Nonaxisymmetric Dynamic Instabilities of Rotating Polytropes. I. The Kelvin Modes

Abstract

We study the dynamic instabilities of rotating polytropes in the linear regime using an approximate Lagrangian technique and a more precise Eulerian scheme. We consider nonaxisymmetric modes with azimuthal dependence proportional to exp (imϕ), where m is an integer and ϕ is the azimuthal angle, for polytropes with a wide range of compressibilities and angular momentum distributions. We determine stability limits for the m = 2-4 modes and find the eigenvalue and eigenfunction of the most unstable m-mode for given equilibrium models. To the extent that we have explored parameter space, we find that the onset of instability is not very sensitive to the compressibility or angular momentum distribution of the polytrope when the models are parameterized by T/| W |. Here T is the rotational kinetic energy, and W is the gravitational energy of the polytrope. The m = 2, 3, and 4 modes become unstable at T/| W | ≈ 0.26-0.28, 0.29-0.32, and 0.32-0.35, respectively, limits consistent with those of the Maclaurin spheroids to within ±0.015 in T/| W |. The only exception to this occurs for the most compressible polytrope we test and then only for m = 4, where instability sets in at T/| W | ≈ 0.37-0.39.