Question

A planet of mass $m$ moves around the sun of mass $M$ in an elliptical orbit. The maximum and minimum distances of the planet from the sun are $r_1$ and $r_2$ respectively. The time period of the planet is proportional to

• A. $(r_1+r_2)$
• B. ${(r_1+r_2)}^{1/2}$
• C. ${(r_1–r_2)}^{3/2}$
• D. ${(r_1+r_2)}^{3/2}$

Answer 1

The correct option is D

${(r_1+r_2)}^{3/2}$

Step 1: Given data

mass of planet $=m$

mass of sun $=M$

Maximum distance of the planet from the sun $=r_1$

A minimum distance of the planet from the sun $=r_2$

Step 2: Explanation for the correct option

Mean of the distance of the planet from the sun$=\frac{(r_1+r_2)}{2}$

According to Kepler’s third law, the square of the time period is proportional to the cube of the semi-major axis.

Therefore,

$$\begin{gathered} T^2\propto {\left[\frac{(r_1+r_2)}{2}\right]}^3 \\ T\propto {\left[\frac{(r_1+r_2)}{2}\right]}^{3/2} \\ T\propto {\left[r_1+r_2\right]}^{3/2} \end{gathered}$$

Hence, option (D) is correct.