Accretion Disks

From Solar Nebula to Quasars

Energy Sources

Two Main Energy SourcesIn AstrophysicsI) Gravitational PotentialEnergyII) 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 falling. The turbines convert kinetic energy to electricity.

Gravitational energy

Black holes generate energy from matter falling into them.

Rotating black holesFor 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. It has a large magnetic field.

2. It is quite small.

3. It must be highly luminous.

4. It cannot emit radio waves.

Lecture Notes 4

Accretion DisksAccretion 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.

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)

NS,BH

Variability = size scale/speed of light

Again, Neutron Stars andBlack Holes likely Candidates (either in an Accretion disk or on the NS surface).

2 10km/c = .6 msc = 1010cm/s

Measuring a Quasar’s Black Hole

Light travel time effects

A B

d = c x t

If photons leave A and B at the same time, A arrives at the observer

a time t ( = d / c ) later.

If an event happens at A and takes a time t, then we see a change over

a timescale t+t. This gives a maximum value for the diameter, d, because we know that our measured

timescale must be larger than the light crossing time.

c = speed of lightd = diameter

• 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

• Total luminosity of disk depends on inner radius of dissipative part of accretion disk

Initial energy (at infinity)

Final energy

Mass flow rate

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

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 non-rotating 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

The accretion disk (AD) can be considered as rings or annuli of blackbody emission.r

2/1

*3

18

3

r

R

r

MGM

Dissipation rate, D(R) is

= blackbody flux

)(4 rT

Disk Temperature

Thus temperature as a function of radius T(R): 4/1

2/1

*3

18

3)(

r

R

r

MGMrT

then for *Rr 4/3** / RrTT

4/1

3*

* 8

3

R

MGMT

and if

Disk Spectrum

Flux as a function of frequency,

Log

Log

*F

Total disk spectrum

Annular BB emission

Black Hole and Accretion DiskFor a non-rotating spherically symetrical BH, the innermost stable orbit occurs at 3rg or :

2min

6

c

GMr

and when *RR 4/3** / RRTT

Fe K Line

Fluorescence line observed in Seyferts – from gas with temp of at least a million degrees.

X-ray

e-

FeK

Radio Galaxies and Jets

Cygnus-A →VLA radio image at = 1.4.109 Hz- the closest powerfulradio galaxy (d = 190 Mpc)

← 3C 236 Westerbork radio image at = 6.08.108 Hz – a radio galaxy of very large extent (d = 490 MPc)

Jets, emanating from a central highlyactive galaxy, are due to relativisticelectrons that fill the lobes

150 kpc

Radio Lobes

5.7 Mpc

Radio Lobes

More about Accretion DisksDisk self-gravitation is negligible so material in differential or

Keplerian rotation with angular velocity K(R) = (GM/R3)1/2

Q

Q

If is the kinematic viscosity for rings of gas rotating, the viscous torque exerted by the outer ring on the inner will be

Q(R) = 2R R2 (d/dR) (1)

where the viscous force per unit length is acting on 2R and = His 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 dR/ring

Each ring has two plane faces of area 4RdR, so the radiative dissipation from the disc per unit area is from (1):

D(R) = Q(R) /4R = ½ R)2 (2)

and since K = (G M/R3)1/2

differentiate and then

D(R) = 9/8 Q(R) M/R3 (3)

•

• •

More about Accretion Disks

•

From a consideration of radial mass and angular momentumflow in the disk, it can be shown (Frank, King & Raine, 3rd ed., sec 5.3/p 85, 2002) that

= (M/3 [1 – (R*/R)1/2]

where M is the accretion rate and from (2) and (3) we thenhave

D(R) = (3G M M/8R3) [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 …)Esin, Narayan et al. (1997 …)