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General Examination forAndrew Abraham

August 31st, 2012

Journal Article: “Characterization of Trajectories Near the Smaller Primary in

Restricted Problem for Applications” by Davis and Howell

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Motivation: Mission Planning

• Most missions planned using 2-body dynamics– Space Station (LEO)– GPS (MEO)– Direct TV (GEO)

• Some are planned using 3-body dynamics– SOHO (Sun-Earth L1)

– GRAIL (Sun-Earth L1 to Lunar Orbit)

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Genesis Mission: 2001

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Keplerian Dynamics

Semi-Major Axis, a

Apse Line

r

Periapsis

Primary

Secondary

θ

Apoapsis

rprp

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• At periapsis velocity vector is orthogonal to radius vector.

• Two constants of motion:– Angular Momentum, h– Total Energy, E

Keplerian Dynamics:Need to Know

r

v

periapsis

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Poincare Maps

X

P1(x) S

• Traditional- X maps to P1(x), P2(x), P3(x),…

• Non-Traditional- X maps only to P1(x)- Map many X in set U

P2(x)

• Poincare Surface of Selection- 2-D plane S- Used to reduce dimensions

X in R3 P(x) on S in R2

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Coordinate System: 3-Body

Non-Inertial!

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Definitions / Assumptions

Assume Define

• m1 ≥ m2 >> m3 • m1 and m2 orbit their Barycenter in

perfectly circular orbits1

1 2

mm m

1 23

1,2

G m mG

r

12T tu

2 1 radGT tu

1,2 1r du

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Energy

2 22 23 3

1 12 2

T m v m x y y x z

1 3 2 3

1,3 2,3

Gmm Gm mVr r

2 2 2

3 1,3 2,3

1 12

E x y y x zm r r

2 2 21,3r x y z 2 2 2

2,3 1r x y z

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Equations of MotionL T V

Euler-Lagrange Eq: 0L d Lx dt x

2

2

Vx y xxVy x yy

Vzz

Equations of Motion:

Lagrangian:

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Simplified EOM

2 212

U V x y Pseudopotential:

2

2

Ux yxUy xy

Uzz

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Lagrange Points

0x x y y Look for stationary points from EOM:

5 such points exist: L1-5

4,51 3,2 4

L

Two “triangular” points exist:

Three “co-linear” points exist but are analytically intractable. Numeric approximation, given μ, is the only way to deal with them:

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2

3

0.8369, 0

1.1557, 0

1.0050, 0

L

L

L

Earth-Moon System

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Lagrange Point Geometry

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Jacobi Energy

2 2 2

1,3 2,3

2 12 2J x y vr r

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Forbidden Regions

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Summary of Article

Items Covered:

• Periapsis Poincare Map• Affect of 3-Body Dynamics on Periapsis• Short Term vs. Long Term Dynamics• yp vs. xp Maps• Titan Mission Example

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Periapsis Poincare MapOriginal State:

Three Conditions:

1. The Poincare Surface of Selection shall be the xy-plane.

2. All trajectories shall have exactly the same Jacobi Energy, J.

3. The initial state shall be the periapsis point of an orbit about P2.

, , , , , ToX x y z x y z

, , , ToX x y x y

2 2 2x y v

Gives direction to the velocity vector

, ToX x y

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Periapsis Poincare Map

State After 1 Revolution State After 6 Revolutions

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Hill Region

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P1 Perturbations

Decrease E, rp Increase e

Increase e, Rotate Apse Line

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yp vs. xp Maps

33 Revolutions, 0.07 0.25oHill p Hillr r r

0or

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Periodicity

CR3BP Coordinates

Inertial Coordinates

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Titan Mission Example

Enter Titan’s Hill Region via L2

Remain Captured After Delta-V Burn

6 Rev. Poincare Map 33 Rev. yp vs. xp Map

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Titan Mission

7.2 m/s Burn at Green Dot: Closes ZVC

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Extension by Abraham

Items Covered:

• List of Possible Extensions• RGB Poincare Map• Programming• Results• Future Work

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Possible Extensions: Abraham

• Numeric Sensitivity– Error Tolerance of Integrator• Affect on: runtime, accuracy

– ODE45 vs. ODE113• Runge-Kutta vs. Adams-Bashforth-Moulton

• Sensitivity to Initial Conditions– w.r.t. to position or Jacobi Energy– Are gradients related? Gravity or psuedopotential

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Possible Extensions: Abraham

• Propagation Time– Sensitivity?– How long should you propagate?– Dependence on J or μ?– Quantify. (Ex: 99% chance Trajectory X won’t impact nor escape moon)

• Spacecraft Dispersions– How vulnerable are the trajectories?– Can the degree of vulnerability be

quantified/mapped?

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RGB Poincare Map

• Working with periapsis is limited– Based on a 2-body definition– Velocity direction may not be orthogonal

• What can replace periapsis?• Average velocity vector

• Can represent any point along trajectory

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How to Assign Colors?

3 Stopping Conditions:1. STOP 1 = Impact Moon = RED2. STOP 2 = Escaped Moon = Blue3. End of Time = Bounded near Moon = Green

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Record Stopping ConditionsLet’s say 36 trajectories are computed per point and the results are:

12 Blue4 Red20 Green

[R, G, B] = [4,20,12]/36

[R, G, B] = [0.111, 0.556, 0.333] on a 0 1 scale

Resulting Color

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First Test

Time = 1tu, Distance Between Pt. = 0.05 du, Angular Separation = 90o, Run-Time = 2min.

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Second Test

Time = 10tu, Distance Between Pt. = 0.05 du, Angular Separation = 90o, Run-Time = 15min.

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Third Test

Time = 10tu, Distance Between Pt. = 0.001 du, Angular Separation = 90o, Run-Time = 10 hours

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Final Test

Time = 10tu, Distance Between Pt. = 0.001 du, Angular Separation = 10o, Run-Time = 30 hours w/ Parallel Computing Using 4-Core Processor

32,000 Pixels,1 Million Trajectories

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Structure

• Flower Petal Pattern– Repetitive– Green where leaves overlap

• Orange = Twice as likely to impact moon as to remain bounded (Orange = 2 red + 1 green)

• Lower left and upper right are same color• Upper left and lower right are same color

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Future Work

• Re-run code w/ better resolution– 1o or even 0.1o angular separation – 10 or 100 million trajectories

• Distributed Computer Cluster– Parallel Computing

• Re-write in C++• Fill in all pixels within Hill Region• Create GUI for manipulation of output

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Future Work

• Restrict the range of velocity vector

• Determine the Maps for various values of J & μ• Make video of map as J evolves from JL1 JL4,5

– Look for bifurcations and chaos – Speaks to spacecraft dispersions

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Questions?

Thank You!