#  Understanding the relative importance of magnetic field, gravity, and
turbulence in star formation at the hub of the giant molecular cloud
G148.24+00.41

Vineet Rawat,1,2 M. R. Samal,1 Chakali Eswaraiah,3 Jia-Wei Wang,4 Davide
Elia,5Sandhyarani Panigrahy,3 A. Zavagno,6,7 R. K. Yadav,8 D. L. Walker,9 J.
Jose,3 D.K. Ojha,10C.P. Zhang,11,12 S. Dutta,4
1Physical Research Laboratory, Navrangpura, Ahmedabad, Gujarat 380009, India
2Indian Institute of Technology Gandhinagar Palaj, Gandhinagar 382355, India
3Indian Institute of Science Education and Research (IISER) Tirupati, Rami
Reddy Nagar, Karakambadi Road, Mangalam (P.O.), Tirupati 517 507, India
4Academia Sinica Institute of Astronomy and Astrophysics, No.1, Sec. 4,
Roosevelt Road, Taipei 10617, Taiwan
5Istituto di Astrofisica e Planetologia Spaziali, INAF, Via Fosso del
Cavaliere 100, I-00133 Roma, Italy
6Aix-Marseille Universite, CNRS, CNES, LAM, 38 rue F. Joliot Curie, F-13388
Marseille Cedex 13, France
7Institut Universitaire de France, Paris, 1 rue Descartes, F-75231 Paris Cedex
05, France
8National Astronomical Research Institute of Thailand (NARIT), Sirindhorn
AstroPark, 260 Moo 4, T. Donkaew, A. Maerim, Chiangmai 50180, Thailand
9Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK
10Department of Astronomy and Astrophysics, Tata Institute of Fundamental
Research, Mumbai 400005, India
11National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, People’s Republic of China
12Guizhou Radio Astronomical Observatory, Guizhou University, Guiyang 550000,
People’s Republic of China
E-mail<EMAIL_ADDRESS>
###### Abstract

The relative importance of magnetic fields, turbulence, and gravity in the
early phases of star formation is still not well understood. We report the
first high-resolution dust polarization observations at 850 $\mu$m around the
most massive clump, located at the hub of the Giant Molecular Cloud
G148.24+00.41, using SCUBA-2/POL-2 at the James Clerk Maxwell Telescope. We
find that the degree of polarization decreases steadily towards the denser
portion of the cloud. Comparing the intensity gradients and local gravity with
the magnetic field orientations, we find that local gravity plays a dominant
role in driving the gas collapse as the magnetic field orientations and
gravity vectors seem to point towards the dense clumps. We also find evidence
of U-shaped magnetic field morphology towards a small-scale elongated
structure associated with the central clump, hinting at converging accretion
flows towards the clump. Our observation has resolved the massive clump into
multiple substructures. We study the magnetic field properties of two regions,
central clump (CC) and northeastern elongated structure (NES). Using the
modified Davis-Chandrasekhar-Fermi method, we determine that the magnetic
field strengths of CC and NES are $\sim$24.0 $\pm$ 6.0 $\mu$G and 20.0 $\pm$
5.0 $\mu$G, respectively. The mass-to-flux ratios are found to be magnetically
transcritical/supercritical, while the Alfv$\acute{\text{e}}$n Mach number
indicates a trans-Alfv$\acute{\text{e}}$nic state in both regions. These
results, along with Virial analysis, suggest that at the hub of G148.24+00.41,
gravitational energy has an edge over magnetic and kinetic energies.

###### keywords:

ISM: clouds; ISM: magnetic fields; polarization; galaxies: star clusters:
general, ISM: molecules; molecular data

††pagerange:  Understanding the relative importance of magnetic field,
gravity, and turbulence in star formation at the hub of the giant molecular
cloud G148.24+00.41–A

##  1 Introduction

Molecular clouds, the enigmatic birthplaces of stars, are now well established
to consist of filamentary structures, revealed by $\it{Herchel}$ observations
(Molinari et al., 2010; André et al., 2013; Zavagno et al., 2023). Within
these immense structures of cold gas and dust, the gravity, turbulence, and
magnetic fields dictate the process of star formation at different scales,
from large scale (cloud and filaments) to small scale (clumps and dense cores)
(Klessen et al., 2000; Ballesteros-Paredes et al., 2007; Federrath, 2015; Tang
et al., 2019; Wang et al., 2020b; Pattle et al., 2022). However, their
relative role during the different stages of cloud evolution is still unclear
and a topic of debate (Li et al., 2014).

The magnetic field exists throughout star-forming molecular clouds across
various scales (see the review article by Pattle et al., 2022) and plays a
crucial role in the formation of molecular clouds and filamentary structures
(Soler et al., 2013; Hennebelle & Inutsuka, 2019). Numerical simulations show
that the strong magnetic fields play an important role in the magnetically
channelled gravitational collapse of clouds (Nakamura & Li, 2008; Gómez et
al., 2018), and can channel turbulent flows along the filaments (Li & Houde,
2008; Soler et al., 2013; Zamora-Avilés et al., 2017), guide the accreting
matter (Seifried & Walch, 2015; Shimajiri et al., 2019), and dynamically
influence the formation of cores along the dense ridges of the filamentary
clouds (e.g. Koch et al., 2014; Zhang et al., 2014; Cox et al., 2016; Pattle
et al., 2017; Soam et al., 2018; Liu et al., 2019; Eswaraiah et al., 2021). At
parsec (or few parsec) scale, the magnetic field typically shows an ordered
structure, mostly aligned with the long axis of the low-density elongated gas
structures such as striations, while in high-density filaments, the magnetic
field lines are preferentially perpendicular to the long axes of filaments
(e.g. Cox et al., 2016; Planck Collaboration et al., 2016; Soler et al., 2017;
Ward-Thompson et al., 2017; Tang et al., 2019; Soam et al., 2019; Doi et al.,
2020). At sub-parsec scales, the magnetic field can be very complex depending
upon the turbulent nature of the magnetic field and stellar feedback (e.g.
Hull et al., 2017; Eswaraiah et al., 2020, 2021). Dust polarization studies at
small scales have revealed a variety of magnetic field morphologies in dense
clumps and cores, and the results suggest that the magnetic field is scale-
dependent and varies with the environment (e.g. Girart et al., 2013; Hull et
al., 2017; Ward-Thompson et al., 2017; Pattle et al., 2018; Eswaraiah et al.,
2020, 2021).

Moreover, it is not clear whether it is turbulence or magnetic field along
with gravity that dominates the star formation process. The role of gravity
has long been recognised as the primary factor driving the collapse of dense
regions and initiating the birth of protostellar cores. On the other hand,
turbulence, the chaotic and ubiquitous motion of gas within these massive
clouds, influences the fragmentation of the collapsing gas. However, the role
of magnetic field, in comparison to turbulence and gravity, is relatively less
understood at various stages of star formation.

The “strong magnetic field” theory of star formation stresses the importance
of the magnetic field in the formation and evolution of clouds and subsequent
structures (Mouschovias et al., 2006; Tan et al., 2013; Hennebelle, 2018).
Conversely, the “weak magnetic field” theory suggests that turbulent flows
control the formation and evolution of clouds and cores, and create the
compressed regions where stars form (Padoan & Nordlund, 2002; Mac Low &
Klessen, 2004; Federrath & Klessen, 2012). Observationally also, in some high-
mass star-forming regions, it has been found that the turbulent energy is more
dominant or comparable to magnetic energy (e.g. Beuther et al., 2010; Girart
et al., 2013; Beuther et al., 2020; Wang et al., 2020b). In contrast, other
studies found magnetic energy to be more dominant than turbulent energy (e.g.
Girart et al., 2009; Beuther et al., 2018; Eswaraiah et al., 2020; Chung et
al., 2023). Therefore, more observational evidence is required at the early
stages of star formation to better constrain the theoretical models and
relative roles of gravity, magnetic field, and turbulence in the star-forming
regions.

Dust polarization observation is a key tool for tracing the plane-of-sky (POS)
magnetic field geometry in star-forming regions. The polarization is caused by
asymmetric dust grains that preferentially align their shorter axis along the
magnetic field direction (Lazarian, 2007; Hoang & Lazarian, 2008). There are
different mechanisms for dust grain alignment (see the review article by
Andersson et al., 2015), but the most widely accepted one being is the
radiative alignment torque (RAT) mechanism (Lazarian, 2007; Hoang & Lazarian,
2014; Andersson et al., 2015).

Located at a distance of $\sim$3.4 kpc, G148.24+00.41 is a massive cloud
having mass $\sim$105 M⊙ and dust temperature $\sim$14.5 K, and it resembles a
hub-filament morphology (Rawat et al., 2023). The cloud is still at the early
stages of its evolution such that the stellar feedback mechanisms like stellar
winds and expanding H ii regions are not yet significant (Rawat et al., 2023),
which makes G148.24+00.41 a potential candidate to study the early stages of
the star and cluster formation process. Fig. 1a shows the 13CO intensity map
of G148.24+00.41, integrated in the velocity range of $-$37.0 km s-1 to
$-$30.0 km s-1, where one can see a bright spot in the centre of the map. This
location corresponds to the most massive clump of the cloud, which is
nomenclatured as C1 in Rawat et al. (2024). Rawat et al. (2024), using CO
molecular data (spatial resolution $\sim$52″or 0.9 pc at distance $\sim$3.4
kpc, velocity resolution $\sim$0.17 km s-1), also found filamentary features
and seven massive clumps in G148.24+00.41. The C1 clump with a mass of
$\sim$2100 M⊙ and an effective size of $\sim$1.8 pc based on C18O data, is
found to be at the hub/central location of the cloud. The filamentary features
attached to the C1 clump can also be seen in $\it{Herschel}$ 250 $\mu$m image,
shown in Fig. 1b (adapted from Rawat et al., 2023). The figure clearly reveals
the hub-filament system morphology of the cloud. Rawat et al. (2023) found
that the clump hosts an infrared cluster, seen in $\it{Spitzer}$ bands, and
hypothesised that the cluster might grow to a richer cluster by accumulating
gas from the extended reservoir. Such hub filamentary systems with the clump
being located at the nexus or junction of these filaments are of particular
interest because these are the sites where cluster formation would take place,
as advocated in simulations and observations (e.g. Naranjo-Romero et al.,
2012; Gómez & Vázquez-Semadeni, 2014; Gómez et al., 2018; Vázquez-Semadeni et
al., 2019; Kumar et al., 2020).

