The Equilibrium and the Stability of the RocheRiemann Ellipsoids
Abstract
The classical problem of Roche (wherein a fluid mass maintains an ellipsoidal form while in synchro nous Keplerian orbit about a rigid sphere) is extended to the case where motions of uniform vorticity exist within the ellipsoid. It is shown that the introduction of such motions gives rise to new types of equilibri um configurations (the RocheRiemann ellipsoids). The case where the vorticity vector ~ lies parallel to the angular velocity vector ~ is considered first, and the problem is then generalized to the case where ~ and ~ lie in one of the principal planes of the ellipsoid. The existence of a minimum distance between the ellipsoid and the point mass (commonly referred to as the Roche limit) is found to be a function of the semimajor axes of the ellipsoid; in fact, this distance maybe zero. The Roche and the Dedekind ellipsoids are shown to be special cases of the RocheRiemann ellipsoids. The stability of the ellipsoids is determined, and oniy some of the allowed equilibrium configurations are stable with respect to small perturbations. The results of these computations are illustrated by means of diagrams in the (a2/al, a3/ai)plane. The existence of curves of bifurcation from ellipsoids with ~ parallel to ~ to ellipsoids where ~ and ~ lie in a principal plane is demonstrated, and it is found that there is an exchange of stability along these curves. I. INTRODUCTIO
 Publication:

The Astrophysical Journal
 Pub Date:
 August 1968
 DOI:
 10.1086/149683
 Bibcode:
 1968ApJ...153..511A