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
  • ZugriffsrechteAuch auzserhalb des TU-Netzes nutzbar

Kapitel

  • PrefaceMoritz, HelmutpdfVII
  • 1. Background Information1
    • 1. Conceptual and Historical Backgroundpdf1
    • 2. Elements of Gravitation and Gravitypdf4
    • 3. Spherical Harmonicspdf8
    • 4. Elements of Ellipsoidal Geometrypdf13
    • 5. Earth Models and Parameterspdf17
  • 2. the Equilibrium Figure of the EarthBasic Theory25
    • 1. External Ellipsoidal Field to First-Order Approximationpdf25
    • 2. Internal Field of a Stratified Spherepdf29
    • 3. Homogeneous EllipsoidFirst-Order Theorypdf34
    • 4. Heterogeneous Ellipsoidpdf37
    • 5. Hydrostatic EquilibriumClairaut`s Equationpdf42
    • 6. Radau`s Transformationpdf46
    • 7. Moments of Inertiapdf49
  • 3. Equilibrium FiguresAlternative Approaches53
    • 2. the Method of Integral Equationspdf53
    • 2. the Geometry of Equilibrium Surfacespdf60
      • 1. Stratification of Equisurfacespdf61
      • 2. Wavre`s Theorempdf63
      • 3. Spherical Stratification as an Exceptionpdf64
      • 4. Impossibility of a Purely Ellipsoidal Stratificationpdf65
      • 5. Another Derivation of Clairaut`s Equationpdf69
      • 6. Concluding Remarkspdf70
    • 3. Stationary Potential Energypdf73
      • 1. Potential Energypdf73
      • 2. Dirac`s and Heaviside`s Functionspdf74
      • 3. a Remarkable Expression for the Densitypdf75
      • 4. Variation of the Potential Energypdf75
      • 5. a General Integral Equationpdf77
  • 4. Second-Order Theory of Equilibrium Figures79
    • 1. Internal Potentialpdf81
      • 2. Change of Variablepdf82
      • 3. Potential of Shell \( E_P \)pdf85
      • 4. Computation of \( K_n(q) \) and \( L_n(q) \)pdf87
      • 5. Gravitational Potential at Ppdf89
      • 1. Potential of Interior \( I_P \)pdf81
    • 2. Clairaut`s and Darwin`s Equationspdf90
      • 1. Internal Gravity Potentialpdf90
      • 2. Clairaut`s Equation to Second Orderpdf93
      • 3. Radau`s Transformationpdf96
      • 4. Darwin`s Equationpdf98
      • 5. Practical Comments and Resultspdf101
    • 3. Derivation from Wavre`s Theorypdf104
      • 1. General Formulas for X and Ypdf105
      • 2. Series Expansionspdf108
      • 3. Basic Equationspdf111
      • 4. Darwin`s Equationpdf112
      • 5. Clairaut`s Equationpdf114
  • 5. the Equipotential Ellipsoid and Its Density Distributions119
    • 1. Ellipsoidal Coordinates and Ellipsoidal Harmonicspdf119
    • 2. the Level Ellipsoida and Its External Fieldpdf126
    • 3. Mass Distributions for the Levelpdf131
      • 1. A Simple Examplepdf137
    • 4. the Maclaurin Ellipsoidpdf140
    • 5. Reduction to a Maclaurin Ellipsoidpdf142
    • 6. Heterogeneous Mass Distributions for the Maclaurin Ellipsoidpdf145
    • 7. Zero-Potential Densitiespdf146
    • 8. Representation by Polynomialspdf149
      • 1. a Fourth-Degree Polynomialpdf153
    • 9. Combined Density Modelspdf155
    • 10. Numerical Considerations and Problemspdf156
    • 11. Potential and Gravity Inside the Ellipsoidpdf163
    • 12. Potential Energypdf166
    • 1. the Spherical Casepdf167
  • 6. Equipotential EllipsoidSecond-Order Approximation171
    • 1. Basic Formulaspdf171
    • 2. Level Ellipsoid and Equilibrium Figurespdf173
    • 3. Equipotential Surfaces and Surfaces of Constant Densitypdf175
    • 4. the Deviationpdf176
    • 5. Numerical Results and Conclusionspdf180
  • 7. Density Inhomogeneities183
    • 1. the Gravitational Inverse Problempdf183
    • 2. Zero-Potential Densitiespdf186
    • 3. Unique Solutionspdf188
    • 4. a General Solutionpdf190
    • 5. Analytical Continuationpdf192
    • 6. Continuous Density Distributions for the Spherepdf194
      • 1. Use of Spherical Harmonicspdf195
      • 2. A Very General Solutionpdf197
      • 3. harmonic Densitiespdf199
      • 4. Zero-Potential Densitiespdf200
      • 5. Remarks on the General Solutionpdf201
      • 6. An Essential Simplificationpdf202
      • 7. Application of Orthonormal Expansionspdf203
    • 7. Lauricella`s Use of Green`s Functionpdf207
      • 1. Application of Green`s Identitypdf207
      • 2. Transformation of Green`s Identitypdf208
      • 3. Lauricella`s Theoremspdf210
      • 4. Green`s Function for the Spherepdf213
      • 5. Stokes` Constants and the Harmonic Densitypdf215
  • 8. Isostasy217
    • 1. Classical Isostatic Modelspdf218
      • 1. the Model of Pratt-Hayfordpdf218
      • 2. the Model of Airy-Heiskanenpdf220
      • 3. Regional Compensation According to Vening Meineszpdf222
      • 4. Attraction of the Compensating Massespdf229
      • 5. Remarks on Gravity Reductionpdf231
    • 2. Isostasy as a Dipole Fieldpdf232
      • 1. Potential of the Topographic Massespdf233
      • 2. Attraction of Topographypdf236
      • 3. Condensation on Sea Levelpdf239
      • 4. Effect of Compensationpdf241
      • 5. Conclusions Regarding Gravity Anomaliespdf243
    • 3. Inverse Problems in Isostasypdf246
      • 1. the Inverse Pratt Problempdf246
      • 2. the Inverse Vening Meinesz Problempdf255
      • 3. Concluding Remarkspdf262
  • Referencespdf265
  • Indexpdf275

