4.1 Comparison of the emission-line fluxes

Table 6 lists the observed and predicted nebular emission line fluxes. Column 4 presents the observed, dereddened intensities of PB8 from García-Rojas et al. (2009), relative to the intrinsic dereddened H$\beta$ flux, on a scale where $I$(H$\beta$)$=100$. The ratios of predicted over observed values from the model MC1 are presented in Column 6. Columns 7-9 present the ratios of predicted over observed values for the normal component, the metal-rich component, and the entire nebula (normal+metal-rich) from the best-fitting model MC2. The same values obtained from the model MC3 are given in Columns 10-12. The majority of the CEL intensities predicted by model MC1 are in reasonable agreement with the observations. However, there are some large discrepancies between the prediction of model MC1 and the observations for ORLs. From the model MC2, it can be seen that the ORL discrepancy between model and observations can be explained by recombination processes of colder metal-rich inclusions embedded in the global H-rich environments.

Figure 6: The predicted over observed flux ratio for the bi-abundance model MC3. The relative contributions of the normal and the metal-rich components to each line flux are shown by black and grey parts, respectively.

As seen in Table 6, the $[$NII$]$ $\lambda$6584 and $[$OIII$]$ $\lambda$5007 line intensities predicted by the models MC1 and MC2 are in excellent agreement with the observations. As both the models MC2 and MC3 have exactly the same density distribution and chemical abundances, we can see how dust grains introduce a $10$ percent increase in the $[$NII$]$ $\lambda$6584 line, which means that nitrogen abundance could be overestimated in some dusty nebulae. The HI line intensities as well as the majority of the HeI line intensities are in reasonable agreement with the observations, discrepancies within 20 percent, apart from the HeI $\lambda$3889, $\lambda$5875 and $\lambda$7065 (around 30 percent). This could be due to high uncertainties of the recombination coefficients of the HeI lines below 5000K (see Porter et al., 2013; Porter et al., 2012). The [OII]$\lambda$7319 and $\lambda$7330 doublets are underestimated by around 50 percent in the model MC1. Recombination processes can largely contribute to the observed fluxes of these lines, which can be estimated by the empirical equation given by Liu et al. (2000) (see equation 2).

There are discrepancies between the predicted intensities of $[$SII$]$ and $[$SIII$]$ lines and the observed values. While the intensities of the $[$SII$]$ lines are predicted to be about 10-20 percent lower than the observations, the intensity of the $[$SIII$]$ $\lambda$6312 line is calculated to be almost twice more than the observed value. Adjusting the sulfur abundance cannot help reproduce $[$SIII$]$ lines, so these discrepancies could be related to either the atomic data or the physical conditions. The predicted intensities of $[$SII$]$ lines were calculated using S$^{+}$ collision strengths from Ramsbottom et al. (1996) incorporated into the CHIANTI database (V 7.0), which is currently used in MOCASSIN. Recently, new S$^{+}$ collision strengths were calculated by Tayal & Zatsarinny (2010), which ignored the effect of coupling to the continuum in their calculations, so their results were estimated to be accurate to about 30 percent or better. We note that the emissivities of $[$SII$]$ $\lambda\lambda$6716,6731 lines calculated by the proEQUIB IDL Library ³, which includes an IDL implementation of the Fortran program EQUIB (Howarth et al., 2016; Howarth & Adams, 1981), show that the collision strengths by Tayal & Zatsarinny (2010) make them about 8 percent lower at the given physical conditions. The predicted $[$SIII$]$ line intensities are perhaps much more uncertain, as there seem to be some errors in the atomic data, as mentioned by Grieve et al. (2014). For example, the emissivity of $[$SIII$]$ 18.68$\mu$m line calculated using the collision strengths from Tayal & Gupta (1999) is about 40 percent higher than the calculation made with Hudson et al. (2012) or Grieve et al. (2014). This issue might be related to the long-standing problem of the sulfur anomaly in PNe (see reviews by Henry et al., 2012).

The predicted intensities of the $[$ArIII$]$ $\lambda\lambda$7136,7751 lines are in agreement with the observations, discrepancies within 20 percent, however, the IR fine-structure $[$ArIII$]$ 8.99 $\mu$m line is predicted to be about 80 percent higher. We used Ar$^{2+}$ collision strengths from Galavis et al. (1995) used by the CHIANTI database (V 7.0). There is another set for Ar$^{2+}$ collision strengths (Munoz Burgos et al., 2009) whose predictions are significantly different and need to be examined carefully. We notice that the emissivities of $[$ArIII$]$ $\lambda\lambda$7136,7751 lines predicted by proEQUIB with the collision strengths from Munoz Burgos et al. (2009) show a discrepancy of about 9 percent in comparison to those calculated with Galavis et al. (1995), whereas there is a 30 percent difference in the $[$ArIII$]$ 8.99 $\mu$m emissivity calculated with the different atomic data.

