such early-type
dwarfs across all the LV hosts. For comparison, the NGVS sample includes 295
dwarfs.

##  3 Methods

In this section, we outline the steps we take in analyzing the NSCs and GCs of
the LV sample. Briefly put, we first model and then subtract out the
underlying light profile of the dwarf. In the image residuals, we look for a
possible NSC. If one is found, we perform a second fit to the galaxy and NSC
simultaneously using a multi-component model. Then, in the residuals of either
the single- or multi-component fit, we look for candidate GCs. Due to the
distance of the dwarf galaxies, GCs will not be spatially resolved. We take
two different approaches to distinguish between actual GC candidates and
contaminating sources (background galaxies or foreground stars). The first is
a simple, statistical background subtraction, and the second is a more
sophisticated likelihood-based approach. This second approach allows us to
explore the radial distribution of GCs and fit for the fraction of dwarfs that
possess GCs as a function of stellar mass. In the following sections, we give
more details for each of these major steps.

Figure 1: Example of the NSC and GC analysis process. The top left shows a
color image of a nucleated dwarf at $D=11.9$ Mpc; the image is $3^{\prime}$ to
a side. The top right shows the residuals after the galaxy light and NSC have
been modelled and subtracted. The red ellipse shows the ellipticity and
effective radius of the dwarf. The dashed blue circle is the annulus within
which we count GC candidates for the simple background subtraction method. The
small green circles denote all the sources that pass the point source
selection criterion (bottom left panel). The small red circles denote the
sources that also pass the color cuts (see Section 3.3.3). Bottom left shows
the difference in aperture magnitudes (6 and 12 pixels) that we use to select
point sources. The red line is the locus of point sources for this cutout.
Bottom right shows the estimated completeness of these data as a function of
$g$-band magnitude. Indicated on the plot is the expected GCLF peak at the
distance of the dwarf. The data used generally extend multiple magnitudes
below the GCLF peak.

###  3.1 Galaxy and Nucleus Photometry

We start the analysis with image cutouts of each dwarf. The cutouts are
generally $\sim 6^{\prime}$ to a side. This corresponds to $\gtrsim 20r_{e}$
for the LV dwarf sample. To model the underlying light profiles of the dwarfs,
we rely on parametric 2D Sérsic profile fits. Since we are focusing only on
the early-type dwarfs, Sérsics are generally quite good models for the surface
brightness profiles.

For each dwarf, we have two bands, either $g$ and $r$ or $g$ and $i$. We fit
the Sérsic profiles in the manner of Carlsten et al. (2019b, 2020a) using
imfit (Erwin, 2015). The $g$-band image is masked for nearby stars and
background galaxies and fit with a Sérsic profile. This initial masking also
includes point sources that are likely NSCs. The $r$ or $i$-band image is then
fit, allowing only the amplitude to change while fixing other Sérsic
parameters.

We subtract this initial single component model from the image and inspect the
residuals for possible NSCs. Following Eigenthaler et al. (2018) and Sánchez-
Janssen et al. (2019), NSCs are determined through a combination of photometry
and visual inspection. We show below that the visual classification of NSCs is
essentially a cut in whether there is a point source within a projected radius
of $r\lesssim r_{e}/8$ of the galaxy’s photocenter. While it is possible some
of the NSC candidates are interloping foreground MW stars or compact
background galaxies, we consider this possibility unlikely. Sánchez-Janssen et
al. (2019) estimate the contamination rate to be $\sim 0.1$% for the NGVS
sample, and we expect a similarly low contamination for the ELVES dwarfs as
well. The median density of point sources of the magnitude and color expected
for NSCs/GCs (see 3.3 below) in the vicinity of the LV dwarf galaxies is $\sim
2$ arcmin-2. Compared to the median effective radius of the ELVES sample of
$0.24$ arcmin, this means the chance of an interloping point source being in
the central $r_{e}/8$ of a dwarf is $\sim 0.5$%, similar to the Sánchez-
Janssen et al. (2019) estimate. Similarly, we do not expect many NSCs to
actually be (non-nuclear) GCs of the dwarf galaxy that are located in the
center just by chance projection. A quick estimate indicates that a GC
randomly located within $2r_{e}$ of a dwarf has a roughly 5% chance of
projecting into the inner $r_{e}/8$ along a random line of sight. Given that
dwarfs in the mass range considered here have few, if any, GCs, this is not a
large concern.

Once we identify an NSC candidate in the residuals to the single-Sérsic fit,
we refit the galaxy (and nucleus) using a double-Sérsic profile fit. At the
distances of the LV dwarfs, we do not expect the NSCs to be resolved in the
ground-based data that we use. However, we find that the double-Sérsic fit
leads to more stable fits than a Sérsic $+$ point source combination since it
is more forgiving of errors in our PSF model (which simply comes from the
cutouts of nearby, unsaturated stars). The second Sérsic profile is limited in
effective radius to $r_{e}<1$′′.

This process is exemplified in Figure 1. This shows an example of a nucleated
dwarf (top left) and the residuals after subtraction of the double-Sérsic
model (top right).

To deal with the fact that sometimes we have $g/r$ and sometimes we have $g/i$
bands, we make use of the conversion

$\displaystyle g-i=1.53(g-r)-0.032$ (1)

which we derive from MIST (Choi et al., 2016) SSP models for a range in age
between 3 and 10 Gyr and metallicities in the range $-2<\mathrm{[Fe/H]}<0$.
Note that this is calculated specifically for the CFHT filter system.

We derive stellar masses of the dwarf satellites and NSCs from integrated
luminosity and color using a color-$M_{\star}/L$ relation from Into &
Portinari (2013), as described more in Carlsten et al. (submitted). We refer
the reader to that paper for more information on how we derive uncertainties
in the photometry. The NSC photometric measurements are given in Appendix C.

We estimate the point source completeness of the various imaging datasets we
use when analyzing the GC systems, as discussed below. Suffice to say here
that in almost all cases the 50% completeness magnitude is $M_{g}>-6$ mag at
the distance of the hosts surveyed. Considering the magnitudes expected for
NSCs ($M_{g}\gtrsim-7.5$ mag; Sánchez-Janssen et al., 2019), we do not expect
that we are missing many NSCs due to shallow imaging. We discuss the matter of
completeness further in §4.

Figure 2: Cumulative distribution of galaxies with a detected source within a
certain fraction of the galaxy’s effective radius. Galaxies classified as
nucleated are shown separately from the overall sample of galaxies. The
vertical lines show where a physical radius of 100 pc is, considering the
average effective radii of galaxies in different mass bins.

###  3.2 NSC Radial Distribution

As stated above in §3.1, to be comparable to contemporary analyses of cluster
dwarfs, the classification of nucleated vs non-nucleated galaxies is largely
based on visual inspection. Below, we show that our nucleation categorization
is essentially identical to that of Sánchez-Janssen et al. (2019) for the NGVS
Virgo sample.

Figure 2 shows the radial distribution (relative to the photometric center of
the host galaxy) of nuclei in the LV sample. The cumulative number of galaxies
with a source within a certain radius (scaled by the effective radius) is
shown for both nucleated galaxies and the whole dwarf sample. The source
catalogs used here are the ones input into the GC analysis, as described below
in §3.3.3. Any dwarf with a source within $<r_{e}/8$ is classified as
nucleated. Thus, the visual classification of nuclei essentially is a cut on
whether there is a point source within the inner $<r_{e}/8$ region of a dwarf.
Since there are no non-nucleated galaxies with point sources within
$<r_{e}/8$, we can quantify our completeness to nuclei using the completeness
of the GC catalogs determined below in §3.3.2.

###  3.3 Catalogs of Candidate GCs

With the galaxy profile modelled and subtracted (either a single or double-
Sérsic model), we move to the analysis of the GC systems of these dwarfs. As
mentioned above, we use two different methods of estimating the GC system of
each galaxy. We know the distance to each dwarf through various methods, but
we have no knowledge of the distance to the various point sources surrounding
the galaxies so we have to account for a significant fraction of background
(and foreground) contamination. The first method we use to estimate the GC
system is a simple, statistical background subtraction. This procedure has
been used in several similar previous works (e.g. Miller & Lotz, 2007; Lim et
al., 2018). The second method uses a more sophisticated Bayesian model fit to
all dwarfs simultaneously. Both methods rely on the same point-source catalogs
of eligible GC candidates in the vicinity of each dwarf. We describe the key
steps in generating these catalogs in this section.

####  3.3.1 Point Source-Galaxy Separation

The first step in counting the GC candidates is to procure a catalog of point
sources in the surroundings of each dwarf. For point source-background galaxy
separation, we measure the magnitude of sources in the image in two different
diameter apertures, 6 and 12 pixels. Point sources, like stars and GC
candidates, will appear as a distinct horizontal locus when plotting the
difference in the aperture magnitudes versus the total magnitude of sources,
which we take from SExtractor’s MAG_AUTO measurement (Bertin & Arnouts, 1996).
This is shown in the bottom left panel of Figure 1. Point sources are those
sources which are close to this horizontal locus. To determine the appropriate
threshold in the aperture magnitude difference to use, we inject artificial
stars into the images using the PSF model. The average recovered difference in
the aperture magnitudes gives the horizontal location of the point source
locus (red line in Figure 1). The vertical width of the accepted region, shown
by the green points, is determined from the $3\sigma$ spread (as a function of
magnitude) in the injected stars. There is a floor in the width of $0.15$ mag.
For the Virgo dwarf sample, we found that we needed a less restrictive point
source-galaxy separation and so the floor in the width is $0.25$ mag. We do
the artificial star injection in both bands but only do the star-galaxy
separation in the redder band (either $r$ or $i$) since the PSF is, on
average, smaller at longer wavelengths (e.g. Carlsten et al., 2018).

####  3.3.2 Point Source Completeness

Using the artificial star injection tests, we also derive the completeness of
the source catalogs as a function of magnitude. We derive completeness curves
for both bands, which are combined into an overall completeness by taking the
lower recovery fraction of the two at each magnitude, assuming a fiducial
color of $g-r=0.4$ or $g-i=0.75$. This overall completeness curve is shown in
the bottom right of Figure 1. The example dwarf shown in the Figure uses
relatively deep CFHT/Megacam data but is the furthest dwarf in the LV sample
($D=11.9$ Mpc). For closer dwarfs, even with the relatively shallow DECaLS
imaging, the $90$% completeness limits are always $\gtrsim 1$ mag fainter than
the expected GC luminosity function (GCLF) peak at the distance of the dwarfs
($M_{g}\sim-7.2$ mag; Jordán et al., 2007; Villegas et al., 2010).

