…are you saying a focus of the ellipse is at the center of mass of the system and not at the center of mass of the other object?.. I can also imagine that, using the center of one of the masses as origin of the reference frame, the other mass moves in an ellipse with the origin at one focus. Could this be right at the same time as your assertion? I’m way to lazy to go figure out how to figure this out.
In a system with only two objects, then the path of each of them could be described as ellipses that have the center of mass of the system as one of the foci. Neither of the objects are themselves at the focus, but it can appear that one is if it is sufficiently larger than the other object.
If you added a third object to the system, much more massive than the first two and at a distance, then you’d find that the first two objects would still orbit their common center of mass in an elliptical fashion, but the major axis of the ellipse would slowly revolve with respect to the third object. You’d also be able to set up a new coordinate system, tied to the new object, in which to describe the paths of the smaller objects. In that new coordinate system the equation describing their paths would be a bit more complex.
If you added a fourth mass, larger than the first two, but much smaller than the third, you’d further influence the wobbling and revolving, and have another coordinate system to use. The equations of motion of the first three objects in this third reference frame would be even more unwieldy than the in the second - but all three sets of equations would describe the same relative motion between all of the objects.
I’ve just described Earth, the Moon, the Sun and Jupiter. You could, if you chose, completely describe the orbits of all four bodies with respect to a coordinate system having the left fromt corner of your driveway as the (0,0,0) point, the x-direction going down your sidewalk, and the y-direction pointing toward your garage door - but why would you? The equation would be valid, but a bear to work with.
It’s much easier to take the most massive body in the system as your reference, and assume either that it doesn’t rotate or that there is a distant point that doesn’t move.
How you define the system depends on what you’re trying to do. If you want to put a satellite in orbit, then the most massive object is the earth. If you want to send one to Mars, then the Earth dominates at first but then it’s the Sun. If You want to go to another galaxy, then after the Sun you need to start paying more attention to the immediate group of stars and the galaxy as a whole.
Sungenis’s challenge is a HUGE red herring. Squabbling over it is a waste of time.