KEPLER
T² = a³, equal areas proven as an identity — Newton’s arithmetic, checked.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A planet, a Sun, and an ellipse with the Sun at one focus — Kepler’s first law, stated in 1609 and never wrong since. The units here are the astronomer’s own: distances in AU, times in years, so the third law becomes the cleanest sentence in physics: T² = a³.
Two laws
Gravity points at the Sun, so it can do work but never torque: energy and angular momentum are both frozen at launch. Everything that follows — the shape, the period, the speeding and slowing — is those two numbers refusing to change.
Solve
At the two apses the velocity is exactly sideways, so L = v·r at both — one equation. Energy at both — a second. Two equations, two speeds: the whole orbit’s kinematics falls out of conservation alone, no calculus required. The third law then prices the period.
Orbit
One full year in a few seconds: the planet tears through perihelion and coasts through aphelion, while the violet wedge — the area swept in the last twelfth of the period — stays stubbornly the same size. Position, velocity, and the clock are all closed forms; nothing here is integrated.
Audit
The tightest books on this site: vis-viva, angular momentum as a raw cross product, and the energy sum, all constant to 1e-12 at sixty points — and the second law proven as an algebraic identity, swept area ≡ ½L·t. Newton showed all three of Kepler’s laws live inside one inverse square; this page checks his arithmetic.