# Ultra-Diffuse Galaxies as Extreme Star-forming Environments I: Mapping Star
Formation in H I-Rich UDGs

Erin Kado-Fong Department of Astrophysical Sciences, Princeton
University,Princeton, NJ 08544, USA  Jenny E. Greene Department of
Astrophysical Sciences, Princeton University,Princeton, NJ 08544, USA  Song
Huang  Department of Astronomy, Tsinghua University, Beijing 100084, China
Andy Goulding  Department of Astrophysical Sciences, Princeton
University,Princeton, NJ 08544, USA Erin Kado-Fong<EMAIL_ADDRESS>
###### Abstract

Ultra-Diffuse Galaxies are both extreme products of galaxy evolution and
extreme environments in which to test our understanding of star formation. In
this work, we contrast the spatially resolved star formation activity of a
sample of 22 H I-selected UDGs and 35 low-mass galaxies from the NASA Sloan
Atlas (NSA) within 120 Mpc. We employ a new joint SED fitting method to
compute star formation rate and stellar mass surface density maps that
leverage the high spatial resolution optical imaging data of the Hyper
Suprime-Cam Subaru Strategic Program (HSC-SSP) and the UV coverage of GALEX,
along with H I radial profiles estimated from a subset of galaxies that have
spatially resolved H I maps. We find that the UDGs have low star formation
efficiencies as a function of their atomic gas down to scales of 500 pc. We
additionally find that the stellar mass-weighted sizes of our UDG sample are
unremarkable when considered as a function of their H I mass – their stellar
sizes are comparable to the NSA dwarfs at fixed H I mass. This is a natural
result in the picture where UDGs are forming stars normally, but at low
efficiencies. We compare our results to predictions from contemporary models
of galaxy formation, and find in particular that our observations are
difficult to reproduce in models where UDGs undergo stellar expansion due to
vigorous star formation feedback should bursty star formation be required down
to $z=0$.

††software: Astropy (Astropy Collaboration et al., 2013; Price-Whelan et al.,
2018), matplotlib (Hunter, 2007), SciPy (jon, 2001), the IPython package
(Pérez & Granger, 2007), NumPy (Van Der Walt et al., 2011)

##  1 Introduction

Figure 1:  A comparison of the integrated properties of the samples derived in
this work against the directly measured result of Leroy et al. (2008) (beige
points). In all panels, the red points show UDGs and the purple points show
NSA dwarfs. From left, we show the relationship between stellar mass and H I
mass, the relationship between H I mass and SFR, and the star-forming main
sequence (SFMS). In the SFMS panel, we also show the results of the SDSS DR7
MPA-JHU added-value catalogs in greyscale (Kauffmann et al., 2003; Brinchmann
et al., 2004; Salim et al., 2007). We find a good agreement between our
results and those from the literature. Although the UDGs have high H I masses
for their stellar mass (left panel), they have relatively little SFR for their
H I mass (middle panel).  Figure 2:  Top: 21 cm line flux ($S_{21}$) versus
redshift for the NSA (purple) and UDG (red) samples. The nominal ALFALFA flux
limit is shown by the dashed black line, while the minimum $S_{21}$ of the UDG
sample is shown in red. The unfilled symbols show the expected $S_{21}$ of
each of the galaxies in the UDG sample if we assume that the distribution over
H I fraction of the underlying UDG population is identical to that of the NSA
dwarfs. Bottom: the distribution over observed H I fractions for the NSA
(filled purple histogram) and UDG (unfilled red histogram) samples. The
hatched red histogram shows the distribution of H I fractions that we would
expect to see from the UDG sample if the distribution over H I fractions is
identical between the UDG population and NSA sample. We thus conclude that the
UDG sample is inconsistent with having the same distribution over $\rm f_{\rm
HI}$ as the NSA sample.

Ultra-diffuse galaxies (UDGs) are dwarf galaxies that are characterized by
large stellar sizes (${\rm R_{\rm eff}}$$>1.5$ kpc) and low surface
brightnesses ($\mu_{0,g}>24$ mag arcsec-2, though exact definitions vary; see
Van Nest et al. 2022 for an overview). These diffuse systems are extreme in
two senses: they lie far above the mass-size relation, making them extreme
products of galaxy evolution, but they are also extreme environments for star
formation due to their low stellar surface densities.

UDGs as extreme products of galaxy evolution is a well-trodden topic both
observationally (see, e.g. Sandage & Binggeli, 1984; McGaugh et al., 1995;
Dalcanton et al., 1997; van Dokkum et al., 2015; Beasley & Trujillo, 2016a, b;
Peng & Lim, 2016; Yagi et al., 2016; Leisman et al., 2017; van Dokkum et al.,
2018; Danieli et al., 2019; Janowiecki et al., 2019; van Dokkum et al., 2019;
Danieli et al., 2021; Gault et al., 2021) and theoretically (e.g. Amorisco &
Loeb, 2016; Di Cintio et al., 2017; Chan et al., 2018; Greco et al., 2018b, a;
Jiang et al., 2019; Liao et al., 2019; Wright et al., 2021; Van Nest et al.,
2022; Zaritsky et al., 2022). However, the formation of the UDG population
remains an open question. Several mechanisms have been proposed to form UDGs
in the field, including stellar expansion from star formation feedback (El-
Badry et al., 2016; Di Cintio et al., 2017; Chan et al., 2018), early mergers
that trigger radial expansion of star formation (Wright et al., 2021), and
preferential UDG formation in the high-spin tail of the halo distribution
(Dalcanton et al., 1997; Liao et al., 2019).

It is observationally challenging to measure robust distances to UDGs due to
their low surface brightnesses; as a result, the majority of UDGs with known
distances are in clusters or groups. To understand the full breadth of the UDG
population, of course, it is necessary to sample this population across a
range of environments. Observational works continue to place new constraints
on such mechanisms (see, e.g. Leisman et al., 2017; Greco et al., 2018b, a;
Janowiecki et al., 2019; Martín-Navarro et al., 2019; Gault et al., 2021;
Kado-Fong et al., 2021; E Greene et al., 2022; Villaume et al., 2022), but as
of yet there has not been a clear convergence in the literature towards one
path – of course, it is also not necessarily the case that all UDGs are formed
via the same mechanism. Towards this end, there has been a push to use H I to
detect and/or measure distances to UDGs (Leisman et al., 2017; Janowiecki et
al., 2019; Karunakaran et al., 2020; Gault et al., 2021) – this allows for
distance determinations in a sample of UDGs that tend towards low density
environments.

It has been shown that although H I-selected UDGs lie on the star-forming main
sequence and H I mass-size relation (Gault et al., 2021), they tend to have
high H I masses (Leisman et al., 2017) – that is, the UDGs are, in an
integrated sense, low in their star formation efficiency as a function of
atomic gas, or SFE(H I). This in and of itself is not surprising – it has been
demonstrated observationally that stellar mass surface density affects the
position of dwarf galaxies with respect to the Kennicutt-Schmidt relation
(Schmidt, 1959; Kennicutt, 1989) in that lower stellar mass surface density
dwarfs lie below the main relation (de los Reyes & Kennicutt, 2019). What has
not yet been ascertained, however, is whether the apparently low SFE(H I) of
the UDGs is a product of their selection, the distribution of their star
formation, or a true physical suppression of SFE(H I).

In order to understand the origin of the low SFE(H I) of the UDGs, we
introduce a joint SED fitting method that utilizes the spatial resolution of
the optical Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) and the
crucial UV coverage of GALEX. We use this method to derive stellar mass and
SFR maps of a sample of H I-selected UDGs from Janowiecki et al. (2019) and a
reference sample of H I-detected galaxies from the NASA Sloan Atlas (Bradford
et al., 2015). We then use these maps, along with estimated H I radial
profiles, to probe the star formation efficiency of these dwarfs down to 500
pc scales. In this work, our main focus will be on characterizing the
properties of the H I-selected UDGs with respect to the “normal” NSA dwarf
sample through a lens aimed largely at galaxy evolution. In Kado-Fong et al.
(2022) (hereafter Paper II) we will shift our focus towards understanding star
formation in H I-rich UDGs, with a particular focus on the role of UDG
structure in the pressure-regulated, feedback-modulated model of star
formation (Ostriker et al., 2010; Kim et al., 2011; Ostriker & Shetty, 2011;
Kim et al., 2013; Kim & Ostriker, 2015; Kim et al., 2017).

Figure 3:  Estimates of the SFR, stellar mass, and H I surface densities in a
globally averaged sense; i.e., the ratio between the integrated measurement
and the isophotal size of the galaxy. UDGs are shown in red and NSA dwarfs in
purple for all cases. In this figure, we exclude all NSA dwarfs with stellar
masses exceeding $\rm M_{\star}=10^{9}M_{\odot}$.

##  2 Datasets

Our dwarf sample consists of two main branches: a high surface brightness,
“normal” dwarf sample drawn from the H I catalog of Bradford et al. (2015) –
hereafter the NSA sample, and an ultra-diffuse galaxy sample with known
distances from the H I catalog of Janowiecki et al. (2019). The UDG sample is
selected to have a maximum distance of $d=120$ Mpc; we enforce the same limit
on the NSA sample. We note that in this work we will refer to the “normal”
galaxy sample as the NSA sample, as these galaxies were not specifically
chosen to exclude low surface brightness galaxies. Rather, their relatively
high surface brightnesses are a result of the observational selection which
lead to their inclusion in the NSA.

