Surface r-modes and burst oscillations of neutron stars

We study the r-modes propagating in steadily mass accreting, nuclear burning, and geometrically thin envelopes on the surface of rotating neutron stars. For the modal analysis, we construct envelope models that are fully radiative or have a convective region. We simply call the former radiative models and the latter convective models in this paper. As the angular rotation frequency Omega is increased, the oscillation frequency omega of the r-modes in the thin envelopes deviates appreciably from the asymptotic frequency omega = 2mOmega/l'(l' + 1) defined in the limit of Omega --> 0, where omega is the frequency observed in the corotating frame of the star, and m and l 0 are the indices of the spherical harmonic function Y-l'(m) representing the angular dependence of the modes. We find that the amplitudes of the fundamental r-modes with no radial nodes of the eigenfunctions are strongly confined to the equatorial region, and omega becomes only weakly dependent on Omega, gathering in a frequency range of omega/2pi less than or similar to 10 Hz, at rapid rotation rates. We also find that the fundamental r-modes in the convective models are destabilized by strong nuclear burning in the convective region. Because of excessive heating by nuclear burning, the corotating-frame oscillation frequency omega of the r-modes in the convective models becomes larger, and hence the inertial-frame oscillation frequency \sigma\ becomes smaller than those of the corresponding r-modes in the radiative models, where sigma = omega -mOmega is negative for the r-modes of positive m. We find that the relative frequency change f = -(sigma(conv) - sigma(rad))/sigma(rad) is always positive and becomes less than similar to0.01 for the fundamental r-modes of l' > \m\ + 1 at \sigma(rad)\/2pi similar to 300 Hz for m = 1 or at \sigma(rad)\/2pi similar to 600 Hz for m = 2, and that we need to consider the r-modes of l' much larger than \m\ for values of f as small as similar to0.001, where sigma(conv) and sigma(rad) denote the oscillation frequencies for the convective and the radiative envelope models, respectively.