1. Effects of Magnetic Coupling on Radiation from Accretion Disc around a Kerr Black Hole
Zhao-Ming Gan, Ding-Xiong Wang and Yang Li
Accepted by MNRAS, astro-ph/

2. Outline The following issues are involved
The “no torque boundary condition” (NTBC) of a standard accretion disk (SAD),
The choice for the inner boundary of SAD ,
The effect of magnetic coupling (MC) on the location of the inner boundary of a thin Keplerian disk,
The effects of MC on the disc radiation .

3. Assumptions
The disc is thin, perfectly conducting and Keplerian, lying in the equatorial plane of a Kerr black hole (BH),
The large scale magnetic field remains constant at the BH horizon, while it varies as a power-law with the disc radius,
No torque is exerted at the inner edge of a thin disc in the MC with a central BH.

4. The expressions for the radiation flux, interior viscous torque and total luminosity of a relativistic thin keplerian disk are (Page & Thorne 1974, Li L.-X.2002 )
where rin is the radius of the inner boundary of the disk, and gin is the extra torque exerted at the inner boundary
Noting that for any realistic case, the radiation of the disk should be always positive.

5. “No Torque Boundary Condition”
It always supposes the “no torque boundary condition” for SAD, which corresponds to
Anyway, from equation (1) and (3) we can see that the boundary torque behaves equivalently as an extra energy source for the accreting matter, which would significantly change the distribution of the radiation and the interior viscous torque of the disk (Agol 2000).

6. Innermost Stable Circular Orbit
It is generally assumed in SAD that the inner boundary of the disk locates at ISCO, where the specific energy and angular momentum attain their minima (Bardeen 1972), i.e.,
What would happen if the inner boundary deviates from ISCO ?

7. We conclude that SAD with an inner edge within ISCO is not stable, while any radius larger than ISCO is possible. It is worth noting that ISCO is not the unique choice for the inner edge of a standard thin disc, only being the smallest radius among all of the possible ones.

8. By virtue of the magnetic field connecting the rotating BH and the disc, energy and angular momentum are transferred between the BH and the disc, just like the energy transportation between a dynamo and a motor (Macdonald & Thorne 1982, Thorne, Price & MacDonald 1986) .

9. Magnetic Coupling
In presence of MC, exchange of energy between the BH and the disk happens. We can derive the radiation flux and the total luminosity of a thin disc by resolving the equations of the conservation of energy and angular momentum
The transfer direction is determined by the difference between the rotation of the BH and that of the disc.

10. It reveals that the inner edge would deviate from ISCO by virtue of the magnetic coupling

11. Criterion for Inner Boundary
Noting the fact that an accretion disc cannot sustain a negative viscous torque, we consider that the inner edge of the disc is the position where the transportation of energy and angular momentum on the disc just begins to exceed the adjustable range of the interior viscous torque (Wang 1995, Krolik 2002).

12. The deviation of rin from rISCO depends on the BH spin as follows: (1) rin>rISCO for a*<0.3594, (2) rin<rISCO for a*> When there is no MC, the inner edge degenerates automatically into ISCO.

13. MC Challenges NTBC ?
Inspecting equations (6) and (8), we find that the MC torque always vanishes at the inner edge, implying that the MC does not disrupt the “no torque boundary condition” (NTBC). However, there is an exception that the closed field lines concentrate at the inner edge of the disc, where the MC torque can be written in the form (Li L.-X.2004)
In this case, the MC behaves equivalently as a non-zero boundary torque.

14. Effects of MC on Disc Radiation
Compared with SAD, the radiation flux is much more concentrated at the inner disc in presence of MC

15. (Wilms 2001)

16. Zhang et al. (1997) firstly invented an approach to measure the BH spin by determining the radius of the inner edge (ISCO) in fitting the spectrum of the X-ray continuum from a thin relativistic disc. And the MC effects on the data fitting behave at least two aspects: (1) the inner edge deviates from ISCO, and (2) the radius of the peak value of the radiation flux is much closer to the inner edge.

17. Thank you !