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    Orbital mechanics

    Kepler's Third Law Calculator

    Solve for orbital period or semi-major axis, including the orbiting body's mass.

    T² = 4π² a³ / (G (M + m))

    Inputs

    Results

    Orbital period
    365.256 days
    365.3 days
    Worked solution
    1. G (M + m) = 1.3271 × 10^20 m³/s²
    2. T = 2π √(a³ / G(M+m)) = 31,558,000 s = 365.3 days

    How it works

    Johannes Kepler found empirically in 1619 that T² ∝ a³ for the planets. Newton's gravity explains why and adds the dependence on mass, which lets astronomers weigh stars and planets from the orbits of their companions.

    In units of years, AU and solar masses the law reduces to T² = a³ / (M + m).

    Variables

    • T — orbital period
    • a — semi-major axis of the relative orbit
    • M, m — central and orbiting masses

    Assumptions and limits

    • Two-body Keplerian motion
    • Including m gives the exact two-body period; for planets around the Sun m is usually negligible

    For education only. These models are simplified and are not suitable for spacecraft navigation or safety-critical engineering. See our scientific accuracy policy.