Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://star.arm.ac.uk/~jsv/complete.pdf Äàòà èçìåíåíèÿ: Mon Apr 2 20:50:05 2007 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 22:01:33 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ngc 2440

Radiation-driven wind models of massive stars


Omslag: Hubble Space Telescope - WFPC2 picture of NGC 3603. Credit: Wolfgang Brandner, Eva K. Grebel, You-Hua Chu, and NASA.

c Copyright 2000 J.S. Vink Printed by Ponsen & Looijen, Wageningen Alle rechten voorbehouden. Niets van deze uitgave mag worden verveelvoudigd, opgeslagen in een geautomatiseerd gegevensbestand, of openbaar gemaakt, in enige vorm, zonder schriftelijke toestemming van de auteur. ISBN 90 393 2559 6


Radiation-driven wind models of massive stars
Stralingsgedreven winden van massieve sterren
met een samenvatting in het Nederlands

Proefschrift
ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof. Dr. H. O. Voorma, ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op maandag 20 november 2000 des middags te 12.45 uur door

Jorick Sandor Vink
geboren op 27 januari 1973 te Goirle


Promotor: Co-Promotor:

Prof. Dr. H.J.G.L.M. Lamers
Sterrenkundig Instituut, Universiteit Utrecht

Dr. A. de Koter
Sterrenkundig Instituut, Universiteit van Amsterdam

Dit proefschrift werd mede mogelijk gemaakt door financiele steun van de Nederlandse Organ¨ isatie voor Wetenschappelijk Onderzoek (NWO).


Contents

Contents
1 Intro 1 .1 1 .2 1 .3 duction Massive stars in the cosmos . . . . . . . . . . . . . The winds from massive stars . . . . . . . . . . . . The status of the radiation-driven wind theory . . . 1.3.1 Achievements of the radiation-driven wind t 1.3.2 Open issues in radiation-driven wind theory 1.4 Studies in this thesis . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . 5 5 6 9 9 10 12 14 17 17 18 18 21 22 24 25 25 26 27 33 34 35 38 38 41 42 43 43 44 46 48 48 50 51 51

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2 The Physics of the line acceleration 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2 Standard radiation-driven wind theory (CAK theory) 2.2.1 The radiative force . . . . . . . . . . . . . . 2.2.2 The equation of motion . . . . . . . . . . . . 2.2.3 The Solution of the equation of motion . . . 2.3 Multiple Scattering . . . . . . . . . . . . . . . . . . 2.4 The unified model . . . . . . . . . . . . . . . . . . . 2.4.1 The model atmospheres . . . . . . . . . . . 2.4.2 The Modified nebular approximation . . . . 2.4.3 The Monte Carlo method . . . . . . . . . . . 2.5 The determination of M in a Unified Wind model . . 2.5.1 The determination of the global mass-loss rate 2.5.2 Self-consistent solutions . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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3 On the nature of the bi-stability jump in the winds of early-type supergiants 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 What determines M and V ? . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The theory of M determination . . . . . . . . . . . . . . . . . . . . 3.2.2 A simple numerical experiment: the sensitivity of M on the subsonic 3.2.3 The effect of an increased M on V . . . . . . . . . . . . . . . . . ........................... 3.3 The method to predict M 3.3.1 Momentum transfer by line scattering . . . . . . . . . . . . . . . . 3.3.2 Scattering and absorption processes in the MC calculations . . . . . 3.3.3 The calculation of the radiative acceleration gL (r) . . . . . . . . . . 3.3.4 The determination of M . . . . . . . . . . . . . . . . . . . . . . . 1

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3 .4 3.5 The model atmospheres . . . . . . . . . . . . . . . . . . . The predicted bi-stability jump . . . . . . . . . . . . . . . 3.5.1 The predicted bi-stability jump in M . . . . . . . . 3.5.2 The predicted bi-stability jump in . . . . . . . . The origin of the bi-stability jump . . . . . . . . . . . . . 3.6.1 The main contributors to the line acceleration . . . 3.6.2 The effect of the Fe ionization . . . . . . . . . . . 3.6.3 The effect of Teff on M . . . . . . . . . . . . . . . 3.6.4 The effect of V . . . . . . . . . . . . . . . . . . . 3.6.5 A self-consistent solution of the momentum equatio 3.6.6 Conclusion about the origin of the bi-stability jump Bi-stability and the variability of LBV stars . . . . . . . . Summary, Discussion, Conclusions & Future work . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 3 4 4 6 7 8 8 0 2 3 5 5 6 7 9 9 0 1 3 5 7 7 8 8 8 1 3 3 4 4 5 6 6

3.6

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New 4.1 4.2 4 .3

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theoretical mass-loss rates of O and B stars Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Method to calculate M . . . . . . . . . . . . . . . . . . . . The predicted mass-loss rates . . . . . . . . . . . . . . . . . 4.3.1 M for supergiants in Range 1 (30 000 Teff 50 000 4.3.2 M at the bi-stability jump around 25 000 K . . . . . 4.3.3 M for supergiants in Range 2 (12 500 Teff 22 500 4.3.4 M at the second bi-stability jump around 12 500 K . The wind momentum . . . . . . . . . . . . . . . . . . . . . 4.4.1 The wind efficiency number . . . . . . . . . . . . 4.4.2 The importance of multiple scattering . . . . . . . . 4.4.3 The Modified Wind Momentum . . . . . . . . . . Mass loss recipe . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Range 1 (30 000 Teff 50 000 K) . . . . . . . . . 4.5.2 Range 2 (15 000 Teff 22 500 K) . . . . . . . . . 4.5.3 The complete mass-loss recipe . . . . . . . . . . . . 4.5.4 The dependence of M on the steepness of the velocity Comparison between theoretical and observational M . . . . 4.6.1 M comparison for Range 1 (27 500 < Teff 50 000 K 4.6.2 Modified Wind momentum comparison for Range 1 (27 500 < Teff 50 000 K) . . . . . . . . . . . . . . 4.6.3 Modified Wind momentum comparison for Range 2 (12 500 Teff 22 500 K) . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary & Conclusions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . and .. .. .. m od B . . . el stars ... ... ... grid as .. .. .. .. 2 a . . . . function ..... ..... ..... ..... of .. .. .. ..

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Mass-loss predictions for O 5.1 Introduction . . . . . . 5.2 Theoretical context . . 5.3 Method to calculate M 5.4 The assumptions of the

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Contents
5.5 The predicted mass-loss rates and bi-stability jumps . . . . . . . . . 5.5.1 The bi-stability jump at Teff 25 000 K . . . . . . . . . . . 5.5.2 Additional bi-stability jumps around 15 000 and 35 000 K . 5.5.3 The origin of the (low Z )jump at Teff 35 000 K . . . . . . The relative importance of Fe and CNO elements in the line accelerat 5.6.1 The character of the line driving at different Z . . . . . . . . 5.6.2 Observed abundance variations at different Z . . . . . . . . The global metallicity dependence . . . . . . . . . . . . . . . . . . Complete mass-loss recipe . . . . . . . . . . . . . . . . . . . . . . Comparison between theoretical M and observations at subsolar Z . 0 Summary & Conclusions . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . at . . . K . . low ... ... ... ... ... ... . . . . io . . . . . . . . . . . n . . . . . . . . . . . at . . . . . . . . . . . l . . . . . . . .. .. .. .. ow .. .. .. .. .. .. .. 101 104 107 108 Z 108 108 110 111 113 115 117 118 121 121 122 124 125 125 126 127 127 129 130 130 131 131 133 134 134 136 137 138 139 139 140 142 144 146 148 149 149 151 151 153

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5.7 5 .8 5.9 5 .1

Research note on the bi-stability jump in the winds of hot stars 5.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 5.11.2 The line driving of CNO . . . . . . . . . . . . . . . 5.11.3 The ionization of carbon around Teff 35 000 K . . 5.11.4 The line acceleration of carbon around Teff 35 000 5.11.5 Summary & Discussion . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The 6.1 6 .2 6.3

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radiation driven winds of rotating B[e] supergiants Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical context . . . . . . . . . . . . . . . . . . . . . . . . . . . . The physics of rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The shape of a rotating star . . . . . . . . . . . . . . . . . . . . 6.3.2 Von Zeipel gravity darkening . . . . . . . . . . . . . . . . . . . 6.3.3 The equation of motion of a line driven wind of a rotating star . 6.3.4 The radiative line forces . . . . . . . . . . . . . . . . . . . . . Solutions of the equation of motion . . . . . . . . . . . . . . . . . . . . 6.4.1 Simplified solutions for non-rotating star . . . . . . . . . . . . 6.4.2 Solution of the equation of motion for the wind of a rotating star 6.4.3 The calculation of Dfd (r) and the continuum correction factor Dc 6.4.4 Solving the equation of motion . . . . . . . . . . . . . . . . . . Application to B[e] winds . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 A typical B[e] supergiant . . . . . . . . . . . . . . . . . . . . . 6.5.2 The overall density properties . . . . . . . . . . . . . . . . . . 6.5.3 Varying L . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotationally induced bi-stability models . . . . . . . . . . . . . . . . . Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bl . . . . ue . . . . Va .. .. .. .. riables .... .... .... ....

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7 Radiation-driven wind models for Luminous 7.1 Introduction . . . . . . . . . . . . . . . . 7.2 The method to calculate M . . . . . . . . 7.3 The assumptions of the LBV-like models . 7.4 The predicted mass-loss rates of LBVs . . 3

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Contents
7.4.1 The effect of the lower masses on M . . . . . . . . . 7.4.2 The effect of helium enrichment on mass loss . . . . 7.4.3 The effect of the nitrogen enrichment on M . . . . . 7.4.4 The complete grid of mass-loss rates for LBVs . . . 7.4.5 Uncertainties in the locations of the bi-stability jumps Comparison between LBV predictions and observations . . . Discussion and Conclusions . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 1 1 1 5 5 5 5 5 5 6 6 4 5 6 8 8 8 1 2

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Summary and P 8.1 Summary 8.2 Prospects References .

rospects 165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 1 1 1 1 1 1 1 6 6 6 7 7 7 7 9 9 9 0 0 1 1

Samenvatting Wat zijn massieve sterren? . . . . De rol van zwaartekracht en gasdru Wat zijn sterwinden? . . . . . . . Stralingsgedreven sterwinden . . . Het probleem voor dit proefschrift Het resultaat van dit proefschrift . Publication List Curriculum vitÔ Dankwoord

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4


Introduction

1

Introduction
Before discussing the motivations for a study of stellar winds from early-type stars, I would like to place the role of massive stars in a broad astrophysical context. Star counts in the solar neighbourhood reveal that there are more low mass stars ­ like the sun ­ than massive stars. The reason for the rarity of massive stars is on the one hand, the short evolutionary timescale for massive stars, and on the other hand the shape of the Initial Mass Function (IMF), as star counts over the past decades have shown the IMF to be rather steep (Salpeter 1955, Scalo 1986). This implies that at the present cosmological epoch Nature seems to favour the formation of lower mass stars over the formation of massive stars.

1.1 Massive stars in the cosmos
Recent theoretical studies suggest that this scenario need not have been the case in the past. In the early universe, when primordial elements left over from the Big Bang were the only constituents of the cosmos, Nature may have operated in the opposite way, and massive stars may have preferentially been formed over low mass stars (e.g. Carr et al. 1984, Larson 1998). Some of these arguments are based on the fact that at extremely low metallicity, the Jeans mass is expected to be higher. Additional evidence for a top-heavy IMF at earlier times comes from arguments that such different IMF can solve a long-standing issue known as the "G dwarf problem" (Pagel & Patchett 1975): `if the IMF has always been as it is today, why don't we observe any extreme-metal poor G stars ?' Recent numerical simulations by Bromm et al. (1999) support the idea of a top-heavy IMF as they present evidence for a primordial IMF favouring the formation of very massive population III stars with masses on the order of 100 M , and higher. The question of how massive these first stars really were, is an open one, which will have to be answered with the help of observations. For instance, future observations with the Next Generation Space Telescope (NGST) of distant stellar populations at high redshifts may either confirm or exclude such a heavy IMF (for a discussion, see Bromm et al. 2000). Note that in case such observations confirm the dominance of massive stars in the early universe then these first stars will probably have produced large amounts of ionizing photons and mechanical energy in the early days of galaxy formation. In this case, massive stars become of relevance to the cosmological question concerning the "reionization" of the universe, referring to the period when the universe changed from being cold and dark to a status in which it became reionized by ultraviolet (UV) radiation (Madau 2000). As long as the question on the nature of the first stars is unanswered, it is valid to ask ourselves what role massive stars play in the present day universe, namely the part of the cosmos that is directly accessible with today's observational techniques. In the following I will argue 5


Chapter 1 that although massive stars are rare, they play an important, and in some respects even a dominant role in the physical conditions of galaxies and the "life cycle" of gas and dust (for a nice example, see the Hubble Space Telescope picture of NGC 3603 on the front page of this thesis). First of all, massive stars play a role in the chemical enrichment of the Interstellar Medium (ISM). Since chemical elements are produced in the interiors of stars with different masses, they enrich the ISM on different timescales. Massive stars especially contribute to the enrichment of oxygen and other heavy elements. Since the lifetimes of massive stars are so short, the recycling of these elements is a very efficient process and the impact on chemical evolution models therefore becomes significant. Apart from their role on the chemical enrichment of galaxies, massive stars play a major dynamical role, as they are responsible for a large amount of momentum and energy input into the ISM due to stellar winds and supernova explosions. See Abbott (1982b) and Leitherer et al. (1992) for an overview. Moreover, as massive stars are hot, with effective temperatures in the range of about 10 000 - 50 000 K, they emit a large amount of UV photons. This radiation ionizes the surrounding nebula and heats the associated H II region. As massive stars are mostly seen grouped in young clusters, wind-blown bubbles around these stars interact with each other and subsequently evolve into superbubbles. These superbubbles are thought to be places for the propagation of new star formation (see e.g. Oey & Massey 1995). As massive stars live for only a short amount of time, evol 107 years, studies of bubbles and superbubbles yield important information on the physical conditions of the poorly-understood process of star formation itself. A final property of massive stars that attracts attention is their intrinsic brightness. Due to this brightness they can in principle be used to derive distances to extra-galactic systems. Kudritzki et al. (1995) have shown that the observed wind momentum is proportional to the stellar luminosity. This implies that by using spectroscopic tools it is possible to determine the stellar luminosity from just the emergent spectrum. The use of this "Wind momentumLuminosity Relation (WLR)" can provide distances to extra-galactic objects, which may give massive stars the status of extra-galactic standard candles. The method of the WLR should be reliable up to distances as far as the Virgo and Fornax clusters of galaxies within the Hubble flow (Kudritzki 1998), and may therefore even help to constrain the Hubble Constant. From the above, it is clear that studies of massive stars play an important role in a broader context, which starts off in the local environments immediately surrounding massive stars, via their combined chemical and dynamical impact on starbursting galaxies, up to even cosmological issues. In the next section, I will concentrate on the role that stellar winds play on their environments as well as on the evolution of the star itself. This will be followed by a short historical overview of radiation-driven winds, including the achievements of the theory as well as its remaining problems. In the last part of this introduction, I will concentrate on these open issues and describe the way in which I will attack these questions in this thesis.

1.2 The winds from massive stars
In this section, I would like to justify this study on stellar winds of massive stars, with special emphasis on why it is useful to compute quantitative mass-loss rates as a function of stellar parameters. The various aspects of stellar winds from massive stars will be divided into four subjects, these are the following: 1. The impact of a wind on its emergent spectrum 6


Introduction 2. The influence of mass loss on stellar evolution 3. The impact of winds on their environment 4. The physics of radiation hydrodynamics Emergent spectra Stellar winds have a pronounced effect on the emergent spectrum. Massive stars show clear signatures of outflow in their UV and optical spectra, in the form of blue-shifted absorption lines, P Cygni profiles and emission lines. The velocity field in the wind affects both the density structure and the transport of radiation in the atmospheres. For this reason it is not sufficient to use hydrostatic model atmospheres for quantitative spectroscopy of massive stars. Instead, one should switch to the use of "unified" models for these objects (Gabler et al. 1989, Hillier 1991, de Koter 1993). In a unified model an artificial distinction between a photospheric "core" and a separate "halo" for the wind region (core-halo), is avoided. Note that a neglect of the stellar wind on the atmospheres of hot stars can lead to substantial errors in its basic stellar parameters. For instance, the derivation of stellar masses from log g determinations, is heavily influenced by winds. Although more than half of the stars are part of a binary system, there are hardly any mass determinations for massive stars from binaries, since massive stars are relatively rare. Therefore mass determinations using spectroscopic tools become necessary. For this reason, it is vital to know the influence of the winds on the emergent spectra. The issue of stellar masses from massive stars is especially intriguing since there is a problem in the astrophysical literature referred to as the "mass discrepancy" of massive stars (Herrero et al. 1992). It has been found that spectroscopically derived masses differ significantly from evolutionary masses. Up to now, the issue is not solved, although the situation has improved somewhat (see Lanz et al. 1996). Evolution The role of mass loss on the evolution of massive stars has been reviewed extensively in the literature (see e.g. Chiosi & Maeder 1986). The main impact of stellar mass loss on massive star evolution is its influence on evolutionary tracks and surface abundances. Additionally, mass loss determines the final mass of the star, and it is this quantity that determines whether a neutron star or a black hole is formed. I do not intend to describe the complete evolution of massive stars, but I would like to stress some evolutionary aspects which are relevant for this study. Basically, the evolution of an isolated star is determined by (1) the initial mass M , (2) its metallicity Z , and (3) initial rotation, the role of which has recently gained attention (e.g. Langer 1998). A schematic overview of the basic evolutionary scenario of single massive stars is presented in Fig. 1.1. It is generally assumed that a massive star evolves off the Main Sequence to lower effective temperature and that this occurs at approximately constant luminosity. In the Hertzsprung-Russell Diagram (HRD) this can be represented by a horizontal shift to the right (see Fig. 1.1). The role of Wolf-Rayet (WR) stars as the final stage of massive stellar evolution, during which the star explodes as a supernova, is well established (Lamers et al. 1991). The evolution of massive stars between their Main Sequence and WR phase, is less secure, though it is generally assumed that massive stars pass through a short, unstable phase, in which the star loses a substantial amount of mass. This unstable stage is referred to as the Luminous Blue Variable (LBV) phase (see Nota and Lamers 1997 for an overview). 7


Chapter 1

Figure 1.1: Hertzsprung-Russell Diagram (HRD) for massive stars. The major evolutionary stages are indicated in the plot: the zero-age main sequence (ZAMS), the Luminous Blue Variable phase (LBV) and the final Wolf-Rayet phase (WR). Additionally the Humphreys-Davidson (HD) limit is displayed. During its life, a massive star loses a considerable amount of mass. For instance, a star with initially 60 M on the Main Sequence is expected to end up as a 6 M WR star (Meynet et al. 1994). A large part of this mass is lost during the LBV and WR stage. Nevertheless, one should note that while the star is still on the Main Sequence, it may already suffer substantial mass loss and this is especially relevant in regard to another topical issue in massive star evolution concerning the possible existence of a Red Supergiant (RSG) phase. Several recent studies have questioned the physical existence of the so-called Humphreys-Davidson (HD) limit (see Fig. 1.1). Above this limit no stars have been found (Humphreys & Davidson 1979). On the one hand, Voors et al. (2000) and Smith et al. (1998) suggest that perhaps even the most massive stars have gone through a RSG phase, implying they may have passed the "forbidden" HD limit. On the other hand, from a recent study of nebulae around LBVs and WR stars Lamers et al. (2000) conclude that these massive stars have not gone through such a phase. Whether or not massive stars go through a RSG phase, is still under debate, but accurate knowledge of mass loss as a function of stellar parameters is certainly expected to help in answering this question, as the outcome critically depends on the exact amount of mass loss in prior phases of evolution. Environments The impact of winds on their environments was already discussed in the previous section. The cumulative effect of winds from massive stars plays a major role in both the chemical and the dynamical evolution of the ISM. This mechanical input from both winds and supernova probably results in energetic outflows from galaxies. Such energetic outflows may be responsible for the phenomenon called "the Galactic fountain" in our host galaxy, and are also observed in star forming galaxies at high redshift (Pettini et al. 1998) as well as in local starbursts (Kunth et al. 8


Introduction 1998). Radiation hydrodynamics The term "radiation hydrodynamics" probably deserves some additional explanation. One may talk about radiation hydrodynamics when radiation plays a dominant role in the energy and momentum balance of an astrophysical plasma. In this thesis mass-loss rates will be determined from the calculation of the radiative acceleration in the winds from massive stars. In the next chapter, it will be shown how a large reservoir of photons, coming from the star is able to "drive" the stellar wind. Note that radiation-driven stellar winds are not the only objects where the physical process of "radiation hydrodynamics" plays a dominant role. There are also other astrophysical objects where large numbers of photons are available that may deposit radiative momentum on matter in large amounts. Such exotic objects are e.g. accretion disks, quasars and Active Galactic Nuclei. The special advantage of the study of massive stars compared to extra-galactic objects, is that stars are relatively close-by, and this implies that high resolution information can be more easily obtained, which may also teach us about the coupling between photons and plasma in a more general astrophysical context.

1.3 The status of the radiation-driven wind theory
The development of the radiation-driven wind theory which has proven to be very successful in explaining the mass loss from massive stars, already started in the 1920's with a series of papers by e.g. Milne (1926). Milne realised that radiation could be coupled to ions which may subsequently eject the ions from the stellar surface. Radiation pressure as a driving mechanism for stellar outflow came back into the picture some 40 years later, when Morton (1967) discovered P Cygni profiles in the spectra of supergiants in Orion indicating substantial mass loss. The first accurate mass-loss determinations from UV lines were made by Lamers & Morton (1976) and Lamers & Rogerson (1978).

1.3.1 Achievements of the radiation-driven wind theory
New theoretical work on stellar winds was started by Lucy & Solomon (1970), who identified line scattering as the mechanism that could drive stellar winds. One should note that these authors predicted mass-loss rates that were too low compared to the observations, as they assumed that only a few optically thick lines were present. The situation improved due to the landmark paper by Castor et al. (1975, hereafter CAK), who included an extensive line list, with 105 lines, and therefore predicted significantly larger values for the mass-loss rate. They realised that these large mass-loss rates could seriously alter evolutionary tracks of massive stars. Moreover, CAK solved the momentum equation of the stellar wind in a self-consistent way and could therefore also predict the terminal flow velocity of the stellar wind. Further modifications of the CAK theory included even more elaborate line lists (Abbott 1982a) and the inclusion of a finite disk correction factor, which allowed photons to stream from the entire stellar disk instead of only radially (Friend & Abbott 1986, Pauldrach et al. 1986). Additionally, occupation numbers were calculated in non-LTE (Pauldrach et al. 1994), further refining the theory. The basic properties of these modern-era CAK-like wind models are the following: the wind is assumed to be spherically symmetric, and homogeneous, i.e. clumps are not taken into 9


Chapter 1 account. In addition, the wind is stationary and therefore mass loss is assumed to be constant in the models. However, wind variability is a well-known phenomenon both observationally as well as theoretically (see Wolf et al. 1998 for an overview). Furthermore, the emission of X-rays for single O stars (Harnden et al. 1979) as well as the presence of black troughs in UV P Cygni line profiles indicate that the winds of O stars are not smooth. In time-independent models structured winds are usually not properly taken into account. Nevertheless, Owocki et al. (1988) have shown the CAK steady-state solution to be a quite good approximation for the time-averaged wind. Other physical ingredients, such as magnetic fields, rotation and multiply scattered photons are usually not included in standard wind models. Although a lot of fundamental theoretical work on all of these items has been carried out over the past decades (see respectively Friend & MacGregor 1984, Friend & Abbott 1986, Puls 1987). It remains to be seen whether these physical ingredients will ultimately be proven to play a significant role. We do not intend to handle all these issues in this thesis, but we will concentrate on the importance of multiple scattering. Finally, I would like to conclude with, in my opinion, the main achievement of CAK linedriven wind theory. CAK wind solutions predict the terminal flow velocity to be proportional to the escape velocity and the mass-loss rate to depend strongly on the stellar luminosity. Obser vations over the past decades have shown that these basic wind parameters, M and V , indeed behave as predicted by CAK. This basic agreement between observations and theory provides strong evidence that the winds from massive stars are driven by radiation pressure and this has given the CAK theory a well-established status in the hot-star community.

1.3.2 Open issues in radiation-driven wind theory
In this work three open issues in radiation-driven wind theory will be investigated, they are: 1. The bi-stability jump 2. The "momentum problem" in radiation-driven winds 3. The metallicity dependence of radiation-driven winds Ad. 1 The bi-stability jump (Pauldrach & Puls 1990) was observed by Lamers et al. (1995) in a large sample of spectra by the International Ultraviolet Explorer (IUE) satellite. This jump is represented by a dramatic decrease in the terminal flow velocity from V 2.6 Vesc for supergiants of types earlier than B1 to V 1.3 Vesc for those later than B1. The jump is displayed in Fig. 1.2. Figure 1.3 displays spectra of several stars at this jump temperature around spectral type B1. The spectra indicate that for those stars where V 2.6 Vesc the wind ionization state is high (i.e. strong C IV and weak C II lines), whereas for those stars where V 1.3 Vesc , the wind ionization state is low. (i.e. strong C II and weak C IV lines). This demonstrates that the steep jump in the terminal velocity is accompanied by a change of the ionization state in the wind. If the ionization in the wind changes so abruptly, one may also expect this to have an effect on the mass-loss rate. However, standard radiation-driven wind theory has not predicted this jump in the terminal velocity, nor is it known what happens to the mass-loss rate at spectral type B1. 10


Introduction

Figure 1.2: The bi-stability jump in winds from early-type stars. Note the clear jump at spectral type B1 (Teff 21 000 K), where the ratio V /Vesc drops from 2.6 to 1.3. A second jump may be present at spectral type A0 (Teff 10 000 K) where V /Vesc drops from 1.3 to 0.7. The data are taken from Lamers et al. (1995).

Ad. 2 The second problem that will be investigated in this study concerns the momentum problem in radiation-driven wind theory. Over the last decade, the predicted mass-loss rates for O stars have shown a persistent discrepancy with the observed mass-loss rates (Lamers & Leitherer 1993, Puls et al. 1996). The discrepancy is displayed in Fig. 1.4. There are both theoretical as well as observational reasons to believe that multiple scattering is important. The best theoretical reason is that the spectra show a considerable line overlap (see e.g. Puls 1987), which subsequently offers photons the possibility to multiply scatter. Observational evidence for the importance of multiple scattering is also present. The most striking example is the "momentum problem" in WR stars. This is a well-known property of the winds of WR stars, as they reveal wind efficiencies that substantially exceed the "singlescattering limit". In this limit, = 1, every stellar photon transfers its momentum just once to the wind material. For WR stars values as high as = 10 have been reported (Schmutz et al. 1989, Willis 1991), which may be explained by multiple scattering of photons in an atmosphere with a strong ionization stratification (Lucy & Abbott 1993, Springmann 1994, Gayley et al. 1995). Apart from these extreme properties in WR winds, note that there is also observation evidence for luminous O stars violating the "single-scattering limit" (see Lamers & Leitherer 1993). Yet, realistic wind models including multiple scattering have only been performed so far for one O star Pup (Abbott & Lucy 1985, Puls 1987), but these simulations suffered from an unrealistic separation between the stellar core and the surrounding wind (core-halo approach). The computation of mass-loss rates over a wide range in stellar parameters, including multi-line effects, certainly seems useful. 11


Chapter 1

Figure 1.3: IUE Spectra of several stars at spectral type B1 show that for those stars where V 2.6 Vesc the wind ionization state is high (strong C IV and weak C II lines), whereas for those stars where V 1.3 Vesc , the wind ionization state is low (strong C II and weak C IV lines). Figure reconstructed from Lamers et al. (1995). Ad. 3 A third item which is more or less an open issue in line-driven wind theory is the dependence of the winds on metallicity. Observational evidence for metallicity dependent stellar wind properties was found by Garmany & Conti (1985), but from the theoretical side only a few models have been computed (Abbott 1982a, Kudritzki et al. 1987, Leitherer et al. 1992). Unfortunately, not only did these models not account for multiple-scatterings, but they also used a core-halo approach (and its associated shortcomings). Hence, a new theoretical study of mass loss as a function of metallicity, is appropriate.

1.4 Studies in this thesis
The studies presented in this thesis address the three open issues that have been described above. To investigate these problems radiation-driven wind models are computed, using Monte Carlo simulations. As has already been noted, standard radiation-driven wind models, from which the models by Pauldrach et al. (1994) and Taresch et al. (1997) represent the current state12


Introduction

Figure 1.4: Comparison between theoretical and observed mass-loss rates for O stars. The data are taken from Lamers & Leitherer (1993). Note the systematic discrepancy between the observations and the theory. The solid line indicates where the points should fall if observations and theory would be in perfect agreement. of-the-art, suffer from certain assumptions, which will be avoided in our approach. The main improvements in the computation of the wind models are to relax the "core-halo" structure, and to handle the photosphere and the wind in a unified way. In addition multi-line effects will be naturally included by performing Monte Carlo simulations. In chapter 2, our approach to calculate radiation-driven wind models and mass-loss rates will be extensively described, starting with the basic physics from standard CAK theory. In chapter 3, the bi-stability jump will be investigated in detail. A series of wind models is calculated to investigate the behaviour of the mass loss as a function of effective temperature across the jump. Furthermore, the physical origin of the jump will be studied. In chapter 4, the grid of wind models is extended and mass-loss rates as a function of stellar parameters are computed. This results in a mass-loss recipe and a comparison with the best available observations. The predictions turn out to be in good agreement with the observed mass-loss rates. In chapter 5, the grid will be extended to metallicities different from the solar value and the mass-loss recipe is also extended to incorporate this dependency. A comparison between our predictions and observed mass-loss rates in the Small Magellanic Cloud with a metallicity of about 1/10 (Z /Z ), yields additional support for our wind models. It will turn out that at very low metallicity the ions that drive the winds, are different from the ions that drive the winds in the Galaxy. In a separate research note, following this chapter, a new bi-stability jump, which is only present in our models at very low metallicity, but at higher temperature, will be studied. In chapter 6, we will investigate radiation-driven winds of rotating stars with respect to the mysterious presence of disks around rapidly rotating B[e] stars. It will be shown that our bi13


Chapter 1 stable wind models are able to induce a density difference between the pole and the equator of a factor of ten. Finally, in chapter 7, the grid of wind models will be extended to LBVs. LBVs have the unique property that during typical variations in radius and temperature they are expected to cross bi-stability jumps, where the line driving suddenly switches, inducing mass-loss changes. This behaviour will be compared to the behaviour of the best-observed LBVs.

References
Abbott D.C., 1982a, ApJ 259, 282 Abbott D.C., 1982b, ApJ 263, 723 Abbott D.C., Lucy L.B., 1985, ApJ 288, 679 Bromm V., Coppi P.S., Larson R.B., 1999, ApJ 527, 5 Bromm V., Kudritzki R.-P, Loeb A., 2000, submitted, astro-ph/0007248 Carr B.J., Bond J.R., Arnett W.D., 1984, ApJ 277, 445 Castor J.I., Abbott D.C., Klein R.I., 1975, ApJ 195, 157 Chiosi C., Maeder A., 1986, ARA&A 24, 329 de Koter A., 1993, PhD thesis at Utrecht University Friend D.B., McGregor K.B., 1984, ApJ 282, 591 Friend D.B., Abbott D.C., 1986, ApJ 311, 701 Gabler R., Gabler A., Kudritzki R.-P., Puls J., 1987, Pauldrach A.W.A., 1989, A&A 226, 162 Gayley, K.G., Owocki S.P., Cranmer S.R., 1995, ApJ 442, 296 Garmany C.D., Conti P.S., 1985, ApJ 293, 407 Harnden F.R. Jr., Branduardi G., Gorenstein P., 1979, ApJ 234, 51 Herrero A., Kudritzki R.-P., Vilchez J.M. et al., 1992, A&A 261, 209 Hillier, J., 1991, in:"Wolf-Rayet stars and Interrelations with other Massive Stars in Galaxies", eds. van der Hucht K.A., Hidayat B, IAU Symp. 143, 59 Humphreys, R., Davidson K., 1979, ApJ 232, 409 Kudritzki R.-P, 1998, in: "Variable and Non-spherical Stellar Winds in Luminous Hot Stars", Lecture notes in Physics, IAU Coll.no. 169, 405 Kudritzki R.-P., Pauldrach A.W.A., Puls J., 1987, A&A 173, 293 Kudritzki R.-P, Lennon D.J., Puls J., 1995, in: "Science with the VLT", eds. Walsh J.R., Danziger I.J., Springer Verlag, p. 246 Kunth D., Mas-Hesse J.M., Terlevich R., et al., 1998, A&A 334, 11 Lamers H.J.G.L.M., Morton D.C., 1976, ApJS 32, 715 Lamers H.J.G.L.M., Rogerson J.B., 1978, A&A 66, 417 Lamers H.J.G.L.M., Leitherer, C., 1993, ApJ 412, 771 Lamers H.J.G.L.M., Maeder A., Schmutz W., Cassinelli J.P., 1991, ApJ 368, 538 Lamers H.J.G.L.M., Snow T.P., Lindholm D.M., 1995, ApJ 455, 269 Lamers H.J.G.L.M., Nota A., Panagia N., Smith L., Langer N., 2000, in press Langer N., 1998, A&A 329, 551 Lanz, T., de Koter A., Hubeny I., Heap S.R., 1996, ApJ 465, 359 Larson R.B., 1998, MNRAS 301, 569 Leitherer C., Robert C., Drissen L.,1992, ApJ 401, 596 Lucy L.B., Solomon P., 1970, ApJ 159, 879 Lucy L.B., Abbott D.C., 1993, ApJ 405, 738 14


Introduction Madau P., 2000, in press, astro-ph/0003096 Meynet G., Maeder A., Schaller G., Schearer D., Charbonel C., 1994, A&AS 103, 97 Milne E.A., 1926, MNRAS 86, 459 Morton D.C., 1967, ApJ 150, 535 Nota A., Lamers H.J.G.L.M., 1997, Luminous Blue Variables: Massive Stars in Transition, ASP Conf.Ser. 83. Oey M.S., Massey P., 1995, ApJ 452, 210 Owocki S.P., Castor J.I., Rybicki G.B., 1988, ApJ 335, 914 Pagel B.E.J., Patchett B,E., 1975, MNRAS, 172, 13 Pauldrach A.W.A., Puls J., Kudritzki R.P., 1986, A&A 164, 86 Pauldrach A.W.A., Puls J., 1990, A&A 237, 409 Pauldrach A.W.A., Kudritzki R.P., Puls J., Butler K., Hunsinger J.,1994, A&A 283, 525 Pettini M., Kellogg M., Steidel C.C., et al., 1998, ApJ 508, 539 Puls J., 1987, A&A 184, 227 Puls J., Kudritzki R.P., Herrero A., et al., 1996, A&A 305, 171 Salpeter, E.E., 1955, ApJ 121, 161 Scalo, J., Fund. Cosm. Phys. 11, 1 Schmutz W., Hamann W.-R., Wessolowski U., 1989, A&A 210, 236 Smith L.J., Nota A., Pasquali A., et al., 1998, ApJ 503, 278 Springmann U., 1994, A&A 289, 505 Taresch, G., Kudritzki, R.P., Hurwitz, M., et al., 1997, A&A 321,531 Voors et al., 2000, in press Willis A.J., 1991, in: "Wolf-Rayet stars in interaction with other massive stars in galaxies", eds. van der Hucht K.A., Hidayat B., IAU Symp 143, 265 Wolf, B., Stahl, O., Fullerton A.W., 1998, Variable and Non-spherical Stellar Winds in Luminous Hot Stars, Lecture notes in Physics, IAU Coll.no. 169

15


Chapter 1

16


The Physics of the line acceleration

2

The Physics of the line acceleration
2.1 Introduction
In this chapter, the basic physical ingredients that play a role in the stellar winds of massive stars will be discussed. The chapter serves as a guideline to the approach that we have used to calculate the radiative acceleration and mass-loss rates of the various wind models. The main goal of this chapter is to show how the mass-loss rate is physically related to the radiative acceleration and how these quantities are computed. In short, the approach is as follows: model atmospheres are calculated to obtain the occupation numbers of the different species, which subsequently serve as an input into a Monte Carlo code in order to compute the radiative acceleration due to photon interactions with the gas. The cumulative effect of these interactions eventually yields the mass-loss rate. The main difficulty of the dynamics of radiation-driven winds is that the line acceleration gL depends on the velocity gradient (dV /dr), but in turn the velocity law V (r) (and therefore also dV /dr) also depends on gL . Due to this non-linear character, the dynamics of line-driven winds are complicated. Fortunately, observational analyses of early-type stars that have been performed over the last decades provide quite accurate information on the values of the terminal flow velocities. These observational values will be used as constraints for the dynamics of the winds. Throughout a large part of this thesis, we will adopt a velocity law V (r) based on these observational constraints and we will only predict mass-loss rates. To check whether such a "global" approach is justified, we have also employed a "self-consistent" approach for some representative wind models. In this approach, we obtain the mass-loss rate and the velocity law simultaneously. It is reassuring to find that the calculated terminal velocities with this method are in good agreement with the initially adopted values from the observational analyses. In addition, the mass-loss rates that are predicted with the global approach turn out to be in excellent agreement with the values of mass loss that follow from these self-consistent computations (chapter 3). In Sect. 2.2, I will briefly describe the most relevant aspects of the standard radiation-driven wind theory, which was developed by Castor, Abbott & Klein (CAK) and later on refined by several others (e.g. Abbott 1982, Pauldrach et al. 1986, Kudritzki et al. 1989). In Sect. 2.3 the physical aspects of one of our main modifications to these standard models, namely the process of "multiple scattering", will be discussed. Then, in Sect. 2.4, I will describe our approach to calculate the radiative acceleration in detail. From here onwards, some limitations from CAK will be avoided, and a new treatment of radiation-driven winds will be presented. The main modifications in our approach compared to CAK are that we do not use the "core-halo" approximation and that we naturally include multi-line effects. Finally, in Sect. 2.5, the method we have followed to compute mass-loss rates will be discussed. The approach to calculate the global mass loss will be presented. This will ultimately be refined in the sense that we will 17


Chapter 2 solve the momentum equation, enabling us to determine values for the mass-loss rate and the terminal velocity in a self-consistent way.

2.2 Standard radiation-driven wind theory (CAK theory)
2.2.1 The radiative force
The general idea of a star losing mass due to a stellar wind is that there is some force directed outwards which is larger than the inward directed gravitational force. In the case of early-type stars this force has been shown to be the radiation force on lines and the continuum. The radiation force depends both on the available amount of flux F that is radiated by the star and on the cross section (opacity) of the particles that may intercept this radiation. I will first consider the radiative force due to continuum opacity only, followed by the more complicated case of the line force. As a first approximation, the radiation is assumed to emerge directly from the star and diffuse radiation and multiple scatterings are not taken into account in this section (as in CAK). The continuum radiative force In hot star winds nearly all of the hydrogen is ionized by the strong radiation field. This implies that there is a large number of free electrons present in the atmospheres of hot stars and it is these free electrons that are the main contributors to the continuum opacity. The radiative acceleration due to scattering of photons on free electrons (Thomson scattering) can be represented by e L e F = (2.1) c 4 r 2 c where F is the radiative flux, L is the total stellar luminosity, c is the speed of light and e ([e ]= cm2 g-1 ) is the absorption coefficient for Thomson scattering. Note that the continuum acceleration displays a 1/r2 dependence on radius. As this is an identical behaviour as for the gravitational acceleration, g
cont

=

GM (2.2) r2 the ratio of the two accelerations, may conveniently be expressed in terms of the luminosityto-mass ratio (L /M ), or the so-called Eddington factor e , both independent of radius. The Eddington factor is given by g
Newton

=

e =

g g

cont

Newton

L L e = = 7.66 10-5 e 4cGM L

M M



-1

(2.3)

where the constants have their usual meaning. This means that in case the H/He ionization content remains constant, e has a constant value for any star with its specific stellar parameters. The requirement for the onset of an outflow is that at a certain radius r, the outward force (e.g. radiative forces) becomes larger than the inward force, i.e. the force due to gravity. For hot star winds, it has been shown that this requirement can be fulfilled by inclusion of the line force (Lucy & Solomon 1970). The line force together with the continuum radiative force is able to overcome the gravitational well of the star and thus "drive" the stellar wind. 18


The Physics of the line acceleration The line force Apart from the overwhelming presence of free electrons in hot stars, there are also bound electrons in the ions of the atmospheric plasma. At specific wavelengths, ions are able to intercept photons coming from the stellar core and produce lines in the observable spectrum. These lines may be optically thick or optically thin, depending on the strengths of the transitions. Below, I will present equations for the line force from both optically thin and optically thick lines for the geometrically simplest (radial) case. For a more general derivation of the line force, the reader is referred to the book by Lamers & Cassinelli (1999). Optically thin line acceleration: In the case that a line with a certain frequency = 0 is optically thin, the line cannot absorb all the flux that is emitted from the core at some particular frequency 0 . The amount of energy per second that is absorbed, is proportional to the number of absorbing particles per unit volume, and consequently the line acceleration of an optically thin line may be represented by g
thin

=

F





Z
line

c

L ( ) d = 4 r 2 c

Z
line

( ) d

(2.4)

where ([ ] = cm2 g-1 ) is the line absorption coefficient, which is determined by atomic physics and can be represented by = e2 me c2 fn
i

(2.5)

where (e2 /me c2 ) is the cross-section of a classical oscillator, f is the oscillator strength, and ni is the number density of atoms of ion i that can absorb the line. Stimulated emission has been neglected in the above equation for simplicity, but it is properly taken into account in the model calculations. () is the profile function for absorption and is normalized to Z ( ) d = 1 (2.6)
-

This function () can be represented by a Doppler profile with a typical width for the thermal and turbulent velocities (Vt ) of the ions. Note that in case the "Sobolev approximation" (see below) is applied, the profile can be approximated with a -function. Equation (2.4) shows that the line acceleration from an optically thin line has an identical radius dependence as the continuum acceleration due to electron scattering (see Eq. 2.1). This implies that although optically thin lines are able to counteract gravity to a certain extent, they are not able to "drive" a stellar wind by themselves. This trick can only be done by optically thick lines. Optically thick line acceleration: In the case of an optically thick line, all flux (around 0 ) will be absorbed, independent of the number of absorbing particles per unit volume. Therefore, the amount of absorption only depends on the fraction of the stellar flux around = 0 that can be absorbed. Therefore, the integral from Eq. (2.4) can simply be replaced by the bandwidth of the flux that is completely absorbed. This bandwidth can be determined as described in the following. 19


Chapter 2 The geometrical size of a line interaction region, r, is determined by both the width of the line, i.e. by Vt , and the velocity gradient (dV /dr) in the wind: r Vt (dV /dr) (2.7)

A narrow interaction region r is obtained if the absorption profile is narrow, i.e. if Vt is small. However, also a steep velocity gradient in the wind yields a narrow interaction region. If the velocity gradient is so large that the physical conditions of the medium do not change significantly within the line interaction region, one may use the so-called Sobolev approximation. The width of the frequency interval is then given by = 0 c dV dr r (2.8)

and as in the Sobolev approximation the width of the line absorption coefficient is assumed to be small, () can be represented by a delta-function. The line acceleration of an optically thick line in the Sobolev approximation may be represented by g
thick

=

F

0 dV L 0 = c c dr 4 r 2 c c



dV dr

(2.9)

Combining the formulae for the optically thin (Eq. 2.4) and optically thick line acceleration (Eq. 2.9) yields a general equation for the line acceleration. This may be represented by g
line

=

F

0 cc



dV dr

1 - e-

S

(µ=1)

(2.10)

where S represents the Sobolev optical depth and where µ = cos . This is the cosine of the direction angle with respect to the radial direction. In case, µ = 1, i.e. for photons moving radially from the star, the Sobolev optical depth is given by S ( µ = 1) =


c 0

dr dV

(2.11)

1, Eq. (2.10) reduces to Eq. (2.9) and that for small Note that at large optical depths, or S -S optical depths, e 1 - S , Eq. (2.4) is retrieved. In addition, one should realize that collisions between the accelerating atoms with the hardly absorbing ions ­ such as hydrogen ­ result in a strong coupling of the whole plasma. This "Coulomb coupling" ensures that the wind can be treated as one fluid. The total line force is simply the summation of the line forces due to all individual lines, both optically thick and optically thin. g
tot line

=

lines



F

0 dV c c dr



1 - e-

(µ=1)

(2.12)

The line acceleration due to all these spectral lines may conveniently be expressed in terms of the radiative acceleration due to electron scattering times a certain multiplication factor M (t ), which is called the force multiplier. gL ( r ) = g
ref elec

ref L M (t ) = e 2 M (t ) 4 r c 20

(2.13)


The Physics of the line acceleration where ref is a reference value for the electron scattering opacity. CAK used a value of ref = e e 0.325 cm2 g-1 . The force multiplier M (t ) can be parameterized in the following ways (CAK, Abbott 1982): ne (2.14) W where ne is the electron density and W is the geometrical dilution factor, which is given by 2 R 1 W (r)= (2.15) 1 - (1 - 2 r M (t ) = K t
-

= kt

-

The parameters k (or K ), and are the so-called force multiplier (or K ) is a measure for the number of lines. The second one, is the distribution of strong to weak lines. If only strong (weak) lines eration, then = 1 (0). t is the optical depth parameter and is given t = eVth (dr/dV )

parameters. The first one, k a constant which describes contribute to the line accelby: (2.16)

where Vth is the mean thermal velocity of the protons. Finally, the parameter represents a value for the ionization in the wind. However, it is also possible to simply include the factor (ne /W ) in the constant K , as shown in Eq. (2.14).

