# Flattening

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The **oblateness**, **ellipticity**, or * flattening* of an oblate spheroid, or

**oblatum**, is a measure of the "squashing" of the spheroid's Geographical pole, towards its equator. If is the distance from the spheroid center to the equator——the

**transverse**radius——and the distance from the center to the pole——the

**conjugate**radius——then .

## First, second and third flattening

The first, primary flattening, *f*, is the versine of the spheroid's **angular eccentricity**, :

- The flattening ratio for Earth is 1:298.257223563
^{[1]}(which corresponds to a radius difference of 21.385 km of the Earth radius 6378.137 - 6356.752 km) and would not be realized visually from space, since the difference represents only 0.335 %. - The flattening of Jupiter (1:16) and Saturn (nearly 1:10), in contrast, can be seen even in a small telescope;
- Conversely, that of the Sun is less than 1:1000 and that of the Moon barely 1:900.

- The flattening ratio for Earth is 1:298.257223563

The amount of flattening depends on

- the relation between gravity and centrifugal force;

and in detail on

- size and density of the celestial body;
- the rotation of the planet or star;
- and the elasticity of the body.

There is also a second flattening, *f' *,

and a third flattening^{[2]}^{[3]}, *f''* (more commonly denoted as "*n*" and first used in 1837 by Friedrich Bessel on calculation of meridian arc length^{[4]}), that is the squared half-angle tangent of :

## Prolate valuations

The above formations apply to an ellipse and oblatum, which is an ellipse rotated about its polar, or conjugate, axis, resulting in . If it is rotated about its equatorial, or transverse, axis, it is a prolate spheroid, or prolatum, where .

With a prolatum, and are reversed in all of the flattening formation elements, except for the denominators of *f* and *f' *, which means their function assignments are reversed^{[5]}:

## Notes

- ↑ H. MORITZ (1979): Report of Special Study Group N° 539 of I.A.G., Fundamental Geodetic Constants, presented at XVII General Assembly og I.U.G.G., Canberra. [PDF]
- ↑ König, R. and Weise, K. H. (1951):
*Mathematische Grundlagen der höheren Geodäsie und Kartographie*, Band 1, Das Erdsphäroid und seine konformen Abbildungen, Springer-Verlag, Berlin/Göttingen/Heidelberg, Sec I.1 - ↑ Ганьшин, В. Н. (1967):
*Геометрия земного эллипсоида*, Издательство «Недра», Москва - ↑ Bessel, F. W. (1837): Bestimmung der Axen des elliptischen Rotationssphäroids, welches den vorhandenen Messungen von Meridianbögen der Erde am meisten entspricht, Astronomische Nachrichten,
**14**, 333-346. [PDF] - ↑ MathWorld: Flattening.