11institutetext: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
53121 Bonn, Germany
11email<EMAIL_ADDRESS>22institutetext: Centre for Astrophysics and
Planetary Science, University of Kent, Canterbury, CT2 7NH, UK
33institutetext: Department of Astronomy, The University of Texas at Austin,
2515 Speedway, Stop C1400, Austin, Texas 78712-1205, USA 44institutetext:
Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de
México, Antigua Carretera a Pátzcuaro # 8701, Ex-Hda. San José de la Huerta,
Morelia, Michoacán, México C.P. 58089 55institutetext: Shanghai Astronomical
Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030,
Peoples Republic of China

# High-resolution APEX/LAsMA 12CO and 13CO (3-2) observation of the G333 giant
molecular cloud complex : I. Evidence for gravitational acceleration in hub-
filament systems

J. W. Zhou High-resolution APEX/LAsMA 12CO and 13CO (3-2) observation of the
G333 giant molecular cloud complex : I. Evidence for gravitational
acceleration in hub-filament systemsHigh-resolution APEX/LAsMA 12CO and 13CO
(3-2) observation of the G333 giant molecular cloud complex : I. Evidence for
gravitational acceleration in hub-filament systems F. Wyrowski High-resolution
APEX/LAsMA 12CO and 13CO (3-2) observation of the G333 giant molecular cloud
complex : I. Evidence for gravitational acceleration in hub-filament
systemsHigh-resolution APEX/LAsMA 12CO and 13CO (3-2) observation of the G333
giant molecular cloud complex : I. Evidence for gravitational acceleration in
hub-filament systems S. Neupane High-resolution APEX/LAsMA 12CO and 13CO (3-2)
observation of the G333 giant molecular cloud complex : I. Evidence for
gravitational acceleration in hub-filament systemsHigh-resolution APEX/LAsMA
12CO and 13CO (3-2) observation of the G333 giant molecular cloud complex : I.
Evidence for gravitational acceleration in hub-filament systems J. S. Urquhart
High-resolution APEX/LAsMA 12CO and 13CO (3-2) observation of the G333 giant
molecular cloud complex : I. Evidence for gravitational acceleration in hub-
filament systemsHigh-resolution APEX/LAsMA 12CO and 13CO (3-2) observation of
the G333 giant molecular cloud complex : I. Evidence for gravitational
acceleration in hub-filament systems N. J. Evans II High-resolution APEX/LAsMA
12CO and 13CO (3-2) observation of the G333 giant molecular cloud complex : I.
Evidence for gravitational acceleration in hub-filament systemsHigh-resolution
APEX/LAsMA 12CO and 13CO (3-2) observation of the G333 giant molecular cloud
complex : I. Evidence for gravitational acceleration in hub-filament systems
E. Vázquez-Semadeni High-resolution APEX/LAsMA 12CO and 13CO (3-2) observation
of the G333 giant molecular cloud complex : I. Evidence for gravitational
acceleration in hub-filament systemsHigh-resolution APEX/LAsMA 12CO and 13CO
(3-2) observation of the G333 giant molecular cloud complex : I. Evidence for
gravitational acceleration in hub-filament systems K. M. Menten High-
resolution APEX/LAsMA 12CO and 13CO (3-2) observation of the G333 giant
molecular cloud complex : I. Evidence for gravitational acceleration in hub-
filament systemsHigh-resolution APEX/LAsMA 12CO and 13CO (3-2) observation of
the G333 giant molecular cloud complex : I. Evidence for gravitational
acceleration in hub-filament systems Y. Gong High-resolution APEX/LAsMA 12CO
and 13CO (3-2) observation of the G333 giant molecular cloud complex : I.
Evidence for gravitational acceleration in hub-filament systemsHigh-resolution
APEX/LAsMA 12CO and 13CO (3-2) observation of the G333 giant molecular cloud
complex : I. Evidence for gravitational acceleration in hub-filament systems
T. Liu High-resolution APEX/LAsMA 12CO and 13CO (3-2) observation of the G333
giant molecular cloud complex : I. Evidence for gravitational acceleration in
hub-filament systemsHigh-resolution APEX/LAsMA 12CO and 13CO (3-2) observation
of the G333 giant molecular cloud complex : I. Evidence for gravitational
acceleration in hub-filament systems

(Received xxx / Accepted xxx)

###### Abstract

Context. Hub-filament systems are suggested to be the birth cradles of high-
mass stars and clusters.

Aims. We investigate the gas kinematics of hub-filament structures in the G333
giant molecular cloud complex using 13CO (3$-$2) observed with the APEX/LAsMA
heterodyne camera.

Methods. We apply the FILFINDER algorithm to the integrated intensity maps of
the 13CO $J$=3$-$2 line to identify filaments in the G333 complex, and we
extract the velocity and intensity along the filament skeleton from moment
maps. Clear velocity and density fluctuations are seen along the filaments,
allowing us to fit velocity gradients around the intensity peaks.

Results. The velocity gradients fitted to the LAsMA data and ALMA data agree
with each other over the scales covered by ALMA observations in the ATOMS
survey ($\textless$ 5 pc). Changes of velocity gradient with scale indicate a
”funnel” structure of the velocity field in PPV space, indicative of a smooth,
continuously increasing velocity gradient from large to small scales, and thus
consistent with gravitational acceleration. The typical velocity gradient
corresponding to a 1 pc scale is $\sim 1.6$ km s-1 pc-1. Assuming free-fall,
we estimate a kinematic mass within 1 pc of $\sim$ 1190 M⊙, which is
consistent with typical masses of clumps in the ATLASGAL survey of massive
clumps in the inner Galaxy. We find direct evidence for gravitational
acceleration from comparison of the observed accelerations to those predicted
by free-fall onto dense hubs with masses from millimeter continuum
observations. On large scales, we find that the inflow may be driven by the
larger scale structure, consistent with hierarchical structure in the
molecular cloud and gas inflow from large to small scales. The hub-filament
structures at different scales may be organized into a hierarchical system
extending up to the largest scales probed, through the coupling of
gravitational centers at different scales.

Conclusions. We argue that the ”funnel” structure in PPV space can be an
effective probe for the gravitational collapse motions in molecular clouds.
The large scale gas inflow is driven by gravity, implying that the molecular
clouds in G333 complex may be in the state of global gravitational collapse.

###### Key Words.:

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

##  1 Introduction

To understand the formation of hierarchical structures in high-mass star
formation regions, it is critical to measure the dynamical coupling between
density enhancements in giant molecular clouds and gas motion of their local
environment (McKee & Ostriker, 2007; Motte et al., 2018; Henshaw et al.,
2020). Such studies may distinguish between competing concepts for high-mass
star formation, such as monolithic collapse of turbulent cores in virial
equilibrium (McKee & Tan, 2003; Krumholz et al., 2007), competitive accretion
in a protocluster environment through Bondi-Hoyle accretion (Bonnell et al.,
1997, 2001), turbulence-driven inertial-inflow (Padoan et al., 2020) and
gravity-dominated global hierarchical collapse (Vázquez-Semadeni et al., 2009;
Ballesteros-Paredes et al., 2011; Hartmann et al., 2012; Vázquez-Semadeni et
al., 2017, 2019), as discussed in Zhou et al. (2022). High-resolution
observations show that density enhancements are organized in filamentary gas
networks, especially in hub-filament systems. In such systems, converging
flows are funneling matter into the hub through the filaments. Many case
studies have suggested that hub-filament systems are the birth cradles of
high-mass stars and clusters (Peretto et al., 2013; Henshaw et al., 2014a;
Zhang et al., 2015; Liu et al., 2016b; Yuan et al., 2018; Lu et al., 2018;
Issac et al., 2019; Dewangan et al., 2020; Liu et al., 2022; Zhou et al.,
2022). Numerical simulations of colliding flows and collapsing clumps reveal
velocity gradients along the dense filamentary streams converging toward the
hubs (Wang et al., 2010; Gómez & Vázquez-Semadeni, 2014; Smith et al., 2016;
Padoan et al., 2020). In observations, velocity gradients along filaments are
often interpreted as evidence for gas inflow along filaments (Kirk et al.,
2013; Liu et al., 2016b; Yuan et al., 2018; Williams et al., 2018; Chen et
al., 2019, 2020; Pillai et al., 2020; Zhou et al., 2022).