####  3.3.3 Color and Magnitude Cuts

We implement broad color cuts on the catalog of point sources that roughly
encapsulate the expected range of GCs. We measure colors using the aperture
photometry but with aperture correction derived from the artifical star
injection process. For dwarfs with $g/r$ imaging, we restrict to sources with
$g-r$ in the range $[0.1,0.9]$ and for dwarfs with $g/i$ imaging, we restrict
to sources with $g-i$ in the range $[0.2,1.1]$. These ranges largely encompass
the color spread of GCs in Fornax dwarfs found in Prole et al. (2019) and the
blue peak of GCs around massive Fornax galaxies found in D’Abrusco et al.
(2016). Note that since we are focusing just on dwarfs, we expect the GC
systems to be fairly metal poor and, hence, blue. We have tried larger ranges
in color but find that the results remain largely the same, albeit with lower
signal to noise due to higher contamination levels.

We also cut in luminosity, restricting sources to the range $-9.5<M_{g}<-5.5$
mag which will encompass the entire expected GCLF.

These cuts are exemplified in Figure 1. The green circles in the top right
panel show all the point sources while the small red circles denote the
sources that satisfy the color and magnitude cuts.

We can then use these cleaned point-source catalogs to directly determine the
overall stacked GCLF or GC radial profile (see below) or to determine an
estimate for the number of GCs for each individual dwarf, as we describe in
the next two sections.

###  3.4 GC Abundance: Simple Background Subtraction

With the catalog of GC candidates in hand, our first method of inferring GC
abundance is to perform a simple background subtraction to estimate the number
of GCs around each dwarf. We count the number of sources within a circular
annulus of radius $2r_{e}$. The count of sources within the annulus are then
corrected for an estimated background contribution determined from separate
$2r_{e}$ annuli placed around the galaxy at distances $>5r_{e}$ from the
galaxy center. Note that the image cutouts we use are $>20r_{e}$ on a side so
we generally get $>20$ background annuli per galaxy. The distribution of
number counts in the background annuli are used to estimate the uncertainty of
the background subtracted GC counts. We have checked that using $1r_{e}$ or
$3r_{e}$ annuli instead do not alter the main conclusions of this work. With a
larger annulus, the GC abundance measurements are generally of lower signal to
noise.

The main systematic uncertainty involved in this calculation of
$N_{\mathrm{GC}}$ is the unknown radial distribution of GCs for low-mass dwarf
galaxies. While it would be possible to count up the number of sources in a
much larger aperture ($5r_{e}$, for instance) that would capture essentially
all of the GCs, larger apertures mean more background sources. Even if we
subtract out correspondingly larger estimate of the background contribution,
the increased Poisson scatter will cause the measurement to have lower signal
to noise. A relatively common approach in the literature is to assume the
half-number radius of the GC distribution is $\sim 1.5\times$ the galaxy
effective radius (e.g. van Dokkum et al., 2017; Lim et al., 2018). Various
studies of dwarfs in clusters have found that the radial distribution of GCs
in dwarf galaxies is generally less concentrated than the galaxy light profile
(Amorisco et al., 2018; Prole et al., 2019). This result has also been found
for individual, well-studied dwarfs, including the Fornax dSph (Mackey &
Gilmore, 2003), NGC 147/185 (Veljanoski et al., 2013), NGC 6822 (Huxor et al.,
2013; Veljanoski et al., 2015), VCC 1087 in Virgo (Beasley et al., 2006), VCC
1287 in Virgo (Beasley et al., 2016), and DF17 in Coma (Peng & Lim, 2016).

For the fiducial background subtracted results, we assume that the GC radial
distribution is approximated by a Plummer profile with the form:

$\displaystyle\Sigma_{\rm plum}(r;\;r_{\rm GC})=\frac{1}{\pi}\frac{1}{r_{\rm
GC}^{2}(1+r^{2}/r_{\rm GC}^{2})^{2}}$ (2)

with half-number radius of $r_{\mathrm{GC}}=1.5\times r_{e}$. We show this
assumption agrees fairly well with the results from the likelihood-based
method discussed below, although with that method, we find a slightly smaller
value of $r_{\mathrm{GC}}\sim 1.2\times r_{e}$ is more appropriate. In the
end, this assumption does not affect the main conclusions since we assume the
same distribution for the NGVS comparison sample, and so the cluster-to-LV
environment comparison is unchanged555This assertion does assume that the
radial distribution of GCs is the same for dwarfs in clusters as it is for
dwarfs in LV-like environments. While increased tidal stripping due to the
cluster environment could lead to more extended GC systems (Smith et al.,
2013), we do not expect the average cluster dwarf to have experienced much
stripping (e.g. Smith et al., 2015)..

Therefore, we correct the background subtracted results for GCs outside of the
$2r_{e}$ aperture simply by integrating the Plummer profile from $2r_{e}$ to
infinity. For the fiducial choice of $r_{\mathrm{GC}}=1.5\times r_{e}$, this
corresponds to increasing the background subtracted $N_{\mathrm{GC}}$ by about
$50$%. Note that a different assumed value of $r_{\mathrm{GC}}$ would
correspond to a simple scaling up or down of $N_{\mathrm{GC}}$. At this stage,
we also correct for possible luminosity incompleteness in the GC sample by
integrating over the expected GCLF multiplied by the completeness function of
the data as determined above. We take the GCLF to be a Gaussian with mean of
$M_{g}=-7.2$ mag and dispersion of $0.7$ mag (Jordán et al., 2007; Villegas et
al., 2010). We show below that this is quite close to the GCLF properties that
we measure for the LV and Virgo samples. As discussed above, the data
generally extends to multiple magnitudes below the GCLF so these corrections
are small, $\lesssim 5$%.

###  3.5 GC Abundance: Likelihood Method

We complement the simple background subtraction method with a more
sophisticated likelihood-based inference. This method is able to take into
account the magnitude and spatial distribution of candidate GCs, allowing for
a more precise estimate of GC abundance than the simple background
subtraction. Our method is inspired by the methods used by Amorisco et al.
(2018) and Prole et al. (2019), with the primary difference being that we fit
all dwarfs simultaneously. In the mass range studied in this work, dwarfs do
not have many (if any) GCs individually. Thus, we fit for underlying, average
properties of the GC systems for an ensemble of dwarfs.

We model the dwarfs as a mixture of systems with no GCs and systems with a
non-zero GC population. Thus, we primarily fit for the fraction of dwarfs with
GCs and for a relation between average number of GCs (for galaxies with GCs)
and dwarf stellar mass.

We start this procedure with the color restricted catalog of point sources in
the vicinity of each dwarf (Section 3.3 above). The likelihood that a source
(indexed by $i$) of a dwarf galaxy (indexed by $j$) is a GC is

$\displaystyle\mathcal{L}^{i,j}_{\rm GC}\propto N_{\rm
GC}(M_{\star}^{j})\;C^{j}_{\mathrm{mag}}\;\Sigma_{\rm plum}(r^{i};\;r_{\rm
GC})\;\mathcal{N}(m_{g}^{i};\;M_{g}^{0},\;\sigma_{g})$ (3)

where $r^{i}$ is the radial distance of the source (in units of $r_{e}$),
$m_{g}^{i}$ is the source magnitude, and $\mathcal{N}$ is a Gaussian
representing the GCLF with mean at the magnitude $M_{g}^{0}$ and dispersion
$\sigma_{g}$, which are both also fit for in the procedure. $N_{\rm
GC}(M_{\star}^{j})$ is the expected number of GCs for a dwarf with a nonzero
GC system based on the dwarf’s stellar mass, $M_{\star}^{j}$.
$C^{j}_{\mathrm{mag}}$ accounts for the incompleteness of the source catalogs
and is simply

$\displaystyle C^{j}_{\mathrm{mag}}=\int
S^{j}(m)\;\mathcal{N}(m;\;M_{g}^{0},\;\sigma_{g})\;dm$ (4)

Here, $S^{j}(m)$ is the empirically determined overall completeness function
of dwarf $j$ from §3.3.2.

The likelihood a source is a contaminant is then

$\displaystyle\mathcal{L}^{i,j}_{\rm bk}\propto\frac{N^{j}_{\rm tot}-N_{\rm
GC}(M_{\star}^{j})\;C^{j}_{\mathrm{mag}}}{A^{j}}\;\mathcal{F}^{j}(m_{g}^{i})$
(5)

where $N^{j}_{\rm tot}$ is the total number of sources in the cutout for
galaxy $j$ which has area $A^{j}$, and $\mathcal{F}^{j}$ is an empirical
estimate of the background/foreground source luminosity function constructed
from sources beyond $2r_{e}$ from the galaxy.

We do not include color information in the inference. The background sources
appear to have a broader distribution in color than the GCs, but the
difference is not great. As mentioned earlier, we model the ensemble of dwarfs
as a mixture between a sub-population with a nonzero GC system and sub-
population without GCs. The overall likelihood for a given dwarf is then a
mixture between having a GC system and not:

$\displaystyle\mathcal{L}^{j}\propto
f^{j}\prod^{i}\left[\mathcal{L}^{i,j}_{\rm GC}+\mathcal{L}^{i,j}_{\rm
bk}\right]+(1-f^{j})\prod^{i}\mathcal{L}^{i,j}_{\mathrm{bk},\;N_{\mathrm{GC}}=0}$
(6)

$f^{j}\equiv f(M_{\star}^{j})$ is the fraction of dwarfs with a nonzero GC
system at the stellar mass of the dwarf. The second product term comes from
setting $N_{\mathrm{GC}}=0$ in the equations for $\mathcal{L}^{i,j}_{\rm GC}$
and $\mathcal{L}^{i,j}_{\rm bk}$.

The likelihood for the ensemble of dwarfs is then simply

$\displaystyle\mathcal{L}=\prod^{j}\mathcal{L}^{j}$ (7)

The free parameters that are fit for are $N_{\mathrm{GC}}$, $f$,
$r_{\mathrm{GC}}$, $M_{g}^{0}$, and $\sigma_{g}$. $N_{\mathrm{GC}}$ and $f$
are both functions of stellar mass. For $N_{\mathrm{GC}}$, we parameterize it
as a power law

$\displaystyle
N_{\mathrm{GC}}(M_{\star})=1+T_{0}\left(\frac{M_{\star}}{1\times
10^{9}M_{\odot}}\right)^{\alpha}$ (8)

thus contributing two free parameters, $T_{0}$ and $\alpha$. Note there is the
$N_{\mathrm{GC}}=1$ floor in the power law because $N_{\mathrm{GC}}$ is the
average number of GCs for dwarfs with a nonzero GC system. For the GC
occupation fraction, instead of using a parameterized form (like an error
function, for example), we fit for the occupation fraction at five stellar
masses and interpolate it for intermediate masses. We fit for the fraction at
stellar masses of $\log(M_{\star})=[5.5,6.25,7.,7.75,8.5]$, thus contributing
five free parameters. We enforce the occupation fraction to be an increasing
function with stellar mass through the priors. With $M_{g}^{0}$ and
$\sigma_{g}$, this makes 10 total free parameters. The parameters and their
priors are listed in Table 1. All priors are taken to be uniform.

