Editing Clairaut's theorem (gravity)

{{short description|Theorem about gravity}}
{{other uses|Clairaut's formula (disambiguation)}}
[[File:Elipsoid zplostely.png|thumb|200px |Figure 1: An ellipsoid]]
[[File:Gnuplot ellipsoid.svg|thumb|200px|Figure 2: Wireframe rendering of an ellipsoid (oblate spheroid)]]

'''Clairaut's theorem''' characterizes the surface gravity on a viscous rotating [[ellipsoid]] in [[hydrostatic equilibrium]] under the action of its gravitational field and centrifugal force.  It was published in 1743 by [[Alexis Clairaut|Alexis Claude Clairaut]] in a treatise<ref name=RoyalSoc>''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'') [https://archive.org/details/cataloguescient01librgoog/page/n144 <!-- pg=134 quote="Théorie de la figure de la terre". --> From the catalogue of the scientific books in the library of the Royal Society.]</ref> which synthesized physical and geodetic evidence that the Earth is an oblate rotational [[ellipsoid]].<ref name= Torge>{{cite book |title=Geodesy: An Introduction |edition=3rd |author=Wolfgang Torge |page=10 |url=https://books.google.com/books?id=pFO6VB_czRYC&dq=%22Clairaut%27s+theorem%22&pg=PA10 |isbn=3-11-017072-8 |year=2001 |publisher=[[Walter de Gruyter]]   }}</ref><ref name=Routh>{{cite book |author=Edward John Routh |title=A Treatise on Analytical Statics with Numerous Examples |page=154  |year=2001|isbn=1-4021-7320-2 |publisher=Adamant Media Corporation |volume=2 |url=https://books.google.com/books?id=yKmdk4LZxhMC&pg=PA154 }} A reprint of the original work published in 1908 by Cambridge University Press.</ref> 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]].

== 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 {{frac|2|1|2}} minutes per day compared to its rate at Paris.<ref>{{cite book
  | last = Poynting
  | first = John Henry
  |author2=Joseph John Thompson
  | title = A Textbook of Physics, 4th Ed.
  | publisher = Charles Griffin & Co.
  | year = 1907
  | location = London
  | page = [https://archive.org/details/atextbookphysic01thomgoog/page/n29 20]
  | url = https://archive.org/details/atextbookphysic01thomgoog
  }}</ref><ref name="Lenzen1964">{{cite conference
  | first = Lenzen
  | last = Victor F.
  |author2=Robert P. Multauf
  | title = Paper 44: Development of gravity pendulums in the 19th century
  | book-title = United States National Museum Bulletin 240: Contributions from the Museum of History and Technology reprinted in Bulletin of the Smithsonian Institution
  | pages = 307
  | publisher = [[Smithsonian Institution Press]]
  | year = 1964
  | location = Washington
  | url = https://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA307
  | access-date = 2009-01-28}}</ref>  This indicated the [[Gravitational acceleration|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 Philosophiae Naturalis|Principia Mathematica]]'' (1687) in which he outlined his theory and calculations on the shape of the Earth.<ref>[https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookIII-Prop2 Propositions X-XXIV (Motions of celestial bodies and the sea)], Propositions XIX and XX. [https://la.wikisource.org/wiki/Philosophiae_Naturalis_Principia_Mathematica/Liber_III Original Latin].</ref> Newton theorized correctly that the Earth was not precisely a sphere but had an [[Oblate spheroidal coordinates|oblate]] [[ellipsoid]]al shape, slightly flattened at the poles due to the [[centrifugal force]] of its rotation. Using geometric calculations, he gave a concrete argument as to the hypothetical ellipsoid shape of the Earth.<ref>{{Cite book|title = Principia, Book III, Proposition XIX, Problem III|last = Newton|first = Isaac}}</ref>

