Mean motion

Inorbital mechanics,mean motion(represented byn) is theangular speedrequired for a body to complete one orbit, assuming constant speed in acircular orbitwhich completes in the same time as the variable speed,elliptical orbitof 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 commoncenter of mass.While nominally amean,and theoretically so in the case oftwo-body motion,in practice the mean motion is not typically anaverageover 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 currentgravitationalandgeometriccircumstances 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 equationto produce the true position.

Definition

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, or,

$n={\frac {2\pi }{P}},\qquad n={\frac {360^{\circ }}{P}},\quad {\mbox{or}}\quad n={\frac {1}{P}},$

with dimensions ofradiansper unit time,degreesper unit time or revolutions per unit time.

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.

Mean motion and Kepler's laws

Kepler's 3rd law of planetary motionstates,thesquareof theperiodic timeis proportional to thecubeof themean distance,or

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

whereais thesemi-major axisor mean distance, andPis theorbital periodas above. The constant of proportionality is given by

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

whereμis 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,

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

and reducing,

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

which is another definition of Kepler's 3rd law.μ,the constant of proportionality,is a gravitational parameter defined by themassesof the bodies in question and by theNewtonian constant of gravitation,G(see below). Therefore,nis also defined

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

Expanding mean motion by expandingμ,

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

whereMis typically the mass of the primary body of the system andmis 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,M≫m.It is under these circumstances thatmbecomes unimportant and Kepler's 3rd law is approximately constant for all of the smaller bodies.

Kepler's 2nd law of planetary motionstates,a line joining a planet and the Sun sweeps out equal areas in equal times,or

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

for a two-body orbit, where⁠dA/dt⁠is the time rate of change of theareaswept.

See also:Leibniz's notation

Lettingt=P,the orbital period, the area swept is the entire area of theellipse,dA=πab,whereais thesemi-major axisandbis thesemi-minor axisof the ellipse.Hence,

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

Multiplying this equation by 2,

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

From the above definition, mean motionn=⁠2π/P⁠.Substituting,

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

and mean motion is also

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

which is itself constant asa,b,and⁠dA/dt⁠are all constant in two-body motion.

Mean motion and the constants of the motion

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

The first constant, calledspecific angular momentum,can be defined as

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

and substituting in the above equation, mean motion is also

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

The second constant, calledspecific mechanical energy,can be defined,

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

Rearranging and multiplying by⁠1/a²⁠,

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

From above, the square of mean motionn²=⁠μ/a³⁠.Substituting and rearranging, mean motion can also be expressed,

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

where the −2 shows thatξmust be defined as a negative number, as is customary incelestial mechanicsandastrodynamics.

Mean motion and the gravitational constants

Two gravitational constants are commonly used inSolar Systemcelestial mechanics:G,theNewtonian constant of gravitationandk,theGaussian gravitational constant.From the above definitions, mean motion is

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

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

Setting the mass of theSunto unity,M= 1. The masses of the planets are all much smaller,m≪M.Therefore, for any particular planet,

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

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

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

The Gaussian gravitational constantk=√G,therefore, under the same conditions as above, for any particular planet

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

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

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

Mean motion and mean anomaly

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

${\begin{aligned}n&={\frac {M_{1}-M_{0}}{t_{1}-t_{0}}}={\frac {M_{1}-M_{0}}{\Delta t}},\\M_{1}&=M_{0}+n\times (t_{1}-t_{0})=M_{0}+n\times \Delta t\end{aligned}}$

whereM₁andM₀are the mean anomalies at particular points in time, and Δt(≡t₁-t₀) is the time elapsed between the two.M₀is referred to as themean anomaly atepocht₀,and Δtis thetime since epoch.

Formulae

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

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

where

• dis the quantity of time in aday,
• Gis thegravitational constant,
• Mandmare themassesof the orbiting bodies,
• ais the length of thesemi-major axis.

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

${\rm {{\frac {radians}{time\ unit}}\times {\frac {1\ revolution}{2\pi \ radians}}\times }}{\frac {d\ {\rm {time\ units}}}{1{\rm {\ day}}}}={\frac {d}{2\pi }}{\rm {\ revolutions\ per\ day}}$

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

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

therefore the mean motion in revolutions per day is

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

wherePis theorbital period,as above.

See also

• Gaussian gravitational constant
• Kepler orbit
• Mean anomaly
• Mean longitude
• Mean motion resonance
• Orbital elements

External links

• Glossary entrymean motionArchived2017-12-23 at theWayback Machineat the US Naval Observatory'sAstronomical Almanac OnlineArchived2015-04-20 at theWayback Machine