Orbital Period Calculator
T = 2*pi*sqrt(a^3/GM): Kepler's Third Law connects orbital size to orbital period
Kepler did not know why, Newton did 🖖
Kepler found the T^2-a^3 relation empirically. Newton later derived it from gravity, giving T = 2*pi*sqrt(a^3/GM).
The orbiting object's mass drops out 🖖
The formula T = 2π√(a³/GM) contains M, the central body's mass, but not the satellite's. A feather and a fully loaded space station at the same altitude circle Earth in exactly the same time. Orbital period depends only on how big the orbit is and what you orbit, never on how heavy you are.
A tunnel through Earth keeps orbital time 🖖
Drop a ball into a frictionless tunnel bored straight through Earth and it reaches the far side and returns in about 84.5 minutes. That is exactly the period of a satellite skimming Earth's surface. Both are governed by the same √(R³/GM), so the 'gravity train' and the lowest possible orbit share one clock.
Example problems
- ISS - ISS at ~400 km altitude: T ˜ 92 min — circles Earth 16 times a day
- GPS satellite - GPS satellites: T ˜ 12 h — each satellite orbits twice per day
- Geostationary - Geostationary orbit: T ˜ 24 h — appears fixed above one point on Earth
- Moon - Moon around Earth: T ˜ 27.3 days — the origin of the word "month"
- Mercury - Mercury around Sun: T ˜ 88 days — fastest planet
- Earth (around Sun) - Earth around Sun: T ˜ 365.25 days — defines the year
- Mars (around Sun) - Mars around Sun: T ˜ 687 days — nearly two Earth years