N-body Models of Aggregation and Disruption

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N-body Models of Aggregation and Disruption

• Derek C. Richardson
• University of Maryland

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Overview

• Introduction/the N-body problem.
• Numerical method (pkdgrav).
• Application binary asteroids.
• Non-idealized strength models.
• First results YORP spinup of rubble piles
  spin limits with strength.

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Introduction

• Many dynamical processes in the solar system can
  be modeled by gravity and collisions alone.
  E.g.,
• Reaccumulation after catastrophic disruption
  (collisional or rotational).
• Planetary ring dynamics.
• Planet formation.
• Problems well suited to N-body code.

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The N-body problem

• The orbit of any one planet depends on the
  combined motion of all the planets, not to
  mention the actions of all these on each other.
  To consider simultaneously all these causes of
  motion and to define these motions by exact laws
  allowing of convenient calculation exceeds,
  unless I am mistaken, the forces of the entire
  human intellect.
• Isaac Newton, 1687.

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The N-body problem
Cost N (N 1) / 2 O(N2)
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Tree codes

• Reduce computational cost by treating particles
  in groups.

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Tree codes
Replace many summations with single multipole
expansion around center of mass.
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Tree codes

• Reduce computational cost by treating particles
  in groups.
• Error controlled by opening angle criterion and
  order of expansion.

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Tree codes
Use multipole expansion if opening angle ? lt
?crit.
?
?crit
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Tree codes

• Reduce computational cost by treating particles
  in groups.
• Error controlled by opening angle criterion and
  order of expansion.
• Particles organized into systematic hierarchical
  structure.
• Ideally suited for recursive algorithms.

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Tree codes
E.g. Barnes Hut (1986) two-dimensional tree.
Cost O(N log N)
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Reducing cost further

• Parallel methods
• Distribute work among Np processors.
• N-body problem difficultexploit tree.
• Adaptive/hierarchical timestepping
• Focus work on most active particles.
• Good object-oriented code structure.
• Hard-core optimizations.

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Integrating the equations of motion

• Many techniques for solving coupled linear
  ordinary differential equations.
• Most popular
• Runge-Kutta (explicit forward).
• Bulirsch-Stoer (complex/expensive).
• Leapfrog/symplectic methods.
• Preserve phase space volume.
• Timestep adaptability issues.

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Collision detection

• Particles collide when separation distance equals
  sum of radii.

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Collision detection

• Particles collide when separation distance equals
  sum of radii.
• Two approaches
• Predict collisions before they occur.
• Need neighbour-finding algorithm (tree!).
• Detect collisions after they occur.
• Detected by mutual overlap.
• Adaptive timestepping essential.

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Numerical method

• Our group uses pkdgrav
• Parallel k-D tree code.
• k-D split along longest dimension.
• Expand to hexadecapole order.
• Second-order leapfrog integrator.
• Hierarchical timestepping.
• Collisions predicted before they occur.
• Includes bouncing and sliding friction.

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Parallelism in pkdgrav

• master
• controls overall flow

• mdl
• interface between pkdgrav and parallel
  primitives (e.g. mpi)

• pst
• loops over processors

• pkd
• loops over particles on one processor

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Application binary asteroids

• Use N-body code to simulate
• Capture of collisional ejecta in Main Belt.
• Michel et al., Durda et al. collisions that make
  families also make satellites.

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Application binary asteroids
Michel et al. 2001
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Application binary asteroids

• Use N-body code to simulate
• Capture of collisional ejecta in Main Belt.
• Michel et al., Durda et al. collisions that make
  families also make satellites.
• Rotational disruption of gravitational aggregates
  in near-Earth population.
• Tidal disruption.
• YORP thermal spin-up.

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Application binary asteroids
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Tidal disruption vs. YORP

• Tidal disruption makes binaries, but also
  destroys them quickly.
• Binary NEA mean lifetime only 1 Myr.
• YORP thermal effect may form binaries through
  rotational disruption.
• But, some internal strength/cohesion may be
  necessary to prevent material from just
  dribbling away (but that may be OK too!).

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Forming binaries with YORP

• Preliminary investigation
• Slowly spin up various rubble piles.
• Find particles leak away from equator (no
  fission).

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Forming binaries with YORP

• Preliminary investigation
• Slowly spin up various rubble piles.
• Find particles leak away from equator (no
  fission).

Recoil new mobility mechanism?
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Forming binaries with YORP
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Forming binaries with YORP

• May need strength and/or irregular body shape to
  form binaries.
• E.g., contact binary can separate.

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Non-idealized models

• Treating particles as idealized, rigid,
  independent spheres is convenient.
• Components with different shapes may provide more
  realism. E.g.,
• Ellisoidal particles (Roig et al.)
• Polyhedral (Korycansky Asphaug).
• We combine best of both worlds allow spheres to
  fuse together

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Non-idealized models
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Strength model

• Colliding particles/aggregates can
• Stick on contact
• Bounce
• Liberate particle(s) from aggregate(s).
• Outcome currently parameterized by impact speed.

