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Appendix A: Model description

The purpose of this appendix is to give an estimate of the number of H ii regions that can be detected in our survey. This obviously depends on the physical parameters of the H ii regions, their distances, and the sensitivity of our observations. We assumed that:

• the H ii regions are spatially distributed across the Galaxy according to the equation which is the same as Eq. (1) of Mottram et al. (2011), normalized in such a way that ; here R is the galactocentric distance, z the height on the Galactic plane, R_o = 2.2 kpc, R_d = 6.99 kpc, R_h = 1.71 kpc, and z_d = 0.039 kpc (from Mottram et al. 2011);

• the H ii regions are homogeneous Strömgren spheres ionized by stars with luminosities in the range 10³–10⁶ L_⊙;

• the normalized luminosity function () of the ionizing star(s) is described by the power law with L₁ = 10³ L_⊙, L₂ = 10⁶ L_⊙, and β = −0.9, as found by Mottram et al. (2011);

• the normalized electron density distribution () is also described by a power law, i.e., with n_e1 = 30 cm^-3 and n_e2 = 2    ×    10⁴ cm^-3 to span the range of n_e in Fig. 5.

Under these assumptions, the number of Galactic H ii regions per unit volume, dex in luminosity, and dex in electron density is given by the expression (A.1)where N_HII is the total number of H ii regions. There are only two free parameters in this model: the index α and N_HII. To find the corresponding values, we can fit the electron density function obtained from our data that is represented by the points with error bars in Fig. A.1. The number of H ii regions in each n_e bin of this figure can be calculated from the model in the following way. The volume of the Galaxy is divided into a suitable number of cells in cylindrical coordinates. For each of these, Eq. (A.1) is used to calculate the number of H ii regions contained in that cell, spanning all luminosities in the range L₁–L₂, and with density falling in the chosen bin. In this process, only cells falling in the region surveyed by us (i.e., those with 254 < l < 360° and |b| < 10°) were counted. Moreover, for each L the corresponding Lyman continuum, N_Ly, was computed assuming the relationship for a cluster rather than that for a single star (see Sect. 5.1), as it seems unlikely that H ii regions with linear diameters up to 3 pc may be ionized by only one star. From n_e and N_Ly one can calculate the diameter of the Strömgren H ii region, D_H   ii, and the corresponding angular diameter and radio flux measured in the instrumental beam.

To take into account the bias due to the source selection criteria as well as those introduced by the limited sensitivity of the observations, we took into account only the sources that satisfy the following requirements:

• L / (4πd²) > 1.6 L_⊙ kpc^-2, the minimum bolometric flux of the IRAS sources selected by Palla et al. (1991) and Fontani et al. (2005);

• radio flux above 0.9 mJy beam^-1, the mean 3σ sensitivity of our maps;

• angular diameter below 60′′, the maximum size imaged in our interferometric observations.

Finally, to construct the functions in Fig. A.1, we rejected all objects (both in the model and the data) with diameter  ≤ 7″, as these are basically unresolved and the corresponding value of n_e in Table 5 is a lower limit.

The best fit to the electron density function is compared to the data in Fig. A.1. This fit was obtained with χ² minimization by varying the two free parameters of the models over a sufficiently wide range of values. This is illustrated in Fig. A.2, where the χ² is plotted as a function of α and N_HII. The thick contour corresponds to the minimum χ² value (4.64) plus 2.3, which gives the 1σ confidence level according to Table 1 of Lampton et al. (1976). We conclude that the best fit is obtained for N_HII = 15 000 and α = −0.15 ± 0.1 As previously explained, for given n_e and N_Ly, the H ii region size is univocally determined. This means that knowledge of the density function provides us also with the analogous function for the linear diameter, . The latter is shown in Fig. A.3 and compared to the observed distribution. Clearly, the two are quite consistent for all diameters, with some discrepancy for the largest D_H   ii, where the model appears to overestimate the measured value. However, the largest H ii regions are also the most difficult to image with an interferometer, and this may explain this small discrepancy.

Appendix B: Figures

© ESO, 2013