Resonance Overlap and Transport in the Restricted Three-Body Problem...
Transcript of Resonance Overlap and Transport in the Restricted Three-Body Problem...
![Page 1: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/1.jpg)
C����������� Dynamical S�� �����
CA L T EC H
Resonance Overlap andTransport in the Restricted
Three-Body Problem
Shane D. RossControl and Dynamical Systems
California Institute of Technology
Midwest Dynamical Systems SeminarUniversity of Cincinnati, October 5, 2002
![Page 2: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/2.jpg)
Acknowledgements� C. McCord, K. Meyer
� W. Koon, M. Lo, J. Marsden, F. Lekien
� G. Gomez, J. Masdemont
� C. Jaffe, D. Farrelly, T. Uzer, S. Wiggins
� K. Howell, B. Barden, R. Wilson
� C. Simo, J. Llibre, R. Martinez
� E. Belbruno, B. Marsden, J. Miller
� H. Poincare, J. Moser, C. Conley, R. McGehee
2
![Page 3: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/3.jpg)
Outline of talk
� Insight into some dynamical astronomyphenomena can be gained by a restrictedthree-body analysis
� e.g., Jupiter-family comets and scattered Kuiper Beltobjects (under Neptune’s control); near-Earth objects
� By applying dynamical systems methods to the pla-nar, circular restricted three-body problem, several ques-tions regarding these populations may be addressed
� Outline some theoretical ideas
� Several computational results will be shown
� Comparison with observational data is made
� Future directions: other N -particle systems3
![Page 4: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/4.jpg)
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
![Page 5: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/5.jpg)
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 or-bits associated with Lagrange points and mean motionresonances
¤Use these to compute statistical quantities• e.g., probability of resonance transition, escape rates
6
![Page 6: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/6.jpg)
Dynamical astronomy� We want to answer several questions regarding the trans-
port and origin of some kinds of solar system material
◦ How do we characterize the motion of Jupiter-family comets (JFCs)and scattered Kuiper Belt objects (SKBOs)?
◦ 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)?
◦We can answer these questions by considering the phase space
� 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?
4
![Page 7: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/7.jpg)
Jupiter Family Comets◦ JFCs and lines of constant Tisserand parameter,
T =1
a+ 2
√a(1− e2),
an approximation of the Jacobi constant (i.e., C = T +O(µ))
5
![Page 8: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/8.jpg)
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
![Page 9: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/9.jpg)
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).
����� �������� ������� �����
��� ����� ��� �� �!
Jupiter’s orbit
Sun
END
START
First encounter
Second encounter
3:2 resonance
2:3 resonance
Oterma’s orbit
Jupiter
6
![Page 10: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/10.jpg)
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
![Page 11: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/11.jpg)
Viewed in Rotating Frame
Oterma - Rotating Frame
8
![Page 12: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/12.jpg)
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
![Page 13: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/13.jpg)
Scattered Kuiper Belt objects◦ Current SKBO locations in black, with some Tisserand values w.r.t.
Neptune in red (T ≈ 3)
Scattered Kuiper Belt Objects◦ Kuiper Belt objects in green, SKBOs near T = 3
◦ Neptune L2 stable and unstable manifolds in black (around T = 3)
5
6
![Page 14: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/14.jpg)
Scattered Kuiper Belt Objects◦ Seen in inertial space
-200 -150 -100 -50 0 50 100 150 200
-150
-100
-50
0
50
100
150
7
![Page 15: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/15.jpg)
Motion of JFCs and SKBOs� Observation and numerical experiments show chaotic
motion maintaining nearly constant Tisserand parame-ter in the short-term (i.e., a few Lyapunov times, ∼ 102
to 103 years, cf. Tancredi [1995])
� We approximate the short-timescale motion of JFCs andSKBOs as occurring within an energy shell of the re-stricted three-body problem
� Several objects may be in nearly the same energy shell,i.e., all have T s.t. |T − T ∗| ≤ δT for some T ∗, δT
� We analyze the structure of an energy shell to determinelikely locations of JFCs and SKBOs
8
![Page 16: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/16.jpg)
Three-Body Problem
�Circular restricted 3-body problem
� the two primary bodies move in circles; the muchsmaller third body moves in the gravitational field ofthe primaries, without affecting them
� the two primaries could be the Sun and Earth, theEarth and Moon, or Jupiter and Europa, etc.
