11institutetext: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
53121 Bonn, Germany
11email<EMAIL_ADDRESS>22institutetext: Max Planck Institute for
Astronomy, Königstuhl 17, 69117 Heidelberg, Germany 33institutetext:
Department of Physics, P.O.Box 64, FI-00014, University of Helsinki, Finland
44institutetext: National Astronomical Observatory of Japan, National
Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
55institutetext: Shanghai Astronomical Observatory, Chinese Academy of
Sciences, 80 Nandan Road, Shanghai 200030, Peoples Republic of China

# Gas inflows from cloud to core scales in G332.83-0.55: Hierarchical hub-
filament structures and tide-regulated gravitational collapse

J. W. Zhou Gas inflows from cloud to core scales in G332.83-0.55: Hierarchical
hub-filament structures and tide-regulated gravitational collapseGas inflows
from cloud to core scales in G332.83-0.55: Hierarchical hub-filament
structures and tide-regulated gravitational collapse S. Dib Gas inflows from
cloud to core scales in G332.83-0.55: Hierarchical hub-filament structures and
tide-regulated gravitational collapseGas inflows from cloud to core scales in
G332.83-0.55: Hierarchical hub-filament structures and tide-regulated
gravitational collapse M. Juvela Gas inflows from cloud to core scales in
G332.83-0.55: Hierarchical hub-filament structures and tide-regulated
gravitational collapseGas inflows from cloud to core scales in G332.83-0.55:
Hierarchical hub-filament structures and tide-regulated gravitational collapse
P. Sanhueza Gas inflows from cloud to core scales in G332.83-0.55:
Hierarchical hub-filament structures and tide-regulated gravitational
collapseGas inflows from cloud to core scales in G332.83-0.55: Hierarchical
hub-filament structures and tide-regulated gravitational collapse F. Wyrowski
Gas inflows from cloud to core scales in G332.83-0.55: Hierarchical hub-
filament structures and tide-regulated gravitational collapseGas inflows from
cloud to core scales in G332.83-0.55: Hierarchical hub-filament structures and
tide-regulated gravitational collapse T. Liu Gas inflows from cloud to core
scales in G332.83-0.55: Hierarchical hub-filament structures and tide-
regulated gravitational collapseGas inflows from cloud to core scales in
G332.83-0.55: Hierarchical hub-filament structures and tide-regulated
gravitational collapse K. M. Menten Gas inflows from cloud to core scales in
G332.83-0.55: Hierarchical hub-filament structures and tide-regulated
gravitational collapseGas inflows from cloud to core scales in G332.83-0.55:
Hierarchical hub-filament structures and tide-regulated gravitational collapse

(Accepted XXX. Received YYY; in original form ZZZ)

The massive star-forming region G332.83-0.55 contains at least two levels of
hub-filament structures. The hub-filament structures may form through the
”gravitational focusing” process. High-resolution LAsMA and ALMA observations
can directly trace the gas inflows from cloud to core scales. We investigated
the effects of shear and tides from the protocluster on the surrounding local
dense gas structures. Our results seem to deny the importance of shear and
tides from the protocluster. However, for a gas structure, it bears the tidal
interactions from all external material, not only the protocluster. To fully
consider the tidal interactions, we derived the tide field according to the
surface density distribution. Then, we used the average strength of the
external tidal field of a structure to measure the total tidal interactions
that are exerted on it. For comparison, we also adopted an original pixel-by-
pixel computation to estimate the average tidal strength for each structure.
Both methods give comparable results. After considering the total tidal
interactions, for the scaling relation between the velocity dispersion
$\sigma$, the effective radius $R$, and the column density $N$ of all the
structures, the slope of the $\sigma-N*R$ relation changes from 0.20$\pm$0.04
to 0.52$\pm$0.03, close to 0.5 of the pure free-fall gravitational collapse,
and the correlation also becomes stronger. Thus, the deformation due to the
external tides can effectively slow down the pure free-fall gravitational
collapse of gas structures. The external tide tries to tear up the structure,
but the external pressure on the structure prevents this process. The
counterbalance between the external tide and external pressure hinders the
free-fall gravitational collapse of the structure, which can also cause the
pure free-fall gravitational collapse to be slowed down. These mechanisms can
be called ”tide-regulated gravitational collapse.”

###### Key Words.:

Submillimeter: ISM – ISM:structure – ISM: evolution – Stars: formation –
methods: analytical – techniques: image processing

##  1 Introduction

High-resolution observations of high-mass star-forming regions show that
density enhancements are organized in filamentary gas networks, particularly
in hub-filament systems. Hub-filament systems are known as a junction of three
or more filaments. Filaments have lower column densities compared to the hubs
(Myers, 2009; Schneider et al., 2012; Kumar et al., 2020; Zhou et al., 2022).
In such systems, converging flows funnel matter into the hub through the
filaments. Previous studies have suggested that hub-filament systems are birth
cradles of high-mass stars and clusters (Peretto et al., 2013; Henshaw et al.,
2014; Zhang et al., 2015; Liu et al., 2016; Contreras et al., 2016; Yuan et
al., 2018; Lu et al., 2018; Issac et al., 2019; Dewangan et al., 2020; Liu et
al., 2021b, 2022; Kumar et al., 2020; Zhou et al., 2022; Liu et al., 2023; Xu
et al., 2023; Zhou et al., 2023; Yang et al., 2023). Zhou et al. (2022)
studied the physical properties and evolution of hub–filament systems by
spectral lines that were observed in the ATOMS (ALMA Three-millimeter
Observations of Massive Star-forming regions) survey (Liu et al., 2020b). We
found that hub-filament structures can exist at various scales from 0.1 parsec
to several parsec in very different Galactic environments. Below the 0.1
parcsec scale, slender structures similar to filaments, such as spiral arms,
have also been detected in the surroundings of high-mass protostars (Liu et
al., 2015; Maud et al., 2017; Izquierdo et al., 2018; Chen et al., 2020;
Sanhueza et al., 2021; Olguin et al., 2023). Therefore, a scenario was
proposed in Zhou et al. (2022): Self-similar hub-filament structures and
filamentary accretion seem to exist at all scales (from several thousand au to
several parsec) in high-mass star-forming regions. This paradigm of
hierarchical and multi-scale hub-filament structures was generalized from
clump-core scale to cloud-clump scale in Zhou et al. (2023). Hierarchical
collapse and hub-filament structures feeding the central regions are also
described in previous works, such as Motte et al. (2018), Vázquez-Semadeni et
al. (2019), Kumar et al. (2020) and references therein.

