11institutetext: Instituto de Astrofísica e Ciências do Espaço, Universidade
do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal 22institutetext:
Aix Marseille University, CNRS, CNES, LAM - 38 rue Frederic Joliot-Curie,
13388 Marseille, France 33institutetext: AIM, IRFU, CEA, CNRS, Université
Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, F-91191 Gif-sur-
Yvette, France 44institutetext: Faculty of Education & Center for Educational
Development and Support, Kagawa University, Saiwai-cho 1-1, Takamatsu
760-8522, Japan 55institutetext: Department of Physics, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan
55email<EMAIL_ADDRESS>
# Filament coalescence and hub structure in Mon R2††thanks: Based on
observations obtained with $\it{Herschel}$ programme HOBYS ††thanks: Based on
observations obtained with Planck (http://www.esa.int/Planck), an ESA science
mission with instruments and contributions directly funded by ESA Member
States, NASA, and Canada.

Implications for massive star and cluster formation

M. S. N. Kumar 11 D. Arzoumanian 2211 A. Men’shchikov 33 P. Palmeirim 11 M.
Matsumura 44 S. Inutsuka 55

There is growing evidence of the role of hub-filament systems (HFS) in the
formation of stars from low to high masses. As of today, however, the detailed
structures of these systems are still not well described. Here we study the
Mon R2 star-forming region, which has a rich network of filaments joining in a
star cluster forming hub, and aim to understand the hub structure and to
examine the mass fraction residing in the hub and in the filaments, which is a
key factor that influences massive star formation. We conducted a multi-scale,
multi-component analysis of the $\it{Herschel}$ column density maps
(resolution of 18.2′′ or $\sim$0.07 pc at 830 pc) of the region using a newly
developed algorithm getsf to identify the structural components, namely,
extended cloud, filaments, and sources. We find that cascades of lower column
density filaments coalesce to form higher-density filaments eventually merging
inside the hub (0.8 pc radius). As opposed to the previous view of the hub as
a massive clump with $\sim$1 pc radius, we find it to be a network of short
high-density filaments. We analyse the orientations and mass per unit length
(M/L) of the filaments as a function of distance from the hub centre. The
filaments are radially aligned towards the centre of the hub. The total mass
reservoir in the Mon R2 HFS (5 pc $\times$ 5 pc) is split between filaments
(54%), an extended cloud (37%), and sources (9%). The M/L of filaments
increases from $\sim$ 10$M_{\odot}\,$/pc at 1.5 pc from the hub to $\sim$ 100
$M_{\odot}\,$/pc at its centre, while the number of filaments per annulus of
0.2 pc width decreases from 20 to two in the same range. The observed radial
column density structure of the HFS (filament component only) displays a
power-law dependence of $N_{\mathrm{H}_{2}}\propto r^{-2.17}$ up to a radius
of $\sim$2.5 pc from the central hub, resembling a global collapse of the HFS.
We present a scenario where the HFS can be supported by magnetic fields which
interact, merge, and reorganise themselves as the filaments coalesce. We
plotted the plane-of-the-sky magnetic field line geometry using archival
Planck data to support our scenario. In the new view of the hub as a network
of high-density filaments, we suggest that only the stars located in the
network can benefit from the longitudinal flows of gas to become massive,
which may explain the reason for the formation of many low-mass stars in
cluster centres. We show the correlation of massive stars in the region to the
intertwined network-like hub, based on which we updated the implications of
the filaments to clusters (F2C) model for massive star formation.

###### Key Words.:

interstellar medium – star formation – embedded clusters – massive stars hub-
filament systems

##  1 Introduction

The interstellar medium (ISM) is observed to be filamentary both in its atomic
(Heiles, 1979) and molecular (Schneider & Elmegreen, 1979) forms. This view is
firmly pronounced by data obtained with modern observational facilities with
ever-increasing angular resolution and sensitivities (André et al., 2014;
Hacar et al., 2018; Arzoumanian et al., 2019). A hierarchy of filamentary
structures have been named to describe observational data: galactic spines
(Goodman et al., 2014), giant filaments (e.g. Ragan et al., 2014; Zucker et
al., 2018), interstellar filaments (e.g. Arzoumanian et al., 2019), fibres
(e.g. Tafalla & Hacar, 2015; Hacar et al., 2017b), and striations and strands
(e.g. Cox et al., 2016; Bonne et al., 2020). This hierarchy is distinguished
by the physical size and the total mass held within them.

In the nearby ($<$500pc) star-forming regions, filamentary structures span a
range of $\sim$3 orders of magnitude in surface densities (Arzoumanian et al.,
2019). The resulting filament mass functions are proposed to be the origin of
dense core- and stellar-mass functions (André et al., 2019). Velocity coherent
filaments, also known as fibres (Hacar et al., 2017b, 2018) characterised in
NGC1333 and the Orion integral shaped filament (ISF), indicate that the fibre
density increases with the total mass per unit length of filaments.

Figure 1: Column density maps of the Mon R2 star-forming region at two angular
resolutions, produced using getsf. The vertical colour bar is in units of
cm-2. At a distance of 830 pc, the 11.4′′ resolution corresponds to a beam of
0.046 pc.

Velocity coherent dense cores (Barranco & Goodman, 1998) have a radius of
$\sim$0.1 pc, a scale at which the structure decouples itself from the larger-
scale turbulent flows (Goodman et al., 1998) and transits to a coherent sonic
entity. Interestingly, the constant widths of filaments of $\sim$0.1 pc, as
inferred from $\it{Herschel}$ observations of nearby star-forming regions
(André et al., 2014; Arzoumanian et al., 2011, 2019), represent a wide range
in mass per unit length (M/L) of filaments, most remarkably that of a
1000$M_{\odot}\,$ pc-1 filament in NGC6334 (André et al., 2016). This
coincidence makes it tempting to view filaments as cylindrical equivalents of
coherent dense cores. Hacar et al. (2016, 2018) suggest that velocity coherent
fibres may represent the basic structure in star-forming clouds, the fibre
spatial density distinguishing low- and high-mass star-forming regions.
However, there is also no guarantee that structures traced in PPV space
correspond to real structures in PPP space (e.g. Clarke et al., 2018). Given
that there is no well-defined boundary that separates velocity coherence and
that fibres or fibre bundles are identified as filaments, these structures, in
general, may represent a scale at which they decouple themselves from the
cloud-scale turbulence.

Junctions of filaments on the other hand are called hubs, which are high
column density low aspect ratio objects (Myers, 2009; Schneider et al., 2012;
Peretto et al., 2014; Chen et al., 2019; Kumar et al., 2020). Massive star
formation itself appears to take place only in the hubs (Kumar et al., 2020;
Schneider et al., 2012), which are found to be fed by longitudinal flows along
filaments (Peretto et al., 2014; Williams et al., 2018). Kumar et al. (2020)
note a hierarchy of hub-filament systems (HFS) which they speculate to be a
consequence of the hierarchical range in filamentary structures from fibres to
galactic spines. The identification of core- and clump-hubs in NGC6334
(Arzoumanian et al., 2021) is a manifestation of that hierarchy in HFS.

