Prove Kepler's second law of planetary motion

Explanation

Consider a planet moving in an elliptical orbit with sun at one of its foci (S). Let r be the distance of planet from sun and F be the gravitational force on the planet due to the sun, then torque on the planet is `vec(tau) = vec(r ) xx vec(F ) = 0` [`becauses vec(r ) and vec(F )` are oppositely directed]
But `vec(tau)= (d vec(L))/(dt) so vec(tau)= 0`
or `(d vec(L))/(dt)=0`
So `(dt)/(L)=` constant
Suppose the planet moves from P to P. in time `Delta t`
The area of triangular region, SPP. `= Delta vec(A) = (1)/(2) vec(r ) xx vec( P P)`

But `vec(P P)= vec(Delta r) = vec(v) Delta t= (vec(p))/(m) Delta t`
`therefore Delta vec(A) = (1)/(2) vec(r ) xx (vec(P))/(m) Delta t`
`(vec(Delta A))/(Delta t) = (1)/(2m) (vec(r ) xx vec(p)) = (1)/(2m) vec(L) [because vec(r ) xx vec(p) = vec(L)]`
`(Delta vec(A))/(Delta t)`= constant [`because vec(L)` and m are constant]
Thus, the areal velocity of the planet remains constant. That is, the radius vector joining planet to the sun sweeps out equal areas in equal intervals of time.