Hub Formation and Filament-Filament Collision: An Analytical Model

Abstract

Filaments are ubiquitous throughout the Galaxy. Massive star formation is often observed in hub-filament systems, where multiple filaments appear to be interconnected and merging. Filament-filament collisions are therefore a likely triggering mechanism for massive star formation. We derive basic physical properties of filament-filament collisions, such as the collision cross section (CCS), the hub mass, and its mass function, based on a simple cylindrical filament model. We assume a cylindrical filament with length $2p$, full width $2q$, and line-mass $λ_0$, and consider the CCS between two identical filaments. The collision is specified by three vectors: the directions of the colliding filaments ($n_1$ and $n_2$) and the direction of the relative velocity between the two filaments ($n_v=v/|v|$). For the thin filament, $p\gg q$, the CCS is expressed as $S=4p^2|n'_1\times n'_2|$, where $n'_1$ and $n'_2$ represent the directional vectors projected onto a plane perpendicular to the relative velocity $n_v$. As the angle between $n'_1$ and $n'_2$ becomes smaller, the cross section proportional to $p\cdot q$ becomes relatively important. We propose a simple model in which the hub mass is estimated by the overlapping portion of the two colliding filaments. The hub mass function is derived using the CCSs and the geometrically estimated overlapping mass. When the directions and relative velocities of the filaments are isotropically distributed, the mass function expected from a single species of filaments fits well to a power law and the power exponent is $γ_M\simeq -2.96$ ~ $-3.78$. The power exponent of the global hub mass function is the same as that of the line-mass distribution function, $γ_λ\simeq -1.5$. This means that a massive hub is formed by the collision of two massive filaments.

Figures (9)

• Figure 1: Projection of filaments 1 and 2 onto a plane perpendicular to the relative velocity. The impact parameter is denoted by $\bm{b}$. The endpoints of filament 1 are $\bm{p}'_1$ and $-\bm{p}'_1$. The endpoints of filament 2 are $\bm{b}+\bm{p}'_2$ and $\bm{b}-\bm{p}'_2$. If the impact parameter $\bm{b}$ is selected inside the shaded parallelogram in the figure, the two filament vectors $\pm\bm{p}'_1$ and $\bm{b}\pm\bm{p}'_2$ intersect. Alt text: A schematic diagram which contains two directional vectors. We consider the condition in which these vectors intersect each other.
• Figure 2: Projection of finite-width filaments onto a plane perpendicular to the relative velocity (two dotted rectangles). The centers of the filament ends are the same as in figure \ref{['fig.1']}. For filament 1, $\pm \bm{p}'_1$, for filament 2, $\bm{b}\pm\bm{p}'_2$. The conditions for the two rectangles to overlap are considered. See also appendix \ref{['sec:A1']}. Alt text: A schematic diagram which contains two rectangles. We consider the condition in which these rectangles overlap each other. Vertices of the rectangle are labeled UL, UR, LR, and LL clockwise from the top left.
• Figure 3: The mean $\overline{S}$ for various distributions of $\bm{n}_\alpha$. We calculate four models for different sets of distribution functions, as shown in table \ref{['tbl.1']}. The more concentrated the distribution functions we choose, the more significantly the mean $\overline{S}$ decreases in the range $\sigma_\mu \mathrel{ \vcenter{\m@th\f@size4 \ialign{$$\cr <\crcr{ } \sim\crcr}}} 1$. Alt text: A four-line graph. Models A, B, C, and I are shown. Horizontal and vertical axes represent sigma sub mu and S bar over p squared, respectively.
• Figure 4: Comparison of $\overline{S}$ and $\overline{T}$. For the distribution function of Model C (table \ref{['tbl.1']}), the leading term of the CCS, $\overline{S}$ (black dotted line), proportional to $p^2$, and the second term of the CCS, $\overline{T}$ (blue dotted line), proportional to $pq$, are plotted against the standard deviations of the truncated normal distribution $\sigma_\mu$ and $\sigma_\phi$. The filament axis is assumed to follow the distribution specified by $\sigma_{\mu}=\sigma_{\mu_1}=\sigma_{\mu_2}$ and $\sigma_{\phi_1}=\sigma_{\phi_2}=\arcsin\sigma_{\mu}$, which tends to align toward the $y$-axis. The relative velocity field is assumed to be isotropic. When all distributions are isotropic (Model I), the means $\overline{S}$ (black dash-dotted line) and $\overline{T}$ (blue dash-dotted line) are plotted as horizontal lines. For comparison, we assume that the axial ratio is $q/p=0.1$. Alt text: Four-line figure. Horizontal axis represents sigma sub mu and vertical axis represents S bar over p squared and T bar over p squared.
• Figure 5: Mass function (MF) of the hub expected from the filament-filament collisions. MF is plotted against the hub's mass normalized by the total mass of the two filaments $2M_0=4\lambda_0 p$. $d\mathrm{Pr}(M_\mathrm{Hub}>M)/d\log M$ for the CCS of $\Sigma=S$ (solid line) and that for $\Sigma=S+T$ (dotted line) are plotted. When calculating $T$, we assume the axis ratio $q/p=0.1$. The power-law fitting for $\Sigma=S+T$ model is also shown by a dashed line. Alt text: Four-line figure. Horizontal axis represents the hub mass divided over 4 lambda sub 0 p. Vertical axis represents the mass function.