# Convergent Filaments Contracting Towards an Intermediate-mass Prestellar
Core

Zhiyuan Ren1,3, Lei Zhu1,3,7, Hui Shi1,3, Nannan Yue1,2,3, Di Li1,2,3,4 Qizhou
Zhang5, Diego Mardones6, Jingwen Wu1,3, Sihan Jiao1,3, Shu Liu1,3, Gan Luo1,3,
Jinjin Xie1,3, Chao Zhang1,3, Xuefang Xu8
1 National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Road, Chaoyang District, Beijing 100101, China
2 University of Chinese Academy of Sciences, Beijing 100049, China
3 CAS Key Laboratory of FAST, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing, P.R. China, 100012
4 NAOC-UKZN Computational Astrophysics Centre (NUCAC), University of KwaZulu-
Natal, Durban 4000, South Africa
5 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02318, USA
6 Departamento de Astronomia, Universidad de Chile, Casilla 36-D, Santiago,
Chile
7 Chinese Academy of Sciences South America Center for Astronomy
8 Chongqing University, No.174 Shazhengjie, Shapingba, Chongqing, 400044,
China E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Filamentary structures are closely associated with star-forming cores, but
their detailed physical connections are still not clear. We studied the dense
gas in the region of OMC-3 MMS-7 in Orion A molecular cloud using the
molecular lines observed with the Atacama Large Millimeter/submillimeter Array
(ALMA) and the Submillimeter Array (SMA). The ALMA ${\rm N_{2}H^{+}}$ (1-0)
emission has revealed three dense filaments intersected at the center,
coincident with the central core MMS-7, which has a mass of $3.6\,M_{\odot}$.
The filaments and cores are embedded in a parental clump with total mass of
$29\,M_{\odot}$. The ${\rm N_{2}H^{+}}$ velocity field exhibits a noticeable
increasing trend along the filaments towards the central core MMS-7 with a
scale of $v-v_{\rm lsr}\simeq 1.5$ km s-1 over a spatial range of $\sim$20
arcsec ($8\times 10^{3}$ AU), corresponding to a gradient of $40\,{\rm
km\,s^{-1}}\,{\rm pc}^{-1}$. This feature is most likely to indicate an infall
motion towards the center. The derived infall rate ($8\times
10^{-5}\,M_{\odot}$ year-1) and timescale ($3.6\times 10^{5}$ years) are much
lower than that in a spherical free-fall collapse and more consistent with the
contraction of filament structures. The filaments also exhibit a possible
fragmentation, but it does not seem to largely interrupt the gas structure or
the infall motion towards the center. MMS-7 thus provides an example of
filamentary infall into an individual prestellar core. The filament
contraction could be less intense but more steady than the global spherical
collapse, and may help generate an intermediate- or even high-mass star.

###### keywords:

ISM:clouds; ISM: individual objects (MMS-7, Orion); ISM: molecules; stars:
formation; stars: intermediate-mass

††pubyear: 2019††pagerange: Convergent Filaments Contracting Towards an
Intermediate-mass Prestellar Core–LABEL:lastpage

##  1 Introduction

Filaments are widely existing structures in cold dense interstellar medium
(e.g. Ragan et al., 2012; Dirienzo et al., 2015; Wang et al., 2016). They are
observed to have multiple substructures and velocity components, which should
considerably facilitate the mass aggregation and star-forming processes
therein (e.g. Hacar et al., 2013; Henshaw et al., 2014; Shimajiri et al.,
2019). The intersected and converging filaments could have more significant
association with the formation of clustered and massive young stars (Peretto
et al., 2014; Fernández-López et al., 2014; Williams et al., 2018; Chen et
al., 2019; Treviño-Morales et al., 2019; Hu et al., 2021; Liu et al., 2021).
In several representative cases, sharp velocity gradients have been observed
along the filaments towards the central star-forming cores (Peretto et al.,
2013; Kirk et al., 2013; Hacar et al., 2017), which could represent converging
flows feeding young stellar objects (YSOs). Besides the converging motion, the
filaments could also have longitudinal collapse to generate dense clumps and
cores at two ends, as shown in both theoretical (Clarke & Whitworth, 2015) and
observational (Dewangan et al., 2019; Yuan et al., 2020) works.

Although the mass transfer flows are identified from velocity gradients in a
number of regions, their connection to individual star-forming cores are still
not widely confirmed. It is uncertain whether a filament can directly support
the accretion to an YSO or merely influence the core evolution in less direct
ways, such as increasing the perturbation and compressing the gas. An external
mass supply instead of isolated core collapse is also speculated in the
competitive accretion model (Bonnell & Bate, 2006; Clark et al., 2008). But
the detailed process of the gas transfer into the individual star-forming
cores is still largely unknown, and should be explored in subsequent
observations.

Orion A molecular cloud is an ideal target to study the mass transfer and
accretion process. Its main structure is a well-known dense and compact
integral-shaped filament (ISF) elongated from north to south (Johnstone &
Bally, 1999; Salji et al., 2015; Hacar et al., 2018, etc.). It has an average
distance of 390 pc (Großschedl et al., 2018), and a length scale of 6 pc.
There are dozens of dense star-forming cores on the filament structure (Li et
al., 2013; Takahashi et al., 2013; López-Sepulcre et al., 2013; Kirk et al.,
2017). The cloud contains complex substructures which are mostly in a cold and
quiescent state (Hacar et al., 2018; Zhang et al., 2020). Over a spatial scale
of 0.5 pc around the cloud, there are nearly 400 protostars and 3000 more
evolved (Class-II to III) young stars (Megeath et al., 2012). The young stars
have an overall coherent spatial distribution with the molecular cloud (Stutz
& Gould, 2016; Großschedl et al., 2018), and would be influencing the gas
components in both radiative and dynamical ways (Megeath et al., 2012; Da Rio
et al., 2016; Qiu et al., 2018).

MMS-7 is a dense molecular core in Orion A OMC-3 region (Takahashi et al.,
2006). We investigated this region with a joint analysis of different
molecular tracers, including the ${\rm N_{2}H^{+}}$ (1-0) observed with ALMA
and ${\rm C^{18}O}$ (2-1) observed with the Submillimeter Array (SMA). From
these data we can more completely inspect the dense gas around the center as
well as more extended structures. In addition, the ALMA spectra can exhibit
the gas velocity field with a much better sensitivity and spatial resolution
than the previous observations, thereby reveal the interplay between the
filament-based gas motion and the star-forming cores in more details.

