Answer to Question #273140 in Astronomy | Astrophysics for phraser

Question

A satellite is in a circular orbit around the planet saturn. Derive the relationship between the period of the satellite, the mass of Saturn and the radius of orbit

Answer 1

Explanations & Calculations

• Using Newton's law of gravitation for the motion of the satellite around Saturn, this can be derived as follows (mass of Saturn = M, radius of the orbit = r, gravitational constant =G),

$\qquad\qquad \begin{aligned} \small G\frac{Mm}{r^2}&=\small mr\omega^2\\ \small G\frac{M}{r^2}&=\small r\Big(\frac{2\pi}{T}\Big)^2=r.\frac{4\pi^2}{T^2}\\ \small T^2&=\small 4\pi^2.\frac{r^3}{GM}\\ \small T&=\small 2\pi.\sqrt\frac{r^3}{GM} \end{aligned}$