Intracluster stars in simulations with AGN feedback 5 which contain smaller black holes that are less effective in suppressing star formation by their feedback. As a conse- quence, the total stellar mass in our simulated clusters is not nearly as well converged at the resolution of zoom fac- tor 1 as in our runs without AGN. It is, however, reasonably converged at zoom factor 2 (stellar mass about ∼ 20% low) and well converged at zoom factor 3 resolution (about ∼ 6% low). The results of the run with smaller time steps and higher force accuracy agree extremely well with those ob- tained form the corresponding run with our standard nu- merical parameters (see right panel of Fig. 3). We hence conclude that the orbit integration is sufficiently accurate in our simulations. This is important and reassuring, as an insufficient integration accuracy may lead to an artificially enhanced stripping rate of stars out of cluster galaxies. In Fig. 4, we explore the convergence of the properties of the satellite galaxy population in more detail. Here the combined stellar mass of all satellite galaxies whose individ- ual stellar masses exceed M is shown as a function of M for simulations at different resolution. Comparing the curves ob- tained from simulations with zoom factors 1, 2, 3, and 4, we conclude that the satellite galaxy population is reasonably converged for stellar masses larger than ∼ 5 × 10 $^{10}$h − $^{1}$M ⊙ , ∼ 6 × 10 $^{9}$h − $^{1}$M ⊙ and ∼ 2 × 10 $^{9}$h − $^{1}$M ⊙ , at zoom factors 1, 2 and 3, respectively. In other words, the formation, merger and tidal disruption rates of galaxies above these masses should be robust with respect to numerical resolution. Us- ing the double Schechter fit to the observed galaxy stellar mass function from Baldry et al. (2008), we estimate that ∼ 69%, ∼ 17% and ∼ 9% of the total stellar mass are con- tained in galaxies with individual stellar masses below these resolution limits. Thus, at zoom factor 1 the majority of stars resides in poorly resolved galaxies. This explains why we obtain lower values for the mass in satellite galaxies in these runs (see Fig. 3). On the other hand, in the runs with zoom factors 2 and 3, the vast majority of stars resides in galaxies whose abundance is numerically converged. We thus do not expect a significant spurious contribution to the ICL component form underresolved galaxies in these simulations. Overall, these test simulations indicate that the distri- bution of stars in our cluster resimulations with zoom factors 2 and 3 are robust with respect to mass and force resolu- tion, except for maybe a slight underestimation of the total star formation efficiency in zoom factor 2 runs that include AGN. Importantly, we have also verified that our results are invariant when the time step size is reduced or the accuracy of the gravitational force computation is improved. We also investigated how sensitive our results are to changes in the parameters of the star formation model. More pre- cisely, we performed two test simulations, one with a higher gas density threshold for star formation, and another one in which we assumed a larger timescale for the conversion of gas into stars. Figure 2. Velocity distribution of main halo stars of a 2 . 