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

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4. Simple caustics

These caustics are largely the most common as we explain in the next section. So they surely have the most practical interest.

4.1. Shape of the critical curves

Once found the positions of the critical curves, we can carry on our perturbative expansion to discover the shape of these curves. So we put

[EQUATION]

where [FORMULA] is the position of one simple critical curve, found by Eq. (9), and [FORMULA] are the corrections of order [FORMULA]. It is convenient to use the original equation [FORMULA], which can be written in the form

[EQUATION]

starting from Eq. (4)

The expansion of [FORMULA] is

[EQUATION]

The first row is the order -2 and is null according to Eq. (11). The second row is the order -1 and the third row is the order zero. Inserting this expansion in (14), the lowest order equation is of order -2:

[EQUATION]

Being [FORMULA] a simple root, [FORMULA], so that [FORMULA].

The successive terms in the expansion of the equation (14) are of order zero:

[EQUATION]

From this equation we have

[EQUATION]

Then the third order contains the first information on the shape of the critical curve. Eq. (18) tells us that the critical curve at position [FORMULA] is a circle centered on [FORMULA] with radius

[EQUATION]

From the form of [FORMULA] (see Eq. (2)), we see that the closer the critical curve is to some mass, the higher the value of [FORMULA], the smaller the radius of the circle.

If we multiply all masses by a factor [FORMULA], the positions of the critical curves do not change, because [FORMULA] factors out from Eq. (9), but their radii change as [FORMULA]. If we do the same with the positions of the masses instead, the positions of the critical curves scale as [FORMULA] and their radii scale as [FORMULA].

4.2. Caustics

To find the caustics corresponding to the simple critical curves, we just have to put the critical curve, in its obvious parameterization

[EQUATION]

into the lens equation (3) and expand to the third order:

[EQUATION]

We can observe that the lowest order is -1 and is independent of [FORMULA]. It represents the position of the caustic. From the order of this term, we can deduce that these caustics can lie very far from the origin of our system, going to infinity as the distances among the masses are reduced to zero. The successive term is [FORMULA], which is of the first order and represents a correction to the position. Finally, the shape of the caustic is given by the third order.

The cusps of a caustic are characterized by the vanishing of the tangent vector. To find them, we have to require that

[EQUATION]

and solve for [FORMULA]. Taking [FORMULA] from Eq. (21), this equation can be simplified into

[EQUATION]

whose solutions are

[EQUATION]

where [FORMULA] yields the argument of a complex number.

We have three cusps. So, in any close multiple system, having only simple secondary caustics, these caustics have a triangular shape.

Finally, we calculate the area of these caustics. This can be done by the integral

[EQUATION]

where [FORMULA] is the caustic in its clockwise direction. We have

[EQUATION]

The minus in the right member comes from the fact that our parameterization is counterclockwise. The integral only involves complex exponential functions and the result is

[EQUATION]

So the extension of the simple caustics is of the sixth order in the separations among the lenses, justifying the evasive nature of these caustics.

With our expansions, we have attained considerable analytical information about the secondary caustics establishing their shape, the area, the number of cusps in a completely general way. However, since these results are the fruit of perturbative approximations, it is important to discuss their accuracy. So we propose a comparison between our perturbative results and the numerical ones in a typical situation. We consider a system constituted by three lenses disposed as in Fig. 1. According to our previous statement, this system can form, at most, four simple secondary critical curves. For our choice of parameters, we display their positions in the same figure. The caustics produced by these curves are shown in Fig. 2 where they are compared to the numerical ones. We have taken the distances among the masses of this distribution to be one tenth of the total Einstein radius. Even for this not too small value, the positions and the shapes of the secondary caustics are reproduced with a striking accuracy. It is also to be noted that the quality of numerical results is improved thanks to the guide provided by perturbative results.

[FIGURE] Fig. 1. Positions of the four secondary critical curves (indicated by the small dots) for a triple system with masses [FORMULA], [FORMULA], [FORMULA] and positions [FORMULA], [FORMULA], [FORMULA] (indicated by the three crosses).

[FIGURE] Fig. 2. Here are the four caustics produced by the critical curves of Fig. 1 from left to right. The solid curve is the perturbative caustic and the dashed line is the numerical one.

So we see that the analytical formulae derived in this section are very good approximations to the quantitative characteristics of the secondary caustics, proving to be highly reliable.

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© European Southern Observatory (ESO) 2000

Online publication: June 30, 2000
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