Zhou et al. (2022) studied the physical properties and evolution of
hub–filament systems in a large sample of protoclusters that were observed in
the ATOMS (ALMA Three-millimeter Observations of Massive Star-forming regions)
survey (Liu et al., 2020). They found that hub-filament structures can exist
not only in small–scale ($\sim$0.1 pc) dense cores but also in large–scale
clumps/clouds ($\sim$1-10 pc), suggesting that multi-scale hub-filament
systems at various scales are common in regions forming massive stellar
clusters in various Galactic environments. The filaments in clumps observed in
the ATOMS program show clear velocity gradients. The approximately symmetric
distribution of positive and negative velocity gradients strongly indicates
the existence of converging gas inflows along filaments. The observations
confirm that high-mass stars in protoclusters may accumulate most of their
mass through longitudinal inflow along filaments. Velocity and density
fluctuations are discussed in detail in Henshaw et al. (2020), who detected
ubiquitous velocity fluctuations across all spatial scales and galactic
environments and discovered oscillatory gas flows with wavelengths ranging
from 0.3-400 pc that are coupled to regularly spaced density enhancements,
probably formed via gravitational instabilities (Henshaw et al., 2016;
Elmegreen et al., 2018). Furthermore, the locations of some of these density
enhancements spatially correlate with velocity gradient extrema, indicative of
either convergent motion or collapse-induced rotation (Clarke et al., 2016;
Misugi et al., 2019).

In Zhou et al. (2022), we measured the velocity gradients around the intensity
peaks for all observed cores. The statistical analysis found that velocity
gradients are very small at scales larger than $\sim$1 pc, probably suggesting
the dominance of pressure–driven inertial inflow, which can originate either
from large-scale turbulence or from cloud-scale gravitational contraction.
Below $\sim$1 pc, velocity gradients dramatically increase as filament lengths
decrease, indicating that the hub’s or core’s gravity dominates gas infall on
small scales. Due to the FOV limitation of ALMA observation, our previous work
was restricted to scales up to about 5 pc. In this paper, following a similar
method as described in Zhou et al. (2022), we can generalize the results from
the clump-core scale to cloud-clump scale.

Figure 1: Overview of the entire observed field. The enlarged maps in the
first row display several typical hub-filament structures. (a) Background is
the integrated intensity map of 13CO (3-2) for the G333 complex, boxes show
several divided sub-regions. Letters mark the sub-structures studied in this
work. Red contours show the peak emission of 13CO (3-2); (b) An overview
three-color map of the observed field by combining ATLASGAL+Planck 870 $\mu$m
(red) and GLIMPSE 8.0 (green) and 4.5 $\mu$m (blue) emission, including G333
complex and G331 GMC.

Fig.1(b) displays an overview three-color map of the observed field by
combining ATLASGAL+Planck 870 $\mu$m and GLIMPSE 8.0 and 4.5 $\mu$m emission.
Fig.1(a) shows the distribution of newly observed 13CO (3$-$2) emission. The
interesting individual sub-regions are marked and highlighted below. The main
structures of the observed field are the G331 and G333 giant molecular clouds
(GMCs), and the G332 ring structure between them.

The G331 GMC is one of the most massive molecular clouds in the Southern
Galaxy, in the tangent region of the Norma spiral arm (Bronfman et al., 1989).
Using the C18O (1–0) integrated emission, Merello et al. (2013) defined the
central region of the G331 GMC at $l=331.523\degr$, $b=-0.099\degr$, with a
distance of 7.5 kpc. It may harbor one of the most extended and luminous
regions of massive star formation in the Galactic disk. Caswell & Haynes
(1987) determined that the line central velocity of the ionized gas is
$\sim-89$ km s-1 from observations of the H$109\alpha$ and H$110\alpha$
hydrogen recombination lines, which is similar to the peak velocity of the
molecular gas in the GMC. The detection of OH and methanol maser emission
further provides evidence of active star formation in the G331 GMC (Goss et
al., 1970; Caswell et al., 1980; Caswell, 1998; Pestalozzi et al., 2005).

The G333 GMC, centered at $l\sim$ 333.2$\degr$, $b\sim-0.4\degr$, is a
1.2$\degr\times$ 0.6$\degr$ region in the fourth quadrant of the Galaxy at a
distance of 3.6 kpc (Lockman, 1979; Bains et al., 2006). Its gas emission
takes the form of a string of knots. As a part of the Galactic Ring of
molecular clouds at a Galactocentric radius of 3–5 kpc, the G333 molecular
cloud complex contains a diverse sample of molecular regions as well as a
range of high mass star forming clouds, bright HII regions and infrared point
sources, all surrounded by diffuse atomic and molecular gas (Fujiyoshi et al.,
2006; Cunningham et al., 2008; Wong et al., 2008; Lo et al., 2009; Jordan et
al., 2013; Lowe et al., 2014). OH, H2O and CH3OH maser lines as tracers of
high-mass star formation have been detected from various sources in the region
(Caswell et al., 1980, 1995; Breen et al., 2007; Caswell et al., 2011; Breen
et al., 2012). Towards the G333 GMC, Bains et al. (2006) used the Mopra 22-m
radio telescope to identify five distinct velocity features at $-105$, $-90$,
$-70$, $-50$, and $-10$ km s-1 with 13CO (1$-$0) maps. They also found at
least three velocity components ($-55$, $-50$, and $-42$ km s-1) between $-65$
and $-35$ km s-1, with the brightest feature at $-50$ km s-1. Several spiral
arms intersect at $l\sim$ 333$\degr$ along the line-of-sight, i.e., the
Sagittarius-Carina arm, the Scutum-Crux arm and the Norma-Cygnus arm (Russeil,
2003; Russeil et al., 2005; Vallée, 2008). This region may therefore be
suitable to study the formation of molecular clouds under galaxy dynamics.

Between the G333 and G331 GMCs, there is a cavity at $l\sim$ 332$\degr$. The
upper part presents a ring structure within Galactic coordinates
$332.0\degr<l<332.8\degr$ and $-0.3\degr<b<0.4\degr$. The principal emission
of this ring structure is concentrated between the velocity range
$-55\textless v_{\rm LSR}\textless-44$ km s-1 (Romano et al., 2019) and the
average spectral profiles of the CO lines and CI show a peak at $v_{\rm
LSR}=-50$ km s-1. The distance of the ring is determined to be $\sim$3.7 kpc
from the Sun (Romano et al., 2019), thus the ring and G333 GMC are located in
the same region, referred to as the G333 complex in this work.

##  2 Observation

###  2.1 APEX/LAsMA data

We mapped a $3.4\degr\times 1.2\degr$ area centered on
$(l,b)=(332.33\degr,-0.29\degr)$. The observations were conducted between
March and August of 2022 using the APEX telescope (Güsten et al., 2006)111This
publication is based on data acquired with the Atacama Pathfinder EXperiment
(APEX). APEX is a collaboration between the Max-Planck-Institut für
Radioastronomie, the European Southern Observatory and the Onsala Space
Observatory.. The 7 pixel Large APEX sub-Millimeter Array (LAsMA) receiver was
used to observe the $J=3-2$ transitions of 12CO ($\nu_{\text{rest}}\sim
345.796\,$\mathrm{GHz}$$) and 13CO ($\nu_{\text{rest}}\sim
330.588\,$\mathrm{GHz}$$) simultaneously in the upper and lower sideband,
respectively. More details about the receiver are given in Mazumdar et al.
(2021). The local oscillator frequency was set at
$338.190\text{\,}\mathrm{GHz}$ in order to avoid contamination of the 13CO
(3$-$2) lines due to bright 12CO (3$-$2) emission from the image band. The
whole observed region was divided into two parts, the first part is from
$l\sim$ 332$\degr$ to $l\sim$ 334$\degr$, the second part is from $l\sim$
330.6$\degr$ to $l\sim$ 332$\degr$. Each part was further divided into sub-
maps of size
$$10\text{\,}\mathrm{\SIUnitSymbolArcminute}$\times$10\text{\,}\mathrm{\SIUnitSymbolArcminute}$$
each. Observations were performed in a position switching on-the-fly (OTF)
mode. Although the reference-positions were carefully chosen, the reference-
position of the first part still has emission present in the velocity range of
$-$140 to 10 km s-1 (see Appendix B for details about how the issue was
solved).

