# The ALMA Survey of 70 $\mu$m Dark High-mass Clumps in Early Stages (ASHES).
VIII. Dynamics of Embedded Dense Cores

Shanghuo Li  Max Planck Institute for Astronomy, Königstuhl 17, 69117
Heidelberg, Germany  Patricio Sanhueza National Astronomical Observatory of
Japan, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Department of Astronomical Science, SOKENDAI (The Graduate
University for Advanced Studies), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Qizhou Zhang Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA  Garay, Guido Departamento de Astronomía,
Universidad de Chile, Las Condes, Santiago, Chile  Giovanni Sabatini INAF -
Istituto di Radioastronomia - Italian node of the ALMA Regional Centre (It-
ARC), Via Gobetti 101, I-40129 Bologna, Italy  Kaho Morii Department of
Astronomy, Graduate School of Science, The University of Tokyo, 2-21-1, Osawa,
Mitaka, Tokyo 181-0015, Japan  National Astronomical Observatory of Japan,
National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588,
Japan  Xing Lu Shanghai Astronomical Observatory, Chinese Academy of Sciences,
80 Nandan Road, Shanghai 200030, People’s Republic of China  Daniel Tafoya
Department of Space, Earth and Environment, Chalmers University of Technology,
Onsala Space Observatory, 439 92 Onsala, Sweden  Fumitaka Nakamura National
Astronomical Observatory of Japan, National Institutes of Natural Sciences,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Department of Astronomical
Science, SOKENDAI (The Graduate University for Advanced Studies), 2-21-1
Osawa, Mitaka, Tokyo 181-8588, Japan  Natsuko Izumi Academia Sinica Institute
of Astronomy and Astrophysics, 11F of AS/NTU Astronomy-Mathematics Building,
No.1, Section 4, Roosevelt Road, Taipei 10617, Taiwan  Ken’ichi Tatematsu
National Astronomical Observatory of Japan, National Institutes of Natural
Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Fei Li School of
Astronomy and Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing
210023, People’s Republic of China

###### Abstract

We present the dynamical properties of 294 cores embedded in twelve IRDCs
observed as part of the ASHES Survey. Protostellar cores have higher gas
masses, surface densities, column densities, and volume densities than
prestellar cores, indicating core mass growth from the prestellar to the
protostellar phase. We find that $\sim$80% of cores with virial parameter
($\alpha$) measurements are gravitationally bound ($\alpha<$ 2). We also find
an anticorrelation between the mass and the virial parameter of cores, with
massive cores having on average lower virial parameters. Protostellar cores
are more gravitationally bound than prestellar cores, with an average virial
parameter of 1.2 and 1.5, respectively. The observed nonthermal velocity
dispersion (from N2D+ or DCO+) is consistent with simulations in which
turbulence is continuously injected, whereas the core-to-core velocity
dispersion is neither in agreement with driven nor decaying turbulence
simulations. We find a no significant increment in the line velocity
dispersion from prestellar to protostellar cores, suggesting that the dense
gas within the core traced by these deuterated molecules is not yet severely
affected by turbulence injected from outflow activity at the early
evolutionary stages traced in ASHES. The most massive cores are strongly self-
gravitating and have greater surface density, Mach number, and velocity
dispersion than cores with lower masses. Dense cores do not have significant
velocity shifts relative to their low-density envelopes, suggesting that dense
cores are comoving with their envelopes. We conclude that the observed core
properties are more in line with the predictions of “clump-fed” scenarios
rather than those of “core-fed” scenarios.

Unified Astronomy Thesaurus concepts: Infrared dark clouds (787), Star forming
regions (1565), Star formation (1569), Massive stars (732), Protostars (1302),
Interstellar medium (847), Interstellar line emission (844), Protoclusters
(1297)

††facilities: ALMA††software:  CASA (McMullin et al., 2007), Astropy (Astropy
Collaboration et al., 2013), Matplotlib (Hunter, 2007), PySpecKit (Ginsburg &
Mirocha, 2011), SciPy Virtanen et al. (2020), Seaborn (Waskom, 2021), LINMIX
(Kelly, 2007).

##  1 Introduction

High-mass stars ($M\gtrsim 8\,M_{\odot}$) are mainly responsible for the
chemical enrichment and kinetic energy injection into the interstellar matter
(ISM) of galaxies (Kennicutt, 1998; McKee & Ostriker, 2007; Zinnecker & Yorke,
2007). A great progress has been made in investigating the basic properties of
high-mass star-forming regions. However, the formation of high-mass stars is
not yet well understood, in particular their earliest stages of evolution.
High-mass stars are known to form from dense ($\sim$$10^{6}$ cm-3), compact
($\sim$$0.01-0.1$ pc) self-gravitating regions, or ‘cores’, that are embedded
in more extended molecular clumps ($\lesssim$1 pc) which are dense
substructures residing in molecular clouds (10–100 pc; Bergin & Tafalla, 2007;
Zhang et al., 2009, 2015). A gravitationally bound dense core prior to the
protostellar phase is called a prestellar core. A prestellar core has not yet
formed any central protostellar object and is on the verge of gravitational
collapse (Ward-Thompson et al., 1994; Redaelli et al., 2021).

Prestellar cores represent the starting point in the star formation process.
Different high-mass star formation theories predict very different initial
conditions for the prestellar cores. In the “core-fed” or “core-accretion”
models, a molecular cloud will fragment into cores of different masses. The
most massive cores will collapse and form high-mass stars, while the low-mass
cores will evolve into low-mass stars (e.g., McKee & Tan, 2002). In these
models, the high-mass prestellar cores are formed quasi-statically. On the
other hand, in the “clump-fed” scenarios, prestellar cores initially have
masses comparable to the Jeans mass, are subvirialized, and will accrete
material from the reservoir of gas of the parental cloud (e.g., Bonnell et
al., 2001, 2004; Smith et al., 2009; Vázquez-Semadeni et al., 2019; Padoan et
al., 2020; Pelkonen et al., 2021).

For these models, the turbulent feedback in molecular clouds also affects
start formation. Offner et al. (2008a), Krumholz et al. (2005) suggest that
the presence or absence of turbulent feedback is directly related to the star
formation mechanism. If the turbulence is maintained in the cloud, then the
mass of the cores is limited by the initial turbulent compression. On the
other hand, if turbulence decays quickly, then the virial parameter decreases
significantly, and competitive accretion might be possible in the clouds.

Since high-mass stars evolve very quickly, compared to their low-mass
counterparts, and rapidly change their environment, in order to determine
their formation mechanisms, it is necessary to study high-mass star-forming
regions in a very early stage of evolution. The perfect test beds to study
different star formation mechanisms are infrared dark clouds (IRDCs). IRDCs
are molecular clouds that appear dark at mid-infrared wavelengths against the
strong Galactic background emission (Egan et al., 1998; Simon et al., 2006).
They have high column densities ($\sim$1022 cm-2), and are considered to be
the birth places of high-mass stars (Rathborne et al., 2004).

Recently, there have been a handful of observations aimed to determine the
properties of the gas and dust toward IRDCs (e.g., Bontemps et al., 2010;
Sanhueza et al., 2010, 2012, 2013; Palau et al., 2013; Wang et al., 2014;
Sanhueza et al., 2017; Lu et al., 2015; Contreras et al., 2016; Csengeri et
al., 2017; Henshaw et al., 2017; Cyganowski et al., 2017; Pillai et al., 2019;
Li et al., 2019, 2020a, 2021; Barnes et al., 2021). However, most of them lack
sufficient angular resolution to identify individual star-forming cores, or
they target sources with embedded infrared emission, suggesting that active
star formation might have already started.

In this paper, we present sensitive, high-angular resolution ($\sim$0.023 pc)
and high-sensitivity observations of the gas and dust emission within twelve
70 $\mu$m dark IRDCs from the Atacama Large Millimeter/submillimeter (ALMA)
Survey of 70 $\mu$m dark High-mass clumps in Early Stages (ASHES; Sanhueza et
al., 2019). We study the gas kinematics of dense embedded cores revealed by
the continuum emission in Sanhueza et al. (2019). These IRDCs were selected
from the Millimetre Astronomy Legacy Team survey (MALT90; Jackson et al.,
2013; Foster et al., 2013, 2011), which targeted sources from the Atacama
Pathfinder Experiment (APEX) Telescope Large Area Survey of the GALaxy
(ATLASGAL; Schuller et al., 2009; Contreras et al., 2013). They are located at
distances within 5.5 kpc (Whitaker et al., 2017), and have the perfect
condition of cluster-forming clumps in early stages of evolution: they have
large masses ($>500\,M_{\odot}$; Contreras et al., 2017), large surface
densities ($>0.1$ g cm-2), and large volume densities ($>10^{4}$ cm-3)
ensuring that they will form high-mass stars. They also have low dust
temperatures at the clump scale ($\leq 15$ K; Guzmán et al., 2015), and they
appear dark from 3.6 to 70 $\mu$m in Spitzer/Herschel, suggesting that they
have no yet embedded powerful internal heating sources. More details on the
sample selection can be found in Sanhueza et al. (2019). In spite of being IR
dark even up to 70 $\mu$m, most of these twelve IRDCs have deeply embedded
star formation activity, as revealed by “warm core” tracers Sanhueza et al.
(2019) and molecular outflows (Li et al., 2020a; Tafoya et al., 2021; Morii et
al., 2021). For statistical details of the outflow content in these IRDCs, the
reader can refer to Li et al. (2020a). The first work on the detailed
deuterated chemistry, as a first step on a single target, is presented in
Sakai et al. (2022). The chemistry and the CO depletion fraction of all the
detected ASHES cores from the pilot survey (Sanhueza et al., 2019) are
presented in Li et al. (2022) and Sabatini et al. (2022), respectively. The
whole ASHES survey of thirty-nine 70 $\mu$m dark clumps is presented in a
companion paper (Morii, 2023).

In this work, using the molecular line and continuum emission, we study the
properties of embedded dense cores. The observations are described in Section
2. In Section 3 we present our results and analysis. We discuss the
kinematical properties of embedded dense cores in Section 4. Finally, Section
5 presents the summary of our findings.

Table 1: Summary of Clumps Properties.

Name | Abbreviation | Dist. | $M_{\rm gas}$ | $M_{\rm vir}$ | $\alpha$ | $v_{\rm LSR}$ | $\sigma_{\rm obs}$
---|---|---|---|---|---|---|---
|  | (kpc) | 103 ($M_{\odot}$) | 103 ($M_{\odot}$) |  | (km s-1) | (km s-1)
G010.991-00.082 | G10.99 | 3.7 | 2.2$\pm$0.6 | 1.1$\pm$0.2 | 0.5$\pm$0.2 | 29.53 | 1.27$\pm$0.05
G014.492-00.139 | G14.49 | 3.9 | 5.2$\pm$1.1 | 3.0$\pm$0.4 | 0.6$\pm$0.1 | 41.14 | 1.68$\pm$0.05
G028.273-00.167 | G28.27 | 5.1 | 1.5$\pm$0.7 | 0.4$\pm$0.1 | 0.3$\pm$0.1 | 80.00 | 1.34$\pm$0.13
G327.116-00.294 | G327.11 | 3.9 | 0.6$\pm$0.1 | 0.1$\pm$0.1 | 0.2$\pm$0.1 | -58.93 | 0.56$\pm$0.05
G331.372-00.116 | G331.37 | 5.4 | 1.6$\pm$0.2 | 1.3$\pm$0.2 | 0.8$\pm$0.2 | -87.83 | 1.29$\pm$0.05
G332.969-00.029 | G332.96 | 4.4 | 0.7$\pm$0.1 | 0.7$\pm$0.2 | 1.0$\pm$0.3 | -66.63 | 1.41$\pm$0.05
G337.541-00.082 | G337.54 | 4.0 | 1.2$\pm$0.2 | 2.8$\pm$0.7 | 2.4$\pm$0.7 | -54.57 | 2.00$\pm$0.05
G340.179-00.242 | G340.17 | 4.1 | 1.5$\pm$0.3 | 1.8$\pm$0.2 | 1.2$\pm$0.3 | -53.69 | 1.48$\pm$0.05
G340.222-00.167 | G340.22 | 4.0 | 0.8$\pm$0.1 | 4.4$\pm$1.0 | 5.7$\pm$1.7 | -51.30 | 3.04$\pm$0.05
G340.232-00.146 | G340.23 | 3.9 | 0.7$\pm$0.1 | 0.7$\pm$0.2 | 1.0$\pm$0.3 | -50.78 | 1.23$\pm$0.05
G341.039-00.114 | G341.03 | 3.6 | 1.1$\pm$0.2 | 0.7$\pm$0.1 | 0.6$\pm$0.2 | -43.04 | 0.97$\pm$0.05
G343.489-00.416 | G343.48 | 2.9 | 0.8$\pm$0.3 | 0.4$\pm$0.1 | 0.5$\pm$0.2 | -28.96 | 1.00$\pm$0.05

††footnotetext: N2H${}^{+}\,J$ = 1–0 emission was used to derive the velocity
dispersion for the clumps, except for G028.273-00.167, G337.541–00.082, and
G340.222–00.167. The former clump was obtained using NH2D
J${}_{K_{a},K_{c}}=1_{1,1}\rightarrow 1_{0,1}$ emission, and the latter two
clumps were derived from HNC $J$ = 1–0 emission.

