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Accretion Disks
From Solar Nebula to Quasars
Energy Sources
Two Main Energy Sources
In Astrophysics
I) Gravitational Potential
Energy
II) Nuclear Energy
Gravitational energy
The Hoover dam
generates 4
billion kilowatt
hours of power
per year.
Where does the
energy come
from?
Gravitational energy
Water falling down to
the generators at the
base of the dam
accelerates to 80
mph.
The same water
leaving the turbines
moves at only 10
mph.
The gravitational
energy of the water
at the top of the dam
is converted to
kinetic energy by
Gravitational energy
Black holes
generate
energy from
matter falling
into them.
Rotating black holes
For non-rotating black holes:
- event horizon is at the Schwarzschild radius
- inner edge of the disk is at 3 Schwarzschild radii
For maximally rotating black holes:
- event horizon is at ½ Schwarzschild radii
- inner edge of the disk is at ½ Schwarzschild radii
Schwarzschild radius = 3 km (M/MSun)
Luminosity
• Gravitational energy is converted to kinetic
energy as particles fall towards BH
• Efficiency of generators:
– Chemical burning < 0.000001%
– Nuclear burning < 1%
– Non-rotating black hole = 6%
– Rotating black hole = 42%
A quasar varies in brightness by a factor of 2
in 10 days. What does this tell us about the
quasar?
1.
2.
3.
4.
It has a large magnetic field.
It is quite small.
It must be highly luminous.
It cannot emit radio waves.
Accretion Disks
Accretion disks are important in
astrophysics as they efficiently
transform gravitational potential
energy into radiation.
Accretion disks are seen around
stars, but the most extreme disks
are seen at the centre of quasars.
These orbit black holes with masses of ~106-9 M, and radiate
up to 1014 L, outshining all of the stars in the host galaxy.
If we assume the black hole is not rotating, we can describe its
spacetime with the Schwarzschild metric and make predictions
for what we expect to see.
Lecture Notes 4
Intro
•
•
Accretion disks form due to angular-momentum of incoming gas
Once in circular orbit, specific angular momentum (i.e., per unit mass)
is
•
•
So, gas must shed its angular momentum for it to actually accrete…
Releases gravitational potential energy in the process!
Comparison of disk velocity derived from
a 3-d MHD simulation (dotted) with simple
test-particle velocity (solid)… confirms
analytic result that deviations are O[(h/r)2]
Variabilities of GRBs limits models
to compact objects (NS, BH)
Variability =
size scale/speed of light
Again, Neutron Stars and
Black Holes likely
Candidates (either in an
Accretion disk or on the
NS surface).
2 p 10km/c = .6 ms
c = 1010cm/s
NS,
BH
Measuring a Quasar’s Black Hole
Light travel time effects
If photons leave A and B at the same
time, A arrives at the observer a time
t ( = d / c ) later.
A
B If an event happens at A and takes
d=cxt
c = speed of light
d = diameter
a time dt, then we see a change over
a timescale t+dt. This gives a
maximum value for the diameter, d,
because we know that our measured
timescale must be larger than the
light crossing time.
• How does the accreting matter lose its
angular momentum?
• What happens to the gravitational
potential energy of the infalling matter?
How much energy is released
by an accretion disk?
• Consider 1kg of matter in the accretion disk. Further, assume
that…
– The matter orbits in circular paths (will always be approximately
true)
– Centripetal acceleration is mainly due to gravity of central object
(i.e., radial pressure forces are negligible… will be true if the disk is
thin)
• Energy of 1 kg of matter in the accretion disk is..
• So, the total luminosity liberated by accreting a flow of matter is
Initial energy (at
infinity)
Mass flow rate
Final energy
• Total luminosity of disk depends on inner radius of dissipative
part of accretion disk
Question
• It seems like only half of the gravitational potential energy of the
accreting matter is “liberated”. What happens to the other half?
–
–
–
–
A.
B.
C.
D.
It is carried outwards with the angular momentum.
It crosses the inner-edge of the disk as kinetic energy
Its crossed the inner-edge of the disk as thermal energy (heat)
None of the above
What sets the inner edge of the
accretion disk?
• Accretion disk around a star…
– Inner edge set by radius of star, R
– Luminosity of disk is
– Additional
liberated in the boundary
layer between disk and star
• Accretion disk around a black hole
– Inner edge often set by the “innermost stable circular
orbit” (ISCO)
– GR effects make circular orbits within the ISCO
unstable… matter rapidly spirals in
– Risco=6GM/c2 for a non-rotating black hole
• For a disk that extends down to the ISCO for a nonrotating black hole, simple Newtonian calculation
gives…
• More detailed relativistic calculation gives…
• In general, we define the radiative
efficiency of the accretion disk, , as
• For disks extending down to the ISCO, the
radiative efficiency increases as one
considers more rapidly rotating black
holes.
Viscous accretion disks
• What allows the accreting gas to lose its angular momentum?
• Suppose that there is some kind of “viscosity” in the disk
– Different annuli of the disk rub against each other and exchange angular
momentum
– Results in most of the matter moving inwards and eventually accreting
– Angular momentum carried outwards by a small amount of material
• Process producing this “viscosity” might also be dissipative… could
turn gravitational potential energy into heat (and eventually
radiation)
• Consider two
consecutive rings of the
accretion disk.
• The torque exerted by
the outer ring on the
inner ring is
• Viscous dissipation per unit area of disk
surface is given by
• Evaluating for circular Newtonian orbits
(i.e., “Keplerian” orbits),
What gives rise to viscosity?
• Normal “molecular/atomic” viscosity fails to provide
required angular momentum transport by many
orders of magnitude!
• Source of anomalous viscosity was a major puzzle in
accretion disk studies!
• Long suspected to be due to some kind of turbulence
in the gas… then can guess that:
• 20 years of accretion disk studies were based on this
“alpha-prescription”…
• But what drives this turbulence? What are its
properties?
The magnetorotational
instability
• Major breakthrough in 1991… Steve Balbus and
John Hawley (re)-discovered a powerful
magneto-hydrodynamic (MHD) instability
– Called magnetorotational instability (MRI)
– MRI will be effective at driving turbulence
– Turbulence transports angular momentum in
just the right way needed for accretion
• Two satellites connected by a weak spring
provide an excellent analogy for understanding
the MRI.
Accretion Disk Temperature
Structure
r
The accretion disk (AD)
can be considered as
rings or annuli of
blackbody emission.
Dissipation rate, D(R) is
1/ 2


