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 $n\sim1$ 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 $10^{14}$G to $10^{16}$G, we calculate axisymmetric spheroidal magnetic modes whose oscillation frequency is proportional to $B_S$. The typical oscillation frequency of the magnetic modes is $\sim 0.01\times\sqrt{GM/R^3}$ for $B_S\sim 10^{15}$G, where $M$ and $R$ are respectively the mass and radius of the star and $G$ is the gravitational constant. For $M=1.4M_\odot$ and $R=10^6$cm, this frequency is $\sim 20$Hz, which may explain low frequency QPOs found for SGR 1806-204 and SGR 1900+14. We also find modes of frequency $> \sqrt{GM/R^3}$ corresponding to the radial fundamental and first harmonic modes. No unstable magnetic modes are found for axisymmetric spheroidal oscillations of magnetized stars.