Magnetohydrostatic Equilibrium Structure and Mass of Polytropic Filamentary Cloud Threaded by Lateral Magnetic Field

To derive the magnetohydrostatic configuration, we start from the following four equations. First, the polytropic equation is

$$\begin{eqnarray}&&{p}_{g}=K{\rho }^{1+1/N},\end{eqnarray} \tag{ 6 }$$

in which the meaning of the variables is the same as in Equation (5). The equation is based on the assumption that the pressure and the density are connected with the polytropic index. The second equation is a force balance equation between the Lorentz force, gravitational force, and pressure gradient, and is written as

$$\begin{eqnarray}&&\displaystyle \frac{1}{c}{\boldsymbol{j}}\times {\boldsymbol{B}}-\rho {\rm{\nabla }}\psi -{\rm{\nabla }}{p}_{g}=0,\end{eqnarray} \tag{ 7 }$$

where c, j , B , and ψ represent the light speed, electric current density, magnetic flux density, and gravitational potential, respectively. The third equation is Poisson's equation for self-gravity, which is expressed as

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}\psi =4\pi G\rho ,\end{eqnarray} \tag{ 8 }$$

where G is the gravitational constant. The fourth equation is Ampère's law, which is written as

$$\begin{eqnarray}&&{\boldsymbol{j}}=\displaystyle \frac{c}{4\pi }{\rm{\nabla }}\times {\boldsymbol{B}}.\end{eqnarray} \tag{ 9 }$$

We search for a solution for a filament extending infinitely in the z-direction and assume all the physical quantities depend on only (x, y). We introduce a magnetic flux function Φ, from which the magnetic flux density B = (B_x , B_y ) is given as

$$\begin{eqnarray}&&{B}_{x}=-\displaystyle \frac{\partial {\rm{\Phi }}}{\partial y},\end{eqnarray} \tag{ 10a }$$

$$\begin{eqnarray}&&{B}_{y}=\displaystyle \frac{\partial {\rm{\Phi }}}{\partial x}.\end{eqnarray} \tag{ 10b }$$

It should be noted that, in a 2D representation, the magnetic field line is given by a contour line of ${\rm{\Phi }}(x,y)=\mathrm{const}$. From Equation (9), the electric current j is rewritten as

$$\begin{eqnarray}&&{j}_{x}=\displaystyle \frac{c}{4\pi }\displaystyle \frac{\partial {B}_{z}}{\partial y},\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}&&{j}_{y}=-\displaystyle \frac{c}{4\pi }\displaystyle \frac{\partial {B}_{z}}{\partial x},\end{eqnarray} \tag{ 11b }$$

$$\begin{eqnarray}&&{j}_{z}=\displaystyle \frac{c}{4\pi }\left(\displaystyle \frac{\partial {B}_{y}}{\partial x}-\displaystyle \frac{\partial {B}_{x}}{\partial y}\right)=\displaystyle \frac{c}{4\pi }{{\rm{\Delta }}}_{2}{\rm{\Phi }},\end{eqnarray} \tag{ 11c }$$

where Δ₂ is defined as

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{2}\equiv {{\rm{\nabla }}}_{2}^{2}={\partial }^{2}/\partial {x}^{2}+{\partial }^{2}/\partial {y}^{2}.\end{eqnarray} \tag{ 12 }$$

Hereafter, ∇₂ ≡ (∂/∂x, ∂/∂y) represents the two-dimensional differentiation operator. For the polytropic gas, the pressure term of Equation (7) is

$$\begin{eqnarray}&&\displaystyle \frac{1}{\rho }{{\rm{\nabla }}}_{2}{p}_{g}=\displaystyle \frac{K}{\rho }{{\rm{\nabla }}}_{2}{\rho }^{1+\tfrac{1}{N}}=K(N+1){{\rm{\nabla }}}_{2}{\rho }^{1/N}.\end{eqnarray} \tag{ 13 }$$

Using this equation, the force balance Equation (7) becomes

$$\begin{eqnarray}&&\begin{array}{l}x{\rm{ \mbox{-} component}}:-\displaystyle \frac{1}{4\pi }{{\rm{\Delta }}}_{2}{\rm{\Phi }}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial x}-\rho \displaystyle \frac{\partial \psi }{\partial x}\\ \quad -\ K(N+1)\rho \displaystyle \frac{\partial {\rho }^{1/N}}{\partial x}-\displaystyle \frac{1}{8\pi }\displaystyle \frac{\partial {B}_{z}^{2}}{\partial x}=0,\end{array}\end{eqnarray} \tag{ 14a }$$

$$\begin{eqnarray}&&\begin{array}{l}y{\rm{ \mbox{-} component}}:-\displaystyle \frac{1}{4\pi }{{\rm{\Delta }}}_{2}{\rm{\Phi }}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial y}-\rho \displaystyle \frac{\partial \psi }{\partial y}\\ \quad -\ K(N+1)\rho \displaystyle \frac{\partial {\rho }^{1/N}}{\partial y}-\displaystyle \frac{1}{8\pi }\displaystyle \frac{\partial {B}_{z}^{2}}{\partial y}=0,\end{array}\end{eqnarray} \tag{ 14b }$$

and when B_z = 0, these two equations reduce to

$$\begin{eqnarray}&&\begin{array}{l}x{\rm{ \mbox{-} component}}:-\displaystyle \frac{1}{4\pi }{{\rm{\Delta }}}_{2}{\rm{\Phi }}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial x}\\ \quad -\ \rho \displaystyle \frac{\partial }{\partial x}\left[\psi +K(N+1){\rho }^{1/N}\right]=0,\end{array}\end{eqnarray} \tag{ 15a }$$

$$\begin{eqnarray}&&\begin{array}{l}y{\rm{ \mbox{-} component}}:-\displaystyle \frac{1}{4\pi }{{\rm{\Delta }}}_{2}{\rm{\Phi }}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial y}\\ \quad -\ \rho \displaystyle \frac{\partial }{\partial y}\left[\psi +K(N+1){\rho }^{1/N}\right]=0.\end{array}\end{eqnarray} \tag{ 15b }$$