The property of hubs as density-enhanced objects (Kumar et al., 2020) and the
finding that fibre dimensions are self-regulated depending on their intrinsic
gas density (Hacar et al., 2018) hints at a mechanism that may be at work
universally in building the density structure of star-forming regions. If high
M/L filaments are dense bundles of fibres and density enhanced hubs are
junctions of filaments, it is necessary to understand the relation between the
two and which type of junction leads to what structure. Next, it appears that
all massive star formation in the Milky-Way plane covered by the
$\it{Herschel}$ Hi-GAL survey (Molinari et al., 2010) takes place in density
amplified hubs (Kumar et al., 2020), and the hubs are replenished by material
flow from filaments exterior to the hub and constituting the HFS. While the
filaments to clusters (F2C) model of Kumar et al. (2020) assumes that this
flow of matter originates in longitudinal flows (Peretto et al., 2014; Chen et
al., 2019), a global flow of accelerated matter is reported in the Orion ISF
by Hacar et al. (2017a) who argue for a global collapse. If so, it is
necessary to estimate the fraction of material held in the filaments (outside
the hub) of any HFS and to asess if it is significant enough to aid the
massive star formation or enhance cluster formation in the hubs.

To answer these two questions, we chose to study the Mon R2 star-forming
region which has a rich network of filaments merging into a hub (Treviño-
Morales et al., 2019), a nearly face-on geometry, and the region as a whole
displays a column-density probability distribution function N-PDF (Pokhrel et
al., 2016) that is representative of any cluster-forming cloud. Our analysis
is based on a multi-scale multi-component analysis, which uses a new algorithm
getsf (Men’shchikov, 2021b) to separate the elongated filaments from extended
background emission and roundish sources. This is crucial because we are
interested in analysing the properties of the filaments as independent
entities, excluding contributions from the embedded sources and the
surrounding diffuse cloud.

This paper is organised as follows. In Sec. 2 we present the observational
data and column density maps used for the study. Analysis of the data is
described in Sec. 3, which involves spatially decomposing the filament,
extended cloud, and sources (sec. 3.1), classifying the hub and filament
components of the HFS, computing their mass fractions (sec. 3.2), and studying
the filament lengths, angles, and mass-per-unit-length as a function of its
distance to the hub centre (sec. 3.3). Interpretations based on this analysis
are deliberated in Sec. 4, which include coalescence of molecular filaments
(sec. 4.1), the HFS mimicking global collapse (sec. 4.2), the hub-filament
mass fraction (sec. 4.3), the structure of the hub and its implications for
the formation of massive stars (sec. 4.4), and the role of magnetic fields as
necessary support against global collapse (sec. 4.5). A discussion reflecting
on the meaning of our interpretations to the larger picture can be found in
Sec. 5.

##  2 Observational data

Among the nearby (¡ 1 kpc) star-forming regions, Mon R2 ($d$ = 830 pc) is a
hub-filament system with a miniature spiral galaxy appearance (Treviño-Morales
et al., 2019), where the dense central hub is forming a cluster of stars
including the B-type star at the centre of the Mon R2 region (Fig.1). This
large network of filaments is shown to be constrained in a flattened sheet-
like space, based on an analysis of the velocity information from the
observations of molecular emission lines (Treviño-Morales et al., 2019). The
sheet has a low inclination angle of $\sim$30∘ with the plane of the sky
giving its spiral galaxy appearance. In this study, we have produced 18.2 &
11.4′′ high-resolution surface density maps and separated the structural
components of the maps using getsf (Men’shchikov, 2021b) to find filaments,
sources, and the extended non-filamentary emission (hereafter diffuse cloud).
This allowed us to derive an intrinsic surface density structure and the mass
of the individual filaments, free of the contributions from the unrelated
structural components. Our analysis is different from the previous studies
that examined the radial profiles of the overall density (Didelon et al.,
2015) or simple estimates of the diffuse component based on measurements in
regions void of filaments (Treviño-Morales et al., 2019). Here the dense and
diffuse cloud components are separated from the sources by decomposition of
the observed data, allowing us to examine the behaviour of each component and
its impact on the physics of star formation.

The $\it{Herschel}$ data of Mon R2 taken as part of the HOBYS programme (Motte
et al., 2010) provide imaging in five far-infrared bands at 70, 160, 250, 350,
and 500$\rm\,\mu m\,$. Early results on Mon R2 using HOBYS data are described
by Didelon et al. (2015) and Rayner et al. (2017). These images were used to
compute the surface density at the standard resolution of 36′′ and a higher
angular resolution of 18.2′′ (Palmeirim et al., 2013), corresponding to the
resolution of the 250$\rm\,\mu m\,$ image. Using the hires method described by
Men’shchikov (2021b), a 11.4′′ resolution map (corresponding to the
160$\rm\,\mu m\,$ beam) was produced. These maps are displayed in Fig. 1.
Because the 160$\rm\,\mu m\,$ might also contain contributions from the hotter
dust emission, we tested the fidelity of the surface density map at 11.4′′
resolution by smoothing it to 18.2′′ and 36′′ and comparing with the 18.2′′
high-resolution and 36′′ standard resolution maps. While the total surface
density remained conserved, the maps at different angular resolutions were
locally comparable with differences of 10-20%. The column densities, derived
from the spectral energy density (SED) fitting of the $\it{Herschel}$ data
assuming a uniform dust temperature along the line of sight, may be affected
by possible temperature gradients along the line of sight, especially towards
star-forming protostellar sources (Roy et al., 2014; Men’shchikov, 2021b) and
H ii regions. If higher temperature dust is present along the line of sight,
the best-fitted column densities are only underestimated (Anderson et al.,
2012), especially towards the densest filaments and hubs embedded in a warmer
cloud envelope. The structures traced by both the 11.4′′ and 18.2′′ hi-
resolution maps shown in Fig. 1 are almost identical. We traced the structures
using the 11.4′′ map and obtained the column density measurements from the
18.2′′ map.

Figure 2:  Coalescence of filaments: a) Colour map showing the column density
of the filament component that is the sum of filaments detected over scales
from 16′′ up to 258′′ . Groups of coalescing filaments (see Sec. 4.1) are
enclosed by white line boundaries. The black cone encloses an example of a
second step in the coalescence. The red and blue circles (colours used for
convenient visual contrast) identify some of the clear junctions of filaments
within the coalescing groups where the density is visibly enhanced compared to
the incoming structures. b) Individual filament crests identified by getsf are
used to mask the image in panel a. In both panels, the cross symbol at the
centre (black and green in the left and right panels, respectively) indicates
the position of the hub centre and radial origin.

##  3 Analysis

###  3.1 Filaments, sources, and extended cloud

####  3.1.1 Spatial decomposition

The filaments, compact sources, and the large-scale diffuse background were
decomposed on the surface density maps using the getsf algorithm
(Men’shchikov, 2021b, a). The method was developed for extracting filaments
and sources in far-infrared to sub-millimetre observations with the
$\it{Herschel}$ Space Telescope, nevertheless, it applies to any data set. A
unique feature of getsf is the careful separation of the structural components
based on their shapes. In this method, the spatial decomposition of the
observed image is conducted in a range of scales to identify the structural
components and separate them. This allows one to disentangle features over
multiple scales and to separate the sources from filaments and backgrounds.
The main processing steps of getsf are the spatial decomposition of the
observed images, separation of the structural components, flattening of the
detection images of the source and filament components, the combination of the
detection images from different wavebands, detection of sources and filaments,
and their measurements. We applied getsf to the surface density maps of Mon R2
using the 11.4′′ resolution map to detect the structural components and obtain
the measurements on the 18.2′′ map. This way we optimally utilised the angular
resolution in the data to accurately trace the morphological characteristics
while retaining the column density measurements from the 18.2′′ map.

