ON THE IMPLICATIONS OF THE NTH-ORDER VIRIAL EQUATIONS FOR HETEROGENEOUS AND CONCENTRIC JACOBI, DEDEKIND, AND RIEMANN ELLIPSOIDS

The implications of the nth-order virial equations are analyzed for co ncentric heterogeneous ellipsoids with a density distribution of the f orm rho = rho(c) f(m(2)), where m(2) = Sigma(1=1)(3), x(i)(2)/a(i)(2), 0 less than or equal to m(2) less than or equal to 1, and a(i) are th e semiaxes of the external ellipsoid corresponding to m(2) = 1. Soluti ons analogous to Jacobi ellipsoids (with constant angular velocity Ome ga, without vorticity), to Dedekind ellipsoids (with nonuniform vortic ity Z and zero angular velocity), and to Riemann ellipsoids (with cons tant angular velocity and nonuniform vorticity) are explored. It is sh own that only the second- and fourth-order virial equations give nontr ivial results: all the odd-order virial equations are identically sati sfied for ellipsoids rotating around a principal axis of symmetry. The even-order virial equations (sixth, eighth, etc.) are shown to be a c onsequence of the lowest order equations. The entire family of homogen eous and heterogeneous concentric ellipsoids allowed by the virial equ ations is presented, confronted, and contrasted with the known cases i n the literature.