Question

A planet moves around the sun (mass = $M_{s}$ ) in an elliptical orbit such that its minimum and maximum distance from the sun are $r$ and $R$ respectively. The period of revolution of this planet around the sun is

• A. $T=\pi \sqrt{\frac{\left(r + R\right)^{3}}{2 G M_{s}}}$
• B. $T=\pi \sqrt{\frac{\left(r + R\right)^{3}}{3 G M_{s}}}$
• C. $T=\pi \sqrt{\frac{\left(r + R\right)^{3}}{G M_{s}}}$
• D. $T=\pi \sqrt{\frac{2 \left(r + R\right)^{3}}{G M_{s}}}$

Answer 1

Answer: (A)

The length of the semi-major axis of the elliptical orbit of the planet is
$a=\frac{r + R}{2}$
If we assume that there is a hypothetical planet which moves around the sun in a circular orbit of radius $r_{0}=\frac{r + R}{2}$ , then the time period of this hypothetical planet and our given planet will be same. The time period of a planet around the sun in circular orbit is
$T=2 \pi \sqrt{\frac{r_0^3 0}{G M_{\mathrm{S}}}}=2 \pi \sqrt{\frac{\left(\frac{r+R}{2}\right)^3}{G M_{\mathrm{S}}}}$
$T=\pi \sqrt{\frac{\left(r + R\right)^{3}}{2 G M_{s}}}$