2.2.2 The equation of motion
Now that we have found the equations for the radiative acceleration, we can construct the equation of motion. The equation of motion for a stationary stellar wind can be written as the balance between all relevant outward and inward directed forces. The acceleration balance is given by V dV GM 1 dp =- 2 - +g dr r dr
rad

(2.17)

where now grad is the total radiative acceleration, including both continuum and line acceleration. For a stationary wind the mass continuity equation may be applied, which is given by M = 4 r 2 ( r ) V ( r ) (2.18)

Together with an expression for the gas pressure p = R T /µ,where R is the gas constant, T is the temperature, and µ is the mean mass per free particle in units of mH , the equation of motion becomes V dV = dr 2a2 GMeff - 2 +g r r
L

/

1-

a V

2 2

(2.19)

for an isothermal wind, where a is the isothermal sound speed. The mass is expressed as the effective mass Meff = M (1 - e ), which conveniently combines the radiative acceleration by electrons and gravity, as both terms obey the same dependence on radius. The radiative acceleration due to lines is represented separately in Eq. (2.19), as gL . The equation of motion is, mathematically speaking, subject to a singularity. Namely at the point where V (r)= a. Physically, this implies that the point where the velocity equals the sound 21


Chapter 2 speed (sonic point) is the critical point of the wind equation. If the line acceleration gL (r) is known, the equation can be solved numerically. The requirement for a smooth wind solution is that the numerator of Eq. (2.19) equals zero, exactly at the critical (or sonic) point. However, in the Sobolev approximation, as gL is proportional to (dV /dr) (Castor 1974, see also Eq. 2.12), the equation of motion becomes non-linear. In the CAK theory, the Sobolev velocity gradient (dV /dr)Sob is simply set equal to the Newtonian acceleration (dV /dr)Newton , the equation of motion thus becomes non-linear (see Lucy 1998). Using Eqs. (2.13) and (2.14) it follows that V dV GMeff 1 dp e L =- 2 - + K dr r dr 4r2 c e Vth dr dV
-

(2.20)

2.2.3 The Solution of the equation of motion
Having constructed the equation of motion (Eq. 2.20), we proceed to solve this non-linear equation to find a self-consistent solution for a radiation-driven wind. We will start with a simplifying assumption: since the enthalpy term in the energy equation is much smaller than the potential and kinetic energy, we may ignore the gas pressure in the equation of motion. Multiplying both sides of Eq. (2.20) by r2 and using the mass continuity equation (Eq. 2.18), yields dV rV = -GM dr
2

eff

e L K + 4 c

e Vth M dr 4 r2V dV

-

(2.21)

The left-hand side of this equation can be defined as dV (2.22) dr To simplify the right-hand side of Eq. (2.21) as well, we will first put all constant values in a single parameter C. This constant C is defined as D r2V C= e L K 4 c e Vth M 4
-

(2.23)

in which the mass-loss rate M is included. The equation of motion can now be written in a more convenient form C D - D - GM
eff

=0

(2.24)

Note that this equation is valid at all radii in the stellar wind. If we assume that the above equation has only one unique solution for the mass loss, this single solution can be obtained by finding its minimum. Differentiation of Eq. (2.24), followed by finding its zero point, yields d (C D - D - GMeff ) = CD-1 - 1 = 0 dD This results in the key condition for the critical point in CAK theory: Ccrit = 1 D
1-

(2.25)

(2.26)

22


The Physics of the line acceleration Note that similar equations at other points in the wind are equally well legitimate, as Eq. (2.24) is valid for all stellar radii. Let us proceed to solve the equation of motion in the following way. Combining Eqs. (2.24) and (2.26) yields D: GMeff 1- Rewriting Eq. (2.27), using the definition of D from Eq. (2.22), gives D= V dV = (2.27)

GMeff dr (2.28) 1- r2 In case the force multiplier parameter is assumed to be constant over the entire wind regime, the term /(1 - ) can be taken outside the integral:

Z V dV = 1- 0 Additionally, we define the effective escape velocity as
Vesc =

Z V

R


GMeff dr r2

(2.29)

2GMeff (2.30) R Hence, the result of the solution of the equation of motion in its integral form (Eq. 2.29), is that the terminal velocity of the wind is proportional to the effective escape velocity; in other words, the ratio V /Vesc only depends on the value of the constant . V = Vesc 1- The velocity as a function of radius can be represented by a law V (r) = V R 1- r


(2.31)

(2.32)

with equal to 1/2. The mass-loss rate M simply follows from the constant C (Eq. 2.23) in which it was "hidden" up to now. The mass-loss rate in the CAK formalism is thus given by eVth 4 L c
1/

M

=

e 4

1/

1-

1-

( K )

1/

(GM (1 - e ))(-1)/

(2.33)

where all constants have their usual meaning. This simplified solution (Kudritzki et al. 1989) is equal to the full CAK solution in the limit of small sound speed, a Vesc . Note that this self consistent solution of the equation of motion, where M and V are simultaneously determined and represented by Eqs. (2.31) and (2.33), is only valid in the special case that the individual force multiplier parameters, K and , are constant. These standard CAK formulae will later on be used to obtain self-consistent solutions for some of our unified wind models (Sect. 2.5.2). Note that the standard radiation-driven wind models like CAK are subject to some assumptions: 23


Chapter 2 1. A core-halo structure: This means that continuum formation in the wind is neglected. 2. The emergent radiation field is not affected by the wind. Thus backwarming and windblanketing are ignored. 3. Each line interacts only once with "unattenuated" stellar continuum radiation. Multi-line effects are neglected. In a more exact treatment that will be described in what follows, most of these CAK assumptions will be relaxed and the method will be replaced by a more realistic approach using Monte Carlo simulating multi-line transfer.

2.3 Multiple Scattering
Before describing the properties of our approach, it is useful to consider the process of "multiplescattering" itself in some more detail, as it is this effect that plays a dominant role in the resulting models. In fact there are three processes that may all be considered "multiple-scattering": 1. Pure Local Scattering 2. Hemisphere scattering 3. Scattering between resonance zones The first type is where multiple scatterings may occur within a single optically thick line. Each ion in the atmosphere is surrounded by a resonance (Sobolev) region in which in principle a large number of scatterings may take place, as the photon is trapped inside a cavity due to the large line opacity. In our approach, we will assume that these scatterings can be replaced by a single scattering which is coherent in the frame co-moving with the ion. The second type of multiple-scattering has been investigated by Panagia & Macchetto (1982). They considered photons that could bounce back and forth between opposite hemispheres in a stellar wind. However, Monte Carlo simulations by Abbott & Lucy (1985) ­ with realistic line lists ­ have shown that this type of multiple-scattering is basically ineffective as typical path lengths of the photons turn out to be much smaller than (twice) the characteristic radial extension of the wind. The third type of scattering occurs when photons are traveling between different resonance zones. If different lines show a significant "line overlap" ­ in other words if the wavelength separations of the driving lines are less than the Doppler shift of a line < 0V /c ­ photons can be multiply scattered (e.g. Puls 1987). For an instructive illustration of this type of "multiple scattering" in a hot star wind, see Fig. 2.1, taken from Abbott & Lucy (1985). One should note that by allowing the photons to multiply scatter, one may easily exceed the "single-scattering limit" of MV /(L /c) > 1. This is not in conflict with the conservation of momentum. If one considers the momentum vector of the complete system, including both the momentum of the ions and the photons, then it is trivial to see that the total momentum is zero from the start and remains zero during all ion-photon interactions. The only limit that does play a role during the multi-line process is given by the conservation of energy. The number of photon interactions will eventually be limited by the fact that at each scattering the photon "loses" a bit of energy, continuously causing small redshifts at each interaction. 24


The Physics of the line acceleration

Figure 2.1: Photon path in a multiple-scattering process. Note that the included scatterings are not only line scatterings, but continuum processes are also included. The figure is taken from Abbott & Lucy (1985).

2.4 The unified model
The core of our approach is that the loss of radiative momentum is linked to the gain of momentum of the outflowing material. The momentum deposition in the wind is calculated by following the fate of a large number of photon packets that are released from below the photosphere. The calculation of mass loss by this method requires the input of a model atmosphere, before the radiative acceleration grad and M can be calculated with a Monte Carlo (MC) code. The model atmospheres that have been applied in this thesis have been computed with the nonLTE unified Improved Sobolev Approximation code (IS A - W IND) for hot stars with extended atmospheres (de Koter et al. 1993, 1997). The Monte Carlo simulations have been performed with MC - W IND (de Koter et al. 1997), that was tailored for IS A - W IND.

2.4.1 The model atmospheres
As has already been noted, the calculation of the line acceleration requires the radiation field and the occupation numbers to be known. This can easily be seen in Eq. (2.12), which shows that the line acceleration contains the radiation field as well as the Sobolev optical depths of all relevant lines. To be able to compute relevant Sobolev optical depths in our Monte Carlo model, we need the occupation numbers of all abundant ions; these quantities are obtained from the IS A - W IND model atmosphere calculation. Only some relevant points concerning the model atmosphere calculations are made here. The most important feature of the code is that it treats the photosphere and wind in a "unified" manner. This means that there is no artificial separation between photosphere and wind, as in "core-halo" approaches. Its main assumption is the use of an improved version of the Sobolev 25


Chapter 2 approximation. The Sobolev approximation may safely be used when velocity gradients in the atmosphere are large. This is a requirement that is reasonably well fulfilled in the models that will be computed in this thesis. Note that if this condition is not fulfilled, one should switch to the more exact approach, and model the atmosphere using the co-moving frame (CMF) formalism. A disadvantage of CMF calculations is that the computations are very time-consuming. If one is interested in a detailed model spectrum of one particular star, one may turn to the use of a CMF code. However, as our goal requires the calculation of a very large set of wind models, the CMF approach is not feasible from a practical point of view. To return to the code IS A - W IND itself, in the photospheric part of the model atmosphere, the density structure is computed from the solution of the momentum equation, that takes both gas and radiative pressure on electrons into account. The velocity law in the lower part of the wind then simply follows from the density structure via the mass continuity equation (Eq. 2.18). Just below the sonic point, a smooth transition is made to a -type velocity law. The parameter that describes the steepness of the velocity law, , is an input parameter in IS A - W IND that can be given any arbitrary value. The temperature structure in the wind is computed under the assumption of radiative equilibrium in an extended grey LTE atmosphere. The temperature in the wind is not allowed to drop below a certain minimum value Tmin , which is set at Tmin = 1/2 Teff (Drew 1989). The temperature structure is thus simplified. Nevertheless, radiative equilibrium calculations by Hillier (1988) suggest that a grey LTE structure is not too bad an assumption. The chemical species that are explicitly included in the statistical equilibrium calculations are H, He, C, N, O and Si. However, for those elements that have not been explicitly included in the statistical equilibrium equations in IS A - W IND, we need to make an adequate assumption for the ionization and excitation balances. These non-explicit ions include e.g. the iron-group elements, which turn out to be important for the radiative acceleration in the lower part of the wind, where the mass-loss rate is determined (see chapter 3).

2.4.2 The Modified nebular approximation
In a standard nebular approximation the ionization balance is determined by the processes of photoionization and recombination. Abbott & Lucy (1985) defined the balance between the radiative recombinations to all levels and photoionizations from state of the ion under consideration using a modified nebular approximation. The fractions for the stages j and j + 1 are for any particular element, given by N
j+1 ne

competing ionization the ground ionization

N

j

=W

Te TR

1/2

N

j+1 ne

LTE

N

(2.34)
TR

j

where ne and Te are the electron density and temperature, N j and N j+1 are the ion population numbers, TR = TR (r, j) is the radiation temperature of ion j at radial depth r, and W is the dilution factor (as given by Eq. 2.15). The last term in Eq. (2.34) is the LTE ionization ratio for a temperature TR (r, j). The parameter represents the fraction of recombinations that go directly to the ground state. Abbott & Lucy (1985) assumed that the wind is optically thin in the ionizing continua and that the ionizing radiation field is simply that of the stellar core. This standard (core-halo) approach has been modified by Schmutz et al. (1990) to take the continuum formation region into account. As soon as the core-halo approach is dropped, the standard geometrical dilution 26


The Physics of the line acceleration factor W loses its meaning and it should be replaced by a dilution factor W that includes the properties of the diffuse radiation field. In our calculations W is represented by the ratio of the mean intensity J to the Eddington flux H , and is taken to be H (2.35) J Secondly, the parameter is replaced by the more general factor {(1 - )W + }. These two modifications imply that deep in the atmosphere where the radiation field becomes isotropic (H J ) the dilution factor approaches unity, and the ionization fractions are accordingly calculated in LTE, which is a good approximation in an isotropic environment. On the other hand, further out in the wind, the radiation field peaks more and more in the radial direction and one retrieves a standard nebular approximation. In our version of the "modified nebular approximation", the ionization fractions are therefore represented by W =1- N
j+1 ne

N

j

= {(1 - )W + }W

Te TR

1/2

N

j+1 ne

LTE

N

(2.36)
TR

j

The values of TR (r, j) can be obtained by inverting the above equation, for all ionization ratios available from the explicit statistical equilibrium calculations in IS A - W IND. The radiation temperature of an explicit ion is used that has its ionization potential closest to ­ but lower than ­ that of the metal ion of interest. So in the use of the modified nebular approximation, we essentially take advantage of our prior knowledge of the explicitly calculated ionization fractions in the model atmosphere IS A - W IND. We note that the non-explicit elements are not taken into account in the the calculation of the radiation field of the IS A - W IND atmosphere; they are only treated in the computation of the radiative acceleration in the Monte Carlo code. In the calculation of the excitation ratios we follow the procedure of Abbott & Lucy (1985). The excitation states of metastable levels are assumed to be in LTE relative to the ground state. For all other levels "diluted" LTE populations are adopted. These are defined by nu =W n1 n n
u 1 LTE

.
TR

(2.37)

where nu and nl are the population numbers for the upper and lower levels, respectively.

2.4.3 The Monte Carlo method
The processes included in the Monte Carlo simulations The radiative acceleration is calculated by means of a Monte Carlo technique that follows the fate of a certain number of energy packets. Each of these packets represents a number of photons that experience identical interactions. In the calculation of the path of these photons the possibility that they can be scattered, absorbed and re-emitted, or eliminated because they are scattered back into the star, are taken into account. The above is done using the code MC W IND (de Koter et al. 1997) in which the radiative transfer is calculated in the narrow-line limit (Sobolev approximation). The continuum processes included are electron scattering and thermal absorption and emission. The included line processes are photon scattering and photon destruction by collisional de-excitation. 27


Chapter 2 In order to calculate the photon's flight path it is necessary to rank the different options in a proper way. It is not sufficient to determine separate probabilities between the options and then to search for the "largest probability", or equivalently, for the "smallest distance" to the next hypothetical event. The reason is that those events that just slightly miss the necessary probability for them to occur, would be completely neglected. Instead, it is important to treat line and continuum events with their combined optical depths for line plus continuum opacities (see Springmann & Puls 1998). Therefore the MC - W IND code has been improved as described below. The Monte Carlo Game The key point of the Monte-Carlo "game" is that line interactions can only take place at specific points of resonance, whereas continuum processes may occur at any location in the atmosphere. A total number of N photon packets (N is typically about 2 â 105 ) is followed from below the photosphere. The path of each of these packets is followed, starting in the first shell with a Planck weighted, randomly distributed frequency and random flight direction. We follow the photons throughout the photosphere and wind; we allow them to be thermalized and re-emitted according to the radiation field calculated in the model atmosphere code IS A - W IND in such a way that the photon pool represents the physical conditions of the shells. The typical number of shells in our Monte Carlo simulations is 50. When a photon has been released in the first shell, there are a couple of outcomes that can take place. Either the photon simply leaves the first shell (through its inner or outer boundary) or an "event" takes place. This depends on the probability p that a particular event x (line or continuum absorption) can occur. This probability px can be represented by px = e-
x

(2.38)

i.e. it depends on the optical depth x , along the flight path of the photon, of the process under consideration. The combined optical depth tot is given by the summation of the line and continuum optical depths
tot

= c +

L

(2.39)

This combined optical depth is compared with a randomly chosen optical depth, ran , to decide whether or not an event can take place. This random optical depth in MC is represented by
ran

= - ln z

(2.40)

where z is a random number between 0 and 1 (because random number generators usually return values in the range between 0 z < 1, one may instead use ran = - ln(1 - z)). A schematic picture of the Monte Carlo game is given in Fig. 2.2. Continuum events occur with a probability that is a linear function of distance. Line scatterings, on the other hand, can only occur at specific points along the photon's path, namely at those points where the Doppler shift is such that the motion of the atmosphere brings the photon in resonance with a specific spectral line. There are three distances that play a role during the game: the distance to the shell boundary sb (which is fixed), the distance to the next line event sl (which is also fixed) and the distance to a possible continuum event sc . This last distance sc is not fixed, as it is continuously determined during the game. 28


The Physics of the line acceleration

Case A Continuum Event

tot



ran

Distance

Sb

Case B

Line event
tot



ran

Distance

Sb

Figure 2.2: The optical depth summation procedure. The continuum optical linearly with distance, while the optical depth due to lines shows jumps at fixed the photon's path. These locations are determined by the change in the wind along the path of the photon. The amplitude of the jumps reflects the Sobolev this location and in this direction. 29

depth c grows locations along velocity vector optical depth at


Chapter 2 First, the resonance location sl of the very first accessible line from the line list is determined. If this distance sl is larger than the distance to the shell boundary sb , we can already rule out the option of a line event, yet a continuum event might still take place. This can be checked by comparing a random optical depth ran to the total optical depth the photon has already gained during its prior history. Based on this comparison, either a continuum event takes place or the photon simply flies into the next shell, where the game starts again. In case the distance sl is smaller than the distance to the shell boundary, the game goes as presented in Fig. 2.2. The location of the first possible line event is determined and this is multiplied by the total continuum opacity c . The total continuum opacity is the sum of the electron scattering and thermal continuum opacities and is given by c = e ne +
therm

(2.41)

where c ([c]= cm-1 ) and the constants have their usual meaning. This way the distance to a possible continuum event sc is determined. Now one has to decide whether a line or continuum event will take place. This depends on the optical depth summation procedure as presented in Fig. 2.2. Either a continuum event takes place (case A), or a line event occurs (case B). The employed procedure is basically similar to that described by Mazzali & Lucy (1993) for the case of only line and electron scatterings. Note that we also include thermalizations, which were not included in previous work (Abbott & Lucy 1985, Lucy & Abbott 1993, Mazzali & Lucy 1993). It is the unified treatment of the photosphere and wind that allows us to incorporate this additional continuum event. A second random number generator, which is only used after it has been determined that the next event is a continuum one, decides whether it will be an electron scattering or a thermalization. The line list The line list that is employed for the MC simulations presented in this thesis consists of about 105 of the strongest lines of the elements H - Zn, selected from a very extensive line list constructed by Kurucz (1988). Only lines in the relevant part of the spectrum, i.e. in the wavelength ° region between 50 and 7000 A were extracted from this large list. The included ionization stages went up to stage VI. Out of these 105 strong lines, only those that reach a sufficiently large optical depth are expected to contribute to the line acceleration. The criterion that was employed is that the lines are required to reach at least an optical depth equal to 0.005 at some point in the atmosphere. Momentum transfer by line and electron scattering The lines in the MC method are described in the Sobolev approximation. This implies that for scatterings in the frame co-moving with the ions in the wind, the incident and emerging frequencies are both equal to the rest frequency of the line transition 0
in

= 0 =

out

(2.42)

where in and out are the incoming and outgoing frequencies in the CMF. Because the wind is spherically expanding, each ion sees any other particle at any other location in the wind, receding. Therefore, all the photons, both the incident and the emerging ones, are red-shifted 30


The Physics of the line acceleration



in

STAR

µin

v1

out

out

STAR

µ

v2

Figure 2.3: A photon is scattered by an atom. in the CMF. In terms of quantities as seen by an outside observer, the incoming and outgoing frequencies are given by: and
out in

=

in

1-

µinV c

(2.43)

=

out

1-

µoutV c

(2.44)

where in and out are the incident and emergent frequencies for an outside observer; µin and µout are the direction cosines with respect to the radial direction at the scattering point, and V is the radial flow velocity of the scattering ion for an outside observer. Thermal motions of the scattering ions are assumed to be negligible compared to the motion of the outward flow. Note that the same velocity V for the ion before and after the photon interaction is adopted in Eqs. (2.43) and (2.44). This is justified as the change in velocity due to the transfer of momentum from a photon to an ion is very small, i.e. about 101 cm s-1 per scattering. Therefore, the change in in and out is mainly determined by the change in direction angle. Combining Eqs. (2.43) and (2.44) gives the conservation of co-moving frequency in a scattering event (Abbott & Lucy 1985). 31


Chapter 2



in

1-

µin V c

=

out

1-

µ

out

V

c

(2.45)

Because the energy and momentum of a photon are respectively E = h and p = h/c, the equation can be rewritten in the following way: - Eout (2.46) V Eq. (2.46) links the change in radial momentum of a photon interacting with an ion with velocity V to the energy loss of the photon. This shows that momentum and energy are intimately related in this approach. The momentum obtained in Eq. (2.46) represents the radial momentum change of the photon. However, in order to determine the radiative acceleration of the outflowing material, we need to know the momentum transfer from the photons to the ions. For an outside observer, the conservation of radial momentum is given by pin µ
in

-p

out

µ

out

=

E

in

hin hout µin = mV2 + µout (2.47) c c where m is the mass of the moving ion and V1 and V2 are the radial velocities of the ion just before and after the scattering (see Fig. 2.3). For an outside observer: mV1 + and µoutV (2.49) c Again, the change in frequency is dominated by the change in direction angle. So the change in radial velocity per scattering, V = V2 -V1 , is small compared to V , and using Eqs. (2.47), (2.48) and (2.49), it is given by
out in

=

0

1+

µinV c

(2.48)

=

0

1+

V

= =

V2 - V1 h 0 µinV 1+ mc c

µ

in

-

h 0 µoutV 1+ mc c

µ

out

(2.50)

As V

c, Eq. (2.50) becomes V = V2 - V1 =

h 0 (µin - µout ) (2.51) mc This relation describes the velocity increase of the ion depending on the directions µin and µout of the photon. In case µin = µout , then V = 0 as one would expect. The increase in the radial momentum p = m V of the ion under consideration is then given by p = m(V2 - V1 ) = = 32 h 0 (µin - µout ) c hin - hout E = V V

(2.52)


The Physics of the line acceleration where E = Ein - Eout is the loss of radiative energy. This equation shows that the increase in the momentum of the ions can be calculated from the loss of energy of the photons when their path is followed through the wind by means of the Monte Carlo method. Since this relationship links the transfer of momentum of the ions to the transfer of energy, the radial momentum transfer can be derived from the total amount of radiative energy removed in the Monte-Carlo code. Multiplying both sides of Eq. (2.52) by V and realizing that for each scattering V2 V1 so V (V1 + V2 )/2, yields: 1 2 (2.53) m(V2 - V12 ) = hin - hout 2 This states that the gain of kinetic energy of the ions in the radial direction equals the energy loss of the photons. The calculation of the radiative acceleration The radiative acceleration of the wind is calculated by following the fate of the photons emitted from below the photosphere with the MC technique. To this purpose the atmosphere is divided into a large number of concentric, thin shells with radius r and thickness r, containing a mass m ( r ) . The loss of photon energy due to all scatterings that occur within each shell is calculated and yields the total radiative acceleration grad (r) per shell. grad (r)= 1 p( r ) m(r) t (2.54) by the ions in the that shell. Using r2 (r)r,and the 2.52), grad (r) can

where p(r) is the momentum of the ions in the shell. The momentum gained shell is equal to the momentum lost by the photons through interactions in the relationship between m(r) and r for thin concentric shells, m(r)= 4 derived relation between momentum and energy transfer of the photons (Eq. be rewritten as g
rad

(r)=

1 E ( r ) 4r2 (r)r V (r)t

(2.55)

where E (r) is sum of the energy loss of all the photons that are scattered in the shell. Now using mass continuity (Eq. 2.18) and the fact that the total energy transfer E (r) divided by the time interval t equals the rate at which the radiation field loses energy, i.e. E (r)/t = -L(r), the expression for grad (r) becomes g
rad

(r)= -

1 L(r) M r

(2.56)

2.5 The determination of M in a Unified Wind model
For the determination of the mass-loss rates, two distinct approaches will be followed. The mass-loss rates calculated with these two approaches will be referred to as respectively the "global" mass-loss rate and the "self-consistent" mass-loss rate. 33


Chapter 2 As was mentioned before, throughout the most part of this thesis, a velocity law V (r) will be adopted ­ based on observations ­ and we will concentrate on predicting only mass-loss rates. As in this "global" approach the momentum equation is not solved, it is not guaranteed that the solution is locally consistent. To check "local" consistency, we have also employed a "self-consistent" approach for some representative wind models. In this approach, we obtain the mass-loss rate and the velocity law simultaneously from the CAK equations.

2.5.1 The determination of the global mass-loss rate
As we have found an expression for the total radiative acceleration (Eq. 2.56, we are now able to calculate mass-loss rates. For a given set of stellar parameters, the mass loss can be calculated in the following way: 1. For fixed stellar parameters L , Teff , R and Meff several values of the input mass loss inp M (within reasonable bounds predicted by CAK theory) are adopted. 2. For each of these models a wind with a terminal velocity of n times the effective escape velocity is adopted. Vesc is defined by Eq. (2.30). For most models in this thesis a -type velocity law with = 1 is used. This is an appropriate value for OB stars (Groenewegen & Lamers 1989; Haser 1995). Moreover, in chapter 4 it will be shown that the predicted mass loss is essentially insensitive to the adopted value of . 3. For each set of stellar and wind parameters, inp 2.4.1) are calculated for several M values.
IS A - W IND

model atmospheres (see Sect.

4. Then, for each of these models, the radiative acceleration is calculated with the Monte Carlo code MC - W IND as described in Sect. 2.4.3. 5. Finally, for each set of parameters, we check which of the adopted mass-loss rates is globally consistent with the radiative acceleration. The check of consistency is done in the following manner: neglecting the term due to the gas pressure, one can write the equation of motion (Eq. 2.17) as dV GM = - 2 + grad (r) (2.57) dr r Using the derived expression for the radiative acceleration (Eq. 2.56), and integrating the equation of motion (Eq. 2.57) from the stellar surface to infinity, gives the relationship between the total energy gained by the wind material and the energy lost by the radiation field V 1 M (V 2 + Vesc 2 ) = L = 2

shells



Lshell

(2.58)

Note that this equation states that the momentum transfered from the radiation to the wind is used to lift the material out of the potential well and to accelerate the wind to V . L is the total amount of removed radiative energy per second, summed over all the shells. Because V and Vesc are prespecified quantities, the mass-loss rate, M out , can be derived from the total removed radiative luminosity: 34


The Physics of the line acceleration

Figure 2.4: Determination of the mass-loss rate. The point where M inp = M out yields the unique mass-loss rate. Note that the dashed line is the one-to-one relation. The thick, curved line is the best fit to the output mass-loss rates. The stellar parameters in this example are Teff = 30 000 K, (V /Vesc = 2.6), M = 20 M , log (L/L ) = 5.0. The point where the two lines intersect, corresponds to the final mass-loss rate; in this case log M = -6.92.

M

out

=

2 L V + Vesc
2

2

(2.59)

To derive consistent values for the mass-loss rate, M is determined using the condition M=M
inp

=M

out

(2.60)

Only one value of M satisfies this condition for our wind models. This unique value for the global mass-loss rate is determined as is shown in Fig. 2.4. A schematic flow diagram for the determination of mass loss is presented in Fig. 2.5. As noted before, for the set-up of the model atmosphere, V (r) is needed as an input pa rameter. This implies that although the M determination is globally consistent in terms of kinetic wind energy, it is not necessarily locally consistent, because the equation of motion is not solved. This approach will be undertaken in the next section.

2.5.2 Self-consistent solutions
As a test of our calculations, we also solve the momentum equation of line driven wind models in a self-consistent way for some representative models. The approach we take is to combine predicted force multiplier parameters K and from the radiative accelerations computed with MC - W IND with the CAK solution of the dynamics of line driven winds (Eqs. 2.31 and 2.33 in Sect. 2.2). 35


Chapter 2

Stellar Parameters (L,M,T) Input Mass Loss Input Terminal Velocity

ISA-WIND

Density Structure Temperature Structure Occupation numbers

Mass Loss

MC-WIND

Radiative acceleration Output Mass Loss

Figure 2.5: Flow Diagram to obtain a globally consistent mass-loss rate. The radiative acceleration calculated with our Monte Carlo code can be expressed in terms of the force multiplier M (t ) as displayed in panel (a) of Fig. 2.6. For many of the calculated wind models, M (t ) cannot be fit with a simple power-law, as the force multiplier log M (t ) is not always linear in log t (see also Kudritzki et al. 1998). Note that t is the optical depth parameter (see Eq. 2.16). Nevertheless, for the accelerating part of the wind between the sonic point and V 0.5V (see the enlargement in Fig. 2.6b), M (t ) can be accurately expressed in terms of a power-law fit to the optical depth parameter t . For these cases, M (t ) can satisfactory be represented in terms of the fit parameters K MC and MC , as indicated by the dashed line in Fig. 2.6 (b). The total radiative force for the accelerating part of the wind is thus expressed as M
MC

(t ) = K 36

MC -MC

t

(2.61)


The Physics of the line acceleration

Figure 2.6: (a) The Unified Force Multiplier log M (t ) as a function of optical depth log t . (b) M (t ) over the accelerating part of the wind. The dotted line indicates the best-linear fit. The stellar parameters in this example are Teff = 30 000 K, V /Vesc = 2.6, M = 20 M , log L/L = 5.0.

Self-consistent values of V and M can then be found by iterating the solutions of the CAK equations (Eqs. 2.31 and 2.33) for both V and M with our unified force multiplier M MC (t ). Note that the "Modified" CAK solution including the so-called finite disk correction (Pauldrach et al. 1986, Kudritzki et al. 1989) was not applied, as the finite disk and the diffuse radiation field are naturally taken into account in the calculations of gL , M (t ), and k with our Monte Carlo approach (see Sect. 2.4.3). We have added the superscript "MC" to the force multiplier 37


Chapter 2 parameters to avoid confusion with the "classical" force multiplier parameters k and for a point-like source as used by e.g. CAK, Pauldrach et al. (1994). The dynamics of the wind is solved, as the ratio V /Vesc is represented by the CAK formulation: V Vesc = MC 1 - MC (2.62)

Note that the values for MC can be significantly higher than the "classical" values calculated by e.g. Abbott (1982), Pauldrach et al. (1986), and Pauldrach et al. (1994), as in our parameterization of the force multiplier, the finite disk and the diffuse radiation field are already included in the MC -parameter itself. For instance, if MC 0.9, the ratio between terminal and escape velocity is directly derived from Eq. (2.62), i.e. V /Vesc 3. Ultimately, mass-loss rates can also be obtained from the final unified force multiplier parameters K MC and MC using Eq. (2.33). These mass-loss rates should be equal to the mass-loss rates that were calculated with our global Monte Carlo approach discussed in Sect. 2.5.1. It will be shown in chapter 3 that this is indeed the case.

2.6 Summary
In this chapter we have explained our approach of calculating mass-loss rates for winds of earlytype stars. We have explained how the radiative acceleration is able to "drive" a stellar outflow and derived a convenient equation for this acceleration (Eq. 2.56). Its calculation enables us to determine a unique "global" mass-loss rate by requiring that the output mass loss equals the input mass-loss rate. The flow diagram from Fig. 2.5 shows the global mass loss derivation in a schematic way. To obtain a "locally" consistent solution for our wind models, we parameterized our Force Multiplier in a way analogous to that of CAK. However, the finite disk, the diffuse radiation field, and multiple scatterings are naturally included in our force multiplier. The complete procedure to obtain a self-consistent solution for the mass loss and the terminal velocity is summarized in the flow diagram presented in Fig. 2.7. After the calculations with the codes IS A - W IND and MC - W IND have been performed for several input mass-loss rates and terminal velocities, the corresponding force multiplier M MC (t ) can be parameterized in terms of fit parameters K MC and MC . These parameters serve as an input into the CAK solutions (Eqs. 2.31 and 2.33), and by iteration the self-consistent final mass-loss rate ands terminal velocity are found.

References
Abbott D.C., 1982, ApJ 259, 282 Abbott D.C., Lucy L.B., 1985, ApJ 288, 679 Castor J.I., 1974, MNRAS 169, 279 Castor J.I., Abbott D.C., Klein R.I., 1975, ApJ 195, 157 de Koter A., Schmutz W., Lamers H.J.G.L.M., 1993, AAP 277, 561 de Koter A., Heap S.R., Hubeny I., 1997, ApJ 477, 792 Drew J.E., 1989, ApJS 71, 267 38


The Physics of the line acceleration

M

.

inp

V inf

inp

MC

M (t)
Self-Consistent Solution

K

MC MC

M

.

out

V

out inf

Figure 2.7: Flow Diagram explaining how a a self-consistent solution for the mass loss and the terminal velocity is obtained.

Groenewegen M.A.T., Lamers H.J.G.L.M., 1989, A&AS 79, 359 Haser S., PhD thesis at University of Munich Hillier D.J., 1987, ApJS 63, 947 Hillier D.J., 1988, ApJ 327, 822 Hillier D.J., 1990, A&A 231, 111 Kudritzki R.-P., 1998, et al. in: "Boulder-Munich II: properties of hot, luminous stars", ed. Howarth I.D., ASP Conf. Ser., 131, 299 Kudritzki R.-P., Pauldrach A.W.A., Puls J., Abbott D.C., 1989, AAP 219, 205 Kudritzki R.-P., 1998, et al. in: "Boulder-Munich II: properties of hot, luminous stars", ed. Howarth I.D., ASP Conf. Ser., 131, 299 Kurucz R.L., 1988, IAU Trans., 20b, 168 39


Chapter 2 Lamers H.J.G.L.M., Cassinelli J.P., 1999, in: Introduction to Stellar Winds, Cambridge Univ. Press Lucy L.B., 1998, in: Cyclical Variability in Stellar Winds, ESO ASS Proc 22, 16 Lucy L.B., Solomon P., 1970, ApJ 159, 879 Lucy L.B., Abbott D.C., 1993, ApJ 405, 738 Mazzali P.A., Lucy L.B., 1993, A&A 279, 447 Panagia N., Macchetto F., 1982, A&A 106, 266 Pauldrach A.W.A., Puls J., Kudritzki R.P., 1986, A&A 164, 86 Pauldrach A.W.A., Kudritzki R.P., Puls J., Butler K., Hunsinger J., 1994, A&A 283, 525 Puls J., 1987, A&A 184, 227 Schmutz W., Abbott D.C., Russell R.S., Hamann W.-R., Wessolowski U., 1990, ApJ 355, 255 Springmann U., Puls J., 1998, in: "Boulder-Munich II: properties of hot, luminous stars", ed. Howarth I.D., ASP Conf. Ser., 131, 286

40


On the nature of the bi-stability jump in the winds of early-type supergiants

3

On the nature of the bi-stability jump in the winds of early-type supergiants
Jorick S. Vink, Alex de Koter, and Henny J.G.L.M. Lamers Published in A&A

We study the origin of the bi-stability jump in the terminal velocity of the winds of supergiants near spectral type B1. Observations show that here the ratio V /Vesc drops steeply from about 2.6 at types earlier than B1 to a value of V /Vesc =1.3 at types later than B2. To this purpose, we have calculated wind models and mass-loss rates for early-type supergiants in a Teff grid covering the range between Teff = 12 500 and 40 000 K. These models show the existence of a jump in mass loss around Teff = 25 000 K for normal super giants, with M increasing by about a factor five from Teff 27 500 to 22 500 K for constant luminosity. The wind efficiency number = MV /(L /c) also increases drastically by a factor of 2 - 3 near that temperature. We argue that the jump in mass loss is accompanied by a decrease of the ratio V /Vesc , which is the observed bi-stability jump in terminal velocity. Using self-consistent models for two values of Teff , we have derived V /Vesc = 2.4 for Teff = 30 000 K and V /Vesc = 1.2 for Teff = 17 500 K. This is within 10 percent of the observed values around the jump. Up to now, a theoretical explanation of the observed bi-stability jump was not yet provided by radiation driven wind theory. To understand the origin of the bi-stability jump, we have investigated the line acceleration for models around the jump in detail. These models demonstrate that M increases around the bi-stability jump due to an increase in the line acceleration of Fe III below the sonic point. This shows that the mass-loss rate of B-type supergiants is very sensitive to the abundance and the ionization balance of iron. Furthermore, we show that the elements C, N and O are important line drivers in the supersonic part of the wind. The subsonic part of the wind is dominated by the line acceleration due to Fe. Therefore, CNO-processing is expected not to have a large impact on M , but it might have impact on the terminal velocities. Finally, we discuss the possible role of the bi-stability jump on the mass loss during typical variations of Luminous Blue Variable stars. 41


Chapter 3

3.1 Introduction
In this paper we investigate the origin and the consequences of the bi-stability jump of the stellar winds of early-type stars near spectral type B1. This bi-stability jump is observed as a steep decrease in the terminal velocity of the winds from V 2.6Vesc for supergiants of types earlier than B1 to V 1.3Vesc for supergiants of types later than B1 (Lamers et al. 1995). We will show that this jump in the wind velocity is accompanied by a jump in the mass-loss rate with M increasing by about a factor of five for supergiants with Teff between 27 500 and 22 500 K. The theory of radiation driven winds predicts that the mass-loss rates and terminal velocities of the winds of early-type stars depend smoothly on the stellar parameters, with V 3Vesc and M L1.6 (Castor et al. 1975, Abbott 1982, Pauldrach et al. 1986, Kudritzki et al. 1989). This theory has not yet been applied to predict the observed jump in the ratio V /Vesc for supergiants near spectral type B1. The change from a fast to a slow wind is called the bi-st abil it y jump. If the wind momentum MV were about constant across the bi-stability jump, it would imply that the mass-loss rate would increase steeply by about a factor of two from stars with spectral types earlier than B1 to later than B1. Unfortunately, there are no reliable mass-loss determinations from observations for stars later than spectral type B1. So far, a physical explanation of the nature of this bi-stability jump has been lacking. In this paper, we attempt to provide such an explanation and we investigate the change in mass-loss rate that is accompanied by the change in V . The concept of a bi-stability jump was first described by Pauldrach & Puls (1990) in connection to their model calculations of the wind of the Luminous Blue Variable (LBV) star P Cygni (Teff = 19.3 kK). Their models showed that small perturbations in the basic parameters of this star can either result in a wind with a relatively high mass loss, but low terminal velocity, or in a wind with relatively low M , but high V . Their suggestion was that the mechanism is related to the behaviour of the Lyman continuum. If the Lyman continuum exceeds a certain optical depth, then as a consequence, the ionization of the metals shifts to a lower stage. This causes a larger line acceleration gL and finally yields a jump in M . The models of Pauldrach & Puls (1990) for P Cygni show that the wind momentum loss per second, MV , is about constant on both sides of the jump (see Lamers & Pauldrach 1991). So Lamers et al. (1995) put forward the idea that the mass-loss rate for normal stars could increase by about a factor of two, if V decreases by a factor of two, so that MV is constant on both sides of the jump. Whether this is indeed the case, is still unknown. To investigate the behaviour of the mass loss at the bi-stability jump, we will derive mass-loss rates for a grid of wind models over a range in Teff . The main goal of the paper is to understand the processes that cause the bi-stability jump. Although our results are based on complex numerical simulations, we have attempted to provide a simple picture of the relevant physics. We focus on the observed bi-stability jump for normal supergiants. Nevertheless, these results may also provide valuable insight into the possible bi-stable winds of LBVs. It is worth mentioning that Lamers & Pauldrach (1991) and Lamers et al. (1999) suggested that the bi-stability mechanism may be responsible for the outflowing disks around rapidlyrotating B[e] stars. Therefore our results may also provide information about the formation of rotation induced bi-stable disks. The paper is organized in the following way. In Sect. 3.2 we describe the basic stellar wind theory. In particular we concentrate on the question: "what determines M and V ?". We show 42


On the nature of the bi-stability jump in the winds of early-type supergiants that M is determined by the radiative acceleration in the subsonic region. In Sect. 3.3 we explain the method that we use to calculate the radiative acceleration with a Monte Carlo technique and the mass-loss rates of a grid of stellar parameters. Sect. 3.4 describes the properties of the models for which we predict M . In Sect. 3.5 our predicted bi-stability jump in mass loss will be presented. Then, in Sect. 3.6 we discuss the origin of this jump and show that it is due to a shift in the ionization balance of Fe IV to Fe III. Then, we discuss the possible role of the bi-stability jump in M on the variability of LBV stars in Sect. 3.7. Finally, in Sect. 3.8, the study will be summarized and discussed.

3.2 What determines M and V ?
3.2.1 The theory of M determination
Mass loss from early-type stars is due to radiation pressure in lines and in the continuum (mainly by electron scattering). Since the radiative acceleration by line processes is the dominant contributor, the winds are "line-driven", i.e. the momentum of the radiation is transferred to the ions by line scattering or line absorption. Line-scattering and line absorption occur at all distances in the wind, from the photosphere up to distances of tens of stellar radii. So the radiative acceleration of the wind covers a large range in distance. The equation of motion of a stationary stellar wind is V where g
rad

dV GM 1 dp =- 2 - + grad (r) dr r dr

(3.1)

is the radiative acceleration. Together with the mass continuity equation M = 4r2 (r)V (r) (3.2)

and the expression for the gas pressure p = R T /µ, where R is the gas constant, T is the temperature, and µ is the mean mass per free particle in units of mH , we find the equation of motion V dV = dr 2a2 GMeff - 2 +g r r
L

/

1-

a V

2 2

(3.3)

where a is the isothermal speed of sound. For simplicity we have assumed that the atmosphere is isothermal. In this expression the effective mass Meff = M (1 - e ) is corrected for the radiation pressure by electron scattering. gL is the line acceleration. The equation has a singularity at the point where V (r) = a, this critical point is the sonic point. If the line acceleration gL (r) is known as a function of r, the equation can be solved numerically. A smoothly accelerating wind solution requires that the numerator of Eq. (3.3) reaches zero exactly at the sonic point where the denominator vanishes. It should be stated that this critical point (sonic point) at rc 1.025R and Vc 20 km -1 is not the same as the CAK critical point. The CAK critical point is located much further s out in the wind at rc 1.5R and about Vc 0.5V . If the line acceleration gL in Eq. (3.3) were to be rewritten as a function of velocity gradient instead of radius, then one would find the CAK critical point. Pauldrach et al. (1986) showed that if the finite disk correction to the CAK theory is applied, then the Modified CAK critical point moves inward and is located 43


Chapter 3 at rc 1.04R and at Vc 100 km s-1 . This is much closer to the sonic point! Although the (Modified) CAK critical solution may well provide the correct mass-loss rate and terminal velocity, there is concern about its physical reality (see e.g. Lucy 1998 and Lamers & Cassinelli 1999 for a thorough discussion). Lucy (1998) has given arguments favouring the sonic point as the physical more meaningful critical point. We will use the sonic point as the physically relevant critical point. This is the point where the mass-loss rate is fixed. Throughout the paper we will therefore refer to the subsonic part of the wind for the region close to the photosphere where the mass loss is determined, and to the supersonic part for the region beyond the sonic point where the mass-loss rate is already fixed, but the velocity has still to be determined. The critical solution can be found by numerically integrating Eq. (3.3), starting from some lower boundary r0 in the photosphere, with pre-specified values of T0 and 0 and with a trial value of V0 . The value of V0 that produces a velocity law that passes smoothly through the critical point is the correct value. Alternatively, for a non-isothermal wind with a pre-specified T ()-relation, one can integrate inwards from the critical point with an assumed location rc ,and then adjust this value until the inward solution gives a density structure that reaches = 2/3 at the location where T (r)= Teff (e.g. see Pauldrach et al. 1986). The critical solution specifies the values of r0 R , 0 (given by (r0 )= 2/3) and V0 at the lower boundary. This fixes the value of M via the mass continuity equation (Eq. 3.2). Note that M is determined by the conditions in the subsonic region! We will show below that an increase in gL (r) in the subsonic region results in an increase in M . This can be understood because in the subsonic region, where the denominator of Eq. (3.3) is negative, an increase in gL gives a smaller velocity gradient. Integrating from the sonic point inwards to the lower boundary with a smaller velocity gradient, implies that the velocity the 2 lower boundary should be higher and hence the mass-loss rate, M = 4r0 0V0 , must be higher. On the other hand, an increase in gL in the supersonic region, yields a larger velocity gradient and this would directly increase the terminal velocity V . Another way to understand how an increase in M is caused by an increase in gL below the sonic point, is based on the realization that the density structure of the subsonic region is approximately that of a static atmosphere. This can be seen in Eq. (3.1). Since the term V dV /dr is much smaller than the acceleration of gravity, it can approximately be set to zero in the subsonic region. (This is not correct close to the sonic point.) In an isothermal static atmosphere the density structure follows the pressure scaleheight. Adding an extra outward force in the subsonic region results in an increase of the pressure-scaleheight and hence in a slower outward decrease in density. This means that just below the sonic point, where V a, the density will be higher than without the extra force. Applying the mass continuity equation (Eq. 3.2) at the sonic point then shows that the mass-loss rate will be higher than without the extra force in the subsonic region. (See Lamers & Cassinelli 1999 for a thorough discussion).

