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

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.