1. ASEN 5050 SPACEFLIGHT DYNAMICS Orbit Transfers Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 10: Orbit Transfers 1

2. Announcements Homework #4 is due Friday 9/26 at 9:00 am –You’ll have to turn in your code for this one. –Again, write this code yourself, but you can use other code to validate it. Concept Quiz #8 is active after this lecture; due before Wednesday’s lecture. 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. Today’s office hours are at 2:00. Reading: Chapter 6 (SIX, we jumped a few) Lecture 10: Orbit Transfers 2

3. Space News Sunday: MAVEN arrived at Mars! Lecture 10: Orbit Transfers 3

4. Space News Lecture 10: Orbit Transfers 4 Today: Cassini is flying by Titan for the 106 th time. 1400 km altitude, 5.6 km/s V p

5. Space News Then Tuesday: MOM arrives at Mars! MOI: Tuesday at 20:00 Mountain –It will enter occultation at 20:04 –MOI will end at 20:24 –We’ll know if it’s successful around 20:30 –Notice that I write “Tuesday” here. It’ll be Wednesday in India and that keeps throwing me off Aw, time conversions! –Not sure if there will be media coverage. Try http://www.spaceflightnow.com/ or NASA TV. http://www.spaceflightnow.com/ Lecture 10: Orbit Transfers 5

6. ASEN 5050 SPACEFLIGHT DYNAMICS Orbital Maneuvers Prof. Jeffrey S. Parker University of Colorado - Boulder Lecture 10: Orbit Transfers 6

7. 7 Orbital Maneuvers Hohmann Transfer – Walter Hohmann (1880-1945) showed minimum energy transfer between two orbits used two tangential burns.

8. Lecture 10: Orbit Transfers 8 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)

9. 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 10: Orbit Transfers 9 V is highest at perigee, thus energy-changing maneuvers are the most efficient at perigee!

10. Energy Changes Lecture 10: Orbit Transfers 10

11. Hohmann Transfer Example: LEO to GEO: LEO: altitude 185 km, radius 6563.136 km GEO: altitude 35,786 km, radius 42,164 km V LEO = 7.7932 km/sV GEO = 3.0747 km/s V p T = 10.2521 km/sV a T = 1.5958 km/s ΔV 1 = 2.4590 km/sΔV 2 = 1.4788 km/s Total ΔV = 3.9378 km/s Lecture 10: Orbit Transfers 11

12. Hohmann Transfer Lecture 10: Orbit Transfers 12 GEO Moon Radius

13. Hohmann Transfer Lecture 10: Orbit Transfers 13 GEO Moon Radius General radii transfers

14. Lecture 10: Orbit Transfers 14 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.

15. Bi-elliptic Transfer Equations you need: Lecture 10: Orbit Transfers 15 SIMPLE, because all maneuvers are tangential, co-planar.

16. Lecture 10: Orbit Transfers 16 Bi-elliptic Transfer Much longer flight times for bi-elliptic transfer, but sometimes less energy. (Algorithm 37, Example 6-2)

17. Bi-elliptic Transfer LEO – GEO via 100,000 km altitude ΔV ΔV1 = 2.903 km/s ΔV2 = 0.799 km/s ΔV3 = 0.605 km/s Total ΔV: 4.307 km/s –More than Hohmann! Lecture 10: Orbit Transfers 17

18. Bi-elliptic LEO-GEO Lecture 10: Orbit Transfers 18 Moon Radius

19. Bi-elliptic LEO-GEO Lecture 10: Orbit Transfers 19 Moon Radius Hohmann

20. Bi-elliptic Transfer LEO – 250,000 km via 2.4 million km altitude ΔV ΔV1 = 3.192 km/s ΔV2 = 0.329 km/s ΔV3 = 0.327 km/s Total ΔV: 3.849 km/s –More than Hohmann (4.058 km/s)! Lecture 10: Orbit Transfers 20

