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

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

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

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

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,

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

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

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

Expanding mean motion by expanding μ,

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

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.

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

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

for a two-body orbit, where dA/dt is the time rate of change of the area swept.

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,

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

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 n = 2π/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 as a, b, and dA/dt are all constant in two-body motion.

Mean motion and the constants of the motion

Because of the nature of two-body 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]

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

and substituting in the above equation, mean motion is also

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

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

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

Rearranging and multiplying by 1/a²,

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

From above, the square of mean motion n² = μ/a³. Substituting and rearranging, mean motion can also be expressed,

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

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 constant of gravitation and k, the Gaussian 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.

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,

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

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

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

The Gaussian gravitational constant k = √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 and mean anomaly

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

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

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 epoch t₀, and Δt is the time since epoch.