Astrodynamics: Applications and Advanced Topics 978-0120756568

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ASTRODYNAMICS @ACADEMIC PRESS, New York and London

APPLICATIONS ADVANCED

AND

TOPICS

1967

ROBERT M. L. BAKER, JR. Computer Sciences Corporation and University of California Los Angeles, California

"

COPYRIGHT @ 1967, BY ACADEMIC PRESS INC.

ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS, REPRODUCTION IN WHOLE OR IN PART FOR ANY PURPOSE OF THE UNITED STATES GOVERN­ MENT IS PERMITTED. THIS WORK REPRESENTS, IN PART, RESEARCH SPON­ SORED BY THE UNITED STATES AIR FORCE UNDER AF49(638)-495 MONITORED

AT

UCLA BY THE

AIR FORCE OFFICE OF SCIENTIFIC RESEARCH. IT ALSO REPRESENTS SOME RESEARCH SPONSORED BY THE OFFICE OF THE CHIEF SCIENTIST, LOCKHEED­ CALIFORNIA COMPANY AND COMPUTER SCIENCES CORPORATION. ACADEMIC PRESS INC.

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令

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 67-14535

PRINTED IN THE UNITED STATES OF AMERICA

To my wife, Bonnie

PREFACE

吹

This volume represents a modification and extension of the advanced sections of the original edition of An Introduction to Astrodynamics, pub­ lished in 1960. Fresh material is included on the subjects of electromagnetic drag; general relativity; preliminary orbit determination and correction procedure (for example, dealing with both minimum-variance filtering and conventional least squares); extension of variation-of-parameters pre­ diction; regularization techniques; and so forth. Other sections (for example, dealing with applications to lunar and interplanetary trajectories; general perturbations; and so on) while covering standard material, offer a unique arrangement of topics which benefits the teacher in that it has been developed in the actual process of teaching from the original volume over the past seven years. In this regard some ninety-six new exercises have been added. It is the intention of the author to employ the introductory volume, "An Introduction to Astrodynamics," as a prerequisite volume to this advanced volume and it is employed in this fashion in the astronautical courses offered at the University of California at Los Angeles. Thus, the reader will find rather frequent cross reference to the "introductory volume" within the main body of the text. This volume is meant to什11 the gap between the engineering handbooks, which tend rapidly to become outdated, and the more sophisticated texts, which tend either to be over the head of most senior university students or to lack practical value. Emphasis is given to alternative procedures and the limitations of each rather than to a collection of formulas to be utilized in cookbook fashion or committed to memory. Each chapter includes a certain amount of advanced exposition. Elementary or tedious background ma­ terial is presented in the appendixes. Most of the analyses are derived from first principles and, although the emphasis in the text is upon practical ．．

VII

．．．

VIII

PREFACE

PREFACE

application rather than elaborate mathematical proofs, the text does in­ elude over fifty derivations of fundamental relationships. At the end of each chapter a list of key equations is presented and as a further aid to the reader a summary table of the astrodynamic constants more frequently employed in calculations is given on pages 441 and 442. The glossary of terms provides a self-contained key to all of the astronomical nomenclature utilized in the text and each definition includes a reference to the section where the term is treated in the text. (The glossary of symbols, found in the introductory volume, is not repeated since it applies equally well to both volumes.) The reader is not expected to have had a prior background in classical mechanics or astronomy and, for that matter, sufficient prerequisites for the book would be freshman and sophomore courses in physics or engi­ neering, mathematical courses in integral calculus, and a familiarity with the introductory volume. For the more advanced reader the rather com­ prehensive list of references should serve as a key to the literature (up to 1967) for the purpose of further research and self-study. Because an annotated table of contents of the text is available separately, little detail is required in the discussion of the text's scope here. Chapters 1,3, and 4 discuss the orbit·determination and orbit prediction. The、nu­ merical examples found here serve as a guide to the practical utilization of the analysis. It is emphasized that each and every orbit determination procedure includes a complete illustrative numerical example. These are indispensable aids to the proper understanding and implementation of the procedures. Chapter 2 involves a detailed accounting for the various per­ turbative forces that influence spacecraft motion including a very detailed examination of radiation pressure and drag perturbations. Because the text is meant to be a practical one, both for teachers and for practicing aero­ space scientists, the subject of the application of all of the proceeding ma­ terial in this and in the introductory volume is paramount in importance and the last two chapters treat applications to lunar and interplanetary orbits and trajectories. Each chapter contains exercises of various degrees of difficulty, some of which lead to extensions of the,_theory. At the University of California, Los Angeles, the advanced text has been employed in connection with one undergraduate and one graduate course. The undergraduate course in orbit determination utilizes Chapter 1 for the first half of the quarter and a class-project orbit-determination numerical study for the second half. The graduate class draws from se­ lected sections of Chapters 2 through 6. The author wishes to acknowledge the extensive services of Merrilee Vold, who organized and retyped the final manuscript and of Bonnie

