Wind properties of Blue Supergiants
A thesis submitted for the degree of
Doctor of Philosophy
by
Blagovest Petrov, B.Sc, M.Sc.
Armagh Observatory
Armagh, Northern Ireland
&
Faculty of Engineering and Physical Science
School of Mathematics and Physics
The Queen’s University of Belfast
Belfast, Northern Ireland
April 2014
Declaration
This thesis was submitted for evaluation and accepted after examination in accordance with the
requirements for the Degree of Doctor of Philosophy in Physics of the Queen’s University of
Belfast, United Kingdom. I certify that the contents of this thesis is solely my own work, other
than where I have clearly indicated so, and has not been presented for the award of any other
degree, title or fellowship elsewhere.
External Examiner:
Prof. Raman K. Prinja
University College London, United Kingdom
Internal Examiner:
Prof. Simon Jeffery
Armagh Observatory, United Kingdom
Principal Supervisor:
Dr. Jorick Vink
Armagh Observatory, United Kingdom
Ph.D. Candidate:
Blagovest V. Petrov
Student Number: 40054634
Armagh Observatory and Queen’s University of Belfast
i
To my parents who always expected more from me
Acknowledgements
Despite that the Acknowledgements do not belong to the academic part of a thesis, it is perhaps
the most important part of a thesis not only because it is the only thing everyone actually would
read, but also because it enables you to thank all those who have helped in carrying out the
research.
Firstly, I would like to thank my supervisor, Dr. Jorick Vink because over the past 3 years, he
has lost great amount of energy and hence mass (loss), while guiding me throughout my PhD.
His ideas and approach to research have been the “driving” forces in developing this work and
increasing my potential. Therefore, I would like to express my sincere gratitude for his continuous support, patience and many helpful (and exciting!) discussions to which I undoubtedly
owe my current understanding of research. He taught me how to approach problems, what is
essential in my work and how to challenge myself.
I am especially thankful also to my co-supervisor Götz Gräfener for his cmfgen support and
stimulating discussions. An ongoing thank you also goes to Joachim Bestenlehner for the many
serious and seriously funny conversations.
Many thanks to Dr. Joachim Puls for detailed comments and discussions on some parts of
the thesis.
I would like to thank Prof. Raman Prinja. and Prof. Simon Jeffery, for providing me with
constructive feedback on my PhD thesis.
A big thank you also goes to the the fellow students and staff of the Armagh Observatory
for providing an excellent working environment and insights into different fields of astronomy.
I wish to specially thank (by alphabetical order) to:
• Aileen for helping in everything concerning logistics
• Alex, Onur and Shenghua for being wonderful office mates. I had many stimulating conversations with them
• Aswin for educating me on the different types of Whiskeys. I have always been amazed
at the ethics and courtesy of this distinguished gentleman
• Chris, Ruxi, Tugca, and Yani for their friendship
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ii
• Geert for all the funny jokes and for showing me cool stuff with PYTHON
• Juie for the many (and interesting!) insights into the life of guinea pigs
• Kamalam and Will for being excellent housemates
• Maria for her competent opinion on everything
• Mark for being a great Director
• Martin for his high-tech support. He has saved me many headaches, many times!
• Shane for locking up the observatory
I am also gratefull to the Director of the Institute of Astronomy of the Bulgarian Academy of
Sciences , Assoc. Prof. Tanyu Bonev, thanks to whom, I had an excellent working environment
when I was addresing the revisions of the manuscript recommended by the examiners.
To carry out this research I made extensive use of non-LTE radiative transfer code cmfgen
and therefore I am grateful to Dr. John Hillier for providing the cmfgen code to the astronomical
community. I acknowledge also financial support from the Northern Ireland Department of
Culture, Arts and Leisure (DCAL) and the United Kingdom (UK) Science and Technologies
Facilities Council (STFC).
Last, but not least, I am truly grateful to my parents for their support and their confidence in
me.
Blagovest V. Petrov
Created with LATEX
Abstract
The evolutionary state of blue supergiants is still unknown. Stellar wind mass loss is one of the
dominant processes determining the evolution of massive stars, and it may provide clues to the
evolutionary properties of blue supergiants. However, their mass-loss properties are not well
understood. Therefore, in this thesis, we investigate the wind properties of blue supergiants by
means of the non-LTE radiative transfer code cmfgen (Hillier & Miller 1998).
The thesis describes two self-contained pieces of research which are linked through their
connection with the wind properties of blue supergiants. The first involves a detailed analysis of
the Hα line formation over a range in effective temperature between 30 000 and 10 000 K. The
purpose of this analysis is to understand the influence of T eff on the formation of Hα and the
significance of micro-clumping on both sides of the bi-stability jump.
We find a maximum in the Hα equivalent width around 22 500 K. Intriguingly, this is the temperature location of the bi-stability jump. This behaviour is always present in sets of models with
various stellar and wind parameters, and it is characterised by two branches of effective temperature: (i) a hot branch between 30 000 and 22 500 K, where Hα emission becomes stronger with
decreasing T eff ; and (ii) a cool branch between 22 500 and 12 500 K, where the Hα line becomes
weaker. Our models show that this non-monotonic Hα behaviour is related to the optical depth
of the Lyα line, finding that at the “cool” branch the population of the 2nd level of hydrogen
is enhanced in comparison to the 3rd level. This is expected to increase line absorption, leading to weaker Hα flux when T eff drops from 22 500 K downwards. We also show that for late
iii
iv
B supergiants (at T eff below ∼15 000 K), the differences in the Hα line between homogeneous
and clumpy winds becomes insignificant. Moreover, we show that, at the bi-stability jump, Hα
changes its character completely, from an optically thin to an optically thick line, implying that
macro-clumping should play an important role at temperatures below the bi-stability jump. This
would not only have consequences for the character of observed Hα line profiles, but also for
the reported discrepancies between theoretical and empirical mass-loss rates.
The second part of the thesis is devoted to the wind properties of the blue supergiants. As
accurate information of the radiative force could lead to valuable statements about the massloss rates or terminal velocities of blue supergiant winds, in this part of the thesis the physical
ingredients that play a role in the line acceleration are explored. Our calculations confirm the
bi-stability jump in mass-loss rates predicted by Vink et al. (1999). We also show that at temperatures around 10 000 K a second jump in mass-loss rates is produced if the observed velocity
ratios are applied. This jump is caused by Fe iii/Fe ii recombination/ionisation as was suggested
by Vink et al. (1999). For models with half-solar metal abundances the second bi-stability jump
is only produced for models near the Eddington limit, underlying that this jump is important
for wind properties of the LBVs. Understanding the behaviour of the second jump may provide valuable science prospects for late B/A supergiants and LBVs, and therefore, a detailed
investigation of this jump would be valuable.
List of Acronyms
In this thesis the following abbreviations are commonly used.
• Bsg– B-type supergiant
• BSG – Blue supergiant
• CMF – Co-moving frame
• FLAMES – Fibre large array multi element spectrograph
• HRD – Hertzsprung-Russell diagram
• LBV – Luminous blue variables
• LMC – Large Magellanic cloud
• MS – Main sequence
• RGB – Red giant branch
• RSG –Red supergiant
• SMC – Small Magellanic cloud
• SN – Supernova
• VLT – Very large telescope
• YHG – Yellow hypergiant
v
Contents
Declaration
i
Acknowledgements
i
Abstract
iii
Acronyms
v
List of Tables
x
List of Figures
xvii
Publications
xviii
I INTRODUCTION
1
1 Blue supergiants - troublemakers or candles in the dark
2
1.1
1.2
The life-cycle of massive stars . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1
Main-sequence Evolution . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.2
Life after the main-sequence . . . . . . . . . . . . . . . . . . . . . . .
5
1.1.3
Supernovae from Blue Supergiants . . . . . . . . . . . . . . . . . . . .
7
Troublemakers across the Hertzsprung-Russell diagram . . . . . . . . . . . . .
10
vi
CONTENTS
1.3
1.4
vii
Aspects of BSGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3.1
Candles in the dark . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3.2
Wind properties and the bi-stability jump . . . . . . . . . . . . . . . .
15
1.3.3
Mess in the mass loss rates . . . . . . . . . . . . . . . . . . . . . . . .
20
1.3.4
Rotational velocities . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Overview of the Chapters of this Thesis . . . . . . . . . . . . . . . . . . . . .
27
2 Methods
2.1
2.2
2.3
28
Hot star wind diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.1.1
Diagnostics from UV P Cygni lines . . . . . . . . . . . . . . . . . . .
29
2.1.2
Hα line: a conventional mass loss probe for massive stars . . . . . . . .
31
2.1.3
Mass loss from radio . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Wind inhomogeneities: problems and perspectives
. . . . . . . . . . . . . . .
37
2.2.1
Observational history . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.2.2
Theoretical background
. . . . . . . . . . . . . . . . . . . . . . . . .
38
2.2.3
Clumping may reconcile Hα, UV and radio Ṁ determinations? . . . . .
40
Numerical methods: the cmfgen atmosphere code . . . . . . . . . . . . . . . .
41
2.3.1
Main ingredients of the cmfgen code . . . . . . . . . . . . . . . . . . .
42
2.3.2
Other characteristics of cmfgen . . . . . . . . . . . . . . . . . . . . . .
43
2.3.3
Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
II The Physics Behind the Hα Line
3
47
Hα line formation: rise and fall over the bi-stability jump
48
3.1
Method and input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.2
Hα line profile and equivalent width . . . . . . . . . . . . . . . . . . . . . . .
50
CONTENTS
CONTENTS
3.3
Two branches of Hα behaviour . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.3.1
The “hot” branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.3.2
The “cool” branch . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Two possible explanations for the existence of the “cool” branch . . . . . . . .
60
3.4.1
A decrease of n3 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.4.2
An increase of n2 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.5
Lyα and the second level . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.4
4
viii
The effect of clumping
72
4.1
The Hα line in a micro-clumping approach
. . . . . . . . . . . . . . . . . . .
73
4.2
The Hα optical depth in a micro-clumping approach . . . . . . . . . . . . . . .
74
4.3
Impact of macro-clumping on Hα . . . . . . . . . . . . . . . . . . . . . . . .
76
4.4
Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5 Exploration of Hα
79
5.1
Strategy and grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.2
The influence of various parameters on Hα . . . . . . . . . . . . . . . . . . . .
80
5.2.1
The dependence of the Hα line EW on T eff for various Ṁ . . . . . . . .
81
5.2.2
Influence of T eff and Ṁ on Hα line EW and morphology . . . . . . . .
82
5.2.3
Influence of stellar mass . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.2.4
Influence of stellar luminosity . . . . . . . . . . . . . . . . . . . . . .
87
5.2.5
Influence of Metallicity . . . . . . . . . . . . . . . . . . . . . . . . . .
90
5.3
Hα in the context of the bi-stability jump in 3∞ /3esc . . . . . . . . . . . . . . .
92
5.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
CONTENTS
CONTENTS
ix
III Wind properties
96
6 Wind properties of blue supergiants
97
6.1
6.2
An overview of line-driven winds: recall the basic relations . . . . . . . . . . .
97
6.1.1
The momentum equation . . . . . . . . . . . . . . . . . . . . . . . . .
98
6.1.2
Driving forces of the winds in hot massive stars . . . . . . . . . . . . .
99
Ion contributors to the line driving . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.1
CNO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2.2
Iron the wind driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3
Bi-stability jump on trial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4
A second bi-stability jump? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.5
The second bi-stability jump as a function of mass for solar metallicities . . . . 112
6.6
Consequences for LBVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7 Conclusions & Future work
119
7.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Appendices
122
A Where in the wind do Hα photons originate from?
123
B Sobolev approximation
126
C Atomic data and model atoms in (sophisticated models)
130
C.1 Atomic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
C.2 Model atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
CONTENTS
List of Tables
1.1
Supernova classification
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Adopted stellar parameters used in the model grid.
3.2
Atomic data used for our simplistic H + He supergiant models. For each ion, the
. . . . . . . . . . . . . . .
number of full levels, super levels, and bound-bound transitions are provided. .
3.3
8
50
53
Model atoms used in the sophisticated models. For each ion the number of full
levels, super levels, and bound-bound transitions is provided. . . . . . . . . . .
54
5.1
Adopted stellar and wind parameters for the main grid of models. . . . . . . . .
84
5.2
Atomic data used to test the sensitivity of WHα to the adopted model atoms of
the iron ions. For comparison, the initial model atoms are provided as well. . .
87
5.3
Adopted stellar and wind parameters for the additional grid of models. . . . . .
90
6.1
Mass-loss rates at which Qwind = 1 for different model series. . . . . . . . . . . 115
C.1 Atomic data included in our realistic models . . . . . . . . . . . . . . . . . . . 132
x
List of Figures
1.1
– (a) Evolutionary tracks for stars with masses from 15 to 30 M⊙ . (b) Evolutionary tracks for star with mass 25M⊙ with and without mass loss rate. Figure from
El Eid et al. (2004). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
5
Mass-loss predictions as a function of metallicity. With the solid line is expressed the dependence of mass-loss rate of metallicity according to Vink et al.
(2001) and the dotted line shows the predictions of Kudritzki (2002). Figure
from Vink (2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
Observed fractions of each type of SN. Figure from Li et al. (2011). . . . . . .
8
1.4
H-R diagrams of O- and early B-type stars in the fields of the Magellanic Clouds.
Open circles represent foreground stars. Objects with evidence for binarity are
denoted with crosses and the open triangles indicate objects with emission lines.
The evolutionary tracks for models with LMC metallicity (N 11 and NGC 2004)
are obtained from Schaerer et al. (1993), and from Charbonnel et al. (1993) for
SMC metallicity (NGC 330 and NGC 346). Figure from Evans et al. (2006). . .
1.5
11
Modified wind momenta as a function of luminosity for galactic OBA supergiants. Early B: B0 to B1;mid B: B1.5 to B3. Figure from Kudritzki et al. (1999). 14
1.6
The observed bi-stability jump in terminal wind velocities near T eff ≈ 21 000 (at
a spectral type B1) . A second jump may be present at T eff ≈ 10 000 (at spectral
type A0) where v∞ /vesc ratio drops from 1.3 to 0.7. Figure from Lamers et al.
(1995). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7
The ratio of 3∞ /3esc as a function of T eff for BSGs from Crowther et al. (2006)
(left) and from Markova & Puls (2008) (right). . . . . . . . . . . . . . . . . . .
xi
16
17
LIST OF FIGURES
xii
1.8
Predicted bi-stability jump in Ṁ. Figure from Vink et al. (1999). . . . . . . . .
1.9
Mass-loss rate (blue dotted) and rotational velocities of a Galactic 40 M⊙ star
19
which had a initial rotational velocity of 275 km/s on ZAMS, including predicted
bi-stability jump (red solid) and without it (green dashed).Figure from Vink et al.
(2010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.10 Left: rotational velocities vs T eff for Howarth et al. (1997) data-set of Galactic
OB supergiants (red diamonds) and non-supergiants (blue triangles) Right: rotational velocities of LMC supergiants (red asterisks) and non-supergiants (blue
pluses) as a function of T eff . The gray lines indicate LMC evolutionary tracks
with initial vrot = 250 km/s for models with masses = 15, 20, 30, 40 and 60 M⊙ .
The black dots on the tracks illustrate 105 year time-steps. Figure from Vink
et al. (2010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
1.11 Nitrogen abundance as a function of T eff for LMC objects. Figure from Vink
et al. (2010) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.1
Schematic formation of a P Cygni type line profile. Figure from Murdin (2003).
30
2.2
Synthetic Hα line profiles for Bsg model with different Ṁ. The spectra were
computed with cmfgen code (cf. § 2.3). . . . . . . . . . . . . . . . . . . . . . .
2.3
Schematic energy distribution of a star with R⋆ = 10 R⊙ , T eff = 37 500 K and
with free-free emission from a wind of Ṁ = 1 × 10−5 M⊙ yr−1 . Figure from
Lamers & Cassinelli (1999). . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
32
35
Left: Hα line profiles for cmfgen models with parameters as listed in Table 3.1.
Black triangles represent the line profile with the same Q-parameter (Eq. 3.1)
as model C (T eff =12 500 K), but with different Ṁ and R⋆ values. Right: Hα
line EW vs T eff for models with only H (crosses), H+He (circles), and more
sophisticated (triangles) models composed by H, He, C, N, O, Si, P, S, and Fe
with half-solar metal abundances. Red asterisks represent the changes in the Hα
line when the He mass fraction in the pure H+He models is increased to 60%.
Blue squares indicate how the Hα EW behaves as a function of a constant Q value. 51
3.2
Integrated line (circles) and continuum flux (squares) at the wavelength of Hα
for H + He models. Note that the flux represents the flux at a distance of 10 parsec. 52
3.3
Hydrogen ionisation structure for models with various T eff . . . . . . . . . . . .
56
LIST OF FIGURES
LIST OF FIGURES
3.4
xiii
Total number of H atoms in the stellar wind versus T eff . Note that the total
number of H atoms is determined from τross < 2/3. . . . . . . . . . . . . . . .
3.5
57
Changes in the (n3 /n2 ) ratio with T eff . Regions where most of the emergent Hα
photons originate from are represented with a thick solid line (cf. Appendix A).
58
3.6
Spectral energy distribution at the stellar surface (τross = 2/3) of our models. .
59
3.7
Comparison between the H ionisation structure (red dashed line) and the Lyman
continuum optical depth at λ ∼ 900 Å (black solid line) versus the distance
from the stellar photosphere. Solid lines are reserved for the wind optical depth,
whilst the dotted horizontal lines indicate the transition between optically thick
and thin part of the wind in the Lyman continuum (τ = 1). Red colour on the
right-hand side is used for the H ionisation structure. . . . . . . . . . . . . . .
3.8
Wind optical depth at τross = 2/3 in the Lyman (left), Balmer (grey circles), and
Paschen continua (red squares) (right). . . . . . . . . . . . . . . . . . . . . . .
3.9
60
61
Number of photons in the Lyman (blue triangles), Balmer (grey circles), and
Paschen (red squares) continua vs T eff . Right-hand side is a “zoom in” from the
left-hand side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.10 Population levels of H as a function of Rosseland optical depth. The thick
solid lines in the lowermost panel illustrate a linear fit of n1 (black), n2 (red),
and n3 (blue) in the line formation region, i.e. between log(τross ) = −1.77 and
log(τross ) = −2.67. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.11 Non-LTE departure coefficients for the 2nd (solid) and 3rd (dashed) level of H. .
65
3.12 Upper panels: effect of Lyα on the formation of the Hα line: initial Hα profile
(black solid) and the profile from the models in which Lyα transitions were
artificially removed (the red dash-dotted line). Middle panels: changes in the
2nd (dashed) and 3rd (solid) level of H due to the removal of Lyα in model C
(left) and M (right). The plots present the ratio of the populations produced from
the initial model over the populations from the models without Lyα transition.
Lower panels: comparison of the net radiative rate of 2→1 transitions in the
initial (solid black) and the model without Lyα (the red dash-dotted line) (see
§ 3.5 for details). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
LIST OF FIGURES
LIST OF FIGURES
xiv
3.13 Lyα Sobolev optical depth as a function of τross . The region where most of
the emergent Hα photons originate from is shown with thick solid lines (cf.
Appendix A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
69
Left: synthetic Hα line profiles from clumped models with volume filling factor
fV∞ = 0.1. Right: Hα line EW as a function of the effective temperature for
homogeneous (circles) and clumped (squares) models. . . . . . . . . . . . . .
4.2
73
Hα Sobolev optical depth as a function of τross for homogeneous (left) and
clumped (right) models. Sites where most of the emergent Hα photons originates from are set with thick solid lines. White squares represent the point at
which 50% of the line EW is already formed (see Appendix A). . . . . . . . .
5.1
Hα line profiles for sophisticated supergiant models with parameters as listed in
Table 3.1, but different mass-loss rates.
5.2
. . . . . . . . . . . . . . . . . . . . .
81
Hα line EW as a function of T eff for models with different values of Ṁ in units
of 10−6 M⊙ yr−1 . Right hand side is a “zoom in” from the left hand side. . . . .
5.3
75
82
Influence of T eff and Ṁ on the morphology of the Hα line profile. White squares
indicate the positions of the grid-models used. . . . . . . . . . . . . . . . . . .
83
5.4
Influence of the stellar mass on the the Hα line profile. . . . . . . . . . . . . .
85
5.5
Influence of the stellar mass on the Hα line EW in models with strong (right)
and weak (left) winds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.6
Example of Hα line profiles in models with different luminosities. . . . . . . .
88
5.7
Influence of luminosity on Hα morphology. Model series ’L5.5M30’ (left) and
’L5.0M30’(right) are presented. The green solid line (right) indicates the absorption and P-Cygni transition mass-loss rates in model series ’L5.5M30’. . .
5.8
89
Left: Behaviour of WHα in sets of models with different luminosity and mass.
Right: Behaviour of WHα in sets of models with different Ṁ and metal composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
Example of changes in the Hα line profile due to a varying 3∞ /3esc ratio. . . . .
91
5.10 Influence of 3∞ /3esc ratio on Hα line EW. . . . . . . . . . . . . . . . . . . . . .
92
5.9
LIST OF FIGURES
LIST OF FIGURES
xv
5.11 Morphology of the Hα line for models with 3∞ /3esc ratio as known from obser-
vations and half-solar metal abundances. . . . . . . . . . . . . . . . . . . . . .
93
5.12 Comparison of the morphology of Hα in grid of models with solar (right) and
five times lower metal composition (left). 3∞ /3esc = 2.6 for models with T eff ≥
22 500 K 3∞ /3esc = 1.3 for cooler models. . . . . . . . . . . . . . . . . . . . .
6.1
94
Relative contribution of individual ions to the total radiative force for models
with 3∞ /3esc =1.3 and half-solar metallicities. . . . . . . . . . . . . . . . . . . 103
6.2
Left: Qwind vs T eff for models with half-solar metal abundances. The observed
velocity ratios of 3∞ /3esc = 2.6 for T eff ≥ 22 500 K, 3∞ /3esc = 1.3 for T eff ∈
[10 000 K, 20 000 K], and 3∞ /3esc = 0.7 for T eff < 10 000 K are applied. Right:
relative contribution of individual ions to the corresponding work ratio Qwind . . 105
6.3
Change in ionisation balance between Fe iv and Fe iii. . . . . . . . . . . . . . . 106
6.4
Acceleration in units of gravity as a function of τROSS for models from series
’L5.5M40’ with half-solar metallicities. Wind acceleration (red dash-dotted) is
compared to the acceleration from the prescribed velocity law (black solid). . . 106
6.5
Left: contour plot of Qwind as a function of T eff and Ṁ in model series ’L5.5M40’
with 3∞ /3esc = 2. If Qwind ∼ 1 then the radiative acceleration is able to drive
the wind. Right: contour plot of Qwind from the same model series but with
observed ratio of 3∞ /3esc = 2.6 for T eff >∼ 21 000 K and 3∞ /3esc = 1.3 for
T eff <∼ 21 000 K. White squares mark the positions of the of the calculated
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.6
Left: contour plot of the relative contribution of the ions of C,N, and O to Qwind ,
QCNO /Qwind , from model series ’L5.5M40’. Right: contour plot of the relative
contribution of iron to Qwind , QFe /Qwind , for same models. For T eff ≥ 22 500 K
3∞ /3esc = 2.6, whilst for T eff ≤ 20 000 K 3∞ /3esc = 1.3. . . . . . . . . . . . . . 108
6.7
Contour plot of Qwind in Ṁ − T eff plane for model series ’L5.5M30’ (left) and
’L5.75M71’ (right). Models have half-solar metal abundances and parameters
as listed in Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.8
Change in ionisation balance between Fe iii and Fe ii. . . . . . . . . . . . . . . 109
LIST OF FIGURES
LIST OF FIGURES
6.9
xvi
Acceleration in units of gravity as a function of τROSS for models from series
’L5.5M40’ with half-solar metallicities. Wind acceleration (red dash-dotted) is
compared to the acceleration from the prescribed velocity law (black solid). . . 110
6.10 Qwind vs Ṁ for models on both sides of the second bi-stability jump. . . . . . . 111
6.11 Contour plot of Qwind in Ṁ − T eff plane for model series ’L5.5M40’ (left) and
’L5.75M71’ (right). Models have solar metal composition and parameters as
listed in Table 6.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.12 Radiative force provided by Fe ii, Fe iii, CNO, and all ions as a function of λ and
τROSS for models with T eff = 8 800 K (left) and T eff = 20 000 K (right). Models
have solar metal composition. The lowermost panels illustrate the contributions
of the spectral lines to the work ratio obtained by the acceleration from Fe ii (left)
or Fe iii (right). The red line (with ordinate on the right-hand side) presents the
total contribution of spectral lines located in various frequency bins to the work
ratio of Fe ii (left) or Fe iii (right). . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.13 Dependence of the jump in Ṁ between 10 000 and 8 800 K on Eddington factor. The numbers in parentheses show the increase of mass loss rate across the
8 800 / Ṁ 10 000 . . . . . . . . . 116
second bi-stability jump in relative sense, i.e. ṀQ
Qwind =1
wind =1
6.14 Time-dependent Ṁ of AG Car against T eff as derived from Hα analysis by Stahl
et al. (2001).The solid indicate the changes in Ṁ over the period Dec. 1990−
Feb. 1995, when the star increases its visual brightness. The dotted line connects
points when visual brightness is decreasing. Figure from Vink & de Koter (2002). 118
A.1 Model C (T eff =12 500 K). Bottom: grey scale plot of the flux like quantity p ×
I(p) as a function of impact parameterp/R⋆ , where R⋆ is hydrostatic radius.
The figure provides the distribution of the emergent intensity around Hα from
different p. Top: corresponding normalised flux in Hα, directly obtained by
integrating p × I(p) over the range of p. . . . . . . . . . . . . . . . . . . . . . 124
A.2 Hα line EW as a function of τross . The figure illustrates how the EW changes
when outer layers of the star (p/R⋆ < 1) are added. . . . . . . . . . . . . . . . 125
LIST OF FIGURES
LIST OF FIGURES
xvii
B.1 The source function of Hα line SHα
λ over the mean integrated intensity (J) as
function of Hα Sobolev optical depth for simplified H+He (lower panels), sophisticated (upper panels; with atomic data as listed in Table 3.3), homogeneous
(left panels), and clumped (right panels) models. . . . . . . . . . . . . . . . . . 128
LIST OF FIGURES
Publications
A list of publications and talks resulting from the work presented in this thesis is given below.
Refereed Publications
Petrov, B., Vink, J., Gräfener G., On the Hα behaviour of blue supergiants: rise and fall over
the bi-stability jump, 2014, Astronomy & Astrophysics, 565, A62
In preparation
Petrov, B., Vink, J., Gräfener G., Two bi-stability jumps in the wind of blue supergiants and
their application to LBVs,
Conference proceedings
Petrov, B., Vink, J., Gräfener G., The B Supergiant problem and mass loss through Hα., 2013
Massive Stars: From α to Ω
Petrov, B., Vink, J., Gräfener G., The mass loss and nature of B supergiants., 2012, UKGermany National Astronomy Meeting
xviii
’To work creatively he /the artist/ must put flesh into
it, and enjoy it as a lark, or as a fascinating adventure.
How different from the workers in the heavy industry that professional writing has become!’
Ray Bradbury
Part I
INTRODUCTION
1
Chapter 1
Blue supergiants - troublemakers or
candles in the dark
The Earth is a very small stage in a vast cosmic arena. Think of the rivers of blood
spilled by all those generals and emperors so that in glory and triumph they could
become the momentary masters of a fraction of a dot. Think of the endless cruelties
visited by the inhabitants of one corner of this pixel on the scarcely distinguishable
inhabitants of some other corner. How frequent their misunderstandings, how eager
they are to kill one another, how fervent their hatreds. Our posturings, our imagined
self-importance, the delusion that we have some privileged position in the universe,
are challenged by this point of pale light. Our planet is a lonely speck in the great
enveloping cosmic dark. In our obscurity – in all this vastness – there is no hint that
help will come from elsewhere to save us from ourselves.
Carl Sagan, Pale Blue Dot
During the last 3 years it has been a great pleasure for me to sit on the north coast of Northern
Ireland and to admire the utter vastness of the ocean. It makes you think how small and insignificant we are. From this perspective the Earth seems enormously large, but it is not. It is not
readily apparent that the “huge“ Earth is in fact a tiny speck compared to our Sun and that the
Sun is actually a midget in the realm of the stars. Most stars in the Universe are tiny, just like our
2
3
Sun, but some of them are much larger, brighter and more massive. It is not a simple process for
the stars to gather large mass, because there are many factors which influence star formation and
stellar development, and these factors often limit the size and the mass of the stars. In addition
to that, the short evolutionary timescales of massive stars (with M⋆ > 8M⊙ ) makes them rare
objects.
