# Unveiling the gravitationally unstable disc of a massive star-forming galaxy
using NOEMA and MUSE††thanks:  This work is based on observations carried out
under project numbers W16BS and E16AG with the IRAM NOEMA interferometer. IRAM
is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). Based on
observations collected at the European Southern Observatory under ESO
programme 0101.B-0703(A).

Johannes Puschnig,1,2 Matthew Hayes,2 Oscar Agertz,3 Eric Emsellem,4 John M.
Cannon,5Jens Melinder,2 Göran Östlin,2 Christian Herenz,6 Veronica Menacho2
1Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71,
D-53121 Bonn, Germany
2Department of Astronomy, Oskar Klein Centre, Stockholm University, AlbaNova
University Centre, SE-106 91 Stockholm, Sweden
3Lund Observatory, Department of Astronomy and Theoretical Physics, Lund
University, Box 43, SE-221 00 Lund, Sweden
4European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching,
Germany
5Department of Physics & Astronomy, Macalester College, 1600 Grand Avenue,
Saint Paul, MN 55105, US
6European Southern Observatory, Av. Alonso de Córdova 3107, 763 0355 Vitacura,
Santiago, Chile<EMAIL_ADDRESS>(0000-0003-1111-3951)

(2022)

###### Abstract

Using new high-resolution data of CO (2–1), H$\upalpha$ and H$\upbeta$
obtained with the Northern Extended Millimeter Array (NOEMA) and the Multi-
Unit Spectroscopic Explorer (MUSE) at the Very Large Telescope, we have
performed a Toomre $Q$ disc stability analysis and studied star formation, gas
depletion times and other environmental parameters on sub-kpc scales within
the $z\sim 0$ galaxy SDSS J125013.84+073444.5 (LARS 8). The galaxy hosts a
massive, clumpy disc and is a proto-typical analogue of main-sequence galaxies
at $z\sim 1-2$. We show that the massive (molecular) clumps in LARS 8 are the
result of an extremely gravitationally unstable gas disc, with large scale
instabilities found across the whole extent of the rotating disc, with only
the innermost 500 pc being stabilized by its bulgelike structure. The radial
profiles further reveal that – contrary to typical disc galaxies – the
molecular gas depletion time decreases from more than 1 Gyr in the center to
less than $\sim$100 Myr in the outskirts of the disc, supporting the findings
of a Toomre-unstable disc. We further identified and analysed 12 individual
massive molecular clumps. They are virialized and follow the mass-size
relation, indicating that on local (cloud/clump) scales the stars form with
efficiencies comparable to those in Milky Way clouds. The observed high star
formation rate must thus be the result of triggering of cloud/clump formation
over large scales due to disc instability. Our study provides evidence that
“in-situ” massive clump formation (as also observed at high redshifts) is very
efficiently induced by large-scale instabilities.

###### keywords:

galaxies: starburst – galaxies: star formation – galaxies: ISM – galaxies:
kinematics and dynamics – techniques: interferometric – techniques: imaging
spectroscopy

††pubyear: 2022††pagerange: Unveiling the gravitationally unstable disc of a
massive star-forming galaxy using NOEMA and MUSE††thanks:  This work is based
on observations carried out under project numbers W16BS and E16AG with the
IRAM NOEMA interferometer. IRAM is supported by INSU/CNRS (France), MPG
(Germany), and IGN (Spain). Based on observations collected at the European
Southern Observatory under ESO programme 0101.B-0703(A).–A

##  1 Introduction

Several studies based on deep field observations have revealed that at
redshifts $\sim$3 galaxies with total (gas+stars) masses similar to the Milky
Way ($\sim$ $10^{11}$ M⊙) are already in place (Dessauges-Zavadsky et al.,
2017; Elbaz et al., 2018; Tacconi et al., 2018; Cassata et al., 2020). Since
the Universe was then only $\sim$2 Gyr old, these massive objects must have
formed within a very short time, thus requiring very high star formation rates
(SFRs) compared to the $z\sim 0$ Universe. In recent years, observations of
galaxies at redshifts between 0 and 4 have shown that the level of star
formation is mainly dictated by stellar mass and regulated by secular
processes (Popesso et al., 2019). This is manifested in a tight relation
between stellar mass and SFR, the so called main sequence of star forming
galaxies (Brinchmann et al., 2004; Noeske et al., 2007; Daddi et al., 2007;
Elbaz et al., 2007; Peng et al., 2010; Wuyts et al., 2011; Whitaker et al.,
2012; Whitaker et al., 2014; Tomczak et al., 2016). While the slope of the
relation does not vary with redshift, its intercept shifts towards higher SFRs
with increasing lookback time, see for example Wuyts et al. (2011); Rodighiero
et al. (2011). As shown by several studies (Tacconi et al., 2013; Genzel et
al., 2015; Scoville et al., 2017; Wiklind et al., 2019), the evolution of the
main sequence is driven by an increase of the (molecular) gas fraction. As a
result the gas depletion time, defined as the inverse of the SFR per unit
molecular gas mass, remains roughly constant even out to $z\sim$4\. The high
SFRs observed on the main sequence at high-$z$ are thus mainly driven by
increasing gas fractions. Although the strong evolution of the SFR with time
(redshift) is relatively well constrained (Madau & Dickinson, 2014), the
underlying physical mechanisms that drive star formation in gas-rich discs is
still a matter of debate.

Beside their large gas fractions, high-$z$ main-sequence galaxies are observed
to have higher gas velocity dispersions (Förster Schreiber et al., 2009;
Lehnert et al., 2009; Swinbank et al., 2012; Wisnioski et al., 2015) compared
to local spirals. Additionally, their morphologies show extremely massive
clumps, exhibiting considerable fractions of the total mass. Initially, this
was interpreted in the context of “bottom up” structure formation as an
ongoing process of merging. However, with the advent of near-infrared integral
field spectroscopy, in some of the clumpy galaxies, disc structures were found
that are characterized by significant rotation (Genzel et al., 2006, 2008), a
sign for associated structures rather than mergers. However, some fraction of
them may still be ongoing mergers (Weiner et al., 2006; Förster Schreiber et
al., 2009; Puech, 2010; Rodrigues et al., 2017). The observed discs host giant
clumps with masses of $M_{\mathrm{cl}}\lesssim 10^{9}$ M⊙. It has been
proposed that such clumps result from the fragmentation of massive gas discs
driven by gravitational instability (Agertz et al., 2009b; Dekel et al., 2009;
Bournaud et al., 2012; Romeo & Agertz, 2014). Thus, the mode of star formation
in gas-rich systems seems fundamentally different compared to the star
formation within spiral arms as found in most galaxies at $z\sim 0$.

Much numerical work has been undertaken in the last few years to study
gravitational fragmentation scenarios. But in early simulations, inefficient
thermal feedback of supernovae resulted in overcooling, which then enhanced
disc instability and star formation and led to an overproduction of giant
clumps (Ceverino & Klypin, 2009; Agertz et al., 2009b). More recently, various
works within cosmological simulations and simulations of isolated disc
galaxies were again focusing on disc fragmentation at high-$z$, but including
novel feedback recipes which systematically led to less fragmentation even in
massive gas-rich discs and generally lower clump masses in the range
$10^{7}$–$10^{8}$ M⊙ (Tamburello et al., 2015; Behrendt et al., 2015; Moody et
al., 2014; Mandelker et al., 2017; Oklopčić et al., 2017). It was further
proposed that some of the most massive observed star forming clumps might not
be the result of “in-situ” disc fragmentation, but rather they could be
accreted cores of massive satellite galaxies (Mandelker et al., 2017; Oklopčić
et al., 2017). Thus, to date the question of whether massive clumps are a
result of “in-situ” disc fragmentation or the product of accreted cores of
massive satellite galaxies remains unanswered and needs iterations on both
ends, theory and observations. The observational difficulty is that at
high-$z$ the spatial resolution is often too coarse to constrain relevant
physical parameters (turbulence, density, timescales related to star
formation). Only gravitational lensing may help to reveal details about the
clumpy discs at $z\sim 1-3$ (Dessauges-Zavadsky et al., 2019).

Despite the difficulties, recent observations support galaxy-wide disc
instabilities as a cause of the clumpy nature, e.g. Dessauges-Zavadsky & Adamo
(2018) showed that the clump mass function at $z\sim$1–3 follows a power-law
consistent with turbulence being the driving mechanism. Moreover, the DYNAMO
survey (Fisher et al., 2014), targeting extremely rare local clumpy gas-rich
disc galaxies as a proxy for high-$z$ galaxies, revealed clump properties that
favour clump formation induced by galaxy-wide disc instabilities (Fisher et
al., 2017; White et al., 2017; Fisher et al., 2019).

