Derive an expression to show that for satellites in a circular orbit T² ∝ r ³ where T is the period of orbit and r is the radius of the orbit.
For an object to stay in a steady orbit; F=mv2/r where: F is the force on the object towards the centre of the orbit, m is the mass of the object, v is the radial velocity of the object, and r is the radius of the orbit.In the case of a satellite orbiting a planet, all of F is provided by the gravitational force acting on the satellite due to the planet/moon/star. This force is given by Newton's law of gravitation:F = GMm/r2where F is the gravitational force, G is the gravitational constant; 6.67 x 10 -11 Nm2kg-2, M is the mass of the planet/moon/star, m is the mass of the satellite, and r is the distance between the planet/moon/star and the satellite.We can therefore equate these two forces, as F = F, giving;GMm/r2 = mv2/rWe can multiply both sides by r and divide both sides by m to give;GM/r = v2Finally, we need the time period, T, not the velocity, v, therefore we can use v = s/t. In this case, s is the circumference of orbit = 2πr, and t is T, the time period of the orbit. We can write:v = 2πr/TSubstituting this into before gives:GM/r = (2πr/T)2Expanding the brackets, multiplying both sides by T, and multiplying both sides by r gives;GMT2 = 4π2r3
