The flatness , F , of a Planet is a measurement of sound “ellipticity”; a Sphère has a flatness of 0, whereas a infinitely thin disc has a flatness of 1.

A planet in rotation has a tendency natural to be flattened, the Effet centrifuges creating a “equatorial pad”. Mathematically, flatness is given by:

$f=\; \backslash \; mbox\; \{worm\}\; (O\; \backslash !\; \backslash \; varepsilon)\; =2\; \backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{O\; \backslash !\; \backslash \; varepsilon\}\; \{2\}\; \backslash \; right)\; ^2=1-\; \backslash \; cos\; (O\; \backslash !\; \backslash \; varepsilon)\; =\; \backslash \; frac\; \{a-b\}\; \{has\}\; \backslash \; approx\; \backslash \; frac\; \{3\; \backslash \; pi\}\; \{2GT^\; \{2\}\; \backslash \; rho\};\; \backslash ,\; \backslash !$

where $a\; \backslash ,\; \backslash !$ and $b\; \backslash ,\; \backslash !$ is the rays equatorial and polar of planet, respectively, and $o\; \backslash !\; \backslash \; varepsilon\; \backslash ,\; \backslash !$ is the angular eccentricity . The approximation, in the case of validates a fluid planet of uniform Densité, is function of the constant of universal gravitation, $G$, of the Period of rotation $T$ and of the density $\backslash \; rho$.
There is also the second flatness, f' (sometimes indicated as a " N "), that is the tangente² of half-angle:

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

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