Application of angular momentum conservation to Kepler's laws

Kepler's second law states that a line drawn from the Sun to a planet sweeps out equal areas in equal times. This is illustrated in Figure 1.

The angular momentum of the planet is also conserved since it moves fastest when closest to the Sun and slowest when at its greatest distance.

It can easily be shown that the ratio of the maximum and minimum velocities of a planet in orbit is in the inverse ratio to the maximum and minimum distance of the planet from the Sun.

Let the angular velocity of a planet be ω_P at the perihelion (closest point to the Sun) and ω_A at the aphelion (furthest point from the Sun). Let the distance of the planet from the Sun be r_P at perihelion and r_A at aphelion.

The angular momentum of the planet at perihelion is therefore mω_P = mv_Pr_P

and

the angular momentum of the planet at perihelion is therefore mω_A = mv_Ar_A

But by the law of conservation of angular momentum: mv_Pr_P = mv_Ar_A and so

v_Pr_P = v_Ar_A

giving: v_P= v_A[r_A/r_P]