ON THE ROLE OF RESONANCES IN NONRADIAL PULSATORS

Resonances or near resonances are ubiquitous among the excited nonradi al pulsation modes of variable stars and they must play an important r ole in determining their pulsational behavior. Here in a first step at nonlinear asteroseismology we explore some of the consequences of res onances by means of the amplitude equation formalism. We show how pari ty and angular momentum constraints can be used to eliminate many of t he possible nonlinear resonant couplings between modes (and multiplets bf modes), and how the amplitude equations can thus be simplified. Ev en when we may not be able, nor wish, to make an ab initio computation of the values of the coupling coefficients, it is still possible to o btain constraints on the nature of the excited modes if a resonance be tween observed frequencies can be identified. Resonances can cause non linear frequency locking of modes. This means that the observed freque ncies appear in exact resonance even though the linear frequencies are only approximately in resonance. The nonlinear frequency lock, when i t occurs, it does so over a range of departures from linear resonance, and it is accompanied by constant pulsation amplitudes. The locked, n onlinear frequencies can differ noticeably from their nonresonant coun terparts which are usually used in seismology. This is particularly tr ue for multiplets of modes split by rotation. Beyond the regime of the frequency lock, ampli tude and frequency modulations can appear in th e pulsations. Far from the resonance condition one recovers the regime of steady pulsations with nonresonant frequencies for which the seism ological studies, as they are presently carried out, are justified (pr ovided furthermore, of course, that nonlinear frequency shifts are neg ligible). Success in identifying-a resonance in an observed power spec trum depends on the quality of the data. While keeping this limitation in mind, we discuss the possible existence of peculiar resonances in the pulsations of specific variable white dwarfs and delta Scuti stars .