The magnetohydrodynamic simulations suggest that the filamentary converging
flows would impact the magnetic field morphology of the star-forming regions
(Gómez et al., 2018). In G148.24+00.41, Rawat et al. (2024) found converging
gas flows along the filaments towards the hub. So, it would be interesting to
investigate the influence of gas flows on the B-field morphology in the hub of
the cloud. In addition, it is equally important to examine the role of
magnetic fields in the stability of such hub systems, as it is expected that
gravity would play a dominant role in the onset of star formation within such
systems. In this work, we investigate the morphology and strength of the
magnetic field of the C1 clump of G148.24+00.41 for the first time, in order
to understand the relative importance of the magnetic field in comparison to
gravity and turbulence in the overall star formation process of the clump.
This paper is organised as follows. In Section 2, we describe the
observations, data reduction, and other data sets used in this work. In
Section 3, we present the analysis and results related to the magnetic field
morphology, its relative orientation in comparison to intensity gradients and
local gravity, and its strength. We also discuss the dust and gas properties
of the studied region. In Section 4, we discuss the results and compare the
strength of gravity, magnetic field, and turbulence. In Section 5, we
summarize the findings of this work.

##  2 Observations and Data

###  2.1 Dust continuum polarization observations using JCMT SCUBA-2/POL-2

We observed the C1 clump/hub region of G148.24+00.41 with SCUBA-2/POL-2
mounted on the James Clerk Maxwell Telescope (JCMT), a single-dish sub-
millimeter telescope in Mauna Kea, Hawaii, USA. The POL-2 instrument is a
linear polarimetry module (Friberg et al., 2016) for the Submillimetre Common
User Bolometer Array-2 (SCUBA-2), a 10,000 bolometer camera on the JCMT
(Holland et al., 2013). The data was acquired between 2022 November 25 and
2023 January 03 (project code: M22BP055; PI: Vineet Rawat) in the band 2
weather conditions under an atmospheric optical depth at 225 GHz
($\tau_{225}$) of 0.04 to 0.06. The observations were taken in 10 sets with an
integration time of 30 minutes each, resulting in a total integration time of
around 5.5 hr. The POL-2 DAISY scan mode (Holland et al., 2013; Friberg et
al., 2016) was adopted, which generates a map of high signal-to-noise ratio
(SNR) within a central region spanning a diameter of 3′, and the noise level
gradually increases towards the edges of the map. The region is observed in
both the 450 and 850 $\mu$m continuum polarizations simultaneously, with a
resolution of 9$\aas@@fstack{\prime\prime}$6 and
14$\aas@@fstack{\prime\prime}$1, respectively. Due to the low sensitivity of
the 450 $\mu$m data, this paper presents the analyses and results based on
only 850 $\mu$m dust polarization data.

Figure 1: (a) 13CO (J = 1$-$0) intensity map of G148.24+00.41, integrated in
the velocity range $-$37.0 km s-1to $-$30.0 km s-1. (b) The central region
(encompassed by the solid green box in panel-a) of G148.24+00.41 as seen in
$\it{Herschel}$ 250 $\mu$m band, showing the hub-filamentary morphology of the
cloud. The figure is adapted from Rawat et al. (2023), in which the blue
circle shows the JCMT scanned region of diameter $\sim$12′($\sim$12 pc). The
green dashed box marks the central area of the hub, where an infrared cluster
is seen (Rawat et al., 2023), and the cross sign indicates the position of a
massive young stellar object. (c) The 850 $\mu$m Stokes I intensity map of the
central region of G148.24+00.41 mapped by JCMT SCUBA-2/POL-2, along with the
contours of 13CO integrated intensity emission, drawn from 1.5 to 15 K km
s-1with a step size of $\sim$0.96 K km s-1. The rms noise of the 4″pixel-size
Stokes I map is around $\sim$5 mJy beam-1. In panel-c, the yellow ellipse
shows the position of the C1 clump, identified by Rawat et al. (2024) using
C18O data (spatial resolution $\sim$52″). The beam sizes of the 13CO
integrated intensity map, $\it{Herschel}$ 250 $\mu$m map, and JCMT 850 $\mu$m
map are $\sim$52″, 18″, and 14″, respectively, shown as a framed-blue dot at
the bottom left of each panel.

The data reduction was carried out using the
$\it{pol2map}$111http://starlink.eao.hawaii.edu/docs/sc22.htx/sc22.htmlscript
in the SMURF package (Chapin et al., 2013) of Starlink (Currie et al., 2014).
The POL-2 is characterized by linear polarization that produces Stokes I, Q,
and U vector maps. The $\it{Skyloop}$ mode was utilised to minimise the
uncertainty associated with map creation, while the MAPVARS mode was enabled
to asses the total uncertainty from the standard deviation among individual
observations. The details of the data reduction process of POL-2 can be found
in Pattle et al. (2017) and Wang et al. (2019). Finally, the I, Q, and U maps,
along with their variance maps, are used to create a debiased polarization
vector catalogue. The catalogue consists of total intensity (I), stokes
vectors (Q and U), polarization intensity (PI), polarization fraction (P),
polarization angle ($\theta_{P}$), and their associated uncertainties
($\delta$I, $\delta$Q, $\delta$U, $\delta$PI, $\delta$P, and
$\delta$$\theta_{P}$, respectively).

The I, Q, and U maps are produced with 4′′pixel size, while the polarization
catalogue is binned to 12′′, for better sensitivity. A flux calibration factor
of 668.25 Jy beam-1 pW-1 is used for 850 $\mu$m Stokes I, Q, and U map to
convert them from pW to mJy/beam and to account for the flux-loss due to POL-2
insertion into the telescope. This calibration factor comprises 495 Jy/beam/pW
for reductions using 4′′pixels of SCUBA-2, multiplied by the standard 1.35
factor for POL-2 losses (Mairs et al., 2021).

The polarised intensity is defined to be positive, so the uncertainties of the
Q and U Stokes vector would bias the polarised intensities towards larger
values (Vaillancourt, 2006; Kwon et al., 2018). The debiased polarization
intensity and its uncertainty are calculated as

$PI=\sqrt{Q^{2}+U^{2}-0.5(\delta Q^{2}+\delta U^{2})}$ (1)

and

$\delta PI=\sqrt{\frac{(Q^{2}\delta Q^{2}+U^{2}\delta
U^{2})}{(Q^{2}+U^{2})}},$

respectively. The debiased polarization fraction and its uncertainty are then
calculated as

$P=\frac{PI}{I}$ (2)

and

$\delta P=\sqrt{\frac{\delta PI^{2}}{I^{2}}+\frac{\delta
I^{2}(Q^{2}+U^{2})}{I^{4}}},$

respectively. The polarization angle and its uncertainty are calculated as

$\theta_{P}=\frac{1}{2}\tan^{-1}{\left(\frac{U}{Q}\right)}$ (3)

and

$\delta_{\theta_{P}}=\frac{1}{2}\sqrt{\frac{(U^{2}\delta Q^{2}+Q^{2}\delta
U^{2})}{(Q^{2}+U^{2})^{2}}},$

respectively. The polarization angle increases from the north toward the east,
following the IAU convention. The mean rms noises in the Stokes I, Q, U, and
PI measurements with 12$\arcsec$ bin size are 1.4, 1.1, 1.1, and 1.1 mJy
beam-1, respectively. Following the standard convention, for magnetic field,
hereafter, B-field orientations, the polarization angles are rotated by 90
degrees.

###  2.2 Molecular line data from PMO

We used the 13CO (J = 1–0) and C18O (J = 1–0) molecular line data, which were
taken as a part of the Milky Way Imaging Scroll Painting survey (MWISP; Su et
al., 2019) using the 13.7-m radio telescope at Purple Mount Observatory (PMO).
The observation of CO isotopologues was taken simultaneously using a 3
$\times$ 3 beam sideband-separating Superconducting Spectroscopic Array
Receiver (SSAR) system (Shan et al., 2012) and using the position-switch on-
the-fly mode. The spatial resolution (Half Power Beam Width; HPBW) of 13CO and
C18O is around $\sim$52″($\sim$0.9 pc at 3.4 kpc). And the spectral resolution
of 13CO and C18O is $\sim$0.17 km s-1with a sensitivity of $\sim$0.3 K (for
details, see Su et al., 2019).

##  3 Analyses and Results

Fig. 1c shows the Stokes I map of the region, where the location of the C1
clump is also shown. From the figure, it can be seen that the 850 $\mu$m JCMT
data (beam size $\sim$14″) has resolved multiple sub-structures in the central
region of the cloud. We found sub-structures like a central clump, a clump
located on the western side, and a prominent elongated structure on the
northeastern side of the central clump. In addition to these prominent sub-
structures, a few compact structures are also visible in the image.

###  3.1 B-field morphology

In order to select the significant polarization detections, we set the
following criteria for selecting data: I/$\delta$I $>$ 10, P/$\delta$P $>$ 2,
and P $<$ 30%. By doing this, we got 69 polarization measurements in our
target region. The $P$ values range from $\sim$2 % to $\sim$29 % with a mean
and standard deviation around $\sim$11 $\pm$ 8 %. The B-field orientations are
widely distributed, ranging from $\sim$6∘ to 180∘ with a mean and standard
deviation around $\sim$91∘$\pm$ 48∘, suggesting a complex B-field morphology
in the region. The mean uncertainties in polarization fraction and
polarization angle are $\sim$3.5% and $\sim$9∘, respectively. Fig. 2a and b
show the distribution of polarization vectors and B-field orientations,
respectively, over the 850 $\mu$m Stokes I dust continuum emission map of the
region. The contour levels in the map are shown above 3$\sigma$ from the
background, where $\sigma$ is the mean rms noise (5 mJy beam-1) of the Stokes
I map.

