The dynamical and chemical evolution of dwarf spheroidal galaxies with GEAR

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Volume 538 (February 2012)

A&A, 538 (2012) A82

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Issue		A&A Volume 538, February 2012


Article Number		A82
Number of page(s)		28
Section		Extragalactic astronomy
DOI		https://doi.org/10.1051/0004-6361/201117402
Published online		09 February 2012

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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 M_e is the mass ejected by the stellar particle i. Similarly, the mass of element k is (A.2)where w_ij 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.
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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 w_ij M_e is much smaller that m_j, 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 M_tot = 8 × 10⁸ M_⊙, ρ_c,gas = 0.029 m_H/cm³, and r_max = 8 kpc. The parameters for the star formation and supernova feedback are c_⋆ = 0.1 and ϵ_SN = 0.05.
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	Fig. B.2 Effect of supernova efficiency ϵ_SN. The parameters for the initial conditions are M_tot = 3 × 10⁸ M_⊙, ρ_c,gas = 0.022 m_H/cm³, and r_max = 8 kpc. The parameter for the star formation is c_⋆ = 0.03.
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	Fig. B.3 Effect of supernova efficiency ϵ_SN. The parameters for the initial conditions are M_tot = 6 × 10⁸ M_⊙, ρ_c,gas = 0.044 m_H/cm³, and r_max = 8 kpc. The parameter for the star formation is c_⋆ = 0.03.
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	Fig. B.4 Effect of adiabatic time t_ad. The parameters for the initial conditions are M_tot = 3.5 × 10⁸ M_⊙, ρ_c,gas = 0.025 m_H/cm³, and r_max = 8 kpc. The parameters for the star formation and supernova feedback are c_⋆ = 0.05 and ϵ_SN = 0.05.
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	Fig. B.5 Effect of adiabatic time t_ad. The parameters for the initial conditions are M_tot = 7 × 10⁸ M_⊙, ρ_c,gas = 0.053 m_H/cm³, and r_max = 8 kpc. The parameters for the star formation and supernova feedback are c_⋆ = 0.05 and ϵ_SN = 0.05.
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	Fig. B.6 Effect of the star formation parameter c_⋆. The parameters for the initial conditions are M_tot = 3.5 × 10⁸ M_⊙, ρ_c,gas = 0.025 m_H/cm³, and r_max = 8 kpc. The parameters for the supernova feedback is ϵ_SN = 0.05.
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	Fig. B.7 Effect of the star formation parameter c_⋆. The parameters for the initial conditions are M_tot = 7 × 10⁸ M_⊙, ρ_c,gas = 0.053 m_H/cm³, and r_max = 8 kpc. The parameters for the supernova feedback is ϵ_SN = 0.05.
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	Fig. B.8 Effect of the star formation parameter ρ_sfr. The parameters for the initial conditions are M_tot = 3.5 × 10⁸ M_⊙, ρ_c,gas = 0.025 m_H/cm³, and r_max = 8 kpc. The parameters for the star formation and supernova feedback are c_⋆ = 0.05 and ϵ_SN = 0.05.
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	Fig. B.9 Effect of the star formation parameter ρ_sfr. The parameters for the initial conditions are M_tot = 7 × 10⁸ M_⊙, ρ_c,gas = 0.053 m_H/cm³, and r_max = 8 kpc. The parameters for the star formation and supernova feedback are c_⋆ = 0.05 and ϵ_SN = 0.05.
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	Fig. B.10 Effect of varying the number of stars formed per gas particle N_⋆. The parameters for the initial conditions are M_tot = 5 × 10⁸ M_⊙, ρ_c,gas = 0.037 m_H/cm³, and r_max = 8 kpc. The parameters for the star formation and supernova feedback are c_⋆ = 0.025 and ϵ_SN = 0.02.
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	Fig. B.11 Effect of varying the IMF. The parameters for the initial conditions are M_tot = 2 × 10⁸ M_⊙, ρ_c,gas = 0.015 m_H/cm³, and r_max = 8 kpc. The parameters for the star formation and supernova feedback are c_⋆ = 0.05 and ϵ_SN = 0.03.
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	Fig. B.12 Effect of varying the IMF. The parameters for the initial conditions are M_tot = 8 × 10⁸ M_⊙, ρ_c,gas = 0.059 m_H/cm³, and r_max = 8 kpc. The parameters for the star formation and supernova feedback are c_⋆ = 0.05 and ϵ_SN = 0.03.
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	Fig. B.13 Effect of varying the number of neighbors, N_ngb. The parameters for the initial conditions are M_tot = 4.5 × 10⁸ M_⊙, ρ_c,gas = 0.033 m_H/cm³, and r_max = 8 kpc. The parameters for the star formation and supernova feedback are c_⋆ = 0.05 and ϵ_SN = 0.01.
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	Fig. B.14 Effect of the total mass. We vary r_max keeping a constant central gas density ρ_c,gas = 0.015 m_H/cm³. The parameters for the star formation and supernova feedback are c_⋆ = 0.1 and ϵ_SN = 0.05.
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	Fig. B.15 Effect of the total mass. We vary r_max, keeping a constant central gas density ρ_c,gas = 0.044 m_H/cm³. The parameters for the star formation and supernova feedback are c_⋆ = 0.1 and ϵ_SN = 0.05.
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	Fig. B.16 Effect of varying the central gas density, for a constant total mass M_tot = 1 × 10⁸ M_⊙. The parameters for the star formation and supernova feedback are c_⋆ = 0.1 and ϵ_SN = 0.05.
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	Fig. B.17 Effect of varying the central gas density, for a constant total mass M_tot = 5 × 10⁸ M_⊙. The parameters for the star formation and supernova feedback are c_⋆ = 0.1 and ϵ_SN = 0.05.
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© ESO, 2012

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