# Which Molecular Cloud Structures Are Bound?

Neal J. Evans II  Department of Astronomy, The University of Texas at Austin,
2515 Speedway, Stop C1400, Austin, Texas 78712-1205, U.S.A.  Korea Astronomy
and Space Science Institute, 776 Daedeokdaero, Daejeon 305-348, Korea
Humanitas College, Global Campus, Kyung Hee University, Yongin-shi 17104,
Korea  Mark Heyer  Department of Astronomy, University of Massachusetts,
Amherst, MA 01003, USA  Marc-Antoine Miville-Deschênes  Université Paris
Saclay and Université de Paris, CEA, CNRS, AIM, F-91190 Gif-sur-Yvette, France
Quang Nguyen-Luong  The American University of Paris, 2bis, Passage Landrieu
75007 Paris, France  Manuel Merello  Departamento de Astronomía, Universidad
de Chile, Casilla 36-D, Santiago, Chile

###### Abstract

We analyze surveys of molecular cloud structures defined by tracers ranging
from CO $J=1\rightarrow 0$ through 13CO $J=1\rightarrow 0$ to dust emission
together with NH3 data. The mean value of the virial parameter and the
fraction of mass in bound structures depends on the method used to identify
structures. Generally, the virial parameter decreases and the fraction of mass
in bound structures increases with the effective density of the tracer, the
surface density and mass of the structures, and the distance from the center
of a galaxy. For the most complete surveys of structures defined by CO
$J=1\rightarrow 0$, the fraction of mass that is in bound structures is 0.19.
substantial alleviation of the fundamental problem of slow star formation. If
only clouds found to be bound are counted and they are assumed to collapse in
a free-fall time at their mean cloud density, the sum over all clouds in a
complete survey of the Galaxy yields a predicted star formation rate of 46 M⊙
yr-1, a factor of 6.5 less than if all clouds are bound.

interstellar medium, molecular clouds, star formation

††software: astropy (Astropy Collaboration et al. 2013, 2018), GILDAS (Pety
2005; Gildas Team 2013)

##  1 Introduction

The fundamental problem of star formation was first stated by Zuckerman &
Palmer (1974) and Zuckerman & Evans (1974). Simply stated, a theoretical
estimate of the star formation rate in the Milky Way exceeded by a factor of
about 100 the rate averaged over the last Gyr or so, as inferred from
observations. Those early estimates were based on preliminary surveys of the
Galaxy in CO and estimates of the average star formation rate, but the
situation has only worsened after nearly a half century of improved surveys.
Heyer & Dame (2015) estimated a mass of $(1.0\pm 0.3)\times 10^{9}$ M⊙ for the
total molecular gas in the Galaxy. The free-fall time,

$\mbox{$t_{\rm
ff}$}=\bigg{(}\frac{3\pi}{32G\rho}\bigg{)}^{0.5}=3.34\mbox{${}\times
10^{7}$}n^{-0.5}{\rm yr}=\\\ 1.66\mbox{${}\times 10^{7}$}r_{\rm
pc}^{1.5}M_{\mbox{M${}_{\odot}$}}^{-0.5}{\rm yr},$ (1)

where $n$ is the particle density , related to $\rho$ by the mean mass per
particle of 2.37 (Kauffmann et al. 2008). For a characteristic density of 100
cm-3, $\mbox{$t_{\rm ff}$}=3.34\mbox{${}\times 10^{6}$}$ yr, predicting a star
formation rate of 300 M⊙ yr-1. In contrast, observational data indicate much
lower rates. Counting young stars from the GLIMPSE (Benjamin et al. 2003;
Churchwell et al. 2009) survey, Robitaille & Whitney (2010) constrained the
star formation rate to 0.68-1.45 M⊙ yr-1, averaged over the lifetime of
infrared excesses around young stars, roughly the last few Myr. Using a wider
range of indicators, Chomiuk & Povich (2011) derived a star formation rate of
$1.9\pm 0.4$ M⊙ yr-1, which we adopt. The discrepancy has, if anything,
worsened with further research to a factor of 160.

This problem arises because the molecular clouds identified by CO emission are
assumed to be gravitationally bound and collapsing on a free-fall timescale.
This problem is properly called the slowness of star formation (Krumholz et
al. 2014; Padoan et al. 2014) and parameterized by the efficiency per free-
fall time, $\epsilon_{\rm ff}$. which is taken to be 0.01, reflecting the
factor of 100 problem identified by the first references, but the latest
numbers suggest $\mbox{$\epsilon_{\rm ff}$}=0.006$. Another expression of the
slowness is that the depletion time, given by the molecular mass divided by
the star formation rate, is $\mbox{$t_{\rm dep}$}=5\mbox{${}\times 10^{8}$}$
yr, much larger than $t_{\rm ff}$. The depletion time is even longer in other
galaxies, $\mbox{$t_{\rm dep}$}\approx 2$ Gyr (Bigiel et al. 2008).

A somewhat distinct problem is the final efficiency of star formation,
$\epsilon_{\rm sf}$, defined by

${\mbox{$\epsilon_{\rm sf}$}=M_{*}/M_{\rm gas},}$ (2)

where $M_{*}$ is the mass in stars at the end of the event and $M_{\rm gas}$
is the mass of gas at the beginning. While we cannot measure these for an
individual cloud, star clusters that have formed reasonably recently (i.e.,
leaving aside globular clusters) are much less massive than the mass in
clouds. We can compare the mass function of molecular clouds to that of
clusters. The mass functions, expressed per logarithmic unit of mass are
expressed as

$\frac{dN}{d\log M}\propto(M/M_{\rm u,x})^{\Gamma_{\rm x}}$ (3)

where $M$ is the mass of entity $x$, up to a maximum mass $M_{\rm u,x}$. For
Milky Way OB assocations and clusters, $\Gamma_{\rm cluster}$ is $-1.0$ and
$M_{\rm u,cluster}$ is about 6${}\times 10^{4}$ M⊙ (Massey et al. 1995; McKee
& Williams 1997; Portegies Zwart et al. 2010). For molecular clouds, the
values are $\Gamma_{\rm cloud}=-0.6$ and $M_{\rm u,cloud}=6\mbox{${}\times
10^{6}$}$ M⊙ (Williams & McKee 1997). See also Rosolowsky (2005), who give
results in terms of $\gamma=\Gamma_{\rm cloud}+1$, with our definition of
$\Gamma_{\rm cloud}$. The ratio of upper mass limits suggests that
$\mbox{$\epsilon_{\rm sf}$}=0.01$, but the differences in the slopes suggests
that molecular clouds are not the immediate precursors of clusters and
associations. Dense clumps, in contrast, have a similar slope and upper mass
limit to those of clusters and associations. Shirley et al. (2003) found
$\Gamma_{\rm clump}=-1.0$ for dense clumps defined by strong CS
$J=5\rightarrow 4$ emission. Elia et al. (2017) found $\Gamma_{\rm
clump}=-1.05\pm 0.08$ in a sample of $10^{4}$ clumps defined from
submillimeter continuum emission. The upper mass limit for clumps, $M_{\rm
u,clump}$ is about 1${}\times 10^{5}$ M⊙ based on various studies (Shirley et
al. 2003; Svoboda et al. 2016; Urquhart et al. 2018).

Much theoretical ingenuity has been expended to solve the problem of slow star
formation. Magnetic fields may help to “support” clouds, and turbulence can
slow collapse. However, numerical simulations generally found that supersonic
turbulence should decay rapidly, with or without magnetic fields (Stone et al.
1998; Mac Low 1999). Also, turbulence appears to be driven on scales larger
than the cloud, rather than from within the cloud (Larson 1981; Heyer & Brunt
2004; Brunt et al. 2009).

The kind of solution that invokes turbulence to support a cloud has the
problem of the inherent instability of a gravitationally bound entity. It is
hard to imagine an individual cloud avoiding gravitational collapse for a
depletion time that is over 200 times the free-fall time. Indeed, Heyer & Dame
(2015) placed a strong upper limit on cloud lifetime ($t_{\rm cloud}$) of
$10^{8}$ yr and preferred a lifetime of $3\mbox{${}\times 10^{7}$}$ yr, based
on Kawamura et al. (2009), closer to 10 times the canonical value for $t_{\rm
ff}$. Clearly most of the mass in clouds must disperse after a star forming
event. The low value of $\epsilon_{\rm sf}$ suggests that 99.4% is dispersed.
Another approach has been to question the use of the current density to
estimate the free-fall time (Vázquez-Semadeni et al. 2019). A lower initial
density would slow the initial rate of collapse, but the conflict with the
prediction based on current properties of molecular clouds remains.

Recent theoretical work has concentrated on feedback to remove most of the gas
before it forms stars For example, Kim et al. (2018) compared the
effectiveness of radiation pressure and photoionization in disrupting clouds.
Even though they started with marginally bound clouds, the final efficiencies,
$\epsilon_{\rm sf}$, increased strongly with increasing initial cloud surface
density ($\Sigma_{\rm M}$), exceeding 5% by initial $\mbox{$\Sigma_{\rm
M}$}=20$ M⊙ pc-2 (figure 7 of their paper). Bound clouds overproduce stars
even at very low initial $\Sigma_{\rm M}$, even with strong radiative
feedback. Another interesting feature of figure 7 of Kim et al. (2018) is that
$\epsilon_{\rm sf}$ reaches about 0.5 at $\mbox{$\Sigma_{\rm
M}$}\sim\mbox{$10^{3}$}$ M⊙ pc-2, similar to the surface density of the dense
clumps studied by Shirley et al. (2003). If these were the bound entities, the
ratio of maximum cluster mass to initial gas mass would come out about right.
Simulations of unbound clouds look more promising. Kim et al. (2021) have
performed radiative magnetohydrodynamical simulations (RMHD) for clouds with
starting $\mbox{$\Sigma_{\rm M}$}=80$ M⊙ pc-2, reasonably appropriate to the
structures traced by CO. The star formation is limited by feedback from
massive stars. They found that both $\epsilon_{\rm ff}$ and $\epsilon_{\rm
sf}$ decrease with increasing virial parameter ($\alpha_{\rm vir,0}$),
reaching values similar to the observed values for $\mbox{$\alpha_{\rm
vir,0}$}\sim 5$.

In the current situation, it seems prudent to reexamine the assumptions
underlying the fundamental problem: that all molecular clouds are
gravitationally bound and that they are collapsing at free-fall, with the
free-fall time calculated from the mean cloud density of 100 cm-3. The
question of whether the cloud is bound is usually addressed by calculating the
virial parameter, $\alpha_{\rm vir}$ (Bertoldi & McKee 1992). The definition
of this parameter varies. The most common definition assumes virial
equilibrium. Ignoring surface terms and magnetic fields,

$\mbox{$\alpha_{\rm vir}$}=\frac{2E_{kin}}{|E_{g}|},$ (4)

where $E_{g}$ is the gravitational potential energy and $E_{kin}$ is the
kinetic energy. With this definition, the dividing line between bound and
unbound is $\mbox{$\alpha_{\rm vir}$}=2$, where $E_{kin}=|E_{g}|$. Surface
pressure from the surrounding medium would increase $\alpha_{\rm vir}$ while
magnetic fields would decrease $\alpha_{\rm vir}$ (Tan et al. 2013). Because
we cannot measure them, we neglect the largely unknown pressure and magnetic
terms. Then a cloud with $E_{kin}>E_{g}$ would tend to expand or dissipate, so
we take $\mbox{$\alpha_{\rm vir}$}=2$ as our critical value to define a cloud
as bound if $\mbox{$\alpha_{\rm vir}$}<2$ and unbound if $\mbox{$\alpha_{\rm
vir}$}>2$.