Abbildungen

  • 1. The earth ellipsoid as an ellipsoid of revolution with semimajor axis a, semiminor axis b, and flattening fpdf2
  • 2. Illustrating eqs. (1-1) and (1-5)pdf5
  • 3. Spherical and rectangular coordinatespdf9
  • 4. The meridian ellipse as parametrized by geographic ϕ, geocentric latitude ψ, or reduced latitude β latitude pdf14
  • 5. The earth's interiorpdf19
  • 6. Distribution in the earth of density \( ρ (gcm^{(-3)} \), pressure \( p (10^{11}Nm^{(-2)} \) and gravity \( g (ms^{(-2)}) \)pdf20
  • 7. Distribution of density ρ according to the earth model PREMpdf20
  • 8. The PREM model and the approximate models by Bullard and Rochepdf24
  • 1. A spherical shellpdf30
  • 2. Computation point P inside the spherepdf31
  • 3. Ellipsoid and mean spherepdf35
  • 4. Ellipsoidal shellpdf38
  • 5. Illustrating the potential at an interior point Ppdf40
  • 1. Rotation deforms a sphere into a spheroidpdf55
  • 2. The geometry of stratificationpdf61
  • 3. A layer of constant density ( \( \underline{x} \) denotes x )pdf75
  • 1. Ellipsoid and equilibrium spheroidpdf80
  • 2. Illustrating the computation of \( V(P) \)pdf81
  • 3. Illustrating the computation of \( V_i \)pdf82
  • 4. Illustrating the computation of \( V_e \)pdf85
  • 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)pdf102
  • 6. Inverse flattening \( f^{(-1)} \) for two different models of density ρpdf103
  • 7. The normal radius of curvaturepdf106
  • 8. The distance between two neighboring equisurfacespdf107
  • 9. The θ-correctionpdf110
  • 10. Polar radius t and mean radius βpdf115
  • 1. Ellipsoidal coordinatesTop: View from the front ; Bottom: View from abovepdf120
  • 2. The confocal coordinate ellipsoids \( u = const \) and hyperboloids \( \bar{θ} = const.\), together with the reference ellipsoid u = bpdf121
  • 3. A coordinate ellipsoid u = const. and the auxiliary spheres S and σpdf132
  • 4. The focal disc singularity in the ellipsoidal coordinate systemϵ is an arbitrary small number (dimensionless if b = 1)pdf158
  • 5. A spherical shellpdf169
  • 1. A surface of constant density, \( S_1 \), and the corresponding surface of constant potential, \( S_2 \)pdf175
  • 1. Two possible functions \( V_0 \) in one dimensionpdf186
  • 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!)pdf190
  • 3. Two possible functions V in one dimensionpdf191
  • 4. Possible types of singularities of the analytical continuationpdf193
  • 5. Possible choices of the vector xpdf198
  • 6. The powers \( r^n \) (0 ≤ r ≤ 1)pdf200
  • 7. The sum \( x = x^{(1)} + x^{(2)} \) again is of type \( x^{(1)} \)pdf201
  • 8. Representation of the solution x by an arbitrary vector vpdf203
  • 9. Illustrating the method of Green's functionpdf209
  • 10. Kelvin transformation as an inversion in the spherepdf213
  • 11. The point Q lies on the sphere Spdf214
  • 1. Isostasy - Pratt-Hayford modelpdf219
  • 2. Isostasy - Airy-Heiskanen modelpdf220
  • 3. Local and regional compensationpdf222
  • 4. Bending (direct effect, (a)) and thickening (indirect effect, (b)) of an elastic platepdf222
  • 5. The bending curvepdf223
  • 6. The basic coordinate systems xyz and xyhpdf229
  • 7. Illustrating the attraction of the compensating massespdf230
  • 8. Topographic and compensating masses contribute to gravity reductionpdf231
  • 9. Bouguer plate and terrain correctionnote that the effect of both the positive and the negative masses on C is always positivepdf232
  • 10. Topographic and isostatic masses form a dipolepdf233
  • 11. The spherical approximationpdf234
  • 12. The terrain correctionpdf238
  • 13. Spherical equivalent of Fig. 8.10note again the dipole characterpdf241
  • 14. Various points on the sphere that play a role in the theory of Dorman and Lewispdf247
  • 15. Notations for the inverse Vening Meinesz problempdf256