##  2 Observations

The observations were carried out with the Atacama Large
Millimeter/submillimter Array (ALMA), located in the Llano de Chajnantor,
Chile, during the ALMA cycle (3) and (4) (project ID: 2015.1.01539.S, PI:
Sanhueza). These observations covered both the continuum and molecular line
emission toward 12 massive 70 $\mu$m dark high-mass clumps located in the
Galactic plane. Table 1 shows the values for the masses ($M_{\rm gas}$),
virial masses ($M_{\rm vir}$), virial parameter ($\alpha$), source velocity
($v_{\rm LSR}$), and velocity dispersion ($\sigma_{\rm obs}$) of the molecular
line emission for the 12 IRDC clumps obtained from single-dish telescopes
(Contreras et al., 2017; Rathborne et al., 2016), and indicates whether there
is any asymmetries in the line profiles.

The 12 m array observations had between 36 and 48 antennas, with baselines
ranging from 15 to 704 m. Large-scale dust continuum and line emission was
also recovered thanks to the inclusion of the 7 m array and the total power
(TP; only for line emission) antennas. The 7 m array observations consisted of
8–10 antennas, with baselines ranging from 8 to 44 m. Each clump was covered
by a 10 point mosaic using the 12 m array and by a 3 pointing mosaic using the
7 m array, except for G028.27 that was observed with 11 and 5 pointing,
respectively. The angular resolution of the images is $\sim$1$\farcs$2,
corresponding to 4800 au (or 0.023 pc) at the averaged source distance of 4
kpc. The primary beams at 224 GHz of the 12 and 7 m arrays are 25$\farcs$2
(0.49 pc at the averaged source distance of 4 kpc) and 44$\farcs$6 (0.86 pc at
the averaged source distance of 4 kpc), respectively. The maximum scale
recovered scale of the 7 m array at the observed frequency is about 30′′ (0.58
pc at the averaged source distance of 4 kpc).

The receiver was set to the band 6 of ALMA, centered at $\sim$224 GHz in dual
polarization mode. The velocity resolution of the spectral windows ranged
between 0.17 and 1.3 km s-1. In this paper, we mostly focus on the N2D+ (3-2),
DCO+ (3-2), and C18O (2-1) molecular transitions, whereas the H2CO and CH3OH
are also used to study the core-to-core velocity dispersion. The former two
lines (N2D+ and DCO+) and latter three lines (C18O, H2CO, and CH3OH) have
spectral resolutions of 0.17 and 1.3 km s-1, respectively. Calibration of the
observations was carried out using the Common Astronomy Software Applications
(CASA) software package version 4.5.3, 4.6, and 4.7, while imaging was done
using CASA 5.4 (McMullin et al., 2007).

The 12 and 7 m array datasets were concatenated and imaged together using the
CASA 5.4 tclean algorithm. Line cubes were made using the yclean script, which
automatically clean each map channel with custom-made masks (Contreras et al.,
2018). Natural weighting was used. To avoid artifacts due to the complex
structure of the IRDCs emission, we used a multiscale clean, with scale values
of 1, 3, 10, and 30 times the image pixel size of 0$\farcs$2\. The 12 and 7 m
array line emission was combined with the TP observations through the
feathering technique. The achieved line rms is $\sim$0.06 K ($\sim$3.6 mJy
beam-1) and $\sim$0.02 K ($\sim$1.2 mJy beam-1) at a velocity resolution of
0.17 km s-1 and 1.3 km s-1, respectively. More details on the observations can
be found in Sanhueza et al. (2019).

##  3 Results and Analysis

###  3.1 Core Sample

For each clump, Sanhueza et al. (2019) determined their embedded cores using
the astropy dendrogram package (Rosolowsky et al., 2008; Price-Whelan et al.,
2018). A dendrogram was applied to both the emission from the image obtained
using the data of the 12 m array alone, and to the images obtained from the 12
and 7 m arrays combined. Sanhueza et al. (2019) determined that by including
the 7 m array data it was possible to detect fainter cores within the IRDCs,
thus allowing to trace a wider range of core masses. Therefore, for this
paper, we use the cores detected in the images obtained by combining the 12
and 7 m array datasets. In total, there are 294 cores in the 12 clumps.

###  3.2 Core Evolutionary Stage Classification

We have used the classification of cores into prestellar core candidates
(hereafter prestellar core) and protostellar cores defined in Sanhueza et al.
(2019), but further refined in Li et al. (2022). This classification is based
on whether there is a detection of warm gas tracers, i.e., CH3OH
$4_{2,2}-3_{1,2}$ ($E_{u}/k$ = 45.46 K; where $k$ is Boltzmann’s constant),
H2CO $3_{2,2}-2_{2,1}$ ($E_{u}/k$ = 68.09 K), and H2CO $3_{2,1}-2_{2,0}$
($E_{u}/k$ = 68.11 K), or if the cores have outflows detected in the CO, SiO,
and/or H2CO molecular line emission. A core is classified as prestellar core
if it is not associated with emission from any of three aforementioned lines
nor molecular outflow signatures. On the contrary, cores presenting any of
three aforementioned lines and/or molecular outflows are classified as
protostellar. In total, we classified the core sample into 97 protostellar
cores and 197 prestellar cores (category (1)). The protostellar cores can
further classify into three subcategorizes, 1) “outflow core” (category (2))
if it is associated with outflows but without detection of any of three
aforementioned warm lines, 2) “warm core” (category (3)) if it is associated
with any of three aforementioned warm lines but without outflow detection, and
3) “warm & outflow core” (category (4)) if it is associated with both outflows
and any of three aforementioned warm lines (see also Sanhueza et al., 2019; Li
et al., 2022).

We note that, although some cores are classified as protostellar, they still
have no emission detected at infrared wavelengths; thus, they are in an
earlier evolutionary stage than the cores traditionally classified as
protostellar. This classification scheme will be used throughout the paper to
determine whether any of the properties derived for the cores change with
their evolutionary stage.

Table 2: Summary of Dense Cores properties

Clump | Core | $M_{\rm gas}$ | $R$ |  $N_{\rm peak}$(H2) | $\Sigma$ | T${}_{\rm NH_{3}}$ | $\sigma_{\rm tot}$ | $M_{\rm vir}$ | $\alpha$ | $a_{\rm G}$ | $a_{\rm K}$ | SF
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | ($M_{\rm\odot}$) | (pc) | ($\times 10^{23}$cm-2) | (g cm-2) | (K) | (km s-1) | ($M_{\rm\odot}$) |  | (pc Myr-2) | (pc Myr-2) |
G10.99 | 1 | 6.90 | 0.024 | 1.62 | 0.78 | 13.4 | 0.34 | 2.47 | 0.36 | 10.54 | 5.09 | 3
G10.99 | 2 | 1.68 | 0.009 | 1.53 | 1.41 | 12.5 | 0.34 | 0.78 | 0.46 | 18.99 | 13.45 | 1
G10.99 | 3 | 3.10 | 0.013 | 1.57 | 1.27 | 12.1 | 0.47 | 2.14 | 0.69 | 17.19 | 18.49 | 1
G10.99 | 4 | 2.05 | 0.014 | 1.06 | 0.70 | 14.2 | 0.24 | 0.66 | 0.32 | 9.52 | 4.48 | 3
G10.99 | 5 | 3.24 | 0.015 | 1.16 | 0.96 | 10.9 | 0.25 | 0.66 | 0.20 | 13.04 | 4.44 | 0
G10.99 | 6 | 0.60 | 0.007 | 0.82 | 0.78 | 13.2 | … | … | … | 10.57 | … | 3
G10.99 | 7 | 2.87 | 0.016 | 0.75 | 0.72 | 12.2 | 0.27 | 0.98 | 0.34 | 9.69 | 4.58 | 0
G10.99 | 8 | 2.17 | 0.017 | 0.68 | 0.52 | 12.3 | 0.35 | 1.37 | 0.63 | 7.08 | 7.70 | 0
G10.99 | 9 | 1.44 | 0.012 | 0.66 | 0.71 | 11.5 | 0.40 | 1.34 | 0.93 | 9.58 | 14.04 | 0
G10.99 | 10 | 0.92 | 0.013 | 0.55 | 0.35 | 12.8 | 0.34 | 1.04 | 1.12 | 4.79 | 8.97 | 1
G10.99 | 11 | 0.53 | 0.007 | 0.62 | 0.79 | 11.8 | 0.22 | 0.22 | 0.41 | 10.65 | 7.59 | 1

(This table is available in its entirety in a machine-readable form in the
online journal. A portion is shown here for guidance regarding its form and
content.) ††footnotetext: Summary of the molecular line emission and virial
properties derived for all the cores embedded in the 12 IRDCs observed with
ALMA. Columns (1) and (2) show the short name of the parental IRDC and dense
cores, respectively. The gas mass, core radius, peak H2 density, core-averaged
surface density are presented in columns (3)–(6). Column (7) shows the gas
temperature derived from NH3. The total velocity dispersions derived from N2D+
or DCO+, virial mass, and virial parameters are presented in columns (8)–(10),
respectively. Columns (11) and (12) are the $a_{\rm G}$ and $a_{\rm K}$,
respectively. Columns (13) shows the star formation category defined for each
core, where “0” means no star formation signature (prestellar core candidate),
“1” represents the outflow core, “2” is warm core, and “3” means warm and
outflow core.

Figure 1:  Panels (a)–(d) show the histograms of the peak column density
($N_{\rm peak}$(H2)), gas mass ($M_{\rm gas}$), surface density ($\Sigma_{\rm
H_{2}}$), and volume density ($n_{\rm H_{2}}$) for the prestellar and
protostellar cores. The black and red dashed vertical lines indicate the
median values of prestellar and protostellar cores, respectively.

###  3.3 Core Mass and Core Density

We retrieved the rotational excitation temperature ($T_{\rm NH_{3}}$), that is
derived from NH3 (1,1) and (2,2) transition lines (see Appendix B in Li et
al., 2022, for detailed procedure on excitation temperature determination),
from the Complete ATCA111The Australia Telescope Compact Array Census of High-
Mass Clumps (CACHMC; Allingham et al., 2023, in preparation) at about 5′′
angular resolution. The retrieved $T_{\rm NH_{3}}$ is used as excitation
temperature in the calculation of all molecular parameters and core mass,
except for G332.96 that has no available NH3 data. The dust temperature of
12.6 K derived from the clump scale is used for G332.96 (Sanhueza et al.,
2019). The prestellar and protostellar cores have similar $T_{\rm NH_{3}}$,
with the median values of 14.1 and 14.8 K for the prestellar and protostellar
cores, respectively. These values are lower than the typical gas temperature
$\geqslant$ 20 K of protostellar cores reported in previous studies (e.g., Lu
et al., 2014; Billington et al., 2019). This further supports the notion that
these protostellar cores are still at a very early evolutionary phase. The low
temperature in the protostellar cores could be attributed to the fact that
these cores are still at a very early evolutionary stage, and the surrounding
core materials have not been sufficiently warmed up. On other other hand, we
cannot fully rule out the possibility that the NH3 observations have not
enough sensitivity or angular resolution to reveal local small temperature
enhancement toward the protostellar cores.

We have updated the gas mass ($M_{\rm gas}$), peak column density ($N_{\rm
peak}$(H2)), volume density ($n_{\rm H_{2}}$), and surface density
($\Sigma=M_{\rm gas}/(\pi r^{2})$) using the temperatures derived from the NH3
data (see Li et al., 2022). The updated parameters are 0.05–31.66 $M_{\odot}$
for the gas mass, 6.3 $\times$ 1021–5.0 $\times$ 1023 cm-2 for the peak column
density, 1.5 $\times$ 105–1.8 $\times$ 107 cm-3 for the core-averaged volume
density, and 0.05–3.39 g cm-2 for the core-averaged surface density. The
updated parameters are slightly different compared to the previous results
using the clump-scale dust temperatures in Sanhueza et al. (2019). The updated
parameters of each core are tabulated in Table 2 (see also Figure 1).

###  3.4 Properties of the Molecular Line Data

Due to the low temperatures ($<$20 K) and high densities ($>10^{5}$ cm-3)
found in cores at early stages of evolution, there is a high level of
deuteration due to freeze out of the CO molecules into dust grains (e.g.,
Caselli et al., 2002; Tan, 2018; Sabatini et al., 2019, 2022; Li et al.,
2021). This makes the emission from the N2D+ ($J$=3-2; $n_{\rm crit}=1.7\times
10^{6}$ cm-3) and DCO+ ($J$=3-2; $n_{\rm crit}=1.8\times 10^{6}$ cm-3)
molecules almost exclusively associated to the cold and dense gas toward the
cores. Unlike H2CO and CH3OH that are associated with more turbulent gas
components, N2D+ and DCO+ preferentially trace quiescent dense gas (see Li et
al., 2022). Therefore, N2D+ and DCO+ are excellent tracers of the gas velocity
dispersion in cold dense cores, minimizing the contribution from more diffuse
intra-clump gas and/or more turbulent gas related to protostellar activity,
like, for example, molecular outflows.

