Clairaut's Theorem

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Clairaut's Theorem is useful to calculate the gravitational acceleration at any point on the Earth provided its latitude is known. Here the shape of the Earth is considered oblate spheroid.

Transcript of Clairaut's Theorem

  • 10/26/2015 Clairaut'stheoremWikipedia,thefreeencyclopedia

    https://en.wikipedia.org/wiki/Clairaut%27s_theorem 1/3

    Figure1:Anellipsoid

    Figure2:Wireframerenderingofanellipsoid(oblatespheroid)

    Clairaut'stheoremFromWikipedia,thefreeencyclopedia

    NottobeconfusedwithClairaut'srelation.Forthetheoremconcerningsymmetryofsecondderivativesofamapping ,seeSymmetryofsecondderivatives.

    Clairaut'stheoremisageneralmathematicallawapplyingtospheroidsofrevolution.Publishedin1743byAlexisClaudeClairautinhisThoriedelafiguredelaterre,tiredesprincipesdel'hydrostatique,[1]whichsynthesizedphysicalandgeodeticevidencethattheEarthisanoblaterotationalellipsoid,[2][3]itwasinitiallyusedtorelatethegravityatanypointontheEarth'ssurfacetothepositionofthatpoint,allowingtheellipticityoftheEarthtobecalculatedfrommeasurementsofgravityatdifferentlatitudes.

    Contents

    1Formula1.1Somiglianaequation

    2Geodesy3References

    Formula

    Clairaut'sformulafortheaccelerationduetogravitygonthesurfaceofaspheroidatlatitude,was:[4][5]

    where isthevalueoftheaccelerationofgravityattheequator,mtheratioofthecentrifugalforcetogravityattheequator,andftheflatteningofameridiansectionoftheearth,definedas:

    (wherea=semimajoraxis,b=semiminoraxis).

    Clairautderivedtheformulaundertheassumptionthatthebodywascomposedofconcentriccoaxialspheroidallayersofconstantdensity.[6]ThisworkwassubsequentlypursuedbyLaplace,whorelaxedtheinitialassumptionthatsurfacesofequaldensitywerespheroids.[7]Stokesshowedin1849thatthetheoremappliedtoanylawofdensitysolongastheexternalsurfaceisaspheroidofequilibrium.[8][9]Ahistoryofthesubject,andmoredetailedequationsforgcanbefoundinKhan.[10]

    Somiglianaequation

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  • 10/26/2015 Clairaut'stheoremWikipedia,thefreeencyclopedia

    https://en.wikipedia.org/wiki/Clairaut%27s_theorem 2/3

    TheaboveexpressionforghasbeensupplantedbytheSomiglianaequation(afterCarloSomigliana):

    where,

    isthespheroid'seccentricity,squared

    isthedefinedgravityattheequatorandpoles,respectively

    (formulaconstant)

    ForEarth, =9.7803253359ms2 =9.8321849378ms2k=0.00193185265241e2=0.00669437999013:[11][12]

    Geodesy

    Seealso:Theoreticalgravity

    ThespheroidalshapeoftheEarthistheresultoftheinterplaybetweengravityandcentrifugalforcecausedbytheEarth'srotationaboutitsaxis.[13][14]InhisPrincipia,NewtonproposedtheequilibriumshapeofahomogeneousrotatingEarthwasarotationalellipsoidwithaflatteningfgivenby1/230.[15][16]Asaresultgravityincreasesfromtheequatortothepoles.ByapplyingClairaut'stheorem,Laplacewasabletodeducefrom15gravityvaluesthatf=1/330.Amodernestimateis1/298.25642.[17]SeeFigureoftheEarthformoredetail.

    ForadetailedaccountoftheconstructionofthereferenceEarthmodelofgeodesy,seeChatfield.[18]

    References1. FromthecatalogueofthescientificbooksinthelibraryoftheRoyalSociety.

