Transcript Black Hole Accretion

ACCRETION DISKS
AROUND BLACK
HOLES
Ramesh Narayan
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:
Lecture Topics

Lecture 1:


Lecture 2:


Application of the Standard Thin Accretion
Disk Model to BH XRBs
Advection-Dominated Accretion
Lecture 3:

Outflows and Jets
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
Newtonian physics: if pressure increases rapidly enough
towards the interior, an object can counteract its self-gravity
1 dP
GM (r )

, P  P
2
 dr
r
General relativity (TOV eq): pressure does not help
2
3
2
1

P
/

c
1

4

Pr
/
Mc


1 dP
GM 
 2
2
 dr
r
1

2
GM
/
c
r

Pressure=energy=mass=gravity
A Black Hole is
Extremely Simple

Mass: M

Spin: a* (J=a*GM2/c)

Charge: Q (~0)
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
Schwarzschild Metric (G=c=1)
(Non-Spinning BH)
1
 2M  2  2M 
2
ds   1 
dt

1

dr



r
r




 r 2 d 2  r 2 sin 2  d 2
2
One parameter: Mass M
Schwarzschild metric describes space-time around a nonspinning BH
Excellent description of space-time exterior to slowly
spinning spherical objects (Earth, Sun, WDs, etc.)
Non-Spinning BH



All the matter is squeezed into a
Singularity with infinite density (in
classical GR)
Event
Horizon
Surrounding the singularity is the
Event Horizon
Schwarzschild radius:
 M 
2GM
R =
= 2.95 
 km
s
2
C
M 
Singularity
Kerr Metric (Spinning BH)
(Boyer-Lindquist coordinates)
2
2
Mr
4
aMr
sin

 2

 2
2
ds   1 
dtd  dr
 dt 
 



2
2

2
Mra
sin
 2
2
2
2
2
 d   r  a 
sin

d





  r 2  a 2 cos 2  ,   r 2  2Mr  a 2
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





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 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
Two conserved quantities
d
l  rv  r 2
dt
1
1
GM
EN  vr2  v2 
2
2
r
2
1  dr 
l2
GM
 



2  dt 
2r 2
r
2
 dr 

  2  EN  Veff,N (r ) 
 dt 
l2
GM
Veff,N (r )  2 
2r
r
Test Particle Geodesics :
Schwarzschild Metric
x   x  x
Newtonian
d l
 2
dt r
d l
 2
d r
2
2GM l
2M   l   dr 
 dr 
2 
   E  1 
 1  2   dt   2 EN  r  r 2
r  r   
 d 

2
2
 E 2  Veff (r )
dt
E

d 1  2M / r
2
 2  EN  Veff,N (r ) 
E : specific energy, including rest mass
l : specific angular momentum
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
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

Continuum Spectrum (This Lecture)

Relativistically Broadened Iron Line
(Mike Eracleous)

Quasi-Periodic Oscillations
(Ron Remillard)
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
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

Measure the flux F received from the star

Measure the temperature T (from spectrum)

Then, using blackbody radiation theory:
L  4 D F  4 R  T
2
2
4
2
F
R
  =
4
D

T
 

F and T give solid angle of star

If we know D, we directly obtain R
R
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

Ldisk
GM M

2 Rin
Note that the results do
not depend on the
details of the ‘viscous’
stress ( parameter)
3GMM
D( R)  F ( R) 
8 R 3
 Rin 
T ( R )  T* 

R


3/4

Rin 
1 

R 

1/4

Rin 
1 

R 

1/4
 3GMM 
T*  
3 
 8 Rin 
Spectrum of an
accretion disk when it
emits blackbody
radiation from its
surface
Given Ldisk and Tmax we obtain
L1/2
R in  15.5 disk
(cgs units)
2
Tmax
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
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)
A Test of the
Blackbody
Assumption




For a blackbody, L scales as
T4 (Stefan-Boltzmann Law)
H1743-322
Kubota et al. (2002)
McClintock et al. (2008)
BH accretion disks vary a lot
in their luminosity
If a disk is a good blackbody,
L should vary as T4
Looks reasonable
L  A T
4
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
f = Tcol/Teff
H1743-322
Tin4
Davis et al. (2005, 2006)
With color correction
(from Shane Davis),
get an excellent L-T4
trend
Conclusion:
Thermal State is
very good for
quantitative
modeling
 ISCO
f
Spectral hardening factor
4
Teff
f
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 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
GRO J1655-40
 4U 1543-47
 GRS 1915+105
 M33 X-7
 LMC X-3
 (XTE J1550-564)

M33 X-7: Spin
a* = cJ/GM
2
15 total spectra: 4 “gold” + 11 “silver”
Photon counts (0.3 - 8 keV)
Chandra & XMM-Newton
Liu et al. (2008)
4 gold Chandra spectra
a* = 0.77  0.02
Including uncertainties in D, i & M
a* = 0.77  0.05
LMC X-3: Five missions agree!
Steiner et al. (2008)
Further strong evidence for existence of a constant radius!
BH Masses and Spins
Source 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 2
a* = cJ / GM
(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

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




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:




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?
Disk Flux Profile

For an idealized thin Newtonian disk with
zero torque at its inner edge
1/2

 Rin  
4
4  Rin 
F ( R)   Teff (r )   Teff* 
 1  
 
 R    R  
T ( R)  f Teff ( R)
3



No dependence on viscosity parameter 
Analogous results are well-known for a
relativistic disk (Novikov & Thorne 1973)
Suggests no serious uncertainty…
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



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
H
1/2
cs
K
p  c ,
2
s
,

 0,
t
 GM 
v K   KR  

 R 
c2s

 0,    c sH  

K

M  2 RvR 2H  constant
vR

M
dvR
GM
1 dp
  2   2R 
dR
 dR
R
d
d 
3 d 
 R2  

2H

2

R
dR
dR 
dR 


Plus a simple energy equation to ensure
a geometrically thin disk
Torque vs Disk Thickness


For H/R < 0.1, good agreement with idealized thin disk model
True for any reasonable value of 
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 multidimensional MHD simulations, and

Directly check magnetic stress profile

Check viscous energy dissipation profile
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)
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)?





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 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