Lens equation

3.1 Lens equation

The basic setup for such a simplified gravitational lens scenario involving a point source and a point lens is displayed in Figure 2

. The three ingredients in such a lensing situation are the source S, the lens L, and the observer O. Light rays emitted from the source are deflected by the lens. For a point-like lens, there will always be (at least) two images S₁ and S₂ of the source. With external shear – due to the tidal field of objects outside but near the light bundles – there can be more images. The observer sees the images in directions corresponding to the tangents to the real incoming light paths.

Figure 2: Setup of a gravitational lens situation: The lens L located between source S and observer O produces two images S₁ and S₂ of the background source.

In Figure 3 the corresponding angles and angular diameter distances $DL$ , $DS$ , $DLS$ are indicated³ . In the thin-lens approximation, the hyperbolic paths are approximated by their asymptotes. In the circular-symmetric case the deflection angle is given as

where $M (ξ)$ is the mass inside a radius $ξ$ . In this depiction the origin is chosen at the observer. From the diagram it can be seen that the following relation holds:

(for $𝜃$ , $β$ , $&tidle;α ≪ 1$ ; this condition is fulfilled in practically all astrophysically relevant situations). With the definition of the reduced deflection angle as $α(𝜃) = (DLS ∕DS )α&tidle;(𝜃)$ , this can be expressed as:

This relation between the positions of images and source can easily be derived for a non-symmetric mass distribution as well. In that case, all angles are vector-valued. The two-dimensional lens equation then reads:

Figure 3: The relation between the various angles and distances involved in the lensing setup can be derived for the case $&tidle;α ≪ 1$ and formulated in the lens equation (6