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**Accretion Disks Around Black Holes**

Ramesh Narayan

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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:

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**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

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**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

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**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

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**A Black Hole is Extremely Simple**

Mass: M Spin: a* (J=a*GM2/c) Charge: Q (~0)

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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

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**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.)

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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

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**Kerr Metric (Spinning BH) (Boyer-Lindquist coordinates)**

Two parameters: M, a If we replace rr/M, tt/M, aa*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

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**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

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**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…

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Newtonian Gravity

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**Test Particle Geodesics : Schwarzschild Metric**

E : specific energy, including rest mass l : specific angular momentum

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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

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**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

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**Estimating Black Hole Spin**

Continuum Spectrum (This Lecture) Relativistically Broadened Iron Line (Mike Eracleous) Quasi-Periodic Oscillations (Ron Remillard)

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**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!

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**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);…

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**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

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**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

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

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

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**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

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**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)

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**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)

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**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

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**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

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**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

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**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)

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**Movie credit: Chris Reynolds**

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**BH XRBs Analyzed So Far GRO J1655-40 4U 1543-47 GRS 1915+105 M33 X-7**

LMC X-3 (XTE J )

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**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)

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**LMC X-3: Five missions agree!**

Steiner et al. (2008) Further strong evidence for existence of a constant radius!

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**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)

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

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**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?

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**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

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**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

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**How Reliable is the Theoretical Flux Profile?**

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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…

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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,…)

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**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)

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**Height-Integrated Disk Equations**

Plus a simple energy equation to ensure a geometrically thin disk

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**Torque vs Disk Thickness**

For H/R < 0.1, good agreement with idealized thin disk model True for any reasonable value of

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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

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**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)

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**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…

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**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

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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

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