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Kepler's Third Law Calculator

Find how long an orbit takes from its size.

How it is calculated

Kepler's third law links the size of an orbit to its period. In astronomical units, solar masses and years the law reads T² = a³ ÷ M, so T = √(a³ ÷ M). a is the semi-major axis of the orbit in astronomical units (AU, the Earth-Sun distance), M is the mass of the central body in solar masses (1 if empty or not positive) and T is the period in years. The period is also shown in days (years × 365.25). The semi-major axis must be above zero. The formula assumes the orbiting body is far lighter than the central one, which is right for planets around the Sun but only approximate for two similar stars.

Worked example

Mars orbits the Sun at a mean distance of 1.524 AU, and the central mass is 1 solar mass. Cube the distance: 1.524³ = 3.540. Divide by the mass: 3.540 ÷ 1 = 3.540. Square root: √3.540 = 1.88 yr. In days: 1.88 × 365.25 = 687.18 days. Doubling the central mass would shorten the period by a factor of √2, to about 1.33 years.

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Common mistakes

  • Entering distance in kilometres or millions of kilometres. The axis must be in astronomical units.
  • Entering the mass in kilograms or Earth masses instead of solar masses.
  • Using the distance from the Sun at one moment rather than the semi-major axis, which for elliptical orbits is the average of the closest and farthest distances.
  • Forgetting the cube: doubling the distance multiplies the period by about 2.83, not by 2.

About this calculator

This calculator finds an orbit's period from its size using Kepler's third law: T² = 4π²a³/(GM), where a is the semi-major axis of the orbit, M is the mass of the central body, and G is the gravitational constant. In the simplified form for solar-system objects, T² (in years) equals a³ (in AU) when orbiting the Sun.

Astronomers use this to determine orbital periods of planets, moons, exoplanets, and artificial satellites from their orbital distance, and — run in reverse — to infer a central body's mass from an orbiting object's period and distance, which is how astronomers weigh stars and planets they can't put on a scale. Enter the semi-major axis and central mass to get the orbital period.

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