6.1.3 Ionizing source

The central ionizing source of SuWt 2 was modelled using different non-local thermodynamic equilibrium (NLTE; Rauch, 2003) model atmospheres listed in Table 6, as they resulted in the best fit of the nebular emission-line fluxes. Initially, we tested a set of blackbody fluxes with the effective temperature ( $T_{\rm eff}$) ranging from $80\,000$ to $190\,000$ K, the stellar luminosity compared to that of the Sun ( $L_{\star}/{\rm L}_{\bigodot}$) ranging from 50-800 and different evolutionary tracks (Blöcker, 1995). A blackbody spectrum provides a rough estimate of the ionizing source required to photoionize the PN SuWt 2. The assumption of a blackbody spectral energy distribution (SED) is not quite correct as indicated by Rauch (2003). The strong differences between a blackbody SED and a stellar atmosphere are mostly noticeable at energies higher than 54 eV (He II ground state). We thus successively used the NLTE Tübingen Model-Atmosphere Fluxes Package⁴ (TMAF; Rauch, 2003) for hot compact stars. We initially chose the stellar temperature and luminosity (gravity) of the best-fitting blackbody model, and changed them to get the best observed ionization properties of the nebula.

Fig. 9 shows the NTLE model atmosphere fluxes used to reproduce the observed nebular emission-line spectrum by our photoionization models. We first used a hydrogen-rich model atmosphere with an abundance ratio of H:He=8:2 by mass, $\log g =7$ (cgs), and $T_{\rm eff}=140\,000$ K (Model 1), corresponding to the final stellar mass of $M_{\star}=0.605\,{\rm M}_{\bigodot}$ and the zero-age main sequence (ZAMS) mass of $M_{\rm\textsc{zams}}=3\,{\rm M}_{\bigodot}$, where ${\rm M}_{\bigodot}$ is the solar mass. However, its post-AGB age ( $\tau _{\rm post-\textsc {agb}}$) of 7500yr, as shown in Fig. 10 (left-hand panel), is too short to explain the nebula's age. We therefore moved to a hydrogen-deficient model, which includes Wolf-Rayet central stars ([WC]) and the hotter PG 1159 stars. [WC]-type central stars are mostly associated with carbon-rich nebula (Zijlstra et al., 1994). The evolutionary tracks of the VLTP for H-deficient models, as shown in Fig.10 (right-hand panel), imply a surface gravity of $\log g= 7.2$ for given $T_{\rm eff}$ and $L_{\star}$. From the high temperature and high surface gravity, we decided to use `typical' PG 1159 model atmosphere fluxes (He:C:N:O=33:50:2:15) with $T_{\rm eff}=160\,000$ K and $L_{\star}/{\rm L}_{\bigodot}=600$ (Model 2), corresponding to the post-AGB age of about $\tau_{\rm post-\textsc{agb}}=2$5000yr, $M_{\star}=0.64\,{\rm M}_{\bigodot}$ and $M_{\rm\textsc{zams}}=3\,{\rm M}_{\bigodot}$. The stellar mass found here is in agreement with the $0.7\,{\rm M}_{\bigodot}$ estimate of Exter et al. (2010). Fig.9 compares the two model atmosphere fluxes with a blackbody with $T_{\rm eff}=160\,000$ K.

Table 6 lists the parameters used for our final simulations in two different NTLE model atmosphere fluxes. The ionization structure of this nebula was best reproduced using these best two models. Each model has different effective temperature, stellar luminosity and abundances (N/H, O/H and Ne/H). The results of our two models are compared in Tables 7-10 to those derived from the observation and empirical analysis.

Table 8: Mean electron temperatures (K) weighted by ionic species for the whole nebula obtained from the photoionization model. For each element the upper row is for Model 1 and the lower row is for Model 2. The bottom lines present the mean electron temperatures and electron densities for the torus (ring) and the oblate spheroid (inside).

