Diamagnetic screening of the magnetic field in accreting neutron stars — II. The effect of polar cap widening

Recently, we have proposed a model for the screening of the magnetic field of an accreting neutron star by the accreted material flowing from the polar regions towards the equator and sinking there underneath the surface. In this earlier model, it was assumed that the flow pattern remained stationary over time. However, as the surface magnetic field weakens, the accretion takes place over a wider region around the pole, making the flow more radial and isotropic. In the present work, we extend this two-dimensional model to include the time dependence of the flow of the accreted material. The final radial flow is found to be less efficient in screening the magnetic field compared with the initial tangential flow. After an initial phase of rapid decay, the magnetic field slowly reaches an asymptotic value when the accretion becomes nearly isotropic and radial. Assuming the initial extent of the polar cap to be ∼5°–10°, a simple geometric argument suggests that the magnetic field should decay by 3–4 orders of magnitude before stabilizing to an asymptotic value, consistent with the magnetic fields observed in millisecond pulsars.