# The nucleation fraction of Local Volume galaxies

Nils Hoyer1,2, Nadine Neumayer1, Iskren Y. Georgiev1, Anil C. Seth3, and Jenny
E. Greene4
1Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg,
Germany
2Ruprecht-Karls-Universität Heidelberg, Seminarstraße 2, D-69117 Heidelberg,
Germany
3Department of Physics and Astronomy, University of Utah, 115 South 1400 East,
Salt Lake City, UT 84112, USA
4Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544, USA
Contact e-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Nuclear star clusters (NSCs) are a common phenomenon in galaxy centres and are
found in a vast majority of galaxies of intermediate stellar mass
$\approx{10}^{9}\text{\,}\mathrm{M_{\odot}}$. Recent investigations suggest
that they are rarely found in the least and most massive galaxies and that the
nucleation fraction increases in dense environments. It is unclear whether
this trend holds true for field galaxies due to the limited data currently
available. Here we present our results on the nucleation fraction for $601$
galaxies in the Local Volume ($\lesssim$12\text{\,}\mathrm{Mpc}$$). Covering
more than eight orders of magnitude in stellar mass, this is the largest
sample of galaxies analysed in a low-density environment. Within the Local
Volume sample we find a strong dependence of the nucleation fraction on galaxy
stellar mass, in agreement with previous work. We also find that for galaxies
with $M_{\star}<${10}^{9}\text{\,}\mathrm{M_{\odot}}$$, early-type galaxies
have a higher nucleation fraction than late-types. The nucleation fraction in
the Local Volume correlates independently with stellar mass, Hubble type, and
local environmental density. We compare our data to those in galaxy cluster
environments (Coma, Fornax, and Virgo) by compiling previous results and
calculating stellar masses in a homogeneous way. We find significantly lower
nucleation fractions (up to $40\text{\,}\mathrm{\char 37\relax}$) in galaxies
with $M_{\star}\lesssim$e9.5\text{\,}\mathrm{M_{\odot}}$$, in agreement with
previous work. Our results reinforce the connection between globular clusters
and NSCs, but it remains unclear if it can explain the observed trends with
Hubble type and local environment. We speculate that correlation between the
nucleation fraction and cluster environment weakens for the densest clusters
like Coma and Virgo.

###### keywords:

galaxies: general – galaxies: nuclei – galaxies: star clusters: general –
galaxies: clusters: general – galaxies: groups: general

††pubyear: 2021††pagerange: The nucleation fraction of Local Volume galaxies–D

##  1 Introduction

Nuclear star clusters (NSCs) are a common phenomenon in the centres of
galaxies (e.g. Phillips et al., 1996; Böker et al., 2002; Scarlata et al.,
2004; Seth et al., 2006; Georgiev et al., 2009b; Neumayer et al., 2011;
Georgiev & Böker, 2014), but are not always present. The nucleation fraction
of galaxies (denoted as $f_{\mathrm{n}}$) measures the fraction of galaxies
with an NSC. It increases as a function of galaxy stellar mass (denoted as
$M_{\star}$), but seems to hit a peak at
$\approx{10}^{9}\text{\,}\mathrm{M_{\odot}}$ (den Brok et al., 2014; Muñoz et
al., 2015; Sánchez-Janssen et al., 2019; Neumayer et al., 2020).

Early studies considered $f_{\mathrm{n}}$ in samples of galaxies over a
limited range in Hubble type and / or stellar mass in both field (e.g.
Balcells et al., 2007; Baldassare et al., 2014; Carollo et al., 1997, 1998;
Carollo et al., 2002; Lauer et al., 2005) and cluster (Côté et al., 2006; den
Brok et al., 2014; Muñoz et al., 2015; Sánchez-Janssen et al., 2019)
environments. Recently, Sánchez-Janssen et al. (2019, hereafter SJ+19)
investigated NSCs in the core of the Virgo galaxy cluster and combined their
results with ancillary data for high-mass galaxies from Côté et al. (2006). To
compare their results, they added the data of den Brok et al. (2014) for the
Coma and Muñoz et al. (2015) for the Fornax galaxy clusters. They found that
$f_{\mathrm{n}}$ coincides between the Fornax and Virgo galaxy clusters and
that it is elevated for the Coma galaxy cluster, and reason that this
difference is due to differences in host halo mass. Further evidence for the
role of the environment in determining $f_{\mathrm{n}}$ comes from studies
that show that nucleated galaxies are preferentially located at the centres of
galaxy clusters (e.g. Binggeli et al., 1987; Ferguson & Sandage, 1989; Lisker
et al., 2007; Lim et al., 2018; Ordenes-Briceño et al., 2018).

To compare $f_{\mathrm{n}}$ to a low-density environment, SJ+19 visually
inspected and assigned a nuclear classification to $55$ low-mass satellite
galaxies of the Milky Way, M 31, and M 81 using high-resolution Hubble Space
Telescope (HST) data. The resulting $f_{\mathrm{n}}$ is smaller than for all
cluster environments, but the uncertainties on their data points are large and
still compatible with the values for both the Fornax and Virgo galaxy clusters
(see their Figure $2$). At the high-mass end
($M_{\star}\geq${10}^{9}\text{\,}\mathrm{M_{\odot}}$$) Baldassare et al.
(2014) analysed the central regions of $28$ galaxies and report a consistency
of $f_{\mathrm{n}}$ between early-type field galaxies and Virgo galaxy cluster
members of similar stellar mass. However, given the limited sample sizes and
coverage in stellar mass and Hubble type, the significance of the offset
between $f_{\mathrm{n}}$ for field and cluster environments remained unclear.

Carlsten et al. (2021a) investigated the globular cluster and NSC populations
of $177$ low-mass early-type galaxies
($M_{\star}\lesssim$e8.5\text{\,}\mathrm{M_{\odot}}$$) around $27$ massive
hosts in the Local Volume. They confirmed the difference in $f_{\mathrm{n}}$
between the field and cluster environments at the low-mass end. Furthermore,
they were able to show that galaxies in close proximity to their host galaxy
show an elevated $f_{\mathrm{n}}$ compared to galaxies further away for all
stellar masses. This led the authors to question whether the parent halo mass
or distance from the centre of the halo is the stronger influence on the
nucleation fraction.

The Local Volume (LV; $d\lesssim$12\text{\,}\mathrm{Mpc}$$) is a natural place
to study NSC demographics since the sample of galaxies is nearly complete down
to the lowest galaxy masses (Karachentsev et al., 2013) and nuclei are at
least partially resolved (e.g. Pechetti et al., 2020). Despite being an ideal
laboratory for studying $f_{\mathrm{n}}$, thus far there have been no studies
of the nucleation fraction for the complete population of LV galaxies. In this
paper we present our analysis of $f_{\mathrm{n}}$ of galaxies in the LV. The
sample of galaxies and volume is large enough that it provides a good proxy
for the majority of galaxies that live in field and group environments. Based
on the most complete catalogue of galaxies in the LV available today, we both
collect literature nuclear classifications where they are available and
classify other galaxies ourselves based on high-resolution HST data. We use
self-consistent mass-to-light ratios from the literature to calculate galaxy
stellar masses for all environments in a homogeneous way. Our final sample
contains galaxies of all Hubble types over a wide range of stellar mass
($$e2.5\text{\,}\mathrm{M_{\odot}}$\lesssim
M_{\star}\lesssim$e11.5\text{\,}\mathrm{M_{\odot}}$$) and environments, from
isolated field galaxies to rich groups. We confirm that the stellar mass is
the primary indicator for the nucleation of a galaxy and show that the Hubble
type also correlates with $f_{\mathrm{n}}$. Likewise, we find a correlation
between $f_{\mathrm{n}}$ and the local environment of a galaxy. Furthermore,
we study $f_{\mathrm{n}}$ as a function of galactocentric distance to find
that it increases for the LV in the central regions, slightly increased for
the Fornax galaxy cluster, and decreases for the Virgo galaxy cluster. The
latter observation seems to disagree with the literature.

The paper is structured as follows: §2 introduces the galaxy catalogue of the
LV and briefly describes the nuclear classification scheme of its members
without a decisive nuclear classification. Here we also introduce the data
used to calculate $f_{\mathrm{n}}$ for galaxy clusters. In §3 we highlight the
mass determination scheme and show $f_{\mathrm{n}}$ for different environments
as a function of stellar mass, Hubble type, tidal index, and galactocentric
distance. In this section we also fit $f_{\mathrm{n}}$ with a logistic
function. Based on our analysis, we discuss potential consequences for NSC
formation and evolution theories in §4. Finally, we conclude in §5 and present
an outlook for further investigations.

##  2 Data

###  2.1 The Local Volume galaxy sample

One of the earliest catalogues of LV galaxies contained $179$ objects within
$d\lesssim$10\text{\,}\mathrm{Mpc}$$ (Kraan-Korteweg & Tammann, 1979). Over
the years, the number of objects steadily increased and reached $869$ in
$2013$ following the release of the ‘Updated Nearby Galaxy Catalog’ (UNGC;
Karachentsev et al., 2013, hereafter KMK+13). It is updated regularly111The
most recent version can be obtained via
https://www.sao.ru/lv/lvgdb/tables.php. and contains $1246$ objects as of late
February $2021$.

The UNGC is the most complete catalogue of the LV which is complete down to
$M_{B}=$-11\text{\,}\mathrm{m}\mathrm{a}\mathrm{g}$$. For fainter magnitudes
the estimated completeness was between $40\text{\,}\mathrm{\char 37\relax}$
and $60\text{\,}\mathrm{\char 37\relax}$ in $2013$ (see discussion in KMK+13),
where half of the ultra-faint dwarf companions around massive galaxies would
be missing. Indeed, recently many new ultra-faint dwarfs were discovered in
the LV (e.g. Carlsten et al., 2020b; Habas et al., 2020) which are currently
not part of the UNGC. Although these studies include a nuclear classification,
we do not add them to the UNGC as many distance estimates and apparent
magnitudes are missing. However, their results will be used to investigate
potential completeness biases in the UNGC (cf. §A.1).

KMK+13 constructed the UNGC by setting two limitations to potential members:
(1) the distance estimate of a galaxy must be smaller than
$11\text{\,}\mathrm{Mpc}$ or (2) the radial velocity component of its velocity
vector with respect to the Local Group centroid is $\leq
600\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. The latter choice was
motivated by the uncertainty on today’s value of the Hubble parameter and
perturbations by both the Local Void (Tully, 1988) and the nearby Virgo galaxy
cluster. As a result, some galaxies have reliable distance estimates larger
than $11\text{\,}\mathrm{Mpc}$ but are only included because of (2). Since we
use LV galaxies as a proxy for the field environment, we keep these galaxies
in the catalogue if they do not belong to any galaxy cluster.

We removed twelve objects from the UNGC because they are classified as
globular clusters of M 31 (e.g. Huxor et al., 2014; McConnachie et al., 2018).
An additional $13$ galaxies are removed because they contain a ‘Virgo Cluster
Catalog’ (Binggeli et al., 1985) identification number and are, therefore,
classified as Virgo galaxy cluster members. Finally, we also removed ‘Kim 2’
which has been identified as a globular cluster associated to the Milky Way
(Kim et al., 2015).

Consequentially, our updated LV catalogue contains $1220$ galaxies and is
presented in Table 3 (its full version is available online only). It contains
the main identifier, angular coordinates, a distance estimate, apparent
magnitudes in the $B$-, $g$-, $V$-, $r$-, and $K$-bands corrected for Galactic
extinction, $(B-V)_{0}$ and $(g-r)_{0}$ colours, $E(B-V)$, the Hubble type,
three tidal indices and a list of the ten most influential neighbouring
galaxies (cf. §3.4), a mass estimate (cf. §3.1), and a nuclear classification
with literature references, if available.

###  2.2 Galaxy parameters

The UNGC contains some basic galaxy properties including physical coordinates,
a Hubble type value (denoted as $T$), and apparent magnitudes in the $B$\- and
$K_{\mathrm{s}}$-bands. We use the HyperLEDA222http://leda.univ-lyon1.fr/ and
SIMBAD333https://simbad.harvard.edu/simbad/ data bases to supplement the
information from the UNGC for the following reasons:

  1. $\bullet$

to calculate stellar masses by adding magnitudes in the Johnson-Cousins and
SDSS filter bands ($B$, $g$, $V$, $r$, and $K$);

  2. $\bullet$

to provide uncertainties on apparent magnitudes and the Hubble type, which are
currently lacking;

  3. $\bullet$

to update $B$-band magnitudes for galaxies for which those value was estimated
by eye from a comparison to a ‘similar looking’ galaxy (cf. KMK+13);

  4. $\bullet$

to improve the sampling and range of the Hubble Type, currently limited to
$T\in[-3,\,10]$.

We search both data bases using the angular coordinates from the UNGC and
apply a search box and diameter of $10\text{\,}\arcsec$, respectively. The
angular coordinates of five galaxies (Draco, Fornax, ESO 174-1, NGC 55, NGC
5194) differ by more than $100\text{\,}\arcsec$ between the UNGC and the
HyperLEDA data base. For Draco, Fornax, and NGC 55 this difference is due to
their large spatial extent. For NGC 5194 it seems likely that the angular
coordinates of M 51 were used in the UNGC. We are unable to explain the
difference for ESO 174-1. For all five galaxies we adopt the angular
coordinates presented in the HyperLEDA data base.

Parameters from both online data bases and the UNGC are combined in the
following way:

  1. $\bullet$

Main identifier: the main identifier is taken from the UNGC.

  2. $\bullet$

Distance: we adopt the value given in the UNGC if it is reliable, i.e. based
on the TRGB, the luminosity of Cepheids or RR Lyrae stars, or surface
brightness fluctuations. Otherwise, we adopt the modbest value444The modbest
parameter is a weighted average of mod0 taken from a distance catalogue and
the luminosity distance inferred from the redshift. from HyperLEDA and, if
unavailable, adopt the value from SIMBAD. Finally, if no distance estimate is
available in either online data base, we use the unreliable value in the UNGC.

