Orbital Period Calculator (Kepler's Third Law)

Johannes Kepler published his third law of planetary motion in 1619: the square of a planet’s orbital period is proportional to the cube of its orbital radius. Newton later showed why — it follows directly from the inverse-square law of gravity.

The formula

T = 2pi x sqrt(a^3 / (G x M))

Where T is the orbital period in seconds, a is the semi-major axis in meters, G = 6.674 x 10^-11 N·m²/kg² is the gravitational constant, and M is the mass of the central body in kilograms.

Central body masses for reference

Earth: 5.972 x 10^24 kg. Moon: 7.342 x 10^22 kg. Sun: 1.989 x 10^30 kg. Jupiter: 1.898 x 10^27 kg.

Checking against known orbits

The International Space Station orbits at about 408 km altitude, or 6,778 km from Earth’s center. Plugging in: T ≈ 92.6 minutes — matching the real value of 92.68 minutes. The Moon orbits at 384,400 km and completes one orbit in 27.3 days — confirmed by the formula.

Geostationary orbit (where a satellite appears stationary over one point on Earth) sits at exactly 35,786 km altitude, or 42,164 km from Earth’s center, yielding a period of exactly 24 hours.

Elliptical orbits

For elliptical orbits, use the semi-major axis a (half the longest diameter of the ellipse) rather than a circular radius. The period depends only on the semi-major axis, not on the eccentricity. A very elongated comet orbit and a nearly circular orbit with the same semi-major axis have the same period.