The predicted $[$NeII$]$ $\lambda$12.82 $\mu$m and $[$NeIII$]$ $\lambda\lambda$3869,3967 line intensities do not show high discrepancies (less than 20 percent), nevertheless, the calculated intensities of $[$NeII$]$ $\lambda\lambda$15.55,36.02 $\mu$m lines have discrepancies about 26 and 57 percent. The predicted $[$ClIII$]$ $\lambda\lambda$5518,5538 lines are in agreement with the observations, discrepancies less than 25 percent.

Although the [OIII] $\lambda$4363 auroral line is perfectly matched by the model MC3 and discrepancies remain less than 10 percent in the model MC2, there is a notable discrepancy in the [NII] $\lambda$5755 auroral line. This could be due to excitation by continuum fluorescence and/or recombination process. Bautista (1999) found that the [NI]  $\lambda\lambda$5198,5200 lines are efficiently affected by fluorescence excitation in many objects, while [OI] lines were found to be sensitive to fluorescence in colder regions ($\leq 5000$K) or very high radiation fields. Nevertheless, this PN is not known to be surrounded by a photo-dissociation region (PDR) that is responsible for the fluorescence excitation. We notice that García-Rojas et al. (2009) observed the brightest part of the nebula, and excluded the central star contamination and the surrounding potential PDR. Moreover, the absences of the [O I] $\lambda\lambda$6300,6364 lines emitted by neutral O$^{0}$ ion and the [N I] $\lambda\lambda$5198,5200 lines emitted by neutral N$^{0}$ ion in the spectrum presented by García-Rojas et al. (2009) exclude any possibilities of the fluorescence contamination. Hence, there is no strong evidence for any possible fluorescence contributions to the observed fluxes. Alternatively, the recombination contribution to [NII] auroral lines may have some implications, which can be estimated for low-density uniform nebular media (see e.g. Liu et al., 2000).

The recombination contribution to the [NII] $\lambda$5755 line and the [OII] $\lambda\lambda$7320,7330 doublet can be estimated as follows (Liu et al., 2000):

$$\frac{I_{\rm R}(\lambda5755)}{I({\rm H}\beta)}=3.19 \,\, {t^{0.30}} \, \bigg( \frac{\rm N^{2+}}{\rm H^{+}} \bigg)_{\rm ORLs},$$ (1)

$$\frac{I_{\rm R}(\lambda7320+\lambda7330)}{I({\rm H}\beta)}=9.36 \,\, {t^{0.44}} \, \bigg( \frac{\rm O^{2+}}{\rm H^{+}} \bigg)_{\rm ORLs},$$ (2)

where $t\equiv T_{\rm e}/10^4$ is the electron temperature in $10^4$K from Tables 7 and 8 and $({\rm N^{2+}}/{\rm H^{+}})_{\rm ORLs}$ and $({\rm O^{2+}}/{\rm H^{+}})_{\rm ORLs}$ derived from Tables 3, 9 and 10. The recombination contributions to the [NII] $\lambda$5755 auroral line are estimated to be about 12 percent in the model MC1 and 48 percent in the models MC2 and MC3 (including contributions from the metal-rich inclusions). As MOCASSIN does not currently estimate the recombination contributions to the auroral lines (due to the lack of atomic data), we empirically calculated them with equations (1) and (2) and included them in the [NII] $\lambda$5755 line and the [OII] $\lambda\lambda$7320,7330 doublet in Tables 6. We see that the [NII] $\lambda$5755 auroral line in the model MC3 show better agreement, but about 30 percent lower. The uncertainty of this faint line, 14 percent reported by García-Rojas et al. (2009), which could be even higher, may explain this discrepancy. Additionally, it is extremely difficult to evaluate the recombination contribution in the presence of inhomogeneous condensations. The collisional de-excitations of very dense clumps in the nebula can suppress the $\lambda\lambda$6548,6584 nebular lines, but not the auroral lines (Viegas & Clegg, 1994), so the discrepancy between the model and the observation could be related due to unknown inhomogeneous condensations and high uncertainties in the recombination atomic data at low temperatures (i.e., below 5000K in a dense clump). Similarly, the recombination contribution to the [OII]  $\lambda\lambda$7320,7330 doublets are estimated to be about 15 percent in the model MC1 and 53 percent in the models MC2 and MC3. As you see, the predicted [OII]  $\lambda\lambda$7320,7330 doublets are in excellent agreement with the observations.