The goal of ''[[Principia Mathematica Philosophiae Naturalis|Principia]]'' was not to provide exact answers 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 (Finland)|Lapland]] in an attempt to accurately measure a [[meridian arc]].  From such measurements they could calculate the [[eccentricity (mathematics)|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 that his calculations were in error, and he wrote a letter to the [[Royal Society|Royal Society of London]] with his findings.<ref>{{Cite book|title = The Problem of the Earth's Shape from Newton to Clairaut|last = Greenburg|first = John|publisher = [[Cambridge University Press]]|year = 1995|isbn = 0-521-38541-5|location = New York|pages = [https://archive.org/details/problemofearthss1995gree/page/132 132]|url = https://archive.org/details/problemofearthss1995gree/page/132}}</ref> The society published an article in [[Philosophical Transactions of the Royal Society|Philosophical Transactions]] the following year, 1737.<ref>{{Cite journal|last1 = Clairaut|first1 = Alexis|last2 = Colson|first2 = John|date = 1737|title = 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|jstor = 103921|journal = Philosophical Transactions}}</ref> In it Clairaut pointed out (Section XVIII) that Newton's Proposition XX of Book 3 does not apply to the real earth. It stated that the weight of an object at some point in the earth depended only on the proportion of its distance from the centre of the earth to the distance from the centre to the surface at or above the object, so that the total weight of a column of water at the centre of the earth would be the same no matter in which direction the column went up to the surface. Newton had in fact said that this was on the assumption that the matter inside the earth was of a uniform density (in Proposition XIX). Newton realized that the density was probably not uniform, and proposed this as an explanation for why gravity measurements found a greater difference between polar regions and equatorial regions than what his theory predicted. However, he also thought this would mean the equator was further from the centre than what his theory predicted, and Clairaut points out that the opposite is true. Clairaut points out at the beginning of his article that Newton did not explain why he thought the earth was ellipsoid rather than like some other oval, but that Clairaut, and [[James Stirling (mathematician)|James Stirling]] almost simultaneously, had shown why the earth should be an ellipsoid in 1736.

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 theorem says that the acceleration due to gravity ''g'' (including the effect of centrifugal force) on the surface of a spheroid in [[hydrostatic equilibrium]] (being a fluid or having been a fluid in the past, or having a surface near sea level) at latitude {{mvar|φ}} is:<ref name=Ball>[http://www.maths.tcd.ie/pub/HistMath/People/Clairaut/RouseBall/RB_Clairaut.html W. W. Rouse Ball ''A Short Account of the History of Mathematics'' (4th edition, 1908)]</ref><ref name=Rouse2>{{cite book |title=A short account of the history of mathematics | author= Walter William Rouse Ball |page=[https://archive.org/details/ashortaccounthi00ballgoog/page/n359 384] | url=https://archive.org/details/ashortaccounthi00ballgoog | quote=A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball. |year=1901 |publisher=Macmillan |edition=3rd }}</ref>

<math display="block"> g(\varphi) = G_e \left[ 1 + \left(\frac{5}{2} m - f\right) \sin^2 \varphi \right] \, , </math>where 

* <math>G_e</math> is the value of the acceleration of gravity at the equator, 
* <math>m</math> the ratio of the centrifugal force to gravity <math display="inline">\omega^2R_E(\phi=0)/g(\phi=0)</math> at the equator, and 
* <math>f</math> the [[flattening]] of a [[meridian (geography)|meridian]] section of the earth, defined as:

<math display="block">f = \frac {a-b}{a} \, , </math>(where ''a'' = semi-major axis, ''b'' = semi-minor axis).

The contribution of the centrifugal force is approximately <math display="inline">-G_e m\cos^2\varphi,</math> whereas gravitational attraction itself varies approximately as <math display="inline">G_e\left(\frac32 m-f\right)\sin^2\varphi.</math> This formula holds when the surface is perpendicular to the direction of gravity (including centrifugal force), even if (as usually) the density is not constant. (In which case the gravitational attraction can be calculated at any point from the shape alone, without reference to <math>m</math>.) For the earth, <math display="inline">m\approx 1/289,</math> and <math display="inline">\frac52 m\approx 1/116,</math> while <math display="inline">f\approx 1/300,</math> so <math>g</math> is greater at the poles (5.3‰, 0.43% Newton) than on the equator.<ref name="Stokes" />

