Orbital Mechanics for Engineering Students (Aerospace Engineering) 3rd Edition

Book Preface

The purpose of this book is to provide an introduction to space mechanics for undergraduate engineering students. It is not directed toward graduate students, researchers, and experienced practitioners, who may nevertheless find useful review material within the book’s contents. The intended readers are those who are studying the subject for the first time and have completed courses in physics, dynamics, and mathematics through differential equations and applied linear algebra. I have tried my best to make the text readable and understandable to that audience. In pursuit of that objective, I have included a large number of example problems that are explained and solved in detail. Their purpose is not to overwhelm but to elucidate. I find that students like the “teach by example” method. I always assume that the material is being seen for the first time and, wherever possible, I provide solution details so as to leave little to the reader’s imagination. The numerous figures throughout the book are also intended to aid comprehension. All of the more labor-intensive computational procedures are accompanied by the MATLABcode.

I retained the content and style of the second edition. Although I added some new homework problems, I made few if any changes to Chapters 1–11. I corrected all the errors that I discovered or that were reported to me by students, teachers, reviewers, and other readers. Chapter 12 on perturbations is new. The addition of this chapter is accompanied by some new MATLAB scripts in

Appendix D and a new Appendix F.

The organization of the book remains the same as that of the second edition. Chapter 1 is a review of vector kinematics in three dimensions and of Newton’s laws of motion and gravitation. It also focuses on the issue of relative motion, crucial to the topics of rendezvous and satellite attitude dynamics. The new material on ordinary differential equation solvers will be useful for students who are expected to code numerical simulations in MATLAB or other programming languages. Chapter 2 presents the vector-based solution of the classical two-body problem, resulting in a host of practical formulas for the analysis of orbits and trajectories of elliptical, parabolic, and hyperbolic shape. The restricted three-body problem is covered in order to introduce the notion of Lagrange points and to present the numerical solution of a lunar trajectory problem. Chapter 3 derives Kepler’s equations, which relate position to time for the different kinds of orbits. The universal variable formulation is also presented. Chapter 4 is devoted to describing orbits in three dimensions. Coordinate transformations and the Euler elementary rotation sequences are defined. Procedures for transforming back and forth between the state vector and the classical orbital elements are addressed. The effect of the earth’s oblateness on the motion of an orbit’s ascending node and eccentricity vector is examined. Chapter 5 is an introduction to preliminary orbit determination, including Gibbs’ and Gauss’s methods and the solution of Lambert’s problem. Auxiliary topics include topocentric coordinate systems, Julian day numbering, and sidereal time. Chapter 6 presents the common means of transferring from one orbit to another by impulsive delta-v maneuvers, including Hohmann transfers, phasing orbits, and plane changes. Chapter 7 is a brief introduction to relative motion in general and to the two-impulse rendezvous problem in particular. The latter is analyzed using the Clohessy–Wiltshire equations, which are derived in this chapter. Chapter 8 is an introduction to interplanetary mission design using patched conics. Chapter 9 presents those elements of rigid-body dynamics required to characterize the attitude of a space vehicle. Euler’s equations of rotational motion are derived and applied in a number of example problems. Euler angles, yaw, pitch, and roll angles, and quaternions are presented as ways to describe the attitude of a rigid body. Chapter 10 describes the methods of controlling, changing, and stabilizing the attitude of spacecraft by means of thrusters, gyros, and other devices. Chapter 11 is

a brief introduction to the characteristics and design of multistage launch vehicles. Chapter 12 is an introduction to common orbital perturbations: drag, nonspherical gravitational field, solar radiation pressure, and lunar and solar gravity.

Chapters 1–4 form the core of a first orbital mechanics course. The time devoted to Chapter 1 depends on the background of the student. It might be surveyed briefly and used thereafter simply as a reference. What follows Chapter 4 depends on the objectives of the course.

Chapters 5–8 carry on with the subject of orbital mechanics, as does Chaper 12. Chapter 6 on orbital maneuvers should be included in any case. Coverage of Chapters 5, 7, and 8 is optional. However, if all of Chapter 8 on interplanetary missions is to form a part of the course, then the solution of Lambert’s problem (Section 5.3) must be studied beforehand.

Chapters 9 and 10 must be covered if the course objectives include an introduction to satellite dynamics. In that case Chapters 5, 7, and 8 would probably not be studied in depth. Chapter 11 is optional if the engineering curriculum requires a separate course in propulsion, including rocket dynamics.

Finally, Chapter 12 is appropriate for a course devoted exclusively to orbital mechanics with an introduction to perturbations, which is a whole topic unto itself. The important topic of spacecraft control systems is omitted. However, the material in this book and a course in control theory provide the basis for the study of spacecraft attitude control. To understand the material and to solve problems requires using a lot of undergraduate mathematics. Mathematics, of course, is the language of engineering. Students must not forget that Sir Isaac Newton had to invent calculus so he could solve orbital mechanics problems in more than just a heuristic way. Newton (1642–1727) was an English physicist and mathematician whose 1687 publication Mathematical Principles of Natural Philosophy (the Principia) is one of the most influential scientific works of all times. It must be noted that the German mathematician Gottfried Wilhelmvon Leibnitz (1646–1716) is credited with inventing infinitesimal calculus independently of Newton in the 1670s.

In addition to honing their math skills, students are urged to take advantage of computers (which, incidentally, use the binary numeral system developed by Leibnitz). There are many commercially available mathematics software packages for personal computers. Wherever possible, they should be used to relieve the burden of repetitive and tedious calculations. Computer programming skills can and should be put to good use in the study of orbital mechanics. The elementary MATLAB programs referred to in Appendix D of this book illustrate how many of the procedures developed in the text can be implemented in software. All of the scripts were developed and tested using MATLAB version 8.0 (release 2012b). Information about MATLAB, which is a registered trademark of The MathWorks, Inc., may be obtained from.