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Astron. Astrophys. 350, 1007-1017 (1999)

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5. Photo-ionization modelling

From the usual diagnostic diagram relating H[FORMULA]/[S II ] to [S II ] [FORMULA]6717/[FORMULA]6731 (see Sabaddin et al. 1977) we find that M1-67 falls in the photo-ionization dominated region. Therefore we use the general purpose photo-ionization code CLOUDY (Ferland 1996) for our analysis. The ionising radiation fields presented in the previous section are expected to yield different predicted nebular properties, when input into CLOUDY , which can be quantitatively compared with spectroscopic nebular observations taken from Esteban et al. (1991).

5.1. Description of the calculations

We constructed photo-ionization models using CLOUDY (v90.04) as described in Ferland (1996) and Ferland et al. (1998). Comparisons with other photo-ionization codes are provided by Ferland et al. (1995). The nebula are represented by a sphere of variable gas density, n, and filling factor, [FORMULA], with a central cavity that is ionized and heated solely by the UV radiation of a single central star. Nebular fluxes are predicted, given input abundances, flux distributions and physical parameters. We use a volume filling factor of [FORMULA]=0.05 as estimated by Solf & Carsenty (1982). The de-projected radial density profile obtained by Grosdidier et al. (1998) was used, together with an outer (inner) nebula boundary at 50" (8"), which corresponds to 1.25 pc (0.2 pc) using a distance of 5 kpc.

Grosdidier et al. (1998) provided a measurement of the integrated nebular H[FORMULA] flux, after correction for [N II ] contamination. We obtain an integrated de-reddened H[FORMULA] flux using the reddening of [FORMULA]=1.3 mag, determined in Sect. 2.1, namely I(H[FORMULA])=5.45[FORMULA]10- 10 erg cm-2 s-1. In addition, long slit nebular line intensities of Esteban et al. (1991) (their position A) are de-reddened according to this value of [FORMULA]. Although the majority of previous stellar and nebular reddening determinations are in reasonable agreement with our value, Esteban et al. (1991) obtained a substantially lower value of [FORMULA]=0.90 mag for M1-67 using H[FORMULA]/H[FORMULA]. Consequently, the H[FORMULA]/H[FORMULA] ratio, de-reddened according to our stellar reddening of [FORMULA]=1.3 is far from the usual Case B value.

In order to assess the impact of alternative reddening determinations on the nebular properties, we have also calculated the integrated H[FORMULA] flux, following the reddening determination of Esteban et al. (1991), i.e. [FORMULA]=0.9, so that I(H[FORMULA]=2.35[FORMULA]10-10 erg cm- 2 s-1. For this case, long slit nebular line intensities are taken directly from Esteban et al. (1991). Note that identical stellar ionizing flux distributions to those obtained for the higher reddening case are used in this analysis, except that the absolute K-band magnitude is adjusted by 0.15 mag to [FORMULA]6.1 mag, for consistency with the adopted distance of 5 kpc. Results from this alternative approach are very similar to those discussed here for [FORMULA]=1.3 mag.

The electron density at the inner boundary was selected so that the predicted H[FORMULA] flux agreed with observations. For the [FORMULA]=0.9 mag case, typical quantities are [FORMULA]=1,350 cm-3 at the inner boundary (8" or 0.2 pc), and 320 cm-3 at the outer boundary (50" or 1.25 pc). Similar values were measured by Esteban et al. (1991), namely [FORMULA]1,000 cm-3 at a distance of 10-25" and [FORMULA]200 cm-3 for [FORMULA]30" (their region W).

Abundances are assumed to be typical of Galactic H II regions, except for N, O and S, which are adapted from Esteban et al. (1991). For each reddening, electron densities and temperatures were obtained from the de-reddened [S II ] [FORMULA]6716/[FORMULA]6731 and [N II ] [FORMULA]5755/[FORMULA]6584 ratios using the RATIO program (Adams & Howarth, priv. comm.). For [FORMULA]=0.9 we obtained [FORMULA]=[FORMULA] cm-3 and [FORMULA]=6,000 K, while [FORMULA]=[FORMULA] cm-3 and [FORMULA]=6,200 K for [FORMULA]=1.3. Subsequently, ionic abundances were obtained by solving the equations of statistical equilibrium using the EQUIB program (Adams & Howarth, priv. comm.), and are given in Table 4 for each reddening. The usual elemental diagnostics were chosen, [N II ] [FORMULA]6584, [O II ] [FORMULA]3727, [S II ] [FORMULA]6731 and [S III ] [FORMULA]9069, with no ionization correction factors for nitrogen (since O[FORMULA] [FORMULA] O+). Also, we have adopted He/H[FORMULA]0.2, since helium is expected to be moderately enriched. Solar abundance ISM dust grains were assumed, although the precise choice is not critical.