  3. $\bullet$

Magnitudes: From HyperLEDA we adopt total apparent magnitudes in the $B$-,
$V$-, and $K$-bands. From SIMBAD we take the same Johnson-Cousins magnitudes
and additional SDSS magnitudes in the $g$\- and $r$\- bands, if
available555Most of the $g$\- and $r$-band magnitudes stem from the SDSS DR7
(Abazajian et al., 2009) and those are model-fit galaxy magnitudes. See
http://classic.sdss.org/dr7/algorithms/photometry.html for details.. The
addition of magnitudes to the UNGC values is similar as for the distance
values: first we consider HyperLEDA, then SIMBAD, and finally the values
presented in the UNGC.

From both online data bases we include apparent magnitudes and a distance
estimate. In addition, we search for the corrected asymptotic $(B-V)_{0}$
colour in the HyperLEDA data base (catalogued as bvtc) as it is corrected for
internal extinction. Apparent magnitudes are corrected for foreground Galactic
extinction based on the Schlafly & Finkbeiner (2011) re-calibration of the
Schlegel et al. (1998) dust maps, assuming the Fitzpatrick (1999) reddening
law of $R_{V}=3.1$. Our final $(B-V)_{0}$ magnitudes are calculated as
follows: (1) where available, we use the bvtc parameter from HyperLEDA. (2)
where bvtc is unavailable, we calculate $B-V$ using apparent magnitudes from
HyperLEDA, SIMBAD, or the UNGC (see above) and correct the colour for Galactic
extinction. Similarly, the final $(g-r)_{0}$ is calculated using the apparent
magnitudes from SIMBAD and correcting for Galactic extinction. All final
apparent magnitudes and $E(B-V)$ are given in Table 3.

We show the $K$-band luminosity as a function of optical $(B-V)_{0}$ colour in
Figure 1 where galaxies are already split based on their nuclear
classification (cf. §2.3.1).

Figure 1: Logarithmic $K$-band luminosity as a function of $(B-V)_{0}$ colour.
Galaxies are colour-coded based on their nuclear classification into nucleated
(blue), non-nucleated (orange) and unknown (gray). Uncertainties are omitted
for clarity. The right and bottom panels show the Gaussian kernel density
estimates along the projected axes. As indicated by the numbers in the legend,
only part of the Local Volume sample has a $(B-V)_{0}$ colour estimate.

For $842$ galaxies ($\approx 67.6\text{\,}\mathrm{\char 37\relax}$) a direct
measurement of the $K$-band magnitude of a galaxy is unavailable. The $K$-band
magnitude is a good proxy to estimate galaxy stellar masses if optical
$(B-V)_{0}$ or $(g-r)_{0}$ colours are unavailable. Therefore, we calculate
missing apparent $K$-band magnitudes $m_{K}$ by using the apparent $B$-band
magnitude $m_{B}$ and the Hubble type value $T$ via

$\langle m_{B}-m_{K}\rangle_{0}=\begin{cases}$4.10$&\mathrm{for}\hskip
4.30554ptT<$3$\,,\\\ $2.35$&\mathrm{for}\hskip 4.30554ptT>$8$\,,\\\
$4.60$-$0.25$\times T&\mathrm{otherwise}\,,\end{cases}$ (1)

which is taken from the 2MASS Large Galaxy Atlas (Jarrett et al., 2003) and
was already used by KMK+13 for the UNGC666Due to the updated $m_{B}$ and $T$
values, we do not adopt $m_{K}$ from the UNGC, if calculated via Equation 1..
The relation is based on the observation that $B-K$ depends on Hubble type
ranging between $\approx 4$ and $\approx 2$ for early- and late-type galaxies,
respectively (Figure 19 of Jarrett et al., 2003). Note that the scatter of
$B-K$ increases with Hubble type to $\approx 2$ for $T=9$, resulting in large
uncertainties.

The uncertainties for the galaxy parameters in our catalogue are obtained as
follows: depending on the filter band, up to one third of all galaxies have no
uncertainty tabulated on their apparent magnitude. For those cases we adopt an
uncertainty of $0.5\text{\,}\mathrm{mag}$ which equals the value adopted by
KMK+13 for their ‘visually’ derived $B$-band magnitudes. In addition, we
consider uncertainties on $g$\- and $r$-band magnitudes in SIMBAD below
$0.1\text{\,}\mathrm{mag}$ to be unreliable777For high-mass galaxies the
uncertainty on the apparent magnitude often is $\delta
m=$0.001\text{\,}\mathrm{mag}$$ which likely does not take into account
systematic contributions. and adopt a value of $0.1\text{\,}\mathrm{mag}$. For
galaxies without an uncertainty on their Hubble type ($358$ galaxies) we
assume an uncertainty of $1.0$ which seems to be a typical assumption made by
other works like KMK+13 and Jarrett et al. (2003).

To compare the nucleation fraction as a function of stellar mass for different
environments, the galaxy parameters given in the reference papers (see next
sections) are insufficient. For both the Fornax and Virgo galaxy clusters we
consider all galaxies flagged as members by the HyperLEDA data base ($337$ and
$2093$ galaxies, respectively). For the Coma galaxy cluster we use the data
set of den Brok et al. (2014). We search the HyperLEDA and SIMBAD data bases
for apparent magnitudes and Hubble types in the same way as for LV galaxies,
including an estimate of their $K$-band magnitudes via Equation 1, if missing.
For the NGVS sample of SJ+19, we use the absolute $g$\- and $r$\- magnitudes
from their Table 4. We assume the following distances if no measurement is
available: $d_{\mathrm{Coma}}=$35.10\pm 0.55\text{\,}\mathrm{mag}$$ (Ferrarese
et al., 2000), $d_{\mathrm{Fornax}}=$31.51\pm 0.15\text{\,}\mathrm{mag}$$
(Blakeslee et al., 2009), and $d_{\mathrm{Virgo}}=$31.09\pm
0.01\text{\,}\mathrm{mag}$$ (Mei et al., 2007).

###  2.3 Galaxy nuclear classification

####  2.3.1 Local Volume

Roughly a third of the galaxies (451) in the predecessor of the UNGC (called
‘Catalog of Neighboring Galaxies’; Karachentsev et al., 2004) have been
broadly classified as containing a ‘star-like nucleus’ (Karachentsev &
Karachentseva, 2002). The large amount of recent literature and available
high-resolution imaging data in the HST archive prompted us to make a
comprehensive update to the galaxy nucleation classification in the LV as
follows: we performed an ADS888https://ui.adsabs.harvard.edu/ search based on
galaxy (object) name taken from the UNGC and HyperLEDA. The abstract keyword
search contained ‘nuclear star cluster’, ‘NSC’, ‘nuclear cluster’, and
‘nucleus’. To mostly include reliable classifications which are primarily
based on HST data, we limited the publication date to be more recent than
January $1990$. For $310$ galaxies we detected literature sources which are
given in Table 3.

We search through the Hubble Legacy Archive999https://hla.stsci.edu (HLA) to
classify galaxies whose literature classification is ambiguous or non-
existent. Given the small spatial extent of NSCs (effective radius
$\lesssim$50\text{\,}\mathrm{pc}$$, Neumayer et al., 2020, e.g.), we limit our
search to the high-resolution HST cameras ACS, WFC3, and WFPC2, and search
within a radius of $36\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ around the
galaxy coordinates in our catalogue. No restrictions are set on the filter,
but we focused our analysis on red / NIR filters to minimise effects due to
dust extinction. This eases the confirmation of extended sources and thus
allows for a reliable nuclear classification. Including the previously removed
globular clusters of M 31 and Virgo galaxy cluster members, we find available
HLA data for $633$ objects ($\approx$50.8\text{\,}\mathrm{\char 37\relax}$$)
in at least one filter. With the above constraints, no HST data is found for
the remaining $613$ ($\approx$49.2\text{\,}\mathrm{\char 37\relax}$$)
galaxies.

Using the available HLA data, we classify galaxies with ambiguous or without a
nuclear classification into nucleated (‘1’), non-nucleated (‘0’), or ‘remains
unknown’ (‘?’). The latter category is adopted if all available data is either
significantly affected by dust extinction or if a galaxy centre is not within
the HST camera field of view. We do _not_ re-analyse galaxies previously
classified as nucleated, but we did look at all galaxies without an NSC and
with available HST data. The classification process includes a visual
inspection of all available HST data, a determination of the position of the
potential NSC with respect to the photometric centre of the host galaxy at
different isophotal radii, and a verification that the potential NSC is
extended by using an oversampled PSF and two-dimensional fitting techniques.
This scheme is comparable to the one described in Georgiev & Böker (2014) and
will be detailed further in an upcoming paper (Hoyer et al. in prep.). In that
paper we will also discuss the properties of the newly discovered NSCs in the
LV while this paper solely focuses on the nucleation fraction.

In total, using the HLA data for 633 objects we identify 22 new detections of
NSCs. In combination with the compilation of literature classifications, this
results in $145$ nucleated and $456$ non-nucleated galaxies. Currently, $46$
galaxies with available HST data are part of the ‘remains unknown’ category
(‘?’ in the data table) where no nuclear classification could be assigned. All
galaxies and their parameters used in this work are given in Table 3. Their
nuclear classifications are grouped into (‘1’), (‘0’), and (‘?’). For
simplicity, the latter category combines galaxies with insufficient and
without available HST data (the ‘unknown’ distribution of galaxies in Figure
1).

The nuclear classification of galaxy cluster members from the literature
sources is described but _not_ verified in the following sections.

####  2.3.2 Coma galaxy cluster

As part of the Advanced Camera for Surveys Coma Cluster Survey (ACSCCS), den
Brok et al. (2014) used HST ACS F814W images and PSFs created with TinyTim
(Krist, 1993, 1995) to assign a nuclear classification to their data. If a fit
with a central Gaussian source yielded a higher evidence than a fit without a
central source, the galaxy was classified as nucleated. Therefore, we classify
all galaxies as nucleated if an apparent magnitude for the central source of a
galaxy is available in their Table A1. If no apparent magnitude is presented,
we assume that the galaxy is not nucleated. Note that the original data table
contains five duplicates and bad identifiers for some galaxies which have been
removed (Mark den Brok, priv. comm.).

Recently, Zanatta et al. (2021) investigated the core of the Coma galaxy
cluster using HST ACS imaging to classify $66$ elliptical dwarf galaxy
candidates. However, due to a lack of photometric parameters needed to
determine stellar masses, we do not use their data set.

####  2.3.3 Fornax galaxy cluster

Muñoz et al. (2015) used the Next Generation Fornax Cluster Survey (NGFS) to
classify dwarf galaxies in the central regions of the Fornax galaxy cluster.
They used galfit (Peng et al., 2010) to fit the surface brightness of a galaxy
with a Sérsic profile (Sérsic, 1968), and used an additional profile if needed
to fit the NSC.

Turner et al. (2012) analysed $43$ high-luminosity galaxies in the Fornax
galaxy cluster as part of the Advanced Camera for Surveys Fornax Cluster
Survey (ACSFCS). Both studies classify galaxies as either nucleated or non-
nucleated. Two galaxies are part of both surveys and were assigned the same
nuclear classification.

Furthermore, the Fornax Deep Survey (FDS; Iodice et al., 2016) covers the
whole Fornax galaxy cluster and the Fornax A group. While both Venhola et al.
(2018) and Su et al. (2021) present nuclear classifications for the dwarf
elliptical cluster members, their classifications differ for $55$ galaxies
despite using the same photometric data products. In addition, their nuclear
classifications do not agree with Muñoz et al. (2015) for $175$ and $26$
galaxies, respectively. As a result, we do not use the data of Venhola et al.
(2018) at all and only consider the data of Su et al. (2021) for our analysis
of the nucleation fraction versus galactocentric radius (cf. Section 3.5).
Finally, four duplicates have been removed from the FDS: FDS12_0241,
FDS25_0232, FDS25_0296, and FDS4_0098.

####  2.3.4 Virgo galaxy cluster

SJ+19 investigated NSCs in mainly low-mass systems in the core of the Virgo
galaxy cluster in the Next Generation Virgo Cluster Survey (NGVS). For NSC
identification, the authors modelled the galaxy body with a single Sérsic
profile and added a second component for an NSC, if central light excess was
found.

Based on the Advanced Camera for Surveys Virgo Cluster Survey (ACSVCS), Côté
et al. (2006) classified mainly high-luminosity early-type galaxies. Their
classification is split into three classes which contain further subcategories
(their Table 2): nucleated (‘I’), non-nucleated (‘II’), and unknown (‘0’). We
only consider galaxies classified as (‘Ia’) and (‘Ib’) as nucleated as the
authors considered (‘Ic’), (‘Id’), and (‘Ie’) as uncertain.

Lisker et al. (2007) assigned a nuclear classification to $413$ early-type
dwarf galaxies using SDSS imaging data. The authors note that it seems likely
that some galaxies classified as non-nucleated may host a faint NSC but
escaped detection due to high-surface brightness. Therefore, we remove the
classification of galaxies flagged as non-nucleated and ‘bright’ in their
Table 1. Because this choice will artificially elevate (i.e. bias) the
nucleation fraction towards higher values, we only use their data in Section
3.5 where we investigate the nucleation fraction versus galactocentric
distance. In that section we are only interested in general trends of
$f_{\mathrm{n}}$ and not in exact values.

Data tables for the Coma, Fornax, and Virgo galaxy clusters are presented in
Table 4, Table 5, and Table 6, respectively. Their structure is the same as
for the LV (cf. Table 3), however, in addition we include the nuclear
classifications of each survey. The full data tables are available in the
online supplementary material.

##  3 Analysis

###  3.1 Galaxy stellar mass estimate

The stellar photometric mass of a galaxy is often calculated by fitting its
spectral energy distribution (SED) or by using (colour-dependent) mass-to-
light ratios (denoted as $M_{\star}/L$). SED fitting gives a more accurate
stellar mass, as exemplified by the analysis of SJ+19 of more than $400$
galaxies finding a typical uncertainty of $0.15\text{\,}\mathrm{dex}$. This is
considerably smaller than the scatter derived from single colour-based
$M_{\star}/L$ (approximately $0.5\text{\,}\mathrm{dex}$, e.g. McGaugh &
Schombert, 2014), nevertheless, it has been shown to yield very comparable and
unbiased results compared to SED fitting (e.g. Roediger & Courteau, 2015).

