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Astron. Astrophys. 332, 1044-1054 (1998)

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1. Introduction

The physical processes governing line formation in stationary gaseous nebulae are believed to have been well established for a long time (Seaton 1960; Spitzer 1978; Osterbrock 1974, 1989; Aller 1984; Harrington 1989). The determination of chemical abundances, particularly in Planetary Nebulae (PNe) with their relatively high degree of symmetry and small amount of dust grains, has therefore been considered to be rather safe (cf. Kaler 1985; Pottasch 1984a). Discussion of their implications for the theory of stellar evolution has been quite extensive (Iben & Truran 1978; Iben & Renzini 1983; Iben 1984; Renzini & Voli 1981; Marigo et al. 1996). In comparison with the predictions of the stellar evolution theory it is of course of paramount importance to be able to define the accuracy which is attached to each individual determination of chemical abundances, something which, in our opinion, is currently not handled with a sufficient care. We also note that in principle a dynamical treatment is needed, considering the complicated velocity field, the density structure and the occurrence of shocks within a planetary. The importance of an accurate knowledge of the chemical abundances is clear already in connection with the simple, yet very important question, on whether a given PN is carbon or oxygen rich. Only a precise knowledge of the accuracy of the measured C/O abundance ratio permits derivation of meaningful astrophysical conclusions from the observed distribution of C/O in PNe. Similarly an accurate knowledge of the errors of chemical abundances is required when dealing with abundance gradients from PNe in galaxies. It is useful to recall that the determinations of chemical abundances in PNe are done with three methods:

  • [i)] by comparing individual line fluxes with those predicted from detailed photoionization models;
  • ii) by approximating the nebula with one, two or more zones (with [FORMULA], [FORMULA] constant in each zone) and using a proper ionization correction scheme for the unseen ions;
  • iii) with a procedure intermediate between the first two. (a) The ionic abundances are first derived from the observed line fluxes, as in ii); (b) a photoionized model is tailored for the specific nebula, reproducing relevant intensity line ratios, as the one from [FORMULA] / [FORMULA] ions; (c) icf's are then derived from (b) and used to get total abundances from (a).

Technique iii) was used essentially by Aller and associates (Aller & Czyzak 1983; Aller & Keyes 1987) on about 90 PNe. Method i), which is physically the most satisfactory, has been applied to very few objects so far. The bulk of our information rests on method ii) (in its different specifications), used for the determination of abundances in more than 250 galatic PNe. When properly applied, method ii) may produce chemical abundances not far from those of method i) (cf. Pottasch 1984b). Based on a comparison of results from the different methods on the well studied planetary NGC 7662, Pottasch emphasizes: "A tentative conclusion is that careful analysis using the constant [FORMULA], [FORMULA] method will give abundances within 30% to 50% of the results of the detailed nebular models. It is likely that the absolute accuracy of the abundances from nebular models is also limited to 30-50% because of a combination of various factors discussed above."

To assess the general validity of these concepts, a more extended use of method i) is clearly desirable. But there are limitations because this method requires various poorly known information items, including the distance of the object. In other words one is dealing with a multi-parameter problem, with the possibility of non-unique solutions.

The comparison of the abundance of a given ion as derived by lines powered by different physical processes is clearly quite important to assess the correctness of the derived abundances. A well known example is the abundance of [FORMULA], which is much larger if the faint recombination line 4267 [FORMULA] is used, relative to the value derived from the strong collisionally excited 1909 [FORMULA] C III ] semiforbidden doublet. Efforts to explain the discrepancy in terms of some other mechanism, as fluorescence from the central star, contributing to the population of the upper level of the 4267 [FORMULA] transition did not provide a solution. Waiting for an explanation, the general wisdom has been to give credit to the lower values derived from the well observed 1909 C III ] doublet. Recently the same type of difficulty arose again with the abundance of [FORMULA], when detailed calculations of the recombination coefficients of the [FORMULA] permitted lines in the optical became available (Péquignot et al. 1991; Osterbrock et al. 1992; Peimbert et al. 1993; Storey 1994). One possibility invoked to alleviate the problem is a significant role of the so-called "temperature fluctuations", discussed by Peimbert (1967). According to this description, the temperature to be used to deduce the abundance of [FORMULA] (and of other ions as well) is not the one directly derived from the classical 4363/5007 line ratio, but a smaller one. With this lower temperature the deduced abundance is higher, closer to the value from the (temperature independent) recombination lines. Liu et al. (1995) found however that this effect is not sufficient to explain the discrepancy, which amounts up to a factor of five in NGC 7009. On the other hand, recent work by Rubin et al. (1997) making use of far-infrared emission lines, which are quite insensitive to the electron temperature, appears to confirm the oxygen abundance derived for NGC 7009 from the classical UV/optical collisionally excited lines.

All in all, it is evident that substantial work needs to be done to come to a satisfactory understanding of the errors associated with the determinations of chemical abundances in PNe. With the present work, we wish to contribute to the subject by evaluating the errors that can be associated to the determinations made with method ii) just due to the basic assumption of constant electron density and temperature. While this is only a piece of the effort to be made, it will at least serve to give a quantitative idea of errors inherent in method ii) and therefore in the bulk of the available abundances in PNe.

In Sect. 2we illustrate the modeling of the planetary nebulae and the technique used to derive abundances. The results are presented and discussed in Sect. 3. The conclusions follow in Sect. 4.

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Online publication: March 30, 1998