# Simulation of Head-on Collisions Between Filamentary Molecular Clouds
Threaded by a Lateral Magnetic Field and Subsequent Evolution

Raiga Kashiwagi Department of Astronomical Science, School of Physical
Sciences, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1
Osawa, Mitaka, Tokyo 181-8588, Japan  Kazunari Iwasaki Department of
Astronomical Science, School of Physical Sciences, The Graduate University for
Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Center for Computational Astrophysics, National Astronomical Observatory of
Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Division of Science,
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Kohji Tomisaka Department of Astronomical Science, School of
Physical Sciences, The Graduate University for Advanced Studies (SOKENDAI),
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Division of Science, National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

(Accepted July 15, 2023)

###### Abstract

Filamentary molecular clouds are regarded as the place where newborn stars are
formed. In particular, a hub region, a place where it appears as if several
filaments are colliding, often indicates active star formation. To understand
the star formation in filament structures, we investigate the collisions
between two filaments using two-dimensional magnetohydrodynamical simulations.
As a model of filaments, we assume that the filaments are in
magnetohydrostatic equilibrium under a global magnetic field perpendicular to
the filament axis. We set two identical filaments with an infinite length and
collided them with a zero-impact parameter (head-on). When the two filaments
collide while sharing the same magnetic flux, we found two types of evolution
after a merged filament is formed: runaway radial collapse and stable
oscillation with a finite amplitude. The condition for the radial collapse is
independent of the collision velocity and is given by the total line mass of
the two filaments exceeding the magnetically critical line mass for which no
magnetohydrostatic solution exists. The radial collapse proceeds in a self-
similar manner, resulting in a unique distribution irrespective of the various
initial line masses of the filament, as the collapse progresses. When the
total line mass is less massive than the magnetically critical line mass, the
merged filament oscillates, and the density distribution is well-fitted by a
magnetohydrostatic equilibrium solution. The condition necessary for the
radial collapse is also applicable to the collision whose direction is
perpendicular to the global magnetic field.

Molecular clouds (1072), Interstellar filaments (842), Magnetic fields (994),
Magnetohydrodynamical simulations (1966), Star formation (1569)

##  1 Introduction

The early phase of star formation can be understood by studying the
filamentary structures in molecular clouds. Filaments are elongated structures
within the dense parts of molecular clouds, and the Herschel space telescope
showed that they are a fundamental component of these clouds (André et al.,
2010, 2014). Furthermore, prestellar and protostellar cores exist along such
filaments (Könyves et al., 2015; Arzoumanian et al., 2019). Star formation
seems to occur in dense clumps, which have evolved from the filaments. Thus,
filaments play an important role as the birthplace of stars.

Magnetic fields are also essential to understanding the early stages of star
formation in various aspects, such as changing the character of interstellar
turbulence (e.g., Cho et al., 2002; Federrath & Klessen, 2012), modifying the
gas compression by shocks (e.g., Inoue & Fukui, 2013; Iwasaki & Tomida, 2022),
achieving stability against gravitational instability (e.g., Nakano &
Nakamura, 1978), and transferring the angular momentum from a star forming gas
(e.g., Mouschovias & Paleologou, 1980; Tomisaka, 2000). Orientation of the
magnetic field is observed using near-infrared, far-infrared, and millimeter-
wave polarizations. These polarizations are obtained from dust grains aligned
with the interstellar magnetic field, that is, the background starlight in the
near-infrared is polarized parallel to the magnetic field, whereas the thermal
emission from the magnetically aligned dust in the far-infrared and millimeter
wavelengths is polarized perpendicular to the magnetic field. Previous
observations have indicated that the global magnetic field is nearly
perpendicular to the massive filament (Chapman et al., 2011; Palmeirim et al.,
2013; Sugitani et al., 2019; Doi et al., 2020; Arzoumanian et al., 2021), and,
from the Planck all-sky survey, Planck Collaboration Int. XXXV (2016) found
that the interstellar magnetic field is often perpendicular to massive
filaments but is parallel to filaments with a low column density (striations).
This trend is evident in typical molecular clouds such as Taurus, Lupus, and
Chamaeleon–Musca.

One of the formation scenarios of such filaments is explained by the colliding
streams (Inutsuka et al., 2015). Colliding streams form a compressed layer,
which is a flattened sheet-like structure. The physical origin of this stream
corresponds to the expanding Hii region, supernova remnant, or collision of
two clouds. In this context, the filament is formed by the self-gravitational
fragmentation of the sheet-like clouds.

The self-gravitational instability of an equilibrium sheet is well-
established. Perturbations with wavelengths longer than a critical value can
grow, leading to the fragmentation of sheets into filaments. The critical
wavelength of a nonmagnetized sheet is typically a few times larger than its
thickness. Miyama et al. (1987) performed a detailed analysis of the nonlinear
growth of unstable perturbations and found that sheet-like clouds are expected
to fragment into filaments separated by approximately twice the critical
wavelength. The presence of a perpendicular magnetic field tends to stabilize
the sheet. Nakano & Nakamura (1978) showed that if the magnetic field strength
is larger than the critical value ($B_{\mathrm{crit}}=2\pi G^{1/2}\Sigma$,
where $\Sigma$ is the column density of the sheet), the sheet is stable
against gravitational fragmentation. When the magnetic field is uniform and in
the plane of the sheet, its stabilizing effect is limited (Tomisaka & Ikeuchi,
1983), but the direction of the fragmentation is determined by the strength of
the external pressure (Nagai et al., 1998). When the external pressure is
considerably smaller than the central pressure of the sheet, filaments are
formed in the perpendicular direction to the mean magnetic field lines and
have a line mass $\lambda$ heavier than the maximum line mass of a filament
supported only by the thermal pressure $\lambda_{\mathrm{crit}}$ (see Equation
(2) below for further details), as $\lambda\sim 2\lambda_{\mathrm{crit}}$.
Such dense filaments have been observed using the Herschel space telescope as
star-forming filaments (André et al., 2010). By contrast, when the external
pressure is comparable to the central pressure of the sheet, the resulting
filament is parallel to the mean magnetic field, with a line mass being
significantly smaller than the critical one
($\lambda\ll\lambda_{\mathrm{crit}}$). This type of filament may be known as a
“striation” observed using the Herschel telescope (Palmeirim et al., 2013).

Recently, several observations have indicated that active star formation
occurs at the junction or overlap point of the filaments (e.g., André et al.,
2010; Nakamura et al., 2014; Frau et al., 2015; Dewangan, 2019; Tokuda et al.,
2019; Kumar et al., 2020). For example, Nakamura et al. (2014) suggested that
the formation of a star cluster in the Aquila Serpens South region, which is
one of the active star-forming regions nearby the molecular clouds, was
triggered by the collision between three filaments. Kumar et al. (2020)
reported that, in high-mass star-forming regions, all luminous clumps ($L\geq
10^{4}L_{\odot}$) exist in the hub-filamentary structures. These structures
are defined as the junction of the filaments and are believed to be formed
because of the coalescence of massive filaments. Therefore, the hub-
filamentary structures are essential to understanding the massive star
formation.

Based on the findings of these observational studies, there is a need to
perform a theoretical study of the induced star formation caused by the
collision between the filaments. Thus far, only a few theoretical (numerical)
studies have been conducted. Duarte-Cabral et al. (2011) reproduced the star
cluster at the Serpens region through a collision of the cylindrical clouds.
Hoemann et al. (2021) determined the conditions for filament collisions by
comparing the contraction time of a filament with a finite length and the
required time for the filaments to collide. However, these studies did not
consider the magnetic field for simplicity. Star formation is significantly
affected by the magnetic field, for example, shock compression and stability
against gravitational instability. Hence, a magnetohydrodynamical simulation
study of filament collisions is required.

From a theoretical standpoint, in this study, we investigated the conditions
for star formation induced by the collision between the filaments, including
the magnetic field and the subsequent evolution.

In this paper, we developed a numerical simulation for explaining the filament
collision. The remainder of this paper is organized as follows: In section 2,
we briefly summarized the basic theory of the filament. In section 3, we
introduced a model for our numerical calculations In section 4, we detailed
the simulation results obtained from the filament collision In section 5, we
discussed the condition of the radial collapse of the merged filament. In
section 6, we summarized the results and conclusions of this study

##  2 The critical line mass of the filament

The stability of the isolated filaments is important in considering the
initial condition of star formation and is often expressed with the mass per
unit length, that is, line mass. The hydrostatic equilibrium state of a
nonmagnetized infinite cylindrical isothermal cloud has a density distribution
$\rho(r)$, which is given as follows Stodółkiewicz (1963); Ostriker (1964):

$\rho(r)=\rho_{c}\left(1+\frac{r^{2}}{8H^{2}}\right)^{-2},$ (1)

where $\rho_{c}$ is the central density of the cylinder and $H$ is the scale
height, which is expressed as $H=c_{s}/\left(4\pi G\rho_{c}\right)^{1/2}$
using the isothermal sound speed $c_{s}$ and the gravitational constant $G$.
By integrating Equation (1) into the radius $r$ to infinity, the line mass of
the filament reaches a constant value as follows:

$\lambda_{\rm crit}=\int^{\infty}_{0}2\pi\rho rdr=\frac{2c_{s}^{2}}{G}.$ (2)

This constant value is called the “critical line mass” and is often used as a
criterion that determines whether the star formation begins in the filament or
not (Nagasawa, 1987; Inutsuka & Miyama, 1992; André et al., 2010).

Filaments with a larger line mass than $\lambda_{\mathrm{crit}}$ have no
hydrostatic solution, and radial contraction is induced in such filaments
owing to self-gravity. By contrast, less massive filaments than
$\lambda_{\mathrm{crit}}$ have hydrostatic equilibrium configurations.

