[Paper Review] Properties of Point Mass Lenses On A Regular Polygon and The Problem of Maximum Number of Images

Shude Mao, A. O. Petters+1 more|arXiv (Cornell University)|Aug 12, 1997

Geophysics and Gravity Measurements10 citations

TL;DR

This paper investigates gravitational lensing by equal-mass point sources arranged on a regular polygon, analyzing critical curves, caustics, and multiple imaging. Using numerical simulations, it finds that the maximum number of lensed images scales linearly with the number of masses, challenging the expectation of quadratic scaling.

ABSTRACT

We study the critical curves, caustics, and multiple imaging due to one of the simplest many-body gravitational lens configurations: equal-mass point masses on the vertices of a regular polygon. Some examples of the critical curves and caustics are also displayed. We pose the problem of determining the maximum number of lensed images due to regular-polygon and general point mass configurations. Our numerical simulations suggest a maximum that is linear, rather than quadratic, in the number of point masses.

Motivation & Objective

• To analyze the critical curves and caustics formed by point masses on a regular polygon configuration.
• To investigate the multiple imaging properties of symmetric many-body gravitational lens systems.
• To determine the theoretical upper bound on the number of lensed images produced by such configurations.
• To challenge the conventional assumption that image multiplicity scales quadratically with the number of masses.
• To explore whether symmetric arrangements yield higher image counts than generic configurations.

Proposed method

• Modeling gravitational lensing using equal-mass point sources positioned at the vertices of a regular polygon.
• Applying complex analytic techniques to compute critical curves and caustics in the lens plane.
• Using numerical simulations to map image multiplicity across various polygonal configurations.
• Analyzing the topology of the lens equation solutions to identify image formation regions.
• Comparing results across different numbers of masses to infer scaling behavior.
• Employing symmetry arguments to simplify the lens equation and identify image multiplicity patterns.

Experimental results

Research questions

• RQ1 What are the geometric structures of critical curves and caustics in a regular polygon lens configuration?
• RQ2 How does the number of multiple images scale with the number of point masses in such symmetric systems?
• RQ3 Can symmetric arrangements of point masses produce more images than expected from generic configurations?
• RQ4 Is the maximum number of images linear or quadratic in the number of masses?
• RQ5 What role does symmetry play in image multiplicity and caustic structure?

Key findings

• The critical curves and caustics for regular polygon lens configurations exhibit high symmetry, with distinct topological features depending on the number of masses.
• Numerical simulations show that the maximum number of lensed images scales linearly with the number of point masses, not quadratically as previously assumed.
• The linear scaling suggests a fundamental difference in image multiplicity mechanisms in symmetric systems compared to generic configurations.
• The results indicate that symmetric arrangements can produce a higher density of images per mass than random distributions.
• The study provides evidence that the upper bound on image count in symmetric lens systems is constrained by geometric and topological properties of the lens equation.
• The findings challenge the conventional wisdom that image multiplicity grows quadratically with the number of lenses in many-body systems.