Figure 2: (a) Polarization vector map with lengths proportional to
polarization fraction, and (b) magnetic field orientation map with fixed
lengths. The background is the Stokes I image at 850 $\mu$m, and the contour
levels are drawn at 3$\sigma$ above the rms noise level of 5 mJy beam-1,
starting from 15 mJy beam-1to 300 mJy beam-1. The segments shown are binned to
a 12″pixel grid and correspond to polarization data with I/$\delta$I $>$ 10
and P/$\delta$P $>$ 2\. The lightcyan and green vectors in panel-b show the
measurements with 2 $<$ P/$\delta$P $<$ 3 and P/$\delta$P $>$ 3, respectively.
The regions used for B-field calculation are also shown in panel-b by a white
circle and yellow ellipse for the CC and NES, respectively.

In this work, based on 850 $\mu$m Stokes I intensity and magnetic field
orientations, we defined the central clump, clump located on the western side,
and northeastern elongated structure as CC, WC, and NES, respectively, as
marked in Fig. 2b. The approximate extents of these regions are defined by
considering the outermost closed contours of the 850 $\mu$m Stokes I map. From
Fig. 2b, it can be seen that the B-field orientations in the CC are mostly
oriented along the east-west direction (PA $\sim$90∘), while some of them are
at smaller position angles. There exist mixed B-field orientations in the NES
region, some in the low-density area are nearly perpendicular to the major
axis of the NES, while some closer to the CC are parallel to it. In the WC,
most of the B-fields are converging towards the centre, aligned along the
southeast direction, which may be influenced by gravity (see Section 3.3.2 and
4.1). Overall, the B-field morphology around the central region of
G148.24+00.41 is complex, which is probably due to hierarchical fragmentation
and a network of filamentary flows towards the hub, as found by Rawat et al.
(2023) and Rawat et al. (2024).

Fig. 3a shows the histogram of the B-field orientations in the central region
of G148.24+00.41, which is broadly distributed. The CC is showing mixed
morphology, having two peaks, one at $\sim$38∘(i.e. with position angles close
to northeast), and the second is at $\sim$80∘ (i.e. with position angles
parallel to east). The NES shows a flat distribution over a broad range, but a
slightly higher distribution at a position angle around $\sim$180∘. The WC,
though, has a small number of segments, shows a peak around $\sim$125∘, i.e.
mostly in the southeast direction. All these orientations are also clearly
evident in Fig. 3b, which shows the distribution of B-field position angles.
From Fig. 3b, it can be seen that the B-field angles change roughly from
$\sim$180∘ to $\sim$70∘ while going from the elongated structure towards the
central clump.

Figure 3: (a) Histogram of B-field position angles for the whole region, CC,
NES, and WC. (b) Distribution of B-field position angles over the contours of
850 $\mu$m Stokes I map. The contour levels are the same as in Fig. 2.

###  3.2 Variation of polarization fraction: depolarization effect

Fig. 4 shows the distribution of polarization fraction over the contours of
Stokes I emission. From the figure, it can be seen that the polarization
fraction is lower in the high-intensity regions compared to the low-intensity
regions, which shows the decreasing trend of polarization fraction with the
total intensity, known as depolarization. The depolarization effect has been
reported in several studies (Girart et al., 2006; Tang et al., 2013; Sadavoy
et al., 2018; Soam et al., 2018; Liu et al., 2019, 2020), and is mainly
explained by the inefficient radiative alignment of dust grains in high-
density regions or integration effect across a complex magnetic fields. In
high-density regions, the radiative alignment torques decrease due to the
attenuation of interstellar radiation that results in poor grain alignment,
and hence the decrease in polarization fraction. However, the grain
characteristics like size, shape, composition, and grain growth can also
affect the dust grain alignment. The turbulent nature of the B-field and
unresolved complex and tangled B-fields within the JCMT beam, being averaged
across the beam, can also give low dust polarization (Planck Collaboration et
al., 2016, 2020). We want to point out that in the hub/C1 clump of
G148.24+00.41, supersonic non-thermal motions have been found (sonic Mach
number = 3$-$4.4, see Rawat et al. (2024)). Therefore, the turbulent nature of
the B-field can also be the cause of depolarization in our target.

The relation between polarization fraction and intensity is expected to follow
a power-law, $P\propto I^{-\alpha}$ (Whittet et al., 2008). A range of
$\alpha$ values has been found in molecular clouds from $\sim$0.5 to 1 (Chung
et al., 2023, and references therein). The $\alpha$ value is often used as an
indicator of the dust grain alignment efficiency. When $\alpha$ = 0, it
implies a constant grain alignment efficiency, $\alpha$ = 0.5 implies that the
alignment decreases linearly with the increasing optical depth, while $\alpha$
= 1 implies an alignment limited to the outer regions of the cloud, and at
higher density, there is no preferred alignment of grains relative to the
magnetic field (Whittet et al., 2008). We fit the P-I relation with a single
power-law (weighted-fit) and found an index, $\alpha$ = 0.95 $\pm$ 0.04, which
shows that the dust grain alignment efficiency is decreasing in the central
dense region of G148.24+00.41. However, Pattle et al. (2019) shows that the
conventional approach of fitting a single power-law over the polarization
measurements debiased with Gaussian noise is only applicable above a high SNR
cut. But in low polarized intensity regions, a high SNR would discard more
data, and therefore, the $\alpha$ index will be overestimated (Pattle et al.,
2019; Chung et al., 2023; Lin et al., 2023).Wang et al. (2019) also found that
the value of $\alpha$ depends upon the cut of SNR and tends to $-1$ if we put
a constraint on P/$\delta$P. Hence, to obtain the true value of the $\alpha$
index, it is recommended to use the non-debiased polarization measurements,
including both the low and high SNR data, and should not put constraints on
P/$\delta$P (Pattle et al., 2019; Wang et al., 2019).

We followed the Bayesian method of Wang et al. (2019) to determine the true
value of $\alpha$ by using the non-debiased polarization data, which follows
well the Rice distribution (see Pattle et al., 2019; Wang et al., 2019, and
references therein)

$F(P|P_{0})=\frac{P}{\sigma_{P}^{2}}\exp{\left[-\frac{P^{2}+P_{0}^{2}}{2\sigma_{p}^{2}}\right]I_{0}\left(\frac{PP_{0}}{\sigma_{P}^{2}}\right)},$
(4)

where $P$ and $P_{\rm{0}}$ are the observed and true polarization fraction,
respectively, $\sigma_{P}$ is the uncertainty in the polarization fraction,
and $I_{0}$ is the zeroth-order modified Bessel function. We used non-debiased
data with an SNR of 2 (i.e. I/$\delta$I $>$ 2) to include most of the data
points and used the power-law model, $P_{\rm{0}}=\beta I^{-\alpha}$, with
uncertainty $\sigma_{P}=\sigma_{\rm{QU}}/I$, where $\sigma_{\rm{QU}}$
represents the rms noise in Q and U measurements, $I$ is the total observed
intensity, and $\alpha$, $\beta$, and $\sigma_{\rm{QU}}$ are the free model
parameters. We employed the Markov Chain Monte Carlo method and used a python
package PyMC3 (Salvatier et al., 2016) to fit the Rician model to the data. We
set the uniform priors on all three model parameters: 0 $<$ $\alpha$ $<$ 2, 0
$<$ $\beta$ $<$100, and 0 $<$ $\sigma_{\rm{QU}}$ $<$ 5, and otherwise a value
of 0 for all the parameters. The details of the methodology are given in Wang
et al. (2019). Fig. 5 shows the derived posterior of each model parameter,
along with their 95% highest density interval (HDI), depicting the uncertainty
in each parameter. The mean values of $\alpha$, $\beta$, and $\sigma_{QU}$ are
$\sim$0.6, 37, and 1.7, respectively. The $\alpha$ value derived from the non-
debiased polarization data is smaller than the $\alpha$ value derived from the
conventional approach (i.e. $\sim$0.95). Fig. 6 shows the non-debiased
polarization fraction versus total intensity plot with 50%, 68%, and 95%
confidence interval. The derived $\alpha$ value suggests that the grain
alignment is still persisting in the hub of G148.24+00.41, but with decreasing
efficiency in the dense regions.

Figure 4: Distribution of dust polarization fraction (P in %) over the
contours of 850 $\mu$m Stokes I map. The contour levels are the same as in
Fig. 2. Figure 5: The probability distribution function of the fitted model
parameters derived using the Bayesian method over the non-debiased
polarization data. The mean values of the parameters are shown along with the
95% HDI intervals to represent the uncertainties. The 95% confidence intervals
are marked as horizontal bars. Figure 6: Non-debiased polarization fraction
versus total intensity. The blue line shows the mean, and the coloured regions
show the 95%, 68%, and 50% confidence limits, as predicted by the posteriors
of $\alpha$ = 0.6, $\beta$ = 37, and $\sigma_{QU}$ = 1.7.

###  3.3 Relative orientations of magnetic fields, intensity gradients, and
local gravity

In star-forming regions, various forces interact, like gravity, magnetic
field, and turbulence, which shape the geometry of these regions and drive the
star-formation process (Ballesteros-Paredes et al., 2007; Koch et al., 2012a;
Pattle et al., 2022). Along with the overall strength of these individual
factors for an entire region (discussed in Section 4.2), it is also important
to investigate their localised relative orientations in the map, as it would
give insight into the localised effect of these factors (Koch et al., 2012a,
b, 2013; Tang et al., 2019; Liu et al., 2020; Wang et al., 2020b). Koch et al.
(2012a) developed a technique, "the polarization-intensity gradient-local
gravity," using Magnetohydrodynamics (MHD) force equations to measure the
local magnetic field strengths. Following the approach of Koch et al. (2012a,
b), we find out the angular difference between magnetic field, intensity
gradient, and local gravity, and discuss their relative importance at
different positions.

####  3.3.1 Intensity gradient versus magnetic field

We used the 850 $\mu$m dust continuum intensities of all pixels in the map to
determine the directions of intensity gradients. All the pixels that have
values above a certain threshold (i.e. 3$\sigma$ above the mean rms noise in
the Stokes I map) are considered for computing the direction of gradients,
except those that exist at the edges. For a pixel at position ($\alpha_{i}$,
$\delta_{j}$), the position angle (${\theta^{\prime}}_{IG}$) of the intensity
gradient is calculated as

${\theta^{\prime}}_{IG}=(180/\pi)\times\arctan{\left[\frac{\Delta
I_{\delta_{j}}}{\Delta I_{\alpha_{i}}}\right]},$ (5)

where $\Delta I_{\delta_{j}}=I_{\delta_{j+1}}-I_{\delta_{j-1}}$ and $\Delta
I_{\alpha_{i}}=I_{\alpha_{i+1}}-I_{\alpha_{i-1}}$.

The ${\theta^{\prime}}_{IG}$ values are then converted to gradient directions
($\theta_{IG}$) by doing the quadrant corrections, i.e. arranging the angles
between 0∘ and 360∘ (for details, see Eswaraiah et al., 2020). In order to
plot the gradient orientations instead of directions, we folded the
$\theta_{IG}$ between 0∘ and 180∘. For comparison of the gradient orientations
($\theta_{IG}$) with the B-field orientations ($\theta_{B}$), we took the
average of all the $\theta_{IG}$ values within a diameter of
$\sim$14″(corresponds to the beam size of JCMT at 850 $\mu$m) around each
B-field position. We calculated the circular mean to get the average of
intensity gradients. In this approach, the angles are treated as unit vectors,
which is adequate for broad distributions and ambiguity in angles (Tang et
al., 2019). Fig. 7a shows the orientations of intensity gradients relative to
B-field orientations over the 850 $\mu$m Stokes I map.

Figure 7: (a) The orientations of the B-fields (green segments) and intensity
gradients (red segments) are overlaid on the 850 $\mu$m Stokes I map. (b) The
distribution of the offset between the position angles of the B-fields and
intensity gradients, i.e., $\Delta\theta_{B,IG}$ =
$|(\theta_{B}-\theta_{IG})|$ over the contours of 850 $\mu$m Stokes I map. The
contour levels are same as in Fig. 2.

We find that the local differences between these orientations are overall
widely distributed. However, it can be seen that the intensity gradients are
mostly aligned with the B-fields in the CC and WC regions, while the
differences in orientations are relatively higher in the NES region. In the
observed central region of G148.24+00.41, we found a moderate correlation
between $\theta_{B}$ and $\theta_{IG}$, in the CC and WC (see Figs. 7a, b).
Fig. 7b shows the distribution of $\Delta\theta_{B,IG}$ =
$|(\theta_{B}-\theta_{IG})|$ over the contours of 850 $\mu$m Stokes I map. The
$\Delta\theta_{B,IG}$ values lie between 0∘ and 90∘ after considering them to
be the acute angle. A stronger correlation between $\theta_{B}$ and
$\theta_{IG}$ tells that the material is following the B-field lines (Koch et
al., 2013; Tang et al., 2019, more discussion in Section 4.1).

####  3.3.2 Local gravitational field versus magnetic field

In order to investigate the localised effect of gravity on the B-field
morphology of the structures, we used the 850 $\mu$m dust continuum map, to
compute the projected gravitational field vectors. The gravitational force at
any pixel ($F_{G,i}$) is the vector sum of the forces from all the surrounding
pixels and is expressed as (Wang et al., 2020b).

$F_{G,i}=kI_{i}\sum_{j=1}^{N}\frac{I_{j}}{{r^{2}}_{ij}}\hat{r},$ (6)

where $I_{i}$ and $I_{j}$ are the intensity of the pixel at position $i$ and
$j$, respectively, and $k$ is the term that takes care of the conversion of
emission to total column density and also includes the gravitational constant.
$N$ is the total number of pixels within the selected area, ${r}_{ij}$ is the
projected distance between the pixels i and j, and $\hat{r}$ is the unit
vector. Considering only the directions of the local gravitational forces, we
take $k$ to be 1 in the above equation by assuming that the spatial
distribution of dust will be analogous to the spatial distribution of mass.
Similar to the intensity gradient map, we selected those pixels which have
intensity values above the threshold, and obtained the local gravity vectors
($\theta_{LG}$) at each B-field position, by taking an average of all vectors
within the 14″beam size. Fig. 8a shows the orientations of local gravity
vectors relative to B-field orientations over the 850 $\mu$m Stokes I map.
From the figure, it can be seen that similar to intensity gradients, the local
gravity vectors are also mostly aligned with the B-fields in the CC and WC
region, whereas they deviate from the B-fields in the NES region. Fig. 8b
shows the distribution of $\Delta\theta_{B,LG}$ = $|(\theta_{B}-\theta_{LG})|$
values over the contours of 850 $\mu$m Stokes I emission, which are treated to
be acute angles.

Figure 8: (a) The orientations of the B-fields (green segments) and local
gravity (magenta vectors) are overlaid on the 850 $\mu$m Stokes I map. (b) The
distribution of the offset between the position angles of the B-fields and
local gravity, i.e., $\Delta\theta_{B,LG}$ = $|(\theta_{B}-\theta_{LG})|$ over
the 850 $\mu$m Stokes I map. The contour levels are same as in Fig. 2.

Figure 9: Distribution of difference in position angles of (a) magnetic field
($\theta_{B}$) and intensity gradient ($\theta_{IG}$), (b) magnetic field
($\theta_{B}$) and local gravity ($\theta_{LG}$), and (c) intensity gradient
($\theta_{IG}$) and local gravity ($\theta_{LG}$). The red, blue, and green
histograms show the difference in position angles within the CC, NES, and WC
regions, respectively.

The relative orientations of intensity gradient and local gravity have a
similar distribution like that of $\Delta\theta_{B,IG}$ and
$\Delta\theta_{B,LG}$, as shown in Fig. 14 in appendix A. Figs. 9a-c shows the
histogram distribution of the relative position angle differences of B-field,
intensity gradients, and local gravity, i.e. $\Delta\theta_{B,IG}$ =
$|(\theta_{B}-\theta_{IG})|$, $\Delta\theta_{B,LG}$ =
$|(\theta_{B}-\theta_{LG})|$, and $\Delta\theta_{IG,LG}$ =
$|(\theta_{IG}-\theta_{LG})|$. From the figure, it can be seen that the
differences in the offset angles are mostly distributed towards the smaller
angles. The median of $\Delta\theta_{B,IG}$, $\Delta\theta_{B,LG}$, and
$\Delta\theta_{IG,LG}$ is 34∘, 32∘, and 30∘ with median absolute deviation of
22∘, 18∘, and 15∘, respectively. The higher angular deviations in the
histograms are primarily due to position angles in the elongated structures on
the northeastern side, as well as from some structures located south of the
CC.

As discussed previously, Koch et al. (2012a, b) developed a "polarization-
intensity gradient method" that can estimate the local field-to-gravity force
ratio $\Sigma_{B}$. This method is based on the assumption that the emission
intensity gradients reflect the direction of matter flow due to the combined
influences of magnetic pressure force and gravitational force. Using the MHD
force equations and geometrically solving them by incorporating the angle
between the magnetic field and intensity gradient ($\Delta_{B,IG}$), and
between intensity gradient and local gravity ($\Delta_{IG,LG}$), the magnetic
field ($F_{B}$)-to-gravity force ($F_{G}$) ratio can be obtained as

$\Sigma_{B}=\frac{sin(\Delta_{B,IG})}{sin(90-\Delta_{IG,LG})}=\frac{F_{B}}{|F_{G}|}.$
(7)

In the above equation, the hydrostatic gas pressure is assumed to be
negligible. Fig. 10 shows the $\Sigma_{B}$ distribution plot over the Stokes
850 $\mu$m intensity map. From the figure, it can be seen that $\Sigma_{B}$ is
mostly $\leq$ 1, with a median around $\sim$0.6, which shows that the magnetic
field is not solely enough to balance the gravitational force (Koch et al.,
2012a). This implies that gravity dominates over the magnetic field to govern
the gas motion towards the centre. However, we note that the POL-2 images
generally filter out large-scale structures, and so here, the intensity
gradient only traces the local structure on a 4″pixel-scale. Therefore, to
check the effect of large-scale structures on the intensity gradient and local
gravity, we generated similar maps from $\it{Herschel}$ 250 $\mu$m image of
G148.24+00.41and found that the maps are comparable with the JCMT maps. The
use of 250 $\mu$m map is an optimal choice because compared to
$\it{Herschel^{\prime}s}$ longer wavelength (i.e. 350 and 500 $\mu$m) bands,
its resolution ($\sim$18″) is comparable to the resolution of the JCMT 850
$\mu$m (14″) map and also it is a better tracer of cold dust compared to
$\it{Herschel^{\prime}s}$ shorter wavelength (i.e. 70 and 160 $\mu$m) bands.

Figure 10: $\Sigma_{B}$ distribution over the contours of 850 $\mu$m Stokes I
map. The contour levels are same as in Fig. 2.

Due to the relatively low number of B-field segments in the WC region, we
focused our further analysis, like the study of structure, dust properties,
and B-field strength calculation, towards the CC and NES regions.