####  3.1.2 Filament component map

The output from the getsf decomposition includes three component images,
namely filaments, extended emission background, and compact sources, a sum of
which corresponds to the original surface density map. All of these images are
displayed in the appendix Fig. A.1. The image of filaments is a reconstruction
of filaments detected on scales ranging from 16 to 258′′, the upper limit
defined after an initial examination of the features in the input image. The
skeletons tracing crests of the detected filaments on different angular
scales, as well as catalogues containing lengths, widths, surface densities,
and linear densities, are produced by getsf (Men’shchikov, 2021b).

From Figs. 1 & 2, it is evident that most filamentary structures have similar
widths and there are not many structures with largely discrepant widths. This
is because the MonR2 is at a distance of 830 pc and the angular resolution of
11–18′′ cannot resolve physical scales smaller than 0.1 pc. Even though
identification was made over the scales between 16 to 258′′, most of the
filaments and sources are detected on scales below 50′′. A given filament is
often traced on more than one scale depending on its width. Almost all
filaments in the region displayed in Fig. 1 are exhaustively traced by the
scales up to 46′′ corresponding to the physical size of $\sim$0.18 pc. On
scales larger than 46′′ , the features mostly represent the lower-column-
density and larger-scale halo structures surrounding groups of multiple
filaments. To analyse the filaments, we used a set of unique skeletons that
were identified up to an angular scale of 46′′. A mask was created from those
unique skeletons and applied to the filament component image to examine the
crest column densities and their variation in the entire image, as shown in
Fig. 2b. The column densities displayed in Fig. 2a represent only the
contribution from the filaments (dense gas, $A_{V}>8$ mag), after the removal
of sources and the variable diffuse background.

Figure 3: Radial profiles of the azimuthally averaged column density centred
on the Mon R2 hub, as distributed in its structural components. The column
density radial profile of the total map is compared with separated filaments
and diffuse cloud component maps. The fitted curves and their power-law
indices are listed for each component. The fits ware performed for regions
beyond 1 pc, excluding the hub. Error bars represent the standard deviation of
the non-zero values in each pixel annulus. Figure 4: Filament angle analysis.
a) Average filament orientation with respect to the purely radial direction as
a function of distance to the centre of the hub shows that they are radially
aligned. b) Deviation of the filament angle from the purely radial direction
to Mon R2 is an increasing function of distance to Mon R2, suggesting
influence from the hub gravitational potential in aligning the filaments.
Figure 5: Radial variation of filament M/L (red) and the number of filaments
crossing annular circles (black). Red dots show the actual data points for all
skeletons longer than 0.4 pc. The red line represents the average M/L in each
0.2 pc bin and the error bars are their standard deviations. The black line
shows the number of filaments that intersect the circumference of concentric
circles drawn around the HFS as displayed in Fig. A.2. Figure 6: Different
version of Fig. 5 where the number of filaments per unit area (N) is compared
with the M/L. Readers should notice that while both the M/L and N increase
with decreasing distance from the hub centre from 3 pc to 1 pc, in the region
within 1 pc, N flattens out while M/L steeply continues to increase with
decreasing distance before flattening out in the central 0.2-0.3 pc.

###  3.2 Characterising the Mon R2 hub-filament system

Given the circular symmetry of the Mon R2, we have analysed its properties in
radial coordinates, centred on the location of the B star in Mon R2, which is
also the centre of the spiral-like pattern of the column density map.
Throughout the subsequent analysis, the location of this radial origin is
considered to be $\alpha_{0}$ = 06:07:45.6 $\delta_{0}$ = $-$6:23:00.

####  3.2.1 The hub and its relation to the stellar cluster

We first studied the azimuthally averaged radial surface density of the
structural components. The extended background is assumed to represent the
’diffuse cloud’ from now on. The column density radial profile of the non-
decomposed original image as well as the decomposed filaments- and diffuse-
components in a 5 pc $\times$ 5 pc projected region centred on the radial
origin were produced (Fig. 3). From the radial origin ($\alpha_{0}$,
$\delta_{0}$), moving outwards, a one-pixel width azimuthal average was
computed up to a projected radial distance of 2.5 pc. The standard deviation
of the non-zero pixels in the azimuthal average was used as the error at each
radial point. The curves were fitted with power-law functions for the region
with a radius between 0.8 pc and 2 pc, allowing the power-law exponent to be
freely adjusted by the fitting programme. Also, we note that the measurements
for filaments here involve subtracting the diffuse component, hence, the low
values of N${}_{H_{2}}<$10${}^{21}cm^{-2}$ beyond 2.0 pc. The fitted functions
are also plotted in Fig. 3. As evident from this figure, the region interior
to 0.8 pc, defined as the hub, shows a steeper change in the column densities,
whereas the region beyond the hub varies smoothly. Therefore, we define the
radius of the hub Rhub=0.8 pc. We define the inner 0.4 pc as the hub-core for
the analysis here.

The Mon R2 cluster identified by Carpenter et al. (1997) is centred on
06:07:46.2 $-$6:22:51.6, with a full-width-half-maximum size scale of
0.38$\pm$0.03 pc and an estimated central stellar volume density of
$\sim$9000$\pm$1000 stars pc-3. Even though the cluster extent is traced up to
2 pc, the stellar surface density effectively drops to $<$ 50 stars pc-2 at a
radius of 0.5 pc (see Fig. 7 in Carpenter et al., 1997). This radius is
similar to the radius of the hub-core above, suggesting that much of the
observed star formation resides within the hub-core. This region corresponds
to the area represented by the highest-column-density filaments appearing as
white skeletons in Fig. 2b.

####  3.2.2 Mass fraction of the structural components

Table 1: Non-cumulative mass of the different components (filaments, diffuse cloud, and compact sources) as a function of radius. Radius (pc) | 0.4 | 0.4–0.8 | 0.8–1.6 | 1.6–2.5
---|---|---|---|---
| Mass $M_{\odot}\,$
Filament | 837 | 761 | 632 | 345
Diffuse cloud | 108 | 198 | 585 | 855
Sources | 164 | 115 | 103 | 71
Total | 1109 | 1074 | 1320 | 1271
4774$M_{\odot}\,$=100% | Mass fraction %
Filament (53.8) | 17.5 | 15.9 | 13.2 | 7.2
Diffuse cloud (36.7) | 2.3 | 4.2 | 12.2 | 18.0
Sources (9.5) | 3.4 | 2.4 | 2.2 | 1.5

