1. Accretion Disks Around Black Holes
Ramesh Narayan

2. Black Hole Accretion
Accretion disks around black holes (BHs) are a major topic in astrophysics
Stellar-mass BHs in X-ray binaries
Supermassive BHs in galactic nuclei
A variety of interesting observations, phenomena and models
Disks are excellent tools for investigating BH physics:

3. Lecture Topics Lecture 1: Lecture 2: Lecture 3:
Application of the Standard Thin Accretion Disk Model to BH XRBs
Lecture 2:
Advection-Dominated Accretion
Lecture 3:
Outflows and Jets

4. Why Does Nature Form Black Holes?
When a star runs out of nuclear fuel and dies, it becomes a compact degenerate remnant:
White Dwarf (held up by electron degeneracy pressure)
Neutron Star (neutron degeneracy pressure)
Assuming General Relativity, and using the known equation of state of matter up to nuclear density, we can show that there is a maximum mass allowed for a compact degenerate star: Mmax  3M (Rhoades & Ruffini 1974 …)
Above this mass limit, the object must become a black hole

5. A Black Hole is Inevitable
Newtonian physics: if pressure increases rapidly enough towards the interior, an object can counteract its self-gravity
General relativity (TOV eq): pressure does not help
Pressure=energy=mass=gravity

6. A Black Hole is Extremely Simple
Mass: M
Spin: a* (J=a*GM2/c)
Charge: Q (~0)

7. Black Hole Spin
The material from which a BH is formed almost always has angular momentum
Also, accretion adds angular momentum
So we expect astrophysical BHs to be spinning: J = a*GM2/c, 0  a*  1
Spinning holes have unique properties

8. Schwarzschild Metric (G=c=1) (Non-Spinning BH)
One parameter: Mass M
Schwarzschild metric describes space-time around a non-spinning BH
Excellent description of space-time exterior to slowly spinning spherical objects (Earth, Sun, WDs, etc.)

9. Non-Spinning BH
Event Horizon
All the matter is squeezed into a Singularity with infinite density (in classical GR)
Surrounding the singularity is the Event Horizon
Schwarzschild radius:
Singularity

10. Kerr Metric (Spinning BH) (Boyer-Lindquist coordinates)
Two parameters: M, a
If we replace rr/M, tt/M, aa*M, then M disappears from the metric and only a* is left (spin parameter)
This implies that M is only a scale, but
a* is an intrinsic and fundamental parameter

11. Horizon shrinks: e.g., RH=GM/c2 for a*=1
Singularity becomes ring-like
Particle orbits are modified
Frame-dragging --- Ergosphere
Energy can be extracted from BH

12. 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
Fortunately, we have the gas in the accretion disk on circular orbits…

13. Newtonian Gravity

14. Test Particle Geodesics : Schwarzschild Metric
E : specific energy, including rest mass
l : specific angular momentum

15. Circular Orbits
In Newtonian gravity, stable circular orbits are available around a point mass at all radii
This is no longer true in General Relativity
In the Schwarzschild metric, stable orbits allowed only down to r=6GM/c2 (innermost stable circular orbit, ISCO)
The radius of the ISCO (RISCO) depends on BH spin

16. Innermost Stable Circular Orbit (ISCO)
RISCO/M depends on a*
If we can measure RISCO, we will obtain a*
We think an accretion disk has its inner edge at RISCO
Gas free-falls into the BH inside this radius
We could use observations to estimate RISCO

17. Estimating Black Hole Spin
Continuum Spectrum (This Lecture)
Relativistically Broadened Iron Line (Mike Eracleous)
Quasi-Periodic Oscillations (Ron Remillard)

18. Need a Quantitative Model of BH Accretion Disks
Whichever method we choose for estimating BH spin, we need
A reliable quantitative model for the accretion disk: for this Lecture, it is the standard disk model as applied to the Thermal State of BH XRBs
High quality observations
Well-calibrated analysis techniques
And patience, courage and luck!

