1. The formation of stars and planets
Day 4, Topic 2:
Particle motion in disks:
sedimentation, drift
and clustering
Lecture by: C.P. Dullemond

2. Epstein regime, Stokes regime...
Particle smaller than molecule mean-free-path (Epstein, i.e. single particle collisions):
Particle bigger than molecule mean-free-path (Stokes, i.e. hydrodynamic regime).
Complex equations

3. Vertical motion of particle
Vertical equation of motion of a particle (Epstein regime):
Damped harmonic oscillator:
No equator crossing (i.e. no real part of ) for:
(where =material density of grains)

4. Vertical motion of particle
Conclusion:
Small grains sediment slowly to midplane. Sedimentation velocity in Epstein regime:
Big grains experience damped oscillation about the midplane with angular frequency:
and damping time:
(particle has its own inclined orbit!)

5. Vertical motion of small particle

6. Vertical motion of big particle

7. Turbulence stirs dust back up
Equilibrium settling velocity:
Turbulence vertical mixing:

8. Turbulence stirs dust back up
Distribution function:
Normalization:

9. Turbulence stirs dust back up
Time-dependent settling-mixing equation:
Time scales:
Dust can settle down to tsett=tturb but no further.

10. Turbulence stirs dust back up

11. Settling toward equilibrium state

12. Radial drift of large bodies
Assume swinging has damped. Particle at midplane with
Keplerian orbital velocity.
Gas has (small but significant) radial pressure gradient.
Radial momentum equation:
Estimate of dP/dr :
Solution for tangential gas velocity:
25 m/s at 1 AU

13. Radial drift of large bodies
Body moves Kepler, gas moves slower.
Body feels continuous headwind. Friction extracts angular momentum from body:
= friction time
One can write dl/dt as:
One obtains the radial drift velocity:

14. Radial drift of large bodies
Gas slower than dust particle: particle feels a head wind.
This removes angular momentum from the particle.
Inward drift

15. Radial drift of small dust particles
Also dust experiences a radial inward drift, though the
mechanism is slightly different.
Small dust mo
ves with the gas. Has sub-Kepler velocity.
Gas feels a radial pressure gradient. Force per gram gas:
Dust does not feel this force. Since rotation is such that gas is in equilibrium, dust feels an effective force:
Radial inward motion is therefore:

16. Radial drift of small dust particles
Gas is (a bit) radially supported by pressure gradient. Dust not! Dust moves toward largest pressure.
Inward drift.

17. In general (big and small)
Peak at 1 meter (at 1 AU)
Weidenschilling 1980

18. 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)

19. Problem
Radial drift is very fast for meter sized bodies ( 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!

20. 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
Disk surface
Dust midplane layer
Equatorial plane
Nakagawa, Sekiya & Hayashi

21. 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
Toomre number for dust layer:

22. Kelvin-Helmholtz instability
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
Weidenschilling 1977, Cuzzi 1993, Sekiya 1998

23. Kelvin-Helmholtz instability
Vertical stratification
Shear between dust layer and gas above it:
Two ‘forces’:
Shear tries to induce turbulence
Vertical stratification tries to stabilize things
Richardson number:
Ri> = Stable
Ri< = Kelvin-Helmholtz instability: turbulence

24. Kelvin-Helmholtz instability
Ri = 0.07, Re = 300

25. Kelvin-Helmholtz instability

26. Model sequence...
Johansen & Klahr (2006)

27. Equilibrium thickness of layer
Resulting patterns differ for different particle size:
z/h
y/h
centimeter- sized grains
meter-sized bodies
Johansen & Klahr (2006) (see also Sekiya 1998)

28. Particle concentrations in vortices
Klahr & Henning 1997