As discussed in Zhou et al. (2023), the change in velocity gradients with the
scale in the G333 complex indicates that the morphology of the velocity field
in PPV space resembles a ”funnel” structure. The funnel structure can be
explained as accelerated material flowing toward the central hub and
gravitational contraction of star-forming clouds or clumps. To some extent,
the funnel structure gives an indication of the gravitational potential well
that is formed by the clustering.

In this work, we focus on a local region G333-s3a in the G333 complex, called
G332.83-0.55 hereinafter, which is marked in Fig.1 of Zhou et al. (2023), also
shown in Fig.1. G332.83-0.55 presents a typical hub-filament morphology at
$\sim$10 pc. As a cloud-scale hub-filament system, its hub corresponds to the
ATLASGAL clump AGAL332.826-00.549. High-resolution APEX/LAsMA 12CO and 13CO
(3-2) observation allowed us to trace the gas kinematics and gas inflows from
cloud to clump scales in G332.83-0.55. We also zoomed in on the hub region of
G332.83-0.55 using high-resolution ALMA HCO+ (1-0) and H13CO+ (1-0)
observations to investigate gas motions from clump to core scales. By combing
LAsMA and ALMA observations, we could directly recover the hierarchical or
multi-scale hub-filament structures and gas inflows in G332.83-0.55. In
addition, we were also able to characterize the details of the funnel
structure at different scales.

##  2 Observations and data

###  2.1 LAsMA data

The seven pixel Large APEX sub-Millimeter Array (LAsMA) receiver of the APEX
telescope was used to observe the $J=3-2$ transitions of 12CO and 13CO in the
G333 molecular cloud complex. A detailed description of data calibration,
imaging, and product creation procedures is presented in Zhou et al. (2023).
The final data cubes have a velocity resolution of 0.25 km s-1 and an angular
resolution of $19.5\arcsec$ and the pixel size is $6\arcsec$. The final noise
levels of the 12CO (3–2) and 13CO (3–2) data cubes are $\sim$0.32 K and
$\sim$0.46 K, respectively. In this paper, we only focus on the subregion
G332.83-0.55.

###  2.2 ALMA data

We selected one source from the ATOMS survey (Project ID: 2019.1.00685.S; PI:
Tie Liu). The details of the 12m array and 7m array ALMA observations were
summarized in Liu et al. (2020b, 2021a). Calibration and imaging were carried
out using the CASA software package version 5.6 (McMullin et al., 2007); more
details can be found in Zhou et al. (2021). All images have been primary-beam
corrected. The continuum image reaches a typical 1 $\sigma$ rms noise of
$\sim$0.2 mJy in a synthesized beam FWHM size of $\sim 2.2\arcsec$. In this
work, we also use the H13CO+ J=1-0 and HCO+ J=1-0 molecular lines. The final
noise levels of the H13CO+ J=1-0 and HCO+ J=1-0 data cubes are $\sim$0.008 Jy
beam-1 and $\sim$0.012 Jy beam-1, respectively.

###  2.3 Archival data

We also used ATLASGAL+Planck 870 $\mu$m data (ATLASGAL combined with Planck
data), which are sensitive to a wide range of spatial scales at a resolution
of $\sim 21\arcsec$ (Csengeri et al., 2016).

##  3 Results

###  3.1 Connection of gas inflows in LAsMA and ALMA observations

Figure 1: First and second columns: Moment maps of LAsMA 12CO (3$-$2) and 13CO
(3$-$2) emission. The radius of the entire region is 0.1$\degr$ ($\sim$6.28
pc). The big red circle marks the region with a 0.05$\degr$ ($\sim$3.14 pc)
radius, and the small red circle shows the region with a 0.01$\degr$
($\sim$0.63 pc) radius. Third and fourth columns: Moment maps of ALMA 7m+12m
HCO+ J=1-0 and H13CO+ J=1-0 emission. The radius of the entire region is
0.01$\degr$ ($\sim$0.63 pc). Red contours in panels (c) and (d) show the 3mm
continuum emission. Black contours in the second row show ATLASGAL 870$\mu$m
continuum emission. Dashed black lines in panels (b1) and (d1) show the paths
of the PV diagrams in Fig.4. Ellipses in panel (d) represent the dendrogram
structures identified in Sec.3.3. Figure 2:  Hierarchical hub-filament
structures in G332.83-0.55. (a) and (b) Moment 0 maps of LAsMA 13CO (3$-$2)
and ALMA H13CO+ (1-0) emission. (c) and (d) Moment 1 maps of LAsMA 13CO
(3$-$2) and ALMA H13CO+ (1-0) emission. The hub region marked by a red circle
in panels (a) and (c) are enlarged in panels (b) and (d). More detailed
structures can be seen in panels (b) and (d), thanks to the high resolution of
ALMA observation. Lines in color present the filament skeletons identified by
the FILFINDER algorithm.  Figure 3: (a) Normalized velocity and intensity
along the filament LAsMA-f3 in Fig.2(a); here we can see a blueshifted V-shape
velocity structure. (b) Segment of panel (a) around the central hub; here we
can see a V+$\Lambda$-shape velocity structure. (c) and (d) Velocity and
intensity along the segment of LAsMA-f3 in panel (b) compared to the velocity
and intensity along the filament ALMA-f3 in Fig.2(b). Figure 4: PV diagrams of
ALMA H13CO+ (1-0) and LAsMA 13CO (3$-$2) along the paths marked in Fig.1 (d1)
and (b1).  Figure 5: Linkage of large- and small-scale velocity fields in PPV
space. (a) Velocity field revealed by LAsMA data 13CO (3$-$2) on large scales
decomposed by GAUSSPY+. (b) Combined velocity fields traced by 13CO (3$-$2)
and H13CO+ (1-0) at different scales, i.e., panel (a) plus panel (c). (c)
Velocity field revealed by ALMA data H13CO+ (1-0) on small scales decomposed
by GAUSSPY+.

