Orbital Period Calculator (Kepler's Third Law)

Kepler’s Third Law, published by Johannes Kepler in 1619, states that the square of the orbital period is proportional to the cube of the semi-major axis. Isaac Newton later showed why: it follows directly from the inverse-square law of gravity.

For any central body:

T = 2π √(a³ / GM)

Simplified for the Solar System (a in AU, T in years):

T (years) = a (AU)^(3/2)

This works because it is normalized to Earth’s orbit (1 AU, 1 year, central mass = 1 M☉).

Parameters:

• a = semi-major axis (half the longest diameter of the elliptical orbit)
• G = 6.674 × 10⁻¹¹ N·m²/kg²
• M = mass of the central body

Solar system examples (using T = a^1.5):

• Mercury: a = 0.387 AU → T = 0.241 years (88 days)
• Venus: a = 0.723 AU → T = 0.615 years (225 days)
• Mars: a = 1.524 AU → T = 1.881 years (687 days)
• Jupiter: a = 5.203 AU → T = 11.86 years
• Saturn: a = 9.537 AU → T = 29.46 years
• Halley’s Comet: a = 17.8 AU → T = 75.3 years

Why does mass of the orbiting body not matter? The orbital period depends only on the mass of the central body and the semi-major axis. A satellite and the Space Shuttle orbit at the same speed if at the same altitude. (This assumes M_central » M_orbiting — valid for planets around stars and satellites around planets.)

Checking the formula against known orbits

The International Space Station orbits at about 408 km altitude (6,778 km from Earth’s center) and completes one orbit in 92.68 minutes; plug those into T = 2π√(a³/GM) and you get 92.6 minutes. The Moon at 384,400 km gives 27.3 days, the actual sidereal period. Geostationary orbit sits at 35,786 km altitude (42,164 km from Earth’s center), yielding 23.934 hours. That is not a rounding error and it matters: a geostationary satellite matches Earth’s sidereal rotation of 23h 56m 04s, not the 24-hour solar day. The four-minute difference is exactly why the orbit works, since the Earth has to turn a little further each day to face the Sun again. This is where weather and broadcast satellites sit, appearing fixed over one spot. GPS (Global Positioning System) satellites are a different orbit entirely, at 20,180 km with a period of about 11h 58m.

Elliptical orbits

For elliptical orbits, use the semi-major axis a (half the longest diameter of the ellipse) rather than a circular radius. The period depends only on the semi-major axis, not on the eccentricity. A nearly circular orbit and a wildly elongated comet orbit with the same semi-major axis have exactly the same period.