Table 1 lists the computed mass of the individual structural components
distributed in four annular regions centred on the radial origin, which is
also the hub centre. The radii of 0.4 and 0.8 pc represent the central core of
the hub and the hub. The mass reservoir for all structural components listed
in Table 1 were obtained by directly converting the observed column density
(N${}_{H_{2}}$) for each of these components and assuming a distance of 830 pc
to Mon R2 by using the formula Mass = N${}_{H_{2}}\times$
area$\times\,\mu_{H_{2}}\times\,m_{H}$, where $\mu_{H_{2}}$ is the mean
molecular weight per H2 taken as 2.8 and mH is the atomic weight of hydrogen.
These values are generally in agreement with previous estimates (Treviño-
Morales et al., 2019) if we take the differences in the analysed area of the
image into account. However, here, the diffuse cloud and source masses are
more accurately derived owing to the proper decomposition of these components.
We note that we have restricted this table to a radial distance of 2.5 pc
(also true for Figs. 4 and 5), because of the symmetrical field-of-view
centred on Mon R2 which is available from the data used. The cumulative mass
within a circular region of radius 2.5 pc combined from all components is
4774$M_{\odot}\,$, which was considered to be 100% while computing the listed
mass fraction in Table. 1. It can be seen that about 54% of the total mass
reservoir resides in the filaments and nearly 40% in the diffuse cloud. The
diffuse cloud mass is often found to be higher as estimated by the CO
observations (see Sect. 4.3). Considering only the filaments’ component, the
central hub region within the radius of 0.8 pc holds $\sim$ 1598
$M_{\odot}\,$, while $\sim$977 $M_{\odot}\,$is held in the filaments located
in the annular region of 0.8-2.5 pc. This indicates a filament:hub mass ratio
of 0.6:1. However, the filaments extend further beyond to a radius of more
than 4 pc from the centre, so the filaments outside the hub should hold a mass
much larger than 977$M_{\odot}\,$ if not twice that value because the column
density of the filaments decreases as $\sim$r-2.17 (see Fig. 3) away from the
hub. Therefore, it is reasonable to assume that the mass residing in the hub
area is roughly similar to the mass residing in the filaments exterior to the
hub region.

###  3.3 Filament lengths, angles, and line masses

We analyse the orientation of the filaments to examine in detail the
circularly symmetric pattern of the Mon R2 region. The radial origin defined
before is also the centre of the spiral-like pattern of the column-density
map. To assess, how the filaments are oriented around Mon R2, we divided the
region around Mon R2 radially into many concentric annuli with widths of
$dr$=0.2 pc. In each annulus, we calculated the average deviation
$\overline{\delta\theta}$ of the filament orientation $\theta$ from the purely
radial direction from the hub centre, as well as
$\Delta\theta=\delta\theta-\overline{\delta\theta}$ and average
$\overline{\Delta\theta}$. Plotting $\overline{\delta\theta}$ and
$\sqrt{\overline{{\Delta\theta}^{2}}}$ as a function of the radial distance of
the annuli, we note that $\overline{\delta\theta}$ remains close to zero up to
$\sim$3 pc, suggesting that the filaments are oriented radially towards Mon R2
(Fig. 4a). The $\sqrt{\overline{{\Delta\theta}^{2}}}$ is a slowly increasing
function of radius beyond the ’hub’ (Fig. 4b), suggesting that the filaments
deviate from the purely radial direction away from the hub.

Next, the radial variation of the filament mass per unit length (M/L) and the
number density is investigated. For every skeleton, the average of the radial
distances of all skeleton pixels was taken as the filament distance from the
hub centre and origin. The crest column densities for each skeleton were
measured and averaged using the masked skeleton map shown in Fig. 2b. The
total mass of each filament was computed by assuming a 0.1 pc width
(Arzoumanian et al., 2011) and 830 pc distance to Mon R2. The total mass was
divided by skeleton lengths to obtain the M/L of the filaments. An average
value computed in 0.2 pc bins were used to produce the red curve in Figs. 5 &
6, and the error bars are the standard deviation of the average. Similarly,
the total number of filaments in 0.2 pc bins were counted and divided by the
area of the annuli to obtain the number of filaments per unit area that is
shown in Fig. 6. Even though a skeleton may stretch between bins, they were
counted only once and do not overlap between bins, because we consider the
radial distance of all skeletons as the average of the distance of all pixels
of a given skeleton.

##  4 Interpretations

###  4.1 Coalescing filaments

The Mon R2 HFS is a target with a high degree of radial symmetry. Analysis of
the orientation of the filaments shows that they are radially focussed on the
hub centre towards the Mon R2 source (see Fig. 4a and 4b). The average
deviation $\overline{\delta\theta}$ of the filament angles from the purely
radial direction is close to zero near the hub, suggesting that the skeletons
are radially aligned to the hub. The mean deviation from the purely radial
direction $\sqrt{\overline{{\Delta\theta}^{2}}}$ is a slowly increasing
function of the radius, suggesting that the skeletons are less radially
aligned at larger distances from the hub. The visual spiral appearance
(Treviño-Morales et al., 2019), which is prominent only within the hub region,
may be represented by the dip at $\sim$1 pc in Fig. 4a and the change in the
curve direction inside the hub region in Fig. 4b. The region at $\sim$1.5 pc,
where the angles change, is also the radial distance at which the number of
filaments has a maximum (Fig. 5). This may be an indication of the bending of
filaments, possibly due to the rotation of the hub with respect to the outer
areas, because the density profile is smooth and shows no such breaks at
$\sim$1.5 pc (Fig. 3).

In a radially symmetric HFS, if all the filaments were to move inwards towards
the hub centre, the number density of filaments should be very large inside
the hub. In Fig. 5, the radial variation of filament M/L, together with the
number of filaments is shown. The M/L and the number of filaments can be seen
to vary in opposite trends. The M/L of filaments 2–3 pc away from the hub is
typically a few and up to 10 $M_{\odot}\,$ pc-1. From about a radial distance
of 1–1.5 pc, the M/L increases smoothly towards the hub, reaching $\sim$ 100
$M_{\odot}\,$ pc-1 within the central 0.4 pc radius (the core of the hub)
where the curve flattens out. However, the number of filaments decreases
towards the centre of the hub at distances $r\sim$1–1.5 pc. In Fig. 5 the
number of filaments represents the number of skeletons that intersect
concentric circles drawn around the hub as shown in Fig. A.2. This decrease in
filament numbers and increase in M/L towards the hub suggests the coalescence
of lower-density filaments into higher-density filaments. If the filaments did
not coalesce or if they were in a uniform arrangement of filaments, the number
of filaments intersecting the concentric circles would not show the trends of
Fig. 5. A variation of Fig. 5 where the number of filaments per unit area (N),
counted in 0.2 pc bins, is plotted in Fig. 6. Viewing from a larger to a
smaller radius, here again, the M/L and N increase with decreasing radius from
3 pc to 1 pc; however, within 1 pc of the hub, N flattens out while M/L
continues to increase towards the centre, suggesting coalescence.

As seen in Figs. 5 & 6, the effect of an increase in M/L and a reduction in
the number of filaments is prominent within the hub-core of radius 0.4 pc. At
a radius of 1-2 pc, a large number of low M/L filaments are found to be
concentrated which can be seen as a peak in the number of filaments. There is
a hint of a second peak in the number of filaments at 0.6-0.8 pc (depending on
the intervals of concentric circles used to produce the curve), which may be
indicative of coalescence taking place in two steps in this target.