	Ion
Element	I	II	III	IV	V	VI	VII
H	11696	12904
	11470	12623
He	11628	12187	13863
	11405	11944	13567
C	11494	11922	12644	15061	17155	17236	12840
	11289	11696	12405	14753	16354	16381	12550
N	11365	11864	12911	14822	16192	18315	18610
	11170	11661	12697	14580	15836	17368	17475
O	11463	11941	12951	14949	15932	17384	20497
	11283	11739	12744	14736	15797	17559	19806
Ne	11413	11863	12445	14774	16126	18059	22388
	11196	11631	12215	14651	16166	18439	20488
S	11436	11772	12362	14174	15501	16257	18313
	11239	11557	12133	13958	15204	15884	17281
Ar	11132	11593	12114	13222	14908	15554	16849
	10928	11373	11894	13065	14713	15333	16392
	Torus				Spheroid
	$T_{\rm e}[$OIII$]$		$N_{\rm e}[$SII$]$		$T_{\rm e}[$OIII$]$		$N_{\rm e}[$SII$]$
M.1	12187K		105cm${}^{-3}$		15569K		58cm${}^{-3}$
M.2	11916K		103cm${}^{-3}$		15070K		58cm${}^{-3}$

Table 9: Fractional ionic abundances for SuWt 2 obtained from the photoionization models. For each element the upper row is for the torus (ring) and the lower row is for the oblate spheroid (inside).

		Ion
	Element	I	II	III	IV	V	VI	VII
Model 1	H	6.53($-2$)	9.35($-1$)
		3.65($-3$)	9.96($-1$)
	He	1.92($-2$)	7.08($-1$)	2.73($-1$)
		3.05($-4$)	1.27($-1$)	8.73($-1$)
	C	5.92($-3$)	2.94($-1$)	6.77($-1$)	2.33($-2$)	1.86($-4$)	7.64($-16$)	1.00($-20$)
		3.49($-5$)	1.97($-2$)	3.97($-1$)	4.50($-1$)	1.33($-1$)	1.09($-12$)	1.00($-20$)
	N	7.32($-3$)	4.95($-1$)	4.71($-1$)	2.62($-2$)	4.18($-4$)	6.47($-6$)	2.76($-17$)
		1.02($-5$)	1.30($-2$)	3.65($-1$)	3.97($-1$)	1.59($-1$)	6.69($-2$)	6.89($-13$)
	O	6.15($-2$)	4.98($-1$)	4.21($-1$)	1.82($-2$)	7.09($-4$)	1.34($-5$)	7.28($-8$)
		6.96($-5$)	1.26($-2$)	3.31($-1$)	4.03($-1$)	1.69($-1$)	6.00($-2$)	2.42($-2$)
	Ne	3.46($-4$)	6.70($-2$)	9.10($-1$)	2.26($-2$)	3.56($-4$)	4.25($-6$)	2.11($-9$)
		1.39($-6$)	3.32($-3$)	3.71($-1$)	3.51($-1$)	2.05($-1$)	6.55($-2$)	4.49($-3$)
	S	1.13($-3$)	1.67($-1$)	7.75($-1$)	5.52($-2$)	1.15($-3$)	6.20($-5$)	8.53($-7$)
		3.18($-6$)	3.89($-3$)	1.73($-1$)	3.53($-1$)	2.43($-1$)	1.57($-1$)	6.91($-2$)
	Ar	4.19($-4$)	3.15($-2$)	7.51($-1$)	2.10($-1$)	5.97($-3$)	1.13($-3$)	5.81($-5$)
		1.12($-7$)	2.33($-4$)	5.81($-2$)	2.83($-1$)	1.85($-1$)	2.73($-1$)	2.01($-1$)
Model 2	H	7.94($-2$)	9.21($-1$)
		4.02($-3$)	9.96($-1$)
	He	2.34($-2$)	7.25($-1$)	2.51($-1$)
		3.51($-4$)	1.33($-1$)	8.67($-1$)
	C	7.97($-3$)	3.23($-1$)	6.49($-1$)	1.93($-2$)	1.29($-4$)	5.29($-16$)	1.00($-20$)
		4.45($-5$)	2.23($-2$)	4.13($-1$)	4.41($-1$)	1.23($-1$)	1.00($-12$)	1.00($-20$)
	N	1.00($-2$)	5.44($-1$)	4.24($-1$)	2.15($-2$)	2.62($-4$)	2.20($-6$)	9.23($-18$)
		1.31($-5$)	1.52($-2$)	3.84($-1$)	4.07($-1$)	1.50($-1$)	4.40($-2$)	4.34($-13$)
	O	7.91($-2$)	5.29($-1$)	3.78($-1$)	1.40($-2$)	4.27($-4$)	2.05($-6$)	6.62($-11$)
		9.34($-5$)	1.50($-2$)	3.60($-1$)	4.20($-1$)	1.75($-1$)	2.97($-2$)	1.85($-4$)
	Ne	4.54($-4$)	7.35($-2$)	9.09($-1$)	1.73($-2$)	1.41($-4$)	1.94($-8$)	2.25($-14$)
		1.75($-6$)	3.85($-3$)	4.19($-1$)	3.86($-1$)	1.89($-1$)	1.73($-3$)	6.89($-7$)
	S	1.64($-3$)	1.95($-1$)	7.58($-1$)	4.47($-2$)	7.84($-4$)	3.39($-5$)	3.05($-7$)
		4.23($-6$)	4.86($-3$)	1.96($-1$)	3.61($-1$)	2.39($-1$)	1.47($-1$)	5.16($-2$)
	Ar	7.22($-4$)	3.99($-2$)	7.74($-1$)	1.81($-1$)	3.95($-3$)	5.62($-4$)	1.60($-5$)
		1.72($-7$)	3.22($-4$)	7.30($-2$)	3.30($-1$)	1.96($-1$)	2.62($-1$)	1.39($-1$)