Moment maps of various molecular lines are shown in Fig.1. The Moment 0 maps
of 13CO (3$-$2), H13CO+ J=1-0, and HCO+ J=1-0 emission show a typical hub-
filament morphology. The FILFINDER algorithm (Koch & Rosolowsky, 2015) was
used to identify and characterize filaments traced by H13CO+ (1-0) and 13CO
(3$-$2) emission. Following the method described in Zhou et al. (2022) and
Zhou et al. (2023), we used the integrated intensity (Moment 0) maps to
identify filaments. The skeletons of the identified filaments, overlaid on the
Moment 0 maps in Fig.2, are highly consistent with the gas structures. We
selected the filaments through the redshifted and blueshifted peaks (LAsMA-f3
and ALMA-f3), and extracted the velocity and intensity along the filaments, as
shown in Fig.3.

Fig.4 displays the position-velocity (PV) diagrams along the dashed lines in
panels (b1) and (d1) of Fig.1. For H13CO+ J=1-0 emission, feedback from the
central UC-HII region identified in Zhang et al. (2023) has destroyed the
surrounding molecular gas structure (Fig.1(d)). Thus, the diagram looks
broken, but it still has a clear double-peak velocity structure, indicating
that the surrounding gas is still infalling onto the protocluster (Sec.3.2).
In Fig.4 (b), the central part of the PV diagram has a similar morphology to
Fig.4 (a). Large-scale gas structures with clear velocity gradients are
connected to the central velocity jump. Correspondingly, large-scale
filamentary structures connected to the central clump can be seen in Fig.1(a)
and (b); small-scale filamentary structures down to dense core scales are also
presented in Fig.1(c) and (d). These features indicate that cloud-scale gas
inflows extend down to core scales. The inheritance of gas flows on large
scales to small scales is clearly visible in Fig.3(c).

Fig.5(a) displays the PPV map of 13CO (3$-$2) emission decomposed by the fully
automated Gaussian decomposer GAUSSPY+ (Lindner et al., 2015; Riener et al.,
2019); the decomposition was carried out in Zhou et al. (2023). We can see a
funnel structure that can be a probe of gravitational collapse as discussed in
Zhou et al. (2023). At small scales, we can roughly identify a velocity jump
comprised of the redshifted and blueshifted peaks in Fig.2(c), which is
clearly shown in Fig.2(d), thanks to the higher resolution of the ALMA data.

To conclude, G332.83-0.55 shows at least two levels of hub-filament
structures. Fig.1(a) and (b) present the first level from cloud to clump
scales. The second level from clump to core scales is shown in Fig.1(c) and
(d). In a statistical study of hub-filament systems in Zhou et al. (2022), we
found filaments are ubiquitous in proto-clusters, and hub-filament systems are
very common from dense core scales ($\sim$0.1 pc) to clump and cloud scales
($\sim$1-10 pc). These observed features are reproduced in G332.83-0.55 well.

The structures marked by small red circles in Fig.1(a) and (b) present an
approximately spherical morphology due to the limited resolution in single-
dish observation, usually referred to as a ”clump.” However, in the eyes of
ALMA, the spherical clump displays anisotropic and filamentary accretion
structures (hub-filament) in Fig.1(c) and (d). Thus, relative to the
traditional ”infall envelope,” filamentary accretion and inflow structures may
be more realistic in high-mass star-forming clumps.

###  3.2 Infall+rotation motions

In Fig.1(c) and (d), on small scales, H13CO+ J=1-0 and HCO+ J=1-0 emission
present spiral-like structures, indicating the rotation of gas around the
protocluster. PV diagrams in Fig.4 also reveal rotation motions. The PV
diagrams show two peaks separated by several km s-1, and they are
symmetrically located with respect to the gravitational center. The double-
peaked feature on the PV map is often seen in cluster-forming clumps, and this
feature can only be reproduced when the clump has both infall motion and
rotation at the same time, thus strongly implying that the clump is collapsing
with rotation (Ohashi et al., 1997; Shimoikura et al., 2016, 2018; Sanna et
al., 2019; Liu et al., 2020a; Shimoikura et al., 2022). Around the
gravitational center, we can see the blueshifted and redshifted extrema at the
end of the velocity jump in Fig.3, which is comparable with Fig.2 of Sakai et
al. (2014). In the schematic diagram of an infalling and rotating envelope
model for IRAS 04368+2557 (Sakai et al., 2014), the shift of blueshifted and
redshifted peaks can be used to estimate the radius of the centrifugal barrier
$r_{\rm c}$, where the rotation velocity $V_{\rm rotation}$ has its maximum
value. According to equations (1) and (2) in Sakai et al. (2014), we can
estimate the mass of the gravitational center according to $r_{\rm c}$ by

$M_{*}=\frac{V_{\rm rotation}^{2}r_{\rm c}}{2G}.$ (1)

The initial physical quantities in this equation can be estimated from the
velocity jump in Fig.3.

