Magnetic angle evolution in accreting neutron stars

The rotation of a magnetised accreting neutron star (NS) in a binary system is described by its spin period and two angles: spin inclination $\alpha$ with respect to the orbital momentum and magnetic angle $\chi$ between the spin and the magnetic moment. Magnetospheric accretion spins the NS up and adjusts its rotation axis, decreasing $\alpha$ to nearly perfect alignment. Its effect upon the magnetic angle is more subtle and relatively unstudied. In this work, we model the magnetic angle evolution of a rigid spherical accreting NS. We find that the torque spinning the NS up may affect the magnetic angle while both $\alpha$ and $\chi$ significantly deviate from zero, and the spin-up torque varies with the phase of the spin period. As the rotation axis of the NS is being aligned with the spin-up torque, the magnetic axis becomes misaligned with the rotation axis. Under favourable conditions, magnetic angle may increase by $\Delta \chi \sim 15^\circ-20^\circ$. This orthogonalisation may be an important factor in the evolution of millisecond pulsars, as it partially compensates the $\chi$ decrease potentially caused by pulsar torques. If the direction of the spin-up torque changes randomly with time, as in wind-fed high-mass X-ray binaries, both the rotation axis of the NS and its magnetic axis become involved in a non-linear random-walk evolution. The ultimate attractor of this process is a bimodal distribution in $\chi$ peaking at $\chi =0^\circ$ and $\chi = 90^\circ$.