Accretion Disks Around Black Holes

1. Ramesh Narayan Accretion Disks Around Black Holes

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: • 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 obtaina* • 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 Freceived from the star • Measure the temperature T(from spectrum) • Then, using blackbody radiation theory: • F and T give solid angle of star • If we knowD,we directly obtainR 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 • 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) 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

26. H1743-322 A Test of the Blackbody Assumption • For a blackbody, L scales asT4 (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. Tin4 f = Tcol/Teff Davis et al. (2005, 2006) H1743-322 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 Tvaries 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 lumunosityL < 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 J1550-564)

33. M33 X-7: Spin15 total spectra: 4 “gold” + 11 “silver” a* = cJ/GM2 Photon counts (0.3 - 8 keV) 4gold 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 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 a* ~ 0.25 a* = 0.98 - 1.0 a* = 0.65 - 0.75 (a* ~ 0.5) a* = 0.75 - 0.85

37. The sample is still small at this time… • Reassuring that values are between0 and 1(!!) • GRS 1915+105 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 • Good news: • 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