1. ASEN 5050 SPACEFLIGHT DYNAMICS Conversions, f & g, Orbit Transfers Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 9: Conversions, Orbit Transfers 1

2. Announcements Homework #3 is due right now –You must write your own code. –For this HW, please turn in your code (preferably in one text/Word/PDF document) –After this assignment, you may use Vallado’s code, but if you do you must give him credit for work done using his code. If you don’t, it’s plagiarism. Homework #4 is due Friday 9/26 at 9:00 am –You’ll also have to turn in your code for this one. No Quiz over the weekend! Enjoy your weekend. I’ll be at the career fair Monday, so I’m delaying Monday’s office hours to 2:00. Reading: Chapter 6 (SIX, we jumped a few) Lecture 9: Conversions, Orbit Transfers 2

3. Concept Quiz 7 Lecture 9: Conversions, Orbit Transfers 3 ✔ ✔ ✔ ✔ ✔

4. Concept Quiz 7 Lecture 9: Conversions, Orbit Transfers 4

5. Concept Quiz 7 Lecture 9: Conversions, Orbit Transfers 5 The class is 50-50 split on this! Talk it over with your neighbor and convince him/her why you’re right.

6. Concept Quiz 7 Lecture 9: Conversions, Orbit Transfers 6

7. Quizzes Speaking of quizzes, I had a bug in my grade book and only the first two quiz scores were shown. As of now, Quiz 1-6 should be shown. Lecture 9: Conversions, Orbit Transfers 7

8. Space News Sunday: MAVEN arrives at Mars! MOI: this Sunday at 19:37 Mountain LASP is holding a viewing party Lecture 9: Conversions, Orbit Transfers 8

9. Space News Lecture 9: Conversions, Orbit Transfers 9

10. Space News Lecture 9: Conversions, Orbit Transfers 10

11. Space News Then Wednesday: MOM arrives at Mars! MOI: Wednesday at 21:00 Mountain –It will enter occultation at 21:04 –MOI will end at 21:24 –We’ll know if it’s successful around 21:30 Lecture 9: Conversions, Orbit Transfers 11

12. Space News Lecture 9: Conversions, Orbit Transfers 12

13. Challenge #4 If you were to plot the position and velocity of a satellite over time using VNC (Velocity-Normal- Conormal) coordinates, what would you find? –Say, an elliptical orbit Lecture 9: Conversions, Orbit Transfers 13 C V

14. ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate Transformations Prof. Jeffrey S. Parker University of Colorado - Boulder Lecture 9: Conversions, Orbit Transfers 14

15. Principal Axis Rotations Lecture 9: Conversions, Orbit Transfers 15

16. Principal Axis Rotations Lecture 9: Conversions, Orbit Transfers 16 Equation 3-15

17. Express the position and velocity in the perifocal system ( goes through periapse, in the direction of, perpendicular to and, in the orbit plane.) and From Orbital Elements Lecture 9: Conversions, Orbit Transfers 17 PQW

18. and From Orbital Elements Now we simply need to rotate into the geocentric equatorial system. The order of the rotations does matter Algorithm 10 in book. Ex. 2-6 Lecture 9: Conversions, Orbit Transfers 18

19. f and g Series Lecture 9: Conversions, Orbit Transfers 19

20. f and g Series Start by crossing the position vector into the initial velocity vector: The second term is zero, and the other terms are normal to the plane: Differentiating this last equation: Lecture 9: Conversions, Orbit Transfers 20

21. f and g Series Now cross the initial position vector into the position vector: The first term is zero, and the other terms are normal to the plane: Differentiating this last equation: Lecture 9: Conversions, Orbit Transfers 21

22. f and g Series Look at the cross-product: Which can only be true if: A good test! Lecture 9: Conversions, Orbit Transfers 22

23. f and g Series Which gives: Lecture 9: Conversions, Orbit Transfers 23

24. f and g Series So, to summarize, given an initial position and velocity, we can calculate a future position and velocity given the change in the true anomaly  : Which you can test using Example 2-4 in the textbook. Lecture 9: Conversions, Orbit Transfers 24

25. f and g Series: State Transition Matrix We can re-express our f and g series representation: in terms of a state-variable relationship: Lecture 9: Conversions, Orbit Transfers 25

26. f and g Series What uses do these functions have? –Given two states, find the time of flight between them. –Given two states, find an orbit that connects them. Big fan of this application. –Using an iterative technique, such as Newton Raphson, can determine a future state given a current state and a transfer time or transfer angle. Lecture 9: Conversions, Orbit Transfers 26

