I L i1 NASA TECHNICAL NOTE NASA TN D-5239 THEORY OF THE CORRECTION OF CELESTIAL OBSERVATIONS MADE FOR SPACE NAVIGATION OR TRAIN1 by Barnett L. Gadeberg and Kenneth C. White Ames Research Center Moffett Field, Cali$ NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. JANUARY 1970 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. NASA TN D-5239 4. Title and Subtitle 5. Report Date THEORY OF THE CORRECTION OF CELESTIAL OBSERVATIONS January 1970 MADE FOR SPACE NAVIGATION OR TRAINING 6. Performing Organization Code 7. Author(s) 8. Performing Organization Report No. Burnett L. Gadeberg and Kenneth C. White A-2449 9. Performing Orgonization Name and Address 10. Work Unit No. 125-17-05-01-00-21 NASA Ames Research Center Moffett Field, Calif. 94035 11. Controct or Grant No. 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address National Aeronautics and Space Adnlinistration TECHNICAL NOTE !\lashington, D. C. 20546 14. Sponsoring Agency Code 15. Supplementary Notes 16. Abstract This report discusses the theory of the correction of celestial observations, made for space navigation, instrument calibration, or training; as well as new theories concerning window refraction and lunar geon~etricallibrations. It presents, in an easily understood form, essentially all current information on the correctio~lsfor celestial observations, providing a sound basis from which the engineer can derive high-precision observations. The corrections associated with these observations have been listed and classified. Where the theory has been con~pleted,equations and co~istantsare provided, or indicated, by which the reduction may be effected; in some cases, the equations have been recast in a form conve~lientfor machine computation. Where the theory or knowledge was incomplete, developments are provided, or the requirements to develop the theory are made known. 17. Key Wards (Suggested by Author(s)) 18. Distribution Statement Astronomy Unclassified Unlimited Space - Navigation Libration Refraction 19. Security Clossif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Pricef Unclassified Unclassified 48 $3.00 TABLE OF CONTENTS Page SUMMARY ....................... ........ 1 INTRODUCTION .............................. 1 ANALYSIS OF CELESTIAL OBSERVATION CORRECTIONS .............. 3 General Procedure for Processing an Observation ........... 3 Classification of Corrections ..................... 3 Corrections Due to the Finite Velocity of Light (Aberration) ..... 4 Corrections Dependent on Position of Observer (Parallax) ....... 7 Corrections Dependent on Time of Observation ............. 7 Propermotion ............................ 7 Ephemeristime .......................... 8 Earth precession and nutation ................... 8 Lunar and planetary librations .................. 10 Corrections Due to Medium of Transmission (Refraction) ........ 11 Atmosphericrefraction ...................... 11 Window refraction ......................... 12 Corrections for Instrument Read-Out Errors .............. 13 Miscellaneous Corrections ...................... 14 Irradiance ............................ 14 Limbeffects ........................... 14 Semidiameter ........................... 14 Personal equation ......................... 15 PROCEDURES FOR APPLYING CORRECTIONS ................... 16 Order of Making Corrections ..................... 16 Aberration.. (stellar) ...................... 16 Aberration.. (planetary) ..................... 17 Parallax.. (earth-based) ..................... 17 Parallax.. (non-earth-based) ................... 17 Ephemeristime .......................... 18 Earth orientation .. (precession) ................. 18 Librations .. (geometrical and physical) .............. 18 Refraction.. (atmospheric) .................... 19 Refraction.. (window) ....................... 19 Instrument.. (index, arc, filter) ................. 20 Irradiance ............................ 20 Limb effects.. (atmospheric, irregularities) ........... 21 Semidiameter ........................... 21 CONCLUDINGREMARKS ............................ 21 APPENDIX A . DEVELOPMENT OF VECTOR EQUATIONS FOR THE ABERRATION OF LIGHT 22 APPENDIX B . DEVELOPMENT OF EQUATIONS FOR LUNAR GEOMETRICAL LIBRATIONS . 26 APPENDIX C . DEVELOPMENT OF EQUATIONS FOR WINDOW REFRACTION ....... 34 REFERENCES ............................... 