As mentioned in Section 1, previous observations have shown that the filaments
tend to have a magnetic field perpendicular to the long axis. The magnetic
fields affect the stability of the filaments as follows: when the isothermal
filament is threaded by a lateral magnetic field, the critical line mass of
the magnetized filament ($\lambda_{\rm crit,B}$) is obtained through numerical
calculations performed by Tomisaka (2014) and is expressed as a function of
the magnetic flux as,

$\lambda_{\rm crit,B}\simeq 0.24\frac{\Phi}{G^{1/2}}+1.66\frac{c_{s}^{2}}{G},$
(3)

where $\Phi$ represents one-half of the magnetic flux threading a filament per
unit length, which has a dimension of the magnetic flux density of $B$ times
the scale length $L$, that is, $[\Phi]=[B][L]$ (see also Kashiwagi & Tomisaka
(2021) for the equivalent empirical relation for a polytropic gas filament.)
This empirical formula indicates that the magnetized critical line mass
increases linearly by the magnetic flux, and the filament supports a large
line mass by the Lorentz force compared with the nonmagnetized filaments,
$2c_{s}^{2}/G$. 111 In Equation (3), when $\Phi\rightarrow 0$, the magnetized
critical value is smaller than the nonmagnetized value ($\lambda_{\rm
crit,B}<\lambda_{\rm crit}=2c_{s}^{2}/G$). This is because this equation was
obtained using the least square method; however, in practice, the magnetized
critical line masses at small $\Phi$ converge at the nonmagnetized critical
line masses.

This empirical formula (Equation (3)) was obtained by plotting the maximum
line mass of the magnetohydrostatic equilibrium solutions for a given magnetic
flux

$\Phi=R_{0}\cdot B_{0}.$ (4)

A series of equilibrium solutions were obtained for filaments whose mass
distribution against the magnetic flux was considered as a hypothetical
filament with a uniform density $\rho_{0}$ and a radius $R_{0}$, which was
threaded by a uniform magnetic field $B_{0}$ and immersed in an external
pressure of $p_{\mathrm{ext}}$. In this case, an equilibrium configuration was
specified with the hypothetical density $\rho_{0}$ or its line mass
$\lambda=\rho_{0}\pi R_{0}^{2}$, $R_{0}$, and $B_{0}$. As is mentioned in
Tomisaka (2014), the density of the hypothetical filament $\rho_{0}$ was
substituted with the central density $\rho_{c}$, which appeared in the
equilibrium filament.

Based on these results, we investigate the dynamical evolution of
filament–filament collision.

##  3 Methods and models

###  3.1 Basic Equations of Ideal MHD

In this study, we investigated a head-on collision of two identical filaments
penetrated by a magnetic field such that they share the same magnetic flux. To
simulate the filament–filament collision, we solve the following ideal
magnetohydrodynamic equations, which can be written in a conservative form as
follows:

$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\textit{{v}})=0,$ (5)

$\frac{\partial\rho\textit{{v}}}{\partial
t}+\nabla\cdot\left(\rho\textit{{v}}\textit{{v}}-\frac{\textit{{B}}\textit{{B}}}{4\pi}+P_{\rm
tot}\right)=-\rho\nabla\psi,$ (6)

$\frac{\partial\textit{{B}}}{\partial
t}-\nabla\times\left(\textit{{v}}\times\textit{{B}}\right)=0,$ (7)

where $\rho$ is the density, $\bm{v}$ is velocity, $\bm{B}$ is the magnetic
field, $P_{\rm tot}$ is the total pressure, which is the sum of the gas
pressure and the magnetic pressure and is written as,

$P_{\rm tot}=p_{\rm gas}+\frac{B^{2}}{8\pi},$ (8)

, and $\psi$ is the gravitational potential obtained by solving Poisson’s
equation expressed in the following equation:

$\Delta\psi=4\pi G\rho,$ (9)

where $G$ is the gravitational constant. We assumed that the isothermal
filaments are confined within an isothermal hot ambient medium, leading to the
presence of two gases with different temperature. To reproduce this, we
implemented a scalar field, in which the equation of state is given as,

$p_{\rm gas}=S\rho,$ (10)

where $S$ is the scalar field proportional to the temperature, and we assumed
that $S$ evolves according to the advection of gas as follows:

$\frac{\partial S}{\partial t}+(\bm{v}\cdot\mathbf{\nabla})S=0.$ (11)

Further details regarding the scalar field is described in the next section
(see section 3.2).

To solve the basic equations, we used Athena++ (Stone et al., 2020) with the
following setting: the time evolution of these magnetohydrodynamic equations
was solved using the third-order Runge–Kutta time integrator scheme, which was
proposed by Gottlieb et al. (2009), and the Riemann problem was solved using
the HLLD method (Miyoshi & Kusano, 2005). For spatial construction, the
piecewise linear method was applied to the characteristic variables. The
constrained transport method was applied for maintaining the initial
$\nabla\cdot\bm{B}=0$ condition (Evans & Hawley, 1988; Gardiner & Stone,
2008). The multi-grid algorithm was used to solve Poisson’s equation (Equation
(9)) and was implemented in Athena++ by Tomida & Stone (2023).

In addition, we modified the multi-grid module of our two-dimensional
calculations because it was originally designed for three-dimensional
simulations. The Jeans criterion (Truelove et al., 1997) was used to avoid
unexpected numerical fragmentation. We stopped the calculations when the
relation between the Jeans length $L_{\rm J}=\sqrt{\pi c_{s}^{2}/(G\rho)}$ and
the grid size $\Delta x$ breaks the condition of $4\Delta x\leq L_{\rm J}$.

###  3.2 Initial and Boundary Conditions

Figure 1:  An example of the initial conditions: the density and magnetic
field lines of the initial condition of the model $\beta 01\mathrm{M}$ are
plotted (see Table 2). Two identical filaments in magnetohydrostatic
equilibrium are placed close to each other. These filaments have initial
velocities of $V_{0}$ and $-V_{0}$. White lines represent the magnetic field
lines. The external medium is shown in black.

We assume that the filaments are initially in magnetohydrostatic equilibrium
threaded by the global magnetic field, which is perpendicular to their long
axis. This magnetohydrostatic equilibrium solution is obtained by numerical
calculations (Tomisaka, 2014; Kashiwagi & Tomisaka, 2021). We then consider
the initial condition of two colliding filaments, which are formed through
fragmentation from the sheet, in which they shared the same magnetic flux
(Nagai et al., 1998). The filaments are assumed to extend infinitely and
uniformly in the $z$-axis direction in the Cartesian coordinate system
$(x,y,z)$. In the $x-y$ plane, we assume a computational domain of
$-2.5R_{0}\leq x,y\leq 2.5R_{0}$, where $R_{0}$ is the radius of the
hypothetical parent cloud of the filament in magnetohydrostatic equilibrium
(see Equation (4)). The magnetic field outside the filament is considered to
be force-free and converged to the uniform magnetic field of $B_{0}$ with an
increase in the distance from the filament $r\gg R_{0}$, following Tomisaka
(2014). Figure 1 shows one of the initial states assumed in this study. The
two identical filaments are adjacent to each other and moving in $\pm
x$-directions. The global magnetic field is parallel to the $x$-axis.

In our simulation, we assumed that the gas temperature in the molecular cloud
is constant, $T\simeq 5-10\,\mathrm{K}$; thus, the equation of state is
isothermal. However, we consider the possibility that the temperature of the
gas may depend on its origin (filament or external medium). Therefore, we use
the scalar field (Equation (10)) in the equation of state. Initially, we
assume that the filament and the external medium achieved pressure balance at
the surface as follows:

$p_{s}=S_{\mathrm{fil}}\rho_{s}=S_{\mathrm{ext}}\rho_{\mathrm{ext}},$ (12)

where $p_{s}$ is the pressure at the filament surface; $\rho_{s}$ and
$\rho_{\mathrm{ext}}$ represent the density at the surface and that of the
external medium, respectively; and $S_{\mathrm{fil}}$ and $S_{\mathrm{ext}}$
denote values of scalar $S$ inside and outside of the filament, respectively.
We set the density contrast of the external medium and the filament surface as
$\rho_{\mathrm{ext}}/\rho_{s}=0.01$. Therefore, the scalar $S$ of the external
medium is $S_{\mathrm{ext}}=100c_{s}^{2}$ and that of inside the filament is
$S_{\mathrm{fil}}=c_{s}^{2}$, where $c_{s}$ is the sound speed inside the
filament. Finally, in our simulation, we maintain the value of the sound speed
of the external medium at $S_{\mathrm{ext}}^{1/2}=10c_{s}$. In other words, as
the density of the external medium is lower than that of the filament, we
assume that the temperature of the external medium is higher than the
filament, and the initial temperature is maintained after the simulations are
started.

The boundary conditions of the hydrodynamic variables were imposed to be
periodic in the $y$-direction and outflow in the $x$-direction. The boundary
condition of the gravitational potential was assumed to be the Dirichlet
condition, and the potential is calculated using multipole expansion.

###  3.3 Normalization

Table 1: Units used for normalization Unit of pressure | External pressure, $p_{\rm ext}$
---|---
Unit of density | Density at the surface, $\rho_{s}=p_{\mathrm{ext}}/c_{s}^{2}$
Unit of time | Free-fall time, $t_{\rm ff}=(4\pi G\rho_{s})^{-1/2}$
Unit of speed | Isothermal sound speed in the filament, $c_{s}$
Unit of magnetic field strength | $B_{u}=(4\pi p_{\rm ext})^{1/2}$
Unit of length | $L=c_{s}t_{\rm ff}=c_{s}/(4\pi G\rho_{s})^{1/2}$

In this study, physical variables are normalized using the following
quantities: external pressure ($p_{\mathrm{ext}}$), density of the filament
surface ($\rho_{s}$), and sound velocity in the filament ($c_{s}$). The scale
length $L$ is defined by the free-fall time $t_{\mathrm{ff}}$ at the
filament’s surface density $\rho_{s}$ and the isothermal sound speed inside
the filament as $L=c_{s}t_{\mathrm{ff}}=c_{s}/(4\pi G\rho_{s})^{1/2}$. The
physical scales characterizing the system are given in Table 1. We define the
normalized variables as follows: $\bm{v}^{\prime}\equiv\bm{v}/c_{s}$,
$p^{\prime}\equiv p_{\mathrm{gas}}/p_{\mathrm{ext}}$,
$\rho^{\prime}\equiv\rho/\rho_{s}$, $\Phi^{\prime}\equiv\Phi/(B_{u}L)$,
$\bm{B}^{\prime}\equiv\bm{B}/B_{u}$,
$\lambda^{\prime}\equiv\lambda/(\rho_{s}L^{2})$, $x^{\prime}\equiv x/L$, and
$y^{\prime}\equiv y/L$, where the prime represents the normalized variables.
The scalar field is normalized using the isothermal sound speed in the
filament as $S^{\prime}\equiv S/c_{s}^{2}$.