Despite their scarcity, massive stars are very important actors in the Universe. They do not
only produce heavy elements, but also through their strong winds, massive stars release huge
quantities of mechanical energy and spread material in their surroundings. When they run out
of nuclear fuel, massive stars end their lives as supernovae (SNe), which makes them prominent
figures in the entire cosmos. In this way massive stars shape the interstellar medium and cause a
huge impact on the ecology of galaxies.
The role played by massive stars in our understanding of the Universe is paramount. Those
stars are the main culprit responsible for distributing most of the chemical elements in the Universe, including those necessary for the forms of life. As an example, the human body contains
about 65 % oxygen (by mass), 18.5 % carbon, 9.5 % hydrogen and 7 % heavier elements (Frieden
1972). Only hydrogen and helium were created during the primordial nucleosynthesis. Most elements heavier than helium (which comprise about ∼90 % of our body) are ejected into the space
by mass loss from luminous stars and by supernova explosions (Burbidge et al. 1957).
If we are able to understand how massive stars form, live and die, we will be one step closer
to unravelling mysteries about the origin of life. However, a proper understanding of their formation, evolution and death is still beyond current capabilities of modern astrophysics (Langer
2012).
Blue supergiants - troublemakers or candles in the dark
1.1 The life-cycle of massive stars
4
1.1 The life-cycle of massive stars
Stars more massive than about 8 M⊙ have a completely different evolution to less massive stars.
While low mass stars end their evolution producing a white dwarf in the middle of planetary
nebula, the massive stars end their lives in spectacular supernova explosions, leaving behind a
neutron star or black hole. In this section, a brief overview of the evolution of massive stars is
given.
1.1.1 Main-sequence Evolution
During most of their life, massive stars convert hydrogen (H) into helium (He) in their cores
through the nuclear CNO cycle (Bethe 1939). In the CNO cycle four protons are merged into a
4 He
nucleus through catalyst species (carbon, nitrogen and oxygen). As C, N and O nuclei are
highly charged, the protons interacting with them must penetrate a stronger Coulomb potential
and higher velocities are required. Therefore, CNO cycles are relevant for stars with masses
larger than approximately 1.5 M⊙ . Stars burn H for about 90 % of their lifespan. When all H
in the stellar core is transformed into He, the generation of energy stops and the core contracts,
whilst the envelope expands rapidly because of the “mirror principle”(Kippenhahn & Weigert
1990). As a result the star decreases its effective temperature. This transition occurs on a very
short time-scale, thus there is little chance to observe a star in this phase. However, this part of
the Hertzsprung-Russell diagram (HRD) is populated by many B-type supergiants (as will be
discussed in § 1.2).
Blue supergiants - troublemakers or candles in the dark
1.1 The life-cycle of massive stars
5
Figure 1.1: – (a) Evolutionary tracks for stars with masses from 15 to 30 M⊙ . (b) Evolutionary
tracks for star with mass 25M⊙ with and without mass loss rate. Figure from El Eid et al. (2004).
1.1.2 Life after the main-sequence
The post-main-sequence (MS) evolution of massive stars may involve many phases including
blue supergiants (BSG), red supergiants (RSG), luminous blue variables (LBV), and Wolf-Rayet
stars. The evolution depends on many parameters including the total mass, metallicity, magnetic
field, rotational velocity and mass-loss rate.
From models and observations it is known that after the MS, massive stars evolve towards
the red giant branch (RGB) with nearly constant bolometric luminosities (cf. Fig. 1.1a; see also
Smith et al. 2004; El Eid et al. 2004) but that depends not only on mass, but also on the massloss rate. Actually, the stellar mass-loss rate is one of the most important quantities determining
the evolution of massive stars. A precise description of mass loss is required to understand the
evolution of massive stars. As an example, Fig. 1.1b shows that models with mass loss evolve
towards the RGB at lower luminosities in comparison to models without mass loss.
During the core He-burning phase, the main contributors to the luminosity are the He-burning
core and the H-burning shell. El Eid et al. (2004) showed that the H-burning shell in a case
without mass loss is stronger. As a consequence, the helium core has a lower mass in the model
with mass loss. Because of the lighter core, the star that evolves with mass loss should have a
Blue supergiants - troublemakers or candles in the dark
1.1 The life-cycle of massive stars
6
lower luminosity than the evolution of the one without mass loss (Fig. 1.1b)
The centrifugal force associated with rotation balances part of the gravitational force and
could effect the stellar evolution. Rotation “makes” the star slightly lighter than it actually is
and it reduces its effective surface gravity. Thus, one may expect that a rapidly rotating star
has to appear slightly less luminous and cooler than a slowly rotating star with the same mass
(von Zeipel 1924). However, new models for massive stars show the opposite effect: rotation
increases stellar luminosity and T eff because of larger convective cores (Leitherer et al. 2014).
Consequently, rotation would lead to older ages, as the less massive but faster-rotating stars
would have the same luminosities as slowly rotating, but more massive stars. Rapid rotation may
also help to lift material from/close to the core up to the surface and in this way could enhance
the surface metal abundances (see e.g. Heger et al. 2000). This process is most important for
MS evolution, because MS stars have a wide range of rotational velocities (see Hunter et al.
2008, and § 1.3.4). After the MS the stars slow down their rotation due to the increase of their
radius. In addition to that the timescale for post MS evolution is much shorter than the hydrogen
burning timescale. These effects make rotational mixing very inefficient after the MS.
The chemical composition of a star is also a very important parameter controlling the star’s
evolution. The models predict that the mass-loss rate is metallicity dependent: Ṁ ∝ Z m , where
m is in the range 0.47–0.94 (Abbott 1982; Kudritzki et al. 1987; Vink et al. 2001; Kudritzki
2002; Mokiem et al. 2007). In Fig. 1.2 Ṁ is shown as a function of metallicity. On the basis
of Monte Carlo calculations for OB-stars Vink et al. (2001) predicted a value of m = 0.69 (O
stars) and m = 0.64 (BSGs), while Kudritzki (2002) predict a value of m = 0.5 − 0.6. The
evolutionary tracks of the massive stars are significantly affected by mass loss, which is why a
precise description of mass loss and a more accurate value for m are needed to construct reliable
evolutionary models for massive stars.
Depending on the exact wind properties an evolving massive star could return to hotter
surface temperatures and perform a blue-loop on the HRD during the core He burning phase
Blue supergiants - troublemakers or candles in the dark
1.1 The life-cycle of massive stars
7
Figure 1.2: Mass-loss predictions as a function of metallicity. With the solid line is expressed
the dependence of mass-loss rate of metallicity according to Vink et al. (2001) and the dotted
line shows the predictions of Kudritzki (2002). Figure from Vink (2006).
(Georgy et al. 2013a).
When the He-burning phase ends the core contracts until carbon ignition. During subsequent
burning phases, which are much shorter, the temperatures in the core increase so much, that
elements with atomic masses up to iron can be synthesised. When that happens the star explodes
as a supernova (SN).
1.1.3 Supernovae from Blue Supergiants
SNe are probably the most spectacular events in the known Universe and can sometimes be
observed even with the naked eye. They are extremely energetic stellar explosions signalling
the final stage of massive star evolution. SNe could be valuable probes for passed events that
Blue supergiants - troublemakers or candles in the dark
1.1 The life-cycle of massive stars
8
Figure 1.3: Observed fractions of each type of SN. Figure from Li et al. (2011).
Type
Type Ia
Type Ib
Type Ic
Type IIP
Type IIL
Type IIN
Table 1.1: Supernova classification
Characteristics
Lacks hydrogen and presents a singly ionised silicon (Si II) line at 615.0
nm.
Non-ionised helium (He I) line at 587.6 nm and no strong silicon absorption
feature near 615 nm.
Weak or no helium lines and no strong silicon absorption feature near 615
nm.
Reaches a "plateau" in its light curve.
Displays a "linear" decrease in its light curve.
Displays narrow H emission lines.
happened billions of years ago and far away from us – if we can understand them well enough.
There are two main types of SNe according to the absorption lines of different chemical
elements that appear in their spectra. If the spectrum of SNe contains H lines it is classified
as Type II, otherwise it is Type I. Type I can then be further divided into types Ia, Ib and Ic
(Table 1.1). The progenitors of Types Ib and Ic have lost most of their outer envelopes due to
strong stellar winds which can occur for the case of a Wolf-Rayet (W-R) star. These massive
objects show a spectrum that is lacking in H. Type Ib progenitors have ejected most of the H in
their outer atmospheres, while type Ic progenitors have lost both the H and He shells.
Li et al. (2011) showed that type II SNe are marginally the more common SN types (Fig. 1.3).
Most of these SNe are believed to be from RSGs. Kleiser et al. (2011) estimated that the SNe
Blue supergiants - troublemakers or candles in the dark
1.1 The life-cycle of massive stars
9
from BSGs are ≈ 2% of all core-collapse SNe. Nevertheless, there are a number of SNe
produced by likely BSG explosions (SN 1909A; SN 1987A, SN 1998A, SN 2000cb, SN 2005ci,
SN 2006V, SN 2006au, SN 2009ip, and SN 2010mc), we are 100% sure only for SN 1987A.
While SN 1987A is the only confirmed SN with a BSG progenitor, SN 1909A and SN 1998A
have a similar light curve which suggest that they may also have had a BSG as progenitor.
Kleiser et al. (2011) and Taddia et al. (2012) found similarities between SN 2000cb, SN 2005ci,
SN 2006V and SN 2006au and suggested that they might also arise from BSGs. More recently,
Smith et al. (2014) suggested that the progenitors of SN 2009ip and SN 2010mc might be BSGs
as well, as their spectra were found to be very similar to the spectrum of SN 1987A.
SN 1987A SN 1987A was the nearest and brightest SN in the night sky since Kepler’s star of
1604. It was classified as a type II SN because it has hydrogen lines in its spectrum. However,
the progenitor of SN 1987A – was observed as a BSG of spectral type B3 I, with helium core
mass ∼ 6 M⊙ , which corresponds to a main-sequence star with mass of about 20 M⊙ (Nomoto
et al. 1987; Woosley et al. 1987). Such an evolution for a BSG was not expected. Even now,
after more than 25 years, the explosion of SN 1987A is still not well understood. While the
vast majority of Type II SNe are usually produced by RSGs, SN 1987A showed that sometimes
BSGs also explode. We just do not understand why massive stars normally make red supergiants
and then type II SNe but this one, which was the closest, from which have most information,
was a BSG.
The most surprising finding of this event concerns the evolution of the blue progenitor before explosion. Observations show the existence of low-velocity circumstellar shells around
SN 1987A (Fransson et al. 1989), which strongly indicates that the progenitor was in a RSG
phase before its explosion. This was the most difficult aspect of the evolution to explain (Saio
et al. 1988). Even now, three decades after this event, there is no conclusive answer to the
question “why did the progenitor of SN 1987A undergo the blue-red-blue evolution?”
Sher 25: a twin of SN 1987A? Sher 25 is classified as a hot supergiant of spectral type B1 Iab.
Blue supergiants - troublemakers or candles in the dark
1.2 Troublemakers across the Hertzsprung-Russell diagram
10
The star has a circumstellar ring-shaped nebula with a structure which is similar to that of
SN 1987A in spatial extent, velocity and mass (Brandner et al. 1997). Brandner et al. (1997)
noted an enhanced N abundance in the ring nebula around Sher 25 and concluded that the star
should be an evolved BSG that passed through the RSG phase. On this basis, the authors suggested that Sher 25 is possibly a twin of the progenitor of SN 1987A and therefore is expected
to explode within the next few thousand years or even sooner.
At the moment the progenitor evolution of SN 1987A is still not understood but maybe detailed studies of Sher 25 would help. So far, we do not even know the evolutionary status of
BSGs, as will be discussed in the next section. If we do not understand the evolution of the
BSGs, then it is not surprising that we do not understand the evolution of the progenitor of
SN 1987A.
1.2 Troublemakers across the Hertzsprung-Russell diagram
The first attempt to understand the physics behind the light from the stars can be traced back
to Emden (1907). Despite our knowledge about stellar structure and evolution has improved
radically since then, yet, after more than a century our understanding of many observational
properties of massive stars remains incomplete.
Currently, evolutionary models cannot reproduce the distribution of B-type supergiants (Bsgs)
across the HRD. Theory predicts a clear gap between core-hydrogen and blue core helium burning stars. Observations show that the “forbidden” area is populated by B-type supergiants (cf.
Fig. 1.4) whose evolutionary state is still under debate (Fitzpatrick & Garmany 1990; Evans et al.
2006; Vink et al. 2010; Larsen et al. 2011; Georgy et al. 2013b). This gives rise to so called “Blue
Hertzsprung Gap problem” and makes the BSGs “troublemakers” in modern astrophysics.
As a possible solution to the problem, rotational mixing was introduced into stellar evolution
Blue supergiants - troublemakers or candles in the dark
1.2 Troublemakers across the Hertzsprung-Russell diagram
11
Figure 1.4: H-R diagrams of O- and early B-type stars in the fields of the Magellanic Clouds.
Open circles represent foreground stars. Objects with evidence for binarity are denoted with
crosses and the open triangles indicate objects with emission lines. The evolutionary tracks for
models with LMC metallicity (N 11 and NGC 2004) are obtained from Schaerer et al. (1993),
and from Charbonnel et al. (1993) for SMC metallicity (NGC 330 and NGC 346). Figure from
Evans et al. (2006).
models (Meynet & Maeder 1997, 2000; Heger & Langer 2000; Ekström et al. 2012; Georgy
et al. 2013a), which produced a wider main-sequence. However this solution is not conclusive,
partly because binarity and magnetic fields could be important ingredients in rotational mixing
(Brott et al. 2011), partly because the processes which prescribe massive star evolution, such as
mass loss, convection, and efficiency of convective overshooting are not well known.
In addition to that, knowledge of the evolutionary status of the BSGs is also hampered by
the fact that the surface metal abundance is not an unambiguous indicator of the evolutionary
state. This signature is obscure due to the possibility that the metal enrichment results from deep
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
12
mixing during a RSG phase or from rotational induced mixing on the MS enhanced by high
mass loss (Herrero & Lennon 2004). As a result, massive stars with similar metal abundances
(an important indicator of evolutionary state) and similar observational properties might be in
two different stages of their evolution. Therefore, large spectroscopic surveys, such as vltflames i (Evans et al. 2006) and vlt-flames tarantula (Evans et al. 2011), provide more fruitful
approaches to reveal the evolutionary connection between MS and post-MS objects.
Understanding qualitatively the quantities which determine the evolution of massive stars
may provide valuable new insights into the evolutionary properties of BGSs and could transform
them from troublemakers into a “gift from nature” (Kudritzki 1996). To be specific, the stellar
wind mass loss, one of the most important drivers of massive star evolution, is not yet understood. Moreover, understanding BSG mass loss is expected to provide powerful extra-galactic
distance indicators, via the Wind-Momentum-Luminosity relation (Kudritzki et al. 1994, 1999).
1.3 Aspects of BSGs
1.3.1 Candles in the dark
Because of their considerable brightness, BSGs are easy to observe individually in nearby
galaxies. In principle, that makes them ideal candidates for standard candles. Unfortunately,
the photospheres of BSGs are complicated by the presence of strong stellar wind outflows
which contaminate their spectra. The success of the radiation wind theory combined with the
presence of wind lines, led to the realisation that these wind features provide a key to distance
determination via the Wind-Momentum-Luminosity relation (WLR).
The theory of radiation driven winds (Castor et al. 1975; Kudritzki et al. 1989) predicts
that the total mechanical wind momentum rate, Ṁv∞ , should be proportional to the photon
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
13
momentum rate of the photosphere, which is a function of the stellar luminosity:
Ṁ3∞ ∝ L1/αeff R−0.5
⋆ ,
(1.1)
where Ṁ is the mass-loss rate, R⋆ the stellar radius and v∞ is the terminal velocity of the stellar
wind. αeff is dimensionless power law exponent of the order of ≈ 2/3, which corresponds to the
line-strength distribution. Generally, αeff can be expressed as the difference of two dimensionless
numbers, α and δ. The parameter α depends on the optical depth of driving lines and quantifies
the ratio of the line force from optically thick lines to the total one. If the wind was driven by
optically thick lines then α is expected to be 1. If all driving lines were optically thin then α
should be 0. In fact, the radiative acceleration is caused by an assortment of optically thin and
thick lines and therefore 0 < α < 1. The parameter δ takes into account the variation of the
ionisation throughout the wind (Abbott 1982; Kudritzki et al. 1989).
Spectral analysis of wind lines from galactic BSGs confirmed that the theoretically predicted
“modified” wind momentum, Ṁv∞ (R⋆ /R⊙ )0.5 , scales reasonably well with stellar luminosity
(Kudritzki et al. 1989; Puls et al. 1996; Kudritzki et al. 1999). This is shown in Fig. 1.5, where
the adopted WLR is in the form:
log Dmom = log D0 + xlog (L/L⊙ ),
(1.2)
with αeff = 1/x = α − δ. The figure demonstrates that the slope x of the WLR changes with
spectral type. This is an indication that the winds are driven by different ions. It is evident from
the figure that the slope is higher for later spectral types, which is expected if iron (Fe iii and
Fe ii) dominate the wind driving (Vink et al. 1999, § 6).
Kudritzki et al. (1999) noted that the uncertainties in the distances determined from the WLR
appear to be comparable to those obtained from Cepheids, but the advantage of the WLR-method
is that the individual reddening and metallicity can be obtained directly from the spectrum of
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
14
Figure 1.5: Modified wind momenta as a function of luminosity for galactic OBA supergiants.
Early B: B0 to B1;mid B: B1.5 to B3. Figure from Kudritzki et al. (1999).
every object.
The WLR relation requires a precise determination of the stellar mass-loss rates. However,
there are significant discrepancies between theoretical and empirical mass-loss rates (as will
be discussed in § 1.3.3). Moreover, discrepancies exist between different mass-loss diagnostics (Hα, UV and radio: Massa et al. 2003; Puls et al. 2006b; Fullerton et al. 2006), which
may be attributed to distance-dependent clumping and/or porosity effects (Oskinova et al. 2007;
Sundqvist et al. 2010). To use the BSGs as “candles in the dark“, reliable determination of their
real mass-loss rates is required. Therefore, a proper understanding of the conventional mass-loss
diagnostics, such as Hα, is needed and this is the primary motivation of this thesis.
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
15
1.3.2 Wind properties and the bi-stability jump
The winds of BSGs are pushed by radiation pressure in spectral lines. This is possible because
of three simple facts.
1. BSGs are luminous and hot stars and therefore release huge quantities of photons in the
ultraviolet. This is a consequence from Wien’s displacement law for black-body radiation:
λmax =
2.9 × 107 Å.K
.
T (K)
(1.3)
2. In this spectral range, the outer atmospheres of these stars have plenty of absorption lines
with substantial opacity. The opacity of one strong line (e.g. the C IV resonance line
at 1550 Å) could be millions times larger than the electron scattering opacity (Lamers
& Cassinelli 1999). Consequently, the photon-momentum rate provided by the stellar
photosphere is transferred to the ions that have these absorption lines. Finally, the gained
momentum of the ions is shared with the rest of the plasma (protons, electrons, helium
ions) via Coulomb coupling.
3. The third important ingredient of line driven winds is the Doppler effect. Without it, the
absorption in spectral lines would be negligible in the outer layers, because the radiation
from the photosphere at the wavelength of the lines would already be absorbed in the lower
layers of the atmosphere. Therefore the radiative acceleration in the outer wind would be
very weak without the Doppler shift.
Accurate information about radiative acceleration could lead to valuable statements about the
mass-loss rates or terminal velocities of the stellar winds. The reverse is also true: information
about the radiative force can be inferred from the empirical terminal velocities of the winds.
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
16
Figure 1.6: The observed bi-stability jump in terminal wind velocities near T eff ≈ 21 000 (at a
spectral type B1) . A second jump may be present at T eff ≈ 10 000 (at spectral type A0) where
v∞ /vesc ratio drops from 1.3 to 0.7. Figure from Lamers et al. (1995).
Lamers et al. (1995) were the first to show that the empirical terminal velocities are discontinuous near spectral type B1, where the winds with fast velocities (3∞ /3esc ≈ 2.6) switch to
slow winds with 3∞ /3esc ≈ 1.3 for stars cooler than ≈ 21 000 K, i.e, near spectral type B1 (cf.
Fig. 1.6). Later on, Vink et al. (1999) predicted that the drop in the terminal velocities should
be accompanied by an increase (or a jump) in mass-loss rate. This is the so-called bi-stability
jump1 which is still under debate.
On the basis of sophisticated line-blanketed model atmospheres Crowther et al. (2006) found
that the observed bi-stability jump in wind velocities represents a more gradual decrease than
1
The term bi-stability was originally established to imply that the stellar wind can change between two states.
Despite that, the discontinuity displayed in Fig. 1.6 is produced by the bi-stability mechanism and the figure compares
stars in one state with stars in the other state.
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
17
Figure 1.7: The ratio of 3∞ /3esc as a function of T eff for BSGs from Crowther et al. (2006) (left)
and from Markova & Puls (2008) (right).
a jump as suggested by Lamers et al. (1995). Their results are shown in the left-hand side of
Fig.1.7. However, Markova & Puls (2008) found that the early B supergiants have considerably
different wind velocities than the wind velocities of late B supergiants, supporting the jump
scenario. This is shown in right-hand side of Fig. 1.7. Note that between 23 000 and 18 000 K,
a variety of ratios are present, indicating that the bi-stability jump does not occur at specific
temperature. As will be discussed in Chapter 6 (but see also Vink et al. 1999), the bi-stability
jump is sensitive to the ionisation equilibrium (chiefly of Fe iii), and therefore, we anticipate
the temperature of the jump to be different for samples of stars with different luminosities and
masses. Consequently, the exact temperature of the bi-stability jump is somewhat ambiguous
and a transition zone in Fig. 1.7 is expected.
The idea of a bi-stability mechanism was first introduced by Pauldrach & Puls (1990). Based
on model calculations of the wind of the prototype star P Cygni they found that small photospheric changes result either in a wind with relatively low mass loss and high terminal velocity,
or to a wind with relatively high mass loss and low velocity. They suggested that the physics of
the bi-stability mechanism is related to the optical depth of the wind in the Lyman continuum.
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
18
If the Lyman continuum exceeds a certain optical depth (τ & 3), then the Lyman photons are
blocked. As a consequence the metals shift to lower ionisation stages and the radiation force
is produced mainly by a very large number of weak metal lines in the Balmer continuum. This
effect increases the radiative pressure, which finally produces a jump in Ṁ and a drop in v∞ .
The model calculations of Pauldrach & Puls (1990) show that the mass-loss rate and the
terminal velocity are anti-correlated: the jump in mass-loss rate (which is about a factor of 3) is
compensated by a drop in terminal velocity (about a factor of 0.3) (see also Lamers & Pauldrach
1991). This indicates that, independent of the degree of ionisation, approximately the same
fraction of the photon momentum rate provided by the stellar photosphere is transferred to the
mechanical wind momentum Ṁv∞ . Therefore Lamers et al. (1995) suggested that the observed
drop in the wind terminal velocity has to be accompanied by a jump in the mass-loss rate (by
about of factor of 2) around T eff ≃ 21 000 K, in order for Ṁv∞ to be similar on both sides of the
jump.
To investigate the origin of the observed bi-stability jump in wind terminal velocities and
whether there is a jump in Ṁ, Vink et al. (1999) computed a series of radiation driven wind
models in the temperature range between 40 000 and 12 500 K, by means of a Monte Carlo technique. In their models, the mass-loss rate was increased by a factor of about five between 27 500
and 22 500 K (Fig. 1.8). This increase was caused by a change in iron ionisation. More specifically, with the decrease of effective temperature from 27 500 to 22 500 K, the Fe iv ionisation
fraction dropped in favour of Fe iii. As a consequence, Fe iii increased the radiative force below
the sonic point2 and determined the mass-loss rate. This result implies that the bi-stability jump
should be very sensitive to the abundance and ionisation balance of iron.
It is noteworthy that the produced jump in Ṁ by about a factor of 5 from Vink et al. (1999)
was accompanied by a drop in v∞ by about a factor of 2. This difference may exist because Ṁ is
2
The sonic point in Vink et al. (1999) models was defined as the point where the wind speed reaches the isothermal
speed of sound.
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
19
Figure 1.8: Predicted bi-stability jump in Ṁ. Figure from Vink et al. (1999).
determined in the subsonic region, where Fe iii is the important line-driver, whilst v∞ still has to
be determined in the supersonic part of the wind, where C, N and O are important line drivers.
Therefore, C, N and O are expected to have an insignificant effect on Ṁ but to be important for
v∞ . Consequently, the fraction of the momentum of radiation, L/c, which is transferred into the
wind momentum, Ṁv∞ , is not expected to be constant on both sides of the bi-stability jump but
should be dependent on the degree of ionisation.
The results from the models of Vink et al. (1999) are based on only one value of M⋆ , L⋆ and
H/He abundance. However, Ṁ depends on these parameters and further calculations of mass
loss rates for stars with different luminosities and masses will provide valuable information for
the bi-stability jump in terminal velocities and mass-loss rate. Despite all this, the predicted
decrease of v∞ by a factor of ∼2 is observationally confirmed by more recent investigations
(Markova & Puls 2008), which gives confidence that the predicted jump in Ṁ should actually
be real. Unfortunately, there is currently a mess in the empirical mass-loss rates which hampers
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
20
the confirmation or refutation of the predicted jump in Ṁ.
1.3.3 Mess in the mass loss rates
Observations and predictions. Lamers & Leitherer (1993) and Puls et al. (1996) showed that
for O stars the observed mass-loss rates are systematically higher than the values predicted by
radiation-driven wind theory. On the basis of multiple linear regression analysis of Monte Carlo
models, Vink et al. (2000, 2001) derived two mass loss recipes as a function of luminosity, mass,
terminal velocity and metallicity of the star. If these parameters are known from observations,
mass-loss rates could be calculated and compared to the empirical mass-loss rates (but see also
Howarth & Prinja 1989).
Late B supergiants. According to the theoretical predictions by Vink et al. (1999) the decrease
in v∞ over the bi-stability region should be over-compensated by an increase of Ṁ. Therefore,
the late B-type supergiants (later than B1) should have higher wind momenta, Dmom , than earlytype supergiants. In order to test this expectation several works found significant discrepancies
between theoretical and empirical mass-loss rates for late B-type supergiants (Vink et al. 2000;
Trundle et al. 2004; Trundle & Lennon 2005; Crowther et al. 2006; Benaglia et al. 2007; Markova
& Puls 2008). They found that the empirical mass-loss rates from Hα are generally lower than
the predicted values for Bsgs at the cool side of the bi-stability jump. As a consequence, the
empirical wind momentum (from Hα and radio observations), Dmom , for late B supergiants was
found to be systematically lower than predicted. Markova & Puls (2008) and Searle et al. (2008)
found that supergiants later than B2 have wind momenta which are even lower than Dmom from
high temperature predictions (27 500 <T eff <50 000 K)3 . Searle et al. (2008) emphasised that
empirical models for Bsgs likely have an incorrect ionisation structure as they found it challenging to reproduce the optical Hα line simultaneously with key ultraviolet (UV) diagnostics.
3
This is intriguing because according to the predictions on the hot side of the bi-stability jump Ṁ is expected to
be lower than the mass-loss rate at the cool side (cf. Fig. 1.8) and thus Dmom should be lower for hotter stars.
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
21
Despite all these discrepancies, Vink et al. (2000) remarked that for late Bsgs their predictions
agreed reasonably well with observed rates from both radio and Hα in emission, but a large
discrepancy was found when the Hα line was P Cygni shaped or in absorption.
Early B supergiants. The results for early B-type supergiants are controversial. While Vink
et al. (2000) found good agreement between their predictions and ”observed“ mass-loss rates
from Hα and radio for early subtypes, Markova & Puls (2008) noted that their majority of O
supergiants are consistent with the low temperature predictions (12 500 < T eff < 22 500 K) and
most of the early B0-B1.5 supergiants follow the high temperature predictions of Vink et al.
(2000). The result of Markova & Puls (2008) is consistent with the finding of Searle et al.
(2008), who noted that the early Bsgs have higher wind momenta than predicted.
To make the picture even more complex, one should be aware of discrepancies between massloss rates estimated from Hα, UV, and radio observations for OB stars in general (Massa et al.
