Editing Mean motion (section)

===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.