# Feedback and galaxy dynamics: A study of turbulence and star formation in 34
galaxies using the PHANGS survey

Bruce G. Elmegreen11affiliation: Katonah, New York USA<EMAIL_ADDRESS>
###### Abstract

The correlation between interstellar turbulent speed and local star formation
rate surface density, $\Sigma_{\rm SFR}$, is studied using CO observations in
the PHANGS survey. The local velocity dispersion of molecular gas, $\sigma$,
increases with $\Sigma_{\rm SFR}$, but the virial parameter, $\alpha_{\rm
vir}$, is about constant, suggesting the molecular gas remains self-
gravitating. The correlation arises because $\sigma$ depends on the molecular
surface density, $\Sigma_{\rm mol}$, and object cloud mass, $M_{\rm mol}$,
with the usual molecular cloud correlations, while $\Sigma_{\rm SFR}$
increases with both of these quantities because of a nearly constant star
formation efficiency for CO. Pressure fluctuations with $\Delta\Sigma_{\rm
SFR}$ are also examined. Azimuthal variations of molecular pressure, $\Delta
P_{\rm mol}$, have a weaker correlation with $\Delta\Sigma_{\rm SFR}$ than
expected from the power-law correlation between the total quantities,
suggesting slightly enhanced SFR efficiency per molecule in spiral arms.
Dynamical equilibrium pressure and star formation rate correlate well for the
whole sample, as $P_{\rm DE}\propto\Sigma_{\rm SFR}^{1.3}$, which is steeper
than in other studies. The azimuthal fluctuations, $\Delta P_{\rm
DE}(\Delta\Sigma_{\rm SFR})$, follow the total correlation $P_{\rm
DE}(\Sigma_{\rm SFR})$ closely, hinting that some of this correlation may be a
precursor to star formation, rather than a reaction. Galactic dynamical
processes correlate linearly such that $\Sigma_{\rm SFR}\propto(\Sigma_{\rm
gas}R)^{1.0\pm 0.3}$ for total gas surface density $\Sigma_{\rm gas}$ and
galactic dynamical rates, $R$, equal to $\kappa$, $A$, or $\Omega$,
representing epicyclic frequency, shear rate $A$, and orbit rate $\Omega$.
These results suggest important roles for both feedback and galactic dynamics.

##  1 Introduction

Interstellar turbulence is driven by a combination of pressure from young
stars, gravitational and magnetic instabilities, and interactions with
galactic-scale processes such as spiral arms, bars, accretion, and neighboring
galaxies. The proportion of each depends on their relative energy and momentum
densities, which vary from galaxy to galaxy and within each galaxy (see
reviews in Elmegreen & Scalo, 2004; Mac Low & Klessen, 2004; Kritsuk et al.,
2017; Hayward & Hopkins, 2017; Burkhart, 2021).

Star formation is also driven by a combination of processes (see reviews in
McKee & Ostriker, 2007; Krumholz, 2014; Padoan et al., 2014). The rate
generally depends on the gas mass and density, with molecular cloud cores
ultimately undergoing gravitational collapse (Jin et al., 2017; Vázquez-
Semadeni et al., 2019; Rosen et al., 2020). Additional stimuli come from some
of the same processes that generate turbulence, like pressures from young
stars (e.g., Egorov et al., 2023), turbulent flow convergence and cloud
collisions (e.g., Hartmann et al., 2001; Suwannajak et al., 2014; Wu et al.,
2018; Fukui et al., 2021), and spiral shocks (Kim & Ostriker, 2006; Elmegreen
& Elmegreen, 2019). There are also impediments to collapse such as ISM
stabilization by rotation and turbulence (Martig et al., 2008), cloud heating
and disruption by young stars (e.g., Haid et al., 2019; Chevance et al., 2020;
Lu et al., 2020; Grudić et al., 2022), and density restructuring by shear and
cloud collisions in turbulent and galactic-scale flows (Federrath, 2015;
Hayward & Hopkins, 2017, and references therein).

All of these processes interact in complex ways, producing both positive and
negative feedback between turbulence and star formation. For example, low
turbulent speed leads to a more unstable disk, which pumps energy back into
the turbulence directly (Fleck, 1981; Bertin, 1997; Huber & Pfennniger, 2002;
Wada et al., 2002) via swing-amplifier instabilities (Toomre & Kalnajs, 1991)
and the resulting torque-driven disk inflows (Krumholz et al., 2018), and also
increases turbulence indirectly through the resulting enhanced star formation
rate (SFR) and its accompanying pressures (Goldreich & Lynden Bell, 1965).
Computer simulations of these interactions generally agree with observations
on a wide range of scales (e.g. Dobbs et al., 2014; Hung et al., 2019;
Ostriker & Kim, 2022).

The combined processes of turbulence and star formation may be viewed as
comprising two main themes: (1) feedback control of star formation through ISM
pressurization, disk thickening, and stabilization against gravitational
collapse, with the same feedback driving turbulence, and (2) gravity-
controlled star formation through swing-amplified instabilities, spiral-arm
torques, and in-plane accretion tapping galactic potential energy as a source
of turbulence. Both consider molecular cloud internal structure and turbulence
as a result of their formation history and evolution, including collapse and
the smaller-scale effects of feedback. Attempts at comparing these two themes
through their scaling relations are usually inconclusive (e.g., Suwannajak et
al., 2014; Johnson et al., 2018), probably because there is a mixture of them
in most regions and other processes operate too. For example, triggered cloud
and star formation in stellar spiral waves might be considered a third theme,
but it is difficult to recognize in global studies because the net effect on
the SFR is small (Elmegreen & Elmegreen, 1986; Kim et al., 2020) even though
the effect on star formation morphology is large (Elmegreen et al., 2018).

Ostriker et al. (2010) proposed a “pressure-regulated feedback model,” which
predicts that young stars generate turbulence on the scale of the disk
thickness with a velocity dispersion, $\sigma$, that is independent of the SFR
density $\Sigma_{\rm SFR}$ (see also Joung et al., 2009; Faucher-Giguère et
al., 2013). This follows from a balance between the dissipation and generation
rates of vertical momentum when $\Sigma_{\rm SFR}$ scales with the gas
freefall rate at the midplane. Also in this case, $\Sigma_{\rm SFR}$ closely
follows the total pressure (Ostriker & Shetty, 2011; Kim et al., 2011; Shetty
& Ostriker, 2012; Kim et al., 2013; Ostriker & Kim, 2022). This prediction
seems to be at variance with observations inside or among local galaxies,
where the velocity dispersion or kinetic energy density appear to increase
with the SFR (Tamburro et al., 2009; Stilp et al., 2013), but after removing
the average radial profiles of these quantities, Hunter et al. (2021) and
Elmegreen et al. (2022) showed for local dwarf irregulars and spiral galaxies
that the prediction is verified: the H i velocity dispersion and kinetic
energy density in each region of star formation is independent of any excess
or deficit in the SFR compared to the average at that radius. We return to
this method here for the molecular ISM.

Increasing $\sigma$ with $\Sigma_{\rm SFR}$ is found for starbursts and high
redshift galaxies (Lehnert et al., 2009, 2013; Swinbank et al., 2012; Green et
al., 2014; Johnson et al., 2018; Oliva-Altamirano et al., 2018) (but not
always, see Genzel et al. 2011), but the turbulent speeds are much larger than
in local galaxies and could be the result of different processes, such as
accretion energy (Elmegreen & Burkert, 2010), feedback from accretion-driven
star formation (Hung et al., 2019), and gravitational instabilities (Krumholz
et al., 2018).

In contrast to the lack of a correlation between fluctuations in $\sigma$ and
$\Sigma_{\rm SFR}$ for local galaxies, there is a correlation between $\sigma$
and the previous star formation rate. Stilp et al. (2013) found in 17 dwarf
irregular galaxies that H i velocity dispersion correlates with $\Sigma_{\rm
SFR}$ from 30 to 40 Myr ago. Similarly, Hunter et al. (2022) studied 4 dwarfs
and found a correlation between H i dispersion on 400 pc scales and
$\Sigma_{\rm SFR}$ 100 to 200 Myr ago. Hunter et al. (2023) also found a 70 to
140 Myr delay for the correlation in another dwarf and showed it occurred
primarily on a scale of 400 pc. Galaxy simulations suggest the same type of
delay (Orr et al., 2020).

The second prediction of the pressure-regulated feedback model, that
$\Sigma_{\rm SFR}$ scales with total pressure for dynamical equilibrium,
$P_{\rm DE}$ (see Eq. 9 below), has been confirmed by observations in several
studies, where it is written as a power law relation between the two
quantities, with a power $d\log{\Sigma_{\rm SFR}}/d\log{P_{\rm DE}}$ slightly
larger than unity, e.g., 1.21 in Ostriker & Kim (2022). We examine this
relationship again here, getting a similarly good correlation but with a slope
less than unity, $\sim 0.77$. We also find this correlation in the azimuthal
fluctuations of $P_{\rm DE}$ with $\Sigma_{\rm SFR}$, and that the slope of
these fluctuations equals the slope predicted from the power law correlation
between the total quantities. This equivalence implies that the quantities
mutually vary in azimuth according to the same power law as the total
correlation and contain no obvious time delays on the resolution scale (Sect.
3). We also point out that $P_{\rm DE}$ is the sum of two terms that are
commonly used separately for the Kennicutt-Schmidt (KS) (Kennicutt & Evans,
2012) and “Extended” KS (Shi et al., 2018) relations, suggesting that $P_{\rm
DE}(\Sigma_{\rm SFR})$ variations might also be a predecessor to star
formation, like gas in the KS relations.

The influence of galactic-scale processes on SFR has also been examined
numerous times with mixed conclusions. With a galactic-scale rate $R$ for
conversion of gas into stars, we might expect $\Sigma_{\rm
SFR}\propto\Sigma_{\rm gas}R$ for total gas surface density $\Sigma_{\rm
gas}$. Leroy et al. (2008) found a steeper result, $\Sigma_{\rm
SFR}/\Sigma_{\rm gas}\sim R^{2}$ for $R=\Omega$, the orbital rate. Suwannajak
et al. (2014) determine fit parameters for 16 nearby galaxies considering
rates $R$ equal to the orbital rate $\Omega$, and something like the shear
rate, $\Omega(1-0.7\beta)$. (The Oort shear rate used below is
$A=0.5\Omega(1-\beta)$; $\beta$ is the local power-law slope of the rotation
curve, $d\log V_{\rm rot}/d\log R$, or $d\log\Omega/d\log R+1$). Suwannajak et
al. (2014) do not give fitted slopes for these relationships, but assume the
slope is unity and derive the rms deviations. Sun et al. (2023) plotted
$\Sigma_{\rm SFR}$ versus $\Sigma_{\rm mol}R$ for molecular surface density
$\Sigma_{\rm mol}$ instead of total gas surface density and for $R$ equal to
the orbital rate and the cloud-scale free-fall rate. The slopes they found are
shallower than unity because the rates get larger for the inner and denser
regions of galaxies, where $\Sigma_{\rm SFR}$ is larger, and the $\Sigma_{\rm
SFR}(\Sigma_{\rm mol})$ relation alone is about linear without the rate
multiplier. Here we plot $\Sigma_{\rm SFR}$ versus $\Sigma_{\rm gas}R$ for
total gas because galactic dynamical processes act on the total gas, and for
epicyclic, shear, and orbital rates, $R$, finding essentially unity slopes for
all of them.

