The FiniteAmplitude Oscillations of the Maclaurin Spheroids
Abstract
The work of Dirichlet on the oscillations of selfgravitating, incompressible, homogeneous, ellipsoidal configurations has been extended to treat the finiteamplitude oscillations of the Maclaurin spheroids which, in the limit of small amplitude, degenerate into their secondharmonic normal modes. These oscillations preserve the ellipsoidal nature of the surface and generate internal motions having a uniform vorticity. The problem is made tractable by the circumstance that the relevant partial differential equations of motion may be replaced by a set of ordinary differential equations, which have special integrals of the motion for the cases of interest. The results of analytic and numerical integration of these equations are given for the oscillations degenerating into that mode by which the Maclaurin spheroids become unstable. The nature of these oscillations is conveniently represented by trajectories in the (a3,a3)plane (where a5 and a3 are the lengths of two of the semiaxes of the ellipsoidal figure); and it is shown how these trajectories can be interpreted in terms of a particledynamical point of view incorporating such concepts as zerovelocity curves and isolating or nonisolating integrals of the motion. Computations have been made for various choices of the unperturbed configuration, both stable and unstable, and various energies, both negative and positive, of the perturbed configuration. In particular, the results for the case of a highly oblate unstable Maclaurin spheroid suggest that the inclusion of dissipation could result in the ultimate assumption by such a configuration of a new stationary state which is highly prolate about one of its principal axes transverse to the axis of rotation.
 Publication:

The Astrophysical Journal
 Pub Date:
 July 1967
 DOI:
 10.1086/149238
 Bibcode:
 1967ApJ...149..145R