Kepler's Third Law, the Law of Harmonies, shows a precise mathematical link between a planet's orbital period and its average distance from the Sun.

The Law of Harmonies

While Keplerā€™s First and Second Laws describe the motion of a single planet, the Third Law compares different planets. It shows that a planetā€™s ā€œyearā€ is determined by its distance from the Sun.

The Mathematical Relationship

Kepler discovered that the square of a planetā€™s orbital period ($T$) is proportional to the cube of the semi-major axis ($a$) of its orbit:

\[T^2 \propto a^3\]

For our solar system, using Earth years for time and Astronomical Units (AU) for distance:

\[T^2 = a^3\]

What This Means

  • Further = Slower: Planets farther from the Sun travel longer paths and move more slowly.
  • Universal Constant: Newton later showed this law comes from the Law of Universal Gravitation. The ratio $T^2 / a^3$ depends only on the mass of the Sun (or central body).

Interactive: Keplerā€™s Third Law Vocabulary

Match the key terms and symbols to their correct definitions.

Click a term or symbol and then its matching definition. Match all pairs to complete!

šŸ’” Quick Concept Check:

If Planet X is 4 AU away from the Sun, how many Earth years does it take to complete one orbit?

Click to Reveal Answer
Using $T^2 = a^3$, we calculate $T^2 = 4^3 = 64$. Taking the square root, $T = 8$. So Planet X takes 8 Earth years to orbit the Sun.
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