Stability of rotating liquid drops: II. Homogeneously charged or self-gravitating drops

The oscillations and instabilities of a uniformly rotating and homogeneously charged or self-gravitating liquid spheroid with surface tension are investigated. The dispersion relation is obtained by means of an appropriate energy integral better suited to yield the higher modes than the virial method. Criteria are derived for the onset of secular instability that indicate bifurcation toward nonrotational symmetric equilibrium configurations, and of dynamical stability. In the oblate case the validity of the spheroidal approximation can be verified both quantitatively and qualitatively by comparison with two limiting cases: an uncharged or nongravitating rotating drop and a rotating drop without surface tension (astronomical limit). For the prolate spheroids there exists a clear difference between rotating and nonrotating configurations, for the equilibrium as well as for the stability.