Gravitational Lensing by Spherical Lenses

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In this work we introduced
a new proposal to study the gravitational lensing theory by spherical lenses,
starting from its surface mass density ∑(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. Although the
puntual mass is spherically symmetric, its mass density is not continuous so
that its *f*(x) function is discussed
in Appendix 1.

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