3. Results and discussion
From the Tables 2 and 3 it is seen that the abundances calculated following the prescriptions of the presently most elaborate "constant , " method deviate from the input abundances differently for the various elements.
3.1. Optically thin hydrodynamical models
We first consider the optically thin hydrodynamical models A1 to A4, B1 and B2. By comparing the abundances derived from these models with the original input ones we find in sequence A maximum deviations of 10% for helium and by factors of 0.8 to 1.2, 0.4 to 2.2, 0.4 to 0.9, 0.9 to 0.4, 0.7 to 0.06 for oxygen, nitrogen, neon, argon and sulphur, respectively, the largest deviations occurring in the most excited models. If most of these deviations would be attributed to the ionization correction scheme adopted for unseen ions, we might infer that systematically larger icf's should be appropriate for the more excited models.
3.2. Optically thick hydrodynamical models
In the optically thick models A0 and B0 we note larger deviations than in optically thin models. To be precise, in model A0 the observed abundances are underestimated by factors of 4.2, 1.6, 1.3 and 3.4 in helium, oxygen, nitrogen and neon, respectively; they are overestimated by a factor 1.5 in argon, while they turn out to be about correct in sulphur. In model B0 argon and sulphur are not present, because the models of sequence B were computed with six elements. The behaviour of the other elements is similar to the case of A0, namely: 5.2, 3.3, 1.8, 6.9 for He, O, N and Neon respectively.
These larger deviations are due to the approximations intrinsic to the "constant , " method which become worse in these cases. To clarify the matter we illustrate in Figs. 1 and 3 the full behaviour of the physical parameters and of the ionization structure of models A0 and B0.
Consider Fig. 1. A conspicuous density enhancement is present at a radial distance between 1.2 and . This is caused by the well known dynamical behaviour of the D-type ionization front which has formed in the superwind and drives a shock into the neutral gas. The ratio decreases to a value of 0.5 at a radius of about , while reaches 0.5 already at about . Therefore, helium is essentially neutral across the zone of the density enhancement, while hydrogen is still substantially ionized up to the radius of the density enhancement peak. The relative helium abundance He/H is consequently not well approximated by the used expression:
The amount of is negligible in this low excitation model, but the amount of not measurable is not compensated by the unseen , as is assumed in Eq. (9). The expression (9) will therefore underestimate considerably the true helium abundance. Indeed our analysis gives , equal to He/H, a factor 4.2 below the true abundance. Similar considerations enter, together with the role of the icf's correction factors, to explain the deviations found in the other chemical elements.
In model B0 we have an even more pronounced density enhancement in the outer nebula peaking at about just in front of the hydrogen ionization. The ratio drops at about . Then, also in this case, helium is all neutral across the density enhancement where hydrogen is ionized up to the inner part of the density peak. The derived is 0.021 equal to He/H, since is negligible. Expression (9) underestimates here He/H by the factor 5.2. The deviations of the elements He, O, N and Ne all follow the behaviour of the corresponding ones in model A0, at slightly higher amounts. For helium we can improve the calculated He/H in the low excitation models A0 and B0, by adopting the formula (15) by Peimbert and Torres-Peimbert (1977) that they used for the Orion nebula, instead of the above Eq. (9). The resulting He/H are 0.062, 0.080 for the A0dyn and A0equil models respectively, and 0.060, 0.035 for the corresponding B0 models. The discrepancy with the input values is thus reduced, although not completely. From the above it is obvious that the errors inherent in the "constant , " method depend on the specific ionization stratification and density structure of the nebula. Our hydrodynamical models show that especially in the early evolutionary phases the latter is highly dependent on the adopted initial density distribution which in turn depends on the mass loss history on the AGB. For instance, in cases of rather small mass loss rates and/or high stellar luminosities it is to be expected that hydrogen ionization does not proceed via a D-type front at all. Instead, an R-type front should form, expanding supersonically relative to the neutral gas and causing only minor dynamical effects. Currently, no elaborate parameter study using realistic radiation gasdynamical PNe modeling with a large number of initial density distributions and stellar post-AGB tracks exists that adresses these questions. A small-scale study (Schönberner et al., in prep.) indicates, however, that - assuming a stationary outflow - the critical AGB mass loss rate for the formation of a D-type front amounts to about for Blöcker's central star. Since already the computation of sequences A and B required several thousand hours of CPU time, limited computational resources have forced us to refrain from probing abundance errors that would result from the "constant , " method when a larger number of initial density distributions and stellar tracks with different masses would be used.
3.3. Equilibrium versus hydrodynamical models
The differences in the abundances obtained from the full hydrodynamical and the equilibrium models were found to be always small both in the optically thin and in the optically thick models. The equilibrium abundances are almost always larger than the dynamical ones. In no case is the difference greater than 25 %.
As illustrated in Figs. 2 and 4 the temperature behaviour in the two types of models is similar, with in the equilibrium models either equal to or a bit larger than that in the dynamical models, apart from the increase in the latter models at the outer nebular rim, due to heating behind the shock. This is however a local effect which does not affect significantly the abundances deduced with the "constant , " method. A higher electron temperature should result in smaller ionic abundances in collisionally excited heavy ions and in larger ionic abundances in the ions whose levels are dominated by recombination processes, i.e. hydrogen and helium. The resulting ionic abundance of heavy ions relative to hydrogen should then be smaller. The opposite is observed. In reality the different behaviour is already accounted for in our procedure of computing the line intensities from the whole model nebula. Thus when we interpret these line intensities with the "constant , " method, we would not expect any systematic effect due to this cause. We have not been able to determine the origin of these effects which are anyhow rather small.
© European Southern Observatory (ESO) 1998
Online publication: March 30, 1998