Table Of ContentPec
Milan Bursa · Karel
Gravity Field and
Dynamie s of the Earth
With 89 Figures
Springer-Verlag Berlin Heidelberg GmbH
Prof. Dr. MILAN BuRSA
Astronomical Institute
Academy of Seiences of the Czech Republic
Bocni II, 1401
14131 Praha 4
Czech Republic
t
Prof. Dr. KAREL PEt
Charles University
Povltavska 2
180 00 Praha 8
Czech Republic
Translated from the Czech by
Dr. JAROSLAV TAUER
Sleska 86
130 00 Praha 3
Vinohrady
Czech Republic
Title of the Original Czech Edition
Tihove pole a dynamika Zeme
© Academia 1988
ISBN 978-3-642-52063-1
Library of Congress Cataloging-in-Publication Data. Bursa, Milan. [Tihove pole a dynamika
Zeme. Czech] Gravity field and dynamics of the Earth/Milan Bursa, Kare! Pec; [translated by
J. Tauer] p. cm. Includes bibliographical references and index.
ISBN 978-3-642-52063-1 ISBN 978-3-642-52061-7 (eBook)
DOI 10.1007/978-3-642-52061-7
I. Gravity. 2. Geodesy. I. Pec,
Kare!. II. Title. QB33l.B8313 1993 526'.7-dc20 93-4997
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© Springer-Verlag Berlin Heidelberg 1993
Originally published by Springer-Verlag Berlin Heidelberg New York in 1993
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Preface
Since the Czech edition was published four years ago, the authors
have revised the original text tobring it up to date. During these four
years, thanks to satellite altimetry the accuracy of the global descrip
tion of the gravity field (model GEM-T2), of the fundamental astro
geodetic constants, of the principal moments of inertia of the Earth
and, in particular, of their differences, of the precession constant, and
of a number of other dynamical parameters of the Earth have been
improved.
The authors have included most of these improvements in the
revised English edition. They have, of course, also made factual,
formal and other corrections and have modified some of the figures.
Additions to the index and references have also been made.
Praha, Czech Republic M. BuRSA and K. PF:c
August 1993
Contents
Preface ................................... V
Introduction ...............................
1 Fundamentals of Determining the Parameters
Defining the Earth's Gravitational Field by
Satellite Methods ........................... 6
1.1 Introduction ............................... 6
1.2 Satellite Equations of Motion ................. 6
1.3 Perturbing Function and Perturbing Potential .... 23
1.3.1 General Definitions ......................... 23
1.3.2 Perturbing Gravitational Potential of the Earth in
Outer Space ............................... 24
1.3.3 Perturbations due to the Moon and the Sun ...... 27
1.4 Solution of the Perturbed Motion .............. 28
1.5 Transformation of the Perturbing Gravitational
Potential into the Function of the Satellite's
Orbital Elements ........................... 32
1.5.1 Transformation of Potential Rs$ ............... 32
1.5.2 Transformation of Potentials L1 Vs»' L1 Vso ........ 34
1.6 Fundamentals of the Theory of Determining the
Parameters of the Earth's Gravitational Potential by
Satellite Methods ........................... 35
1.6.1 Motion of the Nodal Line due to the Earth's Polar
and Equatorial Flattening .................... 35
1.6.2 Geopotential Coefficients Determined from the
Variation in Satellite Orbital Elements- An Outline.
