AFAL-TR-71-223 SUDAAR NO. 424 STANFORD UNIVERSITY CENTER FOR SYSTEMS RESEARCH » M > <t) t-. D H; Cr in £ tt O M O SOME HIGHER ORDER ANALYSES *• o OF EARTH AND LUNAR ORBITERS HI o by a-o o Robert R. Dasenbrock H- < ,tc a: w O-« so KJ so <*» coo Guidance and Control Laboratory t/5 ^ NJ O MAY 1971 Approved for public release; distribution unlimited. O n This work was performed in association with research sponsored by the National Aeronautics and Space Administration under Research Contract NaG 133 61 X (t /. , 0£~- and the Department of the Air Force under Research Contract F33615-70-C-1637 NOTICE When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever; and the fact that the government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data, is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way -be related thereto. .. Copies of this report should not be returned unless return is required by security considerations, contractual obligations, or notice on a specific document. AFAL-TR-71-223 Department of Aeronautics and Astronautics Stanford University Stanford, California SOME HIGHER ORDER ANALYSES OF EARTH AND LUNAR ORBITERS by Robert R. Dasenbrock Approved for public release; distribution unlimited. SUDAAR No. 424 May 1971 This work was performed in association with research sponsored by the National Aeronautics and Space Administration _ under Research Contract -#&e-*33-6i A/ (L,f-~ &*> - 0£ 0-Q07 and the Department of the Air Force under Research Contract F33615-70-C-1637 FOR E W 0 R D The work herein was under Contract F33615-70-C-1637, Project No. 5102, Task No. 510215 at the Center for Systems Research, Guidance and Control Laboratory, Stanford University, California. This report has been issued as Department of Aeronautics and Astro- nautics SUDAAR NO. 424. This program is monitored by Herb Kohlmann, Air Force Avionics Laboratory (AVNE). Publication of this report does not constitute Air Force approval of the report's findings or conclusions. It is published only for the exchange and stimulation of ideas. This report has been reviewed and is approved for publication. Lt Cornel, USAF Chief, Navigation & Guidance Division ii ABSTRACT This study is concerned with the accurate modelling of the transla- tional behavior of a drag-free satellite in an almost circular near- earth orbit. Specifically, all of the short period and long period posi- tion fluctuations in the satellite's coordinates down to one meter in amplitude are determined. Considered in the equations of motion are the general zonal and tesseral harmonic effects as well as the lunar and solar effects. The results of the study will allow the efficient evaluation of information about the gravity field from special low altitude small eccentricity orbits selected for drag-free operation. A non-canonical approach is followed. A mean orbital plane is chosen in such a way that no short period out-of-plane fluctuations greater than second order occur. This frame of reference undergoes long period changes in inclination and nodal position. The short period radial and cross- track fluctuations are computed as a function of position along the orbit. The in-track fluctuations are determined by the solution of the time equation. These fluctuations are derived from Lagrange's Planetary Equations. The long period and secular rates of the mean elements which define the slowly varying plane of reference are determined to third order in the small quantities. Therefore retained in these long period equations 3 are terms involving J as well as the coupling between J and the £t £ general zonal harmonic coefficient J and the interaction between J ,& ^ and the lunar-solar effects. The mean eccentricity is assumed to be a first order small quantity. The equations of motion are integrated analytically away from criti- cal inclination and tesseral resonance. The resonant situation of a sun- »• synchronous orbit is discussed as a special case. Also considered in this research is the long-period behavior of a lunar orbiter. Of special interest are the effects due to the inclina- tion of the earth's apparent orbit about the moon and those effects de- scribed by the laws of Cassini on the equations of motion. Both resonant 111 and non-resonant low orbits -are 'Cpnsidered. Here the lunar gravity -field dominates the earth perturbation. The near polar orbits are taken as an example of resonance. Long^-term 'Stable positions for the orbit plane are found. The higher .orbits are also discussed. An .analytical solution is presented that is valid for an initially *near circular orbit. This solu- tion giv.es the .orbital lifetime for an unstable orbit,. The Hamiltonian formalism is .used for the lunar orbiter studies. iv CONTENTS Page INTRODUCTION 1 A. Introductory Remarks and Statement of the Problem . 