Table 4. Comparison between observed and predicted nebular properties of M1-67 from CLOUDY , based on various ionizing flux distributions. Nebular line measurements are taken from slit position A of Esteban et al. (1991), de-reddened according to [FORMULA]=1.3 (Sect. 2.1) and given relative to H[FORMULA]=100, including formal errors from Esteban et al., neglecting systematic (calibration) uncertainties. I(H[FORMULA]) represents the de-reddened, integrated H[FORMULA] flux received at the Earth (erg cm-2 s-1), obtained via the H[FORMULA] flux measured by Grosdidier et al. (1998).
[FORMULA] See text for an explanation why the H[FORMULA] flux does not agree with the Case B value using our stellar reddening.

5.2. Results from photo-ionization modelling

Esteban et al. (1993) identified a large discrepancy between the nebular fluxes of M1-67 as determined by unblanketed model atmospheres (Schmutz et al. 1992) and that observed. In particular, they were unable to reproduce the observed [FORMULA] and S+/S[FORMULA] ratio. Closest agreement was achieved for the line blanketed model of R84 (WN9) from Schmutz et al. (1991). We now investigate whether this discrepancy remains, based on improved stellar models, including those for which line blanketing is incorporated. Comparisons between predicted and observed nebular properties are presented in Table 4, including results from both the reddening of Esteban et al. (1991), or the present value, to illustrate the possible range of line strengths and plasma conditions. Lamers et al. (1999) discuss various aspects of the reliability of generally derived nebular conditions and abundances, of relevance to the present comparisons.

5.2.1. CMFGEN

From Table 4, the discrepancy between the observed plasma conditions and photo-ionization modelling from the unblanketed CMFGEN analysis is more acute than Esteban et al. (1993) obtained, for both reddening solutions. (A higher stellar temperature is imposed here by modelling the stellar spectrum, including the allowance for light metals, whereas Esteban et al. did not adopt any stellar constraint).

In this case, the ionization balance in the nebula is much higher than observed, with S[FORMULA]/S+ overestimated by a factor of 40. In addition, the mean [FORMULA] is far too high, with the predicted range from 11,200 K at the inner nebula to 9,500 K at the outer boundary, contrasting sharply with the observed value of 6,200 K. Recalling Fig. 5, a relatively strong flux at energies greater than the S[FORMULA] edge at 356 Å indicates that the predicted nebular ionization balance implies S[FORMULA] [FORMULA] S[FORMULA] [FORMULA] S+. Similarly for oxygen, the strong flux predicted below the O+ edge at 353 Å implies an error in the O[FORMULA]/O+ ratio of order [FORMULA]300! Consequently, [O III ] [FORMULA]5007 is predicted to have an intensity that is greater than H[FORMULA], yet [O III ] [FORMULA]5007 is barely detected in M1-67. Similar problems relate to [N II ] [FORMULA]6584, He I [FORMULA]5876 and [Ar III ] [FORMULA]7135.

In contrast, the line blanketed CMFGEN model, with a much softer extreme UV energy distribution, produces a much lower predicted electron temperature, in the range 6,600-7,900 K, now in reasonable agreement with observations. From Table 4, almost every nebular line is predicted to be within a factor of two of the de-reddened value, with the predicted ionization balance in good agreement with observations.

An explanation for the success of this model can be understood by recalling Fig. 5 in which negligible flux is predicted at energies greater than the He0 edge at 504 Å, and indeed only a small flux beyond the S+ [FORMULA]532 edge. Consequently, [S II ] [FORMULA]6731 and especially [S III ] [FORMULA]9069 are fairly well matched for either case, although the predicted ionization balance is actually marginally too low: S+ [FORMULA] S[FORMULA] [FORMULA] S[FORMULA]. Another important success of this model is that negligible flux is predicted above the O+ 353 Å edge, so that the observed absence of [O III ] [FORMULA]5007 is accounted for.

However, the ionization of the model predicts negligible He I [FORMULA]5876 and [Ar III ] [FORMULA]7135 emission, yet these are both observed, albeit weakly. Note that the precise choice of the He/H abundance ratio in the nebula does not play a major role in the predicted He I [FORMULA]5876 strength, since it is primarily determined by the number of He I ionizing photons. Less crucial deficiencies of this model are that [N II ] [FORMULA]6584 and [O II ] [FORMULA]3727 are predicted to be too strong, although [N II ]/H[FORMULA] (independent of reddening uncertainties) is in good agreement.

