Improved Asymptotic Expressions for the Eigenvalues of Laplace's Tidal Equations

TL;DR

This paper derives improved asymptotic formulas for eigenvalues of Laplace's tidal equations in rotating stars, enhancing accuracy for large spin parameters and aiding the analysis of stellar oscillation modes.

Contribution

It introduces new asymptotic expressions for eigenvalues of Laplace's tidal equations, valid at large spin parameters, with improved accuracy over previous models.

Findings

Eigenvalue expressions have relative accuracy of order q^{-3} for gravito-inertial modes.

Expressions are accurate to order q^{-1} for Rossby and Kelvin modes.

Validated formulas can be used to derive periods and eigenfunctions of Rossby modes.

Abstract

Laplace's tidal equations govern the angular dependence of oscillations in stars when uniform rotation is treated within the so-called traditional approximation. Using a perturbation expansion approach, I derive improved expressions for the eigenvalue associated with these equations, valid in the asymptotic limit of large spin parameter $q$. These expressions have a relative accuracy of order $q^{-3}$ for gravito-inertial modes, and $q^{-1}$ for Rossby and Kelvin modes; the corresponding absolute accuracy is of order $q^{-1}$ for all three mode types. I validate my analysis against numerical calculations, and demonstrate how it can be applied to derive formulae for the periods and eigenfunctions of Rossby modes.