| Abstract
| - Aims. Both empirical evidence and theoretical findings indicate that the stellar winds of massive early-type stars are inhomogeneous, i.e., porous and clumpy. For relatively dense winds, empirically derived mass-loss rates might be reconciled with predictions if these empirical rates are corrected for clumping. The predictions, however, do not account for structure in the wind. To allow for a consistent comparison, we investigate and quantify the effect of clumpiness and porosity of the outflow on the predicted wind energy and the maximal effect on the mass-loss rate of O-type stars. Methods. Combining non-LTE model atmospheres and a Monte Carlo method to compute the transfer of momentum from the photons to the gas, the effect of clumping and porosity on the energy transferred from the radiation field to the wind is computed in outflows in which the clumping and porosity stratification is parameterized by heuristic prescriptions. Results. The impact of structure in the outflow on the wind energy is complex and is a function of stellar temperature, the density of gas in the clumps, and the physical scale of the clumps. If the medium is already clumped in the photosphere, the emergent radiation field will be softer, slightly increasing the wind energy of relatively cool O stars (30 000 K) but slightly decreasing it for relatively hot O stars (40 000 K). More important is that as a result of recombination of the gas in a clumped wind the line force increases. However, because of porosity the line force decreases, simply because photons may travel in-between the clumps, avoiding interactions with the gas. If the changes in the wind energy only affect the mass-loss rate and not the terminal velocity of the flow, we find that the combined effect of clumpiness and porosity is a small reduction in the mass-loss rate if the clumps are smaller than 1/100th the local density scale height Hρ. In this case, empirical mass-loss determinations based on H α fitting and theory match for stars with dense winds ( Ṁ ≳ 10 -7 M⊙ yr -1) if the overdensity of gas in the clumps, relative to the case of a smooth wind, is modest. For clumps larger than 1/10th Hρ, the predicted mass-loss rates exhibit almost the same dependence on clumpiness as do empirical rates. We show that this implies that empirical and predicted mass-loss rates can no longer be matched. Very high overdensities of gas in clumps of such large size may cause the predicted Ṁ to decrease by a factor of from 10 to 100. This type of structure is likely not to be the cause of the “weak-wind problem” in early-type stars, unless a mechanism can be identified that causes extreme structure to develop in winds for which Ṁ ≲ 10 -7 M⊙ yr -1 (weak winds) that is not active in denser winds.
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