7 Discussion

From inspection of Table 1, the lensing clusters do not peak around any preferred redshift, as might be expected if clusters are evolving with redshift (Bartelmann 1993; Wu 1993). It is apparent that although we have a substantial number of clusters at high redshift (5 with z$\,\gt\,$0.5), completeness may be a problem beyond z $\,\sim\,$ 0.35-0.4. The apparent increase in the lensing fraction with redshift is no doubt a selection bias, resulting from the fact that we can only see the most X-ray luminous clusters at the high redshifts, and these very luminous clusters are likely to be more massive, thus increasing our chances of finding lenses. This effect is illustrated by the dramatic increase in the lensing fraction with increasing $L_{\rm x}$. Three of the 5 EMSS clusters with $L_{\rm x}$$\,\gt\,$10⁴⁵ ergs^-1, and none of the 15 EMSS clusters with $L_{\rm x}$$\,<\,$4 10⁴⁴ ergs^-1 contain giant arcs. The giant arc lensing fraction for clusters having $L_{\rm x}$$\,\gt\,$4 10⁴⁴ ergs^-1 is 30%. Although the statistics are poor because the number of high-$L_{\rm x}$ clusters is small, it appears that high X-ray luminosity does indeed point to the most massive clusters. We find no evidence of lensing in the 2 clusters beyond z $\,\sim\,$ 0.7. One could interpret this as a real cutoff, thus constraining the redshift distribution of the background sources. The more likely explanation, however, is that our images for these clusters are simply not deep enough. For clusters beyond z $\,\sim\,$ 0.7, the sources will be at z$\,\gt\,$1-1.5 and consequently will be very faint with only the most luminous ($\gt\gt L^\ast$) galaxies visible. Deeper images will reveal faint arcs, if they exist, in these clusters as is the case of MS1137+6625, at z=0.78, where a system of giant arcs present in the core of the cluster is revealed by a 8700 s long exposure in R-band taken with the Keck II telescope (see Fig. 1 in Clowe et al.1998).

A diagnostic of the mass density profile is provided by the arc widths (Hammer 1991). The observed width ($w_{\rm arc}$) of a lensed arc is related to the intrinsic source width ($w_{\rm s}$) through the following relation, $w_{\rm arc}={1\over 2}\,w_{\rm s}\,(1-K_{\rm arc})^{-1}$.All of the information on the mass density profile is contained in the so-called matter term $K_{\rm arc}$ which can vary from 0 to 0.5 for compact, singular point mass, singular isothermal sphere or r^1/4 profiles, or from 0.5 to 1 for a non-singular isothermal sphere. Therefore, if we assume that the arcs are drawn from a population of field galaxies whose intrinsic widths are more-or-less constant, then we see that more compact lenses produce thinner arcs. The majority of the arcs in this sample are thin, often unresolved or marginally resolved, even in $0\hbox{$.\!\!^{\prime\prime}$}5-0\hbox{$.\!\!^{\prime\prime}$}7$ seeing. Thus, we find that the majority of the clusters must have compact cores. Obviously, the assumptions about the field galaxy sizes are critical in this interpretation. Hammer (1991) and Hammer et al.(1993) argue that faint field galaxies are in general resolved or marginally resolved with intrinsic FWHM of order $0\hbox{$.\!\!^{\prime\prime}$}5-1''$ (Lilly et al.1991; Tresse et al.1993; Colless et al.1994), in agreement with our conclusions.

In addition to the arc widths, the arc locations, curvatures and orientations also depend on the mass density profile. The arc radius of curvature can be considered to be an upper limit to the cluster core radius (Bartelmann et al.1995). Furthermore, Miralda-Escudè (1993a) and Grossman & Saha (1994) show that when arcs trace ellipses whose major axes are aligned with the cluster major axis (i.e. the arcs are perpendicular to the cluster major axis), then the cluster mass density profile must be steeper than isothermal. If arcs trace ellipses aligned with the cluster minor axis, then the mass density profile is shallower than isothermal.

When we examine the giant arcs in our sample, we find none that are perpendicular to the cluster minor axis. In four of the giant arc clusters, the optical galaxy distribution is clearly elongated, and in these four clusters, the arcs are orthogonal to a radius vector that is more-or-less aligned with the optical (and in MS0451-0305, the X-ray as well) major axis. The other four giant arc clusters have no obvious optical axis of symmetry. Similarly, the arclets and candidate arcs also tend to be normal to the cluster major axis, in the cases where a cluster has a clear axis of symmetry. From this evidence, we conclude that in most our examples of lensing, the mass density profiles appear to be steeper than isothermal. Of course, in arriving at this conclusion, we are assuming that the optical galaxies are tracing the total cluster mass. We also point out, however, that several of the clusters (e.g. MS2137-23 and MS0839+29) may contain radial gravitational images, which requires a cluster density profile with a finite core radius (e.g. Grossman & Saha 1994; Mellier et al.1993).