� the smaller body could be a spacecraft or asteroid
� we consider the planar and spatial problems
� there are five equilibrium points in the rotating frame,places of balance which generate interesting dynamics
8
![Page 17: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/17.jpg)
Three-Body Problem� Equations of motion:
x− 2y = −U effx , y + 2x = −U eff
y
where
U eff = −(x2 + y2)
2− 1− µ
r1− µ
r2.
� Have a first integral, the Hamiltonian energy, given by
E(x, y, x, y) =1
2(x2 + y2) + U eff(x, y).
� Energy manifolds are 3-dimensional surfaces foliatingthe 4-dimensional phase space.
� This is for the planar problem, but the spatial problemis similar.
11
![Page 18: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/18.jpg)
Regions of Possible Motion
�Effective potential
� In a rotating frame, the equations of motion describea particle moving in an effective potential plus a corio-lis force (goes back to the work of Jacobi, Hill, etc.)
Ueff
S JL1 L2
ExteriorRegion (X)
Interior (Sun) Region (S)
JupiterRegion (J)
ForbiddenRegion
Effective potential Level set shows accessible regions
12
![Page 19: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/19.jpg)
Partition the Energy Surface
¥Restricted 3-body problem (planar)
¤ Partition the energy surface: S, J, X regions
S JL1 L2
ExteriorRegion (X)
Interior (Sun) Region (S)
JupiterRegion (J)
ForbiddenRegion
Position Space Projection21
![Page 20: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/20.jpg)
Tubes in the 3-Body Problem� Stable and unstable manifold tubes
• Control transport through the neck.
x(rotatingframe)
y(r
ota
tin
gf
ram
e)
L2 Periodic
OrbitStable
Manifold
Unstable
Manifold
Sun
Stable
Manifold
Forbidden Region
Unstable
Manifold
Jupiter
Forbidden Region
L2
24
![Page 21: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/21.jpg)
Motion within energy shell� For fixed µ, an energy shell (or energy manifold) of en-
ergy ε is
M(µ, ε) = {(x, y, x, y) | E(x, y, x, y) = ε}.The M(µ, ε) are 3-dimensional surfaces foliating the4-dimensional phase space.
9
![Page 22: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/22.jpg)
Poincare surface-of-section� Study Poincare surface of section at fixed energy ε:
Σ(µ,ε) = {(x, x)|y = 0, y = f (x, x, µ, ε) < 0}reducing the system to an area preserving map on theplane. Motion takes place on the cylinder, S1 × R.
z
P(z)
Poincare surface-of-section and map P10
![Page 23: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/23.jpg)
Connected chaotic component◦ The energy shell has regular components (KAM tori) and irregular com-
ponents. Large connected irregular component is the “chaotic sea.”
11
![Page 24: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/24.jpg)
Movement among resonances◦ The motion within the chaotic sea is understood as the movement of
trajectories among resonance regions (see Meiss [1992] and Schroer andOtt [1997]).
12
![Page 25: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/25.jpg)
Movement among resonances
Schematic of two neighboring resonance regions from Meiss [1992]
13
![Page 26: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/26.jpg)
Movement among resonances◦ Confirmed by numerical computation.
◦ Shaded region bounded by stable and unstable invariant manifolds of anunstable resonant (periodic) orbit.
Movem
ent
am
ong
resonances
◦T
hisis
confirmed
bynum
ericalcom
putation.
◦Shaded
regionbounded
bystable
andunstable
invariantm
anifoldsof
anunstable
resonant(p
eriodic)orbit.
13
14
![Page 27: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/27.jpg)
Homoclinic tangle◦ Unstable/stable manifolds of periodic points understood as the back-
bone of the dynamics. This is the homoclinic tangle glimpsed by Poincare.
15
![Page 28: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/28.jpg)
Resonance region◦ Unstable/stable manifolds up to “pip” (cf. Wigging [1992]) denote the
boundary of the resonance region.
16
![Page 29: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/29.jpg)
Resonance region◦ Neighboring resonance regions indeed overlap, leading to complicated
mixing.
17
![Page 30: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/30.jpg)
Transport quantities• Lobe dynamics; following intersections of stable and unsta-
ble invariant manifolds of periodic orbits (Wiggins et al.)
19
![Page 31: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/31.jpg)
Transport quantities� These methods are preferred over the “brute force”
solar system calculations seen in the literature since theyare based on first principles.
