Astron. Astrophys. 348, 311-326 (1999)

3. Caustics and perturbative analysis

In the solar system, the mass ratios between planets and the sun are always less than one thousandth. Jupiter is . Other planets are even less: Earth is . With these numbers, it is natural to expect that the presence of planets should cause only little perturbations to the single lens case. Upon this consideration the perturbative hypothesis is based. In this and the following section the ratios between planetary and stellar masses will play the role of perturbative expansions parameters. We shall see that in most cases a first order expansion is sufficient to get very reliable results.

Let's turn to the caustics of planetary systems. First I shall examine the modifications induced in the star's Einstein ring and consequently the central caustic. Then I shall deal with planetary caustics.

3.1. Central caustic

Of course, the starting point for the study of critical curves is the equation , which can easily be written in polar coordinates:

Here . Expanding this equation to the first order in , we get:

The zeroth order solution is simply , i.e. the Einstein ring. Let's write the first order solution as:

with . Substituting in (5) and expanding to the first order in . The zeroth order solution cancels and is found solving the remaining first degree equation:

where is to the zeroth order.

By very few steps we have found the perturbation of the Einstein ring in a very simple way. The parametric equation of the central caustic is soon found by applying the lens Eq. (1) and expanding again to the first order in :

Of course, perturbative results are characterized by precise limits of validity. In our case we see that when the denominators in (7) vanish the perturbation diverges. This is not allowed by our assumption that must be very small with respect to the unity. Those denominators represent the distance between the planet and the general point of the unperturbed star's Einstein ring. So we expect the perturbative theory to fail in those portions of the critical curve which are very near to one of the planets at least. We can understand this "failure" if we look back at Fig. 2b: when the planet is close to the star's Einstein ring, there is only one critical curve which is somewhat different from the ring in the proximity of the planet. For some values of , the radial coordinate describing the critical curve assumes also more than one value; this situation cannot be described by a first order approximation, where, as we saw, the perturbation solves a first degree equation.

The most interesting aspect of Eqs. (7) and (8a) is that they are comprehensive of the action of the whole planetary system: they are valid for an arbitrary number of planets, not only the classically investigated case of the single planet. So these formulas enjoy a very high generality and can be used in more realistic contexts. We also note that the contributions coming from different planets superpose without interfering. This is an obvious consequence of the first order approximation; if I had included second order terms, I would have found "interaction" between planets. These "interaction" terms are thus not relevant in a first approximation.

Now let's compare the perturbative caustics with those found by classical numerical algorithms to test the validity of the perturbative approach. In Fig. 3 I show the results for the case of two Jovian planets placed in several positions. When the planets are far enough from the Einstein ring (Fig. 3a, 3d), the caustic found according to (8a) is entirely identical to the numerical one. Letting one of the planets approach the Einstein ring, a small deviation begins appearing in the region coming from the portion of the Einstein ring that is closest to the planet. This deviation manifests itself in the size of the largest cusp. For Jovian planets these discrepancies unveil at distances from the star's Einstein ring of the order of a tenth of the Einstein radius. These first encouraging results become much better in the case of Earth - like planets. We expect the range of validity of perturbative results to be increased for this kind of planets, because of their smaller mass. Fig. 4 shows that for these little planets things go very well down to a hundredth of the Einstein radius.

Fig. 3a-d. Central caustic for Jovian planets. The caustics on the left are those found perturbatively, while those on the right are obtained numerically. Little differences occur when one of the planets is close to the star's Einstein ring.

Fig. 4. Central caustics for Earth - like planets. The perturbative ones are on the left and the numerical on the right.

So the perturbative method is likely to provide reliable results at the first order already. Moreover, it is not to be forgotten that, in principle, the approximations can be improved pushing farther the perturbative expansion.

3.2. Planetary caustics

Planetary caustics are usually studied considering the planet as a point-lens with an external shear due to the star's gravitational field (Schneider, Ehlers & Falco, 1992). This kind of lens was introduced by Chang & Refsdal (1979; 1984) in a cosmological context. However, this lens is valid in planetary systems only to the lowest non-trivial order in . Therefore a correct study should only retain the lowest order terms in the critical curves equation, so that Chang & Refsdal's caustics are a suitable approximation at the lowest order only and not beyond.

In order to complete the discussion of caustics in planetary systems and study their features properly, in this subsection I derive planetary caustics from perturbative hypothesis paying full attention to the order of each term. The situation for planetary caustics is rather different from that of the central caustic. There is no zeroth order solution to start from, since their very presence is perturbative. Nevertheless this is not a great problem: in fact we shall just search for the lowest order solution of the critical curves equation.