The calculation of $E_{g}$ depends on cloud shape and structure (Bertoldi &
McKee 1992; Kauffmann et al. 2013) but these effects can be considered as
correction factors for the expression for a uniform density sphere:

$E_{g}=\frac{-3GM^{2}}{5R}.$ (5)

where M and R are the cloud mass and radius. The kinetic energy is estimated
from

$E_{kin}=\frac{3\mbox{$\sigma_{v}$}^{2}M}{2}$ (6)

where $\sigma_{v}$ is the one-dimensional velocity dispersion averaged over
the cloud.

Unbound clouds may of course contain bound structures. Measuring $\alpha_{\rm
vir}$ in the complex structures of the ISM is more problematic, as surface
terms are likely to be important (Mao et al. 2020). Kim et al. (2021) compared
the virial parameters from their model clouds to those that would be derived
from observations. They found that the methods used by observers are more
likely to overestimate the boundedness of the cloud, but generally by factors
of 2 or less. They caution that neglect of tidal effects in crowded regions
would make the overestimation worse. This effect was indeed apparent in
larger-scale simulations (Mao et al. 2020), which found that many regions that
were apparently bound, based on the criteria that $\mbox{$\alpha_{\rm
vir}$}<2$ were in fact unbound. In particular, the boundedness of large
massive structures was more likely to be overestimated using the simpler
estimations available to observers.

Given our simple approach to the virial parameter, ignoring external pressure,
magnetic fields, and tides, we use the term “bound” as shorthand for
gravitationally bound. A cloud could be unbound in our definition, but
contained or “bound” by external pressure. Our focus is on whether gravity or
turbulence is dominant. If the results of the simulations discussed above
apply, our criteria for boundedness will be more generous than reality. We
calculate virial parameters from

$\mbox{$\alpha_{\rm
vir}$}=\frac{2E_{kin}}{|E_{g}|}=1160\mbox{$\sigma_{v}$}^{2}R_{pc}/M_{\mbox{M${}_{\odot}$}}=\\\
209\mbox{$\Delta v$}^{2}R_{pc}/M_{\mbox{M${}_{\odot}$}},$ (7)

where $M_{\mbox{M${}_{\odot}$}}$ is the actual mass of the cloud in solar
masses.

Estimating the actual mass of the cloud can be done in a number of ways, but
the original source of the fundamental problem relied on observations of the
lowest rotational transition of CO. The problem is that this line is usually
very optically thick. Many words have been expended explaining why this can
work, including an implicit assumption that the clouds are in virial
equilibrium (e.g., Bolatto et al. 2013). If this were the only argument, the
whole discussion would be circular, but we will take the CO-derived masses as
independent of the assumption of virial equilibrium. The correlation of CO
emission with dust extinction via the $X({\rm CO})$ factor provides an
independent basis for its use in determining column density (Dickman 1978;
Pineda et al. 2010; Heiderman et al. 2010). While column density may be
underestimated at high column density and overestimated at low column density,
both by factors of 2 or more, Bolatto et al. (2013) argued that variations in
$X$(CO) average out, so $L$(CO) may reflect mass better than the intensity
traces column density for a particular line of sight. The relation is
$\mbox{$M_{\rm CO}$}=\mbox{$\alpha_{\rm CO}$}\mbox{$L_{\rm CO}$}$. We follow
Bolatto et al. (2013) in taking $\alpha_{\rm CO}$ of 4.3 M⊙(K km s-1 pc2)-1.

Samples using 13CO $J=1\rightarrow 0$ to get mass usually combine the 13CO
data with CO data to derive the optical depth and then column density of 13CO,
$N_{\rm 13}$, assuming that the excitation temperatures of CO and 13CO are
equal and that the partition function can be approximated by assuming that the
excitation temperature applies to all levels (commonly referred to as the
“LTE” assumption). The excitation temperature can be found from the CO
brightness temperature. Because we use CO data with low spatial resolution, we
instead assume an excitation temperature of 8 K for all lines of sight (Roman-
Duval et al. 2010). We follow the description by Ripple et al. (2013) but note
the error in their equation 4, in which the exponential factor should be in
the denominator. Conversion to the column density of H2 follows

$N({\rm H_{2}})=N_{\rm 13}\Bigl{[}\frac{{\rm H_{2}}}{\rm
CO}\Bigr{]}\Bigl{[}\frac{\rm CO}{\mbox{${}^{13}$CO}}\Bigr{]}.$ (8)

We have used the latest number for H2/CO of 6000 (Lacy et al. 2017). Because
the new measurement directly relates the abundance of H2, not including
helium, to that of CO through infrared absorption observations, we use the
mean molecular weight of $\mbox{$\mu_{\rm H_{2}}$}m_{\rm H}$, with
$\mbox{$\mu_{\rm H_{2}}$}=2.8$ to convert to mass (Kauffmann et al. 2008). We
have corrected data to use these values. Unless otherwise noted, when
analyzing the 13CO $J=1\rightarrow 0$ data, we account for a Galactic gradient
in ${\rm CO}/{\rm{}^{13}CO}$. We apply the elemental isotope gradient in the
Galaxy most recently derived by Jacob et al. (2020),

$\frac{{}^{12}{\rm C}}{{}^{13}{\rm C}}=5.87\mbox{$R_{\rm gal}$}+13.25,$ (9)

where $R_{\rm gal}$ is the distance of the cloud from the Galactic Center in
kpc.

##  2 Samples

The sets of data used in this paper are listed in Table 1. We consider
structures defined by CO emission, 13CO emission, and dust emission,
supplemented by molecular line data. From the catalogs in the papers, we
extracted the information on size, mass, mass surface density ($\Sigma_{\rm
M}$) and virial parameter ($\alpha_{\rm vir}$), or the parameters needed to
calculate them. In most cases, some pruning of the catalog entries was
necessary. The cuts used are explained below.

###  2.1 Structures Defined by CO $J=1\rightarrow 0$ Emission

The samples in this category rely on masses measured from the luminosity of
the $J=1\rightarrow 0$ transition of CO. The conversion from luminosity to
mass ($\alpha_{\rm CO}$) differed somewhat among the studies; to standardize
them, the value of $M_{\rm CO}$ in the Milky Way samples was calculated using
$\mbox{$\alpha_{\rm CO}$}=4.3$ M⊙(K km s-1 pc2)-1 (Bolatto et al. 2013).

Table 1: Sample List

Name | Tracer | Reference | Notes
---|---|---|---
CO-Rice | CO $J=1\rightarrow 0$ | Rice et al. (2016) | 1
CO-MD | CO $J=1\rightarrow 0$ | Miville-Deschênes et al. (2017) | 2
CO-CMZ | CO $J=1\rightarrow 0$ | Oka et al. (2001) | 3
CO-OGS | CO $J=1\rightarrow 0$ | Heyer et al. (2001) | 4
CO-Sun | CO $J=1\rightarrow 0$ | Sun et al. (2020) | 5
13CO-GRS |  13CO $J=1\rightarrow 0$ | Roman-Duval et al. (2010) | 6
13CO-EXFC55-100 |  13CO $J=1\rightarrow 0$ | Roman-Duval et al. (2016) | 7
13CO-EXFC135-195 |  13CO $J=1\rightarrow 0$ | Roman-Duval et al. (2016) | 8
13CO-SEDIGISM |  13CO $J=2\rightarrow 1$ | Duarte-Cabral et al. (2021) | 9
Herschel | submillimeter$+$ NH3 | Merello et al. (2019) | 10

Notes: 1. Recovers 25% of emission in the survey by Dame et al. (2001).
2\. Recovers 98% of emission in the survey by Dame et al. (2001).
3\. Attempted to restrict to CMZ.
4\. Outer Galaxy Survey: $\mbox{$\langle\mbox{$R_{\rm gal}$}\rangle$}=11.8\pm
1.3$
5\. Survey of 66 galaxies.
6\. Galactic Ring Survey: $\mbox{$\langle\mbox{$R_{\rm gal}$}\rangle$}=5.2\pm
1.0$
7\. Exeter-FCRAO Survey for $55<l<100$: $\mbox{$\langle\mbox{$R_{\rm
gal}$}\rangle$}=10.3\pm 1.9$
8\. Exeter-FCRAO Survey for $135<l<195$: $\mbox{$\langle\mbox{$R_{\rm
gal}$}\rangle$}=12.9\pm 2.1$
9\. Covers $R_{\rm gal}$ from 0.6 to 15.7 kpc: $\mbox{$\langle\mbox{$R_{\rm
gal}$}\rangle$}=5.5\pm 1.8$
10\. Based on the Herschel Infrared Galactic Plane Survey

New catalogs of clouds based on decomposition of the CO surveys are now
available (Rice et al. 2016; Miville-Deschênes et al. 2017). These are
particularly interesting as they used the data from Dame et al. (2001) that
was featured by Heyer & Dame (2015) but analyzed it in two different ways.
These two studies, labeled CO-Rice and CO-MD in Table 1, take different
approaches to breaking the large scale CO emission in the Galaxy into clouds,
and we will discuss the effect of those differences in §4.1.

Rice et al. (2016) used a dendrogram analysis to identify 1064 clouds. They
recovered 2.5${}\times 10^{8}$ M⊙, about 25% of the total molecular mass of
the Galaxy. Table 3 of Rice et al. (2016) provides a radius, $\sigma_{v}$, and
$M_{\rm CO}$, the mass determined from the CO luminosity, which we use
directly. The mass surface density was determined from $M_{\rm CO}$ and the
radius. No mass or size information was given for clouds without clear
resolution of the kinematic distance ambiguity, and one cloud was assigned
zero size and mass in Table 3 of Rice et al. (2016); after removing these,
1037 clouds remained for our analysis.

Miville-Deschênes et al. (2017) used a Gaussian decomposition and hierarchical
cluster analysis to assign essentially all the CO emission in the Dame survey
to 8107 clouds comprising $1.2\mbox{${}\times 10^{9}$}$ M⊙. Their online table
provides radii and masses for both near and far kinematic distances, along
with a code for resolution of the kinematic distance ambiguity, which we use
to select the best radius and mass. Miville-Deschênes et al. (2017) calculated
a radius from the two projected radii by $r=(r_{\rm max}r_{\rm min}r_{\rm
min})^{1/3}$, arguing that the depth was more likely equal to the smaller size
projected on the sky. They also provide $\sigma_{v}$, which we use in
calculating $\alpha_{\rm vir}$. The masses were based on summing column
densities, using $X_{\rm CO}=2\mbox{${}\times 10^{20}$}$ cm-2 (K km s-1)-1,
which corresponds to $\mbox{$\alpha_{\rm CO}$}=4.3$ M⊙(K km s-1 pc2)-1
(Bolatto et al. 2013). The catalog contains a small number of clouds with
nominal Galactocentric radii greater than 30 kpc and a large number of low-
mass objects with very high $\alpha_{\rm vir}$. To select against these, we
require that $R_{\rm gal}<30$ kpc, $\mbox{$M_{\rm CO}$}>1$ M⊙ and
$\mbox{$\alpha_{\rm vir}$}<\mbox{$\alpha_{\rm max}$}$, where $\alpha_{\rm
max}$ is an arbitrarily chosen maximum value. Entry 2 in Table 2 required
$\mbox{$\alpha_{\rm vir}$}<100$, while entry 3 used $\mbox{$\alpha_{\rm
vir}$}<20$.

