Relate an orbit's period, size and central mass with T = 2π√(a³ / GM) — enter the semi-major axis and the mass being orbited to get the orbital period, or solve for the axis or mass. Use AU, years and solar masses if you like.
Kepler's third law relates the orbital period T to the semi-major axis a of the orbit and the mass M being orbited: T² = 4π²a³ / (GM), or T = 2π√(a³/GM). Bigger orbits take longer; more massive central bodies make orbits faster.
Switch the units to AU, years and solar masses for solar-system problems, where the law simplifies to T² = a³ for objects orbiting the Sun. Enter any two quantities to find the third.
It states that the square of an orbital period is proportional to the cube of the semi-major axis: T² = 4π²a³/(GM). Larger orbits have proportionally longer periods.
Rearrange to M = 4π²a³/(GT²). Set 'Solve for' to Central mass and enter the orbit's period and semi-major axis — this is how astronomers weigh stars and planets.
Only in units of years and AU for objects orbiting the Sun, where GM/4π² = 1. For other central masses use the full T² = 4π²a³/(GM).
It is half the longest diameter of the elliptical orbit — the average of the closest and farthest distances from the central body. For a circle it equals the radius.