Download Radiation-driven wind models of massive stars

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:
Prof. Dr. H.J.G.L.M. Lamers
Sterrenkundig Instituut, Universiteit Utrecht
Co-Promotor:
Dr. A. de Koter
Sterrenkundig Instituut, Universiteit van Amsterdam
Dit proefschrift werd mede mogelijk gemaakt door financiële steun van de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
Contents
Contents
1 Introduction
1.1 Massive stars in the cosmos . . . . . . . . . . . . . . . .
1.2 The winds from massive stars . . . . . . . . . . . . . . .
1.3 The status of the radiation-driven wind theory . . . . . .
1.3.1 Achievements of the radiation-driven wind theory
1.3.2 Open issues in radiation-driven wind theory . . .
1.4 Studies in this thesis . . . . . . . . . . . . . . . . . . . .
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 Ṁ and V∞ ? . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The theory of Ṁ determination . . . . . . . . . . . . . . . . . . . . . .
3.2.2 A simple numerical experiment: the sensitivity of Ṁ on the subsonic gL
3.2.3 The effect of an increased Ṁ on V∞ . . . . . . . . . . . . . . . . . . .
3.3 The method to predict Ṁ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 Ṁ . . . . . . . . . . . . . . . . . . . . . . . . .
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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 Ṁ 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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Contents
3.4
3.5
3.6
3.7
3.8
4
5
The model atmospheres . . . . . . . . . . . . . . . . . . . .
The predicted bi-stability jump . . . . . . . . . . . . . . . .
3.5.1 The predicted bi-stability jump in Ṁ . . . . . . . . .
3.5.2 The predicted bi-stability jump in η . . . . . . . . .
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 Ṁ . . . . . . . . . . . . . . . .
3.6.4 The effect of V∞ . . . . . . . . . . . . . . . . . . . .
3.6.5 A self-consistent solution of the momentum equation
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. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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New theoretical mass-loss rates of O and B stars
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Method to calculate Ṁ . . . . . . . . . . . . . . . . . . . . . . . .
4.3 The predicted mass-loss rates . . . . . . . . . . . . . . . . . . . . .
4.3.1 Ṁ for supergiants in Range 1 (30 000 ≤ Teff ≤ 50 000 K) . .
4.3.2 Ṁ at the bi-stability jump around 25 000 K . . . . . . . . .
4.3.3 Ṁ for supergiants in Range 2 (12 500 ≤ Teff ≤ 22 500 K) . .
4.3.4 Ṁ at the second bi-stability jump around 12 500 K . . . . .
4.4 The wind momentum . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 The wind efficiency number η . . . . . . . . . . . . . . . .
4.4.2 The importance of multiple scattering . . . . . . . . . . . .
4.4.3 The Modified Wind Momentum Π . . . . . . . . . . . . . .
4.5 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 Ṁ on the steepness of the velocity law β
4.6 Comparison between theoretical and observational Ṁ . . . . . . . .
4.6.1 Ṁ 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) . . . . . . . . . . . . . . . . . .
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Summary & Conclusions . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mass-loss predictions for O and B stars as a function of metallicity
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Theoretical context . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Method to calculate Ṁ . . . . . . . . . . . . . . . . . . . . . .
5.4 The assumptions of the model grid . . . . . . . . . . . . . . . .
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Contents
5.5
The predicted mass-loss rates and bi-stability jumps . . . . . . . . . . . . . . . 101
5.5.1 The bi-stability jump at Teff ' 25 000 K . . . . . . . . . . . . . . . . . 104
5.5.2 Additional bi-stability jumps around 15 000 and 35 000 K . . . . . . . 107
5.5.3 The origin of the (low Z) jump at Teff ' 35 000 K . . . . . . . . . . . . 108
5.6 The relative importance of Fe and CNO elements in the line acceleration at low Z108
5.6.1 The character of the line driving at different Z . . . . . . . . . . . . . . 108
5.6.2 Observed abundance variations at different Z . . . . . . . . . . . . . . 110
5.7 The global metallicity dependence . . . . . . . . . . . . . . . . . . . . . . . . 111
5.8 Complete mass-loss recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.9 Comparison between theoretical Ṁ and observations at subsolar Z . . . . . . . 115
5.10 Summary & Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Research note on the bi-stability jump in the winds of hot stars at low metallicity
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 K . . . . . . . . .
5.11.5 Summary & Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 The radiation driven winds of rotating B[e] supergiants
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Theoretical context . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 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 . . . . . . . . . . . . . . . . . . . . .
6.4 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 . . . . . . . . . . . . . . . . . .
6.5 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? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Rotationally induced bi-stability models . . . . . . . . . . . . . . . . .
6.7 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Radiation-driven wind models for Luminous Blue Variables
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 The method to calculate Ṁ . . . . . . . . . . . . . . . . .
7.3 The assumptions of the LBV-like models . . . . . . . . . .
7.4 The predicted mass-loss rates of LBVs . . . . . . . . . . .
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Contents
7.5
7.6
8
7.4.1 The effect of the lower masses on Ṁ . . . . . . . . . .
7.4.2 The effect of helium enrichment on mass loss . . . . .
7.4.3 The effect of the nitrogen enrichment on Ṁ . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Summary and Prospects
165
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.2 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Samenvatting
Wat zijn massieve sterren? . . . . .
De rol van zwaartekracht en gasdruk
Wat zijn sterwinden? . . . . . . . .
Stralingsgedreven sterwinden . . . .
Het probleem voor dit proefschrift .
Het resultaat van dit proefschrift . .
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Publication List
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Curriculum vitæ
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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, τevol ∼ 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. Observations over the past decades have shown that these basic wind parameters, Ṁ 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 η that substantially exceed the “singlescattering limit”. In this limit, η = 1, every stellar photon transfers its momentum just once to
the wind material. For WR stars values as high as η = 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 ζ 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
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Introduction
Madau P., 2000, in press, astro-ph/0003096
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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
σe L∗
σe F
=
(2.1)
c
4πr2 c
where F is the radiative flux, L∗ is the total stellar luminosity, c is the speed of light and
σe ([σe ] = 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,
gcont =
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 Γe , both independent of radius. The
Eddington factor is given by
gNewton =
Γe =
gcont
gNewton
L∗
M∗ −1
L∗ σe
−5
=
= 7.66 10 σe
4πcGM∗
L
M
(2.3)
where the constants have their usual meaning. This means that in case the H/He ionization
content remains constant, Γe 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 ν = ν0 is
optically thin, the line cannot absorb all the flux that is emitted from the core at some particular
frequency ν0 . 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
gthin =
Fν ◦
c
Z
L∗
κν φ(ν)dν =
4πr2 c
line
Z
line
κν φ(ν)dν
(2.4)
where κν ([κν ] = cm2 g−1 ) is the line absorption coefficient, which is determined by atomic
physics and can be represented by
κν ρ =
πe2
me c2
f ni
(2.5)
where (πe2 /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. φ(ν) is the profile function for absorption and is normalized to
Z ∞
φ(ν)dν = 1
(2.6)
−∞
This function φ(ν) 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 δ-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 ν0 )
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 ν = ν0 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 ∆ν of the frequency interval is then given by
ν0 dV
∆ν =
∆r
(2.8)
c dr
and as in the Sobolev approximation the width of the line absorption coefficient is assumed to
be small, φ(ν) can be represented by a delta-function. The line acceleration of an optically thick
line in the Sobolev approximation may be represented by
Fν◦ ν0 dV
L∗ ν0 dV
gthick =
=
(2.9)
c c dr
4πr2 c c dr
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
Fν◦ ν0 dV gline =
(2.10)
1 − e−τS (µ=1)
c c dr
where τS represents the Sobolev optical depth and where µ = cos θ. 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
dr
c
τS (µ = 1) = κν
(2.11)
ν0 dV
Note that at large optical depths, or τS 1, Eq. (2.10) reduces to Eq. (2.9) and that for small
optical depths, e−τS ' 1 − τS , 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.
gtot
line =
∑
lines
Fν◦ ν0 dV c
c dr
1 − e−τ(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) =
gref
elec
L∗
σref
M(t) = e 2 M(t)
4πr c
20
(2.13)
The Physics of the line acceleration
ref
where σref
e is a reference value for the electron scattering opacity. CAK used a value of σe =
0.325 cm2 g−1 .
The force multiplier M(t) can be parameterized in the following ways (CAK, Abbott 1982):
M(t) = K t −α = k t −α
n δ
e
(2.14)
W
where ne is the electron density and W is the geometrical dilution factor, which is given by


s
2
R∗ 
1 
W (r) =
(2.15)
1 − (1 −
2
r
The parameters k (or K), α and δ are the so-called force multiplier parameters. The first one, k
(or K) is a measure for the number of lines. The second one, α is a constant which describes
the distribution of strong to weak lines. If only strong (weak) lines contribute to the line acceleration, then α = 1 (0). t is the optical depth parameter and is given by:
t = σeVth ρ(dr/dV )
(2.16)
where Vth is the mean thermal velocity of the protons. Finally, the parameter δ represents a
value for the ionization in the wind. However, it is also possible to simply include the factor
(ne /W )δ 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 d p
=− 2 −
+ grad
dr
r
ρ dr
(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
Ṁ = 4πr2 ρ(r) V (r)
(2.18)
Together with an expression for the gas pressure p = R ρT /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
2
2a
dV
GMeff
a2
V
=
− 2 + gL
1− 2
(2.19)
dr
r
r
V
/
for an isothermal wind, where a is the isothermal sound speed. The mass is expressed as
the effective mass Meff = M∗ (1 − Γe ), 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
dV
GMeff 1 d p σe L∗
V
=− 2 −
+
K
dr
r
ρ dr 4πr2 c
dr
σe Vth ρ
dV
−α
(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
σe L∗ K
rV
= −GMeff +
dr
4πc
2
σe Vth Ṁ dr
4π r2V dV
−α
(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
σe L∗ K
C =
4πc
σe Vth
Ṁ
4π
−α
(2.23)
in which the mass-loss rate Ṁ is included. The equation of motion can now be written in a more
convenient form
C Dα − D − GMeff = 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 Dα − D − GMeff ) = αCDα−1 − 1 = 0
dD
This results in the key condition for the critical point in CAK theory:
Ccrit =
1 1−α
D
α
22
(2.25)
(2.26)
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:
α
GMeff
(2.27)
1−α
Rewriting Eq. (2.27), using the definition of D from Eq. (2.22), gives
α
GMeff
V dV =
dr
(2.28)
1−α
r2
In case the force multiplier parameter α is assumed to be constant over the entire wind regime,
the term α/(1 − α) can be taken outside the integral:
Z ∞
Z V∞
α
GMeff
V dV =
dr
(2.29)
1−α
r2
0
R∗
Additionally, we define the effective escape velocity as
r
2GMeff
Vesc =
(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 α.
r
V∞
α
=
(2.31)
Vesc
1−α
The velocity as a function of radius can be represented by a β law
D =
V (r) = V∞
R∗
1−
r
β
(2.32)
with β equal to 1/2.
