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

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5. Multiple caustics

In this section, we consider the case where [FORMULA] is a multiple root of Eq. (9). The parameters space of a system with n lenses is [FORMULA] 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 z. This is one constraint equation, then the points of the parameters space producing multiple roots constitute a [FORMULA] dimensional hypersurface, thus having measure zero. For this reason, the occurrence of multiple roots is relatively rare. Anyway, very interesting features emerge, justifying a detailed study of these particular cases.

5.1. Critical curves

Suppose that [FORMULA] is a root with multiplicity p. We have to find the correct order of the perturbation to insert in the equation [FORMULA], representing the shape of our critical curve. According to the equivalence (12), the [FORMULA] with [FORMULA] are null. Then, we put


with the order (that we shall indicate by q) of [FORMULA] to be found. We only assume that q be higher than one. Then, the expansion of [FORMULA] is


The [FORMULA] term is of order [FORMULA] and the first term to be non-null is that for [FORMULA]. When we put this expansion in the equation [FORMULA], the first non-null term is


having order [FORMULA]. If this order is less than zero, we just get from [FORMULA] that [FORMULA], but if the order of this term is zero, then the zero order expansion of [FORMULA] also involves another term (equal to 1):


and the equation gives the non-trivial solution


This happens when the order of [FORMULA] is [FORMULA]. This is consistent with the result of the previous section, because, for [FORMULA], [FORMULA]. For [FORMULA], we have that the first non trivial order is the second and, for [FORMULA], it is the order [FORMULA]. When p increases, the order of this perturbation decreases, approaching 1 as a limit. This means that at high multiplicities, the perturbative expansion becomes always less accurate, requiring ever more terms for an adequate description of the caustics. Anyway, the main characteristics of the caustics can be derived retaining just the first correction and that is what we shall do.

The critical curve just derived is again a circle with radius


becoming greater with the multiplicity.

5.2. Caustics

We take, as before, the parameterization


for the critical curve, with r given by Eq. (33). Putting this expression into the lens equation (3) and expanding to the [FORMULA] order, we get


The [FORMULA] order is the first depending on [FORMULA] 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 [FORMULA] cusps. This is a very interesting result, because the caustic assumes the shape of a regular polygon with [FORMULA] 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 [FORMULA]. 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 [FORMULA] can be done. The number of cusps become infinite and, from Eq. (35), we see that the caustic becomes a circle of radius r. In fact, the area becomes [FORMULA].

5.3. An example: a double caustic in a triple lens

Now we shall practically see how our formulae work in the case of a multiple caustic. We consider three masses: [FORMULA], [FORMULA] and [FORMULA]. We fix the positions of the first two: [FORMULA], [FORMULA]; but we let the third free for the moment. We simultaneously solve Eq. (9) and its derivative with respect to [FORMULA], for the two unknowns [FORMULA] and [FORMULA]. 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 [FORMULA]-axis. We choose one of them: [FORMULA]. The double root is in [FORMULA]. 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 [FORMULA], while the radius of the two simple critical curves is [FORMULA], according to Eq. (19).

[FIGURE] Fig. 3. Critical curves in the event of a double root. The masses are indicated by the crosses. The double critical curve is the one at the bottom-center, while the other two are on the top-left and top-right and are hardly visible in this picture.

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 [FORMULA]:


[FIGURE] Fig. 4. Caustic corresponding to the double critical curve of Fig. 3. The dashed curve is the numerical caustic and the solid line is the perturbative one.

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, [FORMULA] tends to zero, yielding a diverging r for the simple critical curves, according to Eq. (19). The transition with the formation of the double critical curve is thus not reproduced. Guided by perturbative approximations, the break of the double caustic, when [FORMULA] moves out from the position [FORMULA], can be investigated numerically. The results are shown in Fig. 5.

[FIGURE] Fig. 5. Critical curves and caustics for [FORMULA] close to [FORMULA]. The left column shows the critical curves and the right column the caustics. The thick lines are the numerical curves and the thin lines are the perturbative ones. The choice of [FORMULA] in cases a, b and c are given in the text.

In case a, [FORMULA], i.e. we have moved the third mass towards the others. The critical curve breaks in the horizontal direction. Looking just at the thick line in Fig. 5a2, representing the numerical caustic, we see that the top cusp and the bottom cusp develop a butterfly geometry. At some critical value, these butterflies touch and the two resulting triangular caustics move away along the horizontal direction. We have displayed in the same plot the perturbative caustics too. Obviously, they are simple caustics, so they cannot show the butterfly geometry but they can help in understanding how the separation occurs. We also notice that the simple caustics cover the area of the numerical transition double caustic very well constituting a significant approximation anyway.

In case b, [FORMULA], so that the third mass is farther from the others. Now the critical curve breaks in the vertical direction and so does the caustic. The left and the right cusps transform into butterflies. These butterflies are slightly distorted by the fact that the resulting simple caustics have different sizes: the one on the top is smaller than the other.

In case c, [FORMULA]. We have displaced the third mass in the horizontal direction. The critical curve breaks diagonally and so does the caustic. But this time the transition occurs with a simple beak-to-beak singularity rather than with butterflies. While in the previous situations the two simple caustics in the last step of the separation touch with a fold, here they touch with a cusp.

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Online publication: June 30, 2000