Numerical models of force-free black-hole magnetospheres

This paper develops numerical models of stationary, axisymmetric, force-free black hole magnetospheres, based on the theory originally developed by Blandford & Znajek and reformulated and extended by Macdonald & Thorne. The structure of such a magnetosphere is determined by a single scalar ‘stream function’ satisfying a nonlinear, second-order partial differential ‘stream equation’ on a region bounded by the black hole horizon, the accretion disk, and an outer boundary beyond which the force-free condition breaks down. The stream equation is solved numerically, using an iterative relaxation method, for three different poloidal magnetic field configurations: (1) (roughly) radial magnetic field; (2) (roughly) uniform magnetic field; and (3) (roughly) paraboloidal magnetic field. The second and third cases also include a force-free gap between the inner edge of the disk and the horizon, with which the horizon may exchange magnetic flux. For the chosen boundary conditions, it is found that the poloidal field structure does not change greatly as the hole and the field are spun up.