# Resolved Maps of the CO-to-H2 Conversion Factor in 41 Nearby Galaxies

I-Da Chiang (江宜達) Institute of Astronomy and Astrophysics, Academia Sinica,
No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan  Karin M. Sandstrom
Department of Astronomy & Astrophysics, University of California, San Diego,
9500 Gilman Drive, La Jolla, CA 92093, USA  Jérémy Chastenet Sterrenkundig
Observatorium, Universiteit Gent, Krijgslaan 281 S9, B-9000 Gent, Belgium
Alberto D. Bolatto Department of Astronomy, University of Maryland, College
Park, MD 20742, USA  Eric W. Koch Center for Astrophysics $|$ Harvard &
Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Adam K. Leroy
Department of Astronomy, The Ohio State University, 4055 McPherson Laboratory,
140 West 18th Avenue, Columbus, OH 43210, USA  Center for Cosmology and
Astroparticle Physics, 191 West Woodruff Avenue, Columbus, OH 43210, USA
Jiayi Sun (孙嘉懿) Department of Physics and Astronomy, McMaster University, 1280
Main Street West, Hamilton, ON L8S 4M1, Canada  Canadian Institute for
Theoretical Astrophysics (CITA), University of Toronto, 60 St George Street,
Toronto, ON M5S 3H8, Canada  Department of Astrophysical Sciences, Princeton
University, 4 Ivy Lane, Princeton, NJ 08544, USA  Yu-Hsuan Teng Center for
Astrophysics and Space Sciences, Department of Physics, University of
California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA  Thomas G.
Williams Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117
Heidelberg, Germany  Sub-department of Astrophysics, Department of Physics,
University of Oxford, Keble Road, Oxford OX1 3RH, UK

###### Abstract

We measure the CO-to-H2 conversion factor ($\alpha_{\mathrm{CO}}$) in 41
galaxies at 2 kpc resolution, using dust surface density inferred from far-
infrared emission as a tracer of the gas surface density and assuming a
constant dust-to-metals ratio. In total, we have $\sim 940$ and $\sim 660$
independent measurements of $\alpha_{\mathrm{CO}}$ for CO (2-1) and (1-0),
respectively. The mean values for $\alpha_{\mathrm{CO~{}(2-1)}}$ and
$\alpha_{\mathrm{CO~{}(1-0)}}$ are $6.9^{+3.9}_{-4.5}$ and
$3.1^{+1.6}_{-1.7}$, respectively. The CO-intensity-weighted mean for
$\alpha_{\mathrm{CO~{}(2-1)}}$ is 4.37, and 2.58 for
$\alpha_{\mathrm{CO~{}(1-0)}}$. We examine how $\alpha_{\mathrm{CO}}$ scales
with several physical quantities, e.g. star-formation rate (SFR), stellar
mass, and interstellar radiation field (ISRF). Among them, the strength of the
ISRF and the resolved specific SFR have the strongest anti-correlation with
spatially resolved $\alpha_{\mathrm{CO}}$. No strong correlations are found
between galaxy-averaged $\alpha_{\mathrm{CO}}$ and galaxy properties. We also
find that $\alpha_{\mathrm{CO}}$ decreases with stellar mass surface density
($\Sigma_{\star}$) in the dense regions ($\Sigma_{\star}\geq 200~{}{\rm
M_{\odot}~{}pc^{-2}}$), following the power-law relations
$\alpha_{\mathrm{CO~{}(2-1)}}\propto\Sigma_{\star}^{-0.5}$ and
$\alpha_{\mathrm{CO~{}(1-0)}}\propto\Sigma_{\star}^{-0.3}$. The power-law
index is insensitive to the assumed D/M. Meanwhile, the discrepancy between
the power-law index for CO (2-1) and (1-0) data indicates a variation in the
CO line ratio traced by $\Sigma_{\star}$. We interpret the decrease in
$\alpha_{\mathrm{CO}}$ with increasing $\Sigma_{\star}$ as a result of higher
velocity dispersion compared to isolated, self-gravitating clouds due to the
additional gravitational force from stellar sources, which leads to the
reduction of $\alpha_{\mathrm{CO}}$. The decrease in $\alpha_{\mathrm{CO}}$ at
high $\Sigma_{\star}$ is important for accurately assessing molecular gas
content and star-formation efficiency in the centers of galaxies, which bridge
“MW-like” to ”starburst-like” conversion factors.

dust, extinction – galaxies: ISM – infrared: ISM – radio lines: ISM – ISM:
molecules

††facilities: ALMA, Herschel, IRAM:30m, VLA, WISE, WSRT††software:  astropy
(Astropy Collaboration et al., 2013, 2018, 2022), matplotlib (Hunter, 2007),
numpy & scipy (van der Walt et al., 2011),

##  1 Introduction

Star formation is fueled by the molecular gas in the interstellar medium
(ISM). Thus, observing the molecular ISM is essential for studies of star
formation and galaxy evolution. Unfortunately, the most abundant molecule in
the ISM, H2, is not directly observable in many cases in the the cold
molecular ISM due to its high transition energies ($h\nu/k_{\mathrm{B}}\sim
510~{}\mathrm{K}$) for the lowest rotational levels. As a result, low-$J$ CO
emission lines are the most widely used tracer for the molecular ISM, as the
second most abundant molecule with strong millimeter rotational lines that can
be excited at typical temperatures in molecular clouds. The standard practice
is to use a CO-to-H2 conversion factor $\alpha_{\rm CO}$, as follows

$\Sigma_{\mathrm{mol}}=\left\\{\begin{array}[]{l}\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)},~{}\mathrm{for}~{}\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)\\\
\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)},~{}\mathrm{for}~{}\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)\end{array}\right.\,.$
(1)

where $\Sigma_{\mathrm{mol}}$ [$\rm M_{\odot}~{}pc^{-2}$] is the surface
density of molecular ISM, $\alpha_{\rm CO}$ [$\rm M_{\odot}~{}pc^{-2}({\rm
K~{}km~{}s^{-1}})^{-1}$] is the “CO-to-H2 conversion factor”, and $W_{\rm CO}$
[${\rm K~{}km~{}s^{-1}}$] is the integrated intensity of the CO emission at
rest frame frequency. The “(1  $-$ 0)” and “(2  $-$ 1)” symbols represent the
CO transition which $\alpha_{\rm CO}$ is derived for and $W_{\mathrm{CO}}$ is
measured, i.e. the CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ transition at $\sim
115~{}\mathrm{GHz}$ ($\lambda\sim 2.6~{}\mathrm{mm}$)111In column density
units, the standard MW conversion factor is
$X_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}=2\times 10^{20}\,\rm
cm^{-2}\,(K\,km\,s^{-1})^{-1}$. In mass surface density units with a factor of
1.36 for He included, this is equivalent to
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}=4.35$ ${\rm
M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}$. and the CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ transition at $\sim 230~{}\mathrm{GHz}$
($\lambda\sim 1.3~{}\mathrm{mm}$), respectively.

CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ had been the most frequently measured CO
transition, and $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ is
thus the fiducial case for $\alpha_{\rm CO}$ in the literature. Meanwhile, CO
emission measured with modern instruments, e.g. ALMA, throughout the last
decade is overwhelmingly CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$, and directly
deriving $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ has attained
its own importance. Thus we will present both
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ and
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ in this work.

With the precise measurements of CO emission from modern instruments, our
understanding of $\alpha_{\rm CO}$ has become the factor that limits our
ability to precisely study the molecular ISM and star formation in nearby
galaxies. Observations have shown at least two main trends in the variation of
$\alpha_{\rm CO}$. The first one is that $\alpha_{\rm CO}$ tends to increase
at lower metallicity or lower dust-to-gas ratios (e.g. up to $\sim
1~{}\mathrm{dex}$ higher than the MW value at $\sim 0.2$ solar metallicity,
Israel, 1997b; Leroy et al., 2011). This enhanced $\alpha_{\rm CO}$ is often
explained by the decrease in CO emission relative to the cloud mass defined by
H2 as shielding for CO weakened at lower metallicity (Papadopoulos et al.,
2002; Grenier et al., 2005; Wolfire et al., 2010; Planck Collaboration et al.,
2011; Madden et al., 2020). This phenomena is often known as the “CO-dark
gas.”

The second trend is that $\alpha_{\rm CO}$ appears to be lower in the central
$\sim\mathrm{kpc}$ of some galaxies (it can be a factor 5–10 times lower,
Israel, 2009a, b, 2020; Sandstrom et al., 2013; Teng et al., 2022, 2023). It
is also observed to be lower in (ultra-)luminous infrared galaxies (U/LIRGs,
Downes et al., 1993; Downes & Solomon, 1998; Papadopoulos et al., 2012;
Herrero-Illana et al., 2019). This trend towards lower $\alpha_{\rm CO}$ in
galaxy centers and starbursts likely results from a combination of higher gas
temperature, larger line width, and lower CO optical depth, which breaks the
relationship between molecular cloud mass and line width that one would expect
from isolated, virialized clouds (Shetty et al., 2011; Bolatto et al., 2013).
This phenomena is often referred to as the “starburst conversion factor.”
Because galaxy centers and U/LIRGs tend to be bright in CO and thus easily
observed, understanding this starburst conversion factor is important to
making best use of a wide range of extragalactic observations in
characterizing the star formation efficiency, gas dynamics and H i-to-H2
transition conditions.

Bolatto et al. (2013) proposed a formula for conversion factor treating the
CO-dark gas and the starburst trend independently and simultaneously. This
formula aimed to predict both the spatially resolved measurements from
Sandstrom et al. (2013, including galaxy centers) and the (U)LIRGs
measurements in Downes & Solomon (1998). The formula reads:

$\frac{\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}}{1~{}{\rm
M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}}=\\\ \rm
2.9\times\exp\left(\frac{0.4}{Z^{\prime}}\right)\times\left\\{\begin{array}[]{ll}\left(\Sigma_{\rm
Total}^{100}\right)^{-0.5}&,\,\Sigma_{\rm Total}^{100}\geq 1\\\
1&,\,\Sigma_{\rm Total}^{100}<1\end{array}\right.\,.$ (2)

Here $\rm Z^{\prime}$ is the metallicity relative to the solar value, which
traces the CO-dark gas effect; $\Sigma_{\rm Total}^{100}$ is the total surface
density ($\Sigma_{\mathrm{Total}}=\Sigma_{\mathrm{gas}}+\Sigma_{\star}$) in
unit of $100~{}\mathrm{M_{\odot}~{}pc^{-2}}$, which is the proposed
observational tracer and threshold for regions where the decrease of
$\alpha_{\rm CO}$ occurs, i.e. galaxy centers and (U)LIRGs. The authors found
that with a threshold at $\Sigma_{\rm Total}^{100}\geq 1$ (a threshold related
to self-gravitating giant molecular clouds), the relation $\alpha_{\rm
CO}\propto\left(\Sigma_{\rm Total}^{100}\right)^{-0.5}$ reproduces the trend
found in galaxy centers and ULIRG samples. A similar formula was also
suggested by Ostriker & Shetty (2011), where the authors suggested a power-law
relationship between $\alpha_{\rm CO}$ and $\Sigma_{\mathrm{gas}}$ to describe
the decrease in $\alpha_{\rm CO}$ needed for their simulations to match
observations. They showed that a relation of $\alpha_{\rm
CO}\propto\Sigma_{\mathrm{gas}}^{-0.5}$ over the surface density range
$10^{2}$ to $10^{3}$ $\rm M_{\odot}~{}pc^{-2}$ brings the observations and
simulations into agreement.

To better characterize how $\alpha_{\rm CO}$ depend on local environments,
spatial resolved measurements of $\alpha_{\rm CO}$ are required. To measure
$\alpha_{\rm CO}$, one needs to measure $\Sigma_{\mathrm{mol}}$ independent of
(single-line) CO intensity, and then divide it by the measured CO intensity.
This could be achieved by several methodologies, e.g. using virial mass
estimates (see the review in McKee & Ostriker, 2007), modelling multiple
spectral lines (e.g. Cormier et al., 2018; Teng et al., 2022, 2023),
converting $\gamma$-ray emission (e.g. Abdo et al., 2010; Ackermann et al.,
2012), and tracing gas mass with dust mass (Israel, 1997a, b; Leroy et al.,
2007, 2009b, 2011; Bolatto et al., 2011; Planck Collaboration et al., 2011;
Sandstrom et al., 2013; Schruba et al., 2017; den Brok et al., 2023). Based on
existing resources, most of the methods are not practical for a survey in tens
of nearby galaxies due to the requirement in target brightness or total
observing time. The most feasible methodology is to use dust as a tracer for
gas mass, where dust mass can be derived from infrared (IR) data observed with
Herschel (Pilbratt et al., 2010), WISE (Wright et al., 2010) and Spitzer.

In this work, we measure $\alpha_{\rm CO}$ with dust as the tracer for total
gas mass. We use dust masses derived from modeling the far-IR spectral energy
distribution (SED) to trace the total gas mass. The key assumption we make is
a constant fraction of heavy elements locked in the solid phase, i.e. a dust-
to-metals ratio (D/M), which allows us to convert measurements of dust surface
density, H i surface density and metallicity into molecular gas mass. The
assumption of approximately constant D/M is supported by dust evolution models
(Dwek, 1998; Hirashita & Kuo, 2011; Asano et al., 2013; Feldmann, 2015) and
kpc-scale measurements (Issa et al., 1990; Leroy et al., 2011; Draine et al.,
2014; Vílchez et al., 2019; Chiang et al., 2021) in high-metallicity
($12+\log({\rm O/H})\gtrsim 8.2$) galaxy disks, matching the region of
interest in this work. In simulations (e.g. Aoyama et al., 2020; Choban et
al., 2022), an approximately constant D/M results from efficient dust growth
in the ISM, i.e. the majority of the refractory elements are locked in solid
grains quickly. Although there are also studies that found variations in D/M
with both depletion measurements (Jenkins, 2009; Jenkins & Wallerstein, 2017;
Roman-Duval et al., 2019) and emission measurements (Roman-Duval et al., 2014,
2017; Chiang et al., 2018; De Vis et al., 2019), no widely agreed-upon
prescription for the environmental dependence of D/M has not been found thus
far. The other reason we assume a constant D/M is that we anticipate the
variation of D/M ($\leq 2$ times) to be smaller in comparison to that of
$\alpha_{\rm CO}$ (up to $\sim 10$ times) in normal galaxy disks.

