Kepler's Third Law Calculator

Kepler's Third Law, published in 1619, states that the square of a planet's orbital period (T²) is proportional to the cube of its semi-major axis (a³). In the Solar System, when T is measured in Earth years and a in astronomical units, this ratio equals exactly 1 for all planets: T² = a³.

The simplified form for Solar System bodies is T² = a³, where T is the orbital period in years and a is the semi-major axis in AU. The general form is T² = (4π² / GM) × a³, where G is the gravitational constant (6.674×10⁻¹¹ N·m²/kg²) and M is the mass of the central body. This general form applies to any two-body orbital system.

It implies that planets farther from the Sun take disproportionately longer to complete an orbit — not just longer, but the period grows faster than the distance. Neptune, about 30 AU from the Sun, takes roughly 165 years to orbit, compared to Earth's 1 year at 1 AU. The law reveals a deep mathematical harmony in orbital mechanics.

No — Kepler's Third Law applies to any orbiting body, including moons, artificial satellites, comets, asteroids, and even binary star systems. The key is using the correct central mass in the general equation. For Earth's satellites, for example, you would use Earth's mass instead of the Sun's mass.

By rearranging the general form, M = (4π² × a³) / (G × T²). If you measure any planet's orbital period T (in seconds) and semi-major axis a (in meters), you can plug those values into this formula along with G = 6.674×10⁻¹¹ to calculate the Sun's mass. Earth's orbit gives approximately 1.989×10³⁰ kg.

Mars has a semi-major axis of about 1.524 AU. Applying Kepler's Third Law: T = √(1.524³) = √(3.540) ≈ 1.881 years, or roughly 686 Earth days. This matches observed data very closely, demonstrating the precision of Kepler's law.

An astronomical unit (AU) is defined as the average distance between the Earth and the Sun, approximately 149.6 million kilometers (1.496×10¹¹ meters). It is the standard unit for expressing distances within the Solar System and is especially useful for Kepler's Third Law calculations.

Yes. To calculate the orbital parameters of a moon, use the general equation T² = (4π² / GM_planet) × a³, where M_planet is the mass of the planet being orbited. The same principle applies — the farther a moon orbits from its planet, the longer its orbital period.