3GMM
 R*  

 
1  
3
8pr
  r  
= blackbody flux
4
 T (r )
Disk Temperature
Thus temperature as a function of radius
T(R):
1/ 4
1/ 2
 3GMM
T (r )  
3
8
p
r


  R* 
1   
  r 

 3GMM
and if T*  
3
 8pR* 
then for
r  R*



 

1/ 4



T  T* r / R* 
3 / 4
Disk Spectrum
Flux as a function of frequency, n
Log n*Fn
Total disk spectrum
Annular BB emission
Log n
Black Hole and Accretion Disk
For a non-rotating spherically symetrical BH, the
innermost stable orbit occurs at 3rg or :
6GM
rmin  2
c
3 / 4
and when R  R* T  T* R / R* 
Fe Ka Line
Fluorescence line observed in Seyferts –
from gas with temp of at least a million
degrees.
FeKa
X-ray
e-
Radio Galaxies and Jets
Cygnus-A →
VLA radio image at
n = 1.4.109 Hz
- the closest powerful
radio galaxy
(d = 190 Mpc)
150 kpc
Radio Lobes
← 3C 236 Westerbork radio image
Radio Lobes
5.7 Mpc
at n = 6.08.108 Hz – a radio
galaxy of very large extent
(d = 490 MPc)
Jets, emanating from a central highly
active galaxy, are due to relativistic
electrons that fill the lobes
More about Accretion Disks
Disk self-gravitation is negligible so material in
differential or
Keplerian rotation with angular velocity WK(R) =
If n is the 3kinematic
viscosity
1/2
(GM/R )
for rings of gas rotating,
Q
the viscous torque
exerted by the outer
Q
ring on the inner will be
Q(R) = 2pR nS R2 (dW/dR) (1)
where the viscous force per unit length is acting on 2pR and
S= Hr is the surface density with H (scale height) measured
in the z direction.
More about Accretion Disks
(Cont.)
•
The viscous torques cause energy dissipation of Q W dR/ring
Each ring has two plane faces of area 4pRdR, so the radiative
dissipation from the disc per unit area is from (1):
•
•
D(R) = Q(R) W/4pR = ½ n S RW)2 (2)
and since
W  WK = (G M/R3)1/2
differentiate and then
D(R) = 9/8 n S Q(R) M/R3 (3)
More about Accretion Disks
From a consideration of radial mass and angular momentum
flow in the disk, it can be shown (Frank, King & Raine, 3rd
ed., sec 5.3/p 85, 2002) that
•
n S = (M/3p [1 – (R*/R)1/2]
•
where M is the accretion rate and from (2) and (3) we then
have
•
D(R) = (3G M M/8pR3) [1 – (R*/R)1/2]
and hence the radiation energy flux through the disk faces is
independent of viscosity
• Jet power scales with
accretion disk power
Qjet = qj/l · Ldisk
• Model applicable to
– quasars
– LLAGN
– X-ray binaries
• Below a critical
accretion rate, disks
may become
radiatively inefficient
(and become advection
dominated: ADAFs,
BDAFs, CDAFs …).
• At lower accretion rates
disks become less and
less prominent.
Esin, Narayan et al. (1997 …)