Formula and Method for Kepler's Third Law
Kepler's Third Law describes how long an orbiting body takes to complete one revolution. Johannes Kepler observed empirically in 1619 that for the planets orbiting the Sun, the square of the orbital period is proportional to the cube of the semi-major axis: T² ∝ a³. Isaac Newton later derived this from the law of gravitation and generalized it to any two bodies orbiting their common center of mass: T² = 4π²a³ / [G(M + m)], where M and m are the masses of the central and orbiting bodies, a is the orbit's semi-major axis, G is the gravitational constant, and T is the orbital period. This calculator solves that equation for T, then derives the mean orbital speed from the result.
How the calculation works
Newton's form comes from setting the gravitational force between two masses equal to the centripetal force needed to keep a body on a circular path, then extending the result to elliptical orbits using the semi-major axis a in place of the radius. Enter the central body's mass M (and, if it is not negligible, the orbiting body's mass m), pick the units for each, then enter the semi-major axis a and its unit. The calculator converts everything to SI units (kilograms and meters), computes T = 2π√(a³ / [G(M+m)]) in seconds, and converts that period to days and years for easier reading. It also reports the mean orbital speed v = 2πa / T = √(G(M+m) / a), which is exact for a circular orbit and a close approximation of the time-averaged speed for a mildly elliptical one.
Working with units
- For anything orbiting the Sun, using astronomical units (AU) for distance, years for period, and solar masses (M☉) for mass is convenient because G·M☉ works out to almost exactly 4&pi² in those units, so the formula reduces to T² ≈ a³ when the orbiting body's mass is negligible.
- 1 AU = 1.495978707 × 10ⁿ¹ m, 1 solar mass (M☉) = 1.98847 × 10³⁰ kg, and 1 Earth mass (M⊕) = 5.9722 × 10²⁴ kg.
- The gravitational constant is G = 6.6743 × 10⁻¹¹ N·m²/kg² — always convert M, m, and a to kilograms and meters before combining them with G in SI units.
Real-world applications
- Solar system mechanics: predicting a planet's, comet's, or asteroid's orbital period from its measured distance from the Sun.
- Exoplanet science: astronomers use the transit or radial-velocity period together with Kepler's Third Law to estimate a planet's orbital distance and the host star's mass.
- Satellite and mission design: engineers choose a semi-major axis (and therefore an altitude) that gives a satellite the period required for its mission, such as a 24-hour geostationary orbit.
- Binary star systems: measuring the orbital period and separation of two stars lets astronomers calculate the combined mass of the pair.