(3) Some simple things about gravity, shape of the Earth spherical harmonics, potential, Laplace equation, Poisson equation perturbations, geoid, mechanics

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(1)A lot of complicated things about gravity, shape of the Earth spherical harmonics, potential, Laplace equation, Poisson equation perturbations, geoid, mechanics ... Frédéric Chambat École Normale Supérieure de Lyon Laboratoire de Géologie. Doctoral school, Barcelonette, 2019.

(2) Some complicated things about gravity, shape of the Earth spherical harmonics, potential, Laplace equation, Poisson equation perturbations, geoid, mechanics ... Frédéric Chambat École Normale Supérieure de Lyon Laboratoire de Géologie. Doctoral school, Barcelonette, 2019.

(3) Some simple things about gravity, shape of the Earth spherical harmonics, potential, Laplace equation, Poisson equation perturbations, geoid, mechanics ... Frédéric Chambat École Normale Supérieure de Lyon Laboratoire de Géologie. Doctoral school, Barcelonette, 2019.

(4) Some simple things about the shape of the Earth Frédéric Chambat École Normale Supérieure de Lyon Laboratoire de Géologie. Doctoral school, Barcelonette, 2019.

(5) The spherical Earth Frédéric Chambat École Normale Supérieure de Lyon Laboratoire de Géologie. Doctoral school, Barcelonette, 2019.

(6) The spherical Earth (and planets, asteroids, moons, dwarf planets) Frédéric Chambat École Normale Supérieure de Lyon Laboratoire de Géologie. Doctoral school, Barcelonette, 2019.

(7) The spherical Earth I. How do we know ? Greece. I. How do we measure ? Egypt. I. Why is it so ? Hawaï. I. When is it so ? Mars. I. What a spherical model does mean ?. I. Why (again) is it so ? Lichtenstein.

(8) I. The spherical Earth : How do we know, how do we measure ?.

(9) a. Measure the local curvature. Aristote (≈ 384 – 322 BC) thinks the Earth is spherical. Eratosthene (≈ 276 – 194 BC) measures the radius..

(10) Triangulation (Fernel, Snellius). Maupertuis, La Figure de la Terre 1738 (1736 expedition in Sweeden). http ://dutarte.club.fr/Siteinstruments/Maupertuis.htm.

(11) Triangulation nowadays.

(12) b. Satellite measure. Geodetic satellite.

(13) a(satellite) = g(satellite) + small various forces.

(14) Gravitationnal potential of the Earth   2 X R ` m m GM  R 0 0 U(r , θ, λ) = 1+ U2 Y2 (θ, λ) + U` Y` (θ, λ) r r r `,m. with GM = 3.986000979(40) × 1014 m3 s−2 : spherical part U20 ' 10−3 : elliptical part U`m . 10−6 : others. The spherical Earth is a good model up to 10−3 . The ellipsoidal Earth is a good model up to 10−6 ..

(15) Spherical harmonic functions Spherical harmonics are the equivalent of Fourier series on the sphere : they apply to a function of the two variables θ and λ of the spherical reference frame. Their property is that any function h(θ, λ) can be broken down into the form :. θ. h(θ, λ) =. λ Représentation. Comment retrouver l’allure d’une harmonique sphérique ? I. Ylm s’annule sur 2m méridiens et l Exemples : #. s’annule sur Y!20" s’annule sur - 0 cercle méridien 2 ⇥-0 2=parallèles 0 méridiens, et 2 0 = 2 parallèles :. m. m. h` Y` (θ, λ),. Historique Mesure et observation Analyse spatiale Variabilité temporelle. Harmoniques sphériques. I. ∞ X ` X `=0 m=−`. ". m parallèles. sursur " s’annule Y!22 s’annule cercles méridiens 2 ⇥-- 202= 4 méridiens, parallèles et 2 2 = 0 parallèle :. ". !Y$ s’annule sur 42 s’annule sur - 2 cercles méridiens 2⇥ = 4 méridiens, - 2 2 parallèles et 4 2 = 2 parallèle :. Examples of spherical harmonics. The color indicates the value of the function : red for positive values, blue for negative values.. where ` is the degree of the harmonic, m is the order, h`m are the coefficients of development, and Y`m are the spherical harmonic functions that thus form a basis for the functions defined on the sphere. The Y`m oscillate in θ and λ. They cancel each other out on m meridian circles, and ` − m parallel (fig.). For example, the first harmonic is constant : Y00 = 1, and to represent a flattened Earth at the poles, we use Y20 , which only cancels out on two parallels : 0. Y2 (θ) =. 3 cos2 θ − 1 2. ..

