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Gravitation · Foundations

Kepler's Three Laws

Before Newton explained why planets move as they do, Johannes Kepler discovered three precise rules describing how they move, drawn from decades of observation. Each law below is shown as a live animation.

All three animations share one clock
I

The Law of Ellipses

The orbit is an ellipse with the Sun at one focus
0.50
Every planet traces an ellipse, not a circle, with the Sun fixed at one focus (the other focus is empty). An ellipse is the set of points whose two focal distances add to a constant, so as the planet moves r₁ + r₂ = 2a always holds — watch the two coloured radii change while their sum stays fixed. Drag the slider to stretch the orbit: at e = 0 it's a circle; larger e means a more elongated ellipse.
II

The Law of Equal Areas

A line to the Sun sweeps equal areas in equal times
The orbit is divided into eight wedges of equal area, each swept in one-eighth of the orbital period. Because the areas are equal but their shapes are not, the planet must move faster when close to the Sun (near perihelion the wedge is short and fat) and slower when far away (near aphelion it is long and thin). This is really conservation of angular momentum in disguise. The glowing wedge marks the slice being swept right now.
III

The Harmonic Law

The period squared is proportional to the semi-major axis cubed
Planets on larger orbits travel slower and have farther to go, so they take dramatically longer to circle the Sun. Kepler found the exact relationship T² ∝ a³: the four planets on the left orbit at the correct relative speeds, and the plot on the right shows their T² plotted against a³ falling exactly on a straight line through the origin — the ratio T² / a³ is the same for every orbit.
One clock, three laws: the orbit's shape, the sweep of its radius, and its period are all fixed by gravity alone.
About this experiment ›

What you are looking at

Three animations driven by one shared clock. Each planet's position is computed the proper way — solving Kepler's equation numerically each frame, so the speeds you see are physically correct:

M = E − e·sin E  →  r = a(1 − e·cos E)

The three laws

I — Ellipses. Each orbit is an ellipse with the Sun at one focus; the two focal radii always satisfy r₁ + r₂ = 2a.

II — Equal areas. The Sun–planet line sweeps equal areas in equal times. This is angular momentum conservation: with gravity pulling along the line to the Sun, there is no torque, so

dA/dt = L / 2m = constant

III — Harmonic law. Newton's gravity makes the period depend only on the semi-major axis:

T² = (4π² / GM) · a³

Why it matters

Kepler distilled these rules from Tycho Brahe's naked-eye data decades before calculus existed. Newton then showed all three follow from a single inverse-square force — the first great unification in physics, and the same mathematics that plans every interplanetary mission today.