Numerical Results .......................... 38
1.7 The Geocentric Gravitational Constant. ......... 42
1.8 Resonance Phenomena ...................... 45
1.9 Geostationary Satellites ...................... 47
2 The Earth's Gravity Field and lts Sources ........ 51
2.1 Introduction ............................... 51
2.2 Gravitational and Gravity Potentials ........... 52
VIII Contents
2.3 Transformation of the Gravitational Potential and
Potential ofCentrifugal Forces Under Rotation ofthe
Coordinate System. Transformation of Geopotential
Coefficients .............................. . 55
2.4 Gravity in Outer Space ..................... . 57
2.5 Listing's Geoid ............................ . 62
2.5.1 Monge's Figure of the Geoid ................. . 62
2.5.2 Geometrical Properties of the Geoid ........... . 63
2.5.3 The Earth's Triaxial Ellipsoid ................ . 77
2.5.4 Determination of the Coefficients in the Harmonie
Development of the Geoid's Radius-Vector and of the
Geopotential Scale Factor R0 ................ . 89
2.5.5 Power Series of the Geoid's Radius-Vector ...... . 94
2.6 True Gravity Anomalies .................... . 97
2.7 Structure of the Gravitational Field over the
Northern and Southern Hemispheres .......... . 101
2.8 Theory of the Order of Flattening ............. . 106
2.8.1 Clairaut's Theory of the External Field ......... . 109
2.8.2 Interna! Gravitational Field of the Hydrostatic Earth.
Clairaut's Differential Equation ............... . 112
2.9 Interna! Sources of the Gravitational field ....... . 117
2.9.1 Physical Interpretation of the Geopotential
Coefficients. Tensor of Inertia ................ . 118
2.9.2 Transformation of the Coordinate System into the
Principal Axes of the Earth's lnertia Tensor ..... . 121
2.10 Density Models of the Earth ................. . 124
2.10.1 Mean Spherically Symmetrical Models of the Earth 125
2.11 Lateral Density Variations ................... . 130
2.11.1 Integral Density Equations .................. . 131
2.11.2 Analytical Density Model for a Spherically
Asymmetrical Earth ........................ . 131
2.11.3 Powers x" Developed into a Series of Shifted
Legendre Polynomials ...................... . 132
2.11.4 System of Algebraic Equations for the Density Model
Coefficients. Compatibility Conditions for the Mean
Spherical Model. .......................... . 133
2.11.4.1 Algebraic Equations for Coefficients F)~ of the
Density Variations Model ................... . 133
2.11.4.2 Total Mass of the Earth .................... . 135
2.11.5 Moments of Inertia ........................ . 136
3 Fundamentals of the Earth's Rotation Dynamics .. . 141
3.1 Introduction. ............................. . 141
3.2 Fundamental Relations of the Earth's Rotation
Dynamics, Euler's Dynamic and Kinematic
Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Contents IX
3.3 The Earth's Rotation Dynamics in the Absence of
External Moments; Euler's Free Nutation ........ 148
3.4 Liouville's Equations ........................ 156
3.5 Polar Motion; Variations in the Angular Velocity
of the Earth's Rotation. Numerical Results ....... 162
3.6 Dynamics of the Earth's Rotation and the Problem of
Defining Time ............................. 178
3.7 Effect of the Deceleration of the Earth's Rotation on
the Observed Ephemerides of Orbiting Bodies .... 184
3.8 Problem of Realization of the Reference Coordinate
System in the Earth's Rotation Dynamics ........ 186
3.9 Fundamentals of the Dynamics of the Earth's
Precession and Nutation ..................... 189
3.9.1 Force Function of the Earth-Moon-Sun System .. 189
3.9.2 Right-Hand Sides of Euler's Dynamic Equations as
Functions of the Gravitational Perturbations due
to the Moon and the Sun .................... 198
3.10 Approximate Solution for the Precession-Nutation
Motion Under Equal Equatorial Moments of
Inertia .................................... 202
3.11 Numerical Results .......................... 204
4 The Earth's Tides. Tidal Deformation of the
Earth's Crust .............................. 207
4.1 Introduction ............................... 207
4.2 Tide-Generating Potential of a Perfectly Rigid
Earth .................................... 209
4.3 Tide-Generating Potential of a Perfectly Elastic
Earth .................................... 220
4.4 Additional Potential in Outer Space due to the
Earth's Tidal Deformation .................... 223
4.5 Effect of the Moon's Motion on the Tide-Generating
Potential ................................. 227
4.6 Components of Tidal Forces .................. 229
4.7 Love Numbers and Methods of Determining Them 237
4.8 The Precession-Nutation Torque of Tidal Forces .. 239
4.9 The Secular Love Number. ................... 240
5 The Earth's Deformations and Variations in the
Earth's Rotation ............................ 245
5.1 Introduction ............................... 245