(16) And inside the Earth, how do we known its stratification is not cubic ?.

(17) Body and surface waves. Earthquakes recorded by the International Seismic Center : 1960 to 2015. 22000 seismic ISC stations..

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(19) 33000 seismogramms, M>5, Componants : vertical (blue), longitudinal (green), transverse (red). Astiz, 1996..

(20) ModesFree propres (ou (normal Oscillations oscillations modes) libres).

(21) Fourier transform of the seismogram.

(22) Free oscillations (normal modes) : a solution of wave equation ∂2p = c 2 ∇2 p ∂2t 3 kinds of solution : I. p(x, z, t) = e i(ωt−αx−γz).

(23) Free oscillations (normal modes) : a solution of wave equation ∂2p = c 2 ∇2 p ∂2t 3 kinds of solution : I. p(x, z, t) = e i(ωt−αx−γz). P,S body waves x. z.

(24) Free oscillations (normal modes) : a solution of wave equation ∂2p = c 2 ∇2 p ∂2t 3 kinds of solution : I. p(x, z, t) = e i(ωt−αx−γz). P,S body waves x. z. I. p(x, z, t) = e i(ωt−αx) f (z).

(25) Free oscillations (normal modes) : a solution of wave equation ∂2p = c 2 ∇2 p ∂2t 3 kinds of solution : I. p(x, z, t) = e i(ωt−αx−γz). P,S body waves x. z. I. p(x, z, t) = e i(ωt−αx) f (z). Rayleigh, Love surface waves x. z.

(26) Free oscillations (normal modes) : a solution of wave equation ∂2p = c 2 ∇2 p ∂2t 3 kinds of solution : I. p(x, z, t) = e i(ωt−αx−γz). P,S body waves x. z. I. p(x, z, t) = e i(ωt−αx) f (z). Rayleigh, Love surface waves x. z. I. p(x, z, t) = e iωt f (x, z).

(27) Free oscillations (normal modes) : a solution of wave equation ∂2p = c 2 ∇2 p ∂2t 3 kinds of solution : I. p(x, z, t) = e i(ωt−αx−γz). P,S body waves x. z. I. p(x, z, t) = e i(ωt−αx) f (z). Rayleigh, Love surface waves x. z. I. p(x, z, t) = e iωt f (x, z). Free oscillations x. z.

(28) Some modes. Fonctions propres de qq radiaux.

(29) The fundamental radial mode (0 S0 , T=20 min) has low attenuation. OscillationsSeismogram libres 11 jours le séisme de Sumatra 11 après days after the Sumatra earthquake : 1 jour. Mode 0S0. Période 1200s. www.geologie.ens.fr/~madariag.

(30) Zoom between 0.2 and 1 :.

(31) Asphericity is small.

(32) II. Why is the Earth spherical, why not a cube, why in hydrostatic equilibrium ?.

(33) II. Why is the Earth spherical, why not a cube, why in hydrostatic equilibrium ?. How high can a mountain be ?.

(34) The geophysicist mountain. ρg h σxx. σzz.

(35) The geophysicist mountain. ρg h σxx. σzz. Deviatoric stress τ = σzz − σxx = ρgh − 0.

(36) The geophysicist mountain. ρg h σxx. σzz. Deviatoric stress τ = σzz − σxx = ρgh − 0 Consequence : The stress is more extensive underneath the relief.

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(38) Commemorative limestone plaque. Abbaye des Récollets, Béthune, Pas-de-Calais (André Lardon, http ://www2.ac-lyon.fr/enseigne/biologie)..

(39) Pli-faille de Saint-Rambert en Bugey, et dans du gypse.

(40) Température Elastic - Ductile - Brittle. – p.7/34. Figure 0:. à pression constante, T◦ change pour roche calcaire.

(41) Strength of rocks. 5.8 Differential stress, deviatoric stress and some implications 1200. rential stress, deviatoric stress e implications. " s3. 1000. 800. Strength (sdiff) (MPa). t of stress increases downwards from the surhe lithosphere. How much stress can a rock efore deformation occurs? The reference states ncrease in stress all the way to the center of the k-forming minerals undergo phase changes orphic reactions in response to this increase, a deviation from the reference state for rocks y fracturing or shearing. Anderson gave us an the relative orientation of the tectonic stresses he style of faulting (close to the surface). We o know how faulting occurs in the lithosphere. ot the absolute level of stress, but rather the etween the maximum and minimum princithat causes the rock to fracture or flow. This s called the differential stress:. 600. 400. Quartzite Granite (Ultra)mafics 200. Dolomite. ð5:10Þ. Limestone Marble. static stress (Figure 5.7) the principal stresses l, and we have 100. ð5:11Þ. lithostatic model itself provides no differenthe lithosphere, regardless of depth of burial. niaxial-strain reference state of stress the. sv. ð5:12Þ. . erential stress becomes. 200. 300. Confining pressure (MPa). Figure 5.16 The strength of various rock types, plotted against confining pressure (burial depth). The data indicate that the strength of the brittle crust increases with depth, and that the absolute strength depends on lithology (mineralogy). Data compiled from a range of sources.. Differential stress at any given point in the Earth is Fossen, Structural Geology, 2010 limited by the strength of the rock itself. Any attempt to. 93.

