[email protected]/books/cds270/270_4b.pdf · between pair of periodic orbits. found a large...

Dynamical Systemsand Space Mission Design

Jerrold Marsden, Wang Sang Koon and Martin Lo

Shane RossControl and Dynamical Systems, Caltech

[email protected]

� Resonance Transition of Comets: Outline

I Outline of Lecture 4B:

•Resonance transition seen in comets such as Oterma.•Mixed phase space of 3-body problem:•Mean motion resonance “islands” imbedded in chaotic “sea.”• Exterior and interior resonances connected by Lyapunov orbit

stable & unstable manifold tubes, the dynamical channels.• Future work: transition between planets, belts, etc.

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ensional

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ensional

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rotatingfram

e)

Sun

3:2

2:3 2:3

3:2

Sun

Jupiter'sorbit

Jupiter'sorbit

L1 L22:3 resonance

3:2 resonance

Jupiter

Jupiter encounter,transition event

� Jupiter Comets: Oterma

I Some Jupiter comets perform a rapid transition from theoutside to the inside of Jupiter’s orbit.

ICaptured temporarily by Jupiter during transition.

I Exterior (2:3 resonance). Interior (3:2 resonance).

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x (AU, Sun-centered inertial frame)

y(A

U,Sun-centeredinertialframe)

Jupiter’s orbit

Sun

END

START

First encounter

Second encounter

3:2 resonance

2:3 resonance

Oterma’s orbit

Jupiter

� Jupiter Comets: Previous Works

I Belbruno/B. Marsden [1997]

I Lo/Ross [1997] :

• Jupiter comets Oterma, Gehrels 3, etc. in Sun-Jupiter rotat-ing frame follow stable and unstable invariant mani-folds of the equilibrium points L1 and L2.

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ame)

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Jupiter’s orbit-10 -8 -6 -4 -2 0 2 4 6 8 10

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S J


y (A

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upite

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Oterma’s orbit

END

START

S

� Jupiter Comets: Planar CR3BP Model

I Use planar circular restricted 3-body problem as initial model:

• Simplest 3-body model, easiest to analyze.• Comets of interest are mostly heliocentric, but

their perturbation is dominated by Jupiter’s gravitation.• Their motion is nearly in Jupiter’s orbital plane (i < 5◦), and

Jupiter’s small eccentricity (0.0483) plays little role duringresonance transition.

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-1

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1


y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

mS = 1 - µ mJ = µ

S J

Jupiter's orbit

L2

L4

L5

L3 L1

comet

� Jupiter Comets: Heteroclinic Connection

IMore recently, Koon/Lo/Marsden/Ross [2000]:

• Found heteroclinic connection between pair of periodic orbits.

• Found a large class of orbits near this (homo/heteroclinic) chain.

• Comets can follow these dynamical channels in rapid transi-tion between interior and exterior Hill’s regions.

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Jupiter's Orbit


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ame)


y (A

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upite

r ro

tatin

g fr

ame)

Jupiter

SunHomoclinic

Orbit

Homoclinic

Orbit

L1 L2L1 L2

Heteroclinic

Connection

� Jupiter Comets: Following Dynamical Channels

• For instance, consider the comet Oterma from 1910 to 1980.

• The average Jacobi constant for Oterma during its transition isC = 3.030± 0.005 (computed at Jupiter encounter).

•We can compute a homoclinic-heteroclinic chain forC = 3.030 (shown in black on the left).

• Overlaying the chain, we plot Oterma’s orbit in red (at right).

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1980

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y (A

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upite

r ro

tatin

g fr

ame)

SunL1 L2

Jupiter's orbit

Jupiter

3:2 resonance

2:3 resonance

Jupiter

(a) (b)

Oterma'sorbit

� Jupiter Comets: Rapid Transition Mechanism

•Rapid transition between the interior and exterior regions ispossible via the L1 & L2 periodic orbit stable & unstable man-ifold tubes (containing transit orbits) and their intersections.

• This was a surprising result. Some believed that a third degreeof freedom was necessary for such a transition, or that “Arnolddiffusion” was invloved.

• But as we have seen, only the planar CR3BP, the simplestmodel of 3-body gravitational dynamics, is necessary.