2003; Bouret et al. 2005; Puls et al. 2006b; Fullerton et al. 2006), which may be due to distancedependent wind clumping and/or porosity effects (Oskinova et al. 2007; Sundqvist et al. 2010,
2011; Muijres et al. 2011; Šurlan et al. 2012).
Generally, the predictions underestimate the mass-loss rates derived from Hα for early Bsgs,
while later subtypes are overestimated. The question about this discrepancy will be raised again
in Chapter 3, where a detailed analysis of the Hα line over the temperature range of the bistability jump is provided. So far it is worth to mention that a local maximum was uncovered by
Benaglia et al. (2007) and Markova & Puls (2008) in radio and Hα mass-loss rates at the location
of the bi-stability jump, which support qualitatively the existence of a bi-stability jump in Ṁ.
However, to confirm the bi-stability jump in mass-loss rate a larger sample of stars is required.
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
22
Figure 1.9: Mass-loss rate (blue dotted) and rotational velocities of a Galactic 40 M⊙ star which
had a initial rotational velocity of 275 km/s on ZAMS, including predicted bi-stability jump (red
solid) and without it (green dashed).Figure from Vink et al. (2010).
1.3.4 Rotational velocities
On the MS, O-type stars are observed as rapid rotators with 3 sin i up to ∼600 km/s (Howarth
et al. 1997; Vink et al. 2010; Ramírez-Agudelo et al. 2013). Such rapid rotation can influence
the evolution of massive stars because of it will affect the effective gravity, mixing of chemical
elements, and mass-loss rate. Stellar evolution models (Langer 1998, and references therein)
generally assume that rotating massive stars have an enhanced mass-loss via the following relation:
ṀΩ = Ṁ(3rot =0)
1
1−Ω
!ξ
, ξ ≈ 0.43
(1.4)
where Ω = 3rot /3crit with 32crit = GMeff /R⋆ . The effective mass Meff = M⋆ (1 − Γe ) accounts for
the radiation pressure due to electron scattering, with
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
23
Γe =
σe L ⋆
.
4πcGM⋆
(1.5)
Although the validity of Eq. 1.4 is questionable (cf. e.g. Owocki et al. 1996), an increase
of mass-loss rate with rotation may still remain valid, as the mass flux is proportional to the
effective gravity (Owocki et al. 1996). If rapid rotation has large influence on mass-loss rates,
then the evolution of massive stars would depend on their rotational properties (see however
Müller & Vink 2014).
When Ω approaches unity (Ω-limit), according to Eq. 1.4, M˙Ω highly increases and the star
slows down effectively via angular momentum loss at a rate:
J˙ = βR2⋆ ω M˙Ω,
(1.6)
where ω is the angular velocity of the star, Ṁ is the mass-loss rate of the wind, R⋆ is the stellar
radius and β is parameter which depends on the mass loss geometry. For spherical mass loss
β= 2/3; if the mass is lost only from the equator then β = 1 (see e.g. Lamers 2004).
In a rapidly rotating star with angular velocity ω0 , the layers with largest specific angular
momentum are the closest to the stellar surface . These layers have initial angular momentum
j0 = ω0 R2⋆ . If a small amount of mass is taken away from the stellar surface, the lost mass
carries away its specific angular momentum and due to local angular momentum conservation
the layers close to the surface will spin down. Meanwhile, the core will still have the same initial
angular velocity. However a momentum transport from the core will again increase the surface
angular velocity somewhat, but to ω1 < ω0 . This behaviour will slow down the rotation of the
star.
Langer (1998) already showed that the mass-loss rate could be increased by an order of
magnitude via rotation if the Ω-limit is reached for a 60 M⊙ star ( ṀΩ ≈ 10−5 ). The time spent
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
24
Figure 1.10: Left: rotational velocities vs T eff for Howarth et al. (1997) data-set of Galactic OB
supergiants (red diamonds) and non-supergiants (blue triangles) Right: rotational velocities of
LMC supergiants (red asterisks) and non-supergiants (blue pluses) as a function of T eff . The
gray lines indicate LMC evolutionary tracks with initial vrot = 250 km/s for models with masses
= 15, 20, 30, 40 and 60 M⊙ . The black dots on the tracks illustrate 105 year time-steps. Figure
from Vink et al. (2010).
near Ω-limit depends strongly on the initial rotation rate. A rapidly rotating star would reach the
Ω-limit earlier than a slowly rotating star and consequently would lose more mass and angular
momentum (see Langer 1997, 1998, for details).
However, if the Ω-limit is not reached and if the mass-loss rate is truly increased at the bistability jump, then the stars at the cooler side of the jump should lose more angular momentum
than the stars at the hotter side. Thus, one may expect a drop in rotational velocities at the
bi-stability jump.
In order to test this idea, Vink et al. (2010) modelled the resulting rational velocities of a
Galactic 40 M⊙ star, including the predicted bi-stability jump and without it. Their results are
illustrated in Fig. 1.9. The figure displays a severe drop in rotational velocity when the mass-loss
rate is increased due to the bi-stability jump. If the mass-loss rate is not increased due to the
bi-stability jump, the star remains rapidly rotating.
To test the potential existence of such wind induced ”braking” at the bi-stability jump, Vink
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
25
et al. (2010) examined the rotational properties of OB stars from Howarth et al. (1997) and the
flames objects (Evans et al. 2008).
Howarth et al. (1997) measured the projected rotational velocities, 3 sin i, for 373 OB stars
and found that there are no O supergiants in the sample with 3 sin i < 65 km s−1 . Whilst
the late O/early B supergiants rotated rapidly (with 3 sin i as high as ∼250 kms), all late Bsgs
(T eff < 22 000 K) were slow rotators (with 3 sin i<100 km/s). The 3 sin i values from their dataset are displayed in Fig. 1.10 (left). The figure reveals a steep drop in 3 sin i for stars cooler than
∼ 22 000 K. Vink (2008) and Vink et al. (2010) also established a general lack of fast rotating
Bsgs (3 sin i > 50 km s−1 ) in LMC (Fig. 1.10, right). In both data-sets, the observed temperature
where 3 sin i drops steeply is at the temperature where the “bi-stability braking” is predicted to
occur. Therefore, Vink et al. (2010) suggested that the slow-rotating Bsgs could naturally be
explained as MS stars, if they lose their angular momentum via an increased mass-loss rate due
to the bi-stability jump. They also pointed out that this mechanism would be efficient if the
stars spend a significant amount of time on the MS. However, Hunter et al. (2008) argued on the
basis of high-resolution vlt-flames data that the slowly rotating Bsgs in the LMC and SMC are
post-MS objects, although their large numbers would remain unexpected. While the former hypothesis received some support by the apparently brightest SN in the telescopic era, SN 1987A,
bi-stability braking for stars with initial masses above 40 M⊙ was confirmed by Markova et al.
(2014). This raises again the issue of the evolutionary state of BSGs.
The two populations scenario
Vink et al. (2010) found the vast majority of slowly rotating supergiants to be N enriched (cf.
Fig. 1.11), which implies that these objects might be evolved. Therefore, they suggested an
alternative scenario in which the cooler and slowly rotating supergiants might form an entirely
separate, non core hydrogen-burning population. That population might the be product of binary evolution (although this would normally be expected to lead to rapidly rotating objects) or
Blue supergiants - troublemakers or candles in the dark
1.3 Aspects of BSGs
26
Figure 1.11: Nitrogen abundance as a function of T eff for LMC objects. Figure from Vink et al.
(2010)
maybe blue-loop stars. However, the N might be a misleading diagnostic as far as evolution is
concerned. The enhanced N does not necessarily imply a post-RSG status (see e.g. Meynet &
Maeder 2000). On the main sequence rotational mixing (enhanced by mass loss) may lead to N
enrichment (Herrero & Lennon 2004).
In this it remains unclear how the cooler supergiants lost their angular momentum and why
the drop in 3 sin i is at the temperature of the bi-stability jump. It is possible that the observed
sample of slowly rotating Bsgs is a result of both scenarios. Vink et al. (2010) argued that “The
strongest argument for the two population scenario is the large N abundances of the BSGs,
whilst the strongest argument for bi-stability braking is that the drop is observed at the correct
location”. Currently, there is not enough information to conclude which one is correct.
Blue supergiants - troublemakers or candles in the dark
1.4 Overview of the Chapters of this Thesis
27
1.4 Overview of the Chapters of this Thesis
In view of the properties of the BSGs discussed in previous sections, this thesis focuses on two
main problems:
• The Hα line as a mass-loss indicator:
We were especially interested to understand the influence of T eff on the formation of Hα
and the significance of micro-clumping on both sides of the bi-stability jump.
• Wind properties of BSGs
In order to better understand the wind properties of massive stars, we have investigated
also the physical ingredients that play a role in the line acceleration. The origin and the
properties of the second bi-stability jump in Ṁ were studied in detail. The existence of
the first bi-stability jump in Ṁ was confirmed as well.
The thesis is organised as follows:
• Chapter 2: a brief presentation of the methods for deriving mass-loss rates from massive
stars. and of the cmfgen code.
• Chapter 3: qualitative study of the temperature dependence of the Hα line
• Chapter 4: the effects of micro-clumping on the Hα line and the importance of macroclumping.
• Chapter 5: a quantitative analysis of the Hα line in realistic BSG models and the effect of
luminosity, stellar mass, and the bi-stability jump in 3∞ /3esc .
• Chapter 6: a study of ∼ 21 000 K and ∼ 10 000 K bi-stability jumps.
• Chapter 7: summarises the results and discusses future work.
Blue supergiants - troublemakers or candles in the dark
Chapter 2
Methods
2.1 Hot star wind diagnostics
Information about stellar-wind structure and mass loss is hidden in the line profiles. Our knowledge about stellar-wind mass loss (and massive-star evolution) depends powerfully on how well
we are able to interpret these line profiles. The line profiles can be either in absorption, pure
emission (formed by recombination) or a composition of both – P Cygni-type line profiles.
While P Cygni lines are formed by line scattering, emission lines result from recombinations.
The mass-loss rates can be inferred mainly by two types of lines: i) optical and near-infrared
recombination lines (Hα, Brα); ii) P Cygni-type resonance lines.
Section 2.1.3 discusses, a third type of measurement which provides reliable mass-loss diagnostics, namely: iii) infrared, millimetre or radio excesses due to free-free emission. However, massive stars are weak radio sources and therefore these diagnostics are only applicable to
nearby massive stars. Thus, the main mass-loss tracers for massive stars in nearby galaxies are
line profiles.
Each of these three type of measurements probe different parts of the stellar wind: from the
28
2.1 Hot star wind diagnostics
29
dense and rapidly accelerating region, where recombination lines are formed, through the entire
wind (UV resonance lines) to the very distant regions where the terminal velocity is reached
(free-free emission).
The following sections are based on Lamers & Cassinelli (1999) and Puls et al. (2008) and
therefore for a more detailed review, the reader is referred to these references (but see also
Kudritzki & Puls 2000).
2.1.1 Diagnostics from UV P Cygni lines
The idea to exploit the strengths of ultraviolet (UV) P Cygni lines as a mass loss tracer goes
back to Lamers & Morton (1976). This was the first diagnostic to be used in order to infer
reliable wind-densities. UV P Cygni resonance lines are the most sensitive probes of mass loss
and are therefore used to derive mass-loss rates of stars with weak winds (10−9 M⊙ yr−1 < Ṁ <
10−7 M⊙ yr−1 ). For stronger winds, however, most of the “strategic” P Cygni lines (C iv, N iv,
N v, Si iv for late O/early-B and C ii for late-B stars) are usually saturated and thus the derivation
of Ṁ is impossible. In this case, they provide a lower limit of Ṁ and only terminal velocities and
the shape of the velocity law can be inferred. Yet, phosphorus being less abundant than C, N, O
by about a factor of 102 − 103 , provides diagnostics (P v λ1118, 1128 Å) which can be applied
to stars with mass-loss rates larger than ∼ 10−7 M⊙ yr−1 .
Unsaturated UV lines: If P Cygni line are unsaturated, the mass-loss rate can be inferred by
a comparison of observed and predicted line profiles with different ion densities ni (r). When
the predicted profile matches the observed one, the distribution ni (r) is inferred and Ṁ can be
calculated from the following equation:
ni (r) =
ni (r) nE (r) nH (r)
nH (r) Ṁ
,
ρ(r) = qi (r)AE
nE (r) nH (r) ρ(r)
ρ(r) 4πr2 3(r)
(2.1)
Methods
2.1 Hot star wind diagnostics
30
Figure 2.1: Schematic formation of a P Cygni type line profile. Figure from Murdin (2003).
where qi = ni /nE is the ionisation fraction of the ion producing the line, AE = nE /nH is the
abundance of the element with respect to H and the ratio nH /ρ is a function of the metallicity
(for solar metallicity nH /ρ ≈ 4.43 × 1023 atoms g−1 ; Lamers & Cassinelli 1999). In Eq. 2.1 the
density ρ is expressed as a function of Ṁ and 3(r) via the continuity equation:
Ṁ = 4πr2 ρ(r)3(r).
(2.2)
However, in case of unsaturated line profiles, the terminal velocity, 3∞ , can not be derived
accurately as the absorption extends only to a certain velocity, which is smaller than the terminal
velocity, 3∞ . In order to derive the 3∞ accurately, the line profiles have to be saturated.
Saturated Profiles: Strongly-saturated P Cygni line profiles are very sensitive to the velocity
law and are therefore expected to provide the most accurate information of the wind velocity
Methods
2.1 Hot star wind diagnostics
31
structure. The velocity law and the terminal velocity of the wind can be derived from their blueshifted absorption components. The violet zero-intensity absorption edge of a saturated P Cygni
line profile is extended to a Doppler velocity of about −3∞ and therefore it is a measure for 3∞ .
The velocity law can be determined from the shape of the emission peak. Figure 2.1 gives an
example of a schematic determination of 3∞ by this method.
In this way the terminal velocity and velocity law are very well derived from P Cygni profiles
(Lamers et al. 1995). However, the determination of Ṁ is much more complicated as it depends
on the element abundances as well as the inferred ionisation fractions which are not well known
because of their sensitivity to non-LTE and clumping effects. Fortunately, the wind emission in
Hα can be used as an alternative mass-loss indicator. The advantage is that the the hydrogen
abundance is usually less uncertain and the ionisation corrections for Hα are much simpler.
2.1.2 Hα line: a conventional mass loss probe for massive stars
The Hα wind emission, observed in Early-type stars, is predominantly fed by recombinations.
The efficiency of the recombination process is proportional to the square of the density ρ (unlike
the scattering process, which scales linearly with density), because it involves collisions between
ions and electrons. This suggests that the emission from Hα originates from the lower layers
of the wind close to the star, where the density is high. It is expected that most of the wind
acceleration occurs in this region. Consequently, recombination lines are sensitive to the velocity
field. Thus, from the shape of the emission peak the steepness of velocity law can be determined.
At present, the Hα line is most oft-used mass-loss diagnostic in OB supergiant regime, because it is detectable for a large numbers of stars and does not require space-bound observations
as do UV P Cygni diagnostics. Moreover, the strong velocity dependence of Hα combined with
high-resolution spectroscopy could provide valuable information about velocity fields and density structures in LBVs and SN progenitors (Groh & Vink 2011).
Methods
2.1 Hot star wind diagnostics
32
log ( LL⊙⋆ ) = 5.50;
M⋆
M⊙
= 40 [M⊙ ]; Teff = 22500 [K];
2.4
Ṁ =1.0e-07
Ṁ =2.5e-07
Ṁ =5.0e-07
Ṁ =7.5e-07
Ṁ =1.0e-06
Ṁ =1.5e-06
2.2
2
1.8
F/Fc
1.6
1.4
1.2
1
0.8
0.6
0.4
−20
−10
0
∆λ [Å]
10
20
Figure 2.2: Synthetic Hα line profiles for Bsg model with different Ṁ. The spectra were computed with cmfgen code (cf. § 2.3).
In general, the mass-loss rates from Hα emission are derived by matching of synthetic line
profiles to the observed profiles. However, the calculation of synthetic spectra is not a simple
task. It requires knowledge of the radiation field in order to account for non-LTE effects. A
proper treatment of the line-blanketing and the wind extension is also required. The atmosphere
models provide valuable guidance to account for those effects (cf. § 2.3), although there are
issues related with the ionisation in the wind (see e.g. Searle et al. 2008).
In hot stars, the Hα line (unlike UV P Cygni lines) is expected to have a nearly constant
source function throughout the wind, because the involved levels are predominantly determined
through recombinations. For cooler stars however, as shown in § 3.5, the lower level of Hα is
prevented from recombining to the ground state and thus it becomes an effective ground state of
H. Consequently, the line source function is dominated by line scattering the Hα line behaves
Methods
2.1 Hot star wind diagnostics
33
like a resonance line, displaying a P Cygni type profile (cf.left-hand side of Fig. 3.1; see also
Puls et al. 1998).
While UV resonance lines are used to measure the mass-loss rates of massive stars with relatively weak winds, recombination lines like Hα are traditionally used as mass-loss diagnostics
for stars with dense winds. Figure 2.2 demonstrates the dependence of the Hα line on Ṁ in
the case of a Bsg. The line is transformed from pure emission into pure absorption when the
wind-density is decreased. At lower mass-loss rates ( Ṁ . 10−8 M⊙ yr−1 for late O/early B-type
stars), Hα becomes insensitive to mass loss, because the wind emission vanishes. In this case,
the shape of Hα line is dominated by the rotational broadening and the velocity structure of the
wind can not be determined, as the rotational broadening corrupts all clues concerning the wind
emission.
Complications
Variability and rotation: It is well documented that B- and A-type supergiants (Abt 1957;
Rosendhal & Wegner 1970; Kaufer et al. 1996) and O-type stars (Markova et al. 2005) show
variations in their optical spectrum with typical time scales from a few days to several months.
A significant line-profile variability in Hα might have important consequences for the derived
wind parameters (see e.g. Kudritzki et al. 1999). In addition, OB-stars have significant rotational
velocities 3rot which should broaden the wind emission. Thus, the value of 3rot and the differential
rotation may strongly influence the shape and behaviour of the Hα line profiles. Only very strong
and broad lines are barely influenced by the rotation. Therefore, the UV lines are less affected
by the rotation.
The blue-ward He ii blend: Additionally, in early-type stars, a He ii λ6560 line contaminates
the Hα line profile. As the He ii line is mostly in absorption, the Hα line may look like P Cygni
type and may be difficult to fit the observed spectrum (see e.g. Herrero et al. 2000). In that
Methods
2.1 Hot star wind diagnostics
34
case, a reliable result could still be obtained from the red side of Hα because of the weaker
contamination in comparison with the blue wing.
Wind inhomogeneities: The quantitative analysis of the mass-loss rates derived from Hα is
challenged by the presence of small-scale wind-inhomogeneities, which influences the ρ2 dependent diagnostics. If the general assumption of smooth winds in the models is violated, the
analysis of the Hα line profiles might lead to large systematic errors in the derived mass-loss
rates. Further details and consequences are discussed in § 2.2 and in Chapter 4.
2.1.3 Mass loss from radio
Another method to derive Ṁ is based on an excess of (far)infrared (IR), (sub)millimetre and
radio continua. This approach is conceptually different from the methods of UV resonance lines
or recombination lines. It is based on rather simple processes and does not require accurate
information about the ionisation or temperature structure of the winds. At present, only this
method provides mass-loss rates based on: i) the distance to the star d; ii) the radio flux density
fν ; iii) the terminal velocity of the wind 3∞ . The basic idea is to measure the excess flux relative
to the flux expected from the stellar photosphere if the star did not have a wind. The excess flux
is emitted by free-free or bound-free processes in the wind and the mass-loss rate is proportional
to it.
The free-free opacity increases at longer wavelengths as kν ∝ λ2 and therefore the corresponding radius of the photosphere, Rref , increases at IR/radio wavelengths. The monochromatic luminosity, Lν , of the star increases at longer wavelengths as well. This is because in the
Eddington-Barbier relation, Lν is given by the thermal emission from the surface at optical depth
τν =
1
3
Cassinelli & Hartmann (1977); Lamers & Cassinelli (1999):
Methods
2.1 Hot star wind diagnostics
35
Figure 2.3: Schematic energy distribution of a star with R⋆ = 10 R⊙ , T eff = 37 500 K and with
free-free emission from a wind of Ṁ = 1 × 10−5 M⊙ yr−1 . Figure from Lamers & Cassinelli
(1999).
Lν ≈ 4πr2 (τν = 1/3)πBν (T(τν = 1/3)).
The effective radius is at τν =
1
3
instead of τν =
2
3
(2.3)
because an extended stellar wind adds weight
to emission from small optical depth (Castor 1974). Finally, a star with an ionised wind is
expected to have an excess of radiation if T wind ≈ T eff . A schematic energy distribution of the
stellar photosphere and free-free emission is shown in Fig. 2.3.
The relation between Ṁ and the monochromatic radio/infrared flux density fν (in Janskys) is
Methods
2.1 Hot star wind diagnostics
36
given by Wright & Barlow (1975):
Ṁ = 0.095
µ3∞ ( fν d2 )3/4
.
√
Z ygν ν
(2.4)
Here d is the distance of the star in kpc, Ṁ is in M⊙ yr−1 , ν is in Hz and 3∞ is in km s−1 , µ is
the mean molecular weight of the ions, Z the rms charge of the atoms (Z = 1 for a singly ionised
gas). Ṁ weakly depends on wind temperature via the gaunt factor, gν . For radio wavelengths,
gν is given by (Allen 1973, page 103):
gν ≈ 10.6 + 1.9 log T − 1.26 log νZ.
(2.5)
The temperature structure is usually not well known and therefore this is an advantage.
Early-type stars generally have weak flux densities at radio frequencies, and therefore useful
radio observations can be obtained only for nearby stars with dense winds. Benaglia et al.
(2007) derived mass-loss rates from continuum radio observations of nearby supergiants with
effective temperature around the bi-stability jump. The sample was composed of 19 supergiants
with firm radio detections (up to 3 mJy) and additional 11 sources with signal typically below
0.3 mJy. Their results showed possible existence of a local maximum in wind efficiency around
21 000 K in line with predictions. Despite this, large discrepancies (by a factor of a few) between
empirical and predicted mass-loss rates were found. Therefore, a larger sample of stars around
the bi-stability jump, with firm radio detections is required to confirm the bi-stability jump.
Free-free emission scales with ρ2 and thus a presence of significant clumping will cause the
radio method to overestimate mass-loss rates.
Methods
2.2 Wind inhomogeneities: problems and perspectives
37
2.2 Wind inhomogeneities: problems and perspectives
2.2.1 Observational history
There is observational evidence that the winds of hot massive stars are inhomogeneous. In the
following, we summarise various observational problems which are likely related to inhomogeneous winds.
Spectral variability: A possible source of the spectra variations discussed in § 2.1.2 might be
the presence of inhomogeneities in the wind, which are believed to result from instabilities in
the stellar wind itself (Owocki et al. 1988; Owocki & Puls 1999; Owocki 2014) or sub-surface
convection zones (Cantiello et al. 2009). The UV line profile variability (e.g. Prinja 1992) is attributed to a small-scale wind structures. However, “discrete absorption components”, detected
in absorption troughs of unsaturated P Cygni profiles from O/early B stars (Howarth & Prinja
1989; Kaufer et al. 1996) and in late B-supergiants Bates & Gilheany (1990), are believed to be
associated with the presence of large-scale structures (co-rotating interaction regions) and the
stellar rotation.
Spectropolarimetric variability: Lupie & Nordsieck (1987) detected variations of the position
angles of the polarization in a sample of ten OB supergiants. They suggested that the origin of
these variations might be related to the existence of density enhancements within the wind.
Weak electron scattering wings: Hillier (1991); Hamann & Koesterke (1998) found that the
electron scattering wings of strong recombination lines in W-R stars are weaker than predicted
by smooth models. This implies that the real electron densities in the wind should be lower than
assumed by homogeneous models. Thus electron scattering wings can indicate the clumping
properties in hot stars with dense winds.
Inconsistency between mass loss diagnostics: several studies (Massa et al. 2003; Bouret et al.
2005; Puls et al. 2006b; Fullerton et al. 2006) found systematic discrepancies between mass-loss
Methods
2.2 Wind inhomogeneities: problems and perspectives
38
rates derived from Hα, UV and radio observations for OB stars. As noted earlier, the different
mass-loss diagnostics probe different parts of the stellar wind. Therefore, if the inner part of
the wind has a different structure from the outer wind, different diagnostics should give different
values for Ṁ. In addition, diagnostics that are linearly dependent on the density (UV P Cygni
resonance lines) are insensitive to clumping, whilst recombination lines and free-free emission
are ρ2 dependent processes and would tend to overestimate the mass-loss rate if the wind is
clumped (see Sect. 2.2.2, but see also Oskinova et al. 2007; Sundqvist et al. 2010; Muijres et al.
2011).
2.2.2 Theoretical background
The above discussion strongly implies that stellar winds are not homogeneous. An approach to
account for the wind-inhomogeneities has been made by Hamann & Koesterke (1998); Hillier &
Miller (1999); Puls et al. (2006b), and it is based on the assumption that the wind-inhomogeneities
are small in comparison to the mean free path of the photons. This is the so-called “microclumping” approximation.
The standard micro-clumping approach is based on the hypothesis that the wind consists of
small-scale over-density “clumps” which are optically thin. These clumps are assumed to have
an enhanced density ρ+ compared to the average density ρ by a clumping factor fcl :
ρ+ = fcl ρ , where ρ = Ṁ/(4πr2 3).
(2.6)
The inverse of the clumping factor corresponds to a volume filling factor fV in such a way that
ρ = fV ρ+ (assuming that the volume between the clumps is void). Observations indicate that the
clumping factor is distance dependent fcl (r) (Puls et al. 2006b; Liermann & Hamann 2008).
As the density inside the clumps is enhanced by fcl , the opacity and the emissivity of the
Methods
2.2 Wind inhomogeneities: problems and perspectives
39
processes inside the clumps are given by k f = kc (ρ+ ) = kc ( fcl ρ) and j f = jc ( fcl ρ). In clumped
winds the mean opacity and emissivity are given by:
k = k f fV = kc ( fcl ρ) fV =
kc ( fcl ρ)
jc ( fcl ρ)
and j = j f fV = jc ( fcl ρ) fV =
.
fcl
fcl
(2.7)
The volume filling factor is the fractional volume “ fV ” which clumps occupy and therefore the
mean opacity or emissivity is equal to the opacity or emissivity of the clumps multiplied by the
fractional volume fV .
The processes contributing to the emissivity and opacity scale with different powers of the
density. For processes which are linearly dependent on density (such as UV P Cygni resonance
lines), fV and fcl cancel and the mean opacity and emissivity of a clumped wind is the same as
in a smooth wind. Therefore, UV diagnostics are insensitive to clumping. Of course, clumping
might change the ionisation balance and thus indirectly affect UV lines.
However, for processes which scale with the square of the density, mean opacities and emissivities are effectively enhanced by a factor fcl compared to homogeneous winds. Consequently,
the mass-loss rates measured from emission diagnostics (recombination lines, free-free ) are
p
lower by a factor of fcl than corresponding mass-loss rates assuming homogeneous winds.
If the wind-inhomogeneities are optically thick, the micro-clumping approximation can not
be justified as the photons are absorbed or scattered by the clumps and they may leak only
through gaps between those clumps. In that case the mean opacity and emissivity are affected
by the distribution, the size and the geometry of the clumps. Unfortunately, this makes the full
non-LTE radiative transfer-simulation very difficult. Therefore, in non-LTE radiative-transfer
codes, such as cmfgen, micro-clumping is implemented only as a first approximation.
Methods
2.2 Wind inhomogeneities: problems and perspectives
40
2.2.3 Clumping may reconcile Hα, UV and radio Ṁ determinations?
Several investigations (Figer et al. 2002; Crowther et al. 2002; Hillier et al. 2003; Bouret et al.
2003; Markova et al. 2004; Repolust et al. 2004; Bouret et al. 2005; Puls et al. 2006b,a; Fullerton
et al. 2006) found difficulties to reconcile the mass-loss rates derived from different type of
measurements using homogeneous models.
On the basis of smooth models, Fullerton et al. (2006) found significant discrepancies between mass-loss rates derived from P v lines and the corresponding Hα or radio mass-loss rates
for a sample of 40 Galactic O-type stars. The major result from their investigation is that massloss rates obtained from fits to Hα emission lines or radio observations are systematically higher
than those derived from P v by factors of ∼ 130 (between types O7 and O9.7) and ∼ 20 for types
between O4 to O7. Note that these value were derived under the assumption that optically thin
clumping does not change the ionisation balance of P v. If that assumption is true, this would
imply fcl ≈ 10 000 in the cooler temperature regime.
