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Cerenkov

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With the recent interest in the JWST orbiting at the L2 Lagrange point of the Earth - Moon system, I was wondering about the dynamics of the Pluto - Charon system. Specifically, the barycentre of that system.

This barycentre lies at a point in space between these two bodies. Does this mean that it can be considered as a stable or semi-stable region, similar to the classical Lagrange points, L1 to L5? Effectively an L6?

Also, since the system's barycentre is displaced from the interior of Pluto, does this mean that points L1 to L5 are proportionally displaced too? Or is there some else going on that I'm unaware of?

Thank you for any help given.

Cerenkov.