Dusty winds -- I. Self-similar solutions

We address the dusty wind problem, from the point where dust formation has been completed and outward. Given the 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 the 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 τV. The self-similarity implies general scaling relations among mass-loss rate (M˙), luminosity (L) and terminal velocity (v∞). Systems with different M˙, L and v∞ but the same combination M˙/L3/4 necessarily have also the same . For optically thin winds we find the exact analytic solution, including the effects of radiation pressure, gravitation and (sub and supersonic) dust drift. For optically thick winds we present numerical results that cover the entire relevant range of optical depths, and summarize all correlations among the three global parameters in terms of τV. In all winds, M˙ with a proportionality constant that depends only on grain properties. The optically thin end of this universal correlation, , has been verified in observations; even though the wind is driven by radiation pressure, the luminosity does not enter because of the dominant role of dust drift in this regime. The correlation is . At a fixed luminosity, M˙ is not linearly proportional to τV, again because of dust drift. The velocity–luminosity correlation is , explaining the narrow range of outflow velocities displayed by dusty winds. Eliminating τV produces , 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 attained at , with proportionality coefficients derived from A and B.