Which statement correctly describes Kepler's Third Law?

Which statement correctly describes Kepler's Third Law?

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

Answer: (C)

Kepler’s Third Law expresses that for bodies orbiting the same central mass, the orbital period grows with the size of the orbit according to T^2 ∝ a^3. This means that as the semi-major axis a increases, the square of the orbital period increases in proportion to the cube of a, so longer orbits take disproportionately longer to complete. The specific case often used in astronomy is that, with units where a is in astronomical units and T in years, the proportionality becomes an equality: T^2 = a^3. Among the options, the statement that the square of the orbital period is proportional to the cube of the semi-major axis captures the correct general relationship. The other options either imply the wrong dependence (T^2 ∝ a^2 would give T ∝ a) or state an equality in a broader context where a constant depends on the central mass, or claim independence from a, which is not true.

Kepler’s Third Law expresses that for bodies orbiting the same central mass, the orbital period grows with the size of the orbit according to T^2 ∝ a^3. This means that as the semi-major axis a increases, the square of the orbital period increases in proportion to the cube of a, so longer orbits take disproportionately longer to complete. The specific case often used in astronomy is that, with units where a is in astronomical units and T in years, the proportionality becomes an equality: T^2 = a^3. Among the options, the statement that the square of the orbital period is proportional to the cube of the semi-major axis captures the correct general relationship. The other options either imply the wrong dependence (T^2 ∝ a^2 would give T ∝ a) or state an equality in a broader context where a constant depends on the central mass, or claim independence from a, which is not true.