# Deep Hubble Space Telescope Photometry of LMC and Milky Way Ultra-Faint
Dwarfs:
A careful look into the magnitude-size relation

Hannah Richstein Department of Astronomy, University of Virginia, 530
McCormick Road, Charlottesville, VA 22904 USA  Nitya Kallivayalil Department
of Astronomy, University of Virginia, 530 McCormick Road, Charlottesville, VA
22904 USA  Joshua D. Simon Observatories of the Carnegie Institution for
Science, 813 Santa Barbara Street, Pasadena, CA 91101, USA  Christopher T.
Garling Department of Astronomy, University of Virginia, 530 McCormick Road,
Charlottesville, VA 22904 USA  Andrew Wetzel Department of Physics and
Astronomy, University of California, One Shields Avenue, Davis, CA 95616, USA
Jack T. Warfield Department of Astronomy, University of Virginia, 530
McCormick Road, Charlottesville, VA 22904 USA  Roeland P. van der Marel Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
The William H. Miller III Department of Physics & Astronomy, Bloomberg Center
for Physics and Astronomy,
Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA
Myoungwon Jeon School of Space Research, Kyung Hee University, 1732 Deogyeong-
daero,
Yongin-si, Gyeonggi-do 17104, Republic of Korea  Department of Astronomy &
Space Science, Kyung Hee University, 1732 Deogyeong-daero,
Yongin-si, Gyeonggi-do 17104, Republic of Korea  Jonah C. Rose Department of
Astronomy, University of Florida, 211 Bryant Space Science Center,
Gainesville, FL 32611, USA  Center for Computational Astrophysics, Flatiron
Institute, 162 Fifth Avenue, New York, NY 10010, USA  Paul Torrey Department
of Astronomy, University of Virginia, 530 McCormick Road, Charlottesville, VA
22904 USA  Anna Claire Engelhardt George Mason University, Department of
Physics and Astronomy, 4400 University Dr.,
MSN: 3F3, Fairfax, VA 22030, USA  Gurtina Besla Steward Observatory,
University of Arizona, 933 N. Cherry Ave, Tucson, AZ 85719, USA  Yumi Choi
NOIRLab, 950 N. Cherry Ave, Tucson, AZ 85719, USA  Marla Geha Department of
Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520, USA  Puragra
Guhathakurta Department of Astronomy and Astrophysics, University of
California Santa Cruz,
University of California Observatories, 1156 High Street, Santa Cruz, CA
95064, USA  Evan N. Kirby Department of Astronomy, California Institute of
Technology, 1200 E. California Blvd., MC 249-17,
Pasadena, CA 91125, USA  Ekta Patel Department of Physics and Astronomy,
University of Utah, 115 South 1400 East,
Salt Lake City, Utah 84112, USA  Elena Sacchi Leibniz-Institut für Astrophysik
Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany  INAF–Osservatorio di
Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti 93/3, I-40129
Bologna, Italy  Sangmo Tony Sohn Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, MD 21218, USA

###### Abstract

We present deep Hubble Space Telescope (HST) photometry of ten targets from
Treasury Program GO-14734, including six confirmed ultra-faint dwarf galaxies
(UFDs), three UFD candidates, and one likely globular cluster. Six of these
targets are satellites of, or have interacted with, the Large Magellanic Cloud
(LMC). We determine their structural parameters using a maximum-likelihood
technique. Using our newly derived half-light radius ($r_{h}$) and $V$-band
magnitude ($M_{V}$) values in addition to literature values for other UFDs, we
find that UFDs associated with the LMC do not show any systematic differences
from Milky Way UFDs in the luminosity-size plane. Additionally, we convert
simulated UFD properties from the literature into the $M_{V}-r_{h}$
observational space to examine the abilities of current dark matter (DM) and
baryonic simulations to reproduce observed UFDs. Some of these simulations
adopt alternative dark matter models, thus allowing us to also explore whether
the $M_{V}-r_{h}$ plane could be used to constrain the nature of DM. We find
no differences in the magnitude-size plane between UFDs simulated with cold,
warm, and self-interacting dark matter, but note that the sample of UFDs
simulated with alternative DM models is quite limited at present. As more
deep, wide-field survey data become available, we will have further
opportunities to discover and characterize these ultra-faint stellar systems
and the greater low surface-brightness universe.

##  1 Introduction

Within the cosmological paradigm of Cold Dark Matter (CDM), structures in the
Universe that we observe today were formed through hierarchical evolution
(e.g., White & Rees, 1978; Peebles, 1984; Frenk et al., 1988). The most
massive dark matter (DM) halos grew by accreting less massive objects, a
phenomenon that has left evidence in many forms, such as galaxy clusters and
the satellite population of our own Milky Way (MW). While the two most massive
satellites of the MW, the Large and Small Magellanic Clouds (LMC, SMC), have
been known for millennia, the advent of wide-field optical imaging surveys
with digital CCD detectors has enabled the discovery of dwarf satellites
orders of magnitude fainter than the Magellanic Clouds.

Among these galaxies are ultra-faint dwarfs (UFDs), which were first
discovered after the Sloan Digital Sky Survey (SDSS) began (e.g., Willman et
al., 2005), and over 60 candidates have been discovered since (e.g., Belokurov
et al., 2006; Bechtol et al., 2015; Drlica-Wagner et al., 2015; Kim & Jerjen,
2015; Koposov et al., 2015a; Laevens et al., 2015a; Homma et al., 2016;
Torrealba et al., 2016a). While UFDs seem to be an extension of classical
dwarf galaxies ($M_{\star}\approx 10^{5-7}M_{\odot}$), as a sub-class they
have been defined as having $L\leq 10^{5}L_{\odot}$ and $M_{\star}\approx
10^{2-5}M_{\odot}$ (e.g., Bullock & Boylan-Kolchin, 2017; Simon, 2019). UFDs
have the oldest, least chemically enriched stellar populations and could be
the “fossils” of the first galaxies, having formed at least $\sim$80% of their
stellar mass by the end of reionization ($z\sim 6$; Madau & Dickinson, 2014)
with little-to-no star-formation activity since (e.g., Bovill & Ricotti, 2009;
Brown et al., 2014; Weisz et al., 2014; Sacchi et al., 2021).

The infall history of these UFDs is encoded in their present-day dynamics.
While some information can be extracted with only line-of-sight velocities
(e.g., Rocha et al., 2012; Carlin et al., 2019; Garling et al., 2020), full
6-D kinematics like those provided by the Gaia mission (Gaia Collaboration et
al., 2016) allow for full orbital reconstruction. This in turn allows us to
search for signatures of group infall. With Gaia proper motions (PMs), a
subset of UFDs have been linked kinematically with the LMC (e.g., Kallivayalil
et al., 2018; Patel et al., 2020; Erkal & Belokurov, 2020), and could thus be
considered “satellites of satellites.” These UFDs have occupied a different
environment than the non-LMC associated dwarf galaxies for much of their
existence.

For example, the LMC-associated UFDs had been in proximity to a moderately
massive ($\gtrsim 10^{11}M_{\odot}$) galaxy before their infall into the MW
halo. Additionally, with the LMC being on its first infall (e.g., Besla et
al., 2007; Kallivayalil et al., 2013), these satellites may have been farther
away from the proto-MW during the epoch of reionization ($z\sim 6$) than the
MW satellites that were captured prior to the LMC’s infall. Both sets (the
non-LMC-associated and LMC-associated) of UFDs would have been distant enough
to be at least outside the virial radius of the MW (Wetzel et al., 2015;
Santistevan et al., 2023).

While essentially all UFDs appear to have formed the majority of their
present-day stellar mass by the end of reionization (e.g., Brown et al., 2014;
Weisz et al., 2014), UFDs that inhabited different environments during
reionization could have different quenching times due to the inhomogeneity of
reionization (Dawoodbhoy et al., 2018; Ocvirk et al., 2020; Sorce et al.,
2022; Kim et al., 2023). As such, comparing the star formation histories
(SFHs) of UFDs that inhabited different environments during reionization could
offer insights into this epoch. Splitting the UFDs into subgroups based on
their LMC association, Sacchi et al. (2021) found tantalizing signs of
differences in the SFHs. The LMC-associated UFDs continued forming stars for,
on average, more than 500 Myr after the non-LMC-associated galaxies had
completed their star formation. This could hint at the galaxies’ early
environment having an impact on their evolution.

There are some properties of galaxies that we might expect to be largely
invariant to environment and more dependent on their DM halo mass. For
example, the existence of a stellar-to-halo-mass relation has been well-
established (e.g., Behroozi et al., 2010). However, for halo masses below
$\sim 10^{10}M_{\odot}$ the relation has been more difficult to characterize
(e.g., Brook et al., 2014; Garrison-Kimmel et al., 2014, 2017; Read et al.,
2017; Munshi et al., 2021). It is not immediately clear then, whether equal
mass halos in an LMC versus MW environment would (without any disruptive
events) evolve to host galaxies of similar stellar mass. Stellar mass is
inferred from a galaxy’s luminosity, but with such a small sample size, it
could be difficult to interpret how meaningful the differences between
populations are.

The goal of this study is thus two-fold: 1) to compare LMC and MW UFDs in an
observed space, namely the absolute magnitude ($M_{V}$) versus half-light
radius ($r_{h}$) plane, and 2) to explore whether simulated UFDs can offer
insights about what we observe and what we might expect to observe as newer
facilities come online. While reviewing the state of simulated UFDs, we also
compare simulation results from baryonic runs based on alternative dark matter
models, particularly as they relate to predictions of cored or cuspy density
profiles.

The “core-cusp” problem arose from the mismatch of the predicted “cuspy”
density profiles from cold DM-only simulations and the implied “cored”
profiles from observed rotation curves (e.g., Flores & Primack, 1994; Moore,
1994), and has been one of the classical challenges to the CDM model (see
e.g., Bullock & Boylan-Kolchin, 2017; Sales et al., 2022, for a review). While
solutions have since been presented that are consistent with CDM (e.g.,
Navarro et al., 1996; Gelato & Sommer-Larsen, 1999; Read & Gilmore, 2005; Oh
et al., 2011; Pontzen & Governato, 2012; Chan et al., 2015; Lazar et al.,
2020; Orkney et al., 2021), we take this opportunity to examine and compare
the predictions coming from alternative DM models on UFD mass scales.

Here, we choose to revisit the seven targets from Sacchi et al.
(2021)–Horologium I (Hor I), Hydra II (Hya II), Phoenix II (Phe II), Reticulum
II (Ret II), Sagittarius II (Sgr II), Triangulum II (Tri II), and Tucana II
(Tuc II)–as well as add three additional targets from the Hubble Space
Telescope (HST) Treasury Program GO-14734 (PI: N. Kallivayalil) that could be
associated with the LMC: Grus II (Gru II), Horologium II (Hor II), and Tucana
IV (Tuc IV) (Battaglia et al., 2022; Correa Magnus & Vasiliev, 2022). We note
that Hor II has not been spectroscopically confirmed to be a dark-matter
dominated galaxy and that Sgr II is most likely a globular cluster (Longeard
et al., 2021; Baumgardt et al., 2022). We include Sgr II to remain consistent
with the original Sacchi et al. (2021) sample and will refer to all of the
included targets as UFDs for ease of discussion.

The structure of this paper is as follows. In Section 2, we describe the
observations and photometry process. In Sections 3 and 4, we discuss our
methods of structural analysis and absolute magnitude calculations, then
examine our results in the context of past literature. Section 5 includes our
comparison of LMC-associated and non-LMC associated UFDs, as well as a
comparison of observed and simulated UFDs. We conclude in Section 6.

Table 1:
Summary of Primary ACS Observations and Field Completeness Satellite Name | UT Date | Filter | Exposure Time (s) | 50% (VegaMag) | 90% (VegaMag)
---|---|---|---|---|---
Grus II (Gru II) | 2016 Nov 02 | F606W | 4 $\times$ 1130 | 27.8 | 27.0
| 2016 Oct 08 | F814W | 4 $\times$ 1130 | 26.8 | 26.0
Horologium I (Hor I) | 2017 Jul 31 | F606W | 4 $\times$ 1131 | 27.8 | 27.0
| 2017 Aug 01 | F814W | 4 $\times$ 1131 | 26.8 | 26.0
Horologium II East (Hor II) | 2016 Sep 26 | F606W | 4 $\times$ 1131 | 28.1 | 27.3
| 2016 Sep 26 | F814W | 4 $\times$ 1131 | 27.1 | 26.3
Horologium II West (Hor II) | 2017 Mar 22 | F606W | 4 $\times$ 1131 | 28.0 | 27.2
| 2017 Mar 22 | F814W | 4 $\times$ 1131 | 27.0 | 26.2
Hydra II (Hya II) | 2016 Dec 13 | F606W | 4 $\times$ 1186 | 27.6 | 26.9
| 2016 Dec 16 | F814W | 4 $\times$ 1186 | 26.6 | 25.9
Phoenix II (Phe II) | 2017 May 08 | F606W | 4 $\times$ 1131 | 27.8 | 27.0
| 2017 May 08 | F814W | 4 $\times$ 1131 | 26.8 | 26.0
Reticulum II (Ret II) | 2017 Nov 18 | F606W | 4 $\times$ 1145 | 27.4 | 26.2
| 2017 Nov 28 | F814W | 4 $\times$ 1145 | 26.4 | 25.2
Sagittarius II (Sgr II) | 2016 Oct 19 | F606W | 4 $\times$ 1127 | 27.5 | 26.8
| 2016 Oct 01 | F814W | 4 $\times$ 1127 | 26.8 | 26.1
Triangulum II East (Tri II) | 2016 Dec 31 | F606W | 4 $\times$ 1137 | 27.7 | 27.0
| 2016 Dec 03 | F814W | 4 $\times$ 1137 | 26.7 | 26.0
Triangulum II West (Tri II) | 2017 Jun 22 | F606W | 4 $\times$ 1137 | 27.7 | 26.9
| 2017 Jun 22 | F814W | 4 $\times$ 1137 | 26.7 | 25.9
Tucana II Northeast (Tuc II) | 2017 Jun 27 | F606W | 4 $\times$ 1138 | 28.1 | 27.3
| 2017 Jun 29 | F814W | 4 $\times$ 1138 | 27.1 | 26.3
Tucana II Northwest (Tuc II) | 2017 Aug 26 | F606W | 4 $\times$ 1138 | 27.9 | 27.1
| 2017 Aug 31 | F814W | 4 $\times$ 1138 | 26.9 | 26.1
Tucana II Southeast (Tuc II) | 2017 Aug 21 | F606W | 4 $\times$ 1138 | 28.0 | 27.2
| 2017 Aug 25 | F814W | 4 $\times$ 1138 | 27.0 | 26.2
Tucana II Southwest (Tuc II) | 2018 Mar 01 | F606W | 4 $\times$ 1138 | 27.5 | 26.7
| 2018 Mar 03 | F814W | 4 $\times$ 1138 | 26.5 | 25.7
Tucana IV North (Tuc IV) | 2017 Apr 06 | F606W | 4 $\times$ 1140 | 28.0 | 27.3
| 2017 Apr 06 | F814W | 4 $\times$ 1140 | 27.0 | 26.3
Tucana IV South (Tuc IV) | 2016 Sep 28 | F606W | 4 $\times$ 1140 | 27.9 | 27.2
| 2016 Sep 28 | F814W | 4 $\times$ 1140 | 26.9 | 26.2

##  2 Photometry of Hubble Space Telescope Data

###  2.1 Target Selection

As explained above, we choose to study the seven UFDs whose SFHs were
characterized in Sacchi et al. (2021). In this work the seven UFDs were
separated into LMC (Hor I, Phe II, Ret II) and non-LMC associated UFDs (Hya
II, Sgr II, Tri II, Tuc II) on the basis of orbit models (Patel et al., 2020)
constructed with Gaia DR2 proper motions for the UFDs (Kallivayalil et al.,
2018; Erkal & Belokurov, 2020). Previous studies (e.g., Jethwa et al., 2016;
Sales et al., 2017) also supported some of these classifications, but where
there were disagreements, the results of Patel et al. (2020) were adopted as
they used the most updated PM information (Gaia DR2) available at the time.

Since then, Battaglia et al. (2022), Correa Magnus & Vasiliev (2022), and Pace
et al. (2022), among others, have published studies using Gaia eDR3 (Gaia
Collaboration et al., 2021) to improve the PM measurements and orbital models
for several of these UFDs. Battaglia et al. (2022) support the classification
of Hor I, Phe II, and Ret II as “highly likely former satellites of the LMC,”
and adds Hor II and Tuc IV as “potential” former satellites and Gru II as a
recently captured LMC satellite. Correa Magnus & Vasiliev (2022) also present
orbital models which suggested Tuc IV and Gru II were recently captured by the
LMC, while Hor II and Ret II only show a moderate likelihood of having been
associated with the LMC in the past. Other suspected/confirmed LMC satellites
include Carina II, Carina III, and Hydrus I (Kallivayalil et al., 2018; Erkal
& Belokurov, 2020; Patel et al., 2020; Battaglia et al., 2022; Correa Magnus &
Vasiliev, 2022), however we do not present photometry for them here as they
were not observed as part of the Treasury program.

