Dusty winds: I. Self-similar solutions

We address the dusty wind problem, from the point where dust formation has been completed and outward. Given grain properties, both radiative transfer and hydrodynamics components of the problem are fully defined by four additional input parameters. The wind radiative emission and the shape of its velocity profile are both independent of the actual magnitude of the velocity and are determined by just three dimensionless free parameters. Of the three, only one is always significant---for most of phase space the solution is described by a set of similarity functions of a single independent variable, which can be chosen as the overall optical depth at visual \tV. The self-similarity implies general scaling relations among mass loss rate (\Mdot), luminosity ($L$) and terminal velocity (\vf). Systems with different \Mdot, $L$ and \vf but the same combination $\Mdot/L^{3/4}$ necessarily have also the same $\Mdot\vf/L$. ... Eliminating \tV produces $\vf^3 = A \Mdot(1 + B \Mdot^{4/3}/L)^{-1.5}$, where $A$ and $B$ are coefficients that contain the only dependence of this universal correlation on chemical composition. At a given $L$, the maximal velocity of a dusty wind is $v\sub{max} \propto L^{1/4}$ attained at $\Mdot \propto L^{3/4}$, with proportionality coefficients derived from $A$ and $B$.