Editing Mean motion (section)

===Mean motion and the constants of the motion===
Because of the nature of [[two-body problem|two-body motion]] in a [[conservation law|conservative]] [[gravitational field]], two aspects of the motion do not change: the [[angular momentum]] and the [[mechanical energy]].

The first constant, called [[Specific relative angular momentum|specific angular momentum]], can be defined as<ref name="Vallado30"/><ref>{{cite book
 | last1 = Bate
 | first1 = Roger R.
 | last2 = Mueller
 | first2 = Donald D.
 | last3 = White
 | first3 = Jerry E.
 | title = Fundamentals of Astrodynamics
 | publisher = Dover Publications, Inc., New York
 | isbn = 0-486-60061-0
 | date = 1971
 | page = [https://archive.org/details/fundamentalsofas00bate/page/32 32]
 | url-access = registration
 | url = https://archive.org/details/fundamentalsofas00bate/page/32
 }}</ref>
:<math>h = 2\frac{\mathrm{d}A}{\mathrm{d}t},</math>

and substituting in the above equation, mean motion is also
:<math>n = \frac{h}{ab}.</math>

The second constant, called [[specific orbital energy|specific mechanical energy]], can be defined,<ref name="Vallado27">Vallado, David A. (2001). p. 27.</ref><ref /name="BMW28">Bate, Roger R.; Mueller, Donald D.; White, Jerry E. (1971). p. 28.</ref>
:<math>\xi = -\frac{\mu}{2a}.</math>

Rearranging and multiplying by {{sfrac|1|''a''<sup>2</sup>}},
:<math>\frac{-2\xi}{a^2} = \frac{\mu}{a^3}.</math>

From above, the square of mean motion ''n''<sup>2</sup>&nbsp;=&nbsp;{{sfrac|''μ''|''a''<sup>3</sup>}}. Substituting and rearranging, mean motion can also be expressed,
:<math>n = \frac{1}{a}\sqrt{-2\xi},</math>

where the −2 shows that ''ξ'' must be defined as a negative number, as is customary in [[celestial mechanics]] and [[astrodynamics]].