NONAXISYMMETRIC DYNAMIC INSTABILITIES OF ROTATING POLYTROPES - I - THE KELVIN MODES

We study the dynamic instabilities of rotating polytropes in the linea r regime using an approximate Lagrangian technique and a more precise Eulerian scheme. We consider nonaxisymmetric modes with azimuthal depe ndence proportional to exp (im phi), where m is an integer and phi is the azimuthal angle, for polytropes with a wide range of compressibili ties and angular momentum distributions. We determine stability limits for the m = 2-4 modes and find the eigenvalue and eigenfunction of th e most unstable m-mode for given equilibrium models. To the extent tha t we have explored parameter space, we find that the onset of instabil ity is not very sensitive to the compressibility or angular momentum d istribution 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\ approximate to 0.26-0.28, 0.29-0.32, and 0.32-0.35, respectivel y, limits consistent with those of the Maclaurin spheroids to within /- 0.015 in T/\W\. The only exception to this occurs for the most comp ressible polytrope we test and then only for m = 4, where instability sets in at T/\W\ approximate to 0.37-0.39. The eigenfunctions for the fastest growing low in-modes are similar to those of the Maclaurin sph eroid eigenfunctions in that they do not show large vertical motions, are only weakly dependent on z, and increase strongly in amplitude as the equatorial radius of the spheroid is approached. The polytrope eig enfunctions are, however, qualitatively different from the Maclaurin e igenfunctions in one respect: they develop strong spiral arms. The spi ral arms are stronger for more compressible polytropes and for polytro pes whose angular momentum distributions deviate significantly from th ose of the Maclaurin spheroids. Nevertheless, our approximate Lagrangi an method, which explicitly assumes nonspiral Maclaurin-like trial fun ctions, yields reasonable estimates for the pattern periods and e-fold ing times of unstable m = 2 modes even for highly compressible and str ongly differentially rotating polytropes. Comparisons for m = 2 betwee n the linear analyses in this paper and nonlinear hydrodynamic simulat ions give excellent quantitative agreement in eigenfunctions, pattern speeds, and e-folding times for the dynamically unstable modes.