Optically Thick Clumps - Not the Solution to the Wolf-Rayet Wind Momentum Problem?

The hot star wind momentum problem η = Ṁ3∞/(L/c) ≫ 1 is revisited, and it is shown that the conventional belief, that it can be solved by a combination of clumping of the wind and multiple scattering of photons, is not self-consistent for optically thick clumps. Clumping does reduce the mass loss rate Ṁ, and hence the momentum supply, required to generate a specified radio emission measure ε, while multiple scattering increases the delivery of momentum from a specified stellar luminosity L. However, in the case of thick clumps, when combined the two effects act in opposition rather than in unison since clumping reduces multiple scattering. From basic geometric considerations, it is shown that this reduction in momentum delivery by clumping more than offsets the reduction in momentum required, for a specified ε. Thus the ratio of momentum deliverable to momentum required is maximal for a smooth wind and the momentum problem remains for the thick clump case. In the case of thin clumps, all of the benefit of clumping in reducing η lies in reducing Ṁ for a given ε so that extremely small filling factors f ≈ 10−4 are needed. It is also shown that clumping affects the inference of Ṁ from radio ε not only by changing the emission measure per unit mass but also by changing the radio optical depth unity radius Rrad, and hence the observed wind volume, at radio wavelengths. In fact, for free-free opacity ∝ n2, contrary to intuition, Rrad increases with increasing clumpiness.