The detected filaments appear in a range of lengths and column densities all
of which are radially converging to the central hub, where the column density
reaches its peak values (Fig. 2a). The white cone-like boundaries were grouped
by eye to identify groups of lower-density filaments that appear to culminate
at the tip of the cone which marks the beginning of a higher-density filament.
Within these groups of filaments, there are junctions with higher column
densities, which are marked with red and blue circles. Each of the groups
marked by the white boundaries represents cascades of lower-density filaments
which appear to funnel into the higher-density filaments, which in turn funnel
into even higher-density filaments. We interpret that the coalescence is
taking place in at least two steps. The white cones represent the first step
of coalescence, and an example of the second step is marked by the black cone
in Fig. 2a. From Fig. 2b, these steps are reflected in the three density
ranges represented by the greenish yellow colour that leads into red-coloured
skeletons which in turn funnel into the short white skeletons at the centre.
At each step, these colour changes can be read out as changes in column
density that are found to increase by a factor of $\sim$5–10 as the skeleton
length decreases. In Fig. A.3, we display radial profiles of the azimuthally
averaged column density within three groups (marked in Fig. A.2) of coalescing
filaments. These profiles show several jumps, but in all three profiles, the
slopes change once at 0.8 pc and again at 0.2 pc. At larger radii, the changes
are difficult to discern, however, groups 1 and 3 display a common bump at
$\sim$1.4 pc. The two peaks in the number of filaments at $\sim$1.5 pc and at
0.8 pc in Fig. 5 likely correspond to the changes apparent in Fig. A.3. The
net change in surface density along these cascades is from
$\sim$5$\times$1020cm-2 to $\sim$2$\times$1023cm-2 in the densest portion at
the centre just below the Mon R2 B-type star. The density jumps become
prominent in the direction of the radial converging point inside the central
hub ($r=$0.8 pc). In Fig. 3, the steep and smooth rise in the filament
component as a function of radius shows that the column density of filaments
is gradually increasing towards the centre, at least in the inner 2.5 pc
radius of the cloud. This is in agreement with the description of cascades of
low column density filaments funnelling towards higher-density filaments.

Given that the filament networks are observed in projection, not all aligned
features are coalescing. The identified groups are the regions, where the
observed column density of the coalesced filaments are roughly equal or larger
than the sum of the column densities of the filaments that are merging. The
containment of the filaments within the sheet-like cloud enhances the
probability of viewing several coalescing junctions, as evidenced by the
measurable density jumps that are marked by circles in Fig. 2a. The incidence
of filament merging may be enhanced due to the radial alignment towards the
hub. Hence, based on the radial variations of the number of filaments and M/L,
and that of column density junctions and filament angles, the analysed data
are best described by coalescence of molecular filaments in Mon R2.

###  4.2 Hub-filament system mimicking global collapse

Figure 7: Hub is a high-density filament network along which massive stars
form. Left: Column density of the filament component of the hub region at
11.4′′ resolution. The most massive sources known in the region, namely IRS1,
IRS2, IRS3, IRS4, and IRS5, are marked. The blue and maroon arrows indicate
the filaments leading to the junctions in which IRS3 and IRS5 are embedded,
respectively. The red square corresponds to the cluster (radius = 0.5 pc)
(Carpenter et al., 1997), which is zoomed in on in the right panel. Right:
Deep $K$ band (2$\rm\,\mu m\,$ ) image overlaid with contours of the filament
column density image shown in the left panel. Readers should notice the
striking correspondence between the infrared dark patches and the column
density contours. The IRS sources are marked by red dots.

Fig. 3 shows the averaged radial column density profile of the observed image,
filaments, and diffuse components in a 5 pc$\times$ 5 pc projected region
centred on the Mon R2 hub. The annular region excluding the hub ($r$ = 0.8 pc)
and extending up to 2 pc can be well fitted by power-law functions. The column
density of the filaments and diffuse cloud components can be fitted by
N${}_{H_{2}}\propto r^{-2.17}$ and N${}_{H_{2}}\propto r^{-0.38}$,
respectively, where $r$ is the radius from the hub centre, whereas the
observed image shows an N${}_{H_{2}}\propto r^{-1.35}$ relation. Sources
contribute a small fraction to the total column density (mass fraction of
9–15%, see Table. 1), so they are not plotted. The line-of-sight averaging
affects the measured column densities and temperatures, and therefore
intrinsically one expects lower temperatures and higher densities (Roy et al.,
2014). Therefore, the radial profile of the surface density of the filament
component may be steeper than the observed N${}_{H_{2}}\propto r^{-2.17}$
relation, but not shallower. Given that the filaments in Mon R2 are
constrained in a sheet-like volume, almost in the plane of the sky, the
observed filament N${}_{H_{2}}$ profile represents the volume density profile
$\rho(r)$. Such profiles are indicative of an isothermal sphere in equilibrium
or at the verge of collapse (Ward-Thompson et al., 1994). Therefore it appears
that the dense gas (filamentary) component of the hub-filament system,
implying a $\rho(r)$ profile steeper than $r^{-2}$, is mimicking the
conditions for gravitational collapse on parsec scales.

Such a result has been previously suggested by the velocity gradients (Hacar
et al., 2017a) observed in the Orion ISF, especially towards the BN/KL region.
The directional orientation of the velocity gradients towards the
gravitational potential of the hub led those authors (Hacar et al., 2017a,
2018) to suggest that the global collapse scenario may be at work in the Orion
ISF. Such behaviour is in some ways similar to that expected from the models
of global hierarchical collapse (Vázquez-Semadeni et al., 2017). This global
collapse represents motions of overall dense gas in the hub-filament system
towards the hub, similar to the coalescing filaments moving towards the hub in
Mon R2 (see Sect. 4.5). This collapse is different from the accretion of
matter through longitudinal flows (Peretto et al., 2014; Williams et al.,
2018) responsible for replenishing the hub, as represented in the F2C model of
Kumar et al. (2020). However, both scenarios suggest a flow of material
towards the hub, which is of interest for the formation of massive stars in
the hub. This prompts us to estimate the fraction of material in an HFS that
is held in the filaments.

###  4.3 Filament to hub mass ratio

As shown in Sec. 3.2.2, the filament to hub mass ratio is roughly unity. This
estimate, however, depends on the tracer used to calculate the mass. This is
evident from Table 2 of Treviño-Morales et al. (2019), where the estimates
using CO lines suggest that the hub, which is at a radius of 1 pc by their
definition, holds a significantly lower mass (1600$M_{\odot}\,$) compared to
the filaments (3200$M_{\odot}\,$) suggesting a filament:hub ratio of 2:1. On
the contrary, their dust estimates show that this ratio is inverted with
filament (2500$M_{\odot}\,$):hub (3600$M_{\odot}\,$) of 1:1.4 (Treviño-Morales
et al., 2019). The lower mass estimate from CO lines may be a direct
consequence of the line opacity and the removal of gas from the warmer and
denser core of the hub. The filament reservoir is important for massive star
formation because massive stars are known to derive mass through longitudinal
flows. If the currently B-type young star (IRS 1) in the Mon R2 core region
were to accrete further mass at rates of 10-3-10-4$M_{\odot}\,$yr-1, as
estimated from the observed longitudinal flows (Treviño-Morales et al., 2019),
it can grow to become an O star, according to the F2C scenario (Kumar et al.,
2020).

