# Ultra-Diffuse Galaxies as Extreme Star-forming Environments II:
Star Formation and Pressure Balance in H I-Rich UDGs

Erin Kado-Fong Department of Astrophysical Sciences, Princeton
University,Princeton, NJ 08544, USA  Chang-Goo Kim Department of Astrophysical
Sciences, Princeton University,Princeton, NJ 08544, USA  Jenny E. Greene
Department of Astrophysical Sciences, Princeton University,Princeton, NJ
08544, USA  Lachlan Lancaster Department of Astrophysical Sciences, Princeton
University,Princeton, NJ 08544, USA Erin Kado-Fong<EMAIL_ADDRESS>
###### Abstract

In addition to occupying the extreme, diffuse tail of the dwarf galaxy
population, Ultra-Diffuse Galaxies (UDGs) are themselves a key laboratory in
which to study star formation in extreme low-density environments. In the
second paper of this series, we compare the spatially resolved star formation
activity of 22 H I-selected UDGs and 21 “normal” dwarf galaxies within 120 Mpc
to predictions within the pressure-regulated, feedback-modulated (PRFM) theory
of star formation. To do so, we employ a joint SED fitting method that allows
us to estimate star formation rate and stellar mass surface density from UV-
optical imaging. We find that the PRFM framework extends successfully to the
UDG regime – although the UDGs in our sample show unusually low star formation
rate surface densities given their H I content, this low star formation
efficiency can be naturally explained by the diffuse structure of the UDGs. In
fact, when cast in the PRFM framework, the relationship between midplane
pressure and star formation in the UDG sample is in good agreement not only
with the “normal” dwarf reference sample, but also with measurements from more
massive galaxies. Our results suggest that despite their low star formation
efficiencies, the H I-rich UDGs need not be forming stars in an exotic manner.
We also find that the UDGs are likely ${\rm H_{2}}$-poor compared even to the
overall dwarf population.

††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

Star formation and galaxy evolution are intrinsically linked processes; a full
understanding of one is not possible in the absence of a theory of the other.
In addition to the most definitional link – that star formation grows stellar
mass while consuming gas – the resulting feedback from star formation also has
a direct impact on the structure of the galaxy’s interstellar medium (ISM) via
processes including supernovae, UV radiation, and stellar winds (see, e.g. Kim
et al., 2013, 2017; Girichidis et al., 2018; Kannan et al., 2019; Kim et al.,
2021; Lancaster et al., 2021). In order to understand how and why galaxies
evolve the way that they do – especially at the low-mass end, where star
formation feedback is expected to play an outsized role (see, e.g. Silk, 1997;
Dekel & Silk, 1986; El-Badry et al., 2016; Behroozi et al., 2019; Hu, 2019;
Dashyan & Dubois, 2020; Smith et al., 2020) – we must understand the self-
regulatory process of star formation. Likewise, to understand the environments
in which star formation proceeds, we must understand the arc of galaxy
evolution through cosmic time.

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
NASA Sloan Atlas (NSA, our “normal” dwarf comparison sample – see text)
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).

In both the fields of star formation and galaxy evolution, we seek out
“extreme” cases in order to best test and stretch our understanding of the
underlying physical processes at work. Ultra-diffuse galaxies (UDGs) are dwarf
galaxies characterized by large stellar sizes and low surface brightnesses.
Though exact definitions vary, UDGs are typically required to have effective
radii greater than 1.5 kpc and central surface brightnesses fainter than 24
mag arcsec-2 (see Van Nest et al. 2022 for an overview of UDG definitions).
These diffuse galaxies are extreme as both a product of galaxy evolution and
as an environment in which stars form. Indeed, the study of UDGs as an extreme
sector of the dwarf galaxy population has enjoyed both extensive observational
(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; Greco et al., 2018b, a; 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; Greene et al., 2022) and
theoretical (e.g. Amorisco & Loeb, 2016; Di Cintio et al., 2017; Chan et al.,
2018; Jiang et al., 2019; Liao et al., 2019; Wright et al., 2021; Van Nest et
al., 2022) study. However, less work has been devoted to their star formation
properties.

UDGs are extreme environments for star formation due to their presumably low
stellar mass surface densities and likely shallow potential wells (for UDGs in
the field, see, for example, Leisman et al. (2017) and Kong et al. (2022) –
typical halo masses for UDGs in clusters are a topic of significant debate,
see, e.g., van Dokkum et al. 2015; Beasley & Trujillo 2016a; van Dokkum et al.
2019; Sales et al. 2020; Saifollahi et al. 2021). These conditions are a
marked departure from Solar Neighborhood-like or even outer (Milky Way) disk-
like conditions that serve as the fiducial environment for many models of star
formation (Krumholz et al., 2009; Ostriker et al., 2010). Mapping out the star
formation activity in UDGs thus serves as a test of the extensibility of these
star formation models to new and extreme diffuse conditions.

In the first paper of this series (Kado-Fong et al., 2022, hereafter referred
to as Paper I) we demonstrated that UDGs form stars inefficiently relative to
their H I surface densities (where SFE(H I)$\equiv\Sigma_{\rm SFR}/\Sigma_{\rm
HI}$) on scales down to 500 pc. However, a study of the atomic gas alone is
insufficient to understand whether this low SFE(H I) is unexpected given the
diffuse structure of the UDGs. In Paper I we demonstrated that the star
formation in UDGs is different from normal dwarfs, but in this paper we will
consider whether that difference in star formation can be explained by our
understanding of the physical processes that drive and modulate star
formation.

To this end, we turn to the pressure-regulated, feedback-modulated model of
star formation that has been developed over a series of works (Ostriker et
al., 2010; Kim et al., 2011; Ostriker & Shetty, 2011; Kim et al., 2013; Kim &
Ostriker, 2015) and most recently condensed into Ostriker & Kim (2022). This
theory establishes a link between star formation rate surface density and
weight ($\mathcal{W}$) by considering the role of star formation feedback in
maintaining the structure and energy density (pressure) of the ISM. This
physical relationship provides us with a method that links the two notable
features of the UDGs in this work – their unusual structure and their low star
formation rate surface density. With this framework in hand, we will endeavour
to explain the surprising – or perhaps expected – nature of the star formation
in H I-rich UDGs.

We will structure the paper as follows, and note that readers familiar with
either Paper I or PRFM star formation may feel free to skip certain sections.
In Section 2 we will summarize the data and methods that we utilized in Paper
I to produce the initial data products used in this analysis. We will then
give an abbreviated overview of PRFM star formation theory in Section 3,
before moving onto a discussion of its application to the dwarfs at hand in
Section 4. Finally, we will discuss the implications of the analysis on our
understanding of the structure and star formation of UDGs in Section 5

Figure 2:  A schematic layout of the fitting technique used in this work, as
introduced in Paper I. At left, we show a schematic representation of the
optical data and UV data in panels a and c, respectively. In this analysis,
the UV data are at much lower spatial resolution than the optical data. We
then illustrate the global SED fit of the optical and UV data in b and d; the
apertures here are held fixed over the UV and optical data. We then divide the
optical data into regions in which regional SEDs are measured (panel e). 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 of panel d.

##  2 Datasets

Readers familiar with Paper I of this series will find that this content has
been covered in greater detail in Paper I, and may skip to Section 3. We
provide the most salient points of our sample here for those readers not
familiar with the first paper of this series, but encourage those readers with
an interest in the methods to refer to Paper I for a more in-depth discussion
of the analysis therein.

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 NSA sample is
drawn from the NASA Sloan-Atlas, a catalog of nearby galaxies re-analyzed from
the Sloan Digital Sky Survey DR8 (Aihara et al., 2011; Blanton et al., 2011).
The UDG sample was 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

Let us first define the UDG sample. We draw our galaxies from the H I-selected
catalog of Janowiecki et al. (2019), selecting those which have 5-band imaging
from the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP, Aihara et al.,
2019, 2022). Drawing from an H I-selected sample, at least in the regime of
the low surface brightness UDGs, comes with two key advantages for the study
of star formation in low-density conditions. First, the galaxies selected via
H I observations tend to be relatively isolated, which is quite key given the
large influence that environment – and particularly the presence of massive
galaxies – plays on star formation in low-mass galaxies (see, e.g. Geha et
al., 2012; Carlsten et al., 2022).  We find that one UDG in our sample,
AGC227965, is quenched (SFR$<10^{-3}$ M⊙ yr-1) presumably 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 star
formation models does not apply to this system due to the ongoing interaction.
Secondly, and perhaps most importantly for the low surface brightness regime,
an H I-selected sample has redshift measurements from the 21 cm line, which
allows us to determine distances to relatively isolated UDGs. This selection
results in a total sample of 21 H I-selected UDGs.

In order to make a fair assessment of the star formation activity of the UDGs,
we also draw a reference sample of 32 NASA-Sloan Atlas (NSA) dwarfs at $d<120$
Mpc with H I measurements from Bradford et al. (2015) to act as a “normal”
dwarf reference sample. For this work, we consider the 21 NSA dwarfs with
stellar masses no greater than $\rm M_{\star}=3\times 10^{8}\ M_{\odot}$ (the
maximum stellar mass covered by the UDG sample). This results in a median
stellar mass of $\langle\rm\log_{10}(M_{\star}/M_{\odot})\rangle_{50}=8.0$ for
the NSA sample and $\langle\rm\log_{10}(M_{\star}/M_{\odot})\rangle_{50}=7.9$
for the UDG sample.