The organization of this paper is as follows. Section 2 considers turbulent
motions in the molecular medium where stars form and feedback is strongest. We
examine PHANGS data (Leroy, et al., 2021) for correlations between CO velocity
dispersion $\sigma$, molecular surface density $\Sigma_{\rm mol}$, molecular
cloud mass $M_{\rm mol}$, and local star formation rate, $\Sigma_{\rm SFR}$,
including also the dimensionless virial parameter $\alpha_{\rm vir}$. We study
both excursions around the average radial profiles, as done previously for H i
to measure local effects (Hunter et al., 2021; Elmegreen et al., 2022), and
the total quantities (Sect. 3) to compare with the Larson (1981) laws for
molecular clouds and conventional star-formation efficiencies in molecular gas
(Krumholz et al., 2019). Because individual molecular clouds are seldom
resolved in other galaxies, we consider both the pixel and object versions of
the PHANGS data compiled by Sun et al. (2022). The pixel version gives
molecular quantities measured on 150 pc scales and averaged over 1.5 kpc
hexagons covering a galaxy disk, and the object versions are from fits to
individual clouds, also averaged over the 1.5 kpc hexagons. We make a
distinction between galaxies with strong and weak spiral arms, as determined
by the range of azimuthal variations in star formation rate density. We
include 17 galaxies of each type, all with stellar masses above $2\times
10^{10}\;M_{\odot}$, to get enough data points in each disk.

To examine the second aspect of the two models mentioned above, we consider
variations in molecular pressure and total dynamical pressure with the local
star formation rate, again relative to their azimuthal averages and in total
(Sect. 4.1). Similarities between the $P_{\rm DE}(\Sigma_{\rm SFR})$ relation,
which suggest that feedback influences $P_{\rm DE}$ because of star formation,
and the inverse but equivalent relation $\Sigma_{\rm SFR}(P_{\rm DE})$, which
suggests that star formation follows the gas and associated dynamical rates,
are discussed in Section 4.2. Finally, we consider how $\Sigma_{\rm SFR}$
varies locally and in total with the three main gas dynamical rates for large-
scale processes, namely, the epicyclic, shear and orbital rates (Sect. 5). The
conclusions are in Section 6.

##  2 Parameter variations with azimuth in the PHANGS data for large galaxies

The PHANGS survey (Leroy, et al., 2021) includes galaxy-wide maps of H i,
$\Sigma_{\rm SFR}$ as determined from several methods, and CO(2-1) using
several formulae for CO-to-gas conversion. Viral parameters, interstellar
pressures and other secondary quantities are also tabulated. Here we use
version “v4p0_public_release” in tabular form
111https:/www.canfar.net/storage/list/phangs/RELEASES/Sun_etal_2022, as
explained in Sun et al. (2022). The $\alpha_{\rm CO}$ conversion used is for
the fiducial case, which has a metallicity dependence. An $\alpha_{\rm CO}$
that depends inversely with velocity dispersion (Chiang et al., 2023) might be
better, but the difference should not be large (Teng et al., 2023).

Figure 1 shows correlated fluctuations between several quantities for the
grand-design galaxy NGC 4321, shown in the insert. These are all fluctuations
in azimuth, meaning that the average value of the quantity at that radius has
been subtracted. In the various panels, the excess molecular surface density
$\Sigma_{\rm mol}$ is shown versus the excess SFR density, $\Sigma_{\rm SFR}$
(lower left), the excess CO velocity dispersion $\sigma$ is shown versus the
excess $\Sigma_{\rm SFR}$ (upper left) and $\Sigma_{\rm mol}$ (upper right),
and the excess virial parameter $\alpha_{\rm vir}$ is shown versus the excess
$\Sigma_{\rm SFR}$ (lower left). For the discussion, we denote excess
quantities by the symbol $\Delta$ and we denote a correlation of A versus B by
A(B). The figure shows a positive correlation for
$\Delta\sigma(\Delta\Sigma_{\rm SFR})$ in the sense that regions with higher
than the azimuthally-averaged $\Sigma_{\rm SFR}$ at their radius also have
higher $\sigma$. We note there are not many regions with low
$\Delta\Sigma_{\rm SFR}$ or low pixel-wise measurements of $\Delta\Sigma_{\rm
mol}$ (blue dots in the top-right) for NGC 4321, although there are sub-zero
$\Delta\sigma$ values. This is because star formation is highly concentrated
in a small fraction of each annulus where there are spiral arms, so the
increases above average are strong but atypical and the decreases below
average are close to the average. The other galaxies studied here do not
generally show such an asymmetry in $\Delta\Sigma_{\rm SFR}$.

The $\Delta\sigma(\Delta\Sigma_{\rm SFR}))$ correlation in the top-left panel
might be a signature for turbulence enhancement by star formation feedback,
but the other panels suggest that feedback contributions to $\sigma$ are
minor: $\Delta\Sigma_{\rm mol}(\Delta\Sigma_{\rm SFR})$ (bottom left) is the
usual molecular KS law and $\Delta\sigma(\Delta\Sigma_{\rm mol})$ (top right)
is one of the usual Larson Laws for a fixed pixel size, so the
$\Delta\sigma(\Delta\Sigma_{\rm SFR})$ correlation could be driven by pre-
existing turbulence through the Larson Law, with regions of higher SFR
following the presence of more molecular mass according to the KS law. This
interpretation will be quantified in several ways in Section 3.

The excess virial parameter in the lower right of Figure 1 is not increasing
as $\Delta\Sigma_{\rm SFR}$ increases, which might be expected if the
molecular gas is dispersing because of feedback. For NGC 4321,
$\Delta\alpha_{\rm vir}$ decreases a little with increasing $\Delta\Sigma_{\rm
SFR}$. This lack of increase for $\alpha_{\rm vir}$ does not alone imply that
feedback does not increase the molecular velocity dispersion, because feedback
could agitate the GMCs which then expand a little as they absorb the
additional energy, while maintaining their near-virial signature. However,
GMCs cannot satisfy both the Larson Laws, which include the condition
$\alpha_{\rm vir}\sim$constant, and increase their velocity dispersions or
radii without increasing their masses, and star formation does not increase a
cloud’s mass. We return to this point also in Section 3 where the slopes of
the Larson power-law relations in our data are shown to be the same as the
slopes of the excursion relations for azimuthal variations.

Figure 1 shows a clear difference in two panels between the values derived
from individual, deconvolved clouds (red points) and value derived inside 150
pc pixels regardless of whether they center on a cloud or not (blue points).
Because of this difference, we maintain this distinction throughout the paper.
The difference manifests as a steeper slope for $\Delta\Sigma_{\rm
mol}(\Delta\Sigma_{\rm SFR})$ and a shallower slope for
$\Delta\sigma(\Delta\Sigma_{\rm mol})$ for the objects than the pixels. This
is probably because $\Sigma_{\rm mol}$ for the pixels is averaged over cloud
and intercloud emissions inside each 150 pc region, giving it a smaller range
than for $\Sigma_{\rm mol}$ peaked-up on each cloud.

The large variations in $\Delta\Sigma_{\rm SFR}$ in Figure 1, amounting to a
variation from $-0.005$ to +0.02$\;M_{\odot}$ pc-2 Myr-1, are mostly the
result of azimuthal variations in the SFR and molecular cloud density with the
spiral phase: both are much higher in the arms. This is in contrast to the
case for a flocculent galaxy, NGC 2775, where the PHANGS values are shown in
Figure 2. Note that the scale for $\Delta\Sigma_{\rm SFR}$ is blown-up by a
factor of 10 compared to Figure 1, and the scale for $\Delta\Sigma_{\rm mol}$
is blown up by a factor of 2. This difference is sensible because flocculent
galaxies have much more uniform SFRs around in azimuth than galaxies with
strong spiral density waves like NGC 4321, where the star formation is
concentrated in the arms. In what follows, the strong-arm and weak arm
galaxies will be distinguished by different colored symbols or different
figures.

Now we consider a larger sample of galaxies from the PHANGS survey. We
examined plots like these for all of the galaxies in release 4 at the link
given above and found that galaxies with stellar masses less than around
$2\times 10^{10}\;M_{\odot}$ had too few CO observations to make good
correlations. We also found that several galaxies with larger masses than this
had too few CO points also. These were eliminated from further study here.
Table 1 lists all three classes of galaxies: strong-arm with a relatively
large range in $\Delta\Sigma_{\rm SFR}$, weak-arm or flocculent galaxies with
a relatively small range in $\Delta\Sigma_{\rm SFR}$, and galaxies with too
few CO observations to be useful here, all with $M_{\rm star}>2\times
10^{10}\;M_{\odot}$.

Figure 3 shows the same quantities as in Figures 1 and 2 but now for the full
sample using pixel values, distinguishing broad-range $\Delta\Sigma_{\rm SFR}$
with blue symbols and narrow-range $\Delta\Sigma_{\rm SFR}$ with red symbols.
The differences between these galaxy types in all the panels are clear, but
within each panel, there are tight correlations for each one. Best fit
correlations are shown by the lines, as determined with bivariable linear fits
because there is a dispersion along each axis. We limit each fit to the ranges
of variables shown in the figure, as some galaxies, particularly near their
nuclei, have much larger variations in some quantities that are far off the
scales here. (In a bivariable fit, we first fit $y=A_{1}+B_{1}x$ for abscissa
$x$ and ordinate $y$ values and then fit $x=A_{2}+B_{2}y$ for the other
direction. The bivariate fit is $y=A_{3}+B_{3}x$ for
$A_{3}=(A_{1}-A_{2}/B_{2})/2\pm(A1+A2/B2)/2$ and
$B_{3}=(B_{1}+1/B_{2})/2\pm(B1-1/B2)/2)$.) The fits are given in Table 2. With
this type of fitting, the inverse correlation, $B(A)$, has a slope equal to
the inverse of the direct correlation, $A(B)$.