3.2.2 A simple numerical experiment: the sensitivity of M on the subsonic gL
A simple numerical experiment serves to demonstrate the dependence of M acceleration in the subsonic region. We start with an isothermal model of the of Meff = 20M , R = 16.92R , Teff = 25 000 K, Twind = 0.8Teff = 20 000 K. the line acceleration gL (r) in such a way that it produces a stellar wind with a 1.86 10-7 M yr-1 and with a -type wind velocity law 44 on the radiative wind from a star We then specify mass-loss rate of


On the nature of the bi-stability jump in the winds of early-type supergiants

Figure line is bumps 1.0135

3.1: Extra "bumps" on the radiative acceleration gL (r) below the sonic point. The solid gL (r) of the model without a "bump". The dotted lines show gL (r) with the adopted with peakheights of 150, 300 and 500 cm s-2 . The cross indicates the sonic point at R .

V (r) = V (1 -

R ) r



(3.4)

where = 1 and V = 1500 km s-1 . (This gL (r) is found by solving Eq. (3.3) with this fixed velocity law). This model is very similar to one of the models near the bi-stability jump that we will calculate in detail in Sect. 3.5. As a lower boundary we choose the point where = 10-10 gcm-3 at r0 = R . Figure 3.1 shows the resulting variation of gL (r). Adopting this variation of gL (r) and solving the momentum equation with the condition that the solution goes smoothly through the sonic point, we retrieve the input mass-loss rate and input velocity law, as one would expect. The sonic point is located at rc = 1.0135R where V = 16.6 km s-1 , and where gL (r)= 1.63 103 cm s-2 . Let us study what happens to M and V if we change the line acceleration in the subsonic region. To this purpose we add a Gaussian "bump" to gL (r). This bump is characterized by g
bump L

(r ) = g

peak L

exp -

z-z z

p

2

(3.5)

where z = 1/{(r/R) - 1}, z p = 150 describes the location of the peak at r/R = 1.0067 and z = 30 gives the width of the bump (r 0.0015R). The line acceleration with the extra bumps is shown in Fig. 3.1. The solution of the momentum equation, with the condition that it passes smoothly through the sonic point, gives the velocity at the lower boundary and hence the mass-loss rate. The upper panel of Fig. 3.2 shows the resulting mass-loss rates as a function of the peak value of 45


Chapter 3

Figure 3.2: The effect of increasing the line acceleration in the subsonic region on M (upper panel) and a simple derivation of its effect on V (lower panel). The horizontal axis gives the peak peak-value, gL ,of the bump in gL (r) in the subsonic region (i.e. the bumps in Fig. 3.1). the bump in the line acceleration in the subsonic region. We see that as the line acceleration in the subsonic region increases, M increases.

3.2.3 The effect of an increased M on V
Once M is fixed by the processes in the subsonic region, the radiative acceleration in the supersonic region then determines the terminal velocity V that the wind will reach. This can easily be seen in the following way. Integrating the momentum equation (Eq. 3.1) in the supersonic region from the critical point rc to infinity, and ignoring the influence of the gas pressure, gives

Z
so

rc

gL (r) dr =

1 1 Vesc 2 + V 2 2

2

(3.6)

V

2

Z
2

rc

gL (r) dr - Vesc

2

(3.7)

Here we have used the observed property that V a and that rc - r0 R ,so rc R . Eq. (3.7) says that V is determined by the integral of gL (r) in the supersonic region. The radiative acceleration in the supersonic part of the wind will decrease as M is forced to increase by an increase in the radiative acceleration in the subsonic part of the wind. This is because the optical depth of the optically thick driving lines, which is proportional to the density in the wind, will increase. Thus an increase in M results in an increase of the line optical depth. This results in a decrease of gL in the supersonic region, which gives a lower terminal velocity V . We will estimate this effect below. 46


On the nature of the bi-stability jump in the winds of early-type supergiants Assume that the radiative acceleration by lines depends on the optical depth in the wind, as given by CAK theory (Castor et al. 1975). e L - kt (3.8) 4 r 2 c where k and are constants and ge is a reference value describing the acceleration due to e L electron scattering. It is given by ge = 4r2c . The optical depth parameter is gL (r) = ge M (t ) = t = eVth (dr/dV ) (3.9)

where Vth is the mean thermal velocity of the protons. Let us define ginit (r) as the radiative L acceleration in the supersonic part of the initial wind model, i.e. without the increased massloss rate due to the bump in the subsonic region, and gL (r) as the radiative acceleration of the model with the increased M . From Eqs. (3.8) and (3.9) with Eq. (3.2) we find that gL (r)= g
init L

r2VdV /dr (r2VdV /dr)init



init M M



(3.10)

where the superscript "init" refers to the initial model. Let us now compare the terminal velocities of the initial model without the bump, to that with the increased mass-loss rate due to the bump, in a simple but crude way, by solving the momentum equation in the supersonic part of the wind. If we neglect the terms due to the gas pressure and due to the gravity, the momentum equation in the supersonic part of the wind reduces to dV (3.11) gL (r ) dr Solving the equation for the initial model and the model with the increased M results in the following expression V dV V dr dV V dr
init

init M M

/(-1)

(3.12)

So the ratio between the terminal velocities of the models with and without the increased massloss rate is V in V it init M M
/(2-2)

M init M

-3/4

(3.13)

where we adopted = 0.60 (Pauldrach et al. 1986) for the last expression. We see that V will -3/4 when the mass-loss rate increases. The result is shown in the lower decrease roughly as M panel of Fig. 3.2. We realize that this numerical test is a drastic simplification of the real situation: (a) we have assumed an isothermal wind; (b) we have taken the lower boundary at a fixed density; (c) we have ignored possible changes in the ionization of the wind due to changes in M and (d) we have ignored the role of the gas pressure and of gravity in estimating the change in V . However, this simple test serves the purpose of explaining qualitatively that the mass-loss rate depends on the radiative acceleration in the subsonic part of the wind only, and that an increase in the 47


Chapter 3 mass-loss rate due to an increase of gL in the subsonic region will also be accompanied by a decrease in V . In the rest of the paper, we will quantitatively calculate radiative accelerations and mass-loss rates with a method which will be described in Sect. 3.3. Thus, an increase in the radiative acceleration in the subsonic region of the wind results in an increase of M and a decrease in V . So, in order to understand the origin of the bi-stability jump of radiation driven winds, and to predict its effect on M and V , we should pay close attention to the calculated radiative acceleration in the subsonic part of the wind.

3.3 The method to predict M
In order to understand the nature of the bi-stability jump, we calculate a series of radiation driven wind models for supergiants in the range of Teff = 12 500 to 40 000 K. The calculation of the radiative acceleration of the winds requires the computation of the contributions of a very large number of spectral lines. To this end, we first calculate the thermal, density and ionization structure of a wind model computed with the non-LTE expanding atmosphere code IS A - W IND (de Koter et al. 1993)(for details, see Sect. 3.4). We then calculate the radiative acceleration by following the fate of a very large number of photons that are released from below the photosphere into the wind, by means of a Monte Carlo technique. In this section, we describe the basic physical properties of the adopted Monte Carlo (MC) technique which was first applied to the study of winds of early-type stars by Abbott & Lucy (1985). Then, we describe the calculation of the radiative acceleration by lines with the MC method, and finally the method for calculating theoretical mass-loss rates.

3.3.1 Momentum transfer by line scattering
The lines in the MC method are described in the Sobolev approximation. This approximation for the line acceleration is valid if the physical conditions over a Sobolev length do not change significantly, i.e. 1 df | | f dr 1 dV | | Vt dr (3.14)

where f is any physically relevant variable for the line driving, e.g. density, temperature or ionization fraction. Vt is a combination of thermal and turbulent velocities. Eq. (3.14) shows that the validity range of the Sobolev approximation is in practice somewhat arbitrary, since it depends on the value of the turbulent velocity which is poorly known. Nevertheless, the Sobolev approximation is often used (e.g. Abbott & Lucy 1985) and we will also adopt it in calculating the line acceleration and mass loss, mainly because of computational limitations. We cannot exclude that due to the use of the Sobolev approximation we may predict quantitatively inaccurate line accelerations below the sonic point. However, if an exact treatment would be followed, then this is expected to have a systematic effect on the line acceleration for all models. Therefore, we do not expect our conclusions regarding the origin of the bi-stability jump to be affected. The Sobolev approximation implies that for scatterings in the frame co-moving with the ions in the wind (co-moving frame, CMF), the incident and emerging frequencies are both equal to the rest frequency of the line transition 0 in the CMF. 48


On the nature of the bi-stability jump in the winds of early-type supergiants



in

= 0 =

out

(3.15)

where in and out are the incident and emerging frequencies in the CMF. In terms of quantities seen by an outside observer, these two CMF frequencies are given by: and µoutV ) (3.17) c where in and out are the incident and emergent frequencies for an outside observer; µin and µout are the direction cosines with respect to the radial flow velocity of the photons at the scattering point and V is the radial flow velocity of the scattering ion for an outside observer. Thermal motions of the scattering ions are assumed to be negligible compared to the motion of the outward flow. Note that we adopted the same velocity V for the ion before and after the photon interaction (Eqs. 3.16 and 3.17). This is justified since the change in velocity due to the transfer of momentum from a photon to an ion is very small, i.e. about 101 cm s-1 per scattering.. Therefore, the change in in and out is mainly determined by a change in direction angle. Combining Eqs. (3.16) and (3.17) gives the conservation of co-moving frequency in a scattering event (Abbott & Lucy 1985).
out in

= in (1 -

µinV ) c

(3.16)

=

out

(1 -

µin V µout V ) = out ( 1 - ) (3.18) c c Because the energy and momentum of a photon are E = h and p = h/c, the equation can be rewritten in the following way: in ( 1 - - Eout (3.19) V Eq. (3.19) links the change in radial moment um of a photon in an interaction with an ion with velocity V to the energy loss of the photon. In order to determine the line acceleration gL we will need to derive the momentum transfer from the photons to the ions in the wind. For an outside observer, the conservation of radial momentum is: pin µ
in

-p

out

µ

out

=

E

in

hin hout µin = mV2 + µout (3.20) c c where m is the mass of the moving ion and V1 and V2 are the radial velocities of the ion just before and after the scattering. For an outside observer: mV1 + and µoutV ) (3.22) c Again, the change in frequency is dominated by the change in direction angle. So the change in radial velocity per scattering, V = V2 - V1 , is small compared to V and is given by
out in

= 0 ( 1 +

µinV ) c

(3.21)

= 0 ( 1 +

49


Chapter 3

V

= =

V2 - V1 µinV h 0 (1 + )µ mc c

in

-

µoutV h 0 (1 + )µ mc c

out

(3.23)

Since V

c, Eq. (3.23) becomes V = V2 - V1 =

h 0 (µin - µout ) (3.24) mc This relation describes the velocity increase of the ion depending on the directions µin and µout of the photon. In case µin = µout then V = 0, as one would expect. The increase in the radial momentum p = m V of the scattering ion is now given by: p = m(V2 - V1 ) = = h 0 (µin - µout ) c E hin - hout = V V

(3.25)

where E = Ein - Eout is the loss of radiative energy. This equation shows that the increase in the momentum of the ions can be calculated from the loss of energy of the photons when these are followed in their path through the wind by means of the Monte Carlo method. Multiplying both sides of Eq. (3.25) by V and using the fact that for each scattering V2 V1 so V (V1 + V2 )/2, gives: 1 2 m(V2 - V12 ) = hin - hout (3.26) 2 Equation (3.26) says that the gain of kinetic energy of the ions in the radial direction equals the energy loss of the photons.

3.3.2 Scattering and absorption processes in the MC calculations
The radiative acceleration as a function of distance is calculated by means of the MC technique by following the fate of the photons using the program MC - W IND (de Koter et al. 1997). In the calculation of the path of the photons we have properly taken into account the possibility that the photons can be scattered or absorbed & re-emitted due to true absorption or eliminated because they are scattered back into the star. The radiative transfer in MC - W IND is calculated in the Sobolev approximation. Multiple line and continuum processes are included in the code. The continuum processes included are electron scattering and thermal absorption and emission. The line processes included are photon scattering and photon destruction by collisional de-excitation. In deciding whether a continuum or a line event takes place, we have improved the code in the following way: The key point of the Monte-Carlo "game" is that line processes can only occur at specific points in each shell of the stellar wind, whereas continuum processes can occur at any point. The correct way of treating the line and continuum processes is by comparing a random optical depth value to the combined optical depth for line and continuum processes along the photon's path. First, this combined optical depth is compared to a random number to decide whether a continuum or a line process takes place. This first part of the treatment is basically the same as described by 50


On the nature of the bi-stability jump in the winds of early-type supergiants Mazzali & Lucy (1993) for the case of line and electron scatterings only. Now, after it has been decided that the process will be a continuum process, a second random number is drawn to decide which continuum process will take place, an electron scattering or absorption.

3.3.3 The calculation of the radiative acceleration gL (r)
The radiative acceleration of the wind was calculated by following the fate of the photons emitted from below the photosphere with the MC technique. To this purpose the atmosphere is divided into a large number of concentric, thin shells with radius r, thickness r containing a mass m(r). The loss of photon energy due to all scatterings that occur within each shell are calculated to retrieve the total line acceleration gL (r) per shell. The total line acceleration per shell summed over all line scatterings in that shell equals 1 p( r ) (3.27) m(r) t where p(r) is the momentum of the ions in the shell. The momentum gained by the ions in the shell is equal to the momentum lost by the photons due to interactions in that shell. Using the relationship between m(r) and r for thin concentric shells, m(r)= 4r2 (r)r,and the derived relation between momentum and energy transfer of the photons p = E /V (Eq. 3.25), gL (r) can be rewritten as gL (r)= 1 E ( r ) (3.28) 4r (r)r V (r)t where E (r) is sum of the energy loss of all the photons that are scattered in the shell. Now using mass continuity (Eq. 3.2) and the fact that the total energy transfer E (r) divided by the time interval t equals the rate at which the radiation field loses energy, -L(r), i.e. E (r)/t = -L(r), the expression for gL (r), which is valid for each shell, simply becomes (Abbott & Lucy 1985) gL (r)=
2

1 L(r) (3.29) M r The line list that is used for the MC calculations consists of over 105 of the strongest lines of the elements H - Zn from a line list constructed by Kurucz(1988). Lines in the wavelength region ° between 50 and 7000 A are included in the calculations with ionization stages up to stage VI. 5 photon packets, distributed over the spectrum at the lower boundary of Typically about 2 10 the atmosphere were followed for each model, i.e. for each adopted set of stellar and wind parameters. For several more detailed models we calculated the fate of 2 107 photon packets. The wind was divided in about 50-60 concentric shells, with many narrow shells in the subsonic region and wider shells in supersonic layers. The division in shells is essentially made on the basis of a Rosseland optical depth scale. Typical changes in the logarithm of this optical depth are about 0.13. gL (r)= -

3.3.4 The determination of M
We predict the mass-loss rates for a grid of model atmospheres to study the behaviour of M near the bi-stability jump. For a given set of stellar parameters we calculate the mass loss in the 51


Chapter 3 following way: 1. For fixed values of L, Teff , R and Meff we adopt several values of the input mass loss M (within reasonable bounds predicted by CAK theory).
inp

2. For each model we adopt a wind with a terminal velocity of 1.3, 2.0 or 2.6 times the effective escape velocity, given by Vesc = 2GM R
eff

(3.30)

A -type velocity law with = 1 was adopted, appropriate for OB stars (Groenewegen & Lamers 1989; Puls et al. 1996) 3. For each set of stellar and wind parameters we calculate a model atmosphere with IS A W IND (see Sect. 3.4). This code gives the thermal structure, the ionization and excitation structure and the population of the energy levels of all relevant ions. 4. For each model the radiative acceleration was calculated with the uses the Monte Carlo method described above.
MC - W IND

program that

5. For each set of stellar parameters and for each adopted value of V we check which one of the adopted mass-loss rates is consistent with the radiative acceleration. This consistency was checked in the following way: Neglecting the term due to the gas pressure, one can write the equation of motion in the following way: dV GMeff (3.31) = - 2 + gL (r ) dr r Using the expression for the line acceleration (Eq. 3.29) and integrating the equation of motion (Eq. 3.31) from the stellar surface to infinity gives V

Z 1 2 2 gL (r)dr (3.32) M (V + Vesc ) = L = M 2 R L = L(r), is the total amount of radiative energy, summed over all the shells, that is lost in the process of line-interaction and is transfered into kinetic energy of the ions as given in Eq. (3.26). Equation (3.32) states that the momentum transfered from the radiation into the wind is used to lift the mass loss out of the potential well and to accelerate the wind to V . Only one value of M will satisfy this equation (Lucy & Abbott 1993). This is the predicted mass-loss rate.
We note that Eq. (3.32) only describes the"global" consistency of the mass-loss rate with the radiative acceleration. For the set-up of the model atmosphere the velocity law V (r) is needed as input. This means that although the M calculation is globally consistent in terms of kinetic wind energy, the velocity is not necessarily locally consistent, since the equation of motion is not solved. Instead, we have used observed values for V and for the velocity law. Since the total amount of radiative energy in Eq. (3.32) is mainly determined in the supersonic region, where the Sobolev approximation is an excellent approximation, L is accurately calculated. This implies that if one adopts the correct values for the terminal velocity, one may predict accurate values for M ! 52


On the nature of the bi-stability jump in the winds of early-type supergiants

3.4 The model atmospheres
The calculation of the mass-loss rates by the method described in the previous section requires the input of a model atmosphere, before the radiative acceleration and M can be calculated. The model atmospheres used for this study are calculated with the most recent version of the non-LTE unified Improved Sobolev Approximation code IS A - W IND for stars with extended atmospheres. For a detailed description of this code we refer to de Koter et al. (1993, 1997). Here, we just make a few relevant remarks. IS A - W IND treats the atmosphere in a unified manner, i.e. no artificial separation between photosphere and wind is assumed. This is distinct from the so-called "core-halo" approaches. In the photosphere the density structure follows from a solution of the momentum equation taking into account gas and radiative pressure on electrons. The velocity law follows from this density structure via the mass continuity equation. Near the sonic point, a smooth transition is made to a -type velocity law for the supersonic part of the wind (see Eq. 3.4). The temperature structure in the wind is computed under the assumption of radiative equilibrium in an extended grey LTE atmosphere. The temperature in the wind is not allowed to drop below a certain minimum value Tmin = 1/2 Teff (Drew 1989). Finally, the chemical species included explicitly in the non-LTE calculations are H, He, C, N, O and Si. The complexity of the model atoms is similar to that used by de Koter et al. (1997). For the iron-group elements, which are important for the radiative acceleration, we calculate the ionization/excitation equilibrium in the modified nebular approximation (see Schmutz 1991). In this representation the ionization equilibrium is given by N
j+1 ne

N

j

= {(1 - )W + }W

Te TR

1/2

N

j+1 ne

LTE

N

(3.33)
TR

j

where ne and Te are the electron density and temperature, N j and N j+1 are the ion population numbers, TR = TR (r, j) is the radiation temperature of ion j at radial depth r, and W is a geometrical dilution factor as defined by Schmutz et al. (1990). The last factor of Eq. (3.33) is the LTE ionization ratio for a temperature TR (r, j). The parameter , introduced by Abbott & Lucy (1985), represents the fraction of recombinations going directly to the ground state. The values of TR (r, j) are obtained by inverting the above equation, using all 19 ionization ratios available from the IS A - W IND calculation. The radiation temperature of an explicit ion is used that has its ionization potential closest (but lower) to that of the metal ion of interest. For instance, the N II/III ratio is used to define the ionization equilibrium of Fe III/IV. The excitation state of metastable levels is assumed to be in LTE relative to the ground state. For all other levels we adopt "diluted" LTE populations, defined by nu =W n1 n n
u 1 LTE

.
TR

(3.34)

where nu and nl are the excitation population numbers for the upper and lower levels. Clearly, the simplified treatment of the iron-group metals is prompted by the computationally intensive nature of the problem at hand. It needs to be improved in the future, but we do not expect that our conclusions regarding the nature of the bi-stability jump would be affected. (We return to this in the discussion in Sect. 3.8). 53


Chapter 3

3.5 The predicted bi-stability jump
Using the procedure as described in Sect. 3.3.4, we calculated mass-loss rates for stars with a luminosity of L = 105 L and a mass of M = 20M . The models have effective temperatures between 12 500 and 40 000 K with a stepsize of 2500 K. These parameters are approximately those of OB supergiants, for which Lamers et al. (1995) found the bi-stability in V . We calculated M for wind models with a -type velocity law with = 1 (Eq. 3.4) for three values of the V /Vesc = 2.6, 2.0 and 1.3. Lamers et al. (1995) found that V /Vesc 2.6 for stars of types earlier than B1, and V /Vesc 1.3 for stars of types later than B2. For the determination of Vesc we used the effective mass Meff = 17.4 M , with e = 0.130. The stellar parameters for the calculated grid are indicated in Table 3.1. The models are calculated for solar metallicities.

3.5.1 The predicted bi-stability jump in M
The results are listed in Table 3.1. This Table gives the values of Teff , R , Vesc and M for each temperature and for the three values of V /Vesc . We also give the value of the wind efficiency factor , which describes the fraction of the momentum of the radiation that is transferred to the ions L MV = (3.35) c The fraction of the photon energy that is transferred into kinetic energy of the ions is also listed (in column 8). The values for this energy efficiency number L/L are a factor of about 10-3 smaller than the wind momentum efficiency number , which is given in column (7). This is because a photon transfers a large fraction of its momentum during a scattering, but only a very small fraction (of order V /c) of its energy. The last column of Table 3.1 marks three models that will be discussed in more detail in Sect. 3.6. The results are plotted in Fig. 3.3. For each of the three values of V /Vesc the value of M is decreasing for decreasing Teff between 40 000 and 30 000 K and also between 22 500 and 12 500 K. Between about Teff = 27 500 K and Teff = 20 000 K (slightly dependent on V /Vesc ) the mass loss increases with decreasing Teff . These increments in M roughly coincide in Teff with the observed bi-stability jump in V /Vesc near spectral type B1, at about 21 000 K. For the ratio of V /Vesc = 2.6, the increase in M between model A and B equals 45 %. We know from the observations that V /Vesc jumps from 2.6 at the hot side of 21 000 K to 1.3 at the cool side of 21 000 K (Lamers et al. 1995). Including this observed jump in V /Vesc in the mass-loss predictions, provides an even steeper increase in M from models A and B to the smaller value of V /Vesc = 1.3, as is shown in the lower part of Fig. 3.3. This figure shows an increase in M of about a factor of five between Teff = 27 500 and 20 000 K. This is our prediction for a bi-stability jump in M . The exact position of Teff of the bi-stability jump in Fig. 3.3 is somewhat ambiguous, since V is adopted from observations, and does not directly follow from our models. For a discussion on the exact position of the jump in Teff , see Sect. 3.8. To test the sensitivity of our predictions of mass-loss rates for different shapes of the velocity law, we calculated another series of models with = 1.5 . Since the differences are only about 10 %, we conclude that the predicted mass-loss rates are only marginally sensitive to the shape of the adopted velocity law. 54


On the nature of the bi-stability jump in the winds of early-type supergiants

V Vesc 1.3

Teff

R



Vesc

V

log M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

.095 .097 .146 .192 .254 .302 .174 .126 .136 .207 .255 .350 .056 .088 .139 .191 .264 .203 .125 .128 .155 .220 .247 .325 .045 .067 .114 .184 .184 .129 .098 .115 .143 .194 .233 .266

L/L (in 10-3 ) 0.103 0.126 0.221 0.332 0.493 0.653 0.414 0.326 0.382 0.626 0.826 1.210 0.073 0.138 0.254 0.398 0.620 0.530 0.360 0.400 0.527 0.801 0.969 1.356 0.070 0.126 0.248 0.458 0.517 0.403 0.337 0.430 0.579 0.845 1.089 1.327

model

2.0

2.6

(K) (R ) (km s-1 ) (k 12 500 67.7 310 15 000 47.0 380 17 500 34.5 440 20 000 26.4 500 22 500 20.9 560 25 000 16.9 630 27 500 14.0 690 30 000 11.8 750 32 500 10.0 810 35 000 8.6 880 37 500 7.6 940 40 000 6.6 1000 12 500 67.7 310 15 000 47.0 380 17 500 34.5 440 20 000 26.4 500 22 500 20.9 560 25 000 16.9 630 27 500 14.0 690 30 000 11.8 750 32 500 10.0 810 35 000 8.6 880 37 500 7.6 940 40 000 6.6 1000 12 500 67.7 310 15 000 47.0 380 17 500 34.5 440 20 000 26.4 500 22 500 20.9 560 25 000 16.9 630 27 500 14.0 690 30 000 11.8 750 32 500 10.0 810 35 000 8.6 880 37 500 7.6 940 40 000 6.6 1000

m s-1 ) (M /yr) 410 - 6.32 490 - 6.39 570 - 6.28 650 - 6.22 730 - 6.15 810 - 6.12 900 - 6.40 980 - 6.58 1060 - 6.58 1140 - 6.43 1220 - 6.37 1300 - 6.26 630 - 6.74 750 - 6.62 880 - 6.49 1000 - 6.41 1130 - 6.32 1250 - 6.48 1380 - 6.73 1500 - 6.76 1630 - 6.71 1750 - 6.59 1880 - 6.57 2000 - 6.48 810 - 6.95 980 - 6.85 1140 - 6.69 1300 - 6.54 1460 - 6.59 1630 - 6.79 1790 - 6.95 1950 - 6.92 2120 - 6.86 2280 - 6.76 2440 - 6.71 2600 - 6.68

C

B A

Table 3.1: Stellar parameters of the grid of calculated models. log (L/L )= 5.0, M = 20M , e = 0.130, Meff = 17.4 M , = 1, solar metallicity.

55


Chapter 3

Figure 3.3: Upper panel: The calculated mass-loss rates M as a function of Teff for three values of the ratio V /Vesc . The values for V /Vesc are indicated in the lower left corner. The stellar parameters are log L/L = 5.0, M = 20 M and = 1.0; all models are calculated for solar metallicities. Lower panel: The predicted bi-stability jump in M from models with the observed ratios of V /Vesc = 2.6 for Teff > 21, 000 K and V /Vesc = 1.3 for Teff < 21, 000 K, as indicated in the lower left corner.

3.5.2 The predicted bi-stability jump in
Another view at these results can be obtained by plotting the wind efficiency factor . Figure 3.4 shows the behaviour of as a function of Teff for the same grid of models as was presented for the mass-loss rates in the upper panel of Fig. 3.3. 56


On the nature of the bi-stability jump in the winds of early-type supergiants

Figure 3.4: The wind efficiency number = MV /(L /c) as a function of Teff for three values of the ratio V /Vesc . These values are indicated in the upper right corner. Note the steady decrease of to lower temperatures, except the jump of about a factor 2 or 3 near 25 000 K. Fig. 3.4 clearly shows that is not a constant function of Teff . The overall picture shows that for the three values of V /Vesc , decreases as Teff decreases. This is probably due to the fact ° that the maximum of the flux distribution shifts to longer wavelengths. At > 1800 A there ° are significantly less lines than at < 1800 A. Therefore, radiative acceleration becomes less effective at lower Teff . In the ranges of 40 000 < Teff < 30 000 and 20 000 < Teff < 12 500 K, is almost independent of the adopted value for V /Vesc . This means that the behaviour of is intrinsically present in the model calculations and does not depend on the values adopted for V /Vesc . In the range of 30 000 < Teff < 20 000 K, the situation is reversed. now increases by a factor of 2 to 3. This means that the wind momentum loss, MV is not constant over the jump, but instead, jumps by a factor of 2 - 3 also. Since V drops by a factor of about two, M is expected to jump by a factor of about five, which was already shown in the lower panel of Fig. 3.3. The behaviour of as a function of Teff is not exactly the same for the three different series of models. First, the size of the jump is different. Second, the jump occurs at somewhat different temperatures. This is no surprise, since the ionization equilibrium does not only depend on T , but on as well, a smaller value of the velocity V , means a larger density in the wind. Hence, the jump is expected to start at a larger value of Teff for a smaller value of V /Vesc . This behaviour for the position of Teff of the jump can be seen in M in Fig. 3.3 and in in Fig. 3.4.

3.6 The origin of the bi-stability jump
In the previous section we have shown that the mass-loss rate increases around Teff = 25 000 K. The next step is to investigate the physical process that causes the bi-stability jump. Therefore, 57


Chapter 3 we will look into the details of the line acceleration gL (r) for three models around the bistability jump. For these models (A, B and C in Table 3.1 and Fig. 3.3) we made improved Monte-Carlo calculations, using 2 â 107 packets of photons, to derive more details about the radiative acceleration. First, we will investigate the line acceleration gL (r) of the model at the hot side of the bistability jump. This model A with Teff = 27 500 K and V /Vesc = 2.6, is our basic model. Then, we will compare model A to model B that has the same V /Vesc , but is situated on the cool side of the bistability jump, where Teff = 25 000 K. By comparing models A and B, we can investigate the intrinsic increase in M of 45 % in our model calculations due to the lower Teff . Thenextstepis tocompare gL (r) of model B and model C which also has Teff = 25 000 K, but a smaller ratio V /Vesc = 1.3. By comparing model B and C, we can obtain information about the effects of a jump in V . Finally, we check our approach for self-consistency by simultaneously calculating the mass-loss rate and terminal velocity.

3.6.1 The main contributors to the line acceleration
Model A has a mass-loss rate of log M = -6.95. The behaviour of the line acceleration as a function of the distance from the stellar surface, gL (r) is shown in Fig. 3.5. The sonic point is reached at a distance of 1.025 R . It is clear that most of the line driving is produced far beyond the sonic point. But, as was explained in Sect. 3.2 the important region that determines the mass-loss rate is bel ow the sonic point. Therefore, the part of the atmosphere around the sonic point is enlarged in Fig. 3.5(b). To investigate the origin of the jump, it is useful to know which elements are effective line drivers in which part of the stellar wind. Therefore, extra Monte-Carlo calculations were performed. The first extra Monte-Carlo simulation was performed with a line list containing only Fe lines. The second one was performed with a line list containing the lines of the elements C, N and O. Figure 3.5(b) shows that Fe is the main line driver below the sonic point. C, N and O, are important line drivers in the supersonic part of the wind, which can be seen in 3.5(a). C, N and O contribute roughly 50 % of the line acceleration in the supersonic part of the wind. Not indicated here, but relevant to mention is that Si, Cl, P and S are other important line drivers in the supersonic part of the wind. Ni was found not to be an important line driver in any part of the stellar wind at all. The mass-loss rate is determined by the radiative acceleration below the sonic point, and the terminal velocity is determined by the radiative acceleration in the supersonic part of the wind. So our results show that the mass-loss rates of hot star winds are mainly determined by the radiation pressure due to Fe! The terminal velocities are mainly determined by the contributions of C, N and O.

3.6.2 The effect of the Fe ionization
To understand the origin of the bi-stability jump in M , we investigate the line acceleration due to Fe. The ionization balance of Fe for models A and B is plotted in Fig. 3.6, top and bottom respectively. The right hand figures show the enlargement of the ionization balance in the region near the sonic point. In Model A (Teff =27 500 K) Fe V has a maximum around x = 1.004, which can be seen in Fig. 3.6 (b). Then, due to the outward decreasing temperature, Fe V decreases 58


On the nature of the bi-stability jump in the winds of early-type supergiants

Figure 3.5: The line acceleration of model A (Teff = 27 500 K and V /Vesc = 2.6), from 1 to 15 R (left) and around the sonic point (right). (a) The solid line shows the total gL as a function of the distance. The dashed line is the contribution by C, N and O only. The dotted line shows the contribution by Fe lines. Some values for the velocity are indicated on top of the figure. (b) The region around the sonic point is enlarged. The sonic point is reached at x = 1.025. Note the bump in the gL (r) just below the sonic point, which is largely due to Fe lines. in favour of Fe IV, which peaks around x = 1.008. Next, one may expect Fe IV to decrease in favour of Fe III. However, around x = 1.013 Fe IV re-ionizes due to a decrease of the density . In this region of the atmosphere, where dV /dr is rapidly increasing, the effect of the decreasing is larger than the effect of the decreasing T . Fig. 3.6 (b) clearly shows that Fe IV is the dominant ionization stage in the subsonic region of the stellar wind. In the region just below the sonic point, the ionization fraction of Fe IV is 90 - 100 % whereas that of Fe III is less than 10 %. However, this does not necessarily mean that Fe IV is the main line driver. To investigate the contribution to the line acceleration gL of the different ionization stages of Fe some extra Monte-Carlo simulations were performed. One simulation included only the lines of Fe III, another simulation included just the lines of Fe IV. The results for gL for Fe III and Fe IV are plotted in Fig. 3.7. It is surprising to note that, although Fe IV is the dominant ionization stage throughout the wind, most of the driving is contributed by Fe III. Below the sonic point Fe III is clearly the most important iron line driver (see Fig. 3.7(b)). From the data shown in Figures 3.6 and 3.7 we conclude that the mass-loss rate of winds from stars with Teff 27 500 K is mainly determined by the radiative acceleration due to Fe III 59


Chapter 3

Figure 3.6: The ionization fraction of Fe as a function of distance. The upper panels are for model A and the lower panels for model B. (a) Fe ionization for model A from x = 1 to 15. (b) Model A, enlarged around sonic point. (c) Fe ionization for model B from x = 1 to 15. (d) Model B, enlarged around sonic point. lines. This suggests that the bi-stability jump is mainly due to changes in the ionization balance of Fe. We test this hypothesis in the next section.

3.6.3 The effect of Teff on M
In the previous section we have shown that the mass loss of model A is dominated by radiative acceleration due to Fe III lines. In this section we investigate changes in the radiative accelera tion due to Fe as Teff decreases. This may explain the increase of M near the bi-stability jump. To this purpose we compare the ionization and gL of models A and B in detail. The ionization balance of model B is shown in Fig. 3.6(c) and (d). It shows that, due to a lower temperature, the decrease of the Fe IV fraction drops to smaller values than for model A, which was shown in Fig. 3.6(b). The ionization fraction of Fe III below the sonic point in the case of model B is up to almost 40 %. To see whether this extra amount of Fe III can cause the increase in the line acceleration, we must look at gL of Fe for model B. Since model A and B have different Teff at the same L , they have a different radiative surface flux. The radiative acceleration will be proportional to this flux. In order to compare the values of gL of the two models, we scale the results to a flux of a Teff = 25 000 K model. So 60


On the nature of the bi-stability jump in the winds of early-type supergiants

Figure 3.7: The contribution of several Fe ions to gL as a function of distance from the stellar surface for model A. (a) The full distance range of x = 1 to 15. (b) The region around the sonic point is enlarged. The legend indicates the ionization stage. Some values for the velocity are indicated on the top of the figure. Note that the strongest contribution to gL below the sonic point is due to Fe III, although the ionization fraction of this ion is less than 10 %.

g

norm L

=g

L

25000 Teff

4

(3.36)

Since Teff 4 R models.

-2

for constant luminosity, this is also a scaling to the Newtonian gravity of the

Figure 3.8 shows the normalized gL of Fe for the models A (top) and B (bottom). The right hand figures show an enlargement of the region near the sonic point. It shows that for model B gL of Fe III around the sonic point is more than a factor two larger than for model A (see Figs. 3.8(b) and (d)). This extra amount of Fe III in model B causes an increase in the total gL in the subsonic part of the wind also, as can be seen in Fig. 3.9(b). We conclude that the increase in mass loss from model A to B is due to the larger radiative acceleration (compared to the gravity) of model B by a larger ionization fraction of Fe III below the sonic point. 61


Chapter 3

Figure 3.8: Normalized gL of Fe as a function of distance from the stellar surface A and B. (a) Normalized gL for the different Fe ionization stages of model indicates the ionization stage. Some values for the velocity are indicated on figure. (b) model A, enlarged around the sonic point. (c) Normalized gL for t ionization stages of model B. (d) model B, enlarged around sonic point.

for the models A. The legend the top of the he different Fe

3.6.4 The effect of V
Now the effect of gL on V will be examined. Therefore, Model B is compared to model C. We remind that models B and C have the same Teff , and hence the same radiative flux and gravity, but model C has a twice as small value of V /Vesc as model B. Figure 3.9(a) shows the normalized gL for models A, B and C. As expected, gL (r) for model C is significantly smaller than gL (r) R for models A and B. This is obviously due to the smaller value of V . The integral gL (r) dr in R Fig. 3.9(a) for model A and B is larger than for model C. The values of gL (r) dr for the models are 2.34 â 1016 and 1.92 â 1016 cm2 s-2 for models A and B respectively, and 6.12 â 1015 cm2 s-2 for model C. Using Eq. (3.7) and the values of Vesc from column (4) in Table 3.1, the output values for V can be obtained from the values of the integral of gL . The derived output values for V for the models are V = 2050, 1860 and 920 km s-1 respectively for the models A,B and C. These values are equal within 10 % to the input values for V which were indicated in column (5) of Table 3.1. We can conclude that a smaller value for V is indeed consistent with R a smaller value of the integral gL (r) dr. However, this is not an independent check, since the calculated line acceleration of optically thick lines (in the Sobolev approximation) is inversely proportional to the Sobolev optical depth which is proportional to (dV /dr)-1 . Hence, assuming 62


On the nature of the bi-stability jump in the winds of early-type supergiants

Figure 3.9: (a) The normalized total gL for the models A, B and C as a function of distance. Notice the much smaller radiative acceleration in the supersonic region of model C compared to models A and B. (b) An enlargement of the region around the sonic point. The sonic point is located around x = 1.025 r/R . Notice also the much smaller radiative acceleration in the subsonic region of model C compared to models A and B. This is due to the smaller value of V /Vesc for model C. Teff (kK) 17.5 30.0 V Vesc 2.0 2.0 V Vesc 1.5 2.5 V Vesc 1.3 2.4 V Vesc 1.2 2.4
MC

0

1

2

3

k

MC

V Vesc 1.2 2.4

4

log M

CAK

log M

MC

0.58 0.85

0.2065 0.0076

(M /yr) -6.21 -6.86

(M /yr) -6.27 -6.90

Table 3.2: Force multipliers and consistent models. log (L/L )= 5.0, M = 20M , e = 0.130, Meff = 17.4 M , = 1, solar metallicity.

a smaller terminal velocity will automatically result in a smaller calculated line acceleration.

3.6.5 A self-consistent solution of the momentum equation
In earlier sections we have demonstrated that the mass loss around the bi-stability jump increases. As we have used observed values for the ratio V /Vesc in our model calculations, we 63


Chapter 3 have not yet provided a self-consistent explanation of the observed bi-stability jump in V /Vesc . As a consistency test of our calculations and an attempt to explain the observed jump in the ratio V /Vesc , we proceeded to solve the momentum equation of line driven wind models around the bi-stability jump. The approach we take is to combine predicted force multiplier parameters k and (see below) from the Monte Carlo calculation with the analytical solution of line driven winds from CAK. We calculated the line acceleration gL for several models with different Teff using the Monte Carlo method. The values of gL were expressed in terms of the force multiplier M (t ) (Eq. 3.8). Following CAK we tried to express M (t ) in terms of a power-law fit of the optical depth parameter t (Eq. 3.9). We found that in the range 20 000 Teff 27 500, M (t ) is not accurately fit by a power-law, since the ionization changes over this critical range in Teff . Fortunately, just outside this temperature region, M (t ) can be accurately represented in terms of k and , i.e. M
MC

(t ) = k

MC -MC

t

(3.37)

Therefore, we have calculated models with effective temperatures just below (Teff = 17 500 K) and just above (Teff = 30 000 K) this critical temperature range. Self-consistent values of V and M were thus found in the following way: 1. We started with an assumed ratio of V /Vesc = 2.0 (See column (2) in Table 3.2). 2. The force multipliers M MC (t ) were calculated and a power-law fit of the type Eq. (3.37) was derived. The fit was found to be excellent in the important part of the wind between the sonic point and V 0.5V . This yielded values of MC and kMC . Next, the mass loss and terminal velocity were simultaneously calculated from these MC and kMC parameters using the CAK solution of the momentum equation. Note that the solution with the finite disk correction (Pauldrach et al. 1986) was not applied, since this is already properly taken into account in the values of MC and kMC calculated in the Monte Carlo technique (see Sect. 3.3). The superscript, MC, to the force multiplier parameters was added to avoid confusion with k and for a point-like source used by e.g. Kudritzki et al. (1989). The ratio V /Vesc can be derived from the simple CAK formulation: V = Vesc MC 1 - MC

(3.38)

The value for MC for the model of 30,000 K is significantly higher than values for that were calculated before (e.g. Pauldrach et al. 1986), since the finite disk is already included in the MC -parameter! 3. The new calculated terminal velocity ratio V /Vesc (column (3) of Table 3.2) was used in the next iteration. 4. New mass-loss rates were calculated from the MC approach using the procedure as explained in Sect.3.3.4. The mass-loss rates are equal within 15 % to the mass-loss rates that can be calculated using the expression for M of CAK using MC and kMC . 5. The above procedure (step 1. through 4.) was repeated until convergence was reached. After four iterations, the ratio V /Vesc did not change anymore. The intermediate values of V /Vesc are given in columns (3), (4) and (5) of Table 3.2. The final value for the ratio 64


On the nature of the bi-stability jump in the winds of early-type supergiants V /Vesc is given in column (8). For the hot model (Teff = 30 000 K) the final ratio V /Vesc equals 2.4; for the cool model (Teff = 17 500 K) V /Vesc = 1.2. These values are within 10 % of the observed values of V /Vesc , i.e. 2.6 and 1.3 respectively. 6. CAK mass-loss rates were also calculated from the resulting final force multiplier parameters kMC and MC (given in columns (6) and (7) of Table 3.2 and the final mass-loss rates are given in column (9) of this Table. Note that the values of M are only marginally different from the mass-loss rates that were calculated from the Monte Carlo approach (column (10) of Table 3.2). In summary; we have self-consistently calculated values of V and M of two models located at either side of the bi-stability jump. We have found a jump in terminal velocity V /Vesc of a factor of two, similar as observed by Lamers et al. (1995). Moreover, the mass-loss rates calculated from the CAK formulation are consistent with those obtained from our Monte Carlo approach. This implies that the origin of the observed change in the ratio V /Vesc of a factor of two around spectral type B1 is identical to the predicted jump in mass-loss rate of a factor of five due to the recombination of Fe IV to Fe III.