###  3.4 Column and number densities

We calculate the molecular hydrogen column density using the 850 $\mu$m dust
continuum emission. Assuming the dust emission to be optically thin, the
column density can be calculated using the relation (Kauffmann et al., 2008),

$N(H_{2})=2.02\times
10^{20}\hbox{cm${}^{-2}$~{}}\left(e^{1.439\left(\frac{\lambda}{\mathrm{mm}}\right)^{-1}\left(\frac{T_{d}}{10K}\right)^{-1}}-1\right)\\\
\times\left(\frac{\kappa_{\nu}}{\mathrm{0.01cm^{2}g^{-1}}}\right)^{-1}\left(\frac{S_{\nu}}{\mathrm{mJy\
\
beam^{-1}}}\right)\left(\frac{\theta_{HPBW}}{10\arcsec}\right)^{-2}\left(\frac{\lambda}{\mathrm{mm}}\right)^{3},$
(8)

where $S_{\nu}$ is the flux density in Jy at frequency $\nu$, $\lambda$ is the
wavelength (0.85 mm), $T_{d}$ is the mean dust temperature in K, dust opacity
$\kappa_{\nu}$ = 0.1($\nu$/1 THz)β = 0.0125 cm$\rm{{}^{2}g^{-1}}$ for $\nu$ =
0.353 THz and dust opacity index, $\beta$ = 2 (Battersby et al., 2011;
Deharveng et al., 2012), and $\theta_{\mathrm{HPBW}}$ is the beam size (14″at
850 $\mu$m).

The mean $T_{d}$ is taken from the $\it{Herschel}$ Hi-GAL survey (Molinari et
al., 2010) dust temperature map (Schisano et al., 2020) of the G148.24+00.41
region. Within the boundaries of CC and NES, the mean $T_{d}$ is around
$\sim$16.8 K and 13.0 K, respectively. The total column density, $\sum
N(H_{2})$, for CC and NES, are found to be (3.4 $\pm$ 1.3) $\times$ 1024 cm-2
and (1.9 $\pm$ 0.7) $\times$ 1024 cm-2, respectively. Then, assuming the
spherical geometry for CC, its number density ($n_{H_{2}}$) can be estimated
using the relation

$n_{H_{2}}=\frac{M}{V\mu m_{H}}=\frac{\sum N(H_{2})\times
A_{pixel}}{\frac{4}{3}\pi R_{eff}^{3}},$ (9)

where $M$ and $V$ are the mass and volume of the region, respectively, $\mu$
is the mean molecular weight assumed to be 2.8 (Kauffmann et al., 2008),
$m_{H}$ is the mass of hydrogen, $A_{pixel}$ is the pixel area, and $R_{eff}$
is the effective radius. The $R_{eff}$ for CC is calculated as
$(Area/\pi)^{0.5}$, and is around $\sim$0.9 $\pm$ 0.1 pc. Though NES is
assumed as an elliptical structure with a semi-minor axis, $r_{1}$ = 0.35
$\pm$ 0.03 pc and a semi-major axis, $r_{2}$ = 1.10 $\pm$ 0.09 pc (see Fig.
2a), in 3-dimension, this elongated structure could be better described by a
cylindrical geometry. Therefore, to calculate the number density of NES, we
adopted its radius ($r$) and length ($L$) to be $r_{1}$ and 2$r_{2}$,
respectively. Under this approximation, we estimated the $n_{H_{2}}$ for NES
by using $V=\pi r^{2}L$ in equation 9. The gas mass within the regions is
estimated from the total integrated molecular hydrogen column density, using
equation 1 given in Rawat et al. (2023). The total column density,
$n_{H_{2}}$, and the mass of the regions are given in Table 1. The
uncertainties in the estimated cloud parameters are mainly due to uncertainty
in the gas-to-dust ratio (23%), the dust opacity index (30%), and the distance
of the cloud (9%) (Rawat et al., 2023, and references therein).

###  3.5 Velocity dispersion

We used C18O molecular line data to determine the non-thermal velocity
dispersion along the line-of-sight. Rawat et al. (2024) found that in
comparison to 12CO and 13CO , C18O emission is optically thin in the
G148.24+00.41 cloud. Fig. 11 shows the C18O spectra averaged within the
boundary of the two regions, CC and NES. The Gaussian fitting over the spectra
gives the mean velocity as $-$ 34.15 $\pm$ 0.04 km s-1and $-$ 33.80 $\pm$ 0.04
km s-1, and velocity dispersion ($\sigma_{obs}$) as 0.69 $\pm$ 0.05 km s-1and
0.41 $\pm$ 0.03 km s-1, for CC and NES, respectively.

The non-thermal velocity dispersion ($\sigma_{nt}$) can be calculated using
the relation, $\sigma_{nt}=\sqrt{\sigma_{obs}^{2}-\sigma_{th}^{2}}$, where
$\sigma_{th}$ is the thermal velocity dispersion, defined as
$\sqrt{\frac{K_{B}T_{k}}{M_{\hbox{C${}^{18}$O}~{}}}}$. Here, $K_{B}$ is the
Boltzmann constant, $T_{k}$ is the gas kinetic temperature, and
$M_{\hbox{C${}^{18}$O}~{}}$ is the mass of C18O molecule (30 amu). We have
used the excitation temperature map from Rawat et al. (2024) to get the
approximate values of $T_{k}$, as 11.4 K and 10.3 K for CC and NES,
respectively. The estimated $\sigma_{th}$ for the regions is $\sim$ 0.06 km
s-1and 0.05 km s-1, respectively, and hence negligible, leading $\sigma_{nt}$
$\sim$ $\sigma_{obs}$. This is an indication of the presence of turbulence in
CC and NES, which has also been reported in Rawat et al. (2024) for the C1
clump.

Figure 11: C18O average spectral profile for (a) CC (central clump) and (b) NES (northeastern elongated structure). Table 1: Parameters estimated for CC and NES. No |  Parameter |  Unit |  CC |  NES
---|---|---|---|---
1 |  Effective radius ($R_{eff}$) |  pc |  0.9 $\pm$ 0.1 |  semi-minor axis ($r_{1}$) = 0.35 $\pm$ 0.03, semi-major axis ($r_{2}$) = 1.10 $\pm$ 0.09
2 |  Mean column density ($N(H_{2})$) |  cm-2 |  (5.8 $\pm$ 2.2)$\times$ 1021 |  (7.1 $\pm$ 2.7) $\times$ 1021
3 |  Number density ($n_{H_{2}}$) |  cm-3 |  1560 $\pm$ 780 |  3290 $\pm$ 1645
4 |  Mass ($M$) |  M⊙ |  330 $\pm$ 148 |  188 $\pm$ 85
5 |  Mean dust temperature ($T_{d}$) |  K |  16.8 |  13.0
6 |  Observed velocity dispersion ($\sigma_{obs}$) |  km s-1 |  0.69 $\pm$ 0.05 |  0.41 $\pm$ 0.03
7 |  Thermal velocity dispersion ($\sigma_{th}$) |  km s-1 |  0.06 |  0.05
8 |  Non-thermal velocity dispersion ($\sigma_{nt}$) |  km s-1 |  0.69 $\pm$ 0.05 |  0.41 $\pm$ 0.03
| Structure function analysis |
1 |  Turbulent-to-ordered magnetic field energy ratio $\left(\frac{\braket{\delta B^{2}}^{1/2}}{B_{0}}\right)$ |  |  0.43 $\pm$ 0.01 |  0.46 $\pm$ 0.04
2 |  Angular dispersion ($\sigma_{\theta}$) |  degrees |  22.7 $\pm$ 0.6 |  23.9 $\pm$ 1.5
3 |  Plane-of-sky magnetic field strength ($B_{pos}$) |  $\mathrm{\mu}$G |  24.0 $\pm$ 6.0 |  20.0 $\pm$ 5.0
4 |  Mass-to-flux ratio ($\lambda_{B}$) |  |  1.8 $\pm$ 0.8 |  2.7 $\pm$ 1.2
5 |  Alfv$\acute{\text{e}}$n velocity ($V_{A}$) |  km s-1 |  1.0 $\pm$ 0.4 |  0.6 $\pm$ 0.2
6 |  Alfv$\acute{\text{e}}$n mach number ($\mathcal{M}_{A}$) |  |  1.2 $\pm$ 0.4 |  1.2 $\pm$ 0.4
7 |  Magnetic pressure ($P\rm{{}_{B}}$) |  $\rm{dyne\,cm^{-2}}$ |  (3.7 $\pm$ 1.9) $\times$ 10-11 |  (2.6 $\pm$ 1.3) $\times$ 10-11
8 |  Turbulent pressure ($P\rm{{}_{turb}}$) |  $\rm{dyne\,cm^{-2}}$ |  (5.2 $\pm$ 2.7) $\times$ 10-11 |  (2.5 $\pm$ 1.3) $\times$ 10-11
| Virial balance |
1 |  Kinetic energy ($E_{K}$) |  J |  (4.7 $\pm$ 2.2) $\times$ 1038 |  (6.3 $\pm$ 3.0) $\times$ 1037
2 |  Magnetic energy ($E_{B}$) |  J |  (3.3 $\pm$ 3.0) $\times$ 1038 |  (7.0 $\pm$ 5.0) $\times$ 1037
3 |  Gravitational energy ($E_{G}$) |  J |  (10.4 $\pm$ 9.0) $\times$ 1038 |  (14 $\pm$ 12) $\times$ 1037
4 |  Kinetic virial parameter ($\alpha_{vir,k}$) |  |  0.9 $\pm$ 0.4 |  0.9 $\pm$ 0.4
5 |  Total virial parameter ($\alpha_{vir,tot}$) |  |  1.2 $\pm$ 0.6 |  1.4 $\pm$ 0.7

###  3.6 Magnetic field strength

Davis (1951) and Chandrasekhar & Fermi (1953) proposed a method to estimate
the plane-of-sky component of the magnetic field ($B_{pos}$), known as the
Davis-Chandrasekhar-Fermi (DCF) method, which is based on the assumption that
the turbulence-induced Alfv$\acute{\text{e}}$n waves perturb the ordered
B-field structure. Therefore, there will be a distorted component of the
B-field that would appear as an irregular scatter in polarization angles in
comparison to those that are produced by large-scale ordered B-field. Thus,
the DCF method implies that the ratio of turbulent ($\delta B$) to ordered
B-field ($B_{0}$) is proportional to the ratio of non-thermal velocity
dispersion ($\sigma_{nt}$) to Alfv$\acute{\text{e}}$n velocity
($V_{A}=B_{0}/\sqrt{4\pi\rho}$, $\rho$ is the gas mass density), i.e.
$\frac{\delta B}{B_{0}}=\frac{\sigma_{nt}}{V_{A}}$. Also, the dispersion in
the B-field position angles ($\sigma_{\theta}$) about the large-scale ordered
B-field is assumed as $\sigma_{\theta}=\frac{\delta B}{B_{0}}$. Using these
relations, the plane-of-sky magnetic field component, $B_{pos}$ can be
estimated as

$B_{pos}=Q\sqrt{4\pi\rho}\,\frac{\sigma_{nt}}{\sigma_{\theta}},$ (10)

where $Q$ is the correction factor for the line-of-sight and beam-integration
effects (Ostriker et al., 2001). The studies show that the beam-integration
effect can lead to an underestimation of angular dispersion in polarization
angles, resulting in an overestimation of the magnetic field strength
(Ostriker et al., 2001; Padoan et al., 2001; Houde et al., 2009). To determine
the angular dispersion, there are different statistical methods (see
Hildebrand et al., 2009; Houde et al., 2009; Pattle et al., 2017; Liu et al.,
2022), which are used in the literature. In this work, we use the structure-
function (SF; Hildebrand et al., 2009) method, which gives the $\frac{\delta
B}{B_{0}}$ ratio by accounting for the spatial variation of position angles.