By taking the inner product of Equation (15) and B = (B_x , B_y ),

$$\begin{eqnarray}&&\rho ({\boldsymbol{B}}\cdot {{\rm{\nabla }}}_{2})\left[\psi +K(N+1){\rho }^{1/N}\right]=0\end{eqnarray} \tag{ 16 }$$

is required in the direction parallel to the magnetic field lines. This means the quantity ψ + K(N + 1)ρ^1/N is constant along a magnetic flux tube as

$$\begin{eqnarray}&&\psi +K(N+1){\rho }^{1/N}=H({\rm{\Phi }}),\end{eqnarray} \tag{ 17 }$$

where H(Φ) is the Bernoulli constant, which is a function dependent only on Φ. Along a magnetic tube given by a constant Φ, the density is calculated from the gravitational potential

$$\begin{eqnarray}&&\rho ={\left[\displaystyle \frac{H({\rm{\Phi }})-\psi }{K(N+1)}\right]}^{N},\end{eqnarray} \tag{ 18 }$$

and then Equation (15) is rewritten as

$$\begin{eqnarray}&&-\ \displaystyle \frac{1}{4\pi }{{\rm{\Delta }}}_{2}{\rm{\Phi }}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial x}=\rho \displaystyle \frac{\partial H}{\partial x}=\rho \displaystyle \frac{{dH}}{d{\rm{\Phi }}}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial x},\end{eqnarray} \tag{ 19a }$$

$$\begin{eqnarray}&&-\ \displaystyle \frac{1}{4\pi }{{\rm{\Delta }}}_{2}{\rm{\Phi }}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial y}=\rho \displaystyle \frac{\partial H}{\partial y}=\rho \displaystyle \frac{{dH}}{d{\rm{\Phi }}}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial y}.\end{eqnarray} \tag{ 19b }$$

Similarly to the method in Tomisaka (2014), we assume that forces are balanced inside the filament (Equation (19)). Outside the filament, we assume that a force-free magnetic field (the electric current j = 0) and hot tenuous gas (ρ = 0) exist. The extended gas confines the filament with its pressure (external pressure p_ext). Thus, the right-hand side of Equation (19) vanishes outside the filament. Finally, we can derive the two basic equations. Equation (19) leads to

$$\begin{eqnarray}\quad -{{\rm{\Delta }}}_{2}{\rm{\Phi }}=\left\{\begin{array}{cl}4\pi \rho \displaystyle \frac{{dH}}{d{\rm{\Phi }}} & (\mathrm{inside}\,\mathrm{the}\,\mathrm{filament}),\\ 0 & (\mathrm{outside}\,\mathrm{the}\,\mathrm{filament}),\end{array}\right.\end{eqnarray} \tag{ 20 }$$

and with use of Equation (18), Poisson's Equation (8) is rewritten as

$$\begin{eqnarray}\quad {{\rm{\Delta }}}_{2}\psi =\left\{\begin{array}{cl}4\pi G{\left[\displaystyle \frac{H({\rm{\Phi }})-\psi }{K(N+1)}\right]}^{N} & (\mathrm{inside}\,\mathrm{the}\ \,\mathrm{filament}),\\ 0 & (\mathrm{outside}\,\mathrm{the}\ \,\mathrm{filament}).\end{array}\right.\end{eqnarray} \tag{ 21 }$$

We find the equilibrium state by solving these two second-order differential equations simultaneously via the self-consistent field method, but we need to know the value of H(Φ) at each magnetic field line.

Here, we introduce a mechanism to derive H(Φ). In this paper, we assume that a large-scale magnetic field runs along the y-direction. We assume vertical symmetry at y = 0. Then, a line mass Δλ that is contained between two magnetic field lines, Φ and Φ + ΔΦ, is expressed as

$$\begin{eqnarray}&&{\rm{\Delta }}\lambda ({\rm{\Phi }})=2{\int }_{0}^{{y}_{s}({\rm{\Phi }})}{\int }_{x(y,{\rm{\Phi }})}^{x(y,{\rm{\Phi }}+{\rm{\Delta }}{\rm{\Phi }})}\rho (x,y){dxdy}\end{eqnarray} \tag{ 22a }$$

$$\begin{eqnarray}&&=\ 2{\int }_{0}^{{y}_{s}({\rm{\Phi }})}\displaystyle \frac{\rho }{(\partial {\rm{\Phi }}/\partial x)}{\rm{\Delta }}{\rm{\Phi }}{dy}\end{eqnarray} \tag{ 22b }$$

$$\begin{eqnarray}&&=\ 2{\int }_{0}^{{y}_{s}({\rm{\Phi }})}{\left[\displaystyle \frac{H({\rm{\Phi }})-\psi }{K(N+1)}\right]}^{N}{\left(\partial {\rm{\Phi }}/\partial x\right)}^{-1}{\rm{\Delta }}{\rm{\Phi }}{dy},\end{eqnarray} \tag{ 22c }$$

where y_s represents the y-coordinate of the filament surface and H(Φ) stays constant in the integration. This leads to the problem of finding an appropriate set of H(Φ) and y_s (Φ) while at the same time satisfying the following equation:

$$\begin{eqnarray}&&\displaystyle \frac{d\lambda }{d{\rm{\Phi }}}=2{\int }_{0}^{{y}_{s}({\rm{\Phi }})}{\left[\displaystyle \frac{H({\rm{\Phi }})-\psi }{K(N+1)}\right]}^{N}{\left(\partial {\rm{\Phi }}/\partial x\right)}^{-1}{dy}.\end{eqnarray} \tag{ 23 }$$

The left side of this equation is given as a model of the mass-to-flux ratio distribution, which is called mass loading.

We assume the central density of the filament as ρ_c and the potential as ψ_c . Then, Equation (17) gives the value of the Bernoulli constant for the central magnetic field line coinciding with the y-axis, which is specified by Φ = 0, as

$$\begin{eqnarray}&&H({\rm{\Phi }}=0)=K(N+1){\rho }_{c}^{1/N}+{\psi }_{c}.\end{eqnarray} \tag{ 24 }$$

Equation (23) uses Equation (24) to obtain the mass loading on the central magnetic field line Φ = 0 as follows:

$$\begin{eqnarray}&&{\left.\displaystyle \frac{d\lambda }{d{\rm{\Phi }}}\right|}_{{\rm{\Phi }}=0}=2{\int }_{0}^{{y}_{s}}{\left[{\rho }_{c}^{1/N}+\displaystyle \frac{{\psi }_{c}-\psi }{K(N+1)}\right]}^{N}{\left(\partial {\rm{\Phi }}/\partial x\right)}^{-1}{dy},\end{eqnarray} \tag{ 25 }$$

where the upper boundary y_s is given as a point where ψ = ψ_s . The surface potential is written as

$$\begin{eqnarray}&&{\psi }_{s}={\psi }_{c}+K(N+1)({\rho }_{c}^{1/N}-{\rho }_{s}^{1/N}),\end{eqnarray} \tag{ 26 }$$

where the ρ_s is the density at the filament surface in the equilibrium state (see Figure 1(b)). Equation (25) indicates that, from a set of potentials Φ and ψ, the mass loading for the central magnetic field line ${\left.d\lambda /d{\rm{\Phi }}\right|}_{{\rm{\Phi }}=0}$ is obtained as a function of ρ_c .

