$C_{\mathrm{fb}}\approx 3$. In this case, most of the
$\left<P_{\mathrm{turb,\,\theta pc}}\right>$ and
$\Sigma_{\mathrm{SFR,\,1kpc}}$ measurements in the disk sample are consistent
with $p_{\star}/m_{\star}=10^{3}\text{--}10^{4}\;\mbox{$\rm km\,s^{-1}$}$.

There are many caveats related to this approximate estimation of
$C_{\mathrm{fb}}$. The spatial and temporal distribution of gas and star
formation could change the amount of momentum injected into the gas, and the
spatial scales on which most momentum is deposited also matters. Given these
complications, a logical next step on this topic will be a detailed, multi-
scale comparison between observations of gas and star formation at high
resolution and numerical simulations with realistic feedback prescriptions.

An alternative explanation for the high
$P_{\mathrm{turb}}/\Sigma_{\mathrm{SFR}}$ ratio measurements in Figure 11
(left panel) is that other turbulence driving mechanisms might also play a
role, at least within a subset of our sample. For example, many measurements
in bulge or bar regions show the highest $P_{\mathrm{turb}}$ at fixed
$\Sigma_{\mathrm{SFR}}$ (orange symbols in Figure 11, left panel). While part
of this can be explained by beam smearing effects, we also expect bar-induced
radial inflow and the corresponding conversion of gravitational potential
energy to be an additional source of turbulence in the ISM (e.g., through
shocks near the center of the stellar bar; see Binney et al., 1991; Sormani et
al., 2015). More quantitative comparisons between our results and other
turbulent driving mechanisms will be carried out in future works.

####  7.1.2 Equilibrium Pressure versus SFR Surface Density

The right panel in Figure 11 shows kpc-scale SFR surface density,
$\Sigma_{\mathrm{SFR,\,1kpc}}$, as a function of kpc-scale dynamical
equilibrium pressure, $P_{\mathrm{DE,\,1kpc}}$. We find a strong correlation
between the two quantities for all measurements in disk regions. This supports
the idea that in steadily star-forming disks, the three dimensional star and
gas distribution on kpc-scale largely determines the average SFR surface
density on the same scale. In any part of the disk where the ISM weight is
higher, the steady-state SFR must also increase for feedback to maintain the
pressure that matches the dynamical equilibrium pressure.

To quantify this observed correlation, we derive a best-fit power-law relation
for all disk measurements, using the OLS bisector method in logarithmic space:

$\frac{\Sigma_{\mathrm{SFR}}}{10^{-3}\;\mbox{$\rm
M_{\odot}\,yr^{-1}\,kpc^{-2}$}}=3.2\,\left(\frac{P_{\mathrm{DE}}}{10^{4}\;\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}}\right)^{0.84}~{}.$
(33)

The rms scatter in $\Sigma_{\mathrm{SFR}}$ around this relation is 0.20 dex.
The statistical errors estimated from bootstrapping are 0.01 for the slope and
0.01 dex for the normalization, but we expect systematic errors (on both
$\Sigma_{\mathrm{SFR,\,1kpc}}$ and $P_{\mathrm{DE,\,1kpc}}$) to be a larger
source of uncertainty on the fitting results.

The same correlation has been observed in various types of galaxies by many
previous studies (e.g., Leroy et al., 2008; Genzel et al., 2010; Ostriker et
al., 2010; Ostriker & Shetty, 2011; Herrera-Camus et al., 2017; Fisher et al.,
2019). To put our measurements into context, we also include in Figure 11 two
other datasets derived also from observations of nearby galaxies. Leroy et al.
(2008) (brown crosses) measure this relation in a sample of 23 nearby
galaxies, with each independent measurement representing one kpc-wide radial
bin in a galaxy. Herrera-Camus et al. (2017) (green plus symbols) investigate
the same relation in the Hi dominated regions in 31 KINGFISH galaxies
(Kennicutt et al., 2011), treating each kpc-sized region independently.

These previous works adopted slightly different assumptions when deriving
$P_{\mathrm{DE}}$ from observables. To make a fair comparison between our
measurements and theirs, our $P_{\mathrm{DE,\,1kpc}}$ measurements shown in
Figure 11 are instead derived by assuming
$\sigma_{\mathrm{gas,\,z}}=11\;\mbox{$\rm km\,s^{-1}$}$, following exactly the
same prescription in Leroy et al. (2008) and Herrera-Camus et al. (2017) (also
see Section 6.2). We find overall consistency, but our measurements
concentrate more towards the high $P_{\mathrm{DE}}$, high
$\Sigma_{\mathrm{SFR}}$ end. This is because the PHANGS-ALMA CO observations
primarily target the star-forming inner part of high mass disk galaxies, while
the Herrera-Camus et al. (2017) sample focuses on the Hi-dominated outer disks
on purpose, and the Leroy et al. (2008) sample includes many dwarf galaxies.

In addition to previous observational results, we also show in Figure 11 the
predicted $P_{\mathrm{DE}}$–$\Sigma_{\mathrm{SFR}}$ relation from a
hydrodynamic simulation (Kim et al., 2013, black dashed line in the right
panel):

$\frac{\Sigma_{\mathrm{SFR}}}{10^{-3}\;\mbox{$\rm
M_{\odot}\,yr^{-1}\,kpc^{-2}$}}=1.8\,\left(\frac{P_{\mathrm{DE}}}{10^{4}\;\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}}\right)^{1.13}~{}.$
(34)

This relation has a steeper slope than our best-fit power-law relation
(Equation 33), while it agrees with previous observations in the low
$\Sigma_{\mathrm{SFR}}$ regime. We point out that the simulations in Kim et
al. (2013) cover a $\Sigma_{\mathrm{SFR}}$ range from $10^{-4}$ to
$10^{-2}\;\mbox{$\rm M_{\odot}\,yr^{-1}\,kpc^{-2}$}$, whereas our sample
covers from $10^{-3}$ to $10^{-1}\;\mbox{$\rm M_{\odot}\,yr^{-1}\,kpc^{-2}$}$.
The shallower slope found in our sample might then reflect some systematic
change in the properties of the ISM in massive galaxies and inner disk
environments, relative to the ISM in dwarf galaxies and/or outer disk
environments.

A shallower $P_{\mathrm{DE}}$–$\Sigma_{\mathrm{SFR}}$ relation in high
$\Sigma_{\mathrm{SFR}}$ environments is seen in several previous works. In the
Leroy et al. (2008) sample, some hint of a shallower
$\Sigma_{\mathrm{SFR}}$–$P_{\mathrm{DE}}$ relation is visible from the high
$\Sigma_{\mathrm{SFR}}$ measurements. For a sample of local ultra luminous
infra-red galaxies (ULIRGs) and high-redshift star-forming galaxies, Ostriker
& Shetty (2011) found a $\Sigma_{\mathrm{SFR}}$–$P_{\mathrm{DE}}$ relation
with a slope of 0.95 at
$P_{\mathrm{DE}}>10^{5}\,\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}$, which is
closer to our result101010Ostriker & Shetty (2011) assume that the gas self-
gravity term dominates in $P_{\mathrm{DE}}$, and that
$\alpha_{\mathrm{CO}}\propto I_{\mathrm{CO}}^{-0.3}$. More recently, Fisher et
al. (2019) report a much shallower slope of 0.77 at $P_{\mathrm{DE}}\gtrsim
10^{5}\,\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}$ for a sample of local
turbulent disk galaxies111111The $P_{\mathrm{DE}}$ estimates in Fisher et al.
(2019) assume a Galactic $\alpha_{\mathrm{CO}}$ and use the ionized gas
velocity dispersion for $\sigma_{\mathrm{gas,\,z}}$. Newly obtained CO
velocity dispersion for the same galaxies implies systematically lower
$P_{\mathrm{DE}}$ (by $\sim$0.3 dex; D. Fisher, priv. comm.), but no
significant change in the $\Sigma_{\mathrm{SFR}}\text{--}P_{\mathrm{DE}}$
relation slope., which are believed to resemble typical star-forming galaxies
at $z\sim 1\text{--}2$. Future studies on the ISM in more local ULIRGs and
galaxy centers can provide better constraints on the slope of the
$\Sigma_{\mathrm{SFR}}$–$P_{\mathrm{DE}}$ relation at the high
$\Sigma_{\mathrm{SFR}}$ end, and potentially help unveil the physics
regulating star formation in the “starburst” regime (see e.g., Thompson et
al., 2005; Ostriker & Shetty, 2011; Shetty & Ostriker, 2012; Crocker et al.,
2018; Krumholz et al., 2018).

###  7.2 Link to the Molecular-to-Atomic Gas Ratio

Following early suggestions by Elmegreen (1989), the ISM dynamical equilibrium
pressure has also been viewed as a determinant of the molecular/atomic phase
balance in the ISM. In this scenario, $P_{\mathrm{DE}}$ relates closely to the
molecular-to-atomic gas ratio, $R_{\mathrm{mol}}$, and thus influences the
fraction of the ISM in the dense, star-forming phase. This idea has been
tested by many subsequent works (e.g., Wong & Blitz, 2002; Blitz & Rosolowsky,
2006; Leroy et al., 2008), and commonly adopted as a prescription for
determining the molecular gas fraction in semi-analytic models of galaxy
evolution (Lagos et al., 2018, e.g.,). Here we report the observed scaling
relation between molecular-to-atomic ratio and the ISM dynamical equilibrium
pressure across our sample.

