Keplers derde wet van planetaire beweging

Consider a planet orbiting the Sun in an elliptical orbit. If the planet takes time T to complete one orbit around the Sun, its angular velocity is expressed as two pi divided by T.

The centripetal force acting on the orbiting planet is expressed as the products of its mass, square of its angular velocity, and its average distance from the Sun.

Substituting for omega, the centripetal force equation can be obtained in terms of the planet's orbital period and the orbit's semi-major axis.

The Sun's gravitational force provides the centripetal force for the planet to orbit around the Sun. Substituting and rearranging the terms, square of orbital period equals the product of a constant term and the cube of the orbit's semi-major axis.

Therefore, the square of a planet's orbital period is proportional to the cube of the orbit's semi-major axis. This is popularly known as Kepler's third law of planetary motion. The proportionality constant, which involves the gravitational constant and the Sun's mass, was derived later when Newton stated his law of gravitation.