Therefore, we calculate the galaxy photometric stellar mass using the $B$-,
$g$-, $V$-, and $r$–band magnitudes from our catalogue via the
$M_{\star}/L$-colour relation
$\log_{10}(M_{\star}/L)_{\lambda}=\alpha_{\lambda}+\beta_{\lambda}\times(\mathrm{colour})$,
where $\lambda\in[V,\,r]$ and $\mathrm{colour}\in[(B-V)_{0},\,(g-r)_{0}]$. We
chose these two colours because (1) they are most readily available for the
majority of LV galaxies, and (2) although rigorously investigated and debated
in the literature, which colour is the ideal proxy for the stellar mass of a
galaxy, McGaugh & Schombert (2014) and Du et al. (2020) find that $(B-V)_{0}$
and $(g-r)_{0}$, respectively, are among the best proxies and are less
sensitive to the uncertain contribution of thermally pulsating AGB stars. This
is not surprising because both $(B,\,V)$ and $(g,\,r)$ filter transmission
curves are very close to each other.

The works of McGaugh & Schombert (2014) and Du et al. (2020) build upon
earlier studies by Bell et al. (2003), Portinari et al. (2004), Zibetti et al.
(2009), Into & Portinari (2013), and Roediger & Courteau (2015) and re-
calibrate their relations to ensure self-consistency within the model: as
pointed out by McGaugh & Schombert (2014), NIR luminosities are over-predicted
compared to optical luminosities leading to different $M_{\star}/L$ and,
therefore, to different stellar mass estimates. All coefficients
$(\alpha_{V},\,\beta_{V})$ and $(\alpha_{r},\,\beta_{r})$ that we use for the
calculation of stellar masses are listed in Table 1. We also give
$M_{\star}/L_{K}$ from McGaugh & Schombert (2014) for an assumed
$(B-V)_{0}=$0.6\text{\,}\mathrm{mag}$$ as they show that a stellar mass
estimate from the NIR luminosity is only weakly dependent on colour. Also, the
scatter of $M_{\star}/L$ is expected to be smaller in the NIR compared to that
in the optical (Bell & de Jong, 2001; Portinari et al., 2004). The assumption
of $(B-V)_{0}=$0.6\text{\,}\mathrm{mag}$$ seems to be justified for the LV
(cf. Figure 1).

All five publications differ in their model by either using a different
initial mass function or stellar population models, both of which influence
$M_{\star}/L$ and contribute to the systematic bias and uncertainty. Bell et
al. (2003) and Roediger & Courteau (2015) test their models on galaxies of
different Hubble type whereas the other three studies mainly focus on late-
type galaxies. However, the difference between calculated stellar masses from
different models (typically $\leq 0.3\text{\,}\mathrm{dex}$) is often below
the uncertainty of each value (typically $\approx 0.4\text{\,}\mathrm{dex}$).
To minimise the bias from adopting a specific $M_{\star}/L$, we use the
relations from all publications, i.e. the coefficients in Table 1 and
depending on the filter availability, we calculate up to nine stellar masses
per galaxy from the two colours101010Four masses estimates stem from
$(B-V)_{0}$ and five from $(g-r)_{0}$, respectively.. We remind again that if
the bvtc colour is unavailable, only then we use the calculated optical
$(B-V)_{0}$ colour.

For each calculated stellar mass we determine its formal uncertainty by error
propagation of the uncertainty on the $M_{\star}/L$, its photometry, the
absolute magnitude of the Sun111111Assumed to be $0.04\text{\,}\mathrm{mag}$;
http://mips.as.arizona.edu/~cnaw/sun.html, and the distance measurement. The
uncertainty on $M_{\star}/L$ is assumed to be $0.3\text{\,}\mathrm{dex}$ for
all mass-to-light ratios. This value roughly corresponds to the dispersion in
stellar mass-to-light ratio between McGaugh & Schombert (2014) and Du et al.
(2020) for the relation of Zibetti et al. (2009) caused by different reference
stellar masses. We note that this uncertainty of $0.3\text{\,}\mathrm{dex}$
dominates the stellar mass error-budget for most galaxies. The final stellar
mass, as reported in our data tables, are calculated as the uncertainty
weighted mean from the masses computed from each $M_{\star}/L$. The final
uncertainty equals the standard error of the weighted mean. To avoid
systematic differences between environments due to different calculations of
stellar mass performed by other studies, we re-calculate stellar masses for
all Coma, Fornax, and Virgo galaxies using the same approach (cf. Section
3.1.1). Depending on the data set, the mean standard deviations between
individual mass estimates ranges between $0.09\text{\,}\mathrm{dex}$ and
$0.12\text{\,}\mathrm{dex}$.

Note that there are exceptions to this scheme. Some galaxies have unreliable
apparent magnitudes that result in poor stellar mass estimates. For these
galaxies we adopt the stellar mass based on $M_{\star}/L_{K}$ given in Table
1. The masses for the Milky Way and the Sag dSph galaxy are assumed to be the
following:
$M_{\star,\,\mathrm{MW}}=$6.43\text{\times}{10}^{10}\text{\,}\mathrm{M_{\odot}}$$
(McMillan, 2011) and $M_{\star,\,\text{Sag
dSph}}=$1\text{\times}{10}^{8}\text{\,}\mathrm{M_{\odot}}$$ (Vasiliev &
Belokurov, 2020). Although SJ+19 calculate stellar masses using the more
accurate SED fitting technique, we will use our mass estimates for further
analysis to avoid a systematic bias in mass estimates between different
environments. In the following section we show that our calculated stellar
masses are consistent with their values.

Table 1: Parameters $(\alpha_{V},\,\beta_{V})$ and $(\alpha_{r},\,\beta_{r})$
are used to calculate stellar masses and are taken from McGaugh & Schombert
(2014) and Du et al. (2020), respectively. The authors re-calibrate the
initial parameters provided by Bell et al. (2003); Portinari et al. (2004);
Zibetti et al. (2009); Into & Portinari (2013), and Roediger & Courteau (2015)
to ensure self-consistency. ‘BC03’ and ‘FSPS’ refer to the models of Bruzual &
Charlot (2003) and Conroy et al. (2009).

$\alpha_{V}$ | $\beta_{V}$ | $\alpha_{r}$ | $\beta_{r}$ | $M_{\star}\ /\ L_{K}$ | Ref.(c)
---|---|---|---|---|---
Revised(a) |  Revised(b) | Original |  Revised(a)
$-0.628$ | $1.305$ | $-0.306$ | $1.097$ | $0.73$ | $0.60$ | (1)
$-0.654$ | $1.290$ | – | – | $0.50$ | $0.60$ | (2)
$-1.075$ | $1.837$ | $-0.840$ | $1.654$ | $0.21$ | $0.50$ | (3)
$-0.900$ | $1.627$ | $-0.663$ | $1.530$ | $0.41$ | $0.54$ | (4)
– | – | $-0.792$ | $1.629$ | – | – | (5)
– | – | $-0.647$ | $1.497$ | – | – | (6)

  * (a)

Revised by McGaugh & Schombert (2014) to ensure self-consistency.

  * (b)

Revised by Du et al. (2020) to ensure self-consistency.

  * (c)

(1) Bell et al. (2003); (2) Portinari et al. (2004); (3) Zibetti et al.
(2009); (4) Into & Portinari (2013); (5) Roediger & Courteau (2015, BC03); (6)
Roediger & Courteau (2015, FSPS).

####  3.1.1 Consistency of stellar masses with previous studies

To test the robustness of our stellar mass estimates, we compare our values to
the literature values of Eigenthaler et al. (2018) for the NGFS, Su et al.
(2021) for the FDS, Peng et al. (2008) and Neumayer et al. (2020) for the
ACSVCS, and SJ+19 for the NGVS. The difference between our mass estimates and
the literature values is shown in Figure 2. A thick dashed line gives the mean
of the difference and shaded regions give the $1\sigma$ interval. For
reference, the solid line at ${10}^{9}\text{\,}\mathrm{M_{\odot}}$ highlights
the peak in nucleation fraction found by SJ+19.

A comparison with the data of Eigenthaler et al. (2018) for the NGFS (top
panel; orange colour) shows the largest scatter with the $1\sigma$ interval
covering a range of $\approx 1\text{\,}\mathrm{dex}$. The mean of the
difference has a value of $\approx 0.53\text{\,}\mathrm{dex}$ and is
significantly different from zero. The authors base their mass estimates on
non-revised the stellar mass-to-light ratios from Bell et al. (2003). In
comparison, the agreement with the stellar mass estimates of Su et al. (2021)
for the FDS (top panel; blue colour) is better with a mean difference of
$\approx-0.15\text{\,}\mathrm{dex}$.

Likewise, the comparison to the data of SJ+19 (bottom panel; light violet
colour) gives a similar value for the mean difference
($\approx-0.13\text{\,}\mathrm{dex}$). The comparison for the ACSVCS mass
estimates of Peng et al. (2008) (bottom panel; medium violet colour) results
in a similar value but with opposite sign ($\approx
0.16\text{\,}\mathrm{dex}$). Finally, a comparison to the mass estimates of
Neumayer et al. (2020) (bottom panel; dark violet colour) shows the best
agreement with a difference of only $\approx-0.03\text{\,}\mathrm{dex}$.

As a result, it is important to use a homogeneous approach to estimate stellar
masses for galaxies in different environments, especially at the low-mass end.
In other words, adopting stellar mass estimates from different literature
sources introduces different systematic biases which then affect our analysis
of $f_{\mathrm{n}}$. Although the overall agreement with literature estimates
is below the typical uncertainty on stellar masses ($\approx
0.3\text{\,}\mathrm{dex}$ in log-space) we use our mass estimates for all
environments.

Figure 2: Comparison between stellar mass estimates in the literature and our
values. Top panel: Comparison to Eigenthaler et al. (2018) for the NGFS and Su
et al. (2021) for the FDS (both Fornax galaxy cluster). Bottom panel:
Comparison to Peng et al. (2008) and Neumayer et al. (2020) for the ACSVCS,
and Sánchez-Janssen et al. (2019) for the NGVS (all Virgo galaxy cluster). The
means of the logarithmic mass differences are shown with dashed lines and the
$1\sigma$ interval is highlighted with a shaded region. One-dimensional
Gaussian kernel density estimates present the density of each distribution
(right panels). Typical uncertainties on the stellar masses range between
$0.3\text{\,}\mathrm{dex}$ and $0.5\text{\,}\mathrm{dex}$ but are omitted for
clarity.

###  3.2 Nucleation fraction as a function of galaxy stellar mass

The nucleation fraction is calculated as the number of nucleated galaxies
divided by the total number of classified galaxies. Here we examine its
dependence on the galaxy stellar mass (see also e.g. Figure $3$ in Neumayer et
al., 2020) including our LV galaxy measurements. We choose uniform binning
across the whole mass range (unlike SJ+19) with a bin width of
$0.5\text{\,}\mathrm{dex}$.

In Figure 3 we show the nucleation fraction ($f_{\mathrm{n}}$) as a function
of logarithmic stellar mass ($\log_{10}\,M_{\star}$) for the LV (green
diamonds) and the galaxy clusters Coma (dark gray triangles), Fornax (light
gray squares), and Virgo (orange circles), respectively. A
$68\text{\,}\mathrm{\char 37\relax}$ confidence interval121212Based on the
Agresti-Coull interval; Agresti & Coull, 1998. of a binomial distribution is
shown with colour-shaded areas and the number of galaxies per mass bin is
given at the top. Surveys and publications from which we obtained data for the
galaxy clusters are indicated in the caption. The total number of galaxies per
environment are indicated in the legend. Mass bins containing fewer than seven
galaxies are shown with open symbols and without the shaded confidence
intervals. A vertical dashed line at ${10}^{9}\text{\,}\mathrm{M_{\odot}}$
indicates the peak of the nucleation fraction as identified by SJ+19. Colour-
coded vertical lines at the bottom of the panel show the mean mass of
classified galaxies split by environment. In addition to the previously
mentioned reference papers for each galaxy cluster, we add $4$ and $25$
classified galaxies from Georgiev & Böker (2014) for the Fornax and Virgo
galaxy clusters, respectively, to include late-type galaxies in galaxy
clusters and to increase the statistical significance at the high-mass end.
The data of Lisker et al. (2007) and Su et al. (2021) are not included.

We see that, regardless of environment, $f_{\mathrm{n}}$ is a strong function
of $M_{\star}$ and peaks between ${10}^{9}\text{\,}\mathrm{M_{\odot}}$ and
$e9.5\text{\,}\mathrm{M_{\odot}}$. A direct comparison between the different
$f_{\mathrm{n}}$ suggest an environmental dependence, i.e. at fixed
$M_{\star}$, $f_{\mathrm{n}}$ is higher for denser galaxy cluster environments
(up to $\approx 40\text{\,}\mathrm{\char 37\relax}$ in the mass range
$e7.5\text{\,}\mathrm{M_{\odot}}$ to ${10}^{9}\text{\,}\mathrm{M_{\odot}}$ for
the Fornax and Virgo galaxy clusters). Differences beyond
$e9.5\text{\,}\mathrm{M_{\odot}}$ may be attributed to low statistics. We
investigate possible selection biases due to classified versus unclassified
galaxies and non-detections of nuclei due to the resolution limits of HST in
§A. We note that our results may differ from those presented in SJ+19 due to
different mass calculations being used to ensure consistency between the
samples (see previous section).

Independent of environment, nucleation seems to start in the range
${10}^{5}\text{\,}\mathrm{M_{\odot}}e5.5\text{\,}\mathrm{M_{\odot}}$ and peaks
between ${10}^{9}\text{\,}\mathrm{M_{\odot}}$ and
$e9.5\text{\,}\mathrm{M_{\odot}}$. Despite poor number statistics at the
highest masses, nucleation beyond $\approx e10.5\text{\,}\mathrm{M_{\odot}}$
is quite rare, regardless of environment. This suggests that nucleation at
these stellar masses may not be dependent on environment, which could be due
to the emergence and influence of supermassive black holes (cf. §4.2) and the
fact that the gas accretion to (re-)build NSCs after mergers cannot be
sustained in these high-mass galaxies (e.g. Antonini et al., 2015).

