5 Photoionization modeling

The 3-D photoionization code MOCASSIN (version 2.02.67; Ercolano et al., 2008; Ercolano et al., 2005; Ercolano et al., 2003b) was used to study the best-fitting model for Abell48. The code has been used to model a number of PNe, for example NGC 3918 (Ercolano et al., 2003a), NGC 7009 (Gonçalves et al., 2006), NGC 6302 (Wright et al., 2011), and SuWt 2 (Danehkar et al., 2013). The modeling procedure consists of defining the density distribution and elemental abundances of the nebula, as well as assigning the ionizing spectrum of the central star. This code uses a Monte Carlo method to solve self-consistently the 3-D radiative transfer of the stellar radiation field in a gaseous nebula with the defined density distribution and chemical abundances. It produces the emission-line spectrum, the thermal structure and the ionization structure of the nebula. It allows us to determine the stellar characteristics and the nebula parameters. The atomic data sets used for the calculation are energy levels, collision strengths and transition probabilities from the CHIANTI database (version 5.2; Landi et al., 2006), hydrogen and helium free-bound coefficients of Ercolano & Storey (2006), and opacities from Verner et al. (1993) and Verner & Yakovlev (1995).

The best-fitting model was obtained through an iterative process, involving the comparison of the predicted H$\beta$ luminosity $L_{{\rm H}\beta}$ (ergs${}^{-1}$ ), the flux intensities of some important lines, relative to H$\beta$ (such as $[$ O III$]$ $\lambda$ 5007 and $[$ N II$]$ $\lambda$ 6584), with those measured from the observations. The free parameters included distance and nebular parameters. We initially used the stellar luminosity ( $L_{\star}=6000$ L $_{\bigodot}$ ) and effective temperature ( $T_{\rm eff}=70$ kK) found by Todt et al. (2013). However, we slightly adjusted the stellar luminosity to match the observed line flux of $[$ O III$]$ emission line. Moreover, we adopted the nebular density and abundances derived from empirical analysis in Section 4, but they have been gradually adjusted until the observed nebular emission-line spectrum was reproduced by the model. The best-fitting $L_{{\rm H}\beta}$ depends upon the distance and nebula density. The plasma diagnostics yields $N_{\rm e} = 750$ -1000cm$^{-3}$ , which can be an indicator of the density range. Based on the kinematic analysis, the distance must be less than 2 kpc, but more than 1.5 kpc due to the large interstellar extinction. We matched the predicted H$\beta$ luminosity $L({\rm H}\beta)$ with the value derived from the observation by adjusting the distance and nebular density. Then, we adjusted abundances to get the best emission-line spectrum.

5.1 The ionizing spectrum

The hydrogen-deficient synthetic spectra of Abell48 was modeled using stellar model atmospheres produced by the Potsdam Wolf-Rayet (PoWR) models for expanding atmospheres (Gräfener et al., 2002; Hamann & Gräfener, 2004). It solves the non-local thermodynamic equilibrium (non-LTE) radiative transfer equation in the co-moving frame, iteratively with the equations of statistical equilibrium and radiative equilibrium, for an expanding atmosphere under the assumptions of spherical symmetry, stationarity and homogeneity. The result of our model atmosphere is shown in Fig.5. The model atmosphere calculated with the PoWR code is for the stellar surface abundances H:He:C:N:O = 10:85:0.3:5:0.6 by mass, the stellar temperature $T_{\rm eff}$ =70kK, the transformed radius $R_{\rm t}=0.54$ R ${}_{\bigodot}$ and the wind terminal velocity $v_{\infty}=1000$  kms$^{-1}$ . The best photoionization model was obtained with an effective temperature of 70kK (the same as PoWR model used by Todt et al., 2013) and a stellar luminosity of $L_{\rm\star}/$ L $_{\bigodot}$ = 5500, which is close to $L_{\star}/$ L $_{\bigodot}$ = 6000 adopted by Todt et al. (2013). This stellar luminosity was found to be consistent with the observed H$\beta$ luminosity and the flux ratio of $[$ O III$]$ /H$\beta$ . A stellar luminosity higher than 5500L $_{\bigodot}$ produces inconsistent results for the nebular photoionization modeling. The emission-line spectrum produced by our adopted stellar parameters were found to be consistent with the observations.

Table 8: Input parameters for the MOCASSIN photoionization model.

