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We improve the model presented in Contini & Gu 2020 that describes the radialmass distribution of brightest cluster galaxies (BCGs) and the diffusecomponent also known as intra-cluster light (ICL), by assuming that the globalBCG+ICL radial mass distribution follows the sum of three profiles: a Jaffe andan exponential profiles for the bulge and disk of the BCG, respectively, and amodified version of an NFW profile for the ICL. We take advantage of a widesample of BCG+ICL systems simulated with our state-of-art semi-analytic modelto: (a) investigate the reliability of our BCG+ICL distribution by looking atseveral scaling relations between the BCG+ICL stellar mass within differentapertures and the total BCG+ICL/halo mass, at different redshift; (b) make aprediction of the distance where the radial distribution transitions from BCGto ICL dominated. We find that our model nicely reproduces all the observedscaling relations investigated at the present time with a compelling degree ofprecision, but slightly biased-low with respect to observations at higherredshifts ($z\gtrsim 0.5$). The transition radius predicted by our model is ingood agreement with recent observational results, and spans a range between$\sim 15$ kpc and $\sim 100$ kpc. It mostly depends on the morphology of theBCG, whether it is bulge or disk dominated, on the amount of ICL with respectto the bulge and/or disk, and on the dynamical state of the group/cluster.

Brightest Cluster Galaxies and Intra-Cluster Light: Their Mass Distribution in the Innermost Regions of Groups and Clusters