E

Fig. 2.12 The theoretical gravitational amplification curve as a function of time, for a simple lens

the lens where the flux of the star is amplified by a factor of 1.34, is a circle, whose radius is precisely the Einstein radius,4 as defined by Eq. (2.29).

This method has been used since 1995 by various groups (EROS, MACHOS, OGLE) to try to detect dark matter in the Galaxy. The theory to be verified is that dark matter consists of massive, compact, and non-luminous objects, whose overall mass could explain the difference between the mass deduced from the dynamics of the Galaxy, and that which is directly observable (stars and gas).5

For planet hunters, the interesting case is when the deflector is accompanied by one or more planets (a multiple lensing event). The structure of the amplification zone is then no longer symmetrical. Because of the presence of planets, a line appears between the star and the planet, known as a caustic, where the gravitational amplification is theoretically infinite for point sources such as stars. In the case of a simple lens (without a planet), the only point of infinite gravitational amplification is the position of the lens itself, i.e., when the source, lens, and observer are perfectly aligned. In the case of a multiple lens, the apparent intensity of the source undergoes significant variations when the apparent path of the source relative to the lens, approaches or crosses the caustic.6 It is the structure of the gravitational-amplification curve (Fig. 2.13) and, more especially, the study of its artefacts (number, duration, and intensity) that enables us to determine the characteristics of the planet, or planets (mainly the mass and projected angular distance), by use of a model that incorporates a significant number of parameters that may be adjusted to the data. The error bars associated with this method are rather large.

4 The Einstein radius is also the radius of the circle that is the image of the source produced by the deflecting object, when the source, the deflector, and the observer are perfectly aligned.

5 Calculations actually reveal an enormous mass-deficit (a problem which is known in cosmology as the 'missing-mass problem'). A possible solution to this problem was to consider the existence of a vast number of brown dwarfs, which were not detectable by direct observation. This explanation, however, does not appear to be a convincing solution.

6 The most spectacular effects occur when the apparent path of the source star crosses the caustic, giving a gravitational-amplification curve that is extremely variable, depending on the exact case. The most frequent case is that when the path approaches the caustic, but does not cross it. In such a case, we observe anomalies in the characteristic Gaussian curve of a simple lens.