2.2. Model fitting
2.2.1. Philosophy of model fitting
A widely spread opinion is that photoionization model fitting provides the most accurate abundances. This would be true if the constraints were sufficiently numerous (not only on emission line ratios, but also on the stellar content and on the nebular gas distribution) and if the model fit were perfect (with a photoionization code treating correctly all the relevant physical processes and using accurate atomic data). These conditions are never met in practise, and it is therefore worth thinking, before embarking on a detailed photoionization modelling, what is the aim one is pursueing.
Two opposite situations may arise when trying to fit observations with a model.
The first one occurs when the number of strong constraints is not sufficient, especially when no direct Te indicator is available. Then various models may be equally well compatible with the observations. For example, from a photoionization model analysis Ratag et al. (1997) derive an O/H ratio of 2.2 10-4 for the PN M 2-5. However, if one explores the range of acceptable photoionization models one finds two families of solutions (see Stasinska 2002). The first has O/H 2.4 10-4, the second has O/H 1.2 10-3! The reason for such a double solution is simply the behaviour of [O III] 5007 / H or [O II] 3727 / H with metallicity, as explained in Sect. 1.3. Note that both families of models reproduce not only the observed line ratios (including upper limits on unobserved lines) but also the nebular size and total H flux.
The other situation is when, on the contrary, one cannot find any solution that reproduces at the same time the [O III] 4363/5007 line ratio and the constraints of the distribution of the gas and ionizing star(s) (e.g. Peña et al. 1998, Luridiana et al. 1999, Stasinska & Schaerer 1999). The model that best reproduces the strong oxygen lines has a different value of O/H than would be derived using an empirical electron-temperature based method. The difference between the two can amount to factors as large as 2 (Luridiana et al. 1999). It is difficult to say a priori which of the two values of O/H - if any - is the correct one.
The situation where the number of strong constraints is large and everything is satisfactorily fitted with a photoionization model is extremely rare. One such example is the case of the two PNe in the Sgr B2 galaxy, for which high signal-to-noise integrated spectra are available providing several electron temperature and density indicators with accuracy of a few %. Dudziak et al. (2000) reproduced the 33 (resp. 27) independent observables (including imagery and photometry) with two-density component models having 18 (resp. 14) free parameters for Wray 16-423 (resp. He 2-436). Still, the models are not really unique. The authors make the point that they can reproduce the present observations with a range of values for C/H and T. Yet, the derived abundances are not significantly different from those obtained from the same observational data by Walsh et al. (1997) using the empirical method. The only notable difference is for sulfur whose abundance from the models is larger by 50%, and for nitrogen whose abundance from the models is larger by a factor of 2.8 in the case of He 2-436. This apparent discrepancy for the nitrogen abundance actually disappears if realistic error bars are considered for the direct abundance determinations (rather than the error bars quoted in the papers). Indeed, the fact that the nebular gas is rather dense, with different density indicators pointing at densities from 3 × 103 cm-3 up to over 105 cm-3 introduces important uncertainties in the temperature derived from [N II] 5755/6584 due to collisional deexcitation. It must be noted that realistic error bars on abundances derived from model fitting are extremely difficult to obtain, since this would imply the construction of a tremendous number of models, all fitting the data within the observational errors.
To summarize, abundances are not necessarily better determined from model fitting. However, model fitting, if done with a sufficient number of constraints, provides ionization correction factors relevant for the object under study that should be more accurate than simple formulae derived from grids of photoionization models. This could be called a "hybrid method" to derive abundances. Such a method was for example used by Aller & Czyzak (1983) and Aller & Keyes (1987) to derive the abundances in a large sample of Galactic planetary nebulae, and is still being used by Aller and his coworkers. It must however be kept in mind that if photoionization models do not reproduce the temperature sensitive line ratios, this actually points to a problem that has to be solved before one can claim to have obtained reliable abundances.
Ab initio photoionization models are sometimes used to estimate uncertainties that can be expected in abundance determinations from empirical methods. For example Alexander & Balick (1997) and Gruenwald & Viegas (1998) explored the validity of traditional ionization correction factors in the case of spatially resolved observations. A complete discussion of uncertainties should also take into account uncertainties in the atomic data and the effect of a simplified representation of reality by photoionization models.