Johannes Kepler (1609, 1619) discovered three laws governing how planets move — empirically, from Tycho Brahe’s painstaking observations. Newton later proved all three follow from gravity and calculus. They apply to any orbiting body: planets, moons, comets, spacecraft, binary stars.

The three laws

Law 1 (Ellipses): Every orbit is an ellipse with the central body at one focus.

A circle is a special ellipse with eccentricity $e = 0$. Earth’s orbit has $e = 0.017$ (nearly circular). Mercury has $e = 0.206$ (noticeably oval). Comets can have $e > 0.99$ (extreme ellipses, barely captured).

Law 2 (Equal Areas): A line from the orbiting body to the central body sweeps out equal areas in equal times.

This means the body moves faster near periapsis (closest approach) and slower near apoapsis (farthest point). It’s a direct consequence of conservation of angular momentum.

Law 3 (Harmonic Law): The square of the orbital period is proportional to the cube of the semi-major axis:

$$T^2 = \frac{4\pi^2}{GM} a^3$$

For the solar system, this means outer planets have much longer years: Earth takes 1 year, Jupiter takes 12, Neptune takes 165.

Watch the planet speed up near periapsis (right) and slow down near apoapsis (left). The swept areas (coloured wedges) are all equal despite covering different arc lengths.

From empirical to fundamental

Kepler discovered these laws by fitting curves to data — he had no theory of why they held. Newton showed (1687) that all three laws are mathematical consequences of an inverse-square gravitational force. This was the birth of mathematical physics: a single equation ($F \propto 1/r^2$) explaining the motion of everything in the solar system.

Kepler’s laws are exact for two bodies. Real solar systems have perturbations from other planets, but the two-body solution is the foundation of all orbital mechanics.