Clairaut's theorem

Clairaut's theorem is a general mathematical law giving the surface gravity on a viscous rotating ellipsoid in equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise[1] which synthesized physical and geodetic evidence that the Earth is an oblate rotational ellipsoid.[2][3] It was initially used to relate the gravity at any point on the Earth's surface to the position of that point, allowing the ellipticity of the Earth to be calculated from measurements of gravity at different latitudes. Today it has been largely supplanted by the Somigliana equation.

Not to be confused with Clairaut's relation in differential geometry or Clairaut's theorem on equality of mixed partials in calculus.

Figure 1: An ellipsoid

Figure 2: Wireframe rendering of an ellipsoid (oblate spheroid)

History

Although it had been known since antiquity that the Earth was spherical, by the 17th century evidence was accumulating that it was not a perfect sphere. In 1672 Jean Richer found the first evidence that gravity was not constant over the Earth (as it would be if the Earth were a sphere); he took a pendulum clock to Cayenne, French Guiana and found that it lost ​2 ¹⁄₂ minutes per day compared to its rate at Paris.[4][5] This indicated the acceleration of gravity was less at Cayenne than at Paris. Pendulum gravimeters began to be taken on voyages to remote parts of the world, and it was slowly discovered that gravity increases smoothly with increasing latitude, gravitational acceleration being about 0.5% greater at the poles than at the equator.

British physicist Isaac Newton explained this in his Principia Mathematica (1687) in which he outlined his theory and calculations on the shape of the Earth. Newton theorized correctly that the Earth was not precisely a sphere but had an oblate ellipsoidal shape, slightly flattened at the poles due to the centrifugal force of its rotation. Since the surface of the Earth is closer to its center at the poles than at the equator, gravity is stronger there. Using geometric calculations, he gave a concrete argument as to the hypothetical ellipsoid shape of the Earth.[6]

The goal of Principia was not to provide exact answer for natural phenomena, but to theorize potential solutions to these unresolved factors in science. Newton pushed for scientists to look further into the unexplained variables. Two prominent researchers that he inspired were Alexis Clairaut and Pierre Louis Maupertuis. They both sought to prove the validity of Newton's theory on the shape of the Earth. In order to do so, they went on an expedition to Lapland in an attempt to accurately measure the meridian arc. From such measurements they could calculate the eccentricity of the Earth, its degree of departure from a perfect sphere. Clairaut confirmed that Newton's theory that the Earth was ellipsoidal was correct, but his calculations were in error, and wrote a letter to the Royal Society of London with his findings.[7] The society published an article in Philosophical Transactions the following year in 1737 that revealed his discovery. Clairaut showed how Newton's equations were incorrect, and did not prove an ellipsoid shape to the Earth.[8] However, he corrected problems with the theory, that in effect would prove Newton's theory correct. Clairaut believed that Newton had reasons for choosing the shape that he did, but he did not support it in Principia. Clairaut's article did not provide a valid equation to back up his argument as well. This created much controversy in the scientific community.

It was not until Clairaut wrote Théorie de la figure de la terre in 1743 that a proper answer was provided. In it, he promulgated what is more formally known today as Clairaut's theorem.

Formula

Clairaut's formula for the acceleration due to gravity g on the surface of a spheroid at latitude φ, was:[9][10]

${\displaystyle g(\varphi )=G_{e}\left[1+\left({\frac {5}{2}}m-f\right)\sin ^{2}\varphi \right]\ ,}$

where ${\displaystyle G_{e}}$ is the value of the acceleration of gravity at the equator, m the ratio of the centrifugal force to gravity at the equator, and f the flattening of a meridian section of the earth, defined as:

${\displaystyle f={\frac {a-b}{a}}\ ,}$

(where a = semimajor axis, b = semiminor axis).

Clairaut derived the formula under the assumption that the body was composed of concentric coaxial spheroidal layers of constant density.[11] This work was subsequently pursued by Laplace, who relaxed the initial assumption that surfaces of equal density were spheroids.[12] Stokes showed in 1849 that the theorem applied to any law of density so long as the external surface is a spheroid of equilibrium.[13][14] A history of the subject, and more detailed equations for g can be found in Khan.[15]

Somigliana equation

The above expression for g has been supplanted by the Somigliana equation (after Carlo Somigliana):

${\displaystyle g(\varphi )=G_{e}\left[{\frac {1+k\sin ^{2}\varphi }{\sqrt {1-e^{2}\sin ^{2}\varphi }}}\right],\,\!}$

where,

• ${\displaystyle e^{2}={\frac {a^{2}-b^{2}}{a^{2}}}}$ is the spheroid's eccentricity, squared;
• ${\displaystyle G_{e},G_{p}}$ is the defined gravity at the equator and poles, respectively;
• ${\displaystyle k={\frac {bG_{p}-aG_{e}}{aG_{e}}}}$ (formula constant);

