Theoretical gravity

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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

 \ g_{\phi}= 9.780327\left( 1 + 0.00516323\sin^2(\phi) + 0.00002269\sin^4(\phi) \right)\,\frac{\mathrm{m}}{\mathrm{s}^2}

The term 0.00516323 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.

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