3.6.6 Conclusion about the origin of the bi-stability jump
From the results and figures presented above we conclude that the mass-loss rate of early-B supergiants near the bi-stability jump is mainly determined by the radiative acceleration by iron. Although Fe IV is the dominant ionization stage in the atmosphere of stars near 25 000 K, it is Fe III that gives the largest contribution to the subsonic line acceleration. This is due to the number of effective scattering lines and their distribution in wavelengths, compared to the energy distribution from the photosphere. This implies that the mass-loss rates of B-supergiants are very sensitive to the ionization equilibrium of Fe in the upper photosphere. Our models show that the ionization fraction of Fe III increases drastically between Teff = 27 500 and 25 000 K. This causes an increase in the line acceleration below the sonic point and in turn increases the mass loss near the bi-stability jump.

3.7 Bi-stability and the variability of LBV stars
Luminous Blue Variables (Conti 1984) are massive stars undergoing a brief, but important stage of evolution. During this period they suffer severe mass loss with M values of up to -4 M yr-1 . LBVs are characterized by typical variations in the order of V of 1 to 2 magni10 tudes. Nevertheless, the total bolometric luminosity of the star L seems to be about constant. The reason for the typical LBV variations is still unknown. For reviews see Nota & Lamers (1997). Leitherer et al. (1989) and de Koter et al. (1996) have shown that it must be the actual radius of the star that increases during these typical variations. Therefore, Teff decreases during the variations, if L is about constant. In this paper, we have calculated the mass-loss behaviour for normal OB supergiants as a function of Teff . Despite many differences between OB supergiants and LBVs, we can retrieve valuable information about the behaviour of M during a typical LBV variation by investigating the M behaviour of normal OB supergiants, since both types of stars are located in the same part of the HRD. Our calculations can be used as a tool to understand 65


Chapter 3 the mass loss changes of an LBV in terms of changes in Teff during such a typical variation (see also Leitherer et al. 1989). Observations of LBVs show that for some LBVs that undergo typical variations M is in creasing from visual minimum to maximum, while for others it is the other way around: M is during an LBV variation is not a complete decreasing. This "unpredictable" behaviour of M surprise, if one considers our M values as a function of Teff . We have found that in the ranges Teff = 40 000 - 30 000 K and Teff = 20 000 - 12 500 K, M decreases for a decreasing Teff , whereas in the interval between Teff = 30 000 - 20 000 K, M increases for a decreasing Teff . This shows that whether one expects an increasing or decreasing M during an LBV variation depends on the specific range in Teff between visual minimum and maximum. This was already suggested by Lamers (1997), albeit a constant value of was anticipated. Our present calculations cannot be used to model the observed LBV variations, because we have assumed solar metallicities, whereas the LBVs are known to have an enhanced He and N abundance (e.g. Smith et al. 1994). Moreover, since most LBVs have already suffered severe mass loss in the past, their L /M ratio will be higher than for normal OB supergiants. This means that LBVs are closer to their Eddington limit, which one may expect to have an effect on M also. These combined effects explain the lack of a consistent behaviour of M for LBV variations so far. Especially since it is not sure that L really remains constant during the variations (see Lamers 1995).

3.8 Summary, Discussion, Conclusions & Future work
We have investigated the nature of the observed jump in the ratio V /Vesc of the winds of supergiants near spectral type B1. Calculations for wind models of OB supergiants show that around Teff = 25 000 K the mass loss rate M jumps due to an increase in the line acceleration of Fe III below the sonic point. This jump in M is found in three different series of models. In all cases, the wind efficiency number = MV /(L /c) increases significantly, by about a factor of 2 to 3, if Teff decreases from about 27 500 K to about 22 500 K. Observations show that the ratio V /Vesc drops by a factor of two around spectral type B1. Applying these values for V /Vesc , we predict a bi-stability jump in M is expected to increase by about this factor between 27 500 and of about a factor of five. So M 22 500 K. We have argued that the mass loss is determined by the radiative acceleration in the subsonic part of the wind, i.e. below r 1.03R . We found that this radiative acceleration is dominated by the contribution of the Fe III lines. Therefore M is very sensitive to both the metal abundance and to the ionization equilibrium of Fe. Our models show that the ionization fraction of Fe III and the subsonic radiative acceleration increases steeply between Teff = 27 500 and 25 000 K. This explains the calculated increase in M in this narrow temperature range. The exact temperature of the bi-stability jump is somewhat ambiguous. Observations indicate that the jump occurs around spectral type B1, corresponding to Teff 21 000 K (Lamers et al. 1995). If one would not completely trust the value of V /Vesc for the star HD 109867 (number 91 in Lamers et al. (1995)), because of its relatively large error bar, then Teff of the observed jump can easily occur at a few kK higher. In fact we cannot expect the bi-stability jump to occur at one and the same temperature for all luminosity classes, because the jump is sensitive to the ionization balance (mainly of Fe III) in the subsonic region of the wind and hence to the gravity of the star. Our models predict that the jump will occur near Teff 25 000 K. However, 66


On the nature of the bi-stability jump in the winds of early-type supergiants this is sensitive to the assumptions of the models: the adopted masses and luminosities and to the assumption of the modified nebular approximation for the calculation of the ionization equilibrium of iron (see Sect. 3.4). A more consistent treatment of the ionization and excitation equilibrium of the Fe-group elements may have two effects: i) M predicted from L may alter, and ii) Teff at which the ionization ratio of e.g. Fe III/IV flips, may shift. Nevertheless, in view of the very encouraging results using the modified nebular approximation in the modeling of UV metal-line forests (de Koter et al. 1998), we expect the error in Teff at which the dominant ionization of Fe switches from IV to III to be at most a few kK. Furthermore, if a more consistent treatment would yield a change in M this would most likely produce a systematic shift. Since we are essentially interested in relative shifts in M , we do not expect that our conclusions regarding the nature of the bi-stability jump would be affected. It is relevant to mention that Leitherer et al. (1989) calculated atmospheric models for LBVs and suggested that the recombination of iron group elements from doubly to singly ionized stages, which according to them, occurs around Teff = 10 000 K, can explain M increases when LBVs approach their maximum states. We have found a Fe III/IV ionization/recombination effect around Teff = 25 000 K for normal supergiants. We also anticipate that somewhere, at a lower value of Teff a similar ionization/recombination effect will occur for Fe II/III, causing a second bi-stability jump. Lamers et al. (1995) already mentioned the possible existence of a second bi-stability jump around Teff = 10 000 K from their determinations of V /Vesc , but the observational evidence for this second jump was meagre. Possibly, this second jump is real and we anticipate that this second jump could very well originate from a Fe II/III ionization/recombination effect. Furthermore, we have shown that the elements C, N and O are important line drivers in the supersonic part of the wind, whereas the subsonic part of the wind is dominated by the line acceleration due to Fe. 1 Therefore, we do not expect CNO-processing to have a large impact on M , but it might have impact on the terminal velocities. Finally, we would like to add that our calculations for M around Teff = 25 000 K have only been performed for one value of M , L and H/He abundance. M is expected to depend on these stellar parameters, so calculating mass-loss rates for a wider range of stellar parameters will provide valuable information on the size of the bi-stability jump in V and M and will allow us to constrain its amplitude and location in the HRD.

References
Abbott D.C., 1982, ApJ 259, 282 Abbott D.C., Lucy L.B., 1985, ApJ 288, 679 Castor J.I., Abbott D.C., Klein R.I., 1975, ApJ 195, 157 Conti P.S., 1984, in: Maeder A., Renzini A. (eds.) Proc. IAU Symp. 105, Observational Tests of Stellar Evolution Theory, Kluwer, Dordrecht, p. 233 de Koter A., Schmutz W., Lamers H.J.G.L.M., 1993, AAP 277, 561 de Koter A., Lamers H.J.G.L.M., Schmutz W., 1996, A&A 306, 501 de Koter A., Heap S.R., Hubeny I., 1997, ApJ 477, 792
When this study was finished we received a preprint by Puls et al. (1999) who have independently found that the Fe-group elements control the line acceleration in the inner wind part, whereas light ions dominate the outer part.
1

67


Chapter 3 de Koter A., Heap S.R., Hubeny I., 1998, ApJ 509, 879 Drew J.E., 1989, ApJS 71, 267 Groenewegen M.A.T., Lamers H.J.G.L.M., 1989, A&AS 79, 359 Kudritzki R.-P., Pauldrach A.W.A., Puls J., Abbott D.C., 1989, AAP 219, 205 Kurucz R.L., 1988, IAU Trans., 20b, 168 Lamers H.J.G.L.M., 1995, in: Astropysical Applications of Stellar Pulsations, ASP Conf.Ser. 83, 176 Lamers H.J.G.L.M., 1997, in: Luminous Blue Variables: Massive Stars in Transistion, ASP Conf.Ser. 120, 76 Lamers H.J.G.L.M., Cassinelli J.P., 1999, in: Introduction to Stellar Winds, Cambridge Univ. Press, Chapter 3 Lamers H.J.G.L.M., Pauldrach A.W.A., 1991, A&A 244, L5 Lamers H.J.G.L.M., Snow T.P., Lindholm D.M., 1995, ApJ 455, 269 Lamers H.J.G.L.M., Vink J.S., de Koter A., Cassinelli J.P., 1999, in: Variable and Non-Sperical Stellar Winds in Luminous Hot Stars, 159 Leitherer C., Schmutz W., Abbott D.C., Hamann W.R., Wessolowski U., 1989, ApJ 346, 919 Lucy L.B., 1998, in: Cyclical Variability in Stellar Winds, ESO ASS Proc 22, 16 Lucy L.B., Abbott D.C., 1993, ApJ 405, 738 Mazzali P.A., Lucy L.B., 1993, A&A 279, 447 Nota A., Lamers H.J.G.L.M., 1997, Luminous Blue Variables: Massive Stars in Transition, ASP Conf.Ser. 83 Pauldrach A.W.A., Puls J., 1990, A&A 237, 409 Pauldrach A.W.A., Puls J., Kudritzki, R.P., 1986, A&A 164, 86 Puls J., Kudritzki R.P., Herrero A., et al., 1996, A&A 305, 171 Puls J., Springmann U., Lennon, M, A&A submitted Schmutz W., 1991. In: Stellar Atmospheres: Beyond Classical Models, eds. Crivellari L., Hubeny I., Hummer D.G., NATO ASI Series C, Vol. 341, 191 Schmutz W., Abbott D.C., Russell R.S., Hamann W.-R., Wessolowski U., 1990, ApJ 355, 255 Smith L.J., Crowther P.A., Prinja R.K., 1994, A&A 281, 833

68


New theoretical mass-loss rates of O and B stars

4

New theoretical mass-loss rates of O and B stars
Jorick S. Vink, Alex de Koter, and Henny J.G.L.M. Lamers Accepted by A&A

We have calculated mass-loss rates for a grid of wind models covering a wide range of stellar parameters and have derived a mass-loss recipe for two ranges of effective temperature at either side of the bi-stability jump around spectral type B1. For a large sample of O stars, it is shown that there is now good agreement between these new theoretical mass-loss rates that take multiple scattering into account and observations. Agreement between the observed and new theoretical wind momenta increases confidence in the possibility to derive distances to luminous stars in distant stellar systems using the Wind momentum Luminosity Relation. For the winds of the B stars there is an inconsistency in the literature between various mass-loss rate determinations from observations by different methods. One group of M determinations of B stars does follow the new theoretical relation, while another group does not. The lack of agreement between the observed mass-loss rates derived by different methods may point to systematic errors in mass-loss determinations from observations for B stars. We show that our theoretical mass-loss recipe is reliable and recommend it be used in evolutionary calculations.

4.1 Introduction
In this paper we present new theoretical mass-loss rates M for a wide range of parameters for galactic O and B stars, taking multiple scattering into account. These predictions for M are compared with observations. The goal of the paper is to derive an accurate description of mass loss as a function of stellar parameters. Early-type stars have high mass-loss rates, which substantially affects their evolution (e.g. Chiosi & Maeder 1986). The winds of early-type stars are most likely driven by radiation 69


Chapter 4 pressure in lines and in the continuum. The radiation-driven wind theory was first developed by Lucy & Solomon (1970) and Castor et al. (1975) (hereafter CAK). At a later stage the theory was improved by Abbott (1982), Friend & Abbott (1986), Pauldrach et al. (1986) and Kudritzki et al. (1989). During the last decade, the radiation-driven wind theory has been compared with the most reliable mass-loss determinations from observations that are available: mass loss determined from radio data and from the analysis of H line profiles. Both Lamers & Leitherer (1993) and Puls et al. (1996) came to the conclusion that the theory of radiation-driven winds shows a systematic discrepancy with the observations. For O stars the radiation-driven wind theory predicts systematically lower values for mass loss than have been derived from observations. Since the discrepancy increases as a function of wind density, it is possible that the reason for this is an inadequate treatment of "multiple scattering" in the current state of radiation-driven wind theory. It has been suggested (e.g. by Lamers & Leitherer 1993) that the "momentumproblem" that has been observed in the dense winds of Wolf-Rayet stars is the more extreme appearance of this discrepancy seen in the winds of normal O-type stars. Because the observed mass-loss rates for O type supergiants are typically a factor of two higher than the values predicted by radiation-driven wind theory, evolutionary models would be significantly affected if theoretical values were adopted. It is obvious that an accurate description of mass loss is of great importance to construct reliable evolutionary tracks for massive stars. Abbott & Lucy (1985) and Puls (1987) have investigated the importance of "multiple scattering" relative to "single scattering" for the winds of O stars. Abbott & Lucy found an increase in M of a factor of about three for the wind of the O supergiant Puppis if multiple scattering was applied in a Monte Carlo simulation. We will use a similar Monte Carlo technique in which multiple scatterings are taken into account to calculate mass-loss rates for a wide range of stellar parameters throughout the upper part of the Hertzsprung-Russell Diagram (HRD). In Sect. 4.2, the approach to calculate massloss rates will be briefly described, while in Sect. 4.3, a grid of wind models and mass-loss rates will be presented. A clear separation of the HRD into two parts will be made. The first range is that on the "hot" side of the bi-stability jump near spectral type B1, where the ratio of the terminal velocity to the effective escape velocity at the stellar surface (V /Vesc ) is about 2.6; the second range is that on the "cool" side of the jump where the ratio suddenly drops to about 1.3 (Lamers et al. 1995). At the jump the mass-loss rate is predicted to change dramatically due to a drastic change in the ionization of the wind (Vink et al. 1999). In Sect. 4.4, the theoretical wind momentum will be studied and in Sect. 4.5 fitting formulae for the mass-loss rate will be derived by means of multiple linear regression methods: this yields a recipe to predict M as a function of stellar parameters. In Sect. 4.6 these predicted mass-loss rates will be compared with observational rates. We will show that for O stars theory and observations agree if "multiple scattering" is properly taken into account. Finally, in Sects. 4.7 and 4.8 the study will be discussed and summarized.

4.2 Method to calculate M
The basic physical properties of the adopted Monte Carlo (MC) method to predict mass-loss rates are similar to the technique introduced by Abbott & Lucy (1985). The precise method was extensively described in Vink et al. (1999). The core of the approach is that the total 70


New theoretical mass-loss rates of O and B stars series logL no (L ) 1 2 3 4 5 6 7 8 9 10 11 12 4.5 5.0 M


e 0.055 0.041 0.130 0.087 0.068 0.274 0.206 0.165 0.325 0.434 0.326 0.386

M

eff

Teff (kK) 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 12.5 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0

V Vesc 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6

(M ) 15 20 20 30 40 30 40 50 45 60 80 120

(M ) 14.2 19.2 17.4 27.4 37.3 21.8 31.8 41.8 30.4 34.0 53.9 73.7

5.5

5.75 6.0 6.25

Table 4.1: Parameters for the 12 (L , M ) series of wind models. For details about the model assumptions and grid spacing, see text.

loss of radiative energy is linked to the total gain of momentum of the outflowing material. The momentum deposition in the wind is calculated by following the fate of a large number of photons that are released from below the photosphere. The calculation of mass loss by this method requires the input of a model atmosphere, before the radiative acceleration and M can be calculated. The model atmospheres used for this study are calculated with the most recent version of the non-LTE unified1 Improved Sobolev Approximation code (IS A - W IND) for stars with extended atmospheres. For details we refer the reader to de Koter et al. (1993,1997). The chemical species that are explicitly calculated in nonLTE are H, He, C, N, O and Si. The iron-group elements, which are important for the radiative acceleration and M , are treated in the modified nebular approximation (see Schmutz 1991).

4.3 The predicted mass-loss rates
Using the procedure summarized in Sect. 4.2, we have calculated mass-loss rates for 12 values of Teff in the range between 12 500 and 50 000 K. For every effective temperature a grid of 12 series of models for galactic stars was calculated with luminosities in the range log (L /L ) = 4.5 - 6.25 and masses in the range M = 15 - 120 M . For these 144 models, mass-loss rates were calculated for three values of the ratio V /Vesc , yielding a total number of 432 models. The parameters for all series of models are indicated in Table 4.1. In Fig. 4.1 the luminosities and effective temperatures of the models are indicated with asterisks on top of evolutionary tracks to show the coverage of the model grid over the upper HRD. To study the mass-loss dependence on different stellar parameters (L, M and Teff ) separately, a wide range of parameters was chosen, this implies that some of the models in Fig. 4.1 have positions to the left of the main
1 ISA - WI ND

treats the photosphere and wind in a unified manner. This is distinct from the so-called "core-halo"

approaches.

71


Chapter 4

Figure 4.1: Coverage of the calculated wind models over the HRD. The crosses indicate the model values of log L/L and Teff . Evolutionary tracks from Meynet et al. (1994) are shown for several initial masses, which are indicated in the plot. The Zero Age Main Sequence (ZAMS) is also shown. sequence. We enumerate the assumptions in the model grid: 1. The models are calculated for solar metallicities (Allen 1973). 2. The stellar masses in the grid of models were chosen in such a way that they are representative for the evolutionary luminosities of the tracks from the Geneva group (Meynet et al. 1994). To investigate the dependence of M on M , a number of smaller and larger values for M was also chosen (see column (3) in Table 4.1). 3. The grid was constructed in a way that e < 0.5 (see column (4) in Table 4.1), where e is the ratio between the gravitational acceleration and the radiative acceleration due to electron scattering. e is given by: e = L Le = 7.66 10-5 e 4cGM L M M
-1

(4.1)

where e is the electron scattering cross-section (its value is taken as determined in Lamers & Leitherer 1993) and the other constants have their usual meaning. For values of e > 0.5, the stars approach their Eddington limit and the winds show more dramatic mass-loss behaviour. In this study, stellar parameters for these "Luminous Blue Variablelike" stars are excluded to avoid confusion between various physical wind effects. 4. All series of models from Table 4.1 have effective temperatures between 12 500 and 50 000 K, with a stepsize of 2 500 K from 12 500 to 30 000 K and a stepsize of 5 000 K, starting from 30 000 up to 50 000 K. 72


New theoretical mass-loss rates of O and B stars series log L no (L ) 1 2 3 4 5 6 7 8 9 10 11 12 4.5 5.0 M (M ) 15 20 20 30 40 30 40 50 45 60 80 120 log(M ) 0.78 0.61 0.83 0.87 0.73 0.76 0.81 0.82 0.77 0.76 0.76 0.77 Teff (K) 23 22 26 25 25 26 26 25 25 25 26 25
jump

jump (g cm-3 ) -14.82 -15.13 -14.22 -14.68 -14.74 -13.89 -14.13 -14.40 -13.93 -13.66 -13.89 -13.87

5.5

5.75 6.0 6.25

750 500 250 000 000 250 250 000 000 000 250 000

Table 4.2: Bi-stability jump characteristics for the 12 (L , M ) series of wind models.

5. We calculated M for wind models with a -type velocity law for the accelerating part of the wind: R V (r) = V 1 - (4.2) r Below the sonic point, a smooth transition from this velocity structure is made to a the velocity that follows from the photospheric density structure. A value of = 1 was adopted in the accelerating part of the wind. This is a typical value for normal OB supergiants (see Groenewegen & Lamers 1989; Haser et al. 1995; Puls et al. 1996). At a later stage models for other values will be calculated and it will be demonstrated that the predicted M is essentially insensitive to the adopted value of (see Sect. 4.5.4). 6. The dependence of M on various values of V was determined. Lamers et al. (1995) found that the ratio V /Vesc 2.6 for stars of types earlier than B1, and drops to V /Vesc 1.3 for stars later than type B1. Therefore, we calculated mass-loss rates for various input values of this ratio, namely 1.3, 2.0 and 2.6 to investigate the mass loss as a function of this parameter, similar to that in Vink et al. (1999). For the determination of Vesc , the effective mass Meff = M (1 - e ) was used. Meff is given in column (5) of Table 4.1.

4.3.1 M for supergiants in Range 1 (30 000 Teff 50 000 K)
The results for the complete grid of all the 12 (L , M ) series are plotted in the individual panels of Fig. 4.2. Note that for each calculated point in the grid, several wind models had to be calculated to check which adopted mass-loss rate was consistent with the radiative acceleration (see Lucy & Abbott 1993). This yields predicted, self-consistent values for M (see Vink et al. 1999). For each (L , M ) set and for each value of V /Vesc , we found that the mass loss decreases for decreasing effective temperature between 50 000 and 27 500 K. The reason for this fall-off 73


Chapter 4

Figure 4.2: The calculated mass-loss rates M as a function of Teff for the grid of 12 (L , M ) series for three values of the ratio V /Vesc . The values for V /Vesc are indicated in the lower part of the first panel. The stellar parameters L and M are indicated in the upper part of each panel. The thin dotted lines connect the calculated mass-loss rates. The thick solid lines indicate two multiple linear regression fits to the calculated values (see Sect. 4.5). All models were calculated for solar metallicities. 74


New theoretical mass-loss rates of O and B stars is essentially that the maximum of the flux distribution gradually shifts to longer wavelengths. ° Since there are significantly less lines at roughly > 1800 A than at shorter wavelength, the line acceleration becomes less effective at lower Teff , and thus M decreases.

4.3.2 M at the bi-stability jump around 25 000 K
Between about Teff = 27 500 and 22 500 K the situation is reversed: in this range the mass loss increases drastically with decreasing Teff . These increments in M coincide both in Teff and in size of the M jump with the bi-stability jump that was studied by Vink et al. (1999). They showed that the origin of the jump is linked to a shift in the ionization balance of iron in the lower part of the wind and that it is this element that dominates the line driving at the base of the wind. Below Teff 25 000 K, Fe IV recombines to Fe III and as this latter ion is a more efficient line driver than Fe IV, the line acceleration in the lower part of the wind increases. This results in an upward jump in M of about a factor of five and subsequently a drop in V . The drop in V was predicted to be a factor of two, which is confirmed by determinations of V of OB supergiants from ultraviolet data by Lamers et al. (1995). A comparison between the spectral type of the observed bi-stability jump and the effective temperature of the predicted jump, was made in Vink et al. (1999). Since we know from both theory and observations that the ratio V /Vesc jumps from 2.6 at the hot side of the jump to 1.3 at the cool side of the jump, we can predict the jump in mass loss for all 12 (L , M ) series of models. The size of the jump is defined as the difference between the minimum M at the hot side of the jump (where V /Vesc = 2.6) and the maximum M at the cool side (where V /Vesc = 1.3) in Fig. 4.2. The size of the predicted jump in M (log M ) is about a factor of five to seven. Table 4.2 tabulates is indicated in column (4) of Table 4.2: M additional characteristics for the models at the bi-stability jump. The jump in mass loss around Teff 25 000 K is not exactly the same for all series of models: the jump occurs at somewhat different effective temperatures. This is no surprise, since the ionization equilibrium does not only depend on temperature, but on density as well. A smaller value of the ratio V /Vesc leads to a larger density in the wind. Hence, the jump is expected to start at higher Teff for smaller V /Vesc . This behaviour for the position of Teff of the jump is confirmed by all individual panels in Fig. 4.2. To understand the behaviour of the bi-stability jump as a function of the other stellar parameters, i.e. M and L , we will compare the wind characteristics of the 12 series of models around the bi-stability jump in some more detail. First we define a characteristic wind density at 50 % of the terminal velocity V of the wind: . For a standard velocity law with = 1, this characteristic wind density is given by = M 8R2V (4.3)

For all 12 series of models this characteristic density is plotted vs. the effective temperature of the jump. This is done for both the minimum M (at the hot side of the jump) and the maximum (at the cool side of the jump). Figure 4.3 shows the location of the bi-stability jump in terms M of Teff as a function of . The characteristic densities and effective temperatures for the cool side of the jump are indicated with "diamond" signs and with "plus" signs for the hot side. As expected, for all 12 models the minimum M corresponds to a relatively low and relatively high Teff , whereas the maximum M corresponds to a relatively high , but low Teff . Note that 75


Chapter 4

Figure 4.3: Characteristic and Teff of the bi-stability jump around Teff = 25 000 K. An explanation for the different symbols is given in the legend of the plot. The solid line represents a linear fit through the average jump parameters log and Teff . the effective temperature at minimum and maximum mass loss is not a very smooth function of wind density. This is due to our choice of resolution in effective temperature of the grid. We have checked whether the obtained minima and maxima were indeed the extreme mass-loss values by calculating extra models at intermediate values of Teff . The minimum and maximum M values obtained with the initial grid resolution were found to be similar to those determined with a the finer resolution. We thus concluded that the initial resolution of the grid was justified. The "filled circles" represent the average values of Teff and for the "jump" model for each (L , M ) series. The "jump" model is a hypothetical model between the two models where M is maximal and minimal. The solid line indicates the best linear fit through these averages. The relation between the jump temperature (in kK) and log is given by: Teff
jump

= 49.1 (± 9.2) + 1.67 (± 0.64) log

(4.4)

The average temperature and density of the jump are given in columns (5) and (6) of Table 4.2. jump Note that the range in Teff is relatively small; all 12 series of models have jump temperatures in the range between 22.5 < Teff < 26 kK. Figure 4.4 shows the behaviour of the characteristic density log as a function of e . Again this is done for both the cool and hot side of the jump, and for the average between them. As expected, log increases as e increases. Since the average characteristic wind density at the jump shows an almost linear dependence on e , a linear fit through the average densities is plotted. This is the solid line in Fig. 4.4. The relation between log and e is given by: log = -14.94 (± 0.54) + 3.2 (± 2.2) e (4.5)

From the quantities L and M it is now possible to estimate log using Eq. (4.5) and subjump sequently to predict Teff using Eq. (4.4). Later on this will be used as a tool to connect two 76


New theoretical mass-loss rates of O and B stars

Figure 4.4: Characteristic at the bi-stability jump as a function of e . An explanation for the different symbols is given in the legend of the plot. The solid line indicates a linear fit through the average jump parameters for log . fitting formulae for the two ranges in Teff at either side of the bi-stability jump (see Sect. 4.5).

4.3.3 M for supergiants in Range 2 (12 500 Teff 22 500 K)
Figure 4.2 shows that at effective temperatures Teff 22 500 K, M initially decreases. This behaviour in the Teff range between 50 000 and 27 500 K. For some series is similar to the M (dependent on the adopted L /M ) the mass loss decreases until our calculations end at Teff = 12 500. For other series of L and M , the initial decrease suddenly switches to another increase. Vink et al. (1999) already anticipated that somewhere, at lower Teff , a recombination would occur from Fe III to II similar to the recombination from Fe IV to III at 25 000 K. Lamers et al. (1995) already mentioned the possible existence of such a second bi-stability jump around Teff = 10 000 K from their determinations of V /Vesc , but the observational evidence for this second jump is still quite meagre.

4.3.4 Mat the second bi-stability jump around 12 500 K
To understand the characteristics of the "second" bi-stability jump as a function of different stellar parameters (M and L ), we have also studied the models around this second jump in some more detail. Since our model grid is terminated at 12 500 K, it is not possible to determine the maximum M of the second bi-stability jump in a consistent way, similar to that of the first jump discussed in Sect. 4.3.2. Thus, it is not possible to determine the exact size of the second jump in M . Neither is it possible to derive an accurate equation for the position of the second bi-stability jump in Teff (as was done in Eq. (4.4) for the first jump around 25 000 K). Still, it is useful to 77


Chapter 4 determine a rough relationship between the position of the second jump in Teff and the average log by investigating for each model at which temperature the mass-loss rate still decreases and for which models approaching the second bi-stability jump, the mass loss again increases. The relation found between the temperature of the second bi-stability jump and log is determined by eye and is roughly given by: T
jump2

= 100 + 6log

(4.6)

where T jump2 is in kK. From the quantities L and M it is again possible to estimate log using Eq. (4.5) and then to roughly predict T jump2 using Eq. (4.6). This formula will be used for our mass-loss recipe at the low temperature side (see Sect. 4.5).

4.4 The wind momentum
4.4.1 The wind efficiency number
In this section, we present values for the wind efficiency number for the different (L , M ) series. (sometimes called the wind performance number) describes the fraction of the momentum of the radiation that is transferred to the ions in the wind: L MV = (4.7) c Figure 4.5 shows the behaviour of as a function of Teff for the complete grid of models. Figure 4.5 demonstrates that is not constant as a function of Teff . The figure shows that when a star evolves redwards at constant luminosity (from high to low temperature) the momentum efficiency initially decreases until the star approaches the bi-stability jump around 25 000 K, where the wind efficiency suddenly increases by a factor of two to three. Subsequently, below about 22 500 K, decreases again and in some cases (again dependent on L and M )it eventually jumps again at the second bi-stability jump. This overall behaviour of is similar to that of M as showninFig. 4.2. In some of the panels of Fig. 4.5, i.e. in those cases where L /M is large, exceeds the single scattering limit. MV 1 (4.8) L /c This occurs at Teff > 40 000 K and log (L /L ) > 6. It suggests that already for high luminosity OB stars stellar winds cannot be treated in the single scattering formalism. The single-scattering limit which is definitely invalid for the optically thick winds of Wolf-Rayet type stars, is often assumed to be valid for the winds of "normal" supergiants. Here, however, we come to the conclusion that due to multiple scattering, already exceeds unity for luminous, but "normal" OB supergiants, in case log(L/L ) > 6. This was already suggested by Lamers & Leitherer (1993) on the basis of observations.

4.4.2 The importance of multiple scattering
Puls et al. (1996) proposed that the reason for the systematic discrepancy between the observed mass-loss rates and recent standard radiation driven wind models (Pauldrach et al. 1994) was 78


New theoretical mass-loss rates of O and B stars

Figure 4.5: The wind efficiency number as a function of Teff for the grid of 12 (L , M )series for three values of the ratio V /Vesc . The values for V /Vesc are indicated in the legend of the first panel. The stellar parameters are indicated at the top of each panel. All models were calculated for solar metallicities. 79


Chapter 4 caused by an inadequate treatment of multi-line effects in these wind models. To compare our new mass-loss predictions with the most sophisticated prior investigations, it is useful to briefly discuss the most important assumptions that are made in modelling the wind dynamics of OB-type stars. The following four basic choices must be made: 1. One may treat the photosphere and wind in a "core-halo" approximation, or one may not make this distinction and treat photosphere and wind in a "unified" way. This choice must be made twice, i.e. with respect to the calculation of the occupation numbers as well as with respect to the computation of the line force. 2. One may adopt a "single-line" approach, i.e. neglecting effects caused by overlapping lines, or one may follow an approach including "multi-line" effects. 3. One solves the rate equations for all relevant ions explicitly in non-LTE, or one adopts a "nebular type of approach" to calculate the ionization balance. 4. One solves the equation of motion self-consistently, or one derives the wind properties from a global energy argument. Standard radiation driven wind models (CAK, Abbott 1982, Pauldrach et al. 1994) treat the momentum equation in a core-halo approach (1) adopting the single-line approximation (2). Various degrees of sophistication can be applied to determine the occupation numbers. The studies of Pauldrach et al. (1994) and Taresch et al. (1997) represent the current state-of-theart, i.e. they treat all relevant ions explicitly in non-LTE (3) and solve the equation of motion self-consistently (4). Pauldrach et al. (1994) also use a unified method for the calculation of the occupation numbers, but a "core-halo"approach is applied with respect to the line force. Additionally, as line overlap is neglected in the method used by Pauldrach et al. (1994), these models can overestimate the line force as unattenuated photospheric flux is offered to each line, which consequently may produce efficiency numbers larger than unity. Puls (1987) found that for winds of relatively low density (say < 1/2) the inclusion of multi-line effects leads to a reduction of wind momentum compared to the standard model due to backscattering and blocking of photons in the lower part of the wind. For winds of relatively high density (say > 1), such as the dense winds of Wolf-Rayet stars, the situation is likely to be reversed. Here momentum transfer from an extended diffuse field is expected to dominate over the effect of the attenuation of flux in the layers just above the photosphere. This could result in more mass loss compared to the standard radiation driven wind theory (Abbott & Lucy 1985, Springmann 1994). Wolf-Rayet and Of/WN stars profit from a layered ionization structure, which increases the number of lines that can be used for the driving and thus increasing the mass loss (Lucy & Abbott 1993, de Koter et al. 1997). Our method differs in almost all aspects from that of Pauldrach et al. (1994). In our method, photosphere and wind are treated in a unified manner (1) and we properly take multi-scatterings into account with a Monte Carlo technique (2). On the other hand, we derive the level populations of the iron-group elements using (a sophisticated version of) the nebular approximation (3). Finally, we derive the mass loss from a global energy argument (4). This distinct difference of approach implies that a comparison between both methods is difficult. Still, we will address some of the differences in approach by focusing on a star with parameters representative for the O4I(f)-star Puppis, which has been studied in detail by Abbott & Lucy (1985), Puls (1987), Pauldrach et al. (1994) and Puls et al. (1996). 80


New theoretical mass-loss rates of O and B stars e 0.041 0.206 0.434 log L (L ) 4.5 5.5 6.0 M


MS 0.107 0.460 1.07

log M

SS

log M

MS

M MS M SS

(M ) 20 40 60 - 7.87 - 6.46 - 5.76 - 7.72 - 5.95 - 4.97 1.41 3.24 6.17

Table 4.3: The relative importance of multiple (MS) vs. single scattering (SS) for a wind model at Teff = 40 000 K.

We can test the difference between single scattering and multiple scattering by allowing photons to interact with a line only once. Fig. 4.6 show a comparison between the single- and multiple scattering case for three representative wind models at Teff = 40 000 K. The model parameters are given in Table 4.3. For the often studied wind of the O supergiant Puppis, which has a mass-loss rate of M obs = 5.9 â 10-6 M yr-1 (Puls et al. 1996), the observed efficiency number is about 0.6, suggesting that the real efficiency of multiple vs. single scattering is a factor of about four for Puppis (see Fig. 4.6). This is close to the findings of Abbott & Lucy (1985) who found an increase in M by a factor of 3.3 for the wind of Puppis if multiple scattering was taken into account in a Monte Carlo simulation. Note from the figure that at low wind densities, the single- and multiple scattering approach converge, as one would expect. For typical O-stars, which have < 0.5, the mass loss will increase by up to a factor of two when multiple scattering is properly included. The Wolf-Rayet stars, located at the extreme high wind density side, and which in some cases have observed efficiency numbers of factors 10 or even higher, may benefit by factors of up to 50. The reason why Puls (1987) found a reduced mass loss for Puppis when comparing the single-line approach with the multi-line approach is because the single-line approach (which is not the same as the single scattering process) overestimates the line force at the base of the wind, where the mass loss is fixed. However, a similar relative behaviour is not found when we compare the predicted single-line mass loss M = 5.1 â 10-6 M yr-1 of Pauldrach et al. -6 M yr-1 derived from our fitting formula based on (1994) with the value of M = 8.6 â 10 multiple scattering models. It is not possible to exactly pinpoint the cause of this difference, but it is likely to be related to differences in our multi-line treatment and that of Puls (1987). Contrary to Puls (and also contrary to Abbott & Lucy 1985), we do not adopt the core-halo approximation. The formation region of the strong driving lines extends from the photosphere out to the base of the wind. If one assumes an input photospheric spectrum representative of the emergent ultraviolet spectrum as in a core-halo approach, one may overestimate the blocking in the subsonic wind regime which results in a lower mass loss.

4.4.3 The Modified Wind Momentum
Kudritzki et al. (1995) have defined the Wind momentum Luminosity Relation (WLR): MV R
0.5

Lx

(4.9)

where (or MV R0.5 ) is called the "modified wind momentum". Observations of M and V of O supergiants have shown that log is proportional to log L (see e.g. Puls et al. 1996). 81


Chapter 4

Figure 4.6: The efficiency of multiple-scattering for a range of wind densities. MS refers to multiple-scattering, and SS refers to single-scattering. The WLR may in principle be used as a tool to derive distances to galaxies (see Kudritzki et al. 1995). In the theory of line driven winds, the reciprocal value of x equals (Puls et al. 1996): 1/x =
eff

= -

(4.10)
line

Here and are force multiplier parameters, describing the radiative line acceleration g through the stellar wind: g 1 dV dr


line

ne W



(4.11)

where ne is the electron density and W is the dilution factor. corresponds to the power law exponent of the line strength distribution function controlling the relative number of strong to weak lines. If only strong (weak) lines contribute to the line acceleration force, then = 1 (0). The predicted value of is about 0.6. The parameter describes the ionization balance of the wind. Values for this parameter are usually between 0.0 and 0.1. For a detailed discussion of the parameterisation of the line acceleration, see e.g. CAK, Abbott (1982) and Kudritzki et al. (1989). The important point to note here is that possible changes in the slope x as a function of effective temperature reflect the fact that the stellar winds are driven by different sets of ions, i.e. lines of different ions. Figure 4.5 shows that around the bi-stability jump at Teff 25 000 K, increases for decreasing Teff . This implies that one does not necessarily expect a universal WLR over the complete spectral range of O, B and A stars, nor does one expect a constant value of eff or x for different spectral types. 82


New theoretical mass-loss rates of O and B stars

4.5 Mass loss recipe
In this section we present a theoretical mass loss formula for OB stars over the full range in Teff between 50 000 and 12 500 K. The mass-loss rate as a function of four basic parameters will be provided. These parameters are the stellar mass and luminosity, effective temperature and terminal velocity of the wind. To obtain a mass-loss recipe, we have derived interpolation formulae from the grid of M calculations presented in Sect. 4.3. The fitting procedure was performed using multiple linear regression methods to derive dependence coefficients. We have applied this method for the two ranges in Teff separately. The first range is roughly the range for the O-type stars between Teff = 50 000 and 30 000 K. The second range is between Teff = 22 500 and 15 000 K, which is roughly the range for the B-type supergiants. The two relations are connected at the bi-stability jump. We have already derived the jump parameters for different series of models in Sect. 4.3, so we have knowledge about the position of the jump in Teff as a function of stellar parameters. This will be applied in the determination of mass loss for stars with temperatures around the bi-stability jump.

4.5.1 Range 1 (30 000 Teff 50 000 K)
The first range (roughly the range of the O-type stars) is taken from Teff between 50 000 K and 30 000 K. In this range the step size in effective temperature equals 5 000 K. So, for the first range we have five grid points in Teff . Five times 12 series of (L , M ), together with three ratios of (V /Vesc ) yields a total of 180 points in M for the first range. We have found that for on Teff , the fit improved if a second order term (log Teff )2 was taken into the dependence of M account. In fact, this is obvious from the shapes of the plots in the panels of Fig. 4.2. The best fit that was found by multiple linear regression is:

log M =

- 6.697 (±0.061) + 2.194 (±0.021) log(L /105 ) - 1.313 (±0.046) log(M /30) V /Vesc - 1.226 (±0.037) log 2. 0 + 0.933 (±0.064) log(Teff /40000) - 10.92 (±0.90) {log(Teff /40000)} for 27 500 < Teff 50 000K

2

(4.12)

where M is in M yr-1 , L and M are in solar units and Teff is in Kelvin. Note that M is the stellar mass not corrected for electron scattering. In this range V /Vesc = 2.6. Equation (4.12) predicts the calculated mass-loss rates of the 180 models with a root-mean-square (rms) accuracy of 0.061 dex. The fits for the various (L , M ) series are indicated with the thick lines in the panels of Fig. 4.2. Note that some of the panels in Fig. 4.2 seem to indicate that a more accurate fit might have been possible. However, Eq. (4.12) is derived by multiple linear regression methods and thus it provides the mass loss as a function of more than just one parameter. 83


Chapter 4

4.5.2 Range 2 (15 000 Teff 22 500 K)
The second range (roughly the range of the B-type supergiants) is taken from Teff between 22 500 and 15 000 K. In this range the step size in effective temperature equals 2 500 K. For this range, there are four grid points in Teff . Four times 12 series of (L , M ), together with three ratios of (V /Vesc ) yields a total of 144 points in M . In this range the fit did not improve if a second order term in effective temperature was taken into account, so this was not done. The best fit that was found by multiple linear regression for the second range is:

log M =

- 6.688 (±0.080) + 2.210 (±0.031) log(L /105 ) - 1.339 (±0.068) log(M /30) V /Vesc - 1.601 (±0.055) log 2. 0 + 1.07 (±0.10) log(Teff /20000) for 12 500 < Teff 22 500K (4.13)

where again M is in M yr-1 , L and M are in solar units and Teff is in Kelvin. In this range V /Vesc = 1.3. The fitting formula is also indicated by solid lines in the panels of Fig. 4.2. Equation 4.13 predicts the calculated mass-loss rates of the 144 models for this Teff range with an rms accuracy of 0.080 dex. For this second range (12 500 < Teff 22 500 K) the fit is slightly less good than for the first Teff range. This is due to the presence of the second bi-stability jump which already appears in some (L /M ) cases, as was shown in Fig. 4.2. If those models that do show the second bi-stability jump, i.e. stars with high e , are omitted from the sample, the accuracy improves to 0.06 dex. In all cases the rms is < 0.08 dex in log M , which implies that the fitting formulae yield good representations of the actual model calculations. We are aware of the fact that there could be systematic errors in our approach, since we have made assumptions in our modelling. For a discussion of these assumptions, see Vink et al. (1999). Whether there are still systematic errors between the observed mass-loss rates and these new predictions of radiation-driven wind theory, will be investigated in Sect. 4.6.

4.5.3 The complete mass-loss recipe
For stars with effective temperatures higher than 27 500 K, one should apply the mass-loss formula for the first range (Eq. 4.12); for stars with Teff lower than 22 500 K the formula for the second range (Eq. 4.13) is to be used. In the range between 22 500 and 27 500 K, it is not a priori known which formula to apply. This due to the presence of the bi-stability jump. Nevertheless, it is possible to retrieve a reliable mass-loss prediction by using Eqs. (4.4) and (4.5) as a tool to determine the position of the jump in Teff . In predicting the mass-loss rate of stars close to the bi-stability jump, one should preferentially use the observed V /Vesc value to determine the position with respect to the jump. This is a better approach than to use the tools from Eqs. (4.4) and (4.5) to determine the position of the jump. The reason is that errors in the basic stellar parameters may accidently place the star 84


New theoretical mass-loss rates of O and B stars at the wrong side of the jump. A computer routine to calculate mass loss as a function of input parameters is available upon request as well as publicly available2. If V is not available, as is the case for evolutionary calculations, one should adopt the ratio V /Vesc = 2.6 for the hot side of the jump and V /Vesc = 1.3 for the cool side of the jump, in agreement with the analysis by Lamers et al. (1995). Note that the exact Teff of the jump is not expected to have a significant effect on evolutionary tracks calculated with this new massloss description, since the most luminous stars spend only a relatively short time around Teff 25 000 K during their evolution. Since our calculations were terminated at 12 500 K, we are not able to determine the size and the position of the second bi-stability jump. Predicting the mass-loss behaviour below this second jump would therefore be speculative. Yet, for evolutionary tracks the mass loss below 12 500 K is an important ingredient in the evolutionary calculations. We roughly estimate from our grid calculations that for a constant ratio of V /Vesc the increase in M around 12 500 is about a factor of two, similar to that found for the first jump near 25 000 K. Furthermore, observations by Lamers et al. (1995) indicate that for stars around 10 000 K, V /Vesc drops again by a factor of two from V /Vesc 1.3 to about 0.7. It is therefore plausible to expect that the size in M of the second jump is comparable to the size of the first jump. So, M of the second jump is expected to be a factor of five also. We argue that this second jump should also be considered in evolutionary calculations and suggest Eq. (4.13) could be used for effective temperatures below the second jump when the constant in Eq. (4.13) is increased by a factor of five (or log M = 0.70) to a value of -5.99. The mass-loss recipe can be applied for evolutionary calculations until the point in the HRD where line driven winds become inefficient and where probably another mass-loss mechanism switches on for the cooler supergiants (see Achmad et al. 1997). We suggest that in the temperature range below the second jump V /Vesc = 0.7 is adopted.