###  2.2 Observations and Reduction

All ten targets (Table 1) were observed in the F606W and F814W filters of the
HST Advanced Camera for Surveys (ACS; Ford et al., 1998) using the Wide Field
Channel (WFC). Off-target fields were simultaneously observed with the
UV/Visible channel (UVIS) of the Wide Field Camera 3 (WFC3) in order to
characterize the stellar background/foreground distributions. Each ACS/WFC
field-of-view (FOV) is $202\arcsec\times 202\arcsec$, while the WFC3/UVIS off-
fields are $162\arcsec\times 162\arcsec$. As these observations were planned
as first epochs for follow-up PM measurements, each target field had four
dithered exposures in order to facilitate well-sampled templates for
background galaxies. These were taken using two orbits in each filter, with an
integration time per exposure of approximately one half-orbit ($\sim$1100 s).
Table 1 gives the average exposure time in integer seconds for each target.
Hor II, Tri II, and Tuc IV had two pointings, while Tuc II had four pointings,
so as to ensure there were enough stars for robust PM measurements.

The individual dither (flc) images were downloaded from the Mikulski Archive
for Space Telescopes, having already been processed and corrected for charge
transfer inefficiency using the current ACS and WFC3 pipelines (Anderson &
Ryon, 2018; Anderson et al., 2021). The four separate dithers from each filter
were aligned to each other using TweakReg and then combined using the DRIZZLE
algorithm (Fruchter & Hook, 2002) as implemented by the HST software
DrizzlePac (STScI Development Team, 2012; Avila et al., 2015) to create drc
fits files. The segmentation routine from photutils was used to identify and
mask the chip gaps, detector artifacts, and large saturated sources.

###  2.3 DAOPHOT-II

The photometry will be described in detail in the data paper accompanying the
public release of the PSF-fit source catalogs (H. Richstein et al., 2024, in
prep.). We provide a brief overview here.

PSF-fitting photometry was performed on the masked drc files using DAOPHOT-II
and ALLSTAR (Stetson, 1987, 1992). The source lists from F606W and F814W were
matched using DAOMATCH and DAOMASTER to create a preliminary PSF source
catalog. To calibrate our photometry, we first removed DAOPHOT’s intrinsic
zeropoint of 25 mag, then added appropriate VegaMag zeropoints (from the ACS
Zeropoint Calculator for ACS and stsynphot for WFC3) corrected for the
exposure times of our images.

The PSF magnitudes are computed within a finite radius, so to correct for the
infinite aperture case, we applied an aperture correction. To calculate this,
we performed aperture photometry on the same drc images using the photutils
(Bradley et al., 2020) routines DAOStarFinder and aperture_photometry. We used
an aperture with a radius of 0.2″and applied the VegaMag zeropoints and
encircled energy corrections from Bohlin (2016) and Medina et al. (2022) for
ACS and WFC3/UVIS, respectively. The F606W and F814W aperture source lists
were then matched to each other to create an aperture photometry catalog.
Sources were matched between the PSF and aperture photometry catalogs to
determine an aperture correction that was then applied to the DAOPHOT PSF-
derived VegaMag magnitudes.

###  2.4 Artificial Star Tests and Flag Creation

To determine our photometric completeness limits, photometric error functions,
and to aid in developing metrics for star-galaxy separation, we performed
artificial star tests (ASTs) with the ADDSTAR function of DAOPHOT. For each
filter and pointing, we inject 500 artificial stars into the science image at
a time in order to avoid changing the total source density of the frame
significantly. After injecting the artificial stars, we performed the same
ALLSTAR photometry as on the masked drc files. We performed this process 500
times so that we had a total of 250,000 artificial stars per filter and
pointing.

A primary source of contamination in deep photometric studies of MW dwarf
satellites are faint, unresolved background galaxies. As such, essentially all
downstream analyses (e.g., measuring structural parameters) rely on methods to
exclude these contaminants in order to obtain a highly pure sample of stars in
the dwarf galaxy of interest. We created a flag to distinguish between stars
and galaxies based on the standard error (ERR), SHARP, and CHI parameters of
stars recovered from the ASTs. The DAOPHOT ERR value is an estimate of the
star’s standard error, which takes into account the flux, PSF residuals, and
noise properties of the images. SHARP and CHI are quality diagnostics relating
how the model PSF compares to each source.

The SHARP value of an isolated star should be close to zero, while more
extended objects would have SHARP values much greater than zero, and image
defects and cosmic rays would have values much less than zero. The CHI value
represents the ratio between the pixel-to-pixel scatter in the PSF residuals
and the scatter expected from the image noise properties. Stars should scatter
around unity, although brighter stars may have higher values due to saturation
effects.

We used exponential functions to divide between acceptable and poor parameter
values. To begin, we binned the artificial star parameters in magnitude space,
then calculated the median values across the bins. To account for the spread
in parameter space of the observed stars, we added the $3\sigma$ of each bin
to its respective median, and then fit an exponential function to those
values. For the SHARP parameter, we fit two functions, as there was a spread
in both the positive and negative direction. For the negative function, we
subtracted instead of added the $3\sigma$.

We assigned sources their flag values (between 0 and 1, inclusive) based on
where their SHARP, ERR, and CHI values fell in relation to the functional
fits. The ERR and CHI functions acted as upper limits, while the source’s
SHARP value had to fall between the positive and negative function lines to
count as meeting the criteria. If a source met the SHARP and ERR criteria in
both F606W and F814W, it was assigned a 1. If sources did not meet these
criteria in both filters, but did meet at least the SHARP and ERR criteria in
the same filter and the CHI criterion in either filter, they were assigned
flag values between 0.5 and 1 (exclusive). For this analysis, we only
considered sources with flag values greater than 0.5.

Figure 1: Color-magnitude diagrams in the VegaMag system for the ten targets
presented herein. The sources presented are all sources that passed the
quality, completeness, and color cuts imposed for the morphological analysis.
We have colored in gray the sources that are most likely background/foreground
contamination but are used in the structural fitting to determine the
background parameter.

###  2.5 Color-Magnitude Diagrams

The color-magnitude diagrams (CMDs) for each target are shown in Figure 1.
Here, they have been transposed to absolute magnitude space by applying the
dust correction and subtracting the distance modulus listed in Table
LABEL:tab:derived. To correct for the reddening and dust extinction, we used
the Schlegel et al. (1998) maps within the dustmaps module (Green, 2018) and
applied the Schlafly & Finkbeiner (2011) recalibration. The correction was
calculated for each individual star based on its right ascension (RA) and
declination (DEC) values. The sources plotted in the CMDs are those above the
90% completeness limit and with flag values above 0.5. We additionally applied
a straight (vertical) color-cut in CMD space tailored to each target for
background galaxy removal, and have highlighted the likely member stars in
black. We leave the foreground/background sources that pass our color-cut, as
the code that we use for fitting structural parameters can account for a
foreground/background term. The lack of bright red-giant branch stars in some
of the CMDs (e.g., Ret II, Tri II) is due to saturation in the images.

##  3 Structural Analysis

In the years following the discovery of the first UFDs, it became important to
quantify how well one could actually describe the morphology of such diffuse
objects. In a comprehensive exploration of systematics that affect the
measurement of dwarf morphologies with resolved stars, Muñoz et al. (2012)
found three aspects of observational design that influence the measurement
accuracy. They found that morphologies are best measured with observations
covering a large field of view (FOV; $>$3 times the galaxy half-light radius)
to sufficient depth to achieve a central density contrast (relative to the
background and foreground contamination) of $\sim$20 and to measure as many
stars in the dwarf as possible (preferably $>$1000).

In practice, ground-based observatories with large fields-of-view can achieve
sufficient depths to robustly measure the morphologies of the brighter and
closer MW satellites. However, for the faintest and farthest UFDs, these
observatories can be limited by the photometric depth that they can achieve
and their ability to achieve good enough seeing for confident star-galaxy
separation. Although its FOV is limited, HST/ACS can thus have an advantage in
this regime for its superior photometric depth and image resolution, which
allows for better star-galaxy separation and lower background levels.

With the available HST pointings, we are able to measure structural parameters
for six of the ten targets: Hor I, Hor II, Hya II, Phe II, Sgr II, and Tri II.
For Gru II, Ret II, Tuc II, and Tuc IV, our imaging covers only between $\sim
0.5-1\times$ the half-light radii of these objects, preventing us from making
reliable morphological measurements for these objects.

Figure 2: The posterior distributions of the 7-parameter structural fit for
Hor I using the exponential model. The three black vertical dashed lines
represent the 16th, 50th, and 84th percentiles. The blue lines and markers
show the maximum likelihood values.

###  3.1 Fitting 2D Profiles

We employed the same technique used in Drlica-Wagner et al. (2020) and Simon
et al. (2021) that was based on the maximum likelihood algorithm presented in
Martin et al. (2008) to fit 2D profiles of each galaxy. For each target, we
modeled both an exponential and a Plummer (Plummer, 1911) profile, then
performed binned Poisson maximum likelihood fits to the probability density
functions. We had seven free parameters: center position ($x_{0},y_{0}$),
number of stars (richness), 2D projected semi-major axis of the ellipse
containing one-half the surface density111Similar to past morphology studies
(e.g., Martin et al., 2008), assuming there is no mass segregation, the half-
density and half-light radii are equivalent. of the galaxy (elliptical half-
light radius; $a_{h}$), ellipticity222Ellipticity is defined as
$\epsilon=1-b/a$, where $a$ and $b$ are the lengths of the semi-major and
semi-minor axes, respectively. ($\epsilon$), position angle of the semi-major
axis measured from North through East ($\theta$), and foreground/background
surface density (average density of star-like sources in the FOV not belonging
to the galaxy; $\Sigma_{b}$).

As we noted in Richstein et al. (2022), much of the past literature (e.g.,
Martin et al., 2008; Muñoz et al., 2018) reporting structural parameters has
used $r_{h}$ when referring to the elliptical half-light radius, though
$r_{h}$ has also been used to report azimuthally-averaged half-light radius
values (equal to $a_{h}\sqrt{1-\epsilon}$; e.g., Drlica-Wagner et al., 2015).
Thus, we choose to make the distinction here, and in our tables of past
measurements use $a_{h}$ for measured elliptical half-light radii. We use
$r_{h}$ as the azimuthally-averaged half-light radius in our later comparison
to simulations.

We divided the ACS FOV into $4.55\arcsec\times 4.55\arcsec$ bins and applied
masks to account for the ACS chip gap, saturated foreground stars, and
extended, resolved background galaxies and to match the mask applied during
the photometry process. We counted the stars in each bin and used the Markov
chain Monte Carlo (MCMC) ensemble sampler emcee (Foreman-Mackey et al., 2013)
to sample the posterior probability distribution. We refer the reader to
Richstein et al. (2022) for the explicit functional forms333We note that the
in-text equation for $r_{e}$ in Richstein et al. (2022) (§3.1) should be
$r_{e}=a_{h}/1.68$. used for the exponential and Plummer profiles. Figure 2
shows an example corner plot (Foreman-Mackey, 2016) with the posterior
distributions for the exponential fit to Hor I. The corner plots for the
exponential fits of the other targets are shown in Appendix A.

We see three different correlations between pairs of parameters: richness and
background, $a_{h}$ and background, and richness and $a_{h}$. These relations
come from the finite number of stars used in the fit. For example, richness
and background counts both depend on whether individual stars are considered
to be contributing the surface density of the galaxy or not. If more stars
belong to the galaxy, the richness will increase and the background will be
lower. Similarly, a higher richness correlates with a larger $a_{h}$, as a
greater number of stars at extended distances from the center would be
considered part of the galaxy. This $a_{h}$ and richness correlation also
explains the $a_{h}$ and background anti-correlation, as a larger $a_{h}$ with
a higher richness necessitates fewer stars belonging to the background.

As shown in Wheeler et al. (2019), measuring dwarf morphologies from shallow
imaging with surface brightness limits $<30$ mag arcsec-2 can bias size
measurements towards being too compact. For typical dwarfs in our sample, we
are able to trace their light profiles to $\sim 31.2$ mag arcsec-2 such that
we should experience no significant bias in our morphological measurements
from our surface brightness limits. It is more common in our measured sample
to be limited by our imaging FOV, as our imaging for some objects does not
meet the recommended coverage of three times the $a_{h}$ (Muñoz et al., 2012).

###  3.2 Apparent and Absolute V-band Magnitude Calculations

To calculate integrated magnitudes, we transformed the extinction-corrected
F606W and F814W VegaMag values to $V$ and $I$ band, respectively. For the ACS
photometry, we used the Sirianni et al. (2005) conversions and for the WFC3
values, we used the coefficients from Harris (2018).

We applied the same probabilistic model used in Richstein et al. (2022).
Briefly, this entailed creating a Gaussian kernel (for both the ACS and WFC3
fields) with scikitlearn (Pedregosa et al., 2011) Kernel Density and fitting
it to the stars in CMD space. Random samples were generated from these kernels
and the log-likelihood of each sample was computed with respect to the binned
CMD data. We calculated excess flux from non-member sources using a
probabilistic background model created from the WFC3 field. The stellar
density was determined by multiplying the log-likelihoods by the area of the
CMD box and subtracting the off-field (WFC3) model from the on-field (ACS)
model. The integral returned the flux of the stars inside the CMD box, which
we multiplied by a correction factor to account for the flux outside the FOV.
We converted this flux into magnitude space, giving the integrated apparent
magnitude ($m_{V}$).

We performed this calculation within a Monte Carlo simulation to include
magnitude errors of the individual sources and the uncertainties on the FOV
correction incorporating different model parameters. The median integrated
$m_{V}$ values are presented in Table LABEL:tab:derived. To calculate the
absolute integrated magnitude ($M_{V}$), we ran the same Monte Carlo
simulation but included the distance modulus and its associated uncertainty.

As our methodology for measuring the integrated magnitudes of our galaxies
does not account for the luminosity from stars fainter than our observational
completeness limits, we are likely underestimating the total luminosity of our
targets. Using a stellar luminosity function constructed by combining the
Kroupa (2001) initial mass function with a 12 Gyr, [Fe/H]$=-2.2$ PARSEC
(Bressan et al., 2012; Chen et al., 2014; Marigo et al., 2017) isochrone, we
estimate that for the targets considered here, we are missing between
$1$–$10\%$ of their total luminosities with an average of $\sim 3\%$. This
indicates that our integrated magnitudes may be biased fainter by at most
$\sim 0.1$ mag due to our photometric completeness limits. The other sources
of uncertainty (FOV correction, distance) thus dominate the reported
uncertainties.

###  3.3 Mass-to-Light Ratios

We used our measured elliptical half-light radii to calculate updated mass
estimates with velocity dispersion values ($\sigma_{v}$) coming from the
literature (see Table LABEL:tab:derived). We employed the Wolf et al. (2010)
equation:

$M_{1/2}\simeq\frac{4}{G}\sigma^{2}_{v}R_{e}.$ (1)

The $R_{e}$ in this equation is defined as the 2D-projected half-light radius
from elliptical fits, which is equivalent to our $a_{h}$. $M_{1/2}$ is the
mass enclosed within this $R_{e}$.

The $V$-band luminosity was calculated by converting the integrated $V$-band
magnitude using a Solar $M_{V}=4.83$. We divided the $M_{1/2}$ calculated
above by the resulting $L_{V}$ for the mass-to-light ratio. Uncertainties
reported in Table LABEL:tab:derived are the 84th and 16th percentiles from the
Monte Carlo simulation using the individual uncertainties on the mass and
luminosity.

