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Transcript
Katholieke Universiteit Leuven
Faculteit Wetenschappen
Fundamental parameters of B-type stars
Application to a HIPPARCOS sample of B supergiants
and a CoRoT sample of B dwarfs
Karolien Lefever
PROMOTORES:
Prof. Dr. Conny Aerts, K.U.Leuven
Priv. Doz. Dr. Joachim Puls, U.S.M. München
Leuven 2007
Proefschrift voorgelegd tot
het behalen van de graad van
Doctor in de Wetenschappen
MEMBERS OF THE SCIENTIFIC COMMITTEE :
PROMOTORES:
JURY:
Prof. Dr. C. Aerts
Priv. Doz. Dr. J. Puls
Dr. L. Decin
Prof. Dr. A. Herrero
Dr. A. de Koter
Dr. T. Morel
Prof. Dr. G. Neyens
Prof. Dr. C. Waelkens
Katholieke Universiteit Leuven, Belgium
Universitätssternwarte München, Germany
Katholieke Universiteit Leuven, Belgium
Universidad de La Laguna, Tenerife
Universiteit van Amsterdam, The Netherlands
Katholieke Universiteit Leuven, Belgium
Katholieke Universiteit Leuven, Belgium
Katholieke Universiteit Leuven, Belgium
Cover illustration:
Science and Art with GIMP: impression of the stellar winds of young massive stars by Dejan
Vinković.
Acknowledgement:
This thesis was made possible thanks to the financial support of the Department of Physics
and Astronomy and of the Research Council of K.U.Leuven under grants FLOF 10377 and
GOA/2003/04, and was carried out within the Belgian Asteroseismology Group.
Copyright:
c
2007
Faculteit Wetenschappen, Geel Huis, Kasteelpark Arenberg 11, 3001 Leuven
Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt
worden door middel van druk, fotokopie, microfilm, elektronisch of op welke andere wijze ook zonder
voorafgaandelijke schriftelijke toestemming van de uitgever.
All rights reserved. No part of the publication may be reproduced in any form by print, photoprint,
microfilm, electronic or any other means without written permission from the publisher.
ISBN 978-90-8649-130-8
D/2007/10.705/52
To Jan,
for the smile on my face
Dank u wel!
Het einde is in zicht... tijd voor het dankwoord. Het begin van dit boekje, maar het einde van
een vier jaar lange reis. Gezien het voor velen bij het lezen van dit dankwoord zal blijven,
wil ik van de komende bladzijden optimaal gebruik maken om tot iedereen die de voorbije
vier jaar mijn pad gekruist en/of gedeeld heeft een oprecht woordje van dank te richten.
Christoffel, van harte bedankt voor de mooie kansen die ik van u gekregen heb door me als
doctoraatsstudente op het Instituut voor Sterrenkunde te aanvaarden: ik heb genoten van de
vele waarneemruns op La Palma en in Chili, evenals van de verschillende conferenties en
zomerscholen, waar ik heel wat interessante mensen mocht leren kennen en waar ik vaak
geestelijk verrijkt van terugkwam. Bedankt voor uw stille aanwezigheid aan de overkant van
de gang op uren waarop niemand anders zich op het instituut bevond, en voor de vrolijke
speeches op tal van bijzondere aangelegenheden.
Conny, ere wie ere toekomt, ook al ben je nog zo druk bezet, op de momenten dat het echt
nodig was, stond je als promotor altijd klaar om mij van nieuwe wetenschappelijke inspiratie te voorzien1. Dankzij jou heb ik vooral mijn ‘self management competenties’ weten aan
te scherpen: zelfstandigheid, doelgerichtheid, efficiëntie, doorzettingsvermogen en stressbestendigheid. Je ongelooflijk optimistische deadlines hebben me menigmaal aangezet tot
lange avonden en nachten op het instituut. Je op en top gedreven persoonlijkheid, je feminisme2 , en vooral je gezonde portie durf zullen me altijd bijblijven. Bij dat laatste denk
ik dan vooral terug aan onze eerste gezamenlijke trip naar München, wanneer je gezwind
op ijsgladde perrons de trein uitsprong om onze ticketjes voor de trein te valideren3, en de
deuren terug dicht vielen... Opluchting alom toen bleek dat je op het nippertje een andere
deur was kunnen binnenspringen, en je even vrolijk als altijd, zwaaiend met de gevalideerde
ticketjes, terug de coupé binnen stapte. Bedankt, Conny, voor de bemoedigende woorden als
ik het weer eens niet meer zag zitten. Op een of andere manier wist je die momenten altijd
(vrouwelijk) intuı̈tief aan te voelen. Bovenal een dikke merci voor alle tijd en enthousiasme
die je in dit werk gestopt hebt en voor de 101 keren dat je teksten van me nagelezen hebt.
1 De
bewijzen van wijsheid die ik op die manier vergaard heb, zijn geleverd: de eerste grijze haren zijn geteld :-)
denk dat ik op een of andere manier altijd zal blijven percentages vrouwen tellen.
3 Ja, want Geert had eens moeten weten dat we (notabene ongewild) aan het zwartrijden waren...
2 Ik
10
Joachim, as my German supervisor you have spent an enormous amount of time and effort
in this thesis. I must admit that it hasn’t always been easy for me to cope with your typical
German straightforwardness, but somehow, through the years, I have learned to appreciate it
and I really enjoyed working with you. You were always honest and realistic: when things
were bad, I immediately knew, but you were never averse to give me a pat on the back when I
did a good job. I really enjoyed our trip to Neuschwanstein on one of the few non-snowy days
during my stays in Munich. The hour and a half of waiting before our traditional Bavarian
lunch was served4 , was compensated by the delicious fresh salmon we made together for
dinner. Thanks again to you and your family for your hospitality! You were always in for
a good beer or for an enchanting cocktail bar: a nice entertainment after an intensive day of
work. Arno, Philip and Tadziu, thanks for the company during the ‘all except German food’
take-away lunches.
Met veel plezier dank ik ook alle (ex-) collega’s op het instituut, zij zorgden voor de broodnodige (koffie)pauzes en tal van ‘social events’, ook al was mijn deelname het laatste jaar
noemenswaardig geminderd. Een aantal onder jullie wil ik toch nog even speciaal vernoemen, omdat jullie op een meer rechtstreekse manier bijgedragen hebben tot dit werk. Thierry,
merci un millier de fois pour tout l’effort que tu as mis dans la préparation des spectres de
GAUDI. Ta contribution dans le dernier chapitre de cette thèse fut importante. Merci pour
toutes les réponses à mes questions faciles et moins faciles. Leen, bedankt om zo vaak een
gaatje vrij te maken om mee te denken over de beste aanpak, en voor de bemoedigende
woorden af en toe. Bedankt aan ‘de thesisstudenten’ (sinds enkele maanden ook volwaardige
collega’s geworden), en in het bijzonder aan Els en Kristof, die mij gezelschap hielden tijdens
de eindspurt naar de eerste versie van dit boek. Zij gaven me een reden om de chocoladefondue boven te halen of een taartje van de bakker mee te delen. De systeembeheerploeg:
Erik, Jan, Wim, Rik, Bart, bedankt om mij met raad en daad bij te staan bij grote en kleine
computerprobleempjes, al dan niet gerelateerd aan het grote grid uit hoofdstuk 3.
Een speciaal dankjewel voor de mensen die met mij de bureau gedeeld hebben: Bram,
Maarten en Evelien. Bram, op de een of andere manier was je een bron van rust en wijsheid. Bedankt om die af en toe eens (al dan niet bewust) naar uw bureaugenootje door te
stralen. Maarten, bedankt om mij op tijd en stond even af te leiden met verhaaltjes over het
geocachen, en voor het delen van geheimpjes ;-) Ik hoop dat je onze stress gezond en wel zult
doorkomen. Evelien, jij hebt met mij niet alleen de bureau in d’n B en d’n D gedeeld, maar
eveneens de grote eindspurt. Samen hebben we ’ups’ an ’downs’ getrotseerd. Je was naast
een goeie collega ook een heel goede vriendin: bedankt dat je door dik en dun voor me klaar
stond, en voor alle mooie herinneringen.
Wat ik tijdens de laatste jaren bovenal geleerd heb, is dat onderzoek een proces is van vallen
en opstaan, telkens weer met nieuwe moed van nul beginnen. Zwoegen, maar doen zoals
de boer... en hij ploegde voort... Gelukkig waren er altijd vrienden om me heen die klaar
stonden om de ploeg opnieuw in gang te trekken. Annelies en Joris, Evelien, Mario, Katrienu,
Maarten, Wim, Sara, Rachel en aan alle andere mensen die op de een of andere manier mijn
dagen opgevrolijkt hebben: bedankt! Jullie verdienen een standbeeld!
4 Noteworthy
because a group of some 50 musicians, dressed in Lederhosen, had arrived just before us.
11
Bedankt aan alle kotgenootjes door de jaren heen... Brenden, Leen, Tine, Emmanuel, Tim,
Hadeli, Veerle, Nathalie, en alle anderen: ’t was plezant om samen met jullie te ontspannen.
Bedankt dat ik af en toe eens bij jullie mocht binnenspringen of jullie mocht bemoederen met
goede raad bij het koken, en voor de leuke pudding-avondjes!
Mama & papa, Tom & Annemieke, Nele & Johan, oma & opa, ik geef het toe: ik heb jullie
verwaarloosd. Sorry voor al die keren dat ik er niet was ‘omdat ik het te druk had’. Het was
altijd plezant om weer thuis (‘in die andere wereld’) te komen, al was het soms maar voor een
blitsbezoek. Bedankt voor de bemoedigende woorden en jullie geduld. Ik hoop de verloren
tijd binnenkort weer allemaal in te halen.
Kris en Maries, een hele dikke merci voor het ter beschikking stellen van jullie huis tijdens
het schrijven. Het was een hele verademing wat meer ruimte te hebben. Bedank Hobbes van
mij om die ruimte met me te delen en voor de kopjes op tijd en stond. Mark en An, Bart en
Ine, bedankt om af en toe eens te vragen hoe het nu eigenlijk met mijn thesis ging en voor
de vele woorden van steun. Jos en Lieve, bedankt dat ik af en toe eens bij jullie ben mogen
komen eten, wanneer ik geen tijd had om zelf mijn potje te koken.
To all the friends that I have met during my astronomical career, and from whom I know that
I can always rely on them: Francesc, Timi, Elke, Juan, Alexander. I hope that our yearly
‘Alpbach’ reunions may continue forever.
Jan, jou leren kennen is wel het beste wat me in de voorbije 4 jaar overkomen is... Voor jou
heb ik dan ook speciaal het laatste, bijzondere plaatsje in dit dankwoord bewaard. Jij hebt
zoveel voor mij gedaan, dat ik eigenlijk wel een boek vol kan schrijven met dankbetuigingen
aan jou alleen, maar ik zal proberen het kort houden (dit manuscript moet namelijk morgenvroeg al naar de drukker, dus tijd voor een tweede boek is er niet meer ;-)). Ik wil je graag
van harte bedanken voor de vele grote en kleine attenties, in het bijzonder voor al het superlekkere eten dat je voor me gemaakt hebt ‘omdat ik zo minder tijd zou verliezen’, voor het
verwenweekendje in Nederland ‘omdat ik toch nog even zou ontspannen tussendoor’, maar
ook voor het nalezen van mijn tekst op typfouten en het met zorg snijden van de uitnodigingen. Je was de enige tegen wie ik altijd mijn hele verhaal heb kunnen doen, en die altijd even
aandachtig en geı̈nteresseerd bleef luisteren. Bedankt voor je niet aflatende steun, voor de
vele avonden/nachten dat je me vergezeld hebt op het instituut ‘omdat ik het zo misschien
langer kon volhouden’. Bedankt dat je ervoor gezorgd hebt dat de glimlach nooit al te lang
van mijn gezicht verdween. Het minste dat ik dan ook kan doen, is dit boek aan jou opdragen.
Dank u wel!
Tot slot: bedankt aan iedereen die van ver en van dichtbij afgekomen is, om deze historische
dag met mij te delen!
Contents
Contents
i
Introduction
1
1
The fundamental parameters of stars
3
1.1
Spectral classification
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Stellar evolution of massive B-type stars . . . . . . . . . . . . . . . . . . . .
5
1.3
Why study massive stars? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4
Deriving the physical properties of stars using stellar model atmospheres . . .
8
1.4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.4.2
Stellar winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.4.3
Stellar model atmospheres . . . . . . . . . . . . . . . . . . . . . . . 10
1.5
Methodology and fit parameters . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.1
Effective temperature and abundances . . . . . . . . . . . . . . . . . 14
1.5.2
Surface gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.3
Microturbulent velocity . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.4
Macroturbulence and rotation . . . . . . . . . . . . . . . . . . . . . 18
ii
CONTENTS
1.5.5
Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.6
Wind parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6
Stellar oscillations in massive stars . . . . . . . . . . . . . . . . . . . . . . . 22
1.7
Summary of the major results obtained in this thesis . . . . . . . . . . . . . . 25
2 Statistical properties of a sample of periodically variable B-type supergiants
2.1
27
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.1
HIPPARCOS: the discovery of new variables . . . . . . . . . . . . . 28
2.1.2
Variability of massive stars . . . . . . . . . . . . . . . . . . . . . . . 28
2.2
The sample: observations and data reduction . . . . . . . . . . . . . . . . . . 30
2.3
Photometric variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4
Line profile fitting: fit-by-eye procedure . . . . . . . . . . . . . . . . . . . . 33
2.5
2.6
2.4.1
Determination of the rotational velocity . . . . . . . . . . . . . . . . 33
2.4.2
Determination of physical parameters . . . . . . . . . . . . . . . . . 34
Analysis of the 28 sample stars . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5.1
Sample divided in subgroups . . . . . . . . . . . . . . . . . . . . . . 37
2.5.2
Individual discussion of the resulting parameters . . . . . . . . . . . 38
2.5.3
Comparison with other investigations . . . . . . . . . . . . . . . . . 44
Analysis of the 12 comparison stars . . . . . . . . . . . . . . . . . . . . . . 52
2.6.1
Derived Teff calibration . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.2
Individual discussion of the resulting parameters . . . . . . . . . . . 54
2.7
Comments on general problems . . . . . . . . . . . . . . . . . . . . . . . . 57
2.8
Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.8.1
Error estimates for Teff . . . . . . . . . . . . . . . . . . . . . . . . . 59
iii
CONTENTS
2.8.2
2.9
Error estimates for other quantities . . . . . . . . . . . . . . . . . . . 60
Wind momentum-luminosity relation
. . . . . . . . . . . . . . . . . . . . . 61
2.10 Position in the HRD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.11 Pulsations and mass loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.12 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.13 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3
4
A grid of FASTWIND NLTE model atmospheres of B stars with winds
81
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2
Description of the FASTWIND BSTAR06 grid . . . . . . . . . . . . . . . . 82
3.2.1
Grid setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2.2
Location in the HRD . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3
Computational requirements and technical setup . . . . . . . . . . . . . . . . 85
3.4
Grid analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.1
Diagnostic lines and their isocontours of equivalent width . . . . . . 86
3.4.2
Convergence properties of the grid
. . . . . . . . . . . . . . . . . . 86
Spectral analysis of the GAUDI B star sample using a grid based fitting method 93
4.1
CoRoT and GAUDI
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.1
The space mission CoRoT . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.2
The birth of GAUDI . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.3
GAUDI Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2
Responsibilities of the Institute of Astronomy in Leuven . . . . . . . . . . . 96
4.3
AnalyseBstar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.1
Spectroscopic line profile fitting: methods . . . . . . . . . . . . . . . 98
iv
CONTENTS
4.4
4.5
4.6
4.7
Methodology of AnalyseBstar . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4.1
Preparation of the input
4.4.2
Starting values for the fit parameters . . . . . . . . . . . . . . . . . . 112
4.4.3
The cycle of the combined determination of the effective temperature,
microturbulence and abundances . . . . . . . . . . . . . . . . . . . . 113
4.4.4
Macroturbulence vmacro . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.5
Surface gravity log g . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.4.6
Wind parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.4.7
Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Formal tests and comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.5.1
Formal tests of convergence on synthetic spectra . . . . . . . . . . . 128
4.5.2
Testing AnalyseBstar on high-resolution spectra of selected pulsators 132
4.5.3
Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Alternative strategy for a grid-based method . . . . . . . . . . . . . . . . . . 143
4.6.1
Description of the alternative method . . . . . . . . . . . . . . . . . 143
4.6.2
Comparison with AnalyseBstar for ι CMa . . . . . . . . . . . . . . . 146
4.6.3
Discussion points
. . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Results for the GAUDI stars . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.7.1
4.8
. . . . . . . . . . . . . . . . . . . . . . . . 102
Comparison between observed and synthetic equivalent widths . . . . 151
Photometric calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.8.1
Geneva photometric system . . . . . . . . . . . . . . . . . . . . . . 164
4.8.2
Strömgren photometric system . . . . . . . . . . . . . . . . . . . . . 165
4.8.3
Comparison between the two photometric systems . . . . . . . . . . 168
4.8.4
Calibration based on standard stars . . . . . . . . . . . . . . . . . . . 170
v
CONTENTS
4.8.5
Spectroscopy versus photometry . . . . . . . . . . . . . . . . . . . . 173
Conclusions and future perspectives
182
Fundamentele parameters van B-type sterren
188
References
197
Publication list
207
Introduction
Stars seem to be a constant factor in our nightly skies. People often do not realise that
stars evolve and change, and that one day, they will die and fade, some very quietly, some
explosively. We do not have to wait an evolutionary long time to see stellar changes, however.
One can detect stellar variability over a wide variety of time scales and amplitudes. In fact,
at micromagnitude level, no star seems constant anymore.
Stellar variations are of great scientific importance, as they provide insights into stellar structures and lives. Indeed, one way to probe the interior of stars is, indirectly, through the study
of stellar pulsations. In analogy to Earth seismologists, who can extract information about
the Earth interior through the study of Earthquakes, asteroseismologists can probe the internal structure of a star by studying its oscillations. Since different oscillation modes penetrate
into different depths inside the star, frequency differences of the modes reveal high precision
information about the internal layers in the stellar interior. Although stellar evolution theory
is by now fairly well established, knowledge about convection, internal rotation and mixing,
processes which can alter the evolution of a star, is still rather limited. The ultimate goal of
asteroseismology is to improve upon this.
We start this work with an introduction into the fundamental parameters of stars, and in
particular of massive B-type stars, which constitute the central topic of this thesis. We give
a clear picture of the evolution of massive B-type stars, as it is currently understood, and
we stress their importance for the evolution of galaxies and the enrichment of the interstellar
material. We elaborate on stellar winds and the atmosphere code FASTWIND, which will be
exploited for all analyses presented in this thesis. We explain the general methods applied to
derive the physical parameters of a star. Chapter 1 ends with a short introduction into stellar
oscillations.
The B-type range is a very rewarding temperature region to study oscillations, since it hosts
multiple groups of variables, such as slowly pulsating B stars, β Cephei stars, Be stars, luminous blue variables and α Cyg type variables. For a long time, a clear picture of the variability in B-type stars has been hampered by observational bias with respect to time scales and
variability type. With the help of the HIPPARCOS astrometry space mission we have been
able to overcome this problem and, consequently, discovered an impressive amount of new
variables. Thanks to their expertise in B-type variability research, members of the Institute
of Astronomy in Leuven were entrusted with the classification of the new B-type variables.
2
Introduction
This revealed the existence of 29 new supergiants showing periodic (micro)variations seemingly related to the so-called α Cyg type variables. Since our knowledge about this type of
stars and their variations is still rather limited, they constituted a very interesting sample to
be studied. A large-scale high-resolution spectroscopic follow-up study was thus performed
immediately after their discovery. We were involved in the detailed spectroscopic analysis
of these targets. Through the comparison of the observed line profiles with a large set of
theoretically predicted line profiles, we were able to derive the fundamental parameters of
these stars and position them accurately in the Hertzsprung-Russell Diagram. This helped us
to better understand the mechanism behind the variations in this type of star and led to the
suggestion that they are gravity-mode oscillators. These results will be presented in Chapter
2.
A second likewise unbiased but more extended sample concerns all B-type stars observable by the CoRoT space mission. In the framework of the preparation of this (asteroseismology and extrasolar planet search) satellite mission, and in particular with the goal of
making an optimal target selection, the CoRoT consortium took one high-resolution spectrum for each star in the field of view, with visual magnitude below 8th mag, extending to
9.5th mag in the selected fields of the CoRoT main targets. All spectra were gathered in
the GAUDI database (Solano et al. 2005) and subsequently made available through a webinterface (http://sdc.laeff.inta.es/gaudi/). As a partner in the CoRoT team, the Institute of
Astronomy in Leuven was responsible for the determination of the fundamental parameters
(in particular the effective temperature, surface gravity, luminosity, rotation and wind characteristics) of the hot OB stars in the GAUDI database. To cope with this large dataset of about
200 stars not only required a robust, homogeneous and objective method, but also appropriate
models including state-of-the-art physics, representative for a wide variety of stellar properties. This basically required an automatic tool for the spectroscopic analysis of B-type stars,
which was unavailable before this work. To resolve this issue, we have derived an optimal
compromise between computation time and precision of parameters by developing a method
based on a comprehensive and representative grid of atmosphere models for the full B-type
spectral range. Thanks to its fast performance, the Non Local Thermodynamic Equilibrium
atmosphere and line prediction code FASTWIND is the only model code at present which
is suited for this kind of analysis. The grid, which will be a good reference work for future
large-scale spectroscopic analyses, and its properties, are fully described in Chapter 3.
Based on this grid, the first automated method for the quantitative spectroscopic analysis
of B-type stars with stellar winds has been constructed. After an intensive test period on
synthetic data, as well as on high-resolution and extremely high signal-to-noise spectra of β
Cephei and Slowly Pulsating B stars, it was applied to a selection of high-resolution, but much
lower signal-to-noise spectra of the B stars in GAUDI. The physical parameters obtained in
this way were supplied to GAUDI and CoRoTSKY, the tool used by the CoRoT community
to prepare the CoRoT target selection. The GAUDI OB stars constitute a unique sample to
perform statistical studies in order to verify the currently widely used effective temperature
and gravity calibration for B-type stars. This will be discussed in detail in Chapter 4.
We end this dissertation with some general conclusions, a discussion of the results of our
research and some future prospects.
Chapter
1
The fundamental parameters of stars
The studies presented in this thesis pursue the determination of the physical properties of stars
of spectral type B through spectral line fitting. In this first general chapter, we introduce the
B-type stars and their evolution, as well as the methods applied to derive their fundamental
parameters.
1.1 Spectral classification
Since the advent of spectrography in the second half of the 19th century, astronomers have
been classifying stars following the strength of their Balmer lines1 . It soon became clear,
however, that not only the strength of the Balmer lines, but also the strength of all other
lines followed a nice sequence. It was Annie Cannon’s team in Harvard (Cannon & Pickering 1920) who empirically classified the stars in roughly 7 standard classes, constituting the
spectral sequence (and mnemonic) Oh-Be-A-Fine-Girl/Guy-Kiss-Me. This led to the presently
widely used Henry Draper (HD) Catalogue of stars, which now contains some 400,000 stars.
It was Cecilia Payne-Gaposchkin (Payne 1925) who realised that this spectral sequence is
ordered according to the effective temperature, Teff , of the stars and that the line strength is
1 The Balmer lines are emission or absorption lines in the spectrum of hydrogen that arise from transitions between the second (or first excited) state and higher energy states of the hydrogen atom.
4
The fundamental parameters of stars
determined by the Saha-Boltzmann ionisation law. She also showed that stars consist mainly
of hydrogen (∼70%) and helium (∼28%) and only of ∼2% heavier metals.
The spectral sequence has been further refined, with the addition of the spectral subtypes and
luminosity classes. Originally there were 10 subtypes, from 0 (hottest) to 9 (coolest), within
each spectral type. Nowadays, some of these spectral subtypes seem to be redundant, in the
sense that there are hardly any stars designated to them (e.g., B4), while for other subtypes
further refinement was required (e.g., B2.5) due to the very rapid change in effective temperature. Stars from different classes will appear with different colours. The objects of our
research, B-type stars, have effective temperatures between 10,000 and 30,000 K, and appear
visually as blue stars. The luminosity class, which adds a second dimension to the classification scheme, is based on spectral lines which are sensitive to the stellar surface gravity, and
is related to the stellar luminosity L∗ ∝ Teff 4 R∗ 2 . Five classes are considered: dwarfs or
main sequence stars (V), subgiants (IV), giants (III), luminous giants (II) and supergiants (I,
in order of decreasing luminosity: Ia0, Ia, Iab, Ib). Sometimes, additional characters are used
to point towards specific spectral characteristics, such as the ‘e’ for the presence of strong
emission lines (mainly the Balmer lines but also other strong lines).
Spectral characteristics offer a way to classify stars (following their effective temperature and
luminosity/gravity) in a more detailed way. Specific absorption lines can only be observed for
a certain range of temperatures because only in that range the involved atomic energy levels
are populated. The optical spectra of B stars are characterised by strong lines of neutral
helium (and He II2 for stars hotter than B1) and moderate hydrogen lines. Most metallic lines
are absent or weak, except for some absorption lines of the first and/or second (depending on
Teff ) ionisation stages of silicon, magnesium, oxygen and carbon.
Spectral classification by itself reveals little of the physical nature of stars, and how they are
linked together in an evolutionary sense. Such a link is provided by the Hertzsprung-Russell
Diagram (HRD, Russell 1914), which relates stellar luminosity and colour. Alternatively,
and depending on the specific objectives, the luminosity on the y-axis may be replaced by
gravity and the colour by Teff , to result in the so-called log Teff − log g diagram. Fig. 1.1
shows the position of the different spectral classes in the HRD. The HRD helps us to study
the evolutionary lifecycles of stars as will be discussed in the next section.
Since B (and even more the hotter O) stars have extremely high luminosities, their nuclear
fuel is quickly exhausted, i.e., they are extremely short-lived. Therefore they do not get the
chance to move far from their birthplaces in the galactic plane and they are often found in
so-called OB-associations: young and very loose ‘clusters’ of stars, containing 10 to 1,000
massive O and B-type stars.
2 We use the notation according to the standard astronomical convention, i.e. He I stands for neutral helium, He II
represents the first ionisation stage of helium, Si III the second ionisation stage of silicon, etc...
1.2 Stellar evolution of massive B-type stars
5
Figure 1.1: Schematic representation of the different spectral classes in the HRD (luminosity/absolute magnitude versus colour/spectral type/effective temperature). The colour of the
stars in this HRD represents the colour as observed with the naked eye. Figure taken from:
http://www.porpoisehead.net/mysw/stellarium user guide html-0.9.0-1/
1.2 Stellar evolution of massive B-type stars
One of the primary facets of stellar life is mass loss. To some extent, all stars lose mass
through a stellar wind, which in many evolutionary phases can have a decisive impact on
the specific evolutionary path to be chosen (as will be discussed in Section 1.3). The track of
evolution followed by a certain star throughout the HRD will mainly depend on its initial mass
on the zero-age main-sequence (ZAMS) where stars reside as they begin to burn hydrogen.
The main sequence or dwarf stars form a relatively narrow band from the upper left (high
luminosity, high temperature) to the lower right (low luminosity, low temperature) when
plotted according to luminosity and surface temperature on the Hertzsprung-Russell diagram.
6
The fundamental parameters of stars
A star’s position and length of stay on the main sequence depend critically on mass. Although
the chemical composition of the cores changes completely during hydrogen burning, the
dwarfs do not show much change in their characteristics and remain within the main sequence
band for the major part of their lives. They all undergo however a small contraction of the
core and an expansion of the envelope as a result of the change in the mean molecular weight
in the core.
Since the B-type stars encompass a very large mass range, with masses going roughly from
3 to 20 solar masses, not all stars have the same evolutionary scenario. We can make a crude
separation between stars with masses lower than 8 to 10 M⊙ and stars with a higher initial
mass, the precise cut-off in mass being a function of the chemical composition. After the
main sequence phase, when the core hydrogen gets exhausted, the star reacts to this pending
shortage of energy by a short contraction phase. Since the temperature in the core is at
this point still too low to start helium burning, a phase of hydrogen shell burning (around
the helium core) is initiated. As a result of this, the internal density and temperature rise.
A thermal instability occurs and the envelope expands much further while the core is still
contracting. Due to the associated drop in surface temperature the B dwarfs (luminosity
class V) evolve into red giants (luminosity class III) with spectral types G or K. While the
envelope expands, the core further contracts, until helium starts to burn (non-explosively)
and forms carbon and oxygen. This new source of energy stabilises the star. After core
helium exhaustion, helium will burn in a shell around the core. Outside the helium shell,
hydrogen burning will take place. Finally a similar cycle can be followed for helium instead
of hydrogen. Every time the central burning is exhausted, in a next phase, shell burning
will occur. B stars with an initial mass below 8 to 10 M⊙ will have a degenerate CO-core
after the central He-burning phase. In this case the temperature is too low to initiate carbon
burning and the star ends as a carbon-rich white dwarf. B stars with an initial mass above
∼ 10 M⊙ (earlier than B2 on the main sequence) will have a non-degenerate core and will
end the different burning cycles as a supernova with a remaining neutron star or black hole.
In stars with birth masses above ∼ 20 M⊙ (O-type stars), the burning processes happen much
faster and the time on the main sequence is relatively short. Similar to the lower mass B stars,
they will move to the red part of the HRD (i.e. towards lower temperatures), when the central
hydrogen runs out and their cores contract. The stars are going from the blue supergiant
phase to the stable helium-burning red supergiant phase. The winds of blue supergiants are
driven by radiation pressure and can be so severe that, at some point, they will have expelled
their entire outer envelope, and hotter, helium rich inner layers become visible. These objects
evolve further into so-called Wolf-Rayet stars. Wolf-Rayet stars only have a few hundred
thousand years to live before they explode as a supernova. Stars initially more massive than
∼ 60 M⊙ will never evolve into red supergiants, since at some point the outer layers become
unstable, leading to severe mass loss and the unveiling of hotter layers. Finally also these
objects evolve further into Wolf-Rayet stars.
The position of a supergiant in the HRD does not uniquely reveal which phase of its life
the star is currently going through (the star can be moving to the right as well as to the left)
until finally the star ends as a core-collapse supernova. In Fig. 1.2 we show the described
evolutionary tracks for stars with birth masses up to 40M⊙ .
1.3 Why study massive stars?
7
Figure 1.2: HRD with the post main sequence evolutionary tracks for stars with a birth mass
up to 40 M⊙ . The evolutionary tracks were taken from Schaller et al. (1992). B-type stars
occupy the temperature range between roughly 4.0 and 4.48 in log Teff . Figure courtesy:
Conny Aerts.
1.3 Why study massive stars?
Due to their extreme properties, their evolutionary behaviour and their final fate as a supernova explosion, massive stars are at the focus of attention in several fields of astrophysics.
The high-speed outflows through their stellar winds/supernova ejecta and the intense ionising
radiation field emitted from massive stars can lead to the energetic and chemical enrichment,
and the dynamic structuring of the surrounding interstellar medium. In starburst galaxies the
energetic ‘feedback’ from massive stars can lead to the large-scale outflow of the interstellar
medium in the form of a galactic wind. Massive stars are the main sources of ionising photons
creating H II regions. During the supernova explosion, neutron stars and stellar black holes
are formed. Eventually (depending on the rotational speed of the progenitor), a Gamma Ray
Burst emerges, the most energetic cosmic flash.
Massive stars are the only stars which evolve beyond the helium burning phase and thus form
elements heavier than oxygen. In so far, they are the main source of most metals in the
8
The fundamental parameters of stars
Universe3, which are released during stellar outflows and supernova ejecta. In this sense, on
a cosmological timescale, they will also play an unmistakable role in the chemical evolution
of galaxies.
Not only in the current Universe, but also in its (very) early epochs massive stars have played
a crucial role. Due to their intense UV radiation field, the very first generation(s) of these
objects (‘First Stars’, Population III) were responsible for the re-ionisation of the early Universe (Bromm et al. 2001), at least in part (Matteucci & Calura 2005). Moreover, these Very
Massive Stars (with - theoretical - masses of 100 to 300 M⊙ and even beyond) were also
important with respect to an early enrichment of the Universe with metals.
Particularly at present epochs, the individual evolution of these massive stars is, to a large
part (and in combination with rotation), controlled by their stellar winds. Meynet et al. (1994)
have shown that the effect of a factor of two difference in mass-loss rate may significantly
alter the evolutionary track of a star. The effects of mass loss on the internal structure of
massive stars influence the nature of the supernova precursors and their chemical yields, as
well as the general stellar stability throughout evolution (Chiosi & Maeder 1986). In this
sense, accurate values for the actual mass-loss rates from these stars are urgently required.
1.4 Deriving the physical properties of stars using stellar
model atmospheres
1.4.1 Introduction
To understand the evolutionary cycle that a particular star is going through, it is important
to investigate its physical properties and its position in the HRD. The only way to deduce
this information from the star is to decode the radiation it emits in an appropriate way. In
practice, we seek to derive both the stellar and wind characteristics of the star. Stellar properties encompass the effective temperature Teff , the surface gravity log g, the stellar radius R∗ ,
mass M∗ and luminosity L∗ , the micro- and macroturbulent velocities ξ and vmacro (see Sections 1.5.3 and 1.5.4 for a definition), the rotational velocity v sin i and, finally, the chemical
composition (besides the eventual presence of a magnetic field). In this work we will concentrate only on the abundances of the elements helium and silicon, for reasons explained
later on. Wind properties include the mass-loss rate, Ṁ , a measure for the flow acceleration,
β, and the velocity of the wind at an ‘infinite’ distance from the star, v∞ (cf. Section 1.4.2).
Since the challenge of determining those quantities will occupy the major part of this work,
we will elaborate a bit more on this, but first we give a brief summary about stellar winds.
3 leaving
aside SNeIa with their production of iron-group elements in the later phases of cosmic evolution.
1.4 Deriving the physical properties of stars using stellar model atmospheres
9
1.4.2 Stellar winds
Besides radiation (i.e. photons), massive OB stars also lose particles (i.e. atoms, ions and
electrons), constituting their characteristic stellar wind. The two most important parameters
describing the winds of hot stars are the amount of mass lost per unit of time (the massloss rate, Ṁ ) and the velocity of the wind at very large distance from the star (the terminal
wind velocity, v∞ ). The winds of massive stars are driven by strong radiation pressure: the
momentum of the radiation field (i.e. the photons emitted by the star) is transferred to the
components of the stellar plasma (i.e. the elements at the surface of the star) by absorption in
millions of metal lines. It is this radiative line acceleration suffered by the external layers of
the star which creates the outflow. This mechanism works in massive stars since the luminosity is high, i.e., a large number of photons is available. For an efficient line driving, also the
metallicity should not be too small to ensure the presence of ‘enough’ lines with significant
interaction probability4.
Lucy & Solomon (1970) were the first to show that the strong ultraviolet (UV) resonance
lines produce enough radiation pressure to counteract gravity and accelerate the wind to supersonic velocities by radiative line pressure. Five years later, Castor et al. (1975) published
their famous ‘CAK’-theory (Castor, Abbott and Klein), which quantified, to a certain extent,
this theory, pointing out the importance of a mixture of optically thick and thin lines. As
a consequence of this theory, a close coupling between the (kinetic) ‘modified’ stellar wind
momentum Ṁ v∞ R∗ 1/2 and the stellar photonic momentum rate (i.e. the stellar luminosity
L∗ ) is predicted, by means of the so-called Wind momentum-Luminosity Relation or WLR
(Kudritzki et al. 1995; Puls et al. 1996), which can be written as follows:
Ṁ v∞ R∗ 1/2 ∼ (L∗ /L⊙ )x ,
with x the inverse of the slope of the line-strength distribution function corrected for ionisation effects (Puls et al. 2000). This power law may vary as a function of spectral type,
luminosity class and metal content.
Different methods are available to observationally derive the wind properties. Either one
can rely on an estimate for the excess of infrared, millimeter and/or radio radiation due to
free-free (and partly bound-free) processes in the wind (e.g., Bieging et al. 1989; Lamers &
Leitherer 1993; Leitherer et al. 1995), or on the fits of spectral lines sensitive to wind properties, both in the UV (e.g., Lamers et al. 1995; Prinja & Massa 1998; Herrero et al. 2001;
Prinja et al. 2007) and the optical (Hα, e.g., Leitherer 1988; Lamers & Leitherer 1993; Puls
et al. 1996; Herrero et al. 2002; Trundle et al. 2004; Repolust et al. 2004; Massey et al. 2004,
2005; Mokiem et al. 2005, 2006; Lefever et al. 2007). The different spectral ranges scan
different regions of the wind: Hα (in the optical spectral range) is formed in the outer photosphere and lower/intermediate wind region (1-5 R∗ ). Infrared lines are mostly formed above
the stellar photosphere, in the lower/intermediate wind. The broad P Cygni line profiles of
strong, mostly resonance, lines in the UV form in different parts of the expanding atmosphere.
