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2.2 Caustics and Formation of Arcs

It is clear from Fig. 4 that the deflecting potential can generate several pieces of delayed wavefronts which can be eventually observed. The number of images varies with the observer position, or conversely with the source position. In two dimensions, the source plane is generally separated into several regions where the number of images changes. The boundaries of these regions are referred to as caustic lines. The number of images will change suddenly when the source crosses a caustic line, and if the deflecting potential is non-singular , there must be an odd number of images (Burke 1981) within each region. Folds and cusps are the simplest caustics predicted by gravitational optics theory. When a source moves close to a fold caustic, two images will tend to merge with an arc-like geometry. The two merging images will vanish just after the crossing of the fold (Fig. 8(9)). Near cusps, 3 images merge and this is the generic situation whereby the largest (perhaps broken) arcs are formed (Fig. 6 and 8(7)). When multiple images of an extended object are formed, all segments have changes in the orientation of the images relative to the source. These parity changes can be seen on very well-resolved images as mirror-like effects, and are particularly useful for identifying the caustic configuration. Without a peculiar perturbing potential on the specific line of sight, each segment will be a linear mapping of the others.

More marginal or complicated caustics can also generate arcs. Lips caustics appear in elliptical potentials with core radius when the outer caustic shrinks (see Kochanek 1992). Beak-to-beak caustics appear in bimodal potentials when two individual cusp caustics merge (see Kovner 1989).

Figure 6. Formation of giant arcs in fold and cusp catastrophic configurations. The positions of the source with respect to the caustics are shown in the central panel. The corresponding image positions with respect to the critical lines in the image plane are on the left (fold) and on the right panels (cusp). They have an axis direction which is tangent to the critical line (see section 4.5). The relationship between image and source is respectively a parabola (bottom left) and a cubic (bottom right) and therefore 2 or 3 images can be formed respectively.

Critical lines in the image plane correspond to caustic lines in the source plane. They are the loci where the determinant of the amplification tensor is infinite. The magnification diverges for a point source in the immediate vicinity of a caustic line. Very large circular arcs almost perfectly describe the critical lines of deflectors with small ellipticities.

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