In Figure 1 we show the galaxies in our sample 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 obtained via the SED fitting method presented
in Paper I that we will summarize in Section 2.2. In this figure and all
subsequent figures, we show the NSA sample in purple and the UDG sample in
red. 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 find
that the UDGs have high H I masses for their stellar masses, and that they
have low SFRs for their H I masses.

Figure 3:  An example of the fitting process for one of the UDGs (AGC 334349).
Top row: from left, the HSC $gri$-composite image, the SED cluster map, 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 as blue points. Bottom row:
the resultant stellar mass (left) and SFR (right) maps.

###  2.2 SED Fitting

In order to understand the star formation of UDGs down to 500 pc scales, we
must devise a method in which to measure their star formation on the relevant
physical scales. In the absence of spectroscopic or narrow-band measurements –
both of which are hindered, but not rendered impossible, by the low surface
brightness of the UDGs – we turn instead towards a joint UV-optical SED
fitting method that combines the star formation information from GALEX FUV and
NUV with the spatial resolution of the Hyper Suprime-Cam Subaru Strategic
Program (HSC-SSP) optical imaging, which attains a median seeing of 0.77” in
the g band (Aihara et al., 2019). The GALEX imaging is over a factor of 6
lower in spatial resolution, with a FUV and NUV PSF FWHM of 4.2” and 4.9”,
respectively. Convolving the HSC optical imaging to GALEX seeing would clearly
constitute an unreasonable degradation of the optical imaging. We have thus
developed an SED fitting process which jointly models global UV and spatially
resolved optical photometry.

A full description of the SED fitting process and validation can be found in
Paper I and we direct the interested reader to this work. We also present a
schematic layout of the procedure in Figure 2.

First, let us review the typical assumptions and parameters one must choose
when approaching an SED fitting problem. We use a Kroupa initial mass function
(IMF) throughout (Kroupa, 2001) with the Flexible Stellar Population Synthesis
(FSPS) library (Conroy et al., 2009; Conroy & Gunn, 2010). For this work, we
hold stellar and gas-phase metallicity fixed at $Z=0.004$ (approximately one-
third solar). This choice is motivated by both technical and scientific
concerns: first, due to the joint nature of the SED fitting, we would like to
minimize the number of parameters fit for each cluster SED (panel f, Figure
2). Second, the mass-metallicity relation is not well-understood for these
dwarf samples. Direct ($T_{e}$) measurements are limited to nearby ($d\lesssim
20$ Mpc) samples of typically “normal” dwarfs (Lee et al., 2006; Berg et al.,
2012; Jimmy et al., 2015); these direct measurements can show significant
offsets relative to each other and to SED fitting methods used for more
distant 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 find in Paper I that our results
are not significantly impacted by our assumption of a fixed metallicity;
assuming solar metallicity shifts the estimated stellar mass and SFR of our
dwarf samples by $\sim 0.08$ dex on average. The star formation rate surface
densities and stellar mass surface densities estimated using solar metallicity
models are shifted on by less than 0.01 dex on average compared to their
fiducial values.

We also adopt an exponentially declining star formation history

${\rm SFR}(t)\propto e^{-(t-t_{0})/\tau}$ (1)

for all of the models in this work.

Our SED fitting method uses the emcee implementation of the Markov Chain Monte
Carlo method for parameter estimation (Foreman-Mackey et al., 2013). We begin
with a joint fit to the UV and optical global photometry; this allows us to
estimate the reddening in the galaxy, and provides a reasonable point at which
to instantiate walkers in the joint fitting step. We then subdivide the galaxy
into regions no smaller than twice the full width at half-maximum (FWHM) of
the PSF in our PSF-matched optical images. These regions are required to have
a median signal-to-noise ratio of 3 across the five optical bands. Then, we
use K-means clustering in the space of optical photometry to assign these
regions to clusters that have similar SEDs (Figure 3, top middle panel) – this
reduces the number of individual SEDs that we must fit in the joint modeling
step. Finally, we fit the global UV data jointly with the cluster optical SEDs
(Figure 3, middle row): that is, we adopt the likelihood:

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

where $i$ refers to the bandpass index and $j$ refers to the cluster index.
This allows us to jointly fit the spatially resolved optical photometry and
the global, unresolved UV photometry. From this inference, we can then
immediately extricate the stellar mass surface density, given the amplitude of
the cluster SEDs, as well as the star formation rate surface density, which we
estimate by integrating the inferred star formation history over the past 100
Myr (Figure 3, bottom row). We choose 100 Myr given that due to our UV
coverage, we should be most sensitive to star formation averaged over this
timescale.

We finally compute the average stellar mass density and star formation rate
surface density over 500 pc, 1 kpc, and global scales. In this step, we
require that the averaging scale is larger than the scale of the regions in
which we measure the region SEDs (panel e of Figure 2). In practice, this
affects only five galaxies: four are UDGs (AGC 227965, AGC 322019, AGC 198543,
and AGC 238961), and one is a NSA dwarf (NSA ID 17750). The results of these
galaxies are thus shown on 1 kpc and global scales, but not on 500 pc scales –
their exclusion at 500 pc does not constitute a statistically significant
impact on our analysis.

##  3 Background on PRFM theory

We will summarize the basic points of pressure-regulated, feedback-modulated
(PRFM) star formation, but direct the reader to Ostriker & Kim (2022) for an
in-depth discussion of the theory.

For a galaxy disk in a quasi-steady state or in vertical dynamical
equilibrium, the weight (per unit area) of the ISM should be balanced by the
pressure difference between the midplane and the top of the gas layer. Since
the pressure is generally decreasing rapidly along the z-axis (perpendicular
to the disk plane), the midplane pressure must match the overlying weight. The
weight of the ISM, $\mathcal{W}$, is a sum of the contribution by the gas and
by the external components (stars, dark matter),

$\begin{split}\mathcal{W}&=\int_{0}^{z_{\rm max}}dz\ \rho(g_{\rm gas}+g_{\rm
ext})\\\ &=\mathcal{W}_{\rm gas}+\mathcal{W}_{\rm ext},\end{split}$ (3)

where $z_{\rm max}$ refers to the vertical confine of the gas disk, $g_{\rm
gas}$ and $g_{\rm ext}$ are the vertical components of the gravitational field
due to the gas and the external gravitational potential, and $\mathcal{W}_{\rm
gas}$ and $\mathcal{W}_{\rm ext}$ are the corresponding weight components.

Assuming slab geometry, it can be shown that the weight from the gas gravity
is $\mathcal{W}_{\rm gas}=(\pi G\Sigma_{\rm gas}^{2})/2$, where $\Sigma_{\rm
gas}$ is the total gas surface density. If the external gravity dominates, we
can approximate the weight as $\mathcal{W}_{\rm ext}=\Sigma_{\rm
gas}\sqrt{2G\rho_{\rm sd}}\sigma_{\rm eff}$, where $\rho_{\rm sd}$ is the
combined density of stars and dark matter and $\sigma_{\rm eff}$ is the
effective vertical velocity dispersion of the gas. This effective velocity
dispersion includes the turbulent, thermal, and magnetic contributions. To
estimate the weight from observables, we take a simplified form called the
dynamical equilibrium pressure $P_{\rm DE}$ that is obtained by combining the
two weight contributions as introduced in Ostriker & Kim 2022:

$\begin{split}\mathcal{W}&\approx P_{\rm DE}=\frac{\pi G\Sigma_{\rm
HI}^{2}}{2}+\Sigma_{\rm HI}\sqrt{2G\rho_{sd}}\sigma_{\rm eff}.\\\ \end{split}$
(4)

This approximation is good to within $\approx 20$% (see Ostriker & Kim 2022
for a more complete derivation of the full weight expression), and has been
adopted in the literature to estimate the total weight from observable
quantities (see, e.g. Sun et al., 2020). The reader will note that we have
made a notable change to the construction of $\mathcal{W}$. We do not have
molecular gas estimates for our samples, and thus take the base assumption
that these galaxies are H I-dominated and that the molecular gas is a minor
contributor to the total gas mass of the ISM. That is, we assume $\Sigma_{\rm
gas}\approx\Sigma_{\rm HI}$. We assume a $\sigma_{\rm eff}=10{\ \rm km\
s}^{-1}{}$, as in previous works (see, e.g. Sun et al., 2020). This assumption
is reasonable for the dwarf samples at hand as $\sigma_{\rm eff}$ is likely
dominated by the warm gas sound speed.

Having laid out an observational estimate of the weight (that is, the
dynamical equilibrium pressure), which is an estimate of the total midplane
pressure, we now want to make a connection between the total pressure and star
formation activity. Total pressure here is defined as the sum of contributions
from thermal pressure, turbulent pressure (vertical Reynolds stress), and
vertical Maxwell stress as $P_{\rm tot}=P_{\rm th}+P_{\rm turb}+\Pi_{\rm
mag}$. Each of these pressure terms is expected to be linked to star formation
activity: thermal pressure scales largely with the mean FUV field intensity,
turbulent pressure is sourced by momentum injection from supernovae, and
magnetic pressure – though relatively less well understood – is thought to
scale to some degree with turbulent kinetic pressure via galactic dynamo.

