& low energy transfer to the moon dynamical systems, 3 ...koon/research/3body/moonaas.pdf ·...

Dynamical Systems, 3-Body Problem& Low Energy Transfer to the Moon

Koon, Lo, Marsden, and Ross

Wang Sang KoonControl and Dynamical Systems, Caltech

[email protected]

� Acknowledgements

IH. Poincare, J. Moser

IC. Conley, R. McGehee

IC. Simo, J. Llibre, R. Martinez

I E. Belbruno, B. Marsden, J. Miller

IG. Gomez, J. Masdemont

IK. Howell, B. Barden, R. Wilson

I L. Petzold, S. Radu

� Outline of Presentation

IMain Theme

• how to use dynamical systems theory of 3-body problemin space mission design.

IBackground and Motivation:

• NASA’s Genesis Discovery Mission.• Jupiter Comets.

I Planar Circular Restricted 3-Body Problem.

IMajor Results and Some Technical Details.

I Low Energy Transfer to the Moon.

IConclusion and Ongoing Work.

� Genesis Discovery Mission

IGenesis spacecraft will

• collect solar wind from a L1 halo orbit for 2 years,• return those samples to Earth in 2003 for analysis.

IHalo orbit, transfer/ return trajectories in rotating frame.

L1 L2

x (km)

y(k

m)

-1E+06 0 1E+06

-1E+06

-500000

0

500000

1E+06

� Genesis Discovery Mission

IMust return in Utah during daytime.

IReturn-to-Earth portion utilizes heteroclinic dynamics.

L1 L2

x (km)

y(k

m)

-1E+06 0 1E+06

-1E+06

-500000

0

500000

1E+06

� Jupiter Comets

IRapid transition from outside to inside Jupiter’s orbit.

ICaptured temporarily by Jupiter during transition.

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

-8 -6 -4 -2 0 2 4 6 8

-8

-6

-4

-2

0

2

4

6

8

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

I Belbruno/B. Marsden [1997]

I Lo/Ross [1997] :

• Comet in rotating frame follows invariant manifolds.

I Jupiter comets make resonance transition near L1 and L2.

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

x (AU, Sun-Jupiter rotating frame)

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

L4

L5

L2L1L3J

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

-10

-8

-6

-4

-2

0

2

4

6

8

10

S J


y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

Oterma’s orbit

END

START

S

� Planar Circular Restricted 3-Body Problem

I PCR3BP is a good starting model:

• Comets mostly heliocentric, buttheir perturbation dominated by Jupiter’s gravitation.• Their motion nearly in Jupiter’s orbital plane.• Jupiter’s small eccentricity plays little role during transition.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

x (nondimensional units, rotating frame)

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

� Planar Circular Restricted 3-Body Problem

I 2 main bodies: Sun and Jupiter.

• Total mass normalized to 1: mJ = µ, mS = 1− µ.• Rotate about center of mass, angular velocity normalized to 1.

I Choose a rotating coordinate system with (0, 0) at center of mass,S and J fixed at (−µ, 0) and (1− µ, 0).

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

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

� Equilibrium Points (PCR3BP)

I Comet’s equations of motion are

x− 2y = −∂U∂x

y + 2x = −∂U∂y

U = −x2 + y2

2− 1− µ

rs− µ

rj

I Five equilibrium points:

• 3 unstable equilbrium points on S-J line, L1, L2, L3.• 2 equilateral equilibrium points, L4, L5.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

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

� Hill’s Region (PCR3BP)

I Energy integral: E(x, y, x, y) = (x2 + y2)/2 + U(x, y).

I E can be used to determine (Hill’s ) region in position spacewhere comet is energetically permitted to move.

I Effective potential: U(x, y) = −x2+y2

2 − 1−µrs− µrj.

