# Star cluster progenitors are dynamically decoupled from their parent self-
gravitating molecular clouds

Nicolas Peretto1 Andrew J. Rigby 2 Fabien Louvet 3 Gary A. Fuller4,5 Alessio
Traficante6 Mathilde Gaudel7
1 Cardiff Hub for Astrophysics Research & Technology, School of Physics &
Astronomy, Cardiff University, Queens Buildings, The parade, Cardiff CF24 3AA,
UK
2 School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK
3 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
4Jodrell Bank Center for Astrophysics, Department of Physics & Astronomy,
University of Manchester, Oxford Road,Manchester, M13 9PL, UK
5 I. Physikalisches Institut, University of Cologne, Zülpicher Str. 77, 50937
Köln, Germany
6 IAPS-INAF, Via Fosso del Cavaliere, 100, I-00133 Rome, Italy
7 LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne
Université, 75014, Paris, France E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The formation of stellar clusters dictates the pace at which galaxies evolve,
and solving the question of their formation will undoubtedly lead to a better
understanding of the Universe as a whole. While it is well known that star
clusters form within parsec-scale over-densities of interstellar molecular gas
called clumps, it is, however, unclear whether these clumps represent the
high-density tip of a continuous gaseous flow that gradually leads towards the
formation of stars, or a transition within the gas physical properties. Here,
we present a unique analysis of a sample of 27 infrared dark clouds embedded
within 24 individual molecular clouds that combine a large set of
observations, allowing us to compute the mass and velocity dispersion profiles
of each, from the scale of tens of parsecs down to the scale of tenths of a
parsec. These profiles reveal that the vast majority of the clouds, if not
all, are self-gravitating on all scales, and that the clumps, on parsec-scale,
are often dynamically decoupled from their surrounding molecular clouds,
exhibiting steeper density profiles ($\rho\propto r^{-2}$) and flat velocity
dispersion profiles ($\sigma\propto r^{0}$), clearly departing from Larson’s
relations. These findings suggest that the formation of star clusters
correspond to a transition regime within the properties of the self-
gravitating molecular gas. We propose that this transition regime is one that
corresponds to the gravitational collapse of parsec-scale clumps within stable
molecular clouds.

###### keywords:

stars: formation – ISM: kinematics and dynamics

††pubyear: 2015††pagerange: Star cluster progenitors are dynamically decoupled
from their parent self-gravitating molecular clouds–E

##  1 Introduction

Only a few years after the first detection of interstellar carbon monoxide,
Zuckerman & Evans (1974) showed that if all the gas within dense interstellar
clouds were to be freely collapsing as a result of their self-gravity then the
star formation rate in the Milky Way should be $\sim 300$ M⊙/yr, two orders of
magnitude larger than what it actually is ($\sim 2$ M⊙/yr - e.g. Robitaille &
Whitney, 2010). In other words, molecular clouds convert only $\sim 1\%$ of
their mass into stars every cloud free-fall time, making star formation a very
inefficient process (e.g. Krumholz & Tan, 2007). Despite five decades of star
formation research, the physics behind this fundamental property of molecular
clouds remain to be fully understood. Over the years, a number of competing
theories have been developed to explain the low star formation efficiency of
molecular clouds. The main differences between those models reside in both the
fraction of the volume/mass of any molecular cloud that undergoes
gravitational collapse, along with the dynamical state of the gas that does
not. In one family of models, supersonic turbulence is the one mechanism
responsible for defining the mass reservoirs accessible to individual
protostars and, as a result, for setting the stellar initial mass function
(e.g. Padoan et al., 1997; Padoan et al., 2020; Krumholz & McKee, 2005;
Hennebelle & Chabrier, 2008; Hopkins, 2012). In those models, the low star
formation efficiency is explained by the fact that those mass reservoirs
represent only a couple of percents of the molecular gas mass, the rest of the
gas is either unbound or in quasi-static equilibrium and therefore does not
directly participate to star formation. On the other hand, other models
predict that the hierarchical gravitational collapse of molecular clouds is
what drive their evolution (e.g. Hartmann & Burkert, 2007; Ballesteros-Paredes
et al., 2011; Vázquez-Semadeni et al., 2017, 2019) and that massive star
formation benefits from the favourable conditions generated by the global
collapse of dense clumps (e.g. Bonnell & Bate, 2006; Peretto et al., 2007;
Smith et al., 2009). In those models, what limits the efficiency of star
formation is stellar feedback from young low- and high-mass stars, by
stabilising or dispersing most of the molecular cloud’s mass (e.g Nakamura &
Li, 2007; Wang et al., 2010; Dale et al., 2012; Kim et al., 2018; Offner &
Liu, 2018; Grudić et al., 2022). The controversy around which of these two
very different scenarios of star formation describes reality best fuels the
majority of the star formation research for the past 20 years or so.

Figure 1: Images of SDC18.888-0.476. (a): top - Spitzer 8$\mu$m; middle- H2
column density from Herschel observations; bottom - N2H+(1-0) integrated
emission. The contours are identical in all panels, and are those of the H2
column density image. The three thicker white contours are those used to
compute the average N2H+(1-0) spectra displayed in magenta in panel (c). (b):
Multi-colour image of the molecular cloud hosting the SDC18.888-0.476 infrared
dark clump (white: 3.6 $\mu$m, orange: 8 $\mu$m, yellow: 70 $\mu$m, orange:
350 $\mu$m, blue: 1.42 GHz, red: H2 column density). The contours show the H2
column density obtained from the Galactic Ring Survey 13CO(1-0) data. The
thicker white contours are those used to compute the 13CO(1-0)-based spectra
shown in green in panel (c). The plus symbol shows the central position of the
IRDC. (c): Spectra averaged within the highlighted H2 column density contours
in panels (a) and (b). The radius of the region within which the spectra have
been averaged are indicated in each panel. The vertical blue dashed lines show
the systematic clump velocity as measured from N2H+(1-0) . The compilation of
the data presented in this figure summarises all the information used for each
cloud in the study presented here. A similar figure for each remaining IRDC
can be found in Appendix A.

A large number of studies have looked at the gravitational binding of
molecular clouds and their sub-structures within, most often via the
calculation of their virial parameters (e.g. Larson, 1981; Solomon et al.,
1987; Heyer et al., 2009; Roman-Duval et al., 2010; Kauffmann et al., 2013;
Schuller et al., 2017; Miville-Deschênes et al., 2017; Rigby et al., 2019).
Depending on the cloud sample that is being studied, the methods that are
being used, and the interpretation of the data that is being made, the
conclusions range from molecular clouds are: in hydro-static equilibrium,
collapsing, or unbound. As a result, a consensus as yet to be found.

A possibly more insightful analysis of molecular clouds is the study of their
internal virial ratio profiles. Indeed, if there is a scale/density threshold
at which the gravitational binding of clouds change from unbound to bound as a
result of, for instance, stellar feedback, then the virial ratio profiles of
individual clouds should exhibit some breaks at that particular scale. No one
has attempted yet to perform such analysis. Heyer et al. (2009) have measured
the properties of clouds using one single tracer, i.e. 13CO(1-0), at two
different radii and based on those, argued that clouds are likely to be in
quasi-static equilibrium. Traficante et al. (2018, 2020) have also measured
cloud properties at two different radii using, this time, different tracers
and argued that the size–velocity dispersion relationship within parsec-scale
clumps is flatter than what is observed on larger scales.

In this paper, we propose to derive, in a uniform way, the virial ratio
profiles of a sample of molecular clouds from scales of tenths of a parsec, up
to tens of parsecs in order to determine their dynamical state. In Sec. 2 we
present the source selection and observations. Section 3 explains how the
profiles of individual cloud are built. Section 4 presents the models we use
to determine the origin of the observed profile features. In Sec. 5 we discuss
our results while conclusions are laid out in Sec.6.

##  2 Source selection and observations

###  2.1 Sample

We selected a sample of 27 infrared dark clouds (IRDCs) from the Spitzer Dark
Cloud catalogue of Peretto & Fuller (2009). Compared to other cloud samples,
IRDCs have the advantage that their heliocentric distances are better
constrained, with a large majority of IRDCs lying at the near kinematic
distance solution provided by Galactic rotation models (Ellsworth-Bowers et
al., 2013). In this paper, the adopted distances for all IRDCs are the near
kinematic distance solutions from the Reid et al. (2009) model. The selection
criteria for these IRDCs are: (a) The kinematic distance as estimated from
${}^{13}\rm{CO}(1-0)$ GRS data (Roman-Duval et al., 2010) should be $d=4(\pm
1)$ kpc; (b) Selected IRDCs should exhibit a range of aspect ratios, i.e. from
circular to filamentary, as measured from Herschel column density images
(Peretto et al., 2016); (c) Selected IRDCs should exhibit a range of mass and
size as estimated from Herschel column density images; (d) all IRDCs have to
lie beyond $l=15\hbox{${}^{\circ}$}$ in order to be easily observed from the
IRAM 30m telescope. Global properties of the 27 selected clouds can be found
in Table 1. Note that kinematic distances have been recalculated using the
dense gas data presented in this paper, leading in a few cases to a departure
from condition (a). Figure 1a shows one of the selected IRDCs, images of the
remaining 26 can be seen in Appendix A.

Table 1: Infrared dark cloud sample

Cloud ID | Name | Coordinates | Systemic velocity | Distance
---|---|---|---|---
|  | (J2000) | (km/s) | (kpc)
1 | SDC18.624-0.070 | 18:25:10.0 -12:43:45 | +45.6 | 3.50
2 | SDC18.787-0.286 | 18:26:19.0 -12:41:16 | +65.4 | 4.36
3 | SDC18.888-0.476 | 18:27:09.7 -12:41:32 | +66.3 | 4.38
4 | SDC21.321-0.139 | 18:30:32.1 -10:22:50 | +66.5 | 4.24
5 | SDC22.373+0.446 | 18:30:24.5 -09:10:34 | +53.0 | 3.61
6 | SDC22.724-0.269 | 18:33:38.3 -09:11:55 | +73.3 (+105.0) | 4.44
7 | SDC23.066+0.049 | 18:33:08.2 -08:44:53 | +91.8 | 5.11
8 | SDC23.367-0.288 | 18:34:53.8 -08:38:00 | +78.3 (+103.0; +58) | 4.60
9 | SDC24.118-0.175 | 18:35:52.6 -07:55:06 | +80.9 | 4.68
10 | SDC24.433-0.231 | 18:36:41.0 -07:39:20 | +58.4 | 3.75
11 | SDC24.489-0.689 | 18:38:25.7 -07:49:36 | +48.1 | 3.28
12 | SDC24.618-0.323 | 18:37:22.4 -07:32:18 | +43.4 | 3.04
13 | SDC24.630+0.151 | 18:35:38.2 -07:18:35 | +53.2 (+115.0) | 3.51
14 | SDC25.166-0.306 | 18:38:13.0 -07:03:00 | +63.6 | 3.95
15 | SDC25.243-0.447 | 18:38:57.1 -07:02:20 | +59.1 | 3.75
16 | SDC26.507+0.716 | 18:37:07.9 -05:23:58 | +48.3 | 3.21
17 | SDC28.275-0.163 | 18:43:30.3 -04:12:45 | +80.3 | 4.60
18 | SDC28.333+0.063 | 18:42:54.1 -04:02:30 | +79.3 | 4.56
19 | SDC31.039+0.241 | 18:47:03.3 -01:33:50 | +78.2 (+98; +110) | 4.54
20 | SDC34.370+0.203 | 18:53:18.9 +01:24:54 | +57.9 | 3.59
21 | SDC35.429+0.138 | 18:55:30.4 +02:17:10 | +77.0 | 4.67
22 | SDC35.527-0.269 | 18:57:08.6 +02:09:08 | \+ 45.4 | 2.95
23 | SDC35.745+0.147 | 18:56:02.6 +02:34:44 | +83.4 | 5.11
24 | SDC38.850-0.427 | 19:03:46.8 +05:04:03 | +42.2 | 2.81
25 | SDC40.283-0.216 | 19:05:41.2 +06:26:09 | +72.7 | 4.89
26 | SDC47.061+0.257 | 19:16:41.8 +12:39:39 | +57.0 | 4.64
27 | SDC52.723+0.045 | 19:28:34.4 +17:34:17 | +44.1 | 4.49

  * •

Col. 1: IRDC identification number; Col. 2: IRDC name from Peretto & Fuller
(2009); Col. 3: central IRDC coordinates; Col. 4: Systemic LSR velocity of the
clump as estimated from N2H+(1-0), the velocities in between brackets
correspond to the additional components identified in the spectra; Col. 5:
Near kinematic distance as estimated from the Reid et al. (2009) model,
uncertainties on those are typically of the order of 10% to 20%.

###  2.2 Observations

In this study, we exploit four different datasets, each of which is tracing a
specific density regime of molecular clouds and/or giving us access to
different sets of information (mass versus kinematics). In the following, we
describe each of these datasets.

####  2.2.1  N2H+(1-0) data

We observed the 27 infrared dark clouds at the IRAM 30m between the 18th and
24th of June 2013, reaching a total of 42h of telescope time. The weather
conditions were stable with an average sky opacity at 230 GHz of 0.2. We
mapped each region using the 90 Hz EMIR receiver in conjunction with the FTS
spectrometer at 50 kHz spectral resolution, providing a velocity resolution of
0.16 km/s. Primary pointing and focus were performed on Saturn. The pointing
accuracy was $<5\hbox{${}^{\prime\prime}$}$. In this study we focus on the
N2H+(1-0) line, with an angular resolution of 28′′ . All data have been
reduced using the CLASS package, and gridded into 9′′ pixel-size cubes. The
final noise range from 0.09 K to 0.2 K per velocity channel and pixel.

####  2.2.2 Herschel data

We used the PACS (Poglitsch et al., 2010) and SPIRE (Griffin et al., 2010)
Herschel (Pilbratt et al., 2010) data from the Hi-GAL survey (Molinari et al.,
2010). The Hi-GAL data were reduced, as described in Traficante et al. (2011),
using HIPE (Ott, 2010) for calibration and deglitching (SPIRE only), routines
especially developed for Hi-GAL data reduction (drift removal, deglitching),
and the ROMAGAL map-making algorithm. Post-processing on the maps was applied
to help with image artefact removal (Piazzo et al., 2015). In this paper, we
make use of the PACS 160 $\mu$m and SPIRE 250/450/500 $\mu$m data with a
nominal angular resolution of 12′′, 18′′, 25′′, and 36′′, respectively. In
addition, zero-flux levels for every Hi-GAL field have been recovered by
correlating Herschel data with Planck and IRAS data (Bernard et al., 2010).

