  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.

Solar and Stellar Astrophysics