Gravitational arcs as a perturbation of the perfect ring

The image of a point situated at the centre of a circularly symmetric potential is a perfect circle. The perturbative effect of non-symmetrical potential terms is to displace and break the perfect circle. These two effects, displacement and breaking, are directly related to the Taylor expansion of the perturbation at first order on the circle. The numerical accuracy of this perturbative approach is tested in the case of an elliptical potential with a core radius. The contour of the images and the caustic lines are well reproduced by the perturbative approach. These results suggest that the modelling of arcs, and in particular that of tangential arcs, may be simplified by using a general perturbative representation for points located on the circle. This linear perturbative approach is accurate when the gradient of the circular potential is almost linear; this constraint is satisfied when the potential is nearly isothermal.