However, due to the increased density inside the clumps, stronger recombination is expected,
which should change the ionisation balance of phosphorus. Therefore, a constant ionisation fraction of P v is not very likely. To test this idea Puls et al. (2006a) investigate how micro-clumping
affects the ionisation balance of phosphorus. These authors found that, if micro-clumping is
taken into account, phosphorus changes its ionisation balance: in homogeneous models P v was
dominant at O8/7, whereas in clumped models P v dominates at hotter temperatures (O5).
Puls et al. (2006a) reproduced the data from Fullerton et al. (2006) with models which are
highly clumped (with fcl = 144), confirming that mass-loss rates might be lower by an order of
magnitude than the mass-loss rate previously derived from models with smooth winds if microclumping approach were correct.
Puls et al. (2006b) subsequently performed a comprehensive analysis of Hα, IR, mm and
radio fluxes based on a sample of 19 Galactic O-type supergiants with well-known stellar paMethods
2.3 Numerical methods: the cmfgen atmosphere code
41
rameters. In this way they were able to probe the lower and outer parts of the stellar wind in
parallel. They found ṀHα ≈ Ṁradio , for objects with Hα in absorption, whereas for objects with
Hα in emission ṀHα ≈ 2 × Ṁradio . This finding is in agreement with earlier results from Repolust et al. (2004) who also found such principal differences between weak and strong winds, and
suggested clumping a factor of the order of 5 (see also Mokiem et al. 2007).
So far, the analysis of the different mass-loss diagnostics yields a broad spectrum of clumping
factors from a factor of a few up to 100. This suggests that empirical mass-loss rates derived
from recombination lines are overestimated and have to be revised.
However, Oskinova et al. (2007) promote the idea that the mass-loss rate discrepancy might
be explained if the wind clumps become optically thick at certain wavelengths. They showed
that macro-clumping makes the P v resonance lines weaker, whilst Hα (still optically thin for
late O/early B stars) is not affected by macro-clumping.
Therefore, macro-clumping needs to be taken into account in non-LTE radiative transfer
simulations, in order to get the real mass-loss rates from UV diagnostics. However, the treatment
of 3D clumps in model atmosphere simulations is not feasible yet. Therefore, optically thin
recombination lines, such as Hα, offer better prospects for mass-loss diagnostics in the OB
supergiant regime.
2.3 Numerical methods: the cmfgen atmosphere code
In massive stars many physical processes are involved and therefore sophisticated atmosphere
codes are required in order to reproduce their observational properties. The results in this thesis
are based on atmosphere models calculated with cmfgen (Co-Moving Frame GENeral, Hillier
& Miller 1998) code1 . This is a radiative transfer code designed to solve statistical equilibrium
and radiative transfer equations in spherical geometry for Wolf-Rayet stars, OB stars, LBVs and
1
The version of cmfgen employed in the thesis was released on 7 April 2011.
Methods
2.3 Numerical methods: the cmfgen atmosphere code
42
even Supernovae.
2.3.1 Main ingredients of the cmfgen code
One of the greatest advantages of the code is that it makes as few assumptions as possible
and therefore it is one of the most realistic codes devoted to the modelling of massive star
atmospheres. But this is also disadvantage because it makes cmfgen very expensive in terms of
CPU time. With this code is easy to compute non-local thermodynamic equilibrium (non-LTE)
models including the effects of wind extension and line blanketing.
Wind treatment: the intensive radiative force in the atmosphere of massive stars produces a
wind of ionised gas from the star. The stellar wind can extend up to several tens of stellar radii
allowing possible emission from lines well above the stellar surface. Therefore, to compute
realistic synthetic spectra of stars, the inclusion of the stellar wind in models is absolutely necessary. The temperature structure in this region can be significantly different from the effective
temperature.
Non-LTE: in massive stars, the radiation field becomes sufficiently strong and it dominates
collisional processes, which are ruled by the local temperature. The radiative processes are nonlocal in character and tend to destroy the thermodynamic equilibrium (TE). Therefore massive
stars show severe deviations from LTE. When the assumption of LTE fails, statistical equilibrium
equations given by:
N
N
X
dni (r) X
=
n j (r)P ji (r) − ni (r)
Pi j (r) = 0
dt
j,i
j,i
(2.8)
have to be solved in order to calculate the level populations of the different energy levels. Here
N is the total number of levels that are important for the populations ni and the rates Pi j are the
number of transitions per second per particle in state i or j. For a spectral line the excitation Plu
Methods
2.3 Numerical methods: the cmfgen atmosphere code
43
and de-excitation Pul rates are given by:
Plu = Blu J ν0 + Clu
(2.9)
Pul = Aul + Bul J ν0 + Cul ,
(2.10)
where J ν0 is the mean intensity; Aul , Blu , and Bul are the Einstein coefficients, which define
the respective transition probabilities for spontaneous de-excitations, radiative excitation and
stimulated emission. Clu and Cul are the electron collision rates for bound-bound transitions.
Line blanketing: although the abundances of metals are small, they have many more transitions
than H and He. Thus, metals contribute a major part of the total opacities of the atmosphere and
in this way they define the structure of the atmosphere (especially the temperature and velocity
structure) and of course the emerging spectrum. The effect of lines on the continuum energy
distribution, and the effect of line overlap (as well as line blanketing) is incorporated into the
code. Basically line blanketing acts as a “wall” which blocks the radiation and the photons
are more back-scattered towards the inner part of the atmosphere, increasing the temperature
and thus ionisation. This effect implies that a model with metals requires a cooler effective
temperature to obtain the same degree of ionisation as a model without metals. Line blanketing
is especially efficient in hot stars since their emission is mainly in the UV where there are many
bound-free opacities of metals and therefore it is vital for obtaining a reliable synthetic spectrum.
2.3.2 Other characteristics of cmfgen
Super-level concept: The main challenge of line-blanketing effects is the enormous computational effort. In the models with this process included, the radiative field and level populations
are computed in an atmosphere with thousands of transitions and energy levels within the nonLTE approach. Thus the number of statistical equilibrium equations becomes very large. In
Methods
2.3 Numerical methods: the cmfgen atmosphere code
44
order to reduce this number, cmfgen uses the concept of “super-levels” (Anderson 1989). The
essential idea is to combine levels with similar energy in a single super-level and the computations are performed with this super-level. The advantages of using super-levels are that (i) it
allows cmfgen to easily handle complex model atmosphere structures at a reasonable computational cost; (ii) many energy levels of metals can be included to examine the effect of line
blanketing. Unfortunately, the use of super-levels also has two big disadvantages: (i) depending
on the energy levels combined in a super-level, a slight inconsistency is possible between the
radiative and statistical equilibrium equations; and (ii) there is no algorithm for choosing the
optimal number of super-levels.
Radiative-transfer equation: thanks to the inclusion of the three main ingredients, cmfgen
computes detailed and reliable atmosphere models for massive stars. The code iteratively solves
the statistical equilibrium and radiative-transfer equations in the co-moving frame (CMF) to obtain a correct model atmosphere structure. The radiative-transfer equation for spherical geometry
in the CMF is given by:
µ
d ln v(r) i ∂Iν (r, µ)
∂Iν (r, µ) 1 − µ2 ∂Iν (r, µ) νv(r) h
+
−
(1 − µ2 ) + µ2
∂r
r
∂µ
rc
d ln r
∂ν
(2.11)
= −kµ (r)Iν (r, µ) + ην (r),
where r is the radial coordinate, µ the projection of the propagation direction of the beam
(µ =cos θ), ην the emission coefficient and kν the absorption coefficient (Hillier & Miller 1998).
In the CMF, kµ and ην are independent of the direction. Therefore it is convenient to solve the
radiation transport equation in the CMF.
Hydrostatic and velocity structure: cmfgen solves the hydrostatic equation:
dP
= −ρg,
dr
(2.12)
only in the inner wind, where the wind velocity reaches up to 75% of the local speed of sound.
Methods
2.3 Numerical methods: the cmfgen atmosphere code
45
In the outer part of the wind, the code does not solve self-consistently the momentum equation
of the wind and thus a velocity structure has to be assumed. For the accelerating part of the wind
a classical β-law in the form:
R∗ β
3(r) = 3∞ 1 −
,
r
(2.13)
is adopted. Here, R∗ is the hydrostatic radius and 3∞ is the terminal velocity reached at the
farthest part of the atmosphere. The exponent β describes the steepness of the velocity law in
such a way that larger values lead to a flatter velocity law.
Density structure: after the velocity structure is assumed, the local density ρ(r) is simply calculated from continuity equation (Eq. 2.2), as Ṁ is an input parameter.
Steady-state: there are evidences that the wind of massive stars show time-dependent changes,
however, cmfgen solves all the equations under the assumption of steady-state.
Spherical symmetry: cmfgen assumes spherical symmetry
Clumping: in § 2.2 we mentioned that the winds of OB stars are likely inhomogeneous. Therefore clumping is implemented in cmfgen to a first approximation (micro-clumping). The volume
filling factor fV (r) is described by the following exponential law:
fV (r) = fV∞ + (1 − fV∞ )exp(−3(r)/3cl ),
(2.14)
where 3cl is the velocity at which clumping is switched on and fV∞ is the volume filling factor at
the top of atmosphere.
2.3.3 Input parameters
To run a cmfgen model the following input data are needed:
Methods
2.3 Numerical methods: the cmfgen atmosphere code
46
• Stellar parameters: luminosity, effective temperature, log g (mass of the star), chemical
composition.
• Wind parameters: mass-loss rate, terminal velocity, the velocity law (β), the degree of
clumping.
• Model settings: the number of depth points, ionisation stages of elements, levels/superlevels assignments.
When the model converges, the temperature structure, degree of ionisation, level populations
etc. are provided, which allows for a complete investigation. An auxiliary code (cmf_flux)
computes a formal solution of the radiative transfer equation in the observer’s frame, which
facilitates the comparison between synthetic and observed spectra.
Methods
Part II
The Physics Behind the Hα Line
47
Chapter 3
Hα line formation: rise and fall over
the bi-stability jump
In preceding chapters we saw that the evolutionary state of BSGs is still unknown. This issue
might be solved if we are able to derive accurately their Ṁ, as stellar wind mass loss is one of
the dominant processes determining the fate of massive stars. However, as was stated in § 1.3.3,
there is discrepancy between theoretical and empirical mass-loss rates. The key question is
whether this discrepancy is the result of incorrect predictions or alternatively that we may not
understand the mass-loss indicators, such as Hα, well enough. What would one expect to happen
when Fe iv recombines, and Fe iii starts to control the wind driving (Vink et al. 1999)?
Over the last decade a dedicated effort to improve the physics in the Monte-Carlo line-driving
calculations has been made. The wind dynamics is now solved more locally consistently (Müller
& Vink 2008; Muijres et al. 2012), Fe was added to the statistical equilibrium calculations in the
isa-wind model atmosphere (de Koter et al. 1993) rather than treating this important line-driving
element in a modified nebular approximation. The effects of wind clumping and porosity on the
line driving has also been investigated (Muijres et al. 2011). After all these improvements in the
48
3.1 Method and input parameters
49
Monte-Carlo line-driving physics, we have to admit that the basic problem of Ṁvink > ṀHα for
B1 and later supergiants, is still present, and it is time that we also consider the possibility that
it is not the predictions that are at fault, but that we do not understand Hα line formation in Bsgs
sufficiently well to allow for accurate mass-loss determinations from Hα.
Despite the spectral modelling of Bsgs is an active area of research (Zorec et al. 2009; Fraser
et al. 2010; Castro et al. 2012; Clark et al. 2012; Firnstein & Przybilla 2012), the T eff dependence
of the mass-loss rates of BSGs is still uncertain, and a better knowledge is required, especially
as it impacts the question of the evolutionary nature of BSGs and LBVs. The Hα line is an
excellent mass-loss tracer in the OB supergiant regime, and therefore a proper understanding of
its formation is worth having. Therefore, in the following, we investigate the formation of the
Hα line as a function of T eff in the context of the bi-stability jump.
3.1 Method and input parameters
Make everything as simple as possible, but not simpler.
Albert Einstein (1879 - 1955)
In order to study Hα line formation, we calculate a grid of cmfgen models over a range of
temperature and log g appropriate for Bsgs (cf. Table 3.1). Apart from changes in T eff and log g,
Hα line formation is sensitive to luminosity, L⋆ , mass, M⋆ , mass-loss rate Ṁ, clumping, and
velocity structure. For the accelerating part of the wind, we adopt a standard β-type velocity law
with β = 1, whilst a hydrostatic solution is adopted for the sub-sonic region.
To understand first how the Hα line profile changes in a qualitative sense, we use pure H
models, keeping L⋆ , M⋆ , 3∞ /3esc and Ṁ fixed, whilst we vary T eff over a range from 30 000 K
down to 12 500 K. Following Vink et al. (2001), we adopt the parameters listed in Table 3.1.
Note that for fixed L⋆ , the changes in T eff inevitably lead to different R⋆ and 3esc values,
Hα line formation: rise and fall over the bi-stability jump
3.2 Hα line profile and equivalent width
50
Table 3.1: Adopted stellar parameters used in the model grid.
log LL⋆⊙ = 5.5, M⋆ = 40 M⊙ , Ṁ = 2.33 × 10−6 [M⊙ yr−1 ]
T eff [K] R⋆ [R⊙ ] 3∞ = 2 × 3esc [ km
s ] log(g)
30 000
27 500
25 000
22 500
20 000
17 500
15 000
12 500
21
25
30
37
46
61
83
120
1701
1558
1447
1276
1138
998
855
715
3.40
3.25
3.09
2.90
2.70
2.47
2.20
1.88
Q
-6.46
-6.51
-6.59
-6.64
-6.72
-6.80
-6.91
-7.03
model
M
C
which may influence the Hα line equivalent width (WHα ). To account for this, we used an
optical-depth parameter, Q, that was introduced by Puls et al. (1996); see § 3.2 for details. To
keep as many model parameters as possible fixed, we used 3∞ /3esc = 2 – the mean value of
3∞ /3esc ratio at both sides of the bi-stability jump (Lamers et al. 1995; Markova & Puls 2008). It
is more natural to keep the 3∞ /3esc ratio constant rather than the value of 3∞ itself, as the models
have different radii.
It is important to keep in mind that due to metal line blocking and a blend with He ii the Hα
line could be sensitive to changes in He as well as metal abundance. To estimate these effects we
also compare how the WHα behaves in models with different chemical complexities: (i) pure H;
(ii) H + He, with two different He mass fractions (YHe = 0.25 and 0.6); and (iii) line-blanketed
models with half-solar metal abundances.
3.2 Hα line profile and equivalent width
Our systematic examination of the Hα line for supergiant models over T eff range between 30 000
and 12 500 K shows non-monotonic changes in the Hα line profiles with T eff . As shown in
Hα line formation: rise and fall over the bi-stability jump
3.2 Hα line profile and equivalent width
51
25
2.2
2
12 500 K (with Identical Q-parameter)
12 500 K
15 000 K
22 500 K
25 000 K
30 000 K
20
15
Hα
1.6
W
log F/F c
1.8
10
1.4
H+He
5
H
1.2
CNO,Fe,P,S,Si
0
1
H+He60
H+He with constant Q
WHa−WHa
0.8
−50
0.5
0
∆λ[Å]
50
−5
30
25
20
Teff kK
phot
15
Figure 3.1: Left: Hα line profiles for cmfgen models with parameters as listed in Table 3.1.
Black triangles represent the line profile with the same Q-parameter (Eq. 3.1) as model C
(T eff =12 500 K), but with different Ṁ and R⋆ values. Right: Hα line EW vs T eff for models
with only H (crosses), H+He (circles), and more sophisticated (triangles) models composed by
H, He, C, N, O, Si, P, S, and Fe with half-solar metal abundances. Red asterisks represent the
changes in the Hα line when the He mass fraction in the pure H+He models is increased to 60%.
Blue squares indicate how the Hα EW behaves as a function of a constant Q value.
Fig. 3.1, the Hα line becomes stronger when T eff drops from 30 000 to 22 500 K, where the line
reaches its peak value. Below 22 500 K the line becomes weaker when the effective temperature
is further reduced to 12 500 K.
Although we have kept all stellar parameters fixed, the models have different radii and 3∞ .
Note that in the hottest models the radii are up to a factor of 6 smaller, whilst the terminal
velocities are only up to a factor 2.4 higher. To extract the true temperature effect, the different
radii and terminal velocities should be taken into account. One way to do this is through the use
of the wind-strength parameter Q concept introduced by Puls et al. (1996). It was demonstrated
that for O-type stars recombination lines remain unchanged by specific sets of individual values
of Ṁ, v∞ and R⋆ as long as the wind-strength parameter Q:
Q=
Ṁ
,
(v∞ R⋆ )1.5
(3.1)
Hα line formation: rise and fall over the bi-stability jump
3.2 Hα line profile and equivalent width
52
−4.6
−4.8
log(Fλ ) [erg/(cm2 × s)]
−5
−5.2
−5.4
−5.6
−5.8
−6
−6.2
−6.4
30
27.5
25
22.5 20
Teff [kK]
17.5
15
12.5
Figure 3.2: Integrated line (circles) and continuum flux (squares) at the wavelength of Hα for H
+ He models. Note that the flux represents the flux at a distance of 10 parsec.
is invariant and T eff remains unchanged. Here Ṁ is in units of 10−6 M⊙ /yr, v∞ in kms−1 and R⋆
in R⊙ . As an example, in Fig. 3.1 we show that the Hα line profile from model C (red dashed
line) is basically unaffected when the mass-loss rate is decreased by factor of two and the radius
by 22/3 (black triangles).
The right-hand panel of Fig. 3.1 displays how the WHα behaves as a function of T eff for
models with similar parameters, but with different individual Ṁ, which scale in such a way that
Q is constant (blue squares). The peak is still present and this implies that the Hα behaviour
is not due to the changes in v∞ × R⋆ , i.e., it is a real temperature effect. Note that the peak is
slightly shifted towards T eff ∼ 20 000 K.
To gain additional insight, the EW was separated into the line and continuum flux (respectively, illustrated in Fig. 3.2 with red circles and squares). The reason for doing so is that the EW
is a measurement of the ratio of the line flux over the continuum flux, and models with similar
EW may have completely different line fluxes. As can be seen in Fig. 3.2, the coolest model
Hα line formation: rise and fall over the bi-stability jump
3.2 Hα line profile and equivalent width
53
Table 3.2: Atomic data used for our simplistic H + He supergiant models. For each ion, the
number of full levels, super levels, and bound-bound transitions are provided.
Ion
Super levels Full levels b-b transitions
HI
20
30
435
He I
45
69
905
He II
22
30
435
has a line flux seven times larger than the line flux in the hottest model, whilst in Fig. 3.1 it
seems that both models have a similar line strength. The similar WHα in both models is due to a
constantly increasing continuum flux when reducing T eff , and the higher line flux in the coolest
model. The peak in the right-hand panel of Fig. 3.1 results at 22 500 K (and not at 20 000 K as
in Fig. 3.2) because the ratio between the line and continuum fluxes is the largest for that T eff .
Note that the cooler supergiants models have larger radii, which increases the continuum flux
with decreasing T eff .
The right-hand panel of Fig. 3.1 also presents behaviour patterns in models with different
chemical complexities. The simplest set of models (with only H; i.e., without He) presented
with red crosses shows practically identical WHα 1 as the EWs from models including both H
and He (displayed with grey circles). In other words, He does not seem to influence the Hα line.
The reason could be that for this T eff range HeII is diminished and the blueward HeII blend is not
essential, whilst the HeI continuum plays only a minor role at these temperatures. However, the
latter holds only when the He abundance is less than 25% by mass. The red asterisks in the righthand panel demonstrate that the He rich models (comprising 60% He mass fraction) have a factor
2-3 lower WHα . This behaviour is in agreement with the results of Dimitrov (1987) who found
that a high abundance of He produces stronger absorptions in H lines. Despite the quantitative
differences, He is not important for the qualitative behaviour of Hα versus T eff . The line EW
values from the simplistic H+He models (grey circles) are also compared to those determined
1
We have defined the line EW to be positive for an emission line and negative for an absorption line.
Hα line formation: rise and fall over the bi-stability jump
3.2 Hα line profile and equivalent width
54
Table 3.3: Model atoms used in the sophisticated models. For each ion the number of full levels,
super levels, and bound-bound transitions is provided.
Ion
Super levels Full levels b-b transitions
HI
20
30
435
He I
45
69
905
He II
22
30
435
CI
81
142
3426
C II
40
92
903
C III
51
84
600
C IV
59
64
1446
NI
52
104
855
N II
45
85
556
N III
41
82
578
N IV
44
76
497
NV
41
49
519
OI
32
161
2138
O II
54
123
1375
O III
88
170
1762
O IV
38
78
561
OV
32
56
314
O VI
25
31
203
Si II
9
16
37
Si III
33
33
92
Si IV
22
33
185
P IV
30
90
656
PV
16
62
561
S III
24
44
196
S IV
51
142
1504
SV
31
98
798
Fe I
9
33
47
Fe II
275
827
23004
Fe III
104
1433
57972
Fe IV
74
540
13071
Fe V
50
220
2978
Fe VI
44
433
8662
Fe VII
29
153
1247
Hα line formation: rise and fall over the bi-stability jump
3.3 Two branches of Hα behaviour
55
from more realistic chemical models (green triangles), which include C, N, O, Si, S, P, and Fe
(with atomic data as listed in the Table 3.3 and half-solar metal abundances). Whilst there are
quantitative differences due to line blanketing, the qualitative behaviour in WHα is similar in
both sets of models with different complexities. This implies that the reason for their behaviour
is fundamentally driven by the properties of H. Despite their simplification, the models including
H+He only (with atomic data as indicated in Table 3.2), provide an overall picture of the effective
temperature dependence of Hα for Bsgs. It is, therefore, reasonable to take advantage of these
H+He models, using them as a starting point for our investigation.
As an aside, we found that for models with a flatter velocity law (β = 2) or with different mass-loss rates (varying from ∼ 10−7 to 10−5 M⊙ yr−1 ) the resulting WHα behaviour was
qualitatively similar to those presented in Fig. 3.1 with WHα showing a peak at T eff ≃ 22 500 K.
Finally, we investigated the strength of the photospheric contribution to the total Hα EW as
a function of T eff . This may be important because the photospheric absorption can be stronger
at lower temperatures and this may influence of the position of the peak in the Hα line EW
or the shape of the trends shown in Fig. 3.1. In order to extract the photospheric contribution
from the total WHα , we calculated models similar to those listed in Table 3.1, however with very
low mass-loss rate, Ṁ = 10−8 M⊙ yr−1 . Then, we extracted WHα of these models from WHα of
the initial set of H+He models. The resulting WHα is shown in Fig. 3.1 with gray diamonds.
The extraction of the photospheric contribution leads to an almost parallel shift of the line EW
towards higher values. Note that the peak is still present and the behaviour of WHα line is still
qualitatively the same. Thus, the peak in WHα should be mainly determined by the wind.
3.3 Two branches of Hα behaviour
The WHα changes drastically with T eff in all sets of models. In general, we find there to be two
branches in Figs. 3.1 and 3.2: the “hot” and “cool” ones. The hot branch is located between
Hα line formation: rise and fall over the bi-stability jump
3.3 Two branches of Hα behaviour
56
−1
−2
−3
log (H/H+ )
−4
−5
−6
−7
−8
−9
1
12 500 K
17 500 K
22 500 K
25 000 K
30 000 K
0
−1
−2
log (τROSS)
−3
−4
Figure 3.3: Hydrogen ionisation structure for models with various T eff .
30 000 to 22 500 K, where the Hα line emission becomes stronger with decreasing T eff . At
the cool branch, from 22 500 to 12 500 K, the behaviour of the line flux changes in the opposite
direction. This implies that there is a qualitative change in the behaviour of Hα around 22 500 K.
3.3.1 The “hot” branch
The formation of the Hα line involves transitions between the 3rd and 2nd level of H. Therefore,
the emission should be proportional to the total number of H atoms in the 3rd level in the wind
above τROSS = 2/3 (N3 ). As N3 (as do the number of H atoms in other levels) scales with the
fraction of neutral H, we show the H ionisation structure of our models in Fig. 3.3. It is evident
that the wind is mostly ionised in all models and as T eff drops the wind recombines up to 2%.
The fraction of neutral H increases to almost two percent when T eff is reduced to 12 500 K (at
log τross ∼ −1.5, log (H/H+ ) ∼ -1.8). Furthermore, in an absolute sense, the total number of
Hα line formation: rise and fall over the bi-stability jump
3.3 Two branches of Hα behaviour
49
57
N1
N2
N3
48
log(Ni )
47
46
45
44
43
42
30
27.5
25
22.5
20
Teff [kK]
17.5
15
12.5
Figure 3.4: Total number of H atoms in the stellar wind versus T eff . Note that the total number
of H atoms is determined from τross < 2/3.
H atoms in the second level (N2 ) and N3 increase, as illustrated in Fig. 3.4. As a result, the
flux in Hα, which is proportional to N3 , should increase to first order as T eff drops. Therefore,
the trend in Hα on the “hot“ side of Figs. 3.1 and 3.2 can be understood in terms of a simple
recombination effect. Difficulties with such a simple explanation would arise if we were to try
to explain the existence of the cool branch in a similar way. Figure 3.4 illustrates that when T eff
is reduced from 22 500 to 12 500 K, N3 is still increasing and this should probably produce a
stronger Hα line for cooler models. Contrary to expectation, the opposite behaviour of Hα is
produced for this branch.
Hα line formation: rise and fall over the bi-stability jump
3.3 Two branches of Hα behaviour
58
0.4
0.2
log (n 3 /n 2 )
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
1
12500
17500
22500
25000
30000
0
K
K
K
K
K
−1
−2
log τROSS
−3
−4
Figure 3.5: Changes in the (n3 /n2 ) ratio with T eff . Regions where most of the emergent Hα
photons originate from are represented with a thick solid line (cf. Appendix A).
3.3.2 The “cool” branch
On the cool branch, small changes in T eff lead to qualitatively different Hα line profiles. Most
notable is the appearance of a P Cygni profile in Fig. 3.1 when the effective temperature is
reduced from 15 000 to 12 500 K. In fact, these differences between both Hα line profiles contain
major clues to the unexpected changes in the line flux of the models on the cool branch. If Hα
were a pure recombination line only the third level would be relevant. However, as shown later,
Hα increases its optical depth at cooler T eff . It is thus necessary to assess the source function
and the ratio of the number of H atoms per cm3 in the 3rd over the 2nd level (n3 /n2 ).
Figure 3.4 reveals that N3 is constantly increasing when T eff is reduced, however, N2 increases
more steeply. The (n3 /n2 ) ratio has been plotted versus Rosseland optical depth in Fig. 3.5. First
of all, the (n3 /n2 ) ratio is close to unity for the models at the “hot” branch. However, for the
models on the cool branch, n2 becomes significantly larger than n3 , particularly in the outer wind.
Hα line formation: rise and fall over the bi-stability jump
3.3 Two branches of Hα behaviour
59
14
13
logF λ [Erg/(cm 2 s)]
12
11
10
9
8
12 500 K
17 500 K
22 500 K
25 000 K
30 000 K
7
6
3
4
10
10
λ [ Å]
Figure 3.6: Spectral energy distribution at the stellar surface (τross = 2/3) of our models.
Although N1 , N2 , N3 are all increasing as T eff drops, the ratio (n3 /n2 ) is always decreasing, and
the second level becomes more populated than the third level on the cool branch. This leads
to a “dip” in (n3 /n2 ) in the outer wind of the cooler models. The “dip” is quite pronounced,
especially for the coolest model, where n2 is ten times higher than n3 . The increased n2 in the
outer wind produces absorptions for Hα photons emitted close to the photosphere. This naturally
decreases the Hα flux. Hence the question about the decreasing EW over the cool branch, could
be referred to as an issue regarding the behaviour of the (n3 /n2 ) ratio.
Therefore, the next question is, why does n2 become significantly larger than n3 ? Is the “dip”
in (n3 /n2 ) ratio is predominantly caused by an increase of n2 or by a decrease of n3 ? In order
to address these questions, we take a detailed look into the behaviour of the continua and their
effects on individual levels.
