# What’s in a name? Quantifying the interplay between definition, orientation
and shape of UDGs using the Romulus Simulations

Jordan D. Van Nest  Homer L. Dodge Department of Physics & Astronomy,
University of Oklahoma, 440 W. Brooks St., Norman, OK 73019, USA  F. Munshi
Homer L. Dodge Department of Physics & Astronomy, University of Oklahoma, 440
W. Brooks St., Norman, OK 73019, USA  A.C. Wright  Department of Physics &
Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD
21218, USA  Department of Physics & Astronomy, Rutgers, The State University
of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA  M. Tremmel
Department of Astronomy, Yale University, New Haven, CT 06511, USA  A.M.
Brooks  Department of Physics & Astronomy, Rutgers, The State University of
New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA  D. Nagai
Department of Physics, Yale University, New Haven, CT 06520, USA  Department
of Astronomy, Yale University, New Haven, CT 06511, USA  T. Quinn  Astronomy
Department, University of Washington, Box 351580, Seattle, WA, 98195-1580

###### Abstract

We explore populations of ultra-diffuse galaxies (UDGs) in isolated,
satellite, and cluster environments using the Romulus25 and RomulusC
simulations, including how the populations vary with UDG definition and
viewing orientation. Using a fiducial definition of UDGs, we find that
isolated UDGs have notably larger semi-major ($b/a$) and smaller semi-minor
($c/a$) axis ratios than their non-UDG counterparts, i.e., they are more
oblate, or diskier. This is in line with previous results that adopted the
same UDG definition and showed that isolated UDGs form via early, high spin
mergers. However, the choice of UDG definition can drastically affect what
subset of a dwarf population are classified as UDGs, changing the number of
UDGs by up to $\sim 45\%$ of the dwarf population. We also find that a
galaxy’s classification as a UDG is dependent on its viewing orientation, and
this dependence decreases as environmental density increases. Overall, we
conclude that some definitions for UDG used in the literature manage to
isolate a specific formation mechanism for isolated dwarfs, while less
restrictive definitions erase a link to formation mechanism. Thus, how we
define UDG populations must be considered if we want to understand the
formation and evolution of UDGs.

galaxies:dwarf - galaxies:evolution

††journal: ApJ

##  1 Introduction

A significant population of very low surface brightness ($\mu_{0}>24$
mag/arcsec2) dwarf galaxies with large effective radii ($>1.5$ kpc) were
detected in the Coma cluster van Dokkum et al. (2015) and classified as
“ultra-diffuse galaxies” (UDGs). While we have known of the existence of low
surface brightness (LSB) galaxies for some time, hundreds of these
particularly diffuse galaxies have been discovered both in cluster
environments (Koda et al., 2015; Mihos et al., 2015; van der Burg et al.,
2016; Venhola et al., 2017; Danieli et al., 2017) and isolated environments
(Papastergis et al., 2017; Greco et al., 2018; Rong et al., 2017). Though
these galaxies generally have standard dwarf luminosities and metallicities
(Greco et al., 2018; Ferré-Mateu et al., 2018), their sizes are more
comparable to those of L∗ galaxies like the Milky Way. However, other studies
argue that these galaxies consist of the extreme end of the continuum of LSBs
(Tanoglidis et al., 2021), and that their sizes are consistent with those of
standard dwarf galaxies when considering the expected location of the gas
density threshold for star formation (Trujillo et al., 2020; Chamba et al.,
2020).

The recent conclusion that UDGs are so ubiquitous has prompted questions about
their formation. Do they originate in standard dwarf mass dark matter halos,
or are they failed L∗ mass galaxies within larger dark matter halos that for
some reason failed to build up their stellar population? Is their formation
driven by internal processes, such as bursty star formation and supernova
feedback, or external processes, such as tidal interactions and mergers? Or is
it possible that multiple types of UDGs exist, driven by different formation
mechanisms? The globular cluster content of UDGs has resulted in both
constraining UDGs to live in both ‘failed’ L∗ dark matter halos (van Dokkum et
al., 2017; Forbes et al., 2020; Doppel et al., 2021) and and also in standard
dwarf galaxy DM halos (Amorisco et al., 2018; Saifollahi et al., 2020;
Carleton et al., 2021). Some measurements even indicate UDGs with largely
under-massive dark matter halos (van Dokkum et al., 2018; Danieli et al.,
2019; van Dokkum et al., 2019).

Idealised simulations (e.g. Yozin & Bekki, 2015; Dutta Chowdhury et al.,
2019), analytic and semi-analytic models (e.g. Amorisco & Loeb, 2016; Ogiya,
2018; Rong et al., 2017; Carleton et al., 2019), and cosmological simulations
(e.g. Di Cintio et al., 2017; Chan et al., 2018; Jiang et al., 2019; Liao et
al., 2019; Martin et al., 2019) have all been utilized in order to better
understand UDG formation, resulting in multiple channels for their formation
including: dynamical origins, with UDGs forming through tidal heating and/or
stripping from interactions (e.g. Ogiya, 2018; Jiang et al., 2019; Liao et
al., 2019), bursty star formation and supernova feedback resulting in
extended, diffuse stellar distributions and UDG-like properties (e.g. Di
Cintio et al., 2017; Chan et al., 2018; Martin et al., 2019), ram pressure
stripping resulting in a extended, passively evolving stellar populations in
cluster galaxies (Tremmel et al., 2020), and finally, mergers resulting in
isolated UDG progenitors with extended effective radii, high spin, and low
central star formation (Wright et al., 2021).

The above demonstrates that theory predicts multiple formation channels for
UDGs, though conclusions may limited by both physical models in simulations
and limited statistics in observations. However, there is no consensus for
what constitutes the ‘ultra-diffuse’ designation; the questions of ‘how dim’
and ‘how large’ are all answered differently by different groups. Some
identify UDGs by the central surface brightness (e.g. van Dokkum et al., 2015;
Forbes et al., 2020), while others use the effective surface brightness (e.g.
Di Cintio et al., 2017; Cardona-Barrero et al., 2020) or the average surface
brightness within the effective radius (e.g. Koda et al., 2015; Leisman et
al., 2017; Martin et al., 2019). Many groups require the effective radii of
UDGs to be larger than 1.5 kpc (e.g. van Dokkum et al., 2015; Leisman et al.,
2017; Forbes et al., 2020) while others only require them to be larger than 1
kpc (e.g. Di Cintio et al., 2017; Cardona-Barrero et al., 2020). The question
then becomes: are we comparing apples to apples? Is each group identifying the
same population of galaxies as ultra-diffuse? Is this consistent across
environment? If not, does it matter?

