Flattening

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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}\,\!

Contents

[edit] First, second and third flattening

[edit] 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[1][2], f'' (sometimes denoted as “n”, the notation of which first used in 1837 by Friedrich Bessel on calculation of meridian arc length[3]), 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)}.\,\!

[edit] 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}\,\!

[edit] Numerical values for planets

For the Earth modelled by the WGS84 ellipsoid the defining values are[4]

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.

[edit] 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).[5]

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.

[edit] See also

[edit] 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
  4. ↑ WGS84 parameters are listed in the National Geospatial-Intelligence Agency publication TR8350.2 page 3-1.
  5. ↑ Isaac Newton:Principia Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation, available on line at [1]
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