####  2.2.3 13CO(1-0) and 12CO(1-0) data

We used the FCRAO 13CO(1-0) data from the Galactic Ring Survey (GRS - Jackson
et al., 2006) along with the FCRAO UMSB 12CO(1-0) data (Sanders et al., 1986;
Clemens et al., 1986). The GRS data has an angular resolution of 44′′, a
velocity resolution of 0.21 km/s and a one $\sigma$ noise of 0.13K (in
$T_{A}^{*}$ scale). The main beam efficiency of the FCRAO telescope at the
13CO(1-0) frequency is 0.48. All clouds from our sample of 27 IRDCs are
covered by the GRS.

The UMSB 12CO(1-0) data has a nominal angular resolution of 44′′. However, the
data has been sampled on a 3′-grid, which effectively decreases the
resolution. The velocity resolution is 1 km/s, and the one $\sigma$ noise is
0.4K (in $T_{R}^{*}$) scale. In order to be able to convert that into a main
beam temperature one needs first to multiply by $n_{ffs}=0.7$ which converts
the unit back to $T_{A}^{*}$ (Kutner & Ulich, 1981; Sanders et al., 1986) and
then divide by the main beam efficiency 0.48, so effectively multiplying the
UMSB dataset by a (0.7/0.48) factor.

Figure 2: Sketch of molecular cloud configuration and relevant tracers.
Diffuse gas (represented in green) is traced by 13CO(1-0) and dust continuum.
However, only the former is able to disentangle the emission of multiple
clouds along the line of sight by segmenting them in velocity space. In this
paper, we will use both tracers to constrain the mass and morphology of the
clouds on the largest scales. Dense gas (represented in purple) is well probed
by both dust continuum and molecular line tracers such as N2H+(1-0). It is
very rare that two N2H+(1-0) cloud overlap (as the low frequency of multiple
N2H+(1-0) velocity components is showing). Dust continuum can therefore also
be used once a background contamination (from the diffuse gas) has been
removed.

##  3 Mass and velocity dispersion profiles

The goal of this paper is to determine how the ratio of kinetic to
gravitational energy of clouds changes as a function of spatial scale. In
order to observationally measure such ratio, one needs to determine three
quantities: radius, mass, and velocity dispersion. While the cloud mass can
reliably be determined via dust emission observations, no single molecular
line can trace molecular gas velocity dispersion on all scales, either because
of high optical depth or low abundance. We therefore need a combination of
tracers to trace different parts of the cloud. Here, we use 13CO(1-0) to trace
the large scales, more diffuse parts of the clouds, and N2H+(1-0) to trace
their densest parts. Figure 2 shows a simple sketch that illustrates what
tracer we use for what purpose. In the following subsections we describe how
we computed the three required quantities for both the dense and diffuse
regions of the clouds.

###  3.1 Dense gas

####  3.1.1 Herschel column density maps of IRDCs

For the purpose of this study, we computed H2 column density maps using the
method presented in Peretto et al. (2016, referred to as P16 hereafter). That
method consists in using the ratio of the Herschel 160 $\mu$m over 250 $\mu$m
dust emission to measure the temperature of the dust, and then use it, in
combination with the 250$\mu$m image to derive the column density of gas
(assuming a dust to gas mass ratio of 1%) at an angular resolution of 18". For
the purpose of the study presented here, we convolved the column density image
to the same angular resolution as the N2H+(1-0) data, i.e. 28". The assumed
specific dust opacity is
$\kappa_{{\lambda}}=0.1\left(\frac{\lambda}{300\hbox{$\mu$m}}\right)^{\beta}$
cm2 g-1 (Hildebrand, 1983), with $\beta=1.8$ (e.g. Planck Collaboration et
al., 2011; Sadavoy et al., 2016; Rigby et al., 2018).

When computing these maps, we make the assumption of a uniform temperature
along the line of sight. This is of course incorrect but it is not completely
clear though how wrong this assumption is for the structures we are studying.
Since we might expect this assumption of a single temperature to be the most
inaccurate towards the centre of each clump, we decided to compare the mass
profiles of each clumps obtained with P16’s method with that of PPMAP (Marsh
et al., 2015), a bayesian code that derive, from Herschel observations, the
distribution of dust temperatures along the line of sight. Note that we do not
use PPMAP in this paper as it can generate a number of artefacts around bright
protostellar sources, it is computationally expensive, and arising issues are
a lot less straightforward to identify than when using the P16’s method.

On the y-axis of Fig. 3 we show the ratio of the PPMAP over the P16 masses,
radially averaged. On the x-axis of the same figure we show the radial
dispersion of that same ratio, i.e. how much it varies about the average value
as a function of radius (i.e. 0% means that the ratio is radially uniform).
One can see that while, on average, the PPMAP masses are about 20% larger than
the P16 masses, the variations of the mass ratio as a function of radius are
small, and remain below 5% for most clouds, with a maximum standard deviation
of less than 8%. This shows that, while there might be a systematic
uncertainty on the mass of 20%, the shape of the mass profiles derived from
both methods are very much consistent with each other.

Figure 3: Ratio of the PPMAP masses over the P16 masses averaged over their
radial profiles as a function of their mass ratio standard deviation. The
black dashed lines show mass ratios of 1 and 1.5. Each colour corresponds to a
single IRDC whose ID number can be found at the top of the figure (see Table 1
for the corresponding IRDC name). Note that IRDC SDC31.039+0.241 (ID number
19) has been left out as a result of the presence of multiple dense clumps
present along the line-of-sight (see Sec. 3.1.2).

####  3.1.2 N2H+(1-0) as a tracer of Herschel clumps

All 27 IRDCs are detected in N2H+(1-0). For 4 of them ($\sim 15\%$), multiple
clouds with velocities differing by more than 20 km/s have been identified
within the observed field of views. For one of this IRDC (SDC31.039+0.241),
the N2H+(1-0) emission of the different clouds spatially overlap. This cloud
is therefore excluded from the rest of the analysis as the origin of the
corresponding dust continuum emission becomes very uncertain. Regarding the
remaining three clouds (SDC22.724-0.269, SDC23.367-0.288, SDC24.630+0.151), we
only consider the cloud for which the N2H+(1-0) integrated emission best
matches the extinction feature seen in the mid-infrared. The corresponding
velocities are provided in Table 1.

Another 4 IRDCs (SDC24.433-0.231, SDC24.630+0.151, SDC26.507+0.716,
SDC35.527-0.269) show multiple velocity components with velocity differences
lower than 3 km/s, only one of these also exhibits multiple clouds along the
line of sight (SDC24.630+0.151). However, once averaged within column density
contours (see Appendix A), the multiple velocity components are mostly washed
out, and are therefore not a concern in the context of this study. Note that
one of the multiple velocity component cloud, i.e. SDC35.527-0.269, has been
extensively studied in the past at high angular resolution clearly revealing
multiple velocity component structures (e.g. Henshaw et al., 2014).

The morphology of the N2H+(1-0) integrated intensity images are very similar
to that of the H2 Herschel column density maps (see Fig. 1), qualitatively
showing that N2H+(1-0) is a good tracer of the column density structure of
star-forming clouds. In order to quantify the correlation between dust column
density and N2H+(1-0) line emission we produced scatter plots for each cloud
of the H2 column density derived from Herschel, for which the background as
defined by N${}_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ (see next section) has been
subtracted, versus the integrated intensity of N2H+(1-0) (see Fig, 4 for four
representative examples). One can see there is, indeed, a strong linear
correlation between the two quantities, although the slopes can differ from
cloud to cloud. We observe such correlations for all clouds for which there is
enough dynamic range and independent points (i.e. 21/27 clouds). We have also
checked whether the relation provided by Hacar et al. (2018) between H2 column
density, N2H+(1-0) integrated intensities, and temperature hold for our cloud
sample. We can confirm that it does for most of the clumps, but some
significant departures are observed, which can be explained by a variation of
the N2H+ abundance by a factor of 2 or so. In fact, the two branches seen for
SDC28.333+0.063 in Fig. 4 likely correspond to two distinct regions within the
IRDC exhibit different physical conditions. Nevertheless, from this comparison
we can conclude that N2H+ is a good tracer of the dense gas as traced with
Herschel, and therefore that we can reasonably use it to trace the kinematics
of Herschel clumps for the (column) density range we are probing (i.e.
N${}_{\rm{H_{2}}}\geq 10^{22}$ cm-2). As such we do not expect the effect of
using different tracers for mass and kinematics to be a significant issue in
our study (see Traficante et al., 2018; Yuan et al., 2020).

Figure 4: Scatter plots of the background subtracted H2 column density versus
the N2H+(1-0) integrated intensity (in T${}_{a}^{*}$ scale) for four clumps.
In each panel the same linear relation is displayed as a dashed black line.
The correlation between the two quantities is obvious in all clouds, even
though the slope differ from cloud to cloud.

####  3.1.3 Mass and velocity dispersion estimates

The resulting H2 column density maps (see Fig. 1) are contaminated by
foreground and background interstellar structures which are not physically
associated to the cloud. Removing such contributions is not an easy task
(Peretto et al., 2010; Battersby et al., 2011). In the context of this study,
we are mostly interested in the part of the cloud which is seen in N2H+(1-0)
in the IRAM 30m data. Therefore we define the "edge" of the dense part of the
clouds as being the column density contour, $N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$,
that best matches the extent of the N2H+(1-0) integrated intensity map. This
is done by computing the median (along with the 16th and 84th percentiles)
column density value within a ring just outside the N2H+(1-0) integrated
intensity contour of 0.5 K km/s, i.e. our detection limit. The value of
$N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ will then serve as the background column
density of the clump that we will remove from any clump scale mass
measurements (see Table 2 for the individual values of
$N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ and corresponding 16th and 84th
percentiles).

We used the contour-based dendrogram tool from Peretto & Fuller (2009) on the
Herschel column density maps to estimate sizes and masses of connected groups
of pixels lying above a certain column density. In order to be considered for
the analysis those groups of pixels need to be larger than the number of
pixels within an angular resolution element, and need to be part of a
structure whose column density amplitude from local maximum to local minimum
is larger than a predefined threshold, $N_{\rm{H_{2}}}^{\rm{th}}$. The column
density increment we used in our dendrogram analysis is
$\sigma_{N_{\rm{H_{2}}}}=2\times 10^{21}$ cm-2 for all clouds, with
$N_{\rm{H_{2}}}^{\rm{th}}=5\sigma_{N_{\rm{H_{2}}}}$. The starting column
density contour, $N_{\rm{N_{2}H^{+}}}^{\rm{start}}$ (see Table 2) is set to be
larger or equal to $N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$, and is determined by
eye. The reason for not systematically having
$N_{\rm{N_{2}H^{+}}}^{\rm{start}}=N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ is that the
$N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ contour can be more extended than the
coverage of our N2H+(1-0) maps, and therefore, in such cases, the computed
masses would be overestimated. The mass of any identified group of pixels is
then given by:

$M_{\rm{N_{2}H^{+}}}=\Omega_{\rm{pix}}^{\rm{N_{2}H^{+}}}d^{2}\mu_{\rm{mol}}m_{\rm{H}}\sum_{i=1}^{n_{\rm{pix}}^{\rm{N_{2}H^{+}}}}\left(N_{\rm{H_{2}},i}-N_{\rm{N_{2}H^{+}}}^{\rm{edge}}\right)$
(1)

where the sum is on all the $n_{\rm{pix}}^{\rm{N_{2}H^{+}}}$ pixels belonging
to the group of interest, $\Omega_{\rm{pix}}^{\rm{N_{2}H^{+}}}$ is the solid
angle subtended by a pixel, $d$ is the distance to the IRDC, $\mu_{\rm{mol}}$
is the mean molecular weight and is set to 2.33, and $m_{\rm{H}}$ is the mass
of the hydrogen atom. The estimated radius associated to that group is
calculated via the following relation:

$R_{\rm{N_{2}H^{+}}}=\sqrt{\frac{n_{\rm{pix}}^{\rm{N_{2}H^{+}}}\Omega_{\rm{pix}}^{\rm{N_{2}H^{+}}}d^{2}}{\pi}}$
(2)

Projected masses estimated this way will always overestimate the mass enclosed
within the volume of radius $r$ as lower-density material along the line-of-
sight is wrongly associated to that volume (see Sec. 4). In Table 2 we provide
the radius, mass, and aspect ratio of each IRDC at the starting column density
contour $N_{\rm{N_{2}H^{+}}}^{\rm{start}}$. Note that we also give the mass
uncertainties related to the $16^{\rm{th}}-84^{\rm{th}}$ percentiles range of
$N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ values.

In order to estimate the dense gas velocity dispersion of the structures
identified in the dendrogram of the Herschel column density images, we
computed their corresponding N2H+(1-0) spectra, averaged over all
$n_{\rm{pix}}^{\rm{N_{2}H^{+}}}$ pixels that belong to the relevant group. We
then used a python routine, inspired from that of GILDAS/CLASS, that uses the
curvefit minimisation routine in order to fit the 7 hyperfine components of
the N2H+(1-0) transition. The parameters of the fit are: the central velocity
$\varv_{\rm{N_{2}H^{+}}}$, the velocity dispersion of the gas
$\sigma_{\rm{N_{2}H^{+}}}$, the sum of the central opacities of the 7
hyperfine components $\tau_{\rm{N_{2}H^{+}}}$, and the sum of the antenna
temperature peaks of the 7 hyperfine components $T_{\rm{ant}}^{\rm{tot}}$. The
velocity dispersion $\sigma_{\rm{N_{2}H^{+}}}$ is obtained from the best
fitting model. Examples of average N2H+(1-0) and their fit are displayed in
Fig. 1 for SDC18.888 (and in Appendix A for the other clouds). We fitted only
one velocity component to all spectra, even for those showing potential
multiple velocity components as those are almost systematically blended once
the emission is averaged within each contour. The velocity dispersions
$\sigma_{\rm{N_{2}H^{+}}}^{\rm{start}}$ estimated within the column density
contour $N_{\rm{N_{2}H^{+}}}^{\rm{start}}$ are given in Table 2.

