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Kepler's Law

Two planets r...

Question

Two planets revolves around the sun with frequencies $N_{1}$ and $N_{2}$ revolutions per year. If their average radii (orbital) be $R_{1}$ and $R_{2}$ respectively, then $R_{1} / R_{2}$ is equal to:

A

(N_{1} / N_{2})^{2 / 3}

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B

(N_{1} / N_{2})^{3 / 2}

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C

(N_{2} / N_{1})^{2 / 3}

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D

(N_{2} / N_{1})^{3 / 2}

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Solution

The correct option is B $(N_{2} / N_{1})^{2 / 3}$
According to Kepler's law of planetary motion
$T^{2} \propto R^{3}$
If N is frequency then
$N^{2} \propto (R)^{- 3}$
or $\frac{N_{2}}{N_{1}} (\frac{R_{2}}{R_{1}})^{- 3 / 2}$
$\Rightarrow \frac{R_{1}}{R_{2}} = (\frac{N_{2}}{N_{1}})^{2 / 3}$

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