Figure 3: Top panel: Nucleation fraction ($f_{\mathrm{n}}$) as a function of
logarithmic stellar mass ($\log_{10}\,M_{\star}$) for the Local Volume (green
diamonds) and the galaxy clusters Coma (dark gray triangles), Fornax (light
gray squares), and Virgo (orange circles). Shaded areas indicate the
$68\text{\,}\mathrm{\char 37\relax}$ confidence interval and the number of
galaxies per bin is indicated at the top. Open symbols show mass bins with
fewer than seven contributing galaxies. The vertical dashed line shows the
peak of $f_{\mathrm{n}}$ identified by Sánchez-Janssen et al. (2019). Colour-
coded lines at the bottom show the mean mass for each by environment. The mean
mass for the Local Volume and the Virgo galaxy clusters are similar which is
why only the vertical line for the Local Volume is visible. The total number
of galaxies are shown in the legend. For the Coma galaxy cluster, we use the
data set of den Brok et al. (2014) (ACSCCS). For the Fornax galaxy cluster, we
use the data sets of Muñoz et al. (2015) (NGFS), Turner et al. (2012)
(ACSFCS), and four galaxies of Georgiev & Böker (2014). Finally, for the Virgo
galaxy cluster, we use the data sets of Sánchez-Janssen et al. (2019) (NGVS),
Côté et al. (2006) (ACSVCS), and $25$ galaxies of Georgiev & Böker (2014).
Bottom panel: Difference between nucleation fractions as a function of stellar
mass. The horizontal dashed line shows agreement between $f_{\mathrm{n}}$ of
the Local Volume and a galaxy cluster.

####  3.2.1 A generic model

To study the effects of the stellar mass of a galaxy and its environment in
more detail, we use a simple model to fit the nucleation fraction. For
simplicity and the apparent shape of the nucleation fraction, we model it with
a logistic function $L$ with slope $\kappa$ and midpoint (x-axis offset)
$M_{50}$,

$L(\kappa,\,M_{50})=\frac{1}{1+\exp[-\kappa(\log_{10}M_{\star}-M_{50})]}\quad.$
(2)

Note that the amplitude of the logistic function is set to one and
$f_{\mathrm{n}}$ values are only fit to galaxies with
$M_{\star}<$e9.5\text{\,}\mathrm{M_{\odot}}$$ where the behaviour of
$f_{\mathrm{n}}$ with galaxy mass is found to be monotonic in all
environments.

To fit this model to the unbinned data of all environments131313We do not fit
the data for the Coma galaxy cluster due an incompleteness of galaxies at the
low-mass end. See Zanatta et al. (2021) for a recent investigation., we use
the emcee (Foreman-Mackey et al., 2013) package to perform a Markov-Chain
Monte Carlo analysis. Further details as well as a corner plot for the each
data set are presented in Appendix C. The resulting parameters of our model
are presented in Table 2. Figure 4 compares the best fitting model to the
data.

The slope and midpoints of the logistic functions differ between a field and
cluster environment: For the Fornax and Virgo galaxy clusters the parameters
are practically the same while for the LV the midpoint of the logistic
function is shifted to higher masses by $\approx 1\text{\,}\mathrm{dex}$, and
the slope paramater is $10\text{\,}\mathrm{\char 37\relax}$ to
$15\text{\,}\mathrm{\char 37\relax}$ larger (and thus steeper) than in the
cluster environments.

Table 2: Logistic function model parameters $\kappa$, $M_{50}$, and $\tau$ obtained from the MCMC posteriors by fitting Equation 2 to the unbinned data of the respective environments. The uncertainties are the posterior $1\sigma$ percentiles (cf. Figures 18, 19, 20 and 21).  Environment | $\kappa$ | $M_{50}$
---|---|---
All environments | $1.12^{+0.06}_{-0.06}$ | $8.21^{+0.05}_{-0.05}$
Local Volume | $1.40^{+0.14}_{-0.13}$ | $8.63^{+0.07}_{-0.07}$
Fornax galaxy cluster | $1.25^{+0.16}_{-0.14}$ | $7.68^{+0.10}_{-0.09}$
Virgo galaxy cluster | $1.22^{+0.10}_{-0.10}$ | $7.70^{+0.08}_{-0.07}$
Figure 4: Nucleation fraction ($f_{\mathrm{n}}$) as a function of logarithmic
stellar mass ($\log_{10}\,M_{\star}$) for all combined environments (first),
the Local Volume (second), and the Fornax and Virgo galaxy clusters (third and
fourth panels). Data above $M_{\star}=$e9.5\text{\,}\mathrm{M_{\odot}}$$ (gray
shaded area) are excluded from the Markov-Chain Monte Carlo (MCMC) analysis.
The best fit is shown with a dashed colour-coded line. Additionally, we show
$500$ randomly drawn MCMC samples for each analysis. The parameters for the
best fitting results are indicated where $\kappa$ is the slope and $M_{50}$
the midpoint of the logistic function.

###  3.3 Nucleation fraction as a function of Hubble type

Here we examine if there exists a difference between early- and late-type
morphologies. We split the LV data into early- and late-types based on their
Hubble type. All galaxies of the ACSCCS, ACSFCS, ACSVCS, NGFS, and NGVS are of
early-type. We combine data for all galaxy clusters into one $f_{\mathrm{n}}$
and compare it to the LV in Figure 5. Galaxy cluster data are shown with
hexagons and LV data with diamonds. Red colour indicates early- and blue
colour late-type galaxies. Similar to Figure 3, the colour-coded area shows
the confidence interval and galaxies per bin are indicated at the top.

We can see that, considering only early-type galaxies, the significant
difference between the LV and the galaxy clusters persists. For the LV, it
appears that there is an apparent difference in $f_{\mathrm{n}}$ between
early- and late-type galaxies in the mass range
$$e6.5\text{\,}\mathrm{M_{\odot}}$\lesssim
M_{\star}\lesssim${10}^{9}\text{\,}\mathrm{M_{\odot}}$$. We investigate this
difference by calculating a $p$-value for the null hypothesis that both early-
and late-type galaxies have the same $f_{\mathrm{n}}$. We split the data in
this mass range into bins as indicated in Figure 5. The value of
$f_{\mathrm{n}}$ of the underlying galaxy population is unknown and, hence, we
use the joint value of both Hubble types as an estimator. Using this
estimator, for both early- and late-type galaxies we draw $n$ samples from a
binomial distribution where $n$ is the number of galaxies per bin. We repeat
this exercise ${10}^{7}$ times to estimate a $p$-value for each bin using both
Hubble types. Depending on the mass bin, we find $p$-values between $\approx
0.008$ and $\approx 0.18$. Assuming that the null hypothesis is true and that
the data in each mass bin are independent, we can combine all $p$-values into
a single parameter using Fisher’s method (Fisher, 1992). This parameter will
then follow a $\chi^{2}$-distribution where the number of degrees of freedom
equals twice the number of $p$-values ($12$ in our case). Based on this
parameter, we can determine a final $p$-value, again drawing ${10}^{7}$ times
from a $\chi^{2}$-distribution. The final $p$-value is $\approx
6\text{\times}{10}^{-4}$. Thus, it is very likely that early- and late-type
galaxies originate from galaxy populations with different $f_{\mathrm{n}}$.

However, note that there also is some unreliability in the split into early-
and late-types: as already pointed out by KMK+13, early- and late-type dwarf
galaxies (spheroidal and irregular) are often wrongfully classified due to
their low surface brightness. Therefore, some early-type galaxies might have
been classified as an irregular-type galaxy (and vice versa). Given the total
number of galaxies in this mass range, it seems unlikely that this effect can
compensate for the observed discrepancy.

In the literature, Habas et al. (2020) uses $2210$ dwarf galaxies around
massive early-type galaxies in the nearby Universe ($z<0.01$) to find the same
trend, albeit as a function of absolute $g$-band magnitude. In comparison, the
difference is greater in their analysis which is likely related to their
magnitude limited sample. Figure $3$ of Neumayer et al. (2020) shows $1191$
galaxies split into early- and late-type galaxies based on $(g-i)_{0}$ colour.
No significant enhancements are seen there, however, the uniformity of our
current sample provides more reliable data than the collection of literature
values published there.

Figure 5: Nucleation fraction ($f_{\mathrm{n}}$) as a function of logarithmic
stellar mass ($\log_{10}\,M_{\star}$) for the Local Volume (diamonds) and
galaxies in the Coma, Fornax, and Virgo galaxy clusters (hexagons). The Local
Volume data set is split by Hubble type into early- (red) and late-types
(blue). Shaded areas show the $68\text{\,}\mathrm{\char 37\relax}$ confidence
interval and mass bins with fewer than seven galaxies are shown with open
symbols.

###  3.4 Nucleation fraction as a function of tidal index

From the previous sections we have already seen that the nucleation fraction
appears to depend on environment. Early-type galaxy cluster members show an
increased nucleation fraction at low galaxy stellar masses compared to the LV.
Here we examine this environmental dependence in more detail.

To investigate the significance of the local environment, we calculate the
‘tidal index’ (denoted as $\Theta$) which is a measure of the local density
surrounding a galaxy. This parameter was first introduced by Kacharov &
Makarov (1999) and takes into account the distance to and stellar mass of
nearby galaxies. In contrast to the original definition, we constrain the
possibility of a galaxy to become a disturber by requiring that its stellar
mass lies above $5\text{\times}{10}^{9}\text{\,}\mathrm{M_{\odot}}$. The ‘main
disturber’ for the $i$th galaxy can be calculated via

$\Theta_{i}=\max_{j\in[1,\,N]}\log_{10}\bigg{(}\frac{M_{\star,\,j}}{d_{ij}^{3}}\bigg{)}+\mathcal{C},\;\;\forall\,M_{\star,\,j}\geq$5\text{\times}{10}^{9}\text{\,}\mathrm{M_{\odot}}$\quad,$
(3)

where $N$ is the total number of galaxies in the data set, $d_{ij}$ the 3D
distance between the galaxies, and $\mathcal{C}=-11.76$ a constant reference
value chosen such that each galaxy can be divided into isolated
($\Theta_{i}<0$) and clustered ($\Theta_{i}>0$). This parameter can be
expanded to include also effects of more than one disturbing galaxy (KMK+13).
Therefore,

$\Theta_{i,\,s}=\log_{10}\bigg{(}\sum_{j=1}^{s}\frac{M_{\star,\,j}}{d_{ij}^{3}}\bigg{)}+\mathcal{C},\;\;\forall\,M_{\star,\,j}\geq$5\text{\times}{10}^{9}\text{\,}\mathrm{M_{\odot}}$,\;\;s<N\quad,$
(4)

where we sum over the $s$-most influential neighbouring galaxies of the $i$th
galaxy. While this is a calculation of the gravitational tidal index for a
galaxy, it also is a proxy for other environmental effects, i.e. ram pressure
stripping or strangulation.

Using Equations 3 and 4, we calculate $\Theta_{1}$, $\Theta_{5}$, and
$\Theta_{10}$ for each galaxy in the LV and Coma, Fornax, and Virgo galaxy
clusters. While we use 3D positions to calculate the distance between LV
galaxies, we use the angular separation for cluster members as reliable
relative distances are often not available. All three parameter values and the
ten main disturbers are presented Tables 3, 4, 5 and 6, but here we will only
consider $\Theta_{10}$ as it is less sensitive to uncertainties in stellar
mass estimates and distances between galaxies than both $\Theta_{1}$ and
$\Theta_{5}$.

In Figure 6 we show the distribution of classified galaxies for the LV, the
Fornax and Virgo galaxy clusters, and the combined data set. The bottom panel
shows the kernel density estimate of the $\Theta_{10}$ distribution. We use a
Gaussian kernel with a bandwidth of $0.5$. The maximum value of the kernel
density estimate is shown with dashed lines except for the combined data set
where we use a value of zero. Galaxies with $\Theta_{10}$ below this peak
value (denoted as $\Theta_{10}^{\mathrm{peak}}$) are assigned to a ‘loose’
environment group; galaxies with $\Theta_{10}>\Theta_{10}^{\mathrm{peak}}$ are
assigned to a ‘dense’ environment group.

We calculate $f_{\mathrm{n}}$ as a function of $\Theta_{10}$ and present the
results in §B.2. As an example, we show the distribution of NGVS members in
the Virgo galaxy cluster as a function of stellar mass, tidal index, and
angular coordinates in §B.1. However, since it becomes apparent that the
stellar mass influences the nucleation fraction as a function of tidal index,
we choose a different approach: we split each data set into two groups based
on the peak of the kernel density estimate of $\Theta_{10}$ for each
environment.

It is important to note the differences in $\Theta_{10}^{\mathrm{peak}}$
between environments. Kacharov & Makarov (1999) consider a galaxy to be rather
isolated if its $\Theta_{10}$ value is smaller than zero. Therefore, many
galaxies in the LV, which are part of the ‘dense’ environment group, can still
be considered isolated galaxies. For both the Fornax and Virgo galaxy clusters
the values of $\Theta_{10}^{\mathrm{peak}}$ are significantly larger than
zero.

Based on the peak as a function of $\Theta_{10}$, we then determine
$f_{\mathrm{n}}$ for both subgroups as a function of $M_{\star}$ and present
the results in Figure 7. The values of $\Theta_{10}^{\mathrm{peak}}$ are
indicated at the top of each panel. We do not perform this analysis for the
Coma galaxy cluster as the data sample of den Brok et al. (2014) is incomplete
at the low-mass end (see e.g. Madrid et al., 2010; Chiboucas et al., 2011; Lim
et al., 2018; Zanatta et al., 2021); this incompleteness heavily influences
the $\Theta_{10}$ values.

First, we combine the LV and the Fornax and Virgo galaxy cluster environments
to gain the statistical significance to split the combined data sample at
$\Theta_{10}=0$ (first panel). For the combined data set we can clearly see a
difference in $f_{\mathrm{n}}$, which is expected based on the comparison of
the LV with the galaxy clusters in Figure 3. For the LV (second panel), we
find that galaxies in a ‘dense’ local environment have a slightly higher
$f_{\mathrm{n}}$ than in a ‘loose’ local environment. For the Fornax galaxy
cluster (third panel), $f_{\mathrm{n}}$ is elevated for galaxies in a ‘dense’
local environment until
$M_{\star}\approx${10}^{8}\text{\,}\mathrm{M_{\odot}}$$. On the other hand in
the Virgo galaxy cluster, this same trend is not visible, and in fact the
trend is reversed, with loose regions having higher nucleation fractions than
dense ones. We test a potential bias of classifiying ‘non-nucleated high-
surface brightness’ in Lisker et al. (2007) as unclassified by excluding their
entire data set and repeating the analysis. This limits the data beyond
$r\,/\,r_{\mathrm{vir.}}=0.33$ to $\geq{10}^{8}\text{\,}\mathrm{M_{\odot}}$.
Therefore, due to low number statistics, we cannot find significant trends for
the Virgo galaxy cluster.