(42) ρg h σxx. σzz. τ = σzz − σxx.

(43) ρg h σxx. σzz. τ = σzz − σxx τ = ρgh < τmax ≈ 300 MPa.

(44) ρg h σxx. σzz. τ = σzz − σxx τ = ρgh < τmax ≈ 300 MPa hmax =. τmax 3 × 108 ≈ ≈ 10 km. ρg 3000 × 10.

(45) ρg h σxx. σzz. τ = σzz − σxx τ = ρgh < τmax ≈ 300 MPa hmax =. τmax 3 × 108 ≈ ≈ 10 km. ρg 3000 × 10 Hawaï = +4 km -5 km.

(46) ρg h σxx. σzz. τ = σzz − σxx τ = ρgh < τmax ≈ 300 MPa hmax =. τmax 3 × 108 ≈ ≈ 10 km. ρg 3000 × 10 Hawaï = +4 km -5 km Mars...≈ 27 km.

(47) Did you like the geophysicist mountain ?.

(48) Topography is more like this h(x). z x.

(49) Topography is more like this h(x). z x. Equilibrium ∂x σxx + ∂z σxz = 0 ∂x σxz + ∂z σzz = −ρg.

(50) Topography is more like this h(x). z x. Equilibrium ∂x σxx + ∂z σxz = 0 ∂x σxz + ∂z σzz = −ρg. Isostasy : σxz = 0, thus σxx = f (z) σzz = ρg (h(x) − z)..

(51) Topography is more like this h(x). z x. Equilibrium ∂x σxx + ∂z σxz = 0 ∂x σxz + ∂z σzz = −ρg. Isostasy : σxz = 0, thus σxx = f (z) σzz = ρg (h(x) − z). Thus σzz − σxx = ρgh(x) + F (z).

(52) h(x). z x. Thus σzz − σxx = ρgh(x) + F (z).

(53) h(x). z x. Thus σzz − σxx = ρgh(x) + F (z) Consequences :.

(54) h(x). z x. Thus σzz − σxx = ρgh(x) + F (z) Consequences : a. The stress is more extensive underneath the relief ; and max|σzz − σxx | ≥. 1 ρghmax 2.

(55) h(x). z x. Thus σzz − σxx = ρgh(x) + F (z) Consequences : a. The stress is more extensive underneath the relief ; and max|σzz − σxx | ≥. 1 ρghmax 2. b. Same order of magnitude as the geophysicist mountain..

(56) h(x). z x. Thus σzz − σxx = ρgh(x) + F (z) Consequences : a. The stress is more extensive underneath the relief ; and max|σzz − σxx | ≥. 1 ρghmax 2. b. Same order of magnitude as the geophysicist mountain. (c. Isostasy relation σzz =. R. ρg dz = σxx at some depth = cst).

(57) 3 sources of stress. divσ = −ρ g. inside. σ · n = T on borders ¯ The static stress (tectonic stress) depends on : 1. rock weight : density and topography 2. traction below (σxz 6= 0) 3. traction on edges (for regional studies)..

