Kepler's third law — harmonic

Statement

The square of the orbital period T² is proportional to the cube of the semi-major axis a³ — for all planets orbiting the same star:

T² / a³ = 4π² / (G·M) = const

with G the gravitational constant and M the star's mass.

Why is it true?

Kepler found this ratio in 1619 by cross-checking astronomical data — without knowing why. Newton (1687) later proved it follows from the inverse-square law of gravitation.

Kepler III lets astronomers "weigh" stars: measure T and a of a planet or binary star and M falls out — a key tool of modern astronomy.

Applications

Estimating stellar and supermassive-black-hole masses, exoplanet detection (transit + RV), satellite orbit sizing, astronomical distance scales.

Topics that use this theorem

Related theorems

Step-by-step proofs