Kepler's 3rd Law

Kepler observed that planets further from the Sun take longer to orbit the Sun. From observations, Kepler noticed that the period of orbit squared was directly proportional to the radius of orbit cubed.

As the force of gravity between the two objects provides the centripetal force maintaining the circular motion, we begin the derivation by equating these two forces.

The mass of the smaller body, , cancels as it is the smaller body that is orbiting the larger mass, . The bodies actually orbit a common centre of mass, but as the larger body is usually significantly larger, the centre of mass typically falls within the radius of the larger mass. We can then model the smaller body orbiting the centre of the larger.

We can substitute the velocity, , for distance over time using the circumference of a circle for distance travelled and T, the time period, for time as we are modelling for one full orbit.

Substituting for :

as  is a constant for a set of bodies all orbting a common larger object, such as all of the planets orbiting the Sun, in which case  is the mass of the Sun.

Kepler’s law doesn't just apply to the motion of the planets around the Sun. It applies to any set of bodies orbiting a common centre of mass.

For example, Mars has two moons – Deimos and Phobos, that both follow this relationship. But in this case would be the mass of Mars, so the constant is a different number from the Sun and planet system, but a constant nonetheless.

Jupiter has confirmed moons, of which the four largest ones are Io, Europa, Ganymede and Callisto. All moons of Jupiter obey Kepler's law.

here would be the mass of Jupiter.