To further complicate this question, we can consider the orientation of the
galaxy when fitting surface brightness profiles. While observers are limited
to observing galaxies from Earth, it is standard practice in simulations to
orient galaxies to face-on positions before identifying UDGs in order to
maximize the sample size (Di Cintio et al., 2017; Jiang et al., 2019; Liao et
al., 2019; Tremmel et al., 2020; Wright et al., 2021). But what role does
orientation play in identifying a UDG? Does this vary with environment? If a
dependence on orientation exists, it would stem from the morphology of the
galaxies. Intuitively, one would expect a largely spherical galaxy to appear
roughly equivalent at all viewing angles, whereas a ‘disky’ galaxy’s
appearance would be very dependent on orientation. As with the criteria for
being ‘ultra-diffuse’, there is no cohesive understanding about the shapes of
UDG populations. Some groups speculate that isolated UDGs favor a prolate
morphology (e.g. Burkert, 2017; Jiang et al., 2019) while others claim an
oblate-triaxial morphology is preferred (e.g. Rong et al., 2020; Kado-Fong et
al., 2021). Identifying a cohesive shape distribution for UDGs could provide
insight into how they evolve and what formation channels exist.

In this work, we study the populations of galaxies that common definitions of
UDG identify, and whether they are consistent across definitions. We also
explicitly test what effect the orientation of galaxies has on the UDG
populations in the Romulus25 and RomulusC simulations. Using the large sample
of dwarfs and UDGs in the Romulus simulations, which contain varying
environments from field to cluster, we have the ability to explicitly test the
above questions. In doing so, we demonstrate that the physical processes
separating UDGs from the underlying dwarf population can be explicitly
dependent on definition. Thus, it is imperative to consistently identify UDGs
in order to solve the mystery of their origin.

The paper is organized as follows: in Section 2, we detail the Romulus
simulations, and our sample of 1249 resolved dwarf galaxies in isolated,
satellite, and cluster environments, with anywhere from 354 to 990 UDGs
(depending on how they are identified). In Section 3, we study the shapes of
the dwarf galaxies in different environments, and the correlation between a
galaxy’s shape evolution and its status as a UDG. In Sections 4 we explore how
changing the UDG criteria and galaxies’ orientations affect the resultant UDG
population and our results in Section 3. Finally, we summarize our findings in
Section 5.

##  2 The Romulus Simulations

All galaxies analyzed in this work come from the Romulus25 (Tremmel et al.,
2017) and RomulusC (Tremmel et al., 2019) simulations. These are high
resolution cosmological simulations run using ChaNGa, an N-body+Smooth
Particle Hydrodynamics (SPH) code (Menon et al., 2015). ChaNGa implements
several physics models from its predecessor Gasoline (Wadsley et al., 2004);
however, its improved SPH implementation allows for better capture of fluid
instabilities through reduced artificial surface tension (Wadsley et al.,
2017). Physics below the resolution limit, such as stellar formation and
feedback as well as super massive black hole (SMBH) growth and feedback, are
governed by sub-grid prescriptions.

The Romulus25 simulation is a 25 Mpc-per-side uniform volume simulation. The
simulated galaxies match important $z=0$ scaling relations, including the
stellar mass to halo mass relation (Moster et al., 2013) and the stellar mass
to SMBH mass relation (Schramm & Silverman, 2013). Furthermore, Tremmel et al.
(2017) demonstrated the simulation’s ability to produce realistic galaxies
across four orders of magnitude in halo mass, ranging from dwarf galaxies
resolved with over 10,000 particles to groups, while reproducing observations
of high redshift SMBH growth and star formation.

The RomulusC simulation is a cosmological zoom-in simulation of a galaxy
cluster. At $z=0$ the cluster has an $R_{200}$ (the radius at which the
enclosed density is 200 times the critical density of the universe) of 1033
kpc and an $M_{200}$ (the mass enclosed within $R_{200}$) of $1.15\times
10^{14}$ M⊙ . The initial conditions for the cluster were extracted from a 50
Mpc-per-side uniform volume simulation using the re-normalization technique of
Katz & White (1993).

Both of the Romulus simulations are evolved to $z=0$ with a $\Lambda$CDM
cosmology ($\Omega_{0}=0.3086,\Lambda=0.6914,h=0.6777,\sigma_{8}=0.8288$)
following Planck Collaboration et al. (2014). Gravitational interactions are
resolved with a spline force softening length of 350 pc, which is a Plummer
equivalent of 250 pc, that converges to Newtonian force at 700 pc. The
simulations oversample dark matter particles, such that the initial high
resolution dark matter particle count is 3.375 times that of the gas
particles. This oversampling results in dark matter and gas particles with
similar masses, $3.39\times 10^{5}$ M⊙ and $2.12\times 10^{5}$ M⊙
respectively. The similar particle masses aid in reducing the numerical
effects from two-body scattering and energy equipartition, both of which can
lead to spurious growth in galaxy sizes within simulations (Ludlow et al.,
2019). The increased resolution in dark matter also gives the simulations the
ability to track the dynamics of SMBHs within galaxies (Tremmel et al., 2015).
Additionally, the simulations are allowed a more realistic treatment of weak
and strong shocks (Wadsley et al., 2017) through an on-the-fly time-step
adjustment and time-dependent artificial viscosity (Saitoh & Makino, 2009).
The sub-grid parameters have been optimized to create galaxies across a halo
mass range of $10^{10.5-12}$ M⊙, and an updated implementation of turbulent
diffusion (Wadsley et al., 2017) allows for the formation of a realistic metal
distribution within galaxies (Shen et al., 2010) and the intra-cluster medium
(Wadsley et al., 2008; Tremmel et al., 2019; Butsky et al., 2019).

###  2.1 Sub-Grid Physics and Star Formation

To approximate reionization effects, the Romulus simulations include a cosmic
UV background (Haardt & Madau, 2012) with self-shielding from Pontzen et al.
(2008). The simulations implement primordial cooling for neutral and ionized H
and He. This cooling is calculated from H and He line cooling (Cen, 1992),
photoionization, radiative recombination (Black, 1981; Verner & Ferland,
1996), collisional ionization rates (Abel et al., 1997), and bremsstrahlung
radiation. Although the Romulus simulations are high resolution, they lack the
ability to resolve the multiphase interstellar medium (ISM) or track the
creation and annihilation of molecular hydrogen. It has been shown that
simulating metal-line cooling at low resolution and without the presence of
molecular hydrogen physics can lead to over cooling in spiral galaxies
(Christensen et al., 2014), thus the simulations do not include high-
temperature metal line cooling (see Tremmel et al. (2019) for more details on
this omission). Low-temperature metal line cooling is implemented following
Bromm et al. (2001).