Hα line formation: rise and fall over the bi-stability jump
3.4 Two possible explanations for the existence of the “cool” branch
0
10
1
10
2
10
3
10
−1
4
12500 K
log τ
2
−2
−3
−4
0
−2
10
−1
10
0
10
1
10
2
10
3
10
−2
4
17500 K
2
−4
0
−5
−5
−2
−2
−6
−7 −4
−7
−4
−6
−4
2
−4
0
log τ
22500 K
0
−5
30000 K
−5
−2
−2
−3
−6
−7
log (H/H + )
−1
10
log (H/H + )
−2
10
60
−6
−8
−4 −2
−1
10
10
0
1
2
10
10
10
(R-R ⋆ ) [R ⊙ ]
−4 −2
−7
3
−1
10
10
10
0
1
2
10
10
10
(R-R ⋆ ) [R ⊙ ]
3
10
Figure 3.7: Comparison between the H ionisation structure (red dashed line) and the Lyman
continuum optical depth at λ ∼ 900 Å (black solid line) versus the distance from the stellar
photosphere. Solid lines are reserved for the wind optical depth, whilst the dotted horizontal
lines indicate the transition between optically thick and thin part of the wind in the Lyman
continuum (τ = 1). Red colour on the right-hand side is used for the H ionisation structure.
3.4 Two possible explanations for the existence of the “cool” branch
3.4.1 A decrease of n3 ?
In Fig. 3.6 we show spectral energy distributions (SEDs) at the stellar surface. The Lyman
continuum flux is greatly reduced from 30 000 to 12 500 K. Despite that, nothing dramatically
happens with the Lyman continuum around 22 500 K. The changes in the continuum flux are
rather gradual. To understand the behaviour of the Lyman continuum in Fig. 3.7, we compare the
Hα line formation: rise and fall over the bi-stability jump
3.4 Two possible explanations for the existence of the “cool” branch
4.5
0.45
4
0.4
61
0.35
3.5
0.3
log τ
log τ
3
2.5
2
0.2
0.15
1.5
0.1
1
0.5
0.25
τLyman at λ ∼ 900
30
27.5
25
22.5 20
Teff [kK]
17.5
15
12.5
τ
Balmer
0.05
τ
at λ ∼ 3500
Paschen
0
30
27.5
at λ ∼ 8100
25
22.5 20
Teff [kK]
17.5
15
12.5
Figure 3.8: Wind optical depth at τross = 2/3 in the Lyman (left), Balmer (grey circles), and
Paschen continua (red squares) (right).
continuum optical depth at λ ∼ 900 Å (black solid line) to changes in the H ionisation structure
(red dashed line with ordinate in red colour placed on the right side). It is evident that the Lyman
continuum becomes optically thick at distances closer than d ∼ 0.06, 0.25, 0.6 and 7 [R⊙ ] from
the photosphere (roughly τross = 2/3), respectively for the models with T eff = 30 000, 22 500,
17 500, and 12 500 K. The fraction of neutral H at those distances is between ∼ 10−4.9 and 10−4.3 .
The comparison between the Lyman continuum optical depth and H ionisation structure shows
that as soon as neutral H atoms exceed critical values, the Lyman continuum becomes optically
thick. Moreover, the steep increase of neutral H close to the star leads to a significant increase
of the optical depth of the Lyman continuum. Consequently, a large fraction of Lyman ionising
photons are blocked, and the Lyman continuum is no longer the main source of ionisation.
Najarro et al. (1997) studied the appearance of H and He lines in the wind of the LBV
P Cygni, and they found that H recombination, crucial for the Ly continuum and the Lyα optical
depth, blocks the ionising Lyman flux. In their models, the increased n2 was related to the
optical depth of Lyα, giving rise to strong absorption, similar to those produced in our coolest B
supergiant model.
Hα line formation: rise and fall over the bi-stability jump
3.4 Two possible explanations for the existence of the “cool” branch
62
50.75
51
50.5
50.7
50
50.65
49.5
50.6
Log Nλ
Log Nλ
49
48.5
48
50.55
50.5
47.5
50.45
47
50.4
46.5
50.35
46
30
27.5
25
22.5
20
Teff [kK]
17.5
15
12.5
30
27.5
25
22.5
20
Teff [kK]
17.5
15
12.5
Figure 3.9: Number of photons in the Lyman (blue triangles), Balmer (grey circles), and Paschen
(red squares) continua vs T eff . Right-hand side is a “zoom in” from the left-hand side.
As n1 , n2 and n3 are controlled from different continuum ranges, knowledge about the behaviour of Balmer and Paschen fluxes (not only of the Lyman continuum) is required to understand their behaviour. In Fig. 3.6 it is shown that the Balmer and Paschen continuum fluxes
decrease when T eff is reduced. To understand this, we plot the wind optical depth in the Balmer
and Paschen continua in Fig. 3.8 (right panel). Since the cross-section for the photo-ionisation
of a H atom in quantum state n by a photon of wavelength λ is:
σbf = 1.31 × 10−19
λ
1
n5 5000 Å
!3
,
(3.2)
the opacity would be proportional to λ3 . Therefore, we have chosen to plot the wind optical
depth in the Balmer and Paschen continua at wavelengths close to their corresponding jumps
(λ ∼ 3500 and 8100 Å, respectively).
Figure 3.8 shows that the wind optical depth in the Balmer continuum remains fairly constant for all models. Note that when T eff is reduced below 22 500 K, the wind optical depth in
the Paschen continuum is steeply decreased. Models that are cooler than 17 500 K even have
Hα line formation: rise and fall over the bi-stability jump
3.4 Two possible explanations for the existence of the “cool” branch
63
τPaschen < τBalmer . This should produce larger changes in the Paschen flux than in the Balmer
flux. To quantify the reaction of the continuum fluxes Fig. 3.9 shows the number of photons
- Nλ , versus T eff for the Lyman, Balmer, and Paschen continua respectively. The number of
photons is given by:
Nλ = 4πR2⋆
Z
0
λc
πFλ λ
dλ,
hc
(3.3)
where λc is the wavelength boundary of the corresponding continuum series; R⋆ is the stellar
radius. The numbers of photons in the Lyman and Balmer continua are in reasonable agreement
(by a factor of ∼ 4) with those from previous studies (Thompson 1984; Diaz-Miller et al. 1998)
if the different radii are taken into account. Balmer and Paschen fluxes are able to ionise H
atoms respectively from levels 2 and 3, and regulate those levels. It is evident from Fig. 3.9 (left
panel) that these fluxes are about 4 orders of magnitude larger than the Lyman flux on the cool
branch. This is a consequence of high wind optical depth in the Lyman continuum (reported in
Fig. 3.8) and the optically thinner wind in the Balmer and Paschen continua. Hence, the Balmer
and Paschen fluxes are the main sources of ionisation of H atoms in second and third levels over
the cool branch. Figure 3.9 indicates that the total number of photons able to ionise atoms in
level 2 is nearly the same for all models. This would provide nearly the same number of H
atoms ionised from the second level over both branches. Therefore, we do not expect dramatic
changes in n2 due to the Balmer continuum. By contrast, the “Paschen photons” are gradually
increasing in number as T eff is reduced. This is seen in the right panel of Fig.3.9, where we
zoomed in around Nλ for Balmer and Paschen continua. For cooler models, the increasing flux
in the Paschen continuum may thus depopulate more H atoms in the third level.
To understand the key question whether the “dip” in (n3 /n2 ) results solely from an increase
of level 2, or from a decrease of level 3 as well, we show in Fig. 3.10 how their number densities
(and the number density of levels 4 and 10, n4 and n10 ) change with T eff . It is evident from
the plot that when T eff is reduced, n3 behaves in a fashion more similar to the number densities
of the higher levels, e.g., n4 . This is expected if the levels are mainly fed by recombinations.
Hα line formation: rise and fall over the bi-stability jump
3.4 Two possible explanations for the existence of the “cool” branch
5
64
n1
log(ni)
n2
0
n3
30000 K
n4
−5
n10
1/r2
log(ni)
5
0
22500 K
−5
log(ni)
5
2.3x + const
3.4
x+
0
12500 K
4x
con
st
+c
on
st
−5
1
0
−1
−2
log(τross)
−3
−4
Figure 3.10: Population levels of H as a function of Rosseland optical depth. The thick solid
lines in the lowermost panel illustrate a linear fit of n1 (black), n2 (red), and n3 (blue) in the line
formation region, i.e. between log(τross ) = −1.77 and log(τross ) = −2.67.
Therefore, a decrease of level 3 does not occur and consequently the (n3 /n2 ) ratio is not affected
by changes in n3 . It seems that the changes in the Paschen continuum with T eff are not large
enough to cause a decrease of n3 . As a result, n2 should play a major role in the (n3 /n2 ) ratio,
causing the “dip.”
Hα line formation: rise and fall over the bi-stability jump
3.4 Two possible explanations for the existence of the “cool” branch
65
0.6
log(bi )
0.4
30000 K
0.2
log(bi )
0
1.5
1
0.5
22500 K
0
log(bi )
1
0.5
12500 K
0
1
0
−1
−2
log(τross )
−3
−4
Figure 3.11: Non-LTE departure coefficients for the 2nd (solid) and 3rd (dashed) level of H.
3.4.2 An increase of n2 ?
Due to the enormous optical depth in the Ly continuum (reported in Fig. 3.8), the ionising flux
is blocked for the “cool” models and n1 can no longer be affected by the Lyman continuum.
As a result, level 2 becomes the effective ground state, exhibiting a depopulation close to the
photosphere, and an overpopulation in the outer wind in model C (T eff =12 500 K) as shown in
Fig. 3.11. This is similar to the findings of Puls et al. (1998) who concluded that Hα appears like
a P-Cygni line for A supergiants. It should be noted that the peak produced in the departures
Hα line formation: rise and fall over the bi-stability jump
3.5 Lyα and the second level
66
from LTE for the second level (b2 ) in model C occupies the same position as the “dip” in the
(n3 /n2 ) ratio. This coincidence indicates that the “dip” is mainly produced by an increase of n2 .
The difference in behaviour of level 2 compared to higher levels can clearly be seen in
Fig. 3.10, where the number densities of level 3 (blue triangles), 4 (squares) and 10 (green solid
line) are similar for all T eff . For the models on the “hot” branch even n2 behaves in an identical
manner to the number densities of the higher levels. This behaviour is to be expected when the
level populations are solely fed by recombinations, and when they should scale as ρ2 ∼ r−4 (via
the continuity equation). The population of the ground state is, however, affected by ionisations
as well, and it is thus inversely proportional to the dilution factor of the radiation field. The
latter effect makes it increase as distance-squared, and in the outer wind the final dependence
is 1/r2 , as shown in Fig. 3.10. The black asterisks, connected by a dashed line, represent the
number density of the ground state (n1 ), which is decreasing as 1/r2 (illustrated as a blue dashed
line). When T eff is reduced to 12 500 K, n2 diverges from higher-lying energy levels. Now, n2
has smaller slope coefficient in the Hα line formation region compared to higher levels, but still
larger than the slope of the ground state (cf. lowermost panel of Fig. 3.10). Consequently, the
second level of H behaves as a quasi-ground state.
3.5 Lyα and the second level
In order to understand why level 2 diverges from the higher levels when T eff drops, we ran additional models in which we artificially removed the Lyα transition by reducing its oscillator
strength by a huge factor (104 ). The reaction of Hα is displayed in the upper panels of Figure 3.12. It highlights the key role of Lyα. The effects on model M (T eff =22 500 K) are striking:
Hα now switches from a pure emission line into a P Cygni line, and the line flux is decreased
significantly. Furthermore, for the cooler model the removal of Lyα leads to a deeper absorption
component. The middle panels in Fig. 3.12, show changes produced in n2 (dashed) and n3 (solid)
Hα line formation: rise and fall over the bi-stability jump
3.5 Lyα and the second level
67
due to the absence of Lyα – as a function of τross . They demonstrate that the Lyα removal leads
to a tremendous increase of n2 (in comparison to the initial model) in the outer wind, leading to
a stronger absorption component in model C, and the appearance of a P Cygni profile in model
M, where it is noted that the lack of Lyα leads to significant changes in the third level as well:
n3 is surprisingly reduced, and as a result the Hα flux decreases.
To understand the changes in the level populations due to the removal of Lyα, we show its
net radiative rate in the lower panels of Fig. 3.12. This quantity is defined as the difference
between the number of radiative transitions from the upper (second) level to the lower (ground)
state and the number of radiative transitions from the ground level to the second level. As a
result, the net radiative rate is positive when there is a net decay of electrons from the upper
level, and it is negative when there is a net excitation of electrons. It is seen that as soon as
n2noLyα and n3noLyα diverge from their initial populations, at log(τross ) <∼ −1.8 (in model C) and
log(τross ) <∼ −1.5 (in model M), the total Lyα radiative rate in the initial model (black solid
line) is positive2 . Therefore, the line effectively acts as a “drain” for the second level. When we
remove it, the decay of electrons from the second level is suppressed, and n2 is tremendously
increased, as shown in the middle panels.
In other words, by artificially removing Lyα, we can simulate the appearance of P Cygni
profiles for hotter models, showing that the Lyα line is key to the Hα behaviour. We also note
that neither He nor Fe are directly required for achieving this, i.e., it is a pure H effect (as was
shown in Figure 3.1).
Figure 3.13 illustrates how the Lyα Sobolev optical depth changes (cf. Appendix B). The
location where most of the Hα photons originates form (Appendix A) is shown with thicker line
sections. It is evident that, at the hot branch, Lyα is optically thick in the inner Hα forming
region and becomes optically thin in the outer region. Furthermore, the Lyα optical depth at the
2
Note that at log(τross ) <∼ −1.8 and log(τross ) <∼ −1.5 the Hα line starts to form in models C and M (see
Appendix A for details about the Hα forming regions presented in Fig. 3.5).
Hα line formation: rise and fall over the bi-stability jump
3.5 Lyα and the second level
68
Teff = 12 500 K (Model C)
Teff = 22 500 K (Model M)
1.6
2.5
initial model
noLyα
2
log F/Fc
log F/Fc
1.4
1.2
1
1.5
1
0.8
6540
6560
0.5
6580
6540
λ [Å]
6580
1
log(n init /n noLyα )
log(n init /n noLyα )
1
0
−1
−2
n3
n2
0
−1
−2
5e+08
initial model
noLya
1e+07
Nu Aul Zul
6560
λ [Å]
0
0
−1e+07
−2e+07
−5e+08
−3e+07
−4e+07
0
−2
log(τross)
−4
−1e+09
0
−2
−4
log(τross)
Figure 3.12: Upper panels: effect of Lyα on the formation of the Hα line: initial Hα profile
(black solid) and the profile from the models in which Lyα transitions were artificially removed
(the red dash-dotted line). Middle panels: changes in the 2nd (dashed) and 3rd (solid) level of
H due to the removal of Lyα in model C (left) and M (right). The plots present the ratio of the
populations produced from the initial model over the populations from the models without Lyα
transition. Lower panels: comparison of the net radiative rate of 2→1 transitions in the initial
(solid black) and the model without Lyα (the red dash-dotted line) (see § 3.5 for details).
Hα line formation: rise and fall over the bi-stability jump
3.5 Lyα and the second level
69
12500
17500
22500
25000
30000
10
8
K
K
K
K
K
log τLyα
6
4
2
0
−2
−4
1
0
−1
−2
log τROSS
−3
−4
Figure 3.13: Lyα Sobolev optical depth as a function of τross . The region where most of the
emergent Hα photons originate from is shown with thick solid lines (cf. Appendix A)
.
start of the Hα line-formation region is similar for hot models. However, when T eff drops below
22 500 K, the Lyα optical depth at the start of the Hα line-formation region steeply increases,
which continues over the cool branch where Lyα is always optically thick throughout the entire
Hα line-forming region. This means that most photons from the photosphere at the Lyα wavelength do not manage to escape, i.e., they are being scattered or absorbed. As a result, the decay
from the second level is effectively shut off. This can be seen in the lower panels of Fig. 3.12. In
the region where n2 and n3 are affected by Lyα (i.e., for log(τross ) <∼ −1.8 for the cool model
and log(τross ) <∼ −1.5 for the hotter model), the net radiative rate is 2 orders of magnitude lower
for the cooler model, i.e., the second level is depopulated less efficiently by 2 → 1 transitions.
In short, the departure of n2 from the higher-level occupation numbers (when T eff drops from
22 500 K downwards) is related to the Lyα optical depth. When T eff is reduced below 22 500 K,
the Lyα optical depth increases steeply, and level 2 is less efficiently depleted through decay.
Hα line formation: rise and fall over the bi-stability jump
3.6 Conclusions
70
Unlike the second level, level 3 behaves like higher levels, thus the “dip” in (n3 /n2 ) is the result
of an increased level 2 population.
3.6 Conclusions
The behaviour of the Hα line over T eff range between 30 000 and 12 500 K might be characterised by the competition between two processes. Whilst the ”rise“ is the result of simple
recombination (n3 ↑ ), the ”fall“ is due to the intricate behaviour of the second level (n2 ↑ ). As T eff
drops below 22 500 K, the existence of a cool branch may be summarised as follows:
• The high Lyman continuum optical depth makes ionisation from the first level unlikely.
• Lyα becomes optical thick.
• The drain from the second level is suppressed.
• As T eff drops, level 2 diverges from higher levels, and it operates like a ground state.
• Hα changes its character and behaves like a scattering line with a P-Cygni profile.
During the transition from a recombination to a scattering line, the EW decreases because recombination lines have a larger (and basically unlimited) EW, if the mass-loss rate is increased,
whilst a scattering line is confined in its EW as it is dominated by the velocity field. Thus, the
EW has to decrease when the line starts to change its character, i.e., over the cool branch.
It should be stated that the observed Hα line profiles for stars on both sides of the bi-stability
jump are highly variable, i.e. the morphology of the Hα line may vary from pure emission
to P Cygni or absorption in timescales from few days to several months. Such variations are
likely to result from small-scale wind structures (see e.g. Markova et al. 2005) and therefore
the observed column densities have to vary through the epochs. Consequently, the variations in
Hα line formation: rise and fall over the bi-stability jump
3.6 Conclusions
71
Hα should influence only the size of the dispersion expected in Fig. 3.1, if empirical WHα were
derived, whilst the principal processes that play a major role in the Hα line formation should still
be the same as discussed above, i.e. the Hα line behaviour should still depend on the balance
between the recombinations and the efficiency of the Lyα ”drain”.
Even though the proper comparison of our results to the observations is likely to be hampered
by the observed Hα line-profile variabilities, these variabilities are not expected to affect the
general concept of the behaviour of Hα, as the line is formed by recombinations and/or through
the Lyα “drain” - only the temperature and/or the wind-region at which the specific process
govern the line formation would change through the various epochs.
The qualitatively similar Hα behaviour, including just H, and H+He only models, and metalblanketed models suggests that the Hα behaviour is not related to He or metal properties. Although all codes include the physics explained in this chapter, it is interesting that independent
of model complexity, the WHα peaks at the location of the bi-stability jump for all sets of models. This might have consequences for both the physics of the bi-stability mechanism, as well
as the derived mass-loss rates from Hα line profiles, as objects located below the WHα peak
are predicted to be weaker for a similar mass-loss rate, i.e., higher empirical mass-loss rates are
required to reproduce a given WHα if the star is located at a T eff below the peak. Whether this
deeper understanding of WHα over the bi-stability regime would indeed lead to a resolution of
the BSG problem remains to be shown with detailed comparisons of our models and observed
Hα profiles.
Hα line formation: rise and fall over the bi-stability jump
Chapter 4
The effect of clumping
In chapter 3, we have made progress in our understanding of the Hα line over the hot and cool
branches around the bi-stability jump. However, to obtain a more complete picture of Hα line
as a mass-loss diagnostic, we need to know how sensitive Hα is to the clumping on both sides
of the bi-stability jump. Therefore, in this chapter we investigate qualitatively the influence of
clumping on Hα line formation.
Currently, cmfgen takes only optically thin (micro) clumping into account, i.e., the clumps
are assumed to have a dimension smaller than the photon mean free path. The density ρ within
clumps is assumed to be enhanced by a clumping factor, fcl , compared to the wind mean density
ρ̄. We recall from § 2.2.2 that this factor can also be understood in terms of volume filling factor
fV = fcl−1 , assuming that the inter-clump medium is void. Mass-loss diagnostic techniques that
are linearly dependent on density are insensitive to micro-clumping, whilst recombination lines
q
(sensitive to ρ2 ) tend to overestimate the mass-loss rate of a clumped wind by a factor of fV−1 .
However, if the clumps are optically thick, the micro-clumping approach is no longer justified.
72
4.1 The Hα line in a micro-clumping approach
4
70
12500
15000
17500
20000
22500
25000
30000
3.5
60
50
Hα EW [ Å]
3
log(F/F c )
73
2.5
2
40
30
1.5
20
1
10
0.5
−50
0
∆λ[ Å]
50
0
30
25
20
Teff [kK]
15
Figure 4.1: Left: synthetic Hα line profiles from clumped models with volume filling factor
fV∞ = 0.1. Right: Hα line EW as a function of the effective temperature for homogeneous
(circles) and clumped (squares) models.
4.1 The Hα line in a micro-clumping approach
The Hα line emission is a ρ2 dependent process and is therefore sensitive to micro-clumping.
In order to investigate the potential role of micro-clumping we calculated additional models,
identical to the simplistic H+He models from the previous chapter (cf. Table 3.1), but with fV∞ ,
described by the following exponential law:
fV (r) = fV∞ + (1 − fV∞ )exp(−3(r)/3cl ),
(4.1)
where 3cl is the velocity at which clumping is switched on and fV∞ = 0.1. We have chosen the
clumping to start at 3cl = 20 km s−1 , just above the sonic point.
The Hα line profiles are presented in the left panel of Fig. 4.1, where it is shown that clumping enables the asymmetry in Hα to appear at hotter T eff . When clumping was neglected, the
increased level 2 produced an asymmetric line profile at 15 000 K and 12 500 K (cf. Fig. 3.1),
The effect of clumping
4.2 The Hα optical depth in a micro-clumping approach
74
whilst in clumped models with fV∞ = 0.1, the asymmetry is already present at 20 000 K. This
shift towards higher temperatures is caused by the increased mean density of the micro-clumped
winds (in comparison to the mean density of smooth winds), which in turn increases the Lyα
optical depth. Consequently, n2 increases at hotter temperatures. The next question is whether
micro-clumping may have an effect in terms of the cool versus hot branch sequences?
Therefore, in the right panel of Fig. 4.1, we compare how the Hα line EW behaves as a function of T eff for both clumped and unclumped models. It is found that micro-clumping changes
the EW dramatically in the hotter models. Also, at the bi-stability jump location (∼22 500 K)
micro-clumping has a dramatic impact on the Hα EW, where the effect of clumping on Hα line
is largest. However, micro-clumping progressively plays a lesser role towards the cooler edge
of the Bsg regime. The reason for this is that the second level now behaves as a quasi-ground
state, i.e., it scales linearly with density ρ, and remains rather unaffected, whilst the more drastic effects for hotter models are the result of the ρ2 scaling. Nevertheless, clumping transforms
Hα from a pure emission line into a P-Cygni line at 15 000 K. Although micro-clumping is of
quantitative relevance in Bsgs (especially around the bi-stability jump), the existence of an Hα
EW peak remains present in clumped model sequences.
4.2 The Hα optical depth in a micro-clumping approach
In Fig. 4.2, we compare how the Hα optical depth changes with T eff for homogeneous (left)
and clumped (right) models. The line-forming region is indicated by thick lines. White squares
illustrate at which point 50% of the line EW is already formed (see Appendix A). According
to the left plot most of the Hα photons emerge from regions in which the line is optically thin.
Interestingly, when T eff drops from 30 000 to 22 500 K, Hα becomes optically thinner. Below
22 500 K, the line changes its behaviour and becomes optically thicker with decreasing T eff .
Figure 4.2 illustrates that the introduction of micro-clumping would increase the Hα optical
The effect of clumping
4.2 The Hα optical depth in a micro-clumping approach
2
12500
15000
17500
20000
22500
25000
27500
30000
1.5
1
0.5
2
K
K
K
K
K
K
K
K
1
0.5
K
K
K
K
K
K
K
K
0
log(τHα )
log(τHα )
12500
15000
17500
20000
22500
25000
27500
30000
1.5
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−2.5
−2.5
−3
−1
75
−1.5
−2
log(τROSS)
−2.5
−3
−3
−1
−1.5
−2
log(τROSS)
−2.5
−3
Figure 4.2: Hα Sobolev optical depth as a function of τross for homogeneous (left) and clumped
(right) models. Sites where most of the emergent Hα photons originates from are set with thick
solid lines. White squares represent the point at which 50% of the line EW is already formed
(see Appendix A).
depth. It seems that at the bi-stability jump (∼22 500 K) clumping has the largest impact on the
Hα optical depth: it increases by an order of magnitude at the location where 50% of the line
EW is formed (indicated by white squares in Fig. 4.2). At this point, all homogeneous models
are optically thin (left panel), while the clumped models are predominantly optically thick in Hα
(right panel). It appears that when T eff drops the character of Hα changes: from an optically thin
to an optically thick line, although, we kept the mass-loss rate and 3∞ /3esc constant in all models.
Moreover, it is well known from the observations that 3∞ /3esc drops from 2.6 to 1.3 across the
bi-stability jump (Lamers et al. 1995; Markova & Puls 2008). This should produce even sharper
differences in the Hα line on both sides of the bi-stability jump, as the higher velocity ratio on
the hot side is expected to decrease the mean density and thus the Hα line optical depth, whilst
the lower velocity ratio on the cool side would favour higher optical depth of Hα line. It is
worth mentioning that the inclusion of clumping in the models is equivalent to an increase of the
q
mass-loss rate by factor of fcl∞ (in comparison to a homogeneous model)1 . Because of this,
1
We have tested this assumption by comparing the Hα optical depth in a homogeneous model to a clumped model
The effect of clumping
4.3 Impact of macro-clumping on Hα
76
Fig. 4.2 also illustrates how the Hα line optical depth would change if the mass-loss rate were
q
√
increased by a factor of fcl∞ = 10.
All models discussed here have Ṁ = 2.33 × 10−6 M⊙ yr−1 , which is nearly three times larger
than the predicted mass-loss rate for Bsgs around the bi-stability jump. Thus, our homogeneous
models are similar to models with mass-loss rates roughly three times lower, but with clumped
winds (with fcl∞ = 10). Hence, in the context of the predictions, we expect an increase of the
Hα line optical depth around the bi-stability jump analogous to what is shown in Fig. 4.2 (i.e.,
increased by an order of magnitude or more if the mass-loss rate is increased by a factor 5; Vink
et al. 1999).
4.3 Impact of macro-clumping on Hα
Our finding implies that although changes in the Hα optical depth are largest at the cool branch,
the Hα line EW is less sensitive to micro-clumping in the coolest models. The reasons for that
are as follows:
• The Hα optical depth increases when T eff is reduced.
• The Lyα optical depth also increases.
• The second level is prevented from recombining to the ground state.
As a consequence, the second level behaves as an effective ground state. Thus, on the cool
branch the Hα line behaves as an optically thick resonance line and is therefore more sensitive
to macro-clumping.
We thus suggest that macro-clumping may play a major role in Hα line formation on the
cool side of the bi-stability jump. This could significantly affect empirically derived mass-loss
with Ṁ decreased by a factor of
p
fcl∞ .
The effect of clumping
4.4 Discussion and conclusions
77
rates. Šurlan et al. (2013) showed that for O-type supergiants macro-clumping may resolve the
discrepancy between empirical mass-loss rates derived from Hα and PV diagnostics (see also
Sundqvist et al. 2011).
4.4 Discussion and conclusions
In this and the previous chapter, we have made progress in our understanding of the Hα line
over the hot and cool branches around the bi-stability jump. We have not yet discussed whether
our results have a direct bearing on the reported discrepancies between the empirical late-Bsg
mass-loss rates from model atmosphere analyses and Monte Carlo iron-line driving calculations
of Vink et al. (2000). In other words, the problem of the general trend that Ṁvink > ṀHα for B1
and later supergiants.
Previous investigators, in particular Trundle et al. (2004); Trundle & Lennon (2005) and
Crowther et al. (2006) argued that wind clumping would make the discrepancy between ṀHα
and Ṁvink worse, but this conjecture relies on three assumptions: (i) that the character of the
wind clumps (e.g., optically thin versus optically thick) would remain the same at the bi-stability
jump; (ii) that the amount of wind clumping would remain the same; and (iii) that the diagnostic
effects of micro-clumping on the Hα line would be constant with T eff . How likely is it that the
wind structure remains the same when the physics changes at the bi-stability jump?
