Analytical Solutions of Singular Isothermal Quadrupole Lens

Abstract

Using analytical method, we study the Singular Isothermal Quadrupole (SIQ) lens system, which is the simplest lens model that can produce four images. In this case, the radial mass distribution is in accord with the profile of the Singular Isothermal Sphere (SIS) lens, and the tangential distribution is given by adding a quadrupole on the monopole component. The basic properties of the SIQ lens have been studied in this paper, including deflection potential, deflection angle, magnification, critical curve, caustic, pseudo-caustic and transition locus. Analytical solutions of the image positions and magnifications for the source on axes are derived. As have been found, naked cusps will appear when the relative intensity $k$ of quadrupole to monopole is larger than 0.6. According to the magnification invariant theory of the SIQ lens, the sum of the signed magnifications of the four images should be equal to unity \citep{dal98}. However, if a source lies in the naked cusp, the summed magnification of the left three images is smaller than the invariant 1. With this simple lens system, we study the situations that a point source infinitely approaches a cusp or a fold. The sum of magnifications of cusp image triplet is usually not equal to 0, and it is usually positive for major cusp while negative for minor cusp. Similarly, the sum of magnifications of fold image pair is usually neither equal to 0. Nevertheless, the cusp and fold relations are still equal to 0, in that the sum values are divided by infinite absolute magnifications by definition.

12 pages, 2 figures, accepted for publication in ApJL

References in corpus (8)

• Mathematics of Gravitational Lensing: Multiple Imaging and Magnification
• A Universal Magnification Theorem for Higher-Order Caustic Singularities
• Analytic Relations for Magnifications and Time Delays in Gravitational Lenses with Fold and Cusp Configurations
• Multipole Gravitational Lensing and High-order Perturbations on the Quadrupole Lens
• The Fundamental Surface of Quad Lenses
• Fold Lens Flux Anomalies: A Geometric Approach
• Substructure and the Cusp and Fold Relations
• Geometry of Universal Magnification Invariants

Cited by in corpus (3)

• Magnification relations of quad lenses and applications on Einstein crosses
• Cusp Summations and Cusp Relations of Simple Quad Lenses
• Strong lensing cross sections for isothermal models - I. Finite source effects in the circular case