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 ($\mathrm{\ell }=2$) 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} $whilethecentraldensityincreasesinproportionto$ \rho_c \propto (t - t_0) ^{-2} $inthesimilaritysolution\mathrm{.}Thegrowthrate,$ \sigma $isobtainednumericallyasafunctionof$ \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 $\sigma =1\mathrm{/}3$ for any $\gamma$.