Table 1: Summary of parameters fit for in the likelihood-based GC ensemble analysis. All priors are taken to be uniform  Parameter | Range of prior
---|---
$T_{0}$ | [1, 150]
$\alpha$ | [-0.1, 1.4]
$M_{g}^{0}$ | [-7.5, -6.5]
$\sigma_{g}$ | [0.2, 1.2]
$r_{\mathrm{GC}}$ | [0.25, 2.25]
$f_{1}$ | [0, $f_{2}$]
$f_{2}$ | [$f_{1}$, $f_{3}$]
$f_{3}$ | [$f_{2}$, $f_{4}$]
$f_{4}$ | [$f_{3}$, $f_{5}$]
$f_{5}$ | [$f_{4}$, 1]

The posterior distributions are sampled via MCMC. The posterior distributions
are visually inspected and all appear well-behaved and unimodal. The
exceptions are $f_{1}$ and $f_{5}$ which, for both the LV and Virgo samples,
are pushed against the allowed boundaries of 0 and 1, respectively.

###  3.6 Image Simulations

To test both the simple background subtraction and likelihood-based methods,
we perform realistic, end-to-end image simulations of dwarfs with GC systems.
Artificial dwarfs are generated in empty fields in the imaging data for three
hosts that span the range of data quality and depth of the data used. Data for
NGC 4258 and NGC 3379666In particular, DECaLS data is used for NGC 3379 in
this test to represent a worst-case scenario. Most of the dwarfs of NGC 3379
are actually covered by extremely deep HSC data but that is not used in the
simulations. are used to demonstrate a best and worse-case scenario,
respectively, for the ELVES sample. The third host included in the simulations
is Virgo, to show that the methods also work at the somewhat further distance
of Virgo.

We go into more detail of the inputs and recovery performance of the
simulations in Appendix D. Suffice to say here that we are able to robustly
recover the input GC system parameters, including $N_{\mathrm{GC}}$ and the
occupation fraction, within the estimated uncertainties from the posterior
distributions. The simple background subtraction method is able to recover
well the average overall trend in $N_{\mathrm{GC}}$. The simulations are run
with $\sim 200$ dwarfs, which is similar to the number in the observational
datasets, although the simulated dwarfs are given, on average, more GCs than
observed. This leads to more precise recovery than expected for the real
datasets, but we show that the methods are unbiased in this higher-precision
regime.

Much of the previous work in the field of dwarf galaxy GC systems has been
conducted with _HST_. The resolving power of _HST_ makes it very useful for
photometrically identifying GCs in nearby ($D<10$ Mpc) dwarf galaxies without
the need for a background subtraction. This is because the GCs are actually
resolved as star clusters at these distances. However, through these image
simulations and the agreement we get with previous work done with _HST_ (see
below), we show that studying GCs is not infeasible with ground-based data.

###  3.7 NSCs and GCs in the Local Group

The satellites of the MW and M31 must be treated differently from the other
ELVES hosts. Due to the complexities involved in measuring the GC systems, we
only use the LG satellites for the nucleation analysis and not for the GC
analysis, although we do compare with the LG GC abundances for a check on our
method. We take photometry from McConnachie (2012)777Except the stellar mass
of the disrupting Sgr dSph which we take from Forbes et al. (2018a).. The
nucleated LG early-type dwarfs are: Sgr, NGC 147, NGC 205 and M32. NGC 205 and
M32 are above our upper mass limit of $10^{8.5}$M⊙ and thus are not considered
here. The NSC associated with Sgr is M54, and we adopt the stellar mass from
Baumgardt & Hilker (2018). Veljanoski et al. (2013) list a GC being at a
projected separation of 30 pc ($\sim 0.05r_{e}$) from the center of NGC 147,
which would therefore classify as ‘nucleated’ by the criterion used for the LV
dwarfs. We take the stellar mass of this GC/NSC from Veljanoski et al. (2013),
assuming $M_{\star}/L_{V}=1.5$.

##  4 Nuclear Star Cluster Results

In this section and the next, we present the main results of our analysis of
NSCs and GCs in LV dwarfs, starting with the NSC analysis. Throughout, we
focus only on dwarfs in the mass range $5.5<\log(M_{\star}/{\rm
M}_{\odot})<8.5$. The lower limit is roughly the luminosity limit ($M_{V}=-9$
mag) of the survey, and the upper bound is set so that we avoid many of the
highest mass satellites for which our analysis methods start to fail (e.g.
they are invariably poorly fit by Sérsic profiles). There are also very few
higher mass early-type dwarfs in MW-sized halos. In this mass range, we
characterize 40 nuclei from 177 total LV early-type dwarfs, for an overall
nucleation fraction of 0.23.

Figure 3: NSC photometry for early-type dwarfs in the Local Volume. The left
panel gives the absolute $g$ magnitude while the right shows the the color
difference between the nucleus and galaxy. In general, nuclei are slightly
bluer than their host galaxies. The vertical dashed lines in the left panel
show the median 50% point source completeness limit for the LV dwarfs, with
the 16th and $84^{th}$ percentile range shown by the dotted lines.

###  4.1 NSC Luminosity and Color

Figure 3 gives the photometry for the NSCs; NSCs are generally bright with
$M_{g}\sim-9$ mag. The right panel of Figure 3 shows the colors of NSCs
relative to their host galaxies. By normalizing the color by the host galaxy,
we avoid most systematic issues related to the different filter systems used.
For dwarfs with $g/r$ photometry, we multiply their colors by 1.53 (cf
Equation 1) to account for the longer baseline of $g-i$. The nuclei are, on
average, slightly bluer than their host galaxies by about $\sim 0.05$ mag, in
line with the findings of Ordenes-Briceño et al. (2018a) and Sánchez-Janssen
et al. (2019).

Also shown in the left panel of Figure 3 is the distribution of 50% point
source completeness limits for the LV dwarfs from the GC analysis. The data
are generally sensitive to NSCs with $M_{g}>-6$ mag.

Figure 4: NSC stellar mass versus host stellar mass for the LV and Virgo
samples. The errorbar in the left panel shows the characteristic errors in
stellar mass. The right panel shows power law fits for the two samples, along
with a running average in 0.5 dex wide bins of stellar mass.

###  4.2 NSC Scaling Relations

Figure 4 shows the correlation between NSC stellar mass and dwarf-host stellar
mass. It compares the 40 LV dwarf nuclei to 98 Virgo dwarf nuclei found in our
analysis of the NGVS sample. For this plot, we use our own photometry of the
NGVS dwarfs and nuclei to make as direct a comparison as possible. The right
panel shows characteristic errorbars for both stellar mass measurements. We
assume a constant 0.3 dex uncertainty in the NSC stellar mass, which is the
dispersion between the NSC masses we measure for the NGVS sample and those
reported in Sánchez-Janssen et al. (2019).

We fit power laws to the NSC-galaxy stellar mass relation using the linmix
algorithm (Kelly, 2007)888We use the python implementation of linmix written
by J. Meyers: https://github.com/jmeyers314/linmix of the form:

$\displaystyle\log(M_{\star}^{\rm nuc})=a+b\log(M_{\star}^{\rm gal})$ (9)

with intrinsic scatter $\sigma$. The best-fitting values and marginalized
uncertainties for the two samples are given in Table 2.

Table 2: Results of fitting the NSC-galaxy stellar mass relation for LV and Virgo dwarfs in the mass range $5.5<\log(M_{\star}/{\rm M}_{\odot})<8.5$. $\sigma$ denotes the intrinsic scatter in size at fixed stellar mass. Sample | $a$ | $b$ | $\sigma$
---|---|---|---
LV | $2.68^{+1.05}_{-1.01}$ | $0.38^{+0.14}_{-0.14}$ | $0.41^{+0.08}_{-0.07}$
Virgo | $3.0^{+0.5}_{-0.5}$ | $0.36^{+0.07}_{-0.07}$ | $0.37^{+0.05}_{-0.04}$

Both samples give relations somewhat shallower than the oft-found $M_{\rm
nuc}\propto M_{\rm gal}^{0.5}$, although they are consistent within the
uncertainties. We also show a linear fit to the Virgo photometry of Sánchez-
Janssen et al. (2019) which shows fair agreement with our Virgo results. Our
fit to the Sánchez-Janssen et al. (2019) photometry gives a power law slope of
$b=0.44^{+0.08}_{-0.08}$, which is a little steeper than what our Virgo
photometry yields. We note that we use different prescriptions for calculating
stellar masses than done in Sánchez-Janssen et al. (2019) although that
shouldn’t cause an offset in this plot. It is possible that our Virgo results
disagree from the Sánchez-Janssen et al. (2019) results at high galaxy masses
because our method of modelling these galaxies as Nucleus$+$Sérsic is
inadequate for many massive dwarfs which often have more complicated
brightness profiles. There appears to be some evidence that LV nuclei have
lower stellar mass on average at fixed host mass than the Virgo galaxies,
although the discrepancy is only at the $\sim 1-2\sigma$ level. The intrinsic
scatter in the NSC-galaxy stellar mass relation is similar between both
environments.

Figure 5: Left: Nucleation fraction of early-type dwarfs as a function of
stellar mass in different environments in the local universe. The Virgo
results are from Sánchez-Janssen et al. (2019), although we also show the
results from our parallel analysis of the NGVS sample. The Fornax results are
from Eigenthaler et al. (2018) and Coma from den Brok et al. (2014). Right:
The nucleation fraction of LV satellite dwarfs split by their projected
separation from their massive host galaxy. The Virgo results are included in
this panel for reference. In both panels, there is a clear trend in
environment where early-type dwarf satellites in denser environments (either
more massive parent halo or closer to their host) are more frequently
nucleated at fixed stellar mass.

###  4.3 NSC Abundance

Figure 5 shows the nucleation fraction of dwarfs as a function of stellar mass
in the different environments where comparable samples are available. For
Virgo, we include both the results directly from Sánchez-Janssen et al. (2019)
and also from our analysis of the NGVS dwarf sample. They agree quite well,
indicating that many systematic concerns in this process, including the visual
inspection of candidate nuclei and the determination of stellar masses, have a
minor effect. The nucleation fraction from our Virgo analysis is a little
higher at the lowest stellar mass bins. This difference is due to 2-3 low-mass
dwarfs, which we classify as nucleated but Sánchez-Janssen et al. (2019) did
not.

We also show the results for the inner $r\lesssim R_{\mathrm{vir}}/4$ area of
Fornax from the Next Generation Fornax Survey Eigenthaler et al. (2018) and
for the inner $r\lesssim R_{\mathrm{vir}}/6$ area of Coma from den Brok et al.
(2014). We describe how we get stellar masses for these dwarf samples in
Appendix E.