###  2.1 Sample Overview

Here we detail the basic physical properties of the UDG and NSA samples. The
sample consists of 58 galaxies at distances less than 120 Mpc; 22 are UDGs
from Janowiecki et al. (2019) and 35 are NSA galaxies from Bradford et al.
(2015). All sources are required to have clean 5-band HSC imaging, and all
sources have measured H I masses and redshifts due to their selection method.
The sample of Janowiecki et al. (2019) is not explicitly chosen to be in the
field (in contrast to Leisman et al. 2017), but they find that the properties
of their H I-selected UDGs are largely unchanged by environment. We find that
one UDG in our sample, AGC227965, is quenched (SFR$<10^{-3}$ M⊙ yr-1) due to
being a close satellite of MRK1324. We leave this galaxy in our analysis as it
still yields a significant H I detection, but note that our discussion of the
field UDG formation pathways does not apply to this system.

In Figure 1 we show our sample galaxies in stellar mass versus H I mass
(left), SFR versus H I mass (center), and the star-forming main sequence
(SFMS, right). These results are from a customized SED fitting method that we
describe in detail in Section 3, but can be thought of as standard SED fitting
results for the purposes of the figures in this section. In this figure and
all subsequent figures, we show the NSA sample in purple and the UDG sample in
red. The NSA sample reaches higher stellar masses and lower H I masses than
does the UDG sample; when noted, we compare only NSA galaxies with stellar
masses within the observed range of the UDG sample. The results of the Leroy
et al. (2008) sample of nearby galaxies and SDSS spectroscopic value-added
catalog (Kauffmann et al., 2003; Brinchmann et al., 2004; Salim et al., 2007)
are shown in beige and grey, respectively. We see that the UDGs have high H I
masses for their stellar masses and that they have low SFRs for their H I
masses – conversely, the UDGs lie directly on the SFMS.

A clear question arises from this initial analysis: are the H I-selected UDGs
truly drawn from a more H I-rich population than the NSA dwarfs, or are the
higher H I masses simply an effect of observational selection? To test this,
we consider what our expected distribution of UDG H I fractions, $\rm
f_{HI}=M_{HI}/(M_{\star}+M_{HI})$, would be given the 21cm flux limit of the
UDG sample if the $\rm f_{HI}$ distribution of the UDGs were the same as the
NSA sample. To put it another way, we ask the following question: if the UDG
population had the same distribution of H I fractions as the NSA dwarf sample,
what is the distribution of $\rm f_{HI}$ that we would expect to see from an H
I-selected sample?

We show the results of this test in Figure 2. The top panel shows the observed
21 cm line fluxes in solid points (as always, purple for the NSA sample and
red for UDGs), while the unfilled red points show the expected 21 cm flux for
each UDG in the sample if the H I fraction were drawn from the NSA dwarf
distribution over $\rm f_{HI}$. The errorbars in this panel are generated via
bootstrap resampling from the NSA distribution. We estimate the detection
limit of the UDG sample in two ways: first, from the stated completeness limit
of the ALFALFA survey for sources with $1.65<\log_{10}(W_{50}/[\rm km\
s^{-1}])<1.7$ (Haynes et al., 2011, black dashed line), and second, from the
lowest 21 cm flux of the UDG sample (red solid line). These limits are in good
agreement with one another, and we adopt the minimum 21 cm line flux of the
UDG sample as our detection limit for this exercise. The bottom panel of
Figure 2 shows the distribution of observed H I fractions for the NSA (solid
purple histogram) and UDG (unfilled red histogram) samples, as well as the UDG
sample that we would expect to have detected if the UDG distribution over $\rm
f_{HI}$ were the same as that of the NSA sample (hatched dashed red
histogram). Indeed, we find that we would expect to detect a significant
number of $\rm f_{HI}\lesssim 0.75$ UDGs, should they exist. Rather, the
marked H I richness of the UDG sample cannot simply be attributed to
observational limits.

We extend this positioning of H I-rich UDGs as unusual star-forming
environments by considering a globally averaged stellar mass, H I, and SFR
surface density in Figure 3. In this figure, we compute $\rm\Sigma_{\rm
HI,global}$ using the total H I mass and the total area (as defined by the
global elliptical HSC-g aperture, which we will describe Section 3). One can
immediately see that, as expected from their definition, the UDGs have
significantly lower SFR and stellar mass surface densities than the NSA
sample. Intriguingly, however, the UDGs do not also show signs of lower H I
surface densities. Indeed, their H I surface densities span the same range as
their spectroscopic counterparts.

The main thrust of this paper will be to root out the physical culprits of
this apparently lowered SFE(H I). From globally averaged measurements alone,
it is unclear whether the low average star formation rate surface density is a
result of a truly lower SFE(H I), or whether the spatial distribution of star
formation in UDGs differs from normal dwarfs (e.g. a low globally averaged
$\Sigma_{\rm SFR}$ could arise from an old stellar population that is
significantly more extended than the star-forming regions). In order to
distinguish between these two scenarios, we must build an SED fitting method
to constrain local (500 pc scale) SFR and stellar mass surface densities
(Section 3) – we will return to this narrative of star formation efficiency in
Section 5.

Figure 4:  A comparison between our estimates of the H I surface density
radial profiles for the UDG (red solid curves) and NSA (purple solid curves)
samples against spatially resolved H I measurements of dwarfs from the
literature. We show the direct radial profiles of Leroy et al. (2008) in
green, and Sérsic fits inferred for a sample of H I-rich UDGs from the H I
spatial distribution metrics provided by Gault et al. (2021) in orange. We
find that our radial profile estimates are in good agreement with these direct
literature measurements.

###  2.2 UV-Optical Imaging

HSC-SSP. The HSC-SSP imaging boasts wide spatial coverage, high sensitivity (a
point source depth of $i_{\rm HSC}\sim 26$ in its shallowest “Wide” layer),
and high spatial resolution (the worst-seeing band, $g_{\rm HSC}$, has an
average seeing of 0.77″).

For this work, we use the internal data release S21A, which covers 670 deg2 in
5 optical bands. The HSC-SSP data products include two options for the
background subtraction – a local subtraction suited for small, high redshift
sources, and a global subtraction suited for nearby galaxies. We use the
coadd/bg versions of the co-added HSC images, which uses the “global”
background subtraction of the S18A and PDR2 data releases (for an in-depth
discussion of the HSC global background subtraction scheme, see Section 4.1 of
Aihara et al. 2019 and Li et al. 2021). The data have been shown to reach
surface brightness limits of $i_{\rm HSC}\sim 28.5$ mag arcsec-2 for radially
averaged measurements around a known target (Huang et al., 2018; Kado-Fong et
al., 2020).

GALEX. To complement the deep and high resolution optical imaging of the HSC
imaging, we draw upon archival GALEX imaging to measure global UV photometry
of our galaxies. We use the package astroquery to query the Barbara A.
Mikulski Archive for Space Telescopes (MAST) for GALEX FUV and NUV
observations that cover the position of each galaxy. We adopt the average
GALEX PSFs in this work; the FUV and NUV PSFs have full widths at half maximum
(FWHM) of 4.2” and 4.9”,
respectively111http://www.galex.caltech.edu/researcher/techdoc-ch5.html. We
measure our GALEX photometry on the background-subtracted intensity maps
provided by the GALEX pipeline, and estimate uncertainties assuming Poisson
noise for the counts.

For targets with more than one GALEX archival image, we prefer the data
product with a longer total exposure time. Two NSA galaxies (NSA ID 5109 and
30738) do not have FUV imaging; all galaxies have NUV imaging. Of our sample
with FUV imaging, 13 galaxies are drawn from the GALEX All-Sky Imaging Survey
(AIS, $m_{AB}=20.5$) and 35 from the Medium Imaging Survey (MIS,
$m_{AB}=23.5$, Martin et al., 2005). 13 are drawn from guest investigator (GI)
programs; we make an estimate the depth of the GI programs by measuring the
standard deviation of random apertures of apertures the size of the average
GALEX FWHM. The GI imaging is somewhat deeper than the MIS imaging, with a
mean $5\sigma$ depth of $m_{AB}=24.3$. Similarly, 6 galaxies in our sample
have NUV AIS imaging, 40 have MIS imaging, and 9 have GI imaging. We repeat
the same procedure to estimate the depth of the GI imaging and find a mean
$5\sigma$ depth of $m_{AB}=23.4$. We note that the depths estimated here are
not used in the analysis, and are simply meant to contextualize the GALEX
imaging sources for the reader.

###  2.3 H I Measurements

All of the galaxies in the present work have integrated H I measurements from
Janowiecki et al. (2019, UDGs) or Bradford et al. (2015, NSA galaxies).
However, resolved H I measurements exist only for a small subset of the UDGs
identified by Janowiecki et al. (2019) – 12 from Gault et al. (2021), 3 from
Leisman et al. (2017).

