When considering a decreasing density distribution described by a power law with an index q, , the mass included within a radius R is given by (A.1) As far as q< 3, it gives (A.2) or in numerical terms (A.3) The mean density in the sphere of radius R is then given by (A.4) and for a gradient typical of infall, q = 1.5, it gives
We consider here a decreasing density distribution with a power law of index q and a central core of constant density to avoid singularity at the origin and mimic a core envelope using the same definition as in Eq. (8): (A.5) Considering the density, ρi, and the speed, ci, in the ionised medium, the speed of the shell V, from the Rankine-Hugoniot conditions (Eq. (B.5), Minier et al. 2013) we get (A.6) using the density profile of (Eqs. (8)/(A.5)), it transforms into (A.7) Considering an initial Strömgren radius smaller than the core radius rc, Eq. (6) and the photon conservation gives (A.8) Equation (A.7) can then be written as (A.9) This equation can be integrated to give the radius of the shell, rshell, as a function of time with tc the time at which the shell reaches rc: (A.10) Equation (A.9) taken at rc allows defining the speed at this point, (A.11) and can be introduced in Eq. (A.10) (A.12)
Appendix B: Column density line of sight calculation from an envelope of decreasing density with a power law profile
Column density dependance to power law index of decreasing density envelope.
When considering a decreasing density distribution with a power law of index q:, the column density as measured along the y axis at impact parameter a is given by (B.1) with the column density at impact parameter a is then given by (B.2) The analytical integration for integer values of q leads to well-known integrals already calculated by Yun & Clemens (1991). Half integers lead to elliptical integrals, and their tabulated values result in a polynomial expression of . The analytical and numerical values are given in Cols. 2 and 3 of Table B.1. Adopting a value of 1 pc for the impact parameter a, we can derive a relation between the column density and the density via Σ1(a = 1 pc) = 2ρ1I(a = 1 pc). The relation of ρ1 as a function of Σ1 uses the conversion factor ζ corresponding to the appropriate q value and given in Col. 5 of Table B.1: (B.3) The dependence between ζ and q is given at the 5% error level by the following relation (B.4)
Hennebelle et al. (2003) have made numerical simulations to test the influence on density profiles of compression induced by additional external pressure. The compression is characterised (see their Eq. (12)) by a dimensionless factor (φ) that gives the number of sound-crossing times needed to double the external pressure. High φ values correspond to low compression, called subsonic and slow pression increase. Hennebelle et al. (2003) show radial density and velocity profile at five time steps. The three first ones correspond to prestellar phases, well represented by Bonnor-Ebert spheres (Bonnor 1956; Ebert 1955). The fourth step would correspond to the protostellar class 0 phase and the last one to the beginning of the class I phase (see Andre et al. 2000 for classes 0 and I definitions). For a supersonic compression (φ< 1) during the prestellar phase, a density wave crosses the core profile towards the interior. For a sonic or subsonic compression (), the Bonnor-Ebert sphere like profile is smoothly distorted during prestellar phases leading to a single power law in protostellar phases.
Envelope density power law slope dependence on compression at two time step of protostellar stage.
We measured and give in Table C.1 the variation in the density profile power law index (q) as a function of the compression factor φ during the two time steps corresponding to the beginning and the end of Class 0 phase, (steps 4 and 5). During the Class 0 phase, the slope of the density profile does not depend on the compression and remains q = −2, the classical value expected for hydrostatic equilibrium. At the end of the Class 0 phase, the slope of the power law fitting the density profile is correlated to the compression given by the φ value (see Table C.1). It increases from 1.5, the classical value of a free-falling envelope, to 1.8 when φ varies from 10 (virtually no compression) to .1 (strongly supersonic compression). This 20% steepening of the density profile could be one effect that also occurs at a later stage and to larger radii as observed in the MonR2 central UCH ii region, as advocated in Sect. 5.2.
The “collect-and-collapse” scenario proposes that the ionising flux of OB-type stars indeed efficiently sweeps up the gas located within the H ii region extent and develops a shell at the periphery of H ii bubbles (Elmegreen & Lada 1977).
Contribution of the shell to the column density profile of the central UCH ii region. Its density radial extension is given by the two dashed black lines. The column density resulting from the line-of-sight accumulation is illustrated by the black solid line. The shell column density is then convolved with the Herschel beams (coloured dotted lines) to simulate the flux or column density of the shell (coloured solid lines) observable from 6″ (70 μm) to 36″ (500 μm) resolutions.
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We (see also Appendix E) investigate the contribution of this very narrow component to the column density measured with Herschel. This contribution should be especially large at the border of the H ii regions where the line of sight tangentially crosses the shell.
We thus chose to model the density structure of a shell surrounding an H ii region, similar to the one powered by the Mon R2-IRS1 star. We used a density structure that linearly increases with the radius, ρ(r) ∝ r (see Fig. 3) to mimic those suggested by the calculations of H ii region expansion (Hosokawa & Inutsuka 2005, 2006), giving more weight to the shell outer layers, and a thickness of ~0.002 pc as modelled by Pilleri et al. (2013, Fig. 2). Pilleri et al. (2012, 2013) studied the shell of the central Mon R2 UCH ii and considered two concentric slabs with homogeneous density, located at ~0.08 pc around the H ii region. In their model, the photo-dissociation region slab has a density of 2 × 105 cm-3 and a 6.5 × 10-4 pc thickness, while the high-density shell has a 3 × 106 cm-3 density and a 10-3 pc thickness. We used the densities of these slabs to model the inner and outer density values of the shell surrounding the central UCH ii region.
