On the oscillations and stability of rapidly rotating stellar models. III. Zero-viscosity polytropic sequences

The frequencies of oscillation of the lowest "radial" and nonradial modes are calculated for differentially rotating, axisymmetric, stellar models along the generalized polytropic sequences discussed in the preceding paper. The stability of the models to nonaxisymmetric perturbations is discussed and compared with the properties of the classical Maclaurin spheroids. As in the case of the Maclaurin sequence, the models become neutrally stable to a nonaxisymmetric mode when t =- IT/WI, the ratio of kinetic to potential energy, reaches 0.14. 13eyond this point the models are expected to be secularly unstable if dissipative processes are present. Also, when t > 0.25, the models are dynamically overstable to the same mode, which transforms them into triaxial configurations. It is particularly significant that these critical values of t are found to be practically independent of polytropic index and distribution of angular momentum. The critical y (below which stars are unstable to the pulsation mode) declines from 4/3 to approximately 1.23 for models near the point of bifurcation for n = 1.5, and to 1.28 for n = 3. Some astrophysical consequences of the numerical results are discussed, with applications made to the problem of fission, the stability of close binaries, the possible existence of triaxial stars, and the stability of supermassive stars. Subject headings: instabilities - pulsation - rotation