Where are the r-Modes of Isentropic Stars?

Almost none of the r-modes ordinarily found in rotating stars exist, if the star and its perturbations obey the same one-parameter equation of state; and rotating relativistic stars with one-parameter equations of state have no pure r-modes at all, no modes whose limit, for a star with zero angular velocity, is a perturbation with axial parity. Similarly (as we show here), rotating stars of this kind have no pure g-modes, no modes whose spherical limit is a perturbation 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 degenerate zero-frequency subspace. We find that rotation splits the degeneracy to zeroth order in the star's angular velocity Ω, and the resulting modes are generically hybrids, whose limit as Ω → 0 is a stationary current with axial and polar parts. Lindblom & Ipser have recently found these hybrid modes in an analytic study of the Maclaurin spheroids. Since the hybrid modes have a rotational restoring force, they call them "rotation modes" or "generalized r-modes." Because each mode has definite parity, its axial and polar parts have alternating values of l. We show that each mode belongs to one of two classes, axial-led or polar-led, depending on whether the spherical harmonic with lowest value of l that contributes to its velocity field is axial or polar. We numerically compute these modes for slowly rotating polytropes and for Maclaurin spheroids, using a straightforward method that appears to be novel and robust. Timescales for the gravitational-wave driven instability and for viscous damping are computed using assumptions appropriate to neutron stars. The instability to nonaxisymmetric modes is, as expected, dominated by the l = m r-modes with simplest radial dependence, the only modes which retain their axial character in isentropic models; for relativistic isentropic stars, these l = m modes must also be replaced by hybrids of the kind considered here.