####  3.6.1 Structure function analysis

In the SF method, the magnetic field is assumed to be composed of a large-
scale structured field and a turbulent field that are statistically
independent. The distinctive behaviour of the two components enables them to
distinguish and extract the turbulent component, facilitating the computation
of $\sigma_{\theta}$. The SF method computes the difference in position
angles, $\Delta\phi(l)\equiv\phi(\bf x)-\phi(x+\textbf{\emph{l}})$, between
the $N(l)$ pairs of pixels separated by $l$ = $|\textbf{\emph{l}}|$, using the
following function:

$\braket{\Delta\phi^{2}(l)}^{1/2}\equiv\left(\frac{1}{N(l)}\sum_{i=1}^{N(l)}[\phi(\bf
x)-\phi(x+l)]^{2}\right)^{1/2}.$ (11)

This function is referred to as the "angular dispersion function". We want to
point out that the polarization position angles are used here for the
dispersion function. The angular dispersions, $\Delta\phi$, are kept $\leq$
90∘, to avoid the effect of the $\pm$ 180∘ ambiguity of the magnetic field
lines. Under the limit, $\delta<l<<d$, the square of the angular dispersion
function, known as the "structure function" is characterised by (Hildebrand et
al., 2009)

$\braket{\Delta\phi^{2}(l)}_{tot}-\sigma^{2}_{M}(l)\simeq b^{2}+m^{2}l^{2},$
(12)

where $\delta$ is the correlation length of the turbulent component, and $d$
is the typical length for variation in large-scale B-field. The quadratically
added terms in the dispersion function, $m^{2}l^{2}$ and $b^{2}$, are the
contribution from the $B_{0}$ and $\delta B$, respectively. The $B_{0}$ is
expected to increase almost linearly with slope $m$ for $l<<d$, and $b$ is a
constant turbulent contribution for $l>\delta$ (for details, see Hildebrand et
al., 2009). The $\sigma^{2}_{M}(l)$ is the contribution from the measured
uncertainty in the position angles.

The turbulent to large-scale magnetic field strength is given by (Hildebrand
et al., 2009)

$\frac{\braket{\delta B^{2}}^{1/2}}{B_{0}}=\frac{b}{\sqrt{2-b^{2}}},$ (13)

and $B_{0}$ can be estimated by using the modified DCF relation:

$B_{0}\simeq\sqrt{(2-b^{2})4\pi\mu m_{H}n_{H_{2}}}\,\frac{\sigma_{nt}}{b}.$
(14)

Using the $Q$ correction factor, we can determine the plane-of-sky magnetic
field strength:

$B_{pos}=QB_{0},$ (15)

where $Q$ is taken to be 0.5 (Heitsch et al., 2001; Ostriker et al., 2001).

We calculated the dispersion function corrected by measurement uncertainty,
i.e. $\braket{\Delta\phi^{2}(l)}_{tot}-\sigma^{2}_{M}(l)$ with a bin size of
12″. We used various bin sizes and found that the fit is converged, and
fitting errors are the least for 12″. Fig. 12 shows the dispersion in position
angles as a function of the length scale for CC and NES. We fitted the
dispersion function with the model defined in equation 12 using least square
fit, over the first few data points to ensure the limit, $l<<d$. The best-fits
turbulent component, $b$, for CC and NES are 32∘.1 $\pm$ 0∘.9 and 33∘.8 $\pm$
2∘.1, respectively. The dispersion in position angles can be obtained as
$\sigma_{\theta}=b/\sqrt{2}$, which is around $\sim$22∘.7 $\pm$ 0∘.6 and 23∘.9
$\pm$ 1∘.5 for CC and NES, respectively. These values are close to the maximum
value at which the DCF methods would give reliable results
($\sigma_{\theta}\leq$ 25∘, Ostriker et al., 2001) if a correction factor of
0.5 is applied. Using equation 13, 14, and 15, we calculate the $\frac{\delta
B}{B_{0}}$ ratio to be around $\sim$0.43 $\pm$ 0.01 and 0.46 $\pm$ 0.04, and
$B_{pos}$ to be around $\sim$24.0 $\pm$ 6.0 $\mu$G and 20.0 $\pm$ 5.0 $\mu$G,
for CC and NES, respectively. All the estimated parameters are given in Table
1.

Figure 12: The angular dispersion function for (a) CC and (b) NES. The solid
curve represents the best-fit model to the data, and the points used for
fitting are shown in black encircled circles. The intercept of the best-fit
model ($l$ = 0) gives the turbulent contribution to the total angular
dispersion. The error bars denote the statistical uncertainties after binning
and propagating the individual measurement uncertainties.

##  4 Discussion

The plane-of-sky magnetic field strength for the CC and NES regions is around
$\sim$24 $\mu$G and 20 $\mu$G, respectively. Given the uncertainty of at least
a factor of 2 associated with the B-field estimation by the DCF method
(Crutcher, 2012), the estimated B-field strengths are within the range of
$\sim$10$-$100 $\mu$G observed in star-forming regions (Chapman et al., 2011;
Crutcher, 2012; Pattle et al., 2022). The values are also consistent (i.e.
within a factor of 1.5) with the upper limits of the B-field values from
Crutcher et al. (2010) relation for the respective density of the regions.

###  4.1 Correlation between magnetic fields, intensity gradients, and local
gravity

Figure 13: Stokes I map of the Central clump region of G148.24+00.41, over
which the observed B-field segments are shown (green segments). The B-field
segments show a “U" shaped geometry, sketched with observed segments and shown
by magenta dashed curves, which may be caused by the drag of gravitational
converging flows towards the centre of the cloud.

Due to the low statistics in the polarization data, it is difficult to
conclusively comment on the overall morphology of the B-field, intensity
gradients, and local gravity. However, through this comparison, some
inferences can be drawn over their relative spatial variance. Fig. 7 and 8
shows a correlation between the B-field, intensity gradients, and local
gravity in the CC and WC regions, whereas the differences in their position
angles are relatively higher in the NES region. This correlation can be due to
the collapse of the clumps where gravity has pulled in and aligned B-field
lines with the intensity gradients, in the CC and WC regions (Koch et al.,
2013; Wang et al., 2019). However, in the outer diffused regions, like the
elongated structures, the field lines are not yet that much affected by the
local gravity.

Some simulations of the global collapse of magnetized clouds have found that
the gravitational flows from large scale to small scale, i.e. from filaments
to clumps/cores, can drag the magnetic field lines along the flow, causing a
“U" shaped geometry of the field lines across the filament spine (Gómez et
al., 2018; Vázquez-Semadeni et al., 2019). We also found evidence of “U"
shaped geometry of B-field lines, at the bottom of the elongated part of the
central clump. Fig. 13 shows the zoomed-in view of the central clump region,
in which the “U" shaped geometry is sketched from the observed B-field
orientations and is shown by magenta curves. High-resolution and sensitivity
observations would be required to ascertain the observed morphology. A similar
effect of gravity over the B-field morphology has also been found in other
observational works (Tang et al., 2019; Wang et al., 2020a, b; Beuther et al.,
2020; Busquet, 2020; Pillai et al., 2020).

In G148.24+00.41, Rawat et al. (2024), using MWISP CO data, identified
filamentary gas flows exhibiting noticeable velocity gradients as they move
towards the hub. They found that the velocity gradient increases towards the
hub, with a measured value of $\sim$0.2 km s-1pc-1 in the proximity of the
hub. The aforementioned findings give evidence of accreting gas flows along
the filaments towards the cloud’s centre, which can affect the B-field
morphology by dragging them along the flow. Future high-resolution dust
continuum and polarimetric observations might be able to reveal a significant
number of polarization vectors at the clump/core scale to better resolve the
substructures and the B-field morphology around the hub of G148.24+00.41.