Stellar and Nebular			Nebular Abundances
Parameters			Model	Obs.
$T_{\rm eff}$ (kK)	70	He/H	0.120	0.124
$L_{\rm\star}$ (L $_{\bigodot}$ )	5500	C/H  $\times 10^{3}$	3.00	-
$N_{\rm H}$ (cm$^{-3}$ )	800-1200	N/H  $\times 10^{5}$	6.50	4.30
$D$ (kpc)	1.9	O/H  $\times 10^{4}$	1.40	1.59
$r_{\rm out}$ (arcsec)	23	Ne/H  $\times 10^{5}$	6.00	6.36
$\delta r$ (arcsec)	13	S/H  $\times 10^{6}$	6.00	6.73
$h$ (arcsec)	23	Ar/H  $\times 10^{6}$	1.20	1.48

5.2 The density distribution

We initially used a three-dimensional uniform density distribution, which was developed from our kinematic analysis. However, the interacting stellar winds (ISW) model developed by Kwok et al. (1978) demonstrated that a slow dense superwind from the AGB phase is swept up by a fast tenuous wind during the PN phase, creating a compressed dense shell, which is similar to what we see in Fig. 6. Additionally, Kahn & West (1985) extended the ISW model to describe a highly elliptical mass distribution. This extension later became known as the generalized interacting stellar winds (GISW) theory. There are a number of hydrodynamic simulations, which showed the applications of the ISW theory for bipolar PNe (see e.g. Mellema, 1997; Mellema, 1996). As shown in Fig. 6, we adopted a density structure with a toroidal wind mass-loss geometry, similar to the ISW model. In our model, we defined a density distribution in the cylindrical coordinate system, which has the form $N_{\rm H}(r) = N_{0}[ 1 + (r/r_{\rm in})^{-\alpha} ],$ where $r$ is the radial distance from the center, $\alpha$ the radial density dependence, $N_{0}$ the characteristic density, $r_{\rm in} = r_{\rm out}-\delta r$ the inner radius, $r_{\rm out}$ the outer radius and $\delta r$ the thickness.

Figure: Non-LTE model atmosphere flux (solid line) calculated with the PoWR models for the surface abundances H:He:C:N:O = 10:85:0.3:5:0.6 by mass and the stellar temperature $T_{\rm eff}$ =70kK, compared with a blackbody (dashed line) at the same temperature.

The density distribution is usually a complicated input parameter to constrain. However, the values found from our plasma diagnostics ( $N_{\rm e} = 750$ -1000 cm$^{-3}$ ) allowed us to constrain our density model. The outer radius and the height of the cylinder are equal to $r_{\rm out}=23\hbox{$^{\prime\prime}$}$ and the thickness is $\delta r=13\hbox{$^{\prime\prime}$}$ . The density model and distance (size) were adjusted in order to reproduce $I$ (H $\beta)=1.355 \times 10^{-10}$ ergs$^{-1}$ cm$^{-2}$ , dereddened using c(H$\beta$ )=3.1 (see Section 2). We tested distances, with values ranging from 1.5 kpc to 2.0kpc. We finally adopted the characteristic density of $N_{0}=600$ cm$^{-3}$ and the radial density dependence of $\alpha=1$ . The value of 1.90kpc found here, was chosen, because of the best predicted H$\beta$ luminosity, and it is in excellent agreement with the distance constrained by the synthetic spectral energy distribution (SED) from the PoWR models. Once the density distribution and distance were identified, the variation of the nebular ionic abundances were explored.

Figure 6: The density distribution based on the ISW models adopted for photoionization modelling of Abell 48. The cylinder has outer radius of $23\hbox {$^{\prime \prime }$}$ and thickness of $13\hbox {$^{\prime \prime }$}$ . Axis units are arcsec, where 1 arcsec is equal to $9.30\times 10^{-3}$ pc based on the distance determined by our photoionization model.
(A color version of this figure is available in the online journal.)

Table 9: Dereddened observed and predicted emission lines fluxes for Abell 48. References: D13 - This work; T13 - Todt et al. (2013).