(58) World Stress Map 180°. 210°. 240°. 270°. 300°. 330°. 0°. 30°. 60°. 90°. 120°. 150°. 180° 75°. World Stress Map 2016 Hosted by the Helmholtz Centre Potsdam - GFZ German Research Centre for Geosciences Editors: Oliver Heidbach1, Mojtaba Rajabi2, Karsten Reiter3, Moritz Ziegler1,4 1. GFZ German Research Centre for Geosciences, Germany, Section 2.6 Seismic Hazard and Stress Field 2 Australian School of Petroleum, University of Adelaide, Australia 3 Institute of Applied Geosciences, Technical University of Darmstadt, Germany 4 Institute of Earth and Environmental Science, University of Potsdam, Germany. Introduction. The World Stress Map (WSM) is a global compilation of information on the present day crustal stress field. It is a collaborative project between academia and industry that aims to characterize stress patterns and to understand the stress sources. It commenced in 1986 as a project of the International Lithosphere Program under the leadership of Mary-Lou Zoback. From 1995-2008 it was a project of the Heidelberg Academy of Sciences and Humanities headed first by Karl Fuchs and then by Friedemann Wenzel. Since 2009 the WSM is maintained at the GFZ German Research Centre for Geosciences and since 2012 the WSM is a member of the ICSU World Data System.. All stress information is analysed and compiled in a standardized format and quality-ranked for reliability and comparability on a global scale. The stress map displays A-C quality stress data records of the upper 40 km of the Earth’s crust from the WSM database release 2016. Focal mechanism solutions labelled as possible plate boundary events in the database (for details see Heidbach et al., 2010) are 60° not displayed. Further detailed information on the WSM quality ranking scheme, guidelines for the various stress indicators and software for stress map generation and the stress pattern analysis is available at www.world-stress-map.org.. 60°. Stress map displays the orientation of maximum horizontal compressional stress SHmax Method. Quality. borehole breakouts drill. induced frac.. Stress Regime. A. focal mechanism. SHmax is within ± 15°. Normal faulting. SHmax is within ± 20°. B. Strike-slip faulting Thrust faulting. SHmax is within ± 25°. C. Unknown regime. overcoring SV. SV. hydro. fractures geol. indicators. SHmax. 0-40 km. SV. Shmin. NF. SS. Data depth range normal faulting regime Sv > SHmax > Shmin. Shmin. SHmax. strike-slip regime SHmax > Sv > Shmin. Shmin. SHmax. TF. thrust faulting regime SHmax > Shmin > Sv. Citation of this map. 30°. 30°. 0°. 0°. Heidbach, O., Rajabi, M., Reiter, K., Ziegler, M. (2016): World Stress Map 2016, GFZ Data Service, doi:10.5880/WSM.2016.002.. Key references for the WSM project Zoback, M.L., Zoback, M., Adams, J., Assumpção, M., Bell, S., Bergman, E.A., Blümling, P., Brereton, N.R., Denham, D., Ding, J., Fuchs, K., Gay, N., Gregersen, S., Gupta, H.K., Gvishiani, A., Jacob, K., Klein, R., Knoll, P., Magee, M., Mercier, J.L., Müller, B.C., Paquin, C., Rajendran, K., Stephansson, O., Suarez, G., Suter, M., Udías, A., Xu, Z.H., Zhizhin, M. (1989): Global patterns of tectonic stress. Nature 341, 291-298, doi:10.1038/341291a0. Zoback, M.L. (1992): First and second order patterns of stress in the lithosphere: The World Stress Map Project, J. Geophys. Res., 97, 11703-11728, doi: 10.1029/92JB00132. Fuchs, K., Müller, B. (2001): World Stress Map of the Earth: a key to tectonic processes and technological applications. Naturwissenschaften 88, 357-371, doi:10.1007/s001140100253. Sperner, B., Müller, B., Heidbach, O., Delvaux, D., Reinecker, J. & Fuchs, K. (2003): Tectonic stress in the Earth's crust: advances in the World Stress Map project. in New insights in structural interpretation and modelling, pp. 101-116, ed. Nieuwland, D. A. Geological Society, London, doi:10.1144/GSL.SP.2003.212.01.07. Tingay, M., Müller, B., Reinecker, J., Heidbach, O., Wenzel, F. & Fleckenstein, P. (2005): The World