In Sec. 2, we described the observation. In Sec. 3, we presented the filament
and dense core structures and the velocity distribution. The dynamical origin
of the velocity distribution is discussed in detail in Sec. 4, including the
overall dynamical condition in Sec. 4.1, energy scales in Sec. 4.2, the
velocity field of each filament in Sec. 4.3, gravitational instability in Sec.
4.4, and evolutionary state in Sec. 4.5. A summary is given in Sec. 5.

##  2 Observation and Data Reduction

The ${\rm N_{2}H^{+}}$ (1-0) data around MMS-7 is from a large scale mapping
over the Orion A OMC-2,3 region (Zhang et al., 2020, also see Figure 1a). The
ALMA observation of OMC-2/3 (proposal ID 2013.1.00662.S, PI: Diego Mardones)
was carried out during November 2014 and August 2015 in Band 3 using the 12-m
main array and the Atacama Compact Array (ACA). The baselines of the 12-m main
array and the ACA observations range from 13.6 to 340.0 m and 6.8 to 87.4 m,
respectively. We combined the UV datasets of the 12-m array and ACA and
CLEANed it in CASA (McMullin et al., 2007) using the interactive model. The
restored images was corrected by Primary-beam collection. The synthesized
full-width half-maximum beam size is $3\times 3^{\prime\prime}$, the maximum
spatial coverage is $94^{\prime\prime}$. The standard deviation noise level
per channel is $\sigma=7$ mJy beam-1 (corresponding to $T_{\rm b}=0.14$ K).

The SMA data is publicly available in the SMA data archive. The observation is
performed in March 2014 in the compact array. The receiver 0 is used with the
local oscillation frequency tuned at 225 GHz. The three isotopologue CO
$(2-1)$ lines are included in the sidebands. The velocity resolution is 0.7 km
s-1. The sensitivity in the line observation is 0.5 K. The synthesized images
have a beam size of $6^{\prime\prime}.1\times 4^{\prime\prime}.5$ with a
position angle of $39^{\circ}$ to the northeast. The spatial structures
smaller than $30^{\prime\prime}$ can be restored by the SMA baselines.

We also referred to the JCMT/SCUBA $450\,{\rm\mu m}$ continuum image
(Johnstone & Bally, 1999) to inspect the global gas distribution. The image
has a beam size of $9^{\prime\prime}$ and an rms level of 0.13 Jy beam-1.

##  3 Result

###  3.1 The Gas Distribution around MMS-7

Figure 1a shows the integrated emission of the ${\rm
N_{2}H^{+}}~{}(1_{0}-0_{1})$ transition overlaid on the IRAC $8\,{\rm\mu m}$
emission for the entire OMC-2,3 region. The overall gas structures are
presented in a relevant study (Zhang et al., 2020).

Figure 1b shows the zoom-in ${\rm N_{2}H^{+}}$ emission region (white
contours) around MMS-7 overlaid on the Spitzer/IRAC $8\,{\rm\mu m}$ image. The
$8\,{\rm\mu m}$ image shows two point sources, IRAC-2407 and IRAC-2405, which
were presented in Takahashi et al. (2006) and are included in the Orion YSO
catalogue (Megeath et al., 2012). The ${\rm N_{2}H^{+}}$ emission exhibits
three major filament structures extending from MMS-7 to the Northwest (NW),
South, and North, respectively. The three filaments are all closely overlapped
with the infrared-dark region in the $8\,{\rm\mu m}$ image. The SMA 1.3 mm
continuum emission (green contours) exhibits three dense cores. The central
one (MMS-7) coincides with the ${\rm N_{2}H^{+}}$ emission peak. The second
core (denoted as Core-2) overlaps with IRAC-2405. The other three cores,
Core-3, 4, and 5 are located on the NW filament. Core-5 is not fully covered
by our mapping field, but can still be seen to have a higher intensity than
Core-3 and Core-4, thus could be more massive. The NW filament thus has a
similarity with the end clumps observed in other filaments (e.g. Dewangan et
al., 2019; Yuan et al., 2020).

The bright young star IRAC-2407 was found to drive a bi-polar outflow
(Takahashi et al., 2006), which is also labeled in Figure 1b. The outflow is
likely causing a dissipation of the North filament, sweeping it away from the
outflow direction. The central core MMS-7 is displaced from IRAC-2407 and its
outflow, and has no other overlapped IR sources. MMS-7 thus should be an
independent prestellar core. The three ${\rm N_{2}H^{+}}$ filaments could be
originally intersected at MMS-7, but later become separated due to the
presence of two IR sources (YSOs). This overall morphology is comparable to
the frequently observed filament-hub systems despite the different spatial
scales (e.g. Williams et al., 2018; Chen et al., 2019; Treviño-Morales et al.,
2019).

Figure 1c shows the SMA ${\rm C^{18}O}$ (2-1) emission overlapped on the ${\rm
NH_{3}}~{}(1,1)$ emission (gray contours) and kinetic temperature map obtained
from our previous Very Large Array (VLA) observation (Li et al., 2013). The
${\rm C^{18}O}~{}(2-1)$ emission shows a similar spatial distribution with the
1.3 mm continuum. The emission peak overlaps with Core-2, the other two
emission features are near MMS-7 and Core-3, respectively. Although the ${\rm
C^{18}O}$ and ${\rm NH_{3}}$ emission regions do not fully overlap with the
${\rm N_{2}H^{+}}$ filaments, they all have a bulk spatial extension from
Southeast to Northwest that is in parallel with the NW filament. The kinetic
temperature estimated from the ${\rm NH_{3}}$ inversion shows a quite uniform
distribution around $13\pm 2$ K over the MMS-7 region (Li et al., 2013),
suggesting that the dense filaments and cores are not significantly affected
by the stellar heating.

Figure 1d shows the ${\rm N_{2}H^{+}}$ emission overlaid on the dust continuum
emission of the JCMT/SCUBA $450\,{\rm\mu m}$ band. The $450\,{\rm\mu m}$
emission has a similar morphology with the ${\rm N_{2}H^{+}}$ filaments and
meanwhile has a broader and more continuous spatial distribution. The
$450\,{\rm\mu m}$ emission region should represent the parental clump of the
filaments and cores. We estimated the effective radius of the clump using
$A_{\rm clump}=\pi r_{\rm eff}^{2}$, where $A_{\rm clump}$ is the clump area
within the 30%-contour level.