A great deal of work has been devoted to quantifying the relationship between
these pressure terms and star formation activity. The key parameter
quantifying this relation is the “feedback yield”, defined as

$\Upsilon_{\rm X}\equiv\frac{P_{\rm X}}{\Sigma_{\rm SFR}}$ (5)

where X refers to a given pressure component. In particular, Ostriker & Shetty
(2011) showed that the turbulent pressure is related to SFR surface density as
$P_{\rm turb}=p_{\star}/(4m_{\star})\Sigma_{\rm SFR}$, where $p_{\star}$ is
the spherical momentum injection per supernova and $m_{\star}$ is the total
mass of stars formed per star that will become a supernova. For a typical IMF
(e.g., Kroupa, 2001), $m_{\star}\approx 100M_{\odot}$. Recently, 3D supernova-
driven bubble expansion simulations have converged to a characteristic value
of $p_{\star}/m_{\star}\sim 1000-3000{\ \rm km\ s}^{-1}{}$, giving rise to
$\Upsilon_{\rm turb}\sim 250-750{\ \rm km\ s}^{-1}{}$. This value is
insensitive to the background density and metallicity (e.g. Kim & Ostriker,
2015; Kim et al., 2017; Martizzi et al., 2015; Iffrig & Hennebelle, 2015;
Fielding et al., 2018; Oku et al., 2022)

Turning our attention to the thermal pressure, it is established that $P_{\rm
th}$ should scale with the FUV field intensity as the photoelectric heating
(e.g., Bakes & Tielens, 1994; Weingartner & Draine, 2001) is the dominant
heating source, which is dependent on both $\Sigma_{\rm SFR}$ and attenuation
from the ISM. In particular, a scaling relation can be written for
$\Upsilon_{\rm th}$ with respect to solar neighborhood conditions as:

$\begin{split}\Upsilon_{\rm th}(f_{\tau},\Sigma_{\rm
HI},Z_{d}^{\prime})=&240{\ \rm km\ s}^{-1}{}\ \times\\\
&\frac{4.1f_{\tau}/f_{\tau,\odot}}{1+3.1\left(\frac{\Sigma_{\rm
HI}Z_{d}^{\prime}f_{\tau}/f_{\tau,\odot}}{10\ {\rm M_{\odot}}\ {\rm
pc}^{-2}}\right)^{0.4}},\end{split}$ (6)

where we assume a dust metallicity of $Z_{d}^{\prime}=Z/Z_{\odot}=1/3$, in
line with our SED fitting assumptions. We also introduce $f_{\tau}$, which is
the mean attenuation factor of the UV radiation field. Under the assumption of
slab geometry and uniform gas and source distribution, the radiation transfer
solution is

$f_{\tau}\equiv\frac{1-E_{2}(\tau_{\rm FUV}/2)}{\tau_{\rm FUV}},$ (7)

where $E_{2}(n)$ is the second exponential integral and $\tau_{\rm FUV}$ is
the mean FUV optical depth in the direction perpendicular to the disk. We
estimate $\tau_{\rm FUV}$ for our galaxies by computing $A_{\rm FUV}$ assuming
a Calzetti (2013) extinction curve and the $A_{V}$ inferred from our global
SED fits. Ostriker et al. (2010) showed that for solar neighborhood-like
conditions one arrives at $\Upsilon_{\rm th}\approx 240{\ \rm km\ s}^{-1}{}$
(see Equation 15 of Ostriker et al., 2010, or, equivalently, Equation 12 of
Ostriker & Kim 2022). We compute somewhat higher values of $\Upsilon_{\rm th}$
for the present samples due to low $\Sigma$ and low $Z^{\prime}$ than those of
solar neighborhood, with a median value of $\Upsilon_{\rm th}\sim 660{\ \rm
km\ s}^{-1}{}$ at 1 kpc scales.

It is generally expected that the saturation level of the magnetic stress is
set by a fraction of the turbulent stress (e.g., $\Upsilon_{\rm mag}\sim
0.5-1.0\Upsilon_{\rm turb}$; Kim & Ostriker 2015). However, the quantitative
prediction may still depend on the details of galactic dynamo. In this work,
we adopt a fiducial $\Upsilon_{\rm turb}=500{\ \rm km\ s}^{-1}{}$ and a
scaling of $\Upsilon_{\rm mag}=0.75\Upsilon_{\rm turb}$.

We thus arrive at a theoretical PRFM prediction of

$\Upsilon_{\rm tot}=\frac{P_{\rm tot}}{\Sigma_{\rm SFR}}\sim 875{\ \rm km\
s}^{-1}{}+\Upsilon_{\rm th}(f_{\tau},\Sigma_{\rm HI}).$ (8)

Numerical results from the TIGRESS111Three-phase Interstellar medium in
Galaxies Resolving Evolution with Star formation and Supernova feedback
framework (Kim & Ostriker, 2017) validate the theoretical assumptions of the
PRFM theory and calibrate $\Upsilon_{\rm tot}$ as a function of $P_{\rm DE}$
as (Ostriker & Kim 2022, Equation 25c):

$\log_{10}\left(\frac{\Upsilon_{\rm tot}}{{\ \rm km\
s}^{-1}{}}\right)=-0.212\log_{10}\left(\frac{P_{\rm DE}}{k_{B}\ {\rm cm}^{-3}\
K}\right)+3.86.$ (9)

Given a method to compute $\Upsilon_{\rm tot}$ (we will use the numerical
prescription of Equation 9, but will include a comparison to Equation 8 in
Section 4.5), the PRFM prediction for SFR surface density is

$\Sigma_{\rm SFR}=\frac{P_{\rm DE}}{\Upsilon_{\rm tot}},$ (10)

where the right hand is composed of our observable quantity ($P_{\rm DE}$) and
the theoretical/numerical prediction ($\Upsilon_{\rm tot}$).

From here, we will consider whether the markedly low SFE(H I) of our observed
UDG sample can be explained within the framework of PRFM star formation.

##  4 Results

Having provided the reader a brief introduction to PRFM theory, we may now
proceed to the application of this framework to the present sample. Before we
jump headlong into the computation, however, it is of substantial importance
to first justify that the PRFM theory of star formation is applicable to our
dwarf samples, and how we will go about estimating the parameters necessary to
predict star formation within the model framework.

###  4.1 The validity of PRFM assumptions in the present sample

A preliminary question that we must first address before applying the PRFM
model to the sample at hand is whether the equilibrium disk assumptions that
lie at the heart of the theory are satisfied in our low-mass systems. In this
work, we are averaging both spatially (over at least 500 pc) and temporally
(over approximately 100 Myr, given that GALEX FUV is included in the SED
fitting). The equilibrium assumption is valid if either the averaging
timescale or length scale is large enough to average over deviations from
equilibrium. In this section, we will provide an argument that our averaging
timescale is long enough to justify the equilibrium assumption, but we note
that the spatial averaging is also likely sufficient to validate the
equilibrium assumption. Indeed, it should be noted that spatial averaging
allows PRFM theory to be used for star formation tracers that track SFR on
shorter timescales than our time-averaging arguments may otherwise imply (e.g.
H$\alpha$, which traces star formation activity on $\sim\\!10$ Myr
timescales).

Before moving onto these timescale arguments, however, it is worth considering
whether these dwarfs are well-described by gaseous disks. There is substantial
evidence from H I rotation curves that dwarfs (including UDGs) in this mass
range do host H I disks (Hunter et al., 2012; Greco et al., 2018a; Mancera
Piña et al., 2019, 2020; Gault et al., 2021; Mancera Piña et al., 2022). There
are, to the authors’ knowledge, no published analyses of the stellar
kinematics of field UDGs, but there is evidence that UDGs in the field are
“puffy” but largely axisymmetric systems characterized by a mean disk
thickness of $\sim 0.5R_{\rm d}$ where $R_{\rm d}$ is the disk length (Kado-
Fong et al., 2021). We thus generally find long-lived disks for the galaxies
in our sample so that the slab geometry assumption of PRFM is reasonable for
the present work. We then must carefully consider the assumptions of thermal
and dynamical equilibrium due to proposals that dwarfs have generally bursty
star formation histories, and that UDGs in particular may be formed via
particularly bursty histories (Chan et al., 2018; Di Cintio et al., 2017).

The thermal equilibrium assumption of the PRFM model may be disrupted if the
cooling and heating time scales are so long that the majority gas is in out of
equilibrium state. In this case, the gas would not promptly respond to the
change in the heating rate and hence the star formation rate. We can estimate
the validity of this assumption by comparing the time-scale over which we
measure the star formation rate, to the cooling and heating time for these
dwarfs, which quantifies the time it takes to re-establish a thermal
equilibrium. In particular, let us write the cooling and heating time of warm
gas as:

$t_{\rm cool}=\frac{k_{B}T_{w}}{n\Lambda}=\frac{k_{B}T_{w}}{\Gamma},$ (11)

where $k_{B}$ is Boltzmann’s constant, $T_{w}\sim 5000-8000$ K is the
temperature of the warm gas, $n\Lambda$ and $\Gamma$ are the cooling and
heating rate per particle, respectively. With the $\Gamma\sim 5\times
10^{-27}-10^{-26}\,{\rm erg/s}$, appropriate for low-metallicity gas (Wolfire
et al., 2003), we have a cooling time of $t_{\rm cool}\sim 2-6$ Myr. This
cooling time is significantly shorter than the timescale over which we measure
our star-formation rates (100 Myr) – thus, the time scale over which our
measurement is averaging is larger than the cooling time by roughly an order
of magnitude. This indicates that even the smallest regions over which we are
averaging (500 pc) should follow the equilibrium expectation.