Millimeter and radio emission forms in the outer regions of the wind (10-50 R∗ ).
4 Exactly for this reason, the winds of First Stars are comparably weak, since they are driven almost only by a
few number of lines from hydrogen and helium. Population III stars are (almost) metal-free.
10
The fundamental parameters of stars
For O/early B-type stars, typical UV lines are sensitive to mass-loss rates larger than roughly
∼ 10−7 M⊙ /yr.
Though the optical range comprises another line which is strongly sensitive to mass loss,
namely He II 4686, the Hα line is the mass-loss indicator in the optical, due to a number of
advantageous features: it is relatively simple to model thanks to the simple atomic structure
and it does not depend so much on complicating effects such as blanketing, blocking and
X-ray emission5 (but it is affected by clumping, see Section 1.5.6). Moreover, it is present
throughout the complete OB spectral type range, in contrast to He II.
Let us finally point out that the wind does not only affect environment and evolution, but
also the physics of massive star atmospheres themselves, as should be obvious: the density
stratification and the radiative transfer of the outer atmosphere are dominated by the outflow,
in particular by the presence of a transonic velocity field. In so far, stellar winds can have
a significant influence on the spectral energy distributions (SED) and on the emergent line
spectra, and often one cannot avoid accounting for them in the spectral analysis. Even for
stars with a low mass loss, wind effects show up in the infrared. In this dissertation, however,
we will concentrate only on an optical analysis.
1.4.3 Stellar model atmospheres
A detailed quantitative study of the physical properties of a star requires a model for its atmosphere. This includes not only the behaviour of pressure p(r), temperature T (r), density
ρ(r) and velocity v(r) with depth r in the atmosphere, but also a prediction of the emergent
spectrum, which is a direct and observable diagnostic of the underlying atmospheric properties. We thus need a code that can predict how the spectrum would look like when the star
has a certain effective temperature, gravity, etc. This code should be sophisticated enough to
give an accurate approximation of the real observed spectrum, but at the same time it should
be approximative enough to produce a synthetic spectrum within a reasonable computation
time. Indeed, the more physics is included (i.e., the less approximations are made), the longer
the computation of one single model will take. This was exactly the concern of the Munich
research group, led by J. Puls, when they started the development of a new, fast and highprecision code for the NLTE6 prediction of the atmospheres and winds of O, B and A-type
stars: FASTWIND (Fast Analysis of STellar Atmospheres with WINDs, Santolaya-Rey et al.
1997, Puls et al. 2005). Since we consider speed a high priority, we have chosen to use
FASTWIND for our purposes. Although a detailed description of FASTWIND is beyond the
scope of this thesis, and can be found in Santolaya-Rey et al. (1997) and Puls et al. (2005), we
briefly summarise the physics that is accounted for, focusing on the aspects that are important
for the further understanding of this thesis.
5 thought
to arise in shocks and colliding wind-inhomogeneities.
stands for Non Local Thermodynamic Equilibrium. In LTE one assumes that the material is in equilibrium with the local kinetic temperature, but that the radiation may deviate from its thermodynamic value Bν (T).
NLTE thus points towards deviations from this assumption, which mainly become important in hot stars with thin
atmospheres.
6 NLTE
1.4 Deriving the physical properties of stars using stellar model atmospheres
11
‘Unified atmospheres’ with spherical extension and stellar winds
Contrary to codes which neglect outflows and predict the hydrostatic photosphere in a planeparallel approximation, FASTWIND allows for the particles to flow through the sonic point
and escape in the supersonic velocity field of the stellar wind (i.e. the particles follow transonic radial trajectories). Even for stars with negligible winds (where only the stellar properties are left as fit-parameters), differences in the predicted atmospheric structure and the
emerging line profiles between both approaches remain due to the presence of this transonic
velocity field. Indeed, there will always be an enhanced probability that photons can escape
from regions close to the sonic point, and above, independent of the actual mass-loss rate
(Puls et al. 2005). This can change the population of important ions and thus the spectral
energy distribution (SED) considerably (e.g., Gabler et al. 1989).
FASTWIND starts from the standard concept of a stationary, smooth and spherically symmetric atmosphere, neglecting any magnetic field. For the photosphere and the expanding wind,
a different velocity law is adopted due to different physics, and both regimes are connected
by a smooth transition (see the velocity stratification displayed in Fig. 1.3). The inner atmosphere (i.e. the subsonic photosphere) is in pseudo-hydrostatic equilibrium with a velocity
law following from the equation of continuity, i.e. v(r) = Ṁ /4πr2 ρ(r), whereas the outer
expanding atmosphere (i.e. the supersonic wind) follows a so-called β-law:
with
r0 β
,
v(r) = v∞ 1 − b
r
b=1−
v0
v∞
1/β
,
with v0 the velocity at r0 , the ‘transition’ velocity/transition point, as defined in SantolayaRey et al. (1997):
v0 = v(r0 ) = 0.1a(r0 ),
where a is the isothermal speed of sound.
Temperature stratification T (r)
One of the most complex tasks in the construction of a stellar atmosphere model is the computation of the temperature stratification from a rigorous treatment of radiative equilibrium,
which simply states that the total absorbed radiation energy is balanced by the emitted one.
Although they give no direct contribution to the temperature (but only indirectly influence
the temperature via the equations of statistical equilibrium), processes due to scattering (radiative bound-bound transitions) dominate the radiative energy balance numerically, which
makes the computations for the radiative equilibrium really slow. They overrule the real
important transitions, i.e. the collisional bound-bound, bound-free and free-free transitions,
which directly affect the temperature stratification (Kubát et al. 1999). In order to avoid
12
The fundamental parameters of stars
Figure 1.3: Velocity stratification of a typical B star model atmosphere, illustrating the
smooth transition between the velocity law of the pseudo-hydrostatic photosphere, resulting from the equation of continuity, and the β-velocity law in the transonic stellar wind, as
a function of the Rosseland optical depth τR . The transition point is at a tenth of the sound
velocity, which is for this illustrative model approximately 1.5 km/s.
this problem and to enable a fast calculation of a consistent temperature structure, FASTWIND utilises the thermal balance of electrons in the line forming regions and in the optically thin part of the outer atmosphere. This equation considers all effects of electrons
regarding the heating/cooling of the plasma, by accounting for all important interactions, via
collisional bound-bound and bound-free processes, but also via free-free absorption/emission
and radiative ionisation/recombination. For larger optical depths, in the lower atmosphere,
a flux-correction method is applied, which calculates the temperature stratification from the
condition of flux conservation. Of course, this condition has to be valid throughout the complete atmosphere, and can be used as an independent tool to check whether the models have
been constructed in a consistent way or not. Generally, the final models yield a conservation
better than 1%. The region where both methods (the flux-correction method and the thermal
balance of electrons) are connected, typically lies at τR = 0.5 (with τR the Rosseland optical
depth), but is to some extent dependent on mass loss. We consider the T-structure T (r) to be
converged if the relative error in T is less than 0.3%.
Explicit and background elements
A second important update, carried out in the most recent version of FASTWIND, is the
distinction between two groups of chemical elements, i.e. the ‘explicit’ elements and the
‘background’ elements. The explicit elements are calculated with high precision (i.e. by using
detailed atomic models and co-moving frame (CMF) transport for the line transitions), since
1.4 Deriving the physical properties of stars using stellar model atmospheres
13
they will be used as diagnostics in the spectral line synthesis. For our temperature range (i.e.
the B-type stars), these will be hydrogen (H), helium (He) and silicon (Si). The background
elements, on the other hand, are introduced to account for the line blocking and blanketing
effects in the emergent spectrum, the most abundant ones being carbon (C), nitrogen (N),
oxygen (O), neon (Ne), magnesium (Mg), sulfur (S), argon (Ar), iron (Fe) and nickel (Ni).
The calculation of the line-blocking effects is treated in an approximate, statistical way, by
using suitable mean values for the corresponding line opacities. Occupation numbers and
opacities of both the explicit and the most abundant background ions are constrained by
assuming statistical equilibrium. We require the occupation numbers to be converged with
|δn/n| < 0.3%, for all transitions and all radii. The only difference in treatment between
both types of ions is that for the determination of the level populations of the background
ions the Sobolev approximation (Sobolev 1960) is employed in describing the line transfer,
to save, once again, computational effort. When the temperature structure has not converged
or when the convergence of the occupation numbers is not achieved within a certain number
of iterations, we investigate the reason for this failure (see Section 3.4.2).
Alternative atmosphere codes
By now, several research groups have taken up the challenge of developing codes for the
numerical computation of stellar atmospheres of massive stars, some codes being more sophisticated than others. More specifically, besides the above described FASTWIND code,
there are presently four more codes in use for the spectroscopic analysis of hot luminous
stars. These are CMFGEN (Hillier & Miller 1998), PoWR (Gräfener et al. 2002), PHOENIX
(Hauschildt & Baron 1999) and WM-basic (Pauldrach et al. 2001). A thorough comparison
between FASTWIND and the above-mentioned alternative codes CMFGEN and WM-basic
has been performed by Puls et al. (2005). Each of the above-mentioned codes has been designed with specific aims, and due to the fact that they are less approximative at some points,
they also require longer computation time compared to FASTWIND. This implies that only a
limited part of the parameter space and a limited number of stars can be investigated. The fast
performance of FASTWIND, however, allows to spectroscopically analyse large samples of
hot stars within an acceptable time. Meanwhile, a number of spectroscopic investigations of
early-type stars were performed by means of FASTWIND, both in the optical (e.g., Herrero
et al. 2002; Trundle et al. 2004; Repolust et al. 2004; Massey et al. 2004, 2005; Mokiem et al.
2005, 2006, 2007; Lefever et al. 2007) and in the near infrared (Repolust et al. 2005).
As mentioned before, we will use H, He and Si as explicit elements for our studies. Considered in parallel, these three fairly simple elements (with respect to number of atomic levels
and transitions) will give us all the information we need to fully deduce the stellar properties.
The inclusion of CNO and/or iron group elements would slow down the code due to the high
number of transitions to be accounted for. Therefore, the CNO/iron group elements are omitted and treated in a more approximate way, as background elements. In what follows we will
describe our methodology to extract the required information.
14
The fundamental parameters of stars
1.5 Methodology and fit parameters
Our general adopted procedure to find the best approximation to the real stellar and wind
parameters is spectral line fitting. In essence, this is an (iterative) optimisation problem, in
which one tries to find the optimum fit between observed and synthetic line profiles, which
emerge from the underlying stellar atmosphere model. Generally, one starts from a preexisting grid of atmosphere models spanning a sufficiently large and representative part of
parameter space. Depending on the density of this grid and the desired accuracy, one will
need to calculate additional models in order to reproduce the observed line profiles as well as
possible. So far, preference was given to the ‘fit-by-eye’ method to find the best fit (cf. Martins et al. 2005; Crowther et al. 2006; Przybilla et al. 2006; Lefever et al. 2007; Morel 2007).
It can be questioned whether this is really the best way to proceed in terms of objectivity and
efficiency (see, e.g. Mokiem et al. 2005). A more profound discussion concerning this issue
will be given in Chapter 4.
Throughout the next sections, in the displayed examples, we use a reference model with the
following parameters:
Teff
log g
log n(Si)/n(H)
n(He)/n(H)
log Q
=
=
=
=
=
22,000 K
3.5
-4.49
0.15
-13.40
β
v sin i
ξ
vmacro
∆λ/λ
=
=
=
=
=
1.2
40 km/s
10 km/s
30 km/s
30,000
where log Q is a measure for the wind density (see Section 1.5.6 for a definition) and ∆λ/λ
is the spectral resolution. This reference model is always displayed as a full line, whereas a
change ∆ in the parameters is represented by a dashed line when it concerns a change towards
higher value (i.e. ∆ > 0), and by a dotted line when it concerns a change towards a lower
value (i.e. ∆ < 0).
1.5.1 Effective temperature and abundances
Silicon lines have been chosen as a good diagnostic because of their strong sensitivity to
changes in temperature in the B-type spectral range (Becker & Butler 1990). When Si II, Si III
and Si IV can be considered in parallel, they unambiguously define the effective temperature.
Increasing the temperature in the late B-type regime results in an opposite effect regarding
the equivalent width (EW) of Si II and Si III. In the higher temperature region, a similar effect
can be observed between Si III and Si IV. This behaviour is demonstrated in the top panels
of Fig. 1.4, where we show the line profiles for a representative Si line for each of the three
ionisation stages: Si II 4128, Si III 4552 and Si IV 4116. The same effect can be observed
by looking at the isocontours of equivalent width of these lines, as will be shown later in
Fig. 3.2. Using the ionisation balance between two consecutive stages of Si (i.e. Si IV/Si III
for the early B-type stars and Si III/Si II for the mid and late B type stars), rather than single
lines, temporarily eliminates the effect of the Si abundance, which affects all Si lines in the
1.5 Methodology and fit parameters
15
same way (see lower panels of Fig. 1.4). A similar behaviour can be observed between the
different ionisation stages of helium, i.e. He I and He II (see Fig. 1.5). Even though both
ionisation stages can be observed simultaneously only in the earliest B-type stars, He I on its
own also provides us with a good temperature diagnostic for the latest B-type stars, due to
its strong reaction to changes in the temperature (see Fig. 1.6). Outside these regions He I
becomes more or less independent of temperature (see Fig. 1.5), but turns into a good gravity
indicator instead.
1.5.2 Surface gravity
The wings of the Balmer lines are Stark-broadened7 and therefore they strongly react on
changes in the electron density ne , which is (almost) directly coupled to the gravitational acceleration, log g 8 , provided that the centrifugal acceleration and the (photospheric) radiation
pressure are known and correctly calculated, respectively.
Fig. 1.7 shows the influence of a change in gravity on the wings of the Balmer lines. Although all Balmer lines are to some extent affected by a change in gravity, only Hγ and Hδ
will actually be used to determine log g. As long as the mass-loss rate is not too high, also
Hβ can be used. However, as soon as the star exhibits some wind, both Hα and to a lesser
extent Hβ will be affected by this outflow, since they are formed higher in the stellar atmosphere. In many cases, Hǫ is unreliable from the observational point of view. The merging
process in the bluest spectral ranges is really difficult (see item 2 of Section 4.4.1). Although
the situation for the FEROS spectra (cf. Section 4.7) is much better than for the CORALIE
spectra (cf. Section 4.5.2), due to the flatness of the spectral orders, we have omitted Hǫ in
the determination of log g for reasons of homogeneity, and only use it as a consistency check
afterwards.
Since log g strongly depends on the effective temperature (which is derived from independent
diagnostics, see Section 1.5.1), both will be determined in parallel. For fast rotating objects,
the gravity derived from the Balmer wings should be corrected to account for the centrifugal
acceleration due to the rotation as follows:
log g c = log g + (v sin i )2 /R∗ ,
(1.1)
where v sin i ≡ ΩR∗ sin i is the observed rotational velocity, projected along the line of sight
(cf. Section 1.5.4), and R∗ is the radius of the star. For slow B-type rotators this effect is
negligible.
7 The linear Stark effect is the most important pressure broadening mechanism for hydrogenic atoms (i.e., atoms
with degenerate levels) in hot stellar atmospheres, caused by the Coulomb interaction of emitting atoms with charged
particles or particles with a strong permanent electrical dipole moment.
8 We follow the standard convention to list the gravity g ≡ GM/R2 as log g in cgs units.
16
The fundamental parameters of stars
Figure 1.4: Influence of the effective temperature (top) and the silicon abundance (bottom) on
the Si line profiles, for three different ionisation stages: Si II 4128, Si III 4552 and Si IV 4116.
Full line: reference model (p. 14); dashed (∆ > 0) and dotted (∆ < 0) lines represent the
respective changes in Teff and log n(Si)/n(H): ∆Teff = ± 2,000 K (top), ∆log n(Si)/n(H) = ±
0.30 (bottom).
Figure 1.5: Influence of the effective temperature (left) and the helium abundance (right) on
the He line profiles, for two different ionisation stages: He I 4471 and He II 4686. Full line:
reference model (p. 14); dashed (∆ > 0) and dotted (∆ < 0) lines represent the respective
changes in Teff and n(He)/n(H): ∆Teff = ± 2,000 K (left), ∆ n(He)/n(H) = ± 0.05 (right).
1.5 Methodology and fit parameters
17
Figure 1.6: He I lines are good temperature indicators in the cool temperature regime, as
illustrated here for He I 4471 (left) and He I 4922. The variation in effective temperature is
now only 1,000 K, but the reaction is very strong. Full line: reference model (p. 14), but with
the reference temperature adapted to 12,000 K; dashed and dotted: ∆Teff = ±1, 000 K.
Figure 1.7: Influence of the surface gravity on the Stark broadened wings of Hβ, Hγ and Hδ.
Full: reference model (p. 14); dashed and dotted: ∆log g = ±0.5 dex.
1.5.3 Microturbulent velocity
Microturbulence, i.e. small-scale non-thermal motion in the atmosphere, denoted by ξ, is one
of the most significant broadening mechanisms of absorption lines in stellar spectra, which
affects both the line shapes and strengths. In essence, it was introduced in the prediction
of spectral lines to reduce discrepancies between observations and theoretical predictions9.
Villamariz & Herrero (2000) have studied the effect of microturbulence in the line formation
calculations of H and He lines, in the parameter range typical for O and early B stars, and the
effect on the determination of stellar parameters, i.e., on Teff , log g and the He abundance.
They concluded that microturbulence is affecting the derivation of stellar parameters, but its
9 For hot stars, it is still not clear whether this quantity has a physical origin or whether it is just an artefact arising
from the model atmosphere calculations (McErlean et al. 1998). At least for G/F dwarfs, however, Asplund et al.
(2000) have shown, by means of radiation-hydrodynamic simulations of stellar surface convection, that it seems to
be possible to synthesize line profiles in perfect agreement with observations, without relying on various ad-hoc
parameters, such as mixing length, micro- and macro-turbulence, which are required to fit observations by means of
1-D models.
18
The fundamental parameters of stars
Figure 1.8: Effect of the microturbulent velocity on two representative lines: one Si line
(Si II 4128, left) and one He line (He I 6678, right). Full: reference model (p. 14) at ξ =
10 km/s; dashed: ξ = 15 km/s; dotted: ξ = 6 km/s.
effect is comparable to the adopted uncertainties. It can reduce moderate He overabundances
and solve line fit quality differences, but it cannot explain by itself large He overabundances.
Markova & Puls (2007) studied the effect of the microturbulence for mid and late B-type supergiants, and found the effect on the ratios of equivalent widths from Si lines to be minimal,
thus leaving the temperature determination unaffected. Microturbulence seems to vary with
spectral type, being largest for hot objects and diminishing towards later spectral types (cf.
McErlean et al. 1998, Markova & Puls 2007). The strength of the microturbulence can be
determined by comparing the broadening of strong lines versus weak lines, since both react
differently. Actually, the behaviour of the microturbulence is exploited to make sure that the
same abundance would result from each line of a certain ion. Different species, however,
could lead to a different microturbulent velocity, presumably due to stratification effects in
this quantity. Fig. 1.8 shows the effect of the microturbulence on the Si and He lines, for two
representative spectral lines, i.e. Si II 4128 and He I 6678.
1.5.4 Macroturbulence and rotation
Macroturbulent, vmacro , and rotational broadening, v sin i , are in practice very difficult to
decouple. The absorption lines in the spectra of early and mid B-type supergiants (not the
main sequence stars) seem to exhibit a significant, but so far unexplained amount of broadening in addition to rotation. This additional broadening is generally gathered under the term
’macroturbulence’. Although rotation and macroturbulence have a different profile shape (ellipsoidal versus Gaussian profile), often an equally good fit can be obtained by using different
combinations of vmacro and v sin i , when the resolution is lower than vmacro (see Fig. 1.9).
We have found a satisfactory solution to this problem by Fourier transforming weak metal
lines and determining v sin i from the first minimum of this transformation (Simón-Dı́az
et al. 2006).
19
1.5 Methodology and fit parameters
Figure 1.9: Effect of the macroturbulent velocity (left) and the rotational velocity (right) on
Si III 4552. Full: reference model (p. 14) at vmacro = 30 km/s and v sin i = 40 km/s; dashed
and dotted for ∆vmacro = ± 20 km/s (left) and ∆v sin i = ± 20 km/s (right).
1.5.5 Radius
The stellar radius R∗ is not a free parameter, since it is connected with the absolute visual
magnitude MV . For stars for which we know the distance d (e.g. from the parallax π), we
can derive MV from
MV = V0 + 5 − 5 log(d),
where V0 is the dereddened visual magnitude. In practice, for the objects considered in
this thesis, accurate parallax measurements are not available, and distances remain unknown
(except for cluster members). Thus, we have to rely on a calibration for MV as a function of
spectral type or effective temperature, such as the one by Schmidt-Kaler (1982), but then we
have to deal with the large uncertainty of such calibration.
Once we know MV , we can calculate R∗ . For this, we follow the procedure as outlined in
Kudritzki (1980) and calculate the radius from
5 log
R∗
= 29.57 − (MV − Vtheo )
R⊙
(1.2)
by comparing the V-band integrated theoretical fluxes of the best fitting model, given by
Z ∞
−2.5 log
Fλ Sλ dλ,
(1.3)
0
with corresponding absolute visual magnitudes MV . In Eq. (1.3), Fλ is the theoretical stellar
flux of the best fitting model, and Sλ is the V -filter function of Matthews & Sandage (1963).
Since the fluxes Vtheo are only known a posteriori, initially, we have to adopt a “typical”
value for the radius, as calculated from evolutionary models, and update it after the model
calculation. This procedure (including corresponding updates of Ṁ , see Section 1.5.6) is
repeated until the difference in input and output radius becomes negligible.
20
The fundamental parameters of stars
Subsequently also the spectroscopic mass M∗ and luminosity L∗ can be calculated from the
radius, the gravity, and the effective temperature, respectively, as follows:
M∗ = g R∗ 2 /G,
(1.4)
L∗ = 4 π σ R∗ 2 Teff 4 ,
(1.5)
with G the gravitational constant and σ the Stefan Boltzmann constant.
Another way to proceed, would be to calculate the stellar luminosity from the bolometric
correction (B.C.) and the absolute visual magnitude, as follows:
log L/L⊙ = 1.9 − 0.4 (MV + B.C.),
(1.6)
where B.C. can be calculated from calibrations, such as the one by Martins et al. (2005) for
O-type stars, or by Crowther et al. (2006) for B-type stars. Both calibrations nicely connect
in the late O/early B-type regime. The radius can then be determined from Eq. (1.5).
1.5.6 Wind parameters
Once we have fixed the stellar properties, we are ready to determine also the wind characteristics. To obtain the best possible information about these wind properties, we need a spectral
line which is sufficiently affected by the stellar wind. To determine the terminal wind velocity v∞ , ideally, one would want to use UV observations, employing the typical P-Cygni
profile shape (i.e., a profile which is the composite of a blue-shifted absorption trough and
an almost symmetric emission peak) of the UV resonance lines in hot stars. Unfortunately,
P-Cygni lines are only present in the earlier subtypes (if at all). For cool and dense winds, the
optical Hα line can also give a clue about v∞ (cf. Section 1.4.2). For thin winds, however,
Hα is fully in absorption, which makes a reliable estimate for v∞ impossible. Therefore, as a
first estimate, we adopt typical values for v∞ from empirical UV studies, such as the values
determined by Prinja & Massa (1998), with an extrapolation towards later B-types by using
the corresponding A supergiant data provided by Lamers et al. (1995). Both datasets have
been collected in Table 1 of Kudritzki & Puls (2000).
We can also exploit Hα to determine the other two wind characteristics: the velocity-field
exponent β and the wind strength parameter, log Q, with log Q ≡ log Ṁ /(v∞ R∗ )1.5 (Puls
et al. 1996). Indeed, the wind strength parameter log Q, rather than the mass-loss rate Ṁ
itself, will be the real fit parameter. The reason for this is that different combinations of Ṁ ,
v∞ and R∗ , which lead to the same log Q, give rise to almost identical Hα profiles (Puls et al.
1996), and at the same time, all other lines remain (almost) conserved (Puls et al. 2005). The
final mass-loss rate itself is then only determined after v∞ and R∗ are fixed.
Since the adopted radius (the input of the model) can be different from the actual radius (output) as determined from Eq. (1.2), we need to scale the mass-loss rate accordingly. Thanks to
the unique property of log Q (namely almost invariant line profiles for invariant log Q values),
21
1.5 Methodology and fit parameters
Figure 1.10: Effect of the wind parameters log Q (left) and β (right) on the shape of the
resulting (P Cygni) profile of Hα. Full: reference model at log Q = -13.40 and β = 1.2 (see
p. 14) ; dashed and dotted lines at log Q = -13.15 and -14.30 (left) and β = 1.5 and 0.9 (right),
respectively.
this is straightforward:
Ṁactual =
R∗ 1.5
actual Ṁadopted
.
R∗ 1.5
adopted
In Fig.1.10, we show the influence of the wind parameters log Q and β on the shape of the
resulting (P Cygni) profile.
Clumping in hot stellar winds
The mass-loss rates, as derived from the Hα profile, may need to be scaled down, by factors
between ∼2 and 10, due to clumping10. Clumping is the presence of small-scale (outward
moving) inhomogeneities in the stellar wind, which redistribute matter into overdense clumps
and a rarified inter-clump medium. Whereas most spectroscopic analyses of hot stars, aiming
at deriving stellar and wind parameters, rely on the assumption of a globally stationary wind
with a smooth density and velocity stratification (ρ(r) and v(r), respectively), clumping introduces depth-dependent deviations from a smooth density structure and a highly perturbed
velocity field. This implies that any ρ2 -dependent diagnostic, among which also Hα (our
primary mass loss indicator), may be affected. To account for this, the so-called ‘clumping
factor’, fcl , has been introduced in the predictions of stellar winds, with
fcl ≡
< ρ2 >
≥ 1,
< ρ >2
and ρ the density. The most important effect of clumping concerns the actual mass-loss
rate. Compared to diagnostics
√ assuming a smooth wind, the actual mass-loss rate reduces,
Ṁ (actual) ≈ Ṁ (smooth)/ fcl .
10 This quantity
is heavily debated at present, but an average clumping factor of 3 is likely, at least for O-type stars.
22
The fundamental parameters of stars
Puls et al. (2006) point out a physical difference in the clumping properties of weaker and
stronger O-star winds. Whereas the clumping factors for weaker winds remain more or less
constant throughout the wind, the situation is different for denser winds. Considerable clumping may already be present very close to the star, remaining constant over a large volume, and
then decreasing again in the outer wind. Clumping factors in the inner wind (< 2R∗ ) may be
a factor of 3 to 6 larger than in the outer wind (> 10R∗ ).
Since it is not expected that clumping extends below the sonic point, the densities in the photospheric line forming region should be correct, and thus the influence of clumping on the
photospheric parameters should be negligible. In this sense, we can safely say that the inclusion of clumping will hardly make any difference for the results presented in this work. The
majority of the studied targets concern thin wind objects (cf. Chapter 4), where we concentrated on the photospheric rather than on the wind parameters. Only for the studies presented
in Chapter 2, which often include denser wind objects, the effect of clumping might play an
important role, particularly with respect to the mass-loss rates and the WLR. However, we
only expect a qualitative change of our results if clumping is a strong function of spectral
type. So far, however, such a dependency remains to be proven or rejected.
1.6 Stellar oscillations in massive stars
In what follows, we give a short introduction into stellar oscillations. The text is based on
Ausseloos (2005) and lecture notes written by C. Aerts.
Probing stellar interiors
Oscillations are excited inside almost all types of stars during many stages of stellar evolution.
In Fig. 1.11, we show the location of the main classes of oscillating stars in a theoretical
HRD. Two of these classes will play an important role in this thesis: the β Cephei stars and
the Slowly Pulsating B stars (cf. Chapters 2 and 4). Asteroseismology is the field within
astronomy which is dedicated to the study of these oscillations. Through the interpretation of
the oscillation frequency spectra, asteroseimologists try to probe the stellar interior.
Radial oscillations are the simplest oscillations a star can undergo: the star contracts and
expands radially, while preserving spherical symmetry. If the star oscillates in the radial
fundamental mode (i.e. the one with the lowest acoustic eigenfrequency) all mass elements
move simultaneously in the same direction, either up or down. For the radial first overtone,
there is a node in the eigenfunction between the centre and the surface with the mass elements
moving in opposite directions on either side of the node. In general, an oscillation with
radial order n implies n nodes between the centre and the surface of the star (n-th overtone
oscillation). The period of a radial mode is inversely proportional to the square root of the
mean density of the star.
1.6 Stellar oscillations in massive stars
23
Figure 1.11: Location of the main classes of pulsating stars in a theoretical HertzsprungRussell diagram. These classes comprise the β Cephei stars (β Cep), the Slowly Pulsating B
stars (SPB), the δ Scuti stars (δ Sct), the rapidly oscillating Ap stars (roAp), the γ Doradus
stars (γ Dor), the solar-like oscillators, the RR Lyrae stars (RR Lyr), the semi-regulars (SR),
the Mira variables, the subdwarf B variables (sdBV), the variable DB (helium-rich) and DA
(hydrogen-rich) white dwarfs (DBV and DAV respectively), the GW Virginis stars (including
the stars formerly known as PNNV -Planetary Nebulae Nuclei Variables- and DOV -variable
hot DO white dwarfs-) and the Cepheids (Ceph) in the classical instability strip, indicated by
the parallel long-dashed lines. Figure courtesy: Jørgen Christensen-Dalsgaard.
24
The fundamental parameters of stars
A lot of stars do not only oscillate radially, but also transversally (i.e. horizontally). Indeed,
many stars execute a motion in which some regions of the surface expand while others contract. These oscillations are called non-radial oscillations. They are characterised by three
quantum numbers: the radial order n, the degree l and the azimuthal number m. The degree
l represents the number of nodal lines of the eigenfunction on the surface. During the oscillation cycle, mass elements on these lines do not move. The azimuthal number |m| gives the
number of these nodal lines that pass through the rotation axis of the star.
We can divide the oscillations in two different groups on the basis of the dominant restoring force of the oscillations: pressure or p-modes and gravity or g-mode oscillations. The
p-modes are basically acoustic oscillations for which the dominant restoring force is the pressure force. The mass elements oscillate dominantly radially. They exhibit short periods (high
frequencies), of the order of a few hours for main sequence B stars, and usually attain a large
amplitude only in the outer layers of the star. The g-modes have the gravity as the dominant
restoring force and exhibit longer periods (lower frequencies), of the order of a few days for
main sequence B stars. From an asteroseismological point of view, these are the most interesting, since they obtain large amplitudes in the deeper layers of the star. In this respect,
B-type stars are a rewarding temperature region to study, since they can pulsate in p-modes
(e.g., the β Cephei stars) as well as in g-modes (e.g., the Slowly Pulsating B stars or SPBs).
Studying oscillations can help to build up knowledge of what happens in the interiors of stars,
since stars have large convective and rotational motions in their core, which lead to mixing
of chemical species. As long as we do not understand the details of these internal processes,
and along with this the precise extent of the convective core, we cannot accurately predict
their stellar evolution.
Excitation mechanism
For a pulsation mode to be observable, a small perturbation of the stellar equilibrium should
be able to grow in time, i.e. there should be an excitation mechanism acting in the star. The
oscillations in many stars can be explained by the so-called κ-mechanism, which is described
in the remainder of this section.
When the layers of a star are compressed, the density and the temperature increase. Since
the opacity is proportional to the density and inversely proportional to the temperature, it
depends on the conditions whether the opacity will increase or decrease. Since the opacity
is more sensitive to the temperature than to the density, it usually decreases under compression. However, in the partial ionisation zones, part of the work done on the gas produces
further ionisation rather than raising the temperature of the gas. This implies that the density
increase dominates over the temperature increase, resulting in an increase in opacity rather
than a decrease. Therefore, the stellar layer becomes opaque, blocks the energy radiated outwards by the stellar core, and pushes the surface layers further upwards. As the expanding
layer becomes more transparent, the trapped heat will be released and the layer falls down to
restart a new cycle. Similarly, during expansion, the ions in the partial ionisation zone will
1.7 Summary of the major results obtained in this thesis
25
recombine with electrons, rather than decreasing the temperature, and therefore, the opacity
will decrease with decreasing density. For the κ mechanism to work and to make the whole
star oscillate, the partial ionisation zone has to lie at a position where certain conditions are
fulfilled concerning the temperature, luminosity and chemical composition. Therefore, radial
and non-radial oscillations are usually excited in instability strips in the HRD.
1.7 Summary of the major results obtained in this thesis
From what is this described in this first chapter, it should be clear that an understanding of
massive stars is fundamental for astrophysical progress, where a first step into this direction
is to obtain accurate estimates of their stellar and wind parameters. In the following, we will
apply the line profile fitting procedure as introduced in Section 1.5 to determine the basic
physical properties for B-type stars in different phases of their evolution.
Chapter 2 presents a study of a sample of B-type supergiants showing (multi)periodic variability. We have used a thorough fit-by-eye procedure to (re)position these stars in the HRD.
All the sample stars for which we could derive reliable stellar parameters have effective temperatures and gravities consistent with those of stellar models in which gravity modes are
predicted, as they all lie at the high-gravity edge of the (non-radial) high-order gravity-mode
(SPB-type) instability domain. On the basis of their position in the HRD and their multiperiodic behaviour, we propose the variability in our sample of periodically variable B-type
supergiants to be opacity-driven, in analogy to the κ mechanism acting in main sequence
B-type pulsators.
As a part of the preparation of the CoRoT space mission, we were assigned the task to analyse
a sample of some 190 B-type stars. One of the most important stumbling blocks hampering
the analysis of such a large sample of stars, is the lack of an automatic procedure to analyse
mid- and late B-type stars11 , which constitute the majority of the sample. To this end, we
have developed AnalyseBstar, the first automatic procedure to perform line profile fitting for
stars covering the complete B-type spectral range, based on a static, but representative and
dense grid of FASTWIND NLTE atmosphere models. The extensive grid is fully described
in Chapter 3, while the automated procedure is presented in detail in Chapter 4. This method
allows for a fast and objective determination of accurate fundamental parameters for large
samples of B-type stars. To rigorously test the fitting procedure, we first performed formal
tests of convergence on synthetic data, after which we subjected it to a sample of well known
β Cephei and SPB type stars. Our results agree very well with alternative analyses using
different methods, for objects in common. Finally, we applied the method to a subsample
of good quality data within the CoRoT sample. We have used the results for this subsample
to show that the currently existing Geneva and Strömgren calibrations for log g in terms of
photometric colour indices are insufficient to provide us with accurate gravity estimations.
11 For the early B-type
stars, the genetic algorithm developed by Mokiem et al. (2005) may be applied alternatively.
26
The fundamental parameters of stars
Chapter
2
Statistical properties of a sample of
periodically variable B-type
supergiants
Evidence for opacity-driven gravity-mode oscillations
The original version of this chapter was published as Lefever, K., Puls, J. and Aerts, C.,
Astronomy & Astrophysics, 463, 1093-1109 (2007)
In this chapter we present the analysis of our first sample of stars: 40 B-type supergiants
showing microvariations with periods of the order of a few days. This variability seems intermediate between the one of the slowly pulsating B stars near the main sequence and the one
of the class of variable stars called ‘α Cyg type variables’. There are, by now, quite some
members of these two classes. The cause of the periodic variations of our sample of supergiants is still unclear. A prerequisite for a better understanding of the variability mechanism
is a reliable and accurate position of the supergiants in the (log Teff , log g)-diagram. This
enables a comparison with variables for which the cause of variation is known, and with theoretically predicted instability domains. Unfortunately, at the time of the discovery of these
40 stars, only photometric data from the HIgh Precision PARallax COllecting Satellite (HIPPARCOS) were available. Moreover, NLTE atmosphere and wind predictions for massive
28
Periodically variable B supergiants
stars (and especially for B-type stars) were still in a rather early phase of development by the
time that the first spectra of our sample stars were assembled. Thus, one was forced to do the
first analysis using photometric methods, knowing that, especially for log g, this approach
was not accurate (Waelkens et al. 1998). In the last decade, however, a lot of improvement
in the model atmosphere predictions has been achieved. The ‘birth’ of the FASTWIND code,
and the gathering of high-resolution spectra for the whole sample, enabled us to spectroscopically reanalyse the sample in a more profound way and to derive the atmospheric and wind
parameters of the complete sample through the line profile fitting of selected H, He, and Si
line profiles. Our ultimate goal in this chapter is to unravel if the variability of our sample
stars is compatible with opacity-driven non-radial oscillations.