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Strength model

• In addition, bonded aggregates can have a
  size-dependent bulk tensile and/or shear
  strength.
• Particles experiencing stress (relative to center
  of mass) in excess of strength are liberated.
• Global model (no fractures/cracks).

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Strength model
For a demo of the new strength model in action,
see Patricks presentation!
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Testing strength spin limits

• One way to test the strength model is to compare
  with analytical predictions of global failure
  (e.g. Holsapple).
• Found good match for cohesionless models
  (Richardson et al. 2005).
• Science motivation spin-up past critical limit
  could make binaries (e.g. YORP).

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Spin limits preliminary results
Work in progress!
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Summary

• N-body methods allow modeling of complex
  phenomena involving gravity collisions.
• Examples include post-disruption gravitational
  reaccumulation to form binaries families.
• Binaries more work needed to assess YORP
  (including survivability against BYORP!).
• New pkdgrav strength model provides added
  realism/complexity, but needs fracture model.

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Extra Slides
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What is YORP?

• Yarkovsky-O'Keefe-Radzievskii-Paddack effect.
• Irregular bodies reflect/re-radiate solar photons
  in different directions net torque ?
  spin-up/down.

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Results Many Binaries
Close approach distance q
Encounter speed v8

• High rates of production for
• Low q.
• Low v8.
• Rapid spin.
• Large elongation.

Spin period P
Elongation e
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Orbital Properties
Semimajor axis a (50 gt 10 Rp)
Eccentricity e (97 gt 0.1)

• High eccentricity.
• Range of semi-major axis.
• Binary orbit aligned more with approach orbit
  than progenitor spin.
• Retrograde orbits possible.

Spin-orbit angle
Inclination I
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Physical Properties
Size ratio

• Size ratio peaks at 0.10.2 (1051).
• Obliquities
• Primary spin aligned with binary orbit.
• Wide range of secondary spin axes.
• Spin Periods
• Primary has narrow range (3.5 ? 6.0 h).
• Secondary has wide range (4.0 ? 20 h).

Obliquities
Spin periods
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Evolutionary Effects

• Mutual tides damp eccentricity in 110 My.
• Repeated encounters may strip binary.
• NEA population refreshed by MBAs (some of which
  may be binary).
• Thermal effects (YORP) important?

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Steady-state (Monte Carlo) Model

• We know
• Binary production efficiency from tidal
  disruption (Walsh Richardson 2006)
• Planetary encounter circumstances (Bottke et al.
  1994)
• Distribution of NEA lifetimes (Gladman et al.
  2000)
• Shape and spin of source bodies (Harris et al.
  2005)
• Tidal evolution effects (Weidenschilling et al.
  1989)
• Effects of binary encounters with planets (Bottke
  Melosh 1996 this work)
• Small binary MBAs formed in collisional
  simulations (Durda et al. 2004).

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Steady-state (Monte Carlo) Model

• In one timestep
• All asteroids in the simulation are tested for
• End of lifetime (median 10 Myr)
• Close planetary encounter lt 3REarth (one every 3
  Myr).
• All binaries are tested for
• End of lifetime
• Close planetary encounter lt 24REarth explicit
  3-body integration.
• If neither happen, the binary is tidally evolved.
• Removed NEAs/binaries are immediately replaced.
• Fresh asteroids take spin/shape characteristics
  of MBAs, with a variable percentage being
  binaries.
• MBA binaries have characteristics determined from
  the Durda et al. 2004 simulations.

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Steady-state Results

• For 2000 asteroids
• Find 2 binary fraction.
• Binary NEA mean lifetime 1 Myr.
• 93 of removed binaries destroyed by planetary
  encounters.
• MBA initial binary percentage has little effect
  (mean lifetime 0.32 Myr).

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Steady-state Results

• The resultant steady-state binaries
• Have slightly larger semi-major axes than
  observed

Observed
Steady-state
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Steady-state Results

• The resultant steady-state binaries
• Have slightly larger semi-major axes than
  observed
• Mostly have low eccentricities (lt 0.2),
  consistent with observations.

Eccentricity
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A Word About Rubble Piles

• Rubble piles are low-tensile-strength,
  medium-porosity gravitational aggregates.
• In simulations, rubble piles consist of perfectly
  smooth spheres some dissipation.
• Used in a variety of contexts planetesimal
  collisions, tidal disruption, spin-up.
• How do they differ from perfect fluids?

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Rubble Pile Equilibrium Shapes
Mass loss 0 lt 10 gt 10 X initial
condition
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Rubble Pile Equilibrium Shapes
Mass loss 0 lt 10 gt 10 X initial
condition
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YORP Spinup of Rubble Piles
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Resolution Effects
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Classifications
Richardson et al. 2003
Stress response may be predicted by plotting
tensile strength (resistance to stretching) vs.
porosity.
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Strength vs. Gravity
Asphaug et al. 2003
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Aggregates Resist Disruption

• Once shattered, impact energy is more readily
  absorbed at impact site.

Asphaug et al. 1998
Damaged
Coherent
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