Kepler's Laws

Before anyone knew why planets move, Johannes Kepler found three precise rules for how they do, distilled from decades of naked-eye observations. They are the exact solution of the two-body problem, and Newton later showed they follow directly from an inverse-square law of gravity.^[1]

Eccentricity e = 0.50

First law: planets orbit on an ellipse with the Sun at one focus (the other focus is empty — the small dot). Second law: every shaded slice takes the same amount of time to sweep, yet they're the same area — so the planet must move faster when close (perihelion) and slower when far (aphelion). Raise the eccentricity and the speed-up becomes dramatic. Third lawties orbits together: the period squared is proportional to the semi-major axis cubed (T² ∝ a³), which is why outer planets crawl and inner ones race.

First law — the law of ellipses. Each planet's orbit is an ellipse with the Sun at one of the two foci (not the centre). A circle is just the special case of zero eccentricity.

Second law — equal areas. A line from the Sun to the planet sweeps out equal areas in equal intervals of time. This is really conservation of angular momentum in disguise: since the swept area rate $\tfrac{dA}{dt} = \tfrac{L}{2m}$ is constant, the planet must speed up near the Sun and slow down far from it.

Third law — the harmonic law. The square of the orbital period is proportional to the cube of the semi-major axis,

T^2 \propto a^3.

Double an orbit's size and its year grows by a factor of $2^{3/2} \approx 2.8$ . This single relation lets us weigh the Sun, map the solar system, and — applied to stars and galaxies — detect unseen mass, including the planets orbiting other stars.