Both the velocity fields traced by LAsMA and ALMA data show a velocity jump
around the gravitational center. In Fig.3(c), the velocity fields along the
large- and small-scale filaments can be roughly connected, despite the
significant differences in their resolutions (2.5$\arcsec$ and 19.5$\arcsec$).
According to the velocity difference and the separation between blueshifted
and redshifted peaks in Fig.3(c), we roughly estimate $V_{\rm
rotation,ALMA}\approx$3.29 km/s and $r_{\rm c,ALMA}\approx$ 0.21 pc. Then
according to equation.1, the mass of the gravitational center (i.e., the
protocluster) is $\sim$264 M⊙.

As a further test, following the method described in Kauffmann et al. (2008),
the mass can be computed by dust emission using

$\displaystyle M$ $\displaystyle=$ $\displaystyle\displaystyle
0.12\,M_{\odot}\left({\rm e}^{1.439(\lambda/{\rm mm})^{-1}(T/{\rm
10\leavevmode\nobreak\ K})^{-1}}-1\right)$ (2)
$\displaystyle\displaystyle\left(\frac{\kappa_{\nu}}{0.01\rm\leavevmode\nobreak\
cm^{2}\leavevmode\nobreak\ g^{-1}}\right)^{-1}\left(\frac{F_{\nu}}{\rm
Jy}\right)\left(\frac{d}{\rm 100\leavevmode\nobreak\
pc}\right)^{2}\left(\frac{\lambda}{\rm mm}\right)^{3},$

where the opacity is

$\kappa_{\nu}=0.1({\nu}/\textrm{1000\leavevmode\nobreak\
GHz})^{\beta}\leavevmode\nobreak\ {\rm cm}^{2}{\rm g}^{-1}$ (3)

and $\beta$ is the dust emissivity spectral index. Here, we consider
$\beta=1.5$, $F_{\nu}$ is the total flux of 3mm dust emission, $d$ is the
distance to the source, and $\lambda$ is the wavelength. The temperature $T$
takes the value $\sim$31.4 K listed in Table A1 of Liu et al. (2020b). A mass
$\sim$264M⊙ corresponds to the flux of 3mm dust emission $\sim$0.12 Jy. This
value is $\sim$3% of the total flux of 3mm continuum emission in ALMA
observations. As discussed in Keto et al. (2008) and Zhang et al. (2023), 3mm
continuum emission of UC HII regions in high-resolution interferometric
observations should be mostly free–free emission rather than from dust, and
thus the proportion of $\sim$3% for the dust emission is reasonable.

###  3.3 Local dense gas structures

The issues of identifying the structures in a PPV cube by the dendrogram
algorithm are described in Zhou et al. (2024b), and thus we adopted the same
method as in Zhou et al. (2024b) and Zhou et al. (2024a) to identify dense gas
structures. We directly identified hierarchical (sub)structures based on the
2D integrated intensity (Moment 0) map of H13CO+ (1$-$0) emission, and then
extracted the average spectrum of each structure to investigate its velocity
components and gas kinematics. More details can be found in Zhou et al.
(2024a). In this work, we focus on leaf structures, which are local
gravitational centers. According to their averaged spectra, the structures
with absorption features were eliminated first. Following Zhou et al. (2024a),
we only consider type1 (single velocity component) and type3 (blended velocity
component) structures, as marked in Fig.1(d) by ellipses.

We derived the column density by combining H13CO+ J=1-0 and HCO+ J=1-0
emission using the same method described in Sec.3.3 of Zhou et al. (2024a) by
the local thermodynamic equilibrium (LTE) analysis. Then the masses and virial
ratios of all leaf structures were calculated.

###  3.4 Shear and tide from the protocluster

Figure 6: (a) Distribution of the shear parameter $S_{g}$. (b) Distribution of
the velocity dispersion ratio between $\sigma_{t}$ and $\sigma$. (c)
$\sigma-N*R$ relation, where $\sigma$ is the local gas velocity dispersion.
(d) $\sigma_{s}-N*R$ relation, where
$\sigma_{s}=\sqrt{\sigma_{t}^{2}+\sigma^{2}}$. We note that “r” represents the
Pearson correlation coefficient.

The very large velocity difference (velocity jump) around the protocluster in
Fig.3, Fig.4 and Fig.5 may indicate that the surrounding gas structures bear
considerable shear and tides from the protocluster, apart from infall and
rotation motions. Fig.6(c) shows the scaling relations between the velocity
dispersion $\sigma$, the effective radius $R$, and the column density $N$ of
all leaf structures, similar to Zhou et al. (2024b) and Zhou et al. (2024a),
the slope of the $\sigma_{s}-N*R$ relation is $\sim$0.2, which is much smaller
than 0.5 and may indicate a pure free-fall gravitational collapse is slowing
down.

To investigate the shear and tides from the protocluster, here we analogy to
the methods that are adopted on galaxy-cloud scales. For cloud-scale gas
structures in the galaxy, as described in Hunter et al. (1998) and Dib et al.
(2012), the shear parameter $S_{g}$ is defined as

$S_{g}=\frac{\Sigma_{\rm cri}}{\Sigma}.$ (4)

We note that $\Sigma$ is the local gas surface density and $\Sigma_{\rm cri}$
is the critical surface density, which represents the minimal surface density
needed to resist the shear. Following equation.7 of Dib et al. (2012),

$\Sigma_{\rm cri}=\frac{A\sigma\ln(C)}{2\pi G}\,.$ (5)

The $C$ represents the density contrast between leaf structures and their
surroundings. The ratio between the average surface density derived from 13CO
(3$-$2) emission in the FOV of the ALMA observation and the highest surface
density of leaf structures is $\sim 100$, and thus we consider $C$=100. We
note that $\sigma$ is the local gas velocity dispersion and $A$ is the Oort
constant, which measures the local shear level. For a spherical cloud of
radius $r$, with a rotational velocity $V$ at a galactocentric distance
$R_{G}$, $A$ can be calculated as

$A=0.5\left(\frac{V}{R_{G}}-\frac{\mid
V(R_{G}+r)-V(R_{G}-r)\mid}{2r}\right)\,,$ (6)

in the context of molecular clouds, $V$ is determined by the rotation curve of
the galaxy. In our case, for gas structures in the clump around the
protocluster, $V$ was estimated by the Kepler rotation curve, as discussed in
Sec.3.2. The $V=\sqrt{GM_{0}/R^{\prime}}$, with $M_{0}$ being the mass of the
protocluster and $R^{\prime}$ being the distances between gas structures and
the protocluster. Shear is effective at tearing apart the gas structure if
$S_{g}>1$ and it can be ineffective if $S_{g}<1$.

