Chaos-assisted capture in the formation of Kuiper-belt binaries

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Chaos-assisted capture in the formation of
Kuiper-belt binaries

• Sergey Astakhov
• UniqueICs, Saratov, Russia and NIC
  Forschungszentrum Jülich,Germany
• Ernestine Lee
• FivePrime Therapeutics, San Francisco, Calif. USA
• David Farrelly
• Department of Chemistry, Utah State University,
  Logan, Utah, USA

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Why are binaries interesting?
Dynamics N-body problem N 3 Sun-Earth-Moon
1867
Planetary Physics
Deprit, Henrard, and Rom Science, 168, 1569 (1970)
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Currently 24 NEA binaries 26 Main Belt
binaries 22 TNO binaries (K. S. Noll,
Asteroids, Comets, Meteors, 2005)
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(No Transcript)
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Key challenges for binary TNO formation models
(1998 WW31)
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Brief overview of previous models
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Formation three-body dynamics

• Step 1 Capture
• Step 2 Post-capture Keplerization
• Require a source of dissipation or other means of
  energy loss

Zero-velocity (energy) surface in circular RTBP
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Path 1 Weidenschilling 2002Icarus, 106, 190

• Physical collisions between two bodies inside
  Hill sphere of a larger object
• Bodies accrete and remain in orbit around larger
  body
• Requires higher number density of massive objects
  than seems consistent with observations

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Paths 2 3 Goldreich et al. - 2002Nature, 420,
643.

• Binary partners get caught up inside their mutual
  Hill sphere
• Either dynamical friction (L2) or gravitational
  scattering from a 3rd, large body (L3) leads to
  permanent capture
• L2 Not enough small bodies. No mass effect
  predicted.
• L3 Need multiple scattering events to convert
  large captured Hill orbit into a Keplerian orbit
  1- 2 of Hill radius. Large intruders tend to
  destabilize.

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Path 4 Funato et al. - 2004 Nature, 427, 518.
Three-body exchange reactions

• Very unequal mass binary formed through tidal
  dissipation after a close encounter or a giant
  impact small, circular orbit
• Exchange reaction in which a large 3rd body
  intruder exchanges (preferentially) with smaller
  binary partner mass effect
• Neglects Hill sphere
• Leads almost exclusively to eccentricities gt 0.8
• Very large orbits compared to Hill radius

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Model comparison arbitrary units
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Chaos-assisted capture
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Sticky Tori

• Chaotic orbits cling to sticky KAM tori
• Opportunity for capture into nearby tori
• Capture above Lagrange points possible.

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Long-lived chaotic orbits
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4-body Hill equations (Scheeres, 1998) Sun
binary small intruder
(relative to Sun)
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Algorithm
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(i) Form a nascent binary in Hill sphere
prograde retrograde
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Capture in the Hill sphere and stabilization
(ii) Capture in Hill sphere and (iii)
Keplerization by small ( 1 2 binary mass)
intruder scattering
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Statistics of stabilization
Equal mass

• Stabilization is most efficient by small
  intruders (1 of the total binary mass).

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Keplerization through multiple intruder scattering

• After 200 encounters, binary survival
  probabilities were
• 0.103 0.007 (equal masses)
• 0.019 0.002 (m2/m1 0.05)

m2/m1 1 m2/m1 0.05
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Why are equal mass binaries more stable? Intruder
dwell-times in Hill sphere

• Dwell-times ltlt proto-binary lifetime in absence
  of intruders
• Dwell-times somewhat longer for unequal mass
  proto-binaries
• Approximate actual chaotic binary orbit by an
  elliptical bound orbit
• Problem reduces to elliptical restricted
  three-body problem

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• Chaos in the elliptic-restricted
• three-body problem limit

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Model

• Nascent chaotic binary time scales gtgt intruder
  dwell times assume binary elliptical orbit
• Neglect solar tides
• Set intruder mass to zero
• Binary partners are now the primaries
• Intruder is mass-less test particle
• Examine phase-space for intruder scattering using
  FLI maps

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Elliptic RTBP
Fast Lyapunov Indicator
Corresponding SOS in CRTBP

• Chaos-assisted capture robust to moderate
  ellipticity
• Detect chaos using Fast Lyapunov Indicator (FLI)

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• Intruders get stuck in Hill sphere resonances
  unequal masses destabilization
• equal mass binaries undergo rapid intruder
  scattering
• Circular or very elliptical binary orbits
  destabilized

FLI MAPS
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• Capture of Neptunes retrograde moon Triton

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Binary-Planet Scattering

• Agnor Hamilton capture in a 3-body
  binary-Neptune exchange encounter Nature, 441,
  192 (2006).
• Triton approaches Neptune as part of a binary.
• Neptune then exchanges with one of the binary
  partners leaving Triton in an elliptical orbit
  with close approach distance
• q 0.5 aTriton
• 0.15 rHill
• 7 RNeptune
• Keplerization through tidal interactions
• Doesnt consider Sun-Neptune Hill sphere
• Binary has to surmount Lagrange points

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Binary-Neptune scattering solar tides
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Approach speeds
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Outcomes 3Dim orbits projectedon to x-y plane
Hill Sphere

• Capture of one binary partner
• Temporary trapping of both binary partners as
  intact binary
• Escape of both binary partners
• Split-up of binary and temporary trapping of both
  binary partners (rare)
• Collisions with planet
• Numerous close encounters with planet implies
  collisions with satellites may be probable

Y
X
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Rmin Rmax
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Summary
ab

• Small binary semi-major axes (ab 10RN) hard
  binaries and low relative velocities (few
  km/s) binary behaves as a composite and exchange
  is rare
• Binary partner collisions
• Softer binaries long-term trapping and capture
• Opportunity for collisions with inner moons

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Conclusions

• Chaos is important in providing the glue to allow
  otherwise improbable events to occur
• Likely that all proposed KBB capture mechanisms
  play some role, perhaps in combination
• Capture of moons in binary-planet encounters is
  possible but very complex dynamics inside Hill
  sphere results

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Acknowledgements

Andrew Burbanks, University of Portsmouth,
U.K. Funding National Science Foundation