Table 2: IRDCs and parent cloud properties

ID | $N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ | $N_{\rm{N_{2}H^{+}}}^{\rm{start}}$ | $R_{\rm{N_{2}H^{+}}}^{\rm{start}}$ | $M_{\rm{N_{2}H^{+}}}^{\rm{start}}$ | $\sigma_{\rm{N_{2}H^{+}}}^{\rm{start}}$ | $\alpha_{\rm{vir}}^{\rm{N_{2}H^{+}}}$ | $R_{\rm{{}^{13}CO}}^{\rm{start}}$ | $M_{\rm{{}^{13}CO}}^{\rm{start}}$ | $\sigma_{\rm{{}^{13}CO}}^{\rm{start}}$ | $\alpha_{\rm{vir}}^{\rm{{}^{13}CO}}$
---|---|---|---|---|---|---|---|---|---|---
| ($\times 10^{22}$ cm-2) | ($\times 10^{22}$ cm-2) | (pc) | (M⊙) | (km/s) |  | (pc) | ($\times 10^{4}$ M⊙) | (km/s) |
1 | $2.8^{+0.3}_{-0.2}$ | 3.3 | 1.66 | $3239^{+365}_{-546}$ | 0.99 | 0.62 | 12.62 | 14.52 | 5.74 | 3.35
2 | $2.4^{+0.4}_{-0.4}$ | 3.3 | 1.18 | $1807^{+365}_{-365}$ | 1.06 | 0.89 | 21.54 | 48.36 | 4.19 | 0.91
3 | $1.9^{+0.3}_{-0.3}$ | 3.7 | 1.61 | $7882^{+513}_{-513}$ | 1.55 | 0.58 | 21.63 | 47.30 | 3.72 | 0.74
4 | $3.2^{+0.3}_{-0.5}$ | 3.3 | 1.09 | $562^{+395}_{-237}$ | 0.69 | 1.20 | 9.02 | 3.71 | 2.81 | 2.25
5 | $2.3^{+0.4}_{-0.5}$ | 2.9 | 0.63 | $437^{+101}_{-81}$ | 1.13 | 2.03 | 3.46 | 0.31 | 1.39 | 2.61
6 | $3.3^{+0.7}_{-0.5}$ | 4.6 | 0.60 | $533^{+118}_{-164}$ | 0.65 | 0.62 | 11.58 | 11.85 | 4.50 | 2.32
7 | $3.2^{+0.4}_{-0.3}$ | 3.6 | 1.48 | $1536^{+434}_{-579}$ | 0.98 | 1.14 | 16.27 | 30.40 | 5.90 | 2.18
8 | $4.2^{+0.5}_{-0.5}$ | 6.1 | 0.72 | $1575^{+172}_{-172}$ | 1.17 | 0.76 | 17.76 | 27.61 | 3.67 | 1.02
9 | $2.6^{+0.3}_{-0.4}$ | 2.9 | 1.32 | $1318^{+462}_{-346}$ | 1.03 | 1.31 | 6.46 | 2.02 | 1.63 | 1.00
10 | $2.9^{+0.5}_{-0.5}$ | 3.1 | 1.39 | $3282^{+642}_{-642}$ | 1.50 | 1.14 | 5.23 | 2.31 | 2.88 | 2.21
11 | $1.2^{+0.2}_{-0.2}$ | 1.4 | 1.12 | $831^{+167}_{-167}$ | 1.13 | 2.08 | 9.19 | 2.18 | 1.67 | 1.39
12 | $2.0^{+0.3}_{-0.1}$ | 2.5 | 0.82 | $594^{+45}_{-134}$ | 0.85 | 1.26 | 12.67 | 5.85 | 1.63 | 0.68
13 | $2.7^{+0.5}_{-0.3}$ | 4.4 | 0.48 | $472^{+45}_{-75}$ | 1.08 | 1.42 | 6.80 | 3.34 | 3.58 | 3.06
14 | $2.2^{+0.2}_{-0.2}$ | 2.5 | 1.68 | $3622^{+374}_{-374}$ | 0.96 | 0.53 | 8.65 | 3.23 | 1.78 | 1.00
15 | $1.8^{+0.2}_{-0.2}$ | 2.0 | 1.33 | $1233^{+235}_{-235}$ | 0.84 | 0.95 | 13.11 | 13.35 | 4.23 | 2.05
16 | $1.7^{+0.3}_{-0.3}$ | 1.8 | 1.62 | $1474^{+518}_{-518}$ | 1.02 | 1.40 | 7.28 | 3.20 | 1.44 | 0.56
17 | $3.0^{+0.5}_{-0.3}$ | 3.7 | 1.53 | $2687^{+465}_{-775}$ | 1.27 | 1.10 | 31.13 | 98.02 | 4.31 | 0.69
18 | $4.4^{+0.4}_{-0.4}$ | 5.1 | 2.65 | $13954^{+1863}_{-1863}$ | 1.42 | 0.46 | 30.85 | 103.70 | 5.27 | 0.97
20 | $2.7^{+0.5}_{-0.4}$ | 3.5 | 2.15 | $12048^{+1223}_{-1528}$ | 1.34 | 0.39 | 11.33 | 18.13 | 3.79 | 1.05
21 | $2.7^{+0.4}_{-0.3}$ | 3.0 | 2.62 | $7761^{+1366}_{-2276}$ | 1.33 | 0.72 | 11.24 | 4.01 | 3.20 | 3.37
22 | $2.2^{+0.3}_{-0.2}$ | 2.7 | 1.36 | $1499^{+245}_{-367}$ | 0.77 | 0.68 | 8.22 | 3.49 | 1.89 | 0.99
23 | $1.9^{+0.1}_{-0.2}$ | 2.0 | 2.01 | $3490^{+536}_{-268}$ | 1.07 | 0.80 | 11.58 | 7.66 | 3.74 | 2.47
24 | $2.7^{+0.2}_{-0.2}$ | 2.9 | 0.77 | $276^{+79}_{-79}$ | 0.62 | 1.42 | 8.46 | 4.29 | 1.52 | 0.55
25 | $2.1^{+0.2}_{-0.2}$ | 2.2 | 1.81 | $4044^{+432}_{-432}$ | 1.47 | 1.15 | 11.06 | 5.00 | 1.32 | 0.46
26 | $1.7^{+0.5}_{-0.5}$ | 2.2 | 1.35 | $1982^{+601}_{-601}$ | 1.29 | 1.36 | 13.34 | 12.67 | 2.48 | 0.76
27 | $1.7^{+0.2}_{-0.2}$ | 1.7 | 1.37 | $358^{+247}_{-247}$ | 0.67 | 2.20 | 17.31 | 16.24 | 2.37 | 0.71

  * •

Col. 1: IRDC identification number; Col. 2: H2 column density matching the
edge of the $N_{2}$H+(1-0) emission (median, 16th and 84th percentiles); Col.
3: H2 column density from which the dendrogram tree starts; Col. 4: IRDC
radius corresponding to $N_{\rm{N_{2}H^{+}}}^{\rm{start}}$; Col. 5: Gas mass
within $R_{\rm{N_{2}H^{+}}}^{\rm{start}}$, uncertainties reflect the mass
changes when considering the 16th and 84th percentiles of
$N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$; Col. 6: N2H+(1-0) velocity dispersion
estimated within $R_{\rm{N_{2}H^{+}}}^{\rm{start}}$; Col. 7: virial ratio
estimated within $R_{\rm{N_{2}H^{+}}}^{\rm{start}}$; Col. 8: Parent cloud
radius; Col. 9: Gas mass within $R_{\rm{{}^{13}CO}}^{\rm{start}}$; Col. 10:
13CO(1-0) velocity dispersion estimated within
$R_{\rm{{}^{13}CO}}^{\rm{start}}$; Col. 11: virial ratio estimated within
$R_{\rm{{}^{13}CO}}^{\rm{start}}$

###  3.2 Diffuse gas

One key aspect of this study is to estimate cloud properties on a large range
of scales. While we presented in the previous section how we estimate the
properties of the clumps, we want now to connect these to their more diffuse
envelopes. To achieve this we use a combination of the Herschel dust continuum
data and the 13CO(1-0) GRS data (Roman-Duval et al., 2010). While the latter
is primarily used to derive gas velocity dispersion, we also use it to derive
the morphology of the molecular clouds along with the line-of-sight mass
contamination on our Herschel-based mass estimates (see Fig. 2). For this
purpose, we first need to compute 13CO-based H2 column density cubes.

####  3.2.1 13CO column density cubes

The Galactic Ring Survey 13CO(1-0) data have been used to compute H2 column
density cubes. To do this, we followed the exact same procedure as in Roman-
Duval et al. (2010). We give here a short description of this method. Towards
each cloud of our sample, we compute the excitation temperature of the
13CO(1-0) line by using the 12CO(1-0) data from the UMSB survey and assuming
that this line is optically thick. Then we make the further assumption that
the excitation temperature of the 12CO(1-0) line is the same as that of the
13CO(1-0) line. With these assumptions, we can compute the 13CO(1-0)
excitation temperature for every voxel. In some cases, where there are strong
density/temperature gradients not traced by 12CO(1-0) (because it is optically
thick) but traced by 13CO(1-0) the estimated excitation temperature is not
high enough. In such cases ($\sim 1\%$ of the voxels) we artificially increase
the excitation temperature by 10%, enough to get the excitation temperature
larger than the 13CO(1-0) brightness temperature everywhere in our cloud
sample. Combining the excitation temperature with the 13CO(1-0) cube one can
compute the 13CO(1-0) opacity which is then converted into a 13CO column
density, and finally into a H2 column density assuming a constant 13CO
abundance with respect to H2 of $1.8\times 10^{-6}$ for all clouds (Blake et
al., 1987; Langer & Penzias, 1990). This procedure provides us with cubes of
H2 column density for each IRDC. Note that some IRDCs are embedded within the
same molecular clouds, and as a result we end up with 27 IRDCs embedded within
24 individual molecular clouds.

####  3.2.2 Mass and velocity dispersion estimates

For each molecular cloud we estimate two mass profiles, one using our
Herschel-based column density images (see Sec. 3.1.1) and one using the 13CO-
based column density cubes. Each of them are affected by different biases that
can be, at least partially, removed by using the combination of both datasets.
On one hand, far-infrared dust continuum emission of Galactic plane molecular
clouds is mostly optically thin and traces the entire ISM, but suffers from
line-of-sight confusion (see Fig. 2). As a result, one cannot determine
correctly the morphology of the clouds using our ${\it Herschel}$-based column
density images and mass estimates are likely to be overestimated. On the other
hand, 13CO-based column density cubes permit the identification of individual
molecular clouds along the velocity axis, but can suffer from opacity effects
and abundance variations. Because our mass estimates on clump scale are based
on the Herschel column density images, we have decided to do the same for the
diffuse parts of the clouds. However, we use the 13CO-based column density
cubes to determine their morphologies, along with determining the percentage
of multiple cloud line-of-sight mass contamination in our Herschel-based mass
measurements.

Using our 13CO-based column density cubes, we first produced a 2D H2 column
density map by integrating the cube in a 5 to 10 km/s window centred on the
cloud systemic velocity (see Fig. 1 and Appendix A). Then, we ran the same
dendrogram analysis on this map as the one used on the clumps. We then
produced the average13CO-based H2 column density spectra for all identified
groups of connected pixels. While the resulting spectra displayed in Fig. 1
for the SDC18.888-0.476 cloud show a relatively simple (even though non-
Gaussian) single-peaked emission line, most clouds exhibit rather complex
spectra often exhibiting multiple components with overlapping column density
wings (see Appendix A).

With this in mind, we used multiple-Gaussian, up to a maximum of 4 components,
to fit each spectrum using the python curvefit function (we also tested two
other methods - see Appendix B). The mass $M_{\rm{{}^{13}CO}}$ and velocity
dispersion $\sigma_{\rm{{}^{13}CO}}$ are then estimated using the following
equations for the velocity dispersion:

$\sigma_{\rm{{}^{13}CO}}=\sqrt{\sum_{i}w_{i}\left[(\varv_{i}-\bar{\varv})^{2}+\sigma_{i}^{2}\right]}$
(3)

where the sum is over the Gaussian components, and $w_{i}$, $\varv_{i}$, and
$\sigma_{i}$ are the weight, the central velocity and velocity dispersion of
the $i^{\rm{th}}$ component, respectively. The centroid velocity $\bar{\varv}$
is obtained by:

$\bar{\varv}=\sum_{i}w_{i}\varv_{i}$ (4)

And the weights are defined by:

$w_{i}=\frac{m_{i}}{\sum_{i}m_{i}}$ (5)

where $m_{i}$ is the mass resulting from the integration of each individual
Gaussian component, and:

$M_{\rm{{}^{13}CO}}=\sum_{i}m_{i}$ (6)

The velocity dispersion calculated via Eq. (3) includes two terms, i.e. the
velocity dispersion from individual Gaussian components, along with the
component-to-component centroid velocity dispersion. This is justified by the
fact that we are here interested in estimating the entire kinetic energy
budget of the clouds we are analysing. Note also that only the Gaussian
components that we believe belong to the cloud of interest are used for the
determination of the mass and velocity dispersion. Those are identified by
integrating, separately, each 13CO(1-0) emission peak and visually evaluate
what peak best matches the morphology of the embedded IRDC. It is possible
though that different components that we consider as being part of different
molecular clouds are physically interacting with each other via, e. g., cloud-
cloud collision (as it has been argued for the SDC18.624-0.070 cloud -
Dewangan et al., 2018). Such interactions can lead to the creation of
intermediate velocity gas (Haworth et al., 2015; Bisbas et al., 2017) for
which it might become difficult to determine to which cloud it belongs,
potentially leading to large uncertainties in the estimate of
$\sigma_{\rm{{}^{13}CO}}$.

The radius of each dendrogram’s connect group of pixels is given by:

$R_{\rm{{}^{13}CO}}=\sqrt{\frac{n_{\rm{pix}}^{\rm{{}^{13}CO}}\Omega_{\rm{pix}}^{\rm{{}^{13}CO}}d^{2}}{\pi}}$
(7)

where $n_{\rm{pix}}^{\rm{{}^{13}CO}}$ is the number of pixels within each
connected group of pixels and $\Omega_{\rm{pix}}^{\rm{{}^{13}CO}}$ is the
solid angle subtended by a pixel. In parallel to these 13CO-based mass
estimates, we derive Herschel-based ones. For this we use the exact same
connected groups of pixels as those used above, but this time we use our
Herschel-based H2 column density maps to obtain the masses via:

$M_{Hers.}^{\rm{unc}}=\Omega_{\rm{pix}}^{\rm{{}^{13}CO}}d^{2}\mu_{\rm{mol}}m_{\rm{H}}\sum_{i=1}^{n_{\rm{pix}}^{\rm{{}^{13}CO}}}N_{\rm{H_{2}},i}$
(8)

where all parameters are identical to those presented in Eq. (1) and
$M_{Hers.}^{\rm{unc}}$ stands for uncorrected Herschel-based masses. The
reason why those are uncorrected is due to the contamination of the mass
estimates by the presence of multiple clouds along the line-of-sight. One can
correct for this by estimating the fraction $f_{\rm{los}}$ of the total mass
of molecular clouds along the line-of-sight that is locked up within the cloud
of interest. That can be achieved by integrating the 13CO-based H2 column
density spectra across the entire GRS velocity range, along with integrating
the best-fit Gaussian model for the cloud of interest. This can be formulated
as:

$f_{\rm{los}}=\frac{\int_{\rm{model}}N_{\rm{H_{2}}}^{\rm{{}^{13}CO}}d\varv}{\int_{\rm{all}}N_{\rm{H_{2}}}^{\rm{{}^{13}CO}}d\varv}$
(9)

This correction factor can be calculated for each dendrogram group of
connected pixels and then be applied to the uncorrected masses via:

$M_{Hers.}^{\rm{corr}}=f_{\rm{los}}M_{Hers.}^{\rm{unc}}$ (10)

Figure 5 shows a comparison of the ratio of
$M_{\rm{{}^{13}CO}}/M_{Hers.}^{\rm{unc}}$ versus $M_{Hers.}^{\rm{corr}}$ and
$M_{\rm{{}^{13}CO}}/M_{Hers.}^{\rm{corr}}$ versus $M_{Hers.}^{\rm{corr}}$.
This figure clearly shows the vast improvement in the mass agreement once the
correction factor is being applied. After correction, the masses agree within
less than a factor 2 and we see little evidence for significant 13CO
depletion, at least not on those scales. This excellent agreement also
indicates that one can safely use the 13CO(1-0) velocity dispersion
measurements in conjunction with the Herschel-based cloud masses. In the rest
of this paper we will be using the Herschel-based corrected masses.

