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The Elliptical Earth (And Satellites, Asteroids, Planets, Dwarf Planets) from 1672 to 2019

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The Elliptical Earth (And Satellites, Asteroids, Planets, Dwarf Planets) from 1672 to 2019 The elliptical Earth (and satellites, asteroids, planets, dwarf planets) From 1672 to 2019 Frédéric Chambat École Normale Supérieure de Lyon Laboratoire de Géologie Doctoral school, Barcelonette, 2019 ω y ds c g ϕ r= a x a − c 1 = = a − c = 21km a 300 1. How do we measure ? 2. How do we explain ? 3. How do we measure ? 4. How do we explain ? 5. England 1 - France 0 6. How do we measure in 2019? 7. Ellipticity and the interior of planets in 2019 8. How do we explain (Clairaut, My nice theory explained with modern maths, 2019)? Carlo Denis (1941-2019) Born in Luxembourg, 1941. PhD in Sciences, in 1974, at the University of Liège, Belgium. He then pursued his entire career as a professor-researcher at the Institute of Astrophysics and Geophysics in the same university. Among the subjects he worked on, there were two on which he devoted more time and energy: the normal modes for his thesis and the hydrostatic equilibrium figures in the 1970s and 1980s. In both cases, it has developed its own numeric codes. He has supervised one doctoral student (Yves Rogister). He liked to travel (especially in his car), read and listen to classical music. He loved being independent and he was very independent. Carlo Denis wrote third order codes to integrate hydrostatic equation giving the shape of the Earth. He summarized his results in a long and useful paper published in a series: Denis C. (1989). The Hydrostatic Figure of the Earth, Physics and Evolution of the Earth’s Interior 4, Gravity and Low-Frequency Geodynamics (edited by R. Teisseyre), Chap. 3, pp. 111–186. PWN-Polish Scientific Publishers & Elsevier, Amsterdam. https://fr.wikipedia.org/wiki/Figure_de_la_Terre and articles on geodesy. Jean Richer (1672) Map of America, maybe by Claude Bernou, 1681. Colbert introduces the members of the Royal Academy of Sciences to Louis XIV. Fictional scene. Henri Testelin (1616-1695), from Charles Le Brun (1619 – 1690), tapestry cardboard ordered by the king for the "Manufacture des Gobelins", around 1667. 1679 - 71 pp. Richer 1679 One of the most significant Observations I have made, is that of the length of the pendulum oscillating at a second of time, which was shorter in Caïenne than in Paris; because the same measure that had been marked in this place on an iron rod, according to the length that was necessary to make a pendulum oscillating at a second of time, having been brought to France, & compared with that of Paris, their difference was found to be one line & a quarter [2.81 mm], of which that of Caïenne is less than that of Paris, which is 3. feet 8. lines 3/5 [99.39 cm]. This Observation was repeated during ten full months, where there has not been a week during which it was not made several times with great care. The oscillations of the pendulum were very small, & remained very sensitive up to fifty-two minutes, & were compared to those of a very excellent clock, whose oscillations marked the second of time. Pendulum clock, with escapment (Huygens, Horologium, 1658) Video s ` 1 second = π γ s ` 1 second = π γ In Paris: γ = π2` = π2 × 99:39 cm = 9:809 m/s2 dγ d` −2:81 mm = = = −2:8 × 10−3 γ ` 99:39 cm In Cayenne: γ = 9:781 m/s2 z P G x 0 x 1 2 a = ! @ y A 0 In Paris: γ = 9:809 m/s2 In Cayenne: γ = 9:781 m/s2 dγ = −0:028 m/s2 Centrifugal acceleration diminishes the gravity as: !2R cos2 (latitude) Between Paris and Cayenne the difference is −0:019 m/s2 6= −0:028 m/s2 About one third of the measured difference does not come from the centrifugal acceleration z P G x 0 x 1 2 a = ! @ y A 0 gpole 6= gequator ω y Equatorial radiusds a Polar radiusc gc ϕ Two mesurables quantitiesr= a x a−c Flattening: = a g −g Gravity relative difference: pole equator gequator 2 Order of magnitude: ≈ ! a γeq An important dimension-less number, the rotationnal parameter: !2a 1 !2R 1 = ≈ ≈ 3:10−3 ≈ γeq 289 g 300 Someone known 1643-1727 1687, 1713, 1723 1687, 1713, 1723 Emilie du Châtelet (1706-1749) par Maurice Quentin de La Tour 1756-59 1756-59 Emilie du Châtelet (1706-1749) par Maurice Quentin de La Tour Calculation of g Calculation of the attraction g ? Book 1, propositions XC-XCI (p. 228). ω y ds c g ϕ r= a x Calculation of g Calculation of g g Z R 1 Y g = Gρ 2 2πr dr dY 0 d d α d Y R r Calculation of g g Z R 1 Y g = Gρ 2 2πr dr dY 0 d d α d Y Y g = 2πGρ 1 − p dY Y 2 + R2 R r Calculation of g ω y Y R Y c g = 2πGρ 1 − p dY a Y 2 + R2 x ! Z 2c Y gpole = 2πGρ 1 − dY p 2 2 Y =0 Y + R (Y ) with R2(Y ) (c − Y )2 + = 1: a2 c2 ω y Y Calculation of g R c a x calculus calculus calculus... ) 4πGρa 1 g ≈ 1 − pole 3 5 Calculation of g ω y Y calculus calculus calculus... ) R c 4πGρa 1 a g ≈ 1 − x pole 3 5 ... genius of Newton... the attraction at the equator is approximately the attraction at the pole of an ellipsoid whose equatorial radius is the mean of the two radius... ) 4πGρa 2 ::: ) g ≈ 1 − eq 3 5 x dP = ρg dx pole pole c y dP = ρ(g − !2y) dy eq eq a ∆P g c x x pole = pole dPpole = ρgpole d c ∆P g − !2a a c c eq eq y y y dP = ρ(g − !2a ) d a eq eq 1 − 5 a a a = (1 − ) 1 − 2 1 − !2a 5 geq ∆Ppole = ∆Peq ) 5 !2a 1 = = : 4 geq 230 Gravity within a homogeneous ellipsoid is a linear function of coordinates. x g = g axe poles pole c y g = g axe eq eq a ∆Ppole gpole c = 2 ∆Peq geq − ! a a 1 − 5 = (1 − ) 1 − 2 1 − !2a 5 geq ∆Ppole = ∆Peq ) 5 !2a 1 = = : 4 geq 230 x dP = ρg dx pole pole c y 2 ω dPeq = ρ(geq − ! y) dy y a ds c g x x ϕ dP = ρg d c r= a x pole pole c c y y y dP = ρ(g − !2a ) d a eq eq a a a 1 − 5 = (1 − ) 1 − 2 1 − !2a 5 geq ∆Ppole = ∆Peq ) 5 !2a 1 = = : 4 geq 230 ∆Ppole gpole c = 2 ∆Peq geq − ! a a x dPpole = ρgpole dx ω c y ds y 2 dP = ρ(g − ! y) dy c g eq eq a ϕ r= a x x x dP = ρg d c pole pole c c y y y dP = ρ(g − !2a ) d a eq eq a a a ∆Ppole = ∆Peq ) 5 !2a 1 = = : 4 geq 230 ∆Ppole gpole c = 2 ∆Peq geq − ! a a 1 − 5 = (1 − ) 1 − 2 1 − !2a 5 geq x dPpole = ρgpole dx ω c y ds y 2 dP = ρ(g − ! y) dy c g eq eq a ϕ r= a x x x dP = ρg d c pole pole c c y y y dP = ρ(g − !2a ) d a eq eq a a a ∆Ppole gpole c = 2 ∆Peq geq − ! a a 1 − 5 = (1 − ) 1 − 2 1 − !2a 5 geq ∆Ppole = ∆Peq ) 5 !2a 1 = = : x 4 geq 230 dPpole = ρgpole dx ω c y ds y 2 dP = ρ(g − ! y) dy c g eq eq a ϕ r= a x x x dP = ρg d c pole pole c c y y y dP = ρ(g − !2a ) d a eq eq a a a Shape of the Earth Comparison with the measures 7,35 measures 8,60 Newton: a − c = 27 km (instead of 21 km measured now) Mid-term conclusions I Centrifugal force flattens the Earth at the poles a−c I We define flattening by = a !2R 1 I Its order of magnitude is g ≈ 300 I The gravity on the polar axis is higher than that on the equatorial radius, I To balance the pressure in the center, for a homogeneous, ellipsoidal, slightly flattened Earth (Newton) it is necessary that: 5 !2R 1 = = : 4 g 230 1718. Jacques Cassini. Traité de la grandeur et de la figure de la Terre. The Earth is elongated at the poles. 1737. Maupertuis. Report to the Royal Academy of Sciences. The Earth is flattened. Newton was right. Problem: the measures of flattening (≈ 1/178) do not correspond well with Newton’s (1/230). England 1 - France 0 1687. Newton. Philosophiae Naturalis Principia Mathematica. The Earth is flattened at the poles. 1737. Maupertuis. Report to the Royal Academy of Sciences. The Earth is flattened. Newton was right. Problem: the measures of flattening (≈ 1/178) do not correspond well with Newton’s (1/230). England 1 - France 0 1687. Newton. Philosophiae Naturalis Principia Mathematica. The Earth is flattened at the poles. 1718. Jacques Cassini. Traité de la grandeur et de la figure de la Terre. The Earth is elongated at the poles. England 1 - France 0 1687. Newton. Philosophiae Naturalis Principia Mathematica. The Earth is flattened at the poles. 1718. Jacques Cassini. Traité de la grandeur et de la figure de la Terre. The Earth is elongated at the poles. 1737. Maupertuis. Report to the Royal Academy of Sciences. The Earth is flattened.
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