Computation of the Adiabatic Eigenmodes of Stars Using the Riccati Method

Abstract

Abstract In stellar oscillation problems, it is often necessary to compute a large number of stellar eigenmodes with wildly varying radial orders and angular degrees. Conventionally the relaxation method is used for this purpose. This method is, however, fraught with several numerical issues, which make it cumbersome and tedious to write the corresponding computer programs. For this reason we adopt the Riccati method as an alternative, in which the original system of linear equations is transformed into a nonlinear matrix Riccati equation. The dependent variable of the converted equation does not retain the dependent variables of the original linear equations themselves, but keeps the relations among them. This method is particularly useful when we compute eigenmodes, where the amplitudes of the eigenfunctions vary by many orders of magnitude, since the ratios of the amplitudes generally show much less variation than the amplitudes themselves. The Riccati method was successfully applied to the problems of nonadiabatic stellar oscillations in previous studies. Our aim is to do the same for the adiabatic case.