6.1.2 Nebular abundances

All major contributors to the thermal balance of the gas were included in our model. We used a homogeneous elemental abundance distribution consisting of eight elements. The initial abundances of He, N, O, Ne, S and Ar were taken from the observed empirically derived total abundances listed in Table 5. The abundance of C was a free parameter, typically varying between $5\times10^{-5}$ and $8\times10^{-3}$ in PNe. We initially used the typical value of ${\rm C}/{\rm H}=5.5\times10^{-4}$ (Kingsburgh & Barlow, 1994), and adjusted it to preserve the thermal balance of the nebula. We kept the initial abundances fixed while the stellar parameters and distance were being scaled to produce the best fit for the H$\beta$ luminosity and He${}^{2+}$/He${}^{+}$ ratio, and then we gradually varied them to obtain the finest match between the predicted and observed emission-line fluxes, as well as ionic abundance ratios from the empirical analysis.

The flux intensity of He II $\lambda$4686 Å and the He${}^{2+}$/He${}^{+}$ ratio highly depend on the temperature and luminosity of the central star. Increasing either $T_{\rm eff}$ or $L_{\star}$ or both increases the He${}^{2+}$/He${}^{+}$ ratio. Our method was to match the He${}^{2+}$/He${}^{+}$ ratio, and then scale the He/H abundance ratio to produce the observed intensity of He II $\lambda$4686 Å.

The abundance ratio of oxygen was adjusted to match the intensities of $[$O III$]$ $\lambda \lambda$4959,5007 and to a lesser degree $[$O II$]$ $\lambda \lambda$3726, 3729. In particular, the intensity of the $[$O II$]$ doublet is unreliable due to the contribution of recombination and the uncertainty of about 30% at the extreme blue of the WiFeS. So we gradually modified the abundance ratio O/H until the best match for $[$O III$]$ $\lambda \lambda$4959,5007 and O${}^{2+}$/H${}^{+}$ was produced. The abundance ratio of nitrogen was adjusted to match the intensities of $[$N II$]$ $\lambda \lambda$6548,6584 and N${}^{+}$/H${}^{+}$. Unfortunately, the weak $[$N II$]$ $\lambda$5755 emission line does not have a high S/N ratio in our data.

The abundance ratio of sulphur was adjusted to match the intensities of $[$S III$]$ $\lambda$9069. The intensities of $[$S II$]$ $\lambda \lambda$6716,6731 and S${}^{+}$/H calculated by our models are about seven and ten times lower than those values derived from observations and empirical analysis, respectively. The intensity of $[$S II$]$ $\lambda \lambda$6716,6731 is largely increased due to shock-excitation effects.

Figure 10: Hertzsprung-Russell diagrams for hydrogen-burning models (left-hand panel) with $(M_{\rm ZAMS},M_{\star })=$ $(3{\rm M}_{\bigodot}, 0.605{\rm M}_{\bigodot})$, $(3{\rm M}_{\bigodot}, 0.625{\rm M}_{\bigodot})$ and $(4{\rm M}_{\bigodot}, 0.696{\rm M}_{\bigodot})$, and helium-burning models (right-hand panel) with $(M_{\rm ZAMS},M_{\star })=$ $(1{\rm M}_{\bigodot}, 0.524{\rm M}_{\bigodot})$, $(3{\rm M}_{\bigodot}, 0.625{\rm M}_{\bigodot})$ and $(5{\rm M}_{\bigodot}, 0.836{\rm M}_{\bigodot})$ from Blöcker (1995) compared to the position of the central star of SuWt 2 derived from two different photoionization models, namely Model 1 (denoted by $\blacksquare$) and Model 2 ( $\blacktriangledown$). On the right, the evolutionary tracks contain the first evolutionary phase, the VLTP (born-again scenario), and the second evolutionary phase. The colour scales indicate the post-AGB ages ( $\tau _{\rm post-\textsc {agb}}$) in units of $10^3$ yr.

Finally, the differences between the total abundances from our photoionization model and those derived from our empirical analysis can be explained by the $icf$ errors resulting from a non-spherical morphology and properties of the exciting source. Gonçalves et al. (2012) found that additional corrections are necessary compared to those introduced by Kingsburgh & Barlow (1994) due to geometrical effects. Comparison with results from photoionization models shows that the empirical analysis overestimated the neon abundances. The neon abundance must be lower than the value found by the empirical analysis to reproduce the observed intensities of $[$Ne III$]$ $\lambda \lambda$3869,3967. It means that the $icf$(Ne) of Kingsburgh & Barlow (1994) overestimates the unseen ionization stages. Bohigas (2008) suggested to use an alternative empirical method for correcting unseen ionization stages of neon. It is clear that with the typical Ne${}^{2+}$/Ne=O${}^{2+}$/O assumption of the $icf$ method, the neon total abundance is overestimated by the empirical analysis.

Figure 11: The 3 D distributions of electron temperature, electron density and ionic fractions from the adopted Model 2 constructed in $45 \times 45 \times 7$ cubic grids, and the ionizing source being placed in the corner (0,0,0). Each cubic cell has a length of $1.12\times 10^{-2}$ pc, which corresponds to the actual PN ring size.