Rosette Modes of Oscillations of Rotating Stars Caused by Close Degeneracies. I. Axisymmetric Modes

Abstract

Abstract We present an analysis of unique axisymmetric eigenmodes of oscillations in rigidly rotating stars, which have been recently discovered numerically in the frequency region of low-order gravity modes of the polytropic model with index 3, and named as rosette modes after their structure of the eigenfunctions. We show that the appearance of rosette modes in the presence of slow rotation is caused by a close degeneracy in the frequency among several eigenmodes in the non-rotating case. Regarding the effect of the Coriolis force on stellar oscillations as a small perturbation, and applying quasi-degenerate perturbation theory, we can successfully reproduce the structure of rosette modes as a linear combination of several unperturbed eigenmodes, which have almost the same frequency and successive spherical degrees of the same parity. We confirm that the eigenfrequencies and the structure of the eigenfunctions that are determined by the perturbative technique are consistent with the results of more direct numerical computations, which are based on the decomposition of the eigenfunctions into a series of spherical harmonics. We discuss the characteristics of rosette modes in detail.