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. Building on the standard CAK theory of driving by scattering in an ensemble of lines with a power-law distribution of opacity, we first show that the formal divergence of such line-driven mass loss as a star approaches the Eddington limit is actually limited by the ``photon tiring'' associated with the work needed to lift material out of the star's gravitational potential. We also examine such tiring in simple continuum-driven models in which a specified outward increase in opacity causes a net outward acceleration above the radius where the generalized Eddington parameter exceeds unity. When the density at this radius implies a mass loss too close to the tiring limit, the overall result is flow stagnation at a finite radius. Since escape of a net steady wind is precluded, such circumstances are expected to lead to extensive variability and spatial structure. After briefly reviewing convective and other instabilities that also can be expected to lead to extensive structure in the envelope and atmosphere of a star near or above the Eddington limit, we investigate how the porosity of such a structured medium 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 the Eddington limit. 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 models, given roughly by L_*/a_*c (where L_* is the stellar luminosity and c and a_* are the speed of light and the atmospheric sound speed). This is much higher than is typical of line-driven winds but is still only a few percent of the tiring limit. To obtain still stronger mass loss that approaches observationally inferred values near this limit, we draw on an analogy with the power-law distribution of line-opacity in the standard CAK model of line-driven winds and thereby introduce a ``power-law-porosity'' model in which the associated structure has a broad range of scales. We show that for power indices α_p<1, the mass-loss rate can be enhanced over the single-scale model by a factor that increases with the Eddington parameter as Γ^-1+1/α_p. For lower α_p (~0.5-0.6) and/or moderately large Γ (>3-4), such models lead to mass-loss rates that approach the photon-tiring limit. Together with the ability to drive quite fast outflow speeds (of order the surface escape speed), the derived, near-tiring-limited mass loss offers a potential dynamical basis to explain the observationally inferred large mass loss and flow speeds of giant outbursts in η Carinae and other luminous blue variable stars.