Gravity Field and Dynamics of the Earth

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1 Milan Bursa Karel Pec Gravity Field and Dynamics of the Earth With 89 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest

2 Preface v Introduction 1 1 Fundamentals of Determining the Parameters Defining the Earth's Gravitational Field by Satellite Methods Introduction Satellite Equations of Motion Perturbing Function and Perturbing Potential General Definitions Perturbing Gravitational Potential of the Earth in Outer Space Perturbations due to the Moon and the Sun Solution of the Perturbed Motion Transformation of the Perturbing Gravitational Potential into the Function of the Satellite's Orbital Elements Transformation of Potential R Transformation of Potentials AV S p AV SO Fundamentals of the Theory of Determining the Parameters of the Earth's Gravitational Potential by Satellite Methods Motion of the Nodal Line due to the Earth's Polar and Equatorial Flattening Geopotential Coefficients Determined from the Variation in Satellite Orbital Elements - An Outline. Numerical Results The Geocentric Gravitational Constant Resonance Phenomena Geostationary Satellites 47 2 The Earth's Gravity Field and Its Sources Introduction Gravitational and Gravity Potentials 52

3 2.3 Transformation of the Gravitational Potential and Potential of Centrifugal Forces Under Rotation of the Coordinate System. Transformation of Geopotential Coefficients Gravity in Outer Space Listing's Geoid Monge's Figure of the Geoid Geometrical Properties of the Geoid The Earth's Triaxial Ellipsoid Determination of the Coefficients in the Harmonie Development of the Geoid's Radius-Vector and of the Geopotential Scale Factor R Q Power Series of the Geoid's Radius-Vector True Gravity Anomalies Structure of the Gravitational Field over the Northern and Southern Hemispheres Theory of the Order of Flattening Clairaut's Theory of the External Field Internal Gravitational Field of the Hydrostatic Earth. Clairaut's Differential Equation Internal Sources of the Gravitational field Physical Interpretation of the Geopotential Coefficients. Tensor of Inertia Transformation of the Coordinate System into the Principal Axes of the Earth's Inertia Tensor Density Models of the Earth Mean Spherically Symmetrical Models of the Earth Lateral Density Variations Integral Density Equations Analytical Density Model for a Spherically Asymmetrical Earth Powers x" Developed into a Series of Shifted Legendre Polynomials System of Algebraic Equations for the Density Model Coefficients. Compatibility Conditions for the Mean Spherical Model Algebraic Equations for Coefficients FfJ, of the Density Variations Model Total Mass of the Earth Moments of Inertia Fundamentals of the Earth's Rotation Dynamics Introduction Fundamental Relations of the Earth's Rotation Dynamics, Euler's Dynamic and Kinematic Equations 141

4 IX 3.3 The Earth's Rotation Dynamics in the Absence of External Moments; Euler's Free Nutation Liouville's Equations Polar Motion; Variations in the Angular Velocity of the Earth's Rotation. Numerical Results Dynamics of the Earth's Rotation and the Problem of Defining Time Effect of the Deceleration of the Earth's Rotation on the Observed Ephemerides of Orbiting Bodies Problem of Realization of the Reference Coordinate System in the Earth's Rotation Dynamics Fundamentals of the Dynamics of the Earth's Precession and Nutation Force Function of the Earth-Moon-Sun System Right-Hand Sides of Euler's Dynamic Equations as Functions of the Gravitational Perturbations due to the Moon and the Sun Approximate Solution for the Precession-Nutation Motion Under Equal Equatorial Moments of Inertia Numerical Results The Earth's Tides. Tidal Deformation of the Earth's Crust Introduction Tide-Generating Potential of a Perfectly Rigid Earth Tide-Generating Potential of a Perfectly Elastic Earth Additional Potential in Outer Space due to the Earth's Tidal Deformation Effect of the Moon's Motion on the Tide-Generating Potential Components of Tidal Forces Love Numbers and Methods of Determining Them The Precession-Nutation Torque of Tidal Forces The Secular Love Number The Earth's Deformations and Variations in the Earth's Rotation Introduction Dynamics of the Tidal Deceleration of the Earth's Rotation Deformations of the Earth due to the Variations in tbe Earth's Rotation 259

5 X Contents Variations in the Potential of Centrifugal Forces; Perturbing Forces Deformations of Equipotential Surfaces due to Polar Motion for a Perfectly Rigid Earth Deformations of Equipotential Surfaces due to Polar Motion for a Perfectly Elastic Earth Deformations due to Variations in the Earth's Angular Velocity Comparison with Tidal Deformations Dynamics of the Earth's Ellipsoid of Inertia On the Hypothesis of an Expanding Earth Decrease in the Maximum Principal Moment of the Earth's Inertia and Its Effect on Polar Motion Secular Decrease in the Earth's Angular Momentum and Kinetic Energy 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 The Earth in the Solar System Introduction Structure of the Solar System Orbital Elements of the Planets Laplace's Invariable Plane of the Solar System Gravitational Forces Acting on the Earth 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

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