The mass-loss rate Ṁ 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
Ṁ
=
1−α
σeVth σe 1/α 1 − α α
(αK)1/α
4π
4π
α
1/α
L∗
(GM∗ (1 − Γe ))(α−1)/α
c
(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 selfconsistent solution of the equation of motion, where Ṁ 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 α, 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 ∆λ < λ0V∞ /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 ṀV∞ /(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 Ṁ 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 (ISA - WIND) for hot stars with extended
atmospheres (de Koter et al. 1993, 1997). The Monte Carlo simulations have been performed
with MC - WIND (de Koter et al. 1997), that was tailored for ISA - WIND.
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 ISA - WIND 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 ISA - WIND 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 β-type velocity law. The parameter that
describes the steepness of the velocity law, β, is an input parameter in ISA - WIND 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 ISA - WIND, 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 competing
processes of photoionization and recombination. Abbott & Lucy (1985) defined the ionization
balance between the radiative recombinations to all levels and photoionizations from the ground
state of the ion under consideration using a modified nebular approximation. The ionization
fractions for the stages j and j + 1 are for any particular element, given by
N j+1 ne
=ζW
Nj
Te
TR
1/2 N j+1 ne
Nj
LTE
(2.34)
TR
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 ζ 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 0 that includes the
properties of the diffuse radiation field. In our calculations W 0 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 ζ is replaced by the more general factor {(1 − ζ)W 0 + ζ}. 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
W0 = 1 −
1/2 N j+1 ne
N j+1 ne LTE
0
0 Te
= {(1 − ζ)W + ζ}W
Nj
TR
Nj
TR
(2.36)
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 ISA - WIND. 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 ISA - WIND. We note that the non-explicit elements are not taken into
account in the the calculation of the radiation field of the ISA - WIND 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
nu
n1
LTE
.
(2.37)
TR
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 WIND (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 - WIND 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 × 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 ISA - WIND 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−τx
(2.38)
i.e. it depends on the optical depth τx , along the flight path of the photon, of the process
under consideration. The combined optical depth τtot is given by the summation of the line and
continuum optical depths
τtot = τc + τL
(2.39)
This combined optical depth is compared with a randomly chosen optical depth, τran , to decide
whether or not an event can take place. This random optical depth in MC is represented by
τran = − 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 τran = − 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
τ tot
τran
Distance
Sb
Case B
Line event
τ
tot
τ
ran
Distance
Sb
Figure 2.2: The optical depth summation procedure. The continuum optical depth τc grows
linearly with distance, while the optical depth due to lines shows jumps at fixed locations along
the photon’s path. These locations are determined by the change in the wind velocity vector
along the path of the photon. The amplitude of the jumps reflects the Sobolev optical depth at
this location and in this direction.
29
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 τran 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 κc . The total continuum opacity is the sum of the
electron scattering and thermal continuum opacities and is given by
κc = σe ne + κtherm
(2.41)
where κc ([κc] = 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 Å 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 τ 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 ν0
ν0in = ν0 = ν0out
(2.42)
where ν0in and ν0out 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
ν
in
µin
STAR
v1
ν
µ
STAR
out
out
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:
ν0in
µinV
= νin 1 −
c
(2.43)
and
ν0out
= νout
µoutV
1−
c
(2.44)
where νin and νout 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 νin and νout 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
νin
µin V
1−
c
= νout
µout V
1−
c
(2.45)
Because the energy and momentum of a photon are respectively E = hν and p = hν/c, the
equation can be rewritten in the following way:
Ein − 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 µin − pout µout =
hνin
hνout
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:
µinV
νin = ν0 1 +
(2.48)
c
and
µoutV
νout = ν0 1 +
(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
mV1 +
∆V
=
=
V2 −V1
hν0
hν0
µinV
µoutV
1+
µin −
1+
µout
mc
c
mc
c
(2.50)
As V c, Eq. (2.50) becomes
hν0
(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
∆V = V2 −V1 =
∆p = m(V2 −V1 )
=
=
32
hν0
(µin − µout )
c
hνin − hνout
∆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
(2.53)
m(V22 − V12 ) = hνin − hνout
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)
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 through interactions in that shell. Using
the relationship between ∆m(r) and ∆r for thin concentric shells, ∆m(r) = 4πr2 ρ(r)∆r, and the
derived relation between momentum and energy transfer of the photons (Eq. 2.52), grad (r) can
be rewritten as
grad (r) =
1
∑ ∆E(r)
4πr2 ρ(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
grad (r) = −
1 ∆L(r)
Ṁ ∆r
(2.56)
2.5 The determination of Ṁ 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
Ṁ (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 β-type velocity law with β = 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 β.
3. For each set of stellar and wind parameters,
inp
2.4.1) are calculated for several Ṁ values.
ISA - WIND
model atmospheres (see Sect.
4. Then, for each of these models, the radiative acceleration is calculated with the Monte
Carlo code MC - WIND 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
Ṁ (V∞ 2 +Vesc 2 ) = ∆L =
2
∑
∆Lshell
(2.58)
shells
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, Ṁ 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 Ṁ inp = Ṁ 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 Ṁ = −6.92.
Ṁ
out
=
2 ∆L
V∞ +Vesc 2
2
(2.59)
To derive consistent values for the mass-loss rate, Ṁ is determined using the condition
Ṁ = Ṁ inp = Ṁ out
(2.60)
Only one value of Ṁ 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 parameter. This implies that although the Ṁ 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 α from the radiative accelerations computed with
MC - WIND 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 αMC , 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 MC t −α
MC
36
(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 Ṁ can then be found by iterating the solutions of the CAK
equations (Eqs. 2.31 and 2.33) for both V∞ and Ṁ 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), α 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 α 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:
s
V∞
αMC
=
(2.62)
Vesc
1 − αMC
Note that the values for αMC can be significantly higher than the “classical” α 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 αMC -parameter itself. For instance, if αMC ' 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 αMC 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 ISA - WIND
and MC - WIND 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 αMC . 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
.
inp
M
inp
Vinf
MC
M (t)
Self−Consistent
Solution
MC
K
α MC
.
out
M
out
Vinf
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 supergiants, with Ṁ increasing by about a factor five from Teff ' 27 500 to 22 500
K for constant luminosity. The wind efficiency number η = ṀV∞ /(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 Ṁ 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 Ṁ, 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 Ṁ 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
Ṁ ∝ 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-stability jump. If
the wind momentum ṀV∞ 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 Ṁ, 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 Ṁ.
The models of Pauldrach & Puls (1990) for P Cygni show that the wind momentum loss per
second, ṀV∞ , 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 ṀV∞ 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 Ṁ and V∞ ?”. We show
42
On the nature of the bi-stability jump in the winds of early-type supergiants
that Ṁ 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 Ṁ. 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 Ṁ 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 Ṁ and V∞ ?
3.2.1 The theory of Ṁ 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
dV
GM∗ 1 d p
=− 2 −
+ grad (r)
dr
r
ρ dr
(3.1)
where grad is the radiative acceleration. Together with the mass continuity equation
Ṁ = 4πr2 ρ(r)V (r)
(3.2)
and the expression for the gas pressure p = R ρT /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
dV
=
V
dr
2a2 GMeff
− 2 + gL
r
r
a2
/ 1− 2
V
(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 −Γe ) 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
s is not the same as the CAK critical point. The CAK critical point is located much further
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 ρ0 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 (τ)-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 τ = 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∗ , ρ0 (given by τ(r0 ) = 2/3) and V0 at the lower boundary. This fixes the value
of Ṁ via the mass continuity equation (Eq. 3.2). Note that Ṁ 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
Ṁ. 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
lower boundary should be higher and hence the mass-loss rate, Ṁ = 4πr02 ρ0V0 , 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 Ṁ 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 ρ 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 Ṁ on the subsonic
gL
A simple numerical experiment serves to demonstrate the dependence of Ṁ on the radiative
acceleration in the subsonic region. We start with an isothermal model of the wind from a star
of Meff = 20M , R∗ = 16.92R, Teff = 25 000 K, Twind = 0.8Teff = 20 000 K. We then specify
the line acceleration gL (r) in such a way that it produces a stellar wind with a mass-loss rate of
1.86 10−7 M yr−1 and with a β-type wind velocity law
44
On the nature of the bi-stability jump in the winds of early-type supergiants
Figure 3.1: Extra “bumps” on the radiative acceleration gL (r) below the sonic point. The solid
line is gL (r) of the model without a “bump”. The dotted lines show gL (r) with the adopted
bumps with peakheights of 150, 300 and 500 cm s−2 . The cross indicates the sonic point at
1.0135 R∗ .
V (r) = V∞ (1 −
R∗ β
)
r
(3.4)
where β = 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 ρ = 10−10
g cm−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 Ṁ 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
bump
gL
peak
(r) = gL
( )
z − zp 2
exp −
∆z
(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 Ṁ (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, Ṁ increases.
