Classification of the Spheroidal Normal Modes

Abstract

Many spheroidal normal modes of stars are standing waves resulting from interferences of running waves that travel in opposite directions. Radial modes originate from interferences of acoustic waves that are associated with the degree ℓ equal to zero and travel to-and-fro in the A-cavity with a frequency fitting to the dimensions of the cavity. One gets an idea of the spectrum of the fitting frequencies by treating the eigenvalue problem of the radial oscillations as a Sturm–Liouville eigenvalue problem with singular end points. Moreover, the interferences of acoustic waves and internal gravity waves that are associated with a degree ℓ different from zero and travel to-and-fro in the A-or G-cavity lead, for certain frequencies, to standing waves resulting in non-radial modes. Insight into the types of spectra of eigenfrequencies has been provided by an approach of Cowling. Modes arising from interferences of acoustic waves are called p-modes, and those arising from interferences of internal gravity waves, g ⁺-modes. In stars that contain a convectively unstable region, g ⁻-modes appear, which render the global reactions of the star to the local convective instabilities. Besides the p-, g ⁺-, and g ⁻-modes, non-radial f-modes exist, whose origin is related to surface waves, at least for higher degrees.