Kepler's Third Law
🛰️ Orbital mechanics
Question
Is there a pattern in how long orbits take?
Johannes Kepler noticed in 1609 that the planets' orbital periods are not random: the square of the period equals a constant times the cube of the average distance (T² proportional to a³). The astonishing part is that it applies to EVERYTHING - the planets, the Moon, the ISS, your phone's GPS satellites. The same rule, no matter what is orbiting.
The math
T²/a³ = 4π²/(GM). Check it: the Moon takes 423.5 times longer to orbit than the ISS (27.3 days vs 92.9 minutes), so it must orbit 423.5 to the power 2/3 = 56.4 times farther out: 6,791 km × 56.4 = 383,000 km - the Moon's actual distance (384,400 km). From timing alone.
Why it matters
You can weigh the Earth without ever touching it: measure the Moon's orbit and GM falls out of the equation. That is how we know Earth's mass to five significant figures - pure arithmetic on a clock and a distance.