These results conclusively confirm the result found by Sánchez-Janssen et al.
(2019) that nucleation fraction of early-type dwarfs has a significant
secondary dependence on environment. Sánchez-Janssen et al. (2019) just
considered early-type dwarfs in the LG and M81 systems while our sample size
is nearly an order of magnitude greater. The trend of lower density
environments being less conducive to the formation of nuclei extends all the
way down to MW-sized halos.

Very recently, Zanatta et al. (2021) also confirmed this environmental trend
comparing satellites in different mass clusters and lower mass, LV hosts. For
the latter, they used satellites partly from the surveys of Carlsten et al.
(2020a, b). We have greatly increased the statistics for LV satellites since
those works. Additionally, Zanatta et al. (2021) include all confirmed and
‘possible’ satellites (candidate satellites where the SBF distances were
inconclusive) from those works. From our experience with acquiring deeper
follow-up, we suspect roughly half of the latter class are background objects.
Since these background objects will likely not have detectable nuclei, their
inclusion will lower the average nucleation fraction somewhat. In this work,
we only consider fully confirmed satellites and show that, among these
objects, the nucleation fraction in LV hosts is still quite a bit less than in
the galaxy clusters. We discuss what this finding means for the formation of
NSCs below in the Discussion section.

Interestingly, in the right panel of Figure 5, we show that there is an
environmental dependence even within the LV sample. We split the ELVES sample
between dwarfs within 100 projected kpc of their host and those outside. This
radius roughly splits the sample into halves. There is a surprisingly
consistent trend of lower nucleation fraction for the outer LV dwarfs.

While the inner LV nucleation fractions are still slightly below that of the
Virgo sample, the difference is much less. This brings up the intriguing
possibility that the difference seen in the left panel and reported by
Sánchez-Janssen et al. (2019) and Zanatta et al. (2021) is actually more of an
effect of the cluster samples being very centralized and less due to the
parent halo mass in which the dwarfs reside. Unfortunately we do not have the
dwarf samples to conclusively show which affect (parent halo mass or location
of dwarf within parent halo) is dominant. The full NGVS (that covers the whole
virial volume) and final ELVES dwarf samples will be critical in addressing
this. We do note that Ordenes-Briceño et al. (2018b) presented the Fornax
dwarf population over a more extended radial range
($R_{\mathrm{vir}}/4<r<R_{\mathrm{vir}}/2$). The dwarf sample in that radial
range had a similar overall nucleation fraction of $106/384=0.28$ as that
reported for the inner population ($r<R_{\mathrm{vir}}/4$; Eigenthaler et al.,
2018) of $75/258=0.29$. We show in Appendix F that the nucleation fractions at
fixed stellar mass are similar between the two radial ranges, hinting that the
effect is primarily driven by parent halo mass.

Figure 6: Stacked, background subtracted GC luminosity functions and color
distributions for GC candidates in the vicinity of the dwarfs. The top shows
our results for the LV sample while the bottom shows the analogous results for
the Virgo comparison sample. Errorbars show the Poisson uncertainties
considering the overall (including background) number of sources in each
magnitude bin. The gray areas denote the magnitude and color regions outside
of the cuts made on the catalog. The vertical orange line and shaded region
denote the 16th, 50th, and 84th percentile of the 50% completeness magnitudes
for the different dwarfs. In almost all cases, it is more than a magnitude
fainter than the GCLF peak. The right panels also show the distributions of
nuclei color for each environmental sample as the green, dotted histograms.

##  5 GC Results

In this section, we present the results on the GC analysis of the LV dwarf
sample. We start with a discussion of the observed GCLF, then move to the
radial distributions of GCs, and then, finally, present the abundance of GCs
as a function of stellar mass with particular focus on a comparison across
environments.

###  5.1 GC Luminosity Function

Figure 6 shows the overall stacked, background-subtracted GCLF and color
distributions for the LV and Virgo samples. To make these distributions, we
stack the background subtracted histograms for each individual dwarf for
sources within a circular $2r_{e}$ annulus. Note the histograms are not
corrected for possible GCs outside of $2r_{e}$. The errorbars are Poisson
uncertainties given the combined total number of sources (before background
subtraction) in the $2r_{e}$ annuli. The GCLF’s both show a peak at
$M_{g}\sim-7.2$ mag, which we explore in more detail below. The color
distribution for the Virgo dwarfs shows a single peak around $g-i\sim 0.7$,
similar to the findings of Prole et al. (2019) and corresponding to the color
range of the metal-poor subpopulation of GCs found around more massive
galaxies (e.g. Peng et al., 2006). The color distribution for GCs of the LV
dwarfs does not show a clear peak, likely due to limited statistics.

To get better signal-to-noise on the background subtracted LF, in Figure 7, we
show the background subtracted GCLF but restricted only to dwarfs with $N_{\rm
GC}>5$ (as determined by the simple background subtraction method of §3.4). We
show the posteriors of the GCLF parameters from the likelihood-based fit in
the top panel and GCLF curves corresponding to the median posterior values in
the bottom two panels. The GCLF parameters are similar for both dwarf samples
with $M_{g}\sim-7.1$ and dispersions of $\sim 0.6$ mag. Jordán et al. (2007)
and Villegas et al. (2010) both show that the GCLF dispersion decreases for
lower luminosity galaxies, at least for galaxies brighter than
$M_{B}\lesssim-16$ mag. Using Equation 18 of Jordán et al. (2007) and assuming
$\langle M_{B}\rangle\sim-12$ mag for our samples, we’d expect $\mu_{g}\sim
0.34$ mag, smaller than the observed $\sim 0.6$ mag. This indicates that the
dwarf luminosity-GCLF width relation likely flattens out around $M_{B}\sim-16$
mag, the faintest magnitudes probed by Jordán et al. (2007).

The LV dwarf GCLF shown in Figure 7 exhibits a hint of a non-Gaussian tail
extending to faint luminosities. The tail even extends into the magnitude
range where some of the data used will be incomplete ($M_{g}\gtrsim-6$ mag)
and is thus a lower limit to the true LF at those magnitudes. To explore this
in more detail, in Figure 8, we compare the LV dwarf GCLF from Figure 7 with
two other samples of nearby dwarfs. First is the sample of GCs in LG dSphs. We
include GCs of NGC 147, NGC 185, NGC 205, Sagittarius, and Fornax. The
luminosities of the GCs of M31 dwarfs come from Da Costa & Mould (1988) and
Veljanoski et al. (2013). The list of Sgr GCs comes from Law & Majewski (2010)
with luminosities from Harris (1996). The properties of the Fornax GCs come
from Mackey & Gilmore (2003). The second sample of GCs in dwarfs that we
include is the combined sample of nearby ($D\lesssim 6$ Mpc) dwarfs of Sharina
et al. (2005) and Georgiev et al. (2009b). Only early-type dwarfs are
included. These are all satellites of nearby LV massive hosts. Note,
therefore, that many of the Sharina et al. (2005) and Georgiev et al. (2009b)
sample galaxies are included in ELVES, but those works used _HST_ to identify
GCs (thus obviating the need for a background subtraction) and are still an
important comparison to our results. We assume $M_{g}\approx M_{V}+0.2$ for
the GCs.

Both of these other samples show remarkably similar GCLF shapes to the ELVES
result, including a tail to faint luminosities. Indeed, Sharina et al. (2005)
comment on this surplus of faint GCs as they observe it extending down to
$M\sim-5$ mag, the completeness limit of that work. It is unclear if the
dwarfs in the Virgo comparison sample also show this GCLF shape since the
completeness limits of the Virgo data are not as faint. An in-depth
investigation into this GCLF shape and if it is actually different to the more
symmetric and Gaussian GCLFs observed for more massive cluster early-type
dwarfs (e.g. Miller & Lotz, 2007; Jordán et al., 2007) would be an interesting
future venture, particularly after some of the ELVES GCs are confirmed
spectroscopically.

Figure 7: Top: Contours showing the 1$\sigma$ and 2$\sigma$ confidence regions
from the posterior distribution for the GCLF parameters (peak magnitude and
dispersion). Green is for the LV sample, red for Virgo. Middle & Bottom:
Background subtracted GCLFs restricted to dwarfs with significant GC systems
($N_{\rm GC}>5$). Gaussian GCLFs corresponding to the median parameters from
the posterior distributions are shown in dashed red. Figure 8: The LV GCLF
from Figure 7 compared to the GCLF for Local Group early-type satellites and
nearby ($D<6$ Mpc) early-type dwarfs from Georgiev et al. (2009b) and Sharina
et al. (2005). The samples show a pronounced tail towards fainter
luminosities.  Figure 9: Left: Posterior distributions from the fit for
$r_{\mathrm{GC}}$, the GC system half-number radius, from the combined
likelihood-based analysis. Right: Stacked average radial density profiles of
the GC systems. The errorbars are Poisson based on the total number of sources
(before background-subtraction) in each radial annulus. The values for
$r_{\mathrm{GC}}$ are from the posterior distributions in the left panel.
Plummer profiles with $r_{\mathrm{GC}}$ drawn from the posterior distributions
are also shown. Figure 10: Top: Constraints on the GC half-number radius,
$r_{\mathrm{GC}}$, from fitting individual dwarfs with rich GC systems (see
text for definition). The dwarfs are ordered in terms of increasing stellar
mass within each sample. The constraints on $r_{\mathrm{GC}}$ from most of the
individual dwarfs are not tight but almost always $>r_{e}$. Bottom: A
histogram showing the constraints on $r_{\mathrm{GC}}$ from combining the LV
and Virgo dwarfs from the top panel. Also shown are the results from fitting
for $r_{\mathrm{GC}}$ using simulated dwarfs with similar GC abundance as the
real dwarfs (see §3.6). The simulations were generated assuming
$r_{\mathrm{GC}}=1.5\times r_{e}$.

###  5.2 GC Radial Profiles

As discussed in the Methods section, there is significant uncertainty in the
radial profiles of the GC systems of dwarf galaxies. The radial profile of
dwarf GC systems is a very interesting observable, both from the viewpoint of
understanding the formation of the GC system and also the NSCs as they likely
formed, at least in part, through GC inspiral. We return to this point in the
Discussion section below. Also, the GC radial distribution influences the
total GC abundance estimates since we had to assume a certain radial
distribution to extrapolate from number counts in an annulus.

Figure 9 shows the posterior distributions for $r_{\mathrm{GC}}$ from the
combined fits of all dwarfs in each of the LV and Virgo samples.
Unfortunately, neither sample has the statistics to provide tight constraints
on the radial distribution, although $r_{\mathrm{GC}}>r_{e}$ is mildly
preferred for both the LV and Virgo samples. The distribution for the two
samples is similar.