The data were calibrated using a three load chopper wheel method, which is an
extension of the “standard” method used for millimeter observations (Ulich &
Haas, 1976) to calibrate the data to the antenna temperature $T^{*}_{A}$
scale. The data was reduced using the GILDAS
package222http://www.iram.fr/IRAMFR/GILDAS. A velocity range of $-$190 to 60
km s-1 was extracted for each spectrum and resampled to an adequate velocity
resolution of 0.25 km s-1 to reduce the noise. The velocity range $-$140 to 10
km s-1 was masked before fitting a first order baseline to each spectrum. The
reduced, calibrated data obtained from the different scans were then combined
and gridded using a $6\arcsec$ cell size. The gridding process includes a
convolution with a Gaussian kernel with a FWHM size of one-third the telescope
FWHM beam width. The data cubes obtained have a final angular resolution of
$19.5\arcsec$ comparable with the resolution of the ATLASGAL survey.

After the online calibration, the intensities are obtained on the $T^{*}_{A}$
(corrected antenna temperature) scale. Apart from the atmospheric attenuation,
this also corrects for rear spillover, blockage, scattering and ohmic losses.
A beam efficiency value $\eta_{mb}=0.71$ (Mazumdar et al., 2021) was used to
convert intensities from $T^{*}_{A}$ to the main beam brightness temperature
$T_{mb}$. 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 the rest of paper, we
focus on the analysis of the 13CO (3-2) emission. The information from both
lines will be combined in a forthcoming paper.

###  2.2 Archival continuum emission data

The observed region was covered in the infrared by the GLIMPSE survey
(Benjamin et al., 2003). The images of the Spitzer Infrared Array Camera
(IRAC) at 4.5 and 8.0 $\mu$m were retrieved from the Spitzer Archive. The
angular resolutions of the images in the IRAC bands are $\sim 2\arcsec$. 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).