Figure 6: Top panel: Logarithmic galaxy stellar mass ($\log_{10}\,M_{\star}$)
as a function of tidal index ($\Theta_{10}$) for the Local Volume (green
diamonds) and the Fornax (gray squares) and Virgo (orange circles) galaxy
clusters. A Gaussian kernel density estimate with a bandwidth of $0.5$ is
applied to each environment and shown in the bottom panel. The blue solid line
shows the combined data sets. Except for the combined data set, where we set
$\Theta_{10}^{\mathrm{peak}}=$0$$, dashed lines indicate the maxima of the
kernel density estimate. Their values for each environment are given in Figure
7.  Figure 7: Nucleation fraction ($f_{\mathrm{n}}$) as a function of
logarithmic stellar mass ($\log_{10}\,M_{\star}$) for all combined
environments (first), the Local Volume (second), and the Fornax and Virgo
galaxy clusters (third and fourth panels). Data for each environment are split
based on the peak of the kernel density distribution of $\Theta_{10}$ (denoted
as $\Theta_{10}^{\mathrm{peak}}$ at the top of each panel). The combined data
set is split at zero. Galaxies in a ‘loose’ / ‘dense’ environment are shown in
blue / red colour.

###  3.5 Nucleation fraction as a function of galactocentric distance

In this section we investigate the nucleation fraction as a function of
galactocentric distance. In the literature it seems to be established that, in
a cluster environment, nucleated galaxies are more centrally concentrated than
non-nucleated ones (e.g. Binggeli et al., 1987; Ferguson & Sandage, 1989;
Lisker et al., 2007; Ordenes-Briceño et al., 2018). For the LV, the situation
was not clear until very recently when Carlsten et al. (2020a) investigated
dwarf elliptical satellites of massive LV hosts. They found that galaxies with
a projected distance $\leq 100\text{\,}\mathrm{kpc}$ have a higher
$f_{\mathrm{n}}$ than galaxies at larger distances. The authors point out that
this increase could be related to the parent halo mass, but no decisive
conclusions could be made. The aim of this section is, therefore, to analyse
the radial dependence of $f_{\mathrm{n}}$ using the LV and galaxy clusters.

For the LV, we follow the approach by Carlsten et al. (2021a) and calculate
the distance between satellites and their hosts before stacking the results.
We use the $30$ most massive galaxies
($$10.18$\lesssim\log_{10}\,M_{\star}\,/\,\mathrm{M}_{\odot}\lesssim$11.33$$)
in the LV as hosts and select all galaxies with a 3D distance $r$ of
$2\text{\,}\mathrm{Mpc}$. Each galaxy is only selected once at the distance to
the closest of the massive galaxies. This value equals multiple virial radii
of the host galaxies and will be useful for a comparison to the galaxy
clusters. In total, we find $148$ galaxies within $2\text{\,}\mathrm{Mpc}$
around the $30$ hosts.

In the LV, we need an estimate of $r_{\mathrm{vir.}}$ to put all radial
measurements on comparable footing. In the literature only few virial radii
are available and many are estimated via the half-mass radius (e.g. Byun et
al., 2020; Karachentsev et al., 2020b). Because the estimates range typically
between $200\text{\,}\mathrm{kpc}$ and $400\text{\,}\mathrm{kpc}$, we assume a
common value of $300\text{\,}\mathrm{kpc}$ for the virial radii of all hosts.

For the Fornax galaxy cluster we include the data of Su et al. (2021) from the
FDS. In comparison to Muñoz et al. (2015) and the NGFS, the data of the FDS
reaches the virial radius of the NGC 1399 ($\approx 700\text{\,}\mathrm{kpc}$;
Drinkwater et al., 2001). Similar to the NGFS, the NGVS covers only the
central region around M 87 in the Virgo galaxy cluster. To study
$f_{\mathrm{n}}$ at larger distances, we include the data set of Lisker et al.
(2007) who uses SDSS imaging to classify galaxies. As noted by the authors,
NSCs in high-surface brightness galaxies may escape detection. Therefore, we
classify all non-nucleated high-surface brightness galaxies of Lisker et al.
(2007) as unclassified, thus, not impacting the nucleation fraction estimates.
We take the cluster-centric radii as the angular separation from NGC 1399 and
M 87, respectively.

In Figure 8 we show the nucleation fraction as a function of stellar mass
where the data have been split based on their proximity to their host galaxy.
With increasing distance, red, orange, and blue colour show different
populations. The number of galaxies per bin are indicated at the top. In
contrast to previous figures, we also show uncertainties for all data points
(even those with low number statistics). Although only few galaxies reside in
the innermost radial bin ($r\,/\,r_{\mathrm{vir.}}\leq$0.33$$), a trend is
visible in that $f_{\mathrm{n}}$ increases, at the same stellar mass, with
decreasing galactocentric distance. This result is in agreement with Carlsten
et al. (2021a) who split their sample at $r=$100\text{\,}\mathrm{kpc}$$
($r\,/\,r_{\mathrm{vir.}}\approx 0.33$ in our figure). The same trend
continues for the Fornax galaxy cluster where the number statistics are better
than in the LV. It is noteworthy that only galaxies within
$0.33r_{\mathrm{vir.}}$ ($\approx 233\text{\,}\mathrm{kpc}$) have an elevated
$f_{\mathrm{n}}$ and that the other two curves closely follow each other. Our
results for the Fornax galaxy cluster are in agreement with Ferguson & Sandage
(1989) and Muñoz et al. (2015). Ferguson & Sandage (1989) found that nucleated
dwarf ellipticals are concentrated in the centre of both the Fornax and Virgo
galaxy clusters, and Muñoz et al. (2015) found that the surface number density
of nucleated dwarf galaxies increases with decreasing galactocentric radius
while the it remains roughly constant for non-nucleated ones. However,
Carlsten et al. (2021a) compared the results of Muñoz et al. (2015) and
Ordenes-Briceño et al. (2018), those galaxy samples reside at different
galactocentric radii ($r\lesssim 0.25r_{\mathrm{vir}}$ and
$r\,/\,r_{\mathrm{vir}.}\in[0.25,0.5]$, respectively), and found no
significant increase.

For the Virgo galaxy cluster, no clear trend with galactocentric distance is
apparent. This result disagrees with Binggeli et al. (1987), Côté et al.
(2006), and Lisker et al. (2007) who found that nucleated dwarf ellipticals
are more centrally concentrated than non-nucleated ones (i.e. an increase in
$f_{\mathrm{n}}$ with decreasing galactocentric distance). One explanation, as
already pointed out by Côté et al. (2006), could be a seletion bias of these
studies as mainly high-mass
($M_{\star}\geq${10}^{8}\text{\,}\mathrm{M_{\odot}}$$) galaxies were
considered. The difference between $f_{\mathrm{n}}$ due to the position within
the cluster at high masses seems to become insignificant (cf. Figure 8).

Figure 8: Nucleation fraction ($f_{\mathrm{n}}$) versus stellar mass
($\log_{10}\,M_{\star}$) for the Local Volume (left panel), and the Fornax
(middle panel) and Virgo (right panel) galaxy clusters. The data are split
based on their distance to the central host galaxy. With increasing
galactocentric distance, the data are shown in red, orange, and blue colour,
respectively. Numbers at the top of each panel give the number of galaxies per
bin. For the Local Volume, the satellite populations of $30$ massive hosts are
stacked, assuming a common virial radius of $300\text{\,}\mathrm{kpc}$.

##  4 Discussion

Overall, including the LV and cluster samples, we find that $f_{\mathrm{n}}$
positively correlates with large-scale environment which is in agreement with
the recent literature (Sánchez-Janssen et al., 2019; Carlsten et al., 2021a;
Zanatta et al., 2021). However, there are other secondary effects which can
influence the nucleation of a galaxy. In Section 4.1 these effects will be
discussed in detail. In Section 4.2 we elaborate further on the decrease of
$f_{\mathrm{n}}$ at the highest masses
($M_{\star}\geq$e9.5\text{\,}\mathrm{M_{\odot}}$$) which seems to be
independent of environment.

###  4.1 Low-mass galaxies with
$M_{\star}<${10}^{9}\text{\,}\mathrm{M_{\odot}}$$

####  4.1.1 Stellar mass & large-scale environment

Significant differences between $f_{\mathrm{n}}$ for the LV and cluster
environments are observed in the low-mass regime. One possible origin for this
trend stems from enhanced NSC formation in dense environments. NSC formation
has been studied in various papers and two different formation mechanisms were
postulated: (1) for high-mass galaxies
($M_{\star}\geq${10}^{9}\text{\,}\mathrm{M_{\odot}}$$) in-situ star-formation
contributes a significant part to the stellar population(s) of NSCs (see
Neumayer et al., 2020, and references therein). (2) For low-mass galaxies
($M_{\star}<${10}^{9}\text{\,}\mathrm{M_{\odot}}$$) globular cluster (GC)
migration towards the galactic centre is important and constitutes the largest
contributor to NSC populations (e.g. Tremaine et al., 1975; Hartmann et al.,
2011; Turner et al., 2012; Antonini et al., 2015). NSCs in galaxies with
stellar masses $M_{\star}\approx${10}^{9}\text{\,}\mathrm{M_{\odot}}$$ likely
have contributions from both mechanisms (e.g. Fahrion et al., 2021). If GC
migration is the dominant contributor to NSC masses in low-mass galaxies, we
would expect to find an increase in GC abundance in dense environments.
Indeed, Carlsten et al. (2021a) found that, at fixed stellar mass, dwarf
ellipticals in the Virgo galaxy cluster host more GCs than in the LV.
Therefore, GC abundance seems to relate to NSC occupation. Additional
arguments supporting the correlation between NSCs and GCs stem from SJ+19 who
found that the NSC and GC occupation fractions vary similarly with stellar
mass for dwarf early-types in the core of Virgo. Recently, Carlsten et al.
(2021a) showed that this is also true for dwarf early-types in the LV,
suggesting that the connection between NSC and GC occupation fraction is
independent of environment.

Within each environment we find an increase in $f_{\mathrm{n}}$ for galaxies
$\lesssim${10}^{9}\text{\,}\mathrm{M_{\odot}}$$ such that at fixed stellar
mass, the nucleation fraction is elevated for satellite galaxies residing
close to their host galaxies. Therefore, if GC abundance and NSC occupation
correlate with each other, we would expect to find higher GC abundance in
galaxies close to their host galaxy. For galaxy clusters this trend has been
observed (Peng et al., 2008; Lim et al., 2018; Liu et al., 2019), but data is
lacking for the LV.

This interpretation seems to fit to our data for the LV and the literature
data for the Fornax galaxy cluster, however, for the Virgo galaxy cluster we
do not find an increase in $f_{\mathrm{n}}$ with galactocentric distance.
Because GC abundance seems to correlate with galactocentric distance, it seems
unclear why $f_{\mathrm{n}}$ would not. This could be related to (1) selection
biases in removing ‘non-nucleated high-surface brightness’ galaxies from
Lisker et al. (2007) or (2) selection biases in the data sample. However, no
significant differences between $f_{\mathrm{n}}$ at different radii persists
at $\geq{10}^{8}\text{\,}\mathrm{M_{\odot}}$, albeit with small number
statistics. If no selection bias is present, one possibility could be that the
correlation between $f_{\mathrm{n}}$ and the environment weakens once a
certain density is reached. One possibile physical mechanism that could cause
this is tidal heating which could prolong (or stop) the inspiral of GCs into a
galaxy’s centre. Carlsten et al. (2021b) found that the sizes of their dwarf
elliptical galaxies are, on average, $\approx 10\text{\,}\mathrm{\char
37\relax}$ smaller than dwarf ellipticals in the core of the Virgo galaxy
cluster. If tidal heating plays a role, we would also expect that
$f_{\mathrm{n}}$ remains constant (or even decreases) with decreasing
galactocentric distance in the Coma galaxy cluster, which is denser than the
Virgo galaxy cluster. Lim et al. (2018) presented an analysis of ultra-diffuse
galaxies in combination with dwarf elliptical galaxies in the Coma galaxy
cluster to find that $f_{\mathrm{n}}$ increases with decreasing galactocentric
distance (their Figure $8$). However, their data cover a range of $\approx
3\text{\,}\mathrm{Mpc}$ with a bin width of $\approx
0.7\text{\,}\mathrm{Mpc}$. While a global increase in $f_{\mathrm{n}}$ is
expected, we can currently not resolve the regions closer to the cluster
centre, where we would expect the trend to reverse. Such an analysis is not
possible as the data sample of den Brok et al. (2014) is incomplete at the
low-mass end and the data set of Zanatta et al. (2021) 1) lacks photometric
parameters to determine stellar masses and 2) could potentially contain a
significant number of background objects (Carlsten et al., 2021a).

####  4.1.2 Hubble type & local environment

Because both the analyses of the galaxy clusters and that of Carlsten et al.
(2021a) focused on dwarf elliptical galaxies, we have tested whether the
morphological type of a galaxy is important for $f_{\mathrm{n}}$. In Section
3.3 we showed that, at fixed stellar mass, dwarf early-type galaxies have a
higher $f_{\mathrm{n}}$ than dwarf late-type galaxies. The argument presented
in the previous section would, therefore, suggest that dwarf elliptical
galaxies have a higer GC abundance than dwarf irregular galaxies. This
increase in GC abundance is found in both a dense cluster (e.g. Miller et al.,
1998; Seth et al., 2004; Miller & Lotz, 2007; Sánchez-Janssen & Aguerri, 2012)
and field environment (e.g. Georgiev et al., 2008).

