Kapitel

• PrefaceMoritz, HelmutVII
• 1. Background Information​1
  • 1. Conceptual and Historical Background1
  • 2. Elements of Gravitation and Gravity4
  • 3. Spherical Harmonics8
  • 4. Elements of Ellipsoidal Geometry13
  • 5. Earth Models and Parameters17
• 2. the Equilibrium Figure of the EarthBasic Theory​25
  • 1. External Ellipsoidal Field to First-Order Approximation25
  • 2. Internal Field of a Stratified Sphere29
  • 3. Homogeneous EllipsoidFirst-Order Theory34
  • 4. Heterogeneous Ellipsoid37
  • 5. Hydrostatic EquilibriumClairaut`s Equation42
  • 6. Radau`s Transformation46
  • 7. Moments of Inertia49
• 3. Equilibrium FiguresAlternative Approaches​53
  • 2. the Method of Integral Equations53
  • 2. the Geometry of Equilibrium Surfaces60
    • 1. Stratification of Equisurfaces61
    • 2. Wavre`s Theorem63
    • 3. Spherical Stratification as an Exception64
    • 4. Impossibility of a Purely Ellipsoidal Stratification65
    • 5. Another Derivation of Clairaut`s Equation69
    • 6. Concluding Remarks70
  • 3. Stationary Potential Energy73
    • 1. Potential Energy73
    • 2. Dirac`s and Heaviside`s Functions74
    • 3. a Remarkable Expression for the Density75
    • 4. Variation of the Potential Energy75
    • 5. a General Integral Equation77
• 4. Second-Order Theory of Equilibrium Figures​79
  • 1. Internal Potential81
    • 2. Change of Variable82
    • 3. Potential of Shell \( E_P \)85
    • 4. Computation of \( K_n(q) \) and \( L_n(q) \)87
    • 5. Gravitational Potential at P89
    • 1. Potential of Interior \( I_P \)81
  • 2. Clairaut`s and Darwin`s Equations90
    • 1. Internal Gravity Potential90
    • 2. Clairaut`s Equation to Second Order93
    • 3. Radau`s Transformation96
    • 4. Darwin`s Equation98
    • 5. Practical Comments and Results101
  • 3. Derivation from Wavre`s Theory104
    • 1. General Formulas for X and Y105
    • 2. Series Expansions108
    • 3. Basic Equations111
    • 4. Darwin`s Equation112
    • 5. Clairaut`s Equation114
• 5. the Equipotential Ellipsoid and Its Density Distributions​119
  • 1. Ellipsoidal Coordinates and Ellipsoidal Harmonics119
  • 2. the Level Ellipsoida and Its External Field126
  • 3. Mass Distributions for the Level131
    • 1. A Simple Example137
  • 4. the Maclaurin Ellipsoid140
  • 5. Reduction to a Maclaurin Ellipsoid142
  • 6. Heterogeneous Mass Distributions for the Maclaurin Ellipsoid145
  • 7. Zero-Potential Densities146
  • 8. Representation by Polynomials149
    • 1. a Fourth-Degree Polynomial153
  • 9. Combined Density Models155
  • 10. Numerical Considerations and Problems156
  • 11. Potential and Gravity Inside the Ellipsoid163
  • 12. Potential Energy166
  • 1. the Spherical Case167
• 6. Equipotential EllipsoidSecond-Order Approximation​171
  • 1. Basic Formulas171
  • 2. Level Ellipsoid and Equilibrium Figures173
  • 3. Equipotential Surfaces and Surfaces of Constant Density175
  • 4. the Deviation176
  • 5. Numerical Results and Conclusions180
• 7. Density Inhomogeneities​183
  • 1. the Gravitational Inverse Problem183
  • 2. Zero-Potential Densities186
  • 3. Unique Solutions188
  • 4. a General Solution190
  • 5. Analytical Continuation192
  • 6. Continuous Density Distributions for the Sphere194
    • 1. Use of Spherical Harmonics195
    • 2. A Very General Solution197
    • 3. harmonic Densities199
    • 4. Zero-Potential Densities200
    • 5. Remarks on the General Solution201
    • 6. An Essential Simplification202
    • 7. Application of Orthonormal Expansions203
  • 7. Lauricella`s Use of Green`s Function207
    • 1. Application of Green`s Identity207
    • 2. Transformation of Green`s Identity208
    • 3. Lauricella`s Theorems210
    • 4. Green`s Function for the Sphere213
    • 5. Stokes` Constants and the Harmonic Density215
• 8. Isostasy​217
  • 1. Classical Isostatic Models218
    • 1. the Model of Pratt-Hayford218
    • 2. the Model of Airy-Heiskanen220
    • 3. Regional Compensation According to Vening Meinesz222
    • 4. Attraction of the Compensating Masses229
    • 5. Remarks on Gravity Reduction231
  • 2. Isostasy as a Dipole Field232
    • 1. Potential of the Topographic Masses233
    • 2. Attraction of Topography236
    • 3. Condensation on Sea Level239
    • 4. Effect of Compensation241
    • 5. Conclusions Regarding Gravity Anomalies243
  • 3. Inverse Problems in Isostasy246
    • 1. the Inverse Pratt Problem246
    • 2. the Inverse Vening Meinesz Problem255
    • 3. Concluding Remarks262
• References265
• Index275

