Annu. Rev. Astron. Astrophys. 1999. 37:
127-189
Copyright © 1999 by . All rights reserved |

**2.2. Relation with Observable Quantities**

Let us assume that, to first approximation, faint galaxies can be described as ellipses. Their shape can be expressed as a function of their weighted second moments which fully define the properties of an ellipse,

(9) |

where the subscripts *ij* denote the axes (1, 2) of coordinates
in the source and the
image planes,
*S*() is the
surface brightness of the source
and ^{C} is the
center of the source.

Since the surface brightness of the source is conserved through the
gravitational lensing effect
(Etherington 1933),
it is easy to show that, if one assumes that the magnification matrix is
constant across the image (lensed source), the relation between the shape
of the source, *M*^{S} and the lensed
image, *M*^{I} is

(10) |

Therefore, to first approximation, the gravitational lensing effect on a circular source changes its size (magnification) and transforms it into an ellipse (distortion) with axis ratio given by the ratio of the two eigenvalues of the magnification matrix. The shape of the lensed galaxies can then provide information about these quantities. The approximation that the magnification matrix is constant over the image area is always valid in the weak-lensing regime, because the spatial scale variation of the magnification is much larger than the typical size of the lensed galaxies (a few arcseconds). This is not the case when the magnification tends to infinity, but this case is beyond the scope of this review (see Schneider et al 1992, Fort & Mellier 1994).

The relation between the lens quantities described in
Section 2.1 and
the shape parameters of lensed galaxies is not immediately
apparent. Although
_{1}
and
_{2}
describe the
anisotropic distortion of the magnification, they are not directly
related to observables (except in the weak-shear regime). It is
preferable to
use the *reduced complex shear*, **g**, and the complex
polarization (or distortion),
, which is an
observable,

(11) |

because can be
expressed in terms of the observed major and minor axes
*a*^{I} and *b*^{I}
of the image, *I*, produced by a circular source *S*:

(12) |

In this case, the two components of the complex polarization are easily expressed with the second moments:

(13) |

where *Tr(M)* is the trace of the magnification matrix. For
non-circular sources, from Equations (8) and (11) it is possible to
relate the ellipticity of the image
^{I} to
the ellipticity of the lensed source,
^{S}. In
the general case, it depends on the
sign of *Det* (*A*) (that is the position of the source with
respect to the caustic lines) which expresses whether
images are radially or tangentially elongated. In most cases of
interest, *Det*(*A*) > 0 (the external regions, where the
weak lensing regime applies) and:

(14) |

(Seitz & Schneider
1996), but when *Det*(*A*) > 0:

(15) |

Equations 14 and 15 summarize most of the cases that will be discussed in this review.