The most common theoretical model is a rotating Earth ellipsoid of revolution (i.e., a spheroid).
[1] The type of gravity model used for the Earth depends upon the degree of fidelity required for a given problem.
For many problems such as aircraft simulation, it may be sufficient to consider gravity to be a constant, defined as:[2] based upon data from World Geodetic System 1984 (WGS-84), where
If it is desirable to model an object's weight on Earth as a function of latitude, one could use the following:[2]: 41 where Neither of these accounts for changes in gravity with changes in altitude, but the model with the cosine function does take into account the centrifugal relief that is produced by the rotation of the Earth.
On the rotating sphere, the sum of the force of the gravitational field and the centrifugal force yields an angular deviation of approximately (in radians) between the direction of the gravitational field and the direction measured by a plumb line; the plumb line appears to point southwards on the northern hemisphere and northwards on the southern hemisphere.
So the rotational component of change due to latitude (0.35%) is about twice as significant as the mass attraction change due to latitude (0.18%), but both reduce strength of gravity at the equator as compared to gravity at the poles.
For such problems, the rotation of the Earth would be immaterial unless variations with longitude are modeled.
Also, the variation in gravity with altitude becomes important, especially for highly elliptical orbits.
(A shape elongated on its axis of symmetry, like an American football, would be called prolate.)
A similar model adjusted for the geometry and gravitational field for Mars can be found in publication NASA SP-8010.
[4] The barycentric gravitational acceleration at a point in space is given by: where: M is the mass of the attracting object,
For example, the equation above gives the acceleration at 9.820 m/s2, when GM = 3.986 × 1014 m3/s2, and R = 6.371 × 106 m. The centripetal radius is r = R cos(φ), and the centripetal time unit is approximately (day / 2π), reduces this, for r = 5 × 106 metres, to 9.79379 m/s2, which is closer to the observed value.
[citation needed] 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.
For the Geodetic Reference System 1980 (GRS 80) the parameters are set to these values:
The parameters are: In the course of time the values were improved again with newer knowledge and more exact measurement methods.
Harold Jeffreys improved the values in 1948 at: The normal gravity formula of Geodetic Reference System 1967 is defined with the values: From the parameters of GRS 80 comes the classic series expansion: The accuracy is about ±10−6 m/s2.
With GRS 80 the following series expansion is also introduced: As such the parameters are: The accuracy is at about ±10−9 m/s2 exact.
Cassinis determined the height dependence, as: The average rock density ρ is no longer considered.
Since GRS 1967 the dependence on the ellipsoidal elevation h is: Another expression is: with the parameters derived from GRS80: where
In all German standards offices the free-fall acceleration g is calculated in respect to the average latitude φ and the average height above sea level h with the WELMEC–Formel: The formula is based on the International gravity formula from 1967.
The scale of free-fall acceleration at a certain place must be determined with precision measurement of several mechanical magnitudes.