The sample labeled CO-CMZ in Table 1 is from Oka et al. (2001) and focuses on
clouds in the Central Molecular Zone (CMZ) using data obtained with the
Nobeyama Radio Observatory 45-m telescope. Clouds were defined by
topologically closed surfaces in $(l,b,v)$ space. Three different thresholds
were used and some clouds appear multiple times in the catalog, providing a
range of surface densities. The dispersions in all three coordinates were used
to define sizes and linewidths. A distance of 8.5 kpc was used for all clouds.
They accounted for truncation by the threshold by comparing to simulated
clouds and provided corrected values for sizes, velocity dispersions, and CO
luminosities. These are what we use. To calculate the mass surface density, we
take the size as a radius. They also computed virial masses, but we do not use
these. We eliminate clouds determined to be outside the CMZ (labeled D in
their table 1) or confused (labeled with C in their table 1). The use of
$\mbox{$\alpha_{\rm CO}$}=4.3$ M⊙(K km s-1 pc2)-1 in the CMZ is dubious. Oka
et al. (1998) suggested a value 10 times smaller. Elemental abundances
certainly are higher in the CMZ and a correction for that might be
appropriate, as suggested by Sun et al. (2020) for other galaxies.

The sample labeled CO-OGS in Table 1 is a sample of clouds in the outer Galaxy
mapped by the FCRAO 14-m telescope (Heyer et al. 1998) and tabulated by Heyer
et al. (2001). The Table provides $L_{\rm CO}$, $\Delta v$ (FWHM), and
diameter of the structure. Heyer et al. (2001) used $\mbox{$\alpha_{\rm
CO}$}=4.1$ to convert to mass, but we changed to 4.3 for consistency. We
convert diameter to radius for consistency. The linewidth used to calculate
$M_{\rm vir}$ was that of a spectrum averaged over the cloud. These were
almost always larger than an alternative estimate using the dispersion of line
centroids over the cloud, indicating that the clouds are generally not
dominated by systematic motions such as shear. The catalog contains 10156
structures, but many are local. To select outer Galaxy sources, Heyer et al.
(2001) required $\mbox{$v_{LSR}$}<-25$ km s-1. To select against small
structures, they also required $\mbox{$M_{\rm CO}$}>2.5\mbox{${}\times
10^{3}$}$ M⊙. We give values for this cut, but we relax it to $\mbox{$M_{\rm
CO}$}>10$ M⊙ for the figures and most analysis because we are more interested
in completeness in structures than in defining a mass function.

###  2.2 Structures Defined by CO $J=2\rightarrow 1$ Emission in Other
Galaxies

Studies of other galaxies are beginning to provide similar information without
the line-of-sight confusion of Milky Way studies. For other galaxies, we use
the remarkable compilation by Sun et al. (2020) of the PHANGS-ALMA study of
the CO $J=2\rightarrow 1$ transition, labeled CO-Sun in Table 1, which is part
of a larger program including HST observations (Lee et al. 2021). They
collected data on over $10^{5}$ sightlines toward 66 galaxies, resampled to a
uniform spatial resolution of 150 pc, by far the largest data base available,
albeit with very poor spatial resolution by the standards of studies in our
Galaxy. They used a metallicity dependent value for $\alpha_{\rm CO}$, and we
use their values. Their table 2 provides $\Sigma_{\rm M}$ and $\alpha_{\rm
vir}$. We compute $M_{\rm CO}$ from $\Sigma_{\rm M}$ and the fixed size of 150
pc and use their $\alpha_{\rm vir}$.

###  2.3 Structures Defined by 13CO $J=1\rightarrow 0$ Emission

Roman-Duval et al. (2010) provided the sample defined by 13CO $J=1\rightarrow
0$ emission from the Galactic Ring Survey (GRS) by Jackson et al. (2006). The
13CO data were combined with the CO data from the University of Massachusetts-
Stony Brook (UMSB) survey (Sanders et al. 1985; Clemens et al. 1986) to
determine optical depth and 13CO column density. The clouds in the Roman-Duval
et al. (2010) study were actually identified by Rathborne et al. (2009), who
presented a detailed description of the method, using CLUMPFIND (Williams et
al. 1994) and a number of tests to be able to identify both bright, compact
structures and fainter, more diffuse structures. The effective lowest
brightness level was 0.2 K, so they should not have selected regions of
abnormally high density.

For the original analysis by Roman-Duval et al. (2010) of the GRS survey, we
consider these to be structures defined by 13CO. Radius, mass, surface
density, and $\alpha_{\rm vir}$ are provided by their table 1, with 749
entries. Requiring $\mbox{$M_{\rm CO}$}>1$ M⊙, $\mbox{$\Sigma_{\rm M}$}>10$ M⊙
pc-2, and $\mbox{$\alpha_{\rm vir}$}<20$ pruned the sample to 737. The
condition on $\alpha_{\rm vir}$ removed a few extreme outliers. 170 clouds in
the sample were not covered by the UMSB CO survey, so the usual method to
determine the column density of 13CO was impractical. If we exclude those, the
results do not differ substantially. Roman-Duval et al. (2010) used a
different definition of $\alpha_{\rm vir}$, assuming a correction for the
density distribution. For consistency, we recalculated $\alpha_{\rm vir}$ from
Equation 7. These values were larger by a factor of 1.3 on average than those
given in their paper when the same assumptions about abundances were made.

For the 13CO $J=1\rightarrow 0$ samples from other parts of the Galaxy, both
13CO and CO $J=1\rightarrow 0$ were observed simultaneously. These are the
Exeter-FCRAO surveys, which are divided into EXFC55-100 and EXFC135-195, but
our analysis excludes the range of longitude where kinematic distances are
unreliable ($165<l<195$), as described by Roman-Duval et al. (2016). For the
EXFC surveys, Roman-Duval et al. (2016) attempted to identify all voxels with
emission to distinguish diffuse (CO only) from “dense” gas (13CO and CO).
Structures were not identified and no catalogs were provided. We have re-
analyzed all the 13CO $J=1\rightarrow 0$ surveys in a uniform way, using the
CO-defined structures from Miville-Deschênes et al. (2017), with the procedure
detailed in the Appendix. Because the masses are based on all the 13CO data
within the CO-defined structures, we refer to these results as 13CO-CO,
followed by the survey name, such as GRS.

The GRS sample is concentrated in the inner Galaxy; the sub-sample that we
analyze has median($R_{\rm gal}$)$=5.2$ kpc, $\mbox{$\langle\mbox{$R_{\rm
gal}$}\rangle$}=5.2\pm 1.0$ kpc. The EXFC55-100 sample is concentrated closer
to the solar circle, with $7.2<\mbox{$R_{\rm gal}$}<14.5$, median($R_{\rm
gal}$)$=10.7$ kpc, $\mbox{$\langle\mbox{$R_{\rm gal}$}\rangle$}=10.3\pm 1.9$
kpc. The EXFC135-195 data is concentrated in the outer Galaxy, with
$10.3<\mbox{$R_{\rm gal}$}<20.4$, median($R_{\rm gal}$)$=12.5$ kpc,
$\mbox{$\langle\mbox{$R_{\rm gal}$}\rangle$}=12.9\pm 2.1$ kpc. They thus
provide convenient samples for examining the effects of $R_{\rm gal}$.

###  2.4 Structures Defined by 13CO $J=2\rightarrow 1$ Emission

The SEDIGISM survey (Duarte-Cabral et al. 2021) has provided a catalog of
structures defined by emission from the $J=2\rightarrow 1$ transition of 13CO.
They used the SCIMES algorithm described by Colombo et al. (2019) to define
the structures. They determined column densities by assuming a constant ratio
of H2 column density to integrated intensity of 13CO $J=2\rightarrow 1$ of
$X(\mbox{${}^{13}$CO}\ \mbox{$J=2\rightarrow 1$})=1\mbox{${}\times
10^{21}$}\mbox{cm${}^{-2}$}(\mbox{K\ \mbox{km s${}^{-1}$}})^{-1}$ and a mean
molecular weight per H2 of 2.8. Because their choice of $X(\mbox{${}^{13}$CO}\
\mbox{$J=2\rightarrow 1$})$ was empirically based (Schuller et al. 2017), we
did no re-scaling. The sample was pruned to remove those with ambiguous
distances, near edges of maps, or smaller than 114 pixels, roughly re-creating
their “science sample.” The masses and virial parameters in their tables were
used directly, but $\alpha_{\rm vir}$ was checked and agreed with a
calculation from their mass, size, and linewidth information.

###  2.5 Structures Defined by Herschel and NH3

Structures identified from the Herschel and ATLASGAL maps of the Milky Way
were presented by Merello et al. (2019). They meet a number of requirements,
but most relevant to us, they have NH3 observations, which are used to obtain
a linewidth from the inversion transition in the $(J,K)=(1,1)$ level and a
temperature from the ratio of (1,1) and (2,2) lines. We use the sample of
1068, called the “Final Sample” by Merello et al. (2019). They provide mass,
$\Delta v$ (FWHM), and radius, from which we compute $\alpha_{\rm vir}$ and
$\Sigma_{\rm M}$. The masses were computed from a fit to the SED, the
resulting optical depth at 300 µm, and an opacity of 0.1 cm2 g-1 of gas plus
dust. The radius was half the FWHM of the emission at 250 µm.

##  3 Results

For each of the samples, four plots are provided: (1) the mean, median, and
standard deviation of $\alpha_{\rm vir}$ versus mass; (2) the mean, median,
and standard deviation of $\alpha_{\rm vir}$ versus surface density
($\Sigma_{\rm M}$); (3) a histogram of $\log\mbox{$\Sigma_{\rm M}$}$; (4) a
histogram of $\log\mbox{$\alpha_{\rm vir}$}$. Statistics of the relevant
properties are summarized in Table 2. For each sample, Table 2 has the median,
mean, and standard deviation of $\alpha_{\rm vir}$ and the median, mean, and
standard deviation of $\Sigma_{\rm M}$. The table also includes the fraction
of the total number of structures that satisfy $\mbox{$\alpha_{\rm vir}$}<2$.
The fraction of the number ($f(N)$), and the fraction of the total mass
($f(M)$) satisfying these conditions are both tabulated.