27. ASEN 5050 SPACEFLIGHT DYNAMICS Orbital Maneuvers Prof. Jeffrey S. Parker University of Colorado - Boulder Lecture 9: Conversions, Orbit Transfers 27

28. Orbital Maneuvers Orbital maneuvers are used to do many things: –Change a satellite’s orbit Size Shape Orientation –Change the phase of a satellite in its orbit –Rendezvous and/or proximity operations –Avoid collisions (debris) –Change the satellite’s groundtrack –Etc. Lecture 9: Conversions, Orbit Transfers 28

29. Terminology Coplanar maneuvers: no change to the orbit plane; the maneuvers can only change a, e, . Impulsive maneuvers: instantaneous change in velocity: ΔV –Requires an infinitely powerful engine Finite maneuvers: maneuvers that require a duration of time to achieve Ballistic: the trajectory of an object under the effects of only external forces (no maneuver firings). Lecture 9: Conversions, Orbit Transfers 29

30. Lecture 9: Conversions, Orbit Transfers 30 Orbital Maneuvers Tangential Burns: in velocity/anti-velocity vector direction -Do not change velocity orientation, just magnitude -Do not change flight path angle

31. Lecture 9: Conversions, Orbit Transfers 31 Orbital Maneuvers Nontangential: plane changes, orbit rotations

32. Lecture 9: Conversions, Orbit Transfers 32 Orbital Maneuvers Hohmann Transfer – Walter Hohmann (1880-1945) showed minimum energy transfer between two orbits used two tangential burns.

33. Lecture 9: Conversions, Orbit Transfers 33 Hohmann Transfer (math, compute DV and DT)

34. Lecture 9: Conversions, Orbit Transfers 34 Hohmann Transfer Can also be done using elliptical orbits, but must start at apogee or perigee to be a minimum energy transfer. (Algorithm 36, Example 6-1)

35. Hohmann Transfer We just argued that the Hohmann Transfer is (usually) the most energy-efficient orbital transfer. Why? –Consider Elliptical—Elliptical transfer –Tangential Burns –Energy efficiency considerations Lecture 9: Conversions, Orbit Transfers 35 V is highest at perigee

36. Energy Changes Lecture 9: Conversions, Orbit Transfers 36

37. Lecture 9: Conversions, Orbit Transfers 37 Orbital Maneuvers Bi-elliptic Transfer – Uses two Hohmann transfers. Can save  v in some cases. r b must be greater than r final, but can otherwise be optimized.

38. Lecture 9: Conversions, Orbit Transfers 38 Bi-elliptic Transfer Much longer flight times for bi-elliptic transfer, but sometimes less energy. (Algorithm 37, Example 6-2)

39. Lecture 9: Conversions, Orbit Transfers 39 Hohmann vs Bi-elliptic

40. Lecture 9: Conversions, Orbit Transfers 40 One-Tangent Burns

41. Lecture 9: Conversions, Orbit Transfers 41 Orbit Transfer Comparison

42. Changing Orbital Elements Δa  Hohmann Transfer Δe  Hohmann Transfer Δi  Plane Change ΔΩ  Plane Change Δω  Coplanar Transfer Δν  Phasing/Rendezvous (later discussion) Lecture 9: Conversions, Orbit Transfers 42

43. Changing Inclination Δi  Plane Change Inclination-Only Change vs. Free Inclination Change Lecture 9: Conversions, Orbit Transfers 43

44. Changing Inclination Let’s start with circular orbits Lecture 9: Conversions, Orbit Transfers 44 V0V0 VfVf

45. Changing Inclination Let’s start with circular orbits Lecture 9: Conversions, Orbit Transfers 45 V0V0 VfVf

46. Changing Inclination Let’s start with circular orbits Lecture 9: Conversions, Orbit Transfers 46 V0V0 VfVf ΔiΔi Are these vectors the same length? What’s the ΔV? Is this more expensive in a low orbit or a high orbit?

47. Changing Inclination More general inclination-only maneuvers Lecture 9: Conversions, Orbit Transfers 47 Line of Nodes Where do you perform the maneuver? How do V 0 and V f compare? What about the FPA?

48. Changing Inclination More general inclination-only maneuvers Lecture 9: Conversions, Orbit Transfers 48

49. Changing The Node Lecture 9: Conversions, Orbit Transfers 49

50. Changing Argument of Perigee Lecture 9: Conversions, Orbit Transfers 50