In this paper, we present new highly-resolved NOEMA CO (2–1) and MUSE
H$\upalpha$ observations of LARS 8, a clumpy $z\sim 0$ galaxy drawn from the
Lyman Alpha Reference Sample (Östlin et al., 2014; Hayes et al., 2014). Given
the basic properties of the galaxy (see Table 1) with a stellar mass of
$\sim$1011 M⊙ and a SFR of $\sim$30 M⊙ yr-1, LARS 8 resembles main-sequence
galaxies at high-$z$. The LARS galaxy is also known to be rotationally
supported (Herenz et al., 2016; Puschnig et al., 2020), just like face-on disc
galaxies typically observed at high-$z$ (compare Figure 1). Herenz et al.
(2016) and Micheva et al. (2018) further revealed the existence of shells at
large galactocentric radii, caused by a merger event that LARS 8 must have
undergone recently. Using deep high-resolution 21 cm observations, Le Reste et
al. (2022) found a large neutral gas reservoir westwards of the optical galaxy
disk. The galaxy is known to have a relatively high gas fraction of 27
percent, a gas depletion time of $\sim$1.2 Gyr (Puschnig et al., 2020) and a
clumpy morphology. These properties make LARS 8 an ideal laboratory to study
clump formation in a gas-rich disc galaxy.

The paper is organised as follows. In Section 2 we inform about the
spectroscopic observations our results are based on. The methods and tools we
use to convert the observables into physical parameters (e.g. star formation
rates, mass surface densities, dynamical parameters) are outlined in Section
3. The results are presented in Section 4 and subsequently discussed and
compared to related works in 5. Section 6 concludes the paper with a summary.

Throughout the paper, we adopt a cosmology with $H_{\mathrm{0}}$=70,
$\Omega_{\mathrm{M}}$=0.3 and $\Omega_{\mathrm{vac}}$=0.7.

Table 1: Properties of LARS 8 (see Tables 1, 9 and 10 of Puschnig et al. 2020). $\dagger$ The metallicity $Z$ was derived by Östlin et al. (2014) using the $R_{\mathrm{23}}$-$P$ relation and is given in units of $\mathrm{12+log(O/H)}$. ${\ddagger}$ The blue-band diameter of the 25 mag arcsec-2 isophote was derived from SDSS g-band observations using an SQL query on SDSS DR7. Since the SDSS g-band isophote is typically $\sim$1.3 times larger than those measured in the Johnson B band, the g-band diameter was divided by that factor. | $D_{\mathrm{L}}$ | log $M_{*}$ | $Z^{\dagger}$ | SFR | $D_{\mathrm{25}}^{{\ddagger}}$ | $f_{\mathrm{gas}}$
---|---|---|---|---|---|---
| [Mpc] | [M⊙] |  | [M⊙ yr-1] | ["] | [%]
| 167.5$\pm$12 | 10.97$\pm$0.10 | 8.51 | 30$\pm$8 | 30.8 | 27$\pm$16
Figure 1: High-$z$ target from the PHIBBS survey (Tacconi et al., 2013) versus
LARS 8 (Puschnig et al., 2020). The optical morphologies (top panel) as well
as the CO line emission (bottom panel) are remarkably similar.

##  2 Observations and data reduction

###  2.1 NOEMA CO (2–1) cube

We observed LARS 8 in a single pointing under programs W16BS and E16AG with
the IRAM Northern Extended Millimeter Array (NOEMA) using eight antennas in
configurations A and D, providing maximum baselines of $\sim$760 m and
$\sim$180 m respectively. The target line, CO (2–1), was observed with the
WideX correlator (bandwidth $\sim$3.6 GHz) using a tuning frequency of 222.044
GHz, corresponding to the systemic velocity derived from H I observations
(Pardy et al., 2014). We further performed on-the-fly mapping of LARS 8 with
the IRAM 30m telescope under programs 064-15 and 178-15, allowing us to
include short spacing visibility data.

Extended array observations were executed on December 15, 2016 for a total on-
source time of 5.2 hours under good weather conditions with a precipitable
water vapour (PWV) of $\sim$1.8 mm. Compact array observations were executed
on three days during May 2017 for a total on-source time of 6.3 hours under
average weather conditions with a PWV of $\sim$2–3 mm.

The absolute flux scale of the configuration A data was calibrated on LKHA101
using a model flux of 0.54 Jy. The sources 1222+216 and 3C273 were used as
phase and amplitude calibrators. Average polarization mode was chosen for the
amplitude calibration, because the signal was found to be polarized. 3C84 was
used as bandpass calibrator.

The absolute flux scale of the configuration D data obtained on May 8, 2017
was calibrated on MWC349 with a model flux of 1.87 Jy. For the observations
executed on May 2 and May 3, MWC349 data was not available and 3C273 was used
instead, assuming a model flux of 7.65 Jy, as measured on May 8. 1236+077 and
3C273 were used for phase and amplitude calibration, whereas 3C273 was also
used as bandpass calibrator.

All observations were calibrated using the IRAM reduction pipeline
GILDAS/CLIC111http://www.iram.fr/IRAMFR/GILDAS/. Data flagging was performed
manually taking into account tracking errors, pointing and focus offsets as
well as quality assessment through outlier rejection in time versus amplitude
and phase plots as well as large phase discrepancies between the two
polarizations. We remark that for the configuration A observations, for one of
the antennas tracking errors of more than 4" were reported. All baselines
including this antenna were thus rejected.

Merging of the calibrated visibilities and short spacing correction were
performed in the GILDAS/mapping environment, which was also used for imaging.
Robust weighting of 0.5 was found to lead to a good compromise between
sidelobe suppression and spatial resolution, both of which are important for
our science case. Cleaning was done using the Högbom algorithm (Högbom, 1974)
and a central circular support of 6" diameter. The final clean cube has an
r.m.s. noise of 0.8 mJy/beam at a velocity resolution of 10 km/s. The
synthetic beam size is 0.61"x0.37" with a position angle of 36°.

As LARS 8 is substantially extended compared to NOEMA’s $\sim$23" field of
view at the observed frequency, a primary beam correction was finally
performed using the PRIMARY task within the GILDAS/mapping environment.

###  2.2 MUSE observations and data reduction

We observed LARS 8 with the Multi-Unit Spectroscopic Explorer (MUSE; Bacon et
al. 2010) integral field spectrograph, mounted at Unit Telescope 4 of the Very
Large Telescope (VLT). Spectra were obtained on the night of 18 May 2018 under
conditions of new moon, airmass lower than 1.2, and with a V-band seeing of
0$\aas@@fstack{\prime\prime}$8\. We obtained four observations of the main
target, each rotated by 90 degrees compared to the previous to minimize fixed
pattern noise from the image slicers, using integration times of 650 seconds.
Because LARS 8 occupies a major fraction of the MUSE field-of-view, we also
obtained a separate sky frame from an adjacent empty pointing using an
integration time of 120 seconds. Data were reduced using Version 2.6 of the
ESO pipeline, using standard methods and paying special attention to the
removal of low surface brightness emission in strong nebular lines.

##  3 Methods

Here we briefly describe the routines and tools that we used to obtain
physical parameters from the observations. In subsections 3.1–3.3 we explain
how the observed data cubes are prepared for further scientific analysis, i.e.
the convolution to a common resolution and the derivation of moment maps.

Subsections 3.4–3.6 summarize the assumptions and constraints used to convert
the observables into physical quantities such as star formation rates, stellar
and gas surface densities.

In 3.7 and 3.8 the routines for the characterisation of the dynamics of the
galaxy’s gaseous and stellar components are presented. These dynamical
quantities finally allow to constrain the gravitational instability via the
Toomre parameter (see Section 3.9).

We conclude in 3.11 and 3.12 with a brief description of how individual
molecular gas clouds are identified in the NOEMA cube and how we use
previously derived physical quantities to estimate the dynamic equilibrium
pressure.

###  3.1 Convolution of the NOEMA data cube to a common resolution

Given the slightly lower spatial resolution of the optical data cube compared
to our radio data, we convolve the latter to match the resolution of the MUSE
observations. To do so, we first deconvolve the elliptical NOEMA beam from the
circularized target beam (based on MUSE cube). The resulting convolution
kernel is then applied onto the 3D NOEMA cube (plane-by-plane) using the
scipy.signal.convolve package (Virtanen et al., 2020). The circularized
synthetic beam size of the matched-resolution NOEMA CO (2–1) data cube is
0.78". Note that throughout the paper we make use of the native resolution CO
(2–1) data whenever possible (clump identification, Toomre disc stability
analysis). Only plots that include both star formation rates (from
H$\upalpha$) and properties derived from the CO observations are based on the
matched-resolution data.

