The stellar upper-mass limit is highly uncertain. Some studies have claimed there is a universal upper limit of ∼150 M . A factor that is often overlooked is that there might be a significant difference between the present-day and the initial masses of the most massive stars -as a result of mass loss. The upper-mass limit may easily supersede ∼200 M . For these reasons, we present new mass-loss predictions from Monte Carlo radiative transfer models for very massive stars (VMS) in the mass range 40-300 M , and with very high luminosities 6.0 ≤ log(L /L ) ≤ 7.03, corresponding to large Eddington factors Γ. Using our new dynamical approach, we find an upturn or "kink" in the mass-loss versus Γ dependence, at the point where the model winds become optically thick. This coincides with the location where our wind efficiency numbers surpass the single-scattering limit of η = 1, reaching values up to η 2.5. In all, our modelling suggests a transition from common O-type winds to Wolf-Rayet characteristics at the point where the winds become optically thick. This transitional behaviour is also revealed with respect to the wind acceleration parameter, β, which starts at values below 1 for the optically thin O-stars, and naturally reaches values as high as 1.5-2 for the optically thick Wolf-Rayet models. An additional finding concerns the transition in spectral morphology of the Of and WN characteristic He ii line at 4686 Å. When we express our mass-loss predictions as a function of the electron scattering Eddington factor Γ e ∼ L /M alone, we obtain anṀ vs. Γ e dependence that is consistent with a previously reported power lawṀ ∝ Γ 5 e (Vink 2006) that was based on our previous semi-empirical modelling approach. When we expressṀ in terms of both Γ e and stellar mass, we find optically thin winds andṀ ∝ M 0.68 Γ 2.2 e for the Γ e range 0.4 < ∼ Γ e < ∼ 0.7, and mass-loss rates that agree with the standard Vink et al. recipe for normal O stars. For higher Γ e values, the winds are optically thick and, as pointed out, the dependence is much steeper,Ṁ ∝ M 0.78 Γ 4.77 e . Finally, we confirm that the effect of Γ on the predicted mass-loss rates is much stronger than for the increased helium abundance, calling for a fundamental revision in the way stellar mass loss is incorporated in evolutionary models for the most massive stars.

Context. Mass loss from massive stars forms an important aspect of the evolution of massive stars, as well as for the enrichment of the surrounding interstellar medium. Aims. Our goal is to predict accurate mass-loss rates and terminal wind velocities. These quantities can be compared to empirical values, thereby testing radiation-driven wind models. One specific topical issue is that of the so-called "weak-wind problem", where empirically derived mass-loss rates and (modified) wind momenta fall orders of magnitude short of predicted values. Methods. We employ an established Monte Carlo model and a recently suggested new line acceleration formalism to solve the wind dynamics more consistently. Results. We provide a new grid of mass-loss rates and terminal wind velocities of O-type stars, and compare the values to empirical results. Our models fail to provide mass-loss rates for main-sequence stars below a luminosity of log(L/L ) = 5.2, where we appear to run into a fundamental limit. At luminosities below this critical value there is insufficient momentum transferred to the wind in the region below the sonic point in order to kick-start the acceleration of the flow. This problem occurs at almost the exact location of the onset of the weak-wind problem. For O dwarfs, the boundary between being able to start a wind, and failing to do so, is at spectral type O6/O6.5. The direct cause of this failure for O6.5 stars is a combination of the lower luminosity and a lack of Fe v lines at the base of the wind. This might indicate that -in addition to radiation pressure -another mechanism is required to provide the necessary driving to initiate the wind acceleration. Conclusions. For stars more luminous than 10 5.2 L , our new mass-loss rates are in excellent agreement with the mass-loss prescription by Vink et al. (2000, A&A, 362, 295) using our terminal wind velocities as input to this recipe. This implies that the main assumption entering the method of the Vink et al. prescriptions -i.e. that the momentum equation is not explicitly solved for -does not compromise the reliability of the Vink et al. results for this part of parameter space. Finally, our new models predict terminal velocities that are typically 35 and 45 percent larger than observed values. Such over-predictions are similar to those from (modified) CAK-theory.

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 H ρ . In this case, empirical mass-loss determinations based on Hα 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 H ρ , 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 predictedṀ 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 whichṀ < ∼ 10 −7 M yr −1 (weak winds) that is not active in denser winds.