Magnetic field evolution of accreting neutron stars

We study the evolution of the magnetic field of an accreting neutron star in the frozen field and incompressible fluid approximations. The plasma is accreted onto two polar caps and squeezes some of the surface material of the neutron star toward the equator. The frozen B-field is then pushed toward the equator and is eventually buried there. The magnetic field within the polar cap areas, which is defined by the Alfvèn radius, decreases due to the expansion of the polar cap areas resulting from the physical motion of the accreted material, which conserves the magnetic flux. But the decrease of the magnetic field also changes the Alfvèn radius which modifies the size of the polar cap and also affects the decrease of the magnetic flux within the polar caps. Therefore, the magnetic field enclosed by the polar caps appears to decay rapidly with a time scale of ∼ 105 mB/10−3M Ṁ/(1018gs−1) years. As a consequence the magnetic field outside the polar cap is increasing because the total flux of the entire stellar surface is conserved in our approximations. The decrease of the polar cap magnetic field will stop and reach a minimum value ∼ 108G when the magnetic field outside the polar cap reaches Bout ∼ 1015G, which is strong enough to stop the motion of the accretion material across the stellar surface. However, this strong Bout cannot be observed because the accreted matter stopped by this strong field cannot move toward the equator. Instead it moves inward and pulls this field inside the crust with a time scale ∼ 10H5R 6ρ14Ṁ 18 yr. Pulsars accreting similar masses but having very different magnetic field may result from different equations of state.