Figure 4 shows the same parameters for the cloud-object measurements, with
bivariate linear fits also in Table 2. The pixel and object correlations are
about the same in the two figures, and for each, the difference between broad-
range $\Sigma_{\rm SFR}$ (strong spiral arms, blue symbols) and narrow range
$\Sigma_{\rm SFR}$ (weak arms or flocculent structure with red symbols) are
consistent: the weak-arm correlations are steeper.

We show now that the correlation between excess molecular velocity dispersion
and excess star formation rate density is trivially related to two other
correlations: one between excess velocity dispersion and excess molecular mass
and another between excess molecular mass and excess star formation. Writing

$\Delta\sigma=A_{\sigma,{\rm mol}}+B_{\sigma,{\rm mol}}\Delta\Sigma_{\rm mol}$
(1)

and

$\Delta\Sigma_{\rm mol}=A_{\rm mol,SFR}+B_{\rm mol,SFR}\Delta\Sigma_{\rm SFR}$
(2)

from the fits in Table 2, we expect the coefficients in the third correlation,

$\Delta\sigma=A_{\sigma,{\rm SFR}}^{\prime}+B_{\sigma,{\rm
SFR}}^{\prime}\Delta\Sigma_{\rm SFR}$ (3)

to be

$A_{\sigma,{\rm SFR}}^{\prime}=A_{\sigma,{\rm mol}}+B_{\sigma,{\rm mol}}A_{\rm
mol,SFR}$ (4)

and

$B_{\sigma,{\rm SFR}}^{\prime}=B_{\sigma,{\rm mol}}B_{\rm mol,SFR}.$ (5)

Table 2 gives these values also and shows they compare well with the directly
measured correlation between $\Delta\sigma$ and $\Delta\Sigma_{\rm SFR}$. The
correspondence is not exact because of our fitting limits to the ranges in
Figures 3 and 4.

Because star formation follows molecular mass, it is most likely that the
$\Delta\Sigma_{\rm SFR}(\Delta\Sigma_{\rm mol})$ correlation is primary, or
causal, i.e., star formation depends on the molecular gas. In that case,
because more mass corresponds to higher turbulent speeds through intrinsic
molecular cloud correlations (the “Larson Laws”; Larson, 1981), it is likely
that the $\Delta\sigma(\Delta\Sigma_{\rm SFR})$ correlation is secondary and
not causal. If, conversely, larger $\sigma$ were the result of feedback with a
locally larger SFR, then there would have to be an unphysical addition of
molecular mass after star formation begins to increase $\Sigma_{\rm mol}$ up
to the correlated value with $\sigma$ (considering that the virial parameter
is about constant).

##  3 Total parameter variations in the PHANGS survey

We now look at the full values of $\sigma$, $\Sigma_{\rm mol}$, $\Sigma_{\rm
SFR}$ and $\alpha_{\rm vir}$ in our sample galaxies, without subtracting the
average radial profiles. These are first shown in Figure 5 for NGC 4321 with
red and blue symbols again distinguishing between object measurements and
pixel measurements. The correlations for these two measurement types are about
the same although the object values have slightly higher $\sigma$ and
$\Sigma_{\rm mol}$, and slightly lower $\alpha_{\rm vir}$ than the pixel
values, again most likely because of the averaging process in the pixel
measurement.

Figure 6 shows the same quantities for the pixel measurements in our full
survey, distinguishing between strong spiral-arm (blue symbols) and weak
spiral-arm (red) galaxies. Figure 7 shows the object measurements. In each
figure, the full parameter correlations for molecular clouds are about the
same for the two galaxy types, which means that the Larson Law,
$\sigma(\Sigma_{\rm mol})$ (top right), is nearly independent of spiral arms,
as is the efficiency of star formation per unit CO mass (lower left). The
weak-arm galaxies are systematically shifted toward lower $\Sigma_{\rm SFR}$
and $\Sigma_{\rm mol}$, in addition to having a lower range of these values
with azimuth (cf. Figs. 3 and 4). These shifts in full values are presumably
because either the total star formation rates in weak-arm galaxies are smaller
than in strong-arm galaxies, or because the star formation is more spread-out
in weak-arm galaxies with about the same total rate (depending on the galaxy).

The coefficients $C$ and slopes $D$ of the power law correlations in Figures 6
and 7 are given in Table 3, considering equations like $\log y=C+D\log x$.
These are from bivariate fits in log-log space except for $\alpha_{\rm
vir}(\Sigma_{\rm SFR})$, which is in linear-log space, keeping $\alpha_{\rm
vir}$ linear as in the figures.

Now we ask whether the excursions about the azimuthal averages, shown in
Figures 3 and 4, are the result of more or less CO mass inside each measured
region, following the full correlations seen in Figures 6 and 7. We do this
for both the pixel measurements, which have $\Sigma_{\rm mol}$ for molecular
cloud content in a pixel, and the object measurements, which have both
$\Sigma_{\rm mol}$ and average cloud mass, $M_{\rm mol}$, in each measured
region. Because we have not mentioned $M_{\rm mol}$ yet, Figures 8 and 9 show
how the average molecular cloud mass and the azimuthal variations of this
average molecular mass correlate with $\sigma$ and $\Sigma_{\rm SFR}$ for
strong-arm and weak-arm galaxies, respectively. The top right in each figure,
$\sigma(M_{\rm mol})$, is another standard molecular cloud correlation (Larson
Law). The top left depends on the efficiency of star formation per molecular
cloud (although the abscissa is per unit area and the ordinate is not). Fits
to the power laws involving cloud mass are also in Tables 2 and 3. The dashed
lines in Figure 9 are error limits to the slopes (not shown in the other
figures for clarity).

To determine if the full correlations are entirely responsible for the
azimuthal variations (which is a non-trivial result), we determine the slopes
of the full correlations and compare them to the ratios of excursions in the
azimuthal variations. That is, we examine whether the power-law relations for
total quantities in Figures 6 and 7, such as

$\sigma=10^{C_{\sigma,{\rm SFR}}}\Sigma_{\rm SFR}^{D_{\sigma,{\rm SFR}}},$ (6)

produce the linear relations of the excursion quantities in Figures 3 and 4,
correspondingly,

$\Delta\sigma=A_{\sigma,{\rm SFR}}+B_{\sigma,{\rm SFR}}\Delta\Sigma_{\rm
SFR}.$ (7)

For this to be true for the slope (since the intercept cancels out in the
excursion relations), we need

$B_{\sigma,{\rm SFR}}^{\prime}={D_{\sigma,{\rm
SFR}}}\left({{<\sigma>}\over{<\Sigma_{\rm SFR}>}}\right)$ (8)

to equal the measured slope of the excursion, $B_{\sigma,{\rm SFR}}$, where
the brackets, $<>$, denote averages. Equation 8 comes from the derivative of
the power law, setting $\Delta\sigma/\Delta\Sigma_{\rm
SFR}=d\sigma/d\Sigma_{\rm SFR}$ at the average value of each quantity.

Table 4 compares the slope $B_{\sigma,{\rm SFR}}$ of the correlation between
excess velocity dispersion and excess SFR with the slope
$B^{\prime}_{\sigma,{\rm SFR}}$ of the correlation between total dispersion
and SFR from the power laws. The average values used for equation 8 are from
the averages of the logs of these quantities, since the logs were used in the
fit to the power law. The second and sixth columns indicate that
$B_{\sigma,{\rm SFR}}^{\prime}\sim B_{\sigma,{\rm SFR}}$. This result suggests
that the local excess in $\sigma$ in star-forming regions is the result of the
intrinsic properties of molecular clouds, which have normal power-law
relationships in our galaxies. Because these properties are established prior
to and during star formation, the excess $\sigma$ in a region of star
formation cannot be entirely the result of feedback from that star formation.

##  4 Correlations with Pressure

###  4.1 General Pressure Trends

Having examined correlations between molecular cloud turbulence through the
velocity dispersion $\sigma$ and the local SFR, we turn now to correlations
between local pressure $P$ and SFR. If there is significant feedback, then one
might expect $P$ to increase with SFR, especially for the older regions where
feedback has been working for a long time. However, $P$ also increases with
$\Sigma_{\rm mol}$ and $\sigma$, which, as we have seen, increase in a region
as molecular clouds form before star formation begins, so a correlation like
$\Delta P(\Delta\Sigma_{\rm SFR}$) could also be secondary.

Figure 10 shows various correlations between pressure and SFR, with azimuthal
variations in the bottom panels on a linear scale and total parameter values
in the upper panels on a log scale. We consider three measures of pressure:
$P_{\rm P150}$ is the molecular pressure for 150 pc pixels in the PHANGS
tabulation, averaged over 1.5 kpc hexagons; $P_{\rm O150}$ is the molecular
pressure for cloud objects in 150 pc regions, averaged over 1.5 kpc hexagons
(both from the PHANGS tabulation), and $P_{\rm DE}$ is the dynamical
equilibrium pressure from Ostriker & Kim (2022), written in equation 1 of Sun
et al. (2023):

$P_{\rm DE}={{\pi G}\over{2}}\Sigma_{\rm gas}^{2}+\Sigma_{\rm
gas}\left(2G\rho_{\rm star}\right)^{1/2}\sigma_{\rm gas,z}$ (9)

where $\Sigma_{\rm gas}=\Sigma_{\rm mol}+\Sigma_{\rm atom}$ is the total gas
surface density from molecules and atoms, $\rho_{\rm star}$ is the midplane
stellar density, and $\sigma_{\rm gas,z}$ is the effective vertical component
of the gas velocity dispersion, assumed by Sun et al. (2023) and also here to
be a constant $11$ km s-1 (including magnetic and cosmic-ray pressures). All
of these pressures are uncertain because of resolution limits (e.g., cloud
pressure depends on density which depends on radius) and unknown quantities,
such as $\rho_{\rm star}$ and $\sigma_{\rm gas,z}$ in $P_{\rm DE}$. Also, the
nature of molecular gas motions is not known, e.g., whether they are turbulent
or collapsing, compressional or rotational, etc., and a wide combination of
motions could give $\alpha_{\rm vir}\sim 1$ when self-gravity is important.

The bivariate fits to the three $P(\Sigma_{\rm SFR})$ correlations are in
Table 5. The excursion correlations have linear fits in the second and third
columns, and the total correlations have power law fits in the fourth and
fifth columns.

