3.2 Simple Lens Models

The deflection angle for impact parameter relative to a point mass M is given by hat () = 4GM / c22. A source on the optic axis will form an Einstein ring (Chwolson 1924, Einstein 1936) of angular radius

4.

where we have introduced an effective lens distance D. An off-axis source will produce two images on opposite sides of the lens - a magnified image outside the Einstein ring, and one inside with magnification diminishing as (E / )4 for >> E. For source positions E the two images have roughly comparable magnifications. Therefore, the effective cross section for lensing is usually taken to be E2. Using this estimate of the cross section, the optical depth to lensing for a point source at high redshift is of order the fractional mass density 0 of the universe in point mass lenses (Press & Gunn 1973). For extended sources of angular size E, this is also the optical depth corresponding to significant image distortion due to lensing.

The point mass lens is an appropriate model for computing deflections by individual stars and black holes. However, even when the lens is not point-like, one can still use Equation 4 for rough estimates. For instance, once could take the characteristic angular separation of the images (say in a multiply-imaged quasar) to be a measure of 2E and thus estimate the mass M of the lens ``enclosed'' by the images. Further, for non-point-like lenses, the optical depth to lensing for a high redshift source is roughly given by the density parameter E corresponding to the fractional lens mass enclosed within the respective Einstein rings of the lenses.

In the opposite limit, when the mass distribution has a length scale that is much larger than the size of the image region, one can make a multipole expansion of the potential to quadratic order and rotate the coordinate system so that

5.

where the subscripts 1 and 2 refer to components along two orthogonal axes. The associated magnification matrix of such a quadratic lens is

6.

The parameter is the convergence, which measures the isotropic part of the magnification:

7.

where cr is known as the critical density and D' = D (Dds / Ds)2 is a second effective lens distance. Normally a lens will have > cr somewhere in order to produce multiple images, but this is not strictly necessary (Subramanian & Cowling 1986). The parameter is known as the shear and measures the anisotropic stretching of the image. It arises from matter lying outside the beam. Quadratic lenses are used to describe background galaxies surrounding individual stars and sometimes also for galaxy clusters around galaxies.

The intermediate case, when the image separations are comparable with the size of the potential, is harder to treat. For galaxies and clusters, simple spherical models have been often used (e.g. Clark 1972, Gott & Gunn 1974, Young et al. 1980, Yakovlev et al. 1983). These models can be parametrized by the 1D velocity dispersion = 300300 km s-1. For a singular isothermal sphere, the deflection angle is constant and given by hat = 42 / c2 = 2.6" 2300, so that

8.

For source positions < E, there are two images at = ± E. Technically, there is also a third image at = 0, but this has zero magnification because the surface density is singular at the center of the lens. The cross section for multiple-imaging is given by E2.

 Figure 6. Multiple imaging of point sources at fixed redshift by a generic ``elliptical lens''. The solid lines in the left panels are caustics that separate regions in the source plane corresponding to different image multiplicities (1, 3, and 5 as indicated). The inner caustic, sometimes referred to as the tangential caustic, has four cusps connected by fold lines. The outer radial caustic is a pure fold. The outer dashed lines in the right panels are tangential critical curves and the inner ones are radial critical curves. The symbols show representative source positions and the corresponding image locations. When the source is close to a caustic, some of the images are strongly magnified, indicated by large symbols in the image panels. One of the multiple images usually occurs near the center of the lens and is strongly demagnified if the core radius of the lens is small. Among the ``secure'' multiple quasars, Q1413+117 and Q2237+031 correspond to the source position x and Q0142-100 to O in the upper panels. Q0414+053 and Q1115+080 correspond to the triangle and Q0957+561 is midway between O and + in the lower panels. The weak central image has not been seen in any of the observed cases.

Two refinements are often introduced when modeling galaxy and cluster lenses (Bourassa et al. 1973; Bourassa & Kantowski 1975, 1976; Sanitt 1976; Bray 1984; Blandford & Kochanek 1987b; Kovner 1987a). (a) The singularity at the center is removed by introducing a finite core radius c such that the deflection angle is diminished for < c and vanishes for = 0. The surface density c at the center of the lens now is finite, and multiple imaging is possible only if c exceeds the (distance-dependent) critical density cr. When there is multiple imaging, in addition to two images on either side of the lens as in the singular isothermal lens, there is also a third image in the core region which for small c is usually weak in comparison to the other two. (b) It is often essential to break the circular symmetry of the lens. This can be done by introducing a quadrupolar component in either the mass distribution or the potential of the lens, or by adding an external shear to a circularly symmetric lens. We refer to such models as elliptical lenses. Figures 6 and 7 show some image configurations that arise with elliptical lenses in front of point and extended sources.

 Figure 7. Representative arc and ring images of resolved sources produced by an elliptical lens. In each set, the source planes are on the left and the corresponding images are on the right. The long luminous arcs in Abell 370, Cl 2244-02, Abell 963 and other clusters are similar to the case displayed at top right. (The counter-image shown here will not be present for certain choises of the lens parameters; see Narayan & Grossman 1989, Narayan & Wallington 1992a.) The radio rings correspond to the case shown at bottom right, and the incomplete ring in MG1131+0456 at 15 GHz is similar to the example at bottom left.