To determine the properties of the core envelope, defined as the lower-density
gas surrounding a dense core, we used the C18O emission. C18O ($J$=2-1;
$n_{\rm crit}=9.3\times 10^{3}$ cm-3) traces lower-density gas; thus by
determining the properties of the emission observed in the same line of sight
to the cores, we can have an idea of the material that might be associated to
the gas surrounding each core, or core envelope.

For each core detected in the continuum, we extracted the core-averaged
spectrum of the N2D+, DCO+, C18O, H2CO, and CH3OH molecular line emission
within the same area defined for each core by the dendrogram technique in
Sanhueza et al. (2019). The spectral extraction was done for the images
created by combining the 12 and 7 m arrays (hereafter 12m7m or interferometric
images), and for the images created by adding the TP array data to the 12m7m
dataset via the feathering technique (hereafter 12m7mTP or feathered images).
We preformed Gaussian fittings to the core-averaged spectrum for each line,
except for N2D+ that is fitted using hyperfine line structures (hfs); the
detail fitting processes can be found in Li et al. (2022). The best-fit
parameters, including the peak brightness temperature ($I$), the line central
velocity ($v_{\rm LSR}$), and observed velocity dispersion ($\sigma_{\rm obs}$
= FWHMobs/$\rm 2\sqrt{2ln2}$), are presented in Table 3.

The C18O emission, along the line of sight, presents asymmetric profiles
toward some of cores. These line features in our sample can be attributed
mostly to multiple velocity components because the C18O is virtually always
optically thin in this sample (see Sabatini et al., 2022), and these multiple
velocity components match those of N2H+ ($J$=1-0) and ortho-H2D+ (11,0-11,1)
toward G14.49 (the latter two lines are from Redaelli et al., 2022). In order
to fit the C18O emission associated to the cores, the central velocity derived
from the other dense gas tracers (i.e., DCO+, N2D+, CH3OH, or H2CO) was used
to guide the Gaussian fit of the C18O line.

Table 3 summarizes the properties derived from the Gaussian fit for all cores,
and Figure 2 shows examples of the molecular lines emission and the fits for
five selected cores.

Figure 2: Examples of spectra of C18O, N2D+, and DCO+ for five selected cores
in the G14.49. The results of best fit are overlaid on the spectra as red
curves.  Figure 3: Left panel: histogram of the differences between the
velocity dispersion of the DCO+ and N2D+ emission. Right panel: histogram of
the differences between the $v_{LSR}$ of these two molecules. In both panels,
the histogram shows the differences based on the spectra from the 12 and 7 m
images. The blue dashed line indicates the median value in each panel.  Table
3: Summary of N2D+ and DCO+ lines properties

Clump | Core |  |  | N2D+ |  |  |  |  | DCO+ |  |
---|---|---|---|---|---|---|---|---|---|---|---
|  | $T_{a}$ | $v_{\rm LSR}$ | $\sigma_{\rm obs}$ | $\sigma_{\rm nth}$ | $\mathcal{M}$ | Ta | $v_{\rm LSR}$ | $\sigma_{\rm obs}$ | $\sigma_{\rm nth}$ | $\mathcal{M}$
|  | (K) | (km s-1) | (km s-1) | (km s-1) |  | (K) | (km s-1) | (km s-1) | (km s-1) |
G10.99 | 1 | 0.41(0.03) | 29.75(0.03) | 0.28(0.03) | 0.27(0.03) | 1.24(0.14) | 0.33(0.03) | 29.69(0.05) | 0.23(0.12) | 0.21(0.13) | 0.95(0.61)
G10.99 | 2 | … | … | … | … | … | 0.57(0.06) | 29.89(0.09) | 0.28(0.25) | 0.27(0.27) | 1.27(1.29)
G10.99 | 3 | 0.51(0.05) | 29.63(0.05) | 0.44(0.05) | 0.43(0.05) | 2.07(0.25) | 0.32(0.03) | 29.97(0.13) | 0.53(0.38) | 0.52(0.39) | 2.54(1.88)
G10.99 | 4 | … | … | … | … | … | 0.22(0.02) | 30.01(0.10) | 0.14(0.07) | 0.10(0.10) | 0.46(0.43)
G10.99 | 5 | 0.79(0.11) | 29.90(0.03) | 0.18(0.03) | 0.16(0.03) | 0.82(0.22) | 0.71(0.07) | 29.72(0.06) | 0.22(0.13) | 0.20(0.15) | 1.01(0.77)
G10.99 | 6 | … | … | … | … | … | … | … | … | … | …
G10.99 | 7 | 1.74(0.07) | 29.56(0.01) | 0.19(0.01) | 0.17(0.01) | 0.82(0.06) | 0.55(0.05) | 29.41(0.04) | 0.18(0.05) | 0.15(0.05) | 0.73(0.26)
G10.99 | 8 | 0.44(0.06) | 29.83(0.05) | 0.30(0.05) | 0.28(0.05) | 1.35(0.26) | … | … | … | … | …
G10.99 | 9 | 0.66(0.07) | 29.52(0.04) | 0.35(0.04) | 0.34(0.04) | 1.69(0.23) | 0.56(0.06) | 29.46(0.07) | 0.20(0.08) | 0.17(0.09) | 0.87(0.44)
G10.99 | 10 | 0.29(0.05) | 30.46(0.05) | 0.28(0.05) | 0.26(0.06) | 1.23(0.27) | 0.42(0.04) | 30.17(0.05) | 0.11(0.05) | 0.06(0.09) | 0.30(0.44)
G10.99 | 11 | 0.87(0.16) | 29.78(0.03) | 0.13(0.02) | 0.09(0.04) | 0.43(0.18) | … | … | … | … | …

(This table is available in its entirety in a machine-readable form in the
online journal. A portion is shown here for guidance regarding its form and
content.) ††footnotetext: Summary of the molecular line parameters derived for
all the cores embedded in the 12 IRDCs observed with ALMA. Columns (1) and (2)
show the short name of the parental IRDC and dense cores, respectively. The
N2D+ parameters obtained by the hyperfine line fits are presented in columns
(3)–(5), columns (6) and (7) shows the observed velocity dispersion and Mach
number. Columns (8)–(10) shows the DCO+ parameters obtained by the Gaussian
fits. The nonthermal velocity dispersion and Mach number are shown in columns
(11) and (12), respectively. The corresponding uncertainty is given in
parentheses. Dashes denote no available data.

####  3.4.1 Effects of Combining the Single-dish Observations with the
Interferometric Data

We compared the line properties of the spectra obtained from the 12m7m
datasets with the 12m7mTP datasets. We found no difference in the $v_{\rm
LSR}$ between the spectra for any of the three molecules from the present
study (N2D+, DCO+, and C18O), whether we included or excluded TP observations.

For DCO+, the spectra obtained from the feathered images have a velocity
dispersion $\sigma_{\rm obs}$ on average 10% larger than the spectra obtained
from the interferometric data alone. In addition, for 63% of the sample, this
difference is lower than 10%. For the N2D+ data, we also see a similar
behaviour; however, the differences of $\sigma_{\rm obs}$ between the
feathered and nonfeathered data are on average only 3%. This suggests that the
DCO+ line traces more extended gas than the N2D+ emission toward dense cores.

The largest discrepancy was found for the C18O velocity dispersion, where the
differences between the interferometric data and the one including the single-
dish are of the order of $\sim$30%. This is because the C18O traces relatively
lower-density gas compared to the DCO+ and N2D+, such as the low-density
envelope of cores (Walsh et al., 2004; Tychoniec et al., 2021).

The maximum scale recovered by the 7 m array corresponds to $\sim$0.58 pc
(30′′ at the averaged source distance of 4 kpc), which is much larger than the
detected core typical size of $\sim$0.02 pc. Also, the sizes of the cores were
determined from their dust continuum emission, which only have the emission
from the 12 and 7 m arrays. Therefore to have a better comparison between the
properties of the cores and their envelopes, and to avoid the more extended
emission that might be associated with the clump itself (intra-clump gas not
associated with the dense cores), for the rest of our analyses, we will use
the properties of the line emission derived from the data cubes that only
contain the 12 and 7 m arrays.

In addition, we have assessed how much flux is recovered in the 12m7m
continuum images using the single-dish data retrieved from the APEX ATLASGAL
survey at 870 $\mu$m (Schuller et al., 2009). We have scaled the 870 $\mu$m
flux to 1.3 mm flux assuming a dust emissivity spectral index of $\beta$ =
1.5; $F_{\rm 1.3mm,exp}=F_{\rm 0.87mm}\,(1.3/0.87)^{-(\beta+2)}$. The
continuum images recover about 10–33% ($F_{\rm 1.3mm,12m7m}/F_{\rm
1.3mm,exp}$) of single-dish continuum flux, with a mean value of 21% (see also
Sanhueza et al., 2019). For the DCO+ and N2D+ lines, their 12m7m data can
recover about 79% and 92% of the 12m7mTP flux (see also Li et al., 2022). This
suggests that the continuum images recover less extended emission than the
DCO+ and N2D+ images made using the 12m7m data.

###  3.5 Gas Properties of the Cores

In general, the emission from the N2D+ and/or DCO+ traces well the dust
continuum core in the ASHES observations. For the 294 cores detected in the 12
clumps, we detected DCO+ emission toward 116 cores (40%) and N2D+ emission
toward 54 cores (18%). We detected both DCO+ and N2D+ emission toward 41 cores
(14%). In total, 129 cores (44%) were detected in either DCO+ and/or N2D+.

More than half of dust continuum cores present no detectable emission from
either N2D+ or DCO+. Comparing the physical properties of the cores with
detected and undetected deuterated species, we found that the ones without
emission from deuterated species have lower integrated fluxes in their
continuum emission ($\langle S_{\rm cont,nodeu}\rangle=1.5$ mJy, compared to
$\langle S_{\rm cont,deu}\rangle=4.4$ mJy), smaller masses ($\langle M_{\rm
gas,nodeu}\rangle=0.95$ $M_{\odot}$, compared to $\langle M_{\rm
gas,deu}\rangle=2.87$ $M_{\odot}$), and lower surface densities
($\langle\Sigma_{\rm nodeu}\rangle=0.36$ g cm-2, compared to
$\langle\Sigma_{\rm deu}\rangle=0.70$ g cm-2).

Since the emission from the deuterated species is rather weak, with the
signal-to-noise in both species going up to 10, the lack of detection of
deuterated species could be due to either the sensitivity of our observations
or their abundances being too low in the gas phase, or both. Given that we
want to compare the properties of the gas and the dust, we only focus on those
cores that show emission in both the dust continuum and molecular lines.

The average values of the core line velocity dispersion ($\sigma_{\rm obs}$)
are similar in both molecules, with $\sigma_{\rm obs,N_{2}D^{+}}=0.23$ km s-1
and $\sigma_{\rm obs,DCO^{+}}=0.32$ km s-1. For the 41 cores that showed
emission in both molecules, we compared the observed velocity dispersion
($\sigma_{\rm obs}$) and the central velocity ($v_{\rm LSR}$). Figure 3 shows
the differences of these two quantities between DCO+ and N2D+. In general, the
central velocity differences have a normal distribution centered around 0.05
km s-1. The DCO+ lines are slightly broader than N2D+ lines, which could be
due to unresolved hyperfine structure. The velocity dispersion of DCO+ is
derived from the single Gaussian fit, whereas N2D+ is derived from the hfs
fit. In two cases (these two cores are not shown in Figure 3), the central
velocity differences are up to 2.10 km s-1 (G14.49-#6) and 0.77 km s-1
(G14.49-#11). This is most likely because their local environments have been
significantly affected by the nearby strong protostellar outflow (see Figure 1
of Li et al., 2022).

Figure 4:  Violin plots of the $\mathcal{M}$ distributions for each line. The
shape of each distribution shows the probability density of the data smoothed
by a kernel density estimator. The blue horizontal bars from the top to bottom
in each violin plot represent the maximum, mean, and minimum values,
respectively.