Star formation (SF) in the Romulus simulations is a stochastic process
governed by sub-grid models. In simulations of this resolution (Stinson et
al., 2006), SF is regulated by parameters that determine the physical
requirements for gas to become star forming, the efficiency of the SF, and the
coupling of supernova (SN) energy to the ISM:

  * •

Gas must have a minimum density of n${}_{\text{SF}}=0.2\text{cm}^{-3}$ and a
temperature of no greater than T${}_{\text{SF}}=10^{4}$ K in order to form
stars.

  * •

The probability $p$ of a star particle forming from a gas particle with
dynamical time $t_{\text{dyn}}$ is given by:

$p=\frac{m_{\text{gas}}}{m_{\text{star}}}(1-e^{-c_{\text{SF}}\Delta
t/t_{\text{dyn}}})$ (1)

where $c_{\text{SF}}$ is the star-forming efficiency factor (here set to 0.15)
and $\Delta t$ is the formation timescale (here set to $10^{6}$ years).

  * •

The fraction of the canonical 1051 erg SN energy coupled to the ISM is
$\epsilon_{\text{SN}}=0.75$.

When star particles are formed, they have a mass equivalent to 30% of the
initial gas particle mass, i.e., M${}_{\star}=6\times 10^{4}$ M⊙, and
represent a single stellar population with a Kroupa (2001) initial mass
function. Stars whose masses are in the range of 8-40 M⊙ undergo Type II
supernovae which are implemented using ‘blastwave’ feedback following Stinson
et al. (2006). The SN injects thermal energy into nearby gas particles, where
the adiabatic expansion phase is replicated by temporarily disabling cooling.
Mass and metal diffusion into the ISM follow Shen et al. (2010) and Governato
et al. (2015). Stars whose mass exceed 40 M⊙ are assumed to collapse directly
into a black hole.

The Romulus simulations include a novel implementation of black hole physics
(Bellovary et al., 2010; Tremmel et al., 2015, 2017, 2018b, 2018a, 2019). To
ensure that SMBHs are seeded in gas collapsing faster than the star formation
or cooling time scales, they only form in regions of pristine (Z $<3\times
10^{-4}$ Z⊙ for Romulus25 and Z $<10^{-4}$ Z⊙ for RomulusC), dense
($n>15n_{\text{SF}}$), cool ($9.5\times 10^{3}<$ T $<10^{4}$ K) gas. The seed
mass is $10^{6}$ M⊙ and most SMBHs form within the first Gyr of the
simulation. An implementation of a dynamical friction sub-grid model (Tremmel
et al., 2015) allows the tracking of SMBH orbits as they move freely within
their host galaxies. Growth of SMBHs is modeled via a modified Bondi-Hoyle
accretion formalism that accounts for gas supported by angular momentum.
Feedback from an active galactic nucleus is approximated by converting a
fraction (0.2%) of accreted mass into thermal energy and distributing it to
the surrounding gas particles.

###  2.2 Halo Identification

The Romulus simulations use the Amiga Halo Finder (AHF; Knollmann & Knebe,
2009) to identify dark matter halos, sub-halos, and their associated baryonic
content. AHF uses a spherical top-hat collapse technique to determine the each
halo’s virial mass (Mvir) and radius (Rvir), following a procedure similar to
Bryan & Norman (1998). Halos are considered resolved if they have a virial
mass of at least $3\times 10^{9}$ M⊙, corresponding to a dark matter particle
count of $\sim 10^{4}$, and a stellar mass of at least $10^{7}$ M⊙,
corresponding to a star particle count of $\sim 150$, at z=0. Galaxies are
considered resolved dwarf galaxies if they meet the above criteria, as well as
having a stellar mass less than 109 M⊙. When calculating stellar masses, we
use photometric colors following Munshi et al. (2013) to better represent
values inferred from typical observational techniques. In RomulusC we identify
a sample of 201 dwarf galaxies in a cluster environment. In Romulus25 we have
a sample of 377 dwarf galaxies in a satellite environment, where a galaxy is
classified as a satellite if its center is within the virial radius of a
larger halo. We identify a sample of 671 dwarf galaxies in an isolated
environment in Romulus25\. The criteria for being isolated follows from Geha
et al. (2012), and requires being neither a satellite of another halo nor
within 1.5 Mpc of any galaxy with M${}_{\star}>2.5\times 10^{10}$ M⊙. Galaxies
are analyzed using Pynbody (Pontzen et al., 2013) and tracked across time
using Tangos (Pontzen & Tremmel, 2018), both of which are publicly available.

###  2.3 UDG Identification

The process of identifying UDGs follows the previous studies of UDGs in the
Romulus simulations (Tremmel et al., 2020; Wright et al., 2021). When
analyzing a galaxy, we first orient it in a face-on position based on the
angular momentum. This calculation is performed on the gas particles within
the inner 5 kpc of the galaxy, or the star particles when less than 100 gas
particles are present. Once oriented, we generate azimuthally averaged surface
brightness profiles with 300 pc sampling (the spatial resolution of the
Romulus simulations). We generate profiles in Johnson B, V, and R bands, and
calculate g-band surface brightness from the B and V bands following Jester et
al. (2005). We then fit a Sérsic profile through all points brighter than 32
mag/arcsec2, the general depth of sensitive observations (Trujillo & Fliri,
2016; Borlaff et al., 2019). The Sérsic profile is defined as

$\mu(r)=\mu_{\text{eff}}+2.5(0.868n-0.142)\left(\left(\frac{r}{r_{\text{eff}}}\right)^{1/n}-1\right)$
(2)

where $\mu(r)$ is the surface brightness at radius r, r${}_{\text{eff}}$ is
the effective radius (half-light radius), $\mu_{\text{eff}}$ is the effective
surface brightness (surface brightness at the effective radius), and $n$ is
the Sérsic index (Sérsic, 1963). When fitting a Sérsic profile we allow the
effective surface brightness to range between 10-40 mag/arcsec2, the effective
radius to range between 0-100 kpc, and the Sérsic index to range between
0.5-16.5. In Section 4 we explore different definitions for UDGs, but in
Section 3 we first adopt our fiducial definition. For our fiducial definition,
a halo is considered ultra-diffuse if its central g-band surface brightness is
dimmer than 24 mag/arcsec2 and its effective radius is larger than 1.5 kpc,
following van Dokkum et al. (2015).

