R-modes of a neutron star with a magnetic dipole field

We study r-modes of a rotating magnetized neutron star, assuming a magnetic dipole field whose axis is aligned with the axis of rotation. We approach the problem by applying a singular perturbation theory to the oscillations of rotating stars. In this treatment, we divide the star into a thin surface magnetic layer and a non-magnetic core. We integrate linearized ideal magnetohydrodynamic (MHD) equations in the surface magnetic layer and non-magnetic oscillation equations in the core, and match the two integrations at the interface to obtain a complete solution. For a polytropic neutron star model of mass M = 1.4 M
⊙
and radius R = 10
⁶
cm, the magnetic dipole field becomes effective on the modal properties of the r-modes only when the field strength B
S
is much greater than 10
¹⁴
G. We also find that the damping effects caused by very short magnetic perturbations in the surface layer are not important for the r-mode instability of rapidly rotating neutron stars if the field strength B
S
is smaller than 10
¹²
G.