To estimate the radial profiles of the H I in our UDG and NSA dwarf samples,
we leverage the empirical result that UDGs and “normal” dwarfs are similar in
their H I structure – that is, UDGs lie on the H I mass-size relation and have
typical average H I surface densities (Gault et al., 2021). We use this result
to estimate Sérsic profile parameters for our sample via a parameter
estimation from the spatially resolved measurements of Hunter et al. (2021)
(who report Sérsic parameter fits for their sample), as well as the spatially
resolved measurements of the UDGs in Gault et al. (2021)222here we solve for
the Sérsic parameters of their galaxies using the given measures of H I
structure. We perform a least squares fit to the Sérsic index, $n$, central
surface density, $\Sigma_{0,\rm HI}$ and effective radius, $\rm R_{eff,HI}$,
of the literature samples as a function of H I mass; only $\rm R_{eff,HI}$
shows a significant correlation with H I mass (i.e. the H I mass-size
relation). This allows us to estimate Sérsic parameters for our galaxies as a
function of their H I mass. As these structural measures are not well-
constrained, we will account for the uncertainties induced by our estimated H
I profiles to demonstrate to the reader that these estimates are indeed
reasonable.

First, although the Sérsic profile has three free parameters, our knowledge of
the total H I mass allows us to determine the value of one of these parameters
given estimates for the other two. We therefore choose to fix the H I central
surface density via the analytic solution for the total H I mass of a 1D
Sérsic profile, i.e.:

$\begin{split}{\rm M}_{\rm HI}&=2\pi\int_{0}^{\infty}\Sigma_{\rm HI}(R)RdR\\\
&=2\pi\Sigma_{0}n((\frac{1}{h})^{1/n})^{-2n}\Gamma(2n).\\\ \end{split}$ (1)

We find that the $\Sigma_{0}$ values derived from a fit to the literature data
and the $\Sigma_{0}$ values derived using the total H I mass are in good
agreement, with the latter method resulting in $\Sigma_{0}$ values that are on
average $\sim 8\%$ higher than the former.

Second, to account for the substantial uncertainty induced by our lack of
spatially resolved H I, we include an uncertainty in the parameter estimation
of the Sérsic profiles equivalent to the standard deviation of the literature
parameter values for reference (literature) galaxies within 0.5 dex of that of
the target (NSA/UDG) galaxy in total H I mass. If no galaxies are sufficiently
close in H I mass, we increase the tolerance from 0.5 dex to up to 0.9 dex,
but note that because the reference sample of Hunter et al. (2021) is
systematically lower mass than our sample, this only affects the most massive
($\log_{10}\rm M_{HI}>10$) galaxies in our sample. We then estimate the
uncertainty on the radial H I surface density profile via Monte Carlo
resampling assuming that the uncertainties are well-describe by normal
distributions.

The resulting H I surface density profiles are shown in Figure 4; as above,
red curves indicate UDGs and purple curves indicate NSA galaxies. The shaded
regions span the 16th to 84th percentiles of the Monte Carlo simulation result
for each galaxy. The directly measured H I profiles of Leroy et al. (2008) are
shown in green, while the Sérsic profiles inferred from the structural
measurements of Gault et al. (2021) are shown in orange. We find that our
inferred profiles are in good agreement with the shape, normalization, and
scatter of the directly measured profiles, with the exception of one UDG
(AGC238764, which is not a part of our sample) that has an anomalously high
Sérsic index ($n=1.98$; almost all dwarfs in this mass range are characterized
with sub-exponential Sérsic indices, see Hunter et al. 2021). There is, to our
knowledge, only one H I-rich UDG that has spatially resolved H I data at a
high enough resolution to probe the clumpiness and substructure of the H I
(Mancera Piña et al., 2022). These data, which have a physical beam size of
$\sim 2.9$ kpc, show significant H I substructure that is not captured in our
simple radial profile estimation approach. However, the deviations from the
radial profile that these azimuthal variations cause are comparable to the
uncertainty we have already attributed to our radial profiles (see right panel
of Figure 1 of Mancera Piña et al. 2022). We thus find it unlikely that we are
overestimating the precision of the H I profiles, though we of course note
that obtaining high resolution H I maps would dramatically improve the
precision and accuracy of our $\Sigma_{\rm HI}$ estimates.

Figure 5:  A schematic layout of the fitting technique used in this work. At
left, we represent the high-resolution optical data and low-resolution UV data
(panels a and c, respectively). Panels b and d represent the global fit
between the optical and UV data; in this step, the apertures are matched
between the optical and UV data. Panel e shows the spatial division of the
optical data into regions in which regional SEDs are measured. Panel f shows
the clustering of these spatial regions into three representative clusters;
these cluster SEDs will be fit jointly with the global UV photometry (panel
d).  Figure 6:  An example of the fitting process for one of the NSA galaxies
(NSA ID 5449). Top row: from left, the HSC $gri$-composite image, the SED
cluster map (analogous to panel f of Figure 5), and the joint UV-optical SED
fit. The joint model SED is the result of fitting the spatially resolved
(cluster) optical photometry and global UV photometry. The black curves show
the model spectra, while the orange points show the model photometry for each
filter. The red points show the global UV-optical photometry. The model
fitting and figure fluxes are computed as HSC counts. Middle row: optical-only
results for the individual cluster SEDs. The color of the title text
corresponds to the color of the region in the top middle panel; the format is
equivalent to the joint SED fit (top right), with the addition of the SEDs
measured in individual spatial regions (i.e. regional SEDs) as blue points.
Here, the red points show the cluster SEDs (i.e. the sum of the regional SEDs)
rather than the global photometry. Bottom row: the resultant stellar mass
(left) and SFR (right) maps.

##  3 SED Fitting

The key technical challenge of this work is inferring the star formation and
stellar mass properties of the dwarf sample at sub-galactic scales. We
accomplish this by modeling the high spatial resolution HSC optical data in
conjunction with the UV GALEX measurements to constrain star formation
activity in a consistent way with the UV data.

The novelty of our approach is to leverage both the high spatial resolution of
HSC and the expanded wavelength coverage of GALEX to infer local star
formation surface density properties of our dwarfs via a joint fit to
localized optical photometry and global UV photometry. Here, we detail the
fitting process employed in this work.

###  3.1 Stellar Population Models

We use the Flexible Stellar Population Synthesis (FSPS) models (Conroy et al.,
2009; Conroy & Gunn, 2010) with a fixed stellar and gas-phase metallicity of
$Z=0.004$ along with a Kroupa (2001) initial mass function to generate our
synthetic spectra.  We choose a single value for the model metallicity as the
mass-metallicity relation is not well-understood for these dwarf samples.
Direct ($T_{e}$) measurements of nearby dwarfs (Lee et al., 2006; Berg et al.,
2012; Jimmy et al., 2015) can show significant offsets relative to each other
and to metallicity estimates from SED fitting to more distant dwarf samples
(Bellstedt et al., 2021). We thus choose to adopt a typical value following
the literature compilation of (Bellstedt et al., 2021) for a galaxy of
$M_{\star}\sim 10^{8}M_{\odot}$. We also check the impact of adopting solar
metallicity models in Appendix B, and find that our SED fitting results are
shifted by less than $0.1$ dex on average.  The library includes nebular
emission, which may contribute significantly to the broadband flux of the
dwarf galaxies in our sample. We adopt an exponentially declining star
formation history

${\rm SFR}(t)\propto e^{-(t-t_{0})/\tau};$ (2)

We choose this relatively rigid star formation history (two free parameters,
the onset of star formation $t_{0}$ and the timescale of decay $\tau$) in
order to achieve reasonable computation times when the number of regions to be
fit is high. Such a rigid choice of the SFH functional form has the potential
to bias SFH recovery, and so we note that the main goal of our analysis is to
diagnose the relative change in star formation surface density and star
formation efficiency between normal, high surface brightness dwarfs and ultra
diffuse galaxies, not to provide comprehensive stellar population synthesis
properties for the UDGs. We will return to this point more comprehensively in
Section 4 when we consider validation tests for this method.

The composite stellar population (CSP) model library spans ages between 2 Myr
and 12 Gyr and timescales ($\tau$) between 0.3 Gyr and 10 Gyr. Each axis
consists of 50 logarithmically spaced values. We pre-compute both the UV-
optical spectra and the fluxes through the GALEX and HSC bands for each model
in the library. During fitting, we linearly interpolate between pre-computed
library fluxes to obtain the predicted GALEX and HSC fluxes for arbitrary
choices of $t_{0}$ and $\tau$. Because we use pre-computed library fluxes at a
fixed distance of 10 Mpc, this induces a small k-correction error in the
galaxy sample. However, the most distant galaxies in our sample are at 120
Mpc, and so the k-correction error induced is negligible.

