Light deflection in the gravitational field of a solar system body with finite distance of source and observer for sub-micro-arcsecond astrometry

Abstract

The effect of deflection of a light signal that propagates through the gravitational field of a solar system body at rest is considered in the case that the source and the observer are at a finite distance from the body. The observer is assumed to be located somewhere nearby the Earth, for instance at Lagrange point L2 of the Sun-Earth system, while the spatial position of the celestial light source is arbitrary. The gravitational fields in the exterior of the bodies are described by the full set of time-dependent mass-multipoles and spin-multipoles of these bodies. So the gravitating bodies can be of arbitrary shape, inner structure and rotational motion. The unit tangent vector of the light trajectory at the position of the observer is determined in the 1PN and 1.5PN approximation of the post-Newtonian (PN) scheme. A simplified expression for the unit tangent vector is obtained, where all terms are neglected that together contribute less than 10 nano-arcseconds in light deflection for all astrometric configurations between source, body and observer. It is shown that in the case of an axi-symmetric body the unit tangent vector of the light ray as well as the light deflection are given in terms of Chebyshev polynomials. This fact allows for determining the effect of light deflection in the gravitational fields of the bodies up to any order of the mass-multipoles and spin-multipoles. It is shown that the total light deflection, that is the angle of light deflection where source and observer are located at infinite spatial distance from the body, represents an upper limit of the effect of light deflection. These investigations are aiming at astrometric measurements on the sub-micro-arcsecond level of accuracy.