Figure 5: Comparison between the ratio of 13CO-based and Herschel-based
uncorrected/correct (top/bottom) cloud masses for all clouds, at all radii, as
a function of the corrected Herschel-based cloud masses.

###  3.3 Combined profiles

Figure 6: Profiles of all 26 infrared dark clouds and their parent molecular
clouds from our sample. On the top row the purple points represent those for
which the clump scale velocity dispersion has been measured using N2H+(1-0),
and the green points are those for which the cloud scale velocity dispersion
has been measured using 13CO(1-0). The middle and bottom rows show the same
data point as the top row but each individual cloud/clump has a unique colour
so that one can track their profiles. Half of the clouds have been plotted in
each for clarity. (left): mass profiles $m(r)$; (middle): velocity dispersion
profiles $\sigma_{\rm{tot}}(r)$; (right): virial ratio profiles
$\alpha_{\rm{vir}}(r)$.

In this paper, we adopt a top-down approach by which, for every column density
contours, we only analyse the one group of connected pixels that covers the
position of the IRDC Herschel-based column density peak. As a result, sibling
clumps that might be part of the same molecular clouds as our IRDC sample are
not separately analysed, even though they contribute to the mass and velocity
dispersion of the dendrogram structures that encompass both them and the
targeted IRDC.

Figure 6 shows the mass profiles $m(r)$ and the velocity dispersion profiles
$\sigma_{\rm{tot}}(r)$ for the 26 clumps of our sample and their parent
molecular clouds. For the measurement on clump scales (the purple lines) we
have $m(r)=M_{\rm{N_{2}H^{+}}}\left(R_{\rm{N_{2}H^{+}}}\right)$, while for the
measurements on cloud scale (the green lines), we have
$m(r)=M_{\rm{{}^{13}CO}}\left(R_{\rm{\rm{{}^{13}CO}}}\right)$. Also, the
velocity dispersion $\sigma_{\rm{tot}}$ is the total (thermal+turbulent) line-
of-sight velocity dispersion of the gas and is estimated using the observed
velocity dispersion via:

$\sigma_{\rm{tot}}^{2}=\sigma_{\rm{line}}^{2}+k_{B}T\left(\frac{1}{\mu_{\rm{mol}}m_{\rm{H}}}-\frac{1}{m_{\rm{mol}}}\right)$
(11)

where $\sigma_{\rm{line}}$ is the observed velocity dispersion of the gas as
inferred from the observation of a given molecular line (N2H+(1-0) for the
purple points, and 13CO(1-0) for the green points), $T$ is the gas
temperature, $m_{\rm{mol}}$ is the mass of the observed molecule (here
$m_{\rm{mol}}=m_{\rm{N_{2}H+}}=m_{\rm{{}^{13}CO}}=29m_{\rm{H}})$,
$\mu_{\rm{mol}}$ is the molecular weight which is here taken to be 2.33, and
$m_{\rm{H}}$ is the mass of the hydrogen atom. We here assume a gas
temperature of 15 K for all clouds, which is the average temperature measured
within infrared dark clouds (e.g. Peretto et al., 2010; Battersby et al.,
2011). The impact of that assumption is negligible for most velocity
dispersion measurements, and only have a measurable impact for velocity
dispersions $\leq 1$ km/s. Finally, the right-hand-side panel of Fig. 6 shows
the corresponding viral ratio profiles $\alpha_{\rm{vir}}(r)$. Virial ratios,
defined as $\alpha_{\rm{vir}}=2E_{\rm{K}}/|E_{\rm{G}}|$ with $E_{K}$ the
kinetic energy of the gas and $E_{\rm{G}}$ its gravitational energy, provide a
zero-order measure of a cloud dynamical state. While the kinetic energy of a
cloud can be relatively easily estimated, the estimate of its gravitational
energy usually requires to make simplifying assumptions on the morphology and
density profile of the cloud. Bertoldi & McKee (1992) have evaluated
$E_{\rm{G}}$ in case of different power law densities and different cloud
aspect ratios. They show that for cloud with aspect ratios lower than 10 (as
it is the case in this study) $|E_{\rm{G}}|$ is only decreased by a maximum of
8% compared to the spherical case. However, for clouds that have power law
density such as $\rho\propto r^{-\gamma}$ with $\gamma=2$, $|E_{\rm{G}}|$ is
increased by 67%. The impact of the density gradient on $|E_{\rm{G}}|$ is
stronger than the non-sphericity of the cloud. For simplicity, most studies of
the virial ratio of molecular clouds usually approximate them as uniform
density spheres, which is also what we will do, and discuss correction factors
later. In this case, one can show that the virial ratio $\alpha_{\rm{vir}}(r)$
is given by:

$\alpha_{\rm{vir}}(r)=5\frac{\sigma_{\rm{tot}}^{2}r}{Gm}$ (12)

Figure 6 shows a number of important features. In the following we will
discuss those separately.

####  3.3.1 Observed mass profiles

The mass profiles presented in Fig. 6 spread over 4 orders of magnitude in
mass and 2 orders of magnitude in radius. The masses estimated on cloud scale
(the green lines) and clump scale (the purple lines) mostly connect at a scale
of about 2 pc, which corresponds to the maximum extent of the N2H+(1-0)
emission and half the resolution ($3^{\prime}$) of the 13CO-based column
density images that we use to derive the morphology of the clouds. Note that,
even though derived from the same Herschel-based H2 column density maps, the
masses on clump and cloud scales do not produce continuous mass profiles. The
reason for this is that we are removing the column density background
$N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ to every clump scale mass measurements so
that the velocity dispersion estimate is that of the measured gas mass.
Finally, when looking at shapes of the profiles, we notice that the clump
scale mass profiles are more curvy and and exhibit shallower gradients than
those from the more diffuse parts.

####  3.3.2 Observed velocity dispersion profiles

The velocity dispersion profiles presented in Fig. 6 are the most striking.
First, there is a clear discontinuity between the velocity dispersion
measurements obtained on clump scale and those obtained on cloud scale. Having
such different measurements clearly indicates that there is a systematic bias
in the method that is being used to perform those measurements. Second, the
shapes of the profiles are also strikingly different. While on the largest
scale, the velocity dispersion mostly decreases with decreasing radius, on the
smallest scale, the velocity dispersion profiles are mostly flat. This is very
different from a typical Larson-type relation (Larson, 1981) for which we
would expect the velocity dispersion to decrease down to the sonic-scale at
about 0.1 pc.

####  3.3.3 Observed virial ratio profiles

Since the virial ratio profiles presented in Fig. 6 are built from the mass
and velocity profiles, they carry similar features. For instance, the virial
ratios present a discontinuity at the around $r=2$ pc, which is the
consequence of the discontinuity observed in the velocity dispersion profiles.
Note, however, that this discontinuity is attenuated as a result of the
slightly larger masses estimated from the cloud scale measurements at that
radius. Also, it is pretty clear that for most of the clouds, the virial
ratios on the large scales (green lines) increase as the radius decreases.
This trend has already been observed by Hernandez & Tan (2015) who interpreted
it as a sign of CO depletion. Finally, the virial ratios estimated on clump
scales (purple lines) present a curvy shape, which is the direct reflection of
the curvy mass profiles observed on the same scales.

Figure 7: (a) Sketch illustrating measurement biases once a 3D cloud is being
projected onto the plane of sky. In this case, a spherical cloud (on the
right) whose volume density increases towards the centre has a projected
column density profile that is represented by the plot on the left. The
bjjective mass estimate within projected radius $R$ will be that represented
by the green shaded area, which includes material that is not part of the
volume of the sphere of radius $R$ (region of the cloud that is barred). (b):
Normalised mass profiles of a spherical cloud with 3 different density profile
$\rho\propto r^{-\gamma}$, with $\gamma=(2.0,1.5,1.0)$. The solid lines show
the masses as observed, while the dashed lines show the real mass enclosed
within a given radius. (c): Normalised velocity dispersion profiles of a
spherical with the same density profiles as in (b) and with a velocity
dispersion profile $\sigma\propto r^{\beta}$ with $\beta=0.5$.

###  3.4 Uncertainties

There are a number of uncertainties that we need to consider when interpreting
the profiles presented in Fig. 6. First, there are uncertainties that do not
affect the shape of the profiles but do impact their overall scaling. One
example of such uncertainty is the distance to the clouds which is typically
10% to 20% (Reid et al., 2009). If our IRDC-hosting cloud are not located at
the near distance though then the distance could be 4 times larger for some
clouds (see Fig. 48). That uncertainty will impact the mass and radius
measurements uniformly across the profile of an individual cloud. Second,
there are uncertainties that can potentially impact the shape of individual
profiles. Regarding the mass profiles, the assumption of a single temperature
along the line-of-sight could potentially have an impact on the shape of the
observed profiles. However, as we have shown in Sec. 3.1.3, the impact on the
shape of the profile is minimal, while the impact on the absolute mass values
can be impacted by $20\%$ on average. Another uncertainty is related to the
dust emissivity, i.e. $\kappa_{\lambda}$, we used when computing the Herschel-
based H2 column density maps. In this study we used the same dust emissivity
law for the clump scale and cloud scale measurements. It is however well known
that dust emissivity changes with density and temperature (e.g. Ysard et al.,
2015; Sadavoy et al., 2016). At this point we have no means to set strong
constraints on this particular aspect of dust property uncertainties, but the
law we adopted has been shown to be compatible with dust emission in both the
more diffuse (Planck Collaboration et al., 2011) and denser (Rigby et al.,
2018) gas environments. Also, as it can be seen in Fig. 5, the Herschel-based
masses are within a factor of two of the 13CO-based masses which use a
completely different set of assumptions. This suggests that, if dust
properties do change across the radial profiles of molecular clouds, this does
not have a dramatic effect on our mass estimates. Finally, the uncertainty
related to our choice of $N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ (see Sec. 3.1.3)
has a direct impact on the clump scale mass estimates with a $\sim 10\%$ to
$\sim 30\%$ uncertainty for most clumps (see Table 2). This fractional mass
uncertainty is not constant across the clump radial profiles and therefore
might affect the mass profile shape. However, after computing the clump mass
profiles with a representative range of $N_{\rm{N_{2}H^{+}}}^{\rm{edge}}$ we
can confirm that it barely impacts their overall shapes.

Regarding uncertainties on the velocity dispersion, the N2H+(1-0) and
13CO(1-0) measurements differ. Indeed, the N2H+(1-0) velocity dispersion
measurements are very well constrained, and have uncertainties that are of the
order of $\sim 0.1$ km/s. This implies that the flat velocity dispersion
profiles observed on clump scale are very robust. Uncertainties on the
13CO(1-0) velocity dispersion measurements are a lot more variable from cloud-
to-cloud depending on how complex the 13CO(1-0) spectra are. For the simple
cases, such as SDC18.888-0.476 (see Fig. 1) the uncertainty is of the order of
$\sim 0.2$ km/s, however is can be as high as $\sim 1$ km/s in more complex
cases such as SDC18.624-0.070 (see Fig. A1). These larger uncertainties are
also reflected by the large differences in velocity dispersion measurements
when using different evaluation methods (see Fig. 46).

Overall, while the inherent uncertainties on the different quantities
presented in Fig. 6 might shift the profiles up and down, their shapes are
fairly robust and are likely to be a true representation of how the projected
mass, velocity dispersion, and virial ratio profiles of clumps and clouds
behave.

##  4 Spherical models

As discussed above, the profiles displayed in Fig. 6 present a number of
characteristic features. Before interpreting them one needs to be aware a few
biases that exist and that we may be able to quantify. First, masses, as
presented in the left-hand-side panel of Fig. 6, have been computed using the
bijective mass estimates (Rosolowsky et al., 2008). Such masses are always
overestimated as a consequence of cloud material lying along the line-of-sight
which is not part of the closed volume of radius $r$ (see Fig. 7). The impact
of using the bijective method to estimate masses at different radius is
illustrated in Fig. 7b. Second, the velocity dispersion measurements are being
done on spectra that also include the same unrelated line-of-sight material
which may have larger or smaller velocity dispersion than the gas lying within
the volume of interest. Depending on the exact shape of the combined density
and velocity dispersion profiles, this might lead to over or underestimated
observed velocity dispersions (see Fig. 7c). Also, the clump and cloud scale
measurements are derived at different angular resolutions, and a background
column density has been subtracted to the former and not to the latter. The
impact of all those on the observed profiles is unclear. The purpose of the
models presented in the rest of this section is to quantify the impact of
projection on the mass and velocity dispersion measurements in relation to the
observed profiles (for a similar approach on core scale see Singh et al.,
2021). We do not attempt to fit the profiles of individual clouds as spherical
clouds are not a good representation of the complex density structures of
molecular clouds.

Figure 8: Mass, velocity dispersion, and virial ratio profiles (from left to
right), for three different $\gamma$ values (from top to bottom:
$\gamma=[1.0,1.5,2.0]$). The grey lines are the same observed data points as
presented in Fig. 6. The red, orange, and gold lines are three different
spherical models with three different normalisations such that they cover the
range of masses and velocity dispersions as measured on parsec scale (i.e. on
scales representative of $R_{\rm{N_{2}H^{+}}}^{\rm{start}}$ in Table 2). All
models have the same velocity dispersion profile exponent, i.e. $\beta=0.5$.

###  4.1 Single power-law profiles

We first consider models with single density and velocity dispersion power-law
such as:

$\rho(r)=\rho_{0}\left(\frac{r}{r_{0}}\right)^{-\gamma}$ (13)

$\sigma(r)=\sigma_{0}\left(\frac{r}{r_{0}}\right)^{\beta}$ (14)

where $\rho_{0}$, $r_{0}$ and $\sigma_{0}$ are normalisation constants. For a
given pair of $\gamma$ and $\beta$ values, we numerically construct a
spherical cloud of a given mass and radius that we then project on the plane-
of-the-sky in order to construct mass surface density maps (see Appendix C for
more details). We do this last operation twice, once up to radius
$R_{\rm{edge}}^{\rm{{}^{13}CO}}$ and once up to radius
$R_{\rm{edge}}^{\rm{N_{2}H^{+}}}$, $R_{\rm{edge}}^{\rm{{}^{13}CO}}$ and
$R_{\rm{edge}}^{\rm{N_{2}H^{+}}}$ being the radii at which
${}^{13}\rm{CO}$(1-0) and $\rm{N_{2}H^{+}}$(1-0) emission becomes
undetectable. We then convolve each mass surface density images at the
resolution of our observations. Finally, we integrate both mass surface
density images at various radii to derive their projected mass profiles.
Regarding the velocity dispersion profile, we first weight the velocity
dispersion at each radius, in 3D, by the local mass density. We then project
this quantity onto the plane-of-the-sky, and then integrate the resulting maps
at various radii. Finally, we divide these profiles by the corresponding mass
profiles in order to obtain projected mass-weighted velocity dispersion
profiles.