In measuring the longitudinal flow rates, Treviño-Morales et al. (2019)
conclude that the main (higher-density) filaments transport mass to the
central hub at a rate that is four times higher than the secondary (lower-
density) filaments to the main filaments. If we consider the coalescence
effect in the radial direction, where the main filaments are composed of
higher-density structures, that implies that longitudinal flow rates may be
simply proportional to the density of the filament. Comparing the annuli in
Table 1, the relative mass held in the dense filaments is the highest in the
hub and slowly declines away from the hub. The diffuse cloud displays a
reverse trend. Because the column density correlates with star formation
efficiency, the number of sources also (naturally) follows the trend of the
dense gas filaments. Sources in these column density maps trace both pre-and
proto-stellar cores. The listed mass fraction (%) of sources in Table 1 is,
therefore, a rough indicator of the core+star formation efficiency within the
respective annular regions. It can be seen that the source formation
efficiencies in the central hub-core with a radius of 0.4 pc and in the
annular region of 1.6–2.5 pc radius are 3.4% and 1.5%, respectively.

###  4.4 Structure of hubs and formation of massive stars

Myers (2009), in his original definition of hubs, identified them as high-
column density low-aspect-ratio objects similar to star-forming clumps. Even
after several subsequent higher-resolution observations (Schneider et al.,
2012; Peretto et al., 2014; Williams et al., 2018; Chen et al., 2019; Treviño-
Morales et al., 2019; Kumar et al., 2020), the view of the hub as a massive
dense clump has remained. This is because the farther away ($>$ 1 kpc) objects
have mostly been studied by higher spatial resolution observations and the
relatively nearby region Mon R2 has been studied with beams of 36′′, before
this work. From Figs. 1 and 2, and subsequent identification of the hub from
Fig. 3 as the region with a radius of 0.8 pc, it is found to be a complex
network of short, intertwined, high-column density filaments that coincides
with the embedded young stellar cluster.

This new view of the hub structure has important consequences for our
understanding of high-mass star formation. Even though matter can be
transported and fed to the stars inside the hub through longitudinal flows, in
our new view, whether a star will benefit from it and gain mass depends on
whether the stellar seed is embedded inside one of the dense filaments within
the hub network. Therefore, only some, and not all, of the stellar seeds
within the hub can become massive stars. This hypothesis is already supported
by the existing data. In Fig. 7a, we display the filaments component map at
11.4′′resolution, overlaid with the most massive and luminous sources known in
Mon R2, namely IRS1–5 (Hackwell et al., 1982; Henning et al., 1992). The
positions for some sources were updated using newer astrometry from the
Submillimeter Array (SMA) observation of the central region (Dierickx et al.,
2015). The central one-parsec region shown by the red circle in Fig. 7a is
zoomed in on in Fig. 7b, where a $K$-band image from the UKIDSS (Lawrence et
al., 2007) Galactic Cluster Survey (GCS) is overlaid with contours of the
column density from the image on the left. It is evident from these images
that the most massive sources are located on the filament network grid and not
in the cavities between them. The luminosities of IRS1, IRS2, IRS3, IRS4, and
IRS5 sources are 2000, 100, 3000, 800, and 300 $\rm L_{\odot}\,$ respectively
(Hackwell et al., 1982; Henning et al., 1992; Didelon et al., 2015). Both IRS1
(the well-known B star) and IRS2 (bright $K$-band star) are continuum sources
displaying free-free emission. IRS3 is the highest-luminosity object and a
previously known hot molecular core (Boonman & van Dishoeck, 2003), driving a
powerful massive outflow (Dierickx et al., 2015). IRS5 is also a young object
with a rich chemistry (Dierickx et al., 2015). IRS4 is the lowest-luminosity
object in the region. IRS1 is located at the highest column density point in
the region. IRS3 and IRS5 are the youngest and most massive sources, located
in two separate coalescing junctions identified by the blue and maroon
coloured arrows in Fig. 7a. IRS2 is at the centre of the ring-like cavity seen
in the $K$-band image, which corresponds to the compact HII region of Mon R2.

The locations of the massive sources above are well represented by the F2C
scenario (see Fig. 14 of Kumar et al., 2020). Although the results in this
paper move a step further from the simplistic two-filament case depicted by
the F2C model, the highest-density filament of the region (white filament in
Fig. 7a) is a junction of the filaments marked by the blue arrows. IRS1 and
IRS3 represent the older and the younger pair of massive sources from the F2C
model, both competing for the same source of longitudinal flow. IRS1 in
reality represents the massive star embedded in the white filament (Fig. 7a),
corresponding to the optically visible B star which is revealed through a hole
punched out in the densest filament. The ring-shaped compact H II region in
Fig. 7b is driven by this B star. IRS3 is the most luminous object of the
region, located in the exact junction leading to the densest filament. It
drives a massive outflow, and therefore it may at present be accreting through
the direct longitudinal flows fed through the filaments represented by blue
arrows. It is interesting to note that IRS1, IRS3, and IRS4 are equally spaced
in the densest filament and therefore the basic stellar seeds may be the
result of filament fragmentation. However, the evolution of these seeds to
become a massive star depends upon a competition for the mass reservoir that
is present within the densest filament itself and which is arriving through
the longitudinal flows.

The network of intertwined filamentary structures within the hub enhances the
efficiency of feedback dispersal. The lower column density material between
the filaments (with a radial profile of $\Sigma\propto r^{-0.38}$, Fig. 3)
allow the ionising pressure from the young massive stars that formed in the
dense filaments of the grid-like hub to escape out of the system (Dale &
Bonnell, 2011; Kumar et al., 2020). At the same time, the matter accretion
continues along the filaments feeding the stellar seeds of the densest
filaments in the hub that can evolve into massive stars. Together with its
network-like structure, the key to ’holding off’ massive star formation until
the end may come from the magnetic support that threads the dense filaments of
the network (see Sec. 4.5). This allows the build-up of massive cores in very
dense filaments and supports them from collapsing until high densities are
reached so that when the core collapses, it can begin massive star formation
and benefit from additional high accretion rates from longitudinal flows to
grow further. In the hub, the network of the filaments has a range in
densities, albeit high, which can increase with time, and that can lead to a
range in the mass of the massive cores. Depending on the structure of the
network, longitudinal flows feeding these massive cores can also differ. These
differences may be the factor responsible for producing groups of OB stars
with a spectrum of masses in the hub. A good example would be the Trapezium
cluster in Orion, with four stars in the range 15–30 $M_{\odot}\,$that could
have formed from an intertwined network of dense filaments similar to that
observed in Mon R2.

Figure 8: $\it{Herschel}$ column density map at 18.2′′resolution overlaid by
grey streamlines displaying the plane of the sky B fields which were computed
using the 353 GHz observations from the Planck mission. The spatial resolution
of the Planck data is 4.9′(as shown by the grey circle in the lower left of
the image). The Stokes $Q$ and $U$ components were used to obtain the B-field
position angles. The results are displayed as grey streamlines by using
streamplot in matplotlib, as thicker lines for the signal-to-noise ratio, S/N
$>$ 2.5, and thinner ones for 2.0 $<$ S/N $<$ 2.5.

