Vía de entrada y temas existenciales

Continuum porosity formalisms have been developed only for processes depending linearly on the density (the attenuation of X-rays, Thomson scattering). However, for example the continuum based mass-loss diagnostics IR and radio emission depend on the square of the density, and for these diag- nostics the effects of optically thick clumping are completely unexplored (at least to our knowledge). Here we show how an extension of the porosity formalisms developed in the preceding paragraphs to handle alsoρ2processes is, in fact, trivial.

Eq. 3.28 for the effective opacity of a clump ensemble, including non-radial photons, is repeated for convenience,

χeff=µ

1₋e−τcl

h . (3.37)

The clump optical depth is now

τcl= ρ2 clκ2lcl µ = ρ2 smκ2lcl f_V2µ = χh fVµ = χmch µ , (3.38)

where we have used results from Sect. 3.2 and assumed a void inter-clump medium, andχmcis the

opacity in a microclumped model. Thus, the effective opacity may simply be written as

χeff=χmc

1₋e−τcl

τcl

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CHAPTER 3. RADIATIVE TRANSFER IN STOCHASTIC MEDIA AND HOT STAR WINDS - MICROCLUMPING, VOROSITY, AND POROSITY REVISITED

This expression now unifies the porosity formalisms forρ- and ρ2-processes, since for the former

χmc=χsm =hχi. Actually, it illustrates how one should always measure opacity reductions from

optically thick clumping relative to microclumped models, and not relative to smooth ones. We discuss this property further in Chapter 5, for the case of Hα line formation in O star winds (which is aρ2 process).

Let us point out that these porosity formalisms represent a class of techniques that attempt to find an effective opacity in order to capture the essence of the statistical medium (Sect. 3.1). The effective opacity can then be used in the ordinary equations of radiative transfer, rather than aiming to solve for the ensemble averaged intensity directly (which the ‘vorosity’ formalism presented earlier does). There are, however, dangers in applying these type of ‘effective opacity’ methods, as discussed in the book by Pomraning (1991). Essentially what one does is trying to approximate the sum of two exponentials (Eq. 3.11) with only one (Eq. 3.4). Thus the porosity formalisms are inherently approx- imative, except for, in this case, the very limiting case of Eq. 3.25, and the errors introduced can be very hard to estimate. On the other hand, the alternative approach, i.e. to try and find a corresponding expression to Eq. 3.11 but for a more complex medium, certainly is everything but straightforward (and in many cases impossible), as demonstrated in Pomraning (1991). Supposedly the best practical approach to test the simplified approaches rather is to perform suitable Monte-Carlo simulations, and simply compare the results stemming from the different methods (as has been done for resonance and recombination line formation in Chapters 4-5). Note, however, that for at least line diagnostics of hot star winds we cannot assume a pure absorption model, as done in this chapter, but must treat also the emission component (or, equivalently, the source function). An analytic approximation for the emis- sion component in recombination lines formed in clumpy winds is provided in Chapter 5, whereas a corresponding treatment of the re-emission in resonance lines is still to be developed (although we comment on a possible first approximation in Sect. 5.7.2).

Finally, we notice also that whereas the Levermore et al. model was derived for the special case of a Markovian mixture, no assumptions regarding the underlying clump statistics were made in the corresponding techniques developed for hot star winds, suggesting that our basic results may perhaps not be so dependent on the particular statistics of the clumps.

Chapter 4

Mass loss from inhomogeneous hot star

winds

I. Resonance line formation in 2D models

This chapter is a copy of Sundqvist, Puls, & Feldmeier (2010), Astronomy & Astrophysics, 510, A11. The only revision from the original version is that the two appendices here have been added at the end as normal sections (4.9 and 4.10).

4.1

Abstract

The mass-loss rate is a key parameter of hot, massive stars. Small-scale inhomogeneities (clumping) in the winds of these stars are conventionally included in spectral analyses by assuming optically thin clumps, a void inter-clump medium, and a smooth velocity field. To reconcile investigations of different diagnostics (in particular, unsaturated UV resonance lines vs. Hα/radio emission) within such models, a highly clumped wind with very low mass-loss rates needs to be invoked, where the resonance lines seem to indicate rates an order of magnitude (or even more) lower than previously accepted values. If found to be realistic, this would challenge the radiative line-driven wind theory and have dramatic consequences for the evolution of massive stars. We investigate basic properties of the formation of resonance lines in small-scale inhomogeneous hot star winds with non-monotonic velocity fields. We study inhomogeneous wind structures by means of 2D stochastic and pseudo-2D radiation-hydrodynamic wind models, constructed by assembling 1D snapshots in radially indepen- dent slices. A Monte-Carlo radiative transfer code, which treats the resonance line formation in an axially symmetric spherical wind (without resorting to the Sobolev approximation), is presented and used to produce synthetic line spectra. The optically thin clumping limit is only valid for very weak lines. The detailed density structure, the inter-clump medium, and the non-monotonic velocity field are all important for the line formation. We confirm previous findings that radiation-hydrodynamic wind models reproduce observed characteristics of strong lines (e.g., the black troughs) without apply- ing the highly supersonic ‘microturbulence’ needed in smooth models. For intermediate strong lines, the velocity spans of the clumps are of central importance. Current radiation-hydrodynamic models predict spans that are too large to reproduce observed profiles unless a very low mass-loss rate is in- voked. By simulating lower spans in 2D stochastic models, the profile strengths become drastically

