Outflow dynamics of dust-driven wind models and implications for cool envelopes of PNe

The density profiles of cool envelopes of young Planetary Nebulae (PNe) are reminiscent of the final AGB outflow history of the central star, so far as these have not yet been transformed by the hot wind and radiation of the central star. Obviously, the evolution of the mass loss rate of that dust-driven, cool wind of the former giant in its final AGB stages must have shaped these envelopes to some extent. Less clear is the impact of changes in the outflow velocity. Certainly, larger and fast changes would lead to significant complications in the reconstruction of the mass-loss history from a cool envelope's density profile. Here, we analyse the outflow velocity v_{\rm exp} in a consistent set of over 50 carbon-rich, dust-driven and well "saturated" wind models, and how it depends on basic stellar parameters. We find a relation of the kind of v_{\rm exp} \propto (L/M)^{0.6}. By contrast to the vast changes of the mass-loss rate in the final outflow phase, this relation suggest only very modest variations in the wind velocity, even during a thermal pulse. Hence, we conclude that the density profiles of cool envelopes around young PNe should indeed compare relatively well with their recent mass-loss history, when diluted plainly by the equation of continuity.