Fandom

Math Wiki

Flattening

995pages on
this wiki
Add New Page
Talk0 Share

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

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 a is the distance from the spheroid center to the equator——the transverse radius——and b the distance from the center to the pole——the conjugate radius——then f\mbox{lattening}=\quad\frac{a-b}{a}\,\!.

First, second and third flattening

The first, primary flattening, f, is the versine of the spheroid's angular eccentricity, o\!\varepsilon\,\!:

o\!\varepsilon=\arccos\left(\frac{b}{s}\right)=2\arctan\left(\!\sqrt{\frac{a-b}{a+b}}\;\right);\,\!
f=\mbox{ver}(o\!\varepsilon)=2\sin^2\left(\frac{o\!\varepsilon}{2}\right)=1-\cos(o\!\varepsilon)=\frac{a-b}{a};\,\!
  • 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 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' ,

f'=\frac{\mbox{ver}(o\!\varepsilon)}{1-\mbox{ver}(o\!\varepsilon)}
=\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{1-\cos(o\!\varepsilon)}{\cos(o\!\varepsilon)}
=\frac{a-b}{b};\,\!

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 o\!\varepsilon\,\!:

f''=\frac{\mbox{ver}(o\!\varepsilon)}{\mbox{vrc}(o\!\varepsilon)}
=\frac{\sin^2\left(\frac{o\!\varepsilon}{2}\right)}{1-\sin^2\left(\frac{o\!\varepsilon}{2}\right)}
=\tan^2\left(\frac{o\!\varepsilon}{2}\right)
=\frac{1-\cos(o\!\varepsilon)}{1+\cos(o\!\varepsilon)}
=\frac{a-b}{a+b}.\,\!

Prolate valuations

The above formations apply to an ellipse and oblatum, which is an ellipse rotated about its polar, or conjugate, axis, resulting in a>b\,\!. If it is rotated about its equatorial, or transverse, axis, it is a prolate spheroid, or prolatum, where b>a\,\!.
With a prolatum, a\,\! and b\,\! are reversed in all of the flattening formation elements, except for the denominators of f and f' , which means their o\!\varepsilon\,\! function assignments are reversed[5]:

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

Notes

  1. 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]
  2. 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
  3. Ганьшин, В. Н. (1967): Геометрия земного эллипсоида, Издательство «Недра», Москва
  4. 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]
  5. MathWorld: Flattening.
Wikipedia logo
See also the Wikipedia article:

Also on Fandom

Random Wiki