4.3 Fractional ionic abundances

Table 9: Fractional ionic abundances obtained from the photoionization models. For each element the first row is for MC1, the second row is for MC2 and the third row is for MC3.

	Ion
Element	I	II	III	IV	V	VI	VII
H	6.88($-4$)	9.99($-1$)
	8.23($-4$)	9.99($-1$)
	8.59($-4$)	9.99($-1$)
He	1.91($-3$)	9.98($-1$)	1.62($-12$)
	2.39($-3$)	9.98($-1$)	1.38($-12$)
	2.47($-3$)	9.98($-1$)	1.39($-12$)
C	1.19($-5$)	4.15($-2$)	9.56($-1$)	2.39($-3$)	2.70($-16$)	1.00($-20$)	1.00($-20$)
	1.63($-5$)	5.03($-2$)	9.48($-1$)	1.86($-3$)	1.91($-16$)	1.00($-20$)	1.00($-20$)
	1.75($-5$)	5.18($-2$)	9.46($-1$)	1.71($-3$)	1.75($-16$)	1.00($-20$)	1.00($-20$)
N	1.60($-5$)	6.82($-2$)	9.28($-1$)	3.40($-3$)	6.86($-16$)	1.00($-20$)	1.00($-20$)
	2.51($-5$)	8.50($-2$)	9.12($-1$)	2.92($-3$)	5.41($-16$)	1.00($-20$)	1.00($-20$)
	2.69($-5$)	8.74($-2$)	9.10($-1$)	2.80($-3$)	5.21($-16$)	1.00($-20$)	1.00($-20$)
O	6.40($-5$)	9.40($-2$)	9.06($-1$)	2.00($-13$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
	1.06($-4$)	1.31($-1$)	8.68($-1$)	1.82($-13$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
	1.19($-4$)	1.36($-1$)	8.64($-1$)	1.81($-13$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
Ne	1.06($-4$)	2.02($-1$)	7.97($-1$)	7.74($-14$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
	1.78($-4$)	2.61($-1$)	7.39($-1$)	6.04($-14$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
	1.87($-4$)	2.65($-1$)	7.35($-1$)	6.03($-14$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
S	3.01($-7$)	4.72($-3$)	5.81($-1$)	4.13($-1$)	1.90($-3$)	8.26($-16$)	1.00($-20$)
	4.46($-7$)	6.14($-3$)	6.36($-1$)	3.56($-1$)	1.46($-3$)	5.63($-16$)	1.00($-20$)
	4.87($-7$)	6.46($-3$)	6.42($-1$)	3.50($-1$)	1.42($-3$)	5.48($-16$)	1.00($-20$)
Cl	2.68($-6$)	1.85($-2$)	8.91($-1$)	9.02($-2$)	9.99($-15$)	1.00($-20$)	1.00($-20$)
	3.79($-6$)	2.22($-2$)	8.97($-1$)	8.09($-2$)	7.66($-15$)	1.00($-20$)	1.00($-20$)
	4.12($-6$)	2.30($-2$)	8.96($-1$)	8.09($-2$)	5.00($-7$)	1.47($-19$)	1.00($-20$)
Ar	4.47($-7$)	4.11($-3$)	7.21($-1$)	2.75($-1$)	1.56($-13$)	1.00($-20$)	1.00($-20$)
	7.85($-7$)	5.95($-3$)	7.72($-1$)	2.22($-1$)	1.11($-13$)	1.00($-20$)	1.00($-20$)
	8.43($-7$)	6.12($-3$)	7.72($-1$)	2.22($-1$)	1.12($-13$)	1.00($-20$)	1.00($-20$)

Table 10: Fractional ionic abundances obtained from the photoionization model MC2. For each element the first row is for the normal component and the second row is for the H-poor component.

