Orbital Mechanics and Astrodynamics: Techniques and Tools for Space Missions

Preface
Introduction
Contents
Acronyms and Abbreviations
1: Fundamentals of Astrodynamics
	1.1 Introduction
	1.2 Mathematical Models
		1.2.1 Use of Mathematical Models to Solve Physical Problems
		1.2.2 Coordinate Systems
	1.3 Physical Principles
		1.3.1 Kepler´s Laws
		1.3.2 Newton´s Laws
		1.3.3 Work and Energy
		1.3.4 Law of Conservation of Total Energy
		1.3.5 Angular Momentum
	1.4 Fundamental Transformations
		1.4.1 Transformations Between Coordinate Systems
		1.4.2 Orthogonal Transformations
		1.4.3 Euler Angles
		1.4.4 Relative Motion and Coriolis Acceleration
2: Keplerian Motion
	2.1 Introduction
		2.1.1 Orbital Mechanics Versus Attitude Dynamics
		2.1.2 Reducing a Complex Problem to a Simplified Problem
	2.2 Two-Body Problem
		2.2.1 Derivation of the Equation of Motion: The Mathematical Model
		2.2.2 Equation of Motion for the Two-Body System
		2.2.3 Solution of the Equation of Motion
		2.2.4 An Application: Methods of Detecting Extrasolar Planets
	2.3 Central Force Motion
		2.3.1 Another Simplifying Assumption
		2.3.2 Velocity Vector
		2.3.3 Energy Equation
		2.3.4 Vis-Viva Equation
		2.3.5 Geometric Properties of Conic Sections
		2.3.6 Orbit Classification: Conic Section Orbits
		2.3.7 Types of Orbits
		2.3.8 Flight Path Angle
	2.4 Position Versus Time in an Elliptical Orbit
		2.4.1 Kepler´s Equation
		2.4.2 Proving Kepler´s Laws from Newton´s Laws
	2.5 Astronomical Constants
	2.6 Geometric Formulas for Elliptic Orbits
3: Orbital Maneuvers
	3.1 Introduction
	3.2 Statistical Maneuvers
		3.2.1 Trajectory Correction Maneuvers
		3.2.2 Problem Statement for Designing a TCM
		3.2.3 Problem Resolution for Designing a TCM
		3.2.4 Maneuver Implementation
		3.2.5 Burn Models
	3.3 Determining Orbit Parameters
		3.3.1 Parameter Estimation
		3.3.2 Analytical Computations
		3.3.3 Graphical Presentation of Elliptical Orbit Parameters
		3.3.4 Circular Orbits
		3.3.5 Slightly Eccentric Orbits
	3.4 Orbit Transfer and Adjustment
		3.4.1 Single-Maneuver Adjustments
		3.4.2 Hohmann Transfer
		3.4.3 Bi-elliptic Transfer
		3.4.4 Examples: Hohmann Transfer
		3.4.5 General Coplanar Transfer Between Circular Orbits
		3.4.6 Transfer Between Coplanar Coaxial Elliptical Orbits
	3.5 Interplanetary Trajectories
		3.5.1 Hyperbolic Trajectories
		3.5.2 Gravity-Assist Technique of Navigation
		3.5.3 Patched Conics Trajectory Model
		3.5.4 Types and Examples of Interplanetary Missions
		3.5.5 Target Space
		3.5.6 Interplanetary Targeting and Orbit Insertion Maneuver Design Technique
	3.6 Other Spacecraft Maneuvers
		3.6.1 Orbit Insertion
		3.6.2 Plane Rotation
		3.6.3 Combined Maneuvers
	3.7 The Rocket Equation
		3.7.1 In Field-Free Space
		3.7.2 In a Gravitational Field at Launch
		3.7.3 In an Atmosphere
Untitled
4: Techniques of Astrodynamics
	4.1 Introduction
	4.2 Orbit Propagation
		4.2.1 Position and Velocity Formulas as Functions of True Anomaly for Any Value of e
		4.2.2 Deriving and Solving Barker´s Equation, e = 1
		4.2.3 Orbit Propagation for Elliptic Orbits: Solving Kepler´s Equation, 0 < e < 1
		4.2.4 Hyperbolic Form of Kepler´s Equation, e > 1
		4.2.5 Orbit Propagation for All Conic Section Orbits with e > 0: Battin´s Universal Formulas
	4.3 Keplerian Orbit Elements
		4.3.1 Definitions
		4.3.2 Transformations Between Inertial and Satellite Orbit Reference Frames
		4.3.3 Conversion From Inertial Position and Velocity Vectors to Keplerian Orbit Elements
		4.3.4 Conversion from Keplerian Elements to Inertial Position and Velocity Vectors in Cartesian Coordinates
		4.3.5 Alternative Orbit Element Sets
	4.4 Lambert´s Problem
		4.4.1 Problem Statement
		4.4.2 Mission Design Application
		4.4.3 Trajectories/Flight Times Between Two Specified Points
		4.4.4 Mission Design Application (Continued)
		4.4.5 Parametric Solution Tool and Technique
