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

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2. Planetary lensing

A planetary system is nothing but a discrete set of Schwarzschild lenses in a small portion of the space. Fig. 1 shows the typical situation of planetary lensing. The lens plane is defined as a plane orthogonal to the line of sight situated on the barycentre of the lens mass distribution. According to the ordinary theory of lensing (Schneider, Ehlers & Falco, 1992), if the scale of this distribution is much smaller than the distances separating the lens from the source and the observer, then one can deal with the density projected on the lens plane instead of considering the original volume density. This hypothesis is quite verified in real observations where the typical distances are at least of the order of kpc. This consideration can be important in planetary systems where very far planets can eventually have projections enough near to the central star to give perturbations comparable to those of planets placed in more favourable positions. This means that multiple events could be less out of common than one could think (Wambsganns, 1997; Gaudi, Naber & Sackett, 1998). So, it is desirable to preserve the whole planetary system as much as possible before abandoning it for the simplest case of the single planet around the big star. We shall see that the perturbative theory has the considerable advantage of being rather insensitive to the number of planets as regards the difficulty of the problem.

[FIGURE] Fig. 1. Tipical scheme of planetary lensing.

Let's define the length [FORMULA]. [FORMULA] shall denote the coordinates in the lens plane normalized to [FORMULA], while [FORMULA] shall be the coordinates in the source plane normalized to [FORMULA]. [FORMULA] will be the mass of the central star and [FORMULA],..., [FORMULA] will be the masses of the planets. All of these masses are meant to be measured in solar masses. The star will always be placed at the origin, while the projection on the lens plane of the i-th planet will be denoted by [FORMULA]. With these notations, the lens equation reads:

[EQUATION]

In (1), the deviation of light rays due to the star has been explicitly separated by those of the planets. Given a source position [FORMULA], the corresponding [FORMULA] solving the lens equation are called images.

Many interesting properties of this vectorial application can be studied through its Jacobian matrix. In particular, the determinant of this matrix contains nearly all the information about the properties of the images. Let's write the Jacobian determinant for the case of planetary lensing using (1):

[EQUATION]

where [FORMULA] Given an image I at position [FORMULA], the sign of [FORMULA] is called the parity of I. It can be proved that the amplification of the image I is given by

[EQUATION]

We see from this equation that when [FORMULA] is null, the amplification diverges. This is rigorously true for point sources (in ray optics), while for finite (real) sources the integration over the source's surface makes the amplification finite (Witt & Mao, 1994). The points where [FORMULA] vanishes are called critical points, the corresponding points in the source plane through (1) constitute the caustics. So, a point source crossing a caustic will produce images with infinite amplification. Real sources crossing caustics by some of their parts will be highly (but not infinitely) amplified.

The structure of critical curves and caustics provides a substantial description of the general behaviour of the lens (number and roughly location of the images can be established). In microlensing, it is possible to give a qualitative description of light curves just checking whether the track of the source threads some caustic or not.

In Fig. 2 the (numerically obtained) critical curves and the caustics of a star with a single planet are shown. There are three possible cases in such a situation. I recall that for a single point source the critical curve is a ring with radius given by its Einstein radius [FORMULA], while the caustic is a point in the origin of the source plane. When the planet is far beyond the star's Einstein ring, there is only a small perturbation of the two rings which lends finite extension to the originally point - like caustics. This is much more evident in the planetary caustic which is also displaced towards the star. When the planet is in the proximity of the star's Einstein ring, the two critical curves merge and so do the caustics. In the last situation where the planet is internal to the star's Einstein ring, the star's critical curve returns to be very near to a ring while the planet's critical curve turns into two ovals to which a couple of triangular caustics correspond behind the star.

[FIGURE] Fig. 2. There are three possible situations in planetary lensing as regards critical curves, shown in these plots. The smooth round curves are critical curves, while the small cusped curves are the corresponding caustics.

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999
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