###  3.2 Defining Global, Regional, and Cluster SEDs

The first step to executing a joint fit between the optical and UV photometry
is to establish the apertures in which we measure the optical and UV
photometry. To obtain consistent optical photometry, we PSF match the HSC
imaging with the worst seeing. To do so, we use the Fourier transform-based
implementation of photutils with a cosine bell window function ($\alpha=3$) to
compute matching kernels between the other optical bands and the band to which
we will match the PSF. We model the PSFs measured by the HSC pipeline (which
produces PSF postage stamps with spatial extents of about 7″) as two-
dimensional Moffat functions. We choose to simplify the PSF model to Moffat
functions in order to prevent the emergence of imaging artifacts during the
matching process caused by high frequency features in the original pipeline-
produced PSFs. We additionally correct for galactic extinction in all UV and
optical photometry presented in this work using the measurements of Schlafly &
Finkbeiner (2011) via the associated online service (IRSA, 2022).333Galactic
extinction measurements were accessed via
https://irsa.ipac.caltech.edu/applications/DUST/

We then define the global aperture of the galaxy using sep, a python
implementation of the segmentation algorithm used in SExtractor (Bertin &
Arnouts, 1996). In all cases, we use the HSC-g band as the detection band,
with a detection threshold of 5$\sigma$ over the variance image supplied by
the HSC pipeline. We define the elliptical global aperture as

$c_{xx}(x-\bar{x})^{2}+c_{yy}(y-\bar{y})^{2}+c_{xy}(x-\bar{x})(y-\bar{y})=3^{2},$
(3)

where $\bar{x}$ and $\bar{y}$ are the light-weighted central coordinates of
the galaxy and $c_{xx}$, $c_{yy}$, and $c_{xy}$ are the standard SExtractor
ellipse parameters

$\begin{split}c_{xx}&=\frac{\cos^{2}\theta}{a^{2}}+\frac{\sin^{2}\theta}{b^{2}}\\\
c_{yy}&=\frac{\sin^{2}\theta}{a^{2}}+\frac{\cos^{2}\theta}{b^{2}}\\\
c_{xy}&=2\cos\theta\sin\theta(\frac{1}{a^{2}}-\frac{1}{b^{2}}),\\\
\end{split}$ (4)

where $a$ and $b$ are the semi-major and semi-minor axes, respectively, and
$\theta$ is the position angle of the galaxy measured counterclockwise from
the horizontal (in image coordinates). This global aperture step is shown
visually in step b of Figure 5 – the global aperture defined in step b using
the optical data is then applied to the UV imaging to derive the global UV SED
(step d). We choose not to PSF match the optical data to the UV data to
maintain the spatial resolution of the optical data in order to hold the input
images consistent between the global and spatially resolved SED fitting. If we
did perform PSF matching to the GALEX imaging to compute the global optical
photometry, we have verified that the changes would be negligible, with a mean
fractional deviation in flux of 0.03 and a standard deviation of -0.02.

Next, to define the regional boundaries for the optical photometry, we lay a
grid of fixed size over the cutout image, as shown in step e of Figure 5. The
box length of this grid is set to be no less than twice the FWHM of the PSF-
matched images, and is increased iteratively until the median SNR of the SEDs
measured in the regions is at least 3. To remove regions where a reliable SED
cannot be measured at the required spatial scales, we remove the bottom 5%
faintest regions. We also mask regions with $(g-i)>0.8$ and generate
additional masks for the UDGs, whose diffuse structure and large on-sky sizes
make standard deblending routines unreliable. We verify that this pruning does
not affect the total inferred mass and star formation rate in Section 4.

Because the size of the regions is small compared to the size of the galaxy,
it is unsurprising to note that many of the regional SEDs are identical to
each other to within their photometric errors. To decrease the number of
parameters to fit, and to increase the SNR of the SEDs that we fit, we group
the regions (step e) into clusters of similar SEDs (step f). We group the
regions into clusters using K-means clustering, initially with 3 clusters. If
any cluster has a SNR$<10$, we combine the low-SNR cluster with the closest
high-SNR cluster (here defining closeness by the $\ell^{2}$ norm444Where the
$\ell^{2}$-norm is the difference between the SED flux and reference flux
$\sqrt{\sum_{k}^{K}|f_{k}-f_{\rm ref}|^{2}}$ over the $K$ optical bands over
the optical SEDs). Next, we must verify that the cluster SEDs adequately
represent their input regional SEDs. We use the metric
$z^{j}_{b,i}=|f_{b,i}-\bar{f}^{j}_{b}|/\sqrt{\sigma_{f_{b,i}}^{2}+\sigma_{\bar{f}^{j}_{b}}^{2}}$,
where $f_{b,i}$ is the flux in band $b$ measured in region $i$, while
$\bar{f}^{j}_{b}$ is the flux in band $b$ measured in cluster $j$.
$\sigma_{f_{b,i}}^{2}$ and $\sigma_{\bar{f}^{j}_{b}}^{2}$ refer to the
uncertainty on the respective flux measurement. We impose the condition
$z_{b,i}<2$ for all bands $b\in\\{\rm g,r,i,z,y\\}$ and all regions $i$
assigned to cluster $j$ for each cluster in the galaxy. If the clustering
result fails this verification, we rerun the clustering process with an
additional cluster at the K-means step. We allow up to five clusters, at which
point the clustering assignment is accepted regardless of verification status.
In practice, only one galaxy (the UDG AGC223246) is fit with the maximum five
clusters.

Figure 7:  Top: a comparison between the best-fit stellar masses of the NSA
(purple) and UDG (red) samples using the difference between results derived
from the spatially-resolved and global photometry (y-axis) and the global
optical photometry (x-axis). Bottom: the same as above, but for SFR.  Figure
8:  Left: a comparison between the stellar masses of the galaxies in this work
(x-axis) and their catalog NSA stellar masses. The dashed line and shaded
regions show the median, and range between the 16th and 84th percentiles of
the deviation $\log_{10}(\rm M_{\star}/M_{\odot})-\log_{10}(\rm
M_{\star,NSA}/M_{\odot})$. Middle: the same, but for SED fitting-derived SFRs
and FUV-only SFRs as computed from the catalog GALEX measurements using the
SFR(FUV) prescription of Kennicutt (1998). Right: a comparison between SFRs
derived from the H$\alpha$ fluxes measured from SDSS 3” fiber fibers (x-axis)
and the total SFR within the central 3” of our SED fits (y-axis, red). The
dashed black line denotes a 1:1 relation. As in the left panel, the dashed
purple line and shaded purple regions show the median, and range between the
16th and 84th percentiles. SFRs derived from H$\alpha$ measure star formation
over a shorter timescale ($\sim 10$ Myr) than our SED fits (100 Myr), and some
physical scatter is expected as a result; we show the standard deviation (in
logarithmic space) of the difference between H$\alpha$-derived and FUV-derived
SFRs for a set of Local Volume dwarfs from Karachentsev et al. (2021) as a
visual aid to the reader in gauging the scatter.

###  3.3 SED Fitting

With our PSF-matched photometry and apertures in hand, we are now ready to
begin the SED fitting. For all of our fitting, we use the Markov Chain Monte
Carlo implementation of emcee (Foreman-Mackey et al., 2013). We adopt a flat
prior over positive values for the amplitude of each cluster SED, a flat prior
between 0 and 13.6 Gyr for $\tau$ (see Equation 2), and a log-normal prior
over $A_{V}$ distributed as $\log_{10}(A_{V})\sim\mathcal{N}(0,0.1)$. In our
joint fitting step, where multiple amplitudes and $\tau$ values are being
inferred, the prior is the same for each individual parameter.

We first perform a global SED fit using both the optical and UV photometry to
constrain the reddening, $A_{V}$, and to provide an estimate of the total
stellar mass and star formation rate. We adopt a Gaussian likelihood, defined
as:

$\log\mathcal{L}\propto-\frac{1}{2}\sum_{b}^{M}\frac{(f_{b}-\hat{f}_{b})^{2}}{\sigma_{f_{b}}^{2}}+\log(\sigma_{f_{b}}^{2})$
(5)

where $f_{b}$ refers to the flux of the $b^{\rm th}$ band out of $M$ bands
($M=7$ for our fiducial case), $\hat{f}_{b}$ is the predicted model flux, and
$\sigma_{f_{b}}$ is the uncertainty in the $b^{\rm th}$ band. We measure
$\sigma_{f_{b}}$ directly from the variance images of HSC-SSP added in
quadrature to an extra uncertainty of $0.05f_{b}$. In this step we run emcee
for 1500 iterations with 32 walkers, discarding the first 400 steps. The only
parameter from this fit that survives as a final result is the reddening,
which is used to the joint fit as a fixed parameter. We find no difference in
the $A_{V}$ estimated for the UDGs and the NSA galaxies at
$M_{\star}<10^{9}M_{\odot}$: the 50th (16th and 84th) percentile of the
$A_{V}$ distribution is $A_{V}=0.17$ (0.04, 0.37) for the UDGs and
$A_{V}=0.18$ (0.05, 0.38) for the NSA dwarfs at $M_{\star}<10^{9}M_{\odot}$.
Unsurprisingly, the NSA galaxies at higher stellar mass have systematically
higher estimated reddening – $A_{V}=0.67$ (0.55, 0.99).

