Doppler effect in a moving medium

The great increase in the accuracy of Doppler measurements in space requires a rigorous definition of the observed quantity in a moving medium, such as the solar wind. This is usually done in two different ways: in the phase point of view it is the time derivative of the correction to the optical path; in the ray point of view - suitable when the medium is confined to a small part of the ray - the signal is obtained from the deflection produced in the ray. They can be reconciled by using the time derivative of the optical path in the Lagrangean sense, i.e., differentiating from ray to ray. A rigorous derivation of this result requires an understanding, through relativistic Hamiltonian theory, of the delicate interplay between rays and phase; this is accomplished with the help of a general perturbation theorem which generalizes the concept of the Doppler effect as a Lagrangean derivative. Relativistic corrections O(upsilon) due to retardation are obtained, well within the expected sensitivity of Doppler experiments near solar conjunction.