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Astron. Astrophys. 359, 1-8 (2000)

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2. Basics of multiple lensing

We shall study a system of n point-lenses placed at positions [FORMULA] in coordinates normalized to the Einstein radius

[EQUATION]

where [FORMULA] is a reference mass (it can be chosen to be the total mass, the typical mass of a single object or anything else). The source coordinates [FORMULA] are normalized to the scaled Einstein radius [FORMULA]. The masses [FORMULA] of the lenses are measured in terms of [FORMULA].

We introduce the complex coordinate in the lens plane [FORMULA] and the complex source coordinate [FORMULA]. The positions of the masses will be denoted by [FORMULA]. We also introduce the functions

[EQUATION]

The lens equation for our system of n masses reads (Witt 1990)

[EQUATION]

Given a source at position y, the z's solving this equation are the images produced by gravitational lensing.

This map is locally invertible where the determinant of the Jacobian matrix

[EQUATION]

is different from zero. The points where the Jacobian determinant vanishes are arranged in smooth closed curves called critical curves. The images of these points through the lens map (3) in the source plane are called caustics. When a source crosses a caustic, creation or destruction of pairs of images occurs and the magnification diverges (Schneider et al. 1992).

This is all we need to start our search for secondary caustics in close multiple systems. The fundamental hypothesis we make is

[EQUATION]

where M is the total mass of the system. In this way, the distances between pairs of lenses will be very small with respect to the Einstein radius of the lens that we would have if all the masses were concentrated at the origin. This Einstein radius is [FORMULA] in our notation. The relation (5) allows us to consider the [FORMULA]'s as perturbative parameters in a series expansion. Then we can solve the equation [FORMULA] at each order, writing its solutions as series expansions in powers of the perturbative parameters. In this way we shall find the critical curves of this system and study their properties analytically.

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© European Southern Observatory (ESO) 2000

Online publication: June 30, 2000
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