Hauptmenü
• Autor
• Moritz, Helmut
• TitelThe figure of the earth
• Zusatz z. Titeltheoretical geodesy and the earth's interior
• Persistent Identifier
• VerlagsortKarlsruhe
• VerlagWichmann
• Erscheinungsjahr1990
• BeschreibungXI, 279 S.
• Beschreibunggraph. Darst.
• ISBN3-87907-220-5
• Zugriffsrechte

### Kapitel

• PrefaceMoritz, HelmutVII
• 1. Background Information1
• 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 Theory25
• 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 EquilibriumClairauts Equation42
• 6. Radaus Transformation46
• 7. Moments of Inertia49
• 3. Equilibrium FiguresAlternative Approaches53
• 2. the Method of Integral Equations53
• 2. the Geometry of Equilibrium Surfaces60
• 1. Stratification of Equisurfaces61
• 2. Wavres Theorem63
• 3. Spherical Stratification as an Exception64
• 4. Impossibility of a Purely Ellipsoidal Stratification65
• 5. Another Derivation of Clairauts Equation69
• 6. Concluding Remarks70
• 3. Stationary Potential Energy73
• 1. Potential Energy73
• 2. Diracs and Heavisides 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 Figures79
• 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. Clairauts and Darwins Equations90
• 1. Internal Gravity Potential90
• 2. Clairauts Equation to Second Order93
• 3. Radaus Transformation96
• 4. Darwins Equation98
• 5. Practical Comments and Results101
• 3. Derivation from Wavres Theory104
• 1. General Formulas for X and Y105
• 2. Series Expansions108
• 3. Basic Equations111
• 4. Darwins Equation112
• 5. Clairauts Equation114
• 5. the Equipotential Ellipsoid and Its Density Distributions119
• 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 Approximation171
• 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 Inhomogeneities183
• 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. Lauricellas Use of Greens Function207
• 1. Application of Greens Identity207
• 2. Transformation of Greens Identity208
• 3. Lauricellas Theorems210
• 4. Greens Function for the Sphere213
• 5. Stokes` Constants and the Harmonic Density215
• 8. Isostasy217
• 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