Mass loss and the Eddington parameter: a new mass-loss recipe for hot and massive stars

Mass loss through stellar winds plays a dominant role in the evolution of massive stars. In particular the mass-loss rates of very massive stars (VMSs, $> 100\,M_{\odot}$) are highly uncertain. Such stars display Wolf-Rayet spectral morphologies (WNh) whilst on the main-sequence. Metal-poor VMSs are progenitors of gamma-ray bursts and pair instability supernovae. In this study we extended the widely used stellar wind theory by Castor, Abbott & Klein from the optically thin (O star) to the optically thick main-sequence (WNh) wind regime. In particular we modify the mass-loss rate formula in a way that we are able to explain the empirical mass-loss dependence on the Eddington parameter ($\Gamma_{\rm e}$). The new mass-loss recipe is suitable for incorporation into current stellar evolution models for massive and very massive stars. It makes verifiable predictions, namely how the mass-loss rate scales with metallicity and at which Eddington parameter the transition from optically thin O star to optically thick WNh star winds occurs. In the case of the star cluster R136 in the Large Magellanic Cloud we find in the optically thin wind regime $\dot{M} \propto \Gamma_{\rm e}^{3}$ while in the optically thick wind regime $\dot{M} \propto 1/ (1 - \Gamma_{\rm e})^{3.5}$. The transition from optically thin to optically thick winds occurs at $\Gamma_{\rm e, trans} \approx 0.47$. The transition mass-loss rate is $\log \dot{M}~(M_{\odot} \mathrm{yr}^{-1}) \approx -4.76 \pm 0.18$, which is in line with the prediction by Vink & Gr\"afener assuming a volume filling factor of $f_{\rm V} = 0.23_{-0.15}^{+0.40}$.