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Inorbital mechanics,mean motion (represented byn) is theangular speed required for a body to complete one orbit, assuming constant speed in acircular 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 commoncenter of mass. While nominally amean, and theoretically so in the case oftwo-body motion, in practice the mean motion is not typically anaverage 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 currentgravitational andgeometric 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 oforbital elements. This mean position is refined byKepler's equation to produce the true position.

Define theorbital period (the time period for the body to complete one orbit) asP, with dimension of time. The mean motion is simply one revolution divided by this time:^[2][3]

${\displaystyle n=\frac{2\pi \text{radians}}{P}=\frac{{360}^{\circ }}{P}=\frac{1\text{revolution}}{P}}$

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

Kepler's 3rd law of planetary motion states,thesquare of theperiodic time is proportional to thecube of themean distance,^[4] or

${\displaystyle a^3\propto P^2,}$

wherea is thesemi-major axis or mean distance, andP is theorbital period as above. The constant of proportionality is given by

${\displaystyle \frac{a^3}{P^2}=\frac{\mu }{4{\pi }^2},}$

whereμ is thestandard 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,

${\displaystyle \frac{\mu }{4{\pi }^2}=\frac{a^3}{{\left(\frac{2\pi }{n}\right)}^2},}$

and reducing,

${\displaystyle \mu =a^3{n}^2,}$

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 themasses of the bodies in question and by theNewtonian constant of gravitation,G (see below). Therefore,n is also defined as^[7]

${\displaystyle n^2=\frac{\mu }{a^3},\text{or}n=\sqrt{\frac{\mu }{a^3}}.}$

Expanding mean motion by expandingμ,

${\displaystyle n=\sqrt{\frac{G(M+m)}{a^3}},}$

whereM is typically the mass of the primary body of the system andm is the mass of a smaller body.

This is the complete gravitational definition of mean motion in atwo-body system. Often incelestial mechanics, the primary body is much larger than any of the secondary bodies of the system, that is,${\displaystyle M\gg m}$. It is under these circumstances thatm 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 alternatively, usingLeibniz's notation,

${\displaystyle \frac{\mathrm{d}A}{\mathrm{d}t}=\text{constant}}$

for a two-body orbit, where${\displaystyle \frac{\mathrm{d}A}{\mathrm{d}t}}$ is the time-based rate of change of thearea swept.

Letting${\displaystyle t=P}$, the orbital period, the area swept is the entire area of theellipse,${\displaystyle \mathrm{d}A=\pi ab}$, wherea is thesemi-major axis andb is thesemi-minor axis of the ellipse.^[8] Hence,

${\displaystyle \frac{\mathrm{d}A}{\mathrm{d}t}=\frac{\pi ab}{P}.}$

Multiplying this equation by 2,

${\displaystyle 2\left(\frac{\mathrm{d}A}{\mathrm{d}t}\right)=2\left(\frac{\pi ab}{P}\right).}$

From the above definition, mean motion${\displaystyle n=\frac{2\pi }{P}}$. Substituting,

${\displaystyle 2\frac{\mathrm{d}A}{\mathrm{d}t}=nab,}$

and mean motion is also

${\displaystyle n=\frac{2}{ab}\frac{\mathrm{d}A}{\mathrm{d}t},}$

which is itself constant asa,b, and${\displaystyle \frac{\mathrm{d}A}{\mathrm{d}t}}$ are all constant in two-body motion.

Because of the nature oftwo-body motion in aconservativegravitational field, two aspects of the motion do not change: theangular momentum and themechanical energy.

The first constant, calledspecific angular momentum, can be defined as^[8][9]

${\displaystyle h=2\frac{\mathrm{d}A}{\mathrm{d}t},}$

and substituting in the above equation, mean motion is also

${\displaystyle n=\frac{h}{ab}.}$

The second constant, calledspecific mechanical energy, can be defined,^[10][11]

${\displaystyle \xi =-\frac{\mu }{2a}.}$

Rearranging and dividing by${\displaystyle a^2}$,

${\displaystyle \frac{-2\xi }{a^2}=\frac{\mu }{a^3}.}$

From above, the square of mean motion${\displaystyle n^2=\frac{\mu }{a^3}}$. Substituting and rearranging, mean motion can also be expressed,

${\displaystyle n=\frac{1}{a}\sqrt{-2\xi },}$

where the −2 shows thatξ must be defined as a negative number, as is customary forelliptic orbits incelestial mechanics andastrodynamics.

Two gravitational constants are commonly used inSolar System celestial mechanics:G, theNewtonian constant of gravitation andk, theGaussian gravitational constant. From the above definitions, mean motion is

${\displaystyle n=\sqrt{\frac{G(M+m)}{a^3}}.}$

By normalizing parts of this equation and making some assumptions, it can be simplified, revealing the relation between the mean motion and the constants.

For example, setting the mass of theSun to unity (M = 1), the masses of the planets are all much smaller (${\displaystyle m\ll M}$). Therefore, for any particular planet,

${\displaystyle n\approx \sqrt{\frac{G}{a^3}},}$

and also taking the semi-major axis as oneastronomical unit,

${\displaystyle n_{1\text{AU}}\approx \sqrt{G}.}$

The Gaussian gravitational constant${\displaystyle k=\sqrt{G}}$,^{[12][13][note 2]} therefore, under the same conditions as above, for any particular planet,

${\displaystyle n\approx \frac{k}{\sqrt{a^3}},}$

and again taking the semi-major axis as one astronomical unit,

${\displaystyle n_{1\text{ AU}}\approx k.}$

Mean motion also represents the rate of change ofmean anomaly, and hence can also be calculated,^[14]

where${\displaystyle M_1}$ and${\displaystyle M_0}$ are the mean anomalies at particular points in time, and${\displaystyle \mathrm{\Delta }t(\equiv t_1-t_0)}$ is the time elapsed between the two.${\displaystyle M_0}$ is referred to as themean anomaly atepoch${\displaystyle t_0}$, and${\displaystyle \mathrm{\Delta }t}$ is thetime since epoch.

For Earth satellite orbital parameters, the mean motion is typically measured in revolutions perday. In that case,

${\displaystyle n=\frac{d}{2\pi }\sqrt{\frac{G(M+m)}{a^3}}=d\sqrt{\frac{G(M+m)}{4{\pi }^2{a}^3}}}$

where

• d is the quantity of time in aday,
• G is thegravitational constant,
• M andm are themasses of the orbiting bodies, and
• a is the length of thesemi-major axis.

To convert from radians per unit time to revolutions per day, consider the following:

${\displaystyle \frac{\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{s}}{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}}\times \frac{1\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{2\pi \mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{s}}\times \frac{d\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}}{1\mathrm{d}\mathrm{a}\mathrm{y}}=\frac{d}{2\pi }\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{d}\mathrm{a}\mathrm{y}}$

From above, mean motion in radians per unit time is:

${\displaystyle n=\frac{2\pi }{P},}$

therefore the mean motion in revolutions per day is

${\displaystyle n=\frac{d}{2\pi }\frac{2\pi }{P}=\frac{d}{P},}$

whereP is theorbital period, as above.

• Glossary entrymean motionArchived 2017-12-23 at theWayback Machine at the US Naval Observatory'sAstronomical Almanac OnlineArchived 2015-04-20 at theWayback Machine

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