Asymptotic solutions for the case of SIE lens models and application to the quadruply imaged quasar Q2237+0305

Abstract

Considering a small misalignment between a point-like source, a singular isothermal ellipsoid deflector and an observer, we derive to first order simple relations between the model parameters and the lensed image positions, and an expression for the time delay between pairs of opposed images which is analogue to the one previously derived for the case of e − γ models. Combined with the first-order astrometric relations, we retrieve a simple expression for the time delays, in agreement with Witt, Mao & Keeton, which solely depends on the lensed image positions. The real advantage of using the first-order equations when dealing with symmetric gravitational lens systems is to directly test the validity of the adopted lens model without having to perform any accurate numerical fit. In this paper, we present in detail the calculations which lead to those relations between the singular isothermal ellipsoid lens model parameters and the lensed image positions. In addition, we model the well-known Einstein cross Q2237+0305 with three families of models: e − γ, singular isothermal ellipsoid and non-singular isothermal ellipsoid plus shear, using a genetic algorithm from the Qubist Optimization Toolbox. We conclude that although the non-singular isothermal ellipsoid plus shear model shows the best agreement between the calculated and the observed image positions (��x �= 0.0026 arcsec), the more simple singular isothermal ellipsoid also leads to quite satisfactory and acceptable results (��x �= 0.0059 arcsec).