Screening of the Magnetic Field of Disk Accreting Stars

An analytical model is developed for the screening of the external magnetic field of a rotating, axisymmetric neutron star due to the accretion of plasma from a disk. The decrease of the field occurs due to the electric current in the infalling plasma. The deposition of this current carrying plasma on the star's surface creates an induced magnetic moment with a sign opposite to that of the original magnetic dipole. The field decreases independent of whether the star spins-up or spins-down. The time-scale for an appreciable decrease (factor of $>100$) of the field is found to be $\sim 1.6 \times 10^7$ yr, for a mass accretion rate $\dot{M}=10^{-9} M_\odot/$yr and an initial magnetic moment $\mu_i = 10^{30}{\rm G cm}^3$ which corresponds to a surface field of $10^{12} $G if the star's radius is $10^6$ cm. The time-scale varies approximately as $\mu_i^{3.8}/\dot{M}^{1.9}$. The decrease of the magnetic field does not have a simple relation to the accreted mass. Once the accretion stops the field leaks out on an Ohmic diffusion time scale which is estimated to be $ > 10^9$ yr.