Following previous studies, we use the kpc-scale dynamical equilibrium
pressure $P_{\mathrm{DE,\,1kpc}}$ to trace the average ambient pressure in the
ISM. To allow a quantitative comparison with previous results, we again use
the $P_{\mathrm{DE,\,1kpc}}$ values estimated by assuming a fixed
$\sigma_{\mathrm{gas,\,z}}=11\,\mbox{$\rm km\,s^{-1}$}$ (similar to Section
7.1.2). We determine the molecular-to-atomic gas ratio from the ratio of our
measured molecular and atomic gas surface densities on kpc-scale:

$R_{\mathrm{mol,1kpc}}\equiv\Sigma_{\mathrm{mol,\,1kpc}}/\Sigma_{\mathrm{atom,\,1kpc}}~{}.$
(35)

Figure 12:  Molecular to atomic gas ratio $R_{\mathrm{mol,1kpc}}$ as a
function of $P_{\mathrm{DE,\,1kpc}}$. The best-fit power-law relation for the
disk measurements (blue solid line) has a slope of $\alpha=1.02$, and it
crosses the $R_{\mathrm{mol}}=1$ threshold at $P_{0}=2.1\times
10^{4}\,\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}$. Given the systematic
uncertainties associated with the choice of methodology, our best-fit
$R_{\mathrm{mol}}$–$P_{\mathrm{DE}}$ relation is consistent with those
reported in previous studies (black lines; Blitz & Rosolowsky, 2006; Leroy et
al., 2008).

We show the relation between $P_{\mathrm{DE,\,1kpc}}$ and
$R_{\mathrm{mol,1kpc}}$ in Figure 12. Our sample spans nearly two orders of
magnitude in $R_{\mathrm{mol,\,1kpc}}$, with most measurements clustering
around or above the atomic-to-molecular transition threshold (i.e.,
$R_{\mathrm{mol,\,1kpc}}=1$). We find a positive and statistically significant
correlation between $R_{\mathrm{mol,\,1kpc}}$ and $P_{\mathrm{DE,\,1kpc}}$
across our whole sample (Spearman’s rank correlation coefficient $\rho=0.58$;
corresponding p-value $\ll 0.001$). This strong, positive correlation is
qualitatively consistent with previous observations (Wong & Blitz, 2002; Blitz
& Rosolowsky, 2006; Leroy et al., 2008), even though the adopted CO-to-H2
conversion factor, stellar mass-to-light ratio, and stellar disk geometry vary
among studies.

We perform an OLS bisector fit on all our disk measurements over the range
$0.1<R_{\mathrm{mol,\,1kpc}}<10$, following Leroy et al. (2008). This yields a
best-fit power-law relation (blue solid line in Figure 12) of

$R_{\mathrm{mol,\,1kpc}}=\left(\frac{P_{\mathrm{DE,\,1kpc}}}{2.1\times
10^{4}\;\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}}\right)^{1.02}~{}.$ (36)

The scatter in $R_{\mathrm{mol,\,1kpc}}$ around this relation is 0.36 dex. The
formal statistical errors in the fit are small: 0.02 for the slope $\alpha$,
and 0.01 dex for the threshold pressure $P_{0}=2.1\times
10^{4}\,\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}$ (at which the ISM transitions
from being predominantly atomic to molecular, or vice versa). However, when
varying the choice of $\alpha_{\mathrm{CO}}$ prescriptions and other
assumptions, the best-fit $\alpha$ and $P_{0}$ appear systematically uncertain
by $\sim 0.20$ and $\sim 0.15$ dex, respectively. Given this level of
systematic uncertainty, Equation 36 quantitatively agrees with the
$R_{\mathrm{mol}}$–$P_{\mathrm{DE,\,1kpc}}$ relations reported in previous
works ($\alpha=0.73\text{--}1.05$, $P_{0}=1.5\text{--}4.5\times
10^{4}\,\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}$; see Wong & Blitz, 2002; Blitz
& Rosolowsky, 2006; Leroy et al., 2008, also see black lines in Figure 12).

##  8 Summary

For a sample of 28 nearby star-forming disk galaxies, we estimate the pressure
needed to support the ISM against its own weight at a range of spatial scales,
taking into account the combined gravity of all gas components and the stellar
disk. We compare this estimated “dynamical equilibrium pressure” to the
observed turbulent pressure in molecular gas. This tests the common hypothesis
that the ISM in star-forming galaxy disks is in dynamical equilibrium.

For this purpose, we create a multi-wavelength dataset, which includes high
resolution PHANGS-ALMA $\mathrm{CO}\,(2\text{--}1)$ imaging data (A. K. Leroy
et al., in preparation) and GMC catalogs (E. Rosolowsky et al., in
preparation; A. Hughes et al., in preparation), new and archival Hi 21 cm data
(D. Utomo et al., in preparation), processed Spitzer IRAC 3.6 µm data
($\mathrm{S^{4}G}$; Querejeta et al., 2015), and combined GALEX near-UV and
WISE mid-IR data (Z0MGS; Leroy et al., 2019). These data provide us with a
comprehensive picture of the molecular gas distribution and kinematics on
$\sim 60$–$120$ pc scales (i.e., cloud scales), and the distribution of gas
mass, stellar mass, and star formation rate on $\sim 1$ kpc scale.

We divide the 28 galaxies in our sample into 1,762 independent, kpc-sized
hexagonal regions, covering the PHANGS-ALMA CO footprint. In each kpc-sized
region, we use the high resolution CO data to calculate the mass-weighted mean
turbulent pressure in molecular gas, $\left<P_{\mathrm{turb,\,\theta
pc}}\right>$ ($\theta=60$ and $120$ pc; see Section 4.1). We then compare
$\left<P_{\mathrm{turb,\,\theta pc}}\right>$ to the required pressure to
balance the weight of the ISM disk in the galaxy gravitational potential,
a.k.a., the dynamical equilibrium pressure, $P_{\mathrm{DE}}$.

Building on earlier works (Hughes et al., 2013a; Schruba et al., 2019), we
compute two different measures of $P_{\mathrm{DE}}$. One measure, widely
adopted in previous studies, represents the expected mean mid-plane ISM
pressure averaged over all gas, and implicitly assumes no bound sub-
structures. The other measure considered in this work represents the pressure
expected within individual resolved molecular structures on 60–120 pc scales,
allowing them to be self-gravitating.

The first estimate assumes uniform gas surface density within each kpc-sized
aperture. This provides a measure of equilibrium pressure on kpc-scale,
$P_{\mathrm{DE,\,1kpc}}$ (see Section 4.2). With this measure, we find that:

  1. 1.

$P_{\mathrm{DE,\,1kpc}}$ ranges from $10^{3}$ to
$10^{6}\,\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}$ across our sample. This
agrees well with previous estimates of the average ISM pressure in galaxy
disks. The lower bound roughly corresponds to Solar Neighborhood-like
environments, whereas the higher bound corresponds to conditions found in gas-
rich galaxy centers.

  2. 2.

The mass-weighted turbulent pressure $\left<P_{\mathrm{turb,\,\theta
pc}}\right>$ ranges from $10^{4}$ to
$10^{7}\,\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}$.
$\left<P_{\mathrm{turb,\,\theta pc}}\right>$ correlates with
$P_{\mathrm{DE,\,1kpc}}$, but it almost always exceeds the
$P_{\mathrm{DE,\,1kpc}}$ estimate in the same region (Figure 1). At 120 pc
scale, we measure an average $\left<P_{\mathrm{turb,\,\theta
pc}}\right>/P_{\mathrm{DE,\,1kpc}}$ ratio of $\sim 2.8$ across our sample.
That is, molecular gas appears highly over-pressurized compared to the mean
$P_{\mathrm{DE,\,1kpc}}$ calculated assuming a smooth ISM distribution in each
kpc-sized aperture.

  3. 3.

We fit a relation between $\left<P_{\mathrm{turb,\,\theta pc}}\right>$ and
$P_{\mathrm{DE,\,1kpc}}$ (Equation 20) that can be used to predict cloud-scale
molecular gas properties from kpc-resolution observation of distant galaxies,
from low resolution galaxy simulations, or from analytic or semi-analytic
models of star-forming galaxy disks.

In reality, the molecular ISM is highly clumped, and self-gravity plays an
important role in dynamical equilibrium. Indeed, the molecular gas
distribution traced by PHANGS-ALMA on 60–120 pc scales does display this rich
substructure. The presence of this substructure means that
$P_{\mathrm{DE,\,1kpc}}$ under-estimates the required pressure for the clumpy
molecular gas to achieve dynamical equilibrium, because it does not account
for the enhancement of gas self-gravity in over-densities.

We thus introduce a formalism which explicitly incorporates knowledge about
molecular gas clumping, and considers the self-gravity of the molecular gas,
the external gravitational potential, and the pressure in the ambient ISM in a
unified framework. This provides a measure of the equilibrium pressure on
cloud scales, $\left<P_{\mathrm{DE,\,\theta pc}}\right>$ (see Section 4.3).

