Circular Orbits

Circular orbits represent one possible solution to the equations of motion. For an satellite of mass m, we relate the energy and radius of a circular orbit around a planet of mass M >> m.

Energy conservation:    E = U + K.

U =
- G m M
        and         K = ½ mv2

Applying Newton's Second Law:   ΣF = ma   →   G m M/ r2 = m v2/r.

So the orbital radius and orbital speed are related by:   v2r = G M

Using this result, the kinetic energy is: K = ½mv2 = G m M/ (2 r) = ½ |U|.

Finally, the total energy is:
E =
-G m M
G m M
-G m M
< 0

Negative total energy corresponds to a bound system. Just like an electron is bound to a proton in a hydrogen atom with a negative binding energy, a satellite is bound to the Earth, energy must be added to move the satellite to infinity.