Transcript Lecture 13 - BH 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
• 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)
T  R    M d  J  R   J 0   M d J  R    M d GMR
– dissipation per unit area is then

d
T d  3GMM d
   πr R
dz 
3
dR
2 R dR
4 R

– 3 x binding energy, because of viscous dissipation
Thin Disk Radiation
• if dissipated heat all radiated away, then
3GMM d
4
 rad  2 T 
4 R3
• this gives temperature distribution T ~ R3/4
• Integrating over the disk gives spectrum
2
• around a BH, energy release is ~ 0.1c M d
• Observed luminosities from, e.g. Sag A* appear
4 2
to be as low as 10 c M d
• 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
And now for something completely
different...
Ruden
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
Ruden
determines initial
conditions.
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):

2G  m1  m2 
ve2 
    a1  a2  1 
, where ve 
2
a1  a2
  V  
so a planet accreting small planetesimals will have
2
2


v
2
e
 lV   a 1 
, with p'mal density  l
2
dt
  V  
dm p
Planet Growth
• Orderly growth by planetesimal accretion has
long time scale:
Ruden 99
• 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
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
Gmp GM *  rH 
mp 
3
3
 2    rH  r 

2
rH
r  r 
 M* 
• 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 density Σ from
2
3
planet at rp
r
mp 
2 4 p  
Tt f  p rp   

 H   M* 
2
• Viscous torque
2
2 4 H 
Tv  3r  3 r  
 r 
• Gap opened if Tt > Tv which means
52
mp
3  H 
1/ 2

  
M*
f  r 
• In solar system this is about 75 M  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
1.
2.
3.
4.
5.
6.
7.
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
Lyot