Question

The period of a planet around sun is $27$ times that of the earth. The ratio of radius of planet's orbit to the radius of the earth's orbit is

• A. $4$
• B. $9$
• C. $64$
• D. $27$

Answer 1

The correct option is B

$9$

Step 1: Given Data:

Let the period of the earth is $T_e$

The period of a planet around the sun is $T_p$

$T_p=27T_e$

Step 2: Formula Used:

According to Kepler’s law of periods,” The square of the time period of revolution of a planet around the sun in an elliptical orbit is directly proportional to the cube of its semi-major axis”.

That is-

$T^2\propto a^3$

Step 3: Calculating the ratio of the radius of the planet's orbit to the radius of the earth's orbit:

Let the average radius of the planet from the sun is $R_p$

The radius of the earth is $R_e$

The ratios can be as follows-

$\frac{{T_e}^2}{{T_p}^2}=\frac{{R_e}^3}{{R_p}^3}$

$T_p=27T_e$

Putting the value

$\begin{array}{rcl}\frac{{T^2}_e}{{\left(27\right)}^2{T^2}_e}& =& \frac{{R^3}_e}{{R^3}_p}\\ \frac{{R^3}_e}{{R^3}_p}& =& {\left(27\right)}^2\\ \frac{R_e}{R_p}& =& {\left(3^3\right)}^{\frac{2}{3}}\\ R_p& =& 3^2{R}_e\\ R_p& =& 9R_e\\ & & \end{array}$

Hence option B is the correct answer.