Orbital Period Calculator

Orbital period is the time required for one complete revolution, and for a light body circling a much heavier one it depends on only two quantities: the semi-major axis and the central mass. The Newtonian form of the third law of Kepler gives T equal to 2 pi times the square root of a cubed divided by GM. Because period goes as the three-halves power of distance, quadrupling the semi-major axis multiplies the period by eight. Eccentricity does not appear at all, so a circular orbit and a highly elongated ellipse sharing the same semi-major axis have identical periods, even though the ellipse spends most of its time far out and races through periapsis. Speed at any instant follows the vis-viva relation, v squared equal to GM times the quantity 2 over r minus 1 over a. Familiar values anchor the scale. In solar units the law collapses to period in years squared equals semi-major axis in astronomical units cubed, giving Mercury 88 days at 0.387 AU and Jupiter 11.86 years at 5.20 AU. Around Earth, the International Space Station near 420km altitude takes about 93 minutes, GPS satellites at 20,200km altitude take 11 hours 58 minutes, and geostationary orbit at 35,786km altitude matches the sidereal day of 23 hours 56 minutes. The formula assumes a point-mass primary and a negligible satellite mass; for comparable masses the sum of both must be used. Real satellites also feel the equatorial bulge of Earth, atmospheric drag below roughly 600km, and lunar and solar perturbations, all of which shift observed periods measurably.

Johannes Kepler derived the relationship from the naked-eye positional observations of Tycho Brahe, accurate to roughly one to two arcminutes and the finest data available before the telescope. Kepler published the first two laws in Astronomia Nova in 1609 after years of work on the stubborn orbit of Mars, but the period-distance relation eluded him for another decade. He recorded finding it in May 1618 and published it in Harmonices Mundi in 1619, where it sits among his broader attempt to locate musical harmony in planetary motion. In that original form the law was purely empirical: a ratio that happened to hold across the six then-known planets, with no account of why. Isaac Newton supplied the reason in the Principia in 1687, demonstrating that an inverse-square attraction produces elliptical orbits and that the constant of proportionality contains the mass of the central body. That converted a descriptive rule into a measuring instrument, since weighing the Sun, the planets, and later binary stars became a matter of observing a period and a separation. The technique still dominates. Edmond Halley applied the same framework in 1705 to predict the return of the comet now named for him, and the Kepler space telescope, launched by NASA in 2009, used transit timing together with the third law to convert measured periods into orbital distances for more than 2,600 confirmed exoplanets.