Figure 1: The $c/a$ axis ratios plotted against the $b/a$ axis ratios for the
dwarf galaxies’ stellar distributions at z=0 in various environments. The
populations are separated into UDGs and non-UDGs according to Section 2.3. The
bold points represent the medians of the populations, while the error bars
show the 25${}^{\text{th}}$ to 75${}^{\text{th}}$ percentiles. Isolated
galaxies, in general, are ‘diskier’ in morphology than the satellite and
cluster galaxies, though isolated UDGs are the most ‘disky’ population. Figure
2: The isolated galaxies’ stellar $b/a$ and $c/a$ axis ratios as a function of
time elapsed since their final major merger ($t=0$). The shaded regions
represent the 25${}^{\text{th}}$ to 75${}^{\text{th}}$ percentiles of the UDG
(red) and non-UDG (blue) populations. Intermediate and high mass UDGs evolve
to higher $b/a$ values and lower $c/a$ values than their non-UDG counterparts.
Figure 3: The spin parameter, $\lambda$, for isolated galaxies separated into
UDGs and non-UDGs for the fiducial definition. The top row contains galaxies
that finished merging within the first 5 Gyr of the simulation, while to
bottom row contains galaxies with major mergers after 5 Gyr. The shaded
regions represent the 25${}^{\text{th}}$ to 75${}^{\text{th}}$ percentiles.
Intermediate and high mass, early merging UDGs evolve to higher spin values
after merging. While intermediate and high mass late merging galaxies evolve
to similarly high spins, they do not typically result in UDGs.

##  3 Shapes of UDGs and non-UDGs

To study possible differences between UDG and non-UDG morphology in different
environments, we analyzed the $b/a$ and $c/a$ axis ratios for all dwarf
galaxies, which correspond to the major axis ratios when looking at the
galaxies from face-on and edge-on orientations, respectively. Hence, a
$b/a\approx c/a\approx 1$ would be a spherical distribution, a $b/a\gg c/a$
would be an oblate (or ‘disky’) distribution, and a $b/a\approx c/a<1$ would
be a prolate distribution. Axis ratios are derived from the eigenvalues of the
shape tensor, which is calculated by dividing the moment of inertia tensor by
the total mass. Shape tensors are generated using star particles which are
separated into radial bins. The values presented in this work come from the
stellar particle bin located at twice the half-light radius.

Figure 1 shows the stellar $c/a$ axis ratios plotted against the stellar $b/a$
axis ratios for dwarf galaxies in all environments at z=0. In general, we see
that the isolated galaxies reside closer to the lower-right hand corner of the
plot, indicating a more disk-like structure. We find that in the cluster and
satellite environments, the $b/a$ and $c/a$ axis ratios for UDGs are in good
agreement with non-UDGs, however, the isolated galaxies show a notable
disparity between the two populations. In general, the isolated UDG population
is more oblate than the underlying dwarf population.

Our results seem to contrast those of Jiang et al. (2019), who found that
field UDGs are more prolate than non-UDGs. While we are unsure why our results
differ, there are some probable causes that can be ruled out. In this work we
perform shape measurements using stellar particles in a bin around 2 half-
light radii, while Jiang et al. (2019) used all stellar particles within 1
half-light radius. However, if we calculate our shapes in a similar method, we
still find that our isolated UDGs are more oblate than the non-UDGs. Another
possible explanation for our differing results is the Romulus simulations’
inability to form cores. The Romulus simulations do not resolve the multiphase
interstellar medium, thus stars must be allowed to form in gas that is
relatively diffuse. As a result, the galaxies lack the bursty, clustered
central star formation required to produce feedback strong enough create a
core in the dark matter halo (e.g., Dutton et al., 2020). However, in
performing our shape analysis on the “Marvel-ous Dwarfs” zoom simulations
(Munshi et al., 2021), we find results consistent with the isolated galaxies
in Romulus, where UDGs are more oblate. Since the “Marvel-ous” suite has the
resolution to form cores, this suggests that core formation (or lack thereof)
is not influencing our shape analysis in Romulus. We note, however, that a
recent study by Kado-Fong et al. (2021) found that the inferred 3-dimensional
shapes of observed LSB galaxies are well-characterized by oblate spheroids,
with $c/a$ values higher than those of high mass dwarfs with thick disks. The
LSB galaxies in the Romulus simulations (defined as having $r_{\text{eff}}>1$
kpc and $\bar{\mu}_{\text{eff,g}}>24.3$ mag/arcsec2) were found to be in good
agreement with the observations (see Figure 8 in Kado-Fong et al. (2021)).

This disparity in the isolated UDG and non-UDG morphologies suggests that the
two populations have different formation histories. Wright et al. (2021) found
that isolated dwarfs in Romulus25 all undergo similar numbers of mergers, both
major and minor, that UDGs are primarily the products of early ($>8$ Gyr ago)
major mergers. Here we consider what effect this has on the shape evolution of
the isolated galaxies. Figure 2 shows the stellar $b/a$ and $c/a$ axis ratios
for the isolated galaxies as a function of time elapsed since a galaxy’s last
major merger. A time of $t=0$ represents the time of a halo’s final major
merger, with a time of $t<0$ being the time preceding the final merger, and a
time of $t>0$ being the time elapsed since the merger. The time $t=0$ is
chosen to be the instant that the virial radii of the merging halos first
overlap, following Hetznecker & Burkert (2006). The galaxies are split into
columns according to stellar mass. The bold lines represent the populations’
median values, while the shaded regions cover the 25${}^{\text{th}}$ to
75${}^{\text{th}}$ percentiles. Note that all of the UDGs in the high mass
bin, and most other UDGs, undergo their last major merger (LMM) within the
first 5 Gyr of the simulation while the LMM for non-UDGs are much more spread
out in time. After their LMM, both UDGs and non-UDGs experience a gradual
increase in $b/a$. Since UDGs more typically have early LMMs this could result
in a z=0 $b/a$ disparity between dwarfs and UDGs as seen in Figure 1. However,
when scaling to LMM time we see that intermediate and high mass UDGs still
evolve to higher $b/a$ values than even their early-merging non-UDG
counterparts, suggesting that something in addition to early mergers is
driving a change in shapes. The divergence from this trend in the lowest mass
bin is likely a combination of over-quenching due to resolution effects and
that these galaxies are genuinely dispersion supported.

