C����������� Dynamical S�� �����

CA L T EC H

A Statistical Theoryof Transition and Collision

ProbabilitiesShane D. Ross

Control and Dynamical Systems

California Institute of Technology

International Conference on Libration Point Orbits and ApplicationsParador d’Aiguablava, Girona, Spain, 10-14 June 2002

Acknowledgements� G. Gomez, M. Lo, J. Masdemont

� J. Marsden, W. Koon

� C. Jaffe, D. Farrelly, T. Uzer

� C. Simo, 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

2

Overview

�Motivation

� Planetary science

� Comet motion, collisions

� Chaotic dynamics, intermittency

�Transport theory

� Restricted three-body problem

� Predictions of theory

� Comparison with observational data?

3

Motivation

�Planetary science

� Current astrophysical interest in understanding thetransport 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-bodyproblem may provide a first-order answer.

� The “interaction” of several three-body systemsincreases the complexity.

4

Jupiter Family Comets

�Physical example of intermittency

� We consider the historical record of the cometOterma 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

5

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).

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Jupiter’s orbit

Sun

END

START

First encounter

Second encounter

3:2 resonance

2:3 resonance

Oterma’s orbit

Jupiter

6

Viewed in Rotating Frame� Oterma’s orbit in rotating frame with some invariant

manifolds of the 3-body problem superimposed.

y (

rota

tin

g)

x (rotating)

L1

y (

rota

tin

g)

x (rotating)

L2

Jupiter

Sun

Jupiter

Boundary of EnergySurface

Homoclinic Orbits

1980

1910

Oterma'sTrajectory

HeteroclinicConnection

Oterma

7

Viewed in Rotating Frame

Oterma - Rotating Frame

8

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 DynamicsTransport through a bottleneck in phase space; intermittency

10

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

11

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 orbitsassociated with libration points• e.g, planar Lyapunov orbits

� Use these to compute statistical quantities• e.g., probability of resonance transition, escape rates

12

Partition the Phase SpaceRegion A Region B

13

Partition the Phase Space

�Systems with potential barriers

� Electron near a nucleus with crossed electric andmagnetic fields• See Jaffe, Farrelly, and Uzer [1999]

Nucleus

"Bound" "Free"

Potential Configuration Space

14

Partition the Phase Space� Comet near the Sun and Jupiter

• Some behavior similar to electron!

Ueff

JupiterSun

Potential Configuration Space

15

Partition the Phase Space

�Partition is specific to problem

� We desire a way of describing dynamical boundariesthat represent the “frontier” between qualitativelydifferent types of behavior

�Example: motion of a comet

� motion around the Sun

� motion around Jupiter

16

Statement of Problem� Following Wiggins [1992]:

� Suppose we study the motion on a manifoldM� SupposeM is partitioned into disjoint regions

Ri, i = 1, . . . , NR,

such that

M =

NR⋃i=1

Ri.

� At t = 0, region Ri is uniformly coveredwith species Si

� Thus, species type of a point indicates theregion in which it was located initially

17

Statement of Problem� Statement of the transport problem:

Describe the distribution of speciesSi, i = 1, . . . , NR, throughout the regionsRj, j = 1, . . . , NR, for any time t > 0.

R1

R2

R3

R4

18

Statement of Problem� Some quantities we would like to compute are:

Ti,j(t) = amount of species Si contained in region Rj

Fi,j(t) =dTi,j

dt (t) = flux of species Si into region Rj

R1

R2

R3

R4

19

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

20

Restricted 3-Body Prob.

�Planar circular case

� Partition the energy surface: S, J, X regions

S JL1 L2

ExteriorRegion (X)

Interior (Sun) Region (S)

JupiterRegion (J)

ForbiddenRegion

Position Space Projection21

Equilibrium Region

�Look at motion near the potentialbarrier, i.e. the equilibrium region

S JL1 L2

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L2

Position Space Projection

22

Local Dynamics

� For fixed energy, the equilibrium region ' S2 × R.

• Stable/unstable manifolds of periodic orbit define mappingsbetween bounding spheres on either side of the barrier

PeriodicOrbit

p.o.