19. Continuum Method: Basic Idea
Measure Radius of the Hole in the disk by estimating the area of the bright inner disk using X-ray Data in the Thermal State: LX and TX
Zhang et al. (1997); Shafee et al. (2006); Davis et al. (2006); McClintock et al. (2006); Middleton et al. 2006; Liu et al. (2008);…

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

21. Measuring the Radius of the Disk Inner Edge
Here we want 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 (inclination) get RISCO
From RISCO get a*
RISCO
RISCO

22. Note that the results do not depend on the details of the ‘viscous’ stress ( parameter)

23. Spectrum of an accretion disk when it emits blackbody radiation from its surface

24. Blackbody-Like Thermal Spectral State
BH XRBs are sometimes found in the Thermal State (or High Soft State)
Soft blackbody-like spectrum, which is consistent with thin disk model
Only a weak power-law tail
Perfect for quantitative modeling
XSPEC: diskbb, ezdiskbb, diskpn, KERRBB, BHSPEC

25. Blackbody-Like Spectral State in BH Accretion Disk
LMC X-3: Beppo-SAX
(Davis, Done & Blaes 2006)
Up to 10 keV, the only component seen is the disk
Beyond that, a weak PL tail
Perfect for estimating inner radius of accretion disk  BH spin
Just need to estimate LX, TX (and NH) from X-ray continuum
Use full relativistic model (Novikov-Thorne 1973; KERRBB, Li et al. 2005)

26. A Test of the Blackbody Assumption
For a blackbody, L scales as T4 (Stefan-Boltzmann Law)
BH accretion disks vary a lot in their luminosity
If a disk is a good blackbody, L should vary as T4
Looks reasonable
Kubota et al. (2002)
McClintock et al. (2008)

27. Spectral Hardening Factor
Disk emission is not a perfect blackbody
Need to calculate non-blackbody effects through detailed atmosphere model
True also for measuring radii of stars
Davis, Blaes, Hubeny et al. have developed state-of-the-art models
Mike Eracleous’s Lecture

28. Thermal State is very good for quantitative modeling
Tin4
f = Tcol/Teff
Davis et al. (2005, 2006) H
With color correction (from Shane Davis), get an excellent L-T4 trend f
Spectral hardening factor f
Conclusion:
Thermal State is very good for quantitative modeling
 ISCO
Teff4

29. BH Spin From Spectral Fitting
Start with a BH disk in the Thermal State
Given the X-ray flux and temperature (from spectrum), obtain the solid angle subtended by the disk inner edge: (RISCO/D)2 = C (F/Tmax4)
More complicated than stellar case since T varies with R, but functional form of T(R) is known
From RISCO/(GM/c2), estimate a*
Requires BH mass, distance and disk inclination
Most reliable for thin disk: low lumunosity L < 0.3 LEdd

30. Relativistic Effects Doppler shifts (blue and red) of the orbiting gas
Gravitational redshift
Deflection of light rays
Modifies what observer sees
Causes self-irradiation of the disk
Energy release should be calculated according to General Relativity (different from Newtonian)
Powerful and flexible modeling tools available to handle all these effects: KERRBB (Li et al. 2005) BHSPEC (Davis)

31. Movie credit: Chris Reynolds

32. BH XRBs Analyzed So Far GRO J1655-40 4U 1543-47 GRS 1915+105 M33 X-7
LMC X-3
(XTE J )

33. M33 X-7: Spin 15 total spectra: 4 “gold” + 11 “silver”
a* = cJ/GM2
Photon counts ( keV)
4 gold Chandra spectra
a* = 0.77  0.02
Including uncertainties in D, i & M
a* = 0.77  0.05
Chandra & XMM-Newton
Liu et al. (2008)

