destroyed subhalos, to correct the normalization of our original subhalo mass function.

Our scheme is based on the fact that most of the luminosity of the central galaxy is built up by merging with the satellite galaxies brought in by these subhaloes. In other words, the central (BCG) optical galaxy is made up of the galaxies that have been ”merged away” – disappeared from the original distributions. This is consistent with what is known of the size, shape and colour properties of central galaxies. We therefore assume that the fraction of mass in these destroyed subhalos (with respect to the total halo mass), is given by the ratio of the central galaxy luminosity to the total luminosity of the halo:

$$\label{eq:destfracdef}
f_{dest}=\frac{m_{dest}}{M}=\frac{L_{cent}}{L_{total}} \, .$$

For a given $L(M)$ relation, which sets the luminosity of both the central and satellite galaxies as a function of the halo/subhalo mass, the previous equation can then be solved for $f_{dest}$ as a function of halo mass, since the total luminosity is going to be a function of only it and the total mass: $L_{total}=L_{cent}(M)+(1-f_{dest})L_{sat,max}(M)$, where $L_{sat,max}$ is the maximum contribution of the satellites for when $f_{dest}=0$. This is given by:

$$\label{eq:lsatmax}
L_{sat,max}(M)=\int_0^{0.5 M} L_{ref}(m) 