Table 1: Properties of ATLASGAL clumps associated with the intensity peaks of 13CO (3$-$2) emission (orange circles in Fig. 7), which are treated as hubs in this work. $\Delta$V and L represent the velocity gradient and the length of the filament. As shown in Fig. 7, there are two velocity gradients around one intensity peak, marked as ’1’ and ’2’. However, here we only reserve the good fittings, thus many hubs only have the velocity gradient of one side. Clump | log(M) | Radius | Tdust | log(Lum) | vLSR |  $\Delta$V1 | L1 |  $\Delta$V2 | L2
---|---|---|---|---|---|---|---|---|---
| ($\mathrm{M_{\odot}}$) | ($\mathrm{pc}$) | ($\mathrm{K}$) | ($\mathrm{L_{\odot}}$) | ($\mathrm{km\,s^{-1}}$) | ($\mathrm{km\,s^{-1}\,pc^{-1}}$) | ($\mathrm{pc}$) | ($\mathrm{km\,s^{-1}\,pc^{-1}}$) | ($\mathrm{pc}$)
AGAL332.094$-$00.421 | 3.155 | 0.865 | 28.4 | 4.789 |  $-$56.5 |  $-$2.97 | 1.4 | 1.9 | 1.1
AGAL332.312$-$00.556 | 2.947 | 0.831 | 21.4 | 3.835 |  $-$51.9 | 0.9 | 1.9 |  $-$1.76 | 0.6
AGAL332.351$-$00.436 | 3.034 | 0.935 | 18.3 | 3.483 |  $-$47.6 |  |  |  $-$0.58 | 6.0
AGAL332.411$-$00.471 | 2.618 | 0.502 | 19.3 | 2.694 |  $-$55.8 | 0.3 | 6.0 |  $-$0.49 | 5.0
AGAL332.467$-$00.522 | 3.162 | 0.848 | 19.8 | 3.864 |  $-$52.6 | 0.2 | 3.0 |  $-$2.25 | 0.8
AGAL332.606$-$00.854 | 2.569 | 0.537 | 19.8 | 3.088 |  $-$56.2 |  $-$0.53 | 2.0 | 2.0 | 1.0
AGAL332.647$-$00.609 | 3.549 | 1.35 | 28.3 | 5.195 |  $-$49.2 |  $-$2.7 | 2.0 | 3.7 | 1.0
AGAL332.694$-$00.612 | 3.184 | 0.813 | 26.6 | 4.466 |  $-$47.6 |  $-$7.84 | 1.0 | 9.7 | 1.0
AGAL332.694$-$00.612 | 3.184 | 0.813 | 26.6 | 4.466 |  $-$47.6 | 3.3 | 0.8 |  $-$4.59 | 1.0
AGAL332.726$-$00.621 | 2.407 | 0.294 | 24.8 | 3.456 |  $-$49.7 |  $-$3.2 | 0.7 | 1.5 | 0.8
AGAL332.751$-$00.597 | 2.964 | 1.038 | 24.7 | 3.937 |  $-$53.1 |  |  | 0.8 | 1.8
AGAL332.762$-$00.641 | 2.32 | 0.623 | 22.4 | 3.313 |  $-$50.4 |  $-$4.14 | 0.6 | 0.9 | 1.7
AGAL332.826$-$00.549 | 3.724 | 1.367 | 31.4 | 5.543 |  $-$57.3 |  $-$0.58 | 6.7 | 2.7 | 1.0
AGAL332.826$-$00.549 | 3.724 | 1.367 | 31.4 | 5.543 |  $-$57.3 |  $-$2.65 | 2.5 | 3.1 | 1.0
AGAL332.826$-$00.549 | 3.724 | 1.367 | 31.4 | 5.543 |  $-$57.3 |  |  |  $-$0.86 | 8.5
AGAL332.892$-$00.569 | 2.595 | 0.485 | 19.7 | 3.19 |  $-$57.3 |  $-$1.49 | 1.5 | 0.9 | 2.0
AGAL332.866$-$00.587 | 2.364 | 0.169 | 23.3 | 3.113 |  $-$58.1 |  |  |  $-$1.59 | 1.0
AGAL332.962$-$00.679 | 3.305 | 0.779 | 22.5 | 4.225 |  $-$48.5 | 2.0 | 3.0 |  $-$1.53 | 5.5
AGAL332.969$-$00.737 | 2.785 | 0.64 | 13.5 | 2.525 |  $-$55.6 |  $-$3.7 | 1.0 | 3.9 | 1.0
AGAL332.995$-$00.519 | 2.511 | 0.467 | 18.2 | 2.593 |  $-$52.8 |  |  | 3.3 | 1.0
AGAL333.001$-$00.436 | 2.848 | 0.865 | 35.7 | 4.989 |  $-$55.4 |  $-$3.4 | 0.7 | 1.0 | 2.5
AGAL333.013$-$00.466 | 2.821 | 0.606 | 34.5 | 4.968 |  $-$53.3 |  $-$1.74 | 1.2 | 3.4 | 0.6
AGAL333.014$-$00.521 | 2.986 | 1.056 | 22.6 | 4.152 |  $-$53.2 |  $-$2.58 | 1.0 | 1.3 | 2.0
AGAL333.053+00.029 | 3.017 | 1.056 | 29.2 | 4.398 |  $-$45.3 | 1.1 | 1.4 |  $-$0.42 | 2.1
AGAL333.071$-$00.399 | 3.281 | 0.969 | 19.1 | 3.744 |  $-$53.3 |  |  | 0.9 | 1.2
AGAL333.071$-$00.399 | 3.281 | 0.969 | 19.1 | 3.744 |  $-$53.3 |  $-$1.14 | 1.5 | 0.2 | 9.0
AGAL333.089$-$00.352 | 2.347 | 0.692 | 28.1 | 3.95 |  $-$52.7 |  |  | 0.6 | 2.5
AGAL333.094$-$00.524 | 2.728 | 0.623 | 26.5 | 3.966 |  $-$57.6 |  $-$1.64 | 1.5 | 0.8 | 3.5
AGAL333.103+00.087 | 2.071 | 0.169 | 17 | 2.119 |  $-$44.9 | 0.3 | 2.0 |  $-$0.26 | 3.0
AGAL333.103$-$00.502 | 3.251 | 1.367 | 25.5 | 4.601 |  $-$55.8 |  $-$0.81 | 6.0 | 1.3 | 1.0
AGAL333.129$-$00.559 | 3.843 | 1.402 | 16.1 | 3.983 |  $-$56.4 |  $-$0.8 | 4.0 | 2.4 | 2.0
AGAL333.134$-$00.431 | 3.959 | 1.506 | 32 | 5.823 |  $-$51.9 |  $-$3.3 | 1.3 | 2.0 | 1.7
AGAL333.169$-$00.431 | 2.959 | 0.831 | 23 | 4.052 |  $-$50.8 |  $-$0.63 | 2.2 | 2.0 | 1.3
AGAL333.248+00.054 | 3.27 | 1.454 | 26.3 | 4.706 |  $-$46.9 |  |  |  $-$0.81 | 5.5
AGAL333.264$-$00.291 | 2.229 | 0.485 | 27.4 | 3.951 |  $-$52.2 |  |  |  $-$3.4 | 0.8
AGAL333.284$-$00.387 | 3.678 | 1.211 | 32.1 | 5.435 |  $-$51.6 |  |  | 0.7 | 3.0
AGAL333.284$-$00.387 | 3.678 | 1.211 | 32.1 | 5.435 |  $-$51.6 |  $-$1.66 | 1.8 | 2.5 | 1.0
AGAL333.308$-$00.366 | 3.561 | 1.246 | 34 | 5.544 |  $-$50.2 |  $-$4.51 | 0.4 | 0.9 | 3.8
AGAL333.308$-$00.366 | 3.561 | 1.246 | 34 | 5.544 |  $-$50.2 |  |  | 2.8 | 1.0
AGAL333.308$-$00.366 | 3.561 | 1.246 | 34 | 5.544 |  $-$50.2 |  $-$2.5 | 1.2 | 2.8 | 0.7
AGAL333.308$-$00.366 | 3.561 | 1.246 | 34 | 5.544 |  $-$50.2 |  $-$0.32 | 6.0 | 0.2 | 6.0
AGAL333.314+00.106 | 2.901 | 0.71 | 23.3 | 3.922 |  $-$47.7 | 2.4 | 1.3 |  $-$0.88 | 2.7
AGAL333.466$-$00.164 | 3.427 | 1.038 | 24.6 | 4.591 |  $-$42.5 | 2.3 | 1.5 |  $-$1.57 | 5.0
AGAL333.483$-$00.246 | 3.133 | 0.623 | 18 | 3.255 |  $-$50.4 |  |  |  $-$1.57 | 5.0
AGAL333.499$-$00.329 | 2.163 | 0.433 | 23.9 | 3.244 |  $-$50.2 |  |  |  $-$2.18 | 1.0
AGAL333.506$-$00.101 | 2.653 | 0.744 | 19.7 | 3.096 |  $-$42.8 |  |  | 1.3 | 2.0
AGAL333.524$-$00.269 | 3.488 | 1.54 | 22.5 | 4.252 |  $-$48.8 |  $-$0.42 | 3.5 | 4.9 | 1.3
AGAL333.524$-$00.269 | 3.488 | 1.54 | 22.5 | 4.252 |  $-$48.8 |  $-$1.41 | 0.7 |  |
AGAL333.539$-$00.242 | 2.93 | 0.727 | 22.8 | 3.873 |  $-$48.9 |  $-$0.79 | 2.5 | 1.5 | 1.0
AGAL333.566$-$00.296 | 2.982 | 1.09 | 21.7 | 3.881 |  $-$46.7 |  $-$2.56 | 1.9 | 1.3 | 3.3
AGAL333.651$-$00.334 | 2.832 | 0.796 | 18.2 | 3.083 |  $-$47.1 |  $-$1.51 | 3.5 | 1.6 | 1.0
AGAL333.683$-$00.256 | 2.83 | 0.883 | 28.3 | 4.18 |  $-$48.1 |  $-$2.83 | 1.0 | 0.9 | 1.5
AGAL333.696$-$00.231 | 2.519 | 0.398 | 29.7 | 3.653 |  $-$46.7 |  |  |  $-$0.29 | 4.8
AGAL333.774$-$00.256 | 2.993 | 0.692 | 15.5 | 3.061 |  $-$49.5 |  $-$2.7 | 1.3 | 1.0 | 0.7
AGAL331.706$-$00.266 | 2.674 | 0.661 | 16.1 | 2.631 |  $-$49 |  |  |  $-$0.92 | 1.6
AGAL331.856$-$00.126 | 3.05 | 0.947 | 18.2 | 3.539 |  $-$50.1 |  |  |  $-$0.39 | 3.2
Table 2: Remaining part of Table 1. Usually one clump corresponds to one hub. Here 2 or 3 clumps separated by the gap indicate that the 2 or 3 clumps together form a hub. Clump | log(Mass) | Radius | Tdust | log(Lum) | vLSR |  $\Delta$v1 | L1 |  $\Delta$v2 | L2
---|---|---|---|---|---|---|---|---|---
| ($\mathrm{M_{\odot}}$) | ($\mathrm{pc}$) | ($\mathrm{K}$) | ($\mathrm{L_{\odot}}$) | ($\mathrm{km\,s^{-1}}$) | ($\mathrm{km\,s^{-1}\,pc^{-1}}$) | ($\mathrm{pc}$) | ($\mathrm{km\,s^{-1}\,pc^{-1}}$) | ($\mathrm{pc}$)
AGAL332.166+00.126 | 2.515 | 0.541 | 14.4 | 2.316 |  $-$49.9 |  $-$0.65 | 2.0 | 0.7 | 3.5
AGAL332.226$-$00.124 | 2.892 | 0.556 | 15.4 | 2.579 |  $-$50.8 |  $-$2.63 | 1.0 | 1.6 | 2.0
AGAL332.226$-$00.124 | 2.892 | 0.556 | 15.4 | 2.579 |  $-$50.8 |  $-$1.57 | 1.5 | 2.9 | 0.7
AGAL332.241$-$00.044 | 3.226 | 0.917 | 17 | 3.526 |  $-$48.1 |  $-$1.45 | 1.0 | 3.0 | 1.0
AGAL332.254$-$00.056 | 2.862 | 0.421 | 14.3 | 2.675 |  $-$47.7 |  |  | 0.8 | 1.7
AGAL332.296$-$00.094 | 3.15 | 0.932 | 24 | 4.303 |  $-$48.2 | 0.6 | 3.0 |  $-$0.36 | 2.5
AGAL332.317+00.177 | 2.576 | 0.406 | 13.6 | 2.223 |  $-$48.6 |  |  | 2.0 | 0.5
AGAL332.352$-$00.116 | 2.513 | 0.601 | 27.9 | 3.887 |  $-$49.8 |  $-$0.39 | 5.0 | 0.5 | 2.0
AGAL332.442$-$00.139 | 2.723 | 0.421 | 18.3 | 2.669 |  $-$51.3 |  |  | 0.7 | 2.0
AGAL332.469$-$00.131 | 2.643 | 0.556 | 20.3 | 3.088 |  $-$50.5 | 1.8 | 1.2 |  $-$1.3 | 1.2
AGAL332.544$-$00.124 | 2.857 | 1.037 | 32.5 | 4.591 |  $-$47.5 |  |  |  $-$1.92 | 2.4
AGAL332.604$-$00.167 | 3.185 | 1.052 | 17.7 | 3.371 |  $-$46.2 |  |  |  $-$0.89 | 1.6
AGAL332.141$-$00.466 | 2.66 | 0.571 | 24.2 | 3.495 |  $-$57 |  |  |  |
AGAL332.144$-$00.469 | 2.711 | 0.883 | 24.5 | 4.203 |  $-$58 |  |  | 3.0 | 0.8
AGAL332.141$-$00.446 | 2.883 | 0.71 | 27.7 | 4.76 |  $-$56.3 |  |  |  |
AGAL332.147$-$00.439 | 2.371 | 0.169 | 26.9 | 3.974 |  $-$57.9 |  |  |  |
AGAL332.156$-$00.449 | 3.239 | 1.004 | 30 | 5.078 |  $-$55.6 |  |  | 3.0 | 0.9
AGAL332.252$-$00.539 | 2.821 | 0.45 | 17.8 | 2.608 |  $-$52.9 |  |  |  |
AGAL332.281$-$00.547 | 3.133 | 0.71 | 16.9 | 3.299 |  $-$52.2 |  $-$1.73 | 1.0 | 2.2 | 1.0
AGAL333.521$-$00.241 | 2.713 | 0.415 | 20 | 2.93 |  $-$48.8 |  |  |  |
AGAL333.524$-$00.269 | 3.488 | 1.54 | 22.5 | 4.252 |  $-$48.8 | 2.3 | 1.0 |  $-$1.36 | 1.0
AGAL332.442$-$00.139 | 2.723 | 0.421 | 18.3 | 2.669 |  $-$51.3 |  |  |  |
AGAL332.469$-$00.131 | 2.643 | 0.556 | 20.3 | 3.088 |  $-$50.5 |  $-$0.74 | 2.0 | 0.7 | 4.5
AGAL332.751$-$00.597 | 2.964 | 1.038 | 24.7 | 3.937 |  $-$53.1 |  |  |  |
AGAL332.774$-$00.584 | 3.103 | 1.038 | 23 | 4.183 |  $-$55.2 |  $-$1.74 | 3.3 | 5.6 | 1.0