Next, we compute the flux in the $N$ SED clusters defined as above. At this
stage, the model consists of a set of $N$ models across $M$ bands. The
likelihood is computed as follows:

$\begin{split}\log\mathcal{L}\propto&-\frac{1}{2}\sum_{i\in
UV}\frac{(\sum_{j}^{N}f_{i,j}-\sum_{j}^{N}\hat{f}_{i,j})^{2}}{\sum_{j}^{N}\sigma_{f_{i,j}}^{2}}+\log(\sum_{j}^{N}\sigma_{f_{i,j}}^{2})\\\
&+-\frac{1}{2}\sum_{j}^{N}\sum_{i\in
Opt}\frac{(f_{i,j}-\hat{f}_{i,j})^{2}}{\sigma_{f_{i,j}}^{2}}+\log(\sigma_{f_{i,j}}^{2}),\end{split}$
(6)

where $i$ refers to the bandpass index and $j$ refers to the cluster index. In
other words, we retain a Gaussian likelihood wherein the UV bands (FUV and
NUV) are compared against the sum of the regional models, whereas the optical
bands ($Opt$ in Equation 6; g,r,i,z, and y) are compared against the
appropriate regional models. In this step, the walkers are initialized with
the positions drawn from the posterior of the global SED fit; the amplitude of
each cluster SED is additionally multiplied by the mean ratio between the flux
in the cluster SED and the global SED over the optical bands. In this step, we
run emcee for 20000 iterations with at least 32 or $2N+1$ walkers, discarding
the first 5000 steps.

From this process we estimate the total stellar mass and star formation rate
for each cluster SED. We then convert these to stellar mass and star formation
rate surface densities assigned to each regional SED by multiplying by the
ratio of the flux in the regional SED and the flux in its corresponding
cluster SED, and dividing by the physical area subtended by the region.

We show an example of the fitting procedure for the galaxy NSA 5449 in Figure
6. We also show an abbreviated gallery of fitting results in Appendix A for a
set of galaxies that span the morphologies seen in the NSA and UDG samples.

Figure 9:  A comparison between radial profiles in stellar mass surface
density (left) and star formation rate surface density (right) from our SED
fitting method (solid curves) as compared to direct measurements from (Leroy
et al., 2008, dashed curves). The color of the curves show the integrated
stellar mass of the galaxy in all cases. We find that there is good agreement
between our radial profiles and the directly measured profiles in both
amplitude and extent.

##  4 Method Validation

In order to apply the framework constructed in the previous section to our
galaxy samples, we must first take the necessary step of validating that our
method is able to accurately deliver the quantities of interest to this study
– namely, maps of the star formation and stellar mass surface densities down
to the spatial resolution of HSC.

An equally important step is to lay out the quantities that we are not
attempting to constrain with this modeling. The age, stellar metallicity, and
detailed (flexible) star formation history of the galaxies are all parameters
that may be estimated from SED fitting. However, it has been shown that the
exponentially declining tau model is an imperfect model of star formation
histories at large, both due to its overall rigidity and that it, by
definition, ties early and late star formation (Simha et al., 2014; Carnall et
al., 2019). Thus, we stress that our main goals, and the subjects of our
verification tests, will be the estimation of the star formation rate and
stellar mass surface densities, and not a detailed interrogation of the star
formation history and enrichment of the galaxies in our sample.

###  4.1 Intra-sample Comparisons

We first consider an internal comparison between the star formation rate and
stellar masses measured from the global UV+optical SED fitting (step b and
step d of Figure 5) and the integrated star formation rate and stellar masses
obtained by summing the properties inferred from the cluster SED fitting (step
f). In Figure 7, we show the integrated estimates of stellar mass (top) and
star formation rate (bottom) for the UDGs (red) and NSA galaxies (purple). In
all cases, we find good internal consistency between the initial SED fit on
the global photometry and the final model SED of the spatially resolved
method.

We additionally compare the stellar mass and SFR estimates obtained from our
spatially resolved SED fitting method to the catalog NSA stellar masses and
FUV-only SFRs in Figure 8. The FUV-only SFRs are computed from the catalog
GALEX measurements using the prescription of Kennicutt (1998), i.e.

${\rm SFR}(L_{FUV})=1.4\times 10^{-28}\frac{L_{FUV}}{\rm erg\ s^{-1}\ Hz}.$
(7)

Both the catalog stellar mass estimates and the FUV-only SFRs are also in good
agreement with the results of our spatially resolved fit.

Although we lack truly spatially-resolved spectroscopic measurements for the
galaxies in our sample, we do have access to one quasi-resolved measurement
for the NSA sample – the SDSS 3” fiber spectra. Because our galaxies exceed
the SDSS fiber size, the SDSS spectroscopic measurements do contain
information about the spatial distribution of star formation within the
galaxy, albeit in a very limited capacity. In the right panel of Figure 8 we
show a comparison between SFRs derived from the SDSS fiber spectroscopy of the
NSA galaxies and SFRs we compute by integrating our SFR maps over SDSS-like
apertures. The SDSS spectroscopy we reference are 3” diameter fiber spectra;
we use the NSA catalog line fluxes in this comparison. In particular, we use
the catalog H$\alpha$ flux to compute H$\alpha$-derived star formation rates
within the SDSS aperture following Calzetti (2013) with a Kroupa (2001) IMF:

${\rm SFR}(L_{H\alpha})=5.5\times 10^{-42}\frac{L_{H\alpha}}{\rm erg\
s^{-1}}.$ (8)

Although the 100 Myr averaged SFRs we produce via our SED fitting are not
completely analogous to the H$\alpha$ SFRs derived from the SDSS spectra
(which probe timescales closer to $\lesssim 10$ Myr), we expect that the two
should be well-correlated. Indeed, we find that the star formation rates we
compute from the central three arcseconds of our star formation rate maps are
in good agreement with the SDSS spectroscopic measurements, with a scatter on
the same order as the physical variation between UV and H$\alpha$ SFRs
measured from Local Volume dwarfs (Karachentsev et al., 2021, red line in
right panel).

###  4.2 Inter-sample Comparisons to the Literature

We also find good agreement between our results and literature measures of the
integrated stellar mass, SFR, and $\rm A_{V}$ for the two low surface
brightness galaxies presented in Greco et al. (2018a), though we do not
include these galaxies in this work because they lack H I measurements.

Finally, we compare to spatially resolved measurements of stellar mass, SFR,
and H I in nearby samples of dwarfs from the literature. In Figure 9, we show
a comparison between radial profiles of stellar mass surface density (left)
and SFR surface density (right) as derived from our SED fitting method as
compared to the directly measured results of Leroy et al. (2008). Our results
are shown as solid curves, the Leroy et al. (2008) profiles are shown as
dashed curves; all curves are colored by total stellar mass. Encouragingly, we
find good agreement in both the shape and normalization between our inferred
radial profiles and the directly measured results of Leroy et al. (2008),
indicating that our method is able to recover not just the total amount of
stellar mass and SFR, but the distribution of these quantities.

Figure 10:  A comparison between H I surface density estimates and SFR surface
density measurements averaged over 500 pc, 1 kpc, and global scales. In all
panels, we only show NSA galaxies that are within the stellar mass range of
the UDG sample (i.e. $\rm M_{\star}<3\times 10^{8}\ M_{\odot}$). For visual
clarify, we show a random subset of (at most) 70 points for each sample with
individual errorbars and colored by sample source; we show the full sample as
the underlying grey scatter. Here, we demonstrate that the low SFE(H I) that
characterizes the globally averaged properties of the UDGs persists down to
sub-kpc scales.  Figure 11:  A comparison between various measures of galaxy
mass and galaxy size; the top row shows size as a function of H I mass, while
the bottom row shows size as a function of stellar mass. The left column shows
size as measured by the HSC-$g$ band half-light radius, while the right column
shows size as measured by the half stellar mass radius. In each panel, the
purple dashed line shows the least squares power law fit to the NSA galaxies.
The red dashed line shows the least squares fit to the UDG sample when the
index of the power law is held fixed to that of the NSA dwarf fit. The shaded
regions span the 16th to 84th percentiles as estimated from bootstrap
resampling. As one expects, the UDG sample is large in size for their stellar
masses. However, their half-stellar mass radii are unremarkable for their H I
masses.  Figure 12:  Optical color versus stellar mass-to-light ratios in the
HSC-$g$ band computed for our UDG (red) and NSA (purple) samples. We find no
significant difference in the color-mass relation between the two samples, and
additionally find the samples to be reasonably well-described by color-mass
relations from the literature (Into & Portinari 2013 as the dashed line, Bell
et al. 2003 as the dotted line).  Table 1: Selected SED fitting results

