1. MEASURING SPIN PARAMETERS OF STELLAR-MASS BLACK HOLES Ramesh Narayan

2. Spin: Fundamental Property of a Black Hole Mass: M Spin: a * (J=a * GM 2 /c) Charge: Q (~0) From the point of view of BH physics, the spin of a BH is very fundamental

3. Importance in Astrophysics Free energy associated with BH spin may be responsible for relativistic jets BH spin values give a handle on angular momenta of progenitor stars Gamma-ray bursts and BH spin? SMBH spin and galaxy merger history BH spin affects gravitational wave signals

4. Mass is Easy, Spin is Hard Mass can be measured in the Newtonian limit using test particles (e.g., stellar companion) at large radii Spin has no Newtonian effect To measure spin we must be in the regime of strong gravity, where General Relativity operates Need test particles at small radii So we must use the gas in the accretion disk…

5. Estimating Black Hole Spin X-Ray Continuum Spectrum  Relativistically Broadened Iron Line – Quasi-Periodic Oscillations

6. Our Team Jeff McClintockRamesh Narayan Charles Bailyn, Shane Davis, Lijun Gou, Akshay Kulkarni, Li-Xin Li, Jifeng Liu, Jon McKinney, Jerry Orosz, Bob Penna, Mark Reid, Ron Remillard, Rebecca Shafee, Danny Steeghs, Manuel Torres, Jack Steiner, Sasha Tchekhovskoy, Yucong Zhu

7. Innermost Stable Circular Orbit (ISCO) R ISCO /M depends on the value of a * If we can measure R ISCO, we will obtain a * Note factor of 6 variation in R ISCO Especially sensitive as a *  1

8. The Basic Idea Accretion disk has a dark central “hole” with no radiation Measure radius of hole by estimating area of the bright inner disk

9. Measuring the Radius of a Star Measure the flux F received from the star Measure the temperature T * (from spectrum) R*R*

10. Measuring the Radius of the Disk Inner Edge We want the radius of the “hole” in the disk emission Same principle as for a star From X-ray data we obtain F X and T X   (bright) Knowing distance D and inclination i we get R ISCO (some (geometrical factors) From R ISCO /M we get a * Need to be careful: focus on Thermal Dominant (TD) data Zhang et al. (1997); Li et al. (2005); Shafee et al. (2006); McClintock et al. (2006); Davis et al. (2006); Liu et al. (2007); Gou et al. (2009,2010); Steiner et al. (2010)… R ISCO

11. Note that the result does not depend on the details of the ‘viscous’ stress (  parameter) Shakura- Sunyaev model R in =6M

12. For a blackbody, L scales as T 4 (Stefan-Boltzmann Law) BH accretion disks vary a lot in their luminosity If a disk is a perfect blackbody, L should exactly as T 4 Good, but not perfect… Kubota, Done et al. (2002,…) McClintock et al. (2008) A Test of the Blackbody Assumption H1743-322

13. Tin 4 T eff 4 f = T col /T eff Davis et al. (2005, 2006) Conclusion: Thermal State is very good for quantitative modeling Spectral hardening factor f f After including the color correction, we get an excellent L-T 4 trend H1743-322

14. General Relativistic Disk Model: Novikov & Thorne (1973) L(r) peaks at a different radius for each value of the dimensionless BH spin parameter a * Therefore, the observed spectrum depends on a * This is what enables us to estimate a * from observations

15. Summary of the Method We fit the X-ray continuum spectrum We include all relativistic effects We focus on “good data” (TD), where the emission is mostly blackbody so that we can model the spectrum reliably   To convert “solid angle” measurement to an estimate of R ISCO, we measure independently D, i Then, knowing M, we calculate R ISCO /M, and thus obtain an estimate of a *

16. LMC X-3: 1983 - 2009 L / LEdd LMC X-3

17. LMC X-3: 1983 - 2009 L / LEdd LMC X-3 Thick Disk Hard State

18. LMC X-3: 1983 - 2009 L / LEdd Rin Steiner et al. (2010) 403 spectra (assuming M=10M , i=67 o ) LMC X-3

19. New Result: XTE J1550-564 (Steiner, Reis et al. 2010) M=9.10  0.61, D=4.38  0.5, i=74.7  3.8 (Orosz et al. 2010) Spin estimate from continuum fitting: a * ~ 0.35 (90% cl: -0.11, +0.71) Spin estimate from iron line fitting: a * ~ 0.55 (1  limits: 0.33, 0.70) Combined spin estimate: a* = 0.49 (1  limits: 0.29, 0.62)

20. XTE J1550-564: Outburst Light Curve 1998-1999

21. X-ray continuum spectral fits and residuals for a TD (“gold”) and an SPL (“silver”) observation

22. Estimates of disk inner edge R in and BH spin parameter a * from all suitable TD (“gold”) and SPL/Intermediate (“silver”) observations