###  3.2 MUSE line extraction of H$\alpha$, H$\beta$ and continuum subtraction

From the reduced MUSE cube, we first extract a fixed spectral range around the
observed H$\upalpha$ and H$\upbeta$ lines, using $z$=0.0382531 as the redshift
and an extraction window of $\pm$420 km/s, centered on the systemic line-
center (corresponding to $z$). We ensured that the [N II] lines are outside
the extracted line window of H$\upalpha$. In order to define the continuum
level at each line, individual spectral windows were defined blueward and
redward of each emission line, after manual inspection of the spectral cube.
For H$\upalpha$, suitable windows were found between -2500 and -1500 km/s and
from 3000 to 4000 km/s. H$\upbeta$ continuum levels were evaluated between
-4000 and -2000 km/s as well as within the range of 2000 and 4000 km/s. The
continuum correction for each line was then performed via subtraction of a
linear fit, obtained from regression (using the python lmfit package) of the
flux within the given velocity intervals.

###  3.3 Moment maps of CO (2–1) and optical emission lines

We generate moment zero maps of CO (2–1), H$\upalpha$ and H$\upbeta$ via
summation of the flux in masked channels, using the approach of “dilated
masking”. In the NOEMA cube, peak channels were identified that have a more
than 4-sigma strong signal in at least three adjacent channels. The mask was
then expanded in velocity space as long as the flux in two adjacent channels
was above a 2-sigma limit. Additionally, we only allow connected spatial
regions that cover at least the size of the synthetic beam of our
observations. Moment maps of H$\upalpha$ and H$\upbeta$ were created in a very
similar manner, i.e. we identified channels with 4-sigma peaks and
subsequently grow the mask down to a level of 2-sigma. However, given the
lower spectral resolution of the MUSE data cubes, we allow to mask even single
channels in velocity space rather than a number of adjacent ones.

First and second moment maps were created using the same masks, with moment
one being the intensity-weighted mean velocity found under the masked channels
and moment two being the intensity-weighted r.m.s. velocity scatter. Moment
maps are shown in Figures 2.

The uncertainties of our moment maps are calculated via Gaussian error
propagation using the r.m.s. outside the line masks as an estimate for the
uncertainty of each masked channel.

Figure 2: Maps derived from NOEMA CO (left column), MUSE H$\upalpha$ (middle
column) and VLA HI 21cm (right column; Le Reste et al. 2022) data cubes. The
top row moment-0 maps are given in units of Jy km s-1 beam-1 on a logarithmic
scale for CO, 10-20 erg s-1 cm-2 on a logarithmic scale for H$\upalpha$ and
1021 cm-2 on a linear scale for HI. The moment-1 and moment-2 maps in the
middle and bottom rows are given in km s-1.

###  3.4 Star formation rates from MUSE H$\alpha$

In order to obtain the intrinsic, extinction-corrected H$\upalpha$ flux, we
calculate the dust attenuation from the Balmer decrement using the Cardelli et
al. (1989) attenuation law and assume case B recombination at 104 K and an
intrinsic, theoretical H$\upalpha$/H$\upbeta$ ratio of 2.86. The average
extinction E(B-V) within an aperture of 5 arcsec radius enclosing the center
of the galaxy – and thus covering the main part of the NOEMA field-of-view –
is 0.7 mag with a maximum value of 1.2 mag in the central pixel and values as
low as 0.2 mag in the outer region.

We first convert the extinction-corrected H$\upalpha$ flux from erg/s/cm2 into
the corresponding luminosity ($L_{\mathrm{H\upalpha}}$) in erg/s using a
luminosity-distance of 167.5 Mpc. The star formation rates (SFRs) in units of
M⊙ yr-1 per pixel are then calculated using the calibration of Calzetti et al.
(2012): SFR=5.5 10-42 $L_{\mathrm{H\upalpha}}$. These SFRs are then converted
into surface densities in units of M⊙ yr-1 kpc-2 ($\Sigma_{\mathrm{SFR}}$)
taking into account the cosine correction factor ($\mathrm{cos}\ i$) for the
galaxy inclination $i$ of 50∘ that we found from the rotation curve (see
Section 3.7).

###  3.5 Stellar surface density

A stellar mass map of the galaxy is constructed by performing a pixel spectral
energy distribution fit using the HST FUV and optical broad band data from the
LARS project. The fitting code “the Ly$\upalpha$ eXtraction Software” (Östlin
et al. 2014, Melinder et al. in preparation) uses two single stellar
populations with four free parameters: stellar mass for the two components,
stellar age, and stellar extinction (only one of the populations have a
varying age and extinction, the other one is kept at an age of 10 Gyrs and an
E(B-V)s of 0). The fit is performed for each pixel (or spatial bin) to produce
maps of stellar continuum fluxes, mass, age, and extinction. The uncertainties
on the stellar masses are estimated within the code using Monte Carlo
simulations, in which random noise (corresponding to the r.m.s. in each pixel
after drizzling) is added to the originally measured value. The final
uncertainty is then the standard deviation obtained from the measurements in
all Monte Carlo simulations. For details on the code and the data used for
LARS 8 we refer the reader to Östlin et al. (2014). To find the stellar
surface density radial profile we co-add the stellar mass maps of the two
components and measure the mean mass surface density in elliptical annuli that
exactly match those used for the NOEMA and MUSE data. Finally, the derived
mean surface densities ($\Sigma_{*}$) were corrected for inclination using the
same quantities as for $\Sigma_{\mathrm{SFR}}$. The scale length of the
stellar disc, $l_{*}$, was derived from the stellar mass map by fitting an
exponential function to the inclination corrected mass profile.

###  3.6 Molecular gas surface density, depletion time and gas fraction

We convert our measured CO (2–1) fluxes $S_{\mathrm{CO}}$ (in units of Jy km
s-1 beam-1) to CO luminosities using the definition of
$L^{\prime}_{\mathrm{CO}}$ by Solomon & Vanden Bout (2005):

$L^{\prime}_{\mathrm{CO(2-1)}}=3.25\ 10^{7}\ S_{\mathrm{CO(2-1)}}\
\nu_{\mathrm{obs}}^{-2}\ D_{\mathrm{L}}^{2}\ (1+z)^{-3}$ (1)

$L^{\prime}_{\mathrm{CO(2-1)}}$ is then given in K km s-1 pc2, $z$ is the
redshift, $D_{\mathrm{L}}$ the luminosity distance in Mpc and
$\nu_{\mathrm{obs}}$ the observed frequency in GHz. For the conversion from
$L^{\prime}_{\mathrm{CO(2-1)}}$ to molecular gas masses we first need to down-
convert to the luminosity of the J=1–0 line ($L^{\prime}_{\mathrm{CO(1-0)}}$),
for which we assume a line ratio CO(2–1)/(1–0) of 0.7, which is typically
observed in several types of galaxies (Saintonge et al., 2017; den Brok et
al., 2021). Subsequent multiplication with the conversion factor
$\alpha_{\mathrm{CO}}$ finally leads to the molecular gas masses
($M_{\mathrm{H_{2}}}$). Here we use $\alpha_{\mathrm{CO}}$ = 4.5 that we
derived previously using a metallicity-dependent approach (Puschnig et al.,
2020). This value is similar to $\alpha_{\mathrm{CO}}$ in the Milky Way
(Bolatto et al., 2013). We stress that our choice of $\alpha_{\mathrm{CO}}$ is
based on a galaxy-wide average. A lower conversion factor might be applicable
in the center of the galaxy due to lower CO optical depths driven by a large
velocity dispersion. However, to date, no data (e.g. 13CO) is available to
assess any radial trend of the conversion factor in LARS 8. Again, the final
gas mass surface density map ($\Sigma_{\mathrm{{H_{2}}}}$) was corrected for
the inclination of the galaxy. The molecular gas depletion time
$\mathsf{\uptau_{depl}}$ and the gas fraction $f_{\mathrm{gas}}$ were
calculated in the following way:

$\tau_{\mathrm{{depl}}}\ =\ \frac{M_{\mathrm{{H_{2}}}}}{SFR}$ (2)
$f_{\mathrm{{gas}}}\ =\ \frac{M_{\mathrm{{H_{2}}}}}{M_{\mathrm{{H_{2}}}}\ +\
M_{*}}$ (3)

###  3.7 Molecular gas rotation curve analysis

We use 3DBAROLO (Di Teodoro & Fraternali, 2015) to derive the galaxy rotation
curve from the NOEMA CO (2–1) data cube. The software iteratively fits 3D
tilted-ring models to the cube and solves in each ring for inclination,
position angle (PA), rotation velocity and velocity dispersion. We ran the
software several times to experiment with input parameters such as pixel
coordinates of the kinematic center, fixating systemic velocity, inclination
and/or PA. Despite the fact that the algorithm robustly constrained
inclination and PA we ultimately decided to fix the two parameters (for each
ring) to 50 and 160 degrees respectively, while leaving the position of the
kinematic center as free parameter. The free parameters derived for each 0.61"
wide ring (i.e. the major axis of the beam) are thus the rotation velocity,
the velocity dispersion ($\sigma_{g}$) and the coordinates of the kinematic
center. A summary of the results is shown in Table 2. The position-velocity
diagram and the smoothed and interpolated rotation curve obtained using these
constraints are shown in Figures 3 and 4. A comparison between observed and
modeled quantities is found in Appendix 13.

