Prove the Kepler’s second law of planetary motion.

Proof of Kepler’s Second Law:
Consider a planet moving in an elliptical orbit with the Sun at one of the foci. Let r be the position vector of the planet relative to the Sun, and F be the gravitational force exerted on the planet by the Sun. The torque τ exerted on the planet by this force about the Sun is given by:

[τ = r x F = 0]

Since the torque is zero, the angular momentum of the planet is conserved, meaning the planet’s areal velocity remains constant. Therefore, the radius vector joining the planet to the Sun sweeps out equal areas in equal time intervals.

This proves Kepler’s second law of planetary motion.