Evolution of planetary nebulae - I. An improved synthetic model

We present a new synthetic model to follow the evolution of a planetary neb ula (PN) and its central star, starting from the onset of AGB phase up to t he white dwarf cooling sequence. The model suitably combines various analyt ical prescriptions to account for different (but inter-related) aspects of planetary nebulae, such as: the dynamical evolution of the primary shell an d surrounding ejecta, the photoionisation of H and He by the central star, the nebular emission of a few relevant optical lines (e.g. H beta; He II la mbda 4686; [O III] lambda 5007). Particular effort has been put into the an alytical description of dynamical effects such as the three-winds interacti on and the shell thickening due to ionisation (i.e. the thin-shell approxim ation is relaxed), that are nowadays considered important aspects of the PN evolution. Predictions of the synthetic model are tested by comparison wit h both findings of hydrodynamical calculations, and observations of Galacti c PNe. The sensitiveness of the results to the model parameters (e.g. trans ition time, mass of the central star, H-/He-burning tracks, etc.) is also d iscussed. We briefly illustrate the systematic differences that are expecte d in the luminosities and lifetimes of PNe with either H- or He-burning cen tral stars, which result in different "detection probabilities" across the H-R diagram, in both H beta and [OIII] lambda 5007 lines. Adopting reasonab le values of the model parameters, we are able to reproduce, in a satisfact ory way, many general properties of PNe, like the ionised mass{nebular radi us relationship, the trends of a few main nebular line ratios, and the obse rved ranges of nebular shell thicknesses, electron densities, and expansion velocities. The models naturally predict also the possible transitions fro m optically-thick to optically-thin configurations (and vice versa). In thi s 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 rel ation. These predictions are supported by observational indications by Mend ez et al. (1992). Another interesting result is that the change of slope in the electron density{nebular radius relation at R-ion similar to0.1 pc, po inted out by Phillips (1998), is also displayed by the models and may be in terpreted as the result of the progressive convergence of the PNe to the co ndition of constant ionised mass. Finally we would like to remark that, tha nks to its computational agility, our synthetic PN model is particularly su itable to population synthesis studies, and it represents the basic ground from which many future applications will be developed.