We present an axially symmetric formula to calculate the probability of finding gravitational arcs in galaxy clusters, being induced by their massive dark matter haloes, as a function of clusters redshifts and virial masses. The formula includes the ellipticity of the clusters dark matter potential by using a pseudo-elliptical approximation. The probabilities are calculated and compared for two dark-matter halo profiles, the Navarro, Frenk and White (NFW) and the Non-Singular-Isothermal-Sphere (NSIS). We demonstrate the power of our formulation through a Kolmogorov-Smirnov (KS) test on the strong lensing statistics of an X-ray bright sample of low redshift Abell clusters. This KS test allows to establish limits on the values of the concentration parameter for the NFW profile ($c_Delta$) and the core radius for the NSIS profile (rc), which are related to the lowest cluster redshift ($z_{rm cut}$) where strong arcs can be observed. For NFW dark matter profiles, we infer cluster haloes with concentrations that are consistent to those predicted by $Lambda$CDM simulations. As for NSIS dark matter profiles, we find only upper limits for the clusters core radii and thus do not rule out a purely SIS model. For alternative mass profiles, our formulation provides constraints through $z_{rm cut}$ on the parameters that control the concentration of mass in the inner region of the clusters haloes. We find that $z_{rm cut}$ is expected to lie in the 0.0--0.2 redshift, highlighting the need to include very low-$z$ clusters in samples to study the clusters mass profiles.
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