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 Issue A&A Volume 538, February 2012 A82 28 Extragalactic astronomy https://doi.org/10.1051/0004-6361/201117402 09 February 2012

## Online material

### Appendix A: Metal ejection and energy conservation

In this appendix, we describe precisely how the velocities of the neighboring stellar particles i are modified in order to improve the conservation of energy.

The mass received by the particle j is (A.1)where Me is the mass ejected by the stellar particle i. Similarly, the mass of element k is (A.2)where wij is (A.3)The final mass the the stellar particle i is (A.4)As the mass of particles changes, to conserve the total energy, it is necessary to modify the velocities of the particles. We neglect to correct the change in potential energy. Before the mass redistribution, the energy of the particles involved is (A.5)and after, it is (A.6)where we have assumed that the particle i does not change its velocity. If are chosen such that (A.7)the summing over j gives (A.8)

 Fig. A.1 Evolution of the relative energy as a function of time. The green curve corresponds to the model presented in Fig. 3. It includes the velocity correction while the blue one is similar but uncorrected. Open with DEXTER

which implies that (A.9)

The new velocity for each particle is deduced from Eq. (A.7) (A.10)The modification of the square of the velocity is caused by chages in two terms. The velocity is first decreased, owing to the increase in the mass of the particle. The second term corresponds to the decrease in energy of particle i caused by its decrease in mass. As in practice wij   Me is much smaller that mj, the change in velocity is small. Only the norm of the velocity is affected during the modification of the velocities.

The effect of this correction on the total energy is estimated by running the same simulation used in Fig. 3, but with the correction switched off. The comparison of the evolution of the relative energy is given in Fig. A.1. The improvement is about 20% over 7   Gyr, while its increase in CPU time is insignificant.

We did not observe any improvement in the linear momentum conservation. This is related to the poor conservation of the treecode method that does not fulfill Newton’s third law and dominates the error.

### Appendix B: Free parameters

 Fig. B.1 Effect of the random number seed. The parameters for the initial conditions are Mtot = 8    ×    108   M⊙, ρc,gas = 0.029   mH/cm3, and rmax = 8   kpc. The parameters for the star formation and supernova feedback are c⋆ = 0.1 and ϵSN = 0.05. Open with DEXTER

 Fig. B.2 Effect of supernova efficiency ϵSN. The parameters for the initial conditions are Mtot = 3    ×    108   M⊙, ρc,gas = 0.022   mH/cm3, and rmax = 8   kpc. The parameter for the star formation is c⋆ = 0.03. Open with DEXTER

 Fig. B.3 Effect of supernova efficiency ϵSN. The parameters for the initial conditions are Mtot = 6    ×    108   M⊙, ρc,gas = 0.044   mH/cm3, and rmax = 8   kpc. The parameter for the star formation is c⋆ = 0.03. Open with DEXTER

 Fig. B.4 Effect of adiabatic time tad. The parameters for the initial conditions are Mtot = 3.5    ×    108   M⊙, ρc,gas = 0.025   mH/cm3, and rmax = 8   kpc. The parameters for the star formation and supernova feedback are c⋆ = 0.05 and ϵSN = 0.05. Open with DEXTER

 Fig. B.5 Effect of adiabatic time tad. The parameters for the initial conditions are Mtot = 7    ×    108   M⊙, ρc,gas = 0.053   mH/cm3, and rmax = 8   kpc. The parameters for the star formation and supernova feedback are c⋆ = 0.05 and ϵSN = 0.05. Open with DEXTER

 Fig. B.6 Effect of the star formation parameter c⋆. The parameters for the initial conditions are Mtot = 3.5    ×    108   M⊙, ρc,gas = 0.025   mH/cm3, and rmax = 8   kpc. The parameters for the supernova feedback is ϵSN = 0.05. Open with DEXTER

 Fig. B.7 Effect of the star formation parameter c⋆. The parameters for the initial conditions are Mtot = 7    ×    108   M⊙, ρc,gas = 0.053   mH/cm3, and rmax = 8   kpc. The parameters for the supernova feedback is ϵSN = 0.05. Open with DEXTER

 Fig. B.8 Effect of the star formation parameter ρsfr. The parameters for the initial conditions are Mtot = 3.5    ×    108   M⊙, ρc,gas = 0.025   mH/cm3, and rmax = 8   kpc. The parameters for the star formation and supernova feedback are c⋆ = 0.05 and ϵSN = 0.05. Open with DEXTER

 Fig. B.9 Effect of the star formation parameter ρsfr. The parameters for the initial conditions are Mtot = 7    ×    108   M⊙, ρc,gas = 0.053   mH/cm3, and rmax = 8   kpc. The parameters for the star formation and supernova feedback are c⋆ = 0.05 and ϵSN = 0.05. Open with DEXTER

 Fig. B.10 Effect of varying the number of stars formed per gas particle N⋆. The parameters for the initial conditions are Mtot = 5    ×    108   M⊙, ρc,gas = 0.037   mH/cm3, and rmax = 8   kpc. The parameters for the star formation and supernova feedback are c⋆ = 0.025 and ϵSN = 0.02. Open with DEXTER

 Fig. B.11 Effect of varying the IMF. The parameters for the initial conditions are Mtot = 2    ×    108   M⊙, ρc,gas = 0.015   mH/cm3, and rmax = 8   kpc. The parameters for the star formation and supernova feedback are c⋆ = 0.05 and ϵSN = 0.03. Open with DEXTER

 Fig. B.12 Effect of varying the IMF. The parameters for the initial conditions are Mtot = 8    ×    108   M⊙, ρc,gas = 0.059   mH/cm3, and rmax = 8   kpc. The parameters for the star formation and supernova feedback are c⋆ = 0.05 and ϵSN = 0.03. Open with DEXTER

 Fig. B.13 Effect of varying the number of neighbors, Nngb. The parameters for the initial conditions are Mtot = 4.5    ×    108   M⊙, ρc,gas = 0.033   mH/cm3, and rmax = 8   kpc. The parameters for the star formation and supernova feedback are c⋆ = 0.05 and ϵSN = 0.01. Open with DEXTER

 Fig. B.14 Effect of the total mass. We vary rmax keeping a constant central gas density ρc,gas = 0.015   mH/cm3. The parameters for the star formation and supernova feedback are c⋆ = 0.1 and ϵSN = 0.05. Open with DEXTER

 Fig. B.15 Effect of the total mass. We vary rmax, keeping a constant central gas density ρc,gas = 0.044   mH/cm3. The parameters for the star formation and supernova feedback are c⋆ = 0.1 and ϵSN = 0.05. Open with DEXTER

 Fig. B.16 Effect of varying the central gas density, for a constant total mass Mtot = 1    ×    108   M⊙. The parameters for the star formation and supernova feedback are c⋆ = 0.1 and ϵSN = 0.05. Open with DEXTER

 Fig. B.17 Effect of varying the central gas density, for a constant total mass Mtot = 5    ×    108   M⊙. The parameters for the star formation and supernova feedback are c⋆ = 0.1 and ϵSN = 0.05. Open with DEXTER