2.1 Introduction
2.1.1 HIPPARCOS: the discovery of new variables
One of the most successful space missions of the European Space Agency (ESA) was HIPPARCOS, launched in 1989. Its prime goal was to measure the distances of 120 000 stars in
the solar neighbourhood with unprecedented precision of 2 milli-arcsecond for the parallax.
This was achieved by measuring each star on average 100 times during the 3.3 years lifetime
of the mission. These time series are very different from those obtained with ground-based
instruments, in the sense that the stars were not at all selected as targets on the basis of their
variability, i.e. the input catalogue was completely unbiased. In this way, the HIPPARCOS
mission has led to the discovery of thousands of new variable stars, among which numerous
ones with periods of the order of days. Such variables are very hard to find from groundbased data since these suffer severely from one-day aliasing1 . Among these new variables,
29 periodically variable B-type supergiants were discovered (Waelkens et al. 1998).
2.1.2 Variability of massive stars
The photometric behaviour of different kinds of evolved massive stars were analysed in detail from HIPPARCOS data. Van Leeuwen et al. (1998) performed an analysis of 24 known
B- to G-type variable supergiants and found periods of tens of days for each of them, in
agreement with previous ground-based studies. Marchenko et al. (1998), on the other hand,
focused on the HIPPARCOS variability of 141 O-type and Wolf-Rayet stars and noted the
remarkable variety of variability, with very diverse causes, within that sample. The study of
Waelkens et al. (1998) is quite different in this respect, as they came up with a sample of 29
new variable B supergiants exhibiting clear periodic variations with amplitudes typically between 1 and a few tens of millimagnitude at relatively short periods of one to a few days. This
1 Certain periodicities in the observation times give rise to ‘false’ alias frequencies in the observed frequency
spectrum. More particular, the one-day alias is introduced due to the fact that one can only observe during the night.
2.1 Introduction
29
chapter contains a follow-up study of the latter sample. Our goal is to evaluate the suggestion by Waelkens et al. (1998) that these periodically variable B-type supergiants experience
oscillations excited by the opacity mechanism, in analogy to main sequence B stars.
The suggestion of the occurrence of non-radial oscillation modes in massive supergiants is
not new. Microvariations with amplitudes between a hundredth and a tenth of a magnitude
in the visual, and periods ranging from some 5 to 100 d, have been found in numerous supergiants of spectral type OBA, termed α Cyg variables (Sterken 1977, 1983; Burki et al. 1978;
van Genderen et al. 1989; Lamers et al. 1998; van Genderen 2001). Burki (1978) considered
a sample of 32 B- to G-type supergiants and derived an empirical semi-period-luminositycolour (PLC) relation (see his Eq. 5), from which he suggested the variability to be caused by
oscillations rather than mass loss. Lovy et al. (1984) indeed managed to come up with a theoretical PLC relation in agreement with the observed one for this type of stars. However, only
40% of the variable supergiants have periods compatible with the radial fundamental mode,
while the majority must exhibit a different kind of oscillation mode. Kaufer et al. (1997)
made an extensive high-resolution spectroscopic study of 6 BA-type supergiants, which they
monitored for several years. They concluded that the variability patterns are extremely complex and point towards cyclic variations in the radial velocities. From CLEANing (Roberts
et al. 1987) the periodograms of the radial-velocity curves, they derived multiple periods
and assigned them to non-radial oscillations because the travelling features in the dynamical
spectra turned out to be incompatible with the rotational periods of the stars.
Glatzel & Kiriakidis (1993) interpreted the periodic variability of supergiants with masses
above 40 M⊙ in terms of strange-mode instabilities and showed the classical opacity mechanism to be unimportant in such objects. Such strange modes are caused by a strong enhancement in the opacity in the second partial ionisation layer of helium (i.e. in the zone in which
both He II and He III occur) and of the heavy iron-group elements (mainly Fe). The radial
modes that are predicted have amplitudes that are much larger than those found for the classical radial oscillators. However, detailed modelling of the observations has not yet been done
in terms of strange modes. Glatzel et al. (1999) subsequently performed detailed calculations
for stars with M = 64 M⊙ and predicted irregular velocity and luminosity variations with
time scales of 1 to 20 d. They also proposed that such massive stars undergo pulsationallydriven mass loss. It is not clear how their result will change for stars in the mass range of 10
to 30 M⊙, which is the transition region from low to large mass-loss rates due to line driving.
As the kappa mechanism is so successful in explaining the variability of many types of stars,
the periodic variations of the B supergiants found by Waelkens et al. (1998) might therefore
still be due to the classical opacity mechanism, since the instability strips of the β Cephei
stars and the Slowly Pulsating B stars (SPBs) were found to extend to higher luminosities
shortly after the discovery paper (Pamyatnykh 1999; Balona & Dziembowski 1999).
Waelkens et al. (1998) positioned the new periodic B supergiants in the HRD on the basis
of multicolour photometric calibrations (accurate parallaxes were not available) and found
them to be situated between the SPBs and previously known α Cyg-type variables (see their
Fig. 2). Oscillations were not predicted in that part of the HRD. Given the uncertainty in the
effective temperature and luminosity, Aerts (2000a) tried to improve upon the fundamental
parameter determination by constructing the spectral energy distribution of the stars as a
30
Periodically variable B supergiants
better diagnostic to derive their effective temperature and gravity. This did not improve the
large uncertainties on the latter parameters, however. Nevertheless, the sample selected by
Waelkens et al. (1998) remained the most valuable one to observationally investigate the
occurrence of gravity modes in supergiant stars, because it is unbiased and not selected to
be observed with HIPPARCOS on the basis of variability. For this reason, we conducted an
extensive spectroscopic campaign for the sample, with the goal to perform line-profile fitting
as a way to estimate the fundamental parameters of the stars with unprecedented precision.
In this chapter we report upon the analysis of these data, and the position of the stars with
respect to the instability strip of gravity modes.
The questions we will address are the following. We elaborate further on the HIPPARCOS
data to search for multiperiodic variability, which is a natural property of non-radial oscillators. From selected H, He, and Si line profiles, we derive the physical parameters (effective
temperature, gravity, luminosity, mass loss, rotational velocity, etc.) of the stars in the sample
by Waelkens et al. (1998) using high-quality spectroscopic data. From this, we derive their
position in the HRD with high precision and check if the stars lie within the instability strips
of gravity mode oscillations predicted by Pamyatnykh (1999) and Saio et al. (2006). Further,
we search for correlations between the physical parameters and the photometric variability.
In particular, we investigate if there is any connection between the observed peak-to-peak
amplitude and frequency of the light variability, the wind density, and the rotation of the
stars. Finally, we investigate the wind momentum-luminosity relation (WLR) for the complete B-type spectral range.
2.2 The sample: observations and data reduction
We selected all southern stars of luminosity class I or II brighter than 9th magnitude in the
sample by Waelkens et al. (1998), which fitted the observation window of the assigned telescope time. This concerns 21 stars. To this we added 10 more B-type supergiant variables
from the Catalogue of Periodic Variables of HIPPARCOS (ESA 1997), in such way that the
complete sample fully covers the B-type spectral range. Waelkens et al. (1998) were unable
to assign these 10 objects to one of the five considered classes (i.e. the β Cephei stars, SPBs,
chemically peculiar stars, B-type supergiants, Be stars), see also Kestens (1998), although
our period analysis and the spectral types clearly point towards multiperiodic variability. The
31 targets (for spectral types, see Table 2.7 and the discussion in Section 2.6.1) were added to
the long-term spectroscopic monitoring programme of periodic B stars conducted at Leuven
University (Aerts et al. 1999). The spectra of the stars were gathered with the high resolution
CES2 spectrograph attached to the ESO CAT3 telescope at La Silla (Chile) during numerous
observing runs spread over two years. For most targets, we obtained two exposures of the Hα
line in different seasons (to check for its variability), one of the Hγ line, one of the He I 6678Å
line, and one of the He I 4471Å line. Besides these, we observed one silicon line for each star
with the goal of obtaining an accurate temperature estimate. Depending on spectral type, this
2 Coudé
3 Coudé
Echelle Spectrograph
Auxiliary Telescope
2.3 Photometric variability
31
is either the Si II 4130Å doublet (late B-type) or the Si III 4560Å triplet (early B-type). It is
noteworthy that this assembly of spectral lines is not optimal for a high-precision determination of the stellar parameters, as explained in Chapter 1 and elaborated on in Chapter 4. This
is due to our choice to observe all stars at the same level within the allocated observing time.
For this reason, the analysis of the sample stars required a non-standard approach, unlike the
method we will adopt in Chapter 4.
The spectra were reduced in the standard way, i.e., by means of flatfielding, wavelength calibration through Th-Ar exposures, and rectification of the continuum through a cubic spline
fit. The resolution is 70,000 and the exposure times ranged from 3 to 50 minutes, resulting in high signal-to-noise ratios of about 250. It became immediately evident that the three
stars HD 71913, HD 157485, and HD 165812 were misclassified in the Bright Star Catalogue
(BSC) as supergiants. They turned out to be new β Cephei stars. These have been studied in
detail by Aerts (2000b) and are not included here. This finally led to 28 sample stars.
To assess the importance of having periodic light variability in our sample, we have additionally selected 12 bright B supergiants from the BSC, again chosen to cover the complete
B-type spectral range. These variables were not classified as periodic variable by the HIPPARCOS classification algorithm. While for some of these bright objects stellar parameters
are available in the literature, we have invested in collecting their spectra as well, to treat
these stars in the same way as the sample stars.
2.3 Photometric variability
The HIPPARCOS data of the 40 targets (28 sample stars and 12 comparison stars) were
subjected to detailed period search by means of the Scargle (1982) and Phase Dispersion
Minimisation (Stellingwerf 1978) methods. In Fig. 2.1, we show the phase diagrams for
the dominant frequency for two representative cases. The detailed results of the frequency
analyses are provided in Table 2.1. For most targets, we recovered the main period found
by Waelkens et al. (1998), but not for all of them. For six stars, the first harmonic of the
main frequency was also needed to obtain an accurate fit to the HIPPARCOS data (see e.g.,
Fig. 2.1 and Col. ’H’ of Table 2.1). We found evidence for multiperiodicity for eleven stars.
The detected periods range from 1.15 to 25 d, with only four stars having a period longer than
10 d. Thus we confirm that most of the 28 sample stars have periods that are significantly
shorter than the ones of α Cyg variables. We also find short periods (less than 10 days) for
some comparison stars, though they were not classified as periodic by the HIPPARCOS team.
Seven out of twelve comparison stars have considerably lower peak-to-peak variations than
most sample stars, whereas the other five seem to have periods and amplitudes comparable to
the ones detected in our target sample.
32
Periodically variable B supergiants
Table 2.1: Results of the period analyses for the 28 sample stars (upper part) and for the 12 comparison
stars (lower part). The periods Pi are expressed in days and have an accuracy better than 0.001 d. Note
that P2 is derived from the residuals, after prewhitening with P1 and that the total variance reduction
(column ‘v.r.’, expressed in %) is obtained from a harmonic fit to the data with both periods. When
the frequency’s first harmonic is present in the lightcurve, the label ‘yes’ occurs in column ‘H’. The
differences between the largest and the smallest observed magnitude are indicated by the observed
peak-to-peak values ∆HP,obs and are given in mmag, just like the amplitudes Ai and their 1σ error,
σAi .
HD
P1
A1 ± σA1
H
v.r.
47240
51110
53138
54764
68161
80558
89767
91024
91943
92964
93619
94367
94909
96880
98410
102997
105056
106343
108659
109867
111990
115363
141318
147670
148688
154043
168183
170938
1.730
2.315
24.390
2.695
16.949
1.695
1.153
2.398
6.452
14.706
4.310
7.937
16.949
2.475
1.453
2.688
2.899
3.650
5.076
4.484
2.890
3.077
1.466
5.435
6.329
2.874
2.105
5.618
29 ± 4
61 ± 6
45 ± 5
17 ± 4
25 ± 2
38 ± 6
28 ± 4
36 ± 6
24 ± 4
43 ± 5
27 ± 6
48 ± 5
37 ± 6
46 ± 6
97 ± 8
29 ± 6
41 ± 8
27 ± 6
22 ± 4
32 ± 3
30 ± 4
38 ± 8
16 ± 2
66 ± 4
46 ± 6
30 ± 7
60 ± 5
94 ± 18
no
no
no
no
no
no
no
no
no
no
no
yes
no
yes
yes
yes
no
yes
no
no
no
no
no
no
no
no
no
yes
60%
53%
49%
33%
68%
42%
34%
42%
40%
45%
24%
58%
36%
47%
53%
33%
36%
38%
30%
46%
26%
22%
35%
68%
61%
34%
49%
61%
46769
58350
64760
75149
86440
106068
111904
125288
149038
157038
157246
165024
0.1122
6.6313
2.8090
1.2151
6.1996
4.2644
3.3389
8.0906
0.6390
3.6430
1.1811
2.7693
9±2
47 ± 6
11 ± 2
33 ± 5
9±2
42 ± 5
30 ± 4
9±2
19 ± 3
48 ± 7
10 ± 2
6±1
no
no
no
no
yes
no
no
no
no
no
no
no
40%
39%
34%
43%
51%
46%
38%
21%
61%
68%
40%
28%
P2
A2 ± σA2
H
v.r.
Total v.r.
3.690
35 ± 4
no
45%
(73%)
5.814
26 ± 5
no
25%
(57%)
2.119
36 ± 5
no
43%
(71%)
4.329
1.256
27 ± 5
25 ± 4
no
no
34%
28%
(73%)
(54%)
8.696
2.976
7.299
3.906
40 ± 8
23 ± 4
51 ± 8
23 ± 4
yes
yes
yes
no
45%
37%
46%
31%
(75%)
(58%)
(67%)
(58%)
4.785
20 ± 3
no
30%
(63%)
1.845
24 ± 4
no
42%
(79%)
1.8447
2.2143
0.2371
25.1889
19.1205
9±2
20 ± 4
5±2
24 ± 4
19 ± 3
no
no
no
no
no
33%
34%
22%
31%
32%
(56%)
(63%)
(62%)
(64%)
(58%)
1.5432
0.1281
0.8455
27 ± 3
9±2
6±1
no
no
no
50%
40%
24%
(84%)
(64%)
(46%)
△HP,obs
70
150
100
50
50
90
100
100
70
110
100
100
110
120
230
110
130
110
80
80
120
120
50
110
90
80
150
110
22
145
28
100
36
110
110
38
37
100
20
24
2.4 Line profile fitting: fit-by-eye procedure
33
Figure 2.1: The HIPPARCOS lightcurve folded according to the dominant frequency for the
stars HD 96880 (B1 Ia) and HD 98410 (B2.5 Ib/II). The dots are the observations and the full
line is a least-squares fit for the indicated frequency and its first harmonic.
2.4 Line profile fitting: fit-by-eye procedure
2.4.1 Determination of the rotational velocity
The projected rotational velocity, v sin i , was found from the automated tool developed by
Simón-Dı́az et al. (2006), which is based on a Fourier method to separate the effects of rotational broadening and macroturbulence (which can be significant in B-type supergiants, cf.
Ryans et al. 2002). This method was first described by Gray (1973) and reviewed in Gray
(1978). It is discussed in detail more recently by Piters et al. 1996 (see also item 6 in Section 4.4.1). Weak metallic lines are the best diagnostic to derive v sin i , since they are free
from saturation effects and least affected by collisional broadening. We have the following
lines at our disposal: either Si II 4128-4130Å or Si III 4552-4567-4574Å (depending on spectral type), Mg II 4481Å (in the same spectral order as He I 4471Å) and, for slow rotators,
also C II 6578-6582Å near Hα. Besides these primary diagnostics, also other, even weaker
metallic lines can be used. Table 2.2 lists our results for v sin i , its standard deviation and
the number of lines used. For the 12 comparison objects, not all selected orders were measured, and hence only a few lines could be used. For five stars appropriate metallic lines
were lacking. Three out of five (HD 64760, HD 157246, and HD 165024) have blended metal
lines due to their fast rotation. In this case we adopted a mean value for v sin i as provided
by SIMBAD. The other two objects are HD 51110 and HD 147670. For HD 51110, the He I
lines were too weak and peculiar, whereas for HD 147670 only Hα had been secured. No
value is given in SIMBAD either. The occurrence of asymmetries in the line profiles may reveal the presence of time-dependent pulsational broadening (see, e.g., Aerts & De Cat 2003).
In the current sample, clear asymmetries in the Si lines were detected only for HD 54764,
HD 92964, and HD 109867 . Most probably, they are related to binarity (HD 54764) or to a
large mass-loss rate (HD 92964), which both affect the photospheric lines in an asymmetric
way. Only for HD 109867, we can speculate about a relation between line asymmetry and
pulsational variability.
34
Periodically variable B supergiants
Table 2.2: Projected rotational velocities, v sin i , and their standard deviation (in km/s) for
all 40 sample stars, determined via the Fourier transform of metallic lines. When no metallic
lines or only blended lines are available, the corresponding values from SIMBAD are given
in italics. Col. “n.o.l.” gives the number of lines used to determine the mean value of v sin i
and its standard deviation.
HD v sin i s.d. n.o.l.
HD v sin i s.d. n.o.l.
46769
68
5
2 102997
39
8
13
47240
94
9
6 105056
61 21
10
51110
n/a
106068
26
5
6
53138
38
4
14 106343
44
7
14
54764
108 15
10 108659
29
5
12
50 14
16
58350
37
5
4 109867
64760
220
111904
32
9
4
68161
17
2
20 111990
36
6
6
55 12
8
75149
30
8
4 115363
25
4
4
80558
28
5
17 125288
32
3
8
86440
20
6
5 141318
89767
47
6
12 147670
n/a
91024
25
6
23 148688
50 11
13
57
4
2
91943
48
7
12 149038
92964
31
6
17 154043
37
9
8
93619
47 13
11 157038
41
5
4
94367
31
4
15 157246
302
94909
64 10
11 165024
95
124 16
7
96880
44
9
14 168183
98410
31
3
10 170938
51
6
10
2.4.2 Determination of physical parameters
To investigate the position of our sample stars in the HRD on a more solid basis, we determine the fundamental parameters of the stars from the gathered high-resolution spectroscopic
follow-up data. For this purpose, we use the non-LTE, spherically symmetric model atmosphere code FASTWIND, which enables us to derive the atmospheric and wind parameters
(as discussed in Chapter 1).
As stated earlier, for most stars two hydrogen lines (Hα and Hγ), two He I lines (the triplet
line 4471Å and the singlet line 6678Å), and one silicon multiplet (Si III 4552-4567-4574 for
the early types, up to B2, and Si II 4128-4130 for the later spectral types) have been observed.
The choice to measure these specific lines has not been made randomly, but is based on their
well-known specific dependency on one or more of the basic parameters. The model atom for
silicon used in this investigation is the same as the one used and described by Trundle et al.
(2004) in their analysis of SMC B supergiants.
2.4 Line profile fitting: fit-by-eye procedure
35
Figure 2.2: Isocontour levels of equivalent line width (expressed in Å), assuming solar Si
abundance and negligible mass loss. Left: Si II 4128 - a good temperature indicator up to
20,000 K. Right: Si III 4552 - shows a maximum in equivalent width around 23,000K. From
this maximum, theoretical line profiles can behave similarly towards lower and higher temperatures, causing a dichotomy in the determination of the effective temperature, when Si II
or Si IV are lacking.
Figure 2.3: Isocontour levels of equivalent line width (expressed in Å) for He I 4471
(n(He)/n(H) = 0.1, negligible mass loss). In the cool B-type regime, He I 4471 is a perfect temperature indicator as the isocontours are almost vertical. At higher temperatures, this
line changes into a good diagnostic for the gravity.
In Figs 2.2 and 2.3, we show the isocontour levels of equivalent line width of Si II 4128,
Si III 4552, and He I 4471 and their dependence on the effective temperature and surface
gravity based on an extensive grid of synthetic supergiant models (see below). Note that this
is a different grid than the one described in Chapter 3. These figures show that Si II is a very
good temperature indicator for B-type stars with an effective temperature below 20,000 K.
From then on, Si III takes over as a temperature diagnostic (Fig. 2.2). Used in parallel, both
silicon multiplets could be used to infer information about the silicon abundance. Since,
however, we always have only one of the two at our disposal, we adopted a solar silicon
abundance for our study (log (Si/H) = -4.45 by number (cf. Grevesse & Sauval (1998) and
36
Periodically variable B supergiants
references therein), which has changed only marginally to log (Si/H) = -4.49 in the recent
update of the solar composition (Asplund et al. 2005).
The silicon abundance in B stars is heavily disputed. Depending on the sample and on the
method used, values range from roughly solar (Gies & Lambert 1992; Gummersbach et al.
1998; Rolleston et al. 2000) to a depletion by typically 0.3 dex (Kilian 1992; Kilian et al.
1994; McErlean et al. 1999; Daflon & Cunha 2004), in both cases with variations by ± 0.2
dex. Analyses of individual objects by Urbaneja (2004) and Przybilla et al. (2006) indicate
a rather large scatter, again in the same range of values. In view of this uncertainty, and the
fact that Crowther et al. (2006) in their analysis of galactic B supergiants found no obvious
problems in using solar values, we also adopted this value. We will report on the influence of
this assumption on the final outcome later on.
He I 4471 serves several purposes: for early B-types it is a good gravity indicator (with a
certain sensitivity to temperature as well), whereas for the later B-types (Teff < 15,000 K) it
becomes progressively independent of gravity, but can be used to constrain the temperatures
perfectly (Fig. 2.3). In those cases where the effective temperatures (from Si) and the gravity
(from Hγ, see below) are well defined, both He I lines (He I 4471 and He I 6678) are useful
to constrain the helium content, as well as to check for the overall consistency of the results,
which in any case is the primary purpose of the second He I (singlet) line. The recent debate
on the difficulty of using He I singlet lines as diagnostic tools, due to subtle line overlap
effects of the He I resonance line in the far UV (Najarro et al. 2006), is (almost) irrelevant in
the present context, since it concerns only spectral types B0 and hotter.
Of course, He I 4471 is not our primary gravity indicator. As usual, we rely on the Balmer
line wings for this purpose, in our case, particularly the wings of Hγ. Since the Hα line
is formed further out in the atmosphere, it is affected by the stellar wind, and, for appreciable wind densities, displays the typical emission or P Cygni type profile. Depending on
the mass-loss rate, Ṁ , the velocity law exponent, β, and the terminal wind velocity, v∞ ,
the profile will have a more pronounced emission profile, a steeper red wing or a broader
absorption. Note that for lower wind densities, only the core of Hα is refilled by wind emission, and the errors regarding the derived mass-loss rates become larger. We used a “by
eye” line profile fitting procedure to find the best-fitting synthetic FASTWIND spectrum for
the observed line profiles of each object, to derive their fundamental parameters. The synthetic profiles are convolved with a rotational profile with appropriate v sin i (cf. Table 2.2),
whereas the macroturbulence vmacro (well visible in the wings of the Si-lines) is considered
as an additional free parameter, determined in parallel with the fit and accounting for a typical
radial-tangential distribution (Gray 1978).
In a first step, we derive coarse parameters by using an extensive grid of synthetic models,
with effective temperatures ranging from 10,000 K up to 30,000 K (taking steps of 1,000 K)
and appropriate gravities with steps of 0.1 dex in logg (systematically shifting the gravity
range to higher values for higher effective temperatures). We consider the grid steps in Teff
and log g as a rough estimate (see Section 2.8) for their uncertainty. For each Teff /log g grid
point, six different values (equidistant in log g, with appropriate boundaries) for the wind
strength parameter Q have been calculated. As a first guess, we adopted a “typical” radius
2.5 Analysis of the 28 sample stars
37
for each grid point, as explained in Section 1.5.5, and used the observed terminal velocities
of massive hot supergiants from Prinja & Massa (1998) to initialise v∞ . In combination
with the predescribed Q-values, this led to a wide spread in mass-loss rates. As mentioned
above, all models were calculated for the “old” solar silicon abundance. We also considered
solar values for the helium content (n(He)/n(H) = 0.10). However, these values were adapted
whenever required. Finally, all profiles have been calculated for three different values of the
microturbulent velocity, ξ, namely 5, 10, and 15 km/s.
After having derived a coarse fit by comparison of the observed line profiles with the synthetic
spectra from our model grid, we further refined the stellar and wind parameters (in particular,
Ṁ and β) to obtain the best possible fit. Note that the microturbulent velocity was also adapted
when necessary. We considered this quantity to be spatially constant and identical for all
investigated lines, i.e., we assumed that ξ does not follow any kind of stratification throughout
the atmosphere (see Sections 1.5.3 and 2.8).
In a last step, the actual radius (contrasted to the adopted one) was estimated in an iterative
process, by comparing the V-band integrated theoretical fluxes with corresponding absolute
visual magnitudes MV (see Section 1.5.5). These in turn were taken from the calibrations
by Schmidt-Kaler (1982), except for HD 168183 and HD 111904, which are known cluster
members and hence we were able to derive MV from their distance. This procedure (including
corresponding updates of Ṁ ) was repeated until the difference in input and output radius
became negligible (usually one or two iterations were sufficient).
We recall that the derived gravities are contaminated by centrifugal effects. To obtain the
“true” gravities, needed to calculate, e.g., the masses and to find the appropriate positions in
the log Teff − log g diagram (see Section 2.10), one has to apply a “centrifugal correction”
with respect to the derived rotational velocities (Repolust et al. 2004, and references therein,
see also Eq. (1.1)). These corrections have been applied, and the corresponding values can be
found in Tables 2.7 and 2.5 as entry gcorr . For our further analysis, we used exclusively these
values.
2.5 Analysis of the 28 sample stars
2.5.1 Sample divided in subgroups
Due to the restricted number of available spectral lines and because different regimes of the
parameter space lead to different accuracies, given the available diagnostic, we subdivide our
sample in three groups of stars, depending on the reliability of the derived stellar parameters
(mainly Teff and log g). The first group (hereafter “group I”) comprises sample stars for
which we consider the results to be very reliable. The second group (hereafter “group II”)
constitutes objects that suffer from the following “defect”. From Fig. 2.2 (right panel), it is
obvious that there will be models at each side of the “peak” around 23,000 K that produce
similar Si III 4552 (and 4567-4574) profiles. Since these lines are our major temperature
38
Periodically variable B supergiants
indicator for the early-type stars, this leaves us with two possibilities for Teff , and only the
strengths of additional Si II or Si IV lines would allow for a conclusive answer. For sample
stars of spectral type B1-B2, only the Si III multiplet is available to us. In this case, we make
the appropriate choice between the high and the low temperature solution relying either on
He I 4471 (which still has a certain sensitivity on the temperature, but requires an assumption
of the Helium abundance) or on its spectral subtype (which we infer from SIMBAD or the
recent literature). Due to the restricted reliability of this approach, all corresponding objects
are collected in “group II”. We discuss this approach in detail, for the prototypical example
of HD 54764, in Section 2.5.2 (Group II objects). The effective temperatures and gravities
derived for group I and II objects will finally be used to obtain a new calibration of Teff as
a function of spectral subtype. The third group (hereafter “group III”) contains stars for
which we have no means to derive accurate stellar parameters, either because the objects are
rather extreme or suffer from additional effects not included in our atmospheric models, either
because of their peculiar spectrum that complicates a reliable fit, or a combination of both.
Therefore we classify these stars as “unreliable”, and their parameters should be considered
with caution. Apart from these three groups, we define a fourth group (hereafter “group IV”)
consisting of the twelve comparison stars. For these objects, at most three lines have been
observed (He I 4471, Hγ, and Hα), which, in combination with our new Teff -calibration (see
Section 2.6.1), will be used to estimate effective temperatures, surface gravities, and wind
parameters.
For most of the sample stars we observed two Hα profiles about one year apart, to obtain an
impression of the wind variability (for a detailed investigation, see Markova et al. 2005). We
modelled each Hα profile separately by fixing all parameters except the mass-loss rate. The
resulting two values for Ṁ were averaged to compute the wind strength parameter log Q and
the mean wind density that are required for our further investigations in Section 2.11.
In what follows we will discuss the individual fits of the most noticeable sample stars, group
by group. The fits can be found in Section 2.12 at the end of this chapter to keep the current
text well structured and readable. A summary of the derived stellar and wind parameters of
all B-type supergiants is presented in Tables 2.7 to 2.6.
2.5.2 Individual discussion of the resulting parameters
The order in which we discuss the objects is based on their spectral type as adopted in this
study (see discussion of individual objects below and also Section 2.6.1). Within each spectral
type, stars are ordered following their HD number. This order refers to the order of the objects
in Figs 2.16 to 2.23 as well. HD 94909, HD 91943, HD 148688 and HD 53138 will be omitted
here, since they will be discussed more thoroughly in Section 2.5.3. All other omitted objects
require no explicit comments.
2.5 Analysis of the 28 sample stars
39
Stars discarded from the sample
We have good reasons to remove HD 51110 (B9 Ib/II) from our sample. From our spectra, it
immediately turned out that HD 51110 is very He-weak (all He lines are almost absent). We
ascribe this chemical abundance anomaly to the gravitational settling of helium. This is only
expected to show up in the stable, non-convective atmospheres of the higher gravity stars
(white dwarfs and sub-dwarfs), as hypo- thesized by Michaud et al. (1983) and confirmed by
Fabbian et al. (2005) for hot Horizontal Branch stars in NGC 1904. The extreme strength
of the observed Hγ and Hα profiles are consistent with this hypothesis. Thus, we consider
this star to be misclassified and hence unimportant for our study. For HD 147670 (B8/B9 II)
only Hα was measured. Since we cannot derive stellar parameters from a single line, we are
forced to exclude it from our sample as well.
Group I objects
The first group of periodically variable B-type supergiants consists of those objects for which
we were able to derive accurate results, on the basis of our assumptions, as discussed in
Sects. 2.4.2 and 2.8. They are the following (in order of spectral type):
HD 168183 (O9.5 Ib) was already found to be a short-period (single-lined) binary, exhibiting large variations in radial velocity between consecutive nights (from −52 up to 92 km/s)
and having a period of about 4 days (Bosch et al. 1999). From our data we confirm both
the short period and the huge radial velocity changes, ranging from −60 km/s to 80 km/s.
The fact that HD 168183 is a member of the open cluster NGC 6611 allowed us to derive
its radius as 19 R⊙ (from d = 2.14 ± 0.10 – Belikov et al. 1999; V = 8.18, AV = 2.39 –
Hillenbrand et al. 1993). Several different spectral types are mentioned for this star: B1 Ib/II
(SIMBAD), B0 III (Evans et al. 2005), O9.5 I (Hillenbrand et al. (1993), confirmed by Bosch
et al. (1999)). We accept O9.5 I as the “true” spectral type, relying on our spectroscopically
derived temperature of 30,000 K, which is in agreement with the observational Teff scale for
O-type supergiants provided by Martins et al. (2005). From the Hα profile, we conclude
that this object is of luminosity class Ib. The peculiar feature seen in the Si III 4574 line is a
known instrumental defect, which is also present in the spectrum of HD 170938.
HD 115363 (B1 Ia) The observations of all 40 B-type supergiants were done near the end
of the lifetime of the CAT telescope. Unfortunately, at the time when it was completely closed
down, in September 1998, we did not have Si measured for this target yet. He I 4471, which
is a reasonable temperature estimator in this temperature range (at least if we adopt a solar
helium content), gives us a good fit at a temperature of 20,000 K. HD 115363 is very similar to
HD 170938, which has the same spectral type and for which we do have Si III measured. For
the latter object, we find exactly the same temperature and gravity. This makes us confident
that the derived parameters are credible. Note also the excellent fit of the Hα emission.
40
Periodically variable B supergiants
HD 170938 (B1 Ia) Although the emission peak observed in the Hα profile of HD 170938
is lower than the one observed in HD 115363, marking a difference in wind properties (with
mass-loss rates differing within a factor of about two to four), all other stellar parameters are
identical in both B1 Ia supergiants. Note that we detect the same instrumental defect in the
Si III 4574 line of this star, as in the spectrum of HD 168183.
HD 109867 (B1 Iab) is mainly worth mentioning because of the huge changes in radial
velocity, extending from -70 to 20 km/s. Si III appears to be asymmetric (see Section 2.4.1).
HD 154043 (B1 Ib) For this poorly known supergiant we are able to obtain a convincing
fit. This is one of the few stars that does not show any radial velocity changes between the
different line measurements. The radial velocity amounts to -20 km/s.
HD 106343 (B1.5 Ia) Apart from the fact that the blue part of Hα does not perfectly fit
(both in the absorption trough and in the wings), we can be quite confident about the stellar
parameters. The derived parameters confirm previous results from Lamers et al. (1995) and
Bianchi et al. (1994) obtained by means of UV spectroscopy.
HD 111990 (B2 Ib) For this supergiant, part of a double system, only one Hα profile has
been observed, which can be fitted very nicely. The P Cygni profile shows only a very small
emission peak, which points to a moderate wind density. Note that from this star on we switch
from the analysis of Si III to the analysis of Si II.
HD 92964 (B2.5 Iae) This object is one of the few sample stars that exhibits a clear asymmetry in the line profile, especially visible in Hγ. Since we are able to obtain an acceptable fit
without further assumptions, we ascribe this asymmetric behaviour to the strong wind that is
affecting the photospheric lines. The He I 6678 line requires a macroturbulent velocity which
is twice as large as the one we derive from the Si lines (see also Section 2.7).
HD 102997 (B5 Ia) The wind strength of the first measurement is lower than in the second
observation, turning the P Cygni profile with a partly refilled absorption wing into a pure
emission profile. Unfortunately we are unable to correctly reproduce this pure emission.
This is the first object out of two (together with HD 108659) for which we find an effective
temperature larger than predicted from our calibration (see further discussion at the end of
Section 2.6.1).
HD 108659 (B5 Ib) Quite unknown among the supergiants, there is only one previous temperature estimate for this object, derived by Waelkens et al. (1998) using photometric mea-
2.5 Analysis of the 28 sample stars
41
surements (Teff = 11,750 K). Our spectral fits indicate a much higher temperature, at least
by 4,000 K. Although it seems that the derived mass-loss rate is too low (synthetic cores too
deep), a further increase in Ṁ would result in a small red emission peak that is not observed.
Also for this second B5 target, we find a temperature larger than expected for this spectral
type (see further, Section 2.6.1).
HD 80558 (B6 Iab) As indicated especially by the second Hα profile, the wind of HD 80558
might show a non-spherical distribution, if one regards its shape as the beginning of a doublepeaked structure. So far, the results of our wind analysis can only be considered as a very
rough estimate.
HD 91024 (B7 Iab) From this spectral type on, He I 4471 becomes strongly sensitive to
changes in effective temperature (see Fig. 2.3), at the expense of its reaction to changes in
surface gravity. Thus, Teff is easily fixed at 12,500 K, with a gravity of log g = 1.95 following
from the Hγ line wings. Note that for this object the observed Si II components are of different strengths (contrary to many other objects for which both lines are predicted with different
strength but observed with equal strength), and thus could be fitted without any compromise
(see Section 2.7). HD 91024 has a moderate wind, which is slightly refilling the wings of Hα.
By inspection of the first Hα profile, we see an almost flat red wing (at continuum level),
with a very steep decline into absorption. This also occurs for HD 106068, and cannot be
represented by our synthetic profiles, at least at the inferred gravities.
Group II objects
We recall from Section 2.4.2 that, for some stars, the lack of Si II and/or Si IV prohibits us
from choosing between two equally well fitting models at different positions in parameter
space. To break this dichotomy, we proceed as follows. First of all, we aim at a good fit
of He I 4471, which is a reasonable temperature indicator, though best suited only at the
latest spectral types (below 15,000 K, see Fig. 2.3). Still, we have to keep in mind that this
line depends on the (unknown) helium abundance. By combining the He I 4471 diagnostics
(using solar helium abundances) with knowledge about the spectral type of the star, we are
then able to obtain some clue about which temperature is the correct one.
HD 54764 (B1 Ib/II) We take HD 54764 as a prototypical example to illustrate the dichotomy problem in some detail. Relying on both the Si III triplet and the He I 4471 line,
we are able to find two very different models that both give a reasonable match with the observations: Teff = 19,000 K, log g = 2.45 and Teff = 26,000 K, log g = 2.9, respectively. By
comparing the corresponding synthetic lines, one can hardly tell the difference (see Fig. 2.4).
Lyubimkov et al. (2004) derived values for Teff , log g, and AV for 102 stars, based on their
colour indices. Although photometric methods rapidly lose their predictive capacity in the
42
Periodically variable B supergiants
Figure 2.4: Spectral line fits for HD 54764 - a prototypical example for supergiants belonging
to group II: two models, located in completely different parameter domains (regarding Teff
and log g), produce similar line profiles. Full line: cool model (Teff = 19,000 K, log g = 2.4);
dashed: hot model (Teff = 26,000 K, log g = 2.9). As is clear from this figure, there is hardly
any difference between both profiles.
massive star domain, it is quite interesting to have a look at their results for HD 54764, which
are Teff = 25,500 ± 800 K and log g = 3.56 ± 0.17 (note the rather optimistic error bars).