Figure 9: Same as Fig. 8 but for broken power-law profiles. Each row now
corresponds to a different $\beta_{\rm{out}}$ value (from top to bottom:
$\beta_{\rm{out}}=[0.3,0.5,0.7]$). For all models, $\beta_{\rm{in}}$,
$\gamma_{\rm{in}}$, and $\gamma_{\rm{out}}$ are fixed to [0.0, 2.0,1.5],
respectively.

In the models presented here, there are essentially four free parameters, i.e
$\gamma$, $R_{\rm{edge}}^{\rm{N_{2}H^{+}}}$ ($=r_{0}$),
$R_{\rm{edge}}^{\rm{{}^{13}CO}}$ , and $M_{\rm{edge}}^{\rm{N_{2}H^{+}}}$, for
the mass profiles, and an additional two free parameters, i.e. $\beta$, and
$\sigma_{\rm{edge}}^{\rm{N_{2}H^{+}}}$ ($=\sigma_{0}$), for the velocity
dispersion profiles. The parameter $\rho_{0}$ is derived from $\gamma$,
$R_{\rm{edge}}^{\rm{N_{2}H^{+}}}$, and $M_{\rm{edge}}^{\rm{N_{2}H^{+}}}$ and
is, thus, not a free-parameter of the models. As already mentioned, the
purpose of those models are not to find a set of best parameters for each
individual clouds, but rather to understand the trends that are present in the
cloud sample. With that in mind, Fig. 8 shows a set of 9 models against the
observed profiles. The normalisation of those models is such they match the
range of mass and velocity dispersion at parsec-scales. Each row corresponds
to a different $\gamma$ value but the same $\beta$ value. In each row, the
three panels correspond to the mass, velocity dispersion, and virial ratio
profiles. There are a number of important features in those models that we can
notice straight away. First, regarding the mass profiles, one can see that the
cases $\gamma=1$ and $\gamma=2$ over-predict and under-predict, respectively,
the mass of the clouds on the largest scales. We also notice that, while the
$\gamma=1.5$ case provides a better overall agreement with the observed
profiles, the profile shapes provided by the cases $\gamma=2$ and $\gamma=1$
seem to give a better match to the inner and outer parts, respectively, of the
observed profiles. We also notice that we successfully reproduce the curved
shape of the inner parts of the profiles.

Moving on to the velocity dispersion profiles displayed in Fig. 8 it is clear
that the simple 1D models represented here manage to reproduce the velocity
dispersion discontinuity, in particular for the $\gamma=1$ case. The reason
for this is that, for that series of models, a larger fraction of the mass is
at low density where the velocity dispersion is the largest and as a result of
the projection, the mass weighted velocity dispersion is overestimated by a
large factor, up $\sim 3$ in the $\gamma=1$ case. Also, while the $\gamma=1$
case manages to reproduce in a satisfactory way the shape and amplitude of the
outer velocity dispersion profile, it cannot entirely match the observed flat
velocity dispersion profiles in the inner regions. And this gets worse when
considering steeper density profiles, i.e. $\gamma=1.5$ and $\gamma=2$.

Finally, looking at the virial ratio profiles in the last column of Fig. 8, we
notice that none of them are completely satisfactory when compared to the
observed profiles, even though one could argue that the shallower density
models do better than the steeper ones.

From this comparison between single-power law models and observed profiles, it
seems clear that most of the observed features can be reproduced, at least to
some extent, providing strong evidence that projection biases are mostly
responsible for them. This comparison also shows that the single power-law
models are limited and do not allow us to reproduce both the the inner and
outer parts of the observed mass and velocity dispersion profiles. Most
noticeable is the velocity dispersion profiles for which the flat inner
profiles are clearly different from the outer profile shapes.

Figure 10: Violin plots of the virial ratios obtained on the largest scales in
both N2H+(1-0) (magenta) and 13CO(1-0) (green), along with that obtained by
Miville-Deschênes et al. (2017) in 12CO(1-0) (yellow) for the same sample of
clouds. Each violin plot is located, along the x-axis, at the median radius
value of each group. The median virial ratio value for each group is
represented by a coloured circular symbol with a black edge, while the 16th
and 84th percentile ranges are represented by vertical solid black lines. We
have also overplotted the corresponding individual measurements as coloured
circular symbols. The horizontal black dashed lines show virial ratio values
$\alpha_{\rm{vir}}=1$ while the shaded area show the region of energy
equipartition for density profile indices between $\gamma=1$ and $\gamma=2$.

###  4.2 Broken power-law profiles

In this section, we extend the models presented above from single power-law
profiles to broken power-law profiles. More explicitly, we set two power-law
exponents for both the density and velocity dispersion profiles defined as:

$\gamma_{\rm{in}}\hskip 5.69046pt{\rm and}\hskip
5.69046pt\beta_{\rm{in}}\hskip 5.69046pt{\rm for}\hskip 5.69046ptr<r_{0}\\\
\\\ \gamma_{\rm{out}}\hskip 5.69046pt{\rm and}\hskip
5.69046pt\beta_{\rm{out}}\hskip 5.69046pt{\rm for}\hskip 5.69046ptr>r_{0}$
(15)

The method used to create the profiles is identical to that presented in the
previous section. Based on our single power-law models, it seems that the
inner density profiles are, on average, steeper than the outer ones.
Therefore, in the series of models presented in Fig. 9 we used
$\gamma_{\rm{in}}=2$ and $\gamma_{\rm{out}}=1.5$. Regarding the velocity
dispersion profiles it seems clear that the velocity dispersion on clump scale
is rather flat, with apparent very little variation in the profiles. On the
other hand, the velocity dispersion profiles on cloud scale are diverse, both
in terms of shape and normalisation. Therefore, each row in Fig. 9 corresponds
to a different value of $\beta_{\rm{out}}$, while $\beta_{\rm{in}}$ is fixed
to 0 for all models. In that figure one can see that we now reproduce rather
well the average shape and magnitude of the mass profiles, and similarly for
the velocity dispersion profiles on clump scales. One can also see that we do
reproduce well some of the velocity dispersion profiles on cloud scale,
although we fail in reproducing the low-mass high velocity dispersion profiles
that populate the top part of the velocity and virial ratio profiles. This is
where our simple 1D models reach their limitations. Indeed, if one looks at
the large scale mass distribution of those clouds (via the 13CO-based H2
column density maps - see Fig. 1 and Appendix A), one can see that the clump
we are focussing on does not dominate the mass on those scales, with many or
more sibling clumps being present in the same parent cloud. As a result, the
large scale velocity dispersion measured towards the clumps of interest is mot
likely driven by the presence of its siblings. This cannot be reproduced with
spherical models. Nevertheless, what this is showing is that lower gas density
layers of high velocity dispersion gas surround those clumps, generating steep
velocity dispersion discontinuities in their profiles.

##  5 Discussion

###  5.1 Self-gravitating molecular clouds

The question of the gravitational binding of molecular clouds has been, and
still is, the subject of numerous debates (e.g. Heyer et al., 2009; Dobbs et
al., 2011; Ballesteros-Paredes et al., 2011; Miville-Deschênes et al., 2017;
Vázquez-Semadeni et al., 2019). Here, we have all the necessary information to
check wether the clouds we selected are gravitationally bound or whether there
is a scale at which they switch from being bound to unbound. The profiles
displayed in Fig. 6 show that observed virial ratios for our cloud sample are
nearly systematically below $\alpha_{\rm{vir}}=3$ at all radii. Most of the
exceptions correspond to the measurements made at the smallest radii of the
13CO(1-0)-based profiles. As our models showed (see Fig. 8) increased virial
ratios with decreasing radii can be reproduced when large layers of high
velocity dispersion gas lay along the line-of-sight and contaminates the
measurements. As a result the most reliable measurements are those obtained on
the largest scales. Figure 10 shows the distributions of virial ratios
obtained at those largest scales for both N2H+(1-0) and 13CO(1-0)
measurements. For uniform density spheres, the transition between
gravitationally bound and unbound gas occurs at $\alpha_{\rm{vir}}=2$, while
for clouds with density profiles such as $\rho\propto r^{-1}$, $\rho\propto
r^{-1.5}$, and $\rho\propto r^{-2}$, the limit moves up to 2.2, 2.5, and 3.3,
respectively. Correction factors regarding the non-spherical shape of clouds
are less than 8% as long as the aspect ratio of the clouds is lower than 10
(Bertoldi & McKee, 1992), which is the case for all clouds in the sample. For
non-uniform velocity dispersion profiles correction factors also exist
(Miville-Deschênes et al., 2017), but these are of the order of 5% for the
diffuse parts of the cloud and non-existent for the dense part (see Appendix
D). Figure 10 reveals that $\sim 85\%$ of 13CO-based measurements, and 100$\%$
of the N2H+-based measurements have $\alpha_{\rm{vir}}\leq 2.5$. Taken at face
value, this demonstrates that the vast majority of the molecular clouds, if
not all, from our sample is self-gravitating on all scales, from tenths of a
parsec up to several tens of parsecs.

Figure 11: Distributions of virial ratios as estimated by Miville-Deschênes et
al. (2017) for the 24 clouds presented here (blue histogram) and their entire
cloud population (orange histogram). Figure 12:  The profiles are the same as
those presented in the upper row of Fig. 6. In addition, we have added the
12CO(1-0) data points as presented in MD17 (transparent orange symbols) and
once corrected by a factor of 2 in mass (yellow symbols). The blue dashed-
lines show Larson’s laws. Figure 13: The profiles are the same as those
presented in the upper row of Fig. 6, only restricted to the 12 clouds in
common with H09. In addition, we have added the H09’s measurements for these
12 clouds, for both radii measurements. We also overplot Larson’s relations as
dashed blue lines, and Solomon’s size-velocity dispersion relation in the
middle panel.

How does this fit with studies such as that of Miville-Deschênes et al. (2017,
MD17 hereafter), claiming that most molecular clouds are unbound? In order to
answer that question we searched for the MD17 counterparts of all 24 molecular
clouds from our sample and compared the distributions of the virial ratio (as
estimated by MD17) to the entire MD17 cloud population. Figure 11 shows that
our 24 IRDC-hosting molecular clouds are amongst the most gravitationally
bound clouds from the MD17 sample, oversampling the low virial ratio tail of
the distribution. So the fact that all the clouds studied in the present paper
are gravitationally bound is not in contradiction with the MD17 results. We
also notice that the virial ratio values plotted in Fig. 11 and estimated by
MD17 are larger than those we have estimated ourselves for the same sample of
clouds. This difference could be real as MD17 computed their cloud properties
from a lower gas density tracer, that is 12CO(1-0), or it could be due to
systematics in the way properties are calculated. We investigated this by
reporting the MD17 values of radius, mass, velocity dispersion, and virial
ratio for all 24 clouds and added them to our observational profiles. Figure
12 (transparent orange circular symbols) shows an overall good agreement
between out data points and those from MD17. However, while the radii reported
by MD17 are larger, the masses are very similar to those we report on smaller
radii. This somehow suggests that either we have overestimated our masses or
MD17 have underestimated their 12CO masses. In MD17 they used a standard
X${}_{\rm{CO}}=2\times 10^{20}$cm-2 (K km-1)-1 factor to convert integrated
12CO intensities into H2 column densities. As Barnes et al. (2015) showed,
this standard conversion factor typically underestimates column densities by a
factor of $\sim 2$ for resolved molecular clouds. Taking into account this
change in XCO would put the MD17 masses more in line with ours (see Fig 12
yellow circular symbols). In addition to this mass correction, one can wonder
whether one should apply one to velocity dispersion measurements as well.
Indeed, 12CO(1-0) is typically optically thick above H2 column densities of
few $10^{20}$ cm-2, which means that mass and velocity dispersion measurements
could be overestimated and underestimated, respectively. The effect on the
velocity dispersion though is probably only of the order of 20% (e.g. Hacar et
al., 2016), but as the result of the $\sigma^{2}$ dependency of the virial
ratio, a small correction factor on the velocity dispersion can lead to a
significant difference on the virial ratios. However, as it can be seen in
Fig. 12, the velocity dispersion measurements obtained by MD17 are in good
agreement with ours, and we therefore do not believe that there is a
systematic under-estimation of 12CO velocity dispersion for the clouds we are
looking at. The corresponding distribution of virial ratios has also been
reported onto Fig. 10 showing that 85% of the 12CO-based virial ratio
measurements are below 2.5, which is identical to the 13CO-based virial ratio
measurements. Overall, Figs. 12 and 10 show that even on scales of 100 pc, the
vast majority of clouds from our sample is self-gravitating.

Figure 14: Colour and contours are the 13CO-based H2 column density images of
three of our clouds that are in common with H09’s sample. The white dashed
boxes show the area used by H09 to derive the large-scale masses represented
as cyan symbols in Fig. 13. Figure 15: Heyer’s plots (left): measurements from
H09 on large (blue) and small (orange) scales. The points highlighted in cyan
and yellow with black edges are those clouds in common with our sample;
(right): Measurements from our study on large (green) and small (magenta)
scales. The clouds in common with H09’s sample are highlighted with light
green and pink symbols with black edges. On each panel, the ellipses show the
$1\sigma$, $2\sigma$, $3\sigma$ ellipses for each distributions, where
$\sigma$ is the standard deviation. Lines of constant virial ratios of 1 and 3
are shown as black dashed lines.

###  5.2 Larson’s, Solomon’s, and Heyer’s relations

Probably one of the most influential studies on the observational
characterisation of dynamical state of molecular clouds is that by Richard
Larson in 1981. In that study, they found, mostly from using 13CO(1-0) data
from the literature at the time, that the averaged cloud properties follow a
number of relationships such that: $\sigma\propto r^{\beta}$ with
$\beta=0.38$, and $\rho\propto r^{-\gamma}$ with $\gamma=1.1$. A third
relation, consequence of the first two, is that molecular clouds have virial
ratios close to unity and $\alpha_{\rm{vir}}\propto r^{-\delta}$ with
$\delta=0.14$. Larson’s size-velocity dispersion relation has been interpreted
as an evidence for turbulence-regulated gas dynamics, since $\beta=1/3$ is
what one expects for incompressible Kolmogorov-like turbulence. However, these
relations have later been revised by Solomon et al. (1987, S87 hereafter) who
found a steeped size-velocity dispersion relation with $\beta\simeq 0.5$. They
suggested that such index is the direct consequence of the virialisation of
individual molecular clouds at nearly constant mass surface densities. Heyer
et al. (2009, H09 hereafter) reanalysed S87 cloud sample using 13CO(1-0) GRS
data and determined that, even though cloud properties are compatible with
being in virial equilibrium at all radii, the change in the internal mass
surface density of clouds result in a different size-velocity dispersion
relation to that proposed by Larson’s and Solomon’s.

