Question

The Martian satellite Phobos travels in an approximately circular orbit of radius $\mathbf{9}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathit{m}$with a period of$\mathbf{7}\mathbf{}\mathit{h}\mathbf{}\mathbf{39}\mathbf{}\mathit{m}\mathit{i}\mathit{n}$ .Calculate the mass of Mars from this information.

Short answer

The mass of the Mars is $6.5\times {10}^{23}\text{kg}$.

Step-by-step solution

Step 1: Given

Radius,$R=9.4\times {10}^{6}m$

Period,$7hours39\text{minutes}=\text{2.754}\times {\text{10}}^{\text{4}}\text{s}$

Determining the concept

Using the formula for Kepler’s third law, we can find themass of Mars from the given radius and period. According to Kepler’s third law, the squares of the orbital periods of the planets are directly proportional to the cubes of the semi major axes of their orbits.

Formula is as follow:

$T^2=\left(\frac{4{\pi }^2}{GM}\right)R^3$

where, T is time, G is gravitational constant, M is mass and R is radius.

Determining themass of Mars

Now,

$$\begin{gathered} T^2=\left(\frac{4{\pi }^2}{GM}\right)R^3 \\ M=\frac{4{\pi }^2{R}^3}{G{T}^2} \\ \frac{4{\pi }^2{\left(\text{9.4}\times {\text{10}}^{\text{6}}\text{m}\right)}^3}{G{\left(\text{2.754}\times {\text{10}}^{\text{4}}\text{s}\right)}^2} \end{gathered}$$

Hence, the mass of the Mars is $6.5\times {10}^{23}kg$.

Therefore, using the formula for Kepler’s third law, the mass of Mars from the given radius and period can be found.