In a galactic context, the overall gas and stellar distribution can contribute
to confine or disrupt the gas structures. For a cloud at a distance $R_{G}$
from the galactic center, the edge of the cloud from the galactic center is at
a distance of $R_{G}$+$r$. The tidal acceleration in the cloud is $\sim T*r$,
following Stark & Blitz (1978),

$T=\frac{V^{2}}{R_{G}^{2}}-\frac{\partial}{\partial
R_{G}}\left(\frac{V^{2}}{R_{G}}\right)\,.$ (7)

Inserting the Keplerian rotation curve, $T=3GM_{0}/R^{\prime 3}$, which is
similar to equation.8. Then the effective velocity dispersion from the tide
for a structure is $\sigma_{t}^{2}\sim$ $T*R^{2}$, where $R$ is the effective
radius of the structure.

In Fig.6, $S_{g}<<1$ and $\sigma_{t}^{2}/\sigma^{2}<<1$; except for two
structures adjacent to the protocluster, the results seem to deny the
importance of the shear and tide from the protocluster. However, the analysis
here is actually not entirely appropriate. For a leaf structure, it also bears
the tides from other gas structures, not only the protocluster. In fact, it
should be N-body tidal interactions; therefore, a complete tidal calculation
would be more appropriate.

###  3.5 Tidal field

Figure 7: Different components of the projected tidal field in the FOV of the
ALMA observation.

To fully consider the tidal interactions, we directly derived the tidal field
according to the density distribution. Only the projected surface density map
is available in the observations. To align with other projected physical
quantities derived from the observations, we also calculated the 2D projected
tide. We note that 3D calculations can refer to Ganguly et al. (2024) and Li
(2024). Calculating the gravitational potential field $\phi(x,y)$ based on the
surface density map $\Sigma(x,y)$ has been discussed in Gong & Ostriker
(2011); He et al. (2023). The tidal tensor described in Appendix.A has only
two eigenvalues in the case of 2D, that is to say $\lambda_{i}$ (i=1,2). As
described in Appendix.A, the tidal tensor of a structure includes both self-
gravity and external tides, that is,
$\mathbf{T}=\mathbf{T}_{\mathrm{ext}}+\mathbf{T}_{\mathrm{int}}$. The
decomposition of the tidal tensor is discussed in detail in Appendix.A. For
three limiting cases, the external tides can be measured by $\lambda_{1}$,
$\lambda_{2}$, and $(\lambda_{2}-\lambda_{1})/2$ and they are called Td,1,
Td,2, and Td,3, respectively. Fig.9 displays different components of the tidal
tensor.

###  3.6 Structure of G332.83-0.55

Figure 8: (a) Comparison of $E_{\rm a,int}$ and $E_{\rm g}$ of dendrogram
structures. (b) Distribution of the average tidal strength in each
computation.

As shown in Fig.1 of Zhou et al. (2023), G332.83-0.55 is part of a large-scale
shell structure. Thus, it may have a sheet structure. In combining Fig.1(b)
and Fig.5(a), the single direction gas inflows on large scales also indicate
that the structure on large scales should be approximately 2D, and should be
face-on. Thus, we can observe the unidirectional converging flows. Close to
the center, the structure becomes thicker and thicker. The redshifted and
blueshifted components may originate from the forward and backward gas motions
in the hub region. However, strong rotation motions on small scales may also
explain their origin. The 3D and 2D structures of G332.83-0.55 may not be very
different, given its limited thickness. Therefore, 2D-projected tidal field
strength should be comparable to those for 3D ones, and equation.19 derived
from the 3D tidal tensor in Appendix.A should still be effective for a 2D
situation, which is confirmed in Fig.8(a).

###  3.7 Pixel-by-pixel computation

Below we explain how we tested the above tidal tensor analysis using an
original method.

####  3.7.1 Maximum computation

Figure 9: Computed structure first masked from the density map. The external
tides from all outside material sustained by a point in the computed structure
was calculated pixel by pixel and is illustrated by orange lines. (a)
Background representing the column density map derived from ATLASGAL+Planck
870 $\mu$m data. The material in the FOV of the ALMA observation (white
region) was masked. (b) Background representing the column density map derived
from H13CO+ J=1-0 and HCO+ J=1-0 lines from the ALMA observation by the LTE
analysis. Figure 10: Edge-on view of G332.83-0.55. The calculated gas
structure is in red, and all external material apart from the calculated
structure is in blue. (I) The gas distribution in G332.83-0.55 can be
everywhere. (II) In 2D computation, all molecular gases are assumed to be
concentrated on the same plane. Thus, we obtained a maximal estimate for the
tidal strength. (III) Taking a characteristic scale height of the molecular
gas as 0.03 pc, all external material apart from the calculated structure were
moved vertically to 0.03 pc away. Thus we obtained a minimal estimate of the
tidal strength.

For a structure A, located at a distance $R$ from the center of another
structure B, of mass $M$, the tidal strength sustained by structure A due to
the external gravity from structure B is

$T\approx\frac{2GM}{R^{3}}.$ (8)

The total tidal strength at a pixel measures the total deformation due to the
external tides at that point. A gas structure is neither a rigid body nor
point mass, but it is a parcel of gas with malleability. Most gas structures
are also not spherically symmetric, and their morphologies can be highly
complex. Although the external gravity from all directions may cancel each
other out in a structure, the tides inside the structure due to the external
gravity do not. Therefore, we used the scalar superposition to measure the
overall tidal strength in the structure from the external gravity. Tidal
tearing may be an important cause for the deformation and complex morphology
of gas structures, as discussed in Sec.4.1.