� 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 J

S JS J

Case 1 : C>C1 Case 2 : C1>C>C2

Case 4 : C3>C>C4=C5Case 3 : C2>C>C3

� The Flow near L1 and L2

I For energy value just above that of L2,Hill’s region contains a “neck” about L1 & L2.

I Comet can make transition through these equilibrium regions.

I 4 types of orbits:periodic, asymptotic, transit & nontransit.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1


y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

S JL1

Zero velocity curve bounding forbidden

region


y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

L2

(a) (b)

exteriorregion

interiorregion

Jupiterregion

forbiddenregion

L2

� Major Result (A): Heteroclinic Connection

I Found heteroclinic connection between pair of periodic orbits.

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

I Comet can follow these channels in rapid transition.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Sun

Jupiter's Orbit


y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)


y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

Jupiter

SunHomoclinic

Orbit

Homoclinic

Orbit

L1 L2L1 L2

Heteroclinic

Connection

� Major Result (B): Existence of Transitional Orbits

I Symbolic sequence used to label itinerary of each comet orbit.

IMain Theorem: For any admissible itinerary,e.g., (. . . ,X,J; S,J,X, . . . ), there exists an orbit whosewhereabouts matches this itinerary.

I Can even specify number of revolutions the comet makesaround Sun & Jupiter (plus L1 & L2).

S J

L1 L2

X

� Major Result (C): Numerical Construction of Orbits

I Developed procedure to construct orbitwith prescribed itinerary.

I Example: An orbit with itinerary (X,J; S,J,X).

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

x (rotating frame)

y(rotatingfram

e)

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Jupiter

Forbidden Region

Forbidden Region

L1 L2

Sun

x (rotating frame)

y(rotatingfram

e)

(X,J,S,J,X)Orbit

� Details: Construction of (J,X; J,S,J) Orbits

I Invariant manifold tubes separate transit from nontransit orbits.

IGreen curve (Poincare cut of L1 stable manifold).Red curve (cut of L2 unstable manifold).

I Any point inside the intersection region ∆J is a (X; J,S) orbit.

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

x (rotating frame)

y (r

otat

ing

fram

e)

JL1 L2

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

y (rotating frame)

∆J = (X;J,S)

(X;J)

(;J,S)

y (r

otat

ing

fram

e).

Forbidden Region

Forbidden Region

StableManifold

UnstableManifold

UnstableManifold Cut

StableManifold Cut

Intersection Region

� Details: Construction of (J,X; J,S,J) Orbits

I The desired orbit can be constructed by

• Choosing appropriate Poincare sections and• linking invariant manifold tubes in right order.

X

S J

U3

U2

U1U4

� Low Energy Transfer to the Moon

I Traditional transfer from Earth to Moon isby Hohmann transfer. See Apollo mission.

I 2 body Keplerian ellipse from Earth to Moon. Need 2 ∆V s.

Earth

Moon's Orbit

V2∆

∆V1

Transfer

Ellipse

� Low Energy Transfer to the Moon

I In 1991, Muses-A did not have enough propellantto reach Moon by Hohmann transfer.

I Belbruno/Miller designed a Sun-assistedEarth-to-Moon transfer with ballistic capture at Moon.

I Similar techniques used by Japanese team to save mission.

(from Belbruno and Miller [1993])

� Two Coupled 3-Body Systems

IWe provide a theoretical basis and a numerical procedurefor constructing such ballistic capture transfer.

I By considering Sun-Earth-Moon-SC 4-body systemas 2 coupled 3-body systems.

