Cusp Summations and Cusp Relations of Simple Quad Lenses

TL;DR

This paper analyzes five simple quad lens models, deriving analytical expressions for cusp summations and relations, revealing how these quantities depend on lens parameters and deviations from circular symmetry, with implications for gravitational lensing observations.

Contribution

It provides the first analytical calculation of cusp summations and relations for these lens models, highlighting their dependence on lens parameters and deviations from circular symmetry.

Findings

Cusp summations are always positive for sources on major cusps.

Cusp summations can be positive or negative for sources on minor cusps.

Lenses approaching circular symmetry have infinite cusp summations with opposite signs.

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.