Question

Io, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of 4.22 3 105 km. From these data, determine the mass of Jupiter.

Short answer

Thus the required mass of Jupiter is$M=1.12\times {10}^{22}kg$

Step-by-step solution

Step 1:

It is important to know the gravitational force because it is the force that allows orbiting to exist. A central body exerts a gravitational force on the orbiting body to keep in it orbit. Centripetal force is also important, as this is the force responsible for circular motion.

Step 2:

Given:

Orbital period$T=1.77days$

$T=1.77\times 24\times 3600=152928\sec$

Orbitalradius$r=4.22\times {10}^{5}m$

We know that, for circular orbit net force toward the center will be equals to centripetal force,

$$\begin{gathered} \frac{GMm}{r^2}=\frac{m{v}^2}{r} \\ M=\frac{v^{2}r}{G}...........\left(1\right) \\ T=\frac{2\pi r}{v}..........\left(2\right) \\ {T}^2=\frac{4{\pi }^2{r}^2}{v^2}..........\left(3\right) \end{gathered}$$

From equation (1) and (2), we get

$$\begin{gathered} \frac{M}{T^2}=\frac{r}{G4{\pi }^2{r}^2} \\ M=\frac{T^2}{4G{\pi }^2{r}^2} \\ M=\frac{{\left(152928\right)}^2}{6.67\times {10}^{-11}\times 4\times {\left(3.14\right)}^2\times {\left(4.22\times {10}^5\right)}^2} \\ M=1.12\times {10}^{22}kg \end{gathered}$$