21. Bi-elliptic 185 km – 250,000 km Lecture 10: Orbit Transfers 21 Moon Radius Hohmann

22. Lecture 10: Orbit Transfers 22 Hohmann vs Bi-elliptic

23. Lecture 10: Orbit Transfers 23 One-Tangent Burns

24. Lecture 10: Orbit Transfers 24 Orbit Transfer Comparison

25. Changing Orbital Elements Δa  Hohmann Transfer Δe  Hohmann Transfer Δi  Plane Change ΔΩ  Plane Change Δω  Coplanar Transfer Δν  Phasing/Rendezvous Lecture 10: Orbit Transfers 25

26. Changing Inclination Δi  Plane Change Inclination-Only Change vs. Free Inclination Change Lecture 10: Orbit Transfers 26

27. Changing Inclination Let’s start with circular orbits Lecture 10: Orbit Transfers 27 V0V0 VfVf

28. Changing Inclination Let’s start with circular orbits Lecture 10: Orbit Transfers 28 V0V0 VfVf

29. Changing Inclination Let’s start with circular orbits Lecture 10: Orbit Transfers 29 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?

30. Changing Inclination More general inclination-only maneuvers Lecture 10: Orbit Transfers 30 Line of Nodes Where do you perform the maneuver? How do V 0 and V f compare? What about the FPA?

31. Changing Inclination More general inclination-only maneuvers Lecture 10: Orbit Transfers 31

32. Changing The Node Lecture 10: Orbit Transfers 32

33. Changing The Node Lecture 10: Orbit Transfers 33 Where is the maneuver located? Neither the max latitude nor at any normal feature of the orbit! There are somewhat long expressions for how to find u initial and u final in the book for circular orbits. Lambert’s Problem gives easier solutions.

34. Changing Argument of Perigee Lecture 10: Orbit Transfers 34

35. Changing Argument of Perigee Lecture 10: Orbit Transfers 35

36. Changing Argument of Perigee Lecture 10: Orbit Transfers 36 Which ΔV is cheaper?

37. Lecture 8: Orbital Maneuvers 37 Circular Rendezvous (coplanar) Target spacecraft; interceptor spacecraft

38. Lecture 8: Orbital Maneuvers 38 Circular Rendezvous (coplanar)

39. How do we build these? Determine your phase angle, φ Determine how long you want to spend performing the transfer –How many revolutions? Build the transfer Compute the ΔV Lecture 8: Orbital Maneuvers 39

40. How do we build these? Compute the ΔV Lecture 8: Orbital Maneuvers 40

41. Lecture 8: Orbital Maneuvers 41 Example 6-8 This should be +20°

42. Lecture 8: Orbital Maneuvers 42 Example 6-8 This should really be an absolute value (one maneuver is in-track, one is anti-velocity) Should be positive

43. Conclusions Better to use as many revolutions as possible to save fuel. Trade-off is transfer duration If you perform the transfer quickly, be sure to check your periapse altitude. Lecture 8: Orbital Maneuvers 43

44. Lecture 8: Orbital Maneuvers 44 Circular Coplanar Rendezvous (Different Orbits)

45. Lecture 8: Orbital Maneuvers 45 Circular Coplanar Rendezvous (Different Orbits) Use Hohmann Transfer The “wait time”, or time until the interceptor and target are in the correct positions: Synodic Period: π – α L

46. Lecture 8: Orbital Maneuvers 46 Example 6-9

47. Lecture 8: Orbital Maneuvers 47 Example 6-9 I think this should be pi – alpha, not alpha – pi (see Fig 6-17)

48. Announcements Homework #4 is due Friday 9/26 at 9:00 am –You’ll have to turn in your code for this one. –Again, write this code yourself, but you can use other code to validate it. Concept Quiz #8 is active after this lecture; due before Wednesday’s lecture. 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. Today’s office hours are at 2:00. Reading: Chapter 6 (SIX, we jumped a few) Lecture 10: Orbit Transfers 48