Given the challenges in measuring spatial resolved $\alpha_{\rm CO}$, there
have been few studies with large samples of resolved measurements in galaxies.
Sandstrom et al. (2013) looked at the overlap of IR from KINGFISH (Kennicutt
et al., 2011), CO from HERACLES (Leroy et al., 2009a), and H i from THINGS
(Walter et al., 2008) and made measurements of $\alpha_{\rm CO}$ in 26
galaxies with $\sim$40″resolution elements, which is the resolved study with
one of the largest sample size. Most other studies in the field, e.g. Hunt et
al. (2015), Accurso et al. (2017) and COMING (Yasuda et al., 2023), focused on
galaxy-scale $\alpha_{\rm CO}$. Moreover, a survey of $\alpha_{\rm CO}$ at
fixed physical scale, which allows us to evaluate the environmental dependence
of $\alpha_{\rm CO}$ fairly, is also missing as previous spatially resolved
studies (e.g. Leroy et al., 2011; Schruba et al., 2012; Sandstrom et al.,
2013) tend to perform their analysis at fixed angular resolution. In this
work, we will measure $\alpha_{\rm CO}$ across 41 nearby galaxies at a fixed
$\sim$2 kpc resolution. This study is made possible by several surveys of
resolved CO intensities in the past two decades: the Nobeyama CO Atlas of
nearby galaxies (CO Atlas, Kuno et al., 2007), the HERA CO Line Extragalactic
Survey (HERACLES, Leroy et al., 2009a), the CO Multi-line Imaging of Nearby
Galaxies project (COMING, Sorai et al., 2019), the Physics at High Angular
resolution in Nearby Galaxies project (PHANGS-ALMA, Leroy et al., 2021), and
recent IRAM 30m observations (PI: A. Schruba; see Leroy et al., 2022).

This paper is presented as follows. In Section 2, we explain our methodology
for deriving $\alpha_{\rm CO}$ from the data. We describe the data sets
necessary for this work and how we constrain other physical quantities from
observations in Section 3. We present the $\alpha_{\rm CO}$ measurements and
their correlations with local and galaxy-integrated conditions in Section 4.
In Section 5, we investigate an observed power-law relation between
$\alpha_{\rm CO}$ and $\Sigma_{\star}$ in the high-surface-density regions,
and provide a prescription for $\alpha_{\rm CO}$ based on our findings. We
discuss the physical interpretations of our results and how it compares to the
literature findings in Section 6. Finally, we summarize our findings in
Section 7.

##  2 Calculating $\alpha_{\rm CO}$

To calculate $\alpha_{\rm CO}$, we first estimate $\Sigma_{\mathrm{mol}}$
without using CO emission and an adopted conversion factor. In this study, we
use a dust-based strategy to estimate $\Sigma_{\mathrm{mol}}$ by assuming a
value for the fraction of metals locked in solid phase, i.e. the dust-to-
metals ratio (D/M). The D/M is defined as:

${\rm D/M}=\frac{\Sigma_{\mathrm{dust}}}{Z\times\Sigma_{\mathrm{gas}}},$ (3)

where $\Sigma_{\mathrm{gas}}=\Sigma_{\mathrm{atom}}+\Sigma_{\mathrm{mol}}$ is
the total neutral gas surface density, $\Sigma_{\mathrm{mol}}$ is the
molecular gas surface density, and the metallicity $Z$ converts
$\Sigma_{\mathrm{gas}}$ to a “metal mass surface density”. By replacing
$\Sigma_{\mathrm{mol}}$ in the above equation with the definition of
$\alpha_{\rm CO}$ in Eq. 1, we have:

$\alpha_{\rm
CO}=\Big{(}\frac{\Sigma_{\mathrm{dust}}}{Z\times\mathrm{(D/M)}}-\Sigma_{\mathrm{atom}}\Big{)}{W_{\mathrm{CO}}^{-1}},$
(4)

where $\Sigma_{\mathrm{dust}}$, Z, $\Sigma_{\mathrm{atom}}$ and
$W_{\mathrm{CO}}$ are measurable quantities, thus, by assigning a value of
D/M, we can then calculate $\alpha_{\rm CO}$ with our data set.

Dust-based $\alpha_{\rm CO}$ measurements in the literature usually have
formulae similar with Eq. 3 but with different assumptions. For example,
Israel (1997b) and Leroy et al. (2009b) assumed a fixed dust-to-gas ratio
(D/G) in their sample galaxies to derive $\alpha_{\rm CO}$. On the other hand,
Sandstrom et al. (2013, also see (), () and ()) assumed that the D/G remains
approximately constant in a certain spatial region, e.g. kpc scale or entire
galaxy. With this assumption, the authors are able to derive $\alpha_{\rm CO}$
by minimizing the scatter in D/G in a group of nearby pixels, treating
$\alpha_{\rm CO}$ and D/G as free parameters. This method has the advantage in
not forcing the value of D/G and the disadvantage in sacrificing the spatial
resolution.

Generally speaking, we could assume different D/M values in each pixel when we
apply Eq. 4. However, despite the previous efforts on studying the evolution
of D/M (De Vis et al., 2019; Aoyama et al., 2020; Péroux & Howk, 2020; Chiang
et al., 2021; Choban et al., 2022; Roman-Duval et al., 2022), we do not have a
well established prescription of how D/M depends on local environments, or the
prescription is not more accurate than simply assuming a constant D/M. Studies
have shown that at $12+\log({\rm O/H})>8.2$, the D/M falls in the range222For
studies that only measure D/G measurements, we quote the D/M value calculated
at $Z_{\odot}$. between $0.4\mbox{--}0.7$, e.g. $0.5$ (Jenkins, 2009, taking
$F_{\star}=1$), $0.72$ (Leroy et al., 2011), $0.46$ (Rémy-Ruyer et al., 2014),
$0.68$ (Draine et al., 2014), $0.7$ (Feldmann, 2015), $0.56$ (Chiang et al.,
2018), and $0.40\mbox{--}0.58$ (Chiang et al., 2021). Meanwhile, studies have
shown that in galaxy centers, $\alpha_{\rm CO}$ can vary by a factor of 10
(Sandstrom et al., 2013; Bolatto et al., 2013; Israel, 2020; Teng et al.,
2022, 2023), which is a significantly larger dynamic range than D/M. In the
following, we will take a constant $\mathrm{D/M}=0.55$ (mean of 0.4–0.7) as
the fiducial case for deriving $\alpha_{\rm CO}$. A possible drawback by
assuming a constant D/M is that D/M has been shown to decrease toward lower
metallicity (e.g. Rémy-Ruyer et al., 2014; Hirashita & Kuo, 2011; Chiang et
al., 2018; De Vis et al., 2019). Thus, the assumed D/M value that is
appropriate for galaxy centers is likely too high for low-metallicity regions
(usually the outer disks), resulting in an underestimation of $\alpha_{\rm
CO}$ (Eq. 4) in the outer disk. We will discuss the case of a varying D/M with
a toy model in Appendix A. We will also discuss the possible uncertainties in
$\Sigma_{\mathrm{dust}}$ derivation in Section 3.

Practically, we blank data points with
$\frac{\Sigma_{\mathrm{dust}}}{Z\times\mathrm{(D/M)}}<\Sigma_{\mathrm{atom}}$
in Eq. 4, which causes negative $\Sigma_{\mathrm{mol}}$. This happens in
regions where the S/N of both CO and dust emission are low, while the H i 21
cm line is well detected, i.e. $\Sigma_{\mathrm{atom}}$-dominated regions.

###  2.1 CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ and CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ cases

In this multi-wavelength, multi-galaxy study, we do not always have both the
lowest-$J$ CO emission lines for all the target galaxies. Studies have been
using CO line ratios to convert the intensity between CO emission lines. For
an in-depth discussion on low-$J$ CO line ratios, we refer the readers to
Leroy et al. (2022). In the literature, perhaps the most frequently used
method to treat different CO line coverage is converting everything to
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ with a constant CO line
ratio (e.g. Sandstrom et al., 2013; Sun et al., 2020; Chiang et al., 2021).
This method allows us to uniformly use
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ for calculating
$\Sigma_{\mathrm{mol}}$ in the study. Theoretically, the line ratio to vary
with excitation conditions like gas temperature and linewidth. Thus, we expect
$R_{21}$ to trace the local environmental conditions. Taking the line ratio
for $W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ and
$W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ as an example:

$R_{21}\equiv\frac{W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}}{W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}},$
(5)

where $R_{21}$ is usually falls in the range of 0.3 to 0.9 with mean value
$\sim 0.65$ for normal star-forming galaxies, and it is expected to be higher
in galaxy centers (Leroy et al., 2009a, 2022, 2023; den Brok et al., 2021;
Yajima et al., 2021).

In this work, however, we will not adopt the simple strategy as our fiducial
case because most of our target galaxies have CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ data, that
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ has attained its own
importance due to modern observations, and that the variation in $R_{21}$ is
non-negligible. We will present four solutions of $\alpha_{\rm CO}$ in
parallel, two without any conversions between CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ and CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$,
and two with different prescriptions of the line ratio:

  1. 1.

$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ calculated with
$W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ data only.

  2. 2.

$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ calculated with
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ data only.

  3. 3.

$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ calculated with
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ data, plus
$W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ for galaxies without
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ data, converted with a
constant $R_{21}$.

  4. 4.

$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ calculated with
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ data, plus
$W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ for galaxies without
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ data, converted with an
environment-dependent $R_{21}$.

For the third method, we adopt the constant $R_{21}=0.65$ from Leroy et al.
(2022). For the last method, we adopt the MIR-dependent formula suggested by
Leroy et al. (2023), namely

$R_{21}=0.62\left(\frac{I_{\mathrm{WISE4}}}{1\,{\rm
MJy\,sr\textsuperscript{-1}}}\right)^{0.22}.$ (6)

We follow the suggestion in Leroy et al. (2023) and cap $R_{21}$ at
$R_{21}=1$. This formula in general agrees with the finding that $R_{21}$
scales with MIR intensity or $\Sigma_{\mathrm{SFR}}$ with a power-law index
$\sim 0.15$–$0.2$ (den Brok et al., 2021; Yajima et al., 2021; Leroy et al.,
2022, 2023).

##  3 Data

We measure $\alpha_{\rm CO}$ in 41 nearby galaxies in this study. To measure
$\alpha_{\rm CO}$ with our dust-based methodology (Section 2), the data sets
required are dust surface density ($\Sigma_{\mathrm{dust}}$, from IR SED
modelling), CO low-$J$ rotational line integrated intensity
($W_{\mathrm{CO}}$), atomic gas surface density ($\Sigma_{\mathrm{atom}}$,
from H i 21 cm line emission), and metallicity ($Z$, from gas-phase oxygen
abundance in H ii regions). We first select our sample galaxies from the dust
catalog of $z$=0 Multiwavelength Galaxy Synthesis ($z$0MGS) (Leroy et al.,
2019; Chastenet et al., 2021, J. Chastenet et al. in preparation). From this
large sample, we pick the 49 galaxies with both low-$J$ CO rotational line and
H i data available, including our own new H i data sets. We design our study
with a common resolution of 2 kpc, which draws a limit in sample selection at
distance $\sim 20~{}$Mpc since the worst resolution data in our sample usually
have angular resolution around $20\arcsec$. High-inclination ($>80^{\circ}$)
targets are also excluded. The selection yields 41 galaxies in our sample. We
list the properties of these galaxies in LABEL:tab:samples.

