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

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4. Microlensing

Among the numerous forms of gravitational lensing, microlensing is surely one of the most relevant since it opens the possibility of probing the galactic structure through a directly gravitational investigation.

Microlensing occurs when the images of a given source, produced by a small lens, are too close (typically less than [FORMULA] arcsecs) to be separated by our telescopes. As we cannot see but a point image of the source, the only way to notice a lensing effect is through a variation of the total light flux coming from the observed source. For a point lens mass, this variation was found by Paczynski (1986) who first thought of galactic microlensing as a new observable astronomical phenomenon. For a planetary system the anomalies in amplification patterns do not enjoy a full analytical description. Our aim is to use a perturbative approach to solve this problem and find analytical light curves for stars accompanied by their planets.

4.1. Paczynski's curve

Before considering the problem of planetary microlensing, it is useful to review the steps to be followed in order to get amplification light curves in the event of a single mass (Schneider, Ehlers & Falco, 1992). This will help us in fixing the problems to be faced. In this case, the lens equation takes a very simple aspect:


The total amplification is found by summing the amplification of all images. So the first step is to find these images, i.e. the lens equation is to be inverted. Here the task is quite easy, because (27) reduces to a second degree equation whose solutions are:


The next step is to compute the amplifications corresponding to each of these images. According to (3), these are:


It is interesting to study the properties of the two images to discover their physical essence (Blandford & Narayan, 1986). The image [FORMULA] has positive parity; in the limit of vanishing lensing effect [FORMULA] [FORMULA] tends to [FORMULA] and its amplification becomes unitary. Thus [FORMULA] reduces to the usual image of the source in the absence of lensing. In what follows I'll refer to it as the principal image. [FORMULA] has negative parity and in the limit of low lensing goes as [FORMULA], while its amplification is always [FORMULA]. I shall call it secondary image as it disappears when the lensing effect is not present. Both images are aligned on the line connecting the source and the lens: the principal image is always external to the Einstein ring, while the secondary one is internal to it.

Now we have to sum up the two amplifications to obtain the so - called amplification map:


This function tells us the amplification corresponding to any given position of the source relatively to the lensing object. Of course it only depends on the distance because of the symmetry of the lens.

The final step is to make the source move along a rectilinear trajectory to obtain the complete light curves corresponding to the passage of a massive lens near the line of sight of the source (obviously it makes no difference who is moving, what counts is only the relative motion). The distance [FORMULA] is:


where b is the impact parameter (the closest approach distance) and [FORMULA] is the projection of the relative speed in a plane orthogonal to the line of sight.

A typical light curve is shown in Fig. 7. The height of the maximum is found by substituting the impact parameter in (30). It becomes infinite as b tends to zero. Real sources have finite extensions implying integration processes smoothing the peak of the curve (Witt & Mao, 1994). This cut - off becomes evident when [FORMULA] is comparable to the source radius.

[FIGURE] Fig. 7. Typical microlensing curve for a point lens mass.

4.2. The problem of planetary microlensing

In principle, the procedure for attaining microlensing light curves for multiple lenses is the same just expounded for a single point lens. First we should invert the lens application, second we have to compute the amplification of all the images, then sum up to have the amplification map and finally introduce the motion of the source relatively to the lens system. But if we write the lens equation for a star with just one planet placed in [FORMULA]:


we at once see that the inversion is not possible since one must surrender at a fifth degree equation which does not allow to find the images produced by such a lens.

A glance at the numerical results can indicate us which way is to be taken in the inversion of the lens application (32). When the source is outside the caustics, only three images are present (see Fig. 8). One of them is outside all critical curves and approaches the source when the latter is far enough from the lens. This is indeed the principal image. Another image is inside the star's critical curve. It is easy to understand that when the mass of the planet vanishes this image becomes the star's secondary image. The last image is near the planet (inside the planetary critical curve when the planet is external to the star's Einstein ring). I shall refer to this as the planetary image. It is clear that the presence of the planet slightly perturbs the principal and secondary image of the star, so that their position can be found applying perturbation theory to Paczynski's images. The planetary image is completely perturbative, since it is not present in the zeroth order situation in which the planet is absent, and must be treated separately. When the source threads a caustic, two new images are formed with opposite parities whose effects are similar to those of the planetary image.

[FIGURE] Fig. 8. In presence of a planet, a source (here represented by a four-cornered star) placed outside the caustics generates three images (the three little circles in the figure).

So, the perturbative analysis is likely to be the key to solve the problem of planetary microlensing. In the following two subsections I will use it to discover the images and their amplification. Finally, I will build amplification light curves and compare them with their numerical counterparts.

