How does Kepler's third law connect a planet's distance from the Sun with its orbital period?

Johannes Kepler’s third law, also known as the harmonic law, establishes a precise mathematical relationship between a planet's orbital period and its distance from the Sun.

The law states that the ratio of the squares of the periods of revolution ($P^2$) to the cubes of the mean distances (or the semimajor axes, $a^3$) is constant for all planets orbiting the Sun. In simpler terms, the square of a planet's orbital period is proportional to the cube of its mean distance from the Sun.123

When using standard units—where time is measured in years and distance is measured in astronomical units (AU)—the relationship can be expressed as:

$${\tau }^2=a^3$$

In this formula, $\tau$ represents the orbital period and $a$ represents the semimajor axis of the planet's elliptical orbit. This law allows astronomers to calculate a planet's distance from the Sun if its orbital period is known, or conversely, to determine the period if the distance is known.2