Applying this formalism, we find that:

  1. 4.

Accounting for the enhanced gas self-gravity due to clumping at small scales,
we estimate that the pressure needed to support the gas is
$\left<P_{\mathrm{DE,\,\theta
pc}}\right>=10^{4}$–$10^{8}\,\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}$,
systematically higher than the kpc-scale estimates of
$P_{\mathrm{DE,\,1kpc}}$.

  2. 5.

$\left<P_{\mathrm{turb,\,\theta pc}}\right>$ and $\left<P_{\mathrm{DE,\,\theta
pc}}\right>$ are nearly equal across most regions in our sample (Figure 2).
This is consistent with the idea that molecular clouds have internal pressures
close to the value needed to balance the sum of their own internal weight and
the weight of the ambient atomic gas.

  3. 6.

In our sample, the self-gravity of cloud-scale molecular gas structures
accounts for $\sim 33-70\%$ of the total $\left<P_{\mathrm{DE,\,\theta
pc}}\right>$. For the molecular gas at high internal pressure ($\gtrsim
10^{5}\,\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}$),
$\left<P_{\mathrm{DE,\,\theta pc}}\right>$ is more likely to be dominated by
the self-gravity term. Gas at low internal pressure is more likely to be
affected by ambient pressure and/or external gravity (Figure 4). However, we
observe large scatter about this general trend, and we see examples for both
scenarios across all environments.

We explore the systematic effects associated with key assumptions (Section 6).
We (1) vary the adopted CO-to-H2 conversion factors in a reasonable range, (2)
choose different assumptions for the stellar disk geometry and/or gas vertical
velocity dispersion, and (3) adopt an alternative, cloud statistic approach
utilizing the PHANGS-ALMA CPROPS cloud catalogs. We find that varying these
assumptions changes the range of the derived pressure estimates, but does not
affect our qualitative conclusion of a strong, nearly unity correlation
between $\left<P_{\mathrm{DE,\,\theta pc}}\right>$ and
$\left<P_{\mathrm{turb,\,\theta pc}}\right>$. Our quantitative measurements do
show mild variations due to these systematic effects, which we report in Table
4.

Based on our analysis and tests of systematic effects, our most general
conclusion is that:

  1. 7.

A close-to-unity $\left<P_{\mathrm{turb,\,\theta
pc}}\right>$–$\left<P_{\mathrm{DE,\,\theta pc}}\right>$ relation holds across
different physical regimes, and is robust against many systematic effects. In
other words, the molecular gas in the disk regions of nearby, massive, star-
forming galaxies does appear to be in or near a state of dynamical
equilibrium.

Beside testing the assumption of dynamical equilibrium, we also investigate
the driving mechanism of this equilibrium. We find that the observed kpc-scale
SFR surface density $\Sigma_{\mathrm{SFR}}$ shows a strong correlation with
both $\left<P_{\mathrm{turb,\,\theta pc}}\right>$ and $P_{\mathrm{DE,\,1kpc}}$
(Figure 11). The ratio between $\left<P_{\mathrm{turb,\,\theta pc}}\right>$
and $\Sigma_{\mathrm{SFR,\,1kpc}}$ is generally consistent with the expected
range of momentum injection from supernova feedback, if one considers the
clumping of star formation on small scales. The nearly linear relationship
between $\Sigma_{\mathrm{SFR,\,1kpc}}$ and $P_{\mathrm{DE,\,1kpc}}$ is
consistent with the feedback-regulated scenario, and in quantitative agreement
with previous observational and theoretical studies where the parameter
regimes overlap. We show that $P_{\mathrm{DE,\,1kpc}}$ correlates positively
with the molecular to atomic ratio, $R_{\mathrm{mol,\,1kpc}}$ (Figure 12). The
best-fit relation we find (Equation 36) is consistent with those reported in
previous studies (Blitz & Rosolowsky, 2006; Leroy et al., 2008).

We publish the estimated $P_{\mathrm{turb}}$, $P_{\mathrm{DE}}$,
$\Sigma_{\mathrm{SFR}}$, and $R_{\mathrm{mol}}$ across our full sample in
machine-readable form (see Table 5 in Appendix C). We encourage the use of
this dataset as a benchmark for future observations and numerical simulations.

In the near future, it will be possible to extend these measurements from the
disks of massive spiral galaxies to a wider set of environments. Sensitive CO
observations targeting dwarf galaxies and the outer disks of spiral galaxies
will reveal whether a similar equilibrium holds in the atomic gas- and/or
external pressure-dominated regimes. A more careful treatment of in-plane
motions and a better modeling of the three-dimensional distribution and
kinematics of the stellar component will provide better understanding of
dynamical equilibrium in early-type galaxies, galaxy bulges, and central
regions.

We thank the anonymous referee for helpful comments that improved the quality
of the paper. J.S. would like to thank R. Herrera-Camus and C. Murugeshan for
kindly sharing their data, and D. Fisher and T. A. Thompson for helpful
discussions. This work was carried out as part of the PHANGS collaboration.
The work of J.S., A.K.L., and D.U. is partially supported by the National
Science Foundation under Grants No. 1615105, 1615109, and 1653300. The work of
E.C.O. was partly supported by NASA under ATP Grant NNX17AG26G. A.H., C.N.H.,
and J.P. acknowledge funding from the Programme National “Physique et Chimie
du Milieu Interstellaire (PCMI)” of CNRS/INSU with INC/INP, co-funded by CEA
and CNES, and from the “Programme National Cosmology et Galaxies (PNCG)” of
CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES. E.R. acknowledges the
support of the Natural Sciences and Engineering Research Council of Canada
(NSERC), funding reference number RGPIN-2017-03987. E.S., C.F., and T.S.
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 694343). J.M.D.K. and M.C. gratefully acknowledge funding from
the Deutsche Forschungsgemeinschaft (DFG) through an Emmy Noether Research
Group (grant number KR4801/1-1) and the DFG Sachbeihilfe (grant number
KR4801/2-1). J.M.D.K. gratefully acknowledges funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme via the ERC Starting Grant MUSTANG (grant agreement
number 714907). F.B. acknowledges funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 726384). S.C.O.G. acknowledges support by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
138713538 – SFB 881 (“The Milky Way System”, sub-projects B01, B02, B08), and
by the Heidelberg cluster of excellence EXC 2181-390900948 “STRUCTURES: A
unifying approach to emergent phenomena in the physical world, mathematics,
and complex data”, funded by the German Excellence Strategy. A.U. acknowledges
support from the Spanish funding grants AYA2016-79006-P (MINECO/FEDER) and
PGC2018-094671-B-I00 (MCIU/AEI/FEDER). This paper makes use of the following
ALMA data: ADS/JAO.ALMA# 2012.1.00650.S, 2015.1.00925.S, 2015.1.00956.S,
2017.1.00392.S, 2017.1.00886.L, and 2018.1.01321.S. ALMA is a partnership of
ESO (representing its member states), NSF (USA), and NINS (Japan), together
with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO, and NAOJ. The National Radio Astronomy Observatory is a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. This work is based in part on
observations made with NSF’s Karl G. Jansky Very Large Array (Legacy ID: AS
1303, AS1387, AS1434, AU157). VLA is also operated by the National Radio
Astronomy Observatory. This work is based in part on observations made with
the Australia Telescope Compact Array (ATCA). ATCA is part of the Australia
Telescope National Facility, which is funded by the Australian Government for
operation as a National Facility managed by CSIRO. This work is based in part
on observations made with the Spitzer Space Telescope, which is operated by
the Jet Propulsion Laboratory, California Institute of Technology under a
contract with NASA. This work is based in part on observations made with the
Galaxy Evolution Explorer (GALEX). GALEX is a NASA Small Explorer, whose
mission was developed in cooperation with the Centre National d’Etudes
Spatiales (CNES) of France and the Korean Ministry of Science and Technology.
GALEX is operated for NASA by the California Institute of Technology under
NASA contract NAS5-98034. This publication makes use of data products from the
Wide-field Infrared Survey Explorer (WISE), which is a joint project of the
University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology, funded by the National
Aeronautics and Space Administration. We acknowledge the usage of the
Extragalactic Distance Database121212http://edd.ifa.hawaii.edu/index.html
(Tully et al., 2009), the HyperLeda database131313http://leda.univ-lyon1.fr
(Makarov et al., 2014), the NASA/IPAC Extragalactic
Database141414http://ned.ipac.caltech.edu, and the SAO/NASA Astrophysics Data
System151515http://www.adsabs.harvard.edu.

##  Appendix A Dynamical Equilibrium on Cloud Scales

In Section 4.3, we determine the cloud-scale dynamical equilibrium pressure in
molecular gas, $\left<P_{\mathrm{DE,\,\theta pc}}\right>$, by adding up the
weight of molecular clouds and the ambient atomic gas in the total
gravitational potential. Here we provide a detailed derivation for each of the
constituent terms included in this calculation.