    (http://books.google.com/books?id=3owAAAAAYAAJ&pg=PA134&lpg=PA134&dq=%22Th%C3%A9orie+de+la+figure+de+la+terre%22&source=web&ots=an0JWH3C8&sig=BMkuXfZEsK3p0tzrZ1Jvfcy7hmw&hl=en&sa=X&oi=book_result&resnum=10&ct=result)

    2. WolfgangTorge(2001).Geodesy:AnIntroduction(3rded.).WalterdeGruyter.p.10.ISBN3110170728.3. EdwardJohnRouth(2001).ATreatiseonAnalyticalStaticswithNumerousExamples.Vol.2.Adamant

    MediaCorporation.p.154.ISBN1402173202.Areprintoftheoriginalworkpublishedin1908byCambridgeUniversityPress.

    4. W.W.RouseBallAShortAccountoftheHistoryofMathematics(4thedition,1908)(http://www.maths.tcd.ie/pub/HistMath/People/Clairaut/RouseBall/RB_Clairaut.html)

    5. WalterWilliamRouseBall(1901).Ashortaccountofthehistoryofmathematics(3rded.).Macmillan.p.384.

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  • 10/26/2015 Clairaut'stheoremWikipedia,thefreeencyclopedia

    https://en.wikipedia.org/wiki/Clairaut%27s_theorem 3/3

    6. Poynting,JohnHenryJosephJohnThompson(1907).ATextbookofPhysics,4thEd.London:CharlesGriffin&Co.pp.2223.

    7. IsaacTodhunter.AHistoryoftheMathematicalTheoriesofAttractionandtheFigureoftheEarthfromtheTimeofNewtontothatofLaplace.Vol.2.ElibronClassics.ISBN1402117175.Reprintoftheoriginaleditionof1873publishedbyMacmillanandCo.

    8. OsmondFisher(1889).PhysicsoftheEarth'sCrust.MacmillanandCo.p.27.9. JohnHenryPoynting&JosephJohnThomson(1907).ATextbookofPhysics.C.Griffin.p.22.

    10. NASAcasefileOntheequilibriumfigureoftheearthbyMohammadA.Khan(1968)(http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690003446_1969003446.pdf)

    11. DepartmentofDefenseWorldGeodeticSystem1984ItsDefinitionandRelationshipswithLocalGeodeticSystems,NIMATR8350.2,3rded.,Tbl.3.4,Eq.41(http://earthinfo.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf)

    12. Eq.2.57inMITEssentialsofGeophysicsOpenCourseWarenotes(http://ocw.mit.edu/courses/earthatmosphericandplanetarysciences/12201essentialsofgeophysicsfall2004/lecturenotes/ch2.pdf)

    13. JohnP.Vinti,GimJ.Der,NinoL.Bonavito(1998).OrbitalandCelestialMechanics.Progressinastronauticsandaeronautics,v.177.AmericanInstituteofAeronauticsandAstronautics.p.171.ISBN1563472562.

    14. ArthurGordonWebster(1904).TheDynamicsofParticlesandofRigid,Elastic,andFluidBodies:beinglecturesonmathematicalphysics.B.G.Teubner.p.468.

    15. IsaacNewton:PrincipiaBookIIIPropositionXIXProblemIII,p.407inAndrewMottetranslation.16. SeethePrincipiaonlineatAndrewMotteTranslation

    (http://ia310114.us.archive.org/2/items/newtonspmathema00newtrich/newtonspmathema00newtrich.pdf)17. Table1.1IERSNumericalStandards(2003)(ftp://tai.bipm.org/iers/convupdt/chapter1/icc1.pdf))18. AverilB.Chatfield(1997).FundamentalsofHighAccuracyInertialNavigation.Volume174inProgressin

    AstronauticsandAeronautics.AmericanInstituteofAeronauticsandAstronautics.Chapter1,PartVIIIp.7.ISBN1563472430.

    Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Clairaut%27s_theorem&oldid=685926789"

    Categories: Geodesy GlobalPositioningSystem Navigation Surveying PhysicstheoremsGravimetry

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