Invariant Manifolds and Transport in the Three-Body Problem
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1 Dynamical S C C A L T E C H Invariant Manifolds and Transport in the Three-Body Problem Shane D. Ross Control and Dynamical Systems California Institute of Technology Classical N-Body Systems and Applications University of Warwick, April 2002
2 Acknowledgements W. Koon, M. Lo, J. Marsden G. Gómez, J. Masdemont C. Jaffé, D. Farrelly, T. Uzer K. Howell, B. Barden, R. Wilson C. Simó, J. Llibre, R. Martinez E. Belbruno, B. Marsden, J. Miller L. Petzold, S. Radu H. Poincaré, J. Moser C. Conley, R. McGehee 2
3 Outline Transport theory Time-independent N-body Hamiltonian systems Examples: ionization of Rydberg atoms restricted three-body problem 3
4 Chaotic Dynamics Transport through a bottleneck in phase space; intermittency 4
5 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, correlation functions Applications: Atomic ionization rates Chemical reaction rates Comet and asteroid escape rates, resonance transition probabilities, collision probabilities 5
6 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 Look at stable and unstable manifold of periodic orbits associated with Lagrange points and mean motion resonances Use these to compute statistical quantities e.g., probability of resonance transition, escape rates 6
7 Transport Theory of current astrophysical interest for understanding the transport of solar system material eg, how ejecta gets from Mars to Earth how likely is Shoemaker-Levy 9-type collision 7
8 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 8
9 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 2:3 resonance Second encounter Jupiter!"$#%'& START Sun 3:2 resonance END First encounter Jupiter s orbit 9
10 Viewed in Rotating Frame Oterma s orbit in rotating frame with some invariant manifolds of the 3-body problem superimposed. Homoclinic Orbits Oterma's Trajectory Oterma y (rotating) 1980 Sun L 1 Jupiter L 2 y (rotating) 1910 Jupiter Heteroclinic Connection x (rotating) Boundary of Energy Surface x (rotating) 10
11 Partition the Phase Space Reactants Products 12
12 Partition the Phase Space Systems with potential barriers Electron near a nucleus with crossed electric and magnetic fields "Bound" "Free" Nucleus Potential Configuration Space 13
13 Partition the Phase Space Comet near the Sun and Jupiter U eff Sun Jupiter Potential Configuration Space 14
14 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 15
15 Statement of Problem Suppose we study the motion on a manifold M Suppose M is partitioned into disjoint regions such that R i, i = 1,..., N R, M = N R i=1 R i. To keep track of the initial condition of a point, we say that initially (at t = 0) region R i is uniformly covered with species S i. Thus, species type of a point indicates the region in which it was located initially. 16
16 Statement of Problem Statement of the transport problem: Describe the distribution of species S i, i = 1,..., N R, throughout the regions R j, j = 1,..., N R, for any time t > 0. R 1 R 2 R 4 R 3 17
17 Statement of Problem Some quantities we would like to compute are: T i,j (t) = the total amount of species S i contained in region R j at time t F i,j (t) = dt i,j dt (t) = the flux of species S i into region R j at time t R 1 R 2 R 4 R 3 18
18 Hamiltonian Systems Time-independent Hamiltonian H(q, p) N degrees of freedom Motion constrained to a (2N 1)-dimensional energy surface M E corresponding to a value H(q, p) = E = constant Symplectic area is conserved along the flow p dq = dp dq = constant L A 19
19 Poincaré Section Suppose there is another (2N 1)-dimensional surface Q that is transverse (i.e., nowhere parallel) to the flow in some local region. The Poincaré section S is the (2N 2)-dimensional intersection of M E with Q. 20
20 Example for N=2 Restricted 3-body problem (planar) Partition the energy surface: S, J, X regions Forbidden Region Exterior Region (X) S L 1 J L 2 Interior (Sun) Region (S) Jupiter Region (J) Position Space Projection 21
21 Equilibrium Region Look at motion near the potential barrier, i.e. the equilibrium region S J L1 L2 L2 Position Space Projection 22
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 Asymptotic Orbits Transit Orbits Spherical Cap of Transit Orbits Asymptotic Circle Nontransit Orbits p.o. Spherical Zone of Nontransit Orbits Periodic Orbit Cross-section of Equilibrium Region Equilibrium Region 23
23 Transition State Theory This is related to the transition state theory of the chemical literature. Wiggins, Wiesenfeld, Jaffé, and Uzer [2001] extend transition state theory to higher dimensional systems. Interesting connection between chemical and celestial dynamics! 24
24 Tubes in the 3-Body Problem Stable and unstable manifold tubes Control transport through the potential barrier. Forbidden Region Stable Manifold Jupiter Unstable Manifold Sun L 2 y (rotating frame) Stable Manifold Forbidden Region x (rotating frame) L 2 Periodic Orbit Unstable Manifold 25
25 Flux between Regions Tubes of transit orbits are the relevant objects to study Tubes determine the flux between regions F i,j (t). Net flux is zero for volume-preserving motion, so we consider the one-way flux Example: F J,S (t) = volume of trajectories that escape from the Jupiter region into the Sun region per unit time. 26
