Evolution of Planetary Nebulae I. An improved synthetic model

We present a new synthetic model to follow the evolution of a planetary nebula (PN) and its central star, starting from the onset of AGB phase up to the white dwarf cooling sequence. The model suitably combines various analytical prescriptions to account for different (but inter-related) aspects of planetary nebulae, such as: the dynamical evolution of the primary shell and surrounding ejecta, the photoionisation of H and He by the central star, the nebular emission of a few relevant optical lines (e.g. Hβ ; He II lambda 4686; [O III] lambda 5007). Particular effort has been put into the analytical description of dynamical effects such as the three-winds interaction and the shell thickening due to ionisation (i.e. the thin-shell approximation is relaxed), that are nowadays considered important aspects of the PN evolution. Predictions of the synthetic model are tested by comparison with both findings of hydrodynamical calculations, and observations of Galactic PNe. The sensitiveness of the results to the model parameters (e.g. transition time, mass of the central star, H-/He-burning tracks, etc.) is also discussed. We briefly illustrate the systematic differences that are expected in the luminosities and lifetimes of PNe with either H- or He-burning central stars, which result in different ``detection probabilities'' across the H-R diagram, in both Hβ and [OIII] lambda5007 lines. Adopting reasonable values of the model parameters, we are able to reproduce, in a satisfactory way, many general properties of PNe, like the ionised mass-nebular radius relationship, the trends of a few main nebular line ratios, and the observed ranges of nebular shell thicknesses, electron densities, and expansion velocities. The models naturally predict also the possible transitions from optically-thick to optically-thin configurations (and vice versa). In this context, our analysis indicates that the condition of optical thinness to the H continuum plays an important role in producing the observed ``Zanstra discrepancy'' between the temperatures determined from H or He II lines, as well as it affects the mass-increasing part of the ionised mass-radius relation. These predictions are supported by observational indications by Méndez et al. (\cite{Mendez92}). Another interesting result is that the change of slope in the electron density-nebular radius relation at R_ion ~ 0.1 pc, pointed out by Phillips (\cite{Phillip98}), is also displayed by the models and may be interpreted as the result of the progressive convergence of the PNe to the condition of constant ionised mass. Finally we would like to remark that, thanks to its computational agility, our synthetic PN model is particularly suitable to population synthesis studies, and it represents the basic ground from which many future applications will be developed.