As illustrated by Figure 14, we approximate the ISM in a galaxy disk as
comprised of two components: 1) a thin, clumpy layer of molecular gas near the
disk mid-plane, which includes many denser molecular clouds in it, and 2) a
smooth, plane-parallel atomic gas outer layer “sandwiching” the molecular gas
layer. We further assume that the vertical scale height of the molecular gas
($H_{\mathrm{mol}}$) and the atomic gas ($H_{\mathrm{atom}}$) are both much
smaller than that of the stellar disk ($H_{\mathrm{\star}}$), with
$H_{\mathrm{mol}}<H_{\mathrm{atom}}$. With this setup, we can consider an
arbitrary molecular cloud in the molecular layer. To compute the equilibrium
pressure at the center of the cloud, we first integrate the weight of the
cloud from its center to its edge along the vertical direction, and then
integrate the weight of the atomic gas above the cloud.

Figure 13:  Figure 14:  A sketch showing the adopted geometrical model for the
dynamical equilibrium calculation. We consider an arbitrary molecular cloud
(dark blue) with radius $R_{\mathrm{cloud}}$ and volume density
$\rho_{\mathrm{cloud}}$. This molecular cloud lives in a molecular gas layer
(light blue), which has a half width of $H_{\mathrm{mol}}=R_{\mathrm{cloud}}$
and an average volume density of $\bar{\rho}_{\mathrm{mol}}$ outside the
molecular cloud in focus. The molecular layer is “sandwiched” by an outer
atomic gas layer (light gray), which has a half width of
$H_{\mathrm{atom}}>H_{\mathrm{mol}}$ and an integrated surface density of
$\Sigma_{\mathrm{atom}}$. This entire multi-phase ISM is centered near the
disk mid-plane (dotted black line), and its overall scale height is much
smaller than that of the galaxy stellar disk.

###  A.1 The Weight of a Molecular Cloud

The weight of a cloud, $\mathcal{W}_{\mathrm{cloud}}$, includes three
constituent parts:

$\mathcal{W}_{\mathrm{cloud}}=\mathcal{W}_{\mathrm{cloud}}^{\mathrm{self}}+\mathcal{W}_{\mathrm{cloud}}^{\mathrm{ext\text{-}mol}}+\mathcal{W}_{\mathrm{cloud}}^{\mathrm{star}}~{}.$
(A1)

These three terms represent the weights due to the cloud’s own self-gravity
($\mathcal{W}_{\mathrm{cloud}}^{\mathrm{self}}$), to the gravity of external
molecular gas outside the cloud
($\mathcal{W}_{\mathrm{cloud}}^{\mathrm{ext\text{-}mol}}$), and to the gravity
of stars ($\mathcal{W}_{\mathrm{cloud}}^{\mathrm{star}}$). By symmetry, the
outer atomic gas layers exert no gravity on any structure in the molecular
layer.

For the cloud self-gravity term, we integrate the weight of a sphere with
radius $R_{\mathrm{cloud}}$ and constant density161616Given our assumption
that turbulent motions dominate the pressure budget inside molecular clouds,
these clouds could still achieve an internal pressure-gravity balance by
having a scale-dependent turbulent velocity field. $\rho_{\mathrm{cloud}}$:

$\displaystyle\mathcal{W}_{\mathrm{cloud}}^{\mathrm{self}}$
$\displaystyle=\int_{0}^{R_{\mathrm{cloud}}}\rho_{\mathrm{cloud}}\frac{G\rho_{\mathrm{cloud}}(4/3)\pi
r^{3}}{r^{2}}\,\mathrm{d}r$
$\displaystyle=\frac{2\pi}{3}G\rho_{\mathrm{cloud}}^{2}R_{\mathrm{cloud}}^{2}$
$\displaystyle=\frac{3\pi}{8}G\Sigma_{\mathrm{cloud}}^{2}~{}.$ (A2)

The last step re-expresses $\mathcal{W}_{\mathrm{cloud}}^{\mathrm{self}}$ in
terms of the cloud surface density $\Sigma_{\mathrm{cloud}}\equiv
M_{\mathrm{cloud}}/(\pi
R_{\mathrm{cloud}}^{2})=(4/3)\rho_{\mathrm{cloud}}R_{\mathrm{cloud}}$.

For the external molecular gas gravity term, we approximate the distribution
of all the molecular gas outside the cloud as a slab centered at the disk mid-
plane. This slab has constant volume density $\bar{\rho}_{\mathrm{mol}}$
anywhere outside the cloud, and has zero density within the extent of the
cloud (i.e., it has a spherical “hole”). For simplicity, we further assume
that the vertical half width of this slab is equal to the cloud radius,
$H_{\mathrm{mol}}=R_{\mathrm{cloud}}$. The weight of the cloud due to the
gravity of this component is thus:

$\displaystyle\mathcal{W}_{\mathrm{cloud}}^{\mathrm{ext\text{-}mol}}$
$\displaystyle=\mathcal{W}_{\mathrm{cloud,slab}}-\mathcal{W}_{\mathrm{cloud,hole}}$
$\displaystyle=\int_{0}^{R_{\mathrm{cloud}}}\rho_{\mathrm{cloud}}\,(4\pi
G\bar{\rho}_{\mathrm{mol}}z)\,\mathrm{d}z-\int_{0}^{R_{\mathrm{cloud}}}\rho_{\mathrm{cloud}}\frac{G\bar{\rho}_{\mathrm{mol}}(4/3)\pi
r^{3}}{r^{2}}\,\mathrm{d}r$ $\displaystyle=2\pi
G\bar{\rho}_{\mathrm{mol}}\rho_{\mathrm{cloud}}R_{\mathrm{cloud}}^{2}-\frac{2\pi}{3}G\bar{\rho}_{\mathrm{mol}}\rho_{\mathrm{cloud}}R_{\mathrm{cloud}}^{2}$
$\displaystyle=\frac{4\pi}{3}G\bar{\rho}_{\mathrm{mol}}\rho_{\mathrm{cloud}}R_{\mathrm{cloud}}^{2}$
$\displaystyle=\frac{\pi}{2}G\bar{\Sigma}_{\mathrm{mol}}\Sigma_{\mathrm{cloud}}~{}.$
(A3)

The last step re-expresses
$\mathcal{W}_{\mathrm{cloud}}^{\mathrm{ext\text{-}mol}}$ in terms of the cloud
surface density $\Sigma_{\mathrm{cloud}}$ and the average surface density of
the slab
$\bar{\Sigma}_{\mathrm{mol}}=\bar{\rho}_{\mathrm{mol}}\,(2H_{\mathrm{mol}})=\bar{\rho}_{\mathrm{mol}}\,(2R_{\mathrm{cloud}})$.

For the stellar gravity term, given that $R_{\mathrm{cloud}}\ll
H_{\mathrm{\star}}$, we treat the stellar disk as having a uniform density
near the mid-plane, $\rho_{\mathrm{\star}}$. We thus have:

$\displaystyle\mathcal{W}_{\mathrm{cloud}}^{\mathrm{star}}$
$\displaystyle=\int_{0}^{R_{\mathrm{cloud}}}\rho_{\mathrm{cloud}}\,(4\pi
G\rho_{\mathrm{\star}}z)\,\mathrm{d}z$ $\displaystyle=2\pi
G\rho_{\mathrm{\star}}\rho_{\mathrm{cloud}}R_{\mathrm{cloud}}^{2}$
$\displaystyle=\frac{3\pi}{2}G\rho_{\mathrm{\star}}\Sigma_{\mathrm{cloud}}R_{\mathrm{cloud}}~{}.$
(A4)

###  A.2 The Weight of the Outer Atomic Gas Layer

The weight of the atomic gas layer, $\mathcal{W}_{\mathrm{atom}}$, also
consists of three parts:

$\mathcal{W}_{\mathrm{atom}}=\mathcal{W}_{\mathrm{atom}}^{\mathrm{self}}+\mathcal{W}_{\mathrm{atom}}^{\mathrm{mol}}+\mathcal{W}_{\mathrm{atom}}^{\mathrm{star}}~{}.$
(A5)

These terms represent the weights due to the atomic layer’s self-gravity
($\mathcal{W}_{\mathrm{atom}}^{\mathrm{self}}$), to the gravity of the inner
molecular gas layer ($\mathcal{W}_{\mathrm{atom}}^{\mathrm{mol}}$), and to the
gravity of stars ($\mathcal{W}_{\mathrm{atom}}^{\mathrm{star}}$). The
calculation of the first and last terms is very similar to that of
$P_{\mathrm{DE,\,1kpc}}$ in Section 4.2. We simply quote the results here:

$\displaystyle\mathcal{W}_{\mathrm{atom}}^{\mathrm{self}}$
$\displaystyle=\frac{\pi}{2}G\Sigma_{\mathrm{atom}}^{2}$ (A6)
$\displaystyle\mathcal{W}_{\mathrm{atom}}^{\mathrm{star}}$
$\displaystyle=\Sigma_{\mathrm{atom}}\sqrt{2G\rho_{\mathrm{\star}}}\,\sigma_{\mathrm{atom,\,z}};$
(A7)

Note that the second expression adopts the assumption that
$H_{\mathrm{mol}}\ll H_{\mathrm{atom}}$.

