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

Kepler’s third law establishes a precise mathematical relationship between a planet’s orbital period and its mean distance from the Sun. It states that the square of the orbital period ($P^2$) is proportional to the cube of the planet's mean distance ($d^3$) from the Sun.1

In this relationship, the ratio of the square of the period to the cube of the mean distance is a constant for every planet orbiting the same central body. When using units of years for the period and astronomical units (AU) for the mean distance (the semimajor axis of the orbit), the law is expressed as:1

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

In this formula, $\tau$ represents the orbital period and $a$ represents the semimajor axis. 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. Mathematically, this relationship is derived from the principles of celestial mechanics and Newton's law of gravity, confirming that the motion of planets is governed by the mass of the central body and the geometry of their elliptical orbits.234