Relativistic r-Modes in Slowly Rotating Neutron Stars: Numerical Analysis in the Cowling Approximation

We investigate the properties of relativistic r-modes of slowly rotating neutron stars by using a relativistic version of the Cowling approximation. In our formalism, we take into account the influence of the Coriolis-like force on the stellar oscillations but ignore the effects of the centrifugal-like force. For three neutron star models, we calculate the fundamental r-modes with l^'=m=2 and 3. We find that the oscillation frequency σ of the fundamental r-mode is given in a good approximation by σ~κ₀Ω, where σ is defined in the corotating frame at spatial infinity and Ω is the angular frequency of rotation of the star. The proportional coefficient κ₀ is only weakly dependent on Ω, but it strongly depends on the relativistic parameter GM/c²R, where M and R are the mass and the radius of the star. All the fundamental r-modes with l^'=m computed in this study are discrete modes with distinct regular eigenfunctions, and they all fall in the continuous part of the frequency spectrum associated with Kojima's equation. These relativistic r-modes are obtained by including the effects of rotation higher than the first order of Ω so that the buoyant force plays a role, the situation of which is quite similar to that for Newtonian r-modes.