What is Kepler's Third Law?

Kepler's Third Law states T² ∝ a³: the square of an orbital period is proportional to the cube of the semi-major axis. The exact form is T = 2π√(a³ / GM), where G = 6.674 × 10⁻¹¹ N·m²/kg² and M is the mass of the central body.

What is the semi-major axis?

For an elliptical orbit, the semi-major axis is half the longest diameter. For a circular orbit it equals the orbital radius — the average distance between the orbiting body and the central body.

How do I find the semi-major axis if I know the period?

Rearrange Kepler's law: a = ∛(GM × T² / 4π²). Enter the known period in this calculator's "Axis from Period" mode and it computes the result automatically.

Can I use this for moons and artificial satellites?

Yes. Select Earth (or any planet) as the central body. For the Moon use Earth's mass (~5.972 × 10²⁴ kg) and the Moon's semi-major axis (~384,400 km). For a custom satellite, enter Earth's mass and the desired orbital altitude plus Earth's radius (6,371 km).

What units can I enter?

For the axis you can choose AU, km, or meters. For the period you can choose years, days, or seconds. The calculator converts internally before applying the formula.

Does this account for the mass of the orbiting body?

The simplified form T = 2π√(a³/GM) assumes the orbiting body is much less massive than the central body. For two bodies of comparable mass, the full form uses the sum of both masses, but for planets, moons, and satellites the approximation is accurate to better than 0.1%.

What is the geostationary orbit radius?

A geostationary satellite has T = 1 sidereal day (86,164 s). Solving for the axis gives a ≈ 42,164 km from Earth's center — about 35,786 km above the surface. You can verify this with the 'Axis from Period' mode using Earth as the central body.

Kepler's Third Law Calculator

Solve Kepler's Third Law in both directions — enter the semi-major axis to get the orbital period, or enter the period to get the semi-major axis, for any central body.

Kepler's Third Law

Solve For

Central Body

Semi-Major Axis

1 AU = 149,597,870.7 km

Quick Fill — Solar System

T = 2π √(a³ / GM)
a = ³√(GM · T² / 4π²)
G = 6.674 × 10⁻¹¹ N·m²/kg²

Select a preset or enter values, then click Calculate.

Orbital Period

T² ∝ a³

Input Values Used

Central body:

Body mass:

Semi-major axis:

Solar System Comparison

Body	Axis (AU)	Period (days)	Scale

Summary

Kepler's Third Law states that the square of a body's orbital period is proportional to the cube of its semi-major axis: T² = 4π²a³ / (GM). This calculator solves the law in both directions — given the semi-major axis you get the period, and given the period you get the semi-major axis. Preset central bodies (Sun and Earth) are included, and you can enter any custom mass to model moons, exoplanets, or artificial satellites.

How it works

Choose a solve direction: Period from axis, or Axis from period.
Select a central body preset (Sun or Earth) or choose Custom and enter the body mass in kilograms.
Enter the known value — semi-major axis in AU, km, or m; or period in years, days, or seconds.
The calculator applies T = 2π√(a³/GM) or its rearrangement a = ∛(GMT²/4π²).
Results are shown in multiple units with a Solar System comparison so you can sanity-check the answer.

Use cases

Verify planetary orbits using Kepler's third law for physics or astronomy coursework.
Find the semi-major axis of a satellite given only its orbital period.
Calculate where a planet or moon must orbit to have a specific period.
Model geostationary or Molniya orbits by solving for axis from a target period.
Compute hypothetical exoplanet orbits in habitable-zone studies.
Cross-check telescope or space mission parameters for educational projects.
Prepare for astronomy olympiads or space science competitions.
Demonstrate the T² ∝ a³ relationship visually with the Solar System bar chart.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu