Kepler's Third Law Calculator

What the Kepler's third law calculator does

Kepler's third law ties an orbit's period to its size. This calculator uses it to find the semi-major axis of an orbit from the central body's mass and the orbital period, with an option to include the orbiting body's mass too.

Below is what Kepler's third law is, the equation behind it, how it lets astronomers weigh stars and find planets, and a worked example.

How to use it

1. Enter the central body's mass, such as a star, in kilograms, Sun masses, or other units.
2. Enter the orbital period, the time for one orbit. In advanced mode, add the orbiting body's mass.
3. Press Calculate for the semi-major axis, the size of the orbit, or Reset to clear it.

What Kepler's third law is

Kepler's third law is one of the great discoveries of astronomy: it states that the square of a planet's orbital period is proportional to the cube of its orbital size. In plain terms, planets farther from the Sun take longer to orbit, and the law gives the exact mathematical relationship between how far out a planet is and how long its year lasts. Johannes Kepler found it in 1619 by poring over decades of careful observations, years before Newton explained why it holds.

The law applies to any orbit under gravity, not just planets around the Sun. Moons around planets, satellites around the Earth, and planets around distant stars all obey it. Its power lies in connecting two things that are easy to confuse but turn out to be rigidly linked: the time an orbit takes and the distance it spans. Given one, plus the mass at the centre, you can find the other. This calculator uses the law to find the size of an orbit from its period.

The equation it uses

In its precise form, Kepler's third law relates the period and the semi-major axis through the masses involved. Solved for the orbital size, it reads:

a = ∛(G (M + m) T² ÷ (4π²))

Here a is the semi-major axis, the average size of the orbit, T is the orbital period, G is the gravitational constant, and M and m are the central and orbiting masses. The cube root reflects the law's core relationship, that the cube of the distance tracks the square of the period. The calculator combines the mass and period in this way and takes the cube root to give the orbit's size. In simple mode the orbiting body's mass is treated as negligible, which is accurate for a planet around a star.

How it lets us weigh the heavens

Kepler's third law is far more than a way to relate period and distance; it is the primary tool astronomers use to measure the masses of distant objects. Rearranged, the law gives the mass of a central body from the period and size of something orbiting it. By watching how long a moon takes to circle a planet, or a planet to circle a star, and how big that orbit is, astronomers can weigh the central body, even one trillions of kilometres away that they can never visit.

This is how we know the mass of the Sun, of Jupiter, of distant stars, and even of the supermassive black hole at the centre of our galaxy, whose mass was found by tracking the orbits of stars whipping around it. Almost everything we know about the masses of celestial objects traces back to this one law. The calculator works in the direction of finding orbit size, but the same relationship, turned around, is what underlies our entire cosmic weighing scale.

Finding worlds around other stars

Kepler's third law is central to the modern hunt for planets beyond our Solar System. When astronomers detect a planet orbiting another star, often by the tiny, regular dip in starlight as the planet passes in front, what they measure directly is the orbital period. Kepler's law then converts that period, together with the star's mass, into the size of the planet's orbit, telling us how far the planet sits from its star.

That distance matters enormously, because it determines how much warmth the planet receives, and thus whether it might lie in the habitable zone where liquid water, and perhaps life, could exist. A planet with a short period orbits close and is scorching; one with a long period orbits far and is frozen; somewhere in between lies the temperate band. So this centuries-old law, applied to faint flickers of distant starlight, is one of the keys to judging which newfound worlds might be hospitable. The calculator performs exactly the conversion at the heart of that work.

Units and precision

The calculator takes the central mass in kilograms, Earth masses, or Sun masses, and the period in units from seconds to years, which suits orbits from satellites to planets. It returns the semi-major axis in units from metres to parsecs, including astronomical units, the natural measure for planetary orbits. The calculation is an exact application of Kepler's third law, and results carry several significant figures.

A worked example

Take a planet orbiting a star of one solar mass with a period of one year, the Earth's own situation.

Kepler's third law gives a = ∛(G M T² ÷ (4π²)) ≈ 1 astronomical unit, which is Earth's distance from the Sun, as it must be. A planet with a four-year period around the same star would orbit at about 2.5 astronomical units, farther out, since the distance grows as the two-thirds power of the period. The law turns a simple timing measurement into a distance.

Questions people ask

What is Kepler's third law?

That the square of an orbital period is proportional to the cube of the orbit's size. Planets farther from the Sun take longer to orbit, in a precise mathematical relationship.

How do you use it to find orbit size?

Use a = ∛(G(M+m)T²/(4π²)), combining the central mass and the period. The cube root reflects that distance cubed tracks period squared.

How does it let astronomers measure mass?

Rearranged, the law gives the central body's mass from the period and size of something orbiting it. This is how we weigh the Sun, planets, distant stars, and even black holes.

How does it help find exoplanets?

Astronomers measure an exoplanet's orbital period, then use Kepler's law and the star's mass to find its orbital distance, which reveals whether the planet lies in the habitable zone.

References

A quick note on where the physics comes from. Kepler's third law and its use in measuring masses are standard astronomy, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. NASA describes Kepler's laws and exoplanet detection. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, University Physics Volume 1, Section 13.5, Kepler's Laws of Planetary Motion. https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion
2. HyperPhysics, Kepler's Laws. http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html
3. NASA Exoplanet Exploration, How We Find and Characterize Planets. https://science.nasa.gov/exoplanets/how-we-find-and-characterize/