## 5. Multiple causticsIn this section, we consider the case where is a multiple root of Eq. (9). The parameters space of a system with n lenses is dimensional, since each mass adds three parameters (its mass and its coordinates in the lens plane). Four parameters can be eliminated by considering equivalent those systems differing by a global translation or rotation and/or by a global scale factor. Thus, for example, the binary lens is completely characterized by the mass ratio and the separation between the lenses. The requirement of a double root in Eq. (9) translates into the
vanishing of the derivative of this equation with respect to ## 5.1. Critical curvesSuppose that is a root with
multiplicity with the order (that we shall indicate by The term is of order and the first term to be non-null is that for . When we put this expansion in the equation , the first non-null term is having order . If this order is less than zero, we just get from that , but if the order of this term is zero, then the zero order expansion of also involves another term (equal to 1): and the equation gives the non-trivial solution This happens when the order of is
. This is consistent with the result
of the previous section, because, for
, .
For , we have that the first non
trivial order is the second and, for
, it is the order
. When The critical curve just derived is again a circle with radius becoming greater with the multiplicity. ## 5.2. CausticsWe take, as before, the parameterization for the critical curve, with The order is the first depending on and determines the shape of the caustic. To understand this shape, we calculate the cusps as in the previous section. The equation for the cusps is and its solutions are Now we have cusps. This is a very interesting result, because the caustic assumes the shape of a regular polygon with curved sides. The area of the multiple caustic can be calculated in the same way as for the simple one. We just give the result: It is of order . So, increasing the multiplicity from 1 to infinity, the order of the area lowers from 6 to 2 and the extension of the caustic becomes ever more important. Some other consideration about the limit for
can be done. The number of cusps
become infinite and, from Eq. (35), we see that the caustic becomes a
circle of radius ## 5.3. An example: a double caustic in a triple lensNow we shall practically see how our formulae work in the case of a multiple caustic. We consider three masses: , and . We fix the positions of the first two: , ; but we let the third free for the moment. We simultaneously solve Eq. (9) and its derivative with respect to , for the two unknowns and . We find six possible positions of the third mass, giving rise to a double root of the positions equation. None of them is a triple root. Two of these positions are on the -axis. We choose one of them: . The double root is in . In Fig. 3, we see that the critical curve in this point is much greater than the other two. In fact, the radius of the double critical curve, calculated by Eq. (33), is , while the radius of the two simple critical curves is , according to Eq. (19).
In Fig. 4, we show the caustic generated by this double critical curve. The geometry is correctly predicted by our perturbative expansion: there are four cusps in a double caustic. We see that the approximation is less accurate than before, as we anticipated in our discussion about the order of the perturbation. However, for double caustics, it is not so difficult to add another term to the perturbative expansion and reach the same accuracy of the simple caustics. The third order term in the critical curve depends on :
The successive term in the caustic is Double caustics, and, more generally, multiple caustics, are formed by the union of small caustics, in some sense. Another interesting question is: what happens if we change the parameters in the neighbourhood of our particular choice producing the double root? We expect the double critical curve to separate into two smaller ovals and the double quadrangular caustic to break into two triangular ones; but this can happen in different ways. In this regime, the perturbative caustics are simple. However, as
the parameters tend to give the double root,
tends to zero, yielding a diverging
In case In case In case © European Southern Observatory (ESO) 2000 Online publication: June 30, 2000 |