Question

The mass of the “planet” Pluto was not known until it was discovered to have a moon. Explain how this enabled an estimate of Pluto’s mass.

Short answer

The mass of planet Pluto is $m\text{'}=\frac{4{\pi }^2{r}^3}{G{T}^2}$.

Step-by-step solution

Step 1. Understanding the gravitational force and centripetal force

In this problem, the relation of gravitational force and centripetal force will be used in order to estimate the mass of the plant Pluto.

Step 2. Formula of centripetal force and gravitational force

The relation of the centripetal force is given by,

$F=\frac{m{v}^2}{r}$

Here, m is the mass of the moon, v is the velocity of the moon and r is the distance.

The relation of the gravitational force from Pluto is given by,

$F=\frac{Gmm\text{'}}{r^2}$

Here, $m\text{'}$is the mass of the planet and G is the gravitational constant.

Step 3. Estimating the mass of the planet Pluto

On equating the above two relations.

$\begin{array}{c}\frac{m{v}^2}{r}=\frac{Gmm\text{'}}{r^2}\\ v^2=\frac{Gm\text{'}}{r}\\ m\text{'}=\frac{v^{2}r}{G}\end{array}$

The relation of velocity of the moon is given by,

$v=\frac{2\pi r}{T}$

Here, T is the time period.

On plugging the values in the above relation.

$\begin{array}{l}m\text{'}=\frac{{\left(\frac{2\pi r}{T}\right)}^{2}r}{G}\\ m\text{'}=\frac{4{\pi }^2{r}^3}{G{T}^2}\end{array}$

Thus, $m\text{'}=\frac{4{\pi }^2{r}^3}{G{T}^2}$is the mass of the Pluto.