4.5.4 The dependence of M on the steepness of the velocity law
To test the sensitivity of our predictions of mass-loss rates on different shapes of the velocity law, we have calculated series of models for = 0.7, 1.0 and 1.5. This is a reasonable range for OB stars, see Groenewegen & Lamers 1989; Puls et al. 1996). The adopted stellar parameters for this test are L = 105 L and M = 20 M . We have calculated M for all of the above values for wind models with the three values V /Vesc = 2.6, 2.0 and 1.3. From the results shown in Fig. 4.7 we derived for the dependence of M on the adopted value of : log M = C + 0.112 (±0.048) log(/1.0) (4.14)

where C is a constant. This relation is valid for the range between = 0.7 - 1.5. Since the dependence on this parameter is significantly smaller than that on the other parameters, L , M , Teff and V /Vesc , as was found in Eqs. (4.12) and (4.13), we have omitted the dependence from the mass loss recipe. We have just presented the dependence in this section for the sake of completeness, but we can conclude that the predicted mass-loss rates are only marginally sensitive to the shape of the adopted velocity law. One could argue that a dependence on M could be of significance for more extreme series of models. This was tested, but it turned out that for the high e series, the dependence is also insignificant, i.e. deviations of predicted M
2

see: www.astro.uu.nl/jvink/

85


Chapter 4

Figure 4.7: Dependence of M on the shape of the velocity law, for three values of = 0.7, 1.0 and 1.5, as is indicated in the lower left corner of the plot. The values for V /Vesc are indicated in the upper left corner of the plot. For other stellar parameters, see text. are less than M < 0.03 dex. This shows that we can safely omit the dependence on M in the mass-loss recipe for the O and B stars.

4.6 Comparison between theoretical and observational M
4.6.1 M comparison for Range 1 (27 500 < Teff 50 000 K)
An extended compilation of observed mass-loss rates of early-type OBA stars is obtained by Lamers et al. (2000; in preparation). Since both the ultraviolet and the infrared method do not yet yield reliable rates, only mass-loss rates based on radio free-free emission and emission of H have been considered. The H mass-loss rates and their stellar parameters are from: Herrero et al. (2000); Kudritzki et al. (1999); Lamers & Leitherer (1993) (these H equivalent width values are corrected with the curve of growth method from Puls et al. 1996); Puls et al. (1996); Scuderi et al. (1992), Scuderi (1994), and Scuderi & Panagia (2000). The radio rates are from the compilation of Lamers & Leitherer (1993); from Leitherer et al. (1995) and Scuderi et al. (1998). The observed terminal velocities are from the same papers. These were mainly determined from P Cygni profiles. The stellar masses are derived from evolutionary tracks of Meynet et al. (1994). For a critical discussion of the observed mass-loss rates and for the selection of the most reliable data, see Lamers et al. (in preparation). For all these stars with known observational mass-loss rates and stellar parameters, we have determined theoretical M values with the mass-loss recipe that was derived in Sect. 4.5. A starto-star-comparison between these predicted mass-loss rates and those derived from observations is presented in Fig. 4.8. In this plot only the stars above the bi-stability jump (where Teff 27 500 K) are included. The mass-loss rates from Puls et al. (1996) are indicated with a 86


New theoretical mass-loss rates of O and B stars

Figure 4.8: Comparison between theoretical and observational M (both radio data and H) for the O stars. The Puls et al. (1996) H rates; H rates from other determinations, and radio mass-loss rates are indicated with different symbols. The dashed line is a one-to-one relation. different symbol (filled circle), because these are obtained from a homogeneous set, and are analyzed with the most sophisticated wind models. Note that the outlier at log Mobs - 7.4 is the star Oph (HD 149757) for which Lamers & Leitherer (1993) reported that the mass-loss rate is uncertain. The errors in Fig. 4.8 can be due to several effects. There is an error in the theoretical fitting formula, though this error is only 0.061 dex (see Sect. 4.5.1). There could also be systematic errors due to assumptions in the modelling. Furthermore, there could be systematic errors in the mass-loss determinations from observations. Such systematic effects may for instance occur if the clumping factor in the wind changes with distance to the central star. This because the H and radio emission originate from distinctly different regions in the stellar wind. However, Lamers & Leitherer (1993) have shown that for a significant sample of O stars there is good agreement between the radio and the H mass-loss rates. The random errors in the observational mass-loss rates are due to uncertainties in the stellar parameters and in the mass-loss determinations. We tentatively estimate the intrinsic errors in the observed mass-loss rates from the radio and H method to be on the order of 0.2 - 0.3 dex (see Lamers et al. in preparation). This means that for a star-by-star comparison between observations and theory one would expect a scatter around the mean which is a combination of the theoretical and observed uncertainties. This error is on the order of 0.3 dex. The scatter between observational and theoretical mass-loss rates for the O stars from Fig. 4.8 that was 87


Chapter 4 actually derived, equals 0.33 dex ( 1 ) for the complete set and is 0.24 dex for the Puls et al. set. This is an expected scatter and it implies that we do not find a systematic discrepancy between observations and our predictions for the O star mass-loss rates. Contrary to earlier comparisons between observations and theory where systematic discrepancies have been reported (see Lamers & Leitherer 1993, Puls et al. 1996), here we find that there is agreement between our predictions and the mass-loss rates derived from observations for the O-type stars. The essential difference between previous studies and the present one is that in our treatment of the theory of line driven winds, we consistently take into account effects of "multiple-scattering" in the transfer of momentum from the radiation field to the wind. We find systematic agreement between observed and theoretical mass-loss rates for a large sample of O stars. This result implies that physical effects that were not incorporated in our models, such as magnetic fields and stellar rotation, is not expected to influence the mass-loss rates of O stars significantly.

4.6.2 Modified Wind momentum comparison for Range 1 (27 500 < Teff 50 000 K)
Instead of comparing just the mass-loss rates it is useful to compare (modified) wind momenta derived from observations and theory. In earlier studies, e.g. Lamers & Leitherer (1993), and Puls et al. (1996), wind momenta have been plotted versus the wind efficiency number . Comparisons between observed and theoretical wind momenta as a function of could yield important information about the origin of the systematic discrepancy between theory and observations. However, since these two quantities (wind momentum and wind efficiency number) both contain the mass-loss rate, they are not independent. Therefore, no such comparison is made here. Instead, the wind momenta are plotted versus the stellar luminosity, to compare the observational and theoretical WLR. We divide the Teff range into two parts. First, we examine the wind momenta for stars where Teff 27 500 K, later on we will also compare the cooler stars. Figure 4.9 shows the modified wind momentum as a function of stellar luminosity for the sample of stars with known observational mass-loss rates. The upper panel shows these modified wind momentum values for the theoretical mass-loss rates and a linear best fit through these theoretical data (dotted line). Note that the "theoretical" WLR only contains the theoretical M , the included values for V and R were taken from observations. The theoretical WLR is:
theory

=

-12.12 (± 0.26) + 1.826 (± 0.044) log(L/L ) for Teff 27 500 K (4.15)

Since the slope of the WLR of Eq. (4.15) has a slope of x = 1.826, the derived theoretical value for eff (Eq. 4.10) that follows, is: 1 = 0.548 x This corresponds well to predicted values of the force multiplier parameter ( 0.10, see e.g. Pauldrach et al. 1994).
eff

=

(4.16) 0.66 and

88


New theoretical mass-loss rates of O and B stars

Figure 4.9: Upper panel: The theoretical modified wind momentum expressed in M /yr km/s R 0.5 for the stars in the first Teff range (27 500 < Teff 50 000 K). The dotted line indicates the best linear fit. Lower panel: The observational modified wind momentum for these stars. The dotted line indicates the same theoretical linear fit, as in the upper panel.

The lower panel of Fig. 4.9 shows that both the WLR for the Puls et al. (1996) data and that for the other methods/authors, follow the same relationship, both in agreement with the theoretical WLR. The dotted line is again the theoretical best linear fit. We conclude that for the range of the O stars, there is good agreement between theoretical wind momenta and those determined from observations. The scatter between theoretical and observational modified wind momenta is only 0.06 (1 ). 89


Chapter 4 The good agreement between the observational and theoretical wind momenta adds support to the possibility to derive distances to luminous, hot stars in extragalactic stellar systems using the WLR. In practice the technique may be hampered by e.g. the fact that O stars are mostly seen in stellar clusters and cannot be spatially resolved in distant stellar systems. This is one of the reasons why the visually brighter B-type and especially the A-type supergiants located in the field are expected to be better candidates in actually using the WLR as a distance indicator (see Kudritzki et al. 1999). Comparison between the theoretical and observational WLR for the winds of B and A type supergiants is thus essential to investigate whether the slope of the WLR is the same for different spectral ranges. This is not expected, since the winds of different spectral types are driven by lines of different ions (see Vink et al. 1999; Puls et al. 2000).

4.6.3 Modified Wind momentum comparison for Range 2 (12 500 Teff 22 500 K)
Figure 4.10 shows the modified wind momentum as a function of luminosity for both theory and observations for the stars in the second range (12 500 Teff 22 500 K). A best fit through the theoretically derived WLR is indicated with a dotted line in both panels. The theoretical WLR for this Teff range is:
theory

=

-12.28 (± 0.23) + 1.914 (± 0.043) log(L/L ) for 12 500 Teff 22 500 K (4.17)

Since the slope of the WLR for this range is slightly higher than that for the O star range, the predicted value for eff is somewhat lower (see Sect. 4.4.3), namely: 1 = 0.522 (4.18) x The lower panel of Fig. 4.10 indicates the observed modified wind momenta (the dotted line contains the theoretical mass-loss rates). For this second Teff range (12 500 Teff 22 500 K) the plot in the lower panel reveals a large scatter in the observed data. Comparison of these observations with our predictions shows that within the subset of radio mass-loss rates there does not appear to be a systematic discrepancy. Also, those H profiles which are fully in emission (the filled symbols in the lower panel of Fig. 4.10), i.e. the profiles that within the H method most likely provide the most reliable mass-loss rates, do not show a systematic difference with the radio rates. The picture becomes different for stars that show H to be P Cygni shaped (grey symbols in lower panel of Fig. 4.10) or fully in absorption (open symbols). Although the measurements of Scuderi (1994,2000) remain reasonably consistent, those by Kudritzki et al. (1999) are discrepant in that at log L/L 5.8 these values start to diverge from the other observed rates, such that below log L/L 5.6 a systematic difference of about a factor of 30 results between different sets of observed mass-loss rates. An investigation of the origin of these systematic differences in observed B star wind momenta is beyond the scope of this paper. We will address this issue in a separate study (Lamers
eff

=

90


New theoretical mass-loss rates of O and B stars et al., in preparation). Here we just note that the large scatter in the observed H data implies that there is either a dichotomy in the wind-momenta of B-stars (as suggested by Kudritzki et al. 1999) or that there exist systematic errors in the mass-loss determinations from H for B stars. The systematic discrepancies for the observed B star wind momenta imply that we cannot currently compare our predictions with observed data in the most meaningful way, since the data are not consistent and thus a fair comparison with our predictions cannot be conclusive. In addition, it may be meaningful to further investigate the validity of assumptions in our method of predicting the mass-loss rates of B-type stars (see e.g. Owocki & Puls 1999). Still, we note that the most reliable rates (from radio and pure H emission profiles) appear to be consistent with our predictions. The upper panel of Fig. 4.10 reveals that most of the models in the second Teff range (12 500 Teff 22 500 K) lie above the theoretical fit for the models from the first Teff range (Teff 27 500 K). This is due to the increase in the mass-loss rate at the bi-stability jump of a factor of five. The models with 12 500 Teff 22 500 K are, however, only slightly above the fit for the O star models (Teff 27 500 K), as at the bi-stability jump the terminal velocity V drops by a factor of two.

4.7 Discussion
We have shown that our predictions of mass loss for O stars, using Monte Carlo simulations of energy loss during photon transport in non-LTE unified wind models, yields good agreement with the observed values. This demonstrates that an adequate treatment of "multiple scattering" in radiation-driven wind models resolves the discrepancy between observations and theory that had been reported earlier. The agreement between observed and theoretical wind momenta of O stars adds support to the method of deriving distances to distant stellar systems using the WLR. The comparison between the predicted and observed values of the modified wind momentum for the B stars is not conclusive. A good comparison between the observations and our predictions for the B star regime needs to await an explanation of the discrepancies in the observed B star mass-loss rates. This issue will be addressed in a separate study. Our models predict a jump in mass loss of about a factor of five around spectral type B1. An important point that supports this prediction is the following. Vink et al. (1999) have calculated the mass-loss rate and V for winds at both sides of the bi-stability jump in a self-consistent way for models with log(L/L ) = 5.0 and M = 20 M . These self-consistent calculations showed a jump in M of a factor of five and a simultaneous drop in V /Vesc of a factor of two. This drop in V /Vesc has been observed (Lamers et al. 1995). This gives support to our prediction that the mass-loss rate at spectral type B1 increases by the predicted amount. Since there is good agreement between observed mass-loss rates by different methods and the new theoretical mass-loss rates for the O-type stars, whereas there is inconsistency between the observed mass-loss rates from different authors for the B-type stars, this may point to the presence of systematic errors in mass-loss determinations from observations for B stars. Because our predictions for the O stars agree with observations and our models also predict the bi-stability jump around spectral type B1, we believe that our theoretical mass-loss predictions are reliable and suggest they be used in new evolutionary calculations of massive stars. 91


Chapter 4

4.8 Summary & Conclusions
1. We have calculated a grid of wind models and mass-loss rates for a wide range of stellar parameters, corresponding to masses between 15 and 120 M . 2. We have derived two fitting formulae for the mass-loss rates in two ranges in Teff at either side of the bi-stability jump around 25 000 K. A mass-loss recipe was derived that connects the two fitting formulae at the bi-stability jump. 3. There is good agreement between our mass-loss predictions that take multiple scattering into account, and the observations for the O stars. There is no systematic difference between predicted and observed mass-loss rates. 4. A comparison between observed and predicted wind momenta of O-type stars also shows there is good agreement. This adds support to the use of the WLR as a way to derive distances to luminous O stars in distant stellar systems. 5. For the observed mass-loss rates of B stars there is an inconsistency between by different authors and/or methods. One group of M determinations of follow the theoretical relationship, while another group does not. This lack between the observed mass-loss rates of B stars may point to systematic observed values. rates derived B stars does of agreement errors in the

6. Since our new theoretical mass-loss formalism is successful in explaining the observed mass-loss rates for O-type stars, as well as in predicting the location (in Teff ) and size (in V /Vesc ) of the observed bi-stability jump, we believe that our predictions are reliable and suggest that our recipe be used in new evolutionary calculations for massive stars. A computer routine to calculate mass loss is available upon request as well as publicly available at the address www.astro.uu.nl/jvink/.

References
Abbott D.C., 1982, ApJ 259, 282 Abbott D.C., Lucy L.B., 1985, ApJ 288, 679 Achmad L., Lamers H.J.G.L.M., Pasquini L., 1997, A&A 320, 196 Allen C.W., 1973, Astrophysical quantities, Athlone Press Castor J.I., Abbott D.C., Klein R.I., 1975, ApJ 195, 157 Chiosi C., Maeder A., 1986, ARA&A 24, 329 de Koter A., Schmutz, W., Lamers, H. J. G. L. M., 1993, A&A 277, 561 de Koter A., Heap S.R., Hubeny I., 1997, ApJ 477, 792 Groenewegen M.A.T., Lamers H.J.G.L.M., 1989, A&AS 79, 359 Haser S., Lennon D.J., Kudritzki R.-P., 1995, A&A 295, 136 Herrero A., Puls, J., Villamariz, M.R., 2000, A&A 354, 193 Kudritzki R.-P., Pauldrach A.W.A., Puls J., Abbott D.C., 1989, AAP 219, 205 Kudritzki R.-P, Lennon D.J., Puls J., 1995, in: "Science with the VLT", eds. Walsh J.R., Danziger I.J., Springer Verlag, p. 246 Kudritzki R.-P, Puls J., Lennon D.J., et al., 1999, A&A 350, 970 Lamers H.J.G.L.M., Leitherer, C., 1993, ApJ 412, 771 92


New theoretical mass-loss rates of O and B stars Lamers H.J.G.L.M., Snow T.P., Lindholm D.M., 1995, ApJ 455, 269 Lamers H.J.G.L.M., Nugis T., Vink J.S., de Koter A., 2000, in: "Thermal and ionization aspects from hot stars", eds. Lamers H.J.G.L.M., Sapar A., ASP Conf Ser 204, p. 395 Leitherer C., Chapman J., Korabalski B., 1995, ApJ 450, 289 Lucy L.B., Solomon P., 1970, ApJ 159, 879 Lucy L.B., Abbott D.C., 1993, ApJ 405, 738 Meynet G., Maeder A., Schaller G., Schearer D., Charbonel C., 1994, A&AS 103, 97 Owocki S.P., Puls J., 1999, ApJ 510, 355 Pauldrach A.W.A., Puls J., Kudritzki R.P., 1986, A&A 164, 86 Pauldrach A.W.A., Kudritzki R.P., Puls J., Butler K., Hunsinger J.,1994, A&A 283, 525 Puls J., 1987, A&A 184, 227 Puls J., Kudritzki R.P., Herrero A., et al., 1996, A&A 305, 171 Puls J., Springmann U., Lennon M., 2000, A&AS 141, 23 Schmutz W., 1991, in: "Stellar Atmospheres: Beyond Classical Models", eds. Crivellari L., Hubeny I., Hummer D.G., NATO ASI Series C, Vol. 341, 191 Scuderi S., 1994, "Properties of winds of early type stars", thesis, Univ. of Catania Scuderi S., Panagia N., 2000, in: "Thermal and ionization aspects from hot stars", eds. Lamers H.J.G.L.M., Sapar A., ASP Conf Ser 204, p. 419 Scuderi S., Bonanno G., Di Benedetto R., Sparado D., Panagia N., 1992, A&A 392, 201 Scuderi S., Panagia N., Stanghellini C., Trigilio C., Umana C., 1998, A&A 332, 251 Springmann U., 1994, A&A 289, 505 Taresch G., Kudritzki, R.P., Hurwitz, M., et al., 1997, A&A 321, 531 Vink J.S., de Koter A., Lamers H.J.G.L.M., 1999, A&A 350, 181

93


Chapter 4

Figure 4.10: Upper panel: The theoretical modified wind momentum expressed in M /yr km/s R 0.5 for the second range (12 500 Teff 22 500 K). The dotted line indicates the best linear fit for this range. The solid line indicates the theoretical WLR for stars in the range 27 500 < Teff 50 000 K. Lower panel: The observational modified wind momentum for stars in this Teff range (12 500 Teff 22 500 K). The different sources of the observations are indicated in the upper left corner. The dotted line again indicates the theoretical linear fit for the stars in the second range (12 500 Teff 22 500 K).

94


Mass-loss predictions for O and B stars as a function of metallicity

5

Mass-loss predictions for O and B stars as a function of metallicity
Jorick S. Vink, Alex de Koter, and Henny J.G.L.M. Lamers Submitted to A&A

We have calculated a grid of massive star wind models and mass-loss rates for a wide range of metal abundances between 1/100 Z /Z 10. We have found that in the range between about 1/30 Z /Z 3 the mass loss vs. metallicity dependence is constant and is given by M Z 0.85 . This M (Z ) relation completes the Vink et al. (2000) mass-loss recipe with an additional parameter Z . Although it is derived that stellar mass loss is a constant function of Z over a large range in metal content, one should be aware of the presence of bistability jumps at specific temperatures. Here the character of the line driving changes drastically due to recombinations of dominant metal species resulting in jumps in the mass loss. We have investigated the physical origins of these jumps and have derived formulae that combine mass loss recipes for both sides of such jumps. As observations of different galaxies show that the ratio Fe/O varies with metallicity, we make a distinction between the metal abundance Z derived on the basis of iron or oxygen lines. Our mass-loss predictions are successful in explaining the observed massloss rates for Galactic and Small Magellanic Cloud O-type stars, as well as in predicting the observed Galactic bi-stability jump. Hence, we believe that our predictions are reliable and suggest that our mass-loss recipe be used in future evolutionary calculations of massive stars at different metal abundance. A computer routine to calculate mass loss is publicly available.

5.1 Introduction
In this paper we predict the rate at which mass is lost due to stellar winds from massive O and B type stars as a function of metal abundance: M = f (Z ). The calculations are based on state-ofthe-art modelling. The model description takes into account momentum transfer of radiation to 95


Chapter 5 gas in a way that photons are allowed to interact with ions in the wind more than just once. In a previous study, Vink et al. (2000) have calculated wind models including this effect of "multiple scattering" for Galactic early-type stars. They have shown that these predictions agree with the observations for Galactic O stars, which resolved a persistent discrepancy between observed and theoretical mass-loss rates (Lamers & Leitherer 1993, Puls et al. 1996). Metallicity is a key parameter that controls many aspects of the formation and the evolution of both stars and galaxies. For instance, the overall chemical enrichment of the interstellar medium (ISM) is a strong function of metallicity. Secondly, the relative importance of stellar winds compared to Supernova explosions depends on Z in the sense that stellar winds become more important with increasing metallicity (Leitherer et al. 1992). Since chemical elements are produced in stars with different masses, they enrich the ISM on different timescales. Massive stars mainly contribute to the enrichment of oxygen, other -elements and iron. Therefore, these elements are ejected on short timescales. Although carbon and nitrogen are also produced in massive stars, their main contribution comes from longer-lived intermediate mass stars. This implies that if the star formation history and the initial mass function are considered, metallicity is expected to cause a "differential" chemical enrichment of the ISM in different galaxies. Recent models of the chemical evolution versus redshift in the Universe predict that metallicity shows a stronger dependence on the local density (i.e. galaxy mass) than on redshift (Cen & Ostriker 1999). Hence, galaxies with high and low metal abundances are expected to be found at all cosmological distances. These models reasonably predict the range in metal abundance that has been observed. The metallicity reaches as high as 10 times the solar value Z in central regions of active galactic nuclei and quasars (Artymowicz 1993, Hamann 1997), but is only about 1/50 Z for the blue compact dwarf galaxy IZw18 (Sargent & Searle 1970, Izotov & Thuan 1999). Such low metallicity may imply that blue compact dwarf galaxies only experience their first episode of star formation. Based on the observed range in Z , we will study the mass loss properties of massive stars within the representative metallicity range of 1/100 Z /Z 10. The driving mechanism of the winds of massive early-type stars is radiation pressure on numerous spectral lines (Castor et al. 1975, hereafter CAK; Abbott 1982, Pauldrach et al. 1986, Vink et al. 2000). It is important to know which lines are actually responsible for the acceleration of the winds. As hydrogen and helium only have very few lines in the relevant spectral range in which early-type stars emit most of their radiation, it is mainly lines of the metals that are responsible for the line driving. This thus implies that the stellar wind strengths are expected to depend on metal abundance. Observational evidence for metallicity dependent stellar wind properties was found by Garmany & Conti (1985) and Prinja (1987). They found that the terminal flow velocity of the stellar wind in the Magellanic Cloud stars was lower than that of Galactic stars. The authors attributed this difference to an under-abundance of metals in the Magellanic Clouds compared to the Galaxy. The quantitative dependence of M on Z was theoretically studied by CAK, Abbott (1982) and Kudritzki et al. (1987). These studies have shown that the M (Z ) relation is expected to behave as a power-law: MZ
m

(5.1)

with predictions for the index m ranging between about 1/2 (Kudritzki et al. 1987) to 0.94 (Abbott 1982). Since these results were based on radiation-driven wind models that did not 96


Mass-loss predictions for O and B stars as a function of metallicity take into account the effect of "multiple scattering", a new investigation of the M vs. Z relation, is appropriate. Especially since Eq. (5.1) is widely used in evolutionary calculations for massive stars, usually adopting m = 1/2 (e.g. Meynet et al. 1994). We will use our "Unified Monte Carlo" method (Vink et al. 2000) to predict mass-loss rates of early-type stars over a wide range in metallicities and stellar parameters. In this approach, multiple scatterings are consistently taken into account and an artificial separation between the stellar photoshere and wind (core-halo) is avoided. The main question we will address is: 'What is the dependence of stellar mass loss on metal abundance ?'. In Sects. 5.3 and 5.4, the method to calculate mass-loss rates and the adopted assumptions will be described. In Sect. 5.5, the resulting wind models and mass-loss rates will be presented. The relative importance of Fe and CNO elements to the line force will be discussed in Sect. 5.6. In Sects. 5.7 and 5.8 the dependence of the mass-loss rate on metallicity will be determined. This completes the Vink et al. (2000) mass-loss recipe to predict M as a function of stellar parameters with an additional Z dependence. It will be shown that over a large parameter space, the M (Z ) dependence is constant, but that at specific temperatures, one needs to take the presence of so-called bi-stability jumps into account. In Sect. 5.9 these mass-loss predictions will be compared with observed mass-loss rates for the Large Magellanic Cloud and the Small Magellanic Cloud. Finally, in Sect. 5.10, the study will be summarised.

5.2 Theoretical context
In this section we will discuss the basic physical processes that may play a role in determining the dependence of mass loss on metal abundance. We will describe the expected effects in terms of CAK theory. However, in our detailed predictions (Sect. 5.5), we will not use this formalism, but extend on it by including multiple scattering effects. In CAK theory the line acceleration is conveniently expressed in units of the force multiplier M (t ) and is given by (CAK, Abbott 1982): ne (5.2) W where ne is the electron density and W is the geometrical dilution factor. The parameters k, and are the so-called force multiplier parameters. The first one, k, is a measure for the number of lines. The second parameter, , describes the distribution of strong to weak lines. If only strong (weak) lines contribute to the force, then = 1 (0). The predicted value of for O-type stars is typically 0.6 (Abbott 1982, Kudritzki et al. 1989). The parameter describes the ionization in the wind. Its value is usually 0.1. Finally, t is the optical depth parameter, given by: M (t ) = kt
-

t = eVth (dr/dV )

(5.3)

where Vth is the mean thermal velocity of the protons and e is the electron scattering crosssection. Abbott (1982) and Puls et al. (2000) have shown that the CAK force-multiplier parameter k is dependent on the metallicity in the following way: k (Z ) Z 97
1-

(5.4)


Chapter 5 Kudritzki et al. (1989) have calculated analytical solutions for radiation-driven wind models that include the finite cone angle effect. The scaling relation for the mass-loss rate that was derived, is proportional to Mk where
eff 1/eff

(5.5) (5.6)

= -

This implies that M is expected to depend on metallicity in the following way: MZ with m= 1- - (5.8)
m

(5.7)

Since a typical value for m is (1 - 0.6)/(0.6 - 0.1) = 0.8, one would expect an almost linear (m 0.8) dependence of M on Z , instead of the square-root (m = 1/2) dependence that was found by Kudritzki et al. (1987). Note that Leitherer et al. (1992) indeed derived such an almost linear (m 0.8) dependence of M on Z , however, multi-line transfer was not taken into account in these calculations either. We note that a pure power-law dependence of M on Z over the entire parameter space, is questionable. It may be expected that for a certain metallicity range Eq. (5.1) provides a useful representation of the mass loss vs. metallicity relation, but that at some minimum and maximum Z , deviations from a power-law may occur. For instance, deviations at high metallicity may occur when mass loss is so efficient that densities in the wind are so high that all relevant Fe lines become saturated. Hence, at some point, an increase in metallicity may no longer cause a substantial increase in mass loss and subsequently a flattening of the M (Z ) relation is expected. Deviations at low metallicity, with subsequently low mass loss, may occur when only weak iron lines remain present. Other abundant ions, such as those of C, N, and O, which normally have far fewer effective driving lines than Fe, may start to dominate the driving because their main lines are still strong. Again a shallower slope of the M (Z ) relation is anticipated. A second important item in the calculations of mass loss at different Z , is the possible presence of one or more "bi-stability" jumps at different Z . For Galactic metallicities, at an effective temperature of 25000 K, the mass loss is predicted to increase dramatically by a factor of about five. The effect of this jump on terminal velocity has observationally been found by Lamers et al. (1995). The origin for this jump is related to a recombination from Fe IV to III in the lower part of the wind (Vink et al. 1999). Since the ionization equilibrium does not only depend on temperature, but also on density, one may expect a shift in the position of this bi-stability jump as a function of Z . Moreover, at lower metallicity, other abundant ions, such as those of CNO, may start to dominate the wind driving, implying there could be additional bi-stability jumps at different Z due to recombinations of one of these elements. In this paper we will therefore concentrate on three main issues: firstly, the global dependence of the mass-loss rate on Z ; secondly, the presence and position of bi-stability jumps for different Z , and, thirdly, the relative importance of Fe and CNO elements at low metal abundance. 98


Mass-loss predictions for O and B stars as a function of metallicity

5.3 Method to calculate M
The mass-loss rates are calculated with a Monte Carlo (MC) method that follows the fate of a large number of photons from below the stellar photoshere through the wind and calculates the radiative acceleration of the wind material. The core of the approach is that the total loss of radiative energy is coupled to the momentum gain of the outflowing material. Since the absorptions and scatterings of photons in the wind depend on the density in the wind and hence on the mass-loss rate, it is possible to find a consistent model where the momentum of the wind material is exactly equal to the radiative momentum that has been transferred. The method is similar to the technique introduced by Abbott & Lucy (1985). The precise characteristics of our Unified MC approach have been described in Vink et al. (1999). The essential ingredients and the assumptions of our approach have extensively been discussed in Vink et al. (2000). The calculation of mass loss by this method requires the input of a model atmosphere, before the radiative acceleration and mass loss can be calculated with the MC approach. The model atmospheres used for this study are calculated with the non-LTE unified Improved Sobolev Approximation code (IS A - W IND) for stars with extended atmospheres. For details of the model atmosphere we refer the reader to de Koter et al. (1993, 1997). The chemical species that are explicitly calculated in non-LTE are H, He, C, N, O and Si. The iron-group elements, which are important for the radiative driving and consequently for M , are treated in a generalised version of the "modified nebular approximation" (Schmutz 1991). The line list that is used for these MC calculations consists of over 105 of the strongest transitions of the elements H - Zn extracted from the line list constructed by Kurucz (1988). ° Lines in the wavelength region between 50 and 7000 A are included in the calculations with ionization stages up to stage VI. The wind was divided into about 50-60 concentric shells, with many narrow shells in the subsonic region and wider shells in supersonic layers. The division in shells is essentially made on the basis of the Rosseland optical depth scale, with typical changes in the logarithm of the optical depth of about 0.13. For each set of model parameters a certain number of photon packets is followed. For Galactic metallicities this number is typically about 2 105 (see Vink et al. 2000) At lower Z , and consequently at lower mass-loss rates, however, the typical amount of photon packets has to be increased, to keep up good statistics, as one is shooting photons through a less dense wind. Consequently, photon packets experience smaller numbers of line interactions. We found that as long as there were typically 100 line scatterings in each supersonic shell, the derived mass loss was reasonably accurate, i.e. log (M ) < 0.05. < 1/30) the line driving mechanism becomes very At extremely low metallicities (Z /Z inefficient and accurate wind solutions can only be obtained for the highest stellar luminosities, i.e. log L/L > 6. Hence, the lowest Z models (Z /Z = 1/100) will only be calculated for L/L = 6 (see Sect. 5.5).

5.4 The assumptions of the model grid
For every Z , the mass-loss rate was calculated for 12 values of Teff in the range between 12 500 and 50 000 K. The abundances of the metallicity grid are given in Table 5.1. Z is the total metallicity content of all elements heavier than helium. Throughout the paper we will indicate the absolute value of the metals with Z and the value of metallicity relative to the Sun by Z /Z , adopt99


Chapter 5 (Z /Z ) 1/30 1/10 1/3 1 3 X 0.758 0.752 0.733 0.68 0.52 Y 0.242 0.246 0.260 0.30 0.42 Z 0.00063 0.0019 0.0063 0.019 0.057

Table 5.1: Adopted abundances of the wind models.

ing Z = 0.019 (Anders & Grevesse 1989). For every value of Z , the helium and hydrogen abundances, Y and X respectively, need be adjusted accordingly. X is simply given by X = 1- Y - Z For Y we adjust the abundances in the following way Y = Yp + Y Z Z (5.10) (5.9)

where Yp is the primordial helium abundance and (Y /Z ) is an observed constant, discussed below. We enumerate the assumptions in the model grid: 1. Following Schaller et al. (1992) we adopt a primordial helium abundance of Yp = 0.24 (Audouze 1987) and a (Y /Z ) ratio of 3 (Pagel 1992). The scaled solar metallicities were take from Allen (1973). 2. All models have effective temperatures between 12 500 and 50 000 K with a stepsize of 2 500 K in the range 12 500 - 30 000 K and a stepsize of 5 000 K for the range between 30 000 and 50 000 K. 3. To investigate whether the dependence of M on Z is universal for different luminosity and mass, it is calculated for three different values of the Eddington factor e . This is the ratio between the gravitational and radiative acceleration due to electron scattering and is given by: L Le e = = 7.66 10-5 e 4cGM L M M
-1

(5.11)

where e is the electron scattering cross-section per unit mass (its dependence on Teff and composition is described in Lamers & Leitherer 1993). The other constants have their usual meaning. The values for e are given in column 1 of Table 5.2. The corresponding luminosities and masses are given in columns 2 and 3 of the same table. 4. Also the dependence of M on the adopted ratio of the terminal flow velocity over the escape velocity, V /Vesc , was determined. Lamers et al. (1995) found that for Galactic supergiants the ratio V /Vesc 2.6 for stars of types earlier than B1 and drops to V /Vesc 1.3 for stars later than type B1. Therefore, we have calculated mass-loss rates for input 100


Mass-loss predictions for O and B stars as a function of metallicity e 0.130 0.206 0.434 logL (L ) 5.0 5.5 6.0 M


Teff (kK)

(Z /Z ) Range

V Vesc

(M ) 20 40 60

12.5 - 50.0 1/30 - 3 1.3 - 2.6 12.5 - 50.0 1/30 - 3 1.3 - 2.6 12.5 - 50.0 1/100 - 10 1.3 - 2.6

Table 5.2: Adopted stellar and wind Parameters for the set of unified models.

ratios of V /Vesc of 1.3, 2.0 and 2.6 to investigate the mass loss for different values of this ratio. We are aware that these ratios V /Vesc may vary for different metallicity. However, our goal here is to determine the dependence of mass loss on different stellar parameters, including V /Vesc . If new observations with e.g. the Far Ultraviolet Spectroscopic Explorer show that the observed values of V at other Z are significantly different from Galactic values, the predicted mass-loss rates can easily be scaled to accommodate the new values of V /Vesc . 5. We have calculated M for wind models with a -type velocity law for the accelerating part of the wind: R V (r) = V 1 - (5.12) r Below the sonic point, a smooth transition from this velocity structure is made to a velocity that follows from the photospheric density structure. Vink et al. (2000) have shown that the predicted mass-loss rate is essentially insensitive to the adopted value of . A value of = 1 was adopted for the supersonic velocity law. The total grid thus contains 540 models. Note that for each calculated point in the grid, several wind models had to be calculated to derive the mass-loss rate that is consistent with the radiative acceleration (see Lucy & Abbott 1993). This results in accurate and self-consistent values for M (see Vink et al. 1999).

5.5 The predicted mass-loss rates and bi-stability jumps
The calculated mass-loss rates are shown in the different panels of Fig. 5.1 and most results are also given in Table 5.3. They show bi-stability jumps superimposed on an overall behaviour where M decreases for decreasing Teff . The reason for this M decrease is that the maximum of the flux distribution gradually shifts to longer wavelengths. Since there are significantly less ° lines at roughly > 1800 A than at shorter wavelength, the line acceleration becomes less effective at lower Teff , and thus the mass loss decreases. However, most of the panels of Fig. 5.1 show bi-stability jumps, where the mass loss drastically increases. Before we can investigate the overall dependence of metallicity on mass loss, we need to describe the positions of these bi-stability jumps in Teff . 101


Chapter 5
log M (M yr-1 )
e

logL



M



V Vesc 2.6

Teff (kK) 50 45 40 35 30 50 45 40 35 30 27.5 25 22.5 20 17.5 15 12.5 22.5 20 17.5 15 12.5 50 45 40 35 30 50 45 40 35 30 27.5 25 22.5 20 17.5 15 12.5 22.5 20 17.5 15 12.5

1/100 Z /Z ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­

1/30 Z /Z ­ ­ ­ ­ -7.98 -7. -7. -8. -8. -7. -7. -7. -7. -7. -7. -7. -8. -7. -7. -7. -7. -7. -7. -7. -7. -7. -7. -6. -7. -7. -7. -6. -7. -7. -7. -6. -6. -7. -7. -6. -6. -6. -6. -7. 79 93 16 45 74 71 76 75 71 66 88 10 49 43 50 53 71 30 30 45 74 10 97 02 10 33 96 04 09 07 97 88 03 35 76 61 69 82 06

1/10 Z /Z -7. -7. -7. -7. -7. -7. -7. -7. -7. -7. -7. -7. -7. -7. -7. -7. -7. -6. -6. -7. -7. -7. -6. -7. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. 48 56 68 56 45 25 35 47 31 31 40 42 40 24 24 42 61 96 99 06 22 41 91 12 74 92 80 56 65 73 70 70 78 79 62 52 59 78 96 27 28 40 51 78

1/3 Z /Z -7. -7. -7. -7. -7. -6. -6. -7. -6. -7. -7. -7. -6. -6. -6. -6. -7. -6. -6. -6. -6. -7. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -5. -5. -6. -6. -6. 03 12 18 09 19 88 91 01 93 08 12 04 84 72 88 98 27 55 53 63 85 04 36 41 47 37 58 20 22 26 27 41 48 38 12 11 17 35 70 82 88 02 13 26

1 Z /Z -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. 68 63 68 76 92 46 47 48 59 76 73 48 32 41 49 62 74

3 Z /Z -6. -6. -6. -6. -6. -6. -5. -6. -6. -6. -6. -6. -5. -6. -6. -6. -6. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -4. 23 22 29 45 60 01 97 05 29 38 26 01 99 06 12 15 13 75 83 83 79 72 53 45 53 77 90 28 24 35 60 67 56 34 33 40 43 43 31 12 18 11 03 92

10 Z /Z ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­

(L ) 0.130 5.0

(M ) 20

2.0

1.3

-6.15 -6.22 -6.28 -6.39 -6.32 -5. -5. -5. -6. -6. -5. -5. -5. -5. -6. -6. -5. -5. -5. -5. -5. -6. 97 95 95 06 25 76 73 75 90 10 01 75 66 75 86 93 09

0.206

5.5

40

2.6

2.0

1.3

-5.44 -5.52 -5.59 -5.67 -5.65

Table 5.3: Predicted mass-loss rates for different metallicities. 102


Mass-loss predictions for O and B stars as a function of metallicity

Figure 5.1: The calculated mass-loss rates M as a function of Teff for five range Z /Z = 1/30 - 3. The metal content is indicated in the legend at the panel. Upper five panels (a)-(e): e = 0.130 (log L/L = 5.0). Lower five 0.206 (log L/L = 5.5). The values for (V /Vesc ) are indicated in the legend the last panel (j). 103

metallicities in the upper part of each panels (f)-(j): e = at the lower part of


Chapter 5
log M (M yr-1 )
e

logL



M



V Vesc 2.6

Teff (kK) 50 45 40 35 30 50 45 40 35 30 27.5 25 22.5 20 17.5 15 12.5 22.5 20 17.5 15 12.5

1/100 Z /Z -6. -6. -6. -7. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. -6. 81 80 86 16 78 42 47 58 78 47 50 60 52 33 36 54 71 24 06 09 29 49

1/30 Z /Z -6. -6. -6. -6. -6. -6. -6. -5. -6. -6. -6. -6. -6. -5. -6. -6. -6. -5. -5. -5. -5. -6. 31 59 16 27 21 17 35 98 11 07 16 24 11 93 01 17 35 77 70 80 98 13

1/10 Z /Z -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. 84 87 95 95 90 67 69 73 74 80 99 92 63 59 74 90 99 36 37 52 65 75

1/3 Z /Z -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -5. -4. -5. -5. -5. -4. 46 45 41 47 57 25 22 23 28 44 51 38 13 19 33 42 48 91 00 09 07 80

1 Z /Z -5. -4. -4. -5. -5. -4. -4. -4. -4. -5. -5. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. 07 99 97 05 29 86 76 76 88 14 19 95 78 83 90 85 51 55 63 59 28 30

3 Z /Z -4.57 -4.55 -4.59 -4.78 -4.94 -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. 42 42 47 65 82 68 44 45 54 48 25 19 29 38 19 06 10

10 Z /Z -4.31 -4.31 -4.42 -4.60 -4.52 -4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -3. -3. -4. -4. -3. -3. -3. 23 24 32 47 38 23 11 17 26 11 94 99 10 12 97 91 95

(L ) 0.434 6.0

(M ) 60

2.0

1.3

Table 5.3: Continued: Predicted mass-loss rates for different metallicities.

5.5.1 The bi-stability jump at Teff

25 000 K

All panels show a bi-stability jump around Teff 25 000 K. Here, Fe IV recombines to Fe III and as the latter ion is a more efficient line driver than the first, the acceleration in the lower part of the wind increases. This results in an upward jump in M of about a factor of five and subsequently a drop in V of about a factor 0.5 (Vink et al. 1999). Since we know from both theory and observations that the Galactic ratio V /Vesc jumps from 2.6 at the hot side of the jump to 1.3 at the cool side of the jump, we can estimate the size of the jump in mass loss for the different metallicities. The size of the jump is defined as the difference between the minimum M at the hot side of the jump (where V /Vesc = 2.6) and the maximum M at the cool side (where V /Vesc = 1.3). The size of the predicted jump in M (i.e. log M ) is indicated in the last column of Table 5.4. For most models M is about a factor of five to seven. There is no clear trend with metallicity. The position of the jump for different Z shifts somewhat in Teff , since the ionization equilibrium does not only depend on temperature, but also on density and therefore on mass loss and thus on metallicity as well. To handle the influence of the metallicity on the position of the bi-stability jump in Teff , we compare the characteristics of the wind models around the bistability jump. We will discuss this behaviour for the case of the highest wind densities (e = 104


Mass-loss predictions for O and B stars as a function of metallicity

Figure 5.1: Continued: Series of M (Z ) calculations with e = 0.434 (log L/L = 6.0). The calculated mass loss as a function of Teff for seven metallicities in the range Z /Z = 1/100 - 10. The metal abundance is indicated in the legend at the upper part of each panel (a-g). The values for (V /Vesc ) are indicated in the legend of the last panel (g). 0.434), as for these models, the statistics in the Monte-Carlo code are the best (see Sect. 5.3). Nevertheless, the uniformity is checked for the other series of e also. As in Vink et al. (2000), is defined as the characteristic wind density at 50 % of the terminal velocity of the wind. For a standard velocity law with = 1, this characteristic wind density is given by = M 8R2V (5.13)

Figure 5.2 shows the behaviour of the characteristic density as a function of Z . This is done for both the minimum M (at the hot side of the jump) and the maximum M (at the cool side of the jump). The characteristic densities for the cool side of the jump are indicated with "diamond" signs and with "plus" signs for the hot side. The "filled circles" represent the logarithmic average values of for the "jump" model for each metallicity. The "jump" model is a hypo thetical model between the two models where M is maximal and minimal. As expected, log increases as the metallicity increases. Because the average density at the jump shows a linear 105


Chapter 5 e 0.130 (Z /Z ) 1/30 1/10 1/3 1 3 1/30 1/10 1/3 1 3 1/100 1/30 1/10 1/3 1 3 10 log(M ) 0.75 0.77 0 .8 3 0 .8 6 0.66 0.63 0.81 0 .8 1 0 .8 1 0.72 0.71 0.74 0.76 0 .7 6 0 .6 8 0.43

logL (L ) 5.0

M (M ) 20

0.206

5.5

40

0.434

6.0

60

Table 5.4: The size of the bi-stability jump around 25 000 K for different Z .

Figure 5.2: Characteristic density at the bi-stability jump around 25 000 K as a function of Z . An explanation for the different symbols is given in the legend. The solid line indicates the best linear fit through the average jumps parameters for log .

106


Mass-loss predictions for O and B stars as a function of metallicity

Figure 5.3: Characteristic density log and Teff of the bi-stability jump around Teff = 25 000 K. An explanation for the different symbols is given in the legend. The solid line represents the best linear fit through the average jump parameters log and Teff . dependence on log (Z /Z ), a linear fit is plotted. This is the solid line in Fig. 5.2. The relation is given by: log = -13.636 (± 0.029) + 0.889 (± 0.026) log(Z /Z )

(5.14)

Figure 5.3 shows the effective temperature of the bi-stability jump as a function of . Again this is done for both the cool and hot side of the jump and for the average. The solid line indicates the best linear fit through these averages. The relation between the jump temperature (in kK) and log is given by: Teff
jump

= 61.2 (± 4.0) + 2.59 (± 0.28) log

(5.15)

It is now possible to estimate for any Z using Eq. (5.14) and subsequently to predict the position of the jump in Teff using Eq. (5.15).