Finding out how our study compares to those mentioned above and understanding
where the differences come from is fundamental if one wants to settle the
question of the dynamical states of molecular clouds. Interestingly, half of
our cloud sample (12/24) is common to both H09 and S87’s samples, and since
H09 used the same 13CO data we use here, one can make a direct one-to-one
comparison. The first property we compare is the distance used for all 12
clouds. As it can be seen in Fig. 48, for half of the clouds the distances
match, while for the other half they do not. The latter group of clouds have
been assigned the far distance by S87 and H09. Even though they have
recalculated the kinematics distances, H09 have kept the near/far distance
ambiguity solutions provided by S87. Looking in detail, these 6 clouds with
far distances have been assigned so based on the fact that: i. they best fit
the S87 size versus velocity dispersion relation; ii. They best match the
scale-height of the molecular layer for that position and velocity range.
These are both very questionable criteria. All clouds here host IRDCs, and it
has been shown that 90% of IRDCs are located at the near distance (Ellsworth-
Bowers et al., 2013). This would suggest that maybe one of the 12 clouds
presented here is indeed at the far distance, but it is very unlikely that the
6 are. In the rest of the comparison we set the distance to all 12 clouds to
that we give in Table 1.

Figure 16: Velocity dispersion profiles. The models displayed in each of the
three panels correspond to three different clump masses and radii, but all
with the same density profile $\gamma=2$. The yellow, orange, and red solid
lines correspond to models with velocity dispersion profile index $\beta=0$,
$\beta=\pm 0.1$, and $\beta=\pm 0.2$, respectively. The observed profiles are
represented with thin coloured lines.

Figure 13 compares the profiles of the 12 clouds as we measured them with
H09’s measurements (after distance correction). In H09, each cloud has two
measurements taken at different radii, both measured using 13CO. The cyan
symbols represent the large scale measurements and the yellow symbols the
small scale ones. Compared to our measurements of the same clouds, we can see
that, at large radii, both H09’s masses and velocity dispersions tend to be
underestimated. On small scales though, the masses are similar but the
velocity dispersions are overunderestimateestimated. Before interpreting these
discrepancies, one needs to understand the differences in the measurements
themselves. For the large-scale mass measurements H09 used the original
rectangular boxes that S87 used to measure their own masses. Those boxes where
defined based the location and extent of the 12CO(1-0) emission peaks derived
from low-resolution high-noise maps. Figure 14 show three representative
examples of such boxes overlaid on top of the cloud column density images. One
can see that, with the exception of the biggest clouds, the boxes do not match
the cloud morphologies, sometime missing the column density peaks, and often
covering regions where no, or little, column density is present. The net
impact of this is, for a given effective radius (defined as the radius of the
disc having the same area as the box), the mass is heavily underestimated.
This problem mostly disappears for small scale mass measurements as H09 have
for those used the contours of the column density maps (as we did). Regarding
the velocity dispersion measurements on small scales, H09 overestimate them as
a result of the same projection effect that is responsible for over-estimating
our own 13CO velocity dispersion measurements. On large scale H09
underestimate the velocity dispersion most likely because of the unadapted
velocity window used to compute their 1st order moment. However, we cannot
test this since velocity windows used for the integration by H09 are not
provided.

One particular plot that has been used by H09, and many others since, to
support the picture of virialised clouds on all scales is one that plots the
mass surface density $\Sigma_{\rm{gas}}$ of the clouds versus the parameter
$p=\sigma_{v}/\sqrt{R}$. On the left panel of Fig. 15 we reproduced the figure
from H09, the clouds in common with our studies being highlighted with
different colours (cyan and yellow) and with black edges. In this panel, it is
quite clear that the large scale data points (the blue squares) are at lower
mass surface densities than the small scales points (orange). The distribution
of these points stretch across more than two orders of magnitude along lines
of constant virial ratios between 1 and 3. On the same figure, the right-hand-
side panel shows the same quantities for the common sample of clouds with
properties as derived in this paper (here we used the values displayed in
Table 2). We can see that the large scale points (green) completely overlap
with the small scale points (magenta). When compared with Heyer’s quantities
for the same clouds we see that the spread is reduced by one order of
magnitude. This is a direct consequence of the measurement biases explained
above. In fact, whether we look at the large scale measurements obtained on
scales of 20 pc to 60 pc or measurements obtained on scales between 1.5pc and
5pc, the data points are located within a very similar area of the plot. This
is a direct consequence of the density profile of the clouds being close to
$\rho\propto r^{-1}$ on those scales and velocity dispersion profiles close to
$\sigma\propto r^{0.5}$. We also notice the quasi-absence of points below a
100 M⊙/pc-2. As noted by Schruba et al. (2019), molecular clouds with lower
mass surface density are non-self-gravitating. As our comparison with the MD17
virial ratio distribution shows, we are here biased towards the most self-
gravitating clouds of the Milky Way population, it is therefore consistent to
have nearly no measurements with mass surface density below 100 M⊙/pc-2.

###  5.3 Dynamically decoupled clumps

As the comparison of our observed profiles and spherical models have shown,
the discontinuity in the observed velocity dispersion profiles is most likely
the result of the combination of projection effects and a genuine change of
the velocity dispersion profile index from $\beta\simeq 0.5$ on large scales
to $\beta\simeq 0$ on small scales. However, one can wonder how sensitive the
observed velocity dispersion profiles are to the exact value of $\beta$ as the
clumps only have a limited number of angular resolution elements in them. To
test this, we built spherical models of varying $\beta$ index in order to set
some constraints on the range of values compatible with our observations.
Figure 16 displays the observed clump-scale velocity profiles along with
different spherical models. Each panel correspond to models of the same mass
and radius, but with different velocity dispersion profiles. One can see that
the different profiles are better resolved for the largest clouds, as
expected. With this models in hands, it is also clear that $|\beta|<0.2$ in
all clumps, confirming the fact that the clump velocity dispersion profiles
are flat and significantly different from Larson’s profile.

The velocity dispersion discontinuity observed in Fig. 6 between the N2H+(1-0)
and the 13CO(1-0) measurements is, according to our models, the result of
foreground/background layers of low-density and high-velocity dispersion gas
that contaminate the 13CO(1-0) velocity dispersion measurements at small
radii. If this interpretation of the observed profiles is correct, measuring
the gas velocity dispersion with a line emission that traces intermediate gas
densities should bridge, to some extent, the observed velocity dispersion
discontinuity. To test this conjuncture, we used the CHIMPS 13CO(3-2) survey
data (Rigby et al., 2016). Indeed, being a higher transition line, 13CO(3-2)
is optically thinner and less extended than 13CO(1-0), making it a good tracer
of intermediate gas densities. However, only 8 of our clouds have been covered
by CHIMPS, amongst which one shows clear sign of self-absorption and has
therefore been discarded. Figure 17 shows the velocity dispersion profiles of
6 of the 7 remaining clouds (one has been left out for a matter of figure
readability). The 13CO(3-2) line has been fitted following the exact same
procedure as for the 13CO(1-0) line. On this figure, we can see that the
13CO(3-2) velocity dispersion systematically lies at intermediate values
between that of the other two tracers. Also, in most cases, the 13CO(3-2)
profiles nicely make the bridge between the denser and more diffuse gas.
Altogether, these profiles further support our interpretation that clumps are
dynamically decoupled from their parent molecular clouds.

Figure 17:  Velocity dispersion profiles of a sub-sample of 6 clouds. The
magenta and green lines are the same of those plotted in Fig. 6 for those 6
clouds. The blue lines show the velocity dispersion profiles obtained from
CHIMPS 13CO(3-2) emission.

As discussed in Sec. 5.1 the vas majority of the clouds studied here are self-
gravitating on all scales. This means that the observed change in the velocity
dispersion profiles cannot be the result of the gas switching from a non self-
gravitating state to a self-gravitating state. Because clouds are self-
gravitating, it can only be in two states: either it is collapsing, or in
quasi-static equilibrium, with the nature of the stabilising agent left to be
determined. One possibility would be for instance that the clouds, as proposed
by Vázquez-Semadeni et al. (2019), are collapsing on all scales. However, the
collapse in these models is scale-free, and there is no evidence for a
transition regime at any scale. It is important to note that feedback, such as
protostellar outflows (e.g. Duarte-Cabral et al., 2012; Hsieh et al., 2023),
are not included in such simulations which could play a role in generating a
change in the velocity dispersion profiles of collapsing clouds. It could also
be that, as the specific angular momentum increases during the collapse,
clumps become somewhat supported by rotation (e.g. Lee & Hennebelle, 2016).
However, this seems incompatible with a steeper clump density profile, and no
systematic observation of rotation motion is observed in these clumps (Peretto
et al. in prep.). A third and preferred possibility is that clouds are stable
on the largest scales and that they collapse on clump scale. Indeed, both
density ($\gamma=2$) and velocity dispersion profiles ($\beta=0$) derived from
our 1D modelling of the clumps are asymptotic solutions to a spherical
isothermal non-free-falling collapsing cloud with initial uniform density
(Larson, 1969; Penston, 1969), and as noted by these authors the self-similar
nature of the solution means that it may apply to any structure (i.e.
protostellar core, clump, cloud). Note however that what we observe is the
velocity dispersion profile and not the infall velocity profile. Even though
we do expect a relationship between the two, it is not clear whether both are
expected to have the exact same index. In a recent study, Gómez et al. (2021)
showed that $\gamma=2$ naturally arises in collapsing cores that accrete from
their surroundings. There is also now plenty of evidence for clump collapse
and clump accretion (e.g. Peretto et al., 2006, 2007; Peretto et al., 2013,
2014, 2020; Schneider et al., 2010; Traficante et al., 2018; Williams et al.,
2018; Schwörer et al., 2019; Barnes et al., 2019; Anderson et al., 2021; Rigby
et al., 2021; Bonne et al., 2022; Zhou et al., 2022; Xu et al., 2023). A
possible agent that might be able to stabilise the clouds on the largest
scales is stellar feedback. For instance, in Watkins et al. (2019), it has
been shown that stellar feedback from embedded O stars does not impact much
the dynamical properties of the dense gas that has already been assembled, but
does clearly modify the structure of the larger scale clouds. This is
compatible with the observed change in velocity dispersion profiles presented
here. Even though most clumps in our study do not have any embedded HII
regions associated to them, injection of momentum and energy within the more
diffuse cloud could come from nearby sites of massive star formation. Another
possible agent that could stabilise the cloud is magnetic field. An increasing
number of studies suggest that magnetic fields are dynamically
important/dominant in the low density regions of molecular clouds. A
transition in the relative orientation between magnetic field and the density
gradients of interstellar structures has been interpreted as evidence for a
change in the dominant energy source, from magnetic energy on large scale to
gravitational energy on clump scale (e.g. Soler et al., 2013, 2016; Chen et
al., 2016; Planck Collaboration et al., 2016a, b; Tang et al., 2019;
Arzoumanian et al., 2021). In Vela C, Fissel et al. (2019) determined that
this change of relative orientations occur at a number density of $n\sim
10^{3}$ cm-3. The change in the velocity dispersion profiles we observe in our
sample could then be the dynamical counterpart of that "magnetic" transition.
Figure 18 shows the same plot as in Fig. 15 in which the largest cloud scale
measurements and all the clump scale measurements are shown. On that plot it
becomes obvious that clumps behave differently than their parent clouds. The
clump mass surface densities increase over two order of magnitudes along lines
of constant virial ratios except towards the most central points where $p$ and
the virial ratios increase. To our knowledge no theoretical equivalent to the
plots we are producing here exists. However, a scenario in which parsec-scale
clumps are collapsing while their parent molecular clouds are in quasi-static
equilibrium seems to intuitively match what we see.

Figure 18: Same as Fig. 15 with the addition of all measurements made from
N2H+(1-0) (purple lines). The shaded area show the region of energy
equipartition for density profile indices between $\gamma=1$ and $\gamma=2$.

Another important aspect of Fig. 6 is the relatively large range of velocity
dispersion profile indices on cloud scale (the green lines). They range from
being flat $\beta\sim 0$ to relatively steep $\beta>0.5$. The fact that
molecular clouds of several $10^{4}$ M⊙ on tens of parsec-scale may have
velocity dispersions that are barely above 1 km/s clearly shows that Larson’s
relation is just a statistical average over clouds of very different dynamical
states. In fact, it is possible that the range of velocity dispersion profiles
correspond to different evolutionary stages in the formation and evolution of
molecular clouds and clumps within. Any scenario that attempt to explain the
dynamical decoupling of clumps needs to do so in the context of this observed
variety of large-scale velocity dispersion profiles.

##  6 Summary and conclusion

We performed the analysis of 27 IRDC embedded within 24 molecular clouds. We
computed the mass, velocity dispersion, and virial ratio profiles of each of
them using three different datasets: Herschel-derived H2 column density maps,
GRS 13CO(1-0)-derived H2 column density cubes, and N2H+(1-0) data cubes. The
combination of these data allowed us to probe both the dense and diffuse parts
of the clouds, with radii from $\sim 0.2$ pc up to $\sim 30$ pc. Using 1D
power-law models we can explain the origin of the different features observed
in those profiles and we conclude that: 1. the vast majority of cluster-
forming molecular clouds are self-gravitating on all scales; 2. the diffuse
part of the cloud has a shallow density profile ($\gamma\sim 1$) that steepens
($\gamma\sim 2$) in the densest parts on a couple of parsec scale; 3. the
velocity dispersion profile switches, for most clouds, from $\beta\sim 0.5$ in
the diffuse part of the clouds to $\beta\sim 0$ in the denser parts. We
discuss the possible interpretation of such a decoupling of the clumps from
their surrounding cloud and conclude that the observations are best explained
by a universal global collapse of dense clumps embedded within stable
molecular clouds, even though we cannot completely rule out a scenario in
which the entire cloud collapses, with small-scale feedback, such as
protostellar outflows, impacting the gas kinematics on clump scales. We also
notice that the velocity dispersion profiles on molecular cloud scales (i.e.
$>2$ pc) show a large variety of $\beta$ values, some very far from the
standard Larson’s relation, which might be linked to their evolution since the
time of their formation.

