A Statistical Theory of Transition and Collision. Probabilities. Shane D. Ross. Control and Dynamical Systems. California Institute of Technology. International ...
C A L T E C H C Dynamical S
A Statistical Theory of Transition and Collision Probabilities Shane D. Ross Control and Dynamical Systems California Institute of Technology
International Conference on Libration Point Orbits and Applications Parador d’Aiguablava, Girona, Spain, 10-14 June 2002
Acknowledgements G. G´omez, M. Lo, J. Masdemont J. Marsden, W. Koon C. Jaff´e, D. Farrelly, T. Uzer C. Sim´o, J. Llibre, R. Martinez K. Howell, B. Barden, R. Wilson E. Belbruno, B. Marsden, J. Miller B. Farquhar, V. Szebehely, C. Marchal J. Moser, C. Conley, R. McGehee
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Overview Motivation Planetary science Comet motion, collisions Chaotic dynamics, intermittency
Transport theory Restricted three-body problem Predictions of theory Comparison with observational data?
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Motivation Planetary science Current astrophysical interest in understanding the transport and origin of solar system material: • How likely is Oterma-like resonance transition? • How likely is Shoemaker-Levy 9-type collision with Jupiter? – or an asteroid collison with Earth? • How does impact ejecta get from Mars to Earth?
Statistical methods applied to the three-body problem may provide a first-order answer. The “interaction” of several three-body systems increases the complexity. 4
Jupiter Family Comets Physical example of intermittency We consider the historical record of the comet Oterma from 1910 to 1980 • first in an inertial frame • then in a rotating frame • a special case of pattern evocation
Similar pictures exist for many other comets
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Jupiter Family Comets • Rapid transition: outside to inside Jupiter’s orbit. ◦ Captured temporarily by Jupiter during transition. ◦ Exterior (2:3 resonance) to interior (3:2 resonance). Oterma’s orbit
Second encounter
2:3 resonance Jupiter
!
START
Sun 3:2 resonance
END
First encounter
Jupiter’s orbit
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Viewed in Rotating Frame Oterma’s orbit in rotating frame with some invariant manifolds of the 3-body problem superimposed. Oterma's Trajectory
Homoclinic Orbits
1980
Sun
L1
L2 Jupiter
y (rotating)
y (rotating)
Oterma
Heteroclinic Connection 1910
Jupiter
x (rotating)
x (rotating) Boundary of Energy Surface
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Viewed in Rotating Frame
Oterma - Rotating Frame
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Collisions with Jupiter Shoemaker Levy-9: similar energy to Oterma • Temporary capture and collision; came through L1 or L2
Possible Shoemaker-Levy 9 orbit seen in rotating frame (Chodas, 2000) 9
Chaotic Dynamics Transport through a bottleneck in phase space; intermittency
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Transport Theory Chaotic dynamics =⇒ statistical methods Transport theory Ensembles of phase space trajectories • How long or likely to move from one region to another? • Determine transition probabilities
Applications: • Comet and asteroid collision probabilities, resonance transition probabilities, transport rates
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Transport Theory Transport in the solar system For objects of interest • e.g., Jupiter family comets, near-Earth asteroids, dust
Identify phase space objects governing transport View N -body as multiple restricted 3-body problems Consider stable/unstable manifolds of bounded orbits associated with libration points • e.g, planar Lyapunov orbits
Use these to compute statistical quantities • e.g., probability of resonance transition, escape rates
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Partition the Phase Space Region A
Region B
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Partition the Phase Space Systems with potential barriers Electron near a nucleus with crossed electric and magnetic fields • See Jaff´e, Farrelly, and Uzer [1999]
"Bound"
"Free"
Nucleus
Potential
Configuration Space 14
Partition the Phase Space Comet near the Sun and Jupiter • Some behavior similar to electron!
Ueff
Sun
Potential
Jupiter
Configuration Space 15
Partition the Phase Space Partition is specific to problem We desire a way of describing dynamical boundaries that represent the “frontier” between qualitatively different types of behavior
Example: motion of a comet
motion around the Sun motion around Jupiter
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Statement of Problem Following Wiggins [1992]: Suppose we study the motion on a manifold M Suppose M is partitioned into disjoint regions Ri, i = 1, . . . , NR, such that M=
NR [
Ri .
i=1
At t = 0, region Ri is uniformly covered with species Si Thus, species type of a point indicates the region in which it was located initially 17
Statement of Problem Statement of the transport problem: Describe the distribution of species Si, i = 1, . . . , NR, throughout the regions Rj , j = 1, . . . , NR, for any time t > 0.
R1
R2 R3
R4
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Statement of Problem Some quantities we would like to compute are: Ti,j (t) = amount of species Si contained in region Rj dTi,j Fi,j (t) = dt (t) = flux of species Si into region Rj
R1
R2 R3
R4
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Probabilities For some problems, probability more relevant • e.g., probability = 0 implies event should not occur
Test this on celestial mechanics problems of interest
R1
R2 R3
R4
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Restricted 3-Body Prob. Planar circular case Partition the energy surface: S, J, X regions Exterior Region (X)
Forbidden Region
S
L1 J
L2
Interior (Sun) Region (S)
Jupiter Region (J)
Position Space Projection 21
Equilibrium Region Look at motion near the potential barrier, i.e. the equilibrium region &' )(! +**,
S
L1 J
L2
!