5.2 Dynamics of the Tidal Deceleration of the Earth's
Rotation .................................. 245
5.3 Deformations of the Earth due to the Variations in
the Earth's Rotation 259
•••••••••••••••• 0 •••••• 0
X Contents
5.3.1 Variations in the Potential of Centrifugal Forces;
Perturbing Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 259
5.3.2 Deformations of Equipotential Surfaces due to Polar
Motion for a Perfectly Rigid Earth . . . . . . . . . . . . . 267
5.3.3 Deformations of Equipotential Surfaces due to Polar
Motion for a Perfectly Elastic Earth . . . . . . . . . . . . 270
5.3.4 Deformations due to Variations in the Earth's
Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
5.3.5 Comparison with Tidal Deformations. . . . . . . . . . . 273
5.4 Dynamics of the Earth's Ellipsoid of Inertia . . . . . . 274
5.5 On the Hypothesis of an Expanding Earth . . . . . . . 277
5.6 Decrease in the Maximum Principal Moment of the
Earth's Inertia and Its Effect on Polar Motion . . . . 279
5.7 Secular Decrease in the Earth's Angular Momentum
and Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . 284
5.8 Long-Term Variations in the Earth's Gravity Field
due to Variations in the Earth's Rotation Vector
and in the Second Zonal Geopotential Coefficient . 286
6 The Earth in the Solar System . . . . . . . . . . . . . . . . . 288
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
6.2 Structure of the Solar System . . . . . . . . . . . . . . . . . 288
6.3 Orbital Elements of the Planets. . . . . . . . . . . . . . . . 290
6.4 Laplace's Invariable Plane of the Solar System. . . . 296
6.5 Gravitational Forces Acting on the Earth. . . . . . . . 302
6.6 Orbital Elements of the Earth and Their Variations
in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Appendix A: Current Representative Values of the Parameters
of Common Relevance to Astronomy, Geodesy and
Geodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
List of the Most Important Symbols . . . . . . . . . . . . . . . . . . . 323
Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Introduction
Earth and space sciences are developing very rapidly thanks to contemporary
methods of satellite research and to modern measuring and computer facilities.
The origin of modern satellite methods dates back to the launehing of the first
satellite, Sputnik I, in 1957; since then satellite methods have developed very
quickly and have had a profound effect on Earth and space sciences. New
independent scientific disciplines have originated, e.g. space geodesy and space
meteorology, and satellite methods have, especially in geophysics, yielded new
and fundamental information about the structure and dynamics of the Earth's
magnetosphere. In celestial mechanics this has led to the development of
improved theories of satellite motion, and in geodesy it has contributed signifi
cantly to accurate determination of the parameters of the figure of the Earth and
of the external gravitational field. The data obtained from satellite observations
are used in studying inhomogeneities in the Earth's interior and related dynamic
processes.
Thanks to satellite methods, the Earth's external gravitational field is known
with remarkable accuracy and high resolution, second to no other geophysical
quantity subject to lateral variations. The fundamental global parameters of the
Earth have been determined with high accuracy: the equatorial radius, the
ftattening of the Earth's reference ellipsoid, and the product of the Earth's mass
and gravitational constant, referred to as the geocentric gravitational constant,
GM. Satellite methods also provide data on the position of the axis of rotation
with an accuracy better than 10 cm, and enable accurate monitaring of the
motion of the Earth's pole. The external gravitational field has been described in
terms of geopotential coefficients (spherical harmonic coefficients) up to degree
n = 180 and this enables the equipotential surfaces of the geopotential, i.e. also
the geoid, to be mapped in considerable detail. The current accuracy of global
long-wave components of the geoid, corresponding to geopotential coefficients
up to degree n = 4, is 8 cm and the overall accuracy, inclusive of the other
known terms of higher degrees in the determination of the geoid, is about 1m.
Although the accuracy of the higher short-wave components of the geoid is
lower, it is nevertheless sufficient to locate geoid slopes with !arge gradients,
remarkable in that they are nearly identical with tectonically active areas on the
Earth's surface in which stress is accumulated and released; this is reflected in the
occurrence of tectonic earthquake foci. The accuracy with which the shape of
the gravimetric geoid has been determined can be checked by another, practic
ally independent method, namely satellite altimetry, which consists in measuring