Figure 3: NOEMA CO (2–1) position-velocity diagram along the major axis (top panel) and minor axis (bottom panel) of LARS 8. The yellow-brown points indicate the radial bins of the derived rotation curve. Figure 4: Smoothed and interpolated rotation curve (before inclination correction) derived for LARS 8 from the NOEMA CO (2–1) cube. Table 2: Kinematic properties for elliptical rings. | Radius
---
["]
| vrot
---
[km/s]
| $\sigma_{g}$
---
[km/s]
| i
---
[°]
| PA
---
[°]
0.305 | 154$\pm$17 | 11$\pm$4 | 50 | 160
0.915 | 154$\pm$11 | 28$\pm$7 | 50 | 160
1.525 | 142$\pm$7 | 20$\pm$4 | 50 | 160
2.135 | 151$\pm$7 | 17$\pm$4 | 50 | 160
2.745 | 176$\pm$12 | 17$\pm$6 | 50 | 160
3.355 | 215$\pm$12 | 11$\pm$7 | 50 | 160
3.965 | 223$\pm$10 | 9$\pm$5 | 50 | 160
4.575 | 210$\pm$9 | 6$\pm$5 | 50 | 160
5.185 | 240$\pm$15 | 8$\pm$4 | 50 | 160
5.795 | 272$\pm$30 | 7$\pm$4 | 50 | 160

###  3.8 Stellar velocity dispersion

We use the Penalized Pixel-Fitting method (pPXF) developed by Cappellari &
Emsellem (2004); Cappellari (2017) to measure stellar velocity dispersion
($\sigma_{*}$). We used a python wrapper developed in the course of the
PHANGS-MUSE survey by F. Belfiore and I. Pessa (Belfiore et al., 2022;
Emsellem et al., 2022) and based on the gist package (Galaxy IFU Spectroscopy
Tool; Bittner et al. 2019). As required by pPXF, we first resample the MUSE
data to a logarithmic wavelength axis using a channel size of 50 km s-1.
Following Emsellem et al. (2022), this channel size is sufficient to Nyquist
sample the line spread function of MUSE for wavelengths at approximately 7000
Å, while over-sampling it at the blue end. In order to avoid strong sky
residuals the wavelength range for fitting was limited to 4850–7000 Å. In the
following we briefly describe the fitting routine as implemented in pPXF. For
the stellar continuum fitting the E-MILES simple stellar population models of
Vazdekis et al. (2016) are used in combination with a Chabrier (2003) initial
mass function, BaSTI isochrones (Pietrinferni et al., 2004), eight ages
(0.15–14 Gyr), and four metallicities ([Z/H] = [-1.5, -0.35, 0.06, 0.4]).
Thus, a total number of 32 templates are used. Spectral ranges of strong
ionised gas emission lines are masked using a width of $\pm$400 km s-1. Since
E-MILES offers a higher resolution than our MUSE data, the templates are
convolved to the spectral resolution of our data, using an appropriate
wavelength-dependent kernel. We fitted four moments of the line-of-sight
velocity distribution: velocity, velocity dispersion, h3 and h4. To derive the
stellar kinematics we make use of additive Legendre polynomials (12th order,
in the spectral direction), and no multiplicative polynomials. The
uncertainties on the kinematic parameters are formal errors as given by pPXF.

In the literature, the stellar velocity dispersion in galaxies is often
estimated from the stellar surface density following the prescription of Leroy
et al. (2008):

$\sigma_{*}\ =\ 1.67\ \sqrt{\frac{2\ \pi\ G\ l_{*}}{7.3}}\ \Sigma_{*}^{0.5},$
(4)

with $\Sigma_{*}$ being the observed stellar surface density in SI units (kg
m-2), and $l_{*}$ being the stellar scale length (=630 pc measured via fitting
of an exponential profile to the data) in $m$. The underlying assumptions of
the equation are the following: the exponential stellar scale height h∗ of the
galaxy does not vary with radius, and $h_{*}$ is related to the stellar scale
length $l_{*}$ via $l_{*}$/$h_{*}$=7.3$\pm$2.2, i.e. the flattening ratio
measured by Kregel et al. (2002). It is further assumed that the disc is
isothermal in the z-direction and hydrostatic equilibrium then allows one to
derive $\sigma_{*}$ from the observed stellar surface density $\Sigma_{*}$ and
the estimated stellar scale height. Finally a fixed ratio of 0.6 between the
radial and vertical component of the velocity dispersion is assumed, which is
reasonable for most late-type galaxies (Shapiro et al., 2003). We refer the
reader to the appendix of Leroy et al. (2008) for more details.

A comparison of radial averages of the velocity dispersion derived from MUSE
and estimated as explained above is shown in Figure 5. We find for LARS 8 that
in particular in the central region where the velocity dispersion is highest,
the estimates overshoot the true (MUSE-based) values by up to approximately 40
percent.

Figure 5: Comparison of radial averages of the velocity dispersion derived
from MUSE (y-axis) and estimated using the prescription of Leroy et al.
(x-axis). The vertical errorbars show the propagated uncertainties of the
measurements in each radial ring.

###  3.9 Toomre $Q$ disc stability

As gravitational instability is believed to hold a key part in driving gas
turbulence (Agertz et al., 2009a; Krumholz & Burkhart, 2016), we consider a
theoretical framework to evaluate this instability. One of the most common
ways of quantifying this instability is Toomre’s $Q$ parameter (Toomre, 1964),
which governs the stability of a smaller patch inside a disc system. The
Toomre parameter for an axisymmetric, fluid disc with a differential rotation,
can be determined by analysing the response of the disc to a small
perturbation. The growth of this perturbation is driven by gravity, expressed
as a surface density wave function. By evaluating the dispersion relation,
first shown by Safronov (1960), Toomre (1964) found the condition:

$Q_{g}\ =\ \frac{\kappa\ \sigma_{g}}{\pi\ G\ \Sigma_{g}}>1$ (5)

for the disc being locally stable against graviational collapse. In this work
we compute the epicyclic frequency $\kappa$ as $1.41\
\frac{v(r_{\mathrm{gal}})}{r_{\mathrm{gal}}}\ \sqrt{1+\beta}$ and $\beta\ =\
\frac{d\log\ v(r_{\mathrm{gal}})}{d\log\ (r_{\mathrm{gal}})}$. For $Q_{g}$<1,
the disc is locally unstable. In this equation, $\kappa$ is the epicyclic
frequency, $\Sigma_{g}$ is the gas surface density, $\sigma_{g}$ is the gas
velocity dispersion (from the rotation curve analysis) and $G$ is the
gravitational constant. The physical meaning of $\kappa$ can be thought of as
the rotational support against collapse, $\sigma$ is the pressure support
against collapse and $\Sigma$ sets the level of self-gravity driving the
instability. An implementation of the method to compute $Q_{g}$ is provided
via GitHub by Puschnig (2020)222https://github.com/astrojohannes/toomreQ,
including a working example. The above method can be expanded to a disc filled
with star particles and differ only slightly from the approach of a fluid:

$\centering Q_{*}\ =\ \frac{\kappa\ \sigma_{*}}{3.36\ G\
\Sigma_{*}}\@add@centering$ (6)

Combining the Toomre parameters for stars and gas is a necessary step to
determine the stability of a multi-component disc, which is the case for most
galaxies. We assume that $\kappa$ is the same for both the gaseous and the
stellar disc, i.e. that gas and stars follow the same rotation. We stress that
the combined $Q$ is derived such that it also obeys the instability criterion
of $Q\sim$1\. There have been several different approaches to combining $Q$
parameters and an extensive look into different methods was done by Romeo &
Falstad (2013). In this paper, we use the approximation of Romeo & Falstad
(2013):

For $Q_{*}>Q_{g}$ (gas dominated regime):

$\frac{1}{Q}=\frac{1}{Q_{g}}+\frac{CF}{Q_{*}}$ (7)

For $Q_{*}<=Q_{g}$ (star dominated regime):

$\frac{1}{Q}=\frac{CF}{Q_{g}}+\frac{1}{Q_{*}}$ (8)

The correction factor CF is given for both cases via:

$\frac{2\sigma_{*}\sigma_{g}}{\sigma^{2}_{*}+\sigma^{2}_{g}}$ (9)

###  3.10 Uncertainty of the Toomre Q parameter measurement

In order to assess the uncertainty of the derived Toomre $Q$ parameter, we
propagate our measurement uncertainties, i.e. the final variance of the Toomre
$Q_{g}$ parameter is given by the sum of the following products:

  * •

squared partial derivatives of $Q_{g}$ with respect to $\kappa$ times the
square of the propagated uncertainty of $\kappa$,

  * •

squared partial derivatives of $Q_{g}$ with respect to $\sigma_{g}$ times the
square of the uncertainty of $\sigma_{g}$,

  * •

squared partial derivatives of $Q_{g}$ with respect to $\Sigma_{g}$ times the
squared uncertainty of $\Sigma_{g}$.