While UDGs are more likely to experience an earlier LMM, this does not fully
explain the discrepancy in shapes. This is consistent with Wright et al.
(2021), who found that while isolated UDGs were largely influenced by early
mergers, that alone was not sufficient to explain their formation. In fact,
Wright et al. (2021) found that mergers were not only earlier on average, but
also tended to produce a larger increase in specific angular momentum. In
Figure 3, we examine the evolution in halo spin, now also splitting the galaxy
population based on early and late-time mergers. Here the spin parameter is
the Bullock et al. (2001) spin. Examining the early mergers (top row), we find
that mergers which result in intermediate and massive UDGs at z$=0$ have more
angular momentum which results in a significant increase in halo spin
beginning at the last major merger. Galaxies which experience late last major
mergers also see an increase in spin, but this is much less likely to produce
an UDG at z$=0$. We find that UDGs experience less of an increase in $c/a$
post-LMM; however we note that the slight difference in the $c/a$ shape post-
LMM may be attributed to the differences in spin of UDG and non-UDG LMMs. Thus
we confirm that the combination of early and high spin mergers is required to
form isolated UDGs as demonstrated in Wright et al. (2021), and that this is
the driving force behind the difference in shapes between isolated UDGs and
isolated dwarfs. To summarize: we show that post-LMM, a dwarf galaxy’s $b/a$
value increases as the galaxy relaxes, such that those with earlier mergers
(UDGs) have larger b/a values by z$=0$ (see Fig. 2). For UDGs, $c/a$ remains
lower for UDGs when compared to dwarf galaxies because their mergers have
higher spin (see Figs 2 and 3). As such, we predict that observations of
stellar shape can be used to identify major merger history of dwarf galaxies
and a formation channel for UDGs (early, high spin mergers).

##  4 Effects of Changing Definition and Orientation

In Section 3, we showed how the unique formation mechanism for isolated UDGs
identified in Wright et al. (2021) leads to differences in UDG and non-UDG
morphology. During our study, we found that using alternate UDG criteria can
cause these differences to disappear, removing any indication of the
underlying formation mechanisms. To study the effect that the UDG definition
has on both the above results and the resultant UDG population as a whole, we
analyze our dwarf sample using multiple sets of UDG criteria.

###  4.1 Definitions of UDG

To illustrate this point, we consider five different definitions for ultra-
diffuse galaxy. Each of these definitions are used in recent literature, and
are summarized in Table 1:

  * •

G0-F: This definition comes from van Dokkum et al. (2015). It requires the
central g-band surface brightness to be fainter than 24 mag/arcsec2 and the
effective radius to be larger than 1.5 kpc. While performing the Sérsic fits
for this definition, we allow the Sérsic index to vary as a free parameter.
This is the definition outlined in Section 2.3 and used in the results
presented in Section 3.

  * •

G0-1: This definition is the same as G0-F, but when performing the Sérsic
fits, the Sérsic index is fixed at a value of 1, keeping in line with the
original method in van Dokkum et al. (2015).

  * •

R0: This definition comes from Forbes et al. (2020). It requires the central
R-band surface brightness to be fainter than 23.5 mag/arcsec2 and the
effective radius to be larger than 1.5 kpc. While performing the Sérsic fits
for this definition, we allow the Sérsic index to vary as a free parameter.

  * •

RE: This definition comes from Di Cintio et al. (2017). It requires the
effective R-band surface brightness to be fainter than 23.5 mag/arcsec2 and
the effective radius to be greater than 1 kpc. Additionally, the authors
require the R-band absolute magnitude to be within -16.5 to -12. While
performing the Sérsic fits for this definition, we allow the Sérsic index to
vary as a free parameter. We note that the authors define the effective
surface brightness as L/(2$\pi r^{2}_{\text{eff,R}}$) rather than
$\mu_{\text{R}}$($r_{\text{eff}}$) (the value returned by the Sérsic fit), so
we adopt this method as well when identifying UDGs with this definition. Using
$\mu_{\text{R}}$($r_{\text{eff}}$) results in a slight increase in the
obtained value for $\mu_{\text{eff}}$ ($\approx 1$ mag/arcsec2), but only
results in a slight difference in the number of UDGs ($\approx 0-4$% of each
environments’ populations) and has no bearing on the results of this work.

  * •

RĒ: This definition comes from van der Burg et al. (2016). It requires the
average R-band surface brightness within the effective radius to be between
24-26.5 mag/arcsec2 and the effective radius to be larger than 1.5 kpc. While
performing the Sérsic fits for this definition, we allow the Sérsic index to
vary as a free parameter.

Name | Surface Brightness [$\frac{\text{mag}}{\text{arcsec}^{2}}$] | r${}_{\text{eff}}$ [kpc] | Sérsic Index | Absolute Magnitude | Reference
---|---|---|---|---|---
G0-F |  $\mu_{\text{g}}$(0) $>$24 | r${}_{\text{eff}}>$ 1.5 | Free | N/A | van Dokkum et al. (2015)
G0-1 |  $\mu_{\text{g}}$(0) $>$24 | r${}_{\text{eff}}>$ 1.5 | n = 1 | N/A | van Dokkum et al. (2015)
R0 |  $\mu_{\text{R}}$(0) $>$23.5 | r${}_{\text{eff}}>$ 1.5 | Free | N/A | Forbes et al. (2020)
RE |  $\mu_{\text{eff,R}}>$23.5 | r${}_{\text{eff}}>$ 1 | Free |  $-12>$ M${}_{\text{R}}>-16.5$ | Di Cintio et al. (2017)
RĒ | 26.5 $>\langle\mu_{\text{R}}(r_{\text{eff}})\rangle>$ 24 | r${}_{\text{eff}}>$ 1.5 | Free | N/A | van der Burg et al. (2016)
Table 1: A summary of the UDG definitions considered in this work.

###  4.2 Definition Effects

Definition | Environment | N${}_{\text{UDG,faceon}}$ | Percentage
---|---|---|---
G0-F | Isolated | 134 | $19.97\pm 1.73$
Satellite | 142 | $37.67\pm 3.16$
Cluster | 122 | $60.70\pm 5.50$
G0-1 | Isolated | 241 | $35.92\pm 2.31$
Satellite | 230 | $61.01\pm 4.02$
Cluster | 167 | $83.08\pm 6.43$
R0 | Isolated | 125 | $18.63\pm 1.67$
Satellite | 125 | $33.16\pm 2.97$
Cluster | 104 | $51.74\pm 5.07$
RE | Isolated | 454 | $67.66\pm 3.18$
Satellite | 309 | $81.96\pm 4.66$
Cluster | 168 | $83.58\pm 6.45$
RĒ | Isolated | 194 | $28.91\pm 2.08$
Satellite | 180 | $47.75\pm 3.56$
Cluster | 132 | $65.67\pm 5.72$
Table 2: The number of galaxies that are identified as a UDG at face-on
orientation for each environment and UDG definition. The number is also given
as a fraction of the environment’s total dwarf population with Poisson error.