Variations in star formation can also drive large-scale changes in the
dynamical equilibrium structure of the galaxy (El-Badry et al., 2016; Orr et
al., 2019). We can quantify the degree to which dynamical equilibrium is a
valid assumption for the system under consideration here by comparing the
time-scale over which we measure the SFR to the vertical crossing time of the
disks, which generally quantifies the time over which excess kinetic energy is
dissipated in the galaxy (Ostriker et al., 2001). Generally, we can write
$t_{\rm cross}\approx H_{\rm gas}/\sigma_{\rm eff}$, where $H_{\rm gas}$ is
the gas scale height. We estimate the gas scale height as

$\begin{split}H_{\rm gas}=\frac{2\sigma_{\rm eff}^{2}}{\pi G\Sigma_{\rm
HI}+2\sigma_{\rm eff}\sqrt{2G\rho_{sd}}},\end{split}$ (12)

which is equivalent to equation 5 of Ostriker & Kim (2022) except that we
assume $\Sigma_{\rm gas}\approx\Sigma_{\rm HI}$ and take $W\approx P_{\rm
DE}$. This estimate results in $t_{\rm cross}\approx 30\,{\rm Myr}$ for the
samples at hand – again, significantly smaller than the timescale over which
we are measuring star formation activity.

###  4.2 Estimating $\rho_{\rm sd}$

Having addressed these assumptions, we now proceed to the sample at hand. In
order to estimate $P_{\rm DE}$ (Equation 4), we will need an estimate for the
midplane density of stars ($\rho_{\star}$) and dark matter ($\rho_{\rm dm}$),
$\rho_{\rm sd}=\rho_{\star}+\rho_{\rm dm}$. We estimate the stellar mass
density at the midplane as $\rho_{\star}=\Sigma_{\star}/(2H_{\star})$, where
$H_{\star}$ is the disk scale height and $\Sigma_{\star}$ is the stellar mass
under the assumption of an exponential density profile.

Figure 4:  The distribution over $\rho_{sd}$ for our fiducial dark matter halo
assumptions (see text) for the NSA dwarfs (purple) and UDGs (red) at 500 pc, 1
kpc, and global scales from left to right. The shaded regions are bounded by
the 16th to 84th percentiles of the distribution. The NSA dwarfs are
characterized by systematically higher stellar$+$dark matter densities, which
is unsurprising given that the UDGs are characterized by low stellar surface
densities.

We obtain this estimate statistically by using the measured scale lengths of
our galaxies in conjunction with the three-dimensional shape distributions
inferred by Carlsten et al. (2021) for the NSA dwarf sample and by Kado-Fong
et al. (2021) for the UDG sample. We assume in both cases that the three-
dimensional shape distribution of the galaxy sample measured at one effective
radius is the same as the distribution measured at the scale length
$l_{\star}$. That is to say that we assume $C/A{}\approx H_{\star}/l_{\star}$
where $C/A{}$ indicates the ratio between the smallest and largest principle
axes of the ellipsoid that describes the galaxy shape distribution at one
effective radius, and $H_{\star}$ and $l_{\star}$ are the stellar scale height
and length, respectively. Both the UDGs (Kado-Fong et al., 2021) and the
“normal” dwarfs (Kado-Fong et al., 2020) are well-characterized by exponential
surface brightness profiles, meaning that ${\rm R_{\rm eff}}$$\approx
0.6l_{\star}$ assuming a constant stellar mass-to-light ratio. We can then
estimate the probability distribution function of $H_{\star}$ as:

$\begin{split}{\rm Pr}[H_{\star}]=&(2\pi\sigma_{C/A{}}^{2})^{-1/2}\times\\\
&\exp\left[\frac{-(H_{\star}/l_{\star}-\mu_{C/A{}})^{2}}{(2\sigma_{C/A{}}^{2})}\right]\end{split}$
(13)

where $\mu_{C/A{}}$ and $\sigma_{C/A{}}$ are the inferred mean and standard
deviation of the bivariate Gaussian used to describe that intrinsic shape
distribution in Carlsten et al. (2021) and Kado-Fong et al. (2021).

We use the Carlsten et al. (2021) 3D shapes because of the overlap in stellar
mass between the two samples, but we note that their sample is comprised only
of satellite galaxies. We therefore also compute $\Sigma_{\rm SFR}$
predictions using the 3D intrinsic shape distribution of Kado-Fong et al.
(2020), which is incomplete at the relevant stellar masses but includes field
galaxies, and find no significant difference in the SFR predictions between
the assumptions of intrinsic shape distribution.

We opt to not implement inclination corrections for our sample because the
stellar disks are not thin and because empirical results suggest that there is
significant variation in the three-dimensional shapes and thicknesses of the
dwarf stellar disks (Kado-Fong et al., 2020; Carlsten et al., 2021); however,
we compute inclination estimates using the mean disk height-to-length ratio
following Holmberg (1958) and find that our results would not qualitatively
change if we did implement such an inclination correction. We note that
adopting an inclination correction would systematically lower the stellar mass
surface density and star formation rate surface density estimates in this work
by an average of $\sim 34\%$ below fiducial values, though we stress that this
value is highly uncertain due to the assumptions made in the application of
the Holmberg (1958) correction. Furthermore, because the feedback yield
$\Upsilon_{\rm tot}$ is the ratio between the midplane pressure and the star
formation rate surface density, the effect of an inclination correction
applied to both the stellar mass and star formation rate surface density
estimates should be lessened.

Figure 5:  The distribution over $P_{\rm DE}$ for the NSA dwarfs (purple) and
UDGs (red) at 500 pc, 1 kpc, and global scales from left to right. The shaded
regions at 500 pc and 1 kpc are bounded by the 16th to 84th percentiles of the
distribution. We find no significant variation in $P_{\rm DE}$ as a function
of spatial scale, but find that the NSA dwarfs have systematically higher
dynamical equilibrium pressures than the UDG dwarfs.

We now move on to the estimation of the dark matter density, $\rho_{dm}$.
Here, we must adopt a dark matter halo profile for each galaxy. This component
is by far the most uncertain ingredient in the star formation prescription –
not only do we not have any constraint on the individual dark matter halos in
which these galaxies live, the overall stellar-to-halo mass relation is highly
uncertain for low-mass galaxies. In order to take into account the scatter and
uncertainty in the stellar-to-halo mass relation at these masses, we compute
the predicted star formation rate using three different dark matter halos: a
fiducial best-guess halo mass using the stellar-to-halo mass relation of the
FIRE-2 simulations (which predicts halo masses between $1.6\times 10^{10}$ and
$6\times 10^{10}$ M⊙ for the sample at hand, Hopkins et al., 2018), a halo
with $M_{h}=10^{12}M_{\odot}$, and a halo with $M_{h}=10^{9}M_{\odot}$. We
assume an NFW profile for all dark matter halos in this work (Navarro et al.,
1997). We assign a concentration based on the concentration-mass relation of
Child et al. (2018). We assign the uncertainty due to the dark matter density
to be the difference between $\rho_{dm}$ estimated for a very massive
($M_{h}=10^{12}M_{\odot}$) and very low-mass ($M_{h}=10^{9}M_{\odot}$) halo,
as it is unlikely that any of our galaxies are living in halos with a stellar-
to-halo mass ratio more extreme than 0.1 (in the case of the extreme low-mass
halo) or 0.0001 (in the case of the extreme high-mass halo). For our fiducial
halo assumptions, the combined stellar and dark matter density tends to be
dominated by dark matter, as is expected for these low-mass galaxies (see,
e.g. Oh et al., 2011). The UDGs tend to be more dark matter-dominated, with a
median $\rho_{\star}/\rho_{sd}=0.14$ compared to the mildly dark matter-
dominated NSA dwarfs (median $\rho_{\star}/\rho_{sd}=0.46$).

We show the distribution over $\rho_{sd}$ for our fiducial dark matter halo
assumption in Figure 4. We find that the NSA dwarfs are characterized by
systematically higher values of $\rho_{sd}$, which is unsurprising given that
the UDGs are characterized by relatively low stellar surface densities.

###  4.3 $P_{\rm DE}$ Estimates

Having estimated both $\Sigma_{\rm HI}$ and $\rho_{sd}$, we may now arrive at
an estimate of the dynamical equilibrium pressure ($P_{\rm DE}$). We compute
this quantity following Equation 4.

Before considering the full PRFM prediction, it is of interest to first
consider how $P_{\rm DE}$ varies between the NSA and UDG samples. We show the
overall distribution of $P_{\rm DE}$ at each spatial scale used in this work
(500 pc, 1 kpc, and global) for each sample in Figure 5. We find that there is
no significant variation in $P_{\rm DE}$ distribution as a function of spatial
scale, but that the UDGs tend to have lower dynamical equilibrium pressures
than do the NSA dwarfs. This result is as expected due to the low $\rho_{sd}$
values of the UDGs (see Figure 4). The relatively low stellar$+$dark matter
densities (and relatively similar H I surface densities) of the UDGs also
imply that $\mathcal{W}_{\rm gas}$ is a larger contributor to $\mathcal{W}$ in
the UDGs than it is in the NSA dwarfs.