5.5.2 Additional bi-stability jumps around 15 000 and 35 000 K
In many of the panels in Fig. 5.1 one can see more than just one bi-stability jump. In cases for high mass loss at relatively high Z , an additional jump is visible at Teff 15 000 K (see e.g. panel (e) in Fig. 5.1). Leitherer et al. (1989) calculated atmospheric models for Luminous Blue Variables (LBVs) and found a recombination of iron group elements from doubly to singly ionised stages, which may explain mass-loss variability when LBVs change from minimum to maximum visual brightness phase (de Koter et al. 1996). Vink et al. (2000) also found this 107


Chapter 5 jump around 15 000 K and attributed it to a recombination of Fe III to Fe II. Possibly this jump is related to the drop in the ratio V /Vesc from 1.3 to about 0.7 around spectral type A0 as identified by Lamers et al. (1995) on the basis of observed values for V . For the lower mass-loss rates at relatively low metallicity, at about Teff 35 000 K, another drastic increase in M occurs (e.g. panel (f) with Z /Z = 1/30 in Fig. 5.1). The origin of this 35 000 K jump, which appears only at low Z , will be discussed in Sect. 5.5.3. In order to express the mass-loss behaviour as a function of metal content, it is obvious that all these jumps need to be accounted for. Since these additional jumps are only present in a few cases, the relationships can only be given as rough estimates. For the jump at Teff 15 000 K: T15kK = 43 + 1.9 log For the jump at Teff 35 000K: T35kK = 192 + 10.4log
jump jump

(5.16)

(5.17)

In both cases the jump temperature is in units of kK. It is again possible to estimate log using Eq. (5.14) and then to roughly predict the positions of these additional bi-stability jumps in effective temperature using Eqs. (5.16) and (5.17). Later on these will be referred to when the complete mass-loss recipe is presented (Sect. 5.8).

5.5.3 The origin of the (low Z) jump at Teff

35 000 K

Intuitively, one might attribute the jump at 35 000 K in models of low metal abundance (say Z /Z 1/30) to the recombination of Fe V to Fe IV. This in analogue to the jump at 25 000 K, due to the recombination of Fe IV to Fe III. However, in the next section we will show that this is not the case, since at lower Z the relative contribution of Fe vs. CNO in the line acceleration decreases (see also Puls et al. 2000). Instead, the low Z jump at Teff 35 000 K turns out to be caused by a recombination from carbon IV to carbon III (see Vink 2000). To summarise the physical origin of the jump: C III has more lines in the crucial part of the spectrum than C IV, therefore C III is a more efficient driving ion causing the increase in mass loss at the bi-stability jump around 35 000 K at low Z . Whether this is also accompanied by a change in terminal velocity is an open question that may be answered if V determinations at very low Z become available.

5.6 The relative importance of Fe and CNO elements in the line acceleration at low Z
5.6.1 The character of the line driving at different Z
Vink et al. (1999) have shown that for Galactic wind models around 25 000 K the elements C, N and O are important line drivers in the supersonic part of the wind, whereas the subsonic part of the wind is dominated by the line acceleration due to Fe. As the mass-loss rate is determined by the radiative acceleration below the sonic point, and the terminal velocity is determined by the acceleration in the supersonic part, these results imply that for Galactic wind models M is essentially set by Fe lines, whereas V is determined by the lighter elements, i.e. mainly by CNO. 108


Mass-loss predictions for O and B stars as a function of metallicity

Figure 5.4: The relative contribution to the line acceleration for models with (V /Vesc ) = 2.0, log L /L =5.5 and M = 40M . The solid lines show the contribution of Fe lines. The dotted line is the contribution by C, N and O. The dashed line shows the contribution by H and He lines. (a) and (b) give the contribution for solar Z at resp. V = Vsound and at V = 0.5V . (c) and (d) give the contribution for (Z /Z ) = 1/30 at resp. V = Vsound and at V = 0.5V. To study the origin of the additional (low Z ) jump around 35 000 K, it becomes necessary to investigate the relative importance of the species at low metallicity. To this end, additional Monte Carlo calculations were performed. One simulation was performed with a line list containing only Fe lines. A second calculation was done with a list of lines of CNO only, and finally a third simulation was performed with the lines of H and He. Figure 5.4 shows the relative importance for the line acceleration of these elements as a function of effective temperature for different parts of the wind, i.e. at V = Vsound and at V = 0.5V . Panel (a) and (b) indicate the fractions in the acceleration at solar metallicity. Panel (c) and (d) present the same, but for the low metallicity models, i.e. Z /Z = 1/30. Figure 5.4 (a) shows that at solar Z , Fe dominates the line acceleration around the sonic point, where the mass-loss rate is fixed. However, this relative importance of iron decreases for increasing Teff . Figure 5.4 (c) shows that at the low metallicity, CNO already dominate the acceleration in the region around the sonic point. This implies that at low Z , CNO determine both the terminal velocity by dominating the supersonic line acceleration in Fig. 5.4 (d), as well as the mass loss by dominating the line acceleration around V = Vsound . The only exception occurs at low effective temperature (Teff = 20 000 K), where Fe still plays an important role in setting the mass loss. 109


Chapter 5 Z
theory

(Z ) 1 1/3 1/10 1/30 1/100

Dominant elements that set M Fe Fe Fe CNO CNO

[O/Fe] [O/Fe]

obs ZOxygen

obs ZFe

(Z ) 0 0 0 + 0.4 dex + 0.4 dex 1 1 1 1/30 1/100

(Z ) 1 1 1 1/75 1/250

Table 5.5: Conversion table for the observed differential abundance variations between oxygen and iron.

These considerations thus explain why the high Teff jump at low Z is not caused by a recombination effect of iron, instead it turns out to be caused by a recombination of a CNO element, in this case C IV to C III (Vink 2000).

5.6.2 Observed abundance variations at different Z
Now we will make a distinction between the metal abundance Z derived on the basis of stellar iron and nebular oxygen lines. The reason for this distinction is that observations to study the chemical evolution of galaxies have shown that the ratio of Fe/O varies with metallicity. Determinations of heavy-element abundances for metal poor blue compact galaxies (Izotov & Thuan 1999) as well as observations of Galactic halo stars (Pagel & Tautvaisiene 1995 and references therein) show a significant overabundance of O/Fe of about 0.4 dex with respect to the Sun. These observed differential abundance variations between oxygen and iron could signifi cantly alter our mass-loss predictions in the case M were set by Fe over the full range in Z . However, we have shown that at low Z , the mass loss is mainly determined by CNO instead of by Fe. Since the observed metallicity is mostly determined from nebular oxygen lines rather than from iron lines, this implies that our mass loss recipe will still yield the proper mass-loss rates. Only in those cases where the observed metallicity were determined from stellar iron lines instead of from nebular oxygen lines, one would need to transform the observed iron abundance obs (ZFe ) to our adopted metallicity (Z theory ). This can easily be done according to the scaling relations given in Table 5.5. The first column of this table indicates the metallicity that has been adopted in the wind models. The second column shows for each Z which elements dominate the line driving around the sonic point, where the mass loss is set. The third column represents the observed abundance variation between oxygen and iron compared to the sun. For relatively high Z (Z /Z > 1/10), there is hardly any observed difference between the oxygen and iron abundances. As said, for very low Z (Z /Z < 1/30), this observed difference is about 0.4 dex. Because at low Z mass loss is mainly set by CNO, the observed oxygen abundances are the same as the adopted Z in the wind models (column 4), whereas in case iron lines were to be analysed, one should convert the iron abundance to our adopted Z theory , by comparing column 5 and column 1. 110


Mass-loss predictions for O and B stars as a function of metallicity

5.7 The global metallicity dependence
Now we can determine the global M (Z ) dependence over a wide range in metallicity. This (Z ) will be determined for the three e values separately. If the dependencies were identical M for different e , then we might simply add the metallicity dependence to the mass-loss recipe that was derived by Vink et al. (2000) for Galactic stars. Fig. 5.5 shows the M (Z ) behaviour for the three cases of e . To avoid complications due to the presence of the bi-stability jumps, we use models where Teff is above all of the identified jumps, i.e. at Teff = 50 000 K. In case e = 0.130, the linear fit is taken in the metallicity range between Z /Z is 1/10 and 3, as the model at Z /Z = 1/30 is influenced by the low Z bi-stability jump. This is why we have excluded this from the fit. The best linear fit is thus given by log M = -6.439 (±0.024) + 0.842 (±0.039) log(Z /Z ) for e = 0.130 (5.18)

In case e = 0.206, the models at Teff = 50 000 K are not influenced by the low Z jump and a linear fit is taken over the full metallicity range of Z /Z = 1/30 - 3. The best fit is given by log M = -5.732 (±0.028) + 0.851 (±0.033) log(Z /Z ) for e = 0.206 (5.19)

Finally, in case e = 0.434, the M (Z ) dependence is studied over an even wider metallicity range: Z /Z is 1/100 - 10. For this relatively high value of e it is computationally easier to calculate mass loss at the extremely low value Z /Z = 1/100. The mass-loss rate at extremely high metallicity (Z /Z = 10) is determined for a somewhat different abundance ratio than the standard one that was used throughout the paper given by Eq. (5.10). The helium abundance is now kept constant (at Y = 0.42, see Table 5.1) increasing the metal fraction from three to ten times solar. It was checked whether the results are dependent on this choice of Y, but this turned out not to be the case. One may expect the M (Z ) relation to flatten at some high Z value due to saturation of iron lines (see Sect. 5.2). The lowest panel in Fig. 5.5 shows that this is indeed the case. However, this only happens above Z /Z = 3. It implies that over the range from about Z /Z = 1/30 - 3, mass loss is a constant function of metallicity, i.e. M vs. Z behaves as a power-law. The linear fit for the highest value of e is determined from the range Z /Z = 1/30 - 3. The best fit is given by log M = -4.84 (±0.020) + 0.878 (±0.023) log(Z /Z ) for e = 0.434 111 (5.20)


Chapter 5

Figure 5.5: The M (Z ) dependence for three cases of e . In all three panels, the dashed lines indicate the best linear fit through the models at different Z . Note that at e = 0.130 the lowest Z model is not included in the fit, due to the presence of a bi-stability jump. All models have Teff = 50 000 K and (V /Vesc ) = 2.0. The values of e are indicated in the legends. 112


Mass-loss predictions for O and B stars as a function of metallicity Combining Eqs. (5.18), (5.19) and (5.20) for the three different values of e , we find that over the metallicity range from 1/30 Z /Z 3 there is a constant power law for Teff = 50 000 K with M Z 0.86 . We have done similar analyses for the other effective temperatures in our model grid, some of these were affected by a bi-stability jump, but on average, these jumps cancelled out. The average power-law index factor m was found to be m =0.85 (± 0.10). This implies that there is a constant, universal power-law that is given by MZ
m



Z

0.85

for 1/30 Z /Z

3

(5.21)

Note that the derived power-law dependence factor m = 0.85 with our Unified Monte Carlo approach yields a significantly larger metallicity dependence than the value of m = 1/2 that was derived by Kudritzki et al. (1987) and has since been used in many evolutionary calculations (e.g. Langer 1991, Maeder 1992, Schaller et al. 1992, Meynet et al. 1994, Vassiliadis & Wood 1994, Vanbeveren D. 1995, Iben et al. 1996, Deng et al. 1996).

5.8 Complete mass-loss recipe
In this section we present the "complete" theoretical mass loss formula for OB stars over the range in Teff between 50 000 and 12 500 K and the range in Z between 1/30 and 3 times Z . The mass-loss rate as a function of five basic parameters will be provided. These parameters are M , L , Teff , V /Vesc , and Z . First, some relationships for the bi-stability jumps have to be connected. The position of this jump in Teff now depends both on the metallicity Z (this paper) and on the luminosity-to-mass ratio, i.e. e (Vink et al. 2000). The characteristic density for the bi-stability jump around Teff 25 000 K can be determined by smoothly combining Eq. (5.14) from the present paper with Eq. (4) from Vink et al. (2000). The joint result is given by log = -14.94 (±0.54) + 0.85 (±0.10) log(Z /Z ) + 3.2 (±2.2) e

(5.22)

The positions (in Teff ) of the several bi-stability jumps can now be found using Eqs. (5.15), (5.16), (5.17) and (5.22). We will divide our mass-loss recipe into two parts, taking into account only the bi-stability jump around 25 000 K, since this jump is present at all metallicities in all panels of Fig. 5.1. If one wants a mass-loss rate for relatively high metallicity, say Z /Z > 1, for low temper atures, Teff < 15 000 K, one should take into account the presence of the Fe III/II jump, and follow the strategy that was described in Vink et al. (2000). One may simply use Eq. (5.24; below) below the Fe III/II jump, but one should increase the constant by a factor of five (or log M = 0.70) to a value of -5.99. The recipe can then be used until the point in the HertzsprungRussell Diagram (HRD) where line driven winds become inefficient (see Achmad & Lamers 1997). We suggest that below the Fe III/II jump V /Vesc = 0.7 (Lamers et al. 1995) is adopted. 113


Chapter 5 If one needs a mass-loss rate for low metallicity, say Z /Z < 1/30, at high temperatures Teff > 35 000 K, one should be aware of the carbon jump and a similar strategy may be followed. Note that this jump is only present for cases where the wind density is weak, i.e. for stars with a relatively low luminosity. One can decrease the constant in Eq. (5.23; below) by a factor of five (or log M = 0.70) to a value of -7.40. In case one does not know the value for V such as is the case for evolutionary calculations, one would like to know the appropriate change in terminal velocity at the low Z jump. Leitherer et al. (1992) have calculated the dependence of V on Z and have found that V Z 0.13 . Such a trend with metallicity has been confirmed by observations in the Magellanic Clouds, however, what happens to V /Vesc at extremely low Z is still an open question. We stress that if the observed values for V at very low Z turn out to be very different from the Galactic values, our mass-loss predictions can simply be scaled to accommodate the proper values of V /Vesc and our recipe will still yield the corresponding mass-loss rates. Now we can present the complete mass-loss recipe including the metallicity dependence. This can be done by simply adding the constant Z dependence from Eq. (5.21) to the multiple linear regression relations from the Vink et al. (2000) recipe. We are indeed allowed to do so, as the M (Z ) dependence was found to be independent of other investigated stellar parameters (see Sect. 5.7). For the hot side of the bi-stability jump 25 000 K, the complete recipe is given by: log M = - 6.697 (±0.061) + 2.194 (±0.021) log(L /105 ) - 1.313 (±0.046) log(M /30) V /Vesc - 1.226 (±0.037) log 2. 0 + 0.933 (±0.064) log(Teff /40000) - 10.92 (±0.90) {log(Teff /40000)} + 0.85 (±0.10) log(Z /Z ) for 27 500 < Teff 50 000K

2

(5.23)

where M is in M yr-1 , L and M are in solar units and Teff is in Kelvin. In this range the Galactic ratio of V /Vesc = 2.6. As was noted in Sect. 5.4, if the values for V at other Z are different from these Galactic values, then the mass-loss rates can easily be scaled accordingly. For the cool side of the bi-stability jump, the complete recipe is log M = - 6.688 (±0.080) + 2.210 (±0.031) log(L /105 ) - 1.339 (±0.068) log(M /30) V /Vesc - 1.601 (±0.055) log 2. 0 + 1.07 (±0.10) log(Teff /20000) + 0.85 (±0.10) log(Z /Z )

114


Mass-loss predictions for O and B stars as a function of metallicity for 12 500 Teff 22 500K (5.24)

where again M is in M yr-1 , L and M are in solar units and Teff is in Kelvin. In this range the Galactic ratio of V /Vesc = 1.3. In the critical temperature range between 22 500 Teff 27 500 K, either Eq. (5.23) or Eq. (5.24) should be used depending on the position of the bi-stability jump given by Eq. (5.15). A computer routine to calculate mass loss as a function of stellar parameters is publicly available1.

5.9 Comparison between theoretical M and observations at subsolar Z
Now we will compare our mass-loss predictions for different Z with the most reliable observational rates presently available. Unfortunately, there are only substantial samples available in the literature for the relatively nearby Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC). The metallicity of the LMC is only slightly smaller than the Galactic one and its absolute value is not accurately known. What complicates a meaningful comparison is that there are differences in the observed stellar and nebular abundances. Additionally, there are abundance gradients present in these galaxies, which makes a good comparison between our predicted M (Z ) dependence and the observed mass-loss rates of the LMC sample rather difficult. As the metallicity difference between the Galaxy and the SMC is significantly larger, we should be able to test our predictions in a more meaningful way with the observed rates of the SMC sample. Following Kudritzki et al. (1987), we did not adopt the individual abundance patterns quoted for the Clouds (e.g. Dufour 1984). Instead we simply scaled down all abundances by a constant factor adopting: ZLMC = 0.28 Z ZSMC = 0.10 Z

(5.25)

We are aware that the differential metal abundances in the Clouds could be different from the Galaxy due to a different stellar evolution at lower Z . However, we expect these effects to be of relatively minor importance, since the mass-loss rate at these metallicities (Z > 1/10 Z ) is still mainly determined by iron. The upper panel of Fig. 5.6 shows the comparison between the observed LMC mass-loss rates and the theoretical values from our mass-loss recipe. The scatter between observations and theory can be attributed to errors in the stellar parameters and the mass-loss determinations, but may also be due to differential metal abundance patterns in the LMC. Note that there is a systematic difference between the two sets of mass-loss determinations themselves (Puls et al. 1996 vs. de Koter et al. 1997, 1998). The possible systematic differences between these two sets have been discussed in de Koter et al. (1998). Nevertheless, both samples show an offset with respect to our predictions. This could in principle be due to systematic errors in our predictions. However, since there is good agreement between observations and our predictions for a large sample of Galactic supergiants (Vink et al. 2000), we do not expect this to be the
1

see: www.astro.uu.nl/jvink/

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Chapter 5

Figure 5.6: Comparison between theoretical and observational M for O stars in the LMC. The upper panel is for an adopted ZLMC = 0.28 Z and the lower panel is for an adopted metallicity ZLMC = 0.8 Z . The Puls et al. (1996) H rates and de de Koter rates are indicated with different symbols. The dashed lines are one-to-one relations, tools for convenient comparison between observations and theory

case. Perhaps the systematic offset is due to a too low assumed Z for the LMC. Haser et al. (1998) analysed individual O stars in the LMC and found metallicities significantly higher for these stars than usually derived from nebular abundance studies. Adopting the Haser et al. value of Z = 0.8 Z derived for the LMC O star SK-67o 166, for the whole LMC sample, there is much better agreement between our predictions and the observed mass-loss rates (see the lower panel 116


Mass-loss predictions for O and B stars as a function of metallicity

Figure 5.7: Comparison between theoretical and observational M for O stars in in the SMC with the adopted abundance of ZSMC = 0.10 Z . The dashed line is the one-to-one relation, a tool for convenient comparison between observations and theory. in Fig. 5.6). The scatter between observational and theoretical mass-loss rates decreases from 0.65 dex (1 ) for the upper panel of Fig. 5.6 to only 0.36 dex for the lower panel of the figure. Figure 5.7 shows the comparison between observed mass-loss rates and our predictions for the sample of the SMC stars. The figure shows a reasonable agreement between predictions and observations. We admit that there is quite a large scatter (0.55 dex) for which there may be several reasons. The important point at this stage is that the comparison with the SMC data yields good average agreement and thus yields support to the reliability of our mass loss recipe at metallicities other than solar. For a test of our mass-loss recipe at extremely low Z , say Z /Z < 1/10, we will have to await new Hubble Space Telescope (HST) observations of some relatively nearby low metallicity galaxies.

5.10 Summary & Conclusions
We have presented predictions of mass-loss rates for O and B stars over a wide range of metallicities. The calculations take the important effect of multiple line scattering into account in a consistent manner, using a "Unified Monte Carlo approach". It is shown that there is a constant universal metallicity dependence over a wide range of metal abundance, given by M Z 0.85 ,but that one needs to take into account some specific positions in the HRD where recombinations of Fe or CNO ions may cause the mass loss to increase dramatically and produce "bi-stability" jumps. It will be a challenge for the future to test our mass-loss recipe at extremely low Z in local starbursting galaxies, where the difference in mass-loss rate compared to the solar neighbourhood can be significant. We can summarise the main results of the paper as follows: 117


Chapter 5 1. We have calculated a grid of wind models and mass-loss rates for a wide range of metallicities, covering 1/100 Z /Z 10. 2. We have found that the mass loss vs. metallicity dependence behaves as a power-law with M Z 0.85 . This is in contrast to an often applied square-root dependence of mass loss on Z . 3. Although the M (Z ) reaction is a constant function of Z , one should be aware of the presence of bi-stability jumps, where the character of the line driving changes drastically due to a change in the wind ionization resulting in jumps in mass loss. We have investigated the physical origins of these jumps and derived formulae that connect mass loss recipes at opposite sides of such bi-stability jumps. Additionally, we have made a distinction between the metal abundance derived from iron and from oxygen lines, since observations of different galaxies have shown that the [Fe/O] abundance ratio varies with metallicity. 4. As our mass-loss predictions are successful in explaining the observed mass-loss rates for Galactic and Small Magellanic Cloud (Fig. 5.7) O-type stars, as well as in predicting the observed Galactic bi-stability jump, we believe that they are reliable and suggest that our mass-loss recipe be used in future evolutionary calculations, also at different Z . A computer routine to calculate mass loss is publicly available at the address www.astro.uu.nl/jvink/.

References
Abbott D.C., 1982, ApJ 259, 282 Abbott D.C., Lucy L.B., 1985, ApJ 288, 679 Achmad L., Lamers H.J.G.L.M., Pasquini L., 1997, A&A 320, 196 Allen C.W., 1973, Astrophysical quantities, University of London, Athlone Press Anders E., Grevesse N., 1989, Geochim. Cosmochim. Acta 53, 197 Artymowicz P., 1993, PASP 105, 1032 Audouze J., 1987, Observational Cosmology, IAU Symp. 124, eds. A. Hewitt et al., Reidel Publ. p. 89 Castor J.I., Abbott D.C., Klein R.I., 1975, ApJ 195, 157 Cen R., Ostriker J.P., 1999, ApJ 519, 109 de Koter A., Schmutz W., Lamers H.J.G.L.M., 1993, A&A 277, 561 de Koter A., Lamers H.J.G.L.M., Schmutz W., 1996, A&A 306, 501 de Koter A., Heap S.R., Hubeny I., 1997, ApJ 477, 792 de Koter A., Heap S.R., Hubeny I., 1998, ApJ 509, 879 Deng L., Bressan A., Chiosi C., 1996, A&A 313, 145 Dufour R., 1984, IAU Symp. 108, p. 353 Garmany C.D., Conti P.S., 1985, ApJ 293, 407 Hamann F., 1997, ApJ 109, 279 Haser S.M., Pauldrach A.W.A., Lennon D.J., 1998, A&A 330, 285 Iben I.Jr., Tutukov A.V., Yungelson L.R., 1996, ApJ 456, 750 Kudritzki R.-P., Pauldrach A.W.A., Puls J., 1987, A&A 173, 293 Kudritzki R.-P., Pauldrach A.W.A., Puls J., Abbott D.C., 1989, AAP 219, 205 Kurucz R.L., 1988, IAU Trans., 20b, 168 118


Mass-loss predictions for O and B stars as a function of metallicity Izotov Y.I., Thuan T.X., 1999, ApJ 511, 639 Lamers H.J.G.L.M., Leitherer C., 1993, ApJ 412, 771 Lamers H.J.G.L.M., Snow T.P., Lindholm D.M., 1995, ApJ 455, 269 Langer N., 1991, A&A 252, 669 Leitherer C., Schmutz W., Abbott D.C., Hamann W.R., Wessolowski U., 1989, ApJ 346, 919 Leitherer C., Robert C., Drissen L.,1992, ApJ 401, 596 Lucy L.B., Abbott D.C., 1993, ApJ 405, 738 Maeder A., 1992, A&A 264, 105 Meynet G., Maeder A., Schaller G., Schearer D., Charbonel C., 1994, A&AS 103, 97 Pagel B.E.J. Simonson E.A., Terlevich R.J., Edmunds M.G., 1992, MNRAS 255, 325 Pagel B.E.J., Tautvaisiene G., 1995, MNRAS 276, 505 Pauldrach A.W.A., Puls J., Kudritzki R.P., 1986, A&A 164, 86 Prinja R., 1987, MNRAS 228, 173 Puls J., 1987, A&A 184, 227 Puls J., Kudritzki R.P., Herrero A., et al., 1996, A&A 305, 171 Puls J., Springmann U., Lennon M., 2000, A&AS 141, 23 Sargent W.L.W., Searle L., 1970, ApJL, 162, 155 Schaller G., Schaerer D., Meynet G., Maeder A., 1992, A&AS 96, 269 Schmutz W., 1991, in: "Stellar Atmospheres: Beyond Classical Models", eds. Crivellari L., Hubeny I., Hummer D.G., NATO ASI Series C, Vol. 341, 191 Vanbeveren D., 1995, A&A 294, 107 Vassiliadis E., Wood P.R., 1994, ApJS 92, 125 Vink J.S., 2000, PhD thesis at Utrecht University Vink J.S., de Koter A., Lamers H.J.G.L.M., 1999, A&A 350, 181 Vink J.S., de Koter A., Lamers H.J.G.L.M., 2000, accepted by A&A

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Research note on the bi-stability jump in the winds of hot stars at low metallicity

Reasearch Note on The bi-stability jump in the winds of hot stars at low metallicity

We have investigated the origin of the hot bi-stability jump in the winds of low metallicity stars. This jump in mass-loss rate by a factor of about five at low metallicity was found by Vink et al. (2000) in wind models around Teff 35 000 K. We have investigated the relative importance in the line driving of the three CNO elements at low metal abundance. It turns out that carbon is the dominant line driving element around the hot jump. From detailed calculations of the line acceleration it is concluded that the jump in the mass loss around 35 000 K at low metallicity is caused by a recombination from C IV to C III. Finally, we speculate that this bi-stability jump in mass loss of hot stars in a low metal environment may show capricious wind behaviour around Teff 35 000 K, and possibly give rise to O stars with rotation-induced outflowing disks, e.g. O[e] stars.

5.11.1 Introduction
In this note, we study the nature of the bi- stability jump at high temperature that was found in the wind models at low metallicity by Vink et al. (2000). The goal is to explain the origin of the predicted jump in mass loss by studying aspects of the line driving mechanism at low metal abundance. At Teff 21 000 K, a bi-stability jump in the ratio of the terminal velocity over the escape velocity was found for Galactic stars by Lamers et al. (1995). This jump is accompanied by a jump in the mass loss caused by a recombination of the dominant line driving element iron, from Fe IV to Fe III (Vink et al. 1999). For the winds of Galactic stars C, N and O are important line drivers in the supersonic part of the wind, whereas the subsonic part of the wind is dominated by the line acceleration of iron (Vink et al. 1999). As mass loss is determined by the radiative acceleration below the sonic point, and as the terminal velocity is determined by the radiative acceleration in the supersonic part of the wind, M is determined by the radiation pressure of Fe lines, whereas V is determined by the lighter elements, i.e. mainly CNO. 121


Research note on the bi-stability jump in the winds of hot stars at low metallicity

Figure 5.8: The relative CNO contributions to the 2.0, log L /L = 5.5 and M = 40M . The solid the dotted line that of oxygen, and the dashed line (a) and (b) give the contribution for solar Z at V = (d) give the contribution for Z /Z = 1/30, at V =

line acceleration for models with V /Vesc = lines show the contribution of carbon lines, represents nitrogen (see legend in panel b). Vsound and V = 0.5V respectively, (c) and Vsound and V = 0.5V, respectively.

Vink et al. (2000) performed Monte Carlo simulations with separate line lists to investigate the relative importance of Fe and CNO at both Galactic and lower metal abundance. In contrast to the Galactic wind models, at low metal abundance (Z /Z 1/30) ­ and relatively high > 30 000 K) ­ CNO dominates the line driving throughout the entire effective temperature (Teff wind. This implies that in such an environment CNO does not only determine the terminal velocity, but also sets the mass-loss rate. As the mass loss is determined in the wind region around the sonic point, and as, for low Z , this region is dominated by CNO driving, it is expected that the origin of this hot bi-stability jump is due to a drastic increase in the ability of the line driving due to one of the elements of C, N, or O rather than Fe. Here, we study the relative importance of these three elements to determine which of them causes the jump at high effective temperature.

5.11.2 The line driving of CNO
Figure 5.8 shows the relative importance of these three elements (C, N, and O) as a function of effective temperature for different parts of the wind, i.e. at V = Vsound and at V = 0.5V . Panel (a) and (b) indicate the fractions in line acceleration for the different elements at solar 122


Research note on the bi-stability jump in the winds of hot stars at low metallicity

Figure 5.9: The ionization fraction of carbon (see legend in panel a) as a function of radial distance in the wind for models with V /Vesc = 2.0, log L /L = 5.5 and M = 40M . The upper panels are for the hot (35 000 K) model A and the lower panels are for the cool (30 000 K) model B. (a) carbon ionization for model A from x = 1 to 15. (b) Model A, enlargement around sonic point. (c) Carbon ionization for model B from x = 1 to 15. (d) Model B, enlargement around sonic point.

metallicity. Panel (c) and (d) present the same, but at a metallicity Z /Z = 1/30. Figure 5.8 (a) shows that at solar Z , CNO is relatively unimportant around the sonic point, where the mass-loss rate is fixed. This situation changes in the supersonic part of the wind, where CNO starts to become more important. Such distinct difference between CNO and Fe driving is not present at low metal-content, where CNO also dominates the radiative acceleration in the region around the sonic point (see Fig. 5.8 c and Fig. 5.8 d). The following trend can be seen in these two panels (c and d): at Teff > 40 000 K, oxygen is a more important line driver than nitrogen and carbon, whereas at lower Teff , carbon starts to take over a substantial fraction of the line driving. Between 40 000 and 30 000 K ­ the temperature region of the hot bi-stability jump ­ the line acceleration due to carbon increases dramatically. This suggests that the hot jump at low Z is caused by a drastic increase in the line driving of carbon. This hypothesis will be tested below. 123


Research note on the bi-stability jump in the winds of hot stars at low metallicity

Figure 5.10: Normalised gL of carbon as a function of distance from the stellar surface for the models A and B where V /Vesc = 2.0, log L /L = 5.5 and M = 40M . (a) Normalised gL for the different carbon ionization stages of model A. (b) model A, enlarged around the sonic point. (c) Normalised gL for the different carbon ionization stages of model B. (d) model B, enlarged around sonic point. Some values for the velocity are indicated on the top of the figure. The legend in panel (b) indicates the ionization stage.

5.11.3 The ionization of carbon around Teff 35 000 K

If the hypothesis posed above is correct, then the line acceleration of carbon is expected to change dramatically around Teff 35 000 K. Such drastic behaviour may be induced by a change in the carbon ionization around this temperature. Hence, we will first investigate the ionization balance of this element on both sides of the jump. Model A is the "hot" model at 35 000 K, and model B is the relatively "cool" model at 30 000 K. The carbon ionization stratification for the two models is displayed in Fig. 5.9, top and bottom panel respectively. The right hand panels show an enlargement of the region around the sonic point. In the "hot" model A the ionization fraction of C IV displays a peak just below the sonic point (Fig. 5.9 b). In the "cool" model B it is C III that reaches a maximum ionization just below the sonic point (see Fig. 5.9 d). As C III has more lines than C IV it is a more efficient driving ion. This will be shown below. 124


Research note on the bi-stability jump in the winds of hot stars at low metallicity

5.11.4 The line acceleration of carbon around Teff 35 000 K
Additional Monte Carlo simulations were performed around the hot bi-stability jump. One simulation included only the lines of C IV, a second included only lines of C III. Because the "hot" and "cool" model (resp. A and B) have different Teff for the same luminosity, they have different radiative surface fluxes as well. As the radiative acceleration is proportional to the flux, we need to scale the line acceleration gL of these two models to the same flux in order to compare their line acceleration. We scale gL to the flux of the hot model in the following way: g
norm L

=g

L

35000 Teff

4

(5.26)

As Teff 4 R-2 for constant luminosity, this is also a scaling to the gravity of the models. Figure 5.10 shows the normalised gL of carbon for the "hot" model A (top) and the "cool" model B (bottom). The right hand panels ­ (b) and (d) ­ display enlargements of the region near the sonic point. Figure 5.10 (b) shows that for the hot model the carbon line acceleration is dominated by C IV. It also shows that for the cooler model (panel d) gL of C III around the sonic point is substantially larger than for the hotter model (panel b). Hence, the increase in the line acceleration of C III causes an increase in the total gL around the sonic point, which subsequently causes the jump in mass loss.

5.11.5 Summary & Discussion
We have studied the origin of the hot bi-stability jump in the winds of low metallicity stars. First, the relative importance in the line driving of the three CNO elements at low metal abundance was investigated. From calculations of the line acceleration, it is concluded that the jump in mass loss in our models around 35 000 K, at Z /Z = 1/30, is caused by a change in the line acceleration of carbon, when this element recombines from C IV to C III. The nature of this bi-stability jump is somewhat similar to the Galactic jump that was shown to be caused by a recombination from Fe IV to Fe III (Vink et al. 1999). Therefore, it may be expected that this hot bi-stability jump in mass loss results in a drop in the ratio of V /Vesc , analogous to the observed bi-stability jump around spectral type B1 for Galactic supergiants (Lamers et al. 1995). However, the ionization balance from C IV could be influenced by shocks (see Pauldrach et al. 1994). Therefore, the predicted temperature at which C IV recombines to C III, neglecting shock ionization, may possibly be erroneous. So, whether this carbon recombination really occurs 35 000 K, remains to be validated. We suggest that the existence of the hot bi-stability jump can be tested by observations of V in low metal abundance environments. This can be done by UV observations of hot stars in Local Group galaxies with Z /Z < 1/30 using e.g. the Hubble Space Telescope. Lamers & Pauldrach (1991) suggested that bi-stability could induce a density difference between pole and equator of a rapidly rotating star. Pelupessy et al. (2000) showed that this mechanism can explain the formation of disks around B[e] stars. If this rotationally induced bi-stability mechanism offers the possibility to form these disks, the two necessary ingredients for this concept to work are: (1) the bi-stability jump around spectral type B1 and (2) rapid rotation of the star. Bi-stability as a necessary ingredient of the mechanism for the formation of 125


Research note on the bi-stability jump in the winds of hot stars at low metallicity
disks around rapidly rotating B[e] stars is especially attractive as it naturally explains the reason why disks are only found around supergiants with the "B" spectral type. We may speculate that if the presence of disks around B[e] stars is related to the proximity of these rotating stars to the bi-stable wind limit, perhaps, at lower metallicity, the hot bi-stability jump may induce an increase in the mass flux from the equator for rapidly rotating supergiants with effective temperatures on the order of about 35 000 K. This may result in the formation of disks around rapidly rotating low metal O stars, and thus lead to the presence of "O[e]" stars in galaxies with low metal content. However, as the mass-loss rate is predicted to decrease drastically with metal abundance, M Z 0.85 (Vink et al. 2000), the wind densities are also expected to be lower, which would complicate the formation of a disk. Yet, capricious wind behaviour for O stars around this jump can certainly be expected.

References
Lamers H.J.G.L.M., Pauldrach A.W.A., 1991, A&A 244, 5 Lamers H.J.G.L.M., Snow T.P., Lindholm D.M., 1995, ApJ 455, 269 Pauldrach A.W.A., Kudritzki R.P., Puls J., Butler K., Hunsinger J., 1994, A&A 283, 525 Pelupessy I., Lamers H.J.G.L.M., Vink J.S., 2000, A&A 359, 695 Vink J.S., de Koter A., Lamers H.J.G.L.M., 1999, A&A 350, 181 Vink J.S., de Koter A., Lamers H.J.G.L.M., 2000, submitted to A&A

126


The radiation driven winds of rotating B[e] supergiants

6

The radiation driven winds of rotating B[e] supergiants
Inti Pelupessy, Henny J.G.L.M. Lamers, and Jorick S. Vink Published in A&A

We have formulated the momentum equation for sectorial line driven winds from rotating stars including: (a) the oblateness of the star, (b) gravity darkening (von Zeipel effect), (c) conservation of angular momentum, (d) line driving specified by the force multiplier parameters (k, , ), (e) finite disk correction factors for an oblate star with gravity darkening for both the continuum and the line driving. The equations are solved numerically. We calculated the distribution of the mass flux and the wind velocity from the pole to the equator for the winds of B[e]-supergiants. Rotation decreases the terminal velocity in the equatorial region but hardly affects the wind velocity from the poles; it enhances the mass flux from the poles while the mass flux from the equator remains nearly the same. These effects increase with increasing rotation rates. We also calculated models with a bi-stability jump around 25 000 K, using force multipliers recently calculated with a Monte Carlo technique. In this case the mass flux increases drastically from the pole to the equator and the terminal velocity decreases drastically from pole to equator. This produces a density contrast in the wind (equator)/(pole) of about a factor 10 independent of the rotation rate of the star. We suggest that the observed density contrast of a factor 102 of the disks of B[e] stars may be reached by taking into account the wind compression due to the trajectories of the gas above the critical point towards the equatorial plane.

6.1 Introduction
In this paper we study the effects of rotation on the radiation driven winds of early-type supergiants. We will focus on the explanation for the occurrence of disks around fast rotating B[e] supergiants. 127


Chapter 6 B[e] supergiants, also designated sgB[e] stars (Lamers et al. 1998), are B type supergiants that exhibit forbidden emission lines in their optical spectra. The observations of hybrid spectra, i.e. spectra with broad UV P Cygni features and narrow emission lines and dust emission of B[e] supergiants in the Magellanic Clouds led to the proposal of a disk wind model by Zickgraf et al. (1985). This model postulates a dense disk of outflowing material in a fast line driven wind to explain the observed characteristics of the spectra of these stars (see Zickgraf 1992). The stellar wind at the equator is about ten times slower than that at the pole. Also, the wind at the equator is about a hundred times denser than at the pole. Additional evidence for a twocomponent outflow has been obtained from polarimetric measurements, e.g. by Zickgraf & Schulte-Ladbeck 1989. The precise mechanism behind the formation of these disks is still a mystery. It is however clear that the origin of an axisymmetric wind structure such as a disk may well be connected to the fast rotation of a star. Two theories using rotation in a different way have been considered for the formation of these disk winds: ­ (1) The Wind Compressed Disk (WCD) model of Bjorkman & Cassinelli (1993), that invokes the kinematics of the winds from rotating stars. The streamlines of the gas in the wind from both hemispheres of a rapidly rotating star cross in the equatorial plane. The concentration of the gas and the shock in the equatorial plane produce an outflowing equatorial disk, with a thickness on the order of a few degrees. Owocki & Cranmer (1994) have argued that the motion to the equatorial plane may be counteracted by the radiation force perpendicular from the plane due to lines. This "wind-inhibition effect" may not be effective in the winds with a strong density gradient in the equatorial direction. We return to this in Sect. 6.7. ­ (2) The rotationally induced bi-stability model (RIB) of Lamers and Pauldrach (1991) invokes an increase in the mass flux from the equator and a decrease in the equatorial wind velocity compared to the poles from the bi-stability jump. This jump in mass flux and in wind velocity is due to the temperature difference between the pole and the equator of a fast rotating B[e] supergiant with gravity darkening. The jump will occur for stars with effective temperatures between 20 000 and 30 000 K. For a detailed explanation of both models, see Lamers & Cassinelli (1999) (hereafter ISW ), chapter 11. In reality, both effects, i.e. the wind compression and the rotation induced bi-stability, may be operating together and amplifying one another in the line driven winds of rapidly rotating early-B stars. The theory of radiation driven winds for non-rotating stars was developed by Castor et al. (1975) (hereafter CAK) and predicts the mass loss rate and V for spherical winds. The influence of rotation on line driven winds was investigated by Friend & Abbott (1986). They found an increase in the mass-loss rate at the equator and a corresponding decrease in V , but these authors did not consider the effects of gravity darkening and the oblate shape of the star. Cranmer & Owocki (1994) described the finite disk correction factor in case of an oblate star. In this paper we will modify the line driven wind theory, including these effects of oblateness and gravity darkening as well as the rotational terms in the equation of motion. This will then be applied to rapidly rotating B[e] supergiants to investigate whether these effects can explain the occurrence of disks. We will also apply recent bi-stability calculations to study the effect of the RIB-model to explain the occurrence of outflowing disks around rapidly rotating B[e] supergiants. In Sect. 6.2 we describe the theoretical background of the radiation driven wind theory and the bi-stability effect. In Sect. 6.3 we derive the equations for the winds from oblate rotat128


The radiation driven winds of rotating B[e] supergiants ing stars with a temperature gradient between poles and equator due to gravity darkening. In Sect. 6.4 we describe the method for solving the equations and for calculating the winds from rotating stars. These calculations will be applied to B[e] supergiants in Sect. 6.5. In Sect. 6.6 we will investigate the effect of bi-stability on rotating stars and Sect. 6.7 concludes with a summary of this work and a discussion on the formation of disks of B[e] supergiants.

6.2 Theoretical context
The computation of the dynamics of line driven stellar winds is a complicated problem in radiative hydrodynamics. It involves the simultaneous solution of the equations of motion, the rate equations and the radiative transfer equations in order to calculate the radiative acceleration. For hot luminous stars the line forces are the most important driving forces in the wind. CAK have shown that the line acceleration can be parameterized in terms of the optical depth parameter t (dr/dV ) as gline kt , where k, and are parameters that depend on composition and temperature of the wind. In this expression k is a measure of the number of lines and is a measure of the distribution of the line strengths with = 0 or 1 for a pure mix of optically thin or thick lines respectively. In this paper we adopt the CAK formalism and simplify the equations of motion by assuming a stationary, radial flow of a viscousless fluid. The possible influence of magnetic fields will be ignored. These simplifications are subject to the following restrictions (see e.g. Abbott 1980): · Both observations and calculations show that line driven stellar winds are not stationary. Even a wind in a stationary solution will develop shocks (see e.g. Lucy 1982). However, the time-averaged structure follows the stationary state quite well (see Owocki et al. 1988, and Feldmeier 1999). Therefore we will restrict ourselves to stationary models. · A sectorial model will be adopted. This means that for every latitude a one dimensional problem will be solved. Wind compression as in the WCD model will thus be neglected, but the main effect of wind compression is expected to redistribute the mass loss and not to change the total mass that is lost from the rotating star. · The absence of viscosity is a good approximation at the high density of line driven winds (see CAK). The representation of the line force by a simple power law may seem to be a gross simplification of the underlying physics of myriad line absorption processes, but it can be shown to hold for a homogeneously distributed mix of optically thick and thin lines (Abbott 1982, Gayley 1995). Calculations of CAK and more recently in NLTE by Vink et al. 1999) of realistic model atmospheres confirm this to be generally true to good accuracy for the important part of the wind (from about the sonic point up to a few stellar radii). Therefore, the complex physics of ion populations can be ignored in the investigation of the various effects of rotation on the stellar wind. However, there is one notable exception which is called the bi-stability jump. This refers to a jump in V around the temperature of 21000 K where V /vesc climbs from 1.3 (lower temperature) to 2.6 (higher temperature). This jump was observed by Lamers, Snow and Lindholm (1995) in B supergiants. It is linked to a shift in ionization states of Fe in the lower part of the wind. The line driving in the lower part of the wind is dominated by iron. Below about 129


Chapter 6 Teff 25000 K, Fe IV recombines to Fe III and since Fe III is a more efficient line driver than Fe IV, the wind structure changes dramatically (Vink et al. 1999). The bi-stability jump may also occur in the temperature difference between the pole and the equator of a fast rotating star with gravity darkening. Therefore two sets of force multiplier parameters for the wind will be adopted to reflect the sudden change in ionization states (a high-temperature set for the pole and a low-temperature set for the equator).

6.3 The physics of rotation
Intuitively it is clear that rotation has an effect on the shape of a star and on the motion of the gas in the wind. An additional effect that the rotation of a star can have is the darkening of the equatorial regions of a star via the von Zeipel effect. These effects will modify the wind of a star considerably and we will incorporate them in the line driven wind theory for rotating stars.

