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.^{[1]} The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively samesized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of twobody motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the twobody 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 constantlychanging, 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.
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Contents
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,
with dimensions of radians per unit time, degrees per unit time or revolutions per unit time.^{[2]}^{[3]}
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,^{[4]} or
where a is the semimajor axis or mean distance, and P is the orbital period as above. The constant of proportionality is given by
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,
and reducing,
which is another definition of Kepler's 3rd law.^{[3]}^{[5]} μ, the constant of proportionality,^{[6]}^{[note 1]} is a gravitational parameter defined by the masses of the bodies in question and by the Newtonian gravitational constant, G (see below). Therefore, n is also defined^{[7]}
Expanding mean motion by expanding μ,
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 twobody system. Often in celestial mechanics, the primary body is much larger than any of the secondary bodies of the system, that is, 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 2nd law of planetary motion states, a line joining a planet and the Sun sweeps out equal areas in equal times,^{[6]} or
for a twobody orbit, where dA/dt is the time rate of change of the area swept.
Letting dt = P, the orbital period, the area swept is the entire area of the ellipse, dA = πab, where a is the semimajor axis and b is the semiminor axis of the ellipse.^{[8]} Hence,
Multiplying this equation by 2,
From the above definition, mean motion n = 2π/P. Substituting,
and mean motion is also
which is itself constant as a, b, and dA/dt are all constant in twobody motion.
Mean motion and the constants of the motion
Because of the nature of twobody motion in a conservative gravitational field, two aspects of the motion do not change: the angular momentum and the mechanical energy.
The first constant, called specific angular momentum, can be defined as^{[8]}^{[9]}
and substituting in the above equation, mean motion is also
The second constant, called specific mechanical energy, can be defined,^{[10]}^{[11]}
Rearranging and multiplying by 1/a^{2},
From above, the square of mean motion n^{2} = μ/a^{3}. Substituting and rearranging, mean motion can also be expressed,
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 Newtonian gravitational constant and k, the Gaussian gravitational constant. From the above definitions, mean motion is
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 = 1. The masses of the planets are all much smaller, m ≪ M. Therefore, for any particular planet,
and also taking the semimajor axis as one astronomical unit,
The Gaussian gravitational constant k = √G,^{[12]}^{[13]}^{[note 2]} therefore, under the same conditions as above, for any particular planet
and again taking the semimajor axis as one astronomical unit,
Mean motion and mean anomaly
Mean motion also represents the rate of change of mean anomaly, and hence can also be calculated,^{[14]}
where M_{1} and M_{0} are the mean anomalies at particular points in time, and t is the time elapsed between the two. M_{0} is referred to as the mean anomaly at epoch, and t is the time since epoch.
Formulae
For Earth satellite orbital parameters, the mean motion is typically measured in revolutions per day. In that case,
where
 d is the quantity of time in a day,
 G is the gravitational constant,
 M and m are the masses of the orbiting bodies,
 a is the length of the semimajor axis.
To convert from radians per unit time to revolutions per day, consider the following:
From above, mean motion in radians per unit time is:
therefore the mean motion in revolutions per day is
where P is the orbital period, as above.
See also
Notes
 ^ Do not confuse μ, the gravitational parameter with μ, the reduced mass.
 ^ The Gaussian gravitational constant, k, usually has units of radians per day and the Newtonian gravitational constant, G, is usually given in the SI system. Be careful when converting.
References
 ^ Seidelmann, P. Kenneth; Urban, Sean E., eds. (2013). Explanatory Supplement to the Astronomical Almanac (3rd ed.). University Science Books, Mill Valley, CA. p. 648. ISBN 9781891389856.
 ^ Roy, A.E. (1988). Orbital Motion (third ed.). Institute of Physics Publishing. p. 83. ISBN 0852742290.
 ^ ^{a} ^{b} Brouwer, Dirk; Clemence, Gerald M. (1961). Methods of Celestial Mechanics. Academic Press. pp. 20–21.
 ^ Vallado, David A. (2001). Fundamentals of Astrodynamics and Applications (second ed.). El Segundo, CA: Microcosm Press. p. 29. ISBN 1881883124.
 ^ Battin, Richard H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. American Institute of Aeronautics and Astronautics, Inc. p. 119. ISBN 1563473429.
 ^ ^{a} ^{b} Vallado, David A. (2001). p. 31.
 ^ Vallado, David A. (2001). p. 53.
 ^ ^{a} ^{b} Vallado, David A. (2001). p. 30.
 ^ Bate, Roger R.; Mueller, Donald D.; White, Jerry E. (1971). Fundamentals of Astrodynamics. Dover Publications, Inc., New York. p. 32. ISBN 0486600610.
 ^ Vallado, David A. (2001). p. 27.
 ^ Bate, Roger R.; Mueller, Donald D.; White, Jerry E. (1971). p. 28.
 ^ U.S. Naval Observatory, Nautical Almanac Office; H.M. Nautical Almanac Office (1961). Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. H.M. Stationery Office, London. p. 493.
 ^ Smart, W. M. (1953). Celestial Mechanics. Longmans, Green and Co., London. p. 4.
 ^ Vallado, David A. (2001). p. 54.
External links
 Glossary entry mean motion at the US Naval Observatory's Astronomical Almanac Online