Kappa calculus: Difference between revisions

Revision as of 06:53, 16 September 2013

Template:No footnotes 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

\ g_{\phi }=\left(9.780327+0.0516323\sin ^{2}(\phi )+0.0002269\sin ^{4}(\phi )\right)\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}

The term 0.0516323 is called gravity flattening (abbreviated β). 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 Template:Clarify to modern values of Template:Val, Helmert only Template:Val. The Helmert formula is

\ g_{\phi }=\left(9.8061999-0.0259296\cos(2\phi )+0.0000567\cos ^{2}(2\phi )\right)\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}

A slightly different formula for g as a function of latitude is the WGS (World Geodetic System) 1984 Ellipsoidal Gravity Formula:

\ 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}}}

The difference between the WGS-84 formula and Helmert's equation is less than 0.68 ppm or Template:Val.

Literature

Karl Ledersteger: Astronomische und 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.

Template:Geophysics-stub

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+{{no footnotes|date=November 2011}}
+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]] ''&phi;''; the actual formula is
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+:<math> \ g_{\phi}= \left( 9.780327 + 0.0516323\sin^2(\phi) + 0.0002269\sin^4(\phi) \right)\,\frac{\mathrm{m}}{\mathrm{s}^2} </math>
+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}}

Kappa calculus: Difference between revisions

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