Whereas this effective temperature would perfectly match our hotter model, the gravity is
certainly far too high, which is immediately reflected in the Balmer line wings. Unfortunately,
we could not detect any He II or Si IV lines in the observed spectral range to unambiguously
decide on Teff .
If HD 54764 is really of spectral type B1, then the cooler model would be the more plausible
one. Note that we find similar temperatures for all other B1 supergiants, in agreement with
the Teff calibration by Lennon et al. (1993). HD 54764 is one of the three supergiants within
our sample that exhibits clear asymmetries in its line profiles, in this case most likely due to
an optical companion (Abt & Cardona 1984).
HD 47240 (B1 Ib) is another example of finding two models with completely different
stellar parameters, both fitting the Si III lines very well: one model with Teff = 19,000 K and
log g = 2.4, and a second one with Teff = 24,000 K and log g = 2.8. Again, the He I lines are
also too similar to allow for a clear distinction.
In contrast to the first group II object, we do have a complete FEROS spectrum at our disposal
for HD 47240, provided in GAUDI. The drawback of this spectrum is the very low signal-tonoise ratio (SNR, which is the ratio of the signal S to the noise N), so that the Si II and Si IV
lines, although being very different in strength, disappear in the noise. From He II 4686, on
the other hand, the hotter solution can be ruled out with high confidence, since in that case
the line prediction is much stronger than the actual observed strength. Again, also for this
object, its spectral type points towards the cooler solution. The complete comparison with
the FEROS observations is given in Fig. 2.5. It is worth mentioning that this star might be
a binary and that the Hα morphology (double-peaked structure) is typical for a fast rotator
observed almost equator-on (Morel et al. 2004). Note that we observe this structure also for
the fast rotators HD 64760 and HD 157246.
2.5 Analysis of the 28 sample stars
43
Figure 2.5: FEROS spectrum of HD 47240: synthetic line profiles at Teff = 19, 000 K and
log g = 2.4.
HD 141318 (B2 II) For this object, the differences between the parameters of the cool and
the hot solutions are not as large as for the previous cases, namely (Teff , log g) = (20,000 K,
2.9) and (22,000 K, 3.2), respectively. Although the forbidden component of He I 4471 fits
slightly better for the hotter model, the Si III triplet is better represented for the cooler one.
Note that an intermediate model at 21,000 K does not give a good fit. From its spectral type
then, we prefer the cooler solution. The different temperatures indicated by the He I and the
Si III lines is probably just an abundance effect, due to the fact that we assumed solar values.
Group III objects
Group III constitutes those three stars for which we cannot claim a similar accuracy as obtained for the previous two groups, either because they are somewhat extreme, or because of
their peculiar spectrum that complicates a reliable fit, or a combination of both. All three objects belong to the group of ten stars that were added to the sample of Waelkens et al. (1998),
44
Periodically variable B supergiants
and are possibly chemically peculiar stars. We have classified these stars as “unreliable”, and
the derived parameters have to be considered with caution.
HD 105056 (B0 Iabpe) is a nitrogen-rich, carbon-deficient supergiant, exhibiting very strong
emission lines, up to twice the continuum level. It has been classified as an ON9.7 Iae supergiant in several studies. Its peculiar nature and the extremely dense wind hamper a correct
modelling of the photospheric lines (e.g., Hγ), since these are severely contaminated by the
wind. Although nitrogen enrichment usually goes along with the enrichment of helium, we
used a compromise solution for He I 4471 and He I 6678 at normal abundance, guided by the
temperature we found from fitting the Si III triplet.
HD 98410 (B2.5 Ib/II) Similar to the previous object, also HD 98410 is an extreme supergiant, with a very strong wind (Hα completely in emission) and strongly refilled photospheric
lines. Because of the restricted wavelength range around each line, the normalisation of the
Hα profile became more problematic than usual, thus increasing the uncertainty of the derived mass-loss rate. The difficulty to fit Hα and Hγ in parallel might point to the presence of
strong clumping in the wind (e.g., Repolust et al. 2004), and the actual mass-loss rate might
be considerably lower than implied by Hα. Taken together with the peculiar shape of one of
these profiles (which might be explained by an equatorially enhanced wind), we “classify”
our analysis as unreliable.
HD 68161 (B8 Ib/II?) Although the spectroscopically derived spectral type is B8 Ib/II,
Eggen (1986) mentioned that photometry indicates a different luminosity class, namely B8
IV. HD 68161 has been considered by Paunzen & Maitzen (1998) as a probable, chemically
peculiar variable star, in particular a variable star of the α2 CVn type. If so, this star should
be a main sequence star of spectral type later than B8p (consistent with the photometrically
derived spectral type), exhibiting a strong magnetic field and abnormally strong lines of Si
among other elements. Since for this star we could not observe the Si lines, we cannot confirm
this conjecture. Note, however, that the derived gravity is not so different from HD 46769,
which is a “normal” B8 Ib supergiant, and that it is also in agreement with typical gravities
for these objects. Due to the discussed uncertainties we add this object to our list of unreliable
cases.
2.5.3 Comparison with other investigations
We compared the results of our analysis with corresponding ones from similar investigations,
in particular those by Crowther et al. (2006, hereafter CR06), McErlean et al. (1999), and
Kudritzki et al. (1999) for five stars in common.
2.5 Analysis of the 28 sample stars
45
Comparison with Crowther et al. (2006)
Our sample has four targets in common with the sample of Galactic early B supergiants
studied by CR06: HD 94909 (B0 Ia), HD 91943 (B0.7 Ib), HD 148688 (B1 Ia), and HD 53138
(B3 Ia). They used the alternative NLTE model atmosphere code CMFGEN (Hillier & Miller
1998) to derive the physical parameters and wind properties of these stars. Compared to their
low dispersion CTIO4 and intermediate dispersion JKT5 /INT6 spectra, we have the advantage
of the very high resolution CES data. On the other hand, CR06 have complete spectra at
their disposal (kindly provided to us by P. Crowther). To compare the spectra with each
other, we first downgraded our synthetic spectral lines by convolving with an appropriate
Gaussian. Subsequently, we verified whether the best model we found from a limited number
of lines also provides a good fit to the additional H, He, and Si lines in the complete spectrum.
CR06 used a solar Si abundance as well and 0.20 by number for the helium abundance. The
complete comparison is summarised in Table 2.3. In this procedure, we only re-determined
the mass-loss rates from the low/intermediate-resolution spectra of CR06 to fit the Hα profile.
They do not differ significantly from the values derived from our data.
HD 94909 (B0 Ia) From the CTIO spectrum of this star, CR06 estimated the effective temperature to be 27,000 K and log g 2.9. It is impossible to fit our Si III lines with such a
high temperature, because they become far too weak. Instead, we derive an effective temperature of 25,000 K, in combination with a log g of 2.7. When we compare our (degraded)
best-fitting model with the CTIO spectrum (Fig. 2.6), we see that Si IV is predicted too weak,
which explains the higher temperature found by CR06. By exploring the neighbouring parameter space, it turned out that we cannot simultaneously fit Si III and Si IV, and we suggest
that this star is overabundant in Si.
HD 91943 (B0.7 Ib) For this star, only a few additional lines (besides those measured by
us) are available due to the high noise level. In Fig. 2.7 we show Si III 4552-4567-4574
for consistency, together with lines of an additional ionisation stage of Si (Si II 4128 and
Si IV 4089/4116/4212) and helium (He I 4026, He I 4387, and He II 4200). There might be a
problem with the normalisation of the Si IV 4212 and Si IV 4089 profiles, but still it is clear
that the strength of the observed and theoretical profiles agree satisfactory. In view of the low
dispersion, we obtain a reasonable fit, which gives us confidence in our results.
HD 148688 (B1 Ia) This star is often used as a comparison star in UV studies as a galactic
counterpart for early B-type supergiants in M31, M33, or the SMC (Bresolin et al. 2002;
Urbaneja et al. 2002; Evans et al. 2004, respectively). It is one out of a few that show no
radial-velocity changes. We have only a few additional lines available in the CTIO spectrum. It is very encouraging that they all nicely confirm our results (see Fig. 2.8). To fit the
4 Cerro
Tololo Inter-american Observatory
Kapteyn Telescope
6 Isaac Newton Telescope
5 Jacobus
46
Periodically variable B supergiants
Si III triplet, we need an effective temperature of 1,000 K lower than the one suggested by
CR06.
HD 53138 (B3 Ia) can surely be named one of the most “popular” B-type supergiants studied until now. Let us first concentrate on our high resolution spectrum (Fig. 2.18 in Section 2.12). Clear variations in the wind outflow are registered. In the first measurement of Hα,
the P Cygni profile has only a tiny absorption trough and a considerable emission, whereas
the second profile indicates a much lower wind density. This star is one of the objects that
show the discrepancy between predicted and actual line strength of the Si II doublet components (Section 2.7) for which we adopted a compromise solution. In this way, our best-fitting
model gives an effective temperature of 18,000 K and a log g of 2.25.
When degrading the resolution of this best-fitting model to the resolution of the full spectrum
provided by CR06 (originating from the LDF Atlas7 ), we find a discrepancy in the Si III
triplet (see Fig. 2.9). This discrepancy can be resolved by decreasing the temperature in
combination with either a lower microturbulent velocity or a depletion in Si, since Si IV 4089
also seems to be a bit too strong. To cure this problem, we can go down as far as 17,000 K
(with log g = 2.15 and ξ = 10 km/s), which is still 1,500 K higher than the value derived by
CR06 (Teff = 15,500 K, log g = 2.05). To find an explanation for this difference, we had a
closer look at their spectral line fits. Though the overall fit is good, the Si lines do not match
perfectly. In particular, at their value of Teff = 15,500 K, the Si II 4128-4130 profiles are
predicted too strong, whereas Si III 4552-4567-4574 is predicted too weak. This disagreement
in the ionisation balance suggests that the effective temperature should be somewhat higher.
We use the 17,000 K model, which is at the lower bound of the quoted error range of our
original analysis, as a solution to all discussed problems. Note that the fits and the parameters
provided in Table 2.7 refer to this model at 17,000 K.
A summary of the main parameters resulting from both studies is given in Table 2.3. Interestingly, in all cases, the low/intermediate resolution Hα profile observed by CR06 lies amidst
our two Hα profiles, so that the inferred mass-loss rates are very similar and the variability is
not large (cf. Section 2.11). The differences in radii obtained by us and CR06 are of similar
order. On the other hand, the β-values implied by our fits are generally larger than the ones
from CR06. Accounting additionally for the moderate differences in Teff for the first three
objects (in these cases, our values are lower), this explains the lower log Q values we found.
For HD 53138, on the other hand, the β values are similar, whereas our effective temperature
is higher, explaining the higher log Q value.
Comparison with McErlean et al. (1999) and Kudritzki et al. (1999)
By means of the plane-parallel, hydrostatic, NLTE atmosphere code TLUSTY (Hubeny &
Lanz 2000), McErlean et al. (1999) deduced the photospheric parameters and CNO abundances of 46 Galactic B supergiants. Effective temperatures were mostly obtained from the
7 http://www.ast.cam.ac.uk/STELLARPOPS/hot
stars/spectra lib/mw library/mw lib rary index.html
47
Relative intensity
2.5 Analysis of the 28 sample stars
Figure 2.6: Comparison between synthetic line profiles (downgraded) from our best-fitting
models (black) and the low/intermediate dispersion spectrum of CR06 (grey) for the early
B-type supergiant HD 94909.
Periodically variable B supergiants
Relative intensity
48
Figure 2.7: Same as for Fig. 2.6 but now for HD 91943.
Relative intensity
2.5 Analysis of the 28 sample stars
Figure 2.8: Same as for Fig. 2.6 but now for HD 148688.
49
Periodically variable B supergiants
Relative intensity
50
Figure 2.9: Same as for Fig. 2.6 but now for HD 53138.
51
2.5 Analysis of the 28 sample stars
Table 2.3: Comparison between the fundamental parameters derived in this study and by
CR06, McErlean et al. (1999) - ME99, and Kudritzki et al. (1999) - K99 for the objects in
common.
HD
94909
91943
148688
53138
58350
Teff
(kK)
25.0
27.0
24.0
24.5
21.0
22.0
17.0
15.5
18.5
18.5
13.5
16.0
log g
(cgs)
2.7
2.9
2.7
2.8
2.5
2.6
2.15
2.05
2.35
2.30
1.75
2.10
MV
-6.9
-6.4
-5.95
-6.3
-6.9
-6.8
-7.0
-7.3
-7.0
R∗ log L/L⊙ v∞
β log Q
ξ
Ref.
(R⊙ )
(km/s)
(km/s)
36
5.65
1450 1.8 -12.96 20 this study
25.5
5.49
1050 1.5 -12.34 10 CR06
23
5.19
1400 2.5 -13.36 15 this study
26.8
5.35
1470 1.2 -13.01 10 CR06
42
5.49
1200 3.0 -12.90 15 this study
36.7
5.45
725 2.0 -12.39 15 CR06
50
5.27
490 2.5 -13.20 10 this study
65
5.34
865 2.0 -13.57 20 CR06
5.04
10 ME99
39.6
5.22
620 2.5 -13.61 40 K99
65
5.10
250 2.5 -13.17 12 this study
5.36
15 ME99
ionisation balance of Si. Two of their objects are in common with our sample (HD 53138
and HD 58350, a B5Ia Group IV object). The comparison is given in Table 2.3. The effective
temperature for HD 53138 from McErlean et al. (1999) compares well with our high temperature solution for this object, i.e., differs significantly from the much lower value derived by
CR06 (see above), and also the gravities are consistent.
Regarding HD 58350, we have adopted a temperature consistent with the calibrations provided in the next section, which is significantly lower than the value found by McErlean
et al. (1999). The complete JKT spectrum allows for an increase in temperature by roughly
1,000 K. Finally, on the basis of the temperature scale derived by McErlean et al. (1999),
Kudritzki et al. (1999) analysed the wind properties of a sample of early/mid B and A supergiants, by means of a previous, unblanketed version of FASTWIND. Their value for the
wind-strength parameter of HD 53138 coincides with the value provided by CR06, i.e., is
lower than our result. We regard this agreement/disagreement as not conclusive, since (i) the
Hα spectra used by Kudritzki et al. (1999) are different (less emission), and (ii) the analysis
was based on unblanketed models, which in certain cases can lead to differences in the derived mass-loss rate (cf. Repolust et al. 2004, their Fig. 21). A further, more comprehensive
comparison will be given in Section 2.9, with respect to modified wind-momenta. Summarising the results from this comparative study, we conclude that, despite the multidimensional
parameter space we are dealing with and the interdependence of the parameters, we are able
to derive rather accurate values from only a few selected lines (of course, within our assumptions, in particular regarding the Si abundance). This enables us to provide (and to use) new
calibrations for the effective temperatures, which will be discussed in the next section.
52
Periodically variable B supergiants
2.6 Analysis of the 12 comparison stars
2.6.1 Derived Teff calibration
Thanks to the (almost) complete coverage of the B star range and the large number of objects
available, we are able to derive a Teff calibration as a function of spectral type, which we
subsequently use to derive the temperatures of our group IV objects. The spectral types
were taken from the literature. In case different assignments were given, we have provided
arguments why we prefer one above the other (Section 2.6.2). Ideally, one would want to
re-assign spectral types from our high-quality data as in, e.g., Lennon et al. (1993). However,
one needs a fair number of spectral lines to do this in a safe way. Since we have only a few H,
He, and Si lines, we preferred to use the spectral classifications from the literature, keeping
in mind that some of them may not be very refined. We confirmed or adopted the luminosity
class on the basis of the strength of Hα.
In Fig. 2.10 we see that the effective temperature follows a systematic decrease with spectral
type. To derive the temperature calibration, we merged our results for group I/II objects with
those from CR06 (who used assumptions similar to ours in their analysis) and added two
more objects from Przybilla et al. (2006) at the low temperature edge. In contrast to the
errors inherent to our analysis8 , which are identical to those from CR06, these two objects
(HD 34085 = β Ori (B8 Iae:), HD 92207 (A0 Iae)) could be analysed in a very precise way
by exploiting the complete spectrum, with resulting errors of only ± 200 K.
By performing a polynomial fit to these data (including the quoted errors), we derive the
following effective temperature scale for B-type supergiants,
Teff (in K) = 26522.63 − 5611.21x + 817.99x2 − 42.74x3 ,
with x the spectral subclass. Note that we have included both luminosity subclasses (Ia and
Ib) to obtain this fit, and that the inclusion of the objects by CR06 changed the results only
marginally compared to a regression using our and Przybilla et al.’s (2006) data alone. The
obtained standard error for this regression is ±1,500 K.
The three group III objects in this figure match our derived calibration very well, although
we considered their parameters as unreliable. The B0 star HD 105056 lies at exactly the
same position as HD 94909, just at the lower edge of our error bar. HD 68161 (B8 Ib/II?)
will be excluded from our further analysis, due to problems regarding its classification (see
individual discussion in Section 2.5.2).
As the Ia supergiants have more mass loss than Ib objects, because of their larger luminosity,
they suffer more from line blanketing and mass-loss effects. This is why supergiants of
luminosity class Ia are expected to appear cooler than the ones with luminosity class Ib.
Comparing the two in Fig. 2.10, this is indeed exactly what we observe: all Ib objects (filled
symbols) seem to lie at higher temperatures than the corresponding Ia ojects (open symbols)
8 Generally,
∆T = 1,000 K, except for the two B5 objects with ∆T = 2,000 K (see text).
2.6 Analysis of the 12 comparison stars
53
Figure 2.10: Teff as a function of spectral type for the sample B-type supergiants: group I
(black circles), group II (small grey filled circles), and group III (asterisks). For group I, we
subdivided according to luminosity class: Ia (open symbols) and Ib (filled symbols). Triangles denote the early B Ia supergiants from CR06, and squares two late-type Ia supergiants
with very precise parameters from Przybilla et al. (2006). The dotted line represents the effective temperature scale of Lennon et al. (1993) and the full line shows our newly derived
effective temperature scale based on group I/II stars in addition to the objects from CR06 and
Przybilla et al. (2006). The grey area denotes the standard deviation of our regression fit.
of the same spectral type, except for one star (HD 154043 - B1 Ib) which may have too low
an effective temperature, as shown by Markova & Puls (2007). Our temperature scale agrees
well with the one provided by Lennon et al. (1993). The largest differences occur between
B0 and B2, where our scale lies roughly 1,500 K higher, mostly due to the objects analysed
by CR06.
Let us point out a problem concerning the two B5 objects (with identical Teff ). We see in
Fig. 2.10 that both temperatures lie clearly above the errors of the calibration. We can still
fit the Si II lines well by decreasing the temperature (from 16,000 K to 14,500 K), when
we reduce ξ by 4 km/s (see Section 2.8.1), which would still be acceptable for these latetype objects. However, with such a low value, a simultaneous fit to the He and Si lines
becomes impossible, violating our general fitting strategy. Thus, if the lower temperature
would be the actual one, this might be due to two reasons: either He and Si have different
microturbulent velocities, or Si is underabundant in both B5 targets (see also Section 2.8).
Note that the calibration itself remains rather unaffected by these objects, since we used
larger error estimates of ∆T = 2, 000 K here.
54
Periodically variable B supergiants
Table 2.4: Effective temperature calibration for B-supergiants, used to derive the effective
temperatures for the group IV objects.
SpT
O 9.5
B0
B 0.5
B1
B 1.5
B2
B 2.5
Teff
29,500
26,500
23,900
21,600
19,800
18,200
16,900
SpT
B3
B4
B5
B6
B7
B8
B9
Teff
15,800
14,400
13,500
13,000
12,600
12,100
11,100
Finally, by means of this calibration (for particular values, see Table 2.4), we are able to
derive the fundamental parameters for the group IV comparison objects. Using the He I 4471
line as a double check for a consistent effective temperature, the surface gravity is obtained
from Hγ and the wind properties from Hα. All results are summarised in Tables 2.5 and 2.6.
2.6.2 Individual discussion of the resulting parameters
Group IV is the group of 12 bright comparison stars, selected from the Bright Star Catalogue,
previously not known to exhibit any periodic variability. For these objects at most three lines
have been observed: He I 4471, Hγ, and Hα, which will be used to estimate the required
stellar and wind parameters, in combination with our Teff calibration for B-supergiants (see
Section 2.6.1 and Table 2.4).
HD 149038 (O9.7 Iab) While SIMBAD lists this star as a B0 supergiant, a spectral type
of O9.7 Iab has been suggested by Walborn & Bohlin (1996), Lamers et al. (1999), Maı́zApellániz et al. (2004), and Fullerton et al. (2006). Recent revisions of stellar parameter
calibrations in the O-type regime (Martins et al. 2005) predict Teff ≈ 30500 K, log g ≈ 3.2,
R ≈ 22.1 R⊙ , and log L/L⊙ ≈ 5.57 for an O9.5 supergiant. Fullerton et al. (2006) used these
values and also revised the distance of this star. They found it to be situated at 1.0 kpc instead
of 1.3 kpc as stated earlier by Georgelin et al. (1996). The radio mass-loss rate, log Ṁradio
(at time of observation), could be constrained as being less than -5.51 ± 0.18 M⊙ /yr, with a
terminal velocity of v∞ = 1750 km/s and β = 1.0. Our calibration predicts somewhat lower
effective temperatures at O9.7, but the general agreement between our values and those stated
above is convincing.
HD 64760 (B0.5 Ib) is a very fast rotating supergiant. With a projected rotational velocity
of more than 220 km/s, it is likely being observed at a very high inclination, i.e., almost
equator-on. Due to its particularly interesting wind structure, HD 64760 is amongst the best
studied early B-type supergiants (Prinja et al. 2002; Kaufer et al. 2002, 2006, being the most
2.6 Analysis of the 12 comparison stars
55
recent investigations). Its richness in spectral features led to conclusive evidence for the
existence of a corotating two and four armed spiral structure, suggested to be originating
from stream collisions at the surface and perturbations in the photosphere of the star (Kaufer
et al. 2002, 2006). Hα consists of a double-peaked structure, with a blueward and a somewhat
stronger, redward shifted emission peak around the central absorption (as in Fig. 2 of Kaufer
et al. 2002). This star is clearly a Ib supergiant, with a moderate mass-loss rate refilling the
photospheric absorption. Of course, we are not able to reproduce this double-peaked profile,
and only the gravity can be derived (together with some crude estimate for Ṁ ) for the adopted
effective temperature, Teff = 24,000 K, which is log g = 3.2. From the MV calibration, we
finally have R∗ = 24 R⊙ . These parameters are similar to those reported by Howarth et al.
(1998).
HD 157246 (B1 Ib) This is another example of a star with a rotationally modulated wind,
similar to HD 64760. We find a projected rotational velocity of 275 km/s. The Hα profile
shows the typical blue- and redward shifted emission peaks, which are about equal in height.
They suggest that the wind is equatorially compressed, see also Prinja et al. (1997). With only
two lines at our disposal (i.e., Hα and Hγ), we can only suggest that the stellar parameters
are consistent with those reported until now (Prinja et al. 2002).
HD 157038 (B1/2 IaN) is enriched in nitrogen and helium (n(He)/n(H) ≈ 0.2 according
to our analysis), and the observed lines can be fitted reasonably well with a model at Teff =
20,000 K and log g = 2.3.
HD 165024 (B2 Ib) As for HD 157246, we have only two hydrogen lines at our disposal,
and we can derive only log g and an estimate of the wind properties. As we have no direct
means to derive the rotational velocity, we can only give a range for vmacro , depending on the
adopted v sin i from the literature. The most reasonable solution is a combination of v sin i
= 120 km/s and vmacro = 50 km/s (considering the macroturbulence derived for objects of
similar spectral type).
HD 58350 (B5 Ia) or η CMa is a well-studied B-type supergiant. By means of the NLTE
atmosphere code TLUSTY (Hubeny & Lanz 2000), McErlean et al. (1999) have also analysed this star (cf. Section 2.5.3). Their best-fitting model has a temperature of 16,000 K,
whereas previous temperature estimates were lower, between 13,000 and 14,000 K. From
our calibrations (Table 2.4), we find a typical value of 13,500 K at B5, which is consistent
with these lower values and also gives an acceptable fit. On the other hand, the JKT spectrum
(from the online LDF atlas) shows that the effective temperature might be actually higher by
1,000 K (still within the quoted error bars), since Si II is too strong and Si III too weak (see
Fig. 2.11).
56
Periodically variable B supergiants
Figure 2.11: Spectral line fits for the JKT spectrum of HD 58350. Gray: observed JKT
spectrum (Lennon et al. 1993), black: predictions at Teff = 13, 500 K and log g = 1.75.
HD 106068 (B8 Iab) This rarely studied bright B supergiant shows exactly the same feature
in Hα as we have found for HD 91024 (group I): a very flat red wing, with a sudden steep
decrease into the (blue-shifted) absorption. Of course, we cannot fit such a profile.
HD 46769 (B8 Ib) For B8 objects, our calibration gives an effective temperature of 12,100 K,
which is also required to fit the He I 4471 line (remember its sensitivity to Teff in this temperature domain). Models with a temperature increased by 1,000 K, compared to our calibration,
would give too strong a He I 4471 line, and vice versa for lower temperatures.
HD 111904 (B9 Ia) is another poorly known object from the BSC. Contrary to HD 106068,
the flat wing is now in the blue part of the line and not on the red side, i.e., there is some
(unknown) refilling mechanism. The star, being a member of the open star cluster NGC 4755
in Crux, allows us to derive an MV of -7.38 ± 0.15 (Slowik & Peterson 1995) (consistent
with our calibration within the adopted errors), and hence a radius of 95 R⊙ .
2.7 Comments on general problems
57
2.7 Comments on general problems
We noticed a certain discrepancy between the two lines of the Si II 4128-4130 doublet. Theory predicts the Si II 4130 line to be somewhat stronger than the Si II 4128 line, since the
gf value of the second component is roughly a factor 1.5 larger than of the first component
(different atomic databases give very similar results). However, in most (but not all) cases,
we observe an equal line strength for both lines. While further investigation is needed, we
approached this problem by finding a compromise solution in the final fit. The related errors
are discussed in Section 2.8.
Thanks to the high spectroscopic resolution, a problem with the forbidden component in the
blue wing of He I 4471 could be identified. It appears that this (density dependent) component
is often (however not always) predicted too weak for the early-type stars (< B1) and too
strong for the later type stars (> B2). For the hottest stars, this might be explained by a fairly
strong O II blend at this position, which cannot be synthesized with the present version of
FASTWIND, using ‘only’ H/He/Si as explicit elements.
For some stars, He I 6678 would need a higher macroturbulence, vmacro , than the other lines.
The clearest example of this situation is given by HD 92964, but, to a lesser extent, it also
occurs in HD 89767, HD 94909, HD 93619, HD 96880, and HD 106343, all of them earlytype stars. Their He I 6678 ‘fits’ show that there must be an additional broadening mechanism
that we do not understand at present.
In a number of cases we were not able to reproduce the shape of the Hα profile, mainly for
two reasons. On the one hand, the assumption of spherical symmetry adopted in FASTWIND
(and almost all other comparable line-blanketed NLTE codes9 ) prohibits the simulation of
disks or wind compressed zones in the case of large rotational speeds. On the other hand, we
neglected the effects of wind clumping (small-scale density inhomogeneities redistributing
the matter into dense clumps and an almost void interclump medium, see, e.g., Puls et al.
(2006) and references therein), which can have a certain effect on the shape of the Hα profile
and on the absolute value of the mass-loss rate. Recent findings by Crowther et al. (2006) have
indicated that this effect is rather small in B-type supergiants, though. Even if the detailed
shape of Hα does not match, the error in the derived (1-D) mass-loss rate remains acceptable,
due to the strong reaction of the profile on this parameter, at least if the wind densities are not
too low. For such low wind densities, the discussed processes do not lead to any discrepancy
with the observed profile shape, since only the core of Hα becomes refilled.
We stress that, in this kind of analysis, a reliable normalisation of the spectra is of crucial
importance. An incorrect normalisation of the silicon lines, e.g., leads to errors in the derived effective temperatures, which will propagate into the derived surface gravities.10 Errors
occurring in the normalisation of Hγ additionally enlarge the error in log g, whereas an erroneous rectification of Hα affects the Q value and thus the mass-loss rate. Although we were
9 see
Zsargó et al. (2006) for recent progress regarding a 2-D modelling
preserve the Hγ profile, changing the temperature by 1,000 K requires a simultaneous change in gravity by
roughly 0.1 dex.
10 To
58
Periodically variable B supergiants
restricted to few selected orders (thus cutting out the largest part of the available continuum),
the remaining spectral windows were generally sufficient to obtain a correct normalisation
thanks to the high SNR that was obtained. For pure emission Hα profiles, on the other hand,
this was more difficult, due to the large width of the profiles.
A reliable derivation of terminal velocities, v∞ , turned out to be possible only for a restricted
number of stars. As already explained, we adopted the values determined by Prinja & Massa
(1998) from the UV as a first estimate, with an extrapolation towards later B-types by using
the corresponding A supergiant data provided by Lamers et al. (1995). Both data sets have
been collected in Table 1 of Kudritzki & Puls (2000). By means of these values, for most
of the objects such a good fit in Hα (and other lines) had been obtained in the first instance
that further alterations seemed to be unnecessary. Only for HD 92964, the adopted v∞ -value
might be too low, which could explain the mismatch of He I 6678. However, the first Hα profile of this object is in complete emission, and the second one displays only a tiny absorption
dip, which makes it difficult to derive a reliable value. Thus, also for this star, we adopted the
UV-value.
On the other hand, by alternatively deriving v∞ from fitting the shape of Hα, we generally
found values that are either similar or lower than the ones from the UV. Though the agreement
was extremely good for some objects, for others a discrepancy by more than a factor of two
was found, resulting in a mean difference of Hα and UV terminal velocity of 45%. Note that
the fits generally improved when adopting the UV values. Thus we conclude that it is not
possible to precisely estimate v∞ from Hα alone, at least in a large fraction of cases, where
the typical error by such an approach is given by a factor of two.
2.8 Error estimates
Thanks to the high quality of our spectra, fitting errors due to resolution limitations or instrumental noise do not play a role in our analysis. The major problem encountered here is the
very restricted number of available lines, and the involved assumptions we are forced to apply
(particularly regarding the Si abundance and the microturbulent velocity, see below). Apart
from this principal problem, the major source of errors is due to our “eye-fit” procedure (contrasted to automated methods, e.g., Mokiem et al. 2005, see also Chapter 4), which is initiated
by manually scanning our pre-calculated grid (see Section 2.4.2). The effective temperatures
and gravities did not need refinement once a satisfactory solution had been found from the
grid, after tuning the mass-loss rates and the velocity exponents. Therefore, the grid steps reflect the errors on those parameters. In practice, this means that typical errors are of the order
of 1,000 K in Teff and 0.1 in log g, which is – for later spectral types – somewhat larger than
possible under optimal conditions, i.e., if many more lines were available. Finer step sizes
or further fine-tuning of the models with respect to effective temperature, on the other hand,
was regarded as irrelevant, due to the consequences of our assumptions regarding abundance
and microturbulence.
2.8 Error estimates
59
2.8.1 Error estimates for Teff
Whenever many lines from one ion are present, the microturbulence can be specified with
a high precision given a “known” abundance. A few lines from different ionisation stages,
on the other hand, allow for a precise temperature estimate, since in this case the ratios of
lines from different stages are (almost) independent of abundance (which is the reason why
spectral classification schemes use these ratios). Missing or incomplete knowledge becomes
a major source of uncertainty if only a few lines from one ionisation stage are available. We
assess the different effects one by one.
Influence of Si abundance
Concentrating first on Si, a star with depleted abundance will display, at a given temperature, weaker Si lines in all ionisation stages, and vice versa, if Si is enhanced. Thus, if the
line strengths decrease with increasing temperature (as for Si II, see upper panel of Fig. 2.2),
the effective temperature would be overestimated if the actual abundance is lower than the
assumed solar value, and underestimated for increasing line strength with temperature (e.g.,
for the low temperature region of Si III, cf. lower panel of Fig. 2.2). To check the quantitative consequences of this uncertainty, we calculated, for three different temperatures (15,000,
20,000, and 25,000 K), various models that are depleted and enhanced in Si by a factor of
two (thus comprising the lower values discussed in the literature, see Section 2.4.2), and investigated how much the derived temperature would change. When changing the effective
temperature of the model, one has to change the surface gravity in parallel, to preserve the
Hγ profile. For late-type stars at 15,000 K (where only Si II is available), such a depletion/enhancement in Si corresponds to a decrease/increase of Teff by 2,000K (and log g by
0.2). At 25,000 K, which is a representative temperature for the early-type objects for which
we only have the Si III triplet, the effect was found to be identical. At 20,000 K, the effect depends on whether we have Si II or Si III at our disposal, and the overall effect is a bit smaller.
If we have Si II, we again find an overestimation, now by 1,500 K, if the star is depleted in
Si, but assumed to be of solar composition. If we have Si III, which is still gaining in strength
in this temperature regime, the effective temperature would be underestimated by 1,500 K.
In conclusion, due to the uncertainties in the Si abundance, we expect that our temperature
scale might systematically overestimate the actual one (except for those group II objects that
rely on Si III, where an underestimation is possible), by 1,500 K to 2,000 K if the average
abundance were actually 0.3 dex lower than solar.
Influence of ξ
One might argue that a Si depletion is not present in our sample, since in almost all cases our
secondary temperature diagnostic, He I, was fitted in parallel with Si without further problems. However, we have no independent check of the He content (only one ion available). In
most cases, the He I line profiles were consistent with solar abundance, but strongly evolved
60
Periodically variable B supergiants
objects have processed material and should have a larger He content (see, e.g., the corresponding discussion in Crowther et al. (2006)). Even if one would regard the consistency
between Si and He as conclusive, it depends on one of our additional assumptions, namely
that the microturbulent velocities are constant with height, i.e., identical for He and Si lines11 .
Although there is no clear indication in the present literature that this hypothesis is wrong,
a stratified microturbulent velocity seems plausible. In such a case (i.e., different ξ for He
and Si), we would no longer have a clear handle on this quantity, and due to the well-known
dependence of line strength on this quantity (for Si lines, see, e.g., Trundle et al. 2004, Urbaneja 2004; for He lines, McErlean et al. 1998), an additional source of error would be
present. From test calculations, it turned out that a change of 1,000 K (which is our nominal
error in Teff ) corresponds to a change of ξ by roughly 4 to 5 km/s in the Si lines.
For the few objects with low macroturbulent velocity, vmacro , and low rotational speed, we
were able to directly ‘measure’ ξ, thanks to the high resolution of our spectra. In these cases,
the profiles become too narrow in the wings and too strong in the core, when decreasing ξ
(and vice versa when increasing ξ). This behaviour cannot be compensated for by changing
Teff . For most of the objects, however, such precise information is washed out by vmacro .
In the majority of cases, we were able to obtain satisfactory fits (He in parallel with Si) by
keeping typical values available in our grid, which are 15 km/s for early-type objects and
10 km/s for late-type objects. Changing ξ by more than 2 to 3 km/s would destroy the fit
quality of either Si or He. In conclusion, we are confident about the derived values of ξ
(and thus of the temperatures), provided that the He and Si lines are affected by a similar
microturbulent broadening, i.e., that stratification effects are negligible.
Influence of the Si II problem
As mentioned in Section 2.7, there is a discrepancy between the two lines of the Si II 41284130 doublet for most of our late-type objects. By allowing for a compromise solution (in
which Si II 4128 is predicted as too weak and Si II 4130 as too strong), we minimise the
error. Indeed, to fit either of the two lines perfectly, we would have to change the effective
temperature by roughly 500 K, which is well below our nominal error.
2.8.2 Error estimates for other quantities
Although the fit-/modelling-error in log g is ±0.1 dex for a given effective temperature, log g
itself varies with Teff (as already mentioned, typically by 0.1 for ∆T = 1,000 K), so that for
a potentially larger error in Teff (due to under-/over- abundances of Si) the gravity also needs
to be adapted.
The errors for the other parameters follow the usual error propagation (for a detailed dis11 For most of our objects, the analysed He lines are stronger than the Si lines, i.e., they are formed above the Si
lines.
