Kepler's Third Law

Orbital period from the semi-major axis and back, comparing orbits, and satellite periods by Kepler's third law.

What to find:

the orbital period around the Sun from the semi-major axis

the semi-major axis from the period

the period of a second body (comparing orbits)

the semi-major axis of a second body (comparing orbits)

the period of a satellite around a planet or the Sun

For an Earth satellite the orbit radius is the Earth's radius (6371 km) plus the altitude.

Test yourself

The Kepler's third law calculator finds the orbital period of a planet from the semi-major axis of its orbit and back, compares the orbits of two bodies and computes the period of a satellite around the Earth, Moon, Mars, Jupiter, Saturn or the Sun.

Kepler's third law

^T₁²/_T₂² = ^a₁³/_a₂³

The squares of the orbital periods of bodies orbiting the same central body are in the ratio of the cubes of their semi-major axes. For planets it is convenient to use years and astronomical units: then T² = a³.

Newton's form

Newton showed that the period depends on the mass M of the central body:

T = 2π · √(a³ / GM)

This is how satellite periods are calculated. The orbit radius is measured from the planet's center: for an Earth satellite it is 6371 km plus the altitude.

Example 1. Jupiter: a = 5.2 AU, T = √(5.2³) ≈ 11.86 years.

Example 2. Geostationary orbit: a = 42,164 km, T = 2π · √(42,164³ / 398,600) ≈ 23.93 h — a sidereal day.