Accretion Disk Torqued by a Black Hole

Abstract

If a Kerr black hole is connected to a disk rotating around it by a magnetic field, the rotational energy of the Kerr black hole provides an energy source for the radiation of the disk in addition to disk accretion. The black hole exerts a torque on the disk, which transfers energy and angular momentum between the black hole and the disk. If the black hole rotates faster than the disk, energy and angular momentum are extracted from the black hole and transferred to the disk. The energy deposited into the disk is eventually radiated away by the disk, which will increase the efficiency of the disk. If the black hole rotates slower than the disk, energy and angular momentum are transferred from the disk to the black hole, which will lower the efficiency of the disk. With suitable boundary conditions, quasi-steady state solutions are obtained for a thin Keplerian disk magnetically coupled to a Kerr black hole. By "quasi-steady state" we mean that any macroscopic quantity at a given radius in the disk slowly changes with time: the integrated change within one rotation period of the disk is much smaller than the quantity itself. We find that the torque produced by magnetic coupling propagates outward only in the disk, and the total radiation flux of the disk is a superposition of the radiation flux produced by magnetic coupling and that produced by accretion. Interestingly, a disk magnetically coupled to a rapidly rotating black hole can radiate without accretion: the total power of the disk comes from the rotational energy of the black hole. With a simple example that the magnetic field touches the disk at a single radius, we show that the radial radiation profile produced by magnetic coupling can be very different from that of a standard accretion disk: the emissivity index is significantly larger, and most radiation can come from a region that is closer to the center of the disk. While the shape of the radiation flux curve sensitively depends on the extension of the magnetic field in the disk, the spectral signature of magnetic coupling can be robust. The limitations of our model are briefly discussed, which include the assumption of a weak magnetic field, the ignorance of the instabilities of the disk and the magnetic field, and the ignorance of the radiation captured by the black hole and the radiation returning to the disk.