Figure 3: Digitized Sky Survey cutout images from Aladin showing the relative
size of our measured half-light ellipses ($a_{h}$; semi-major axis) and HST
fields-of-view. North is up and East is to the left for all panels. The ACS
FOV in blue has a side length of 202″; the WFC3 FOV in red has a side length
of 162″. For targets with multiple ACS pointings, the corresponding WFC3 off-
field is labeled to match the ACS label, not the WFC3 relative physical
positions. While our FOV/$a_{h}$ ratios are less than the Muñoz et al. (2012)
recommendation ($\sim$3), we are generally able to recover $a_{h}$ values
within 2$\sigma$ agreement of past literature measurements. Adopted and
Derived Parameters
---
Parameter |  $(m-M)_{0}$a | Distance |  $\sigma_{v}$b,c | $r_{h}$ | $r_{h}$ | $m_{V}$ | $M_{1/2}$ | ($M/L_{V}$)1/2 | Ref.
Units | (mag) | (kpc) | (km s-1) | (arcmin) | (pc) | (mag) | ($10^{6}$ $M_{\odot}$) | ($M_{\odot}/L_{\odot}$) |
Hor I | 19.5 $\pm 0.2$ | 79${}^{+8}_{-7}$ | 4.9${}^{+2.8}_{-0.9}$ | 1.46 $\pm 0.14$ | 34${}^{+5}_{-4}$ | 16.12${}^{+0.06}_{-0.09}$ | 0.88${}^{+1.2}_{-0.35}$ | 890${}^{+1200}_{-380}$ | 1,2
Hor II | 19.46 $\pm 0.2$ | 78${}^{+8}_{-7}$ | … | 1.43 $\pm 0.17$ | 32 $\pm 5$ | 17.39 $\pm 0.10$ | … | … | 3
Hya II | 20.64 $\pm 0.16$ | 134$\pm 10$ |  $<$3.6,$<$4.5 | 2.26${}^{+0.37}_{-0.34}$ | 88${}^{+16}_{-14}$ | 15.80${}^{+0.11}_{-0.22}$ |  $<$2.0 |  $<$670 | 4,5
Phe II | 19.85${}^{+0.16}_{-0.17}$ | 93$\pm 7$ | 11.0 $\pm 9.4$ | 1.18${}^{+0.20}_{-0.18}$ | 32${}^{+6}_{-5}$ | 17.00${}^{+0.13}_{-0.09}$ | 4.8${}^{+11.6}_{-3.7}$ | 7800${}^{+18000}_{-6500}$ | 6,7
Sgr II | 19.2 $\pm 0.2$ | 69${}^{+7}_{-6}$ | 1.7 $\pm 0.5$ | 1.82${}^{+0.08}_{-0.09}$ | 37 $\pm 4$ | 13.82${}^{+0.06}_{-0.04}$ | 0.10${}^{+0.07}_{-0.05}$ | 17${}^{+12}_{-8}$ | 8,9
Tri II | 17.27 $\pm 0.11$ | 28${}^{+2}_{-1}$ |  $<$3.4,$<$4.2 | 3.34${}^{+0.60}_{-0.61}$ | 28 $\pm 5$ | 15.97${}^{+0.23}_{-0.25}$ |  $<$0.48 |  $<$3200 | 10,11
Gru II | 18.71 $\pm 0.1$ | 55$\pm 3$ |  $<$1.9,$<$2.0 | 5.24${}^{+0.50}_{-0.48}$ | 84$\pm 9$ | … |  $<$0.35 |  $<$330 | 12,13
Ret II | 17.5 $\pm 0.1$ | 32${}^{+2}_{-1}$ | 2.97${}^{+0.43}_{-0.35}$ | 3.98${}^{+0.53}_{-0.57}$ | 37 $\pm 5$ | … | 0.48${}^{+1.5}_{-1.1}$ | 640${}^{+200}_{-150}$ | 8,14
Tuc II | 18.8 $\pm 0.2$ | 58${}^{+6}_{-5}$ | 3.8${}^{+1.1}_{-0.7}$ | 9.83${}^{+1.66}_{-1.11}$ | 170${}^{+37}_{-33}$ | … | 2.9${}^{+2.0}_{-1.1}$ | 2100${}^{+1400}_{-780}$ | 1,15
Tuc IV | 18.36${}^{+0.18}_{-0.19}$ | 47$\pm 4$ | 4.3${}^{+1.7}_{-1.0}$ | 7.26${}^{+1.14}_{-0.88}$ | 100${}^{+17}_{-15}$ | … | 2.2${}^{+1.8}_{-1.1}$ | 3100${}^{+2900}_{-1600}$ | 13
Figure 4: The 1D surface density profiles of Hor I (top row), Hor II (middle),
and Hya II (bottom) plotted against the elliptical radius, $R_{ell}$. The left
column shows the surface density measurements of the annularly-binned data, at
increments of 0.1$a_{h}$ for the respective models (exponential, cyan;
Plummer, magenta). Uncertainties are derived from Poisson statistics. The
plotted curves represent the 1D profiles produced with the best-fit parameters
of the 2D data. Floating horizontal lines represent the $a_{h}$ measurements
and their uncertainties. The middle column shows our exponential fits and 1D
surface density measurements compared to existing literature measurements. The
1D profiles from literature fits are normalized to have an area under the
curve equal to ours. The right column is the same as the middle, except for
the Plummer model rather than the exponential. Note the agreement, despite the
surface density profiles being fit to 2D-binned data rather than the 1D-binned
data also shown here. Figure 5: The same as Figure 4, for the targets Phe II
(top row), Sgr II (middle), and Tri II (bottom).

##  4 Results

###  4.1 Comparison of Measured Systems

Here, we report and compare our results to the literature for Hor I, Hor II,
Hya II, Phe II, Sgr II, and Tri II. Figure 3 shows the relative sizes of our
measured half-light radii against the observed fields-of-view. We have
overplotted a cyan ellipse encompassing 50% of the surface density of each
target (according to our best-fit exponential model) on a Digitized Sky Survey
(DSS; Morrison, 1995) optical cutout of the region of interest from Aladin
(Bonnarel et al., 2000). Our ACS footprints are shown in blue, with the WFC3
off-fields in red.

We also discuss their classification status (e.g., star cluster or ultra-faint
galaxy) and their possible association with the LMC. UFD candidates are
typically confirmed to be galaxies based on velocity dispersion measurements
determined spectroscopically (e.g., Simon & Geha, 2007), as higher values
suggest large mass-to-light ratios. The presence of a metallicity dispersion
has also been used to support classification as a galaxy, as it suggests more
than one period of star formation (Willman & Strader, 2012). Particularly
relevant here is the work of Fu et al. (2023), which estimated photometric
metallicity distribution functions (MDF) and dispersion values for a few of
the objects in our sample using narrow-band HST imaging. Their analysis showed
good agreement with smaller samples of spectroscopic measurements, and we
include a discussion of their results for the applicable targets here.

Another classification technique that has recently been applied to UFDs relies
on stellar mass segregation (Baumgardt et al., 2022). Energy equipartition
should cause high-mass stars to become more centrally concentrated within a
cluster, while the low-mass stars move further out. The timescale for this
process is longer in stellar systems that inhabit dark matter halos. As such,
systems which show no sign of stellar mass segregation are more likely to be
galaxies than star clusters. We report the Baumgardt et al. (2022)
classification alongside or in lieu of the spectroscopic determination for
completeness.

For brevity, we note here that four of our targets (Hor I and Phe II
(discussed here), Ret II and Tuc II (discussed in Section 4.2)) were
discovered concurrently by Koposov et al. (2015b) and Bechtol et al. (2015)
using data from the first year of the Dark Energy Survey (DES). Koposov et al.
(2015b) reported azimuthally-averaged radii ($r_{h}$) rather than elliptical
semi-major axis lengths ($a_{h}$) for their physical sizes.444We note that the
$r_{maj}$ values given in Koposov et al. (2015b) are the exponential scale
lengths, $a_{h}/1.68$ In our written comparisons, we present these $r_{h}$
values but also include the physical $a_{h}$ that we calculated using their
$\epsilon$ values. The uncertainties reported on $a_{h}$ in Tables 4, 7, 11,
and 12 are from a Monte Carlo using the Koposov et al. (2015b) $r_{h}$,
$\epsilon$, and ($m-M$) values.

For all of the targets in our paper except for Hya II, Moskowitz & Walker
(2020) reported “circularized” projected half-light radii555Moskowitz & Walker
(2020) fit both a 1- and a 3-component Plummer; we choose to report the best-
fitting values from the 1-component model as that model is most similar to the
Plummer function that we adopted. (our $r_{h}$) in arcminutes. To ease
comparison, we converted the Moskowitz & Walker (2020) fit $r_{h}$ to an
$a_{h}$ in arcminutes using their reported $\epsilon$. We denote $a_{h}$
values derived using the exponential model as $a_{h,\mathrm{exp}}$ and those
from the Plummer model as $a_{h,\mathrm{p}}$.

Figures 4 and 5 show the 1D surface density model profiles for the six UFDs
that we measured, as well as the binned surface density values. Note that
these 1D profiles were not fit to the binned data, but rather the result of
flattening the best-fit 2D model profiles. The x-axis is the elliptical
radius, $R_{ell}$, which can be described as the radial distance when the
major axis is aligned with the position angle (see Richstein et al. (2022) Eq.
3 for the full mathematical expression). The left column shows only the
measurements from this work, while the middle and right column compare our
findings to past literature values for the exponential and Plummer models,
respectively. We normalized the areas-under-the-curve from the literature to
our area values.

####  4.1.1 Horologium I

Both Koposov et al. (2015b) and Bechtol et al. (2015) were unable to constrain
a position angle for Hor I, and Koposov et al. (2015b) were only able to give
an upper limit on its ellipticity using the discovery data. The two teams also
reported different radii: Koposov et al. (2015b) estimated Hor I to have a
$\sim 1\farcm 3$ azimuthally-averaged radius ($a_{h,\mathrm{exp}}\sim 1\farcm
4$), while Bechtol et al. (2015) reported a Plummer radius of $\sim 2\farcm 4$
with large uncertainties. The next set of published structural parameters for
Hor I was from Muñoz et al. (2018), who fit both Plummer and exponential
models to photometry that they performed on the public DES data. Muñoz et al.
(2018) measured an exponential half-light radius of $\sim 1\farcm 7$, and a
Plummer half-light radius of $\sim 1\farcm 6$. The $a_{h,\mathrm{p}}$ of
Moskowitz & Walker (2020) was $\sim 1\farcm 7$.

Our $a_{h,\mathrm{exp}}=1\farcm 70^{+0.16}_{-0.15}$ and
$a_{h,\mathrm{p}}=1\farcm 61\pm 0.13$ agree within $1\sigma$ with all past
literature measurements. Using the previous Muñoz et al. (2018)
$a_{h,\mathrm{exp}}=1\farcm 71$ as a benchmark, the FOV/$a_{h}$ ratio would be
just short of 2, though Hor I was one of the targets for which over 1000
sources were used in the structural fitting. Our $\theta$ and $\epsilon$
values for both the exponential and Plummer fits additionally agree within
1$\sigma$ with the Muñoz et al. (2018). Our empirically-derived $M_{V}=-3.4\pm
0.2$ mag is also in agreement with past literature values.

Shortly after publishing the discovery paper, Koposov et al. (2015a) reported
on follow-up data of Ret II and Hor I taken with the Very Large Telescope
(VLT) FLAMES-GIRAFFE spectograph. For Hor I, they confirmed five member stars
and found a kinematic dispersion of $4.9^{+2.8}_{-0.9}$ km s-1, corresponding
to a mass-to-light ratio of $\sim$570 $M_{\odot}/L_{\odot}$. Though Koposov et
al. (2015a) also found a low metallicity dispersion (0.17 dex), they find that
the low [Fe/H]$=-2.76$ and large mass-to-light ratio made it consistent with
the other UFD galaxies known at the time.

A more detailed spectroscopic analysis based on three members stars using VLT
FLAMES-UVES and the MIKE spectograph on the Magellan-Clay Telescope was
published by Nagasawa et al. (2018). Their team found a similarly low average
[Fe/H] $\sim-2.6$. The Baumgardt et al. (2022) mass-segregation measurement
was not significant, supporting the classification of Hor I as a dwarf galaxy.
From HST narrow-band imaging of 27 stars, Fu et al. (2023) calculated an
average [Fe/H]$=-2.79^{+0.12}_{-0.13}$ dex and a
$\sigma_{\rm[Fe/H]}=0.56^{+0.11}_{-0.09}$ dex. Removing a possible higher-
metallicity outlier, the $\sigma_{\rm[Fe/H]}$ dropped to
$0.41^{+0.11}_{-0.10}$ dex. Either metallicity spread is consistent with Hor I
being a galaxy.

####  4.1.2 Horologium II

Kim & Jerjen (2015) discovered Hor II in the DES Y1A1 images, measuring an
$a_{h,\mathrm{exp}}=2\farcm 09^{+0.44}_{-0.41}$ ($r_{\mathrm{h}}$ in their
paper). The only other structural parameter measurement is from Moskowitz &
Walker (2020); using DES DR1, they fit $a_{h,\mathrm{p}}\sim 2\farcm 5$, which
agreed within 1$\sigma$. Hor II is one of our targets that had two telescope
pointings, giving us approximately a combined FOV of $3\farcm 3\times 6\farcm
7$. The observations were oriented for maximal coverage along the semi-major
axis, making our FOV/$a_{h}$ ratio (with the Kim & Jerjen (2015) value as a
reference) slightly over 3 along that direction.

Our $a_{h,\mathrm{exp}}=1\farcm 63\pm 0.18$ measurement is just outside of
being within $1\sigma$ agreement with the Kim & Jerjen (2015) value. Our
$a_{h,\mathrm{p}}=1\farcm 69^{+0.18}_{-0.17}$ is within $1.7\sigma$ of the
Moskowitz & Walker (2020) value. Our exponential $\theta$ and $\epsilon$
values agree with the Kim & Jerjen (2015) within 1.5 and 1.6$\sigma$,
respectively. For the Plummer model, our $\theta$ and $\epsilon$ are within 1
and 1.1$\sigma$ of the Moskowitz & Walker (2020) values. Our integrated
$M_{V}=-2.1\pm 0.2$ mag is 0.5 magnitudes fainter than the $M_{V}=-2.6$
presented by Kim & Jerjen (2015), but still within $1.8\sigma$ agreement.

There has not yet been a spectroscopic study of Hor II, but Baumgardt et al.
(2022) found no evidence for mass segregation and suggested that Hor II is
likely a dwarf galaxy.

####  4.1.3 Hydra II

Martin et al. (2015) discovered Hya II serendipitously in data from the Survey
of the Magellanic Stellar History (SMASH; Nidever et al., 2017), which used
the Dark Energy Camera (DECam) on the Blanco 4m telescope. They reported an
$a_{h,\mathrm{exp}}=1\farcm 7$ ($r_{h}$ in their paper), with the caveat that
the chip gaps and a bright foreground star near the center of the galaxy added
uncertainty to the measurement. The Muñoz et al. (2018) measurement from the
DECam data gave an $a_{h,\mathrm{exp}}\sim 1\farcm 7$ and an
$a_{h,\mathrm{p}}\sim 1\farcm 5$.

Our best-fit $a_{h,\mathrm{exp}}$ is $2\farcm 70^{+0.41}_{-0.38}$, and our
$a_{h,\mathrm{p}}$ is $2\farcm 13^{+0.42}_{-0.35}$, which are within 2 and
1.4$\sigma$ of the Muñoz et al. (2018) values, respectively. Hya II is another
target for which we have over 1000 sources, however, the saturated star
mentioned in the Martin et al. (2015) discovery paper is also near the center
of our frame. With the Muñoz et al. (2018) $a_{h,\mathrm{exp}}=1\farcm 65$ as
a reference, our FOV/$a_{h}$ is $\sim$2\. This discrepancy is thus likely due
to the limited FOV, the position of the chip gap, and the saturated star. As
this field was not optimized with structural measurements as a science goal,
we consider the Muñoz et al. (2018) measurement based on a wider FOV to be
more reliable.

From our fit, we measured a center approximately $45\arcsec$ from the original
Martin et al. (2015) value. To investigate whether this was significant, or an
artifact from our smaller FOV combined with the saturated star, we downloaded
the SMASH DR2 object catalog (Nidever et al., 2021) for the appropriate
region. We applied a distance cut to only consider sources within $6\arcmin$
of the Martin et al. (2015) center, as well as quality cuts for
$|\mathrm{SHARP}|\leq 0.4$ and $\chi<2$. Additionally, we required
$\texttt{prob}\neq 0$, where prob is the average Source Extractor (Bertin &
Arnouts, 1996) “stellaricity probability value.” We ran our fitting code on
the SMASH data and recovered a center that was $\sim 40\arcsec$ from the
Martin et al. (2015) value, but within $1.4\sigma$ of our newly-derived HST
value. It is possible that the shallower nature of the discovery data led to a
different stellar density center than is fit by the two more recent, deeper
data sets. Our $M_{V}=-4.6^{+0.2}_{-0.3}$ mag is consistent with the previous
measurements from Martin et al. (2015) and Muñoz et al. (2018).

Kirby et al. (2015b) presented Keck II/DEIMOS spectroscopy for 13 member stars
of Hya II. They reported 90 and 95% confidence levels for the upper limits of
the velocity dispersion corresponding to upper limits on the mass-to-light
ratios of 200 and 315 $M_{\odot}/L_{\odot}$, respectively. Kirby et al.
(2015b) were able to marginally resolve a metallicity dispersion of $\sim$0.40
dex, and suggest that this supports tentative classification as a galaxy. This
dispersion paired with the large physical radius more strongly supports the
dwarf galaxy nature of Hya II, although Simon (2019) does not list Hya II
among the spectroscopically confirmed UFDs. More recently, Fu et al. (2023)
measured the MDF of Hya II from HST imaging and reported an average [Fe/H] of
$-3.08^{+0.11}_{-0.12}$ dex and a $\sigma_{\rm[Fe/H]}=0.33^{+0.12}_{-0.13}$
dex based on 30 stars. Including a more metal-rich possible outlier, the
$\sigma_{\rm[Fe/H]}$ increased to $0.47^{+0.13}_{-0.12}$ dex.

####  4.1.4 Phoenix II

From the DES data for Phe II, Koposov et al. (2015b) reported an azimuthally-
averaged $r_{h}$ of $1\farcm 3$ ($a_{h,\mathrm{exp}}\sim 1\farcm 7$), and
Bechtol et al. (2015) published an $a_{h,\mathrm{p}}$ value of $\sim 1\farcm
2$, although Bechtol et al. (2015) were unable to constrain a position angle
or ellipticity. Using the same DES data, Muñoz et al. (2018) measured similar
values, with $a_{h,\mathrm{exp}}\sim 1\farcm 6$ and $a_{h,\mathrm{p}}\sim
1\farcm 5$. Mutlu-Pakdil et al. (2018) took follow-up data with Megacam at the
Magellan Clay telescope, reaching $\sim$2-3 magnitudes deeper, and reported an
exponential $a_{h,\mathrm{exp}}\sim 1\farcm 5$. The most recent literature
measurement from Moskowitz & Walker (2020) used DES DR1 and fit
$a_{h,\mathrm{p}}\sim 1\farcm 9$.

