Question

Find the mass of Jupiter based on data for the orbit of one of its moons, and compare your result with its actual mass.

Short answer

Jupiter’s mass is ${\text{1.90×10}}^{\text{27}}\text{\hspace{0.17em}kg}$.

The calculated value agrees with the actual value for the mass of Jupiter.

Step-by-step solution

Definition of Force

A force is a factor that can impact an object's motion. A force can cause a mass item to accelerate (e.g., from a standstill).

Calculation of the centripetal force on the Jupiter’s moon Europa due to revolution

The gravitational force between the two objects is given by,

$F_g=\frac{G{m}_1{m}_2}{r^2}$

Where G is the universal gravitation constant, r is the distance between the two objects, and ${\text{m}}_{\text{1}}{\text{,m}}_{\text{2}}$are the masses of two bodies, respectively.

In angular motion, the centripetal acceleration is given,

$\begin{array}{c}{F}_c=m{a}_c\\ =m{\omega }^{2}r\end{array}$

As the moon revolves around Jupiter, so there is the only force that is the gravitational force which provides centripetal force to the moon for revolution. Therefore, the centripetal force on the moon is equal to the gravitational force between Jupiter and its moon.

Mathematically, it can be written as,

$\begin{array}{c}{F}_g=F_c\\ \frac{G{m}_{earth}{m}_{sun}}{r^2}=m_{earth}{\omega }^{2}r\\ m_{sun}=\frac{{\omega }^2{r}^3}{G}\end{array}$

Calculate the mass of Jupiter

In 3.55 days, the Europa completes one revolution of Jupiter. So, the angular velocity of the Europa is,

$\begin{array}{l}\omega =\frac{1\text{\hspace{0.33em}rev}}{3.55\text{\hspace{0.33em}days}}\\ \omega =\frac{1\text{\hspace{0.33em}rev}}{3.55\text{\hspace{0.33em}d}}\times \frac{2\pi \text{\hspace{0.33em}rad}}{1\text{\hspace{0.33em}rev}}\times \frac{1\text{\hspace{0.33em}d}}{24\text{\hspace{0.33em}hr}}\times \frac{1\text{\hspace{0.33em}hr}}{3600\text{\hspace{0.33em}s}}\\ \omega =2.05\times {10}^{-5}\frac{\text{rad}}{\text{s}}\end{array}$

The distance of Jupiter from one of its moons, Europa, in Table 6.2 is $r=6.71\times {10}^{8}m$.

So, the mass of the Jupiter,

$\begin{array}{c}{m}_{sun}=\frac{{\omega }^2{r}^3}{G}\\ =\frac{{\left(2.05\times {10}^{-5}\right)}^2\times {\left(6.71\times {10}^8\right)}^3}{6.673\times {10}^{-11}}\\ m_{sun}=1.90\times {10}^{27}\text{\hspace{0.33em}kg}\end{array}$

Hence, the mass of Jupiter based on the data from one of its moons, Europa orbit is ${\text{1.90×10}}^{\text{27}}\text{\hspace{0.33em}kg}$, which is very close to Jupiter’s actual mass that is ${\text{1.898×10}}^{\text{27}}\text{\hspace{0.33em}kg}$. So, the calculated value agrees with the actual value for the mass of Jupiter.