6 Photoionization model

Figure: 3-D isodensity plot of the dense torus adopted for photoionization modelling of SuWt 2. The torus has a homogeneous density of 100 cm${}^{-3}$, a radius of $38.1$arcsec from its centre to the tube centre, and a tube radius of $6.9$arcsec. The less dense oblate spheroid has a homogeneous density of 50 cm${}^{-3}$, a semi-major axis of $31.2$arcsec and a semi-minor axis of $6.9$arcsec. Axis units are arcsec, where 1 arcsec is equal to $1.12\times 10^{-2}$ pc based on the distance determined by our photoionization models.

We used the 3 D photoionization code MOCASSIN (version 2.02.67) to study the ring of the PN SuWt 2. The code, described in detail by Ercolano et al. (2003a); Ercolano et al. (2008); Ercolano et al. (2005), applies a Monte Carlo method to solve self-consistently the 3 D radiative transfer of the stellar and diffuse field in a gaseous and/or dusty nebula having asymmetric/symmetric density distribution and inhomogeneous/homogeneous chemical abundances, so it can deal with any structure and morphology. It also allows us to include multiple ionizing sources located in any arbitrary position in the nebula. It produces several outputs that can be compared with observations, namely a nebular emission-line spectrum, projected emission-line maps, temperature structure and fractional ionic abundances. This code has already been used for a number of axisymmetric PNe, such as NGC 3918 (Ercolano et al., 2003b), NGC 7009 (Gonçalves et al., 2006), NGC 6781 (Schwarz & Monteiro, 2006), NGC 6302 (Wright et al., 2011) and NGC 40 (Monteiro & Falceta-Gonçalves, 2011). To save computational time, we began with the gaseous model of a $22\times22\times3$ Cartesian grid, with the ionizing source being placed in a corner in order to take advantage of the axisymmetric morphology used. This initial low-resolution grid helped us explore the parameter space of the photoionization models, namely ionizing source, nebula abundances and distance. Once we found the best fitting model, the final simulation was done using a density distribution model constructed in $45 \times 45 \times 7$ cubic grids with the same size, corresponding to 14175 cubic cells of length 1 arcsec each. Due to computational restrictions on time, we did not run any model with higher number of cubic cells. The atomic data set used for the photoionization modelling, includes the CHIANTI database (version 5.2; Landi et al., 2006), the improved coefficients of the H I, He I and He II free-bound continuous emission (Ercolano & Storey, 2006) and the photoionization cross-sections and ionic ionization energies (Verner & Yakovlev, 1995; Verner et al., 1993).

Figure: Comparison of two NLTE model atmosphere fluxes (Rauch, 2003) used as ionizing inputs in our two models. Red line: H-rich model with an abundance ratio of H:He=8:2 by mass, $\log g =7$ (cgs) and $T_{\rm eff}=140\,000$ K. Blue line: PG 1159 model with He:C:N:O=33:50:2:15, $\log g =7$ and $T_{\rm eff}=160\,000$ K. Dashed green line: the flux of a blackbody with $T_{\rm eff}=160\,000$ K.

The modelling procedure consists of an iterative process during which the calculated H$\beta$ luminosity $L_{{\rm H}\beta}$(ergs${}^{-1}$), the ionic abundance ratios (i.e. He${}^{2+}$/He${}^{+}$, N${}^{+}$/H${}^{+}$, O${}^{2+}$/H${}^{+}$) and the flux intensities of some important lines, relative to H$\beta$ (such as He II $\lambda$4686, $[$N II$]$ $\lambda$6584 and $[$O III$]$ $\lambda$5007) are compared with the observations. We performed a maximum of 20 iterations per simulation and the minimum convergence level achieved was 95%. The free parameters included distance and stellar characteristics, such as luminosity and effective temperature. Although we adopted the density and abundances derived in Sections 4 and 5, we gradually scaled up/down the abundances in Table 5 until the observed emission-line fluxes were reproduced by the model. Due to the lack of infrared data we did not model the dust component of this object. We notice however some variations among the values of $c({\rm H}\beta )$ between the ring and the inner region in Table 2. It means that all of the observed reddening may not be due to the ISM. We did not include the outer bipolar lobes in our model, since the geometrical dilution reduces radiation beyond the ring. The faint bipolar lobes projected on the sky are far from the UV radiation field, and are dominated by the photodissociation region (PDR). There is a transition region between the photoionized region and PDR. Since MOCASSIN cannot currently treat a PDR fully, we are unable to model the region beyond the ionization front, i.e. the ring. This low-density PN is extremely faint, and not highly optically thick (i.e. some UV radiations escape from the ring), so it is difficult to estimate a stellar luminosity from the total nebula H$\beta$ intrinsic line flux. The best-fitting model depends upon the effective temperature ( $T_{\rm eff}$) and the stellar luminosity ($L_{\star}$), though both are related to the evolutionary stage of the central star. Therefore, it is necessary to restrict our stellar parameters to the evolutionary tracks of the post-AGB stellar models, e.g., `late thermal pulse', `very late thermal pulse' (VLTP), or `asymptotic giant branch final thermal pulse' (see e.g. Miller Bertolami et al., 2006; Blöcker, 1995; Herwig, 2001; Schönberner, 1983; Iben & Renzini, 1983; Vassiliadis & Wood, 1994). To constrain $T_{\rm eff}$ and $L_{\star}$, we employed a set of evolutionary tracks for initial masses between $1$ and $7{\rm M}_{\bigodot}$ calculated by Blöcker (1995, Tables 3-5). Assuming a density model shown in Fig. 8, we first estimated the effective temperature and luminosity of the central star by matching the H$\beta$ luminosity $L({\rm H}\beta)$ and the ionic helium abundance ratio He${}^{2+}$/He${}^{+}$ with the values derived from observation and empirical analysis. Then, we scaled up/down abundances to get the best values for ionic abundance ratios and the flux intensities.