Our $a_{h,\mathrm{exp}}=1\farcm 58^{+0.26}_{-0.23}$ is consistent with the
three past literature measurements. Our $a_{h,\mathrm{p}}=1\farcm
5^{+0.20}_{-0.17}$ is also consistent with the past Plummer profile fits.
Broadly, our $\theta$ and $\epsilon$ values are consistent with past
measurements, with the exception of Moskowitz & Walker (2020). Using the Muñoz
et al. (2018) $a_{h,\mathrm{exp}}=1\farcm 60$, our FOV/$a_{h}$ is 2.1. We
calculate $M_{V}=-2.9^{+0.2}_{-0.1}$ mag, consistent with all past literature
measurements.

VLT/FLAMES spectroscopy for Phe II was presented in Fritz et al. (2019), who
measured a velocity dispersion of $11.0\pm 9.4$ km s-1 and an intrinsic
metallicity spread of 0.33 dex for the five stars that the authors identified
as members. Their assessment of Phe II as a galaxy agreed with the previous
suggestion by Mutlu-Pakdil et al. (2018), who based their classification on
its ellipticity and position in the size-luminosity plane. Baumgardt et al.
(2022) found no significant mass segregation, supporting the dwarf galaxy
classification; they noted, however, that because of its large relaxation
time, there is still the possibility that Phe II could be a star cluster. Fu
et al. (2023) report a mean [Fe/H]$=-2.36^{+0.18}_{-0.16}$ dex and a
$\sigma_{\rm[Fe/H]}=0.41^{+0.22}_{-0.17}$ dex from 10 HST stars, and paired
with the mass segregation information, supported the classification of Phe II
as a galaxy.

####  4.1.5 Sagittarius II

Laevens et al. (2015b) reported the discovery of this satellite from the
Panoramic Survey Telescope and Rapid Response System (Pan-STARRS 1, PS1;
Chambers et al., 2016) survey data, and measured $a_{h,\mathrm{exp}}\sim
2\arcmin$ ($r_{h}$ in their paper). This satellite’s nature (cluster versus
galaxy) was ambiguous at the time of discovery leading to the dual name of
Sagittarius II/Laevens 5, however, all subsequent papers adopted the Sgr II
naming. Mutlu-Pakdil et al. (2018) presented deeper Magellan Megacam imaging
with an $a_{h,\mathrm{exp}}$ of $\sim 1\farcm 6$, which agreed within the
uncertainties with that of Laevens et al. (2015b). Longeard et al. (2020) took
deep observations with the Canada-France-Hawaii Telescope (CFHT) MegaCam and
presented $a_{h,\mathrm{exp}}\sim 1\farcm 7$ ($r_{\mathrm{h}}$ in their
paper). Moskowitz & Walker (2020) used Pan-STARRS DR1 (Chambers et al., 2016)
and fit a larger $a_{h,\mathrm{p}}\sim 1\farcm 9$.

Our $a_{h,\mathrm{exp}}=1\farcm 94\pm 0.08$ is larger than recent
measurements, though consistent with Laevens et al. (2015a). We differ at a
level of $\sim 2.6\sigma$ with Longeard et al. (2020) and Mutlu-Pakdil et al.
(2018). Based on the Longeard et al. (2020) $a_{h,\mathrm{exp}}=1\farcm 7$,
our FOV/$a_{h}$ is a little under 2. Similarly to Mutlu-Pakdil et al. (2018)
and Longeard et al. (2020), we also find very little ellipticity, with
$\epsilon<0.03$ and $<0.09$ at the 95% confidence level. Additionally, our
$a_{h,\mathrm{p}}=1\farcm 85^{+0.07}_{-0.08}$ is consistent with the
measurement of Moskowitz & Walker (2020).

Reported position angle values vary widely, which could be related to the low
ellipticity leading to the lack of a clear semi-major axis. Our measured
$\theta$ values agree within 0.1 and 0.6$\sigma$ to the Longeard et al. (2020)
and Moskowitz & Walker (2020) values, respectively, however our
$\epsilon<0.09$ suggest that $\theta$ may not be meaningful here. Our
$M_{V}=-5.3\pm 0.2$ mag is consistent with Laevens et al. (2015a) and Longeard
et al. (2020), but not with the Mutlu-Pakdil et al. (2018) $M_{V}=-4.2\pm 0.1$
mag.

Simon et al. (2019) presented a spectroscopic study of Sgr II based on
Magellan/IMACS data, reporting a velocity dispersion of 1.6 $\pm 0.3$ km s-1
and a metallicity dispersion $<0.08$ dex at the 95% confidence limit,
suggesting that Sgr II could be a globular cluster. From Keck II/DEIMOS data,
Longeard et al. (2020) measured a velocity dispersion of 2.7${}^{+1.3}_{-1.0}$
km s-1 and a combined photometric and spectroscopic metallicity dispersion of
0.12${}^{+0.03}_{-0.02}$ dex. Longeard et al. (2021) used new VLT/FLAMES data
combined with the DEIMOS spectra, measuring a smaller velocity dispersion of
1.7 $\pm 0.5$ km s-1 from 113 stars. From 15 stars, they calculated an
unresolved metallicity dispersion less than 0.20 dex at the 95% confidence
limit. Longeard et al. (2021) thus concluded that Sgr II is a globular
cluster. Baumgardt et al. (2022) additionally found a high level of mass
segregation, supporting the star cluster classification. While Fu et al.
(2023) do not present their analysis on Sgr II, they state that their results
are consistent with the globular cluster status.

####  4.1.6 Triangulum II

This was the first discovered ultra-faint satellite from the PS1 data by
Laevens et al. (2015a), who then took follow-up imaging with the Large
Binocular Cameras at the Large Binocular Telescope (LBT). Unable to determine
whether this satellite was a cluster or dwarf galaxy, Laevens et al. (2015a)
gave it the double name of Laevens 2 and Triangulum II. The initial
$a_{h,\mathrm{exp}}$ value from Laevens et al. (2015b) (reported as $r_{h}$)
was $3\farcm 9$, although they included the caveat that with the low contrast
of the stellar overdensity to the background, deeper data would provide a
stronger measurement.

Carlin et al. (2017) presented further follow-up deep imaging from the Subaru
Hyper Suprime-Cam that reached $\sim$2 magnitudes deeper than the discovery
data and increased the overdensity-to-background contrast. They derived an
$a_{h,\mathrm{exp}}$ of $2\farcm 5$, consistent within 1.5$\sigma$ to the
Laevens et al. (2015b) value. Muñoz et al. (2018) derived a similar
exponential $a_{h,\mathrm{exp}}\sim 2\farcm 3$ and a slightly smaller
$a_{h,\mathrm{p}}\sim 2$ using Pan-STARRS data from the discovery team.
Moskowitz & Walker (2020) measured an $a_{h,\mathrm{p}}\sim 1\farcm 8$ from
the Pan-STARRS data.

Tri II had two ACS pointings, giving us an FOV of $\sim 3\farcm 3\times
6\farcm 7$, and a maximum possible FOV/$a_{h}$ value of $\sim 2.9$ along the
semi-major axis. We measured $a_{h,\mathrm{exp}}=3\farcm 50^{+0.59}_{-0.65}$,
larger, but within 1.4$\sigma$ agreement with Carlin et al. (2017) and Muñoz
et al. (2018); our value is smaller than the Laevens et al. (2015a)
measurement, though consistent within $0.4\sigma$. Our
$a_{h,\mathrm{p}}=3\farcm 25^{+0.51}_{-0.59}$ measurement is larger but within
$1.7\sigma$ agreement with the Muñoz et al. (2018) Plummer measurement and
$2.2\sigma$ agreement with the Moskowitz & Walker (2020) value.

Our $\theta$ and $\epsilon$ values are consistent with both the Carlin et al.
(2017) and Moskowitz & Walker (2020) measurements. The $\theta$ and $\epsilon$
values from our exponential models are within 1.3 and 1.6$\sigma$ of the Muñoz
et al. (2018); for the Plummer model, they are within 0.9 and 0.6$\sigma$,
respectively. Our fits are most consistent with the larger $a_{h}$ and smaller
ellipticity presented in the Laevens et al. (2015b) discovery paper. The low
number of stars in this galaxy and the slight gap between the two ACS
pointings may have affected our fits, and we advise adopting either the Carlin
et al. (2017) or Muñoz et al. (2018) values. The integrated $M_{V}=-1.3\pm
0.2$ mag that we measure is in $1\sigma$ agreement with the three past
literature determinations.

Numerous spectroscopic studies of Tri II have been published (Kirby et al.,
2015a; Martin et al., 2016; Venn et al., 2017; Kirby et al., 2017; Ji et al.,
2019), often with results hinting at a dwarf galaxy nature, but with no
definitive conclusions. The most recent spectroscopic study by Buttry et al.
(2022) combined new MMT/Hechtochelle data with the existing Keck II/DEIMOS set
and presented an upper limit on the velocity dispersion of 3.4 km s-1 at the
95% confidence level. Additionally, with new Gaia proper motion information
confirming two questionable member stars used in the Kirby et al. (2017)
metallicity dispersion measurement, Buttry et al. (2022) used their combined
data to report a value of 0.46${}^{+0.37}_{-0.09}$ dex; this large spread
lends stronger support to a Tri II galaxy classification. Additionally, the
Baumgardt et al. (2022) study found no mass segregation and supported this
conclusion.

Figure 6: Similar to Figure 3, for the four targets that we present photometry
for but do not measure structural parameters. North is up and East is to the
left in all panels. The ACS FOV in blue has a side length of 202″; the WFC3
FOV in red has a side length of 162″. For targets with multiple ACS pointings,
the corresponding WFC3 off-field is labeled to match the ACS label, not the
WFC3 relative physical positions.

###  4.2 Review of Non-Measured Systems

For Gru II, Ret II, Tuc II, and Tuc IV, we give a brief review of past
morphological studies. Figure 6 shows the coverage of our HST observations
relative to past literature measurements of the target half-light ellipses.

####  4.2.1 Grus II

Gru II was the ultra-faint candidate with the highest sigma detection
discovered in the second year of DES data, and had no significant ellipticity
or asphericity (Drlica-Wagner et al., 2015). As such, there would not be a
corresponding $a_{h}$; Drlica-Wagner et al. (2015) reported $r_{h}=6\arcmin$.
Using the DES Y3A2 catalog,666DES Y3A2 refers to an internal DES release. The
public DES DR1 was based on this (Abbott et al., 2018). Simon et al. (2020)
repeated the structural analysis and recovered an $a_{h,\mathrm{p}}$ (referred
to as $r_{1/2}$) of $5\farcm 9$ and an upper limit on the ellipticity of 0.21.
The Moskowitz & Walker (2020) $a_{h,\mathrm{p}}$ measurement from DES DR1
(Abbott et al., 2018) data was larger, at $\sim 7\farcm 8$, yet within 1.7 and
2.5$\sigma$ of the Drlica-Wagner et al. (2015) and Simon et al. (2020) values,
respectively.

We did not fit structural parameters for Gru II, as past measurements of its
size have found $a_{h}\sim 6\arcmin$ (Drlica-Wagner et al., 2015; Moskowitz &
Walker, 2020; Simon et al., 2020), and our single ACS pointing only extends to
$\sim 0.25a_{h}$, which is insufficient for a robust morphology measurement.

The first spectroscopic observation of Gru II was performed by Simon et al.
(2020), who identified 21 member stars with Magellan/IMACS spectroscopy. They
were unable to clearly resolve a velocity dispersion, but did report 90 and
95.5% upper limits. Additionally, Simon et al. (2020) did not detect a
metallicity spread, however based on the low metallicity ([Fe/H]=$-2.51\pm
0.11$ dex) and large size, they concluded Gru II is likely a dwarf galaxy.
Using DOLPHOT photometry of HST data, Baumgardt et al. (2022) did not find
significant mass segregation among the member stars, leading them to also
conclude that Gru II is likely a dwarf galaxy.

####  4.2.2 Reticulum II

From the DES data, in their discovery papers, Koposov et al. (2015b) measured
an $r_{h}$ of $\sim 3\farcm 6$ ($a_{h,\mathrm{exp}}\sim 5\farcm 7$) and
Bechtol et al. (2015) measured an $a_{h,\mathrm{p}}\sim 6\arcmin$. Muñoz et
al. (2018) fit slightly smaller values of $a_{h,\mathrm{exp}}\sim 5\farcm 4$
and $a_{h,\mathrm{p}}\sim 5\farcm 5$ to the same DES data. Mutlu-Pakdil et al.
(2018) used deeper (by $\sim$2-3 mags) data from Megacam to fit an
$a_{h,\mathrm{exp}}\sim 6\farcm 3$, in agreement with the original Bechtol et
al. (2015) measurement. The Moskowitz & Walker (2020) $a_{h,\mathrm{p}}$ from
DES DR1 was slightly larger, at $\sim 6\farcm 6$. We did not fit structural
parameters for Ret II, as the ACS FOV covers approximately 0.25$a_{h}$.

Three independent spectroscopic studies of Ret II were published in the months
following the discovery papers (Simon et al., 2015; Walker et al., 2015;
Koposov et al., 2015a), all concluding that Ret II had the kinematic and
spectroscopic properties consistent with being a dwarf galaxy. Simon et al.
(2015) presented Magellan/M2FS, VLT/GIRAFFE, and Gemini South/GMOS
spectroscopy; from 25 member stars, they calculated a velocity dispersion of
$3.3\pm 0.7$ km s-1 and a metallicity dispersion of $0.28\pm 0.09$ dex. Walker
et al. (2015) confirmed 17 member stars and reported a velocity dispersion of
3.6${}^{+1.0}_{-0.7}$ km s-1 and a metallicity dispersion of
0.49${}^{+0.19}_{-0.14}$ dex using Magellan/M2FS data. Lastly, Koposov et al.
(2015a) measured a velocity dispersion of 3.22${}^{+1.64}_{-0.49}$ km s-1 and
a metallicity dispersion of 0.29${}^{+0.13}_{-0.05}$ dex from the VLT/GIRAFFE
data.

The most recent spectroscopic study by Ji et al. (2023) updated these values
to be 2.97${}^{+0.43}_{-0.35}$ km s-1 and 0.32${}^{+0.10}_{-0.07}$ dex for the
velocity and metallicity dispersion, respectively. Baumgardt et al. (2022)
applied their mass-segregation method to Ret II and found results in agreement
with the previous studies, however they noted that due to the large relaxation
time, the mass-segregation method alone would not be able to clearly determine
the classification. Fu et al. (2023) used HST photometric metallicities from
76 stars to derive an average [Fe/H]$=-2.64^{+0.10}_{-0.11}$ dex and
$\sigma_{\rm[Fe/H]}=0.72^{+0.09}_{-0.08}$ dex. This large metallicity
dispersion would strongly support a galaxy classification. They note the large
discrepancy with past spectroscopic studies and invite follow-up observations
for better membership determination.

####  4.2.3 Tucana II

The discovery parameters for Tuc II from Koposov et al. (2015b) and Bechtol et
al. (2015) were quite different, with the former reporting an azimuthally-
averaged $r_{h}$ of $9\farcm 8$ ($a_{h,\mathrm{exp}}\sim 12\farcm 9$) and the
latter publishing an $a_{h,\mathrm{p}}$ of $7\farcm 2$. This discrepancy could
be due in part to different CMD masking techniques for member selection. While
Bechtol et al. (2015) note that there was some elongation in the outer regions
of the detected overdensity, they suggest that the distortion was likely
noise-related and show that after their CMD-masking, the overdensity is much
rounder. Figure 12 of Koposov et al. (2015b) shows the CMD-masked stars in the
Tuc II field with the elongation still visible. Moskowitz & Walker (2020) fit
DES data for Tuc II and recovered $a_{h,\mathrm{p}}\sim 13\farcm 5$ pc,
consistent with the $a_{h,\mathrm{exp}}$ of Koposov et al. (2015b) within
1$\sigma$.

Our FOV included four ACS pointings, for a maximum field-width of less than
$7\arcmin$. We did not fit the structural parameters as though literature
$a_{h}$ measurements varied from $\sim 7\arcmin-13\farcm 5$, our FOV cover at
most the inner one-half of the galaxy.

Using eight probable member stars with spectroscopy from the Michigan/Magellan
Fiber System (M2FS), Walker et al. (2016) measured a velocity dispersion for
Tuc II of 8.6${}^{+4.4}_{-2.7}$ km s-1 and a metallicity dispersion of
0.23${}^{+0.18}_{-0.13}$ dex and concluded that it is a dwarf galaxy. The
velocity dispersion has since been remeasured by Chiti et al. (2023), who used
16 member stars and calculated a value of 3.8${}^{+1.1}_{-0.7}$ km s-1.

####  4.2.4 Tucana IV

Tuc IV had the largest angular size of the ultra-faint satellite candidates
discovered in the DES year-two data, with an azimuthally-averaged
$r_{\mathrm{h}}=9.1^{\prime}$ ($a_{h,\mathrm{p}}\sim 11\farcm 8$; Drlica-
Wagner et al., 2015). The analysis using DES DR1 by Moskowitz & Walker (2020)
measured a slightly smaller $a_{h,\mathrm{p}}\sim 9\farcm 2$, consistent
within 1.3$\sigma$ with the Drlica-Wagner et al. (2015) value. From the DES
Y3A2 catalog, Simon et al. (2020) published the most recent structural
parameters for Tuc IV, reporting $a_{h,\mathrm{p}}\sim 9\farcm 3$. Our two ACS
pointings would cover approximately the inner one-third of the target, so we
did not attempt to characterize the morphology with our HST data.

