ACCRETION DISKS AROUND BLACK

HOLES

ACCRETION DISKS AROUND BLACK

HOLES

Ramesh Narayan

Black Hole AccretionBlack 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:

Lecture TopicsLecture Topics

Lecture 1: Application of the Standard Thin

Accretion Disk Model to BH XRBs Lecture 2:

Advection-Dominated Accretion Lecture 3:

Outflows and Jets

Why Does Nature Form Black Holes?

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

A Black Hole is Inevitable

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

2

1 ( ),

dP GM rP P

dr r

2 3 2

2 2

1 / 1 4 Pr /1

1 2 /

P c McdP GM

dr r GM c r

A Black Hole is Extremely Simple

A Black Hole is Extremely Simple

Mass: M

Spin: a* (J=a*GM2/c)

Charge: Q (~0)

Black Hole SpinBlack 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

Schwarzschild Metric (G=c=1)

(Non-Spinning BH)

Schwarzschild Metric (G=c=1)

(Non-Spinning BH)

12 2 2

2 2 2 2 2

2 21 1

sin

M Mds dt dr

r r

r d r d

One parameter: Mass MSchwarzschild metric describes space-time around a non-spinning BHExcellent description of space-time exterior to slowly spinning spherical objects (Earth, Sun, WDs, etc.)

Non-Spinning BHNon-Spinning BH

All the matter is squeezed into a Singularity with infinite density (in classical GR)

Surrounding the singularity is the Event Horizon

Schwarzschild radius:Singularity

Event Horizo

n

2GR = = 2.95 km

s 2M M

MC

Kerr Metric (Spinning BH)

(Boyer-Lindquist coordinates)

Kerr Metric (Spinning BH)

(Boyer-Lindquist coordinates)

22 2 2

2 22 2 2 2 2

2 2 2 2 2

2 4 sin1

2 sinsin

cos , 2

Mr aMrds dt dtd dr

Mrad r a d

r a r Mr a

Two parameters: M, aIf 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, buta* is an intrinsic and fundamental parameter

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

Mass is Easy, Spin is Hard

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…

Newtonian Gravity

Newtonian Gravity

2

2 2

2 2

2

2

N eff,N

2

eff,N 2

1 1

2 2

1

2 2

2 (

Two conserved quantit

)2

ies

)

(

N r

dl rv r

dtGM

E v vr

dr l GM

dt r r

drE V r

dt

l GMV r

r r

Test Particle Geodesics :

Schwarzschild Metric

Test Particle Geodesics :

Schwarzschild Metric

2

2 22

2

2eff

21 1

( )

1 2 /

x x x

d l

d r

dr M lE

d r r

E V r

dt E

d M r

2

2 2

N 2

N eff,N

22

2 (

Newtonian

)

d l

dt r

dr GM lE

dt r r

E V r

E : specific energy, including rest massl : specific angular momentum

Circular OrbitsCircular 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

Innermost Stable Circular Orbit (ISCO)

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

Estimating Black Hole Spin

Estimating Black Hole Spin

Continuum Spectrum (This Lecture)

Relativistically Broadened Iron Line

(Mike Eracleous)

Quasi-Periodic Oscillations

(Ron Remillard)

Need a Quantitative Model of BH Accretion

Disks

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!

Continuum Method: Basic Idea

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);…

Measuring the Radius of a Star

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

2 2 4

2

4

4 4

R F=

D T

L D F R T

R

Measuring the Radius of the Disk Inner Edge

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*

RISCORISCO

diskin2

GM ML

R

in

3

1/43/4

in in*

1/4

* 3in

3( ) ( ) 1

8

( ) 1

3

8

RGMMD R F R

R R

R RT R T

R R

GMMT

R

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

disk max

1/2disk

in 2max

Given and we obtain

R 15.5 (cgs units)

L T

L

T

Spectrum of an accretion disk when it emits

blackbody radiation from its

surface

Blackbody-Like Thermal Spectral State

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

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)

Blackbody-Like Spectral State in BH Accretion DiskLMC X-3: Beppo-SAX

(Davis, Done & Blaes 2006)

Up to 10 keV, the only component seen is the diskBeyond that, a weak PL tail

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)

A Test of the Blackbody Assumption

4L A T

H1743-322

Spectral Hardening Factor

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

Tin4

Teff4

f = Tcol/Teff

Davis et al. (2005, 2006)

Conclusion: Thermal State is

very good for quantitative

modeling ISCO

Spectral hardening factor f

f

With color correction (from Shane Davis), get an excellent L-T4

trend

H1743-322

BH Spin From Spectral Fitting

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

Relativistic EffectsRelativistic 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)

Movie credit: Chris Reynolds

BH XRBs Analyzed So Far

BH XRBs Analyzed So Far

GRO J1655-40 4U 1543-47

GRS 1915+105 M33 X-7 LMC X-3

(XTE J1550-564)

4 gold Chandra spectraa* = 0.77 0.02

Including uncertainties in D, i & M

a* = 0.77 0.05

M33 X-7: Spin15 total spectra: 4 “gold” + 11

“silver”

M33 X-7: Spin15 total spectra: 4 “gold” + 11

“silver”

a*

= c

J/G

M2

Photon counts (0.3 - 8 keV)

Chandra & XMM-NewtonLiu et al. (2008)

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

Further strong evidence for existence of a constant radius!

Steiner et al. (2008)

BH Masses and SpinsBH Masses and SpinsSource Name BH Mass (M) BH Spin (a*)

LMC X-3 5.9—9.2 ~0.25

XTE J1550-564 8.4—10.8 (~0.5)

GRO J1655-40 6.0—6.6 0.7 ± 0.05

M33 X-7 14.2—17.1 0.77 ± 0.05

4U1543-47 7.4—11.4 0.8 ± 0.05

GRS 1915+105 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)

Spin

Parameter

a* = cJ / GM2

(0 < a* < 1)

a* = 0.77 0.05

a* = 0.98 - 1.0

a* = 0.65 - 0.75

a* = 0.75 - 0.85

(a* ~ 0.5)

a* ~ 0.25

The sample is still small at this time… Reassuring that values are between 0 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?

Primordial vs Acquired Spin

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

Good News/Bad News on Continuum Fitting

Method

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

How Reliable is the Theoretical Flux Profile?

How Reliable is the Theoretical Flux Profile?

Disk Flux ProfileDisk 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…

3 1/24 4 in in

eff eff*

eff

( ) ( ) 1

( ) ( )

R RF R T r T

R R

T R f T R

However,…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,…)

Check: Hydrodynamic Model

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)

Height-Integrated Disk Equations

Height-Integrated Disk Equations

R

2RR 2

2 3

M 2 Rv 2H constant

dv GM 1 dpv R

dR dRR

d d dM R 2H 2 R

dR dR dR

1/2

2ss K K

K

2s

sK

c GMH , p c , v R

R

c0, 0, c H

t

Plus a simple energy equation to ensure a geometrically thin disk

Torque vs Disk Thickness

Torque vs Disk Thickness

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

CaveatCaveat

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

3D GRMHD Simulation of a Thin Accretion Disk3D 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)

GRMHD Simulation Results

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…

What is the Effect on F(R)?

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

Bottom LineBottom 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