I Better seen in Sun-Earth rotating frame.

y

∆V

L2L1

Earth

Moon'sOrbit

Sun

BallisticCapture

Sun-Earth Rotating Frame

x

y

Inertial Frame

Moon'sOrbit

Earth

BallisticCapture

∆V

x


I Find position/velocity for spacecraft

• integrating forward, SC guided by Earth-Moon manifoldand get ballistically captured at Moon;• integrating backward, SC hugs Sun-Earth manifolds

with a twist and return to Earth.

x

y

L2 orbit

Sun

Lunar Capture

Portion

Earth Targeting Portion

Using "Twisting"

Moon's

Orbit

Earth

L2

Maneuver (∆V)at Patch Point


I In Sun-Earth rotating frame, we have

• Sun-Earth libration point portion.• Lunar ballistic capture portion.

x

ySun

Earth-Moon-S/C

System

Moon Earth

L2


Sun-Earth-S/C

System

� Lunar Ballistic Capture Portion

I Stable manifold tube providestemporary ballistic capture mechanism by the Moon.

I Picking a point inside stable manifold cut and integrating for-ward, spacecraft gets ballistic capture by Moon.

Earth L2 orbit

Moon

ForbiddenRegion

Tube of TransitOrbitsBallistic

CaptureOrbit

x

y

2.2 2.3 2.4 2.5 2.6 2.7 2.8-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

r

r.

Poincare Cutof Stable Manifold

Ballistic CaptureInitial Condition

Transit Orbits

Poincare Section

� Lunar Ballistic Capture Portion

I By saving (on-board) fuel for lunar ballistic capture portion,this design uses less fuelthan Earth-to-Moon Hohmann transfer.

x (rotating frame)

y (r

ota

ting f

ram

e)

Passes throughL2 Equilibrium Region

Earth L2

Ends inBallisticCapture

TrajectoryBegins

Inside Tube

Moon

� Sun-Earth Libration Point Portion

I Pick initial condition outside Poincare cut,backward integrate to produce a trajectory:

• hugs unstable manifold back to L2 region with a twist,• hugs stable manifold back towards Earth.

x

y

Earth

L2 Orbit

Sun

Earth Backward Targeting Portion

Using "Twisting" Near Tubes

L2

Moon'sOrbit

0 0.001 0.002 0.003 0.004 0.005 0.006

0.06

0.05

0.04

0.03

0.02

0.01

0

0.01

y(S

un-E

arth

rot

atin

g fr

ame)

.

y (Sun-Earth rotating frame)

Moon's

Orbit

Sun-Earth L2Orbit UnstableManifold Cut

Earth

InitialCondition

Poincare Section

� Sun-Earth Libration Point Portion

I Amount of twist depends sensitively on distance from manifold,can change dramatically with small ∆V .

IWith small ∆V ,can target back to (200 km) Earth parking orbit.

q1

UnstableManifold

P-1(q2)P-1(q

1)

q2

StableManifold

Earth

Strip SPre-Imageof Strip S

y-position x-position

y-position

Earth

L2 orbit

Sun

Earth Targeting

Using "Twisting"

L2

Poincare Section

y-velocity

� Connecting Two Portions

I Recall: Sun-Earth-Moon-SC 4-body systemas 2 coupled 3-body systems.

x

y

L2 orbit

Sun

Lunar Capture

Portion

Earth Targeting Portion

Using "Twisting"

Moon's

Orbit

Earth

L2



I Vary phase of Moon until Earth-Moon L2 manifold cutintersects Sun-Earth L2 manifold cut.

0 0.001 0.002 0.003 0.004 0.005 0.006

0.06

0.05

0.04

0.03

0.02

0.01

0

0.01

y(S

un-E

arth

rot

atin

g fr

ame)

.


Sun-Earth L2Orbit UnstableManifold Cut

Poincare Sectionin Sun-Earth Rotating Frame Earth-Moon L2

Orbit StableManifold Cutwith Moon at

Different Phases

Earth L2 orbit

Moon

x

y

StableManifoldTube

Stable Manifold Tubein Earth-Moon Rotating Frame

A

BB A


I Pick initial condition in region

• in interior of green curve• but in exterior of red curves.

x

yL2 orbit

Sun

Lunar Capture

Portion

Earth Backward Targeting

Portion Using "Twisting"

Moon's

Orbit

Earth

L2

0 0.001 0.002 0.003 0.004 0.005 0.006

0.06

0.05

0.04

0.03

0.02

0.01

0

0.01

y(S

un-E

arth

rot

atin

g fr

ame)

.


