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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 >> 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

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

- 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

1999

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

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):

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

- 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

- 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 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.

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 techniques

Lyot

- 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)

habitable

zone

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