The gravitational lens equation near cusps
Abstract
The behaviour of the gravitational lens mapping near cusps is studied, both analytically and numerically, paying particular attention to magnification probabilities. We demonstrate that the three images of a point source inside a cusp satisfy the relation that the sum of the magnifications of the two images with the same parity equals, up to a sign, the magnification of the third image (of opposite parity). This property will then be used to show that the asymptotic magnification crosssection for point sources, in the limit μ_S_ approaches infinity, derived previously for folds only, is also valid in the presence of cusps. The next order term of such an expansion, which is due to sources just outside of cusps, is derived. We apply these relations to a special gravitational lens model and show that these asymptotic relations are indeed very good approximations for the largeμ_S_ crosssections. For the study of the magnification of extended sources near cusps, we generalize the rayshooting method to allow for very small sources. The magnification crosssections for extended sources are then Compared to those for point sources. A magnification contourplot for extended sources near a cusp is obtained. Since the largest magnifications of sources occur near cusps, this paper may directly apply to studies of the amplification bias in source counts.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 July 1992
 Bibcode:
 1992A&A...260....1S
 Keywords:

 Catastrophe Theory;
 Gravitation Theory;
 Gravitational Lenses;
 Point Sources;
 Computational Astrophysics;
 Polar Cusps;
 Astrophysics