Woah it's new to me to hear about barycenter, I think it's one of the accepted explanations. The Center of gravity is always to look at if we need to balance the object, in terms of the sun and earth the mass isn't equal that's why the center of gravity is located at the sun which had a greater mass.
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Of course, Barycenter theory is most important. All objects that exist in space obey the Barycenter theory