44

CHAPTER 4. MASS LOSS FROM INHOMOGENEOUS HOT STAR WINDS I. RESONANCE LINE FORMATION IN 2D MODELS

reduced, and are consistent with higher mass-loss rates. To simultaneously meet the constraints from strong lines, the inter-clump medium must be non-void. A first comparison to the observed Phospho- rus V doublet in the O6 supergiantλ Cep confirms that line profiles calculated from a stochastic 2D model reproduce observations with a mass-loss rate approximately ten times higher than that derived from the same lines but assuming optically thin clumping. Tentatively this may resolve discrepancies between theoretical predictions, evolutionary constraints, and recent derived mass-loss rates, and sug- gests a re-investigation of the clump structure predicted by current radiation-hydrodynamic models.

4.2

Introduction

Mass loss through supersonic stellar winds is pivotal for the physical understanding of hot, massive stars and their surroundings. A change of only a factor of two in the mass-loss rate has a dramatic effect on massive star evolution (Meynet et al., 1994). Winds from these stars are described by the line-driven wind theory (Castor et al., 1975; Pauldrach et al., 1986), which traditionally assumes the wind to be stationary, spherically symmetric, and homogeneous. Despite this theory’s apparent success (e.g., Vink et al., 2000), evidence for an inhomogeneous and time-dependent wind has over the past years accumulated, recently summarized in the proceedings from the workshop ‘Clumping in hot star winds’ (Hamann et al., 2008) and in a general review of mass loss from hot, massive stars (Puls et al., 2008b).

That line-driven winds should be intrinsically unstable was already pointed out by Lucy & Solomon (1970), and was later confirmed first by linear stability analyses and then by direct, radiation- hydrodynamic modeling of the time-dependent wind (e.g., Owocki & Rybicki, 1984; Owocki et al., 1988; Feldmeier, 1995; Dessart & Owocki, 2005), where the line-driven (or line-deshadowing) insta- bility causes a small-scale, inhomogeneous wind in both density and velocity.

Direct observational evidence of a small-scale, clumped stellar wind has, for O-stars, so far only been given for two objects,ζ Pup and HD 93129A (Eversberg et al., 1998; L´epine & Moffat, 2008). Much indirect evidence, however, has arisen from quantitative spectroscopy, where the standard way of deriving mass-loss rates from observations nowadays is via line-blanketed, non-LTE (LTE: local thermodynamic equilibrium) model atmospheres that include a treatment of both the photosphere and the wind. Wind clumping has been included in such codes (e.g., CMFGEN (Hillier & Miller, 1998), PoWR (Gr¨afener et al., 2002), FASTWIND (Puls et al., 2005)) by assuming statistically distributed optically thin density clumps and a void inter-clump medium, while keeping the smooth velocity law. The major result from this methodology is that any mass-loss rate derived from smooth models and density-squared diagnostics (Hα, infra-red and radio emission) needs to be scaled down by the square root of the clumping factor (which describes the over density of the clumps as compared to the mean density, see Sect. 4.3.2). For example, Crowther et al. (2002), Bouret et al. (2003), and Bouret et al. (2005) have concluded that a reduction of ‘smooth’ mass-loss rates by factors 3. . .7 might be necessary. Furthermore, from a combined optical/IR/radio analysis of a sample of Galactic O-giants/supergiants, Puls et al. (2006) derived upper limits on observed rates that were factors of 2. . .3 lower than previous Hαestimates based on a smooth wind.

On the other hand, the strength of UV resonance lines (‘P Cygni lines’) in hot star winds depends linearly on the density and is therefore not believed to be directly affected by optically thin clumping. By using the Sobolev with exact integration technique (SEI; cf. Lamers et al. 1987) on the unsaturated Phosphorus V (PV) lines, Fullerton et al. (2006) for a large number of Galactic O-stars derived rates