Clairaut derived the formula under the assumption that the body was composed of concentric coaxial spheroidal layers of constant density.<ref>{{cite book | last = Poynting | first = John Henry |author2=Joseph John Thompson | title = A Textbook of Physics | edition = 4th | publisher = Charles Griffin & Co. | year = 1907 | location = London | pages = [https://archive.org/details/atextbookphysic01thomgoog/page/n32 22]–23 | url = https://archive.org/details/atextbookphysic01thomgoog }}</ref> This work was subsequently pursued by [[Pierre-Simon Laplace|Laplace]], who assumed surfaces of equal density which were nearly spherical.<ref name="Stokes" /><ref name="Todhunter">{{cite book |author=Isaac Todhunter |title=A History of the Mathematical Theories of Attraction and the Figure of the Earth from the Time of Newton to that of Laplace |date=January 1999 |volume=2 |publisher=Elibron Classics |isbn=1-4021-1717-5 |url=https://books.google.com/books?id=blZ_Tar9IRMC&dq=%22Clairaut%27s+theorem%22&pg=PA62 }} Reprint of the original edition of 1873 published by Macmillan and Co.</ref> The English mathematician [[Sir George Stokes, 1st Baronet|George Stokes]] showed in 1849<ref name="Stokes">{{cite journal |last1=Stokes |first1=G. G. |title=On attractions, and on Clairaut's theorem |journal=The Cambridge and Dublin Mathematical Journal |date=1849 |volume=4 |pages=194–219 |url=https://archive.org/stream/cambridgeanddub03unkngoog#page/n201/mode/2up}}</ref> that the theorem applied to any law of density so long as the external surface is a spheroid of equilibrium.<ref name="Fisher">{{cite book |title=Physics of the Earth's Crust |author=Osmond Fisher |page=27 |url=https://books.google.com/books?id=o8oPAAAAIAAJ&dq=%22Clairaut%27s+theorem%22&pg=PA27 |year=1889 |publisher=[[Macmillan and Co.]] }}</ref><ref name="Poynting">{{cite book |title=A Textbook of Physics |author1=John Henry Poynting |author2=Joseph John Thomson |url=https://archive.org/details/bub_gb_TL4KAAAAIAAJ |quote=Clairaut's theorem. |page=[https://archive.org/details/bub_gb_TL4KAAAAIAAJ/page/n32 22] |year=1907 |publisher=C. Griffin }}</ref> A history of more recent developments and more detailed equations for ''g'' can be found in Khan.<ref name="Khan">Mohammad A. Khan (1968) [https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690003446_1969003446.pdf NASA case file ''On the equilibrium figure of the earth'']. ''ntrs.nasa.gov/archive'' [https://web.archive.org/web/20100528083748/https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690003446_1969003446.pdf Archived] on 2010-05-28.</ref>

The above expression for ''g'' has been supplanted by the [[Somigliana equation]] (after [[Carlo Somigliana]]).

==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.<ref name=Vinti>{{cite book |title=Orbital and Celestial Mechanics |series=Progress in astronautics and aeronautics, v. 177 |author1=John P. Vinti |author2=Gim J. Der |author3=Nino L. Bonavito |page=171 |url=https://books.google.com/books?id=-dXzdYHvPgMC&dq=Earth+spheroid+centrifugal+date:1990-2008&pg=PA171 |isbn=1-56347-256-2 |year=1998 |publisher=[[American Institute of Aeronautics and Astronautics]]}}</ref><ref name=Webster>{{cite book |title=The Dynamics of Particles and of Rigid, Elastic, and Fluid Bodies: being lectures on mathematical physics |author=Arthur Gordon Webster |year=1904 |publisher=[[B.G. Teubner]] |url=https://archive.org/details/dynamicsparticl03websgoog |page=[https://archive.org/details/dynamicsparticl03websgoog/page/n488 468] }}</ref> In his ''Principia'', [[Isaac Newton|Newton]] proposed the equilibrium shape of a homogeneous rotating Earth is a rotational ellipsoid with a flattening ''f'' given by 1/230.<ref name=Newton>Isaac Newton: ''Principia'' Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation.</ref><ref name="Principia">{{Cite book |last=Newton |first=Isaac |url=https://archive.org/details/newtonspmathema00newtrich |title=Newton's Principia : the mathematical principles of natural philosophy |last2=Chittenden |first2=N. W. (Life of Sir Isaac Newton) |last3=Adee publ. |first3=Daniel |last4=Motte transl. |first4=Andrew |last5=Hill |first5=Theodore Preston |date= |publisher=Early American mathematics books CU-BANC |others=University of California Libraries |year=1846 |location=New York}}</ref>  As a result, gravity increases from the equator to the poles. By applying Clairaut's theorem, [[Pierre-Simon Laplace|Laplace]] found from 15 gravity values that ''f'' = 1/330: ''g''<sub>NP</sub>/''G''<sub>eq</sub> − 1 = 5.6‰. A modern estimate is 1:298.25642.<ref>[https://tai.bipm.org/iers/convupdt/chapter1/icc1.pdf IERS Numerical Standards (2003)] ''tai.bipm.org'' Table 1.1</ref> See [[Figure of the Earth]] for more detail.

For a detailed account of the construction of the [[Reference ellipsoid|reference Earth model]] of geodesy, see Chatfield.<ref name=Chatfield>{{cite book |title= Fundamentals of High Accuracy Inertial Navigation |url=https://books.google.com/books?id=2hJTDpT2U1UC&dq=frame+coordinate+%22state+of+motion%22&pg=PA7
|isbn=1-56347-243-0 |year=1997 |author=Averil B. Chatfield |publisher=American Institute of Aeronautics and Astronautics  |series=Volume 174 in ''Progress in Astronautics and Aeronautics'' |no-pp= true |pages= Chapter 1, Part VIII p. 7  }}</ref>

==References==
{{reflist}}

[[Category:Eponymous theorems of physics]]
[[Category:Geodesy]]
[[Category:Navigation]]
[[Category:Surveying]]
[[Category:Gravimetry]]
[[Category:Ellipsoids]]