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

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1. Introduction

In the last years, the prediction and the observation of many microlensing events are gathering ever more interest in gravitational lensing. The typical light amplification curve for these events, found by Paczynski in 1986, has been observed by several astronomical collaborations in observation campaigns toward the bulge of our galaxy (Udalski et al., 1993a; Alard 1997; Alard & Guibert 1997), the Large Magellanic Cloud and the Small Magellanic Cloud (Alcock et al., 1993; Aubourg et al., 1993), the spiral arms and Andromeda galaxy (Tomaney & Crotts, 1996; Ansari et al., 1997; Melchior et al., 1999). Recently, observations toward globular clusters have even been suggested (Jetzer, Strassle & Wandeler, 1998). Beyond proving the correctness of Paczynski's predictions, the observation of microlensing events provides a very cunning instrument for the investigation of the halo of our (and/or some other) galaxy.

Together with events strictly following Paczynski's curve, some events showing deviations from the standard behaviour have been detected. Each of these deviations has found some interpretation (Finite source cut-off (Witt & Mao, 1994; Alcock et al., 1998), blending (Sutherland, 1998), parallax effect (Gould, 1992; Alcock et al., 1995), binary lens (Schneider & Weiss, 1986; Mao & Paczynski, 1991; Udalski et al., 1993b). Indeed, the most intriguing of these deviations is the one induced by a binary (or multiple) lens.

The study of light amplification curves produced by multiple lenses has not yet been performed analytically because of the difficulties in the inversion of the lens application. Anyway, these curves can be obtained numerically by using some inversion algorithm (inverse ray shooting) (Wambsganns, 1997). From the analytical point of view, only the caustics of a general binary lens have been studied in some detail (Witt & Petters, 1993). The lack of analytical results utilizable in microlensing constitutes an irksome obstacle in the complete interpretation of multiple microlensing events.

A particularly interesting case of multiple Schwarzschild lenses is formed when one of the masses is much biggest than others (Mao & Paczynski, 1991; Gould & Loeb, 1992). This is the situation of a typical planetary system where a central star is surrounded by its planets bearing masses thousand or million times smaller. The perturbations on Paczynski's curve induced by the presence of a (even Earth - like) planet are in principle detectable by collaboration teams exploiting world wide telescopic networks (Peale, 1997; Sackett, 1997). Then microlensing could become a new efficient method for the detection of small planets in extra - solar systems. This justifies the major interest in this field that is growing in the last months. Preliminary calculations on the probability of detection of planets have been made (Gould & Loeb, 1992; Bolatto & Falco, 1994) and great efforts are lavished on the problem of extraction of planetary parameters by approximate models (Gaudi & Gould, 1996). It is easy to imagine how the availability of an analytical expression for light curves could help the researches in this field.

The aim of this work is to describe planetary effects perturbatively, exploiting the very little ratios of the masses of the planets with respect to the star mass (Gould & Loeb in 1992 first pioneered this kind of approach). In Sect. 2 the lens equation and other usual objects are specified for the case of planetary systems. In Sect. 3, by means of perturbative theory, I derive the complete structure of critical curves and caustics of a general (not only binary) planetary system; position of planetary caustics and cusps are also found. In Sect. 4 the problem of the inversion of lens application is faced and resolved; consequently, analytical microlensing light curves for planetary events thus obtained are shown.

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999