IC0342 | 3.5 | 31.0 | 42.0 | 10.1 | 4.4 | 10.2 | 5 | CO Atlas |  $\cdots$ | EveryTHINGS | $e.$
---|---|---|---|---|---|---|---|---|---|---|---
NGC0253 | 3.7 | 75.0 | 52.5 | 14.4 | 4.7 | 10.5 | 5 | CO Atlas | PHANGS-ALMA | $c.$ | $f.$
NGC0300 | 2.1 | 39.8 | 114.3 | 5.9 | 2.0 | 9.3 | 6 |  $\cdots$ | PHANGS-ALMA | $c.$ | $g.$
NGC0598 | 0.9 | 55.0 | 201.0 | 8.1 | 2.4 | 9.4 | 5 |  $\cdots$ | $a.$ | $d.$ | $g.$
NGC0628 | 9.8 | 8.9 | 20.7 | 14.1 | 3.9 | 10.2 | 5 | COMING | PHANGS-ALMA | THINGS | PHANGS-MUSE
NGC0925 | 9.2 | 66.0 | 287.0 | 14.3 | 4.5 | 9.8 | 6 |  $\cdots$ | HERACLES | THINGS | $g.$
NGC2403 | 3.2 | 63.0 | 124.0 | 9.3 | 2.4 | 9.6 | 5 |  $\cdots$ | HERACLES | THINGS | $g.$
NGC2841 | 14.1 | 74.0 | 153.0 | 14.2 | 5.4 | 10.9 | 3 | COMING |  $\cdots$ | THINGS | $f.$
NGC3184 | 12.6 | 16.0 | 179.0 | 13.6 | 5.3 | 10.3 | 5 | CO Atlas | HERACLES | THINGS | $g.$
NGC3198 | 13.8 | 72.0 | 215.0 | 13.0 | 5.0 | 10.0 | 5 | COMING | HERACLES | THINGS | $f.$
NGC3351 | 10.0 | 45.1 | 193.2 | 10.5 | 3.1 | 10.3 | 3 | CO Atlas | PHANGS-ALMA | THINGS | PHANGS-MUSE
NGC3521 | 13.2 | 68.8 | 343.0 | 16.0 | 3.9 | 11.0 | 3 | CO Atlas | PHANGS-ALMA | THINGS | $f.$
NGC3596 | 11.3 | 25.1 | 78.4 | 6.0 | 1.6 | 9.5 | 5 |  $\cdots$ | PHANGS-ALMA | PHANGS-VLA | $f.$
NGC3621 | 7.1 | 65.8 | 343.8 | 9.9 | 2.7 | 10.0 | 6 |  $\cdots$ | PHANGS-ALMA | THINGS | $g.$
NGC3627 | 11.3 | 57.3 | 173.1 | 16.9 | 3.6 | 10.7 | 3 | CO Atlas | PHANGS-ALMA | THINGS | PHANGS-MUSE
NGC3631 | 18.0 | 32.4 | -65.6 | 9.7 | 2.9 | 10.2 | 5 | CO Atlas | $b.$ | EveryTHINGS | $f.$
NGC3938 | 17.1 | 14.0 | 195.0 | 13.4 | 3.7 | 10.3 | 5 | COMING | HERACLES | HERACLES-VLA | $f.$
NGC3953 | 17.1 | 61.5 | 12.5 | 15.2 | 5.3 | 10.6 | 4 |  $\cdots$ | $b.$ | EveryTHINGS | $f.$
NGC4030 | 19.0 | 27.4 | 28.7 | 10.5 | 2.1 | 10.6 | 4 | COMING |  $\cdots$ | EveryTHINGS | $f.$
NGC4051 | 17.1 | 43.4 | -54.8 | 14.7 | 3.7 | 10.3 | 3 | CO Atlas | $b.$ | EveryTHINGS | $f.$
NGC4207 | 15.8 | 64.5 | 121.9 | 3.5 | 1.4 | 9.6 | 7 |  $\cdots$ | PHANGS-ALMA | PHANGS-VLA | $f.$
NGC4254 | 13.1 | 34.4 | 68.1 | 9.6 | 2.4 | 10.3 | 5 | CO Atlas | PHANGS-ALMA | HERACLES-VLA | PHANGS-MUSE
NGC4258 | 7.6 | 68.3 | 150.0 | 18.8 | 5.9 | 10.7 | 4 | COMING |  $\cdots$ | HALOGAS | $g.$
NGC4321 | 15.2 | 38.5 | 156.2 | 13.5 | 5.5 | 10.7 | 3 | CO Atlas | PHANGS-ALMA | HERACLES-VLA | PHANGS-MUSE
NGC4450 | 16.8 | 48.5 | -6.3 | 13.3 | 4.3 | 10.7 | 2 |  $\cdots$ | $b.$ | EveryTHINGS | $f.$
NGC4496A | 14.9 | 53.8 | 51.1 | 7.3 | 3.0 | 9.6 | 6 |  $\cdots$ | PHANGS-ALMA | EveryTHINGS | $f.$
NGC4501 | 16.8 | 60.1 | -37.8 | 21.1 | 5.2 | 11.0 | 3 | CO Atlas |  $\cdots$ | EveryTHINGS | $f.$
NGC4536 | 16.2 | 66.0 | 305.6 | 16.7 | 4.4 | 10.2 | 3 | CO Atlas | PHANGS-ALMA | HERACLES-VLA | $f.$
NGC4569 | 15.8 | 70.0 | 18.0 | 21.0 | 5.9 | 10.8 | 2 | CO Atlas | PHANGS-ALMA | HERACLES-VLA | $f.$
NGC4625 | 11.8 | 47.0 | 330.0 | 2.4 | 1.2 | 9.1 | 9 |  $\cdots$ | HERACLES | HERACLES-VLA | $g.$
NGC4651 | 16.8 | 50.1 | 73.8 | 9.5 | 2.4 | 10.3 | 5 |  $\cdots$ | $b.$ | EveryTHINGS | $g.$
NGC4689 | 15.0 | 38.7 | 164.1 | 8.3 | 4.7 | 10.1 | 5 | CO Atlas | PHANGS-ALMA | EveryTHINGS | $f.$
NGC4725 | 12.4 | 54.0 | 36.0 | 17.5 | 6.0 | 10.8 | 1 |  $\cdots$ | HERACLES | HERACLES-VLA | $f.$
NGC4736 | 4.4 | 41.0 | 296.0 | 5.0 | 0.8 | 10.3 | 1 | CO Atlas | HERACLES | THINGS | $f.$
NGC4941 | 15.0 | 53.4 | 202.2 | 7.3 | 3.4 | 10.1 | 1 |  $\cdots$ | PHANGS-ALMA | EveryTHINGS | $f.$
NGC5055 | 9.0 | 59.0 | 102.0 | 15.5 | 4.2 | 10.7 | 4 | CO Atlas | HERACLES | THINGS | $f.$
NGC5248 | 14.9 | 47.4 | 109.2 | 8.8 | 3.2 | 10.3 | 3 | CO Atlas | PHANGS-ALMA | PHANGS-VLA | $f.$
NGC5457 | 6.7 | 18.0 | 39.0 | 23.4 | 13.5 | 10.3 | 5 | CO Atlas | HERACLES | THINGS | $g.$
NGC6946 | 7.3 | 33.0 | 243.0 | 12.1 | 4.4 | 10.5 | 5 | CO Atlas | HERACLES | THINGS | $g.$
NGC7331 | 14.7 | 76.0 | 168.0 | 19.8 | 3.7 | 11.0 | 4 | COMING | HERACLES | THINGS | $f.$
NGC7793 | 3.6 | 50.0 | 290.0 | 5.4 | 1.9 | 9.3 | 6 |  $\cdots$ | PHANGS-ALMA | THINGS | $g.$

Note. — (2) Distance (from EDD Tully et al., 2009); (3-4) inclination angle
and position angle (Sofue et al., 1999; de Blok et al., 2008; Leroy et al.,
2009a; Muñoz-Mateos et al., 2009; Meidt et al., 2009; McCormick et al., 2013;
Makarov et al., 2014; Lang et al., 2020); (5) isophotal radius (Makarov et
al., 2014); (6) effective radius (Leroy et al., 2021); (7) logarithmic global
stellar mass (Leroy et al., 2019); (8) numerical Hubble stage T; (9)
References of CO $J=1\to 0$ observations (“ $\cdots$ ” means no CO $J=1\to 0$
data adopted in this work): CO Atlas Kuno et al. (2007); COMING (Sorai et al.,
2019); (10) References of CO $J=2\to 1$ observations (“ $\cdots$ ” means no CO
$J=2\to 1$ data adopted in this work): HERACLES Leroy et al. (2009a); PHANGS-
ALMA (Leroy et al., 2021); $a.$ M33 data from Gratier et al. (2010); Druard et
al. (2014); $b.$ New HERA data (P.I.: A. Schruba; presented in Leroy et al.,
2022); (11) References of Hi observations: THINGS (Walter et al., 2008);
HALOGAS (Heald et al., 2011); HERACLES-VLA (Schruba et al., 2011); PHANGS-VLA
(P.I. D. Utomo; I. Chiang et al. in preparation); EveryTHINGS (P.I. K. M.
Sandstrom; I. Chiang et al. in preparation); $c.$ Puche et al. (1991); $d.$
Koch et al. (2018); (12) References of $12+\log({\rm O/H})$ measurement:
PHANGS-MUSE (Emsellem et al., 2022; Santoro et al., 2022); $e.$ private
communication with K. Kreckel (see Chiang et al., 2021); $f.$ using the
empirical formula described in Section 3.1; $g.$ data from Zurita et al.
(2021) compilation.

###  3.1 Physical Quantities and Data Sets

Dust properties. We obtain dust properties by fitting the dust emission
spectral energy distribution (SED) to the Draine & Li (2007) physical dust
model. The details of the IR data processing and dust SED fitting are reported
in Chastenet et al. (2021) and J. Chastenet et al. (in preparation). We
briefly summarize the process below.

We obtain the dust emission SED in the IR observed by two space telescopes:
the 3.4, 4.6, 12 and 22 µm bands with the Wide-field Infrared Survey Explorer
(WISE, Wright et al., 2010), and the 70, 100, 160 and 250 µm bands with the
Herschel Space Observatory (Pilbratt et al., 2010). The Herschel and WISE maps
are first convolved to SPIRE 250 PSF and then to a 21″ Gaussian PSF using the
SPIRE 250-to-Gauss-21″ kernel from Aniano et al. (2011). The 21″ PSF is the
“moderate” Gaussian from Aniano et al. (2011) that provides relatively high
angular resolution without amplifying image artifacts. Finally, these maps are
convolved to the desired resolution: a Gaussian PSF with spatial resolution
corresponding to FWHM of 2 kpc.

After convolving the IR maps, we fit the dust SED with the Draine & Li (2007)
physical dust model (with the dust mass correction derived in Chastenet et
al., 2021). The complete set of data products from the fitting includes the
maps of dust mass surface density ($\Sigma_{\mathrm{dust}}$), interstellar
radiation field (the minimum radiation field $U_{\mathrm{min}}$ and the
fraction of dust heated by the power-law radiation field $\gamma$) and the
fractional dust mass in the form of polycyclic aromatic hydrocarbons ($\rm
q_{PAH}$). The maximum radiation field is fixed at $U_{\rm max}=10^{7}$ and
the power-law index for radiation field distribution is fixed at $\alpha=2$.
From the fitted $U_{\mathrm{min}}$ and $\gamma$, we can derive the dust-mass-
averaged radiation field $\overline{U}$, which is the fiducial tracer for
radiation field in this work.

We also note that we assume fixed dust properties in our dust SED fitting
throughout this study, which is the most frequently adopted strategy in the
literature. The accuracy of $\Sigma_{\mathrm{dust}}$ estimates, derived by
fitting the IR SED with dust emission models, can be affected by variations in
the dust opacity. Interstellar dust grains are not uniform in their chemical
composition, size distribution, and shape, leading to variations in their
opacity (e.g. Draine & Li, 2007; Hirashita & Voshchinnikov, 2014; Draine &
Hensley, 2021). In the MW, Stepnik et al. (2003) found that the dust opacity
increases by $\sim 3$ times from the diffuse ISM to the dense clouds. The
authors argued that the increase in dust opacity is resulted from the deficit
of small grains due to grain-grain coagulation. In extra-galactic studies, it
is challenging to measure the variation in opacity of interstellar dust since
it is degenerate with the environmental dependence of $\alpha_{\rm CO}$ and
D/M. Moreover, most the mechanisms that affects dust opacity, e.g. grain-grain
coagulation and ice mantles, are smoothed out in kpc-scale extra-galactic
studies (Galliano et al., 2018), meaning that its variation is likely less
observable than the other degenerate factors like $\alpha_{\rm CO}$ and D/M.
We note that there are extra-galactic studies that attribute all the
variations in dust and gas properties to dust opacity to evaluate its
variation, e.g. Clark et al. (2019) found that dust opacity change by a factor
$\sim 2$ within M74 and $\sim 5$ within M83.

Atomic gas surface density. We trace the atomic gas surface density
($\Sigma_{\mathrm{atom}}$) with the H i 21 cm integrated intensity
($I_{\textsc{H\,i}}$), assuming the opacity is negligible (e.g., Walter et
al., 2008):

$\frac{\Sigma_{\mathrm{atom}}}{1~{}\rm
M_{\odot}~{}pc^{-2}}=1.36\times(1.46\times
10^{-2})\times\frac{W_{\textsc{H\,i}}}{1~{}{\rm
K\,km\,s\textsuperscript{-1}}}~{},$ (7)

where the $1.36$ factor accounts for the mass of helium.

We obtain $W_{\textsc{H\,i}}$ from both literature and new data, as listed in
LABEL:tab:samples. The two new H i surveys are the EveryTHINGS survey (P.I. K.
M. Sandstrom; I. Chiang et al. in preparation) and the PHANGS-VLA survey (P.I.
D. Utomo). The EveryTHINGS survey targets nearby galaxies with Herschel
photometric data but without high-resolution H i observations, while the
PHANGS-VLA survey focuses on galaxies in the Physics at High Angular
resolution in Nearby Galaxies (PHANGS) project333http://phangs.org/. Both
surveys have their data observed with the C- and D-configurations of Karl G.
Jansky Very Large Array (VLA)444The VLA is operated by the National Radio
Astronomy Observatory (NRAO), which is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc., which yield an angular resolutions in the range of $20\arcsec$ to
$30\arcsec$. Both surveys provide new high-sensitivity 21 cm H i observations
in tens of nearby galaxies. We did not include WHISP (Swaters et al., 2002)
data because the galaxies that only have WHISP data have resolution coarser
than 2 kpc after convolving to a circular PSF.

CO low-$J$ rotational lines. The integrated intensity of CO low-$J$ rotational
lines traces the molecular gas surface density (Eq. 1), and is key to this
study. We use the compilation of CO mapping assembled by Leroy et al. (2022,
2023) from several publicly available CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ and
CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ data:

  * •

CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ data from the COMING survey (Sorai et
al., 2019) and the CO Atlas (Kuno et al., 2007).

  * •

CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ data from HERACLES (Leroy et al., 2009a),
the PHANGS-ALMA survey (Leroy et al., 2021), and a new survey observed by the
IRAM-30m focused on the Virgo Cluster (P.I. A. Schruba; processed in Leroy et
al., 2022).

The source of CO data for each galaxy is listed in LABEL:tab:samples, where CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ and CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ are
listed separately. All these literature measurements focus on the
${}^{12}\mathrm{C}^{16}\mathrm{O}$ isotopologue, hereafter CO for simplicity.

Surface densities of stellar mass and SFR. We trace the surface densities of
stellar mass and SFR ($\Sigma_{\star}$ and $\Sigma_{\mathrm{SFR}}$,
respectively) using the data and conversion formulae presented in the $z$0MGS
survey (Leroy et al., 2019). We utilize the $z$0MGS compilation of the
background-subtracted WISE (Wright et al., 2010) $\lambda\sim 3.4$ and 22 µm
(hereafter WISE1 and WISE4, respectively) data and the Galaxy Evolution
Explorer (GALEX, Martin et al., 2005) $\lambda\sim 154~{}\rm nm$ (hereafter
FUV) data.

We use WISE1 data to trace stellar mass surface density ($\Sigma_{\star}$)
with:

$\frac{\Sigma_{\star}}{1~{}\rm M_{\odot}~{}pc^{-2}}=3.3\times
10^{2}\left(\frac{\Upsilon^{3.4}_{\star}}{0.5~{}\mathrm{M_{\odot}~{}L_{\odot}^{-1}}}\right)\frac{I_{\rm
WISE1}}{1~{}{\rm MJy\,sr\textsuperscript{-1}}}~{},$ (8)

where $\Upsilon^{3.4}_{\star}$ is the $\Sigma_{\star}$-to-WISE1 mass-to-light
ratio. The value of $\Upsilon^{3.4}_{\star}$ is calculated from the galaxy-by-
galaxy SFR-to-WISE1 ratio, a “specific SFR-like” quantity, with the
prescription calibrated in Appendix A of Leroy et al. (2019).

We use FUV and WISE4 data to trace the SFR surface density
($\Sigma_{\mathrm{SFR}}$) also with the conversion formula suggested by
$z$0MGS (Leroy et al., 2019; Belfiore et al., 2023). For galaxies with both
FUV and WISE4 available, we use:

$\frac{\Sigma_{\mathrm{SFR}}}{1~{}{\rm M_{\odot}\,yr}^{-1}\,{\rm
kpc}^{-2}}=\\\ 8.85\times 10^{-2}\frac{I_{\rm FUV}}{1~{}{\rm
MJy\,sr\textsuperscript{-1}}}+3.02\times 10^{-3}\frac{I_{\rm WISE4}}{1~{}{\rm
MJy\,sr\textsuperscript{-1}}}~{}.$ (9)

For NGC3953 and NGC4689, where only WISE4 is available, we use:

$\frac{\Sigma_{\mathrm{SFR}}}{1~{}{\rm M_{\odot}\,yr}^{-1}\,{\rm
kpc}^{-2}}=3.81\times 10^{-3}\frac{I_{\rm W4}}{1~{}{\rm
MJy\,sr\textsuperscript{-1}}}~{}.$ (10)

For both WISE and GALEX maps, we blank the regions with foreground stars
identified in the $z$0MGS database. We interpolate the intensities in the
blanked region with a Gaussian kernel FWHM = 22.5″(the adopted WISE and GALEX
maps have FWHM = 15″) with the function interpolate_replace_nans in
astropy.convolution. This interpolation is done on the maps before any
convolution, reprojection, or unit conversion. Regarding the WISE maps, this
treatment is only done for the maps used for calculating $\Sigma_{\star}$ and
$\Sigma_{\mathrm{SFR}}$. For the WISE maps used in dust SED fitting, we refer
the readers to J. Chastenet et al. (in preparation).

