The beginning of the problem is fairly simple: A system of two bodies will orbit around their mutual barycenter, which is a point defined by their respective masses. Often that point is inside the larger body, but sometimes it’s in space (i.e. Pluto and Charon, or binary stars).
Let’s add some more bodies to the equation, and my question: Why don’t man-made Earth satellites orbit the Earth-Moon barycenter too? Isn’t the E-M barycenter the functional Center of Mass of the entire Earth system? Why don’t we see our satellites orbiting way out of center with the Earth since the E-M barycenter is well away from the very middle of the Earth?
I suspect if we launched a satellite out to an orbit of a million miles or so, that satellite may orbit the Earth-Moon system barycenter and not the Earth’s physical center. But then the question is how far away until that takes place? Is there a transition?