Question

Saturn has an orbital period of 29.46 years. In two or more complete sentences, explain how to calculate the average distance from Saturn to the sun and then calculate it.

Answer 1

With M = mass of Sun, balance centripetal etc forces on Saturn to get \(F = \dfrac{GMm}{r^2} = m \omega^2 r \implies r = \sqrt[3] {\dfrac{GM}{\omega^2}}\)

And \(\omega = \dfrac{2 \pi}{T} \implies r = \sqrt[3] {\dfrac{GM T^2}{4 \pi^2}}\)

Plug relevant constant values (from Wiki), be sure to use correct units (years to seconds etc), and you should get \(r \approx 1.43 \times 10^{9}\) km, which is v close to what Wiki says it is. It will be an elliptical orbit so this way is an estimation anyways.

But wordifying that sounds like a horrible job :( so good luck