Global Nonradial Instabilities of Dynamically Collapsing Gas Spheres

Self-similar solutions provide good descriptions for the gravitationalcollapse of spherical clouds or stars when the gas obeys a polytropic equationof state, $p=K\rho^\gamma$ (with $\gamma\le 4/3$). We study the behaviors ofnonradial perturbations in the similarity solutions of Larson, Penston andYahil, which describe the evolution of the collapsing cloud prior to coreformation. Our global stability analysis reveals the existence of unstablebar-modes ($l=2$) when $\gamma\le 1.09$. In particular, for the collapse ofisothermal spheres, which applies to the early stages of star formation, the$l=2$ density perturbation relative to the background, $\delta\rho({\bfr},t)/\rho(r,t)$, increases as $(t_0-t)^{-0.352}\propto \rho_c(t)^{0.176}$,where $t_0$ denotes the epoch of core formation, and $\rho_c(t)$ is the cloudcentral density. Thus, the isothermal cloud tends to evolve into an ellipsoidalshape (prolate bar or oblate disk, depending on initial conditions) as thecollapse proceeds. In the context of Type II supernovae, core collapse isdescribed by the $\gamma\simeq 1.3$ equation of state, and our analysisindicates that there is no growing mode (with density perturbation) in thecollapsing core before the proto-neutron star forms, although nonradialperturbations can grow during the subsequent accretion of the outer core andenvelope onto the neutron star. We also carry out a global stability analysisfor the self-similar expansion-wave solution found by Shu, which describes thepost-collapse accretion (``inside-out'' collapse) of isothermal gas onto aprotostar. We show that this solution is unstable to perturbations of all$l$'s, although the growth rates are unknown.Comment: 28 pages including 7 ps figures; Minor changes in the discussion; To be published in ApJ (V.540, Sept.10, 2000 issue