3.2.3 The effect of an increased Ṁ on V∞
Once Ṁ 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
∞
rc
gL (r) dr =
1
1
Vesc 2 + V∞ 2
2
2
(3.6)
so
V∞ ' 2
2
Z
∞
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 Ṁ 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 Ṁ 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).
σe L∗ −α
kt
(3.8)
4πr2 c
where k and α are constants and ge is a reference value describing the acceleration due to
σe L∗
electron scattering. It is given by ge = 4πr
2 c . The optical depth parameter is
gL (r) = ge M(t) =
t = σeVth ρ(dr/dV )
(3.9)
where Vth is the mean thermal velocity of the protons. Let us define ginit
L (r) as the radiative
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 Ṁ. From Eqs. (3.8) and (3.9) with Eq. (3.2) we find that
2
α ( init )α
Ṁ
r
V
dV
/dr
gL (r) = ginit
(3.10)
L
2
init
(r V dV /dr)
Ṁ
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 Ṁ results in the
following expression
V
dV
V
'
dr
( init )α/(α−1)
dV init Ṁ
V
dr
Ṁ
(3.12)
So the ratio between the terminal velocities of the models with and without the increased massloss rate is
(
V∞
'
V∞init
init
Ṁ
Ṁ
)α/(2−2α)
'
Ṁ
Ṁ
init
−3/4
(3.13)
where we adopted α = 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 Ṁ
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 Ṁ 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 Ṁ 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 Ṁ 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 Ṁ
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
ISA - WIND (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
1 dV
|
| |
|
f dr
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 ν0 in the CMF.
48
On the nature of the bi-stability jump in the winds of early-type supergiants
ν0in = ν0 = ν0out
(3.15)
where ν0in and ν0out 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:
ν0in = νin (1 −
µinV
)
c
(3.16)
and
µoutV
)
(3.17)
c
where νin and νout 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 νin and νout 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).
ν0out = νout (1 −
µin V
µout V
) = νout ( 1 −
)
(3.18)
c
c
Because the energy and momentum of a photon are E = hν and p = hν/c, the equation can be
rewritten in the following way:
νin ( 1 −
Ein − Eout
(3.19)
V
Eq. (3.19) links the change in radial momentum 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 µin − pout µout =
hνin
hνout
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 +
νin = ν0 (1 +
µinV
)
c
(3.21)
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
νout = ν0 (1 +
49
Chapter 3
∆V
=
=
V2 −V1
µinV
µoutV
hν0
hν0
(1 +
)µin −
(1 +
)µout
mc
c
mc
c
(3.23)
Since V c, Eq. (3.23) becomes
hν0
(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:
∆V = V2 −V1 =
∆p = m(V2 −V1 )
=
=
hν0
(µin − µout )
c
∆E
hνin − hνout
=
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
m(V22 − V12 ) = hνin − hνout
(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 - WIND (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 - WIND 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) = 4πr2 ρ(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) =
∑ ∆E(r)
(3.28)
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) =
1
4πr2 ρ(r)∆r
1 ∆L(r)
(3.29)
Ṁ ∆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 Å are included in the calculations with ionization stages up to stage VI.
Typically about 2 105 photon packets, distributed over the spectrum at the lower boundary of
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 Ṁ
We predict the mass-loss rates for a grid of model atmospheres to study the behaviour of Ṁ
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 Ṁ
(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
r
2GMeff
Vesc =
(3.30)
R∗
A β-type velocity law with β = 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 ISA WIND (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 MC - WIND program that
uses the Monte Carlo method described above.
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)
Ṁ (V∞ +Vesc ) = ∆L = Ṁ
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 Ṁ 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 Ṁ 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 β 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 Ṁ!
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 Ṁ 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 ISA - WIND 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.
ISA - WIND 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 β-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
= {(1 − ζ)W + ζ}W
Nj
Te
TR
1/2 N j+1 ne
Nj
LTE
(3.33)
TR
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 ζ, 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 ISA - WIND 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
nu
n1
LTE
.
(3.34)
TR
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 Ṁ for wind models with a β-type velocity law with β = 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 Γe = 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 Ṁ
The results are listed in Table 3.1. This Table gives the values of Teff , R∗ , Vesc and Ṁ for each
temperature and for the three values of V∞ /Vesc . We also give the value of the wind efficiency
factor η, which describes the fraction of the momentum of the radiation that is transferred to the
ions
L∗
ṀV∞ = η
(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 η, 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 Ṁ
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 Ṁ 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 Ṁ 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 Ṁ 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
Ṁ of about a factor of five between Teff = 27 500 and 20 000 K. This is our prediction for a
bi-stability jump in Ṁ.
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 β = 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
2.0
2.6
Teff
R∗
Vesc
V∞
log Ṁ
(K)
(R ) (km s−1 ) (km s−1 ) (M /yr)
12 500 67.7
310
410 - 6.32
15 000 47.0
380
490 - 6.39
17 500 34.5
440
570 - 6.28
20 000 26.4
500
650 - 6.22
22 500 20.9
560
730 - 6.15
25 000 16.9
630
810 - 6.12
27 500 14.0
690
900 - 6.40
30 000 11.8
750
980 - 6.58
32 500 10.0
810
1060 - 6.58
35 000
8.6
880
1140 - 6.43
37 500
7.6
940
1220 - 6.37
40 000
6.6
1000
1300 - 6.26
12 500 67.7
310
630 - 6.74
15 000 47.0
380
750 - 6.62
17 500 34.5
440
880 - 6.49
20 000 26.4
500
1000 - 6.41
22 500 20.9
560
1130 - 6.32
25 000 16.9
630
1250 - 6.48
27 500 14.0
690
1380 - 6.73
30 000 11.8
750
1500 - 6.76
32 500 10.0
810
1630 - 6.71
35 000
8.6
880
1750 - 6.59
37 500
7.6
940
1880 - 6.57
40 000
6.6
1000
2000 - 6.48
12 500 67.7
310
810 - 6.95
15 000 47.0
380
980 - 6.85
17 500 34.5
440
1140 - 6.69
20 000 26.4
500
1300 - 6.54
22 500 20.9
560
1460 - 6.59
25 000 16.9
630
1630 - 6.79
27 500 14.0
690
1790 - 6.95
30 000 11.8
750
1950 - 6.92
32 500 10.0
810
2120 - 6.86
35 000
8.6
880
2280 - 6.76
37 500
7.6
940
2440 - 6.71
40 000
6.6
1000
2600 - 6.68
η
∆L/L
0.095
0.097
0.146
0.192
0.254
0.302
0.174
0.126
0.136
0.207
0.255
0.350
0.056
0.088
0.139
0.191
0.264
0.203
0.125
0.128
0.155
0.220
0.247
0.325
0.045
0.067
0.114
0.184
0.184
0.129
0.098
0.115
0.143
0.194
0.233
0.266
(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
Table 3.1: Stellar parameters of the grid of calculated models.
log (L/L ) = 5.0, M = 20M , Γe = 0.130, Meff = 17.4 M , β = 1, solar metallicity.
55
C
B
A
Chapter 3
Figure 3.3: Upper panel: The calculated mass-loss rates Ṁ 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 β = 1.0; all models are calculated for solar
metallicities. Lower panel: The predicted bi-stability jump in Ṁ 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 η
Another view at these results can be obtained by plotting the wind efficiency factor η. Figure
3.4 shows the behaviour of η 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 η = ṀV∞ /(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 η to lower temperatures, except the jump of about a factor 2 or 3 near 25 000 K.
Fig. 3.4 clearly shows that η is not a constant function of Teff . The overall picture shows that
for the three values of V∞ /Vesc , η decreases as Teff decreases. This is probably due to the fact
that the maximum of the flux distribution shifts to longer wavelengths. At λ > 1800 Å there
are significantly less lines than at λ < 1800 Å. 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,
η is almost independent of the adopted value for V∞ /Vesc . This means that the behaviour of η
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. η now increases by
a factor of 2 to 3. This means that the wind momentum loss, ṀV∞ 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, Ṁ
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 η 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 ρ as well, a smaller value of the velocity V∞ , means a larger density ρ 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 Ṁ in Fig. 3.3 and in η 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 × 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 Ṁ of 45 % in our model calculations due to the lower Teff .
The next step is to compare 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 Ṁ = −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 below 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 Ṁ, 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 ρ.
In this region of the atmosphere, where dV /dr is rapidly increasing, the effect of the decreasing
ρ 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 Ṁ
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 acceleration due to Fe as Teff decreases. This may explain the increase of Ṁ 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 %.
gnorm
L
= gL
25000
Teff
4
(3.36)
Since Teff 4 ∝ R−2
∗ for constant luminosity, this is also a scaling to the Newtonian gravity of the
models.
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 for the models
A and B. (a) Normalized gL for the different Fe ionization stages of model A. The legend
indicates the ionization stage. Some values for the velocity are indicated on the top of the
figure. (b) model A, enlarged around the sonic point. (c) Normalized gL for the different Fe
ionization stages of model B. (d) model B, enlarged around sonic point.
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
R than gL (r)
for models A and B. This is obviously due to the smaller value of V∞ . The
R integral gL (r) dr in
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 × 1016 and 1.92 × 1016 cm2 s−2 for models A and B respectively, and 6.12 × 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
R conclude that a smaller value for V∞ is indeed consistent with
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
0
V∞
Vesc
1.5
2.5
1
V∞
Vesc
1.3
2.4
2
V∞
Vesc
1.2
2.4
3
αMC
0.58
0.85
kMC
0.2065
0.0076
V∞
Vesc
1.2
2.4
4
log Ṁ CAK
log Ṁ MC
(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 , Γe = 0.130, Meff = 17.4 M , β = 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 α (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 α, i.e.
