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SURVEY and COMPARATIVE ANALYSIS of CURRENT GEOPHYSICAL MODELS by Jesco Von Puttkamer George CI Marshdl2 Space Fzight Center Marshal4 Ala

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SURVEY and COMPARATIVE ANALYSIS of CURRENT GEOPHYSICAL MODELS by Jesco Von Puttkamer George CI Marshdl2 Space Fzight Center Marshal4 Ala NASA TECHNICAL NOTE --NASA T'J u163 e. / LOAN COW: RETURN TO AFWL IWLIL-2) KJRTLUNDAFB, N MEX SURVEY AND COMPARATIVE ANALYSIS OF CURRENT GEOPHYSICAL MODELS by Jesco von Puttkamer George CI Marshdl2 Space FZight Center Marshal4 Ala NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. MAY 1969 TECH LIBRARY KAFB, NM 0333853 NASA TN D-5163 SURVEY AND COMPAFUTIVE ANALYSIS OF CURRENT GEOPHYSICAL MODELS By Jesco von Puttkamer George C. Marshall Space Flight Center Mar shall, Ala. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sals by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00 TABLE OF CONTENTS Page SUMMARY .......................................... 1 INTRODUCTION ..................................... 1 GENERAL BACKGROUND AND DISCUSSION OF THE PROBLEM ..... 3 The Potential Function of the Earth .................... 3 The Spherical Harmonics. .......................... 4 The Harmonics Coefficients ......................... 7 The Theorem of Clairaut ........................... 9 TheGeoid.. ................................... 13 Determination of the Harmonics Coefficients ............. 15 Krause's Theory of the Geoid Surface .................. 18 ANALYSISOFMODE LS................................. 21 Model 1 (J2-Only) ............................... 21 Model 2 (NASA Standard) .......................... 25 Model 3 (New J2, J3, J4) ........................... 31 Higher-Order Bodies with Equator Symmetry ............. 34 Model 4a (Triaxial Ellipsoid) ................... 34 Model 4b (Symmetrical Geoid) .................. 37 "Best" Geoid Bodies .............................. 39 Model 5a ( ltBest'fGeoid with Sectorial Term) ........ 39 Model 5b (''Best" Geoid without Sectorial Term) ...... 44 COMPARISONOFMODELS .............................. 44 CONCLUSIONS ...................................... 50 APPENDIX A. COMPUTATION OF LEGENDRE POLYNOMIALS. .... 55 APPENDIX B. ALTERNATE EXPRESSION FOR THE GRAVITY ACCELERATION .......................... 56 iii LIST OF ILLUSTRATIONS Figure Title Page I. First Sectorial Harmonic (due to J2 2) at Equator ..... 8 2. Second Spherical Harmonics Terms in Earth Geoid .... 10 3. Third and Fourth Order Spherical Harmonics Terms in Earth Geoid .............................. I1 4. Fifth and Sixth Order Spherical Harmonics Terms in Earth Geoid .............................. 12 5. Height of J2-Spheroid (Model I) over Reference Ellipsoid of Flattening 1/298.222 (Model 3) ......... 26 6 Heights of Models 1, 2 and 3 (Spheroids) over "Best" Geoid without Sectorial Term ................... 45 7. Heights of Model 2 Spheroid, and of lfBestltGeoid Without Sectorial Term, over Standard Ellipsoid (1/298.3) ................................ 47 8. Heights of Model 3-SpheroidY and of "Best" Geoid without Sectorial Term, over Ellipsoid of Flattening 1/298.222 (Model 3) ........................ 48 9. Heights of "Best" Geoid (Model 5) over Triaxial Ellipsoid of Flattening 1/298.184, Along Prime Meridian ................................ 49 10. Heights of "Best" Geoid (Model 5) over Triaxial Ellipsoid of Flattening 1/298. 184, Along Prime Meridian.. ............................... 51 11. Deviations of Gravities of Models I, 2, 3 and 4 from Gravity of "Best" Geoid (Model 5) , Along Prime Meridian.. ............................... 52 iv LIST OF TABLES Table Title Page I . Model1 ................................ 27 I1 . Model2 ................................ 29 III. Model3 ................................ 33 IV . Models 4a and 4b ........................... 38 V . Geoid ................................... 43 V DEFINITION OF SYMBOLS Symbol Definition coefficients in the series expansion for the geoid radius A,, 2 coefficient of the sectorial harmonic in the geoid expansion B2 coefficient in the series expansion of the geoid radius C numerical coefficients in the geopotential function n, m f flattening or oblateness f ellipticity (or flattening) of the equator e gravity at the equator mean equatorial gravity GM gravitational parameter of the earth ( =p ) 0 J coefficients of zonal harmonics in geopotential n J coefficients of tesseral harmonics in geopotential E, in coefficients of sectorial harmonics in geopotential oblateness coefficients ( older notation) m centrifugal factor at the equator also: index in series notation pm(sin @) associated Legendre polynomials of degree n and'order m n P (sin@) Legendre polynomials of nth degree n vi DEFINITION OF SYMBOLS (Concluded) Symbol Definition equatorial radius of the earth R e - mean equatorial radius of the earth R e polar radius of the earth north polar radius of the