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 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 centre, $\alpha$ 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.

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 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.

Table 9: Dereddened observed and predicted emission-line fluxes for Abell 48. References: D13 - this work; T13 - Todt et al. (2013). Uncertain and very uncertain values are followed by `:' and `::', respectively. The symbol `*' denotes blended emission lines.

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