Mon. Not. R. Astron. Soc., 453, 1404-1427 (2015/October-3)
On resonances in the pulsations of stars - II. Canonical perturbation theories.
Abstract (from CDS):
This is a study of stellar pulsations that are dominated by the non-linear interaction of a pair of nearly degenerate modes of infinitesimal pulsation. We describe two examples in which the equations that govern the adiabatic, non-linear pulsations of a star admit of Hamiltonian formulations, and we construct canonical perturbation theories for the solution of the canonical equations of motion in those examples. The primary example is a model of non-linear pulsations described in an earlier paper, in which we have represented the pulsations as quasi-homologous oscillations of a compressible, heterogeneous spheroid. The tensor virial equations of the second order and an equation representing an integral form of the first law of thermodynamics govern the pulsations of that model. The second example is a Hamiltonian representation of stellar pulsations of the kind originally formulated by J. Woltjer. In these examples the pulsations are quasi-periodic in two or more degrees of freedom. Two degrees of freedom characterize the non-linear interaction of the nearly degenerate modes of infinitesimal pulsation. The period of the motion in one of those degrees of freedom is a non-linear counterpart of the beat period of a superposition of the two nearly degenerate modes. It appears that episodes of this non-linear beat phenomenon must occur during the evolution of β Cephei stars.