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Astron. Astrophys. 318, 687-699 (1997)

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2. Basic equations

The distortion of images of background galaxies depends on the dimensionless surface mass density of the lens, which is the physical surface mass density [FORMULA] divided by the critical surface mass density [FORMULA]. If we now consider the background sources to be distributed in redshift, then the critical surface mass density depends on the redshift z of the source:

[EQUATION]

Here [FORMULA] and [FORMULA] are the angular diameter-distances from the observer to the lens at redshift [FORMULA] and to the source at redshift z, and [FORMULA] is the angular diameter-distance from the lens to the source. Defining

[EQUATION]

we obtain for the dimensionless surface mass density [FORMULA] at angular position [FORMULA] for a source at redshift z

[EQUATION]

The function [FORMULA] relates the `lensing strength' for a source with redshift z to that of a hypothetical source at `infinite redshift', and its form depends on the geometry of the universe. For an Einstein-de Sitter universe we have

[EQUATION]

In particular, for sources with redshift smaller than that of the lens, the `lensing strength' vanishes. For the rest of this paper, we consider a single cluster lens at redshift [FORMULA], and drop the second argument of w, i.e., [FORMULA].

Since the shear [FORMULA] is related linearly to the surface mass density, its dependence on source redshift is the same as for [FORMULA], so the [FORMULA]. The magnification [FORMULA] of an image with position [FORMULA] and a source redshift z then becomes

[EQUATION]

The cluster is non-critical for sources at redshift z if [FORMULA] [FORMULA] everywhere; it is non-critical for all source redshifts if [FORMULA] for all [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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