We calculated the gas column density and core mass from the dust continuum
emission using the gray-body emission model Hildebrand (1983),

$\displaystyle S_{\nu}$ $\displaystyle=\kappa_{\nu}B_{\nu}(T_{\rm d})\Omega\mu
m_{\rm H}N_{\rm tot}$ (1) $\displaystyle=\frac{\kappa_{\nu}B_{\nu}(T_{\rm
d})M_{\Omega}}{D^{2}},$

wherein $S_{\nu}$ is the flux density at the frequency $\nu$ within solid
angle $\Omega$, $N_{\rm tot}=N({\rm H_{2}}+{\rm HI})$ is the total column
density, $B_{\nu}(T_{\rm d})$ is the Planck function of the dust temperature
$T_{\rm d}$. We assumed that $T_{\rm d}$ is comparable to the ${\rm NH_{3}}$
kinetic temperature of $T=13$ K. $\mu$=$2.33$ is the mean molecular weight
(Myers & Benson, 1983), $m_{\rm H}$ is the atomic hydrogen mass, and
$\kappa_{\nu}$ is the dust opacity, which is related to the frequency as
$\kappa_{\nu}=\kappa_{\rm 230GHz}(\nu/{\rm 230GHz})^{\beta}$. The reference
value of $\kappa_{\rm 230GHz}=0.009$ cm2 g-1 is provided by the dust model for
the grains with coagulation for $10^{5}$ years with accreted ice mantles at a
density of $10^{6}\,{\rm cm^{-3}}$ (Ossenkopf & Henning, 1994), and the
average index of $\beta=1.9$ in Orion A (Sadavoy et al., 2016) is adopted for
the calculation. We calculated $N_{\rm tot}$ and $M_{\rm core}$ for the three
dense cores, MMS-7, Core-1, and Core-2 from the SMA 1.3 mm continuum emission.
We measured the core area $A_{\rm core}$ above the detection limit, and
derived the effective core radius from the from the expression of $A_{\rm
core}=\pi r_{\rm eff}^{2}$. The finally adopted core radius is also
deconvolved with the beam size, i.e. $r_{\rm core}^{2}$=$r_{\rm
eff}^{2}-r_{\rm beam}^{2}$. The parental clump mass is estimated from the
$450\,{\rm\mu m}$ emission also using Equation 1. We also derived the mass of
each filament assuming that the its mass ratio to MMS-7 is equal to ${\rm
N_{2}H^{+}}$ intensity ratio, that is $M_{\rm fil}/M_{\rm MMS-7}=S_{\rm
fil}({\rm N_{2}H^{+}})/S_{\rm MMS-7}({\rm N_{2}H^{+}})$. The physical
parameters are shown in Table 1.

Figure 1e shows the ${\rm N_{2}H^{+}}$ (1-0) and ${\rm C^{18}O}$ spectra at
the MMS-7 center. We fitted the ${\rm N_{2}H^{+}}$ hyperfine components (HFCs)
using Pyspeckit111Toolkit for fitting and manipulating spectroscopic data in
python. program, wherein the physical parameters of the HFCs are adopted from
Caselli et al. (1995). The molecular column density can be estimated as
(Caselli et al., 2002; Henshaw et al., 2014):

$\displaystyle N_{\rm mol}$ $\displaystyle=\frac{8\pi}{\lambda^{3}A_{\rm
ul}}\frac{g_{l}}{g_{u}}\times\frac{1}{J(T_{\rm ex})-J(T_{\rm
bg})}\times\frac{1}{1-\exp(-h\nu/k_{\rm B}T_{\rm ex})}$ (2)
$\displaystyle\times\frac{Q_{\rm rot}(T_{\rm ex})}{g_{l}\exp(-El/k_{\rm
B}T_{\rm ex})}\int T_{\rm b}{\rm d}v,$

wherein $A_{\rm ul}$ is the Einstein coefficient (Schöier et al., 2005),
$g_{u}$ and $g_{l}$ are the statistical weights of the upper and lower states,
respectively; $J_{\nu}(T)$ is the equivalent Rayleigh-Jeans temperature.
$Q_{\rm rot}(T)$ is the partition function. In calculation, we also assume
that $T_{\rm ex}$ is similar with the ${\rm NH_{3}}$ kinetic temperature.

Figure 1f shows the column density profile along the NW filament. The sampling
direction is indicated by the solid line in Figure 1d. The $N_{\rm tot}$
profile is estimated from the $450\,{\rm\mu m}$ continuum. It has a peak value
of $N_{\rm tot}=4.1\times 10^{23}$ cm-2 at MMS-7 and a FWHM width of $\sim
10^{\prime\prime}$. The surrounding gas has average $N_{\rm tot}=1.4\times
10^{23}$ cm-2 on the NW filament (offset$>10^{\prime\prime}$) and $1.0\times
10^{23}$ cm-2 out of the filaments. The $N({\rm N_{2}H^{+}})$ and $N_{\rm
tot}$ exhibit similar spatial profiles, suggesting that ${\rm N_{2}H^{+}}$ and
$450\,{\rm\mu m}$ continuum emissions could trace the nearly same gas-and-dust
components.

Figure 1g shows the molecular abundance ($X_{\rm mol}$) profile along the NW
filament estimated from $X_{\rm mol}=N_{\rm mol}/N_{\rm tot}$. The ${\rm
N_{2}H^{+}}$ abundance varies between 0.4 to $1.2\times 10^{-9}$, and the
three cores have quite similar values of $X({\rm N_{2}H^{+}})=(0.8-1.0)\times
10^{-9}$. In comparison, the ${\rm C^{18}O}$ abundance is peaked at Core-2 and
rapidly declines towards MMS-7. The abundances of individual cores are
presented in Table 1. Core-2 has high $X({\rm C^{18}O})$ probably due to its
association with the protostar IRAC-2405. MMS-7 and Core-3 have no detectable
IR counterparts thus could be younger and have a certain CO depletions. MMS-7
in particular has the highest mass but the lowest ${\rm C^{18}O}$ abundance,
suggesting the absence of stellar heating.

###  3.2 The Velocity Distribution

Figure 2a shows the velocity field overlaid on the integrated ${\rm
N_{2}H^{+}}$ emission region. The velocity is measured by fitting the ${\rm
N_{2}H^{+}}~{}(1_{0}-0_{1})$ transition with a gaussian profile pixel by
pixel. The velocity field shows a noticeable increasing trend along the NW
filament towards the center, with a variation from $v_{\rm lsr}=11$ to 9.2 km
s-1. The Southern and Northern filaments also exhibit less prominent but
similar velocity gradient, with $v_{\rm lsr}$ variation from 10.75 to 9.75 km
s-1 towards the edges near the center.

