There are a lot of images and animations on the internet depicting two bodies orbiting around their common barycenter. The barycenter is defined as the (let's say right) focus of the ellipse.
If we define the right focus as the star instead of the barycenter, then the right focus will move in a tiny ellipse around the barycenter just as the star does.
First question: What does physics say about the left focus? Will it stay put or also move around?
Second question: If the focuses are moving around, then won't the orbit eccentricity expand and contract at different points in the orbit, and also result in some kind of precession?
(I have put together a geometric simulation where the right and left foci of the orbit move around in small ellipses, in opposite directions, resulting in a wobbly orbit with "precession"--sort of; and am trying to figure out how this translates into orbital mechanics of planets).