Oskinova et al. (2007) showed for O star winds that, if clumps become optically thick, and the
wind becomes porous, clumping might in fact underestimate empirical mass-loss rates. Our results suggest that Hα may become optically thick below the bi-stability jump, whilst conversely
it may remain optically thin for hotter objects, implying that previous empirical mass-loss rates
below the bi-stability jump could be underestimated, whilst those from hotter stars could be
correct or slightly overestimated as a result of micro-clumping.
The effect of clumping
4.4 Discussion and conclusions
78
Interestingly, Prinja & Massa (2010) did not find apparent difference in the incidence of
macro-clumping for stars located on both sides of the bi-stability jump. However, their result
were on the basis of Si iv λλ 1400 resonance line doublets.
It might also be relevant that the modelled Hα lines in the work of Trundle et al. (2004) and
Trundle & Lennon (2005) do not reproduce the observed Hα line shapes. This suggests that the
underlying model used in these analyses might not be correct. Moreover, the “derived” values
for β are much higher than predicted, and may be an artifact of an inappropriate modelling
procedure in case macro-clumping would be relevant. It would thus be worthwhile for future
investigations if the effects seen in Figs. 4.1 and 4.2 could indeed explain the reported mass-loss
discrepancies.
The effect of clumping
Chapter 5
Exploration of Hα
A complex system that works is invariably found to have evolved from a simple
system that worked. A complex system designed from scratch never works and
cannot be patched up to make it work. You have to start over with a working
simple system.
John Gall, Systemantics: How Systems Really Work and How They Fail
5.1 Strategy and grid
In preceding chapters, we have obtained the temperature dependence of the Hα line formation
around the bi-stability jump. The effect of clumping was shown as well. It should be clear that
these results are qualitative in nature, as they are based on simplified H+He models. However, a
proper understanding of the Hα line as a mass-loss diagnostic requires analysis of more sophisticated models. Therefore, in this chapter, we shall examine in detail the behaviour of the Hα line
in models which include ions of C, N, O, Ne, Mg, Al, Si, S, P, Ar, Ca and Fe (with atomic data as
presented in Appendix C.1). In these models the accelerating part of the wind is described by a
79
5.2 The influence of various parameters on Hα
80
standard β-velocity law (cf. Eq. 2.13), which is appended to the hydrostatic structure just below
the sonic point. Typical values for β = 0.8...1.5 have been found for OB stars by Haser et al.
(1995); Markova & Puls (2008); Garcia et al. (2014), and therefore we use β = 1 (see however
Crowther et al. 2006).
As the Hα line is sensitive not only to T eff , but also to L⋆ , M⋆ , Ṁ, clumping, etc., throughout
this chapter we shall explore the influence of various stellar and wind parameters on the Hα
line formation. For this purpose, we have calculated a grid of models over a range in T eff ,
L⋆ and M⋆ appropriate for Bsgs. Most of these models are calculated with half-solar metal
abundances1 . However, in § 5.2.5 and § 5.3 the behaviour of Hα in grid of models with different
metal abundances is shown. As was already discussed in § 2.2, the observations indicate that
the winds of hot massive stars are not homogeneous. Therefore, the outcome of the following
is based only on models with non-homogeneous wind structure having a volume filling factor
fv∞ = 0.1 and 3cl = 30 km s−1 (Eq. 2.14).
5.2 The influence of various parameters on Hα
In the following section, we investigate how the behaviour of the Hα line profile and EW depends
on Ṁ, , M⋆ , L⋆ , and metallicity. To understand the influence of these parameters we first kept
3∞ /3esc fixed. In § 5.3 however, we present the behaviour of the Hα line in grids of models
with 3∞ /3esc = 2.6 for T eff ≥22 500 K and 3∞ /3esc = 1.3 for models with T eff between 20 000 and
10 000 K. It is essential to calculate such grids of models as the observations indicate that the
velocity ratio drops from 2.6 for T eff > 20 000 K to 1.3 at cooler temperatures (Lamers et al.
1995; Crowther et al. 2006; Markova & Puls 2008; Garcia et al. 2014).
1
The scaled solar metallicities were taken from Asplund et al. (2009)
Exploration of Hα
5.2 The influence of various parameters on Hα
81
Ṁ =1.5×10−6 [M⊙ yr−1 ]
1.1
2.2
1
2
0.9
1.8
0.8
1.6
F/Fc
F/Fc
Ṁ =0.1×10−6 [M⊙ yr−1 ]
0.7
0.6
0.5
0.4
1.4
1.2
12500
20000
22500
25000
30000
−20
12500
20000
22500
25000
30000
1
0.8
0
λ [Å]
20
−20
0
λ [Å]
20
Figure 5.1: Hα line profiles for sophisticated supergiant models with parameters as listed in
Table 3.1, but different mass-loss rates.
5.2.1 The dependence of the Hα line EW on T eff for various Ṁ
In order to study the role of Ṁ in the formation of the Hα line, we have calculated a grid of
sophisticated models analogous to those discussed in Chapters 3 and 4, i.e., over a range in T eff
from 30 000 to 12 500 K, with log (L⋆ /L⊙ ) = 5.50, M⋆ = 40 M⊙ , 3∞ /3esc = 2 and Ṁ = 0.1, 0.25,
0.5, 0.75, 1.00, 1.50 (in units of 10−6 M⊙ yr−1 ).
Figure. 5.1 shows that the Hα line profile turns from absorption into pure emission (or
P Cygni type) when the mass-loss rate is increased. Note that in the models with weak winds
( Ṁ = 0.1 × 10−6 M⊙ yr−1 ), the non-monotonic behaviour of Hα line is not re-produced. This
implies that there is a limit of the mass-loss rate Ṁlim below which the peak in Hα EW is not
formed. The reason is that the peak in Hα is a wind feature. Thus, in weak winds the emission
of Hα is very small and it is therefore “hidden” in its photospheric absorption.
The value of Ṁlim can be inferred from Fig. 5.2, where it is shown how the Hα line EW
changes when Ṁ is decreased. The peak is present for models with Ṁ ' 0.25 × 10−6 M⊙ yr−1 .
Exploration of Hα
5.2 The influence of various parameters on Hα
82
log L = 5.5; M⋆ = 40 [M⊙ ]; Γe = 0.20
20
0.10
0.25
0.50
0.75
1.00
1.50
10
3
2
Hα EW [Å]
15
Hα EW [Å]
log L = 5.5; M⋆ = 40 [M⊙ ]; Γe = 0.20
4
5
1
0
−1
0
−5
−2
30
25
20
Teff
15
10
−3
30
25
20
Teff
15
10
Figure 5.2: Hα line EW as a function of T eff for models with different values of Ṁ in units of
10−6 M⊙ yr−1 . Right hand side is a “zoom in” from the left hand side.
Below this value it seems the photospheric absorption dominates the wind emission and the Hα
line EW is a gradually decreasing function with T eff . This is a consequence from the increasing
value of 3∞ (when T eff rises), which would produce broader absorption in hotter models. Thus
Hα EW has to decrease with T eff .
Figure 5.2 demonstrates that the peak of the Hα line EW is shifted towards higher temperatures when the mass-loss rate is increased. This shift can be understood because the increased
mass-loss rate would increase the mean density, which in turn increases the Lyα optical depth.
Consequently, the second level of H is highly increased and overtakes the recombination effect
at hotter temperatures.
5.2.2 Influence of T eff and Ṁ on Hα line EW and morphology
Figure 5.3 presents the dependence of the Hα line EW and morphology on T eff and Ṁ for model
series ’L5.5M40’ (cf. Table 5.1). Coloured parts of the figure indicate when Hα is in absorption
Exploration of Hα
5.2 The influence of various parameters on Hα
log L⋆ = 5.5; M⋆ = 40; v∞ /vesc = 2.0
1.5
1.5
Emission
PCy
g
4
0
8
4
0.5
0
0
PCy
g
6
1
4
10
6
Ṁ (10−6 M⊙ yr−1)
8
Ṁ (10−6 M⊙ yr−1)
log L⋆ = 5.5; M⋆ = 40; v∞ /vesc = 2.0
18
8
83
Emission
1
2
6
2
0.5
2
0
2
0
0
Absorption
0
30
25
0
20
Teff (kK)
Absorption
15
10
0
30
25
0
20
Teff (kK)
15
10
Figure 5.3: Influence of T eff and Ṁ on the morphology of the Hα line profile. White squares
indicate the positions of the grid-models used.
(orange), P Cygni (blue) or pure emission (white). We have defined the line to be in emission
when its EW is positive, whilst negative EWs are representative for absorption. In the coolest
models however (T eff ≤12 500 K), the Hα line is directly transformed into P Cygni emission
when Ṁ is increased. Therefore for the coolest models we have defined Hα to be P Cygni type
when the equivalent width of its red shifted part, EWred , (i.e., for λ > 6564.5 Å) is positive. If
EWred is negative, then the line is defined to be in absorption. In the right panel of Fig. 5.3, we
show contours of EWred with solid lines, whilst in the left panel the solid lines indicate contours
of the total EW of Hα line. The yellow squares show the value of EW contours.
The mass-loss rate at which Hα transforms from absorption into emission line, Ṁtr , is progressively increased for models hotter than 15 000 K. Qualitatively, this trend resembles the
changes associated with the behaviour of the Hα line EW (displayed in Fig. 5.2), i.e., the hottest
(but also coolest) models would require higher Ṁ than the models with intermediate temperatures in order to produce similar Hα line EW. Thus a minimum in Ṁtr around 22 500 K is
expected.
Exploration of Hα
5.2 The influence of various parameters on Hα
84
Table 5.1: Adopted stellar and wind parameters for the main grid of models.
Z = 0.5 × Z⊙ , log LL⋆⊙ = 5.50, 3∞ /3esc = 2; fcl = 10; 3cl = 30 km s−1
M⋆
(M⊙ )
40
30
20
Γe
T eff
(K)
0.20
0.26
0.39
12 500 – 30 000
12 500 – 30 000
12 500 – 30 000
Ṁ
(10−6 M⊙ yr−1 )
0.1 – 1.5
0.1 – 1.5
0.1 – 1.5
model
series
L5.5M40
L5.5M30
Figure 5.3 indeed shows that a minimum in Ṁtr is formed, however at cooler temperatures:
15 000 − 12 500 K. This can be understood because the values of Ṁtr are defined when Hα line
is weak (no emission), and thus it is relevant to associate the behaviour of Ṁtr only with the
behaviour of the Hα line EW in models with weak emission, i.e., to the set of models with
Ṁ = 0.25 × 10−6 M⊙ yr−1 . In these models, the Hα line EW exhibits a maximum for T eff ≈
15 000 − 17 500 K (not at 22 500 K) and thus Ṁtr has to increase at hotter than 17 500 − 15 000 K
temperatures, which is illustrated in Fig. 5.3. Note that for contours with larger EW the minimum
shifts toward hotter temperatures.
5.2.3 Influence of stellar mass
The next relevant question is to understand how stellar mass impacts the formation of the Hα
line. To investigate this question, in Fig. 5.4 we compare Hα line profiles from models with
different masses. Following Vink et al. (2001), we selected M⋆ = 40, 30 and 20 M⊙ and kept
other parameters constant (cf. Table 5.1)2 . Fig. 5.4 shows that the Hα line becomes stronger but
also narrower as the gravity is reduced. Note that in the cooler model the absorption component
becomes deeper.
This behaviour is prescribed by the strategy we employ to investigate the influence of stellar
mass on the Hα line. To be specific, for fixed stellar and wind parameters, a decrease of stel2
Stellar mass in cmfgen is updated by changing log g.
Exploration of Hα
5.2 The influence of various parameters on Hα
log L⋆ =5.5 [L⊙ ]; Ṁ =1.5×10−6 [M⊙ yr−1 ]
2.5
85
log L⋆ =5.5 [L⊙ ]; Ṁ =1.5×10−6 [M⊙ yr−1 ]
2.5
Γe = 0.39
Γe = 0.26
2
Γe = 0.26
Teff =20000 K
Γe = 0.20
Γe = 0.20
2
F/Fc
Teff =12500 K
F/Fc
Γe = 0.39
1.5
1.5
1
0.5
−10
0
∆λ [Å]
10
1
−20
−10
0
∆λ [Å]
10
20
Figure 5.4: Influence of the stellar mass on the the Hα line profile.
lar mass leads to a lower value for 3esc and thus 3∞ (if 3∞ /3esc ratio is kept constant, 3∞ would
√
decrease by factor R when M⋆ is reduced by a factor R). As 3∞ decreases, the entire velocity structure of the wind shifts towards lower velocities and therefore the Hα photons should
originate from regions with lower velocities. Thus, the line wings are expected to become narrower. In addition, the density must increase (according to the continuity equation, Eq. 2.2) and
therefore the intensity of the Hα line has to increase as well.
On the other hand, lowering gravity must decrease the electron pressure Pe (as Pe ∝∼ g1/3 ;
see Gray 2005, for details). According to the Saha equation (Saha 1921),
Ni+1 2kT Zi+1 2πme kT
=
Ni
Pe Zi
h2
!3/2
e−χi /kT ,
(5.1)
the neutral hydrogen (and thus intensity of Hα) must decrease when Pe is reduced.
Finally, when M⋆ is decreased we have two opposing effects: an increase in density (due
to the decrease of 3∞ ) and a decrease of neutral hydrogen (due to the lower gravity). After all,
Exploration of Hα
5.2 The influence of various parameters on Hα
86
logL=5.5; Mdot = 0.25E06 [Msun/yr];
2
log L = 5.5; Ṁ =1.50×10−6 [M⊙ yr−1 ];
20
15
Hα EW [Å]
WHα EW [A]
1
0
−1
M20; f=0.1; Vcl=30 km/s
M30; f=0.1; Vcl=30 km/s
M40; f=0.1; Vcl=30 km/s
M40; noCL
M40; f=0.1, Vcl=100km/s
M40;other Fe model atoms
−2
−3
30
25
20
Teff
15
10
10
M20; Γe = 0.39
M30; Γe = 0.26
M40; Γe = 0.20
5
0
30
25
20
Teff
15
10
Figure 5.5: Influence of the stellar mass on the Hα line EW in models with strong (right) and
weak (left) winds.
the intensity of Hα increases when M⋆ is reduced by factor of R, because the density structure
√
√3
increases with factor of R, whilst the neutral hydrogen is reduced by factor of R (because
Pe ∝∼ g1/3 ). Note that when M⋆ is decreased, the narrower line wings are compensated by an
increase of line-core emission and this may cause non-monotonic changes in the behaviour of
the Hα line EW.
Figure 5.5 illustrates the changes in the Hα line EW caused by reducing the gravity in models
with strong (right) and weak (left) winds. As evident, the maximum in the Hα line EW is
always present, but in the stronger winds the Hα line changes its EW non-monotonically, whilst
in weaker winds the Hα line successively increases its EW. Note that the trend of successive
increases of the line EW in the models with weak winds would tend to reduce the values of Ṁlim
in less massive stars.
At the different temperatures we choose different level assignments for the involved species
and therefore it is important to estimate how sensitive WHα is to the adopted model atoms. In
the thesis, other assumptions, such as the value of the volume filling factor fV , or the velocity
Exploration of Hα
5.2 The influence of various parameters on Hα
87
Table 5.2: Atomic data used to test the sensitivity of WHα to the adopted model atoms of the
iron ions. For comparison, the initial model atoms are provided as well.
Ion
Super/Full levels
Initial
Test
Fe ii
17/218
275/827
Fe iii 136/1500 136/1500
Fe iv 74/540 100/1000
Fe v
17/57
34/352
at which clumping starts 3cl , are also made, and thus the influence of these assumptions needs
to be estimated. To do that, we calculated additional test models from L5.5M40 series with
T eff = 17 500 K . These models have different clumping parameters ( fV = 1 or 3cl = 100 km/s)
or different super-level assignments of the iron ions (cf. Table 5.2). Fig. 5.5 shows that the
Hα line is sensitive to parameters such as the velocity at which clumping is switched on (gray
square) or the degree of clumping (grey triangle), whilst the choice for the model atoms seems
to be less important.
Note that the effects of the clumping were already discussed in Chapter 4, however, here we
estimate the influence of clumping on WHα again, in order to stress that the behaviour of the Hα
line is more sensitive to assumptions regarding clumping (such as the degree of the clumping or
the velocity at which clumping stars) than to assumptions concerning model atoms. Thus, we
do not expect the various model atoms used at the different temperature ranges (cf. Appendix C)
to cause significant differences in the behaviour of the Hα line.
5.2.4 Influence of stellar luminosity
To investigate whether the dependence of the Hα line EW on T eff is universal for different lu minosity, we have calculated series of models with log LL⋆⊙ = 5.00, whilst the other stellar and
wind parameters were kept fixed. (cf. Table 5.3).
Exploration of Hα
5.2 The influence of various parameters on Hα
88
M⋆ =30.0 [M⊙ ]; Ṁ =0.75×10−6 [M⊙ yr−1 ]
2.2
M⋆ =30.0 [M⊙ ]; Ṁ =0.75×10−6 [M⊙ yr−1 ]
1.6
2
1.4
Teff =12500
Teff =22500
1.8
F/Fc
F/Fc
1.2
1
1.6
1.4
1.2
0.8
log L⋆ = 5.5
log L⋆ = 5.0
−10
0
λ [Å]
10
log L⋆ = 5.5
log L⋆ = 5.0
1
0.8
−20
0
λ [Å]
20
Figure 5.6: Example of Hα line profiles in models with different luminosities.
It should be stated that varying Ṁ at fixed L⋆ will characterise the Hα line profile as a function
of wind density, whilst varying L⋆ at fixed Ṁ is less straightforward as the wind density depends
√
on Ṁ and R⋆ (R⋆ ∝ L⋆ ).
Figure 5.6 shows that the Hα line becomes stronger and broader when L⋆ is decreased. The
reason is that the decrease in L⋆ results in an increase of log g, which in turn, as we already
discussed in § 5.2.3, must increase the intensity of Hα. Despite, that in less luminous models a
lower density structure is favoured by higher 3∞ , the effect of increased log g is dominant and
finally the line emission rises. Note that in the cooler model, the absorption component occurs
at shorter wavelengths which is a consequence of the increased 3∞ .
Figure 5.7 illustrates the overall influence of luminosity on the morphology of Hα in the
T eff − Ṁ plane. When we reduce L⋆ , the Hα line forms a P Cygni profile in models with lower
Ṁ. Note also that, with respect to model series ’L5.5M30’ Ṁtr is notably reduced in models
with T eff > 17 500 K.
The impact of luminosity on WHα presented in the left panel of Fig. 5.8. As the decrease
Exploration of Hα
5.2 The influence of various parameters on Hα
log L⋆ = 5.5; M⋆ = 30; v∞ /vesc = 2.0
Ṁ (10−6 M⊙ yr−1)
Ṁ (10−6 M⊙ yr−1)
1
0.5
g
Cy
P-
Emission
log L⋆ = 5.0; M⋆ = 30; v∞ /vesc = 2.0
1.5
g
Cy
P-
1.5
89
Emission
1
0.5
Absorption
Absorption
0
0
30
25
20
Teff (kK)
15
30
25
20
Teff (kK)
15
Figure 5.7: Influence of luminosity on Hα morphology. Model series ’L5.5M30’ (left) and
’L5.0M30’(right) are presented. The green solid line (right) indicates the absorption and PCygni transition mass-loss rates in model series ’L5.5M30’.
Ṁ =0.50×10−6 [M⊙ yr−1 ];
log L⋆ =5.50; M⋆ =40 [M⊙ ]
12
Hα EW [Å]
Hα EW [Å]
8
10
5
0
L5.0M6.3; Γe = 0.39
L5.0M30; Γe = 0.08
L5.5M20; Γe = 0.39
L5.0M20; Γe = 0.12
−5
30
25
20
Teff
15
=1
.0
10
Ṁ
15
Z⊙ /2
Z⊙
6
4
2
˙ =
M
0
Ṁ =0.1
−2
10
30
0.5
25
20
Teff
15
10
Figure 5.8: Left: Behaviour of WHα in sets of models with different luminosity and mass. Right:
Behaviour of WHα in sets of models with different Ṁ and metal composition.
Exploration of Hα
5.2 The influence of various parameters on Hα
90
Table 5.3: Adopted stellar and wind parameters for the additional grid of models.
Z = 0.5 × Z⊙ , log L⋆ = 5.00, 3∞ /3esc = 2; fcl = 10; 3cl = 30 km s−1
M⋆
(M⊙ )
40
30
20
9.5
6.3
Γe
0.08
0.13
0.26
0.39
T eff
(K)
12 500 – 30 000
12 500 – 30 000
12 500 – 30 000
12 500 – 30 000
12 500 – 30 000
Ṁ
(10−6 M⊙ yr−1 )
0.1 – 0.75
0.1 – 0.75
0.1 – 0.75
0.1 – 0.75
0.1 – 0.75
model
series
L5.0M30
of L⋆ leads to broader line wings and stronger Hα emission, it is expected that WHα increases
its EW when L⋆ is reduced. In order to compare the influence of luminosity with the impact
of the stellar mass in the plot we also provide the WHα from model series with log L⋆ = 5.00
and M⋆ = 30, 20 and even 6.3 M⊙ (same Γe as model series ’L5.5M20’). Note that reducing
L⋆ by a factor of three, changes WHα more than reducing M⋆ by a factor of five. This indicates
that luminosity has a stronger influence on the Hα line EW than the stellar mass. This can be
understood because varying L⋆ changes the line wings and the intensity of Hα in one direction,
whilst the changes in Hα caused by varying M⋆ are compensated by each other (narrower wings,
stronger emission).
5.2.5 Influence of Metallicity
The presence of numerous spectral lines that blend together may ’block’ the continuum and
make it difficult, or even impossible, to identify the stellar continuum. In this way the spectral
lines could influence the SED of the star. As H and He only have few lines, it is mainly the lines
of the metals that can influence significantly the SED of a star. In general, line blocking makes
the star appear redder.
If metal abundance of a star is increased, then the number of absorbers is increased as well.
Consequently, more stellar flux is blocked. As the flux is blocked, a certain number of photons
Exploration of Hα
5.2 The influence of various parameters on Hα
log ( LL⊙⋆ ) = 5.50;
1.8
M⋆
M⊙
log ( LL⊙⋆ ) = 5.50;
= 40
Ṁ =0.75 ×10 −6 [M⊙ yr−1]
Teff =15000 K
M⋆
M⊙
1.5 Ṁ =0.75×10 −6 [M⊙ yr−1]
v∞
vesc
=2.0
v∞
vesc
=1.3
Teff =22500 K
1.4
F/Fc
1.6
F/Fc
91
1.4
1.2
= 40
v∞
vesc =2.0
v∞
vesc =2.6
1.3
1.2
1.1
1
1
−10
−5
0
λ [Å]
5
10
−20
0
λ [Å]
20
Figure 5.9: Example of changes in the Hα line profile due to a varying 3∞ /3esc ratio.
are back-scattered and the temperature in the deeper layers increases, whilst the temperature in
the outer layers decreases. The former effect is referred to as ’back warming’, whilst the latter
as ’surface cooling’. The combined effects of back warming and surface cooling are usually
referred to as line ’blanketing effects’. These effects make the relation between local temperature
and T eff less straightforward. Note that the back warming effect should enhance the ionisation
in the deeper layers of the atmosphere due to the increased temperature.
The right panel of Fig. 5.8 compares the behaviour of the Hα line EW in a series of models with different metallicity. Note that the influence of metallicity is negligible in the models
with weak winds. In denser winds however, the metals are able to back-scatter more photons
and the ionisation of H in the deeper layers increases with metallicity. Consequently, the Hα
line is weaker in models with more metals. In addition, the back-scattering absorb photons at
wavelengths around Hα which has to decrease further the emission from Hα line.
Exploration of Hα
5.3 Hα in the context of the bi-stability jump in 3∞ /3esc
92
M⋆
log ( LL⊙⋆ ) = 5.50; M
= 40; Ṁ =0.25×10−6 [M⊙ yr−1 ]log ( LL⊙⋆ ) = 5.50;
⊙
12
4
v∞
vesc = 1.3, 2.6
v∞
3
vesc = 2.0
10
1
0
−1
6
4
2
v∞
vesc
v∞
vesc
−2
0
−3
= 40; Ṁ =1×10−6 [M⊙ yr−1 ]
8
Hα EW [Å]
Hα EW [Å]
2
M⋆
M⊙
30
25
20
Teff [kK]
15
10
30
= 1.3, 2.6
= 2.0
25
20
Teff [kK]
15
10
Figure 5.10: Influence of 3∞ /3esc ratio on Hα line EW.
5.3 Hα in the context of the bi-stability jump in 3∞/3esc
The results from previous subsections were obtained for fixed 3∞ /3esc ratio. However, we know
from observations that 3∞ /3esc drops from about 2.6 at the hot side of the bi-stability jump (for
T eff >21 000 K) to 1.3 at the cool side (Lamers et al. 1995; Crowther et al. 2006; Markova &
Puls 2008; Garcia et al. 2014). The different wind properties on both sides of the bi-stability
jump may influence the behaviour of the Hα line and therefore it is relevant to investigate the
behaviour of the Hα line EW and morphology in model series with appropriate values of 3∞ /3esc
ratio. Therefore, in this section we present the behaviour of Hα line in models with 3∞ /3esc =2.6
for T eff ≥ 22 500 and 1.3 for 20 000 ≥ T eff ≥10 000 K. The other parameters were kept fixed.
Examples of Hα line profiles from such models are presented in Fig. 5.9 which reveals that
the applied increase of 3∞ /3esc to 2.6 generally decreases the intensity of the Hα line. The
opposite effect holds when the velocity ratio is decreased to 1.3. This is because, according
to the continuity equation (Eq. 2.2), higher velocities lead to lower densities and thus to less
emission. Another thing to note is that the cooler model in Fig. 5.9 forms a P Cygni type profile
Exploration of Hα
5.3 Hα in the context of the bi-stability jump in 3∞ /3esc
log ( LL⊙⋆ ) = 5.50;
2
M⋆
M⊙
93
= 40; Z=Z⊙ /2
6
10
Ṁ (10−6 M⊙ yr−1)
10
1.5
6
10
Emission
1
6
6
2
PCy
g
10
2
0
2
0.5
6
0
Absorption
0
30
25
0
2
0
20
Teff (kK)
15
10
Figure 5.11: Morphology of the Hα line for models with 3∞ /3esc ratio as known from observations and half-solar metal abundances.
when the velocity ratio is reduced. This behaviour together with the narrower line wings (due
to lower velocities) would tend to compensate the stronger Hα emission from the models over
the cool branch. Precisely this is seen in the right panel of Fig. 5.10, where the influence of the
bi-stability jump in 3∞ /3esc on the WHα is shown. Whereas on the hot branch the increase of
3∞ /3esc leads to a shift of WHα towards lower values, on the cool branch WHα is less affected by
changes in 3∞ /3esc . Consequently, the peak in the WHα becomes sharper and it is now located at
a slightly cooler temperature ∼20 000 K.
For weaker winds, however, on the cool branch WHα is more sensitive to changes in 3∞ /3esc
(cf. left panel of Fig. 5.10). The reason is that in tenuous winds Hα does not form a P Cygni type
profile, which would normally compensate the stronger line emission when the velocity ratio
is decreased 3 . Thus, the increased emission of Hα can be compensated only by narrower line
wings. Finally, it is interesting to note that for weak winds, the signature of the bi-stability jump
3
To form a P Cygni feature, the wind has to be dense enough, in order for Lyα to become optically thick and to
suppress the drain from the second level. For hotter temperatures the wind is more ionised than cooler temperatures
and therefore Lyα is optically thinner. Thus, the wind density (or mass-loss rate) at which Hα forms a P Cygni feature
has to be higher for hotter temperatures.
Exploration of Hα
5.3 Hα in the context of the bi-stability jump in 3∞ /3esc
log ( LL⊙⋆ ) = 5.50;
M⋆
M⊙
94
log ( LL⊙⋆ ) = 5.50;
= 40; Z=Z⊙ /5
10
10
6
6
10
Emission
10
1
1.5
P- 2
Cy
g
0
6
10
2
6
2
0
0.5
2
6
0
Absorption
0
30
25
Ṁ (10−6 M⊙ yr−1)
Ṁ (10−6 M⊙ yr−1)
1.5
10
6
PCy
2 g
Emission10
1
6
2
6
0
0.5
0
2
2
0
Absorption
0
20
Teff (kK)
= 40; Z=Z⊙
6
0
2
M⋆
M⊙
15
10
0
30
25
0
20
Teff (kK)
15
10
Figure 5.12: Comparison of the morphology of Hα in grid of models with solar (right) and five
times lower metal composition (left). 3∞ /3esc = 2.6 for models with T eff ≥ 22 500 K 3∞ /3esc =
1.3 for cooler models.
in 3∞ /3esc is seen as a jump in Hα line EW in the left panel of Fig. 5.10.
