Radiative transfer code tests

In this subsection we describe some of the verification tests of our MC radiative transfer code that we have made. The MC-1D version was first applied on spherically symmetric winds, comparing profiles from smooth, stationary winds to profiles calculated using the well-established CMF (cf. Mihalas et al. 1975; Hamann 1981) and SEI methods, and profiles from time-dependent RH winds to profiles calculated using the Sobolev method developed in POF. Thereafter we applied the MC-2D version on models in which all lateral slices had the same radial structure, comparing the results to the MC-1D version.

First we calculated line profiles for smooth, 1D winds. We have verified that for low9 values of vt,

profiles from all the methods described above agree perfectly, whereas for higher values the MC-1D and CMF give identical results but the SEI deviates significantly, especially for a medium-strong line (see Fig. 4.14, upper panel). This is due to the hybrid nature of the SEI technique, which approximates the source function with its local Sobolev value but carries out the exact formal integral. Because of this, the method does not account for the increasing amount of photons close to line center that

9 _{For a typical terminal velocity value v}

4.10. THE EFFECTIVE ESCAPE RATIO 73

are backscattered into the photosphere when the resonance zone grows and overlaps with the lower boundary.10 Consequently the re-emitted flux in this region is higher when calculated via the SEI than when calculated via the CMF or MC methods. These discrepancies between the CMF and SEI are quite well documented and discussed (e.g., Hamann, 1981; Lamers et al., 1987), however we still emphasize that one should exercise caution when applying the SEI method with high microturbulence on wind resonance lines. Especially today, when increased computer-power enables us to compute fast solutions using both methods, the CMF is preferable.

Next we calculated line profiles for structured, 1D winds. Profiles computed with all three methods agreed for weak and intermediate lines. For strong lines, the agreement between MCS-1D and the method from POF, which uses a Sobolev source function accounting for multiple-resonance points, was satisfactory. However, minor discrepancies between Sobolev and non-Sobolev treatments oc- curred for the strong line also when no microturbulent velocity was applied (see Fig. 4.14), as opposed to the smooth case.

Finally we performed a simple test of our MC-2D code by applying it on models in which all lateral slices had the same radial structure, i.e., the wind was still spherically symmetric and all observers ought to see the same spectrum. We confirmed that indeed so was the case, both for smooth and structured models (in Fig. 4.14 the latter case is demonstrated).

4.10

The effective escape ratio

We define the ratio of the velocity gap∆v between two clumps (see Fig. 4.6 in the main paper) and the thermal velocity vtas

η≡∆_vv

t

(4.21) In the following, we derive an expression forη, for the wind geometry used throughout this paper. If

∆vtot=∆v+|δv|is the velocity difference between two clump centers, we may write (omitting the

absolute value signs here and in the following)

∆v=∆vtot−δv= ∆vtot ∆vtot,fi∆ vtot,fi− δv δv_βδvβ, (4.22)

where we have normalized the arbitrary velocity intervals to the correspondingβ intervals.β suffixes are used to denote parameters of a smooth velocity law. For notational simplicity we write

ξ1= ∆ vtot ∆vtot,fi , ξ2= δ v δv_β. (4.23)

Assuming radial photons,∆v may be approximated by

∆v_≈∂vβ

∂r ∆rtot,fi(ξ1−ξ2

δr_β

∆rtot,β

), (4.24)

10_{Remember that neither the SEI nor the CMF, as formulated here, include a transition to the photosphere, but treat the}

74

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

with the notations of r following those of v. The volume filling factor for the geometry in use is fv≡ Vcl Vtot ≈ r₁2δr r2₂∆rtot (4.25) with Vclthe volume of the clump, Vtotthe total volume, and r1 ≈ r2the radial points associated with

the beginning of the clump and the ICM. Using Eq. 4.25 and∆rtot=vβδt (see Sect. 4.3.2), we obtain ∆v_≈∂vβ

∂r vβδt(ξ1−ξ2fv), (4.26)

and forη, using the radial Sobolev length of a smooth flow Lr = vt/(∂vβ/∂r), η≈vβδt(ξ1_L−ξ2fv)

r

. (4.27)

In our models ξ1 is not given explicitly, but is on the order of unity, because we distribute clumps

according to the underlying smoothβ =1 velocity law. Thus we approximate

η≈vβδt(1_L−ξ2fv)

r

. (4.28)

We notice that the porosity length h as defined by Owocki et al. (2004) is h=l/fv, where l is the length

associated with the clump. For the geometry used here this becomes h_≈δr/fv≈vβδt. Hence, using ξ2=1 for a smooth velocity field,ηrepresents the porosity length corrected for the finite size of the

Chapter 5

Mass loss from inhomogeneous hot star

winds

II. Constraints from a combined optical/UV study

This chapter is a copy of Sundqvist, Puls, Feldmeier, & Owocki (2010), submitted to Astronomy & Astrophysics (A&A) in September 2010. Due to comments and suggestions by the referee is the version presented here slightly different from the version finally published in A&A. Also in this chapter is the original appendix added at the end as a normal section (Sect. 5.9).

