LECTURE NOTES ON SPACE MECHANICS Dr. P K Mohanta Associate Professor AERONAUTICAL ENGINEERING INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal – 500043, Hyderabad 1 Contents 1 Introduction to Space Mechanics .................................................................................................... 5 1.1 Basic concepts ................................................................................................................................... 5 1.2 The solar system ................................................................................................................................ 5 1.2.1 Classify the planets of our solar system. ................................................................................... 5 1.2.2 What are the Characteristics of Small Rocky Planets ............................................................... 7 1.2.3 What are the Characteristics of Gas Giants ............................................................................... 7 1.2.4 Meteorite vs. Meteoroid ............................................................................................................ 9 1.2.5 Comets..................................................................................................................................... 10 1.3 Reference frames and coordinate systems ...................................................................................... 12 1.3.1 Introduction ............................................................................................................................. 12 1.3.2 Spherical trigonometry ............................................................................................................ 16 1.4 The celestial sphere ......................................................................................................................... 18 1.4.1 Terrestrial coordinate systems ................................................................................................. 22 1.5 The ecliptic ...................................................................................................................................... 24 1.6 Motion of vernal equinox ................................................................................................................ 24 1.7 Sidereal time.................................................................................................................................... 26 1.8 Solar Time ....................................................................................................................................... 27 1.9 Standard Time ................................................................................................................................. 27 1.10 The earth‘s atmosphere ................................................................................................................... 27 1.11 Greenwich mean sidereal time ........................................................................................................ 28 1.12 The many body problem ................................................................................................................. 28 1.12.1 KEPLER'S LAWS .................................................................................................................. 28 1.12.2 Newton's Laws of Motion. ...................................................................................................... 29 1.13 Lagrange-Jacobi identity ................................................................................................................. 31 1.14 The circular restricted three body problem ..................................................................................... 31 1.15 Libration points ............................................................................................................................... 31 1.16 Relative Motion in the N-body problem ......................................................................................... 33 2 Unit-II Two Body Problem ........................................................................................................... 34 2.1 Equations of motion ........................................................................................................................ 34 2.1.1 Newton's Laws of Motion and Universal Gravitation ............................................................. 34 2.2 General characteristics of motion for different orbits ..................................................................... 37 2.3 Relations between position and time for different orbits ................................................................ 42 2.4 Expansions in elliptic motion .......................................................................................................... 42 2.5 Orbital Elements.............................................................................................................................. 42 2.5.1 Types Of Orbits ....................................................................................................................... 44 2.6 Relation between orbital elements and position and velocity ......................................................... 46 2 2.7 Launch vehicle ascent trajectories ................................................................................................... 46 2.7.1 Launch of a Space Vehicle ...................................................................................................... 46 2.8 General aspects of satellite injection ............................................................................................... 49 2.8.1 Satellite Launch Procedure...................................................................................................... 50 2.9 Dependence of orbital parameters on in-plane injection parameters .............................................. 52 2.10 Launch vehicle performances.......................................................................................................... 53 2.11 Orbit deviations due to injection errors ........................................................................................... 53 3 Unit-III Perturbed Satellite Orbit .................................................................................................. 54 3.1 Special and general perturbations ................................................................................................... 54 3.1.1 Orbit Perturbations .................................................................................................................. 54 3.1.2 Third-Body Perturbations ........................................................................................................ 54 3.1.3 Perturbations due to Non-spherical Earth ............................................................................... 55 3.2 Cowell‘s Method ............................................................................................................................. 58 3.3 Encke‘s method ............................................................................................................................... 59 3.4 Method of variations of orbital elements ........................................................................................ 61 3.5 General perturbations approach ...................................................................................................... 61 3.6 Two-dimensional interplanetary trajectories ................................................................................... 62 3.7 Fast interplanetary trajectories ........................................................................................................ 62 3.8 Three dimensional interplanetary trajectories ................................................................................. 62 3.9 Launch of interplanetary spacecraft ................................................................................................ 62 3.10 Trajectory about the target planet.................................................................................................... 62 4 Unit-IV Ballistic Missile Trajectories ........................................................................................... 63 4.1 The boost phase ............................................................................................................................... 63 4.1.1 Terminology ............................................................................................................................ 63 4.2 The ballistic phase ........................................................................................................................... 64 4.3 Trajectory geometry ........................................................................................................................ 66 4.4 Optimal flights ................................................................................................................................ 72 4.5 Time of flight .................................................................................................................................. 77 4.6 Re-entry phase ................................................................................................................................. 80 4.7 The position of the impact point ..................................................................................................... 80 4.8 Influence coefficients .....................................................................................................................