# What is the orbital velocity of Pluto around the Pluto-Charon barycenter?

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?

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.