We show $\mathcal{W_{\rm gas}}/\mathcal{W}$ in Figure 6 as computed from
Equation 4 where $\mathcal{W}\approx P_{\rm DE}$ and $\mathcal{W}_{\rm
gas}=(\pi G\Sigma_{\rm HI}^{2})/2$. Each panel shows the dynamical equilibrium
pressure versus the fraction of $\mathcal{W}$ contributed by $\mathcal{W}_{\rm
gas}$. From left to right, the panels show measurements at 500 pc, 1 kpc, and
global scales. For visual clarify, we only show errorbars for a random subset
of the regions measured in this work. The colored points show a random sample
of UDG regions (red) and NSA regions (purple) with their errorbars included;
the full sample is shown by the grey scatter. The tracks visible in the left
and middle panels are a result of the median H I profiles we assume for the
galaxies; the uncertainty in these profiles result in a wider dispersion than
is implied by the tracks (as shown by the vertical extent of the errorbars).
Indeed, we find that $\mathcal{W}_{\rm gas}$ accounts for a larger fraction of
the total weight in the UDGs than it does in the NSA dwarfs.

Figure 6:  The contribution of self-gravity to the total midplane weight
($\rm\mathcal{W}_{gas}/\mathcal{W}$) as a function of dynamical equilibrium
pressure ($P_{\rm DE}$) as a function of spatial scale (500 pc to global
average, from left to right). We show individual errorbars for a random sample
of regions from the UDG sample (red) and the NSA sample (purple); the full
sample is shown in grey. Note that the tracks visible in the left and middle
panels are a result of the median assumed H I profiles; the uncertainty in the
H I profiles result in a larger dispersion in
$\rm\mathcal{W}_{gas}/\mathcal{W}$ than is implied by the tracks alone (as is
shown by the errorbars of the colored points). We find that the external
weight ($\mathcal{W}_{\rm ext}$) is relatively less important in the UDGs as
compared to the NSA dwarfs.

###  4.4 PRFM predictions

With the assumptions validated and parameters estimated, we may now move on to
predicting $\Sigma_{\rm SFR}$ from the PRFM framework for our observed
galaxies. We base our predictions upon the numerically estimated form of
$\Upsilon_{\rm tot}$ presented by Ostriker & Kim (2022) using the TIGRESS
simulations, as discussed in Equation 9.

In the top row of Figure 7, we show the relationship between $\Sigma_{\rm
SFR}$ and H I surface density for our NSA dwarfs (purple) and UDGs (orange)
averaging over 500 pc regions, 1 kpc regions, and the full area of the galaxy.
As was shown in more detail in Paper I, the UDGs form stars at consistently
lower efficiencies (relative to their H I content) relative to the NSA dwarfs.

We then show the relation between the observed and predicted star formation
rate surface densities in the bottom row of Figure 7 for 500 pc, 1 kpc, and
global averages. We find that the PRFM prescription succeeds in reproducing
the star formation rate surface density measurements of the UDGs at all
spatial scales probed, which we will explore quantitatively via a comparison
between the predicted and measured values of $\Upsilon_{\rm tot}$ in the
following section. The prescription under-predicts $\Sigma_{\rm SFR}$ for the
most vigorously star-forming regions of the NSA sample – this under-prediction
can be understood if one considers that the NSA dwarfs may have considerable
stores of molecular hydrogen (see, e.g. Leroy et al., 2008; de los Reyes &
Kennicutt, 2019). The discrepancy is most obvious in the globally averaged
estimates, which is due to the global averages acting as a $\Sigma_{\rm
SFR}$-weighted average (where the most vigorously star-forming regions
dominate the total mean signal). We also see that the H I-only PRFM
predictions perform best for the NSA dwarfs at 500 pc scales, which is
consistent with the picture where much of the galaxy is dominated by atomic
hydrogen (over molecular hydrogen as has been suggested previously in, e.g.
Leroy et al. 2008). Nonetheless, we find that PRFM theory well-describes the
majority of the regions within both galaxy samples and thus naturally predicts
the low SFE(H I) of the H I-rich UDGs as a consequence of their low ISM
weight.

###  4.5 Empirical estimates of $\Upsilon_{\rm tot}$

Finally, though we have adopted $\Upsilon_{\rm tot}$ as a way to predict
$\Sigma_{\rm SFR}$, it is also informative to re-frame the computation as a
comparison between the theoretically predicted values for $\Upsilon_{\rm tot}$
(Equation 8 and Equation 9) and empirically-derived $\Upsilon_{\rm tot}$ as
the ratio between our observed $\Sigma_{\rm SFR}$ measurements and $P_{\rm
DE}$ estimates. In this way, we can quantitatively compare the relationship
between star formation and dynamical equilibrium pressure in our dwarf samples
and similar measurements of more massive galaxy samples from the literature
within the context of theoretical expectations.

Because the theoretical $\Upsilon_{\rm tot}$ depends explicitly upon
$\tau_{\rm FUV}$ (Equation 7), we compute a running median predicted
$\Upsilon_{\rm tot}$ for our sample as a function of $P_{\rm DE}$ in bins of
width 0.1 dex. The shaded region in each panel spans the 16th to 84th
percentiles of the theoretical $\Upsilon_{\rm tot}$ prediction over the same
domain.

We show such a comparison in Figure 8. At left, we show $\Sigma_{\rm SFR}$
versus $P_{\rm DE}$ at 500 pc, 1 kpc, and global scales with the predictions
using $\Upsilon_{\rm tot}$ over-plotted (Equation 8 solid, Equation 9 dashed).
At right we show the empirical estimate for $\Upsilon_{\rm tot}$ as a function
of $P_{\rm DE}$, again with both the theoretical and numerical predictions for
$\Upsilon_{\rm tot}$ shown in brown. Here we show the median value of
$\Upsilon_{\rm tot}$ for the UDG and NSA samples as the red and purple points,
respectively, and depict the range between the 16th and 84th percentiles in
both $P_{\rm DE}$ and $\Upsilon_{\rm tot}$ as solid unfilled rectangles of the
same color. For the reader’s convenience, we tabulate these values in Table 1.
For comparison, we also show empirical $\Upsilon_{\rm tot}$ estimates inferred
from the literature results of PHANGS (orange dot-dashed box, Sun et al.,
2020), EDGE-CALIFA (blue dot-dashed box, Barrera-Ballesteros et al., 2021),
and KINGFISH (green dot-dashed box, Herrera-Camus et al., 2017). We show the
same literature results in each panel, but note that these measurements are
taken on the scale of $\sim 1$ kpc. We also note that the literature
measurements take ${\rm H_{2}}$ into account in their computation of $P_{\rm
DE}$.

Directing the reader’s attention first to the top and middle rows (500 pc
scales and kiloparsec scales, respectively), we see that both the empirically-
derived $\Upsilon_{\rm tot}$ measured from the UDG sample and NSA sample are
in good agreement with the predictions of Ostriker & Kim (2022). We moreover
see that the empirically-derived $\Upsilon_{\rm tot}$ of both dwarf samples at
kiloparsec and lower scales are in good agreement with the measured
$\Upsilon_{\rm tot}$ values of the literature samples of more massive galaxies
(which do have ${\rm H_{2}}$ incorporated in their $P_{\rm DE}$ estimates).
This is to say that, as quantitatively measured by $\Upsilon_{\rm tot}$, the
relationship between our estimate of ISM weight and star formation appears to
be relatively constant between “extreme” (low density) dwarfs, “normal”
dwarfs, and their much more massive counterparts. When we consider globally-
averaged values, we find that the median empirical $\Upsilon_{\rm tot}$
measured for the NSA dwarfs drops significantly – this may be suggestive of a
significant store of ${\rm H_{2}}$ in these NSA dwarfs, which we will discuss
further in Section 5.2.

Encouragingly, we find that the empirically-measured $\Upsilon_{\rm tot}$
values derived from the dwarf samples on $\leq 1$ kpc scales are in good
agreement with both the theoretical and numerical predictions for
$\Upsilon_{\rm tot}$ presented in Ostriker & Kim (2022). However, a unified
analysis of a sample that includes high $P_{\rm DE}$ regions ($P_{\rm DE}$
$\sim 10^{5}k_{B}$ K cm-3) would be necessary to expand quantitatively upon
the numerical expectation of a $P_{\rm DE}$ dependence beyond the simple
literature comparison presented in this work.