In this paper, for the purpose of comparison with the isothermal model, we assume the following mass-loading distribution:

$$\begin{eqnarray}&&\displaystyle \frac{d\lambda }{d{\rm{\Phi }}}={\left.\displaystyle \frac{d\lambda }{d{\rm{\Phi }}}\right|}_{{\rm{\Phi }}=0}{\left[1-{\left(\displaystyle \frac{{\rm{\Phi }}}{{{\rm{\Phi }}}_{\mathrm{cl}}}\right)}^{2}\right]}^{1/2},\end{eqnarray} \tag{ 27 }$$

which is the same assumed in Tomisaka (2014). In this equation, Φ_cl represents the magnetic flux threading the unit length of the filament and Φ_cl = R₀ · B₀, where R₀ and B₀ represent the initial radius of the filament and magnetic field strength, respectively, of the initial uniform magnetic field. This is realized when a uniform-density cylindrical filament is threaded with a uniform magnetic field, where −Φ_cl ≤ Φ ≤ Φ_cl represents the magnetic field line threading the filament, while Φ < −Φ_cl and Φ > Φ_cl represent the magnetic field lines not threading the filament (see Figure 1). In this paper, we refer to this filament with uniform density and uniform magnetic field as the "parent" cloud, which gives the mass loading in the filament in equilibrium. That is, we assume that the mass loading is determined by the parent cloud and is conserved by flux freezing.

For the magnetic field lines Φ ≠ 0, the equation becomes

$$\begin{eqnarray}&&\begin{array}{l}{\left.\displaystyle \frac{d\lambda }{d{\rm{\Phi }}}\right|}_{{\rm{\Phi }}=0}{\left[1-{\left(\displaystyle \frac{{\rm{\Phi }}}{{{\rm{\Phi }}}_{\mathrm{cl}}}\right)}^{2}\right]}^{1/2}=2{\displaystyle \int }_{0}^{{y}_{s}({\rm{\Phi }})}\\ \quad \times \,{\left[{\rho }_{s}^{1/N}+\displaystyle \frac{{\psi }_{s}(x({\rm{\Phi }},{y}_{s}),{y}_{s})-\psi (x({\rm{\Phi }},y),y)}{K(N+1)}\right]}^{N}\\ \quad \times \,{\left(\partial {\rm{\Phi }}/\partial x\right)}^{-1}{dy}.\end{array}\end{eqnarray} \tag{ 28 }$$

For a given Φ, y_s (Φ) is chosen to satisfy the above equation, which is achieved with use of the bisection method of nonlinear equations.

In this paper, physical variables are normalized with their quantities at the filament surface. We regard the following three quantities as fundamental: the external pressure p_ext, the surface density ρ_s , and the isothermal sound speed c_ss at the filament surface. Then, for example, the scale length (L) is defined by the freefall time (t_ff) at the filament surface density (ρ_s ) and the isothermal sound speed (c_ss) at the surface of the filament, as L = c_ss t_ff. From the polytropic equation, the external pressure p_ext and the surface density ρ_s are related as

$$\begin{eqnarray}&&{p}_{\mathrm{ext}}=K{\rho }_{s}^{1+1/N}.\end{eqnarray} \tag{ 29 }$$

The physical scales characterizing the system are given as in Table 1. We define the normalized variables as

$$\begin{eqnarray}&&p^{\prime} \equiv {p}_{g}/{p}_{\mathrm{ext}},\end{eqnarray} \tag{ 30a }$$

$$\begin{eqnarray}&&\rho ^{\prime} \equiv \rho /{\rho }_{s},\end{eqnarray} \tag{ 30b }$$

$$\begin{eqnarray}&&\psi ^{\prime} \equiv \psi /{c}_{\mathrm{ss}}^{2}=\psi /(K{\rho }_{s}^{1/N}),\end{eqnarray} \tag{ 30c }$$

$$\begin{eqnarray}&&{\rm{\Phi }}^{\prime} \equiv {\rm{\Phi }}/({B}_{u}L),\end{eqnarray} \tag{ 30d }$$

$$\begin{eqnarray}&&x^{\prime} \equiv x/L,\end{eqnarray} \tag{ 30e }$$

$$\begin{eqnarray}&&y^{\prime} \equiv y/L,\end{eqnarray} \tag{ 30f }$$

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{2}^{{\prime} }\equiv {{\rm{\Delta }}}_{2}\cdot {L}^{2},\end{eqnarray} \tag{ 30g }$$

$$\begin{eqnarray}&&\lambda ^{\prime} \equiv \lambda /({\rho }_{s}{L}^{2}).\end{eqnarray} \tag{ 30h }$$

where the prime represents the normalized variables. The density is normalized as

$$\begin{eqnarray}&&\rho ^{\prime} ={\left[\displaystyle \frac{H^{\prime} -\psi ^{\prime} }{N+1}\right]}^{N}.\end{eqnarray} \tag{ 31 }$$

Poisson's equation is rewritten as

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{2}^{{\prime} }\psi ^{\prime} =\rho ^{\prime} ={\left[\displaystyle \frac{H^{\prime} ({\rm{\Phi }}^{\prime} )-\psi ^{\prime} }{N+1}\right]}^{N},\end{eqnarray} \tag{ 32 }$$

while Poisson's equation for magnetic flux function is given as

$$\begin{eqnarray}&&-{{\rm{\Delta }}}_{2}^{{\prime} }{\rm{\Phi }}^{\prime} =\displaystyle \frac{1}{2}\rho ^{\prime} \displaystyle \frac{{dH}^{\prime} }{d{\rm{\Phi }}^{\prime} }.\end{eqnarray} \tag{ 33 }$$

The mass loading on the central magnetic field is given as

$$\begin{eqnarray}&&{\left.\displaystyle \frac{d\lambda ^{\prime} }{d{\rm{\Phi }}^{\prime} }\right|}_{{\rm{\Phi }}=0}=2{\int }_{0}^{{y}_{s}^{{\prime} }}{\left[{{\rho }_{c}^{{\prime} }}^{1/N}+\displaystyle \frac{{\psi }_{c}^{{\prime} }-\psi ^{\prime} }{N+1}\right]}^{N}{\left(\displaystyle \frac{\partial {\rm{\Phi }}^{\prime} }{\partial x^{\prime} }\right)}^{-1}{dy}^{\prime} ,\end{eqnarray} \tag{ 34 }$$

and Equation (23) reduces to

$$\begin{eqnarray}&&\displaystyle \frac{d\lambda ^{\prime} }{d{\rm{\Phi }}^{\prime} }=2{\int }_{0}^{{y}_{s}^{\prime} }{\left[1+\displaystyle \frac{{\psi }_{s}^{{\prime} }-\psi ^{\prime} }{N+1}\right]}^{N}{\left(\displaystyle \frac{\partial {\rm{\Phi }}^{\prime} }{\partial x^{\prime} }\right)}^{-1}{dy}^{\prime} .\end{eqnarray} \tag{ 35 }$$

