If you want to put a satellite into a geostationary orbit, meaning that it stays above the same position over the Earth at all times, at what altitude (above sea level) must you have it orbiting?

Answer 1

The centripetal force CF ωacting on a object of mass m rotating with angular velocity ω at distance r from the center of the Earth is given by

CF=mωr²

For circular orbits, the centripetal force must be equal to the gravitational force GF given by

GF=γmM/r²

where γ is the gravitational constant (numerical value γ≈6.6·10^-11), M is the mass of the Earth (numerical value M≈5.97·10²⁴)

You therefore must impose the above quantity to be equal, thus

mωr²=γmM/r²

and therefore, r=(γM/ω)^1/3

In order to obtain the geostationary orbit, you use the angular velocity of the Earth (numerical value ω≈7,2921·10⁻⁵).

If you plug in all the numerical values (you can also check that the units will be meters!) you will obtain the geostationary radius r≈4.2·10⁷ m which is roughly 42.000 km.

Best,

Davide