2.9 Wind momentum-luminosity relation
61
cussion, see Markova et al. 2004 and Repolust et al. 2004), being mainly dependent on the
uncertainty in the stellar radius, which in turn depends on the validity of the used calibration
of the absolute visual magnitude from Schmidt-Kaler (1982). A precise error estimate of the
latter quantity is difficult to obtain, but at least for early and mid Ia supergiants (until B3 Ia)
we can compare this calibration with the results from CR06 (Section 2), who derived MV
values either directly from cluster membership arguments or from average subtype magnitudes of Magellanic Cloud stars. Comparing their results with ours, we find similar average
values, with a 1-σ scatter of
∆MV ≈ ±0.43 mag,
which will be adopted in the following, also for the later spectral types and the other luminosity classes. From this number and the error in Teff , the error in the radius becomes
∆ log R/R⊙ ≈ ±0.088,
which corresponds to 22%, and is consistent with the differences in the radii derived by us
and by CR06, see also Table 2.3. We subsequently find a typical uncertainty in the luminosity
of
∆ log L/L⊙ ≈ 0.22. . .0.19
for Teff = 12,000 K. . . 25,000 K, respectively. The wind-strength parameter, log Q, can be
determined with rather high precision. Adopting a combined contribution of fit error and uncertainty in Teff of ±0.05 (Repolust et al. 2004, their Section 6.2), and an additional contribution of ±0.1 accounting for the temporal variability (Tables 2.8 and 2.6, and Section 2.11),
we find
∆ log Q ≈ ±0.11.
The precision in ∆ log Ṁ amounts to
∆ log Ṁ ≈ ±0.24,
if we estimate the error in v∞ as 30%. Finally, the error in the derived wind-momentum rate
is
∆ log Dmom ≈ ±0.34,
i.e., somewhat larger than the error in log L.
2.9 Wind momentum-luminosity relation
In Fig. 2.12 we present the position of our galactic B supergiants in the wind momentumluminosity diagram (with log Ṁ v∞ (R∗ /R⊙ )0.5 the modified wind momentum rate). The
presence of such a relation (with wind momenta being a power law of luminosity, and exponents depending on spectral type and metallicity) is a major prediction of the theory of
line-driven winds (for details, see, e.g., Kudritzki & Puls 2000), and has been used in recent years to check our understanding of these winds. To compare our results with earlier
findings, we provide results from different investigations relevant in the present context. In
62
Periodically variable B supergiants
particular, the theoretical predictions from Vink et al. (2000) for objects with Teff > 23,000 K
and 12,500 K < Teff < 23,000 K, respectively, are displayed, where the difference is related
to an almost sudden change in the ionisation equilibrium of Fe around Teff ≈ 23,000 K (from
Fe IV to Fe III), the so-called bi-stability jump (Vink et al. 2000, and references therein). Due
to this change and below this threshold, the line acceleration is predicted to increase in the
lower and intermediate wind, because of the increased number of driving lines being available. Three distinctive lines refer to the findings from Kudritzki et al. (1999), who derived
these relations from observed wind momenta of early B, mid B, and A supergiants, respectively. The wind momenta of our sample objects have been overplotted, by using averaged
mass-loss rates (from our two Hα profiles, see also Section 2.11), for group I/II and IV objects. Objects with disk-like features are indicated by “d”, and typical error bars are displayed
in the lower right. Finally, we have noted objects with Hα in absorption, to indicate that the
wind momenta of these objects are subject to errors somewhat larger than typical, of the order
of 0.3 dex, due to problems in deriving reliable values for the velocity field exponent, β (cf.
Puls et al. 1996 and Markova et al. 2004). The top and bottom figure allow us to compare
our findings with the displayed predictions and observations, both for objects as a function of
spectral type (top) and as a function of their position with respect to the predicted bi-stability
jump (bottom).
First, let us point out that differences in their behaviour as a function of luminosity class
are minor, and that there is no obvious difference between group I/II and group IV objects,
i.e., the wind-momenta for periodic pulsators and comparison objects behave similarly. From
the top figure, we see the following: the position of the early B-types (B0/1) is consistent
with both the theoretical predictions and the findings from Kudritzki et al. (1999), except
for two objects with uncertain positions and one object with disk-like features. Late B-type
supergiants12 (B6 and later) nicely follow the observed relation for A supergiants, but are
located below the theoretical predictions. Only for mid B supergiants (B2. . . 5), we find a
difference to earlier results. Whereas Kudritzki et al. (1999) have derived a very strict relation
for all B2/3 supergiants of their sample, located considerably below the relation for early-type
objects (a finding that still lacks theoretical explanation), our sample follows a non-unique
trend. Though high luminosity objects (with log L/L⊙ > 5) behave similarly to the sample
studied by Kudritzki et al. (1999) (with a somewhat smaller offset), lower luminosity objects
follow the (theoretical) hot star trend, but might also be consistent with the low Teff relation
when accounting for the larger errors in Dmom (Hα absorption objects). Note that a detailed
discussion and comparison of these findings have meanwhile been given by Markova & Puls
(2007).
The bottom panel displays our sample objects as a function of Teff , differentiating for objects
with Teff > 23, 000 K and objects with 12,500 K < Teff < 23,000 K, i.e., with temperatures
below and above the predicted bi-stability jump. Cooler objects with Teff <12,500 K (no
predictions present so far) are indicated by triangles. The situation is similar to that above:
almost all hotter objects follow the predicted trend, and also a large number of cooler objects
follows this trend or the alternative one. Additionally, however, there are a number of cool
objects that lie below both predictions, particularly at intermediate luminosities with 4.7 <
log L/L⊙ < 5.4.
12 Note
that this is the first investigation with respect to this class of objects.
2.10 Position in the HRD
63
Interestingly, there is only one “real” outlier, an Hα absorption object of group IV (HD 157038),
which is an evolved star that is significantly enriched in He and N. Note that the three objects with disk-like features display a rather “normal” wind-momentum rate, though one of
those (HD 157246) lies at the lower limit of the complete sample. Nevertheless, we will omit
the latter stars (as well as group III objects) from all further analyses regarding mass loss to
obtain clean results.
Summarising the results from above, no obvious differences compared to earlier findings
could be identified within our complete sample (pulsators and comparison objects), with the
only exception being that the unique trend for mid B-type supergiants claimed by Kudritzki
et al. (1999) could not be confirmed by us: part of these objects seem to behave “normally”,
whereas another part shows a momentum deficit. Note also that our findings are not contradictory to those of CR06 (keeping in mind that their sample only included B supergiants not
later than B3).
2.10 Position in the HRD
After having derived the atmospheric parameters with satisfactory precision, we are now
in a position to tackle the major objective of this chapter, namely to try and clarify if the
opacity mechanism could be held responsible for the variability of our targets, as proposed
by Waelkens et al. (1998) and Aerts (2000a). To do so, we calculated the luminosities of
the targets from the effective temperatures and stellar radii derived from our analyses. When
comparing the relative difference in temperature and luminosity between our values and those
derived by Waelkens et al. (1998), we notice that, in general, effective temperatures agree
fairly well. The typical differences are less than 5% in logarithmic scale, with a maximal
difference of 10%. On the other hand, the changes in luminosity are significant (up to 40%
in the log, see Fig. 2.13). Since the temperatures agree so well, the luminosity difference is
mainly due to the difference in stellar radii. As discussed in Section 2.8.2, the typical errors
of our luminosities are of the order of ∆log L/L⊙ ≈ 0.19. . .0.22, so that at least half of the
displayed difference should be attributed to problems with the values provided by Waelkens
et al. (1998). This is not surprising, since these authors used photometric calibrations for
main-sequence stars because they had no spectroscopic data available. We are confident that
our values are more trustworthy, and that the estimated error bars are reliable.
In the top panel of Fig. 2.14 we compare the position of the variable B-type supergiants
in the HRD from spectroscopic and photometric derivations and place them among known
variable supergiants available in the literature. It shows that the new position of the sample
supergiants is quite different from the one obtained by Waelkens et al. (1998). The stars now
lie at the low-luminosity region of the known α Cyg variables. Also shown are the theoretical
instability strips of opacity-driven modes in β Cep and SPB stars, derived from main sequence
(pre-TAMS13) models (Pamyatnykh 1999), and post-TAMS model predictions for ℓ = 1 and
2 for B stars up to 20 M⊙ (Saio et al. 2006). The former strips do not encompass our targets
13 The
TAMS is the Terminal Age Main Sequence
64
Periodically variable B supergiants
Figure 2.12: Wind momentum-luminosity relation (WLR) for our sample of B supergiants.
Only class I,II (large symbols) and IV objects (small symbols) have been included, and the
mass-loss rates from both Hα profiles were averaged. Bold solid/dashed: Theoretical predictions from Vink et al. (2000) for objects with Teff > 23,000 K and 12,500 K < Teff <
23,0000 K, respectively. Dashed-dotted, dashed-dotted-dotted, and dotted lines are the “observed” relations from Kudritzki et al. (1999), for early and mid B supergiants and A supergiants, respectively.
Top: results from this study, objects denoted as a function of spectral type. Filled circles:
B0/1; open circles: B2. . . B5; triangles: B6 and later.
Bottom: as top figure, but objects denoted as a function of Teff . Filled circles:
Teff >23,000 K; open circles: 12,500 K < Teff < 23,000 K; triangles: Teff <12,500 K.
Note that the theoretical bi-stability jump is predicted at 23,000 K. Typical error bars are indicated at the lower right. Overplotted crosses denote objects with Hα in absorption, which
have a larger error in log Q than the other objects, due to the uncertainty regarding β. “d”
denotes three objects with disk-like features (HD 47240, HD 64760, and HD 157246).
2.10 Position in the HRD
65
Figure 2.13: The relative difference in luminosity against the relative difference in effective temperature between photometrically (Waelkens et al. 1998) and spectroscopically
(this study) determined stellar parameters (with the relative difference defined as δx =
(xLefever − xWaelkens)/xLefever).
at first sight, whereas the latter do, at least for all stars hotter than 15,000 K. A far more
appropriate comparison between the position of these strips and the sample stars is achieved
from a (log Teff , log g) plot. Indeed, the position of the targets in such a diagram is much
more reliable because it is free from uncertainties on the radii and quantities derived thereof.
Such a plot is provided in the bottom panel of Fig. 2.14. It encapsulates the major result of
this study.
We see that all group I stars fall very close to the higher-gravity limit of the predicted
pre-TAMS instability strip and within the post-TAMS instability strip predictions of gravity
modes in evolved stars. This implies that the observed periodic variability is indeed compatible with non-radial gravity oscillations excited by the opacity mechanism. The photometric
period(s) found for our sample stars, ranging from one day to a few weeks, and the high
percentage of multiperiodic stars, fully support this interpretation. The position of the comparison stars in this (log Teff , log g) diagram is also interesting. In Section 2.3 we argued
that five among these so-called non-periodically variable supergiants, show amplitudes and
periods comparable to the ones of the sample stars. We see that these five objects also lie at
the same higher-gravity limit of the instability strip. From the remaining seven objects, with
much lower amplitude variability, there are three that have very high (projected) rotational
velocities, which result in lower amplitudes (De Cat 2002), namely HD 64760, HD 157246,
and HD 165024. Together with HD 149038, which nicely falls within the SPB instability
domain, we suggest these nine targets to be newly discovered oscillators in gravity modes,
unlike the remaining three objects HD 46769, HD 125288, and HD 86440, which lie, even
when taking into account the error bar in Teff , completely outside the predicted strip.
66
Periodically variable B supergiants
Figure 2.14: Position of the sample stars in the log Teff -log L/L⊙ and the log Teff -log g diagram. The positions, as derived from spectroscopy (this study) are marked by circles (filled group I; grey - group II; open - group III). In the top figure, we additionally show the position
of the sample stars as derived by Waelkens et al. (1998) based on photometry by asterisks
(arrows indicate a lower limit), and some well-studied periodically variable supergiants reported in the literature (Burki 1978; van Genderen 1985; Lamers et al. 1998; van Leeuwen
et al. 1998) by triangles. In the lower figure, we have added the group of comparison stars
(group IV). They are represented by squares. The dotted lines represent the ZAMS (four
initial ZAMS masses - in M⊙ - are indicated) and TAMS. Theoretical instability domains for
low-order p-modes (β Cephei - thick solid line) and high-order g-modes (SPB stars - dashed
lines) for main-sequence models are shown (Pamyatnykh 1999), together with post-TAMS
model predictions for ℓ = 1 (grey dotted) and ℓ = 2 (black dotted) g-modes for B stars with
masses up to 20 M⊙ (Saio et al. 2006).
2.11 Pulsations and mass loss
67
2.11 Pulsations and mass loss
Glatzel et al. (1999) suggested the occurrence of pulsationally-driven mass loss in supergiants
to be due to strange-mode instabilities on theoretical grounds. While we found evidence for
non-radial gravity modes in our sample of “low-mass” stars, it is worthwile to investigate if
these modes play a role in the mass loss. If this is the case, one expects a correlation between
the frequency and/or amplitude of the oscillations and the mass-loss rate. In view of the
normal behaviour of our sample stars in terms of the WLR relation, we can anticipate this
relation to be weak, at best. In Fig. 2.15, we plot the photometric variability parameters as a
function of (i) the wind density, log ρS , at the sonic point, (ii) the projected rotational velocity,
v sin i, and (iii) the ratio Ω of v sin i to the critical velocity14 . We have chosen to consider
the wind density at the sonic point (i.e., log ρS = log(Ṁ /(4πR∗2 cS ), with cS the isothermal
sound speed), because this is the lowermost place where mass loss is initiated and oscillations
could have an effect. The Spearman rank correlation coefficients, r, computed for group I and
II stars (indicated in each panel in Fig. 2.15) reveal only two mild correlations related to the
mean wind density. Except for a similarly mild correlation between Ω and the peak-to-peak
variation, no other connection between rotation and photometric variability can be detected.
At first, a positive trend between the observed amplitude of variation and the mean wind
density is visible. At the same time, we notice a tendency that higher mass-loss is present
when the detected frequencies are lower, i.e., the oscillation periods are longer. In general,
p
the oscillation periods scale as the square root of the inverse mean density (P ∼ 1/ρ )
within the star so this downward trend may suggest that the role of oscillations in helping to
increase the mass loss is more evident in more evolved stars. However, it concerns only weak
correlations, which are not significant in a statistical sense. Moreover, the trends weaken if
we add the five comparison stars with significant amplitudes.
Prinja & Howarth (1986) investigated the variability in UV P Cygni profiles of early-type stars
and concluded that variability in Ṁ occurs at the 10% level, on time scales of a day or longer.
Changes of a factor of two or larger were never observed. Recently, Markova et al. (2005)
investigated the wind variability in 15 O type supergiants, using Hα as a signature. They
found variations of the order of 4% with respect to the mean value of Ṁ for stars with strong
winds and of ± 16% for stars with weak winds. The ratio of maximum to minimum massloss rate over the time interval for their sample ranges from 1.08 to 1.47 (with a mean ratio of
1.22), with a tendency for weaker winds to show larger changes. Both numbers are fully in
agreement with Prinja & Howarth (1986). For our sample supergiants, we find that the ratio
of maximum to minimum mass-loss rate15 ranges from 1.05 up to 1.88, with a mean ratio for
the whole sample of 1.31, which is in agreement with Prinja & Howarth (1986), in the sense
that these numbers do not exceed a factor two. On the other hand, the maximum relative
differences amount to 60%, with an average of 22% (without distinction between weak or
strong winds), which is somewhat higher than what is expected for “normal” O supergiants.
This might support the idea of a connection between the mass loss and the variability in these
stars.
14 Calculated
15 But
here in the spherical approximation, since most of our objects do not rotate too fast.
remember that we have only two observations at our disposal, separated by about one year.
68
Periodically variable B supergiants
Figure 2.15: Frequency and peak-to-peak variation as a function of the mean wind density,
log ρS , at the sonic point, v sin i and Ω = v sin i/vcrit . Symbols have the same meaning
as in Fig 2.14 (bottom). Spearman rank correlation coefficients, r, are calculated for the
periodically variable B supergiants of group I and II joined. The two-sided significance is
given between braces.
At present we cannot conclude definitively to have found evidence for mass-loss enhancement
through oscillations. We merely find an indication that higher photometric amplitudes seem
to be accompanied by a higher wind density and a higher relative change in the mass-loss
rate. The validity of these suggested connections would merit from further study through
a more refined time-resolved analysis of both the oscillation and wind properties of a few
selected targets.
2.12 Discussion and summary
We derived stellar and wind parameters of 40 B-type supergiants using the NLTE atmosphere
code FASTWIND and high-resolution, high signal-to-noise spectroscopic data of selected H,
He, and Si lines. For the majority of the objects (excluding group III), these parameters are
accurate within the discussed assumptions, and reliable error bars have been estimated as
well. The primary aim of our study was to investigate if the origin of the variability found
2.12 Discussion and summary
69
in HIPPARCOS data of 28 of these stars could be gravity-mode oscillations excited by the
opacity mechanism that is also at work in main sequence B stars. To assess this suggestion
made by Waelkens et al. (1998), we needed to achieve accurate values of the effective temperature and gravity of the sample stars to compare them with those of stellar models for which
theoretical computations predict oscillations. The conclusion is clear cut: all the sample stars
for which we could derive reliable stellar parameters have Teff and log g values consistent
with those of stellar models in which gravity modes are predicted. They are all situated at the
high log g limit of the instability strip of gravity modes in evolved stars, covering the entire
range in spectral type B. In view of the large fraction of stars with multiperiodic behaviour,
we suggest our sample stars to be opacity-driven non-radial oscillators.
Our study involved a sample of twelve comparison B supergiants that were not selected to
be periodically variable by the HIPPARCOS team. Nevertheless, we found that nine of these
behave similarly to our original sample stars and we suggest them to be new pulsators with
gravity modes as well. We thus end up with a sample of 37 non-radial pulsators in the upper
part of the HRD. The occurrence of opacity-driven oscillations in that area was recently
proven to be correct for the B2 Ib/II star HD 163899 from a 37-day ultra-precise MOST
space-based photometric lightcurve (Saio et al. 2006). This star was found to oscillate in 48
frequencies with amplitudes below 4 mmag. Our findings and those by Saio et al. (2006) open
up the area of asteroseismology to the upper-part of the HRD and lead to excellent prospects
for fine-tuning of evolutionary models from the ZAMS towards the supernova stage from the
seismic sounding of B supergiant interiors. To achieve this, one needs to obtain long-term
high-precision time series in photometry and spectroscopy to perform mode identification,
and derive accurate stellar parameters in the way acquired in this thesis.
We find marginal evidence for a connection between the photometric amplitudes and the
wind density, in a sense that the oscillations may help the line driving at the base of the
wind. Firm conclusions in this direction require further detailed study of a few sample stars,
though. Since our sample stars display similar WLRs as found for “normal” B-supergiants
(if there is something like “normal” at all), the enhancement of the mass loss cannot be
very strong. Interestingly, however, we also found one (important) difference compared to
previous evidence. Whereas the results from Kudritzki et al. (1999) strongly implied that all
mid B-type supergiants display a WLR that lies significantly below the theoretically predicted
one, our findings show that there is no such unique (and problematic) correlation. Part of our
sample objects show this dilemma indeed (in particular those with log L/L⊙ > 5), whereas
lower luminosity objects were found to be rather consistent with theory. In this investigation
we also derived, for the first time, wind momenta of late-type B-supergiants, which follow
the trend of A-supergiants surprisingly well.
While trying to achieve a high-precision estimate of the stellar and wind parameters through
line-profile fitting, we came across several interesting features and results. The most important one is that we found ways to achieve good estimates for the parameters, despite the
limited number of available lines, thanks to the high predictive power for different parameter estimates of each of the available lines. We reached this conclusion because we could
compare our analyses with similar ones based on more spectral lines available in the literature. This also led to a new effective temperature calibration in good agreement with previous
70
Periodically variable B supergiants
ones. We used this calibration to provide a practical recipe to compute Teff as a function of
spectral type.
Appendix: Resulting parameters and line profile fits
We list the stellar and wind parameters of all sample supergiants, subdivided by group, in Tables 2.7 and 2.8. Table 2.7 lists the stellar parameters. Spectral types are taken from SIMBAD
or from the recent literature. The visual magnitudes, V, are from SIMBAD as well. Gravities
as obtained from the line fits are denoted by geff , and those approximately corrected for the
centrifugal acceleration by gcorr (see discussion on page 37). Absolute visual magnitudes
MV come from the calibrations by Schmidt-Kaler (1982). They are used to derive the stellar
radii (in combination with the theoretical fluxes). Luminosities log L/L⊙ are derived from
Teff and R∗ /R⊙ . Table 2.8 gives the wind parameters and the different velocities (turbulence
and rotation). For those sample stars that have been observed in Hα twice (typically one year
apart), we provide the derived wind parameters for both observations (when both coincide,
this is indicated by (2)). The wind-strength parameter, log Q, is in the same units as in Table 2.3, Ṁ is in units of M⊙ /yr and the modified wind momentum rate, Dmom , in cgs. All
velocities are given in km/s and have their usual meaning. Question marks for vmacro denote
those cases where the line profiles could be fitted without any macroturbulent velocity. We
also indicate the type of Hα emission: e/i/a indicate emission/intermediate/ absorption profiles, respectively. Tables 2.5 and 2.6 list the same parameters, but now for the comparison
stars. On the pages following these tables, we show the line profile fits corresponding to the
numbers in these tables.
71
2.12 Discussion and summary
Table 2.5: Stellar parameters of the comparison stars. For an explanation of the notation: see
text.
V
Teff log g eff log g corr MV R∗ /R⊙ log L/L⊙ n(He)/n(H)
(mag) (K) (cm/s2 ) (cm/s2 )
HD SpT
number
149038
64760
157246
157038
165024
75149
58350
86440
125288
106068
46769
111904
O9.7Iab
B0.5Ib
B1Ib
B1/B2IaN
B2Ib
B3Ia
B5Ia
B5Ib
B6Ib
B8Iab
B8Ib
B9Ia
4.91
4.23
3.31
6.41
3.66
5.47
2.40
3.50
4.35
5.95
5.79
5.80
28000
24000
21500
20000
18000
16000
13500
13500
13000
12000
12000
11100
2.90
3.20
2.65
2.30
2.30
2.05
1.75
2.25
2.40
1.70
2.55
1.55
2.91
3.27
2.94
2.31
2.39
2.06
1.77
2.25
2.40
1.72
2.57
1.56
-6.45
-5.95
-5.8
-6.9
-5.7
-7.0
-7.0
-5.4
-5.35
-6.2
-5.2
-7.38
25
24
25
43
26
39
65
31
31
50
30
95
5.53
5.23
5.08
5.42
4.80
4.95
5.10
4.45
4.39
4.66
4.22
5.09
0.20
0.10
0.10
0.20
0.10
0.10
0.15
0.10
0.10
0.10
0.10
0.10
Table 2.6: Wind parameters and velocities of the comparison stars. For an explanation of the
notation: see text.
HD Emission log Q
Ṁ
v∞
β
number
(M⊙ /yr) (km/s)
149038
64760
157246
157038
165024
75149
58350
86440
125288
106068
46769
111904
i
i
i
e
a
i
i
a
a
i
a
i
-12.58
-13.16
-14.11
-13.25
-15.25
-13.43
-13.17
-13.64
-13.23
-13.35
-13.09
-13.25
0.23E-05
0.42E-06
0.30E-07
0.50E-06
0.18E-08
0.10E-06
0.14E-06
0.40E-07
0.40E-07
0.44E-07
0.37E-07
0.12E-06
1700
1400
1000
1000
840
500
250
470
250
200
200
175
1.3
0.8
1.5
3.0
1.5
2.5
2.5
0.9
0.9
2.0
1.0
3.0
log Dmom
29.09
28.25
26.97
28.31
25.68
27.29
27.25
26.81
26.54
26.59
26.40
27.11
ξ vmacro v sin i
(km/s) (km/s) (km/s)
15
75
57
15
?
230
15
?
275
15
40
41
15 85-50 95-120
15
60
30
12
40
37
10
?
20
10
30
25
8
25
26
12
35
68
10
?
32
72
Periodically variable B supergiants
Table 2.7: Stellar parameters of the sample supergiants. For an explanation of the notation:
see text.
Group
I
II
III
HD SpT
number
V
Teff
(mag) (K)
log geff log gcorr MV R∗ /R⊙ log L/L⊙ n(He)/n(H)
(cm/s2 ) (cm/s2 )
168183 O9.5Ib
8.26 30000
3.30
3.32
-5.86
19
5.42
0.10
89767 B0Ia
7.23 23000
2.55
2.56
-6.40
30
5.35
0.20
94909 B0Ia
7.35 25000
2.70
2.71
-6.90
36
5.65
0.10
93619 B0.5Ib
91943 B0.7Ib
6.98 26000
6.69 24000
2.90
2.70
2.90
2.71
-5.95
-5.95
22
23
5.30
5.19
0.10
0.10
96880 B1Ia
115363 B1Ia
7.62 20000
7.82 20000
2.40
2.40
2.41
2.41
-6.90
-6.90
43
43
5.42
5.42
0.10
0.10
148688 B1Ia
5.33 21000
2.50
2.51
-6.90
42
5.49
0.10
170938 B1Ia
7.92 20000
2.40
2.41
-6.90
43
5.42
0.10
109867 B1Iab
6.26 23000
2.60
2.61
-6.90
38
5.56
0.10
154043 B1Ib
106343 B1.5Ia
7.11 20000
6.24 20000
2.50
2.50
2.51
2.50
-5.80
-6.90
26
42
4.98
5.40
0.10
0.10
111990 B1/B2Ib
92964 B2.5Iae
6.77 19500
5.40 18000
2.55
2.10
2.55
2.10
-5.70
-6.95
25
48
4.91
5.33
0.10
0.10
53138 B3Ia
3.00 17000
2.15
2.16
-7.00
50
5.27
0.20
102997 B5Ia
6.55 16000
2.00
2.01
-7.00
55
5.25
0.10
108659 B5Ib
80558 B6Iab
7.31 16000
5.91 13500
2.30
1.75
2.31
1.76
-5.40
-6.20
26
45
4.60
4.78
0.10
0.10
91024 B7Iab
7.62 12500
1.95
1.95
-6.20
50
4.74
0.10
94367 B9Ia
5.27 11500
1.55
1.57
-7.10
77
4.97
0.10
47240 B1Ib
6.18 19000
2.40
2.48
-5.80
27
4.93
0.15
54764 B1Ib/II
141318 B2II
6.06 19000
5.77 20000
2.45
2.90
2.53
2.90
-5.80
-4.80
26
16
4.90
4.56
0.10
0.10
105056 B0Iabpe
7.34 25000
2.70
2.71
-6.90
40
5.75
0.10
98410 B2.5Ib/II 8.83 17000
2.10
2.11
-5.60
27
4.73
0.15
68161 B8Ib/II?
2.40
2.40
-5.60
43
4.30
0.10
5.66 10500
73
2.12 Discussion and summary
Table 2.8: Wind parameters and velocities of the sample supergiants. For an explanation of
the notation: see text.
HD Emission log Q
Ṁ
number
(M⊙ /yr)
168183
a
89767
e
94909
e
93619
91943
i
i
96880
115363
e
e
148688
e
170938
e
109867
i
154043
106343
a
i
111990
92964
i
e
53138
i
102997
e
108659
80558
a
i
91024
i
94367
i
47240
i
54764
141318
a
a
105056
e
98410
e
68161
a
v∞ β log Dmom
ξ
vmacro v sin i
(km/s)
(km/s) (km/s) (km/s)
-13.58
-13.46
-13.09
-13.17
-12.77
-12.96
-12.88
-13.28
-13.55
-13.46
-12.98
-12.86
-12.90
-13.01
-13.21
-13.26
-13.38
-13.36
-13.49
-13.07
-13.04
-13.26
-13.14
-13.19
-13.09
-13.26
-13.01
-12.88
-13.36
-13.37
-13.20
-13.09
-13.14
-12.81
-12.93
0.15E-06
0.20E-06
0.85E-06
0.70E-06
0.20E-05
0.13E-05
0.75E-06 (2)
0.30E-06
0.16E-06
0.40E-06 (2)
0.12E-05
0.16E-05
0.14E-05
0.11E-05
0.71E-06
0.63E-06
0.50E-06
0.53E-06
0.20E-06
0.52E-06
0.55E-06
0.14E-06
0.28E-06
0.25E-06
0.31E-06
0.21E-06
0.23E-06
0.31E-06
0.58E-07 (2)
0.50E-07
0.75E-07
0.95E-07
0.85E-07
0.24E-06
0.18E-06
1700 1.3
-13.41
-13.26
-14.07
-13.76
0.17E-06
1000 1.5
0.24E-06
0.30E-07 (2) 900 1.5
0.30E-07
900 1.5
27.74
27.89
26.93
26.83
-12.60
-12.50
-12.71
-12.58
-14.42
0.40E-05
1600 3.0
0.51E-05
0.30E-06
500 2.0
0.41E-06
0.30E-08 (2) 200 0.9
29.40
29.51
27.69
27.82
25.39
1600 2.5
1450 1.8
1470 1.2
1400 2.5
1200 2.2
1200 3.0
1200 3.0
1200 3.0
1400 2.5
1300 2.0
800 2.0
750
520
1.8
3.0
490
2.5
325
1.5
470
250
1.0
1.5
225
1.5
175
1.0
27.84
27.97
28.67
28.58
29.04
28.85
28.51
28.10
27.83
28.29
28.77
28.89
28.83
28.73
28.54
28.49
28.43
28.46
27.92
28.23
28.25
27.52
27.80
27.75
27.83
27.66
27.54
27.67
26.94
26.72
26.89
26.97
26.93
27.36
27.24
10
60-90
124
15-20 75-80
47
20
75
64
15
15
75
82
47
48
15
15
65
50
44
55
15
40
50
15
55
51
15
70
50
13
15
65
65
37
44
15
15
62
50
36
31
15
45
38
12
50
39
10
11
40
45
29
28
8
30
25
10
28
31
15
55
94
10
10
70
50
108
32
15
?
61
15
70
31
10
?
17
74
Periodically variable B supergiants
Hα 1
Hα 2
Hγ
He I 4471
He I 6678
Si III
HD 168183 (O9.5 Ib)
HD 89767 (B0 Iab)
HD 94909 (B0 Ia)
HD 93619 (B0.5 Ib)
HD 91943 (B0.7 Ib)
HD 96880 (B1 Ia)
Figure 2.16: Spectral line fits for the periodically variable B-type supergiants with reliable
parameters: group I.
75
2.12 Discussion and summary
Hα 1
Hα 2
Hγ
He I 4471
He I 6678
Si III
HD 115363 (B1 Ia)
HD 148688 (B1 Ia)
HD 170938 (B1 Ia)
HD 109867 (B1 Iab)
HD 154043 (B1 Ib)
HD 106343 (B1.5 Ia)
Figure 2.17: Spectral line fits for the periodically variable B-type supergiants with reliable
parameters: group I (continued).
76
Periodically variable B supergiants
Hα 1
Hα 2
Hγ
He I 4471
He I 6678
Si II
HD 111990 (B2 Ib)
HD 92964 (B2.5 Iae)
HD 53138 (B3 Ia)
HD 102997 (B5 Ia)
HD 108659 (B5 Ib)
HD 80558 (B6 Iab)
Figure 2.18: Spectral line fits for the periodically variable B-type supergiants with reliable
parameters: group I (continued).
77
2.12 Discussion and summary
Hα 1
Hα 2
Hγ
He I 4471
He I 6678
Si II
HD 91024 (B7 Iab)
HD 94367 (B9 Ia)
Figure 2.19: Spectral line fits for the periodically variable B-type supergiants with reliable
parameters: group I (continued).
Hα 1
Hα 2
Hγ
He I 4471
He I 6678
Si III
HD 54764 (B1 Ib/II)
HD 47240 (B1 Ib)
HD 141318 (B2 II)
Figure 2.20: Spectral line fits for the periodically variable B-type supergiants with a potential
Teff dichotomy: group II. Only our favoured solution is displayed (see text).
78
Periodically variable B supergiants
Hα 1
Hα 2
Hγ
He I 4471
He I 6678
Si
HD 105056 (B0 Iabpe)
HD 98410 (B2.5 Ib/II)
HD 68161 (B8 Ib/II?)
Figure 2.21: Spectral line fits for the periodically variable B-type supergiants with parameters to be considered as indicative numbers only: group III.
Hα
Hγ
He I 4471
Hα
Hγ
HD 149038 (O9.7 Iab)
HD 64760 (B0.5 Ib)
HD 157246 (B1 Ib)
HD 157038 (B1/2 IaN)
He I 4471
Figure 2.22: Spectral line fits for the comparison stars (previously not known to exhibit any
periodic variability). No Si available.
79
2.13 Acknowledgements
Hα
Hγ
He I 4471
Hα
Hγ
HD 165024 (B2 Ib)
HD 75149 (B3 Ia)
HD 58350 (B5 Ia)
HD 86440 (B5 Ib)
HD 125288 (B6 Ib)
HD 106068 (B8 Iab)
HD 46769 (B8 Ib)
HD 111904 (B9 Ia)
He I 4471
Figure 2.23: Spectral line fits for the comparison stars (continued).
2.13 Acknowledgements
We thank P. Crowther for providing us with the CTIO/JKT spectra for the stars we had in
common, and A.A. Pamyatnykh and H. Saio for providing us with the theoretical instability
domains of B stars. We thank T. Vanneste, B. Nicolai, and L. Eylenbosch for their contribution to this study in the framework of their Master Theses and Drs. P. De Cat, L. Decin, and
J. De Ridder for their contribution to the data gathering.
80
Periodically variable B supergiants
Chapter
3
A grid of FASTWIND NLTE model
atmospheres of B stars with winds
Partly based on Lefever, K.; Puls, J.; Aerts, C.; “A Grid of FASTWIND NLTE Model
Atmospheres of Massive Stars”, Proceedings of the conference ‘The Future of Photometric, Spectrophotometric and Polarimetric Standardisation’, held on 8-11 May, 2006
in Blankenberge, Belgium, published in ASP Conference Series, Vol. 364, 2007, p.545553, and edited by C. Sterken.
In this chapter, we describe in detail the properties of the BSTAR06 grid of NLTE atmosphere
models, which we calculated with FASTWIND. It has been set up to cope, in a fast way, with
the high number of B-type stars in the CoRoT preparatory archive. To this end it will be
combined with an automated tool for the spectral synthesis of B-type stars (Chapter 4). However, it has been designed in such way that it can easily be exploited in future spectroscopic
analyses, for a wide variety of purposes and stars.
82
A grid of FASTWIND NLTE model atmospheres of B stars with winds
3.1 Introduction
The advent of high-resolution, high signal-to-noise spectroscopy in the nineties led to an
important enhancement of the interest of the scientific community in spectroscopic research,
and in particular in the relatively poorly understood massive stars, besides exoplanets. The
establishment of continuously better instrumentation and the improvement in quality of the
obtained spectroscopic data was the start of a consecutive series of studies, which showed
that our knowledge in the massive star domain increased rapidly. In this respect, it is not
surprising to note that this is exactly the period in which people from several groups started to
upgrade their atmosphere prediction code for such stars, see, e.g., FASTWIND - SantolayaRey et al. (1997), CMFGEN - Hillier & Miller (1998), PHOENIX - Hauschildt & Baron
(1999), WM-Basic - Pauldrach et al. (2001), the Kiel/Potsdam code - Gräfener et al. (2002).
Since the detailed spectroscopic analysis of individual objects through fit-by-eye is a rather
time-consuming (and boring!) job, scientists refrained from analysing large samples and
never tackled more than a few tens of stars at once. Therefore, unfortunately, knowledge in
the B-type regime still relies on small number statistics and we could thus gain from large
sample studies. As a first step towards improvement it would be useful to dispose of an automatic procedure for the spectroscopic analysis of B stars. One option would be to choose
for automatic high-precision analyses through neural networks or genetic algorithms. Mokiem et al. (2005) have chosen for this approach for the automatic analysis of O and early
B-type stars. However, in the mid and late B-type region, one can no longer rely on He as a
temperature indicator. In this region Si (in its different ionisation stages) becomes the most
appropriate temperature indicator. In the ideal case, simply including the Si lines in the genetic algorithm of Mokiem et al. (2005) would be a logical extension. However, this is not as
simple as it may seem. Not only does Si depend on multiple parameters, it is also structurally
much more complicated than the simplest H and He atoms, with many more transitions. We
anticipated the inclusion of Si to slow down the genetic search process for each individual
target tremendously. Therefore, in view of the time constraints we were facing and the large
sample of target stars, we have opted for a grid-method as an alternative approach. This offers
a good compromise between effort, time and precision if an appropriate grid has been set up.
It is in this context that we felt the need for a comprehensive grid of representative atmosphere
models and corresponding synthetic spectra. In what follows we will describe the setup of
the grid and our adopted strategy along this process.
3.2 Description of the FASTWIND BSTAR06 grid
3.2.1 Grid setup
The BSTAR06 grid has been set up to cover the complete parameter space of B-type stars.
