Force-Free Magnetosphere of an Accretion Disk -- Black Hole System. II. Kerr Geometry

We consider a stationary axisymmetric force-free degenerate magnetosphere of a rotating Kerr black hole surrounded by a thin Keplerian infinitely-conducting accretion disk. We focus on the closed-field geometry characterize by a direct magnetic coupling between the disk and the hole's event horizon. We first argue that the hole's rotation necessarily limits the radial extent of the force-free link on the disk surface: the faster the hole rotates, the smaller the magnetically-connected inner part of the disk has to be. We then show that this is indeed the case by solving numerically the Grad--Shafranov equation--the main differential equation describing the structure of the magnetosphere. An important element in our approach is the use of the regularity condition at the inner light cylinder to fix the poloidal current as a function of the poloidal magnetic flux. As an outcome of our computations, we are able to chart out the maximum allowable size of the portion of the disk that is magnetically connected to the hole as a function of the black hole spin. We also calculate the angular momentum and energy transfer between the hole and the disk that takes place via the direct magnetic link. We find that both of these quantities grow rapidly and that their deposition becomes highly concentrated near the inner edge of the disk as the black hole spin is increased.