4.3. Principal and secondary image

Paczynski's images (28a) are the starting point for our expansion and will be generically indicated by the symbol [FORMULA]. Let's write the position of the images to the first order in [FORMULA] as the sum of Paczynski's image and a small perturbation [FORMULA]:


With this position, in the lens equation expanded to the first order in [FORMULA]


the planetary term no longer contains the perturbation [FORMULA]. Putting:


and bringing these terms to the left members, we re-gain the structure of the Schwarzschild lens Eq. (27) in the variable [FORMULA] for the source position [FORMULA]. The planetary induced perturbation can be thus read as a shift in the source position. [FORMULA] has the same expression as [FORMULA] evaluated in [FORMULA] instead of [FORMULA]. The perturbation [FORMULA] is found by expanding [FORMULA] to the first order in [FORMULA]:


The upper signs stand for the principal image and the lower for the secondary. The expansion parameter [FORMULA] appears through [FORMULA].

Now the position of the principal and secondary image are known. The most delicate operation is done and the door to the planetary microlensing is open at last. What remains is only mechanic computation without any conceptual difficulties.

The amplification of each image is found by the general formula (3) expanded to the first order in [FORMULA] (I drop the zero from [FORMULA] to simplify notation):


This is the sought formula for the amplification of the images. Paczynski's amplification multiplies the main brackets containing the sum of all perturbations following the zero order solution represented by the unity. Two kinds of perturbations can be recognized: the first is caused by the previously found shift in the image positions [FORMULA]; the second is the consequence of the change of the function [FORMULA] produced by the presence of the planetary term in the lens equation. I have dropped the modulus from the main brackets because its content is always positive since the perturbations are smaller than unity (except for the zones where perturbative method is no longer valid).

As usual, the validity of perturbation theory is limited to the regions where perturbations are enough small to make sense. So it is necessary a careful examination of the denominators of all perturbative terms. The shift terms present the distance of the zeroth order image from the origin raised to the sixth power. There's no problem for the principal image which is always far beyond the Einstein ring, but this is not true for the secondary image. However the "failure" rises in the limit of vanishing lensing where the amplification of this image is so low to be totally masked by the amplification of the principal image. When the amplification of the secondary image begins to become important, the distance from the origin is largely sufficient to eliminate all the problems and have fine perturbations. The shift [FORMULA] becomes infinite when the source passes through the origin; so the region very near the origin is the first to avoid. The displacement [FORMULA] diverges when the zeroth order image approaches the planet as could easily be foreseen for a first order perturbation theory. As regards the terms coming from the alteration of [FORMULA], there's nothing new; the prescriptions are the same as those from the other terms.

In sum we are allowed to use these amplification formulae for all source positions being not too near the origin or generating images too close to the planet. This hardly happens when the source is internal to the caustics. We'll see that very reliable results can be obtained up to very short distances from the centres of the caustics.

4.4. Planetary image

As previously announced, in this subsection I shall deal with the third image. The presence of this image is absolutely tied to that of the planet. Anyway, Paczynski's images can still provide a good starting point for our analysis. In fact, if the planet is very far from the star, it too will behave as a single lens. In this case, the planetary image is nothing but the secondary Paczynski's image for a very low mass. In this limit, its distance from the planet is of order [FORMULA]. So, in our perturbative expansion, we are encouraged to search for images with distances from the planet of order [FORMULA]. Let [FORMULA] be the position of the planetary image. We have:


with [FORMULA] of order [FORMULA]. Saving only the lowest order, the lens application reads:


These equations can easily be solved. The solution is:


where [FORMULA] is the zeroth order position of the planetary caustic already rising in former discussions.

As ever the amplification is calculated by expanding (3). The lowest order result is:


Notice how this formula is much more simple than other images amplification.

The denominators in these expressions vanish when [FORMULA]. Consequently the perturbative method fails when the source is very close to the centre of the planetary caustic.

4.5. Perturbative light curves in planetary microlensing

Once we have found the amplification for each image, in order to obtain the microlensing amplification map we must sum up the components coming from the three images. However, we see that the contribution to the total amplification of the planetary image is of the second order in [FORMULA]. Since we are only considering first order corrections to Paczynski's curve, this contribution is to be ignored. Therefore, from now on, we shall confine ourselves to the principal and secondary images only.