The molecular gravity term, however, is slightly different from the version in
the $P_{\mathrm{DE,\,1kpc}}$ calculation. As the entire molecular gas inner
layer is assumed to be “sandwiched” by the atomic gas outer layer in this
calculation, the latter feels the full gravity of the former, such that:

$\mathcal{W}_{\mathrm{atom}}^{\mathrm{mol}}=\pi
G\bar{\Sigma}_{\mathrm{mol}}\Sigma_{\mathrm{atom}}~{}.$ (A8)

In reality, the molecular and atomic medium (or at least the cold atomic
phase) are often well mixed, so this estimate represents an upper limit for
the true $\mathcal{W}_{\mathrm{atom}}^{\mathrm{mol}}$ value.

###  A.3 The Total Weight

Combining all the above derivations together, we have:

$\displaystyle\mathcal{W}_{\mathrm{total}}=$
$\displaystyle\,\mathcal{W}_{\mathrm{cloud}}+\mathcal{W}_{\mathrm{atom}}$
$\displaystyle=$
$\displaystyle\,\Sigma_{\mathrm{cloud}}\left(\frac{3\pi}{8}G\Sigma_{\mathrm{cloud}}+\frac{\pi}{2}G\bar{\Sigma}_{\mathrm{mol}}+\frac{3\pi}{2}G\rho_{\mathrm{\star}}R_{\mathrm{cloud}}\right)$
$\displaystyle+\Sigma_{\mathrm{atom}}\left(\frac{\pi}{2}G\Sigma_{\mathrm{atom}}+\pi
G\bar{\Sigma}_{\mathrm{mol}}+\sqrt{2G\rho_{\mathrm{\star}}}\,\sigma_{\mathrm{atom,\,z}}\right)~{}.$
(A9)

This gives the estimated total weight of a molecular cloud and the atomic gas
above it in a galaxy disk, or equivalently, the required pressure in this
molecular cloud to keep it under dynamical equilibrium.

##  Appendix B Stellar Disk Flattening Ratio

In this paper, we assume a constant stellar disk flattening ratio
$R_{\mathrm{\star}}/H_{\mathrm{\star}}=7.3$ when estimating the stellar mass
volume density near the disk mid-plane (see Section 4.2). This value is
suggested by Kregel et al. (2002), and has been widely adopted in recent
studies on similar topics (e.g., Leroy et al., 2008; Ostriker et al., 2010;
Gallagher et al., 2018b).

Kregel et al. (2002) derived this average
$R_{\mathrm{\star}}/H_{\mathrm{\star}}$ value from careful analysis of 34
nearby, edge-on galaxies. To provide an improved estimate with better
statistics, here we do a similar calculation for a much larger sample of 313
edge-on galaxies selected from the $\mathrm{S^{4}G}$ sample (Sheth et al.,
2010).

We select 313 edge-on galaxies from the $\mathrm{S^{4}G}$ parent sample based
on the structural decomposition results published by Salo et al. (2015). We
pick galaxies in which Salo et al. (2015) fit the shape of an edge-on disk
component (a.k.a., “edgedisk”) that accounts for at least 50% of the light of
the galaxy. We then calculate disk flattening ratios based on the exponential
scale length and scale height of this flux-dominating “edgedisk” component,
using the measurement in their data table. Note that in their structural
decomposition analysis, Salo et al. (2015) assumed exponential density profile
along both the radial and vertical direction. This is different from the
isothermal vertical density profile assumed in this work (see Section 4.2).
Rather than redoing their entire structural decomposition analysis, we instead
use 0.5 times their (exponential) scale height values to approximate the scale
height we would have measured assuming isothermal profile. This correction
factor of 0.5 comes from the fact that the mid-plane stellar volume density is
$\rho_{\mathrm{\star}}=\Sigma_{\mathrm{\star}}/4H_{\mathrm{\star}}$ for an
isothermal profile, and
$\rho_{\mathrm{\star}}=\Sigma_{\mathrm{\star}}/2H_{\mathrm{\star}}$ for an
exponential profile (see e.g., van der Kruit, 1988).

Figure 16 shows the histogram of $R_{\mathrm{\star}}/H_{\mathrm{\star}}$
measured from the 313 $\mathrm{S^{4}G}$ edge-on galaxies. We find that the
median and 16–84% range of this distribution (black dot with an error bar) is

$R_{\mathrm{\star}}/H_{\mathrm{\star}}=7.3^{+2.6}_{-1.9}~{}.$ (B1)

This agrees perfectly with the $R_{\mathrm{\star}}/H_{\mathrm{\star}}=7.3$
value (blue dashed line) suggested by Kregel et al. (2002), and supports the
appropriateness of adopting this $R_{\mathrm{\star}}/H_{\mathrm{\star}}$ value
for nearby disk galaxies. The 16–84% range of this distribution can also be
translated to a 1$\sigma$ scatter of 0.13 dex in logarithmic space. This is
the corresponding systematic uncertainty associated with this fixed
$R_{\mathrm{\star}}/H_{\mathrm{\star}}$ ratio in all derived quantities that
depend linearly on it (e.g., $\rho_{\mathrm{\star,\,1kpc}}$).

Figure 15:  Figure 16:  Distribution of stellar disk flattening ratios
($R_{\mathrm{\star}}/H_{\mathrm{\star}}$), as derived for 313 edge-on galaxies
based on the $\mathrm{S^{4}G}$ galaxy structural decomposition catalog (Salo
et al., 2015). The black dot with an error bar shows the median and 16–84%
range of the same distribution:
$R_{\mathrm{\star}}/H_{\mathrm{\star}}=7.3^{+2.6}_{-1.9}$. This agrees well
with the $R_{\mathrm{\star}}/H_{\mathrm{\star}}=7.3$ value derived by Kregel
et al. (2002, blue dashed line) in a much smaller galaxy sample.

##  Appendix C Table of Key Measurements

Table 5 presents a collection of the key measurements derived in this work.
Due to the space limit, in the print version we only include measurements in
10 apertures in the galaxy NGC 628. The full data table, which includes all
measurements across the entire sample, is available in the online journal.

The content of the individual columns is as follows. (1) ‘Galaxy’: name of the
host galaxy; (2) ‘inDisk’: whether each row corresponds to a “disk” aperture
(see Section 3.2); (3) ‘fCO120pc’: CO flux recovery fraction on 120 pc scale
(see Section 3.1 and Figure 2a); (4) ‘Pturb120pc’: turbulent pressure in the
molecular gas, measured on 120 pc scale (see Section 4.1 and Figure 2a); (5)
‘PDE120pc’: dynamical equilibrium pressure estimated on 120 pc scales (see
Section 4.3 and Figure 2a); (6) ‘fCO60pc’: CO flux recovery fraction on 60 pc
scales (see Section 3.1 and Figure 2b); (7) ‘Pturb60pc’: turbulent pressure in
the molecular gas, measured on 60 pc scales (see Section 4.1 and Figure 2b);
(8) ‘PDE60pc’: dynamical equilibrium pressure estimated on 60 pc scales (see
Section 4.3 and Figure 2b); (9) ‘PDEkpc’: dynamical equilibrium pressure
estimated on kpc scales (see Section 4.2 and Figure 1); (10) ‘PDEkpc11’:
dynamical equilibrium pressure estimated on kpc scales, assuming
$\sigma_{\mathrm{gas,\,z}}=11\,\mbox{$\rm km\,s^{-1}$}$ (see Section 7.1.2,
Figure 11b and Figure 12); (11) ‘SigSFRkpc’: star formation rate surface
density estimated on kpc scales (see Section 3.3.2 and Figure 11); (12)
‘Rmolkpc’: molecular-to-atomic gas ratio estimated on kpc scales (see Section
7.2 and Figure 12).

