**B.6.3. Cross Sections**

The basic quantity we need for any statistical analysis is the
cross section of the lens for producing the desired lensing effect
(e.g. multiple images, two images, bright images...). The simplest cross
section is the multiple imaging cross section of the SIS lens -
the angular area on the source plane in which a source will produce two
lensed images. We know from Eqns. B.21 and B.22 that the source
must lie within Einstein radius *b* of the lens center to produce
multiple images, so the cross section is simply
_{SIS} =
*b*^{2}. Since
the Einstein radius
*b* = 4(_{v} /
*c*)^{2} *D*_{ls} / *D*_{s}
depends on the velocity dispersion and
redshift of the lens galaxy, we will need a model for the distribution of
lenses in redshift and velocity dispersion to estimate the
optical depth for lensing. If we are normalizing directly to stellar
dynamical measurements of lenses, then we will also need a dynamical
model (e.g. the Jeans equations of
Section B.4.9) to relate the
observed stellar velocity dispersions to the characteristic dark matter
velocity dispersion
_{v} appearing
as a parameter of the SIS model. We can also compute cross sections for
obtaining different image morphologies.
For example, in Eqn. B.32 we calculated the caustic boundaries for the
four-image region of an SIS in an external shear
. If we
integrate to find the area inside the caustic we obtain the four-image
cross section

(B.104) |

while (provided
|| <
1/3) the two-image cross section is
_{2} =
_{SIS} -
_{4}
_{SIS}. If the shear
is larger, then the tips of astroid caustic extend beyond the
radial (pseudo-)caustic and the lens has regions producing two images,
three images in the disk geometry
(Fig. B.18), and
four images with no simple expression for the cross sections.
There are no analytic results for the singular isothermal ellipsoid
(Eqn. B.37 with *s* = 0), but we can power expand the cross
section as a series in the ellipticity to find at lowest order that

(B.105) |

for a lens with axis ratio
*q* = 1 - ,
while the total cross section is
_{SIS} =
*b*^{2}
(e.g. Kochanek
[1996b],
Finch et al.
[2002]).
As a general rule, a lens of ellipticity
is roughly
equivalent to a spherical lens in an external shear of
/3.
According to the cross sections, the fraction of four-image lenses should
be of order
_{4} /
_{SIS} ~
^{2}
~ ( / 3)^{2}
~ 0.01 rather than the observed 30%. Most of this difference is a
consequence of the different magnification biases of the two image
multiplicities.

There is an important subtlety when studying lens statistics with models
covering a range of axis ratios, namely that the definition of the
critical radius *b* in (say) the SIE model (Eqn. B.37) depends
on the axis ratio and exactly what quantity you are holding fixed in
your calculation (see Keeton, Kochanek & Seljak
[1997],
Keeton & Kochanek
[1998],
Rusin & Tegmark
[2001],
Chae
[2003]).
For example, if we compare a singular
isothermal sphere to a face on Mestel disk with the same equatorial
circular velocity, the Einstein radius of the disk is 2 /
smaller
than the isothermal sphere because for the same circular velocity a
disk requires less mass than a sphere. Since we usually count galaxies
locally and translate these counts into a dynamical variable, this means
that lens models covering a range of ellipticities must be normalized in
terms of the same dynamical variables as were used to count the galaxies.

Much early effort focused of the effects of adding a finite core radius
to these standard models (e.g. Blandford & Kochanek
[1987b],
Kochanek & Blandford
[1987],
Kovner
[1987a],
Hinshaw & Krauss
[1987],
Krauss & White
[1992],
Wallington & Narayan
[1993],
Kochanek
[1996a]).
The core radius *s* leads to an evolution of the
caustic structures (see Part 1, Blandford & Narayan
[1986])
with the ratio between the core radius and the
critical radius *s* / *b*. Strong lenses with *s* /
*b* << 1 act like
singular models. Weak, or marginal, lenses with *s* / *b* ~ 1 have
significantly reduced cross sections but higher average magnifications
such that the rising magnification bias roughly balances the diminishing
cross section to create a weaker than expected effect of core
radii on the probability of finding a lens (see Kochanek
[1996a]).
As the evidence that lenses are effectively singular
has mounted, interest in these models has waned, and we will not discuss
them further here. There is some interest in these models as a probe
of large separation lenses due to groups and clusters where a finite
core radius is replaced by effects of the shallow
*r*^{-1}
NFW density cusp, and we will consider this problem in
Section B.7 where we discuss large
separation lenses.