I.Derivation of the effect and radial motion

Context. Very precise planned space astrometric missions and recent improvements on imaging capabilities require a detailed review of the assumptions of classical astrometric modeling. Aims. We show that Light-Travel Time must be taken into account to model the kinematics of astronomical objects in nonlinear motion, even at stellar distances. Methods. A closed expression to include Light-Travel Time in the actual astrometric models with nonlinear motion is provided. Using a perturbative approach the expression of the Light-Travel Time signature is derived. We propose a practical form of the astrometric modelling to be applied in astrometric data reduction of sources at stellar distances(d > 1pc). Results. We show that the Light-Travel Time signature is relevant at μas accuracy (or even at mas) depending on the time span of the astrometric measurements. We explain how information about the radial motion of a source can be obtained. Some estimative numbers are provided for known nearby binary systems Conclusions. In the light of the obtained results, it is clear that this effect must be taken into account to interpret any kind of precise astrometric measurements. The effect is particularly interesting in measurements performed by the planned astrometric space missions (GAIA, SIM, JASMINE, TPF/DARWIN). Finally an objective criterion is provided to quickly evaluate whether the Light-Travel Time modeling is required for a given source or system.