The Quadruple Image Configurations of Asymptotically Circular Gravitational Lenses

At a Glance

Metadata	Details
Publication Date	2022-09-01
Journal	The Astronomical Journal
Authors	Chirag Falor, Paul L. Schechter
Analysis	Full AI Review Included

Executive Summary

Model Simplification: The research simplifies the complex 7-parameter model for quadruply lensed quasars (quads) by analyzing the configurations of asymptotically circular gravitational lenses (ACLE), reducing the salient degrees of freedom to two.
Core Equation (ACLE): A quartic equation, the Asymptotically Circular Lens Equation (ACLE), is derived. This equation governs the angular positions of the four images and is parameterized by a single complex quantity (W).
Dimensionality and Stability: The space of quadruple configurations is shown to be two-dimensional, bounded by a unit astroid. These configurations remain stable even as the lens ellipticity vanishes, provided the source position relative to the diamond caustic is held constant.
Geometric Derivation: The ACLE is derived from the limiting case of the Witt-Wynne geometric construction, where the Wynne ellipse becomes a circle. The image positions are the intersections of this circle and a Witt hyperbola.
Invariance and Generalization: New “semi-astroidal” coordinates (causticity ζ and astroidal angle α) are introduced. These coordinates are invariant under “scronching” (stretching/squeezing), allowing ACLE solutions to be mapped directly to non-circular potentials like the Singular Isothermal Sphere with External Shear (SIS+XS).
Configuration Invariant: The Kassiola-Kovner configuration invariant is shown to be exactly zero for any lens satisfying the ACLE, and the surface derived from this invariant closely resembles the Fundamental Surface of Quads (FSQ).
Application Beyond Lensing: The ACLE framework is adapted to model Solar System “occultation flashes,” showing that these configurations differ subtly from observed quasar quads.

Technical Specifications

The paper focuses on theoretical constraints and dimensionalities rather than physical material properties.

Parameter	Value	Unit	Context
Salient Degrees of Freedom (Quads)	2	Dimensions	Reduced model space for asymptotically circular lenses (ACLE).
ACLE Solution Dimensionality	2	Dimensions	The space of image configurations is governed by the complex parameter W.
Astroid Boundary Condition	p^2/3 + q^2/3 < 1	Unitless	Condition for the complex parameter W (p + iq) to yield four distinct image solutions.
SIQP Quadrupole Amplitude Range	-1/3 ≤ ε ≤ 1/3	Unitless	Range of quadrupole amplitudes that share the same angular configurations as the ACLE.
FSQ Arc Midpoint (θ₁₂, θ₃₄)	(2π/3, 2π/3)	Radians	Position on the Fundamental Surface of Quads (FSQ) where the two closest images merge (θ₂₃ = 0).
Occultation Flash Deviation (Mean)	-0.455	Degrees	Mean deviation of measured eccentric angle difference (η₂₃) from the calculated value for Saturn occultation data.
Occultation Flash Deviation (Std Dev)	1.04	Degrees	Standard deviation of eccentric angle difference (η₂₃) for Solar System occultation flashes.

Key Methodologies

The core methodology involves geometric construction and algebraic analysis of limiting cases for gravitational lens potentials.

Witt-Wynne Construction (Limiting Case): The image configurations are derived by inverting the Witt-Wynne recipe and taking the limiting case where the Wynne ellipse has vanishing ellipticity, resulting in a unit circle.
Image Position Determination: The four image positions are found at the intersections of the unit circle (Wynne’s circle) and a rectangular hyperbola (Witt’s hyperbola) that passes through the center of the circle (the source position).
ACLE Derivation: The geometric intersection condition is translated into the Asymptotically Circular Lens Equation (ACLE), a quartic polynomial in e^iθ, parameterized by the complex quantity W, which relates to the position of the hyperbola’s center.
Parameterization via Astroidal Coordinates: The 2D solution space is defined using “semi-astroidal” coordinates:
- Causticity (ζ): The relative displacement of the source toward the astroid boundary (ζ = 1).
- Astroidal Angle (α): The angle the source makes with respect to the symmetry axis, defined such that it is invariant under scronching.
De-scronching Technique: To model non-circular potentials (like SIS+XS or SIEP), the image configuration of the corresponding ACLE solution is calculated and then “scronched” (stretched/squeezed) according to the lens’s shear (γ) or semi-ellipticity (ε). This process maintains the astroidal coordinates (ζ and α).
Magnification Ratio Calculation: Relative flux ratios (magnifications) are derived by calculating the determinant of the Jacobian for the SIQP and SIS+XS potentials in the asymptotically circular limit, showing that the SIS+XS magnification is twice that of the SIQP for the same configuration.

Commercial Applications

This research is foundational theoretical astrophysics, primarily applicable to astronomical observation and modeling.

Application Area	Description
Astrophysical Modeling Software	Provides robust, simplified analytical solutions (ACLE) for gravitational lensing software used to model quadruply lensed quasars (quads), improving efficiency over full 7-parameter numerical models.
Strong Lensing Surveys	The derived Fundamental Surface of Quads (FSQ) acts as a critical constraint for validating observed image configurations in large astronomical surveys.
Lens Potential Characterization	The invariant astroidal coordinates (ζ, α) offer a robust method for characterizing the source position relative to the caustic, allowing direct comparison of lens systems governed by different physical potentials (e.g., SIQP vs. SIS+XS).
Planetary Science and Occultations	The ACLE framework is adapted for analyzing atmospheric effects observed during stellar occultations by Solar System bodies (e.g., Saturn), providing a tool for calculating image configurations formed by refraction.
Theoretical Physics Research	Generalizes the understanding of caustic formation, showing that the angular behavior of perturbations, rather than the specific mass distribution, dictates the image configurations for nearly circular potentials.

View Original Abstract

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,” parameterized by a single complex quantity. Several alternative parameterizations are examined. Relative magnifications of the images are derived. When a nonvanishing 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.”

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Original Source

DOI: https://doi.org/10.3847/1538-3881/ac80bc