Stress Map Project 'Present-day Stress in Sedimentary Basins' initiative: building a valuable public resource to understand tectonic stress in the oil patch, The Leading Edge, 24, 1276-1282. Heidbach, O., Reinecker, J., Tingay, M., Müller, B., Sperner, B., Fuchs, K. & Wenzel, F. (2007): Plate boundary forces are not enough: Second- and third-order stress patterns highlighted in the World Stress Map database, Tectonics, 26, TC6014, doi:10.1029/2007TC002133. Heidbach, O., Tingay, M., Barth, A., Reinecker, J., Kurfeß, D. & Müller, B. (2010): Global crustal stress pattern based on the World Stress Map database release 2008, Tectonophysics 482, 3-15, doi:10.1016/j.tecto.2009.07.023. Heidbach, O., Rajabi, M., Reiter, K., Ziegler, M. and the WSM Team (2016): World Stress Map Database Release 2016, GFZ Data Services, doi:10.5880/WSM.2016.001.. References of used data and software This map contains of a number of datasets: Plate boundaries are from the global plate model PB2002 (Bird, 2003), topography and bathymetry from Smith and Sandwell (1997). Stress maps are produced with CASMI (Heidbach and Höhne, 2008) which is based on GMT from Wessel and Smith (1998). Bird, P. (2003): An updated digital model for plate boundaries, Geochem. Geophys. Geosyst., 4 (3), 1027, doi:10.1029/2001GC000252. Heidbach, O., Höhne, J. (2008): CASMI - a tool for the visualization of the World Stress Map data base. Computers & Geosciences, 34, 783-791, doi:1016/j.cageo.2007.06.004. Wessel, P., Smith, W.H.F. (1998): New, improved version of Generic Mapping Tools released, Eos Trans., 79 (47), 579, doi:10.1029/98EO00426. Smith, W.H.F., and Sandwell, D.T. (1997): Global sea floor topography from satellite altimetry and ship depth soundings, Science, 277, 1956-1962, doi:10.1126/science.277.5334.1956.. -30°. -30°. Major contributors to the WSM database release 2016 Australasian Stress Map Project, Geofon Cataloge, The Global CMT Cataloge, European-Mediterranean Regional CMT Solutions Cataloge, Ikon Science Adelaide, DGMK, NAGRA, NECSA, PETROM, BP, Schlumberger, CHEVRON-Texaco, Fennoscandian Rock Stress Database, Wintershall, Shell, Karasu, PTT, Eni, RWE-Dea, WEG, Daleel Petroleum, Premier Oil Adams, J. Ágústsson, K. Alt, R. Al-Zoubi, A.S. Andreoli, M. Árnadóttir, S. Ask, D. Ask, M. Assumpcao, M. Barth, A. Babyyev, G. Balfour, N. Baptie, B. Barr, M. Batchelor, T. Becker, A. Bell, S. Bergerat, F. Bergman, E. Bluemling, P. Bohnhoff, M. Bonjer, K.-P. Bosworth, W. Bratli, R. Brereton, R. Brudy, M. Bungum, H. Chatterjee, R. Colmenares, L. Connolly, P. Cornet, F. Custodio, S. Deichmann, N. Delvaux, D.. Denham, D. Ding, J. Doeveny, P. Enever, J. Feijerskov, M. Fellgett, M.W. Finkbeiner, T. Fleckenstein, P. Fuchs, K. Furen, X. Gay, N. Gerner, P. Gough, D.I. Gowd, T.N. Grasso, M. Gregersen, S. Grünthal, G. Gupta, H. Guzman, C. Gvishiani, A. Haimson, B. Hanssen, T.H. Hauk, C. Heidbach, O. Hergert, T. Hersir, G.P. Hickman, S. Hillis, R. Horvath, F. Hu, X. Jacob, K. Jarosinski, M. Jianmin, D. Jurado, M.J.. King, R. Kingdon, A. Kjorholt, H. Klein, R. Knoll, P. Kropotkin, P. Kurfeß, D. Larsen, R. Lindholm, C. Logue, A. López, A. Lund, B. Lund-Snee, J. Magee, M. Mariucci, M.T. Marschall, I. Mastin, M. Maury, V. Mercier, J. Mildren, S. Montone, P. Mularz-Pussak, M. Müller, B. Negut, M. Oncescu, M.C. Paquin, C. Pavoni, N. Pierdominici, S. Pondrelli, A. Ragg, S. Rajabi, M. Rajendran, K. Reinecker, J. Reiter, K.. Reynolds, S. Röckl, T. Roth, F. Rummel, F. Schmitt, D. Schoenball, M. Sebrier, M. Sherman, S. Sperner, B. Stephansson, O. Stromeyer, D. Suarez, G. Suter, M. Tingay, M. Tolppanen, P. Townend, J. Tsereteli, N. Udias, A. van Dalfsen, W. van Eijs, R. Van-Kin, L. Wenzel, F. Williams, J. Wiprut, D. Wolter, K. Xu, Z. Yunga, S. Zhizhin, M. Zhonghuai, X. Ziegler, M. Zoback, M. Zoback, M.-L.. -60°. -60°. Mercator projection, equatorial scale 1:46,000,000 180°. 210°. 240°. 270°. 300°. 330°. 0°. 30°. 60°. 90°. 120°. 150°. -70° 180°.

