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.

Read more about this topic:  Kepler's Laws Of Planetary Motion

Famous quotes containing the word law:

    I do not believe in lawyers, in that mode of attacking or defending a man, because you descend to meet the judge on his own ground, and, in cases of the highest importance, it is of no consequence whether a man breaks a human law or not. Let lawyers decide trivial cases.
    Henry David Thoreau (1817–1862)

    Most magazines have that look of being predestined to be left which one sees on the faces of the women whose troubles bring them to the Law Courts.
    Rebecca West (1892–1983)