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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 ωP at the perihelion (closest point to the Sun) and ωA at the aphelion (furthest point from the Sun). Let the distance of the planet from the Sun be rP at perihelion and rA at aphelion.

The angular momentum of the planet at perihelion is therefore mωP = mvPrP

and

the angular momentum of the planet at perihelion is therefore mωA = mvArA

But by the law of conservation of angular momentum: mvPrP = mvArA and so

vPrP = vArA

giving: vP= vA[rA/rP]

 
 
 
© Keith Gibbs 2013