Kepler's Second Law: The Law of Equal Areas

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

This law describes how fast a planet moves at different points in its orbit. Unlike the First Law (which describes the shape), the Second Law describes the dynamics: the planet speeds up as it approaches the Sun and slows down as it moves away.

What This Means in Practice

Imagine drawing a line from the Sun to the planet and tracking the area this line sweeps out over, say, 30 days. According to the Second Law, the area swept in any 30-day period is the same, no matter where the planet is in its orbit.

Near perihelion (closest to the Sun), the planet is moving fast. The swept region is short and wide (like a fat, short pie slice). Near aphelion (farthest from the Sun), the planet moves slowly. The swept region is long and thin (a tall, narrow pie slice). Both slices have the same area.

Connection to Angular Momentum

The Second Law is a direct consequence of the conservation of angular momentum. Because gravity is a central force (always directed toward the Sun), it exerts no torque on the planet, so the planet's angular momentum stays constant. When the planet is closer (smaller r), it must move faster (larger v) to maintain the same angular momentum (L = m × r × v).

Discovery

Kepler actually discovered this law before the First Law, during his analysis of Mars in the early 1600s. He published both in Astronomia Nova (1609). He initially used the equal-areas rule as a computational tool while still trying to fit circular orbits, only later realizing it applied to the elliptical orbit he eventually found.

Hands-On

You can explore this law visually using the orbit simulator or the equal areas demonstration.