M MC (t) = kMC t −α
MC
(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 Ṁ 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 αMC and kMC . Next, the mass loss
and terminal velocity were simultaneously calculated from these αMC 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 αMC 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 α 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:
s
V∞
=
Vesc
αMC
1 − αMC
(3.38)
The value for αMC for the model of 30,000 K is significantly higher than values for α
that were calculated before (e.g. Pauldrach et al. 1986), since the finite disk is already
included in the αMC -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 Ṁ of CAK using αMC 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 αMC (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 Ṁ 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 Ṁ 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 Ṁ values of up to
10−4 M yr−1 . LBVs are characterized by typical variations in the order of ∆V of 1 to 2 magnitudes. 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 Ṁ during a typical LBV
variation by investigating the Ṁ 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 Ṁ is increasing from visual minimum to maximum, while for others it is the other way around: Ṁ is
decreasing. This “unpredictable” behaviour of Ṁ during an LBV variation is not a complete
surprise, if one considers our Ṁ 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, Ṁ decreases for a decreasing Teff ,
whereas in the interval between Teff = 30 000 − 20 000 K, Ṁ increases for a decreasing Teff .
This shows that whether one expects an increasing or decreasing Ṁ 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 η 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 Ṁ also. These combined effects explain the lack of a consistent behaviour of Ṁ 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 massloss rate Ṁ jumps due to an increase in the line acceleration of Fe III below the sonic point. This
jump in Ṁ is found in three different series of models. In all cases, the wind efficiency number
η = ṀV∞ /(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 Ṁ
of about a factor of five. So Ṁ is expected to increase by about this factor between 27 500 and
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 Ṁ 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 Ṁ 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) Ṁ 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 Ṁ this would most likely produce a systematic shift. Since we are essentially
interested in relative shifts in Ṁ, 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 Ṁ 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 Ṁ, but it might have impact on the terminal velocities.
Finally, we would like to add that our calculations for Ṁ around Teff = 25 000 K have only
been performed for one value of M∗ , L∗ and H/He abundance. Ṁ 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 Ṁ 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
1 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.
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 Ṁ 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 Ṁ for a wide range of parameters for
galactic O and B stars, taking multiple scattering into account. These predictions for Ṁ 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 Hα 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 Ṁ of a factor of about three for the wind of the O supergiant ζ 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 Ṁ 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 Ṁ
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
5.5
5.75
6.0
6.25
M∗
Γe
(M )
15
20
20
30
40
30
40
50
45
60
80
120
0.055
0.041
0.130
0.087
0.068
0.274
0.206
0.165
0.325
0.434
0.326
0.386
Meff
Teff
(M )
(kK)
14.2
19.2
17.4
27.4
37.3
21.8
31.8
41.8
30.4
34.0
53.9
73.7
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
12.5 - 50.0
V∞
Vesc
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
1.3 - 2.6
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 Ṁ 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 (ISA - WIND) 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 Ṁ, 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 - WIND
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 Ṁ 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 Γe <
∼ 0.5 (see column (4) in Table 4.1), where
Γe is the ratio between the gravitational acceleration and the radiative acceleration due to
electron scattering. Γe is given by:
L
M −1
Lσe
−5
= 7.66 10 σe
Γe =
4πcGM
L
M
(4.1)
where σe 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 Γe > 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
5.5
5.75
6.0
6.25
M∗
(M )
15
20
20
30
40
30
40
50
45
60
80
120
jump
log(∆Ṁ)
Teff
(K)
hρijump
(g cm−3 )
0.78
0.61
0.83
0.87
0.73
0.76
0.81
0.82
0.77
0.76
0.76
0.77
23 750
22 500
26 250
25 000
25 000
26 250
26 250
25 000
25 000
25 000
26 250
25 000
-14.82
-15.13
-14.22
-14.68
-14.74
-13.89
-14.13
-14.40
-13.93
-13.66
-13.89
-13.87
Table 4.2: Bi-stability jump characteristics for the 12 (L∗ , M∗ ) series of wind models.
5. We calculated Ṁ for wind models with a β-type velocity law for the accelerating part of
the wind:
R∗ β
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 β = 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 β values will be calculated and it will be demonstrated that the predicted
Ṁ is essentially insensitive to the adopted value of β (see Sect. 4.5.4).
6. The dependence of Ṁ 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 − Γe ) was used. Meff is given in column (5) of Table 4.1.
4.3.1 Ṁ 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 Ṁ (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 Ṁ 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 λ >
∼ 1800 Å than at shorter wavelength, the
line acceleration becomes less effective at lower Teff , and thus Ṁ decreases.
4.3.2 Ṁ 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 Ṁ coincide both in Teff and
in size of the Ṁ 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 Ṁ 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 Ṁ at the hot side of the jump (where V∞ /Vesc = 2.6) and the maximum Ṁ
at the cool side (where V∞ /Vesc = 1.3) in Fig. 4.2. The size of the predicted jump in Ṁ (log ∆Ṁ)
is indicated in column (4) of Table 4.2: ∆Ṁ is about a factor of five to seven. Table 4.2 tabulates
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:
hρi. For a standard velocity law with β = 1, this characteristic wind density is given by
hρi =
Ṁ
8πR2∗V∞
(4.3)
For all 12 series of models this characteristic density hρi is plotted vs. the effective temperature
of the jump. This is done for both the minimum Ṁ (at the hot side of the jump) and the maximum
Ṁ (at the cool side of the jump). Figure 4.3 shows the location of the bi-stability jump in terms
of Teff as a function of hρi. 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 Ṁ corresponds to a relatively low ρ and relatively
high Teff , whereas the maximum Ṁ corresponds to a relatively high ρ, but low Teff . Note that
75
Chapter 4
Figure 4.3: Characteristic hρi 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 hρi 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
Ṁ 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 hρi for the “jump” model for
each (L∗ , M∗ ) series. The “jump” model is a hypothetical model between the two models where
Ṁ 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 hρi is given by:
jump
Teff
= 49.1 (± 9.2) + 1.67 (± 0.64) loghρi
(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 hρi as a function of Γe .
Again this is done for both the cool and hot side of the jump, and for the average between them.
As expected, log hρi increases as Γe increases. Since the average characteristic wind density at
the jump shows an almost linear dependence on Γe , a linear fit through the average densities is
plotted. This is the solid line in Fig. 4.4. The relation between log hρi and Γe is given by:
loghρi = −14.94 (± 0.54) + 3.2 (± 2.2) Γe
(4.5)
From the quantities L∗ and M∗ it is now possible to estimate log hρi 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 hρi at the bi-stability jump as a function of Γe . 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 hρi.
fitting formulae for the two ranges in Teff at either side of the bi-stability jump (see Sect. 4.5).
4.3.3 Ṁ for supergiants in Range 2 (12 500 ≤ Teff ≤ 22 500 K)
Figure 4.2 shows that at effective temperatures Teff ≤ 22 500 K, Ṁ initially decreases. This
is similar to the Ṁ behaviour in the Teff range between 50 000 and 27 500 K. For some series
(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 Ṁ at 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
Ṁ 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 Ṁ.
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 hρi 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 hρi is
determined by eye and is roughly given by:
T jump2 = 100 + 6 loghρi
(4.6)
where T jump2 is in kK. From the quantities L∗ and M∗ it is again possible to estimate log hρi
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 η
In this section, we present values for the wind efficiency number η for the different (L∗ , M∗ )
series. η (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∗
ṀV∞ = η
(4.7)
c
Figure 4.5 shows the behaviour of η as a function of Teff for the complete grid of models.
Figure 4.5 demonstrates that η 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 η 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, η 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 η is similar to
that of Ṁ as shown in Fig. 4.2.
In some of the panels of Fig. 4.5, i.e. in those cases where L∗ /M∗ is large, η exceeds the
single scattering limit.
ṀV∞
≥1
(4.8)
L∗ /c
This occurs at Teff >
∼ 6. It suggests that already for high luminosity
∼ 40 000 K and log (L∗ /L ) >
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, η 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.
η≡
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 η 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 η <
∼ 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 η >
∼ 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 ζ 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
Γe
0.041
0.206
0.434
log L∗
M∗
(L )
(M )
4.5
5.5
6.0
20
40
60
ηMS
log Ṁ SS
log Ṁ MS
Ṁ MS
Ṁ SS
0.107
0.460
1.07
- 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 ζ Puppis, which
has a mass-loss rate of Ṁ obs = 5.9 × 10−6 M yr−1 (Puls et al. 1996), the observed efficiency
number is about η ' 0.6, suggesting that the real efficiency of multiple vs. single scattering
is a factor of about four for ζ Puppis (see Fig. 4.6). This is close to the findings of Abbott &
Lucy (1985) who found an increase in Ṁ by a factor of 3.3 for the wind of ζ 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 η <
∼ 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 ζ 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 Ṁ = 5.1 × 10−6 M yr−1 of Pauldrach et al.
(1994) with the value of Ṁ = 8.6 × 10−6 M yr−1 derived from our fitting formula based on
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 Π
Kudritzki et al. (1995) have defined the Wind momentum Luminosity Relation (WLR):
x
Π ≡ Ṁ V∞ R0.5
∗ ∝ L∗
(4.9)
where Π (or ṀV∞ R0.5
∗ ) is called the “modified wind momentum”. Observations of Ṁ and V∞
of O supergiants have shown that log Π 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 = αeff = α − δ
(4.10)
Here α and δ are force multiplier parameters, describing the radiative line acceleration gline
through the stellar wind:
gline ∝
1 dV
ρ dr
α ne δ
W
(4.11)
where ne is the electron density and W is the dilution factor. α 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 α = 1 (0).