Understanding the origin of the observed low star-formation efficiency (SFE)
in molecular clouds is one of the main goals of star formation research. A low
SFE involves a scale/density-dependent dynamical state of the gas in which
most of a cloud mass is not directly involved in the formation of stars.
Observationally, the existence of a star formation threshold has been
discussed in the context of the study of nearby star-forming clouds (e.g. Lada
et al., 2010; Heiderman et al., 2010; Pokhrel et al., 2020). However, so far,
no studies has searched for direct evidence of a transition regime in the
dynamical properties of the gas within individual molecular clouds. The work
presented here clearly suggests that such transition regime does exist.
Because parsec-scale clumps are believed to be the direct progenitors of star
clusters (e.g. Krumholz et al., 2019), our results hence suggest that star
cluster formation is not a scale-free process.

Our results also carry a number of key questions and implications. First, we
here do not explain what the trigger of the clump collapse is, whether it is
the result of a gravitationally instability or the diffusion of magnetic
fields, or any other mechanism. We also do not explain what is the main agent
that counter-balance gravity in the diffuse parts of the clouds. These
questions will have to be answered if one wants to derive a comprehensive
scenario for the formation of star clusters. Also, one implication of our
results is the fact that star formation is likely to be mostly confined to
these parsec-scale collapsing clumps. Therefore their properties define the
initial conditions for cluster formation, and understanding the link, on one
side, between the properties of clumps and that of their associated
protostellar population, and on the other side, between the global population
of Galactic clumps and the star formation rate and efficiency of the Milky Way
remains a fundamental challenge.

## Acknowledgements

NP and AJR acknowledges the support of STFC consolidated grant number
ST/N000706/1 and ST/S00033X/1. FL acknowledges support by the Marie Curie
Action of the European Union (project MagiKStar, Grant agreement number
841276). G.A.F acknowledges support from the Collaborative Research Centre
956, funded by the Deutsche Forschungsgemeinschaft (DFG) project ID 184018867.
This work is based on observations carried out under project number 023-13
with the IRAM 30m telescope. IRAM is supported by INSU/CNRS (France), MPG
(Germany) and IGN (Spain).

## Data Availability

The Herschel and GRS data used in this article are already publicly available
on their respective survey webpages. The IRAM 30m N2H+(1-0) data can be
provided upon reasonable request to N. Peretto.

## References

  * Anderson et al. (2021) Anderson M., et al., 2021, MNRAS, 508, 2964
  * Arzoumanian et al. (2021) Arzoumanian D., et al., 2021, A&A, 647, A78
  * Ballesteros-Paredes et al. (2011) Ballesteros-Paredes J., Hartmann L. W., Vázquez-Semadeni E., Heitsch F., Zamora-Avilés M. A., 2011, MNRAS, 411, 65
  * Barnes et al. (2015) Barnes P. J., Muller E., Indermuehle B., O’Dougherty S. N., Lowe V., Cunningham M., Hernandez A. K., Fuller G. A., 2015, ApJ, 812, 6
  * Barnes et al. (2019) Barnes A. T., et al., 2019, MNRAS, 486, 283
  * Battersby et al. (2011) Battersby C., et al., 2011, A&A, 535, A128
  * Bernard et al. (2010) Bernard J.-P., et al., 2010, A&A, 518, L88+
  * Bertoldi & McKee (1992) Bertoldi F., McKee C. F., 1992, ApJ, 395, 140
  * Bisbas et al. (2017) Bisbas T. G., Tanaka K. E. I., Tan J. C., Wu B., Nakamura F., 2017, ApJ, 850, 23
  * Blake et al. (1987) Blake G. A., Sutton E. C., Masson C. R., Phillips T. G., 1987, ApJ, 315, 621
  * Bonne et al. (2022) Bonne L., Peretto N., Duarte-Cabral A., Schmiedeke A., Schneider N., Bontemps S., Whitworth A., 2022, A&A, 665, A22
  * Bonnell & Bate (2006) Bonnell I. A., Bate M. R., 2006, MNRAS, 370, 488
  * Chen et al. (2016) Chen C.-Y., King P. K., Li Z.-Y., 2016, ApJ, 829, 84
  * Clemens et al. (1986) Clemens D. P., Sanders D. B., Scoville N. Z., Solomon P. M., 1986, ApJS, 60, 297
  * Dale et al. (2012) Dale J. E., Ercolano B., Bonnell I. A., 2012, MNRAS, 424, 377
  * Dewangan et al. (2018) Dewangan L. K., Ojha D. K., Zinchenko I., Baug T., 2018, ApJ, 861, 19
  * Dobbs et al. (2011) Dobbs C. L., Burkert A., Pringle J. E., 2011, MNRAS, 413, 2935
  * Duarte-Cabral et al. (2012) Duarte-Cabral A., Chrysostomou A., Peretto N., Fuller G. A., Matthews B., Schieven G., Davis G. R., 2012, A&A, 543, A140
  * Ellsworth-Bowers et al. (2013) Ellsworth-Bowers T. P., et al., 2013, ApJ, 770, 39
  * Fissel et al. (2019) Fissel L. M., et al., 2019, ApJ, 878, 110
  * Gómez et al. (2021) Gómez G. C., Vázquez-Semadeni E., Palau A., 2021, MNRAS, 502, 4963
  * Griffin et al. (2010) Griffin M. J., et al., 2010, A&A, 518, L3+
  * Grudić et al. (2022) Grudić M. Y., Guszejnov D., Offner S. S. R., Rosen A. L., Raju A. N., Faucher-Giguère C.-A., Hopkins P. F., 2022, MNRAS, 512, 216
  * Hacar et al. (2016) Hacar A., Alves J., Burkert A., Goldsmith P., 2016, A&A, 591, A104
  * Hacar et al. (2018) Hacar A., Tafalla M., Forbrich J., Alves J., Meingast S., Grossschedl J., Teixeira P. S., 2018, A&A, 610, A77
  * Hartmann & Burkert (2007) Hartmann L., Burkert A., 2007, ApJ, 654, 988
  * Haworth et al. (2015) Haworth T. J., et al., 2015, MNRAS, 450, 10
  * Heiderman et al. (2010) Heiderman A., Evans II N. J., Allen L. E., Huard T., Heyer M., 2010, ApJ, 723, 1019
  * Hennebelle & Chabrier (2008) Hennebelle P., Chabrier G., 2008, ApJ, 684, 395
  * Henshaw et al. (2014) Henshaw J. D., Caselli P., Fontani F., Jiménez-Serra I., Tan J. C., 2014, MNRAS, 440, 2860
  * Hernandez & Tan (2015) Hernandez A. K., Tan J. C., 2015, ApJ, 809, 154
  * Heyer et al. (2009) Heyer M., Krawczyk C., Duval J., Jackson J. M., 2009, ApJ, 699, 1092
  * Hildebrand (1983) Hildebrand R. H., 1983, QJRAS, 24, 267
  * Hopkins (2012) Hopkins P. F., 2012, MNRAS, 423, 2037
  * Hsieh et al. (2023) Hsieh C.-H., et al., 2023, arXiv e-prints, p. arXiv:2302.03174
  * Jackson et al. (2006) Jackson J. M., et al., 2006, ApJS, 163, 145
  * Kauffmann et al. (2013) Kauffmann J., Pillai T., Goldsmith P. F., 2013, ApJ, 779, 185
  * Kim et al. (2018) Kim J.-G., Kim W.-T., Ostriker E. C., 2018, ApJ, 859, 68
  * Krumholz & McKee (2005) Krumholz M. R., McKee C. F., 2005, ApJ, 630, 250
  * Krumholz & Tan (2007) Krumholz M. R., Tan J. C., 2007, ApJ, 654, 304
  * Krumholz et al. (2019) Krumholz M. R., McKee C. F., Bland -Hawthorn J., 2019, ARA&A, 57, 227
  * Kutner & Ulich (1981) Kutner M. L., Ulich B. L., 1981, ApJ, 250, 341
  * Lada et al. (2010) Lada C. J., Lombardi M., Alves J. F., 2010, ApJ, 724, 687
  * Langer & Penzias (1990) Langer W. D., Penzias A. A., 1990, ApJ, 357, 477
  * Larson (1969) Larson R. B., 1969, MNRAS, 145, 271
  * Larson (1981) Larson R. B., 1981, MNRAS, 194, 809
  * Lee & Hennebelle (2016) Lee Y.-N., Hennebelle P., 2016, A&A, 591, A30
  * Marsh et al. (2015) Marsh K. A., Whitworth A. P., Lomax O., 2015, MNRAS, 454, 4282
  * Miville-Deschênes et al. (2017) Miville-Deschênes M.-A., Murray N., Lee E. J., 2017, ApJ, 834, 57
  * Molinari et al. (2010) Molinari S., et al., 2010, A&A, 518, L100+
  * Nakamura & Li (2007) Nakamura F., Li Z.-Y., 2007, ApJ, 662, 395
  * Offner & Liu (2018) Offner S. S. R., Liu Y., 2018, Nature Astronomy, 2, 896
  * Ott (2010) Ott S., 2010, in Mizumoto Y., Morita K. I., Ohishi M., eds, Astronomical Society of the Pacific Conference Series Vol. 434, Astronomical Data Analysis Software and Systems XIX. p. 139 (arXiv:1011.1209)
  * Padoan et al. (1997) Padoan P., Nordlund A., Jones B. J. T., 1997, MNRAS, 288, 145
  * Padoan et al. (2020) Padoan P., Pan L., Juvela M., Haugbølle T., Nordlund Å., 2020, ApJ, 900, 82
  * Penston (1969) Penston M. V., 1969, MNRAS, 144, 425
  * Peretto & Fuller (2009) Peretto N., Fuller G. A., 2009, A&A, 505, 405
  * Peretto et al. (2006) Peretto N., André P., Belloche A., 2006, A&A, 445, 979
  * Peretto et al. (2007) Peretto N., Hennebelle P., André P., 2007, A&A, 464, 983
  * Peretto et al. (2010) Peretto N., et al., 2010, A&A, 518, L98+
  * Peretto et al. (2013) Peretto N., et al., 2013, A&A, 555, A112
  * Peretto et al. (2014) Peretto N., et al., 2014, A&A, 561, A83
  * Peretto et al. (2016) Peretto N., Lenfestey C., Fuller G. A., Traficante A., Molinari S., Thompson M. A., Ward-Thompson D., 2016, A&A, 590, A72
  * Peretto et al. (2020) Peretto N., et al., 2020, MNRAS, 496, 3482
  * Piazzo et al. (2015) Piazzo L., Calzoletti L., Faustini F., Pestalozzi M., Pezzuto S., Elia D., di Giorgio A., Molinari S., 2015, MNRAS, 447, 1471
  * Pilbratt et al. (2010) Pilbratt G. L., et al., 2010, A&A, 518, L1+
  * Planck Collaboration et al. (2011) Planck Collaboration et al., 2011, A&A, 536, A25
  * Planck Collaboration et al. (2016a) Planck Collaboration et al., 2016a, A&A, 586, A136
  * Planck Collaboration et al. (2016b) Planck Collaboration et al., 2016b, A&A, 586, A138
  * Poglitsch et al. (2010) Poglitsch A., et al., 2010, A&A, 518, L2+
  * Pokhrel et al. (2020) Pokhrel R., et al., 2020, arXiv e-prints, p. arXiv:2005.05466
  * Reid et al. (2009) Reid M. J., et al., 2009, ApJ, 700, 137
  * Rigby et al. (2016) Rigby A. J., et al., 2016, MNRAS, 456, 2885
  * Rigby et al. (2018) Rigby A. J., et al., 2018, A&A, 615, A18
  * Rigby et al. (2019) Rigby A. J., et al., 2019, A&A, 632, A58
  * Rigby et al. (2021) Rigby A. J., et al., 2021, MNRAS, 502, 4576
  * Robitaille & Whitney (2010) Robitaille T. P., Whitney B. A., 2010, ApJ, 710, L11
  * Roman-Duval et al. (2010) Roman-Duval J., Jackson J. M., Heyer M., Rathborne J., Simon R., 2010, ApJ, 723, 492
  * Rosolowsky et al. (2008) Rosolowsky E. W., Pineda J. E., Kauffmann J., Goodman A. A., 2008, ApJ, 679, 1338
  * Sadavoy et al. (2016) Sadavoy S. I., Stutz A. M., Schnee S., Mason B. S., Di Francesco J., Friesen R. K., 2016, A&A, 588, A30
  * Sanders et al. (1986) Sanders D. B., Clemens D. P., Scoville N. Z., Solomon P. M., 1986, ApJS, 60, 1
  * Schneider et al. (2010) Schneider N., Csengeri T., Bontemps S., Motte F., Simon R., Hennebelle P., Federrath C., Klessen R., 2010, A&A, 520, A49
  * Schruba et al. (2019) Schruba A., Kruijssen J. M. D., Leroy A. K., 2019, ApJ, 883, 2
  * Schuller et al. (2017) Schuller F., et al., 2017, A&A, 601, A124
  * Schwörer et al. (2019) Schwörer A., et al., 2019, A&A, 628, A6
  * Singh et al. (2021) Singh A., et al., 2021, ApJ, 922, 87
  * Smith et al. (2009) Smith R. J., Longmore S., Bonnell I., 2009, MNRAS, 400, 1775
  * Soler et al. (2013) Soler J. D., Hennebelle P., Martin P. G., Miville-Deschênes M. A., Netterfield C. B., Fissel L. M., 2013, ApJ, 774, 128
  * Soler et al. (2016) Soler J. D., et al., 2016, A&A, 596, A93
  * Solomon et al. (1987) Solomon P. M., Rivolo A. R., Barrett J., Yahil A., 1987, ApJ, 319, 730
  * Tang et al. (2019) Tang Y.-W., Koch P. M., Peretto N., Novak G., Duarte-Cabral A., Chapman N. L., Hsieh P.-Y., Yen H.-W., 2019, ApJ, 878, 10
  * Traficante et al. (2011) Traficante A., et al., 2011, MNRAS, 416, 2932
  * Traficante et al. (2018) Traficante A., Fuller G. A., Smith R. J., Billot N., Duarte-Cabral A., Peretto N., Molinari S., Pineda J. E., 2018, MNRAS, 473, 4975
  * Traficante et al. (2020) Traficante A., Fuller G. A., Duarte-Cabral A., Elia D., Heyer M. H., Molinari S., Peretto N., Schisano E., 2020, MNRAS, 491, 4310
  * Vázquez-Semadeni et al. (2017) Vázquez-Semadeni E., González-Samaniego A., Colín P., 2017, MNRAS, 467, 1313
  * Vázquez-Semadeni et al. (2019) Vázquez-Semadeni E., Palau A., Ballesteros-Paredes J., Gómez G. C., Zamora-Avilés M., 2019, MNRAS, 490, 3061
  * Wang et al. (2010) Wang P., Li Z.-Y., Abel T., Nakamura F., 2010, ApJ, 709, 27
  * Watkins et al. (2019) Watkins E. J., Peretto N., Marsh K., Fuller G. A., 2019, A&A, 628, A21
  * Williams et al. (2018) Williams G. M., Peretto N., Avison A., Duarte-Cabral A., Fuller G. A., 2018, A&A, 613, A11
  * Xu et al. (2023) Xu F.-W., et al., 2023, arXiv e-prints, p. arXiv:2301.01895
  * Ysard et al. (2015) Ysard N., Köhler M., Jones A., Miville-Deschênes M. A., Abergel A., Fanciullo L., 2015, A&A, 577, A110
  * Yuan et al. (2020) Yuan Y., Krumholz M. R., Burkhart B., 2020, MNRAS, 498, 2440
  * Zhou et al. (2022) Zhou J.-W., et al., 2022, MNRAS, 514, 6038
  * Zuckerman & Evans (1974) Zuckerman B., Evans N. J. I., 1974, ApJ, 192, L149