###  3.4 Parameters

In this study, we consider three parameters: the field strength $B_{0}$, the
total line mass $\lambda_{\mathrm{tot}}$, and the collision speed
$V_{\mathrm{int}}$. The former two characterize the magnetohydrostatic state
of each of the filaments, while $V_{\mathrm{int}}$ specifies the condition of
the collision.

The nondimensional form of the field strength $B_{0}$ is given by
$B^{\prime}_{0}=B_{0}/(4\pi p_{\mathrm{ext}})^{1/2}=\sqrt{2}\beta_{0}^{-1/2}$,
where $\beta_{0}=8\pi p_{\mathrm{ext}}/B_{0}^{2}$ is the plasma beta of the
external gas far from the filaments. In this study, $\beta_{0}$ is used
instead of $B_{0}$ to specify the field strength. We examine three plasma beta
values: $\beta_{0}=1$ (weak), $\beta_{0}=0.1$ (fiducial), and $\beta_{0}=0.01$
(strong).

The total line masses are determined based on the magnetic critical line mass,
whose nondimensional form is given by

$\lambda^{\prime}_{\rm crit,B}\simeq 3.04\Phi^{\prime}+20.8,$ (13)

where the nondimensional magnetic flux
$\Phi^{\prime}=R_{0}^{\prime}\sqrt{2/\beta_{0}}$ is a function of
$R_{0}^{\prime}$ and $\beta_{0}$, that is,
$\lambda_{\mathrm{tot}}^{\prime}/2\leq\lambda_{\mathrm{crit,B}}^{\prime}$. For
simplicity, the radius of the parental cylinder $R_{0}^{\prime}$ is set to
$2$. The magnetic critical line masses for the three $\beta_{0}$ are
$\lambda^{\prime}_{\rm crit,B}(\beta_{0}=1)=1.17\lambda^{\prime}_{\rm crit}$,
$\lambda^{\prime}_{\rm crit,B}(\beta_{0}=0.1)=1.91\lambda^{\prime}_{\rm
crit}$, and $\lambda^{\prime}_{\rm
crit,B}(\beta_{0}=0.01)=4.25\lambda^{\prime}_{\rm crit}$, where
$\lambda^{\prime}_{\mathrm{crit}}$ is the nondimensional form of the critical
line mass of Equation (2) as
$\lambda^{\prime}_{\mathrm{crit}}=\lambda_{\mathrm{crit}}/(\rho_{s}L^{2})=8\pi$.
We examine three types of line masses for the respective magnetic strengths,
in which the filament in models L (massive), M (intermediate), and S (less
massive) has a line mass of $0.9$, $0.75$, and $0.5$ times as large as the
magnetically critical line mass,
$\lambda_{\mathrm{crit,B}}^{\prime}(\beta_{0})$, respectively. We also
calculated additional models of S’
(0.6$\lambda_{\mathrm{crit,B}}^{\prime}(\beta_{0})$) and M’
(0.7$\lambda_{\mathrm{crit,B}}^{\prime}(\beta_{0})$), if necessary.

For calculating the collision speed $V_{\mathrm{int}}$, the cases with
$V_{\mathrm{int}}=c_{\mathrm{s}}$ and $10c_{\mathrm{s}}$ are considered.

In short, the head-on collision was reproduced using the plasma beta value
($\beta_{0}$), total line mass ($\lambda^{\prime}_{\mathrm{tot}}$), and
initial relative velocity ($V^{\prime}_{\rm int}$) as the parameters. In this
study, we simulate 16 different models, shown in Table 2. Hereafter, we omit ′
with normalized variables, unless the quantities are misunderstood as
dimensional variables.

Table 2: Model parameters

$\mathrm{Model}^{(1)}$ | ${\beta_{0}}^{(2)}$ | ${\lambda_{\rm tot}/\lambda_{\rm crit}}^{(3)}$ | ${V_{\rm int}/c_{s}}^{(4)}$ | ${\rho_{c}/\rho_{s}}^{(5)}$ | ${\rho_{\rm lim}/\rho_{s}}^{(6)}$ | ${N_{\rm mesh}}^{(7)}$ | $\mathrm{Result}^{(8)}$
---|---|---|---|---|---|---|---
$\beta 1$S | 1 | 1.17 | 1 | 3.30 | $4.13\times 10^{5}$ | $4096\times 4096$ | stable
$\beta 1$S’ | 1 | 1.40 | 1 | 5.00 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 1$M | 1 | 1.76 | 1 | 11.0 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 1$L | 1 | 2.11 | 1 | 46.90 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 01$S | 0.1 | 1.91 | 1 | 6.80 | $4.13\times 10^{5}$ | $4096\times 4096$ | stable
$\beta 01$S_highV | 0.1 | 1.91 | 10 | 6.80 | $4.13\times 10^{5}$ | $4096\times 4096$ | stable
$\beta 01$S’ | 0.1 | 2.29 | 1 | 10.90 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 01$M | 0.1 | 2.87 | 1 | 24.50 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 01$M_highV | 0.1 | 2.87 | 10 | 24.50 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 01$L | 0.1 | 3.44 | 1 | 81.0 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 001$S | 0.01 | 4.25 | 1 | 37.0 | $4.13\times 10^{5}$ | $4096\times 4096$ | stable
$\beta 001$S’ | 0.01 | 5.20 | 1 | 62.50 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 001$M | 0.01 | 6.38 | 1 | 140.0 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 001$L | 0.01 | 7.65 | 1 | 406.2 | $1.65\times 10^{6}$ | $8192\times 8192$ | radial collapse
$\beta 1$S’$\perp$ | 1 | 1.40 | 1 | 5.0 | $1.03\times 10^{5}$ | $2048\times 2048$ | stable
$\beta 1$M’$\perp$ | 1 | 1.64 | 1 | 8.20 | $1.03\times 10^{5}$ | $2048\times 2048$ | radial collapse

  * •

Notes. The table columns are as follows: (1) Name of the model, (2) plasma
beta value, (3) the total line mass normalized by the thermal critical line
mass, (4) initial relative velocity, (5) central density of the initial
filament, (6) maximum density due to numerical restriction, which is
determined by the Jeans criterion that is $\rho^{\prime}_{\mathrm{lim}}\simeq
1.66\times
10^{6}(L^{\prime}_{\mathrm{box}}/10)^{-2}(N_{\mathrm{grid}}/8192)^{2}$, when
the numerical box size $L^{\prime}_{\mathrm{box}}=10$ is divided into
$N_{\mathrm{grid}}\times N_{\mathrm{grid}}=8192\times 8192$ grid points, (7)
number of grid points, and (8) dynamical state of the merged filament. In all
the models, the numerical box size is set to $L^{\prime}_{\mathrm{box}}=10$.

##  4 Results

As shown in Table 2, our results showed that the outcome of filament
collisions can be divided into two categories: a radial collapse model and a
stable model. The radial collapse model is characterized by a monotonic
increase in the central density of the merged filament and indicates the
global collapse of the system. For the stable models, the merged filament does
not collapse globally but oscillates around the magneto-hydrostatic state. We
focus on the typical evolution of the models by explaining the radial collapse
model, first.

###  4.1 Radial Collapse Model

|
---|---
|
Figure 2:  Time evolution of the radial collapse model $\beta 01$M. This figure shows the cross-section of the colliding filaments on the $x-y$ plane, and the plot area is restricted to $\pm 2.5$ from the box size ($\pm 5$). Each panel shows the cross-section at different epochs: $t=0.350$ (a), $0.670$ (b), $0.710$ (c), and $0.735$ (d). The color scale represents the density, and the white lines are the magnetic field lines.  |
---|---
|
Figure 3:  The one dimentional profiles on the $x$\- and $y$-axes for the
radial collapse model $\beta$01M at four different epochs: $t=0.350$ (red),
$0.670$ (yellow), $0.710$ (green), and $0.735$ (blue). The density and
velocity profiles on the $x$\- and $y$-axes are shown in (a) and (b),
respectively, in which the solid lines represent the density distributions
($\rho(x)$ in (a) and $\rho(y)$ in (b)) in the upper panels, whereas these
solid lines represent the velocity profiles ($v_{x}$ in (a) and $v_{y}$ in
(b)) in the lower panels. The gas and magnetic pressure profiles on the $x$\-
and $y$-axes are shown in (c) and (d), respectively. In (c) and (d), the solid
and dashed lines in the upper panel represent the gas and magnetic pressure,
respectively. The solid lines in the lower panels represent the plasma beta
values.

In this section, we describe the typical evolution of the radial collapse
model. The model with $\beta_{0}=0.1$,
$\lambda_{\mathrm{tot}}=2.87\lambda_{\rm crit}$, and $V_{\rm int}=1$ is
selected as the fiducial model and corresponded to $\beta 01$M in Table 2.

Figure 2 shows the images of the time evolution of the filament–filament
collision. In Figure 2(a), at $t=0.350$, the collision of two filaments
resulted in the formation of a sheet-like dense structure, which is elongated
in the $y$-direction and with a thickness of $\sim 0.1$ measured on the
$x$-axis at the center of the merged filament, referred to as the shocked
region.

Figures 3(a) and (b) show the density and velocity profiles on the $x$\- and
$y$-axes, respectively. In the upper panel of Figure 3(a), a shock front is
visible at $|x|\simeq 0.06$ in the density profile. This is visualized as the
edge of the shocked region, which extends in the $y$-direction in Figure 2(a).
Along the $y$-axis, shock fronts have not yet formed at $t=0.350$ (see the
upper panel of Figure 3(b)). By comparing the density profiles on the $x$\-
and $y$-axes, we can observe that the density distribution of the merged
filament in the $y$-direction ($|y|\simeq 1.5$) is wider than that in the
$x$-direction ($|x|\simeq 0.06$). In the lower panel of Figure 3(a), the
velocity distribution ($v_{x}$) at $t=0.350$ shows that a global inflow to the
shocked region occurs in the $x$-direction outside the shocked region
($|x|\gtrsim 0.06$). On the $y$-axis, the velocity distribution ($v_{y}$) does
not show such a global inflow; however, an accretion flow is newly formed by
the self-gravity near $|y|\simeq 1$ (see the lower panel of Figure 3(b)).

