Kepler's Laws of Planetary Motion - Second Law

Second Law

"A line joining a planet and the Sun sweeps out equal areas during equal intervals of time."

In a small time the planet sweeps out a small triangle having base line and height .
The area of this triangle is given by
and so the constant areal velocity is

Now as the first law states that the planet follows an ellipse, the planet is at different distances from the Sun at different parts in its orbit. So the planet has to move faster when it is closer to the Sun so that it sweeps equal areas in equal times.

The total area enclosed by the elliptical orbit is

.

Therefore the period

satisfies

or

where

is the angular velocity, (using Newton notation for differentiation), and

is the mean motion of the planet around the Sun.