The quadrupole gravitational lens

An asymmetric lens made of a spherically symmetric galaxy and a quadrupole tidal part allows a very simple pictorial examination of the multiplicity and positions of the images, for any mass density profile. The lens equations can be reduced to one equation in one variable. In the special case of a singular isothermal sphere with a tide (SIST), the lens properties are described by five parameters. Two parameters determine the angular scale and orientation, while three determine the configuration of the images. A comprehensive investigation of the SIST lens is performed, including the variation of the configurations with the parameter values, the critical surfaces in the parameter space, and the extremal solutions. Analytic expressions are derived for the SIST parameters, and additional image positions, in the case when the intensity ratio and positions relative to the galaxy for two of the images are given as input. Quadrupole lenses with a point mass and a pseudoisothermal sphere as the main part are also described in terms of direction diagrams. A direct relation between deflection profile and rotational velocity is derived.