[Paper Review] Stellar mass-loss near the Eddington limit. Tracing the sub-photospheric layers of classical Wolf-Rayet stars

G. Gräfener, J. S. Vink|arXiv (Cornell University)|Sep 24, 2013

Stellar, planetary, and galactic studies61 references22 citations

TL;DR

This paper investigates the sub-photospheric structure of Wolf-Rayet stars near the Eddington limit using a novel semi-empirical method to infer sonic-point conditions. It finds a universal wind condition with $ P_{\text{rad}}/P_{\text{gas}} \approx 80 $, indicating optically thick, radiatively driven winds that may drive clumping via non-linear strange-mode instabilities, explaining observed radius inflation and wind clumping in late-type WC stars.

ABSTRACT

Towards the end of their evolution hot massive stars develop strong stellar winds and appear as emission line stars, such as WR stars or LBVs. The quantitative description of the mass loss in these important pre-SN phases is hampered by unknowns such as clumping and porosity due to an in-homogeneous wind structure, and by an incomplete theoretical understanding of optically thick stellar winds. In this work we investigate the conditions in deep atmospheric layers of WR stars to find out whether these comply with the theory of optically thick winds, and whether we find indications of clumping in these layers. We use a new semi-empirical method to determine sonic-point optical depths, densities, and temperatures for a large sample of WR stars of the carbon (WC) and oxygen (WO) sequence. Based on an artificial model sequence we investigate the reliability of our method and its sensitivity to uncertainties in stellar parameters. We find that the WR stars in our sample obey an approximate relation with P_rad/P_gas~80 at the sonic point. This 'wind condition' is ubiquitous for radiatively driven, optically thick winds, and sets constraints on possible wind/envelope solutions affecting radii, mass-loss rates, and clumping properties. Our results suggest that the presence of an optically thick wind may force many stars near the Eddington limit to develop clumped, radially extended sub-surface zones. The clumping in these zones is most likely sustained by the non-linear strange-mode instability, and may be the origin of the observed wind clumping. The properties of typical late-type WC stars comply with this model. Solutions without sub-surface clumping and inflation are also possible but demand for compact stars with comparatively low mass-loss rates. These objects may resemble the small group of WO stars with their exceptionally hot stellar temperatures and highly ionized winds.

Motivation & Objective

• To determine the physical conditions in the sub-photospheric layers of Wolf-Rayet stars, particularly near the sonic point, where direct observation is impossible.
• To assess whether optically thick, radiatively driven winds impose universal constraints on stellar structure, especially regarding radius inflation and wind clumping.
• To investigate whether observed wind clumping in WC stars originates from instabilities in radially extended, inflated sub-surface layers near the Fe-opacity peak.
• To differentiate between compact, low-clumping solutions (like WO stars) and extended, clumped solutions (like late-type WC stars) based on wind conditions.
• To evaluate the sensitivity of the method to uncertainties in stellar parameters and validate its reliability using artificial model sequences.

Proposed method

• A semi-empirical method is developed to infer sonic-point optical depth, density, and temperature from observed wind momenta, bypassing direct modeling of the entire wind structure.
• The method relies on the assumption that winds are radiatively driven and optically thick, using the relation between wind momentum and fundamental stellar parameters.
• Two approaches are applied: one based on detailed wind modeling and another model-independent method using observable wind momentum to estimate sonic-point conditions.
• An artificial model sequence of WC stars is constructed to test the method’s robustness and sensitivity to uncertainties in luminosity, mass, radius, and mass-loss rate.
• Theoretical wind models are used to validate the method’s accuracy in recovering true sonic-point conditions under varying physical assumptions.
• The analysis includes a comparison of solutions with and without sub-surface clumping to assess their viability under the observed wind condition $ P_{\text{rad}}/P_{\text{gas}} \approx 80 $.

Experimental results

Research questions

• RQ1 What physical conditions prevail in the sub-photospheric layers of Wolf-Rayet stars near the Eddington limit, particularly at the sonic point?
• RQ2 Does the observed wind condition $ P_{\text{rad}}/P_{\text{gas}} \approx 80 $ at the sonic point constrain the structure of optically thick winds in massive stars?
• RQ3 Can the observed wind clumping in WC stars be explained by instabilities in radially extended, inflated sub-surface layers near the Fe-opacity peak?
• RQ4 How do different assumptions about stellar parameters affect the inferred sonic-point conditions and the validity of the wind condition?
• RQ5 What are the implications of the wind condition for the radii, mass-loss rates, and clumping properties of Wolf-Rayet stars, especially in the context of the 'radius problem'?

Key findings

• The paper identifies a near-universal wind condition $ P_{\text{rad}}/P_{\text{gas}} \approx 80 $ at the sonic point across a large sample of Galactic and LMC WC/WO stars, indicating optically thick, radiatively driven winds.
• This wind condition is robust and only weakly sensitive to uncertainties in stellar parameters, as confirmed by testing on an artificial model sequence.
• The condition implies that optically thick winds naturally emerge near the Eddington limit and impose a strong outer boundary condition on the stellar envelope.
• Two viable solutions exist: a compact, high-temperature solution (resembling WO stars) without sub-surface clumping, and an extended, low-temperature solution (like late-type WC stars) that requires clumping to sustain the wind.
• The observed clumping factors in WR winds match those predicted by the model, suggesting that wind clumping may originate from non-linear strange-mode instabilities in inflated sub-surface layers near the Fe-opacity peak.
• The results support the hypothesis that radius inflation and wind clumping in WC stars are linked to the same physical mechanism—instabilities driven by radiation pressure near the Eddington limit—offering a unified explanation for the 'radius problem' and wind clumping.