1. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Orbital Mechanics Energy and velocity in orbit Elliptical orbit parameters Orbital elements Coplanar orbital transfers Noncoplanar transfers Time and flight path angle as a function of orbital position Relative orbital motion (“proximity operations”) © 2003 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu

2. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Energy in Orbit Kinetic Energy Potential Energy Total Energy <--Vis-Viva Equation

3. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Implications of Vis-Viva Circular orbit (r=a) Parabolic escape orbit (a tends to infinity) Relationship between circular and parabolic orbits

4. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Some Useful Constants Gravitation constant  = GM –Earth: 398,604 km 3 /sec 2 –Moon: 4667.9 km 3 /sec 2 –Mars: 42,970 km 3 /sec 2 –Sun: 1.327x10 11 km 3 /sec 2 Planetary radii –r Earth = 6378 km –r Moon = 1738 km –r Mars = 3393 km

5. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Classical Parameters of Elliptical Orbits

6. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Basic Orbital Parameters Semi-latus rectum (or parameter) Radial distance as function of orbital position Periapse and apoapse distances Angular momentum

7. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND The Classical Orbital Elements Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993

8. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND The Hohmann Transfer v perigee v1v1 v apogee v2v2

9. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND First Maneuver Velocities Initial vehicle velocity Needed final velocity Delta-V

10. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Second Maneuver Velocities Initial vehicle velocity Needed final velocity Delta-V

11. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Limitations on Launch Inclinations

12. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Differences in Inclination

13. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Choosing the Wrong Line of Apsides

14. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Simple Plane Change v perigee v1v1 v apogee v2v2 v2v2

15. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Optimal Plane Change v perigee v1v1 v apogee v2v2 v2v2 v1v1

16. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND First Maneuver with Plane Change  i 1 Initial vehicle velocity Needed final velocity Delta-V

17. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Second Maneuver with Plane Change  i 2 Initial vehicle velocity Needed final velocity Delta-V

18. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Sample Plane Change Maneuver Optimum initial plane change = 2.20°

19. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Bielliptic Transfer

20. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Coplanar Transfer Velocity Requirements Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993

21. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Noncoplanar Bielliptic Transfers

22. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Calculating Time in Orbit

23. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Time in Orbit Period of an orbit Mean motion (average angular velocity) Time since pericenter passage åM=mean anomaly

24. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Dealing with the Eccentric Anomaly Relationship to orbit Relationship to true anomaly Calculating M from time interval: iterate until it converges

25. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Patched Conics Simple approximation to multi-body motion (e.g., traveling from Earth orbit through solar orbit into Martian orbit) Treats multibody problem as “hand-offs” between gravitating bodies --> reduces analysis to sequential two-body problems Caveat Emptor: There are a number of formal methods to perform patched conic analysis. The approach presented here is a very simple, convenient, and not altogether accurate method for performing this calculation. Results will be accurate to a few percent, which is adequate at this level of design analysis.

26. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Example: Lunar Orbit Insertion v 2 is velocity of moon around Earth Moon overtakes spacecraft with velocity of (v 2 -v apogee ) This is the velocity of the spacecraft relative to the moon while it is effectively “infinitely” far away (before lunar gravity accelerates it) = “hyperbolic excess velocity” v apogee v2v2

27. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Planetary Approach Analysis Spacecraft has v h hyperbolic excess velocity, which fixes total energy of approach orbit Vis-viva provides velocity of approach Choose transfer orbit such that approach is tangent to desired final orbit at periapse

28. Space Systems Laboratory – University of Maryland Project Diana  V Requirements for Lunar Missions

29. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Hill’s Equations (Proximity Operations) Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993

30. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Clohessy-Wiltshire (“CW”) Equations

31. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND “V-Bar” Approach Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001

32. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND “R-Bar” Approach Approach from along the radius vector (“R- bar”) Gravity gradients decelerate spacecraft approach velocity - low contamination approach Used for Mir, ISS docking approaches Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001

33. Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND References for Lecture 3 Wernher von Braun, The Mars Project University of Illinois Press, 1962 William Tyrrell Thomson, Introduction to Space Dynamics Dover Publications, 1986 Francis J. Hale, Introduction to Space Flight Prentice-Hall, 1994 William E. Wiesel, Spaceflight Dynamics MacGraw-Hill, 1997 J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993