Effects of variations of parallel angular velocity and vorticity on the oscillations of compressible homogeneous rotating ellipsoids

Abstract

Earlier work on the oscillations of an ellipsoid is extended to investigate the behaviour of a nonequilibrium compressible homogeneous rotating gaseous ellipsoid, with the components of the velocity field as linear functions of the coordinates, and with parallel angular velocity and uniform vorticity. The dynamical behaviour of the ellipsoid is obtained by numerically integrating the relevant differential equations for different values of the initial angular velocity and vorticity. This behaviour is displayed by the (a ₁,a ₂) and (a ₁,a ₃) phase plots, where thea _i's (i = 1, 2, 3) are the semi-diameters, and by the graphs ofa ₁,a ₂,a ₃, the volume, and the angular velocity as functions of time.

The dynamical behaviour of the nonequilibrium ellipsoid depends on the deviation of the angular momentum from its equilibrium value; for larger deviations, the oscillations are more nonperiodic with larger amplitudes.

An initially ellipsoidal configuration always remains ellipsoidal, but it cannot become spheroidal about its rotation axis, though it may become spheroidal instantaneously about either one of the other two principal axes.

For an ellipsoid approaching axisymmetry about its axis of rotation, the angular velocity can suddenly increase by a large amount. Thus if an astrophysical object can be modelled by a nonequilibrium ellipsoid, it may occasionally undergo sudden large increases of angular velocity.