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 ∼ 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 ∼ 10¹⁶ G, we find discrete normal toroidal modes which form 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 ∼ 10¹⁶ 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 quasi-periodic oscillations (QPOs) based on the magnetic toroidal modes may be possible if the field strength of the star is as strong as B_p ∼ 10¹⁶ G.