###  4.5 Magnetic fields: Critical support against global collapse

The apparent global collapse inferred from the radial dependence of the
densities (Sect. 4.2) roughly extends to a diameter of 4-5 pc in Mon R2, while
the velocity gradients found in the central regions of OMC1 (Hacar et al.,
2017b) extends to about 2 pc size. In both cases, the signatures of collapse
are directed towards a hub located in a larger molecular cloud complex. These
examples may indeed be representative of a more general scenario extending to
other star-forming regions with prominent HFS. Evidence presented by Treviño-
Morales et al. (2019) suggests that Mon R2 in all likelihood is a flattened
sheet-like structure. If the OMC 1 also represents a sheet-like structure
viewed edge-on, it is possible that the results here and that presented by
Hacar et al. (2017b), are pointing to a similar phenomenon. In Mon R2, the
radial orientations of the filaments and the collapse features over scales of
4-5 pc may be viewed through multiple scenarios, including for example: a)
formation of HFS in a sheet-like cloud compressed by a propagating shock
front, with filaments converging towards the hubs (Inoue et al., 2018), or b)
two clouds that are part of a larger virialised cloud complex colliding with
each other resulting in filament formation, and the lateral contraction of the
compressed layer in an inside-out manner leading to radial alignment of the
filaments and filament coalescence (Balfour et al., 2015, 2017). In the F2C
model (Kumar et al., 2020), stage I requires filament overlaps to take place
to form hubs, allowing all possible scenarios such as a) and b) above to
operate. It should be noted that the observed collapse features, both in Mon
R2 and OMC1, are confined to the inner regions of a much larger sheet-like
cloud, and not to the entire cloud, as may be misunderstood by the term
’global collapse’.

In Mon R2, the central region studied here is represented by star formation
efficiencies of 5-10% (Pokhrel et al., 2016). This was obtained by assuming a
gas-to-star density relation (Gutermuth et al., 2011), which is very similar
to other nearby star-forming regions (Pokhrel et al., 2016, 2020) in Gould’s
belt. More interestingly, the observed Kennicutt-Schmidt-like relation in this
cloud and other clouds has been argued to be linked to a tight linear
correlation between the star formation rate per unit area and their gas
surface density normalised by the gas free-fall time (Pokhrel et al., 2021).
The 5-10% efficiency in Mon R2 is higher than the global cloud-scale star
formation efficiencies that are typically 2-3% and free-fall times of 0.5-2
Myr for nearby molecular clouds (Evans et al., 2014; Lada et al., 2013).
However, if the gas reservoir in these collapsing central regions is turned
into stars with an efficiency similar to that of core-to-star efficiency
(Alves et al., 2007), the expected star formation efficiencies are
$\sim$30-50% which would be much higher than the estimated 5-10%. Given that
such high star-formation rates implied by accelerated collapses are unknown,
not only in Mon R2 but for all well-studied nearby star-forming regions, it
appears that despite the observed signatures, the HFS is at least partially
supported against the global collapse. Such support can be provided by either
higher turbulence or magnetic fields. The observed line widths of various
molecules in both Mon R2 (Treviño-Morales et al., 2019) and Orion ISF (Hacar
et al., 2018) do not suggest any unusual non-thermal support against such a
global collapse. The non-thermal support is minimal (filaments in virial
equilibrium (Arzoumanian et al., 2013)) in these high-density regions, as
reported by the observations of fibres in the Orion ISF (Hacar et al., 2018).
Therefore, the critical support needed against the global collapse of the Mon
R2 HFS must come from magnetic fields. Here, we suggest a simplistic scenario
depicting the role of magnetic fields. Assuming that the B field is frozen
into the filaments quasi-statically, as the filaments come together and
coalesce if the field also merges and reorganises itself, it can provide the
necessary support against the global collapse. This scenario is at the outset
well represented by the Planck observations (Fig. 8).

Throughout much of the region, the B fields are orthogonal in most filaments,
and not so much in a few, for example the northern main arm (Fig. 8). Because
the fields are projected onto the sky (2D) and obtained with a large beam of
4.9′, we may expect relatively more inclined projections for some places in
Mon R2, with its inclination of 30∘ to the line of sight. Results prior to the
Planck mission using starlight polarisation (Palmeirim et al., 2013), from
Planck (Planck Collaboration Int. XXXII, 2016; Planck Collaboration Int.
XXXIII, 2016; Planck Collaboration Int.XXXIV, 2016) and those combining the
starlight and dust polarisation (Cox et al., 2016), show that B fields are
parallel to low-column density filaments and perpendicular to high-column
density filaments. More recent observations (after 2016) at higher angular
resolutions, using JCMT and SOFIA facilities (Pattle et al., 2019; Santos et
al., 2019; Pillai et al., 2020; Arzoumanian et al., 2021), show that this is
not exactly the case, especially for high-column density filaments. The B
field inside the filaments (when resolved) shows some variations and it is not
always perpendicular. Pillai et al. (2020) also found similar results to the
post Planck studies and suggest the presence of two transitions, one from
parallel to perpendicular at A${}_{v}\sim$7-8 mag (similar to Planck results
Soler et al. (2016)) and a transition from perpendicular to parallel for
A${}_{v}>$21 mag, where the B field was dragged by the flow and gravity along
the filaments, compatible with the simulations of Gómez et al. (2018).

In Mon R2, the filaments identified by our analysis are all well above
A${}_{v}\sim$7 mag and they coalesce to form higher column density filaments
towards the hub. Therefore, to the first order, all dense filaments have
magnetic fields perpendicular to their axes in Fig. 8, and less orthogonal
orientations between field lines and filaments can be viewed as effects of
projection. If the B fields are frozen to the filaments quasi-statically,
then, as the filaments coalesce and move towards the radial centre of the hub,
they come closer, resulting in the interaction of B fields frozen to
individual filaments. This can lead to field distortions and reorganisations
naturally. The dip in the field lines between the two eastern arms in Fig. 8
may be hinting at such interactions. Although the B field may be dragged by
flows within some filaments, it is not a ubiquitous feature and depends upon
local conditions and the strength of longitudinal flows. On the contrary, the
frozen-in-fields with a largely orthogonal orientation to the dense filaments
are ubiquitously observed (Palmeirim et al., 2013; Cox et al., 2016; Soler et
al., 2016). It is unknown if the B fields are smoothly connected between the
filaments throughout the cloud, such as the results by many numerical
simulations (e.g. Gómez et al., 2018; Inoue et al., 2018). Given that the
nature of the magnetic field in the low-density regions between filaments is
unknown so far, field interaction of the frozen-in fields between filaments
must be investigated. Mon R2 is an excellent target to study if the changes in
field direction take place because of the flow-induced entrainments, field
interactions, or a combination of both.

The magnetic field structures within the hub regions are relatively uncertain,
especially in the light of the results that the hub is a network of short
dense filaments, as opposed to the idea of a uniform clump. B-field mapping of
the hub with higher spatial resolutions can shed light on its configuration.
However, we suggest that the overall field direction would result in hourglass
shapes, perpendicular to a flattened hub (a network of high-density filaments
in a sheet-like space). This inference is guided by comparisons of the
observations of known HFS. In the nearby Ophiuchus region (Pattle et al.,
2019; Santos et al., 2019), the polarisation data provide a spatial resolution
of $\sim$0.01 pc. If we view Oph A as an edge-on hub in the shape of an arc,
then the magnetic fields are oriented perpendicular to the arc (Fig. 2 of
Santos et al., 2019). In NGC6334 (Arzoumanian et al., 2021), the average B
field runs perpendicular to the main clump-hub at 1 pc scale (similar in size
to the hub in Mon R2), and a similar situation is in the main hub of Serpens
(Pillai et al., 2020). Such field configurations of the hubs are crucial to
provide strong support against the collapse of the hub itself, instead of
allowing it to accrete large amounts of material from the surrounding
filaments (a secondary reservoir in the F2C model), therefore aiding the
formation of massive stars within the hub (Kumar et al., 2020).

