When did we discover that our sun does not orbit its own axis but a barycenter? It's commonly known that the center of gravity of our solar system is not the suns axis, and, that the same goes for other orbital systems like Pluto and Charon, and so on. When was it discovered that our sun has an orbit around a barycenter?
Newton explained the motions of planets in the solar system in Principia in 1687.

This amongst many other things "defines the very slow motion of the Sun relative to the solar-system barycenter;" (https://en.m.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica ).

Since Newton came up with the ideas of gravity and barycentres, surely this is the answer? (Though, I note there is some debate that the inverse square law may have been around and discussed during the previous decade).

Ideas that the Sun moved were of course commonplace before then.