Simon et al. (2020) published the first spectroscopic study of Tuc IV,
significantly resolving a velocity dispersion of 4.3${}^{+1.7}_{-1.0}$ km s-1.
From eight RGB stars, they measured a metallicity dispersion of
0.18${}^{+0.20}_{-0.18}$ dex. Simon et al. (2020) calculated a mass-to-light
ratio of 3100${}^{+2900}_{-1600}$ $M_{\odot}/L_{\odot}$, indicating a dark-
matter dominated nature, and thus spectroscopically confirmed Tuc IV as a
galaxy.

##  5 Discussion

###  5.1 Uses for HST Imaging in Structural Analysis

The HST MW UFD Treasury Program (GO-14734) contains a wealth of data, and
precision photometry is essential for optimizing its value. Here, we have
presented a subset of the data and an example use case, with a focus on
targets that have been kinematically linked to the LMC. While a space
telescope with a small FOV would not be the premier choice for morphological
studies, these types of analyses are still beneficial as ancillary uses for
the data. The depth and resolution of the images have given us the ability to
characterize the inner surface densities, and in some cases, recover a full
set of structural parameters in agreement with past literature. Five out of
twelve structural fits (6 targets, 2 models) for $a_{h}$ fall within 1$\sigma$
of previously measured values, and all fall within $3\sigma$ even though we do
not meet the suggested FOV and number of sources criteria (e.g., Muñoz et al.,
2012).

One of the advantages of high-resolution, space-based imaging is the ability
to more clearly distinguish stars from galaxies, as the PSFs are not affected
by the atmosphere. This can be important for determining whether apparent
structural overdensities from ground-based observations are meaningful or if
they are from contaminating background sources.

An illustrative example of this is when the analysis of ground-based, Megacam
imaging of Leo V seemed to suggest the existence of a debris stream (Sand et
al., 2012) and follow-up space-based observations showed otherwise. Mutlu-
Pakdil et al. (2019) presented HST/ACS imaging centered on the overdensity in
the Leo V field, and showed that almost half of the photometric sources were
background galaxies. The higher-precision photometry also revealed that many
of the true stars from the overdensity were unlikely to be Leo V members.

In the case of Hercules, however, photometry of off-center HST/ACS fields
contained a main sequence consistent with the stellar population identified in
the central pointing, supporting the existence of Hercules’ elongation (Mutlu-
Pakdil et al., 2020). The Mutlu-Pakdil et al. (2019, 2020) studies both relied
on archival imaging of the central regions of their target UFDs, much like the
observations we have presented here. Their results highlight how we should be
cautious in our interpretations of apparent low surface-brightness features,
yet at the same time, how important follow-up, space-based imaging is in order
to gain more insight into what the structure of a satellite can tell us.

###  5.2 The Importance of UFD Structural Characterization

As UFDs are thought to reside in the least massive DM halos observed to be
hosting galaxies (e.g., Bullock & Boylan-Kolchin, 2017; Simon, 2019), their
mass accretion histories and their sensitivity to tidal forces have
implications for their role in hierarchical structure formation (e.g., Mutlu-
Pakdil et al., 2020). While we cannot observe directly how UFDs have evolved
to their present state, their morphologies may still reflect major events that
disturbed their original mass distributions. For example, early morphology
studies noted that UFDs were more elliptical than classical dwarfs, with a
possible explanation being that these less massive galaxies were more
susceptible to tidal effects from the MW (e.g., Martin et al., 2008). While
updated measurements from a larger sample size showed that there is little
evidence to support a difference between the two populations in this respect
(Sand et al., 2012; Simon, 2019), the existence of UFD “tidal features” (and
the need for a physical explanation) remain (e.g., Simon & Geha, 2007; Sand et
al., 2009; Muñoz et al., 2018; Mutlu-Pakdil et al., 2020).

One particularly interesting case is that of Tuc II, one of our limited-FOV
targets with only photometry presented. Chiti et al. (2021) reported the
discovery of member stars out to $\sim$9 times the Bechtol et al. (2015)
$a_{h}$ and noted that the orbital parameters of Tuc II made tidal disruption
an incompatible explanation. Soon after, simulation groups were able to
reproduce UFD “tidal features” without invoking tidal heating from a more
massive host galaxy (Tarumi et al., 2021; Goater et al., 2023). For example,
assembly from less massive “building block galaxies” (Tarumi et al., 2021) as
well as late-time dry accretion (Goater et al., 2023) can produce extended
stellar halos and more elliptical stellar distributions. There has also been
more interest in the stellar halos of dwarf galaxies, including how they could
be built up through dwarf-dwarf mergers (Deason et al., 2022) and how to
systematically search for them observationally (e.g., Filion & Wyse, 2021;
Jensen et al., 2023; Longeard et al., 2023).

Figure 7: Absolute $V$-band magnitude, $M_{V}$, versus azimuthally-averaged
half-light radius, $r_{h}$, for UFDs, UFD candidates, and the globular cluster
Sag II (green circle). Golden circles represent the six LMC-associated UFDs,
while the blue circles are the non-LMC confirmed UFDs. The three dark gray
markers (1 circle, 2 stars) represent the UFDs that are either thought to be
recently captured or formerly associated with the LMC, which were not used in
the trendline fits. Circles with bold black outlines (with labels) show
measurements from this work. Star symbols are targets (with italicized labels)
for which we present photometry, but do not fit structural parameters. Silver
markers with errorbars are non-confirmed UFDs, which were not used in the
trendline fits. The solid gold (dark blue) line represents the median value
from the Monte Carlo simulations for the LMC (non-LMC) UFD $M_{V}-r_{h}$
relation. The dashed continuation of the gold line references the
extrapolation of the LMC-associated trendline. The gold and dark dark blue
shaded regions cover the 16th to 84th percentiles from the Monte Carlos for
the respective sets. The cyan shaded region represents the 16th to 84th
percentiles of lines fit to the six galaxy, non-LMC Monte Carlo simulation as
described in Section 5.3. The light green band is where the yellow and cyan
regions overlap, and the darkest shaded region is where all three uncertainty
spaces overlap. Though the fit trendlines diverge, the overlapping uncertainty
bands suggest no statistical difference.

###  5.3 LMC vs. Non-LMC UFDs in the $M_{V}-r_{h}$ Plane

Sacchi et al. (2021) found tantalizing hints of differences between the SFHs
of the LMC- versus non-LMC-associated UFDs, which inspired us to investigate
whether the two UFD subgroups showed differences in other observational
planes. One could postulate that environmental effects from the density
differences in the early Universe or more recently, the MW halo, could have
affected the two groups of satellites differently, perhaps leading to either
more compact or extended morphologies based on length of time in the MW halo
and pericenter distance.

To explore whether there were any differences between the two subgroups
apparent in the $M_{V}-r_{h}$ parameter space, we fit a line for each set of
UFDs to broadly characterize the relationship. To account for the large
uncertainty spaces, we drew $M_{V}$ and $r_{h}$ values for each galaxy in the
two sets (LMC and non-LMC) using the reported values as the Gaussian mean and
the reported uncertainties as the Gaussian sigmas. Additionally, we assume an
intrinsic scatter in $M_{V}$ at fixed $r_{h}$. This is modeled using a
Gaussian with a sigma set to the root-mean-square scatter of the data around
the best-fit line, with a separately drawn value added to each $M_{V}$. For
each set of redrawn values, we fit a linear regression model. We performed
this procedure 10,000 times per set and show the median $M_{V}$ value of the
evaluated linear functions for $r_{h}$ from 0 to 200 pc in Figure 7 (gold and
dark blue solid lines for the LMC and non-LMC UFDs, respectively). The gold
and dark blue shaded regions encompass the 16th to 84th percentiles of $M_{V}$
values evaluated across the $r_{h}$ space.

While the median trendlines lines overlap at magnitudes fainter than $-3.5$,
at brighter magnitudes they begin to diverge. This seems to agree with the
Erkal & Belokurov (2020) finding that at fixed luminosity, likely LMC
satellites have slightly smaller $r_{h}$ than MW satellites, which they
postulated could be due to different tidal environments. To test the
significance of this comparison, we ran a Monte Carlo that drew 10,000 sets of
six random non-LMC associated UFDs with $M_{V}$ and $r_{h}$ values drawn using
the previously described Gaussian technique. We fit a linear regression to
each set of six galaxies and over-plotted the region that encompasses the 16th
to 84th percentiles in light blue on Figure 7. This region overlaps the LMC
satellite trendline heavily and encompasses the space around five out of the
six LMC UFDs, suggesting that the divergence of the median $M_{V}-r_{h}$
relation is not statistically significant.

Figure 8: Absolute $V$-band magnitude, $M_{V}$, versus azimuthally-averaged
half-light radius, $r_{h}$. The dashed light and dark gray diagonal lines
represent constant surface brightness values of 28 and 30 mag arcsec-2,
respectively. The dash-dotted horizontal light gray line marks
$M_{V}\leq-7.7$, often used as the delineation between classical and ultra-
faint dwarfs (Simon, 2019). Classical dwarfs (black inverted triangles) are
labelled. Measurements of classical dwarfs, UFDs (black squares), and
ambiguous satellites (open diamonds) are shown as black markers with error
bars. LMC-associated satellites are additionally marked with a black square
frame. Globular and “ultra-faint” clusters are shown as gray points and x’s,
respectively. All other markers are from simulations and colored by their
baryonic particle mass resolution, except for the TNG points, which have a
mass resolution $\sim 10^{4}M_{\odot}$. If different environments were used
within the same simulation grouping, we have marked the difference using black
outlines. Simulated points using alternative DM models are shown as open
symbols. The Macciò et al. (2019) WDM galaxies are connected to their
corresponding CDM galaxies with coral dashed lines. Advances in simulations
have led many groups to be able to generate galaxy analogs in and around the
observed UFD space. Full references for the data are given in Appendix B, and
references for the simulations are in Table LABEL:tab:simtable.  Simulation
Descriptions
---
Simulation Set |  $N_{\rm gal}$ a | Hydro. Code | Baryonic Res. | Halo Massc, $z=0$ | Environment | Ref.
|  |  | ($M_{\odot}$) | ($M_{\odot}$) |  |
CDM
DC Justice League | 36 | CHANGA | 994 | $1.1\times 10^{6}-3.7\times 10^{9}$ | MW-like | 1
EDGE | 31 | RAMSES-RT | $20-300$ | $8.0\times 10^{8}-3.7\times 10^{9}$ | Isolated | 2, 3, 4, 5
FIRE-2 + CDM | 42 | GIZMO | $30-250$ | $7.0\times 10^{8}-9.0\times 10^{9}$ | Isolated | 6, 7, 8, 9
FIRE-2, Jahn 2022 | 25 | GIZMO | 880 | $(2.9-6.1)\times 10^{8}$ | LMC-like | 6, 10, 11
GEAR | 19 | GEARb | 380 | $3.0\times 10^{8}-2.4\times 10^{9}$ | Isolated | 12
Jeon 2017 | 6 | Modified GADGET | 465 | $(1.5-4.0)\times 10^{9}$ | Field | 13
Jeon 2021 | 42 | Modified GADGET | 65 | $(2.0-2.5)\times 10^{8}$ | Field | 14, 15
LYRA | 4 | AREPO | 4 | $3.0\times 10^{8}-2.8\times 10^{9}$ | Isolated | 16
Macciò 2017, Cent. | 9 | GASOLINE | $40,135$ | $(1.7-5.7)\times 10^{9}$ | Isolated overdensity | 17
Macciò 2017, Sat. | 6 | GASOLINE | $90,305$ | $4.5\times 10^{8}-1.1\times 10^{10}$ | Isolated overdensity | 17
Macciò 2019 (CDM) | 4 | GASOLINE-2 | $2.4\times 10^{3}-1.9\times 10^{4}$ | $(2.9-9.1)\times 10^{9}$ | Isolated overdensity | 18
Marvel-ous Dwarfs | 32 | CHANGA | 420 | $2.6\times 10^{8}-4.5\times 10^{9}$ | Local Volume | 19
TNG (CDM) | 47 | AREPO | $4.2\times 10^{4}$ | $2.2\times 10^{7}-2.8\times 10^{9}$ | Field | 20
SIDM
FIRE-2 + SIDM | 4 | GIZMO | 500 | $8.1\times 10^{9}$ | Isolated | 21
TNG (ETHOS/SIDM) | 52 | AREPO | $4.2\times 10^{4}$ | $1.9\times 10^{6}-4.2\times 10^{9}$ | Field | 22
WDM
FIRE-2 + WDM | 3 | GIZMO | 500 | $(6.7-8.5)\times 10^{9}$ | Isolated | 23
Macciò 2019 (WDM) | 3 | GASOLINE-2 | $2.4\times 10^{3}-1.9\times 10^{4}$ | $3.1\times 10^{9}-1.3\times 10^{10}$ | Isolated overdensity | 18

###  5.4 Observed and Simulated UFDs in the $M_{V}-r_{h}$ Plane

Baryonic simulations have advanced to the extent that some are now able to
resolve galaxy formation in the ultra-faint regime. We present a subset of
these simulated UFD-analogs plotted in $M_{V}$ versus $r_{h}$ in Figure 8
along with observed UFDs, UFD candidates, and star clusters.

As simulation groups often report stellar mass ($M_{\star}$) and 3D, half-mass
radii rather than $M_{V}$ and 2D, projected half-light radii, respectively,
some literature values had to be converted. When $M_{V}$ and 2D $r_{h}$ values
were provided from the simulations, we plotted those. Otherwise, to place the
simulated galaxies in a plane where the quantities are more readily
comparable, we converted $M_{\star}$ to $M_{V}$ using a stellar-mass-to-light
ratio of 2, which is reasonable for an ancient stellar population (e.g.,
Martin et al., 2008; Simon, 2019). To convert the 3D radius values, we used
the relation presented in Wolf et al. (2010):

$R_{e}\simeq\frac{3}{4}r_{1/2},$ (2)

where $R_{e}$ is the 2D, projected half-light radius and $r_{1/2}$ is the 3D,
deprojected half-light radius. Assuming that all simulated galaxies have
roughly spherical mass distributions, $R_{e}$ is then equivalent to $r_{h}$.
The simulated galaxies are colored according to the reported baryonic mass
resolutions, except for the TNG points, as these were much lower resolution
with star particle masses $\sim 10^{4}M_{\odot}$. Most of the simulations
followed the CDM model, but there are some that used warm dark matter (WDM)
and self-interacting dark matter (SIDM); these are shown with open markers.
Basic properties of the simulations used are listed in Table
LABEL:tab:simtable.

For the data, we convert $a_{h}$ values from the literature to $r_{h}$ using
the reported $\epsilon$ values. Plotted error bars are generated from a Monte
Carlo using both the uncertainties on the $a_{h}$ and $\epsilon$ values. To
simplify the plot, we do not show error bars on the globular cluster values
from the literature. Star clusters are classified here as “ultra-faint” if
they have $M_{V}\geq-3$ (e.g., Cerny et al., 2023a). Literature values for
classical dwarfs, UFDs, and ambiguous satellites are shown in black. For the
purposes of this plot, we are not highlighting our new measurements but have
included an extra square frame around the LMC-associated sample.

We have also included simulated galaxies from LYRA (Gutcke et al., 2022) even
though no UFDs are produced, as this model created classical dwarf analogs
similar to those from the other simulations considered. The LYRA set also has
the highest resolution, with a baryonic particle mass of 4 $M_{\odot}$ and is
likely going to produce UFD analogs in future studies.

We also note that the TNG galaxies (Rose et al., 2023, J. Rose et al. 2024, in
prep.) may not be considered direct UFD analogs due to limitations from their
use of a pressurized equation of state. These galaxy formation prescriptions
are identical to those used in IllustrisTNG (e.g., Pillepich et al., 2018), a
well-known, large volume cosmological simulation that has been shown to be
consistent with observed size relations for galaxies with $M_{\star}\gtrsim
10^{9}\ M_{\odot}$ and to have good numerical convergence (Genel et al.,
2018). While the TNG galaxies have much lower resolution than the other
simulations, we include them because they offer a comparison between samples
from both CDM and alternative DM models with the same baryonic physics. The
ETHOS alternative DM models (Vogelsberger et al., 2016) included contain both
suppression to the initial matter power spectrum in the form of dark acoustic
oscillations, and SIDM with a self-consistently chosen velocity-dependent
cross-section. As the TNG ETHOS galaxies include more than the standard SIDM
(which would use the CDM initial matter power spectrum), we will refer to them
as ETHOS/SIDM.

####  5.4.1 Do simulated UFDs match observations?

While there are many different metrics for determining how well simulations
are producing UFDs (e.g., velocity dispersions, metallicity distribution
functions), here, we choose to examine agreement in the $M_{V}-r_{h}$ plane.
From Figure 8, it can be seen that the Jeon et al. (2021a, b) simulations
(higher resolution, magenta x’s) produce the highest number of UFD analogs
that overlap with observed MW UFDs. EDGE (Agertz et al., 2020; Prgomet et al.,
2022), FIRE-2 (Jahn et al., 2022; Wetzel et al., 2023) using both CDM and WDM
(Bozek et al., 2019), GEAR (Revaz & Jablonka, 2018; Sanati et al., 2023),
Macciò et al. (2017, 2019), and the Marvel-ous Dwarfs (Munshi et al., 2021)
groups have also produced some UFDs that match the bright end of the observed
population.