(59) Andes.

(60) III. When is a body spherical ?.

(61) hmax =. τmax . ρg. Critical radius ?.

(62) hmax =. τmax . ρg. Critical radius ? g=. GM 4 = πG ρR R2 3.

(63) hmax =. τmax . ρg. Critical radius ? GM 4 = πG ρR R2 3 ⇒ 3τmax hmax = 4πG ρ2 R. g=.

(64) hmax =. τmax . ρg. Critical radius ? GM 4 = πG ρR R2 3 ⇒ 3τmax hmax = 4πG ρ2 R. g=. The body is spherical when (say) hmax < R/4 i.e. when R > Rc with.

(65) hmax =. τmax . ρg. Critical radius ? GM 4 = πG ρR R2 3 ⇒ 3τmax hmax = 4πG ρ2 R. g=. The body is spherical when (say) hmax < R/4 i.e. when R > Rc with s r 3τmax 3 × 3 × 108 ≈ ≈ 7 × 105 m = 700 km. Rc = 2 πG ρ π × 6.7 × 10−11 × (3000)2.

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(67) Photographs of Ceres by the Hubble Space Telescope in 2005 with a resolution of about 30 km. The first and last images are separated by 2 h 20 min from the last..

(68) Ceres seen par Dawn space probe on the 19th february 2015. R ≈ 450 km.

(69) Vesta vue par Dawn. Its shape can be fit by an ellipsoid of radii 280, 272, 227 (±12) km. The mean density is 3.8 ± 0.6 gm/cm3 . For this density Vesta’s shape is close to that of a Maclaurin spheroid with superposed variations of ≈15 km.animation.

(70) Camilla, by light curves inversion. 285 x 205 x 170 ± 20 km..

(71) Eros, space probe NEAR Shoemaker, 1996-2001. 33 x 13 x 13 km. Rotation : 4,8 h..

(72) Mimas Saturnian moon R = 198 km Shaped like a sphere Relief ≤ 5 km Mostly water ice Cassini-Huygens mission 2010s’. Proteus Second-largest Neptunian moon R = 210 km Shaped like an irregular polyhedron Relief ' 20 km Mostly water ice Voyager 2, 1989.

(73) « The International Astronomical Union therefore resolves that planets and other bodies in our Solar System, except satellites, be defined into three distinct categories in the following way : (1) A planet is a celestial body that : (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit. (2) A dwarf planet is a celestial body that : (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, (c) has not cleared the neighbourhood around its orbit, and (d) is not a satellite. (3) All other objects, except satellites, orbiting the Sun shall be referred to collectively as Small Solar System Bodies. Footnotes : - The eight planets are : Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. - Pluto is a "dwarf planet" by the above definition and is recognised as the prototype of a new category of trans-Neptunian objects. ».

(74) IV. What is a spherical model with respect to the Earth ?. Spherical = spherically symmetric.

(75) Density (pink, in g/cm3 ), gravity (green, in m/s2 ), mass (blue, in 1024 kg), pressure (cyan, in 1011 Pa) and gravitational energy (red) derived from the PREM model. The density and gravity functions use the left scale. The mass, pressure and gravitational energy (1026 MJ), use the right. The PREM model of the Earth provides the following numbers : Binding Energy = 24.84608 x 1025 MJ, Pressure at center of Earth = 363.65 GP, Surface Gravity = 9.8129 m/s2 , Radius = 6371 km. Ref : http ://www.preearth.net/worlds-collide.html.

(76) We may define an average parameter as : 1 ρ0 (r ) = ρ(r , θ, λ) = 4πr 2. Z ρ(r , θ, λ) dS Sphere. Several questions. We use physical equations. An example : r µ VS = ρ is transposed as : s VS. =. µ ρ. Strictly, the only thing we know is r VS. =. µ ρ.

(77) Can we use physical relation between means ? Define ρ(r , θ, λ) =. ρ(r , θ, λ). + δρ(r , θ, λ). µ(r , θ, λ) =. µ(r , θ, λ). + δµ(r , θ, λ). Observations tells that δρ ρ and δµ µ. Definition tells that < δρ >= 0 and < δµ >= 0. Thus VS. = = = =. q. µ. qρ. <µ>+δµ <ρ>+δµ <µ> δµ 1 + 2<µ> <ρ>. 2 2 δρ δµ δρ δµ δρ − 2<ρ> + O <µ> + O <ρ> + O <µ> <ρ> q 2 2 <µ> δρ δµ δρ δµ + O <ρ> + O <µ> 1 + O <µ> <ρ> <ρ> q. Fortunetly, the interior of the Earth is nearly spherical : δρ <ρ>. ≈%. δµ <µ>. ≈%. then s VS. =. <µ> −4 −3 1 + Second order terms < 10 − 10 ? <ρ>. In a spherically symetric Earth, parameters and physical relation are only know up to 10−4 − 10−3 in relative value..

(78) Around the interfaces. Radius of the interface : Z 1 r0 = r (θ, λ) dS 4πr 2 Sphere Around the interface, once have to refine that : Z 1 ρ0 (r ) = ρ(r , θ, λ) dS 2 4πr Sphere.

