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Appendix A: Effect of the numerical resolution on the energy budget through the shock. Case of a 0.01 M_⊙ dense core

	Fig. A.1 Radial profiles of various first core properties during the collapse of a 0.01 M⊙ clump for a core central density ρc = 1.8 × 10-11 g cm-3, for calculations done with 4500 cells (dotted red line) and 18 000 cells (solid black line). From top left to bottom right: a) density; b) gas temperature; c) velocity; d) entropy; e) optical depth; f) luminosity.
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	Fig. A.2 Normalized energy balance as a function of the mass for the calculations with 18 000 cells (left) and 4500 cells (right), when the central density reaches ρc = 1.8 × 10-11 g cm-3. Scales are logarithmic and normalized to the rate of change of the total energy d(Ek + Ep + Ui) /dt.
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In contrast to the hydrodynamical case, the structure of a radiative shock can extend over large distances, depending on the optical properties of the material. For an optically thin material, the photon mean-free path is long, so the shock structure is very extended compared to the viscous scale. In this work (see Sect. 3), we present numerical calculations of dense core collapse, using a fixed resolution in mass; i.e., the mesh is not refined in the large gradient zones. Although this Lagrangean description is well-suited to the hydrodynamical shocks, thanks to the artificial viscosity scheme, it may encounter difficulties in the case of radiation-hydrodynamical flows, in particular in the optically thin region (upstream region, outside the first core).

In this appendix, we present the results of the collapse of a 0.01 M_⊙ dense core, using the same initial ratio of thermal energy over gravitational energy as in Sect. 3 (α ~ 0.97). To investigate the effect of the numerical resolution, we performed calculations with 4500 cells and 18 000 cells, using the FLD model.

Figure A.1 shows the density, gas temperature, velocity, entropy, optical depth, and luminosity radial profiles for the two calculations at a central density ρ_c ~ 1.6 × 10^-11 g cm^-3. Although there are some significative differences in the radiative precursor region (i.e. the transition region between optically thin and thick regions, where 2 < τ < 0.5) and in the estimate of the first core radius (~10%), the entropy, density, velocity, and luminosity jumps are about the same. In both calculations,

the shock is supercritical, and the amount of energy radiated away is about the same (L = 1.5 × 10^-2 with 4500 cells, and L = 1.44 × 10^-2 with 18 000 cells). This means that the overall properties of the first accretion shock, including its global energy budget remain correctly calculated even at low resolution. However, using 18 000 cells, we see that the spike in the gas temperature is resolved and that the radiative precursor length is much shorter. On the other hand, the central entropy within the first core is the same in both cases, indicating that the cooling of the first core by radiation is not affected by the lack of resolution in the radiative shock.

Figure A.2 shows the normalized energy balance at ρ_c = 1.8 × 10^-11 g cm^-3 for the calculations run with 18 000 cells (left) and 4500 cells (right). The figures display the rate of change of potential energy dE_p /dt, kinetic energy dE_k /dt, internal energy dU_i /dt, total energy d(E_k + E_p + U_i) /dt, and the work done by thermal pressure and radiative flux (L_rad + 4πr²Pu). The total energy equation reads as (A.1)First, we see from Fig. A.2 that the radiative pressure exerts a negligible work compared to the thermal pressure one. Comparing the energy balance of the two calculations, we see that it is globally the same, which confirms that the calculations done with 4500 cells receive the correct features of the first core accretion shock, even though the radiative structure of this shock is not resolved.

© ESO, 2011