Figure 7:  Top Row: a comparison between H I surface density estimates and SFR surface density measurements at 500 pc (left column), 1 kpc (middle column), and global scales (right column). In all panels, we only show NSA galaxies that are within the stellar mass range of the UDG sample. For visual clarity, 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. Bottom row: a comparison between the observed SFR surface densities and the PRFM predictions. Again, red points show UDGs while purple points show NSA dwarfs.  Figure 8:  Left: an estimate of the dynamical equilibrium pressure ($P_{DE}$) for our sample. As in Figure 7, we show a random sample of points with their individual errorbars and colors (red for the UDG sample, purple for the NSA sample) and the full sample in grey. We overplot the theoretical (Equation 8) and numerical (Equation 9) predictions of Ostriker & Kim (2022) as the dashed and solid lines, respectively. Right: an empirical estimate of $\Upsilon_{\rm tot}$ ($=P_{\rm DE,obs}/\Sigma_{\rm SFR,obs}$) as a function of dynamical equilibrium pressure. We compare to kpc-scale observations from PHANGS (orange dot-dashed, Sun et al., 2020), EDGE-CALIFA (blue dot-dashed, Barrera-Ballesteros et al., 2021), and KINGFISH (green dot-dashed, Herrera-Camus et al., 2017), as well as to the theoretical predictions of Ostriker & Kim (2022).  Table 1: 16th, 50th, and 84th percentile values of $P_{\rm DE}$ and $\Upsilon_{\rm tot}$ for the UDG and NSA samples at 500 pc, 1 kpc, and global scales. |  |  $P_{\rm DE}^{16}$ |  $\mathbf{P_{\rm DE}^{50}}$ |  $P_{\rm DE}^{84}$ |  $\Upsilon_{\rm tot}^{16}$ |  $\mathbf{\Upsilon_{\rm tot}^{50}}$ |  $\Upsilon_{\rm tot}^{84}$
---|---|---|---|---|---|---|---
Sample | Scale |  $\rm\left[\frac{10^{3}K}{k_{B}\ cm^{3}}\right]$ |  $\rm\left[\frac{10^{3}K}{k_{B}\ cm^{3}}\right]$ |  $\rm\left[\frac{10^{3}K}{k_{B}\ cm^{3}}\right]$ |  $\rm\left[10^{3}\frac{km}{s}\right]$ |  $\rm\left[10^{3}\frac{km}{s}\right]$ |  $\rm\left[10^{3}\frac{km}{s}\right]$
UDGs | 500 pc | $1.32_{-0.30}^{+0.18}$ | $3.42_{-0.22}^{+0.28}$ | $6.60_{-0.22}^{+0.23}$ | $1.01_{-0.18}^{+0.12}$ | $2.13_{-0.14}^{+0.17}$ | $4.02_{-0.26}^{+0.33}$
| 1 kpc | $1.38_{-0.26}^{+0.12}$ | $3.26_{-0.18}^{+0.21}$ | $6.29_{-0.17}^{+0.20}$ | $1.17_{-0.20}^{+0.13}$ | $2.38_{-0.13}^{+0.11}$ | $4.27_{-0.24}^{+0.28}$
| global | $2.46_{-0.33}^{+0.29}$ | $3.69_{-0.31}^{+0.27}$ | $5.36_{-0.37}^{+0.39}$ | $1.08_{-0.15}^{+0.13}$ | $1.79_{-0.16}^{+0.16}$ | $2.75_{-0.26}^{+0.26}$
NSA | 500 pc | $3.03_{-0.43}^{+0.37}$ | $6.06_{-0.28}^{+0.33}$ | $10.55_{-0.66}^{+0.86}$ | $0.57_{-0.07}^{+0.06}$ | $1.44_{-0.10}^{+0.12}$ | $3.16_{-0.27}^{+0.35}$
dwarfs | 1 kpc | $2.86_{-0.53}^{+0.45}$ | $6.02_{-0.31}^{+0.34}$ | $10.42_{-0.55}^{+0.76}$ | $0.48_{-0.07}^{+0.06}$ | $1.27_{-0.11}^{+0.11}$ | $2.86_{-0.27}^{+0.30}$
| global | $4.94_{-0.54}^{+0.52}$ | $7.37_{-0.58}^{+0.58}$ | $10.65_{-0.83}^{+1.26}$ | $0.29_{-0.06}^{+0.07}$ | $0.73_{-0.08}^{+0.10}$ | $1.33_{-0.15}^{+0.18}$

##  5 Discussion

###  5.1 Star Formation Efficiency in UDGs

In Paper I, we demonstrated that UDGs host relatively low star formation rate
surface densities given their apparent H I surface densities. In this work, we
have gone on to consider whether this low efficiency star formation is
surprising in the context of contemporary theories of galactic star formation.
As we have detailed in the preceding section, we find that the pressure-
regulated, feedback-modulated (PRFM) theory of star formation is remarkably
successful at predicting the low $\Sigma_{\rm SFR}$ and SFE(H I) of the UDGs.
The star formation in these “extreme” galaxies, then, can be well-described
within the same framework of PRFM star formation as can the “normal” (NSA)
dwarfs and even more massive galaxies – the difference in their star formation
behavior can be quantitatively attributed to the shallower gravitational
potential due to the characteristically diffuse stellar structure of the UDGs.

The bottom panels of Figure 7 demonstrate that the PRFM model is highly
successful at reproducing the star formation rate surface densities of the
UDGs, even without considering the surface density contribution of molecular
gas. Furthermore, Figure 8 shows that the NSA dwarfs and UDGs lie on the same
relation between $P_{\rm DE}$ (dynamical equilibrium pressure) and
$\Sigma_{\rm SFR}$ as do massive galaxies. These successes indicate that H
I-rich UDGs, which are largely in the field, need not be forming stars in an
exotic manner, as has been suggested for globular cluster-rich UDGs in high-
density environments (Danieli et al., 2021).

Beyond a direct comparison of observed and predicted star formation rate
surface densities, the PRFM framework allows us to quantitatively explore the
relationship between star formation and galaxy structure in our samples. We
make observational estimates of $\Upsilon_{\rm tot}$ in Section 4.5 to compare
our results to both theoretical expectations and observational measurements of
higher-$P_{\rm DE}$ systems. We find that the median value of $\Upsilon_{\rm
tot}$ is about $1000-2000{\ \rm km\ s}^{-1}{}$ (see Table 1). This is
consistent with theoretical expectations and similar to previous literature
results (though there is significant variation between literature samples, as
shown in Figure 8). Our UDG sample is characterized by somewhat higher values
of $\Upsilon_{\rm tot}$ and lower values of $P_{\rm DE}$ compared to the NSA
sample, which is also consistent with the expectation that the momentum
injection from supernovae and FUV heating are more efficient in low density
environments due to weaker cooling and less shielding. However, as we will
discuss further in the next section, the $P_{\rm DE}$ estimates of the NSA
sample may be biased low due to the neglected contribution from ${\rm H_{2}}$.
Thus, the main takeaway from the empirical $\Upsilon_{\rm tot}$ results should
be the agreement between theoretical predictions and the low-SFE(H I) UDG
sample, rather than the variation between the NSA and UDG samples. The
concordance between midplane pressure and star formation rate surface
densities in the UDG sample also implies that H I-rich UDGs are unlikely to
support bursty and concentrated star formation at $z=0$, which has been
suggested to form UDGs and/or link them to blue compact dwarfs (Di Cintio et
al., 2017; Sánchez Almeida et al., 2018). However, a broader view of the
mechanisms that trigger and sustain concentrated bursts of star formation in
low-mass galaxies is needed to further explore the link between these
populations.

Having established the concordance between the results at hand and both
theoretical & observational results from the literature, we can also examine
the implication of the composition of $P_{\rm DE}$ on predictions of
$\Sigma_{\rm SFR}$. Although $\mathcal{W}_{\rm gas}$ is relatively more
important for the UDGs (than it is for the NSA dwarfs), we also note that
$\mathcal{W}_{\rm ext}\geq\mathcal{W}_{\rm gas}$ for all of the galaxies and
length scales probed. This indicates that the external gravitational potential
(from stars and dark matter) plays a significant role in setting the star
formation timescale; that is, gas surface density alone is not sufficient to
fully predict $\Sigma_{\rm SFR}$. Indeed, a correlation between dwarf stellar
mass surface density and divergence from the Kennicutt-Schmidt relation has
been demonstrated in samples of nearby galaxies (de los Reyes & Kennicutt,
2019, see also the top row of Figure 7). Altogether, these results paint a
picture in which galaxy structure, rather than gas availability alone, plays a
key role in regulating star formation in low-mass systems.

Figure 9:  The estimated ${\rm H_{2}}$ fraction (where f${}_{\rm
H_{2}}=\Sigma_{\rm H_{2}}/[\Sigma_{\rm H_{2}}+\Sigma_{\rm HI}]$) for the
regions in our sample at 500 pc (left), 1 kpc (middle), and global scales
(right) when we assume a constant ${\rm H_{2}}$ depletion time of $\tau_{\rm
dep}=\Sigma_{\rm H_{2}}/\Sigma_{\rm SFR}=1.8$ Gyr. The red histograms shows
the UDGs, while the purple histograms show the NSA dwarfs. We find that while
the distribution in f${}_{\rm H_{2}}$ peaks at f${}_{\rm H_{2}}<0.2$ for both
the UDGs and NSA galaxies, the NSA dwarfs are characterized by a tail to high
molecular gas fractions. This finding is consistent with the under-prediction
of the H I-only PRFM estimate for the highly star-forming regions of the NSA
galaxies (as shown in Figure 7).  Figure 10:  A comparison between the
expected CO (1-0) luminosities of our NSA (purple) and UDG (red) dwarf samples
and real measurements from the literature. We estimate $\rm L_{CO1-0}$ by
assuming a constant ${\rm H_{2}}$ depletion time of $\tau_{\rm dep}=1.8$ Gyr
and bounding our estimates from $\alpha_{CO}$ measurements of the LMC and SMC
(Schruba et al., 2011). We find that the expected CO luminosities of the
galaxies in the present sample are comparable to measurements made of galaxies
at lower distances, but lie well below literature measurements of galaxies at
similar distances.