The uncertainty of $Q_{*}$ is calculated in an analogue way, but with
derivatives of $Q_{*}$ with respect to $\sigma_{*}$ and $\Sigma_{*}$ and using
the uncertainties on these parameters. For reference, we show the exact
formulas we used below:

$\mathrm{var}(Q_{g})=\frac{\sigma_{g}^{2}\ \kappa^{2}\
\mathrm{unc}(\Sigma_{g}^{2})}{\pi^{2}\
G^{2}\Sigma_{g}^{4}}+\frac{\sigma_{g}^{2}\ \mathrm{unc}(\kappa^{2})}{\pi^{2}\
G^{2}\ \Sigma_{g}^{2}}+\frac{\kappa^{2}\
\mathrm{unc}(\sigma_{g}^{2})}{\pi^{2}\ G^{2}\ \Sigma_{g}^{2}}$ (10)
$\mathrm{var}(Q_{*})=\frac{\sigma_{*}^{2}\ \kappa^{2}\
\mathrm{unc}(\Sigma_{*}^{2})}{3.36^{2}\
G^{2}\Sigma_{*}^{4}}+\frac{\sigma_{*}^{2}\ \mathrm{unc}(\kappa^{2})}{3.36^{2}\
G^{2}\ \Sigma_{*}^{2}}+\frac{\kappa^{2}\
\mathrm{unc}(\sigma_{*}^{2})}{3.36^{2}\ G^{2}\ \Sigma_{*}^{2}}$ (11)
$\mathrm{var}(\kappa)=\frac{1.9881\
\mathrm{unc}(v(r_{\mathrm{gal}}))^{2}}{r_{\mathrm{gal}}^{2}}$ (12)

Uncertainty for $Q_{*}>Q_{g}$ (gas dominated regime):

$\mathrm{var}(Q_{\mathrm{tot}})=\frac{CF^{2}\
\mathrm{unc}(Q_{*})^{2}}{Q_{*}^{4}\
\big{(}\frac{CF}{Q_{*}}+\frac{1}{Q_{\mathrm{gas}}}\big{)}^{4}}+\frac{\mathrm{unc}(Q_{g})^{2}}{Q_{\mathrm{gas}}^{4}\
\big{(}\frac{CF}{Q_{*}}+\frac{1}{Q_{\mathrm{gas}}}\big{)}^{4}}$ (13)

Uncertainty for $Q_{*}<=Q_{g}$ (star dominated regime):

$\mathrm{var}(Q_{\mathrm{tot}})=\frac{CF^{2}\
\mathrm{unc}(Q_{\mathrm{gas}})^{2}}{Q_{\mathrm{gas}}^{4}\
\big{(}\frac{CF}{Q_{\mathrm{gas}}}+\frac{1}{Q_{*}}\big{)}^{4}}+\frac{\mathrm{unc}(Q_{*})^{2}}{Q_{*}^{4}\
\big{(}\frac{CF}{\mathrm{gas}}+\frac{1}{Q_{*}}\big{)}^{4}}$ (14)

The individual uncertainties that occur in the equations above are estimated
in the following way. Uncertainties of the kinematics parameters derived with
3DBAROLO are obtained via exploration of the parameter space around the best
fit solutions using an MCMC approach (Iorio et al., 2017). Hence, the
uncertainties for the gas velocity dispersion, rotation velocity and $\kappa$
should be robust and statistically significant measures.

We stress that asymmetric drift correction is negligible in our case, because
the rotation velocities (see Table 2) of LARS 8 are more than ten times higher
than the velocity dispersions (de Blok et al., 2008; Iorio et al., 2017). The
uncertainties on the final rotation curve are thus equal to the uncertainties
of the rotation velocity.

Please refer to the individual sections on gas and stellar masses for a
description of how their uncertainties were estimated.

###  3.11 Molecular clumps: identification, virial mass and virial parameter

We apply CPROPSTOO (Williams et al., 1994; Rosolowsky & Leroy, 2011; Leroy et
al., 2015), an IDL package that is available through
GitHub333https://github.com/akleroy/cpropstoo and was developed to identify
and measure properties of molecular clouds or clumps in fits data cubes. In
particular, CPROPSTOO corrects for the effects of beam convolution and
sensitivity when measuring physical properties such as masses or sizes of
identified clouds, allowing to make unbiased (beam-independent) measurements.
Using a growth-curve analysis on the observed emission line, the algorithm
thus extrapolates the measurements to values one would expect in the case of
perfect sensitivity. Additionally, CPROPSTOO corrects for finite resolution in
both the velocity and spatial domain. This is done via de-convolution of the
telescope beam and the width of a spectral channel from the measured cloud
size and line width. For more details, we refer the reader to the
aforementioned publications. Here we report the main parameters for the
find_local_max task that we applied for clump identification: delta=2, /snr,
minpix=20, minarea=2, minvchan=2, friends=4, specfriends=2.

The virial mass depends on measurements of the size and the observed line
width (due to turbulence) of the cloud. If these quantities are known and the
radial density profile is given, then following Solomon et al. (1987), the
mass of the cloud under the assumptions of virial equilibrium and spherical
symmetry can be calculated:

$M_{\mathrm{vir}}=\frac{3(5-2\gamma)}{G(3-\gamma)}\Delta v^{2}R$ (15)

The virial mass $M_{\mathrm{vir}}$ is then given in M⊙ and depends on the
radial density distribution exponent $\gamma$, the linear cloud size (R) in
parsec and the full width at half-maximum (FWHM) of the line in km/s
($\Delta$v). Taking the frequently assumed $\gamma$=1 radial density
distribution exponent, corresponding to a cloud radial density profile $\rho\
\propto\ r^{-1}$ (MacLaren et al., 1988; Hughes et al., 2010), the above
equation can be re-written as:

$M_{\mathrm{vir}}=1040\sigma^{2}R$ (16)

The units are M⊙, km s-1 and pc for Mvir, $\sigma$ and $R$ respectively. In
this equation the numerical coefficient accounts for the radial density
profile, the conversion factor between FWHM and velocity dispersion
($\Delta$v=2.35$\sigma$) and the gravitational constant. Departures from
virial equilibrium can be expressed via the virial parameter,
$\alpha_{\mathrm{vir}}$:

$\alpha_{\mathrm{vir}}=\frac{2K}{U}=\frac{5\sigma^{2}R}{GM_{\mathrm{lum}}}=1.12\frac{M_{\mathrm{vir}}}{M_{\mathrm{lum}}}$
(17)

where $K$ and $U$ denote the kinetic energy and self-gravitational potential
energy respectively. The quantity Mlum is the luminous molecular mass
converted from low-J CO intensities using a conversion factor. Virialized
clouds without surface pressure or magnetic support have
$\alpha_{\mathrm{vir}}$=1, while both marginally bound clouds and clouds in
free-fall collapse share energy equipartition ($K=U$) and thus have
$\alpha_{\mathrm{vir}}\sim$2 (Ballesteros-Paredes et al., 2011; Camacho et
al., 2016; Ibáñez-Mejía et al., 2016; Sun et al., 2018).

###  3.12 Dynamical equilibrium pressure

ISM pressure plays a crucial role in many theories of star formation (e.g.
Ostriker & Shetty 2011), as it determines the gas density distribution (Helfer
& Blitz, 1997; Usero et al., 2015; Bigiel et al., 2016; Gallagher et al.,
2018). Following Elmegreen (1989) we estimate the mid-plane dynamic
equilibrium pressure, $P_{\mathrm{de}}$, using the following prescription:

$P_{\mathrm{de}}\ =\ \frac{\pi\ G\ \Sigma_{\mathrm{gas}}^{2}}{2}\ +\
\Sigma_{\mathrm{gas}}\ \sqrt{2\ G\ \rho_{*}}\ \sigma_{\mathrm{gas}}$ (18)

Here, $\Sigma_{\mathrm{gas}}$ is the total gas surface density, including the
atomic and molecular component. Since our study only covers the central part
of the galaxy, i.e. the high-density regime, in which most atomic gas is
readily converted to molecular gas, we may only consider
$\Sigma_{\mathrm{H_{2}}}$ instead. The vertical velocity dispersion of the gas
is denoted as $\sigma_{\mathrm{gas}}$ and the parameter $\rho_{*}$ is the mass
volume density of stars and dark matter at the mid-plane, which we estimate
following van der Kruit (1988) using the relation: $\rho_{*}=\Sigma_{*}/(2\
h_{*})$, with the disc scale height $h_{*}$. $P_{\mathrm{de}}$ then expresses
the pressure needed to balance the vertical gravity on the gas in the galaxy
disc. The first term reflects the gas self-gravity, the second term reflects
the weight of the gas in the potential well of the stars. Since the stellar
potential in LARS 8 exceeds the gas self-gravity, we expect the second term to
be dominant.

