Axisymmetric spheroidal modes of neutron stars magnetized with poloidal magnetic fields

We calculate axisymmetric spheroidal modes of neutron stars magnetized with an axisymmetric poloidal magnetic field. We use polytropes of the indices n similar to 1 as background equilibrium models of neutron stars where we ignore the deformation due to the magnetic fields. For a poloidal magnetic field, axisymmetric normal modes of non-rotating stars are decoupled into spheroidal modes and toroidal modes, and we can treat spheroidal modes separately from toroidal modes. For the surface field strength B-S ranging from 10(14) to 10(16) G, we calculate axisymmetric spheroidal magnetic modes whose oscillation frequency is proportional to BS. The typical oscillation frequency of the magnetic modes is similar to 0.01 x root GM/R-3 for B-S similar to 10(15) G, where M and R are, respectively, the mass and radius of the star and G is the gravitational constant. For M = 1.4 M-circle dot and R = 10(6) cm, this frequency is similar to 20 Hz, which may explain low-frequency quasi-periodic oscillations found for SGR 1806-204 and SGR 1900+14. We also find modes of frequency greater than or similar to root GM/R-3 corresponding to the radial fundamental and first harmonic modes. No unstable magnetic modes are found for axisymmetric spheroidal oscillations of magnetized stars.