Theoretical gravity

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 of 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 by a relative simple formula, which is called the International Gravity Formula. It 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^{[clarification needed]} to modern values of 250 m, Helmert only 70 m. 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 6.8×10⁻⁷ m·s⁻².

See also

• reference ellipsoid
• gravity anomaly

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.