Table 2: Statistics

Sample | Number | $\alpha_{\rm vir}$ |  $\Sigma_{\rm M}$ (M⊙ pc-2) | $f(N)$ | $f(M)$ | $f_{\rm c}(N)$ | $f_{\rm c}(M)$ | Note
---|---|---|---|---|---|---|---|---
|  | Med | Mean | Std. | Med | Mean | Std. |  |  |  |  |
CO-Rice | 1037 | 2.44 | 3.45 | 3.34 | 21.2 | 38.3 | 43.4 | 0.38 | 0.73 | 0.33 | 0.41 | 1
CO-MD | 7516 | 7.58 | 15.20 | 18.53 | 21.9 | 41.1 | 53.9 | 0.07 | 0.19 | 0.06 | 0.09 | 2
CO-MD | 5776 | 5.49 | 6.91 | 4.70 | 31.4 | 49.4 | 58.1 | 0.09 | 0.19 | 0.08 | 0.10 | 3
CO-CMZ | 110 | 3.26 | 4.45 | 3.46 | 1311.3 | 1703.2 | 1271.0 | 0.29 | 0.71 | 0.00 | 0.00 | 4
CO-OGS | 380 | 2.27 | 2.51 | 1.25 | 29.3 | 35.4 | 20.2 | 0.41 | 0.72 | 0.40 | 0.56 | 5
CO-OGS | 3714 | 6.15 | 7.22 | 4.79 | 17.7 | 21.0 | 11.7 | 0.06 | 0.58 | 0.06 | 0.46 | 6
CO-Sun | 102788 | 3.47 | 4.57 | 4.10 | 21.6 | 49.9 | 144.3 | 0.21 | 0.35 | 0.18 | 0.15 | 7
13CO(1-0) | 737 | 1.03 | 1.95 | 2.76 | 82.0 | 85.5 | 39.8 | 0.72 | 0.95 | 0.58 | 0.55 | 8
13CO-CO-GRS | 289 | 2.86 | 3.83 | 3.02 | 80.7 | 103.7 | 84.0 | 0.31 | 0.46 | 0.17 | 0.19 | 9
13CO-CO-EXFC55-100 | 105 | 0.99 | 1.92 | 2.24 | 25.9 | 36.1 | 32.8 | 0.62 | 0.73 | 0.58 | 0.70 | 10
13CO-CO-EXFC135-195 | 98 | 1.37 | 2.30 | 3.00 | 21.1 | 24.2 | 13.0 | 0.63 | 0.75 | 0.63 | 0.75 | 11
13CO(2-1) | 6658 | 1.25 | 1.94 | 2.66 | 74.7 | 87.7 | 49.0 | 0.73 | 0.79 | 0.62 | 0.41 | 12
Herschel | 1067 | 0.29 | 0.58 | 1.08 | 2497.7 | 3249.8 | 3032.2 | 0.96 | 1.00 | 0.00 | 0.00 | 13

Notes: 1. Data from Rice et al. (2016). Clouds with ambiguous distance
assignments and one cloud with zero mass were eliminated.
2\. Data from Miville-Deschênes et al. (2017) using criteria $\mbox{$M_{\rm
CO}$}>1$ M⊙, $R_{\rm gal}<30$ kpc, and $\mbox{$\alpha_{\rm vir}$}<100.$
3\. Data from Miville-Deschênes et al. (2017) using criteria $\mbox{$M_{\rm
CO}$}>1$ M⊙, $R_{\rm gal}<30$ kpc, and $\mbox{$\alpha_{\rm vir}$}<20.$
4\. Data from Oka et al. (2001), considering only clouds in CMZ
5\. Data from Heyer et al. (2001), using criterion $\mbox{$M_{\rm
CO}$}>2.5\mbox{${}\times 10^{3}$}$ M⊙
6\. Data from Heyer et al. (2001), using criterion $\mbox{$M_{\rm
CO}$}>1.0\mbox{${}\times 10^{1}$}$ M⊙
7\. Data from Sun et al. (2020)
8\. GRS Data from Roman-Duval et al. (2010), updated abundance, some selection
9\. Re-analysis of GRS 13CO data within CO clouds
10\. Re-analysis of EXFC 55-100 13CO data within CO clouds
11\. Re-analysis of EXFC 135-195 13CO data within CO clouds
12\. Data from Duarte-Cabral et al. (2021)
13\. Data from Merello et al. (2019), using criteria $\mbox{$M_{\rm dust}$}>1$
M⊙

We will compare the criterion for the virial parameter to the surface density
criterion for star formation. In local clouds, star formation is strongly
concentrated in regions with $\mbox{$\Sigma_{\rm M}$}\geq 120$ M⊙ pc-2
(Heiderman et al. 2010; Lada et al. 2010, 2012; Evans et al. 2014). For
simplicity, we refer to structures with $\mbox{$\Sigma_{\rm M}$}<120$ M⊙ pc-2
as clouds and those with $\mbox{$\Sigma_{\rm M}$}\geq 120$ M⊙ pc-2 as clumps
and the plots of $\alpha_{\rm vir}$ versus $\Sigma_{\rm M}$ include a line
demarcating that boundary. We emphasize that this definition of “clump” does
not include constraints often used to define clumps, such as size information,
and we use the terminology only for simplicity. This usage of “clump” is also
distinguished from “dense clumps” as identified in lines of much higher
effective density, as in the Herschel sample or samples studied by Wu et al.
(2010) and others. We use bins of 0.3 in log $\Sigma_{\rm M}$, on either side
of 2.08, corresponding to $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2. Similarly, we
summarize the boundedness with labels of “bound” or “unbound” on either side
of a line at $\mbox{$\alpha_{\rm vir}$}=2$ in the plots. Finally, the fraction
of the number ($f_{\rm c}(N)$) and mass ($f_{\rm c}(M)$) that satisfy both
$\mbox{$\alpha_{\rm vir}$}<2$ and $\mbox{$\Sigma_{\rm M}$}<120$ M⊙ pc-2 are
tabulated, the subscript (c) indicating that they apply only to structures we
have called clouds.

###  3.1 Structures Defined by CO $J=1\rightarrow 0$ Emission

Figure 1:  (Upper Left) The mean and standard deviation of the virial ratio is
plotted versus the logarithm of the clump mass, for the catalog of Rice et al.
(2016). Median values are plotted in a magenta dashed line. (Upper Right) The
same quantities are plotted versus the mass surface density. The horizontal
line at $\mbox{$\alpha_{\rm vir}$}=2$ demarcates nominally unbound clouds
above the line from nominally bound clouds below the line. The vertical line
in the right panel indicates $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2. (Lower
left) The histogram of values of log $\alpha_{\rm vir}$, with a vertical red
line at $\mbox{$\alpha_{\rm vir}$}=2.0$. The number of structures in each
category are given in parentheses. (Lower right) The histogram of values of
log $\Sigma_{\rm M}$, with a vertical red line at $\mbox{$\Sigma_{\rm
M}$}=120$ M⊙ pc-2.

We plot $\alpha_{\rm vir}$ calculated from the data in Rice et al. (2016) in
Figure 1. The mean $\alpha_{\rm vir}$ declines with mass, so that
$\mbox{$\langle\mbox{$\alpha_{\rm vir}$}\rangle$}<2$ for log$\mbox{$M_{\rm
CO}$}>5.7$. The mean $\alpha_{\rm vir}$ also declines with $\Sigma_{\rm M}$,
so that $\mbox{$\langle\mbox{$\alpha_{\rm vir}$}\rangle$}<2$ for
log$\mbox{$\Sigma_{\rm M}$}>1.7$. The histograms indicate that most structures
are clouds and most are

Figure 2:  (Upper Left) The mean and standard deviation of the virial ratio is
plotted versus the logarithm of the clump mass, for the catalog of Miville-
Deschênes et al. (2017) after selecting only those catalog entries with
$\mbox{$\alpha_{\rm vir}$}<100$ and $\mbox{$M_{\rm CO}$}>1$ M⊙. Median values
are plotted in a magenta dashed line. (Upper Right) The same quantities are
plotted versus the mass surface density. The horizontal line at
$\mbox{$\alpha_{\rm vir}$}=2$ demarcates nominally unbound clouds above the
line from nominally bound clouds below the line. The vertical line in the
right panel indicates $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2. (Lower left) The
histogram of values of log $\alpha_{\rm vir}$, with a vertical red line at
$\mbox{$\alpha_{\rm vir}$}=2.0$. The number of structures in each category are
given in parentheses. (Lower right) The histogram of values of log
$\Sigma_{\rm M}$, with a vertical red line at $\mbox{$\Sigma_{\rm M}$}=120$ M⊙
pc-2.

The catalog of Miville-Deschênes et al. (2017), as noted in §2, contains some
outliers with very high nominal $\alpha_{\rm vir}$, so we exclude those
greater than a maxiumum value, $\alpha_{\rm max}$. Entries in Table 2 are
given for two choices, $\mbox{$\alpha_{\rm max}$}=100$ and $\mbox{$\alpha_{\rm
max}$}=20$. The mean and standard deviation are plotted versus mass in Figure
2. As we saw with the catalog of Rice et al. (2016), the
$\langle\mbox{$\alpha_{\rm vir}$}\rangle$ values decline with $M_{\rm CO}$. In
this case, they reach $\mbox{$\langle\mbox{$\alpha_{\rm vir}$}\rangle$}=2$
only for the most massive bin. The mean and median values of $\alpha_{\rm
vir}$ decline with $\Sigma_{\rm M}$, but never drop below $\mbox{$\alpha_{\rm
vir}$}=2$. The histograms indicate that the vast majority of structures are
unbound clouds. The fraction of the mass that is bound is relatively
insensitive to the choice of $\alpha_{\rm max}$, at about 0.19 for all
structures. The fact that these results differ subtantially from those of Rice
et al. (2016) will be discussed in §4.1.

Figure 3:  The catalog of Oka et al. (2001) is plotted. (Upper Left) The mean
and standard deviation of the virial ratio is plotted versus the logarithm of
the clump mass. Median values are plotted in a magenta dashed line. (Upper
Right) The same quantities are plotted versus the mass surface density. The
horizontal line at $\mbox{$\alpha_{\rm vir}$}=2$ demarcates nominally unbound
clouds above the line from nominally bound clouds below the line. The vertical
line indicates $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2. (Lower left) The
histogram of values of log $\alpha_{\rm vir}$, with a vertical red line at
$\mbox{$\alpha_{\rm vir}$}=2.0$. The number of structures in each category are
given in parentheses. (Lower right) The histogram of values of log
$\Sigma_{\rm M}$, with a vertical red line at $\mbox{$\Sigma_{\rm M}$}=120$ M⊙
pc-2.

Figure 3 shows that the structures in the CMZ in the catalog of Oka et al.
(2001) are generally unbound even though they almost all have
$\mbox{$\Sigma_{\rm M}$}>120$ M⊙ pc-2, satisfying the local criterion for star
formation. Clearly, the definition of a clump based on the local surface
density criterion for active star formation does not work in the CMZ. While
most structures are unbound, the tendency for higher mass structures to be
bound results in a relatively high $\mbox{$f(M)$}=0.71$, with the caveat that
these masses may well be overestimated (§2).

Figure 4:  The catalog of Heyer et al. (2001) is plotted. The masses were
limited to $\mbox{$M_{\rm CO}$}>2.5\mbox{${}\times 10^{3}$}$ M⊙ for
completeness and $\mbox{$v_{LSR}$}<-25$ km s-1 to avoid distance ambiguity.
(Upper Left) The mean and standard deviation of the virial ratio is plotted
versus the logarithm of the clump mass. Median values are plotted in a magenta
dashed line. (Upper Right) The same quantities are plotted versus the mass
surface density. The horizontal line at $\mbox{$\alpha_{\rm vir}$}=2$
demarcates nominally unbound clouds above the line from nominally bound clouds
below the line. The vertical line indicates $\mbox{$\Sigma_{\rm M}$}=120$ M⊙
pc-2. (Lower left) The histogram of values of log $\alpha_{\rm vir}$, with a
vertical red line at $\mbox{$\alpha_{\rm vir}$}=2.0$. The number of structures
in each category are given in parentheses. (Lower right) The histogram of
values of log $\Sigma_{\rm M}$, with a vertical red line at
$\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2.