=12em

Table 5: Table of Key Measurements

Galaxy | inDisk | fCO120pc | Pturb120pc | PDE120pc | fCO60pc | Pturb60pc | PDE60pc | PDEkpc | PDEkpc11 | SigSFRkpc | Rmolkpc
---|---|---|---|---|---|---|---|---|---|---|---
|  |  | $[\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}]$ | $[\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}]$ |  | $[\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}]$ | $[\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}]$ | $[\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}]$ | $[\mbox{$k_{\mathrm{B}}\rm\,K\,cm^{-3}$}]$ | $[\mbox{$\rm M_{\odot}\,yr^{-1}\,kpc^{-2}$}]$ |
NGC0628 | 0 | 0.983 | 8.131e+04 | 1.717e+05 | 0.919 | 1.545e+05 | 1.581e+05 | 6.593e+04 | 1.048e+05 | 1.903e-02 | 1.749e+01
NGC0628 | 0 | 0.962 | 4.969e+04 | 1.017e+05 | 0.896 | 1.280e+05 | 1.255e+05 | 3.600e+04 | 6.047e+04 | 1.207e-02 | 9.148e+00
NGC0628 | 0 | 0.925 | 5.763e+04 | 1.011e+05 | 0.861 | 1.298e+05 | 1.136e+05 | 4.345e+04 | 6.780e+04 | 1.288e-02 | 6.907e+00
NGC0628 | 0 | 0.923 | 1.195e+05 | 1.489e+05 | 0.829 | 2.743e+05 | 1.959e+05 | 3.832e+04 | 5.569e+04 | 1.383e-02 | 7.988e+00
NGC0628 | 0 | 0.922 | 8.926e+04 | 1.288e+05 | 0.838 | 1.778e+05 | 1.388e+05 | 3.792e+04 | 5.817e+04 | 1.156e-02 | 9.519e+00
NGC0628 | 0 | 0.966 | 1.095e+05 | 1.460e+05 | 0.866 | 2.306e+05 | 1.768e+05 | 3.416e+04 | 5.070e+04 | 1.043e-02 | 8.183e+00
NGC0628 | 0 | 0.869 | 4.755e+04 | 7.255e+04 | 0.721 | 9.907e+04 | 8.932e+04 | 3.018e+04 | 4.894e+04 | 1.113e-02 | 8.417e+00
NGC0628 | 1 | 0.817 | 1.530e+04 | 2.377e+04 | 0.491 | 3.771e+04 | 2.922e+04 | 1.776e+04 | 3.061e+04 | 6.582e-03 | 5.322e+00
NGC0628 | 1 | 0.812 | 1.451e+04 | 2.383e+04 | 0.500 | 3.025e+04 | 2.799e+04 | 2.136e+04 | 3.360e+04 | 7.466e-03 | 3.542e+00
NGC0628 | 1 | 0.819 | 2.781e+04 | 4.654e+04 | 0.662 | 7.480e+04 | 6.645e+04 | 2.127e+04 | 3.508e+04 | 8.680e-03 | 4.681e+00
… | … | … | … | … | … | … | … | … | … | … | …

Note. — This table is available in its entirety in the PHANGS CADC storage.

