The Inner Boundary Condition for a Thin Disk Accreting Into a Black Hole

Contrary to some recent claims the 'no torque inner boundary condition' as applied at the marginally stable orbit is correct for geometrically thin disks accreting into black holes. A transition from a thin accretion disk to a stream freely falling into a black hole was a topic of many papers about two decades ago. That ancient work is very well reviewed in the introduction of Abramowicz & Kato (1989). It was well established that there is a transition from a subsonic radial accretion flow in the nearly Keplerian disk to a transonic flow near the marginally stable orbit at the radius r in ≈ r ms , and a free fall into the central black hole for r < r in. The free fall proceeds with a conservation of angular momentum, hence the streamlines are spiral. It was shown that the 'no torque inner boundary condition' is an excellent approximation at r in. The reason was simple: no information could propagate upstream in the supersonic region inwards of r in. and the many references therein. They claim that magnetic torques provide a strong interaction between the stream and the disk, and generate a strong torque at r in. This is a puzzling result, as the flow is transonic near r in according to KGA, and the radial infall becomes supersonic inwards of r in. The subsonic accretion for r > r in is not discussed by KGA, but their disks seem to be described by more or less standard 'alpha models', with the α parameter due to tangled magnetic fields, following the work of Balbus & Hawley (1998). While the differential rotation winds up the magnetic field lines and perhaps gives rise to a dynamo, the magnetic energy density is kept in equilibrium, at the level roughly α times the gas and/or radiation