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

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 divided by the critical surface mass density . 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:

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

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

The function 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

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 , and drop the second argument of w, i.e., .

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

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

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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