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Context. Radiation-driven mass loss is key to our understanding ofmassive-star evolution. However, for low-luminosity O-type stars there are bigdiscrepancies between theoretically predicted and empirically derived mass-lossrates (called the weak-wind problem). Aims. We compute radiation-line-drivenwind models of a typical weak-wind star to determine its temperature structureand the corresponding impact on ultra-violet (UV) line formation. Methods. Wecarried out hydrodynamic simulations of the line-deshadowing instability (LDI)for a weak-wind star in the Galaxy. Subsequently, we used this LDI model asinput in a short-characteristics radiative transfer code to compute syntheticUV line profiles. Results. We find that the line-driven weak wind issignificantly shock heated to high temperatures and is unable to cool downeffciently. This results in a complex temperature structure where more thanhalf of the wind volume has temperatures significantly higher than the stellareffective temperature. Therefore, a substantial portion of the weak wind willbe more ionised, resulting in a reduction of the UV line opacity and thereforein weaker line profiles for a given mass-loss rate. Quantifying this, we findthat weak-wind mass-loss rates derived from unsaturated UV lines could beunderestimated by a factor of between 10 and 100 if the high-temperature gas isnot properly taken into account in the spectroscopic analysis. This offers atentative basic explanation for the weak-wind problem: line-driven weak windsare not really weaker than theoretically expected, but rather a large portionof their wind volume is much hotter than the stellar effective temperature.

Shock-heated radiation-driven outflows as a solution to the weak-wind problem of late O-type stars