6.3.1 The shape of a rotating star
The shape of a uniformly rotating star with all its mass concentrated in the core is determined by the equipotential surfaces of the potential in a rotating frame (Roche model): 1 1 22 2 (6.1) + x sin () x2 where x = r/Req , is the co-latitude ( = 0 at the pole) and = Vrot,eq /Vcrit . is of course independent of the longitude . Note that 1 + 0.52 = Req /Rpole . So the maximum oblateness is Req /Rpole = 3/2. The critical (break-up) velocity is defined in this paper as: (x, , )=
2 Vcrit = GM eff /R eq

(6.2)

where M

eff

is the effective mass of the star, M
eff

= M (1 - e )

(6.3)

which is the reduced mass due to radiation pressure by electron scattering (see below). The surface of a rotating star is implicitly given by Eq. (6.1). Solving this expression for x() is equivalent to the solution of a 3rd degree polynomial, which results in 2 + 2 x()= 2 sin 3 sin Because of the difference in radius Req = 1 + 0.52 Rpole (6.5) 1 arcsin 3 3 3 sin (2 + 2 )3/2 (6.4)

and the difference in rotational velocity between pole and equator, the radial escape velocity at the equatorial region is smaller than at the pole Vesc =
eq

1 - 0.52 pol Vesc 1 + 0.52 130

e

(6.6)


The radiation driven winds of rotating B[e] supergiants

Figure 6.1: Gravity darkening for different rotation rates as a function of co-latitude We can then roughly estimate the effect of rotation on V as this value has an almost linear dependence on Vesc (see later in Sect. 6.4).

6.3.2 Von Zeipel gravity darkening
The von Zeipel theorem (1924) for distorted stars states that the radiative flux from a point on the star is proportional to the local effective gravity:
4 F (, )= B Teff () geff ()

(6.7)

with B the Boltzmann constant. As geff = -, we can write down the flux of a rotating star as a function of the co-latitude . This (lengthy) expression for the flux as a function of is plotted in Fig. 6.1 for a few different values of .

6.3.3 The equation of motion of a line driven wind of a rotating star
The rotation of the star can induce a component of the flow of the matter in the wind. In this study a possible component is neglected as was discussed in Sect. 6.2, but the motion in the longitudinal direction must still be taken into account. In the absence of forces that can exert a torque on the wind, the equation of motion for V is given by the conservation of angular momentum. Note however that there could be a torque from the line forces themselves, which is dependent on the velocity gradient of the gas. We assume that there are no external torques, so conservation of angular momentum gives: R r The equation of motion for the radial direction becomes: V (r)= Vrot (R ) 131 (6.8)


Chapter 6

V

2 dV GM 1 dp 2R + 2+ - Vrot 3 - g dr r dr r

rad

=0

(6.9)

where V is the radial velocity and M is the stellar mass. The conservation of mass for a non-spherical sectorial wind can be written as 4r2 (, r)V (, r)= Fm () (6.10)

where Fm () is the "local mass loss rate," i.e. the total mass loss rate if the solution for this latitude where valid for a spherical star. The total mass loss rate from the star is M=

Z
0

/2

Fm () sin()d

(6.11)

The equation of motion together with the conservation of mass governs the dynamics of the wind given the equation of state, p = a2 (a is the isothermal sound speed), and the radiative acceleration grad . Using the conservation of mass, the pressure term can be rewritten as: 1 dp 1 da2 2a2 a2 dV = - - dr dr r V dr (6.12)

As in the CAK-theory the temperature structure is specified a priori as a function of r. (The results depend only very weakly on the chosen temperature structure; see Pauldrach et al. 1986). The radiative acceleration consists of two components. The continuum component due to the electron scattering, ge , and a second component, gL , due to line scattering and absorption processes. The continuum acceleration is given in terms of e : ge = e F GM = e 2 c r (6.13)

where e is the electron opacity, L is the stellar luminosity and F is the radiation flux. For a homogeneous spherical star, e is given by1 : e L (6.14) 4GM In the more general case of a non-homogeneous, non-spherical star the continuum acceleration can be defined as a correction to the 'classical' continuum acceleration of Eq. (6.14): e = e = D
c

e L 4GM

(6.15)

where Dc is the continuum correction factor which is given by: 4 r Dc = L
2

I
disk

I ( , ) d

(6.16)

The line acceleration gL (r, V, V ) has a more complicated form, since it is a function of distance r as well as the velocity V and velocity gradient dV /dr. Combining the equation of motion
Note we define e as the ratio between the continuum force and gravity and not as the ratio between continuum force and the critical radiation force for a rotating star
1

132


The radiation driven winds of rotating B[e] supergiants (Eq. 6.9) with the rewritten pressure and continuum acceleration terms and multiplying by r2 , one finds: F (r, V, V ) 1 - a V
2 2

r2V

2 dV 2R - Vrot dr r

+GM (1 - e ) - 2a2 r - r2 gL = 0

(6.17)

This equation is valid for each co-latitude . Following CAK, the critical point of this equation is found by imposing the singularity condition (where the subscript c indicates values at the critical point): F =0 (6.18) V c and the regularity condition: F r +V
c

F V

=0
c

(6.19)

Note that the critical point is not the sonic point rs (V (rs) = a) due to the fact that there is an additional dependence on V in the line acceleration gL (Abbott 1980; ISW chapter 3.3).

6.3.4 The radiative line forces
The line forces are described within the framework of CAK in the Sobolev approximation. This means that the intristic line absorption profiles are considered to be infinitely sharp. Then the line force becomes a function of local properties of the wind only. The acceleration due to an ensemble of lines is given by the summation of the contributions of all individual lines with rest frequencies l . gL is given by (Castor 1974): gL =


l

l c

I
I
l

1 - e-l µd l

(6.20)

where l is the absorption coefficient per gram for the l th line, µ = cos , Il the intensity at the c dr rest frequency l and l is the Sobolev optical depth of l th line, defined as l = l l ( dV ). The integration is performed over the complete visible disk of the star. Following CAK, the line acceleration can be rewritten in terms of the force multiplier as a function of the optical depth parameter t . Where t is defined as: t = refVth e dr dV (6.21)

with Vth being the thermal velocity of the protons and ref is some reference value for the e electron scattering ref = 0.325 cm2 g-1 (see ISW, chapter 8). The line acceleration in terms of e the continuum acceleration is given by: gL = ge M (t ) (6.22)

If the star is assumed to be a point source, M (t ) can be approximated by a simple power law parameterization (CAK, Abbott 1982): M
point

(t ) = kt 133

-

ne W



(6.23)


Chapter 6 where the (ne /W ) accounts for the effect of the electron density on the ionization balance in the wind. Here ne is the electron density in units of 1011 cm-3 and W is the geometric dilution factor. Friend & Abbott (1986) and Pauldrach et al. (1986) showed the importance of the finite disk correction on the line force in case of an extended source. This finite disk correction factor Dfd is given by (CAK): 1 M (t ) Dfd = = Mpoint (t ) N where r dV LR - 1 and N = (6.25) V dr r This expression is completely general: the actual shape of the star and its intensity distribution enter through the integration domain and the I (, ) dependence. However, here we assume the intensity I to be locally isotropic, i.e. we neglect limb darkening. The line acceleration gL including all the neccesary correction factors is now given by: = e gL = 4
1-

I

(1 + ) I ( , ) 1 + µ2



µd

(6.24)

kL (VthFm ) r2 c

-

D

fd

dV rV dr
2



ne W



(6.26)

Combining Eqs. (6.17) and (6.26) gives the full equation of motion for the sectorial wind of a rotating star
2 a2 2 dV 2R + GM (1 - e ) - 2a2 r - Vrot rV V2 dr r e 1- kL ne 2 dV - rV (Vth Fm ) Dfd - 4 c W dr

0 = 1-

(6.27)

6.4 Solutions of the equation of motion
For line driven winds of non-rotating stars solutions of the equation of motion have been retrieved by e.g. CAK and Pauldrach et al. (1986). These will be presented to serve as an illustration of the solution of the equation of motion and as a basis to interpret the more complicated results from the full equation of motion (Eq. 6.27).

6.4.1 Simplified solutions for non-rotating star
The point source approximation If the star is considered to be a point soure the integral of the line acceleration in Eq. (6.20) collapses to one point. In this case Dfd disappears from Eq. (6.26) and the line acceleration C becomes a simple function gL = r2 (r2V dV ) . The solution of Eq. (6.17) was found by CAK dr and can easily be explained by neglecting the gas pressure terms (a2 ). In this case Eq. (6.17) reduces to 134


The radiation driven winds of rotating B[e] supergiants


r2V

dV dV = GM (1 - e )+ C r2V dr dr

(6.28)

where C is a constant containing the mass loss rate. The equation is solved by imposing uniqueness of the solution (Kudritzki et al. 1989) which then fixes the value of the constant and thus M . The solution for the mass loss and velocity law is given by 4 e eVth 4 ne W


M=

1-
1

1-

( k )

1

L c

{GM (1 - e )} R r
0.5

-1

(6.29)

V (r)= V 1 - with V = Vesc = 1-

(6.30)

2GM (1 - ) ( 1 - ) R

(6.31)

This simplified solution is equal to the full CAK solution, in the limit of small sound speed, a << Vesc (see also ISW chapter 8). Simple finite disk correction In case of a homogeneous spherical star the finite disk correction factor Dfd (Eq. 6.24) can be calculated analytically (CAK):

D

fd

Z1 2 = (1 - µ ) µ (1 + )+1 - = (1 - µ2 )( +

(1 + ) µd µ 1 + µ2 (1 + µ2 )+1 1)(1 + )

(6.32)

Including the finite disk correction results in an increase in V , a decrease in M and a modification of the simple scaling laws that were found in the original CAK approach (see Friend & Abbott 1986; Pauldrach et al. 1986). The decrease of the mass loss rate compared to the point source case is due to the decrease in gL close to the star where M is determined. Close to the stellar photosphere the finite disk correction is smaller than one, viz. Dfd (r = R )= 1/(1 + ) < 1. The accompanying increase in V of typically a factor of two is due to two effects: (1) a reduc tion of M results in a smaller amount of material to be accelerated and (2) far from the photosphere, the correction factor Dfd becomes larger than 1. The resulting dependence between V and Vesc is approximated by Friend &Abbott (1986): V /Vesc 2. 2 1- 135 Vesc 3 km/s 10
0.2

(6.33)


Chapter 6

Figure 6.2: Geometry for the calculation of the correction factors Dfd and Dc . A point on the ray ~ with co-latitude sees a different limb angle for every (not drawn; it is the angle around the line from stellar center to observer, not to be confused with the stellar longitude). The integrals in Dfd and Dc are over 0 < < 2 and 0 < < . These are rewritten to integrals over the star centered angles (0 < < ) and = . The stellar radius and flux are given as a function of , the co-latitude, which can be calculated for every and : = ( , ). They also found a relation for the modified mass loss rate: M 0. 5 M
CAK

Vesc 3 km/s 10

-0.3

(6.34)

6.4.2 Solution of the equation of motion for the wind of a rotating star
The solution of Eq. (6.27) gives the mass loss rate, or rather the local mass loss rate Fm (),and velocity structure of the wind. The solution is complicated however by the presence of Dfd . This is an integral of a velocity-dependent function times the surface intensity over the visible section of the star. The appearance of the star varies throughout the wind in intensity distribution as well as in shape, as the star is no longer spherical and Teff is a function of latitude. Since the analytic solution would be cumbersome, a numerical approach of the solution is chosen. This is done with the analytic solutions of simple, i.e. the non-rotating, models in mind for comparison. The numerical solution of Eq. (6.27) is relatively straightforward in case the function Dfd is a given function of r. Therefore we solve Eq. (6.27) with Dfd (r) in an iterative way as follows: 1. A law for V (r) is assumed, viz. V (r)= (1 - R /r)0.5 . 2. Dfd (r) is calculated using this velocity law. Note that Dfd is not dependent on the actual value of V (r) but on the velocity gradient. 3. Eq. (6.27) is solved using this correction factor Dfd (r), obtaining a new velocity law V (r). 136


The radiation driven winds of rotating B[e] supergiants 4. Step 2-3 are repeated until convergence is reached. Typically three iterations are sufficient, since further iterations changed the obtained values by less than 0.5 %.

6.4.3 The calculation of Dfd (r) and the continuum correction factor Dc
The calculation of the correction factor Dfd from the velocity law (step 1. in the scheme described above) is performed by numerical evaluation of Eq. (6.24). This is a non-trivial task, since both the integrand (Eq. 6.7) and the shape of the visible "disk" (Eq. 6.4) are dependent on r. ~ The main parameters needed to obtain Dfd for a fixed co-latitude are: the stellar surface R() and the surface temperature T (), both given as a function of the co-latitude , as well as the velocity law V (r). It is convenient to change the integration variables in Eq. (6.24) from the wind centered coordinates and to the star centered coordinates and . For the definition of these coordinates, see Fig. 6.2. Note that here is not the stellar longitude anymore, but the ~ angle of rotation around the line from a point (r, ) in the wind to the stellar center. 1 Dfd = N = 1 N

Z
0

2

Z
0

()

I
()

(1 + ) 1 + cos2




cos sin d d (6.35) d d d d

Z
0

2

Z
0

I

(1 + ) 1 + cos2

cos sin

where I is the frequency-integrated intensity I = I ( , ) T ( ) The angles are related to one another via cos = r - R( , ) cos (r - R( , ) cos )2 +(R( , ) sin )
2 4

(6.36)

(6.37)

with the stellar surface given by R( , )= R( ),and cos = sin cos sin + cos cos (6.38)

Equation 6.35 can be integrated, considering that the limb angle () can be determined numerically, yielding the finite disk correction for the line acceleration of an arbitrary shaped star. The correction factor Dc for the electron acceleration (defined in Eq. 6.16) similarly becomes: Dc = 1 N

Z
0

2

Z
0



I cos sin

d d d d

(6.39)

This correction is also included in the solution of the equation of motion. 137


Chapter 6
=0 Fm 3 2. 2. 2. 3. 3. 35 67 95 33 83 = /8 Fm 2. 2. 3. 3. 64 88 23 70 0. 0. 0. 0. 63 62 62 62 = /4 Fm 2. 2. 2. 3. 53 70 95 29 0. 0. 0. 0. 63 62 62 60 = 3/8 Fm 2. 2. 2. 2. 42 46 53 61 0. 0. 0. 0. 63 63 64 64 = /2 Fm 2. 2. 2. 2. 36 34 27 11 0. 0. 0. 0. 64 64 66 70

0 0.3 0.4 0.5 0.6

V 1. 1. 1. 1. 1.

2

0. 0. 0. 0. 0.

4

V 1. 1. 1. 1. 40 40 40 40

V 1. 1. 1. 1. 35 31 27 21

V 1. 1. 1. 1. 30 23 14 04

V 1. 1. 1. 1. 29 20 11 01

40 43 44 46 48

63 63 63 63 63

Table 6.1: Properties of a typical rotating B[e] star model 1 (1) The adopted stellar parameters are: Teff = 20000 K, L = 105.5 L , M = 40M , R = 47R , = 0.565, k = 0.32, = 0.02, solar abundances. (2): the terminal velocity V is in 103 km/s (3): the mass flux Fm in 10-6 M yr-1 (4): the velocity law parameter is obtained from a nonlinear fit of V (r) from 1.1 R to 10 R .

6.4.4 Solving the equation of motion
To solve the equation of motion it is neccesary to determine the rc , Vc and Vc , or, equivalently, rc , Vc and V0 = V (R ), as well Apart from Eqs. (6.18) and (6.19) and the equation of motion constraint for the continuum optical depth e at the stellar radiu 1986) to uniquely determine these quantities: conditions at the critical point: as the local mass loss rate Fm . (Eq. 6.27) itself the additional s is neccesary (Pauldrach et al.

e ()=

Z

R

e dr =

2 3

(6.40)

If the condition a large error in fact that at the a2 R2 H = GMeff << R (this should condition for V0

for V0 is used one may adopt any small value, say V0 = 0.1 km/s, since even V0 will only yield a relatively small error in V and Fm . This is due to the surface of the star V (r) and thus also vary exponentially with scale height R . This means that the radius where Eq. (6.40) is fulfilled is very close to be checked a posteriori). In our calculations we have adopted this boundary in the solution of the full momentum equation.

A first guess for the local mass loss rate obtained from the CAK solution is used to integrate the equation of motion from the stellar surface outward. This gives an approximate value for the critical point where r2V dV = 0 if Fm (guess) < Fm (solution). This approximate value for dr the critical point can then be used to consecutively improve the guessed value of Fm using Eqs. (6.18) and (6.19). This iterative procedure is terminated when the local mass loss rate converges. The calculation is then completed by integrating past the critical point on the solution which extends to infinity (see CAK or Abbott 1980 for a description of the topology of the solutions). To deal with a numerical singularity at the sonic point the equations are solved for r(V ) rather than for V (r). 138


The radiation driven winds of rotating B[e] supergiants
L (L ) 104.5 0 0.3 0.4 0.5 0.6 0 0.3 0.4 0.5 0.6 0 0.3 0.4 0.5 0.6 0 0.3 0.4 0.5 0.6 Vinf 3 km s-1 ) (10 2.84 2.91 2.97 3.04 3.11 Vinf 3 km s-1 ) (10 2.06 2.11 2.15 2.19 2.24 Vinf 3 km s-1 ) (10 1.40 1.43 1.44 1.46 1.48 Vinf 3 km s-1 ) (10 0.65 0.62 0.59 0.55 0.49
pole pole pole pole

Fm -8 M yr-1 ) (10 3.01 3.37 3.67 4.08 4.62 Fm -7 M yr-1 ) (10 2.53 2.84 3.11 3.46 3.93 Fm -6 M yr-1 ) (10 2.35 2.67 2.95 3.33 3.83 Fm -5 M yr-1 ) (10 3.93 4.97 6.00 7.79 11.1
pole pole pole

pole

Vinf 3 km s-1 ) (10 2.60 2.42 2.22 2.00 Vinf 3 km s-1 ) (10 1.88 1.75 1.61 1.46 Vinf 3 km s-1 ) (10 1.29 1.20 1.11 1.01 Vinf 3 km s-1 ) (10 0.60 0.58 0.56 0.56
eq eq eq

eq

(10-

8

Fm M yr-1 )

eq

eq pole

(10-

8

M M yr-1 )

3.04 3.03 2.98 2.82 Fm M yr-1 )
eq

1 1.01 1.01 1.00 0.95
eq pole

3.01 3.18 3.32 3.50 3.71 M M yr-1 )

L (L ) 105.0

(10-

7

(10-

7

2.55 2.54 2.49 2.35 Fm M yr-1 )
eq

1 1.00 1.00 0.98 0.91
eq pole

2.53 2.67 2.79 2.94 3.15 M M yr-1 )

L (L ) 105.5

(10-6

(10-6

2.36 2.34 2.27 2.11 Fm M yr-1 )
eq

1 0.98 0.95 0.90 0.80
eq pole

2.35 2.47 2.56 2.71 2.89 M M yr-1 )

L (L ) 106.0

(10-5

(10-5

3.80 3.61 3.24 2.64

1 0.78 0.62 0.41 0.21

3.93 4.25 4.54 5.01 5.76

Table 6.2: Properties of rotating B[e]1 models: with different values of L (1): The adopted stellar parameters are: M = 40M , R = 47R , Teff = 20000 K.

6.5 Application to B[e] winds
6.5.1 A typical B[e] supergiant
We have calculated models for rotating B[e] stars using the above described method. For the first models we adopted the following stellar parameters, which are typical for B[e] stars: L = 105.5 L , M = 40M , R ,pole = 47R , Teff 20 000 and solar abundances. The effective temperature is defined as Teff = (L /B S)0.25 where S is the total surface of the distorted star. The following force multiplier parameters from Pauldrach et al. (1986) were adopted to 139


Chapter 6
M (M ) 20 0 0.3 0.4 0.5 0.6 0 0.3 0.4 0.5 0.6 Vinf 3 km s-1 ) (10 0.80 0.80 0.80 0.79 0.79 Vinf 3 km s-1 ) (10 1.82 1.85 1.89 1.91 1.95
pole pole

Fm -6 M yr-1 ) (10 5.66 6.64 7.49 8.74 10.5 Fm -6 M yr-1 ) (10 1.55 1.76 1.92 2.16 2.46
pole

pole

Vinf 3 km s-1 ) (10 0.73 0.68 0.64 0.60 Vinf 3 km s-1 ) (10 1.66 1.56 1.43 1.31
eq

eq

(10-

6

Fm M yr-1 )

eq

eq pole

(10-

6

M M yr-1 )

5.67 5.58 5.32 4.76 (10- Fm M yr-1 )
eq

1 0.94 0.88 0.75 0.53
eq pole

5.66 5.69 6.00 6.38 6.81 (10- M M yr-1 )

M (M ) 60

6

6

1.56 1.55 1.51 1.41

1 0.99 0.98 0.93 0.85

1.55 1.55 1.59 1.68 1.75

Table 6.3: Properties of rotating B[e]1 models: with different values of M . (1): The adopted stellar parameters are: L = 105.5 L ; R = 47R ; Teff = 20000 K. describe the line acceleration: = 0.565, k = 0.32, = 0.02. An overview of the results for this generic B[e] star rotating at = 0, 0.3, 0.4, 0.5 and 0.6 (higher where not possible with our method) times the critical rotation speed is given in Table 6.1. The table shows for different co-latitudes and different rotation velocities the values of V , the local mass loss rate Fm () and the value of the velocity law parameter , obtained by fitting the calculated velocity structure from 1.1 R to 10 R to a -law. We see that the value of does not change much. Fig. 6.3 shows the velocity structure of the wind and the location of the critical point for = 0, 0.3 and 0.6. Note that the critical point is very close the star. This is true for all finite disk corrected solutions. Figure 6.4 shows the behaviour of V versus Fm for various rotational speeds and latitudes. As expected, V at the equator is smaller than at the pole because V scales roughly with Vesc and due to the smaller value of Vesc at the equator. Figure 6.5 shows the latitude dependence of the local mass-loss rate and of the terminal velocity from equator to pole, for different rotational speeds, indicated by . At the pole ( = 0) V is about constant for various rotational speeds. The mass-loss rate at the equator decreases slightly with increasing rotation. The increase of the mass loss rate at the equator as calculated by Pauldrach et al. (1986) and Friend & Abbott (1986) is offset by a decrease in mass loss rate due to the smaller radiative flux at the equator due to the von Zeipel effect. The mass loss rate at the pole increases strongly with increasing rotation rate due to the increase in luminosity at the pole. This is because the total luminosity of the star is fixed and the smaller luminosity at the equator must therefore be compensated by a higher luminosity at the pole.

6.5.2 The overall density properties
The overall effect of rotation is to increase the mass loss rate of a star. This can be seen in Fig. 6.5 where the overall mass-loss rate (i.e. Fm integrated over the stellar surface) is plotted vs. the rotational velocity. The mass-loss rate of the models with 4.5 log L /L 5.5 vary 140


The radiation driven winds of rotating B[e] supergiants

Figure 6.3: The velocity law V (r) at the pole and equator of B[e] wind models for different rotation rates. Top panel: = 0.3, bottom panel: = 0.6. The crosses indicate the location of the critical point of the wind. Note that the equatorial curves in the rotating models start at a radius r/Rpole > 1 due to the oblateness of the star (see text for stellar parameters).

with as 1 + 0.642 . Figure 6.6 shows the resulting density contrast (far from the star) between pole and equator eq pole pole eq ( eq /pole = Fm V /Fm V ). The densities at pole and equator are essentially the same: the smaller V at the equator is offset by the larger mass loss rate Fm at the pole. 141


Chapter 6

Figure 6.4: The latitude dependence of the mass-loss rate and V of a typical B[e] star, for various rotational speeds. (See text for stellar parameters).

Figure 6.5: Mass loss rates for different rotation speeds. The drawn line is given by: 1 + 0.642

6.5.3 Varying L
We have also calculated models for different luminosities, (Table 6.2) and different stellar masses (Table 6.3). The mass-loss rate for the different luminosities is also plotted in Fig. 6.5. The density contrast for various luminosities is plotted in Fig. 6.6. We see that the density at the pole compared to the equator increases for increasing luminosity and the effect on the mass-loss 142


The radiation driven winds of rotating B[e] supergiants

Figure 6.6: The density contrast eq /

pole

for various values of log L /L .

rate is only visible for the highest luminosity. This is an effect of the continuum acceleration Dc . Figure 6.7 illustrates the effect of the continuum acceleration on the forces in the wind for a star with high luminosity (e = 0.76). The net effect of the continuum radiation pressure and the gravity, i.e. GM (1 - e )/r2 , is an inward force in the equatorial region and an outward force in the polar region where the flux is higher. For small luminosity the variation in e through Dc is not important because e itself is small. For high luminosity L /(GM ) increases and thus the effect of Dc on e becomes noticeable (this is with all other properties of the star fixed). In Eq. (6.31) we see that a larger value of Dc at the pole causes a relatively larger decrease (through e )in Vesc and thus in V , whereas the mass loss rate at the equator will increase less because of the smaller Dc . This is only a qualitative explanation and the actual dependence of eq /pole does not follow the relation suggested by Eqs. (6.29) and (6.31). We have found that the actual values of eq /pole could be approximated quite well (within 10 percent) with the following relation:

eq /

pole



1 - e ( = 0, r/Req = 1) 1 - e ( = /2, r/Req = rmin )

1.5

(6.41)

where the r

min

1.2 is the radius where Dc reaches its minimum value.

So we see that although rotation alone modifies the wind structure considerably, it cannot be responsible for the density contrast between the equatorial and polar wind observed in B[e] stars, in case only radial effects are considered. We conclude that at least one other physical effect must be responsible for the formation of disks of rotating B[e] stars. The two most likely additional effects are the bi-stability and the flow of the wind material towards the equator. 143


Chapter 6

Figure 6.7: Vector plot of the gravity (Fg ) and the continuum radiation force (Fe ). The vectors indicate strength and direction of r2 (Fg + Fe ) on an arbitrary scale for a star with e = 0.76 and = 0.6. Note that line forces and centrifugal forces are not included.

6.6 Rotationally induced bi-stability models
In the section above we have shown that rotation alone cannot explain the observed large density contrast between the poles and the equator of B[e] stars. One of the possible additional mechanisms to enhance this contrast is the bi-stability jump. To investigate whether the bi-stability jump observed in normal B-supergiants can explain the formation of disks around B[e] stars, we have calculated bi-stable wind models for a fast rotating typical B[e] supergiant. Spherical, NLTE wind models for normal B supergiants have been calculated by Vink et al. (1999). From these models CAK-like force multiplier parameters have been derived using a Monte-Carlo method to simulate photon-gas interactions. These models show the occurence of a bi-stability jump around Teff 25000 K. For two models on either side of the bi-stability jump (Teff = 17 500 and 30 000 K), the CAK parameters were determined by fitting a power law from about the sonic point to about 0.5V (from t = 10-2 to t = 10-4 in optical depth, t is defined in Eq. (6.21) to the calculated force multiplier values. The resulting values for k, and are listed in Table 6.4. Note that the parameter is taken equal to zero as it was not possible to extract it explicitly from the models. Its effect is hidden in the constant k. As expected, the values for the force multiplier parameters are quite different for the two cases, because the ionization has changed dramatically over the 144


The radiation driven winds of rotating B[e] supergiants Teff k (K) 17500 0.57 0.45 0.0 30000 0.06 0.65 0.0 Table 6.4: The force multiplier parameters for a bi-stable wind

Figure 6.8: The terminal velocity V for a star with a bi-stability jump as a function of colatitude. Notice the drastic drop by about a factor 3 around =600 , due to the bi-stability jump. (See text for the adopted stellar parameters bi-stability jump. The predicted bi-stability jump occurs around Teff 25 kK. Therefore we calculated a model for a rotating B[e] star with the following properties: Teff = 25000 K, L = 105 L , M = 17.5M and solar abundances. This implies that the pole of the rapidly rotating star will be hotter than 25 kK and the equator will be cooler than 25 kK. For the pole, the force multipliers k and from the hot Monte Carlo model of 30 000 K were used, whereas for the equator, k and from the cool model of 17 500 K were used. The resulting V () and mass loss rates Fm () are plotted in Fig. 6.8 and Fig. 6.9 for = 0.6. Clearly visible is the drastic decrease in V and the drastic increase of Fm towards the equator. This occurs around a co-latitude (angle from the pole) of about 60 degrees, where Teff 25 000 K. (The precise location of the jump depends on the effective temperature of the star). Since both Fm and V have a equator/pole contrast of about a factor of three, the resulting density contrast is about a factor of 10 in a disklike region with a half opening angle of 30 degrees. Our calculations show that the density contrast in the wind between pole and equator due to the bi-stability jump is significantly larger than without the bi-stability jump. Yet, the calculated 145


Chapter 6

Figure 6.9: The mass loss rate Fm of a star with a bi-stability jump as a function of co-latitude. Notice the steep jump in mass loss rate near = 600 . (See text for the adopted stellar parameters) value of about a factor 10 is not sufficient to explain the observed density contrast of a factor 100. So another mechanism is needed to enhance the density contrast even further. This is most likely the wind compression mechanism (Bjorkman & Cassinelli 1993; Bjorkman 1998).

6.7 Summary and discussion
We have modified the line driven wind theory for rotating stars by including the effects of oblateness and gravity darkening as well as rotational terms in the equation of motion. We considered a sectorial wind, i.e. we neglected the effect of the motion of the wind towards the equator or towards the pole. This assumption is justified close to the stellar surface below the critical point of the momentum equation. This implies that our method will predict about the correct distibution of mass-loss rates from the star as a function of stellar latitude, but it may not be accurate enough to predict the velocity and density distribution further away from the star, if motions in the direction become important. The equation of motion was solved using an iterative numerical scheme, that includes the conservation of angular momentum and the correction factors to the radiative acceleration by lines and by the continuum, due to the non-spherical shape of the star and due to the latitude dependence of the radiative flux. This method was applied to study its possible effects on the formation of disks around fast rotating B[e] supergiants. The models with constant force multiplier parameters k, and show a decrease of both the mass loss rate and the terminal velocity from pole to equator. This is mainly due to two effects: the reduction of the escape velocity from pole to equator, resulting in a higher terminal velocity at the pole, and the reduction of the radiative flux from the pole to the equator due to gravity darkening, which results in a decrease in mass loss rate at the equator. 146


The radiation driven winds of rotating B[e] supergiants For a star with a fixed luminosity and a fixed polar radius, the temperature at the pole increases and the temperature at the equator decreases with increasing rotation rate. The terminal velocity of the wind from the poles is almost independent of Vrot but at the equator V decreases with increasing rotation rate. The mass-loss rate at the pole increases with increasing Vrot , due to the increase in Teff at the poles, but the mass loss rate from the equator is almost independent of the rotation. The combination of these effects alone produce a density contrast between the polar and the equatorial wind of a factor eq /pole 1, except for fast rotating stars with high luminosity (in which case eq /pole < 1). Our results confirm quantatively the results obtained by Maeder (1999), for the latitudinal dependence of Fm and are in general agreement for expected behaviour of the global mass-loss rate in the case of luminosities far from the Eddington limit. For high luminosities we find a strong polar outflow that would not be easily ofset by a change in k and in disagreement with Maeder (1999). The difference between the polar wind and the equatorial wind is strongly enhanced when the bi-stability of radiation driven winds between Teff 20 000 and 30 000 K is taken into account. In this case the force multipliers k and of a rapidly rotating star change drastically with stellar latitude if the pole is hotter than 25 000 K and the equator is cooler. In Sect. 6.6 we have applied the newly calculated force multipliers from Vink et al. (1999) above and below the bi-stability jump to the models of rotating B[e] supergiants. One might argue that it is not allowed to apply the force multipliers of (bi-stable) spherical wind models to aspherical winds of rotating stars. However, we have shown that a disk formed through bi-stability will be sufficiently thick (typically about 30 degrees for a star with an effective temperature equal to about 25 000) to make these spherical models reasonable approximations for the conditions in the wind close to the star, where the mass loss rate is determined. We find that rotationally induced bi-stability models of B[e] stars reach a density contrast of about a factor 10 between the dense equatorial wind and the less dense polar wind. This is less than the factor 102 that is observed (see eg. Bjorkman 1998). The extra density increase is most likely due to the wind compression. The gas that leaves the photosphere from a rotating wind, will follow an orbit in a tilted plane defined by the local rotation vector and the center of the star. If the rotational velocity is large or the wind velocity is small, this orbit will cross the equatorial plane where the streamlines of the wind from different stellar latitudes cross. The resulting shock will compress the gas into a thin outflowing disk with an opening angle of only a few degrees (Bjorkman & Cassinelli 1993; ISW chapter 11). We have not considered this wind compression in our model. However it likely to occur in rotationally induced bi-stable winds because the wind velocity in the equatorial plane is about a factor three smaller than from the poles, thus facillitating the wind compression. Owocki & Cranmer (1994) have argued that the flow of the wind towards the equatorial plane, predicted in the WCD-theory, may be offset by a -component of the line acceleration towards the polar regions. This "disk inhibition" mechanism operates in wind models with constant force multipliers k and . However, it is not clear that this mechanism is sufficiently strong to overcome the wind compression in the RIB-model. This is because k and change with latitude in the RIB-model and there is a strong density gradient in the wind from the equator to the poles (Owocki et al. 1998, Puls et al. 1999). The combination of the RIB and the WCD mechanism offers the best possibility for explaining the disks of B[e]-supergiants. Whether the compression is strong enough to explain the observed high density contrast between the polar and the equatorial wind remains to be 147


Chapter 6 seen. It can be calculated by combining the solutions of the wind momentum equation of rotating oblate stars with gravity darkening (derived in this paper), and the calculation of the resulting trajectories of the wind. The combination of the rotation induced bi-stability model and the wind compressed disk model is promising for explaining the disks of B[e] stars because the RIB-mechanism explains the increased mass loss and the small velocity from the equatorial regions and the WCD-mechanism explains the strong compression of the disk.

References
Abbott D.C., 1980, ApJ 242, 1183 Abbott D.C., 1982, ApJ 259, 282 Bjorkman J.E. 1998 in B[e] stars, eds A.M. Hubert & C. Jaschek Kluwer: Dordrecht, ASSL 233, p 189 Bjorkman J.E., Cassinelli J.P., 1993, ApJ 409, 429 Cassinelli 1998 in B[e] stars, eds A.M. Hubert & C. Jaschek Kluwer: Dordrecht, ASSL 233, 177 Castor J.I., 1974, MNRAS 169, 279 Castor J.I., Abbott D.C., Klein R.I., 1975, ApJ 195, 157 Cranmer S.R., Owocki S.P., 1994, ApJ 440, 308 Feldmeier A., 1999 in Variable and non-spherical stellar winds in luminous hot stars, Lecture Notes in Physics, eds. B. Wolf et al., Springer: Heidelberg, p. 285 Friend D.B., Abbott D.C., 1986, ApJ 311, 701 Gayley K.G., 1995, ApJ 454, 410 Kudritzki R.P.,Pauldrach A., Puls J.,Abbott D.C., 1989, A&A 219, 205 Lamers H.J.G.L.M., Cassinelli J.P., 1999, Introduction to Stellar Winds, Cambridge Univ. Press. (ISW ) Lamers H.J.G.L.M., Pauldrach A., 1991,A&A 244, 5 Lamers H.J.G.L.M., Snow T.P., Lindholm D.M., 1995, ApJ 455, 269 Lamers H.J.G.L.M., Zickgraf F.-J., de Winter D., Houziaux L., Zorec J., 1998, A&A 340, 117 Lucy L.B., 1982, ApJ 255, 286 Marlborough J.M., Zamir M., 1984, ApJ 276, 706 Maeder A., 1999, A&A 347, 185 Owocki S.P., Castor J.I., Rybicki G.B., 1988, ApJ 335, 914 Owocki S.P., Cranmer R.C., 1994, ApJ 424, 887 Owocki S.P., Cranmer R.C., Gayley K.G., 1998, Ap&SS 260, 149 Pauldrach A., Puls J., Kudritzki R.P.,1986, A&A 164, 86 Puls J., Petrenz P., Owicki S.P. 1999 in Variable and non-spherical stellar winds in luminous hot stars, Lecture Notes in Physics, eds. B. Wolf et al., Springer: Heidelberg, p. 131 Vink J.S., de Koter A., Lamers H.J.G.L.M., 1999, A&A 350, 181 von Zeipel H., 1924, MNRAS 84, 665 Zickgraf F.-J., 1992, in Astronomical CCD Observing and Reduction Techniques, ed. Howell S.B. , ASP Conference Series Vol. 22, 75 Zickgraf F.-J., Schulte-Ladbeck R.E., 1989, A&A 214, 274 Zickgraf F.-J., Wolf B., Stahl O., Leitherer C., Klare G., 1985, A&A 143, 421

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Radiation-driven wind models for Luminous Blue Variables

7

Radiation-driven wind models for Luminous Blue Variables
We have calculated radiation-driven wind models for stellar parameters that are typical for Luminous Blue Variables (LBVs). We have investigated the effects of (1) lower masses and (2) modified abundances on the mass-loss rates of these stars in comparison with normal OB supergiants. The main change in mass loss turns out to be an increase in the mass loss for the LBVs due to their lower stellar masses. Typically, a decrease in the stellar mass by a factor of two results in an increase of the mass-loss rate by about a factor of four. Additionally, we have found that an increased nitrogen abundance is relatively unimportant, but that the helium enrichment does change the mass-loss properties somewhat (on the order of 0.1 dex in log M ). Furthermore, as we have shown that for the normal OB supergiants the observed values for the mass loss are in good agreement with our predicted mass-loss rates, we tentatively estimate a stellar mass M for LBVs by comparing observed mass-loss values with our model predictions. This is certainly helpful, as masses of LBVs are not well determined. A comparison between our theoretical models and the observations is hampered by uncertainties in the observations of LBVs. Moreover, only a few LBVs are known, and out of these, only a handful have been monitored. Nevertheless, a qualitative comparison between the observed mass-loss behaviour of five relatively well-studied LBVs in the Galaxy and the LMC with our predictions has been attempted and the result suggests that we can qualitatively explain the observed mass loss variations of LBVs during their excursions in the Hertzsprung-Russell Diagram.

7.1 Introduction
Luminous Blue Variables (LBVs) are evolved massive stars undergoing a brief but violent stage of stellar evolution. They are characterized by so-called "typical LBV-variations" (also called "moderate-" or "S Dor variations") in visual brightness with amplitudes on the order of one to two magnitudes on a time scale of a few years to a decade (Humphreys & Davidson 1994, Lamers 1995). During these variations the stars make horizontal excursions in the HertzsprungRussell Diagram (HRD) as they expand in radius at approximately constant bolometric lumi149


Chapter 7 nosity between visual minimum and maximum phase. Leitherer et al. (1989) and de Koter et al. (1996) have computed atmospheric models for LBVs and these studies have shown that the stellar radius itself increases during a typical variation rather than that an increased mass loss causes the formation of a "pseudo-photosphere". The physical reason for the LBV expansions is, as yet, unidentified (see Nota & Lamers 1997 for overviews). The observed mass-loss behaviour of the LBVs during their variation cycle is rather intriguing. The analysis by Leitherer et al. (1989) for the Large Magellanic Cloud (LMC) star R 71 indicates that as the star expands, its mass loss increases. Intuitively, such behaviour may be anticipated as the star gets closer to its Eddington limit (see e.g. Lamers & Fitzpatrick 1988). However, Stahl et al. (1990) found the surprising result that for R 110, another LBV in the LMC, the mass loss behaviour is the exact opposite: as the star expands, and consequently Teff decreases, the mass-loss rate drops. This implies that there is no general correlation between mass loss and stellar temperature (see Leitherer 1997). Calculations of radiation-driven wind models for "normal" winds of O and B stars have shown that wind models with smaller Teff generally experience smaller mass loss for constant L and M (Vink et al. 2000a). This is due to a shift of the maximum of the spectral energy distribution to longer wavelengths, where less lines are present, and consequently the line acceleration becomes less effective. Nevertheless, superimposed on this behaviour there are bi stability jumps present, where M jumps upward due to recombinations of important line-driving elements. For instance, around Teff = 25 000 K, the mass loss increases by a factor of about five, due to a recombination of Fe IV to Fe III (Vink et al. 1999). Observational evidence for bi-stability jumps has also been found: Lamers et al. (1995) found an abrupt discontinuity in the terminal wind velocities around spectral type B1 and probably discovered a second jump around A0. This second jump is likely due to the recombination of Fe III to Fe II (Leitherer et al. 1989, Vink et al. 1999). The LBVs have the special property that their effective temperature changes significantly on a typical timescale of years. In view of the above mentioned mass loss behaviour of OB supergiants as a function of Teff , one may expect the mass loss of LBVs to be variable during their Teff changes. Whether an increase or decrease in mass loss is expected to occur, simply depends on the exact temperatures of the star with respect to the positions of the bi-stability jumps (see also de Koter 1997, Lamers 1997). Therefore, the surprising difference in mass loss behaviour of R 71 versus R 110 during their variation cycles may for both of them be explained in terms of their range in effective temperatures with respect to the location of the two bi-stability jumps (i.e. at spectral types B1 and A0). We can test this hypothesis by computing radiation-driven wind models with parameters representative of LBVs and compare the predicted mass-loss behaviour with the observed mass loss during their variation cycles. As LBVs are evolved stars, they have already lost a considerable fraction of their initial mass during their prior evolution. Hence, their (L /M ) ratio is higher than for normal, less evolved OB supergiants. Additionally, as massive stars convert hydrogen into helium due to the CNO cycle, the LBV abundances are expected to be different from solar, if these products of nucleosynthesis have reached the stellar surface. Observations and predictions indicate that LBVs have been enriched in the chemical elements of helium and nitrogen, and are depleted in carbon and oxygen (e.g. Smith et al. 1994, Najarro et al. 1997). In this study, we will therefore investigate the changes in mass loss due to two effects: 1. the lower masses for LBVs compared to OB stars 2. the modified abundances of LBVs compared to normal OB stars 150


Radiation-driven wind models for Luminous Blue Variables For the mass loss predictions we will use a Monte Carlo approach, which has proven to be successful in predicting the observed mass-loss rates for the "normal" winds of O stars. This extensive grid of computations has resulted in a recipe which provides M as a function of the stellar parameters L , M , Teff and V /Vesc for OB supergiants. Here, we extend on these calculations by including the winds that experience more dramatic LBV-like behaviour. The expected changes in wind properties could be higher mass loss for the LBVs than for the normal supergiants, as LBVs have lower masses. The main issue we want to address in this study is whether the observed mass-loss behaviour of LBV excursions in the HRD can be explained by our radiation-driven wind models. In Sects. 7.2 and 7.3, the method to calculate mass-loss rates and the adopted assumptions will be described. In Sect. 7.4, we will present wind models and mass-loss rates for stellar parameters that are typical for LBVs. In Sect. 7.5 these M predictions will be compared with the observed mass-loss behaviour during the variation cycles of the five best observed LBVs in the Galaxy and the LMC. We conclude with a summary and a discussion (Sect. 7.6).

7.2 The method to calculate M
The properties of our Monte Carlo method to predict mass-loss rates have been described extensively in previous papers (Abbott & Lucy 1985, de Koter et al. 1997, Vink et al. 1999, 2000a) For details , the reader is referred to these papers. The core of the approach is that the total loss of radiative energy is converted to kinetic energy of the outflow. The momentum deposition in the wind is calculated by following the fate of a large number of photon packets that are released from below the photosphere. The main advantage of the method is the natural way in which the process of "multiple scattering" is accounted for. In addition we employ a unified approach, i.e. the photosphere and wind merge in a gradual way. This implies that a "core-halo" approach is avoided. The calculation of mass loss with our method requires the prior computation of a model atmosphere. These models have been calculated with the non-LTE unified Improved Sobolev Approximation code IS A - W IND for hot stars with winds. For details on IS A - W IND, we refer the reader to de Koter et al. (1993, 1997). The chemical species that are explicitly calculated in non-LTE are H, He, C, N, O and Si. The iron-group elements, which are important for the radiative driving and consequently for the mass loss, are treated in a generalized version of the "modified nebular approximation" (see Schmutz 1991).