There are several points to make from Figure 10 and Table 5. First, the
correlations between molecular cloud pressures ($P_{\rm P150}$ and $P_{\rm
O150}$) and pressure excursions ($\Delta$-values) with $\Sigma_{\rm SFR}$ and
its excursions are steeper for weak spirals than strong spirals. For example,
the weak-spiral cloud objects have a $P(\Sigma_{\rm SFR})$ correlation with a
slope of $\sim 16900$ in the table (in units of $10^{4}$ K
cm${}^{-3}/(M_{\odot}$ pc-2 Myr${}^{-1})$), while for strong spirals the slope
is $\sim 7210$, a factor of 2.3 smaller. For molecular pixels, the ratio
between the two slopes is $\sim 15500/5730=2.7$. Similarly for the power-law
relations for total quantities, the powers are $\sim 1.92$ and $\sim 1.59$ for
cloud objects in weak and strong and spirals, respectively, and $\sim 2.5$ and
$\sim 2.0$ for pixel measurements in weak and strong spirals.

The reason for this arm strength difference is not apparent. Another way of
describing it is that a given increase in $\Sigma_{\rm SFR}$ has a greater
impact on $P_{\rm P150}$ and $P_{\rm P150}$ in a weak spiral arm than a strong
spiral arm, or, alternatively, the molecular cloud pressure has a greater
impact on $\Sigma_{\rm SFR}$ in strong arms than weak arms. In this latter
interpretation, the arm would be doing something extra to the gas to stimulate
additional star formation. That is, stronger spiral arms may have a higher
efficiency of star formation in a given molecular cloud than weaker spiral
arms. This is a sensible interpretation, but it implies that molecular cloud
variations, including the pressure variations that accompany different
molecular clouds, are driving the azimuthal star formation variations, rather
than that star formation is pressurizing, more or less, the molecular clouds.

A second point is that azimuthal variations in molecular cloud pressure,
$\Delta P_{\rm P150}$ and $\Delta P_{\rm O150}$, are much larger in ratio to
the SFR variations than the derivative of the total cloud pressures with
$\Sigma_{\rm SFR}$. To see this in Table 5, compare the $B$ values to the
$B^{\prime}$ values just below them for the O150 and P150 pressures; i.e.,
$B>>B^{\prime}$. This result differs from the $\Delta\sigma(\Delta\Sigma_{\rm
SFR})$ correlations discussed in the previous section, which always followed
from the power law relations for the total quantities, $\sigma(\Sigma_{\rm
SFR})$. Perhaps the reason for the difference in pressure correlations is that
the total pressure variations in the power law contain a radial dependence and
a galaxy-to-galaxy variation, which would introduce significant variations in
the background stellar gravity. Then the derivative of total pressure with SFR
density is diluted by the background stars and becomes smaller than the
pressure fluctuations in azimuth, which are nearly pure gas.

A third point is that the spiral-strength differences in cloud pressure
mentioned above contrast with the dynamical pressure, $P_{\rm DE}$, and its
azimuthal variations $\Delta P_{\rm DE}$. The power-law slope of $P_{\rm
DE}(\Sigma_{\rm SFR})$ is about the same in both spiral arm types (1.25 and
1.33). Also, the ratio of excursion pressure to excursion SFR is about equal
to the predicted value from the slope of the total-pressure power law: 979
compared to 909 for strong spirals and 882 compared to 1180 for weak spirals.
These results imply that dynamical pressure and SFR follow each other closely
for both azimuthal and radial variations and from galaxy to galaxy, unlike the
molecular pressures.

Fourth, the azimuthal variations in dynamical pressure are $\sim 10$ times
smaller than the azimuthal variations in molecular cloud pressure (compare the
y-axis limits in the lower right of Figure 10 with the limits in the lower
left and center). Thus, the slope of the $\Delta P_{\rm DE}(\Delta\Sigma_{\rm
SFR})$ correlation is $\sim 10$ times smaller than the slopes of the molecular
cloud correlations. This is true even though the total pressure ranges are
about the same (top panels). This result suggests that molecular clouds come
and go in regions of more or less star formation, and the molecular cloud
pressures have large positive and negative excursions in these regions, but
the dynamical pressure around them hardly changes. This difference could again
be the result of the disk stellar contribution to the dynamical pressure,
whereas molecular cloud pressures contain substantial self-gravity.

Fifth, the molecular cloud pressures are 10 or more times larger than the
dynamical pressures at high $\Sigma_{\rm SFR}$ (comparing the top three panels
in Fig. 10). We can quantify this from the fits to the power laws in Table 5.
At the high-end of the SFR, $\Sigma_{\rm SFR}=0.1\;M_{\odot}$ pc-2 Myr-1, the
three pressures are, on average: $P_{\rm O150}\sim 1.05\times 10^{3}$, $P_{\rm
P150}\sim 1.58\times 10^{3}$ and $P_{\rm DE}\sim 0.145\times 10^{3}$ for
strong-arm spirals, in units of $10^{4}$ K cm-3. These numbers are $5.89\times
10^{3}$, $22.4\times 10^{3}$ and $0.295\times 10^{3}$ for the weak-arm
spirals. Clearly, $P_{\rm O150}$ and $P_{\rm P150}$ are much larger than
$P_{\rm DE}$. At the low-end of the SFR, where $\Sigma_{\rm
SFR}=10^{-3}\;M_{\odot}$ pc-2 Myr-1, all of these pressures are about the
same, ranging from 0.2 to 0.9 in units of $10^{4}$ K cm-3. These differences
again are explained if molecular cloud pressure is dominated by self-gravity.

Sixth, the molecular cloud pressures increase with $\Delta\Sigma_{\rm SFR}$,
unlike what we would expect if feedback energy puffs up a GMC at fixed mass
and gives it a lower surface density, which corresponds to a lower GMC
pressure and a lower virialized velocity dispersion. This observation suggests
again that GMC properties, including pressure and velocity dispersion, are
precursors to star formation tied to the cloud mass, and that $\Sigma_{\rm
SFR}$ follows the cloud mass too.

In summary, the molecular pressure is up to 10 times higher than the dynamical
pressure everywhere, and it is $\sim 10$ times more sensitive to local
$\Sigma_{\rm SFR}$ variations than the dynamical pressure. This is probably
because molecular clouds are at higher pressures than the average ISM as a
result of their strong internal gravities. This is consistent with virial
parameters of order unity in star-forming clouds. If their boundary pressures
were larger than their self-gravitating pressures, then $\alpha_{\rm vir}$
would be much larger than 1.

###  4.2 Reflections on $P_{\rm DE}$: Feedback or Precursor to Star Formation?

The trend for $P_{\rm DE}$ in Figure 10 may be compared with expectations from
previous work. The slopes of the $P_{\rm DE}(\Sigma_{\rm SFR})$ correlations
found here, namely, $1.25\pm 0.30$ for strong spirals and $1.33\pm 0.40$ for
weak spirals, differ from the slopes in Ostriker & Kim (2022) and Sun et al.
(2023). Ostriker & Kim (2022) considered the inverse relationship and obtained
from simulations $\log\Sigma_{\rm SFR}=1.21\log(P_{\rm DE}/k_{\rm B})-7.66$
for $\Sigma_{\rm SFR}$ in units of $M_{\odot}$ pc-2 Myr-1. Rearranging terms,
that would be $\log(P_{\rm DE}/k_{\rm B})=C+D\log(\Sigma_{\rm SFR})$ where
$C=7.66/1.21=6.33$ and $D=1/1.21=0.83$. In table 5, we get $C=7.41$
(converting here to pressure in $k_{\rm B}$ instead of $10^{4}k_{\rm B}$ in
the table) and $D=1.25$ for strong spirals and $C=7.80$, $D=1.33$ for weak
spirals. Although the fitted slopes differ (0.83 compared to 1.3 here), all of
these values give about the same $P_{\rm DE}$ for a typical $\Sigma_{\rm
SFR}=0.01\;M_{\odot}$ pc-2 Myr-1; i.e., Ostriker & Kim (2022) would get
$\log(P_{\rm DE}/k_{\rm B})=4.67$ from their fit, and we get $\log(P_{\rm
DE}/k_{\rm B})=4.91$ and $5.14$ for strong and weak spirals. In their
comparison with observations (Fig. 15) forcing a unit slope, Ostriker & Kim
(2022) suggest $\Sigma_{\rm SFR}=2.07\times 10^{-7}P_{\rm DE}$, which gives
$\log(P_{\rm DE}/k_{\rm B})=4.68$ at $\Sigma_{\rm SFR}=0.01$ for their choice
of feedback yield. Actually, the observations they plot appear to have a
shallower slope than 1.21 for $\Sigma_{\rm SFR}(P_{\rm DE})$, more like unity,
which brings it closer to the slope in Figure 10 here.

Equation 9 contains several assumptions, so the value of the slope could
change with different assumptions. We used a star formation rate from FUV and
WISE $22\mu$ emission, and as a check, redid all the calculations using
$H\alpha$ plus WISE $22\mu$ (both tabulated in release 4 of PHANGS data). The
resultant slope for $P_{\rm DE}(\Sigma_{\rm SFR})$ differed only in the 3rd
decimal place, which is a negligible difference compared to the uncertainty.
Also uncertain is the CO to mass conversion factor, $\alpha_{\rm CO}$, but the
one we use is not much different than others as long as reasonable variations
with metallicity, velocity dispersion or other co-varying parameters are
considered.

More important to the $P_{\rm DE}(\Sigma_{\rm SFR})$ slope is the assumption
about stellar scale height, $H_{\rm star}$, which enters into the stellar
density $\rho_{\rm star}$ during conversion from the stellar surface density.
Ostriker & Kim (2022) use a constant $H_{\rm star}$ in each of two sets of
simulations, while the PHANGS data we use assumes $H_{\rm star}$ increases
with galactocentric radius as in Figure 6 of Sun et al. (2020). Going from a
constant $H_{\rm star}$ to an increasing $H_{\rm star}$ with radius lowers
$P_{\rm DE}$ at low $\Sigma_{\rm SFR}$, steepening the $P_{\rm DE}(\Sigma_{\rm
SFR})$ slope. This makes the inverse Ostriker & Kim (2022) slope of 0.83 (see
above) larger (in the direction of our slope) if variable $H_{\rm star}$ were
used by them. Sun et al. (2023) fit PHANGS data to a slope for $d\Sigma_{\rm
SFR}/dP_{\rm DE}$ equal to 0.93 in the fiducial case, which corresponds to
1.07 for the inverse correlation in our Figure 10. This is closer to our value
of $\sim 1.3$, and Sun et al. (2023) assume Hstar increases with radius as
well. We note also that our slope of $1.3$ for $P_{\rm DE}(\Sigma_{\rm SFR})$
is similar to that found by Fisher et al. (2019) (i.e., $1/0.75=1.3$) for H i
in redshift $z\sim 1$ turbulent disks.

