Gravitational Lensing by Spherical Lenses

TL;DR

This paper introduces a new method for analyzing gravitational lensing by spherical lenses using a function-based approach that simplifies the derivation of lens properties and applies to various profiles like SIS, NIS, and NFW.

Contribution

The paper presents a novel differential equation framework that directly relates the lens surface mass density function to lens properties, streamlining analysis of spherical gravitational lenses.

Findings

Derived a differential equation linking surface mass density to lens properties.

Applied the method to SIS, NIS, and NFW profiles to demonstrate its versatility.

Analyzed critical curves, caustics, and magnifications using the new approach.

Abstract

In this work we introduced a new proposal to study the gravitational lensing theory by spherical lenses, starting from its surface mass density $\Sigma(x)$ written in terms of a decreasing function $f$ of a dimensionless coordinate $x$ on the lens plane. The main result is the use of the function $f(x)$ to find directly the lens properties, at the same time that the lens problem is described by a first order differential equation which encodes all information about the lens. SIS and NIS profiles are used as examples to find their functions $f(x)$. Using the Poisson equation we find that the deflection angle is directly proportional to $f(x)$, and therefore the lens equation can be written in terms of the function and the parameters of the lens. The critical and caustic curves, as well as image formation and magnification generated by the lens are analyzed. As an example of this method, the properties of a lens modeled by a NFW profile are determined. Altough the puntual mass is spherically symmetric, its mass density is not continuous so that its $f(x)$ function is discussed in the Appendix A.