##  3 Results

###  3.1 Velocity components

Figure 2:  Average spectra of 13CO (3$-$2) for the entire region, red dashed
profiles represent the components of multi-Gaussian fitting.  Figure 3:
ATLASGAL clumps distribution and velocity components of the entire observed
field. (a) Background is the integrated intensity map of 13CO (3$-$2), color-
coded ”+” mark different ATLASGAL clump clusters A (green), B (orange), C
(red) and D (magenta); (b) The moment-1 map of 13CO (3$-$2) for the entire
region in the overall velocity range; (c), (d) and (e) The moment-1 map of
13CO (3$-$2) for three peaks marked in Fig. 2. Figure 4: The velocity
distribution of the entire observed field in PPV space decomposed by GAUSSPY+
along the Galactic latitude and longitude. Figure 5: Background images show
the moment 0 maps of 13CO (3-2). Lines in color present the filament
skeletons. Orange circles show the ATLASGAL clumps, the size of the circle
represents the clump radius. Red ’+’ marks the intensity peak of 13CO (3-2)
emission in Fig. 7. Figure 6: The intensity and velocity distribution of sub-
regions in PPV space based on the decomposition of GAUSSPY+. For other sub-
regions, see Fig. 13 and Fig. 14 in Sec. A. Figure 7: Two selected filaments
are used to illustrate the fitting of the velocity gradient. Upper panel:
Velocity gradients are fitted in the ranges defined by the red vertical dashed
lines, and straight lines show the linear fitting results. Lower panel: Blue
and red dotted lines show the normalized velocity and intensity, respectively.
Orange circles mark the intensity peaks of 13CO (3$-$2) emission associated
with ATLASGAL clumps. For other filaments, see Fig. 15, Fig. 16, Fig. 17, Fig.
18 and Fig. 19 in Sec. A. Figure 8:  Statistical analysis of all fitted
velocity gradients. (a) Velocity gradient versus the length over which the
gradient has been estimated for all sources in ATOMS survey (red dots) and
current LAsMA observation (black ’+’). The color lines show free-fall velocity
gradients for comparison. For the free-fall model, yellow, blue, cyan and
green lines denote masses of 5 M⊙, 500 M⊙, 5000 M⊙ and 50000M⊙, respectively;
(b), (c) and (d) Blow-up maps with lengths $\textless$ 1 pc (small scale),
$\sim$ 1 – 7.5 pc (middle scale) and $\textgreater$ 7.5 pc (large scale) in
panel (a).  Figure 9: Schematic diagram of the ‘funnel’ structure in PPV space
and on PP plane. PP plane shows a multi-scale hub-filament structure in a
molecular cloud, which can turn to a ‘funnel’ structure in PPV space. Red
arrow represents the gas inflow. Figure 10:  Statistical analysis of the
velocity gradients listed in Table 1. (a) The probability distribution of
velocity gradients measured around 1 pc scale; (b) Mass distribution of
ATLASGAL clumps listed in Table 1; (c) and (d) The same with Fig. 8, but only
consider the velocity gradients listed in Table 1. Figure 11: Correlation
between the left and right terms in Eqn. 1 revealed by the values of $M$, $L$
and $\nabla V$ listed in Table 1. Blue dotted line is the line of equality of
the left and right terms. There are two cases in Table 1, a dense clump has
two or more filaments leading to it, or only one filament with a gradient can
be identified (the rows with only one entry for the gradient). The number of
clumps in two cases is 49 and 25, and marked by black and red ’+’,
respectively.

The average emission spectrum of 13CO (3$-$2) coming from the entire observed
field is shown in Fig. 2, together with Gaussian curves used to fit the
average spectrum. The fitting is carried out using the GAUSSPY+ software
(Riener et al., 2019), which will be introduced in detail below. We have
divided the spectrum into three velocity components [$-$120, $-$76] km s-1,
[$-$76, $-$60] km s-1 and [$-$60, $-$35] km s-1 in which emission is
prominent, marked as peak1, peak2 and peak3. From Fig. 3, we can see the
velocity components peak1 and peak2 represent the main two parts of the G331
GMC, called G331 GMC-blue and G331 GMC-red. peak3 shows a very extended
structure all over the observed region with the G333 GMC and G332 ring as the
main parts. There are four main ATLASGAL clump clusters marked by A, B, C and
D in Fig.3(a) (Urquhart et al., 2018, 2022), they well correspond to different
velocity components shown in the moment-1 maps of 13CO (3$-$2) (Fig.3(b)).
peak1 and peak2 associate with the clump clusters C and D. Both A and B clump
clusters are located in the velocity component peak3. peak1 is relatively
independent of peak2 and peak3, but the dominant features in peak2 and peak3
appear have some overlap, yet they can be separated. Hence, we fit Gaussian
functions in order to minimise the effects of cross-contamination. peak2 and
peak3 can be distinguished by the full width at half maximum (FWHM). In this
paper, we mainly focus on peak3 with a velocity range [$-$60, $-$35] km s-1,
which represents a complete shell structure shown in Fig. 3(e). The
association between peak1, peak2 and peak3 will be discussed in detail in a
forthcoming paper.

In order to study the structure of the molecular cloud complex in position-
position-velocity space, we applied the fully automated Gaussian decomposer
GAUSSPY+ (Lindner et al., 2015; Riener et al., 2019), which can decompose the
complex spectra of molecular lines into multiple Gaussian components. The most
important four parameters in the algorithm are the first and second smoothing
parameters $\alpha_{1}$ and $\alpha_{2}$, signal-to-noise (SNR) and
significance criterion. To determine the smoothing parameters $\alpha_{1}$ and
$\alpha_{2}$, sets with 500 randomly selected spectra from the 13CO (3$-$2)
cube were used to train GAUSSPY+. The default values of SNR and significance
are 3 and 5, but considering that the moment-1 map needs a 5$\sigma$ threshold
to show a clear velocity field, we adjust the value of SNR to 5. Other
parameters for the decomposition are set to the default values provided by
Riener et al. (2019).

As shown in Fig. 4, the velocity components separated by the average spectrum
is consistent with the decomposition of GAUSSPY+, with the three main velocity
components well separated in PPV space. Different from peak1 and peak2, for
peak3, no matter the projection on Galactic longitude or latitude, we can see
a large-scale continuous structure, indicating the structures in peak3 are
physically connected. Especially, the overall structure of peak3 consists of a
large-scale shell structure extended from $l=330.6\degr$ to $l=334\degr$. It
may be generated by a large-scale compression, such as the shock from
supernova explosion events, which will be investigated in a forthcoming paper.

The entire G333 complex seems to have fragmented into several independent
molecular clouds, they are treated as sub-regions marked in Fig. 1, such as
s3. Sub-structures in each sub-region marked by letters in Fig. 1(a) are
divided by the bright emission of 13CO (3-2) shown in red contours, such as
s3a. From Fig. 1(b) and Fig. 3(a), we can see the dense parts of 13CO (3$-$2)
emission tightly correlate with the infrared bright regions. The overlap of
different velocity components in sub-regions 5 and 7 is significant, as shown
in Fig.3(a). However, these two sub-regions only include a few structures in
the velocity range [$-$60, $-$35] km s-1, thus they are subordinate in G333
complex.