The predicted value of α is about 0.6. The parameter δ 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, η 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 αeff 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 Ṁ 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 Ṁ for the first range. We have found that for
the dependence of Ṁ on Teff , the fit improved if a second order term (log Teff )2 was taken into
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 Ṁ =
− 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)}2
for 27 500 < Teff ≤ 50 000K
(4.12)
where Ṁ 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 Ṁ. 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 Ṁ =
− 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 Ṁ 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 Γe , are omitted from the sample, the
accuracy improves to ' 0.06 dex. In all cases the rms is <
∼ 0.08 dex in log Ṁ, 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 Ṁ 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 Ṁ
of the second jump is comparable to the size of the first jump. So, ∆Ṁ 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 ∆Ṁ =
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 Ṁ on the steepness of the velocity law β
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 β = 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 Ṁ for all of the above β
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 Ṁ on the adopted value
of β:
log Ṁ = C + 0.112 (±0.048) log(β/1.0)
(4.14)
where C is a constant. This relation is valid for the range between β = 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 β dependence
from the mass loss recipe. We have just presented the β 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 β dependence on Ṁ
could be of significance for more extreme series of models. This was tested, but it turned out
that for the high Γe series, the β dependence is also insignificant, i.e. deviations of predicted Ṁ
2 see:
www.astro.uu.nl/∼jvink/
85
Chapter 4
Figure 4.7: Dependence of Ṁ on the shape of the velocity law, for three values of β = 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 ∆Ṁ <
∼ 0.03 dex. This shows that we can safely omit the β dependence on Ṁ in the
mass-loss recipe for the O and B stars.
4.6 Comparison between theoretical and observational Ṁ
4.6.1 Ṁ 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 Hα have been considered. The Hα mass-loss rates and their stellar parameters are from:
Herrero et al. (2000); Kudritzki et al. (1999); Lamers & Leitherer (1993) (these Hα 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 Ṁ 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 Ṁ (both radio data and Hα) for
the O stars. The Puls et al. (1996) Hα rates; Hα 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 Ṁobs ' - 7.4 is
the star ζ 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
Hα 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 Hα 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 Hα 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 σ) 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 η.
Comparisons between observed and theoretical wind momenta as a function of η 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 Ṁ, the included values for
V∞ and R∗ were taken from observations. The theoretical WLR is:
Πtheory =
−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 αeff (Eq. 4.10) that follows, is:
1
= 0.548
(4.16)
x
This corresponds well to predicted values of the force multiplier parameter (α ' 0.66 and δ '
0.10, see e.g. Pauldrach et al. 1994).
αeff =
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 σ).
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:
Πtheory =
−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 αeff 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 Hα profiles
which are fully in emission (the filled symbols in the lower panel of Fig. 4.10), i.e. the profiles
that within the Hα 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 Hα
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
αeff =
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 Hα 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 Hα 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 Hα 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 Π 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 Ṁ 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 rates derived
by different authors and/or methods. One group of Ṁ determinations of B stars does
follow the theoretical relationship, while another group does not. This lack of agreement
between the observed mass-loss rates of B stars may point to systematic errors in the
observed values.
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 Ṁ ∝ Z 0.85 . This
Ṁ(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 Btype stars as a function of metal abundance: Ṁ = 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 α-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 Ṁ on Z was theoretically studied by CAK, Abbott (1982)
and Kudritzki et al. (1987). These studies have shown that the Ṁ(Z) relation is expected to
behave as a power-law:
Ṁ ∝ Z 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 Ṁ 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 Ṁ as a function of stellar
parameters with an additional Z dependence. It will be shown that over a large parameter
space, the Ṁ(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):
M(t) = k t −α
n δ
e
(5.2)
W
where ne is the electron density and W is the geometrical dilution factor. The parameters k,
α and δ are the so-called force multiplier parameters. The first one, k, is a measure for the
number of lines. The second parameter, α, describes the distribution of strong to weak lines. If
only strong (weak) lines contribute to the force, then α = 1 (0). The predicted value of α for
O-type stars is typically 0.6 (Abbott 1982, Kudritzki et al. 1989). The parameter δ describes
the ionization in the wind. Its value is usually δ ∼ 0.1. Finally, t is the optical depth parameter,
given by:
t = σeVth ρ(dr/dV )
(5.3)
where Vth is the mean thermal velocity of the protons and σe 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 1−α
97
(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
Ṁ ∝ k1/αeff
(5.5)
αeff = α − δ
(5.6)
where
This implies that Ṁ is expected to depend on metallicity in the following way:
Ṁ ∝ Z m
(5.7)
with
m =
1−α
α−δ
(5.8)
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 Ṁ 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 Ṁ on Z, however, multi-line transfer was not taken into
account in these calculations either.
We note that a pure power-law dependence of Ṁ 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 Ṁ(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 Ṁ(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 Ṁ
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 (ISA - WIND) 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 Ṁ, 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 Å 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 (∆Ṁ) <
∼ 0.05.
<
At extremely low metallicities (Z/Z ∼ 1/30) the line driving mechanism becomes very
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
Y
Z
0.758
0.752
0.733
0.68
0.52
0.242
0.246
0.260
0.30
0.42
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
Y = Yp +
Z
∆Z
(5.9)
(5.10)
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 Ṁ on Z is universal for different luminosity and
mass, it is calculated for three different values of the Eddington factor Γe . This is the
ratio between the gravitational and radiative acceleration due to electron scattering and is
given by:
L
M −1
Lσe
−5
Γe =
= 7.66 10 σe
4πcGM
L
M
(5.11)
where σe 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 Γe 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 Ṁ 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
Γe
0.130
0.206
0.434
logL∗
M∗
Teff
(Z/Z )
(L )
(M )
(kK)
Range
5.0
5.5
6.0
20
40
60
V∞
Vesc
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 Ṁ for wind models with a β-type velocity law for the accelerating
part of the wind:
R∗ β
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 β. A
value of β = 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 Ṁ (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 Ṁ decreases for decreasing Teff . The reason for this Ṁ decrease is that the maximum of
the flux distribution gradually shifts to longer wavelengths. Since there are significantly less
lines at roughly λ >
∼ 1800 Å 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
Γe
0.130
0.206
logL∗
M∗
(L )
(M )
5.0
20
5.5
40
V∞
Vesc
log Ṁ(M yr−1 )
Teff
1/100
1/30
1/10
1/3
1
3
10
(kK)
Z/Z
Z/Z
Z/Z
Z/Z
Z/Z
Z/Z
Z/Z
2.6
50
45
40
35
30
–
–
–
–
–
–
–
–
–
-7.98
-7.48
-7.56
-7.68
-7.56
-7.45
-7.03
-7.12
-7.18
-7.09
-7.19
-6.68
-6.63
-6.68
-6.76
-6.92
-6.23
-6.22
-6.29
-6.45
-6.60
–
–
–
–
–
2.0
50
45
40
35
30
27.5
25
22.5
20
17.5
15
12.5
–
–
–
–
–
–
–
–
–
–
–
–
-7.79
-7.93
-8.16
-8.45
-7.74
-7.71
-7.76
-7.75
-7.71
-7.66
-7.88
-8.10
-7.25
-7.35
-7.47
-7.31
-7.31
-7.40
-7.42
-7.40
-7.24
-7.24
-7.42
-7.61
-6.88
-6.91
-7.01
-6.93
-7.08
-7.12
-7.04
-6.84
-6.72
-6.88
-6.98
-7.27
-6.46
-6.47
-6.48
-6.59
-6.76
-6.73
-6.48
-6.32
-6.41
-6.49
-6.62
-6.74
-6.01
-5.97
-6.05
-6.29
-6.38
-6.26
-6.01
-5.99
-6.06
-6.12
-6.15
-6.13
–
–
–
–
–
–
–
–
–
–
–
–
1.3
22.5
20
17.5
15
12.5
–
–
–
–
–
-7.49
-7.43
-7.50
-7.53
-7.71
-6.96
-6.99
-7.06
-7.22
-7.41
-6.55
-6.53
-6.63
-6.85
-7.04
-6.15
-6.22
-6.28
-6.39
-6.32
-5.75
-5.83
-5.83
-5.79
-5.72
–
–
–
–
–
2.6
50
45
40
35
30
–
–
–
–
–
-7.30
-7.30
-7.45
-7.74
-7.10
-6.91
-7.12
-6.74
-6.92
-6.80
-6.36
-6.41
-6.47
-6.37
-6.58
-5.97
-5.95
-5.95
-6.06
-6.25
-5.53
-5.45
-5.53
-5.77
-5.90
–
–
–
–
–
2.0
50
45
40
35
30
27.5
25
22.5
20
17.5
15
12.5
–
–
–
–
–
–
–
–
–
–
–
–
-6.97
-7.02
-7.10
-7.33
-6.96
-7.04
-7.09
-7.07
-6.97
-6.88
-7.03
-7.35
-6.56
-6.65
-6.73
-6.70
-6.70
-6.78
-6.79
-6.62
-6.52
-6.59
-6.78
-6.96
-6.20
-6.22
-6.26
-6.27
-6.41
-6.48
-6.38
-6.12
-6.11
-6.17
-6.35
-6.70
-5.76
-5.73
-5.75
-5.90
-6.10
-6.01
-5.75
-5.66
-5.75
-5.86
-5.93
-6.09
-5.28
-5.24
-5.35
-5.60
-5.67
-5.56
-5.34
-5.33
-5.40
-5.43
-5.43
-5.31
–
–
–
–
–
–
–
–
–
–
–
–
1.3
22.5
20
17.5
15
12.5
–
–
–
–
–
-6.76
-6.61
-6.69
-6.82
-7.06
-6.27
-6.28
-6.40
-6.51
-6.78
-5.82
-5.88
-6.02
-6.13
-6.26
-5.44
-5.52
-5.59
-5.67
-5.65
-5.12
-5.18
-5.11
-5.03
-4.92
–
–
–
–
–
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 Ṁ as a function of Teff for five metallicities in the
range Z/Z = 1/30 - 3. The metal content is indicated in the legend at the upper part of each
panel. Upper five panels (a)-(e): Γe = 0.130 (log L/L = 5.0). Lower five panels (f)-(j): Γe =
0.206 (log L/L = 5.5). The values for (V∞ /Vesc ) are indicated in the legend at the lower part of
the last panel (j).
