Theoretical gravity


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

Formulas

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:
in which is the gravity as a function of the geographic latitude φ of the position whose gravity is to be determined, denotes the gravity at the equator, and the coefficients and are parameters that must be selected to produce a good global fit to true gravity.
Using the values of the GRS80 reference system, a commonly used specific instantiation of the formula above is given by:
Using the appropriate double-angle formula in combination with the Pythagorean identity, this can be rewritten in the equivalent forms
Up to the 1960s, formulas based on the Hayford ellipsoid and of the famous German geodesist Helmert were often used. The difference between the semi-major axis of the Hayford ellipsoid and that of the modern WGS84 ellipsoid is ; for Helmert's ellipsoid it is only.
A more recent theoretical formula for gravity as a function of latitude is the International Gravity Formula 1980, also based on the WGS80 ellipsoid but now using the Somigliana equation:
where,
providing,
A later refinement, based on the WGS84 ellipsoid, is the WGS 1984 Ellipsoidal Gravity Formula:
The difference with IGF80 is insignificant when used for geophysical purposes, but may be significant for other uses.

Literature