Pluto and Charon

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The velocity vector of the center of mass of an isolated system, as measured in an inertial coordinate frame, cannot change except from an external influence.

As the planets in a Solar System move around and change places the velocity vector of the barycenter of that sytem will not change as observed from an inertial reference frame.

If you are basing your coordinate frame on one of the revolving planets then it is not an inertail frame and the barycenter will be seen to wobble all around.
Thanks Bill. I think I’m getting it, now.

In the seesaw case it is not an isolated system. As the four little people move around they are pushing the seesaw legs against the Earth and moving the Earth slightly.
The see-saw analogy won’t work because it fails to make the big guy move, unlike our solar system’s Sun. As the planets align opposite the Sun, the Sun will have moved farther from the barycenter, which remains fixed, thus balancing the system. We can see this in the Pluto-Charon animation, not that it’s accurate.
 
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Feb 26, 2025
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The entire Pluto system is interesting on a wider level, but not least is the Pluto/Charon barycentre - which depends on the masses of dwarf planet and satellite - or are they joint dwarf planets?

Pluto's moon Charon sports a red polar cap that forms as seasons change | Space.com Forums

Cat :)
Ah, the Pluto-Charon system—what a fascinating cosmic duo! When we talk about the Pluto/Charon barycenter, we're essentially discussing the point around which both objects orbit each other. What makes this so intriguing is that the barycenter isn't located inside Pluto, as you'd typically expect with a planet and its moon. Instead, it sits in the space between them, which is a pretty big deal. This happens because Charon is relatively massive compared to Pluto—about 1/8th of Pluto's mass—so their gravitational dance is more like a partnership than a traditional planet-moon relationship.

Now, as for whether they're joint dwarf planets—that's a bit of a semantic puzzle. Officially, Pluto is classified as a dwarf planet, and Charon is considered its largest moon. But because of their unique barycenter and the way they interact gravitationally, some astronomers have playfully suggested they could be thought of as a "binary dwarf planet system." It's not an official classification, but it does capture the spirit of their relationship. They’re kind of like cosmic siblings, locked in a mutual orbit that blurs the line between planet and moon.

So, while the International Astronomical Union (IAU) hasn't given them the joint title, the Pluto-Charon system challenges our traditional definitions and reminds us that nature often defies our neat categorizations. It’s a humbling reminder of how much we still have to learn about the cosmos—and how wonderfully weird it can be.
 

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