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ponent. However, by also including the binding energies of the star particles in the analysis, the method can be ex- tended to allow such an individual assignment, as discussed in full detail in Dolag et al. (2009). 

In short, we use this approach in the following way. We start with the two component Maxwell-Boltzmann fit ob- tained from Method 3 . Then we use an iterative algorithm that assigns star particles according to their binding energy either to the BCG or to the ICL, with the iteration repeated until the BCG and ICL velocity distributions roughly match the fitted Maxwell-Boltzmann components. We start the it- eration with some value r grav for the size of the BCG’s grav- itating mass distribution. Then, the gravitational potential of all simulation particles with a distance r < r grav from the cluster centre is computed. All main halo star particles that are bound in this potential are assigned to the BCG in this iteration step, while unbound main halo stars are as- signed to the ICL. Next, the velocity dispersion of the ICL component is calculated and compared to the velocity dis- persion of the two Maxwell-Boltzmann fits. If the velocity dispersion of the ICL is too large, r grav is reduced to un- bind more of the slowly moving particles. Otherwise, r grav is increased. We stop the iteration once the ICL velocity dis- persion agrees with the velocity dispersion of the higher dis- persion Maxwell-Boltzmann component to within 1%. Fig. 2 displays the resulting velocity distributions of the BCG and ICL components obtained in this way and compares them to the corresponding Maxwell-Boltzmann fits. 

In Fig. 1, we compare the dynamical definition of BCG and ICL obtained with Method 4 to the analysis of the sur- face brightness profiles used in Method 2 . For the shown cluster the agreement between these two completely different methods is remarkably good. However, in some other clus- ters the differences are admittedly larger (see also Fig. 6). 

We note that we can obtain a BCG and ICL mass by all of these four methods. However, only Method 1 and Method 4 allow us to individually assign each main halo star particle to one of these two components. 

## 2.3 Robustness with respect to numerical resolution and integration accuracy 

Before we apply the methods introduced above to our full cluster and group sample, we investigate whether our satel- lite galaxy, BCG and ICL stellar masses are robust with re- spect to numerical resolution by analysing three halos that were simulated at several different mass and force resolu- tions. We also test our integration accuracy with one fur- ther cluster simulation where we used two times smaller time steps and a higher force accuracy setting in our tree code. 3 

Fig. 3 shows the masses of all stars, of all main halo stars (i.e. BCG+ICL), and of the BCG in these test simu- lations for a group with M crit 200 = 4 . 4 × 10 $^{13}$h − $^{1}$M ⊙ and a cluster with mass M crit 200 = 1 . 5 × 10 $^{14}$h − $^{1}$M ⊙ . We computed these masses within r crit $_{500}$, based on resimulations with zoom factors 1, 2, and 3, without inclusion of AGN physics. The BCG masses obtained with all the four different methods introduced in Sect. 2.2 are shown. The total stellar masses 

<img src='content_image/437045.jpg'>

are well converged even in our lowest resolution simulations without AGN. However, the mass of stars in satellite galax- ies, which is given by the difference between the results shown for all stars and for the BCG+ICL component, is not fully converged at the resolution of zoom factor 1. On the other hand, the results for zoom factors 2 and 3 for the BCG+ICL and the satellite galaxy mass agree very well, in- dicating that for zoom factor 2 and larger good convergence is achieved. 

The BCG masses that are inferred with the different analysis methods can sometimes slightly differ from each other. This will be discussed in more detail in Section 3.1.2. Here, we want to assess the numerical convergence of the re- sults obtained by each individual method. The BCG masses obtained by using a cutoff radius ( Method 1 ) or the surface brightness profile analysis ( Method 2 ) are already roughly converged even at the zoom factor 1 resolution. The results for Method 3 and Method 4 on the other hand, which both rely on fitting the velocity distribution of main halo stars, seem to be somewhat more easily affected by low numerical resolution, because the masses obtained for zoom factor 1 are larger than those obtained from the higher resolution runs. However, at least for the group shown in the left panel of Fig. 3, this seems partly due to a slightly different timing of a rather massive subhalo that is just passing by close to the cluster center in the low resolution simulation, so it is not really clear how generic this effect is. In any case, we find that the BCG masses have robustly converged at zoom factors 2 and larger. 

However, we also find that the resolution requirements to obtain converged results for the total stellar mass in runs with AGN feedback are somewhat higher. This is because at higher resolution star formation shifts to smaller halos, 

