4 Results

   
4.1 Convergence procedure

Table 1 lists the parameters of the best two-component model and the observable used to constrain each parameter. In the following we describe the convergence procedure of this model.

The [O III] lines are collisionaly de-excited in component 2 and trace mainly component 1, while the other density diagnostic lines are emitted by both components of the nebula. Therefore the [O III] 51.8/88.3$~\mu$m ratio is used to constrain the density of component 1. The density of component 2 is constrained by the 6 cm radio flux density. The ratio of the two covering factors is determined by fitting the Br$\alpha$ line flux, the sum beeing fixed to 1.

Figure 2 shows the four available [Xⁱ]/[Xⁱ⁺¹] line ratios (divided by the corresponding observed values) versus the effective temperature of the ionizing star, namely [Ar II] 6.98$~\mu$m/[Ar III] 8.98$~\mu$m, [S III] 18.7$~\mu$m/[S IV] 10.5$~\mu$m, [Ne II]12.8 $\mu$m/[Ne III]15.5 $\mu$m and [Ne II]127.8 $\mu$m/[N III]57.3 $\mu$m. The number of ionizing photons is kept constant by adjusting the number of ionizing stars with fixed luminosity, while the effective temperature is varied. The models used for this plot are all 2-component models. In the range of $T_{{\rm eff}}$ considered here (27 to 35 kK) the sensitivity of those ratios to the effective temperature is very high (see also Stasinska & Schaerer 1997, and Paper II). Within a range of $\sim$2500 K, the four ratios provide the same constraint on the effective temperature. However, the [Ar II] 6.98$~\mu$m/[Ar III] 8.98$~\mu$m ratio gives a higher effective temperature than the three other diagnostics (see Sect. 5.6). We will therefore use the [N II] 121.7$~\mu$m/[N III] 57.3$~\mu$m ratio to determine the effective temperature, as it is: 1) independent of the ISO beam size since both lines were observed with LWS, and 2) emitted only by component 1, since it is collisionaly de-excited in component 2. An effective temperature of $\sim$ 30⁺²_-1 kK is then derived.

Figure 2: Variation of the four [Xi]/[Xi+1] ratios with the effective temperature. The ratios correspond to: Ar (solid), S (dot), Ne (dashed) and N (dot-dashed) and have been divided by their corresponding observed values.

 

 

Table 1: Parameters of the best model and derived gas properties. For some parameters, the observable used for the main constraint is given.

Parameters common for both components
Effective temp (kK)	29.7	from [N II] 121.7$~\mu$m/[N III] 57.3$~\mu$m
Luminosity in log( $L/L_{\odot}$)	5.861	from borderline in CoStar models grid (see Sect. 5.8)
Number of stars	1.5	from 2 cm flux density (see Sect. 5.5)
Inner radius (1017 cm)	3.	from imaging
Dust/gas ratio	10-5	from ISO continuum emission
[He]/[H] (in numbers)	0.1
[C]/[H]	1.00 $\times 10^{-4}$
[N]/[H]	1.97 $\times 10^{-4}$	from [N II] 121.8$~\mu$m + [N III] 57.3$~\mu$m
[O]/[H]	4.55 $\times 10^{-4}$	from [O III] 51.8 + 88.3$~\mu$m
[Ne]/[H]	1.70 $\times 10^{-4}$	from [Ne II] 12.8$~\mu$m + [Ne III] 15.5$~\mu$m
[S]/[H]	2.25 $\times 10^{-5}$	from [S III] 18.7$~\mu$m + [S IV] 10.5$~\mu$m
[Ar]/[H]	5.00 $\times 10^{-6}$	from [Ar II] 6.98$~\mu$m + [Ar III] 9.0$~\mu$m
Parameters for:	Component 1	Component 2
Inner $n_{\rm H}$ (cm-3)	640. from $r_{[O {\sc iii}]}$	52000. from 6 cm flux density
Covering factor	36% and	64% from Br$\alpha$
Filling factor	1.00	1.00
Properties for:	Component 1	Component 2
Mean $n_{\rm e}$ (cm-3)	680.	57000.
Constant pressure (cgs)	9.7 $\times 10^{-10}$	1.1 $\times 10^{-7}$
Thickness (1017 cm)	27.2	0.15
Mass ($M_{\odot}$)	29.	0.73
Mean $T_H {\sc ii}{}$ (K)	5520	7230
inner U	8.8 $\times 10^{-1}$	5.7 $\times 10^{-3}$

¹ This luminosity multiplied by the number of stars leads to $1.8 \times 10^{49}$ ionizing photons ( $N_{\rm Lyc}$).

The total luminosity is the product of the CoStar model luminosity by the number of stars, and the number of ionizing photons is adjusted to reproduce the radio flux density at 2 cm. A range of stars with different spectral types is presently not considered. With the available CoStar models, there is an upper limit for the star luminosity beyond which no more models are available and which corresponds to the post main sequence or Wolf-Rayet star regime. The degeneracy of the CoStar model luminosity by the number of stars is discussed in Sect. 5.5.

