Analytic growth rate of gravitational instability in self-gravitating planar polytropes

TL;DR

This paper develops an analytical method to calculate the growth rate of gravitational instability in self-gravitating, plane-stratified polytropic fluids, applicable to astrophysical structures like sheets and filaments.

Contribution

It introduces a novel analytical approach for determining gravitational instability growth rates in stratified polytropic media, extending the understanding of stability in astrophysical systems.

Findings

Analytical expressions match numerical results with high accuracy.

Method applies to various boundary conditions and polytropic exponents.

Provides a general framework for stability analysis in astrophysics.

Abstract

Gravitational instability is a key process that may lead to fragmentation of gaseous structures (sheets, filaments, haloes) in astrophysics and cosmology. We introduce here a method to derive analytic expressions for the growth rate of gravitational instability in a plane stratified medium. We consider a pressure-confined, static, self-gravitating fluid of arbitrary polytropic exponent, with both free and rigid boundary conditions. The method we detail here can naturally be generalised to analyse the stability of more complex systems. Our analytical results are in excellent agreement with numerical resolutions.