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Science

Kepler's Third Law (Orbital Period) Calculator

Kepler's Third Law says the square of an orbital period T is proportional to the cube of the semi-major axis a, with proportionality 4π²/(G·M). Enter a central mass (Sun, Earth or custom) plus a semi-major axis to get the orbital period T for a planet, moon or spacecraft — or flip the mode to recover a from a known period.

Result

Other units

Minutes
Hours
Days
Years (Julian)
Mean orbital speed

Formula

T = 2π · √(a³ / (G · M))

Uses CODATA 2018 value G = 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻² and 1 AU = 1.49598 × 10¹¹ m. Assumes the two-body limit with the orbiting body much less massive than the primary — a good approximation for planets, moons and spacecraft. Use the semi-major axis a for elliptical orbits; for circular orbits a is just the orbital radius. Real periods differ slightly due to perturbations from other bodies and relativistic corrections.

Formula

T = 2π · √(a³ / (G · M)); invert to a = ∛(G · M · T² / (4π²))

Frequently asked

Why does Earth come out to almost exactly 365.25 days?

When a = 1 AU = 1.49598 × 10¹¹ m and M = M☉ = 1.989 × 10³⁰ kg, T = 2π·√(a³/(G·M)) ≈ 3.156 × 10⁷ s ≈ 365.25 days. That's exactly the definition of a Julian year — astronomers chose 365.25 days as a clean time unit (it's what powers the "light-year"). NASA's quoted sidereal year of 365.256 days differs because (a) Earth itself adds a tiny bit of mass making the effective G·(M+m) slightly larger, and (b) other planets perturb the orbit. The tool's idealised two-body answer of ~365.25 days agrees with NASA to four significant figures.

Why is a geostationary satellite period 23 h 56 min, not 24 h?

A geostationary satellite must keep step with Earth's rotation relative to the distant stars, and that's a sidereal day of 23 h 56 min 4 s — not the 24 h solar day. The extra ~4 minutes in a solar day come from Earth also orbiting the Sun by ~1° per day, so the planet must rotate that extra degree before the Sun is overhead again. Plug 23.9345 h and M = M⊕ into this tool and you get a ≈ 42 164 km — i.e. 35 786 km altitude, the ITU-standard GEO height.

What if it's a binary star or a satellite that isn't much lighter than its primary?

The full two-body form is T² = 4π²·a³ / (G·(M + m)), i.e. you use the total mass. When m/M < 0.001 the error is well below 0.05 %; for the Earth–Moon system m/M ≈ 0.0123 so the idealised version is off by ~0.6 %. For planet–Sun systems the approximation is rock solid. Binary stars and near-equal pairs like Pluto–Charon need both bodies counted, and you also need to remember that each body orbits the common centre of mass rather than the other body.

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