A Porosity-Length Formalism for Photon-Tiring-Limited Mass Loss from Stars Above the Eddington Limit

We examine radiatively driven mass loss from stars near and above the Eddington limit (Ledd). We begin by reviewing the instabilities that are expected to form extensive structure near Ledd. We investigate how this "porosity" can reduce the effective coupling between the matter and radiation. Introducing a new "porosity-length'' formalism, we derive a simple scaling for the reduced effective opacity, and use this to derive an associated scaling for the porosity-moderated, continuum-driven mass loss rate from stars that formally exceed Ledd. For a simple super-Eddington model with a single porosity length that is assumed to be on the order of the gravitational scale height, the overall mass loss is similar to that derived in previous porosity work. This is much higher than is typical of line-driven winds, but is still only a few percent of the photon tiring limit--for which the luminosity becomes insufficient to carry the flow out of the gravitational potential. To obtain still stronger mass loss that approaches observationally inferred values near this limit, we introduce a power-law-porosity model in which the associated structure has a broad range of scales. We show that the mass loss rate can be enhanced by a factor that increases with the Eddington parameter Gamma, such that for moderately large Gamma (> 3-4), mass loss rates could approach the photon tiring limit. Together with the ability to drive quite fast outflow speeds, the derived mass loss could explain the large inferred mass loss and flow speeds of giant outbursts in eta Carinae and other LBV stars.