Angular Momentum and Orbits

The physical explanation for Kepler's Second Law is the conservation of angular momentum. This page explains why.

What Is Angular Momentum?

Angular momentum (L) is a measure of rotational motion. For a planet of mass m moving with velocity v at distance r from the Sun, the angular momentum is:

L = m × r × v × sin(θ)

where θ is the angle between the velocity vector and the position vector. For a planet in orbit, this simplifies to L = m × r × v_⊥, where v_⊥ is the component of velocity perpendicular to the line connecting the planet to the Sun.

Why Is It Conserved?

Angular momentum is conserved whenever the net torque on a system is zero. Gravity pulls the planet directly toward the Sun (a central force), so it exerts no torque about the Sun. Therefore, the planet's angular momentum about the Sun never changes.

Connection to the Second Law

Because L is constant:

When the planet is closer to the Sun (small r), it must move faster (large v_⊥) to keep L constant.
When the planet is farther (large r), it moves slower (small v_⊥).

The area swept out per unit time is dA/dt = L / (2m), which is constant. This is exactly Kepler's Second Law.

Newton proved this rigorously in the Principia. It's important to note that the equal-areas law holds for any central force, not just gravity. What makes gravity special is that it also produces elliptical orbits (First Law) and the T² ∝ a³ relationship (Third Law).

Try the equal areas demonstration to see this visually.