ID | RA | Dec | Dist | $M_{\rm HI}$ | $M_{\star}$ | SFR | $\rm A_{V}$ | $R_{\rm 1/2,\star}$ | $R_{\rm 1/2,g}$
---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [Mpc] | [$10^{8}\ \rm M_{\star}$] | [$10^{8}\ \rm M_{\star}$] | [$10^{-3}\ \rm M_{\star}\ yr^{-1}$] | [mag] | [kpc] | [kpc]
NSA16938 | 140.433 | 4.374 | 85.81 | $13\pm 2.8$ | $5_{-0.58}^{+0.66}$ | $40_{-1.5}^{+1.9}$ | $0.15_{-0.099}^{+0.12}$ | $1.9\pm 0.38$ | $1.8\pm 0.32$
NSA11919 | 132.078 | 1.263 | 27.67 | $1.3\pm 0.42$ | $1.3_{-0.092}^{+0.19}$ | $5.7_{-0.24}^{+0.35}$ | $0.15_{-0.1}^{+0.17}$ | $1.7\pm 0.19$ | $1.9\pm 0.1$
NSA16993 | 143.969 | 4.017 | 81.05 | $36\pm 4.2$ | $51_{-9}^{+7.8}$ | $800_{-10}^{+8.5}$ | $0.64_{-0.051}^{+0.059}$ | $2.7\pm 0.48$ | $3.5\pm 0.3$
NSA13797 | 174.755 | 1.336 | 24.92 | $1.9\pm 0.58$ | $0.43_{-0.034}^{+0.042}$ | $35_{-1.1}^{+1.4}$ | $0.079_{-0.035}^{+0.036}$ | $0.71\pm 0.12$ | $0.6\pm 0.093$
AGC334353 | 345.863 | 1.704 | 71.40 | $12\pm 1.9$ | $1.2_{-0.18}^{+0.19}$ | $12_{-1.2}^{+1.1}$ | $0.1_{-0.071}^{+0.11}$ | $2.1\pm 0.28$ | $2.5\pm 0.27$
AGC334349 | 345.277 | 1.998 | 51.10 | $5.2\pm 0.83$ | $0.82_{-0.11}^{+0.12}$ | $10_{-0.78}^{+0.76}$ | $0.087_{-0.064}^{+0.097}$ | $1.4\pm 0.2$ | $1.9\pm 0.19$
AGC103435 | 3.838 | 1.075 | 28.50 | $1.9\pm 0.4$ | $0.14_{-0.012}^{+0.015}$ | $8.6_{-0.15}^{+0.19}$ | $0.18_{-0.058}^{+0.059}$ | $1.5\pm 0.11$ | $1.8\pm 0.11$
AGC198540 | 141.308 | 3.313 | 63.30 | $13\pm 1.8$ | $1.4_{-0.23}^{+0.42}$ | $13_{-0.8}^{+0.68}$ | $0.14_{-0.092}^{+0.11}$ | $1.9\pm 0.26$ | $2\pm 0.24$

Note. — Table 1 is published alongside this article in its entirety in the
machine-readable format. A portion is shown here for guidance regarding its
form and content.

##  5 Results

With our methodology in hand, we may now return our focus to the physical
question at hand. In Section 2.1, we demonstrated that although the UDGs have
low globally averaged stellar mass and SFR surface densities, their H I
surface densities are comparable to those of the NSA galaxies. Now, we can
consider the cause of the low SFE(H I) of the UDGs – and in particular whether
this inefficiency persists to sub-kpc scales – as well as the relationship
between UDG structure and H I content. For the interested reader, we include a
table of selected SED fitting results for the sample in Table 1.

###  5.1 The Efficiency and Distribution of Star Formation in UDGs

In the top row f Figure 10, we show the relation between the estimated
$\Sigma_{\rm HI}$ and $\Sigma_{\rm SFR}$ at 500 pc, 1 kpc, and global scales
from left to right (we note that, for reasons of visual continuity, the bottom
right panel of Figure 10 is identical in content to the bottom panel of Figure
3). We compute 1 kpc and 500 pc average stellar mass surface density and SFR
surface density maps by placing a grid of length 1 kpc or 500 pc (at the
distance of the galaxy) over the stellar mass and SFR maps produced via SED
fitting. We do not compute average quantities over length scales less than the
regional SED box size for each galaxy. For the 1 kpc and 500 pc regions, the H
I surface density is estimated using the central position of the region and
the Sérsic profile estimated for the galaxy as described in Section 2.3. From
this figure, the reader may immediately notice that the shift in $\Sigma_{\rm
SFR}$ persists down to 500 pc; there are no regions in the area-weighted
census of the UDG distribution of SFR surface density that match the
vigorously star-forming SFR surface densities observed in the NSA dwarfs.

The persistence of the discrepancy of star formation activity outcomes as a
function of H I surface density indicates that it is not simply a matter of
the distribution of star formation that drives the apparently low star
formation efficiency (as a function of atomic gas) in the UDG sample. Rather,
it appears to be a generic property of star formation in our sample of UDGs.
The question, then, of whether a more comprehensive understanding of star
formation can explain this low star formation efficiency will naturally arise
to the reader. This question is the central topic of Paper II, in which we
will examine the impact of the UDGs’ peculiar stellar structure on their star
formation activity within the framework of pressure-regulated, feedback-
modulated star formation (Ostriker et al., 2010; Kim et al., 2011; Ostriker &
Shetty, 2011; Kim et al., 2013; Kim & Ostriker, 2015; Kim et al., 2017).

We note that the minimum region size over which we can measure $\Sigma_{\rm
SFR}$ and $\Sigma_{\star}$ is set by the distance of the galaxy and the
angular scale over which we measure our regional SEDs (panel e, Figure 5). In
particular, the galaxies for which we are able to measure 500 pc averages tend
to be the closest galaxies in our samples, and are thus skewed towards lower
stellar masses. However, we find that this effect does not impact the results
presented here, as the majority of galaxies affected are more massive than the
UDGs in this sample. In particular, there are five galaxies within the stellar
mass range of the UDGs that are affected. Four are themselves UDGs (AGC
227965, AGC 322019, AGC 198543, and AGC 238961), and one is a NSA dwarf (NSA
ID 17750).

###  5.2 The H I Properties of the UDG Sample

In addition to their star formation efficiency, it is of interest to
understand the relationship between the stellar structure of these H I-rich
UDGs and their integrated gas properites. In Figure 11, we show the mass-size
relation of the UDG and NSA samples for two definitions of “mass” and “size”
(i.e. four total combinations). The left column shows sizes measured as g-band
half-light radii, while the right column shows sizes measured as half-mass
radii, where we refer to the radius that contains half the stellar mass of the
galaxy as the half-mass radius. The half-light, half-mass, and half-SFR radii
reported in this work are measured over the same area of the galaxy as our
derived stellar mass and SFR maps (i.e. only in pixels where the stellar mass
surface density is nonzero). We use the PSF-matched HSC images with the same
machinery as the half-mass and half-SFR radius measurement. The half-light
radii reported in this work are thus not completely analogous to effective
radii measured using other methods (e.g. Sersic fitting). In particular,
though there is no systematic offset between the two methods
($\bar{\Delta}R/R=-0.05$ for the full sample, and -0.06 (-0.04) for the NSA
(UDG) subset), there is significant scatter ($\Sigma_{\Delta R/R}=0.15$ for
the full sample, and 0.13 (0.19) for the NSA (UDG) subset). We estimate the
uncertainty in our half-mass and half-light radius measurements by adding in
quadrature the physical size of the PSF FWHM and the standard deviation of the
radii due to the uncertainty in the stellar mass fits/photometry,
respectively. The top row shows H I mass, while the bottom shows stellar mass.
In essence, one can think of the bottom left panel as the “canonical” mass-
size relation (stellar mass versus light-weighted size).

When one considers the size of the UDGs as a function of their integrated
stellar mass, it is immediately apparent that they are systematically larger
than the NSA sample in both their light-weighted and mass-weighted sizes. This
is unsurprising, of course, given that UDGs are selected for their low surface
brightnesses and large physical sizes. When one considers the UDGs as a
function of their integrated H I mass, however, the picture changes. In
particular, we see no significant difference in the relation between the H I
masses and half-mass (stellar) radii of the UDG and NSA samples. There is,
however, some evidence that the half-light radii of the UDGs remain elevated
with respect to the NSA sample at fixed H I mass.

###  5.3 The Color-Mass Relation of UDGs

Though not the main focus of this work, we are able to directly compare the
color-mass relation of gas-rich UDGs and “normal” dwarfs as a byproduct of the
SPS fitting presented above. Standard color-mass relations (e.g. Bell et al.,
2003; Into & Portinari, 2013) have been assumed in the literature for color-
based estimates of UDG stellar mass (cf Mancera Piña et al., 2019a;
Karunakaran et al., 2020; Marleau et al., 2021), but it has not been directly
shown that these optical color relations are applicable to the UDG population.
In particular, the canonical color-mass relations are typically calibrated for
and tested on largely massive galaxies; it is thus of practical importance for
both the UDG community and general dwarf galaxy community to test the
applicability of these relations on the extreme pockets of the dwarf
population (e.g. UDGs, Blue Compact Dwarfs, etc.).

In Figure 12 we show the global (i.e. light-weighted) g-band mass-to-light
relations of the dwarfs in this work (as usual, red points indicate the UDG
sample while purple points indicate the spectroscopic dwarf sample). We
calculate stellar mass-to-light ratios by computing the $\rm g_{HSC}$
luminosity and stellar mass from the global optical SED of each galaxy (panel
b in Figure 5). The $\rm g_{HSC}-r_{HSC}$ colors are measured using the same
SED used to derive the stellar mass-to-light ratio.

We find that there is no offset between the H I-selected UDG sample and the
NSA dwarf sample, and that the samples are well-described by a relation that
lies between the Into & Portinari (2013) and Bell et al. (2003) color-mass
relations. For completeness, we list our ordinary least squares parameters and
standard errors here, where we define
$\eta\equiv(M_{\star}/L_{g})(L_{\odot}/M_{\odot})$. We avoid the standard
choice of $\Upsilon$ as it is reserved for a parameter linking star formation
and midplane pressure in Paper II.

