In orbital mechanics, mean motion (represented by n) is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of two-body motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the two-body assumption. It is rather the instantaneous value which satisfies the above conditions as calculated from the current gravitational and geometric circumstances of the body's constantly-changing, perturbed orbit. Mean motion is used as an approximation of the actual orbital speed in making an initial calculation of the body's position in its orbit, for instance, from a set of orbital elements. This mean position is refined by Kepler's equation to produce the true position.
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:
n = 2 π radians P = 360 ∘ P = 1 revolution P {\displaystyle n={\frac {2\pi \,{\text{radians}}}{P}}={\frac {360^{\circ }}{P}}={\frac {1\,{\text{revolution}}}{P}}}
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 3rd law of planetary motion states, the square of the periodic time is proportional to the cube of the mean distance, or
a 3 ∝ P 2 , {\displaystyle {a^{3}}\propto {P^{2}},}
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
a 3 P 2 = μ 4 π 2 , {\displaystyle {\frac {a^{3}}{P^{2}}}={\frac {\mu }{4\pi ^{2}}},}
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,
μ 4 π 2 = a 3 ( 2 π n ) 2 , {\displaystyle {\frac {\mu }{4\pi ^{2}}}={\frac {a^{3}}{\left({\frac {2\pi }{n}}\right)^{2}}},}
and reducing,
μ = a 3 n 2 , {\displaystyle \mu =a^{3}n^{2},}
which is another definition of Kepler's 3rd law. μ, the constant of proportionality, is a gravitational parameter defined by the masses of the bodies in question and by the Newtonian constant of gravitation, G (see below). Therefore, n is also defined as
n 2 = μ a 3 , or n = μ a 3 . {\displaystyle n^{2}={\frac {\mu }{a^{3}}},\quad {\text{or}}\quad n={\sqrt {\frac {\mu }{a^{3}}}}.}
Expanding mean motion by expanding μ,
n = G ( M + m ) a 3 , {\displaystyle n={\sqrt {\frac {G(M+m)}{a^{3}}}},}
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