1. From Solar Nebula to Quasars
Accretion Disks
From Solar Nebula to Quasars

2. Energy Sources Two Main Energy Sources In Astrophysics
Gravitational Potential
Energy
II) Nuclear Energy

3. Gravitational energy
The Hoover dam generates 4 billion kilowatt hours of power per year.
Where does the energy come from?

4. 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 falling. The turbines convert kinetic energy to electricity.

5. Gravitational energy
Black holes generate energy from matter falling into them.

7. 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)

8. Luminosity
Gravitational energy is converted to kinetic energy as particles fall towards BH
Efficiency of generators:
Chemical burning < %
Nuclear burning < 1%
Non-rotating black hole = 6%
Rotating black hole = 42%

9. A quasar varies in brightness by a factor of 2 in 10 days
A quasar varies in brightness by a factor of 2 in 10 days. What does this tell us about the quasar?
It has a large magnetic field.
It is quite small.
It must be highly luminous.
It cannot emit radio waves.

10. 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

11. 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!

12. Explicitly shown here dotted line is derived from the sim and the solid line is the expected test-particle velocity
Track each other to a few parts in 1000
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]

13. 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

14. 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 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.
d = c x t
c = speed of light
d = diameter

15. How does the accreting matter lose its angular momentum?
What happens to the gravitational potential energy of the infalling matter?

16. 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..

17. So, the total luminosity liberated by accreting a flow of matter is
Total luminosity of disk depends on inner radius of dissipative part of accretion disk
Initial energy (at infinity)
Mass flow rate
Final energy

18. Question
It seems like only half of the gravitational potential energy of the accreting matter is “liberated”. What happens to the other half?
A. It is carried outwards with the angular momentum.
B. It crosses the inner-edge of the disk as kinetic energy
C. Its crossed the inner-edge of the disk as thermal energy (heat)
D. None of the above

19. 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

20. For a disk that extends down to the ISCO for a non-rotating black hole, simple Newtonian calculation gives…
More detailed relativistic calculation gives…

21. 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.

22. 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)

23. Consider two consecutive rings of the accretion disk.
The torque exerted by the outer ring on the inner ring is

24. Viscous dissipation per unit area of disk surface is given by
Evaluating for circular Newtonian orbits (i.e., “Keplerian” orbits),

25. 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?

26. 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.

28. Accretion Disk Temperature Structure
The accretion disk (AD) can be considered as rings or annuli of blackbody emission. r
Dissipation rate, D(R) is
It is assumed that the disk is geometrically-thin and optically-thick in the z-direction. Thus each annular element of the disk radiates roughly as a blackbody with a temperature T(R) , where :
s x T^4(R) = D(R)
where D(R) is the dissipation rate and s x T^4 is the blackbody flux.
R* is the radius of the black hole (or compact object).
Dissipation through the disk is independent of the viscosity in the disk – and the dissipation rate is the energy flux through the faces of the thin disk.
Thus if the disk is optically-thick in the z-direction, we are justified in assuming that the dissipation rate is equivalent to the blackbody emission.
= blackbody flux

29. Disk Temperature Thus temperature as a function of radius T(R): and if
Substituting the blackbody flux equation into the dissipation equation gives the temperature of the disk as a function of radius. At radii larger than the radius of the compact body, the temperature is given by the equation shown. Note that the temperature decreases with radius with a power –0.75.
and if
then for

30. Disk Spectrum Flux as a function of frequency, n Total disk spectrum
Log n*Fn
The total disk spectrum is the sum of the emission from each of the annuli that make up the disk.
The emission is dominated by the hottest regions ie from the annuli closest to the black hole.
3. At long wavelengths therefore, the spectrum has the form of the Rayleigh-Jeans tail where the flux rises with the square of the frequency
4. At short wavelengths, the Wien form dominates and the flux falls with e^-nu.
Annular BB emission
Log n

31. Black Hole and Accretion Disk
For a non-rotating spherically symetrical BH, the
innermost stable orbit occurs at 3rg or :
and when
The distance to the inner edge of the accretion disk is proportional to the mass of the central black hole.
The temperature, on the other hand, decreases as the radius increases.
Thus the inner edges of large mass black holes are relatively cool, while those of low mass black holes are relatively hot.
This means that disks around black holes in AGN are cooler than those around Galactic black hole candidates.

32. Fe Ka Line
Fluorescence line observed in Seyferts – from gas with temp of at least a million degrees.
X-ray
FeKa e-
1. An X-ray photon collides with an Fe ion, removing an electron from an inner K or L shell. The ion may return to a lower energy state by emitting an electron from a higher shell (this is known as the ‘Auger effect’) – or by a radiative transition. The relative probability of a radiative transition is known as the fluorescence yield.
2. The energy of the Kalpha line depends on the number of electrons present and it increases as the line becomes more highly ionized, reaching 6.9keV for FeXXVI.
3. The ionization state observed indicates gas temperatures which are relatively cool (about a million degrees) and the strength (ie the equivalent width) is quite high, indicating that the gas producing the Fe line has a high covering factor.
4. Fe Kalpha emission from an accretion disk fits these observations very well, providing a high covering factor to the source of X-rays (probably the accretion disk corona) without obscuring our line of sight.
5. It also has the right temperature in the inner regions, where the line is thought to originate.

33. Radio Galaxies and Jets
150 kpc
Radio Lobes
Cygnus-A →
VLA radio image at
n = Hz
the closest powerful
radio galaxy
(d = 190 Mpc)
5.7 Mpc
Radio Lobes
← 3C 236 Westerbork radio image
at n = 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

34. More about Accretion Disks
Disk self-gravitation is negligible so material in differential or
Keplerian rotation with angular velocity WK(R) = (GM/R3)1/2
If n is the kinematic viscosity
for rings of gas rotating,
the viscous torque
exerted by the outer
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
= Hr is the surface density with H (scale height) measured
in the z direction. Q

35. 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) • •

36. 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 • • •

37. Jet power scales with accretion disk power Qjet = qj/l · Ldisk
Model applicable to
quasars
LLAGN
X-ray binaries

38. At lower accretion rates disks become less and less prominent.
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 …)