$\begin{split}\log_{10}\eta_{\rm all}&=1.59\pm 0.14(g-r)-0.65\pm 0.04\\\
\log_{10}\eta_{\rm UDG}&=1.65\pm 0.21(g-r)-0.66\pm 0.06\\\ \log_{10}\eta_{\rm
NSA}&=1.55\pm 0.20(g-r)-0.64\pm 0.06,\end{split}$ (9)

the exact calibration of this relation is, of course, dependent on both the
photometric bands used and IMF assumptions. We in particular note that Into &
Portinari (2013) and Bell et al. (2003) are calibrated for SDSS g and r bands,
which are similar but not identical to HSC g and r bands). Regardless, we have
demonstrated that the UDGs do not lie systematically off of the color-mass
relation of the NSA dwarfs, and that both samples are in relatively good
agreement with previous relations from the literature.

##  6 Discussion

Taken altogether, we find that our sample of H I-rich UDGs are characterized
by low star formation efficiencies (where SFE(H I) $\equiv\Sigma_{\rm
SFR}/\Sigma_{\rm HI}$) down to sub-kpc scales, and that the UDGs are generally
incapable of supporting the vigorous star formation seen in the “normal” NSA
dwarfs. Intriguingly, although the UDGs are large in light-weighted size for
their total stellar mass, their stellar mass-weighted sizes are typical for
their H I masses. Finally, the stellar mass-to-light ratios of the UDGs appear
consistent with those of the NSA dwarf sample.

###  6.1 Implications for UDG Formation

The results presented in this work suggest that the H I-rich Ultra-Diffuse
Galaxies are unable to efficiently convert gas to stars or host vigorous star
formation. Such low star formation rate surface densities indicate that UDGs
will continue to trail behind the stellar mass build-up of the general dwarf
population. This finding, however, does not alone answer the question of why
certain galaxies fail to trigger vigorous star formation. Several mechanisms
have been put forth in the literature to explain the population of field UDGs.
For each formation model, we will first discuss the points in which our
observations agree with the model predictions before turning towards points of
potential conflict.

Feedback-driven expansion. We first consider the picture in which field UDGs
are formed via starburst-driven outflows which cause potential fluctuations
that in turn lead to net expansion of the stellar population (El-Badry et al.,
2016; Di Cintio et al., 2017; Chan et al., 2018). We note to the reader that
in this section we deal with results from both the Feedback In Realistic
Environments (FIRE, Hopkins et al. 2014, and FIRE-2, Hopkins et al. 2018) and
Numerical Investigation of a Hundred Astrophysical Objects (NIHAO, Wang et al.
2015) simulations; though both simulations point to feedback-driven expansion
as the culprit of UDG formation, the feedback implementation between the two
simulations is not identical.

In the NIHAO simulation, such UDGs are expected to retain systematically
higher H I fractions with respect to “normal” dwarfs, in good agreement with
our results (Di Cintio et al., 2017). However, two points of potential
conflict emerge. First, the radial expansion of the UDG stellar population
should enforce a radial age gradient; that is, the older stars should
experience more expansion due to having gone through a larger number of
inflow/outflow cycles. This picture suggest a relatively centrally
concentrated distribution of star formation in UDGs, where the stars
experience net expansion while the gas goes through cyclical phases of inflow
and outflow. Such behavior is in qualitative conflict with the observation
that UDG star formation is both consistently lower than and more radially
extended than star formation in the NSA dwarfs (see lower panels of Figure 9).
However, to our knowledge a detailed study of the star formation extendedness
and efficiency has not been done for UDGs created by feedback-driven
expansion. Further analysis is necessary to confirm that feedback-driven
expansion cannot reproduce this effect (via, e.g. the substantial expansion of
stars within 100 Myr of formation).

However, another point of conflict exists between our observations and the
feedback-driven expansion picture. Namely, in both FIRE and NIHAO the marked
expansion of the UDGs is driven by particularly prolonged and bursty star
formation histories (Di Cintio et al., 2017; Chan et al., 2018). In this work,
we have shown that the lack of vigorous star formation in UDGs is linked to
their low stellar mass surface densities; that is, their low SFR surface
densities are explainable by their stellar and gaseous structure. Conversely,
then, the structure of the UDGs makes it difficult to envision a scenario in
which they are able to power frequent and vigorous bursts of star formation at
late times. This does not, of course, rule out UDG formation via star
formation feedback if the star formation behavior of UDGs was markedly
different at high redshift than it is in nearby UDGs. Indeed, such a
transition has been predicted in models such as Trujillo-Gomez et al. (2022).
However, these new observations provide new constraints on models which
predict star formation feedback to remain vigorous down to $z=0$.

Early mergers. Another proposed mechanism for field UDG formation is the
result of an early merger (more than 8 Gyr ago) that caused an increase in
total spin as well as a decrease in central SFR and a redistribution of star
formation to large radii (Wright et al., 2021). Like our observed galaxies,
the galaxies in the ROMULUS25 simulation are H I-rich for their stellar mass
and lie on the integrated star-forming main sequence. However, we do not find
the lower central specific star formation that Wright et al. (2021) cite as a
significant observable manifestation of this formation channel; indeed, the
distribution of sSFR of the UDGs in our sample is not statistically different
than that of the NSA sample at any radius.

High-spin halos. Finally, we consider the proposal that UDGs are formed out of
the high spin tail of the halo distribution (see, e.g. Dalcanton et al., 1997;
Amorisco & Loeb, 2016; Rong et al., 2017; Di Cintio et al., 2019; Liao et al.,
2019). This formation scenario predicts that UDGs will host ordered H I disks,
in line with resolved measurements from the literature (Mancera Piña et al.,
2019b, 2020; Gault et al., 2021). However, this formation scenario also
predicts that, at fixed stellar mass, UDGs at $z=0$ should have generally
lower central volumetric gas densities and more radially extended profiles
than high surface brightness galaxies (Liao et al., 2019). This is seemingly
at odds with our results, though a comparison to H I surface density (rather
than total gas density) is necessary to quantitatively demonstrate such a
conflict. However, we note that neither the early merger scenario nor the
high-spin halo scenario are in conflict with the most salient point of our
analysis – that UDGs, once set on a path towards low-efficiency star
formation, are likely to continue to fall behind the stellar mass build-up of
the general dwarf population.

###  6.2 The Stellar Structure of Dwarfs at fixed $M_{\rm HI}$

In addition to a comparison with established theoretical models, it is
informative to consider what fully empirical statement we may make about the
relationship between the H I-rich UDGs and “normal” NSA dwarfs. In the
preceding sections, we have demonstrated that the stellar half-mass radii of
the UDGs are typical for their H I masses (see Figure 11) – that is, the UDGs
are not offset from the mass-size plane for a particular casting of mass (H I
mass) and size (stellar half-mass radius). This is contrasted, naturally, to
the offset in the normal mass-size plane (stellar mass and half light radii)
by which UDGs are definitionally characterized.

A concordance in the H I-mass-stellar-size relation would naturally arise if
the UDGs were forming stars normally with respect to the spatial distribution
of gas, but at decreased star forming efficiency. To illustrate this idea, let
us presume that the spatial distribution of H I in UDGs is normal with respect
to high surface brightness galaxies, as has been shown for a number of the
UDGs in our sample by Gault et al. (2021). As hinted to in the introduction,
the driving mechanism behind the low star formation efficiency of the UDGs
will be further explored in Paper II.

In this picture, a galaxy that will become a UDG will form stars over the same
spatial distribution as a galaxy that will become a high surface brightness
galaxy, but at lower $\Sigma_{\rm SFR}$ – thus becoming a galaxy with a lower
total stellar mass. The concordance of the UDG and NSA samples in the H I
mass-stellar size plane is then a natural consequence of the comparatively
lower star formation efficiency of the UDGs. Thus, one mechanism by which to
generate the “normal” sizes of the UDGs with respect to their H I masses is
for present-day UDGs to have assembled most of their stellar mass via the low
SFE(H I) star formation that characterizes their behavior at $z=0$.

##  7 Conclusions

Due to their large sizes and low stellar mass surface densities, UDGs are both
an extreme outcome of galaxy formation and an extreme environment for star
formation. H I-selected UDGs are systematically more gas-rich than their
“normal” dwarf analogs (see Figure 2), and there is evidence that some UDGs
have assembled the majority of their stars from extremely dense clumps
(Danieli et al., 2021). Understanding the way in which star formation proceeds
in UDGs is thus of key interest to both the galaxy evolution and star
formation communities.

In this work, we have presented a method that allows us to compute star
formation rate and stellar mass surface density maps over galaxies with known
distances using spatially resolved optical data and global UV photometry. We
use this method to explore the star formation activity in a sample of nearby
(d $<120$ Mpc) H I-detected ultra-diffuse galaxies dwarf galaxies from
Janowiecki et al. (2019), along with a NASA Sloan Atlas reference sample of
“normal” dwarfs with H I measurements from Bradford et al. (2015). The samples
in this work allow us to compare and contrast the star formation behavior of
the UDGs against the NSA dwarfs, providing new clues to the evolutionary
pathway of these H I-rich UDGs as well as new tests for star formation theory
in extreme (low density) environments that we will explore in Paper II.