For Earth, ${\displaystyle G_{e}}$ = 9.7803253359 ms⁻²; ${\displaystyle G_{p}}$ = 9.8321849378 ms⁻²; k = 0.00193185265241 ; e² = 0.00669437999013:[16] [17]

${\displaystyle g(\varphi )=9.7803253359\left[{\frac {1+0.00193185265241\sin ^{2}\varphi }{\sqrt {1-0.00669437999013\sin ^{2}\varphi }}}\right]\,\mathrm {ms} ^{-2}.}$

Geodesy

See also: Theoretical gravity

The spheroidal shape of the Earth is the result of the interplay between gravity and centrifugal force caused by the Earth's rotation about its axis.[18][19] In his Principia, Newton proposed the equilibrium shape of a homogeneous rotating Earth was a rotational ellipsoid with a flattening f given by 1/230.[20][21] As a result, gravity increases from the equator to the poles. By applying Clairaut's theorem, Laplace found from 15 gravity values that f = 1/330. A modern estimate is 1/298.25642.[22] See Figure of the Earth for more detail.

For a detailed account of the construction of the reference Earth model of geodesy, see Chatfield.[23]

References

1. Théorie de la figure de la terre, tirée des principes de l'hydrostatique (Theory of the shape of the earth, drawn from the principles of hydrostatics) From the catalogue of the scientific books in the library of the Royal Society.
2. Wolfgang Torge (2001). Geodesy: An Introduction (3rd ed.). Walter de Gruyter. p. 10. ISBN 3-11-017072-8.
3. Edward John Routh (2001). A Treatise on Analytical Statics with Numerous Examples. Vol. 2. Adamant Media Corporation. p. 154. ISBN 1-4021-7320-2. A reprint of the original work published in 1908 by Cambridge University Press.
4. Poynting, John Henry; Joseph John Thompson (1907). A Textbook of Physics, 4th Ed. London: Charles Griffin & Co. p. 20.
5. Victor F., Lenzen; Robert P. Multauf (1964). "Paper 44: Development of gravity pendulums in the 19th century". United States National Museum Bulletin 240: Contributions from the Museum of History and Technology reprinted in Bulletin of the Smithsonian Institution. Washington: Smithsonian Institution Press. p. 307. Retrieved 2009-01-28.
6. Newton, Isaac. Principia, Book III, Proposition XIX, Problem III.
7. Greenburg, John (1995). The Problem of the Earth's Shape from Newton to Clairaut. New York: Cambridge University Press. pp. 132. ISBN 0-521-38541-5.
8. Clairaut, Alexis; Colson, John (1737). "An Inquiry concerning the Figure of Such Planets as Revolve about an Axis, Supposing the Density Continually to Vary, from the Centre towards the Surface". Philosophical Transactions. JSTOR 103921.
9. W. W. Rouse Ball A Short Account of the History of Mathematics (4th edition, 1908)
10. Walter William Rouse Ball (1901). A short account of the history of mathematics (3rd ed.). Macmillan. p. 384. A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
11. Poynting, John Henry; Joseph John Thompson (1907). A Textbook of Physics, 4th Ed. London: Charles Griffin & Co. pp. 22–23.
12. Isaac Todhunter. A History of the Mathematical Theories of Attraction and the Figure of the Earth from the Time of Newton to that of Laplace. Vol. 2. Elibron Classics. ISBN 1-4021-1717-5. Reprint of the original edition of 1873 published by Macmillan and Co.
13. Osmond Fisher (1889). Physics of the Earth's Crust. Macmillan and Co. p. 27.
14. John Henry Poynting; Joseph John Thomson (1907). A Textbook of Physics. C. Griffin. p. 22. Clairaut's theorem.
15. NASA case file On the equilibrium figure of the earth by Mohammad A. Khan (1968)
16. Department of Defense World Geodetic System 1984 ― Its Definition and Relationships with Local Geodetic Systems, NIMA TR8350.2, 3rd ed., Tbl. 3.4, Eq. 4-1
17. Eq. 2.57 in MIT Essentials of Geophysics OpenCourseWare notes
18. John P. Vinti; Gim J. Der; Nino L. Bonavito (1998). Orbital and Celestial Mechanics. Progress in astronautics and aeronautics, v. 177. American Institute of Aeronautics and Astronautics. p. 171. ISBN 1-56347-256-2.
19. Arthur Gordon Webster (1904). The Dynamics of Particles and of Rigid, Elastic, and Fluid Bodies: being lectures on mathematical physics. B.G. Teubner. p. 468.
20. Isaac Newton: Principia Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation.
21. See the Principia on line at Andrew Motte Translation
22. Table 1.1 IERS Numerical Standards (2003))
23. Averil B. Chatfield (1997). Fundamentals of High Accuracy Inertial Navigation. Volume 174 in Progress in Astronautics and Aeronautics. American Institute of Aeronautics and Astronautics. Chapter 1, Part VIII p. 7. ISBN 1-56347-243-0.