		4.4.6 A Fundamental Problem in Astrodynamics
	4.5 Celestial Mechanics
		4.5.1 Introduction
		4.5.2 Legendre Polynomials
		4.5.3 Gravitational Potential for a Distributed Mass
		4.5.4 The n-Body Problem
		4.5.5 Disturbed Relative Two-Body Motion
		4.5.6 Sphere of Influence
	4.6 Observational Basics of Orbit Determination
		4.6.1 Station Locations
		4.6.2 Station Coordinates
	4.7 Time Measures and Their Relationships
		4.7.1 Introduction
		4.7.2 Universal Time
		4.7.3 Atomic Time
		4.7.4 Dynamical Time
		4.7.5 Sidereal Time
		4.7.6 Julian Days
		4.7.7 What Time Is It in Space?
		4.7.8 Additional Definitions of Time
5: Non-Keplerian Motion
	5.1 Introduction
	5.2 Perturbation Techniques
		5.2.1 Perturbations
		5.2.2 Special Perturbations
		5.2.3 Osculating Ellipse
	5.3 Variation of Parameters Technique
		5.3.1 In-Plane Perturbation Components
		5.3.2 Out-of-Plane (or Lateral) Perturbation Component
		5.3.3 Summary
	5.4 Oblateness Effects
		5.4.1 Potential Function for an Oblate Body
		5.4.2 Oblateness
		5.4.3 Precession of the Line of Nodes
	5.5 An Alternate Form of the Perturbation Equations
		5.5.1 Radial, Transverse, and Out-of-Plane (RTW) Coordinate System
		5.5.2 Perturbation Equations of Celestial Mechanics
	5.6 Primary Perturbations for Earth-Orbiting Spacecraft
	5.7 Satellite Orbit Paradox
		5.7.1 Introduction
		5.7.2 Keplerian Orbit
		5.7.3 Orbit Paradox
		5.7.4 Three Applications
	5.8 ``Zero G´´
6: Spacecraft Rendezvous
	6.1 Introduction
	6.2 Phasing for Rendezvous
		6.2.1 Alternative Transfer Orbits
	6.3 Example: Apollo 11 Ascent from the Moon
	6.4 Terminal Rendezvous
		6.4.1 Equations of Relative Motion for a Circular Target Orbit
		6.4.2 Hill´s Equations
		6.4.3 Solutions for the Hill-Clohessy-Wiltshire Equations
		6.4.4 Example: Standoff Position to Avoid Collision with the Target Vehicle
		6.4.5 Spacecraft Intercept or Rendezvous with a Target Vehicle
		6.4.6 Summary of a Terminal Rendezvous Maneuver Sequence
	6.5 Examples of Spacecraft Rendezvous
		6.5.1 Space Shuttle Discovery´s Rendezvous with the International Space Station
		6.5.2 Mars Sample Return Mission
	6.6 General Results for Terminal Spacecraft Rendezvous
		6.6.1 Particular Solutions (f  0)
		6.6.2 Target Orbits with Non-zero Eccentricity
		6.6.3 Highly Accurate Terminal Rendezvous
		6.6.4 General Algorithm
7: Navigation and Mission Design Techniques and Tools
	7.1 Introduction
	7.2 Online Ephemeris Websites: ssd and cneos
	7.3 Maneuver Design Tool
		7.3.1 Flight Plane Velocity Space (FPVS)
		7.3.2 Maneuver Design Examples
		7.3.3 Maneuver Considerations
		7.3.4 Algorithm for Computing Gradients in FPVS
	7.4 Free-Return Circumlunar Trajectory Analysis Techniques
		7.4.1 Introduction
		7.4.2 Apollo Program
		7.4.3 Free-Return Circumlunar Trajectory Analysis Method 1
			7.4.3.1 Objective and Modeling Assumptions
			7.4.3.2 Trajectory Design Tool: Michielsen Chart
			7.4.3.3 Free-Return Circumlunar Trajectories and Applications to the Apollo Missions
			7.4.3.4 Direct Versus Retrograde Trajectories
		7.4.4 Free-Return Circumlunar Trajectory Analysis Method 2
			7.4.4.1 Introduction
			7.4.4.2 Model Assumptions (Simplifications) for Free-Return Circumlunar Trajectory Parametric Plots
			7.4.4.3 Sphere of Influence
			7.4.4.4 Moon´s Orbit
			7.4.4.5 Fixed Conditions
			7.4.4.6 Additional Assumptions Used Selectively
			7.4.4.7 Free-Return Circumlunar Trajectory Selection Procedure
			7.4.4.8 Trajectory Design Tool: Penzo Parametric Plots for Circumlunar Free-Return Trajectory Analyses
			7.4.4.9 Category (1): Pericynthion Distance
			7.4.4.10 Category (2): Time of Flight
			7.4.4.11 Category (3): Fixed Pericynthion Altitude
			7.4.4.12 Category (4): Azimuth and Inclination
			7.4.4.13 Category (5): Maneuver Angle and Touchdown Latitude
			7.4.4.14 Category (6): Touchdown Longitude
			7.4.4.15 Category (7): Some Additional Parameters
			7.4.4.16 Application of the P2 Plots to the Apollo Missions