∆J = (X;J,S)

Intersection Region

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

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y (r

otat

ing

fram

e)

JL1 L2

Forbidden Region

Forbidden Region

Poincare

section

StableManifold

UnstableManifold

StableManifold Cut

UnstableManifold Cut-0.6

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0

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y

(;J,S)

(X;J)

UnstableManifold Cut

StableManifold Cut

(;J,S)

(X;J)

y0 0.01 0.02 0.04 0.05 0.06 0.07 0.08

y0.03

y

� Rapid Transition Mechanism: Resonance Transition

• The tubes are a generic transport mechanism connecting theinterior and exterior Hill’s regions.

•We wish to understand their role in transport between interior andexterior mean motion resonances.

• e.g., we shall try to explain in more precise terms the sense inwhich Oterma transitions between the 3:2 and 2:3 mean motionresonances with Jupiter.

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ensional

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is)

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ensional

units,

inertial

fram

e)


y(n

ondim

ensional

units,

rotatingfram

e)

Sun

3:2

2:3 2:3

3:2

Sun

Jupiter'sorbit

Jupiter'sorbit

L1 L22:3 resonance

3:2 resonance

Jupiter



• For the Sun-Jupiter system (µ = 0.0009537), we can construct ahomoclinic-heteroclinic chain with Jacobi constant similar to thatof Oterma during its recent Jupiter encounters (C = 3.030).

• The chain is a union of orbits: interior region orbit homoclinicto L1 periodic orbit, exterior region orbit homoclinic to L2 peri-odic orbit, and heteroclinic connection between the L1 & L2periodic orbits.

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Jupiter's Orbit


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Jupiter

SunHomoclinic

Orbit

Homoclinic

Orbit

L1 L2L1 L2

Heteroclinic

Connection


•We choose this chain because its homoclinic orbits are (1,1)-type.

• Limiting to (1,1)-type means, for this particular energy regime,that two different resonance connections are possible; 3:2 to 2:3,and 3:2 to 1:2. This will be explained later. We choose 3:2 to 2:3,since this matches Oterma’s orbit.

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Jupiter's Orbit


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y (A

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ame)

Jupiter

SunHomoclinic

Orbit

Homoclinic

Orbit

L1 L2L1 L2

Heteroclinic

Connection


• Our main theorem tells us that in the vicinity of this chain, thereexists an orbit whose symbolic sequence (. . . , J,X, J, S, J, . . . ) isperiodic and has the central block itinerary (J,X, J, S, J).

• This orbit transitions between the interior and exterior regions(the neighborhood of the 3:2 and 2:3 resonances, in particular).We call this kind of itinerary a resonance transition block.

Orbits Passing FromExterior to Interior Region

L1 p.o. StableManifold Cut

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

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(X;J,S)

(X;J)

(;J,S)

y

y .

(;J,S,J)

(J,X;J,S,J)

(J,X;J)

y

y .

L2 p.o. UnstableManifold Cut

Poincare Section in the Jupiter Region


• This orbit makes a rapid transition from the exterior to theinterior region and vice versa, passing through the Jupiter region.It will repeat this pattern eternally.

•While an orbit with this exact itinerary is very fragile, the structureof nearby orbits whose symbolic sequences have the same centralblock itinerary, namely (J,X, J, S, J), is quite robust.



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(X;J)

(;J,S)

y

y .

(;J,S,J)

(J,X;J,S,J)

(J,X;J)

y

y .




• An example orbit with central block (J,X, J, S, J) is shown below.

•We will study how the dynamical channels near the chain connectthe 3:2 resonance of the interior region with the 2:3 resonanceof the exterior region.

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ensional

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rotatingfram

e)

Sun

3:2

2:3 2:3

3:2

Sun

Jupiter'sorbit

Jupiter'sorbit

L1 L22:3 resonance

3:2 resonance

Jupiter


� PCR3BP: Perturbation of the Two-Body Problem

• Recall that the PCR3BP is a perturbation of the two-body prob-lem. Hence, outside of a small neighborhood of L1, the trajectoryof a comet in the interior region follows essentially a two-bodyorbit around the Sun.

• In the heliocentric inertial frame, the orbit is nearly elliptical.

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ensional

units,

inertial

fram

e)


y(n

ondim

ensional

units,

rotatingfram

e)

Sun

3:2

2:3 2:3

3:2

Sun

Jupiter'sorbit

Jupiter'sorbit

L1 L22:3 resonance

3:2 resonance

Jupiter


� Heliocentric Orbits: Mean Motion Resonance

• The mean motion resonance of the comet with respect toJupiter is equal to a−3/2 where a is the semi-major axis of theheliocentric elliptical orbit. Recall that the Sun-Jupiter distance isnormalized to be 1 in the PCR3BP.