Figure 2b shows the ${\rm N_{2}H^{+}}~{}(1_{0}-0_{1})$ line width ($\Delta v$)
distribution. It exhibits peak values at MMS-7 center and several other
positions. There are four $\Delta v$ peaks on the NW filament. They are
coincident with intensity peaks. The $\Delta v$ peak on the North filament
also overlaps with the emission peak. The $\Delta v$ peaks have an average
increasing scale of 0.2 km s-1, which is much smaller than the $v_{\rm lsr}$
increasing scale towards the center (Figure 2a). The $v_{\rm lsr}$ and $\Delta
v$ distributions suggest that although the filaments have a bulk motion, they
still remain a quite low internal velocity dispersion.

Figure 2c and 2d show the position-velocity (PV) diagrams along the NW
filament and the direction from North to South, respectively. As shown in
Figure 2c, the velocity profile along the NW filament varies from $v-v_{\rm
sys}$=$-1$ to $-2$ km s-1 in spatial range from offset$=45^{\prime\prime}$ to
$0^{\prime\prime}$. The velocity gradient is not linear but has a steeper
increase as approaching the center. The velocity profile of ${\rm C^{18}O}$
(contours) also shows an increasing trend from offset$=20^{\prime\prime}$ to
$0^{\prime\prime}$, which is roughly in parallel with the ${\rm N_{2}H^{+}}$
emission. Since the ${\rm C^{18}O}$ is closely associated with the 1.3 mm
dense cores, it suggests that the cores should have a coherent motion with the
filaments.

The velocity gradient from South to North is presented in Figure 2d. This
direction partly covers the South and North filaments. The South one extends
from offset=0 to $30^{\prime\prime}$ and the North one from
offset=$-40^{\prime\prime}$ to $-20^{\prime\prime}$. The two filaments also
have a similar velocity gradient of $v-v_{\rm sys}=0$ to $-1.2$ km s-1 towards
the inner region. Since IRAC-2407 and 2405 (Core-3) are located at positions
where the velocity gradient becomes steeper, the velocity features should
therefore suggest the gas motion to have an interaction with these two YSOs,
respectively. Meanwhile the two filaments are not terminated at the YSOs, but
extend over them to the center. They could therefore have a convergent motion
together with the NW filament towards MMS-7.

##  4 Discussion

###  4.1 Dynamical Origin of the Velocity Profiles

As shown in Figure 2c and 2d, the three ${\rm N_{2}H^{+}}$ filaments exhibit
comparable velocity increase towards the center. The ${\rm C^{18}O}$ emission
also partly follows the ${\rm N_{2}H^{+}}$ velocity profile around MMS-7.
Moreover, for each filament, the velocity gradient becomes steeper within an
area of offset=$\pm 20^{\prime\prime}$, as indicated by the vertical dashed
lines. Since the ${\rm C^{18}O}$ and ${\rm N_{2}H^{+}}$ are both dense-gas
tracers, they should reveal the bulk motion of the dense filaments and cores
instead of activities driven by star formation, such as outflow and envelope
expansion. The star-forming activities could also have an inconsistency with
the observed line widths. The ${\rm N_{2}H^{+}}\,(1_{0}-0_{1})$ transition
only has a limited increase from $\Delta v=0.4$ to 0.65 km s-1 towards the
center (Figure 2b), which is much lower than the scale of radial velocity
increase. Since the velocity gradient is clearly seen in all three filaments,
it suggests that the dense gas should have an overall contraction towards the
center due to the self gravity.

Moreover, in order to generate the observed gas motion, there are two
requirements for the gas structure. First, the clump should have a non-
spherical distribution so that a fraction of mass can obtain a bulk motion
towards the direction where the ambient density is lower. Second, the low-
density region should be on the front side of the clump so that the compressed
gas can have a blueshift. This spatial layout is not difficult to be satisfied
in real condition. A schematic view of the gas distribution and motion is
shown in Figure 3.

Although the density profile along the line of sight is largely unknown, its
spatial distribution shown in Figure 3 is still supported by two factors.
First, as seen in Figure 2c and 2d, a fraction of ${\rm C^{18}O}$ emission
around the central position is not associated with the blueshifted ${\rm
N_{2}H^{+}}$ velocity gradient, but has a different velocity of $v_{\rm
lsr}\simeq-1~{}{\rm km\,s^{-1}}$. This component could trace the more
quiescent dense gas behind the blueshift gas. Second, the ${\rm N_{2}H^{+}}$
emission is weakened towards IRAC-2407, and the velocity profile of the
Northern filament also appears to be disturbed by this source (Figure 2d).
These features suggests that this YSO is causing a gas dissipation on the
front side.

###  4.2 Contraction Velocities and Energy Scales

The filament contraction could be essentially driven by the self gravity and
it requires two major physical conditions in this process: (1) the self
gravity should provide sufficient kinetic energy to account for the current
gas motion; (2) the clump should have a gravitational instability to initiate
the contraction.

To make a semi-quantitative calculation, we assume that the gas is contracting
from an initial radius of $r_{0}$ to a smaller radius $r$. The gravitational
energy release in this process can be approximated to be

$\Delta\mathcal{W}=GM_{c}^{2}\left(\frac{1}{r}-\frac{1}{r_{0}}\right),$ (3)

where $M_{c}$ is the contracting gas mass. It should include MMS-7 and its
surrounding gas. As shown in Figure 3b and 3c, the blueshifted gas is mainly
distributed within $20^{\prime\prime}$ around the center. The gas mass in this
area is $5.5\,M_{\odot}$ as calculated from the $450\,{\rm\mu m}$ emission
using Equation 1. The released kinetic energy can be estimated as

$\epsilon\Delta\mathcal{W}=\frac{1}{2}M_{c}[v(r)^{2}-v(r_{0})^{2}],$ (4)

wherein $\epsilon$ represents the efficiency of energy conversion. Assuming
that the contracting gas has a final velocity of $v_{\rm real}$ and an
inclination of $\theta$ from the line of sight, the observed radial velocity
would be $v_{\rm obs}=v_{\rm real}\cos\theta$. From Equation 3 and 4, the
radius and velocity can have a relation of

$v_{\rm obs}(r)^{2}-v_{\rm
obs}(r_{0})^{2}=2GM_{c}\epsilon\cos^{2}\theta(r^{-1}-r_{0}^{-1}).$ (5)

Since the three parameters $\epsilon$, $M_{c}$, and $\theta$ are fully
degenerated, their production on the right side is considered as one
coefficient, $c_{0}=2GM_{c}\epsilon\cos^{2}\theta$. The other two parameters,
$r_{0}$ and $v_{\rm obs}(r_{0})$ are the initial radius and velocity,
respectively. We expect $r_{0}$ to be much larger than the spatial extent of
the contracting gas ($r<20^{\prime\prime}$), and the initial velocity nearly
zero. In this case, the central part of the $v_{\rm obs}(r)$ curve would
approach $v(r)^{2}\propto 1/r$.

