Numerical Simulations of the Light Propagation in the Gravitational Field of Moving Bodies

One of the most subtle points in the modern relativistic models for microarcsecond astrometrical observations is the treatment of the influence of translational motion of gravitating bodies on the light propagation. This paper describes numerical simulations of the light propagation in the gravitational field of moving gravitating bodies as well as summarizes the underlying theory. The simulations include high-precision numerical integrations of both post-Newtonian and post-Minkowskian differential equations of light propagation and a detailed comparison of the results of the numerical integrations with various available approximate analytical formulas. The simulations has been performed both for hypothetical bodies with various parameters of trajectories as well as for all the major bodies of the solar system using the JPL ephemeris DE405/LE405 to calculate their motion. It is shown that for the accuracy of $\sim 0.2$ $\mu$as it is sufficient to use the well-known solution for the light propagation in the field of a motionless mass monopole and substitute in that solution the position of the body at the moment of closest approach between the actual trajectory of the body and the unperturbed light path (as it was first suggested by Hellings (1986)). For a higher accuracy one should use either the post-Newtonian solution for uniformly moving bodies (Klioner & Kopeikin, 1992) or the post-Minkowskian solution for arbitrarily moving bodies (Kopeikin & Schaefer, 1999). For astrometric observations performed from within the solar system these two solutions guarantee the accuracy of $\sim 0.002$ $\mu$as and are virtually indistinguishable from each other.