Figure 2(b) shows that the shocked region becomes denser at $t=0.670$ and is
observed more clearly with time. In addition, the magnetic field lines are
dragged toward the center of the shocked region. Magnetic field lines bend at
the shock front reaching the front in the post-shock (inner) region,
indicating that the shock wave created by the collision is a fast shock.

In the upper panel of Figure 3(a), By comparing the density profiles at
$t=0.350$ (red line) and $t=0.670$ (yellow line), We show that the overall
density of the shocked region increases approximately by a factor of $100$.

The position of the shock front moved from $|x|\simeq 0.06$ to $|x|\simeq
0.13$, indicating that the shock wave is expanding. The shock front, which is
at $|x|\simeq 0.06$ at $t=0.350$, expands to $|x|\simeq 0.13$ at $t=0.670$.

Furthermore, another shock is observed at $|x|\simeq 0.04$, which continues to
expand to the edge of the filament $|y|\lesssim 1.5$, as observed in Figure
3(b). The panel shows that the magnetic field lines bend at the front,
departing from the front in the post-shock region $|x|\lesssim 0.04$. In
addition, the lower panel of Figure 3(c) shows that the plasma beta attains
$\beta\simeq 0.13$ in the pre-shock region $|x|\gtrsim 0.1$, decreases with a
jump passing the fast shock front at $|x|\simeq 0.13$, and increases with a
positive jump at the accretion shock region $|x|\simeq 0.04$, which indicates
that the second shock is a slow shock. In other words, as a consequence of the
supersonic collision of the filaments, a pair of outward-facing shock fronts
are formed (outer fast and inner slow shocks).

On the contrary, along the $y$-axis, although the overall density of the
shocked region increases with time, there is no clear evidence of the presence
of a shock wave (see the upper panel of Figure 3(b)). In the lower panel of
Figure 3(b), the velocity distribution ($v_{y}$) at $t=0.670$ shows that the
central directional velocity component ranges from $|y|\simeq 10^{-2}$ to
$|y|\simeq 2$. The lower panel of Figure 3(b) shows that the gas in
$10^{-2}\lesssim|y|\lesssim 2$ moves toward the center. By comparing the
velocity structures at $t=0.350$ and $t=0.670$, we show that the inflow is
accelerated with time by the effect of self-gravity.

At $t=0.710$, Figure 2(c) shows that the fast shock front, which was formed by
the collision, expands to $|x|\simeq 0.2$. Another slow shock is almost
stalled around $|x|\simeq 0.04$. However, other differences from the previous
epoch are not easily distinguished from the two-dimentional density profiles.
Therefore, to more effectively analyze the evolution after the collision, we
primarily focused on the radial profiles. In the upper panel of Figures 3(a)
and (b), the density profiles at $t=0.710$ (green lines) showed that the
overall density of the shocked region continues to increase compared with the
last epoch with $t=0.670$ (yellow). Although the slow shock facing toward the
$x$-axis was almost stagnant near $|x|\simeq 0.04$, the density profile
illustrated that the accretion shock is newly formed at $|y|\simeq 0.03$ along
the $y$-axis. This accretion shock is formed by the infalling gas as it
reaches the high-density region, and the jump in the density and pressure at
this location indicates the formation of an accretion shock. This indicates
that the gas is continuously accreted by the self-gravity of the shocked
region, leading to an increase in the density and pressure of the region. The
velocity profiles clearly show that the high-density shocked gas, even inside
the accretion shock, contracts as a whole, primarily by the inflow in the
$y$-direction (see the lower panel of Figures 3(a) and (b)).

At $t=0.735$, the 1D distribution of the merged filament near its final state
is represented by the blue solid lines in Figures 3(a) and (b). The shock
front contracts as a whole and reaches $|x|\simeq|y|\simeq 0.025$. Inside
$r\lesssim 5\times 10^{-3}$, the so-called “Hubble-like inflow” was observed
as $v_{x}/x\simeq v_{y}/y\simeq\mathrm{const}$, and a uniform density core is
contractiong in this region.

Next, we focus on the evolution of the magnetic field driven by the collision.
Figures 3(c) and (d) show the time evolution of the gas ($p_{\rm gas}$),
magnetic ($p_{\mathrm{B}}$) pressures, and plasma beta ($\beta\equiv p_{\rm
gas}/p_{\mathrm{B}}$) along the $x$\- and $y$-axes, respectively. The magnetic
pressure is defined as $p_{\rm B}\equiv B^{2}/2$ in our nondimensional
variables. In the upper panels of Figures 3(c) and (d), we showed that the
magnetic pressure within the shocked region is consistently smaller than the
gas pressure throughout the evolution. As an example, at $t=0.735$, the plasma
beta in the central part of the shocked region is $\beta=2.83$, meaning that,
even in the presence of a magnetic field, thermal pressure dominates near the
center of the shocked region.

The collision process leads to the amplification of the magnetic field
strength, as observed by comparing the magnetic pressure of the central part
at $t=0.350$ (red dashed line) and $t=0.735$ (blue dashed line) in the upper
panel of Figures 3(c) and (d), which indicate that the magnetic pressure at
the central part increases from $p_{\rm B}\simeq 30$ to $p_{\rm B}\simeq
2.6\times 10^{5}$, and the amplification reaches a factor $\sim 10^{4}$.

This suggests that the magnetic field strength is amplified by a factor of
$\sim 10^{2}$ owing to the contraction induced by the collision.

In Figure 4, we plot the evolution of maximum density $\rho_{\mathrm{max}}$
for the collapse ($\beta$01M) and the stable models ($\beta$01S). Except for a
special circumstance in the early phase of collision, maximum density is
attained at the center of the merged filament as
$\rho_{\mathrm{max}}=\rho_{c}\equiv\rho(0,0)$. Figure 4 clearly shows that the
central density increases monotonically with time, and the contraction never
stops in the collapse model.

Figure 4:  Maximum density $(\rho_{\mathrm{max}})$ as a function of the
elapsed time $(t)$. The red and blue lines represent the radial collapse
($\beta$01M) and stable ($\beta$01S) models, respectively.

###  4.2 Stable Models

|
---|---
|
Figure 5:  Same as Figure 2, but for the stable model ($\beta 01$S). Each panel shows the cross-section at different epochs: $t=0.900$ (a), $1.300$ (b), $1.900$ (c), and $3.500$ (d).  |
---|---
Figure 6:  Same as Figure 3, but for the stable model ($\beta 01$S). Color
represents different epochs: $t=0.900$ (red), $1.300$ (yellow), $1.900$
(green), and $3.500$ (blue).

For stable runs, we develop the model $\beta 01$S ($\beta_{0}=0.1$,
$\lambda_{\mathrm{tot}}=1.91\lambda_{\mathrm{crit}}$, and $V_{\rm int}=1$).

The dynamical evolution of the stable models is similar to that of the radial
collapse model during the stage at which the shock fronts are sweeping two
colliding filaments. Figure 5 shows four images of the time evolution after
the collision of two filaments. In Figure 5(a), at $t=0.900$, we can observe a
sheet-like structure enclosed with two shock fronts facing outwardly, which is
the same as that observed in the radial collapse model (Figure 2(a)). Figures
6(a) and (b) represent the density and velocity profiles along the $x$\- and
$y$-axes, respectively. The red lines correspond to the same epoch, $t=0.900$,
as observed in Figure 5(a), which reveals that the shock wave is formed only
along the $x$-axis ($|x|\simeq 0.1$). In the lower panel, velocity
distributions (red lines) confirm this finding by showing that the global
inflow to the shocked region is only observed along the $x$-axis.

At $t=1.300$, Figure 5(b) shows that the central part of the shocked region
undergoes contraction and becomes denser, with the magnetic field lines also
being dragged towards the center. In Figure 6, the yellow lines indicate the
density and velocity profiles at $t=1.300$. By comparing with the density
profile of the previous epoch ($t=0.900$), we show that the central density of
the shocked region increased by a factor of $\simeq 6$. In addition, the
density profile along the $y$-axis shows that a new accretion shock is formed
at $|y|\simeq 0.3$. In the lower panel of Figure 6(b), the velocity
distribution further supports this finding as $v_{y}$ indicates the presence
of the accretion shock at $|y|\simeq 0.3$, resulting from the global inflow
occurring outside the new accretion shock ($|y|\gtrsim 0.3$). Early phase
evolution is qualitatively the same as the collapsing model; for example, the
formation of the shocked region initially occurred because of the $x$-axis
global inflow (stage of $t=0.900$) and subsequently because of the $y$-axis
accretion flow (stage of $t=1.300$). However, in the stable model, the
velocity profile on the $x$-axis demonstrated that the shock fronts observed
in Figure 5(a) have swept across the merged filament, the boundary of which is
observed at $|x|\simeq 0.50$, and are traveling in an external low-density
medium (see the yellow line in the lower panel of Figure 6(a)).

In Figure 4, we plotted the evolution of maximum density
$\rho_{\mathrm{max}}(t)$ of this stable model in blue. Figures 5(c) and (d)
depict the structures when $\rho_{\mathrm{max}}$ takes a maximum ($t\simeq
1.9$) and a minimum ($t\simeq 3.5$), respectively. In Figure 6, a comparison
between these density profiles at $t=1.9$ and $3.5$ shows that the density
near the center decreases from $t=1.900$ to $t=3.500$, more than one order of
magnitude; that is, by passing the maximum at $t=1.900$, the shocked region
does not collapse but instead expands, as shown in Figure 5(d). Furthermore,
the magnetic field lines are dragged outward. After the expansion of the
shocked region, the central density of the shocked region begins to increase
again around $t\simeq 3.7$ and subsequently decreases again around $t\simeq
5.0$, as shown in Figure 4. Therefore, this stable model does not exhibit a
global collapse but the shocked region continues to oscillate after the
collision.