1 2 C. Contributions of This Research 3 II PRELIMINARY RESULTS 5 A. The Drag-Free Satellite 5 B. Disturbing Forces 5 C. Choice of Variables 7 D. The Disturbing Function 8 E. Description of the Orbit 14 F. Method of Solution 18 G. The Lagrangian Planetary Equations 19 III THE MAIN PROBLEM FOR A SATELLITE REVOLVING ABOUT A ROTATIONALLY SYMMETRIC EARTH 21 A. The Solution for T 22 B. The Solution for a 23 C. The Solution for p 23 D. The Long Period Equations 24 E. Solution to the Long Period Equations 25 F. The Time Equation 27 G. Solution to the Time Equation 30 IV. THE INCLUSION OF LONGITUDINALLY DEPENDENT FORCES 33 A. The Contribution to T '• • 33 B. The Contribution to a 34 C. The Contribution to p 35 D. The Long Period and Secular Rates . , 35 E. Solution to the Long Period Equations 38 F. Contribution to the Time Equation 43 G. Solution to the Time Equation 44 H. The Dependence upon Eccentricity 46 V. RESONANCE VI. THE LONG PERIOD BEHAVIOR OF A LUNAR ORBITER 55 A. Introduction 55 B. Choice of Reference Frame 57 C. The Delaunay Variables 59 D. The Disturbing Function 60 E. The Lower Orbits 63 1. The Elimination of Time 65 2. The Long Period Terms 68 3. The Near Polar Orbit 70 F. The Higher Orbits 73 1. The Classification of Orbits ' 73 2. Orbit Stability 79 3. The Initially Near Circular Orbit 88 a. Introductory Remarks 88 b. The General Solution 96 VII. SUMMARY AND RECOMMENDATIONS FOR FUTURE RESEARCH 103 A. Concluding Remarks 103 1. Geocentric Perturbation Analysis 103 2. Lunar Orbiter Analysis 105 B. Recommendations for Future Research ...... 105 I APPENDIX 107 REFERENCES 119 VI NOMENCLATURE Symbol Definition a semi-major axis Ai constant B constant constant earth gravity coefficient 'ijmn constant f true anomaly F disturbing function 3 see Eq. (3.28) g perigee position 2 1/2 G L(l-e ) , conjugate to g G critical value of G CRIT h angular momentum, also position of ascending node H G cos i , conjugate to h 3C Hamiltonian inclination inclination of terrestrial path inclination of lunar path with respect to equator M inclination of solar path zonal harmonic J tesseral harmonic J, constant vii K constant K. constant & mean anomaly -0jyj mean motion of moon •#g - mean motion of sun 1/2 L (|_La) , conjugate to £ £ Lagrangian ^ mean anomaly n mean orbital rate HE mean earth rate n« mean solar rate 2 P a(1-e ) , semi-latus rectum (Chapter VI), also short period portion of angular momentum h q short period part of the radial fluctuation p TM semi-major axis of moon's orbit P P perigee height, equal to a (1-e) Fg semi-major axis of earth's orbit R radius of earth R earth component of disturbing function HM lunar component of disturbing function RS solar component of disturbing function S generating function S. components of S S. earth gravity coefficient -t/m U M-KD U* f-HD Vlll a. constant i P in-track fluctuation & latitude e eccentricity of satellite's orbit e , _, critical value of eccentricity OKor IT 1 eM eccentricity of moon's orbit e eccentricity of earth's orbit o I e cos CD T) e sin CD 9 Greenwich mean sidereal time, to t £j O 0 conjugate to earth coordinate 0p |i mass parameter of earth M mass parameter of moon M p. mass parameter of sun S p 1/r a cross-track fluctuation T "angular momentum variable" cp longitude " r Cp phase angle, constant for, a particular J. \|r constant = 0° for H-m even, 90° for £-m odd T} ? nodal position of satellite , Q, nodal position of lunar orbit with respect to the M earth CD argument of pericenter u^ argument, of pericenter of moon's orbit o>o earth rotation rate Oi argument of pericenter of earth's orbit o0 ' denotes differentiation with respect to Q ; in Chapter VI denotes average with respect to Q .h denotes average with respect to g and h denotes Petty mean variable denotes Petty variable with long, period second order fluctuations involving i and h removed denotes average of Hamiltonian. with respect to earth angle 6 £i $$ denotes average of Hamiltonian with respect to g and h ACKNOWLEDGMENT The author would like to thank Professor John V. Breakwell for his many helpful comments and suggestions. The author also wishes to ex- press his gratitude to Mrs.