Kepler Third Law
Calculate orbital period, semi-major axis, or central body mass using Kepler's Third Law of planetary motion: T² = 4π²a³/GM. Free physics calculator for astronomy and astrophysics.
About This Calculator
The Kepler Third Law Calculator lets you compute orbital period, semi-major axis, or central body mass using Kepler's Third Law of planetary motion. This fundamental law of astrophysics states that the square of a planet's orbital period is directly proportional to the cube of its semi-major axis. Our calculator supports both astronomical units (AU, years, solar masses) and SI units (meters, seconds, kilograms), making it useful for students, astronomers, and physics enthusiasts studying orbital mechanics.
The underlying formula is T² = 4π²a³/G(M + m), where T is the orbital period in seconds, a is the semi-major axis in meters, G = 6.67430 × 10⁻¹¹ m³/(kg·s²) is the gravitational constant, M is the mass of the central body, and m is the planet mass (optional). For solar system calculations in astronomical units, this simplifies to T² = a³ / M when T is in Earth years and a is in AU. The calculator handles all conversions internally, so you can work in whichever unit system you prefer.
Regional Notes
Global: Kepler's Third Law is universal and applies to any two bodies in gravitational orbit, including planets around stars, moons around planets, and binary star systems. The gravitational constant G is a universal physical constant, so the calculation is identical worldwide. When using astronomical units (AU, years, solar masses), the formula T² = a³ / M holds for any system where the central body mass is expressed in solar masses. For exoplanet research or precision astrophysics, always use SI units and include the planet's mass.
Frequently Asked Questions
What is Kepler's Third Law of planetary motion?
Kepler's Third Law states that the square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit. Mathematically, T² ∝ a³, or more precisely T² = 4π²a³/G(M + m), where T is the orbital period, a is the semi-major axis, G is the gravitational constant, M is the star mass, and m is the planet mass.
How do I use the Kepler Third Law calculator?
Select what you want to solve for: orbital period, semi-major axis, or central body mass. Enter the known values in the appropriate units (astronomical or SI) and click Calculate. The calculator will instantly compute the missing parameter using Kepler's Third Law formula.
What units does the Kepler law calculator support?
The calculator supports two unit systems: Astronomical units (AU for distance, years for time, solar masses for mass) and SI units (meters, seconds, kilograms). You can switch between them at any time.
Can I include the planet's mass in the calculation?
Yes, check the 'Include planet mass' option to account for the planet's gravitational influence. For most solar system planets, the planet mass is negligible compared to the Sun (Earth is only 0.000003 solar masses), but it matters for binary star systems or exoplanet calculations.
How does Kepler's Third Law apply to binary star systems?
For binary star systems, Kepler's Third Law uses the total mass of both stars: T² = 4π²a³/G(M₁ + M₂). The semi-major axis a is the distance between the two stars. This is why the planet mass option is important — it turns the calculator into a binary system tool.
What is the gravitational constant value used?
The calculator uses G = 6.67430 × 10⁻¹¹ m³/(kg·s²), the CODATA-recommended value of the Newtonian gravitational constant. In astronomical units, this translates to 4π² when using solar masses, AU, and years.
Is Kepler's Third Law free to use?
Yes, this Kepler's Third Law calculator is completely free to use with no registration required. You can also share your calculations via the URL which saves your input values.