3 Photoionization Modeling

The photoionization modeling is performed using the MOCASSIN code (version 2.02.70), described in detail by Ercolano et al. (2005); Ercolano et al. (2003a) in which the radiative transfer of the stellar and diffuse field is computed using a Monte Carlo (MC) method constructed in a cubical Cartesian grid, allowing completely arbitrary distribution of nebular density and chemical abundances. This code has already been used to study some chemical inhomogeneous models, namely the H-deficient knots of Abell 30 (Ercolano et al., 2003b) and the super-metal-rich knots of NGC 6153 (Yuan et al., 2011). To solve the problem of ORL-CEL abundance discrepancies in those PNe, they used a metal-rich component, whose ratios of heavy elements with respect to H are higher than those of the normal component.

To investigate the abundance discrepancies between the ORLs and CELs, we have constructed different photoionization models for PB 8. We run a set of model simulations, from which we finally selected three models, which best reproduced the observations. Our first model (MC1) consists of a chemically homogeneous abundance distribution. Our second model (MC2) is roughly similar, but it includes some dense metal-rich knots (cells) embedded in the density model of normal abundances (see Section3.2). The final model (MC3) includes dust grains to match the Spitzer IR observation (see Section3.4). The atomic data sets used for our models include energy levels, collision strengths, and transition probabilities from Version 7.0 of the CHIANTI database (Landi et al., 2012, and references therein), hydrogen and helium free-bound coefficients of Ercolano & Storey (2006), and opacities from Verner et al. (1993) and Verner & Yakovlev (1995).

The model parameters, as well as the physical properties for the models, are summarized in Table 3, and discussed in more detail in the following sections. The modeling procedure consists of an iterative process, involving the comparison of the predicted emission line fluxes with the values measured from the observations, and the ionization and thermal structures with the values derived from the empirical analysis. The free parameters used in our models should be the nebular abundances, as the nebular density is adopted based on empirical results (García-Rojas et al., 2009), and the stellar parameters based on the model atmosphere study (Todt et al., 2010). However, it is impossible to isolate effects of any parameter from each other, as they are dependent on each other, so we cannot just modify the nebular abundances without slightly adjusting the density distribution and the distance. The nebular ionization structure depends on the gas density and the stellar characteristics, so we fairly adjusted them to obtain the best-fitting models. We adopted the effective temperature of $T_{\rm eff}=52$ kK, stellar luminosity of $L_{\star}=6000{\rm L}_{\odot}$ , and non-local thermodynamic equilibrium (non-LTE) model atmosphere determined by Todt et al. (2010). The optical depth for Lyman-continuum radiation of $\tau($ Ly-c

estimated by Lenzuni et al. (1989) indicated some ionizing radiation fields may escape from the nebular shell, so it could be a matter-bounded PN. Therefore, we attempted to adjust distance and gas density to reproduce the nebular total H $\beta$ intrinsic line flux. It is found that a model with an electron density of about $2550\pm550$ cm $^{-3}$ empirically derived by García-Rojas et al. (2009) can well reproduce the nebular H $\beta$ intrinsic line flux at a distance of 4.9 kpc. We initially used the elemental abundances determined by García-Rojas et al. (2009), but we adjusted them to match the observed nebular spectrum.

**Table 3:** Model parameters and physical properties for the final photoionization models.
	Empirical		Models
		MC1	MC2			MC3
Parameter	CEL	ORL		Normal	Metal-rich	Total	Normal	Metal-rich	Total
$T_{\rm eff}$ (kK)	52		52	52			52
$L_{\star}$ (L $_{\odot}$ )	6000		6000	6000			6000
$R_{\rm in}$ ( $10^{17}$ cm)	-		0.8	0.8			0.8
$R_{\rm out}$ ( $10^{17}$ cm)	-		2.6	2.6			2.6
Filling factor	-		1.000	0.944	0.056	1.000	0.944	0.056	1.000
$\langle$ ${\it N}{\rm (H^{+})}$ $\rangle$ (cm $^{-3}$ )	-		2009	1957	3300	2032	1957	3300	2032
$\langle$ ${\it N}_{\rm e}$ $\rangle$ (cm $^{-3}$ )	$2550\pm550$		2257	2199	4012	2301	2199	4012	2301
$\rho_{\rm d}/\rho_{\rm g}$	-		-			-			0.01
He/H	-	0.122	0.122	0.122	0.20	0.129	0.122	0.20	0.129
C/H $\times$ 10	-	72.25	63.0	63.0	63.0	63.0	63.0	63.0	63.0
N/H $\times$ 10	16.22	31.41 $^{\mathrm{a}}$	11.0	6.1	298.0 $^{\mathrm{a}}$	32.7 $^{\mathrm{a}}$	6.1	298.0 $^{\mathrm{a}}$	32.7 $^{\mathrm{a}}$
O/H $\times$ 10	57.54	146.61 $^{\mathrm{a}}$	40.0	58.7	551.0 $^{\mathrm{a}}$	103.5 $^{\mathrm{a}}$	58.7	551.0 $^{\mathrm{a}}$	103.5 $^{\mathrm{a}}$
Ne/H $\times$ 10	13.49	19.9 $^{\mathrm{a}}$	10.0	15.0	15.0 $^{\mathrm{a}}$	15.0 $^{\mathrm{a}}$	15.0	15.0 $^{\mathrm{a}}$	15.0 $^{\mathrm{a}}$
S/H $\times$ 10	204.17	-	300.0	300.0	300.0	300.0	300.0	300.0	300.0
Cl/H $\times$ 10	2.0	-	1.2	1.6	1.6	1.6	1.6	1.6	1.6
Ar/H $\times$ 10	43.65	-	39.0	45.0	45.0	45.0	45.0	45.0	45.0

$^{\mathrm{a}}$: The ORL empirical abundances were calculated from the ORLs over the total H $^{+}$ emission flux, emitted from both the diffuse gas and metal-rich inclusion, so the empirical abundances of the ORLs cannot be the same as the model metal-rich component, but roughly similar to the mean total abundances of both the metal-rich and normal components.