Kepler Third Law Calculator

Frequently asked questions

What is Kepler's third law?

Kepler's third law states that the square of an orbit's period T is proportional to the cube of its semi-major axis a: T² = 4π²a³/μ, where μ is the central body's standard gravitational parameter (μ = GM). For a circular orbit, a is simply the orbit radius. All orbits around the same body with the same semi-major axis share the same period, whatever their eccentricity.

How do you calculate the orbital period from the orbit radius?

Rearrange to T = 2π√(a³/μ). Example: Earth at 1 AU (a = 149,597,870,700 m) around the Sun (μ = 1.32712440018e20 m³/s²) gives T = 31,558,196 s = 365.256898 days, the Gaussian year. Mean orbital speed is then v = 2πa/T = 29.7847 km/s.

How do you find the semi-major axis from the orbital period?

Use a = (μT²/(4π²))^(1/3). Example: a geostationary orbit must match Earth's sidereal rotation of 86,164.0905 s, giving a = 42,164.17 km from Earth's center (about 35,786 km altitude).

Can Kepler's third law find a planet's mass?

Yes. From a moon's orbit, μ = 4π²a³/T², then mass M = μ/G with G = 6.67430e-11 m³/(kg·s²). This is how planetary masses are measured from their satellites.

Does Kepler's third law work for elliptical orbits?

Yes. The law uses the semi-major axis a, not the closest or farthest distance. A highly elliptical orbit with the same semi-major axis as a circular one has exactly the same period; only the speed along the orbit varies.

Why does this calculator give 27.45 days for the Moon instead of 27.32?

The calculator uses the standard approximation μ = GM of the central body, assuming the orbiter's mass is negligible. The full two-body form is μ = G(M+m). The Moon is 1.23% of Earth's mass, and solar perturbations also act on it, so the real sidereal month is 27.3217 days. For planets and satellites (mass ratio below 0.01%) the simple form is accurate to better than 0.01%.

What is the mean orbital velocity readout?

v = 2πa/T is the average speed over one full orbit. Earth averages 29.7847 km/s around the Sun; the ISS averages 7.6681 km/s at 408 km altitude. Instantaneous speed on an elliptical orbit varies, so this is the orbit-averaged value.

How is this different from escape velocity or a Hohmann transfer?

Kepler's third law gives period, radius and GM of a closed orbit. Escape velocity v_e = √(2GM/R) is the speed needed to leave a body entirely (an open trajectory, no period). A Hohmann transfer computes the delta-v and travel time to move between two orbits. They answer three different questions: how long, how fast to leave, and how to change orbits.

What units does the calculator use?

Period T: seconds, minutes, hours, days, or years (1 Julian year = 365.25 days = 31,557,600 s). Distance a: meters, kilometers, or astronomical units (1 AU = 149,597,870,700 m exactly). Gravitational parameter μ is entered and shown in m³/s², with a km³/s² readout.

Does it work for exoplanets and moons of other planets?

Yes. Pick the Mars μ preset or enter any custom μ for another star or planet. The tool assumes two-body, non-relativistic motion with the orbiter much lighter than the central body, which covers planets, moons, satellites and exoplanets.