In the right panel of Figure 9, we show the average binned radial profile of
GCs. For these stacked profiles, we start with the color and magnitude
restricted catalogs of candidate GCs from §3.3.3. We then calculate the number
density of sources in radial annuli around each dwarf. These curves are
background subtracted by calculating the density of background sources outside
of $5r_{e}$ for each dwarf. The radial profiles are scaled by each galaxy’s
half-light radius and the densities are similarly normalized by $r_{e}$. To
improve the signal-to-noise of the stack, we include only dwarfs in the mass
range $6.5<\log(M_{\star}/{\rm M}_{\odot})<8.5$. We also show Plummer profiles
with $r_{\mathrm{GC}}$ values drawn from the posterior distributions.

Several of the dwarfs have reasonably rich GC systems, and it should be
feasible to fit for $r_{\mathrm{GC}}$ for individual dwarfs. To do this, we
essentially do the likelihood-based abundance inference but with just a single
dwarf. We fix the GCLF parameters to $M_{g}^{0}=-7.2$ mag and $\sigma_{g}=0.7$
mag and assume the dwarf has a non-zero GC system. For this process, we select
dwarfs that show significant GC systems from the simple background subtraction
results. In particular, we select dwarfs with a background-corrected number of
GC candidates greater than 3 in a $1r_{e}$ annulus and 5 in a $2r_{e}$
annulus. Either the $1r_{e}$ or $2r_{e}$ annulus number counts must also be
$>2\sigma$ above the background as estimated by annuli placed in surrounding
regions. When corrected for GCs outside of these annuli, the selected dwarfs
generally have total $N_{\mathrm{GC}}\sim 7-15$.

Figure 10 shows the results of these individual fits. The top panel shows the
constraints from each individual dwarf, ordered by increasing dwarf stellar
mass within each environmental sample. Even among the most GC-rich dwarfs in
the sample, the constraint on $r_{\mathrm{GC}}$ is relatively weak. Still,
almost all of the dwarfs show $r_{\mathrm{GC}}>r_{e}$. The bottom panel shows
the combined histogram of all the $r_{\mathrm{GC}}$ constraints compared to
the $r_{\mathrm{GC}}$ constraints from individually fitting simulated galaxies
(see §3.6) selected by the same abundance criteria. The simulated galaxies are
generated with $r_{\mathrm{GC}}=1.5\times r_{e}$. The similarity between the
$r_{\mathrm{GC}}$ results of the real and simulated galaxies indicate that the
underlying $r_{\mathrm{GC}}$ for the real galaxies is $>r_{e}$, at least for
the GC-rich dwarfs.

Figure 11: GC specific mass frequencies ($T_{N}\equiv N_{\mathrm{GC}}\times
10^{9}M_{\odot}/M_{\star}$) for the LV and Virgo samples. Points are colored
by their surface brightness, after correcting for the average trend in surface
brightness with stellar mass. The triangles indicate dwarfs whose background
subtracted GC abundance came out zero or negative. The dashed black lines show
constant GC abundances of 1, 10, 20, and 30. The GC abundances are measured
within $2r_{e}$ and have not been corrected for GCs outside of this radius.

###  5.3 Individual GC Abundances

In this section and the next, we explore the GC system abundances,
particularly focusing on how it relates to dwarf stellar mass. We focus on the
results for individual dwarfs from the background subtraction method in this
section, and in the next we present the average trends in GC abundance from
the joint likelihood-based inference.

Figure 11 shows our main results for the individual, background-subtracted GC
abundances. In particular, we show the specific mass frequency $T_{N}\equiv
N_{\mathrm{GC}}\times 10^{9}M_{\odot}/M_{\star}$ for the LV and Virgo dwarfs
as a function of dwarf stellar mass. Note that these values of $T_{N}$ have
not been corrected for GCs outside of the $2r_{e}$ annulus we use to count
sources. As mentioned above, if a GC half-number radius of
$r_{\mathrm{GC}}=1.5\times r_{e}$ is assumed, the total abundance would be
$\sim 50$% higher. Also note that many of the GC abundance values are not
whole numbers both due to the background subtraction process and the
correction for magnitude incompleteness (this last correction is $\lesssim 5$%
but will move data points away from the nearest integer). The points are
colored by the dwarfs’ surface brightness after the general trend of
decreasing surface brightness for lower mass dwarfs is removed. There does not
appear to be a noticeable trend in surface brightness residual with GC
abundance.

In this plot, we see the first indication that dwarfs in Virgo have higher GC
abundance than dwarf satellites in lower-mass, MW-sized parent halos in the
LV. The Virgo sample has significantly more ‘rich’ GC systems. If we consider
a threshold GC abundance of 7 (once corrected for GCs outside of the $2r_{e}$
annulus, this corresponds to $\gtrsim 10$), there are 21 Virgo dwarfs (a
fraction of $0.071\pm 0.014$) with abundance above this compared to 4 LV
dwarfs (a fraction of $0.029\pm 0.014$). The GC results for Virgo generally
had much larger uncertainties, but this does not appear to be causing this
difference since the dwarfs above this threshold in both environments had
average S/N$\sim 3$ in the abundance measurement. This is a reasonable
comparison to make since the stellar mass distributions of Virgo vs LV dwarfs
are quite similar (cf. Appendix B).

Figure 12: The average trends in GC abundance and occupation fraction from the
simultaneous fit of all dwarfs in each environmental sample. Each panel shows
the median result from the posterior along with $\pm 1\sigma$ uncertainties.
Left: The average number of GCs for dwarfs with GCs is shown as a function of
stellar mass. Middle: The average occupation fraction of GCs (i.e. fraction of
dwarfs with non-zero GC systems) as a function of stellar mass. Right: The
overall average number of GCs per dwarf, including dwarfs with and without GC
systems. Shown in the points are running means from the simple background
subtraction analyses, which have been corrected for GCs outside the $2r_{e}$
annuli used to count GCs.

###  5.4 Average Trends in GC Abundance

Figure 12 shows the main results from the joint fit for all dwarfs in each of
the environment samples. The left panel shows the average number of GCs for
dwarfs with non-zero GC systems. This was fit as a power law with a floor of
one. The best-fitting power law parameters are given in Table 3 along with the
other best-fitting parameters from the joint fit (cf. §3.5 for details on what
each parameter is). The median posterior values are given along with
marginalized uncertainties. The middle panel of Figure 12 shows the inferred
occupation fraction of GCs as a function of dwarf stellar mass. The occupation
fraction goes from essentially zero at $M_{\star}<10^{6}M_{\odot}$ to $\sim
80$% at $M_{\star}\sim 10^{8.5}M_{\odot}$. This is a similar trend to that
reported in Figure 6 of Sánchez-Janssen et al. (2019) for the NGVS analysis of
Virgo dwarfs, although it appears we infer a somewhat smaller occupation
fraction for intermediate mass ($M_{\star}\sim 10^{7}M_{\odot}$) dwarfs. We
compare the GC occupation fraction trends directly to the nucleation fraction
trends below in the Discussion.

Table 3: Best-fitting parameters from the joint fit.  Sample | Parameter | Best-fitting value
---|---|---
LV | $T_{0}$ | $25.81^{+17.803}_{-11.762}$
| $\alpha$ | $0.945^{+0.268}_{-0.308}$
| $M_{g}^{0}$ | $-7.016^{+0.11}_{-0.114}$
| $\sigma_{g}$ | $0.565^{+0.079}_{-0.072}$
| $r_{\mathrm{GC}}$ | $1.063^{+0.21}_{-0.177}$
| $f_{1}$ | $0.03^{+0.045}_{-0.023}$
| $f_{2}$ | $0.081^{+0.071}_{-0.044}$
| $f_{3}$ | $0.149^{+0.101}_{-0.072}$
| $f_{4}$ | $0.602^{+0.172}_{-0.17}$
| $f_{5}$ | $0.887^{+0.082}_{-0.15}$
Virgo | $T_{0}$ | $31.657^{+17.556}_{-10.172}$
| $\alpha$ | $0.59^{+0.172}_{-0.155}$
| $M_{g}^{0}$ | $-7.094^{+0.072}_{-0.074}$
| $\sigma_{g}$ | $0.52^{+0.055}_{-0.055}$
| $r_{\mathrm{GC}}$ | $1.248^{+0.236}_{-0.183}$
| $f_{1}$ | $0.02^{+0.032}_{-0.015}$
| $f_{2}$ | $0.054^{+0.045}_{-0.03}$
| $f_{3}$ | $0.228^{+0.122}_{-0.098}$
| $f_{4}$ | $0.541^{+0.122}_{-0.11}$
| $f_{5}$ | $0.786^{+0.141}_{-0.154}$

The right panel shows the overall average GC abundance as a function of
stellar mass, including dwarfs with and without GC systems (i.e. the curves
from the left and middle panels multiplied together). We also show the average
results from the simple background subtraction process (which have been
corrected for GCs outside of the $2r_{e}$ annulus). The simple background
subtracted results are a little higher than the average abundance from the
joint fit. This appears to be largely due to assuming
$r_{\mathrm{GC}}=1.5\times r_{e}$ in correcting for GCs outside of the
$2r_{e}$ annulus while a somewhat smaller $r_{\mathrm{GC}}$ is likely more
realistic.

We find that the Virgo dwarfs have notably higher average GC abundance than
the LV dwarfs, particularly in the mass range $10^{7}<M_{\star}<10^{8}$ M⊙.
Interestingly, this seems to be mostly stemming from the difference in GC
abundance for dwarfs with GCs and less a difference in average GC occupation
fraction.

Since the NGVS dwarf sample is from the innermost $r<R_{\mathrm{vir}}/5$
region of Virgo, while the LV samples are generally complete to at least
$r\gtrsim R_{\mathrm{vir}}/2$, the same caveat applies to these GC results as
for the NSC results discussed in §4.3. With the current dwarf samples, it is
not clear whether the difference in GC abundance is primarily an effect of
parent halo mass or the dwarfs’ current location in the parent halo.

In Figure 13, we show the GC abundance results over a larger dynamic range in
stellar mass. In addition to the LV and Virgo samples, we include results from
Peng et al. (2008) which extend to much higher stellar mass. To roughly match
the spatial distribution of the NGVS dwarfs within Virgo, the Peng et al.
(2008) results are restricted to those within a projected 500 kpc of M87
($\sim R_{\mathrm{vir}}/3$). The LV sample is offset below the NGVS sample by
roughly $\sim 0.3$ dex (about a factor of 2), as was shown in Figure 12. The
Peng et al. (2008) results show a similar average relation as our Virgo
results, scaling approximately as $N_{\mathrm{GC}}\propto M^{0.5}_{\star}$
below about $M_{\star}\sim 10^{10.5}$. Above that mass, the relation steepens
to approximately $N_{\mathrm{GC}}\propto M^{1.5}_{\star}$, presumably due to
the increasing importance of accreting GCs from cannibalized satellite
galaxies. We compare our results in more detail to this and other Virgo dwarf
samples in the next section.