## References

  * Accurso et al. (2017) Accurso, G., Saintonge, A., Catinella, B., et al. 2017, MNRAS, 470, 4750, doi: 10.1093/mnras/stx1556
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Benincasa et al. (2016) Benincasa, S. M., Wadsley, J., Couchman, H. M. P., & Keller, B. W. 2016, MNRAS, 462, 3053, doi: 10.1093/mnras/stw1741
  * Bertoldi & McKee (1992) Bertoldi, F., & McKee, C. F. 1992, ApJ, 395, 140, doi: 10.1086/171638
  * Binney et al. (1991) Binney, J., Gerhard, O. E., Stark, A. A., Bally, J., & Uchida, K. I. 1991, MNRAS, 252, 210, doi: 10.1093/mnras/252.2.210
  * Blitz (1993) Blitz, L. 1993, in Protostars and Planets III, 125–161.  http://adsabs.harvard.edu/abs/1993prpl.conf..125B
  * Blitz & Rosolowsky (2004) Blitz, L., & Rosolowsky, E. 2004, ApJ, 612, L29, doi: 10.1086/424661
  * Blitz & Rosolowsky (2006) —. 2006, ApJ, 650, 933, doi: 10.1086/505417
  * Bolatto et al. (2008) Bolatto, A. D., Leroy, A. K., Rosolowsky, E., Walter, F., & Blitz, L. 2008, ApJ, 686, 948, doi: 10.1086/591513
  * Bolatto et al. (2013) Bolatto, A. D., Wolfire, M., & Leroy, A. K. 2013, ARA&A, 51, 207, doi: 10.1146/annurev-astro-082812-140944
  * Bolatto et al. (2011) Bolatto, A. D., Leroy, A. K., Jameson, K., et al. 2011, ApJ, 741, 12, doi: 10.1088/0004-637X/741/1/12
  * Caldú-Primo et al. (2013) Caldú-Primo, A., Schruba, A., Walter, F., et al. 2013, AJ, 146, 150, doi: 10.1088/0004-6256/146/6/150
  * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763, doi: 10.1086/376392
  * Chevance et al. (2020) Chevance, M., Kruijssen, J. M. D., Hygate, A. P. S., et al. 2020, MNRAS accepted, doi: 10.1093/mnras/stz3525
  * Chung et al. (2009) Chung, A., van Gorkom, J. H., Kenney, J. D. P., Crowl, H., & Vollmer, B. 2009, AJ, 138, 1741, doi: 10.1088/0004-6256/138/6/1741
  * Colombo et al. (2014) Colombo, D., Hughes, A., Schinnerer, E., et al. 2014, ApJ, 784, 3, doi: 10.1088/0004-637X/784/1/3
  * Colombo et al. (2019) Colombo, D., Rosolowsky, E., Duarte-Cabral, A., et al. 2019, MNRAS, 483, 4291, doi: 10.1093/mnras/sty3283
  * Comerón et al. (2011) Comerón, S., Elmegreen, B. G., Knapen, J. H., et al. 2011, ApJ, 741, 28, doi: 10.1088/0004-637X/741/1/28
  * Comerón et al. (2012) Comerón, S., Elmegreen, B. G., Salo, H., et al. 2012, ApJ, 759, 98, doi: 10.1088/0004-637X/759/2/98
  * Crocker et al. (2018) Crocker, R. M., Krumholz, M. R., Thompson, T. A., & Clutterbuck, J. 2018, MNRAS, 478, 81, doi: 10.1093/mnras/sty989
  * Crutcher (1999) Crutcher, R. M. 1999, ApJ, 520, 706, doi: 10.1086/307483
  * Donovan Meyer et al. (2013) Donovan Meyer, J., Koda, J., Momose, R., et al. 2013, ApJ, 772, 107, doi: 10.1088/0004-637X/772/2/107
  * Egusa et al. (2018) Egusa, F., Hirota, A., Baba, J., & Muraoka, K. 2018, ApJ, 854, 90, doi: 10.3847/1538-4357/aaa76d
  * El-Badry et al. (2019) El-Badry, K., Ostriker, E. C., Kim, C.-G., Quataert, E., & Weisz, D. R. 2019, MNRAS, 490, 1961, doi: 10.1093/mnras/stz2773
  * Elmegreen (1989) Elmegreen, B. G. 1989, ApJ, 338, 178, doi: 10.1086/167192
  * Elmegreen (1993) —. 1993, ApJ, 411, 170, doi: 10.1086/172816
  * Elmegreen & Parravano (1994) Elmegreen, B. G., & Parravano, A. 1994, ApJ, 435, L121, doi: 10.1086/187609
  * Falgarone et al. (2008) Falgarone, E., Troland, T. H., Crutcher, R. M., & Paubert, G. 2008, A&A, 487, 247, doi: 10.1051/0004-6361:200809577
  * Faucher-Giguère et al. (2013) Faucher-Giguère, C.-A., Quataert, E., & Hopkins, P. F. 2013, MNRAS, 433, 1970, doi: 10.1093/mnras/stt866
  * Feldmann et al. (2012) Feldmann, R., Gnedin, N. Y., & Kravtsov, A. V. 2012, ApJ, 747, 124, doi: 10.1088/0004-637X/747/2/124
  * Field et al. (2011) Field, G. B., Blackman, E. G., & Keto, E. R. 2011, MNRAS, 416, 710, doi: 10.1111/j.1365-2966.2011.19091.x
  * Fisher et al. (2019) Fisher, D. B., Bolatto, A. D., White, H., et al. 2019, ApJ, 870, 46, doi: 10.3847/1538-4357/aaee8b
  * Freeman et al. (2017) Freeman, P., Rosolowsky, E., Kruijssen, J. M. D., Bastian, N., & Adamo, A. 2017, MNRAS, 468, 1769, doi: 10.1093/mnras/stx499
  * Gallagher et al. (2018a) Gallagher, M. J., Leroy, A. K., Bigiel, F., et al. 2018a, ApJ, 868, L38, doi: 10.3847/2041-8213/aaf16a
  * Gallagher et al. (2018b) —. 2018b, ApJ, 858, 90, doi: 10.3847/1538-4357/aabad8
  * Gentry et al. (2019) Gentry, E. S., Krumholz, M. R., Madau, P., & Lupi, A. 2019, MNRAS, 483, 3647, doi: 10.1093/mnras/sty3319
  * Genzel et al. (2010) Genzel, R., Tacconi, L. J., Gracia-Carpio, J., et al. 2010, MNRAS, 407, 2091, doi: 10.1111/j.1365-2966.2010.16969.x
  * Glover & Mac Low (2011) Glover, S. C. O., & Mac Low, M. M. 2011, MNRAS, 412, 337, doi: 10.1111/j.1365-2966.2010.17907.x
  * Grasha et al. (2018) Grasha, K., Calzetti, D., Bittle, L., et al. 2018, MNRAS, 481, 1016, doi: 10.1093/mnras/sty2154
  * Grasha et al. (2019) Grasha, K., Calzetti, D., Adamo, A., et al. 2019, MNRAS, 483, 4707, doi: 10.1093/mnras/sty3424
  * Herrera et al. (2019) Herrera, C. N., Pety, J., Hughes, A., et al. 2019, A&A accepted.  https://arxiv.org/abs/1910.14311
  * Herrera-Camus et al. (2017) Herrera-Camus, R., Bolatto, A., Wolfire, M., et al. 2017, ApJ, 835, 201, doi: 10.3847/1538-4357/835/2/201
  * Herrera-Endoqui et al. (2015) Herrera-Endoqui, M., Díaz-García, S., Laurikainen, E., & Salo, H. 2015, A&A, 582, A86, doi: 10.1051/0004-6361/201526047
  * Heyer & Dame (2015) Heyer, M., & Dame, T. M. 2015, ARA&A, 53, 583, doi: 10.1146/annurev-astro-082214-122324
  * Heyer et al. (2009) Heyer, M., Krawczyk, C., Duval, J., & Jackson, J. M. 2009, ApJ, 699, 1092, doi: 10.1088/0004-637X/699/2/1092
  * Heyer & Brunt (2004) Heyer, M. H., & Brunt, C. M. 2004, ApJ, 615, L45, doi: 10.1086/425978
  * Heyer et al. (2001) Heyer, M. H., Carpenter, J. M., & Snell, R. L. 2001, ApJ, 551, 852, doi: 10.1086/320218
  * Hirota et al. (2018) Hirota, A., Egusa, F., Baba, J., et al. 2018, PASJ, 70, 73, doi: 10.1093/pasj/psy071
  * Hughes et al. (2013a) Hughes, A., Meidt, S. E., Colombo, D., et al. 2013a, ApJ, 779, 46, doi: 10.1088/0004-637X/779/1/46
  * Hughes et al. (2013b) Hughes, A., Meidt, S. E., Schinnerer, E., et al. 2013b, ApJ, 779, 44, doi: 10.1088/0004-637X/779/1/44
  * Ibáñez-Mejía et al. (2017) Ibáñez-Mejía, J. C., Mac Low, M.-M., Klessen, R. S., & Baczynski, C. 2017, ApJ, 850, 62, doi: 10.3847/1538-4357/aa93fe
  * Iffrig & Hennebelle (2015) Iffrig, O., & Hennebelle, P. 2015, A&A, 576, A95, doi: 10.1051/0004-6361/201424556
  * Isobe et al. (1990) Isobe, T., Feigelson, E. D., Akritas, M. G., & Babu, G. J. 1990, ApJ, 364, 104, doi: 10.1086/169390
  * Johnson et al. (2015) Johnson, K. E., Leroy, A. K., Indebetouw, R., et al. 2015, ApJ, 806, 35, doi: 10.1088/0004-637X/806/1/35
  * Kennicutt et al. (2011) Kennicutt, R. C., Calzetti, D., Aniano, G., et al. 2011, PASP, 123, 1347, doi: 10.1086/663818
  * Keto & Myers (1986) Keto, E. R., & Myers, P. C. 1986, ApJ, 304, 466, doi: 10.1086/164181
  * Kim et al. (2011) Kim, C.-G., Kim, W.-T., & Ostriker, E. C. 2011, ApJ, 743, 25, doi: 10.1088/0004-637X/743/1/25
  * Kim & Ostriker (2015a) Kim, C.-G., & Ostriker, E. C. 2015a, ApJ, 815, 67, doi: 10.1088/0004-637X/815/1/67
  * Kim & Ostriker (2015b) —. 2015b, ApJ, 802, 99, doi: 10.1088/0004-637X/802/2/99
  * Kim & Ostriker (2017) —. 2017, ApJ, 846, 133, doi: 10.3847/1538-4357/aa8599
  * Kim & Ostriker (2018) —. 2018, ApJ, 853, 173, doi: 10.3847/1538-4357/aaa5ff
  * Kim et al. (2013) Kim, C.-G., Ostriker, E. C., & Kim, W.-T. 2013, ApJ, 776, 1, doi: 10.1088/0004-637X/776/1/1
  * Kim et al. (2017) Kim, C.-G., Ostriker, E. C., & Raileanu, R. 2017, ApJ, 834, 25, doi: 10.3847/1538-4357/834/1/25
  * Koyama & Ostriker (2009) Koyama, H., & Ostriker, E. C. 2009, ApJ, 693, 1346, doi: 10.1088/0004-637X/693/2/1346
  * Kreckel et al. (2018) Kreckel, K., Faesi, C., Kruijssen, J. M. D., et al. 2018, ApJ, 863, L21, doi: 10.3847/2041-8213/aad77d
  * Kregel et al. (2002) Kregel, M., van der Kruit, P. C., & de Grijs, R. 2002, MNRAS, 334, 646, doi: 10.1046/j.1365-8711.2002.05556.x
  * Kroupa & Weidner (2003) Kroupa, P., & Weidner, C. 2003, ApJ, 598, 1076, doi: 10.1086/379105
  * Kruijssen & Longmore (2013) Kruijssen, J. M. D., & Longmore, S. N. 2013, MNRAS, 435, 2598, doi: 10.1093/mnras/stt1634
  * Kruijssen et al. (2014) Kruijssen, J. M. D., Longmore, S. N., Elmegreen, B. G., et al. 2014, MNRAS, 440, 3370, doi: 10.1093/mnras/stu494
  * Kruijssen et al. (2019) Kruijssen, J. M. D., Schruba, A., Chevance, M., et al. 2019, Nature, 569, 519, doi: 10.1038/s41586-019-1194-3
  * Krumholz & Burkhart (2016) Krumholz, M. R., & Burkhart, B. 2016, MNRAS, 458, 1671, doi: 10.1093/mnras/stw434
  * Krumholz et al. (2018) Krumholz, M. R., Burkhart, B., Forbes, J. C., & Crocker, R. M. 2018, MNRAS, 477, 2716, doi: 10.1093/mnras/sty852
