Force-Free Magnetosphere of an Accretion Disk -- Black Hole System. I. Schwarzschild Geometry

In this paper I study the magnetosphere of a black hole that is connected by the magnetic field to a thin conducting Keplerian disk. I consider the case of a Schwarzschild black hole only, leaving the more interesting but difficult case of a Kerr black hole to a future study. I assume that the magnetosphere is ideal, stationary, axisymmetric, and force-free. I pay a special attention to the two singular surfaces present in the system, i.e., the event horizon and the inner light cylinder; I use the regularity condition at the light cylinder to determine the poloidal electric current as a function of poloidal magnetic flux. I solve numerically the Grad--Shafranov equation, which governs the structure of the magnetosphere, for two cases: the case of a nonrotating disk and the case of a Keplerian disk. I find that, in both cases, the poloidal flux function on the horizon matches a simple analytical expression corresponding to a radial magnetic field that is uniform on the horizon. Using this result, I express the poloidal current as an explicit function of the flux and find a perfect agreement between this analytical expression and my numerical results.