COUPLED-MODE EQUATIONS AND AMPLITUDE EQUATIONS FOR NONADIABATIC, NONRADIAL OSCILLATIONS OF STARS

A formalism is developed for the study of nonlinear, nonadiabatic, nonradial oscillations of stars. It is shown that the perturbed equation of motion and the perturbed energy equation can be expressed as a system of first-order differential equations for the Lagrangian displacement, the velocity, and the Lagrangian perturbation of specific entropy as the only dependent variables. Coupled-mode equations are then derived, which govern the time-behaviour of the amplitudes of the linear oscillation modes and take into account the nonlinear couplings with other modes. The coupled-mode equations are derived up to the third order in the amplitudes. The slow modulations of the amplitudes of the linear modes caused by the nonlinear coupling terms in the coupled-mode equations are studied by applying the multiple time scales perturbation method. Amplitude equations governing the time-behaviour on longer time scales of the amplitude of a single nonresonant, nonadiabatic, nonradiaI oscillation mode and the amplitudes of two nonadiabatic, nonradial modes with nearly equal frequencies are derived. By means of amplitude equations, amplitudes and frequencies of stellar oscillations can be determined.