Numerical Modeling of Particle Growth and Collective Gas-Grain Interactions

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Planetesimal Formation Numerical Modeling of
Particle Growth, Settling, and Collective
Gas-Grain Interactions S. J. Weidenschilling,
Planetary Science Institute Cambridge, UK,
Sept 2009
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Viktor Sergeyevich Safronov11/XI/1917 -
18/IX/1999

• SSi Monumentum Requiris, Circumspice

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Outline Numerical Models

1. Structure of a particle layer in the midplane of
   a laminar nebula, where settling is in
   equilibrium with shear-generated turbulence
2. Settling with coagulation conditions for growth
3. Accretion of larger bodies effects of initial
   planetesimal size

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• 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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Collective 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

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• 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

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Response 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

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Ekman 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.
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Richardson 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.

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Rossby 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)

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Vertical 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

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Numerical 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

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• 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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• Relative velocities in midplane for a mixture
  of mm- and m-sized bodies, vs. mass fraction in
  m-sized bodies

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Collective 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?

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• 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

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• 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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Results 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

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Models 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

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Impact 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

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• For perfect sticking, large bodies coagulate
  and settle to central plane in a few thousand
  orbital periods

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• ????V 52 m/s, Vturb 0
• S 105 erg/g, Vshat 4.5 m/s, Vc 10 m/s

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• ????V 52 m/s, Vturb 0
• S 106 erg/g, Vshat 14 m/s, Vc 32 m/s

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• ?V 52 m/s, ? 10-5, Vturb 5.3 m/s
• S 105 erg/g, Vshat 4.5 m/s, Vc 32 m/s

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• ????V 52 m/s, Vturb 0
• S 104 erg/g, Vshat 1.4 m/s, No Erosion

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Results 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?

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Is 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

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• 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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Gravitational 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

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• d0 1 km bump is at 10 km

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• d0 100 km embryos too small

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• d0 10 km too many asteroids 10 km?

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d0 0.1 km
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Still 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!