The number of galaxies that are identified as UDGs are given in Table 2 for
each environment and definition. The process of identification follows from
Section 2.3 for each definition (i.e., we first orient each galaxy to a face-
on position, but explore orientation effects below), with the appropriate size
and surface brightness restrictions applied. Regardless of definition, the
percent of dwarf galaxies that would be identified as UDGs generally increases
with the density of the environment. This trend with environment matches
predictions from Martin et al. (2019) using the Horizon-AGN simulation (Dubois
et al., 2014). The authors found that UDGs (here defined as
$\langle\mu_{\text{R}}(r_{\text{eff}})\rangle>24.5$) represent a significant
percentage of the galaxy population, and this percentage increases with
environmental density. These results also agree with those from Jackson et al.
(2020), who found that UDGs exist in large numbers in groups and the field
within the NewHorizon Simulation (Dubois et al., 2021), as well as with
observational results (e.g. van der Burg et al., 2017).

Although the trend with environment is ubiquitous across most definitions, the
total number of UDGs identified is not. The G0-F and R0 definitions are much
more restrictive in their UDG identification compared to the RE definition
which identifies the largest population of UDGs in all environments. The G0-1
and RĒ definitions tend to fill in the middle ground between the G0-F/R0 and
RE definitions in terms of the size of the UDG populations.

###  4.3 Orientation Effects

Since all of our definitions of UDG are derived from surface brightness
profiles, a galaxy’s status as a UDG is intrinsically dependent on its viewing
orientation. It is standard practice when searching for UDGs within
simulations to first orient the galaxy to a face-on position (Di Cintio et
al., 2017; Tremmel et al., 2020; Wright et al., 2021). This, in theory, will
maximize the galaxy’s effective radius while minimizing its central surface
brightness, optimizing its likelihood of being identified as a UDG. Here, we
study whether this assumption holds true for all galaxies, and how a galaxy’s
orientation could affect its status as a UDG. Some previous studies (e.g. Chan
et al., 2018; Jackson et al., 2020; Cardona-Barrero et al., 2020) have
considered multiple viewing angles (such as the three primary axes or a span
of inclinations), but here we consider 288 positions from incremental
rotations over 2 axes perpendicular to the line-of-sight (see 4(b) for a
visual guide). The entirety of these positions detail how the galaxy is viewed
from any position in 3-dimensional space.

As we describe in 2.3, when analyzing a galaxy, we first orient it to a face-
on position. After the initial orientation, the galaxy is then rotated by an
angle $\theta$ around an axis perpendicular to the line of sight (the angular
momentum vector in face-on position). Next the galaxy is rotated by an angle
$\phi$ around the axis perpendicular to the line of sight and previous axis of
rotation. An azimuthally averaged surface brightness profile for the galaxy is
generated and fit with a Sérsic profile which is used to determine whether the
galaxy is identified as a UDG at the given orientation. The $\theta$-rotations
and $\phi$-rotations considered in this work span the spaces of [0°,180°) and
[0°,360°) respectively, both in increments of 15°. Since the radial surface
brightness profile of a galaxy is unaltered under rotations around the line of
sight, any ($\theta,\phi$) where $\theta<180$° is analogous to a rotation of
($180\degree+\theta,180\degree-\phi\ [mod360]$) where $[mod360]$ (modulo 360°)
means that if $180\degree-\phi=-x<0$, the value becomes $360\degree-x$. Thus
the $\theta$ ($\phi$) rotation space of [180°,360°) ([0°,360°)) is bijectively
mapped to the [0°,180°) ([0°,360°)) space, and we have analyzed all unique
positions. The top panel of Figure 4(a) illustrates the rotations considered
in this work using a flat, circular disk.

(a) A visual representation of the rotated galaxy positions considered in this
work. The sample ‘galaxy’ is a flat, circular disk with one face blue and the
opposite face red. The disk has a single dark colored radius and multicolored
vertices to help visualize the rotations.

(b) Right: An example of the rotation dependent UDG status of an isolated
galaxy from Romulus25 with M${}_{\star}=3.26\times 10^{8}$ M⊙. The grid shows
whether the galaxy is identified a UDG (red) or a standard dwarf (blue) at the
given rotation under the G0-F definition. Left: Example g-band surface
brightness profiles and their associated Sérsic fits of the isolated galaxy
for ($\theta,\phi$) rotations of ($0\degree,0\degree$; i.e., face-on) and
($90\degree,0\degree$; i.e., edge-on). Fit parameters are given in the upper
right hand corners of the surface brightness plots. To the left of the plots
are UVI images of the galaxy at the two orientations showing all features
brighter than 32 mag/arcsec2(the region being fit by the Sérsic profile).

Figure 4: A visual representation of the galaxy rotations considered in this
work, along with an example galaxy.

The number of galaxies that are identified as UDGs when considering any
orientation are given in Table 3 for each environment and definition. The
percent of dwarf galaxies that are UDGs still increases with environmental
density, but the number of UDGs has increased by a significant margin in
almost every case. The value $\Delta$ in Table 3 represents the number of UDGs
gained when considering orientations other than face-on, i.e. the number of
galaxies that are not classified as a UDG at a face-on orientation but are
classified as such at some other orientation. Although this value decreases
with environmental density, it is present in all definitions, indicating that
by analyzing all galaxies in a face-on position, one is likely to omit a
significant fraction of galaxies that could potentially be identified as UDGs.
This fraction can be quite large in the isolated environment, ranging from
7-20% of isolated dwarf galaxies, but more minor in the denser environments,
ranging from 2-17% in satellites and 0-13% in the cluster. This also hints at
isolated galaxies’ classifications as UDGs being more sensitive to
orientation, a result we would expect based on their more ‘disky’ morphology
seen in Figure 1.

Def. | Env. | N${}_{\text{UDG,any}}$ | Percentage |  $\Delta_{\mathrm{face-on}}$ (%)
---|---|---|---|---
G0-F | Isolated | 191 | $28.5\pm 2.1$ | 29.5
Satellite | 186 | $49.3\pm 3.6$ | 23.7
Cluster | 148 | $73.6\pm 6.1$ | 17.6
G0-1 | Isolated | 379 | $56.5\pm 2.9$ | 36.4
Satellite | 293 | $77.7\pm 4.5$ | 21.5
Cluster | 186 | $92.5\pm 6.8$ | 10.2
R0 | Isolated | 197 | $29.4\pm 2.1$ | 36.5
Satellite | 176 | $46.7\pm 3.5$ | 28.0
Cluster | 129 | $64.2\pm 5.7$ | 19.4
RE | Isolated | 502 | $74.8\pm 3.3$ | 10.0
Satellite | 320 | $84.9\pm 4.7$ | 3.4
Cluster | 168 | $83.6\pm 6.5$ | 0.0
RĒ | Isolated | 260 | $38.8\pm 2.4$ | 25.4
Satellite | 224 | $59.4\pm 4.0$ | 19.6
Cluster | 155 | $77.1\pm 6.2$ | 14.8
Table 3: The number of galaxies that are identified as UDGs at any orientation
for each environment and UDG definition. The number is also given as a
fraction of the environment’s total dwarf population with Poisson error. The
value $\Delta_{\mathrm{face-on}}$ is the fraction of UDGs that were not
identified using the face-on orientation, as used in Table 2. Figure 5: Color
grids representing the UDG populations for each environment and definition of
UDG. Each point on the grid represents the UDG population at that particular
orientation. The value plotted is the number of galaxies that are identified
as UDGs at that orientation as a percentage of the total number of galaxies
that are identified as UDGs at any orientation (i.e., the “Total UDG Count”
for that environment and definition). In all environments and definitions, the
UDG count increases as you move from edge-on positions to face-on ones. In
each definition, there is the most rotation dependence (color contrast) in the
isolated environment and the least in the cluster environment.

