Title: Numerical Modeling of Particle Growth and Collective Gas-Grain Interactions
1Planetesimal Formation Numerical Modeling of
Particle Growth, Settling, and Collective
Gas-Grain Interactions S. J. Weidenschilling,
Planetary Science Institute Cambridge, UK,
Sept 2009
2Viktor Sergeyevich Safronov11/XI/1917 -
18/IX/1999
- SSi Monumentum Requiris, Circumspice
3Outline Numerical Models
- Structure of a particle layer in the midplane of
a laminar nebula, where settling is in
equilibrium with shear-generated turbulence - Settling with coagulation conditions for growth
- Accretion of larger bodies effects of initial
planetesimal size
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5- Solid bodies are not supported by the pressure
gradient - Any solid body must move relative to the gas
- Particles move toward higher pressure, i.e.,
generally inward - Radial and transverse velocities are
size-dependent relative velocity between any two
bodies can be easily calculated - if they are
isolated
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7Collective Effects
- Particles settle to form a dense layer in the
nebular midplane - Gravitational instability and collisional
coagulation both require (for different reasons)
this layer to be much denser than the gas - In such a layer, gas is dragged by the particles,
and no longer moves at the pressure-supported
velocity, but closer to the Kepler velocity - Relative velocities depend not just on sizes, but
on the entire population of particles
8- Shear between the layer and the surrounding gas
causes turbulence - The thickness of the layer and its vertical
structure are determined by the balance between
settling and turbulent diffusion - Shear-induced turbulence may halt settling at
densities too low for gravitational instability
9Response of the Gas to Particle Loading
- Nakagawa et al. (1986) derived coupled equations
particles move inward as gas moves outward - Angular momentum is conserved equal and
opposite radial mass fluxes of particles and gas - The solution assumes laminar flow, with local
balance of momentum at any given level - However, turbulent viscosity causes significant
vertical transport of momentum, affecting the
radial and transverse velocities of the gas - The momentum flux must balance overall, but does
not hold locally
10Ekman Length
A characteristic length scale for the thickness
of a turbulent boundary layer of a disk rotating
in a fluid is the Ekman length LE , defined as
LE (?t / ?K)1/2 where ?t is the
turbulent viscosity, and ? ?K is the rotation
frequency. After Cuzzi et al. (1993) we take ?t
(?V/Re)2/?K, where Re 102 is a critical
Reynolds number. This implies LE
?V/(Re?K) where ?V is the velocity difference
between the midplane and large Z. Turbulence
decays over a distance LE.
11Richardson Number
- The Richardson number (Ri) is a measure of the
stability of a stratified shear flow. If a fluid
element is displaced vertically, work is done
against gravity and buoyancy, while kinetic
energy is extracted from the flow due to the
mismatch of velocity due to the shear. Ri is
dimensionless, defined as -
- The flow becomes turbulent if Ri lt 0.25.
12Rossby and Stokes numbers
- In weak turbulence (Ri 1/4), the eddy frequency
??is imposed by the systems rotation frequency,
?? - In strong turbulence (Ri ltlt 1/4), eddies have
higher frequency, ??????Ro ?K, where Ro 10-102
is the Rossby number - The Stokes number St te /?? where te is the
response time to the drag force - Particle random velocity Vturb /(1St)
13Vertical Transport of Momentum by Turbulent
Viscosity (Youdin Chiang 2004)
- Particle concentration is greater nearer the
central plane, so rotation is faster - If the layer is turbulent, viscosity is
significant - The vertical velocity gradient causes upward flow
of angular momentum - Gas in the midplane flows inward, while that near
the surface of the layer flows outward - Inward and outward mass fluxes are equal, but
higher particle concentration in midplane yields
net inflow of particles, if particles and gas are
perfectly coupled
14Numerical Modeling of Particle Layer
- Divide layer into a series of levels, with
assumed particle abundance at t 0 - Compute radial and transverse velocities of gas
due to particle-gas momentum exchange, using
Nakagawa model assume additive for more than one
size - Average velocities over Ekman length
- Assume turbulent velocity is proportional to the
velocity difference between local gas and
particle-free gas, and a function of Ri - Turbulence propagates between levels, decaying
exponentially over Ekman length - Assume eddy timescale is a function of Ri such
that ??varies from ?K to 2 Ro ?K as Ri
approaches zero
15- Compute stress tensor and gas radial velocity due
to vertical shear - Add gas radial velocities due to particle-gas
momentum exchange - Solve for particle radial and transverse
velocities, generalized to include radial motion
of gas - Compute net radial mass fluxes of particles
(inward) and gas (outward) should have equal
magnitudes - If particle flux is too large (small), increase
(decrease) gas velocity due to particle-gas
momentum exchange and solve again until fluxes
balance - Distribute particles vertically by settling and
diffusion - Iterate until a steady state is reached
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25- Relative velocities in midplane for a mixture
of mm- and m-sized bodies, vs. mass fraction in
m-sized bodies
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27Collective Effects and Impact Velocities
- Increased Vrel between small particles due to
shear-induced turbulence - Impact speeds of small particles onto m-sized
bodies decreased due to smaller ?V ( 10 m/s
instead of 50 m/s) - Continued growth to gt m-size results in
decoupling, increased ?V, higher impact speeds
again for small onto large - Growth may be limited by erosion, unless gt
m-sized bodies can accrete each other?
