Kepler's Second Law, The Law of Areas

Definition

A line joining any planet to the Sun sweeps out equal areas in equal times.

When the planet is closer to the sun, it has greater angular velocity that when farther, resulting in the effect below. This law is identical with the law of Conservation of Angular Momentum:

In the equation above, we analyze the areas as triangles or .

Assuming is at rest, the angular momentum of is:

Where the axis is perpendicular to the plane of orbit (In/Out of the screen). Therefore:

Because and are constants, is also a constant, verifying Kepler’s Second Law.