Astron. Astrophys. 359, 1-8 (2000)

## 3. Number and positions of secondary critical curves

Close multiple lenses have two classes of critical curves: the main critical curve, resulting from the deformation of the Einstein ring of the total mass lens, and the secondary critical curves, forming inside the distribution of the masses.

Effectively, if we multiply the equation by the quantity , we get a complex equation in z and :

At the zero order, putting all 's to zero, this equation becomes

This equation has the solution , that is the Einstein ring of the total mass lens. Taking this solution as the starting point of a perturbative expansion, we get the main caustic. The details of this calculation are in (Bozza 2000). But the presence of the solution indicates that also this value can be taken as the starting point for another expansion. This is just the value we shall take to find the secondary critical curves.

Having observed the zero order situation, we can start our perturbative approach, searching for the first order solution. Then we write the solution z as a series expansion:

where is of the first order in . Stopping at the first order, we put in Eq. (6). We see that the first term becomes of order , while the second is of order . Then the latter dominates the first and Eq. (6) is equivalent to

This is a polynomial equation of degree . Then, for a system of n close lenses, there are, at most, points where the Jacobian determinant vanishes (at the first order in ), corresponding to secondary critical curves. This is the first main result of our work. It is consistent with the binary lens, since two secondary critical curves are predicted by this formula.

Eq. (9) can be solved analytically for two and three lenses, otherwise we have to resort to simple numerical methods. In Sect. 6, we shall specify these and the following results for the binary lens where a manageable expression for the positions of the critical curves is available. For the triple lens, the analytical solutions are too cumbersome to allow a detailed study.

Now, we have a straightforward way to calculate the positions of the secondary critical curves for an arbitrary configuration of close multiple lenses. Then, we can avoid the traditional blind sampling of the Jacobian determinant on the lens plane and reach, by this new method, the full efficiency.

We take the generical solution of Eq. (9) as the first order term of our expansion. From now on, we use the notation

As both and are of the first order, according to our perturbative expansion, has all denominators of order k and then it is of order .

To continue our study we do not need an analytical expression for . We shall just use the fact that is a solution of Eq. (9), that is equivalent to say that

Of course, we have to distinguish between simple roots of Eq. (9) and roots of higher multiplicity. Remembering that the derivative of is proportional to , we have the equivalence between the following statements:

We shall treat separately the caustics coming from simple roots (hereafter called simple caustics) and the caustics coming from multiple roots (hereafter multiple caustics).

© European Southern Observatory (ESO) 2000

Online publication: June 30, 2000