A constant effective gaseous vertical velocity dispersion was also assumed for
equation 9. This effective dispersion is from a combination of thermal and
turbulent motions, with magnetic and cosmic ray contributions. If $\sigma_{\rm
gas,z}$ systematically decreases with radius instead, which seems reasonable
as observed turbulent speeds tend to do this (Stilp et al., 2013), that would
decrease $P_{\rm DE}$ in the outer regions where $\Sigma_{\rm SFR}$ is low,
and increase the slope of $P_{\rm DE}(\Sigma_{\rm SFR})$ even more, taking it
further from unity.

Because of these uncertainties and differing assumptions, we do not consider
the slope of $P_{\rm DE}(\Sigma_{\rm SFR})$ to be significantly different from
other values in the literature. Nevertheless, we note that if the slope is
systematically different from unity, then the feedback yield from star
formation, $P_{\rm DE}/\Sigma_{\rm SFR}$ (Ostriker & Kim, 2022), would have to
vary with pressure or SFR in about the same way for both strong and weak arm
galaxies (Fig. 10 top-right), and also as these quantities vary around in
azimuth in any one galaxy (Fig. 10 bottom-right), considering the slopes of
$P_{\rm DE}(\Sigma_{\rm SFR})$ and $\Delta P_{\rm DE}(\Delta\Sigma_{\rm SFR})$
are the same (Table 5).

Equation 9 contains two terms that each resemble a Kennicutt-Schmidt relation.
The first term depends only on $\Sigma_{\rm gas}$, like the original KS
relation (Kennicutt & Evans, 2012), and the second term depends only on
$\Sigma_{\rm gas}(\rho_{\rm star})^{1/2}$ (because $\sigma_{\rm gas,z}$ is
assumed to be constant), which is analogous to the Extended KS relation
discussed in Shi et al. (2018), where $\Sigma_{\rm SFR}=10^{-4.76}(\Sigma_{\rm
gas}\Sigma_{\rm stars}^{0.5})^{1.09}$ (see also Shi et al., 2011). As a check
on the KS relation for our sample, we evaluated the power law correlation for
$\Sigma_{\rm SFR}(\Sigma_{\rm gas})$ for the strong and weak spiral case and
for both together (not shown). Writing the coefficient and power again as $C$
and $D$, we get for the strong spirals: $C=-4.024\pm 0.470$, $D=1.359\pm
0.348$, for the weak spirals: $C=-4.019\pm 0.564$, $D=1.295\pm 0.510$, and for
all together: $C=-4.067\pm 0.424$, $D=1.373\pm 0.337$. These three fits are
all very similar to each other, illustrating the robustness of the KS
relation, and the slopes are about the same as in the conventional KS relation
for total gas, i.e., $\sim 1.4$.

To investigate these two terms in $P_{\rm DE}$ further, we plot histograms in
Figure 11 of the ratio of the first to the second term for all the weak spiral
(red) and strong spiral (blue) data. Generally the second term dominates (see
also Fisher et al., 2019), especially in galaxies with weak spiral arms. This
means that the $\Sigma_{\rm SFR}(P_{\rm DE})$ relation is analogous to the
Extended KS law, especially if $H_{\rm star}$ varies with radius more slowly
than the surface densities.

Figure 12 plots the excess and total relationships between the individual
components in $P_{\rm DE}$ and $\Sigma_{\rm SFR}$. The solid lines are
bivariate fits for these components, and the dashed lines are fits to the
relationships with the full $P_{\rm DE}$ values, from Figure 10. Fitted values
are in Table 6. The excess relationships for the individual terms (bottom
panels) are each shallower than the excess relationship for the full $P_{\rm
DE}$, for trivial reasons of arithmetic. That is, if $P_{\rm
DE,1}=A_{1}+B_{1}\Sigma_{\rm SFR}$ and $P_{\rm DE,2}=A_{2}+B_{2}\Sigma_{\rm
SFR}$ are the relationships for each term in $P_{\rm DE}$, then $P_{\rm
DE}\equiv P_{\rm DE,1}+P_{\rm DE,2}=(A_{1}+A_{2})+(B_{1}+B_{2})\Sigma_{\rm
SFR}$ is the relationship for the full $P_{\rm DE}$, having a slope equal to
the sum of the slopes of the individual ones.

We note from the top panels of Figure 12 that the first term in $P_{\rm DE}$
has a steeper power law relationship with $\Sigma_{\rm SFR}$ than the second
term. In Table 6, the powers for the first term are 1.58 and 1.81 for strong
and weak spirals, respectively. If this were the KS relation, then $P_{\rm
DE,1}\propto\Sigma_{\rm gas}^{2}$ would be proportional to $\Sigma_{\rm SFR}$
to the power 2/1.4=1.4. The slopes of the power-law relationships between the
second term in $P_{\rm DE}$ and $\Sigma_{\rm SFR}$ are 1.11 and 1.23 for
strong and weak spirals, which are shallower than the slopes of the total
relationships in Figure 10 and comparable to the slope of the Extended KS law
when written with $\Sigma_{\rm SFR}$ as the dependent variable.

The slopes of the excursions in the bottom panels of Figure 12, written as $B$
in Table 6, are consistent with the slopes derived from the derivatives of the
power laws in the top panels, written as $B^{\prime}$ in Table 6. This implies
that each term in $P_{\rm DE}$ always follows $\Sigma_{\rm SFR}$ closely,
i.e., in radius and from galaxy to galaxy, and in azimuthal variations. This
is the same as the total $P_{\rm DE}$, where $B$ and $B^{\prime}$ were about
the same in Table 5, but it differs from the molecular pressures, $P_{\rm
O150}$ and $P_{\rm P150}$, which have larger excursions in azimuth than
consistent with their global relationships (Sect. 4).

The tight correlation between $\Sigma_{\rm SFR}$ and $P_{\rm DE}$, as well as
with each term in $P_{\rm DE}$, seems consistent with feedback control of
total pressure, but it has to do this without affecting molecular clouds much
(Sect. 2), and the correlations are similar to the KS relation, which is not
usually considered a result of feedback. If we think of $P_{\rm DE}$ as
following $\Sigma_{\rm SFR}$, then feedback comes to mind, but if we think of
the same relationship with $\Sigma_{\rm SFR}$ following the gas and star
surface density in a KS relation, then cloud formation comes to mind. Is
$P_{\rm DE}$, through its two terms that each resemble a KS relation, a
precursor to star formation, or is $P_{\rm DE}$ a result of it because of
feedback?

For an equilibrium model, both would be true, but star formation and gas
surface density are only in equilibrium on sufficiently long spatial and
temporal timescales. The close correlation for azimuthal variations found
here, especially for strong spiral arms where they presumably reflect the
arm/interarm contrast, suggest the components of $P_{\rm DE}$ are a precursor
to $\Sigma_{\rm SFR}$. For example, if most of the regions with locally high
$\Sigma_{\rm SFR}$, which also have high molecular masses (Sect. 2), are too
young for supernovae to have put in the full yield of feedback momentum, then
the terms in $P_{\rm DE}$ would have to be precursors. This is consistent with
the discussion in the Introduction about evidence for increased H i velocity
dispersions in regions of former star formation, 50-100 Myr ago (Stilp et al.,
2013; Hunter et al., 2022, 2023). If feedback takes that long to show up in
the pressure, then the tight correlation found here for azimuthal fluctuations
would seem to imply that star formation is following the gas mass, which is
the $\Sigma_{\rm SFR}(P_{\rm DE,1})$ and $\Sigma_{\rm SFR}(P_{\rm DE,2})$ way
of looking at it.

Figure 3 in Ostriker & Kim (2022) shows $P_{\rm DE}$ and $\Sigma_{\rm SFR}$ as
functions of time. $\Sigma_{\rm SFR}$ fluctuates by factors of 10 to 100 over
time in various models, while $P_{\rm DE}$ varies by less than a factor of 2.
This is consistent with the small excursion in $P_{\rm DE}$ in the bottom-
right panel of Figure 10 and the bottom panels of Figure 12.

##  5 Turbulence and Pressure from ISM Gravity and Spiral Dynamics

Aside from the question of whether the components of $P_{\rm DE}$ are a
precursor or a result of star formation, a different consideration is that a
third process, such as spiral arm dynamics or shear, might drive them both.
These processes act on the total gas at a rate that depends on local galaxy
dynamics.

For example, if the effective Toomre parameter $Q$ is regulated by galactic
dynamics to be about constant, which seems to be the case (Romeo & Mogotsi,
2017), then the velocity dispersion is partially regulated by swing
amplification and local gravitational collapse. The effective dispersion,
considering also magnetic and cosmic ray pressures that contribute to the
stability of the ISM, is then given by $\sigma_{\rm Q}={{\pi G\Sigma_{\rm
Q}Q}/{\kappa}}$ for some effective surface density $\Sigma_{\rm Q}$ and
effective $Q$, generally involving both stars and gas, and for epicyclic
frequency $\kappa$. Combining this with the disk thickness, $H=\sigma_{\rm
Q}^{2}/(\pi G\Sigma_{\rm H})$, where $\Sigma_{\rm H}$ is the combined gas,
star, and dark matter surface density inside the gas layer, and converting the
resultant midplane gas density, $\rho_{\rm gas}=0.5\Sigma_{\rm gas}/H$, to a
dynamical rate $(G\rho_{\rm gas})^{0.5}$ for total gas surface density
$\Sigma_{\rm gas}$, gives the expectation that

$\Sigma_{\rm SFR}\sim\epsilon_{\rho}\Sigma_{\rm gas}(G\rho_{\rm
gas})^{0.5}=\epsilon_{\rm\rho}\Sigma_{\rm gas}\kappa\left({{\Sigma_{\rm
gas}\Sigma_{\rm H}}\over{2\pi\Sigma_{\rm
Q}^{2}Q^{2}}}\right)^{0.5}\sim\epsilon_{\kappa}\Sigma_{\rm gas}\kappa.$ (10)

The square root term was absorbed into the generic efficiency for this
density-based expression, $\epsilon_{\rm\rho}$, as a global efficiency,
$\epsilon_{\kappa}$. In general, we would expect also

$\Sigma_{\rm SFR}\sim\epsilon_{\rm A}\Sigma_{\rm
gas}A\sim\epsilon_{\Omega}\Sigma_{\rm gas}\Omega$ (11)

for various other dynamical rates, the Oort rate of shear, $A$, and the
orbital rate, $\Omega$. Conversions between these gas dynamical processes and
the SFR involve the dimensionless efficiencies, $\epsilon$.

