1. ASEN 5050 SPACEFLIGHT DYNAMICS Two-Body Motion Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 3: The Two Body Problem 1

2. Announcements I’m away again this Friday (in a wedding this time) –Setting up STK Lab 1. You have the skills to start this one! –But we don’t have any lab space at 9am. –So…we will need you to go to ITLL 1B10 or 2B10 sometime from 1- 5pm Friday (or another time) when they’re free to run the lab. –Alan will be there to help answer questions. Homework #2 is due Friday 9/12 at 9:00 am –D2L or under my door (ECNT 418). I’ll have Alan pick them up around 9:05 or 9:10 so don’t be late! Concept Quiz #4 will be available at 10:00 am, due W at 8. Reading: Chapters 1 and 2 Lecture 3: The Two Body Problem 2

3. Space News SpaceX launched another successful Falcon 9. AsiaSat 6 is now preparing to operate in GEO at 120 deg longitude. Lecture 3: The Two Body Problem 3

4. The A-Train Lecture 3: The Two Body Problem 4

5. Today Talk about why Kepler’s Equations matter and why/how they are useful. For instance: –Say we’re given a state: –Where will the satellite be in 10 minutes? Lecture 4: The Two Body Problem 5

6. Derivation of Kepler’s Equation If t is given: If is given: Another Useful Relation: M = n(t - t p ) Lecture 4: The Two Body Problem 6

7. Example Using Mean/Eccentric Anomalies For a satellite in an Earth orbit with h A =3000 km and h P =300 km, how long does it take to go from an altitude of 1000 km to one of 2000 km? Lecture 4: The Two Body Problem 7

8. Example Using Mean/Eccentric Anomalies For a satellite in an Earth orbit with h A =3000 km and h P =300 km, how long does it take to go from an altitude of 1000 km to one of 2000 km? Plug in r = (6378 + 1000) km and (6378 + 2000) km to determine the Eccentric Anomaly at those critical moments. Compute the time between them. Lecture 4: The Two Body Problem 8

9. Example Using Mean/Eccentric Anomalies For a satellite in an Earth orbit with h A =3000 km and h P =300 km, how long does it take to go from an altitude of 1000 km to one of 2000 km? Find the time of flight between E 1 =61.22  to E 2 =105.03  Lecture 4: The Two Body Problem 9 Between r A and r P

10. Example Using Mean/Eccentric Anomalies For a satellite in an Earth orbit with h A =3000 km and h P =300 km, how long does it take to go from an altitude of 1000 km to one of 2000 km? Find the time of flight between E 1 =61.22  to E 2 =105.03  Lecture 4: The Two Body Problem 10

11. Example Using Mean/Eccentric Anomalies What is the altitude of this same satellite 10 minutes past apogee? At apoapse, M 1 =180 . 10 minutes past apoapse: Solve for E 2 Lecture 4: The Two Body Problem 11

12. Example Problems 1) An Earth satellite is observed to have a perigee height of 100 km and an apogee height of 600 km. Find the period (and e). Lecture 4: The Two Body Problem 12

13. Example Problems 1) An Earth satellite is observed to have a perigee height of 100 km and an apogee height of 600 km. Find the period (and e). Lecture 4: The Two Body Problem 13

14. Example Problems 1) An Earth satellite is observed to have a perigee height of 100 km and an apogee height of 600 km. Find the period (and e). Lecture 4: The Two Body Problem 14

15. Example Problems 1) An Earth satellite is observed to have a perigee height of 100 km and an apogee height of 600 km. Find the period (and e). Lecture 4: The Two Body Problem 15

16. Example Problems 2) How many days each year is the Earth farther from the Sun than 1AU? (1 AU=149,597,870 km) Lecture 4: The Two Body Problem 16

17. Example Problems 2) How many days each year is the Earth farther from the Sun than 1AU? (1 AU=149,597,870 km) Lecture 4: The Two Body Problem 17

18. Example Problems 3) Neglecting the eccentricity of Neptune’s orbit, how many years in each Pluto orbit is Pluto closer to the Sun than Neptune? Lecture 4: The Two Body Problem 18