The outer Galaxy sample (OGS) from the catalog of Heyer et al. (2001) shows a
similar behavior. Most are unbound, but more than half the mass is in bound
structures despite having $\mbox{$\Sigma_{\rm M}$}<120$ M⊙ pc-2, with
$\mbox{$f(M)$}=0.72$ (Table 2). To some extent, this result depends on the
requirement that $\mbox{$M_{\rm CO}$}>2.5\mbox{${}\times 10^{3}$}$ M⊙ imposed
by Heyer et al. (2001) to ensure completeness. Since completenesss is less
important for our purposes, we relaxed that criterion to $\mbox{$M_{\rm
CO}$}>10$ M⊙ to see the effect. While $f(N)$ dropped to 0.06,
$\mbox{$f(M)$}=0.58$, still substantial because most of the bound mass was in
the higher mass structures.

###  3.2 Structures Defined by CO $J=2\rightarrow 1$ Emission in Other
Galaxies

Figure 5:  The mean and standard deviation of the virial ratio is plotted in
bins versus the logarithm of the clump mass, for The catalog of Sun et al.
(2020) is plotted. (Upper Left) The mean and standard deviation of the virial
ratio is plotted versus the logarithm of the clump mass. Median values are
plotted in a magenta dashed line. (Upper Right) The same quantities are
plotted versus the mass surface density. The horizontal line at
$\mbox{$\alpha_{\rm vir}$}=2$ demarcates nominally unbound clouds above the
line from nominally bound clouds below the line. Vertical lines indicate
$\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2 and $\mbox{$\Sigma_{\rm M}$}=1$ g cm-2.
(Lower left) The histogram of values of log $\alpha_{\rm vir}$, with a
vertical red line at $\mbox{$\alpha_{\rm vir}$}=2.0$. The number of structures
in each category are given in parentheses. (Lower right) The histogram of
values of log $\Sigma_{\rm M}$, with a vertical red line at
$\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2.

We were able to use all the catalog entries from table B1 of Sun et al.
(2020), save one entry with a negative value for $\alpha_{\rm vir}$. Sun et
al. (2020) themselves found an area-weighted median $\mbox{$\alpha_{\rm
vir}$}=3.5$ with a 1-$\sigma$ spread of 0.6 dex, based on over $10^{5}$ pixels
of size 150 pc in 66 galaxies. The mass-weighted median was smaller at 2.7.
All the caveats about neglect of tidal effects, etc. apply quite strongly to
studies with 150 pc resolution. Independent analysis of the data in their
table B1 confirms the median value and finds a mean value of
$\mbox{$\langle\mbox{$\alpha_{\rm vir}$}\rangle$}=4.7\pm 4.0$. The averages of
$\alpha_{\rm vir}$ in Figure 5 all lie above $\mbox{$\alpha_{\rm vir}$}=2$,
but they approach that value for the most massive data points. A similar
pattern applies to the plot of $\alpha_{\rm vir}$ versus $\Sigma_{\rm M}$.
While many points have very high $\alpha_{\rm vir}$, the fraction with
$\mbox{$\alpha_{\rm vir}$}<2$ is $\mbox{$f(N)$}=0.21$ and the fraction of the
total mass in those structures is $\mbox{$f(M)$}=0.35$, reflecting the usual
trend for lower $\alpha_{\rm vir}$ in more massive clouds. The median and mean
surface densities are 22 and 50 M⊙ pc-2, but with a maximum at
$\mbox{$\Sigma_{\rm M}$}=9.7\mbox{${}\times 10^{3}$}$ M⊙ pc-2. The last value
is clearly more characteristic of dense clumps, not clouds. All points above
log $\mbox{$M_{\rm CO}$}=6.43$ M⊙ correspond to $\mbox{$\Sigma_{\rm M}$}>120$
M⊙ pc-2 or $\mbox{$A_{V}$}>8$ mag. Vertical lines on Figure 5 indicate surface
densities of 120 M⊙ pc-2 and 1 g cm-2. The second of these corresponds to the
criterion for massive star formation (Krumholz & McKee 2008). The fact that
$\alpha_{\rm vir}$ drops to values near two as the surface density approaches
this value is interesting. To have such high surface density averaged over a
region of 150 pc is quite remarkable. If $\Sigma_{\rm M}$ is restricted to be
less than 120 M⊙ pc-2, the fraction of the total mass with $\mbox{$\alpha_{\rm
vir}$}<2$ becomes 0.15.

###  3.3 Structures Defined by 13CO $J=1\rightarrow 0$ Emission

To use data from all three surveys and to compare methods for structure
identification, we have reanalyzed the three data sets, using the CO clouds
defined by Miville-Deschênes et al. (2017) to constrain the 13CO voxels
included. The detailed method is explained in the Appendix. However, we begin
with the original catalog, but with updated abundance assumptions. These
results will allow a comparison between different cloud identification
methods.

####  3.3.1 GRS: Original Sample

Figure 6:  (Upper Left) The mean and standard deviation of the virial ratio is
plotted versus the logarithm of the clump mass, for the catalog of Roman-Duval
et al. (2010). after selecting only those catalog entries with
$\mbox{$\Sigma_{\rm M}$}>10$ and $\mbox{$M_{\rm CO}$}>1$ M⊙, and
$\mbox{$\alpha_{\rm vir}$}<20$. Median values are plotted in a magenta dashed
line. (Upper Right) The same quantities are plotted versus the mass surface
density. The horizontal line at $\mbox{$\alpha_{\rm vir}$}=2$ demarcates
nominally unbound clouds above the line from nominally bound clouds below the
line. The vertical line in the right panel indicates $\mbox{$\Sigma_{\rm
M}$}=120$ M⊙ pc-2. (Lower left) The histogram of values of log $\alpha_{\rm
vir}$, with a vertical red line at $\mbox{$\alpha_{\rm vir}$}=2.0$. The number
of structures in each category are given in parentheses. (Lower right) The
histogram of values of log $\Sigma_{\rm M}$, with a vertical red line at
$\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2.

The tabulated masses and surface densities for the 13CO GRS data from Roman-
Duval et al. (2010) originally used values of H2/CO of 12500 and
$\mbox{$\mu_{\rm H_{2}}$}=2.37$. As discussed in §1, we instead use the
current best values for H2/CO and $\mu_{\rm H_{2}}$, resulting in masses and
surface densities lower by a factor of 0.567 and virial parameters higher by a
factor of 1.76. For a first analysis, (entry 8 in Table 2), we retain the
assumption from Roman-Duval et al. (2010) of a constant CO/13CO of 45. The
usual quantities are plotted in Figure 6 and the results are in Table 2. The
majority of the sample would be classified as clouds, with $\mbox{$\Sigma_{\rm
M}$}<120$ M⊙ pc-2, but about 1/6 are denser. The fraction of mass in bound
structures, $\mbox{$f(M)$}=0.95$, but the fraction for clouds is much less,
$\mbox{$f_{\rm c}(M)$}=0.55$.

####  3.3.2 GRS: New Analysis

From now on, we use the CO-defined structures and the 13CO-derived properties,
with the abundance assumptions described at the end of §1, including the
variation of ${\rm CO}/{\rm{}^{13}CO}$ with $R_{\rm gal}$. We begin with a re-
analysis of the GRS survey in 13CO (Roman-Duval et al. 2010). The structures
are defined in the catalog of Miville-Deschênes et al. (2017), but only for a
sub-sample. The sub-sample of CO clouds is somewhat biased towards higher
$\Sigma_{\rm M}$ and lower $\alpha_{\rm vir}$ than the full catalog:
$\mbox{$\langle\mbox{$\Sigma_{\rm M}$}\rangle$}=95\pm 64$ M⊙ pc-2, nearly
twice that for the full catalog; $\mbox{$\langle\mbox{$\alpha_{\rm
vir}$}\rangle$}=5.62\pm 5.28$, 0.81 times that for the full catalog.

Figure 7:  (Upper Left) The mean and standard deviation of the virial ratio is
plotted versus the logarithm of the clump mass, for the catalog produced for
the GRS in combination with CO. Median values are plotted in a magenta dashed
line. (Upper Right) The same quantities are plotted versus the mass surface
density. The horizontal line at $\mbox{$\alpha_{\rm vir}$}=2$ demarcates
nominally unbound clouds above the line from nominally bound clouds below the
line. The vertical line in the right panel indicates $\mbox{$\Sigma_{\rm
M}$}=120$ M⊙ pc-2. The mass and surface density are the total (gas+dust)
determined from submm continuum emission. (Lower left) The histogram of values
of log $\alpha_{\rm vir}$, with a vertical red line at $\mbox{$\alpha_{\rm
vir}$}=2.0$. The number of structures in each category are given in
parentheses. (Lower right) The histogram of values of log $\Sigma_{\rm M}$,
with a vertical red line at $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2.

The analysis of the GRS data with the method described in the Appendix is
listed as entry 9 of Table 2 and the usual plots are in Figure 7. The main
difference from entry 8 is that the clouds were identified in the CO analysis
and the 13CO data were taken only within the boundaries of those clouds. The
change in cloud definition makes the mean $\alpha_{\rm vir}$ larger and $f(M)$
smaller by substantial amounts ($\mbox{$f(M)$}=0.46$).

The fraction of the mass of the CO-defined structure contained in the 13CO
emission ($M_{\rm 13}/M_{\rm CO}$) is a valuable guide to the fraction of
possibly bound structures within unbound CO clouds. For the GRS sample,
$\mbox{$\langle\mbox{$M_{\rm 13}/M_{\rm CO}$}\rangle$}=0.22\pm 0.08$ with a
median $\mbox{$M_{\rm 13}/M_{\rm CO}$}=0.21$.

####  3.3.3 EXFC55-100

Figure 8:  (Upper Left) The mean and standard deviation of the virial ratio is
plotted versus the logarithm of the clump mass, for the catalog produced for
the EXFC55-100 sample in combination with CO and using new abundances. Median
values are plotted in a magenta dashed line. (Upper Right) The same quantities
are plotted versus the mass surface density. The horizontal line at
$\mbox{$\alpha_{\rm vir}$}=2$ demarcates nominally unbound clouds above the
line from nominally bound clouds below the line. The vertical line in the
right panel indicates $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2. The mass and
surface density are the total (gas+dust) determined from submm continuum
emission. (Lower left) The histogram of values of log $\alpha_{\rm vir}$, with
a vertical red line at $\mbox{$\alpha_{\rm vir}$}=2.0$. The number of
structures in each category are given in parentheses. (Lower right) The
histogram of values of log $\Sigma_{\rm M}$, with a vertical red line at
$\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2.

For this sample, most structures are identified as clouds, but they are mostly
bound, despite having low $\Sigma_{\rm M}$. The usual plots are in Figure 8
and the statistics are given as entry 10 in Table 2. The fraction of the CO
mass recovered by the 13CO emission, $\mbox{$\langle\mbox{$M_{\rm 13}/M_{\rm
CO}$}\rangle$}=0.79\pm 0.80$ with a median $\mbox{$M_{\rm 13}/M_{\rm
CO}$}=0.61$. These values are much higher than those in the GRS.