Table 1. Units used for Normalization

Unit of pressure	External pressure, pext
Unit of density	Density at the surface, ρs
Unit of time	Freefall time, ${t}_{\mathrm{ff}}={\left(4\pi G{\rho }_{s}\right)}^{-1/2}$
Unit of speed	${c}_{\mathrm{ss}}={\left({p}_{\mathrm{ext}}/{\rho }_{s}\right)}^{1/2}={\left(K{\rho }_{s}^{1/N}\right)}^{1/2}$
Unit of magnetic field strength	${B}_{u}={\left(8\pi {p}_{\mathrm{ext}}\right)}^{1/2}$
Unit of length	$L={c}_{\mathrm{ss}}{t}_{\mathrm{ff}}={c}_{\mathrm{ss}}/{\left(4\pi G{\rho }_{s}\right)}^{1/2}={\left(K{\rho }_{s}^{-1+1/N}/4\pi G\right)}^{1/2}$
Unit of temperature	Temperature at the surface, Ts

Notes. Physical variables are normalized with their quantities at the filament surface. Because the freefall timescale (t_ff) and scale length (L) are defined on the filament surface, the unit of speed is taken as the isothermal sound speed (c_ss) at the surface of the filament.

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The two Poisson equations require boundary conditions. We impose the Dirichlet boundary condition on the outer numerical boundary given below. Far from the origin, we assume that the gravitational potential ψ converges to that realized for a line mass λ placed at the origin as

$$\begin{eqnarray}&&\psi =2G\lambda \mathrm{log}r.\end{eqnarray} \tag{ 36 }$$

The outer boundary condition for the magnetic potential is expressed as

$$\begin{eqnarray}&&{\rm{\Phi }}={B}_{0}x,\end{eqnarray} \tag{ 37 }$$

in which we assume that the magnetic field is connected to the uniform magnetic field with strength B₀ far from the center. We then normalize the two potentials. The normalized value $\lambda ^{\prime} =\lambda /{\rho }_{s}{L}^{2}$ is used to reduce Equation (36) to

$$\begin{eqnarray}&&\psi ^{\prime} =\displaystyle \frac{\lambda ^{\prime} }{2\pi }\mathrm{log}r^{\prime} .\end{eqnarray} \tag{ 38 }$$

The line mass λ is given as

$$\begin{eqnarray}&&\lambda =2{\int }_{0}^{{{\rm{\Phi }}}_{\mathrm{cl}}}\displaystyle \frac{d\lambda }{d{\rm{\Phi }}}d{\rm{\Phi }}\end{eqnarray} \tag{ 39a }$$

$$\begin{eqnarray}&&=\,2{\left.\displaystyle \frac{d\lambda }{d{\rm{\Phi }}}\right|}_{{\rm{\Phi }}=0}{\int }_{0}^{{{\rm{\Phi }}}_{\mathrm{cl}}}{\left[1-{\left(\displaystyle \frac{{\rm{\Phi }}}{{{\rm{\Phi }}}_{\mathrm{cl}}}\right)}^{2}\right]}^{1/2}d{\rm{\Phi }}\end{eqnarray} \tag{ 39b }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{\pi }{2}{{\rm{\Phi }}}_{\mathrm{cl}}{\left.\displaystyle \frac{d\lambda }{d{\rm{\Phi }}}\right|}_{{\rm{\Phi }}=0}.\end{eqnarray} \tag{ 39c }$$

If we know the ratio of mass to magnetic flux at the center, ${\left.d\lambda /d{\rm{\Phi }}\right|}_{{\rm{\Phi }}=0}$, we obtain the boundary value of $\psi ^{\prime}$ after calculating λ using Equations (38) and (39). The magnetic field potential is normalized as

$$\begin{eqnarray}&&{\rm{\Phi }}^{\prime} ={\beta }_{0}^{-1/2}x^{\prime} ,\end{eqnarray} \tag{ 40 }$$

where β₀ is a ratio of the external pressure p_ext to the magnetic pressure ${B}_{0}^{2}/8\pi$ and defined as

$$\begin{eqnarray}&&{\beta }_{0}\equiv \displaystyle \frac{{p}_{\mathrm{ext}}}{{B}_{0}^{2}/8\pi }.\end{eqnarray} \tag{ 41 }$$

After the normalization, a solution is specified by four nondimensional parameters: ${{\rm{\Phi }}}_{\mathrm{cl}}^{{\prime} }$, β₀, ${\rho }_{c}^{{\prime} }\equiv {\rho }_{c}/{\rho }_{s}$, and N (see Figure 1). The nondimensional magnetic flux ${{\rm{\Phi }}}_{\mathrm{cl}}^{{\prime} }$ is given as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{cl}}^{{\prime} }={R}_{0}^{{\prime} }\cdot {B}_{0}^{{\prime} }={R}_{0}^{{\prime} }\cdot {\beta }_{0}^{-1/2},\end{eqnarray} \tag{ 42 }$$

where ${R}_{0}^{{\prime} }$ is defined as the initial radius of uniform filament R₀ normalized by the scale length L. Hereafter, we omit the prime, which indicates normalized quantities, unless the meaning is unclear.

We solved Poisson's equation with the conjugate gradient method preconditioned with incomplete Cholesky factorization (ICCG). The number of grid points was chosen to be 641 × 641 or 1281 × 1281 and the grid spacing was chosen to be Δx = Δy = 0.1/16, 0.1/8, or 0.1/4. The outer numerical boundaries are placed at x = y = ±4 in the models with R₀ = 1 and 2, and at x = y = ±8 in those with R₀ = 5. We summarize the model parameters in Table 2.

Table 2. Model Parameters and Maximum Supported Line Mass

Model	R0	β0	${\rho }_{c\mathrm{Max}}$				Φcl	${\lambda }_{\max }$
			N = −3	−5	−10	−100		N = −3	−5	−10	−100
R1β1	1	1	500	103	103	200	1	9.957	13.42	17.54	23.90
R1β0.5	1	0.5	500	103	103	500	1.41	11.06	14.46	18.84	25.60
R1β0.1	1	0.1	500	103	103	103	3.16	16.30	20.51	25.31	32.52
R1β0.05	1	0.05	500	103	103	103	4.47	19.83*	25.27	30.54	38.03
R2β1	2	1	103	103	103	100	2	14.36	17.77	21.57	26.84
R2β0.5	2	0.5	103 †	103	103	500	2.83	17.22	20.94	24.96	31.08
R2β0.1	2	0.1	103 †	103	103	103	6.32	29.91	34.63	39.81	47.02
R2β0.05	2	0.05	103 †	103	103	103	8.94	38.78*	44.75	50.86	58.74
R5β1	5	1	103 ◊	103 •	103 •	103 •	5	27.72	31.48	35.32	40.98
R5β0.5	5	0.5	500 ◊	103 •	103 •	103 •	7.07	35.90	40.22	44.64	50.74
R5β0.1	5	0.1	500 ◊	103 •	103 •	103 •	15.8	68.61	76.14	82.59	91.37
R5β0.05	5	0.05	500 ◊	103 •	103 •	103 •	22.4	89.18*	99.81*	110.6	120.5

