Which formula expresses Kepler's Third Law?

Which formula expresses Kepler's Third Law?

• A. T^2 = a^3
• B. T^2 / a^2 = constant
• C. T^2 / a^3 = constant
• D. T^3 / a^2 = constant

Answer: (C)

Kepler's Third Law says that the time it takes to orbit (the period) and the size of the orbit are tied by a specific scaling: the square of the period grows with the cube of the orbit’s size. In formula form, T^2 is proportional to a^3 for any body orbiting the same central mass. This can be written as T^2 = k a^3, where k is a constant that depends on the central mass and the units you’re using. Rearranging gives T^2 / a^3 = k, a universal constant for that system. In the Sun–planets case, k equals 4π^2/GM_sun. If you choose units where k = 1 (like astronomical units for distance and years for time), you get T^2 = a^3, but that’s a special unit choice rather than a general rule. The other forms misstate the scaling (for example, T^2 ∝ a^2 would mean T grows with the square root of a, not with the cube of a), so the correct expression is T^2 / a^3 = constant.

Kepler's Third Law says that the time it takes to orbit (the period) and the size of the orbit are tied by a specific scaling: the square of the period grows with the cube of the orbit’s size. In formula form, T^2 is proportional to a^3 for any body orbiting the same central mass. This can be written as T^2 = k a^3, where k is a constant that depends on the central mass and the units you’re using. Rearranging gives T^2 / a^3 = k, a universal constant for that system. In the Sun–planets case, k equals 4π^2/GM_sun. If you choose units where k = 1 (like astronomical units for distance and years for time), you get T^2 = a^3, but that’s a special unit choice rather than a general rule. The other forms misstate the scaling (for example, T^2 ∝ a^2 would mean T grows with the square root of a, not with the cube of a), so the correct expression is T^2 / a^3 = constant.