As such a grid shall also be a good starting point for a (follow-up) detailed spectroscopic
3.2 Description of the FASTWIND BSTAR06 grid
83
analysis of massive stars, it has been constructed as representative and dense as possible
within a reasonable computation time.
The atmospheric models are calculated only with solar Si, since a change in Si abundance
by a factor two will not affect the model, e.g., its density and temperature. After this model
has been converged, the occupation numbers are converged, at first for solar abundance. This
gives in total 88,305 models/profile sets. Based on these models, the Si abundance is changed,
by a factor ±2, and the iteration continued in order to obtain new occupation numbers within
few additional iterations. This gives two additional sets of profiles, 2*88,305, so that in total
264,915 profile sets are calculated.
We considered 33 effective temperature gridpoints, ranging from 10 000 K to 32 000 K, in
steps of 500 K below 20 000 K and in steps of 1 000 K above it. In this way we will be able
to deal with all stars with spectral types between early A and late O.
As we will analyse massive stars in different evolutionary stages, from main sequence up to
supergiant phase, the surface gravities comprise the range of log g = 4.5 down to 80% of the
Eddington limit, in steps of 0.1, resulting in a mean number of 28 values at each effective
temperature point.
For each (Teff , log g)-gridpoint, we have adopted one ‘typical’ value for the radius, R∗ ,
keeping in mind that a rescaling to the ‘real’ value is required once concrete objects are
analysed (cf. Section 1.5.5). In most cases, the actual radius can then be determined from the
visual magnitude, the distance of the star and its reddening, or from the theoretical fluxes of
the best fitting model. As a first approximation for the grid, the input radius R∗ and the mass
M∗ are determined from interpolation between evolutionary tracks, so that the grid is fully
consistent with stellar evolution theory.
The chemical composition has been chosen to be representative for the typical environment
of massive stars. As we consider only H, He and Si explicitly, we have varied only the helium
and silicon abundance, whereas for the remaining background elements (responsible, e.g.,
for radiation pressure and line-blanketing), we have adopted a solar composition, following
Asplund et al. (2005). For helium, three different abundances have been incorporated: He/H
= 0.10, 0.15 and 0.20 by number. As discussed, e.g., in Section 2.4.2, the silicon abundance
in B stars is still subject to discussion. Depending on sample and method, values range from
solar to a depletion by typically 0.3 dex, both with variations by ± 0.2 dex. Therefore, also
for silicon three abundance values have been adopted, i.e., the solar value (log n(Si)/n(H) =
-4.49 by number, Asplund et al. 2005) and an enhancement and depletion by a factor of two,
i.e., log n(Si)/n(H) = -4.19 and -4.79 by number, as mentioned above.
Since our grid should enable the analysis of stars of different luminosity classes and thus
wind-strengths, we incorporated seven different values for the wind-strength parameter, log Q
(cf. Puls et al. 1996). As for the radius, we were forced to assume a ‘typical’ value for the terminal wind velocity, v∞ , in order to reduce the extent of the grid. To this end, terminal wind
velocities for supergiants have either been interpolated from an existing, but rather crude grid
of late O/early B-type stars, either estimated from observed values (Kudritzki & Puls 2000,
84
A grid of FASTWIND NLTE model atmospheres of B stars with winds
0.5
1.0
1.5
log g (cgs)
2.0
2.5
3.0
3.5
4.0
4.5
35
30
25
20
Teff (1000 K)
15
10
Figure 3.1: Location of the BSTAR06 grid models (dots) in the (Teff , log g)-plane, compared
to standard evolutionary models, not taking into account rotation, core overshooting nor stellar wind (Pamyatnykh 1999) for stars with initial stellar masses from 4 Ṁ up to 40 Ṁ (from
right to left). The ZAMS and TAMS are indicated by thick solid lines.
amongst others). For non-supergiants, we used a similar scaling relation as Kudritzki & Puls
(2000), i.e., v∞ = C · vesc -see their equation 9- but with C = 2.5 for Teff ≥ 24 000K, an
interpolation between 1.4 and 2.5 for 20 000K < Teff < 24 000K and an interpolation between 1.0 and 1.4 for lower temperatures. By fixing R∗ and v∞ in this way, we end up with
a wide spread in mass-loss rates for each predescribed Q-value. The wind velocity law is
determined by the β-exponent, for which we considered 5 values in the grid: 0.9, 1.2, 1.5,
2.0 and 3.0 for the most extreme cases.
Finally, for calculating the NLTE model atmospheres we used a microturbulent velocity, ξ, of
8, 10 and 15 km/s for the temperature regimes Teff < 15 000K, 15 000K ≤ Teff < 20 000K
and Teff ≥ 20 000K, respectively, whereas for all synthetic line profiles (from all models)
microturbulent velocities of 6, 10, 12, 15 km/s have been used, with an additional value of 3
km/s for Teff ≤ 20 000K and 20 km/s for Teff > 20 000K.
3.2.2 Location in the HRD
In Fig. 3.1 we display and compare the position of our models in the HRD with evolutionary
tracks of stars with initial stellar masses between 0.4 and 40 M⊙ . Obviously, we completely
cover the evolutionary sequences of these stars, from main sequence to supergiant phase,
within the temperature range of B-type stars.
3.3 Computational requirements and technical setup
85
3.3 Computational requirements and technical setup
Computer infrastructure Since the computation of one FASTWIND model takes on average around half an hour on a modern computer, one can easily calculate that one would need
some 180 CPU months to finish the computation of the full grid of 264 915 models. For this
reason our institute purchased a dedicated Linux cluster of 5 dual-core, dual-processor computers (3800 MHz processors, sharing 4 Gb RAM memory and 8 Gb swap memory), which
amount to 20 dedicated CPU processors. In the ideal case, this would reduce the required
CPU time down to about 9 months. To further reduce this computation time, we additionally
used 40 additional institute CPUs (8 of 3800MHz and 32 of 3400MHz) whenever available,
in consultation with the other users at our institute. Keeping in mind that some models needed
more time to reach convergence and the fact that the calculations were necessarily interrupted
a few times for technical reasons, our calculations were finally finished after a period of seven
months. Note that this was only possible thanks to the very fast performance of the FASTWIND code, which could never have been achieved with a code like, e.g., CMFGEN, since it
is much more time-consuming. A terabyte disk, connected to a Solaris 10 host pc, was needed
to store the full grid, which finally filled 60% of the disk space. In order to avoid unnecessary
repetition of the computation of such a huge grid, we offer BSTAR06 to the community for
further research1.
Technical setup For the small supergiant grid discussed in Chapter 2, we were able to
gather all models in one directory, and the naming was simple. The name of the model
included information about the wind strength, the effective temperature and the surface gravity. In this way a model named A2025 would indicate a model with a weak wind, an effective temperature of 20 000K and a log g value of 2.5. As our new grid is very extensive, it can technically not be justified anymore to put all models in one single directory. A
simple listing of all models would already make the system crash. Therefore, we have reconsidered our naming strategy and we have chosen a tree structure, in which the name of
each subdirectory reveals information about the fundamental parameters involved, as follows:
grid/T20000/g22/B15/He10Si449, where “grid” is the parent directory. Subdirectories have
the following meaning: “T20000” stands for an effective temperature of 20 000K, “g22” for
an effective gravity log g of 2.2, “B15” gives information about the wind properties (both log
Q and β) and “He10Si449” about the abundances of helium and silicon. The character in
the name of the wind subdirectory refers to log Q (see Table 3.1 for their meaning) and the
number is ten times β. “He10Si449” means a solar He-abundance n(He)/n(H) = 0.10 and a
solar Si-abundance log n(Si)/n(H) = -4.49 (Asplund et al. 2005). Each model is linked to a
unique number as a reference.
1 Grid
available upon request from [email protected]
86
A grid of FASTWIND NLTE model atmospheres of B stars with winds
Table 3.1: Meaning of the characters in the wind subdirectory.
letter
O
a
A
b
B
C
D
log Q
-14.30
-14.00
-13.80
-13.60
-13.40
-13.15
-12.70
meaning
negligible wind
very weak wind
weak wind
weak to normal wind
normal wind
moderate wind
strong wind
3.4 Grid analysis
3.4.1 Diagnostic lines and their isocontours of equivalent width
Thanks to the broad scope of the BSTAR06 grid compared to the smaller (supergiant) grid
presented in the previous chapter, we are able to show the new isocontours for the equivalent
width (EW) of some selected lines spread over a vast part of the (Teff , log g)-plane (Fig. 3.2).
They are based on the complete model grid and represent the dependence of each line on
both the effective temperature and the gravity. Each of these selected lines will have its own
significance in the development of the automatic procedure, as will be discussed in the next
chapter. Obviously the discussion of the isocontours written down in Section 2.4.2 remains
valid and will not be repeated here.
3.4.2 Convergence properties of the grid
After seven months of almost uninterrupted calculation to finish the grid, one is still not ready
to actually use the grid. First we have to assure ourselves that the models are correct. We have
to make sure that there are no defects left due to system crashes during the calculation period.
It is also important to check if the requisite of flux conservation is fulfilled (better than 2%)
and if the models converged properly. We divided the models in six groups following their
convergence properties.
When the transition between the photosphere and the wind is not smooth, but has a kink (this
can happen, e.g., when the model lies too close to the Eddington limit), then, in the worst
case, the code stops abruptly and no hydrodynamic model atmosphere is created (group 0),
or, in the more optimistic case that the hydrodynamic model is created, the calculations for the
temperature correction do not converge (group 1), i.e. the last relative temperature correction
(δT /T ) is larger than 0.3% (cf. Section 1.4.3).
3.4 Grid analysis
87
Figure 3.2: Isocontours in the (Teff , log g) plane for several diagnostic lines. Isocontours
shown are for weak winds only. Left panel: Si II 4128 (top), Si III 4552 (middle) and
Si IV 4116 (bottom). Right panel: He I 4471 (top), He II 4541 (middle) and Hγ (bottom).
88
A grid of FASTWIND NLTE model atmospheres of B stars with winds
If the temperature structure converged well2 , we additionally need to check the convergence
of the occupation numbers of both the explicit and the most abundant background ions. The
convergence rate (δn/n) should be below 0.003 as well3 . In case this requirement is fulfilled, we attribute these models to group 6, which contains the best models. On the other
hand, there are certain situations, in which we can still consider the model as converged, even
though the required limit of 0.003 is not attained. This can happen when a) δn/n is monotonically decreasing or oscillating below the 5% level (group 5). The simple explanation for this
is that the number of predefined iterations was insufficient, and enlarging it usually solves the
problem. b) δn/n is oscillating between very similar numbers above 0.05 (group 4): usually
the numbers between which the oscillatory behaviour occurs, stay within a deviation of 5 to
10%. We subdivided this group in 3 subgroups following the oscillatory behaviour: 4c and 4b
are models oscillating between numbers which typically differ less than, respectively, 5 and
10%, 4c is the group for which the difference is larger. The latter group is considered as diverged, the former two are considered as converged. Since only the high levels (i.e. the higher
occupation numbers) oscillate, and not the lower levels, this does not affect our diagnostics,
which justifies our decision to consider them as converged. The remaining two groups are
filled with worse models, namely with those where δn/n is monotonically increasing below
0.05, i.e. pure divergence (group 3) or when there is simply no clear pattern, i.e. divergence
or last values decrease, but above 0.05 (group 2).
An overview of the requisites to attribute an object to one or another group can be found in
Table 3.2. Groups 4 (b and c) to 6 can be considered as ‘good’ models. For all other models,
we have performed several improvements in an attempt to achieve the convergence of these
models. Most problems seemed to occur for the coolest dwarfs. It can happen that, under
certain circumstances, for these low Teff , high log g models, the electron density oscillates.
We cured this problem by forcing the electron densities for these models (i.e. Teff ≤ 11 500,
log g ≥ 2.5 and Teff = 12 000 K, log g ≥ 3.0) into LTE, which is the correct solution
anyway. All other calculations were left in NLTE. For the remainder, and in the cases where
this did not solve the divergence, a slight change in β, as well as a larger radial grid mesh,
mostly brought a solution, so that finally only a small percentage of the grid remained nonconverged.
The attribution of the models to the different groups after these modifications were carried
out can be found in Table 3.3. In Fig. 3.3, we display the location of the non-converged
BSTAR06 grid models (of group 0 to 3) in the (Teff , log g) plane, after the modifications
were carried out. This figure shows that mostly cool models remain diverged. In Figs 3.4
and 3.5, we display the absolute distribution of the models as a function of a few of the
fundamental parameters. When looking at the absolute numbers, one must keep in mind that,
in general, there are fewer grid models towards hotter temperature regimes, because of the
smaller gravity range. To account for this, we show, in the right panels of Figs 3.4 and 3.5,
the relative distribution within each group, represented as a cumulative distribution function
(cdf).
2 This is usually obtained after some 30 to 40 iterations. As the temperature update is done only every two
iteration cycles, this means that convergence is normally reached after some 15 to 20 temperature updates.
3 The convergence should be reached within a predefined number of iterations. Although 70 iterations is usually
more than sufficient, we have nevertheless considered 100 iterations as a save limit.
89
3.4 Grid analysis
Table 3.2: Models divided in groups following their convergence behaviour, from worst case
scenario (group 0) to best case scenario (group 6).
group
0
1
2
3
4a
4b
4c
5
6
δT /T
δn/n
> 0.003
< 0.003
< 0.003
< 0.003
> 0.05
< 0.05
> 0.05
< 0.003
< 0.05
< 0.003
< 0.003
behaviour
no pattern
monotonic increase
oscillatory (> 0.10)
oscillatory (> 0.05, ≤ 0.10)
oscillatory (≤ 0.05)
monotonic decrease
or oscillatory
decision
no model created
T not converged
no decision
divergence
divergence
convergence
convergence
convergence
perfect convergence
Table 3.3: Breakdown of the models into the different convergence groups, for
log n(Si)/n(H) = -4.49 (solar), -4.79 (depleted) and -4.19 (enhanced).
group
0
1
2
3
4a
4b
4c
5
6
solar
number percentage
631
0.71 %
556
0.63 %
147
0.17 %
0
0.00 %
372
0.42 %
146
0.17 %
1155
1.31 %
13356
15.13 %
71942
81.47 %
depleted
number percentage
724
0.82 %
180
0.20 %
229
0.26 %
30
0.03 %
250
0.28 %
144
0.16 %
854
0.97 %
10163
11.51 %
75731
85.76 %
enhanced
number percentage
772
0.87 %
351
0.40 %
279
0.32 %
26
0.03 %
351
0.40 %
96
0.11 %
677
0.77 %
9530
10.79 %
76223
86.32 %
We find that all solar Si ‘group 0’ models have an effective temperature below 12 000 K,
mainly with gravities between 2.6 and 3.6 (see Fig. 3.4). The 10 500 K model seems to cause
most problems. A very similar behaviour can be observed for the ‘group 1’ objects. For the
group 3 models, it seems that mainly low temperatures, in combination with either very low
or very high mass loss rates, causes the models to diverge (see Fig. 3.5). For these diverged
models, we have to live with the fact that, at this position, we will have a missing model in
the grid. This mainly occurs for giant and subdwarf models at 10 500 and 11 500 K.
Since they start from the same model structure, it is obvious that, if the model was not created
for the solar model, then also not for the enhanced or depleted ones. For the other groups
it seems that, sometimes, the non-solar models switch from one group to another, but the
general distribution is very similar to the solar one. Since there are more group 6 models in
the depleted/enhanced case than in the solar case (mostly coming from group 4(b,c) and 5),
this means that a restart with different Si abundances implies a better convergence cycle, i.e.
a faster converging than for the solar starting model. This means that the starting model must
90
A grid of FASTWIND NLTE model atmospheres of B stars with winds
Figure 3.3: Location of the non-converged BSTAR06 grid models in the (Teff , log g) plane
(i.e. groups 0 to 3). Black filled circles represent the solar models, grey and open circles
represent the models that belong to this group even though the solar variant does not belong
to this group.
have been almost converged, except for certain less important levels.
Now that we have set up the grid and thoroughly checked its convergence properties, we
are ready to take the next steps towards an automatic procedure for line profile fitting in
Chapter 4.
3.4 Grid analysis
91
Figure 3.4: Absolute and cumulative (relative) distribution of the BSTAR06 ‘group 0’ (top
box) and ‘group 1’ (bottom box) models as a function of Teff and log g. Full black lines
represent the solar Si grid, grey lines the Si enhanced models and dashed lines the Si depleted
models.
92
A grid of FASTWIND NLTE model atmospheres of B stars with winds
Figure 3.5: Absolute and cumulative (relative) distribution of the BSTAR06 ‘group 2’ (top
box) and ‘group 3’ (bottom box) models as a function of Teff and log Q. Full black lines
represent the solar Si grid, grey lines the Si enhanced models and dashed lines the Si depleted
models.
Chapter
4
Spectral analysis of the GAUDI B
star sample using a grid based fitting
method
Lefever, K.; Puls, J.; Morel, Th.; Briquet, M.; Decin. L; Aerts, C., 2007, AnalyseBstar:
an automated tool for the high-accuracy estimation of fundamental parameters for B
type stars - Application to the B-type stars in GAUDI (in preparation, to be submitted
to A&A)
In this chapter, we present the first automatic tool for the spectroscopic analysis of B-type
stars, covering the complete B-type spectral range: AnalyseBstar. We discuss our choice for
a grid method, and outline the basic properties of the algorithm. After thorough testing, we
apply the method to a sample of B-type stars, observed in preparation of the CoRoT space
mission. The wide spread in spectral types and the accuracy achieved for the fundamental
parameters, allows us to verify the currently widely used photometric calibrations in Teff and
log g.
94
GAUDI B star sample
4.1 CoRoT and GAUDI
4.1.1 The space mission CoRoT
December 27, 2006, 14:28 GMT: literally an explosive minute in the lives of every European
asteroseismologist and planetary scientist. The French-led European space mission CoRoT
(Baglin et al. 2002) has finally (and successfully!) been launched on top of a Soyuz 2.1b
carrier rocket from the ground station in Baikonour. After launch, CoRoT (COnvection,
ROtation, and planetary Transits) was placed on a circular, polar orbit, at an altitude of about
900 kilometers above the Earth atmosphere, allowing for continuous observations of two large
and opposite regions in the sky. The viewing zone is in the equatorial direction, perpendicular
to the orbital plane, thus avoiding any perturbation by the Earth straylight. The first target
field is towards Orion (the anticenter field, at right ascension 6h 50m , in the winter). As
the Earth rotates around the Sun, the telescope will repoint from time to time, in order to
keep pointing to the same field. After 150 days, when the Sun starts to interfere with the
observations, the spacecraft will turn 180 degrees towards the centre of the Milky Way (the
centre field, at right ascension 18h 50m , in the summer). Within each of these two large
regions, many different fields are selected to be monitored. CoRoT has a nominal lifetime of
2.5 years. During this period it will perform uninterrupted high-cadence (32s) high-precision
photometry (on µmag level!) per field (2.8 x 2.8◦ ) during 5 months. This never achieved
precision raises high expectations for the quality of the data and for a breakthrough in our
understanding of the stellar interiors. Indeed, one of the main objectives of the mission
is to perform asteroseismology of more than one hundred dwarfs with the goal to extract
more information from their interior. A few of the brightest among them are monitored
intensively for up to five months, whereas, simultaneously, up to nine fainter dwarfs per field
are observed. Additionally, CoRoT will provide high-accuracy photometry of thousands of
fainter objects, down to V = 16 with a cadence of 512s. In this way one hopes to achieve
information for a wide diversity of stars, covering the HRD as completely as possible.
But CoRoT has more in store. A second main objective is the search for exoplanets. Using
the transit-method, it will be attempted to detect extrasolar planets down to the size of our
own Earth.
Space-based photometry has this advantage over ground-based photometry, that it can monitor almost continuously, in this way avoiding to a very large extent any one-day alias problems. Moreover one is no longer disturbed by absorption from the Earth’s atmosphere. In
addition to these advantages, also the achieved signal-to-noise ratios are several orders of a
magnitude better than can be obtained using ground-based facilities.
4.1.2 The birth of GAUDI
The observational setup of CoRoT (observations of a small number of bright stars over a very
long period) makes target selection a crucial issue. To fully profit from the CoRoT data, it
95
4.1 CoRoT and GAUDI
Figure 4.1: Artist’s view of the satellite mission CoRoT. Courtesy: CNES
http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=29381
is of major importance to assess the properties of the potential targets and thus to gather as
much a priori information as possible on them. Double-lined spectroscopic binaries (SB2s)
should be avoided, because disentangling the oscillation information of both components
from their composite frequency spectra is a very complicated process. Also, the identification of all kinds of peculiarities in both photometry and spectroscopy is necessary. Besides
this, it is important to dispose of an accurate estimate for the physical parameters of the star
(rotation, effective temperature, surface gravity, etc.), in addition to the seismic information
that will be obtained, in order to fully understand their evolution (as discussed in Chapter 1).
Due to the acute shortage of available information on the potential CoRoT targets, the need
for additional data was high. Therefore, an ambitious ground-based observing program for
more than 1500 objects was set up, under the leadership of C. Catala at Meudon, to obtain
Strömgren photometry as well as high-resolution spectroscopy. These data were collected in
an extended catalogue, maintained at LAEFF (Laboratorio de Astrofı́sica Espacial y Fı́sica
Fundamental) and baptised GAUDI: Ground-based Asteroseismology Uniform Database Interface (Solano et al. 2005). Meanwhile, various research groups from different countries
have gathered forces to take up the huge work of analysing and interpreting all the data. Various methods have been used, each of them valid within a certain parameter range, or for a
certain type of objects. These parameters (if necessary provided with additional comments)
have been incorporated in the GAUDI database. A hierarchy between the various methods
has been adopted, to provide users with the best value available for each object. The information gathered in this database is finally fed into CoRoTSKY, the dedicated tool to prepare
and schedule the observations for CoRoT1 .
1 http://smsc.cnes.fr/COROT/A
corotsky.htm
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GAUDI B star sample
4.1.3 GAUDI Data
Strömgren photometry Each summer and winter, from 2000 until 2004, two weeks of
observation time at the OSN (Observatory Sierra Nevada) were completely devoted to the
CoRoT ground-based photometry programme.
In these runs Strömgren-Crawford uvbyβ and Ca II H&K photometry were obtained for all
potential targets. These data have been used to derive estimates of their effective temperatures, surface gravities and metalicities (Amado et al. 2004).
Echelle spectroscopy Most spectroscopic observations were obtained by using high-resolution echelle spectrographs on different sites: FEROS (La Silla, R ∼ 48,000), ELODIE
(France, R ∼ 42,000) and SARG (La Palma, R ∼ 46,000), complemented with some additional spectra obtained by CORALIE (La Silla, R ∼ 50,000), GIRAFFE (South-Africa,
R ∼ 39,000) and the coudé spectrograph at the 2m telescope in Tautenburg (Germany, R ∼
35,000).
4.2 Responsibilities of the Institute of Astronomy in Leuven
As a partner in the CoRoT team, ‘we’ were in charge of the detailed spectroscopic analysis
of the massive O and B-type stars. Since for these stars (except for the Be stars), no other adequate methods were available, our method (detailed spectroscopic analysis) has been ranked
first on the hierarchy list of confidence2.
The very first step to be considered, before really getting started, is to make a (rough) preselection of the targets:
– Be stars were a priori excluded. Be stars are main-sequence or slightly evolved stars, i.e.
non-supergiants. Usually they are fast rotators, surrounded by an equatorially concentrated envelope (disk) fed by discrete mass-loss events caused by yet unknown mechanisms. In the HRD, early Be stars are located at the lower border of the instability domain
of the β Cephei stars, while mid and late Be stars reside among the SPB stars. The presence of a disk causes the Balmer lines (especially Hα) to be in emission (generally, but
not necessarily, double-peaked, depending on the inclination angle). Since also the line
profiles of other atoms and the continuum level can be severely affected, they need a special approach. The FASTWIND code is not developed to treat these stars properly and
therefore we exclude them from our sample. They are the responsibility of the CoRoT Be
star working group led by A.-M. Hubert and C. Neiner: see Neiner et al. (2005), Frémat
et al. (2006) for the results of their investigations.
2 http://corot.oamp.fr/book/VI.3.pdf
4.2 Responsibilities of the Institute of Astronomy in Leuven
97
Figure 4.2: Breakdown by spectral type of the 190 B-type stars with spectra available in the
GAUDI database.
– Also the double-lined spectroscopic binaries (SB2) were a priori excluded from our sample, since their combined spectra make an accurate fundamental parameter estimation impossible as long as their flux ratios (and therefore their individual masses) are not known.
Also these stars need a special approach, which is the responsibility of the Binaries Thematic Team led by C. Maceroni & I. Ribas.
– The spectra with low SNR (below ±100) have to be separated from the ones with high
SNR, since bad quality data can never lead to good quality (i.e. accurate) fundamental
parameters.
– Fast rotators should be identified. Fast rotation smears out the spectral line features and
causes severe blending. Diagnostic lines become strongly affected, which makes it impossible to correctly estimate the fundamental parameters. This of course depends on
how fast the star is rotating (mainly problematic if v sin i is above 200 km/s) and on the
spectral type of the star, since the number and the strength of diagnostic lines depend on
it.
After the exclusion of the Be stars and the spectroscopic binaries, we were left with 190 Btype and 11 O-type stars. In Fig. 4.2 we show the subdivision of the B-type stars following
their spectral type (which is taken from the astronomical database SIMBAD). It is immediately clear that the number of B8 and B9 type stars is very large compared to the number of
stars at earlier spectral types. Since spectral classification relies on spectral characteristics,
these 54 B8 stars, or these 77 B9 stars, are all expected to have very similar spectra (apart
from different rotational or turbulent broadening), which means that also the resulting physical parameters are expected to be similar. Instead of analysing all stars separately, we deem it
a better approach to select a few representative cases with the best quality data to perform the
detailed spectroscopic analysis. Besides saving a lot of time, this has two more advantages:
1) it keeps our standard of high-accuracy analyses high and 2) these targets will be ideally
suited to check the validity of the currently widely used fundamental parameter calibrations.
Considering the extended sample of the GAUDI B stars to be analysed, it is no longer realistic
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GAUDI B star sample
to perform all analyses by hand. Since no automatic tool for the spectral analysis of B-type
stars has been designed so far, we considered the time ripe to do so. In the near future, such
tool will especially prove very useful once more large-scale observing programs are being set
up, or when more and more observations of B-type stars are being gathered, and a fast and
accurate acquisition of their stellar parameters is required. Note that it is possible to apply this
procedure also for other objects (besides B stars), once suitable alterations are carried out. In
what follows we will discuss the steps considered and undertaken towards an automatic tool.
4.3 AnalyseBstar: an automatic tool for the
spectroscopic analysis of B-type stars
We present in this section the first design of an automatic method for the spectroscopic analysis of B stars with winds, which we have baptised “AnalyseBstar”. It is based on the extensive
grid of NLTE atmosphere models presented in Chapter 3 and is developed to treat large samples of stars in a homogeneous way. In contrast to a fit-by-eye method (as used for the sample
of pulsating B supergiants in Chapter 2) an automated method is not only homogeneous, but
also objective and more robust. We could not apply this to the supergiant sample of Chapter 2,
because we had insufficient spectral lines per target.
Note that for objects without winds, an automatic tool to determine simultaneously the effective temperature, gravity, microturbulent velocity and silicon abundance, has been developed
by S. Becker, using the line formation codes DETAIL, SURFACE and ANALYS of Giddings
(1981). The method is based on the isocontours of EW (and their intersection) for different
silicon and Balmer lines in the (Teff , log g)-plane (Becker & Butler 1990; Becker 1991).
4.3.1 Spectroscopic line profile fitting: methods
The detailed spectroscopic analysis of B-type stars has always been a very time consuming
job for two main reasons. The first reason is the large parameter space covered by the photospheric and wind parameters that has to be explored. The second reason is the fact that there
exists no automatic method to derive these parameters in an objective way. Fitting is still
done by using a thorough fit-by-eye method, which requires the calculation of a few tens to a
few hundreds of models per star. Fit-by-eye is a time consuming job, since new models can
only be calculated after inspection of the previous generation of models.
To cope with the large dataset of ground-based observations of CoRoT, we investigate the
possibility of automated fitting. Spectral line fitting is a clear example of an optimisation
problem: the goal is to minimise the difference (or to maximise the correspondence) between
a given observed stellar spectrum and a theoretically predicted spectrum emerging from a stellar atmosphere model. To do so, one needs to search through the parameter space spanned by
the free parameters of the stellar atmosphere model, to find the best fit (i.e. global optimum).
There are several options for the way to proceed.
4.3 AnalyseBstar
99
Fit-by-eye method
This method is the most widely spread method for performing spectral line analyses. Starting
from knowledge about spectral type and previously published values, if any, one makes a
rough guess of the initial parameters by comparing accordingly chosen grid models with the
observations. Exploiting knowledge about the specific dependency of the theoretical line
profiles on one or more of the basic parameters, one tries to obtain an optimum fit in an
iterative way through the calculation of additional models that better match the observational
data. In order to clearly see the influence of the change of a certain parameter on the line
profiles, the best strategy is to change only one or very few parameters at the same time.
However, this strategy implies a few disadvantages, such as 1) it should be avoided to solve
a multidimensional problem in a onedimensional way. A real global optimum can only be
found by allowing all parameters to vary in parallel; 2) the amount of models to be calculated
for one star can become very large. Though FASTWIND models may be calculated very
fast, this way of working remains very time consuming. Scientists unfortunately do not have
eternal time to work on one object which implies that a) analysing large numbers of objects
is out of the question, and b) the investigated part of parameter space is necessarily restricted.
A good initial guess and good knowledge of how each parameter influences the strength, the
width or the shape of a line are thus indispensable. The latter, however, can only be built
up through experience. Altogether, the risk is high that the solution that appears optimal in
the eyes of the person who performs the fit in the above way, is maybe not the global but
only a local optimum. As suggested by its name, the method severely relies on individual
eyes, which induces that the method risks of being subjective. Since there exists no standard
procedure to follow when performing a fit-by-eye method, anyone else disposing over the
same spectrum, might follow a different sequence of changing parameters and might as well
find different resulting parameters. To overcome these drawbacks, it is necessary to automate
the line profile fitting procedure. This is possible in at least two ways, which we will described
in the next two sections.
Genetic algorithm
As a first option for an automated fit procedure, one may think of using a genetic algorithm
to search for an optimal solution (e.g., Mokiem et al. 2005, see below). A genetic algorithm
(GA) is only one of the different search techniques developed within the field of artificial
intelligence (AI). The algorithm is inspired by evolutionary biology (genetics) to which it
thanks its name, and from which it inherited its terminology and techniques, such as mutation, selection, inheritance, and recombination. It is generally used to find solutions to an
optimisation and search problem. In our specific problem to find an optimum fit to an observed spectrum resulting from a set of atmosphere models, the stellar and wind parameters
play the role of genomes, which, combined, uniquely define the characterising chromosome
of each atmosphere model (so-called individual). Evolution towards a better solution usually
starts from a population of randomly generated individuals, in our case a rough grid of atmosphere models, and happens in generations. Using the DNA information of each of the
models in the current generation, new generations are created through recombination and/or
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GAUDI B star sample
mutation, where the goodness-of-fit acts as the selection mechanism, following the principle
‘survival of the fittest’: the better a model fits, the higher the chance for it to be chosen as
a propagator. The new population is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when a satisfactory fitness level has been achieved.
The main improvement compared to the fit-by-eye procedure, is the fact that this method does
no longer require human intervention, thus excluding any human bias and saving a lot of time
in the calculation of ‘bad’ models. Indeed, the selection mechanism makes sure that every
new generation consists of “fitter” solutions than the parent generation. Moreover, cleverly
written genetic algorithms are able to jump out of local optima and to find a global optimum
in a multidimensional parameter space. On the other hand the genetic algorithm approach
also has a few disadvantages. A large population of models is required for each generation,
which usually takes a huge amount of processing time. Thousands of models need to be
calculated for one single object. The information obtained through the analysis of one object,
is usually lost when treating another object, i.e. for each new object the evolution starts from
an initial guess. Parameters are drawn more or less randomly, without taking into account any
knowledge of how the parameters influence the line profiles, whereas using this experience
could considerably fasten up the fitting process.
A grid-based method
A third option for finding the best fit between theoretical and observational line profiles is
grid-based and is different from the previous automated method in this respect that the method
makes use of an existing grid without calculating additional models. This requires the grid to
be comprehensive, as dense as possible and representative for the kind of objects one wants
to analyse. The ‘subjective’ eye has to be replaced by an ‘objective’ intelligent computer programme. Through the fit-by-eye analysis of more and more different objects, one gains more
and more experience in the line profile fitting procedure. One gradually develops a ‘feeling’
for the reaction of the model and of the predicted line profiles on a change in fundamental
parameters, and slowly but surely one can even predict how large a difference is needed to
improve the fit. This human experience, as well as knowledge of the studied object need
to be translated into computer code. Similar to the fit-by-eye method, this algorithm will
follow an iterative scheme, but now in a reproducible way. Fitness is no longer judged by
visual inspection, but is fixed by a goodness-of-fit parameter. Starting from a first guess for
the fundamental parameters, based on spectral type and/or published information, improved
solutions are derived by simply comparing line profiles resulting from well-chosen existing
gridmodels to the observed line profiles. The algorithm terminates once the fit quality cannot
be improved anymore by modifying the model parameters.
4.3 AnalyseBstar
101
Justification of our choice for the grid method
The first approach, the fit-by-eye approach, is, for our purposes, out of question. The method
is a long-winded process of trial and error, and is not the appropriate choice for handling large
samples of stars. Of course, we have to keep in mind that any automatic procedure will at
some point and to some extent still require some human intervention. However, this should
be reduced as much as possible to guarantee a global and objective solution.
Mokiem et al. (2005) put effort in writing an automated tool for the spectroscopic analysis
of massive stars. Their method was developed for the quantitative analysis of O- and early
B-type stars with stellar winds. Our aim is to develop a complementary method to cover
the complete B-type spectral range. Mokiem et al. (2005) opted for the genetic algorithm
approach, which is the most accurate one, but is also very time-consuming. In first instance,
they analysed 7 O-type stars in the young open cluster Cyg OB2 and 5 other Galactic stars
with their method (Mokiem et al. 2005). They also applied their automatic method to 31 O
and early B-type stars in the Small Magellanic Cloud (Mokiem et al. 2006), and 28 O and
early B-type stars in the Large Magellanic Cloud (Mokiem et al. 2007). Essentially, Mokiem
et al. (2005) linked FASTWIND to the genetic algorithm based optimisation routine PIKAIA
from Charbonneau (1995). They determined the stellar parameters by using continuum normalised hydrogen and helium lines, assigning weights to them to account for line blends
with species that are not taken into account, or to account for potential problems from the
theoretical side. For example, the He I singlets are predicted weaker by CMFGEN than by
FASTWIND in the temperature range between 36,000 and 41,000 K for dwarfs and between
31,000 and 35,000 K for supergiants (see Najarro et al. 2006 for the origin of this discrepancy). Mokiem et al. (2005) accounted for this discrepancy by introducing a lower weighting
factor for these He I singlets for stars in these ranges. Going down in temperature from the O
and early B-type stars towards the mid and late B-type stars, requires other diagnostic lines
besides hydrogen and helium. We need to complement these with (continuum normalised)
silicon lines to determine the effective temperature. However, the inclusion of the Si lines
in the genetic algorithm approach is not straightforward. Silicon is structurally much more
complicated than the H and He atoms, with many more transitions to account for. Moreover,
it simultaneously depends on many parameters. A genetic algorithm approach would require
a sophisticated weighting scheme for the different Si lines and frequency points. For reasons
discussed in Section 3.1, we have opted for a grid-method as an alternative approach.
Of course, we are well aware of the disadvantages inherent to this method:
1. Codes are developing quickly and continuously. Since the method makes use of a predefined grid, this implies that, as soon as a new version/update of the code is released, the
grid and consequently the derived fundamental parameters are out-of-date. This can happen when, e.g., atomic data are improved. The advantage, on the other hand, is that the
line-profile fitting method itself will remain and the job is done with the computation of
a new grid. This is of course a huge work, recalling that, for the grid described in Chapter 3, seven months were needed to compute the grid, without even taking into account
the amount of time spent on the grid setup before and the grid checking after the actual
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GAUDI B star sample
computation. Thanks to the fast performance of the FASTWIND code, however, this is
not really an insurmountable problem.
2. Whereas a genetic algorithm uses a dynamic grid, changing with every new generation and
evolving towards the final best fitting model, a grid-based method fully relies on a static
grid and no additional models are computed. Consequently, the quality of the final best fit
and the accuracy of the final physical parameters are fully determined by the density of the
grid. This underlines, again, the necessity of a grid which is as dense as possible, allowing
for interpolation. On the other hand, the grid-method has another plus-point speaking in
its advantage: it is fast! Whereas the analysis with the genetic algorithm approach can
easily take a few days for one target star, a well-designed grid-method will, on average,
be finished within half an hour, precisely because no additional model computations are
required.