We compared the $v(r)$ function and velocity profile of the filaments as shown
in Figure 4. For each filament, one can let $v(r)$ curve well coincide with
the velocity profile in the range of $r<20^{\prime\prime}$ only by adjusting
$c_{0}$. The three $v(r)$ curves have comparable $c_{0}$ values of
$c_{0}/2G=(0.6\pm 0.3)\,M_{\odot}$. The contracting mass is then constrained
to be

$\displaystyle M_{c}$
$\displaystyle=\frac{c_{0}}{2G}\frac{1}{\epsilon\cos^{2}\theta}$ (6)
$\displaystyle=(5.0\pm
2.5)\left(\frac{\epsilon}{0.5}\right)^{-1}\left(\frac{\cos\theta}{\cos
60^{\circ}}\right)^{-2}M_{\odot}.$

From this equation we can satisfy the observed mass scale $5.5\,M_{\odot}$
using reasonable $\theta$ and $\epsilon$ values. Although the individual
parameters are still uncertain, the calculation shows that the gravitational
contraction could release sufficient kinetic energy to account for the
observed gas motion.

###  4.3 Velocity Variations

Despite the agreement in energy scales, the velocity distribution still has
two deviation features from the modeled $v(r)$ curves that should be further
examined. First, the theoretical $v(r)$ function will approach infinity
towards the center, while the observed velocity profiles are terminated at
final velocities around the center. For the NW filament (Figure 4b), the
velocity profile reaches the maximum value in the area of $r<4$ arcsec around
the center. This area is comparable to the size of MMS-7 core ($r=3.7$
arcsec). As an reasonable explanation this energy release of
$\epsilon\Delta\mathcal{W}$ could be largely transferred to MMS-7, and
accelerating this core as an entire body to the final velocity.

Second, although the three filaments have similar velocity increasing features
towards the center, their final velocities still have a difference. This could
be also attributed to IRAC-2405 and 2407. As shown in Figure 4d, the North
filament becomes weaker at IRAC-2407, but still connected to MMS-7, and the
gas structure well follows the $v(r)$ curve. This suggests that the gas motion
is still towards the center and would reach the same velocity with MMS-7. The
South filament, according to its velocity gradient, could have a similar
connection to MMS-7 except for a more significant interruption by IRAC-2406
(Core-2). The two YSOs could have dissipated the surrounding ${\rm
N_{2}H^{+}}$ component but not largely affect the global inward motion towards
MMS-7. For the NW filament, its outer part ($r>20^{\prime\prime}$) could also
be affected by the local dense cores. As seen in Figure 4b, the emission
region is largely deviated from the $v(r)$ curve in the spatial range between
Core-3 and Core-4 with a scale of $\Delta v\simeq 1.0$ km s-1, suggesting that
the gas motion could be affected by these two cores.

Although the filaments have these systemic velocity variations, the ${\rm
N_{2}H^{+}}$ $(1_{0}-0_{1})$ line profile is usually single-peaked and has
relatively small line width of $\Delta v<0.65$ km s-1 all over the region.
There are no evident multiple velocity components in MMS-7 region. The three
filaments thus should all be one-dimensional compact filaments. Multiple
velocity components in a filament would indicate intertwined substructures
therein, and their formation could depend on initial conditions of turbulence
and density fluctuations in a molecular cloud as well as magnetic field. In
Orion A, the incidence of multiple velocity components vary from region to
region, and may have an overall increasing trend towards the OMC-2 and Orion
KL regions (Hacar et al., 2018, Figure B.3 therein), wherein the major
fraction of dense gas over the ISF is being assembled.

###  4.4 Clump and Filaments: Instabilities

In order to initiate the contraction, the clump should have a gravitational
instability. To estimate this, we adopted the modified virial mass $M_{\rm
vir}^{B}$ (Bertoldi & McKee, 1992; Pillai et al., 2011; Sanhueza et al., 2017;
Liu et al., 2018), which takes the density profile, thermal, turbulent, and
magnetic pressure into account, estimated as

$M_{\rm vir}^{B}=\frac{5R_{\rm eff}}{G}\left(\sigma_{\rm
tot}^{2}+\frac{\sigma_{A}^{2}}{6}\right),$ (7)

where $R_{\rm eff}$ is the effective radius, $\sigma_{\rm tot}$ is the total
velocity dispersion, $\sigma_{A}$ is the Alfvénic velocity. It is determined
by the magnetic field strength $B$ and average gas density $\overline{\rho}$
as $\sigma_{A}=B/\sqrt{4\pi\overline{\rho}}$. Here we adopt the average value
of $B=0.19$ mG in OMC-3 (Poidevin et al., 2010) and the average density of the
$450\,{\rm\mu m}$ clump. The velocity dispersion consists of the thermal and
non-thermal parts:

$\displaystyle\sigma_{\rm tot}$ $\displaystyle=\sqrt{\sigma_{\rm
th}^{2}+\sigma_{\rm nt}^{2}}$ (8) $\displaystyle=\sqrt{\frac{k_{\rm B}T_{\rm
kin}}{\mu m_{\rm H}}+\frac{\Delta v^{2}}{8\ln 2}-\frac{k_{\rm B}T_{\rm
kin}}{m_{\rm mol}}},$

wherein the thermal part is

$\sigma_{\rm th}=\sqrt{\frac{k_{\rm B}T_{\rm kin}}{\mu m_{\rm H}}},$ (9)

and the non-thermal part is

$\sigma_{\rm nt}=\sqrt{\frac{\Delta v^{2}}{8\ln 2}-\frac{k_{\rm B}T_{\rm
kin}}{m_{\rm mol}}},$ (10)

and $m_{\rm mol}$ is the molecular mass. In calculation we adopted a kinetic
temperature of $T_{\rm kin}=13$ K as measured from the ${\rm NH_{3}}$
inversion lines (Li et al., 2013, also see Fig. 1c).