103
Chapter 5
Γe
0.434
logL∗
M∗
(L )
(M )
6.0
60
V∞
Vesc
log Ṁ(M yr−1 )
Teff
1/100
1/30
1/10
1/3
1
3
10
(kK)
Z/Z
Z/Z
Z/Z
Z/Z
Z/Z
Z/Z
Z/Z
2.6
50
45
40
35
30
-6.81
-6.80
-6.86
-7.16
-6.78
-6.31
-6.59
-6.16
-6.27
-6.21
-5.84
-5.87
-5.95
-5.95
-5.90
-5.46
-5.45
-5.41
-5.47
-5.57
-5.07
-4.99
-4.97
-5.05
-5.29
-4.57
-4.55
-4.59
-4.78
-4.94
-4.31
-4.31
-4.42
-4.60
-4.52
2.0
50
45
40
35
30
27.5
25
22.5
20
17.5
15
12.5
-6.42
-6.47
-6.58
-6.78
-6.47
-6.50
-6.60
-6.52
-6.33
-6.36
-6.54
-6.71
-6.17
-6.35
-5.98
-6.11
-6.07
-6.16
-6.24
-6.11
-5.93
-6.01
-6.17
-6.35
-5.67
-5.69
-5.73
-5.74
-5.80
-5.99
-5.92
-5.63
-5.59
-5.74
-5.90
-5.99
-5.25
-5.22
-5.23
-5.28
-5.44
-5.51
-5.38
-5.13
-5.19
-5.33
-5.42
-5.48
-4.86
-4.76
-4.76
-4.88
-5.14
-5.19
-4.95
-4.78
-4.83
-4.90
-4.85
-4.51
-4.42
-4.42
-4.47
-4.65
-4.82
-4.68
-4.44
-4.45
-4.54
-4.48
-4.25
-4.19
-4.23
-4.24
-4.32
-4.47
-4.38
-4.23
-4.11
-4.17
-4.26
-4.11
-3.94
-3.99
1.3
22.5
20
17.5
15
12.5
-6.24
-6.06
-6.09
-6.29
-6.49
-5.77
-5.70
-5.80
-5.98
-6.13
-5.36
-5.37
-5.52
-5.65
-5.75
-4.91
-5.00
-5.09
-5.07
-4.80
-4.55
-4.63
-4.59
-4.28
-4.30
-4.29
-4.38
-4.19
-4.06
-4.10
-4.10
-4.12
-3.97
-3.91
-3.95
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 Ṁ 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 Ṁ at the hot side of the jump (where V∞ /Vesc = 2.6) and the
maximum Ṁ at the cool side (where V∞ /Vesc = 1.3). The size of the predicted jump in Ṁ (i.e.
log ∆Ṁ) is indicated in the last column of Table 5.4. For most models ∆Ṁ 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 (Γe =
104
Mass-loss predictions for O and B stars as a function of metallicity
Figure 5.1: Continued: Series of Ṁ(Z) calculations with Γe = 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 Γe also.
As in Vink et al. (2000), hρi is defined as the characteristic wind density at 50 % of the
terminal velocity of the wind. For a standard velocity law with β = 1, this characteristic wind
density is given by
hρi =
Ṁ
8πR2∗V∞
(5.13)
Figure 5.2 shows the behaviour of the characteristic density as a function of Z. This is done
for both the minimum Ṁ (at the hot side of the jump) and the maximum Ṁ (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 hρi for the “jump” model for each metallicity. The “jump” model is a hypothetical model between the two models where Ṁ is maximal and minimal. As expected, log hρi
increases as the metallicity increases. Because the average density at the jump shows a linear
105
Chapter 5
Γe
logL∗
(L )
M∗
(M )
0.130
5.0
20
0.206
5.5
40
0.434
6.0
60
(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(∆Ṁ)
0.75
0.77
0.83
0.86
0.66
0.63
0.81
0.81
0.81
0.72
0.71
0.74
0.76
0.76
0.68
0.43
Table 5.4: The size of the bi-stability jump around 25 000 K for different Z.
Figure 5.2: Characteristic density hρi 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 hρi.
106
Mass-loss predictions for O and B stars as a function of metallicity
Figure 5.3: Characteristic density log hρi 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 hρi 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:
loghρi
=
−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 hρi. 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 hρi is given by:
jump
Teff
= 61.2 (± 4.0) + 2.59 (± 0.28) loghρi
(5.15)
It is now possible to estimate hρi 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 Ṁ 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:
jump
T∼15kK = 43 + 1.9 loghρi
(5.16)
jump
(5.17)
For the jump at Teff ' 35 000K:
T∼35kK = 192 + 10.4 loghρi
In both cases the jump temperature is in units of kK. It is again possible to estimate log hρi
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 Ṁ 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
Dominant elements
(Z )
that set Ṁ
1
1/3
1/10
Fe
Fe
Fe
1/30
1/100
CNO
CNO
[O/Fe]
[O/Fe]
obs
ZOxygen
obs
ZFe
(Z )
(Z )
0
0
0
1
1
1
1
1
1
+ 0.4 dex
+ 0.4 dex
1/30
1/100
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 significantly alter our mass-loss predictions in the case Ṁ 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 ) to our adopted metallicity (Z theory ). This can easily be done according to the scaling re(ZFe
lations 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 Ṁ(Z) dependence over a wide range in metallicity. This
Ṁ(Z) will be determined for the three Γe values separately. If the dependencies were identical
for different Γe , 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 Ṁ(Z) behaviour for the three cases of Γe . 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 Γe = 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 Ṁ =
−6.439 (±0.024)
+ 0.842 (±0.039) log(Z/Z )
for Γe = 0.130
(5.18)
In case Γe = 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 Ṁ =
−5.732 (±0.028)
+ 0.851 (±0.033) log(Z/Z )
for Γe = 0.206
(5.19)
Finally, in case Γe = 0.434, the Ṁ(Z) dependence is studied over an even wider metallicity
range: Z/Z is 1/100 - 10. For this relatively high value of Γe 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 Ṁ(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. Ṁ vs. Z behaves as a power-law. The linear
fit for the highest value of Γe is determined from the range Z/Z = 1/30 - 3. The best fit is
given by
log Ṁ =
−4.84 (±0.020)
+ 0.878 (±0.023) log(Z/Z )
for Γe = 0.434
111
(5.20)
Chapter 5
Figure 5.5: The Ṁ(Z) dependence for three cases of Γe . In all three panels, the dashed lines
indicate the best linear fit through the models at different Z. Note that at Γe = 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 Γe 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 Γe , 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 Ṁ ∝ 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
Ṁ ∝ Z 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. Γe (Vink et al. 2000). The characteristic density hρi 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
loghρi = −14.94 (±0.54)
+ 0.85 (±0.10) log(Z/Z )
+ 3.2 (±2.2) Γe
(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 temper15
000
K,
one
should
take
into
account
the
presence
of
the
Fe III/II jump, and
atures, Teff <
∼
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
∆Ṁ = 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
35
000
K,
one
should
be
aware
of
the
carbon
jump
and
a
similar
strategy may be followed.
Teff >
∼
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 ∆Ṁ = 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 Ṁ(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 Ṁ =
− 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)}2
+ 0.85 (±0.10) log(Z/Z )
for 27 500 < Teff ≤ 50 000K
(5.23)
where Ṁ 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 Ṁ =
− 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 Ṁ 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 Ṁ 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 Ṁ(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/
115
Chapter 5
Figure 5.6: Comparison between theoretical and observational Ṁ 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) Hα 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 Ṁ 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 σ) 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 Ṁ ∝ 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
Ṁ ∝ Z 0.85 . This is in contrast to an often applied square-root dependence of mass loss
on Z.
3. Although the Ṁ(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
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119
Chapter 5
120
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, Ṁ 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 line acceleration for models with V∞ /Vesc =
2.0, log L∗ /L = 5.5 and M∗ = 40M . The solid lines show the contribution of carbon lines,
the dotted line that of oxygen, and the dashed line represents nitrogen (see legend in panel b).
(a) and (b) give the contribution for solar Z at V = Vsound and V = 0.5V∞ respectively, (c) and
(d) give the contribution for Z/Z = 1/30, at V = 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
effective temperature (Teff >
∼ 30 000 K) – CNO dominates the line driving throughout the entire
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:
gnorm
L
= gL
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, Ṁ ∝ 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, α, δ), (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 ρ(equator)/ρ(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 ∼ ρ(dr/dV ) as gline ∼ kt α , where k, α and δ are parameters that depend on composition and temperature of the wind. In this expression k is a measure of the number of lines
and α is a measure of the distribution of the line strengths with α = 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 2 2 2
(6.1)
+ ω x sin (θ)
x 2
where x = r/Req , θ is the co-latitude (θ = 0 at the pole) and ω = Vrot,eq /Vcrit . Φ is of course
independent of the longitude φ. Note that 1 + 0.5ω2 = Req /Rpole . So the maximum oblateness
is Req /Rpole = 3/2. The critical (break-up) velocity is defined in this paper as:
Φ(x, θ, φ) =
2
Vcrit
= GM?eff /Req
(6.2)
where M?eff is the effective mass of the star,
M?eff = M? (1 − Γe )
(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(θ) is equivalent to the solution of a 3rd degree polynomial, which results in
!)
