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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 >> c s is good. thin disk approximation

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disk topics black hole disks planet formation

Disk Topics: Black Hole Disks, Planet Formation

12 May 2003

Astronomy G9001 - Spring 2003

Prof. Mordecai-Mark Mac Low

black hole accretion disks
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 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
thin disk radiation
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?
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
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
Non-Radiative Accretion Flow
  • Hawley & Balbus (2002) simulate non-radiative MHD flow numerically, finding outflow and unsteady, slow, accretion
planet formation in disks



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.


grain dynamics
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
  • 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):
planet growth
Planet Growth
  • Orderly growth by planetesimal accretion has long time scale:
  • Velocity dispersion Δvmust 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
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
gas accretion
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
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
Gap Formation
  • Tidal torque on disk with surface densityΣ from planet at rp
  • Viscous torque
  • Gap opened if Tt > Tvwhich means
  • In solar system this is about 75 or roughly Saturn’s mass.
  • 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)
  • 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 techniques
Search techniques


  • Kepler: space-based transit search
  • COROT: same
  • Doppler: 3m/s ground-based
  • SIM = Space Interferometry Mission
  • FAME = next ESA astrometry mission
  • ground based transit search
  • Lyot = AO + coronagraph (BRO)