Answers to the question How did Kepler determine the orbital period of Mars? describing careful observations centuries ago got me thinking.

What was the first analysis of observations that directly demonstrated that the Sun and the outer giant planets were moving around a common barycenter rather than all planets rotating around a fixed Sun?

I am not sure if First observation that the movement of a planet in its orbit was affected by another planet? is a completely separate question, or if this confirmation happened all at once, but I've currently asked it separately.

Answers to the question How did Kepler determine the orbital period of Mars? describing careful observations centuries ago got me thinking.

What was the first analysis of observations that directly demonstrated that the Sun and the outer giant planets were moving around a common barycenter rather than all planets rotating around a fixed Sun?

I am not sure if First observation that the movement of a planet or asteroid in its orbit was affected by another planet? is a completely separate question, or if this confirmation happened all at once, but I've currently asked it separately.