	7.5 Welcome Home
		7.5.1 Apollo 11 Comes Home
		7.5.2 Apollo 13 Returns Home
8: Changing from Mission Design to Flight Operations
	8.1 Introduction
	8.2 Determining Precision Maneuver Parameters
		8.2.1 Galileo Spacecraft
		8.2.2 Software Overview
		8.2.3 Information Flow
		8.2.4 Application to the First Trajectory Correction Maneuver
		8.2.5 Application to Other Interplanetary Maneuvers
	8.3 Use of Numerical Integration Algorithms
		8.3.1 Use of Numerical Integration Algorithms in Flight Operations
		8.3.2 Procedure for Using a MATLAB ODE Solver
	8.4 Changes in Operational Procedures
	8.5 Avoiding a Collision with Another Spacecraft or a Celestial Object
	8.6 End-of-Mission Navigation Activities
	8.7 Conclusions
		8.7.1 Conclusions About Determining Precision Maneuver Parameters
		8.7.2 Conclusions About the Use of Numerical Integration Algorithms
		8.7.3 Conclusions About Changes in Operational Procedures
		8.7.4 Conclusions About Avoiding a Collision with Another Spacecraft or a Celestial Object
		8.7.5 Conclusions About End-of-Mission Activities of the Navigation Team
9: Further Study
	9.1 Introduction
	9.2 Topics for Continuing Study in Orbital Mechanics, Astrodynamics and Related Fields
		9.2.1 Mission Analysis and Design
		9.2.2 Launch
		9.2.3 Entry, Descent, and Landing
		9.2.4 Aerogravity Assist
		9.2.5 Determining Gravity-Assist Trajectories
		9.2.6 Orbit Determination
		9.2.7 Optical Navigation
		9.2.8 Autonomous Navigation
		9.2.9 Spacecraft Attitude Dynamics
		9.2.10 Spacecraft Attitude Determination and Control
		9.2.11 Constellations of Spacecraft
		9.2.12 Formation Flying
		9.2.13 Circular, Restricted Three-Body Problem
		9.2.14 Restricted Three-Body Problem
		9.2.15 Lagrange Points and the Interplanetary Superhighway
		9.2.16 A Four-Body Trajectory Design
		9.2.17 Solar Sailing
		9.2.18 Cyclers
		9.2.19 Spacecraft Propulsion
		9.2.20 Advanced Spacecraft Propulsion
Appendix A. Brief Review of Vector Analysis
	.0 A.1 Introduction
	.0 A.2 Vectors and Scalars
	.0 A.3 Dot and Cross Product of Vectors
	.0 A.4 Derivative of a Vector Function
	.0 A.5 Gradient
	.0 A.6 Curl
	.0 A.7 Integral of a Vector Function
Appendix B. Student Projects
	.0 B.1 Trajectory Propagation Project
		.0.0 B.1.1 MATLAB References
		.0.0 B.1.2 Project Statement
		.0.0 B.1.3 Rules for Completing This Project
		.0.0 B.1.4 Programming Guidelines
		.0.0 B.1.5 Final Report
		.0.0 B.1.6 Input
	.0 B.2 Online Ephemeris Project for an Asteroid
		.0.0 B.2.1 Background for Asteroids
		.0.0 B.2.2 Project Statement
			Part 1: Questions About the Asteroid
			Part 2: Other Questions/Problems Related to Course Material:
	.0 B.3 Online Ephemeris Projects on PHAs, NEOs, and Other Celestial Objects
		.0.0 B.3.1 Project Statement
			Part 1: Questions about PHAs, NEOs, and the Torino Impact Hazard Scale
			Part 2: Computation of Rise, Transit, and Set Times for Earth-Based Observatories
			Part 3: Questions About Trojan Asteroids
			Part 4: Use of Orbit Diagrams in the cneos Website
Appendix C. Additional Penzo Parametric Plots
	.0 C.1 Objectives
	.0 C.2 Injection Velocities and the PME Angle
	.0 C.3 Moon-Phase Parameters: v, EMP Angle at SoI Exit, and the B-Plane
Appendix D. Mission Design Plots for 2024-2033 Mars Opportunities
	.0 D.1 Introduction
	.0 D.2 Mission Design Curves for the Mars Opportunities in 2024-2033
		.0.0 D.2.1 Earth-Mars Trajectories for the 2024 Opportunity
		.0.0 D.2.2 Mars-Earth (Return) Trajectories for the 2024 Mars Opportunity
		.0.0 D.2.3 Mars-Earth Trajectories for the 2026 Mars Opportunity
		.0.0 D.2.4 Earth-Mars Trajectories for the 2026 Mars Opportunity
		.0.0 D.2.5 Earth-Mars Trajectories for the 2028 Mars Opportunity
		.0.0 D.2.6 Earth-Mars Trajectories for the 2031 Mars Opportunity
		.0.0 D.2.7 Earth-Mars Trajectories for the 2033 Mars Opportunity
Answers to Selected Exercises
References
Index