Second order singular pertubative theory for gravitational lenses

The extension of the singular perturbative approach to the second order is presented in this paper. The general expansion to the second order is derived. The second order expansion is considered as a small correction to the first order expansion. Using this approach it is demonstrated that the second order expansion is reducible to a first order expansion via a re-definition of the first order pertubative fields. Even if in practice the second order correction is small the reducibility of the second order expansion to the first order expansion indicates a degeneracy problem. In general this degeneracy is hard to break. A useful and simple second order approximation is the thin source approximation which offers a direct estimation of the correction. The practical application of the corrections derived in this paper are illustrated by using an elliptical NFW lens model. The second order pertubative expansion provides a noticeable improvement, even for the simplest case of thin source approximation. To conclude it is clear that for accurate modelisation of gravitational lenses using the perturbative method the second order perturbative expansion should be considered. In particular an evaluation of the degeneracy due to the second order term should be performed, for which the thin source approximation is particularly useful.