Analytical ray-tracing in planetary atmospheres

Context. Ground-based astro-geodetic observations and atmospheric radio occultations are two examples of observational techniques requiring a scrutiny analysis of atmospheric refraction. In both cases, the measured changes of the observables are geometrically related to changes in the photon path through the refractive profile of the crossed medium. Therefore, having a clear knowledge of how the refractivity governs the photon path evolution is of prime importance to clearly understand observational features. Aims. We analytically performed the integration of the photon path and the light time of rays traveling across a non-spherically symmetric planetary atmosphere. Methods. Assuming that the atmospheric refraction evolves linearly with the Newtonian potential, we derived an exact solution to the equations of geometrical optics. By varying the solution's arbitrary constants of integration, we reformulated the equation of geometrical optics into a new set of osculating equations describing the constants' evolution following any changes in the refractive profile. We have highlighted the capabilities of the formalism, carrying out five realistic applications in which we derived analytical expressions. Finally, we assessed the accuracy by comparing the solution to results from a numerical integration of the equations of geometrical optics in the presence of a quadrupolar moment (J2). Results. Analytical expressions for the light time and the refractive bending are given with relative errors at the level of one part in 108 and one part in 105, for typical values of the refractivity and J2 at levels of 10-4 and 10-2, respectively. Conclusions. The establishment of the osculating equations for the ray propagation has two main advantages. Firstly, it provides an easy and comprehensive geometrical picture for interpreting the photon path. Secondly, it allows the analytical solving of the ray propagation in the presence of non-radial dependencies in the refractive profile.