On the Separation Between the Brightest Cluster Galaxy and the Intracluster Light

We present a model-based framework to define an aperture separation between the brightest cluster galaxy (BCG) and the intracluster light (ICL). Using the intrinsic BCG and ICL components predicted by the semi-analytic model \texttt{FEGA25}, we determine, for each halo, the aperture radius that minimizes the bias in the recovered ICL mass. The optimal radius is obtained by balancing the BCG stellar mass lying outside the aperture with the ICL mass enclosed within it. At $z=0$, the optimal aperture in physical units increases with halo mass approximately as $r_{\rm cut}\propto M_{\rm halo}^{0.28}$, close to the expected virial scaling. When expressed in units of the virial radius, the aperture becomes nearly independent of halo mass, with $r_{\rm cut}/R_{\rm vir}\simeq 0.045$ at $10^{14}M_\odot$. Applying the resulting aperture prescriptions, we recover the intrinsic BCG mass to better than 8\% (lowest percentile), while the ICL mass is recovered with median values very close to unity, and an average scatter of $\pm 3\%$. Extending the analysis to $z=1$ and $z=2$, we find that the mass trends remain similar, with the slope and intercept of $r_{\rm cut}/R_{\rm vir}$ that stay stable. Robustness tests show that the inferred aperture is most sensitive to the ICL concentration and BCG size, but the main trends are preserved. Our results provide a physically motivated aperture definition for connecting observational BCG--ICL measurements to intrinsic model components.