Through the above analysis, we see that the G331 GMC and the G333 complex
overlap on the sky, but they have different velocities and lie at very
different distances. Given that the G333 region is closer to us, we focus the
remainder of the paper on it.

###  3.2 Identification of filaments

The first step is to identify and characterize filaments in the G333 complex.
Following the method described in Zhou et al. (2022), we use the Moment 0 maps
(the integrated intensity maps) of 13CO (3$-$2) to identify filaments in the
G333 complex using the FILFINDER algorithm (Koch & Rosolowsky, 2015). The
velocity intervals for making Moment 0 maps are determined from the averaged
spectra of each sub-region, marked by vertical dotted lines in Fig. 12.
Moreover, a threshold of 5$\sigma$ is applied to make the moment maps, which
can reduce the noise contamination effectively. The skeletons of identified
filaments overlaid on the moment-0 maps of 13CO (3-2) line emission are shown
in Fig. 5. They are highly consistent with the gas structures traced by 13CO
(3-2) as seen by eye, indicating that the structures identified in FILFINDER
are reliable. Then for each sub-region, we break the filamentary network into
several filaments. This step is necessary to calculate the offset along the
filament before fitting the velocity gradient. To show the velocity field
along the filament more clearly, we try to make each filament long enough.

Considering that each sub-region includes many sub-structures, it is not
surprising that their average spectra will show multiple velocity peaks. In
Fig. 12, the average spectra of G333-s3 clearly shows multiple components. We
separate out each peak in the line profile, and find that each peak indeed
corresponds to a sub-structure. This situation may also exist in other sub-
regions. However, generally these sub-structures are well spatially separated
and thus do not have a significant impact on the moment maps analysis.

Fig. 6 displays the GAUSSPY+ decomposition of each sub-region in PPV space.
From there, we can see the main structures of each sub-region are connected in
PPV space and thus unlikely to be contaminated by velocity components from
unrelated foreground or background cloud emission. Moreover, after we extract
the velocity and intensity information along the filament skeletons, if there
is overlap of unrelated velocity components, we should find anomalies in Fig.
7, such as a sudden break at a certain position of the skeleton. However, that
is not the case in our analysis. Furthermore, we only analyze the variations
of velocity and intensity along the filament skeletons with a one pixel width,
which can also effectively avoid potential overlap. In Zhou et al. (2022), we
found that multiple velocity components are common in hub-filament systems,
especially in hub regions. Furthermore, the observed samples in Zhou et al.
(2022) with or without multiple velocity components show similar results, such
as the velocity gradient estimated from moment maps.

To say the least, we can assume that most of the pixels above the filaments
have a single velocity component or have a dominant velocity component, and
then we compare the fitting results with the free-fall model and previous
results from ALMA data, if they are consistent, indicating that the hypothesis
is reasonable. That is the case in our work. Conversely, if the overlap of
velocity components is significant, the fitted velocity gradient should be
free without the regularities shown in our work, because the overlap of
unrelated velocity components is random. From Fig. 7, we can see good
correlations between velocity and density fluctuations, which is consistent
with previous studies Hacar & Tafalla (2011); Henshaw et al. (2014b, 2016);
Liu et al. (2019); Henshaw et al. (2020). The variation of the velocity
gradient with the scale shown in Fig. 8 is also consistent with the results in
Zhou et al. (2022).

###  3.3 Velocity gradients

The kinematical features of hub-filament systems can be revealed by the
velocity gradients along the filaments. However, it is difficult to directly
carry out the measurement in PPV space due to the complex gas motion shown in
Fig. 4. In Zhou et al. (2022), we investigate the presence of hub-filament
systems in a large sample of 146 proto-clusters using H13CO+ J=1-0 molecular
line in ATOMS survey. The strongest intensity peaks of H13CO+ emission
coincide with the brightest 3 mm cores or hub regions in those hub-filament
systems. We find that filaments are ubiquitous in proto-clusters, velocity and
density fluctuations along these filaments are seen. We firstly estimated two
overall velocity gradients between velocity peaks and valleys at the two sides
of the center of the gravitational potential well, and ignored local velocity
fluctuation. We also derived additional velocity gradients over smaller
distances around the strongest intensity peaks of H13CO+ emission (see Fig. 6
in Zhou et al. 2022). In this work, the same method is used to derive the
velocity gradients from the LAsMA data. Then the fitted velocity gradients
derived from the LAsMA and ALMA data are combined for a statistical analysis,
as shown in Fig. 8.

Fig. 7 shows the intensity-weighted velocity (Moment 1) and integrated
intensity (Moment 0) of 13CO (3$-$2) line emission along the skeletons of
selected filaments with intense velocity and density fluctuations along these
filaments. Generally, the peaks of the density fluctuations are associated
with ATLASGAL clumps as shown in Fig. 5. The density and velocity fluctuations
along filaments may indicate converging gas flows coupled to regularly spaced
density enhancements that probably form via gravitational instabilities
(Henshaw et al., 2020). In Fig. 7, we can see the typical V-shape velocity
structure around the intensity peaks, which we fit as velocity gradients on
both sides of the intensity peak marked by the straight lines. Fig. 7(a) shows
a more complex velocity field with more repetitive segments than the simple
structure in Fig. 7(b). Gas kinematics of other long filaments in Fig. 5 are
similar to Fig. 7(a). For the long filaments, except for the local velocity
fluctuations or gradients, they also exhibit velocity gradients on larger
scales, which are fitted by ignoring the local velocity fluctuations.
Generally, large-scale velocity gradients are associated with several
intensity peaks.

###  3.4 Hub-filament structure

Each sub-region marked in Fig. 1(a) contains many sub-structures, which are
nearly separated from each other, except for several crowded regions. Red
contours from $\sim$50% of the integrated intensity of the 13CO (3-2) emission
show the high density parts of various structures. In Fig. 1, several typical
hub-filament structures are highlighted in the first row, see also Fig. 5 for
more details. We can see filamentary structures connected to high-density
hubs. Other sub-structures in the entire observed region have more or less
similar morphology. Some differences are expected, since the appearance of
hub-filament morphology relies on the projection angle and the resolution of
observation. Moreover, most structures have the typical kinematic features of
hub-filament systems, as discussed in next sections. For example, the peaks of
intensity (the hubs) are associated with the converging velocities, indicating
that the surrounding gas flows may be converging to dense structures.
Especially, velocity gradients increase at small scales.

Orange circle in Fig. 7 marks the intensity peak of 13CO (3$-$2) emission
associated with an ATLASGAL clump where the distance between the central
coordinate of ATLASGAL clump and the intensity peak of 13CO (3$-$2) emission
is less than the effective radius of ATLASGAL clump. These associated clumps
are treated as hubs in this work, their properties are listed in Table 1. In
the second part of Table 1, some of the peaks have more than one corresponding
ATLASGAL clumps and we have added those together, as indicated by a space in
the table. We focus on their mass, effective radius and the corresponding
velocity gradients. However, as shown in Fig. 7, there are also many peaks of
13CO (3$-$2) emission without corresponding clumps, thus not included in the
table, but they are similar to the peaks associated with clumps. Generally,
large-scale velocity gradients involve many intensity peaks with or without
corresponding clumps, thus it is difficult to estimate the properties of their
hubs, therefore these are not given in the table.

###  3.5 Funnel structure

If we compare the velocity gradients of molecular gas to iron filings, just as
the iron filings can reflect the shape of a magnetic field, the distribution
of velocity gradients may to some extent reflect the shape of the force field
that plays the dominant role in the molecular clouds. Below, we will
investigate the distribution of the molecular velocity gradients in PPV space.
In the case of gravitational free-fall onto a central point mass, velocities
change as $v\propto r^{-1/2}$ ($r=\sqrt{x^{2}+y^{2}}$). In the case of
collapse driven by the gravity of a gaseous mass with a certain density
profile, the radial dependence of the infall speed depends on the slope of the
density profile (Gómez et al., 2021). In either case, this results in a
‘funnel’ morphology of the velocity field in PPV space, as illustrated in Fig.
9. The exterior cloud velocities show only gentle variations (small velocity
gradient, funneling material from cloud to clump), while the velocities in the
interior part near the center change dramatically with scales (large velocity
gradient, funneling material from clump to core). We note that the schematic
in Fig. 9 is oversimplified. Generally a molecular cloud contains many clumps
inside, and a clump also contains many cores. Here, we choose only one clump
and one core to demonstrate the gas inflow and the variation of the velocity
gradient from the molecular cloud scale to the core scale. A more realistic
case is shown in Fig. 6, here we indeed find most of sub-structures show the
expected funnel structures in PPV space. Particularly, close to the intensity
peak, the velocity gradient increases steeply, also reflected in Fig. 7.

