ASEN 5050 SPACEFLIGHT DYNAMICS Intro to Perturbations

1. ASEN 5050SPACEFLIGHT DYNAMICSIntro to Perturbations Prof. Jeffrey S. Parker University of Colorado – Boulder

2. Announcements • STK LAB 2 • Alan will be in ITLL 2B10 Fri 2-3 • STK Lab 2 will be due 10/17, right when the mid-term exam starts. • Homework #5 is due right now! • CAETE by Friday 10/17 • Homework #6 will be due Friday 10/17 10/24 • CAETE by Friday 10/24 10/31 • Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29) • Take-home. Open book, open notes. • Once you start the exam you have to be finished within 24 hours. • It should take 2-3 hours.

3. Concept Quiz 11

4. Concept Quiz 11

5. Concept Quiz 11

6. Concept Quiz 11

7. Space News • Anyone watch the ISS? • Anyone see the lunar eclipse? • There’s an event in 1.5 weeks that is directly related to the lunar eclipse we just had. Anyone have an idea what the event is?

8. LADEE’s Mission to the Moon • Earth phasing orbits, followed by lunar phasing orbits Credit: NASA/Goddard

9. LADEE’s Mission to the Moon • Lunar Orbit Credit: NASA/Ames / ADS

10. LADEE’s Mission to the Moon • Lunar orbit perturbations Credit: NASA/Ames / ADS

11. ASEN 5050SPACEFLIGHT DYNAMICSPerturbations Prof. Jeffrey S. Parker University of Colorado – Boulder

12. Orbital Perturbations • You’ll notice that LADEE’s orbit is not strictly conical. • So far, we’ve only considered orbital solutions to the two-body problem • Point-masses • In reality, nothing is ever in orbit about a point-mass without any other perturbations • (even in an orbit about a black hole!) • The two-body relationship is typically the dominant orbital dynamic. Everything else is a small perturbation • Realistic gravitational masses • Other gravitating bodies • Atmospheric drag • Solar radiation pressure • Spacecraft effects • Even relativity and other subtle effects.

13. Perturbation Discussion Strategy • We know the 2-body problem *really well!* • Introduce the 3-body and n-body problems • We’ll cover halo orbits and low-energy transfers later • Numerical Integration • Introduce aspherical gravity fields • J2 effect, sun-synchronous orbits • Introduce atmospheric drag • Atmospheric entries • General perturbation techniques • Further discussions on mean motion vs. osculating motion.

14. Gravitational Perturbations • Start by considering the effects of other gravitating bodies. • Recall the two-body equation of motion: which is a differential equation describing the motion of msat WRT m. • How would this change if we had multiple gravitating bodies?

15. 3-Body Problem

16. 3-Body Problem • We want to know the position vector of the satellite relative to the Earth over time

17. 3-Body Problem • We want to know the position vector of the satellite relative to the Earth over time Be cognizant of the signs – the signs are defined according to how the vectors are drawn!

18. 3-Body Problem • We want to know the position vector of the satellite relative to the Earth over time Indirect Effect Direct Effect

19. n-Body Problem • The equation of motion for the position vector of a satellite in the presence of n bodies. • … relative to Body “1” (Earth?)

20. Full 2-Body Problem • How about the perturbations that result in being in orbit about a non-spherical body? Images from Park, Werner, and Bhaskaran, “Estimating Small-Body Gravity Field from Shape Model and Navigation Data”, Journal of Guidance, Control, and Dynamics, Vol. 33, No. 1, Jan – Feb 2010.

21. Dynamical Analysis z M2 M1 x y

22. Dynamical Analysis z M3 M1 M2 x y

23. Dynamical Analysis z M3 ~ 0 M1 and M2 follow conic trajectories about their COM M1 M2 x y

24. Dynamical Analysis z M3 ~ 0 y M1 and M2 follow circular orbits about their COM M1 M2 x Synodic Frame

25. Dynamical Analysis y M3 ~ 0 M2 Planar motion M1 x Synodic Frame

26. Building Solutions to the n-Body Problem • We have more degrees of freedom than we have integrals of motion! • Conic sections are no longer solutions. • Most common method used to build solutions to the n-Body problem is to take initial conditions and integrate them forward in time. • Build a trajectory using knowledge of the equations of motion.

27. Numerical Integration • Say we have a state (pos, vel) and some equations of motion. Accelerations due to 2-body, n-body, etc.

28. Numerical Integration • We want to recover the spacecraft’s trajectory using knowledge of the derivative of its state over time. • If we were to accurately integrate the derivative function over time, using the spacecraft’s initial state as the constant of motion, then we could recover its trajectory. • Lots of ways to do this. Some are better than others!

29. Numerical Integration • Euler integration Actual Trajectory

30. Numerical Integration • How do we improve it? • Take smaller time-steps • Take smarter steps Actual Trajectory

31. Euler Integration Example

32. Euler Integration Example

33. Euler Integration Example

34. Euler Integration Example

35. Euler Integration Example

36. Euler Integration Example

37. Higher order terms • Here’s what we just tried: • What about this modification?: • That would be better! • But really hard to implement in a general sense.

38. Higher order terms • Here’s what we just tried: • How about a correction term. Here’s a second-order scheme, usually referred to as a midpoint method: Actual Trajectory

39. Midpoint Integration Example Midpoint Euler

40. Midpoint Integration Example Midpoint Euler Note: this does take 2x as many derivative function calls, but the improvement is better than just doubling Euler’s!

41. Runge-Kutta Integrators • Runge-Kutta integration • 4th order Runge-Kutta “RK4” or “The Runge-Kutta method” Weighted average correction system, related to Simpson’s Rule

42. Example RK4

43. Example RK4

44. Example RK4

45. Example RK4

46. Example RK4

47. Example RK4

48. Example RK4

49. Example RK4

50. Example RK4