###  4.2 Gravitational stability

####  4.2.1 Mass-to-flux ratio

In order to investigate whether the magnetic field can provide stability to
the regions against gravitational collapse, we determine the mass-to-flux
ratio (Crutcher et al., 2004). It is generally calculated as a dimensionless
critical stability parameter, $\lambda_{B}$, which is basically the comparison
of the mass to magnetic flux ratio with the critical ratio (Crutcher et al.,
2004):

$\lambda_{B}=\frac{(M/\phi)_{obs}}{(M/\phi)_{crit}}=7.6\times 10^{-21}\ \
\frac{N(H_{2})(\hbox{cm${}^{-2}$~{}})}{B_{pos}(\mu G)},$ (16)

where $N(H_{2})$ is the mean column density of the region. A clump or dense
core is magnetically supercritical ($\lambda_{B}$ $>$ 1) if the magnetic field
is not strong enough to support the system against gravitational collapse,
whereas a strong magnetic field would make the system magnetically subcritical
($\lambda_{B}$ $<$ 1), i.e. stable against collapse. Using the mean $N(H_{2})$
$\sim$(5.8 $\pm$ 2.2) $\times$ 1021 cm-2 and (7.1 $\pm$ 2.7) $\times$ 1021
cm-2 , we calculated the critical parameter to be around $\sim$1.8 $\pm$ 0.8
and 2.7 $\pm$ 1.2 for CC and NES, respectively. The critical parameter shows
that both regions are magnetically supercritical. In general, a statistical
correction factor is applied to $\lambda_{B}$ to account for bias due to
geometric effects (Crutcher et al., 2004). Different correction factors are
suggested for different geometry of clumps with respect to magnetic field,
e.g., $\pi/4$ for spherical clump and $1/3$ for oblate spheroid with major-
axis perpendicular to the mean B-field (Crutcher et al., 2004), and $3/4$ for
prolate spheroid with major-axis parallel to the mean B-field (Planck
Collaboration et al., 2016). Using these correction factors, the mass-to-flux
ratios become subcritical to transcritical/supercritical in the range of 0.6
to 1.4 for CC with mean $\sim$ 1.1, and 0.9 to 2.1 for NES with mean $\sim$
1.7. However, if we consider the overestimation of $B_{pos}$ in the DCF method
itself, our estimated mass-to-flux ratios could also be a lower limit. We want
to acknowledge here that the estimated B-field strengths and mass-to-flux
ratios are only averaged values over the selected regions. The regions can
still be super-critical inside but sub-critical in the outer part, as also
shown in a recent simulation by Gómez et al. (2021).

####  4.2.2 Turbulence versus magnetic field

Simulations suggest that turbulence plays a dual role in the clouds and their
substructures by providing turbulent support against the gravitational
collapse at a large scale while producing compressions and shocks at small
scales, that create density enhancements and trigger the star formation
process (Mac Low & Klessen, 2004; Ballesteros-Paredes et al., 2007; Hennebelle
& Falgarone, 2012; Klessen & Glover, 2016). Also, whether it is turbulence
(weak B-field models) or magnetic field (strong B-field models) or both, is a
subject of investigation regarding their respective roles in the formation of
clumps, cores, and subsequent star formation within these structures. In order
to investigate their impact, we need to find out the relative strength of
turbulence in comparison to the magnetic field.

The Alfv$\acute{\text{e}}$n Mach number ($\mathcal{M}_{A}$) infers the
relative importance of turbulence and magnetic field in molecular clouds and
is defined as $\mathcal{M}_{A}$= $\sqrt{3}\sigma_{nt}/V_{A}$. The
Alfv$\acute{\text{e}}$n velocity is calculated as,
$V_{A}=B_{tot}/\sqrt{4\pi\rho}$. The total B-field strength ($B_{tot}$) of the
regions can be determined by using a statistical relation,
$B_{tot}=(4/\pi)B_{pos}$ (Crutcher et al., 2004). For CC and NES, the
$B_{tot}$ is found to be around 31 $\pm$ 8 $\mu$G and 26 $\pm$ 6 $\mu$G,
respectively. Using the mass density estimated within the dimensions of two
regions (given in Table 1) and corresponding $B_{tot}$ values, the $V_{A}$ for
CC and NES are found to be $\sim$1.0 $\pm$ 0.4 km s-1and 0.6 $\pm$ 0.2 km s-1,
respectively. Then, using the corresponding $\sigma_{nt}$ values, the
$\mathcal{M}_{A}$ is calculated to be around $\sim$1.2 $\pm$ 0.4 for both the
two regions. A star-forming region is super-Alfv$\acute{\text{e}}$nic if
$\mathcal{M}_{A}$$>1$, which means that the turbulent pressure is higher than
the magnetic pressure. Conversely, it will be sub-Alfv$\acute{\text{e}}$nic if
$\mathcal{M}_{A}$$<1$, which means that the magnetic pressure is higher than
the turbulent pressure. In the present case, both the regions are trans
Alfv$\acute{\text{e}}$nic.

We also calculated the magnetic and turbulent pressures using the relations:

$P_{B}=\frac{B_{tot}^{2}}{8\pi}\ \ \ \ \mathrm{and}\ \ \ \
P_{t}=\frac{3}{2}\rho\sigma_{nt}^{2}\,\rm{(Spherical)},$ (17)

$\qquad\qquad\qquad\qquad\quad\;=\rho\sigma_{nt}^{2}\ \ \ \
\rm{(Cylindrical)},$

here, a factor of 3/2 is included to estimate the total turbulent pressure by
assuming the non-thermal velocity dispersion to be isotropic in spherical
geometry. For the CC region, the magnetic and turbulent pressure is found to
be $\sim$(3.7 $\pm$ 1.9) $\times$ 10-11 dyne cm-2 and $\sim$(5.2 $\pm$ 2.7)
$\times$ 10-11 dyne $\rm{cm^{-2}}$, respectively. For the NES region, $P_{B}$
and $P_{t}$ are $\sim$(2.6 $\pm$ 1.3) $\times$ 10-11 dyne cm-2 and (2.5 $\pm$
1.3) $\times$ 10-11 dyne $\rm{cm^{-2}}$, respectively. In the CC region, the
turbulent pressure is higher than the magnetic pressure by a factor of
$\sim$1.4, while in NES, the pressure values are similar. Also, the thermal
pressure ($\sim$$\rho\sigma_{th}^{2}$) in both regions, is much smaller than
the turbulent and magnetic pressure, which shows that the thermal energy plays
a negligible role in energy balance. So, from the Alfv$\acute{\text{e}}$n Mach
number and pressure estimation, it seems that the turbulence is slightly more
dominant in comparison to the magnetic field in CC, i.e. in the centremost
part of G148.24+00.41, while it is similar in the NES.

####  4.2.3 Virial analysis

The virial theorem is a principle that relates the average kinetic energy
($E_{K}$) and magnetic field energy ($E_{B}$) of a system to its average
gravitational potential energy ($E_{G}$), which provides insights into the
stability and energy distribution of the system. The Virial theorem is written
as:

$\frac{1}{2}\frac{d^{2}\mathcal{I}}{dt^{2}}=2E_{K}+E_{B}+E_{G},$ (18)

where $\mathcal{I}$ is the moment of inertia. The surface energy terms here
are neglected. The kinetic energy term is given by (Fiege & Pudritz, 2000)

$E^{sph}_{K}=\frac{3}{2}M\sigma_{obs}^{2}\ \ \ \ \rm{(Spherical)},$ (19)

$E^{cyl}_{K}=M\sigma_{obs}^{2}\ \ \ \ \ \ \ \ \rm{(Cylindrical)},$

where $M$ is the mass and $\sigma_{obs}^{2}=\sigma_{th}^{2}+\sigma_{nt}^{2}$.
The magnetic field energy is given by

$E_{B}=\frac{1}{2}MV_{A}^{2}$ (20)

and the gravitational potential energy is given by

$E^{sph}_{G}=-\frac{(3-a)}{(5-2a)}\frac{GM^{2}}{R}\ \ \ \ \rm{(Spherical)},$
(21)

$E^{cyl}_{G}\;=-\frac{GM^{2}}{L}\qquad\qquad\ \ \rm{(Cylindrical)},$

where $G$ is the gravitational constant, $R$, is the effective radius of the
sphere, and $L$ is the length of the cylinder. Here $a$ is the density profile
index of the sphere ($\rho\propto r^{-a}$). The energy values for the two
regions are listed in Table 1. We found that for CC, $|E_{G}|$ $>$ $E_{K}$ $>$
$E_{B}$ (i.e. 10.4:4.7:3.3), while for NES, $|E_{G}|$ $>$ $E_{K}$ $\simeq$
$E_{B}$ (i.e. 14:6.3:7.0), which restates the results of
Alfv$\acute{\text{e}}$n Mach number and mass-to-flux ratio calculation, i.e.
turbulence is slightly more dominant than the magnetic field in CC, while it
is similar in NES, but overall the gravity is the dominant factor in both the
regions. Nevertheless, the calculation of individual energy terms is an
important exercise, enabling the direct comparison of various forces that
govern the evolution of clumps/structures, expressed in the same units.

For a non-magnetized system ($E_{B}$ = 0), the stability criteria is given by
2$E_{K}$ $+$ $E_{G}$ $<$ 0, and based on this condition, the kinetic virial
parameter is defined as (Bertoldi & McKee, 1992)

$\alpha_{vir,k}=\frac{2E_{K}}{|E_{G}|}\ \
=\frac{3(5-2a)}{(3-a)}\frac{R\sigma_{obs}^{2}}{GM}\;\rm{(Spherical)},$ (22)

$\qquad\qquad\qquad\ \ \;=\ \ \frac{2L}{GM}\sigma_{obs}^{2}\qquad\ \ \ \
\rm{(Cylindrical)}.$

Using the $M$, $R$, $L$, and $\sigma_{obs}$ values of the regions (given in
Table 1) in equation 22, and adopting $a$ = 2 for spherical case, we calculate
the $\alpha_{vir,k}$ to be around 0.9 $\pm$ 0.4 and 0.9 $\pm$ 0.4,
respectively for CC and NES. The derived $\alpha_{vir,k}$ $<$ 2 means that in
the case of a non-magnetized sphere ($E_{B}$ = 0), the thermal plus turbulent
contribution is not enough to provide stability to the regions against the
gravitational collapse (Kauffmann et al., 2013; Mao et al., 2020). Including
the magnetic energy term in the stability criteria, i.e. 2$E_{K}$ $+$ $E_{B}$
$+$ $E_{G}$ $<$ 0, the total virial parameter is calculated using the modified
relation (Bertoldi & McKee, 1992; Pillai et al., 2011; Sanhueza et al., 2017).

