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(C) The semi-major axis $a$ of an elliptical orbit is the average of the maximum distance $(r_{max} = R)$ and minimum distance $(r_{min} = R/3)$:$a =

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$\frac{32}{27} \cdot \frac{\pi^2}{GM}$

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(C) The semi-major axis $a$ of an elliptical orbit is the average of the maximum distance $(r_{max} = R)$ and minimum distance $(r_{min} = R/3)$:$a = \frac{R + R/3}{2} = \frac{4R/3}{2} = \frac{2R}{3}$According to Kepler's Third Law,the period of revolution $T$ is given by:$T = 2\pi \sqrt{\frac{a^3}{GM}}$Substituting the value of $a$:$T = 2\pi \sqrt{\frac{(2R/3)^3}{GM}} = 2\pi \sqrt{\frac{8R^3}{27GM}}$Squaring both sides:$T^2 = 4\pi^2 \cdot \frac{8R^3}{27GM} = \frac{32\pi^2}{27GM} \cdot R^3$Comparing this with $T^2 = \alpha R^3$,we get:$\alpha = \frac{32\pi^2}{27GM}$

$A$ planet of mass $m$ moves around the Sun along an elliptical path with a period of revolution $T$. During the motion,the planet's maximum and minimum distance from the Sun is $R$ and $\frac{R}{3}$ respectively. If $T^2 = \alpha R^3$,then the magnitude of constant $\alpha$ will be

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