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 tau (v). The self-similarity implies general scaling relations among mass-loss rate ((M) over dot), luminosity (L) and terminal velocity (nu (infinity)). Systems with different (M) over dot, L and nu (infinity) but the same combination (M) over dot/L-3/4 necessarily have also the same (M) over dot nu (infinity)/L. 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 tau (v). In all winds, (M) over dot proportional to nu (3)(infinity) (1 + tau (v))(1.5) with a proportionality constant that depends only on grain properties. The optically thin end of this universal correlation, (M) over dot proportional to nu (3)(infinity), 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 (M) over dot-L correlation is (M) over dot proportional to (L-tauv)(3/4)(1 + tau (v))(0.105). At a fixed luminosity, (M) over dot is not linearly proportional to tau (v), again because of dust drift. The velocity-luminosity correlation is nu (infinity) proportional to (L tau (v))(1/4)(1 + tau (v))(-0.465), explaining the narrow range of outflow velocities displayed by dusty winds. Eliminating tau (v) produces nu (3)(infinity) = A(M) over dot(1 + B(M) over dot(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(max) proportional toL(1/4) attained at (M) over dot proportional to L-3/4, with proportionality coefficients derived from A and B.