## 1. IntroductionThe binary Schwarzschild lens is one of the most intensively studied model. In fact, in a relatively simple way, it shows many features that are observed in general gravitational lenses, such as the formation of multiple images, giant arcs and a not trivial critical behaviour. The first study about the binary lens with equal masses was made by Schneider & Weiß (1986). They derived the critical curves and the caustics showing that three possible topologies are present depending on the distance between the two lenses. Erdl & Schneider (1993) extended these results to a generic mass ratio of the two lenses. Witt & Petters (1993) reached the same results using complex notation. In some limits, Dominik (1999) enlightened the connection between the caustics of the binary lens and other models, such as the Chang-Refsdal lens (Chang & Refsdal 1979; 1984) and the quadrupole lens. The critical curves and the caustics of multiple lenses can develop very complicated structures, so that the attempts to gain some information about them have been very few. However there is a great interest in this problem for its applications in particular situations, such as planetary systems (Gaudi et al. 1998), rich clusters of galaxies and microlensing of quasars by individual stars in the haloes of the lensing galaxies (Chang & Refsdal 1979; Kayser et al. 1988). In some special situations, the critical curves of multiple lenses
can be derived by perturbative methods, referring to the single
Schwarzschild lens as the starting point for series expansions (Bozza
1999; Bozza 2000). These methods work very well in planetary systems,
for a lens very far from the others and systems where mutual distances
are very small with respect to the total Einstein radius. In the first
two cases, the complete caustic structure has been derived and the
connections with other models have been showed. In the last case, only
the central caustic coming up from the deformation of the total
Einstein ring has been studied. Besides this main curve, there are
many small critical curves forming among the masses. The caustics
generated by these curves generally lie far from the centre of mass
and can have some influence on sources distant from the mass
distribution. Moreover, they move very quickly as the parameters of
the system change (Schramm et al. 1993) constituting the most
problematic feature to control in numerical simulations. For these
reasons they are sometimes dubbed In this paper we use complex notation to face the problem of secondary caustics of close multiple lenses. In this way we can study them as deeply as the other caustics, completing the previous works. We shall see that different classes of secondary caustics can be recognized, showing different geometries. After some review of multiple lensing in Sect. 2, in Sect. 3 we calculate the number and the position of secondary critical curves for an arbitrary number and configuration of lenses. Then, in Sect. 4, we treat the simple caustics and in Sect. 5 the multiple caustics (the distinction will be explained at the end of Sect. 3). In Sect. 6 we specify our formulae for the binary case and in Sect. 7 we give the summary. © European Southern Observatory (ESO) 2000 Online publication: June 30, 2000 |