Galaxy clusters as lenses 413Figure 14.11. Hubble Space Telescope image of the cluster Abell 2218. Beside arcs
around the two centres of the cluster, many arclets can be seen (NASA HST Archive).
A remarkable phenomenon is the occurrence of so-calledradial arcsin
galaxy clusters. These areradiallyrather than tangentially elongated, as most
luminous arcs are. They are much less numerous (examples: MS 2137, Abell
370). Their position has been interpreted in terms of the turnover of the mass
profile and a core radius∼ 20 h−^1 kpc has been deduced, quite independent of
any details of the lens model. There are, however, other mass profiles which
can produce radial arcs, and have no flat core; even singular density profiles can
explain radial arcs [61]. Such singular profiles of the dark matter are consistent
with the large core radii inferred from x-ray emission.
14.6.1 Weak lensing
There is a population of distant blue galaxies in the universe whose spatial density
reaches 50–100 galaxies per square arc minute at faint magnitudes. The images
of these distant galaxies are coherently distorted by any forground cluster of
galaxies. Since they cover the sky so densely, the distortions can be determined
statistically (individual weak distortions cannot be determined, since galaxies are
not intrinsically round). Typical separations between arclets are∼(5–10)′′and
this is much smaller than the scale over which the gravitational cluster potential
changes appreciably.
Starting with a paper by Kaiser and Squires [62], a considerable amount of
theoretical work on various parameter-free reconstruction methods has recently
been carried out. The main problem consists in making an optimal use of limited
noisy data, without modelling the lens. For reviews see [63, 64]. The derivation
of most of the relevant equations becomes much easier when using a complex