ix

Baker, Joan Boyle, Linda Foltz, and Janet Clodfelter, who typed many of the rough drafts. P. R. Peabody, Kurt Forster, Geza Gedeon, Bernard Cohlan, Dave Pierce, Ted Moyer, Joseph Ball, Donald Lamar, Pedro Escobal, Paul Koskela, Edward Pitkin, J. L. Junkins, W. E. Nally, Robert R. Lockry, and Howard Dielmann provided me with technical material and constructive suggestions for many of the subsections. Professor Samuel Herrick, especially, has contributed both directly, through specific sugges­ tions and theories, and indirectly, through the years of training that he has provided the author and many of the other contributors to the text. Los Angeles, California

R.M.L.B., Jr.

1.4 FINAL PROCESSING OF OBSERVATIONAL DATA

IMPROVEMENT ORBIT DETERMINATION AND

88 where

叭

2

Wz2 ．

． ．

W今

W(

， ． WN

2

·left as an exercise ed least-squares form is ght wei this f o n atio c ifi ver e Th for the reader. by the coefficient of 11归 Next divide each equation through

..·C(t) Ap. t) A,,, ( Opd 勹 p, (� I S 丿 ｀ 如 ) Llpn 妇(�) /1p, + (�) Llpz ... + (�) Llp; +···+ (� : . 处｀』 Op, C 』 Opd 4 (�;:) Ap, C(-U:) Ap. +

卜

rn

如=

(�!勹 (?) Ap 1 +

+

． ．

+

.

:

. Llp, +· · +

．

(?) Llp; + ...+(五） Llpn, ．

妨N

can be expressed as: which are the same as Eq. (l.l04), and 3

...(d虾I饥） ... 贪

or symbolically

5 ? 5, ' 5 ( t

．．（。也fop;) ... ． ． ．

AL1x=L1z. ？义戈 ． F…:�： ；; § ， ， 芝戈＄ ．． ．．． ．二 ＿ 咖 ＿ ＿咖

above through by A气 To form a square matrix, multiply the ATA Llx= AT Liz or T Llx = (ATA)-1A Liz q. (!.113). by inversion, which is the same as E

89

In matrix notation it is also possible to define the already mentioned standard deviations of the parameters, api'This is accomplished by ex­ tracting the inverse matrix of the least-squares normal equations (Eqs. ( 1.107)]. Such a matrix is termed the "covariance" matrix and is a free by­ product of least-squares inversion. Of course, the confidence one has in the api is dependent upon the correctness of the assigned weights and the validity of the assumption that the observations are uncorrelated and only contain random errors. Because of inevitable systematic errors these叨 lead to an overly optimistic estimate of the accuracy of the parameters. It is emphasized that this formulation of the least-squares procedure points up the fact that the old observed-minus-computed residuals do not need to be "remembered" by the computer as the new residuals are simply accumulated in the A勹z matrix elements.* Also, one needs only (n2 + n)/2 matrix elements in the matrix as the other elements are redundant (that is, it is a symmetric matrix); thus least squares is eminently suitable for high-speed digital computers, and not necessarily large ones, as the com­ puter's size is governed by the number of p/s (parameters) that we wish to improve, not the number of observations. It might be useful to launch into the entirely analogous filter theory approach; but since filters are more applicable to navigation, we shall hold such a discussion in abeyance until Chapter 5. A word of caution should be added to the foregoing discussion. It is not ordinarily "easy " to invert the least-squares equations, that is, (ATA)主 This difficulty arises in spite of the fact that we usually have a very large number of residuals and hence a strongly over-determined system of equa­ tions.This problem, as already noted, is the result of the quality, not the quantity of data—quality here not in the sense of accuracy (in which sense it was used previously), but in the sense of proper correlation to the param­ eters whose improvement is being sought. As an almost trivial example, consider the fact that no matter how many times or how accurately one measured the geocentric equatorial x and y coordinates of a satellite, one could never precisely define its z coordinate. In actually computing (A1'A)-1, one often scales the problem in such a way that the longest diagonal of the matrix consists of ones. If the off­ diagonal elements were nearly zero, then the matrix inversion would be "strong." In practice such elements may have values as high as 0.95 and the inversion is therefore usually quite "weak," that is, the solution exhibits but few significant figures. As already noted, the suggestion by Anderle * A1'A elements are, of course, changed if improved partials are formed.

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