ARC STATISTICS AND CONSTRAINTS ON CLUSTER LENSES

We investigate the statistical distributions of lengths and widths of gravitational arcs found in cluster lenses. Two lens models are considered: the nonsingular isothermal sphere (NSIS) and a new model we call ''quasi-de Vaucouleurs'' (QDEV), whose surface density resembles the de Vaucouleurs profile but is simpler for analytic study. Lensing cross sections are derived and convolved with a power-law distribution of source sizes. We apply selection criteria on the length-to-width ratio and on the length of arcs to our computation, since only arcs with L/W greater than or similar to 3 and L greater than or similar to 2.0'' should enter observational data sets. Comparing the model predictions with data on 85 arcs, we constrain the scaled core radii, r(c)/b, of the lenses, where b measures the lens scale. In the NSIS lens, scaled core radii must be r(c)/b less than or similar to 0.1. In the QDEV lens, the scaled core radius has both upper and lower limits, 0.5 < r(c)/b < 0.6, excluding the point lens. These limits are only moderately sensitive to the slope of the distribution of source sizes. The strongest constraints come from the width analysis, where we fit the observed fraction of arcs (5%) with widths W greater-than-or-equal-to 1.3''. We derive a formula for the radii of curvature of arcs in any lens with a quadrupole perturbation. Analysis of the observed radii of curvature of giant arcs favors cluster profiles that are steeper than isothermal. We find that future observations of arc widths and of the unlensed sizes of faint field galaxies will be most useful for constraining the radial profile of cluster lenses.