Practical Astrodynamics

Preface
Contents
1 The Two-Body Problem
	1.1 Position of the Problem
	1.2 The Conic Sections and Their Geometrical Properties
	1.3 The Elliptic Orbits
	1.4 The Hyperbolic and Parabolic Trajectories
	1.5 The Lambert Problem
	1.6 Transfer Times for Elliptic, Parabolic, and Hyperbolic Trajectories
	1.7 A Unified Form of Lambert’s Equations
	1.8 An Example of Solution of Lambert’s Problem Using Universal Variables
	1.9 The Classical Orbital Elements
	1.10 Orbital Elements Defined for Any Orbit
	1.11 The Lagrangian Coefficients f, g, f′, and g′ in Closed Form
	1.12 The Lagrangian Coefficients f and g in Time Series
	1.13 Canonical Units
	1.14 The n-Body Problem
	1.15 The Halo Orbits
	References
2 Orbit Determination from Observations
	2.1 Position of the Problem
	2.2 Topocentric Co-ordinate Systems
	2.3 Orbit Determination from a Single Radar Observation
	2.4 The Measurement of Time in Astronomy
	2.5 Orbital Elements from Angle and Range Measurements
	2.6 Orbital Elements from Three Measurements of Angles (Method of Gauss)
	2.7 Orbital Elements from Three Measurements of Angles (Method of Laplace)
	2.8 Improvement in Orbit Determination by Differential Correction
	2.9 Improvement in Orbit Determination by Weighted Least-Squares Estimation
	2.10 Numerical Solution of the Least-Squares Estimation Problem
	2.11 The Kalman Filter
	2.12 Numerical Methods for Kalman Filtering
	2.13 The Unscented Kalman Filter
	2.14 The Square-Root Unscented Kalman Filter
	2.15 The Minimax Filter
	2.16 A More Robust Unscented Kalman Filter
	References
3 The Central Gravitational Force and Its Perturbations
	3.1 The System of Forces Acting on an Earth Satellite
	3.2 The Perturbation Due to the Non-spherical Earth
	3.3 The Changes of Orientation of the Earth Axis
	3.4 The Change of Co-ordinates Due to Precession
	3.5 The Change of Co-ordinates Due to Nutation
	3.6 The Change of Co-ordinates Due to the Rotation of the Earth
	3.7 The Change of Co-ordinates Due to Polar Motion
	3.8 The Fundamental Reference Systems
	3.9 The Frame-Bias Matrix
	3.10 The Co-ordinate Transformation, Based on the Equinox, Between the Celestial and Terrestrial Reference Systems
	3.11 The Co-ordinate Transformation, Based on the Non-rotating Origins, Between the Celestial and Terrestrial Reference Systems
	3.12 The Co-ordinate Transformation, According to the GOCE Standards, Between the Celestial and Terrestrial Reference Systems
	3.13 The Luni-Solar Perturbation
	3.14 The Position of the Perturbing Body
	3.15 The Position of the Perturbing Body from NASA/JPL Ephemeris Files
	3.16 The Radiation Pressure Due to the Sun
	3.17 The Eclipse Factor
	3.18 The Radiation Pressure Due to the Earth
	3.19 The Atmospheric Drag
	3.20 The Lifetime of an Earth Satellite Subject to Atmospheric Drag
	3.21 The Fundamental Properties of the Earth Atmosphere
	3.22 Atmospheric Density Models
	3.23 The Angular Velocity of the Atmosphere
	3.24 The Relativistic Perturbations
	3.25 The Perturbations Due to Continuous Low-Thrust Propulsion
	References
4 Impulsive Orbital Manoeuvres
	4.1 Position of the Problem
	4.2 Engines and Propellants for High-Thrust Rockets
	4.3 Launch Windows
	4.4 Range Safety
	4.5 Ascent Trajectories
	4.6 Insertion into Orbit
	4.7 Rendezvous Manoeuvres
	4.8 Rendezvous-Compatible Orbits
	4.9 Intermediate Orbits for Rendezvous
	4.10 The Hill–Clohessy–Wiltshire Equations
	4.11 The Hill–Clohessy–Wiltshire Equations Applied to Rendezvous Manoeuvres
	4.12 Hohmann Transfer Manoeuvres
	4.13 Bi-Elliptic Transfer Manoeuvres
	4.14 Change of Orbital Plane
	4.15 Change of the Position of a Spacecraft in Its Orbit
	4.16 Change of the Apsidal Line of an Orbit
	4.17 Drag Make-up Manoeuvres for Satellites in Low-Altitude Orbits
	4.18 Manoeuvres for Geostationary Satellites
	4.19 De-orbiting Manoeuvres
	References
5 Interplanetary Trajectories
	5.1 Position of the Problem
	5.2 The Hohmann Ellipse Approximation
	5.3 The Departure and Arrival Times
	5.4 The Spheres of Influence
	5.5 The Patched-Conic Approximation
	5.6 The Departure of a Spacecraft from a Planet
	5.7 The Arrival of a Spacecraft at a Planet
	5.8 The Flight of a Spacecraft Past a Planet
	5.9 The Gravity Assist