##  4 Results

The methods outlined in the previous section enable us to quantify clump/cloud
properties (Sections 4.1–4.2) as well as star formation relations and radial
trends in LARS 8 (Section 4.3). Finally, the gravitational instability of the
disc is shown in Section 4.4.

###  4.1 Identification of molecular clumps in LARS 8

Applying CPROPSTOO on our native resolution NOEMA CO (2–1) data cube with a
channel width of 10 km/s, we could identify 12 molecular clumps in total (see
Figures 6 and 7). The unbiased properties of the identified molecular clumps
are summarized in Table 3. Their masses range from 108.1 to 109.3 M⊙, covering
linear (extrapolated) diameters between $\sim$600–2000 pc. Clump 7 was found
to be the most massive one, located in the very center of the galaxy. The
channel maps between approximately $-$180 km s-1 and $+$210 km s-1 in Figure 7
may suggest that CPROPSTOO failed in associating extended gas to clumps in the
central region of the galaxy. This is because the linewidth in the central few
pixels of the galaxy is extremely wide (more than 300 km-1) due to beam
smearing. Much higher resolution (spatially and spectrally) would be needed to
identify individual clumps in that part of the galaxy.

Table 3: Properties of the identified molecular clumps. The clumps in Figures 6 and 7 are identified by the IDs as given in the Table below. See also Equations 16 and 17.  ID | $R_{\mathrm{eff}}$ | $\log\ M_{\mathrm{lum}}$ | $\alpha_{\mathrm{vir}}$ | $\sigma$ | $v_{\mathrm{pos}}$
---|---|---|---|---|---
| [pc] | M⊙ |  | [km s-1] | [km s-1]
1 | 601 | 8.20 | 1.2 | 16.6 | -151
2 | 322 | 8.14 | 1.0 | 19.2 | -126
3 | 913 | 8.55 | 0.9 | 16.8 | -135
4 | 343 | 8.59 | 1.0 | 30.7 | -92
5 | 340 | 9.23 | 0.8 | 59.7 | -73
6 | 320 | 8.40 | 1.4 | 31.1 | -63
7 | 1023 | 9.26 | 0.5 | 27.8 | 4
8 | 469 | 8.51 | 1.7 | 32.0 | 53
9 | 436 | 8.76 | 1.1 | 34.5 | 56
10 | 609 | 8.89 | 0.4 | 20.5 | 135
11 | 480 | 8.57 | 1.0 | 26 | 164
12 | 436 | 8.21 | 1.9 | 24.7 | 181
Figure 6: NOEMA CO (2–1) channel maps of the northern part of LARS 8, showing
velocities between -180 and +10 km/s with identified CPROPSTOO structures
(shown as contours). Properties of the identified clumps are summarized in
Table 3. Figure 7: NOEMA CO (2–1) channel maps of the southern part of LARS 8,
showing velocities between 20 and +210 km/s with identified CPROPSTOO
structures (shown as contours). Properties of the identified clumps are
summarized in Table 3.

###  4.2 Mass-size relation for the massive clumps

Figure 8 compares the derived masses and sizes of the molecular clumps
identified in LARS 8 to the literature compilation of Nguyen-Luong et al.
(2016), which contains giant molecular clouds (GMCs) of the Milky Way with
sizes smaller than 10 pc, molecular cloud complexes (MCCs) with sizes between
10 and 1000 pc, as well as galaxies and structures larger than 1 kpc typically
found at high redshift. Note that the identified structures or clumps in LARS
8 are resolved, i.e. their deconvolved diameters are at least as wide as the
beam major axis. We thus conclude from Figure 8 that the clumps of diffuse
molecular gas in LARS 8 are in fact scaled-up versions of the MCCs in the
literature. In the mass-size relation they populate the range between MCCs and
structures identified at high redshifts. However – despite an ongoing massive
star formation process in LARS 8 – the clumps follow the same trend between
mass and size. This finding implies a universal (constant) diffuse molecular
mass surface density, even in highly star-forming galaxies such as LARS 8. The
elevated star formation rates must thus result from processes within the large
diffuse molecular reservoirs we identified in CO (2–1). It might be that
either the structures contain more over-densities (e.g. traced by HCN) or that
the star formation is in some way more efficient. The latter is supported by
observations of Messa et al. (2019), who have derived sizes and properties of
clumps identified from very high-resolution UV photometry. They find that the
range of clump sizes in LARS 8 is similar to those in normal star forming
galaxies or at high redshift, i.e. 15–200 pc. However, the star formation
rates per UV clump are higher and fall between those observed in the local and
high-$z$ Universe. Also, a combination of both more dense clumps and higher
efficiency per clump may apply.

Figure 8: Mass-size relation for the massive clumps identified in LARS 8 (cyan
points) with the ID numbers as given in Table 3. The clumps are compared to
the literature compilation of Nguyen-Luong et al. (2016), that is based on GMC
(plus signs) data of Onishi et al. (2002); Heyer et al. (2009); Maruta et al.
(2010); Roman-Duval et al. (2010); Evans et al. (2014); Shimajiri et al.
(2015), MCC (stars) data of Rosolowsky (2007); Murray (2011); Wei et al.
(2012); Miura et al. (2012, 2014); Donovan Meyer et al. (2013); García et al.
(2014) and galaxies (diamonds) from Leroy et al. (2013); Tacconi et al.
(2013); Genzel et al. (2010).

###  4.3 Radial profiles and KS relation

Figure 9 shows inclination-corrected, elliptical profiles of several
quantities we have derived, centered on the maximum stellar surface density.
The plots show that while the molecular gas surface density declines
relatively smoothly from the center outwards, the stellar surface density is
peaked in the innermost $\sim$500 pc. This peak may represent a bulgelike
structure that is about to form, similar to observations in high redshift
discs (Elmegreen et al., 2009) and as predicted by numerical simulations, e.g.
in Elmegreen et al. (2008). Thereby, gas-rich disc galaxies show disc
instabilities that first trigger clump formation. These clumps (and other disc
matter) move inwards and merge, forming a bulge (or bulgelike-clump) that is
characterized by a Sersic index n=4 (like a classical bulge) and rotation. See
Rasekh et al. (2022) for a compilation of Sersic profiles for LARS galaxies.

In contrast, the star formation rate density is highest in a ring-like
structure located at a radius of $\sim$1.2 kpc. The lowering of the SFR
towards the inner kiloparsec in combination with the low molecular gas
fraction suggests that some process has quenched star formation in the center,
e.g. AGN feedback. Alternatively, it might be that the extinction correction
underestimates the true SFRs in the innermost parts, where H$\upalpha$ becomes
optically thick. However, this would not explain the relatively low gas
fraction in the center.

Moreover, Figure 9 reveals that the molecular gas depletion time,
$\mathsf{\uptau_{depl}}$, strongly declines from more than 1 Gyr in the center
to $\sim$100 Myr in the outer parts of the disc. This contrasts normal star-
forming galaxies that typically have roughly constant (Bigiel et al., 2011) or
even radially increasing gas depletion time scales (Leroy et al., 2008). This
behaviour is further suggested by (some) gravity-driven theoretical models of
star formation, e.g. Krumholz et al. (2012) argue that in the regime of normal
star-formation the GMCs are basically decoupled from the rest of the ISM. The
depletion time is then mainly set by the internal properties and processes of
the GMCs – that are roughly constant in normal Milky-Way-like clouds – rather
than by the large-scale behavior of the ISM. Krumholz et al. (2012) further
argue that in starbursts (with a Toomre $Q$ parameter $\sim$1) the depletion
time should be set by the orbital (dynamical) time. However, given the fact
that the orbital time increases with radius (flat rotation) one would expect
from such theory that the depletion time increases with radius. This is not
observed in LARS 8.

The molecular Kennicutt-Schmidt relation for LARS 8 is presented in Figure 10.
Each point in the plot represents an independent measurement (line-of-sight)
that we calculated from the mean value within 2x2 bins (using numpy reshape).
It is seen that the measurements of individual lines-of-sight exhibit a
relatively large scatter within a range of roughly one order of magnitude.
However, the central region forms an interesting feature that is characterised
by a roughly constant star formation rate density while the molecular gas
surface density varies by up to an order of magnitude, with a mean gas
depletion time around $\sim$1 Gyr.

