  A formula is derived for the combined motional and gravitational Doppler
effect in general stationary axisymmetric metrics for a photon emitted parallel
or antiparallel to the assumed circular orbital motion of its source. The same
formula is derived from eikonal approximation and Killing vector approaches to
elucidate connections between observational astronomy and modern Relativity.
The formula yields expected results in the limits of a moving or stationary
source in the exterior Kerr and Schwarzschild metrics and a moving source in
flat space.
