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2.4 The Non Singular Squeezed Isothermal Potential

For cluster of galaxies, the simplest 2-dimensional potential currently used to model multiple images was introduced by Kovner (1987a, b) and Blandford & Kochanek (1987). It is a non-singular squeezed isothermal sphere of the form:

Equation 12 (12)

where xi X and xiY are scaled to the core radius xic , epsilon = (a2 - b2) / (a2 + b2), and phi0 = 4 pi (sigma1D / c)2 xic Dds / Ds.

The one dimensional velocity dispersion sigma1D can be related to the observed line of sight velocity dispersion of cluster galaxies sigmaobs. For a circular potential this relation is simple: sigma21D = 4 / 3 sigma2obs (Mellier et al. 1993).

Figure 7a Figure 7b
Figure 7c Figure 7. Formation of an Einstein ring and two opposite arcs by a singular isothermal sphere. The left panel shows the position of the source in the source plane with respect to the optical axis. The central and the right panels show the images for the case of perfect alignment and for a small offset between the source and the lens position.

An elliptical lens is astigmatic. It produces 2 caustics : one internal caustic with a diamond shape responsible for the largest arcs, and a pseudo-elliptical external caustic that is responsible for a possible radial image (Fig. 8). This external caustic exists only if the potential has a core radius, that is to say if the surface mass density is approximately flat with no singularity for xi = 0 (Sigma (xi = 1) = 1 / 2 Sigma (xi = 0).)

Figure 8
Figure 8. Behavior of the gravitational distortion induced by an elliptical potential as a function of source position. The top left panel shows the shape of the source in the source plane. The second panel shows 10 positions of the source in the source plane (referenced from 1 to 10) with respect to a simulated cluster lens. The full drawn lines are the inner and outer caustics. Panels 3 to 12 show the inner and outer critical lines and the shape of the image(s) of the lensed source. Note positions 6 and 7 which correspond to cusp catastrophes, and position 9 which is a typical fold catastrophe. On the fifth panel we see two inner merging images forming a typical radial arc (from Kneib 1993, PhD thesis).

When the arcs are broken into several images it is sometimes possible to observe in exceptional seeing the inversion of images (parity change) predicted by the modeling. This is a clear signature of a gravitational lens effect and it has been observed in the clusters Cl2244-02 (Fig. 6) and Cl0024+1654 (Kassiola, Kovner & Fort 1992).

Most of the luminous arc configurations actually observed can be reproduced fairly well with this mathematically simple elliptical potential. Some typical examples of the possible geometry of arc systems are given in the panels of Fig. 8. However, the model is acceptable only for small ellipticity. When the axis ratio b/a is larger than 0.5, the isodensity contours of the potential tend toward a funny peanut shape which has no physical meaning. Kassiola & Kovner (1993a) have now proposed a new analytic expression for the potential with elliptical mass distributions that are realistic at large ellipticities. They can eventually be adjusted to any distribution of light in the cluster center (smoothed galaxy light distribution, X-rays, faint visible halos of giant central galaxies, etc...).

Last but not least, possible naked cusp configurations like hyperbolic-umbilic catastrophes can form ``uglier'' images with very strong amplification (Miralda-Escudé 1993a). Such images result from the merging of 5 images, but have not been observed to date, despite a reasonable probability of occurrence.

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