Notes. The column ${\rho }_{c\mathrm{Max}}$ indicates that the solutions are obtained between ρ_c = 2 and ${\rho }_{c\mathrm{Max}}$. Numbers marked with the symbol * represent lower limits of ${\lambda }_{\max }$. The symbols †, ◊, and • represent respective grid spacings of 0.1/16, 0.1/8, and 0.1/4. In the models with † and ◊, the number of grid points is 1281 × 1281, and that of the model with • is 641 × 641. The rest of the models are calculated with grid points of 641 × 641 and a grid spacing of 0.1/8.

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We verified our calculation by solving an approximate isothermal equilibrium state with the polytropic method and assuming N = −100 and −1000. From Equation (5), the polytropic indices N = −100 and N = −1000 correspond to the polytropic exponents γ (N = −100) = 0.99 and γ (N = −1000) = 0.999, both of which are close to the isothermal case of γ = 1. We compare the equilibrium state of the isothermal (Tomisaka 2014) and polytropic filaments (N = −100 and −1000) while paying attention to the line mass. In this comparison, the other parameters, i.e., the radius of the parent cloud R₀ = 2 and the plasma beta β₀ = 0.1, are constant. When the central density is ρ_c = 10³, the line mass of the polytropic filament is λ_N=−100 = 47.02 and λ_N=−1000 = 47.96, while the isothermal one is λ_iso = 48.07. When the central density is ρ_c = 10², the corresponding line masses are λ_N=−100 = 43.31, λ_N=−1000 = 44.00, and λ_iso = 44.08.

The line mass of polytropic filaments is slightly lower than the isothermal one. However, it is clearly shown that the line mass converges to the isothermal value when N moves to −∞. Thus, our calculation reproduces a line mass close to the isothermal one when the polytropic index N → −∞. Hereafter, we assume the results with N = −100 to be the isothermal model.

In this section, we compare the density profile and the line mass of the polytropic (N = −3) and isothermal (N = −100) filaments. In the comparison, other parameters of these filaments, the radius of the parent cloud R₀ = 1 and the plasma beta β₀ = 0.1 are constant.

First, we begin with the density distribution of the equilibrium state. We show the cross sections of the polytropic filaments in Figure 2(a)–(c) and the isothermal filaments in panels (d)–(f). In this paper, we refer to the x- and y-coordinates of the cloud surface crossing the x- and y-axes as the "width" x_s and "height" y_s of the filament, respectively. It is shown that both of these filaments become flatter as the central density increases: the height of the filament shrinks, while the width remains almost unchanged. This is due to the character of the Lorentz force, which works in the perpendicular direction to the magnetic field line but does not work in the parallel direction. Because extra force to support the filament is working in the direction perpendicular to the magnetic field, the isodensity contours of the cross section shrink mainly in the y-direction and appear flat.

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Figure 2 shows that the cross section of the polytropic filament is flatter than the isothermal one when two with the same central density are compared. However, the outer part's gas scale height in the y-direction of the polytropic filament is nearly equal to that of the isothermal one. For example, panel (b) shows that the polytropic filament has a height of y_s ≃ 0.39 on the symmetric y-axis. However, the surface inflates outward, and the height of the surface reaches ≃0.60 near x ≃ 0.80. Thus, this polytropic filament has a maximum height of ∼0.60. In contrast, the corresponding isothermal model (panel (e)) does not show such inflation (y_s ≃ 0.64 and maximum height ≃0.66). As is shown, the maximum height of the polytropic filament is nearly the same as that of the isothermal filament. This can be understood by accounting for the temperature near the surface of the polytropic filament, which is not very different from the temperature of the isothermal filament, although the polytropic filament has a lower central temperature compared with the isothermal filament.

Figure 3 shows the density profiles on the x- and y-axes. In particular, the density profile on the y-axis clearly shows the effect of different N values. Figure 3 shows that the density distribution is divided into two parts: an inner core with an almost constant density and an outer envelope in which the density decreases with increasing distance from the center. This figure shows that both the isothermal and the polytropic filaments have power-law envelopes, except for the envelopes cutting along the x-axis for the ρ_c = 10 models. In terms of the distance to the surface from the center, the polytropic filament is more compact than the isothermal one in the y-direction. In addition, the ρ(y) distribution indicates that the density slope is shallower than the isothermal slope. Although these two results seem to be inconsistent, this is natural if we consider that the compactness of a polytropic filament comes from the fact that it has a smaller core than an isothermal filament.

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In contrast, density profiles on the x-axis are almost identical. For example, the height of the polytropic filament on the y-axis is 40% smaller than the isothermal height, while the x-axis width is 20% wider than the isothermal width, when we compare density distributions with the same central density ρ_c = 500.

Next, we pay attention to the difference between the line mass of each filament. For the central densities of ρ_c = 10, 100, and 500, the line masses of the polytropic and the isothermal filaments are obtained to be λ_N=−3 = 10.36, 15.10, 16.30, and λ_N=−100 = 17.90, 29.47, 32.18, respectively. In both the polytropic and isothermal models, the line mass increases as the central density increases. Meanwhile, comparison of filaments with the same central density shows that the polytropic filament is less massive than the isothermal one. This is explained by the fact that the central pressure, which supports the filament against self-gravity, of the polytropic filament is smaller than that of the isothermal one. For example, comparing two models with ρ_c = 100, the central pressure of the polytropic model is only p_c = 100^2/3, which is only ∼21.5% of the central pressure of the isothermal model. Note that these properties come from the nature of the negative-indexed polytropic gas that is immersed in the same ambient gas pressure.

In this section, we address the effect of the radius of the parent cloud R₀, which controls the magnetic flux threading the filament. The other parameters are constant at N = −3 and β₀ = 0.1.

Figure 4 shows the cross sections of the models of R₀ = 2 [(a) − (c)] and R₀ = 5 [(d) − (f)] for respective central densities ρ_c = 10, 100, and 500 (the model with R₀ = 1 is shown in the upper row of Figure 2). When ρ_c = 500, the height y_s of the filament on the y-axis is equal to y_s = 0.238 (R₀ = 1), y_s = 0.213(R₀ = 2), and y_s = 0.188(R₀ = 5) for the three different R₀ values, respectively. In contrast, the half-width x_s on the x-axis is equal to x_s = 0.888 (R₀ = 1), x_s = 1.75(R₀ = 2), and x_s = 4.41(R₀ = 5), respectively. Thus, the aspect ratio x_s /y_s for R₀ = 1, 2, and 5 is equal to x_s /y_s = 3.73, 8.22, and 23.5, respectively. Thus, the aspect ratio is an increasing function of R₀.

