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 dierent (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. H ;H eii4686; (O iii)5007). Particular eort has been put into the analytical description of dynamical eects 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 ndings 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 dierences that are expected in the luminosities and lifetimes of PNe with either H- or He-burning central stars, which result in dierent \detection probabilities" across the H-R diagram, in both H and (Oiii)5007 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 ex- pansion velocities. The models naturally predict also the possible transitions from optically-thick to optically-thin congurations (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 aects the mass-increasing part of the ionised mass-radius relation. These predictions are supported by observational indications by M endez et al. (1992). Another interesting result is that the change of slope in the electron density{nebular radius relation at Rion 0:1 pc, pointed out by Phillips (1998), 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.