� Reveal generic structures; give deeper insight.
20
![Page 32: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/32.jpg)
Escape rates
�Obtain rates and probabilities� One can compute the rate of escape of asteroids
temporarily captured by Mars.• Jaffe, Ross, Lo, Marsden, Farrelly, and Uzer [2002]
� Statistical approach• similar to chemical dynamics, see Truhlar [1996]
� Consider an asteroid (or other body) in orbit aroundMars (perhaps impact ejecta) at a 3-body energy suchthat it can escape toward the Sun.
� Interested in rate of escape of such bodies at a fixedenergy, i.e. FM,S(t)
21
![Page 33: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/33.jpg)
¥ Hill’s Region (PCR3BP)
I To fix energy value E is to fix height of plot of U(x, y).Contour plots give 5 cases of Hill’s region.
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
S J S M
S JS J
Case 1 : C>C1 Case 2 : C1>C>C2
Case 4 : C3>C>C4=C5Case 3 : C2>C>C3
![Page 34: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/34.jpg)
Escape rates� Mixing assumption: all asteroids in the Mars region at
fixed energy are equally likely to escape.
�
Escape rate =flux out of Mars region
Mars region phase space volume
=area of escaping orbits
area of chaotic region
22
![Page 35: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/35.jpg)
Escape rates
23
![Page 36: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/36.jpg)
Escape rates� Compare with Monte Carlo simulations of 107,000 par-
ticles
• randomly selected initial conditions at constant energy
24
![Page 37: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/37.jpg)
Transition Rates� Theory and numerical simulations agree well.◦Monte Carlo simulation (dashed) and theory (solid)
0 10 20 30 40Number of Periods
1
0.9
0.8
0.6
Surv
ival
Pro
babi
lity
29
![Page 38: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/38.jpg)
Steady state distribution� If the planar, circular restricted three-body problem is
ergodic, then a statistical mechanics can be built (cf.ZhiGang [1999]).
� Recent work suggests there may be regions of the energyshell for which the motion is ergodic, in particular the“chaotic sea” (Jaffe et al. [2002]).
� This suggests we compute the steady state distribu-tion of some observable for particles in the chaotic sea;a simple method for obtaining the likely locations of anyparticles within it.
25
![Page 39: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/39.jpg)
Steady state distribution� Assuming ergodicity,
limt→∞
1
t
∫ t
0
A(x, y, px, py)dτ =∫A(x, y, px, py)
C|∂H∂py|dpxdxdy,
where A(x, y, px, py) is any physical observable (e.g.,semimajor axis), one can finds that the density function,ρ(x, px), on the surface-of-section, Σ(µ,ε), is constant.
� We can determine the steady state distribution of semi-major axes; define N(a)da as the number of particlesfalling into a → a+da on the surface-of-section, Σ(µ,ε).
26
![Page 40: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/40.jpg)
Steady state distribution� SKBOs should be in regions of high density.
Steady state distribution� SKBOs should be in regions of high density.
18
27
![Page 41: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/41.jpg)
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
![Page 42: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/42.jpg)
Tubes in the 3-Body Problem� Stable and unstable manifold tubes
• Control transport through the neck.
x(rotatingframe)
y(r
ota
tin
gf
ram
e)
L2 Periodic
OrbitStable
Manifold
Unstable
Manifold
Sun
Stable
Manifold
Forbidden Region
Unstable
Manifold
Jupiter
Forbidden Region
L2
24
![Page 43: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/43.jpg)
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
![Page 44: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/44.jpg)
Collision Probabilities
�Collision probabilities
−8−6 −4
−2 02
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
y
py
![Page 45: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/45.jpg)
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
![Page 46: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/46.jpg)
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
![Page 47: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/47.jpg)
Conclusion and Future Work
�Transport in the solar system
� Approximate some solar system phenomena using therestricted 3-body problem
� Circular 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 and 3-d.o.f.
36
![Page 48: Resonance Overlap and Transport in the Restricted Three-Body Problem …sdross/talks/midwest-ross-2002.pdf · 2006. 8. 28. · How probable is a Shoemaker-Levy 9-type collision with](https://reader036.fdocuments.us/reader036/viewer/2022071423/611db39ee6ccdb071d46b4a3/html5/thumbnails/48.jpg)
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, 89
• 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