Question about Kepler's 1st law and barycentric orbits

[RisingSun361]

Kepler's 1st law states, "All planets move about the sun in elliptical orbits, having the sun as one of the foci". But the orbit must also be barycentric. So, technically speaking, is the center of mass of the system actually at the focus, rather than the sun? And the sun itself orbits this focus in a small elliptical orbit?

And in a multi-planet solar system, would the focus/center of mass move as different planets change their positions?

And in a binary star system, do the two ellipses overlap each other in such a way that two foci overlap each other at one point, the center of mass?

 

[SteamKing]

Kepler's 1st law states, "All planets move about the sun in elliptical orbits, having the sun as one of the foci". But the orbit must also be barycentric. So, technically speaking, is the center of mass of the system actually at the focus, rather than the sun? And the sun itself orbits this focus in a small elliptical orbit?

And in a multi-planet solar system, would the focus/center of mass move as different planets change their positions?

And in a binary star system, do the two ellipses overlap each other in such a way that two foci overlap each other at one point, the center of mass?

Have a read about barycentric coordinates:

http://en.wikipedia.org/wiki/Barycentric_coordinates_(astronomy)