Circular Kepler orbit

A circular Kepler orbit at radius $r$ has speed $v_K=\sqrt{\mu /r}$ and mean motion $n=\sqrt{\mu /r^3}$.

A satellite remaining above a fixed point of a uniformly rotating spherical planet must have a prograde equatorial circular Kepler orbit with angular speed $2\pi /T$. Balancing gravitational acceleration and centripetal acceleration gives the displayed radius and speed $v=2\pi r/T$. The satellite's own mass cancels. A physically realizable external orbit additionally requires this radius to exceed the planet's radius.

A geostationary orbit is a prograde equatorial circular Kepler orbit whose orbital period equals one sidereal day, so its satellite remains above a fixed location on Earth's equator. It is the terrestrial case of a planet-stationary orbit. Its radius follows from Newton's law of universal gravitation and circular acceleration: $r=(G{M}_E{T}^2/(4{\pi }^2){)}^{1/3}$, approximately $42164\mathrm{km}$ from Earth's centre. Its altitude above the surface is smaller by Earth's radius.