Editing Mean motion (section)

==Definition==
Define the [[orbital period]] (the time period for the body to complete one orbit) as ''P'', with dimension of time. The mean motion is simply one revolution divided by this time, or,
:<math>n = \frac{2\pi}{P}, \qquad n = \frac{360^\circ}{P}, \quad \mbox{or} \quad n = \frac{1}{P},</math>

with dimensions of [[radian]]s per unit time, [[Degree (angle)|degrees]] per unit time or revolutions per unit time.<ref>{{cite book 
 | last = Roy
 | first = A.E.
 | title = Orbital Motion
 | publisher = [[Institute of Physics Publishing]]
 | year = 1988
 | ISBN=0-85274-229-0
 | edition = third
 | page =83
}}</ref><ref name="BrowerClemence">{{cite book 
 | last1 = Brouwer
 | first1 = Dirk
 | last2 = Clemence
 | first2 = Gerald M.
 | title = Methods of Celestial Mechanics
 | url = https://archive.org/details/methodsofcelesti00brou
 | url-access = registration
 | publisher = [[Academic Press]]
 | year = 1961
 | pages = [https://archive.org/details/methodsofcelesti00brou/page/20 20–21]
}}</ref>

The value of mean motion depends on the circumstances of the particular gravitating system. In systems with more [[mass]], bodies will orbit faster, in accordance with [[Newton's law of universal gravitation]]. Likewise, bodies closer together will also orbit faster.

===Mean motion and Kepler's laws===
[[Kepler's laws of planetary motion#Third law|Kepler's 3rd law of planetary motion]] states, ''the [[Square (algebra)|square]] of the [[Orbital period|periodic time]] is proportional to the [[Cube (algebra)|cube]] of the [[Semi-major axis|mean distance]]'',<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 = second
 | page = 29
}}</ref> or
:<math>{a^3} \propto {P^2},</math>

where ''a'' is the [[semi-major axis]] or mean distance, and ''P'' is the [[orbital period]] as above. The constant of proportionality is given by

:<math>\frac{a^3}{P^2} = \frac {\mu}{4\pi^2}</math>

where ''μ'' is the [[standard gravitational parameter]], a constant for any particular gravitational system.

If the mean motion is given in units of radians per unit of time, we can combine it into the above definition of the Kepler's 3rd law,
:<math>\frac {\mu}{4\pi^2} = \frac{a^3}{\left(\frac{2\pi}{n}\right)^2},</math>

and reducing,
:<math>\mu = a^3n^2,</math>

which is another definition of Kepler's 3rd law.<ref name="BrowerClemence"/><ref>{{cite book
 | last = Battin
 | first = Richard H.
 | title = An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition
 | publisher = American Institute of Aeronautics and Astronautics, Inc.
 | isbn = 1-56347-342-9
 | date = 1999
 | page = 119
}}</ref> ''μ'', the constant of proportionality,<ref name=Vallado31>Vallado, David A. (2001). p. 31.</ref><ref group=note>Do not confuse ''μ'', the [[standard gravitational parameter|gravitational parameter]] with ''μ'', the [[reduced mass]].</ref> is a gravitational parameter defined by the [[mass]]es of the bodies in question and by the [[Gravitational constant|Newtonian constant of gravitation]], ''G'' (see below). Therefore, ''n'' is also defined<ref name="Vallado53">Vallado, David A. (2001). p. 53.</ref>
:<math>n^2 = \frac{\mu}{a^3}, \quad \text{or} \quad n = \sqrt{\frac{\mu}{a^3}}.</math>

Expanding mean motion by expanding ''μ'',
:<math>n = \sqrt{\frac{ G( M  + m ) }{a^3}},</math>

where ''M'' is typically the mass of the primary body of the system and ''m'' is the mass of a smaller body.

This is the complete gravitational definition of mean motion in a [[two-body problem|two-body system]]. Often in [[celestial mechanics]], the primary body is much larger than any of the secondary bodies of the system, that is, {{nowrap|''M'' ≫ ''m''}}. It is under these circumstances that ''m'' becomes unimportant and Kepler's 3rd law is approximately constant for all of the smaller bodies.

[[Kepler's laws of planetary motion#Second law|Kepler's 2nd law of planetary motion]] states, ''a line joining a planet and the Sun sweeps out equal areas in equal times'',<ref name="Vallado31"/> or
:<math>\frac{\mathrm{d}A}{\mathrm{d}t} = \text{constant}</math>

for a two-body orbit, where {{sfrac|d''A''|d''t''}} is the time rate of change of the [[area]] swept. 

{{See also|Leibniz's notation}}
Letting ''t''&nbsp;=&nbsp;''P'', the orbital period, the area swept is the entire area of the [[ellipse]], d''A''&nbsp;=&nbsp;{{pi}}''ab'', where ''a'' is the [[semi-major axis]] and ''b'' is the [[semi-minor axis]] of the ellipse.<ref name="Vallado30">Vallado, David A. (2001). p. 30.</ref> Hence,
:<math>\frac{\mathrm{d}A}{\mathrm{d}t} = \frac{\pi ab}{P}.</math>

Multiplying this equation by 2,
:<math>2 \left( \frac{\mathrm{d}A}{\mathrm{d}t} \right) = 2 \left( \frac{\pi ab}{P} \right).</math>

From the above definition, mean motion ''n''&nbsp;=&nbsp;{{sfrac|2{{pi}}|''P''}}. Substituting,
:<math>2\frac{\mathrm{d}A}{\mathrm{d}t} = nab,</math>

and mean motion is also
:<math>n = \frac{2}{ab}\frac{\mathrm{d}A}{\mathrm{d}t},</math>

which is itself constant as ''a'', ''b'', and {{sfrac|d''A''|d''t''}} are all constant in two-body motion.

===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]].

===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>

===Mean motion and mean anomaly===
Mean motion also represents the rate of change of [[mean anomaly]], and hence can also be calculated,<ref>Vallado, David A. (2001). p. 54.</ref>
:<math>\begin{align}
n &= \frac{M_1 - M_0}{t_1 - t_0} = \frac{M_1 - M_0}{\Delta t}, \\
M_1 &= M_0 + n \times (t_1 - t_0) = M_0 + n \times \Delta t 
\end{align}</math>

where ''M''<sub>1</sub> and ''M''<sub>0</sub> are the mean anomalies at particular points in time, and &Delta;''t'' (&equiv; ''t''<sub>1</sub>-''t''<sub>0</sub>) is the time elapsed between the two.  ''M''<sub>0</sub> is referred to as the ''mean anomaly at [[Epoch (astronomy)|epoch]]'' ''t''<sub>0</sub>, and &Delta;''t'' is the ''time since epoch''.