Kepler's Third Law
Relates orbital period to semi-major axis.
Formula
Variables
Example
Earth’s a = 1 AU and T = 1 year satisfy T² = a³ exactly.
How it works
Kepler’s Third Law says that for objects orbiting the same central body, the orbital period grows with the 3/2 power of the orbit size: bigger orbits take much longer to complete. In the full Newtonian form, the constant depends on the central mass because gravity provides the centripetal force that keeps the orbit bound; the equation is exact for a two-body system when one mass dominates and the orbit is treated as Keplerian. The semi-major axis is the orbit’s average size, so for circular orbits it is just the radius, while for ellipses it captures the overall scale of the path.
When to use it
Use this when you need to find an orbital period or orbital size around a star, planet, or moon, especially in exoplanet studies, satellite dynamics, or binary systems. It is also the starting point for estimating the mass of a central object from the motion of an orbiting body.
Worked example, step by step
Find the orbital period of a satellite around Earth with semi-major axis a = 26,600 km. Given: G = 6.674 × 10^-11 m^3 kg^-1 s^-2, M = 5.972 × 10^24 kg Convert a to meters: a = 26,600 km = 2.66 × 10^7 m Use T^2 = (4π^2 / GM) a^3 Compute a^3 = (2.66 × 10^7)^3 = 1.881 × 10^22 m^3 Compute GM = (6.674 × 10^-11)(5.972 × 10^24) = 3.986 × 10^14 m^3 s^-2 Compute T^2 = (4π^2 × 1.881 × 10^22) / (3.986 × 10^14) ≈ 1.86 × 10^9 s^2 Take the square root: T ≈ 4.31 × 10^4 s Convert to hours: T ≈ 4.31 × 10^4 / 3600 ≈ 12.0 h Final answer: the satellite’s orbital period is about 12.0 hours.
Common mistakes to avoid
- Using the orbital radius instead of the semi-major axis for an elliptical orbit
- Forgetting to convert units so that a is in meters and M is in kilograms
- Applying the formula to systems where the orbiting body’s mass is not negligible without using the two-body version
Did You Know?
Kepler's third law let astronomers weigh the Sun and planets long before any spacecraft flew.
Share this formula
More in Astronomy & Space
View allEscape Velocity
IntermediateMinimum speed to escape a body's gravity.
Orbital Velocity
IntermediateSpeed of a circular orbit.
Schwarzschild Radius
AdvancedEvent horizon radius of a black hole.
Hubble's Law
AdvancedRecession velocity of galaxies with distance.
Stefan–Boltzmann Law
AdvancedLuminosity of a star from radius and temperature.
Redshift
AdvancedFractional wavelength shift of light.