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For the nonmagnetic model of β₀ = ∞, we expected the cross section to be round. However, the shape of the cross section is flat in the above models. The magnetic field supports the filament in the x-direction, but does not play a role in the y-direction. The average ratio of the Lorentz force to the thermal pressure force is equal to 3.27 (R₀ = 1), 6.25(R₀ = 2), and 15.3(R₀ = 5) for ρ_c = 500, respectively, measured on the x-axis. Thus, the Lorentz force is stronger than the thermal pressure, especially for the model with R₀ = 5. In addition, comparing three models with ρ_c = 10, we found an aspect ratio x_s /y_s = 1.25 (R₀ = 1), 2.889(R₀ = 2), and 8.083(R₀ = 5) for the three different R₀ values, respectively. Models of ρ_c = 100 indicate an aspect ratio x_s /y_s = 2.350 (R₀ = 1), 5.407(R₀ = 2), and 15.167(R₀ = 5), respectively. This shows that the aspect ratio increases as the central density increases when R₀ is the same. Figure 5 shows the density profiles on the x- and y-axes. The density profile on the y-axis is more compact than that on the x-axis, and the filament is flat. Comparison of the models with the same central density shows that the slope of the density profile on the y-axis is almost the same for the three different R₀ values. In contrast, the density profiles on the x-axis are not the same. The core radius on the x-axis increases as R₀ increases, and as a result, the distance to the surface also increases with increasing R₀.

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Next, we examine the difference in the line mass. Figure 6 shows the relation between the line mass and the central density for various N and R₀ values with constant plasma beta β₀ = 0.1. Comparison of models with the same central density and polytropic index shows that the line mass increases as R₀ increases. This suggests that the supported line mass is controlled by the magnetic flux Φ_cl = R₀ · B₀. This property is also valid for other polytropic indices. Figure 6 also shows that the line mass decreases with increasing polytropic index from N = −100 to −3, which is also discussed in Section 3.1.

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Next, we compare the models with different β₀ values. Other parameters are fixed: N = −3, R₀ = 2, and ρ_c = 100.

Figure 7 shows the cross section of the equilibrium state. Each panel corresponds to a different β₀ value: (a) β₀ = 1, (b) β₀ = 0.5, (c) β₀ = 0.1, and (d) β₀ = 0.05. When β₀ is small and the magnetic field is strong, the magnetic field line retains its initial shape. Because the filament is sufficiently supported by the strong magnetic field, the surface of the filament is close to R₀ on the x-axis (x_s ≃ R₀).

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Figure 8 shows the density profiles on the x- and y-axes for the same models shown in Figure 7. The density profile on the y-axis is slightly affected by β₀. In contrast, the density profile on the x-axis becomes steep in the models with low β₀. The strong Lorentz force extends the core radius, but the width x_s is not strongly affected by β₀. Thus, the thickness of the envelope shrinks, which makes the slope steep.

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Figure 9 shows the relation of the line mass and the central density for filaments with N = −3 (△) and N = −100 (◦), in which different line colors represent different β₀ values. All the models have the same R₀ = 2. Comparison of magnetized and nonmagnetized polytropic filaments (△for N = −3) indicates that the line mass of the magnetized filament is heavier than the nonmagnetized one (black symbols and solid curve). Results for the isothermal filaments (N = −100) are the same. The line mass of the polytropic filament with N = −3 (△) is smaller than that with N = −100 (◦) when models with the same β₀ and ρ_c are compared. This reflects the fact that the line mass of the negative-indexed polytropic filament is less massive than that for the isothermal filament. However, when the magnetic field is strong, the line mass of the magnetized N = −3 filament is even larger than that of the nonmagnetized isothermal filament (N = −100: gray symbols and solid curve).

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Section 3.1 shows that the line mass decreases as N increases from −100 to −3. In Section 3.2, it is shown that the line mass increases with increasing R₀. The line mass increases with decreasing β₀, as shown in Section 3.3. Thus, we expect the line mass to be determined by the magnetic flux Φ_cl ≡ R₀ · B₀ and N. Here, we discuss how the maximum line mass ${\lambda }_{\max }$ is expressed by the magnetic flux Φ_cl.

The maximum line mass ${\lambda }_{\max }$ represents the maximum allowable line mass of a filament that is in equilibrium. When the line mass λ exceeds ${\lambda }_{\max }$, there is no equilibrium state and we call the filament "supercritical." In contrast, a filament with $\lambda \lt {\lambda }_{\max }$ is called "subcritical," and its solution is discussed in the previous section. For a review, see André et al. (2014).

Although from its definition, ${\lambda }_{\max }$ is calculated as the slope ${\left(\partial \mathrm{log}\lambda /\partial \mathrm{log}{\rho }_{c}\right)}_{N,{R}_{0},{\beta }_{0}}=0$, considering numerical errors, we regard ${\lambda }_{\max }$ to be achieved when ${\left(\partial \mathrm{log}\lambda /\partial \mathrm{log}{\rho }_{c}\right)}_{N,{R}_{0},{\beta }_{0}}\lt 0.05$ is satisfied. ³

Figure 10 plots ${\lambda }_{\max }$ against Φ_cl for various N values. This shows that ${\lambda }_{\max }$ increases with Φ_cl, and the slope seems almost the same at large Φ_cl ≳ 10. We approximated the curves by using a function ${\lambda }_{\max }=\sqrt{{\lambda }_{0,\max }^{2}+A{{\rm{\Phi }}}_{\mathrm{cl}}^{2}}$, where ${\lambda }_{0,\max }$ corresponds to the maximum line mass of a nonmagnetized polytropic filament, and A represents the slope, which is determined by fitting. The value of A for various N values is obtained as A = 32.0 (N = −100), 26.4(N = −10), 23.5(N = −5), and 18.9(N = −3). From these four points, A is fitted as A = 31.5 + 39.0/N, and the accuracy of this fitting is within 5%. Thus, the normalized line mass is expressed with ${\lambda }_{0,\max }^{{\prime} }$ and ${{\rm{\Phi }}}_{\mathrm{cl}}^{{\prime} }$ as

$$\begin{eqnarray}&&{\lambda }_{\max }^{{\prime} }\simeq \sqrt{\lambda {{\prime} }_{0,\max }^{2}+\left(31.5+\displaystyle \frac{39.0}{N}\right){{{\rm{\Phi }}}_{\mathrm{cl}}^{{\prime} }}^{2}},\end{eqnarray} \tag{ 43 }$$

