Mean motion ($n$)

Mean motion is the average orbital angular frequency. A Kepler orbit about gravitational parameter $\mu$ has $n^2{a}^3=\mu$.

The orbital period is the time required to complete one orbit. It is related to mean motion by $T=2\pi /n$.

For a circular orbit, the angular rate about the central body equals $2\pi$ divided by its orbital period. Matching it to planetary spin is necessary for a planet-stationary orbit, which additionally requires a prograde equatorial circular geometry.

For a spherical body of radius $R$, mass $M$ and mean density $\bar{\rho }=3M/(4\pi R^3)$, a low circular orbit has period $P\simeq 2\pi \sqrt{R^3/(GM)}$, by equating the gravitational acceleration from Newton's law of universal gravitation to the centripetal acceleration. Substitution proves the displayed relation. Thus similar low-orbit periods indicate similar mean densities, even for bodies with very different masses and radii. At altitude $h$, the right side is multiplied by $(1+h/R{)}^3$; the simple density inference requires $h\ll R$ and neglects rotation, nonsphericity and other gravitating bodies.

The synodic period is the interval between successive repetitions of the relative orbital configuration of two bodies. For coplanar prograde mean motions $n_1,n_2$, it is $2\pi /|n_1-n_2|$.