Synchronous circular orbit ($r=(GM{T}^2/4{\pi }^2{)}^{1/3}$)

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.