Kepler's Laws of Planetary Motion

Core Idea

Kepler's three empirical laws describe planetary orbits: (1) Planets move in ellipses with the Sun at one focus. (2) A line from the Sun to a planet sweeps equal areas in equal times (consequence of angular momentum conservation). (3) The square of the orbital period is proportional to the cube of the semi-major axis: T² ∝ a³ (specifically T² = 4π²a³/GM). Newton derived all three from his law of universal gravitation, showing they are not independent empirical facts but consequences of deeper physics.

How It's Best Learned

Derive Kepler's third law for circular orbits from Newton's law of gravitation and centripetal force. Then generalize to ellipses by replacing r with the semi-major axis a. Use Kepler's second law to explain why planets move fastest at perihelion.

Common Misconceptions

• Thinking planetary orbits are perfect circles: they are ellipses, though many planets have low eccentricity that makes them nearly circular.
• Confusing Kepler's third law: it is T² ∝ a³, not T ∝ a or T² ∝ a².
• Believing Kepler's laws only apply to planets: they apply to any two-body gravitational system (moons, comets, binary stars).

Explainer

From your work on orbital mechanics, you know that a satellite in a circular orbit maintains constant speed because gravity provides exactly the centripetal force needed. Kepler's laws generalize this: real orbits are not circles but ellipses, and the geometry of the ellipse encodes everything about how speed and position vary over the orbit.

An ellipse has two foci. The First Law states that the Sun sits at one focus — not the center — of each planet's elliptical orbit. This means the planet's distance from the Sun varies continuously. The closest point is called perihelion; the farthest is aphelion. For most planets the eccentricity is small (Earth's is about 0.017), so the orbit looks nearly circular, but the Sun is still slightly off-center. For comets, eccentricity can be close to 1, producing very elongated orbits that sweep far from the Sun.

The Second Law — equal areas in equal times — is a direct consequence of conservation of angular momentum, which you already know. As a planet approaches perihelion, it speeds up; as it recedes toward aphelion, it slows. The reason is the same as for a spinning ice skater pulling her arms in: as radius decreases, angular velocity must increase to conserve L = mrv. Near perihelion, the planet is moving fastest; near aphelion, slowest. The swept-area law is an elegant geometric expression of this: to cover equal areas in equal times, the planet must move along the arc faster when close to the focus and slower when far.

The Third Law, T² = 4π²a³/GM, is the most quantitatively powerful. To derive it for a circular orbit: set gravitational force equal to centripetal force, GMm/r² = mv²/r, and substitute v = 2πr/T. Solving gives T² = 4π²r³/GM, which is already the right form with r = a for circular orbits. For ellipses, the same result holds with the semi-major axis *a* replacing *r*, a result that requires calculus to prove in full generality. The practical power is enormous: if you know the orbital period of any body in the solar system, you can immediately determine the semi-major axis of its orbit, and vice versa. This is how the distances of the planets were first calculated — using timing from telescope observations before spacecraft were ever possible.