Chaotic Motion in the Solar System: Mapping the Interplanetary Transport Network

Transcription

1 C C Dynamical A L T E C S H Chaotic Motion in the Solar System: Mapping the Interplanetary Transport Network Shane D. Ross Control and Dynamical Systems, Caltech shane W.S. Koon (CDS), F. Lekien (CDS), M. Lo (JPL), J. Marsden (CDS) C. Jaffé (WVU), T. Uzer (G Tech), D. Farrelly (Utah State) Colloquium, Department of Physics and Astronomy, California State University, Long Beach, February 24, 2003

2 Motivation Apply dynamical systems theory to determine the transport of minor bodies throughout the solar system. Insert movie of asteriods 2

3 Important Tools Mechanical systems with symmetry; conserved quantities and reduction. For chaotic regimes of motion, the phase space has structures mediating transport. Theory of tube dynamics developed to study the motion of certain Jupiter-family comets (Koon, Lo, Marsden, SDR). Use the theory (Rom-Kedar, Wiggins, Haller,...) as well as the mangen software for lobe dynamics computations developed by Francois Lekien. Transport calculations for Mars impact ejecta, comets, Kuiper-belt objects, etc. 3

4 Transport Theory Chaotic dynamics = statistical methods e.g., transport through bottlenecks in phase space; intermittency 4

5 Transport Theory Ensembles of phase space trajectories Divide phase space into regions appropriately. How long to move from one region to another? Determine average transition rates. R 1 R 2 R 4 R 3 Boundaries between regions are partial barriers to transport. 5

6 Transport Theory Applications: Geophysical fluid dynamics Chemical reaction rates 6

7 Transport Theory Comet and asteroid transport rates between appropriately defined regions; rates/probabilities of collision with a planet. Insert movie of moon formation collision 7

8 Transport Theory Transport in the solar system For minor bodies of interest e.g., comets, Kuiper-belt objects, asteroids Identify phase space objects governing transport Model N-body system as restricted 3-body systems Assumption: Only one 3-body interaction dominates at a time e.g., comet-sun-p 1 -P 2 system modeled as comet-sun-p 1 and comet-sun-p 2 8

9 Transport Theory Insert pages from Marsden pres. 9

10 Transport Theory Insert pages from Marsden pres. 10

11 Motion within Energy Shell For fixed µ, an energy shell (or energy manifold) of energy ε is M(µ, ε) = {(x, y, ẋ, ẏ) E(x, y, ẋ, ẏ) = ε}. The M(µ, ε) are 3-dimensional surfaces foliating the 4-dimensional phase space. 11

12 Poincaré Surface-of-Section Study Poincaré surface of section at fixed energy ε: Σ (µ,ε) = {(x, ẋ) y = 0, ẏ = f(x, ẋ; µ, ε) > 0} reducing the system to an area preserving map on the plane. Exterior Realm Poincare Section Forbidden Realm Interior Realm z Particle P(z) Planetary Realm Poincaré surface-of-section and map P 12

13 Chaotic Sea The energy shell has regular (KAM tori) and irregular components. Large connected irregular component, the chaotic sea. 0.4 X Velocity (Nondim.) X (Nondim.) 13

14 Transport in 3-Body Problem Unstable resonances: Periodic orbits form a dynamical back-bone, via their unstable and stable manifolds. Physically, these manifolds correspond to orbits undergoing repeated close encounters with the smaller primary, e.g., Jupiter "! #:; <1%' ' +-/.& 1123, <4 "! #$ &%' )( *!,+-/.& Unstable resonances and their manifolds. 14

15 Transport in 3-Body Problem On a Poincaré section, consider the unstable and stable manifolds of unstable periodic orbits p 1 p 2 p 3 Unstable and stable manifolds in red and green, resp. 15

16 Transport in 3-Body Problem Intersection of unstable and stable manifolds define boundaries. p 1 q 1 q 4 q 2 q 3 q 5 p 2 p 3 q 6 16

17 Transport in 3-Body Problem These boundaries divide the phase space into regions. p 1 R 2 q 1 q 4 q 2 R 1 q 3 R 3 q 5 p 2 p 3 R 4 q 6 R 5 17

18 Lobe Dynamics Transport between regions is computed via lobe dynamics. L 2,1 (1) R 2 q 0 f (L 1,2 (1)) f -1 (q 0 ) q 1 p i L 1,2 (1) f (L 2,1 (1)) p j R 1 18

19 Dynamical Astronomy Compute transport between regions, e.g., transport between mean motion resonances, rates of ejecta escape from a planet, etc. Some questions of interest How probable is a Shoemaker-Levy 9-type collision with Jupiter? Or an asteroid collision with Earth (e.g., KT impact 65 Ma)? How likely is a transition from outside a planet s orbit to inside (e.g., the dance of comet Oterma with Jupiter)? Harder questions How does impact ejecta get from Mars to Earth? How does an SKBO become a comet or an Oort Cloud comet? Find features common to all exo-solar planetary systems? 19

20 Movement btwn Resonances We can compute manifolds which naturally divide the phase space into resonance regions. Identify Semimajor Axis Argument of Periapse Unstable and stable manifolds in red and green, resp. 20

21 Movement btwn Resonances Transport and mixing between regions can be computed. Identify R 3 Semimajor Axis R 2 R 1 Argument of Periapse Four sequences of color coded lobes are shown. 21

22 Movement btwn Resonances Transport and mixing between several resonances can be computed. Identify Semimajor Axis Resonance Region B } R 3 }R 2 Resonance Region A }R 1 Argument of Periapse 22

