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

摘要

The implications of the nth-order virial equations are analyzed for concentric heterogeneous ellipsoids with a density distribution of the form ρ = ρ_c f(m~2), where m~2 = Σ_(1 = 1)~3 x_i~2/a_i~2, 0 ≤ m~2 ≤ 1, and a_i are the semiaxes of the external ellipsoid corresponding to m~2 = 1. Solutions analogous to Jacobi ellipsoids (with constant angular velocity Ω, without vorticity), to Dedekind ellipsoids (with nonuniform vorticity Z and zero angular velocity), and to Riemann ellipsoids (with constant angular velocity and nonuniform vorticity) are explored. It is shown that only the second- and fourth-order virial equations give nontrivial results: all the odd-order virial equations are identically satisfied for ellipsoids rotating around a principal axis of symmetry. The even-order virial equations (sixth, eighth, etc.) are shown to be a consequence of the lowest order equations. The entire family of homogeneous and heterogeneous concentric ellipsoids allowed by the virial equations is presented, confronted, and contrasted with the known cases in the literature.