4.4 Methodology of AnalyseBstar
In what follows, we will give a detailed description of AnalyseBstar: the preparation of all
input for the programme, the iteration cycle for the determination of each physical parameter,
the assumptions made, ... In Fig. 4.3 we present a flowchart of the full programme, with
references to the appropriate section, for the description of each step. The reader is advised
to follow this context diagram throughout the further description of AnalyseBstar in this
section. The AnalyseBstar programme is written in the Interactive Data Language (IDL),
which allows for interactive manipulation and visualisation of data. It fully relies on the
extensive grid of model atmospheres and the emerging line profiles, described in Chapter 3.
We use continuum normalised hydrogen, helium and silicon lines to derive the photospheric
properties of the star and the characteristics of its wind.
4.4.1 Preparation of the input
No automatic method can fully replace the ‘by-eye’ procedure, since there will always remain a few steps for which human intervention is required. This is mainly the case for the
preparation of the spectra and the input for the actual programme, including:
1. Observation of peculiarities
The first thing one should do when starting to analyse a star, is simply look at the spectrum and inspect whether any abnormal behaviour can be detected, such as line behaviour
resulting from nebulae, disks or other peculiarities. These are things that can only be
detected through inspection by eye.
4.4 Methodology of AnalyseBstar
Figure 4.3: Context diagram of AnalyseBstar. For a detailed explanation, see text.
103
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GAUDI B star sample
2. Merging and normalisation of the spectra
Echelle spectra are characterised by the fact that the spectrum is cut into pieces (i.e. orders)
by a diffraction grating. In general, the different orders have a small overlapping region.
This can be used to merge the orders into one spectrum. The merging, however, is a
very delicate and non-trivial process. The edges of the orders are often noisy, due to the
decrease in sensitivity of the CCD near the edges. These parts are useless and should
be removed. Moreover, some spectral lines fall right on the edge of the spectral order,
resulting in a cut-off within the spectral line. Lines that are cut-off in this way, in the
overlap region of either of the two orders, are not reliable enough to derive accurate stellar
information from them. They can, in the best case, only be used as a consistency check
afterwards.
Once the merging process is finished, the spectra are ready to be normalised. The most
frequently applied method is by interactive interpolation of continuum regions. The spectrum should be normalised over a large wavelength coverage, where sufficient continuum
regions are present. If not, line depths can get corrupted, which will further propagate in
the fundamental parameters derived from these lines. In our case the normalisation will
especially prove to be very important for the determination of the gravity, which depends
sensitively on the Balmer line wings.
GAUDI sample: The standard FITS binary tables of the FEROS and ELODIE spectra in
the database contain information about both the normalised spectrum, resulting from the
pipeline reduction of the instrument, and the unnormalised spectrum. Since the quality
of both the merging and the normalisation turned out to be insufficient for our detailed
analyses, we have redone this time-consuming job for all spectral ranges around the diagnostic lines. This part of the work was done by T. Morel. To ensure a smooth merging,
first the order edges (where the SNR is really poor) were cut off. Then, the orders were
scaled relative to the counts in the overlap region, before both were actually merged (see
Fig. 4.4). The merging is generally much better for FEROS spectra than for ELODIE
spectra. The reason for this lies in the fact that the FEROS orders are well corrected for
the blaze function and are therefore ‘flat’, which makes a smooth connection rather easy.
On the contrary, the correct reconnection of the ELODIE orders is very hard due to the
incomplete removal of the blaze function during the standard reduction. They are concave, rather than flat, which causes a sawtooth-shaped continuum, which abruptly jumps
at the wavelengths where two consecutive orders connect. Also the above mentioned edge
effects are stronger for the ELODIE spectra than for the FEROS spectra, which leads to,
e.g., discontinuities in the blue part. This cannot be completely removed, not even after
cutting off the extremities. The continuum rectification is performed over a broad spectral
range using cubic splines or sometimes Legendre polynomials for the fit. The fitting is an
iterative process in which pixels that deviate by more than a certain threshold (indicated
by the user) from the fit are excluded at each iteration.
For the few available SARG data, only the normalised spectra have been inserted into the
database and we failed to get hands on the raw data. The continuum rectification was really
poor, especially Hβ suffers a lot from this (see Fig. 4.5). The drop-off in the line wings
is too sudden, and is not at all as expected from a smooth velocity distribution. We have
observed a similar line behaviour in the red wings of the Hα profiles for the B supergiants
HD 91024 and HD 106068 (see 2.5.2 and 2.6.2) and in the blue wing of HD 111904 (see
4.4 Methodology of AnalyseBstar
105
2.6.2). The refill of the wing of Hα can be explained by the wind. However, Hβ only
becomes severely affected when the wind is very strong (which is not the case as we
have concluded from the inspection of Hα), and the line shape is different from what is
observed here, and merely affects the line core. We think the rectification through a cubic
spline has been made too deep into the wing, resulting in the observed sharp decrease, too
narrow wings and a weaker line, since the inner part of the line will be pulled upwards.
Other line profiles are affected in a similar way. We have overplotted the observed Hβ
profile in Fig. 4.5 with what would be a more reasonable shape of the wings. Also the
wavelength coverage is insufficient for our purposes. The spectral range is confined to
[4620, 6845] Å with a gap of 250 Å between 6100 Å and 6350 Å. Moreover, due to bad
merging, the regions [5760, 5800] Å and [5350, 5385] Å cannot be used. This means that
almost all diagnostic He and Si lines are lost. Therefore, we excluded all SARG spectra
from our sample.
We want to put special emphasis on the fact that the huge and precise task of merging
and normalising all spectra has been done by the same person and in a uniform way using
IRAF3 , which is important for assuring a homogeneous treatment of the sample.
3. Line selection
The line selection is done in a automatic way. The lines that will be used in the spectral
analysis will be read directly from a file. It is the user who decides which lines will be
used, by removing or adding lines to this file. As a standard procedure we compute a
number of line profiles in the optical (see below) from the underlying model atmosphere,
complemented with a few lines from the Paschen and Brackett hydrogen line series in the
infrared. The latter infrared lines will not be used in this study, but can contain important
information for future users. These standard line profiles can directly be used. Computing
additional line profiles is, however, straightforward.
We distinguish between two types of lines:
– The first group of lines will be used for the prediction of the physical parameters
during the spectral line fitting procedure. These are:
Hα, Hβ, Hγ and Hδ
He I 4026, He I 4387, He I 4471, He I 4713, He I 4922 and He I 6678
He II 4541 and He II 4686
Si II 4128-4130 and Si II 5041-5056
Si III 4552-4567-4574, Si III 4716, Si III 4813-4819-4829 and Si III 5739
Si IV 4116 and Si IV 4212
Note that we have excluded He II 4200. This line can savely be used for O-type stars,
but not for B-type stars. At least for dwarfs, it only appears at temperatures above ±
25,000 K, where it is still overruled by N II, while at hotter temperatures, He II 4200
is for 50% blended by N III4 .
3 IRAF (Image Reduction and Analysis Facility) is distributed by the National Optical Astronomy Observatories,
operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the
National Science Foundation, USA.
4 See http://www.lsw.uni-heidelberg.de/cgi-bin/websynspec.cgi
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GAUDI B star sample
Figure 4.4: Illustration of the merging and normalisation process for two orders of the
FEROS spectrum of HD 170580, concentrated around Hγ, which is used for deriving log g.
Upper panel: orders 9 (black) and 10 (dark grey) before merging. Bad parts of both orders are
cut off to obtain the merged spectrum (light grey), which has been moved downwards with
flux value 1 for reasons of clarity. Lower panel: After the merging, the normalisation is done
and a continuum rectified spectrum is obtained (black). We have overplotted the pipeline
normalised spectrum (grey), to illustrate why the renormalisation was necessary.
4.4 Methodology of AnalyseBstar
107
Figure 4.5: Example of the bad normalisation of the SARG spectra for Hβ. The observed
and continuum rectified spectrum of HD 48347 (black) is overplotted with an appropriate
theoretical spectrum, accounting for the presence of Stark-broadening and a smooth velocity
distribution (grey).
– The second, smaller group contains additional lines which will not be used in the
fitting procedure, for reasons of uncertainties, either in the theoretical predictions,
either in the observed spectrum (due to problems in merging or normalising). Their
goodness-of-fit is only checked to investigate systematic differences between theory
and observations. This can be useful input for considerations regarding further improvements of atomic data and/or atmospheric models.
These are: Hǫ, He I 4010, 4120, 4140 and Si IV 4950.
4. Signal-to-noise measurement
Since the signal-to-noise ratio SNR can vary a lot with wavelength, we have chosen to
calculate the local SNR for each line separately, by using a continuum region close to
the line. This is done manually through the indication of the left and right edges of the
continuum interval. The SNR is then given by the mean flux Fc divided by the standard
deviation σ(Fc ) of the flux in this interval, i.e., SNR = Fc /σ(Fc ).
5. Equivalent width determination
The equivalent widths of the He and Si lines are measured by a Gaussian non-linear least
squares fit to the observed line profiles, using the Levenberg-Marquardt algorithm. We
have thoroughly tested whether this was a justifiable approach, by comparing the equivalent widths resulting from a Gaussian fit from a proper integration of the line pixel values
themselves for different line profiles of different stars. It seemed that, indeed, the difference between both approaches is minimal. The measurement of the observed equivalent
widths is in this way reduced to a simple integral of the Gaussian profile.
The Gaussian absorption profile (emission profiles need a different treatment and are
therefore omitted so far) can be written as
"
2 #
1 λ − A2
f (λ) = 1 − A0 . exp − .
,
2
A1
108
GAUDI B star sample
where A0 is the line depth, A1 is the half-width-at-half-maximum and A2 is the centre of
the line profile.
The begin and end wavelength of the line can be identified in two ways: the manual
identification is done by simple mouse-clicks, the automatic identification is done by the
determination of the inflection points nearest to the laboratory central wavelength, through
the calculation of smoothed second order derivatives. We have developed a routine which
is very similar to the underlying technique in the ARES code of Sousa et al. (2007). The
ARES code (Automatic Routine for line Equivalent widths in stellar Spectra5 ) has recently
been developed for the automated determination of equivalent widths of the absorption
lines present in stellar spectra, and has been tested mainly for cool stars for measuring
the EW of Fe lines. The basic principles of our code are the same as ARES. However,
slight modifications have been carried out to meet our own specific requirements. The
ARES code, e.g., uses a second order polynomial fit to determine the position of the local
continuum over a small wavelength range around the, already rectified, line profile, for
which the EW is required. We considered a first order polynomial (i.e. a linear fit) to
be a better approach, since a second order polynomial might over- or underestimate the
equivalent width of the line.
An automated method for the EW determination is powerful for spectra with a very high
SNR. However, for our sample of GAUDI stars, it appeared insufficient due to the noise
level of our data. Therefore, for our sample, we kept the manual method, so that we still
have control on this process. To simplify the identification of the begin and end wavelength of the line by manual mouse-clicks and to avoid measuring the EW of other strong
lines which lie, by coincidence, very close to the desired line profile (i.e. to avoid misidentification), we have indicated all transitions of the considered ion (i.e., all transitions for
Si, when a Si line is considered, or, all transitions for He, when a He line is considered),
which have a gf -value6 above 10−3 (see upper panel Fig. 4.6). This is especially important when the considered lines are very weak.
For He I lines, a Gaussian profile is not always the best approximation due to a strong
forbidden component in the blue wing of certain lines. Therefore, in the case of He I lines,
the user can decide, by looking at the fit, whether the Gaussian approximation for these
lines is good enough or if (s)he prefers the equivalent width to be measured as the integral
of the observed profile itself.
The main source of uncertainty in the EW determination is introduced by the noise and
its influence on the exact position of the continuum within the noise. To account for this
we have shifted the continuum level up and down by a factor (1 ± 1/SNR), performed
a new Gaussian fit through both shifted profiles and measured the difference in EW (see
lower panel Fig. 4.6). Note that, by considering a constant factor over the full line profiles
(using the signal at continuum level), we implicitly give an equal weight to each wavelength point, even
p though, in reality, 1/SNR may be slightly larger in the cores, since it is
proportional to 1/S, with S the obtained signal at these wavelengths. So, the noise is
one source of errors to be taken into account. When the user deems it necessary, (s)he can
apply a correction for the continuum level by a local rerectification, e.g. in the case that
the normalisation performed over a large wavelength range is locally a bit offset. If the
5 http://www.astro.up.pt/∼sousasag/ares/
6 The
gf -value is the product of the statistical weight g and the oscillator strength f .
109
4.4 Methodology of AnalyseBstar
offset is different on the blue and red side of the line, then the factor should be wavelength
dependent. Therefore, we allow for a linear rerectification by a factor aλ + b. A third
factor is the error introduced by using a Gaussian fit to estimate the EW.
The total error on the EW can then be estimated as follows (since the individual error are
independent from each other):
p
∆EW = (∆EWSNR )2 + (∆EWcont )2 + (∆EWfit )2 .
∆EWSNR is the error in EW with respect to pure photon noise statistics. According to
Vollmann & Eversberg (2006) it equals
s
Fc (∆λ − EW )
∆EWSNR = 1 +
,
SNR
F
where Fc is the average continuum flux, measured outside the line, and F the average flux
in the line, which covers a spectral range ∆λ.
∆EWcont is the difference in the equivalent width (of the Gaussian fits) when the local
continuum is shifted by a factor (1 ± 1/SNR) (aλ + b). Due to the noise, the interactive
continuum identification is a non-unique and subjective process. By introducing the factor
(1 ± 1/SNR), we account for small shifts in the continuum level. On the other hand, local
rerectifications are accounted for by introducing the factor (aλ + b).
∆EWfit is the uncertainty in the Gaussian profile which includes the error on the line
depth A0 as well as on the half-width-at-half-maximum A1 in the integral.
Since the fitted EW can be written as
√
EW = − 2π A0 A1 ,
the error follows as
∆EWfit =
√ q
2π (∆A0 )2 A21 + (∆A1 )2 A20 .
6. Determination of v sin i
The determination of the rotational velocity is the last step in the preparation of the input
for AnalyseBstar. As mentioned in Section 2.4.1, we use the semi-automatic tool developed by Simón-Dı́az et al. (2006) to derive reliable projected rotational velocities. The
method is based on the fact that only the rotational broadening function has zeros in the
Fourier transform, in contrast to, e.g., the broadening due to macroturbulence and the instrumental profiles, which are Gaussian. As the convolutions with the different broadening
profiles are transformed into multiplications in Fourier space, one can simply divide the
Fourier transform of the observed profile by the transform of the broadening functions that
have no zeroes. The frequency of the first minimum in the residual transform corresponds
to the rotational velocity. Care has to be taken at high frequencies (low v sin i ), since
also zeroes from the microturbulence are introduced there (Gray 1973; Simón-Dı́az et al.
2006).
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GAUDI B star sample
Figure 4.6: Illustration of the EW determination for He I 4026 (for a simulated spectrum).
Top: All He transitions in the considered wavelength interval, with a log gf down to −3,
are indicated as vertical lines. This makes it easier for the user to correctly identify the line,
which could sometimes be a problem in case of noisy spectra or very weak lines. Through
simple mouse-clicks, the user indicates the begin and end wavelength of the line. Bottom:
Within the indicated wavelength interval, a Gaussian fit to the observed line profile is made
to determine the EW of the line (thick grey profile). Also indicated are: the centre of the
line (vertical line) and the shift in flux, upwards and downwards (dashed line profiles), used
to account for the noise level in the determination of the EW. Also to these line profiles a
Gaussian fit is made, but these are omitted here for the sake of clarity.
4.4 Methodology of AnalyseBstar
111
Figure 4.7: Illustration of the determination of v sin i from the first minimum in the Fourier
transform of the selected line profile, as described by Simón-Dı́az et al. (2006). The user can
make several attempts and finally decide which v sin i gives the best match. The difference
in the slope of the first decay gives an indication of the macroturbulence, which in this case
is absent. The Fourier transform of the observed line profile is indicated in black. The user
indicates the first minimum in the black profile, which gives the projected rotational velocity.
The Fourier transform of the rotational profile at this vsini is indicated in grey.
This Fourier Transform (FT) method seems to be a very powerful tool to separate the
effects of rotational broadening and macroturbulence. We refer to Simón-Dı́az & Herrero
(2007) for a thorough discussion.
We use the following least blended, metallic lines for the determination of v sin i : Si II 5041,
Si III 4567, 4574, 4813, 4819, 4829, 5739, C II 4267, 6578, 6582, 5133, 5145, 5151 and
O II 4452.
The task of the user is twofold:
1) identify each line (if visible in the spectrum), i.e., indicate the line region, the continuum
level and the line centre. For the sake of comfort, the line region for each line is selected
automatically and all transitions of this certain species (Si, C or O) with a log gf value
above 10−3 are indicated. 2) select the first zero. The user can make up to ten trials and
then decide which v sin i -value is the most appropriate one (see Fig. 4.7). In this way,
we have obtained a reliable v sin i -value for each selected line. The projected rotational
velocity v sin i of the star is then calculated as the mean of these values, and the standard
deviation as its uncertainty.
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GAUDI B star sample
4.4.2 Starting values for the fit parameters
Once the preparation is finished, we can start with the automatic procedure to derive an
accurate value for all physical parameters. This procedure follows an iterative scheme, in
which the fundamental parameters are derived in the following order:
– The effects of the effective temperature, the Si abundance and the microturbulence on the
Si lines are separated.
– The He abundance is fixed from the helium lines.
– The macroturbulent velocity is determined from well-chosen Si lines.
– The surface gravity is determined from the wings of Hγ, Hδ and, in the case of weak
winds, also Hβ.
– The wind parameters log Q and β are determined in parallel using Hα.
Obviously, to start the iterative procedure, we need a good initial guess for each of these
parameters. This initial value can either be user supplied or standard. If no value is set by the
user, then the following initial values are considered:
– The initial effective temperature will be determined from the spectral type of the star.
– We start from the highest gravity in the grid (i.e. log g = 4.5), but immediately improve
this value to a more realistic guess by running the subprocedure for the determination of
the gravity.
– The initial abundance for He is taken as solar, and for Si, we consider the ‘typical’ value
for B-type dwarfs, i.e. log n(Si)/n(H) = -4.79 (cf. Section 4.5.2).
– We initialise the wind parameters at the lowest values, which means negligible wind, at
log Q = -14.30 and β = 0.9. These are reasonable assumptions for dwarfs, which comprise
the largest part of our sample.
– The macroturbulent velocity is initially zero, whereas for the microturbulent velocity we
have taken a medium value of all possibilities, i.e., 10 km/s.
It is important to choose these initial parameters as well as possible, since they will fully
determine how fast the method will converge to the final solution. This could make the
difference between three minutes and a quarter of an hour of computation time.
4.4 Methodology of AnalyseBstar
113
4.4.3 The cycle of the combined determination of the effective temperature, microturbulence and abundances
The silicon lines serve multiple purposes. By using a well-defined cycle, which is based on
the procedure outlined in Urbaneja (2004), we are able to separate the effects of the effective
temperature, the Si abundance and the microturbulence on the line profiles, and derive an
accurate value for them. Our method automatically determines from the observed spectrum
which lines will be used for this purpose. The lines should be well visible (i.e. clearly distinguishable from the noise in the continuum) and the relative error on their equivalent width is
typically below 10 to 15%, dependent on the combination of the temperature and the specific
line considered. Only lines with a relative error of less than 75% on their equivalent width
will be considered in the following procedures. Such large errors are a rare exception and
only occur for the weakest lines, which almost disappear into the noise level.
Basically, we can discern two different cases, namely when multiple and consecutive ionisation stages of silicon are available, or when there is only one ionisation stage of silicon. Both
need a different approach, as we explain now.
1) Multiple ionisation stages of Si available
Effective temperature
Since the Si abundance affects all Si lines in a similar way, we can, at first instance, put aside
the effect of the Si abundance by considering the EW ratios of two different Si ionisation
stages. For each effective temperature point in the grid, the ratios of the observed equivalent
widths of Si IV to Si III and/or Si III to Si II are compared to those obtained by the model grid
of Chapter 3, given a certain log g and wind parameters (see Fig. 4.8). Whereas we consider
both the effects of microturbulence and silicon abundance simultaneously in AnalyseBstar,
we have separated them here in the figure (top versus bottom) for the sake of clarity in the
presentation. From this figure, it is immediately clear that a model with an effective temperature of 18,000 K is at this point of the iteration cycle the best solution. The effects of
both the microturbulence and the Si abundance become larger towards the hotter temperature
regime, due to the fact that there is hardly any Si II left in this region. The same effect occurs
towards the cool temperature regime, where there is less and less Si III. Consequently, the
observed effect on the line ratio is mainly due to the effect on one of both ionisation stages
alone. For each combination of Si IV/III and/or Si III/II, this gives us a range of ‘acceptable’
effective temperatures for which the observed EW ratio is reproduced, within the observed
errors (indicated as grey symbols in Fig. 4.8). From the different line ratios, slightly different
temperatures may arise. In the assumption that log g and the wind parameters are exact, the
combination of all these possibilities constitute the set of acceptable effective temperatures.
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GAUDI B star sample
Microturbulence and silicon abundance
Once we have a list of acceptable temperatures, we can derive for each temperature the best
microturbulent velocity and the best Si abundance. To this end, we use the “curve of growth”
method, which describes how the line strength increases with the number of atoms producing
the line.
In what follows, we describe the full process, step by step:
Step 1: Deriving the abundances for each microturbulence, line per line
In this first step, we compare, for each Si line, the observed EW of the line to the theoretically
predicted EW, for each combination of Si abundance and microturbulence in the grid (see
Fig. 4.9). For each microturbulent velocity, we determine, by simple interpolation, which Si
abundance would reproduce the exact value of the observed EW. The upper and lower errors
for the Si abundance are derived from similar interpolation for EW ± ∆ EW. This is done
for each Si line separately. In Fig. 4.9, we only show a few selected lines. Several more are
used in case they are clearly present in the spectrum. In the example shown, no Si IV was
observed, so only Si II and Si III were considered.
Step 2: Deriving the mean abundance for each microturbulence and the slope of the
best fit
In a second step we compare, for each microturbulence, the abundances (with the derived
uncertainties) found in step 1 to the observed EW of each line (see Fig. 4.10). Since a star
can only have one Si abundance, we should find the same Si abundance from all the different
Si lines, i.e. the slope of the best fit to all the lines should be zero. From the different
abundances derived from each line, we can calculate the mean abundance.
Step 3: Deriving the ‘interpolated’ microturbulent velocity
In a third step, we investigate the change of the slope when jumping from one microturbulence
to the other (see upper panel Fig. 4.11). Through linear interpolation, we find the microturbulence for which the slope would be zero (i.e. for which the Si abundance derived from each
line separately would be the same). This is the estimated ‘interpolated’ microturbulence.
Step 4: Deriving the ‘interpolated’ abundance
Using now the relation between the microturbulence and the mean abundance derived in step
2, we interpolate to find the estimated ‘interpolated’ Si abundance at the estimated ‘interpolated’ microturbulence found in step 3 (see lower panel Fig. 4.11).
Step 5: Deriving the ‘interpolated’ effective temperature
Now that we have obtained the estimated ‘interpolated’ values for the microturbulence and Si
abundance, we can go back to Fig. 4.8 and interpolate in two dimensions to derive a value for
the effective temperature, which reproduces the observed EW. The mean of the effective tem-
4.4 Methodology of AnalyseBstar
115
peratures from each line ratio will then be accepted as a value for the ‘interpolated’ effective
temperature of the star.
Step 6: Determination of the ‘closest’ grid values
For the next steps in the automatic procedure, we will need the closest grid values to these
estimated ‘interpolated’ values rather than the real values themselves. These closest values
will constitute a new entry in the list of possibilities. If the real value for the Si abundance or
microturbulence falls exactly between two grid points, then both are added to the list.
We repeat these steps for each of the ‘possible’ grid temperatures, initially derived from
Fig. 4.8 (grey symbols), and obtain in this way a set of new possible solutions that optimally
reproduce the Si lines.
Helium abundance
For each (Teff , ξ, log n(Si)/n(H)) combination, found in the above described way, we can now
determine the He abundance, denoted as n(He)/n(H). As for the Si abundance, we have only
three different values to consider. We apply a similar method as for the Si abundance determination, in the sense that we derive the best-suited abundance from each line separately, and
take the mean value as the ‘interpolated’ value. Figs 4.12 and 4.13 illustrate this process. Note
that we intrinsically assume that the microturbulence is the same throughout the atmosphere,
i.e. that there is no radial stratification. In this way, the microturbulence, necessary to account
for the broadening in the He lines, will be assumed to be the same as the one derived from
the Si lines. Note also that the uncertainty in the derived He abundance will be larger when
the He abundance is lower than 0.10. Indeed, in this case we are namely forced to extrapolate
to a region where it is unclear how the dependence of the equivalent width with abundance
will change. The decrease towards abundances lower than solar may be steeper or follow a
logarithmic trend, in which case the predicted values would be underestimated. In this case
we can only state that the solar abundance is an upper limit of the true abundance.Values
lower than the primordial helium abundance of ∼0.10 can no longer be considered physical,
except when diffusion effects start to play a role and helium settles. In that case, the helium
abundance observed at the stellar surface could be lower. Such chemically peculiar stars are
referred to as He weak stars and are usually high-gravity objects. Helium settling was not
taken into account in our models.
2) Only one ionisation stage of Si available
If we do not have two consecutive stages of Si (e.g. for the late type stars, where we only
have Si II), we have to follow a different procedure. Fortunately for the late type stars, He I
is very sensitive to changes in effective temperature (see Fig. 2.3). Therefore, we can use the
joint predictive power of He I and Si II to derive, in first instance, the plausible values for
Teff . We compare the fits of all combinations of Si abundance, microturbulence and effective
temperature, and select only the best ones.
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GAUDI B star sample
Figure 4.8: Synthetic simulation of the effect of the microturbulence (top) and the Si abundance (bottom) on the logarithmic EW ratios of silicon lines of different ionisation stages.
This example gives the (synthetic) EW ratio of Si III 5739 to Si II 4130 against a selected
range of effective temperatures. The ratios are evaluated for the different possibilities of the
microturbulence, for fixed Si abundance (upper panel: ξ = 3, 6, 10, 12, 15 and 20 km/s, respectively triangle up, square, diamond, circle, asterisk, triangle down) and for the different
possibilities of the Si abundances, for fixed microturbulence (log n(Si)/n(H) = -4.79, -4.49
and -4.19, respectively circle, triangle up and square). The horizontal lines show the observed EW ratio and its corresponding uncertainty region. Acceptable EW ratios, which fall
within these boundaries, are indicated in grey.
4.4 Methodology of AnalyseBstar
117
Figure 4.9: Step 1 (for each Teff and for each line): line per line, for each microturbulent
velocity, the abundance that reproduces the observed EW of this line is derived. The different
abundances for each microturbulent velocity are connected by the dashed-dotted lines. The
observed EW interval is indicated with horizontal grey lines. Crosses indicate, for each microturbulent velocity (from low EW to high EW: 3, 6, 10, 12, 15 km/s), the values derived for
the Si abundance (log Si/H by number, and the upper and lower limit), found through linear
interpolation. The displayed example is a synthetic simulation.
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GAUDI B star sample
Figure 4.10: Step 2 (for each microturbulence): we plot the Si abundances and their uncertainties for all 6 lines for which we derived these values in step 1, as a function of the
observed EW. The least squares fit to the abundances are shown in grey. The microturbulent
velocity for which the slope of this fit is zero (i.e. equal abundance from each line) gives an
estimate of the ‘interpolated’ microturbulent velocity. The displayed example is a synthetic
simulation.
119
4.4 Methodology of AnalyseBstar
Figure 4.11: Step 3 & 4: the ‘interpolated’ microturbulent velocity and the accompanying
‘interpolated’ Si abundance (indicated in grey) are derived from the position where the slope
of the best fit (step 2) is zero. The silicon abundances, given in the lower panel, are the mean
abundances derived in step 2. The displayed example is a synthetic simulation.
The criterion for selection is based on the following, standard, expression of the loglikelihood
function (see Decin et al. (2007) for a broader theoretical explanation)
l≡
NX
lines
j=1
"
√
1
−ln(σ j ) − ln( 2 π) −
2
EWobs,j − EWsyn,j
σj
2 #
.
(4.1)
In essence, this formula gives the squared deviations between the observed and synthetic
equivalent widths (EWobs and EWsyn ), summed over all lines considered (Nlines ), and taking
into account the observed error σj on the EW (assuming that the theoretically predicted value
is perfect). In this way the loglikelihood function is very similar to a simple chi-square
criterion. The model for which the loglikelihood distribution reaches its maximum, will fit
the observed data best. The distribution allows for a certain deviation from this maximum,
by taking into account the observed errors on the equivalent width. The criterion determining
which models will be taken into account for the further analysis and which not, is set by the
number of free parameters p and the required significance level α. If we want to determine
only the effective temperature (i.e. one free parameter) from the equivalent widths of the lines
of He and Si, within a 95% confidence interval, then the criterion can be written as follows:
2 (lmax − l) ≤ χ21 (0.05)
(4.2)
with lmax the maximum loglikelihood, l the loglikelihood of any other model and χ21 (0.05)
the chi-square distribution with 1 degree of freedom, evaluated for the commonly used 5%
120
GAUDI B star sample
Figure 4.12: For each helium line, the most suited helium abundance is derived through interpolation. The theoretically predicted values are shown as filled circles, the interpolated
values as crosses. The horizontal lines indicate the observed equivalent width and the observed errors. The displayed example is a synthetic simulation.
significance level. Every Teff (and according model), for which the EWs of the synthetic line
profiles satisfy this criterion will be considered as a possible solution for Teff . Generally, in
this way, the temperature is very well constrained. We will further elaborate on the use and
suitability of this loglikelihood function in the next section.
Since we have only one stage of each ion, we have no means to derive any information
about the abundances of Si nor He. Moreover, it is no longer possible to follow the same
scheme as before, to derive the microturbulence. Therefore, we assume that we know at
least one of these three parameters. Although the He lines are not only dependent on the
He abundance, but also on the microturbulence, it are still the Si lines alone that are used
to fix the value for ξ (besides the Si abundance of course). This means that one can derive
only one parameter from the He lines, but two have to be fixed from the Si lines. Therefore,
the best approach is to assume we know one of those two parameters that are fixed from
the Si lines, and consequently derive the other two free parameters. We have chosen to fix
the Si abundance. We will consider all three possibilities for the Si abundance, although we
attach more credence to the lowest abundance (log n(Si)/n(H) = -4.79), since this appears
4.4 Methodology of AnalyseBstar
121
Figure 4.13: The real helium abundance is calculated as the mean over all lines. The abundance derived from each line separately is represented by the filled circles, while the arrows
show the derived errors. The best linear fit is indicated as the dotted line, while the horizontal
full line shows the mean value. The displayed example is a synthetic simulation.
to be the more ‘typical’ abundance for B-type dwarfs (cf. Section 4.5.2). For each initially
accepted Teff (from the loglikelihood function), and for each Si abundance, we search for the
best value for ξ from the Si lines (see Fig. 4.14), and consequently, using this microturbulence
allows us then to fix also the best He abundance (in the same way as before). Note that, again,
we assume that ξ is the same throughout the atmosphere, and thus for each type of ion. We do
not allow microturbulences higher than 20 km/s, which is, in view of the low temperatures, a
quite conservative and safe upper limit.
Since the surface gravity is closely related to the effective temperature, we will need to redetermine log g for every new Teff . However, before determining log g, we first determine the
macroturbulent velocity since this plays a certain role for the determination of log g as well,
as will be shown in Section 4.4.5.
4.4.4 Macroturbulence vmacro
The effect of the macroturbulence on the line profile is similar to that of the rotation, in this
sense that it only changes the shape of the line profile, but not the strength (i.e. the EW). The
macroturbulence considered is characterised by a radial-tangential model (Gray 1975). In this
model the velocity field is modelled by assuming that a fraction (Ar ) of the material is moving
radially while the complementary fraction moves tangentially (At ). In our model for vmacro ,
we use the standard assumption that the radial and the tangential components are equal in
122
GAUDI B star sample
Figure 4.14: Once a set of possible effective temperatures is fixed through the application
of the loglikelihood criterion, we can determine for each of these temperatures (in this case:
10,500 and 11,000 K) the right microturbulence, under the assumption that we know the Si
abundance. Resulting combinations are shown for log n(Si)/n(H) = -4.19 (open circles), 4.49 (grey filled circles) and -4.79 (black filled circles). The displayed example is a synthetic
simulation.
fraction. Assuming additionally that the distribution of the velocities in the radial and the
tangential directions are Gaussian and that the specific intensity is independent of the angular
coordinate, the effect of macroturbulence can be introduced in the synthetic line profiles by
means of a convolution with a function, which consists of a Gaussian and an additional term
proportional to the error function (see equation (4) in Gray 1975). Only weak lines should
be used to determine vmacro . For B-type stars, silicon lines are the best suited, since they
most obviously show the presence of vmacro in their wings. The user decides which Si lines
should be used throughout the full process. To determine the strength of vmacro , we convolve
the Si profiles of the (at that time) best fitting model with different values of vmacro 7 . The
consecutive values considered for the convolution are chosen using a ‘bisection’ method,
with initial steps of 10 km/s. The bisection continues as long as the stepsize is larger than
or equal to 0.1 km/s, which will be the final precision of the vmacro determination. For each
considered vmacro -value, we compute its loglikelihood in order to quantify the difference
between the wings of the obtained synthetic profile and the observed line profile, as follows:
"
2 #
n
X
√
1 yi − µi
,
(4.3)
−ln(σ) − ln( 2 π) −
l≡
2
σ
i=1
with n the considered number of wavelength points within the line profile, yi the flux at
wavelength point i in the wings of the observed line profile, µi the flux at point i of the
synthetic profile (convolved with the vmacro under consideration), and σ the noise, i.e. 1/SNR,
7 At
this point the profile is already convolved with the appropriate rotational and instrumental profiles.
4.4 Methodology of AnalyseBstar
123
Figure 4.15: Determination of the macroturbulent velocity of β CMa for three different Si
lines. The panels should be read from left to right as follows.
Left: The left and right edge of the line region (i.e. outer most dots at continuum level) are
indicated by the user at the start of the procedure. The blue and red edges (grey vertical lines)
are determined as the point where the flux is half of the minimal intensity. They determine
how far the wings extend towards the line centre. The black dots represent the points that are
finally used to determine the macroturbulence.
Middle: Obtained loglikelihood as a function of vmacro (see Eq. (4.3)). Note the broad maximum in the loglikelihood distribution, which gives rather large error bars.
Right: The fit with the best vmacro is shown as the grey profile.
of the considered line profile. The considered wings extend from the position where the
intensity is half as deep as the core, until the inflection points at continuum level (which were
identified during the EW determination).
The same procedure is repeated for all considered line profiles, and the mean, derived from
the different lines, gives then the final vmacro . The error is set by the standard deviation. From
Fig. 4.15, it will be clear that the shape of the loglikelihood distribution near the maximum can
be quite broad, which indicates that the errors in vmacro with respect to the loglikelihood are
significant. This might be related to the fact that, so far, we have used only the information
contained in the line wings to determine vmacro . It needs to be investigated in the future
whether the errors can be reduced when using also the line core information.
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GAUDI B star sample
4.4.5 Surface gravity log g
The surface gravity log g is the result of fitting the Balmer lines. We will mainly use Hγ
and Hδ for this job, possibly complemented with Hβ in case of weak winds (i.e. if log Q ≤
−13.80). Since Hǫ is somewhat blended, and the merging is not always reliable for this region
(see earlier), this line will not be used as a primary gravity determinator, but will only serve
as an additional check (together with Hβ, in case of stronger winds). Since we want to define
the profile only from the extreme wings down to the strongest curvature (see discussion of
the different contributions below), we have to account for certain mechanisms, which affect
the core of the Balmer lines to some extent. Therefore, we will have to exclude the inner ∆λ
which will be the minimum of the following three ∆’s:
1. ∆1 = (λ1 - λ0 ) with λ0 the rest wavelength and λ1 the wavelength where the intensity is
half as deep as the core. We do this on both sides and use the minimum. This gives a good
guess about the position where under typical conditions the curvature is strongest.
2. ∆2 = λ0 . (v sin i + vmacro )/2c (with c the speed of light) takes into account the influence
of the rotational and macroturbulent velocity on the line core. The factor 2 enters due to
the fact that approximately only half of the width is folded into the core.
