User:Cffk/sandbox/Theoretical Gravity



In geodesy and geophysics, theoretical gravity or normal gravity is an exact solution for gravity for an idealized model of the Earth. In this model all the mass in contained within an ellipsoid of revolution which rotates about its polar axis. The mass distribution is such that gravity is normal to the surface of the ellipsoid; i.e., gravity potential is constant on the ellipoidal &mdash; it is a level ellipsoid. Theoretical gravity is the underlying model for more accurate models of the Earth's gravity.

In this article, the term gravity refers to the sum of gravitational attraction and the centrifugal force. The exposition below is taken from
 * W. A. Heiskanen and H. Moritz, Physical Geodesy (Freeman, San Francisco, 1967).

The theoretical gravity model is specified by 4 parameters
 * the mass M within the ellipsoid,
 * the equatorial radius a of the ellipsoid,
 * the polar semi-axis b of the ellipsoid,
 * the rotation rate &omega; of the ellipsoid.

The solution of the for potential was found by Somigliana (1929) and is expressed in ellipsoidal coordinates, u, &beta;, &lambda;. These are related to Cartesian coordinates, X, Y, Z, by

\begin{align} X &= \sqrt{u^2 + E^2} \cos\beta \cos\lambda, \\ Y &= \sqrt{u^2 + E^2} \cos\beta \sin\lambda, \\ Z &= u \sin\beta. \\ \end{align} $$ where
 * $$ E = \sqrt{a^2-b^2}. $$

The level ellipsoid is given by u = b; on the ellipsoid &beta; is the parametric latitude; &lambda; is the longitude.

The normal potential exterior to the ellipsoid is given by

U(u,\beta) = \frac{GM}E \tan^{-1}\frac Eu + \frac12 \omega^2 a^2 \frac q{q_0} \biggl( \sin^2\beta - \frac13 \biggr) + \frac 12 \omega^2(u^2 + E^2) \cos^2\beta, $$ where

\begin{align} q &= \frac12 \biggl[\biggl(1 + 3\frac{u^2}{E^2}\biggr) \tan^{-1}\frac Eu - 3 \frac uE\biggr],\\ q_0 &= \frac12 \biggl[\biggl(1 + 3\frac{b^2}{E^2}\biggr) \tan^{-1}\frac Eb - 3 \frac bE\biggr]. \end{align} $$

The acceleration due to gravity is given by
 * $$ \gamma = \nabla U. $$

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

where,


 * $$a,b$$ are the equatorial and polar semi-axes, respectively;
 * $$e^2=\frac{a^2-b^2}{a^2}$$ is the spheroid's eccentricity, squared;
 * $$g_e,g_p$$ is the defined gravity at the equator and poles, respectively;
 * $$k=\frac{bg_p-ag_e}{ag_e}$$ (formula constant);

providing,


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

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


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

(where $$g_p$$ = 9.8321849378 ms−2)

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

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.