# Theoretical gravity

In geodesy and geophysics, theoretical gravity or normal gravity is an approximation of the true effective or apparent gravity on Earth's surface by means of a mathematical model representing (a physically smoothed) Earth. The most common model of a smoothed Earth is an Earth ellipsoid, or, more specifically, an Earth spheroid (i.e., an ellipsoid of revolution).

Various, successively more refined, formulas for computing the theoretical gravity are referred to as the International Gravity Formula, the first of which was proposed in 1930 by the International Association of Geodesy. The general shape of that formula is:

${\displaystyle g(\phi )=g_{e}\left(1+A\sin ^{2}(\phi )-B\sin ^{2}(2\phi )\right),}$

in which g(φ) is the gravity as a function of the geographic latitude φ of the position whose gravity is to be determined, ${\displaystyle g_{e}}$ denotes the gravity at the equator (as determined by measurement), and the coefficients A and B are parameters that must be selected to produce a good global fit to true gravity.[1]

Using the values of the GRS80 reference system, a commonly used specific instantiation of the formula above is given by:

${\displaystyle g(\phi )=9.780327\left(1+0.0053024\sin ^{2}(\phi )-0.0000058\sin ^{2}(2\phi )\right)\,\mathrm {ms} ^{-2}.}$[1]

Using the appropriate double-angle formula in combination with the Pythagorean identity, this can be rewritten in the equivalent forms

{\displaystyle {\begin{aligned}g(\phi )&=9.780327\left(1+0.0052792\sin ^{2}(\phi )+0.0000232\sin ^{4}(\phi )\right)\,\mathrm {ms} ^{-2},\\&=9.780327\left(1.0053024-.0053256\cos ^{2}(\phi )+.0000232\cos ^{4}(\phi )\right)\,\mathrm {ms} ^{-2},\\&=9.780327\left(1.0026454-0.0026512\cos(2\phi )+.0000058\cos ^{2}(2\phi )\right)\,\mathrm {ms} ^{-2}.\end{aligned}}\,\!}

Up to the 1960s, formulas based on the Hayford ellipsoid (1924) and of the famous German geodesist Helmert (1906) were often used.[citation needed] The difference between the semi-major axis (equatorial radius) of the Hayford ellipsoid and that of the modern WGS84 ellipsoid is 251 m; for Helmert's ellipsoid it is only 63 m.

A more recent theoretical formula for gravity as a function of latitude is the International Gravity Formula 1980 (IGF80), also based on the WGS80 ellipsoid but now using the Somigliana equation:

${\displaystyle g(\phi )=g_{e}\left[{\frac {1+k\sin ^{2}(\phi )}{\sqrt {1-e^{2}\sin ^{2}(\phi )}}}\right],\,\!}$

where,[2]

• ${\displaystyle a,b}$ are the equatorial and polar semi-axes, respectively;
• ${\displaystyle e^{2}={\frac {a^{2}-b^{2}}{a^{2}}}}$ is the spheroid's eccentricity, squared;
• ${\displaystyle g_{e},g_{p}}$ is the defined gravity at the equator and poles, respectively;
• ${\displaystyle k={\frac {bg_{p}-ag_{e}}{ag_{e}}}}$ (formula constant);

providing,

${\displaystyle g(\phi )=9.7803267715\left[{\frac {1+0.001931851353\sin ^{2}(\phi )}{\sqrt {1-0.0066943800229\sin ^{2}(\phi )}}}\right]\,\mathrm {ms} ^{-2}.}$[1]

A later refinement, based on the WGS84 ellipsoid, is the WGS (World Geodetic System) 1984 Ellipsoidal Gravity Formula:[2]

${\displaystyle g(\phi )=9.7803253359\left[{\frac {1+0.00193185265241\sin ^{2}(\phi )}{\sqrt {1-0.00669437999013\sin ^{2}(\phi )}}}\right]\,\mathrm {ms} ^{-2}.}$

(where ${\displaystyle g_{p}}$ = 9.8321849378 ms−2)

The difference with IGF80 is insignificant when used for geophysical purposes,[1] but may be significant for other uses.