1. Ramesh Narayan (McClintock, Shafee, Remillard, Davis, Li)

2. Black Holes are Extremely Simple Mass: M Spin: a * =a/M (J=a * GM 2 /c) (Electric Charge: Q) Many BH masses have been measured Obvious next frontier: Measure BH spin (much harder) Beyond that: Test the Kerr Metric (even harder)

3. Innermost Stable Circular Orbit (ISCO) In GR, stable circular orbits are allowed only down to an innermost radius R ISCO (effect of strong gravity) R ISCO /M depends on a * (quite a large effect) An accretion disk terminates at R ISCO, and gas falls freely onto the BH inside this radius Disk emission has a ‘hole’ of radius R ISCO at center If we measure the size of the hole we will obtain  a *

4. Measuring the Radius of a Star Measure the flux F received from the star Measure the temperature T (from spectrum) Then, assuming blackbody radiation: F and T give solid angle of star If we know distance D, we directly obtain R R

5. Measuring the Radius of the Disk Inner Edge We want to measure the radius of the ‘hole’ in the disk emission Same principle as before From F and T get solid angle of hole Knowing D and i get R ISCO From R ISCO and M get a * Zhang et al. (1997); Li et al. (2005); Shafee et al. (2006); McClintock et al. (2006); Davis et al. (2006);… R ISCO

6. Estimates of Spin Obtained with this Method Systema*a* Reference GRO J1655-40 0.65-0.75 Shafee et al. (2006) 4U1543-47 0.7-0.8 Shafee et al. (2006) GRS 1915+105 0.98-1.0 McClintock et al. (2006, astro-ph/0606076) LMC X-3 <0.26 Davis et al. (2006)

7. How to Get Reliable Results? Should have good estimates of M, D, i Should include all relativistic effects (Doppler beaming, grav. redshift, ray deflections, Li et al. 2005: KERRBB) The system should be in the high soft state: thermal blackbody radiation, with very little power-law (>90% of the flux in the thermal component) Deviations from blackbody (parameter f) should be estimated via a disk atmosphere model Need accurate theoretical profiles of disk flux F(R) and temperature T(R)

8. GRS 1915+105 in the High Soft State Gierlinski & Done (2002) Kubota et al. (2004)

9. Spectral Hardening Factor Disk emission is not a perfect blackbody Spectral temperature T of the emitted radiation is generally larger than effective temperature: T=f T eff Using disk atmosphere model, can estimate f (Shimura & Takahara 1995; Davis et al. 2006) Results are robust, provided most of the viscous energy is released below the photosphere (it is not necessary to know exact vertical profile, value of  ) Safe assumption in high soft state

10. Viscous Energy Dissipation Profile Well-known result for an idealized thin Newtonian disk with zero torque at inner edge (analogous results for PW or GR disk) Completely independent of viscosity  !!

11. However,… The theoretical model makes a critical assumption: torque vanishes at the inner edge (ISCO) of the disk (Shakura & Sunyaev 1973) Afshordi & Paczynski (2003) say this is okay for a thin disk, but not for a thick disk Krolik, Hawley, et al. say there is always substantial torque at ISCO, and energy generation inside ISCO Gierlinski et al. (1999)

12. Torque vs Disk Thickness Hydrodynamic height- integrated  -disk model with full dynamics (radial velocity, pressure, sonic radius, non-Keplerian,…) For H/R < 0.1 (L<0.3L Edd ), good agreement with idealized thin disk model Less good at large  but still pretty good Bottom line: stick to low luminosities: L < 0.3L Edd Shafee et al. (2007)

13. GRS 1915+105 Spin Estimate Limiting ourselves to L<0.3L Edd, we obtain a robust result: a * =0.98—1.0 Insensitive to how we model the power-law tail Insensitive to , torque Insensitive to uncertainties in M, D, i Can explain discrepancy with Middleton et al. (2006) McClintock et al. (2006)

15. Estimates of Spin Systema*a* Reference GRO J1655-40 0.65-0.75 Shafee et al. (2006) 4U1543-47 0.7-0.8 Shafee et al. (2006) GRS 1915+105 0.98-1.0 McClintock et al. (2006, astro-ph/0606076) LMC X-3 <0.26 Davis et al. (2006)

16. Discussion All four a* values are between 0 and 1 (!!) Spins of XRB BHs evolve very little via accretion  BHs are born with a wide range of spin values GRS 1915+105 (a*  1) is a near-extreme Kerr BH – any connection to its relativistic jets? Was GRS 1915+105 a GRB when it was formed? Other methods of estimating spin (QPOs) could be calibrated using the present method Would also test the Kerr metric… Can we estimate spins of Supermassive BHs?