The fact that observed UFD properties are able to be matched by high
resolution simulations of isolated UFDs could suggest that environmental
effects from the MW have not played a critical role in their size evolution.
Additionally, the LMC-associated UFDs are matched by these isolated
simulations. Their more recent infall ($\sim$2 Gyr ago), paired with their
lack of differences in the $M_{V}-r_{h}$ space from the non-LMC-associated
UFDs, further supports that the presence of the MW has had little overall
consequence on the bulk of the current UFD size distribution.

Interestingly, in a study of the extended stellar populations of UFDs (i.e.,
member stars beyond four times $r_{h}$), Tau et al. (2024) found that at least
10 satellites with external populations had a wide range of magnitude and
pericenter distances. This too suggests that interactions with the MW are
$not$ what dominates the distribution of UFD stars that we observe today.

Despite the overlap of simulations with the brighter UFDs, most simulations
are unable to produce analogs of the most compact observed UFDs ($r_{h}\sim
30$ pc; e.g., Revaz, 2023). Rather, they are creating UFDs with half-light
radii up to an order of magnitude larger than observed. Revaz (2023)
speculates that this could be due to resolution limits, spurious numerical
heating, or improper feedback prescriptions, although the widespread nature of
the pattern suggests there is no singular factor. In their own work using DM-
only simulations, Revaz (2023) found that mini-halo mergers played a large
role in the expansion of the simulated UFD sizes.

Conversely, it could be said that simulations are producing galaxies that are
more diffuse than currently known observationally. This disparity can be
revisited as new observatories come online and we expand our knowledge of the
low surface-brightness universe. Discoveries of larger, low surface-brightness
UFDs could resolve the current discrepancy.

The simulations whose data we show in Figure 8 have a wide range of mass
resolutions and distinct implementations of subgrid physics, which are
necessary to form galaxies in a cosmological simulation, as the range of
scales (e.g., individual supernovae to Mpc-sized volumes) would be too
computationally expensive otherwise. Despite using different combinations of
free parameters, many simulation groups are still able to produce UFD analogs
that fall within the same $M_{V}-r_{h}$ space. For example, simulated UFDs
with $r_{h}$ between $\sim 100-300$ pc and $M_{V}\leq-4$ are produced by the
DC Justice League (Applebaum et al., 2021), EDGE (Agertz et al., 2020; Orkney
et al., 2021; Rey et al., 2022), FIRE-2 (Jahn et al., 2022; Wetzel et al.,
2023), GEAR (Sanati et al., 2023), Jeon et al. (2017), Macciò et al. (2017,
2019), and the Marvel-ous Dwarfs (Munshi et al., 2021).

The consistently most compact UFDs ($r_{h}<100$ pc) come from Jeon et al.
(2021a, b). Paired with their low luminosities ($M_{V}\geq-4$), these
simulations are quite successful in reproducing the observed distribution of
the and LMC- and non-LMC-associated UFDs in the magnitude-size plane. These
high-resolution simulations were run with a customized version of gadget
(Springel, 2005) which used smoothed particle hydrodynamics with a gas
particle mass of 63 $M_{\odot}$. A variable softening length was used for the
gas particles with a minimum length of $\sim 3$ pc. Individual Population III
stars could be born out of pristine gas, while Population II stars were formed
out of metal-enriched gas in complexes no less massive than 500 $M_{\odot}$ to
emulate the birth of stars out of molecular clouds.

The UFDs from the EDGE simulations (Agertz et al., 2020; Rey et al., 2020;
Orkney et al., 2021; Prgomet et al., 2022) are slightly larger and more
luminous than those of Jeon et al. (2021a, b), but still fainter and more
compact than many of the other simulated galaxies. Their properties relative
to the Jeon et al. (2021a, b) galaxies are reasonable as the EDGE UFDs
inhabited DM halos that were up to an order of magnitude more massive than
those hosting the UFDs in Jeon et al. (2021a, b) (see Table
LABEL:tab:simtable). The UFDs from the EDGE simulations track the observed
magnitude-size relation well, overlapping with the LMC- and non-LMC-associated
UFDs at the bright end ($M_{V}\leq-4$). These high-resolution simulations were
run with the ramses-rt adaptive mesh refinement code (Teyssier, 2002; Rosdahl
et al., 2013; Rosdahl & Teyssier, 2015) at a variety of resolutions and with
many variations in the baryonic physics.

In the UFD regime, besides the baryonic mass resolution, one could also
consider the targeted halo mass at $z=0$ for simulations as a possible factor
in the final galaxy size. For example, in a study specifically focused on
reproducing the compactness of observed UFDs, Revaz (2023) used dark-matter-
only (DMO) simulations to study how the initial size and later merger history
of UFD building blocks affected the $z=0$ size of UFDs. They concluded that
simulated UFDs with $r_{h}<30$ pc can form from single mini-halos with masses
smaller than $4\times 10^{8}M_{\odot}$ at $z=6$ and initial sizes $<15$ pc.

Jeon et al. (2021a, b) were able to hydrodynamically simulate UFDs on this
size scale with halo masses $<2.5\times 10^{8}M_{\odot}$. A few other
simulations had halo masses below this (e.g., Applebaum et al., 2021; Rose et
al., 2023, J. Rose et al., 2024, in prep.) but were unable to achieve the
compactness. For Applebaum et al. (2021), this could be due to the difficulty
in simulating UFDs in addition to a MW-mass scale galaxy, while for TNG, their
subgrid physics choices (as discussed further below) limited their minimum
compactness.

The simulated galaxies that lie furthest from the observed sample and away
from the bulk of the other simulations are the highest resolution FIRE-2
galaxies from Wheeler et al. (2019), which have very low surface brightnesses
($\mu_{V}\gtrsim 30$ mag arcsec-2), and the TNG galaxies in both the CDM and
ETHOS/SIDM cosmologies (Rose et al., 2023, J. Rose et al. 2024, in prep.),
which have very large sizes ($r_{h}\gtrsim 10^{3}$ pc) (though it should be
remembered that we are not expecting TNG to be producing true UFD analogs).

The FIRE-2 galaxies in this group have $\sim 30M_{\odot}$ baryonic particle
resolution and were presented in Wheeler et al. (2019). Their larger sizes are
not due to mass resolution or the force-softening scale, and the authors
suggest that only the bright cores of UFDs are being captured by current
observatories. When Wheeler et al. (2019) applied a surface brightness
detection limit of $<$30 mag arcsec-2 and remeasured the apparent $r_{h}$ and
$M_{\star}$, their simulated analogs moved into the same space as observed
UFDs.

As we were able to trace the morphologies of the dwarfs in our sample to an
average surface brightness of $\mu_{V}=31.2$ mag arcsec-2, we expect that our
measurements were not significantly biased by our imaging depth (see also the
discussion at the end of Section 3.1). However, new observational facilities
are set to come online in the next decade that should reach or exceed our
photometric depth while covering the entirety of the dwarfs (e.g., the Vera C.
Rubin Observatory/LSST, Ivezić et al. 2019; and the Roman Space Telescope,
Spergel et al. 2015). As our HST fields are limited in their FOV, it will be
interesting to see what these observatories may reveal about the extremities
of these dwarf galaxies.

Of note, the Wheeler et al. (2019) FIRE-2 galaxies employed the same subgrid
physics model as other FIRE-2 galaxies (e.g., the Jahn et al. 2022) in Figure
8 that are less diffuse for their magnitude. The two differences between these
sets of FIRE-2 simulations were their mass resolutions and the environments in
which the galaxies were simulated. The Wheeler et al. study considered UFDs in
isolated dwarf galaxy halos, while the work of Jahn et al. examined UFDs
around an LMC-mass halo. Environment has previously been shown to affect DM
halo properties such as concentration, where halos that formed earlier in
cosmic time, when the Universe was more dense, have more highly concentrated
density profiles (e.g., Wechsler et al., 2002). Additionally, for galaxies
with the same peak halo masses, those that formed in more dense environments
tended to have higher stellar masses than those that were isolated
(Christensen et al., 2023).

It has not yet been explored whether there is a correlation for simulated UFD
size ($r_{h}$) or light profile concentration with relation to environment or
formation time. As the target UFDs that we fit structural parameters for all
had surface density profiles that were well described by exponential and/or
Plummer profiles, they showed no signs of variation. A larger sample of
observed UFDs with fewer FOV constraints could be used in tandem with
simulations to investigate if the formation environment could have an impact
on present-day morphological properties.

Focusing now on the TNG simulations, both sets of galaxies occupy similar
spaces in the $M_{V}-r_{h}$ plane, suggesting that the different DM models
might not have a large effect on observable size. We also see that they tend
to have larger $r_{h}$ values, especially for the magnitude space that they
occupy. This is due to how they implement their SF and pressurization of the
multiphase ISM (e.g., Pillepich et al., 2018).

The IllustrisTNG model is unable to resolve SNe and uses a pressurized
equation of state that does not allow the gas to cool below 104 K, thus
prohibiting collapse on smaller physical scales. As such, there is a limit to
how compact the TNG dwarf galaxies can be, motivating the need for simulations
based on more resolved subgrid physics with explicit treatments of the ISM to
explore alternative DM models.

While the TNG simulated galaxies shown in Figure 8 should not be used for
observational predictions, they are a useful marker for seeing what well-
tested cosmological models produce at these scales. The similarity in sizes
for the TNG galaxies produced using CDM versus ETHOS/SIDM could also suggest
that the galaxy formation model is dominant over the DM prescription.

####  5.4.2 Alternative dark matter models

In Figure 8, there is a subset of points that come from simulations based on
alternate DM models, such as WDM (e.g., Macciò et al., 2019; Bozek et al.,
2019) and SIDM (e.g., Robles et al., 2017, J. Rose et al. 2024, in prep.).
Within the $M_{V}-r_{h}$ plane, there are not extreme differences apparent in
the faint dwarf analogs created with different DM models. Below, we present a
brief overview of the more in-depth literature studies that have compared
these DM models on mass scales relevant to UFDs.

To leverage DMO simulations, we can first characterize any strong differences
present between the DM models without the presence of baryons. Then, we can
see if those differences persist in the fully hydrodynamic runs, or if the
addition of baryons causes observational degeneracies. Comparisons of cold DMO
simulations to observations of dwarf spiral galaxies previously led to the
“core-cusp” problem, where measured rotation curves implied the presence of
cores, rather than the predicted steep density profiles ($\rho\propto r^{-1}$)
(e.g., Flores & Primack, 1994; Moore, 1994). While there have since been
explanations for this difference consistent with CDM (such as time-varying
gravitational potentials, see e.g., Pontzen & Governato, 2012; Dutton et al.,
2016; Orkney et al., 2021), alternative DM models have also been proposed to
address the “core-cusp” and other challenges that $\Lambda$CDM seemingly faced
on small ($<1$ Mpc) scales (see Bullock & Boylan-Kolchin, 2017, for a review).

For example, SIDM-only simulations have been able to produce dwarf halos with
constant-density cores ($r_{c}\sim 1$ kpc) (e.g., Burkert, 2000; Davé et al.,
2001; Rocha et al., 2013; Robles et al., 2017). WDM-only simulations have also
produced less centrally dense halos (e.g., Lovell et al., 2012; Shao et al.,
2013), although some simulations predict cores closer to $\sim 10$ pc (Macciò
et al., 2012), which would be insufficient for resolving the core-cusp
problem. These three DM models thus have distinct signatures present when
there are no baryons involved. As we rely on observations of baryonic matter
for science, however, we must see what effect the addition of baryons has on
these simulations.

Here, we focus on baryonic simulations that used fixed subgrid physics for
both CDM and alternative DM runs.777When discussing the simulations, unless
specified as DM-only, we are referring to the baryonic simulations based on
the DM models. In comparing UFD analogs in CDM versus SIDM cosmologies, both
FIRE-2 (Robles et al., 2017) and TNG (ETHOS/SIDM) (J. Rose et al. 2024, in
prep.) found similar $M_{\star}-r_{h}$ relations between the two models.
Additionally, cores were difficult to form in CDM, with only the most massive
halo forming a core in the FIRE-2 runs and no halos forming cores in the TNG
(CDM) runs (Rose et al., 2023). The SIDM halos did form cores, although the
UFD core sizes and profiles were not significantly different from those formed
in the SIDM-only runs. This suggests that the baryons had minimal effects on
the DM structure of the dwarf galaxies considered. Robles et al. (2017)
suggested that the observation of field dwarf with a DM core size similar to
its $r_{h}$ and a stellar mass $\lesssim 3\times 10^{6}M_{\odot}$ could
support the need for a new DM paradigm.

For the simulated dwarfs in CDM versus WDM, Bozek et al. (2019) (using FIRE-2
subgrid physics) and Macciò et al. (2019) (using Numerical Investigation of a
Hundred Astrophysical Objects (NIHAO) subgrid models; Wang et al., 2015) both
found that the WDM galaxies formed the bulk of their stellar mass in the past
$\sim$4 Gyr. Bozek et al. (2019) proposed that observations of UFDs with more
than 80% of their stellar mass formed in the last 4 Gyr would support the WDM
model.

Both groups also found that in general, the final WDM density profiles of
halos hosting galaxies were less centrally dense than the halos in CDM,
although baryonic feedback processes had a greater effect overall in lowering
the central densities in the CDM runs; greater stellar mass and star formation
efficiency were also correlated with greater central density reduction. Macciò
et al. (2019) additionally predicted WDM halos hosting galaxies with
$M_{\star}<10^{6}$ would retain their cuspy profiles. Thus, the authors
suggested that the unambiguous detection of a core in a UFD would require a
more drastic change to the standard model. Bozek et al. (2019) used the WDM
resonantly-produced sterile neutrino (RPSN) particle model while Macciò et al.
(2019) chose a WDM particle mass of 3 keV, however both concluded that
observables such as galaxy mass, shape, size, and velocity dispersion would
not be useful in distinguishing between a CDM or WDM universe. These two
studies produced very few simulated galaxies with $M_{V}>=-10$, however, so it
would be informative to have a larger sample size of galaxies from WDM
simulations specifically targeting the UFD-mass regime.

Finally, Fitts et al. (2019)888The simulated data unique to this study are not
in Figure 8 as they were not available. compared the effects of FIRE-2
baryonic physics on WDM, SIDM, and CDM, as well as two models combining SIDM
and WDM. The authors found that the simulated dwarfs across their models had
similar stellar half-mass radii to observed dwarf galaxies in the Local Field,
and that the $r_{1/2}-M_{\star}$ relation (where $r_{1/2}$ is the 3D half-mass
radius) did not show different trends based on the DM model used. Similarly to
the comparisons above, Fitts et al. (2019) found that the addition of baryonic
physics made the DM density profile differences that were evident in the DMO-
simulated halos less apparent.

In their chosen comparison plane of $V_{1/2}$ (the circular velocity at
$r_{1/2}$) versus $r_{1/2}$, all five models (CDM, SIDM, WDM, and two
SIDM+WDM) yielded similar results that agreed with observations. However, the
alternate DM runs produced dwarfs with lower central densities than those in
CDM for $M_{\star}\sim 10^{6}M_{\odot}$. Fitts et al. (2019) suggested that
measurements of circular velocities for the inner regions ($<400$ pc) of less
massive dwarfs could be a strong discriminator for CDM versus an alternative
DM. The authors concluded that a larger sample size of simulated dwarfs in
this mass regime will be needed for a more thorough exploration.

The studies discussed above used the FIRE-2, IllustrisTNG, and NIHAO subgrid
models, but it would be informative to see how the baryonic physics
implementations of other simulation groups interact with alternative DM models
on faint dwarf galaxy mass scales. These could provide additional
observational probes for differentiating between the proposed natures of DM.

####  5.4.3 The low surface-brightness future

As the next generation of observing facilities come online (e.g., Euclid,
Racca et al. 2016; Vera C. Rubin Observatory/LSST, Ivezić et al. 2019; Roman
Space Telescope, Spergel et al. 2015), we can expect even more low surface-
brightness stellar populations to be discovered. For example, Mutlu-Pakdil et
al. (2021) simulated the expected performance of the Subaru Hyper Suprime-Cam
(Miyazaki et al., 2018) and Rubin, forecasting that we should obtain a “secure
census” of galaxies out to 1.5 Mpc with $M_{V}\lesssim-7$ with $\mu_{V,0}\sim
30$ mag arcsec-2. Similar investigations for resolved stellar streams with
Roman (Pearson et al., 2019) and ultra-diffuse galaxies (UDGs) with LSST depth
(Newton et al., 2023) have also been conducted, and the authors are optimistic
about the populations awaiting discovery within our Local Volume. Pearson et
al. (2019) predicted that we will be able to detect resolved stellar streams
from globular clusters out to $\sim$3.5 Mpc, and Newton et al. (2023)
predicted that there are $\sim$12 isolated UDGs within 2.5 Mpc of the Local
Group that should be observable in future deep surveys.

As an additional note, from examining this section of the $M_{V}-r_{h}$ plane,
there appears to be a deficit in the number of observed MW satellites with an
$M_{V}$ between $-6$ and $-8$. This is the same feature in the luminosity
function that Bose et al. (2018) suggested may represent the dividing line
between dwarfs that were quenched by reionization and those that were able to
continue forming stars at later times. Manwadkar & Kravtsov (2022) also
highlight this “break” in their GRUMPY semi-analytic model and speculated that
as the census of Local Volume satellite galaxies grows more complete, we can
probe reionization by further characterizing the abundance of UFDs and how
sensitive they must have been to the ionizing radiation from that epoch.

