General Examination for Andrew Abraham August 31 st , 2012

1. General Examination forAndrew AbrahamAugust 31st, 2012 Journal Article: “Characterization of Trajectories Near the Smaller Primary in Restricted Problem for Applications” by Davis and Howell

2. 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)

3. Genesis Mission: 2001

4. Secondary Keplerian Dynamics r θ Periapsis Semi-Major Axis, a Apoapsis Primary Apse Line rp rp

5. Keplerian Dynamics:Need to Know • At periapsis velocity vector is orthogonal to radius vector. • Two constants of motion: • Angular Momentum, h • Total Energy, E v r periapsis

6. Poincare Maps 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 X in R3 P(x) on S in R2 P2(x) X • Poincare Surface of Selection • 2-D plane S • Used to reduce • dimensions

7. Coordinate System: 3-Body Non-Inertial!

8. Definitions / Assumptions Assume Define • m1 ≥ m2 >> m3 • m1 and m2 orbit their Barycenter in perfectly circular orbits

9. Energy

10. Equations of Motion Lagrangian: Euler-Lagrange Eq: Equations of Motion:

11. Simplified EOM Pseudopotential:

12. Lagrange Points Look for stationary points from EOM: 5 such points exist: L1-5 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: Earth-Moon System

13. Lagrange Point Geometry

14. Jacobi Energy

15. Forbidden Regions

16. 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

17. Periapsis Poincare Map Original State: Three Conditions: The Poincare Surface of Selection shall be the xy-plane. All trajectories shall have exactly the same Jacobi Energy, J. The initial state shall be the periapsis point of an orbit about P2. Gives direction to the velocity vector

18. Periapsis Poincare Map State After 1 Revolution State After 6 Revolutions

19. Hill Region

20. P1 Perturbations Increase e, Rotate Apse Line Decrease E, rp Increase e

21. yp vs. xp Maps 33 Revolutions,

22. Periodicity CR3BP Coordinates Inertial Coordinates

23. Titan Mission Example 6 Rev. Poincare Map 33 Rev. yp vs. xp Map Remain Captured After Delta-V Burn Enter Titan’s Hill Region via L2

24. Titan Mission 7.2 m/s Burn at Green Dot: Closes ZVC

25. Extension by Abraham Items Covered: • List of Possible Extensions • RGB Poincare Map • Programming • Results • Future Work

26. 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

27. 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?

28. 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

29. How to Assign Colors? 3 Stopping Conditions: • STOP 1 = Impact Moon = RED • STOP 2 = Escaped Moon = Blue • End of Time = Bounded near Moon = Green

30. Record Stopping Conditions Let’s say 36 trajectories are computed per point and the results are: 12 Blue 4 Red 20 Green [R, G, B] = [4,20,12]/36 [R, G, B] = [0.111, 0.556, 0.333] on a 0  1 scale Resulting Color

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

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

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

34. Final Test 32,000 Pixels, 1 Million Trajectories Time = 10tu, Distance Between Pt. = 0.001 du, Angular Separation = 10o, Run-Time = 30 hours w/ Parallel Computing Using 4-Core Processor

35. 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

36. 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

37. 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

38. Questions? Thank You!