Kepler's Third Law expresses a relation between the orbital period P and the semi-major axis a. Which of the following is correct?

• A. The square of the orbital period is proportional to the cube of the semi-major axis.
• B. The orbital period is proportional to the cube of the semi-major axis.
• C. The semi-major axis equals the orbital period.
• D. The square of the semi-major axis is proportional to the orbital period.

Answer: (A)

Kepler's Third Law describes how the time it takes to orbit and the size of the orbit are linked. The key idea is that the square of the orbital period P grows with the cube of the orbit’s size a. In math form, for an object orbiting a much more massive central body, P^2 = (4π^2/GM) a^3, where G is the gravitational constant and M is the central mass. This means if you move to a larger orbit, the period increases, not linearly but in proportion to the cube root when you look at a or P, specifically P ∝ a^(3/2). For example, doubling the semi-major axis makes the period increase by 2^(3/2) ≈ 2.83 times.

This is why the statement that the square of the orbital period is proportional to the cube of the semi-major axis is the correct description. It’s not the orbital period itself being proportional to a^3, nor is the semi-major axis equal to the period, nor is the square of the semi-major axis proportional to the period.