Earth-Moon L2Orbit StableManifold Cut

InitialCondition

Sun-Earth L2Orbit

Unstable Manifold CutInitial

Condition

Poincare Section


IWith slight modification ( a 34 m/s ∆V at patch point),this produces a solution in bicircular 4-body problem.

I Since capture at Moon is natural (zero ∆V ),amount of on-board ∆V needed is lowered (by about 20%).

x

y

Earth

Sun

Lunar CapturePortion

Earth Targeting PortionUsing "Twisting"

2L

Moon’sOrbit


� Conclusion

I Review: theory of libration point dynamics of 3-body system.

• Invariant manifold structure determinesmaterial transport (comets) in 3-body system,• It can be used in space mission design.

X

S J

L1L2

U3

U2U1

U4

U3

U2

J

Unstable

Stable

Stable

StableUnstable

Unstable

Stable

Unstable

� Conclusion

I Reveal the dynamics for “Low Energy Transfer to Moon.”

• Tubular regions, regions exterior to manifolds,and manifolds themselves all may be used.• Can pick and choose a variety of trajectories

to suit almost any purpose at hand.

x

y

L2 orbit

Sun

Lunar Capture

Portion

Earth Backward Targeting

Portion Using "Twisting"

Moon's

Orbit

Earth

L2

0 0.001 0.002 0.003 0.004 0.005 0.006

0.06

0.05

0.04

0.03

0.02

0.01

0

0.01

y(S

un-E

arth

rot

atin

g fr

ame)

.


Earth-Moon L2Orbit StableManifold Cut

InitialCondition

Sun-Earth L2Orbit

Unstable Manifold CutInitial

Condition

Poincare Section

� Ongoing Work: Extension to 3 Dimensions

I Find chains/dynamical channels for 3D periodic orbits,use them for low fuel deployment of spacecraft.

I Understand phase space geometry near L1 & L2;use it to design/control constellations of spacecraft.

0.820.84

0.860.88

0.90.92

0.940.96

0.981

1.02−0.02

0

0.02

0.04

0.06

0.08

0.1−0.010

0.01

x

z

y

� Ongoing Work: Coupling Two 3-Body Systems

I To understanddynamics governing transport between adjacent planets.

I Preliminary result on a “Petit Grand Tour” of Jupiter’s moons.

• 1 orbit around Ganymede.• 4 orbits around Europa, etc.• Less than half of Hohmann transfer.

L2

Ganymede

L1

Jupiter

Jupiter

Europa’sorbit

Ganymede’sorbit

Transferorbit

∆V

L2

Europa

Jupiter

� Ongoing Work: Coupling Two 3-Body Systems

I Using differential correction,can utilize this trajectory as initial guess

• to find 3-dimensional “Petit Grand Tour” trajectory,• with full solar system model.

IOptimize trajectory

• by applying optimal control (e.g., COOPT),• with continuous (low) thrust.

L2

Ganymede

L1

Jupiter

Jupiter

Europa’sorbit

Ganymede’sorbit

Transferorbit

∆V

L2

Europa

Jupiter

� Ongoing Work: 4 or More Body Problems

I Interplanetary transport and distribution of material.

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

1OrbitalEccentricity

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

� References and Other Informations

IKoon, W.S., M.W. Lo, J.E. Marsden and S.D. RossHeteroclinic connections between periodic orbits andresonance transitions in celestial mechanics,Chaos, vol. 10, (2000) pp. 427-469.

• http://www.cds.caltech.edu/ ˜ koon/• http://ojps.aip.org/chaos/

IKoon, W.S., M.W. Lo, J.E. Marsden and S.D. RossLow Energy Transfer to the Moon.

I Email: [email protected]

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