Specific SFR. With the measurements of $\Sigma_{\mathrm{SFR}}$ and
$\Sigma_{\star}$, we calculate the specific SFR (sSFR) as:

$\frac{\rm sSFR}{\rm 1~{}yr^{-1}}=1\times
10^{-6}\times\frac{\Sigma_{i}\Sigma_{\mathrm{SFR}}}{1~{}{\rm
M_{\odot}\,yr}^{-1}\,{\rm
kpc}^{-2}}\big{(}\frac{\Sigma_{i}\Sigma_{\star}}{1~{}\rm
M_{\odot}~{}pc^{-2}}\big{)}^{-1},$ (11)

where $\Sigma_{i}$ is the summation over pixels in a galaxy. Meanwhile, we
calculate the spatially resolved sSFR (rsSFR) as:

$\frac{\rm rsSFR}{\rm 1~{}yr^{-1}}=1\times
10^{-6}\times\frac{\Sigma_{\mathrm{SFR}}}{1~{}{\rm M_{\odot}\,yr}^{-1}\,{\rm
kpc}^{-2}}\big{(}\frac{\Sigma_{\star}}{1~{}\rm
M_{\odot}~{}pc^{-2}}\big{)}^{-1}.$ (12)

Metallicity. We use the abundance of oxygen, $12+\log({\rm O/H})$, to trace
the metallicity ($Z$). We assume a fixed abundance pattern, i.e. a constant
oxygen-to-total-metal mass ratio. The conversion from $12+\log({\rm O/H})$ to
$Z$ then becomes:

$Z=0.0134\times 10^{12+\log({\rm O/H})-8.69},$ (13)

where 0.0134 and 8.69 are the adopted $Z_{\odot}$ and $12+\log({\rm
O/H})_{\odot}$, respectively (Asplund et al., 2009).

We calculate $12+\log({\rm O/H})$ for each pixel as a function of the
galactocentric distance by adopting the radial gradient of $12+\log({\rm
O/H})$ derived from measurements in H ii regions. We use data from two
surveys: the PHANGS-MUSE survey (Emsellem et al., 2022; Groves et al., 2023)
and the Zurita et al. (2021) compilation. We use the Pilyugin & Grebel (2016)
S-calibration555Pilyugin & Grebel (2016) utilizes the
$S_{2}=I_{\rm[S~{}II]}\lambda 6717+\lambda 6731/I_{\rm H\beta}$,
$N_{2}=I_{\rm[NII]}\lambda 6548+\lambda 6584/I_{\rm H\beta}$, and
$R_{3}=I_{\rm[OIII]}\lambda 4959+\lambda 5007/I_{\rm H\beta}$ line ratios to
measure gas-phase $12+\log({\rm O/H})$. (hereafter PG16S) as the fiducial
calibration for $12+\log({\rm O/H})$. PG16S is a calibration method showing
good agreement with direct metallicity measurements (Croxall et al., 2016;
Kreckel et al., 2019). Since PG16S only relies on lines covered by MUSE, the
$12+\log({\rm O/H})$ measurement can be expanded to the full PHANGS-MUSE
dataset in our future works.

For galaxies in the PHANGS-MUSE survey, we adopt radial $12+\log({\rm O/H})$
gradients presented in Santoro et al. (2022), which are calculated with the
PG16S calibration. For galaxies only appear in the Zurita et al. (2021)
emission data compilation, we use the Zurita et al. (2021) data to calculate
the PG16S $12+\log({\rm O/H})$ and then fit the radial $12+\log({\rm O/H})$
gradient in these galaxies. We only consider galaxies that have at least 5
measurements spanning at least $0.5R_{25}$ in the Zurita et al. (2021) data
table.

For galaxies without measurements of $12+\log({\rm O/H})$ in either Zurita et
al. (2021) or Santoro et al. (2022), we use the two-step strategy proposed in
Sun et al. (2020) to estimate their $12+\log({\rm O/H})$. First, we use a
mass-metallicity relation (later fitted to the existing PG16S data mentioned
above) to predict $12+\log({\rm O/H})$ at one effective radius
($R_{\mathrm{e}}$) in a given galaxy. Second, we extend the prediction with a
radial gradient of $-0.1~{}\mathrm{dex}/R_{\mathrm{e}}$ suggested by Sánchez
et al. (2014). We characterize the mass-metallicity with a function of the
form:

$12+\log({\rm O/H})=a+bxe^{-x},$ (14)

where $x=\log(M_{\star}/\mathrm{M}_{\odot})-11.5$ (see Sánchez et al., 2019,
and references therein). $a$ and $b$ are free parameters. We fit the function
with $12+\log({\rm O/H})$ at one $\rm R_{e}$ from galaxies with the available
measurements listed in LABEL:tab:samples. The best-fit parameters are
$a=8.56\pm 0.02$ and $b=0.010\pm 0.002$, which are robust under resampling.
Meanwhile, the typical uncertainty in the $12+\log({\rm O/H})$ data used for
fitting is $\sim 0.013$ dex. However, the root mean square error
($\Delta_{\mathrm{rms}}$) between the best fit and data is 0.13 dex, meaning
that there is still intrinsic scatter in $12+\log({\rm O/H})$ that is not
explained by the mass-metallicity relation and the adopted radial gradients,
e.g. the azimuthal variations (Williams et al., 2022). We use this fitted
relation to derive the radial metallicity gradient of galaxies without
metallicity measurements with the $M_{\star}$ and $R_{\mathrm{e}}$ listed in
LABEL:tab:samples.

Studies have reported that the PG16S calibration could result in $12+\log({\rm
O/H})$ value lower than other calibrations (e.g. De Vis et al., 2019).
Aligning with that, there are also studies reporting that the $12+\log({\rm
O/H})$ calibrated with PG16S in Orion Nebula and other H ii regions in the
solar neighborhood have values $\sim 0.2$ dex lower than the solar reference
value (e.g. Esteban et al., 2022). This effect could lead to an underestimate
of Z’ and thus an overestimate in $\alpha_{\rm CO}$ with our methodology. For
consistency with the direct metallicity calibration used in Chastenet et al.
(2021) for calibrating dust opacity, we will use PG16S in this work.

###  3.2 Uniform Data Processing

The analyses in this work are done at a common resolution of 2 kpc for all
data. For H i, CO, $\Sigma_{\star}$ and $\Sigma_{\mathrm{SFR}}$, we convolve
them to a circular Gaussian PSF with a FWHM corresponding to 2 kpc, using the
astropy.convolution package (Astropy Collaboration et al., 2013, 2018, 2022).
The images are then reprojected onto a grid with a pixel size of one third of
the FWHM (that is, we oversample at roughly the Nyquist sampling rate) with
the astropy affiliated package reproject. The convolution and reprojection of
the dust maps are done in J. Chastenet et al. (in preparation). They convolve
the IR maps into the final resolution using kernels from Aniano et al. (2011).
Note that the convolution is done before SED fitting for dust properties. The
galactocentric radius and metallicity are directly calculated on the final
pixel grids. All the surface density and surface brightness quantities
presented in this work have been corrected for inclination, listed in
LABEL:tab:samples.

###  3.3 S/N Mask and Completeness

S/N Mask. For statistical quantities that only involve $\alpha_{\rm CO}$, e.g.
mean values and percentiles, we masked out the low signal-to-noise ratio (S/N)
pixels. Specifically, we blank pixels with $\mathrm{S/N}<1$ in Herschel, WISE
or CO.

Completeness. For statistical quantities that involve $\alpha_{\rm CO}$ and
another quantity ($X$), e.g. correlations and linear regression, we calculate
with data with high ($>50\%$) completeness in $X$ only as the trend confidence
criteria. The completeness for data with $X_{i}\leq X<X_{f}$, or
$[X_{i},~{}X_{f})$, is defined as:

$Completeness\equiv\frac{{\rm
num~{}of~{}high~{}S/N~{}pixels~{}with}~{}[X_{i},~{}X_{f})}{{\rm
num~{}of~{}all~{}pixels~{}with}~{}[X_{i},~{}X_{f})},$ (15)

where the definition of “high-S/N” is the same as the S/N mask earlier this
subsection. We show the completeness and the $50\%$ threshold in our data set
in Figure 1. Across all quantities, the CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$
data has a better completeness coverage than the CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ data.

Figure 1: The completeness of our data set at each $\alpha_{\rm CO}$-quantity
pair. The $50\%$ completeness is marked with a horizontal black line. In the
statistical calculations, e.g. linear regression and correlation, we only use
data in the bins with completeness $>50\%$.

##  4 Results

Table 1: Statistics of $\alpha_{\rm CO}$ measurements. All the $\alpha_{\rm CO}$ values are in unit of [${\rm M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}$ ]. Galaxy | $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ | $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$
---|---|---
Mean | W-mean(a) | 16th–84th %tile(b) | N${}_{\mathrm{pix}}^{(c)}$ | N${}_{\mathrm{pix,100}}^{(d)}$ | Mean | W-mean(a) | 16th–84th %tile(b) | N${}_{\mathrm{pix}}^{(c)}$ | N${}_{\mathrm{pix,100}}^{(d)}$
IC0342 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 2.08 | 1.85 | 1.52–2.61 | 168 | 28
NGC0253 | 11.37 | 4.21 | 3.06–20.55 | 108 | 37 | 2.02 | 1.64 | 1.32–3.06 | 39 | 31
NGC0300 | 7.66 | 7.66 | 6.99–8.28 | 5 | 0 | 3.78† | 3.79† | 3.42–4.11† | 5† | 0†
NGC0598 | 7.07 | 6.77 | 5.29–8.97 | 41 | 0 | 3.97† | 3.88† | 3.1–4.94† | 41† | 0†
NGC0628 | 5.19 | 5.19 | 4.16–6.06 | 172 | 57 | 2.57 | 2.6 | 2.01–3.14 | 172 | 57
NGC0925 | 1.29 | 1.25 | 0.66–1.93 | 16 | 0 | 0.69† | 0.68† | 0.34–1.08† | 16† | 0†
NGC2403 | 0.67 | 0.67 | 0.22–1.07 | 12 | 0 | 0.43† | 0.43† | 0.13–0.68† | 12† | 0†
NGC2841 | 16.91 | 13.22 | 8.9–29.34 | 174 | 109 | 7.63 | 7.35 | 5.49–9.71 | 131 | 92
NGC3184 | 2.34 | 2.58 | 0.97–3.6 | 367 | 56 | 1.56 | 1.63 | 0.89–2.22 | 289 | 56
NGC3198 | 2.44 | 2.35 | 1.11–3.92 | 77 | 0 | 1.45 | 1.36 | 0.85–2.08 | 51 | 0
NGC3351 | 5.81 | 4.38 | 3.73–7.73 | 82 | 54 | 2.34 | 2.3 | 2.02–2.64 | 81 | 54
NGC3521 | 5.94 | 4.96 | 3.83–7.87 | 273 | 146 | 2.86 | 2.76 | 2.24–3.55 | 182 | 126
NGC3596 | 9.28 | 8.32 | 6.63–12.09 | 40 | 16 | 6.5† | 6.1† | 4.98–8.01† | 40† | 16†
NGC3621 | 3.6 | 3.68 | 2.98–4.35 | 52 | 19 | 2.61† | 2.71† | 2.0–3.28† | 52† | 19†
NGC3627 | 3.59 | 3.02 | 2.5–4.96 | 176 | 143 | 1.45 | 1.34 | 1.05–1.82 | 176 | 143
NGC3631 | 5.58 | 3.73 | 2.41–8.66 | 207 | 17 | 1.85 | 1.66 | 1.08–2.71 | 169 | 17
NGC3938 | 3.72 | 3.95 | 1.79–5.05 | 343 | 88 | 2.27 | 2.41 | 1.41–3.05 | 300 | 88
NGC3953 | 11.47 | 9.4 | 7.02–15.83 | 534 | 98 | 6.12† | 5.47† | 4.16–8.01† | 534† | 98†
NGC4030 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 4.76 | 3.82 | 3.22–6.2 | 207 | 135
NGC4051 | 11.51 | 8.14 | 6.58–15.96 | 394 | 42 | 4.53 | 3.94 | 2.88–6.41 | 200 | 36
NGC4207 | 3.29 | 3.26 | 2.82–3.75 | 4 | 4 | 2.57† | 2.56† | 2.25–2.88† | 4† | 4†
NGC4254 | 2.44 | 2.92 | 1.14–3.47 | 235 | 89 | 1.54 | 1.83 | 0.56–2.2 | 231 | 89
NGC4258 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.3 | 1.1 | 0.79–1.76 | 66 | 61
NGC4321 | 5.62 | 4.23 | 3.88–7.19 | 459 | 199 | 3.5 | 3.01 | 2.61–4.15 | 286 | 198
NGC4450 | 6.51 | 4.15 | 2.47–11.31 | 144 | 118 | 3.0† | 2.19† | 1.22–5.14† | 144† | 118†
NGC4496A | 2.47 | 2.33 | 0.89–3.64 | 42 | 0 | 1.45† | 1.38† | 0.52–2.17† | 42† | 0†
NGC4501 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 5.99 | 4.9 | 3.87–8.06 | 336 | 235
NGC4536 | 3.88 | 1.75 | 0.96–6.77 | 164 | 23 | 1.12 | 1.09 | 0.37–1.89 | 82 | 23
NGC4569 | 6.06 | 3.2 | 2.49–9.37 | 139 | 83 | 3.34 | 2.32 | 1.41–5.25 | 127 | 83
NGC4625 | 2.24 | 2.17 | 1.06–3.08 | 14 | 0 | 1.39† | 1.33† | 0.61–1.86† | 14† | 0†
NGC4651 | 2.34 | 2.57 | 1.01–3.72 | 55 | 54 | 1.84† | 2.0† | 0.69–2.97† | 55† | 54†
NGC4689 | 8.01 | 6.69 | 5.2–10.62 | 132 | 40 | 3.6 | 3.35 | 2.75–4.16 | 124 | 40
NGC4725 | 8.97 | 7.77 | 4.61–12.65 | 387 | 146 | 4.33† | 3.8† | 2.06–6.25† | 387† | 146†
NGC4736 | 1.64 | 1.31 | 0.94–2.32 | 30 | 30 | 0.87 | 0.81 | 0.6–1.17 | 14 | 14
NGC4941 | 6.37 | 5.6 | 3.37–9.53 | 48 | 31 | 3.89† | 3.73† | 2.32–5.59† | 48† | 31†
NGC5055 | 6.61 | 5.77 | 4.9–8.08 | 327 | 120 | 3.57 | 3.51 | 3.1–4.12 | 157 | 112
NGC5248 | 9.06 | 5.7 | 5.24–12.47 | 218 | 87 | 4.11 | 3.3 | 2.85–5.38 | 196 | 87
NGC5457 | 4.6 | 4.47 | 3.3–6.15 | 455 | 50 | 2.47 | 2.4 | 2.02–3.02 | 316 | 50
NGC6946 | 2.09 | 2.01 | 1.34–2.74 | 486 | 121 | 1.56 | 1.42 | 1.14–1.99 | 343 | 121
NGC7331 | 13.29 | 8.65 | 6.64–20.72 | 538 | 106 | 4.53 | 4.06 | 3.14–6.31 | 275 | 105
NGC7793 | 0.92 | 0.97 | 0.57–1.2 | 6 | 0 | 0.58† | 0.61† | 0.34–0.75† | 6† | 0†
Overall | 6.9 | 4.37 | 2.39–10.83 | 6956 | 2183 | 3.1 | 2.58 | 1.41–4.68 | 4718 | 2081
Overall† |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 3.43† | 2.69† | 1.48–5.42† | 6118† | 2567†
Figure 2: The distribution of measured
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$. The mean value of
each type of measurement is marked in vertical lines with the corresponding
color.

