Kepler's Third Law Calculator
Apply Kepler's third law in its full Newtonian form T² = (4π²/GM)a³ to find an orbital period from a semi-major axis, or the axis from a period.
Reviewed October 7, 2026 Runs in your browser — inputs are never stored
Formula
T² = (4π² ÷ GM) × a³ · or · a = ∛(GMT² ÷ 4π²)
This tool provides educational conversions of astronomical distances only. It is not a navigation, mission-planning, or ephemeris service. For professional or spacecraft work, use official IAU, NASA, or observatory data products.
About the Kepler's Third Law Calculator
The Kepler’s third law calculator relates a planet’s orbital period to the size of its orbit. Kepler found that the square of the orbital period is proportional to the cube of the semi-major axis; for bodies orbiting the Sun, with the period in years and the distance in astronomical units, P² = a³. Enter one value and the calculator finds the other, which makes it useful for checking planetary data and for astronomy coursework.
How to use it
- 1Choose whether to solve for period (from axis) or axis (from period).
- 2Pick the central body and enter the known value with its unit.
- 3Press Calculate to get the unknown quantity.
Worked example
around the Sun, an orbit at 1 AU has a period of 1 year. At 4 AU (roughly Ceres' orbit), the period is 8 years — because T² = a³ gives T = √(4³) = 8.
How it works
Kepler\'s third law
Kepler found that T² ∝ a³ for planets around the Sun. Newton showed the full form T² = (4π² ÷ GM) × a³, where M is the central mass. This lets astronomers weigh stars and planets from orbiting bodies.
Special case: Sun, AU, years
When a is in AU and T in years around the Sun, the constant becomes 1, so T² = a³ and T = a^(3/2).
Who it is for
Astronomy students, orbital mechanics learners, and educators teaching planetary motion.
Why it helps
Kepler's third law links orbital period and distance — the backbone of how we measure masses of stars and planets from orbiting bodies.
Common use cases
- Deriving a planet's year from its orbital radius
- Finding the orbit radius needed for a target period (e.g. geostationary)
- Teaching the relationship T² ∝ a³
- Cross-checking orbital mechanics homework
Tips for better results
- For the Sun in AU and years, T² = a³ exactly — a quick check.
- Switch the central body to Earth to design satellite orbits.
- The law assumes a two-body system; perturbations add small corrections.
- Use the period→axis mode to find the altitude for a desired period.
Common mistakes to avoid
- Using the wrong central mass (e.g. Earth mass for a heliocentric orbit).
- Forgetting that T scales with a^(3/2), not linearly.
- Mixing units (AU with seconds without converting).
Frequently asked questions
A:It states that the square of an orbital period is proportional to the cube of the semi-major axis: T² ∝ a³. Newton's form is T² = (4π²/GM)a³.
A:It lets you compute a period from an orbit size (or vice versa) and, by inverting it, determine the mass of the central body from the orbit of a companion.
A:Yes. The semi-major axis (average orbital radius) is the relevant distance, so the law holds for ellipses, not just circles.
A:Yes. It runs in your browser on CalcWisee with no sign-up required.
Reviewed for accuracy · Last updated October 7, 2026
The astronomical unit is taken as exactly 149,597,870,700 metres per the IAU 2012 definition. Related light-time and light-year factors use the defined speed of light and the IAU Julian year where applicable.
CalcWisee Editorial Team
Calculator development and formula review
CalcWisee is an independent site founded by Ayman. Every calculator is built from a published formula or official definition, tested against hand-worked examples before it goes live, and corrected promptly when a reader reports an issue to ayman@calcwisee.com.
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References & sources