We found that 71% of the cores show a difference in the DCO+ and N2D+ central
velocity $<0.17$ km s-1, which is smaller than the spectral resolution of our
data, and 94% of all cores shows a difference $<0.34$ km s-1. Thus, the
emission from these two molecules is likely arising from the same physical
location in the cores. Therefore, for the analysis of this paper, we will use
the velocity dispersion obtained from the DCO+ emission, and in the case that
DCO+ is not detected, we will use the velocity dispersion obtained from the
N2D+ line. The measured velocity dispersion have a mean value
$\langle\sigma_{\rm obs}\rangle=0.32$ km s-1, ranging between 0.1 and 0.77 km
s-1. The nonthermal velocity dispersion is given by $\sigma_{\rm
nt}^{2}=\sigma_{\rm obs,int}^{2}-\sigma_{\rm th,m}^{2}$, where $\sigma_{\rm
obs,int}$ is the intrinsic observed velocity dispersion after removing the
smearing effect due to the channel width using $\sigma_{\rm
obs,int}=\sqrt{(\rm FWHM_{\rm obs}^{2}-FWHM_{\rm ch}^{2})/8ln2}$ =
$\sqrt{\sigma_{\rm obs}^{2}-\rm FWHM_{\rm ch}^{2}/8ln2}$, $\sigma_{\rm obs}$
is the observed velocity dispersion, FWHMch is the channel width, and
$\sigma_{\rm th,m}=(k_{\rm B}T/\mu m_{H})^{1/2}=9.08\times 10^{-2}\,{\rm
km\,s^{-1}}\rm\left(\frac{T}{K}\right)^{0.5}\mu^{-0.5}$ is the thermal
velocity dispersion. The value of $\mu$ for both DCO+ and N2D+ is 30. The
sound speed $c_{\rm s}$ can be estimated using a mean molecular weight per
free particle of $\mu_{\rm p}$ = 2.37, which assumes a typical interstellar
abundance of H, He, and metals (Kauffmann et al., 2008). For the cores,
$\sigma_{\rm th,m}$ ranges between 0.04 and 0.08 km s-1, with a mean value of
0.06 km s-1, and $\sigma_{\rm nt}$ ranges between 0.03 and 0.77 km s-1, with a
mean value $\sigma_{\rm nt}=0.27$ km s-1.

###  3.6 Gas properties of the envelopes

The C18O emission is extended and filamentary in the clumps. In our sample,
91% of the cores have C18O emission, and we speculate that C18O is likely
depleted in the cores where there is no detection (Sabatini et al., 2022).
Using the core systemic velocity defined by other dense gas tracers (i.e.,
N2D+, DCO+, H2CO, or CH3OH), we were able to fit a Gaussian to the C18O line
profiles to 267 out of the 294 cores. C18O was observed in a spectral window
with coarser spectral resolution (1.3 km s-1), which is in general still
narrower than the derived typical FWHM (2.2 km s-1) for this transition.
However, the coarser spectral resolution might make us overestimate the true
internal velocity dispersion of the gas (inside cores). For these reasons, we
focused our analysis on the deuterated species, and used the parameters
derived from the C18O only as a reference, to compare the properties of the
core’s gas with the properties of their envelopes. The velocity dispersions
for the C18O emission, as expected, have larger values than the ones found for
the deuterated species, ranging from $\sigma_{\rm obs,C^{18}O}=0.4$ to 3.4 km
s-1, with a mean value of 0.9 km s-1.

###  3.7 Virial mass and virial parameter

To determine the gravitational state of the cores, we calculated their virial
parameter $\alpha=M_{\rm vir}/M_{\rm gas}$, where $M_{\rm vir}$ is the virial
mass, and $M_{\rm gas}$ is the mass derived from the dust continuum emission.
The virial parameter, $\alpha$, is commonly used to assess the gravitational
state of molecular clouds. A value of $\alpha<1$ suggests that a core is
gravitationally unstable to collapse, and turbulence alone cannot maintain its
stability (Bertoldi & McKee, 1992). Nonmagnetized cores with $\alpha\sim 1$
and $\alpha<2$ are considered to be in hydrostatic equilibrium and
gravitationally bound, respectively. A value of $\alpha>2$, on the other hand,
suggests that the core is gravitationally unbound (Kauffmann et al., 2013),
and if we ignore additional support mechanisms such as the magnetic field or
external pressure, then the core might be a transient object.

The virial mass was calculated using the observation measurements following
(Bertoldi & McKee, 1992):

$M_{\rm vir}=\frac{5}{a\beta}\frac{\sigma_{\rm tot}^{2}R}{G},$ (1)

where $R$ is the core radius, $a=(1-b/3)/(1-2b/5)$ is the correction factor
for a power-law density profile $\rho\propto R^{-b}$, $\beta=({\rm
arcsin}\,e)/e$ is the geometry factor (see Fall & Frenk, 1983; Li et al.,
2013, for detailed derivation), $\sigma_{\rm tot}$ is the total velocity
dispersion of the gas in the core, and $G$ is the gravitational constant. The
eccentricity $e=\sqrt{1-f_{\rm int}^{2}}$ is calculated by the intrinsic axis
ratio, $f_{\rm int}$, of the dense cores. The $f_{\rm int}$ can be estimated
from observed axis ratio, $f_{\rm obs}$, with $f_{\rm
int}=\frac{2}{\pi}\,f_{\rm obs}\,\mathcal{F}_{1}(0.5,0.5,-0.5,1.5,1,1-f_{\rm
obs}^{2})$ (Fall & Frenk, 1983), where $\mathcal{F}_{1}$ is the Appell
hypergeometric function of the first kind. For the dense cores, the derived
$\beta$ ranges from 1.0 to 1.4, with a mean value of 1.2. Here we adopted a
typical density profile index $b=1.6$ for all dense cores (e.g., Beuther et
al., 2002; Butler & Tan, 2012; Palau et al., 2014; Li et al., 2019). The
velocity dispersion is given by $\sigma_{\rm tot}^{2}=\sigma_{\rm
nt}^{2}+c_{\rm s}^{2}$, and it reflects a combination between the nonthermal
motions of the gas within the core, $\sigma_{\rm nt}$, and the thermal motion
of the particle mean mass, $c_{\rm s}^{2}$. $c_{\rm s}^{2}$ ranges between
0.16 and 0.28 km s-1, with a mean value of 0.23 km s-1.

We found that the values of the virial masses range from 0.22 to 8.61
$M_{\odot}$, and the mean value is 1.89 $M_{\odot}$ for all the 129 cores
analyzed. The virial parameter $\alpha=M_{\rm vir}/M_{\rm gas}$ ranges from
0.15 to 8.91, with a mean value of 1.39. Seventy-three out of 129 cores have
virial parameters below 1, 28 cores have virial parameters between 1 and 2,
and 28 cores have $\alpha$ larger than 2. The derived values of the virial
mass and virial parameter are summarized in Table 2.

Table 4: Median and mean values of derived core properties for each category

Name | Prestellar | Protostellar | All
---|---|---|---
|  |  | Outflow Core | Warm Core | Warm&outflow Core |  Suma |  |
| Median | Mean | Median | Mean | Median | Mean | Median | Mean | Median | Mean | Median | Mean
$T_{\rm NH_{3}}$ | 14.1 | 14.2 | 12.8 | 13.8 | 14.8 | 14.3 | 15.3 | 14.9 | 14.8 | 14.4 | 14.4 | 14.3
$M$ | 0.63 | 1.15 | 1.26 | 2.18 | 0.69 | 2.00 | 3.11 | 4.66 | 1.41 | 3.11 | 0.77 | 1.79
$n_{\rm H_{2}}$ | 1.11 | 1.58 | 1.59 | 2.62 | 1.26 | 2.82 | 3.59 | 5.55 | 2.40 | 3.87 | 1.32 | 2.34
$N_{\rm peak}$(H2) | 3.01 | 3.58 | 5.40 | 7.26 | 3.35 | 5.55 | 11.40 | 15.57 | 6.32 | 9.95 | 3.58 | 3.73
$\Sigma$ | 0.29 | 0.34 | 0.46 | 0.60 | 0.30 | 0.54 | 0.96 | 1.27 | 0.64 | 0.85 | 0.35 | 0.51
$R$ | 0.012 | 0.014 | 0.014 | 0.015 | 0.012 | 0.013 | 0.013 | 0.015 | 0.013 | 0.014 | 0.013 | 0.014
$M_{\rm vir}$ | 1.05 | 1.67 | 1.31 | 1.33 | 1.28 | 2.29 | 2.00 | 2.42 | 1.58 | 2.18 | 1.25 | 1.89
$\alpha$ | 1.01 | 1.54 | 0.52 | 0.61 | 1.88 | 2.35 | 0.40 | 0.53 | 0.58 | 1.19 | 0.87 | 1.39
$\mathcal{M}_{\rm N_{2}D^{+}}$ | 0.9 | 0.8 | 1.0 | 1.1 | 0.8 | 0.9 | 1.0 | 1.0 | 0.9 | 1.0 | 0.9 | 1.0
$\mathcal{M}_{\rm DCO^{+}}$ | 1.2 | 1.3 | 1.4 | 1.3 | 1.6 | 1.7 | 1.5 | 1.5 | 1.5 | 1.6 | 1.3 | 1.4
$\mathcal{M}_{\rm C^{18}O}$ | 3.1 | 3.8 | 3.8 | 4.2 | 4.0 | 4.4 | 3.6 | 4.0 | 3.8 | 4.2 | 3.3 | 3.9

††footnotetext: Notes. The unit for $M_{\rm gas}$, $n_{\rm H_{2}}$, $N_{\rm
peak}$ (H2), $\Sigma$, $R$, and $M_{\rm vir}$ are $M_{\odot}$, $\times 10^{6}$
cm-3, $\times 10^{22}$ cm-2, g cm-2, pc, and $M_{\odot}$, respectively.

aafootnotetext: All protostellar cores include outflow core, warm core, and
warm and outflow core.

Figure 5:  Panel (a): upper panel shows the number of sources per mass bin,
and the bottom panel shows a box plot of the values of the virial parameter
per bin. In the box plot, the mean value of the virial parameter is shown with
a horizontal line. Panels (b) and (c): $\alpha$ versus mass for all the cores
where molecular emission from either N2D+ or DCO+ was detected. Panel (b)
shows the cores color coded by their evolutionary stage classification (Li et
al., 2022), and panel (c) shows the cores color coded by their parental
molecular cloud. The solid line in both figures shows the linear regression to
all the point in the plot, gives a slope of -0.61 $\pm$ 0.06, and the gray
shadowed area shows the 1$\sigma$ confidence interval for the fit. The
Spearman’s rank test returns a coefficient of $r_{s}$ = -0.68.

###  3.8 Mach Number

To analyze the amount of turbulence within the cores, we calculated the one-
dimensional (1D) turbulent Mach number ($\mathcal{M}=\sigma_{\rm nt}/c_{\rm
s}$). The $\mathcal{M}$ shows no significant difference between the prestellar
and protostellar cores for C18O, DCO+, and N2D+ (see Figure 4), indicating
that the nonthermal motions of the molecular gas traced by C18O, DCO+, or N2D+
have not yet been significantly influenced by protostellar activity. There are
98% (53/54) and 82% (95/116) of $\mathcal{M}$ derived from N2D+ and DCO+
smaller than 2, respectively. N2D+ and DCO+ are typical tracers of cold and
dense gas that are closely related to core formation. The transonic
$\mathcal{M}$ suggests that the cold and dense gas in the cores have been so
far weakly affected by protostellar feedback (see also Li et al., 2020b),
albeit a small fraction (14%) of embedded cores is associated with outflow
activity. Therefore, these 70 $\mu$m dark clumps are ideal targets for
investigating the initial conditions of massive star and cluster formation.

Overall, $\sigma_{\rm obs}$ and $\mathcal{M}$ of C18O are systemically higher
than N2D+ and DCO+. This is because the latter two deuterated species are
preferentially tracing cold and dense gas, whereas C18O probes relatively less
dense gas.

For all cores, the $\mathcal{M}$ derived from DCO+ or N2D+ ranges from 0.1 to
4.6, with the mean and median values of 1.4 and 1.3, respectively. We found
that 44 cores show subsonic motions ($\mathcal{M}\leqslant$1), and 64 cores
present transonic motions (1$<\mathcal{M}\leqslant$2). This suggests that the
motions in most of the deuterated cores (108/129 = 84%) are subsonic or
transonic dominated.

##  4 Discussion

###  4.1 Core mass growth

As shown in Figure 1, the protostellar cores have higher column densities, gas
masses, volume densities, and surface densities than those from prestellar
cores (see also Table 4). A similar trend is found in the individual clumps,
except for G331.37, G340.17, and G340.22. The latter two clumps are still at
very early evolutionary phases as evidenced by the nondetection of molecular
outflows. The G331.37 hosts 1 relatively massive (7.96 $M_{\odot}$) prestellar
core, while the rest of the prestellar and protostellar cores have comparable
mass ($<$ 4 $M_{\odot}$).

These results suggest that the protostellar cores are more massive and denser
than the prestellar cores, indicating a core mass growth from the prestellar
to protostellar stages (e.g., Kong et al., 2021; Takemura et al., 2022; Nony
et al., 2023). In addition, Li et al. (2020b) found that the more massive
cores have a longer accretion history than the less massive cores. A
Kolmogorov-Smirnov (KS) test, which returns a probability ($p$-value) of two
samples being drawn from the same population, was used to compare the peak
column density, gas mass, volume density, and surface density of the
prestellar cores with those of the protostellar cores. If the $p$-value is
much smaller than 0.05, we can reject the null hypothesis that the two samples
are drawn from the same parent distribution (Teegavarapu, 2019). The KS test
reveals that the $p$-value is 2.5$\times$10-13, 1.6$\times$10-6,
1.6$\times$10-6, and 6.8$\times$10-12 for peak column density, gas mass,
volume density, and surface density, respectively, indicating that the
prestellar cores and the protostellar cores in our sample are drawn from
significantly different populations ($p$-value $\ll$ 0.05).