Line	Observed		Predicted
	D13	T13
$I$ (H$\beta$ )/10 $^{-10}\,\frac{\rm erg}{\rm cm^{2}s}$	1.355	-	1.371
H$\beta$ 4861	100.00	100.00	100.00
H$\alpha$ 6563	286.00	290.60	285.32
H$\gamma$ 4340	54.28:	45.10	46.88
H$\delta$ 4102	-	-	25.94
He I 4472	7.42:	-	6.34
He I 5876	18.97	20.60	17.48
He I 6678	5.07	4.80	4.91
He I 7281	0.58::	0.70	0.97
He II 4686	-	-	0.00
C II 6462	0.38	-	0.27
C II 7236	1.63	-	1.90
$[$ N II$]$ 5755	0.43::	0.40	1.20
$[$ N II$]$ 6548	26.09	28.20	26.60
$[$ N II$]$ 6584	87.28	77.00	81.25
$[$ O II$]$ 3726	128.96:	-	59.96
$[$ O II$]$ 3729	*	-	43.54
$[$ O II$]$ 7320	-	0.70	2.16
$[$ O II$]$ 7330	-	0.60	1.76
$[$ O III$]$ 4363	-	3.40	2.30
$[$ O III$]$ 4959	99.28	100.50	111.82
$[$ O III$]$ 5007	319.35	316.50	333.66
$[$ Ne III$]$ 3869	38.96	-	39.60
$[$ Ne III$]$ 3967	-	-	11.93
$[$ S II$]$ 4069	-	-	1.52
$[$ S II$]$ 4076	-	-	0.52
$[$ S II$]$ 6717	7.44	5.70	10.30
$[$ S II$]$ 6731	7.99	6.80	10.57
$[$ S III$]$ 6312	0.60::	-	2.22
$[$ S III$]$ 9069	19.08	-	16.37
$[$ Ar III$]$ 7136	10.88	10.20	12.75
$[$ Ar III$]$ 7751	4.00::	-	3.05
$[$ Ar IV$]$ 4712	-	-	0.61
$[$ Ar IV$]$ 4741	-	-	0.51

5.3 The nebular elemental abundances

Table 8 lists the nebular elemental abundances (with respect to H) used for the photoionization model. We used a homogeneous abundance distribution, since we do not have any direct observational evidence for the presence of chemical inhomogeneities. Initially, we used the abundances from empirical analysis as initial values for our modeling (see Section 4). They were successively modified to fit the optical emission-line spectrum through an iterative process. We obtain a C/O ratio of 21 for Abell48, indicating that it is predominantly C-rich. Furthermore, we find a helium abundance of 0.12. This can be an indicator of a large amount of mixing processing in the He-rich layers during the He-shell flash leading to an increase carbon abundance. The nebulae around H-deficient central stars typically have larger carbon abundances than those with H-rich central stars (see review by De Marco & Barlow, 2001). The ${\rm O}/{\rm H}$ we derive for Abell48 is lower than the solar value ( ${\rm O}/{\rm H}=4.57\times 10^{-4}$ ; Asplund et al., 2009). This may be due to that the progenitor has a sub-solar metallicity. The enrichment of carbon can be produced in a very intense mixing process in the He-shell flash (Herwig et al., 1997). Other elements seem to be also decreased compared to the solar values, such as sulfur and argon. Sulfur could be depleted onto dust grains (Sofia et al., 1994), but argon cannot have any strong depletion by dust formation (Sofia & Jenkins, 1998). We notice that the N/H ratio is about the solar value given by Asplund et al. (2009), but it can be produced by secondary conversion of initial carbon if we assume a sub-solar metallicity progenitor. The combined (C+N+O)/H ratio is by a factor of 3.9 larger than the solar value, which can be produced by multiple dredge-up episodes occurring in the AGB phase.

Figure 7: Top: Electron density and temperature as a function of radius along the equatorial direction. Bottom: Ionic stratification of the nebula. Ionization fractions are shown for helium, carbon, oxygen, argon (left panel), nitrogen, neon and sulfur (right panel).
(A color version of this figure is available in the online journal.)

Table: Fractional ionic abundances for Abell48 obtained from the photoionization model.

	Ion
Element	I	II	III	IV	V	VI	VII
H	3.84($-2$ )	9.62($-1$ )
He	3.37($-2$ )	9.66($-1$ )	1.95($-6$ )
C	5.43($-4$ )	1.73($-1$ )	8.18($-1$ )	8.93($-3$ )	1.64($-15$ )	1.00($-20$ )	1.00($-20$ )
N	1.75($-2$ )	1.94($-1$ )	7.79($-1$ )	8.98($-3$ )	2.72($-15$ )	1.00($-20$ )	1.00($-20$ )
O	4.32($-2$ )	2.60($-1$ )	6.97($-1$ )	1.18($-7$ )	3.09($-20$ )	1.00($-20$ )	1.00($-20$ )
Ne	9.94($-3$ )	3.88($-1$ )	6.03($-1$ )	1.12($-13$ )	1.00($-20$ )	1.00($-20$ )	1.00($-20$ )
S	6.56($-5$ )	8.67($-2$ )	6.99($-1$ )	2.12($-1$ )	2.42($-3$ )	1.66($-15$ )	1.00($-20$ )
Ar	2.81($-3$ )	3.74($-2$ )	8.43($-1$ )	1.17($-1$ )	1.02($-13$ )	1.00($-20$ )	1.00($-20$ )