23. Probability distribution of spin of J1550 (Steiner et al. 2010) CF Method: Includes errors in M, D, I Fe line method Combined

24. BH Spin Measurements via Continuum-Fitting Shafee et al. (2006); McClintock et al. (2006); Davis et al. (2006); Liu et al. (2007,2009); Gou et al. (2009,2010); Steiner et al. (2010) Source NameBH Mass (M  )BH Spin (a * ) A0620-006.3—6.90.10 ± 0.20 LMC X-35.9—9.2~0.25 XTE J1550-5648.5—9.70.50±0.20 GRO J1655-406.0—6.60.70 ± 0.05 M33 X-714.2—17.10.77 ± 0.05 4U1543-477.4—11.40.80 ± 0.05 LMC X-19.0—11.60.92 ± 0.06 GRS 1915+10510—180.99 ± 0.01

25. A Major Issue NT model assumes that the torque vanishes at the ISCO (Shakura & Sunyaev 1973) But magnetic fields could produce significant torque at and inside the ISCO (Krolik 1999; Gammie 1999) Afshordi & Paczynski (2003), Shafee et al. (2008) showed that the effect is not important for a THIN hydrodynamic disk But what about an MHD disk?

26. 3D GRMHD Simulations of Thin Accretion Disks Shafee et al. (2008), Penna et al. (2010) Self-consistent MHD simulations (HARM: Gammie, McKinney & Toth 2003) All GR effects included h/r ~ 0.05 — 0.08 (thin!!) Very few other thin disk simulations: Reynolds & Fabian (2008); Noble, Krolik & Hawley (2009, 2010) a * =0 a * =0, 0.7, 0.9, 0.98 Kulkarni et al. (2010)

27. a * =0.9; i=15 o, 45 o, 75 o a * =0, 0.7, 0.9; i=75 o

28. Modeling Error Due to Deviations From the Novikov-Thorne Model The systematic error due to assuming the NT model is less than the statistical error due to measurement uncertainties The simulated disks correspond to L/L Edd ~ 0.5 Measurements are, however, made using data at L  0.3 L Edd  True systematic errors will be less than the above values Kulkarni et al. (2010) a * =0a * =0.7a * =0.9a*=0.98 i=15 o 0.040.720.900.985 i=45 o 0.060.730.910.986 i=75 o 0.170.800.930.991

29. BH Spin Measurements via Continuum-Fitting Shafee et al. (2006); McClintock et al. (2006); Davis et al. (2006); Liu et al. (2007,2009); Gou et al. (2009,2010); Steiner et al. (2010) Source NameBH Mass (M  )BH Spin (a * ) A0620-006.3—6.90.10 ± 0.20 LMC X-35.9—9.2~0.25 XTE J1550-5648.5—9.70.50±0.20 GRO J1655-406.0—6.60.70 ± 0.05 M33 X-714.2—17.10.77 ± 0.05 4U1543-477.4—11.40.80 ± 0.05 LMC X-19.0—11.60.92 ± 0.06 GRS 1915+10510—180.99 ± 0.01

30. Disk Inclination Is BH spin aligned with orbit vector? We assume this in order to estimate i X-ray polarimetry will help: GEMS Population synthesis studies look hopeful (Fragos et al. 2010) Li, Narayan & McClintock (2009)

31. Importance of BH Spin Of the two parameters, mass and spin, spin is more fundamental Mass is merely a scale – just tells us how big the BH is Spin fundamentally affects the basic properties of space-time around a BH More than simple re-scaling BH spin may power relativistic jets…

32. Relativistic Jets in GRS 1915+105 GRS 1915+105 GRS 1915+105 has an extreme value of spin: a*=0.98-1 Also spectacular relativistic jets Blobs of material are seen to flow out with v = 0.92c Could the relativistic ejections be connected to the BH spin?

33. BH Spin Values vs Relativistic Jets Shafee et al. (2006); McClintock et al. (2006); Davis et al. (2006); Liu et al. (2007,2009); Gou et al. (2009,2010); Steiner et al. (2010) Source NameBH Mass (M  )BH Spin (a * ) A0620-00 (J)6.3—6.90.10 ± 0.20 LMC X-35.9—9.2~0.25 XTE J1550-564 (J)8.5—9.70.50±0.20 GRO J1655-40 (J)6.0—6.60.70 ± 0.05 M33 X-714.2—17.10.77 ± 0.05 4U1543-477.4—11.40.80 ± 0.05 LMC X-19.0—11.60.92 ± 0.06 GRS 1915+105 (J)10—180.99 ± 0.01

34. Summary BH spin measurement by continuum fitting is now a well-developed technique Measurement errors are quantifiable (except for disk inclination) Systematic model errors are under control XTE J1550-564: Consistent spin estimates from continuum-fitting & Fe-line methods BH spin is perhaps not very important for relativistic jets…