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Gravitational instability is a key process that may lead to fragmentation ofgaseous structures (sheets, filaments, haloes) in astrophysics and cosmology.We introduce here a method to derive analytic expressions for the growth rateof gravitational instability in a plane stratified medium. We consider apressure-confined, static, self-gravitating fluid of arbitrary polytropicexponent, with both free and rigid boundary conditions. The method we detailhere can naturally be generalised to analyse the stability of more complexsystems. Our analytical results are in excellent agreement with numericalresolutions.

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