where ${\lambda }_{0,\max }^{{\prime} }$ for various N values is ${\lambda }_{0,\max }^{{\prime} }(N=-100)\,=23.2$, ${\lambda }_{0,\max }^{{\prime} }(N=-10)=17.5$, ${\lambda }_{0,\max }^{{\prime} }(N=-5)=14.4$, and ${\lambda }_{0,\max }^{{\prime} }(N=-3)=12.0$. Because the line mass and magnetic flux are normalized by ${c}_{\mathrm{ss}}^{2}/4\pi G$ and ${c}_{\mathrm{ss}}^{2}/{\left(G/2\right)}^{1/2}$, as in Table 1, we obtain a dimensional form of Equation (43) as

$$\begin{eqnarray}&&{\lambda }_{\max }\simeq \sqrt{{\left(\tfrac{{\lambda }_{0,\max }^{{\prime} }(N)}{8\pi }\right)}^{2}{\left(\tfrac{2{c}_{\mathrm{ss}}^{2}}{G}\right)}^{2}+{\left[{\left(0.10+\tfrac{0.12}{N}\right)}^{1/2}\tfrac{{{\rm{\Phi }}}_{\mathrm{cl}}}{{G}^{1/2}}\right]}^{2}}.\end{eqnarray} \tag{ 44 }$$

Although the critical mass-to-flux ratio of the magnetized isothermal filamentary cloud is ${\left({G}^{1/2}\lambda /{{\rm{\Phi }}}_{\mathrm{cl}}\right)}_{\mathrm{crit}}=0.24$ (Tomisaka 2014), the value for the negative-indexed polytropic filament (N = −3) is approximately equal to 0.24. Finally, Equation (44) is rewritten as

$$\begin{eqnarray}&&{\lambda }_{{\rm{m}}{\rm{a}}{\rm{x}}}\simeq \sqrt{{\left(\frac{{\lambda }_{0,{\rm{m}}{\rm{a}}{\rm{x}}}(N)}{{M}_{\odot }{\rm{p}}{{\rm{c}}}^{-1}}\right)}^{2}+{\left[5.9{\left(1.0+\frac{1.2}{N}\right)}^{1/2}\left(\frac{{{\rm{\Phi }}}_{{\rm{c}}{\rm{l}}}}{1\mu {\rm{G}}{\rm{p}}{\rm{c}}}\right)\right]}^{2}}\,\,{M}_{\odot }\,{{\rm{p}}{\rm{c}}}^{-1},\end{eqnarray} \tag{ 45 }$$

where the maximum line mass of a nonmagnetized polytropic filament at each N value is obtained as

$$\begin{eqnarray}{\lambda }_{0,\max }(N)=\left\{\begin{array}{ll}15.5{\left({c}_{\mathrm{ss}}/190{\rm{m}}{{\rm{s}}}^{-1}\right)}^{2}\,{M}_{\odot }\,{\mathrm{pc}}^{-1} & \,(\mathrm{for}\ N=-100),\\ 11.7{\left({c}_{\mathrm{ss}}/190{\rm{m}}{{\rm{s}}}^{-1}\right)}^{2}\,{M}_{\odot }\,{\mathrm{pc}}^{-1} & \,(\mathrm{for}\ N=-10),\\ 9.6{\left({c}_{\mathrm{ss}}/190{\rm{m}}{{\rm{s}}}^{-1}\right)}^{2}\,{M}_{\odot }\,{\mathrm{pc}}^{-1} & \,(\mathrm{for}\ N=-5),\\ 8.0{\left({c}_{\mathrm{ss}}/190{\rm{m}}{{\rm{s}}}^{-1}\right)}^{2}\,{M}_{\odot }\,{\mathrm{pc}}^{-1} & \,(\mathrm{for}\ N=-3).\end{array}\right.\end{eqnarray} \tag{ 46 }$$

Thus, we derived an empirical formula of the maximum line mass for magnetized polytropic filaments as Equation (45). For example, an N = −3 filament shows that the magnetic contribution for the line mass becomes dominant when ${{\rm{\Phi }}}_{\mathrm{cl}}\gt 4.6\,\mathrm{pc}\,\mu {\rm{G}}{\left({c}_{\mathrm{ss}}/190{\rm{m}}{{\rm{s}}}^{-1}\right)}^{2}$.

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Figures 3, 5, and 8 show the density profiles of models with various ρ_c , N, R₀, and β₀ values. The density profile on the y-axis is almost the same for three different R₀ and four different β₀ values. The slope of the profile on the y-axis becomes shallow when we increase N from −100 to −3. Thus, the slope of the density profile is controlled only by the polytropic index N in the y-direction, where the Lorentz force does not work.

This can be understood as follows. In the y-direction, because the Lorentz force does not play a role, the density distribution is governed by the pressure distribution (and self-gravity). A negative temperature gradient from the surface to the center has the effect of extending the envelope, and the temperature gradient increases from an isothermal equation of state (N = −100) to a polytropic one (N = −3).

Conversely, the slope on the x-axis becomes shallow only for the models with a weak magnetic field (Figure 8), while the slope slightly changes with R₀ (Figure 5).

We pay attention to these characteristics when considering how to reproduce the observed column density profile. To characterize the density of the axisymmetric filament, Plummer-like profiles are often used, such as Equation (4). Accordingly, the observed column density distribution is fitted with the function

$$\begin{eqnarray}&&\sigma (r)=\displaystyle \frac{{\sigma }_{0}}{{\left[1+{\left(r/{R}_{f}\right)}^{2}\right]}^{(p-1)/2}},\end{eqnarray} \tag{ 47 }$$

which is also a Plummer-like function, where σ₀, R_f , and p—the central column density, the core radius, and the density slope parameter, respectively—are three fitting parameters (Nutter et al. 2008; Arzoumanian et al. 2011). We obtain the column density by integrating the numerical solution of the density distribution as

$$\begin{eqnarray}&&{\sigma }_{\parallel }(x)=2{\int }_{y=0}^{{y}_{s}(x)}\rho (x,y){dy},\end{eqnarray} \tag{ 48a }$$

$$\begin{eqnarray}&&{\sigma }_{\perp }(y)=2{\int }_{x=0}^{{x}_{s}(y)}\rho (x,y){dx},\end{eqnarray} \tag{ 48b }$$

where σ_∥(x) and σ_⊥(y) represent the column densities observed from the parallel and perpendicular directions, respectively, with respect to the magnetic field.

As shown in Equation (1), the column density profile of an isothermal filament in hydrostatic equilibrium follows p = 4 (Stodółkiewicz 1963). However, as is summarized in Section 1, Herschel observations indicate that almost all the filaments follow p ≃ 2. Although some researchers have argued that the cylindrical dynamical contraction explains the observed shallow column density slope of p ≃ 2 (e.g., Kawachi & Hanawa 1998), in the present paper, we investigate whether a hydrostatic filament having a negative temperature gradient forms the observed shallow column density slope.

