Gravitational Instability of Shocked Interstellar Gas Layers
  In this paper we investigate gravitational instability of shocked gas layers using linear analysis. An unperturbed state is a self-gravitating isothermal layer which grows with time by the accretion of gas through shock fronts due to a cloud-cloud collision. Since the unperturbed state is not static, and cannot be described by a self-similar solution, we numerically solved the perturbation equations and directly integrated them over time. We took account of the distribution of physical quantities across the thickness. Linearized Rankine-Hugoniot relations were imposed at shock fronts as boundary conditions. The following results are found from our unsteady linear analysis: the perturbation initially evolves in oscillatory mode, and begins to grow at a certain epoch. The wavenumber of the fastest growing mode is given by $k=2\sqrt{2\pi G\rho_\mathrm{E} {\cal M\mit}}/c_\mathrm{s}$, where $\rho_\mathrm{E}, c_\mathrm{s}$ and $\cal M\mit$ are the density of parent clouds, the sound velocity and the Mach number of the collision velocity, respectively. For this mode, the transition epoch from oscillatory to growing mode is given by $t_g = 1.2/\sqrt{2\pi G\rho_\mathrm{E} {\cal M\mit}}$. The epoch at which the fastest growing mode becomes non-linear is given by $2.4\delta_0^{-0.1}/\sqrt{2\pi G \rho_\mathrm{E}{\cal M\mit}}$, where $\delta_0$ is the initial amplitude of the perturbation of the column density. As an application of our linear analysis, we investigate criteria for collision-induced fragmentation. Collision-induced fragmentation will occur only when parent clouds are cold, or $\alpha_0=5c_\mathrm{s}^2 R/2G M < 1$, where $R$ and $M$ are the radius and the mass of parent clouds, respectively. 
Kazunari Iwasaki and Toru Tsuribe