The Effect of Porosity on X‐Ray Emission‐Line Profiles from Hot‐Star Winds

Abstract

We investigate the degree to which the nearly symmetric form of X-ray emission lines seen in Chandra spectra of early-type supergiant stars could be explained by a possibly porous nature of their spatially structured stellar winds. Such porosity could effectively reduce the bound-free absorption of X-rays emitted by embedded wind shocks, and thus allow a more similar transmission of red-vs. blue-shifted emission from the back vs. front hemispheres. To obtain the localized self-shielding that is central to this porosity effect, it is necessary that the individual clumps be optically thick. In a medium consisting of clumps of size ℓ and volume filling factor f , we argue that the general modification in effective opacity should scale approximately as κ ef f ≈ κ/(1 + τ c ), where, for a given atomic opacity κ and mean density ρ, the clump optical thickness scales as τ c = κρℓ/f . For a simple wind structure parameterization in which the 'porosity length' h ≡ ℓ/f increases with local radius r as h = h ′ r, we find that a substantial reduction in wind absorption requires a quite large porosity scale factor, h ′ ∼ > 1, implying large porosity lengths h ∼ > r. The associated wind structure must thus have either a relatively large scale ℓ ∼ < r, or a small volume filling factor f ≈ ℓ/r ≪ 1, or some combination of these. We argue that the relatively small-scale, moderate compressions generated by intrinsic instabilities in line-driving are unlikely to give such large porosity lengths. This raises questions about whether porosity effects could play a significant role in explaining nearly symmetric X-ray line profiles, leaving again the prospect of instead having to invoke a substantial (ca. factor 5) downward revision in the assumed mass-loss rates.

Long term variability of Cygnus X-1

“…A toy model for a focussed line-driven wind without clumps (Gies & Bolton 1986b) does not describe the data well and we attribute the differences to the absence of clumping in the model that would lead to the observed strong variations of N H at the same orbital phase. Qualitatively matching the observed N H distributions at a given orbital phase to a toy clumpy wind model (Owocki & Cohen 2006;Sundqvist et al 2012) without a focussed wind and with spherical clumps results in the best match for models with porosity length, h ∞ , on the order of the stellar radius, R * , at orbital phases φ orb ≈ 0.5, in agreement with results from (effectively) single O stars (Leutenegger et al 2013). Our models exclude both much lower and much higher porosities.…”

“…A.2). Based on previous theoretical and observational investigations of single O stars, it is reasonable that h ∞ should be comparable to or smaller than R * (Dessart & Owocki 2003;Owocki & Cohen 2006;Nazé et al 2013). This is also consistent with the results of Leutenegger et al (2013), who found that the spectrum of the single O supergiant ζ Pup could be fit with models having porosity lengths h ∞ ≤ R * .…”

“…The latter is most conveniently parameterized in terms of the number of clumps per wind flow time N =Ṅt flow , with t flow ≡ R * / ∞ . To relate these parameters to the porosity length h (essentially, the mean free path between clumps; see Owocki & Cohen 2006), we note that h(r) = h ∞ (r)/ ∞ , where the terminal porisity length is…”

The X-ray spectrum ofδOrionis observed by LETGS aboardChandra

“…Only when the wind-clump collision is almost perpendicular to the line of sight is the radiation observable and spread over a narrow velocity range as derived from the HWHM. The effects of clumping and porosity on X-Ray emission-line profiles from hot-star winds have been described extensively by Owocki & Cohen (2006) and Oskinova et al (2006).…”

Structure in the winds of O-Type stars: observations and inferences

“…The character of the radiation transfer in the wind changes once the clumps become optically thick, because (a) radiation is trapped within the clumps; and (b) channels open up between the clumps that permit radiation from the star to leak out; i.e., the wind becomes "porous". The leakage implied by porosity reduces the effective optical depth of the wind (Owocki & Cohen 2006;Oskinova et al 2007) in a particular transition. Oskinova et al (2007) recognized that resonance transitions become optically thick before other transitions, and consequently a clumped wind could be a porous medium for the P v doublet but not for Hα.…”