28- Inward flux of m-sized bodies causes net
outward flux of small grains carried with gas
fast outflow in dense sublayer exceeds radial
drift in the thicker layer
29- Streaming instability and pileups? Effective
drift velocity varies inversely with surface
density only for sizes gt decimeter or surface
density a few times nominal
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31Results of Equilibrium Layer Models
- Relative velocities are not simple functions of
particle sizes, but depend on the ensemble size
distribution and abundances - Inward and outward flows of gas and particles at
different Z, even for a single size - Multiple sizes result in strong stratification,
size sorting - A small fraction of mass in large bodies can
cause net outward flow of small particles, with
radial mixing
32Models with Coagulation and Settling
- Vertical layering up to 2 scale heights, finer
resolution near central plane - Logarithmic size bins from d 10-4 cm fractal
density to 1 cm - Relative velocities thermal, differential
settling, turbulence (alpha, also shear-generated
in midplane) - Gravitational stirring (Safronov number) and
cross-section for large bodies
33Impact Outcomes
- Impact strength S (erg/g) if energy density gt S,
target is disrupted shattering velocity Vshat
(2S)1/2 - If projectile much smaller than target its mass
added to target mass of escaping ejecta
proportional to impact energy - Excavation parameter Cex such that
- mex 0.5 Cex mpV2
- Transition from net gain to erosion at critical
velocity Vc (2/Cex)1/2
34- For perfect sticking, large bodies coagulate
and settle to central plane in a few thousand
orbital periods
35- ????V 52 m/s, Vturb 0
- S 105 erg/g, Vshat 4.5 m/s, Vc 10 m/s
36- ????V 52 m/s, Vturb 0
- S 106 erg/g, Vshat 14 m/s, Vc 32 m/s
37- ?V 52 m/s, ? 10-5, Vturb 5.3 m/s
- S 105 erg/g, Vshat 4.5 m/s, Vc 32 m/s
38- ????V 52 m/s, Vturb 0
- S 104 erg/g, Vshat 1.4 m/s, No Erosion
39Results of Coagulation/Settling Modeling
- Settling/growth timescale for large bodies
few x 103 orbital periods - If impacts of small particles result in net
erosion above a critical velocity Vc lt ?V, growth
can be halted - If Vturb is small (? lt 10-5), growth is
possible even for very low impact strength (104
erg/g), if erosion is limited (experimental
data?) - Critical parameter Velocity threshold for net
gain/loss when a small particle hits a much
larger one?
40Is There an Observational Constraint on Sizes of
Original Planetesimals?
- Canonical km-sized planetesimals from Goldreich
and Ward (1972) model for gravitational
instability of a dust layer no reason to prefer
that size - Asteroid belt experienced accretion of large
embryos, dynamical depletion, and 4 Gy of
collisional evolution, but may retain some trace
of its primordial size distribution
41- Morbidelli et al. (2009)
- Excitation and depletion of early asteroid belt
requires accretion of large ( 104 km) embryos
within a few My - Present-day size distribution shows excess of
100 km bodies relative to a power law of
equilibrium slope had to have formed early - Survival of Vestas crust implies early belt was
deficient in bodies 10 - 100 km - Can accretion produce these features from some
initial characteristic size?
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43Gravitational Accretion Code
- Multiple zones of semimajor axis 15 zones from 2
to 3.5 AU - Collisions for bodies in overlapping orbits
impact rates and velocities - Gravitational stirring of eccentricities and
inclinations - Accretion, cratering, disruption depend on impact
energy - Logarithmic diameter bins fragments below
minimum size are lost
44 45- d0 100 km embryos too small
46- d0 10 km too many asteroids 10 km?
47 d0 0.1 km
48Still To Do
- Vary initial size from 0.1 km add dispersion
about the mean - Add planetesimals over some interval, instead of
instantaneously - Test behavior if mass ground down into small (lt
10 m) fragments is recycled into new
planetesimals instead of lost -
- Planetesimals may have started small!