The Erez-Rosen metric and the role of the quadrupole on light propagation

The gravitational field of a static body with the quadrupole moment is described by an exact solution found by Erez and Rosen. Here, we investigate the role of the quadrupole in the motion, deflection and lensing of a light ray in the above metric. The standard lensing observables such as image positions and magnification have been explicitly obtained in the weak-field and small quadrupole limit. In this limit, the spacetime metric appears as the natural generalization to quadrupole corrections of the metric form adopted also in current astrometric models. Hence, the corresponding analytical solution of the inverse ray tracing problem and the consistency with other approaches are also discussed.