The Oscillations of a Rotating Star.

Rayleigh's principle is generalized to give approximately the periods of oscillation of a rotating star when those of the corresponding nonrotating star are known. Ledoux's formula for the fundamental period of radial oscillation follows from ours if his approximations are made. More exact numerical results for the fundamental period are obtained in certain cases, and the error involved by making Ledoux's approximations is shown to be not small. Formulae applying to oscillations of the overtone radial and nonradial types are also derived. I In 1941, Pekeris and Ledoux' applied Rayleigh's principle to derive approximate values for the fundamental period of radial oscillation of a nonrotating star. They showed that quite good approximations to the period could be obtained by assuming simple formulae for the displacement during the oscillation; in many cases an adequate approximation is found by assuming a dilation and compression which are uniform throughout the star. Some of their results were later derived independently by Bhatna- gar.2 Ledoux and others3 have also considered the "radial" oscillations of rotating stars (i.e., oscillations such that, in the absence of rotation, the displacement would be wholly radial). Their methods did not depend on Rayleigh's principle, which requires generaliza- tion before application to rotating systems.4 Ledoux's methods were based on certain other general principles of dynamics; in applying them he again assumed a uniform dila- tion and compression, a little modified by the effect of rotation. Other workers operated by less accurate methods. In this paper Rayleigh's principle will be applied to the oscillations of rotating stars. The discussion is not limited to "radial" oscillations, though numerical applications are, perforce, made oniy for these. Ledoux's results obtained by assuming uniform dilation are shown to be identical with those obtained by Rayleigh's principle, with a similar assumption. The error involved in making such an assumption is, however, found to be larger in the rotating than in the nonrotating case