Question

What would the mass of the Sun have to be if Pluto (The outermost “planet: most of the time) were to have the same orbital speed that Mercury (the innermost planet) has now? Use data from Appendix C, express your answer in terms of the Sun’s current mass,${\mathit{M}}_{\mathbf{s}}$and assume circular orbits.

Short answer

The mass of the Sun under the given circumstances is$102M_S$

Step-by-step solution

Given Information

Consider that the Pluto has the same orbital speed that Mercury have now.

Write the formula for the gravitational mass as:

From the formula of the gravity force$F_{grav}$ is as follows:

${\mathit{F}}_{\mathbf{g}\mathbf{r}\mathbf{a}\mathbf{v}}\mathbf{=}\frac{\mathbf{G}\mathbf{M}\mathbf{m}}{{\mathbf{r}}^{\mathbf{2}}}$…… (1)

Determine the mass of the Sun under the given condition.

Consider the equation$M\infty v^2$ then equation (1) becomes,

$\begin{array}{c}{F}_{grav}=\frac{GMm}{r^2}\\ =\frac{m{v}^2}{r}\end{array}$

Determine the mass of the sun as:

${M_s}^{\text{'}}={\left(\frac{v_{mercury}}{v_{pluto}}\right)}^2{M}_s$

${M_s}^{\text{'}}={\left(\frac{47.9\raisebox{1ex}{$\text{km}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.}{4.74\raisebox{1ex}{$\text{km}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.}\right)}^2{M}_s$

Therefore, the mass of the Sun under the given circumstances is$102M_S$