What is Kepler's third law?

Kepler's third law states that the square of an orbit's period is proportional to the cube of its semi-major axis: T² ∝ a³. The full formula is T = 2π√(a³ / GM), where G is the gravitational constant (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 of the ellipse. For a circular orbit it equals the orbital radius. It is the standard measure of orbit size used in all celestial mechanics calculations.

Why does orbital period increase with distance?

Because of the cube-root relationship in Kepler's law, doubling the semi-major axis increases the period by a factor of 2^(3/2) ≈ 2.83. Distant planets travel a longer path and also move more slowly because gravity is weaker there.

How do I calculate a geostationary orbit?

A geostationary satellite has a period equal to Earth's rotation: 86,164 seconds (one sidereal day). Rearranging Kepler's law gives a = ∛(GM × T² / 4π²) ≈ 42,164 km from Earth's center, or about 35,786 km above the surface.

Does this formula work for moons and satellites?

Yes. Replace M with the mass of the planet the moon or satellite orbits. For example, to find the period of the International Space Station, use Earth's mass (~5.972 × 10²⁴ kg) and the ISS orbital radius (~6,787 km).

What are the limitations of this calculator?

The formula assumes a two-body system with a much smaller orbiting body and a nearly circular (or average elliptical) orbit. It does not account for perturbations from other bodies, general relativistic effects, or atmospheric drag, which matter for very precise calculations.

Orbital Period Calculator

Enter the semi-major axis and central body mass to instantly compute the orbital period using Kepler's third law.

Orbital Parameters

Solar System Preset

Semi-Major Axis (km)

Distance from center of the central body

Central Body Mass (kg)

Use scientific notation: 1.989e30

Formula: T = 2π √(a³ / GM)
G = 6.674 × 10⁻¹¹ N·m²/kg²

Select a preset or enter custom values, then click Calculate.

Earth

T = 2π√(a³/GM)

— days

Orbital Period

vs. Earth (365.25 days)

Seconds

Hours

Days

Years

Input Values Used

Semi-major axis:

Central mass:

Solar System Comparison (planets around Sun)

Planet	Period (days)	Bar

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Summary

Enter the semi-major axis and central body mass to instantly compute the orbital period using Kepler's third law.

How it works

Select a Solar System preset to load known orbital data, or choose Custom to enter your own values.
For custom input, enter the semi-major axis in kilometers and the central body mass in kilograms.
The calculator applies Kepler's third law: T = 2π × sqrt(a³ / (G × M)).
Results are shown in seconds, hours, days, and years for easy comparison.
A comparison bar shows how the period stacks up against Earth's orbital period (365.25 days).

Use cases

Verify planetary periods using Kepler's third law for physics homework.
Calculate the orbital period of an artificial satellite at a given altitude.
Compare orbital periods across the Solar System for astronomy education.
Model hypothetical exoplanet orbits in habitable zone calculations.
Illustrate why outer planets move more slowly than inner planets.
Compute geostationary orbit conditions (T = 24 hours).
Prepare for astronomy olympiad or space science competitions.
Explore how orbital radius affects period for science fair projects.

Frequently Asked Questions

Last updated: 2026-09-22 · Reviewed by Nham Vu