We find that the globally averaged SFR surface densities of the UDGs are low
compared to their globally averaged H I surface densities (see Figure 3); this
offset could be explained either by a truly lower SFE(H I) in the UDGs, or by
a difference in the relationship between the distribution of the UDGs’ stellar
mass and SFR. Our results indicate that the SFE(H I) of the UDGs remains lower
than the NSA dwarfs down to 500 pc scales (see Figure 10 and Section 5.1) –
the UDGs are indeed less efficient at converting atomic gas (to molecular gas)
to stars. In the second paper of this series, Kado-Fong et al. (2022), we will
consider the role of the galactic structure of the UDGs on their star
formation. Indeed, we find that the diffuse structure of the UDGs can
naturally explain their low star formation rate surface densities and
efficiencies. When the structure of the UDGs is considered (rather than the
gas content alone), we find that their star formation is quantitatively
consistent with star formation in both the NSA sample and in more massive
galaxies. As such, we direct the interested reader to Paper II of this paper
series.

Returning to the results at immediate hand, we also consider the relationship
between the stellar size and H I mass of the UDGs, as compared to the NSA
galaxies (Section 5.2). We find that although the UDGs have large stellar
sizes (both light-weighted and mass-weighted) for their total stellar mass,
their stellar sizes are largely unremarkable as a function of their H I mass.
This superficially surprisingly result can be explained naturally when we
consider the previous conclusions of this work: the UDGs have fairly normal H
I structure and properties, and so the spatial distribution of their star
formation is set by those normal gas properties. However, because they form
stars at a significantly lower SFE(H I), they form fewer stars overall and
thus become large for their total stellar mass.

The low efficiency and low star formation rate surface densities of the UDGs
indicate that the UDGs will continue to lag behind the general dwarf
population in their stellar mass build-up, and thus remain diffuse relative to
“normal” dwarfs. However, this does not answer the question of why present-day
UDGs first entered the evolutionary path to become UDGs at $z=0$. While this
work alone cannot provide a definitive answer to this question, we may
consider the implications of our results on contemporary ideas of UDG
formation. The spatially extended star formation of the UDGs and apparent
inability to power vigorous star formation (high SFR surface densities) are in
conflict with the idea of UDG formation via the feedback-driven radial
expansion of stars for the picture in which feedback-driven expansion
continues even when stellar mass surface density is low. Formation mechanisms
which set certain halos down the path towards UDG formation at early times
(e.g. the high spin halo scenario or early merger scenario) can more naturally
explain the low SFE(H I) cycle seen in the sample of UDGs presented in this
work.

A natural extension of this work will be to obtain a larger number of
spatially resolved H I measurements of the UDGs in this sample, with a
particular focus on the most nearby galaxies in the sample (such that we may
maximize the physical spatial resolution). Higher resolution UV-optical data
will also allow us to directly probe the physical scales relevant to star
formation and cluster populations in these galaxies. A further exploration
into the star formation and ISM of these UDGs is crucial to further our
understanding of UDGs as an extreme environment for star formation, as well as
our understanding of how these extreme objects form.

The authors thank Eve Ostriker, Chang-Goo Kim, and Lachlan Lancaster for
insightful comments and discussion that have greatly improved this manuscript.
We also thank the anonymous referee for their review of this work, which
improved the quality of the manuscript. The research of EKF was supported by
the Porter Ogden Jacobus Fellowship. JEG gratefully acknowledges support from
NSF grant AST-2106730. Based in part on data collected at the Subaru Telescope
and retrieved from the HSC data archive system, which is operated by Subaru
Telescope and Astronomy Data Center at National Astronomical Observatory of
Japan. The Hyper Suprime-Cam (HSC) collaboration includes the astronomical
communities of Japan and Taiwan, and Princeton University. The HSC
instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
Funding was contributed by the FIRST program from Japanese Cabinet Office, the
Ministry of Education, Culture, Sports, Science and Technology (MEXT), the
Japan Society for the Promotion of Science (JSPS), Japan Science and
Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK,
ASIAA, and Princeton University. This paper makes use of software developed
for the Large Synoptic Survey Telescope. We thank the LSST Project for making
their code available as free software at http://dm.lsst.org The Pan-STARRS1
Surveys (PS1) have been made possible through contributions of the Institute
for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the
Max-Planck Society and its participating institutes, the Max Planck Institute
for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial
Physics, Garching, The Johns Hopkins University, Durham University, the
University of Edinburgh, Queen’s University Belfast, the Harvard-Smithsonian
Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network
Incorporated, the National Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space Administration under
Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA
Science Mission Directorate, the National Science Foundation under Grant No.
AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE)
and the Los Alamos National Laboratory. Some of the data presented in this
paper were obtained from the Mikulski Archive for Space Telescopes (MAST).
STScI is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST
data is provided by the NASA Office of Space Science via grant NNX13AC07G and
by other grants and contracts. Based on observations made with the NASA Galaxy
Evolution Explorer. GALEX is operated for NASA by the California Institute of
Technology under NASA contract NAS5-98034. The specific observations analyzed
can be accessed via https://doi.org/10.17909/h19r-ab55 (catalog
https://doi.org/10.17909/h19r-ab55). This research has made use of the
NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration. This research has made use of
the VizieR catalogue access tool, CDS, Strasbourg, France. , pandas (McKinney,
2010, 2011), Astroquery (Ginsburg et al., 2019), extinction (Barbary, 2021)

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##  Appendix A Fitting Gallery

Here we include an abbreviated gallery of fitting results for the NSA and UDG
samples. We choose three galaxies from each sample with their median star
formation rate surface density and median stellar mass surface density maps;
these galaxies are chosen to roughly span the morphologies seen in the
samples.

Figure 13:  A sample of three example galaxies from the NSA sample, as
labeled. From left, we show the HSC-SSP gri-composite RGB image, the median
star formation rate surface density map, and the median stellar mass surface
density map. These galaxies are chosen to span the range of morphologies
present in the sample.  Figure 14:  A sample of three example galaxies from
the UDG sample, as labeled. From left, we show the HSC-SSP gri-composite RGB
image, the median star formation rate surface density map, and the median
stellar mass surface density map. These galaxies are chosen to span the range
of morphologies present in the sample.

##  Appendix B Additional Validation of the SED Fitting

Due to the novelty of our joint-fitting approach, we describe three additional
validation steps here. First, we show a typical cornerplot for one of the
jointly fitted SEDs (the example shows NSA7167) in Figure 17. We find that the
inferred parameters are strongly peaked and unimodal, though there is a
degeneracy between the age and amplitude (stellar mass) inferred for each
cluster SED. This is a physically expected effect since an increase in the age
leads to a drop in the mass-to-light ratio of the stellar population, meaning
that more stellar mass is necessary to achieve the same luminosity. We find no
significant degeneracy between parameters inferred for different cluster SEDs.

Next, we show in Figure 16 a comparison between the SED fitting results using
the global optical photometry and those derived using the cluster SEDs. We
find no evidence for a systematic shift between the global and joint fitting
methods. Indeed, the total stellar mass and star formation rate inferred via
the joint fit method is in good agreement with those estimated via the global
fit method, as shown by the text in each panel.

Finally, as stated in the main text, we assume a fixed metallicity in the SED
fitting presented in this paper. There is some evidence that low surface
brightness galaxies may have higher than average gas-phase metallicities
(Greco et al., 2018a; Sánchez Almeida et al., 2018), though we caution that
these results contain only three low surface brightness galaxies, and that
these low surface brightness galaxies are an optically-selected sample (as
opposed to the H I-selected UDGs in the sample discussed in this work). We
find that assuming solar metallicity models shifts the total star formation
rate and stellar mass estimates by less than 0.1 dex compared to the fiducial
estimates. The stellar masses estimated using solar metallicity models are
slightly lower (with a median difference between solar metallicity and
fiducial values of -0.08 dex), and the star formation rates are slightly
higher (with a median difference of 0.09 dex). This shift is expected – the
higher metallicity models are redder at fixed age, meaning that the best-fit
stellar populations will be on average slightly younger at fixed color.

Figure 15:  A cornerplot of the parameter inference for NSA7167. As expected,
although the 1D distribution of the parameters are strongly peaked and
unimodal, there is some degeneracy in the tails of the distribution between
the age and amplitude of each component. This can be understood naturally by
considering that as the age of a component increases, the mass-to-light ratio
drops, meaning that the amplitude of that component (i.e. the stellar mass)
must increase to account for the same luminosity.  Figure 16:  A comparison of
the global SED fits (left column) and joint SED fits (right column) for three
galaxies in our sample. In each panel, the red points show the observed fluxes
in FUV, NUV, gHSC, rHSC, iHSC, zHSC, and yHSC. Each black curve and set of
orange points show the spectrum and synthetic photometry, respectively, of one
draw from the posterior. The text shows the median stellar mass and SFR
estimates for each fit; uncertainties correspond to the 16th and 84th
percentiles of the posterior.  Figure 17:  A comparison between the total
stellar mass (left) and star formation rate (right) estimates derived from our
spatially resolved SED fitting if we assume a metallicity of $Z=0.012$ (as
opposed to our fiducial value of $Z=0.004$). In both panels, the grey filled
histogram shows the distribution over all galaxies – the unfilled red and
purple histograms show the distribution over the UDG and NSA samples,
respectively.