Stability of Dynamically Collapsing Gas Sphere

Abstract

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 $\gamma$.

submitted to PASJ. 7 pages including 6 figures. This paper is also available at http://www.a.phys.nagoya-u.ac.jp/~hanawa/dpnu9922/dpnu9922.html

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