In Sections 3.4 and 3.5 we also showed that $f_{\mathrm{n}}$ in galaxies
correlates with galaxy density and inversely correlates with distance from a
massive host. Given that early-type galaxies are typically found in denser
environments near massive hosts, it is unclear if the dominant factor in
determining $f_{\mathrm{n}}$ is from galaxy morphology or environment, or if
both contribute independently. In Figure 9 we show the distribution of
classified early- and late-type dwarf galaxies in the LV separated by colour.
To increase number statistics, we use the tidal index instead of the
galactocentric distance141414Using the galactocentric distance leads to the
same conclusions, albeit the trends are not as clear.. We limit the galaxy
sample to a mass range
$M_{\star}\in[${10}^{6}\text{\,}\mathrm{M_{\odot}}$,\,${10}^{9}\text{\,}\mathrm{M_{\odot}}$]$.
We find that, with the exception of the highest $\Theta_{10}$ bins where the
number statistics are poor, early-type galaxies are nucleated more often than
their dwarf late-type counterparts. This increase is not related to stellar
mass as no significant difference, at fixed tidal index, could be determined
between both populations. Furthermore, dwarf elliptical galaxies reside in
denser environments than dwarf irregular galaxies. This obsevation suggests
that the Hubble type, irrespective of stellar mass and environment, correlates
with $f_{\mathrm{n}}$.

Figure 9: Distribution of classified early- (red) and late-type (blue colour)
galaxies in the Local Volume, limited to
$M_{\star}\in[${10}^{6}\text{\,}\mathrm{M_{\odot}}$,\,${10}^{9}\text{\,}\mathrm{M_{\odot}}$]$,
as a function of tidal index ($\Theta_{10}$) which indicates the local
environmental density. We use a Gaussian kernel density estimate with a
bandwidth of $2$ to indicate the nucleation fraction ($f_{\mathrm{n}}$; dashed
lines). The higher nucleation fraction of early-type galaxies relative to
late-types suggests galaxy morphology plays a role in setting the nucleation
fraction independently of environment.

###  4.2 Intermediate- and high-mass galaxies with
$M_{\star}\geq${10}^{9}\text{\,}\mathrm{M_{\odot}}$$

At the highest masses we were able to see that $f_{\mathrm{n}}$ decreases
beyond $M_{\star}\geq$e9.5\text{\,}\mathrm{M_{\odot}}$$ (cf. Figures 3 and 5).
It is speculated that this drop is due to the build-up of massive early-type
galaxies due to galaxy mergers during which NSCs get disrupted via dynamical
heating of the supermassive black holes (e.g. Côté et al., 2006). The
difference between early- and late-type galaxies becomes clearer if we combine
the data sets of Georgiev & Böker (2014) and the LV for late-type galaxies as
no such sharp drop appears (cf. Figure 11).

Antonini et al. (2015) studied the evolution of NSCs and massive black holes
in high-mass galaxies. Using two semi-analytical models named Cluster Inspiral
Model (CliN) and Galaxy Formation Model (GxeV), they were able to investigate
$f_{\mathrm{n}}$. We compare their models with observational results for
early- and late-type galaxies in the top and bottom panels of Figure 10,
respectively. We combine early-type galaxies of all environments (LV and Coma,
Fornax, and Virgo galaxy clusters) and late-types of the LV and the data
sample of Georgiev & Böker (2014) to gain statistical significance.

For early-types (left panel) we find that, although their models overestimate
$f_{\mathrm{n}}$ at all masses, the decline of $f_{\mathrm{n}}$ is similar
between their models and the observations. The slope of the CliN model seems
to decrease with a steeper slope, but this model only takes into account
cluster migration as an NSC formation scenario. We note that the authors found
a better agreement between the CliN model and the data sets of Erwin & Gadotti
(2012); Neumayer & Walcher (2012); Turner et al. (2012), and Scott & Graham
(2013), however, they assumed $M_{\star}/L_{K}=0.8$ based on Bell & de Jong
(2001) which results in an overestimation of stellar masses (cf. Table 1). For
the late-type sample we find better agreement between their models and the
observations. This is impressive given that the identification of nucleation
in their models is uncertain since they are unable to compare the density
profile of the NSC and the host galaxy.

Finally, once Antonini et al. (2015) disabled dynamical heating from the
central supermassive black hole, they found that $f_{\mathrm{n}}$ stays
constant at $100\text{\,}\mathrm{\char 37\relax}$ for both morphologies.
Therefore, it seems likely that this effect is a dominant mechanism of NSC
destruction.

Figure 10: Nucleation fraction ($f_{\mathrm{n}}$) as a function of logarithmic
stellar mass ($\log_{10}M_{\star}$) for early-type (left) and late-type (right
panel) galaxies across all environments (colour-coded hexagons) and the two
models CliN (squares) and GxeV (circles) of Antonini et al. (2015). The models
where dynamical heating from supermassive black holes is disabled show a
constant $f_{\mathrm{n}}$ at $100\text{\,}\mathrm{\char 37\relax}$ (not shown
here).

##  5 Conclusions

In this work we investigated the nucleation fraction ($f_{\mathrm{n}}$) of
Local Volume (LV) galaxies as a proxy for field galaxies in a low-density
environment. Based on the ‘Updated Nearby Galaxy Catalog’ (Karachentsev et
al., 2013) and ancillary data from the HyperLEDA and SIMBAD data bases, we
calculated stellar masses for galaxies in a homogeneous way and combine our
nuclear classification with the literature to construct $f_{\mathrm{n}}$ as a
function of galactic stellar mass, Hubble type, and tidal index. Our catalogue
contains $601$ classified galaxies across eight orders of magnitude in stellar
mass. With the addition of literature data for members of the Coma, Fornax,
and Virgo galaxy clusters and a homogeneous calculation of stellar masses for
all environments, we obtain the following results:

  1. $\bullet$

We identify $22$ new NSCs in LV galaxies. Their photometric and structural
parameters will be analysed in a future paper.

  2. $\bullet$

As a function of stellar mass ($M_{\star}$) we find that $f_{\mathrm{n}}$ is
significantly lower in the LV compared to that in galaxy clusters (cf. Figure
3). This holds true at masses up to $\approx e9.5\text{\,}\mathrm{M_{\odot}}$
and is most significant between $e7.5\text{\,}\mathrm{M_{\odot}}$ and
$e8.5\text{\,}\mathrm{M_{\odot}}$ reaching a difference in $f_{\mathrm{n}}$ of
$\approx 40\text{\,}\mathrm{\char 37\relax}$. These observations confirm
previous investigations of more limited studies (e.g. Sánchez-Janssen et al.,
2019; Carlsten et al., 2021a; Zanatta et al., 2021).

  3. $\bullet$

Regardless of environment, nucleation seems to start between
${10}^{5}\text{\,}\mathrm{M_{\odot}}$ and $e5.5\text{\,}\mathrm{M_{\odot}}$
and seems to peak at $\approx e9.5\text{\,}\mathrm{M_{\odot}}$. Poor
statistics at the high-mass end does not allow us to set tight constrains on
the decline of nucleation at the high mass end, however, it seems likely that
this is the case at $\approx e11.5\text{\,}\mathrm{M_{\odot}}$.

  4. $\bullet$

We fit a logistic function to $f_{\mathrm{n}}$ of each environment below
$e9.5\text{\,}\mathrm{M_{\odot}}$ using a Markov-Chain Monte Carlo analysis.
We find that the slope and midpoint of the logistic function has higher values
for the LV than for both the Fornax and Virgo galaxy clusters, indicating that
nucleation starts at lower stellar masses and rises more steeply with mass in
dense environments.

  5. $\bullet$

At fixed stellar mass, LV dwarf early-types have a higher $f_{\mathrm{n}}$
than dwarf late-type galaxies.

  6. $\bullet$

In addition to a large-scale environmental dependence (field versus cluster),
$f_{\mathrm{n}}$ also correlates with local environment: at fixed stellar
mass, LV satellite galaxies in close proximity of their host galaxy have
higher $f_{\mathrm{n}}$ than distant galaxies. This trend is also observed for
the Fornax galaxy cluster but is not found in Virgo.

  7. $\bullet$

The galactic stellar mass is the dominant factor that determines the
nucleation of galaxies. We also find a correlation between $f_{\mathrm{n}}$
and both the Hubble type and local environmental density (or galactocentric
distance). Both effects of Hubble type and local environment seem to
contribute independently to $f_{\mathrm{n}}$.

  8. $\bullet$

Our results further strengthen the evidence that globular cluster (GC)
abundance and $f_{\mathrm{n}}$ in galaxies below
${10}^{9}\text{\,}\mathrm{M_{\odot}}$ are correlated. It is unclear whether
this connection is sufficient to explain $f_{\mathrm{n}}$. For the Virgo
galaxy cluster, no correlation could be found between the nucleation fraction
and the local environment of a galaxy. We speculate that, if no selection bias
is present, one contributing factor could be the tidal heating of galaxies
which could hinder (or completely stop) the inspiral of GCs into a galaxy’s
centre. Furthermore, it is unclear whether GC abundance of dwarf irregulars in
the LV correlates with local environmental density.

  9. $\bullet$

At the high-mass end, $f_{\mathrm{n}}$ drops for all environments and
nucleation seems to stop at
$M_{\star}\approx$e11.5\text{\,}\mathrm{M_{\odot}}$$. A comparison to the
numerical study of Antonini et al. (2015) reveals that the likely reason for
the decline of $f_{\mathrm{n}}$ is dynamical heating from merging supermassive
black holes (SMBHs) during galaxy mergers.

Follow-up investigations are clearly required to solve the remaining open
questions. In particular, observational studies should investigate the
dependence of $f_{\mathrm{n}}$ on the local environment of the host galaxy in
the centre of the Coma galaxy cluster. Theoretical and numerical
investigations are also important to constrain the relative influence of the
stellar mass, the Hubble type, and the local environmental density on
$f_{\mathrm{n}}$. Additionally, the correlation between GC abundance and
environment would benefit from further investigation.

Another interesting connection can be made between NSC and SMBHs at the high-
mass end where only few classified galaxies reside. A sophisticated analysis
of NSCs and their $f_{\mathrm{n}}$ in high-mass galaxies may help to constrain
the decline of nucleation in this mass regime and allow for a more detailed
study of the interplay between NSCs and central SMBHs.

## Data availability

The observational data underlying this article were accessed from the Hubble
Legacy Archive151515See Footnote 8. and the Mikulski Archive for Space
Telescopes161616https://archive.stsci.edu/. The data underlying this article
are available in the article and in its online supplementary material.

## Acknowledgements

This research is based on observations with the NASA/ESA Hubble Space
Telescope, obtained at the Space Telescope Science Institute, which is
operatedby AURA, Inc., under NASA contract NAS5-26555. This research has made
use of the Updated Nearby Galaxy Catalog (Karachentsev et al., 2013); the
HyperLEDA data base (Makarov et al., 2014); the SIMBAD data base (Wegner et
al., 2000); NASA’s Astrophysics Data System (ADS); Astropy (Collaboration,
2013, 2018); NumPy (https://numpy.org/); dustmaps (Green, 2018); Matplotlib
(Hunter, 2007); SciPy (Virtanen et al., 2020); AstroQuery (Ginsburg et al.,
2019); emcee (Foreman-Mackey et al., 2013); and corner (Foreman-Mackey, 2016).

NH thanks Joel Roediger for the communication of star-forming galaxies in the
NGVS sample and Mark den Brok for details on his data table for the Coma
galaxy cluster. The authors would like to thank the anonymous referee for a
detailed and constructive report.

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##  Appendix A Selection effects

###  A.1 Biases due to incompleteness

The UNGC is incomplete at the low stellar mass end (KMK+13). We test if our
sample is biased by comparing our results to the analysis of Carlsten et al.
(2020b) and Habas et al. (2020) who investigate dwarf galaxy populations
around high-mass field galaxies.

Carlsten et al. (2020b) detected $155$ dwarf elliptical galaxy candidates
around ten primary host galaxies in the Local Volume from which $93$ are
considered new detections and are _not_ part of the most recent iteration of
the UNGC. Although we do not calculate stellar masses for their galaxy sample
as accurate distance estimates are unavailable for $66.5\text{\,}\mathrm{\char
37\relax}$ (103 out of 155 galaxies; Carlsten et al., 2020a), they noted that
their galaxy candidates are similar to known satellites in the Local Group,
i.e. $M_{\star}\lesssim${10}^{7}\text{\,}\mathrm{M_{\odot}}$$. The authors
argued that their sample has a completeness of $>90\text{\,}\mathrm{\char
37\relax}$ and, based on visual inspection of exposures, found $19$ nucleated
and $90$ non-nucleated galaxies
($f_{\mathrm{n}}\approx$19.3\text{\,}\mathrm{\char 37\relax}$$).

Habas et al. (2020) identified $2210$ dwarf galaxies from which
$75\text{\,}\mathrm{\char 37\relax}$ were assigned an early-type morphology.
Similar to Carlsten et al. (2020b), they argued that, based on scaling
relations, their sample is comparable to dwarf galaxies in the Local Group and
the Fornax and Virgo galaxy clusters (again,
$M_{\star}\lesssim${10}^{7}\text{\,}\mathrm{M_{\odot}}$$). The authors found
$f_{\mathrm{n}}\approx$28.2\text{\,}\mathrm{\char 37\relax}$$ for dwarf early-
types and $f_{\mathrm{n}}\approx$9.4\text{\,}\mathrm{\char 37\relax}$$ for
irregular-type galaxies. Both values are in excellent agreement with our
results as most of the early- and irregular-type galaxies in the Local Volume
are system of similar mass compared to their samples (see also Figure 5).
Thus, our data are unbiased for low-mass early- and irregular-type galaxies.