Using the mask of the dendrogram structure, we divided the density map into
two parts, that is, the structure itself and all other materials apart from
this structure. For the external tides from all outside material sustained by
a point in structure A, a simple calculation involves using equation.8 to
directly calculate the tidal strength at that point pixel by pixel, as
illustrated in Fig.9(b). The average tidal strength of all the pixels in
structure A can then represent the average tidal strength sustained by the
structure.

####  3.7.2 Minimal computation

As illustrated in Fig.9(b), the above computation is limited to a 2D plane. In
equation.8, the tidal strength is very sensitive to the distance. If the
external structure and the calculated structure are not in the same plane, the
2D calculation underestimates the distance between them and thus overestimates
the tidal strength (maximum computation, Tmax). However, we only have the 2D
projection in the observation. The 3D structure of the molecular gas is
unknown, but we can consider an extreme case. Using the average effective
radius ($\sim$0.03 pc) of all leaf structures as the characteristic scale
height of G332.83-0.55, all external material apart from the calculated
structure was moved vertically to 0.03 pc away, as illustrated in Fig.10(c).
This is unrealistic, given the continuity of the matter distribution in a
cloud. As a limiting case, the estimated tidal strength should be smaller than
the actual value (minimal computation, Tmin,s).

The other minimal computation (Tmin,v) is the vector superposition of tides,
which allows the tidal interactions to directly cancel each other out. Using
the calculated point in the target structure as the coordinate origin to build
a 2D coordinate system, all external tides were decomposed to the coordinate
axes. Then we calculated the overall tidal strength at the calculated point
after performing vector superposition along each axis.

####  3.7.3 Tide from the envelope

As shown in Fig.2, the dendrogram structures are embedded in a larger-scale
gas environment. The material outside of the FOV of ALMA observation may also
produce considerable external tides (Tout) on small-scale dendrogram
structures. Using the H2 column density derived from ATLASGAL+Planck 870
$\mu$m in Zhou et al. (2024b), the matter in the FOV of ALMA observation was
first masked, as shown in Fig.9(a). Then we calculated the external tides at
each pixel in the FOV. The averaged tidal strength of all pixels is $\sim$1.4
$\times 10^{-25}$ s-2. Compared with the tidal strength in Fig.8(b), Tout is
very small and therefore can be ignored.

####  3.7.4 Comparison

Fig.8(b) shows that the components Td,1, Td,2, and Td,3 (whose derivation is
detailed in Appendix.A) are comparable. Especially, they are also comparable
with Tmin,s and Tmin,v. As expected, Tmax is significantly larger than others.
For Td,1, Td,2, Td,3, Tmin,s, and Tmin,v, Tmin,v is slightly larger than
others and the value of Tmin,s is in the middle. As a compromise, we use the
value of Tmin,s to represent the external tides in the subsequent analysis.

##  4 Discussion

###  4.1 Tide-regulated gravitational collapse

Figure 11: Comparison between the structural energies, i.e., the gravitational
potential energy Eg, the kinetic energy Ek, the tidal energy Et, and the
external pressure energy Ep.

The average strength of the external tides Tmin,s of a structure was used to
measure the deformation of the structure due to the external gravity. The
effective velocity dispersion from the external tides of the structure is
$\sigma_{t}^{2}\approx T_{\rm min,s}*R^{2}$, and the corresponding energy

$E_{\rm t}\approx\frac{1}{2}M*T_{\rm min,s}*R^{2}.$ (9)

The gravitational potential energy $E_{\rm g}$ and internal kinetic energy
$E_{\rm k}$ were calculated in the same way as presented in Zhou et al.
(2024b).

The slowing down of a pure free-fall gravitational collapse as revealed by the
shallow $\sigma_{\rm s}-N*R$ relation means that there are some physical
processes working against gravitational collapse, such as tidal tearing from
external gravity. The effect of the external tides on the gravitational
collapse of gas structures can also be reflected in the scaling relation.
After adding the contribution of the tidal interactions to the observed
velocity dispersion, the effective velocity dispersion $\sigma_{\rm
s}=\sqrt{\sigma_{\rm t}^{2}+\sigma^{2}}$, then the $\sigma_{\rm s}-N*R$
relation was fitted again. As shown in Fig.6(d), now the slope is close to
0.5, and the correlation also gets stronger. Thus, the deformation due to the
external tides may effectively slow down the pure free-fall gravitational
collapse of gas structures.

However, as shown in Fig.11(a), the sum of Ek and Et is notably larger than
Eg. The external tidal strength may be overestimated. The velocity dispersion
is a manifestation of gas perturbations due to many physical processes,
including gravitation collapse, tidal tearing, and turbulence. Thus, Ek should
be overestimated. Moreover, when $E_{\rm k}+E_{\rm t}>E_{\rm g}$, it does not
mean that the structure cannot be gravitationally bound. A structure is
deformed by external tides from the surrounding material, and at the same
time, the surrounding material also provides the structure with a confining
pressure.

Previous studies have suggested that the external pressure provided by the
larger-scale molecular gas might help to confine dense structures in molecular
clouds (Spitzer, 1978; McKee, 1989; Elmegreen, 1989; Ballesteros-Paredes,
2006; Kirk et al., 2006; Dib et al., 2007; Lada et al., 2008; Pattle et al.,
2015; Kirk et al., 2017; Li et al., 2020; Zhou et al., 2024b). External
pressure can have various origins; here we mainly consider the external
pressure from the ambient cloud for each decomposed structure using

$P_{\rm cl}=\pi G\bar{\Sigma}\Sigma_{r},$ (10)

where $P_{\rm cl}$ is the gas pressure, $\bar{\Sigma}$ is the mean surface
density of the cloud, and $\Sigma_{\rm r}$ is the surface density at the
location of each structure (McKee, 1989; Kirk et al., 2017). The external
pressure energy can be calculated as

$E_{\rm p}=-4\pi P_{\rm cl}R^{3}.$ (11)

As shown in Fig.11(b), after considering the external pressure, $E_{\rm
k}+E_{\rm t}<E_{\rm g}+E_{\rm p}$. The external tide tries to tear up the
structure, but the external pressure on the structure prevents this process
from occurring. The counterbalance between the external tide and external
pressure hinders the gravitational collapse of the structure, which can also
cause a pure free-fall gravitational collapse to slow down.

