Probing the Structure of Lensing Galaxies with Quadruple Lenses: The Effect of the “External” Shear

We study a general elliptical potential of the form ψ(x + y/q) (0 < q ≤ 1) plus an additional shear (with an arbitrary direction) as models for the observed quadruple lenses. It is shown that a minimum additional shear is needed even just to reproduce the observed positions alone. We also obtain the dependence of the axial ratio, q, on the orientation of the major axis of potential. A general relation also exists between the shear, the position angle and axial ratio of the lensing galaxy. The relation shows a generic degeneracy in modelling quadruple lenses. In particular, it shows that only the ratio of the ellipticity, ǫ ≡ (1−q2)/(1+q2), to the magnitude of shear, γ can be determined. All these results are valid regardless of the radial profile of the potential. Our formalism applies when the galaxy position is observed, which is the case for seven of the eight known quadruple lenses. Application to these seven cases reveals two quadruple lenses CLASS 1608+656 and HST 12531–2914, requiring highly significant shear with magnitude ≈ 0.2. For HST 12531–2914, there must be a misalignment between the major axis of light and the major axis of potential (mass). We conclude that detailed modelling of quadruple lenses can yield valuable quantitative information about the shape of lensing galaxies and their dark matter halos.