Question

If a satellites is revolving close to a planet of density `rho` with period `T`, show that the quantity `rho T^(2)` is a universal constant.

Answer 1

When a satellite is orbiting close to Earth, then
`(GMm)/(R^(2)) = m R omega^(2)`
or `(G(4)/(3) pi R^(3) rho m)/(R^(2)) = mR (4 pi^(2))/(T^(2))`
or `rho T^(2) = (3 pi)/(G) = a` constant.