Figure 13: We compare the inferred GC abundances for the LV and Virgo samples
to the Peng et al. (2008) results over a larger dynamic range in galaxy
stellar mass. Our results for both environments show a similar scaling to the
Peng et al. (2008) results of approximately $N_{\mathrm{GC}}\propto
M^{0.5}_{\star}$. Above about $M_{\star}\sim 10^{10.5}$, the relation steepens
to approximately $N_{\mathrm{GC}}\propto M^{1.5}_{\star}$.  Figure 14: Left: A
comparison between the ELVES sample and other, _HST_ based, surveys of LV
dwarfs. We also include the LG dSph satellites with these latter surveys.
Right: The Virgo sample analyzed in this work shown next to two comparable
analyses: Miller & Lotz (2007) and Peng et al. (2008). We find that the sample
of Miller & Lotz (2007) shows lower GC abundance at fixed stellar mass than
the sample analyzed in this work. To test if this is an environmental effect
we show the results of Peng et al. (2008) both including all galaxies and also
restricting to those within a projected 500 kpc ($\sim R_{\rm vir}/3$) to M87.
The inner galaxies show higher GC abundance, appearing to be an extrapolation
of the trend found in the current work, indicating the discrepancy with Miller
& Lotz (2007) could feasibly be environmental. The error regions show errors
in the mean, not the intrinsic scatter.

###  5.5 Comparison to Previous Work

In this section, we compare our results for the average GC abundance of ELVES
and NGVS dwarfs to other comparable analyses of LV and Virgo dwarfs. We find
that our inferred GC abundances are similar to what has been found before for
dwarfs in these different environments.

In the left panel of Figure 14, we show the LV results compared to the
combined results of the LG dSphs and the surveys of Sharina et al. (2005) and
Georgiev et al. (2009b). As mentioned above, the Sharina et al. (2005) and
Georgiev et al. (2009b) surveys include many of the ELVES satellites but are
done with _HST_ (obviating the need for a statistical background
subtraction999Recall that these samples are for dwarfs within $D\lesssim 6$
Mpc, so GCs are resolved.) and are important comparison samples for the ELVES
results. The references for the LG GC systems were given in §5.1. All GC
systems are restricted to GCs with $M_{V}<-6$ or $M_{g}<-6$, depending on what
is available, to match the approximate magnitude limit of most ELVES dwarfs.
For reference, the Fornax dSph has four GCs above this limit while the Sgr
dSph has only 2. We describe how we determine stellar masses for these
reference samples in Appendix E. The average GC abundance of the LG satellites
and Sharina et al. (2005) and Georgiev et al. (2009b) surveys agrees well with
the LV results presented here.

In the right panel of Figure 14 we compare the trends of average GC abundance
versus dwarf stellar mass for three Virgo samples: our analysis of the NGVS
dwarfs, the _HST_ survey of Lotz et al. (2004) and Miller & Lotz (2007) and
the ACS Virgo Cluster Survey sample (Côté et al., 2004; Peng et al., 2008). We
detail the sources of the stellar masses for these dwarf samples in Appendix
E. For the dwarfs identified as nucleated by Miller & Lotz (2007), we subtract
their reported GC abundance by one since our GC abundances never include the
NSC.

Interestingly, both the Miller & Lotz (2007) and complete Peng et al. (2008)
(dashed blue) samples show noticeably smaller GC abundance at fixed stellar
mass than the NGVS sample (although still consistent with our NGVS results at
$\sim 2\sigma$). The NGVS sample analyzed in this work is concentrated at the
very core of Virgo ($\lesssim R_{\mathrm{vir}}/5$) while the Miller & Lotz
(2007) sample is distributed all throughout the virial volume of Virgo so this
could feasibly be an environmental effect. To test this, we show the Peng et
al. (2008) sample restricted to those within a projected 500 kpc ($\sim R_{\rm
vir}/3$) to M87101010Restricting this sample to
$R_{\mathrm{proj}}<R_{\mathrm{vir}}/5$ leads to too few statistics. (solid
blue line). As found by Peng et al. (2008), the inner dwarfs show notably
higher GC abundance at fixed stellar mass. The centrally concentrated
subsample lines up well with the abundance trend of the NGVS sample. This is
suggestive that the difference with Miller & Lotz (2007) is indeed
environmental. We note, however, that Miller & Lotz (2007) searched for an
environmental trend within their sample and found no trend in specific
frequency with distance from M87, in seemingly direct conflict with the
results of Peng et al. (2008).

##  6 Discussion

In this paper, we have systematically presented many properties of the NSCs
and GC systems of dwarf galaxies in the Local Volume, paying close attention
to how these properties are the same as or different from star clusters of
dwarfs in other environments, particularly in nearby galaxy clusters. With
much improved statistics over what was possible in the past, we have confirmed
previous findings indicating that dwarf galaxies are more likely to be
nucleated and host richer GC systems, on average, in denser environments.
These trends apply even to dwarfs in the halos of MW-sized galaxies. However,
other than the abundance of the star clusters, we find that other properties
of NSCs and GCs, like luminosity and color, are overall similar across
environment.

In the Introduction, we averred that extending the axis of environment to
studies of star clusters in dwarfs would open a new perspective on star
cluster formation and how it relates to dwarf galaxy evolution. In this
section, we discuss our findings in relation to modern theories of GC and NSC
formation. Overall, we find that our observations are well explained, at least
qualitatively, by these models. First, we explore the cause of the GC
abundance-environment trend and, second, we discuss the NSC properties and
their relations to GCs.

###  6.1 Effect of Environment on GC Abundance

Previous observational studies have generally shown that dwarf galaxies in the
denser, inner regions of clusters have higher GC abundances (Peng et al.,
2008; Lim et al., 2018; Liu et al., 2019) than dwarfs in the cluster
outskirts. In this work, we have confirmed this effect and shown that it
extends to even lower density environments, such as the halos of MW-sized
galaxies. We will discuss three possible origins of this trend inspired by
modern models of GC formation. The first is that dwarfs in denser environments
experience increased efficiency of cluster production. The second is the idea
that dwarfs in denser environments experience reduced levels of cluster
destruction. Finally, the third is that dwarfs in denser environments form
fewer field stars, leading to more prominent GC systems.

####  6.1.1 Increased GC Production

The idea of ‘biased GC formation’, first explored in detail by Peng et al.
(2008), is that subhalos ending up in a cluster core collapse earlier and form
their stars at higher star formation rate (SFR) surface densities compared to
subhalos in the cluster outskirts. Higher SFR densities and the associated
higher pressures and densities in the ISM are more conducive to the formation
of bound stellar clusters, leading to a high cluster formation efficiency
(Goddard et al., 2010; Kruijssen, 2012) and, hence, higher GC specific
frequencies. This trend whereby earlier forming galaxies end up hosting larger
populations of GCs is found in numerous recent models (e.g. Mistani et al.,
2016; Pfeffer et al., 2018; Carleton et al., 2021). Evidence for this also
comes from the elevated [$\alpha$/Fe] ratios of Virgo dwarf galaxies with high
specific frequencies (Liu et al., 2016), indicating these galaxies formed
their stars in a short interval, feasibly accompanied with elevated SFR
surface densities and, hence, cluster formation efficiency.

This biased formation scenario implies that dwarf ETGs in the cores of Virgo
and Fornax are, generally, not newly quenched dIrrs that recently fell into
the cluster from the field. This has been argued based on GC abundance for
more massive dwarfs (Sánchez-Janssen & Aguerri, 2012), and here we show this
must apply to lower mass dwarfs with high GC abundance as well. The gas-rich
progenitors of these cluster dwarfs would not necessarily be similar to dIrrs
that are observed in the field today. However, in a companion paper (Carlsten
et al., submitted), we show that the early-type Virgo dwarf population in the
mass range we consider in this work is very similar in its structural scaling
relations (particularly the mass-size relation) to field dIrrs (this has also
been shown by, e.g., Boselli et al., 2008; Venhola et al., 2019). Evidently,
if the elevated GC abundance in cluster dwarfs is due to an early, intense
phase of star-formation with a high cluster formation efficiency, this did not
leave a correspondingly-large impact on the structure of the dwarf.

####  6.1.2 Decreased GC Destruction

An alternative way to have higher modern day GC abundance in cluster dwarfs is
for these dwarfs to experience lower levels of GC destruction than analogous
dwarfs in less dense environments. If GCs are the descendants of star clusters
similar to the YMCs currently observed in the local universe, then the initial
cluster mass function (ICMF) of GCs will be similar to the mass function of
nearby YMCs; namely a power law with slope $\sim-2$ and exponential cutoff at
the high mass end (e.g. Zhang & Fall, 1999; Larsen, 2009). To get the
observed, peaked GC mass function (GCMF) from this ICMF requires significant
levels of disruption and mass loss, particularly among the lowest mass
clusters. Various estimates have put the total amount of mass loss of a GC
system during its evolution at $\gtrsim 90$% (see Forbes et al., 2018b, for
discussion).

Various dynamical processes have been proposed for the destruction of low mass
clusters (e.g. Fall & Zhang, 2001; Baumgardt & Makino, 2003). The critical
constraint is that the turn-over mass of the GCMF is essentially universal
across different mass hosts and galactocentric radii (Vesperini et al., 2003).
In part to explain this, Elmegreen (2010) and Kruijssen (2015) outline a two-
phase model for GC evolution. In the first phase, the newly born GCs are
efficiently destroyed via tidal shocks due to GMCs present in the gas-rich
environment where the GCs are born. Due to merging at high-redshift and/or
quenching of the host galaxy, GCs are moved to the outer regions, away from
the GMCs. In this second phase, the GCs slowly evolve via two-body relaxation
and evaporation.

Based on this model, Kruijssen (2015) makes the prediction that dwarfs in
denser environments would experience shorter rapid-disruption phases due a
combination of increased mergers or earlier quenching through ram-pressure
stripping.

We see evidence for the GC relocation central to this theory within our
analysis of the GC radial distributions. While we were unable to place
stringent constraints on the GC spatial distribution, Figure 10 shows that our
fits prefer the GC distributions to be on average more spatially extended than
the stellar body of the dwarfs, as is commonly found in the literature (see
§3.4). Since GCs will form preferentially where the ISM is densest, it seems
very unlikely that the GCs formed at the locations we observe them today. As
discussed by Kruijssen (2015) and Lamers et al. (2017), early mergers are one
way to explain the location of GCs. After a merger, the gas will dissipatively
sink back to the galactic center, forming new generations of stars while the
GCs will be stuck in the galaxy’s outskirts. Similarly, Leung et al. (2020)
argue for a merger origin for the spatially extended current configuration of
GCs in the Fornax dSph.

