Linear stability analysis of self-gravitating granular gas

The linear stability of granular gas that reflects the contribution of self-gravitational force of mass density perturbations is investigated in order to clarify the condition of competition between clustering instability and Jeans instability. It is found that the condition depends on three parameters: the mass density $\rho_0$, the collision rate $\omega_0$, and the rate of energy loss per collision $\epsilon$. When $\sqrt{G\rho_0}\ll\epsilon\omega_0$, clustering instability dominates, while when $\epsilon\omega_0\ll\sqrt{G\rho_0}$, Jeans instability dominates. These instabilities are characterized by the decrease and increase, respectively, of the temperature.