Relativistic formulation of coordinate light time, Doppler and astrometric observables up to the second post-Minkowskian order

Given the extreme accuracy of modern space science, a precise relativistic modeling of observations is required. In particular, it is important to properly describe light propagation through the Solar System. For two decades, several modeling efforts based on the solution of the null geodesic equations have been proposed, but they are mainly valid only for the first-order post-Newtonian approximation. However, with the increasing precision of ongoing space missions such as Gaia, GAME, BepiColombo, JUNO, and JUICE, we know that some corrections up to the second order have to be taken into account for future experiments. We present a procedure to compute the relativistic coordinate time delay, Doppler, and astrometric observables avoiding the integration of the null geodesic equation. This is possible using the time transfer function formalism, a powerful tool providing key quantities such as the time of flight of a light signal between two point events and the tangent vector to its null geodesic. Indeed, we show how to compute the time transfer functions and their derivatives (and thus range, Doppler, and astrometric observables) up to the second post-Minkowskian order. We express these quantities as quadratures of some functions that depend only on the metric and its derivatives evaluated along a Minkowskian straight line. This method is particularly well adapted for numerical estimations. As an illustration, we provide explicit expressions in static and spherically symmetric space-time up to second post-Minkowskian order. Then we give the order of magnitude of these corrections for the range and Doppler on the BepiColombo mission and for astrometry in a GAME-like observation.