Collective Instabilities of Self-Gravitating Systems

The instability modes of self-gravitating systems are investigated in comparison with electron oscillations in a plasma for following four cases; one system of a collisionless or a hydrodynamical description, two systems with a relative velocity of a collisionless and a hydrodynamical one, two collisionless ones and two hydrodynamical ones. The fundamental unstable mode is the Jeans instability but this mode is modified for coexistence of two systems as follows: The critical Jeans length becomes longer due to the stabilization effect of relative velocity. Moreover, when the relative velocity is higher than a certain value, each system becomes independently unstable for the Jeans mode and they appear as overstable modes. On the other hand, the Landau instability occurs only in the system of a collisionless and a hydrodynamical description in contrast with a plasma case, where the Landau mode appears in two-collisionless system as well. Then, the condition for the occurrence of the Landau mode is obtained as the existence of the distribution function with velocity dispersion and the repulsive force (or effect). It relation to the density wave theroy of spiral arms, the Landau growing mode in a stellar and gaseous system is discussed