The Transition Region between Brightest Cluster Galaxies and Intra-Cluster Light in Galaxy Groups and Clusters

We take advantage of a state-of-art semi analytic model of galaxy formation, and the model presented in \citet{contini21a}, to investigate the mass distribution of Brightest Cluster Galaxies (BCGs) and Intra-Cluster Light (ICL) by addressing two points: (1) the region of transition between a BCG dominated distribution and an ICL dominated one, and; (2) the relation between the total BCG+ICL mass and the ICL one alone. We find the transition radius to be independent of both BCG+ICL and halo masses, with an average of 60$\pm$40 kpc, in good agreement with previous observational measurements, but given the large scatter, it can be considered as a sort of physical separation between the two components only on cluster scale. From the analysis of $M_{ICL}-M_{BCG+ICL}$ relation, we build a method able to extract the ICL mass directly from the knowledge of the BCG+ICL one. Given the large scatter on low mass systems, such method under/overpredicts the true value of the ICL in a significant way, up to a factor of three in the worst cases. On the other hand, for $\log M_{BCG+ICL}>12$ or $\log M_{Halo}>14$, the difference between the true value and the one extracted from the $M_{ICL}-M_{BCG+ICL}$ relation ranges between $\pm$30\%. We therefore suggest this relation as a reliable test for observational works aiming to isolate the ICL from the BCG, for systems hosted by haloes on cluster scale.