Question

A Saturn year is $29.5$ times the earth year. How far is the saturn from the sun if earth is $1.50 \times {10^8}$ km away from the sun ?

Answer 1

Hint To find the value we should keep in mind the relative comparison of the radius of the two planets. Apart from that the proportionality equations of the radius and distance should also be considered.

 Complete step-by-step solution:The distance of earth from the sun is ${r_e} = 1.5 \times {10^{11}}m$
The time period of the earth is ${T_e}$
The distance of saturn from the sun is ${r_s}$
The time period of the saturn is ${T_s} = 29.5{T_e}$
By implementing the Kepler’s law of planetary motion we will get
$T = \sqrt {4{\pi ^2}{r^3}/GM} $
Which concludes the relation
$\Rightarrow {r_s}^3/{r_e}^3 = {T^2}_s/{T_e}^2$
$\Rightarrow {r_s} = {r_e}{({T_s}/{T_e})^{2/3}}$
$ = 14.32 \times {10^{11}}m$

$\therefore$ The saturn is $14.32 \times {10^{11}}m$ away from the sun.

Note: While calculating the value the comparison of radius and time period is necessary. Along with that the subsequent formulas, the derivation of correct and precise relations are important for their part. These questions need step by step analysis.