####  3.3.4 EXFC135-195

Figure 9:  (Upper Left) The mean and standard deviation of the virial ratio is
plotted versus the logarithm of the clump mass, for the catalog produced for
the EXFC135-195 sample in combination with CO and using new abundances. Median
values are plotted in a magenta dashed line. (Upper Right) The same quantities
are plotted versus the mass surface density. The horizontal line at
$\mbox{$\alpha_{\rm vir}$}=2$ demarcates nominally unbound clouds above the
line from nominally bound clouds below the line. The vertical line in the
right panel indicates $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2. The mass and
surface density are the total (gas+dust) determined from submm continuum
emission. (Lower left) The histogram of values of log $\alpha_{\rm vir}$, with
a vertical red line at $\mbox{$\alpha_{\rm vir}$}=2.0$. The number of
structures in each category are given in parentheses. (Lower right) The
histogram of values of log $\Sigma_{\rm M}$, with a vertical red line at
$\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2.

As for EXFC55-100, most structures are identified as clouds, but they are
mostly bound, despite having very low $\Sigma_{\rm M}$. The usual plots are in
Figure 9 and the statistics are given as entry 11 in Table 2. For the
EXFC135-195 sample, $\mbox{$\langle\mbox{$M_{\rm 13}/M_{\rm
CO}$}\rangle$}=0.76\pm 0.34$ with a median $\mbox{$M_{\rm 13}/M_{\rm
CO}$}=0.66$. These values are much higher than those in the GRS and similar to
those of the EXFC55-100 sample.

####  3.3.5 Summary of Consequences of New Analysis

The main effect of the different method of structure identification is to
decrease the fraction of mass in bound structures. In the inner Galaxy, most
become unbound, but 46% of the mass is still in bound structures. For larger
$R_{\rm gal}$, the structures become less dense but more bound. For the outer
Galaxy (EXFC135-195) sample, $\mbox{$f(M)$}=0.75$ even though
$\mbox{$\langle\mbox{$\Sigma_{\rm M}$}\rangle$}=24$ M⊙ pc-2. This trend
reflects a strong decrease in velocity dispersion with $R_{\rm gal}$. The
fraction of the mass recovered by 13CO ($M_{\rm 13}/M_{\rm CO}$) is low in the
inner Galaxy, but quite high in the other two samples.

###  3.4  13CO $J=2\rightarrow 1$

Figure 10:  (Upper Left) The mean and standard deviation of the virial ratio
is plotted versus the logarithm of the clump mass, for the catalog of Duarte-
Cabral et al. (2021), after selecting only those catalog entries with good
distances, not on edges and at least 114 pixels. Median values are plotted in
a magenta dashed line. (Upper Right) The same quantities are plotted versus
the mass surface density. The horizontal line at $\mbox{$\alpha_{\rm vir}$}=2$
demarcates nominally unbound clouds above the line from nominally bound clouds
below the line. The vertical line in the right panel indicates
$\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2. (Lower left) The histogram of values of
log $\alpha_{\rm vir}$, with a vertical red line at $\mbox{$\alpha_{\rm
vir}$}=2.0$. The number of structures in each category are given in
parentheses. (Lower right) The histogram of values of log $\Sigma_{\rm M}$,
with a vertical red line at $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2.

The situation for the $J=2\rightarrow 1$ transition of 13CO (Duarte-Cabral et
al. 2021), shown in Figure 10, is different. Most clouds defined in this
sample are nominally bound, even for relatively low mass and $\Sigma_{\rm M}$.
However, $\langle\mbox{$\alpha_{\rm vir}$}\rangle$ is lower and $f(M)$ is
higher than for the $J=1\rightarrow 0$ in the GRS sample with the new
analysis, but comparable for the other $J=1\rightarrow 0$ samples.

###  3.5 Stuctures Defined by Herschel and NH3

Figure 11:  (Upper Left) The mean and standard deviation of the virial ratio
is plotted versus the logarithm of the clump mass, for the catalog of Merello
et al. (2019). Median values are plotted in a magenta dashed line. (Upper
Right) The same quantities are plotted versus the mass surface density. The
horizontal line at $\mbox{$\alpha_{\rm vir}$}=2$ demarcates nominally unbound
clouds above the line from nominally bound clouds below the line. The vertical
lines in the right panel indicate $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2 and
$\mbox{$\Sigma_{\rm M}$}=1$ g cm-3. The mass and surface density are the total
(gas+dust) determined from submm continuum emission. (Lower left) The
histogram of values of log $\alpha_{\rm vir}$, with a vertical red line at
$\mbox{$\alpha_{\rm vir}$}=2.0$. The number of structures in each category are
given in parentheses. (Lower right) The histogram of values of log
$\Sigma_{\rm M}$, with a vertical red line at $\mbox{$\Sigma_{\rm M}$}=120$ M⊙
pc-2.

Compact structures identified by their submm emission with Herschel have very
low virial parameters, strongly indicative of being gravitationally dominated
and $\mbox{$f(M)$}=1.0$. Figure 11 plots the virial parameter versus the
logarithm of clump mass from an analysis by Merello et al. (2019). Only the
clumps with the lowest mass and $\Sigma_{\rm M}$ are unbound. While most of
these contain star formation, even the 157 “prestellar” clumps are strongly
gravitationally dominated, with $\mbox{$\langle\mbox{$\alpha_{\rm
vir}$}\rangle$}=0.57$, almost identical to the value for the full sample. The
values for $f_{\rm c}(N)$ and $f_{\rm c}(M)$ of zero reflect the fact that the
very few structures in the sample with surface densities small enough to be
considered clouds are all unbound.

##  4 Discussion

In this section, we will explore how the boundedness of structures depends on
methods and properties. We include sub-sections on the method used to identify
structures, the tracer used to define the structures and measure the mass, the
location in the Galaxy, mass of the structure, and the surface density of the
structure.

###  4.1 Dependence on Method Used to Identify Structure

The first issue to consider is the fact that the results from two different
decompositions of the CO $J=1\rightarrow 0$ survey by Dame et al. (2001) show
quite different results. The difference is clear from the Figures, but it can
be encapsulated in the values of $f(M)$: the structures identified by Rice et
al. (2016) have $\mbox{$f(M)$}=0.73$ while those of Miville-Deschênes et al.
(2017) have $\mbox{$f(M)$}=0.19$.

Clearly, the choice of method to identify structures plays a major role in the
different results. We can borrow the terminology of taxonomists to distinguish
lumpers and splitters. In our context, lumpers would aggregate emission into
larger structures, while splitters would break emission into smaller
structures. Rice et al. (2016) used a dendrogram analysis to identify 1064
clouds, accounting for about 25% of the molecular mass in the Galaxy. They
note that their method avoids merging unrelated clouds into pseudo-clouds. In
this sense, they are splitters. Comparing their catalog to that of Dame et al.
(1986) (based on an early version of the first quadrant survey), they recover
all of the same clouds, but some are split and less massive. They use the
size-linewidth relation and latitude distribution to break the kinematic
distance ambiguity. In contrast, Miville-Deschênes et al. (2017) might
reasonably be called lumpers. They performed a Gaussian decomposition of each
spectrum and then lumped them together by a clustering procedure. This process
led to larger linewidths in the clouds (median $\mbox{$\sigma_{v}$}=3.6$ km
s-1) than in the individual Gaussian components (median
$\mbox{$\sigma_{v}$}=1.65$ km s-1). Interestingly, this method allowed them to
identify 98% of the CO emission with 8107 clouds. They found that the size-
linewidth relation has too much scatter, so they use the relation
$\mbox{$\sigma_{v}$}\propto(R\mbox{$\Sigma_{\rm M}$})^{0.5}$, where $R$ is the
cloud size, to resolve the kinematic distance ambiguity. While the mass also
increased by this process, the net effect was probably to produce larger
$\alpha_{\rm vir}$. Were they lumpers or splitters? In the sense that they
aggregated velocity components, they were lumpers, but they decomposed four
times as much of the CO emission into 8 times as many clouds as Rice, making
them splitters. Lumping in velocity and splitting in number of clouds is
likely to lead to larger $\alpha_{\rm vir}$. However, the fact that the
scaling relation mentioned above holds indicates that unrelated gas is not
being lumped in pseudo-clouds.

Another difference between the results of the two methods is apparent in the
x-axis values for the plots of $\alpha_{\rm vir}$ versus $M_{\rm CO}$. The
plot for Rice et al. (2016) starts at $\log\mbox{$M_{\rm CO}$}=3.5$, while
that of Miville-Deschênes et al. (2017) starts at $\log\mbox{$M_{\rm
CO}$}=0.5$. The latter catalog contains many more clouds of lower mass.
However, this does not explain the difference in $f(M)$ because the total mass
is still dominated by the massive clouds.

Probably the main factor in the difference is the completeness in accounting
for the CO emission of the Galaxy. Because Rice et al. (2016) account for only
25% of the CO emission, their catalog is less suitable for addressing the
question of slow star formation on the Galactic scale, where the problem was
defined by taking the total mass of molecular gas from the CO survey and
dividing by a free-fall time.

In this context, the data from Sun et al. (2020) are interesting as no method
is used to define structures. Instead only the mass within a 150 pc region is
tabulated. The fact that $\mbox{$f(M)$}=0.35$ and $\mbox{$f_{\rm
c}(M)$}=0.15$, averaged over many galaxies and galactocentric radii, even with
some very high values for $\Sigma_{\rm M}$, suggests that the most likely
value for $f(M)$ is relatively small for structures measured by CO, regardless
of which of the two transitions is used.

The importance of the identification method is also apparent in the difference
between the results from the original structures in the Roman-Duval et al.
(2010) GRS sample (entry 8 in Table 2) and the new analysis, where structures
were defined by the CO catalog of Miville-Deschênes et al. (2017) (entry 9 in
Table 2).

These differences suggest that procedures like CLUMPFIND or dendrogram
analyses tend to split structures or minimize low level, extended emission.
For addressing the global issue of slow star formation, procedures that
account for all the emission are necessary.

###  4.2 Dependence on the Tracer Used to Define Structure

We begin with the extremes: the samples of Miville-Deschênes et al. (2017) and
Sun et al. (2020) showing low values for $f(M)$ for mass measured from CO
emission, versus the sample of Merello et al. (2019) showing
$\mbox{$f(M)$}=1.0$ for structures traced by Herschel peaks and NH3
linewidths. The distinction between clouds, primarily traced by CO, and dense
clumps, primarily traced by peaks in submillimeter emission and lines of
species rarer than CO, has been based on this difference, but the analysis of
$\alpha_{\rm vir}$ strongly supports this distinction. The much higher mean
and median $\Sigma_{\rm M}$ for the Herschel-NH3 data also indicate that these
are clearly distinct structures.

What about the structures probed by 13CO? They clearly lie between the
extremes of clouds and dense clumps. For the GRS sample with our new analysis,
$\mbox{$f(M)$}=0.46$ and $\mbox{$\langle\mbox{$\alpha_{\rm
vir}$}\rangle$}=3.83$ and $\mbox{$\langle\mbox{$\Sigma_{\rm
M}$}\rangle$}=104$. For the sample defined by 13CO $J=2\rightarrow 1$, from
(Duarte-Cabral et al. 2021), $\mbox{$f(M)$}=0.79$ and
$\mbox{$\langle\mbox{$\alpha_{\rm vir}$}\rangle$}=1.94$, even though
$\mbox{$\langle\mbox{$\Sigma_{\rm M}$}\rangle$}=87.7$ M⊙ pc-2, lower than that
of 13CO $J=1\rightarrow 0$ in the GRS, but higher than that of 13CO
$J=1\rightarrow 0$ in the other two surveys.

