[Paper Review] Secondary caustics in close multiple lenses

V. Bozza|arXiv (Cornell University)|Feb 14, 2000

Intraocular Surgery and Lenses1 references3 citations

TL;DR

This paper develops a perturbative analytical framework using complex analysis to study secondary caustics in close multiple gravitational lenses—systems where multiple point masses are separated by distances much smaller than the total Einstein radius. It derives the number, positions, shapes, cusps, and areas of these secondary caustics for arbitrary configurations, revealing diverse geometries including triangular, butterfly, and beak-to-beak singularities, with exact formulae for the binary case including caustic area and cusp positions.

ABSTRACT

We investigate the caustic structure of a lens composed by a discrete number of point-masses, having mutual distances smaller than the Einstein radius of the total mass of the system. Along with the main critical curve, it is known that the lens map is characterized by secondary critical curves producing small caustics far from the lens system. By exploiting perturbative methods, we derive the number, the position, the shape, the cusps and the area of these caustics for an arbitrary number of close multiple lenses. Very interesting geometries are created in some particular cases. Finally we review the binary lens case where our formulae assume a simple form.

Motivation & Objective

• Understand the formation and structure of secondary caustics in close multiple lens systems where mutual distances are small compared to the total Einstein radius.
• Characterize the number, positions, shapes, and cusps of secondary critical curves and their associated caustics in multi-lens systems.
• Derive quantitative analytical expressions for the area and geometric features of secondary caustics, especially in the binary lens limit.
• Explain the transition mechanisms between different caustic geometries (e.g., butterfly, beak-to-beak) as system parameters vary.
• Provide a complete analytical description of secondary caustics, extending beyond the central caustic in perturbative lensing models.

Proposed method

• Use complex notation to represent lens positions and source coordinates, enabling analytical treatment of the lens equation and Jacobian determinant.
• Apply perturbative expansion in the small parameters $ z_i $, representing lens positions normalized to the total Einstein radius, to solve $ \det J = 0 $ order by order.
• Derive the critical curves as solutions to $ \det J = 1 - |S_2(z)|^2 = 0 $, where $ S_2(z) = \sum_{i=1}^n \frac{m_i}{(z - z_i)^2} $, and map them to the source plane to obtain caustics.
• Classify secondary caustics into simple (triangular) and multiple (butterfly, beak-to-beak) types based on the nature of the singularities in the perturbative solutions.
• Use the perturbative expansion to derive explicit formulae for the shape, cusp positions, and area of secondary caustics in the binary lens case.
• Validate results numerically by comparing perturbative caustics with full numerical solutions for specific configurations.

Experimental results

Research questions

• RQ1 How many secondary critical curves and caustics emerge in a system of n close point masses, and what determines their number?
• RQ2 What are the positions, shapes, and geometric configurations (e.g., triangular, butterfly, beak-to-beak) of secondary caustics in close multiple lenses?
• RQ3 How do the cusps and topology of secondary caustics evolve as the lens configuration changes, particularly during transitions?
• RQ4 What is the analytical expression for the area of secondary caustics, and how does it depend on mass ratios and separations?
• RQ5 How do the perturbative results compare with numerical simulations in capturing the full caustic structure, especially during bifurcations?

Key findings

• The number of secondary caustics in a system of n close point masses is determined by the number of solutions to the perturbative critical curve equation, with distinct classes emerging based on the configuration.
• Secondary caustics are generally triangular in shape when simple, but can develop complex geometries such as butterfly or beak-to-beak singularities during parameter-dependent transitions.
• Triangular caustics have cusps located at angles $ \theta_k = -\arg\left[\pm i(\sqrt{m_1} \pm i\sqrt{m_2})^4\right] + \frac{2k\pi}{3} $, $ k=0,1,2 $, in the binary case.
• The area of the secondary caustics in the binary lens case is $ A = \frac{\pi m_1 m_2 a^6}{8(m_1 + m_2)^4} $, reaching a maximum of $ \frac{\pi a^6}{32M_{\text{tot}}^2} $ for equal masses.
• Numerical simulations confirm that perturbative caustics accurately reproduce the area and topology of the full numerical caustics, even during complex transitions like butterfly formation.
• Multiple caustics can exhibit up to infinitely many cusps in certain configurations, with the transition between geometries governed by the relative motion of the lens masses.