Non-axisymmetric low frequency oscillations of rotating and magnetized neutron stars

We investigate non-axisymmetric low frequency modes of a rotating and magnetized neutron star, assuming that the star is threaded by a dipole magnetic field whose strength at the stellar surface, $B_0$, is less than $\sim 10^{12}$G, and whose magnetic axis is aligned with the rotation axis. For modal analysis, we use a neutron star model composed of a fluid ocean, a solid crust, and a fluid core, where we treat the core as being non-magnetic assuming that the magnetic pressure is much smaller than the gas pressure in the core. Here, we are interested in low frequency modes of a rotating and magnetized neutron star whose oscillation frequencies are similar to those of toroidal crust modes of low spherical harmonic degree and low radial order. For a magnetic field of $B_0\sim 10^7$G, we find Alfv\'en waves in the ocean have similar frequencies to the toroidal crust modes, and we find no $r$-modes confined in the ocean for this strength of the field. We calculate the toroidal crustal modes, the interfacial modes peaking at the crust/core interface, and the core inertial modes and $r$-modes, and all these modes are found to be insensitive to the magnetic field of strength $B_0\ltsim10^{12}$G. We find the displacement vector of the core $l^\prime=|m|$ $r$-modes have large amplitudes around the rotation axis at the stellar surface even in the presence of a surface magnetic field $B_0\sim10^{10}$G, where $l^\prime$ and $m$ are the spherical harmonic degree and the azimuthal wave number of the $r$-modes, respectively. We suggest that millisecond X-ray variations of accretion powered X-ray millisecond pulsars can be used as a probe into the core $r$-modes destabilized by gravitational wave radiation.