Abbildungen

• 1. The earth ellipsoid as an ellipsoid of revolution with semimajor axis a, semiminor axis b, and flattening f2
• 2. Illustrating eqs. (1-1) and (1-5)5
• 3. Spherical and rectangular coordinates9
• 4. The meridian ellipse as parametrized by geographic ϕ, geocentric latitude ψ, or reduced latitude β latitude 14
• 5. The earth's interior19
• 6. Distribution in the earth of density \( ρ (gcm^{(-3)} \), pressure \( p (10^{11}Nm^{(-2)} \) and gravity \( g (ms^{(-2)}) \)20
• 7. Distribution of density ρ according to the earth model PREM20
• 8. The PREM model and the approximate models by Bullard and Roche24
• 1. A spherical shell30
• 2. Computation point P inside the sphere31
• 3. Ellipsoid and mean sphere35
• 4. Ellipsoidal shell38
• 5. Illustrating the potential at an interior point P40
• 1. Rotation deforms a sphere into a spheroid55
• 2. The geometry of stratification61
• 3. A layer of constant density ( \( \underline{x} \) denotes x )75
• 1. Ellipsoid and equilibrium spheroid80
• 2. Illustrating the computation of \( V(P) \)81
• 3. Illustrating the computation of \( V_i \)82
• 4. Illustrating the computation of \( V_e \)85
• 5. Density ρ, mean density \( ρ_mD \), and η (above) and flattening f and deviation κ (below) as function of the average radius q = Rβ (in kilometers)102
• 6. Inverse flattening \( f^{(-1)} \) for two different models of density ρ103
• 7. The normal radius of curvature106
• 8. The distance between two neighboring equisurfaces107
• 9. The θ-correction110
• 10. Polar radius t and mean radius β115
• 1. Ellipsoidal coordinatesTop: View from the front ; Bottom: View from above120
• 2. The confocal coordinate ellipsoids \( u = const \) and hyperboloids \( \bar{θ} = const.\), together with the reference ellipsoid u = b121
• 3. A coordinate ellipsoid u = const. and the auxiliary spheres S and σ132
• 4. The focal disc singularity in the ellipsoidal coordinate systemϵ is an arbitrary small number (dimensionless if b = 1)158
• 5. A spherical shell169
• 1. A surface of constant density, \( S_1 \), and the corresponding surface of constant potential, \( S_2 \)175
• 1. Two possible functions \( V_0 \) in one dimension186
• 2. The potentials \( V_P \), \( V_H \) and \( V_P \) negative arguments are for the symmetry of the figure only (negative r are without geometric meaning!)190
• 3. Two possible functions V in one dimension191
• 4. Possible types of singularities of the analytical continuation193
• 5. Possible choices of the vector x198
• 6. The powers \( r^n \) (0 ≤ r ≤ 1)200
• 7. The sum \( x = x^{(1)} + x^{(2)} \) again is of type \( x^{(1)} \)201
• 8. Representation of the solution x by an arbitrary vector v203
• 9. Illustrating the method of Green's function209
• 10. Kelvin transformation as an inversion in the sphere213
• 11. The point Q lies on the sphere S214
• 1. Isostasy - Pratt-Hayford model219
• 2. Isostasy - Airy-Heiskanen model220
• 3. Local and regional compensation222
• 4. Bending (direct effect, (a)) and thickening (indirect effect, (b)) of an elastic plate222
• 5. The bending curve223
• 6. The basic coordinate systems xyz and xyh229
• 7. Illustrating the attraction of the compensating masses230
• 8. Topographic and compensating masses contribute to gravity reduction231
• 9. Bouguer plate and terrain correctionnote that the effect of both the positive and the negative masses on C is always positive232
• 10. Topographic and isostatic masses form a dipole233
• 11. The spherical approximation234
• 12. The terrain correction238
• 13. Spherical equivalent of Fig. 8.10note again the dipole character241
• 14. Various points on the sphere that play a role in the theory of Dorman and Lewis247
• 15. Notations for the inverse Vening Meinesz problem256