Modal Analysis of Stellar Nonradial Oscillations by an Asymptotic Method

Modal property of stellar nonradial oscillations is investigated mathematically by using an asymptotic method. Basic equations for linear adiabatic nonradial oscillations are written in the form of a pair of turning-point equations. They are solved in terms of Airy functions in the region containing only one turning point, and eigenfunctions are obtained by a mathematical method similar to that used in the potential- well problem in quantum mechanics. Eigenvalue conditions for simple gravity modes, simple acoustic modes, and modes with the dual character are obtained; they are expressed in forms similar to Bohr-Sommerfeld's quantization rule in the cases of simple modes. It is confirmed that dual character modes can clearly be classified into two types of pseudo- modes by their main trapped regions. The mode mixing is then shown to occur when a resonance condition is met. It is shown that the mode mixing does not lead to the degeneracy of eigenvalues but it results in the "avoided crossing." The effect of wave leakage upon the temporal damping rate is also formulated for eigenmodes having the progressive-wave character at one of the boundaries.