(79) For vectors and tensors The spherical average is less simple. It is first necessary to make, at each point, the tensor invariant to any rotation around the radial axis. Then, the spherical average is made component by component.. Let’s define the rotation matrix around the radial axis :   1 0 0 α  0 cos ψ − sin ψ  Qi = 0 sin ψ cos ψ The spherical average of a vector, with spherical component v i is : Z Z 2π 1 1 Q αi v i (r , θ, λ) dψ dS v0α (r ) = 4πr 2 2π Sphere 0.

(80) The spherical average of, say, a second-order tensor, with spherical component σ ij is : Z Z 2π 1 1 σ0αβ (r ) = Q αi Q βj σ ij (r , θ, λ) dψ dS 4πr 2 2π Sphere 0 etc... for the fourth order elastic tensor c ijkl . The only non zero components are v0r (r ) =. 1 4πr 2. Z. v r (r , θ, λ) dS Sphere. Z 1 σ rr (r , θ, λ) dS 4πr 2 Sphere Z 1 σ0θθ (r ) = σ0λλ (r ) = (σ θθ + σ λλ )/2 (r , θ, λ) dS 4πr 2 Sphere σ0rr (r ) =. c ijkl depends on five independant components A, C , F , L, N (transverse isotropy). ⇒ We know which parameters shall/may include a spherical seismic elastic Earth model : 1. ρ (gravity is deduced from its definition) 2. A, C , F , L, N (wave speeds are deduced from them) 3. σ rr − σ θθ (the stress is deduced from equilibrium equation).

(81) Data for spherical models I. Radius R = 6371.000 + .230 = 6371.230(10) km.. I. Mass M = 5.97218(60) × 1024 kg I I I I. I. →ρ. GM relative precision : 10−8 G 10−2 Cavendish (1798) 2.10−5 CODATA (2018) 10−4 I prefer (2019). Inertia cefficient (spherical) I /MR 2 = 0.330690(9). → ρ. → A, C , F , L, N. I. P, S waves travel times, surface wave phase velocity. I. Free oscillations frequencies → A, C , F , L, N, ρ, σ rr − σ θθ .. And maybe some ’theoretical’ constraints as I. Birch relation Vp ≈ a + bρ. I. Adams-Williamson relation (isentropic). I. Elastic anisotropy is small : A, C , F , L, N ≈ K , µ. I. Stress anisotropy is small : σ rr − σ θθ ≈ 0.. dρ dr. 2. ≈ − ρKg ..

(82) grations were carried out by using a density model in a discretized form, the Runge-Kutta matlab routine ode45 and the matlab spline interpolator interp1 to refine the sampling.. J4 = −2.96 × 10−6 .. (21). Table 1. Data for reference Earth model. The values in parenthesis are the uncertainties referred to the last figures of the nominal values. Data Observeda Physical mean radius Geocentric gravitational constantb Angular velocity Rotationnal factor Gravitational constant Mass Inertia ratiob,c Inertia coefficientb,c Degree 2 zonal potential coefficientb,d,e Degree 4 zonal potential coefficiente. Symbol. Value (uncertainty). Unit. Relative uncertainty. R GM " m G M I /M I /M R 2 J 2 |obs−corr J4. 6.371 230 (10) 3.986 000 979 (40) 7.292 115 0 (1) 3.450 162 (16) 6.674 28 (67) 5.972 18 (60) 1.342 354 (31) 0.330 690 (9) 1.082 604 6 (5) −1.620 (1). 106 m 1014 m3 s−2 −5 10 rad s−1 10−3 10−11 m3 kg−1 s−2 1024 kg 1013 m2. 1.6 × 10−6 1.0 × 10−8 1.4 × 10−8 4.7 × 10−6 1.0 × 10−4 1.0 × 10−4 2.3 × 10−5 2.6 × 10−5 4.6 × 10−7 6.2 × 10−4. k J 21 J2 J 2 − J 21 J4. 0.932 33 (9) 1.072 3 (1) 1.071 2 (1) −1.085 (3) −2.96 (3). Hydrostatic (this study) Fluid degree two Love number Degree 2 zonal potential coefficient, first-ordere Degree 2 zonal potential coefficient, second-ordere Difference of second- and first-ordere Degree 4 zonal potential coefficiente a From. Chambat & Valette (2001) with modifications explained in text. atmosphere. ratio of the spherical model that is closest to the Earth. d Without direct and hydrostatic indirect permanent tide. e J and J are scaled with GM given in this table and a = 6 378 137 m. 2 4. 10−3 10−6. 10−3 10−3 10−6 10−6. 1 × 10−4 1 × 10−4 1 × 10−4 3 × 10−3 1 × 10−2. bWithout c Inertia. ⃝ C 2010 The Authors, GJI C 2010 RAS Geophysical Journal International ⃝. Chambat et al, 2010.