###  5.2  Molecular Hydrogen in UDGs

In this work we have shown that the PRFM model of star formation does a
remarkably good job of accounting for the star formation rate surface density
of the UDGs, despite the fact that we assume that the contribution of
molecular hydrogen to the overall weight of the system is negligible. In
general and quite separate from the discussion of UDGs, galaxies in this mass
range are thought to be relatively poor in ${\rm H_{2}}$ overall – a recent
study suggests that nearby dwarfs with stellar masses less than
$\log_{10}(M_{\star}/M_{\odot})=8.5$ have a median ${\rm H_{2}}$ fraction of
$f_{\rm H_{2}}\equiv\Sigma_{\rm H_{2}}/(\Sigma_{\rm H_{2}}+\Sigma_{\rm
HI})=0.15$ (de los Reyes & Kennicutt, 2019). However, studies have also
suggested that dwarfs may be dominated by ${\rm H_{2}}$ in their central
regions, despite being ${\rm H_{2}}$-poor in a spatially-averaged sense (Leroy
et al., 2008). Despite their overall small ${\rm H_{2}}$ content, dwarfs are
also thought to preferentially host their molecular gas at small radii,
allowing ${\rm H_{2}}$ to contribute significantly to or even dominate the
weight of the ISM near the center of the galaxy (Leroy et al., 2008).

In order to understand what we may say about the ${\rm H_{2}}$ content in
UDGs, we first consider the impact of our H I-only approach on the NSA sample.
In Figure 7, we see that the H I-only PRFM model significantly under-predicts
$\Sigma_{\rm SFR}$ for the NSA dwarfs, particularly when we consider the star
formation rate surface density averaged over kiloparsec or larger scales. At
500 pc scales, the bulk of the regions are well-represented by their PRFM
predictions, though a tail towards under-predicted $\Sigma_{\rm SFR}$ at high
$P_{\rm DE}$ remains. This can be understood if one considers that the average
$\Sigma_{\rm SFR}$ measured within any given region of 500 pc, 1000 pc, or
greater size is essentially a SFR-weighted average. At global scales, the most
vigorously star-forming regions dominate the average $\Sigma_{\rm SFR}$. At
500 pc, these vigorously star-forming regions are siloed into a relatively
small number of points. This transition from a roughly SFR-weighted to a
roughly area-weighted scheme can also be seen in the behavior of the median
$\Upsilon_{\rm tot}$ estimated for the NSA galaxies as a function of region
spatial scale in the right column of Figure 8. At 500 pc, the median empirical
$\Upsilon_{\rm tot}$ computed from the NSA dwarfs is in good agreement with
the prediction from the TIGRESS simulations of Ostriker & Kim (2022). When we
consider the global averages, however, the median empirical $\Upsilon_{\rm
tot}$ lies well below the simulation prediction. This behavior is consistent
with the picture where molecular gas makes a significant contribution to the
weight in the most vigorously star-forming areas of the NSA dwarfs.

Having established the possible impact on neglecting the ${\rm H_{2}}$
contribution in the NSA dwarfs, we may now consider the UDGs. First, unlike
the NSA dwarfs, the PRFM prediction tends to slightly over-predict the star
formation rate surface densities of the UDGs as a function of $P_{\rm DE}$.
This implies that, holding the dark matter halo properties constant (a point
we will return to in the following section), the absence of the weight from
the ${\rm H_{2}}$ component does not strongly affect the $\Sigma_{\rm SFR}$
predictions for the UDGs. This is despite the fact that, as shown in Figure 6,
the self-gravity of the ISM comprises a larger component of the total midplane
weight of the UDGs. The UDGs should thus be relatively more sensitive to the
effect of neglecting ${\rm H_{2}}$ gas, as the $\mathcal{W}_{\rm gas}$ scales
as the square of the gas surface density (as opposed to $\mathcal{W}_{\rm
ext}$, which scales linearly with the gas surface density). Thus, we find that
unlike the NSA dwarfs, the UDGs in our sample are unlikely to host large
molecular gas stores.

One simple exercise we can undertake to test the idea that ${\rm H_{2}}$ is a
less important contributor to $P_{\rm DE}$ in UDGs than in the NSA galaxies is
by a direct estimate of M${}_{\rm H_{2}}$ from the star formation rate via an
assumed depletion time $\tau_{\rm dep}$. Observational support for a constant
${\rm H_{2}}$ depletion time of $\tau_{\rm dep}\sim 1-3$ Gyr has emerged from
observational work (see, e.g. Bigiel et al., 2008; Leroy et al., 2008; Bigiel
et al., 2011; Schruba et al., 2011, 2012; Leroy et al., 2021), though there is
evidence for variation within that range as a function of environment (Utomo
et al., 2017). For this test we will assume a constant depletion time of
$\tau_{\rm dep}\equiv\Sigma_{\rm H_{2}}/\Sigma_{\rm SFR}=1.8$ Gyr following
the results of Schruba et al. (2011).

Using this assumed ${\rm H_{2}}$ depletion time, we may directly estimate the
expected ${\rm H_{2}}$ surface density from our measured star formation rate
surface densities. We can then compute the molecular gas fraction in each
region, as shown in Figure 9, where the ${\rm H_{2}}$ fraction is defined as
f${}_{\rm H_{2}}=\Sigma_{\rm H_{2}}/(\Sigma_{\rm H_{2}}+\Sigma_{\rm HI})$. We
indeed find that the UDG expected ${\rm H_{2}}$ fractions are quite low, with
a median expected molecular gas fraction of f${}_{\rm H_{2}}=0.09$ and a 99th
percentile expected fraction of f${}_{\rm H_{2}}=0.31$. Conversely, although
the median NSA dwarf region has a relatively low expected ${\rm H_{2}}$
fraction (f${}_{\rm H_{2}}=0.24$), there is a significant tail to ${\rm
H_{2}}$-dominated regions – the 99th percentile expected fraction of the NSA
sample is f${}_{\rm H_{2}}=0.83$.

Using these ${\rm H_{2}}$ surface density estimates within the PRFM prediction
framework would clearly be circular, as the quantity of interest ($\Sigma_{\rm
SFR}$) is the very quantity that we use to estimate $\Sigma_{\rm H_{2}}$.
However, this exercise does provide a separate (though we note, not entirely
independent) consistency check with our previous claim that the molecular gas
stores of the UDGs are a subdominant contributor to the overall gas surface
density.

####  5.2.1 Outlooks on ${\rm H_{2}}$ Detection

Clearly, a direct estimation of the ${\rm H_{2}}$ content of the UDGs in our
sample would be immensely powerful in determining whether they are ${\rm
H_{2}}$-depleted, and whether their SFE(${\rm H_{2}}$) is consistent with that
of “normal” galaxies. There are two substantial technical hurdles to gain such
an estimation: first, the previously-discussed low predicted ${\rm H_{2}}$
masses of the UDGs, and second, the substantial uncertainty in converting
between a direct measurement (e.g. of CO(1-0)) and ${\rm H_{2}}$ mass in a
UDG-like environment.

Given a depletion time $\tau_{\rm dep}$, the relation between the CO (1-0)
luminosity and ${\rm H_{2}}$ mass can be written as

$\alpha_{\rm CO(1-0)}=\frac{M_{\rm H_{2}}}{\rm L_{CO(1-0)}}=\tau_{\rm
dep}\frac{\rm SFR}{\rm L_{CO(1-0)}}.$ (14)

As above, we assume an ${\rm H_{2}}$ depletion time of 1.8 Gyr. It is both
observed and theoretically expected that $\alpha_{\rm CO}$ should be sensitive
to environmental factors such as (but not necessarily limited to) metallicity,
with $\alpha_{\rm CO}$ increasing with decreasing metallicity (e.g. Schruba et
al., 2011; Bolatto et al., 2013; Gong et al., 2020). Even if a CO detection
can be made, it is therefore also necessary to spectroscopically determine the
metallicity in order to obtain an estimate of the ${\rm H_{2}}$ mass, which is
in itself observationally difficult given the low surface brightnesses and on-
sky sparsity of the UDGs. Previous exercises with similar samples have
required moderate (1 hr) integration times with wide (2″) slits on the 8.1m
Gemini South telescope in order to yield gas-phase metallicity measurements
(Greco et al., 2018a).

In the absence of a gas-phase metallicity measurement, we can make a simple
estimate of the expected CO (1-0) luminosity using existing $\alpha_{\rm CO}$
measurements of our most nearby massive dwarfs, the Large and Small Magellanic
Clouds (LMC and SMC, respectively). We adopt the $\alpha_{\rm CO}$
measurements of the LMC and SMC of Schruba et al. (2011) – as the more metal-
rich system, adopting the $\alpha_{\rm CO}$ measured for the LMC results in
higher CO (1-0) luminosities than does adopting the value measured from the
SMC. In Figure 10, we show the expected CO (1-0) luminosities of the galaxies
in our sample as a function of luminosity distances given the assumptions
detailed above. We additionally show a selection of observed CO (1-0)
measurements from the literature for a selection of samples that include low-
mass and low-metallicity objects (Schruba et al., 2011; Amorín et al., 2016;
Kepley et al., 2016; Gao et al., 2022). We find that although the predicted CO
(1-0) luminosities are comparable to $\rm L_{CO1-0}$ measurements of nearby
dwarfs, the predicted luminosities are significantly lower than literature
measurements made at the distance range of the sample. As a small aside, it
may be initially surprising to the reader that the distribution of CO
luminosities in the UDG sample is close to that of the NSA dwarfs, given that
the ${\rm H_{2}}$ fractions of the UDG sample is much lower. Because we are
estimating the mass in molecular Hydrogen directly from the integrated star
formation rate (see Equation 14), and because the stellar mass range of the
samples are chosen to be the same, this is essentially a restatement of the
result that the UDGs lie on the star-forming main sequence (right panel of
Figure 1). The lower ${\rm H_{2}}$ fractions estimated for the UDG sample are
then a result of elevated H I masses (at fixed stellar mass) seen in the UDGs
versus the NSA dwarfs (left panel of Figure 1).

