Halo Radius (Splashback Radius) of Groups and Clusters of Galaxies on small scales

We report the results of a study of the distribution of galaxies in the projection along the radius ($R\le 3R_{200c}$) for 157~groups and clusters of galaxies in the local Universe (0.01 < $z$ < 0.10) with line-of-sight velocity dispersions~200~km~s${}^{-1}$ < $\sigma$ < 1100~km~s${}^{-1}$. We introduce a new observed boundary for the halos of clusters of galaxies, which we identify with the splashback radius $R_{\mathrm{s}\mathrm{p}}$. We also identified the core of groups/clusters of galaxies with the radius$R_c$. These radii are determined by the observed integrated distribution of the number of galaxies as a function of squared angular radius from the center of the group/cluster, which (usually) coincides with the brightest galaxy. We found for the entire sample that the boundary of dark matter $R_{\mathrm{s}\mathrm{p}}$ for groups/clusters of galaxies is proportional to the radius $R_{200\mathrm{c}}$ of the virialized region. We measured the mean radius $⟨R_{\mathrm{s}\mathrm{p}}⟩=1.14\pm 0.02$~Mpc for groups of galaxies ($\sigma \le 400$~km~s${}^{-1}$) and $⟨R_{\mathrm{s}\mathrm{p}}⟩=2.00\pm 0.07$~Mpc for clusters of galaxies ($\sigma >400$~km~s${}^{-1}$). The mean ratio of radii is $⟨R_{\mathrm{s}\mathrm{p}}\mathrm{/}{R}_{200\mathrm{c}}⟩=1.40\pm 0.02$}, or $⟨R_{\mathrm{s}\mathrm{p}}\mathrm{/}{R}_{200\mathrm{m}}⟩=0.88\pm 0.02$.