TransitOrbits

AsymptoticOrbits

NontransitOrbits

Spherical Capof Transit Orbits

Spherical Zoneof Nontransit Orbits

Asymptotic Circle

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.

x (rotating frame)

y (rotating frame)

L2 Periodic

OrbitStable

Manifold

Unstable

Manifold

Sun

Stable

Manifold

Forbidden Region

Unstable

Manifold

Jupiter

Forbidden Region

L2

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

25

Transition Probabilities� Example: Comet transport between outside and inside

of Jupiter (i.e., Oterma-like transitions)

ExteriorRegion

InteriorRegion

JupiterRegion

ForbiddenRegion

StableManifold

UnstableManifold

JupiterSun

Rapid Transition

x (rotating frame)

y (rotating frame)

L2

(a) (b)

x (rotating frame)

y (rotating frame)

StableManifold

UnstableManifold

UnstableManifold

StableManifold

L2L1

CaptureOrbit

Jupiter

26

Transition Probabilities� Consider Poincare section intersected by both tubes.

� Choosing surface {x = constant; px < 0}, we look atthe canonical plane (y, py).

x (rotating frame)

y (

rota

ting f

ram

e)

L1 L2

Poincare

section

Stable�����e

Unstable�����e Unstable�����

e Slice

Stable�����e Slice

y

py

Position Space 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.

Unstable�����e Slice

Stable�����e Slice

y

py Comet Orbits Passing �����

Exterior to Interior Region

Stable�����e Slice

Unstable��� �e Slice

Canonical Plane (y, py)

28

Transition Probabilities� Jupiter family comet transitions: X → S, S → X

−1.518 −1.517 −1.516 −1.515 −1.514 −1.513 −1.5120

10

20

30

40

50

60

Energy

Tra

nsi

tio

n p

rob

abili

ty (

%)

Transition Probability for Jupiter Family Comets

L1L2

Oterma

29

Collision Probabilities� Low velocity impact probabilities

� Assume object enters the planetary regionwith an energy slightly above L1 or L2

• eg, Shoemaker-Levy 9 and Earth-impacting asteroids

x

Comes fromInterior Region

Collision!

Example Collision Trajectory

30

Collision Probabilities

�Collision probabilities◦ Compute from tube intersection with planet on Poincare section

◦ Planetary diameter is a parameter, in addition to µ and energy E

← Diameter of planet →

31

Collision Probabilities

�Collision probabilities

−8 −6 −4 −2 0 2 4 6 8

x 10−5

−1.5

−1

−0.5

0

0.5

1

1.5

Poincare Section: Tube Intersecting a Planet

CollisionNon−Collision

← Diameter of planet →32

Collision Probabilities

−1.518 −1.517 −1.516 −1.515 −1.514 −1.513 −1.5120

1

2

3

4

5

Energy

Co

llisi

on

pro

bab

ility

(%

)Collision Probability for Jupiter Family Comets

L1L2

SL933

Collision Probabilities

−4 −3.5 −3

x 10−4

0

1

2

3

4

5

6

7

8

9

10

Energy (scaled)

Col

lisio

n pr

obab

ility

(%

)Collision Probability for Near−Earth Asteroids

L1L2

34

Collision Probabilities

0.995 1 1.005 1.01 1.015 1.02

−0.01

−0.005

0

0.005

0.01

x

yAsteroid Collision Orbit with Earth

Earth

L2

35

Conclusion and Future Work

�Transport in the solar system

� Approximate some solar system phenomena using therestricted 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 multiplethree-body problems

36

References• Jaffe, C., S.D. Ross, M.W. Lo, J. Marsden, D. Farrelly, and T. Uzer [2002]

Statistical theory of asteroid escape rates. Physical Review Letters, toappear.

• Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2001] Resonance andcapture of Jupiter comets. Celestial Mechanics and Dynamical Astronomy81(1-2), 27–38.

• Gomez, 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 spacemission design. AAS/AIAA Astrodynamics Specialist Conference.

• Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2000] Heteroclinic con-nections between periodic orbits and resonance transitions in celes-tial mechanics. Chaos 10(2), 427–469.

For papers, movies, etc., visit the website:http://www.cds.caltech.edu/∼shane

The End37