Table 4 also shows that warm and outflow cores tend to be more massive and
denser than the remaining type of cores. The warm and outflow cores associated
with molecular outflows and warm line(s) emission could be considered as the
most evolved objects among the identified cores. This further supports that
the cores have grown in mass and density over time.

###  4.2 Stability

Several observational biases could lead to underestimate the virial parameter.
For instance, the observed line width could be underestimated when a
particular molecular line preferentially traces molecular gas with densities
above the critical density of the line transition, resulting in an
underestimation of the virial parameter (Traficante et al., 2018b). In our
case, this effect is likely insignificant owing to the critical density of the
transition lines used in this study typically being 1-2 orders of magnitude
larger than the effective excitation density because of the effect of
radiative trapping (Shirley, 2015); the critical density is $\sim$106 cm-3 for
DCO+ and N2D+. Therefore, both DCO+ and N2D+ can effectively trace dense gas
within the cores. On the other hand, the virial parameter could be
overestimated if gas bulk motions and background emission are not properly
considered in the calculation of the velocity dispersion (e.g., Singh et al.,
2021). We computed the velocity dispersion from the core-averaged spectrum for
each core, bulk motions (if present) already included as they could increase
the velocity dispersion over the core. In addition, the interferometric
observations of dense gas tracers filter out the large-scale emission and,
therefore, are barely affected by the contamination of background emission.
Overall, the above-mentioned observational biases do not significantly affect
our results.

Figure 6: Left panel: virial parameter versus surface density. Right panel:
virial parameter versus peak column density. The blue solid line in both
figures shows the linear regression to all the point in the plot, and the gray
shadowed area shows the 1$\sigma$ confidence interval for the fit. The best
fit gives a slope -1.22 $\pm$ 0.15 and -1.17 $\pm$ 0.13 for $\alpha$–$\Sigma$
and $\alpha$–$N_{\rm H_{2}}$, respectively. The Spearman’s rank coefficients
are shown in the lower left of each panel.

####  4.2.1 Are massive cores more unstable?

Comparing the virial parameter of the cores with their masses, we find a
correlation between these two quantities. The value of $\alpha$ appears to
decrease with increasing core mass (see Figure 5), with a Spearman’s rank
correlation222Spearman rank correlation test is a nonparametric measure of the
monotonicity of the relationship between two variables. The correlation
coefficient $|r_{s}|\geqslant 0.5$ means strong correlation,
$0.5>|r_{s}|\geqslant 0.3$ means moderate correlation, $0.3>|r_{s}|\geqslant
0.1$ means weak correlation, and $0.1>|r_{s}|$ means no correlation (Cohen,
1988). If the $p$-value of the correlation is not less than 0.05, the
correlation is not statistically significant. coefficient of $r_{s}$ = -0.68
($p$-value=1.7$\times 10^{-18}$). Considering that the virial parameter is
mathematically inversely proportional to the cores’ mass, a slope of -1 in a
log–log plot would suggest that the trend seen is not physically meaningful. A
linear regression333We fit a linear regression mode to the data using the
LINMIX (Kelly, 2007), which is a hierarchical Bayesian linear regression
routine that accounts for errors on both variables. The LINMIX is used for the
linear regression fitting throughout the paper.  between the $\log{\alpha}$
and $\log{M_{\rm gas}}$, gives a slope of -0.61 $\pm$ 0.06, indicating that
the trend between the stability of the cores and their mass is real, and not
due to an observational bias given by the correlation between the two core
properties. The observed trend of $\alpha$ and $M_{\rm gas}$, $\alpha\sim
M_{\rm gas}^{-0.61\pm 0.06}$, is consistent with the expectations for
pressure-confined cores, in which $\alpha$ should depend on mass as
$\alpha\sim M_{\rm gas}^{-2/3}$ (e.g., Bertoldi & McKee, 1992). The
$\log{\alpha}-\log{M_{\rm gas}}$ correlation indicates that the most massive
cores tend to be more gravitationally unstable. In contrast, the less massive
cores are gravitationally unbound and may eventually disperse if no other
mechanism(s) can help to confine them, such as external pressure (Li et al.,
2020a). A similar inverse relationship between the virial parameter and the
core mass has also been seen in other IRDCs (e.g., Li et al., 2019, 2020a).

As shown in Figure 6, there is a strong inverse relationship between the
virial parameter and the core surface density, with a Spearman’s rank
coefficient of $r_{s}$ = -0.70 ($p$-value=6.4$\times 10^{-20}$), and the best
fit returns a slope of -1.22 $\pm$ 0.15. This indicates that higher surface
density cores appear to be more gravitationally unstable. We also find a
strong trend of virial parameters and column density, with the higher column
density tending to show a lower virial parameter (see Figure 6). The
Spearman’s rank test returns a coefficient of $r_{s}$ = -0.75
($p$-value=1.8$\times 10^{-24}$), and the best fit gives a slope of -1.17
$\pm$ 0.13. The inverse $\alpha-N_{\rm H_{2}}$ relation suggests that the
higher-density cores tend to be more gravitationally unstable. Overall, these
results suggest that the more massive and denser cores tend to be more
gravitationally unstable.

Given the angular resolution of our observations, it is possible that with
observations at higher angular resolution the most massive cores could further
fragment into smaller objects. Assuming that the cores are not fragmented, and
ignoring the effect of additional support (such as magnetic fields), we
suggest that cores with masses M${}_{\rm core}>4$ $M_{\odot}$ in each clump
are strongly self-gravitating ($\langle\alpha\rangle\,=0.37<1$) and prone to
collapse, except for core #8-G28.27, #1-G332.96, and #4-G340.22 that have
$\alpha$ of 1.6, 1.4, and 1.1, respectively.

The most massive core in all the 12 IRDCs is core #1 in G340.23, with a mass
of 31.66 $M_{\odot}$. For the remaining IRDCs, the most massive cores have
masses of $\sim$10 $M_{\odot}$. If these cores will form high-mass stars, then
while collapsing they might still be accreting material from their environment
in order to have enough mass ($>27$ $M_{\odot}$, assuming a star formation
efficiency of 30%) to form a $>$8 $M_{\odot}$ star. Indeed, this is happening
for one of the cores of G331.37 (core #1), where additional observations of
the HCO+ molecular gas show that this subvirialized core is accreting material
at a high rate from its environment (Contreras et al., 2018). If this core in
G331.37 continues to accrete at its current rate ($2.0\times 10^{-3}$
$M_{\odot}$ yr-1), it is expected that in a freefall time (3.3 $\times$ 104
yr) the core will have 82 $M_{\odot}$, gathering a large mass reservoir and
allowing the formation of high-mass stars. Although this is only one example,
we may expect similar accretion rates in several ASHES cores given their low
virial parameters and massive gas reservoir in their natal clumps (see Table
1). In addition, the filamentary structures connected to cores can also
transport material from the parental clump onto cores and thus further
increase their masses. For example, the mass flow rate along a filamentary
structure feeding a protostellar core in one of the ASHES targets, G14.49, is
about $2.2\times 10^{-4}$ $M_{\odot}$ yr-1 (Redaelli et al., 2022). Such
accretion rates along filamentary structures are not unusual in high-mass
star-forming regions (e.g., Sanhueza et al., 2021).

####  4.2.2 Does core stability change with the evolutionary stage of cores?

Prestellar and protostellar cores have a mean virial mass of $\langle M_{\rm
vir}\rangle=1.67$ $M_{\odot}$ and $\langle M_{\rm vir}\rangle=2.18$
$M_{\odot}$, respectively. Overall, protostellar cores have higher virial mass
than that of prestellar cores, indicating that the virial mass tends to
increase with the evolutionary stage of cores. This is likely because the
velocity dispersion toward protostellar cores is slightly larger than that of
prestellar cores; the mean velocity dispersion derived from N2D+ or DCO+ is
0.28 and 0.32 km s-1 for prestellar and protostellar cores, respectively. On
the other hand, the virial parameter of protostellar cores is slightly lower
than the prestellar cores as shown in Figure 5. The prestellar cores have a
mean virial parameter of $\langle\alpha\rangle=1.5$, while the protostellar
cores have a mean value of $\langle\alpha\rangle=1.2$. This is because the
protostellar cores have relatively higher gas mass than the prestellar cores.
The lower value of the virial parameter, ignoring additional forces of
support, for protostellar cores is in agreement with the scenario where
protostars might be embedded in them, and thus the gravitational collapse has
already started in these regions. Figures 5 and 6 suggest that the virial
parameter decreases with mass and density. In Section 4.1, we also find that
both mass and density of cores increase from the prestellar to the
protostellar phase. These observational results indicate that the virial
parameter tends to decrease with the evolutionary stage of cores, and
therefore cores become more gravitationally unstable with evolution.

Figure 7: Left panel: the Mach number ($\sigma_{\rm nt}/c_{\rm s}$) of the
cores, obtained from DCO+ or N2D+ molecular line emission for each clump. The
total row shows the mean value of the velocity dispersion for the clumps that
have a measurement in both the prestellar and protostellar sample. Right
panel: core-to-core velocity dispersion of the cores $v_{\rm LSR}$ obtained
from DCO+ or N2D+ molecular line emission for each clump. The total row shows
the mean value of the core-to-core velocity dispersion for the clumps that
have a measurement in both the prestellar and protostellar sample. The color
scale shows the number of cores considered to determine the velocity
dispersion. ‘0’ means only one available core. Dashes denote no available
core.

Figure 8:  Panel (a): upper panel shows the number of sources per mass bin,
and the bottom panel shows a box plot of the values of the Mach number per
bin. In the box plot the mean value of the Mach number is shown with a
horizontal line. Panel (b): Mach number versus mass for all the cores where
molecular emission from N2D+ or DCO+ was detected. The blue solid line shows
the linear regression to all the points in the plot (slope of 0.17 $\pm$ 0.04)
and the gray shadowed area shows the 1$\sigma$ confidence interval for the
fit. The Spearman’s rank test gives a coefficient of $r_{s}$ = 0.33,
suggesting a moderate correlation between Mach number and gas mass. Panel (c):
nonthermal velocity versus core radius for all the cores where molecular
emission from N2D+ or DCO+ was detected. The correlation between nonthermal
velocity and core radius is weak, with a spearman’s rank coefficient of
$r_{s}$ = 0.28. The black dashed line shows the the original Larson relation
with $\sigma\propto R^{0.38}$ (Larson, 1981), and the red dashed line is the
revised Heyer & Brunt (2004) relation with $\sigma\propto R^{0.56}$. The blue
solid line indicates the best fitting result with $\sigma\propto R^{0.27}$.

###  4.3 Turbulence

####  4.3.1 Global turbulence in IRDCs

The origin of turbulence in molecular clouds is still not well understood, as
well as its behavior with cloud evolution. Molecular cloud formation and
destruction can be a dynamical process, in which turbulence is transient,
decaying quickly on timescales comparable to the cloud lifetime (e.g.,
Elmegreen, 2000; Hartmann, 2001; Dib et al., 2007). Alternatively, the
turbulence can decay slowly in a quasi-equilibrium process, in which
turbulence is fed into the molecular cloud by, for example, protostellar
outflows, HII regions, or external cloud shearing (e.g., Shu et al., 1987;
McKee, 1999; Krumholz & Bonnell, 2007; Nakamura & Li, 2007).

Simulations of isothermal molecular clouds with and without a continuous
injection of energy have shown differences in the properties of the core gas
internal velocity dispersion and the relative motions (i.e., the core-to-core
centroid velocity dispersion) between the cores (Offner et al., 2008a, b).
Therefore, observations of these properties can in principle be used to
analyze the degree of turbulence within molecular clouds.

In the driven scenario, i.e., where there is a continuous injection of energy
into the cloud, the simulations show that, in a clump, prestellar cores have
in average $\sigma_{\rm nt}<1.5c_{\rm s}$, and there is a small increment of
this value for protostellar cores (Offner et al., 2008b, a). In the decay
scenario, where there is no additional injection of energy to the cloud,
prestellar cores have in average $\sigma_{\rm nt}<1.0c_{\rm s}$, and it
increases to larger values for protostellar cores (Offner et al., 2008b, a).
This is because, as the turbulence decays, the cloud will start to collapse,
and material will fall into the clump gravitational potential.

In our observations, the mean velocity dispersion, derived from the N2D+ or
DCO+, between the prestellar and protostellar cores increases slightly from
$\sigma_{\rm nt}\sim(1.33\pm 0.04)c_{\rm s}$ to $\sigma_{\rm nt}\sim(1.47\pm
0.06)c_{\rm s}$. This trend is also seen in 7 out of 9 clumps that have
available $\sigma_{\rm nt}$ from N2D+ or DCO+. G327.11 and G343.48 do not show
this trend. There are three clumps (G331.37, G340.17, G340.22) with no
available $\sigma_{\rm nt}$ from N2D+ or DCO+ for the protostellar sample. The
velocity dispersion in the protostellar cores is similar or slightly smaller
than those of prestellar cores in the latter two clumps. In addition, the mean
velocity dispersion obtained from C18O also shows a small increment from
prestelllar to protostellar stages in a clump. The statistics for the velocity
dispersion of DCO+ and N2D+ for each clump are presented in Appendix A (see
Figure 11). Overall, the measured core-averaged $\sigma_{\rm nt}$ does not
increase significantly from prestellar to protostellar phases, suggesting that
our observations are consistent with simulations where there is a continuous
injection of energy, consistent with the driven scenario.