Figure 12:  The value of $f(M)$ versus $n_{\rm eff}$ of the tracer. The higher
point for 13CO $J=1\rightarrow 0$ is for the average of the two EXFC surveys,
while the lower point is from the analysis in this paper for the GRS survey.
Both points for 13CO $J=1\rightarrow 0$ use the structures in the CO catalog
of Miville-Deschênes et al. (2017).

The trend with tracer is that $\alpha_{\rm vir}$ decreases (structures are
more likely to be bound) as the tracer changes from CO to 13CO $J=1\rightarrow
0$ to 13CO $J=2\rightarrow 1$ to Herschel-NH3. This can be related to a trend
in the characteristic density of the material being traced. The characteristic
density probed by each tracer can be crudely estimated from the effective
density of the tracer used to define the structure. The effective density
($n_{\rm eff}$), defined by Evans (1999) and developed further by Shirley
(2015), is the density of particles needed to produce a 1 K km s-1 observed
line for an assumed column density and kinetic temperature. Because effective
densities for CO and 13CO were not calculated by those references, we
calculated them using the on-line tool RADEX (van der Tak et al. 2007)
assuming $\mbox{$T_{K}$}=10$ K and column density corresponding to
$\mbox{$A_{V}$}=1$ mag, so $N({\rm CO})=1\mbox{${}\times 10^{17}$}$ cm-2 and
$N(\mbox{${}^{13}$CO})=1.7\mbox{${}\times 10^{15}$}$ cm-2. The resulting
effective densities are 15, 60, 800, 3800 cm-3 for CO $J=1\rightarrow 0$, CO
$J=2\rightarrow 1$, 13CO $J=1\rightarrow 0$, 13CO $J=2\rightarrow 1$,
respectively. The comparable density for the Herschel sample is less clear.
The structures were identified from dust continuum emission, but NH3
observations of both (1,1) and (2,2) inversion transitions were required in
order to estimate the kinetic temperature, $T_{K}$. The effective density for
the (2,2) line at $\mbox{$T_{K}$}=15$ K is $\mbox{$n_{\rm
eff}$}=1.6\mbox{${}\times 10^{4}$}$ cm-3 (Shirley 2015), similar to the
density of about 2${}\times 10^{4}$ cm-3 needed to make $T_{K}$ close to the
dust temperature, $T_{d}$, as found by Merello et al. (2019). We take
$\mbox{$n_{\rm eff}$}=1.6\mbox{${}\times 10^{4}$}$ for the Herschel sample.

The values for $f(M)$ are plotted versus these effective densities in Figure
12. We use only the results for entries 3, 7, 9, 12, and 13, along with the
average of 10 and 11 (the EXFC samples), from Table 2 as most representative
of structure selection by each tracer. Clearly the fraction of the mass in
bound structures increases rapidly with $n_{\rm eff}$, reaching values about
0.8 for densities near the effective density of 13CO $J=2\rightarrow 1$ and
unity for the Herschel sample. The data roughly follow a line in the semilog
plot, but the two points for 13CO $J=1\rightarrow 0$ suggest that location in
the Galaxy also matters.

###  4.3 Dependence on Location in the Galaxy

The contrast between the inner Galaxy (CMZ) (Oka et al. 2001) and the outer
Galaxy (OGS) (Heyer et al. 2001) is perhaps the most striking. The structures
identified in the CMZ have very high surface densities
($\mbox{$\langle\mbox{$\Sigma_{\rm M}$}\rangle$}=1700$ M⊙ pc-2), but
$\mbox{$f(M)$}=0.71$. In contrast the structures in the OGS have very low
surface densities ($\mbox{$\langle\mbox{$\Sigma_{\rm M}$}\rangle$}=17.7$ M⊙
pc-2), but $\mbox{$f(M)$}=0.58$ for entry 6, with the more inclusive lower
limit on cloud mass. With the criterion of $\mbox{$\Sigma_{\rm M}$}<120$ M⊙
pc-2, none of the structures in the CMZ defined by CO are bound (both $f_{\rm
c}(N)$ and $f_{\rm c}(M)$ are zero), while $\mbox{$f_{\rm c}(M)$}=0.46$ or
0.56 in the OGS, depending on the mass limits used. Clearly, the local
criterion of $\mbox{$\Sigma_{\rm M}$}=120$ M⊙ pc-2 for boundedness, based on a
star formation threshold, is inappropriate for these parts of the Galaxy. In
the CMZ, a clump would need to be defined with a much higher $\Sigma_{\rm M}$,
higher than 3000 M⊙ pc-2. In the outer Galaxy, the opposite applies.

A difficulty with this comparison is that very different criteria were used to
define structures in the CMZ compared to the disk or outer Galaxy. A high
threshold was needed to isolate structures in the highly confused CMZ. Using
the same procedure for all $R_{\rm gal}$, Miville-Deschênes et al. (2017) did
not find a drop in median of $\alpha_{\rm vir}$ at large $R_{\rm gal}$, but
did see a rise inside a few kpc. The sample of other galaxies (Sun et al.
2020) also uses a common method for all radii. There are 1562 data points in
their catalog with $\mbox{$R_{\rm gal}$}<0.5$ kpc, similar to the definition
of the CMZ in the Milky Way. In that restricted sample, the median, mean, and
standard deviation of $\Sigma_{\rm M}$ are 130, 375, 670 M⊙ pc-2, much higher
than the full sample, but still less than those for the CMZ. The median, mean,
and standard deviation of $\alpha_{\rm vir}$ are 5.45, 8,28, 9.03, about twice
those for the full sample, and larger than those for the CMZ, while the
fractions in bound structures are $\mbox{$f(N)$}=0.11$ and
$\mbox{$f(M)$}=0.27$, substantially smaller than for the full sample or for
the CMZ of the Milky Way. The data from other galaxies supports the idea that
a much higher surface density is needed to produce bound structures in the
inner regions of galaxies and suggests that most of the molecular gas there is
unbound despite very high surface densities.

Comparing the GRS and EXFC samples in 13CO $J=1\rightarrow 0$ reveals a
similar pattern. More mass is in bound structures despite the lower
$\Sigma_{\rm M}$ in the outer parts of the Galaxy.

###  4.4 Dependence on Mass

A trend common to all the samples is a decrease in $\alpha_{\rm vir}$ with
increasing mass. At some level, this trend is a result of a trend of
decreasing $\alpha_{\rm vir}$ with increasing $\Sigma_{\rm M}$. This is most
obvious for the sample of Sun et al. (2020), where the size of the region
considered is constant, so the trend with mass is really a trend with surface
density.

Contrary evidence that the most massive clouds may be unbound can be seen in
Figure 8 from Dame et al. (1986) who consider the 26 most massive complexes in
the first quadrant of the Milky Way. About half have linewidths larger than
those expected from virial equilibrium. A more recent analysis of the most
massive complexes by Nguyen-Luong et al. (2016) found that most were unbound.
While some are engaged in what they called “mini-starbursts” which could be
unbinding the cloud, those sources do not have a very different distribution
of virial parameters. It is these very large complexes that will dominate
studies of CO in other galaxies. They also dominate the total mass of
molecular gas in our Galaxy because the cloud mass function slope is less than
2, unlike the case for stars. However, the mass function peaks at much lower
masses in the outer Galaxy, as shown in Figure 7 of Miville-Deschênes et al.
(2017). A similar trend is seen for molecular clouds in M33 (Rosolowsky 2005).

###  4.5 Dependence on Mass Surface Density

The decrease in median and mean $\alpha_{\rm vir}$ with increasing
$\Sigma_{\rm M}$ is clear from the relevant plots for most samples that cover
the relevant range of $\Sigma_{\rm M}$. The distinction between clouds and
clumps adopted for discussion in this paper is $\mbox{$\Sigma_{\rm M}$}=120$
M⊙ pc-2. To explore this connection further, we have calculated the mass
fraction in bound structures in bins of 0.3 in log $\Sigma_{\rm M}$ for
samples that cover the transition well. As shown in Figure 13, these all show
a strong increase in the fraction of mass that is bound as $\Sigma_{\rm M}$
increases.

Figure 13:  The value of $f(M)$ is plotted versus log $\Sigma_{\rm M}$ for the
CO samples of Sun et al. (2020), Miville-Deschênes et al. (2017), Heyer et al.
(2001), and the 13CO $J=1\rightarrow 0$ sample for the GRS, with structures
identified by CO (entry 9 in Table 2).

The dependence on surface density is also reflected in the values of $f_{\rm
c}(N)$ and $f_{\rm c}(M)$ in Table 2. These are generally smaller than the
$f(N)$ and $f(M)$ computed for all structures. Only a small fraction of the
gas in structures we have called clouds is bound. This statement applies both
to Milky Way structures and to those in other galaxies.

The decline of virial parameter with mass may in part be an artifact of
observational biases in the way that the linewidth is measured (Traficante et
al. 2018). They point out that the tracers used for determining the velocity
dispersion may be weighted more heavily to denser regions, with smaller
dispersions that do not reflect the full dispersion of the entire structure.
This effect would be most noticeable in samples for which the tracers of mass
and velocity dispersion are different. The effect might contribute to the very
low values of $\alpha_{\rm vir}$ and the strong trend with mass in the
Herschel sample.

###  4.6 Caveats and Uncertainties

As discussed extensively in §1, there are substantial uncertainties in
computing the virial parameter. For any given structure, the uncertainty in
$\alpha_{\rm vir}$ is likely to be a factor of 2 or 3. Analyses of the
uncertainties in estimating cloud mass, velocity dispersion, etc. can be found
in Beaumont et al. (2013). Masses from CO luminosities have uncertainties, as
discussed in the introduction, so that we expect uncertainties of at least a
factor of 3, and these may be systematic uncertainties, at least in the case
of $M_{\rm CO}$. Gong et al. (2018) argue that X(CO) may be overestimated
based on “observing” simulations. They get a value of X(CO) about half the
standard value of 2${}\times 10^{20}$. If that is correct, masses from CO
would be overestimated and virial parameters underestimated. Finally, we re-
emphasize the dependence on the method used to define a cloud.

For 13CO, our newer analysis accounts for the latest measurements of the
isotopic ratio, ${\rm CO}/{\rm{}^{13}CO}$, as a function of $R_{\rm gal}$ and
the latest value of H2/CO. On the other hand, we have made no correction for
changes in the CO abundance with metallicity, which is known to decrease with
$R_{\rm gal}$. Deharveng et al. (2000) found a gradient in the oxygen
abundance gradient over the range of 5-15 kpc: $12+\log{\rm O/H}=(-3.95\pm
0.49)\mbox{${}\times 10^{-2}$}\mbox{$R_{\rm gal}$}+(8.82\pm 0.05)$. The effect
of metallicity on CO emission is not simple, but extragalactic observers
commonly use a correction factor: $\mbox{$\alpha_{\rm CO}$}=4.35Z^{-1.6}$ M⊙(K
km s-1pc2)-1 (Sun et al. 2020 and references therein), where $Z$ is the
metallicity relative to solar. If we applied the same correction factor, CO
clouds at 3 kpc would have twice higher $\alpha_{\rm vir}$, while those at 15
kpc would have about one-third the value of $\alpha_{\rm vir}$. Such a
correction would strengthen the already strong trend for clouds to be unbound
in the inner Galaxy, where most of the molecular mass resides, and to be bound
in the outer Galaxy.