23 Oceanic Interlude The software used to compute transport by lobe dynamics, namely MANGEN, comes from a study of ocean dynamics. Interesting: there are analogs of navigating by invariant manifolds in the ocean. Adaptive Ocean Sampling Network (AOSN-II) Princeton: Naomi Leonard, Clancy Rowley, Eddie Forelli, Ralf Bachmayer,... Caltech: Chad Couliette, Francois Lekien, Jerry Marsden, Shawn Shadden MIT: George Haller 23

24 Oceanic Interlude Insert movie of parcels 24

25 Oceanic Interlude Insert movie of parcels w/ mfds 25

26 Tube Dynamics Back to the 3-body problem... Must also consider tube dynamics! Tubes in the energy surface lead toward and away from bottlenecks. Conley, McGehee (1960s) Koon, Lo, Marsden, SDR (2000s) 26

27 Tube Dynamics For example, points reach the exit in U 1 and are transported via a tube to the entrance of U 2. Poincare Section U 1 Poincare Section U 2 z 5 U 2 f 2 Entrance z 4 f 2 z 3 Earth L 1 f 12 U 1 f 1 f 1 z 0 z 2 Exit z 1 Tube dynamics: going from one Poincaré section to another. 27

28 Tube Dynamics Poincaré sections in different realms (U 1 through U 4 ) are linked by phase space tubes. The projection of the tubes on the configuration space appear as strips. Stable Unstable Stable U 3 Unstable U 4 S U 3 P P U 1 Stable U 2 L L 1 2 Unstable Stable U 2 Unstable Unstable and stable manifolds in red and green, resp. 28

29 Resonances and Tubes Resonances and tubes are linked It has been observed that the tubes of capture orbits are coming from certain resonances. Koon, Lo, Marsden, SDR [2001] 29

30 Jupiter Family Comets Jupiter Family Comets A physical example of the link between resonances and tubes 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 30

31 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). 31

32 Viewed in Rotating Frame Oterma s orbit in rotating frame with some invariant manifolds of the 3-body problem superimposed. oterma-rot.qt 32

33 Viewed in Inertial Frame oterma-iner.qt 33

34 Resonances and Tubes Poincaré section: tube cross-sections are closed curves Particles inside curves move toward or away from Jupiter 34

35 Resonances and Tubes Same Poincaré section: a resonance region is plotted 2:3 exterior resonance region 35

36 Resonances and Tubes Regions of overlap occur complex dynamics! Regions of overlap occur 36

37 Escape Rates Applications to dynamical astronomy One can compute the rate of escape of particles temporarily captured by Mars, e.g. asteroids or impact ejecta liberated from the Martian surface. Jaffé, SDR, Lo, Marsden, Farrelly, and Uzer [2002] Mars with temporarily captured asteroids. 37

38 Escape Rates Consider a particle at an energy such that it can escape sunward. Using a statistical approach used in transition state theory (developed by chemists), the rate of escape can be estimated. Sun Mars Case 2 : E 1 <E<E 2 38

39 Escape Rates Mixing assumption: all asteroids in the chaotic sea surrounding Mars are equally likely to escape. Escape rate = log(1 p), where Area of exit sunward p = Area of chaotic sea Tori Bounding the Chaotic Sea Chaotic Sea with Area A Exit to Interior Realm with Area F 39

40 Escape Rates This is a particularly simple situation ( Markovian ) Compare this rate with one obtained from a Monte Carlo simulations of 107,000 particles at randomly selected initial conditions at the same energy. 40

41 Escape Rates Theory and numerical simulations agree well Monte Carlo simulation (dashed) and theory (solid) 1 Survival Probability Number of Periods 41

42 Scattered Kuiper Belt Objects Some scattered Kuiper Belt Objects (SKBOs) in inertial space. 42

43 Scattered Kuiper Belt Objects uiper Belt objects in green, SKBOs near T = 3 eptune L 2 stable and unstable manifolds in black (around T = 3) Current SKBO locations in black, with some approximate curves of constant energy in the Sun-Neptune-SKBO in red. 43

44 Steady State Distribution If the planar, circular restricted three-body problem is approximately ergodic, then a statistical mechanics can be built (cf. ZhiGang [1999]). Recent work suggests there may be regions of the energy shell for which the motion is nearly ergodic, in particular the chaotic sea (Jaffé et al. [2002]). This suggests we compute the steady state distribution of some observable for particles in the chaotic sea; a simple method for obtaining the likely locations of any particles within it. 44

45 Steady State Distribution Assuming ergodicity, lim t 1 t t 0 A(x, y, p x, p y )dτ = A(x, y, p x, p y ) C H py dp xdxdy, where A(x, y, p x, p y ) is any physical observable (e.g., semimajor axis), one can finds that the density function, ρ(x, p x ), on the surface-of-section, Σ (µ,ε), is constant. We can determine the steady state distribution of semimajor axes; define N(a)da as the number of particles falling into a a+da on the surface-of-section, Σ (µ,ε). 45

46 eady state distribution Steady State Distribution KBOs should be in regions of high density. SKBOs should be in regions of high density. 46

47 Selected References Dellnitz, M., O. Junge, W.S. Koon, F. Lekien, M.W. Lo, J.E. Marsden, K. Padberg, R. Preis, S. Ross, & B. Thiere [2003], Transport in Dynamical Astronomy and Multibody Problems, in preparation. Ross, S.D. [2003], Statistical theory of interior-exterior transition and collision probabilities for minor bodies in the solar system, International Conference on Libration Point Orbits and Applications, Girona, Spain. Jaffé, C., S.D. Ross, M.W. Lo, J. Marsden, D. Farrelly, & T. Uzer [2002], Statistical theory of asteroid escape rates, Physical Review Letters 89, Koon, W.S., M.W. Lo, J.E. Marsden & S.D. Ross [2001], Resonance and capture of Jupiter comets. Celest. Mech. & Dyn. Astron. 81(1-2), 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 websites: shane/ 47