Kepler's Third Law Calculator

What Is Kepler's Third Law?

Kepler's Third Law of Planetary Motion states that the square of an orbital period is proportional to the cube of the orbital radius. In Newtonian form, the law is expressed as T² = (4π² / GM) × r³, where T is the orbital period, r is the orbital radius from the center of the central body, M is the central mass, and G = 6.6743 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant.

This calculator applies the Newtonian form of Kepler's third law to solve for the orbital period T, the orbital radius r, or the central mass M given any two of the three quantities. It works for any two-body orbit — planets around the Sun, moons around a planet, or satellites around Earth.

How to Use the Kepler's Third Law Calculator

Select what you want to solve for using the solve-for toggle: orbital period, orbital radius, or central mass. Enter the known values in any supported units (meters/kilometers/miles for distance, kilograms/grams/pounds for mass, seconds/minutes/hours/days/years for time). The calculator uses the Newtonian form T = 2π × √(r³ / (G × M)) and automatically handles unit conversions.

For quick reference, the calculator includes a table of central body masses for the Sun, planets, and Moon from NASA's Planetary Fact Sheet. Click any row to load that body's mass into the calculator.

Key Concepts

Kepler's Third Law is a direct consequence of Newton's gravitational law combined with the requirement that gravity supplies the centripetal force for a circular orbit. The proportionality constant 4π²/GM is fixed for any given central body, so all satellites of that body — regardless of mass — follow the same T² ∝ r³ curve. This is why Earth's relation 1² = 1³ defines the astronomical unit.

The law applies in the limit where the orbiting body's mass is much smaller than the central mass. For comparable masses, the full two-body formula uses the sum M₁ + M₂. For elliptical orbits, use the semi-major axis a in place of r.

Applications

• Spacecraft Mission Planning: Computing transfer orbit periods and timing rendezvous with target bodies.
• Astronomy: Estimating the mass of a planet, star, or galaxy from the orbital period and radius of a satellite.
• Exoplanet Discovery: Deriving orbital radius from the observed period and host star mass.
• Geostationary Orbit Design: Solving for the radius that gives Earth-orbiting satellites a 23h 56m period.
• Education: Demonstrating the same gravitational law governs falling apples and planetary motion.