Question

A Saturn year is 29.5 times the earth year. How far is the Saturn from the sun if the earth is 1.50×10⁸ km away from the sun ?

Answer 1

Distance of the Earth from the Sun, re = 1.5 × 10⁸ km = 1.5×10¹¹ m
Time period of the Earth = T_e 
Time period of Saturn, Ts = 29. 5 T_e
Distance of Saturn from the Sun = r_s
From Kepler’s third law of planetary motion, we have
\(T = (\frac{4 \pi^2 r^3 }{ GM})^{\frac{1}{2}}\)

For Saturn and Sun, we can write 

\(\frac{rs^3}{ re^3 }= \frac{Ts^2 }{ Te ^2}\)

\(r_s = r_e(\frac{T_s}{ T_e})^{\frac{2}{3}}\)

= 1.5 x 10¹¹ \((\frac{29.5T_e}{T_e})^{\frac{2}{3}}\)

= 1.5 x 10¹¹\( (29.5)^{\frac{2}{3}}\)

= 1.5 x 10¹¹ x 9.55
= 14.32 x 10¹¹ m 

Hence, the distance between Saturn and the Sun is 1.432 x 10¹² m .