Using Newton's law of universal gravitation, show that T^2 is proportional to r^3 (where T is the orbital period of a planet around a star, and r is the distance between them).
(Lets consider a simple planetary system composed of a planet orbiting a star. the gravitational force between the two is given by F=(GMm)/(r2). Assuming the planet also moves in a circular orbit, we can consider the centripetal force, F=mω2r. As both gravitational and centripetal forces act in the same direction, we can equate them to find (GMm)/(r2)=mω2r.
We note that 'm' cancels and we can divide through by 'r' to arrive at GM/r3=ω2. ω is simply angular frequency given by ω =2π/T. Substituting this into our expression we find that GM/r3= 4π2/T2.After some simple rearranging, we note that T2 =(4π2r3)/(GM). So T2 is indeed proportional to r3 . This simple statement is known as Kepler's third law of planetary motion.
