Kepler's Third Law Calculator
Run backwards: how far out, from how long it takes.
Work out Kepler's Third Law. Run backwards: how far out, from how long it takes. Shows the working, not just the answer.
Semi-major axis
5.20122 AU
7.7809 × 10^11 m · orbital speed 13.06 km/s
This runs Kepler's third law backwards: given how long something takes to orbit, how far out is it? Cube-rooting T²M gives the semi-major axis directly in astronomical units. It is how the masses of distant systems are measured. Watch a moon or a companion star go round, time it, measure the separation, and the central mass falls out — which is the only way most masses in astronomy are ever known. The semi-major axis is half the long axis of the ellipse, not the radius, because orbits are ellipses rather than circles. For nearly circular orbits like most planets the difference is small; for a comet on a long ellipse it is enormous, and the period still depends only on the semi-major axis.
How the Kepler's Third Law Calculator works
Kepler's third law inverted: given an orbital period and a central mass, how far out is the orbit? This is how the masses and distances of most systems in astronomy are actually measured.
Also known as: orbital distance from period · how far out is this orbit · measure a star's mass from an orbit · semi major axis calculator
Frequently asked questions
How do I find orbital distance from period?
Cube-root the period squared times the central mass, in years and solar masses, and the answer comes out in astronomical units. The constant is 1 in those units.
How are stellar masses measured?
Almost always this way. Watch a companion star or a planet orbit, time it, measure the separation, and the central mass falls out of Kepler's third law. There is no other general method.
What is a semi-major axis?
Half the longest diameter of the elliptical orbit. For a nearly circular orbit it is essentially the radius; for a comet on a long ellipse it is very different, and the period depends only on it.
Does this work for moons and satellites?
Yes, with the right central mass and consistent units. The law applies to any two-body orbit, which is why it works equally for the Moon around Earth and for exoplanets around distant stars.
Why is it called the third law?
Kepler stated three: orbits are ellipses with the star at a focus, equal areas are swept in equal times, and the period-distance relation. The third came a decade after the first two, from a search for numerical harmony in the planetary distances.
Put this calculator on your own site
Free to use, on any site, commercial or not. Paste this where you want it to appear. It is a plain iframe, so it works in WordPress, Squarespace, Wix, Webflow, Ghost and anything else that accepts HTML.
<iframe src="https://www.thecalclibrary.com/embed/keplers-third-law-calculator" width="100%" height="640" style="border:1px solid #e2e8f0;border-radius:12px" loading="lazy" title="Kepler's Third Law Calculator"></iframe>
<p style="font:13px/1.5 system-ui,sans-serif;margin:6px 0 0;color:#64748b">Powered by <a href="https://www.thecalclibrary.com/keplers-third-law-calculator" style="color:#64748b">Kepler's Third Law Calculator</a> from The Calc Library</p>The only condition is that the credit line below the frame stays in place. That one line is what pays for the tool being free — it is how anyone else finds it.
Related calculators
Orbital Period Calculator
In solar masses and AU, the constant disappears.
OpenStellar Parallax Calculator
Distance in parsecs is one over the parallax. No constant.
OpenStellar Luminosity Calculator
Temperature to the fourth power dominates everything.
Open