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Orbital Period Calculator (Kepler's Third Law)

Find an orbital period or semi-major axis from Kepler's third law T = 2π√(a³/GM), for any central mass in kilograms, Earth masses, or solar masses.

About this tool

The Orbital Period Calculator is a free, in-browser astronomy tool built on Kepler's third law in its Newtonian form, T = 2π√(a³/GM). Given the semi-major axis a of an orbit and the mass M of the central body it returns the orbital period T; switch modes and it inverts the relation to find the semi-major axis a = ∛(GM·T²/4π²) from a known period instead. G = 6.674×10⁻¹¹ N·m²/kg² is the gravitational constant.

Everything is computed locally in your browser, so nothing you enter leaves your device. The axis can be given in astronomical units, kilometres, or metres, and the central mass in solar masses (1 M☉ = 1.989×10³⁰ kg), Earth masses, or kilograms, letting you handle planets around the Sun, moons around a planet, or satellites around Earth with the same tool.

For a body orbiting the Sun the law simplifies to the familiar T(years) = √(a(AU)³): at a = 1 AU the period is 1 year, and at a = 4 AU it is √64 = 8 years. Period results are shown in seconds, days, and years, and the tool also reports the mean orbital speed v = 2πa/T for a circular orbit — about 29.8 km/s for Earth.

Frequently asked questions

What is Kepler's third law formula used here?
The Newtonian form T = 2π√(a³/GM), where a is the semi-major axis, M the central mass, and G the gravitational constant. Solving for distance gives a = ∛(GM·T²/4π²).
Why does a = 1 AU around the Sun give 1 year?
For a solar-mass central body the law reduces to T(years) = √(a(AU)³). At a = 1 AU that is √1 = 1 year; at a = 4 AU it is √64 = 8 years. The calculator reproduces this from the full formula.
What central masses can I use?
Any positive mass — solar masses (1 M☉ = 1.989×10³⁰ kg), Earth masses (5.972×10²⁴ kg), or kilograms. That covers planets around stars, moons around planets, and artificial satellites around Earth.
What is the mean orbital speed shown?
It is v = 2πa/T, the average speed for a circular orbit of radius a. Earth's works out to about 29.8 km/s. For eccentric orbits it is an average; actual speed varies along the path.

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