Context. It is well-established that the winds of carbon-rich AGB stars (carbon stars) can be driven by radiation pressure on grains of amorphous carbon and collisional transfer of momentum to the gas. This has been demonstrated by different numerical wind models including time-dependent dust formation, where it has been assumed that the dust grains that form never grow to sizes comparable to wavelengths around the stellar flux maximum (or beyond), which simplifies the treatment of grain opacities considerably. It is not clear, however, if this small particle approximation (SPA) is always valid.
Aims. In this paper we explore the effects of relaxing the SPA by considering a few less severe approximations for the radiation pressure efficiency, which include the effects of grain sizes. The purpose of the study is mainly to establish when the SPA can be applied and to quantify the possible errors that may occur when it does not hold.
Methods. We have computed wind models with time-dependent dust formation and grain-size dependent opacities, where (1) the radiation pressure efficiency is approximated using grain sizes based on various means of the grain size distribution, and (2) where the problem is simplified by assuming a single dust-grain size.
Results. It is shown that in critical cases, the effect of grain sizes can be significant. Mass-loss rates may increase by a factor of two, or more, and wind speeds may be an order of magnitude higher. Furthermore, all models with grain-size dependent opacities that have resultant winds appear to have much lower degrees of dust condensation, compared to corresponding SPA models. Consequently, the "dust-loss rates" are much lower in the new models. However, for well-developed dust-driven winds, where the dust formation has saturated, the effect of grain sizes on the mass-loss rate and wind speed is almost negligible.
Conclusions. We conclude that the SPA is, under many circumstances, a reasonable simplification in models of carbon star mass loss. However, critical cases do exist, where especially the effects of momentum transfer due to scattering become significant. It is therefore uncertain whether previous results for winds in the transition regions (between no dust-driven mass loss and well-developed winds) are quantitatively correct. However, we argue that the SPA is a reasonable simplification for strong dust-driven winds. Furthermore, we note that there are other effects (not yet included in the model) than those of grain-size dependent opacities, which may become important in the critical wind regime.