We now turn to the "royal way" of analysing emission-line spectra: photoionization modelling. We will show that photoionization modelling is an art that requires not only photoionization codes, but also a certain dose of common sense.

Most of the codes listed below have been intercompared at the Lexington conference "Spectroscopic Challenges of Photoionized Plasmas" in 2000. The results of the code comparisons can be found in Pequignot et al. (2001).

• CLOUDY by Gary Ferland and associates computes models for ionized nebulae and photodissociation regions (PDRs). It is regularly updated, well documented and widely used. It is available at http://www.nublado.org.

• MAPPINGS by Michael Dopita plus Kewley, Evans, Groves, Sutherland, Binette, Allen and Leitherer computes models for photoionized nebulae and for planar shocks. It can be found at http://www.ifa.hawaii.edu/~kewley/Mappings.

• XSTAR by Tim Kallman computes models for photoionized regions with special attention to the treatment of X-rays. It can be found at http://heasarc.nasa. gov/lheasoft/xstar/xstar.html.

• CLOUDY-3D by Christophe Morisset is a pseudo-3D code based on CLOUDY. It allows quick modelling of 3D nebulae and visualization (including computation and visualization of line profiles). It can be found at http://132.248.1.102/Cloudy_3D.

• MOCASSIN by Barbara Ercolano is a full 3D Monte Carlo photoionization code that also treats dust transfer in an accurate manner. It is available from [email protected] (see also http://hea-www.harvard.edu/~bercolano).

Many other, independent photoionization codes are mentioned in the literature (some of them benchmarked in Pequignot et al. 2001), but have not been made available for public access. This is especially the case of hydrodynamical codes that include the physics of photoionization and the computation of line emission. In such codes, of course, the physics of radiation is treated in a simplified manner, since a simultaneous treatment of the macrophysics and the microphysics requires tremendous computing power.

1.4.2 Why do we construct photoionization models? There can be many different reasons for building photoionization models. For example, one might want to

• check the sensitivity of observable properties to input parameters

• compute a grid of models for easy interpretation of a certain class of objects

• calculate ionization-correction factors

• derive the chemical composition of a given nebula

• estimate characteristics of the ionizing source

• probe stellar-atmosphere model predictions in the far ultraviolet.

1.4.3 How should one proceed? Each of the problems above requires a specific approach. It is always worth spending some time on finding the best way to achieve one's goal. For example, if one wants to derive the chemical composition of a nebula by means of emission-line fitting, it is not sufficient to find one solution. One must explore the entire range of possible solutions, given the observational constraints. This is not always easy.

If the aim is to interpret one object, or a given class of objects, the first step is to collect all the observational constraints needed for this purpose. This includes monochromatic images as well as line intensities in various wavelength ranges and with different apertures. Also it is important to characterize the ionizing sources as well as possible from the observations (visual magnitude, spectral type in the case of a single star, age of the ionizing stellar population in the case of a large collection of coeval stars).

Then, one must define a strategy. How will one explore a parameter space? How will one deal with error bars? How will one test the validity of a model?

Finally, one must evaluate the result of the investigation. Was the goal achieved? If not, what does this imply? In fact, this aspect is often overlooked; nevertheless, it is potentially very instructive and is an incentive for progress in the field.

In the following we present some commented examples.

1.4.4 Abundance derivation by tailored model fitting The general procedure would be something like the following sequences.

(1) Define the input parameters: the characteristics of the ionizing radiation field (luminosity, spectral energy distribution); the density distribution of the nebular gas; the chemical composition of the nebular gas; the distance.

(2) Use an appropriate photoionization code and compute a model.

(3) Compare the outputs with the observations (corrected for extinction by intervening dust): the total observed Ha flux; the Ha surface-brightness distribution and the angular size of the Ha-emitting zone; the line intensities etc. ...

(4) Go back to (1) and iterate until the observations are satisfactorily reproduced. Here "satisfactorily" means that all the observational data are reproduced within acceptable limits, those limits taking into account both the observational errors and the approximations of the model. These "limits" should be set by a critical analysis of the situation, before running the models. It may be that one finds no solution. This is by no means a defeat. It is actually an important result too, which tells us something about the physics of the object. But, in order to be useful, such a result must be convincing, in other words the reason why no solution can be found must be clearly explained.

For a good-quality model fitting, it is important to do the following.

(1) Use as many observational constraints as possible (not only line-intensity ratios).

(2) Keep in mind that the importance of the constraint has nothing to do with the strengths of the lines. For example He II A4686/Hp, which is of the order of a few per cent at most in HII regions, indicates the presence of photons with energy above 54.4 eV, which are not expected in main-sequence, massive stars (unless they are part of an X-ray binary system); [O Ill] A4363/5007 and [NII] A5755/6584 indicate whether the energy budget is well reproduced. It is too often ignored that photoionization models compute not only the ionization structure of the nebulae, but also compute their temperature, which is essential in predicting the strengths of the emission lines.

(3) Recall that some constraints are not independent. For example, if [O III] A5007/Hp is fitted by the model, then [O Ill] A4959/Hp should be fitted as well because the [O Ill] A5007/4959 ratio is fixed by atomic physics, since both lines originate from the same level.^ In no case can the fact that both lines are fitted at the same time be taken as a success of the model. On the other hand, if they are not, this may indicate an observational problem, e.g. that the strong [O III] A5007 line is saturated. As a matter of fact, many observers use precisely this ratio as a check of the accuracy of their observations.

(4) Choose a good estimator for the "goodness of fit", e.g. avoid using a \2~ minimization technique without being convinced that this is the most-appropriate test in the case under study. All the observables should be fitted (within limits defined a priori).

(5) Try to visualize the model-result comparison as much as possible. Examples among many possibilities can be found in Stasiñska & Schaerer (1999) or in Stasiñska et al. (2004).

Outcomes from model-fitting. A priori, the most satisfactory situation is when all the observations are fitted within the error bars. This may imply (but does not necessarily show) that the model abundances are the real abundances. If the constraints are insufficient, the model abundances may actually be very different from the true ones.

Quite often, some of the observations cannot be fitted. This means either that the observations are not as good as was thought, or that the model does not represent the object well. Some assumptions in the modelling may be incorrect. For example the nebula has a geometry different from the assumed one, the stellar ionizing radiation field is not well described, or an important heating mechanism is missing from the model. In such a situation, the chemical composition is generally not known with the desired accuracy, a fact too often overlooked.

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