Instabilities in a self-gravitating granular gas
  A kinetic and hydrodynamic descriptions are developed in order to analyze the
instabilities of a granular gas in the presence of a gravitational field. In
the kinetic description the Boltzmann equation is coupled with the Poisson
equation, while in the hydrodynamic description the Poisson equation is coupled
with the balance equations of mass density, hydrodynamic velocity and
temperature for an Eulerian fluid. In the background solution for both
descriptions the fluid is at rest with constant mass density and gravitational
potential while the temperature depends on time through Haff's law. In the
kinetic description the perturbed distribution function and gravitational
potential in the Fourier space are related to time dependent small amplitudes.
In the hydrodynamic description the perturbed mass density, hydrodynamic
velocity and temperature in the Fourier space are functions of time dependent
small amplitudes. From the analysis of the system of coupled differential
equations for the amplitudes for the two descriptions the time evolution of the
density contrast -- a parameter that indicate where there are local
enhancements in the matter density -- is determined. The solutions depend on
two parameters, one is the mean free path of the gas particles and another
Jeans' wavelength, which is a function of the gravitational constant, mass
density and speed of sound of the gas. It is shown that instabilities due to
the inelastic collisions occur when the Jeans and the perturbation wavelengths
are larger than the mean free path, while Jeans' instabilities due to the
gravitational field happen when the mean free path and the perturbation
wavelength are larger than Jeans' wavelength.