Figure 9: Top panel: Elliptical inclination-corrected profiles of stellar
(blue diamonds), molecular (cyan circles) and SFR surface densities (black
squares). Bottom panel: Same as top panel for the molecular gas depletion time
($\mathsf{\uptau_{depl}}$), the H$\upalpha$-based star formation rate surface
density ($\Sigma_{\mathrm{SFR}}$) and the molecular gas fraction
($f_{\mathrm{gas}}$). Figure 10: Resolved molecular Kennicutt-Schmidt relation
for LARS 8, based on SFRs from extinction-corrected H$\upalpha$ and molecular
masses from CO (2–1), both corrected for inclincation. Each point corresponds
to a measurement in a $\sim$650 pc sized region/pixel. The colors indicate
distance from the center.

###  4.4 Disc stability - Toomre $Q$ analysis

Using the smoothed rotation curve (see Figure 4) derived from the NOEMA CO
(2–1) data cube, and subsequent calculation of the $\beta$-parameter and the
epicyclic frequency $\kappa$, the Toomre $Q$ parameters for the molecular gas
($Q_{\mathrm{gas}}$), the stellar component ($Q_{*}$) and the combined total
instability parameter ($Q_{\mathrm{tot}}$) could be computed as a function of
galactocentric radius (see Figure 11). Note that we have centered the
previously discussed radial profiles on the stellar peak, while here we (have
to) use the kinematic center. Between these two we find an offset of $\sim$0.8
arcsec or $\sim$650 pc. Such offset is also found in numerical simulations of
Elmegreen et al. (2008) during the phase of the formation of a central
bulgelike-clump. We stress that the overall shape of the radial profiles does
not change if the kinematic center is used instead.

Figure 11 reveals that only the innermost $\sim$500 pc of LARS 8 are stable.
This central stability is mainly driven by high values of $\kappa$ due to the
extremely steep rise in rotation velocity that causes very high $\beta$
values. Note that although we cannot (kinematically) resolve the central
$\sim$500 pc, i.e. we cannot distinguish between rotation and dispersion (beam
smearing), it is still possible to compute $\kappa$. The plot further shows
that the outskirts of the disc are unstable over large scales, with values of
$Q_{\mathrm{tot}}$ well below the critical limit of 1. Such highly unstable
discs are not observed in normal star-forming disc galaxies (Leroy et al.,
2008), but seem typical for the clumps observed in massive high-$z$ discs
(Genzel et al., 2011; Wisnioski et al., 2012; Mieda et al., 2016). The
relatively high star formation rate surface densities observed in LARS 8 over
large scales are thus likely the result of enhanced disc fragmentation due to
$Q_{\mathrm{tot}}<<$1\. These instabilities thus trigger the formation of
massive stellar and molecular clumps.

However, it seems that purely gravity-driven theoretical models of star
formation do not reproduce our observations, in particular e.g. Krumholz et
al. (2012) predict for galaxies in the Toomre regime (as LARS 8) a positive
correlation between the molecular gas depletion time and the orbital period.
As explained, this is not observed in LARS 8.

Other models assume that the star formation process is self-regulated and thus
leads to pressure balance in the ISM. In particular, the star-forming system
is then in balance between feedback processes from star formation and the
external pressure. In case of a disc galaxy the relevant pressure is then
$P_{\mathrm{de}}$, the dynamical equilibrium pressure. Based on that, e.g.
Ostriker & Shetty (2011) predict a linear relation between the star formation
rate surface density and the ISM pressure. We test this prediction in the next
section.

Figure 11: LARS 8 disc stability from radial Toomre $Q$ analysis. Regions with
$Q<$1 (light red shaded area) are considered unstable. The total instability
parameter ($Q_{\mathrm{tot}}$) is shown as black curve with the grey shaded
area indicating its uncertainty. The contribution of the stellar and the
gaseous component are shown by the blue and cyan curves respectively.

##  5 Discussion

We showed in the previous section that the galactic disc of LARS 8 is highly
unstable, in particular at radii outwards of $\sim$500 pc. The Toomre $Q$
parameter is found to be significantly lower than one (see Figure 11) and we
conclude that the formation of the observed massive molecular and stellar
clumps is driven by fragmentation of the disc rather than accretion of
external mass or merging. Contrarily, the central region of LARS 8 was found
to be different. It has a Toomre $Q$ parameter greater than one and is thus
stable, it has a relatively low gas fraction and a low star formation rate
density, and it has a depletion time of more than $\sim$1 Gyr (which is much
longer than within the disc).

Utomo et al. (2017) studied the molecular gas depletion time as a function of
local environment in 52 non-AGN disc galaxies drawn from the EDGE-CALIFA
(Sánchez et al., 2012; Bolatto et al., 2017) survey. They find that galaxies
with increased central stellar surface densities (relative to the disc)
typically show a decrease in $\mathsf{\uptau_{depl}}$ in the center. As
stellar surface density is the determining factor for ISM pressure, Utomo et
al. (2017) claim that the observed shorter central gas depletion times are a
consequence of higher external pressure that facilitates cloud collapse. In
the center of LARS 8 we also observe an increase in stellar surface density
compared to the disc, but at the same time – for the center – we find longer
molecular gas depletion times. Additionally, our radial plots (Figure 9) show
that the star formation rate surface density sharply drops towards the center,
while the molecular gas surface density in LARS 8 decreases relatively
smoothly from the center to the outskirts.

Some process in the center must therefore lead to quenching of star formation.
We find evidence that shear is mainly responsible for the suppression of star
formation in the center, as we see that $\kappa$ increases by a factor of
$\sim$4 between a radius of $\sim$1 kpc towards the innermost central region.
Also the gas velocity dispersion increases in that radial regime (from $\sim$1
kpc to the center), however only by roughly 50 percent. Thus, it is mainly
shear that causes $Q>>1$ in the center. Feedback from supernovae (that drive
the gas velocity dispersion) thus plays only a minor role (if any) for the
suppression of star formation. In fact our kinematic results (see Table 2)
even suggest that the gas velocity dispersion drops towards the innermost
region. This observation further rules out feedback from SNe, but we caution
that the measurement of velocity dispersion in the center is relatively
uncertain due to beam smearing caused by the steep increase of the rotation
curve. However, further support against star formation quenching due to SNe is
found from stellar population synthesis performed by Melinder et al (in
preparation). They show that the central $\sim$500 pc of LARS 8 are dominated
by old stars with ages $>$1 Gyr.

We also compare our observations to the feedback models of e.g. Ostriker et
al. (2010) or Faucher-Giguère et al. (2013), which are based on a balance
between energy injected through feedback and disc vertical pressure. The
models predict an inverse relation between $\mathsf{\uptau_{depl}}$ and the
vertical gas velocity dispersion. Such relation was previously observed by
Fisher et al. (2019) in a set of massive and highly turbulent discs. However,
from our data of the central region of LARS 8 we cannot test any such
correlation, because of spatial resolution and beam smearing that makes
measurements of the velocity dispersion extremely challenging.

Given our current data, we thus conclude that shear is the most likely cause
for the relatively low star formation rates in the center and the long
depletion times. As we mentioned in the previous section, it might also be the
case that the computed star formation rates in the center are somewhat
spurious due to the relatively high extinction found in this region. We stress
that the total galaxy-wide SFR from extinction-corrected H$\upalpha$ is in
fact 50 percent higher than the SFR we previously derived in Puschnig et al.
(2020) from far infrared measurements. This might be an indication that the
Balmer-decrement method overestimates the true fluxes/SFRs rather than
underestimating it. On the other hand, the discrepancy between infrared and
H$\upalpha$ based SFRs may be the result of a star formation history with a
recent burst (to which H$\upalpha$ is more sensitive).

In contrast, the environmental properties of the outer disc of LARS 8 are
different, in particular we find that the disc is highly unstable. The radial
profiles in Figure 9 further revealed that the molecular gas depletion time in
LARS 8 decreases with the galactic radius. This behaviour is contrary to what
is typically observed in nearby disc galaxies in which
$\mathsf{\uptau_{depl}}$ either stays flat or slightly increases with radius
(Leroy et al., 2008). Note that models of star formation in stable discs, e.g.
Krumholz et al. (2012), predict exactly such behaviour for GMCs that are
basically decoupled from the large-scale ISM. In these models star formation
is mainly dictated by local properties rather than large-scale effects.
Additionally, Semenov et al. (2017) showed that for regular, local spiral
galaxies, $\mathsf{\uptau_{depl}}$ is $\sim$1–2 Gyr due to the long time the
gas spends in the non-star-forming phase, while only a small fraction of the
gas is converted into stars within a short time.

