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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 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 flatteningEdit

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 valuationsEdit

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}\,\!

NotesEdit

  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.
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