To further investigate whether isolated UDGs are less robust to orientation,
we look at the different environments’ total UDG population at each specific
orientation. Figure 5 summarizes these results with a color grid for each
environment and UDG definition. Each point on the grid represents the total
number of galaxies that are identified as ultra-diffuse at that orientation as
a fraction of the total number of UDGs in that environment
(N${}_{\text{UDG,any}}$ from Table 3). In all definitions, the isolated UDG
populations demonstrate much stronger orientation dependence than their denser
environment counterparts. The more restrictive definitions, G0-F and R0, show
the strongest orientation dependence, with the percentages varying by $\sim
55-60$% in the isolated environment. In contrast, the least restrictive
definition, RE, also appears the be the least orientation dependent, with the
percentages varying by $\sim 15$% in the isolated environment and essentially
0% in the cluster environment (there are only 2 cluster galaxies that are
identified as RE-UDGs at some orientations, but not all). The RE definition,
in general, seems to be identifying the highest fraction of dwarf galaxies as
UDGs at all orientations, which is expected from its $\Delta$ values in Table
3 being the smallest (especially when considering that the RE
N${}_{\text{UDG,any}}$ values were typically larger than other definitions).
Again, the G0-1 and RĒ definitions seem to fill out the middle ground, both in
terms of orientation dependence and the average percentage of UDGs retained
across orientations. In all cases, the UDG populations are maximized in the
positions that are more face-on than edge-on. This suggests that if simulators
were to analyze all galaxies from the same orientation in order to identify
UDGs, choosing face-on would likely produce the largest UDG population, but we
have shown that allowing for any orientation will produce the maximum possible
number of UDGs.

Figure 6:  Top: Normalized histograms of what percent of the time a galaxy
would be identified as a UDG when oriented randomly. Galaxies that are never
considered ultra-diffuse are not included. From left to right, the UDG
definitions used were G0-F, RĒ, and RE. In the G0-F definition (the most
restrictive), a large number of galaxies in all environments exist in the
0-20% bins, indicating a high rotation dependence. In the RE definition (the
least restrictive), most galaxies exist in the 95-100% bin, indicating little
to no rotation dependence. Bottom: Surface brightness versus effective radii
plots, including dotted lines marking the selection criteria for UDGs.
Galaxies are plotted using their face-on values. Points labeled as UDG (red)
are those that are identified as such at any orientation, while points
labelled as non-UDGs (blue) are never classified as such. As above, the UDG
definitions used were G0-F, RĒ, and RE. Blue “X”s denote galaxies that violate
the R-band absolute magnitude restriction under the RE definition. Different
definitions identify different subsections of the dwarf population to be UDGs,
with the RE definition selecting a notably larger area.

In addition to studying how the statistics of UDG populations vary with
orientation, we analyze how the individual galaxies vary under rotation with
each definition, as well as how the definitions change the population of
galaxies identified as ultra-diffuse. Figure 6 shows this analysis for the
G0-F, RĒ, and RE definitions (the R0 and G0-1 definitions are approximated
here by the G0-F and RĒ definitions respectively). The top row shows the
normalized distributions of individual galaxies’ UDG percentages, i.e., what
percent of the time a galaxy would be identified as a UDG if its orientation
was drawn at random (cos($\theta$) uniform in [-1,1] and $\phi$ uniform in
[0,2$\pi$]). In the G0-F and RĒ definitions, the isolated galaxies tend to
occupy the low-percentage space, particularly 0-15%, while a large fraction of
satellite and cluster galaxies occupy the 95-100% bin. The RĒ definition shows
very similar results to the G0-F definition, but sees some galaxies in all
environments shift from the lower percentages to the higher ones. In contrast,
the RE definition sees most galaxies in all environments occupying the 95-100%
bin, mirroring the lack of orientation dependence seen in Figure 5. This
implies that if a galaxy is identified as a UDG by the RE definition at some
orientation, it is likely to be identified as such at any orientation.

The bottom row shows all of our dwarfs in surface brightness versus effective
radius space with their face-on values. Also shown are the boundaries for UDG
classification. The RE definition, while being the most robust to orientation,
demarcates the largest area of our dwarf population as ultra-diffuse. A large
number of galaxies are not identified as RE-UDGs only because they violate the
absolute magnitude restriction.

Using the individual galaxies’ UDG percentages, we can construct mass
functions. Figure 7 shows the UDG mass functions for each environment using
the G0-F definition, assuming each galaxy is oriented randomly. Rather than
using the binary choice of UDG or non-UDG (adding a 1 or 0 to the mass
function), a galaxy’s contribution is determined by its UDG percentage (the
value plotted in the top row of Figure 6) and the resultant mass function is
normalized to the size of the environment’s dwarf population. The dotted lines
are the ‘maximum’ mass functions, representing the idealized situation where
every galaxy that could be identified as a UDG at some orientation is counted.
The bottom plot shows the fraction of the maximum mass function that is
occupied by the random orientation mass function. In the isolated and cluster
environments, the N/N${}_{\text{max}}$ percentage is relatively constant
between 107 and $10^{8}$ M⊙, after which it starts to decrease. The satellite
fraction steadily increases until $3\times 10^{8}$ M⊙ where the sample size
approaches zero. When switching from the maximum mass function to one that
accounts for random orientations, we see the greatest difference in the
isolated UDGs as demonstrated in the bottom panel of Figure 7. This is
consistent with our shape results: compared to the cluster and satellite UDGs,
isolated UDGs have a less spherical shape (more oblate)- which implies that
orientation will alter their classification. The more spherical populations
are more robust to this, again demonstrated in the bottom panel of Figure 7.
None of the environments yield perfectly spherical populations thus all
populations have mass functions that depend on rotation. This is important to
consider when comparing simulations to observations. Since simulations
typically assemble their UDG population from face-on analysis, it may not be
appropriate to compare the resultant UDG mass function to one from
observations where galaxies are oriented randomly.

