Adiabatic Perturbations in Homologous Conventional Polytropic Core Collapses of a Spherical Star

TL;DR

This paper analyzes non-radial adiabatic perturbations in homologous stellar core collapses with a focus on stability and mode behavior, revealing conditions for various instabilities relevant to supernovae.

Contribution

It provides a detailed eigenvalue and eigenfunction analysis of oscillation modes during core collapse, highlighting new instability criteria for g-modes based on adiabatic exponents.

Findings

p- and f-modes remain stable during collapse

g- modes become unstable under certain conditions

High-order g-modes are unstable in collapsing cores

Abstract

We perform a non-radial adiabatic perturbation analysis on homologous conventional polytropic stellar core collapses. The core collapse features a polytropic exponent $\Gamma=4/3$ relativistic gas under self-gravity of spherical symmetry while three-dimensional perturbations involve an adiabatic exponent $\gamma$ with $\gamma\neq\Gamma$ such that the Brunt-V$\ddot{\rm a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ buoyancy frequency ${\cal N}$ does not vanish. With proper boundary conditions, we derive eigenvalues and eigenfunctions for different modes of oscillations. In reference to stellar oscillations and earlier results, we examine behaviours of different modes and the criterion for instabilities. The acoustic p$-$modes and surface f$-$modes remain stable. For $\gamma<\Gamma$, convective instabilities appear as unstable internal gravity g$^{-}-$modes. For $\gamma>\Gamma$, sufficiently low-order internal gravity g$^{+}-$modes are stable, whereas sufficiently high-order g$^{+}-$modes, which would have been stable in a static star, become unstable during self-similar core collapses. For supernova explosions, physical consequences of such inevitable g$-$mode instabilities are speculated.