Where are ther‐Modes of Isentropic Stars?

Almost none of the r-modes ordinarily found in rotating stars exist, if thestar and its perturbations obey the same one-parameter equation of state; androtating relativistic stars with one-parameter equations of state have no purer-modes at all, no modes whose limit, for a star with zero angular velocity, isa perturbation with axial parity. Similarly (as we show here) rotating stars ofthis kind have no pure g-modes, no modes whose spherical limit is aperturbation with polar parity and vanishing perturbed pressure and density.Where have these modes gone? In spherical stars of this kind, r-modes and g-modes form a degeneratezero-frequency subspace. We find that rotation splits the degeneracy to zerothorder in the star's angular velocity $\Omega$, and the resulting modes aregenerically hybrids, whose limit as $\Omega\to 0$ is a stationary current withaxial and polar parts. Because each mode has definite parity, its axial andpolar parts have alternating values of $l$. We show that each mode belongs toone of two classes, axial-led or polar-led, depending on whether the sphericalharmonic with lowest value of $l$ that contributes to its velocity field isaxial or polar. We numerically compute these modes for slowly rotatingpolytropes and for Maclaurin spheroids, using a straightforward method thatappears to be novel and robust. Timescales for the gravitational-wave driveninstability and for viscous damping are computed using assumptions appropriateto neutron stars.Comment: 54 pages including 4 ps tables and 11 PSTeX figures, AAS latex. To appear in the Astrophysical Journa