Laurence Dixon responded to the In the News piece on barycentric orbits on March 29th by making the point that under Newton's laws of gravity two bodies will orbit around their common centre of mass = their barycentre. In that way the blog site in that piece was somewhat misleading as orthodox astronomers use this fact all the time when interpreting binary stars. In the solar system the barycentre of any one planet with the Sun is withing the Sun's radius, but not at its centre. Similarly, the Earth – Moon barycentre is within the radius of the Earth. However, the barycentre is stationary and the whole of the Earth rotates around it without the barycentre moving. When we introduce more than two bodies, such as the planets of the solar system, the barycentre still remains in the Sun and therefore the diagram on the blog is inside out.
When all the planets are strung out in a straight line on one side of the Sun the barycentre is no longer inside the Sun. When they are in a straight line with Jupiter on one side of the Sun and the planets on the other side the barycentre is momentarily at the centre of the Sun.