7 Conclusion

We have constructed a photoionization model for the nebula of Abell 48. This consists of a dense hollow cylinder, assuming homogeneous abundances. The three-dimensional density distribution was interpreted using the morpho-kinematic model determined from spatially resolved kinematic maps and the ISW model. Our aim was to construct a model that can reproduce the nebular emission-line spectra, temperatures and ionization structure determined from the observations. We have used the non-LTE model atmosphere from Todt et al. (2013) as the ionizing source. Using the empirical analysis methods, we have determined the temperatures and the elemental abundances from CELs and ORLs. We notice a discrepancy between temperatures estimated from $[$O III$]$ CELs and those from the observed HeI ORLs. In particular, the abundance ratios derived from empirical analysis could also be susceptible to inaccurate values of electron temperature and density. However, we see that the predicted ionic abundances are in decent agreement with those deduced from the empirical analysis. The emission-line fluxes obtained from the model were in fair agreement with the observations.

We notice large discrepancies between HeI electron temperatures derived from the model and the empirical analysis. The existence of clumps and low-ionization structures could solve the problems (Liu et al., 2000). Temperature fluctuations have been also proposed to be responsible for the discrepancies in temperatures determined from CELs and ORLs (Peimbert, 1967; Peimbert, 1971). Previously, we also saw large ORL-CEL abundance discrepancies in other PNe with hydrogen-deficient CSs, for example Abell 30 (Ercolano et al., 2003a) and NGC 1501 (Ercolano et al., 2004). A fraction of H-deficient inclusions might produce those discrepancies, which could be ejected from the stellar surface during a very late thermal pulse (VLTP) phase or born-again event (Iben & Renzini, 1983). However, the VLTP event is expected to produce a carbon-rich stellar surface abundance (Herwig, 2001), whereas in the case of Abell 48 negligible carbon was found at the stellar surface (C/He =  $3.5\times10^{-3}$ by mass; Todt et al., 2013). The stellar evolution of Abell 48 still remains unclear and needs to be investigated further. But, its extreme helium-rich atmosphere (85 per cent by mass) is more likely connected to a merging process of two white dwarfs as evidenced for R Cor Bor stars of similar chemical surface composition by observations (Clayton et al., 2007; García-Hernández et al., 2009) and hydrodynamic simulations (Zhang & Jeffery, 2012; Staff et al., 2012; Menon et al., 2013).

We derived a nebula ionized mass of $\sim0.8$ M $_{\bigodot}$. The high C/O ratio indicates that it is a predominantly C-rich nebula. The C/H ratio is largely over-abundant compared to the solar value of Asplund et al. (2009), while oxygen, sulphur and argon are under-abundant. Moreover, nitrogen and neon are roughly similar to the solar values. Assuming a sub-solar metallicity progenitor, a 3rd dredge-up must have enriched carbon and nitrogen in AGB-phase. However, extremely high carbon must be produced through mixing processing in the He-rich layers during the He-shell flash. The low N/O ratio implies that the progenitor star never went through the hot bottom burning phase, which occurs in AGB stars with initial masses more than 5M $_{\bigodot}$ (Karakas & Lattanzio, 2007; Karakas et al., 2009). Comparing the stellar parameters found by the model, $T_{\rm eff}$=70kK and $L_{\rm\star}/$L $_{\bigodot}$= 5500, with VLTP evolutionary tracks from Blöcker (1995), we get a current mass of $\sim 0.62 {\rm M}_{\bigodot}$, which originated from a progenitor star with an initial mass of $\sim 3 {\rm M}_{\bigodot}$. However, the VLTP evolutionary tracks by Miller Bertolami & Althaus (2006) yield a current mass of $\sim 0.52 {\rm M}_{\bigodot}$ and a progenitor mass of $\sim1{\rm M}_{\bigodot}$, which is not consistent with the derived nebula ionized mass. Furthermore, time-scales for VLTP evolutionary track (Blöcker, 1995) imply that the CS has a post-AGB age of about $\sim$9000 yr, in agreement with the nebula's age determined from the kinematic analysis. We therefore conclude that Abell 48 originated from an $\sim3$ M $_{\bigodot}$ progenitor, which is consistent with the nebula's features.