Figure 13 plots the quantities given by equations 10 and 11 for both total
values (top panels) and excursions around the azimuth (bottom panels). We use
the PHANGS tabulations for our 34 galaxies with $\Sigma_{\rm gas}=\Sigma_{\rm
mol}+\Sigma_{\rm atom}$, again divided into 17 strong-arm and 17 weak-arm
cases. The dynamical rates come from the PHANGS circular velocities,
galactocentric radii, and rotation curve slopes, $\beta$. The total
correlations are fitted to power laws and the excursion relations fitted
linearly, with fitting parameters in Table 7. The slope of the excursion
relation, $B$, has been multiplied by $100$ in the Table for clarity
(equivalent to making the efficiency in units of 0.01 for these excursions).
The coefficient of the power law fit, $C$, has not assumed this efficiency
normalization; to do so would add 2 to each $C$.

The slopes of the power law correlations, $D$, in Table 7 are essentially 1.0,
meaning that $\Sigma_{\rm SFR}$ tracks well the total gas surface density
multiplied by a local galactic dynamical rate. All the fits are about the
same, so there is no preferred rate indicated. This unity slope should not be
compared with analogous correlations in Sun et al. (2023), who got a sublinear
slope when plotting $\Sigma_{\rm SFR}$ versus $\Sigma_{\rm mol}\Omega$. The
main difference is that we use the total gas because that is what galactic-
scale dynamical processes act upon. The efficiency factors, $\epsilon$, are
given by $10^{C}$ for power law coefficients $C$ in Table 7. They are 0.52%,
1.84% and 0.78% for dynamical rates $\kappa$, $A$ and $\Omega$.

The slopes of the excursions in Table 7, listed as $B$, average to a factor of
1.8 more than the derivatives of the power-law relations, listed as
$B^{\prime}$. This factor is about that same as for $P_{\rm O150}$ and $P_{\rm
P150}$ in Table 5, and may be for the same reason: star formation scales with
gas density more sensitively in azimuthal perturbations (e.g., spiral arms)
than the overall average. This implies a local factor of $\sim 2$ enhancement
of the star formation efficiency per unit total gas in spiral arms.

##  6 Conclusions

Correlations between the surface density of the star formation rate,
$\Sigma_{\rm SFR}$, and various properties of molecular and total gas were
evaluated for 17 galaxies with strong spiral arms and 17 galaxies with weak
arms. Arm strength was determined by the amplitude of the excursions in
$\Sigma_{\rm SFR}$ measured around in azimuth. Data from the PHANGS survey
were used.

Azimuthal variations in $\Sigma_{\rm SFR}$ closely follow the azimuthal
variations in molecular cloud mass, suggesting a fairly uniform star formation
efficiency on the scale of 1.5 kpc. The CO velocity dispersion also follows
the molecular cloud mass according to the usual cloud scaling laws. Thus we
observe an increase in CO velocity dispersion with increased star formation
rate, but this is the result of the usual scaling laws between velocity
dispersion and molecular cloud mass prior to star formation, rather than the
result of feedback from star formation affecting the molecular velocity
dispersion (Sects. 2 and 3).

Azimuthal variations in $\Sigma_{\rm SFR}$ are larger for given internal cloud
pressure variations in strong arm galaxies than weak arm galaxies, suggesting
that molecular clouds of the same pressure produce a higher star formation
rate in strong spiral arms than in weak arms. An alternate interpretation is
that molecular cloud pressure (e.g., from feedback) is higher for a given
excursion in star formation rate in a weak arm galaxy than a strong arm galaxy
(Sect. 4).

Azimuthal variations in internal molecular cloud pressure are several times
higher in proportion to star formation rate variations than are the total
cloud pressure variations with the total star formation rate. This could be a
result of background stars generally influencing the total cloud pressure over
a wide range of galactic radii and galaxy types, while cloud pressure
variations at a given radius for a particular galaxy are more affected by
internal cloud gravity. In contrast, azimuthal variations in the total
dynamical equilibrium pressure, $P_{\rm DE}$, follow variations in
$\Sigma_{\rm SFR}$ in the same way as the total $P_{\rm DE}$ follows the total
$\Sigma_{\rm SFR}$, suggesting a tight relation between the two quantities in
all environments (Sect. 4).

The internal molecular cloud pressure varies in azimuth ten times more
strongly with the star formation rate than the dynamical equilibrium pressure
does, and the internal cloud pressure is ten times higher than this
equilibrium pressure for high star formation rates, reflecting the self-
gravitational contribution to internal cloud pressure (Sect. 4).

The dynamical equilibrium pressure is composed of two terms, with a first
depending only on the total gas surface density and a second depending on this
surface density as well as the volume stellar density. The first term alone
would be analogous to the usual Kennicutt-Schmidt relation, although with a
steeper power law since the square of $\Sigma_{\rm gas}$ enters rather than
the 1.4 power in KS. The second term alone is analogous to the Extended KS
relation, especially if the galactic disk thickness is assumed to vary less
than the surface density. Thus, the good correlation between $P_{\rm DE}$ and
$\Sigma_{\rm SFR}$ could be viewed either as a result of star formation
feedback pressurizing the interstellar gas to an equilibrium value, or star
formation following the presence of interstellar gas, with a collapse rate
supplemented by the background stellar density. The physical origin of the
relationship is interpreted differently if we write $\Sigma_{\rm SFR}$ or
$P_{\rm DE}$ as the dependent variable. The tightness of the correlation
between them, i.e., for all radial and galactic variations and for all
azimuthal variations, combined with the observation and concept that feedback
acts on the gas over a significant time span, such as 50 to 100 Myr, while
collapse to star formation seems to follow quickly and universally from the
presence of dense gas, suggests an interpretation in which excess gas leads
rapidly to excess star formation, and $P_{\rm DE}$ is primarily a proxy for
the presence of gas.

The star formation rate scales linearly with the product of the total gas
surface density and various galactic rates, i.e., the epicyclic rate, the
shear rate, and the orbital rate, suggesting an important role for galactic
dynamics in controlling star formation (Sect. 5).

Helpful comments by the referee are acknowledged.

