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 falling. The turbines

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

/

M

Sun )

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.

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 ~10 up to 10

14 6-9 M

 , and radiate

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 = 10 10 cm/s

NS, BH

Measuring a Quasar’s Black Hole

A

Light travel time effects

d = c x t

c = speed of light d = diameter If photons leave A and B at the same time, A arrives at the observer a time t ( = d / c ) later.

B

If an event happens at A and takes a time

d

t, then we see a change over a timescale t+

d

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.

• 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. 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 – R isco =6GM/c 2 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

r

Structure

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

Dissipation rate, D(R) is  3

GM

 8 p

r

3    1 

R

*

r

 1 / 2   

=

blackbody flux  

T

4 (

r

)

Disk Temperature

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

T

(

r

)     3

GM

 8 p

r

3     1 

R r

*  1 / 2     1 / 4 and if then for

T

*    3

GM

 8 p

R

* 3    1 / 4

r



R

*

T



T

* 

r

/

R

*   3 / 4

Disk Spectrum

Flux as a function of frequency, n

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 3r g or :

r

min  6

GM c

2 and when

R



R

*

T



T

* 

R

/

R

*   3 / 4

Fe K

a

Line

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

X-ray FeK

a

e-

Radio Galaxies and Jets

Cygnus-A → VLA radio image at n = 1.4.10

9 Hz - the closest powerful radio galaxy (d = 190 Mpc)

150 kpc

Radio Lobes Radio Lobes

5.7 Mpc

← 3C 236 Westerbork radio image at n = 6.08.10

8 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 If n is the kinematic viscosity (GM/R ) 1/2 for rings of gas rotating, the viscous torque

Q

W K (R) = exerted by the outer ring on the inner will be

Q

Q(R) = 2 p R nS R 2 (d W /dR) (1) where the viscous force per unit length is acting on 2 p R and S = H r 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 4 p RdR, so the radiative dissipation from the disc per unit area is from (1): D(R) = Q(R) W /4 p R = ½ n S  R W ) 2 (2) and since W  W K = (G M/R 3 ) 1/2 differentiate and then D(R) = 9/8 n S Q(R) M/R 3 (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, 3 rd ed., sec 5.3/p 85, 2002) that n S • = (M/3 p [1 – (R * /R) 1/2 ] • where M is the accretion rate and from (2) and (3) we then have • D(R) = (3G M M/8 p R 3 ) [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 …)