2 × 10 14 h − 1 M$_{⊙}$ cluster. The red solid histogram shows the velocity distribution of all main halo stars. The smooth red solid curve is the two component Maxwell-Boltzmann fit used in Method 3 and 4 . The smooth green dashed and blue dotted curves indicate the individual Maxwell-Boltzmann distributions corresponding to the BCG and ICL. The green dashed and blue dotted histograms show the velocity distributions of the BCG and ICL as identified by Method 4 . The simulation has been performed without AGN feedback. GLYPH<BC> GLYPH<BH>GLYPH<BC>GLYPH<BC>GLYPH<BC> GLYPH<BD>GLYPH<BC>GLYPH<BC>GLYPH<BC>GLYPH<BC> GLYPH<BD>GLYPH<BH>GLYPH<BC>GLYPH<BC>GLYPH<BC> GLYPH<BE>GLYPH<BC>GLYPH<BC>GLYPH<BC>GLYPH<BC> GLYPH<BC> GLYPH<BH>GLYPH<BC>GLYPH<BC> GLYPH<BD>GLYPH<BC>GLYPH<BC>GLYPH<BC> GLYPH<BD>GLYPH<BH>GLYPH<BC>GLYPH<BC> GLYPH<BE>GLYPH<BC>GLYPH<BC>GLYPH<BC> GLYPH<BE>GLYPH<BH>GLYPH<BC>GLYPH<BC> GLYPH<D2> GLYPH<D9>GLYPH<D1> GLYPH<CQ> GLYPH<CT>GLYPH<D6> GLYPH<D3>GLYPH<CU> GLYPH<D7>GLYPH<D8>GLYPH<CP>GLYPH<D6> GLYPH<D4>GLYPH<CP>GLYPH<D6>GLYPH<D8>GLYPH<CX> GLYPH<D0>GLYPH<CT>GLYPH<D7>$^{GLYPH<D4>}$$^{GLYPH<CT>GLYPH<D6>}$$^{GLYPH<CQ>GLYPH<CX>GLYPH<D2>}$ GLYPH<DA> GLYPH<CT>GLYPH<D0>GLYPH<D3>  GLYPH<CX>GLYPH<D8> GLYPH<DD> v GLYPH<CJ>GLYPH<CZ>GLYPH<D1>GLYPH<BB>GLYPH<D7>℄ GLYPH<CP>GLYPH<D0>GLYPH<D0> GLYPH<D1>GLYPH<CP>GLYPH<CX>GLYPH<D2> GLYPH<CW>GLYPH<CP>GLYPH<D0>GLYPH<D3> GLYPH<D7>GLYPH<D8>GLYPH<CP>GLYPH<D6>GLYPH<D7> GLYPH<BU>GLYPH<BV>GLYPH<BZ> GLYPH<C1>GLYPH<BV>GLYPH<C4> GLYPH<BC> 1 × 10 12 2 × 10 12 3 × 10 12 4 × 10 12 5 × 10 12 GLYPH<BD> GLYPH<BE> GLYPH<BF> GLYPH<D7>GLYPH<D8>GLYPH<CT>GLYPH<D0>GLYPH<D0>GLYPH<CP>GLYPH<D6>$^{GLYPH<D1>GLYPH<CP>GLYPH<D7>GLYPH<D7>}$$^{GLYPH<CJ>}$ M ⊙ /h ℄ GLYPH<DE>GLYPH<D3> GLYPH<D3>GLYPH<D1> GLYPH<CU>GLYPH<CP> GLYPH<D8>GLYPH<D3>GLYPH<D6> GLYPH<BC> 2 × 10 12 4 × 10 12 6 × 10 12 8 × 10 12 1 × 10 13 GLYPH<BD> GLYPH<BE> GLYPH<BE> GLYPH<BF> GLYPH<D7>GLYPH<D8>GLYPH<CT>GLYPH<D0>GLYPH<D0>GLYPH<CP>GLYPH<D6>$^{GLYPH<D1>GLYPH<CP>GLYPH<D7>GLYPH<D7>}$$^{GLYPH<CJ>}$ M ⊙ /h ℄ GLYPH<DE>GLYPH<D3> GLYPH<D3>GLYPH<D1> GLYPH<CU>GLYPH<CP> GLYPH<D8>GLYPH<D3>GLYPH<D6> GLYPH<B4>GLYPH<CP>GLYPH<B5> GLYPH<B4>GLYPH<CQ>GLYPH<B5> GLYPH<CP>GLYPH<D0>GLYPH<D0> GLYPH<D7>GLYPH<D8>GLYPH<CP>GLYPH<D6>GLYPH<D7> GLYPH<BU>GLYPH<BV>GLYPH<BZ>GLYPH<B7>GLYPH<C1>GLYPH<BV>GLYPH<C4> GLYPH<BU>GLYPH<BV>GLYPH<BZ>GLYPH<B8> GLYPH<C5>GLYPH<CT>GLYPH<D8>GLYPH<CW>GLYPH<BA> GLYPH<BD> GLYPH<BU>GLYPH<BV>GLYPH<BZ>GLYPH<B8> GLYPH<C5>GLYPH<CT>GLYPH<D8>GLYPH<CW>GLYPH<BA> GLYPH<BE> GLYPH<BU>GLYPH<BV>GLYPH<BZ>GLYPH<B8> GLYPH<C5>GLYPH<CT>GLYPH<D8>GLYPH<CW>GLYPH<BA> GLYPH<BF> GLYPH<BU>GLYPH<BV>GLYPH<BZ>GLYPH<B8> GLYPH<C5>GLYPH<CT>GLYPH<D8>GLYPH<CW>GLYPH<BA> GLYPH<BG> 2.4 Dependence on the star formation model The idea behind using a higher gas density threshold was that this might lead to smaller and denser galaxies, which should then be less prone to tidal stripping. This in turn could reduce the amount of ICL in the simulated clus- ters. In one of our tests, we therefore increased the thresh- c ○ 0000 RAS, MNRAS 000 , 000–000 Figure 3. BCG, BCG+ICL and total stellar mass found within r crit 500 in a 4 . 4 × 10 13 h − 1 M$_{⊙}$ group ( left panel ) and a 1 . 5 × 10 14 h − 1 M$_{⊙}$ cluster ( right panel ) simulated without AGN at dif- ferent numerical resolutions. Results are shown for zoom factor 1, 2, and 3 resolutions (see Table 1) as well as for the four different methods for determining a BCG mass (see Sect. 2.2). In the right panel , two zoom factor 2 simulations are shown, one with the standard parameters ( a ) and one with smaller time steps and a more accurate force computation ( b ). Results are well converged for zoom factors 2 and larger.