  We study Doppler effects in curved space-time, i.e. the frequency shifts
induced on electromagnetic signals propagating in the gravitational field. In
particular, we focus on the frequency shift due to the bending of light rays in
weak gravitational fields. We consider, using the PPN formalism, the
gravitational field of an axially symmetric distribution of mass. The zeroth
order, i.e. the sphere, is studied then passing to the contribution of the
quadrupole moment, and finally to the case of a rotating source. We give
numerical estimates for situations of physical interest, and by a very
preliminary analysis, we argue that analyzing the Doppler effect could lead, in
principle, in the foreseeable future, to the measurement of the quadrupole
moment of the giant planets of the Solar System.