Figure 7:  Top: The mass functions for G0-F UDGs when all dwarfs are oriented
randomly, normalized to the size of the environment’s dwarf population. The
dotted lines are the maximum situation where every galaxy that could be
identified as a UDG is oriented as such. Bottom: The UDG mass functions for
randomly oriented galaxies divided by the maximum number mass function for
each environment. The difference between the maximum and random orientation
functions is larges in the isolated environment, where the UDGs are most
rotation dependent.  Figure 8: The stellar $c/a$ axis ratios plotted against
the $b/a$ axis ratios for the galaxies’ stellar distributions at z=0 in all
environments. The populations are separated into UDGs and non-UDGs under the
RE definition. The bold points represent the medians of the populations, while
the error bars show the 25${}^{\text{th}}$ to 75${}^{\text{th}}$ percentiles.
Under the RE definition, the disparity in isolated UDG and non-UDG shapes seen
in Figure 1 is no longer present.

###  4.4 Effects on Previous Results

We have shown that altering the criteria for UDGs and accounting for different
orientations can have a large effect on the resultant UDG population. This
effect means that various groups studying UDGs under varying definitions could
potentially be looking at vastly different populations. This could make
reaching a cohesive understanding of UDGs (including formation mechanisms and
whether they exclusively reside in dwarf dark matter halos) difficult, if not
impossible.

As an example: in Section 3, we found a disparity between isolated UDG and
non-UDG morphologies which, after investigation, revealed the formation
mechanism of early, high-spin mergers. This disparity, highlighted in Figure
1, was present when identifying UDGs according to the G0-F definition.
However, recreating Figure 1 under the different definitions can provide
drastically different results. Figure 8 shows the $c/a$ vs $b/a$ axis ratios
for the dwarf halos with UDGs identified via the RE definition. We see now
that the isolated galaxies show no disparity between UDGs and non-UDGs, with
their medians and percentiles lying nearly on top of one another. Together,
Figures 6 & 8 indicate that the RE definition is permissive to the extent that
the UDG and non-UDG populations are effectively homogenized, thus no
underlying formation mechanisms or physics would be unique to the UDG
population.

##  5 Conclusions

We have selected 1249 dwarf galaxies in isolated, satellite, and cluster
environments from the Romulus25 and RomulusC simulations. We analyze the
shapes of these galaxies, as well as the resultant UDG population under 5
definitions and 288 orientations. Our results can be summarized as follows:

  * •

A morphological disparity exists between isolated UDGs’ and non-UDGs’ stellar
distributions, with UDGs having notably larger $b/a$ and smaller $c/a$ axis
ratios, both differing by $\approx 0.1$ (see Figure 1). This more oblate-
triaxial morphology is in agreement with Rong et al. (2020) and Kado-Fong et
al. (2021).

  * •

Analysis of the isolated galaxies’ morphological history shows that UDGs are
primarily the products of early mergers (see Figure 2). Though further
investigation shows that early mergers alone are not sufficient; high spins
are also required (see Figure 3).

  * •

Changing the criteria for being ultra-diffuse can result in largely different
UDG populations, with the percent of dwarfs that are identified as UDGs
ranging from 19-65%, 33-70%, and 50-83% in the isolated, satellite, and
cluster environments, respectively (see Table 2).

  * •

A galaxy’s status as a UDG is dependent on its orientation, and this
dependence is strongest in the isolated environment (see Figures 4 & 5).
Additionally, more restrictive definitions (like G0-F and R0) result in a
stronger orientation dependence (see Figures 5 & 6). This orientation
dependence also manifests when comparing the mass functions of all potential
UDGs to those identified from a random orientation (see Figure 7).

  * •

Although it is standard practice to identify UDGs from a face-on orientation,
we find a significant number of galaxies that are identified as UDGs at some
orientation, but do not meet the criteria when face-on. Depending on the UDG
definition, there are 48-138, 8-63, and 0-26 such galaxies in isolated,
satellite, and cluster environments, respectively (see Table 3).

  * •

Changing the definition of UDG can change or obscure the underlying physics of
the resultant UDG population. The unique formation mechanism of early, high-
spin mergers in isolated UDGs under the restrictive definitions (like G0-F and
R0) is not highlighted under the less restrictive RE definition where the UDG
and non-UDG populations are effectively homogenized (See Figures 1 & 8).

We have shown that the population of dwarfs that are considered UDGs, and
whether the population exhibits a morphological distinction or formation
channel from non-UDGs, is drastically dependent on choice of definition. With
all of the outstanding questions around UDG formation, it is clear that
different conclusions can be reached when identifying separate subsets of
dwarfs as ‘ultra-diffuse’. In order for more advanced questions (like those of
formation and evolution) to be answered cohesively, we must first come to a
cohesive methodology for identifying UDGs. More restrictive definitions (like
G0-F and R0) seem to identify a subset of isolated dwarf galaxies with unique
shape histories, indicating a specific channel of formation through early,
high spin mergers, while less restrictive definitions identify such a large
population of UDGs that the UDG and non-UDG populations are roughly identical.
In short, it seems that whether UDGs are a true physical phenomenon or simply
a product of definition depends on the definition itself.

FDM and JDV acknowledge support from the University of Oklahoma and also
acknowledge that Oklahoma’s land is the traditional territory of the Osage,
Cherokee, Choctaw, Chikasaw and Patowatomi Peoples who have lived here since
time immemorial. MT is supported by an NSF Astronomy and Astrophysics
Postdoctoral Fellowship under award AST-2001810. The Romulus simulations are
part of the Blue Waters sustained-petascale computing project, which is
supported by the National Science Foundation (awards OCI-0725070 and
ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the
University of Illinois at Urbana–Champaign and its National Center for
Supercomputing Applications. This work is also part 12 of a Petascale
Computing Resource Allocations allocation support by the National Science
Foundation (award number OAC-1613674). This work also used the Extreme Science
and Engineering Discovery Environment (XSEDE), which is supported by National
Science Foundation grant number ACI-1548562. This work was partially performed
at the Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-1607611 The authors thank Jenny Greene and Erin Kado-Fong
for useful discussions related to this manuscript. The authors also thank
Nicole Sanchez for use of a HECC allocation. The python packages matplotlib
(Hunter, 2007), numpy (van der Walt et al., 2011), pynbody (Pontzen et al.,
2013), and tangos (Pontzen & Tremmel, 2018) were used in the analysis.

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