Point-like lens and source

Figure 12.1: The source (S), lens (L) and observer (O) are all aligned. Light from S is deflected by an angle $\alpha$ toward O. Because of symmetry, O sees the source S as ring of radius $R_E$ centred around L.

Figure 12.2: The source (S), lens (L) and observer (O) are now misaligned, and O sees two images of S, one of which is depicted here while the other one lies on the other side of L.

Suppose the light source (S), lensing mass $M$ (L) and observer (O) are exactly aligned, as in Fig.12.1. Also assume they are all point like (i.e. not extended like for example a galaxy). Light rays from S are deflected by L over an angle $\alpha$ toward O. Because of symmetry of the situation, O sees a ring of light around L: an Einstein ring. Of course the figure is not to scale: $\alpha$ should be small if we want to apply the lensing equation.

Introducing the distances between observer and lens $D_{OL}$, observer and source $D_{OS}$, and lens and source $D_{LS}$, we can compute $R_E$ and the angle $\theta_E$ under which O sees the ring. Since $\psi+\theta_E=\alpha$ we get, using the lensing equation and the figure,

$$\theta_E = {R_E\over D_{OL}}$$

$$\psi = {R_E\over D_{SL}}$$

$$\alpha = {4GM\over R_E\,c^2}\,,$$ (12.4)

and hence

$$R_E^2 = {4GM\over c^2}\,{D_{OL} (D_{OS}-D_{OL})\over D_{OS}}\,.$$ (12.5)

If O, L and S are not aligned, as in Fig.12.2, then O does not see a ring. Denote the angular position of S from the line OL by $\theta_S$, and the angle between S and its image I by $\theta_I$. Then from the figure you see that $\alpha(D_{OS}-D_{OL})=D_{OS}(\theta_I-\theta_S)$. A bit of juggling and using the lensing equation gets you to:

$$\theta_I^2-\theta_I\theta_S = \theta_E^2\,.$$ (12.6)

This is a quadratic equation for $\theta_I$ for a given $\theta_S$ and so there will be two images, at positions $\theta_{I,\pm}$.

Often, the change in position will be very small and not observable. But besides being lensed, an image will also be magnified (or de-magnified). So when the displacement is too small to see two images, you'll only see one but magnified by total magnification of each image separately. Interestingly, for a small fraction of peculiar alignments, $A$ may diverge, and so potentially you can get very large amplifications.