Figure 5.11 shows the morphology of Hα in the T eff − Ṁ plane for models with half-solar
metal abundances and velocity ratios as above, i.e., 3∞ /3esc = 2.6 for T eff ≥ 22 500 K and
3∞ /3esc = 1.3 for models with T eff between 20 000 and 10 000 K4 . The figure shows that decreasing the velocity ratio on the cool branch produces P Cygni type profile for lower Ṁ, whilst
an increase of velocity in the models over the hot branch produces absorptions at higher Ṁ.
Apart from these differences and the sharper peak in the Hα line EW, the qualitative behaviour
of Hα remains the same.
Similar plots are shown in Fig. 5.12, but for models with solar (right) and five times lower
(left) metallicities. On a general scale, increased metallicity increases the absorption area of Hα,
i.e., the line switches from absorption into emission at higher Ṁ. The reason is that the increased
fraction of metals is expected to absorb more Hα photons and thus higher Ṁ are required in
4
Note that in the figure the absorption area is illustrated with green colour (not with orange), in order to distinguish
the model series with fixed velocity ratio from the models with ratio as known from observations.
Exploration of Hα
5.4 Conclusions
95
order Hα to switch from absorption into emission. Additionally, the model with T eff = 17 500 K
require higher Ṁ in order to form P Cygni type profile, whilst the cooler models form P Cygni
type profiles at similar Ṁ for both metallicities.
5.4 Conclusions
We find that the maximum in WHα is present in set of models with various stellar parameters
and chemical compositions. However, in models with Ṁ below a limiting value, Ṁlim , the
maximum in Hα is not formed. This implies that the maximum is a wind feature and therefore
it is dependant on the wind density and on the processes which influence that density. In models
with 3∞ /3esc = 2, the maximum of the Hα line EW alter its position when Ṁ is varied because
the density is affected and the ”Lyα drain“ is initiated at different T eff . However, in models with
velocity ratio as known from observations, the location of the peak is less sensitive to changes
in Ṁ as Hα peaks always at T eff = 20 000 K in the presented range of Ṁ.
On the observational side, it would be interesting to find out whether such a maximum really
exists. As in model series with velocity ratio taken from observations, the maximum is sharper,
the confirmation or refutation of the predicted peak should not be difficult. As luminosity has a
strong influence on the Hα line (as illustrated in Fig. 5.8), accurate knowledge of stellar luminosity is required for such an investigation.
Exploration of Hα
Part III
Wind properties
96
Chapter 6
Wind properties of blue supergiants
6.1 An overview of line-driven winds: recall the basic relations
The atmospheres of stars are not in hydrostatic equilibrium. Instead stars are loosing material
due to an outward directed force, which is larger than gravitational attraction. In the winds
of hot stars this force is driven by the absorption of photons in atomic transitions. A photon
from the photosphere can be absorbed in an atom/ion if its energy is the same as the energy
required to excite the bound electron to a higher energy level. As the new state of the ion is
very short, the excited electron falls back to the ground state and the photon is re-emitted. In
this process, the average radial momentum of the photons is transferred to the absorbing ion.
Finally, the accelerating ion shares the gained momentum with the surrounding field particles
through Coulomb coupling. The condition for Coulomb coupling is given as:
L ⋆ 3∞
< 5.9 × 1016 ,
Ṁ
(6.1)
where L⋆ is in L⊙ , the terminal velocity 3∞ in km s−1 , and Ṁ in M⊙ yr−1 (Lamers & Cassinelli
1999).
97
6.1 An overview of line-driven winds: recall the basic relations
98
Continuum processes also contribute to the radiative force. The strength of this contribution
depends on the density and the number of photons. As hot stars have strong radiation fields,
nearly all hydrogen atoms in their winds are ionised and therefore the electron density is large.
Thus, the acceleration due to electron scattering becomes relevant. While the acceleration due
to Thomson scattering can be important in the deepest layers of the wind, in the outer layers
the radiative force is dominated by line interactions. The dominant contributor to the overall
radiative acceleration are line transitions because of their resonant nature and the key role of the
Doppler shift.
The first quantitative description of line-driven winds was given by Lucy & Solomon (1970).
However, they derived mass-loss rates which were too low (10−7...−10 M⊙ yr−1 ), because only
a few strong UV resonance lines were considered in their models. Later on Castor, Abbott &
Klein (1975), hereafter called ’CAK’, developed a formalism to treat the line acceleration due
to an ensemble of lines. This allowed them to predict mass-loss rates which are 2 orders of
magnitude higher than the derived by Lucy & Solomon (1970). The theory was then refined by
several other (e.g. Abbott 1982; Pauldrach et al. 1986; Kudritzki et al. 1989; Vink et al. 2000)
The following briefly summarises the main ingredients of the theory of line-driven winds. It
is partly based on Lamers & Cassinelli (1999) and therefore, for a more detailed discussion, the
reader is referred to that book.
6.1.1 The momentum equation
The outflow of a line-driven wind is driven by the interplay between gravity and radiative acceleration. Thus, the equation of motion of the wind can be written as:
3
d3
GM⋆ 1 dp
=− 2 −
+ ge + gtot
L ,
dr
ρ r
r
(6.2)
Wind properties of blue supergiants
6.1 An overview of line-driven winds: recall the basic relations
99
where the first term describes the acceleration field, which is specified by the adopted velocity
law. Second term is the local acceleration of gravity at radius r. The third term describes the
acceleration due to the gas pressure p = RρT/µ, where µ is the mean atomic weight of the
particles in units of mH and R is the gas constant. ge is the radiative force due to scattering by
free electrons and gtot
L is the total radiative acceleration due to all bound-bound transitions.
6.1.2 Driving forces of the winds in hot massive stars
6.1.2.1
The continuum acceleration
The radiative acceleration due to electron scattering is given by:
ge =
σe L ⋆
,
4πr2 c
(6.3)
where σe is the interaction cross section for electron scattering in units cm2 g−1 , which is formally named the mass absorption coefficient, but it is also commonly called opacity. The opacity
depends on the wind ionisation and abundance:
σe = σT
ne
,
ρ
(6.4)
with σT = 6.65×10−25 cm2 , which is the cross section for Thomson scattering and ne the number
density of electrons.
The continuum acceleration can be expressed as a function of gravitational acceleration via
the classical Eddington factor:
Γe =
σe L ⋆
ge
=
.
ggrav 4πcGM⋆
(6.5)
Wind properties of blue supergiants
6.1 An overview of line-driven winds: recall the basic relations
100
Since both accelerations have the same 1/r2 dependence on radius and the electron scattering
opacity in an ionised medium is constant, the classical Eddington factor has a characteristic
value for each star.
However, it is important to realise that the Eddington factor (Γ) describes the ratio between
total radiative force and gravitational acceleration. Thus, the definition of Γ includes the opacities due to all lines and continua, kν , which is radius dependent:
Γ=
gtot
kν (r)L⋆
L + ge
=
.
ggrav
4πcGM⋆
(6.6)
The difference between Γe and Γ depend on the metal composition and on the ionisation structure. If radiative acceleration is comparable to gravitational force, Γ → 1 (Eddington limit;
Eddington 1921), the star becomes unstable and gravitationally unbound.
6.1.2.2
The line acceleration
The prime challenge to solve the equation of motion comes from the line acceleration term.
Important quantities that determine the strength of the radiation force are the stellar flux, the
cross section of the particles that may interact with this flux, the chemical composition and
the degree of ionisation of the wind. While the continuum acceleration may be conveniently
expressed in terms of luminosity, in CAK theory (but see also Abbott 1982) the total acceleration
due to all spectral lines can be expressed in terms of continuum acceleration via the following
power law:
gtot
ne δ
L
,
= M(t) = Kt−α 10−11
ge
W
(6.7)
where M(t) is the so-called force-multiplier which represents the amount by which line acceleration is larger than continuum acceleration. The force-multiplier parameter, K, is proportional
Wind properties of blue supergiants
6.1 An overview of line-driven winds: recall the basic relations
101
to the number of contributing lines. The parameter, t, is a dimensionless optical depth given by:
t = σe 3th ρ
dr
,
d3
(6.8)
where 3th is the thermal velocity of particles. The dimensionless parameter α is a measure of
δ
the ratio between optically thick to optically thin lines. The term 10−11 ne /W , where W is the
geometrical dilution factor of the radiation field and ne is the electron density in units of cm−3 ,
accounts for changes in ionisation of the wind1 .
In reality, the line acceleration is more complicated because it is tightly related to the physical
properties of the absorbing lines, i.e., their number and the probability for absorption of a photon.
The probability for an absorption is given by the absorption coefficient for a single line transition
kνL (in cm2 /g) between a lower l and upper level u:
kνL (∆ν)
!
πe2 nl
nu gl
=
fl
φ(∆ν),
1−
me c ρ
nl gu
(6.9)
where fl is the oscillator strength of the transition, the nl and nu are the number densities of the
ions in the corresponding levels with statistical weights gl and gu . The profile function φ(∆ν)
describes the width of the absorption profile, where ∆ν = ν − νL is the frequency range of the
photons which can be absorbed.
cmfgen calculates kνL for all lines. The sum of kνL over all contributing lines N, will give the
total line opacity kν of all relevant ions:
kν =
N
X
kνL (∆ν),
(6.10)
1
1
The parameters α and δ were already introduced in § 1.3.1
Wind properties of blue supergiants
6.2 Ion contributors to the line driving
102
and the total radiative acceleration by spectral lines can be calculated:
gtot
L =
1
c
Z
0
∞
Fν kν dν,
(6.11)
where the stellar flux Fν and kνL are computed on a relevant frequency grid.
Metal lines are responsible for the most of the line driving as they contribute mostly to kν .
This implies that the line force is expected to depend on metal abundances and the ionisation
throughout the wind.
Clumping also influences the line opacity and therefore any degree of wind clumping should
affect the line acceleration. If the wind is comprised of optically thin clumps, then porosity
effects can be neglected, and the line force increases simply because the recombination rate of
the gas is higher in comparison to a smooth wind. However, if the clumps are optically thick,
then the porosity effects become important and the line force decreases because the photons
may travel in-between the clumps and interact less with the wind (see Muijres et al. 2011, for a
detailed discussion).
6.2 Ion contributors to the line driving
In the following sections we investigate the dependence of the line acceleration on T eff and
chemical composition. The first step is to identify the main contributors to the radiative acceleration. To do that, we calculated Eq. 6.11 for the different ions. Basically, this gives the relative
contribution of individual ions to the total line acceleration. In Fig. 6.1, we present the relative
contribution of individual ions to the total radiative acceleration for models from the ’L5.5M40’
series (with parameters summarised in Table 6.1). Close to the star most of the radiative force in
both models is provided by H and electron scattering (blue dashed)2 . In the outer wind regions
2
Radiative acceleration of H is mainly determined by bound-free processes.
Wind properties of blue supergiants
6.2 Ion contributors to the line driving
Teff =10000 K; L5.5M40T10
Teff =20000 K; L5.5M40T20
0.5Vinf
0.5Vinf
HI
SiII
FeII
FeIII
CNO
HI+ESEC
Ṁ =5e-07
0.8
0.6
0.4
0.8
0.2
0
1
HI
FeIII
CNO
HI+ESEC
Ṁ =5e-07
rad
rad
gION
/gtot
rad
rad
gION
/gtot
103
0.6
0.4
0.2
0
−1
−2
log τROSS
−3
0
1
−4
0
−1
−2
log τROSS
−3
−4
Figure 6.1: Relative contribution of individual ions to the total radiative force for models with
3∞ /3esc =1.3 and half-solar metallicities.
of the hotter model, Fe iii is the most important line driver, whilst for the cooler model, the contribution of Fe iii is not so important. It should be stated that the presented contributions to the
total radiative force are distance dependent and therefore it is not clear which of the ions provide
most of the global wind acceleration.
To understand this, we investigate which ions contribute mostly to the work ratio Qwind
(Gräfener et al. 2002; Gräfener & Hamann 2005):
Qwind =
Wwind = Ṁ
Z
∞
R⋆
Wwind
, with
Lwind
!
!
1 2 GM⋆
1 dp
.
dr and Lwind = Ṁ 3∞ +
grad −
ρ r
2
R⋆
(6.12)
(6.13)
Wwind is the work performed by the radiative and gas pressure, whilst Lwind is the prescribed
mechanical wind luminosity. Practically, the work ratio tells us whether radiative acceleration
Wind properties of blue supergiants
6.2 Ion contributors to the line driving
104
provides enough force to drive the wind.
The left panel of Fig. 6.2 shows Qwind for model series ’L5.5M40’ as a function of T eff .
As can be seen from the figure, the value of Qwind is decreasing when T eff is reduced between
30 000 and 25 000 K and also between 20 000 and 10 000 K. Between 22 500 and 20 000 K a
discontinuity in Qwind is produced. The reasons are twofold: (i) a change in Fe ionisation; and
(ii) the applied lower velocity ratio for the models at the cool side of the B-supergiant domain,
whilst the velocity ratio is increased at the hot edge.
6.2.1 CNO
The right-hand side of Fig. 6.2 displays the relative contribution of individual ions to Qwind .
For simplicity only the contribution of important ions is presented. It is evident that on the hot
side of bi-stability jump (T eff > 22 500 K) ions of C, N, and O contribute mostly to Qwind (and
thus to the total line acceleration). However, when T eff is reduced from 25 000 to 20 000 K, the
contribution of the of ions C, N, and O to Qwind decreases in a favour of iron (Fe iii).
6.2.2 Iron the wind driver
Figure 6.3 displays that when T eff is reduced from 25 000 to 22 500 K, Fe iv decreases in favour
of Fe iii. Even though Fe iv is still the dominant ionisation stage in the cooler model, most of the
radiative force of iron comes from Fe iii (cf. right panel of Fig. 6.2). When T eff is further reduced
to 20 000 K, Fe iii becomes the dominant ionisation stage and now provides about 40% of the
prescribed wind luminosity. The reader should be aware that we have prescribed 3∞ /3esc = 1.3
for models with T eff between 20 000 and 10 000 K (in line with observations). Thus, the models
on the cool edge of the bi-stability jump would achieve more “easily” the prescribed wind velocities than those models at the hot side, where the velocities are higher. Consequently, between
Wind properties of blue supergiants
6.2 Ion contributors to the line driving
105
L5.5M40; Ṁ =5e-07 [M⊙ yr−1 ]
1.2
IRON
FeII
FeIII
FeIV
CNO
HYD+ESEC
HYD
0.6
1
0.5
Q
QION /QTOT
0.8
0.6
0.4
0.3
0.2
0.4
0.1
0.2
30
25
20
Teff [kK]
15
10
30
25
20
Teff [kK]
15
10
Figure 6.2: Left: Qwind vs T eff for models with half-solar metal abundances. The observed
velocity ratios of 3∞ /3esc = 2.6 for T eff ≥ 22 500 K, 3∞ /3esc = 1.3 for T eff ∈ [10 000 K, 20 000 K],
and 3∞ /3esc = 0.7 for T eff < 10 000 K are applied. Right: relative contribution of individual ions
to the corresponding work ratio Qwind .
22 500 and 20 000 K, a jump in Qwind is produced. Note that this jump has to be accompanied
by a jump in Ṁ as well, because the model with T eff = 20 000 K would be able to drive stronger
wind. This is in agreement with previous studies (Vink et al. 1999, 2001), although cmfgen predicts a jump at T eff ≃ 20 000 K, whilst Monte-Carlo calculations predict the jump at somewhat
higher temperatures, at T eff ≃ 25 000 K.
As was already discussed in Chapter 3, Monte-Carlo calculations now have an improved
line driving treatment and therefore a discordance in temperature of 5 000 K between cmfgen
and Monte-Carlo calculations is particularly intriguing. Such large discordance may underline
fundamental differences between the assumptions regarding the treatment of the ionisation in
both codes, or the problem might be caused by differences in the atomic data which both codes
use. Currently, the origin of the underlying differences in Monte-Carlo simulations and cmfgen
in causing 5 000 K temperature discordance in the bi-stability jump remains unclear.
Wind properties of blue supergiants
6.3 Bi-stability jump on trial
106
−6
−7
−8
−9
−10
0
−2
log τROSS
−5
log Fen+ /N(total)
log Fen+ /N(total)
−5
Fe III
Fe IV
−4
Teff =20000 K;
−6
−7
−8
−9
−10
0
−2
log τROSS
Fe III
Fe IV
−4
log Fen+ /N(total)
Teff =22500 K;
Ṁ = 5e-07; L5.5M40T25; Z=0.5×Z⊙
−5
−6
−7
−8
−9
−10
0
−2
log τROSS
Fe III
Fe IV
−4
Figure 6.3: Change in ionisation balance between Fe iv and Fe iii.
Qwind =1.01
0.5
0.5
0
−0.5
0
Teff =20000 K; Ṁ =5e-07;
1
log (a/g)
log (a/g)
1
v∞
vesc =2.6
v∞
vesc =2.0
Qwind =0.80
0.5
0
−0.5
−2
log τROSS
β-law
gtot
−4
Teff =15000 K; Ṁ =2.5e-07;
1
log (a/g)
Teff =30000 K; Ṁ =2.5e-07;
1.5
−1
0
v∞
vesc =1.3
Qwind =0.96
0
−0.5
−2
log τROSS
β-law
gtot
−4
−1
0
−2
log τROSS
β-law
gtot
−4
Figure 6.4: Acceleration in units of gravity as a function of τROSS for models from series
’L5.5M40’ with half-solar metallicities. Wind acceleration (red dash-dotted) is compared to
the acceleration from the prescribed velocity law (black solid).
6.3 Bi-stability jump on trial
cmfgen does not currently calculate mass-loss rates. Instead, Ṁ is required as an input parameter.
Nevertheless, the Qwind ratio enables us in some way to estimate for which value of Ṁ a specific
model is able to drive the wind. We are aware that cmfgen does not solve the hydrodynamic
equations of the wind. Even though, the Qwind ratio is still a meaningful criterion whether or
not stellar winds could be driven, as the velocity structure is prescribed3 . A relevant question is
whether the prescribed and acquired wind accelerations are comparable for models with Qwind ≈
3
We assume that if Qwind << 1 then the wind can not be driven, whilst if Qwind ∼ 1 then the model could drive
such a wind.
Wind properties of blue supergiants
6.3 Bi-stability jump on trial
107
2
1.5
1.5
6
0.
1
1 .5
30
0 .6
0.
8
0.6
0.5
0 .8
1
1.5
0.
6
0.8
1 .5
25
0.8
1
20
15
Teff kK
0 .8 1
1
0.8
0.5
1
0 .6
Ṁ [10−6 M⊙ yr−1 ]
0.6
1
0 .6
Ṁ [10−6 M⊙ yr−1 ]
0.6
2
10
30
1 .5
25
1
20
15
Teff kK
6
0.
0 .8
10
Figure 6.5: Left: contour plot of Qwind as a function of T eff and Ṁ in model series ’L5.5M40’
with 3∞ /3esc = 2. If Qwind ∼ 1 then the radiative acceleration is able to drive the wind. Right:
contour plot of Qwind from the same model series but with observed ratio of 3∞ /3esc = 2.6 for
T eff >∼ 21 000 K and 3∞ /3esc = 1.3 for T eff <∼ 21 000 K. White squares mark the positions of
the of the calculated models.
1.
Figure 6.4 compares the wind acceleration according to the prescribed velocity law to the
radiative acceleration obtained for models with Qwind ≈ 1. The radiative force obtained is in
reasonable agreement with the acceleration given by the prescribed velocity law. An inspection
of the figure shows that, despite the deficit of radiative force in the inner and outermost part of
the wind, the value of Qwind is of order unity. It seems that the intermediate part of the wind, is
important for the global energy budget of the wind. Note that in this region the obtained radiative
acceleration is very similar to the prescribed acceleration.
To find out for which Ṁ our models acquire Qwind ≈ 1, we investigate the behaviour of Qwind
as a function of Ṁ and T eff . In the left panel of Fig. 6.5 we present a contour plot of Qwind
depending on T eff and Ṁ in model series ’L5.5M40’ with 3∞ /3esc = 2. The figure demonstrates
that for a constant velocity ratio, between 22 500 and 20 000 K, Ṁ at which the radiative force
Wind properties of blue supergiants
6.3 Bi-stability jump on trial
108
Ṁ [M⊙ yr−1 ]
0.3
0 .4
0 .3
0 .3
20
Teff kK
0 .2
0 .1
5
25
0 .0 5
0.5
0.2
0.3
30
0.4
0 .1
0.2
0.2
0.4
0.0
0.5
1
0 .4
0.3
1
0 .1
Ṁ [M⊙ yr−1 ]
1.5
0 .5
3
0.
1.5
0 .4
0.5
2
0 .1
0.2
2
15
10
30
25
20
Teff kK
15
10
Figure 6.6: Left: contour plot of the relative contribution of the ions of C,N, and O to Qwind ,
QCNO /Qwind , from model series ’L5.5M40’. Right: contour plot of the relative contribution
of iron to Qwind , QFe /Qwind , for same models. For T eff ≥ 22 500 K 3∞ /3esc = 2.6, whilst for
T eff ≤ 20 000 K 3∞ /3esc = 1.3.
is able to drive the wind is increased by factor of about two. Moreover, if the observed velocity
ratios are applied (i.e. 3∞ /3esc = 2.6 for T eff ≥ 22 500 K and 3∞ /3esc = 1.3 for T eff between
20 000 and 10 000 K), then Ṁ is increased by about a factor of four (cf. right-hand side of
Fig. 6.5). If the stellar luminosity is increased twice or the stellar mass is reduced to 30 M⊙ , then
Ṁ increases even more: by about of a factor of five (cf. Fig. 6.7). On the basis of Fig. 6.2 we
confirm that Fe iii is indeed responsible for this jump.
In Fig. 6.6, we show contour plots of the relative contribution of the ions of C, N, and O
(left), and iron (right) to the total Qwind ratio. It is interesting to note that with the increase of
Ṁ the contribution of C, N, and O to Qwind is decreased in favour of iron. When Ṁ is increased
about three times in the models with T eff ∼ 20 000 − 17 500 K, the contribution of iron (chiefly
Fe iii) to Qwind is increased by about 25% (QFe /Qwind increases from 0.4 to 0.5). This is partly
because in stronger winds the recombinations of Fe iv to Fe iii are favoured, partly because with
Ṁ increases also the absolute number of iron ions.
Wind properties of blue supergiants
109
2
1.5
1.5
0 .8
0.8
1
0.5
0.6
0 .8
0 .6
0 .8
1
0.6
0.6
0.6
1
1 .5
0.8
1
1 .5
30
1
0 .8
1
0.8
0.5
0.
6
1
1
0.8
1
Ṁ [10−6 M⊙ yr−1 ]
0.6
0.8
0.6
Ṁ [10−6 M⊙ yr−1 ]
0.8
2
0 .6
6.4 A second bi-stability jump?
25
0 .8
1
1 .5
20
15
Teff kK
1.5
10
30
25
20
15
Teff kK
10
Figure 6.7: Contour plot of Qwind in Ṁ − T eff plane for model series ’L5.5M30’ (left) and
’L5.75M71’ (right). Models have half-solar metal abundances and parameters as listed in Table 6.1.
−6
−7
−8
−9
−10
v∞ /vesc =1.3
0
−2
log τROSS
Fe II
Fe III
−4
log Fen+ /N(total)
log Fen+ /N(total)
−5
Teff =8800 K;
−5
−6
−7
−8
−9
−10
v∞ /vesc =0.7
0
−2
log τROSS
Fe II
Fe III
−4
log Fen+ /N(total)
Teff =9000 K;
L5.5M40T10; Ṁ = 2.5e-07; Z=0.5×Z⊙
−5
−6
−7
−8
−9
−10
v∞ /vesc =0.7
0
−2
log τROSS
Fe II
Fe III
−4
Figure 6.8: Change in ionisation balance between Fe iii and Fe ii.
6.4 A second bi-stability jump?
Lamers et al. (1995); Vink et al. (1999) suggested that there might be a second jump in Ṁ
near 10 000 K. This has not been studied in detail yet and itself provides new insights into the
evolutionary properties of B/A supergiants and LBVs. To investigate whether such a jump really
exists, we have calculated an additional set of models with T eff = 9 000 and 8 800 K4 . For star
4
Unfortunately, below 8 800 K, a self-consistent hydrostatic solution in the hydrostatic part of wind was not
Wind properties of blue supergiants
6.4 A second bi-stability jump?
Qwind =0.47
0.5
0
−0.5
−1
0
Teff =9000 K; Ṁ =2.5e-07;
1
log (a/g)
log (a/g)
0.5
v∞
vesc =1.3
v∞
vesc =0.7
Qwind =0.57
0.5
0
−0.5
−2
−4
log τROSS
β-law
gtot
−6
Teff =8800 K; Ṁ =2.5e-07;
1
log (a/g)
Teff =10000 K; Ṁ =2.5e-07;
1
110
−1
0
v∞
vesc =0.7
Qwind =0.65
0
−0.5
−2
−4
log τROSS
β-law
gtot
−6
−1
0
−2
−4
log τROSS
β-law
gtot
−6
Figure 6.9: Acceleration in units of gravity as a function of τROSS for models from series
’L5.5M40’ with half-solar metallicities. Wind acceleration (red dash-dotted) is compared to
the acceleration from the prescribed velocity law (black solid).
with T eff . 10 000 K Lamers et al. (1995) found that 3∞ /3esc = 0.7 and therefore, we used such
velocity ratio for those models. However, the terminal velocities of these objects were measured
with an accuracy between 10% and 20% and therefore the reader should be aware that adopted
value of 3∞ /3esc = 0.7 might be uncertain.
Nevertheless, we consider the adopted value as reasonable because: (i) in our coolest models
the ions of Fe ii provide most of the line acceleration, which is agreement with previous investigations (e.g. Vink et al. 1999; Vink & de Koter 2002), and therefore Fe ii could influence 3∞
(and Ṁ); and (ii) the temperature range where Fe ii becomes the main line-driver is between
∼10 000 and 9 000 K, which is the temperature range where Lamers et al. (1995) suspected that
there might be a second bi-stability jump.
Figure 6.8 shows that when T eff is reduced from 10 000 to 8 800 K, Fe iii recombines to Fe ii,
similarly to the recombination/ionisation of Fe iv/iii shown in Fig. 6.3. Note that at the coolest
model, Fe iii is not fully recombined to Fe ii. Whereas in the inner part of the wind Fe ii is
the dominant ionisation stage, in the outer wind Fe iii is still the dominant ion. Even so, Fe ii
contributes most to the total acceleration provided by iron as shown in right panel of Fig. 6.2. If
T eff is further reduced to ∼ 8 000 K we anticipate Fe ii to become the dominant ion throughout
obtained and therefore our grid stops at 8 800 K.
Wind properties of blue supergiants
6.4 A second bi-stability jump?
111
Teff =10000 K; Z=Z⊙ /2
1.3
1.05
1.2
1
1.1
0.95
1
Qwind
Qwind
Teff =8800 K; Z=Z⊙ /2
1.1
0.9
0.9
0.85
0.8
0.8
0.7
0.75
0.7
0
0.6
L5.5M40
L5.5M30
L5.5M20
1
2
Ṁ (10−6 M⊙ yr−1 )
L5.5M30
L5.5M20
3
0.5
0.05
0.1
0.15
0.2
Ṁ (10−6 M⊙ yr−1 )
0.25
Figure 6.10: Qwind vs Ṁ for models on both sides of the second bi-stability jump.
the wind and to provide an even larger fraction of the radiative acceleration.
Figure 6.9 compares the obtained and prescribed wind accelerations for models across the
second bi-stability jump. The prescribed velocity structure of the wind (with β = 1) is not
locally consistent with the acquired wind acceleration, but for different velocity laws one might
obtain better agreement. Therefore, it is still plausible to investigate the second bi-stability jump
in these models.
