# The Gravity Field of the Earth

## 1st Edition

### From Classical and Modern Methods

**Authors:**Michele Caputo

**Editors:**J. Van Mieghem

**eBook ISBN:**9781483222387

**Imprint:**Academic Press

**Published Date:**1st January 1967

**Page Count:**216

## Description

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

References

Author Index

Subject Index

## Details

- No. of pages:
- 216

- Language:
- English

- Copyright:
- © Academic Press 1967

- Published:
- 1st January 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