Global Nonradial Instabilities of Dynamically Collapsing Gas Spheres

Abstract

Self-similar solutions provide good descriptions for the gravitational collapse of spherical clouds or stars when the gas obeys a polytropic equation of state, $p=Kρ^γ$ (with $γ\le 4/3$). We study the behaviors of nonradial perturbations in the similarity solutions of Larson, Penston and Yahil, which describe the evolution of the collapsing cloud prior to core formation. Our global stability analysis reveals the existence of unstable bar-modes ($l=2$) when $γ\le 1.09$. In particular, for the collapse of isothermal spheres, which applies to the early stages of star formation, the $l=2$ density perturbation relative to the background, $δρ({\bf r},t)/ρ(r,t)$, increases as $(t_0-t)^{-0.352}\propto ρ_c(t)^{0.176}$, where $t_0$ denotes the epoch of core formation, and $ρ_c(t)$ is the cloud central density. Thus, the isothermal cloud tends to evolve into an ellipsoidal shape (prolate bar or oblate disk, depending on initial conditions) as the collapse proceeds. In the context of Type II supernovae, core collapse is described by the $γ\simeq 1.3$ equation of state, and our analysis indicates that there is no growing mode (with density perturbation) in the collapsing core before the proto-neutron star forms, although nonradial perturbations can grow during the subsequent accretion of the outer core and envelope onto the neutron star. We also carry out a global stability analysis for the self-similar expansion-wave solution found by Shu, which describes the post-collapse accretion (``inside-out'' collapse) of isothermal gas onto a protostar. We show that this solution is unstable to perturbations of all $l$'s, although the growth rates are unknown.