An additional piece of information can be obtained by studying the core-to-
core velocity dispersion. For simulations where no additional sources of
turbulence are injected, and therefore the turbulence is decaying, the
relative motions between protostellar cores have a higher dispersion compared
to the prestellar cores (Offner et al., 2008b). On the other hand, if the
turbulence is driven, i.e., energy is continuously injected into the
simulations, the prestellar cores have a higher core-to-core velocity
dispersion than the protostellar cores (Offner et al., 2008b).

The core-to-core velocity dispersion ($\Delta v_{\rm LSR}$) is computed from
the standard deviation of the centroid velocity ($v_{\rm LSR}$) of all of the
dense cores within each clump. The calculations were performed for clumps
having more than two core centroid velocities available. We calculated the
core-to-core centroid velocity dispersion between the cores for each clump
from the $v_{\rm LSR}$ of the Gaussian fits to the DCO+ and N2D+ emission.
Figure 7 shows a heat-map of the velocity dispersion for all the cores, and
for the cores classified as prestellar and protostellar. We find that in three
clumps the core-to-core velocity dispersion is comparable between protostellar
and prestellar cores (i.e., G28.27, G343.48, and G327.11), which is neither
consistent with the driven nor the decaying turbulence simulations. The core-
to-core velocity dispersion of prestellar cores is higher than that of
protostellar cores in three clumps (i.e., G10.99, G332.96, and G340.23), which
is consistent the driven turbulence scenario. On the other hand, we also find
that the core-to-core velocity dispersion of prestellar cores is smaller than
that of protostellar cores in other three clumps (i.e., G14.49, G337.54, and
G341.03), which is consistent with the decaying turbulence scenario. The core-
to-core velocity dispersion variation among the clumps might reflect
differences in their internal turbulence, which possibly varies clump by clump
and/or related to the local cloud environments. The statistics for the core-
to-core velocity dispersion derived from DCO+ and N2D+ for each clump are
presented in Appendix A (see Figure 12). Overall, the prestellar cores have a
smaller core-to-core velocity dispersion compared to the protostellar cores if
we take the mean value of all the ASHES clumps (Figure 7).

The observations presented here, corresponding to the pilot survey (Sanhueza
et al., 2019), give some support to the scenario in which the global
turbulence of IRDCs is driven, but a larger sample is required before to draw
firm conclusions on the relative merits of driven or decaying turbulence. We
expect to address this point again using the complete survey in the future.

Figure 9: Modified Larson relationship, also called Heyer relationship,
between the cloud line widths, radius, and surface density. Top and right axes
express the same quantities in term of acceleration parameters
($a_{G},a_{K}$). Green stars and blue circles show the prestellar and
protostellar cores, respectively. Cyan triangles present the values for the
dense cores in high-mass star formation regions (Peretto et al., 2006).
Diamonds show the values for the massive 70 $\mu$m dark clumps (Traficante et
al., 2018a). Crosses present the values obtained for the GMCs in the Galactic
plane (Heyer et al., 2009).

####  4.3.2 Turbulence within the cores

To determine the turbulence of the gas within the cores, we analyzed the
nonthermal velocity dispersion of the molecular emission detected from each
core, and their Mach number (see Section 3.8).

The Mach number, which provides an indication of the level of gas turbulence,
shows a moderate positive correlation with the mass of the cores (see panel
(b) of Figure 8), with a Spearman’s rank correlation of $r_{s}$ = 0.33
($p$–value=1$\times 10^{-4}$). A similar positive correlation has also been
seen in the IR-dark, high-mass star-forming region NGC6334S (Li et al.,
2020a). Also, there is a small increment (11%) in the mean Mach numbers from
prestellar cores, with $\langle\mathcal{M}\rangle=1.33\pm 0.04$, to
protostellar cores, with $\langle\mathcal{M}\rangle=1.47\pm 0.06$.

The nonthermal velocity dispersion derived from the deuterated species also
slightly increases from the prestellar to the protostellar stage. The
variation is small (14%) from the prestellar cores, with ($\langle\sigma_{\rm
nt}\rangle=0.28$ km s-1), to the protostellar cores, with ($\langle\sigma_{\rm
nt}\rangle=0.32$ km s-1). The nonthermal velocity dispersion of the envelope
traced by C18O remains essentially the same, with only a small variation of
12%, between the prestellar ($\langle\sigma_{\rm nt,C^{18}O}\rangle=0.83$ km
s-1) and protostellar ($\langle\sigma_{\rm nt,C^{18}O}\rangle=0.93$ km s-1)
cores. This suggests that the variation of turbulence within the cores from
the prestellar to the protostellar phase is negligible in the very early
stages of high-mass star formation studied in ASHES, likely because not enough
time has passed from the protostellar cores to inject turbulence into the
surrounding medium (core and envelope). This is consistent with the outflow
analysis made on the ASHES sample by Li et al. (2020b); they presented a
detailed study of the energy input given by the outflows and found that the
outflow-induced turbulence contribution to the internal clump turbulence at
the current epoch is very limited.

Studies from more evolved high-mass star-forming regions (e.g., Sánchez-Monge
et al., 2013), however, have shown larger differences between the prestellar
and protostellar cores’ line widths ($\sim$80% differences); the mean observed
velocity dispersion is 0.42 and 0.76 km s-1 for the prestellar and
protostellar cores, respectively. Larger values seen in more evolved objects
seem to be associated with the effect of turbulence injected by outflow
activity (e.g., Liu et al., 2020b). This difference could be explained by the
fact that the ASHES IRDCs are in a very early stage of evolution (70 $\mu$m
dark), and what we define as protostellar cores in this sample corresponds to
an intermediate stage, between prestellar cores and more evolved protostellar
cores that already have significant emission at infrared wavelengths. This
suggests that the outflow of these early protostellar cores is not yet
significantly affecting the turbulence within the cores (see also Li et al.,
2020b).

The lack of a significant increment in the turbulence within the cores as they
evolve suggests that the star formation activity in our protostellar cores is
very recent (see also Figure 7). It is likely that the effect of the new,
recently formed protostars have not had sufficient time and sufficient energy
to considerably affect their natal cocoons at the core scale (a few 1000 au),
and their effect may be localized at much smaller scales than those we can
resolve with the current observations. The massive cores tend to have larger
radii (see Figure 14), whereas the nonthermal velocity dispersion has no
significant change with the core radius (see Figure 8). This suggests that the
larger Mach number found in the massive cores is not because of the spectrum
averaged over the larger core radius. Therefore, the increase in turbulence
with the core mass might not be the result of core evolution, but rather a
natural difference between low- and high-mass cores. The high-mass cores show
relatively lower virial parameters, indicating that the high-mass cores would
need additional support, even though it is not sufficient to maintain
equilibrium, to counterbalance gravity, slowing down the collapse on times
larger than the free-fall time.

####  4.3.3 Are observed line widths dominated by gravitational collapse?

If a cloud is undergoing global gravitational collapse, then the line widths
can be dominated by the inward motions produced by this collapse. This can
happen at both cloud and/or core scales (Vázquez-Semadeni et al., 2009;
Ballesteros-Paredes et al., 2011). Indeed, Heyer et al. (2009) showed that the
dynamics of molecular clouds can be dominated by the global gravitational
collapse and presented a scaling relation that extends the initial relation of
Larson (1981), between the clumps’ line velocity dispersion and radius. The
scaling relation proposed by Heyer et al. (2009) shows that the ratio
$\sigma/R^{0.5}$ is proportional to the surface density, $\Sigma$, of the
clouds.

The effect of the global collapse on the nonthermal motions can be estimated
by a gravitational parameter that depends on the surface density ($\Sigma$) of
the cloud and/or core, $a_{\rm G}=\pi G\Sigma/5$, compared to a kinetic
parameter that depends on the line width ($\sigma$) and radius (R) of the
cloud and/or core $a_{\rm K}=\sigma_{\rm tot}^{2}/R$ (Traficante et al.,
2018a). These two parameters are related to a virial parameter ($\alpha_{a}$)
via $a_{\rm K}=\alpha_{\rm a}a_{\rm G}$, whose interpretation is different
than the usual virial parameter derived from the ratio between virial mass and
total gas mass (Section 3.7). If the gravitational collapse dominates over the
kinetic energy in the cloud, then $a_{\rm K}<a_{\rm G}$.

We computed these parameters for the clumps and their embedded cores from the
ASHES survey, and compared them with the values found by Traficante et al.
(2018c) in a sample of clumps, by Peretto et al. (2006) for a sample of cores
embedded in a high-mass star-forming region, and by Heyer et al. (2009) for a
sample of giant molecular cloud (GMC) in the Galactic plane (see Figure 9),
which includes GMCs (A1) and their high-density area within 1/2 maximum
isophote of H2 column density (A2).

There are 47 out of 129 cores with a value of $\alpha_{a}$ smaller than 1 in
the ASHES survey. In general, the cores have values of $a_{\rm G}$ and $a_{\rm
K}$ leading to values of $\alpha_{\rm a}$ between 0.1 and 5.0, and the dense
clumps have a similar distribution. On the other hand, the GMCs have
relatively larger $\alpha_{\rm a}$ compared to the clumps and the cores.

To determine any real correlation, we calculated the Spearman’s correlation
coefficient between $\sigma_{\rm obs}/R^{0.5}$ and $\Sigma$ (or $a_{\rm G}$
and $a_{\rm K}$) for the prestellar and protostellar cores. The correlation
coefficient for prestellar cores is $r_{s}$ = 0.05 ($p$–value=0.65),
suggesting that there is no linear correlation between these values. For
protostellar cores, the value of the correlation coefficient is $r_{s}$ = 0.27
($p$–value=0.04), suggesting a moderate linear trend. This suggests that for
prestellar cores the observed line widths are not dominated by gravity.
However, once star formation begins, the gravitational collapse plays a role
in the dynamics of the cores and thus contributes to the line widths. This is
also consistent with the low value of the “standard” virial parameter found
for the protostellar cores in which the role of gravitational potential energy
with respects to the kinematic energy becomes more important (Section 4.2.2).

####  4.3.4 The first Larson relation: velocity dispersion-size relation

Larson (1981) found an empirical power-law relationship between global
velocity dispersion, $\sigma_{\rm obs}$ (km s-1), and cloud size, $R$ (pc), of
molecular clouds, with a slope of 0.38, $\sigma_{\rm obs}\propto R^{0.38}$.
This correlation resembles the turbulent cascade of energy, and thus, it is
considered as an indication that turbulence dissipates from large spatial
scale down to smaller spatial scales. The $\sigma_{\rm obs}$–$R$ relation has
also been found in the other studies of different molecular clouds, but with a
different slope (e.g., 0.56; Heyer & Brunt, 2004). On the other hand, the
relation seems to break in some studies of star-forming regions, either with a
significantly lower slope, e.g., $\sigma_{\rm obs}\propto R^{0.21}$ obtained
from Orion A and B (Caselli et al., 1995), or without correlation (Plume et
al., 1997; Ballesteros-Paredes et al., 2011; Traficante et al., 2018a).

To explore the variation of $\sigma_{\rm obs}$–$R$ relation in different
spatial scales, we have retrieved data from Li et al. (2020a) and Lu et al.
(2018) for a sample of embedded cores in high-mass star-forming regions, from
Ohashi et al. (2016) for a sample of high-mass clumps and embedded cores, from
Traficante et al. (2018c) for a sample of high-mass clumps, and from Heyer et
al. (2009) for a sample of GMCs. From the top panel of Figure 10, we note that
there is a weak correlation ($r_{s}$ = 0.27, and $p$–value=0.002) between
$\sigma$ and R for ASHES cores only, with a slope of 0.24 $\pm$ 0.08 that is
similar to the values of 0.2–0.3 reported in Caselli et al. (1995) and Shirley
et al. (2003). For ASHES cores only, the sample covers a limited range of
radii, which could break down the $\sigma_{\rm obs}$–$R$ relation.

As shown in the middle panel of Figure 10, if the clumps and relatively larger
size cores are included, the $\sigma_{\rm obs}$ and $R$ show a strong
correlation ($r_{s}$ = 0.82, and $p$–value=4.5$\times 10^{-114}$), with a
slope of 0.46 $\pm$ 0.01 that is between 0.38 and 0.56. On the other hand, if
the GMCs are included, the correlation between $\sigma_{\rm obs}$ and R
becomes even stronger ($r_{s}$ = 0.88, and $p$–value=2.7$\times 10^{-208}$),
but however, the slope (0.35 $\pm$ 0.01) becomes more flattened. The change in
the slope between including or excluding GMCs could be due to intrinsic
turbulent variation from GMC scale to clump and/or core scale or observational
bias since the measurements are obtained with different molecular lines and
different kind of telescopes. The GMCs values were derived from 13CO, while
clumps and/or cores were obtained from more dense gas tracers (i.e., NH3,
DCO+, and N2D+). Unfortunately, we can not conclude which effect drives the
variation in the slope.