Gravity can play multiple roles in cloud evolution. Inside dense regions, it
drives fragmentation and collapse, and outside these regions, it can suppress
the low-density gas from collapsing through extensive tidal forces and drive
accretion (Li, 2024). This work confirms that tidal interactions can
significantly influence the dynamics and kinematics of gas structures and
regulate their gravitational collapse. In Fig.14(c) and Fig.15(c) of Zhou et
al. (2024b), in the G333 complex, for 3608 gas structures on cloud-clump
scales, the slopes of the $\sigma-N*R$ relations are $\sim$0.19 and
$\sim$0.26, respectively. In the Fig.5(a) of Zhou et al. (2024a), for 2420
dense gas structures on clump-core scales in 64 ATOMS sources (clumps) at
different evolutionary stages, the slope of the $\sigma-N*R$ relation is
$\sim$0.27. According to the above analysis, these shallow slopes
(significantly smaller than 0.5) may indicate extensive tide-regulated
gravitational collapse from cloud to core scales, which will be discussed in
detail in future work.

###  4.2 Gravitational focusing

In Fig.4(b) and Fig.5(b), on large scales, the funnel structure is consistent
with the schematic diagram in Fig.9 of Zhou et al. (2023). However, it looks
as though the bottom of the funnel splits into two parts in Fig.4(b) and
Fig.5(b). A velocity jump appears in the center comprised of the redshifted
and blueshifted components, which may be attributed to the infalling and
rotating motions around the protocluster. If we turn over the redshifted
component and let it merge with the blueshifted component, then we obtain the
funnel structure in Fig.9 of Zhou et al. (2023).

Similar to lights converging to the focus, now the gas inflows are converging
to the gravitational center, that is, the hub region of G332.83-0.55. Thus,
there is a vivid ”gravitational focusing” process, which also provides a clue
as to the formation of the hub-filament structure. As shown in the Fig.1 of
Zhou et al. (2023), the entire G333 complex has a large-scale shell structure.
G332.83-0.55 is a fragment on the shell, which should have a sheet
configuration initially; for more details, readers can also refer to Sec.3.6.
Then the subsequent gravitational focusing process yields the hub-filament
structure. In Fig.1(b), the hub-filament structure is decoupling from the
surrounding gas environment due to gravitational collapse.

##  5 Summary

1\. In high-resolution APEX/LAsMA 12CO and 13CO (3-2) observations,
G332.83-0.55 has a clear hub-filament morphology at $\sim$10 pc, and its hub
corresponds to the ATLASGAL clump AGAL332.826-00.549. We zoomed in on the hub
region of G332.83-0.55 using higher-resolution ALMA HCO+ (1-0) and H13CO+
(1-0) observations, and found that it still presents a hub-filament structure.
Thus, G332.83-0.55 contains at least two levels of hub-filament structures.
High-resolution single-dish and ALMA observations provide the opportunity to
trace gas inflows from cloud to core scales.

2\. G332.83-0.5 is collapsing with rotation. Around the gravitational center
(i.e., the protocluster), both the velocity fields traced by LAsMA and ALMA
observations show velocity jumps comprised of redshifted and blueshifted
peaks. According to the velocity difference and the separation between
blueshifted and redshifted peaks, we roughly estimated the rotation velocity
$V_{\rm rotation,ALMA}\approx$3.29 km/s and the centrifugal barrier $r_{\rm
c,ALMA}\approx$ 0.21 pc. Then the mass of the gravitational center (i.e., the
protocluster) was estimated to be $\approx$264 M⊙, which agrees with the mass
estimated by 3mm dust emission.

3\. We directly identified local dense gas structures based on the 2D
integrated intensity (Moment 0) map of H13CO+ J=1-0 emission using the
dendrogram algorithm, and then extracted the average spectra to fit the
velocity dispersion. The slope of the $\sigma-N*R$ relation for leaf
structures is $\sim$0.2, which is much smaller than 0.5 of the pure free-fall
gravitational collapse. This may indicate that a pure free-fall gravitational
collapse is slowing down.

4\. We investigated the effect of shear and tides from the protocluster on the
surrounding local dense gas structures. The results seem to deny the
importance of shear and tides from the protocluster. However, for a gas
structure, it also bears the tides from other gas structures and the general
gas distribution, not only the protocluster. To fully consider the tidal
interactions, we derived the tide field according to the density distribution,
and then used the average strength of the external tides of a structure to
measure the total tidal interactions on the structure. For comparison, we also
adopted an original pixel-by-pixel computation to estimate the average tidal
strength of each structure. Both methods give comparable calculation results.

5\. After considering the tidal interactions for all structures, the slope of
the $\sigma-N*R$ relation is close to 0.5, and the correlation also gets
stronger. Thus, the deformation due to the external tides can effectively slow
down the pure free-fall gravitational collapse of gas structures. The external
tide tries to tear up the structure, but the external pressure on the
structure prevents this process from occurring. The counterbalance between an
external tide and external pressure hinders the free-fall gravitational
collapse of the structure, which can also cause a pure free-fall gravitational
collapse to slow down. These mechanisms can be called ”tide-regulated
gravitational collapse.”

