Kepler's First Law: The Law of Ellipses

Statement: The orbit of each planet is an ellipse with the Sun at one focus.

This was the most radical of Kepler's three discoveries. Every astronomer before him—from Ptolemy to Copernicus—had assumed that celestial orbits must be circles, or combinations of circles. Kepler, working with Brahe's data on Mars, showed that no combination of circles could match the observations. The orbit was an ellipse.

What Is an Ellipse?

An ellipse is a closed curve defined by two points called foci (singular: focus). For any point on the ellipse, the sum of the distances to the two foci is constant. An ellipse has a semi-major axis (a), which is half the longest diameter, and a semi-minor axis (b), which is half the shortest diameter. See Ellipse Geometry Explained for the full treatment.

The eccentricity (e) measures how elongated the ellipse is. If e = 0, the ellipse is a perfect circle. If e is close to 1, the ellipse is very elongated. Most planetary orbits have low eccentricity: Earth's is about 0.0167, meaning its orbit is nearly circular. Mars has e ≈ 0.0934, which made it just eccentric enough for Kepler to detect the elliptical shape.

The Sun at One Focus

The Sun does not sit at the center of the ellipse. It sits at one of the two focal points. The other focus is an empty point in space. This off-center placement is what causes the planet's distance from the Sun to vary over the course of an orbit. The closest point to the Sun is called perihelion; the farthest is aphelion (see the glossary).

How Kepler Discovered It

Published in Astronomia Nova (1609), the First Law emerged from Kepler's painstaking analysis of Mars's orbit. He tried over 70 different circular-orbit models before finally accepting that the orbit was an ellipse. He later wrote that he felt as if he had been at war with Mars for years.

Why It Matters

The First Law applies not only to planets but to any object orbiting another under gravity: moons, asteroids, comets, artificial satellites, and even stars orbiting each other in binary systems. It was later derived from Newton's law of gravity, which showed that an inverse-square force law naturally produces elliptical orbits.