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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.

Gravitational lensing is one of the greatest achievements of General Relativity and is one of the most useful tools of galactic astronomy, not only because the distortion of background sources carries information from the mass distribution deflecting light (called lens), but also it provides a direct test of cosmological theories [

The deflection angle of the light, as well as the image multiplicities [

Of course, due to the intrinsic ellipticity of a cluster or a galaxy, it is not physically possible to model such systems using a spherical profile. However, computer simulations suggest that the dark matter halo present in these systems can be described by a spherical mass distribution^{1} [

For a basic and comprehensive reference on gravitational lensing see [

Suppose a spherically symmetric mass profile lying at a distance^{2}, acting as a gravitational lens on the light emitted by a source at a distance

where

where

with this, the convergence is defined by

where

Moreover, the Poisson equation relates the convergence and the deflection potential of the lens

which, for a spherically symmetric mass distribution can be expressed as

where

thus, from Equation (1.7) can be found that

or, by Equation (1.4)

Now, since

this assumption is made in order to use the fundamental theorem of calculus in the integral expression of the deflection angle, Equation (1.10), so that

that is

The anterior result shows that for a spherically symmetric mass profile, whose surface mass density can be written in the form of Equation (1.2), the deflection angle is proportional to the function

The lens equation, which relates the image and source positions,

which can be written in terms of the

Joining the results given above, the

which comes from inserting Equation (1.11) in Equation (1.4), according to the initial condition (1.3). Thus, the problem is reduced to solve the first-order ordinary differential Equation (1.16) for

A spherical model widely used in the gravitational lensing theory is the singular isothermal sphere (SIS) [

where

with

The

To find the deflection angle, make the product

One generalization of the SIS model is frequently used with a finite core

Through a procces similar to the SIS, we can found

and

where

Since gravitational lensing conserves the surface brightness, the magnification of an image is defined as the ratio between the solid angles of the image and the source. Namely

from Equation (1.11) and Equation (1.15), this is

Equation (1.28) implies that the magnification has two singularities in

Noting that the magnification Equation (1.28) can be written in terms of the convergence

whereby

and from Equation (1.4), the shear is

This expression allows to calculate

and that generated by a NIS profile

where we made use of Equation (1.24) and Equation (1.25).

Now, recognizing that

and

the shear can be written in terms of the deflection potential of a mass distribution with spherical symmetry, as

Here, the definition of the

The critical curves are those points

or

but, from Equation (1.4) and Equation (1.31)

and

Thus, the critical curves are the level contours of the

Equation (1.42) are not associated forming a system, thus, given

with

In general, the image multiplicity depends on the source position with respect to the caustic circle, changing in two as the source crosses through it. Moreover caustics depends on the critical curves and on the increase or decrease of

• If

• If

• If

In the case where the lens produces three images the source is inside the caustic circle, that is

or

with

Now suppose the source located in the first quadrant of a cartesian coordinate system in whose center lies the lens; namely

Function f(x) for two lens models, the singular isothermal sphere and the Non-singular isothermal sphere (NIS). In the case of SIS xf(x) is constant, while f(x) is decreasing. In the NIS model xf(x) is increased, unlike f(x)

through Equation (1.47)

This implies

therefore, the images

the angle of images will be

that is, the images

Meanwhile, the third image

and since

If the lens produces only two images, they are diametrically opposed lying on the line lens-source. And, if the lens produces only one image, it will be located at the same angle of the source.

The lensing effects of the NFW profile have been widely studied [

Suppose a gravitational lens modeled by a NFW profile [

where the so called scale radius

The NFW mass density Equation (1.54) expressed in terms of

this expression is according with the results found in [

Equation (1.16) and Equation (1.55) leads to the differential equation

finding that

where we use Equation (1.3). Therefore

The deviation angle can be calculated through Equation (1.13)

where the

The behavior of the lens equation is shown in

•

•

•

•

In

From Equation (1.31), we find

Shear Equation (1.62) is a continuous and decreasing function over the range

An horizontal line, , determines the critical curves when crossed with the functions, Equation (1.58), and, Equation (1.59). Note that in

Lens equation by a NFW model, Equation (1.61). As shown, the image positions will depend on the magnitud of the source and the parameters of the profile, r_{s}, ρ_{c} and δ_{k}, represented by C, Equation (1.56). The local maxi- ma and minima (red and blue lines), corresponds to the radius of the critical circles

and

Lens equation by a NFW model for C = 0.1. In x_{c}_{1} ≠ 0, the function intercepts the horizontal axis and takes its minimum value. In x_{c}_{2} the function takes its local maximum in. If the images will be located outside the circle of radius x_{r}

Behavior of the points x_{c}_{1} and x_{c}_{2} where the lens equation takes its maximum values. Radius of the caustic circle associated to x_{c}_{2}, as a function of C. This curves were found numerically

to the outer critical circle.

At the same time, the magnification, given by Equation (1.28) though Equation (1.58) and Equation (59) is plotted in

In this paper we introduce a new proposal to study the gravitational lens effect by a spherically symmetric mass distribution. The main result is the use of a new function

Image positions of a point source as a function of C. The critical curves x_{c}_{1} (blue) and x_{c}_{2} (red) divides the source plane in three regions of image formation, that is, depending of the source position we can found up to 3 images. In agreement to Section (1.5), solid black lines represents the position of the first image (α), dotted lines, the second one (β), and the dashed lines, the third (γ), for each case of. The three images are associated as follows: each of the curves from left to right and under x_{c}_{1} is associated with one curve from top to bottom above x_{c}_{1}. If the image position approaches to zero, i.e., then the Einstein Ring of radius x_{c}_{1} is formed, and the images and go to zero, as we can see from the plot

Magnification of images in the lens plane for two values of C. The asymptotes will form in x_{c}_{1} and x_{c}_{2} (in each curve from right to left, respec- tively). Simulation of the image formation for a circular lens, magnification, critical and caustic curves generated by a lens modeled through a NFW profile is available online

lens which contains all the physical information of the lens and also is a function of the cosmological model.

The importance of the method described in this paper is that if you resolve Equation (1.16) for

In the case where the convergence is not a continuous function of the space, the differential Equation (1.16) can still be used to find the

We apply the method to a lens modelled by the NFW profile and found explicitly the function

R. Hurtado is grateful with Y. Villota for some helpful suggestions that improved the presentation of the paper and the Universidad Nacional de Colombia for financial support.