47 THEORY OF THE CORRECTION OF CELESTIAL OBSERVATIONS MADE FOR SPACE NAVIGATION OR TRAINING By Burnett L. Gadeberg and Kenneth C. White Ames Research Center SUMMARY This report discusses the theory of the correction of celestial observations, made for space navigation, instrument calibration, or training; as well as new theories concerning window refraction and lunar geometrical librations. It presents, in an easily understood form, essentially all current information on the corrections for celestial observations, providing a sound basis from which the engineer can derive high-precision observations. The corrections associated with these observations have been listed and classified. Where the theory has been completed, equations and constants are provided, or indicated, by which the reduction may be effected; in some cases, the equations have been recast in a form convenient for machine com- putation. Where the theory or knowledge was incomplete, developments are provided, or the requirements to develop the theory are made known. INTRODUCTION One of the requirements of a space navigation system is that celestial observations (measurements) of some kind be provided periodically , from which the state vector of the vehicle may be updated or determined directly. The types of measurements that may be classified as celestial observations are limited by standard usage in astronomy to angular measure- ments between celestial bodies. The body on which the observer is situated is not included, except when consideration is given to sextant horizon-altitude measurements. In general, however, we will be considering celestial observa- tions as angular measurements between various combinations of stars and bodies of the solar system, all situated some distance from the observer. Four types of observations are considered in this discussion. They are the measurements between: two stars ; a star and the limb of a body displaying an extended disk; a star and some permanent marking on the surface, commonly known as a landmark; or two limbs of the same body or two different bodies. Any of these types of observations may be made with one setting of a sextant or two settings of a theodolite. All four types of observations are of value. To provide position information for space navigation, an observation must include at least one body within the solar system; however, star-star obser- vations are very useful for instrument calibration and for the training of observers. Measurements made between a star and the limb of an extended disk are useful to provide position information. Observations made between a star and a landmark also provide position information, but they may be more accurate when the destination body is close by. Sights between the two limbs of the same body are useful in providing distance information from the body when its apparent diameter is large. Such measurements have associated errors, which may be classified as either random or systematic. Accurate utilization of the data requires that these errors be compensated as precisely as possible. In the case of random errors, this compensation can be accomplished only by probabilistic means such as a weighted least squares (ref. 1) or a sequential data-processing method in the form of a Kalman filter (ref. 2). Studies describing such opti- mum compensation for random errors ignore the systematic errors on the premise that they are known and, therefore, can be corrected. However, the proper correction of systematic errors is not trivial and requires detailed attention to the theory. There has been much previous work, largely by astronomers, on the systematic errors in celestial observations (refs. 3 and 4). This work, how- ever, has been directed to satisfying particular needs, and does not consti- tute a complete theory for the problems peculiar to space navigation. One purpose of this report is to develop such a theory insofar as it is possible at this time, and to present mathematical techniques suitable for application to specific space-navigation problems . In some cases, astronomers ' previous work can be used directly, or with only minor modifications. Other cases require additional analysis, provided herein. Where there is insufficient knowledge for the development of precise correction formulas, the requirements of the correction theory are outlined in general terms. For most cases in space (position) navigation, the angle desired as an input to the system is the "geometric" angle as seen by the observer, where "geometric" angle means the instantaneous angle between the centers of the masses of the two bodies unaffected by distance, the speed of light, or the velocity of the observer. Since all the bodies of the universe are in a state of dynamical motion, and the angles between them are constantly changing for any observer, a measure of the time of observation is an essential factor. Even under ideal circumstances, an observer rarely can measure the desired angle. His own velocity causes aberration of the light, the apparent position of the body differs from the true position and ephemeris position due to the finite time of transmission of