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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