5.1

Abstract

Mass loss is essential for massive star evolution, and thereby also for the variety of astrophysical applications relying on its predictions. However, mass-loss rates currently in use for hot, massive stars have recently been seriously questioned, mainly because of the effects of wind clumping. We investigate the impact of clumping on diagnostic ultra-violet resonance and optical recombination lines often used to derive empirical mass-loss rates of hot stars. Optically thick clumps, a non-void inter-clump medium, and a non-monotonic velocity field are all accounted for in one single model. The line formation is first theoretically studied, after which an exemplary multi-diagnostic study of an O-supergiant is performed. We use 2D and 3D stochastic and radiation-hydrodynamic wind models, constructed by assembling 1D snapshots in radially independent slices. To compute synthetic spectra, we develop and use detailed radiative transfer codes, for both recombination lines (solving the ‘formal integral’) and resonance lines (using a Monte-Carlo approach). In addition, we propose an analytic method to model these lines in clumpy winds, which does not rely on optically thin clumping. The importance of the ‘vorosity’ effect for line formation in clumpy winds is emphasized. Resonance lines are generally more affected by optically thick clumping than recombination lines. Synthetic spectra calculated directly from present-day, radiation-hydrodynamic wind models of the line-driven instability are unable to reproduce strategic optical and ultra-violet lines in the Galactic O-supergiant

λ Cep. Using our stochastic wind models, we obtain consistent fits essentially by increasing the clumping in the inner wind. A mass-loss rate is derived that is approximately two times lower than what is predicted by the line-driven wind theory, but much higher than the corresponding rate derived when assuming optically thin clumps. Our analytic formulation for line formation is used to demon-

76

CHAPTER 5. MASS LOSS FROM INHOMOGENEOUS HOT STAR WINDS II. CONSTRAINTS FROM A COMBINED OPTICAL/UV STUDY

strate the potential importance of optically thick clumping in diagnostic lines in so-called weak wind stars, and to confirm recent results that resonance doublets may be used as tracers of wind structure and optically thick clumping.

We confirm earlier results that a re-investigation of the structures in the inner wind predicted by line- driven instability simulations is needed. Our derived mass-loss rate for λ Cep suggests that only moderate reductions of current mass-loss predictions for OB-stars are necessary, but nevertheless prompts investigations on feedback effects from optically thick clumping on steady-state, NLTE wind models used for quantitative spectroscopy.

5.2

Introduction

Massive stars are fundamental in many fields of modern astrophysics. In the present Universe, they dynamically and chemically shape their surroundings and the inter-stellar medium by their output of ionizing radiation, energy and momentum, and nuclear processed material. In the distant Universe, they dominate the ultra-violet (UV) light from young Galaxies. Indeed, massive stars may be regarded as ‘cosmic engines’ (Bresolin et al., 2008). Hot, massive stars possess strong and powerful winds that affect evolutionary time scales, chemical surface abundances, and luminosities. In fact, changing the mass-loss rates of massive stars by only a factor of two has a dramatic effect on their overall evolution (Meynet et al., 1994). The winds from these stars are described by the radiative line-driven wind theory, in which the standard model (based on the pioneering works by Lucy & Solomon, 1970; Castor et al., 1975) assumes the wind to be stationary, spherically symmetric, and homogeneous. Despite this theory’s apparent success (e.g., Vink et al., 2000), theoretical as well as observational evidence for an inhomogeneous, time-dependent wind has over the past years become overwhelming (for a comprehensive summary, see Puls et al., 2008b).

Direct simulations of the time-dependent wind have confirmed that the so-called line-driven instability causes a highly structured wind in both density and velocity (Owocki et al., 1988; Feldmeier, 1995; Dessart & Owocki, 2005). Much indirect evidence of such small-scale inhomogeneities (clumping) has arisen from quantitative spectroscopy. Clumping has severe consequences for the interpretation of observed spectra, with the inferred mass-loss rates particularly affected. When deriving mass-loss rates from observations, wind clumping has traditionally been accounted for by assuming optically thin clumps and a void inter-clump medium, while keeping a smooth velocity field. Results based on this microclumping approach have, for example, led to a downward revision of empirical mass-loss rates from Wolf-Rayet (WR) stars by roughly a factor of three (reviewed in Crowther, 2007).

However, for O stars, highly clumped winds with very low mass-loss rates must be invoked in order to reconcile investigations of different diagnostics within the microclumping model. The most alarming example was the phosphorusV(PV) UV analysis by Fullerton et al. (2006), which indicated reductions of previously accepted values by an order of magnitude (or even more), with dwarfs, giants, and supergiants all affected (but see also Waldron & Cassinelli 2010, who argued thatXUVradiation could seriously alter the ionization fractions of PV). Such low mass-loss rates would be in stark contrast with the predictions of line-driven wind theory, and have dramatic consequences for the evolution of, and feedback from, massive stars. Naturally, the widely discrepant values inferred from different observations and diagnostics drastically lower the reliability of mass-loss rates currently in use, and an explanation is urgently needed. A key question is: Does the microclumping model fail to deliver accurate empirical rates under certain conditions?