26 Transition Rates Fluxes give rates and probabilities Jaffé, Ross, Lo, Marsden, Farrelly, and Uzer [2002] computed the rate of escape of asteroids temporarily captured by Mars. RRKM-like statistical approach similar to chemical dynamics, see Truhlar [1996] Consider an asteroid (or other body) in orbit around Mars (perhaps impact ejecta) at a 3-body energy such that it can escape toward the Sun. Interested in rate of escape of such bodies at a fixed energy, i.e. F M,S (t) 27
27 Transition Rates RRKM assumption: all asteroids in the Mars region at fixed energy are equally likely to escape. Then flux across potential barrier Escape rate = Mars region phase space volume Compare with Monte Carlo simulations of 107,000 particles randomly selected initial conditions at constant energy 28
28 Transition Rates Theory and numerical simulations agree well. Monte Carlo simulation (dashed) and theory (solid) 1 Survival Probability Number of Periods 29
29 Transition Rates More exotic transport between regions Look at the intersections between the interior of stable and unstable tubes on the same energy surface. Could be from different potential barriers. Poincare Section Tube B Tube A 30
30 Transition Rates Example: Comet transport between outside and inside of Jupiter (i.e., Oterma-like transitions) Exterior Region Forbidden Region Rapid Transition y (rotating frame) Sun Interior Region x (rotating frame) Jupiter Region (a) L 1 Stable Manifold Jupiter L 2 Unstable Manifold 31
31 Transition Rates Look at Poincaré section intersected by both tubes. Choosing surface {x = constant; p x < 0}, we look at the canonical plane (y, p y ). y (rotating frame) Poincare section py Stable e Slice L 1 L Stable Unstable 2 e eunstable e Slice x (rotating frame) Position Space Canonical Plane (y, p y ) y 32
32 Transition Rates Relative canonical area gives relative flux of orbits. Under RRKM assumptions, can compute probability of transition from one region to another. Stable e Slice Stable e Slice py Comet Orbits Passing Exterior to Interior Region Unstable e Slice Canonical Plane (y, p y ) y Unstable e Slice 33
33 Mixing By keeping track of the intersections of the tubes, one can describe the mixing of different regions (T i,j (t)). It can get complicated! Example: atomic transition rates (Jaffé, Farrelly, Uzer [1999]) 34
34 Collision Collision probabilities computed from tube intersection with planet on Poincaré section Diameter of planet 35
35 Collision Statistical approach to low velocity impact probabilities eg, Shoemaker-Levy 9 and Earth crossers Example Collision Trajectory x Collision! Comes from Interior Region 36
36 Conclusion and Future Work Transport in the solar system View solar system as many restricted 3-body problems Planar restricted 3-body problem Stable and unstable manifold tubes of periodic orbits can be used to compute statistical quantities of interest Asteroid escape problem: first application of RRKM-like statistical approach to celestial mechanics Theory and numerical simulation agree well 37
37 Conclusion and Future Work Future Work Extend planar results to spatial problem theory makes computation in high dimensions much easier Transport between mean motion resonances Slow migration between resonances leading to temporary capture by or close encounter with a planet. Transport between planets e.g., comets switching control from Saturn to Jupiter Consider drag-perturbed case e.g., interplanetary dust particles 38
38 Transport between MMRs Transport rates between mean motion resonances (MMRs) can be computed via a lobe dynamics approach (see Wiggins [1992]). Several statistical quantities of interest can be computed as a function of planetary mass and particle energy. average trapping time in a p : q MMR flux entering p : q MMR from p : q MMR 39
39 Transport between MMRs We can compute the resonance regions Unstable periodic orbit P1 Unstable manifold of P1 Argument of perihelion (degrees) Stable manifold of P1 Unstable periodic orbit P2 (b) A stable island for the resonance region of P2 Semimajor axis Resonance region of P1 (a) Resonance region of P2 Overlapping region 40
40 Transport between MMRs A direct transition from a p : q to a p0 : q0 MMR is possible only if the exit lobe of a p : q turnstile overlaps with the entry lobe of a p0 : q0 turnstile. Unstable periodic orbit P1 Unstable periodic orbit P2 Unstable manifold of P1 Argument of perihelion (degrees) Stable manifold of P1 (b) A stable island for the resonance region of P2 Semimajor axis Resonance region of P1 (a) Resonance region of P2 Overlapping region 41
41 MMRs and Close Encounters Poincaré section: plot resonance regions 2:3 exterior MMR with Jupiter 42
42 MMRs and Close Encounters Same section: tube cross-sections are closed curves Particles inside curves move toward or away from Jupiter 43
43 MMRs and Close Encounters Regions of overlap lead to/from close encounters Regions of overlap occur 44
44 Transport between Planets Comets transfer between the giant planets eg, jumping between tubes of Saturn and Jupiter Semimajor Axis of Comet Smirnova-Chernykh (AD ) Inertial Plot of Comet Smirnova-Chernykh 9 Saturn 10 Saturn Semimajor Axis (AU) Smirnova-Chernykh Jupiter y (AU, inertial frame) Sun Jupiter Year -10 Smirnova-Chernykh x (AU, inertial frame) 45
45 Circumstellar Dust Clouds Drag-perturbed case important for planet-finding 1 y (AU, Sun-Earth Rotating Frame) (A Sun Earth x (AU, Sun-Earth Rotating Frame) Earth's Orbit Relative Particle Density 46
46 References Jaffé, 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), Gómez, 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), For papers, movies, etc., visit the website: shane The End 47
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