In total, we measure resolved $\alpha_{\rm CO}$ values across 41 galaxies,
including $\sim 940$ and $\sim 660$ independent measurements666In our data
products, the pixel size is 1/3 of the FWHM of the Gaussian PSF. Thus, the
number of independent measurements is smaller than the number of pixels listed
in Table 1. from CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ and CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ data, respectively. We summarize the
measurements in Table 1 and the distribution of $\alpha_{\rm CO}$ in Figure 2.
For each galaxy, we report the mean, CO-intensity-weighted mean, 16th-84th
percentiles and number of pixel-by-pixel measurements of $\alpha_{\rm CO}$.

The mean value for $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ are 6.9 and 3.1 ${\rm
M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}$ with 1-$\sigma$ spanning 2.39–10.83
and 1.41–4.68 ${\rm M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}$, respectively.
The mean $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ corresponds
to $\sim 0.7\alpha_{\rm CO}^{\rm MW}$, whereas the mean
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ corresponds to $\sim
1.03\alpha_{\rm CO}^{\rm MW}$ assuming a constant $R_{21}=0.65$.

Besides the simple mean, we also calculate the $W_{\rm CO}$-weighted mean,
which better reflects the $\alpha_{\rm CO}$ value to be adopted for data at
coarser resolution. The $W_{\rm CO}$-weighted mean for
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ are 4.37 and 2.58,
respectively. Both values are lower than the simple mean, which indicates that
the $\alpha_{\rm CO}$ values are lower in brighter regions.

When we include CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ data for galaxies without
CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ measurements with a variable $R_{21}$
(values with † in Table 1), the mean and $W_{\rm CO}$-weighted mean of
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ increase to 3.43 and
2.69, respectively, indicating that galaxies without CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ measurements in this data set tend to have
larger $\alpha_{\rm CO}$. This is also visualized in the bottom panel of
Figure 2. Also, we find that the distribution of
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ does not differ much
between the two $R_{21}$ prescriptions adopted in this work: the fixed
$R_{21}=0.65$ and the $I_{\rm W4}$-dependent $R_{21}$.

###  4.1 Correlations with Local Conditions

Table 2: The correlation and linear regression between pixel-by-pixel
$\alpha_{\rm CO}$ measurements and local physical quantities.
$\log\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$, CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ Only
---
$x^{(1)}$ | $\rho$ |  $p$-value | $m^{(1)}$ | $b^{(1)}$ | $\Delta_{\mathrm{rms}}$
$\log W_{\mathrm{CO}}$ | -0.46 | $<0.05$ | -0.33 | 0.74 | 0.34
$12+\log({\rm O/H})$ | -0.37 | $<0.05$ | -1.59 | 14.21 | 0.36
$\log\Sigma_{\mathrm{dust}}$ | -0.19 | $<0.05$ | -0.17 | 0.49 | 0.37
$\log\overline{U}$ | -0.65 | $<0.05$ | -1.02 | 0.97 | 0.31
$\log\Sigma_{\star}$ | -0.19 | $<0.05$ | -0.15 | 0.96 | 0.37
$\log\Sigma_{\mathrm{SFR}}$ | -0.53 | $<0.05$ | -0.42 | -0.27 | 0.34
$\log$ rsSFR | -0.46 | $<0.05$ | -0.42 | -3.53 | 0.35
$R_{g}/R_{25}$ | 0.3 | $<0.05$ | 0.45 | 0.51 | 0.36
$\log\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ Only
$x$ | $\rho$ |  $p$-value | $m$ | $b$ | $\Delta_{\mathrm{rms}}$
$\log W_{\mathrm{CO}}$ | -0.19 | $<0.05$ | -0.13 | 0.47 | 0.29
$12+\log({\rm O/H})$ | -0.19 | $<0.05$ | -0.93 | 8.33 | 0.29
$\log\Sigma_{\mathrm{dust}}$ | 0.07 | $<0.05$ | 0.09 | 0.52 | 0.26
$\log\overline{U}$ | -0.3 | $<0.05$ | -0.54 | 0.58 | 0.28
$\log\Sigma_{\star}$ | 0.12 | $<0.05$ | 0.09 | 0.24 | 0.29
$\log\Sigma_{\mathrm{SFR}}$ | -0.15 | $<0.05$ | -0.11 | 0.16 | 0.29
$\log$ rsSFR | -0.42 | $<0.05$ | -0.45 | -4.1 | 0.28
$R_{g}/R_{25}$ | 0.13 | $<0.05$ | 0.3 | 0.32 | 0.26
$\log\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, w/
$R_{21}(I_{\mathrm{W4}})$
$x$ | $\rho$ |  $p$-value | $m$ | $b$ | $\Delta_{\mathrm{rms}}$
$\log W_{\mathrm{CO}}$ | -0.26 | $<0.05$ | -0.19 | 0.52 | 0.29
$12+\log({\rm O/H})$ | -0.27 | $<0.05$ | -1.4 | 12.36 | 0.29
$\log\Sigma_{\mathrm{dust}}$ | -0.05 | $<0.05$ | -0.04 | 0.42 | 0.28
$\log\overline{U}$ | -0.37 | $<0.05$ | -0.64 | 0.62 | 0.29
$\log\Sigma_{\star}$ | 0.02 | 0.25 | 0.01 | 0.42 | 0.3
$\log\Sigma_{\mathrm{SFR}}$ | -0.22 | $<0.05$ | -0.15 | 0.07 | 0.3
$\log$ rsSFR | -0.48 | $<0.05$ | -0.54 | -4.99 | 0.29
$R_{g}/R_{25}$ | 0.2 | $<0.05$ | 0.44 | 0.29 | 0.28
$\log\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, w/ $R_{21}=0.65$
$x$ | $\rho$ |  $p$-value | $m$ | $b$ | $\Delta_{\mathrm{rms}}$
$\log W_{\mathrm{CO}}$ | -0.26 | $<0.05$ | -0.19 | 0.52 | 0.29
$12+\log({\rm O/H})$ | -0.28 | $<0.05$ | -1.6 | 14.08 | 0.3
$\log\Sigma_{\mathrm{dust}}$ | -0.05 | $<0.05$ | -0.06 | 0.4 | 0.28
$\log\overline{U}$ | -0.39 | $<0.05$ | -0.68 | 0.64 | 0.29
$\log\Sigma_{\star}$ | -0.01 | 0.68 | -0.01 | 0.48 | 0.31
$\log\Sigma_{\mathrm{SFR}}$ | -0.24 | $<0.05$ | -0.17 | 0.04 | 0.3
$\log$ rsSFR | -0.5 | $<0.05$ | -0.58 | -5.39 | 0.3
$R_{g}/R_{25}$ | 0.22 | $<0.05$ | 0.5 | 0.28 | 0.28
Figure 3: Our measured
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$, measured with
$W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ data only, as a function of
environmental parameters. The orange circles show the median of binned data.
The orange line shows the linear regression between
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and the quantity on
the horizontal axis. The empty circles indicate bins where the quantity in the
horizontal axis is incomplete, i.e. less than 50% of the pixels have S/N ¿ 1
in $W_{\mathrm{CO}}$ and Herschel bands (see Section 3.3). The dashed vertical
line shows the 50% boundary. The gray shaded region shows the hexagonal-binned
pixel-by-pixel measurements. The gray horizontal line shows the value of
$\alpha_{\rm CO}^{\rm MW}$, converted to
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ assuming
$R_{21}=0.65$.  Figure 4: Similar with Figure 3, but for
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ measurements. The
three colors of the circles and lines show the three cases indicated in the
legend. CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ Only: measured with
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ data only; w/ $R_{21}=0.65$
and w/ $R_{21}(I_{\mathrm{W4}})$: measured with
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ and
$W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ data for galaxies without
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$. The background graded
region is plotted for the “CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ Only”
measurements.

We measure the pixel-by-pixel correlations between $\alpha_{\rm CO}$ and
several parameters describing local physical conditions. These results are
summarized in Table 2 and visualized in Figure 3-Figure 4. Note that the
correlations and linear regressions are calculated with data in the complete
zone only (with data completeness $>50\%$; see Section 3.3 for details).

As shown in Table 2, most quantities have significant correlations ($p$-value
¡ 0.05) with $\alpha_{\rm CO}$ except $\Sigma_{\star}$. $\Sigma_{\star}$ has
significant correlations with
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ from CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ data only and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ from CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ only, but not when we combine CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ and CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$
data. This is likely due to the negative
$\Sigma_{\star}$-to-$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$
and the positive
$\Sigma_{\star}$-to-$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$
correlations, although both of which are weak.

$\log\overline{U}$ has the strongest negative correlation with
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$, meaning that
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ decreases toward
regions with stronger interstellar radiation. This is consistent with the
picture that $\alpha_{\rm CO}$ decreases with higher gas temperature and
larger line width (Bolatto et al., 2013). It is also the case that a higher
$\log\overline{U}$ might correspond to a lower $\Sigma_{\mathrm{dust}}$ as a
caveat of our fitting methodology (equivalent to “fixing $\beta$” in modified
blackbody models, see Shetty et al., 2009a, b). However, since we do not see a
strong
$\Sigma_{\mathrm{dust}}$-to-$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$
correlation, the above effect should be minor. Several other quantities also
show moderate (negative) correlations with
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$, i.e.
$\Sigma_{\mathrm{SFR}}$ and rsSFR. Studies have shown strong correlations
between $\log\overline{U}$ and $\Sigma_{\mathrm{SFR}}$ (e.g. Hirashita &
Chiang, 2022; Chiang et al., 2023, J. Chastenet et al. in preparation).
Another quantity that shows moderate correlation is $W_{\mathrm{CO}}$.
$\alpha_{\rm CO}$ is expected to anti-correlate with $W_{\mathrm{CO}}$ due to
either external pressure or other dynamical effects (e.g. Bolatto et al.,
2013; Sun et al., 2018). Note that since we use $\Sigma_{\mathrm{dust}}$ to
trace total gas and $\Sigma_{\mathrm{dust}}$ anti-correlates with
$\overline{U}$ in dust SED fitting, part of the anti-correlation between
$\overline{U}$ and $\alpha_{\rm CO}$ might be artificial due to dust SED
fitting.

Compared to $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$, the
correlations between $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$
and local conditions are overall weaker. $\log~{}{\rm rsSFR}$ has the
strongest correlation with
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ in all three $R_{21}$
cases, and $\log\overline{U}$ is the second strongest.

For all combinations of $\alpha_{\rm CO}$ and local conditions, we perform
linear regression with the functional form:

$\log\alpha_{\rm CO}=mx+b,$ (16)

where $x$ is the local condition and both $m$ and $b$ are free parameters, all
of which are listed in Table 2. In the same table, we also report the root-
mean-square error ($\Delta_{\mathrm{rms}}$) between the measured and fitted
$\log\alpha_{\rm CO}$. For
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$, all the formulas have
$0.3<\Delta_{\mathrm{rms}}<0.4$. For
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, most formulas have
$\Delta_{\mathrm{rms}}$ between 0.25 and 0.3.

One quantity that is often used for parametrizing $\alpha_{\rm CO}$ is
$12+\log({\rm O/H})$. In our measurements, $\alpha_{\rm CO}$ has moderate to
weak correlation with $12+\log({\rm O/H})$ in all cases. The slope from linear
regression ($m$) is $-1.59$ for
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and $-0.93$ to $-1.6$
for $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$. These values fall
at the lower end of literature values ($\sim-1.6$ to $-2.0$, Schruba et al.,
2012; Bolatto et al., 2013; Hunt et al., 2015; Accurso et al., 2017; Sun et
al., 2020). Note that our data set is less suitable for an in-depth study on
$12+\log({\rm O/H})$ since $>80\%$ of our data is concentrated in a small
dynamic range of $8.4\leq 12+\log({\rm O/H})\leq 8.6$.

Overall, we have shown that
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ has stronger
correlations with local conditions than
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$. Among the local
quantities, $\log\overline{U}$ and $\log~{}{\rm rsSFR}$ usually have the
strongest correlations with $\alpha_{\rm CO}$. We do not see a strong
correlation coefficient between $\alpha_{\rm CO}$ and $12+\log({\rm O/H})$,
one of the most frequently used quantity to model $\alpha_{\rm CO}$. However,
our fitted $\log\alpha_{\rm CO}$-to-$12+\log({\rm O/H})$ slopes overlap with
literature values.

###  4.2 Correlation with Galaxy-Averaged Quantities

Figure 5:  Scaling relations between galaxy-averaged
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and environmental
parameters. The errorbars show the 16th-84th percentiles in each galaxy. All
data points are calculated with
$W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ data only. The correlation
coefficients and $p$-values are labeled at the lower left in each panel,
highlighting the significant ($p$-value$<0.05$) ones. Figure 6: Similar with
Figure 5 but for $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$
measurements. The orange points show the measurements with
$W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ data only, and the purple
points show measurements with
$W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ data with a
$I_{\mathrm{W4}}$-dependent $R_{21}$. The fixed $R_{21}$ results are similar
to the ones from $I_{\mathrm{W4}}$-dependent $R_{21}$. They are not displayed
for clarity.

