Collective Instabilities and Waves for Inhomogeneous Stellar Systems. I. The Necessary and Sufficient Energy Principle
Abstract
In this paper an energy principle is derived that gives a necessary and sufficient condition for the linear stability of inhomogeneous collisionless nonrelativistic stellar systems whose distribution function is a monotonic decreasing function of the energy. Although this energy principle is equivalent to several other variational principles, its derivation is made without any reference to normal modes or their completeness. This is regarded as a more satisfactory basis for the energy principle, since in general the normal modes are singular and form a continuous set whose completeness is difficult to treat. The energy principle is given in a different form which brings out the important role of constraints on the trial functions in giving a necessary condition for stability. The energy principle is applied to the case of an equilibrium with plane-parallel symmetry. It is shown that for any such slab equilibrium there exists a critical horizontal wavenumber such that all perturbations with larger horizontal wavenumbers are stable, but the slab is unstable to some perturbation at each of the smaller wavenumbers. This is an instability similar to the classical Jeans instability, hut for a collisionless system rather than a fluid.
- Publication:
-
The Astrophysical Journal
- Pub Date:
- May 1970
- DOI:
- 10.1086/150448
- Bibcode:
- 1970ApJ...160..471K