Table 10: Integrated ionic abundance ratios for the entire nebula obtained from the photoionization models.

		Model 1		Model 2
Ionic ratio	Empirical	Abundance	Ionic Fraction	Abundance	Ionic Fraction
He${}^{+}$/H${}^{+}$	4.80($-2$)	5.308($-2$)	58.97%	5.419($-2$)	60.21%
He${}^{2+}$/H${}^{+}$	3.60($-2$)	3.553($-2$)	39.48%	3.415($-2$)	37.95%
C${}^{+}$/H${}^{+}$	-	9.597($-5$)	23.99%	1.046($-4$)	26.16%
C${}^{2+}$/H${}^{+}$	-	2.486($-4$)	62.14%	2.415($-4$)	60.38%
N${}^{+}$/H${}^{+}$	1.309($-4$)	9.781($-5$)	40.09%	1.007($-4$)	43.58%
N${}^{2+}$/H${}^{+}$	-	1.095($-4$)	44.88%	9.670($-5$)	41.86%
N${}^{3+}$/H${}^{+}$	-	2.489($-5$)	10.20%	2.340($-5$)	10.13%
O${}^{+}$/H${}^{+}$	1.597($-4$)	1.048($-4$)	40.30%	1.201($-4$)	42.44%
O${}^{2+}$/H${}^{+}$	1.711($-4$)	1.045($-4$)	40.20%	1.065($-4$)	37.64%
O${}^{3+}$/H${}^{+}$	-	2.526($-5$)	9.72%	2.776($-5$)	9.81%
Ne${}^{+}$/H${}^{+}$	-	6.069($-6$)	5.47%	6.571($-6$)	5.92%
Ne${}^{2+}$/H${}^{+}$	1.504($-4$)	8.910($-5$)	80.27%	9.002($-5$)	81.10%
Ne${}^{3+}$/H${}^{+}$	-	1.001($-5$)	9.02%	1.040($-5$)	9.37%
S${}^{+}$/H${}^{+}$$^{a}$	2.041($-6$)	2.120($-7$)	13.50%	2.430($-7$)	15.48%
S${}^{2+}$/H${}^{+}$	1.366($-6$)	1.027($-6$)	65.44%	1.013($-6$)	64.55%
S${}^{3+}$/H${}^{+}$	-	1.841($-5$)	11.73%	1.755($-7$)	11.18%
Ar${}^{+}$/H${}^{+}$	-	3.429($-8$)	2.54%	4.244($-8$)	3.14%
Ar${}^{2+}$/H${}^{+}$	1.111($-6$)	8.271($-7$)	61.26%	8.522($-7$)	63.13%
Ar${}^{3+}$/H${}^{+}$	4.747($-7$)	3.041($-7$)	22.52%	2.885($-7$)	21.37%
Ar${}^{4+}$/H${}^{+}$	-	5.791($-8$)	4.29%	5.946($-8$)	4.40%
Ar${}^{5+}$/H${}^{+}$	-	7.570($-8$)	5.61%	7.221($-8$)	5.35%

$^{a}$ Shock excitation largely enhances the S${}^{+}$/H${}^{+}$ ionic abundance ratio.