• The comet is said to be in p:q resonance with Jupiter ifa−3/2 ≈ p/q, where p and q are small integers. In the heliocentricinertial frame, the comet makes roughly p revolutions aroundthe Sun in q Jupiter periods.

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fram

e)


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units,

rotatingfram

e)

Sun

3:2

2:3 2:3

3:2

Sun

Jupiter'sorbit

Jupiter'sorbit

L1 L22:3 resonance

3:2 resonance

Jupiter


� Canonical Coordinates: Delaunay Variables

• To study the process of resonance transition, we shall use a set ofcanonical coordinates, called Delaunay variables, which makethe study of the two-body regime of motion particularly simple.

• Delaunay variables in the rotating coordinates are denoted (l, g, L,G).G = [a(1 − e2)]1/2 is the angular momentum. L is related to thesemi-major axis a, byL = a1/2, hence encodes the mean motionresonance (with respect to Jupiter in the Sun-Jupiter system).

xiner

yiner

.

perihelion

fixedapsidal line

aphelion .

xrot

yrot

Sun.

a

g_

ae

comet

� Canonical Coordinates: Delaunay Variables

• Both l and g are angular variables defined modulo 2π.

• g is the argument of perihelion relative to the rotating axis.

• l is the mean anomaly, the ratio of the area swept out by theray from the Sun to the comet starting from its perihelion passageto the total area.

• Szebehely [1967], Abraham & Marsden [1978], Meyer & Hall [1992].

xiner

yiner

.

perihelion

fixedapsidal line

aphelion .

xrot

yrot

Sun.

a

g_

ae

comet

� Interior Resonances: U1 Poincare Section (L,g)

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.90

50

100

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250

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350g(degrees)

_

.

Γu,SL1,1

PS

∆S

Γs,SL1,1

L =√

a

3:25:3 4:32:17:3

� Interior Resonances: U1 Poincare Section

• The striking thing is that the first cuts of the stable and unstablemanifolds intersect exactly at the region of the 3:2 resonance.

• Their intersection ∆S contains all the orbits that have come fromthe Jupiter region J into the interior region S, gone around theSun once (in the rotating frame), and will return to the Jupiterregion. In the heliocentric inertial frame, these orbits are nearlyelliptical outside a neighborhood of L1.

• They have a semi-major axis which corresponds to 3:2 resonance byKepler’s law (i.e., a−3/2 = L−3 ≈ 3/2). Therefore, any Jupitercomet which has an energy similar to Oterma’s and whichcircles around the Sun once in the interior region must be in3:2 resonance with Jupiter.

� Mixed Phase Space: Stable “Islands” & Chaotic “Sea”

• The black background points on the U1 Poincare section reveal thecharacter of the interior region phase space for this energy surface.

•Mixed phase space of stable periodic and quasiperiodicinvariant tori “islands” embedded in bounded chaotic “sea.”

• The families of stable tori, where a “family” denotes thosetori islands which lie along a strip of nearly constant L, correspondto mean motion resonances. The size of the tori island corre-sponds to the dynamical significance of the resonance. The numberof tori islands equals the order of the resonance (e.g., 3:2 is order1, 5:3 is order 2).

• In the center of each island, there is a point corresponding to anexactly periodic, stable, resonant orbit. In between the stable is-lands of a particular resonance (i.e., along a strip of nearly constantL), there is a saddle point corresponding to an exactly periodic,unstable, resonant orbit. In the figure, the manifold intersectionregion ∆S is centered on this saddle point for the 3:2 resonance.

� Connection Between Interior and Exterior Resonances

• A subset of the interior resonance intersection region ∆S is con-nected to exterior resonances through a heteroclinic intersection inthe Jupiter region. This small blue strip inside ∆S is part ofthe dynamical channel connecting interior and exterior resonances,and is thus the resonance transition mechanism we seek.

P(∆)

∆S

StableManifold

UnstableManifold

3:2 Resonance

� Exterior Resonances: U4 Poincare Section (L,g)

1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.320

50

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L =

g(degrees)

_

Γu,XL2,1

PX

∆X

Γs,XL2,1

.

√a

1:23:4 3:52:3


•We show the first exterior region Poincare cuts of the stable andunstable manifolds of an L2 periodic orbit with the U4 sectionon the same energy surface.

• Notice that the first cuts of the stable and unstable manifoldsintersect at two places; one of the intersections is exactly at theregion of the 2:3 resonance, the other is at the 1:2 resonance.