Besides kpc-scale variations, we also test how galaxy-averaged
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ vary between galaxies,
and how their variations correlate with galaxy-averaged properties. The
results are visualized in Figure 5-Figure 6. We use the symbol
$\text{<}X\text{>}$ to represent the simple averaged value of quantity $X$ in
each galaxy, i.e. $\Sigma_{i}^{\rm gal}X_{i}/\Sigma_{i}^{\rm gal}$, where
$\Sigma_{i}^{\rm gal}$ is the summation over all pixels in a galaxy. There are
two quantities which we mark with $X_{\mathrm{gal}}$ and do not apply simple
averages: (1) $\log\overline{U}_{\mathrm{gal}}$ is calculated as
$\log\big{(}\Sigma_{i}^{\rm gal}\Sigma_{{\rm
dust},i}\overline{U}_{i}/\Sigma_{i}^{\rm gal}\Sigma_{{\rm dust},i}$) to
reflect the dust-mass-weighted averaged ISRF; (2) sSFR is calculated as
$\Sigma_{i}^{\rm gal}\Sigma_{{\rm SFR},i}/\Sigma_{i}^{\rm
gal}\Sigma_{\star,i}=SFR/M_{\star}$. Within each galaxy, we only consider data
points within the S/N mask (Section 3.3). The errorbars in Figure 5-Figure 6
show the 16th-84th percentiles of the corresponding quantity. In Figure 6, we
also include $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$
calculated with CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ data with a
$I_{\mathrm{W4}}$-dependent $R_{21}$. We only include one $R_{21}$
prescription here for clarity.

We report the correlation coefficients and the corresponding $p$-values in
each panel of Figure 5-Figure 6. Unlike the kpc-scale case, most quantities
have insignificant correlations with $\alpha_{\rm CO}$. The two quantities
with significant and moderate anti-correlations with $\alpha_{\rm CO}$ are
$\log\overline{U}_{\mathrm{gal}}$ and $\log~{}\mathrm{sSFR}$, both of which
are reported to have strongest correlations in the kpc-scale case.

##  5 Power-law dependence of the conversion factor on stellar mass surface
density

Figure 7: Measured $\alpha_{\rm CO}$ plotted against stellar mass surface
density. The left panels show the
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ measurements while the
right panels show the $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$
measurements. Top panels: the measured $\alpha_{\rm CO}$ in each galaxy, where
the solid lines show the median in each $\Sigma_{\star}$ bin. Middle panels:
the overall binned median of $\alpha_{\rm CO}$, where the errorbars show the
16th- and 84th-percentiles. Bottom panels: similar to the middle panels, but
the y-axis is normalized to the $\alpha_{\rm CO}$ value at
$\Sigma_{\rm\star,T}=100~{}\rm M_{\odot}~{}pc^{-2}$ (shown in vertical dashed
line) in each galaxy. The galaxies with $\alpha_{\rm CO}$ increasing with
$\Sigma_{\star}$ are removed in the bottom panels. The 2d histogram in all
panels shows the overall data distribution.

In the $\alpha_{\rm CO}$ prescription proposed in Bolatto et al. (2013), the
authors use a power law with $\Sigma_{\mathrm{Total}}$ (=$\Sigma_{\star}$
+$\Sigma_{\mathrm{gas}}$) to trace the changes in $\alpha_{\rm CO}$ due to gas
temperature and line width and a threshold in $\Sigma_{\mathrm{Total}}$ to
trace where the effects become important. Inspired by their work and motivated
by the necessity of improving $\alpha_{\rm CO}$ prescriptions in galaxy
centers, we examine whether a similar functional form applies to our
$\alpha_{\rm CO}$ measurements. With the WISE full-sky observations, the
resolved $\Sigma_{\star}$ for all nearby galaxies is widely available, which
makes this kind of prescription easy to apply. In this study, we will focus on
the $\alpha_{\rm CO}$-to-$\Sigma_{\star}$ relation instead of
$\Sigma_{\mathrm{Total}}$ because our data set is mostly
$\Sigma_{\star}$-dominated (50% with
$\Sigma_{\mathrm{gas}}/\Sigma_{\mathrm{Total}}<0.2$; 99.5% with
$\Sigma_{\mathrm{gas}}/\Sigma_{\mathrm{Total}}<0.5$).

We present the correlations between $\alpha_{\rm CO}$ and $\Sigma_{\star}$ in
Figure 7 for both $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$
(left panels) and $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$
(right panels). In the top panels, we present the profile of measured
$\alpha_{\rm CO}$ versus $\Sigma_{\star}$ in each galaxy. For
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$, most galaxies have
their $\alpha_{\rm CO}$ anti-correlate with $\Sigma_{\star}$ at
$\Sigma_{\star}>100~{}\rm M_{\odot}~{}pc^{-2}$ aside from a few exceptions. It
is similar for $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, but
with a flatter $\alpha_{\rm CO}$-to-$\Sigma_{\star}$ slope. At the
low-$\Sigma_{\star}$ end, some galaxies still have negative $\alpha_{\rm
CO}$-to-$\Sigma_{\star}$ correlations while the others have strong positive
correlations. In the middle panels, we show the collective behavior across
galaxies using a binned average as a function of $\Sigma_{\star}$, with each
bin spanning $\sim 0.1$ dex in $\Sigma_{\star}$.

We find that in regions with high-$\Sigma_{\star}$, $\alpha_{\rm CO}$
generally decreases with $\Sigma_{\star}$, which is consistent with the
negative power-law index in the Bolatto et al. (2013) formula. There appears
to be galaxy-to-galaxy variation in the value of $\alpha_{\rm CO}$, but good
agreement in the rate of how fast $\alpha_{\rm CO}$ decreases with
$\Sigma_{\star}$. To better illustrate this phenomena, we normalize
$\alpha_{\rm CO}$ in each galaxy at a threshold $\Sigma_{\rm\star,T}\equiv
100~{}\rm M_{\odot}~{}pc^{-2}$ (a threshold inspired by Bolatto et al., 2013)
and show the normalized $\alpha_{\rm CO}$ in the bottom panel of Figure 7. The
normalization in each galaxy ($\alpha_{\rm CO,gal,T}$) is defined as the
median $\alpha_{\rm CO}$ of pixels with their $\Sigma_{\star}$ within
$\Sigma_{\rm\star,T}\pm 0.05$ dex. Note that we exclude the galaxies with
$\alpha_{\rm CO}$ increasing with $\Sigma_{\star}$ in this visualization (NGC
4536 and NGC 4651 for $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$;
NGC 3521, NGC 4254 and NGC 4596 for
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$).

In the remainder of this section, we will focus on analyzing the scaling
relation between $\alpha_{\rm CO}$ and $\Sigma_{\star}$ in a sub-sample of
galaxies with at least 5 measurements with
$\Sigma_{\star}>\Sigma_{\rm\star,T}$ (27 galaxies for CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ and 24 galaxies for CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$, see the $\rm N_{pix,100}$ column in Table
1). We use a power law to characterize this scaling relation:

$\log\frac{\alpha_{\rm CO}}{\alpha_{\rm
CO,gal,T}}=a\times\log\frac{\Sigma_{\star}}{\Sigma_{\rm\star,T}}+b,~{}\Sigma_{\star}\geq\Sigma_{\rm\star,T},$
(17)

where $a$ (the power-law index) and $b$ (the offset) are free parameters.
Since both $\alpha_{\rm CO}$ and $\Sigma_{\star}$ are normalized in the
formula, we expect $b\sim 0$ (and $b_{\mathrm{gal}}\sim 0$) if $\alpha_{\rm
CO}$ monotonically decreases with $\Sigma_{\star}$. By default, we fit Eq. 17
with all data. When fitting data in individual galaxies only, we will describe
the parameters as $a_{\mathrm{gal}}$ and $b_{\mathrm{gal}}$.

We exclude data from galaxies that do not satisfy the following criteria: (1)
at least 5 measurements at $\Sigma_{\star}>\Sigma_{\rm\star,T}$; (2) spanning
at least 0.1 dex in $\Sigma_{\star}$ at $\Sigma_{\star}>\Sigma_{\rm\star,T}$;
(3) $a_{\mathrm{gal}}>0$; (4) $b_{\mathrm{gal}}>0.1$. Since all criteria are
$\Sigma_{\rm\star,T}$-dependent, we expect the definition of sub-sample space
to vary with $\Sigma_{\rm\star,T}$. We will visualize the galaxies not
satisfying the last two criteria in Section 5.2.

With the fiducial setting, i.e. $\mathrm{D/M}=0.55$ and
$\Sigma_{\rm\star,T}=100~{}\rm M_{\odot}~{}pc^{-2}$, the fitting yields
$a=-0.49^{+0.07}_{-0.07}$ and $b=0.03^{+0.01}_{-0.01}$ with
$\Delta_{\mathrm{rms}}=0.15~{}\mathrm{dex}$ for CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$; $a=-0.30^{+0.06}_{-0.06}$ and
$b=0.01^{+0.01}_{-0.01}$ with $\Delta_{\mathrm{rms}}=0.13~{}\mathrm{dex}$ for
CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$. The small $b$ values, which are
consistent with our expectations, indicate that the $\alpha_{\rm
CO}$-to-$\Sigma_{\star}$ relation matches with the picture of a simple power
law. The difference between the $a$ values for CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ and CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$
data is consistent with the finding that $R_{21}\propto
I_{\mathrm{MIR}}^{-0.2}$ (Leroy et al., 2023). The uncertainties of $a$ and
$b$ are estimated from 1000 rounds of bootstrap resampling. In each round, we
select 27 (24) galaxies for CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ (CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$) data with replacement from our sample
galaxies to fit the power law. We then take the difference between the best-
fit parameter and the 84th- (16th-) percentile from the 1000 bootstraps as the
upper (lower) uncertainty.

We will measure how the $\alpha_{\rm CO}$-to-$\Sigma_{\star}$ relation varies
according to the adopted D/M and $\Sigma_{\rm\star,T}$ in the remainder of
this section, and show that the results are also robust under the change of
other adopted quantities, e.g. applying internal variations of D/M, in
Appendix A.

###  5.1 Dependence of the power-law index on adopted D/M and
$\Sigma_{\rm\star,T}$

Figure 8: The dependence of fitted power-law index and offset on assumed D/M.
The cyan points show results for
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$, and the orange points
show results for $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$. For
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, we only show results
from $W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ data only since
fitting results including $W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$
data have mininal difference. We highlight the region with D/M inferred from
literature, i.e. $0.4\leq\mathrm{D/M}\leq 0.7$, while the empty circles show
results outside that range. Top: the power-law index ($a$). The dashed lines
show the mean value of $a$ in the range of $0.4\leq\mathrm{D/M}\leq 0.7$.
Middle: the offset ($b$). The dotted line shows $b=0$, where expect the
fitting result to be if $\alpha_{\rm CO}$ monotonically decreases with
$\Sigma_{\star}$. Bottom: the $\Delta_{\mathrm{rms}}$ of each fit.

To test the robustness of our results for potentially different dust
properties, we expand the assumed D/M from the single value (0.55) assumed in
the previous section to the possible range of D/M, i.e. $0.1\leq\text{D/M}\leq
1$. We do not go to even lower D/M values because our methodology implies the
existence of certain amount of dust. We derive $\alpha_{\rm CO}$ and fit the
$\alpha_{\rm CO}$-to-$\Sigma_{\star}$ power-law relation at each assumed D/M.
The results are shown in Figure 8. We highlight the fit results with
$0.4\leq\text{D/M}\leq 0.7$, which is the D/M value inferred from literature
introduced in Section 2. Same as in the previous calculations, the
uncertainties of parameters $a$ and $b$ in the fitting parameters are
estimated from 1000 rounds of bootstrap resampling.

As shown in the top panel of Figure 8, we find that the power-law index ($a$)
is invariant with assumed D/M throughout the range we examine for both
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$. The average $a$ in
the range of $0.4\leq\mathrm{D/M}\leq 0.7$ is -0.51 and -0.30 for
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, respectively. The
statistical uncertainties in $a$ for both CO transitions are around $\pm 0.1$
dex. The result implies that as long as the D/M stays roughly constant within
each galaxy, we can recover similar behavior in the $\Sigma_{\star}$
dependence of $\alpha_{\rm CO}$ with a power law

Due to the nature of the definition of $b$ in Eq. 17, we expect $b\sim 0$.
This is seen in most D/M values we examine as $|b|$ stay below 0.025, shown in
the middle panel of Figure 8. This indicates that the power law
parameterization reasonably fits the observed data regardless of the assumed
D/M value. However, the $b$ values for
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ seems biased toward
the positive end. This could result from a steeper $\log$$\alpha_{\rm
CO}$-to-$\log$$\Sigma_{\star}$ slope toward higher $\Sigma_{\star}$, which
results in a positive offset in the power law at relatively lower
$\Sigma_{\star}$. In the bottom panel, we show the $\Delta_{\mathrm{rms}}$
value of each fit as an indicator of goodness of fit. All fits have
$\Delta_{\mathrm{rms}}$ below 0.2 dex, and the fits around
$0.4\leq\mathrm{D/M}\leq 0.7$ have $\Delta_{\mathrm{rms}}\sim 0.15$ dex.

Figure 9: The dependence of fitted power-law index and offset on adopted
threshold $\Sigma_{\rm\star,T}$. The cyan points show results for
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$, and the orange points
show results for $\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$. For
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, we only show results
from $W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ data only since
fitting results including $W_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$
data have mininal difference. Top: the power-law index ($a$). The dashed lines
show the average $a$ values with $\Sigma_{\rm\star,T}=100~{}\rm
M_{\odot}~{}pc^{-2}$ in the range of $0.4\leq\mathrm{D/M}\leq 0.7$. These are
the same lines as Figure 8. Middle: the offset ($b$). The dotted line shows
$b=0$, where expect the fitting result to be if $\alpha_{\rm CO}$
monotonically decreases with $\Sigma_{\star}$. Bottom: the
$\Delta_{\mathrm{rms}}$ of each fit. All the calculation here are done with
$\text{D/M}=0.55$.

We further test if the chosen threshold in stellar mass surface density,
$\Sigma_{\rm\star,T}$, will affect the fitting results. We fix
$\text{D/M}=0.55$ and fit the power-law relation at $\Sigma_{\rm\star,T}$
ranges from 30 to 300 $\rm M_{\odot}~{}pc^{-2}$. The results are shown in
Figure 9.

