Growth of Perturbation in Gravitational Collapse and Accretion

When a self-gravitating spherical gas cloud collapses or accretes onto a central mass, the inner region of the cloud develops a density profile ρ ∝ r−3/2 and the velocity approaches free-fall. We show that in this region, nonspherical perturbations grow with decreasing radius. In the linear regime, the tangential velocity perturbation increases as r−1, while the Lagrangian density perturbation, ∆ρ/ρ, grows as r−1/2. Faster growth occurs if the central collapsed object maintains a finite multiple moment, in which case ∆ρ/ρ increases as r−l, where l specifies the angular degree of the perturbation. These scaling relations are different from those obtained for the collapse of a homogeneous cloud. Our numerical calculations indicate that nonspherical perturbations are damped in the subsonic region, and that they grow and approach the asymptotic scalings in the supersonic region. The implications of our results to asymmetric supernova collapse and to black hole accretion are briefly discussed. Subject headings: hydrodynamics – accretion – supernovae