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University Microfilms, a XEROX Company, Ann Arbor, Michigan .72-4480 FAJEMIROKUN, Francis Afolabi, 1941- APPLICATION OF NEW OBSERVATIONAL SYSTEMS FOR SELENODETIC CONTROL. The Ohio State University, Ph.D., 1971 Geodesy University Microfilms, A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED APPLICATION OF NEW OBSERVATIONAL SYSTEMS FOR SELENODETIC CONTROL DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University by Francis Afolabi Fajemirokun, B. Sc., M. Sc. The Ohio State University 1971 Approved by l/m /• A dviser Department of Geodetic Science PLEASE NOTE: Some Pages have indistinct p rin t. Filmed as received. UNIVERSITY MICROFILMS To Ijigbola, Ibeolayemi and Oladunni ACKNOW LEDGEME NTS The author wishes to express his deep gratitude to the many persons, without whom this work would not have been possible. First and foremost, the author is grateful to the Department of Geodetic Science and themembers of its staff, for the financial support and academic guidance given to him during his studies here. In particular, the author wishes to thank his adviser Professor Ivan I. Mueller, for his encouragement, patience and guidance through the various stages of this work. Professors Urho A. Uotila, Richard H. Rapp and Gerald H. Newsom served on the author’s reading committee, and offered many valuable suggestions to help clarify many points. The author has also enjoyed working with other graduate students in the department, especially with the group at 231 Lord Hall, where there was always an atmosphere of enthusiastic learning and of true friendship. The author is grateful to the various scientists outside the department, with whom he had discussions on the subject of this work, especially to the VLBI group at the Smithsonian Astrophysical Observatory in Cambridge, Ma s sachu s s etts . A large amount of computer time needed for the study was provided by the Instruction and Research Computer Center (IRCC) of The Ohio State University, and the author was able to make full use of its facilities through the assistance of Mr. John M. Snowden of the IRCC. Partial support for this study was also furnished by the Mapping Sciences Laboratory of the National Aeronautics and Space Administration, Manned Spacecraft Center, Houston, Texas under Contract No. NAS 9-9695 and the technical direction of Mr. James L Dragg. The author is greatly indebted to his wife, Dorothy, and to his children - Bola and Wale - for their patience, support, encouragement and love which sustained him during his years of concentrated study here. Finally, the author wishes to thank Miss Barbara Beer who, with excep­ tional skill and expertise, typed most of this work. He also wishes to thank Mrs. Evelyn Rist who, outside her regular work, helped in typing parts of this presentation. VITA June 20, 1941 Born in Ile-Oluji, Nigeria 1966 B. Sc. Land Surveying, University of Nigeria, Nsukka, Nigeria 1968 Research Assistant, Department of Geodetic Science, The Ohio State Univer­ sity, Columbus 1968 M. Sc., The Ohio State University, Columbus 1968-1969 Teaching Associate, Department of Geodetic Science, The Ohio State University, Columbus 1969-1971 Research Associate, Department of Geodetic Science, The Ohio State University, Columbus PUBLICATIONS "Some Investigations on the Systematic Differences Between the Fundamental Star Catalog. " M. Sc. Thesis, The Ohio State University, Columbus, 1968, FIELDS OF STUDY Geometric Geodesy Professor R. H. Rapp Map Projections Professor R. H. Rapp Satellite Geodesy Professor 1.1. Mueller Gravimetric Geodesy Professor 1.1. Mueller Geodetic Astronomy Professor 1.1. Mueller Adjustment Computations Professor U. A. Uotila Photogrammetry P ro fe sso rs S. K. Ghosh and D. C. Merchant. TABLE OF CONTENTS Page ACKNOW LEDGEME N T ...................................................................................... ii VITA .................................................... iv LIST OF TABLES .............................................................................................. viii LIST OF FIGURES .............................................................................................. ix 1. IN TR O D U CTIO N .......................................................................................... 1 1.1 General Discussion ..................... 1 1.2 Review of Past M ethods ........................................................... 3 1. 3 Future Observational Systems for Selenodetic Control ....................................................................... 5 2. APPLICATION OF LASER RANGING TO SELENODETIC CONTROL........................... ............................. 11 2.1 Historical Background ............................................................... 11 2. 