Self-Similar Evolution of Wind-Blown Bubbles with Mass-loading by Conductive Evaporation

We present similarity solutions for adiabatic bubbles that are blown by winds having time dependent mechanical luminosities and that are each mass-loaded at a rate per unit volume proportional to T r, where T is the temperature, r is the distance from the bubble center, and λ is a constant. In the limit of little mass loading a similarity solution found by Dyson (1973) for expansion into a smooth ambient medium is recovered. For solutions that give the mass of swept-up ambient gas to be less than the sums of the masses of the wind and the evaporated material, λ ∼> 4. The Mach number in a shocked mass-loaded wind shows a radial dependence that varies qualitatively from solution to solution. In some cases it is everywhere less than unity in the frame of the clumps being evaporated, while in others it is everywhere greater than unity. In some solutions the mass-loaded shocked wind undergoes one or two sonic transitions in the clump frame. Maximum possible values of the ratio of evaporated mass to stellar wind mass are found as a consequence of the evaporation rates dependence on temperature and the lowering of the temperature by mass-loading.Mass-loading tends to reduce the emissivity in the interior of the bubble relative to its limb, whilst simultaneously increasing the central temperature relative to the limb temperature.