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 σ̄ of the fundamental r-mode is given in a good approximation by σ̄ K ₀Ω, where σ̄ is defined in the corotating frame at spatial infinity and Ω is the angular frequency of rotation of the star. The proportional coefficient K₀ 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 eigen-functions, 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.