Cusp summations and cusp relations of simple quad lenses

Abstract

We review five often used quad lens models, each of which has analytical solutions and can produce four images at most. Each lens model has two parameters, including one that describes the intensity of non-dimensional mass density, and the other one that describes the deviation from the circular lens. In our recent work, we have found that the cusp and the fold summations are not equal to 0, when a point source infinitely approaches a cusp or a fold from inner side of the caustic. Based on the magnification invariant theory, which states that the sum of signed magnifications of the total images of a given source is a constant, we calculate the cusp summations for the five lens models. We find that the cusp summations are always larger than 0 for source on the major cusps, while can be larger or smaller than 0 for source on the minor cusps. We also find that if these lenses tend to the circular lens, the major and minor cusp summations will have infinite values, and with positive and negative signs respectively. The cusp summations do not change significantly if the sources are slightly deviated from the cusps. In addition, through the magnification invariants, we also derive the analytical signed cusp relations on the axes for three lens models. We find that both on the major and the minor axes the larger the lenses deviated from the circular lens, the larger the signed cusp relations. The major cusp relations are usually larger than the absolute minor cusp relations, but for some lens models with very large deviation from circular lens, the minor cusp relations can be larger than the major cusp relations.

Magnification relations of quad lenses and applications on Einstein crosses

“…With increasing of the q, the areas of the positive parts in Fig. 4(a) and (c) decrease, and when q 0.68, the area of the positive part is 0 (Chu, Li & Lin 2015).…”

“…We also calculate the cross relations and distance ratios for the other three lens models which are introduced in detail in our previous work (Chu, Li & Lin 2015). The two quantities of the three models also do not depend on the Einstein radii.…”

“…1, if a point source moves to the cusp from the inner side of the tangential caustic, three of the images will merge together near the critical curve. The three close images have an asymp-totic magnification relation (Blandford & Narayan 1986;Schneider & Weiss 1992;Schneider, Ehlers & Falco 1992;Mao 1992;Chu, Li & Lin 2015)…”

“…One of two images is a minimum and the other one is a saddle. The fold image pair also has an asymptotic magnification relation (Blandford & Narayan 1986;Schneider, Ehlers & Falco 1992;Mao 1992;Keeton, Gaudi & Petters 2005;Chu, Li & Lin 2015)…”

“…If the point source infinitely approaches the fold line, the magnifications of the two images will approach infinities, and the fold relation R fold will also be close to 0. In general, when a point source lies just on the cusp point or fold line, the numerators Scusp and S fold are usually not equal to 0 (Chu, Lin & Yang 2013;Chu, Li & Lin 2015).…”