I am trying to determine the frequency redshift when a signal is sent between two spacecraft orbiting a star at different radii. I have not been able to find a formula for this, I can only find formula for a signal emitted at a particular radius from a star and received at infinity. Both spacecraft are well outside the Schwartzchild radius, so I think I only need the Newtonian limit.
I think what I need is:
Fr = Fs/(1+(Pr-Ps)/c^2)
Fr = Frequency at receiver Fs = Frequency at source Pr = Gravitational Potential at receiver Ps = Gravitational Potential at source c = speed of light
I determine Gravitational Potential as: P = -GM/r
Is it correct to use the difference in Potential in this way? Is the rest of the equation correct?
I suspect there may be something wrong because if the receiving ship immediately repeats the signal back to the sending ship, the above formula does not result in the final signal having the same frequency as the original transmission, i.e. the red shift in one direction does not cancel the blue shift in the other direction. I would think it should.