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.