Overall, our results suggest that the relation between $\sigma_{\rm obs}$ and
$R$ persists from GMC scale down to core scale (Figure 10), and the slope is
between 1/3 and 1/2, the former is expected for turbulence-dominated
solenoidal motions, and the latter expected for shock-dominated turbulence
(Kolmogorov, 1941; Galtier & Banerjee, 2011; Federrath, 2013). Our results
also indicate that caution is required in interpreting the investigation of a
sample with a small range of radii because it could break down the
$\sigma_{\rm obs}$–$R$ relation due to the small dynamical range in the
spatial size.

Figure 10: Observed velocity dispersion $\sigma_{\rm obs}$ as a function of
the radius. From upper to bottom, panels show the $\sigma_{\rm obs}$ versus
radius for dense cores, clumps and dense core, and GMCs down to core,
respectively. The blue solid line shows the linear regression to all the
points in the plot, and the gray shadowed area shows the 1$\sigma$ confidence
interval for the fit. The best fit returns slopes of 0.24 $\pm$ 0.08, 0.46
$\pm$ 0.01, and 0.35 $\pm$ 0.01 for upper, middle, and bottom panels,
respectively. The Spearman’s rank coefficients between observed velocity and
radius are presented in the lower right corner in each panel. The black dashed
line shows the the original Larson relation with $\sigma_{\rm obs}\propto
R^{0.38}$ (Larson, 1981), and the red dashed line is the revised Heyer & Brunt
(2004) relation with $\sigma_{\rm obs}\propto R^{0.56}$.

###  4.4 The relationship between the embedded cores and parental clump

####  4.4.1 Influence of the parental clump properties on the cores properties

We analyze whether the global properties of the clumps, such as their mass,
virial mass, surface density, velocity dispersion, and virial parameter, have
any impact on the properties of their embedded cores. To quantify any
correlation, we calculated the Spearman’s correlation coefficients between
different parameters of the clumps and the cores.

We found a strong correlation between the mass of the clump and the fraction
of protostellar cores ($r_{s}=0.50$, and $p$–value=0.095). There is also a
strong anticorrelation between the surface density of the clumps and the
average value of the core’s virial parameter ($r_{s}=-0.76$, and
$p$–value=0.004), suggesting that, at a higher surface density, clumps
encompass more gravitationally bound cores within it. There is no correlation
between the number of embedded cores and the clump velocity dispersion
($r_{s}=-0.05$, and $p$–value=0.89), indicating no significant relationship
between the turbulence within the clumps and their level of fragmentation. We
found a moderate correlation between the global virial parameter of the clump
and the mean value of the virial parameter of the cores ($r_{s}=0.36$, and
$p$–value=0.25). We also found a weak correlation ($r_{s}=-0.29$, and
$p$–value=0.35) between the virial parameter of the clump and the fraction of
cores per clump that are gravitationally bound ($\alpha<2$), and a weak
correlation ($r_{p}=-0.21$ and $p$–value=0.51) is seen between the virial
parameter of the clump and the number of embedded cores.

We also analyzed whether the velocity dispersion of the clumps increased with
the number of embedded protostellar cores, as one may expect if the turbulence
is injected from the protostellar objects into the surrounding medium.
However, we find a very weak correlation between the number of protostellar
cores and $\sigma_{\rm clump}$ ($r_{s}=-0.11$, and $p$–value=0.74). Therefore,
the turbulence injected by the protostellar cores in form of outflows is not
enough to increase the overall turbulence in the clumps at these early stages
of evolutions. This is consistent with the results of Li et al. (2020b).

Overall, we found no strong correlation between the large-scale properties of
the clumps with the small-scale properties of the cores, except for the mass
and surface density of the clump that may have an impact on the properties of
their embedded cores.

####  4.4.2 The kinematical relationship between cores and the parental clump

On average, the core-to-core velocity dispersion ($\Delta v_{\rm lsr}$ =
0.5–2.9 km s-1) derived from the C18O line shows the highest value among all
detected lines, except for CO. The core-to-core velocity dispersion is 0.4–1.8
km s-1 for H2CO and 0.2–1.6 km s-1 for CH3OH, both higher than those of the
N2D+ and DCO+ lines; 0.2–1.1 km s-1 for N2D+ and 0.2–1.0 km s-1 for DCO+. The
latter two lines have comparable $\Delta v_{\rm lsr}$ in most of the clumps.
The discrepancy of $\Delta v_{\rm lsr}$ in different lines is most likely
because they trace different gas components (e.g., H2CO, CH3OH, CO and its
isotopologues are easily affected by protostellar outflows, Li et al., 2020b,
2022) The core-to-core velocity dispersions derived from N2D+ and DCO+ are
comparable to that found in low-mass star-forming regions, such as the core-
to-core N2H+ velocity dispersion for Perseus ($\sim$0.4 km s-1; Kirk et al.,
2010).

The comparison between core-to-core motions and the global motions of the
clumps can be used to evaluate how the dense cores are connected to the lower-
density gas in the clumps (e.g., Kirk et al., 2010). For example, if the dense
cores are connected to the large-scale motions, then the dispersion of the
core centroid velocities should be similar to the global velocity dispersion
of the natal clump. On the other hand, we expect to see a much smaller core-
to-core velocity dispersion than the global velocity dispersion of the natal
clump if the dense cores are kinematically detached from the large-scale
motions within the clump. The core-to-core velocity dispersions obtained from
the N2D+ and DCO+ lines are about 4 times smaller than the derived C18O
velocity dispersion ($\sigma_{\rm obs,C^{18}O}$) over the clump that is also
about 2–3 times higher than the core-to-core velocity dispersions derived from
H2CO and CH3OH. These results suggest that the dense cores are kinematically
detached from the large-scale motions (e.g., large-scale gas flows).

###  4.5 The relative motions of dense cores and their envelopes

Simulations of star formation suggest that the mass of a star could be
determined by the motions of its path through the natal molecular cloud
(Bonnell et al., 2001). The relative motions between dense cores and their
associated envelopes can be examined using suitable molecular lines, e.g., a
high-density gas tracer can be used to probe the dense core, and a low-density
gas tracer can be utilized to measure the envelope. In this work, we used N2D+
and DCO+ (hereafter deuterated species) as our high-density core tracers,
because both molecules have high critical densities ($1.7-1.8\times
10^{6}\,\rm cm^{-3}$), and used C18O as the low-density envelope tracer
because it has a low critical density ($9.3\times 10^{3}\,\rm cm^{-3}$). The
line-center velocity ($v_{\rm LSR}$) differences between deuterated species
and C18O range from 0.24 to 1.85 km s-1 (see also Tabel 5), with mean and
median values of 0.64 and 0.41 km s-1, respectively. Our observed values are
greater than that of $\sim$0.1 km s-1 measured toward low-mass star-forming
regions (Walsh et al., 2004). The derived differences in line-center
velocities are smaller than the deuterated species and C18O line widths (FWHM;
see Table 5), which is similar to the results of low-mass star-forming regions
(e.g., Walsh et al., 2004). The mean and median values of line-center velocity
differences for each clump and the line widths of deuterated species and C18O
are tabulated in Table 5. The line-center velocity differences are expected to
be comparable to or even somewhat greater than the C18O line width if cores
move ballistically from their birth sites (Walsh et al., 2004, and reference
therein). The observed small difference indicates that the dense cores are
comoving with their envelopes rather than moving ballistically. In the
framework of Bonnell et al. (2001), this result might imply that the embedded
dense cores do not obtain large amounts of mass by accreting material as they
move through the lower-density environment in their natal molecular cloud.

Table 5: Line-center velocity difference and line width

Clump |  $v_{\rm LSR,deu}$ \- $v_{\rm LSR,C^{18}O}$ |  FWHMdeu |  FWHM${}_{\rm C^{18}O}$
---|---|---|---
| median | mean | median | mean | median | mean
G10.99 | 0.30 | 0.41 | 0.67 | 0.71 | 1.78 | 1.95
G14.49 | 0.70 | 1.19 | 0.63 | 0.80 | 2.43 | 2.65
G28.23 | 0.20 | 0.37 | 1.01 | 1.09 | 2.61 | 3.29
G327.11 | 1.54 | 1.85 | 0.68 | 0.68 | 1.81 | 2.42
G331.37 | 0.17 | 0.30 | 0.70 | 0.74 | 1.43 | 1.60
G332.96 | 0.37 | 0.41 | 0.96 | 1.02 | 1.60 | 1.73
G337.54 | 0.72 | 0.99 | 0.48 | 0.64 | 1.78 | 2.17
G340.17 | 0.73 | 0.73 | 0.99 | 0.99 | 2.15 | 2.63
G340.22 | 0.24 | 0.30 | 0.75 | 0.79 | 2.03 | 2.53
G340.23 | 0.27 | 0.41 | 0.87 | 0.94 | 2.26 | 2.48
G341.03 | 0.28 | 0.46 | 0.58 | 0.63 | 1.91 | 1.87
G343.48 | 0.09 | 0.24 | 0.47 | 0.57 | 1.45 | 1.57
minimum | 0.09 | 0.24 | 0.47 | 0.57 | 1.43 | 1.57
maximum | 1.54 | 1.85 | 1.01 | 1.09 | 2.61 | 3.29
median | 0.29 | 0.41 | 0.69 | 0.77 | 1.86 | 2.29
mean | 0.47 | 0.64 | 0.73 | 0.80 | 1.94 | 2.24

††footnotetext: Notes. The unit is km s-1.

###  4.6 Properties of the most massive cores

We found 6 cores that are relatively massive (10.80–31.66 $M_{\odot}$). On
average, these cores have a core nonthermal velocity dispersion 1.7 times
larger than the remaining lower mass cores (0.49 km s-1 compared to 0.29 km
s-1). On the other hand, the most massive cores have envelopes, traced by
C18O, with a nonthermal velocity dispersion 1.4 times larger than the values
of the lower mass cores (1.22 km s-1 compared to 0.86 km s-1). The most
massive cores are strongly self-gravitating ($\langle\alpha\rangle=0.37$) with
a higher surface density ($\langle\Sigma\rangle=1.50$ g cm-2) and higher peak
column density ($\langle N_{\rm peak}(\rm H_{2})\rangle=2.02\times 10^{23}$
cm-2), compared to the lower mass cores ($\langle\alpha\rangle=1.44$,
$\langle\Sigma\rangle=0.49$ g cm-2, and $\langle N_{\rm peak}(\rm
H_{2})\rangle=5.38\times 10^{22}$ cm-2). The most massive cores also have a
mean Mach number of $\langle\mathcal{M}\rangle=2.5$ which is larger than those
of lower mass cores of $\langle\mathcal{M}\rangle=1.3$.

The derived properties for the most massive cores in the ASHES sample are not
only different from the lower mass cores in the same sample but also different
from the cores embedded in low-mass star-forming regions, in terms of mass,
surface density, and velocity dispersion (e.g., Ophiuchus, Serpens, or
Perseus; Myers & Benson, 1983; Caselli et al., 2002; Kirk et al., 2006; Enoch
et al., 2008; Friesen et al., 2009; Bovino et al., 2021). For low-mass star-
forming regions, the typical core masses range between 0.1 and 10 $M_{\sun}$
with a typical surface density of $<$ 1.0 g cm-2 (e.g. Hogerheijde et al.,
1999; Kirk et al., 2006; Enoch et al., 2008). In addition, the core nonthermal
velocity dispersion is larger in our cores compared to those cores (typical
size $\sim$0.06 pc) seen in low-mass star-forming regions (typical nonthermal
velocity dispersion of $\lesssim$0.2 km s-1, e.g. Caselli et al., 2002;
Friesen et al., 2009; Chen et al., 2019). We conclude that the physical
properties of the most massive cores in IRDCs are inconsistent with cores that
will only form low-mass protostars. Based on the evidence presented in this
work, we suggest that, at least, the most massive cores in ASHES are likely to
be the seeds of future high-mass stars.

###  4.7 Comparison with high-mass star formation models

Two high-mass star formation scenarios, “core-fed” and “clump-fed”, have been
proposed in previous theoretical studies (see Section 1). In this section, we
compare our observational results with four representative high-mass star
formation models in terms of the core’s dynamics or properties.

In the “core-fed” model, the mass of the star is determined by the mass
reservoir of its parental core, which gathers the mass in a prestellar stage
(McKee & Tan, 2002). Therefore, the high-mass prestellar core ($\gtrsim$ 30
$M_{\odot}$), as the cornerstone of this model (is known as “turbulent core
accretion model,” McKee & Tan, 2002), must exist in order to form a high-mass
star. However, based on our results, we do not find the high-mass prestellar
core over 197 prestellar cores toward 12 massive dense clumps. The most
massive prestellar core detected has a mass of 12.89 $M_{\odot}$. In addition,
we find that the majority of identified cores have virial parameters below the
critical value of 2 for a nonmagnetized cloud, which is inconsistent with the
quasi-equilibrium configuration proposed by the “core-fed” model (McKee & Tan,
2002). This indicates that the “core-fed” model cannot explain the virial
parameters of dense cores seen in our observations. We note that the limited
amount of studies including estimations of the magnetic field strength in the
virial equilibrium analysis indicate that the most massive cores still remain
(strongly) subvirialized (e.g., Beuther et al., 2018; Liu et al., 2020a; Morii
et al., 2021; Sanhueza et al., 2021).

