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Appendix A: Kinetic energy injection

	Fig. A.1 Total kinetic energy for the uniform simulations. The thin straight lines correspond to the analytical trends described in Sect. 3.1.
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	Fig. A.2 Kinetic energy injection: comparison between turbulent simulations and our model. The vertical lines correspond to the time at which the gas starts leaving the computational domain (from left to right: border, outside, inside).
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It is also worth studying the amount of kinetic energy that is injected into the ISM during the supernova explosions. Figure A.1 shows the total kinetic energy as a function of time. As with the momentum plots described before, we see the adiabatic phase where energy is conserved (the kinetic energy being a constant fraction of the total energy in this phase), the shell formation, and the snowplow phase where the energy of the hot dense shell is radiated away, approximately following the momentum-conserving snowplow model E_K ∝ t^{− 3/4}. The ratio between total and kinetic energy is about 0.2 − 0.3 in the adiabatic phase in good agreement with Sedov’s model.

For the momentum-conserving snowplow model, the shell radius evolves with time as R_s ∝ t^1/4. This stems from the fact that p ∝ R³v, while v = dR/ dt and p is nearly constant. Therefore, the kinetic energy can be approximated by (A.1)where E_K,51 is the integrated kinetic energy in 10⁵¹ erg, E₅₁ is the initial supernova energy in 10⁵¹ erg, t₄ is the age of the remnant in 10⁴ yr, and t_tr,4 is the transition time in 10⁴ yr.

Figure A.2 shows the evolution of kinetic energy in the turbulent case, compared to our simple model. The injected kinetic energy roughly corresponds to the uniform case in the Sedov-Taylor phase, and stays in the same order of magnitude in the radiative phase.

Appendix B: Density distributions

	Fig. B.1 Density probability distribution just before the explosion and at two later times. Top panel: case without supernova and bottom panel: inside run. While significant differences are seen in the diffuse gas distribution, the high-density tail is largely unchanged by the supernova explosions.
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Figure B.1 shows the density probability distribution functions in the runs without supernova (top panel) and with a supernova inside (bottom panel). As can be seen, a high-density power law with a slope between −1 and −3/2 develops. Such a power law has been found in simulations including gravity and turbulence (e.g., Kritsuk et al. 2011) and is due to the collapse itself. The supernova does not change the high-density part of the distribution, but produces very diffuse gas and hot material. This is even more clearly seen in Fig. 6, which shows the mass above various thresholds as a function of time in the four runs (without supernova, outside, border, and inside).

Appendix C: Influence of the magnetic field

	Fig. C.1 Integrated radial momentum for the uniform MHD simulations. The thin straight lines correspond to the analytical trends described in Sect. 3.1.
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We now study the impact that a magnetic field can have on the supernova remnant influence on the ISM. We proceed as for the hydrodynamical case starting with the uniform configuration, and then investigate the turbulent case.

Appendix C.1: Uniform case

Figure C.1 shows the differences between momentum injection with an ambient uniform magnetic field (taken to be 5 μG) and the model for n = 1 and n = 10 cm^-3. The values in the final stage are almost unchanged by the presence of magnetic field. This is relatively unsurprising since as seen from the turbulent simulations, the final momentum is relatively insensitive to density variations and complex geometry. In particular, the magnetic field does not alter very significantly the shock structure as long as it remains adiabatic, nor does it modify the cooling rate.

Appendix C.2: Turbulent case

To include a magnetic field in the turbulent cloud runs, we introduce a uniform field initially. Its intensity is about 5 μG, which corresponds for the 10⁴M_⊙ cloud to an initial mass-to-flux ratio of about 10. The initial magnetic field is, however, significantly amplified before the supernova is introduced.

	Fig. C.2 MHD case. Mass above densities thresholds 10,100,1000 cm-3 in the case without supernova (top panel) and outside (second panel), border (third panel) and inside (bottom panel) runs.
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	Fig. C.3 Integrated radial injected momentum: comparison between turbulent MHD and our model. The vertical lines correspond to the first matter outflow for each case (from left to right: border, outside, inside).
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Appendix C.2.1: Impact of the supernova remnant on the cloud

The evolution (not shown for conciseness) of the magnetized cloud is very similar to the hydrodynamical case. The supernova hot gas quickly escapes through the low-density material, whereas the high-density clumps are pushed away more slowly. A difference is that the propagation of the supernova in the diffuse medium is no longer spherical because it propagates more easily along the magnetic field lines as noted in earlier works (e.g., Tomisaka 1998).

Figure C.2 shows the mass above 3 density thresholds as a function of time for the four MHD runs. The run without supernova (top panel) is very similar to the corresponding hydrodynamical run (top panel of Fig. 6). In particular, the mass above 10³ cm^-3 is almost identical in the two runs.

In the two MHD runs outside and border (second and third panels), the mass of gas above 10³ cm^-3 is smaller than in the hydrodynamical case by a factor of about 10 − 20%. The same is true for the inside run where it is seen that the mass above 10³ cm^-3 rapidly drops below 5000 M_⊙.

This shows that the presence of a magnetic field tends to enhance the influence supernovae have on molecular clouds, probably because the magnetic field exerts a coupling between the different fluid particles within molecular clouds. Therefore, as some gas is pushed away by the high pressure supernova remnant, more gas is entrained.

Appendix C.2.2: Momentum injection

Figure C.3 shows the evolution of the radial momentum (with respect to the supernova center) with time in the turbulent case with magnetic field. The evolution is very similar to the hydrodynamical case displayed in Fig. 5 (top panel) and reasonably well described by the simple model presented in Sect. 3.1. This confirms the result of the uniform density runs about the weak influence magnetic field has on the total momentum delivered to the ISM.

In addition to the value of the total momentum, it is important (as discussed above) to quantify the momentum distribution as a function of density. Figure C.4 shows the momentum injected for the three density thresholds as a function of time for the MHD runs. These results should be compared with Fig. 7. There are more differences than for the total momentum, particularly in the inside run for which it is seen that the momentum delivered at high density just after the supernova explosion is about 20 to 30% higher than in the hydrodynamical case. This is in good agreement with the results obtained for the mass evolution (Fig. C.2).

	Fig. C.4 MHD case. Evolution of momentum for densities above thresholds 10,100, and 1000 cm-3 in the outside, border, and inside cases.
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Altogether, the various MHD runs presented in this section reveal that the magnetic field does not modify the total amount of momentum injected by supernovae into the ISM. It has a modest

impact on the momentum that is injected into the dense gas, and therefore on the impact that supernovae may have in limiting star formation, if the supernova lies inside the dense gas when it explodes. The magnetic field has almost no impact if the supernova explodes outside the denser regions.

© ESO, 2015