Disk Topics: Black Hole Disks, Planet Formation

12 May 2003

Astronomy G9001 - Spring 2003

Prof. Mordecai-Mark Mac Low

Black Hole Accretion Disks

• In protostellar accretion disks, radiation is always efficient, and the assumption Ωr >> cs is good.– thin disk approximation

• Now turn to compact objects– deeper potential wells produce higher

temperatures – far more energy must be lost to radiation – Some observed supermassive BHs have little

radiation (Sag A* is the classic example)– How does accretion proceed?

Thin Disk Dissipation• Thin disk approximation

• ν = αcs2/Ω (or πrφ = αP) prescription for viscosity

• classic radiative disk (Shakura & Sunyaev 1973, Novikov & Thorne 1973)– viscous heating balances radiative cooling– steady mass inflow gives torque (Sellwood)

– dissipation per unit area is then

– 3 x binding energy, because of viscous dissipation

0d d dR M J R J M J R M GMR T

3

3

2 4d

r

d d GMMR dz

dR R dR R

Tπ

Thin Disk Radiation

• if dissipated heat all radiated away, then

• this gives temperature distribution T ~ R3/4

• Integrating over the disk gives spectrum

• around a BH, energy release is ~

• Observed luminosities from, e.g. Sag A* appear to be as low as

• How is BH accreting so much mass without radiating?

43

32

4d

rad

GMMT

R

20.1 dc M

4 210 dc M

ADAF/CDAF• Narayan & Yi (1992) and others proposed that the

energy is advected into the BH before it can be radiated: advection dominated accretion flow

• Numerical models made clear that the extra energy produces a convectively unstable entropy gradient in the radial direction, as well as unbinding some of the gas entirely

• convection dominated accretion flow proposed as elaboration of ADAF– outward convective transport balances inward

viscous transport, leaving disk marginally stable– analogous to convective zone in stars

Problems with ADAF/CDAF

• Balbus (2000) points out that convection and MRI cannot be treated as independent forces– instead a single instability criterion must be found– this reduces to the MRI, so no balance exists

• Balbus & Hawley (2002) analyze non-radiative MHD flows. – convectively unstable modes overwhelmed by MRI– balanced transport implies that convection recovers

energy produced by viscous dissipation, resulting in a dissipation-free flow: but this violates 2nd Law of Thermodynamics!

Non-Radiative Accretion Flow• Hawley & Balbus (2002)

simulate non-radiative MHD flow numerically, finding outflow and unsteady, slow, accretion

And now for something completely different...

Planet Formation in Disks• Solar planets formed

from protoplanetary disk with at least 0.01 M of gas (Minimum Mass Solar Nebula)

• Observed disks have comparable masses

• Disk evolution determines initial conditions.

Ruden

Ruden1999

Grain Dynamics• Gas moves on slightly sub-Keplerian orbits due to

radial pressure gradient

• Grains move on Keplerian orbits– grains with a < 1 cm feel drag FD = – (4/3) πa2ρcs(Δv)

– coupling time tc = m Δv / FD , so small Ωtc = aρd / Σ means particles drop towards star, large remain.

• Vertical settling also depends on Ωtc

– vertical gravity gz = (z/r)GM* / R2 = Ω2z

– settling time ts = z / vz = Ω-1 (Ωtc)-1 = Σ / (aρd Ω)

– small grains with Ωtc << 1 take many orbits to settle

– coagulation vital to accumulate mass in midplane

Planetesimals• Big enough to ignore gas drag over disk lifetime• How do they accumulate from dust grains?

– gravitational instability requires very cold disk with Δv ~ 10 cm s-1 (Goldreich & Ward)

– shear with disk enough to disrupt most likely– Collisional coagulation main alternative (Cuzzi et al 93)

• Planetesimals collide to form planets– gravitational focussing gives cross-section (Safronov):

22 1 2

1 2 21 2

22

l2

21 , where

so a planet accreting small planetesimals will have

1 , with p'mal density

ee

p el

G m mva a v

a aV

dm vV a

dt V

Planet Growth• Orderly growth by planetesimal accretion has

long time scale:

• Velocity dispersion Δv must remain low to enhance gravitational focussing.

• Dynamical friction transfers energy from large objects to small ones– large objects have lowest velocity dispersion and so

largest effective cross sections.– collisions between them lead to runaway growth

Ruden 99

Final Stages of Solid Accretion• Runaway growth continues until material has

been cleared out of orbits within a few Hill radii– Hill radius determined by balance between gravity of

planet and tidal force of central star

• Protoplanet sizes reach 5–10% of final masses• Final accumulation driven by orbital dynamics of

protoplanets– major collisions of planet-sized objects an essential

part of final evolution– random events determine details of final configuration

of solid planets

3 3*2 2

*

p pHH

H

Gm mGM rr r

r r r M

Gas Accretion• Above critical mass of 10–15 M planetary

atmospheres no longer in hydrostatic equilibrium– heating comes from p’mal impacts– increasing heating required to balance radiative

cooling of denser gas atmospheres (Mizuno 1980)

– collapse of atmosphere occurs until heating from gravitational contraction balances cooling

– rapid accretion can occur

• Final masses determined either by:– destruction of disk by photoevaporation or tides– gap clearing in gaseous disk

Gap Formation & Migration• Giant planets

exert tidal torques on surrounding gas, repelling it and forming a gap in disk.

• Disk also exerts a torque on the planet, causing radial migration.

Gap Formation• Tidal torque on disk with surface density Σ from

planet at rp

• Viscous torque

• Gap opened if Tt > Tv which means

• In solar system this is about 75 or roughly Saturn’s mass.

23

2 4

*

p pt p p

r mT f r

H M

22 2 43 3v

HT r r

r

5 21/ 2

*

3pm H

M f r

M

Observations• Disk Observations

– spectral energy distributions• density distribution• gaps and inner edges

– dust disks (β Pic, Vega)• Poynting-Robertson clears in much less than t*

• presence of dust disk indicates colliding planetesimals

– Proplyds [Protoplanetary disks], seen in silhouette

• Indirect Dynamical Observations– radial velocity searches

• need accurate spectroscopy: calibrator (iodine) in optical path

– radial distance changes: pulsar timing– astrometry: next generation likely productive (SIM)

Observations• Microlensing of planet

– superposes spike on stellar amplification curve– can also shift apparent position of star

• Direct detections– transits

• photometry - eclipse of star (or of planet!)

• transmission spectroscopy of atmosphere

– direct imaging• adaptive optics

• interferometry

• coronagraphs (+ AO = Oppenheimer @ AMNH)

Search techniques1. Kepler: space-based

transit search2. COROT: same3. Doppler: 3m/s

ground-based4. SIM = Space

Interferometry Mission

5. FAME = next ESA astrometry mission

6. ground based transit search

7. Lyot = AO + coronagraph (BRO)

habitable zone

Lyot