The formation of stars and planets

Day 4, Topic 2:

Particle motion in disks:sedimentation, drift

and clustering

Lecture by: C.P. Dullemond

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

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)

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!)

Vertical motion of small particle

Vertical motion of big particle

Turbulence stirs dust back up

Equilibrium settling velocity:

Turbulence vertical mixing:

Turbulence stirs dust back up

Distribution function:

Normalization:

Turbulence stirs dust back up

Time-dependent settling-mixing equation:

Time scales:

Dust can settle down to tsett=tturb but no further.

Turbulence stirs dust back up

Settling toward equilibrium state

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

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

Radial drift of large bodies

Gas slower than dust particle: particle feels a head wind.This removes angular momentum from the particle.Inward drift

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

Radial drift of small dust particles

Gas is (a bit) radially supported by pressure gradient. Dust not! Dust moves toward largest pressure.Inward drift.

In general (big and small)

Weidenschilling 1980

€

−dr

dt

Peak at 1 meter (at 1 AU)

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)

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!

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

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:

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

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

Kelvin-Helmholtz instability

Ri = 0.07, Re = 300

www.riam.kyushu-u.ac.jp/ship/STAFF/hu/flow.html

Kelvin-Helmholtz instability

Model sequence...

Johansen & Klahr (2006)

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:

Particle concentrations in vortices

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Klahr & Henning 1997