Kepler's Third Law Calculator
Find an orbital period from the semi-major axis (around the Sun).
Kepler's third law links a planet's orbital size to its period.
How the Math Works
Kepler's Third Law states that the square of an orbiting object's period (T) is proportional to the cube of its semi-major axis (a) around the Sun. When using years for T and astronomical units (AU) for a, the relationship simplifies to T² = a³. This means that doubling the distance from the Sun increases the orbital period by a factor of 2.83 (2³⁄²), illustrating how orbital speed decreases with distance. The law assumes no other gravitational influences and works precisely for objects orbiting the Sun.
Practical Applications
To calculate a planet's orbital period, take the square root of the cube of its semi-major axis. For example, a planet 8 AU from the Sun would have a period of √(8³) = √512 ≈ 22.6 years. Conversely, if a satellite's period is known, rearrange the formula to solve for a: a = ∛(T²). This is invaluable for astronomers predicting planetary positions, designing satellite orbits, or verifying celestial mechanics models. It also simplifies complex orbital calculations in educational settings by eliminating gravitational constants.
Day-to-Day Use
Understanding orbital periods helps explain phenomena like seasonal cycles and satellite coverage. For instance, GPS satellites require precise orbital periods to maintain their positions relative to Earth, though they orbit Earth rather than the Sun. Kepler's Law also guides space mission planning, such as determining travel times to other planets. Additionally, it underpins public interest in astronomy—knowing Jupiter's 12-year orbit, for example, helps explain its changing visibility patterns in our night sky over decades.
Worked example
Mars at 1.524 AU → about 1.88 years.
FAQ
Does it work for other stars?
Yes, with the full form T² = 4π²a³/(GM) using the central mass.