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.
Highlights
Gravitational lensing is one of the greatest achievements of General Relativity and is one of the most useful tools of galactic astronomy, because the distortion of background sources carries information from the mass distribution deflecting light, and it provides a direct test of cosmological theories [1]-[4].The deflection angle of the light, as well as the image multiplicities [5] and its magnifications, depends on the properties of the lens
Computer simulations suggest that the dark matter halo present in these systems can be described by a spherical mass distribution1 [9], and in this sense, we shall describe our method to the NFW profile
One generalization of the SIS model is frequently used with a finite core x0, that is the non-singular isothermal sphere (NIS), which is more realistic for modeling galaxies
Summary
Gravitational lensing is one of the greatest achievements of General Relativity and is one of the most useful tools of galactic astronomy, because the distortion of background sources carries information from the mass distribution deflecting light (called lens), and it provides a direct test of cosmological theories [1]-[4]. The deflection angle of the light, as well as the image multiplicities [5] and its magnifications, depends on the properties of the lens. Due to the symmetry of these profiles, the relation between the properties of the lens-source system and its observables is reduced to a one-dimensional equation, which provides some important results from a general point of view of the theory, including image position, distortion and magnification. For a basic and comprehensive reference on gravitational lensing see [1] [10] [11]
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