Derive an expression to show that for satellites in a circular orbit T^2 ∝ r^3 where T is the period of orbit and r is the radius of the orbit.

This question is concerned with balancing forces. First, we must consider what forces are acting on the satellites. What is stopping the satellite from shooting off into space and what is preventing it from falling into the object it is orbiting. In this case, the two forces acting on it are a gravitational force and a centripetal force. Since it is a circular orbit we know both of these are equal at all times. Hence we must balance these two forces:
Gravitational force = GmM/r^2Centripetal force = mr(2PiT)^2
Hence, mr(2Pi/T)^2 = GmM/r^2.r(2Pi/T)^2 = GM/r^2 (cancelled the equal mass m)r(4Pi^2)/T^2 = GM/r^2 (expand out the bracket).r^3(4Pi^2) = GM*T^2 (rearrange T and r)We are looking for a proportionality, hence we can remove any constants. Here,Pi,G and M are all constants. Hence, we are left with T^2 is proportional to r^3