Editing Gaussian gravitational constant (section)

===Gauss's constant and Kepler's third law===
The Gaussian constant is closely related to [[Kepler's laws of planetary motion#Third law|Kepler's third law of planetary motion]], and one is easily derived from the other. Beginning with the full definition of Gauss's constant,

:<math>k=\frac{2\pi ab}{P\sqrt{M+m}\sqrt{p}},</math>

where
*{{mvar|a}} is the [[semi-major axis]] of the [[elliptical orbit]],
*{{mvar|b}} is the [[semi-minor axis]] of the elliptical orbit,
*{{mvar|P}} is the [[orbital period]],
*{{mvar|M}} is the [[mass]] of the primary body,
*{{mvar|m}} is the mass of the secondary body, and
*{{mvar|p}} is the [[Ellipse#Latus rectum|semi-latus rectum]] of the elliptical orbit.

From the geometry of an [[ellipse]], the semi-latus rectum, {{mvar|p}} can be expressed in terms of {{mvar|a}} and {{mvar|b}} thus: {{mvar|''p'' {{=}} {{sfrac|''b''<sup>2</sup>|''a''}}}}.<ref name="Smart77-99"/> &nbsp; Therefore,

:<math>\sqrt{p}=\frac{b}{\sqrt{a}}.</math>

Substituting and reducing, Gauss's constant becomes

:<math>k=\frac{2\pi}{P}\sqrt{\frac{a^3}{M+m}}.</math>

From [[orbital mechanics]], {{math|{{sfrac|2π|''P''}}}} is just {{mvar|n}}, the [[mean motion]] of the body in its orbit.<ref name="Smart77-100"/> Hence,

:<math>\begin{align}
k&=n\sqrt{\frac{a^3}{M+m}},\\[8pt]
k^2&=\frac{n^2a^3}{M+m},\\[8pt]
k^2(M+m)&=n^2a^3,
\end{align}</math>

which is the definition of Kepler's third law.<ref name="Smart77-101"/><ref>{{cite book | last = Vallado | first = David A. | title = Fundamentals of Astrodynamics and Applications | publisher = Microcosm Press |location = El Segundo, CA | year = 2001 | isbn = 1-881883-12-4 | edition=2nd|page=31}}
</ref> In this form, it is often seen with {{mvar|G}}, the [[gravitational constant|Newtonian gravitational constant]] in place of {{math|''k''<sup>2</sup>}}.

Setting {{math|''a'' {{=}} 1}}, {{math|''M'' {{=}} 1}}, {{math|''m'' ≪ ''M''}}, and {{mvar|n}} in [[radians]] per [[day]] results in {{math|''k'' ≈ ''n''}}, also in units of radians per day, about which see the relevant section of the [[Mean motion#Mean motion and the gravitational constants|mean motion]] article.