An intriguing alternative way of getting GCs into the outskirts of their hosts
is suggested by the GCs of the LG dwarf NGC 6822 which show a suggestive
alignment and spatial distribution (see Fig. 1 of Huxor et al., 2013)
consistent with the old diffuse stellar halo (Battinelli et al., 2006) of that
dwarf. Many dwarfs are observed to have old/red stellar halos (e.g. Kado-Fong
et al., 2020), which are thought to have formed primarily through stellar
feedback (Stinson et al., 2009). It is unclear if GCs could experience the
same gradual orbital expansion due to stellar feedback as stars. In the end,
whether it is mergers or stellar feedback that relocates the GCs, the end
result of protecting them from disruption in the gas-rich inner regions of the
dwarfs would be the same. Given the importance of GC destruction in setting
the universality of the GCLF, it is unclear if this mechanism could reproduce
the similarity in the GCLF that we observe between the Virgo and LV dwarfs
(cf. Fig 7). Quantitative modelling would be required to address this.

####  6.1.3 Decreased Field Star Formation

The final explanation for the environmental trend in GC abundance we will
discuss is that dwarfs in denser environments experience reduced field star
formation efficiencies due to earlier quenching. Closely linked to this is the
idea that GC systems constitute a constant mass fraction of the total (i.e.
halo) mass of a galaxy (e.g. Blakeslee et al., 1997; Spitler & Forbes, 2009;
Georgiev et al., 2010; Hudson et al., 2014; Harris et al., 2017). Thus, at a
given halo mass, if the dwarfs in denser environments form fewer stars due to,
for example, earlier or more efficient quenching than dwarfs in less dense
environments, then they could have more abundant GC systems at fixed stellar
mass. We note, however, there is significant uncertainty in whether the
linearity between GC system mass and halo mass is applicable to the mass range
($M_{\mathrm{halo}}\lesssim 10^{11}$M⊙) considered in this work (Forbes et
al., 2018a; Bastian et al., 2020).

With that caveat said, essentially this comes down to the question of whether
the stellar-to-halo mass relation (SHMR) is different for dwarfs in different
environments. One of the best constraints available on the SHMR is simply the
observed abundances of satellites of different stellar masses. Grossauer et
al. (2015) used the NGVS dwarf catalog to infer the SHMR applicable to the
Virgo core. In Carlsten et al. (2020b), we showed that the SHMR of Garrison-
Kimmel et al. (2017) was able to well reproduce the observed abundance of LV
satellite systems. Interestingly, the SHMR inferred by Grossauer et al. (2015)
is quite different than that of Garrison-Kimmel et al. (2017), predicting
lower stellar mass at fixed halo mass by nearly an order of magnitude. More
work needs to be done to gauge the severity of the difference. Taking it at
face value, however, it would indicate that Virgo dwarfs on average do form
fewer stars at fixed halo mass and, hence, might explain why they have more
GCs at fixed stellar mass. However, we do note that the SHMR is notoriously
unconstrained at these mass scales with different observables and simulation
projects giving quite disparate constraints.

One interesting feature from Figure 14 is that the $N_{\mathrm{GC}}-M_{\star}$
relation has the same slope between the Virgo and LV samples. Thus, if the
difference in abundance is due to a constant
$M_{\mathrm{GC}}/M_{\mathrm{halo}}$ fraction and different SHMRs in the two
environments, then the SHMRs must also have the same slope at the low-mass
end, with just an offset between them. This would seem to require some fine-
tuning, and we note that the SHMRs of Grossauer et al. (2015) and Garrison-
Kimmel et al. (2017) do not have the same slope. It is unclear how significant
scatter in the SHMR111111And even different amounts of scatter in dwarfs in
Virgo versus the LV environments could affect this, but a detailed exploration
would have to take this into account.

This proposed mechanism would imply that early-type dwarfs in clusters would
be, on average, older than in MW-sized halos. The current observations are
unclear on this. The age distribution of cluster dEs appears to be quite a
mixed bag (e.g. Koleva et al., 2009), as does the age distribution of LV
satellite dwarfs (e.g. Weisz et al., 2014, 2019). This is further complicated
by the fact that even if cluster dwarfs did on average quench earlier, they
might build up stellar mass more rapidly before quenching due to, for example,
increased gas availability in a denser environment (Joshi et al., 2021).

In closing this section, we note that the higher GC abundance at fixed stellar
mass in clusters cannot be due to increased stripping of stars in the cluster
environment. GCs have at least as extended spatial distributions as the stars,
and the GCs would be stripped first (e.g. Smith et al., 2015), leading to
lower GC abundance at fixed stellar mass in cluster dwarfs.

####  6.1.4 Summary

We have discussed three possible causes for the increased GC abundance in
denser environments: (1) preferential formation of GCs in early collapsing
subhalos (increased GC production), (2) shorter duration of an early, rapid-
disruption phase in denser environments due to increased merger rates or
earlier quenching (decreased GC destruction), and (3) lower efficiency of
forming field stars in cluster dwarfs due to earlier quenching (decreased
field star formation). Various forms of these ideas have been proposed to
explain the original observations of Peng et al. (2008). We confirm the
environmental dependence found in that work and show that it extends across
environments spanning several orders of magnitude in parent halo mass.
Although, due to the different radial extent of the Virgo and LV samples, it
is not completely clear whether this dependence is primarily an effect of the
parent halo mass or due to the LV dwarfs being, on average, further out in
their hosts’ virial volumes.

Likely all three of these mechanisms are, to varying degrees, responsible for
the increased GC abundance in denser environments. To figure out which
mechanism(s) is dominant, quantitative models would be required. When
discussing the increased GC production vs decreased disruption, it is possible
that the similarity in GCLF across environments could help elucidate which
process dominates. In-depth photometric comparisons between cluster dwarfs and
LV dwarfs could reveal interesting differences in average stellar population
properties, perhaps illuminating option (3). Additionally, analyses of cluster
dwarf samples that are complete to a large fraction of the cluster virial
radius would elucidate whether the environmental dependence we observe is
primarily an effect of parent halo mass or the dwarfs’ location in the parent
halo. This could further clarify which of these mechanisms is most important
in setting the GC abundance.

###  6.2 Connection Between NSCs and GCs

In this section, we discuss the NSC and GC observations presented in this
paper in the context of whether a physical connection exists between these two
types of clusters. In this work, we showed that the presence of both GCs and
NSCs exhibit a significant secondary dependence with host environment, highly
suggestive of a connection between GCs and NSCs. Here, we assert that these
observations are naturally explained if the inspiral of GCs is the dominant
formation pathway of NSCs in dwarfs of this mass range.

As discussed in the introduction, proposed formation mechanisms of NSCs are
broadly classed into GC-inspiral and in-situ star formation. Support for the
GC origin of NSCs comes from the general similarity between NSCs and GCs,
including the fact that NSCs almost always have stellar mass above that of a
single GC. Both analytic calculations and simulation results have shown that
massive GCs starting within a few kpc of the galaxy center can feasibly
inspiral within a Hubble time. Additionally, this method has strong support
from the fact that many NSCs in dwarf ($M_{\star}<10^{9}$ M⊙) galaxies have
lower metallicity than the surrounding host galaxy (Paudel et al., 2011;
Spengler et al., 2017; Neumayer et al., 2020; Johnston et al., 2020; Fahrion
et al., 2020, 2021), similar to old, metal-poor GCs which are generally found
to be more metal-poor than the host dwarf galaxy (e.g. Larsen et al., 2012;
Fensch et al., 2019). Finally, some have argued that the flattened radial
distribution of surviving GCs in dEs compared to the stellar distribution is
indicative that some of the GCs have decayed into the center and merged (Lotz
et al., 2001).

On the other hand, it is possible that NSCs are created through central star-
formation. Gas can be diverted to the centers of galaxies through mergers and
other ways (e.g. Mihos & Hernquist, 1994; Milosavljević, 2004; Bekki et al.,
2006). Support for this mechanism comes largely from the presence of young
stars in many NSCs which is otherwise hard to explain in the GC-inspiral
paradigm (Walcher et al., 2006; Rossa et al., 2006; Monaco et al., 2009; Seth
et al., 2010; Paudel et al., 2011; Carson et al., 2015; Spengler et al., 2017;
Johnston et al., 2020). The significantly flattened NSCs observed in galaxies
of relatively high stellar mass (Seth et al., 2006; Spengler et al., 2017)
also points to in-situ star formation.

As stated in the introduction, there is a growing consensus that the GC-
inspiral mechanism dominates at low ($M_{\star}<10^{9}$M⊙) masses while in-
situ star formation dominates at higher masses (Turner et al., 2012; Sánchez-
Janssen et al., 2019; Neumayer et al., 2020; Fahrion et al., 2021). This is
particularly true if we separately consider how NSCs form versus how they grow
(Neumayer et al., 2020). NSCs in dwarf galaxies almost certainly started as
GC-like clusters. This would explain the absence of NSCs of lower mass than
typical GCs and their similarities in size and density to GCs. As we show in
Figure 3, we are complete to nuclei down to $M_{g}\sim-5$ mag for many of the
LV galaxies, yet the observed NSC luminosity function drops off steeply
fainter than $M_{g}\sim-7$ mag.

After forming, the NSCs could have grown either through merging of other GCs
or through in-situ star formation. The NSCs with lower metallicity than the
dwarf hosting them likely just grew via GC merging while NSCs with young
stellar populations must have had some in-situ star formation. As a dwarf is
being quenched via ram pressure stripping either in the cluster environment or
in the halo of a MW-like host, the central regions would be the last to
quench, as evinced by the blue cores of many dEs (e.g. Urich et al., 2017;
Hamraz et al., 2019). Thus, it is not surprising that many NSCs have a
subpopulation of young stars. This idea is exemplified elegantly in the
stellar populations of M54, the Sagittarius dSph’s NSC (Siegel et al., 2007;
Alfaro-Cuello et al., 2019). A young metal-rich (age $\sim 2$ Gyr, [Fe/H]$\sim
0$) population is superimposed on an old metal-poor population (age $\sim 12$
Gyr, [Fe/H]$\sim-1.5$). The old stars likely originated as one or more
globular clusters while the young population was believably formed when Sgr
was stripped of its gas as it was accreted by the MW.

The results of the current study fit well into this picture. We confirm that
there is a strong correspondence between GCs and NSCs in low mass early-type
dwarfs. We show one example of this in Figure 15 which directly compares the
NSC occupation fraction (i.e. nucleation fraction) from Figure 5 with the GC
occupation fraction from Figure 12. Broadly speaking, there are clear
similarities in the occupation fraction between the two types of clusters with
both types of clusters going from occupation fractions of essentially zero at
$M_{\star}<10^{6}$ M⊙ to $\sim 60-80$% at $M_{\star}=10^{8.5}$ M⊙. This is
similar to the result found by Sánchez-Janssen et al. (2019) for their
analysis of the NGVS sample. The errorbars on the inferred GC occupation
fraction are quite large so it is hard to compare the GC curves to the NSC
curves in detail, but it does appear that the GC occupation fractions for both
environments are less discrepant than the nucleation fractions, suggestive
that there is more to forming a nucleus than simply the presence of GCs. We
note, however, that Sánchez-Janssen et al. (2019) found the NGVS GC occupation
fraction to more closely track the trend in nucleation fraction. It is unclear
if this is due to that team using more than just two bands in analyzing the
NGVS sample or a difference in methodology.

