Ohmic Decay of Magnetic Fields due to non-spherical accretion in the Crusts of Neutron Stars

We consider magnetic field evolution of neutron stars during polar-cap accretion. The size of the polar cap increases as the field decays, and is set by the last open field line before the accretion disk. Below the polar cap we find the temperature to be so high that electron-phonon scattering dominates the conductivity. Outside the polar cap region, the temperature is such the conductivity is dominated by temperature independent impurity scattering which can be a few orders of magnitude larger than the electron-phonon conductivity. The time-scale for field decay is therefore initially given by impurity scattering dominated conductivity. When the field strength has been reduced to ∼ 10 gauss the accretion is spherical and the time scale for field decay is given by the smaller electron-phonon scattering conductivity. The field strength is now reduced rapidly compared to before and this could be a reason for there being no pulsars known with field strengths below 10 gauss. We also investigate the evolution of multipoles at the neutron star surface. We find that contribution from higher-order multipoles are at most 30 % to that of the dipole mode.