Editing Mean motion (section)

===Mean motion and the gravitational constants===
Two gravitational constants are commonly used in [[Solar System]] celestial mechanics: ''G'', the [[Gravitational constant|Newtonian constant of gravitation]] and ''k'', the [[Gaussian gravitational constant]]. From the above definitions, mean motion is
:<math>n = \sqrt{\frac{ G( M  + m ) }{a^3}}\,\!.</math>

By normalizing parts of this equation and making some assumptions, it can be simplified, revealing the relation between the mean motion and the constants.

Setting the mass of the [[Sun]] to unity, ''M''&nbsp;=&nbsp;1. The masses of the planets are all much smaller, {{nowrap|''m'' ≪ ''M''}}. Therefore, for any particular planet,
:<math>n \approx \sqrt{\frac{G}{a^3}},</math>

and also taking the semi-major axis as one [[astronomical unit]],
:<math>n_{1\;\text{AU}} \approx \sqrt{G}.</math>

The Gaussian gravitational constant ''k''&nbsp;=&nbsp;{{sqrt|''G''}},<ref>{{cite book 
 | last1 = U.S. Naval Observatory
 | first1=Nautical Almanac Office
 | last2 = H.M. Nautical Almanac Office
 | title = Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac
 | publisher = H.M. Stationery Office, London
 | year = 1961
 | page = 493
}}</ref><ref>{{cite book 
 | last = Smart
 | first = W. M.
 | title = Celestial Mechanics
 | publisher = Longmans, Green and Co., London
 | year = 1953
 | page=4
}}</ref><ref group=note>The [[Gaussian gravitational constant]], ''k'', usually has units of radians per day and the [[Gravitational constant|Newtonian constant of gravitation]], ''G'', is usually given in [[International System of Units|SI units]]. Be careful when converting.</ref> therefore, under the same conditions as above, for any particular planet
:<math>n \approx \frac{k}{\sqrt{a^3}},</math>

and again taking the semi-major axis as one astronomical unit,
:<math>n_{1\text{ AU}} \approx k.</math>