(83) Near-end conclusions I. Gravity induces more extensive stress underneath relief. I. Gravity and limited strengh of rocks prevent important topography. I. This is also true inside the Earth. I. When the radius of the "planet" is greater than several hundreds of km, topography is "small". I. In the planets, deviatoric stress is small. I. i.e. equilibrium is (nearly) hydrostatic. I. It’s a condition to be a planet or a dwarf planet. I. Spherically symmetric models are not that easy to think and construct. I. A planet (hydrostatic) with no rotation is spherical. Let’s proove the last item. Now I’ll be hydrostatic....

(84) Earth Surface. 1s 10 y 10 ky 106 y Hydrostatic Fred Chambat. Viscous Andréa Tommasi. Multiactivity guy Jean Braun. Ductile and britle Laurent Jolivet. Viscolelastic Giorgio Spada. Elastic Ved Lekic. Barcelonette PT-diagram. Pressure. Earth Center. 109 y Time Scale.

(85) V. A non-rotating, and alone in space, fluid is spherical ?.

(86) Why is the reference for zero mountain spherical ? What is the shape of a non-rotating fluid in space ? A simple question, a difficult answer.

(87) Why is the reference for zero mountain spherical ? What is the shape of a non-rotating fluid in space ? A simple question, a difficult answer : the Lichtenstein’s theorem (and the Liapounov theorem).

(88) Liapounov’s theorem (1884) Let, alone in space, be a fluid mass, that is non-rotating, and self-gravitating. If this body is homogeneous and in stable hydrostatic equilibrium, then its shape is a sphere..

(89) [the proof is quite long, and is in :].

(90) Lichtenstein’s theorem (1918) Let, alone in space, be a fluid mass, that is uniformly rotating, and self-gravitating. If this body is stratified and in hydrostatic equilibrium, then it admits a plane of symmetry that is orthogonal to the rotation axis (and contains the center). Consequence The theorem is true even if the rotation vanishes. Thus : Let, alone in space, be a fluid mass, that is non-rotating, and self-gravitating. If this body is stratified and in hydrostatic equilibrium, then every axis is a rotation axis with null rotation, any plane containing the center is a symmetry plane : its shape (and stratification) is a sphere..

(91) Hydrostatic equilibrium. gradp = ρ gradW ∇2 W = −4πG ρ + 2ω 2 . p = 0 at surface.

(92) Property : In an hydrostatic body, equipotential sufaces are equipressure and equidensity surfaces. Interfaces are such surfaces. Proof gradp = ρ gradW. (1). gradρ ∧ gradW = 0. Taking the curl of (gradp)/ρ shows that they are also equidensity surfaces. Take the jump JK of (1), it yields Decompose. JgradpK = JρK gradW. ∂p n + gradT p, ∂n use the fact that the pressure is continuous : gradp =. then. JgradT pK = gradT JpK = 0,. ∂p Kn = JρK gradW ∂n that is : gradW // n i.e. an interface is an equipotential surface (and equidensity and equipressure). The surface of a lake is horizontal. J.

(93) Proof of Lichtenstein’s theorem for an homogeneous planet. gradp = ρ gradW = grad(ρW ). ⇒ p = ρW + cst. ∇2 W = −4πG ρ + 2ω 2 . p = 0 at surface. ⇒ W = cst.

(94) Proof of Lichtenstein’s theorem for an homogeneous planet. Potentials : W (gravity ) = U(gravitationnal) + Q(rotationnal). Hydrostatic equilibrium, ’the surface is horizontal’ : W = cst on the surface. Definition : z0 = max(z(x, y )). x,y. Hypothesis, the equator is not a plane : ∃(x, y ) | z(x, y ) < z0 ..

(95) For each column.

(96) For each column. U− > U+.

(97) Thus, in this configuration. We have proved U − > U + , while equilibrium implies U − = U + ..

(98) For an heterogeneous fluid. Definition : z0 = max(zi (x, y )). x,y ,i.

(99) The spherical Earth I. How do we know ? Greece. I. How do we measure ? Egypt. I. Why is it so ? Hawaï. I. When is it so ? Mars. I. What a spherical model does mean ?. I. Why (again) is it so ? Lichtenstein What is the intruder in the list ?.

(100) Conclusion : I hope you will look at spherically symmetric models differently.. References : sorry, I don’t see any reference for these stuff....

(101)