L2
#" $! %
Position Space Projection 22
Local Dynamics For fixed energy, the equilibrium region ' S 2 × R. • Stable/unstable manifolds of periodic orbit define mappings between bounding spheres on either side of the barrier Transit Orbits
Spherical Cap of Transit Orbits
Asymptotic Orbits
Nontransit Orbits
Asymptotic Circle
p.o.
Spherical Zone of Nontransit Orbits Periodic Orbit
Cross-section of Equilibrium Region
Equilibrium Region 23
Tubes in the 3-Body Problem Stable and unstable manifold tubes • Control transport through the potential barrier. Forbidden Region
Stable Manifold
Unstable Manifold
Jupiter
Sun y (rotating frame)
L2
Stable Manifold
L2 Periodic Orbit
Unstable Manifold Forbidden Region x (rotating frame) 24
Transition Probabilities Transport btwn non-adjacent regions Consider intersections between the interior of tubes — the transit orbits connecting regions. Tube A and Tube B from different potential barriers. Poincare Section
Tube A Tube B
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Transition Probabilities Example: Comet transport between outside and inside of Jupiter (i.e., Oterma-like transitions)
y (rotating frame)
Exterior Region
Forbidden Region
Rapid Transition
Sun Jupiter Interior Region
L1 Jupiter Region
Stable Manifold
L2 Unstable Manifold
x (rotating frame) (a) 26
Transition Probabilities Consider Poincar´e section intersected by both tubes.
ection s e r ca Poin
Stable
e Slice
py
y (rotating frame)
Choosing surface {x = constant; px < 0}, we look at the canonical plane (y, py ).
L1
Stable Unstable e e
L2
Unstable
e Slice
x (rotating frame)
Position Space
y
Canonical Plane (y, py ) 27
Transition Probabilities Canonical area ratio gives the conditional probability to pass from outside to inside Jupiter’s orbit. • Assuming a well-mixed connected region on the energy mfd. Stable
e Slice
Stable
e Slice
py
Comet Orbits Passing
Exterior to Interior Region
Unstable
e Slice
Unstable
e Slice
y
Canonical Plane (y, py ) 28
Transition Probabilities Jupiter family comet transitions: X → S, S → X Transition Probability for Jupiter Family Comets 60
L1 L2 Transition probability (%)
50
40
30
20
10
0
−1.518
−1.517
−1.516
−1.515
−1.514
−1.513
−1.512
Energy Oterma 29
Collision Probabilities Low velocity impact probabilities Assume object enters the planetary region with an energy slightly above L1 or L2 • eg, Shoemaker-Levy 9 and Earth-impacting asteroids Example Collision Trajectory
x
Collision!
Comes from Interior Region 30
Collision Probabilities Collision probabilities ◦ Compute from tube intersection with planet on Poincar´e section ◦ Planetary diameter is a parameter, in addition to µ and energy E
← Diameter of planet →
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Collision Probabilities Collision probabilities Poincare Section: Tube Intersecting a Planet 1.5
Collision Non−Collision
1
0.5
0
−0.5
−1
−1.5 −8
−6
−4
−2
0
2
4
6
8 −5
x 10
← Diameter of planet → 32
Collision Probabilities Collision Probability for Jupiter Family Comets L1 L2 Collision probability (%)
5
4
3
2
1
0
−1.518
−1.517
−1.516
−1.515
Energy
−1.514
−1.513
−1.512
SL9 33
Collision Probabilities Collision Probability for Near−Earth Asteroids 10 9
L1 L2
Collision probability (%)
8 7 6 5 4 3 2 1 0
−4
−3.5
Energy (scaled)
−3 −4
x 10
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Collision Probabilities Asteroid Collision Orbit with Earth
0.01
Earth
y
0.005
L2
0
−0.005
−0.01 0.995
1
1.005
1.01
1.015
1.02
x 35
Conclusion and Future Work Transport in the solar system Approximate some solar system phenomena using the restricted 3-body problem Planar restricted 3-body problem • Stable and unstable manifold tubes of libration point orbits can be used to compute statistical quantities of interest • Probabilities of transition, collision
Theory and observation agree
Future studies to involve multiple three-body problems 36
References • Jaff´e,
C., S.D. Ross, M.W. Lo, J. Marsden, D. Farrelly, and T. Uzer [2002] Statistical theory of asteroid escape rates. Physical Review Letters, to appear.
• Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2001] Resonance
and capture of Jupiter comets. Celestial Mechanics and Dynamical Astronomy 81(1-2), 27–38.
• G´omez, G., W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont and S.D. Ross [2001] Invariant manifolds, the spatial three-body problem and space mission design. AAS/AIAA Astrodynamics Specialist Conference.
• Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2000] Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10(2), 427–469.
For papers, movies, etc., visit the website: http://www.cds.caltech.edu/∼shane
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