Table 1: Galaxy Sample, $M_{\rm star}>2\times 10^{10}\;M_{\odot}$ Broad $\Delta\Sigma_{\rm SFR}$aaBroad $\Sigma_{\rm SFR}$ typically corresponds to galaxies with strong spiral arms | Narrow $\Delta\Sigma_{\rm SFR}$bbNarrow $\Sigma_{\rm SFR}$ typically corresponds to galaxies with weak spiral arms or flocculent arms | Too Few CO Obs.
---|---|---
NGC 628 | NGC 1559 | NGC 253 | NGC 1097 | ESO 97-G13
NGC 1566 | NGC 1792 | NGC 1300 | NGC 1365 | NGC 1317
NGC 2566 | NGC 2903 | NGC 1433 | NGC 1512 | NGC 3626
NGC 2997 | NGC 3059 | NGC 1546 | NGC 2775 | NGC 4293
NGC 3627 | NGC 4254 | NGC 3351 | NGC 3507 | NGC 4459
NGC 4303 | NGC 4321 | NGC 3521 | NGC 4457 | NGC 4477
NGC 4535 | NGC 5236 | NGC 4536 | NGC 4548 | NGC 4596
NGC 5248 | NGC 5643 | NGC 4569 | NGC 5134 | NGC 5128
NGC 6300 |  | NGC 6744 |  | NGC 7743
Table 2: Linear FitsaaNumbers rounded to 3 significant figures in all tables.  to Excursion Correlations, $\Delta y=A+B\Delta x$, in Figures 3, 4, 8 and 9 Galaxy Type/ | Pixels | Objects
---|---|---
Correlation | A | B | A | B
Strong Spirals |  |  |  |
$\Delta\sigma(\Delta\Sigma_{\rm SFR})$ | $0.079\pm 0.082$ | $324\pm 188$ | $0.125\pm 0.114$ | $349\pm 259$
$\Delta\sigma(\Delta\Sigma_{\rm mol}[\Delta\Sigma_{\rm SFR}])$bbConsistency check in the proposed secondary correlation, $\Delta\sigma(\Delta\Sigma_{\rm SFR})$, using primary correlations $\Delta\sigma(\Delta\Sigma_{\rm mol})$ and $\Delta\Sigma_{\rm mol}(\Delta\Sigma_{\rm SFR})$, which occur for both pixel and object measurements (Figs. 3,4).  | 0.084 | 310 | 0.124 | 338
$\Delta\sigma(\Delta M_{\rm mol}[\Delta\Sigma_{\rm SFR}])$ccConsistency check in the proposed secondary correlation, $\Delta\sigma(\Delta\Sigma_{\rm SFR})$, using primary correlations $\Delta\sigma(\Delta M_{\rm mol})$ and $\Delta M_{\rm mol}(\Delta\Sigma_{\rm SFR})$, which exist only for object measurements (Figs. 8,9).  |  |  | 0.103 | 323
$\Delta\sigma(\Delta\Sigma_{\rm mol})$ | $0.006\pm 0.009$ | $0.047\pm 0.008$ | $0.112\pm 0.092$ | $0.027\pm 0.017$
$\Delta\Sigma_{\rm mol}(\Delta\Sigma_{\rm SFR})$ | $1.65\pm 1.23$ | $6600\pm 3090$ | $0.451\pm 3.377$ | $12800\pm 6480$
$\Delta\alpha_{\rm vir}(\Delta\Sigma_{\rm SFR})$ | $-0.098\pm 0.084$ | $-240\pm 217$ | $-0.150\pm 0.104$ | $-287\pm 254$
$\Delta\sigma(\Delta M_{\rm mol})$ |  |  | $0.164\pm 0.126$ | $3.39\pm 1.62$
$\Delta M_{\rm mol}(\Delta\Sigma_{\rm SFR})$ |  |  | $-0.018\pm 0.036$ | $95.5\pm 57.5$
Weak Spirals |  |  |  |
$\Delta\sigma(\Delta\Sigma_{\rm SFR})$ | $-0.062\pm 0.006$ | $871\pm 680$ | $-0.034\pm 0.002$ | $1220\pm 1120$
$\Delta\sigma(\Delta\Sigma_{\rm mol}[\Delta\Sigma_{\rm SFR}])$bbConsistency check in the proposed secondary correlation, $\Delta\sigma(\Delta\Sigma_{\rm SFR})$, using primary correlations $\Delta\sigma(\Delta\Sigma_{\rm mol})$ and $\Delta\Sigma_{\rm mol}(\Delta\Sigma_{\rm SFR})$, which occur for both pixel and object measurements (Figs. 3,4).  | -0.030 | 876 | -0.033 | 2270
$\Delta\sigma(\Delta M_{\rm mol}[\Delta\Sigma_{\rm SFR}])$ccConsistency check in the proposed secondary correlation, $\Delta\sigma(\Delta\Sigma_{\rm SFR})$, using primary correlations $\Delta\sigma(\Delta M_{\rm mol})$ and $\Delta M_{\rm mol}(\Delta\Sigma_{\rm SFR})$, which exist only for object measurements (Figs. 8,9).  |  |  | 0.036 | 2650
$\Delta\sigma(\Delta\Sigma_{\rm mol})$ | $0.004\pm 0.018$ | $0.086\pm 0.030$ | $0.021\pm 0.042$ | $0.068\pm 0.055$
$\Delta\Sigma_{\rm mol}(\Delta\Sigma_{\rm SFR})$ | $-0.394\pm 0.105$ | $10200\pm 6830$ | $-0.782\pm 0.131$ | $33200\pm 31000$
$\Delta\alpha_{\rm vir}(\Delta\Sigma_{\rm SFR})$ | $0.009\pm 0.013$ | $-9300\pm 9290$ | $-0.052\pm 0.020$ | $-1890\pm 1850$
$\Delta\sigma(\Delta M_{\rm mol})$ |  |  | $-0.007\pm 0.022$ | $6.31\pm 4.08$
$\Delta M_{\rm mol}(\Delta\Sigma_{\rm SFR})$ |  |  | $0.007\pm 0.010$ | $419\pm 404$
Table 3: Power Law Fits $y=Cx^{D}$ to Total Correlations in Figures 6, 7, 8 and 9 Galaxy Type/ | Pixels | Objects
---|---|---
Correlation | C | D | C | D
Strong Spirals |  |  |  |
$\sigma(\Sigma_{\rm SFR})$ | $1.66\pm 0.28$ | $0.471\pm 0.127$ | $1.59\pm 0.42$ | $0.394\pm 0.194$
$\sigma(\Sigma_{\rm mol})$ | $0.051\pm 0.055$ | $0.425\pm 0.041$ | $0.022\pm 0.358$ | $0.410\pm 0.207$
$\Sigma_{\rm mol}(\Sigma_{\rm SFR})$ | $3.79\pm 0.57$ | $1.103\pm 0.254$ | $3.99\pm 0.76$ | $1.03\pm 0.35$
$\alpha_{\rm vir}(\Sigma_{\rm SFR})$aaThis was fit to a linear-log relation, $\alpha_{\rm vir}=C+D\log(\Sigma_{\rm SFR})$, as shown in the figures. | $14.1\pm 11.2$ | $5.45\pm 5.02$ | $-25.2\pm 26.1$ | $-12.2\pm 11.9$
$\sigma(M_{\rm mol})$ |  |  | $-1.60\pm 0.73$ | $0.351\pm 0.110$
$M_{\rm mol}(\Sigma_{\rm SFR})$ |  |  | $9.117\pm 0.964$ | $1.13\pm 0.44$
Weak Spirals |  |  |  |
$\sigma(\Sigma_{\rm SFR})$ | $2.23\pm 0.70$ | $0.646\pm 0.266$ | $2.08\pm 0.92$ | $0.546\pm 0.359$
$\sigma(\Sigma_{\rm mol})$ | $0.045\pm 0.062$ | $0.495\pm 0.065$ | $-0.030\pm 0.415$ | $0.496\pm 0.292$
$\Sigma_{\rm mol}(\Sigma_{\rm SFR})$ | $4.28\pm 0.96$ | $1.25\pm 0.37$ | $4.42\pm 1.27$ | $1.17\pm 0.50$
$\alpha_{\rm vir}(\Sigma_{\rm SFR})$aaThis was fit to a linear-log relation, $\alpha_{\rm vir}=C+D\log(\Sigma_{\rm SFR})$, as shown in the figures. | $22.1\pm 19.1$ | $7.66\pm 7.25$ | $-659\pm 660$ | $-258\pm 258$
$\sigma(M_{\rm mol})$ |  |  | $-2.06\pm 1.36$ | $0.426\pm 0.212$
$M_{\rm mol}(\Sigma_{\rm SFR})$ |  |  | $9.99\pm 1.94$ | $1.39\pm 0.75$
Table 4: Comparison Between the Excess Correlation Slope and Predictions from the Power Law Fits to the Total Correlations  Data Type | Excess Correlation | Power Law | Average Values | Predicted Excess Correlation
---|---|---|---|---
| SlopeaaFrom Table 2., $B_{\sigma,{\rm SFR}}$ | SlopebbFrom Table 3., $D_{\sigma,{\rm SFR}}$ | $<\sigma>$ | $<\Sigma_{\rm SFR}>$ | Slope, $B_{\sigma,{\rm SFR}}^{\prime}$
|  |  | km s-1 |  $M_{\odot}$ pc-2 Myr-1 |
Strong Spirals, Pixels | 324 | 0.471 | 4.11 | $5.85\times 10^{-3}$ | 330
Strong Spirals, Objects | 349 | 0.394 | 5.41 | $6.56\times 10^{-3}$ | 324
Weak Spirals, Pixels | 871 | 0.646 | 3.43 | $2.37\times 10^{-3}$ | 936
Weak Spirals, Objects | 1220 | 0.546 | 4.81 | $2.73\times 10^{-3}$ | 960
Table 5: Linear Fits to Excursion Correlations ($\Delta y=A+B\Delta x$) and Power-Law Fits to Total Correlations ($\log y=C+D\log x$) between $P$aaPressure $P$ in units of $10^{4}$ K cm-3 and $\Sigma_{\rm SFR}$bbStar formation rate density, $\Sigma_{\rm SFR}$, in units of $M_{\odot}$ pc-2 Myr-1 in Figure 10 Galaxy Type/ | Acc$A$ and $B$ are such that $\Delta P=A+B\Delta\Sigma_{\rm SFR}$. | Bcc$A$ and $B$ are such that $\Delta P=A+B\Delta\Sigma_{\rm SFR}$. | Cdd$C$ and $D$ are such that $\log P=C+D\log\Sigma_{\rm SFR}$, or $P=10^{C}\Sigma_{\rm SFR}^{D}$. | Ddd$C$ and $D$ are such that $\log P=C+D\log\Sigma_{\rm SFR}$, or $P=10^{C}\Sigma_{\rm SFR}^{D}$.
---|---|---|---|---
Correlation |  |  |  |
Strong Spirals, objects |  |  |  |
$\Delta P_{\rm O150}(\Delta\Sigma_{\rm SFR})$ | $-0.73\pm 2.68$ | $7210\pm 4530$ |  |
$B_{\rm O150}^{\prime}(C_{\rm O150},D_{\rm O150})$ |  | 3330 |  |
$P_{\rm O150}(\Sigma_{\rm SFR})$ |  |  | $4.61\pm 1.11$ | $1.59\pm 0.51$
Strong Spirals, pixels |  |  |  |
$\Delta P_{\rm P150}(\Delta\Sigma_{\rm SFR})$ | $-0.41\pm 1.75$ | $5730\pm 3560$ |  |
$B_{\rm P150}^{\prime}(C_{\rm P150},D_{\rm P150})$ |  | 1830 |  |
$P_{\rm P150}(\Sigma_{\rm SFR})$ |  |  | $5.20\pm 1.04$ | $2.00\pm 0.47$
$\Delta P_{\rm DE}(\Delta\Sigma_{\rm SFR})$ | $-0.154\pm 0.220$ | $979\pm 544$ |  |
$B_{\rm DE}^{\prime}(C_{\rm DE},D_{\rm DE})$ |  | 909 |  |
$P_{\rm DE}(\Sigma_{\rm SFR})$ |  |  | $3.41\pm 0.66$ | $1.25\pm 0.30$
Weak Spirals, objects |  |  |  |
$\Delta P_{\rm O150}(\Delta\Sigma_{\rm SFR})$ | $-2.53\pm 0.56$ | $16900\pm 15600$ |  |
$B_{\rm O150}^{\prime}(C_{\rm O150},D_{\rm O150})$ |  | 4180 |  |
$P_{\rm O150}(\Sigma_{\rm SFR})$ |  |  | $5.69\pm 2.24$ | $1.92\pm 0.88$
Weak Spirals, pixels |  |  |  |
$\Delta P_{\rm P150}(\Delta\Sigma_{\rm SFR})$ | $-1.15\pm 0.21$ | $15500\pm 12800$ |  |
$B_{\rm P150}^{\prime}(C_{\rm P150},D_{\rm P150})$ |  | 2090 |  |
$P_{\rm P150}(\Sigma_{\rm SFR})$ |  |  | $6.85\pm 2.21$ | $2.50\pm 0.84$
$\Delta P_{\rm DE}(\Delta\Sigma_{\rm SFR})$ | $-0.000\pm 0.009$ | $882\pm 475$ |  |
$B_{\rm DE}^{\prime}(C_{\rm DE},D_{\rm DE})$ |  | 1180 |  |
$P_{\rm DE}(\Sigma_{\rm SFR})$ |  |  | $3.80\pm 1.05$ | $1.33\pm 0.40$
Table 6: Linear Fits to Excursion Correlations ($\Delta y=A+B\Delta x$) and Power-Law Fits to Total Correlations ($\log y=C+D\log x$) between 1st and 2nd terms of $P_{\rm DE}$aaPressure $P$ in units of $10^{4}$ K cm-3 and $\Sigma_{\rm SFR}$bbStar formation rate density, $\Sigma_{\rm SFR}$, in units of $M_{\odot}$ pc-2 Myr-1 in Figure 12 Galaxy Type/ | Acc$A$ and $B$ are such that $\Delta P=A+B\Delta\Sigma_{\rm SFR}$. | Bcc$A$ and $B$ are such that $\Delta P=A+B\Delta\Sigma_{\rm SFR}$. | Cdd$C$ and $D$ are such that $\log P=C+D\log\Sigma_{\rm SFR}$, or $P=10^{C}\Sigma_{\rm SFR}^{D}$. | Ddd$C$ and $D$ are such that $\log P=C+D\log\Sigma_{\rm SFR}$, or $P=10^{C}\Sigma_{\rm SFR}^{D}$.
---|---|---|---|---
Correlation |  |  |  |
Strong Spirals, pixels |  |  |  |
$\Delta P_{\rm DE,1}(\Delta\Sigma_{\rm SFR})$ | $-0.015\pm 0.142$ | $678\pm 339$ |  |
$B_{\rm DE,1}^{\prime}(C_{\rm DE,1},D_{\rm DE,1})$ |  | 410 |  |
$P_{\rm DE,1}(\Sigma_{\rm SFR})$ |  |  | $3.68\pm 0.89$ | $1.58\pm 0.40$
$\Delta P_{\rm DE,2}(\Delta\Sigma_{\rm SFR})$ | $0.0775\pm 0.0882$ | $587\pm 297$ |  |
$B_{\rm DE,2}^{\prime}(C_{\rm DE,2},D_{\rm DE,2})$ |  | 496 |  |
$P_{\rm DE,2}(\Sigma_{\rm SFR})$ |  |  | $2.90\pm 0.61$ | $1.11\pm 0.28$
Weak Spirals, pixels |  |  |  |
$\Delta P_{\rm DE,1}(\Delta\Sigma_{\rm SFR})$ | $-0.00001\pm 0.00000$ | $480\pm 296$ |  |
$B_{\rm DE,1}^{\prime}(C_{\rm DE,1},D_{\rm DE,1})$ |  | 384 |  |
$P_{\rm DE,1}(\Sigma_{\rm SFR})$ |  |  | $4.42\pm 1.86$ | $1.81\pm 0.72$
$\Delta P_{\rm DE,2}(\Delta\Sigma_{\rm SFR})$ | $-0.00000\pm 0.00000$ | $673\pm 400$ |  |
$B_{\rm DE,2}^{\prime}(C_{\rm DE,2},D_{\rm DE,2})$ |  | 798 |  |
$P_{\rm DE,2}(\Sigma_{\rm SFR})$ |  |  | $3.42\pm 0.93$ | $1.23\pm 0.36$
Table 7: Linear Fits to Excursion Correlations ($\Delta y=A+B\Delta x$) and Power-Law Fits to Total Correlations ($\log y=C+D\log x$) between $\Sigma_{\rm SFR}$aaStar formation rate density, $\Sigma_{\rm SFR}$, and surface density rates are in units of $M_{\odot}$ pc-2 Myr-1 and various surface density rates in Figure 13 Galaxy Type/ | Abb$A$ and $B$ are such that $\Delta\Sigma_{\rm SFR}=A+B(\Delta\Sigma_{\rm gas}R)$ for rate $R$. $B$ values multiplied by 100. | Bbb$A$ and $B$ are such that $\Delta\Sigma_{\rm SFR}=A+B(\Delta\Sigma_{\rm gas}R)$ for rate $R$. $B$ values multiplied by 100. | Ccc$C$ and $D$ are such that $\log\Sigma_{\rm SFR}=C+D\log(\Sigma_{\rm gas}R)$, or $\Sigma_{\rm SFR}=10^{C}(\Sigma_{\rm gas}R)^{D}$ for rate $R$. | Dcc$C$ and $D$ are such that $\log\Sigma_{\rm SFR}=C+D\log(\Sigma_{\rm gas}R)$, or $\Sigma_{\rm SFR}=10^{C}(\Sigma_{\rm gas}R)^{D}$ for rate $R$.
---|---|---|---|---
Correlation |  |  $(\times 100$) |  |
Strong Spirals, pixels |  |  |  |
$\Delta\Sigma_{\rm SFR}(\Delta\Sigma_{\rm gas}\kappa)$ | $0.00008\pm 0.00024$ | $0.902\pm 0.474$ |  |
$B_{\rm SFR}^{\prime}(C_{S,\kappa},D_{S,\kappa})$ |  | 0.525 |  |
$\Delta\Sigma_{\rm SFR}(\Delta\Sigma_{\rm gas}A)$ | $-0.00019\pm 0.00005$ | $2.11\pm 1.21$ |  |
$B_{\rm SFR}^{\prime}(C_{SA},D_{SA})$ |  | 1.84 |  |
$\Delta\Sigma_{\rm SFR}(\Delta\Sigma_{\rm gas}\Omega)$ | $0.00004\pm 0.00018$ | $1.05\pm 0.53$ |  |
$B_{\rm SFR}^{\prime}(C_{S\Omega},D_{S,\Omega})$ |  | 0.79 |  |
$\Sigma_{\rm SFR}(\Sigma_{\rm gas}\kappa)$ |  |  | $-2.28\pm 0.02$ | $1.00\pm 0.26$
$\Sigma_{\rm SFR}(\Sigma_{\rm gas}A)$ |  |  | $-1.73\pm 0.11$ | $1.07\pm 0.26$
$\Sigma_{\rm SFR}(\Sigma_{\rm gas}\Omega)$ |  |  | $-2.11\pm 0.02$ | $1.02\pm 0.26$
Weak Spirals, pixels |  |  |  |
$\Delta\Sigma_{\rm SFR}(\Delta\Sigma_{\rm gas}\kappa)$ | $-0.00004\pm 0.00001$ | $0.79\pm 0.46$ |  |
$B_{\rm SFR}^{\prime}(C_{S,\kappa},D_{S,\kappa})$ |  | 0.334 |  |
$\Delta\Sigma_{\rm SFR}(\Delta\Sigma_{\rm gas}A)$ | $0.00017\pm 0.00015$ | $2.45\pm 2.09$ |  |
$B_{\rm SFR}^{\prime}(C_{SA},D_{SA})$ |  | 1.32 |  |
$\Delta\Sigma_{\rm SFR}(\Delta\Sigma_{\rm gas}\Omega)$ | $-0.00004\pm 0.00002$ | $1.20\pm 0.70$ |  |
$B_{\rm SFR}^{\prime}(C_{S\Omega},D_{S,\Omega})$ |  | 0.52 |  |
$\Sigma_{\rm SFR}(\Sigma_{\rm gas}\kappa)$ |  |  | $-2.45\pm 0.05$ | $0.903\pm 0.311$
$\Sigma_{\rm SFR}(\Sigma_{\rm gas}A)$ |  |  | $-1.88\pm 0.31$ | $0.968\pm 0.429$
$\Sigma_{\rm SFR}(\Sigma_{\rm gas}\Omega)$ |  |  | $-2.28\pm 0.11$ | $0.928\pm 0.335$