The difference in the radial $\mathsf{\uptau_{depl}}$ profiles between LARS 8
and normal star forming disc galaxies (Romeo & Mogotsi, 2017) is thus the
result of the observed large-scale Toomre instabilities in LARS 8, in which
the ISM is dominated by dozens of supermassive star-forming clouds that
disallow the star forming regions to decouple from the ambient ISM (as they
make up the ISM). Krumholz et al. (2012) also made predictions of
$\mathsf{\uptau_{depl}}$ for starbursts in the Toomre regime ($Q\sim$1), for
which they find that $\mathsf{\uptau_{depl}}$ should mainly be dictated by the
dynamic timescale, i.e. $2r\pi/v_{\mathrm{rot}}$. Our observations, however,
are not in agreement with this prediction of a radially increasing gas
depletion time. We argue that the observed instabilities in LARS 8 are more
violent ($Q<<$1) and thus involve more complex physical processes such as
galaxy-scale shocks or inflows (Barnes, 2004; Teyssier et al., 2010; Powell et
al., 2011) which were omitted by the models of Krumholz et al. (2012).

We now test our observations against models that are based on the assumption
that star formation is self-regulated through a balance between ISM pressure
and feedback. For example, Ostriker & Shetty (2011) and Kim et al. (2013)
predict in their semi-analytic models a (nearly) linear relationship between
the pressure and the star formation rate surface density:
$\Sigma_{\mathrm{SFR}}=4\ f^{-1}\ P_{\mathrm{de}}$. As described in Fisher et
al. (2019) the scaling factor $f$ can be determined from
$\sigma_{\mathrm{gas}}=0.366\ (\tau_{\mathrm{ff}}/\tau_{\mathrm{depl}})\ f$
(Shetty & Ostriker, 2012). This leaves the free-fall timescale
$\tau_{\mathrm{ff}}$ as the only unknown. Krumholz et al. (2012) further
estimate that the range of $\tau_{\mathrm{ff}}$ should be between 1–10 Myr for
starburst galaxies, i.e. in high density regimes. In Figure 12 we plot the
star formation rate densities against pressure for LARS 8 and two comparison
samples. Each point in the plot represents an independent measurement (line-
of-sight) that we calculated from the mean value within 2x2 bins. The dashed
and dotted lines indicate the model predictions for the above mentioned range
in $\tau_{\mathsf{ff}}$ and a fixed gas depletion time of 300 Myr that we
typically find in the disc of LARS 8. The Figure shows that the predicted
linear relation does not fit the data, we rather find evidence for a sub-
linear trend, similar to Fisher et al. (2019). The slope in LARS 8, however,
seems even shallower, in particular in the low-pressure regime. We conclude
that in the outskirts of the observed disc the star formation is out of
equilibrium as described in feedback-regulated star formation models, and is
dictated by large scale instabilities instead.

The importance of the large-scale environment for star formation in LARS 8 is
also reflected by the fact that the virial parameter of the identified diffuse
molecular structures (see Table 3) has values that are roughly identical to
those found in Milky Way GMCs or normal disc galaxies (Sun et al., 2018). Most
clumps are found to be virialized with $\alpha_{\mathrm{vir}}\sim 1$ in which
kinetic and gravitational energy are roughly balanced. This provides further
evidence that on the scale of a few hundred parsec the stars form in a roughly
uniform environment. The high star formation rates observed in LARS 8 must
thus be caused by an increase in the number of clouds that are triggered by
large-scale gravitational instability (with low Toomre $Q$). Hence, the
shorter gas depletion time scale – or higher star formation efficiency –
observed in the outer disc does not imply that on our clump scales the process
of star formation is more efficient, but rather that the formation of
individual clumps is more efficient.

Next, we discuss how the choice of a fixed CO conversion factor
$\alpha_{\mathrm{CO}}$ impacts our findings of the radial trend of
$\mathsf{\uptau_{depl}}$ and the Toomre $Q$ instability of the disc. We know
from the MUSE data that there is a slight increase in metallicity towards the
center of the galaxy. Hence, application of a metallicity-dependent conversion
factor would only lead to a relatively lower value of $\alpha_{\mathrm{CO}}$
in the center than in the outer part of the disc. As a result, this would only
exaggerate the observed trend of decreasing molecular gas depletion with
radius. For the results of our disc stability analysis, the fixed conversion
factor has only minor impacts for two reasons. First, the instabilities are
mainly driven by the stellar component (which is formally also shown in Romeo
& Falstad 2013). Second, it would only lead to slightly higher gas surface
densities in the disc, lowering support of the disc against collapse and thus
resulting in even lower $Q$ values.

However, not only the metallicity impacts the conversion factor. In infrared
galaxies, but also in the centers of nearby galaxies, the nuclear zone is
sometimes found to have lower $\alpha_{\mathrm{CO}}$ caused by hotter
molecular gas, thus higher velocity dispersion (which reduces the CO optical
depth). This is seen e.g. in NGC 6946 (Meier & Turner, 2004). If CO optical
depths in the center of LARS 8 were systematically lower, we would need to use
a lower conversion factor. In the case of a typical ULIRG value
($\alpha_{\mathrm{CO}}\sim 1$), the central depletion time would then drop
from 1.35 Gyr to 300 Myr. At the same time, this would provide even further
support against collapse in the central zone. We plan to resolve this issue
with observations of CO isotopologues in a future study.

Figure 12: Star formation rate surface density (y-axis) versus dynamical
equilibrium pressure (x-axis). Each point corresponds to a measurement in a
$\sim$650 pc sized region/pixel. The colors indicate the distance from the
center.

##  6 Summary and Conclusion

We have obtained new high-resolution NOEMA CO (2–1) and MUSE spectroscopy of
the $z\sim$0 massive, clumpy and gas-rich disc galaxy LARS 8, drawn from the
Lyman Alpha Reference Sample. The NOEMA data was used to study the diffuse
molecular gas content and its kinematics at a resolution of $\sim$400 pc,
while the MUSE data was used to derive extinction-corrected star formation
rates from H$\upalpha$ at a resolution of $\sim$600 pc. This enabled us –
together with readily available HST photometry – to perform a disc stability
analysis using the Toomre $Q$ criterion. The main result is presented in
Figure 11, showing that the disc is highly unstable ($Q<<$1) over large
scales. On the other hand, the center of LARS 8 was found to be stable
($Q>$1).

The NOEMA molecular data cube was further examined with CPROPSTOO, allowing us
to identify and compute physical properties of 12 individual molecular clumps
(Table 3). The clumps are found to be virialized
($\alpha_{\mathrm{vir}}\sim$1) and they follow the mass-size relation (Figure
8).

We have further derived several physical parameters such as the molecular gas
depletion time, the molecular gas fraction and the dynamical equilibrium
pressure. Using our results from the CO-based rotation curve (Figure 4), all
(surface) quantities could be corrected for inclination effects. The radial
(elliptical) profiles are shown in Figure 9. Of particular interest is the
smooth radial decline of the molecular gas depletion time, ranging from more
than 1 Gyr in the center to $\sim$100 Myr in the outer disc. This trend is
outstanding, as in normal star forming galaxies the gas depletion time is
observed to be constant or even slightly increasing with radius. These results
lead to the following conclusions:

  * •

The disc of LARS 8 is highly unstable with $Q<<$1 and has relatively short gas
depletion times. The identified diffuse molecular structures, however, are
virialized and thus similar to GMCs in the Milky Way or nearby galaxies.
Hence, the short gas depletion times in the disc cannot be explained by local
(sub-kpc) effects such as a higher local star formation efficiency, but must
be triggered by large-scale processes that cause the formation of more massive
and denser molecular clumps. The observed short gas depletion times observed
in the disc must thus result from more dense gas being present on sub-clump
scales, i.e. density PDFs shifted towards higher values. We argue that the
high star formation rates observed in LARS 8 are the result of large-scale
Toomre instabilities in the galaxy disc.

  * •

The central region of LARS 8 is Toomre-stable, has the longest gas depletion
time, lower gas fraction and a reduced star formation rate surface density.
Given the fact that the stellar surface density (and thus the ISM pressure) is
found to be highest in the center, we argue that some process must lower star
formation in the central $\sim$500 pc. From our dynamical analysis we find
evidence that shear (and not feedback from SNe) is the main driving mechanism
that suppresses the star formation in the center of LARS 8, as $\kappa$
increases by a factor of 4 from r$\sim$1 kpc to r$\sim$0 kpc.

## Acknowledgements

J.P. acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No.726384/Empire). M.H. is Fellow of the Knut and Alice Wallenberg
Foundation. O.A. acknowledges support from the Knut and Alice Wallenberg
Foundation and from the Swedish Research Council (grant 2019-04659). This
research made use of Astropy, a community-developed core Python package for
Astronomy (Astropy Collaboration et al., 2013).

## Data Availability

The MUSE raw data underlying this article are available in the ESO public
archive at http://archive.eso.org/ and can be accessed with the program ID
0101.B-0703(A). The reduced MUSE data cube and the NOEMA data underlying this
article will be shared on reasonable request to the corresponding author.

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##  Appendix A Observed vs. modeled moment maps

Figure 13: Observed (left column) versus modeled (right column) properties of
LARS 8. The latter were produced with 3DBAROLO.