[Paper Review] Stellar envelope inflation near the Eddington limit. Implications for the radii of Wolf-Rayet stars and luminous blue variables

G. Gräfener, S. P. Owocki+1 more|arXiv (Cornell University)|Dec 8, 2011

Stellar, planetary, and galactic studiesPhysics and Astronomy42 references101 citations

TL;DR

This paper proposes that radiative envelope inflation near the Eddington limit, enhanced by stellar wind clumping, explains the anomalously large radii of Wolf-Rayet (WR) stars and S Doradus-type luminous blue variables (LBVs). Using numerical models and a new analytic formalism, the authors identify a dimensionless parameter $ W $ that governs inflation and discover an instability limit at $ W \geq 1 $, where envelopes become gravitationally unbound. The inclusion of clumping resolves the long-standing WR radius problem, bringing observed radii into agreement with theory.

ABSTRACT

(shortened) It has been proposed that the envelopes of luminous stars may be subject to substantial radius inflation. The inflation effect has been discussed in relation to the radius problem of WR stars, but has yet failed to explain the large observed radii of Galactic WR stars. We wish to obtain a physical perspective of the inflation effect, and study the consequences for the radii of WR stars, and 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. 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. 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. The stellar effective temperatures in the upper 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 SNe Ibc, SNe II, and long GRBs.

Motivation & Objective

• To resolve the long-standing discrepancy between observed and theoretically predicted radii of Galactic Wolf-Rayet (WR) stars.
• To investigate the physical mechanism behind the extreme radius variations in S Doradus-type luminous blue variables (LBVs).
• To develop an analytic framework for envelope inflation that accounts for stellar wind clumping and its impact on structure.
• To determine the conditions under which inflated envelopes become gravitationally unbound, identifying a new instability limit.
• To assess the implications of envelope inflation for the effective temperatures and evolutionary status of massive stars, particularly progenitors of SNe Ibc and long GRBs.

Proposed method

• Numerical stellar structure models are used to simulate stars near the Eddington limit, focusing on chemically homogeneous He-core models representative of WR stars.
• A new analytic formalism is developed to describe envelope inflation, introducing a dimensionless parameter $ W $ that quantifies the ratio of internal to gravitational energy at the envelope base.
• The effect of density inhomogeneities (clumping) in the outer envelope is incorporated via a clumping factor $ D $, which modifies the effective opacity and energy transport.
• The stability of inflated envelopes is analyzed by determining the critical value $ W \geq 1 $, beyond which no static solution exists, indicating gravitational unbinding.
• The model is applied to AG Car, a well-known S Doradus variable, to test its ability to reproduce observed luminosity and radius changes during outbursts.
• Stellar parameters such as mass, radius, luminosity, and Eddington factor $ \Gamma_{\rm e} $ are used to estimate $ W $ from opacity tables and observed data.

Experimental results

Research questions

• RQ1 Can envelope inflation near the Eddington limit explain the anomalously large radii observed in Galactic WR stars?
• RQ2 How does stellar wind clumping influence the extent and stability of radiatively inflated envelopes?
• RQ3 What is the critical condition for gravitational unbinding of inflated envelopes, and how does it relate to the dimensionless parameter $ W $?
• RQ4 Can the S Doradus-type variability of LBVs like AG Car be explained by envelope inflation triggered by changes in rotation or $ W $-parameter?
• RQ5 To what extent does envelope inflation affect the effective temperature and observed HRD position of massive evolved stars, particularly progenitors of SNe Ibc and long GRBs?

Key findings

• The inclusion of clumping in the outer envelope reduces the effective opacity and allows the inflation parameter $ W $ to reach values near unity, enabling the observed large radii in WR stars.
• For $ W \geq 1 $, the stellar envelope becomes gravitationally unbound, indicating a new instability limit that may trigger S Doradus-type outbursts in LBVs.
• Clumping factors $ D $ in the range of 1–16 are required to match observed WR radii, which are consistent with typical values inferred from wind diagnostics.
• The model explains the observed radius increase by a factor of two and luminosity decrease by a factor of 1.5 in AG Car during S Dor cycles via the formation of a dense outer shell of $ \sim 2 \times 10^{-3} M_{\odot} $.
• The energy required to lift this shell to the outer radius is comparable to the observed luminosity decrease, supporting the inflation mechanism as a viable explanation.
• Envelope inflation may significantly alter the observed effective temperatures of massive stars, potentially hiding compact, fast-rotating cores beneath inflated, cooler envelopes, with implications for SN and GRB progenitor models.