Orbital Period Calculator: Kepler's Third Law
Kepler's third law relates an orbit's size to its period: the orbital period T equals 2 pi times the square root of the semi-major axis cubed divided by GM, where G is Newton's gravitational constant and M is the mass of the central body. This elegant relationship allows astronomers to calculate how long any body takes to orbit another simply from the orbital size. Select the central body below, enter the semi-major axis, and the calculator returns the orbital period in seconds, hours, days, and years.
Kepler's third law formula
T = 2π * sqrt(a^3 / GM)
Where T is the orbital period in seconds, a is the semi-major axis in meters, G = 6.674 x 10 to the -11th N m squared per kg squared (Newton's gravitational constant), and M is the mass of the central body in kilograms. The GM product (standard gravitational parameter) is known more precisely than G or M individually, so this calculator uses GM directly from NASA values.
Example orbital periods
Earth around Sun: 365.25 days (1 AU semi-major axis). Mars around Sun: 686.97 days (1.524 AU). Moon around Earth: 27.32 days (384,400 km). The International Space Station: about 92 minutes at 408 km altitude above Earth.
Orbital period: frequently asked questions
What is Kepler's third law?
Kepler's third law states that the square of the orbital period (T) of a planet is proportional to the cube of the semi-major axis (a) of its orbit: T squared is proportional to a cubed. In precise form: T = 2 pi * sqrt(a cubed / GM), where G is the gravitational constant and M is the mass of the central body.
What is the semi-major axis?
The semi-major axis is half the longest diameter of an elliptical orbit. For circular orbits it equals the orbital radius. For Earth's orbit around the Sun, the semi-major axis is approximately 1 AU (149,597,870 km).
Can I use this for orbits around planets?
Yes. Select the central body (Sun, Earth, Jupiter, etc.) and enter the semi-major axis of the satellite's orbit. The calculator uses the standard gravitational parameter (GM) for each body.
What is the orbital period of the Moon?
The Moon orbits Earth with a semi-major axis of about 384,400 km. Using Earth's GM = 3.986 x 10 to the 14th m cubed/s squared, the period is approximately 27.32 days (the sidereal month).
What is the difference between the sidereal and synodic period?
The sidereal period is the time for one complete orbit relative to the fixed stars, which is what Kepler's law calculates. The synodic period is the time between successive identical configurations with the Sun (e.g., full Moon to full Moon), which for the Moon is about 29.53 days.
Official sources
- NASA Planetary Fact Sheets (GM values): nssdc.gsfc.nasa.gov/planetary/factsheet/.
- NIST CODATA: Gravitational constant G: physics.nist.gov.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.