An alternative approach is to measure the dust mass via SED fitting in the
FIR, where emission may be modeled as a modified blackbody with the dust mass,
temperature, and emissivity index as free parameters (see, e.g. Kennicutt et
al., 2011; Cigan et al., 2021; Shivaei et al., 2022). Due to the distance and
mass of the UDGs, it is likely that stacking would be necessary to obtain a
robust measurement (Shivaei et al., 2022). The dust mass can then be used to
infer to the total gas mass via an assumed dust-to-gas ratio, which can then
in turn be used to estimate the ${\rm H_{2}}$ mass given a measured H I mass.
However, like $\alpha_{\rm CO}$, the dust-to-gas ratio is naturally expected
to depend on metallicity; measured dust-to-gas ratios at low metallicities
also exhibit non-linearity and substantial scatter (see Figure 11 of Cigan et
al., 2021). Thus, deriving a reliable dust-to-gas ratio for these galaxies
would be challenging even if metallicity were known.

###  5.3 If UDGs Live in Special Halos

A rather significant uncertainty in our analysis is the unknown nature of the
dark matter halos in which our H I-rich UDGs live. A recent work has shed some
light on this topic by estimating halo profile parameters for a sample of H
I-rich UDGs (Kong et al., 2022). In this work, the authors suggest that UDGs
live in unusually low concentration dark matter halos. Thus, having assumed
“normal” dark matter halos for the UDGs – keeping in mind that “normal” is
itself an uncertain term for the dwarf population – we may now consider how
low concentration halos could impact our results.

There is unfortunately only one galaxy, AGC 242019, which is in both our
sample and the sample of Kong et al. (2022). As an exercise, however, we
estimate the change in the predicted $\Sigma_{\rm SFR}$ if we were to adopt
the median halo concentration measured by Kong et al. (2022) of $c=2.62$,
which is approximately 30% of the median (16th, 84th percentile) fiducial halo
concentration of $c=9.16\ (8.90,9.40)$. We show the impact of this change in
the $P_{\rm DE}$-to-$\Upsilon_{\rm tot}$ plane in the middle panel of Figure
11 – assuming the lower concentration halo results in empirical $\Upsilon_{\rm
tot}$ values about 25% lower than the fiducial results. We do find that
assuming a lower concentration halo nominally brings the UDGs closer in line
with theoretical results. However, for the singular galaxy that does
constitute the overlap between the samples, we find that assuming the halo
profile reported by Kong et al. (2022) actually causes a greater difference
between our observed and predicted $\Sigma_{\rm SFR}$. Indeed, the shift in
$\Sigma_{\rm SFR}$ predictions as a function of halo concentration is more
emblematic of the potential effect of unknown systematics in the $P_{\rm DE}$
(and empirical $\Upsilon_{\rm tot}$) estimates than it is evidence for or
against unusual dark matter halo profiles in the UDGs. There thus yet remains
significant work to be done in order to reduce the uncertainty of the impact
of the UDGs’ dark matter halos.

Figure 11:  The same as the right panel of the middle row of Figure 8 with
different assumptions about the halo properties of our sample. We show our
fiducial results at left, the empirical feedback yields under the assumption
of low concentration halos at middle, and the empirical feedback yields with
the assumption that $\rho_{\rm DM}=0$ at right. We find that our 1 kpc
empirical estimates of the feedback yields under the low concentration halo
and no-dark-matter assumptions are still consistent with the numerical
predictions from TIGRESS. Our results are therefore not inconsistent with low
dark matter fractions or low concentration halos in H I-rich UDGs, though we
stress that the precision of our estimates precludes us from distinguishing
between “normal halo” and “no DM” modes of H I-rich UDG structure. This
exercise also shows that our results are not strongly dependent on our
fiducial choice of a stellar-to-halo mass relation.

It has also been suggested that some H I-rich UDGs are dark matter deficient
based on observations of their neutral gas kinematics (Mancera Piña et al.,
2022). It is therefore also useful to understand how the predictions of PRFM
theory would change if we were to assume a dark matter density of $\rho_{\rm
DM}=0.$ For this exercise, we hold all other estimates fixed to their fiducial
values.

We find that the midplane pressure estimates of the UDGs are reduced on
average to 59% that of their fiducial values. The midplane pressure estimates
of the NSA galaxies, which we compute for completeness, are reduced on average
to 73% that of their fiducial values. These reduced values of $P_{\rm DE}$,
and the empirical feedback yields that they imply, remain consistent with PRFM
predictions at the precision of our estimates, as shown in Figure 11. We thus
do not make a statement about whether the $\rho_{\rm DM}=0$ is a better
descriptor of UDG star formation, but rather note that this exercise also
indicates that the results presented in this work are not strongly affected by
our fiducial choice of halo parameters. This finding is not unexpected given
that our estimate of the dynamical equilibrium pressure goes as the square
root of the sum of the stellar and dark matter densities, and that the self-
gravity term is relatively more important for UDGs (see Figure 6).

##  6 Conclusions

In this series, we have used spatially-resolved SED fitting to explore the
star formation activity in a sample of nearby (d $<120$ Mpc) H I-detected
ultra-diffuse dwarf galaxies from Janowiecki et al. (2019), along with a NASA
Sloan Atlas (NSA) 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 tests for star formation theory in extreme (low density)
environments, as well as new clues to the evolutionary pathway of these H
I-rich UDGs (see Paper I).

As established in the first paper of this series, the UDGs are characterized
by low star formation rate surface densities and star formation efficiencies
(as a function of their atomic gas surface densities) down to 500 pc scales.
In this work, we ask whether the UDGs’ lower SFE(H I) is expected in the
context of contemporary models of star formation.

We consider the framework of pressure-regulated, feedback-modulated star
formation (PRFM), which directly connects star formation and galaxy structure
for disks in equilibrium, in Section 4. This necessitates an exploration of
the dynamical equilibrium pressure in the UDG and NSA dwarf systems. The UDGs
are characterized by lower stellar $+$ dark matter densities (Figure 4), lower
dynamical equilibrium pressures (Figure 5), and a relatively larger
contribution to the overall weight by self-gravity (Figure 6). We indeed find
that the lower SFR surface densities and lower SFE(H I) seen in the UDGs are
well-predicted by the PRFM model; that is, the relationship between midplane
pressure and star formation rate surface density ($\Upsilon_{\rm tot}$) is the
same for the UDGs in our sample as it is for the NSA dwarfs, or indeed even
much more massive galaxies.

This holds true for the UDGs despite the fact that we neglect any weight
contributions from ${\rm H_{2}}$ in this analysis. We find that at globally
averaged scales (which one can think of as the SFR-weighted limit), H I-only
predictions underestimate star formation in the NSA dwarfs, but that the same
H I-only predictions are in good agreement with the NSA dwarfs at 500 pc
scales (which one may roughly think of as approaching an area-weighted
average). This implies that ${\rm H_{2}}$ is an important contributor to the
weight in the regions of the most vigorous star formation, but is a minority
component at large – a suggestion that is in agreement with previous results
(see, e.g. Leroy et al., 2008; de los Reyes & Kennicutt, 2019). Star formation
in the UDGs, however, is well-described or even somewhat overestimated by the
H I-only PRFM predictions at all spatial scales, suggesting that the UDGs may
be ${\rm H_{2}}$-poor even compared to similarly (stellar) massive “normal”
dwarfs.

As referenced in Paper I, a clear extension of this work will be to measure
spatially resolved H I in a larger sample of field UDGs. In this work, we have
also considered routes to estimate ${\rm H_{2}}$ mass in the UDGs, but note
that obtaining a detection via either CO or FIR dust emission will likely be
resource-intensive due to the low predicted ${\rm H_{2}}$ fractions and likely
low metallicities of the UDGs. The uncertainty in the $X_{CO}$ factor and/or
gas-to-dust ratio in these galaxies further increases the uncertainty in
obtaining a ${\rm H_{2}}$ mass from such a detection. Nevertheless, a greater
understanding of the ISM in UDGs is a key path towards understanding the
landscape of star formation in these unusual objects, as well as towards
understanding their utility as laboratories in which to study extreme low-
density star formation.

The authors thank Eve Ostriker, Song Huang, and Andy Goulding for insightful
comments and discussion that have greatly improved this manuscript. The
research of EKF was supported by the Porter Ogden Jacobus Fellowship. JEG
gratefully acknowledges support from NSF grant AST-2106730. The work of CGK
was supported by NASA ATP grant 80NSSC22K0717. 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. 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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