(
√
√
2
3 3 ω sin θ
2+ω
1
x(θ) = 2 √
arcsin
(6.4)
sin
3
(2 + ω2 )3/2
3 ω sin θ
Because of the difference in radius
Req
= 1 + 0.5ω2
Rpole
(6.5)
and the difference in rotational velocity between pole and equator, the radial escape velocity at
the equatorial region is smaller than at the pole
s
1 − 0.5ω2 pole
eq
Vesc =
Vesc
(6.6)
1 + 0.5ω2
130
The radiation driven winds of rotating B[e] supergiants
Figure 6.1: Gravity darkening for different rotation rates ω as a function of co-latitude θ
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(θ, φ) = σB Teff
(θ) ∝ geff (θ)
(6.7)
with σB the Boltzmann constant. As geff = −∇Φ, we can write down the flux of a rotating star
as a function of the co-latitude θ. This (lengthy) expression for the flux as a function of θ is
plotted in Fig. 6.1 for a few different values of ω.
6.3.3 The equation of motion of a line driven wind of a rotating star
The rotation of the star can induce a θ component of the flow of the matter in the wind. In this
study a possible θ component is neglected as was discussed in Sect. 6.2, but the motion in the
longitudinal φ 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 Vφ
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:
Vφ (r) = Vrot (R? )
131
(6.8)
Chapter 6
V
2
dV GM? 1 d p
2 R?
+ 2 +
−Vrot 3 − grad = 0
dr
r
ρ dr
r
(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
4πr2 ρ(θ, r)V (θ, r) = Fm (θ)
(6.10)
where Fm (θ) 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
Ṁ =
Z
π/2
0
Fm (θ) sin(θ)dθ
(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 ρ (a is the isothermal sound speed), and the radiative
acceleration grad . Using the conservation of mass, the pressure term can be rewritten as:
1 d p 1 da2 2a2 a2 dV
=
−
−
ρ dr
ρ 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 Γe :
ge =
σe F
GM?
= Γe 2
c
r
(6.13)
where σe is the electron opacity, L is the stellar luminosity and F is the radiation flux. For a
homogeneous spherical star, Γe is given by1 :
σe L∗
(6.14)
4πGM?
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):
Γe =
Γ0e = Dc
σe L∗
4πGM?
(6.15)
where Dc is the continuum correction factor which is given by:
4πr2
Dc =
L∗
I
disk
I(θ, φ)dΩ
(6.16)
The line acceleration gL (r,V,V 0 ) 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
1 Note we define Γ
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
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:
2
a2
dV
2 R?
Fθ (r,V,V ) ≡ 1 − 2 r2V
−Vrot
V
dr
r
0
+GM? (1 − Γ0e ) − 2a2 r − r2 gL = 0
(6.17)
This equation is valid for each co-latitude θ. 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):
∂Fθ
=0
(6.18)
∂V c
and the regularity condition:
0 ∂Fθ
+ V
=0
∂V c
c
∂Fθ
∂r
(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 0 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 νl . gL is given by (Castor 1974):
gL = ∑
l
κl
c
I
Iνl
1 − e−τνl
µdΩ
τ νl
(6.20)
where κl is the absorption coefficient per gram for the lth line, µ = cos θ, Iνl the intensity at the
dr
rest frequency νl and τνl is the Sobolev optical depth of lth line, defined as τνl = κl ρ νcl ( 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 = σref
e Vth ρ
dr
dV
(6.21)
with Vth being the thermal velocity of the protons and σref
e is some reference value for the
ref
2
−1
electron scattering σe = 0.325 cm g (see ISW, chapter 8). The line acceleration in terms of
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):
δ
−α ne
Mpoint (t) = k t
(6.23)
W
133
Chapter 6
where the (ne /W )δ 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
I
(1 + σ)
I(θ, φ)
1 + σµ2
α
µdΩ
(6.24)
where
r dV
L? R?
− 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(θ, φ) 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:
σ=
gL =
σ 1−α kL
e
?
4π
r2 c
(VthFm )
−α
α ne δ
2 dV
Dfd r V
dr
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
2 R?
0 = 1− 2 r V
+ GM? (1 − Γe ) − 2a2 r −Vrot
V
dr
r
α
σ 1−α kL
n δ dV
e
?
e
r2V
(Vth Fm )−α Dfd
−
4π
c
W
dr
(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
α
becomes a simple function gL = rC2 (r2V dV
dr ) . The solution of Eq. (6.17) was found by CAK
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
α
dV
2 dV
rV
= GM? (1 − Γe ) +C r V
dr
dr
2
(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
Ṁ. The solution for the mass loss and velocity law is given by
1−α
1
4π σe 1 − α α
Ṁ =
(kα) α
σeVth 4π
α
1
nn oδ L α
α−1
e α
?
{GM? (1 − Γe )} α
W
c
R? 0.5
V (r) = V∞ 1 −
r
(6.29)
(6.30)
with
r
V∞ =
α
Vesc =
1−α
s
α 2GM? (1 − Γ)
(1 − α)
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):
Dfd
Z 1
(1 + σ) α
2
=
µdµ
(1 − µ? ) µ? 1 + σµ2
(1 + σ)α+1 − (1 + σµ2? )α+1
=
(1 − µ2? )(α + 1)σ(1 + σ)α
(6.32)
Including the finite disk correction results in an increase in V∞ , a decrease in Ṁ 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 Ṁ is determined. Close to the stellar photosphere the finite disk correction is smaller than one, viz. Dfd (r = R? ) = 1/(1 + α) < 1.
The accompanying increase in V∞ of typically a factor of two is due to two effects: (1) a reduction of Ṁ 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):
α
V∞ /Vesc ≈ 2.2
1−α
135
Vesc
3
10 km/s
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 Θ̃ sees a different limb angle Θ for every φ (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 < φ < 2π and 0 < θ < Θ. These are rewritten to integrals over the star
centered angles θ0 (0 < θ0 < Θ0 ) and φ0 = φ. The stellar radius and flux are given as a function
of θ00 , the co-latitude, which can be calculated for every θ0 and φ: θ00 = θ00 (θ0 , φ).
They also found a relation for the modified mass loss rate:
Ṁ ≈ 0.5 Ṁ CAK
Vesc
3
10 km/s
−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 (θ), 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 β 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 θ̃ are: the stellar surface
R(θ) and the surface temperature T (θ), both given as a function of the co-latitude θ, 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 θ and φ to the star centered coordinates θ0 and φ. For the definition
of these coordinates, see Fig. 6.2. Note that here φ is not the stellar longitude anymore, but the
angle of rotation around the line from a point (r, θ̃) in the wind to the stellar center.
1
Dfd =
N
=
1
N
Z
Z
I
0
0
2π Z Θ0 (φ)
I
0
2π Z Θ(φ)
0
(1 + σ)
1 + σ cos2 θ
(1 + σ)
1 + σ cos2 θ
α
α
cos θ sin θ dθdφ
(6.35)
cos θ sin θ
dθ
dθ0 dφ
dθ0
where I is the frequency-integrated intensity
I = I(θ0, φ) ∝ T (θ00)4
(6.36)
The angles are related to one another via
cos θ = p
r − R(θ0 , φ) cos θ0
(r − R(θ0 , φ) cos θ0 )2 + (R(θ0 , φ) sinθ0 )2
(6.37)
with the stellar surface given by R(θ0 , φ) = R(θ00 ), and
cos θ00 = sin θ0 cosφ sin ω + cos θ0 cos ω
(6.38)
Equation 6.35 can be integrated, considering that the limb angle Θ0 (φ) 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
2π Z Θ0
0
0
I cos θ sin θ
dθ
dθ0 dφ
dθ0
This correction is also included in the solution of the equation of motion.
137
(6.39)
Chapter 6
ω
V∞
0
0.3
0.4
0.5
0.6
1.40
1.43
1.44
1.46
1.48
2
θ=0
Fm 3
β4
V∞
2.35
2.67
2.95
3.33
3.83
0.63
0.63
0.63
0.63
0.63
1.40
1.40
1.40
1.40
θ = π/8
Fm
β
2.64
2.88
3.23
3.70
0.63
0.62
0.62
0.62
V∞
θ = π/4
Fm
β
1.35
1.31
1.27
1.21
2.53
2.70
2.95
3.29
0.63
0.62
0.62
0.60
V∞
1.30
1.23
1.14
1.04
θ = 3π/8
Fm
β
2.42
2.46
2.53
2.61
0.63
0.63
0.64
0.64
V∞
1.29
1.20
1.11
1.01
θ = π/2
Fm
β
2.36
2.34
2.27
2.11
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, α = 0.565, k = 0.32, δ = 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 β 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 conditions at the critical point:
rc , Vc and Vc0 , or, equivalently, rc , Vc and V0 = V (R? ), as well as the local mass loss rate Fm .
Apart from Eqs. (6.18) and (6.19) and the equation of motion (Eq. 6.27) itself the additional
constraint for the continuum optical depth τe at the stellar radius is neccesary (Pauldrach et al.
1986) to uniquely determine these quantities:
τe (θ) =
Z
∞
R?
ρσe dr =
2
3
(6.40)
If the condition for V0 is used one may adopt any small value, say V0 = 0.1 km/s, since even
a large error in V0 will only yield a relatively small error in V∞ and Fm . This is due to the
fact that at the surface of the star V (r) and thus also ρ vary exponentially with scale height
a2 R2∗
H = GM
eff << R? . This means that the radius where Eq. (6.40) is fulfilled is very close to
?
