By Laurel Brown – Updated March 24, 2022
Understanding the period of an orbit is essential for everything from spacecraft navigation to predicting eclipses. While the inclination and eccentricity of an orbit must be derived from long‑term observations, the relationship between the semi‑major axis and the orbital period is well defined by Kepler’s Third Law. Once you know the semi‑major axis—typically listed in astronomical tables—you can compute the period with confidence.
Consult reliable astronomical tables (e.g., JPL Horizons or NASA’s planetary fact sheets) for the semi‑major axis of the body of interest. For planets, this value is the mean distance from the Sun; for satellites, it is the mean distance from the primary planet.
An astronomical unit equals the mean Earth–Sun distance, approximately 93,000,000 mi (150,000,000 km). Express the semi‑major axis in AU to align with Kepler’s formula.
Kepler’s Third Law states that the square of the orbital period (P, in years) equals the cube of the semi‑major axis (a, in AU): \(P^2 = a^3\)
Insert the AU value for a, solve for P by taking the square root, and you’ll obtain the period in years.
For bodies with short periods—such as Mercury or the Moon—express the result in days. Divide the period in years by 365.25. For planets with longer orbits, the year unit is usually sufficient.
If you can’t find semi‑major axis data (common for newly‑discovered comets or man‑made satellites), you can derive it from a series of precise, time‑stamped observations. Modern orbit‑determination software will fit the data to a Keplerian model and return the period.
Always use the maximum distance (apocenter) when estimating the semi‑major axis. Averaging distances assumes a circular orbit and will underestimate the true semi‑major axis, leading to an incorrect period.
