6 Model results

6.1 Comparison of the emission-line fluxes

Table 9 compares the flux intensities predicted by the best-fitting model with those from the observations. Columns 2 and 3 present the dereddened fluxes of our observations and those from Todt et al. (2013). The predicted emission line fluxes are given in Column 4, relative to the intrinsic dereddened H$\beta$ flux, on a scale where $I($ H$\beta)$ =100. The most emission-line fluxes presented are in reasonable agreement with the observations. However, we notice that the [O II]$\lambda$ 7319 and $\lambda$ 7330 doublets are overestimated by a factor of 3, which can be due to the recombination contribution. Our photoionization code incorporates the recombination term to the statistical equilibrium equations. However, the recombination contribution are less than 30 per cent for the values of $T_{\rm e}$ and $N_{\rm e}$ found from the plasma diagnostics. Therefore, the discrepancy between our model and observed intensities of these lines can be due to inhomogeneous condensations such as clumps and/or colder small-scale structures embedded in the global structure. It can also be due to the measurement errors of these weak lines. The [O II] $\lambda\lambda$ 3726,3729 doublet predicted by the model is around 25 per cent lower, which can be explained by either the recombination contribution or the flux calibration error. There is a notable discrepancy in the predicted [N II]$\lambda$ 5755 auroral line, being higher by a factor of $\sim3$ . It can be due to the errors in the flux measurement of the [N II]$\lambda$ 5755 line. The predicted [Ar III]$\lambda$ 7751 line is also 30 per cent lower, while [Ar III]$\lambda$ 7136 is about 20 per cent higher. The [Ar III]$\lambda$ 7751 line usually is blended with the telluric line, so the observed intensity of these line can be overestimated. It is the same for [S III]$\lambda$ 9069, which is typically affected by the atmospheric absorption band.

Table 11: Integrated ionic abundance ratios for He, C, N, O, Ne, S and Ar, derived from model ionic fractions and compared to those from the empirical analysis.

Ionic ratio	Observed	Model
He$^+$ /H$^+$	0.124	0.116
C$^{2+}$ /H$^+$	2.16($-3$ )	2.45($-3$ )
N$^+$ /H$^+$	1.42($-5$ )	1.26($-5$ )
O$^+$ /H$^+$	5.25($-5$ )	3.63($-5$ )
O$^{2+}$ /H$^+$	1.06($-4$ )	9.76($-5$ )
Ne$^{2+}$ /H$^+$	4.26($-5$ )	3.62($-5$ )
S$^+$ /H$^+$	3.98($-7$ )	5.20($-7$ )
S$^{2+}$ /H$^+$	5.58($-6$ )	4.19($-6$ )
Ar$^{2+}$ /H$^+$	9.87($-7$ )	1.01($-6$ )

6.2 Ionization and thermal structure

The volume-averaged fractional ionic abundances are listed in Table 10. We note that hydrogen and helium are singly-ionized. We see that the O$^{+}$ /O ratio is higher than the N$^{+}$ /N ratio by a factor of 1.34, which is dissimilar to what is generally assumed in the $icf$ method. However, the O$^{2+}$ /O ratio is nearly by a factor of 1.16 larger than the Ne$^{2+}$ /Ne ratio, in agreement with the general assumption for $icf$ (Ne). We see that only 19 per cent of the total nitrogen in the nebula is in the form of N$^{+}$ . However, the total oxygen largely exists as O$^{2+}$ with 70 per cent and then O$^{+}$ with 26 per cent.

The elemental abundances we used for the photoionization model returns ionic abundances listed in Table 11, are comparable to those from the empirical analysis derived in Section 4. The ionic abundances derived from the observations do not show major discrepancies in He$^{+}$ /H$^{+}$ , C$^{2+}$ /H$^{+}$ , N$^{+}$ /H$^{+}$ , O$^{2+}$ /H$^{+}$ , Ne$^{2+}$ /H$^{+}$ and Ar$^{2+}$ /H$^+$ ; differences remain below 18 per cent. However, the predicted and empirical values of O$^{+}$ /H$^{+}$ , S$^{+}$ /H$^+$ and S$^{2+}$ /H$^+$ have discrepancies of about 45, 31 and 33 per cent, respectively.

Fig. 7 shows plots of the ionization structure of helium, carbon, oxygen, argon (left panel), nitrogen, neon and sulfur (right panel) as a function of radius along the equatorial direction. As seen, ionization layers have a clear ionization sequence from the highly ionized inner parts to the outer regions. Helium is 97 percent singly-ionized over the shell, while oxygen is 26 percent singly ionized and 70 percent doubly ionized. Carbon and nitrogen are about $\sim20$ percent singly ionized $\sim80$ percent doubly ionized. The distribution of N$^{+}$ is in full agreement with the IFU map, given in Fig4. Comparison between the He$^{+}$ , O$^{2+}$ and S$^{+}$ ionic abundance maps obtained from our IFU observations and the ionic fractions predicted by our photoionization model also show excellent agreement.

Table 12: Mean electron temperatures (K) weighted by ionic species for the whole nebula obtained from the photoionization model.

	Ion
El.	I	II	III	IV	V	VI	VII
H	9044	10194
He	9027	10189	10248
C	9593	9741	10236	10212	10209	10150	10150
N	8598	9911	10243	10212	10209	10150	10150
O	9002	10107	10237	10241	10211	10150	10150
Ne	8672	10065	10229	10225	10150	10150	10150
S	9386	9388	10226	10208	10207	10205	10150
Ar	8294	9101	10193	10216	10205	10150	10150

Table 12 lists mean temperatures weighted by the ionic abundances. Both [N II] and [O III] doublets, as well as HeI lines arise from the same ionization zones, so they should have roughly similar values. The ionic temperatures increasing towards higher ionization stages could also have some implications for the mean temperatures averaged over the entire nebula. However, there is a large discrepancy by a factor of two between our model and ORL empirical value of $T_{\rm e}$ (HeI$)$ . This could be due to some temperature fluctuations in the nebula (Peimbert, 1967; Peimbert, 1971). The temperature fluctuations lead to overestimating the electron temperature deduced from CELs. This can lead to the discrepancies in abundances determined from CELs and ORLs (see e.g. Liu et al., 2000). Nevertheless, the temperature discrepancy can also be produced by bi-abundance models (Liu, 2003; Liu et al., 2004a), containing some cold hydrogen-deficient material, highly enriched in helium and heavy elements, embedded in the diffuse warm nebular gas of normal abundances. The existence and origin of such inclusions are still unknown. It is unclear whether there is any link between the assumed H-poor inclusions in PNe and the H-deficient central stars.