3.8. The specific case of the helium abundance determination
The determination of the helium abundance follows the same principles as that of other elements. But one is much more demanding about the accuracy. To follow the production of helium in stars, and the evolution of the helium content in galaxies, 10% accuracy is a goal that one would like to achieve. Helium abundances compiled from the literature over the years must be considered with caution, because of the different treatments adopted by various authors. On the other hand, the required accuracy should be reachable with consistent observations and modern data treatments. To determine the primordial helium abundance, Yp, one needs a much better accuracy, since quite different cosmologies are predicted for values Yp differing by a only few percent. From low metallicity giant extragalactic H II regions, Olive et al. (1997) find Yp = 0.234 ± 0.002 while Izotov & Thuan (1998) find Yp = 0.244 ± 0.002. These two estimates are mutually exclusive. Is it possible to say which of the two if any is correct?
The first step is to obtain the intrinsic values of the intensities of the helium and hydrogen lines in an observed spectrum. If the spectrum contains stellar light, as in the case of giant H II regions, one must correct the observed intensities for underlying stellar absorption. The recent evolutionary synthesis models of Gonzalez Delgado et al. (1999) provide a theoretical framework for that. One also has to correct the intensity ratios for reddening, assuming a given reddening "law" and a given intrinsic value of the ratios of the hydrogen line intensities. The latter mainly depends on the electron temperature, which can be estimated from the [O III] 4363/5007 ratio, with a correction due the fact that the O++ region is only a part of the H+ region. Using an appropriate number of lines, one can estimate iteratively the reddening and the correction for underlying absorption (e.g. Izotov & Thuan 1998). However, as commented by Davidson & Kinman (1985) and Sasselov & Goldwirth (1995), and as mentioned in Sect. 3.3, collisional excitation of H Balmer lines may become important, especially in H II regions of high Te. So far, this effect has always been omitted in the determination of the abundance of primordial He. It may induce an overestimation of the reddening, and therefore an underestimation of the He+ abundance derived from He I 5876 by up to 5 % (Stasinska & Izotov 2001). The importance of this effect depends on the abundance of residual H0.
Then, from the corrected ratios of emission lines one has to determine the value of He+/H+, or, to be more exact, of n(He+)dV / n(H+)dV. This assumes that the line emissivities do not vary strongly over the nebular volume. The emissivities depend on Te, and also on ne in the case of some helium lines, due to enhancement by collisional exci helium lines, one can in principle determine iteratively and self-consistently the characteristic temperature and density of the helium line emission, and the relative abundance of He+. The treatment of radiation transfer in the lines remains to be improved and is announced as a next step by Benjamin et al. (1999). However, this is a complex problem: it depends on the velocity field and on the amount of internal dust which selectively absorbs resonant photons. Therefore, one does not expect models to be easily applicable to real objects. However, since this is a second order effect, this is perhaps not too problematic, if one discards the lines likely to be most affected by this process. One must not forget that the emissivities of the H Balmer lines too may be in question, both because of the contribution of collisional excitation mentioned above and because the presence of dust deviates the hydrogen spectrum from case B (see Hummer & Storey 1992). Another problem is to take into account the non uniformity of Te. Sauer & Jedamczik (2002) have computed a grid of photoionization models for this purpose, and introduce the concept of a "temperature correction factor" which they compute in their models. Note, however, that the real temperature structure of nebulae is not obtained from "first principles", as the preceding sections made clear. Therefore, the distribution of Te in real objects has most probably a larger impact than predicted by the models of Sauer & Jedamczik (2002). Peimbert et al. (2002) have adopted a semi-empirical approach, based on the Peimbert's (1967) temperature fluctuation scheme. But the temperature fluctuation scheme may give spurious results in the hypothesis of zones of highly different temperatures, as argued in Sect. 3.5.2.
If He II lines are present in the spectra, they have to be accounted for, to determine n(He++)dV / n(H+)dV. The major uncertainty in that case comes probably from the lack of knowledge of the temperature characterising the emission of He II lines. An additional difficulty is due to the fact that part of the He II emission may be of stellar origin.
The He/H abundance is obtained after considering ionization structure effects. For low values of the mean effective temperature of the radiation field, a zone of neutral helium is present. Unfortunately, no ionization correction formula can be safely applied, since the ionization structure of helium with respect to hydrogen mainly depends on the hardness of the radiation field, while the ionization structure of the heavy elements also strongly depends on the gas distribution (e.g. Stasinska 1980b). In the case of an H II region ionized by very hot stars, photoionization models show that the He+ region may on the contrary extend further than the H+ region (see for example Stasinska 1980b or Sauer & Jedamczik 2002). Whether this is the case for an object under study should be tested by models.
Olive & Skillman (2001) stress the importance of having a sufficient number of observational constraints and of using them in a self consistent manner with a Monte-Carlo treatment of all sources of errors. Unfortunately, the errors on the temperature structure and on the ionization structure of real nebulae are very difficult to evaluate, and this, combined with uncertainties in atomic parameters and deviations from case B theory implies that the uncertainty in derived helium abundances is certainly larger than claimed.