Space Flight Mechanics a.k.a. Astrodynamics MAE 589C Prof. R. H. Tolson Fall 2005 Mondays 9:00 - 11:45 A.M. MAE 589C Space Flight Mechanics a.k.a Astrodynamics August 24, 2005 9:45 pm Table of Contents Section Page Chapter 1 - Coordinate Systems and Time Systems . .1-1 1.1 Introduction . 1-1 1.2 Coordinate Systems . 1-2 1.2.1 Spherical trigonometry . 1-3 1.2.2 Celestial coordinate systems . 1-5 1.2.3 Terrestrial coordinate systems . 1-8 1.3 Time Systems . 1-10 1.3.1 Atomic time . 1-11 1.3.2 Dynamical time . 1-11 1.3.3 Ephemeris time . 1-12 1.3.4 Julian date . 1-12 1.3.5 Sidereal time . 1-13 1.3.6 Universal time . 1-13 1.3.7 UT1, UTC and Pole Location for 1998 . 1-14 1.3.8 Greenwich and local mean sidereal time . 1-15 1.4 Physical Ephemerides . 1-16 1.5 Problems . 1-17 1.6 Astronautics Toolbox . 1-17 1.7 References . 1-18 1.8 Naval Academy Pledge response to being asked for the time . 1-18 Chapter 2 - N-Body Problem . .2-1 2.1 Introduction . 2-1 2.2 Newtonian Mechanics . 2-1 2.2.1 Laws of motion . 2-1 2.2.2 Law of universal gravitation . 2-2 2.2.3 Kinetic and potential energy . 2-3 2.2.4 Linear and angular momentum . 2-5 2.3 Equations of Motion . 2-5 2.4 Integrals of the Motion . 2-6 2.4.1 Conservation of total linear momentum . 2-6 2.4.2 Conservation of total angular momentum . 2-7 2.4.3 Conservation of energy . 2-7 2.5 Planetary Ephemerides . 2-8 2.5.1 General relativity . 2-8 2.5.2 Approximate ephemerides . 2-8 2.6 Problems . 2-9 2.7 Astronautics Toolbox . 2-9 2.8 References . 2-9 Chapter 3 - Two Body Problem . .3-1 3.1 Introduction . 3-1 3.2 Kepler’s Laws . 3-1 iii MAE 589C Space Flight Mechanics a.k.a Astrodynamics August 24, 2005 9:45 pm Table of Contents Section Page 3.3 Integrals of the Two Body Problem . 3-1 3.3.1 Angular momentum. 3-2 3.3.2 Energy. 3-2 3.3.3 In-plane orbit geometry . 3-3 3.3.4 Orbital plane orientation . 3-5 3.3.5 Motion in the orbital plane . 3-6 3.4 Orbital Elements from Initial Position and Velocity . 3-8 3.5 Solution of Kepler's and Barker's Equations . 3-9 3.6 Position and Velocity from Orbital Elements . 3-10 3.7 Expansions for Elliptic Motion . 3-11 3.8 F and G Functions . 3-12 3.9 Coordinate System Rotation . 3-14 3.10 State Propagation . 3-14 3.11 Degenerate, Circular and Nearly Parabolic Orbits . 3-14 3.12 Table of Relationships . 3-16 3.13 Problems . 3-17 3.14 Astronautics Toolbox . 3-17 3.15 References . 3-17 Chapter 4 - Three Body Problem . .4-1 4.1 Introduction . 4-1 4.2 Restricted Problem . 4-1 4.2.1 Jacobi’s integral and Tisserand's criteria . 4-2 4.2.2 Zero velocity surfaces . 4-4 4.2.3 Lagrange points . 4-5 4.2.4 Stability of Lagrange points . 4-6 4.3 Finite Mass Particular Solutions . 4-8 4.3.1 Equilateral triangle solution . 4-9 4.3.2 Straight line solution . 4-10 4.4 Problems.. 4−12 4.5 Astrodynamics Toolbox . 4-12 4.6 References . 4-12 Chapter 5 - Orbital Perturbations . ..