## Gravitational Microlensing Theory The Single Lens Case

The basic physics of gravitational lensing depends only a single input from General Relativity, the deflection angle, a, for a light ray passing a mass, M, with an impact

Fig. 3.1. The geometry of gravitational lens of mass, M, that is offset by a distance, b, from the line of sight to the source. The observer sees two images that are offset by angles, 6\ and 02 from the line of sight to the source star.

parameter, r:

With the lens geometry shown in Fig. 3.1, we have

4GM riDS

in the small angle approximation. If the lens and source are perfectly aligned, the two images merge to form a ring of radius

known as the Einstein ring radius. (0E is the angular Einstein radius.) We can now rewrite the single lens equation as

i ri _ b and it has two solutions: r+— = 0.5(b ± \Jb2 + 4R2E). The lensed images are also magnified, and the magnification of a source of infinitesimal size can be computed using area elements obtained by differentiating eq. 3.4. This yields