Earth ellipsoid
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An Earth ellipsoid is a mathematical figure approximating the shape of the Earth, used as a reference frame for computations in geodesy, astronomy and the geosciences. Various different ellipsoids have been used as approximations.
It is an ellipsoid of rotation, whose short (polar) axis (connecting the two flattest spots called geographical north and south poles) is approximately aligned with the rotation axis of the Earth. The ellipsoid is defined by the equatorial axis a and the polar axis b; their difference is about 21 km or 0,3 per cent. Additional parameters are the mass function J2, the correspondent gravity formula, and the rotation period (usually 86164 seconds).
Many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks. Amongst the different set of data used in national surveys are several of special importance: the Bessel ellipsoid of 1841, the international Hayford ellipsoid of 1924, and (for GPS positioning) the WGS84 ellipsoid.
Historical Earth ellipsoids
The following table lists 9 ellipsoids which (except Clarke's) were the best estimation of the Earth's figure when they were published:
| Name | Equatorial axis (m) | Polar axis (m) | Inverse flattening, \(1/f\,\!\) |
|---|---|---|---|
| Delambre, France 1810 | 6,376,985.0 | 308,6465 | |
| Airy 1830 | 6,377,563.4 | 6,356,256.91 | 299.3249646 |
| Bessel 1841 | 6,377,397.155 | 6,356,078.963 | 299.1528128 |
| Alexander Ross Clarke | 6,378,206.4 | 6,356,583.8 | 294.9786982 |
| Helmert 1906 | 6,378,200.0 | (close to WGS84!) | 298.30 |
| International 1924 | 6,378,388.0 | 6,356,911.9 | 297.0 |
| Krassowski 1940 | 6,378,245.00 | (for Eastern Europe) | 298.3 |
| Internat. 1967 Luzern | 6,378,165.00 | (incl. Sat.Geodesy) | 298.25 |
| WGS 1984 | 6,378,137.00 | 6,356,752.3142 | 298.257223563 |
Mean Earth ellipsoid and reference ellipsoids
A data set which describes the global average of the Earth's surface curvature is called the mean Earth Ellipsoid. It refers to a theoretical coherence between the geographic latitude and the meridional curvature of the geoid. The latter is close to the mean sea level, and therefore an ideal Earth ellipsoid has the same volume as the geoid.
While the mean Earth ellipsoid is the ideal basis of global geodesy, for regional networks a so called reference ellipsoid may be the better choice. When geodetic measurements have to be computed on a mathematical reference surface, this surface should have a similar curvature as the regional geoid. Otherwise the reduction of the measurements would get small distortions.
This is the reason for the "long life" of former reference ellipsoids like the Hayford or the Bessel ellipsoid, despite of the fact that their main axes deviate by several hundred meters from the modern values. Another reason is a juridical one: the coordinates of millions of boundary stones should remain fixed for a long period. If their reference surface would change, the coordinates themselves would also change.
However, for international networks, GPS positioning or astronautics, these regional reasons are less relevant. As the knowledge of Earth's figure is increasingly accurate, the International Geoscientific Union IUGG usually adopts the axes of the Earth ellipsoid to the best available data.
See also
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