Figure 7:  Density profiles on the $x$\- and $y$-axes for the stable model
$\beta 01$S at time $t=6.500$, which corresponds to the epoch after the
oscillation of two cycles. The magenta-colored solid and dashed lines
correspond to the density profiles on the $x$\- and $y$-axes, respectively.
The cyan-colored solid ($\rho_{\rm eq}(x)$) and dashed ($\rho_{\rm eq}(y)$)
lines correspond to the density profiles on the $x$\- and $y$-axes,
respectively, for the magnetohydrostatic equilibrium state (Tomisaka, 2014).

We then focus on the structure of the shocked region of the stable model.
Figure 7 shows the density profile after oscillating for two cycles
($t=6.500$), in which the density profiles are plotted on the $x$\- (magenta
solid line) and $y$-axes (magenta dashed line). In this figure, we also plot
the magnetohydrostatic equilibrium solution of Tomisaka (2014), which has the
same line mass and magnetic flux as the merged filament, represented in cyan.
Figure 7 clearly indicates that the density profiles of the shocked region are
quite similar to those of the magnetohydrostatic equilibrium state.

###  4.3 Effect of Plasma Beta

|
---|---
Figure 8:  Same as Figure 4, but for the comparison of models with different $\beta_{0}$ values. Panels (a) and (b) show the maximum density evolution for the radial collapse and stable models, respectively. Color represents different plasma beta values: $\beta_{0}=1$ (red), 0.1 (blue), and 0.01 (yellow).  |
---|---
Figure 9:  The density profiles on the $x$\- and $y$-axes for the radial collapse models with varying plasma beta values ($\beta_{0}$) are presented in panels (a) and (b), respectively. The colored solid lines show the results of different values of $\beta_{0}$: $\beta_{0}=1$ (red), 0.1 (yellow), and 0.01 (blue). The black solid and dashed curves indicate the hydrostatic equilibrium solutions for the gas disk and filament, respectively.  |  |
---|---|---
Figure 10:  Same as Figure 7, but for the comparison of models with different
$\beta_{0}$ values. Panels (a), (b), and (c) correspond to the models with
$\beta_{0}=1$, $0.1$, and $0.01$, respectively.

In the previous sections 4.1 and 4.2, we show the models with a plasma beta
value of $\beta_{0}=0.1$. Here, we compared the models with plasma beta values
of $\beta_{0}=1$ (models $\beta 1$S and $\beta 1$M) and $0.01$ (models $\beta
001$S and $\beta 001$M).

Figure 8 shows the maximum density evolution of the radial collapse (a) and
stable (b) models. For the radial collapse model (a), the strength of the
initial magnetic field does not affect the outcome of radial collapse.
However, the stable model (b) maintains a stable state while undergoing
oscillations, regardless of the initial plasma beta value. We assumed that the
line masses of intermediate (M) and less-massive (S) filament are 0.75 and 0.5
times as large as the magnetically critical line mass of Equation (13),
respectively. As the magnetically critical line mass increases with a decrease
in plasma beta $\beta_{0}$, line masses of the respective models satisfy the
following relations:
$\lambda(\beta\mathrm{1M})<\lambda(\beta\mathrm{01M})<\lambda(\beta\mathrm{001M})$,
and
$\lambda(\beta\mathrm{1S})<\lambda(\beta\mathrm{01S})<\lambda(\beta\mathrm{001S})$.
The line mass increases as the plasma beta value decreases, leading to a
stronger gravitational force in the merged filament. Consequently, the radial
collapse model shows a shorter timescale, and the stable model exhibits a
shorter oscillation period. We defined a time for the maximum density reaching
$\rho_{\mathrm{lim}}$ as $t_{\mathrm{lim}}$. In the radial collapse model,
$t_{\mathrm{lim}}$ changes from $t_{\mathrm{lim}}=1.103$($\beta 1$M) to
$t_{\mathrm{lim}}=0.443$($\beta 001$M). In the stable model, the oscillation
period $T_{\mathrm{osi}}$ becomes shorter from $T_{\mathrm{osi}}\simeq
4.0$($\beta 1$S) to $T_{\mathrm{osi}}\simeq 1.8$($\beta 001$S).

Next, we study the density structure of both models. Figure 9 shows the
comparison of the density profiles for the radial collapse models with varying
plasma beta values ($\beta_{0}$). The profiles are shown for the final epochs,
which have the same central density of $\rho_{c}=\rho_{\mathrm{lim}}\simeq
1.66\times 10^{6}$. In Figure 9, two analytic solutions are shown, that of the
self-gravitating isothermal plane-parallel gas disk (solid line;
$\rho=\rho_{c}\mathrm{sech}^{2}\left(r\sqrt{\rho_{c}/2}\right)$) and that of
the self-gravitating isothermal cylinder (dashed line; Equation (1)). By
considering the central high-density part $\rho\gtrsim 10^{4}$ in Figure 9(a),
we realize that the density profiles on the $x$-axis are more similar to the
hydrostatic solution of the gas disk (solid line) than the solution of the gas
cylinder (dashed line), regardless of $\beta_{0}$. This is because the Lorentz
force is weak along the $x$-axis, leading to the thermal pressure playing a
dominant role against the self-gravity of the gas disk. On the contrary, in
the central high-density part on the $y$-axis (b), the profiles seem to be
more expanded than that of the isothermal gas cylinder as the plasma beta
value decreased from $\beta_{0}=1$ (red) to 0.01 (yellow) owing to the strong
Lorentz force.

Figure 10 shows the density distribution of models $\beta 1$S (a), $\beta 01$S
(b), and $\beta$001S (c) achieved at the respective final epochs as $t=8.100$,
6.600, and 4.800. Figure 10 shows the comparison of these stable models with
the magnetohydrostatic equilibrium states described in Tomisaka (2014). Of
note, the equilibrium state is selected to have the same magnetic flux and the
same central density as the final state of the corresponding stable model. The
density structures in all the models resemble the equilibrium states; in
particular, in the models with a strong magnetic field (b) and (c), two
profiles are well-matched. Thus, in general, less massive filaments lead to a
merged filament stable with oscillation, and its density profile is highly
similar to that of the equilibrium state.

###  4.4 Effect of Initial Velocity

|
---|---
Figure 11:  Same as Figure 8, but for the comparison of models with different $V_{\rm int}$ values. Maximum density is plotted against the time. Panels (a) and (b) show the results for $\beta$01M and $\beta$01M_highV, and $\beta$01S and $\beta$01S_highV, respectively. Color represents the initial relative velocity as $V_{\rm int}=1$ (red) and $V_{\rm int}=10$ (blue). The plasma beta value is set to $\beta_{0}=0.1$.  |
---|---
Figure 12:  Same as Figure 9, but for the comparison of models $\beta$01M and
$\beta$01M_highV with different initial relative velocities of $V_{\rm int}=1$
(red) and $V_{\rm int}=10$ (blue). Other parameters are the same:
$\lambda_{\mathrm{tot}}=2.87$ and $\beta_{0}=0.1$. Images are captured for the
final epoch with $\rho_{c}=\rho_{\mathrm{lim}}=1.66\times 10^{6}$.
---
Figure 13:  Same as Figure 7, but for a model with an initial relative
velocity of $V_{\rm int}=10$ ($\beta$01S_highV). The density profile at
$t=7.10$ is illustrated. The other parameters are the same as the model
$\beta$01S shown in Figure 7: $\lambda_{\mathrm{tot}}=1.91$ and
$\beta_{0}=0.1$.

In this section, we describe the evaluation of the effect of the initial
velocity on the evolution of filament collision. We compare four models:
$\beta$01M, $\beta$01M_highV, $\beta$01S, and $\beta$01S_highV. In model
$\beta$01M_highV, we assume the same filament (line mass and the magnetic
flux) as model $\beta$01M but with a larger collision velocity,
$V_{\mathrm{int}}=10$. Furthermore, model $\beta$01S_highV is a high-speed
collision model of $\beta$01S. Figure 11 shows the evolution of the maximum
density for these four models. Models $\beta$01M_highV and $\beta$01M exhibit
runaway collapse, and models $\beta$01S_highV and $\beta$01S result in
oscillation in $\rho_{\mathrm{max}}$.

Panel (a) shows the effect of the initial velocity on the time scale of
reaching the radial collapse. We defined a time for the maximum density
reaching $\rho_{\mathrm{lim}}$ as $t_{\mathrm{lim}}$. The time is
approximately equal to $t_{\mathrm{lim}}=0.735$ ($V_{\mathrm{int}}=1$) and
$0.600$ ($V_{\mathrm{int}}=10$), respectively. Thus, we show that the time
scale required to reach radial collapse $t_{\mathrm{lim}}$ decreases as the
initial velocity increases. This is because a higher initial velocity leads to
a more efficient inflow of gas toward the shocked region. In other words, the
time required for the shock wave to sweep the filament becomes shorter
compared with a low-speed collision. However, if the initial velocity is too
large (e.g., $V_{\mathrm{int}}\gg 10$), it is likely for the merged filament
to expand more before undergoing radial collapse due to its momentum. We
confirmed this finding using a model colliding with an extreme collision speed
$V_{\mathrm{int}}=15$ (not shown in the paper). Such high-speed collisions
result in a long time to collapse compared with the case with
$V_{\mathrm{int}}=1$. By contrast, panel (b) shows that, even by considering
the supersonic collision with a high Mach number ($V_{\mathrm{int}}=10$), less
massive filaments of $\beta 01$S_highV model evolve into a stable state and
form a static filament.

Next, we focus on the density structure of the radial collapse and stable
models with a high Mach number ($V_{\mathrm{int}}=10$). Figure 12 shows the
density profile on the $x$\- and $y$-axes of the intermediate-mass model,
$\beta 01$M_highV. By comparing the models with $V_{\rm int}=1$ (red line) and
that with $V_{\rm int}=10$ (blue line), we observed that, despite the high
initial velocity, the filament width of the shocked region remains similar to
that of the result for $V_{\rm int}=1$. In other words, the structure of the
dense part ($\rho\gtrsim 10^{4}$) of the shocked region remains almost
similar, regardless of the initial velocity.

