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Appendix A: DisPerSe and filament fitting

The DisPerSe algorithm (Sousbie 2011) was used with an intensity contrast level of 10²¹ cm^-2 and a low-density threshold of 2  ×  10²² cm^-2. These parameters are taken to trace the crests of the main parts of the dense fronts that can be identified by eye on the column density map. Then we performed transverse column density profiles along a piecewise linear segmentation of the DisPerSe skeleton. We also fitted the inner part of the profiles with Gaussian functions (similarly to Arzoumanian et al. 2011) after subtracting a background set at the minimum of the fitted profile. The level of the background does impact the width of the Gaussian profile. However, other choices of the background level only induce small variations of the width (Peretto et al. 2012). There are two possibilities to determine the full width at half maximum (FWHM), fitting Gaussian profiles at each position along the front and take the mean value or fitting a Gaussian profile on the averaged profiles. We used both strategies and the corresponding deconvolved FWHM values are in good agreement (0.10  ±  0.02 pc for the front 1 at a resolution 12″ and 0.20  ±  0.02 pc for the front 2 at a resolution of 36″ to compare with the values indicated in Fig. 2).

Appendix B: Collect theory in a homogeneous medium

Based on the works of Elmegreen et al. (1977) and Spitzer (1978), we derive an analytical expression for the different parameters of an H ii region shell: its velocity V_shell, its density n_c, its thickness L_shell and its radius r_shell.

For simplicity we suppose that the velocity of the ionisation front (I-front) and shock front (S-front) are the same and are equal to V_shell. The cold medium (n₀, v₀ = 0) is separated from the compressed shell (n_c, v_c) by the S-front. The compressed shell is separated from the ionised gas (n_II, v_II) by the I-front. All the parameters are resumed in the schematic view in Fig. B.1. The isothermal Rankine-Hugoniot conditions in the referential of the cold gas gives with F_γ the ionisation flux at the I-front (taking in account the recombination in the ionised gas). These relations are local and we neglect the curvature of the fronts, besides the shell is assumed stationary in the referential of the cold gas. Since the shock is strongly supersonic, the second equation can be approximated by (B.5)with ζ = 2 for a D-critical I-front (V_shell − v_II = c_II) and ζ varies from 2 to 1 in the weak D phase. The time-scale of the expansion of the shell is much longer than the ionisation time-scale, therefore the equilibrium between ionisation and recombination holds during the expansion

	Fig. B.1 Outline of the collect and collapse scenario with the different variables used in the analytical model.
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	Fig. B.2 Density and velocity profile of a 1D spherical simulation of the ionization of a homogeneous medium in the conditions of RCW 36. The radius is between 0 and 1.5 pc with 2000 cells. The initial density is 3.4  ×  104 cm-3, the ionizing flux 6  ×  1047 ph s-1 and the simulation is run during 720 ky.
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(B.6)with R_s the Strömgren radius at the end of the R-type phase and r_shell the position of the shell at a time t in the D-type phase. Therefore from Eqs. (B.5) and (B.6), we get (B.7)By integration we get the position of the shell and we set ζ to 1 (weak D-type I-front) to get back the result from Spitzer (1978) (B.8)This is the quickest way to obtain the position of the shell that was obtained by Spitzer (1978). The age of the H ii region can be deduced from (B.9)The density in the shell n_c can be obtained with Eq. (B.5) (B.10)and the density in the H ii region (B.11)It is interesting to note that the compression decreases with time. The column density N in the shell is equal to what has been accumulated during the collect phase: n₀r_shell/3 minus what remains in the H ii region: n_IIr_shell/3 (B.12)The thickness of the shell L_shell is then given by N/n_c(B.13)\newpage\noindentThis expression only depends on the shell radius and the Strömgren radius. Thus the shell thickness and radius can be used to infer the initial Strömgren radius and the initial density in the medium. For a shell thickness of 0.1 pc and a shell radius of 0.9 pc, the initial H i density of the medium is 3.4  ×  10⁴ cm^-3. All the other parameters can be deduced from this density (B.14)We compared these results with a one-dimensional spherical simulation performed with the Heracles code (radius between 0 and 1.5 pc with 2000 cells). We took an initial density at 3.4  ×  10⁴ cm^-3, a flux of 6  ×  10⁴⁷ ph s^-1 and run the simulation during 720 ky to reach the observed shell radius of 0.9 pc. The density and velocity profiles are given in Fig. B.2. In the simulation, the averaged ionised-gas density is 170 cm^-3, the compressed gas density 1.1  ×  10⁵ cm^-3, the shell velocity 0.61 km s^-1, and the shell thickness 0.11 pc. The results from our simple analytical approach and the simulation are in a good agreement (no more than 10% difference).

© ESO, 2013