1. ASEN 5050 SPACEFLIGHT DYNAMICS GRAIL Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 20: GRAIL 1

2. Announcements Alan’s office hours are today at 1:00 pm. STK LAB 2, due right now Homework #6 will be due Friday 10/24 (2014 not 2013) –CAETE by Friday 10/31 Mid-term Exam will be uploaded right after class 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: GRAIL and more perturbation fun. Lecture 20: GRAIL 2

3. Final Project Get started on it! Worth 20% of your grade, equivalent to 6-7 homework assignments. Find an interesting problem and investigate it – anything related to spaceflight mechanics (maybe even loosely, but check with me). Requirements: Introduction, Background, Description of investigation, Methods, Results and Conclusions, References. You will be graded on quality of work, scope of the investigation, and quality of the presentation. The project will be built as a webpage, so take advantage of web design as much as you can and/or are interested and/or will help the presentation. Lecture 20: GRAIL 3

4. Final Project Instructions for delivery of the final project: Build your webpage with every required file inside of a directory. –Name the directory “ ” –there are a lot of duplicate last names in this class! –You can link to external sites as needed. Name your main web page “index.html” –i.e., the one that you want everyone to look at first Make every link in the website a relative link, relative to the directory structure within your named directory. –We will move this directory around, and the links have to work! Test your webpage! Change the location of the page on your computer and make sure it still works! Zip everything up into a single file and upload that to the D2L dropbox. Lecture 20: GRAIL 4

5. Space News Lecture 20: GRAIL 5 Comet Siding Spring flies by Mars on SUNDAY Nov 12 (about 4 weeks from now), Rosetta will release Philae to land here…(next slide)

6. Lecture 20: GRAIL 6 ~500 meters in diameter

7. Concept Quiz 12 Lecture 20: GRAIL 7 I’m still waiting for a few CAETE students to complete quiz 12. Then I’ll fix everyone’s grades on that quiz. Don’t worry

8. Mid-Term Logistics Please write your name on the exam Carry “infinite” precision in your math at all steps. At the very end you are welcome to round your answers, but don’t round too much! Document which equations / process you use, and then verify that your math is correct using a computer. Try to learn something I’m not JUST doing this exam to torture you. Lecture 20: GRAIL 8

9. Review Requests (first 43) CW/H x14 Groundtracks x2 Coordinate systems x5 Transformations x7 f and g series x4 Canonical units x2 Odd orbital elements Geocentric/geodetic Kepler’s Equation derivation Lambert’s Problem x3 Synodic period Plane changes / maneuvers not covered in HW x5 EVERYTHING x4 NOTHING x5 Lecture 20: GRAIL 9

10. Orbit Maneuvering Hohmann Transfers Bi-elliptic Transfers Circular Rendezvous Coplanar Rendezvous Changing orbital elements –a, e, r p, r a, P, M, AOP are all coplanar –i, RAAN are plane changes Lambert’s Problem Lecture 20: GRAIL 10

11. Changing Orbital Elements Δa  Hohmann Transfer Δe  Hohmann Transfer Δi  Plane Change ΔΩ  Plane Change Δω  Coplanar Transfer Δν  Phasing/Rendezvous Lecture 20: GRAIL 11

12. Changing Inclination Δi  Plane Change Inclination-Only Change vs. Free Inclination Change Lecture 20: GRAIL 12

13. Changing Inclination Let’s start with circular orbits Lecture 20: GRAIL 13 V0V0 VfVf

14. Changing Inclination Let’s start with circular orbits Lecture 20: GRAIL 14 V0V0 VfVf

15. Changing Inclination Let’s start with circular orbits Lecture 20: GRAIL 15 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?

16. Changing Inclination More general inclination-only maneuvers Lecture 20: GRAIL 16 Line of Nodes Where do you perform the maneuver? How do V 0 and V f compare? What about the FPA?

17. Changing Inclination More general inclination-only maneuvers Lecture 20: GRAIL 17

18. Changing The Node Lecture 20: GRAIL 18

19. Changing The Node Lecture 20: GRAIL 19 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.

20. Changing Argument of Perigee Lecture 20: GRAIL 20

21. Changing Argument of Perigee Lecture 20: GRAIL 21

22. Changing Argument of Perigee Lecture 20: GRAIL 22 Which ΔV is cheaper?

23. Maneuver Combinations General rules to consider: –Energy changes are more efficient when traveling FAST Periapse –Plane changes are more efficient when traveling SLOW Apoapse –Combinations take advantage of vector addition 3+4 = 5 not 7 Some inclination change at periapse is optimal Some energy change at apoapse is optimal (or necessary) Delta-V vector = V final vector – V initial vector –One’s initial and final orbit do not always intersect; if they don’t you have to build a transfer orbit. Lecture 20: GRAIL 23

24. Questions These are the sorts of questions I hope you can answer: –Given orbit X, when does the satellite reach radius R? –Where in orbit Y is the satellite at its maximum latitude? –Compute element XYZ given element ABC –And all of those concept quiz type questions. –There will be math and concepts. Lecture 20: GRAIL 24

25. ASEN 5050 SPACEFLIGHT DYNAMICS GRAIL Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 20: GRAIL 25

26. ASEN 5050 SPACEFLIGHT DYNAMICS Perturbations Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 20: GRAIL 26

27. Perturbation Discussion Strategy 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 Solar Radiation Pressure Introduce atmospheric drag –Atmospheric entries Other perturbations General perturbation techniques Further discussions on mean motion vs. osculating motion. Lecture 20: GRAIL 27 ✔ ✔ ✔