Question
A planet $\left( P_{1} \right)$ is moving around the star of mass 2 M in the orbit of radius R . Another planet ( $P_{2}$ ) is moving around another star of mass 4 M in a orbit of radius 2R. Ratio of time periods of revolution of $P_{2}$ and $P_{1}$ is $\_\_\_\_$ .
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Solution
Sol. [Key Concept: $T \propto \sqrt{R^{3}/M}$; take the ratio for the two star–planet systems.]
From Kepler's law,
$$\begin{matrix} & T^{2} = \frac{4\pi^{2}}{{GM}_{S}}r^{3} \\ & \ \therefore{\text{ }T}_{P_{1}} = \frac{2\pi}{\sqrt{G(2M)}}R^{3/2} \\ & {\text{ }T}_{P_{2}} = \frac{2\pi}{\sqrt{G(4M)}}(2R)^{3/2} \\ & \ \therefore\frac{{\text{ }T}_{P_{2}}}{{\text{ }T}_{P_{1}}} = \frac{(2)^{3/2}}{\sqrt{4}} \times \frac{\sqrt{2}}{1} = \frac{2\sqrt{2} \times \sqrt{2}}{2} = 2 \end{matrix}$$
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