Galaxy clusters as lenses 411Figure 14.9.The light curve of image A of figure 14.3 is advanced by the optimal value
of the time delay, 417 days (taken from T Kundi ́cet al1997 [60]).
Strong lensing requires that the mass density per surface&hastobeinsome
parts of the lens bigger than the critical mass density given by
&≥&cr=c^2 Ds
4 πGDdDds. (14.115)
Indeed, if this condition is satisfied there will be one or more caustics. The
observation of an arc in a cluster of galaxies allows the projected cluster mass
which lies inside a circle traced by the arc, even if no ring-shaped image is
produced to be easily estimated. For an axisymmetric lens, the average surface
mass density within the tangential critical curve is given by&cr. Tangentially
oriented large arcs occur close to the tangential critical curves, and thus the radius
θarcof the circle traced by the arc gives an estimate of the Einstein radiusθE.