We will likely need a greater sample size to determine if this deficit is
anything besides a random fluctuation. Still, we might expect at least one MW
satellite to be discovered with $100<r_{h}<400$ pc, as it would have a surface
brightness above current and projected detection limits. Many hydrodynamic
simulations have been able to produce galaxies in this magnitude range, and
while there have been other Local Group galaxies discovered in that
$M_{V}-r_{h}$ parameter space (e.g., McConnachie, 2012), most of the currently
known MW UFDs seem to be fainter than $M_{V}=-6$.

There are further details to be studied concerning the faint and low surface-
brightness populations that we have recently discovered. For example, there
are still several stellar systems (e.g., Hor II, Hya II, Phe II, Tri II) whose
status as galaxies (rather than star clusters) has yet to be confirmed via
spectroscopy. The Baumgardt et al. (2022) mass segregation technique has
provided an alternative, however, and concluded that Hor II, Phe II, and Tri
II are likely dwarf galaxies. The Fu et al. (2023) HST photometric metallicity
study has also offered support via metallicity dispersion values for the
classification of Hya II and Phe II as galaxies. The targets that we have
labeled “ultra-faint clusters” also merit further studies, as not all have
been conclusively ruled out of being DM-dominated (e.g., Cerny et al., 2021a,
b, 2023b). Additionally, studies of faint dwarf stellar halos (Jensen et al.,
2023) and UFD stars at distant radii (e.g., Tuc II; Chiti et al., 2021)
suggest that deeper imaging of the fields around UFDs could reveal more about
previous tidal interactions or whether they formed through the merger of
smaller stellar systems (e.g., Goater et al., 2023). The depth and areal
coverage of the surveys to come will be critical in our characterization and
understanding of the low surface-brightness universe.

##  6 SUMMARY AND CONCLUSIONS

In this paper, we have presented new PSF photometry for ten targets from the
HST Treasury GO-14734 (PI: N. Kallivayalil): Grus II, Horologium I, Horologium
II, Hydra II, Phoenix II, Reticulum II, Sagittarius II, Triangulum II, Tucana
II, and Tucana IV. We have tested the ability to obtain structural parameter
and $M_{V}$ measurements from HST imaging with limited fields-of-view and
recovered $a_{h}$ measurements within $1\sigma$ of recent literature values
for $\sim$40% of the cases. While complications in the image data made these
measurements difficult for several objects, all of our $a_{h}$ measurements
agree within $3\sigma$ of published values.

We have shown that there is no systematic difference in the $M_{V}-r_{h}$
relationship for UFDs associated with the LMC versus not. We examined the
current state of dwarf galaxy simulations and found that several groups have
been able to create analogs that are broadly consistent with the observed UFDs
in the magnitude-size plane. We also considered whether where observed UFDs
fall in the $M_{V}-r_{h}$ plane would be useful in constraining the nature of
DM. In comparing simulated galaxies from SIDM (Robles et al., 2017, J. Rose et
al. 2024, in prep.), WDM (Bozek et al., 2019; Macciò et al., 2019), and CDM
(see Table LABEL:tab:simtable for full citations) in the $M_{V}-r_{h}$ space,
we found no consistent trends based on the DM cosmology.

As we found no clear discriminator that would support one DM cosmology over
another in the $M_{V}-r_{h}$ plane, we reviewed and compared the predicted
observables from the literature. From examining the results and conclusions of
alternative DM studies from different simulation groups (though see Fitts et
al., 2019, for an outstanding comparison within the FIRE-2 group), two clear
observables that would strongly support a non-CDM cosmology are 1) a UFD SFH
with $\gtrsim 80$% of its stellar mass formed in the last 4 Gyr, which would
agree with predictions from multiple WDM simulations (Bozek et al., 2019;
Macciò et al., 2019), and 2) the presence of a core in a UFD with
$M_{\star}\lesssim 10^{6}M_{\odot}$, as current simulations suggest that UFDs
of that mass would retain their cusps (Robles et al., 2017; Macciò et al.,
2019; Fitts et al., 2019). Nonetheless, current hydrodyanmical simulations
have generally been presenting baryonic solutions for the classical issues
that CDM has faced on small scales (e.g., Sales et al., 2022) and a paradigm
shift in the near-future would require substantial evidence.

Statistically robust samples of simulated dwarfs produced across different DM
models will be necessary to more thoroughly explore the impact that the
underlying DM cosmology has on tangible properties. Additionally, as more
simulation groups explore alternative DM models, we can see if those, in
combination with their subgrid physics implementations, suggest different
observable relations that could be used to constrain DM properties. Finally,
as new observing facilities come online and we begin to go beyond current
survey depth limits, we will have more opportunities to discover and further
characterize stellar populations in this low surface-brightness regime.

Horologium I
---
Parameter | Bechtol et al. (2015) |  Koposov et al. (2015b)a | Muñoz et al. (2018) |  Moskowitz & Walker (2020)b | This Work
$M_{V}$ | $-3.5\pm 0.3$ | $-3.4\pm 0.1$ |  $-3.55$ $\pm 0.56$ | … |  $-3.4$ $\pm 0.2$
Exponential
RA (deg) | … | 43.8820 | 43.8813 $\pm 25.65\arcsec$ | … | 43.8759 $\pm 3\arcsec$
DEC (deg) | … | $-54.1188$ |  $-54.1160$ $\pm 20.2\arcsec$ | … |  $-54.1174$ $\pm 3\arcsec$
$\theta_{\mathrm{exp}}$ (deg) | … | Unconstrained | 53 $\pm 27$ | … | 44 $\pm 7$
$\epsilon_{\mathrm{exp}}$ | … |  $<$0.28 | 0.32 $\pm 0.13$ | … | 0.26 $\pm 0.05$
$a_{h,\mathrm{exp}}$ (arcmin) | … | 1.41${}^{+0.29}_{-0.13}$ | 1.71 $\pm 0.37$ | … | 1.70${}^{+0.16}_{-0.15}$
$a_{h,\mathrm{exp}}$ (pc) | … | 33${}^{+6.8}_{-4.8}$ | 39.3 $\pm 8.5$ | … | 39.2${}^{+5.2}_{-4.7}$
Plummer
RA (deg) | 43.87 | … | 43.8813 $\pm 25.65\arcsec$ | 43.882 | 43.8755 $\pm 3\arcsec$
DEC (deg) | $-54.11$ | … |  $-54.1160$ $\pm 20.2\arcsec$ | $-54.119$ |  $-54.1174$ $\pm 3\arcsec$
$\theta_{p}$ (deg) | Unconstrained | … | 57 $\pm 25$ | 66.59${}^{+13.41}_{-13.65}$ | 44 $\pm 6$
$\epsilon_{p}$ | Unconstrained | … | 0.27 $\pm 0.13$ | 0.14${}^{+0.06}_{-0.05}$ | 0.27 $\pm 0.05$
$a_{h,\mathrm{p}}$ (arcmin) | 2.4${}^{+3.0}_{-1.2}$ | … | 1.59 $\pm 0.31$ | 1.67$\pm 0.10$ | 1.61 $\pm 0.13$
$a_{h,\mathrm{p}}$(pc) | 60${}^{+76}_{-30}$ | … | 36.5 $\pm 7.1$ | 38.6${}^{+4.6}_{-3.9}$ | 37.3${}^{+4.2}_{-4.7}$
Table 4: Compilation of past literature measurements for Horologium I, which
include fits from both the exponential and Plummer models. Our parameters are
quoted from the maximum-likelihood estimation, with the listed uncertainties
corresponding to the 84th and 16th percentiles around the median of the
distribution.

Horologium II
---
Parameter | Kim & Jerjen (2015) |  Moskowitz & Walker (2020)c | This Work
$M_{V}$ | $-2.6^{+0.2}_{-0.3}$ | … | $-2.1\pm 0.2$
Exponential
RA (deg) | 49.1337 $\pm 5\arcsec$ | … | 49.1315 $\pm 4\arcsec$
DEC (deg) |  $-50.0180$ $\pm 5\arcsec$ | … |  $-50.0088$ $\pm 3\arcsec$
$\theta_{\mathrm{exp}}$ (deg) | 127 $\pm 11$ | … | 103${}^{+12}_{-14}$
$\epsilon_{\mathrm{exp}}$ | 0.52${}^{+0.13}_{-0.17}$ | … | 0.23${}^{+0.07}_{-0.08}$
$a_{h,\mathrm{exp}}$ (arcmin) | 2.09${}^{+0.44}_{-0.41}$ | … | 1.63 $\pm 0.18$
$a_{h,\mathrm{exp}}$ (pc) | 47 $\pm 10$ | … | 36.9${}^{+5.1}_{-5.4}$
Plummer
RA (deg) | … | 49.134 | 49.1310 $\pm 4\arcsec$
DEC (deg) | … | $-50.0181$ |  $-50.0090$ $\pm 3\arcsec$
$\theta_{\mathrm{p}}$ (deg) | … | 279.71${}^{+8.25}_{-189.75}$ | 103${}^{+11}_{-14}$
$\epsilon_{\mathrm{p}}$ | … | 0.40 $\pm 0.14$ | 0.23${}^{+0.07}_{-0.08}$
$a_{h,\mathrm{p}}$ (arcmin) | … | 2.54${}^{+0.43}_{-0.53}$ | 1.69${}^{+0.18}_{-0.17}$
$a_{h,\mathrm{p}}$ (pc) | … | 58.1${}^{+13.8}_{-10.7}$ | 38.4${}^{+4.9}_{-5.5}$
Table 5: The same as Table 4, for Horologium II. Hydra II
---
Parameter | Martin et al. (2015) | Muñoz et al. (2018) | This Work
$M_{V}$ |  $-4.8$ $\pm 0.3$ |  $-4.60$ $\pm 0.37$ | $-4.6^{+0.2}_{-0.3}$
Exponential
RA (deg) | 185.4254 | 185.4251 $\pm 13.65\arcsec$ | 185.4279$\pm 4\arcsec$
DEC (deg) | $-31.9853$ |  $-31.9860$ $\pm 13.7\arcsec$ | $-31.9728^{+7}_{-6}\arcsec$
$\theta_{\mathrm{exp}}$ (deg) | 28${}^{+40}_{-35}$ | 13 $\pm 28$ | $-10^{+9}_{-10}$
$\epsilon_{\mathrm{exp}}$ | 0.01${}^{+0.19}_{-0.01}$ | 0.25 $\pm 0.16$ | 0.30${}^{+0.08}_{-0.09}$
$a_{h,\mathrm{exp}}$ (arcmin) | 1.7${}^{+0.3}_{-0.2}$ | 1.65 $\pm 0.39$ | 2.70${}^{+0.41}_{-0.38}$
$a_{h,\mathrm{exp}}$ (pc) | 68 $\pm 11$ | 64.3 $\pm 15.2$ | 105.6${}^{+17.9}_{-16.1}$
Plummer
RA (deg) | … | 185.4251 $\pm 13.65\arcsec$ | 185.4286${}^{+4}_{-3}\arcsec$
DEC (deg) | … |  $-31.9860$ $\pm 13.7\arcsec$ | $-31.9728^{+9}_{-7}\arcsec$
$\theta_{p}$ (deg) | … | 16 $\pm 25$ | $-9^{+8}_{-9}$
$\epsilon_{p}$ | … | 0.24 $\pm 0.16$ | 0.30$\pm 0.09$
$a_{h,\mathrm{p}}$(arcmin) | … | 1.52 $\pm 0.28$ | 2.13${}^{+0.42}_{-0.35}$
$a_{h,\mathrm{p}}$(pc) | … | 59.2 $\pm 10.9$ | 83.3${}^{+17.1}_{-15.1}$
Table 6: The same as Table 4, for Hydra II.

Phoenix II
---
Parameter |  Bechtol et al. (2015)d |  Koposov et al. (2015b)e | Muñoz et al. (2018) | Mutlu-Pakdil et al. (2018) |  Moskowitz & Walker (2020)f | This Work
$M_{V}$ |  $-3.7$ $\pm 0.4$ |  $-2.8$ $\pm 0.2$ |  $-3.30$ $\pm 0.63$ |  $-2.7$ $\pm 0.4$ | … | -2.9${}^{+0.2}_{-0.1}$
Exponential
RA (deg) | … | 354.9975 | 354.9960 $\pm 13.5\arcsec$ | 354.9928 $\pm 8.3\arcsec$ | … | 354.9922 $\pm 3\arcsec$
DEC (deg) | … | $-54.4060$ |  $-54.4115$ $\pm 21.0\arcsec$ |  $-54.4050$ $\pm 5.7\arcsec$ | … |  $-54.4018$ $\pm 3\arcsec$
$\theta_{\mathrm{exp}}$ (deg) | … | 164 $\pm 54$ |  $-19$ $\pm 15$ | 156 $\pm 13$ | … |  $-34$ $\pm 4$
$\epsilon_{\mathrm{exp}}$ | … | 0.47${}^{+0.08}_{-0.29}$ | 0.62 $\pm 0.19$ | 0.4 $\pm 0.1$ | … | 0.44 $\pm 0.6$
$a_{h,\mathrm{exp}}$ (arcmin) | … | 1.38${}^{+0.45}_{-0.20}$ | 1.60 $\pm 0.33$ | 1.5 $\pm 0.3$ | … | 1.58${}^{+0.26}_{-0.23}$
$a_{h,\mathrm{exp}}$ (pc) | … | 34.3${}^{+10.7}_{-6.4}$ | 38.6 $\pm 8.0$ | 37 $\pm 6$ | … | 43.0${}^{+8}_{-7}$
Plummer
RA (deg) | 354.99 | … | 354.9960 $\pm 13.5\arcsec$ | … | 354.998 | 354.9919 $\pm 3\arcsec$
DEC (deg) | $-54.41$ | … |  $-54.4115$ $\pm 21.0\arcsec$ | … | $-54.406$ |  $-54.4019$ $\pm 3\arcsec$
$\theta_{p}$ (deg) | Unconstrained | … |  $-20$ $\pm 18$ | … | 285.28${}^{+15.31}_{-14.54}$ |  $-33$ $\pm 5$
$\epsilon_{p}$ | Unconstrained | … | 0.67 $\pm 0.22$ | … | 0.27${}^{+0.11}_{-0.10}$ | 0.44 $\pm 0.06$
$a_{h,\mathrm{p}}$(arcmin) | 1.2${}^{+1.2}_{-1.2}$ | … | 1.49 $\pm 0.53$ | … | 1.85${}^{+0.30}_{-0.33}$ | 1.50${}^{+0.20}_{-0.17}$
$a_{h,\mathrm{p}}$(pc) | 33${}^{+20}_{-11}$ | … | 36.0 $\pm 12.8$ | … | 45.1${}^{+9.5}_{-7.8}$ | 40.8${}^{+5.5}_{-6.2}$
Table 7: The same as 4, for Phoenix II.

Sagittarius II
---
Parameter | Laevens et al. (2015b) | Mutlu-Pakdil et al. (2018) | Longeard et al. (2020) |  Moskowitz & Walker (2020)g | This Work
$M_{V}$ |  $-5.2$ $\pm 0.4$ |  $-4.2$ $\pm 0.1$ |  $-5.7$ $\pm 0.1$ | … | $-5.3\pm 0.2$
Exponential
RA (deg) | 298.1688 | 298.1647 $\pm 3.0\arcsec$ | 298.16628 $\pm 3.6\arcsec$ | … | 298.1664 $\pm 2\arcsec$
DEC (deg) | $-22.0681$ |  $-22.0651$ $\pm 2.2\arcsec$ |  $-22.89633$ $\pm 3.6\arcsec$ | … |  $-22.0641$ $\pm 2\arcsec$
$\theta_{\mathrm{exp}}$ (deg) | 72${}^{+28}_{-20}$ | Unconstrained | 103${}^{+28}_{-17}$ | … | 98${}^{+43}_{-30}$
$\epsilon_{\mathrm{exp}}$ | 0.23${}^{+0.17}_{-0.23}$ |  $<$0.1 | 0.0, $<$0.12 at 95% CL | … | 0.03,$<0.09$ at 95% CL
$a_{h,\mathrm{exp}}$ (arcmin) | 2.0${}^{+0.4}_{-0.3}$ | 1.6 $\pm 0.1$ | 1.7 $\pm 0.05$ | … | 1.94 $\pm 0.08$
$a_{h,\mathrm{exp}}$ (pc) | 38${}^{+8}_{-7}$ | 32 $\pm 1$ | 35.5${}^{+1.4}_{-1.2}$ | … | 39 $\pm 4$
Plummer
RA (deg) | … | … | … | 298.169 | 298.1664 $\pm 2\arcsec$
DEC (deg) | … | … | … | $-22.068$ |  $-22.0642$ $\pm 2\arcsec$
$\theta_{p}$ (deg) | … | … | … | 74.90${}^{+14.69}_{-13.02}$ | 96${}^{+50}_{-32}$
$\epsilon_{p}$ | … | … | … | 0.24${}^{+0.10}_{-0.09}$ | 0.03,$<0.08$ at 95% CL
$a_{h,\mathrm{p}}$(arcmin) | … | … | … | 1.87${}^{+0.19}_{-0.22}$ | 1.85${}^{+0.07}_{-0.08}$
$a_{h,\mathrm{p}}$(pc) | … | … | … | 36.7${}^{+5.1}_{-4.3}$ | 37 $\pm 4$

Table 8: Same as Table 4, for Sagittarius II.

gThe values reported as $a_{h}$ in arcminutes here are the values given as
$R_{h,1c}$ divided by $\sqrt{1-\epsilon_{1c}}$ in their Table 3, as their
$R_{h}$ is what they refer to as the “circularized projected half-light radii”
equal to $a_{h}\sqrt{1-\epsilon_{1c}}$. Moskowitz & Walker (2020) do not
report fitted RA and DEC values, thus we list the values given in their Table
2.

