Stellar envelope inflation near the Eddington limit

Context. It has been proposed that the envelopes of luminous stars may be subject to substantial radius inflation. The peculiar structure of such inflated envelopes, with an almost void, radiatively dominated region beneath a thin, dense shell could mean that many in reality compact stars are hidden below inflated envelopes, displaying much lower effective temperatures. The inflation effect has been discussed in relation to the radius problem of Wolf-Rayet (WR) stars, but has yet failed to explain the large observed radii of Galactic WR stars. Aims. We wish to obtain a physical perspective of the inflation effect, and study the consequences for the radii of WR stars, and luminous blue variables (LBVs). For WR stars the observed radii are up to an order of magnitude larger than predicted by theory, whilst S Doradus-type LBVs are subject to humongous radius variations, which remain as yet ill-explained. Methods. We use a dual approach to investigate the envelope inflation, based on numerical models for stars near the Eddington limit, and a new analytic formalism to describe the effect. An additional new aspect is that we take the effect of density inhomogeneities (clumping) within the outer stellar envelopes into account. Results. Due to the effect of clumping we are able to bring the observed WR radii in agreement with theory. Based on our new formalism, we find that the radial inflation is a function of a dimensionless parameter W, which largely depends on the topology of the Fe-opacity peak, i.e., on material properties. For W > 1, we discover an instability limit, for which the stellar envelope becomes gravitationally unbound, i.e. there no longer exists a static solution. Within this framework we are also able to explain the S Doradus-type instabilities for LBVs like AG Car, with a possible triggering due to changes in stellar rotation. Conclusions. The stellar effective temperatures in the upper Hertzsprung-Russell (HR) diagram are potentially strongly affected by the inflation effect. This may have particularly strong effects on the evolved massive LBV and WR stars just prior to their final collapse, as the progenitors of supernovae (SNe) Ibc, SNe II, and long-duration gamma-ray bursts (long GRBs)