earth south polar radius of the earth R longest equatorial radius of the earth e0 R shortest equatorial radius of the earth el r radial distance from the center of the earth U potential function h geographic longitude, positive east of Greenwich longitude of the longest equatorial radius ( semi-major axis) gravitational paraineter of the earth factor in equatorial gravity acceleration equation geocentric latitude X factor defined by equation (22) W angular velocity of the earth's rotation N centrifugal factor at the equator W e perturbation parameter in potential function vii SURVEY AND COMPARATIVE ANALYSIS OF CURRENT GEOPHYS ICAL MODELS SUMMARY By using currently accepted constants of geodesy, especially several different sets of oblateness coefficients, seven models of the geometry of the earth and its gravitational field are numerically derived and compared with each other and with a newly developed geopotential based on recent measurements of the secular and periodic perturbations of artificial satellite orbits. Conclusions are drawn which show, among other things, that the currently used NASA Standard Model deviates significantly from the "bestfTgeoid, and that there are better analytical, as well as numerical, models by which improved approximation can be achieved. Also, it is shown that improvements in the numerical values of the used oblateness coefficients, rather than in their number, appear to have little influ­ ence on the values of the gravity, but that they affect the geometries significantly. Increasing the number of zonal harmonics coefficients, however, can lead to considerable improvements in the gravity model, at least in the range studied (up to and including JI4). INTRODUCT ION In the past a number of theories of varying complexity have been developed to describe the figure and the gravitation field of the earth for use in geodetic, astrodynamic, and astronomical calculations. In scrutinizing these theories, a number of discrepancies are manifest which appear to be caused by two factors; i. e. , ( I)inherent inconsistencies in the development of some of the theories, and (2) differences in gravitational and geometrical parameters used by their authors. With regard to the first consideration, while the earth's gravitational potential (i.e. , the potential energy of the earth in relation to the position relative to the earth) is commonly expressed by tke potential function as a series of spherical harmonics, as adopted by the International Astronomical Union (MU)[ 13, the shape of the earth introduced in the geopotential at the earth's surface with the simultaneous assumption that the potential along the sur­ face of the geoid is constant, is that of an ellipsoid which approximates the actual geoid. The flattening of the ellipsoid is usually determined approximatively by the astrogeodetic methods of geometrical geodesy (sometimes also by the less accurate techniques of gravimetry) , using variants of Clairaut's theorem. How­ ever, even if higher-order extensions of the latter are used, they remain only approximations, which become increasingly inaccurate for increasing deviation of the geopotential from that of a regular ellipsoid. The second consideration refers to the fact that, with our increasing knowledge of our globe and its gravity anomalies, the descriptive geographical parameters are subject to change, particularly the number and values of the oblateness coefficients, but also - to a lesser degree - the gravitational param­ eter GM and the geometrical parameters flattening f, and mean equatorial 0' radius E Urgently needed standardization of the principal parameters was e . accomplished by NASA in 1963, but new and additional data have been determined continuously ever since and are being determined from an ever-increasing popu­ lation of artificial satellites in precision-tracked ( Baker-Nunn) orbits of all inclinations. To relate these new observations to the geoid without having to accept the inconsistencies of the second- and third-order Clairaut extensions, a new theory was developed recently at MSFC by H. Krause [a,31 , which further in­ creased the bulk of existing geographic parameters and models. It is the objective of the following analysis to define and to compare seven different geographic models, one of which is the current (but outdated) NASA Standard. These models are based on four different sets of oblateness coefficients, all of which are applicable to present practical work, and also on the fact that for each set of oblateness conditions two body geometries can be defined; i. e. ~ a spheroid composed of superimposed spherical harmonics and a regular ellipsoid approximating this spheroid. In the first part of the analysis, the general background and the nature of the problem is considered in some detail, followed by the derivation and description of the geophysical theories. In the third part, the
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