Triangulum II
---
Parameter | Laevens et al. (2015a) | Carlin et al. (2017) | Muñoz et al. (2018) |  Moskowitz & Walker (2020)h | This Work
$M_{V}$ |  $-1.8$ $\pm 0.5$ |  $-1.2$ $\pm 0.4$ |  $-1.60$ $\pm 0.76$ | … | $-1.4^{+0.3}_{-0.2}$
Exponential
RA (deg) | 33.3225 | 33.3223 $\pm 14.4\arcsec$ | 33.3252 $\pm 14.55\arcsec$ | … | 33.3152 $\pm 8\arcsec$
DEC (deg) | 36.1784 | 36.1719 $\pm 9.7\arcsec$ | 36.1702 $\pm 19.0\arcsec$ | … | 36.1704${}^{+8}_{-9}\arcsec$
$\theta_{\mathrm{exp}}$ (deg) | 56${}^{+16}_{-24}$ | 73 $\pm 17$ | 28 $\pm 19$ | … | 81${}^{+30}_{-39}$
$\epsilon_{\mathrm{exp}}$ | 0.21${}^{+0.17}_{-0.21}$ | 0.3 $\pm 0.1$ | 0.48 $\pm 0.17$ | … | 0.15${}^{+0.12}_{-0.10}$
$a_{h,\mathrm{exp}}$ (arcmin) | 3.9${}^{+1.1}_{-0.9}$ | 2.5 $\pm 0.3$ | 2.34 $\pm 0.58$ | … | 3.50${}^{+0.59}_{-0.65}$
$a_{h,\mathrm{exp}}$ (pc) | 34${}^{+9}_{-8}$ | 21 $\pm 4$ | 20.4 $\pm 5.1$ | … | 29.0${}^{+5.2}_{-5.4}$
Plummer
RA (deg) | … | … | 33.3252 $\pm 14.55\arcsec$ | 33.323 | 33.3155 $\pm 9\arcsec$
DEC (deg) | … | … | 36.1702 $\pm 19.0\arcsec$ | 36.1783 | 36.1691${}^{+10}_{-12}\arcsec$
$\theta_{p}$ (deg) | … | … | 44 $\pm 18$ | 300.72${}^{+27.55}_{-222.65}$ | 79${}^{+28}_{-38}$
$\epsilon_{p}$ | … | … | 0.46 $\pm 0.16$ | 0.26${}^{+0.13}_{-0.10}$ | 0.15${}^{+0.12}_{-0.10}$
$a_{h,\mathrm{p}}$(arcmin) | … | … | 1.99 $\pm 0.49$ | 1.80${}^{+0.30}_{-0.33}$ | 3.25${}^{+0.51}_{-0.59}$
$a_{h,\mathrm{p}}$(pc) | … | … | 17.4 $\pm 4.3$ | 16.0${}^{+3.0}_{-2.7}$ | 27.0${}^{+4.5}_{-4.9}$
Table 9: Same as Table 4, for Triangulum II, except our parameters are quoted
from the median, with the listed uncertainties corresponding to the 84th and
16th percentiles around the median of the distribution.

Grus II
---
Parameter | Drlica-Wagner et al. (2015) |  Moskowitz & Walker (2020)i | Simon et al. (2020)
$M_{V}$ | -3.9 $\pm 0.22$ | … | -3.5 $\pm 0.3$
Plummer
RA (deg) | 331.02 | … | 331.025${}^{32}_{29}$
DEC (deg) | $-46.44$ | $-46.44$ |  $-46.422$ $\pm 22$
$\theta_{\mathrm{p}}$ (deg) | Unconstrained | 67.05${}^{+228.74}_{-26.25}$ | Unconstrained
$\epsilon_{\mathrm{p}}$ | $<0.2$ | 0.12${}^{+0.06}_{-0.05}$ | $<0.21$
$a_{h,\mathrm{p}}$ (arcmin) | 6.0${}^{+0.9}_{-0.5}$ | 7.80${}^{+0.56}_{-0.58}$ | 5.9 $\pm$0.5
$a_{h,\mathrm{p}}$ (pc) | 93 $\pm 14$ | 121.5${}^{15.23}_{-13.89}$ | 94 $\pm 9$

Table 10: Compilation of past literature measurements for Grus II from the
Plummer model fits.

iThe values reported as $a_{h}$ in arcminutes here are the values given as
$R_{h,1c}$ divided by $\sqrt{1-\epsilon_{1c}}$ in their Table 3, as their
$R_{h}$ is what they refer to as the “circularized projected half-light radii”
equal to $a_{h}\sqrt{1-\epsilon_{1c}}$. Moskowitz & Walker (2020) do not
report fitted RA and DEC values, however the literature values that they list
in their Table 2 for this target do not match other studies. Their listed RA
value does match up with other reported DEC values, so we do include that
here.

Reticulum II
---
Parameter |  Bechtol et al. (2015)j |  Koposov et al. (2015b)k | Muñoz et al. (2018) | Mutlu-Pakdil et al. (2018) |  Moskowitz & Walker (2020)l
$M_{V}$ |  $-3.6$ $\pm 0.1$ |  $-2.7$ $0.1$ |  $-3.88$ $\pm 0.38$ |  $-3.1$ $\pm 0.1$ | …
Exponential
RA (deg) | … | 53.9256 | 53.9203 $\pm 24.45\arcsec$ | 53.9493 $\pm 24.8\arcsec$ | …
DEC (deg) | … | $-54.0492$ |  $-54.0513$ $\pm 7.9\arcsec$ |  $-54.0466$ $\pm 9.1\arcsec$ | …
$\theta_{\mathrm{exp}}$ (deg) | … | 71 $\pm 1$ | 62 $\pm 2$ | 68 $\pm 2$ | …
$\epsilon_{\mathrm{exp}}$ | … | 0.59${}^{+0.02}_{-0.03}$ | 0.56 $\pm 0.03$ | 0.6 $\pm 0.1$ | …
$a_{h,\mathrm{exp}}$ (arcmin) | … | 5.66${}^{+0.39}_{-0.22}$ | 5.41 $\pm 0.18$ | 6.3 $\pm 0.4$ | …
$a_{h,\mathrm{exp}}$ (pc) | … | 50${}^{+6}_{-5}$ | 47.2 $\pm 1.6$ | 58 $\pm 4$ | …
Plummer
RA (deg) | 53.92 | … | 53.9203 $\pm 24.45\arcsec$ | … | 53.925
DEC (deg) | $-54.05$ | … |  $-54.0513$ $\pm 7.9\arcsec$ | … | $-54.049$
$\theta_{p}$ (deg) | 72 $\pm 7$ | … | 70 $\pm 2$ | … | 69.51 $\pm 0.80$
$\epsilon_{p}$ | 0.6${}^{+0.1}_{-0.2}$ | … | 0.58 $\pm 0.02$ | … | 0.60 $\pm 0.01$
$a_{h,\mathrm{p}}$(arcmin) | 6.0 $\pm 0.6$ | … | 5.52 $\pm 0.19$ | … | 6.56 $\pm 0.15$
$a_{h,\mathrm{p}}$(pc) | 55 $\pm 5$ | … | 48.2 $\pm 1.7$ | … | 57.9${}^{+5.6}_{-5.3}$

Table 11: Same as Table 4, for Reticulum II.

jThe values reported as $a_{h}$ in arcminutes here are the values given as
$r_{h}$ (deg) in Bechtol et al. (2015) Table 1, multiplied by 60. The values
reported as $a_{h}$ in parsecs here are the values given as $r_{1/2}$ (pc) in
Bechtol et al. (2015) Table 2.
kThe values reported as $a_{h}$ in arcminutes here are the values given as
$r_{\mathrm{maj}}$ in Koposov et al. (2015b) Table 1 multiplied by 1.68, as
their $r_{\mathrm{maj}}$ corresponded to the exponential scale length $r_{e}$.
lThe values reported as $a_{h}$ in arcminutes here are the values given as
$R_{h,1c}$ divided by $\sqrt{1-\epsilon_{1c}}$ in their Table 3, as their
$R_{h}$ is what they refer to as the “circularized projected half-light radii”
equal to $a_{h}\sqrt{1-\epsilon_{1c}}$. Moskowitz & Walker (2020) do not
report fitted RA and DEC values, thus we list the values given in their Table
2.

Tucana II
---
Parameter |  Bechtol et al. (2015)m |  Koposov et al. (2015b)n |  Moskowitz & Walker (2020)o
$M_{V}$ |  $-3.9$ $\pm 0.2$ |  $-3.8$ $\pm 0.1$ | …
Exponential
RA (deg) | … | 342.9796 | …
DEC (deg) | … | $-58.5689$ | …
$\theta_{\mathrm{exp}}$ (deg) | … | 107 $\pm 18$ | …
$\epsilon_{\mathrm{exp}}$ | … | 0.39${}^{+0.10}_{-0.20}$ | …
$a_{h,\mathrm{exp}}$ (arcmin) | … | 12.89${}^{+1.71}_{-1.98}$ | …
$a_{h,\mathrm{exp}}$ (pc) | … | 217${}^{+40}_{-37}$ | …
Plummer
RA (deg) | 343.06 | … | 342.98
DEC (deg) | $-58.57$ | … | $-58.57$
$\theta_{p}$ (deg) | Unconstrained | … | 274.02${}^{+6.81}_{-187.37}$
$\epsilon_{p}$ | Unconstrained | … | 0.34${}^{+0.07}_{-0.08}$
$a_{h,\mathrm{p}}$(arcmin) | 7.2 $\pm 1.8$ | … | 13.48${}^{+1.10}_{-1.20}$
$a_{h,\mathrm{p}}$(pc) | 120 $\pm 30$ | … | 230.1${}^{+31.0}_{-27.7}$

Table 12: Same as Table 4, for Tucana II.

mThe values reported as $a_{h}$ in arcminutes here are the values given as
$r_{h}$ (deg) in Bechtol et al. (2015) Table 1, multiplied by 60. The values
reported as $a_{h}$ in parsecs here are the values given as $r_{1/2}$ (pc) in
Bechtol et al. (2015) Table 2.
nThe values reported as $a_{h}$ in arcminutes here are the values given as
$r_{\mathrm{maj}}$ in Koposov et al. (2015b) Table 1 multiplied by 1.68, as
their $r_{\mathrm{maj}}$ corresponded to the exponential scale length $r_{e}$.
oThe values reported as $a_{h}$ in arcminutes here are the values given as
$R_{h,1c}$ divided by $\sqrt{1-\epsilon_{1c}}$ in their Table 3, as their
$R_{h}$ is what they refer to as the “circularized projected half-light radii”
equal to $a_{h}\sqrt{1-\epsilon_{1c}}$. Moskowitz & Walker (2020) do not
report fitted RA and DEC values, thus we list the values given in their Table
2.

Tucana IV
---
Parameter | Drlica-Wagner et al. (2015) |  Moskowitz & Walker (2020)p | Simon et al. (2020)
$M_{V}$ | -3.5 $\pm 0.28$ | … | -3.0${}^{+0.3}_{-0.4}$
Plummer
RA (deg) | 0.73 | 0.73 | 0.717${}^{+50}_{-76}$
DEC (deg) | $-60.85$ | $-60.85$ | $-60.830^{+36}_{-40}$
$\theta_{\mathrm{p}}$ (deg) | 11 $\pm 9$ | 333.32${}^{+9.26}_{-54.63}$ | 27${}^{+9}_{-8}$
$\epsilon_{\mathrm{p}}$ | 0.4 $\pm 0.1$ | 0.35${}^{+0.07}_{-0.09}$ | 0.39${}^{+0.07}_{-0.10}$
$a_{h,\mathrm{p}}$ (arcmin) | 11.8${}^{+2.2}_{-1.8}$ | 9.20${}^{+0.96}_{-1.05}$ | 9.3${}^{+1.4}_{-0.9}$
$a_{h,\mathrm{p}}$ (pc) | 167${}^{+35}_{30}$ | 129.2${}^{+19.3}_{-16.8}$ | 127${}^{+22}_{-16}$

Table 13: Same as Table 10, for Tucana IV.

pThe values reported as $a_{h}$ in arcminutes here are the values given as
$R_{h,1c}$ divided by $\sqrt{1-\epsilon_{1c}}$ in their Table 3, as their
$R_{h}$ is what they refer to as the “circularized projected half-light radii”
equal to $a_{h}\sqrt{1-\epsilon_{1c}}$. Moskowitz & Walker (2020) do not
report fitted RA and DEC values, thus we list the values given in their Table
2.

## Acknowledgments

HR would like to thank Peter B. Stetson, Tom Brown, Taylor Hoyt, and Rachael
Beaton for useful discussions about DAOPHOT. HR acknowledges support from the
Jefferson Scholars Foundation Dissertation Year Fellowship, the Hilliard
Family, and the Virginia Space Grant Consortium Graduate Research STEM
Fellowship. CTG acknowledges support from the Owens Family Foundation. This
work was based on observations with the NASA/ESA Hubble Space Telescope
obtained from the Mikulski Archive for Space Telescopes at the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Incorporated, under NASA contract NAS5-26555. Support
for HST GO-14734 was provided through a grant from the STScI under NASA
contract NAS5-26555. This research has made use of NASA’s Astrophysics Data
System. Facility: HST (ACS, WFC3) Software: Aladin (Bonnarel et al., 2000);
Astrodrizzle (Fruchter & Hook, 2002; STScI Development Team, 2012; Avila et
al., 2015); Astropy (Astropy Collaboration et al., 2013, 2018); corner.py
(Foreman-Mackey, 2016); DAOPHOT-II (Stetson, 1987, 1992); dustmaps (Green,
2018); emcee (Foreman-Mackey et al., 2013); Jupyter Notebook (Kluyver et al.,
2016); Matplotlib (Hunter, 2007); Numpy (Harris et al., 2020); photutils
(Bradley et al., 2020); scikit-learn (Pedregosa et al., 2011) Scipy (Virtanen
et al., 2020); stsynphot (STScI Development Team, 2020); synphot (STScI
Development Team, 2018)

##  Appendix A Additional Corner Plots

Figure 9: The posterior distributions of the 7-parameter structural fit for
Hor II using the exponential model. The three black vertical dashed lines
represent the 16th, 50th, and 84th percentiles. The blue lines and markers
show the maximum likelihood values. Figure 10: The posterior distributions of
the 7-parameter structural fit for Hya II using the exponential model. The
three black vertical dashed lines represent the 16th, 50th, and 84th
percentiles. The blue lines and markers show the maximum likelihood values.
Figure 11: The posterior distributions of the 7-parameter structural fit for
Phe II using the exponential model. The three black vertical dashed lines
represent the 16th, 50th, and 84th percentiles. The blue lines and markers
show the maximum likelihood values. Figure 12: The posterior distributions of
the 7-parameter structural fit for Sgr II using the exponential model. The
three black vertical dashed lines represent the 16th, 50th, and 84th
percentiles. The blue lines and markers show the median values. Figure 13: The
posterior distributions of the 7-parameter structural fit for Tri II using the
exponential model. The three black vertical dashed lines represent the 16th,
50th, and 84th percentiles. The blue lines and markers show the median values.

##  Appendix B References for Figure 8

We largely pulled from the compilations of Simon (2019) and McConnachie (2012)
(2021 edition) for the UFD and UFD candidate parameters, but we have also
included the more recently discovered satellites. Here we list the individual
references. Majewski et al. (2003); de Jong et al. (2008); Drlica-Wagner et
al. (2015); Koposov et al. (2015b); Drlica-Wagner et al. (2016); Crnojević et
al. (2016); Homma et al. (2016); Torrealba et al. (2016a); Torrealba et al.
(2016b); Carlin et al. (2017); Luque et al. (2017); Homma et al. (2018);
Koposov et al. (2018); Muñoz et al. (2018); Mutlu-Pakdil et al. (2018);
Torrealba et al. (2018); Simon et al. (2020); Cerny et al. (2021a); Cerny et
al. (2021b); Ji et al. (2021); Cerny et al. (2023a); Richstein et al. (2022);
Cerny et al. (2023b); Smith et al. (2023).

The globular cluster parameters come from the compilation Harris (2010), with
newer additions from Muñoz et al. (2018) and Torrealba et al. (2019). The
Harris (2010) compilation draws from the works of the following: Harris & van
den Bergh (1984); Webbink (1985); Peterson & Reed (1987); van den Bergh et al.
(1991); Tucholke (1992); Mallen-Ornelas & Djorgovski (1993); Trager et al.
(1993); Cote et al. (1995); Trager et al. (1995); Harris et al. (1997);
Kaisler et al. (1997); Lehmann & Scholz (1997); Ivanov et al. (2005);