Figure D.1 shows the column density result of the central UCH ii shell calculated with the assumptions above, the idealised profile of this shell (black solid line), flat towards the centre and sharply increasing to the border. The shell was convolved with the Herschel beams (dotted coloured lines) to evaluate its contribution to the fluxes and column density profiles measured towards this H ii region. The border increase due to the shell seen tangentially is completely smeared out for all Herschel wavelengths tracing the cold gas density, i.e. for λ ≥ 160 μm (see Fig. D.1). The column density profiles measured by Herschel have angular resolutions that are too coarse (25″ and 36″), by factors of at least four. On top of that, the mixing along the line of sight with other density components, such as the envelope and background filaments, means that the centre-to-limb contrast of the shell is rarely observable. We thus cannot expect to resolve any shell around compact or ultra-compact H ii regions located at 830 pc. In contrast in the case of the extended northern H ii region, the relative contribution of the shell compared to the envelope is more important, increasing the centre-to-limb contrast and allowing a marginal detection of the shell (see Fig. 8). Therefore, the shell is not a dominant component of the column density structure of H ii regions and their surroundings, but it cannot always be neglected for its complete bubbling.
We first assumed the simple scenario of a bubble expanding in an homogenous envelope where which all the matter initially located within a sphere with a RH ii radius concentrates in the shell located at this very same radius. According to numerical simulations, the H ii region’s swept-up shells have small thicknesses, Lshell< 0.01 pc (e.g. Hosokawa & Inutsuka 2005). The almost total mass transfer from the bubble to the shell thus leads to the approximate equation, (E.1) where ρinitial is the initial constant density of the envelope, and ρshell the shell density, assumed to be homogeneous. Using the relation between ρshell and ρinitial given by Eq. (E.1), the column density of the shell, Σshell, measured along the line of sight towards the centre of the region simply relates to that of the initial gas sphere, Σinitial(r<RH ii), with homogeneous density ρinitial and radius RH ii through (E.2) The column density measured towards the center of H ii regions with fully developed bubbles is thus expected to be divided by three compared to its original value: (E.3) When approaching the border of the H ii region, the line of sight crosses a greater part of the shell, and the column density reaches higher values. However, as shown in Appendix. D, the very small size expected for the shell will result in a beam dilution with very small enhancement that is usually not observable.
In the case of an H ii region of radius RH ii that not fully developed within an envelope of density ρinitial, to reach its external radius, Renv, the contribution of the outer residual envelope, Σenv, needs to be accounted for. The initial column density is now Σinitial(r<Renv) = 2 × Renv × ρinitial. The column density observed towards the developed H ii bubble would thus be (E.4) and the decreasing factor or line-of-sight attenuation factor, η, would be (E.5) It can be noticed that for a fully extended H ii region that reaches the size of the envelope RH ii = Renv, the attenuation factor reaches 3, the value already obtained for the shell alone.
These purely geometrical attenuation factors are weak for the small compact and ultra-compact H ii regions (η ≃ 1.1/1.2), but start to be noticeable for the more extended northern H ii region (η ≃ 1.5). We could estimated the density of the initial protostellar envelope before the H ii region develops, ρinitial, by correcting the mean density ⟨ ρobs ⟩, measured in Sect. 3.4.1 from observed column density through Eq. (3). The density correction due to the geometrical effect of H ii region expansion is obtained by the following relation: ρinitial ~ ⟨ ρobs ⟩ × η.
In the more realistic case of an envelope with a decreasing density gradient, it is expected that the attenuation factor of the column density by H ii bubbles, η, should be greater than in the case of a constant-density envelope and constantly increasing as the H ii region expands.
The mass collected in the shell for the expansion of the H ii region at a radius RH ii is given by Eq. (A.2). The density in the shell of thichness Lshell is, then, (E.6) The column density observed for this shell at small impact parameter, near the H ii region centre is then (E.7) The initial value of the column density was (E.8) This relation is valid only for q < 1 when the value of R1−q near 0 is negligible. The attenuation by the shell is given by (E.9) For constant density envelope (q = 0), Eq. (E.3) is recovered.
To evaluate the attenuation with an existing residual outer envelope we have to calculate the corresponding column density contribution, (E.10) then will give, after some calculations, (E.11) For a shell reaching the size of the envelope RH ii = Renv, Eq. (E.11) leads to Eq. (E.9), which is obtained for the shell alone, and for a constant density envelope (q = 0) it gives Eq. (E.5)
These relations are only valid for q< 1, owing to the singularity at the origin of the density distribution.
To avoid the singularity at the origin we need to use the density profile defined by Eq. (A.5) with a central core of constant density ρc and a size of rc. Following the same calculation as above they give for the mass (E.12) the density in the shell the shell column density near the centre of the H ii region (E.13)
The same calculations as before for an existing residual outer envelope are still applicable here. As shown by Franco et al. (1990), the shell develops only for q< 1.5, so q< 3, and the terms in the equations above are always negligible. Then the calculations give (E.16) for the attenuation, and for rc = 0 we recover Eq. (E.11)
The application to the northern H ii region with appropriate values of the different parameters gives an attenuation factor η ≃ 25, which becomes ~100 when the H ii region expansion reaches the size of the envelope (RH ii = Renv).
The formal treatment of the case of the compact and UCH ii regions would require considering an envelope with two density gradients, but the attenuation would be as efficient as for the northern extended H ii region once expansion of these region occurred.
Column density map showing the azimuthal selection of areas used to characterise the density profiles of the eastern, western, and northern H ii region envelopes. The H ii region bubble, the inner, and the outer envelopes of the central UCH ii region are outlined with dashed circles, whose sizes are given in Tables 1−3.
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Hot regions overall contours of profile extraction on temp. map. Spectral type are taken from Racine (1968).
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© ESO, 2015