• The intersection ∆X contains all the orbits that have come fromthe Jupiter region J into the exterior region X, have gone aroundthe Sun once (in the rotating frame), and will return to the Jupiterregion. Note that ∆X has two components (the 2:3 and 1:2 reso-nance regions).


• In the heliocentric inertial frame, these orbits are nearly ellipticaloutside a neighborhood of L2. They have a semi-major axis whichcorresponds to either 2:3 or 1:2 resonance by Kepler’s law. There-fore, any Jupiter comet which has an energy similar to Oterma’sand which circles around the Sun once in the exterior region mustbe in either 2:3 or 1:2 resonance with Jupiter.

• The background points were generated by a technique similar tothose in the interior resonance Poincare section. They reveal asimilar mixed phase space, but now the resonances are exteriorresonances (exterior to the orbit of Jupiter). We see that the ex-terior resonance intersection region ∆X envelops both the 2:3 andthe 1:2 unstable resonance points.


• A portion of ∆X is connected to interior resonances through a het-eroclinic intersection in the Jupiter region. In particular, the smallblue strip inside the 2:3 intersection region connects to the3:2 intersection region of ∆S (and is its pre-image). We havethus found the resonance transition used by Oterma.

P −1(∆)

∆X

StableManifold

UnstableManifold

2:3 Resonance


•We have referred to a heteroclinic intersection ∆ connecting inte-rior ∆S and exterior ∆X resonance intersection regions. Below, weshow image of ∆X (2:3 resonance portion) and pre-image of ∆S

in the J region. Their intersection ∆ = P (∆X ) ∩ P−1(∆S) con-tains all orbits whose itineraries have central block (J,X; J, S, J),corresponding to at least one transition between the exterior 2:3resonance and interior 3:2 resonance.



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(X;J,S)

(X;J)

(;J,S)

y

y .

y

y .



= (;J,S,J)P −1( )∆S

= (J,X;J)P ( )∆X

= (J,X;J,S,J)∆


• ∆ contains orbits in transition between the 2:3 to 3:2 resonances.

• Comets such as Oterma have passed through analogous regions.

= (;J,S,J)P −1( )∆S

= (J,X;J)P ( )∆X

= (J,X;J,S,J)∆

P −1(∆)

∆X

2:3 Resonance

P(∆)

∆S

3:2 Resonance Resonance Connection

Jupiter Region Exterior RegionInterior Region

� Resonance Connection for Three Degrees of Freedom

• It is reasonable to conclude that, within the full three-dimensionalmodel, Oterma’s orbit lies in an analogous region of phase space.

• It is therefore within the L1 and L2 periodic and quasiperiodicorbit manifold tubes, whose complex global dynamics lead tointermittent behavior, including resonance transition.

•More study is needed for a thorough understanding of the reso-nance transition phenomenon. The tools developed in this course(dynamical channels, symbolic dynamics, etc.) shouldlay a firm theoretical foundation for any such future studies.

= (;J,S,J)P −1( )∆S

= (J,X;J)P ( )∆X

= (J,X;J,S,J)∆

P −1(∆)

∆X

2:3 Resonance

P(∆)

∆S

3:2 Resonance Resonance Connection

Jupiter Region Exterior RegionInterior Region

� Future Work: Extension to Three Dimensions

I Natural extension: apply same methodology to 3D CR3BP.

• Seek homoclinic & heteroclinic orbits associated with 3D periodic“halo” & quasi-periodic “quasi-halo” & Lissajous orbits about L1& L2. Dimension count suggests heteroclinic intersections exist.

• Union would be 3D homoclinic-heteroclinic chains around whichsymbolic dynamics could be used to track a variety of exotic orbits.

•Three-dimensional dynamical channels will provide morecomplete understanding of phase space transport mechanisms.

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� Future Work: Coupling of Two 3-Body Systems

I Dynamics governing transport between adjacent planets.

• Coupled 3-body problem: e.g., comet between Jupiter & Saturn.

• Between the two planets, the comet’s motion is mostly heliocen-tric, but is precariously poised between two competing three-bodydynamics.

• In this region, heteroclinic orbits connecting Lyapunov orbits of thetwo different three-body systems may exist, leading to complicatedtransfer dynamics between the two adjacent planets.

� Comet Transition Between Jupiter and Saturn

I Example: Comet Smirnova-Chernykh undergoes arapid transition from Saturn’s control to Jupiter’s control.