19. Canonical Units 1.Reduce size of numbers 2.More mathematically stable 3.Speed up algorithms 4.Allow different orgs to use standard values 5.Reduce maintenance programming Define distance unit to be one Earth radius: 1 ER = R  = 6378.137 km We want   =1, so define Thus our time unit (TU) is: TU is time for satellite to cover 1 radian in a circular orbit of radius R . Vallado uses canonical units in examples throughout the book. Lecture 4: The Two Body Problem 19

20. Canonical Units: Example 1 Given: A geosynchronous orbit Find: The semi-major axis (a) P = 24 sidereal hours = 86164/806.8 = 106.795869 TU Lecture 4: The Two Body Problem 20

21. (Bate, Mueller, White, 1971) (Vallado, 1997) Orbital Elements Lecture 4: The Two Body Problem 21

22. Orbital Elements Law of Cosines - useful for many angular relationships We’ll start with deriving the eccentricity vector Then inclination, and the other angles Lecture 4: The Two Body Problem 22

23. Orbital Elements From our two-body derivation, we have: Lecture 4: The Two Body Problem 23 (Eq 1-23 from Vallado)

24. Orbital Elements (Vallado, 1997) Lecture 4: The Two Body Problem 24

25. Orbital Elements Now, let’s define our other orbital elements. The inclination, i, refers to the tilt of the orbit plane. It is the angle between and, and varies from 0-180°. Lecture 4: The Two Body Problem 25

26. Orbital Elements The right ascension of the ascending node, , is the angle in the equatorial plane from to the ascending node. The ascending node is the point on the equator where the satellite passes from South to North (opposite for the descending node). The line of nodes connects the ascending and descending nodes. The node vector,, points towards the ascending node and is denoted: The node lies between 0° and 360°. Lecture 4: The Two Body Problem 26

27. Orbital Elements The argument of periapse, , measured from the ascending node, locates the closest point of the orbit (periapse) and is the angle between and. Lecture 4: The Two Body Problem 27

28. Orbital Elements The true anomaly,, is the angle between periapse and the satellite position; thus: ( is positive going away from periapse, negative coming towards periapse.) Lecture 4: The Two Body Problem 28

29. Special Cases Elliptical Equatorial Orbits –  is undefined, so we use true longitude of periapse,, For i = 0°, equivalent to astronomers’ longitude of periapse,, where Lecture 4: The Two Body Problem 29

30. Special Cases Circular Orbits –  is undefined, use argument of latitude, u, where: Lecture 4: The Two Body Problem 30

31. Special Cases Circular Equatorial –  and  undefined Lecture 4: The Two Body Problem 31

32. (Vallado, 1997) Two Line Element Sets Available on class web page. (Can be read by many programs including STK.), are “Kozai” means. B* is a drag parameter. Lecture 4: The Two Body Problem 32

33. Two Line Element Sets Example 116609U86017A93352.53502934.0000788900000010529-334 216609 51.619013.33400005770102.5680257.595015.5911407044786 Epoch: Dec 18, 1993 12h 50min 26.5350 sec UTC Errors can be as large as a km or more. Lecture 4: The Two Body Problem 33

34. Orbital Elements from and (and t) Algorithm 9 in the book First compute the following vectors Compute the energy: Lecture 4: The Two Body Problem 34 p. 112 - 116

35. Orbital Elements from and (and t) Test using Example 2-5 in book Also, Lecture 4: The Two Body Problem 35

36. Review of Two-Body Problem KE PE both relative to one of the bodies Lecture 4: The Two Body Problem 36

37. Review of Two-Body Problem Elliptical Orbits 0 ≤ e < 1 a = semimajor axis b = semiminor axis Flight Path Angle Lecture 4: The Two Body Problem 37

38. Review of Two-Body Problem Lecture 4: The Two Body Problem 38

39. Announcements Friday will be STK Lab 1. –Get those ITLL accounts open. –I’ll introduce this on Wednesday. –So…we will need you to go to ITLL 1B10 or 2B10 sometime from 1- 5pm Friday (or another time) when they’re free to run the lab. –Alan will be there to help answer questions. Homework #2 is due Friday 9/12 at 9:00 am –D2L or under my door (ECNT 418). I’ll have Alan pick them up around 9:05 or 9:10 so don’t be late! Concept Quiz #4 will be available at 10:00 am, due W at 8. Reading: Chapters 1 and 2 Lecture 3: The Two Body Problem 39