To achieve this, I now rewrite in polar coordinates choosing the planet situated in as the origin:

When the planet is very far from the star, we know that its critical curve tends to an Einstein ring with radius . So we search for critical curves solving (9) with . Let's save the lowest order terms only. In this operation, the contributions coming from the other planets are ejected out from the equation. It is convenient to place the star in the usual position . What remains is:

which is biquadratic in r. The solution is:

which verifies our assumption . The parametric equations of caustics can be found in the usual way substituting (11) in the lens application and expanding to the first non trivial order ():

The contributions from the other planets are again of higher order. So the structure of planetary caustics is not affected by the presence of other planets at the lowest order in a perturbative expansion. These formulas can thus be used in a single planet situation as well as in a rich planetary system.

Observe that r goes to infinity as tends to , i.e. when the planet is next to the star's Einstein radius. So the perturbative results will not be valid in this situation. The reason is the same discussed for the central caustic. The merging of the two caustic is not describable in the lowest order perturbative expansion. Moreover, there's another limit to be taken in account. I have eliminated all the terms coming from the other planets because of their higher order. But these terms can become dominant when their denominators are small. This happens when one of these planets is close to the planet we are examining. This is not an exotic situation since we must always remember that what counts is actually the projection of the positions on the lens plane. So planets could be very far apart but have near projections yielding exotic critical curves.

We see that the critical curves traced by (11) have two branches according to the double sign in their expression. For planets external to the star's Einstein ring (), the branch coming from the positive sign is real while the other coming from the negative sign is imaginary for all values of , being . For internal planets (), the denominator is negative and . So the two branches are both real for:

that is in two small regions around and . They are both imaginary elsewhere. All these results are coherent with the behaviour exposed in Fig. 2. We have one planetary caustic for external planets and two disconnected caustics for internal planets.

Fig. 5 shows the comparison with the numerical caustics. In Fig. 5b the discrepancy with the numerical results appears as a loss of symmetry of the numerical caustic which is elongated towards the central star. This effect is not present in the perturbative Chang & Refsdal's one. For internal planets near the star's Einstein ring, the basis of the triangular perturbative caustic is parallel to the star - planet axis(Fig. 5c). So Chang & Refsdal's lens works good until the field can be taken as uniform. When the spherical symmetry becomes important, the caustics begin to differ from perturbative ones. These effects can be taken into account by considering higher order terms in the expansion. These terms would provide the right corrections to the Chang & Refsdal's approximation.

Fig. 5. Planetary caustics for a Jovian planet. The caustics on the left column are perturbative while the ones on the right are numerical.

Eqs. (11) and (12a) can be employed to find interesting characteristics of planetary caustics. For example, let's find the position of the couple of caustics for internal planets. We saw that the critical curves are centered upon and . Consider the first of these (the other is similar at all). Inserting these values of in (11), the possible values of r are:

The point

obtainable by a quadratic mean from the two values, is internal to the critical curve and gives an approximation for its position. Immediately, using the lens equation and expanding to the lowest order terms, we find the position of the caustics:

The first of these is a well known formula (Griest & Safizadeh, 1997). The second completes the information given from the first and allows a complete individuation of the two caustics. Fig. 6 is a plot of the position of the two caustics as a function of the distance of the planet from the star. When the caustics approximately move on the lines:

The two planetary caustics delimitate a region of high de - amplification which can appear in microlensing light curves as negative peaks. The positions of the caustics can give a measure of the size of this region and consequently the size of these negative peaks.

Fig. 6. For planets internal to the star's Einstein ring, the planetary caustics move on this curve.

3.3. Cusps

This subsection concludes the study of the perturbative caustics with the analysis of the position of cusps in these caustics. The position of cusps can be important in several studies such as microlensing itself. In fact cusps are surrounded by a region with an amplification even higher than that of fold singularities. They also define the extension and the shape of the caustic.

Cusps are defined as the points where the tangent vector of the caustic vanishes. In order to find them we must set

and resolve this system of equations for .

Let's start with the central caustic. Eqs. (18) after several steps become:

where

These can simultaneously vanish only if . Explicitly, this equation is:

which, in despite of its cumbersome aspect, can be exactly solved in the case of the single planet where it yields the four solutions:

For planetary caustics, we can proceed in a similar way. Multiplying (18a) by , (18 b) by and subtracting, we have:

Multiplying by r, we get an equation in which is easier to handle:

Inserting (11) and solving, we have:

on the higher branch, and:

on the lower.

For external planets, the higher branch is complete while the lower is absent, so only the four cusps on the higher branch are actually present. For internal planets, the two branches are real only near and . So the higher branch has the two cusps at and , while in the lower one the four cusps (26c) and (26d) are real and the others are imaginary. Summing up we have six cusps distributed in such a way as to form two triangular caustics.

© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999
helpdesk.link@springer.de