###### Acknowledgements.

This publication is based on data acquired with the Atacama Pathfinder
Experiment (APEX) under programme ID M-0109.F-9514A-2022. APEX has been a
collaboration between the Max-Planck-Institut für Radioastronomie, the
European Southern Observatory, and the Onsala Space Observatory. This research
made use of astrodendro, a Python package to compute dendrograms of
Astronomical data (http://www.dendrograms.org/). This paper makes use of the
following ALMA data: ADS/JAO.ALMA$\sharp$2019.1.00685.S. ALMA is a partnership
of ESO (representing its member states), NSF (USA) and NINS (Japan), together
with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO,

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##  Appendix A Tidal tensor

For a gravitational potential field $\Phi$, the tidal tensor $\mathbf{T}$ is
defined as

$\mathrm{T}_{ij}=\partial_{i}\partial_{j}\Phi.$ (12)

The eigenvalues of the tidal tensor contain information regarding the extent
of deformation (compression and disruption) experienced at a specific point
due to the local gravitational field. The diagonalized tidal tensor is given
as

$\displaystyle\mathbf{T}=\begin{bmatrix}\lambda_{1}&0&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{3}\end{bmatrix}.$ (13)

The sign of the eigenvalues ($\lambda_{1},\lambda_{2},$ and $\lambda_{3}$)
indicates whether the given mode is compressive ($\lambda_{i}>0$) or
disruptive ($\lambda_{i}<0$), while the magnitude represents the strength of
the respective compressive and disruptive mode. The trace of the tidal tensor
$\mathrm{Tr(\mathbf{T})}=\sum_{i=1}^{3}\lambda_{i}$ contains the local density
information in the Poisson equation:

$\mathrm{Tr(\mathbf{T})}=\nabla^{2}\Phi=4\pi G\rho.$ (14)

This indicates that the trace is zero when the location where the tidal tensor
is assessed lies outside the distribution of mass.

As was done in Ganguly et al. (2024), the tidal tensor can be split into three
parts: the tidal tensor due to only the matter inside the structure,
$\mathbf{T}_{\mathrm{int}}$, and into a second component due only to the
matter distribution external to the structure, $\mathbf{T}_{\mathrm{ext}}$.
Their sum, due to the entire matter distribution,

$\mathbf{T}=\mathbf{T}_{\mathrm{int}}+\mathbf{T}_{\mathrm{ext}}.$ (15)

Here, $\mathrm{\mathbf{T}}$ represents the net deformation due to both the
structure itself and its surroundings. We note that
$\mathrm{\mathbf{T}_{ext}}$ represents the deformation introduced solely due
to the external materials. The trace of the three parts are the following:

$\displaystyle\mathrm{Tr(\mathbf{T})}=4\pi G\rho_{\mathrm{a}}$ (16)
$\displaystyle\mathrm{Tr(\mathbf{T}_{int})}=4\pi G\rho_{\mathrm{a}}$ (17)
$\displaystyle\mathrm{Tr(\mathbf{T}_{ext})}=0,$ (18)

where $\rho_{\mathrm{a}}$ is the average density of the structure with radius
$R$ and total mass $M$, $\rho_{\mathrm{a}}=M/(4/3\pi R^{3})$.

According to $\mathrm{Tr(\mathbf{T}_{int})}=4\pi G\rho_{\mathrm{a}}$, the
corresponding energy can be estimated by

$E_{\rm
a,int}\approx\frac{1}{2}M*\mathrm{Tr(\mathbf{T}_{int})}*R^{2}\approx\frac{GM^{2}}{R}.$
(19)

Thus, the tidal tensor $\mathbf{T}_{\mathrm{int}}$ represents the self-gravity
of the structure. If we want to estimate the tidal interactions from all
external materials on the structure, we should focus on
$\mathbf{T}_{\mathrm{ext}}$.

The trace in equation.18 indicates that $\mathrm{\mathbf{T}_{ext}}$ due to
only matter distribution external to the structure must contain both
compressive and disruptive modes, corresponding to positive and negative
$\lambda_{i}$, respectively. In contrast, $\mathrm{\mathbf{T}_{int}}$ due to
only the matter inside the structure and $\mathrm{\mathbf{T}}$ due to the
entire matter distribution must contain at least one compressive mode, but it
might also be fully compressive. These rules are also applicable for 2D
projected tidal tensor. According to equation.15, the 2D tidal tensor can be
decomposed as

$\begin{bmatrix}\lambda_{1}&0\\\
0&\lambda_{2}\end{bmatrix}=\begin{bmatrix}-A_{0}&0\\\
0&A_{0}\end{bmatrix}+\begin{bmatrix}A_{1}&0\\\ 0&A_{2}\end{bmatrix};$ (20)

assuming $A_{2}=k*A_{1}$, we have $A_{1}=\frac{\lambda_{1}+\lambda_{2}}{1+k}$,
$A_{2}=\frac{k(\lambda_{1}+\lambda_{2})}{1+k}$, and
$A_{0}=\frac{\lambda_{1}+\lambda_{2}}{1+k}-\lambda_{1}$. There are three
cases:

(1) When $k=1$, $A_{1}=A_{2}=\frac{\lambda_{1}+\lambda_{2}}{2}$,
$A_{0}=\frac{\lambda_{2}-\lambda_{1}}{2}$ (Fig.9(c)).

(2) When $k<<1$, $A_{1}\approx\lambda_{1}+\lambda_{2}$, $A_{2}\approx 0$,
$A_{0}\approx\lambda_{2}$ (Fig.9(b)).

(3) When $k>>1$, $A_{1}\approx 0$, $A_{2}\approx\lambda_{1}+\lambda_{2}$,
$A_{0}\approx-\lambda_{1}$ (Fig.9(a)).

We estimated the averaged strength of the external tidal field for each
structure according to Fig.9(a), (b) and (c), that is, the averaged value of
$|A_{0,a}|$ in each structure. As shown in Fig.8(b), the three cases give
similar estimates of the averaged strength of the external tidal field.