	5.10 Orbital Elements of the Planets
	5.11 General Interplanetary Trajectories
	5.12 The Aerodynamic Assist
	5.13 Trajectories of Vehicles Propelled by Solar Radiation Pressure
	References
6 Numerical Integration of the Equations of Motion
	6.1 Position of the Problem
	6.2 Fundamental Concepts on the Runge–Kutta Methods
	6.3 Runge–Kutta Fourth-Order Methods with Local Truncation Error Control
	6.4 Runge–Kutta Methods with Order Higher Than Four
	6.5 Runge–Kutta–Nyström Methods
	6.6 Step-Size Control with Runge–Kutta–Nyström Methods
	6.7 Special Runge–Kutta Methods
	6.8 Special Runge–Kutta–Nyström Methods
	6.9 Interpolants
	6.10 Symplectic Explicit Special Nyström Methods
	6.11 Performance Comparison for &!blank;Runge–Kutta(–Nyström) Methods
	6.12 Bulirsch-Stoer Methods
	6.13 Multi-step Methods
	6.14 The Adams Method
	6.15 The Störmer-Cowell Method
	6.16 The Gauss-Jackson Method
	6.17 Calculation of the Starting Values
	6.18 Halving the Step Size
	6.19 Integration for Elliptic Orbits of High Eccentricity
	References
7 Dynamics of Rigid Bodies
	7.1 The Motion of Rigid Bodies
	7.2 The Matrix of Inertia
	7.3 Kinetic Energy of a Rigid Body
	7.4 Moment of Inertia of a Rigid Body About an Arbitrary Axis
	7.5 Principal Axes of Inertia
	7.6 Euler’s Equations
	7.7 An Axially Symmetric (I1€=€I2) Rotating Body not Subject to External Moments
	7.8 An Axially Symmetric (I1 = I2) Rotating Body not Subject to External Moments (in Terms of Euler’s Angles)
	7.9 Unsymmetrical Body Not Subject to External Moments (Geometric Solution)
	7.10 Unsymmetrical Body Not Subject to External Moments (Analytic Solution)
	7.11 Elementary Concepts on Elliptic Integrals
	7.12 Stability of the Rotation of a Rigid Body About Its Principal Axes
	7.13 General Motion of a Rigid Body
	References
8 Instruments for Aerospace Navigation
	8.1 Motion of a Symmetric Gyroscope
	8.2 Steady Precession of a Symmetric Gyroscope
	8.3 Precession and Nutation of the Polar Axis of the Earth
	8.4 Small Oscillations of Gyroscopes
	8.5 Oscillations of Gyroscopes About Gimbal Axes
	8.6 Effects Due to the Moments of Inertia of the Gimbals
	8.7 The Gyrocompass
	8.8 The Rate Gyroscope
	8.9 The Rate Integrating Gyroscope
	8.10 High-Precision Gyroscopes
	8.11 Optical Gyroscopes
	8.12 Vibrating Structure Gyroscopes
	8.13 Accelerometers
	8.14 The Stable Platform
	8.15 Inertial Navigation
	References
9 Attitude Stabilisation and Control of Earth Satellites
	9.1 Attitude of Earth Satellites
	9.2 Moments Due to Aerodynamic Forces
	9.3 Moments Due to Electromagnetic Induction
	9.4 Moments Due to Solar Radiation Pressure
	9.5 Moments Due to Gravity Gradient
	9.6 Moments Due to Micrometeorites
	9.7 Comparison of the Magnitudes of the External Moments
	9.8 Single-Spin and Dual-Spin Stabilisation of Satellites
	9.9 Nutation Dampers
	9.10 Gravity-Gradient Stabilisation of Satellites
	9.11 Stabilisation of Satellites by Means of Electromagnetic Induction
	9.12 Stabilisation of Satellites by Means of Reaction Jets
	9.13 Stabilisation of Satellites by Means of Reaction Flywheels
	9.14 Stabilisation of Satellites by Means of Control Moment Gyroscopes
	9.15 Three-Axis Controlled Satellites
	9.16 Attitude Re-orientation of a Satellite by Means of Impulse Coning
	References
10 Dynamics of Spinning Rockets
	10.1 The Motion of a Spinning Rocket
	10.2 Misalignment of the Thrust Vector in Body-Fixed Co-ordinates
	10.3 Misalignment of the Thrust Vector in Inertial Co-ordinates
	10.4 Near-Symmetric Body of Revolution Not Subject to Moments
	10.5 Rockets of Variable Mass
	10.6 Damping Effect of the Exhaust Gas in a Non-spinning Rocket of Variable Mass
	10.7 Euler’s Equations for Spinning Rockets of Variable Mass
	10.8 Angle of Attack of a Rocket
	10.9 The Motion of a Spinning Rocket with Varying Configuration and Mass
	10.10 The Yo-Yo de-Spin Mechanism
	References
11 Performance and Optimisation of Rockets
	11.1 Performance of a Single-Stage Rocket
	11.2 Multi-stage Rockets
	11.3 Optimum Staging for Multi-stage Rockets
	11.4 Optimum Trajectory to Place a Satellite into Orbit
	11.5 Optimum Consumption of Propellant
	11.6 Gravity Turn Trajectories
	11.7 Trajectories of Long-Range Ballistic Missiles
	References