	Ion
Element	I	II	III	IV	V	VI	VII
H	7.86($-4$)	9.99($-1$)
	1.01($-3$)	9.99($-1$)
He	2.28($-3$)	9.98($-1$)	1.53($-12$)
	2.93($-3$)	9.97($-1$)	6.49($-13$)
C	1.59($-5$)	4.86($-2$)	9.49($-1$)	2.05($-3$)	2.22($-16$)	1.00($-20$)	1.00($-20$)
	1.87($-5$)	5.89($-2$)	9.40($-1$)	9.20($-4$)	3.12($-17$)	1.00($-20$)	1.00($-20$)
N	2.49($-5$)	8.49($-2$)	9.12($-1$)	3.06($-3$)	6.13($-16$)	1.00($-20$)	1.00($-20$)
	2.60($-5$)	8.55($-2$)	9.12($-1$)	2.20($-3$)	1.73($-16$)	1.00($-20$)	1.00($-20$)
O	1.16($-4$)	1.27($-1$)	8.73($-1$)	1.92($-13$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
	5.74($-5$)	1.55($-1$)	8.45($-1$)	1.31($-13$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
Ne	1.60($-4$)	2.47($-1$)	7.53($-1$)	6.77($-14$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
	2.72($-4$)	3.32($-1$)	6.68($-1$)	2.29($-14$)	1.00($-20$)	1.00($-20$)	1.00($-20$)
S	4.20($-7$)	5.83($-3$)	6.29($-1$)	3.64($-1$)	1.59($-3$)	6.56($-16$)	1.00($-20$)
	5.74($-7$)	7.74($-3$)	6.75($-1$)	3.17($-1$)	7.82($-4$)	8.85($-17$)	1.00($-20$)
Cl	3.59($-6$)	2.13($-2$)	8.97($-1$)	8.13($-2$)	8.59($-15$)	1.00($-20$)	1.00($-20$)
	4.84($-6$)	2.69($-2$)	8.94($-1$)	7.89($-2$)	2.87($-15$)	1.00($-20$)	1.00($-20$)
Ar	7.21($-7$)	5.50($-3$)	7.60($-1$)	2.35($-1$)	1.26($-13$)	1.00($-20$)	1.00($-20$)
	1.11($-6$)	8.25($-3$)	8.37($-1$)	1.55($-1$)	3.19($-14$)	1.00($-20$)	1.00($-20$)

Table 9 lists the volume-averaged fractional ionic abundances from the neutral (I) to the highly ionized ions (VII) calculated from the three models, where, the first entries for each element are for the chemically homogeneous model MC1, the second entries are for the bi-abundance model MC2, and the third entries are the dusty bi-abundance model MC3. The definition for the volume-averaged fractional ionic abundances was given in Ercolano et al. (2003c). We see that both hydrogen and helium are fully singly-ionized, i.e. neutrals are almost zero percent in the three models. It can be seen that the ionization structure in MC2 is in reasonable agreement with MC1. The elemental oxygen largely exists as O$^{2+}$ with 91 percent and then O$^{+}$ with 9 percent in the model MC1, whereas O$^{2+}$ is about 87 percent and then O$^{+}$ is about 13 percent in the model MC2. Moreover, the elemental nitrogen largely exists as N$^{2+}$ with 93 percent and then N$^{+}$ with 7 percent in the model MC1, whereas N$^{2+}$ is about 91 percent and then N$^{+}$ is about 9 percent in the model MC2. The O$^{+}$/O ratio is about 1.4-1.6 times higher than the N$^{+}$/N ratio, which is in disagreement with the general assumption of N/N$^+$=O/O$^+$ in the ionization correction factor (icf) method by Kingsburgh & Barlow (1994), introducing errors to empirically derived elemental abundances. Our (N$^{+}$/N)/(O$^{+}$/O) ratio is in agreement with the value of 0.6-0.7 predicted by the photoionization model of NGC 7009 implemented using MOCASSIN (Gonçalves et al., 2006). While the assumption N/N$^+$=O/O$^+$ overestimates the N/H elemental abundance, the new icf(N/O) calculated using 1-D photoionization modeling provides a better agreement (Delgado-Inglada et al., 2014). Moreover, the O$^{2+}$/O ratio is about 1.1-1.2 higher than the Ne$^{2+}$/Ne ratio, in reasonable agreement with the assumption for the $icf$(Ne). The ionic fraction of S, Cl and Ar predicted by MC2 are approximately about the values calculated by MC1. The small discrepancies in fractional ionic abundances between MC1 and MC2 can be explained by a small fraction of the metal-rich structures included in MC2.

The volume-averaged fractional ionic abundances calculated from the model MC2 are listed in Table 10, the upper entries for each element in the table are for the normal component and the lower entries are for the metal-rich component of the nebula. It can be seen that the model MC2 predict different ionic fractions of O$^{+}$ for the two components of the nebula, whereas roughly the same value for N$^{+}$. The O$^{+}$/O ratios in the metal-rich component are about 20 percent higher than those in the normal component. This means that that the ionization correction factors (icf) from CELs are not entirely accurate for deriving the elemental abundances from ORLs as adopted by some authors (see e.g. Wang & Liu, 2007).