We reiterate that methods of structure identification remain one of the most
significant sources of uncertainty.

##  5 Consequences for the Star Formation Rate Problem

The results of this analysis have consequences for the problem of the star
formation rate, discussed in the introduction. If only a fraction of the mass
of molecular gas in the Galaxy is in bound structures, the over-prediction of
the star formation rate of 300 M⊙ yr-1, can be decreased substantially. Based
on the catalog of Miville-Deschênes et al. (2017), only 19% of structures
defined by CO $J=1\rightarrow 0$ are bound, decreasing the theoretical star
formation rate to 57 M⊙ yr-1, using the same assumption about the free-fall
time as the value found in §1.

We can make a more accurate estimate by assessing the star formation rate
cloud by cloud, then adding those rates up. If we assume that only clouds with
$\mbox{$\alpha_{\rm vir}$}\leq 2$ form stars in a free fall time based on the
mean density of that cloud, the predicted star formation rate for an
individual cloud is

$\mbox{$SFR_{\rm th}$}=M/\mbox{$t_{\rm ff}$}=6.04\mbox{${}\times
10^{-8}$}M_{\mbox{M${}_{\odot}$}}^{1.5}r_{\rm pc}^{-1.5}$ (10)

Using the catalog of Miville-Deschênes et al. (2017) because it accounts for
all the molecular clouds defined by CO emission, and summing the star
formation rate only over bound clouds, we find a total star formation rate for
the Milky Way of 46.4 M⊙ yr-1, a factor of 6.5 less than the value in the
introduction. This is mostly due to the small fraction of bound clouds, but
there is also a contribution from a longer $t_{\rm ff}$ because the mean
density of the clouds identified by Miville-Deschênes et al. (2017) is less
than 100 cm-3. The resulting depletion time of bound molecular gas is
decreased to $\mbox{$t_{\rm dep}$}=1.0\mbox{${}\times 10^{8}$}$ yr, closer to
the free-fall time, but still considerably larger.

Of course, regions of clouds are likely to be bound even if the entire cloud
is not. The fraction of bound mass could be assessed from comparison of masses
indicated by different tracers. Ideally, we would do the same analysis for a
complete survey of the Galaxy in other tracers, especially 13CO. A similar
calculation for the 13CO GRS survey, using the full sample (entry 8 in Table
2) predicts a star formation rate of 7.4 M⊙ yr-1. However, the existing 13CO
surveys do not cover the whole Galaxy, and the new analysis was possible only
for a subset of the 13CO data, so we cannot do a direct calculation. Instead
we calculate the fraction of the mass of the unbound CO-defined structures
that is found in the bound 13CO-derived properties. This value is $0.22$ for
the GRS, which applies to the inner Galaxy, which contains most of the
molecular mass. The fraction is higher (0.77-0.80) in the outer parts of the
Galaxy. So this material would increase the predicted star formation rate. If
all of the 13CO-derived mass in the unbound CO-defined clouds in the GRS
sample form stars at the fiducial $t_{\rm ff}$, they would contribute another
55 M⊙ yr-1 to the Galactic star formation rate. Clearly, a full sky survey of
13CO would be very valuable.

We have focused on $\epsilon_{\rm ff}$, rather than the final efficiency
($\epsilon_{\rm sf}$) Because the most massive clouds seem most likely to be
bound (subject to the caveats above), the difference between mass functions of
clouds and clusters is, if anything, increased. The resolution to this problem
may lie in the inclusion of tidal forces since most of these very massive
clouds reside in the inner Galaxy.

##  6 Conclusions

We list the main conclusions before discussing them:

  1. 1.

Clearly, the choice of method to identify clouds plays a major role in the
different results.

  2. 2.

The tracer (both molecule and transition) that is used to define structures
has a strong effect on whether those structures are bound.

  3. 3.

At a fixed $R_{\rm gal}$, the fraction of the mass in bound structures
increases rapidly with the effective density of the tracer, suggesting that
regions with larger volume densities contribute more to bound gas.

  4. 4.

Many clouds identified by CO are unbound and most of the mass in those clouds
is unbound.

  5. 5.

Structures defined by dust continuum emission and linewidths from NH3 emission
are almost all bound.

  6. 6.

Structures defined by 13CO emission are more likely to be bound than those
defined by CO emission. However, their boundedness also depends on the method
used to identify them and on assumptions about isotopic and elemental
abundances.

  7. 7.

Boundedness correlates strongly with surface density.

  8. 8.

Structures within 0.5 kpc of a galaxy center have much higher surface density,
but much of the mass is unbound because of higher turbulence. The opposite is
true in outer regions.

  9. 9.

More generally, structures identified by CO or 13CO have lower surface
density, but are more likely to be bound, at larger $R_{\rm gal}$.

  10. 10.

More massive clouds are more likely to be bound, but this trend is partially
confused with the trend with surface density.

  11. 11.

For the most complete catalog of structures traced by CO (Miville-Deschênes et
al. 2017), the fraction of mass in bound structures, $\mbox{$f(M)$}=0.19$,
alleviating, but not eliminating, the fundamental problem of slow star
formation in the Milky Way.

The most important conclusion is that most of the mass traced by CO emission
is in unbound structures. This result was anticipated by theorists, notably
Dobbs et al. (2011), whose title asked “Why are most molecular clouds not
gravitationally bound?”. Their answer involved cloud-cloud collisions and
stellar feedback; shredding and merging resulted in clouds not being well-
defined entities with the same gas over cloud “lifetimes.” Padoan et al.
(2017) argued that supernova feedback kept the interstellar medium
sufficiently stirred up that many structures were unbound even though their
densities were such that they were likely molecular. As mentioned in §1, Kim
et al. (2021) found that unbound clouds are needed to reach the observational
constraints on the star formation rate.

Observers have often found, but seldom emphasized, this result. Roman-Duval et
al. (2016) suggested that gas detected in CO, but not 13CO might be “diffuse,
non-star-forming gas,” but did not explicitly suggest that this gas was
unbound. Figure 16 of Miville-Deschênes et al. (2017) already showed that
almost all the structures in their catalog are unbound, but this result was
not particularly emphasized. Nguyen-Luong et al. (2016) showed that mini-
starburst complexes, which have enhanced star formation rate density, are
actively forming massive stars but are not necessarily bound, suggesting that
being bound is not the major factor regulating their star formation activity.
The idea that structures identified by molecular emission are well-defined,
bound structures, unlike the more diffuse atomic interstellar medium, is long-
standing and creates cognitive dissonance with contrary factual evidence.
Uncertainties are often invoked to argue that clouds are bound even when the
evidence suggests otherwise (see Dobbs et al. 2011 for discussion of this
tendency.). However, there is no good reason to believe that the atomic-
molecular transition is identical to the unbound-bound transition. This would
be plausible if most of the support were thermal, but it is turbulent
(Zuckerman & Evans 1974). The fact that these transitions are close can be
coincidental (see pg. 36 of Elmegreen 1985) and may have led us astray.

We acknowledge stimulating discussions with many colleagues, but especially
with E. Ostriker and J-G. Kim. NJE acknowledges C. Dobbs, A. Burkert, and J.
Pringle for their paper that opened his eyes to the possibility that molecular
clouds are unbound. The authors acknowledge Paris-Saclay University’s Institut
Pascal program “The Self-Organized Star Formation Process” and the
Interstellar Institute for hosting discussions that nourished the development
of the ideas behind this work. We have selected a subset of clouds from the
Miville-Deschênes et al. (2017) catalog which fit into the coverage of the
13CO $J=1\rightarrow 0$ surveys from the Five College Radio Astronomy
Observatory in order to investigate the boundedness of these clouds using a
mostly optically thin tracer of H2 column density. To connect the 13CO
emission to the structures defined by the sum of Gaussian components (Miville-
Deschênes et al. 2017), we developed the following procedure. For a given
cloud, we calculate the fraction of the observed CO emission that is recovered
by the summed Gaussian for each position and velocity channel, $F(X,Y,V)$. A
typical profile of $F$ is flat-topped with values ranging from 0.9-1 in the
core of the line (see Figure 14). The fractional values rapidly decrease
towards the wings but remain non-zero for the full extent of the spectral
axis. To exclude spectral channels beyond the core and wing of a cloud, we set
$F(X,Y,|v-v_{cen}|>dv)=0$ where $v_{cen}$ and $dv$ are the velocity centroid
and full width half maximum from the catalog of Miville-Deschênes et al.
(2017). A 13CO data cube, $T_{13}(x,y,v)$, is constructed from the FCRAO
surveys at the native 48″ angular resolution and 22″ sampling interval to
match the CO coverage of the cloud. The spectral axis of the 13CO data is
smoothed and resampled to 0.25 km s-1 wide channels. At this stage the 13CO
data cube may contain emission features not associated with the CO-defined
cloud at velocities outside of the core of the CO emission (see e.g., the left
panel of Figure 14). To further isolate the 13CO emission, we first resample
the spectral axis of the fraction cube, $F$, to be aligned with the 13CO
spectral axis and then multiply the 13CO spectral axis with the corresponding
fractional spectrum,

$T_{13}(x,y,v)^{\prime}=F(X,Y,v)T_{13}(x,y,v)$ (1)

where $X-dX/2<x<X+dX/2$, $Y-dY/2<y<Y+dY/2$, and $dX$, $dY$ are the 0.125
degree pixel sizes of the CO data cube. The effect of this multiplication is
to suppress any contribution to the integrated 13CO emission from signal
outside of the CO velocity interval, as shown in Figure 14.

The cloud mass is computed from $N(\mbox{{\rm H}${}_{2}$})$ found from
equation 10 in §1 as follows:

$M(\mbox{${}^{13}$CO})=\mbox{$\mu_{\rm H_{2}}$}m_{\rm H}D^{2}\int d\Omega
N(\mbox{{\rm H}${}_{2}$})(l,b)$ (2)

where $\mbox{$\mu_{\rm H_{2}}$}=2.8$, $m_{\rm H}$ is the atomic hydrogen mass,
and $D$ is the distance to the cloud. For each cloud, the velocity dispersion,
$\sigma_{v}$(13CO) and cloud radius are calculated following Miville-Deschênes
et al. (2017). Both $\sigma_{v}$(13CO) and radius are deconvolved from the
native spectral and angular resolutions of the data. We also deconvolved the
CO velocity dispersions and cloud sizes. Clouds are excluded from any
subsequent analyses if either the cloud size or velocity dispersion is not
resolved. Finally, the 13CO-derived virial mass is derived from Equation 8.
Accounting for these selection criteria, there are 289 clouds from the GRS,
105 clouds from the EXFC55-100 survey, and 98 clouds from the EXFC135-195
survey.

Figure 14:  The method used to filter unrelated signal from 13CO. The spectrum
on the left is the original data. The center panel shows the function that
isolates the velocity range that is consistent with that of the CO-defined
structure. The right panel shows the filtered 13CO spectrum that contributes
to the 13CO-defined mass and other properties.

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