Kepler's Third Law: The Harmonic Law

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

T² = k · a³

Where T is the orbital period, a is the semi-major axis, and k is a constant that depends on the mass of the central body being orbited.

What This Means

Planets farther from the Sun take longer to complete an orbit, but not in a simple linear way. The relationship is precise: if you cube the average distance and square the period, the ratio is the same for every planet in the solar system. Mercury, orbiting close to the Sun, has a short period. Neptune, far out, takes about 165 years per orbit. The Third Law connects them all with a single equation.

Using the Law

If distances are measured in AU (astronomical units, where 1 AU = Earth-Sun distance) and periods in years, then for our solar system, k = 1. That is:

T² = a³ (with T in years and a in AU)

You can verify this with solar system data. For example, Jupiter: a ≈ 5.20 AU, so a³ ≈ 140.6. T ≈ 11.86 years, so T² ≈ 140.7. The match is nearly exact.

Try it yourself with the orbital period worksheet.

The Full (Newtonian) Form

When Newton derived the Third Law from his law of universal gravitation, the constant k was revealed to depend on the mass of the central body:

T² = (4π² / G·M) · a³

This is enormously powerful because it means you can use the Third Law to weigh planets and stars if you can observe objects orbiting them.

Discovery

Kepler discovered this law on March 8, 1618, and published it in Harmonices Mundi (1619), a full decade after the first two laws. He was ecstatic, calling it the culmination of his life's work. See also Music of the Spheres for the broader context of that book.

Applications

The Third Law is used today in exoplanet detection, satellite orbit design, Mars mission planning, and binary star analysis.