Moreover, the V-shape velocity structure around the intensity peaks indicates
accelerated material inflowing towards the central hub (Gómez & Vázquez-
Semadeni, 2014; Kuznetsova et al., 2015; Hacar et al., 2017; Kuznetsova et
al., 2018; Zhou et al., 2022), which can be seen in Fig. 7 everywhere. The
V-shape velocity structure can be treated as the projection of the funnel
structure from PPV space to PV plane. Hence, the funnel structures in PPV
space can be an effective probe for the gravitational collapse motions in
molecular cloud.

##  4 Discussion

###  4.1 Characteristic scale $\sim$1 pc

In this work, we have investigated gas motions on sufficiently large scales,
the longest filament $\sim$50 pc, obtaining similar results to previous small-
scale ALMA observations. We fitted a wide range of values of the velocity
gradients, as shown in Fig. 8(a). The velocity gradients are small at large
scales, while they become significantly larger at small scales ($\lesssim 1$
pc), as is the case in the ATOMS survey (Zhou et al., 2022). For both ALMA and
LAsMA data, most of the fitted velocity gradients concentrate on the scales
$\sim 1$ pc, as shown in Fig. 8. This is consistent with the fact that 1 pc is
considered as the characteristic scale of massive clumps (Urquhart et al.,
2018). In Fig. 8, the variation of velocity gradients with scales is
comparable with the expectations from gravitational free-fall, in the sense
that the gradient decreases smoothly with increasing scale. Fig. 10(c) and (d)
show similar results to Fig. 8, but only consider the velocity gradients
associated with ATLASGAL clumps, listed in Table 1. Especially, in Fig. 10(c),
the required central mass in the fitted free-fall model is consistent with the
mass distribution observed in Fig. 10(b).

###  4.2 Evidence for Gravitational Acceleration

Fig. 10(a) displays the probability distribution of velocity gradients
measured around 1 pc (0.8$\sim$1.2 pc), showing that the most frequent
velocity gradient is $\sim 1.6$ km s-1 pc-1. Assuming free-fall,

$\nabla V_{free}=-\frac{d}{dR}\sqrt{\frac{2GM}{R}}=\sqrt{\frac{GM}{2R^{3}}},$
(1)

we estimate the kinematic mass corresponding to 1 pc is $\sim 1190$ M⊙, which
is comparable with the typical mass of clumps in the ATLASGAL survey (Urquhart
et al., 2018). In Fig. 10(b), the peak of the associated clump’s mass
distribution is also around $\sim$ 1000 M⊙. Thus the clump may be a gravity-
dominated collapsing object, also consistent with the survey results that most
Galactic parsec-scale massive clumps seem to be gravitationally bound no
matter how evolved they are (Liu et al., 2016a; Urquhart et al., 2018; Evans
et al., 2021).

Based on Eqn. 1, we can use the values of $M$, $L$ and $\nabla V$ listed in
Table 1 to compare observations to the simple theory. In Fig. 11, the observed
and predicted accelerations do show a clear correlation, strong evidence for
gravity accelerating the gas inflow. The scatter in Fig. 11 is to be expected
because we measure only the projection of the realistic velocity vector and
hence acceleration. For the same reason, we tend to underestimate the observed
acceleration. Moreover, in Table 1, we only selected the ATLASGAL clumps
associated with the intensity peaks of the 13CO (3$-$2) emission (a deviation
within the effective radius of the clump), thus some of them are not the exact
gravitational centers. Despite these caveats, the correlation between
observations and predictions for the denser clumps is quite good. The less
dense clumps tend to lie above the line of equality, suggesting their masses
underestimate the total mass contributing to the gravitational acceleration.

###  4.3 Large-scale inflow driven by gravity

As shown in Fig. 8 and Fig. 10, the velocity gradients fitted to the LAsMA and
ALMA data agree with each other over the range of scales covered by ALMA
observations in the ATOMS survey ($\textless$ 5 pc). Interestingly, the
variations of velocity gradients at small scales ($\textless$ 1 pc), middle
scales ($\sim$ 1-7.5 pc) and large scales ($\textgreater$ 7.5 pc) are
consistent with gravitational free-fall with central masses of $\sim$ 500 M⊙,
$\sim$ 5000 M⊙ and $\sim$ 50000 M⊙. It means the velocity gradients on larger
scales require larger mass to maintain, which is also revealed by the funnel
structure shown in Fig. 9. Indeed, larger masses imply larger scales, that is
to say, the larger scale inflow is driven by the larger scale structure which
may be the gravitational clustering of smaller scale structures (Sec. 4.4),
consistent with the hierarchical structure in molecular clouds and the gas
inflow from large to small scales. Thus the large scale gas inflow may also be
driven by gravity, with the molecular clouds in G333 in the state of globally
gravitational collapse, which is consistent with the argument that these
molecular clouds act as cloud-scale hub-filament structures, see the
description in Sec. 3.4. Moreover, as described in Sec. 3.5, the funnel
structure of the velocity field shown in PPV space also tends to support the
large-scale gravitational collapse of the molecular clouds in G333 complex.

###  4.4 Hierarchical gravitational structure in molecular cloud

The scales of hub-filament structures (cloud-scale) in this work are much
larger than the ones (clump-scale) in Zhou et al. (2022). Each large-scale
hub-filament structure will contain many sub-structures, thus the kinematic
structure will be very complex. Nevertheless, if they are hub-filament
systems, we expect that they have similar kinematic features like the clump-
scale hub-filament system. A hub-filament system must have a common
gravitational center, which will dominate the overall gravitational field,
thus the velocity field. In Fig. 4, although there are many sub-structures in
each sub-region, which cause many velocity and density fluctuations, globally,
the entire large-scale structure still displays the funnel structure. As
discussed in Sec. 3.3 and Sec. 4.3, large-scale velocity gradients always
involve many intensity peaks, and the larger scale inflow is driven by the
larger scale structure, implying the clustering of local small-scale
gravitational structures can act as the gravitational center on larger scale.
To some extent, the funnel structure gives an indication of the gravitational
potential well formed by the clustering.

Both Zhou et al. (2022) and this work find that when the scale is less than
$\sim$1 pc, the gravitationally dominated gas infall can become apparent.
Thus, relatively small-scale hub–filament structures will have better and more
recognizable morphology than large-scale ones due to the strong local
gravitational field. For the large-scale hub–filament structures, background
turbulence is more likely to disturb their morphology due to weaker
gravitational confinement on large scales, thus they are more ”blurred”. Zhou
et al. (2023) suggested for the interpretation of the formation of the large-
scale hub-filament structure in W33 complex, that IRDCs and ATLASGAL clumps in
W33-blue can be treated as small-scale hub–filament structures, W33-blue
itself as a huge hub–filament structure. Thus there may be a self-organization
process from small-scale hub–filament structures to a large-scale one. If the
large scale gravitational center is the clustering of small scale ones,
compared to the latter, large scale gravity center will be more loose, thus
its ability to control the gas motions on large scale is also weaker,
reflected as the loose funnel structure and small velocity gradient, as shown
in this work. In Fig. 6, the small-scale structures indeed have clearer funnel
structures than the large-scale ones, implying the role of gravity in shaping
the funnel structure.

