Axisymmetric oscillations of magnetic neutron stars

We calculate axisymmetric oscillations of rotating neutron stars composed of the surface fluid ocean, solid crust, and fluid core, taking account of a dipole magnetic field as strong as $B_S\sim 10^{15}$G at the surface. The adiabatic oscillation equations for the solid crust threaded by a dipole magnetic field are derived in Newtonian dynamics, on the assumption that the axis of rotation is aligned with the magnetic axis so that perturbations on the equilibrium can be represented by series expansions in terms of spherical harmonic functions $Y_l^m(\theta,\phi)$ with different degrees $l$ for a given azimuthal wave number $m$ around the the magnetic axis. Although the three component models can support a rich variety of oscillation modes, axisymmetric ($m=0$) toroidal $_{l}t_n$ and spheroidal $_ls_n$ shear waves propagating in the solid crust are our main concerns, where $l$ and $n$ denote the harmonic degree and the radial order of the modes, respectively. In the absence of rotation, axisymmetric spheroidal and toroidal modes are completely decoupled, and we consider the effects of rotation on the oscillation modes only in the limit of slow rotation. We find that the oscillation frequencies of the fundamental toroidal torsional modes $_{l}t_n$ in the crust are hardly affected by the magnetic field as strong as $B_S\sim 10^{15}$G at the surface. As the radial order $n$ of the shear modes in the crust becomes higher, however, both spheroidal and toroidal modes become susceptible to the magnetic field and their frequencies in general get higher with increasing $B_S$. We also find that the surface $g$ modes and the crust/ocean interfacial modes are suppressed by a strong magnetic field, and that there appear magnetic modes in the presence of a strong magnetic field.