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Energetics of Black Hole-Accretion Disk System with Magnetic Connection: Limit of Low Accretion Rate

We study the energetics of a black hole-accretion disk system with magnetic connection: a Keplerian disk is connected to a Kerr black hole by a large-scale magnetic field going through the transition region. We assume that the magnetic field is locked to the inner boundary of the disk and corotates with the inner boundary, the accretion rate is low but the accretion from the disk can still provide enough amount of cold plasma particles in the transition region so that the magnetohydrodynamics approximation is valid. Then, the magnetic field is dynamically important in the transition region and affects the transportation of energy and angular momentum. Close to the equatorial plane, the motion of particles is governed by a one-dimensional radial momentum equation, which contains a fast critical point as the only intrinsic singularity. By finding solutions that smoothly pass the fast critical point, we find that a system with a fast rotating black hole and that with a slow rotating black hole behave very differently. For a black hole with $a > a_{\rm cr}\equiv 0.3594 M$, where $M$ is the mass, $a$ the specific angular momentum of the black hole, the spinning energy of the black hole is efficiently extracted by the magnetic field and transported to the disk, increasing the radiation efficiency of the disk by many orders of magnitude. For a black hole with $0\leq a < a_{\rm cr}$, the inner region of the disk is disrupted by the magnetic field and the inner boundary of the disk moves out to a radius where the Keplerian angular velocity of the disk is equal to the spinning angular velocity of the black hole (which is at infinity if the black hole is nonrotating). As a result, the disk may have an extremely low radiation efficiency if $0\leq a/M \ll 1$.