Chaosassisted capture in the Hill 4body problem: Kuiperbelt binaries - PowerPoint PPT Presentation

Title: Chaosassisted capture in the Hill 4body problem: Kuiperbelt binaries


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Chaos-assisted capture in the Hill 4-body
problemKuiper-belt binaries

• Sergey Astakhov
• RD, UniqueICs, Saratov, Russia
• NIC, Forschungszentrum Juelich, Germany

CollaborationDavid Farrelly Utah State
University, USAErnestine Lee Five Prime
Therapeutics, USA
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Hamiltonian dynamics possible outcomes

• periodic orbits
• quasiperiodic trajectories (KAM tori)
• chaotic transients at the edge of stability
  stickiness, gravitational capture
• short-lived scattering trajectories

bound
unbound
Easy come, easy go attractor at infinity.
How do gravitating particles form bound (stable)
configurations?
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Gravitationally bound configurations

• Natural planetary satellites
• Regular moons
• Irregular moons thought to have been
  captured
• Minor bodies with satellites
• Main-belt and near-Earth double asteroids (4
  of the total population)
• Kuiper-belt binary objects
• Double stars and star clusters
• Technology space mission design, optimal
  flights, ballistic capture

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Kuiper-belt (transneptunian) binaries
Kuiper-belt predicted Leonard (1930),
Edgeworth (1943), Kuiper (1951) discovered
Jewitt Luu (1992) first binary detected
Veillet et. al (1998-2002) Now 10 binaries are
known, of which 4 with well characterized orbital
and compositional properties
D. Jewitt The origin of the Kuiper Belt
Binaries is a subject of speculation. Straight
gravitational capture of one KBO by another is
essentially impossible without some process that
can dissipate some of the kinetic energy present
in the initial motions of the bodies
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Equal partners
Binary the 2-body center of mass resides
outside either of the members.

• Mass ratio of Kuiper-belt binaries of order unity
  (mr0.1-1),
• as opposed to that of the main-belt asteroids
  with satellites (mr10-4-10-3)
• Moderate orbital eccentricities (elt0.8)
• Large mutual orbits in Kepler (2-body) limit
• Observational bias for well-separated
• equally bright objects? (Burns, Nature, 2004)
• No common dynamical origin for these properties
  has been proposed

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Current theories of KB binary formation
Physical collisions (Weidenschilling, Icarus,
2002) Temporary capture and subsequent
dissipation by - dynamical friction -
encounter with a larger body (Goldreich et al.,
Nature, 2002) Exchange reactions (Funato et
al., Nature, 2004)
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The Hill sphere
Hill problem a variant of the three-body
problem mass of the Sun dominates over
the two interacting bodies on a heliocentric
orbit Solar perturbations are approximated by
the leading term of the series expansion in
reduced mass of the binary The Hill sphere
domain of mutual gravitation of binary
members To interact, particles must first enter
their Hill sphere If particles leave their Hill
sphere, they are not bound to each other any more
(surface of no return) Bound Kepler orbits
reside deep inside the Hill sphere
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Two key ingredients of a formation model
Formation of a proto-binary in the Hill sphere
A source of dissipation to
explain orbital evolution from large Hill orbits
towards relatively compact Kepler orbits
(keplerization)
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Three key ideas of the proposed mechanism
Formation of proto-binary in the Hill sphere -
chaos-assisted capture temporary trapping
in chaotic layers close to regular structures
(Astakhov et al., Nature, 2003) -
stabilization by encounters with small-mass
intruders Hill four-body problem
(Scheeres, CMDA, 1998) A source of dissipation
to explain orbital evolution from large Hill
orbits towards relatively compact Kepler orbits
(keplerization) - gradual energy loss in
subsequent multiple encounters with
similarly small intruders on a much longer
time scale
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4-body Hill equations of motion (Scheeres 1998)
Three small masses Relative motion of the
binary members Motion of the third body
(intruder) Center-of-mass relation
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Capture in the Hill sphere and stabilization
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Statistics of stabilization
Stabilization is most efficient by small
intruders (1 of the total binary mass) Larger
intruders tend to disrupt chaotic transients
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Keplerization through multiple encounters
After 200 encounters, binary survival
probabilities were 0.103 0.007 (equal
masses) 0.019 0.002 (m1/m2 0.05)
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Elliptic restricted three-body problem
Hamiltonian becomes dependent on the angular
position of the planet Energy is not conserved
dynamics in the extended phase space with more
that 2 degrees of freedom Equations of motion
contain parametric periodic time-dependence Fini
te-time Fast Lyapunov Indicator to visualize
phase space Froeschlé, Guzzo, Lega. Science,
2000.
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Chaos-assisted capture by planets on elliptic
orbits
FLI to map regular and chaotic regions of phase
space
Monte Carlo simulations suggest noticeably lower
capture probability for extrasolars
Astakhov Farrelly. Mon. Not. R. Astron. Soc.
2004
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4-body interactions formation of binary objects

• Kuiper-belt binary objects with their striking
  tendency to form almost symmetric tandems
  following elliptic orbits around common center of
  mass
• Veilet, Nature, 2002
• Weidenschilling, Icarus, 2002
• Goldreich, Lithwick, Sari, Nature,
  2002
• Funato et al., Nature, 2004
• Chaos-assisted capture applies in Hill 4-body
  problem Sun, two binary members, small mass
  intruder
• Formation of quasi-bound chaotic binaries is
  possible on relatively long time scales
• Stabilization by inelastic scattering with small
  intruder switching binary into regular regions of
  phase space

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Avenues

• Capture of Triton into compact retrograde orbit
• McKinnon, Nature, 1984
• Goldreich et al, Science, 1989
• Tsui, Planetary and Space Science, 2002
• Agnor, Hamilton, Nature, 2006
• Extrasolar moons, potentially habitable
• Capture of Trojan objects

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Acknowledgements
Prof. David Farrelly, Dr. Ernestine Lee and Utah
State University, USA Prof. Peter Grassberger
and Forschungszentrum Juelich, Germany Dr.
Daniel Hestroffer and IMCCE, Observatoire de
Paris