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
($\kappa_s \ll 1$) 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
$\kappa_s \ll 1$ 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 $\kappa_s \lesssim
0.1$. For low ellipticities, values up to $\kappa_s \simeq 0.18$ are allowed.
On the other hand, the mapping between PNFW and the ENFW models is valid up to
$\kappa_s \simeq 0.4$. The solutions derived in this work can be used to speed
up numerical codes and ensure their accuracy in the low $\kappa_s$ regime,
including applications to arc statistics and other strong lensing observables.
(Abridged)
