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# Orbital Mechanics for Engineering Students (Aerospace Engineering) 2nd Edition

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## Book Preface

The purpose of this book, like the fi rst edition, is to provide an introduction to space mechanics for undergraduate engineering students. It is not directed towards graduate students, researchers and experienced practitioners, who may nevertheless fi nd useful review material within the book’s contents. The intended readers are those who are studying the subject for the fi rst 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 fi nd that students like the “ teach by example ” method. I always assume that the material is being seen for the fi rst time and, wherever possible, I provide solution details so as to leave little to the reader’s imagination. The numerous fi gures throughout the book are also intended to aid comprehension. All of the more labor-intensive computational procedures are implemented in MATLAB ® code.

CHANGES TO THE SECOND EDITION

Most of the content and style of the fi rst edition has been retained. Some topics have been revised, rearranged or relocated. I have corrected all of the errors that I discovered or that were reported to me by students, teachers, reviewers and other readers. Key terms are now listed at the end of each chapter. The answers in the example problems are boxed instead of underlined. The homework problems at the end of each chapter have been grouped by applicable section. There are many new example problems and homework problems. Chapter 1, which is a review of particle dynamics, begins with a new section on vectors, which are used
throughout the book. Therefore, I thought a brief review of basic vector concepts and operations was appropriate. The chapter concludes with a new section on the numerical integration of ordinary differential equations (ODEs). These Runge-Kutta and predictor-corrector methods, which I implemented in the MATLAB codes rk1_4.m , rkf45.m and heun.m , facilitate the investigation and simulation of space mechanics problems for which analytical, closed-form solutions are not available. Many of the book’s new example problems illustrate applications of this kind. Throughout the text I mostly use the ODE solvers heun.m (fi xed time step) and rkf45.m (variable time step) because they work well and the scripts (see Appendix D) are short and easy to read. In every case I checked their results against two of MATLAB’s own suite of ODE solvers, primarily ode23.m and ode45.m . These general-purpose codes are far more elegant (and lengthy) than the ones mentioned above. They may be listed by issuing the MATLAB type command.

I have added two algorithms to Chapter 2 for numerically integrating the two-body equations of motion: an algorithm for propagating a state vector as a function of true anomaly, and an algorithm for fi nding the roots of a function by the bisection method. The last one is useful for determining the Lagrange points in the restricted three-body problem.

Chapter 4 now includes the material on coordinate transformations previously found in this and other chapters. Section 4.5 includes a more general treatment of the Euler elementary rotation sequences, with emphasis on the classical (3-1-3) Euler sequence and the yaw-pitch-roll (3-2-1) sequence. Algorithms were added to calculate the right ascension and declination from the position vector and to calculate the classical Euler angles and the yaw, pitch and roll angles from the direction cosine matrix. I also moved all discussion of ground tracks into Chapter 4 and offer an algorithm for obtaining the ground track of a satellite from its orbital elements.

Chapter 6 concludes with a new section on nonimpulsive (fi nite burn time) orbital change maneuvers, including MATLAB simulations.

Chapter 7 now includes an algorithm to fi nd the position, velocity and acceleration of a spacecraft relative to an LVLH frame. Also new to this chapter is the derivation of the linearized equations of relative motion for an elliptical (not necessarily circular) reference orbit. New to Chapter 9 is a discussion of quaternions and associated algorithms for use in numericallysolving Euler’s equations of rigid body motion to obtain the evolution of spacecraft attitude. Quaternions can be used with MATLAB’s rotate command to produce simple animations of spacecraft motion. Appendices C and D have changed. The MATLAB script in Appendix C was revised. Appendix D no longer contains the listings of MATLAB codes. Instead, the algorithms are listed along with the world wide web addresses from which they may be downloaded. This edition contains over twice the number of MATLAB M-fi les as did the fi rst.