1800 1850 1900 1950 2000 2050 2100

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imaj

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xis

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)

Semimajor Axis of Comet Smirnova-Chernykh (AD 1800-2100)

Smirnova-Chernykh

Saturn

Jupiter

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(AU

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rtia

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me)

Inertial Plot of Comet Smirnova-Chernykh

Smirnova-Chernykh

Saturn

Jupiter

Sun

� Comet Transition Between Jupiter and Saturn

I Coupled PCR3BP shows near intersections between Lyapunovorbit manifold tubes of Jupiter and Saturn (requiring mild ∆V ).

• Natural continuous thrust of comet outgassing may be enough.

• Longer time integration will likely reveal genuine intersections.

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xdot

(no

ndim

ensi

onal

uni

ts, S

un-J

upite

r ro

tatin

g fr

ame)

Poincare section (y=0,x<0) in Sun-Jupiter rotating frame showing close approach of Jupiter and Saturn manifolds

First cut ofJupiter’s stableL

2 manifold

First cut of Saturn’sunstable L

1 manifold

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x (Sun-Jupiter distances, Sun-centered inertial frame)

y (S

un-J

upite

r di

stan

ces,

Sun

-cen

tere

d in

ertia

l fra

me)

Transfer from Saturn to Jupiter via equil. region manifolds highway

Transfer time= 68 years

∆V = 615 m/s

Saturn

Jupiter

Sun

Comet

� Future Work: Long Time Integration

I Results limited thus far to short time (a few periods of Jupiter).

• Long time integration (millions of Jupiter periods) will revealstatistical information and new phenomena.

• Preliminary results suggest manifold structures associated toL1 and L2 have helped sculpt the solar systemand transport material between the planets.

Comets

Ast

ero

ids

Kuiper Belt Objects

Pluto

Ne

ptu

ne

Ura

nu

s

Sa

turn

Jup

iter

Tro

jan

s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

OrbitalEccentricity

Semi-major Axis (AU)20 25 30 3515105 40 45 50

� Long Time Integration: Jupiter’s L1 Manifolds

•We show U1 Poincare section of Jupiter’s L1 stable & unstablemanifolds for one million iterations (in a vs. g).


•We can also plot this in semimajor axis a vs. eccentricity e.Away from L1, manifold hugs curve given byC = 1

a+2√a(1− e2).


• Note how manifold acts as stability boundary, separating stableasteriods from unstable comets.

� Intermittent Behavior Along Jupiter’s L1 Manifolds

• Time history of semimajor axis a for one million iterations shown.

•Manifold exhibits intermittency, jumping, sticking.

� Comet Distribution Matches Jupiter’s L1 Manifolds

• Taking histogram of Jupiter’s L1 manifolds shows fair agreementwith distribution of Jupiter comets. Same dynamics is at work.

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80

50

100

150

200

250

300

350

400

450

500

(period of orbit)/(Jupiter‘s period)

rela

tive

num

ber

of o

rbits

Short−period comet orbits (~800 yrs, black) and Sun−Jupiter interior L1 manifold (~5,000,000 yrs, green)

� Kuiper Belt and Neptune’s L2 Manifolds

• Just as Jupiter’s manifolds determine asteroid & comet distributionand transport, Neptune’s manifolds may govern the Kuiper belt.

30 40 50 60 70 80 90 100 110 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Kuiper Belt objects (circles) with Neptune’s L2 manifold (black line)

semimajor axis (AU)

orbi

tal e

ccen

tric

ity

� Transport Between Asteriod Belt and Kuiper Belt

• Intersections between L1 and L2 manifold structures between ad-jacent planets may provide a “highway” connecting the asteroidand Kuiper belts, where material can collect.

Comets

Ast

ero

ids

Kuiper Belt Objects

Pluto

Neptu

ne

Ura

nus

Satu

rn

Jupite

rT

roja

ns

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

OrbitalEccentricity


� Conclusion: L1 and L2 Manifolds are Important!

• The invariant manifold structures associated to L1 and L2, as wellas the homoclinic-heteroclinic dynamical channels connectingthem, are fundamental tools that can aid in understandingmechanisms of transport throughout the solar system.

Comets

Ast

ero

ids

Kuiper Belt Objects

PlutoN

eptu

ne

Ura

nus

Satu

rn

Jupite

rT

roja

ns

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

OrbitalEccentricity


[email protected]/books/cds270/270_4b.pdf · between pair of periodic orbits. found a large...

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