Gravitational Instability and Tsallis' Nonextensive Statistics

The concept of Jeans gravitational instability is rediscussed in the framework of the nonextensive statistics proposed by Tsallis. A simple analytical formula generalizing the Jeans criterion is derived by assuming that the unperturbed self-gravitating collisionless gas is described by the $q$-parametrized class of nonextensive velocity distribution. It is shown that the critical values of wavelenght and mass depend explicitly on the nonextensive $q$-parameter. The standard Jeans wavelenght derived for a Maxwellian distribution is recovered in the limiting case $q$=1. For power-law distributions with cutoff, the instability condition is weakneed with the system becoming unstable even for wavelenghts of the disturbance smaller than the standard Jeans lenght $\lambda_J$.