In Figure 11, we show the column density integrated along the direction of the y- and x-axes for various N and β₀ values. Other parameters are constant at R₀ = 2 and ρ_c = 100. We determined three parameters in the Plummer-like function (the slope index p, the column density at the center σ₀, and the core radius R_f in Equation (47)) via the least-squares method. The least squares are calculated only for the region of σ ≥ σ₀/10. This restriction comes from accounting for the dynamic range of the observed column density above the fore- and background column density (Arzoumanian et al. 2019).

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Figure 11(a) and (b) presents the column density profiles of σ_∥(x) that correspond to the profile in the direction in which the Lorentz force is effective. When β₀ = 1, p reaches 2 as N goes from −100 to −3, p(N = −100) = 4.86 → p(N = −3) = 2.48 (panel a). Conversely, p does not show such convergence when β₀ = 0.05. For example, the model with N = −3 (red curve of Figure 11(b)) indicates that the range of the power-law column density distribution is very narrow, and just outside of this, a sharp density decrement is observed. If we try to fit this column density distribution with a Plummer-like function, this gives an artificially large power-law index p.

In conclusion, the slope of the column density profile becomes shallow due to the temperature gradient for a model with a weak magnetic field. In contrast, we found that a strong magnetic field makes R_f large and worsens the fitting with the Plummer-like function (47).

Next, Figure 11(c) and (d) corresponds to σ_⊥(y), which indicates the column density distribution in the direction in which the Lorentz force does not play a role. For both β₀ = 1 (panel c) and β₀ = 0.05 (panel d), the slope index p reaches 2 as N changes from −100 to −3. This resembles the relation obtained for the nonmagnetized polytropic filament studied by Toci & Galli (2015a). In the direction in which the Lorentz force is less important, the negative temperature gradient toward the center plays a role in making the density slope shallow, even in a magnetized filament.

In Figure 11, we plot the column density distributions observed from the y-direction, σ_∥(x), and that from the x-direction, σ_⊥(y). For comparison with observations, in this section we study the dependence of power-law index p and core radius R_f on the line of sight direction θ, which is defined as the angle between the line of sight and the x-axis. Defining the angle between the line of sight and the x-axis as θ, we rotated the density distribution at −θ. With the rotated density distribution integrated along the x-axis, σ_x (y) ≡ ∫ρ dx gives the column density distribution observed from this line of sight. In this case, the column density profiles σ_⊥(y) and σ_∥(x) are obtained when θ = 0° and θ = 90°, respectively. Fitting the rotated σ_x (y) with the Plummer-like function of Equation (47), we obtain the power-law index p and the core size R_f depending on θ, as shown in Figure 12. The parameters of the model are R₀ = 2, N = −3, β₀ = 1, and ρ_c = 100.

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Figure 12 shows that the power-law index p (dashed curve) is restricted to a narrow range of 2.48 ≲ p ≲ 2.73. Thus, if we measure p, the line-of-sight direction is hardly determined. In other words, this seems to explain the reason why filaments commonly have p ≃ 2, even if the line of sight and thus the angle θ must be chosen randomly for each observational target. In contrast, the core size R_f (solid curve) changes smoothly from 0.07 (θ = 0°) to 0.26 (θ = 90°). Because R_f is strongly dependent on θ, we can distinguish whether the line of sight is nearly perpendicular (θ ≃ 0°) or parallel (θ ≃ 90°) to the magnetic field.

We now propose a way to distinguish whether θ = 0° or θ = 90° when the line of sight is perpendicular to the filament long axis. In the nonmagnetized and thus symmetric filament, Equation (47) indicates that the central column density is equal to σ₀ = S ρ_c R_f , where the numerical factor $S\,=2{\int }_{0}^{\infty }{\left(1+{\zeta }^{2}\right)}^{-p/2}d\zeta$ equals S = π for p = 2 and S = π/2 for p = 4. This means that, in the axisymmetric model, the central column density is given as the central density times the scale length of the column density in the direction perpendicular to the line of sight.

In the nonaxisymmetric configuration, we define the effective central density as

$$\begin{eqnarray}&&{\rho }_{c}^{\mathrm{eff}}\equiv \displaystyle \frac{{\sigma }_{0}}{S\cdot {R}_{f}}.\end{eqnarray} \tag{ 49 }$$

Table 3 shows the quantity calculated assuming S = π for the models shown in Figure 11. Because all the models have the same central density ρ_c = 100, ${\rho }_{c}^{\mathrm{eff}}$ derived from σ_∥(x) is smaller than the true ρ_c . For N = −3, ${\rho }_{c}^{\mathrm{eff}}$ derived from σ_⊥(y) is much larger than the true ρ_c , but ${\rho }_{c}^{\mathrm{eff}}$ derived from σ_∥(x) is much smaller than that. When the central density is observationally estimated, for example, by using the critical density of the molecular line transitions, we can compare the ρ_c and ${\rho }_{c}^{\mathrm{eff}}$ estimated from the central column density σ₀ and the column density scale length R_f . For a filament with N = −3, ${\rho }_{c}^{\mathrm{eff}}\gg {\rho }_{c}$ for σ_⊥(y). Therefore, when we observe ${\rho }_{c}^{\mathrm{eff}}\gg {\rho }_{c}$, this indicates that the line of sight is perpendicular to the magnetic field. Conversely, ${\rho }_{c}^{\mathrm{eff}}\ll {\rho }_{c}$ indicates that the line of sight is parallel to the magnetic field. Based on this, when we observe the central density, central column density, and core radius of the filament, we can evaluate the angle between the line of sight and the magnetic field line.

Table 3. Effective Central Density ${\rho }_{c}^{\mathrm{eff}}$

β0	N	σ⊥(y)			σ∥(x)
		Rf	σ0	${\rho }_{c}^{\mathrm{eff}}$	Rf	σ0	${\rho }_{c}^{\mathrm{eff}}$
1	−100	0.26	55	67.4	0.46	37	25.6
1	−10	0.18	63	111	0.38	29	24.3
1	−5	0.12	62	165	0.31	22	22.6
1	−3	0.07	60	273	0.25	15	19.1
0.05	−100	0.24	141	187	0.92	31	10.7
0.05	−10	0.16	145	289	1.01	25	7.88
0.05	−5	0.11	151	437	1.32	19	4.58
0.05	−3	0.07	162	737	3.26	13	1.27

Notes. Here, σ_∥(x) and σ_⊥(y) represent the column density distributions obtained by integrating the density in the directions parallel and perpendicular to the magnetic field, respectively.

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