Stability of Dynamically Collapsing Gas Sphere

We discuss stability of dynamically collapsing gas spheres. We use a similarity solution for a dynamically collapsing sphere as the unperturbed state. In the similarity solution the gas pressure is approximated by a polytrope of $P = K \rho ^\gamma$. We examine three types of perturbations: bar ($\ell = 2$) mode, spin-up mode, and Ori-Piran mode. When $\gamma < 1.097$, it is unstable against bar-mode. It is unstable against spin-up mode for any $\gamma$. When $\gamma < 0.961$, the similarity solution is unstable against Ori-Piran mode. The unstable mode grows in proportion to $| t - t_0 | ^{-\sigma}$ while the central density increases in proportion to $\rho_c \propto (t - t_0) ^{-2}$ in the similarity solution. The growth rate, $\sigma$ is obtained numerically as a function of $\gamma$ for bar mode and Ori-Piran mode. The growth rate of the bar mode is larger for a smaller $\gamma$. The spin-up mode has the growth rate of $\sigma = 1/3$ for any $\gamma$.