The Schrodinger Operator Criterion for the Stability of Galaxies and Gas Spheres

Abstract

A direct proof is given showing that a stellar system is stable whenever the corresponding barotropic gaseous system is secularly stable. The condition is formulated in terms of a Schrödinger equation. It is shown that a large class of spherical steady-state stellar systems is stable to all non-spherical modes of vibration. For spherical modes the Schrödinger operator method is necessary and sufficient for the stability of barotropic spheres but for stellar systems, though correct, it is only sufficient and is not a very powerful method of proving stability. Antonov's method is harder to apply but more powerful. The Schrödinger method is applied to some model clusters. More general and more powerful methods are needed in this field.