The virial mass is sensitive to $\sigma_{\rm tot}$ and could have large
variation if using different molecular lines. Besides the ALMA ${\rm
N_{2}H^{+}}$ (1-0) line, we also considered the ${\rm N_{2}H^{+}}$ line
observed by the NRO 45-meter telescope (Tatematsu et al., 2008), which has a
resolution of $\sim 18$ arcsec (beam size) and could better recover the
extended gas. For the ${\rm N_{2}H^{+}}$, we measured the line width to be
$\Delta v=0.9$ km s-1 at MMS-7 and 1.3 km s-1 over the clump surface within
$r=33^{\prime\prime}$. We also inspected the NRO ${\rm C^{18}O}$ (1-0) lines
(Feddersen et al., 2020) and measured corresponding values to be $\Delta
v=2.2$ and 2.5 km s-1, respectively. The derived $M_{\rm vir}^{B}$ values are
listed in Table 1. The single-dish ${\rm N_{2}H^{+}}$ emission has a similar
spatial morphology with the dust continuum in OMC-3 (Tatematsu et al., 2008,
Figure 3 and 4 therein), and its line width is close to the value observed
with ALMA herein. This suggests that the cold dense gas component maintains a
low turbulent state at both large and small scales, i.e., from the entire
clump to its inner structures. The large virial ratio of $M_{\rm clump}/M_{\rm
vir}^{B}=2.6$ suggests a considerable instability due to the self gravity.

On the other hand, since the dense gas in the clump is concentrated on the
filaments, they should be the major ingredient to determine the clump
stability. We estimated the critical linear mass density for the filament
(Stodólkiewicz, 1963; Ostriker, 1964) using

$\eta_{\rm crit}=\frac{2\sigma_{\rm tot}^{2}}{G}.$ (11)

The observed gas mass per unit length can be estimated from

$\eta_{\rm obs}=M_{\rm fil}/(Z_{\rm fil}/\sin\theta),$ (12)

where $Z_{\rm fil}$ is the filament length. In calculation we assume
$\theta=60^{\circ}$. The derived $\eta$ of each filament is shown in Table 1.
For all three filaments we have $\eta_{\rm obs}/\eta_{\rm crit}$ slightly
exceeding 1.0. With the uncertainty in $\theta$ taken into account, $\eta_{\rm
obs}$ could reach the maximum value at $\theta=90^{\circ}$. Even in this
extreme case, the ratio of $\eta_{\rm obs}/\eta_{\rm obs}$ is still much lower
than the virial ratio ($M_{\rm clump}/M_{\rm vir}^{B}=2.6$). We therefore
expect the clump to have a less instability than in the spherical case.

###  4.5 Evolutionary State and Timescales

We estimated the current infall rate of the clump contraction from its average
radius $r_{\rm in}$, velocity $v_{\rm in}$, and density $n_{c}$ using

$\displaystyle\dot{M}_{\rm in}$ $\displaystyle\simeq 4\pi r_{\rm in}^{2}\mu
m_{0}n_{\rm c}v_{\rm in}$ (13) $\displaystyle=8.0\times
10^{-5}\left(\frac{r_{\rm in}}{4000\,{\rm
AU}}\right)^{2}\left(\frac{n_{c}}{5\times 10^{5}\,{\rm
cm^{-3}}}\right)\left(\frac{v_{\rm in}}{0.6\,{\rm
km\,s^{-1}}}\right)\,M_{\odot}\,{\rm year}^{-1}.$

In calculation we adopted the average radius of $r=10^{\prime\prime}$ where
the velocity gradient becomes significant. At this infall rate, the
contraction can maintain a timescale up to $t_{\rm in}=M_{\rm
clump}/\dot{M}_{\rm in}=3.6\times 10^{5}$ years.

We considered two typical theoretical collapse models to compare the
observation. First, the free-fall collapse represents the most rapid scenario
for the clump evolution. It has a time scale of

$\displaystyle t_{\rm ff}$ $\displaystyle=\left(\frac{3\pi}{32G\mu m_{\rm
H}n_{c}}\right)^{1/2}$ (14) $\displaystyle=5\times
10^{4}\left(\frac{n_{c}}{5.0\times 10^{5}\,{\rm cm^{-3}}}\right)^{-1/2}{\rm
years},$

and consequently an infall rate of

$\displaystyle\langle\dot{M}_{\rm ff}\rangle$ $\displaystyle=M_{\rm
clump}/t_{\rm ff}$ (15) $\displaystyle=6\times 10^{-4}\left(\frac{M_{\rm
clump}}{30\,M_{\odot}}\right)\left(\frac{n_{c}}{5.0\times 10^{5}\,{\rm
cm^{-3}}}\right)^{1/2}\,M_{\odot}\,{\rm year}^{-1}.$

The second case is the longitudinal collapse of a linear filament. The
timescale can be estimated as (Pon et al., 2012; Clarke & Whitworth, 2015)

$\displaystyle t_{\rm fil}$ $\displaystyle=(0.49+0.26A_{\rm O})(G\rho_{\rm
O})^{-1/2}$ (16) $\displaystyle=3.5\times 10^{5}\left(\frac{0.49+0.26A_{\rm
O}}{5.7}\right)\left(\frac{\overline{n}}{1.0\times 10^{6}\,{\rm
cm^{-3}}}\right)^{-1/2}{\rm years},$

wherein $A_{\rm O}\equiv Z_{\rm O}/R_{\rm O}$ is the filament aspect ratio
between its half-length $Z_{\rm O}=Z_{\rm fil}/2$ and average radius. In
Equation 16 the default $t_{\rm fil}$ is estimated using $A_{\rm O}=20$, which
is the value of the NW filament. Comparing the three timescales, we have
$t_{\rm fil}\simeq t_{\rm in}\gg t_{\rm ff}$. It suggests that the current
infall rate is more inclined to the filamentary case. This result shows an
agreement with the estimation of instabilities (Sec. 4.4).

During the filament contraction, the average velocity is estimated to be
$\overline{v}_{\rm fil}=Z_{\rm O}/t_{\rm fil}=0.4$ km s-1, which is comparable
to the observed velocity more distant from the center ($r>10^{\prime\prime}$).
For the inner region ($r<10^{\prime\prime}$), the one-dimensional kinematics
would be less applicable since the three filaments are intersected, and the
end point of each filament is fixed at MMS-7. The three filaments are
contracting towards a common center and would therefore have the observed
steeper velocity increase.

