Discussion
Two planets revolve around the sun in elliptical orbits with their major axes in the ratio 2 : 3. Their periods of revolution around the sun are in the ratio
Answer: D
The periods are in the ratio \(2\sqrt{2} : 3\sqrt{3}\).
Kepler's third law: \(T^{2} \propto a^{3}\), where a is the semi-major axis. The ratio of semi-major axes equals the ratio of major axes, 2 : 3.
\[ \begin{aligned} \frac{T_{1}}{T_{2}} &= \left(\frac{2}{3}\right)^{3/2} \\ &= \frac{2\sqrt{2}}{3\sqrt{3}} \end{aligned} \]
4 : 9 uses the power 2 instead of \(\frac{3}{2}\), and 2 : 3 assumes T is proportional to a. Option C mixes the roots: \(2\sqrt{3} : 3\sqrt{2}\) equals \(\sqrt{2} : \sqrt{3}\), the square root of the ratio.
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