Next, we compare our results to the study of Georgiev & Böker (2014) who
analysed NSCs in $323$ late-type (Hubble type $>3.5$) galaxies which do not
lie edge-on (inclination $\lesssim$88\text{\,}\mathrm{\SIUnitSymbolDegree}$$)
and have a distance estimate $\leq 40\text{\,}\mathrm{Mpc}$. A total of $88$
galaxies ($\approx 27.2\text{\,}\mathrm{\char 37\relax}$) are contained in the
UNGC and are removed from their sample. We also exclude $4$ / $25$ galaxies
which belong to the Fornax / Virgo galaxy cluster ($\approx
1.2\text{\,}\mathrm{\char 37\relax}$ / $\approx 7.7\text{\,}\mathrm{\char
37\relax}$). All other $206$ galaxies are part of a heterogeneous sample from
a mix of local environments (galaxy group, cluster of galaxies) whose
completeness study goes beyond the scope of this paper. In Figure 11 we
compare $f_{\mathrm{n}}$ from their sample (blue plus-signs) to our results
(diamonds; same as in Figure 5). We find that their $f_{\mathrm{n}}$ is larger
than our results only using the Local Volume. Since we do not have detailed
information about the direct environment of these galaxies, we cannot draw any
conclusions why this difference occurs. Note that if you combine the data sets
of Georgiev & Böker (2014) and the UNGC, the difference between
$f_{\mathrm{n}}$ of field galaxies and e.g. the Virgo galaxy cluster shrinks
between ${10}^{8}\text{\,}\mathrm{M_{\odot}}$ and
$e8.5\text{\,}\mathrm{M_{\odot}}$, while beyond
$e9.5\text{\,}\mathrm{M_{\odot}}$ it still persists. Thus, although adding
late-type field galaxies may increase $f_{\mathrm{n}}$ above
${10}^{8}\text{\,}\mathrm{M_{\odot}}$, it is insufficient to explain the
observed differences between galaxy environments.

Figure 11: Nucleation fraction ($f_{\mathrm{n}}$) as a function of logarithmic
stellar mass ($\log_{10}\,M_{\star}$) for the Local Volume (diamonds) and the
late-type sample of Georgiev & Böker (2014) (pentagons). The presentation is
the same as in Figure 5.

We also investigate whether the distribution of classified galaxies represents
the overall distribution of Local Volume galaxies. In Figure 12 we show the
two-dimensional histogram of Local Volume galaxies where bins are colour-coded
based on the fraction of classified galaxies. A high classification fraction
(dark red colour) corresponds to a good representation of the global galaxy
population. In addition, we show the projected histograms of galactic stellar
mass and Hubble type where we show the distributions of classified (red) and
unclassified (black) galaxies. We find that the classification fraction of our
sample is $\geq 50\text{\,}\mathrm{\char 37\relax}$ for most bins, independent
of Hubble type and stellar mass. Note that only S0+ galaxies in the mass range
${10}^{6}\text{\,}\mathrm{M_{\odot}}{10}^{8}\text{\,}\mathrm{M_{\odot}}$ are
poorly represented. However, only $81$ galaxies are contained in these bins
and that is very unlikely that any of the above results are affected.

We repeat this exercise for both the Fornax and Virgo galaxy clusters by
obtaining a list of their members from HyperLEDA. For the Coma galaxy cluster
no such list is available. As described in §2.2, galactic parameters are taken
from SIMBAD and HyperLEDA and stellar masses are calculated as described in
§3.1. Due to redshift-dependent distance estimates of many galaxies in the
Virgo Cluster Catalog which lead to an overestimation of stellar masses, we
limit the data set to galaxies which are also part of the Extended Virgo
Cluster Catalog. This catalogue contains Virgo Cluster Catalog members whose
radial velocity does not exceed
$3000\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ (Kim et al., 2014),
leading to a removal of more than $1000$ galaxies. We only consider nuclear
classifications of Côté et al. (2006), Lisker et al. (2007), Turner et al.
(2012), Muñoz et al. (2015), Sánchez-Janssen et al. (2019), and Su et al.
(2021). Additionally, we add the classification of Georgiev & Böker (2014) to
the data sets of the Fornax and Virgo galaxy clusters ($4$ and $25$ galaxies,
respectively). As a result, we cannot exclude the possibility that a subset of
unclassified galaxies have a classification in the literature and trying to
obtain one is beyond the scope of this paper.

For the Virgo galaxy cluster we find that only a small subset of all known
cluster members are classified by Côté et al. (2006), Lisker et al. (2007),
and Sánchez-Janssen et al. (2019) (cf. Figure 14). While the overall
population of galaxies is poorly represented, the completeness of elliptical
galaxies is $>50\text{\,}\mathrm{\char 37\relax}$ with $M_{\star}$ <
${10}^{8}\text{\,}\mathrm{M_{\odot}}$ and
$M_{\star}\geq${10}^{10}\text{\,}\mathrm{M_{\odot}}$$. However, keep note that
the analysis of Sánchez-Janssen et al. (2019) solely focused on the core of
Virgo and is complete down to
$M_{\star}\approx${10}^{7}\text{\,}\mathrm{M_{\odot}}$$ (Ferrarese et al.,
2020). Despite the addition of the late-type sample of Georgiev & Böker
(2014), late-type spiral galaxies are not well represented. Since
$f_{\mathrm{n}}$ is similar for elliptical galaxies between the Local Volume
and the Virgo galaxy cluster (i.e. Figure 5) and no also no significant
differences can be found in the literature (Figure $3$ in Neumayer et al.,
2020), we suspect that this holds true for late-type galaxies as well and
would therefore expect to find many (potentially) unidentified NSCs.

For the Fornax galaxy cluster we find that the classification fraction is
$>50\text{\,}\mathrm{\char 37\relax}$ for most bins for early-type galaxies
(cf. Figure 13). Similar to the Virgo galaxy cluster, no late-type galaxies
are covered by Turner et al. (2012), Muñoz et al. (2015), and Su et al.
(2021). Furthermore, only four late-type galaxies of Georgiev & Böker (2014)
are Fornax cluster members, leaving a large parameter space unexplored.

In total, the completeness of the Local Volume is $49.3\text{\,}\mathrm{\char
37\relax}$ (601 classified and 619 unclassified galaxies) whereas it is
$90.7\text{\,}\mathrm{\char 37\relax}$ (737 classified and 76 unclassified
galaxies) for the Fornax and $31.9\text{\,}\mathrm{\char 37\relax}$ (776
classified and 1658 unclassified galaxies) for the Virgo galaxy cluster.

Combining these arguments, we conclude that $f_{\mathrm{n}}$ for the Fornax
and Virgo galaxy clusters are representative of early-type galaxies171717For
the Virgo galaxy cluster this is the case for
$M_{\star}\geq${10}^{8}\text{\,}\mathrm{M_{\odot}}$$.. If we assume that
$f_{\mathrm{n}}$ evolves similar to what we observe in the Local Volume, it
would be representative of all cluster members. Based on a classification
fraction of $>50\text{\,}\mathrm{\char 37\relax}$ for galaxies across all
Hubble types and stellar masses and no visible differences in the
distributions of classified and unclassified galaxies, we conclude that our
results well represent the Local Volume, and therefore, the local field galaxy
population.

Figure 12: Two-dimensional histogram of logarithmic stellar mass
($\log_{10}\,M_{\star}$) and Hubble type ($T$). Bins are colour-coded based on
the fraction of classified galaxies: dark red colour corresponds to a high
completeness / classification fraction. The projections along both axes are
indicated with histograms where galaxies are split into classified (red) and
unclassified (black). Additionally, we show the ratio between classified and
unclassified galaxies as a function of stellar mass with dark gray circles in
the right panel.

###  A.2 Completeness fraction for the Fornax and Virgo galaxy clusters

In Figure 13 and Figure 14 we show the two-dimensional histograms of the
Fornax and Virgo galaxy clusters, respectively, that the show completeness
fraction of classified galaxies. More information about the data sets is
presented in the main body of the paper.

Figure 13: Same as in Figure 12 but for the Fornax galaxy cluster.  Figure 14:
Same as in Figure 12 but for the Virgo galaxy cluster.

###  A.3 Bias due to non-detections of nuclei

Due to the compact nature of nuclear star clusters and the resolution limits
of HST, we might wrongfully classify a nucleated galaxy as non-nucleated by
flagging a nuclear star cluster as a central star. We investigate the false-
negative classification bias to investigating the resolution limits of
different camers of HST which have been used to classify galaxies.

For this investigations we create HST PSFs at the centre of the ACS (WFC,
HRC), WFPC2 (WFC, PC), and WFC3 (UVIS, NIR) detectors in the $F814W$
band181818For the WFC3 NIR detector, we use the $F160W$ detector.. We use this
filter because, as pointed out in Section 2.3.1, our classification process
focuses on red / NIR HST exposures to minimise the influence of Galactic and
internal dust obscuration. The light profile of each PSF is then fit with a
two-dimensional Gaussian profile. The width of the PSF is calculated as the
geometric mean of the width of the Gaussian profile.

Similar to Georgiev & Böker (2014), we assume that we are able to distinguish
a star from an extended source if the difference between the instrument PSF
and the extended source is greater than $20\text{\,}\mathrm{\char 37\relax}$.
As argued by the authors, a value of $20\text{\,}\mathrm{\char 37\relax}$ is a
conservative limit. In Figure 15 we show this limit as a function of distance
for each detector. Focus should be put on the ACS WFC (solid black) and WFPC2
WFC (solid blue) lines because these detectors have been used most often in
our analysis pipeline. Note that the resolution limit on the y-axis
corresponds to twice the effective radius of a potential nuclear star cluster.
In addition, we show in the bottom panel the distribution of galaxies
classified as non-nucleated. In our analysis pipeline an NSC is wrongfully
classified as a star if twice its effective radius is below the resolution
limit.

Figure 15: Top panel: Resolution limit of different HST detectors as a
function of galaxy distance. A nuclear star cluster is flagged as a star in
our analysis if twice its effective radius is below the resolution limit.
Bottom panel: Distribution of galaxies classified as non-nucleated.

##  Appendix B Additions regarding the tidal index

###  B.1 Bimodial distribution of the tidal index for the Virgo galaxy cluster

In the top panel of Figure 16 we show the logarithmic stellar mass
$\log_{10}\,M_{\star}$ as a function of the tidal index $\Theta_{10}$ for thre
Virgo galaxy cluster. In the bottom panel we show the spatial distribution.
The data of the NGVS are shown in blue colour. For comparison, we show the
central M 49 and M 87 with a white and magenta star, respectively.

Figure 16: Top panel: Logarithmic stellar mass ($\log_{10}\,M_{\star}$) as a
function of tidal index ($\Theta_{10}$). A small tidal index corresponds to a
‘loose’ local environment. NGVS members are shown in blue colour. Bottom
panel: Spatial distribution of Virgo galaxy cluster members. A white and
magenta star indicate the position of M 49 and M 87, respectively.

###  B.2 Nucleation fraction

In Figure 17 we construct the nucleation fraction $f_{\mathrm{n}}$ as a
function of tidal index $\Theta_{10}$ for all environments combined (first),
the Local Volume (second), and the Coma (third), Fornax (fourth), and Virgo
(fifth panel) galaxy clusters. The top row shows the whole data set whereas
the data are limited to
$M_{\star}\in[${10}^{7}\text{\,}\mathrm{M_{\odot}}$,\,${10}^{9}\text{\,}\mathrm{M_{\odot}}$]$
in the bottom row. The colourmaps differ between both rows and are indicated
in the left panels, respectively.

You can clearly see that $f_{\mathrm{n}}$ depends on $\Theta_{10}$, however, a
significant part of this correlation is due to $M_{\star}$: for the combined
data set the nucleation fraction decreases with increasing tidal index, but
also with decreasing stellar mass. We tried to limit the impact of the stellar
mass in the bottom row where one can see that, for the combined data set,
$f_{\mathrm{n}}$ positively correlates with $\Theta_{10}$. The two data points
at the lowest values of $\Theta_{10}$ highlight that very massive galaxies are
barely influenced by neighbouring galaxies (third and fourth panels). The
effects of $\Theta_{10}$ and $M_{\star}$ on $f_{\mathrm{n}}$ are investigated
in greater detail in Sections 3.2 and 3.4. For the Local Volume we see that
$f_{\mathrm{n}}$ positively correlates with $\Theta_{10}$. For the Coma galaxy
cluster the trend is unclear and we suspect that our calculation of
$\Theta_{10}$ is inaccurate as many low-mass systems are missing. In
comparison, for both the Fornax and Virgo galaxy clusters, we see a slightly
positive correlation bewteen $f_{\mathrm{n}}$ and $\Theta_{10}$.

Figure 17: Nucleation fraction ($f_{\mathrm{n}}$) as a function of tidal index
($\Theta_{10}$). The stellar mass per data point is given by a colourmap. This
colourmap is different for each row and is given in the left panels,
respectively. From left to right: the combined data set from all environments,
the Local Volume, the Coma, Fornax, and Virgo galaxy clusters. Top: The
complete data set for each environment. Bottom: The data are restricted to the
mass range
$M_{\star}\in[${10}^{7}\text{\,}\mathrm{M_{\odot}}$,\,${10}^{9}\text{\,}\mathrm{M_{\odot}}$]$.

##  Appendix C Details on the MCMC analysis

We use the Markov-Chain Monte Carlo (MCMC) capabilities of the emcee package
(Foreman-Mackey et al., 2013) to find the best fitting parameter values of our
nucleation fraction model for all environments. Because we only model the low-
mass galaxy populations, we limit the data set to
$M_{\star}\leq$e9.5\text{\,}\mathrm{M_{\odot}}$$. For all fits we use
uniformative and uniform priors: $\kappa\in[0.0,\,3.0]$,
$M_{50}\in[2.0,12.0]$. The initial parameters for $\kappa$ and $M_{50}$ are
$1.0$ and $8.0$. The logarithmic likelihood function is

$\log\mathcal{L}(\kappa,\,M_{50})=-\frac{1}{2}\sum_{i}\bigg{[}\bigg{(}\frac{y_{i}-\mathrm{L}_{i}(\kappa,\,M_{50})}{\sigma_{i}}\bigg{)}^{2}+\log
2\pi\sigma_{i}^{2}\bigg{]}\quad,$ (5)

where $y_{i}\in\\{0,\,1\\}$ is the nucleation of the $i$th galaxy and
$\mathrm{L}_{i}$ is the Logistic function (cf. Equation 2).

