412 Gravitational lensing
Figure 14.10.Wavefronts in the presence of a cluster perturbation.Inside the so defined circle the surface mass is&cr, and this way, knowing the
redshifts of the lens and the source, one finds the total mass enclosed byθ=θarc
M(< θ)=&crπ(Ddθ)^2 1. 1 × 1014 M(
θ
30 ′′) 2 (
Dd
1Gpc)
, (14.116)
A mass estimate with this procedure is useful and often quite accurate.
If we assume that the cluster can, at least as a first approximation, be
described as a singular isothermal sphere, then using equation (14.84) we obtain
for the dispersion velocity in the cluster
σv 103 km s−^1(
θ
28 ′′) 1 / 2 (
Ds
Dds) 1 / 2
. (14.117)
A limitation of strong lensing is that it is model-dependent and, moreover,
one can only determine the mass inside a cylinder of the inner part of a lensing
cluster. The fact that the observed giant arcs never have a counter-arc of
comparable brightness and even small counter-arcs are rare, implies that the
lensing geometry has to be non-spherical.