During the filament evolution, the local jeans instability may cause the
filament to have a fragmentation. For the NW filament, the fragmentation could
be taking place and leading to the formation of the four dense cores.
Following Wiseman & Ho (1998) and Takahashi et al. (2013), the jeans length of
a cylindrical gas body can be estimated as

$\lambda_{\rm frag}\simeq\frac{20\sigma_{\rm tot}}{\sqrt{4\pi G\rho}}.$ (17)

Adopting the number density and $\Delta v$ values in Table 1, we can derive
$\lambda_{\rm frag}=24^{\prime\prime}$. With the projection effect taking into
account, this distance is in a favorable agreement with the average spacing of
the cores on the NW filament ($\sim 20^{\prime\prime}$).

Third, the filament can also be oscillating along its transverse direction and
this can occur simultaneously during the fragmentation (Clarke et al., 2017;
Stutz & Gould, 2016; Liu et al., 2019). This may also be ongoing in MMS-7
region, since the filaments indeed exhibit a feature of velocity fluctuation.
As seen in Figure 4, the NW and north filaments have a velocity fluctuation
with a scale of 0.2 to 0.3 km s-1 and a spatial scale of $\sim 10$ arcsec
(0.02 pc) as labelled in red lines throughout the emission feature. The
fluctuation has a velocity scale. The fragmentation model (Clarke et al.,
2017) derived a relation between the dominant perturbation wavelength
$\lambda_{\rm D}$ and the time scale $\tau_{\rm crit}$ for the filament to
become supercritical that is $\lambda_{\rm D}\sim 2\tau_{\rm crit}a_{0}$,
wherein $a_{0}$ is the isothermal sound speed. For the NW filament, if
adopting $\lambda_{\rm D}=10^{\prime\prime}$, we can derive $\tau_{\rm
crit}=10^{4}$ years, which is lower than $t_{\rm in}$ for an order of
magnitude. The contrast between the two time scales suggests that the filament
should be actually stable against oscillation, otherwise it would be already
dissipated. Although the oscillation could be taking place, it should be lower
than a certain threshold level to significantly interrupt the global structure
or the contraction towards the center. If there is no other perturbations, the
inflow would be able to transport a considerable fraction of the clump mass
into the center.

##  5 Summary

We investigated the gas structures in Orion OMC-3 MMS-7 region from dust
continuum and molecular lines, in particular the ALMA ${\rm
N_{2}H^{+}}~{}(1-0)$. These tracers revealed key properties of the multi-
filament structures, as specified in four major aspects:

(1) Three dense and compact filaments are observed in ${\rm
N_{2}H^{+}}~{}(1-0)$ emission. They compose the spine structure of the
parental clump and tend to have an intersection at the clump center. There are
at least seven dense cores and stellar objects being formed in this 0.2-pc
region, with the central core MMS-7 being the most massive one
($3.6\,M_{\odot}$). Besides the two infrared sources (IRAC-2405 and 2407), the
other five objects could all be starless or prestellar cores.

(2) The three filaments all exhibit a noticeable velocity gradient towards the
center as measured from the ${\rm N_{2}H^{+}}~{}(1_{0}-0_{1})$ transition. The
radial velocity reaches the maximum blueshift of $v-v_{\rm sys}=-2.0$ km s-1
at the center. In the mean time the filaments have relatively small velocity
dispersion ($\Delta v<0.65$ km s-1). Since the dense gas has no evident star-
forming activities (e.g. shell expansion and outflow), the velocity gradient
should mainly trace the inward gas motion, namely the filament contraction
towards MMS-7.

(3) The infall timescale ($3.6\times 10^{5}$ years) and mass transfer rate
($8\times 10^{-5}\,M_{\odot}\,{\rm year}^{-1}$) are both much less prominent
than those in a spherical free-fall collapse. The infall property and
critical-mass analysis both suggest that the clump instability should mainly
depend on the internal filaments. Although the filaments in particular the NW
one has a likely fragmentation, it does not seem to disturb the global
filament structures and contraction towards MMS-7.

In general, the MMS-7 region shows a similarity with frequently observed
filament-hub systems. And in this case the filamentary contraction towards an
individual prestellar core is more evidently identified. Although there are
multiple dense cores and young stars, the major fraction of the mass would
still be transferred to the central object. This scenario is comparable to the
competitive accretion, and it further implies that the intersection of the
convergent filaments could be the major site to assemble the most massive
core.

##  6 Acknowledgement

This work is supported by the China Ministry of Science and Technology under
State Key Development Program for Basic Research (973 program) No.
2012CB821802, the National Natural Science Foundation of China No. 11403041,
No. 11373038, No. 11373045, Strategic Priority Research Program of the Chinese
Academy of Sciences, Grant No. XDB09010302, and the Young Researcher Grant of
National Astronomical Observatories, Chinese Academy of Sciences.

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Table 1: The physical properties of the gas structures.

Parameters | MMS-7 | Core-2 | Core-3 |  Clumpa
---|---|---|---|---
(Cores/Clump)
Radius [arcsec (AU)]b |  $3.^{\prime\prime}7$ (1440) |  $2^{\prime\prime}.6$ (1020) |  $2^{\prime\prime}.0$ (780) |  $35^{\prime\prime}$ (13650)
Flux (Jy) | 0.315 (1.3 mm) | 0.236 (1.3 mm) | 0.236 (1.3 mm) |  57 ($450\,{\rm\mu m}$)
Mass ($M_{\odot}$) | 3.6 | 1.8 | 0.9 | 29
$T_{\rm kin}$ (K)c | 13 | 13 | 13 | 13
$N_{\rm tot}$ ($10^{23}$ cm-2)d | 9.5 | 11.6 | 9.4 | 4.2
$n$ (cm-3) e | $2.8\times 10^{7}$ | $5.0\times 10^{7}$ | $5.6\times 10^{7}$ | $5.1\times 10^{5}$
$\Delta v$ (km s-1) f |  0.65(${\rm N_{2}H^{+}}$) |  1.75(${\rm C^{18}O}$) |  1.85(${\rm C^{18}O}$) |  0.9(${\rm N_{2}H^{+}}$)
|  1.82(${\rm C^{18}O}$) | $-$ | $-$ |  0.95(${\rm NH_{3}}$)
| $-$ | $-$ | $-$ |  2.2(${\rm C^{18}O}$)
$M_{\rm vir}$ ($M_{\odot}$)g |  0.4(${\rm N_{2}H^{+}}$) | $-$ | $-$ |  11(${\rm N_{2}H^{+}}$)
|  4.65(${\rm C^{18}O}$) |  2.8(${\rm C^{18}O}$) |  2.4(${\rm C^{18}O}$) |  46(${\rm C^{18}O}$)
$X({\rm N_{2}H^{+}})~{}(10^{-9})$ | 0.75 | 1.0 | 0.8 | $-$
$X({\rm C^{18}O})~{}(10^{-8})$ | 0.2 | 1.2 | 1.0 | $-$
(Filaments) | NW | Southern | Northern |
$T_{\rm ex}$ (K)h | 9.5 | 8.5 | 9.0 |
Width (arcsec)i | 5-10 | 4-9 | 4-12 |
$\overline{\Delta v}$ (km s-1) | 0.31 | 0.29 | 0.29 |
Mass ($M_{\odot}$) | 12 | 5 | 8 |
$N_{\rm tot}$ ($10^{23}$ cm-2)j | 1.0-3.6 | 0.9-2.7 | 0.7-3.3 |
$\overline{n}$ (cm-3) | $1.5\times 10^{6}$ | $1.5\times 10^{6}$ | $1.5\times 10^{6}$ |
$\eta_{\rm obs}$ ($M_{\odot}$ pc-1) | $50\pm 10$ | $55\pm 5$ | $60\pm 10$ |
$\eta_{\rm crit}$ ($M_{\odot}$ pc-1) | 45 | 45 | 45 |