7.3 The assumptions of the LBV-like models
We calculate mass-loss rates for two series of LBV models. These series consist of models with two luminosities, respectively (L/L ) = 5.5 and 6.0. These luminosities are representative for the LBVs: AG Car, S Dor, R 127, R 71 and R 110, whose mass loss behaviour will be discussed in Sect. 7.5. Since the masses of LBVs are poorly known, we adopt a realistic range in masses for both series of luminosity models (see Table 7.1). The luminosity-to-mass ratio (L /M ) can be expressed in terms of the Eddington factor e . This factor is defined as the ratio between the gravitational and radiative acceleration due to electron scattering, and is given by 151


Chapter 7 logL (L ) 5.5 6.0 M (M ) 30 ­ 10 35 ­ 25 (L /M ) â103 (L /M ) 10.5 ­31.6 28.6 ­ 40.0 e Teff in (kK) Teff out (kK)

0.18 ­ 0.53 11.0 ­ 30.0 10.5 ­ 30.0 0.48 ­ 0.67 11.0 ­ 30.0 10.2 ­ 30.0

Table 7.1: Parameters for the LBV models.

e =

L Le = 7.66 10-5 e 4cGM L

M M

-1

(7.1)

where the constants have their usual meaning and e is the electron scattering cross-section per unit mass. The value for e depends on the temperature, but also on the chemical composition in the atmosphere (see Lamers & Leitherer 1993). The (L /M ) ratio, which is independent of the value of e and does not depend on the composition of the atmosphere, is given in the third column of Table 7.1. The range in e is given in column (4) of the table. Note that e is considerably smaller than unity, and thus the classical Eddington limit is not exceeded. The reason for this surprisingly low e is that although the masses are lower than for normal OB supergiants, the value for the electron scattering cross-section e is also lower. This due to helium enrichment in LBV atmospheres. We enumerate the assumptions that have been adopted in the model calculations: · The adopted range in masses is indicated in the second column of Table 7.1. For the first luminosity series with (L/L ) = 5.5, mass loss was computed for M = 30, 25, 20, 15, 12 and 10 M . For the second one with (L/L ) = 6.0, the adopted masses were M = 35, 30, and 25 M . · All models have input effective temperatures between 11 000 and 30 000 K with a stepsize of 2 500 K in the range 12 500 - 30 000 K. This input temperature Tin is defined by the relation:
4 L = 4R2 Tin in

(7.2)

where Rin is the inner boundary of the atmospheric model. Rin is typically located at a Rosseland optical depth 20 < R < 25. Note that the actual effective temperature of the model Teff out is somewhat lower and it follows from the model computation. The effective ° temperature is defined at the point where the thermalization depth at 5555 A equals 1/ 3 (see Schmutz et al. 1990 and de Koter et al. 1996 for a detailed discussion). For stars with relatively low M and consequently optically thin winds, like the winds from normal OB supergiants, there is no substantial difference between the input and output Teff . However, in the case of LBVs, which may have M up to 10-4 M yr-1 , there is a certain offset between the input and output value of Teff , which we take into account (see columns 5 and 6 in Table 7.1). Note that the effect is not as dramatic as for the much denser winds of Wolf-Rayet stars. · Both observations as well as theoretical calculations of evolved stars have shown that LBVs are enriched in helium and have a depleted hydrogen abundance (Smith et al. 1994, 152


Radiation-driven wind models for Luminous Blue Variables Element Solar (per mass) LBV (per mass) H He C N O 0.68 0.30 0.029 0.00095 0.0077 0.38 0.60 0.00029 0.0095 0.00077

Table 7.2: The Abundances for solar photospheres and LBVs (due to the CNO cycle)

Meynet et al. 1994, Najarro et al. 1997). In addition to an enrichment of helium, CNO processed material that has reached the surface also increases the amount of nitrogen. This occurs at the cost of oxygen and carbon, which become depleted. However, the total amount of CNO remains constant during the cycle process. The relative changes in the elements CNO have been adopted from the abundance computations for massive stars with rotation induced mixing by Lamers et al. (2000). These modifications have been applied to the solar CNO abundances from Allen (1973). Table 7.2 presents both the initial solar abundances (column 2) as well as the abundances modified by the CNO process (column 3). The total metal abundance is kept constant at the solar value, Z = Z = 0.02 for models computed with all abundances. · The dependence of M on the adopted ratio V /Vesc was also determined. Lamers et al. (1995) found that for Galactic supergiants the ratio V /Vesc 2.6 for stars of types earlier than B1 and that it drops to V /Vesc 1.3 for later spectral types. Possibly this ratio drops again around spectral type A0 to a value of about 0.7. We have calculated mass-loss rates for a range of ratios V /Vesc between 1.3 and 2.6. · Finally, we have calculated M for wind models with a -type velocity law for the supersonic part of the wind: R V (r) = V 1 - (7.3) r Below the sonic point, a smooth transition from such a velocity structure is made to a velocity that follows from the photospheric density structure. It has been shown that the predicted mass-loss rate is insensitive to the adopted value of (Vink et al. 2000a). A value of = 1 was adopted.

7.4 The predicted mass-loss rates of LBVs
As was mentioned before, the main expected changes in wind behaviour of LBVs compared to normal supergiants, could be higher mass-loss rates, because of lower stellar masses. A second difference in mass loss may arise from the changes in chemical composition, i.e. a fair amount of hydrogen is converted into helium, and additionally nitrogen has been enriched at the expense of carbon and oxygen. Nevertheless, Vink et al. (1999) predicted that modifications in the relative abundances in CNO due to nucleosynthesis are unlikely to cause significant changes in the mass loss. The reason is that the mass-loss rate is mainly determined 153


Chapter 7

Figure 7.1: Predicted mass-loss rates dM /dt as a function of Teff for different stellar LBV masses. The other stellar parameters are log L/L = 5.5 and V /Vesc = 1.3. The different masses are indicated in the plot.

by the line acceleration of iron. We thus anticipate that the main differences in wind behaviour between the normal supergiants and the LBVs arise due to (1) a lower stellar mass and (2) a higher helium abundance. Therefore, the emphasis in this section will be given to these two effects. Additionally, we also present a comparison between models with (3) nitrogen enriched atmospheres versus initial abundances.

7.4.1 The effect of the lower masses on M
The first effect to be investigated is a decrease of the stellar mass. A grid of mass-loss rates for several adopted masses ­ but with constant luminosity ­ will be presented. The adopted parameters are typical for a relatively low luminosity LBV, with log L/L = 5.5. Figure 7.1 displays the mass loss behaviour as a function of effective temperature for the different masses. It shows that for all model temperatures, mass loss increases as the mass M decreases. A decrease in M by a factor of two results in an increase in M by about a factor of four. An additional behaviour that is visible in Fig. 7.1, is that the bi-stability jumps shift to higher temperature for lower mass. This is also what one would expect, as the jump depends on the ionization balance in the wind and thus on both the temperature and the density. Since the mass-loss rates are larger for the models with the smaller masses, the densities in these models are also larger. Therefore, the location of the bi-stability jump is expected at a higher effective temperature. 154


Radiation-driven wind models for Luminous Blue Variables

Figure 7.2: The calculated mass-loss rates dM /dt as a function of Teff for initial hydrogen abundance (X = 0.68, Y = 0.30; dashed lines) versus helium-enriched atmospheres (X = 0.38, Y = 0.60; solid lines). The stellar parameters are log L/L = 6.0 and M = 60M . The values for (V /Vesc ) are indicated in the lower legend.

7.4.2 The effect of helium enrichment on mass loss
For an optimum discrimination between the various effects, we now keep the mass fixed, but modify the chemical abundances. We investigate the differences in mass loss when a substantial amount of hydrogen is converted into helium due to nucleosynthesis. We compare the mass-loss rates of typical OB supergiants with helium enriched models (see Table 7.2). For the models we adopt the stellar parameters log(L/L ) = 6.0 and M = 60M . For the normal composition OB supergiants, mass-loss rates were calculated by Vink et al. (2000a). These predictions are compared with the LBV winds from helium-enriched atmospheres. As the mass is kept constant at M = 60M for the reason given above, the stellar mass is somewhat higher than for realistic LBVs. Figure 7.2 displays the comparison between the initial hydrogen-rich and the helium-enriched atmospheres. It displays some differences: 1. At high Teff , the initial hydrogen-rich models experience a slightly lower mass loss than the helium-enriched models. 2. At lower Teff it is the other way around: the hydrogen-rich models show higher mass loss than the helium-enriched models. As the line acceleration causing the mass loss is mainly determined by metals, the fraction of hydrogen and helium to the total line acceleration is negligible. Hence, there must be another reason for the differences in mass loss between hydrogen-rich vs. helium-enriched atmospheres. The origin of these differences is related to changes in the underlying spectral energy distribu tion. As most of the line driving around the sonic point, where M is fixed, is due to Fe, the 155


Chapter 7

Figure 7.3: The calculated mass-loss rates dM /dt as a function of Teff for solar CNO (dashed lines) versus nitrogen enriched atmospheres (solid lines). The stellar parameters are log L/L = 6.0 and M = 30M . The values for (V /Vesc ) are indicated in the lower legend.

relevant question is in which spectral range these iron ions have most of their line transitions, and how well this matches to the spectral region in which the bulk of the flux is emitted. When hydrogen is (partly) replaced by helium, the size of the Lyman jump decreases. This implies that as the amount of helium in the atmosphere increases, more photons are available ° in the Lyman continuum at wavelengths below 912 A. Since the bolometric luminosity of the models is kept constant at log L/L = 6.0, less flux is emitted in the Balmer continuum at ° wavelengths between = 912 and 3646 A. As Vink et al. (1999) have shown that mass loss is mainly dependent on the number of driving lines ­ which depends on the ionization state of Fe ­ in the Balmer continuum, where the bulk of the flux is emitted, one expects lower mass loss at lower Teff for the helium-enriched models. For the hotter models the situation is reversed: for these models ­ that emit more flux in the Lyman continuum ­ the mass loss is higher for the helium-enriched atmospheres.

7.4.3 The effect of the nitrogen enrichment on M
Figure 7.3 shows a comparison in mass loss between models with nitrogen enriched atmospheres (dashed lines) and initial CNO abundances (solid lines). As expected, there is no significant change in the predicted mass loss due to the nitrogen enrichment in LBVs. The reason is that in the case of Galactic abundances, the mass-loss rate from massive stars is mainly set by iron, and not by CNO (Vink et al. 1999). 156


Radiation-driven wind models for Luminous Blue Variables

Figure 7.4: Predicted mass-loss rates dM /dt as a function of Teff for three adopted masses. The masses are given in the plots. All models have log (L/L ) = 6.0. The values for (V /Vesc ) are indicated in the legend of the lowest panel. 157


Chapter 7

7.4.4 The complete grid of mass-loss rates for LBVs
The mass-loss results for LBVs with log (L/L ) = 5.5 have already been displayed in Fig. 7.1. For the higher luminosity series with log (L/L ) = 6.0, the LBV mass-loss predictions are plotted in the three panels of Fig. 7.4 for masses M = 25, 30 and 35 M , respectively. All panels in Fig. 7.4 show the same kind of mass-loss behaviour, including the presence of two bistability jumps around Teff 25 000 K and around Teff 15 000 K. The mass loss dependence on stellar mass is analogous to the log (L/L ) = 5.5 models as shown in Fig. 7.1: M increases as the stellar mass M decreases. Note that for the models with the lower effective temperatures, say Teff < 17 500 K, the mass loss behaviour as a function of the ratio V /Vesc shows an interesting behaviour. As the ratio V /Vesc decreases, and thus the wind density increases, at some point this no longer causes an increase in the mass loss. Note that this "saturation" effect of the mass loss is similar to the flattening of the mass loss versus metallicity power-law dependence M (Z ) at high metal abundance, as found by Vink et al. (2000b). The mass loss computations for LBVs with log (L/L ) = 6.0 in Fig. 7.4 seem to suggest that mass loss cannot exceed a specific maximum value, on the order of M 10-4 M yr-1 . Intriguingly, this predicted maximum value is close to the observed maximum mass-loss rates for LBVs that are discussed in Sect. 7.5.

7.4.5 Uncertainties in the locations of the bi-stability jumps
A problem when comparing our predictions with the observed mass loss behaviour of LBVs is that there is an offset of a few thousand Kelvin between the observed temperatures of the bi-stability jumps derived from spectral types and the temperatures that we find from our model calculations. This problem was noted by Vink et al. (1999): the observed bi-stability jump in normal B supergiants occurs at Teff 21 000 K, while our predictions yield Teff 25 000 K. A number of reasons for this discrepancy was suggested. One of them was that the position of the jump is expected to depend on the density and thus on the (L /M ) ratio. However, subsequent calculations over a wider range in stellar parameters have shown that the location of the 25 000 K jump is only modestly sensitive to these stellar parameters. Therefore, we can exclude this possibility. Another suggested reason for the discrepancy is that we may introduce a systematic error in the ionization balance of iron, due to the use of the "modified nebular approximation" (see Sect. 7.2). Hence, the exact position of the bi-stability jumps in our mass-loss predictions should not be interpreted too strictly. Instead, it would be better to correct the temperatures of the models to the observed values, as these do not depend on any model approximations. Lamers et al. (1995) found that the bi-stability jumps are present at the temperatures 21 000 K, and 10 000 K respectively. Therefore, we will adopt these temperatures for the bi-stability jumps throughout the remainder of the paper. Note that our predicted mass-loss rates are expected to be reliable, since we have shown that these predictions are in good agreement with the observed values for a large sample of O stars.

7.5 Comparison between LBV predictions and observations
There are only a small number of LBVs with mass-loss determinations in the literature (see below). For the Galaxy, these stars are Car, P Cygni, and AG Car. For the LMC, these are 158


Radiation-driven wind models for Luminous Blue Variables the stars: S Dor (= R 88), R 127, R 71 and R 110. As our interest is in the mass loss of LBVs as a function of effective temperature during their variation cycles, we are only interested in those LBVs that show the typical LBV photometric variations. For this reason the Galactic stars Car and P Cygni are omitted from our analyses, as these stars do not show the typical photometric variability. This implies that the remaining sample is very small, i.e. only five stars are left. Their observed properties are summarized in Table 7.3. The most difficult parameter to obtain from an observational analysis of LBVs is the stellar mass. As the masses for LBVs are so poorly known, this severely complicates a comparison between observations and our theoretical predictions. Also, the derived effective temperatures from observations are somewhat insecure. Yet, it is possible to follow an inverse strategy. As our Monte Carlo approach to predict mass loss has proven to be so successful in predicting the observed mass-loss rates for the normal OB stars, we may use the observed mass-loss value to predict LBV masses. For this purpose, we have compared the observed mass-loss rates (taken from the literature and given in the third column of Table 7.3) and compared these ­ with the appropriate luminosity (column 2) ­ to the different mass models displayed in the Figs. 7.1 and 7.4. The estimated masses are presented in the fourth column of the table. Now we can "simulate" the predicted mass-loss behaviour over a typical LBV variation cycle. From the literature, for these five LBVs, the observed temperature behaviour has been obtained and is given in the fifth column of Table 7.3. With our LBV mass-loss predictions from Figs. 7.1 and 7.4 we can predict the mass loss behaviour during the expansion over a variation cycle for all five LBVs. Based on the range in Teff of the variation cycles for the different LBVs and comparing these values of Teff with the 21 000 K jump and the 10 000 K (see Sect. 7.4.5), enables us to predict the mass-loss behaviour, as illustrated in the sixth column of Table 7.3 indicated with "expected behaviour". Finally, these predictions can be compared to the "observed mass-loss behaviour". This is done for all five LBVs individually, and the comparisons are briefly discussed below. AG Car: This LBV is probably the best observed Galactic LBV in terms of its mass loss behaviour. The effective temperature changed between 21 and 15 kK from visual minimum to maximum (Leitherer et al. 1994). By comparing this range in Teff of the variation cycle to the positions of the bi-stability jumps, we anticipate that the mass-loss rate may increase as the temperature drops from 21 to 20 kK, but it expected to decrease gradually between 20 - 15 kK, as no bi-stability jumps are encountered in this temperature range. Such 'irregular' mass behaviour as a function of decreasing Teff or actually a rather steep increase, followed by a gradual decrease, has indeed been observed (Leitherer et al. 1994). We can conclude that the predicted behaviour for AG Car is similar to its observed behaviour. SDor: The LMC star S Dor has been analyzed by Leitherer et al. (1985, 1989) The effective temperature has varied between about 22 and 10 kK. As a function of decreasing temperature, we therefore expect an increase, followed by a gradual decrease, and finally followed by another increase in the mass loss. Unfortunately, the observed behaviour for S Dor is not as well-known as for AG CAR. But the observations by Leitherer et al. (1985) suggest an increase between, Teff 16 and 9 kK, which is typical for S Dor. The star indeed shows such an increase, as expected by the presence of the second bi-stability jump, which is caused by the recombination from Fe III to II (see also Leitherer et al. 1989). 159


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LBV Name AG Car S Dor R 127 R 71 R 110 160

log L (L ) 6.0 5.8 6.1 5.3 5.5

log M obs (M yr-1 ) -4 .1 (average) 4.3 (max) 4.3 (max) 4.7 (max) 5.5 (max)

est M (M )

Teff range Expected behaviour as a (kK) function of decreasing Teff 21 22 30 14 10 - 15 - 10 - 10 - 10 -8 increase,decrease increase,decrease increase,decrease,increase increase decrease

Observed behaviour as a function of decreasing Teff irregular increase irregular increase decrease

30 30 35 12 30

Table 7.3: Observed properties of LBVs. The literature sources are given in the text.


Radiation-driven wind models for Luminous Blue Variables R 127: The LMC star R 127 has extensively been monitored by the Heidelberg group (Stahl et al. 1983, Stahl & Wolf 1986, Wolf et al. 1988). These authors have noted that its mass-loss behaviour has been rather irregular, this is not very surprising, if one looks at the Teff range of its variation cycle between about 30 000 K and 10 000 K. We would anticipate the presence of two bi-stability jumps indeed leading to a rather 'irregular' mass-loss behaviour for R 127. R 71: For the LMC star R 71 the mass loss behaviour is quite similar to that of S Dor. Wind model calculations by Leitherer et al. (1989) suggest an increase over the temperature range between Teff 16 and 9 kK. The observed mass loss behaviour is based on the observations by Wolf et al. (1981): R 71 shows an increase in observed mass loss. This is also expected due to the presence of the second bi-stability jump around 10 000 K. R 110: The observed mass-loss behaviour of the temperature seems, at first sight, quite surprising. imply that the mass loss decreases as a function of behaviour is observed over a range in Teff , which is 10 k, its observed behaviour is not very surprising predictions. LMC star R 110 as a function of effective The observations by Stahl et al. (1990) decreasing Teff . However, as the mass loss entirely below the second jump, i.e. Teff < at all, but even predicted by our mass-loss

7.6 Discussion and Conclusions
We have calculated radiation-driven wind models for stellar parameters that are typical for LBVs. We investigated the effects of (1) lower masses and (2) modified abundances on the mass-loss rates of these stars in comparison with normal OB supergiants. The main change in mass loss turns out to be an increase in the mass loss for the LBVs due to their lower stellar masses. We have found that an increased nitrogen abundance is unimportant, but that the helium enrichment changes the mass-loss properties by a small amount (about 0.1 dex in log M ). For Teff > 25 000 K the mass-loss rates for the helium enriched models are somewhat higher; for Teff < 25 000 K, they are slightly lower. Furthermore, as Vink et al. (2000a) have shown that for the normal OB supergiants the observed values for the mass loss are in good agreement with our predicted mass-loss rates, we tentatively estimate a stellar mass M for LBVs by comparing observed mass-loss values with our model predictions. A comparison between our theoretical models and the observations is hampered by uncertainties in the observations of LBV masses, effective temperatures and mass-loss rates. Moreover, only a few LBVs are known, and out of these, only a handful have been monitored. Nevertheless, a qualitative comparison between the observed mass-loss behaviour of five relatively well-studied LBVs in the Galaxy and the LMC with our predictions has been attempted and the result suggests that our predicted behaviour of M (Teff ) can naturally explain the mass-loss variations that have been observed in LBVs during their Teff excursions. However, we must admit that there may be systematic errors in our predictions of the locations of the bi-stability jumps, i.e. we likely overpredict the effective temperatures of the bi-stability jumps by a few thousand Kelvin. Still, within all observational uncertainties, the observations do certainly not contradict our mass-loss predictions. 161


Chapter 7 We conclude that the available observational resources in the literature are inadequate for a quantitative comparison with our mass-loss predictions for LBVs. Therefore, we suggest that extensive monitoring of some typical LBVs, in addition to detailed modelling of their spectra to obtain mass-loss rates as a function of Teff over their complete variation cycles, is necessary. In addition, stellar evolution models may be calculated with our new mass-loss rates, including the bi-stability jumps in mass loss. Such evolutionary models, complemented with physical effects due to the stellar rotation may illuminate the origin of the typical photometric LBV variations, which may ultimately teach us about this short, but important evolutionary stage itself.

References
Abbott D.C., Lucy L.B., 1985, ApJ 288, 679 Allen C.W., 1973, Astrophysical quantities, University of London, Athlone Press Castor J.I., Abbott D.C., Klein R.I., 1975, ApJ 195, 157 de Koter A., 1997, in " Luminous Blue Variables: Massive Stars in Transition", eds. Nota A., Lamers H.J.G.L.M., 1997, ASP Conf.Ser. 83, 66 de Koter A., Schmutz W., Lamers H.J.G.L.M., 1993, A&A 277, 561 de Koter A., Lamers H.J.G.L.M., Schmutz W., 1996, A&A 306, 501 de Koter A., Heap S.R., Hubeny I., 1997, ApJ 477, 792 Humphreys R.M., Davidson K., 1994, PASP 106, 1025 Lamers H.J.G.L.M., 1995, ASP Conf. Ser. 83, 176 Lamers H.J.G.L.M., 1997, in " Luminous Blue Variables: Massive Stars in Transition", eds. Nota A., Lamers H.J.G.L.M., 1997, ASP Conf.Ser. 83, 76 Lamers H.J.G.L.M., Fitzpatrick E.L., 1988, ApJ 324, 279 Lamers H.J.G.L.M., Leitherer C., 1993, ApJ 412, 771 Lamers H.J.G.L.M., Snow T.P., Lindholm D.M., 1995, ApJ 455, 269 Lamers H.J.G.L.M., Nota A., Panagia N., Smith L., Langer N., 2000, in preparation Leitherer C., 1997, in:" Luminous Blue Variables: Massive Stars in Transition", eds. Nota A., Lamers H.J.G.L.M., 1997, ASP Conf.Ser. 83, 58 Leitherer C., Appenzeller I., Klare G. et al., 1985, A&A 153, 168 Leitherer C., Schmutz W., Abbott D.C., Hamann W.R., Wessolowski U., 1989, ApJ 346, 919 Leitherer C., Allen R., Altner B. et al., 1994, ApJ 428, 292 Meynet G., Maeder A., Schaller G., Schearer D., Charbonel C., 1994, A&AS 103, 97 Najarro F., Hillier D.J., Stahl O., A&A 326, 1117 Nota A., Lamers H.J.G.L.M., 1997, Luminous Blue Variables: Massive Stars in Transition, ASP Conf.Ser. 83. Schmutz W., Abbott D.C., Russell R.S., Hamann W.-R., Wessolowski U., 1990, ApJ 355, 255 Schmutz W., 1991, in: "Stellar Atmospheres: Beyond Classical Models", eds. Crivellari L., Hubeny I., Hummer D.G., NATO ASI Series C, Vol. 341, 191 Smith L.J., Crowther P.A., Prinja R.K., 1994, A&A 281, 833 Stahl O., Wolf B., 1986, A&A 154, 243 Stahl O., Wolf B. Klare G. et al., 1983, A&A 127, 49 Stahl O., Wolf B. Klare G., Juettner A., Cassatella A., 1983, A&A 228, 379 Vink J.S., de Koter A., Lamers H.J.G.L.M., 1999, A&A 350, 181 Vink J.S., de Koter A., Lamers H.J.G.L.M., 2000a, accepted by A&A Vink J.S., de Koter A., Lamers H.J.G.L.M., 2000b, submitted to A&A 162


Radiation-driven wind models for Luminous Blue Variables Wolf B., Appenzeller I., Stahl O., 1981, A&A 103, 94 Wolf B., Stahl O., Smolinski J., Casatella A., 1988, A&AS 74, 239

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8

Summary and Prospects
8.1 Summary
In this thesis, the three open issues in radiation-driven wind theory of OB supergiants that were mentioned in chapter 1, namely the bi-stability jump,the momentum problem in winds of O stars and the metallicity dependence of radiation-driven winds, have been investigated. The results have been summarised in the abstracts of the individual chapters. However, here, extra attention is drawn to the success of our method of predicting mass-loss rates for O stars. As was shown in Fig. 1.4 of the introduction, standard radiation-driven wind theory displayed a systematic discrepancy between the predicted and observed mass-loss rates (Lamers & Leitherer 1993, Puls et al. 1996). The mass-loss properties of O and B stars as a function of various stellar parameters have been computed. The bi-stability jump around spectral type B1 has been investigated and explained in terms of its physics; in addition, a jump in mass loss of a factor of five around this spectral type is predicted. The calculations were extended to OB stars with various masses, luminosities and metallicities; subsequently additional bi-stability jumps were found. The large grid of computations resulted in the derivation of a mass-loss recipe, which predicts mass-loss rates as a function of stellar parameters. These predictions have been compared to the best available observations from the literature and the comparison is here repeated (Fig. 8.1). Figure 8.1 shows that although there is a large scatter in the data, the least-square fit (dashed line) is in sufficient agreement with the one-to-one relation (solid line). This implies that there is no systematic discrepancy between the observed mass-loss rates and the new predictions. This resolves the previously reported issue of the momentum problem in the winds of O stars, and subsequently provides a reliable mass-loss recipe that can be used in e.g. evolutionary calculations for massive stars. In addition, it also provides interesting prospects for future research in the field of stellar winds.

8.2 Prospects
Although the role of Wolf-Rayet (WR) stars as the final stage of massive stellar evolution is well established, there are still many gaps in our knowledge on the evolution of WR stars. One problem is that of the momentum in the winds of WR stars (see chapter 1, Schmutz et al. 1989, Willis 1991). Although wind models including multiple scattering have already been computed for some WR stars (e.g. Lucy & Abbott 1993, Springmann 1994), mass-loss predictions as a function of stellar parameters, similar to the predictions for O stars as were calculated in this thesis, have not yet been performed. We believe that such a study would yield important information on WR stars and the field of stellar evolution as a whole. The success of the method 165


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Figure 8.1: Comparison between the new predictions and observed mass-loss rates for O stars. The data points are discussed in chapter 4. The solid line indicates where the points should fall if predictions of the theory and observations are in perfect agreement. The dashed line shows the least-square fit to the data points. Note that there is no systematic discrepancy anymore. for the O stars, as demonstrated with Fig. 8.1, indicates the feasibility for such a study. Furthermore, a thorough investigation of the mass-loss rates of Central Stars of Planetary Nebulae (CSPNe) could be performed. Lamers et al. (2000) have reported that there is a dichotomy in the observed mass-loss rates from CSPNe. One of the reasons for this dichotomy could be "bi-stable" behaviour of their radiation driven winds, similar to the character in the massive counterparts that were studied in this thesis. Calculations including the important effect of multiple-scattering have not been performed so far, but may be feasible with our models. Apart from an investigation of the reported dichotomy, there is a second reason to study the winds from CSPNe: Winds from CSPNe are observed to have the characteristics of normal O star winds as well as those of WR star winds. As both of these types of winds exist over a large range in effective temperature, a study of CSPNe is ideally suited to a study of the physical differences between the massive counterparts of O and WR star winds as well. A direct comparison between massive O star winds and massive WR stars may be more difficult, as O stars and WR stars are located in very different parts of the HRD, which implies that the ions providing the radiative driving are very different. Therefore, the winds of CSPNe could be the "key" to the understanding of the enigmatic winds of massive WR stars.

References
Lamers H.J.G.L.M., Leitherer, C., 1993, ApJ 412, 771 Lamers H.J.G.L.M., Nugis T., Vink J.S., de Koter A., 2000, in: "Thermal and ionization aspects from hot stars", eds. Lamers H.J.G.L.M., Sapar A., ASP Conf Ser 204, p. 395 166


Summary and Prospects Lucy L.B., Abbott D.C., 1993, ApJ 405, 738 Puls J., Kudritzki R.P., Herrero A., et al., 1996, A&A 305, 171 Springmann U., 1994, A&A 289, 505 Willis A.J., 1991, in: "Wolf-Rayet stars in interaction with other massive stars in galaxies", eds. van der Hucht K.A., Hidayat B., IAU Symp 143, 265

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Samenvatting

Samenvatting
In dit proefschrift is het mechanisme van "stralingsgedreven" winden van massieve sterren onderzocht. Om enigszins te kunnen begrijpen wat dit nu eigenlijk inhoudt, zal er eerst iets verteld worden over wat massieve sterren zijn en daarna zal ik proberen uit te leggen hoe sterwinden ontstaan.

Wat zijn massieve sterren?
Sterren komen voor in allerlei soorten en maten. De meeste sterren in ons universum zijn relatief licht zoals de ster die ons zo na aan het hart ligt: de Zon. Er zijn echter ook sterren die wel 100 keer zo zwaar zijn als de Zon. Zo'n zware ster is niet alleen groot en zwaar, maar ook bijzonder helder. Een zware ster met een gewicht van zo'n 100 zonsmassa's zendt zoveel energie in de vorm van licht (straling) uit dat zo'n massieve ster wel een miljoen keer helderder is dan de Zon. Gelukkig staan deze sterren veel verder weg dan onze buurman, want anders zouden wij op Aarde levend verbranden!

De rol van de zwaartekracht en gasdruk
In de natuurkunde speelt het evenwicht tussen krachten vaak een belangrijke rol. Als er op een lichaam twee krachten werken die tegengesteld gericht zijn, zal het lichaam alleen dan stil blijven staan wanneer deze twee krachten precies even groot zijn. Als echter de ene kracht groter is dan de andere, zal er sprake zijn van een "netto" kracht in een van de twee richtingen. Het lichaam zal nu niet langer in evenwicht kunnen blijven, maar onvermijdelijk in beweging komen. In de sterrenkunde is vaak het evenwicht tussen de naar binnen toe gerichte zwaartekracht en allerlei naar buiten toe gerichte krachten van belang. Zwaartekracht is de kracht die ervoor zorgt dat een appel uit de boom valt, omdat simpelweg alle materie elkaar aantrekt. Daarom ondervindt de appel een neerwaarts gerichte kracht door de aanwezigheid van de "zware" aarde. Een voorbeeld van een naar buiten toe gerichte kracht is de gasdruk. Een gas kun je je voorstellen als een verzameling deeltjes; hoe sneller deze deeltjes bewegen, hoe hoger de bewegingsenergie van het gas is en dit drukt men uit met behulp van het begrip "gasdruk". Winden op aarde komen voort uit het bestaan van gebieden van zogenaamde "hoge" en "lage" gasdruk. Als er zich in Duitsland een hoge drukgebied bevindt en in Frankrijk een gebied van lage druk, dan zullen er gassen in de aardse atmosfeer in de richting van Duitsland naar Frankrijk willen stromen en dit zorgt voor winden in de aardse atmosfeer. De "balans" tussen de naar binnen gerichte zwaartekracht en de naar buiten gerichte gasdruk speelt een essentiele rol bij het vormen van sterren. Iedere "gaswolk" in het heelal bevat een ¨ 169


Samenvatting
bepaalde hoeveelheid bewegingsenergie t.g.v. de warmtebeweging van de deeltjes en dus ook gasdruk. Als nu de dichtheid van het gas op bepaalde plekken toeneemt, dan zal op die plekken de onderlinge zwaartekracht van het gas plaatselijk toenemen. Als de naar binnen gerichte zwaartekracht plaatselijk groter wordt dan de naar buiten gerichte gasdruk, dan zal de gaswolk ter plekke gaan instorten, en kan er zich een gasbol vormen. Als na het instorten van zo'n gaswolk een "stabiele" ster ontstaan is, zullen de temperatuur en de druk in het centrum van de gasbol opgelopen zijn tot extreem hoge waarden. Hierdoor kan er kernfusie optreden in het binnenste van de ster. Bij dit proces worden lichte deeltjes zoals "waterstofatomen" omgezet in zwaardere deeltjes zoals helium, koolstof en zuurstof. Dit zijn de deeltjes waaruit ook wij, de mens, zijn opgebouwd! Bij het proces van kernfusie komt energie vrij die vervolgens naar buiten vervoerd wordt en de energie wordt uitgezonden in de vorm van licht. Dit is de reden waarom we sterren kunnen zien! Nu een ster is geboren en kernfusie processen ondergaat, zal hij voortdurend van structuur ofwel van opbouw veranderen. Tegenwoordig kunnen sterrenkundigen met computers tot soms grote nauwkeurigheid voorspellingen doen over de precieze levensloop van sterren zoals de Zon. De totale levensduur van sterren loopt uiteen van enkele miljoenen tot enkele miljarden jaren. Aan het einde van haar leven zal de brandstof in het centrum van de ster uitgeput raken, dit betekent dat de ster uiteindelijk, net als de mens, zal "sterven". Bij zware sterren zal dit sterven op spectaculaire wijze plaatsvinden in een enorme explosie, deze gigantische knallen worden supernova's genoemd, en na de knal zal er een compact eindproduct overblijven. Voor de experts onder ons: neutronensterren of zwarte gaten.

Wat zijn sterwinden?
Uit het voorafgaande zou men kunnen denken dat de balans van krachten zorgt voor een volledig stabiele ster. Dit is echter niet helemaal waar. Aan de rand van de ster is er namelijk iets bijzonders aan de hand. Je kunt je vast wel voorstellen dat de druk van het gas aan de rand van de ster veel hoger is dan net buiten de ster, waar nauwelijks gasdeeltjes aanwezig zijn. Door dit drukverschil is er dus voortdurend sprake van een "uit evenwicht" situatie. Met andere woorden de naar buitentoe gerichte gasdruk kan de zwaartekracht op de rand van de ster "overwinnen en de buitenlagen van de ster wegblazen. Dit verschijnsel noemt men een "sterrenwind". Ook de Zon heeft zo'n wind: de zonnewind. We merken het bestaan van de zonnewind bijvoorbeeld door het verschijnsel op Aarde dat "Aurora Borealis" oftewel "het Noorderlicht" heet. Dit verschijnsel treedt op wanneer de Zon zeer actief is in het uitstoten van materie.

Stralingsgedreven sterwinden
Tot nu toe hebben we gesproken over gasdeeltjes en gasdruk, maar ook licht kan een "druk" uitoefenen. In de moderne natuurkunde kan licht worden opgevat als een verzameling "lichtdeeltjes" (in natuurkundig vakjargon "fotonen" genoemd) die net als de gewone gasdeeltjes kunnen botsen en zo hun druk over kunnen brengen op materie. Dit betekent dat ook het mechanisme van lichtdruk kan zorgen voor het verlies van materie aan de rand van een ster. Dit noemen we dan "stralingsgedreven"winden. Het mechanisme van stralingsdruk blijkt voor de Zon niet zo belangrijk te zijn. Feitelijk is de Zon hier niet helder genoeg voor. Toch heeft de Zon wel een wind, maar die wordt veroorzaakt 170


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door de gewone "gasdruk". Hoewel de Zon gedurende z'n gehele levensloop materie verliest, blijkt de hoeveelheid materie dusdanig klein te zijn dat dit feitelijk verwaarloosbaar is voor de structuur en de levensloop van de Zon zelf. Volledig anders is deze situatie voor zware sterren. Hoewel massieve sterren "slechts" honderd keer zo zwaar zijn als de Zon, hebben ze wel een miljoen keer zo grote helderheid. Dit enorme reservoir aan lichtdeeltjes kan bij zware sterren zorgen voor een zeer efficiente stral¨ ingsdruk, die de zwaartekracht gemakkelijk kan overwinnen en de buitenlagen van de ster wegblaast. Deze sterrenwind voor zware sterren is dusdanig groot dat de ster gedurende haar gehele leven wel de helft van haar massa kwijtraakt. Je kunt je wel voorstellen dat dit gigantische consequenties heeft voor de interne structuur en levensloop van deze sterren! Bovendien heeft het enorme massaverlies een grote invloed op haar omgeving en de rest van het heelal. Per jaar wordt er namelijk een hoeveelheid materie uitgestoten, die net zo groot is als de massa van de gehele aarde! Het is belangrijk om te beseffen dat de deeltjes die in het verleden in winden van zware sterren uitgezonden werden, namelijk deeltjes van stikstof en zuurstof, ook de samenstelling van de aardse dampkring vormden. Zonder deze sterwinden zou leven op aarde dan ook niet mogelijk zijn!

Het probleem voor dit proefschrift
De ontwikkeling van de stralingsgedreven windtheorie voor zware sterren is niet nieuw. Belangrijke artikelen op dit gebied bestaan feitelijk al vanaf het begin van de jaren '70. In de laatste decennia is de stralingsgedreven windtheorie echter vergeleken met steeds nauwkeuriger waarnemingen, maar de waarnemingen bleken niet te kloppen met de theorie. De hoeveelheid materie die met de stralingsgedreven windtheorie werd voorspeld bleek te laag te zijn in vergelijking met de waarnemingen. Dit probleem is in dit proefschrift onderzocht.

Het resultaat van dit proefschrift
In dit proefschrift zijn intensieve computerberekeningen gedaan om de hoeveelheden materie verlies van zware sterren met behulp van stralingsgedreven windmodellen nauwkeurig te voorspellen. Dat er voor dit soort berekeningen computers nodig zijn, komt doordat er veel verschillen soorten deeltjes zijn en dat deze deeltjes in eigenschappen flink van elkaar verschillen. Daar komt nog eens bij dat de berekening van de straling niet zo maar eventjes in wiskundige formules op papier geschreven kan worden. Essentieel voor de berekeningen van de stralingsdruk in dit proefschrift is dat er nu netjes rekening gehouden is met het feit dat in werkelijkheid de lichtdeeltjes aan de sterrand meerdere malen hun stralingsdruk op de materie kunnen overbrengen en niet slechts e´ keer. Het rekening houden met deze meervoudige botsingen is een ´ en boekhoudkundig zware klus, zodat computerberekeningen onvermijdelijk zijn. De berekingen hebben laten zien dat door netjes rekening te houden met de meervoudige botsingen van de lichtdeeltjes met de gasdeeltjes, de hoeveelheden uitgestoten materie nauwkeurig voorspeld kunnen worden. Het is gelukt om over een groot bereik van sterparameters voorspellingen te doen die nu wel kloppen met de waarnemingen! Nu we in staat zijn om precies uit te rekenen hoeveel materie massieve sterren verliezen, kunnen we beter voorspellen hoe de levensloop van zware sterren eruit zal zien. Bovendien kunnen we nu berekenen met hoeveel materie de sterwinden het universum verrijken, waaruit 171


Samenvatting
weer nieuwe sterren en planeten gevormd kunnen worden. Het moge duidelijk zijn dat er nog veel interessant sterrenkundig onderzoek in het verschiet ligt!

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Publication List

Publication List
Papers in refereed journals:

1. Hot stellar population synthesis from the UV spectrum: the globular cluster M79 (NGC 1904), Vink, J.S., Heap, S.R., Sweigart, A., Hubeny, I., Lanz, T. 1999, Astronomy & Astrophysics, 345, 109 2. On the nature of the bi-stability jump in the winds of early-type stars, Vink, J.S., de Koter, A., Lamers, H.J.G.L.M. 1999, Astronomy & Astrophysics, 350, 181 3. The radiation driven winds of rotating B[e] supergiants Pelupessy, I., Lamers, H.J.G.L.M., Vink, J.S. 2000, Astronomy & Astrophysics, 359, 695 4. New theoretical mass-loss rates of O and B stars Vink, J.S., de Koter, A., Lamers, H.J.G.L.M. 2000, Astronomy & Astrophysics, accepted

Submitted:

5. Mass-loss predictions for O and B stars as a function of metallicity Vink, J.S., de Koter, A., Lamers, H.J.G.L.M. 2000, Astronomy & Astrophysics, submitted

Conference proceedings:

6. Spectroscopic Dating of Very Massive Stars, de Koter, A., Vink, J.S., Lamers, H.J.G.L.M. 1999. In: Spectrophotometric Dating of Stars and Galaxies, ed. I. Hubeny, S.R. Heap, R. Cornett, ASP Conf. Ser., Vol. 192, 32­40

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7. Disks formed by Rotation Induced Bi-stability, Lamers, H.J.G.L.M., Vink, J.S., de Koter, A., Cassinelli, J.P. 1999. In: Variable and non-spherical stellar winds in luminous hot stars, ed. Wolf, B., Fullerton, A., Stahl, O., IAU colloquium 169, Lecture Notes in Physics, 159­166 8. The bi-stability jump of radiation driven winds Vink, J.S., de Koter, A., Lamers, H.J.G.L.M. 2000. In: Thermal and Ionization Aspects of Flows from Hot Stars: Observations and Theory, ASP Conf. Ser., Vol. 204, 427­433 9. The ionization of hot star winds de Koter, A., Vink, J.S., Lamers, H.J.G.L.M. 2000. In: Thermal and Ionization Aspects of Flows from Hot Stars: Observations and Theory, ASP Conf. Ser., Vol. 204, 135­149 10. The dependence of mass loss on the stellar parameters Lamers, H.J.G.L.M., Nugis, T., Vink, J.S., de Koter, A. 2000. In: Thermal and Ionization Aspects of Flows from Hot Stars: Observations and Theory, ASP Conf. Ser., Vol. 204, 395­412

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Curriculum vitÔ

Curriculum vitÔ
Op 27 januari 1973 ben ik geboren te Goirle, maar ik groeide op in Driebergen. Van 1985 tot 1991 volgde ik onderwijs aan het Revius Lyceum te Doorn, waar ik eindexamen deed in de acht vakken: Nederlands, Engels, Duits, economie 1, geschiedenis, scheikunde, wiskunde B en natuurkunde. Hieropvolgend begon ik in 1991 aan de studie natuurkunde in Utrecht. Ik studeerde in 1996 af in de richting Algemene Sterrenkunde. Het laatste jaar van mijn studie bestudeerde ik sterpopulaties in bolvormige sterrenhopen op het Goddard Space Flight Center van de NASA in de buurt van Washington DC. Dit onderzoek stond onder leiding van Dr. Sally Heap en Dr. Allen Sweigert en vanuit Utrecht was Prof. Henny Lamers mijn officiele ¨ begeleider. Op 16 oktober 1996 begon ik als onderzoeker in opleiding (NWO) aan de Universiteit Utrecht onder leiding van Henny Lamers met als onderwerp stralingsgedreven winden van massieve sterren. Het begin was goed met een bezoek aan een workshop op Hawaii. Verder bezocht ik congressen in Heidelberg (1998), Estland (1999) en verschillende Nederlandse Astronomen conferenties. Tevens bezocht ik zomerscholen over steratmosferen (in Brussel 1996) en sterrenkundig waarnemen (OHP, Frankrijk 1998). Naast het doen van onderzoek was ik onderwijs assistent voor de eerstejaars natuurkunde vakken: relativiteitsleer, klassieke mechanica, elektriciteitsleer en kinetische gastheorie. Na mijn promotie, ga ik 2,5 jaar onderzoek doen aan de vorming van sterren op het Imperial College of Science, Technology and Medicine in Londen.

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Dankwoord

Dankwoord
Gelukkig leef ik niet alleen op deze wereld, want dat zou vreselijk saai zijn. Feitelijk heeft een ieder die mij goed kent op zijn of haar eigen wijze een essentiele bijdrage geleverd aan dit ¨ proefschrift. Ik ben jullie dan ook eeuwig dankbaar. Tevens zijn er een aantal mensen die ik juist hier specifiek wil noemen. Allereerst wil ik mijn begeleiders Henny en Alex bedanken: niet alleen heb ik enorm veel van jullie beiden geleerd, ik vond de samenwerking bovendien buitengewoon plezierig. Speciale dank gaat ook uit naar Rubina voor het kritisch doorlezen van mijn manuscript. Also to the members of the reading committee (leescommissie): thanks for your comments! Vervolgens wil ik iedereen uit de M groep bedanken, met name Jeroen en Willem-Jan: heren, bedankt voor de hulp en wetenschappelijke discussies. Inti, dank voor de samenwerking; Thierry thank you for your help with the Fortran program; en alle andere M -ers bedankt voor de gezellige vergaderingen en lekkere taarten. Trudi, dank voor het ijs! Tenslotte wil ik iedereen op het SRON en het Sterrenkundig instituut, zowel in Utrecht als in Amsterdam, vriendelijk danken voor alle interactie die tussen jullie en mij heeft plaatsgevonden in de afgelopen jaren. Marco wil ik bedanken voor zijn hulp in de laatste weken; Ed en Sake voor hun hulp inzake Computer vraagstukken en Eric, Ferdi, Mandy, Marielle, Marten, ¨ Maureen, Pui-Kei, Robert en alle anderen (!!) voor de gezellige drink,- tafeltennis,- en maffia aangelegenheden.

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