Lens_equation

It is necessary to study the physics of lensing for understand the GL events. In the simplest situation, we will have a source located at S, while the observer sits at O. The lens (the simplest one is a point mass, or “Schwarzschild lens”) is located at a distance from O, on the optic axis. and are the angular-diameter distances between lens and source plane and observer and source plane, respectively (see figure 5). It is important to note that angular-diameter distances are not euclidean distances, so in general: , because of Einstein’s universe is not flat.

figure 5: Typical lensing configuration for a point mass lens. The source is located at the position S, while the observer sits at O. The lens is located in a plane a distance from the observer, at , or on the optic axis.

A ligth ray which passes the lens (with mass

) at a distance

is deflected by

, the deflection angle given by equation [3] (note that

). From simple geometry, we can obtain the condition for this ray to reach the observer at O:

which are the two solutions (a,b) for the second degree equation [8]. The angular separation between the images a and b is:

and the angular separation between the source and the lens is related to the image positions by:

The ray-trace equation [8] always has two solutions of opposite sign. This means that the source has an image on each side of the lens.

If source, lens and observer are colinear then a special situation arises. We will have an “Einstein Ring”, because of there is circular symmetry. In that case we have the mathematical condition:

figure 6: When source, lens and observer are colinear, there is circular symmetry, and we can observe an “Einstein Ring”.

We can now define the “Einstein Radius”, like the radius of the “Einstein Ring”:

Now we find a problem: the description of a lens like a point mass is not enough realistic in most situations, because the “Schwarzschild lens” is an idealization. We need to develop a mathematical formalism and find a lens equation suitable to all mass distributions. When we study more general situations, we cannot suppose that the lens is circular-symmetric, so we have to see the angular separations, the impact parameter and the angles involved in the formalism like two dimensional vectors projected onto the lens or source plane. The geometry in the GL event with a general mass distribution is similar to the point mass one, so we can obtain a equation similar to [6], but with vectorial terms:

Because the angles involved in GL events are very small, it is evident, from elemental geometry (see figure 5), that:

Or, in terms of the distance

(see figure 5) from the source to the optical axis:

Given the matter distribution of the lens and the position

of the source, the equation [18] may have more than one solution

. This means that the same source can be seen at several positions at the sky.

Now we have another problem: to know the deflection law

. For geometrically-thin lenses (we can reject the width of the lens in the optical axis, projecting all the mass of the lens onto the lens plane, if we compare it to the huge distances involved in the GL event) the deflection angles of several point masses simply add [SCH92]. Hence we can descompose a general matter distribution into small parcels of mass m_i and write the deflection angle as:

where

describes the position of the ligth ray in the lens plane, and

describes the position of the point mass m_i in the same plane. We can take the continuum limit and find:

where

is the surface element of the lens plane, and

is the surface mass density at position

which results if the volume mass distribution of the deflector is projected onto the lens plane.