Kepler's Laws of Planetary Motion - Third Law

Third Law

"The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."

The third law, published by Kepler in 1619 captures the relationship between the distance of planets from the Sun, and their orbital periods. Symbolically, the law can be expressed as

where is the orbital period of the planet and is the semi-major axis of the orbit. For example, suppose planet A is 4 times as far from the Sun as planet B. Then planet A must traverse 4 times the distance of Planet B each orbit, and moreover it turns out that planet A travels at half the speed of planet B, in order to maintain equilibrium with the reduced gravitational centripetal force due to being 4 times further from the Sun. In total it takes 4×2=8 times as long for planet A to travel an orbit, in agreement with the law (82=43).

This third law used to be known as the harmonic law, because Kepler enunciated it in a laborious attempt to determine what he viewed as the "music of the spheres" according to precise laws, and express it in terms of musical notation.

The third law currently receives additional attention as it can be used to estimate the distance from an exoplanet to its central star, and help to decide if this distance is inside the habitable zone of that star.

The exact relation, which is the same for both elliptical and circular orbits, is given by the equation

for masses M₁ and M₂. In many cases, one of the objects is much more massive than the other, i.e. M₁ >> M₂. In these cases, the relation can be expressed approximately as constant. In the solar system, for example, the Sun's mass is dominant, and the ratio of period squared to semi-major axis cubed is approximately the same for all planets. That ratio is given by

for a sidereal year (yr), and astronomical unit (AU). See the actual figures: attributes of major planets.