Once the above set of parameters is fixed, the abundance of each species is derived by fitting their lines fluxes. As most of the parameters have feedback effects, an iterative process must be applied.

To summarize, we have 17 free parameters (listed in Table 1), namely: effective temperature and luminosity of one star, number of stars, densities of both components, ratio of the covering factors, inner radius of the empty cavity, dust to gas ratio, filling factors of both components, and 7 abundances. On the other hand, we have 16 observables plus the morphology of the source as given by the radio maps.

Some parameters (the inner radius, the dust/gas ratio, the filling factors) cannot be precisely constrained and are set to a reasonable value. Change in their values are discussed in Sect. 5. The abundance of helium and carbon does not have any effect on the results of the model, if remaining within reasonable values.

We finally stay with 11 free parameters to be constrained by 17 observables. In other words, there are 6 observables that are not used to build the model and are therefore entirely predicted, namely: the three [Xⁱ]/[Xⁱ⁺¹] ratios for Ne, Ar and S, the two [S III] 33.6$~\mu$m and [Ne III] 36.0$~\mu$m line fluxes (not used because of their large calibration error, see Papers I and II), and the 21 cm continuum flux density.

   
4.2 Results and predictions

Table 2 lists the observations of the infrared emission lines and the radio continuum flux densities together with the results of the best model. The contributions of the two density components are given separately. Most of the predicted lines fluxes and radio flux densities agree with the observations to within the uncertainties of the measurements. The few predictions which are off by a larger factor are the results of well understood factors which will be explained hereafter.

The three [Xⁱ]/[Xⁱ⁺¹] ratios not used to determine the effective temperature agree within $\sim$1 kK with the [N II] 121.7$~\mu$m/[N III] 57.3$~\mu$m ratio (see Fig. 2). The [S IV] 10.5$~\mu$m line falls in the absorption feature due to silicate. This has been taken into account when using the attenuation law described in Paper II and the model prediction is in very good agreement with the observation. For [Ne III] 15.5$~\mu$m, we can suspect an overprediction of the line flux due to the used of the CoStar models, as discussed by Oey et al. (2000) and in Sect. 3.2 (see also Sect. 5.6).

Note that the predicted 21 cm flux density is lower than the value observed by Kim & Koo (2001), who reported complex extended radio emission toward G29.96. Part of this diffuse emission could be due to gas ionized by members of the stellar cluster other than the main ionizing star of G29.96. This low excitation gas is not taken into account in the present model.

 

 

Table 2: Observations and model predictions. The model fluxes for each component are already multiplied by the ISO SWS aperture correction factor (given in parenthesis for component 1, component 2 being small enough to be entirely seen by both ISO spectrometers) and by the covering factor corresponding of each component.

Line	Line fluxes (10-18 W/cm2)					Model/Observation
	Observations1	Model	Component 1		Component 2
H I 4.05$~\mu$m	$11.4 \pm0.4$	11.4	.900	(.16)	10.5	1.00
[Ar II] 6.98$~\mu$m	$27.1 \pm3.0$	36.3	1.34	(.08)	34.9	1.34
[Ar III] 8.98$~\mu$m	$20.5 \pm1.4$	13.1	2.21	(.23)	10.9	0.64
[S III] 18.7$~\mu$m	$64.7 \pm1.2$	55.5	25.0	(.21)	30.4	0.86
[S III] 33.6$~\mu$m	$44.3^2 \pm 3.0$	23.8	20.8	(.18)	2.95	0.542
[S IV] 10.5$~\mu$m	$7.90 \pm0.8$	8.95	7.70	(.51)	1.25	1.13
[Ne II] 12.8$~\mu$m	$127.5 \pm9.0$	97.8	6.08	(.13)	91.7	0.77
[Ne III] 15.5$~\mu$m	$34.0 \pm1.3$	42.5	15.9	(.43)	26.6	1.25
[Ne III] 36.0$~\mu$m	$3.28^2 \pm 1.0$	2.37	1.18	(.38)	1.18	0.722
[N II] 121.8$~\mu$m	$4.66 \pm0.2$	4.65	4.50		.150	1.00
[N III] 57.3$~\mu$m	$23.0 \pm0.7$	23.1	22.6		.540	1.00
[O III] 51.8$~\mu$m	$47.4 \pm0.2$	47.6	45.6		2.08	1.00
[O III] 88.3$~\mu$m	$19.7 \pm0.4$	19.6	19.4		.210	0.99
Wavelength (freq.)	Continuum flux density (Jy)
2 cm (15 GHz)	$3.90 \pm0.5$	3.90	1.34		2.57	1.00
6 cm (5 GHz)	$3.40 \pm0.2$	3.38	1.53		1.84	0.99
21 cm (1.4 GHz)	$2.60 \pm0.2$	1.88	1.60		.282	0.72

¹ The line fluxes (from Paper I) are corrected for the interstellar extinction using A_K=1.6 and the law described in Paper II. ² Line measured with the SWS band 4 detector, see Paper I.