Question

Let the period of revolution of a planet at a distance $R$ from a star be $T$. Prove that if it was at a distance of $2R$ from the star, its period of revolution will be$\sqrt{8}T.$

Answer 1

According to Kepler’s third law

$T^2\propto a^3$

T is time period of planet and a is semi major axis of its orbit.

So, according to question

$T^2=k{R}^3..........\left(1\right)$

When distance is 2R then

$T^{{}^{\prime }2}=k{\left(2R\right)}^3$

$T^{{}^{\prime }2}=8R^{3}k...........\left(2\right)$

Taking ratios of (1) and (2)

$\frac{T^2}{T^{{}^{\prime }2}}=\frac{k{R}^3}{8R^{3}k}$

$\frac{T^2}{T^{{}^{\prime }2}}=\frac{1}{8}$

$T^{{}^{\prime }2}=8T^2$

$T^{{}^{\prime }}=\sqrt{8}T$