##  Appendix A Summary figures

Figure 19: Same as Fig. 1 for SDC18.624-0.070. Figure 20: Same as Fig. 1 for
SDC18.787-0.286. Figure 21: Same as Fig. 1 for SDC21.321-0.139. Figure 22:
Same as Fig. 1 for SDC22.373+0.446. Figure 23: Same as Fig. 1 for
SDC22.724-0.269. Figure 24: Same as Fig. 1 for SDC23.066+0.049. Figure 25:
Same as Fig. 1 for SDC23.367-0.288. Figure 26: Same as Fig. 1 for
SDC24.118-0.175. Figure 27: Same as Fig. 1 for SDC24.433-0.231. Figure 28:
Same as Fig. 1 for SDC24.489-0.689. Figure 29: Same as Fig. 1 for
SDC24.618-0.323. Figure 30: Same as Fig. 1 for SDC24.630+0.151. Figure 31:
Same as Fig. 1 for SDC25.166-0.306. Figure 32: Same as Fig. 1 for
SDC25.243-0.447. Figure 33: Same as Fig. 1 for SDC26.507+0.716. Figure 34:
Same as Fig. 1 for SDC28.275-0.163. Figure 35: Same as Fig. 1 for
SDC28.333+0.063. Figure 36: Same as Fig. 1 for SDC31.039+0.241. Note that
because over the spacially overlapping N2H+(1-0) clumps, we do not include
this cloud in the analysis and therefore no fit is presented. Figure 37: Same
as Fig. 1 for SDC34.370+0.203. Figure 38: Same as Fig. 1 for SDC35.429+0.138.
Figure 39: Same as Fig. 1 for SDC35.527-0.269. Figure 40: Same as Fig. 1 for
SDC35.745+0.147. Figure 41: Same as Fig. 1 for SDC38.850-0.427. Figure 42:
Same as Fig. 1 for SDC40.283-0.216. Figure 43: Same as Fig. 1 for
SDC47.061+0.045. Figure 44: Same as Fig. 1 for SDC52.723+0.257.

##  Appendix B Velocity dispersion: method comparison

Figure 45: Examples of 13CO-based H2 column density spectra for
SDC18.624-0.070 in black. The multi Gaussian fits are shown as thin yellow
lines, the shaded green area correspond to the components we believe are
associated to the cloud. The velocity intervals for the moments method are
shown as vertical orange ticks, while the velocity intervals for the Peak
method are shown as vertical red ticks.

Figure 46: Comparing cloud properties using different methods for estimating
the velocity dispersion of 13CO(1-0). In the top row we compare the Gaussian
fit results on the x-axis, the method used in the rest of the paper, against
the Peak results on the y-axis. In the bottom row, the x-axes remain the same,
but the y-axis show the Moments results. The black dashed lines represent the
one-to-one relationships, and each colour corresponds to a different cloud,
each cloud having a number of measurements taken at different radii.

In addition to the Gaussian fits presented in the paper, we also tested two
other methods that are often used in the literature. The first of those is a
traditional moments method, referred to as Moments in this Appendix, for which
we compute the 0th, 1st, and 2nd order moments within a given velocity
interval. The 0th order moment is used to compute the mass. The 1st and 2nd
order moments are used to compute the velocity dispersion. The velocity
intervals are determined by eye and are defined so that the cloud main
component and any overlapping emission from overlapping clouds, in velocity
space, are included (see Fig. 45 for an example). The second method we tested,
referred to as Peak in this Appendix, is based on the full-width-at-half-
maximum (FWHM) of the column density spectrum peak. The velocity dispersion is
taken as FWHM/2.35, and the velocity interval over which the mass is
calculated is taken as eight times the dispersion, and is centred in a way
that matches the asymmetry of the peak velocity with respect to the FWHM
velocity interval (see Fig. 45). To obtain the mass we just integrate the
spectra over that velocity range.

Figure 46 compares the resulting masses, velocity dispersions, and virial
ratios as obtained for the three methods (Gaussian fit, Moments, and Peak).
The top row shows the comparison between the Gaussian fit and Peak methods,
while the bottom row shows the comparison between the Gaussian fit and the
Moments method. This shows that, as far as the masses are concerned, the
method used does not make much of a difference, and the reason is that most of
the mass is located within the central few channels that are covered by all
three methods. The main differences between the methods are related to the
estimate of the velocity dispersions. One can see that the Peak method nearly
systematically produces lower velocity dispersions than the Gaussian fit
method. This is expected as that method only focusses on the central peak of
emission and will therefore exclude any large velocity dispersion wings that
might be present. The results of the Peak method are similar to what we would
get by fitting the spectra with single Gaussians. When comparing the Moments
results to those of the Gaussian fits one can see that they are, overall, in
broader agreement, even though the Moments method tend to produce larger
velocity dispersions for some clouds. Those are the clouds for which a
spectrally overlapping components has been excluded from our Gaussian fit
results but included within the Moments one (as for SDC18.624-0.070 presented
in Fig. 45). The virial ratios differences are a direct consequences of the
differences observed in the velocity dispersion measurements, leading to
virial ratios than can be as high as $\sim 50$ in some cases when using the
Moments method and as low as $\sim 0.1$ when using the Peak method.

Overall, we believe that the Gaussian fits method provide better results than
any of the other two as it allows to include large velocity dispersion wings
and exclude, at the same time, components that we know are not physically
related to the cloud of interest. It is definitely the best method to measure
the velocity dispersion of 13CO clouds.

##  Appendix C Models of bijective mass and velocity dispersion estimates

Here we describe how we built the projected version of the spherical models
presented in the paper.

First, let us consider a sphere of radius with the following the following
density profile:

$\rho(r)=\rho_{0}\left(\frac{r}{r_{0}}\right)^{-\gamma}$ (16)

where $r_{0}$ and $\rho_{0}$ are normalisation constants. Then the enclosed
mass within radius $r$, $m_{\rm{real}}(r)$, is given by:

$m_{\rm{real}}(r)=\int_{0}^{r}\rho_{0}\left(\frac{r}{r_{0}}\right)^{-\gamma}4\pi
r^{2}dr$ (17)

$m_{\rm{real}}(r)=4\pi\rho_{0}r_{0}^{\gamma}\frac{r^{3-\gamma}}{3-\gamma}$
(18)

Now we need to derive the observed mass, $m_{\rm{obs}}(r)$, derived from the
observed column density map using the bijection method. The column density is
obtained by:

$N(b)=2\int_{0}^{z_{\rm{max}}}\rho(z)dz$ (19)

where $b$ is the impact parameter, equivalent to the projected radius, $z$ is
the distance along the line of sight from the tangent point of the sphere of
radius $b$, and $z_{\rm{max}}$ is the maximum distance along the line of sight
from the tangent point to the edge of the cloud. Figure 47 summarises these
different variables. The variable $z$ can be expressed as:

$z=\sqrt{r^{2}-b^{2}}$ (20)

with its maximum value $z_{\rm{max}}$ being:

$z_{\rm{max}}=\sqrt{R_{\rm{cloud}}^{2}-b^{2}}$ (21)

where $R_{\rm{cloud}}$ is the cloud radius. Equation 19 can then be written
as:

$N(b)=2\rho_{0}r_{0}^{\gamma}\int_{0}^{z_{\rm{max}}}\left(b^{2}+z^{2}\right)^{-\gamma/2}dz$
(22)

The observed mass within a projected radius $b=r$ is given by:

$m_{\rm{obs}}(r)=2\pi\int_{0}^{r}N(b)bdb$ (23)

Equation 22 can be solved analytically for some specific values of $\gamma$,
but can easily be solved numerically for any value of $\gamma$ as long as
$0\leq\gamma<3$.

Now, let us consider that the same sphere has the following velocity
dispersion profile:

$\sigma(r)=\sigma_{0}\left(\frac{r}{r_{0}}\right)^{\beta}$ (24)

where $\sigma_{0}$ is a normalisation constant. The corresponding mass-
weighted velocity dispersion is given by:

$\overline{\sigma}_{\rm{real}}(r)=\frac{\int_{0}^{r}\sigma(r)\rho(r)4\pi
r^{2}dr}{m_{\rm{real}}(r)}$ (25)

$\overline{\sigma}_{\rm{real}}(r)=\frac{4\pi\sigma_{0}\rho_{0}r_{0}^{\gamma-\beta}}{m_{\rm{real}}(r)}\frac{r^{3-\gamma+\beta}}{3-\gamma+\beta}$
(26)

Now, the observed velocity dispersion $\overline{\sigma}_{\rm{obs}}$ is given
by:

$\overline{\sigma}_{\rm{obs}}(b)=\frac{\int_{0}^{z_{\rm{max}}}\rho(z)\sigma(z)dz}{N(b)}$
(27)

$\overline{\sigma}_{\rm{obs}}(b)=\rho_{0}\sigma_{0}r_{0}^{\gamma-\beta}\frac{\int_{0}^{z_{\rm{max}}}(b^{2}+z^{2})^{(\beta-\gamma)/2}dz}{N(b)}$
(28)

Equation 28 can easily be numerically integrated for different combinations of
$\beta$ and $\gamma$ values. Note that here, we derived the expression of
$\overline{\sigma}_{\rm{obs}}(r)$ and $m_{\rm{obs}}(r)$ in the case of single
power law profiles, but, when extended to broken power laws, the expressions
becomes longer as one integration has to be made for each part of the profile.
The logic behind it being the same as for single power law profiles, we will
not derive their expressions here.

Figure 47: Sketch illustrating the definition of the different variables.

##  Appendix D Impact of non-uniform velocity dispersion profiles on the
virial ratio parameter

As noted by MD17 the non-uniformity of the velocity dispersion within clouds
impact their virial ratio estimates. Here we show the derivation of the
corresponding correcting factor, noted $a_{3}$ in MD17.

The kinetic energy of a spherical cloud of radius $R$ , density profile
$\rho=\rho(R)\left(\frac{r}{R}\right)^{-\gamma}$, and velocity dispersion
profile $\sigma(r)=\sigma(R)\left(\frac{r}{R}\right)^{\beta}$ is given by:

$E_{k}(r)=\frac{3}{2}\int_{0}^{r}\rho(r)\sigma(r)^{2}4\pi r^{2}dr$ (29)

$E_{k}(r)=6\pi\frac{\rho(R)\sigma(R)^{2}R^{\gamma-2\beta}}{3+2\beta-\gamma}r^{3+2\beta-\gamma}$
(30)

Note that here we chose $r_{0}=R$ as it will simplify the calculations. In
Equation 30, $\rho(R)$ and $\sigma(R)$, the density and velocity dispersion at
radius $r=R$ are not observable quantities. But, $\overline{\rho}(R)$ and
$\overline{\sigma}(R)$, the average density and mass weighted velocity
dispersion within radius $r=R$, are. One therefore needs to derive the
relation between $\rho(R)$, $\sigma(R)$ and $\overline{\rho}(R)$,
$\overline{\sigma}(R)$ in order to sub-in the former within the equation of
kinetic energy. For the density we have:

$\overline{\rho}(R)=\frac{\int_{0}^{R}\rho(r)4\pi r^{2}dr}{\int_{0}^{R}4\pi
r^{2}dr}$ (31)

$\overline{\rho}(R)=\rho(R)\frac{3}{3-\gamma}$ (32)

For the velocity dispersion we have:

$\overline{\sigma}(R)=\frac{\int_{0}^{R}\sigma(r)\rho(r)4\pi
r^{2}dr}{\int_{0}^{R}\rho(r)4\pi r^{2}dr}$ (33)

$\overline{\sigma}(R)=\sigma(R)\frac{3-\gamma}{3+\beta-\gamma}$ (34)

We can now sub these expressions in the equation of kinetic energy for $r=R$:

$E_{k}(R)=2\pi\overline{\rho}(R)\overline{\sigma}(R)^{2}R^{3}\frac{(3+\beta-\gamma)^{2}}{(3-\gamma)(3+2\beta-\gamma)}$
(35)

$E_{k}(R)=\frac{3}{2}M\overline{\sigma}(R)^{2}\frac{(3+\beta-\gamma)^{2}}{(3-\gamma)(3+2\beta-\gamma)}$
(36)

The correction factor on the kinetic energy $a_{3}$ (keeping the same notation
as in MD17) resulting from the non-uniform velocity dispersion is therefore
given by:

$a_{3}=\frac{(3+\beta-\gamma)^{2}}{(3-\gamma)(3+2\beta-\gamma)}$ (37)

For $\beta=0.5$ and $\gamma=1$, we obtain a correction factor
$a_{3}=\frac{25}{24}$, which is basically negligible. This is quite different
to the correction factor evaluated by MD17, which gives $a_{3}=\frac{2}{3}$
for the same power law indices. After exchanging with the authors, it has been
found that MD17 wrongly assumed that the observationally measured velocity
dispersion was $\sigma(R)$ as opposed to $\overline{\sigma}(R)$. This mistake
leads to the differences in the estimated correction factors highlighted here.
Since $a_{3}$ is close to unity, we did not take it into account in the virial
ratio estimated presented in this article.

##  Appendix E Properties comparison with H09

In Figure 48 we present the comparison of kinematics distances between this
paper and Heyer et al. (2009). One can see that for half the clouds we have in
common with H09, the distance differ. The reason is that we opted for the near
kinematic distances for all of our clouds, while they opted for the far
kinematic distance for a significant fraction of their sample. However, as
justified in the main body of this paper, it is more likely that all this
IRDC-hosting molecular clouds lie at the near distance.

In Figure 49 we present all velocity dispersion measurements as a function of
radius from H09. As explained in the main body of the paper, the blue points
are cloud-size measurements, while the yellow points are within the FWHM of
the emission peak. The points highlighted with solid black circle are those in
common between H09 and our sample. The dashed lines show L81 and S87’s
relationships.

Figure 48: Distance comparison for the sub-sample of 12 clouds common to our
study, i.e. y-axis, and that of Heyer et al. (2009), i.e. x-axis. Figure 49:
Radius versus velocity dispersion for all Heyer al. 2009 measurements. The
blue symbols show the measurements done within the original boxes from Solomon
et al. 1987. The yellow symbols show the measurements made within the half-
power isophot of the column density peak within that box. The 12 clouds that
are in common between our and Heyer’s study are shown as yellow and cyan
symbols. The dashed and dotted-dashed lines show the Larson’s and Solomon’s
realtions.