3 Photoionization modeling

   
3.1 Photoionization code

The models are performed using the detailed photoionization code NEBU (Morisset & Péquignot 1996; Péquignot et al. 2002) which consistently computes the line fluxes without any hypothesis on the ionization structure of the gas, especially without using ionization correction factors (icf's). However, it does require assumptions about the geometry, density, and pressure structure of the nebula.

The computation is performed in a spherical geometry and at each radius from the central ionizing source, the electron temperature, the electron and ions densities, and the line emissivities are determined solving the ionic and thermal equilibrium equations. The inputs for the model are the description of the ionizing flux, using e.g. an effective temperature and a luminosity (see Sect. 3.2), and the gas distribution (assuming e.g. constant pressure through the shell) with a set of abundances.

The elements taken into account and for which lines fluxes are predicted are: H, He, C, N, O, Ne, Mg, Si, S, Cl, Ar, Fe, Ca and Ni. Self-absorption effects of the radio continuum are computed in a spherical geometry approximation.

Absorption of incoming and diffuse photons by dust is considered, the number ratio of dust grains over the number of hydrogen atoms being a parameter of the model. The optical properties of dust grains were made available to us by Ryszard Szczerba (private communication). The adopted dust composition is 50% "astronomical'' silicates and 50% graphite (Draine & Laor 1993). No quantum heating by very small grains are taken into account in the version of NEBU used for this work.

Figure 1: Emission lines maps of the best model, in grey linear scale. The two components are presented with angular size related to their corresponding covering factors. Contours for the ISO SWS beam profile are superposed, for transmissions of 20, 40, 60, 80% of the flux. In the four latest maps, the LWS aperture size is of order of the images, and not shown. Note that the SWS aperture is not always centered.

   
3.2 Atmosphere models

The ionizing photons distribution is taken from the recent CoStar atmosphere models, which include non-LTE effects, line blanketing, and stellar winds (Schaerer & de Koter 1997). The CoStar atmospheres have been compared to earlier predictions (Kurucz models and plane parallel non-LTE atmosphere models). Their implications on nebular studies have also been discussed by Stasinska & Schaerer (1997) and Schaerer & Stasinska (1999).

The predicted ionizing fluxes have been "tested'' by the following indirect means:

1)

the CoStar SEDs reproduce well the ionic Ne⁺⁺/O⁺⁺ abundance ratios in Galactic H II regions observed from optical and IR data, thereby solving the so-called [Ne  III] problem (Stasinska & Schaerer 1997). This supports the important increase of the ionizing spectra predicted at energies 35-40 eV.

2)

the observed increase of the ratio of the He⁰ over H⁰ ionizing photons between stellar effective temperatures of $\sim$30000 and 40000 K is well reproduced (Schaerer 2000). This quantity, sensitive to the ionizing flux above 13.6 and 24.6 eV, is constrained by optical He  I recombination line measurements in the sample of Kennicutt et al. (2000) of H II regions with known stellar content.

3)

Tailored photoionization models for two nebulae with exciting stars of spectral types O3-O4 and O7 respectively have been constructed by Oey et al. (2000) using CoStar spectra. While overall the spectra are found to yield a good agreement with the observed nebular line ratios for all objects (confirming also the first point), there is some indication of an overprediction of the spectrum above $\sim$35-40 eV for the early-type object (DEM L323). We note, however, that for DEM L243, this discrepancy is not found if a cooler spectrum is adopted (approximately model B2 instead of C2 preferred by Oey et al.), as would be expected for the region whose ionizing flux is likely dominated by the O7If supergiant instead of the O7V((f)) dwarf (cf. Vacca et al. 1996).

For the range of stellar temperatures of interest here ( $T_{\rm eff} \sim$ 28-34 kK; cf. below) only 1 and 2 are directly applicable here. Indeed, at these relatively cool effective temperatures the uncertainties in the CoStar models are expected to be the largest, as stressed by Schaerer & de Koter (1997). Although the constraints on the atmosphere models are still rather limited, we conclude from the above comparisons that the adopted model atmospheres should provide a reasonable description of the ionizing fluxes. Comparisons with other model atmospheres are discussed in Sect. 5.6.

From the 27 CoStar models available on the Web (Schaerer & de Koter 1997), a finer grid of spectral energy distributions was constructed. For any effective temperature $T_{{\rm eff}}$ and luminosity L in the range covered by the CoStar models, an interpolation is performed between the four nearest models of this grid, using the square inverse of the distance in the $\log T_{{\rm eff}}$-$\log L$ plane to determine the weights of the four CoStar models. The four CoStar models are first divided by the corresponding blackbody spectra, then averaged using the weights previously determined and the result is finally multiplied by the blackbody spectrum of the desired $T_{{\rm eff}}$ and L, avoiding the diktat of the most intense CoStar model.

   
3.3 Two-component model

In order to reproduce both $r_{[O {\sc iii}]}$ and the radio flux densities which imply a higher density (see Sect. 2.2), a two-component model is used to describe G29.96. The two components differ by their densities; the lower and higher density components are named component 1 and 2 respectively.

The present model consists of a simple linear combination of two independent runs of NEBU in a spherical case and under isobar approximation. Both components are assumed to be radiation bounded. The gas is supposed to be homogeneously distributed in each component, with an inner radius to reproduce an empty cavity. The coefficient applied to each component to obtain the fluxes of the emission lines is the covering factor, i.e. the angular size over 4 $\pi$ of each component as seen by the central source. The sum of the two covering factors is set to one, i.e. we do not consider that photons could escape from the H II region. In this model, there is no diffuse radiation exchanged between component 1 and 2, and no effects of shielding of one component by the other is taken into account as: component 2 has a very small geometrical thickness compared to component 1, and component 1 is very optically thin at radio wavelengths. A two-component model must be seen as an approximation, describing the two first moments of a certainly more complicated gas distribution.

The most important effect of this 2-density medium is in the localization of the lines emission, which depends on their critical densities. For example, the [O III] and [N III] lines are emitted only by the lower density part of G29.96, because these lines are collisionally de-excited in the densest region (see Paper II for the critical densities of all the lines).

3.4 The ISO beam profiles

Component 1 of the nebula will have a larger geometric extension than component 2 due to its lower density. The beam sizes of the ISO instruments, which vary from 14-33 arcsec for the SWS to $\sim$80 arcsec for the LWS, have a drastic impact when comparing/dividing different emission lines. The beam profiles, from Garcia-Lario (1999), are used to apply a correction to the predicted line fluxes. For each component and line, an intensity map is generated by projection of the line emissivity on a sky plane. An "ISO'' mask is then applied, according to the detector beam corresponding to the line wavelength.

Figure 1 shows the intensity maps obtained for our best model (presented in Sect. 4) for 8 atomic lines. Contours of the corresponding ISO SWS apertures are superimposed on the image. For the four last images corresponding to lines observed by LWS, the aperture size is larger than the image. For most of the lines, only the core is visible with the adopted linear intensity scale. Note that, despite the absence of visible extended emission, for most of the lines the contribution of the extended part is about that of the core. The low density extended component is clearly seen for the low critical density lines for which the contribution of the core is very low (the [N II], [N III], [O III] and [S III] lines). The effect of neglecting the finite aperture size is clearly illustrated. Note also that the profile of the SWS aperture, as described by Garcia-Lario (1999), is not symmetrical. We did not attempt to exactly adjust the orientation of these profiles to the observations of G29.96, as the effect of the asymmetry in the profiles is negligible compared to the effect of departure from spherical symmetry of the object itself.