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Figure 1: Correlations between azimuthal fluctuations in various measured
quantities for the strong-arm galaxy NGC 4321, from the PHANGS survey.
“Excess” refers to the difference between the quantity and the average at that
radius. The molecular surface density, $\Sigma_{\rm mol}$, and star formation
surface density, $\Sigma_{\rm SFR}$, vary together above and below the
azimuthal averages, giving a positive slope in the lower left panel. Excess
velocity dispersion also varies with excess $\Sigma_{\rm SFR}$ (top left) and
with excess $\Sigma_{\rm mol}$ (top right). The virial parameter, $\alpha_{\rm
vir}$ does not change much with $\Sigma_{\rm SFR}$. Red points correspond to
parameter measurements in 150 pc regions centered on individual molecular
clouds, and blue points correspond to parameter measurements in a regular 150
pc pixel grid, both averaged within a 1.5 kpc hexagon, as tabulated by the
PHANGS group.. An image of the galaxy is shown in the insert.  Figure 2:
Correlations between azimuthal fluctuations for the weak-arm galaxy NGC 2775.
The ranges for $\Sigma_{\rm SFR}$ and $\Sigma_{\rm mol}$ are approximately
one-tenth the ranges for NGC 4321, as reflected in the smaller scales on the
figure axes here. Symbol colors are as in Fig. 1. Figure 3: Correlations
between azimuthal fluctuations of various quantities for 34 galaxies in the
PHANGS survey with stellar masses larger than $2\times 10^{10}\;M_{\odot}$.
This figure uses data measured in regular pixels 150 pc in diameter. Red and
blue lines are fits to the trends corresponding to the red and blue points,
which are for weak and strong spirals, respectively. The fitting results are
in Table 2. Figure 4: As in Figure 3 but with data measured in 150 pc regions
for molecular cloud objects and averaged together in 1.5 kpc hexagons. Red and
blue lines are fits to the trends corresponding to the red and blue points,
which are for weak and strong spirals, respectively. The fitting results are
in Table 2. Figure 5: Correlations between total quantities in NGC 4321,
plotted on a log-log scale. The red points are from object measurements and
the blue points are from pixel measurements, both at 150 pc resolution and
average into 1.5 kpc hexagons.  Figure 6: Correlations between total
quantities in all 34 galaxies considered here, using pixel measurements with
150 pc resolution. Weak and strong spirals are distinguished by color, with
lines showing the linear fits in log-log space. The fitting results are in
Tables 2 and 3. Figure 7: Correlations between total quantities using object
measurements with 150 pc resolution. Weak and strong spirals are distinguished
by color, with lines showing the linear fits in log-log space. The fitting
results are in Tables 2 and 3. Figure 8: Correlations between azimuthal
fluctuations involving molecular cloud mass. This figure is for the 17 strong-
arm galaxies in our survey. Fitting parameters are in Table 3. Figure 9:
Correlations between azimuthal fluctuations involving molecular cloud mass.
This figure is for the 17 weak-arm galaxies in our survey. Fitting parameters
are in Table 3. Figure 10: (bottom) Correlations between azimuthal
fluctuations in $\Sigma_{\rm SFR}$ and azimuthal fluctuations in three
measures of pressure: two for molecular clouds using the object- and pixel-
based methods, and one for the dynamical equilibrium pressure, $P_{\rm DE}$.
(top) Correlations between the total quantities represented by their
fluctuations in the bottom panels. Red and blue points and fitting lines are
for galaxies with weak and strong spiral arms. Fitting parameters are in Table
5.  Figure 11: Histograms of the ratio of the first term in $P_{\rm DE}$ to
the second term for weak and strong arm spirals.  Figure 12: (bottom)
Correlations between azimuthal fluctuations in $\Sigma_{\rm SFR}$ and
azimuthal fluctuations in each term of the dynamical pressure, with the first
term on the left and the second term on the right. (top) Same for total
parameter values. Fitting parameters are in Table 6.  Figure 13: Correlations
between azimuthal fluctuations in the bottom row and total values in the top
row for $\Sigma_{\rm SFR}$ as a function of various gas dynamical rates on a
galactic scale, namely the epicycle rate $\kappa$, shear rate $A$, and orbital
rate $\Omega$. The red and blue points and fitting lines are for galaxies with
weak and strong spiral arms. Fitting parameters are in Table 7.