##  5 Discussion

Returning to the original questions posed in Sec. 1, namely, what the
difference is between bundles of fibres forming higher M/L filaments and a
hub, which is a junction of filaments. It appears that fibre and filament
coalescence is the key mechanism that operates in both circumstances.
Essentially, fibres and filaments of gas and dust in a dynamical molecular
cloud coalesce in different ways, depending on the external influencing
factors, such as shock waves and gravitational potentials. In this coalescence
process, it appears that the angle of incidence between filaments forming the
junction and the gravitational potential difference across the junction
control the final product. For example, acute angles are more likely to
produce high linear-density (high M/L) filaments, while obtuse angles may
result in hubs. If one were to suppose that there are competing centres of
gravitational potential formed by two hubs located a few parsecs away, the
potential gradients would impact how the coalescing structures evolve.
Similarly, if a shock wave were to sweep the filaments into a sheet, or be
compressed in a layer during cloud-cloud (Inoue & Fukui, 2013; Fukui et al.,
2021) or bubble-bubble (Inutsuka et al., 2015) collision, it may enhance the
formation of bundles and high M/L filaments. Elaborate studies are required to
study and understand these dynamic conditions.

For example, Hacar et al. (2017b) note that the super-critical fibres in
NGC1333 survive for roughly five free-fall times as independent entities,
which they attributed to the support provided by non-thermal motions. In
contrast, the dense network of fibres observed in the Orion ISF (Hacar et al.,
2018) is essentially located within the boundaries of the hub (radius of 0.8
pc), similar to that in Mon R2. By their very nature, hubs are junctions that
formed as a consequence of dynamical evolution within the cloud; therefore,
they are younger structures compared to the rest of the cloud. So, it is
unclear how long such bundles of velocity coherent fibres (near or within the
hub) can survive before the internal velocity differences diffuse and the
bundle itself becomes velocity coherent leading to a coalesced high-linear-
density filament collapsing onto its axis and fragmenting to form stars
(Inutsuka & Miyama, 1997).

The observational features that mimic global collapses, such as the radial
density profiles in Mon R2 or the steep velocity gradients close to the hub in
Orion ISF (Hacar et al., 2017b), imply that the gravitational potential in
these HFS is significant and influences the surrounding material up to at
least 2 pc. In Mon R2, this hub potential may be responsible for the bulk
contraction of the central cluster as evidenced by kinematic data of stars
(albeit with large errors) (see Table 2 of Kuhn et al., 2019). Also, the
radial focussing of the filaments found here is driven by the gravitational
potential of the hub. If the HFS are in a 2D geometrical configuration of a
layer, both the dense and diffuse components of the clouds will be compressed
into the layer. The hub’s significant gravitational potential can then even
lead to the formation of further filaments close to the hub, by streaming the
diffuse cloud component along the radially directed gravitational field.

When fibres and filaments coalesce to form a more spherical and circular
structure, whether it remains that way or evolves into a different structure
appears to depend on internal density gradients and external forces. In Mon
R2, as marked by the red and blue circles in Fig. 2a, there are several
junctions with density enhancements of low aspect ratios, satisfying the
definition of Myers (2009) to be referred to as a hub. However, these
junctions in Mon R2, especially those that are exterior to the hub, have
strong internal density gradients, that are directed towards the hub;
therefore, they are more likely transitory in the sense that they are the
nodes feeding higher-linear-density filaments, and moving towards the main hub
along with other coalescing structures. This detailed view of the junctions in
Mon R2 and also of those observed in NGC1333 and Orion ISF suggests that they
may have a range in sizes, depending on the number and density of the
filaments forming the junction. The top-level structures resulting from a
hierarchical coalescence in a dynamically evolving cloud correspond to the
main hubs as originally defined in nearby clouds by Myers (2009) and
represented by the HFS clumps studied by Kumar et al. (2020).

In its simplest form, the volume density of cold dense matter controls the
process of star formation, consequently influencing the key observational
properties, as often demonstrated by the studies of the probability density
functions (Kainulainen et al., 2014), such as rho-PDF (volume) and N-PDF
(surface). Filament coalescence may be a key factor in assembling the density
spectrum over several dynamical scales in the lifetime of molecular clouds.
During this time, the density functions can also be influenced by the
gravitational potential of clumps and hubs from within, and by external shock
waves that could restructure the cloud material, for example into sheets,
enhancing the efficiency of coalescence, as in the case of Mon R2.

###### Acknowledgements.

We thank the referee for critical and thorough scrutiny that greatly enhanced
the flow of the manuscript and improved the clarity of the presentation. MSNK
thanks Prof. Anthony Whitworth for stimulating discussions on sheets and
cloud-cloud collisions. MSNK acknowledges the support from FCT - Fundação para
a Ciência e a Tecnologia through Investigador contracts and exploratory
project (IF/00956/2015/CP1273/CT0002). PP acknowledge support from FCT/MCTES
through Portuguese national funds (PIDDAC) by grant UID/FIS/04434/2019. PP
receives support from fellowship SFRH/BPD/110176/2015 funded by FCT(Portugal)
and POPH/FSE(EC). M. M. is supported by JSPS KAKENHI grant No. 20K03276. The
$\it{Herschel}$ data were obtained from the Herschel Science Archive (HSA) and
were obtained through the guaranteed time key programmer HOBYS (PI: F. Motte).
We thank Pierre Didelon for providing the calibration offsets that were
necessary to obtain flux measurements from PACS data.

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##  Appendix A Structural decomposition maps

Figure 9: Structural component decomposition. The images shown are: a)
$\it{Herschel}$ column density map at 18.2′′, b) filaments, c) extended
background, and d) sources. The total flux in the image area is conserved with
the column density map representing a sum of its structural components. The
colour bar is common to all panels. Figure 10: Number of filaments that
intersect with the white circular annuli drawn on the filament-component map
masked by skeletons which were used to plot the black curve in Fig. 5. Figure
11: Radial profiles of azimuthally averaged column density for the three
groups of coalescing filaments identified in Fig. A.2. We note that these are
profiles obtained on the filament-component map masked by detected skeletons.
These groups encompass a larger number of filaments at larger distances while
narrowing down on individual filaments closer to the centre. Readers should
notice that the slopes of the radial profiles changes once at $\sim$0.8 pc and
again at $\sim$0.2 pc. At larger distances from the centre, it is not possible
to discern noticeable effects. The vertical lines at 3 pc, 1.3 pc, and 0.85 pc
correspond to the two outermost red circles (nodes), the red circle just above
the black cone, and the outer boundary of the black cone in Fig. 2a,
respectively. They demarcate the two steps of coalescence suggested in Sec.
4.1