In addition, Figure 13 shows the density profile on the $x$\- and $y$-axes of
the less-massive model ($\beta 01$S_highV). The structure of the shocked
region overlaps with the magnetohydrostatic equilibrium state, similar to that
of $V_{\rm int}=1$ (see Figure 7).

##  5 Discussion

Figure 14: Criteria for the radial collapse displayed on the plane
($\lambda_{\rm tot},\beta_{0}$). The circles and cross points represent the
radial collapse and stable models, respectively. The enclosed symbols
correspond to models with large initial velocities ($V_{\mathrm{int}}=10$).
Color represents different line masses normalized by
$\lambda_{\mathrm{crit,B}}$: $\lambda/\lambda_{\mathrm{crit,B}}=0.9$ (red),
0.75 (green), 0.6 (blue), and 0.5 (black). The solid line corresponds to the
magnetized critical line mass (Tomisaka, 2014). The dashed horizontal line
represents the critical line mass of the nonmagnetized filament
(Stodółkiewicz, 1963; Ostriker, 1964). The dashed line represents the critical
line mass obtained by considering the geometric mean of the lightest line mass
in the radial collapse and stable models. We also show the dimensional
quantities for $\lambda_{\mathrm{tot}}[\mathrm{M_{\odot}{pc}^{-1}}]$(right)
and $B_{0}[\mathrm{\mu G}]$(upper), where the sound speed is
$190\mathrm{m\,s^{-1}}$ at $T=10\mathrm{K}$ and the filament surface number
density is $n_{s}=10^{3}\mathrm{cm^{-3}}$.

###  5.1 Criteria for Radial Collapse

In this study, we attempt to determine the conditions under which the merged
filament becomes unstable in the radial direction by filament collision.
Figure 14 summarizes the results of the collision, indicating whether the
merged filament underwent radial collapse or not. This shows that models with
larger line masses and/or models with a weaker magnetic field (larger
$\beta_{0}$) are likely to experience radial collapse. The outcome does not
appear to depend on the collision velocity, $V_{\mathrm{int}}$. In Figure 14,
we plotted the critical line mass of magnetized filaments,
$\lambda_{\mathrm{tot}}=\lambda_{\mathrm{crit,B}}(\beta_{0})$. The results can
be separated by comparison with the magnetically critical line mass (Equation
(13)). Merged filaments with
$\lambda_{\mathrm{tot}}\gtrsim\lambda_{\mathrm{crit,B}}$ undergo radial
collapse, whereas those with
$\lambda_{\mathrm{tot}}\lesssim\lambda_{\mathrm{crit,B}}$ exhibit stable
oscillation. From this finding, we conclude that the necessary condition for
the occurrence of radial collapse in the filament collision is that the total
line mass must exceed the magnetically supported critical line mass. In other
words, the merged filament should be a magnetically supercritical filament.
Even if the magnetic field is extremely strong, the magnetically supercritical
filament is likely to undergo radial collapse. If the collision velocity is
too fast, as explained in Section 4.4, collapse may be delayed. Therefore, the
collapse appears not to occur within a finite time for the collision with an
ultimately large initial velocity. However, in the case of a filament
collision originated from typical velocity dispersion ($\Delta
v\lesssim\mathrm{a\,few\,km\,s^{-1}}$) in molecular clouds, as considering in
this study, this delayed collapse is not realized.

It is evident that the critical line mass is a dominant factor in determining
the radial instability of the shocked region. This conclusion may be
restricted to a specific scenario of a head-on collision described here. In
such a collision, a merged filamentary structure elongated in the $z$
direction is retained as the shocked region because there is no inclination
between the long axes of two filaments. As a result, we expect the radial
instability of the shocked region to be described by the magnetically critical
line mass (Equation (3)). This finding can help in considering the evolution
of a filament–filament collision in a more general configuration. An elongated
merger of a filament–filament collision is made even by a collision with some
inclinations, although the filaments far from the collision point pass by
freely. When such colliding filaments share the same magnetic flux, the
necessary condition of the gravitational collapse given as the merged filament
(or a hub) is magnetically supercritical.

####  5.1.1 Comparison with Collision between Spherical Clouds

While our calculations did not show a velocity dependence, it has been
reported that in the simulation studies which investigate collisions between
spherical clouds (Gilden, 1984; Nagasawa & Miyama, 1987). They found that the
critical total mass above which gravitational collapse is triggered depends
strongly on the collision velocity. When the collision velocities are low, the
shocked region undergoes collapse if the total mass exceeds the Bonnor-Ebert
mass (Ebert, 1955; Bonnor, 1956), which is the critical mass of the isothermal
equilibrium cloud. On the other hand, if clouds collide with a high velocity,
the shocked region expands and does not show collapse even when the mass
exceeds the Bonnor-Ebert mass. Therefore, head-on collisions of spherical
clouds which exceed the Bonnor-Ebert mass will collapse if the collision
velocity is low, but will not collapse if the collision velocity is too high.
We suspect that this difference is due to the distance dependence of gravity
for spherical ($\propto r^{-2}$) and cylindrical ($\propto r^{-1}$)
coordinates. Thus, in the spherical case, the gas dispersed by the collision
easily escapes the gravity of the shocked region compared to the cylindrical
case.

In addition to this, although, if the collision velocity is too large, the
shocked region will turn to expansion, the shocked region shows collapse even
with the total mass below the Bonner-Ebert mass when the collision velocity is
moderate. This fact in the moderate collision velocity regime comes from that
the critical mass of the sphere depends on the external pressure ($\propto
p_{\mathrm{ext}}^{-1/2}$), and thus the critical mass of the shocked region
decreases due to the increasing external pressure, which is derived from the
collision velocity. On the other hand, the critical line mass of a cylinder is
independent of the external pressure. Consequently, the collision velocity is
less important to the stability of the shocked region in contrast to the
spherical configurations.

Although the filaments have a strong gravity compared to the spheres, which
makes it difficult for the dispersed gases to escape, if we had considered
collisions at higher velocities than those considered in our paper, we might
have seen the velocity dependence of the threshold line mass. However, if the
origin of the collision velocity is inherited from the turbulence of the
molecular clouds, then the collision speed we assumed
($V_{\mathrm{int}}/c_{s}=1-10$) is reasonable.

####  5.1.2 Comparison with Observations

Aquila Serpens south region is one of the typical examples where stars are
formed in the filament (section 1). This region contains three clumps
(filaments) (Tanaka et al., 2013), and the main filament is regarded as a site
where multiple filaments collide with each other and star formation is in
progress in the merged filament (Figure 3 of Nakamura et al., 2014). From the
gas mass of the main filament $M\sim 230M_{\odot}$, the line mass is estimated
as $\lambda_{\mathrm{POS}}\sim 480M_{\odot}\mathrm{pc}^{-1}$ by assuming a
length of 0.48 pc 222 Figure 2 of Tanaka et al. (2013) shows the length of the
main filament and the northern clump to be approximately 4 arcminutes. We
adopted a distance of $415\mathrm{pc}$ to Serpens South, resulting in a length
of $0.48\mathrm{pc}$.  projected on the plane of the sky (POS). The northern
clump has a similar mass ($M\simeq 190M_{\odot}$) and size ($0.48\mathrm{pc}$)
to the main clump but are composed of two filaments, which appear to collide
in the future.

Magnetic field strength is measured using the Davis–Chandrasekhar-Fermi method
(Davis, 1951; Chandrasekhar & Fermi, 1953). Pillai et al. (2020) reported an
estimated magnetic field strength of approximately $870\mu\rm G$ within the
main filament of the region based on far-infrared observations. On the
contrary, Kusune et al. (2019) estimated the magnetic field strength in the
same region to be ranging from $10\mu\mathrm{G}$ to $80\mu\mathrm{G}$ using
near-infrared wavelengths, which trace the magnetic field strength in a
relatively low-density medium. These observations are consistent with our
simulation results, in which the magnetic field strength is amplified by
approximately two orders of magnitude due to the collision compared with the
initial filament.

Thus, in the Aquila Serpens south region, two filaments with
$\lambda_{\mathrm{POS}}\sim 200M_{\odot}\mathrm{pc}^{-1}$ collide with each
other and form a merged filament with $\lambda_{\mathrm{POS}}\sim
400M_{\odot}\mathrm{pc}^{-1}$. Although there is some uncertainty regarding
the magnetic field strength, the magnetic field strength outside the main
filament is estimated to be $B_{\mathrm{POS}}\sim 80~{}\mu\mathrm{G}$, which
corresponds to $\beta_{\mathrm{POS}}=5.4\times 10^{-3}$ for an external
pressure of $p_{\mathrm{ext}}=10^{4}\mathrm{K\,cm}^{-3}$.

As the magnetically critical mass is given as
$\lambda_{\mathrm{crit}}(B_{\mathrm{POS}})\simeq 88M_{\odot}\mathrm{pc}^{-1}$
from equation (3), both the main and north filaments appear to be magnetically
supercritical as
$\lambda_{\mathrm{POS}}>\lambda_{\mathrm{crit}}(B_{\mathrm{POS}})$, where we
assumed $R_{0}=0.2\mathrm{pc}$, $c_{s}=190\mathrm{m\,s}^{-1}$, and
$n_{s}=10^{3}\mathrm{cm}^{-3}$. However, we must also consider the geometrical
projection effect. setting the angle between the line-of-sight and the
filament axis, $\alpha$, the true line mass and magnetic field strength are
expressed with those of the POS as
$\lambda=\lambda_{\mathrm{POS}}\sin{\alpha}$ and
$B_{0}=B_{\mathrm{POS}}/\cos{\alpha}$ (or
$\beta_{0}=\beta_{\mathrm{POS}}\cos^{2}\alpha$). This reduces to
$\lambda/\lambda_{\mathrm{crit}}(B_{0})\simeq\sin\alpha\cos\alpha\left[\lambda_{\mathrm{POS}}/\lambda_{\mathrm{crit}}(B_{\mathrm{POS}})\right]$,
when the second term is ignored in equation (3). As the average of
$\langle\sin\alpha\cos\alpha\rangle$ is 1/3, the correction factor is expected
to be approximately 1/3. After considering the geometrical correction,
although the main filament remains slightly supercritical, the two north
filaments become subcritical. In conclusion, the condition for radial collapse
is satisfied in the collision, which we observed in the Aquila Serpens south
region.

