The Gravity Field of the Earth

The Gravity Field of the Earth

From Classical and Modern Methods

1st Edition - January 1, 1967

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  • Author: Michele Caputo
  • eBook ISBN: 9781483222387

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International Geophysics Series, Volume 10: The Gravity Field of the Earth: From Classical and Modern Methods explores the theory of the gravity field of the earth based on both classical and modern methods. Classical method involves observations of gravity taken over the earth's surface, while the modern method uses observations of variation of orbital elements of artificial satellites caused by the gravity field of the earth. This book is organized into two parts encompassing 12 chapters. Part I describes the solution of physical problems that are treated as Dirichlet problems or solved by means of integral equations. This part also deals with the determination of the geoid form from ground gravity measurements using the Stokes formula. The method of obtaining the Stokes formula by means of an integral equation is also outlined. Part II contains modern mathematical techniques developed to utilize the observations of artificial satellites for geodetic purposes. This book could be used as a textbook for students in the fields of geodesy, geophysics, or astronomy.

Table of Contents

  • Preface

    Part I

    Chapter I. General Theory

    1. Introductory Considerations; the Coordinates

    2. Morera's Functions

    3. Gravity Potential with a Triaxial Ellipsoid as Equipotential Surface

    4. Values of Gravity at the Ends of the Semiaxes

    5. Pizzetti's Theorem

    6. Modulus of the Gravity Vector and the Conditions on the Parameters

    7. Modulus of the Gravity Vector in Terms of the Coordinates

    Chapter II. the Gravity Field of the Biaxial Case

    8. Gravity Potential Having a Biaxial Ellipsoid as Equipotential Surface

    9. The Pizzetti and Clairaut Theorems for the Biaxial Model

    10. The Somigliana Theorem

    11. International Gravity Formula and Other Gravity Formulas

    12. The International Gravity Formula Extended into Space

    13. The Shape of the Earth as Obtained from Gravity Measurements

    14. Spherical Harmonics Expansion of the Potential of the Normal Gravity Field

    15. Dimensions of the Earth as Obtained from Gravity Data and Satellite Data

    16. The Flattening of the Earth's Equator

    Chapter III. The Gravity Field of the Triaxial Case: the Moon

    17. First-Order Theory of the Field Having a Triaxial Ellipsoid as an Equipotential Surface: The Moon

    18. Comparison with the Expansion of the Potential in Terms of the Moments of Inertia

    19. The Shape of the Moon

    20. The Density Distribution within the Moon

    21. Is the Surface of the Moon Equipotential?

    Chapter IV. Gravitational Potential for Satellites

    22. Equations of Motion of a Satellite in the Biaxial Field

    23. The Case of a Prolate Ellipsoid

    24. The Motion of a Satellite in the Field Described in Section 23

    25. Motion of a Satellite in a Nonbiaxial Field

    Chapter V. Determination of the Geoid from Terrestrial Data

    26. The Determination of the Geoid

    27. Brun's Equation and the Equation of Physical Geodesy

    28. A Boundary-Value Problem

    29. Stoke's Formula

    30. The Surface Density Distribution which Gives the Perturbing Potential

    31. Introduction to the Integral Equations Method for Stokes Formula

    32. Stoke's Formula by the Integral Equation Method

    33. Relations between the Spectral Components of the Geoid, of the Potential, and of the Modulus of Gravity

    Chapter VI. The Adjustment of the Parameters of the Field

    34. Problems arising from Satellite Results

    35. The Nonrotating Field

    36. The Adjustment of the Parameters

    Chapter VII. A Simplified Biaxial Model

    37. A Simple, Accurate Model for the Nonrotating Field: Introduction

    38. The Potential of the Simplified Model

    39. Properties of the Simplified Model

    40. The Gravity Vector

    41. The Clairaut and Pizzetti Theorems

    42. Spherical Harmonic Expansion

    43. The Actual Field

    Chapter VIII. Determination of the Geoid from Unreduced Terrestrial Data

    44. The Method of Levallois

    45. The Method of Molodenski

    Chapter IX. Some Geophysical Implications

    46. The Hydrostatic Equilibrium of the Earth

    47. Comparison with Stresses Associated with Regional and Continental Loads

    48. Other Implications

    49. Implications on the Moon

    Part II

    Chapter I. Satellite Motion in a Central Field

    1. Introduction

    2. Equations of Motion in the Plane of the Orbit

    3. The Polar Equation of the Orbit

    4. Elements of the Elliptic Orbit: The True, Eccentric, and Mean Anomalies

    5. Kepler's Equation

    6. Other Elliptic Elements

    7. Relations between the Elliptic Elements

    Chapter II. Satellite Motion in Noncentral Fields

    8. The Nahewirkungsgesetz and the Fernwirkungsgesetz

    9. The Earth Gravitational Potential and the Coordinates of the Satellite

    10. Some Identities to Be Used in the Expression of the Terrestrial Gravitational Potential by Means of Orbital Elements

    11. The Expression of the Legendre Functions by Means of the Orbital Elements

    12. Preliminary Expression of the Earth's Potential by Means of the Orbital Elements

    13. Terrestrial Gravitational Potential Expressed in Orbital Elements

    14. Lagrangian Brackets

    15. Equations of Motion Expressed in Terms of the Orbital Elements and the Lagrangian Brackets

    16. Integrated Changes of the Orbital Elements

    17. Study of the Earth's Polar Flattening

    18. Study of the Flattening of the Earth's Equator

    19. Study of the Third-Order Zonal Harmonic

    20. Nonlinear Perturbations of Zonal Harmonics

    21. The Solution Including the Fourth-Order Terms

    22. Variation of the Orbital Elements

    23. Other Perturbations

    24. Analysis of Satellite Observations

    25. Lunar Satellites

    Chapter III. The Geoid

    26. The Geoid


    Author Index

    Subject Index

Product details

  • No. of pages: 216
  • Language: English
  • Copyright: © Academic Press 1967
  • Published: January 1, 1967
  • Imprint: Academic Press
  • eBook ISBN: 9781483222387

About the Author

Michele Caputo

About the Editor

J. Van Mieghem

Affiliations and Expertise

Royal Belgian Meteorological Institute, Uccle, Belgium

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