In the top panel of Figure 9, we notice that the power-law index ($a$) has a
larger dynamic range than the case where we alter D/M, but the index stays
negative throughout the $\Sigma_{\rm\star,T}$ range we examine. The power-law
index for $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ stays within
$\pm 0.1$ of the fiducial case, and the indices for
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ is consistent with the
fiducial value at $\Sigma_{\rm\star,T}>80~{}\rm M_{\odot}~{}pc^{-2}$. There is
a weak trend that $|a|$ becomes larger toward larger $\Sigma_{\rm\star,T}$.
The small $b$ values indicate that the power-law function form applies in
general. We also show the $\Delta_{\mathrm{rms}}$ values in the bottom panel.
We have $\Delta_{\mathrm{rms}}\leq 0.2$, with a weak trend of smaller
$\Delta_{\mathrm{rms}}$ toward higher $\Sigma_{\rm\star,T}$.

To summarize, the power-law functional form applies to the normalized
$\alpha_{\rm CO}$-to-$\Sigma_{\star}$ relation within the D/M and
$\Sigma_{\rm\star,T}$ ranges we examine. With a fixed $\Sigma_{\rm\star,T}$ at
100 $\rm M_{\odot}~{}pc^{-2}$, we find a invariant power-law index ($a$)
throughout $0<\mathrm{D/M}<1.0$. The average $a$ in the range of
$0.4\leq\mathrm{D/M}\leq 0.7$ is -0.51 and -0.30 for
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, respectively. With a
fixed D/M at 0.55, we find a weak trend of larger $|a|$ toward higher
$\Sigma_{\rm\star,T}$. The power-law index derived with the fiducial setup,
i.e. $\mathrm{D/M}=0.55$ and $\Sigma_{\rm\star,T}=100~{}\rm
M_{\odot}~{}pc^{-2}$, is a good representative of conditions with
$80<\Sigma_{\rm\star,T}\leq 300~{}\rm M_{\odot}~{}pc^{-2}$, with a span
$\sim\pm 0.05$.

###  5.2 Galaxy-to-galaxy variations

In this section, we examine the possible variation of the $\alpha_{\rm
CO}$-to-$\Sigma_{\star}$ relation between individual galaxies, mainly how the
variation in $\alpha_{\rm CO,gal,T}$ (normalization of $\alpha_{\rm CO}$ at
$\Sigma_{\rm\star,T}$, see Section 5) and $a_{\mathrm{gal}}$ (power-law index,
Eq. 17) correlate with galaxy-averaged properties. By understanding what sets
$a_{\mathrm{gal}}$ and $\alpha_{\rm CO,gal,T}$, we can build a prescription of
$\alpha_{\rm CO}$ considering the $\alpha_{\rm CO}$-to-$\Sigma_{\star}$
relation and galaxy-to-galaxy variations. The results are visualized in Figure
10 and Figure 11.

Figure 10: Scaling relations between $a_{\mathrm{gal}}$ (the power-law index),
$\alpha_{\rm CO,gal,T}$ and galaxy-averaged environmental parameters for CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ data. Galaxies that are barred and not
barred are colored in orange and cyan, respectively. The black dashed line in
the $a_{\mathrm{gal}}$ show the $a$ value calculated with overall data. The
correlation coefficients and $p$-values are labeled at the lower left in each
panel, highlighting the significant ($p$-value$<0.05$) ones. Figure 11:
Similar to Figure 10, but for CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ data.

In the upper panels of Figure 10 and Figure 11, we show how the power-law
index ($a_{\mathrm{gal}}$) varies with 7 selected galaxy-averaged properties
and whether the galaxy is barred or not. The set of properties is the same as
the ones in Figure 5 and Figure 6. None of the properties show significant
correlation with $a_{\mathrm{gal}}$. Meanwhile, the standard deviation of
$a_{\mathrm{gal}}$ is 0.44 for CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ and 0.36
for CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$.

In the lower panels of Figure 10 and Figure 11, we show how the normalization
in each galaxy ($\alpha_{\rm CO,gal,T}$) varies with galaxy-averaged
properties. The standard deviation of $\alpha_{\rm CO,gal,T}$ is 0.27 dex for
CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ and 0.26 dex for CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$. For CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$,
$\log\overline{U}_{\mathrm{gal}}$, $\text{<}\Sigma_{\mathrm{SFR}}\text{>}$ and
sSFR show significant correlations with $\alpha_{\rm CO,gal,T}$ with similar
strength. For CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$,
$\log\overline{U}_{\mathrm{gal}}$, $\text{<}\Sigma_{\star}\text{>}$ and sSFR
show significant correlations with $\alpha_{\rm CO,gal,T}$ with similar
strength. We use these significant correlations to fit empirical relations for
$\alpha_{\rm CO,gal,T}$ and summarize the results in Table 3. The fitted
empirical relations do not differ significantly in terms of
$\Delta_{\mathrm{rms}}$. Meanwhile, the fit with $\overline{U}$ has the
smallest777Considering the product of $\log x$ and the uncertainties in $m$.
the statistical uncertainties in the fitted parameters among the parameters
for both CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ and CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$.

Table 3: Dependence of $\alpha_{\rm CO,gal,T}$ on galaxy-integrated
quantities.
$\log\alpha_{\rm\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1),gal,T}$, CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ Only
---
$x^{(1)}$ | $m^{(1)}$ | $d^{(1)}$ | $\Delta_{\mathrm{rms}}$
$\overline{U}_{\rm gal}$ | -0.5$\pm$0.2 | 0.9$\pm$0.1 | 0.15
¡$\Sigma_{\rm SFR}$¿ | -0.4$\pm$0.1 | -0.1$\pm$0.2 | 0.13
sSFR | -0.4$\pm$0.1 | -3.2$\pm$0.8 | 0.11
$\log\alpha_{\rm\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0),gal,T}$, CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ Only
$x$ | $m$ | $d$ | $\Delta_{\mathrm{rms}}$
$\overline{U}_{\rm gal}$ | -0.7$\pm$0.3 | 0.7$\pm$0.1 | 0.18
¡$\Sigma_{\star}$¿ | 0.5$\pm$0.2 | -0.6$\pm$0.5 | 0.19
sSFR | -0.4$\pm$0.1 | -3.3$\pm$1.3 | 0.17

###  5.3 General suggestions for $\alpha_{\rm CO}$

Combining the $\alpha_{\rm CO}$-to-$\Sigma_{\star}$ power-law index in Section
5.1 and the galaxy-to-galaxy normalization ($\alpha_{\rm CO,gal,T}$) in
Section 5.2, we propose a prescription for $\alpha_{\rm CO}$ that works in
environments with $\Sigma_{\star}>100~{}\rm M_{\odot}~{}pc^{-2}$:

$\left\\{\begin{array}[]{l}\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}=10^{-0.5\log(\overline{U}_{\rm
gal})+0.9}\Sigma_{\star}^{-0.51}\\\
\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}=10^{-0.7\log(\overline{U}_{\rm
gal})+0.7}\Sigma_{\star}^{-0.30}\end{array},~{}\Sigma_{\star}\geq 100~{}\rm
M_{\odot}~{}pc^{-2},\right.$ (18)

where we fix $\Sigma_{\rm\star,T}=100~{}\rm M_{\odot}~{}pc^{-2}$ and adopt the
power-law index averaged from $0.4\leq\text{D/M}\leq 0.7$ considering the
possible variation in dust properties. $\alpha_{\rm CO}$ is in unit of ${\rm
M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}$. The $\overline{U}_{\rm
gal}$-dependent normalization could be replaced with other quantities listed
in Table 3, however, as we mention in Section 5.2, the fit with
$\overline{U}_{\rm gal}$ has the smallest uncertainties in fitted parameters.

Meanwhile, one key mechanism that sets $\alpha_{\rm CO}$, the CO-dark gas, is
likely not parameterized by our formula (see Section 6.1). This is because the
CO-dark gas effect is relatively weak in the metallicity span of our sample.
However, both the CO-dark gas and the “starburst $\alpha_{\rm CO}$” effects
should be considered for an $\alpha_{\rm CO}$ prescription to be applied
through all environments. Thus, another suggestion we have is to make a
Bolatto et al. (2013)-style combination of our power-law term with existing
$\alpha_{\rm CO}$ prescriptions tracing the CO-dark gas effect, e.g. Wolfire
et al. (2010), Narayanan et al. (2012), Schruba et al. (2012), Hunt et al.
(2015), Accurso et al. (2017), or Sun et al. (2020). That is, if the adopted
existing metallicity-dependent prescription of $\alpha_{\rm CO}$ is
$\alpha_{\rm CO}^{\mathrm{CO\text{-}dark}}$, we suggest:

$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}=\left\\{\begin{array}[]{ll}\alpha_{\rm
CO}^{\mathrm{CO\text{-}dark}}&,~{}\Sigma_{\star}<100~{}\rm
M_{\odot}~{}pc^{-2}\\\ \alpha_{\rm
CO}^{\mathrm{CO\text{-}dark}}\Sigma_{\star}^{-0.51}&,~{}\Sigma_{\star}\geq
100~{}\rm M_{\odot}~{}pc^{-2}\end{array}\right..$ (19)

Under this functional form, we expect the normalization ($\alpha_{\rm
CO,gal,T}$) to be taken into account by the $\alpha_{\rm
CO}^{\mathrm{CO\text{-}dark}}$ term. For
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$, just replace the
power-law index with $-0.30$.

##  6 Discussion

###  6.1 Interpreting of the environmental dependence of $\alpha_{\rm CO}$

In this section, we will discuss the physical interpretations of the
correlations between $\alpha_{\rm CO}$ and the physical quantities we present
in the previous sections. As we mentioned in Section 1, we expect two main
trends in the variation of $\alpha_{\rm CO}$: (1) the “CO-dark gas” trend,
where $\alpha_{\rm CO}$ increases toward lower metallicity as shielding for CO
weakens; (2) the “starburst conversion factor” trend, where $\alpha_{\rm CO}$
decreases toward galaxy centers and U/LIRGs with the decrease in CO optical
depth or increase in CO excitation.

Regarding the CO-dark gas trend, we observe weak anti-correlation between
$\alpha_{\rm CO}$ and $12+\log({\rm O/H})$ at kpc scale (Section 4.1), and no
correlation at galaxy-averaged scale (Section 4.2). One possible explanation
for the weak correlation is that the statistical significance becomes weaker
with the small dynamic range of our $12+\log({\rm O/H})$ data: 80% of the
$12+\log({\rm O/H})$ measurements fall within a 0.2 dex range from 8.4 to 8.6.
The $\alpha_{\rm CO}$-$\Sigma_{\star}$ relation we present in Section 5 is
unlikely caused by this CO-dark gas effect since the dynamic range in
$12+\log({\rm O/H})$ is even smaller for data above the $\Sigma_{\rm\star,T}$
threshold. Another explanation is that the CO-dark gas effect is weaker at
nearly solar metallicity (e.g. Wolfire et al., 2010; Glover & Mac Low, 2011;
Hunt et al., 2015). Meanwhile, we note that recent simulations show that there
is significant fraction of CO-dark gas ($f_{\mathrm{dark}}$) up to solar
metallicity, e.g. Gong et al. (2018) found $f_{\mathrm{dark}}$ ranges from
26%–79%. These studies found that $f_{\mathrm{dark}}$ correlates with
extinction strength or $W_{\mathrm{CO}}$ (Smith et al., 2014; Gong et al.,
2018, 2020; Hu et al., 2022).

We interpret $\overline{U}$ and (r)sSFR as empirical tracers for regions with
condition similar to U/LIRGs and starburst regions, where the “starburst
conversion factor” trend matters. High-sSFR traces starburst regions, thus the
anti-correlation between $\alpha_{\rm CO}$ and sSFR is consistent with the
picture that $\alpha_{\rm CO}$ becomes lower in starburst regions.
$\overline{U}$ is derived from the intensity of dust emission in the IR, which
therefore traces IR-bright environments. Studies have also argued that
$\alpha_{\rm CO}$ could vary with increased radiation field due to CO removal
(Israel, 1997b; Wolfire et al., 2010; Accurso et al., 2017). However, it is
shown that the CO removal effect should be weak as long as the CO emission is
optically thick (Wolfire et al., 2010; Bolatto et al., 2013).
$\Sigma_{\mathrm{SFR}}$, which simultaneously traces the UV radiation and
starburst regions, only exhibits significant anti-correlations with
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ but not the
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ cases.

We interpret the $\alpha_{\rm CO}$-to-$\Sigma_{\star}$ anti-correlation as the
increase of line width from additional gravity. Bolatto et al. (2013, also see
()) suggested that in high-$\Sigma_{\star}$ environments, the molecular gas
experiences gravitational potential from stellar sources, ending up with a
total pressure larger than virialized clouds. This larger pressure results in
gas line width larger than one would expect from self-gravitating clouds. This
increase in gas line width scales with total mass (stars and gas) in the
system, or $\alpha_{\rm
CO}\propto\big{(}M_{\mathrm{mol}}/(M_{\mathrm{mol}}+M_{\star})\big{)}^{0.5}$.
The above functional form approximates a $\alpha_{\rm CO}$-to-$\Sigma_{\star}$
power law in $M_{\star}$-dominated regions. For the argument to hold, the CO
emission must be optically thick. Bolatto et al. (2013) mentioned that the
only possible structure for molecular gas that satisfies this scenario is an
extended molecular medium. Although additional gravity might also cause an
increase in gas temperature, a recent study has suggested that the effect of
gas temperature on $\alpha_{\rm CO}$ is minor compared to linewidth and
optical depth (Teng et al., 2023).

###  6.2 Comparing to previous $\alpha_{\rm CO}$ surveys

In this section, we compare our measurements to $\alpha_{\rm CO}$ values
obtained in previous dust-based $\alpha_{\rm CO}$ surveys. Since we will cover
studies with both $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ measurements, we will
convert $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ values in
literature and this work to
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ with $R_{21}=0.65$ for
simplicity and uniformity. First, we compare our $\alpha_{\rm CO}$ maps with
Sandstrom et al. (2013). They measured spatially resolved $\alpha_{\rm CO}$ in
26 nearby, star-forming galaxies using a dust-based methodology (also see
Leroy et al., 2011; den Brok et al., 2023; Yasuda et al., 2023). They assume
that the variation of D/G is minimum in a few-kpc-scale “solution pixel”
consisted of 37 pixels in a hexagonal region, and fit D/G and $\alpha_{\rm
CO}$ simultaneously from data by minimizing the variation in D/G. Sandstrom et
al. (2013) used CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ data from HERACLES (Leroy
et al., 2009a) and $R_{21}=0.7$. We rescale their results with $R_{21}=0.65$
for uniformity. Compared to our work, the Sandstrom et al. (2013) metholodogy
has larger degrees of freedom for the spatial variation of D/G and D/M.
Meanwhile, it is more difficult to push their methodology to a larger sample
size at fixed physical resolution.

