Flattening

"Ellipticity" redirects here. For ellipticity in differential calculus, see elliptic operator.

This article is about geometry. For psychopathology, see flattening of affect.

The flattening, ellipticity, or oblateness of an oblate spheroid is a measure of the "squashing" of the spheroid's pole, towards its equator. If a is the distance from the spheroid center to the equator and b the distance from the center to the pole then

$\mathrm{flattening} = \frac {a - b}{a}$

Definition of flattening

First order

The first, primary flattening, ƒ, is the versine of the spheroid's angular eccentricity α, equalling the relative difference between its equatorial radius, a, and its polar radius, b:

$f=\mathrm{ver}(\alpha)=2\sin^2\left(\frac{\alpha}{2}\right)=1-\cos(\alpha)=\frac{a-b}{a};\,\!$

Second and third orders

There is also a second flattening, f' ,

$f'=\frac{2\sin^2(\alpha/2)}{1-2\sin^2(\alpha/2)}=\frac{a-b}{b}$

and a third flattening,^[1][2] f' ' (sometimes denoted as "n" – a notation first used in 1837 by Friedrich Bessel on calculation of meridian arc length^[3] – that is the squared half-angle tangent of α:

$f''=n=\tan^2\left(\frac{\alpha}{2}\right)=\frac{1-\cos(\alpha)}{1+\cos(\alpha)}=\frac{a-b}{a+b};\,\!$

First order flattening of planets

• The flattening of the smoothed Earth's surface in the World Geodetic System (WGS-84) is 1:298.257223563 (which corresponds to a radius difference of 21.385 km (13 mi) 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 amount of flattening depends on

• the relation between gravity and centrifugal force;

and in detail on

• size and density of the celestial body (see Figure of the Earth);
• the rotation of the planet or star;
• and the elasticity of the body.

See also

• Earth ellipsoid, Gravity flattening
• Equatorial bulge, Earth rotation
• Astronomy, Planetology
• Gravitational field, Gravity formula

References

1. ^ 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
2. ^ Ганьшин, В. Н. (1967): Геометрия земного эллипсоида, Издательство «Недра», Москва
3. ^ 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