###  4.5 G333 in Context of Galactic Molecular Clouds

The same region of the sky, with slightly larger area
($332.6\degr<l<333.8\degr$ and $-0.8\degr<b<0.1\degr$), has also been studied
by Nguyen et al. (2015), using CO and 13CO (1-0) emission and archival data.
They call the G333 GMC the RCW 106 complex after the associated HII region.
They determined the following properties for the RCW 106 complex: the total
effective diameter is 183 pc, the mass is 5.9${}\times 10^{6}$ $\rm
M_{\odot}$, and the mass surface density is 220 $\rm M_{\odot}$ pc-2. The
virial parameter for the whole complex is 0.35, making it strongly
gravitationally bound; since the cloud is elongated, they include a linear
analysis that also implies a strongly bound cloud. These properties would make
the G333 GMC stand out as one of the largest, most massive complexes in the
Galaxy (cf Fig. 7 of Miville-Deschênes et al. 2017), and one of the few that
is gravitationally bound (cf Fig. 2 of Evans et al. 2021). However, Nguyen et
al. (2015) took the velocity range of the RCW 106 complex as [-80, -40] km
s-1. As discussed in Sec. 3.1, after a careful decomposition of velocity
components, we find the velocity range [-80, -40] km s-1 also includes G331
GMC; thus the mass of the G333 GMC is overestimated in Nguyen et al. (2015).
The catalog of Miville-Deschênes et al. 2017, restricted to the velocity range
of peak3, yields a mass for the G333 cloud traced by CO 1-0 emission of $\sim$
1.7 $\times$ 106 M⊙, with the surface density $\sim$ 120 $\rm M_{\odot}$ pc-2.
Tracers that are biased toward higher volume densities find progressively
smaller masses. Karnik et al. (2001) made a far-infrared (FIR) 150 and 210
$\mu$m dust emission study of the region, using the canonical gas:dust ratio
of 100, they measure a total GMC mass of $\sim$ 1.8 $\times$ 105 M⊙. This is
consistent with the total mass of $\sim$ 1 $\times$ 105 M⊙ estimated by
Mookerjea et al. (2004) from a 1.2-mm cold dust continuum emission map and the
same gas-to-dust ratio. These estimates show the decreasing fraction of mass
in progressively denser regions.

The lower total mass from CO 1-0 based on the catalog of Miville-Deschênes et
al. (2017) still makes the G333 complex one of the most massive in the Milky
Way. Analysis of the RCW HII region yields a stellar mass of 48${}\times
10^{3}$ $\rm M_{\odot}$; assuming a timescale of 0.2 Myr (the lifetime of an
O7 star), Nguyen et al. (2015) derive a star formation rate so far of 0.25
$\rm M_{\odot}$ Myr-1 and an efficiency ($M_{*}/M_{\rm gas}$) of 0.008. The
very high star formation rate led Nguyen et al. (2015) to describe it as a
”mini-starburst”. It is thus atypical of most molecular clouds but an exemplar
of the very small subset that form the majority of stars in the Galaxy.

##  5 Summary

We investigated the gas kinematics of hub-filament structures in the G333
complex using 13CO (3$-$2) emission. The main conclusions are as follows:

1\. The G333 complex includes three main velocity components, which can be
well separated by the average spectra and the velocity distribution in PPV
space. Here the velocity distribution is derived from the decomposition of
13CO (3-2) emission by GAUSSPY+. The velocity components with velocity ranges
[$-$120, $-$76] km s-1 and [$-$76, $-$60] km s-1 represent the main two parts
of G331 GMC, the third velocity component with velocity range [$-$60, $-$35]
km s-1 shows a very extended shell structure from $l=330.6\degr$ to
$l=334\degr$ with G333 GMC and G332 ring as the main parts.

2\. The entire G333 complex seems to have fragmented into several independent
molecular clouds, with some of the sub-structures in these molecular clouds
showing typical hub-filament structures, which also have the typical kinematic
features of hub-filament systems. The broken morphology of some very infrared
bright structures indicates that the feedback is disrupting the star-forming
regions.

3\. We use the integrated intensity maps of 13CO J=3-2 to identify filaments
in G333 complex by FILFINDER algorithm, and extract the velocity and intensity
along the filament skeleton from moment maps. We find a good correlation
between velocity and density fluctuations, and fit the velocity gradients
around the intensity peaks. The change in velocity gradients with the scale
indicates that the morphology of the velocity field in PPV space resembles a
“funnel” structure. The funnel structure can be explained as accelerated
material inflowing towards the central hub and gravitational contraction of
star-forming clouds/clumps. The typical V-shape velocity structure along the
filament skeleton can be treated as the projection of the funnel structure
from PPV space to PV plane. Hence, the funnel structure in PPV space can be an
effective probe for the gravitational collapse motion in molecular cloud.

4\. We have investigated gas motions on sufficiently large scales, the longest
filament $\sim$50 pc, but obtain similar results to small-scale ALMA
observations. The typical velocity gradient corresponding to a one pc scale is
$\sim 1.6$ km s-1 pc-1. Assuming the free-fall model, we can predict the
gravitational acceleration onto each hub. The observed accelerations correlate
well with the predicted ones and have values comparable to the prediction.
Given the fact that we observe only one component of the acceleration and that
the masses of the hubs are uncertain, the result provides strong evidence for
gravitational acceleration of material flowing into hubs from filaments.

5\. The velocity gradients fitted by LAsMA data and ALMA data agree with each
other over the scales covered by ALMA observations in the ATOMS survey
($\textless$ 5 pc). On large scales, we find that the larger scale inflow is
driven by the larger scale structure, indicating the hierarchical structure in
molecular cloud and the gas inflow from large scale to small scale. The large
scale gas inflow is driven by gravity, implying that the molecular clouds in
G333 may be in the state of globally gravitational collapse. The funnel
structure of velocity field shown in PPV space also tends to support the
large-scale gravitational collapse.

6\. Although there are many sub-structures in each molecular cloud, which
cause ubiquitous velocity and density fluctuations, totally, the entire large-
scale structure still displays the loose funnel structure. Thus the hub-
filament structures at different scales may be organized into a hierarchical
macroscopic system through the coupling of gravitational centers at different
scales.

In short, changes of velocity gradient with scale indicate a ”funnel”
structure of the velocity field in PPV space, indicative of a smooth,
continuously increasing velocity gradient from large to small scales, and thus
consistent with gravitational acceleration.

###### 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 work has
been supported by the National Key R&D Program of China (No. 2022YFA1603100).
Tie Liu acknowledges the supports by National Natural Science Foundation of
China (NSFC) through grants No.12122307 and No.12073061, the international
partnership program of Chinese Academy of Sciences through grant
No.114231KYSB20200009, and Shanghai Pujiang Program 20PJ1415500.

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##  Appendix A Supplementary maps

Figure 12: Average spectra of 13CO (3-2) and 12CO (3-2) for the sub-regions
shown in Fig.5. Vertical dashed lines mark the velocity ranges which are used
to make moment maps. Figure 13: The same with Fig.6. Figure 14: The same with
Fig.6. Figure 15: The same with Fig.7.  Figure 16: The same with Fig.7. Figure
17: The same with Fig.7. Figure 18: The same with Fig.7. Figure 19: The same
with Fig.7.

##  Appendix B Appendix A: Noise and offset details

Figure 20: Average spectra in one beam size of 7 pixels for the off-position. Figure 21: Noise maps of 13CO (3-2) and 12CO (3-2) in our observation. Table 3: Offsets of outer 6 pixels relative to the central pixel. pixel |  $\Delta$l (″) |  $\Delta$b (″)
---|---|---
1 | 0 | 0
2 | 34.53 | -23.08
3 | 36.20 | 19.15
4 | 1.74 | 40.03
5 | -35.78 | 20.46
6 | -35.33 | -19.86
7 | -0.17 | -41.67

The receiver is a hexagonal array of six pixels surrounding a central pixel.
The outer array is separated from the central pixel by $\sim 2$ FWHM. Table.3
lists the offsets of outer 6 pixels relative to the central pixel. Fig.3 shows
the average spectra in one beam size of 7 pixels for the off-position. We can
see the emission mainly concentrates on the velocity range [-50,-15] km s-1,
the absorption feature of one sub-map shown in Fig.22 also appears in the same
velocity range, thus we add the emission of off-position in the velocity range
[-50,-15] km s-1 to each sub-map. As shown in Fig.22, the absorption feature
indeed disappears after adding the emission of off-position.

Figure 22: Average spectra of one sub-map G333D55M-7M.