Flattening

A sphere of radius a compressed to an oblatum (top), and the polar radius b “stretched” to create an prolatum (bottom).

The flattening, ellipticity, or oblateness of an oblate spheroid, or oblatum, 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 $a>b\,\!$; if $a<b\,\!$ then the meridians are the object of flattening and the polar radius is “stretched”, resulting in a prolate spheroid, or prolatum:

$\begin{matrix}\mbox{flattening} &=\frac{a-b}{a}\quad\mbox{(oblate)},\\ &=\frac{b-a}{a}\quad\mbox{(prolate)};\end{matrix}\,\!$

First, second and third flattening

Oblate

The first, primary flattening, f, is the versine of the spheroid's angular eccentricity ("$o\!\varepsilon\,\!$"), equaling the relative difference between its equatorial radius, $a\,\!$, and its polar radius, $b\,\!$:

$o\!\varepsilon=\arccos\left(\frac{b}{a}\right)=2\arctan\left(\!\sqrt{\frac{a-b}{a+b}}\;\right);\,\!$

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

There is also a second flattening, f' ,

$f'=\frac{a-b}{b}= \frac{2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}{1-2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}=\sec(o\!\varepsilon)-1=\frac{\mbox{ver}(o\!\varepsilon)}{1-\mbox{ver}(o\!\varepsilon)};\,\!$

and a third flattening, f'' (sometimes denoted as “n”, the notation of which first used in 1837 by Friedrich Bessel on calculation of meridian arc length), that is the squared half-angle tangent of $o\!\varepsilon\,\!$:

$f''=\frac{a-b}{a+b}=\frac{2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}{2-2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}=\tan^2\left(\frac{o\!\varepsilon}{2}\right)=\frac{\mbox{ver}(o\!\varepsilon)}{\mbox{vrc}(o\!\varepsilon)}.\,\!$

Prolate

With prolate valuations, the radii positions in the numerator switch, but the denominators’ remain the same, resulting in f and f' switching values of the equivalent oblate form:

$o\!\varepsilon=\arccos\left(\frac{a}{b}\right)=2\arctan\left(\!\sqrt{\frac{b-a}{b+a}}\;\right);\,\!$

$\begin{matrix} {}_{}\\\;f&=&\frac{b-a}{a}&=&\frac{2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}{1-2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}&=&\sec(o\!\varepsilon)-1&=&\frac{\mbox{ver}(o\!\varepsilon)}{1-\mbox{ver}(o\!\varepsilon)};\\\\ f'&=&\frac{b-a}{b}&=&2\sin^2\left(\frac{o\!\varepsilon}{2}\right)&=&1-\cos(o\!\varepsilon)&=&\mbox{ver}(o\!\varepsilon);\\\\ f''&=&\frac{b-a}{b+a}&=&\frac{2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}{2-2\sin^2\left(\frac{o\!\varepsilon}{2}\right)}&=&\tan^2\left(\frac{o\!\varepsilon}{2}\right)&=&\frac{\mbox{ver}(o\!\varepsilon)}{\mbox{vrc}(o\!\varepsilon)}.\\ {}^{}\end{matrix}\,\!$

Numerical values for planets

For the Earth modelled by the WGS84 ellipsoid the defining values are

a (equatorial radius): 6378.137 km,

1/f (inverse flattening): 298.257223563,

from which one derives

b (polar radius): 6356.7523142 km,

so that the difference of the major and minor semi-axes is about 21.385 km. (This is only  0.335% of the major axis so a representation of the Earth on a computer screen could be sized as 300px by 299px. Since this would be indistinguishable from a sphere shown as 300px by 300px illustrations invariably greatly exaggerate the flattening.

Other values in the Solar System are Jupiter,  f=1/16; Saturn,  f= 1/10, the Moon  f= 1/900. The flattening of the Sun is less than 1/1000.

Origin of flattening

In 1687 Isaac Newton published the Principia in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid of revolution (a spheroid).

The amount of flattening depends on the size, density and elasticity of the celestial body (see Figure of the Earth), its rotation, and the balance of gravity and centrifugal force.

See also

• Map projection
• Oblate spheroid
• Prolate spheroid