The equilibrium and the stability of the Jeans spheroids

Abstract

The equilibrium and the stability of homogeneous masses distorted by the tidal effects of a secondary (of mass M' at a distance R) are re-examined on the basis of the second-order virial equations. In agreement with known results, it is shown that, under circumstances when the figure of equilibrium is a prolate spheroid, there is a maximum value of π( = GM'/R³) which is compatible with equilibrium. The problem of the small oscillations of these Jeans spheroids is next considered. The characteristic frequencies of oscillation belonging to the second harmonics are determined both in case the mass is considered incompressible and in case it is considered compressible and subject to the gas laws governing adiabatic changes. In the former case, instability sets in when μ attains its maximum value; and in the latter case it sets in before that happens.