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Analytic Solutions for Navarro--Frenk--White Lens Models for Low Characteristic Convergences

The Navarro-Frenk-White (NFW) density profile is often used to model gravitational lenses. For low values of the characteristic convergence ($κ_s \ll 1$) of this model - corresponding to galaxy and galaxy group mass scales - a high numerical precision is required in order to accurately compute several quantities in the strong lensing regime. An alternative for fast and accurate computations is to derive analytic approximations in this limit. In this work we obtain analytic solutions for several lensing quantities for elliptical (ENFW) and pseudo-elliptical (PNFW) NFW lens models on the typical scales where gravitational arcs are expected to be formed, in the $κ_s \ll 1$ limit, establishing their domain of validity. We derive analytic solutions for the convergence and shear for these models, obtaining explicit expressions for the iso-convergence contours and constant distortion curves (including the tangential critical curve). We also compute the deformation cross section, which is given in closed form for the circular NFW model and in terms of a one-dimensional integral for the elliptical ones. In addition, we provide a simple expression for the ellipticity of the iso-convergence contours of the pseudo-elliptical models and the connection of characteristic convergences among the PNFW and ENFW models. We conclude that the set of solutions derived here is generally accurate for $κ_s \lesssim 0.1$. For low ellipticities, values up to $κ_s \simeq 0.18$ are allowed. On the other hand, the mapping between PNFW and the ENFW models is valid up to $κ_s \simeq 0.4$. The solutions derived in this work can be used to speed up numerical codes and ensure their accuracy in the low $κ_s$ regime, including applications to arc statistics and other strong lensing observables. (Abridged)