The Equilibrium and the Stability of the Riemann Ellipsoids. II

Abstract

In this paper we consider ellipsoidal figures of equilibrium (of semi-axes a, a, and a) of homogeneous masses rotating uniformly with an angular velocity Ω and with internal motions having a uniform vorticity ζ (in the rotating frame) in the case that the directions of Ω and ζ do not coincide. Riemann's theorem, that in this case nand t must lie in a principal plane of the ellipsoid, is shown to follow from a consideration of the non-diagonal components of the second-order tensor-virial theorem. The conditions for equilibrium are also derived; and the domains of occupancy of these Riemann ellipsoids in the (a/a, a/a)-plane (on the assumptions, which entail no loss of generality, that Ω and ζ have no components in the x-direction and that a≥a) are explicitly specified. It is shown that the equilibrium ellipsoids are of three types: ellipsoids of type I which occupy the domain 2a≥(a + a) and a≥a≥a; ellipsoids of type II for which a≥2a and a/a (≤1) are limited by a locus along which ∫pdx = 0; and ellipsoids of type III which occupy the domain limited by 2a≤(a-a) and a locus along which Ω = ζ = 0 and a≥a. And quite generally, it is shown that an ellipsoid, represented by a point in the allowed domain of occupancy, is a figure of equilibrium for two different states of motion (Ω, ζ) and (Ω†, ζ†); and that the two resulting configurations are adjoints of one another in the sense of Dedekind's theorem. Ellipsoids of type T may be considered as branching off from the Maclaurin sequence with an odd mode of oscillation neutralized at the point of bifurcation by the choice of Ω and ζ (Ω and ζ being zero). And ellipsoids of type III may be similarly considered as branching off from the ellipsoids of type S (for which the directions of Ω and ζ coincide with the x-axis) along the curve where they are marginally unstable. The stability of the Riemann ellipsoids with respect to oscillations belonging to the second harmonics is also investigated. It is first shown that the characteristic frequencies of oscillation of an ellipsoid and its adjoint are the same; and further that I Ω I and I Ω† I are allowed proper frequencies. The loci along which instability sets in, in the different domains of occupancy, are determined. Of particular interest are the facts that all ellipsoids of type II are unstable; that along the curve where the ellipsoids of type III branch off from ellipsoids of type S, the stability passes from the latter to the former; and that among the ellipsoids of type I there are some very highly flattened ones that are stable. Several statements of Riemann concerning the stability of these ellipsoids are not substantiated by the present detailed investigation. The origin of Riemann's errors is clarified in the paper by Lebovitz following this one.