One consideration is for the two hidden images coming out when the source crosses a caustic. If the event regards the planetary caustic, the two images can be found by carrying further the expansion (38). The new images arise from higher order solutions and their amplifications will also be of higher orders. So we don't worry about them. On the contrary, if the source crosses the central caustic, the new images appear near the star's Einstein ring, far from any possible starting point for a perturbative expansion. As we are not taking them into account, we cannot expect to obtain good results inside the central caustic. Anyway, central caustic crossing events are very improbable, since the extension of this caustic is [FORMULA] times the star's Einstein ring.

Building light curves presents no difficulty. Chosen one source trajectory, it suffices to parameterize [FORMULA] and [FORMULA] in the amplification map properly. This is no longer a function of the radial coordinate because there is no more rotational symmetry.

To account for the finite size of the source a simple numerical integration of the perturbative amplification map on the source area at each point of the trajectory can be performed. The curves thus obtained can be compared to numerical ones given by "inverse ray shooting" algorithm.

All the results I show in this paper regard a system constituted by a star with mass [FORMULA] and a Jovian planet ([FORMULA]). This choice has been made in order to put in better evidence planetary perturbations and to test the perturbative approach in the least favourable situation. Obviously with Earth - like planets things can only go better.

Let's start with an external planet. In Fig. 9 the planet is in [FORMULA]. The trajectory chosen for this first test is shown in Fig. 9a and has impact parameter 0.5. The numerically attained light curve is displayed in Fig. 9b. The source used for this curve has radius 0.045. In a standard observation towards the bulge of the galaxy ([FORMULA], [FORMULA]), this value would correspond to a giant about 43 times greater than the sun. Here the presence of the planet is responsible for the little peak on the left of the maximum of the curve. Fig. 9c represents the perturbative light curve for a point source moving along the same trajectory. If we perform the numerical integration of the perturbative amplification map, as previously said, the perturbative light curve 9d becomes indistinguishable from the numerical one.

[FIGURE] Fig. 9a-d. Besides a star with unitary mass placed in the origin, here is a Jovian planet ([FORMULA]) in [FORMULA]. The trajectory of the source shown in a has impact parameter 0.5. b  is a numerical light curve obtained by "inverse ray shooting" for a source 43 times greater than the sun. c  is the perturbative light curve for a point source and d  is the same curve after a numerical integration over the source extension.

This is a very encouraging result, so let's choose other trajectories to see other tests. In Fig. 10 the position of the planet is the same but the trajectory passes between the planetary caustic and the central caustic at a minimum distance of 0.2. The peak in the numerical curve 10b is very close to the maximum. The point source perturbative curve 10 c presents a sharp peak which assumes the right proportions after the integration in 10d.

[FIGURE] Fig. 10a-d. Same lens system and same figure ordering as before. The impact parameter is 0.2.

At this point, let's see what happens when the source crosses the planetary caustic. In Fig. 11a, the impact parameter 0.4 allows the crossing. The peak in the numerical curve 11b becomes considerably high. In the point source perturbative curve 11c the peak is very sharp (it would diverge at the centre of the caustic [FORMULA]). However, the integration over the source surface still succeeds in reporting this peak to the right size and shape.

[FIGURE] Fig. 11a-d. The lens system is the same again but the impact parameter 0.4 makes the source cross the planetary caaustic.

Now, let's consider an internal planet ([FORMULA]). The region between the couple of planetary caustic is characteristic for its high de-amplification. This produces negative peaks on light curves such as the one shown in Fig. 12b corresponding to the trajectory in Fig. 12a. It is interesting to see that the perturbative method reproduces this situation with the same great accuracy proved in the former situations. As ever, the point source peak in 12c is smoothed by finite source effect in 12d.

[FIGURE] Fig. 12a-d. Now the planet is in [FORMULA]. The three light curves are obtained in the same ways described in Fig. 9. The impact parameter is 0.6.

In Fig. 13 the impact parameter is 0.25 and things go perfectly as previously.

[FIGURE] Fig. 13a-d. Same lens system as in the previous figure. The impact parameter is 0.25.

Finally, let's consider caustic crossing in this case. Fig. 14a shows a trajectory very close to the planetary caustics. The "inverse ray shooting" curve 14b presents a large de-amplification preceded and followed by little positive peaks. The perturbative curve 14c is characterized by the same situation but the de-amplification is so high to make the total amplification become (unphysically) negative. Now let's see what happens with a finite source. Because of its extension, part of the source hits the centre of the caustic [FORMULA] where the perturbative amplification map diverges. This is a hard problem for the numerical integration which becomes very unstable in this zone, so the bottom of the de-amplification region of the light curve 14d cannot be taken as significant. However, we see that things go fairly well even in this extreme situation.

[FIGURE] Fig. 14a-d. Caustic crossing with impact parameter 0.4.

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999