The Quadruple Image Configurations of Asymptotically Circular Gravitational Lenses

TL;DR

This paper analyzes the stable quadruple image configurations of gravitational lenses approaching circularity, deriving solutions from a quartic equation and exploring effects of external shear, with applications to astrophysical phenomena.

Contribution

It introduces the Asymptotically Circular Lens Equation (ACLE) and examines how external shear influences lens configurations, providing a new framework for understanding nearly circular gravitational lenses.

Findings

Quadruple image configurations are stable near circularity.

Configurations can be described by solutions to a quartic equation.

External shear causes predictable stretching and squeezing of configurations.

Abstract

The quadruple image configurations of gravitational lenses with vanishing ellipticity are examined. Even though such lenses asymptotically approach circularity, the configurations are stable if the position of the source relative to the vanishing diamond caustic is held constant. The configurations are the solutions of a quartic equation, an "Asymptotically Circular Lens Equation" (ACLE), parameterized by a single complex quantity. Several alternative parameterizations are examined. Relative magnifications of the images are derived. When a non-vanishing quadrupole, in the form of an external shear (XS), is added to the singular isothermal sphere (SIS), its configurations emerge naturally as stretched and squeezed versions of the circular configurations. And as the SIS+XS model is a good first approximation for most quadruply lensed quasars, their configurations likewise have only 2+1 salient dimensions. The asymptotically circular configurations can easily be adapted to the problem of Solar System "occultation flashes."