Figure 15: The occupation fraction of GCs and NSCs as a function of dwarf
stellar mass. The NSC occupation fraction is the nucleation fraction from
Figure 5 while the GC occupation fraction comes from the joint fit shown in
Figure 12. There are clear similarities between the occupation rates of the
different types of star cluster. The difference in GC occupation between the
two environments is less significant than the difference in NSC occupation.

Additionally, we confirm previously found trends that both the NSC frequency
and GC abundance show a significant environmental dependence, indicating a
strong connection between the two types of clusters. In the context of a
picture where NSCs form primarily through GC inspiral, the decreased NSC
occupation fraction at fixed stellar mass in LV hosts is because these dwarfs
have fewer GCs which can act as the seed for NSC formation. In Figure 4, we
show evidence that NSCs in LV-like environments are lower mass at fixed host
stellar mass. This is consistent with NSC growth being due to GC merging since
the LV dwarfs would have fewer GCs to fuel this growth.

The current study also highlights some interesting open questions in the GC-
NSC connection. Our results suggest that the GC radial profiles are more
extended than the stellar distributions, in line with numerous previous
findings. As we argue above, it seems very unlikely that the GCs formed in
their current locations. Candidate mechanisms for the relocation of GCs
outward include mergers at high redshift and stellar feedback. Clearly, to
understand the formation of NSCs through GC-inspiral and quantitatively test
that model, we must know the initial galactocentric radii of GCs and
understand how they got there. For instance, it is unclear if NSCs formed
before or after GCs were relocated outwards. Similarly, did the NSCs in early-
type dwarfs form before or after the galaxy was quenched (e.g. Guillard et
al., 2016)? Understanding the radial distribution of GCs would likely also be
a prerequisite to understanding why there is roughly equipartition in mass
between NSCs and the surviving GC system for nucleated dwarfs in Virgo (Côté
et al., 2006; Sánchez-Janssen et al., 2019). Getting IFU spectra that allows
us to measure and compare the stellar populations of the NSC, GC system, and
host galaxy (e.g. Fahrion et al., 2019; Johnston et al., 2020) will be
critical in elucidating the GC-NSC connection.

Finally, we note that an interesting avenue to explore in the GC-NSC
connection is the difference of GC-inspiral based NSC formation in different
dark matter profiles (e.g. Leaman et al., 2020). The fact that the Fornax dSph
does not have a NSC even though it has several GCs has led to a contentious
debate about whether it can be hosted in a cuspy profile (e.g. Goerdt et al.,
2006; Meadows et al., 2020; Shao et al., 2020). Exploring this with the large
samples of dwarfs presented here and elsewhere would be an interesting future
direction.

##  7 Conclusions

In this work, we have systematically presented the NSC and GC properties of
early-type dwarfs in the mass range $10^{5.5}<M_{\star}<10^{8.5}$M⊙ in low-
density environments. In particular, we consider a sample of satellite
galaxies of roughly MW-mass hosts in the Local Volume from the ongoing ELVES
Survey. The goal of the survey is to catalog all satellites down to
$M_{V}\sim-9$ and surface brightness of $\mu_{0,V}\lesssim 26.5$ mag arcsec-2
for all massive hosts ($M_{K}<-22.4$ mag) in the Local Volume. While previous
surveys have characterized the GCs and NSCs of early-type dwarfs in clusters
or field late-type dwarfs, the sample we consider here is by far the largest
sample to-date of early-type dwarf satellites in lower-mass parent halos.
Throughout this paper, we analyze data from the Next Gen Virgo Survey (NGVS)
in the same way as for the LV sample, providing a direct comparison between
two nearly volume limited samples of early-type dwarfs in drastically
different environments: the halos of MW-mass galaxies versus the core of the
Virgo cluster.

To briefly describe our methodology, we search for candidate NSCs in our
galaxies through a combination of visual inspection and photometry. Candidate
GCs are identified as point sources in the vicinity of the dwarf. We deal with
contamination from background galaxies and foreground MW stars among the GC
candidates in two main ways: (1) a simple background subtraction and (2) a
likelihood-based joint fit for the average GC system properties across all
dwarfs. Overall we find the two methods to be in agreement.

The main results regarding the nuclei are as follows:

  * •

We find the nucleation fraction of dwarf satellites of LV hosts is
significantly lower than that of dwarfs in clusters (Figure 5), confirming
that there is a noticeable secondary dependence of nucleation on environment,
beyond the primary dependence on stellar mass.

  * •

The NSCs are located within a projected radius of $r\lesssim r_{e}/8$ of the
galaxy center, corresponding to $\lesssim 75$ pc for the average dwarf in the
sample.

  * •

The NSC luminosities show a steep dropoff fainter than $M_{g}\gtrsim-7$ mag
(Figure 3) even though we are complete to point sources down to $M_{g}\sim-5$
mag in many of the LV dwarfs.

  * •

The LV and Virgo samples of NSCs follow a similar relation with dwarf stellar
mass of approximately $M_{\mathrm{NSC}}\propto M_{\mathrm{gal}}^{0.4}$ (Figure
4). The LV NSCs appear to be lower mass at fixed host mass, at the $\sim
1-2\sigma$ level.

The main GC results are as follows:

  * •

Fitting a Gaussian to the GCLF, we find a peak of $M_{g}^{0}\sim-7.1$ mag and
dispersion $\sigma_{g}\sim 0.6$ mag with no discernible difference between the
LV and Virgo samples (Figure 7). The GCLF exhibits a noticeable, non-Gaussian
tail to fainter luminosities, similar to that found for LG dSphs and in
previous GC studies of LV dwarfs with _HST_.

  * •

We do not have the statistics to robustly constrain the radial distribution of
GCs in either environmental sample, but GC systems more extended than the
galaxy light profiles are favored in both samples, particularly among the GC-
rich dwarfs, in agreement with numerous previous findings (Figure 9).

  * •

We find that Virgo has a significantly higher fraction of dwarfs with rich GC
systems. The fraction of dwarfs with $N_{\mathrm{GC}}>10$ is twice as high in
Virgo as in the LV sample (Figure 11).

  * •

From the joint fit, we find that Virgo dwarfs have, on average, higher GC
abundance than the LV dwarfs. This appears to be primarily due to a higher
average GC abundance for dwarfs with GCs, and less due to higher GC occupation
fraction (Figure 12).

  * •

Our results for the Virgo sample agree well with previous results for higher
mass dwarfs in the Virgo core (Figure 14). Our inferred abundances are higher
than previous work considering dwarf samples from throughout the Virgo
cluster, confirming that the local environment of a dwarf in the cluster
impacts its GC abundance (e.g. Peng et al., 2008).

  * •

The GC occupation fraction as a function of dwarf stellar mass broadly follows
the nucleation fraction, but the difference between environmental samples
appears to be less than that for the nucleation rates (Figure 15).

The main result of this paper is that we confirm that environment plays a
strong role in the star cluster properties (primarily their abundance) of low-
mass galaxies. These results strengthen the connection between NSCs and GCs in
low-mass dwarfs. It is very likely that the nuclei of dwarfs in the mass range
studied here form from a GC that spiralled into the galaxy center with later
grow either through further GC inspiral or in-situ star formation. We identify
three possible explanations for the greater cluster abundance in dense
environments often discussed in the literature:

  1. 1.

Increased GC Production: GCs are thought to form in the regions of a galaxy’s
ISM with the highest pressures and densities. Thus, galaxies with high star
formation rate densities are expected to form higher fractions of their stars
in bound clusters. Dwarfs in the cores of galaxy clusters will form in earlier
collapsing subhalos, on average, experiencing higher star formation rate
densities, possibly explaining their higher star cluster abundance (e.g. Peng
et al., 2008; Mistani et al., 2016).

  2. 2.

Decreased GC Destruction: GCs are known to experience significant tidal
processing, with many lower mass clusters being disrupted before the present
day. Of particular importance to GC destruction is tidal shocks from giant
molecular clouds. Dwarfs in denser environments, like cluster cores, would
likely experience earlier quenching or higher rates of merging, effectively
‘saving’ the GCs from destruction (e.g. Kruijssen, 2015).

  3. 3.

Decreased Field Star Formation: Due to dwarfs in cluster cores likely
experiencing earlier quenching, it is possible that the cluster dwarfs have
assembled less stellar mass overall. This would suggest that dwarfs in denser
environments at a certain stellar mass would be hosted by more massive
subhalos, on average, which could lead to higher GC abundances.

It is likely that multiple, or even all, of these effects are at work.
Exploring which physical process dominates, if any, through quantitative
models will be important to explore in future work.

On the observational front, it will be interesting to further explore these
effects with dwarf samples from galaxy clusters that are complete to the
cluster virial radius. Due to the fact that the LV dwarfs are, on average,
further out from their host ($r\sim R_{\mathrm{vir}}/2$) than the cluster
dwarfs ($r\lesssim R_{\mathrm{vir}}/4$), it is not clear whether the
environmental effects we observe are primarily due to the parent halo mass or
more due to the dwarf satellite’s current location in the parent halo.
Addressing this with dwarf samples from clusters that extend to the cluster
virial radius or with an analysis that quantifies the local environment of a
dwarf (e.g. a metric that measures the local tidal field) could greatly help
to illuminate the important physical processes involved in star cluster
formation in dwarf galaxies.

## Acknowledgements

Support for this work was provided by NASA through Hubble Fellowship grant
#51386.01 awarded to R.L.B. by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS 5-26555. J.P.G. is supported by an NSF Astronomy
and Astrophysics Postdoctoral Fellowship under award AST-1801921. J.E.G. is
partially supported by the National Science Foundation grant AST-1713828.
S.G.C acknowledges support by the National Science Foundation Graduate
Research Fellowship Program under Grant No. #DGE-1656466.

Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT
and CEA/IRFU, at the Canada-France-Hawaii Telescope (CFHT) which is operated
by the National Research Council (NRC) of Canada, the Institut National des
Science de l’Univers of the Centre National de la Recherche Scientifique
(CNRS) of France, and the University of Hawaii. This research was made
possible through the use of the AAVSO Photometric All-Sky Survey (APASS)
(Henden & Munari, 2014), funded by the Robert Martin Ayers Sciences Fund and
NSF AST-1412587

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