We run the MCMC for $5000$ iterations using $500$ walkers. The autocorrelation
time varies between $20$ and $40$ steps, and the mean acceptance rate is
$\approx 70\text{\,}\mathrm{\char 37\relax}$. We discard the first $500$ steps
from each run and use a thinning factor of $25$. The final posteriors for the
Local Volume, the Fornax and Virgo galaxy clusters, and the combined data set
are shown in Figures 18, 19, 20 and 21, respectively. All values reported in
Table 2 are the median values of each distribution in the posteriors. The
uncertainties give the $1\sigma$ interval around the median.

Figure 18: Corner plot of the Markov-Chain Monte Carlo analysis for the data
set of the Local Volume. Only galaxies with
$M_{\star}\leq$e9.5\text{\,}\mathrm{M_{\odot}}$$ are considered for the fit.
The median and $1\sigma$ interval are indicated with dashed lines.  Figure 19:
Same as in Figure 18 but for the Fornax galaxy cluster.  Figure 20: Same as in
Figure 18 but for the Virgo galaxy cluster.  Figure 21: Same as in Figure 18
but for the combined data set.

##  Appendix D Data tables

Table 3 presents the first five entries in the data table for the Local
Volume. The data tables for the Coma, Fornax, and Virgo galaxy clusters share
the same format. All data tables are available online in .fits.gz format.

Table 3:  List of galaxies in the Local Volume based on the Updated Nearvy
Galaxy Catalog (Karachentsev et al., 2013) including basic photometric
parameters, the tidal index which characterises the local density of a galaxy,
a stellar mass estimate, and a nuclear classification. Magnitudes are given in
the VEGAmag system, except for the $g$\- and $r$-bands which are given in the
ABmag system. If no reference is provided, the nuclear classification is based
on available Hubble Legacy Archive data. The full data table is available
online.

Name | RA | DE | $d$ |  $d_{\mathrm{M}}$(a) | … | $T$ | $\Theta_{1}$ | $\Theta_{5}$ | $\Theta_{10}$ | MD | $\log_{10}M_{\star}$ |  Nuc.(b) |  Ref.(c)
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [mag] |  |  |  |  |  |  |  |  [M⊙] |  |
A0554+07 | $89.402\,92$ | $7.491\,94$ |  $29.95$ $\pm$ | – | TF | … |  $10.0$ $\pm$ | $1.0$ | $-3.65$ | $-3.12$ | $-2.94$ | … |  $6.70$ $\pm$ | $0.40$ | ? | –
A0952+69 | $149.370\,83$ | $69.272\,22$ |  $27.97$ $\pm$ | $0.17$ | TRGB | … |  $10.0$ $\pm$ | $1.0$ | $0.40$ | $0.40$ | $0.40$ | … |  $6.48$ $\pm$ | $0.39$ | 0 | –
AF7448_001 | $344.897\,08$ | $16.769\,72$ |  $28.49$ $\pm$ | – | TF | … |  $10.0$ $\pm$ | $1.0$ | $-2.82$ | $-2.37$ | $-2.27$ | … |  $6.39$ $\pm$ | $0.43$ | 0 | –
AGC 102728 | $0.089\,17$ | $31.021\,94$ |  $29.74$ $\pm$ | $0.16$ | TRGB | … |  $10.0$ $\pm$ | $1.0$ | $-3.52$ | $-2.91$ | $-2.72$ | … |  $5.73$ $\pm$ | $0.54$ | 0 | –
AGC 112454 | $23.248\,75$ | $14.373\,33$ |  $30.04$ $\pm$ | – | mem | … |  $10.0$ $\pm$ | $2.0$ | $0.05$ | $0.05$ | $0.05$ | … |  $6.73$ $\pm$ | $0.71$ | ? | –
… |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |

  * (a)

Methods, see Karachentsev et al. (2013). In addition, we use distance
estimates from HyperLEDA and SIMBAD.

  * (b)

Nuclear classification: ‘1’ nucleated; ‘0’ non-nucleated; ‘?’ unknown.

  * (c)

References: (1) Baldassare et al. (2014); (2) Barth et al. (2008); (3)
Baumgardt & Hilker (2018); (4) Beifiori et al. (2009); (5) Bellazzini et al.
(2008); (6) Bellazzini et al. (2020); (7) Böker et al. (1999a); (8) Böker et
al. (1999b); (9) Böker et al. (2001); (10) Böker et al. (2002); (11) Böker et
al. (2003); (12) Böker et al. (2004); (13) Butler & Martínez-Delgado (2005);
(14) Calzetti et al. (2015); (15) Carlsten et al. (2020b); (16) Carlsten et
al. (2021a); (17) Carollo et al. (2002); (18) Carretta et al. (2010); (19)
Carson et al. (2015); (20) Cohen et al. (2018); (21) Cole et al. (2017); (22)
Contenta et al. (2018); (23) Das et al. (2012); (24) Davidge & Courteau
(2002); (25) Desroches & Ho (2008); (26) Emsellem et al. (1999); (27) Fahrion
et al. (2020); (28) Ferrarese et al. (2006); (29) Filippenko & Sargent (1989);
(30) Filippenko & Ho (2003); (31) Ganda et al. (2009); (32) Georgiev et al.
(2008); (33) Georgiev et al. (2009b); (34) Georgiev et al. (2009a); (35)
Georgiev & Böker (2014); (36) Gnerucci et al. (2013); (37) González Delgado et
al. (2008); (38) Gordon et al. (1999); (39) Graham & Driver (2007); (40)
Graham (2008); (41) Graham & Spitler (2009); (42) Graham (2012); (43) Graham &
Scott (2013); (44) Hartmann et al. (2011); (45) Ho et al. (1997); (46) Hopkins
et al. (2009); (47) Jones et al. (1996); (48) Kacharov et al. (2018); (49)
Karachentsev et al. (2004); (50) Karachentsev et al. (2020a); (51) Kormendy et
al. (1996); (52) Kormendy & Bender (1999); (53) Kormendy et al. (2010); (54)
Kormendy & Ho (2013); (55) Lauer et al. (2005); (56) Luo et al. (2012); (57)
Martocchia et al. (2020); (58) Matthews et al. (1999); (59) Matthews &
Gallagher (2002); (60) Milosavljević (2004); (61) Milosavljević & Bromm
(2014); (62) Misgeld & Hilker (2011); (63) Mitzkus et al. (2016); (64) Monaco
et al. (2009); (65) Neumayer & Walcher (2012); (66) Nguyen et al. (2017); (67)
Nguyen et al. (2018); (68) Pechetti et al. (2020); (69) Peng et al. (2002);
(70) Phillips et al. (1996); (71) Piqueras López et al. (2012); (72) Puzia &
Sharina (2008); (73) Ravindranath et al. (2001); (74) Rossa et al. (2006);
(75) Sánchez-Janssen et al. (2019); (76) Sarzi et al. (2005); (77) Scott et
al. (2013); (78) Seth et al. (2006); (79) Seth et al. (2010); (80) Smith et
al. (2016); (81) Veljanoski et al. (2013); (82) Walcher et al. (2005); (83)
Yong et al. (2016); (84) Yuan et al. (2019)

Table 4: List of galaxies for the Coma galaxy cluster adopted from den Brok et
al. (2014). The authors classify a galaxy as nucleated (‘1’) if a fit with a
central Gaussian source yielded a higher evidence than a fit without one. The
full data table is available online.

Name | RA | DE | $d$ |  $d_{\mathrm{M}}$(a) | … | $T$ | $\Theta_{1}$ | $\Theta_{5}$ | $\Theta_{10}$ | MD | $\log_{10}M_{\star}$ | Nuc.
---|---|---|---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [mag] |  |  |  |  |  |  |  |  [M⊙] |
COMAi125713.240p272437. | $195.0773$ | $28.0972$ |  $35.10$ $\pm$ | $0.55$ | Ferrarese et al. (2000) | … | – | $1.27$ | $1.32$ | $1.35$ | … |  $8.73$ $\pm$ | $0.11$ | 1
COMAi125828.358p271315. | $194.8829$ | $27.8613$ |  $35.22$ $\pm$ | $0.15$ | HyperLEDA | … | – | $1.72$ | $2.00$ | $2.01$ | … |  $8.59$ $\pm$ | $0.12$ | 1
COMAi125844.089p274647. | $195.1717$ | $28.0451$ |  $35.10$ $\pm$ | $0.55$ | Ferrarese et al. (2000) | … | – | $0.31$ | $0.81$ | $0.86$ | … |  $7.83$ $\pm$ | $0.09$ | 1
COMAi125855.735p274528. | $195.1435$ | $27.9347$ |  $35.10$ $\pm$ | $0.55$ | Ferrarese et al. (2000) | … | – | $1.52$ | $1.91$ | $1.92$ | … |  – $\pm$ | – | 0
COMAi125924.938p275320. | $194.9978$ | $27.9406$ |  $35.10$ $\pm$ | $0.55$ | Ferrarese et al. (2000) | … | – | $3.34$ | $3.35$ | $3.35$ | … |  $8.00$ $\pm$ | $0.15$ | 1
… |  |  |  |  |  |  |  |  |  |  |  |  |  |

  * (a)

Either based on HyperLEDA, SIMBAD, or Ferrarese et al. (2000).

  * (b)

Nuclear classification: ‘1’ nucleated; ‘0’ non-nucleated.

Table 5:  List of galaxies for the Fornax galaxy cluster adopted from the
HyperLEDA data base. The full data table is available online.

Name | RA | DE | $d$ |  $d_{\mathrm{M}}$(a) | … | $T$ | $\Theta_{1}$ | $\Theta_{5}$ | $\Theta_{10}$ | MD | $\log_{10}M_{\star}$ | … |  Nuc.(b) |  Ref.(c)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [mag] |  |  |  |  |  |  |  |  [M⊙] |  |  |
ESO 302-009 | $56.892\,12$ | $-38.576\,48$ |  $31.38$ $\pm$ | $0.64$ | HyperLEDA | … |  $8.0$ $\pm$ | $0.7$ | $-0.64$ | $-0.11$ | $0.03$ | … |  $9.07$ $\pm$ | $0.58$ | … | ? | –
ESO 357-010 | $48.926\,92$ | $-33.542\,69$ |  $31.51$ $\pm$ | $0.15$ | Blakeslee et al. (2009) | … |  $7.9$ $\pm$ | $0.7$ | $-0.45$ | $-0.07$ | $0.04$ | … |  $8.35$ $\pm$ | $0.46$ | … | ? | –
ESO 357-012 | $49.222\,25$ | $-35.541\,36$ |  $31.48$ $\pm$ | $0.25$ | HyperLEDA | … |  $7.0$ $\pm$ | $0.4$ | $0.41$ | $0.76$ | $0.79$ | … |  $8.21$ $\pm$ | $0.38$ | … | 1 | 1
ESO 358-011 | $52.533\,00$ | $-32.475\,25$ |  $31.51$ $\pm$ | $0.15$ | Blakeslee et al. (2009) | … |  $-1.3$ $\pm$ | $4.3$ | $0.99$ | $1.10$ | $1.12$ | … |  $8.10$ $\pm$ | $0.38$ | … | ? | –
ESO 358-020 | $53.738\,71$ | $-32.639\,98$ |  $31.70$ $\pm$ | $0.73$ | HyperLEDA | … |  $9.3$ $\pm$ | $2.4$ | $0.02$ | $0.40$ | $0.51$ | … |  $8.67$ $\pm$ | $0.41$ | … | ? | –
… |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |

  * (a)

Either based on HyperLEDA, SIMBAD, or Blakeslee et al. (2009).

  * (b)

Nuclear classification: ‘1’ nucleated; ‘0’ non-nucleated; ‘?’ unknown.

  * (c)

References: (1) Georgiev & Böker (2014); (2) Muñoz et al. (2015); (3) Su et
al. (2021); (4) Turner et al. (2012); (5) Venhola et al. (2018).

Table 6:  List of galaxies for the Virgo galaxy cluster adopted from the
HyperLEDA data base. We removed galaxies which are not part of NGVS sample or
the Extended Virgo Cluster Catalog (Kim et al., 2014). The full data table is
available online.

Name | RA | DE | $d$ |  $d_{\mathrm{M}}$(a) | … | $T$ | $\Theta_{1}$ | $\Theta_{5}$ | $\Theta_{10}$ | MD | $\log_{10}M_{\star}$ | … |  Nuc.(b) |  Ref.(c)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [mag] |  |  |  |  |  |  |  |  [M⊙] |  |  |
IC 0767 | $182.761\,42$ | $12.103\,96$ |  $32.30$ $\pm$ | $0.55$ | HyperLEDA | … |  $-5.0$ $\pm$ | $1.0$ | $1.28$ | $1.41$ | $1.42$ | … |  $9.66$ $\pm$ | $0.12$ | … | ? | –
IC 0768 | $182.948\,33$ | $12.143\,61$ |  $31.09$ $\pm$ | $0.01$ | Mei et al. (2007) | … |  $5.9$ $\pm$ | $1.0$ | $2.18$ | $2.20$ | $2.21$ | … |  $9.14$ $\pm$ | $0.10$ | … | ? | –
IC 0769 | $183.134\,71$ | $12.123\,72$ |  $32.86$ $\pm$ | $0.19$ | HyperLEDA | … |  $4.0$ $\pm$ | $0.3$ | $0.78$ | $1.13$ | $1.17$ | … |  $10.10$ $\pm$ | $0.10$ | … | 1 | 2
IC 0771 | $183.805\,00$ | $13.184\,53$ |  $31.09$ $\pm$ | $0.01$ | Mei et al. (2007) | … |  $5.6$ $\pm$ | $1.3$ | $2.86$ | $2.96$ | $2.98$ | … |  $8.55$ $\pm$ | $0.19$ | … | ? | –
IC 0773 | $184.533\,75$ | $6.139\,57$ |  $31.09$ $\pm$ | $0.01$ | Mei et al. (2007) | … |  $-0.5$ $\pm$ | $1.2$ | $2.14$ | $2.49$ | $2.52$ | … |  $9.21$ $\pm$ | $0.09$ | … | ? | –
… |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |

  * (a)

Either based on HyperLEDA, SIMBAD, or Mei et al. (2007).

  * (b)

Nuclear classification: ‘1’ nucleated; ‘0’ non-nucleated; ‘?’ unknown.

  * (c)

References: (1) Côté et al. (2006); (2) Georgiev & Böker (2014); (3) Lisker et
al. (2007); (4)Sánchez-Janssen et al. (2019).