Kepler Orbital Period Calculator

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.