$a.$ The parental clump of the cores and filaments observed in SCUBA
$450\,{\rm\mu m}$ emission.

$b.$ The average radius of the $5\sigma$ emission region deconvolved with the
beam radius.

$c.$ The ${\rm NH_{3}}$ kinetic temperature (Li et al., 2013).

$d.$ The average total column density over the 5-$\sigma$ the emission region
derived from the continuum emissions, For MMS-7, Core-2, and Core-3, $N_{\rm
tot}$ is derived from the SMA 1.3 mm emission. For the parental clump, it is
derived from the SCUBA $450\,{\rm\mu m}$ emission.

$e.$ The average number density estimated from $M=4\pi/3\bar{R}^{3}\mu m_{\rm
H}n$.

$f.$ The line widths of the different molecular tracers, for the clump, each
value represents the average linewidth over the annulus area from
$r=15^{\prime\prime}$ to $45^{\prime\prime}$.

$g.$ The virial mass calculated using Equation 7.

$h.$ The ${\rm N_{2}H^{+}}$ excitation temperature estimated from the spectral
fitting.

$i.$ The width variation range along the filament. The value is the half-
maximum width measured along the transverse direction.

$j.$ The $N_{\rm tot}$ variation range along the filament.

Figure 1: (a) The integrated map of ALMA ${\rm N_{2}H^{+}}$ $(1_{0}-0_{1})$ in
OMC-2,3 region (yellow) overlaid on the IRAC 8${\rm\mu m}$ red background. (b)
The ${\rm N_{2}H^{+}}$ $(1_{0}-0_{1})$ emission (white contours) and the SMA
1.3 mm continuum (green contours) emission on the IRAC 8${\rm\mu m}$ image.
The ${\rm N_{2}H^{+}}$ contours are 4, 6, 8, 10, 12 times 60 mJy beam-1 km
s-1. The 1.3 mm contours are 0.1, 0.3, 0.5, 0.7, 0.9 times of the peak value
300 mJy beam-1. (c) The integrated ${\rm C^{18}O}~{}(2-1)$ overlaid on the
${\rm NH_{3}}~{}(1,1)$ emission and kinetic temperature map (Li et al., 2013).
The ${\rm C^{18}O}$ contours are 0.1, 0.3, 0.5, 0.7, 0.9 times of the peak
value 3.0 Jy beam-1 km s-1. (d) The ${\rm N_{2}H^{+}}$ $(1_{0}-0_{1})$
emission overlaid on the SCUBA $450\,{\rm\mu m}$ emission. The $450\,{\rm\mu
m}$ contour levels are 30, 50, 70, 90 percent of the peak value, 8.1 Jy
beam-1. The solid line denotes the direction to sample the column density and
abundance profiles. (e) The ${\rm C^{18}O}$ and ${\rm N_{2}H^{+}}$ spectra at
the MMS-7 center. The dashed line is the best-fit profile of the ${\rm
N_{2}H^{+}}$ spectrum. The corresponding physical parameters are listed on the
panel. (f) The column density profiles of ${\rm H_{2}}$, ${\rm N_{2}H^{+}}$,
and ${\rm C^{18}O}$ along the axis of the NW filament as indicated in panel
(d). (g) The abundance profiles of ${\rm C^{18}O}$ and ${\rm N_{2}H^{+}}$
along the same direction.
Figure 2: (a) The systemic velocity distribution of the ${\rm N_{2}H^{+}}$
measured from pixel-by-pixel spectral fitting presented in false-color scale.
The positions of the infrared sources and dense cores are labeled with stars
and plus signs, respectively. (a) The ${\rm N_{2}H^{+}}$ line-width
distribution of the ${\rm N_{2}H^{+}}$ emission region. (c) The Position-
Velocity diagram of the ${\rm N_{2}H^{+}}$ $(1_{0}-0_{1})$ along the NW
filament, labeled as PV-1 direction in Figure 2a. The ${\rm C^{18}O}$ emission
is also plotted in white contours; the contour levels are 0.4, 1.0, 2.0, 3.0,
4.0 K. The projected locations of the dense cores and infrared source
IRAC-2407 on the PV-direction are also labeled with the arrows. (d) The ${\rm
N_{2}H^{+}}$ P-V diagram in the north-south direction, (PV-2 in Figure 2a).
The locations of the other sources are also labeled. The vertical dashed lines
in (c) and (d) denote the offset range wherein the velocity increase becomes
significant.
Figure 3: A schematic view of the two scenarios to explain the observed
velocity distribution around MMS-7. The arrows denote the gas motion
directions, including the contraction towards MMS-7 and the bulk motion
towards the observer. The dashed circle denotes the inner region
($r<20^{\prime\prime}$) wherein the infall motion becomes more significant.
Figure 4: A closer look of the PV plot along the three filaments. (a) The
systemic velocity map showing the PV directions. (b)-(d) The PV plot along the
three directions. For each filament, the offset is measured from the MMS-7
center. The solid line represents the $v(r)$ function that closely goes
through the spine of the PV emission region. The $v(r)$ function is calculated
from Equation (5). The two dashed lines represent the $v(r)$ curves with
$c_{0}$ varying by a factor of 0.5 and 2, respectively. In (b) and (d), the
red solid line denotes the central velocity variation along the PV sampling
direction.