Statistics of Highly Magnified Gravitational Images in Clusters of Galaxies. II. Implications for the Sources
The probability of observing images of background galaxies gravitationally lensed by clusters of galaxies, in the form of elongated arcs, is studied as a function of the arc length. A numerical calculation is performed for isothermal lenses with a spherical component and a quadrupole component, and with finite core radius. The number of arcs is calculated for several models of the number counts and angular size of the sources being lensed, and assuming that the detectability of an arc depends only on its observed surface brightness, but taking into account the reduction of the surface brightness produced by seeing.It is shown that the number of arcs expected in lensing clusters depends mostly on the angular size of the sources, basically because a lower magnification is needed to produce an arc of fixed length for a larger source. From the available observations of the number of arcs, we tentatively conclude that most of the sources should have typical scale-lengths larger than 0".5. This constraint on the angular sizes could be made much more secure with a homogeneous sample of arcs with well-known detection limits, and a precise definition of the arc length in terms of the surface brightness profile. We also show how such a homogeneous sample can be used to infer the number counts of sources that are too faint to be observed in the field.