5.2.2. ISA-wind

Turning to ISA -wind, its energy distribution lies intermediate between the unblanketed and blanketed CMFGEN models, such that the nebular ionization structure is predicted to be somewhat higher than the observed properties, with [FORMULA] overestimated by 40%. Once again, a significant ionizing flux is predicted to lie at energies above the O+ edge, so that the predicted O[FORMULA]/O+ ratio is a factor of [FORMULA]75 times too high, with very strong [O III ] [FORMULA]5007 predicted. Similarly, the flux above the S[FORMULA] and He0 edges are too strong, so that that the S[FORMULA]/S+ ratio is in error by a factor of 20, and He I [FORMULA]5876 is strongly overestimated.

5.3. Influence of uncertain nebular properties

How dependent are the predicted nebular properties on our assumed filling factor, [FORMULA], or geometry? As a test, we have carried out calculations for the blanketed CMFGEN case with [FORMULA]=1.3 mag.

For [FORMULA]=0.025 (a factor of two smaller), the required electron density is [FORMULA] higher, while predicted nebular properties and line strengths properties are barely affected, with the electron temperature unchanged. The greatest influence of a decrease in filling factor is a decrease in the sulphur ionization balance, from S[FORMULA]/S+=1.2 to 0.9, with changes in nebular line strengths of up to 20% for [S II-III ]. Comparable increases in S[FORMULA]/S+ and decreases in electron density are obtained for higher filling factors.

Similarly, we have compared results with a constant, rather than exponentially decreasing, density throughout the nebula. For [FORMULA]=0.05, the required electron density is [FORMULA]=530 cm-3. Again, the influence is generally small, with the electron temperature unchanged and line strengths modified by up to 10%. Once again, the effect on sulphur is greatest, such that the ionization balance is decreased from S[FORMULA]/S+=1.2 to 1.0 and changes in [S II-III ] strengths of up to 20%.

From these calculations, we consider that the nebular predictions are robust, for the blanketed CMFGEN case at least, except that the sulphur line strengths and ionization balance suffer from uncertainties of order [FORMULA]50%. Equivalent tests for the unblanketed CMFGEN case indicate similar uncertainties for the ionization balance of oxygen and nitrogen, although the effect on emission lines from [O II-III ] is minor.

Finally, one further note of caution is necessary. Lamers et al. (1999) emphasise that nebular line fluxes and (absolute) abundances may be sensitive to clumping, of particular concern for the M1-67 nebula. Although beyond the scope of the present work, one would ideally like to carry out a re-evaluation of the nebular abundances and conditions of M1-67, accounting for its highly clumped nature.

5.4. Why are predicted ionizing fluxes different?

A definitive explanation for the difference in ionizing spectra produced by the blanketed CMFGEN and ISA -wind models shortward of [FORMULA]700 Å is not straightforward, given the differences in their treatment of the radiative transfer, blanketing, atomic models and atomic data. Nevertheless, several differences in the alternative techniques deserve consideration.

The most promising reason for the differences is that CMFGEN allows for photons that are absorbed in lines and re-emitted at other (longer) wavelengths. In contrast, no such channel is available in the ISA -wind MC calculations, thereby overestimating the ionizing flux. Other possible explanations, such as photon loss operating in CMFGEN at the He I [FORMULA]584 resonance line, or an incorrect iron ionization balance being used for the MC calculation in ISA -wind, appear to be excluded (Sect. 3.2).

In addition, the number of lines treated in both cases is incomplete, so that blocking will be underestimated, and the predicted ionizing fluxes overestimated. Recall that the number of lines considered by CMFGEN is relatively small, principally treating Fe III-V , while ISA -wind considers many more lines and ionization stages, although this is clearly not complete (solely lines between measured energy levels are included). Depending on the strength of this effect, softer ionizing energy distributions would result, leading to improved agreement with nebular properties for ISA -wind.

In spite of the modelling and nebular difficulties, we conclude that the CMFGEN line blanketed ionizing energy distribution for WR124 provides excellent consistency with the nebular properties of M1-67, and solves the problem that was identified by Esteban et al. (1993) for this system based on unblanketed models. Use of ISA- wind leads to consistent stellar properties for WR124, although the consistency with observed plasma conditions for M1-67 are somewhat poorer than for CMFGEN .

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© European Southern Observatory (ESO) 1999

Online publication: October 14, 1999