Image positions and magnifications

3.4 Image positions and magnifications

The lens equation (6

) can be re-formulated in the case of a single point lens:

Solving this for the image positions $𝜃$ one finds that an isolated point source always produces two images of a background source. The positions of the images are given by the two solutions:

The magnification of an image is defined by the ratio between the solid angles of the image and the source, since the surface brightness is conserved. Hence the magnification $μ$ is given as

In the symmetric case above, the image magnification can be written as (by using the lens equation):

Here we defined $u$ as the “impact parameter”, the angular separation between lens and source in units of the Einstein radius: $u = β∕𝜃E$ . The magnification of one image (the one inside the Einstein radius) is negative. This means it has negative parity: It is mirror-inverted. For $β → 0$ the magnification diverges. In the limit of geometrical optics, the Einstein ring of a point source has infinite magnification⁴ ! The sum of the absolute values of the two image magnifications is the measurable total magnification $μ$ :

Note that this value is (always) larger than one⁵ ! The difference between the two image magnifications is unity: