Inorbital 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 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, perturbedorbit.
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.
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.
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 constant of gravitation, 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 two-body 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.
Letting t = P, the orbital period, the area swept is the entire area of the ellipse, dA = πab, where a is the semi-major axis and b is the semi-minor 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 two-body motion.
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,
Mean motion also represents the rate of change of mean anomaly, and hence can also be calculated,[14]
where M1 and M0 are the mean anomalies at particular points in time, and Δt (≡ t1-t0) is the time elapsed between the two. M0 is referred to as the mean anomaly at epocht0, and Δt is the time since epoch.
^
Seidelmann, P. Kenneth; Urban, Sean E., eds. (2013). Explanatory Supplement to the Astronomical Almanac (3rd ed.). University Science Books, Mill Valley, CA. p. 648. ISBN978-1-891389-85-6.
^Vallado, David A. (2001). Fundamentals of Astrodynamics and Applications (second ed.). El Segundo, CA: Microcosm Press. p. 29. ISBN1-881883-12-4.
^Battin, Richard H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. American Institute of Aeronautics and Astronautics, Inc. p. 119. ISBN1-56347-342-9.
^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.