Kepler's Third Law Calculator

Kepler's third law (the harmonic law) states that the square of the orbital period is proportional to the cube of the semi-major axis: T^2 proportional to a^3. In exact form: T^2 = (4*pi^2 / (G*M)) * a^3, where G = 6.674e-11 N*m^2/kg^2 and M is the central body mass. For the Solar System, using Earth's orbit as reference: (T/1 year)^2 = (a/1 AU)^3.

The semi-major axis (a) is half the longest diameter of an elliptical orbit. For a circular orbit, a equals the radius. For an ellipse, a = (periapsis + apoapsis) / 2, where periapsis is the closest approach and apoapsis is the farthest point. Earth's semi-major axis is 1.000 AU = 149,597,870.7 km.

Rearrange Kepler's third law: M = (4*pi^2 * a^3) / (G * T^2). By measuring the orbital period T and semi-major axis a of any satellite (natural or artificial), you can calculate the mass of the central body. This technique discovered the masses of planets with moons, the Sun, and distant stars in binary systems.

Yes, the generalized form uses the total mass: T^2 = (4*pi^2 * a^3) / (G * (M1 + M2)). Here a is the separation between the two stars, and T is the orbital period. This is used to measure stellar masses of binary stars from astrometric or spectroscopic measurements. It is one of the primary ways stellar masses are determined observationally.

Mercury: 0.387 AU, 87.97 days. Venus: 0.723 AU, 224.7 days. Earth: 1.000 AU, 365.25 days. Mars: 1.524 AU, 686.97 days. Jupiter: 5.203 AU, 11.86 years. Saturn: 9.537 AU, 29.46 years. Uranus: 19.19 AU, 84.01 years. Neptune: 30.07 AU, 164.8 years. All values from NASA planetary fact sheets.