Axisymmetric toroidal modes of magnetized neutron stars

We calculate axisymmetric toroidal modes of magnetized neutron stars with a solid crust. We assume the interior of the star is threaded by a poloidal magnetic field that is continuous at the surface with the outside dipole field whose strength $B_p$ at the magnetic pole is $B_p\sim 10^{16}$G. Since separation of variables is not possible for oscillations of magnetized stars, we employ finite series expansions of the perturbations using spherical harmonic functions to represent the angular dependence of the oscillation modes. For $B_p\sim 10^{16}$G, we find distinct mode sequences, in each of which the oscillation frequency of the toroidal mode slowly increases as the number of radial nodes of the eigenfunction increases. The frequency spectrum of the toroidal modes for $B_p\sim 10^{16}$G is largely different from that of the crustal toroidal modes of the non-magnetized model, although the frequency ranges are overlapped each other. This suggests that an interpretation of the observed QPOs based on the magnetic toroidal modes may be possible if the field strength of the star is as strong as $B_p\sim 10^{16}$G.