  * Lagos et al. (2018) Lagos, C. d. P., Tobar, R. J., Robotham, A. S. G., et al. 2018, MNRAS, 481, 3573, doi: 10.1093/mnras/sty2440
  * Lang et al. (2019) Lang, P., Meidt, S. E., Rosolowsky, E., et al. 2019, ApJ submitted
  * Larson (1981) Larson, R. B. 1981, MNRAS, 194, 809, doi: 10.1093/mnras/194.4.809
  * Leroy et al. (2008) Leroy, A. K., Walter, F., Brinks, E., et al. 2008, AJ, 136, 2782, doi: 10.1088/0004-6256/136/6/2782
  * Leroy et al. (2013) Leroy, A. K., Lee, C., Schruba, A., et al. 2013, ApJ, 769, L12, doi: 10.1088/2041-8205/769/1/L12
  * Leroy et al. (2015) Leroy, A. K., Bolatto, A. D., Ostriker, E. C., et al. 2015, ApJ, 801, 25, doi: 10.1088/0004-637X/801/1/25
  * Leroy et al. (2016) Leroy, A. K., Hughes, A., Schruba, A., et al. 2016, ApJ, 831, 16, doi: 10.3847/0004-637X/831/1/16
  * Leroy et al. (2019) Leroy, A. K., Sandstrom, K. M., Lang, D., et al. 2019, ApJS, 244, 24, doi: 10.3847/1538-4365/ab3925
  * Makarov et al. (2014) Makarov, D., Prugniel, P., Terekhova, N., Courtois, H., & Vauglin, I. 2014, A&A, 570, A13, doi: 10.1051/0004-6361/201423496
  * Martizzi et al. (2015) Martizzi, D., Faucher-Giguère, C.-A., & Quataert, E. 2015, MNRAS, 450, 504, doi: 10.1093/mnras/stv562
  * McGaugh & Schombert (2014) McGaugh, S. S., & Schombert, J. M. 2014, AJ, 148, 77, doi: 10.1088/0004-6256/148/5/77
  * McMullin et al. (2007) McMullin, J. P., Waters, B., Schiebel, D., Young, W., & Golap, K. 2007, in Astronomical Data Analysis Software and Systems XVI, 127.  http://adsabs.harvard.edu/abs/2007ASPC..376..127M
  * Meidt (2016) Meidt, S. E. 2016, ApJ, 818, 69, doi: 10.3847/0004-637X/818/1/69
  * Meidt et al. (2019) Meidt, S. E., Leroy, A. K., Rosolowsky, E., et al. 2019, The Astrophysical Journal, submitted
  * Meidt et al. (2014) Meidt, S. E., Schinnerer, E., van de Ven, G., et al. 2014, ApJ, 788, 144, doi: 10.1088/0004-637X/788/2/144
  * Meidt et al. (2018) Meidt, S. E., Leroy, A. K., Rosolowsky, E., et al. 2018, ApJ, 854, 100, doi: 10.3847/1538-4357/aaa290
  * Miville-Deschênes et al. (2017) Miville-Deschênes, M.-A., Murray, N., & Lee, E. J. 2017, ApJ, 834, 57, doi: 10.3847/1538-4357/834/1/57
  * Mogotsi et al. (2016) Mogotsi, K. M., de Blok, W. J. G., Caldú-Primo, A., et al. 2016, AJ, 151, 15, doi: 10.3847/0004-6256/151/1/15
  * Murugeshan et al. (2019) Murugeshan, C., Kilborn, V., Obreschkow, D., et al. 2019, MNRAS, 483, 2398, doi: 10.1093/mnras/sty3265
  * Narayanan et al. (2012) Narayanan, D., Krumholz, M. R., Ostriker, E. C., & Hernquist, L. 2012, MNRAS, 421, 3127, doi: 10.1111/j.1365-2966.2012.20536.x
  * Oka et al. (2001) Oka, T., Hasegawa, T., Sato, F., et al. 2001, ApJ, 562, 348, doi: 10.1086/322976
  * Orr et al. (2018) Orr, M. E., Hayward, C. C., Hopkins, P. F., et al. 2018, MNRAS, 478, 3653, doi: 10.1093/mnras/sty1241
  * Ossenkopf & Mac Low (2002) Ossenkopf, V., & Mac Low, M. M. 2002, A&A, 390, 307, doi: 10.1051/0004-6361:20020629
  * Ostriker et al. (2010) Ostriker, E. C., McKee, C. F., & Leroy, A. K. 2010, ApJ, 721, 975, doi: 10.1088/0004-637X/721/2/975
  * Ostriker & Shetty (2011) Ostriker, E. C., & Shetty, R. 2011, ApJ, 731, 41, doi: 10.1088/0004-637X/731/1/41
  * Pakmor et al. (2017) Pakmor, R., Gómez, F. A., Grand , R. J. J., et al. 2017, MNRAS, 469, 3185, doi: 10.1093/mnras/stx1074
  * Peng et al. (2010) Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2010, AJ, 139, 2097, doi: 10.1088/0004-6256/139/6/2097
  * Pettini & Pagel (2004) Pettini, M., & Pagel, B. E. J. 2004, MNRAS, 348, L59, doi: 10.1111/j.1365-2966.2004.07591.x
  * Pety et al. (2013) Pety, J., Schinnerer, E., Leroy, A. K., et al. 2013, ApJ, 779, 43, doi: 10.1088/0004-637X/779/1/43
  * Querejeta et al. (2015) Querejeta, M., Meidt, S. E., Schinnerer, E., et al. 2015, ApJS, 219, 5, doi: 10.1088/0067-0049/219/1/5
  * Rahner et al. (2019) Rahner, D., Pellegrini, E. W., Glover, S. C. O., & Klessen, R. S. 2019, MNRAS, 483, 2547, doi: 10.1093/mnras/sty3295
  * Rebolledo et al. (2015) Rebolledo, D., Wong, T., Xue, R., et al. 2015, ApJ, 808, 99, doi: 10.1088/0004-637X/808/1/99
  * Rice et al. (2016) Rice, T. S., Goodman, A. A., Bergin, E. A., Beaumont, C., & Dame, T. M. 2016, ApJ, 822, 52, doi: 10.3847/0004-637X/822/1/52
  * Rosolowsky & Blitz (2005) Rosolowsky, E., & Blitz, L. 2005, ApJ, 623, 826, doi: 10.1086/428897
  * Rosolowsky & Leroy (2006) Rosolowsky, E., & Leroy, A. 2006, PASP, 118, 590, doi: 10.1086/502982
  * Saintonge et al. (2017) Saintonge, A., Catinella, B., Tacconi, L. J., et al. 2017, ApJS, 233, 22, doi: 10.3847/1538-4365/aa97e0
  * Salo et al. (2015) Salo, H., Laurikainen, E., Laine, J., et al. 2015, ApJS, 219, 4, doi: 10.1088/0067-0049/219/1/4
  * Sánchez et al. (2014) Sánchez, S. F., Rosales-Ortega, F. F., Iglesias-Páramo, J., et al. 2014, A&A, 563, A49, doi: 10.1051/0004-6361/201322343
  * Sánchez et al. (2019) Sánchez, S. F., Barrera-Ballesteros, J. K., López-Cobá, C., et al. 2019, MNRAS, 484, 3042, doi: 10.1093/mnras/stz019
  * Sandstrom et al. (2013) Sandstrom, K. M., Leroy, A. K., Walter, F., et al. 2013, ApJ, 777, 5, doi: 10.1088/0004-637X/777/1/5
  * Schinnerer et al. (2013) Schinnerer, E., Meidt, S. E., Pety, J., et al. 2013, ApJ, 779, 42, doi: 10.1088/0004-637X/779/1/42
  * Schinnerer et al. (2019) Schinnerer, E., Hughes, A., Leroy, A., et al. 2019, ApJ, 887, 49, doi: 10.3847/1538-4357/ab50c2
  * Schruba et al. (2019) Schruba, A., Kruijssen, J. M. D., & Leroy, A. K. 2019, ApJ, 883, 2, doi: 10.3847/1538-4357/ab3a43
  * Semenov et al. (2017) Semenov, V. A., Kravtsov, A. V., & Gnedin, N. Y. 2017, ApJ, 845, 133, doi: 10.3847/1538-4357/aa8096
  * Semenov et al. (2018) —. 2018, ApJ, 861, 4, doi: 10.3847/1538-4357/aac6eb
  * Sheth et al. (2010) Sheth, K., Regan, M., Hinz, J. L., et al. 2010, PASP, 122, 1397, doi: 10.1086/657638
  * Shetty & Ostriker (2012) Shetty, R., & Ostriker, E. C. 2012, ApJ, 754, 2, doi: 10.1088/0004-637X/754/1/2
  * Solomon et al. (1987) Solomon, P. M., Rivolo, A. R., Barrett, J., & Yahil, A. 1987, ApJ, 319, 730, doi: 10.1086/165493
  * Sormani et al. (2015) Sormani, M. C., Binney, J., & Magorrian, J. 2015, MNRAS, 449, 2421, doi: 10.1093/mnras/stv441
  * Sormani et al. (2019) Sormani, M. C., Treß, R. G., Glover, S. C. O., et al. 2019, MNRAS, 488, 4663, doi: 10.1093/mnras/stz2054
  * Spitzer (1941) Spitzer, Lyman, J. 1941, ApJ, 94, 232, doi: 10.1086/144328
  * Spitzer (1942) —. 1942, ApJ, 95, 329, doi: 10.1086/144407
  * Sun et al. (2018) Sun, J., Leroy, A. K., Schruba, A., et al. 2018, ApJ, 860, 172, doi: 10.3847/1538-4357/aac326
  * Tamburro et al. (2009) Tamburro, D., Rix, H. W., Leroy, A. K., et al. 2009, AJ, 137, 4424, doi: 10.1088/0004-6256/137/5/4424
  * Thompson et al. (2019) Thompson, K. L., Troland, T. H., & Heiles, C. 2019, ApJ, 884, 49, doi: 10.3847/1538-4357/ab364e
  * Thompson et al. (2005) Thompson, T. A., Quataert, E., & Murray, N. 2005, ApJ, 630, 167, doi: 10.1086/431923
  * Troland & Crutcher (2008) Troland, T. H., & Crutcher, R. M. 2008, ApJ, 680, 457, doi: 10.1086/587546
  * Tully et al. (2009) Tully, R. B., Rizzi, L., Shaya, E. J., et al. 2009, AJ, 138, 323, doi: 10.1088/0004-6256/138/2/323
  * Utomo et al. (2015) Utomo, D., Blitz, L., Davis, T., et al. 2015, ApJ, 803, 16, doi: 10.1088/0004-637X/803/1/16
  * Utomo et al. (2018) Utomo, D., Sun, J., Leroy, A. K., et al. 2018, ApJ, 861, L18, doi: 10.3847/2041-8213/aacf8f
  * van der Kruit (1988) van der Kruit, P. C. 1988, A&A, 192, 117.  http://adsabs.harvard.edu/abs/1988A%26A...192..117V
  * van der Kruit & Searle (1981) van der Kruit, P. C., & Searle, L. 1981, A&A, 95, 105.  http://adsabs.harvard.edu/abs/1981A%26A....95..105V
  * Verschuur (1993) Verschuur, G. L. 1993, AJ, 106, 2580, doi: 10.1086/116827
  * Walch & Naab (2015) Walch, S., & Naab, T. 2015, MNRAS, 451, 2757, doi: 10.1093/mnras/stv1155
  * Walker et al. (2018) Walker, D. L., Longmore, S. N., Zhang, Q., et al. 2018, MNRAS, 474, 2373, doi: 10.1093/mnras/stx2898
  * Walter et al. (2008) Walter, F., Brinks, E., de Blok, W. J. G., et al. 2008, AJ, 136, 2563, doi: 10.1088/0004-6256/136/6/2563
  * Williams et al. (1994) Williams, J. P., de Geus, E. J., & Blitz, L. 1994, ApJ, 428, 693, doi: 10.1086/174279
  * Wilson et al. (2011) Wilson, C. D., Warren, B. E., Irwin, J., et al. 2011, MNRAS, 410, 1409, doi: 10.1111/j.1365-2966.2010.17646.x
  * Wolfire et al. (2010) Wolfire, M. G., Hollenbach, D., & McKee, C. F. 2010, ApJ, 716, 1191, doi: 10.1088/0004-637X/716/2/1191
  * Wong & Blitz (2002) Wong, T., & Blitz, L. 2002, ApJ, 569, 157, doi: 10.1086/339287
  * Yang et al. (2007) Yang, C.-C., Gruendl, R. A., Chu, Y.-H., Mac Low, M.-M., & Fukui, Y. 2007, ApJ, 671, 374, doi: 10.1086/522861
  * Yoachim & Dalcanton (2006) Yoachim, P., & Dalcanton, J. J. 2006, AJ, 131, 226, doi: 10.1086/497970