Low Energy Transfer Trajectories

1. Low Energy Transfer Trajectories Paolo Teofilatto Scuola di Ingegneria Aerospaziale Università di Roma “La Sapienza”

2. Summary • Impulsive transfers in Keplerian field • Earth orbits transfers in non Keplerian field • Weak Stability Boundary lunar transfers • Low energy lunar constellation deployment • Eccentricity effect in interplanetary transfers

3. 1) Impulsive transfers in Keplerian Field • Lawden “primer vector” theory • Cicala-Miele optimization theory via Green’s theorem • Hazelrigg definitive contribution in the 2D case • Other important contributes: Ting, Edelbaum, Rider, Eckel, Marec, Marchal , .... T. Edelbaum: “How many impulses ?” Astronautics and Aeronautics, vol.5, 1967

4. Lawden COAXIAL RULE • If the initial or terminal orbit of a transfer is circular then all the other transfer orbits must be coaxial to the point of entrance or exit on the circular orbit. • Optimal time-open, angle-open, transfers between optimally oriented orbits: coaxial transfer orbits

5. Cicala-Miele application of the Green’s Theorem Space state: two dimensional bounded region R Cost Function:

6. Green’s Theorem R R

7. The minimum is at boundary of R

8. X circles increasing apogee I X*xp=1 F parabolas xp Transfer in Keplerian field xa=apogee distance , X=1/xa xp=perigee distance

9. X F I xp Transfer in Keplerian field n.2 -

10. X I a1 b1 XF F a2 b2 xp Transfers with X geq XF Two impulses are better than four Hohmann strategy is optimal if the constraint is imposed

11. X I XF F xp Transfers with X leq XF Three impulses are better than six Biparabolic transfer is optimal if the constraint is imposed

12. Local Analisys Hohmann vs Bielliptic : local analysis Biparabolic is better than Hohmann if Any bielliptic is better than Hohmann if INCLINATION VARIATION

13. Variation of Inclination

14. Moon assisted Earth orbital transfers: GTO

15. Lunar assisted GTO

16. Lunar assisted GTO with reduced apogee

17. Optimal lunar assisted GTO are in the unstable Earth-Moon region

18. Earth-Moon Zero Velocity Curves

19. WSB: a Low Energy Transferto the Moon(up to 20% more of the final payload mass)

20. The Sun gravity-gradient effect

22. Zero Velocity curves during WSB transfer

23. Zero Velocity curves during WSB transfer

24. Zero Velocity curves during WSB transfer

25. Zero Velocity curves during WSB transfer

26. Zero Velocity curves during WSB transfer

27. Weak StabilityBoundary Trajectories for the deployment of lunar spacecraft constellations • Take advantage of the weak stability dynamics in order to deploy a constellation of lunar spacecraft with a small • Consider a nominal WSB trajectorywith periselenium distance • Consider a cluster of small impulses (10:20 cm/s) from a certain point of the nominal trajectory. • Select those impulses such that the injected spacecraft have a periselenium distance “close” to • Since small variations in initial conditions imply large variations in the final conditions (‘instability’) we may expect rather different lunar orbit parameters with respect to the nominal ones (constellation deployment)

28. Variation Z 6 perturbed trajectories Y X

29. Final parametersOnly one of the 6 burns leads the spacecraft to a periselenium “close” to rp • Nominal parameters (at Moon): • Perturbated parameters (at Moon):

30. Saving

31. Different separation times ? Different separation times Different final parameters There are two families of trajectories having the “same” periselenium distance

32. Different separation times ? Different separation times Different final parameters There are two families of trajectories having the “same” periselenium distance

33. Keplerian Case • Given find the velocities in to reach

34. The two Keplerian ellipses A equal energy curve B Case A apogee Case B

35. Lambert Moon orbit

36. The problem in the restricted 3bp x Y y E X M y x

37. Hadjidemetriou work

38. Case

39. The Earth_Moon Jacobi “constant” Jacobi constant of the exterior Lagrangian point L2

40. Hadjidemetriou curves for

41. Argument of periselenium

42. Effect of planetary eccentricity on ballistic capture in the solar system

43. Jupiter Comets

44. Capture Condition

45. Satellite Eccentricity at capture

46. Conclusion • Global results are at disposal for optimal (low energy) orbit transfers in the Keplerian field • Lower energy transfer orbits can be obtained by a third body (e.g. Moon) gravitational help • The effect of a four body (e.g. Sun) is important in low energy lunar transfer • Planet eccentricity has a role in planetary ballistic capture