In geodesy, the terms "shape of the Earth" or "figure of the Earth" have different meanings depending on the context they are used in and the precision considered. Strictly speaking, the figure of the solid Earth is determined by the topography of the solid Earth, i.e. the boundary between solid Earth on the one side and atmosphere, ocean, ice sheets, glaciers, lakes, and rivers on the other side. In modern geodesy, the term refers more often to the surface of the solid Earth including the hydrosphere and the cryosphere. It is this surface on which most in situ measurements are carried out.
Today we know that the shape of the Earth is very close to that of a oblate rotational ellipsoid flattened along the axis from pole to pole and elongated in the equatorial (circular) plane. A conventional reference ellipsoid is used to approximate this overall shape (see below). Local topography deviates from this reference ellipsoid by about +8,848 m and about -10,911 m (Figure 3.1). With only 17% of the radius, these deviations are small, and a reference ellipsoid is a good approximation for many studies and mathematical modeling.
 |
Figure 3.1: ETOPO1 Global Relief Model. The figure illustrates the difficulties in the ambiguity in the term "shape of the Earth:" For the oceans, a bottom topography is indicated, while for the ice sheets, the surface of the ice masses is given. In this case, the 'topography' excludes the oceans but includes the major ice masses. Figure from http://www.ngdc.noaa.gov/mgg/global/global.html. |
The actual topography of the solid Earth, with or without oceans, terrestrial hydrosphere, and cryosphere included, is too complex for as a basis for mathematical descriptions of many features and processes in the Earth system. Mathematical descriptions of the actual topography would require a prohibitive amount of computations. Therefore, it makes sense to find approximations of the Earth's figure that are mathematically simple and accurate enough for many applications.
The overall mean shape of the Earth has occupied humans since very early times of our civilizations, with significant impact on philosophy, religion, and our view on the nature of the universe. Since more than 2,000 years, geodesy is inherently connected to the development of ideas of what shape the solid Earth might have. Soffel (1989) identified four main phases of geodesy which are defined by our ideas of the Earth's shape (see Lecture 1 for more details). Table 3.1 summarizes these four phases.
Table 3.1: The four main phases of geodesy reflecting transitions in our view on main features of the Earth's shape.
| Phase and duration | Key characteristics | General idea |
|---|---|---|
| A: From 200 BC up to the middle of the 17th century | Radius of a spherical Earth | Simple geometrical form. |
| B: From the middle of the 17th century to the middle of the 19th century | Oblateness of a rotational ellipsoid | Geometrical form resulting from rotational dynamics. |
| C: From the middle of the 19th century to the middle of the 20th century | Geoid | Gravitational field in addition to a purely geometrical form. |
| D: Since the middle of the 20th century | Dynamics of the Earth's surface and relativistic models of the Earth system | Changes in the shape instead of mean shape; dynamical instead of static view. |
The simple geometrical form approximating best the mean shape of the Earth is a rotational ellipsoid. Earth's rotation leads The oblateness of the Earth was
| Reference ellipsoid name | Equatorial radius (m) | Polar radius (m) | Inverse flattening | Where used |
|---|---|---|---|---|
| Maupertuis (1738) | 6,397,300 | 6,363,806.283 | 191 | France |
| Plessis (1817) | 6,376,523.0 | ??? | 308.64 | France |
| Everest (1830) | 6,377,299.365 | 6,356,098.359 | 300.80172554 | India |
| Everest 1830 Modified (1967) | 6,377,304.063 | 6,356,103.0390 | 300.8017 | West Malaysia & Singapore |
| Everest 1830 (1967 Definition) | 6,377,298.556 | 6,356,097.550 | 300.8017 | Brunei & East Malaysia |
| Airy (1830) | 6,377,563.396 | 6,356,256.909 | 299.3249646 | Britain |
| Bessel (1841) | 6,377,397.155 | 6,356,078.963 | 299.1528128 | Europe, Japan |
| Clarke (1866) | 6,378,206.4 | 6,356,583.8 | 294.9786982 | North America |
| Clarke (1878) | 6,378,190 | 6,356,456 | 293.4659980 | North America |
| Clarke (1880) | 6,378,249.145 | 6,356,514.870 | 293.465 | France, Africa |
| Helmert (1906) | 6,378,200 | 6,356,818.17 | 298.3 | |
| Hayford (1910) | 6,378,388 | 6,356,911.946 | 297 | USA |
| International (1924) | 6,378,388 | 6,356,911.946 | 297 | Europe |
| NAD 27 (1927) | 6,378,206.4 | 6,356,583.800 | 294.978698208 | North America |
| Krassovsky (1940) | 6,378,245 | 6,356,863.019 | 298.3 | Russia |
| WGS66 (1966) | 6,378,145 | 6,356,759.769 | 298.25 | USA/DoD |
| Australian National (1966) | 6,378,160 | 6,356,774.719 | 298.25 | Australia |
| New International (1967) | 6,378,157.5 | 6,356,772.2 | 298.24961539 | |
| GRS-67 (1967) | 6,378,160 | 6,356,774.516 | 298.247167427 | |
| South American (1969) | 6,378,160 | 6,356,774.719 | 298.25 | South America |
| WGS-72 (1972) | 6,378,135 | 6,356,750.52 | 298.26 | USA/DoD |
| GRS-80 (1979) | 6,378,137 | 6,356,752.3141 | 298.257222101 | Global ITRS |
| WGS-84 (1984) | 6,378,137 | 6,356,752.3142 | 298.257223563 | Global GPS |
| IERS (1989) | 6,378,136 | 6,356,751.302 | 298.257 | |
| IERS (2003) | 6,378,136.6 | 6,356,751.9 | 298.25642 |
![V=\frac{GM}{r}\left(1+{\sum_{n=2}^{n_{max}}}\left(\frac{a}{r}\right)^n{\sum_{m=0}^n}
\overline{P}_{nm}(\sin\phi)\left[\overline{C}_{nm}\cos m\lambda+\overline{S}_{nm}\sin m\lambda\right]\right),](http://upload.wikimedia.org/math/c/5/9/c5909be357a6c4898d92ff61c8f54b51.png)