There are 17 galaxies that are studied in both Sandstrom et al. (2013) and
this work. We show the $\alpha_{\rm CO}$ measurements from both works as a
function of galacto-centric radius ($R_{\mathrm{g}}$ in terms of $R_{25}$) in
Figure 12. The mean $\alpha_{\rm CO}$ from both works are $\sim 2.8~{}{\rm
M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}$. Similar with Sandstrom et al.
(2013), we find a weak (but significant) positive correlation between
$\alpha_{\rm CO}$ and $R_{\mathrm{g}}$. When we normalize the $\alpha_{\rm
CO}$ in each galaxy by the mean $\alpha_{\rm CO}$ in each galaxy
(¡$\alpha_{\mathrm{CO,~{}gal}}$¿), both works show a flat trend with radius in
mid- to outer-disk. Sandstrom et al. (2013) data shows a more significant
decrease in $\alpha_{\rm CO}$ in the galaxy center.

Figure 12: The relation between measured $\alpha_{\rm CO}$ and galacto-centric
radius in this work and Sandstrom et al. (2013). For both works, we display
$\alpha_{\rm CO}=0.65\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$
for uniformity. We only include measurements from the galaxies that are
included in both works. The vertical dashed line shows the completeness
threshold in $R_{\mathrm{g}}/R_{25}$ in this work. In the top panel, the
horizontal cyan dashed-dotted line shows the mean $\alpha_{\rm CO}$ in
Sandstrom et al. (2013), and the orange dashed line shows the mean
$\alpha_{\rm CO}$ from this work. The two lines are close to each other. In
the bottom panel, the horizontal line shows $\alpha_{\rm
CO}=\text{<}\alpha_{\mathrm{CO,~{}gal}}\text{>}$, where
¡$\alpha_{\mathrm{CO,~{}gal}}$¿ is the mean $\alpha_{\rm CO}$ in each galaxy.
Figure 13: The mean $\alpha_{\rm CO}$ values in each galaxy from this work
(both CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ data with $R_{21}=0.7$ and CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ data), Sandstrom et al. (2013, S13) and the
COMING survey (Sorai et al., 2019; Yasuda et al., 2023, Y23). Circles show
$\alpha_{\rm CO}$ values derived with CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$
data, and triangles show $\alpha_{\rm CO}$ values derived with CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ data. Filled symbols show the results from
this work, and empty symbols show literature values. We only include galaxies
that are measured in at least one of the literature survey. The errorbar for
this work shows the 16th- and 84th-percentiles (Table 1). The mean and
errorbar of previous works are quoted from Table 4 of Sandstrom et al. (2013,
with rescaling for $R_{21}$) and Table 3 of Yasuda et al. (2023).

Another previous survey we include here is the COMING survey (Sorai et al.,
2019; Yasuda et al., 2023). The COMING survey solves D/G and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ simultaneously by
minimizing a $\chi^{2}$ value defined by the difference between
$\big{(}\text{D/G}\times(\Sigma_{\mathrm{atom}}+\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}W_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)})\big{)}$
and $\Sigma_{\mathrm{dust}}$ derived from dust SED fitting. Here, we quote
their “global” result, where the authors fit all data within one galaxy to
retrieve one set of D/G and $\alpha_{\rm CO}$ values.

We compare our measured $\alpha_{\rm CO}$ in each galaxy with Sandstrom et al.
(2013) and the COMING survey (Sorai et al., 2019; Yasuda et al., 2023) in
Figure 13. The $\alpha_{\rm CO}$ measured in the three works are in general
consistent with each other within uncertainties. Our measurements made with CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ and CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$
agree with each other. When there is a difference, it is more often that the
one derived with CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ has a slightly larger
value. There are a few cases where our measured $\alpha_{\rm CO}$ is
significantly lower than the one in Sandstrom et al. (2013), i.e. NGC0925,
NGC3184, NGC3198, and NGC3938. Since the two works use almost identical data
sets for gas emission lines in these targets, we believe the main difference
results from the estimate of $\Sigma_{\mathrm{dust}}$. In this work, we apply
the correction factor for $\Sigma_{\mathrm{dust}}$ derived in Chastenet et al.
(2021) for dust properties fitted with the Draine & Li (2007) model. This can
cause a $\sim 0.5$ dex lower $\alpha_{\rm CO}$ than the one directly derived
from the Draine & Li (2007) model. We also note that in several galaxies with
signatures of active galactic nucleus (AGN; see classification in Kennicutt et
al., 2011), there is larger offset between our measurements and literature,
e.g. NGC3627, NGC4254 and NGC4725; however, there are also galaxies with AGN
show consistent results, e.g. NGC4536, NGC4736 and NGC5055. Thus, having AGN
could not be the only cause for the mismatch, and it is likely that the type
of nuclei activities do not dominate the kpc-scale $\alpha_{\rm CO}$ values
(e.g. Sandstrom et al., 2013). Our measurements made with CO
$(1{\scalebox{0.75}[1.0]{$-$}}0)$ agree with the COMING survey.

##  7 Summary

In this work, we measure the spatially resolved CO-to-H2 conversion factor
($\alpha_{\rm CO}$) in 41 nearby galaxies at 2 kpc resolution. We derive
$\Sigma_{\mathrm{mol}}$ by using a fixed D/M and converting
$\Sigma_{\mathrm{dust}}$ and $Z$ into $\Sigma_{\mathrm{gas}}$, then removing
$\Sigma_{\mathrm{atom}}$ to get $\Sigma_{\mathrm{mol}}$. We calculate
$\alpha_{\rm CO}$ with derived $\Sigma_{\mathrm{mol}}$ and measured
$W_{\mathrm{CO}}$. In total, we have $\sim$940 and $\sim$660 independent
measurements of $\alpha_{\rm CO}$ for CO $(2{\scalebox{0.75}[1.0]{$-$}}1)$ and
CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$ data, respectively. The mean values for
$\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ and
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$ are
$6.9^{+3.9}_{-4.5}$ and $3.1^{+1.6}_{-1.7}~{}{\rm
M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}$, respectively. The CO-intensity-
weighted mean for $\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}$ is
4.37 ${\rm M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}$, and 2.58 ${\rm
M_{\odot}\,pc^{-2}\,(K\,km\,s^{-1})^{-1}}$ for
$\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}$.

We examine how $\alpha_{\rm CO}$ scales with several physical quantities, i.e.
$W_{\mathrm{CO}}$, metallicity, $\Sigma_{\mathrm{dust}}$, ISRF,
$\Sigma_{\star}$, $\Sigma_{\mathrm{SFR}}$ and (r)sSFR. At 2 kpc scale, almost
all quantities have significant local correlation with $\alpha_{\rm CO}$.
Among them, the strength of the ISRF and rsSFR have the strongest anti-
correlation with spatially resolved $\alpha_{\rm CO}$. At galaxy-integrated
scale, none of the quantities have strong correlation with galaxy-averaged
$\alpha_{\rm CO}$. Again, the strength of the ISRF and sSFR are the only
quantities that show significant correlation with $\alpha_{\rm CO}$.

In regions with high surface densities ($\Sigma_{\star}\geq 100~{}\rm
M_{\odot}~{}pc^{-2}$), we find that the $\alpha_{\rm CO}$ decreases with
$\Sigma_{\star}$. Specifically, we find:

$\left\\{\begin{array}[]{l}\alpha_{\textup{CO}\,(2{\scalebox{0.75}[1.0]{$-$}}1)}\propto\Sigma_{\star}^{-0.51}\\\
\alpha_{\textup{CO}\,(1{\scalebox{0.75}[1.0]{$-$}}0)}\propto\Sigma_{\star}^{-0.30}\end{array},~{}\Sigma_{\star}\geq
100~{}\rm M_{\odot}~{}pc^{-2}\right.$ (20)

within D/M=0.4–0.7, the D/M values for inner disk inferred from literature.
The power-law index is insensitive to the assumed D/M, and it is roughly
constant at $\Sigma_{\rm\star,T}>80~{}\rm M_{\odot}~{}pc^{-2}$. It also has
little dependence on the adopted ratio between CO rotational lines. Meanwhile,
the discrepancy between the power-law index for CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ and CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$
data indicates a variation in the CO line ratio traced by $\Sigma_{\star}$.

When fitting the power-law relation within individual galaxies, we find
significant dependence of the normalization of $\alpha_{\rm CO}$ in each
galaxy on several quantities. Among them, the linear regression to
$\log\overline{U}_{\mathrm{gal}}$ has the minimal statistical uncertainties.
Thus, we recommend using $\Sigma_{\star}$ and
$\log\overline{U}_{\mathrm{gal}}$ to predict $\alpha_{\rm CO}$ at
high-$\Sigma_{\star}$ environments.

This decrease of $\alpha_{\rm CO}$ in the high-$\Sigma_{\star}$ region is
likely due to the increased CO brightness with increased line width. The line
width is larger than self-gravitating clouds due to the additional gravity
from stellar sources, and the structure satisfying this scenario is likely an
extended molecular medium. Understanding the decrease in $\alpha_{\rm CO}$ at
high $\Sigma_{\star}$ is important for accurately assessing molecular gas
content and star-formation efficiency in the centers of galaxies and bridges
the “MW-like” to ”starburst” conversion factor.

## Acknowledgments

IC thanks Hiroyuki Hirashita for helpful discussion. IC thanks the National
Science and Technology Council for support through grant
111-2112-M-001-038-MY3, and the Academia Sinica for Investigator Award AS-
IA-109-M02 (PI: Hiroyuki Hirashita). KS and YT acknowledge funding support
from NRAO Student Observing Support Grant SOSPADA-012 and from the National
Science Foundation (NSF) under grant No. 2108081. JS acknowledges support by
NASA through the NASA Hubble Fellowship grant HST-HF2-51544 awarded by the
Space Telescope Science Institute (STScI), which is operated by the
Association of Universities for Research in Astronomy, Inc., for NASA, under
contract NAS5-26555. ADB acknowledges support from the National Science
Foundation through grants AST-2108140 and AST-2307441. EWK acknowledges
support from the Smithsonian Institution as a Submillimeter Array (SMA) Fellow
and the Natural Sciences and Engineering Research Council of Canada. JC
acknowledges support from ERC starting grant #851622 DustOrigin

This work uses observations made with ESA Herschel Space Observatory. Herschel
is an ESA space observatory with science instruments provided by European-led
Principal Investigator consortia and with important participation from NASA.
The Herschel spacecraft was designed, built, tested, and launched under a
contract to ESA managed by the Herschel/Planck Project team by an industrial
consortium under the overall responsibility of the prime contractor Thales
Alenia Space (Cannes), and including Astrium (Friedrichshafen) responsible for
the payload module and for system testing at spacecraft level, Thales Alenia
Space (Turin) responsible for the service module, and Astrium (Toulouse)
responsible for the telescope, with in excess of a hundred subcontractors.

This paper makes use of the VLA data with project codes 14A-468, 14B-396,
16A-275 and 17A-073, which has been processed as part of the EveryTHINGS
survey. This paper makes use of the VLA data with legacy ID AU157, which has
been processed in the PHANGS–VLA survey. The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. This publication makes
use of data products from the Wide-field Infrared Survey Explorer, which is a
joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
National Aeronautics and Space Administration.

This paper makes use of the following ALMA data, which have been processed as
part of the PHANGS–ALMA CO(2–1) survey: ADS/JAO.ALMA#2012.1.00650.S,
ADS/JAO.ALMA#2015.1.00782.S, ADS/JAO.ALMA#2018.1.01321.S,
ADS/JAO.ALMA#2018.1.01651.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ.

This research made use of Astropy,888http://www.astropy.org a community-
developed core Python package for Astronomy (Astropy Collaboration et al.,
2013, 2018, 2022). This research has made use of NASA’s Astrophysics Data
System Bibliographic Services. We acknowledge the usage of the HyperLeda
database (http://leda.univ-lyon1.fr). This research has made use of the
NASA/IPAC Extragalactic Database (NED), which is funded by the National
Aeronautics and Space Administration and operated by the California Institute
of Technology.

##  Appendix A Internal variation of D/M

It is questionable whether D/M is a constant within galaxies, even in galaxy
centers. Observations have found internal variations of D/M within galaxies
(Jenkins, 2009; Roman-Duval et al., 2014, 2017; Chiang et al., 2018; Vílchez
et al., 2019). In these studies, people found a higher D/M toward higher
metallicity or gas surface densities. A varying D/M within galaxies is also
expected by several models (Hou et al., 2019; Li et al., 2019; Aoyama et al.,
2020). However, how to characterize the variation of D/M with local condition
is a topic remained unsolved and it is outside the main scope of this work.

To demonstrate how a varying D/M might affect our results, we define a toy
model with D/M increasing toward galaxy centers. For the galaxy disks
($R_{\mathrm{g}}>R_{\mathrm{e}}$), we assume $\mathrm{D/M}=0.4$. This value is
inspired by several recent studies, e.g. $\sim 0.5$ in Draine et al. (2014),
$0.5\pm 0.1$ in Clark et al. (2016), $0.4\pm 0.2$ in Clark et al. (2019), and
$0.46^{+0.12}_{-0.06}$ in Chiang et al. (2021). For the very center of the
galaxies ($R_{\mathrm{g}}=0$), we assume an efficient dust growth, i.e. all
refractory elements are completely depleted and gaseous elements (e.g. oxygen
and nitrogen) are partially depleted, and adopted $\mathrm{D/M}=0.7$ from
Feldmann (2015). In $R_{\mathrm{e}}\geq R_{\mathrm{g}}\geq 0$, we assume a
smooth transition, that is:

$\mathrm{D/M}=\left\\{\begin{array}[]{ll}0.7-0.3\times
R_{\mathrm{g}}/R_{\mathrm{e}}&,~{}R_{\mathrm{g}}\leq R_{\mathrm{e}}\\\
0.4&,~{}R_{\mathrm{g}}>R_{\mathrm{e}}\end{array}\right..$ (A1)

With this toy model, we find $a=-0.72\pm 0.1$ and $-0.50\pm 0.08$ for CO
$(2{\scalebox{0.75}[1.0]{$-$}}1)$ and CO $(1{\scalebox{0.75}[1.0]{$-$}}0)$,
respectively. These indices are steeper than our fiducial case, i.e. constant
D/M, indicating that the $\alpha_{\rm CO}$-to-$\Sigma_{\star}$ relation
observed in Section 5 is not caused by the variation in dust properties.
Although we do expect an internal variation of D/M, the variation in the
galaxy center predicted in simulations is more gentle than our toy model
(Romano et al., 2022; Choban et al., 2022). Thus the calculation with this toy
model should be interpreted as an extreme case. The constant D/M case and the
toy model case should sandwich the actual indices.

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