34. LMC X-3: Five missions agree!
Steiner et al. (2008)
Further strong evidence for existence of a constant radius!

35. BH Masses and Spins Source Name BH Mass (M) BH Spin (a*) LMC X-3
5.9—9.2
~0.25
XTE J
8.4—10.8
(~0.5)
GRO J
6.0—6.6
0.7 ± 0.05
M33 X-7
14.2—17.1
0.77 ± 0.05 4U
7.4—11.4
0.8 ± 0.05
GRS
10--18
0.98—1
Shafee et al. (2006); McClintock et al. (2006); Davis et al. (2006); Liu et al. (2007); Steiner et al. (unpublished); Gou et al. (unpublished)

36. Spin Parameter a* = cJ / GM2 (0 < a* < 1) a* = 0.77  0.05

37. The sample is still small at this time…
Reassuring that values are between 0 and 1 (!!)
GRS with a*  1 is an exceptional system – has powerful jets (Lecture 3)
Several more BH spins likely to be measured in a few years
But more work needed to establish the reliability of the method
Other methods may also be developed – may be calibrated using the present method
Extend to Supermassive BHs?

38. Primordial vs Acquired Spin
A BH in an X-ray binary does not accrete enough mass/angular momentum to cause much change in its spin after birth
So observed spin indicates the approximate birth spin  ang. mmtm of stellar core (but see Poster by Enrique Moreno-Mendez)
A Supermassive BH in a galactic nucleus evolves considerably through accretion
Expect significant spin evolution

39. Good News/Bad News on Continuum Fitting Method
Only need FX, TX from X-ray data
Theoretical model is conceptually simple and reliable (just energy conservation, no )
Disk atmosphere understood
Bad news:
Need accurate M, D, i: requires a lot of supporting optical/IR/radio observations
MHD effects in the disk unclear/under study

40. How Reliable is the Theoretical Flux Profile?

41. Disk Flux Profile
For an idealized thin Newtonian disk with zero torque at its inner edge
No dependence on viscosity parameter 
Analogous results are well-known for a relativistic disk (Novikov & Thorne 1973)
Suggests no serious uncertainty…

42. However,…
iCritical Assumption: torque vanishes at the inner edge (ISCO) of the disk
Makes sense if ’=0
But what about BH accretion?
Afshordi & Paczynski (2003) claim it is okay for a thin disk
But magnetic fields may cause a large torque at the ISCO, and lead to considerable energy generation inside ISCO (Krolik, Hawley, Gammie,…)

43. Check: Hydrodynamic Model
Steady hydrodynamic disk model with -viscosity
Make no assumption about the torque at the ISCO – solve for it self-consistently
Goal: Find out if standard model is OK
(Shafee et al. 2008)

44. Height-Integrated Disk Equations
Plus a simple energy equation to ensure a geometrically thin disk

45. Torque vs Disk Thickness
For H/R < 0.1, good agreement with idealized thin disk model
True for any reasonable value of 

46. Caveat
The results are based on a hydrodynamic disk model with -viscosity
But ‘viscosity’ in an accretion disk is from magnetic fields via the MRI
Therefore, we should do multi-dimensional MHD simulations, and
Directly check magnetic stress profile
Check viscous energy dissipation profile

47. 3D GRMHD Simulation of a Thin Accretion Disk
Shafee et al. (2008)
512 x 128 x 32 grid
Self-consistent MHD simulation
All GR effects included
h/r ~ 0.05 — 0.1 (thin!!)
Only other thin disk simulation: recent work by Reynolds & Fabian (2008)

49. GRMHD Simulation Results
Angular mmtm profile is very close to that of the idealized Novikov-Thorne model (within 2%)
Not too much torque at the ISCO (~2%)
But dissipation profile F(r) is uncertain…
Overall, looks promising, but…

50. What is the Effect on F(R)?
For a Newtonian disk not very serious
F(R) and Tmax increase
But error in estimate of RISCO is only 5%
No worse than other uncertainties
Expect similar results for a GR disk

51. Bottom Line
We can be cautiously optimistic that the spin estimates obtained from fitting continuum X-ray spectra of BHBs are believable
More MHD simulation work needed
Plenty of hard Observational work ahead