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People are using natural language. And considering the barycenter sits inside the sun or close to it (relatively), it isn't that wrong of a statement.

While you're technically correct, everyone already understood. We could be more technically correct by talking about the galactic barycenter, or even more by talking about the universe as a whole, but the accuracy gains by that added complexity doesn't benefit the conversation

And where's the barycenter located? When keeping an open mind make sure your brain doesn't fall out.

Galactic core at center for other astronomy predictions...