This calculator applies Kepler's third law to find how long a satellite, moon, or planet takes to complete one orbit — or, working backward, what orbital radius produces a target period or velocity. It's built for students checking orbital mechanics homework, hobbyists exploring satellite orbits, and anyone curious why a GPS satellite and a geostationary satellite move at such different speeds.

How the Orbital Period Calculator works

For a circular orbit, gravity supplies exactly the centripetal force needed to keep a body moving in a circle. Setting Newton's law of gravitation (F = GMm/r²) equal to the centripetal force requirement (F = mv²/r) and solving gives the orbital velocity v = √(GM/r). The orbital period then follows from the orbit's circumference divided by that velocity, which simplifies to Kepler's third law: T = 2π√(r³/GM).

The calculator assumes a circular orbit and a central body whose mass dominates the system (the orbiting object's own mass is treated as negligible) — the same simplifying assumptions used for satellites, moons, and most introductory orbital mechanics problems. G is the standard gravitational constant, 6.674×10⁻¹¹ N·m²/kg².

Inputs and what they mean

Central mass is the mass of the body being orbited — Earth's is about 5.972×10²⁴ kg. Orbital radius is measured from the center of that body, not its surface, so a satellite at 400 km altitude above Earth has a radius of Earth's own radius (6,371 km) plus 400 km, or 6,771 km. This distinction trips up a lot of first-time users: altitude and orbital radius are not the same number.

The Solve Radius tab flips the problem: give it a mass and a target period or velocity, and it returns the radius needed to achieve that value. This is the standard way engineers determine, for example, that a geostationary satellite must sit at roughly 35,786 km altitude to match Earth's 23.93-hour rotation.

Limits and edge cases

This calculator only models circular orbits. Real satellite and planetary orbits are usually elliptical, where the period still follows Kepler's third law using the semi-major axis in place of the circular radius — but velocity varies throughout the orbit (fastest at perigee, slowest at apogee) rather than staying constant as it does here. It also ignores atmospheric drag, the gravitational pull of other bodies (like the Moon perturbing a satellite's orbit around Earth), and relativistic corrections, all of which matter for high-precision mission planning. The altitude note assumes an Earth-sized central body (6,371 km radius) purely as an illustrative reference — it isn't meaningful when modeling orbits around other planets, moons, or stars.