In Fig. 6.10, we show the Qwind ratio as a function of Ṁ for models with T eff = 8 800
and 10 000 K. The adopted velocity ratios are 0.7 and 1.3 respectively. Note that in the left
panel the models with M⋆ = 20 M⊙ and M⋆ = 30 M⊙ (high Γ) obtain Qwind = 1 at Ṁ ≈
2.5 × 10−6 M⊙ yr−1 and Ṁ ≈ 0.75 × 10−6 M⊙ yr−1 , whilst the models with M⋆ = 40 M⊙ (lower
Γ) does not acquire Qwind = 1 in the Ṁ range computed. The hotter models obtain Qwind = 1
at Ṁ ≈ 0.16 × 10−6 M⊙ yr−1 and Ṁ ≈ 0.07 × 10−6 M⊙ yr−1 for M⋆ = 20 M⊙ and M⋆ = 30 M⊙
respectively, i.e., between 10 000 and 8 800 K cmfgen predicts a jump in mass-loss rate ( ṀJ ).
According to Fig. 6.2 for the model with T eff = 8 800 K, Fe ii contributes most to the work
Wind properties of blue supergiants
6.5 The second bi-stability jump as a function of mass for solar metallicities
1
1
1
0.5
25
20
15
Teff kK
1.5
0.9
1
1.5
1.5
1.5
0
30
1.5
1
9
10 .
0.9
0.5
Ṁ [10−6 M⊙ yr−1 ]
0.9
1
1
1
1.5
1
2
0 .91
Ṁ [10−6 M⊙ yr−1 ]
2
112
1 .5
10
30
25
20
15
Teff kK
10
Figure 6.11: Contour plot of Qwind in Ṁ − T eff plane for model series ’L5.5M40’ (left) and
’L5.75M71’ (right). Models have solar metal composition and parameters as listed in Table 6.1.
ratio; it provides nearly 65% of Qwind . Thus, the predicted second jump in Ṁ should be caused
by the radiative acceleration provided by Fe ii. This implies that mass-loss rates of late B/A
supergiants and LBVs are sensitive to the ionisation equilibrium of iron. The reader should
keep in mind that Fe ii is not fully recombined at T eff = 8 800 K, and therefore we expect ṀJ to
increase even more at cooler temperatures.
6.5 The second bi-stability jump as a function of mass for solar
metallicities
The magnitude of ṀJ depends on the on the stellar mass and metal composition. To investigate
that, we have calculated a grid of models with solar metal abundances, but with different masses
(or Eddington factors).
Figure 6.11 shows the contour plot of Qwind in Ṁ − T eff plane for model series ’L5.5M40’
(left) and ’L5.75M71’ (right) with solar metal abundances. The observed velocity ratios are
Wind properties of blue supergiants
6.5 The second bi-stability jump as a function of mass for solar metallicities
113
applied. Note that in model series L5.5M40’ between 10 000 and 8 800 K, Ṁ at which Qwind = 1
increases from ∼ 0.07 to ∼ 1.12 × 10−6 M⊙ yr−1 (cf. Table 6.1), whilst for half-solar metallicities
the coolest models do not acquire Qwind = 1 at all (as shown in Fig. 6.10). This implies that
the second jump should be favoured in high metallicity environments, whilst for low metal
abundances, the second jump is relevant only for objects close to the Eddington limit (Γ ∼ 1).
To investigate in detail the origin of the second bi-stability jump, we show the total radiative
acceleration in Fig. 6.12 in units of local gravity (uppermost panels) as a function of λ and τROSS
for models with T eff = 8 800 K (left) and T eff = 20 000 K (right). For comparison the radiative
force due to the spectral lines of the ions of Fe ii, Fe iii, and CNO is displayed as well. In the
cooler model most of the radiative acceleration is provided by Fe ii, whilst in the hotter model
Fe iii is the dominant wind driver.
In order to understand which frequencies are important, in Fig. 6.12 (fifth panel from the
top) we also display the contribution of Fe ii (left) and Fe iii (right) to the work ratio of the
respective ions (blue lines). These contributions are normalised in such a way that the sum over
all frequencies would give unity. The red line (with ordinate in red colour placed on the righthand side) shows the sum of the contributions of Fe ii (or Fe iii in the right panel) located in
different frequency bins. It is evident that about 50% of the acceleration of Fe ii comes from
lines with λ between 2 300 and 2 800 Å, and the lines in the Balmer continuum provide more
than 95% of total acceleration of Fe ii.
To understand the significance of these numbers, in the lowermost panels of the figure, we
present the contribution of Fe ii (left) and Fe iii (right) to the total work ratio, provided by all
ions. In the cooler model 40% of the total acceleration comes from lines with λ between 2 300
and 2 800 Å and the Fe ii lines in Balmer continuum provide about 70% of the total acceleration.
In the hotter model, the Balmer continuum also provides significant fraction of the total radiative
force (about 50%).
Wind properties of blue supergiants
6.5 The second bi-stability jump as a function of mass for solar metallicities
10.0%
−15
3
3.5
log λ [Å]
40.0%
−5
25.0%
−10
10.0%
−15
4
3
3.5
log λ [Å]
4
30.0%
20.0%
−10
10.0%
−15
3
3.5
log λ [Å]
4
log (QλFeIII/Qtot
wind)
30.0%
Bin Contribution
log (QλFe2/Qtot
wind)
40.0%
−5
−5
20.0%
−10
10.0%
−15
3
3.5
log λ [Å]
Bin Contribution
25.0%
−10
55.0%
Bin Contribution
40.0%
−5
0
log (QλFeIII/Qtot
FeIII)
55.0%
Bin Contribution
log (QλFe2/Qtot
Fe2)
0
114
4
Figure 6.12: Radiative force provided by Fe ii, Fe iii, CNO, and all ions as a function of λ and
τROSS for models with T eff = 8 800 K (left) and T eff = 20 000 K (right). Models have solar metal
composition. The lowermost panels illustrate the contributions of the spectral lines to the work
ratio obtained by the acceleration from Fe ii (left) or Fe iii (right). The red line (with ordinate on
the right-hand side) presents the total contribution of spectral lines located in various frequency
bins to the work ratio of Fe ii (left) or Fe iii (right).
Wind properties of blue supergiants
6.5 The second bi-stability jump as a function of mass for solar metallicities
115
Table 6.1: Mass-loss rates at which Qwind = 1 for different model series.
L
L⊙
M⋆
log
series name M⊙
5.75
L5.5M71
5.50
L5.5M40
5.50
L5.5M30
fcl = 0.1; 3cl = 30 km s−1
Γe Ṁ range
T eff
10−6 M⊙ /yr
K
R⋆
R⊙
3∞
3esc
log g
Z = Z⊙
Z = Z⊙ /2
∗
ṀQwind =1 Γ(τ=2/3) ṀQwind =1
10−6 M⊙ /yr
10−6 M⊙ /yr
Γ∗(τ=2/3)
71 0.20 0.25 − 2.00
0.25 − 2.00
0.10 − 2.00
0.10 − 2.00
0.25 − 2.00
0.25 − 2.00
0.25 − 2.00
0.10 − 2.00
0.40 − 2.00
0.20 − 2.00
30 000 28 2.6 3.40
27 500 33 2.6 3.25
25 000 40 2.6 3.09
22 500 49 2.6 2.90
20 000 62 1.3 2.70
17 500 81 1.3 2.47
15 000 111 1.3 2.20
12 500 159 1.3 1.88
10 000 249 1.3 1.50
8 800 – 0.7 1.27
0.70
0.67
0.62
0.58
0.60
0.61
0.58
0.57
0.62
0.74
0.70
0.65
0.45
0.40
2.09
1.79
0.79
0.33
–
1.72
0.67
0.63
0.58
0.55
0.57
0.57
0.55
0.55
0.61
0.73
0.50
0.44
0.40
0.25
1.19
0.84
0.45
0.23
–
–
30 000 21 2.6 3.40
27 500 25 2.6 3.25
25 000 30 2.6 3.09
22 500 37 2.6 2.90
20 000 46 1.3 2.70
17 500 61 1.3 2.47
15 000 83 1.3 2.20
12 500 120 1.3 1.88
10 000 187 1.3 1.50
8 800 242 0.7 1.27
0.71
0.68
0.62
0.59
0.61
0.61
0.58
0.57
0.63
0.75
0.43
0.33
0.24
0.22
1.34
1.07
0.46
0.21
0.07
1.12
0.67
0.63
0.58
0.56
0.57
0.57
0.55
0.55
0.62
0.73
0.31
0.24
0.21
0.17
0.68
0.49
0.25
0.15
–
–
30 0.26 0.10 − 2.00
0.10 − 2.00
0.10 − 2.00
0.10 − 2.00
0.10 − 2.00
0.10 − 2.00
0.10 − 2.00
0.10 − 2.00
0.07 − 2.00
0.20 − 2.00
30 000 21 2.6 3.28
27 500 25 2.6 3.13
25 000 30 2.6 2.96
22 500 37 2.6 2.78
20 000 46 1.3 2.58
17 500 61 1.3 2.34
15 000 83 1.3 2.08
12 500 120 1.3 1.76
10 000 187 1.3 1.37
8 800 242 0.7 1.27
–
–
–
–
–
–
–
–
0.69
0.80
–
–
–
–
–
–
–
–
0.10
2.00
0.75
0.73
0.68
0.64
0.65
0.67
0.63
0.63
0.68
0.79
0.41
0.32
0.23
0.19
1.06
0.71
0.36
0.19
0.07
0.75
40 0.20 0.10 − 1.75
0.10 − 1.75
0.10 − 1.75
0.10 − 1.75
0.10 − 1.75
0.10 − 1.75
0.10 − 1.75
0.10 − 1.75
0.07 − 0.50
0.25 − 1.50
Notes. The physical Eddington factor, Γ, is distance dependent and therefore we show the value of Γ at
the reference radius, where τROSS = 2/3.
Wind properties of blue supergiants
6.6 Consequences for LBVs
116
log L/L⊙ = 5.50; Z=Z⊙
5
4.5
˙
MJUMP
(10−6 M⊙ yr−1 )
(22)
4
3.5
3
2.5
2
(20)
1.5
1
(16)
0.2
0.25
Γe
0.3
0.35
0.4
Figure 6.13: Dependence of the jump in Ṁ between 10 000 and 8 800 K on Eddington factor.
The numbers in parentheses show the increase of mass loss rate across the second bi-stability
8 800 / Ṁ 10 000 .
jump in relative sense, i.e. ṀQ
Qwind =1
wind =1
The magnitude of ṀJ also depends on the Eddington factor. This is illustrated in Fig. 6.13,
where ṀJ as a function of classical Eddington factor is shown. The increasing Γe causes a strong
increase of ṀJ . An increase of Ṁ with ∼ 1 − 4 × 10−6 M⊙ yr−1 across the second bi-stability jump
is significant and, if it is real, we may expect to find different wind properties of objects located
on both sides of this second jump. As will be discussed in the next section, this is especially
relevant for LBVs as they are near the Eddington limit.
6.6 Consequences for LBVs
LBVs are unstable massive stars, located close to the empirical upper-luminosity limit on the
H-R diagram, known as the Humphreys-Davidson limit (Humphreys & Davidson 1994). The
theory suggests that LBVs are in a transitory phase between H-burning O-type stars and Heburning Wolf-Rayet stars (Langer et al. 1994; Maeder & Meynet 2000; Ekström et al. 2012).
However, this has not been supported by recent observations which indicate that LBVs might be
Wind properties of blue supergiants
6.7 Conclusions
117
direct progenitors of SNe (Kotak & Vink 2006; Smith et al. 2007; Gal-Yam & Leonard 2009).
Currently, their nature is still under debate.
In the preceding sections, we were able to understand the dependence of the mass-loss rate
for normal BA supergiants on T eff . Although there are differences between BA and LBV winds,
our calculations may provide important information about the behaviour of Ṁ during a typical
LBV phase. It is well known from observations that LBVs substantially change their T eff in the
range ∼ 8 000 − 30 000 K (van Genderen 2001; Vink 2012) and therefore they can be employed
as unique laboratories to study how the degree of mass loss changes as a function of T eff (Stahl
et al. 2001; Vink & de Koter 2002).
Stahl et al. (2001) was able to derive the mass-loss rate of AG Car (HD 94 910) during different phases over the period 1990−1999. The behaviour of the mass-loss rate for the phase from visual minimum (high T eff ) towards maximum (cooler T eff ) is shown with a solid line in Fig. 6.14.
During this transition, Ṁ increases, drops, and increases again at T eff ≃ 9 500 − 10 000 K. The
changes in Ṁ during an LBV variation can be understood if the behaviour of Ṁ as a function
of T eff predicted by cmfgen is considered. This behaviour was partly explained by Vink & de
Koter (2002), although in their calculations for LBV winds, the second jump occurred at a much
higher temperature T eff ≃ 20 000 K.
6.7 Conclusions
We have investigated the wind properties of BA supergiants. Our calculations confirm the bistability jump in Ṁ around T eff ≃ 25 000 K predicted by Vink et al. (1999) . However, cmfgen
predicts that this jump will occur at a lower temperature T eff ≃ 20 000 K, which is consistent with
observations. So far, the reasons for the different temperature locations of the jumps predicted
by cmfgen and Monte-Carlo calculations remains obscure.
Wind properties of blue supergiants
6.7 Conclusions
118
Figure 6.14: Time-dependent Ṁ of AG Car against T eff as derived from Hα analysis by Stahl
et al. (2001).The solid indicate the changes in Ṁ over the period Dec. 1990− Feb. 1995, when
the star increases its visual brightness. The dotted line connects points when visual brightness is
decreasing. Figure from Vink & de Koter (2002).
We also showed that near T eff ≃ 10 000 K a second jump in Ṁ is produced if the observed
velocity ratio is applied. This jump is caused by Fe iii/Fe ii recombination/ionisation as already
suggested by Vink et al. (1999) and itself represents valuable science prospects for late B/A supergiants and LBVs. For solar metallicities, the second bi-stability jump occurs in all calculated
models series, whilst for half-solar metal abundances the second jump is re-produced only for
models close to the Eddington limit (with Γe > 0.26). This implies that the second bi-stability
jump is relevant for LBVs even in low metallicity environments and can be used as a tool to
better understand the observed variations in Ṁ from LBVs.
Wind properties of blue supergiants
Chapter 7
Conclusions & Future work
7.1 Summary
Of all objects, the planets are those which appear to us under the least varied
aspect. We see how we may determine their forms, their distances, their bulk, and
their motions, but we can never know anything of their chemical or mineralogical
structure; and, much less, that of organised beings living on their surface.
Auguste Comte, The Positive Philosophy, Book II, Chapter 1 (1842)
In 1842, Auguste Comte, a distinguished philosopher, said that humans will never know what
stars are made of. And then, just a few years later spectroscopy was born (Kirchhoff & Bunsen
1860). These passages may be amusing in the light of present knowledge, however we may end
up with similar conclusions if we want to know the real mass-loss rates of late Bsgs.
In Chapter 4, we found that for late Bsgs the Hα line becomes optically thick and therefore
has to be influenced by macro-clumping effects. Thus, to derive more accurate Ṁ from Hα
emission of late B/A supergiants, macro-clumping needs to be taken into account. Unfortunately,
the modelling of macro-clumping is a very difficult task, as in that case knowledge about the
119
7.1 Summary
120
distribution, shape and size of the clumps is required. In that case the parameter space to explore
is enormously increased. Moreover, with such a vast parameter range, the derived Ṁ, even if the
line perfectly fits, could be completely different from the real Ṁ, as there is no guarantee that
the same line profile cannot be obtained with different Ṁ, size, and distribution of the clumps.
An optically thick Hα might imply that previously-derived Hα mass-loss rates of late Bsgs
are underestimated due to an inadequate treatment of clumping. Consequently, the “derived” β
values for late B/A supergiants might be over estimated because of a systematic neglecting of
macro-clumping. Thus, it might be worth to re-analyse the previously investigated Ṁ from Hα
emission in the light of our findings.
We also found that the Hα line behaves similarly in models with different stellar and wind
parameters, always displaying a peak in the line EW at temperature around 20 000 K. As was
explained in Chapter 3, this behaviour is determined by the ratio between the recombination of
H atoms in 3rd level and the efficiency of the “Lyα drain”. Applying the observed velocity ratios
in the models led to a sharper peak, which is expected to facilitate its observational confirmation
or refutation.
In the third part of the thesis, we investigated the wind properties of BSGs. On the basis of
contour plots of the work ratio Qwind , we were able to confirm independently the predicted bistability jump in Ṁ by Vink et al. (1999), but at a somewhat lower temperature T eff ≃ 21 000 K.
In our models, the ions of C, N, and O are the most important line “drivers“ for T eff > 22 500 K,
whilst for temperatures between 20 000 and 12 500 K Fe iii provides most of the driving force.
For temperatures below 10 000 K Fe iii starts to recombine to Fe ii and at T eff = 8 800 K we find
that Fe ii provides nearly 65% of the total line acceleration. This causes a second bi-stability
jump in Ṁ around 10 000 K which is dependent on the Eddington parameter and metallicity.
We found that at half-solar metal abundances a second jump is produced only in models close
to the Eddington limit. Thus, the second jump is relevant for LBVs even in low-metallicity
environments. For late B/A supergiants it only becomes important for solar metal abundances.
Conclusions & Future work
7.2 Future work
121
As the nature of LBVs is still not well understood, a detailed investigation of the second jump
might be valuable.
7.2 Future work
The analysis in this thesis heavily relies on the modelling of BSG winds by means of the cmfgen
code and therefore the results are mostly theoretical. What is now needed is to compare our
findings with the observations, e.g. with the vlt-flames and vlt-flames tarantula surveys.
In the calculated models we were able to account only for micro-clumping. In reality however, macro-clumping plays an important role in Hα line formation, especially for late B supergiants. Therefore, an empirical study including macro-clumping effects is required if we want
to know the real mass-loss rates. Such an investigation might lead also to valuable conclusions
concerning the origin of the clumps. Another open question is how the degree of clumping
changes throughout the wind. Multiwavelength analysis from the UV to the radio is required to
answer that question.
Another aspect of the thesis which requires further consideration concerns the second bistability jump: its origin and its dependence on the Eddington factor, clumping, and chemical
composition. In this thesis, we were not able to investigate the importance of clumping for the
second bi-stability jump. However, such knowledge might be valuable, especially for LBVs,
as they experience outbursts and episodes of enhanced mass loss during which the degree of
clumping might change. Thus, the driving efficiency of iron might be different for specific
temperature at various epochs. Therefore, it is important to quantify the effects of clumping on
both bi-stability jumps. Understanding all that, we might be able to explain some of the observed
variations in Ṁ during the different phases of LBVs.
Conclusions & Future work
Appendices
122
Appendix A
Where in the wind do Hα photons
originate from?
In order to understand the Hα line-formation we show in Fig. A.1 a typical distribution of the
emergent intensity I(p), which is scaled by the impact parameter p (see Dessart & Hillier 2005,
for details). The top panel of the figure represents the line profile, which, for each wavelength
corresponds to the integral over all p of the scaled intensity I(p) presented in the lower panel.
Hence, we are able to identify the contribution at each p to the total line flux.
In other words, the figure provides information about where in the wind most of the emergent
Hα photons originate from. This knowledge enables us to display the ”evolution“ of the Hα line
(EW) if larger p values are added (or removed) in Fig. A.2. Note that in Fig. A.1 the absorption
component emanates for p/R⋆ <=∼ 1 (front of the stellar disc), which is in agreement with the
conventional mechanism for P Cygni line formation.
In Fig. A.2, we present how the Hα line EW changes when the emergent flux at larger p/R⋆
(corresponding to x = r/R⋆ or τross ) is added. Note that the Hα EW is nearly constant for
log(τross ) >∼ −1.5: in that range the line is in absorption, mainly produced by the wind in front
123
124
Figure A.1: Model C (T eff =12 500 K). Bottom: grey scale plot of the flux like quantity p×I(p) as
a function of impact parameterp/R⋆ , where R⋆ is hydrostatic radius. The figure provides the distribution of the emergent intensity around Hα from different p. Top: corresponding normalised
flux in Hα, directly obtained by integrating p × I(p) over the range of p.
of the stellar disc (p/R⋆ <∼ 1), i.e., the Hα photons originate at larger distances. From this
figure, we define the Hα line-formation region as the region in which Hα changes its EW from
10% to 90%. Although it is by no means conclusive that the line forms in this region, most of the
Hα photons (in the observer’s frame) are emitted from this part of the wind, and the behaviour
of the line should depend on the local conditions in that region. Therefore, we investigated the
Hα related quantities ((n3 /n2 ) ratio, τLyα , τHα ) at this side of the wind in different models.
Where in the wind do Hα photons originate from?
125
Hα EW [ Å]
5
0
−5
1
0
−1
−2
log (τ ROSS)
−3
−4
Figure A.2: Hα line EW as a function of τross . The figure illustrates how the EW changes when
outer layers of the star (p/R⋆ < 1) are added.
Where in the wind do Hα photons originate from?
Appendix B
Sobolev approximation
In Chapters 3 and 4 we investigated the Sobolev optical depth of lines in line formation region,
however, we do no comment on how appropriate the application of the Sobolev approximation
is with respect to co-moving frame calculations. Therefore, we now compare the Hα Sobolev
optical depth to relevant quantities calculated in the co-moving frame.
Sobolev approximation - some basic equations
In the Sobolev approximation (Sobolev 1960) the region, where photons can interact with ions
in the wind is restricted to a point, called the Sobolev point. In that case, the line optical depth
depends only on the local conditions and does not require knowledge about optical depth of the
wind below or above the Sobolev point.
In such an approximation the line optical depth in terms of the rest wavelength λ0 is:
τSλ0 =
(χ)sp λ0 /(3/r)sp
,
1 + σ cos2 θ
126
(B.1)
127
where θ is the angle between the path of the photon emitted by the photosphere and the radial
direction, χ the line opacity, 3 and r are respectively the local velocity and the distance of the
Sobolev point to the center of the star. σ is defined by:
σ=
r d3
− 1.
3 dr
(B.2)
The Sobolev optical depth in the tangential direction (cos θ = 0) is:
τSλ0 = (χ)λ0 /(3/r),
(B.3)
whilst in the radial direction (cos θ = 1) it is given by:
τSλ0 = (χ)λ0 /(d3/dr).
(B.4)
If a specific emission line is optically thick in the radial direction, the photons still might escape
in the tangential direction. Thus, whether the line is optically thick or thin would depend on its
optical depth in both directions. Therefore, in Figs. 3.13 and 4.2, we have plotted the minimum
between Sobolev optical depth in radial and tangential directions.
Is the Sobolev approximation valid?
In Sobolev approximation photons emitted by the photosphere can interact with an absorbing ion
only at one point. In reality, the lines may overlap in frequency space. Thus, the radiation from
the photosphere which reaches the Sobolev point of a particular line at a particular distance
might be affected by the presence of other lines. Thereat, the Sobolev approximation is not
strictly valid and must be checked.
In order to check the validity of Sobolev approximation in Fig. B.1 we show the source
Sobolev approximation
128
SOPHCL0.1V20kms
1
0.9
0.9
0.8
0.8
SHα
λ /J
SHα
λ /J
SOPHnoCL
1
0.7
0.6
0.5
−3
0.7
0.6
−2
−1
0
1
0.5
−3
2
−2
−1
Hα
τ Sob
0.9
0.9
0.8
0.8
0.7
0.6
−1
2
0.7
12 500 K
30 000 K
22 500 K
0.6
−2
1
HHECL0.1V20kms
1
SHα
λ /J
SHα
λ /J
HHEnoCL
1
0.5
−3
0
Hα
τ Sob
0
1
Hα
τ Sob
2
0.5
−3
−2
−1
0
1
2
Hα
τ Sob
Figure B.1: The source function of Hα line SHα
λ over the mean integrated intensity (J) as function of Hα Sobolev optical depth for simplified H+He (lower panels), sophisticated (upper panels; with atomic data as listed in Table 3.3), homogeneous (left panels), and clumped (right
panels) models.
function of Hα line SHα
λ over the mean integrated intensity (J) as a function of Hα Sobolev
optical depth. These quantities are calculated in the co-moving frame at wavelengths around
λHα ∓ ∆λ and they provide a measure of whether the Sobolev optical depth is similar to the line
optical depth in co-moving frame. If the line becomes optically thick it is expected that SHα
λ ≈ J.
The Doppler width ∆λ is defined as:
Sobolev approximation
129
1
∆λ
=
λLyα c
r
2kT eff
+ 32turb ,
mh
(B.5)
where mh is the mass of the hydrogen atom, k is the Boltzmann constant, T eff is the effective
temperature of the model and 3turb is the turbulent velocity in the models (20 km/s).
Figure B.1 shows SHα
λ /J for models with different complexity (upper-lower panels) and tembecomes close to unity SHα
peratures. It is evident in all panels, that as soon as τHα
λ ≈ J. This
Sob
gives further credence that the Sobolev optical depth presented in Figs. 3.13 and 4.2 should be
useful.
Sobolev approximation
Appendix C
Atomic data and model atoms in
(sophisticated models)
C.1
Atomic data
The atomic data in cmfgen are stored in formatted ASCII data files, which can be updated when
new data become aviable. The main source of atomic data comes from the Opacity project
(Seaton 1987) and the Iron project Hummer et al. (1993). However, for some CNO elements,
atomic data were used also from Nussbaumer & Storey (1983, 1984), whilst for Fe ii, Fe iii, Fe iv,
and Fe vii data were used also from Nahar (1995); Zhang (1996); Becker & Butler (1995b,a)
respectively.
C.2
Model atoms
The adopted atomic data of all elements included in our model atmosphere calculations in Chapters 5 and 6 are summarized in Table C.1. In order to save computational time, at the different
130
C.2 Model atoms
131
temperature regimes we choose, different (but relevant) level assignments for the ions. For reference are given the adopted level assignments in the initial sophisticated models discussed in
Chapter 3.
Atomic data and model atoms in (sophisticated models)
C.2 Model atoms
132
Table C.1: Atomic data included in our realistic models
Ion
Hi
He i
He ii
CI
C II
C III
C IV
CV
NI
N II
N III
N IV
NV
OI
O II
O III
O IV
OV
O VI
Ne II
Ne III
Ne IV
Mg II
Mg III
AL I
AL II
AL III
AL IV
Si II
Si III
Si IV
Si V
P IV
PV
S II
S III
S IV
SV
SOPH
20/ 30
45/ 69
22/ 30
81/142
40/92
51/84
59/64
–
52/104
45/85
41/82
44/76
41/49
32/161
54/123
88/170
38/78
32/56
25/31
–
–
–
–
–
–
–
–
–
9/16
33/33
22/33
–
30/90
16/62
–
24/44
51/142
31/98
9-10
20/30
45/69
22/30
22/42
104/338
91/209
–
–
22/35
100/267
–
–
–
32/161
137/340
–
–
–
–
42/242
–
–
22/65
41/201
–
37/56
18/50
–
27/53
81/147
–
–
30/90
–
41/171
80/257
–
12.5-20
20/30
45/69
22/30
–
104/338
91/209
19/24
–
–
100/267
41/82
13/23
–
13/29
137/340
165/343
9/16
–
–
42/242
20/51
–
18/36
41/201
—
37/56
18/50
46/107
27/53
81/147
39/50
12/22
30/90
9/15
41/171
80/257
49/138
9/15
22.5-27.5
20/ 30
45/ 69
22/ 30
–
31/68
99/243
59/64
–
–
80/192
41/82
78/124
–
–
106/251
165/343
99/202
–
–
42/242
57/188
–
–
41/201
–
–
18/50
46/107
—–
81/147
55/66
52/203
30/90
9/15
12/33
80/257
69/194
17/36
30-35
20/ 30
45/ 69
22/ 30
–
10,10,18
99/243
59/64
46/73
–
9/17
100/191
200/278
–
–
81/182
165/343
71/138
11/19
–
25/116
57/188
–
–
41/201
–
–
7/12
62/199
–
26/51
55/66
52/203
30/90
9/15
–
41/83
69/194
41/167
Atomic data and model atoms in (sophisticated models)
C.2 Model atoms
133
Table C.1. Continued.
Ion
AR III
AR IV
AR V
Ca II
Ca III
Ca IV
Ca V
Fe I
Fe II
Fe III
Fe IV
Fe V
Fe VI
Fe VII
SOPH
–
–
–
–
–
–
275/827
104/1433
74/540
50/220
44/433
29/153
9-10
29/249
–
–
21/70
41/208
–
–
–
275/827
136/1500
–
–
–
–
12.5-20
29/249
8/22
–
–
41/208
2/3
–
–
17/218
136/1500
74/540
17/67
–
–
22.5-27.5
29/249
29/97
–
–
41/208
33/171
–
–
–
136/1500
100/1000
34/352
–
–
30-35
18/82
41/204
–
–
41/208
39/341
–
–
–
–
100/1000
45/869
55/674
13/50
Notes. For each ion, the number of super levels and full levels are provided.
Atomic data and model atoms in (sophisticated models)
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