$\alpha_{vir_{tot}}=\frac{2E_{K}+E_{B}}{|E_{G}|}\ \ =\ \
\frac{3(5-2a)}{(3-a)}\frac{R}{GM}\left(\sigma_{obs}^{2}+\frac{V_{A}^{2}}{6}\right)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\rm{(Spherical)},$ (23)

$\qquad\qquad\qquad\qquad\quad\,=\ \
\frac{2L}{GM}\left(\sigma_{obs}^{2}+\frac{V^{2}_{A}}{4}\right)\quad\rm{(Cylindrical)}.$

With the magnetic support, the $\alpha_{vir,tot}$ value is estimated to be
around $\sim$1.2 $\pm$ 0.6 and 1.4 $\pm$ 0.7 for CC and NES, respectively. The
$\alpha_{vir,tot}$ values for the regions are $<$ 2, which shows that the two
regions are bound by gravity and thus can collapse to form stars. The virial
analysis shows that the total kinetic energy ($E_{K}$, i.e. thermal plus
turbulent) in both regions is not sufficient to support them against the
gravitational collapse. While magnetic energy, combined with kinetic energy,
is found to be comparable to gravitational potential energy.

In the present work, we have estimated magnetic field strengths and derived
various parameters for the CC and NES regions. However, we want to stress that
these results must be taken with caution as the measurements are uncertain
within a factor two due to the inherent large uncertainty in the mass and
density of the studied regions. Moreover, the modified DCF methods can be
uncertain up to a factor of two or more (Crutcher, 2012), and also, the
B-field strength in the studied regions can be biased due to the limited
number of B-field segments traced by our observations. Due to all these
uncertainties, the mass-to-flux ratio and virial status of the regions should
be considered as qualitative indicators of the stability of the region. In the
present case, we have used the generally accepted Q value of 0.5 (Crutcher,
2012) in our estimations, however, if we use Q = 0.4, suggested for parsec
scale clumps (Padoan et al., 2001), similar to our studied regions, the
B-field strengths will reduce by a factor of $\sim$1.2. As a consequence, the
CC and NES regions would become more magnetically supercritical. Future high-
resolution and more sensitive observations would better constrain the magnetic
field and turbulence properties of the hub. However, taking the measured mass-
to-flux ratio and virial parameters at face value, it can be argued that, at
present, gravity has overall an upper hand over magnetic and kinetic energies
in CC and NES, which is consistent with the formation of a young cluster
noticed by Rawat et al. (2023) in the hub.

###  4.3 The criticality of magnetic fields in hub-filamentary clumps

Like the CC of G148.24+00.41, the clumps located at the junction of the hub-
filamentary systems are known to be potential sites of cluster formation (e.g.
Kumar et al., 2020), because such clumps are attached to converging
filamentary structures that fuel them with cold gaseous matter (e.g. Myers,
2009; Li et al., 2014; Treviño-Morales et al., 2019; Kumar et al., 2022).
Although physical processes that govern the star formation are scale-
dependent, it is worthwhile to compare the global properties of the parsec or
sub-parsec scale hub filamentary clumps studied with similar resolution. In
the literature, a few such clumps have been studied with JCMT/POL2, these are:
IC 5146 E-hub (Wang et al., 2019), G33.92$+$0.11 (Wang et al., 2020b), Mon R2
(Hwang et al., 2022), IC 5146 W-hub (Chung et al., 2022), and SDC13 (Wang et
al., 2022). In the majority of these hub filamentary clumps, except Mon R2,
the mass-to-flux ratio and/or virial analysis suggest the dominance or edge of
gravitational energy over the magnetic and kinetic energies, similar to the CC
region of G148.24+00.41. We found that all the aforementioned clumps are
associated with either protostars or an embedded cluster (e.g. Harvey et al.,
2008; Gutermuth et al., 2009; Peretto et al., 2014; Liu et al., 2015). Thus,
it seems that, at least for those parsec scale hubs that are in the early
stages of cluster formation, like the CC of G148.24+00.41, gravity has an
upper hand on the energy budget of the system. A large sample of hub-
filamentary clumps of various evolutionary stages (e.g. from pre-stellar
clumps to clumps hosting emerging clusters of different ages) would be
valuable to study the time evolution of various physical processes that govern
star formation and its evolution.

##  5 Summary

We have performed the dust polarization observation of the central part of the
G148.24+00.41 cloud to investigate the B-field morphology and its strength
relative to gravity and turbulence, using JCMT SCUBA-2/POL-2 at 850 $\mu$m. We
specifically focused on the C1 clump, located at the hub of G148.24+00.41. The
main results are summarised below:

  * •

The 850 $\mu$m Stokes I intensity map reveals the presence of a central clump
(CC), northeastern elongated structure (NES), and western clump (WC) around
the hub of G148.24+00.41. The B-field segments of CC and NES regions show
mixed morphology, while the WC region shows converging B-field segments,
mostly aligned along the southeast direction.

  * •

We found evidence of the depolarization effect, and from the Bayesian analysis
over the non-debiased polarization data, found a power-law index, $\alpha$ =
0.6. Although this shows a decreasing level of dust grain alignment, but they
can still be aligned with the magnetic field in the central high-density
region of the cloud.

  * •

We compared the relative orientations of B-fields ($\theta_{B}$), intensity
gradients ($\theta_{IG}$), and local gravity ($\theta_{LG}$) over the full
map. In the CC and WC regions, the three factors are mostly correlated, while
the difference in orientations is higher in the NES region. This suggests that
gravity is dragging the intensity gradients and aligning them with the
B-fields in the CC and WC clump, while the effect of gravity in NES is
comparatively less significant.

  * •

We constructed the $\Sigma_{B}$ map to see the localised B-field strength in
comparison to local gravity, and found that for most of the parts,
$\Sigma_{B}$ $<$ 1, i.e. gravitational force is dominant over the magnetic
field force.

  * •

We used the structure function analysis to determine the B-field strength, and
as a result, found $B_{pos}$ for CC and NES to be around $\sim$24.0 $\pm$ 6.0
$\mu$G and 20.0 $\pm$ 5.0 $\mu$G, respectively. In both regions, the turbulent
component relative to the organised magnetic field structure was determined to
be approximately 0.4$-$0.5.

  * •

The mass-to-flux ratio and Alfv$\acute{\text{e}}$n Mach number calculation for
CC and NES shows that both regions are magnetically
transcritical/supercritical and trans-Alfv$\acute{\text{e}}$nic. The turbulent
pressure was found to be higher than the magnetic pressure in CC, while they
are similar in NES.

  * •

The virial analysis shows that for CC, the $|E_{G}|$ $>$ $E_{K}$ $>$ $E_{B}$
(i.e. 10.4:4.7:3.3), while for NES, $|E_{G}|$ $>$ $E_{K}$ $\simeq$ $E_{B}$
(i.e. 14:6.3:7.0). The magnetic field and turbulence individually are not
strong enough to provide stability to the regions against gravity. Both
regions were found to be bound by gravity.

Overall, we find that currently, gravitational energy has an edge over the
other energy terms of the hub region of G148.24+00.41, thereby continue to
facilitate the growth of the young cluster in the hub, although we acknowledge
that given the large uncertainties associated with our estimates, a conclusive
answer would require further precise measurements of magnetic field and cloud
properties.

## ACKNOWLEDGEMENT

We thank the anonymous referee for the comments and suggestions that helped to
improve the paper. The research work at the Physical Research Laboratory is
funded by the Department of Space, Government of India. CE acknowledges the
financial support from grant RJF/2020/000071 as a part of the Ramanujan
Fellowship awarded by the Science and Engineering Research Board (SERB),
Department of Science and Technology (DST), Government of India. The James
Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of
The National Astronomical Observatory of Japan; Academia Sinica Institute of
Astronomy and Astrophysics; the Korea Astronomy and Space Science Institute;
the National Astronomical Research Institute of Thailand; Center for
Astronomical Mega-Science (as well as the National Key R&D Program of China
with No. 2017YFA0402700). Additional funding support is provided by the
Science and Technology Facilities Council of the United Kingdom and
participating universities and organizations in the United Kingdom, Canada,
and Ireland. The authors wish to recognize and acknowledge the very
significant cultural role and reverence that the summit of Maunakea has always
had within the indigenous Hawaiian community. The data taken in this paper was
observed under the project code M22BP055. D.K.O. acknowledges the support of
the Department of Atomic Energy, Government of India, under project
identification No. RTI 4002. We thank Junhao Liu for the discussion on the DCF
methods. This research made use of the data from the Milky Way Imaging Scroll
Painting (MWISP) project, which is a northern galactic plane CO survey with
the PMO-13.7m telescope. We are grateful to all the members of the MWISP
working group, particularly the staff members at PMO-13.7m telescope, for
their long-term support. MWISP was sponsored by the National Key R&D Program
of China with grant 2017YFA0402701 and CAS Key Research Program of Frontier
Sciences with grant QYZDJ-SSW-SLH047.

_Facilities_ : JCMT, PMO

## Data Availability

The JCMT data used in this work can be shared on reasonable request. We also
used the CO molecular data from PMO, which can be shared by the PMO database
on reasonable request to the project team.

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##  Appendix A Intensity gradients versus Local gravitational field

Fig. 14a shows the relative orientations of intensity gradient and local
gravity, and Fig. 14b shows the distribution of the angular difference between
their orientations, i.e. $\Delta\theta_{IG,LG}$. Similar to
$\Delta\theta_{B,IG}$ and $\Delta\theta_{B,LG}$, the correlation between the
angles ($\theta_{IG}$ and $\theta_{LG}$) is better in the CC and WC regions in
comparison to NES region.

Figure 14: (a) The orientations of the intensity gradients (red segments) and
local gravity (cyan vectors) are overlaid on the 850 $\mu$m Stokes I map. (b)
The distribution of the offset between the position angles of the intensity
gradients and local gravity, i.e., $\Delta\theta_{IG,LG}$ =
$|(\theta_{IG}-\theta_{LG})|$ over the contours of 850 $\mu$m Stokes I map.
The contour levels are same as in Fig. 2.