2 Basic Operating Principles of Laser Ranging System s ........................................................................... 13 2.3 Earth-Moon Distance Equations ............................................ 15 2. 31 The "Average" Terrestrial Coordinate S y stem ................................................................................... 16 2. 32 The Selenodetic Coordinate System ...................... 17 2. 33 The Geocentric Coordinates of the Selenocenter ............................................................ 19 2. 34 Parameters of the Orientation of the M o o n ........................................................................... 21 2.35 The Parameters of Earth's Orientation .............. 24 2. 36 Computation of Distances between Points on the Earth and on the Moon ...................................... 26 v Page 2. 4 Formulation of Obsez'vation Equations for Earth-Moon Distances ............................................. 35 2. 5 Observation Equations when Orientation Angles are Obtained through Numerical Integration Process ........................................................................ 60 2 . 6 Sum m ary ......................................................................................... 65 3. APPLICATION OF THE VLBI TO SELENODETIC CONTROL ........................................................................ 66 3.1 Introduction .................................... 66 3. 2 Basic Operating Principles of the V L B I ................................ 68 3. 3 Equations for VLBI Measured Time D elay s ....................... 76 3.31 Earth-Earth VLBI Equations ..................................... 77 3. 32 Moon-Moon VLBI Equations ..................................... 95 3.33 Earth-Moon VLBI Equations ................. 109 3.4 Summary ................................................................................................. 122 4. NUMERICAL INTEGRATION OF THE ORIENTATION OF AN EARTH-FIXED COORDINATE SYSTEM WITH RESPECT TO AN INERTIAL SYSTEM .................................................... 123 4.1 Introduction ..................................................................................... 123 4.2 The Equations of Motion of a Rigid B ody ................................ 126 4. 3 The Rotation of the Earth About its Center of M a s s ...................... 128 4. 31 The Dynamical Equations of M otion .......................... 128 4.32 Euler's Geometric Equations ..................................... 134 4. 33 Differential Equations of Motion of the E a r t h ................................................ 136 4.4 Numerical Integration of the Earth's Orientation Angles ............................................................................ 137 4.41 Calculation of Initial V alues ......................................... 139 4.42 Parameters of Integration ............................................. 143 vi Page 4.5 Adjustment of Initial Condition ................... .......................... 146 4.51 Evaluation of the State Transition and the Parameter Sensitivity Matrices .... 149 4.52 Numerical Fit of the Numerically Integrated Angles to the Calculated Angles .................................. 151 4. 6 Numerical Experiments and Results .................................* 155 4. 61 Fitting the Numerically Integrated Eulerian Angles to the Simulated Angles ............... 157 4. 62 Comparison of Numerically Integrated Eulerian Angles to those Obtained through Classical M ethod ............................................................ 163 5. ADJUSTMENT MODEL FOR LASER AND VLBI OBSERVATIONS.......................................................................................... 172 5.1 Introduction ...................................................... 1 7 2 5.2 Classification of Parameters in the Solution .................. 1 7 4 5. 3 Generalized Adjustment M odel ............................................... 178 5. 31 Formation of Normal Equations .............................. 178 5. 32 Solution of Normal Equations .........................................181 5.4 Numerical Experiments .............................................................. 186 6 . SUMMARY AND CONCLUSIONS............................................................ 195 APPENDIX A: Equations for Earth-Based VLBI Observations of an Artificial Source ......................................................... 199 APPENDIX B: Systematic Errors Affecting Earth-Based Lunar Observations .................. 203 BIBLIOGRAPHY .................................................................................................. 211 vii LIST OF TABLES Page 4.1 Gravitational Constants ......................................................................................146
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