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In [[geodesy]] and [[geophysics]], '''theoretical gravity''' is a means to compare the true gravity on the [[Earth's surface]] with a physically smoothed model. The most common model of a smoothed Earth is the [[Earth ellipsoid]].


Despite the fact that the exact density layers in the [[Earth's interior]] are still unknown, the theoretical [[gravity]] ''g'' of its  [[level surface]] can be computed quite easily by using the ''International Gravity Formula''. This refers to a mean [[Earth ellipsoid]], the parameters of which are set by international convention. It shows the gravity at a smoothed [[Earth's surface]] as a function of [[geographic latitude]] ''φ''; the actual formula is


:<math> \ g_{\phi}= \left( 9.780327 + 0.0516323\sin^2(\phi) + 0.0002269\sin^4(\phi) \right)\,\frac{\mathrm{m}}{\mathrm{s}^2} </math>
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The term 0.0516323 is called [[gravity flattening]] (abbreviated ''&beta;''). As a physically defined form parameter it corresponds to the geometrical [[flattening]] ''f'' of the earth ellipsoid.
 
Up to the 1960s, the formula either of the [[Hayford ellipsoid]] (1924) or of the famous German geodesist [[Helmert]] (1906) was used. Hayford has an axis difference {{clarify|reason=Do you mean the position of the axis of symmetry?|date=October 2011}} to modern values of {{val|250|u=m}}, Helmert only {{val|70|u=m}}. The Helmert formula is
:<math> \ g_{\phi}= \left(9.8061999 - 0.0259296\cos(2\phi) + 0.0000567\cos^2(2\phi)\right)\,\frac{\mathrm{m}}{\mathrm{s}^2} </math>
 
A slightly different formula for ''g'' as a function of latitude is the WGS ([[World Geodetic System]]) 1984 Ellipsoidal Gravity Formula:
:<math> \ g_{\phi}= \left(9.7803267714 ~ \frac {1 + 0.00193185138639\sin^2\phi}{\sqrt{1 - 0.00669437999013\sin^2\phi}} \right)\,\frac{\mathrm{m}}{\mathrm{s}^2} </math>
 
The difference between the WGS-84 formula and Helmert's equation is less than 0.68 [[Parts per million|ppm]] or {{val|6.8|e=-7|u=m·s<sup>−2</sup>}}.
 
==See also==
*[[Gravity anomaly]]
*[[Reference ellipsoid]]
 
== Literature ==
* [[Karl Ledersteger]]: ''Astronomische und [[physical geodesy|physikalische Geodäsie]]''. Handbuch der Vermessungskunde Band 5, 10. Auflage. Metzler, Stuttgart 1969
* B.Hofmann-Wellenhof, [[Helmut Moritz]]: ''Physical Geodesy'', ISBN 3-211-23584-1, Springer-Verlag Wien 2006.
 
[[Category:Gravimetry]]
 
 
{{geophysics-stub}}

Latest revision as of 17:57, 12 October 2014


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