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): Particle bigger than molecule mean-free-path (Stokes, i.e. hydrodynamic regime). Complex equations
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)
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!)
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 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
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:
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 particles Also 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: 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:
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) Peak at 1 meter (at 1 AU) Weidenschilling 1980
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 Disk surface Dust midplane layer Equatorial plane Nakagawa, Sekiya & Hayashi
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:
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
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>0.25 = Stable Ri<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 Resulting patterns differ for different particle size: z/h y/h centimeter- sized grains meter-sized bodies Johansen & Klahr (2006) (see also Sekiya 1998)
Particle concentrations in vortices Klahr & Henning 1997