Stability properties for encounterless self-gravitational stellar gas and plasma

The stability properties of the collisionless plasma and encounterless stellar gas described by the Vlasov equations are studied. The introduction of the multiple Water Bag model allows, for one-dimensional plane geometry, a treatment of the general case and removes some of the difficulties connected with the formulation of the energy variation. From this last result it can be deduced that both plasma and stellar systems steady state described by a monotonically decreasing distributionF() are stable. The demonstration is extended to the spherically symmetric case for self-gravitating gas. Next the constraint of a monotonically decreasingF() is relaxed and it is supposed that the instability appears through the point =0. This is known to be true for some type of plasma instabilities (two streams) but is a simple working hypothesis in the gravitational case. For this marginal mode theN bags equations degenerate into a single wave equation and the stability of the system is given by the sign of the eigenvalues of a Schroedinger type operator. A simple physical picture is obtained for the plasma case where the quantitity (dF/d) dV (the square of the local maximum wavenumber of instability) is introduced. A virtual variation of this quantity indicates if the initial steady state was stable or unstable.