###  5.2 Self-similarity of the Collapsing Model

|
---|---
Figure 15:  Density profiles on the $x$\- and $y$-axes for the radial collapse
models ($\beta 1$M), in which $\beta_{0}=1$ and $V_{\rm int}=1$. The ordinate
denotes $\varrho$ and the abscissa denotes $x$\- and $y$-components of
$\bm{\chi}$.

In this section, we describe the examination of self-similar solutions of a
collapsing filament. This solution has been previously studied to describe the
properties of the collapsing filament in (Kawachi & Hanawa, 1998). In Kawachi
& Hanawa (1998), for similarity coordinates, the defined zooming coordinate
$\bm{\chi}$ and the density in the zooming coordinate $\varrho$ as,

$\bm{\chi}\equiv\frac{\bm{r}}{c_{s}|t-t_{0}|},$ (14) $\varrho(\chi)\equiv 2\pi
G(t-t_{0})^{2}\rho,$ (15)

where $t_{0}$ represents the time when the central density becomes
infinity.333 To determine $t_{0}$, we first examine the time evolution of
$1/\sqrt{\rho_{\mathrm{max}}}$. We identify the part that follows a linear
relation similar to$1/\sqrt{\rho_{\mathrm{max}}}=at+b$. The time at which
$1/\sqrt{\rho_{\mathrm{max}}}=0$ corresponds to $t_{0}=-b/a$.

Figure 15 shows the time evolution of the density profiles written in the
similarity coordinate $(\bm{\chi},\varrho(\bm{\bm{\chi}}))$ for model
$\beta$1M. The color gradation indicates a time sequence as the central
density of the merged filament increases from $\rho_{c}=1460$ (cyan) to
$\rho_{c}=71484$ (magenta). Panel (a) shows the density profiles on the
$x$-axis. Although $\rho_{c}$ varies by a factor of $\sim 50$, the density
profiles appear not to depend on time significantly when the similarity
coordinates are used. Panel (b) shows the density profiles on the $y$-axis.
Compared with the density profiles on the $x$-axis, the density profiles on
the $y$-axis demonstrated an increase in their width as the central density
increased. Thus, although the convergence to the self-similarity depends on
the direction, we expect that the contraction proceeds in a self-similar
manner, especially for the collision direction.

###  5.3 Initial Line Mass Dependence of the Shocked Region

---
Figure 16:  Line mass at the shocked region plotted against the maximum
density. The $y$-axis is $\lambda_{\rho_{\mathrm{max}}/100}$, which is the
line mass that is counted over $\rho_{\mathrm{max}}/100$. The $x$-axis shows
the maximum density. The solid and dashed lines represent the intermediate and
massive line mass models, respectively, and the different colors indicate
models with $\beta_{0}=1$ (red), 0.1 (green), and 0.01 (blue), respectively.
The initial relative velocity is fixed as $V_{\rm int}=1$.

In this section, we describe the examination of the initial line mass
dependence of the shocked region. Here, we assume that the line mass of the
shocked region $\lambda_{\rho_{\mathrm{max}}/100}$ is estimated by integrating
the density larger than $\rho_{c}/100$ as
$\lambda_{\rho_{\mathrm{max}}/100}\equiv\iint_{\rho\geq\rho_{\mathrm{max}}/100}\rho
dxdy$. This assumption aims to follow the evolution of the dense part in the
shocked region. We compare the models with the same plasma beta value
$\beta_{0}$ but different initial line masses $\lambda_{\mathrm{tot}}$ for
three pairs of models: ($\beta 1$M, $\beta 1$L), ($\beta 01$M, $\beta 01$L),
and ($\beta 001$M, $\beta 001$L).

In Figure 16, the line mass of the shocked region
$\lambda_{\rho_{\mathrm{max}}/100}$ is plotted against the maximum density
$\rho_{\mathrm{max}}$. As the maximum density rarely decreases with time,
$\rho_{\mathrm{max}}$ substitutes for the elapsed time $t$. The initial
relative velocity is set to $V_{\rm int}=1$. Although the line mass of the
dense part $\lambda_{\rho_{\mathrm{max}}/100}$ differs initially (left end of
each line), both line masses $\lambda_{\rho_{c}/100}$ come close to almost the
same value during the time evolution for all pairs of different plasma beta
values (right end of each line). In particular, in the models with
$\beta_{0}=1$ (red line), the line mass maintains a constant value
$\lambda_{\rho_{\mathrm{max}}/100}\simeq\lambda_{\mathrm{crit}}$ after the
maximum density exceeds $\rho_{\mathrm{max}}\gtrsim 10^{5}$, regardless of the
initial line mass.

In the model with a small plasma beta value of $\beta_{0}=0.01$, the final
$\lambda_{\rho_{\mathrm{max}}/100}$ of the two models agrees with each other.
However, it is unclear whether $\lambda_{\rho_{\mathrm{max}}/100}$ maintains
the final value for further contraction. Evolution may reach only the
extremely early stage with small beta models; thus, we require further
evaluation to confirm the true convergence in the models with
$\beta_{0}=0.01$. In conclusion, although the degree of convergence depends on
the magnetic field strength, the typical line mass of the high-density portion
of the contracting filament $\lambda_{\rho_{\mathrm{max}}/100}$ does not
depend on the initial line mass of the filament.

|
---|---
Figure 17:  Comparison of the density profiles on the $x$\- and $y$-axes for
the radial collapse models with different initial line masses. A comparison of
the three pairs of models, ($\beta$1M, $\beta$1L), ($\beta$01M, $\beta$01L),
and ($\beta$001M, $\beta$001L), is performed. The vertical axis represents the
relative density to the maximum density at the final epoch,
$\rho/\rho_{\mathrm{max,end}}$. The horizontal axis represents the distance
normalized by the scale-height at the center of the filament,
$\tilde{L}=c_{s}/(4\pi G\rho_{\mathrm{max,end}})^{1/2}$. Color represents the
plasma beta value: red ($\beta_{0}=1$), green ($\beta_{0}=0.1$), and blue
($\beta_{0}=0.01$). The solid and dashed lines represent the models with
medium (M) and heavy (L) line masses. The black line in the left panel
corresponds to the hydrostatic density profile of an isothermal sheet.

Figure 17 shows a comparison of the density profiles realized in the final
state. The density distribution is shown to be normalized by the maximum
density at the final state such that the vertical axis represents
$\rho/\rho_{\mathrm{max}}$. In the horizontal axis, we consider the distance
to be normalized by the scale-length given at the center as
$x^{\prime}=x/\tilde{L}$, where $\tilde{L}=c_{s}/(4\pi
G\rho_{\mathrm{max}})^{1/2}$. Figure 17(a) shows that the dense part of
$\rho/\rho_{\mathrm{max}}\gtrsim 10^{-2}$ has an extremely similar density
distribution along the $x$-axis, which is shown in Figures 9(a) and 12(a). By
contrast, density distribution in the $y$-direction has a relatively poor
similarity. Models $\beta$1M and $\beta$1L have similar distributions in the
$y$-direction. This is valid for models $\beta$01M and $\beta$01L. However,
model $\beta$001M has a wider distribution than $\beta$001L. This implies that
the time evolution is not sufficient for models with $\beta_{0}=0.01$ compared
with the other two models with $\beta_{0}=1$ and 0.1 (Figure 16).

The convergence of $\lambda_{\rho_{\mathrm{max}}/100}$ (Figure 16) and the
density distributions resembling each other (Figure 17) indicate that the
collapse is expressed using the same solution, independent of the initial line
mass after $\rho_{c}$ increases sufficiently. Figure 16 shows that the
difference in $\lambda_{\rho_{\mathrm{max}}/100}$ owing to different
$\beta_{0}$ also decreases as the collapse proceeds. However, it is not clear
whether the collapse is also expressed with a unique solution, irrespective of
$\beta_{0}$, similar to the axisymmetric three-dimensional cloud (Nakamura et
al., 1999).

###  5.4  Fragmentation of the merged filament.

In this section, we describe the investigation of the fragmentation of merged
filaments resulting from head-on collisions. Although our investigation
primarily focused on the radial instability caused by such collisions, the
fragmentation process is essential for the formation of molecular cloud cores
from the filaments. Studies on the fragmentation of magnetized filaments have
been conducted by Stodółkiewicz (1963); Nagasawa (1987); Hanawa et al. (1993);
Nakamura et al. (1993); Fiege & Pudritz (2000), with the assumption of a
magnetic field that is either parallel or helical to the filament axis. By
contrast, Hanawa et al. (2017, 2019) assumed a magnetic field that is
perpendicular to the filament axis.

For the nonmagnetized filament, the maximum growth rate of the gravitational
instability was found to be $|\omega_{\mathrm{max}}|=0.339(4\pi
G\rho_{c})^{1/2}$ by Nagasawa (1987), resulting in a timescale of the maximum
growth as $t_{\mathrm{min}}\equiv 1/|\omega_{\mathrm{max}}|=2.94(4\pi
G\rho_{c})^{-1/2}$. If the dynamical timescale is slower than the timescale of
the gravitational instability and if this continues for a sufficient duration,
fragmentation occurs in the filament. For a filament penetrated with a uniform
magnetic field perpendicular to the axis, 444Notably, although the model of a
uniform magnetic field does not influence the cloud’s radial stability, it is
not in the magnetohydrostatic equilibrium state. the wave number ($k$) and
growth time showing the maximum growth are expressed approximately as $k\simeq
0.15\sqrt{4\pi G\rho_{c}}/c_{s}$, and $t_{\mathrm{min}}\simeq 5.78(4\pi
G\rho_{c})^{-1/2}$ according to Hanawa et al. (2017). As the Lorentz force
suppresses the fragmentation process, the growth rate becomes small, and the
wavelength of the fragmentation becomes longer compared with the nonmagnetized
ones.

