Kepler's Third Law Calculator

Kepler's third law connects the time required to complete an orbit with the orbit's size. Johannes Kepler originally expressed the relationship by observing that the square of a planet's orbital period is proportional to the cube of its semi-major axis. The semi-major axis is half the longest diameter of an elliptical orbit. For a circular orbit it is simply the constant distance between the centers of the two bodies. Newtonian gravity supplies the constant of proportionality and makes the law useful beyond planets orbiting the Sun. The general two-body equation is T squared = 4 pi squared a cubed divided by G times the total mass. Here T is period, a is semi-major axis, G is the gravitational constant, and total mass is the sum of both bodies. Including both masses matters for binary stars and other systems whose components are comparable. For a planet around a star, the planet's mass is usually small enough that omitting it changes the result only slightly. This calculator accepts the primary mass in solar masses, the orbiting-body mass in Earth masses, distance in astronomical units, and time in days. It converts those convenient astronomy units to SI units before evaluating the equation. In period mode it calculates T from a. In semi-major-axis mode it rearranges the same equation and takes the cube root. The displayed orbital velocity is the circular speed at the calculated distance, found from the square root of gravitational parameter divided by radius. The circular-speed result is a helpful reference, but speed varies around an elliptical orbit. An object moves fastest near periapsis and slowest near apoapsis, so a detailed trajectory requires the vis-viva equation and the object's current radius. The calculation also assumes an isolated two-body system with constant masses. Atmospheric drag, thrust, nonspherical gravity, resonances, and perturbations from additional bodies are not modeled. Within those limits, Kepler's law provides an accurate first calculation for planetary systems, artificial satellites, moons, and binary objects.