ORBITAL MECHANICS FOR ENGINEERING STUDENTS

1 ORBITAL MECHANICS forEngineering StudentsTo my parents, Rondo and Geraldine, and my wife, Connie DeeOrbital MECHANICS forEngineering StudentsHoward D. CurtisEmbry-Riddle Aeronautical UniversityDaytona Beach, FloridaAMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORDPARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYOE lsevier Butterworth-HeinemannLinacre House, Jordan Hill, Oxford OX2 8DP30 Corporate Drive, Burlington, MA 01803 First published 2005 Copyright 2005, Howard D. Curtis. All rights reservedThe right of Howard D. Curtis to be identified as the author ofthis work has been asserted in accordance with the Copyright, Design andPatents Act 1988No part of this publication may be reproduced in any material form (includingphotocopying or storing in any medium by electronic means and whether ornot transiently or incidentally to some other use of this publication) withoutthe written permission of the copyright holder except in accordance with theprovisions of the Copyright, Designs and Patents Act 1988 or under the terms of alicence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road,London, England W1T 4LP.

2 Applications for the copyright holder s writtenpermission to reproduce any part of this publication should be addressed tothe publisherPermissions may be sought directly from Elsevier s Science & TechnologyRights Department in Oxford, UK: phone (+44) 1865 843830,fax: (+44) 1865 853333, e-mail: may also complete your request on-line via the Elsevier homepage( ), by selecting Customer Support and then Obtaining Permissions British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryLibrary of Congress Cataloguing in Publication DataA catalogue record for this book is available from the Library of CongressISBN 0 7506 6169 0 For information on all Elsevier Butterworth-Heinemannpublications visit our website at by Charon Tec Pvt. Ltd, Chennai, and bound in Great Britain by Biddles Ltd, King s Lynn, NorfolkContentsPrefacexiSupplements to the textxvChapter1 Dynamics of point , force and Newton s law of s law of derivatives of moving motion20 Problems29 Chapter2 The two-body of motion in an inertial of relative momentum and the orbit energy orbits(e=0) orbits(0<e<1) trajectories(e=1) trajectories(e>1)

3 Lagrange three-body constant96 Problems101 Chapter3 ORBITAL position as a function of since variables134 Problems145 Chapter4 Orbits in three right ascension declination vector and the geocentric equatorial elements and the state between geocentric equatorial andperifocal of the earth s oblateness177 Problems187 Chapter5 Preliminary orbit method of orbit determination from threeposition s coordinate equatorial coordinate horizon coordinate determination from angle and preliminary orbit s method of preliminary orbit determination236 Problems250 Chapter6 ORBITAL Hohmann transfers with a common apse line change maneuvers290 Problems304 Chapter7 Relative motion and motion in of the equations of relative motion Wiltshire rendezvous motion in close-proximity circular orbits338 Problems340 Chapter8 Interplanetary Hohmann of of patched interplanetary trajectories391 Problems398 Chapter9 Rigid-body of translational of rotational of axis s spinning , pitch and roll angles459 Problems463 Chapter10 Satellite attitude of torque-free control despin attitude stabilization530 Problems543 Chapter11 rocket vehicle of thrust staging in field-free multiplier570 Problems578 References and further reading581 AppendixAPhysical data583 AppendixBA road map585 ContentsixAppendixCNumerical integration of then-bodyequations of : solution of Kepler s equation byNewton s.

4 Solution of Kepler s equation for thehyperbola using Newton s of the Stumpff functionsS(z)andC(z) : solution of the universal Kepler sequation using Newton s of the Lagrange coefficientsfandgandtheir time : calculation of the state vector (r,v)given the initial state vector (r0,v0) and thetime lapse : calculation of the ORBITAL elements fromthe state : calculation of the state vector fromthe ORBITAL : Gibbs method of preliminary : solution of Lambert s of Julian day number at 0 hr : calculation of local sidereal : calculation of the state vectorfrom measurements of range, angular position andtheir and : Gauss s method of preliminaryorbit determination with iterative the numerical designation of a month ora planet into its : calculation of the state vector ofa planet at a given : calculation of the spacecraft trajectoryfrom planet 1 to planet 2648 AppendixEGravitational potential energy of a sphere657 Index661 This page intentionally left blank PrefaceThis textbook evolved from a formal set of notes developed over nearly ten yearsof teaching an introductory course in ORBITAL MECHANICS for aerospace engineeringstudents.

