Kepler Orbital Period Calculator
Applies Kepler third law to compute orbital period in years from semi-major axis in AU and total system mass in solar masses.
—
Kepler's third law: orbital period
Kepler's 3rd law says T² = (4π²/GM)·a³. Here T is the orbital period, a the semi-major axis, G the gravitational constant and M the central mass. Inside the solar system, if you measure a in AU and T in years, that whole constant collapses to 1 and you're left with T² = a³. Take Mars: a = 1.52 AU gives T² = 3.51, so T ≈ 1.88 years, which is exactly what we observe. The same law works closer to home. Around Earth (M = 5.972·10²⁴ kg), a satellite sitting at a = 6,781 km comes around every 92 min or so.
Applications
It shows up all over orbital engineering. GPS constellations are designed around it (T = 12 h, a ≈ 26,560 km), as are Starlink LEO satellites (T ≈ 95 min) and the math behind geostationary slots. Astronomers lean on it to characterize exoplanets with Kepler and TESS, reading the period from transits and then backing out the semi-major axis from the 3rd law and the stellar mass. Mission planners use it too, when working out Hohmann transfer windows for Mars and Jupiter.
FAQ
Why does T² = a³ work without units in the solar system? Measuring distance in AU and time in years forces 4π²/GM to come out as 1 when M = 1 solar mass. Step outside the solar system and you have to go back to the full equation with G and M.
Does it work for satellites of Earth? Yes. Just swap the Sun's mass for Earth's (5.972·10²⁴ kg). A satellite at a = 7,000 km then has T ≈ 97 min.
What if the orbit is very elliptical? The 3rd law runs on the semi-major axis a (the mean of perihelion and aphelion). Eccentricity doesn't enter into it, so two orbits with the same a share the same T no matter how stretched out one of them is.
Related Tools
Circular Orbit Period
Compute T = 2π·√(r³/(G·M)) for a circular orbit.
Kepler Orbital Period
Compute orbital period T = 2π√(a³/GM) from semi-major axis (m) and central mass (kg).
Kepler Third Law: Orbital Period
A semi-major axis in astronomical units returns the period around the Sun, T equal to the square root of a cubed. Mars at 1.524 AU orbits in 1.88 years.
Star Distance Modulus Calculator
Computes distance modulus m - M and equivalent distance in parsecs from apparent and absolute star magnitudes.
Ship Roll Period (IMO IS Code)
Computes the natural roll period of a ship using the empirical formula of the IMO International Code on Intact Stability (IS Code 2008), used in the weather criterion and in the inclining experiment. The period is T = 2 × C × beam ÷ square root of GM, with the coefficient C = 0.373 + 0.023 × (beam ÷ draught) − 0.043 × (length ÷ 100), so that rolling gets faster as the metacentric height grows. The number indicates comfort and safety: short periods, below some 8 seconds, reveal a stiff ship that rolls with high acceleration and punishes cargo and crew; long periods indicate a tender ship, with little stability reserve. The IMO empirical form was adopted rather than the pendulum expression T = 2π × radius of gyration ÷ square root of (g × GM), because the roll radius of gyration is rarely known on board — which is exactly what the coefficient C estimates from hull geometry. Enter the beam, the draught, the waterline length and the metacentric height GM.
Ocean Wavelength
Calculate the wavelength of an ocean wave in deep water, L = g·T² ÷ (2π), from the wave period T (s) and gravity g (9.81 m/s²). The result, in meters, is the distance between two successive crests — in deep water, it depends only on the period. Long-period waves (swell from distant storms) have much larger wavelengths than local wind waves. The wavelength sets the depth at which the wave 'feels' the bottom (about L/2), starts to refract and shoal until it breaks. It is a fundamental parameter of linear wave theory and coastal engineering. Enter the wave period.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.