The velocity of P is a prototype for the tangent to the wheel at P. Using the ions doesn't seem to add anything except potential confusion. You don 't seem to have understood the question. Does the observer at O look at the T1 ion beam only where it leaves the gun? Or does the observer at O look at any other part of the T1 ion beam? The observer at O will measure a transverse doppler effect for light directly emitted from T1 and from P, so what extra information is provided by the ion beam? OK, but it's a step removed from the point of the debate... so you'll have to demonstrate that the ion beam velocities match the vectors of interest. More interesting, more complex, more confusing, and less relevant to the point of the debate. Is that what you want?