Non-radial oscillations of the magnetized rotating stars with purely toroidal magnetic fields

We calculate non-axisymmetric oscillations of uniformly rotating polytropes magnetized with a purely toroidal magnetic field, taking account of the effects of the deformation due to the magnetic field. As for rotation, we consider only the effects of Coriolis force on the oscillation modes, ignoring those of the centrifugal force, that is, of the rotational deformation of the star. Since separation of variables is not possible for the oscillation of rotating magnetized stars, we employ finite series expansions for the perturbations using spherical harmonic functions. We calculate magnetically modified normal modes such as $g$-, $f$-, $p$-, $r$-, and inertial modes. In the lowest order, the frequency shifts produced by the magnetic field scale with the square of the characteristic Alfv\'en frequency. As a measure of the effects of the magnetic field, we calculate the proportionality constant for the frequency shifts for various oscillation modes. We find that the effects of the deformation are significant for high frequency modes such as $f$- and $p$-modes but unimportant for low frequency modes such as $g$-, $r$-, and inertial modes.