Foundations of Space Dynamics

Ashish Tewari is a Professor in the Department of Aerospace Engineering at IIT Kanpur. He specializes in flight mechanics and control.

• Preface xiii1 Introduction 11.1 Space Flight 11.1.1 Atmosphere as Perturbing Environment 11.1.2 Gravity as the Governing Force 41.1.3 Topics in Space Dynamics 51.2 Reference Frames and Time Scales 51.2.1 Sidereal Frame 51.2.2 Celestial Frame 81.2.3 Synodic Frame 81.2.4 Julian Date 81.3 Classification of Space Missions 10Exercises 10References 112 Dynamics 132.1 Notation and Basics 132.2 Plane Kinematics 142.3 Newton’s Laws 162.4 Particle Dynamics 172.5 The n-Body Problem 202.6 Dynamics of a Body 242.7 Gravity Field of a Body 272.7.1 Legendre Polynomials 292.7.2 Spherical Coordinates 312.7.3 Axisymmetric Body 342.7.4 Spherical Body with Radially Symmetric Mass Distribution 37Exercises 37References 403 Keplerian Motion 413.1 The Two-Body Problem 413.2 Orbital Angular Momentum 433.3 Orbital Energy Integral 453.4 Orbital Eccentricity 463.5 Orbit Equation 493.5.1 Elliptic Orbit 533.5.2 Parabolic Orbit 563.5.3 Hyperbolic Orbit 563.5.4 Rectilinear Motion 583.6 Orbital Velocity and Flight Path Angle 603.7 Perifocal Frame and Lagrange’s Coefficients 63Exercises 654 Time in Orbit 694.1 Position and Velocity in an Elliptic Orbit 704.2 Solution to Kepler’s Equation 754.2.1 Newton’s Method 764.2.2 Solution by Bessel Functions 784.3 Position and Velocity in a Hyperbolic Orbit 804.4 Position and Velocity in a Parabolic Orbit 844.5 Universal Variable for Keplerian Motion 86Exercises 88References 895 Orbital Plane 915.1 Rotation Matrix 915.2 Euler Axis and Principal Angle 945.3 Elementary Rotations and Euler Angles 975.4 Euler-Angle Representation of the Orbital Plane 1015.4.1 Celestial Reference Frame 1035.4.2 Local-Horizon Frame 1045.4.3 Classical Euler Angles 1065.5 Planet-Fixed Coordinate System 111Exercises 1146 Orbital Manoeuvres 1176.1 Single-Impulse Orbital Manoeuvres 1196.2 Multi-impulse Orbital Transfer 1236.2.1 Hohmann Transfer 1246.2.2 Rendezvous in Circular Orbit 1276.2.3 Outer Bi-elliptic Transfer 1306.3 Continuous Thrust Manoeuvres 1336.3.1 Planar Manoeuvres 1346.3.2 Constant Radial Acceleration from Circular Orbit 1356.3.3 Constant Circumferential Acceleration from Circular Orbit 1366.3.4 Constant Tangential Acceleration from Circular Orbit 139Exercises 141References 1437 Relative Motion in Orbit 1457.1 Hill-Clohessy-Wiltshire Equations 1487.2 Linear State-Space Model 1517.3 Impulsive Manoeuvres About a Circular Orbit 1537.3.1 Orbital Rendezvous 1537.4 Keplerian Relative Motion 155Exercises 1588 Lambert’s Problem 1618.1 Two-Point Orbital Transfer 1618.1.1 Transfer Triangle and Terminal Velocity Vectors 1628.2 Elliptic Transfer 1648.2.1 Locus of the Vacant Focii 1658.2.2 Minimum-Energy and Minimum-Eccentricity Transfers 1668.3 Lambert’s Theorem 1688.3.1 Time in Elliptic Transfer 1698.3.2 Time in Hyperbolic Transfer 1738.3.3 Time in Parabolic Transfer 1758.4 Solution to Lambert’s Problem 1778.4.1 Parameter of Transfer Orbit 1788.4.2 Stumpff Function Method 1798.4.3 Hypergeometric Function Method 185Exercises 188References 1909 Orbital Perturbations 1919.1 Perturbing Acceleration 1919.2 Osculating Orbit 1929.3 Variation of Parameters 1949.3.1 Lagrange Brackets 1979.4 Lagrange Planetary Equations 1999.5 Gauss Variational Model 2099.6 Variation of Vectors 2149.7 Mean Orbital Perturbation 2199.8 Orbital Perturbation Due to Oblateness 2209.8.1 Sun-Synchronous Orbits 2259.8.2 Molniya Orbits 2269.9 Effects of Atmospheric Drag 2279.9.1 Life of a Satellite in a Low Circular Orbit 2289.9.2 Effect on Orbital Angular Momentum 2299.9.3 Effect on Orbital Eccentricity and Periapsis 2319.10 Third-Body Perturbation 2359.10.1 Lunar and Solar Perturbations on an Earth Satellite 2389.10.2 Sphere of Influence and Conic Patching 2439.11 Numerical Methods for Perturbed Keplerian Motion 2469.11.1 Cowell’s Method 2469.11.2 Encke’s Method 246Exercises 250References 25410 Three-Body Problem 25510.1 Equations of Motion 25610.2 Particular Solutions by Lagrange 257Equilibrium Solutions in a Rotating Frame 257Conic Section Solutions 25910.3 Circular Restricted Three-Body Problem 26110.3.1 Equations of Motion in the Inertial Frame 26110.4 Non-dimensional Equations in the Synodic Frame 26310.5 Lagrangian Points and Stability 26710.5.1 Stability Analysis 26810.6 Orbital Energy and Jacobi’s Integral 27010.6.1 Zero-Relative-Speed Contours 27210.6.2 Tisserand’s Criterion 27510.7 Canonical Formulation 27610.8 Special Three-Body Trajectories 27810.8.1 Perturbed Orbits About a Primary 27910.8.2 Free-Return Trajectories 279Exercises 282Reference 28311 Attitude Dynamics 28511.1 Euler’s Equations of Attitude Kinetics 28611.2 Attitude Kinematics 28811.3 Rotational Kinetic Energy 29011.4 Principal Axes 29211.5 Torque-Free Rotation of Spacecraft 29411.5.1 Stability of Rotational States 29511.6 Precession and Nutation 29811.7 Semi-Rigid Spacecraft 29911.7.1 Dual-Spin Stability 30111.8 Solution to Torque-Free Euler’s Equations 30311.8.1 Axisymmetric Spacecraft 30411.8.2 Jacobian Elliptic Functions 30711.8.3 Runge-Kutta Solution 30811.9 Gravity-Gradient Stabilization 312Exercises 32112 Attitude Manoeuvres 32312.1 Impulsive Manoeuvres with Attitude Thrusters 32312.1.1 Single-Axis Rotation 32412.1.2 Rigid Axisymmetric Spin-Stabilized Spacecraft 32612.1.3 Spin-Stabilized Asymmetric Spacecraft 33012.2 Attitude Manoeuvres with Rotors 33012.2.1 Reaction Wheel 33212.2.2 Control-Moment Gyro 33312.2.3 Variable-Speed Control-Moment Gyro 334Exercises 335References 337A Numerical Solution of Ordinary Differential Equations 339A.1 Fixed-Step Runge-Kutta Algorithms 339A.2 Variable-Step Runge-Kutta Algorithms 340A.3 Runge-Kutta-Nyström Algorithms 342References 343B Jacobian Elliptic Functions 345Reference 346Index 347