In the “clump-fed” scenario, the final mass of stars is not determined by the
mass of prestellar cores. In this scenario, the material can be replenished by
funnelling mass from large scales (e.g., clump, cloud) to small scales (e.g.,
dense core). For instance, filaments can continue to provide the mass
reservoir for the growth of stars. Therefore, protostars embedded in cores can
continue to grow in mass through accreting material from their parental clump
and/or cloud. There are three representative “clump-fed” theoretical models,
i.e., competitive accretion scenario (Bonnell et al., 2001, 2004), global
hierarchical collapse (GHC; Vázquez-Semadeni et al., 2017, 2019), and
inertial-inflow model (Padoan et al., 2020), in which protostars would then be
fed from the surrounding clump and/or cloud, in order to accumulate the mass
to form high-mass stars.

In the competitive accretion model, fragmentation produces low-mass stars
while the stars near the center of the gravitational potential well grow from
low to high mass via accretion (Bonnell et al., 2001, 2004). To form a high-
mass star from a low-mass stellar “seed” in this model, the embedded object
maintains approximately a virial parameter of $\alpha$ = 0.5 (Bonnell & Bate,
2006). As shown in Figure 5, most of the detected cores have virial parameters
not around 0.5, which appears to be inconsistent with simulations of
competitive accretion.

On the other hand, we also find some agreements between the simulation and the
observations. We find that the subsonic turbulent motions of cores in
competitive accretion simulations are in agreement with the measurements in
identified cores where turbulence is dominated by subsonic or transonic
motions (see Section 3.8, Padoan et al., 2001). In addition, the line-center
velocity difference between high-density cores is smaller than the N2D+ and
DCO+ line widths, which traces the high-density cores, and certainly smaller
than the C18O line width (see Section 4.5 and Table 5), which traces the low-
density envelopes. The small velocity difference is inconsistent with the
competitive accretion model, in which dense cores have large velocities
relative to the low-density envelopes due to the cores’ movement within the
clump (Ayliffe et al., 2007).

In the GHC scenario, the dense cores form from hierarchical gravitational
fragmentation, in which mass is accreted from parent structures onto their
embedded substructures (Vázquez-Semadeni et al., 2017, 2019). This model can
be considered as an extension of the competitive accretion model. The GHC
model suggests that the nonthermal motions of the molecular clouds are
dominated by infall and by a moderately supersonic turbulent background, which
has a typical sonic Mach number of $\sim$3\. As discussed in Section 4.3.3,
the gravitational collapse indeed plays a role in the dynamics of the
protostellar cores, but not for prestellar cores. The 1D Mach number derived
from C18O is around 3–4. This is similar to the values of the GHC model,
although the Mach number could be larger for the lower-density gas traced by
CO; CO ($n_{\rm crit}\sim 2\times 10^{3}\,\rm cm^{-3}$) has a relatively lower
critical density than that of C18O ($n_{\rm crit}=9.3\times 10^{3}$ cm-3). In
addition, the inverse $\alpha\,\sim\,M_{\rm gas}$ relation seen in our data
can be explained by the GHC model, if the sample satisfies $M\,\propto\,R^{p}$
with $0\,<\,p\,<\,3$; the detected cores show $M\,\propto\,R^{1.89}$ (see
Figure 14). The observations are consistent with the GHC model in this
respect.

In the inertial flow model, the dense cores are originated from turbulent
fragmentation in supersonic turbulence environments, in which high-mass stars
are assembled by large-scale, converging, and inertial flows (Padoan et al.,
2020). The infall rate of dense cores is controlled by the large-scale
inertial inflow rather than the current core mass and density. In general, the
accretion time increases with mass, indicating that more massive stars require
longer accretion times than the less massive ones in order to achieve their
final mass. This is in agreement with the observed outflow properties in the
ASHES sample. We find that the estimated outflow dynamical timescale increases
with the core mass, which indicates that more massive cores have longer
accretion timescales than less massive cores (see Li et al., 2019, 2020b). In
addition, the derived mass distribution of the prestellar core candidates is
comparable to the prediction of mass distribution from the inertial flow model
(Padoan et al., 2020; Sanhueza et al., 2019; Morii, 2023).

In general, ASHES observations are more consistent with the “clump-fed”
scenario rather than with the “core-fed” scenario, although all “clump-fed”
models have their own limitations and cannot fully explain the dynamics of
dense cores found in the observations. A more complete and realistic
theoretical model of high-mass star formation should be able to reproduce (1)
the observed dynamics of dense cores revealed in ASHES, including the subsonic
and transonic turbulence in dense cores, nonequilibrium state of dense cores,
comoving of high-density cores and their low-density envelopes, and
kinematically detached dense cores from large-scale motions; (2) the
characteristics of dense cores found in ASHES (see Sanhueza et al., 2019, and
references therein), including the presence of a large population of low-mass
cores but without high-mass prestellar cores, hierarchical subclustering,
absence of primordial mass segregation, and a slightly shallower core mass
function (CMF) than the slope of the Salpeter’s initial mass function (IMF).

##  5 Conclusions

We analyzed the kinematics and dynamics of 12 IRDCs and their 294 embedded
cores observed as part of the pilot ALMA Survey of 70 $\mu$m dark High-mass
clumps in Early Stages (ASHES). For these, we determined their properties,
including the velocity dispersion, gas mass, virial mass, virial parameter,
Mach number, and core-to-core velocity dispersion. Our main results can be
summarized as follows.

1\. The prestellar and protostellar cores have similar gas temperatures
obtained from NH3, suggesting that the protostellar cores in our sample are
still at a very early evolutionary phase. On the other hand, the gas mass,
column density, and volume density of the protostellar cores are higher than
those of the prestellar cores, likely indicating a core mass growth from the
prestellar to the protostellar stage.

2\. On average, protostellar cores have relatively higher virial masses
compared to prestellar cores. The virial parameter, $\alpha$, decreases with
the mass of the cores. Therefore, the more massive cores are more unstable
against gravitational collapse if one neglects the effects of magnetic field
support. The virial parameter tends to decrease with the evolution of the
cores. The value of $\alpha$ is slightly lower for protostellar cores,
compared to the prestellar cores, suggesting they might be prone to
gravitational instabilities. In addition, the most massive cores ($>$ 10
$M_{\odot}$) are strongly self-gravitating ($\langle\alpha\rangle=0.37$), have
higher nonthermal velocity dispersion ($\langle\sigma_{\rm nt}\rangle=0.49$ km
s-1), higher surface density ($\langle\Sigma\rangle=1.50$ g cm-2), higher peak
column density ($\langle N_{\rm peak}(\rm H_{2})\rangle=2.02\times 10^{23}$
cm-2), and higher Mach numbers ($\langle\mathcal{M}\rangle=2.5$) compared to
the lower mass cores ($\langle\alpha\rangle=1.44$, $\langle\sigma_{\rm
nt}\rangle=0.29$ km s-1, $\langle\Sigma\rangle=0.49$ g cm-2, $\langle N_{\rm
peak}(\rm H_{2})\rangle=5.38\times 10^{22}$ cm-2, and
$\langle\mathcal{M}\rangle=1.3$). The most massive cores in ASHES likely
harbor the seeds of future high-mass stars.

3\. There is a positive correlation between the Mach number and the mass of
the cores. This suggests that the turbulence within the cores increases with
their mass. In addition, most of the deuterated detected cores (84%) are
subsonic or transonic, with Mach number values between $0.1$ and 2.0. The
level of turbulence within the cores, traced by their velocity dispersion and
turbulent Mach number, does not increase significantly with the evolutionary
stage of the core. This suggests that, once star formation begins, the core
remains oblivious to the turbulence generated by outflows at the current
extremely early evolutionary stage traced in these 70 $\mu$m dark IRDCs.

4\. With respect to the core-to-core velocity dispersion, and the core
velocity dispersion, we found that the later is consistent with what is
predicted in simulations of molecular clouds in which turbulence is
continuously injected into the system, while the former is in disagreement
with the driven turbulence and decaying turbulence scenarios.

5\. We also analyzed whether the line width in the IRDCs and cores can be
dominated by the gravitational collapse. We find that the protostellar cores
show a moderate correlation between $\sigma_{\rm obs}/R^{0.5}$ and $\Sigma$,
indicating that the gravitational collapse plays a role in the turbulence of
the protostellar cores and thus contributes to the line width. However, for
the prestellar cores, there is no clear trend, which suggests that their line
width is not dominated by gravity-induced motions.

6\. We also find that the relation between velocity dispersion and size, also
known as the first Larson relation, persists from the GMC scale down to the
core scale. We note that considering a sample with a small range of radii can
easily break down this relation.

7\. Within each clump, the core-to-core velocity dispersion obtained from
dense gas tracers is about 2–4 times smaller than the C18O velocity dispersion
over the entire clump. This indicates that the dense cores are kinematically
detached from the large-scale motions. In addition, the differences of line-
center velocities in low (C18O) and high (N2D+ and DCO+) density tracers are
smaller than the line width of N2D+, DCO+, and C18O lines. This indicates that
the dense cores have small velocity shifts relative to their low-density
envelopes, suggesting that the dense cores are comoving with their envelopes
rather than moving ballistically.

8\. We compare our observational results to existing high-mass star formation
models, including “core-fed” (i.e., turbulent core accretion model) and
“clump-fed” scenarios (i.e., competitive accretion model, global hierarchical
model, inertial-inflow model). We find that the observed core properties do
not match the core properties predicted by the turbulent accretion model. On
the other hand, the competitive accretion, global hierarchical collapse, and
inertial-inflow models can partially explain the observed dynamics of dense
cores (see Section 4.7 for details). In general, the observed cores are more
consistent with the “clump-fed” scenario than the “core-fed” scenario.
However, none of the existing theoretical models can fully explain the
dynamics of dense cores revealed in the ASHES observations. A more complete
and realistic theoretical model of high-mass star formation should reproduce
the dynamics of dense cores revealed in this work, as well as the
characteristics of dense cores presented in Sanhueza et al. (2019).

We thank the anonymous referee for constructive comments that helped to
improve this paper. We thank David Allingham for sharing the NH3 rotational
excitation temperature maps. We thank Yanett Contreras for her important
contribution in shaping this article, contributing with early analysis of the
data and very important insights. P.S. gratefully acknowledges the support
from the NAOJ Visiting Fellow Program to have collaborators visiting the
National Astronomical Observatory of Japan in November-December 2016. P.S. was
financially supported by Grant-in-Aid for Scientific Research (KAKENHI Number
JP22H01271) of Japan Society for the Promotion of Science (JSPS). G.G.
acknowledges support by the ANID BASAL project FB210003. F.L. acknowledges the
support from the National Natural Science Foundation of China grant
(12103024), and the fellowship of China Postdoctoral Science Foundation
2021M691531. Data analysis was in part carried out on the open use data
analysis computer system at the Astronomy Data Center, ADC, of the National
Astronomical Observatory of Japan. This paper makes use of the following ALMA
dataset ADS/JAO.ALMA#2015.1.01539.S. ALMA is a partnership of ESO
(representing its member states), NSF (USA) and NINS (Japan), together with
NRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ.

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##  Appendix A Additional Figures and tables

Figure 11: Nonthermal velocity dispersion of the cores $\sigma_{\rm nt}$,
obtained from their DCO+ (left) and N2D+ (right) molecular line emission for
each clump. The total row shows the mean value of the velocity dispersion for
the clumps that have a measurement in both the prestellar and protostellar
sample. The color scale shows the number of cores considered to determine the
velocity dispersion. Dashes denote no available core.

Figure 12: Core-to core velocity dispersion of the cores $v_{\rm LSR}$
obtained from their DCO+ (left) and N2D+ (right) molecular line emission for
each clump. The total row shows the mean value of the core-to-core velocity
dispersion for the clumps that have a measurement in both the prestellar and
protostellar sample. The color scale shows the number of cores considered to
determine the core-to-core velocity dispersion. ‘0’ means only one available
core. Dashes denote no available core.  Figure 13: Nonthermal velocity
dispersion derived from the emission of the deuterated species as function of
the mass of the cores. In the upper panel is shown the number of cores per
mass bin, and in the lower panel is shown a box plot with the mean and range
of values of the nonthermal velocity dispersion per mass bin.  Figure 14: The
core mass as a function of core radius. The blue solid line shows the linear
regression to all the points in the plot, and the gray shadowed area shows the
1$\sigma$ confidence interval for the fit. The best fit gives a slope of 1.89
$\pm$ 0.11. The Spearman’s rank coefficient is 0.69 with the $p$-value of
1.0$\times 10^{-43}$.