We assume that density is perturbed along the $z$-direction and investigate
whether it can grow and form fragmentations in the merged filament. For the
radial collapse models, the central (maximum) density evolves almost
monotonically, making fragmentation difficult (see Figures 4 and 8). However,
we find that the contraction stops temporally for a certain period, even in
the radial collapse models such as $\beta$001M in Figure 8(a) and
$\beta$01M_highV in Figure 11(a). Here, we studied the possibility of whether
gravitational instability developing during contraction is delayed. For
instance, model $\beta 01$M_highV expands after the collision at approximately
$t=0.3$ (see Figure 11). We found that the central density is maintained at
$\rho^{\prime}_{c}\simeq 250$ during $t^{\prime}_{\mathrm{stall}}\simeq 0.08$,
and the growth time of gravitational instability
$t^{\prime}_{\mathrm{min}}\simeq 0.36(\rho^{\prime}_{c}/250)^{-1/2}$ is longer
than the lag time of contraction ($t_{\mathrm{stall}}<t_{\mathrm{min}}$), and,
even in the phase of maxima at $t\sim 0.16$, $t_{\mathrm{stall}}$ is shorter
than $t_{\mathrm{min}}$. Another radial collapse model ($\beta 001M$) with a
bump is also $t_{\mathrm{stall}}<t_{\mathrm{min}}$ for the entire range of the
simulation.

contrastingly, the merged filaments in the stable models can undergo
fragmentation around the maxima or minima during oscillation, regardless of
the plasma beta value. For example, in the model of $\beta 01$S, the maximum
density is maintained at $\rho^{\prime}_{c}\simeq 226$ for
$t^{\prime}_{\mathrm{stall}}\simeq 0.7$ (Figure 8(b)); thus,
$t^{\prime}_{\mathrm{stall}}>t^{\prime}_{\mathrm{min}}\simeq
0.38(\rho^{\prime}_{c}/226)^{-1/2}$.

###  5.5 Effects of Different Geometries of the Magnetic Field Lines

|  |
---|---|---
|  |
Figure 18: Same as Figure 2, but representing the models in which the
collision direction ($x$-direction) is perpendicular to the global magnetic
field ($y$-direction). This figure shows the cross-section of the colliding
filaments on the $x-y$ plane, and the plot area is restricted to $\pm 4.0$
from the box size ($\pm 5$). The upper (a)-(c) and bottom (d)-(f) panels show
the models of $\beta$1M’$\perp$ and $\beta$1S’$\perp$, respectively.

In this section, we investigate the case where the collision direction of the
filamentary molecular clouds is perpendicular to the direction of magnetic
field lines. We studied the models in which the magnetic field lines ran
parallel to the colliding direction. In reality, this happens in a case where
the magnetic field lines run perpendicular to the colliding direction.

Figure 18 shows the time evolution of models $\beta 1$M’$\perp$ (upper) and
$\beta 1$S’$\perp$ (lower), in which the total line mass is considered as
$\lambda_{\mathrm{tot}}=1.64\lambda_{\mathrm{crit}}$ and
$1.40\lambda_{\mathrm{crit}}$. The other parameters are set as $\beta_{0}=1$
and $V_{\rm int}=1$.

In the model of $\beta$1M’$\perp$, the shocked region begins to collapse,
whereas, in the model of $\beta$1S’$\perp$, the shocked region does not
collapse and is stable while oscillating. As the filament with $\beta_{0}=1$
has a magnetically critical line mass of
$\lambda_{\mathrm{crit,B}}=1.17\lambda_{\mathrm{crit}}$ (Equation (13)), the
total line mass of the two models exceeds the critical line mass as $1.4$
($\beta$1M’$\perp$) and $1.2$ ($\beta$1S’$\perp$) times
$\lambda_{\mathrm{crit,B}}$. Thus, if a collision occurs in the parallel
direction to the magnetic field, both models would undergo the radial collapse
(sec. 5.1) as $\lambda_{\mathrm{tot}}>\lambda_{\mathrm{crit,B}}$. However,
model $\beta 1$S’$\perp$ results in a stable oscillation (Figure 18 lower
panels).

This difference is attributed to the collision direction. In this case, the
magnetic flux of the merged filament become twice as large as that in the
previous models ($\beta 1$S and $\beta 1$M), in which the colliding direction
is parallel to the magnetic field lines. Thus, the expression of the threshold
line mass should be modified by the magnetic field’s geometry. When we
consider that the magnetic field lines run perpendicular to the collision
direction, the magnetically critical line mass (Equation (3)) of the merged
filament is changed as

$\lambda_{\rm crit,B}\simeq
0.24\frac{2\Phi}{G^{1/2}}+1.66\frac{c_{s}^{2}}{G}.$ (16)

We rewrite the above equation in the normalized form as,

$\lambda^{\prime}_{\rm crit,B}\simeq 6.08\Phi^{\prime}+20.8.$ (17)

By adopting this equation, the critical line mass of a merged filament is
estimated as, $\lambda_{\mathrm{crit,B}}=1.51\lambda_{\mathrm{crit}}$ instead
of $1.17\lambda_{\mathrm{crit}}$. Therefore, the total line mass of
$\beta$1M’$\perp$ model $\lambda_{\mathrm{tot}}=1.64\lambda_{\mathrm{crit}}$
exceeds the critical value
$\lambda_{\mathrm{crit,B}}=1.51\lambda_{\mathrm{crit}}$ and the collapse of
the shocked region, even in the perpendicular collision. On the contrary, for
the model $\beta$1S’$\perp$, the total line mass $\lambda_{\mathrm{tot}}=1.17$
is significantly smaller than the critical line mass
$1.51\lambda_{\mathrm{crit}}$. Thus, the merged filament of model
$\beta$1S’$\perp$ is magnetically subcritical
$\lambda_{\mathrm{tot}}<\lambda_{\mathrm{crit,B}}$ and exhibits stable
oscillations. Thus, we conclude that the stability of the perpendicular model
can be explained by considering whether the merged filament is larger than the
magnetically critical line mass if we consider the fact that the magnetic flux
increases twice due to the collision.

##  6 Summary and Conclusions

In this study, we performed two-dimensional MHD simulations of head-on
collisions between two identical isothermal filaments in magnetohydrostatic
equilibrium, in which the magnetic field lines are lateral. Our findings are
summarized as follows:

  1. 1.

We studied 14 models in which two filaments shared the same magnetic flux
tube, and the collision direction was parallel to the global magnetic field.
Supersonic collision induced shock waves that swept the merged filament. A
high-density sheet was formed between the shock fronts, which was formed from
shock compression and accretion flow through self-gravity. The outcome of the
shocked high-density sheet was either a radial collapse, in which the central
density increased infinitely within a finite time scale, or a stable
oscillation with a finite amplitude as a whole.

  2. 2.

The condition of whether the merged filament undergoes radial collapse or
stable oscillation was determined by the initial total line mass of the
colliding filaments. When we observed the occurrence of radial collapse after
the collision, the initial total line mass ($\lambda_{\mathrm{tot}}$) exceeded
the critical line mass for a magnetically supported filament $\lambda_{\rm
crit,B}$, Equation (3) (Tomisaka, 2014). This condition appeared not to depend
on the collision speed.

  3. 3.

When $\lambda_{\mathrm{tot}}\leq\lambda_{\mathrm{crit,B}}$, the merged
filament continued to oscillate. The image of the shocked region resembles the
magnetohydrostatic equilibrium state, regardless of the initial relative
velocity and the magnetic field strength.

  4. 4.

In the collapsing model $\lambda_{\rm tot}>\lambda_{\rm crit,B}$, the density
profile of the dense part of the shocked region was affected by the Lorentz
force. Although the density distribution parallel to the magnetic field
remains similar to a sheet-like structure regardless of the magnetic field
strength, the density profile became broader in the direction perpendicular to
the magnetic field lines as the magnetic field strength increased. Contrary to
this, the density profile of the dense part was not affected by the initial
relative velocity, although the distribution changes with the direction of the
magnetic field.

  5. 5.

In the collapsing model $\lambda_{\mathrm{tot}}>\lambda_{\mathrm{crit,B}}$,
the line mass of the shocked region $\lambda_{\rho_{c}/100}$ decreased with
time and converged to a certain value, which appears to be independent of the
initial line mass. In the collapsing model, the collapse proceeded in a self-
similar manner (Section 5.2).

  6. 6.

We examined the timescale of the fragmentation of the merged filament compared
with the time scale of the maximum growth. Our results indicate that stable
models may fragment during oscillation maxima or minima, whereas radial
collapse models did not provide sufficient time for fragmentation to occur.

  7. 7.

We studied two models in which the magnetic field lines ran in a direction
perpendicular to the collision. In this case, the radial stability of the
shocked region appeared to be given by the same condition, although we must
consider the fact that the merged filament had a twice larger magnetic flux
than the parallel collisions.

In this study, we tried to make the filament magnetically supercritical as a
result of the filament collisions. However, we note that there are several
mechanisms that can trigger the radial collapse in filament systems. For
example, ambipolar diffusion causes the filaments to collapse radially, even
if they are initially subcritical. Over approximately $10t_{\mathrm{ff}}$, the
ambipolar diffusion leads to increase the mass-to-flux ratio and to decrease
in the critical line mass, ultimately resulting in radial collapse, even in
stable models. Furthermore, gas accretion from the surrounding gas onto
filaments works as another mechanism to make filaments supercritical. Although
there are other possible mechanisms for increasing the filament line mass and
initiating star formation other than collisions, the observational fact that
active star formation occurs at the hub, the overlapping part of the two
filaments, indicates the importance of filament collisions.

We would like to express our sincere gratitude to the anonymous referee for
their valuable comments and suggestions to improve this paper. Then, we are
grateful to the following scholars for their valuable discussions regarding
this research: Kumar, M. S. N., Arzoumanian, D., Hanawa, T., and Takiwaki, T.
This study was supported by JST, the establishment of university fellowships
towards the creation of science technology innovation, Grant Number
JPMJFS2136, and by JSPS KAKENHI (Grant Numbers: 22J11106(RK), 19K03929,
19H01938 (KI), and 19K03919(KT)). Numerical computations were performed in
Cray XC50 at the Center for Computational Astrophysics, National Astronomical
Observatory of Japan.

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