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

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 $\gamma \ne \Gamma$ such that the Brunt-V$\ddot{\mathrm{a}}$is$\ddot{\mathrm{a}}$l$\ddot{\mathrm{a}}$ buoyancy frequency $\mathcal{N}$ 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.

5 pages, 4 figures, accepted for publication in MNRAS

References in corpus (7)

• A New Mechanism for Core-Collapse Supernova Explosions
• Multi-Dimensional Explorations in Supernova Theory
• New Self-Similar Solutions of Polytropic Gas Dynamics
• Magnetohydrodynamic Rebound Shocks of Supernovae
• Self-Similar Dynamics of a Relativistically Hot Gas
• Magnetized massive stars as magnetar progenitors
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Cited by in corpus (3)

• Relativistic Self-similar Dynamic Collapses of Black Holes in General Polytropic Spherical Clouds
• Three-dimensional Hydrodynamic Instabilities in Stellar Core Collapses
• Buoyancy instability of homologous implosions