Table 7: Mean electron temperatures (K) weighted by ionic species for the entire nebula. For each element the first row is for MC1 and the second row is for MC2.

	Ion
El.	I	II	III	IV	V	VI	VII
H	7746	7625
	6647	6584
He	7746	7625	7595
	6655	6584	6784
C	7829	7741	7621	7468	7431	7625	7625
	6806	6673	6580	6549	6631	6584	6584
N	7834	7746	7617	7468	7431	7625	7625
	6883	6772	6567	6439	6565	6584	6584
O	7860	7746	7613	7566	7625	7625	7625
	7176	6692	6568	6618	6584	6584	6584
Ne	7812	7720	7601	7564	7625	7625	7625
	6602	6599	6579	6764	6584	6584	6584
S	7855	7783	7685	7541	7411	7390	7625
	6760	6676	6632	6497	6453	6572	6584
Cl	7836	7748	7635	7503	7467	7625	7625
	6747	6658	6598	6408	6612	6584	6584
Ar	7846	7766	7659	7535	7490	7625	7625
	6699	6606	6588	6570	6684	6584	6584

The intensities of the ORLs predicted by the model MC2 and MC3, both bi-abundance models, can be compared to the observed values in Table 6. Figure 6 compares the predicted over observed flux ratio for the model MC3, and shows the relative contributions of the normal and the metal-rich components to each emission-line flux. The agreement between the ORL intensities predicted by the two latter models and the observations is better than those derived from the first model (MC1). The majority of the OII lines with strong intensities are in reasonable agreement with the observations, with discrepancies below 40 percent, except for $\lambda$4649.13, $\lambda$3749.48, $\lambda$4075.86, $\lambda$4132.80 and $\lambda$4153.30. The well-measured NII $\lambda$5666.64, $\lambda$5676.02 and $\lambda$5679.56 lines are in good agreement with the observations, and discrepancies are less than 30 percent. There are some discrepancies in some OII ORLs (e.g. $\lambda$4416.97, $\lambda$4121.46, $\lambda$4906.81) and NII ORLs (e.g. $\lambda$4601.48, $\lambda$4613.87, $\lambda$5931.78), which have weak intensities and higher uncertainties (20-30 percent). However, as seen in Figure 6, the model MC3 has significant improvements in predicting the O II and N II lines having intensities stronger then other ORLs. Particularly, the bi-abundance models MC2 and MC3 provide better predictions for the O II ORLs from the V1 multiplet and the N II ORLs from the V3 multiplet, which have the reliable atomic data. Comparing Figure 6 with Fig. 15 in Yuan et al. (2011) demonstrates that our bi-abundance models of PB8, similar to the photoionization model of NGC6153, better predict the observed intensities of the N II and O II ORLs. The models also reproduce the C II ORLs with discrepancies about 10 percent, except the CII $\lambda$6578 line. The CII $\lambda$4267.2 line is stronger than the other C II lines, and it is not blended with any nearby OII ORLs. The CII $\lambda$6578 line may be blended with nearby lines, so its measured line strength may be uncertainty.

Table 8: Mean electron temperatures (K) weighted by ionic species for the nebula obtained from the photoionization model MC3. For each element the first row is for the normal component, the second row is for the H-poor component, and the third row is for the entire nebula.

	Ion
El.	I	II	III	IV	V	VI	VII
H	7298	7097
	4341	4309
	6719	6640
He	7307	7097	7054
	4343	4309	4310
	6726	6640	6843
C	7436	7295	7088	6795	6739	7098	7098
	4361	4342	4307	4252	4253	4309	4309
	6886	6742	6635	6491	6640	6641	6641
N	7460	7315	7077	6796	6738	7098	7098
	4363	4344	4306	4257	4258	4309	4309
	6959	6840	6622	6414	6580	6641	6641
O	7507	7319	7064	6989	7098	7098	7098
	4364	4346	4302	4302	4309	4309	4309
	7250	6762	6621	6671	6641	6641	6641
Ne	7414	7260	7042	6988	7098	7098	7098
	4361	4340	4294	4295	4309	4309	4309
	6690	6671	6630	6819	6641	6641	6641
S	7466	7349	7186	6937	6713	6678	7098
	4365	4349	4324	4276	4243	4243	4309
	6848	6751	6694	6541	6435	6597	6641
Cl	7444	7302	7113	6875	6720	6721	7098
	4361	4343	4312	4261	4241	4245	4309
	6832	6731	6656	6444	5640	6241	6641
Ar	7468	7336	7148	6929	6861	7098	7098
	4366	4349	4317	4267	4267	4309	4309
	6783	6680	6647	6618	6737	6641	6641