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Charon is so massive relative to Pluto that they both orbit a point that is outside Pluto. The distance from the center of Pluto to the barycenter is given by:

$$r = \frac{a}{1+\frac{m_{pluto}}{m_{charon}}}$$

where

  • $a$ is the distance between the centers of the two bodies,

  • $r$ is the distance from the center of the primary to the barycenter,

  • $m_{pluto}$ and $m_{charon}$ are the masses of Pluto and Charon, respectively.

Plugging in the numbers, the distance between Pluto and the Pluto-Charon system barycenter is $2,126$ km from Pluto's center, or $938$ km above its surface. Is it possible to calculate how fast Pluto orbits around this point?

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Pluto and Charon attract each other with a gravitational force of $$F_G=G \frac{m_\mathrm{pluto}\, m_\mathrm{charon}}{a^2},$$ where $G$ is the gravitational constant.

As Pluto and Charon thus both orbit around a common barycenter, that motion can be seen as constant circular motion of either object around the barycenter. For Pluto therefore, as the question asks, from Newton's Second Law there follows a centripetal force $$F_R=m_\mathrm{pluto}\frac{v^2}{r},$$ where $\frac{v^2}{r}$ is the centripetal acceleration and $v$ the tangential velocity of Pluto.

As the gravitational force causes the orbiting of Pluto around the barycenter, $F_G = F_R$. Solving this equation for $v$ and plugging in the other values then yields Pluto's velocity around the barycenter.

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