R? (this should be checked a posteriori). In our calculations we have adopted this boundary
condition for V0 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
dr = 0 if Fm (guess) < Fm (solution). This approximate value for
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
0.64
0.64
0.66
0.70
The radiation driven winds of rotating B[e] supergiants
L∗
(L )
ω
104.5
pole
pole
eq
eq
Vinf
3
(10 km s−1 )
Fm
−8
(10 M yr−1 )
Vinf
3
(10 km s−1 )
Fm
−8
(10 M yr−1 )
0
0.3
0.4
0.5
0.6
2.84
2.91
2.97
3.04
3.11
3.01
3.37
3.67
4.08
4.62
2.60
2.42
2.22
2.00
3.04
3.03
2.98
2.82
L∗
(L )
ω
Vinf
3
(10 km s−1 )
Fm
−7
(10 M yr−1 )
Vinf
3
(10 km s−1 )
Fm
−7
(10 M yr−1 )
105.0
0
0.3
0.4
0.5
0.6
2.06
2.11
2.15
2.19
2.24
2.53
2.84
3.11
3.46
3.93
1.88
1.75
1.61
1.46
2.55
2.54
2.49
2.35
L∗
(L )
ω
Vinf
3
(10 km s−1 )
Fm
−6
(10 M yr−1 )
Vinf
3
(10 km s−1 )
Fm
−6
(10 M yr−1 )
105.5
0
0.3
0.4
0.5
0.6
1.40
1.43
1.44
1.46
1.48
2.35
2.67
2.95
3.33
3.83
1.29
1.20
1.11
1.01
2.36
2.34
2.27
2.11
L∗
(L )
ω
Vinf
3
(10 km s−1 )
Fm
−5
(10 M yr−1 )
106.0
0
0.3
0.4
0.5
0.6
0.65
0.62
0.59
0.55
0.49
3.93
4.97
6.00
7.79
11.1
pole
pole
pole
pole
pole
pole
eq
eq
eq
Vinf
3
(10 km s−1 )
0.60
0.58
0.56
0.56
eq
eq
eq
Fm
−5
(10 M yr−1 )
3.80
3.61
3.24
2.64
ρeq
ρpole
(10−8
1
1.01
1.01
1.00
0.95
ρeq
ρpole
3.01
3.18
3.32
3.50
3.71
(10−7
1
1.00
1.00
0.98
0.91
ρeq
ρpole
Ṁ
M yr−1 )
2.53
2.67
2.79
2.94
3.15
(10−6
1
0.98
0.95
0.90
0.80
ρeq
ρpole
Ṁ
M yr−1 )
Ṁ
M yr−1 )
2.35
2.47
2.56
2.71
2.89
(10−5
1
0.78
0.62
0.41
0.21
Ṁ
M yr−1 )
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∗ /σB 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 )
ω
20
pole
pole
eq
eq
Vinf
3
(10 km s−1 )
Fm
−6
(10 M yr−1 )
Vinf
3
(10 km s−1 )
Fm
−6
(10 M yr−1 )
0
0.3
0.4
0.5
0.6
0.80
0.80
0.80
0.79
0.79
5.66
6.64
7.49
8.74
10.5
0.73
0.68
0.64
0.60
5.67
5.58
5.32
4.76
M∗
(M )
ω
Vinf
3
(10 km s−1 )
Fm
−6
(10 M yr−1 )
Vinf
3
(10 km s−1 )
Fm
−6
(10 M yr−1 )
60
0
0.3
0.4
0.5
0.6
1.82
1.85
1.89
1.91
1.95
1.55
1.76
1.92
2.16
2.46
1.66
1.56
1.43
1.31
1.56
1.55
1.51
1.41
pole
pole
eq
eq
ρeq
ρpole
(10−6
1
0.94
0.88
0.75
0.53
ρeq
ρpole
Ṁ
M yr−1 )
5.66
5.69
6.00
6.38
6.81
(10−6
1
0.99
0.98
0.93
0.85
Ṁ
M yr−1 )
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: α = 0.565, k = 0.32, δ = 0.02. An overview of the results for
this generic B[e] star rotating at ω = 0, 0.3, 0.4, 0.5 and 0.6 (higher ω where not possible with
our method) times the critical rotation speed is given in Table 6.1. The table shows for different co-latitudes θ and different rotation velocities ω the values of V∞ , the local mass loss rate
Fm (θ) and the value of the velocity law parameter β, obtained by fitting the calculated velocity
structure from 1.1 R? to 10 R? to a β-law. We see that the value of β does not change much.
Fig. 6.3 shows the velocity structure of the wind and the location of the critical point for
ω = 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 ω. At the pole (θ = 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: ω = 0.3, bottom panel: ω = 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 ω as 1 + 0.64ω2 .
Figure 6.6 shows the resulting density contrast (far from the star) between pole and equator
eq pole
pole eq
( ρeq /ρpole = 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 + 0.64ω2
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 ρeq /ρpole 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 (Γe = 0.76). The net effect of the continuum radiation pressure and
the gravity, i.e. GM∗ (1 − Γ0e )/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 Γ0e through Dc is not important because Γe itself is
small. For high luminosity L∗ /(GM? ) increases and thus the effect of Dc on Γ0e 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 Γ0e ) 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 ρeq /ρpole does not follow the relation suggested by Eqs. (6.29) and (6.31). We have found that the actual values of ρeq /ρpole could be
approximated quite well (within 10 percent) with the following relation:
ρeq /ρpole ≈
1 − Γ0e (θ = 0, r/Req = 1)
1 − Γ0e (θ = π/2, r/Req = rmin )
1.5
(6.41)
where the rmin ≈ 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 Γe = 0.76 and
ω = 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, α and δ are listed in Table 6.4. Note that the
δ 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
α
δ
(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 θ=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 α from the hot Monte Carlo model of 30 000 K
were used, whereas for the equator, k and α from the cool model of 17 500 K were used.
The resulting V∞ (θ) and mass loss rates Fm (θ) are plotted in Fig. 6.8 and Fig. 6.9 for ω
= 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 θ (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 θ = 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 θ 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, α and δ 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 ρeq /ρpole ' 1, except for fast rotating stars with high
luminosity (in which case ρeq /ρpole < 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 α 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 α 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 θ-component of the line acceleration
towards the polar regions. This “disk inhibition” mechanism operates in wind models with
constant force multipliers k and α. 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 α 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
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148
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 Ṁ).
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 bistability jumps present, where Ṁ 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 Ṁ 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 Ṁ 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 Ṁ
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 ISA - WIND for hot stars with winds. For details on ISA - WIND, 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 Γe .
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 )
M∗
(M )
(L∗ /M∗ )
×103 (L /M )
5.5
6.0
30 – 10
35 – 25
10.5 –31.6
28.6 – 40.0
Γe
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.
L
M −1
Lσe
−5
Γe =
= 7.66 10 σe
4πcGM
L
M
(7.1)
where the constants have their usual meaning and σe is the electron scattering cross-section per
unit mass. The value for σe 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 σe and does not depend on the composition of the atmosphere, is given in the
third column of Table 7.1. The range in Γe is given in column (4) of the table. Note that Γe
is considerably smaller than unity, and thus the classical Eddington limit is not exceeded. The
reason for this surprisingly low Γe is that although the masses are lower than for normal OB
supergiants, the value for the electron scattering cross-section σe 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:
L∗ = 4πR2in σTin4
(7.2)
where Rin is the inner boundary of the atmospheric model. Rin is typically located at a
Rosseland optical depth 20 <
∼ τR <
∼ 25. Note that the actual effective temperature of the
out
model Teff is somewhat lower and it follows from the model computation. The effective
√
temperature is defined at the point where the thermalization depth at 5555 Å equals 1/ 3
(see Schmutz et al. 1990 and de Koter et al. 1996 for a detailed discussion). For stars with
relatively low Ṁ 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 Ṁ 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 Ṁ 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 Ṁ for wind models with a β-type velocity law for the supersonic part of the wind:
R∗ β
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 β (Vink et al. 2000a). A
value of β = 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 Ṁ
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 Ṁ 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 distribution. As most of the line driving around the sonic point, where Ṁ 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 Å. 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 λ = 912 and 3646 Å. 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 Ṁ
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: Ṁ 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 Ṁ(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 Ṁ ∼ 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 η 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 η 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.
S Dor: 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
Chapter 7
160
LBV
Name
log L∗
(L )
log Ṁ obs
(M yr−1 )
M∗est
(M )
AG Car
S Dor
R 127
R 71
R 110
6.0
5.8
6.1
5.3
5.5
- 4.1 (average)
- 4.3 (max)
- 4.3 (max)
- 4.7 (max)
- 5.5 (max)
30
30
35
12
30
Teff range Expected behaviour as a
(kK)
function of decreasing Teff
21 - 15
22 - 10
30 - 10
14 - 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
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 LMC star R 110 as a function of effective
temperature seems, at first sight, quite surprising. The observations by Stahl et al. (1990)
imply that the mass loss decreases as a function of decreasing Teff . However, as the mass loss
behaviour is observed over a range in Teff , which is entirely below the second jump, i.e. Teff <
∼
10 k, its observed behaviour is not very surprising at all, but even predicted by our mass-loss
predictions.
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 Ṁ). 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 Ṁ(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
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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
163
Chapter 7
164
Summary and Prospects
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
Chapter 8
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
167
Chapter 8
168
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 essentiële 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
Samenvatting
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 efficiënte stralingsdruk, 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 één keer. Het rekening houden met deze meervoudige botsingen is een
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!
172
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, 32–40
173
Publication List
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, 159–166
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, 427–433
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, 135–149
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, 395–412
174
Curriculum vitæ
Curriculum vitæ
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 officiële
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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Curriculum vitæ
176
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 essentiële 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 Ṁ 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 Ṁ-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, Mariëlle, Marten,
Maureen, Pui-Kei, Robert en alle anderen (!!) voor de gezellige drink,- tafeltennis,- en maffia
aangelegenheden.
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