5 These undergraduate STUDENTS had no prior formal experience in the subject,but had completed courses in physics, dynamics and mathematics through differentialequations and applied linear algebra. That is the background I have presumed forreaders of this is by no means a grand, descriptive survey of the entire subject of is a foundations text, a springboard to advanced study of the subject. I focus on thephysical phenomena and analytical procedures required to understand and predict, tofirst order, the behavior of orbiting spacecraft. I have tried to make the book readablefor undergraduates, and in so doing I do not shy away from rigor where it is neededfor understanding. Spacecraft operations that take place in earth orbit are consideredas are interplanetary missions. The important topic of spacecraft control systems isomitted.

6 However, the material in this book and a course in control theory providethe basis for the study of spacecraft attitude brief perusal of the Contents shows that there are more than enough topicsto cover in a single semester or term. Chapter 1 is a review of vector kinematics inthree dimensions and of Newton s laws of motion and gravitation. It also focuses onthe issue of relative motion, crucial to the topics of rendezvous and satellite attitudedynamics. Chapter 2 presents the vector-based solution of the classical two-bodyproblem, coming up with a host of practical formulas for orbit and trajectory analy-sis. The restricted three-body problem is covered in order to introduce the notion ofLagrange points. Chapter 3 derives Kepler s equations, which relate position to timefor the different kinds of orbits.

7 The concept of universal variables is 4 is devoted to describing orbits in three dimensions and accounting for themajor effects of the earth s oblate, non-spherical shape. Chapter 5 is an introductionto preliminary orbit determination, including Gibbs and Gauss s methods and thesolution of Lambert s problem. Auxiliary topics include topocentric coordinate sys-tems, Julian day numbering and sidereal time. Chapter 6 presents the common meansof transferring from one orbit to another by impulsive delta-v maneuvers, includingHohmann transfers, phasing orbits and plane changes. Chapter 7 derives and employsthe equations of relative motion required to understand and design two-impulse ren-dezvous maneuvers. Chapter 8 explores the basics of interplanetary mission 9 presents those elements of rigid-body dynamics required to characterizethe attitude of an orbiting satellite.

8 Chapter 10 describes the methods of controlling,changing and stabilizing the attitude of spacecraft by means of thrusters, gyros andother devices. Finally, Chapter 11 is a brief introduction to the characteristics anddesign of multi-stage launch 1 through 4 form the core of a first ORBITAL MECHANICS course. The timedevoted to Chapter 1 depends on the background of the student. It might be surveyedxixiiPrefacebriefly and used thereafter simply as a reference. What follows Chapter 4 depends onthe objectives of the 5 through 8 carry on with the subject of ORBITAL MECHANICS . Chapter 6on ORBITAL maneuvers should be included in any case. Coverage of Chapters 5, 7 and8 is optional. However, if all of Chapter 8 on interplanetary missions is to form a partof the course, then the solution of Lambert s problem (Section ) must be 9 and 10 must be covered if the course objectives include an introductionto satellite dynamics.

9 In that case Chapters 5, 7 and 8 would probably not be studiedin 11 is optional if the ENGINEERING curriculum requires a separate course inpropulsion, including rocket understand the material and to solve problems requires using a lot of under-graduate mathematics. Mathematics, of course, is the language of must not forget that Sir Isaac Newton had to invent calculus so he could solveorbital MECHANICS problems precisely. Newton (1642 1727) was an English physi-cist and mathematician, whose 1687 publicationMathematical Principles of NaturalPhilosophy( thePrincipia ) is one of the most influential scientific works of all time. Itmust be noted that the German mathematician Gottfried Wilhelm von Leibniz (1646 1716) is credited with inventing infinitesimal calculus independently of Newton inthe addition to honing their math skills, STUDENTS are urged to take advantageof computers (which, incidentally, use the binary numeral system developed byLeibniz).

10 There are many commercially available mathematics software packages forpersonal computers. Wherever possible they should be used to relieve the burden ofrepetitive and tedious calculations. Computer programming skills can and should beput to good use in the study of ORBITAL MECHANICS . Elementary MATLAB programs(M-files) appear at the end of this book to illustrate how some of the procedures devel-oped in the text can be implemented in software. All of the scripts were developedusing MATLAB version and were successfully tested using version (release 13).Information about MATLAB, which is a registered trademark of The MathWorks,Inc., may be obtained from:The MathWorks, Apple Hill DriveNatick, MA, 01760-2098 USATel: 508-647-7000 Fax: 508-647-7101E-mail: text contains many detailed explanations and worked-out examples.