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The formation of stars and planets
Day 4, Topic 2:
Particle motion in disks:sedimentation, drift
and clustering
Lecture by: C.P. Dullemond
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Epstein regime, Stokes regime...
Particle smaller than molecule mean-free-path (Epstein, i.e. single particle collisions):
€
f fric = 43 ρ gascsσ v
Particle bigger than molecule mean-free-path (Stokes, i.e. hydrodynamic regime).
Complex equations
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Vertical motion of particle
€
d2z
dt 2= −ΩK
2 z
Vertical equation of motion of a particle (Epstein regime):
€
− 43 ρ gascs
σ
m
⎛
⎝ ⎜
⎞
⎠ ⎟dz
dt
Damped harmonic oscillator:
€
z(t) = z0eiω t
€
ω =1
2iΓ ± 4ΩK
2 − Γ 2[ ]
€
Γ
No equator crossing (i.e. no real part of ) for:
€
Γ > 2ΩK
€
a <ρ gascs2ξ ΩK
€
m = 4π3 ξ a
3
€
σ =π a2
€
σm
>3
2
ΩK
ρ gascs
(where =material density of grains)
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Vertical motion of particle
Conclusion:
Small grains sediment slowly to midplane. Sedimentation velocity in Epstein regime:
€
vsett =3ΩK
2 z
4ρcs
m
σ
Big grains experience damped oscillation about the midplane with angular frequency:
€
ω =ΩK
and damping time:
€
tdamp ≈1
Γ
(particle has its own inclined orbit!)
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Vertical motion of small particle
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Vertical motion of big particle
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Turbulence stirs dust back up
Equilibrium settling velocity:
Turbulence vertical mixing:
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Turbulence stirs dust back up
Distribution function:
Normalization:
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Turbulence stirs dust back up
Time-dependent settling-mixing equation:
Time scales:
Dust can settle down to tsett=tturb but no further.
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Turbulence stirs dust back up
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Settling toward equilibrium state
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Radial drift of large bodies
Assume swinging has damped. Particle at midplane withKeplerian orbital velocity.
Gas has (small but significant) radial pressure gradient.Radial momentum equation:
€
dP
dr− ρ
vφ2
r= −ρ
GM*
r2
€
dP
dr≅ −2
P
r= −2
ρ cs2
r
Estimate of dP/dr :
€
vφ2 = vK
2 − 2cs2
Solution for tangential gas velocity:
€
(vK - vφ ) ≅cs
2
vK25 m/s at 1 AU
€
dP
dr− ρ
vφ2
r= −ρ
vK2
r
€
−2ρcs
2
r− ρ
vφ2
r= −ρ
vK2
r
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Radial drift of large bodies
Body moves Kepler, gas moves slower.
Body feels continuous headwind. Friction extracts angular momentum from body:
€
dl
dt≈
dlK
dt=
d GM*r
dt
One can write dl/dt as:
One obtains the radial drift velocity:
€
dr
dt≈ −
2(vK - vφ )
τ fricΩK
€
dr
dt≈ −
2cs2
τ fric vK2r€
=1
2
GM*
r
dr
dt
€
=12 ΩK r
dr
dt€
dl
dt= −
(vK − vφ ) r
τ fric
€
τ fric = friction time
€
l = vφ r
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Radial drift of large bodies
Gas slower than dust particle: particle feels a head wind.This removes angular momentum from the particle.Inward drift
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Radial drift of small dust particlesAlso dust experiences a radial inward drift, though the mechanism is slightly different.
Small dust moves with the gas. Has sub-Kepler velocity.Gas feels a radial pressure gradient. Force per gram gas:
€
fgas = −1
ρ
dP
dr≅ 2cs
2
r
Dust does not feel this force. Since rotation is such that gas is in equilibrium, dust feels an effective force:
€
feff = − fgas = −2cs
2
rRadial inward motion is therefore:
€
dr
dt= −2
cs2
rτ fric
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Radial drift of small dust particles
Gas is (a bit) radially supported by pressure gradient. Dust not! Dust moves toward largest pressure.Inward drift.
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In general (big and small)
Weidenschilling 1980
€
−dr
dt
Peak at 1 meter (at 1 AU)
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Fate of radially drifting particles• Close to the star (<0.5 AU for HAe stars; <0.1 AU for TT
stars) the temperature is too hot for rocky bodies to survive. They evaporate.
• Meter-sized bodies drift inward the fastest.• They go through evaporation front and vaporize.• Some of the vapor gets turbulently mixed back outward
and recondenses in the form of dust.
Cuzzi & Zahnle (2004)
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Problem
• Radial drift is very fast for meter sized bodies (102...3 years at 1 AU).
• While you form them, they get lost into evaporation zone.
• No time to grow beyond meter size...
• This is a major problem for the theory of planet formation!
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Massive midplane layer: stop drift
Once Hdust <= 0.01 Hgas the dust density is larger than the gas density. Gas gets dragged along with the dust (instead of reverse).
Gas and dust have no velocity discrepancy anymore: no radial drift
Nakagawa, Sekiya & Hayashi
Equatorial plane
Disk surface
Dust midplane layer
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Goldreich & Ward instability
• Dust sediments to midplane
• When Q<1: fragmentation of midplane layer• Clumps form planetesimals• Advantages over coagulation:
– No sticking physics needed– No radial drift problem
• Problems:– Small dust takes long time to form dust layer (some
coagulation needed to trigger GW instability)– Layer stirred by self-induced Kelvin-Helmholtz turbulence
€
Q =h
r
⎛
⎝ ⎜
⎞
⎠ ⎟M*
π r2 Σdust
Toomre number for dust layer:
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Kelvin-Helmholtz instability
Weidenschilling 1977, Cuzzi 1993, Sekiya 1998
Midplane dust layer moves almost Keplerian (dragging along the gas)
Gas above the midplane layer moves (as before) with sub-Kepler rotation.
Strong shear layer, can induce turbulence.
Turbulence can puff up the layer
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Kelvin-Helmholtz instability
Vertical stratification
Shear between dust layer and gas above it:
Two ‘forces’:1. Shear tries to induce turbulence2. Vertical stratification tries to stabilize things
Richardson number:
€
Ri = −g
ρ
∂ρ
∂z
∂v
∂z
⎛
⎝ ⎜
⎞
⎠ ⎟2
Ri>0.25 = StableRi<0.25 = Kelvin-Helmholtz instability: turbulence
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Kelvin-Helmholtz instability
Ri = 0.07, Re = 300
www.riam.kyushu-u.ac.jp/ship/STAFF/hu/flow.html
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Kelvin-Helmholtz instability
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Model sequence...
Johansen & Klahr (2006)
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Equilibrium thickness of layer
Johansen & Klahr (2006) (see also Sekiya 1998)
z/h
y/h y/h
centimeter- sized grains
meter-sized bodies
Resulting patterns differ for different particle size:
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Particle concentrations in vortices
Zur Anzeige wird der QuickTime™ Dekompressor „YUV420 codec“
benötigt.
Klahr & Henning 1997