3. ∆3 = 3 vtherm λ0 /c represents the effective thermal Doppler core width of the Stark profile. vtherm combines the effect of the thermal velocity vth and the microturbulence ξ, and
can be written as
q
2
2 + ξ2
with
vth
= 2 kB Teff /mH
vtherm = vth
in which kB is the Boltzmann constant and mH the proton mass. The factor 3 in ∆3 arises
from the fact that the Doppler profile
√
2
1
e−(∆λ/∆λD ) ,
π∆λD
with
∆λD
λ0
=
c
r
2 kB Teff
+ ξ2
mH
for ∆λ/∆λD = 3 is smaller than the corresponding Stark profile, such that the Stark
wings dominate over pure Doppler broadening (as discussed, e.g., in Section 4.1 and
Fig. 3 of Repolust et al. (2005)).
Once we have removed the inner part of the line profile, we can compare the observed Balmer
lines with the synthetic line profiles, by considering all possibilities for log g which are available at this given effective temperature gridpoint. We decide which gravity is the best, by
maximising, for each log g, the loglikelihood
2 #
nj "
NX
lines X
√
1 yj,i − µj,i
−ln(σj ) − ln( 2 π) −
,
(4.4)
l≡
2
σj
j=1 i=1
4.4 Methodology of AnalyseBstar
125
with Nlines the number of Balmer lines used, nj the considered number of wavelength points
within each line profile, yj,i the observed flux at point i of line j, µj,i the expected (i.e.
theoretically predicted) flux value, and σj the noise of each separate Balmer line.
With this procedure, we came across the following problem: Hγ and Hδ (our main gravity
indicators) can sometimes be seriously affected by a large amount of blends. If one ignores
this, one will find too high a gravity, since the loglikelihood function will take all these blends
into account (see Fig. 4.16). Therefore, we have to cut away also these blends and make the
wings a bit smoother. In order to avoid a manual intervention, we have also automated this
procedure. This is done by performing a sigma-clipping algorithm which keeps only those
points that have a flux higher than both neighbouring points (the so-called ‘high points’) and
where, at the same time, the difference is less than 1/SNR. After removing all other flux
points, only the so-called ‘line continuum’ is left (see middle panel Fig. 4.16). We interpolate
this continuum to obtain the flux at all original wavelength points, and the innermost part of
the line is added again (i.e. the part between the last blend in the blue wing and the first blend
in the red wing, but obviously still without the central core). Also all original flux points
which deviate by less than 1/SNR from the interpolated ‘line continuum’ are included again.
In this way we keep only the flux points which really determine the shape of the wing to
fit the synthetic Balmer wings, and we are able to determine an accurate value for log g (see
right panel Fig. 4.16). We realize that some points that may be marked as local continuum in
this way, may still be lower than what the real local continuum level would be, because of the
transition of one blend into another. However, this will only be the case for very few points,
which will give no significant weight to the least squares fit.
The finally accepted surface gravity will be this log g which gives the best fit to all selected
Balmer lines simultaneously. Half the gridstep in log g could thus be considered a good error
estimate. However, to account for the coupling with Teff , we consider 0.1 dex as a more appropriate error. Due to stellar rotation, the gravity derived from the Balmer wings is actually
not the real gravity, but the effective gravity. If one wants to calculate e.g. the mass of a star,
one should correct this value with a factor (v sin i )2 /R∗ , as explained in Section 1.5.2.
4.4.6 Wind parameters
A change in mass loss rate will mainly affect the shape of Hα. For cool objects and weak
winds, Hα is nearly ‘photospheric’ and will, in essence, be an absorption profile, with more
or less symmetric components, since the ‘re-filling’ by the wind is low. When the mass
loss rate increases, the particles will scatter more photons in the direction of the observer.
The absorption component becomes stronger, but especially the emission component gains
significant strength. Therefore, in the case of stronger winds, Hα will take the shape of a
typical P Cygni profile. For hot objects, where Hα is dominated by recombination processes,
and high mass-loss rates, the profile may even appear in full emission.
The wind parameter β determines the velocity law, which directly influences the density. The
red wing of Hα is well suited to determine β, since it is formed in the receding part of the
126
GAUDI B star sample
Figure 4.16: Illustration of the effect of blends on the determination of log g, and the clipping
algorithm to improve the fit, for β CMa. Left: The many blends in the wings of the profile
prevent an accurate least squares fit and lead to too broad wings, implying too high a gravity.
Central: From the observed line profile (black), only the ‘high points’ (grey dots), which
deviate less than 1/SNR from their neighbours, are kept, complemented with the full inner
part of the line, i.e., the part between the last blend in the blue wing and the first blend in
the red wing (indicated as grey vertical lines). Right: After the removal of all blends through
sigma clipping (see text), we are able to obtain a good representation of the line wings (grey
profile) and, therefore, to derive a very accurate value for the gravity.
(almost) complete wind, whereas the blue wing (either in emission or absorption) forms in
the small volume just in front of the stellar core, both by absorption and emission effects.
Therefore, errors due to certain assumptions in the standard model (rotation, clumping, etc.)
will be amplified in the blue wing, whereas in the red wing, which consists of the emission
from a much larger volume, most errors are more likely to cancel out. For a fixed mass loss, a
‘slower’ velocity law (i.e. a higher β value) will result in higher densities in the lower atmosphere, close to the star. This enlarges the number of emitted photons with velocities close
to the line centre, resulting in more emission. Around the central wavelength, the absorption
component of the line profile refills and the emission component becomes stronger. Therefore
the slope of the red wing of the P Cygni profile becomes steeper. In this sense, the steepness
of the red wing is a measure for the value of β.
We have estimated the wind parameters log Q and β by comparing the observed Hα profile,
with the different synthetic Hα profiles by making combinations of log Q and β. We have
decided to make the determination of the wind parameters not too sophisticated, by simply
using the best values for log Q and β available in the grid, without interpolating and further
refining. The reason for that is that almost all stars in our sample are dwarfs (they are chosen
like this by the CoRoT-team). In this sense we do not need a very sophisticated analysis of the
wind, since they have at most a weak wind. At the start of the programme, the user is asked
to select the type of Hα profile. Choices are: absorption profile (’a’), intermediate emission
profile with only a small emission peak (’i’) or a stronger emission profile (’e’). Then, for
each log Q-value, the corresponding best β-value is the one that gives the best fit to the red
wing (see above). In the case of a thin wind (i.e. absorption profile) the red wing extends
from the lower most flux point up to the line continuum. In case of a thick wind (i.e. emission
profile), the wing goes from the top of the emission peak, until continuum level. Once we
have selected for each log Q the best β, we decide on the best combination by considering
the best fit to the entire profile.
4.4 Methodology of AnalyseBstar
127
Figure 4.17: (Synthetic simulation) For each different wind strength parameter log Q (7 values, from left to right and from top to bottom: log Q =
−14.30, −14.00, −13.80, −13.60, −13.40, −13.15, −12.70), we search for the best β (in
each panel, the 5 different values for β are indicated: 0.9, 1.2, 1.5, 2.0, 3.0) by comparing
only the red wing (grey part of the profile). Then the synthetic profiles of each best (log Q,
β) combination are compared to the entire Hα profile (bottom right).
128
GAUDI B star sample
4.4.7 Final remarks
At the end of this iteration cycle an acceptance test is performed. When the method can run
through the whole cycle without needing to update any of the fit parameters, the model is
accepted as a good model, and gets the flag ‘2’. If a better model was found (i.e. when one
or more parameters have changed), the initial model is rejected and gets flag ‘0’, whereas the
improved model is added to the list of possibilities, and gets flag ‘1’ (cf. ‘models to check’ in
the flowchart diagram on page 103). As long as the list with possibilities with flag ‘1’ is not
empty, the parameters of the next possibility are taken as new starting values. Models that
did not converge properly are automatically skipped.
Line profile fits are always performed allowing for a small shift in wavelength, or radial
velocity (5 wavelength points in either direction), to prevent flux differences to add up in case
of a small radial velocity displacement.
To the major part, this method uses only the EWs and the far wings of the profiles. Cores are
not used, except for wind features. The profile shape only plays a role for deriving v sin i
and vmacro .
4.5 Formal tests and comparison for high-resolution spectra of selected β Cephei and SPB stars
4.5.1 Formal tests of convergence on synthetic spectra
Before applying AnalyseBstar to real, observed spectra, we first test whether the method is
able to recover the input parameters of synthetic spectra. For this purpose, we have created
several synthetic datasets in various regions of parameter space. We will not dwell on discussing all of them here, but we have chosen three specific examples, each representative for
a different type of star:
- Dataset A: a B0.5 I star with a rather dense stellar wind and a strongly enhanced helium
abundance,
- Dataset B: a B3 III star with a weak stellar wind, with ‘typical’ B star values for the abundances of He and Si (cf. Section 4.5.2),
- Dataset C: a B8 V with a very thin stellar wind.
Each dataset has been convolved with both a rotational and a macroturbulent broadening profile. The projected rotational velocity adopted is 50 km/s for each dataset, characteristic for a
‘typical’ slow rotator. An additional Gaussian, instrumental convolution was carried out, chosen in such a way that it is representative for the high resolution of the FEROS and ELODIE
spectra in our sample. Artificial (normally distributed) noise was added to mimic a real spectrum with a mean local SNR of 150, which is more or less the minimal local SNR obtained
for the GAUDI sample. Finally, we have gone through the full process of the preparation of
the spectra as if it were real data, i.e., defining the EW of all available lines, measuring the
4.5 Formal tests and comparison
129
SNR to account for the errors on the EW and fixing the projected rotational velocity. Since
we are dealing with synthetic data, obviously no normalisation was required.
Table 4.1 lists the input parameters for the three synthetic datasets as well as the output parameters, obtained from the application of AnalyseBstar. First we list the derived ‘interpolated’
values and in the third column, we display the parameters of the closest (best fitting) grid
model, which in the ideal case should be exactly the same as the input parameters. This
depends, however, on how well the equivalent widths were measured, and minor deviations
occur as expected.
Besides these three cases, in which we started from a synthetic model which is one of the grid
points, we have additionally created three synthetic datasets for which the parameters lie in
between different grid points. The input parameters and the results of the analysis have been
added to Table 4.1 as datasets D, E and F.
Although the models given in this table were selected as the best fitting models, the programme additionally came up with some other models, which agree with the input model
within the errors, introduced by the artificial noise and the errors in the determination of the
equivalent widths. In all six cases, the input parameters are well recovered. We considered
numerous other test cases, which are not included in this text, but for which the input parameters were equally well recovered. Slight deviations from the input parameters are as expected.
Also in the cool temperature domain (datasets C and F), where an alternative method is required, making an assumption about the Si abundance, we are still able to deduce reliable
parameter estimates, at least for the synthetic models. Also the derived v sin i -value, which
is difficult to disentangle from the macroturbulent velocity, agrees very well with the inserted
values, irrespective of vmacro . This shows that the Fourier Transform method of Simón-Dı́az
et al. (2006) indeed allows to separate both effects. All together, this gives us confidence
that our method is working fine and that the procedure will also be able to recover the true
physical parameters from real spectral data where the input is known to the star only.
Remark on the macroturbulent velocity
The macroturbulent velocity is the only fit parameter for which significant deviations arise,
if vmacro is low (< 30 km/s). Larger macroturbulences are nicely recovered, however. Especially for datasets C and E, where vmacro is only 10 km/s, the deviation is striking (of the
order of 20 to 30 km/s). To understand this discrepancy, we first had a look at the line profiles, which show that the fit is really good for this vmacro (see Fig 4.18). After verifying the
fit quality, we also verified that the discrepancies do not arise from our applied procedure,
by performing several tests on simulated spectra, in which we left vmacro as the only free
parameter. In all cases, the macroturbulent velocity was recovered very well, with deviations
within 5 km/s.
The (sometimes significant) deviations in vmacro can be understood in two ways. On the one
hand, slight deviations are expected, as the closest grid model and the observed line profiles
do not always match perfectly (due to small differences between the interpolated ‘real’ values
and the closest grid values). On the other hand, at present, we do not include the full line pro-
IN
OUT
interpolated
Dataset A
OUT
grid
IN
OUT
interpolated
Dataset B
OUT
grid
IN
OUT
interpolated
Dataset C
OUT
grid
23,000
2.7
C
2.0
0.20
-4.49
10
50
20
142
10
22,600
2.7
C
2.0
0.18
-4.49
10.2
48 ± 2
29 ± 4
23,000
2.7
C
2.0
0.20
-4.49
10
48 ± 2
29 ± 4
18,000
3.3
A
1.5
0.10
-4.79
15
50
30
494
21
18,100
3.3
b
1.2
0.08
-4.81
15
48 ± 4
33 ± 2
18,000
3.3
b
1.2
0.10
-4.79
15
48 ± 4
33 ± 2
13,000
4.2
O
0.9
0.10
-4.79
6.0
50
10
2344
128
13,000
4.2
a
0.9
0.08
-4.79
7.3
51 ± 1
30 ± 1
13,000
4.2
a
0.9
0.10
-4.79
6.0
51 ± 1
30 ± 1
fit parameter
Teff (K)
log g (cgs)
log Q (char)
β
n(He)/n(H)
log n(Si)/n(H)
ξ (km/s)
v sin i (km/s)
vmacro (km/s)
time (s)
6 checked models
=
Dataset D
21,120
3.98
b
1.42
0.14
-4.85
12
50
40
174
10
20,985
4.0
b
2.0
0.11
-4.80
15.7
52 ± 6
33 ± 5
21,000
4.0
b
1.2
0.10
-4.79
15
52 ± 6
33 ± 5
15,100
1.83
A
2.8
0.08
-4.23
11
50
10
392
26
15,030
1.80
A
3.0
0.09
-4.16
10.6
50 ± 7
39 ± 7
Dataset F
15,000
1.8
A
3.0
0.10
-4.19
10
50 ± 7
39 ± 7
11,880
2.43
a
1.02
0.18
-4.49
9
50
70
1254
95
11,500
2.3
O
0.9
0.20
-4.19
6.3
54 ± 1
67 ± 7
11,500
2.3
O
0.9
0.20
-4.19
6
54 ± 1
67 ± 7
GAUDI B star sample
Teff (K)
log g (cgs)
log Q (char)
β
n(He)/n(H)
log n(Si)/n(H)
ξ (km/s)
v sin i (km/s)
vmacro (km/s)
time (s)
6 checked models
=
Dataset E
130
Table 4.1: Input parameters for the synthetic models (IN) are compared to the actual output parameters (OUT) obtained through the application of AnalyseBstar to these synthetic data. Datasets
A, B and C are models which lie exactly on a grid point, whereas dataset D, E and F are models which lie in between the gridpoints. The ‘interpolated’ values derived for the parameters, as well as the
best fitting (i.e. closest) grid model are given. The parameters in italic are those for which no real interpolation was made, but for which the most representative grid value is chosen. We additionally
list the number of different models checked throughout the procedure and the time (in seconds) elapsed during the run of AnalyseBstar. The longer computation times for the cooler models are due to
the applied method in this range, as will be discussed in Section 4.6.3.
4.5 Formal tests and comparison
131
Figure 4.18: Example of the fit quality for dataset E. The slight discrepancy in the cores
of the Si lines arises from the difference in Si abundance between the interpolated value
(log n(Si)/n(H) = -4.16) and the closest grid value (log n(Si)/n(H) = -4.19)
132
GAUDI B star sample
file information in the prediction of the macroturbulence. As discussed in Section 4.4.4.,
line wings alone are perhaps not sensitive enough to changes in vmacro , and core information
may have to be inserted. This has not been implemented yet, since this requires intensive
investigation and testing, beyond the scope of the present work. We will tackle this problem
in the near future though.
As a conclusion, the values derived for vmacro should be treated with caution, since they
may not be representative for the real value. This does not affect the derivation of the other
parameters, however, since a convolution with the vmacro -profile preserves the EW of the
lines. It is the EW, and not the line profile shape, which is used for the derivation of Teff ,
ξ and log n(Si)/n(H). That other parameters remain unaffected, is clear from the very good
agreement between their input and output values in Table 4.1.
4.5.2 Testing AnalyseBstar on high-resolution spectra of selected pulsators
After we convinced ourselves that AnalyseBstar converges towards the optimum solution
and that it is able to recover the input parameters of synthetic spectra, we can go one step
further and test the procedure on real spectra. Before we move on to the GAUDI dataset,
however, we have tested our method on a selected sample of high-quality, high-resolution
spectra of pulsating B stars, mainly β Cephei and SPB stars, but also of a few other candidate
pulsators which are not yet confirmed. The (mean) spectra of these stars have a very high
signal-to-noise ratio, attained through the addition of a large number of individual exposures
(see, e.g., Morel et al. 2006). The spectra were first put in the laboratory rest frame, before
the weighted (by the SNR ratio) average of the whole time series was created. Because of
the high quality of these spectral time series, the average spectra are ideally suited for testing
AnalyseBstar. The β Cephei stars and two of the SPB stars have been analysed in detail by
Morel et al. (2006)/Morel (2007) and Briquet et al. (2007), respectively. These authors have
used the latest version of the NLTE line formation codes DETAIL and SURFACE (Giddings
1981; Butler 1984), in combination with plane-parallel, fully line-blanketed LTE Kurucz atmospheric models (Kurucz 1992), to determine the atmospheric parameters (negligible wind)
and element abundances.
Figs 4.19 and 4.20 show the fit quality obtained for a representative β Cephei and SPB star.
Tables 4.2, 4.4 and 4.3 list the results of our analysis for respectively the β Cephei stars, the
SPBs and the candidate pulsators. Whenever possible, we compare with the physical parameters found by Briquet & Morel (2007) for the SPBs and Morel (2007) for the β Cephei stars
and the candidate pulsators. In Fig. 4.21 we compare the effective temperatures and the surface gravities, as derived by AnalyseBstar, with the values derived with DETAIL/SURFACE.
The shaded area around the one-to-one relation indicates the uncertainty in Teff , as well as
in log g, on the derived DETAIL/SURFACE values, i.e. 1,000 K in Teff and 0.15 in log g.
The FASTWIND nominal errors (1,000 K in Teff and 0.10 in log g) stay within this area.
This figure shows the perfect agreement between the two independent parameter determinations. FASTWIND and DETAIL/SURFACE clearly give consistent results, which strongly
4.5 Formal tests and comparison
133
indicates that AnalyseBstar is indeed converging properly to the correct values, also for real
stars. In the lower panel of Fig. 4.23, we show the position of the pulsators within the HRD.
All β Cephei stars nicely fall in the corresponding instability strip, whereas there are a few
SPB stars which seem hotter than what is expected from the SPB instability strip.
The v sin i values derived by Morel (2007) are typically slightly larger than those derived
by us. This is readily understood as their value includes not only the rotational, but also
the macroturbulent velocity (as macroturbulence is not included in DETAIL/SURFACE). In
those cases where we derive vmacro = 0 km/s, the v sin i values are in perfect agreement.
As a conclusion, 13 out of 31 sample stars seem to display no or only negligible macroturbulence. Even though we stated earlier that we should be careful with the derived macroturbulences (see Section 4.5.1), we believe that these ‘zero’ values are quite representative,
and that indeed only negligible macroturbulence is present in these stars. Indeed, this ‘zero’
macroturbulence results as the average value from all considered Si lines of the star.
Niemczura & Daszyńska-Daszkiewicz (2005) derived basic stellar parameters (besides metalicities and angular diameters, also effective temperatures and surface gravities) for all β
Cephei stars observed by the International Ultraviolet Explorer (IUE) satellite mission. The
parameters are derived by means of an algorithmic procedure of fitting theoretical flux distributions to the low-resolution IUE spectra and ground-based spectrophotometric observations.
In this way they provide us with a means to compare the basic stellar properties derived from
an optical analysis (this work) with the ones derived from UV studies, at least for those β
Cephei stars in common. Once again, we can be delighted: Fig. 4.22 shows that most temperatures and gravities agree within the errors (grey zone). Only for ξ 1 CMa, the temperature
from the optical analysis is at least 2000 K higher than the one derived from the low-resolution
UV spectra. The gravity, however, agrees really well (3.89 compared to our value of 3.8). In
view of the spectral type, and the effective temperature and gravity derived by Morel (2007),
we can be quite convinced that, in this case, the UV value is offset.
Depletion in Si: real or artefact?
For the majority of our sample objects we find a depletion in silicon, with abundances generally between the solar and depleted grid point. We find a mean abundance of log n(Si)/n(H) =
−4.75 ± 0.06 for the β Cephei stars in our sample. The upper panel of Fig. 4.23 shows the
good agreement with the values derived by Morel (2007) for the sample of β Cephei stars.
Indeed, Morel (2007) also finds abundances which are typically lower than solar (on the average, log n(Si)/n(H) = −4.81 ± 0.08), which gives us confidence that the depletion in these
stars might be real. The abundance of the non-confirmed pulsators, however, leads to a mean
value of log n(Si)/n(H) = −4.65 ± 0.16, which is intermediate, and thus consistent with both
the solar and depleted Si abundance. We have verified that no significant trend exists between
the Si abundance on the one hand and the effective temperature or the microturbulence on the
other hand, since these parameters are derived from the same lines.
134
GAUDI B star sample
Figure 4.19: Example of the obtained fit quality for a hot giant (β Cephei) star: 15 CMa
(B1 III).
4.5 Formal tests and comparison
135
Figure 4.20: Example of the obtained fit quality for a cool main sequence (SPB) star:
HD 215573 (B6 IV). We have left out all line profiles which had zero EW. Note that the
Si II line prediction appears slightly stronger and broader than observed. This is due to the
fact that the real microturbulence is actually lower than the displayed 3 km/s, which is the
lowest grid value available at this grid point.
136
GAUDI B star sample
Table 4.2: Basic fundamental parameters of the sample of β Cephei stars, as derived from
AnalyseBstar. We list the values of the closest grid model, except for the Si abundance, where
the interpolated value is given. In italic, we display the results from the DETAIL/SURFACE
analysis by Morel (2007). Column ’He’ indicates the He-abundance n(He)/n(H), and column ’Si’ indicates the Si abundances log n(Si)/n(H). For a summary of the errors, see Section 4.5.3.
name SpT
Teff
(K)
log g
(cgs)
He
3.80
3.75
3.40
3.60
3.80
3.70
3.50
3.50
3.60
3.65
3.90
3.80
3.80
3.75
3.70
3.75
0.10
ξ 1 CMa B0.5-B1IV 27000
27500
15 CMa B1III
24000
26000
β Cep B1IV
25000
26000
β CMa B1.5III
24000
24000
12 Lac B1.5IV
24000
24500
δ Ceti B1.5-B2IV 23000
23000
γ Peg B1.5-B2IV 23000
22500
ν Eri B1.5-B2IV 23000
23500
0.10
0.10
0.10
0.10
0.10
0.10
0.10
ξ log Q
(km/s) (char)
Si
-4.69
6
-4.87 ± 0.21 6 ± 2
-4.79
12
-4.69 ± 0.30 7 ± 3
-4.70
6
-4.89 ± 0.23 6 ± 3
-4.76
15
-4.83 ± 0.23 14 ± 3
-4.70
10
-4.89 ± 0.27 10 ± 4
-4.80
6
-4.72 ± 0.29
1+3
−1
-4.84
<3
-4.81 ± 0.29
1+2
−1
-4.73
10
-4.79 ± 0.26 10 ± 4
A
O
A
O
a
a
O
O
-
β
v sin i vmacro
(km/s) (km/s)
2.0
9±2
10 ± 2
3.0 34 ± 4
45 ± 3
3.0 26 ± 3
29 ± 2
2.0 19 ± 4
23 ± 2
2.0 39 ± 12
42 ± 4
2.0 14 ± 2
14 ± 1
2.0 10 ± 1
10 ± 1
2.0 21 ± 3
36 ± 3
11
38
24
20
46
0
0
39
-
Table 4.3: Same as for Table 4.2, but now for the hot B pulsators which are not yet confirmed.
For τ Sco, we additionally compare with the values found by Mokiem et al. (2006), using his
genetic algorithm approach. These are indicated with the superscript M .
name
SpT
θ Car
B0Vp
Teff
(K)
31000
31000
τ Sco
B0.2V
32000
31500
31900M
HD 36591 B1V
26000
27000
ǫ CMa
B1.5-B2II/III 22000
23000
γ Ori
B2II-III
22000
22000
ι CMa
B2.5Ib-II
17000
17500
log g
(cgs)
He
4.20
4.20
4.20
4.05
4.15M
3.90
4.00
3.20
3.30
3.60
3.50
2.60
2.75
0.15
0.10
0.12M
0.10
0.10
0.10
0.10
-
Si
-4.38
-4.57 ± 0.23
-4.55
-4.76 ± 0.14
-4.73
-4.75 ± 0.29
-4.69
-4.77 ± 0.24
-4.79
-5.00 ± 0.19
-4.78
-4.82 ± 0.31
ξ log Q β
(km/s) (char)
10
12 ± 4
6
2±2
10.8M
6
3±2
15
16 ± 4
10
13 ± 5
15
15 ± 5
a
B
O
b
O
b
-
v sin i vmacro
(km/s) (km/s)
1.5
108 ± 3
113 ± 8
1.2
10 ± 2
8±2
0.8M
5M
3.0
13 ± 1
16 ± 2
2.0
32 ± 2
28 ± 2
3.0
46 ± 8
51 ± 4
1.2
27 ± 4
32 ± 3
0
0
0
20
37
39
-
137
4.5 Formal tests and comparison
Table 4.4: Same as for Table 4.2, but for the SPBs. For two SPBs, we also show (in italic)
the results of the DETAIL/SURFACE analysis by Briquet & Morel (2007).
HD number SpT
3360 B2IV
Teff
(K)
22000
22000
85953 B2IV
20000
21000
3379 B2.5IV 20000
74195 B3IV
16500
160762 B3IV
19500
25558 B3V
17500
181558 B5III
15000
26326 B5IV
15500
206540 B5IV
13500
24587 B5V
14500
28114 B6IV
14000
138764 B6IV
14500
215573 B6IV
13500
39844 B6V
14500
191295 B7III
13000
21071 B7V
13500
37151 B8V
12500
log g
(cgs)
He
Si
3.80
3.70
3.80
3.80
4.30
3.70
4.10
4.00
4.00
3.60
3.80
4.00
3.50
3.90
3.80
3.70
3.70
3.70
3.80
0.10
-4.76
-4.72 ± 0.30
-4.84
-4.75 ± 0.30
-4.73
-4.79
-4.86
-4.72
-4.79
-4.49
-4.79
-4.79
-4.79
-4.51
-4.49
-4.39
-4.79
-4.79
-4.79
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
ξ log Q
(km/s) (char)
β
v sin i vmacro
(km/s) (km/s)
<3
1±1
6
1±1
<3
<3
3
6
6
<3
3
<3
<3
3
<3
<3
3
3
<3
3.0
3.0
2.0
1.2
3.0
1.2
0.9
1.2
2.0
3.0
3.0
1.5
0.9
0.9
2.0
1.2
1.5
18 ± 2
19 ± 1
30 ± 1
29 ± 2
48 ± 8
18 ± 2
8±2
28 ± 2
17 ± 2
17 ± 1
15 ± 2
25 ± 4
21 ± 4
21 ± 2
8±1
16 ± 1
16 ± 1
22 ± 1
20 ± 2
O
O
O
a
O
O
a
a
O
O
O
O
A
O
O
O
O
13
20
43
18
0
31
0
17
0
21
17
0
0
0
15
0
0
Although this is not the first time that such low abundances have been observed, nobody
seems to have a clear answer to the question why Si should be underabundant. Without
attempting to be complete, we list here a few literature studies, which confirm the depletion in
Si. Kilian (1992) found a mean abundance of log n(Si)/n(H) ∼ −4.75 ± 0.2 for 21 unevolved
B-type stars in the local field and the nearby associations Ori OB1, Sco Cen, and Sgr OB1.
Daflon & Cunha (2004) performed a comprehensive and homogeneous abundance analysis
for 90 non-evolved late O to early B Galactic stars. They found a mean abundance of ∼ -4.80.
Morel et al. (2006) find an unweighted mean value for the Si abundance of −4.83 ± 0.05 for
their sample of nine β Cephei stars, which are prime targets for seismic theoretical modelling.
All nine targets are early B type stars on (or close to) the main sequence. All these studies
find similar depletions for the elements C, O, Mg, Al, S and Fe for the B-type dwarfs. From
these studies, it seems clear that the Sun is indeed more metal-rich than young nearby B
dwarfs.
On the other hand, there are also studies that show the contrary, namely that there exist Btype stars which do have values consistent with the solar abundance, as shown by, e.g., Gies
& Lambert (1992) and Rolleston et al. (2000). Crowther et al. (2006), in their study of early
B-type supergiants, fixed the silicon abundance to solar values and did not report on any
discrepancies which might have resulted, so we presume that their results did not show any
evidence to reject this assumption. Also Markova & Puls (2007) found values which agree
within 0.1 dex with the solar values for seven out of eight B supergiants. For one star they
find an increase in Si with 0.44 dex.
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GAUDI B star sample
The scatter in derived abundances is large, and each determination depends both on the specific method applied and on the assumed atomic data for silicon. N. Przybilla is currently
updating the Si model. Some preliminary studies indicate that Si III 4813-4819-4829 becomes stronger, whereas Si III 4552-4567-4574 remains more or less unaffected. We believe
that the change in the Si III 4800 triplet will not affect the Si abundance to such an extent
that this might cancel the depletion completely, certainly not because these lines are used in
parallel with several other Si lines. Moreover, only for 3 or 4 stars, lines from the Si III 4800
triplet was considered due to the fact that they are often severely affected by noise and/or
blends. Since the Si III 4500 triplet does not change significantly, we do not expect much
difference with the updated atomic data.
There might, however, exist a difference between supergiants and dwarfs, since several studies find values consistent with solar Si abundance (or, no discrepancies arose when assuming
it) for B-type supergiants, whereas there is a vast amount of literature pointing out a clear
depletion in the Si abundance for B-type dwarfs. However, a physical explanation has not
been established so far. We suggest the following one. All stars in our sample are solar
neighbourhood stars. As explained in the introduction (cf. Section 1.1), B-type stars evolve
so quickly that they generally stay near their birthplaces in the galactic plane. This means that
our sample stars must have been born in the vicinity of the Sun. Once a star is born, Si will
not be nuclearly produced until close to the supernova stage, as Si is a product of advanced
nucleosynthesis, which means that the sample stars should have Si abundances which do not
differ significantly from the solar one. The lower silicon abundance must thus be due to diffusion. Gravitational settling due to diffusion effects can indeed imply that the Si sinks towards
the stellar interior, such that the abundance observed at the surface is lower than the actual
one. This would be similar to the accumulation of iron in the driving zone of a 10 M⊙ star, as
suggested by Bourge et al. (2006), as well as an explanation of particular surface abundances
in terms of diffusive effects (Bourge et al. 2007). On the other hand, there apparently exists
a difference between main sequence and supergiant B stars regarding their surface Si abundance. We can explain this with the following scenario. During the main sequence phase,
silicon sinks due to diffusion effects. It accumulates in a region, located supposedly near the
iron opacity bump, where the radiative force overcomes the gravitational force acting on Si.
This means that Si is stuck there, which leads to the observed depletion at the surface. When
the B main sequence star leaves the main sequence, two convection zones are created. The
first one is located near the iron opacity bump, and, as Teff decreases, another one appears
near the He ionisation zone. These two zones are likely to merge and the result of this will
be a dredge-up of Si from its accumulation zone to the surface. In this way, the observed Si
abundance at the surface of B supergiants is restored to values similar to the one of the Sun.
This scenario explains both the observed depletion in main sequence dwarfs and the normal
Si abundance of B supergiants (Arlette Noels, private communication). Theoretical diffusion
computations of this scenario are not yet available.
If this scenario is true, we can be quite confident that this does not change the results presented in Chapter 2, where we adopted solar Si abundances for the analysis of the sample of
periodically variable B-type supergiants. For the majority of the SPBs, on the other hand, we
gave preference to a model with a depleted Si abundance, when a choice needed to be made,
i.e. when only Si II was available.
4.5 Formal tests and comparison
139
Figure 4.21: Comparison of the effective temperatures (top) and surface gravities (bottom)
derived with FASTWIND (AnalyseBstar) with those found by Morel (2007) and Briquet &
Morel (2007), using DETAIL/SURFACE, for the β Cephei stars (black), SPBs (grey) and
pulsators which are not yet confirmed to belong to one of these two classes (open symbols).
Supergiants are indicated as diamonds, giants as triangles and dwarfs as circles. The shaded
area around the one-to-one relation (straight line) denotes the uncertainty on the derived fundamental parameters of Morel (2007) and Briquet & Morel (2007) (i.e. 1,000 K in Teff and
0.15 in log g).
140
GAUDI B star sample
Figure 4.22: Comparison of the effective temperatures (top) and surface gravities (bottom)
derived from detailed spectroscopic analysis in the optical region (this work) with those derived from UV flux distribution fitting by Niemczura & Daszyńska-Daszkiewicz (2005), for
the β Cephei stars in common. The shaded area around the one-to-one relation (straight line)
denotes the uncertainty region of 1,000 K in Teff and of 0.15 in log g.
4.5 Formal tests and comparison
141
Figure 4.23: Top: Position of the considered B-type pulsators in the HRD. Theoretical instability domains are the same as in Fig. 2.14. Symbols have the same meaning as in Fig. 4.21.
Bottom: Si abundances derived from AnalyseBstar, compared to those derived from DETAIL/SURFACE, for the β Cephei stars and the non-confirmed pulsator. The black dotted
lines represent the standard solar abundance of log n(Si)/n(H) = -4.49, the grey dotted lines
represent a depletion and enhancement with 0.3 dex (i.e. -4.79 and -4.19 respectively). Symbols have the same meaning as in Fig. 4.21.
142
GAUDI B star sample
4.5.3 Error estimates
Since our method is fully based on finding the best fit within a predefined grid, the precision and the accuracy of the fundamental parameters is reflected by the grid mesh. Indeed,
under the assumption that all parameters would be independent, and could be determined irrespective of all other parameters, the ‘typical’ errors would be half the grid steps. In reality,
however, the determination of most parameters is to some extent linked to the value for the
others, and their accuracy will thus depend on how well each of these can be determined.
For example, as log g and Teff are closely coupled, an error in the determination of Teff will
automatically propagate to the derived log g-value. This is why we have tried to search for
as many independent diagnostics as possible. We are confident that we have found a good
way to reduce the error maximally, by using the described scheme to decouple Teff , ξ and
log n(Si)/n(H), using two ionisation stages and a sufficient number of lines. When we have
lines of only one ion present, the errors will be much larger and will depend on the assumed
abundance.
In general, a thorough statistically valid error propagation, including the interdependency of
all fundamental parameters in this multidimensional space, is extremely complicated (especially because the errors do not propagate in a linear way), and is certainly beyond the scope
of this thesis. The best way to get an idea of the errors is to use an empirical error propagation, by varying each parameter one by one, as has been done in Chapter 2. On the other
hand, it should be noted that the major uncertainty will come from the models themselves
in the sense that their input physics is an approximation of reality. If, e.g., there would be a
stratified ξ, then the results might be rather different.
For this reason, we avoided a time-consuming error propagation. In practice, we have adopted
the following ‘typical’ errors throughout the remainder of this investigation:
– ∆Teff : The errors on the effective temperature will be different above and below 20,000 K,
due to the different grid step, which is 1,000 K for temperatures above 20,000 K and 500 K
below 20,000 K. Since the derived Teff will to some extent depend on the accuracy of the
determined EW, we consider the grid steps as a good estimate for the errors on Teff .
– ∆log g: Even though half the grid step would give us an accuracy of 0.05 dex, we will,
also for log g consider the grid step of 0.10 dex as the typical error, in view of the strong
coupling with the effective temperature.
– ∆ξ: The grid steps can be considered as representative for the errors on ξ. Their values
depend somewhat on the actual derived values, since the grid microturbulences are not
equidistant. The errors will not exceed 6 km/s.
– ∆log n(Si)/n(H): In view of the crude grid steps in log n(Si)/n(H), we consider half the
grid step, i.e. 0.15, as a reasonable estimate for the error on the derived Si abundance.
– ∆n(He)/n(H): In analogy to the Si abundance, also for the He abundance, half the grid
step, i.e., 0.025 will be taken as a conservative error estimate.
4.6 Alternative strategy for a grid-based method
143
– ∆log Q: Since we restrict to the grid points to find log Q, without further refining, the
absolute errors are about half the grid steps, i.e. generally about 0.10 dex, at least for
strong winds. For weak winds (Ṁ . 10−7 M⊙ /yr or more or less for log Q . −13.80),
the errors soon become significantly larger. Indeed, in case of low mass loss rates, Hα is
nearly ‘photospheric’ and (almost) only the core will be affected.
– ∆β: Likewise the errors in log Q, the error in β is set by approximately half the grid step,
i.e. of the order of 0.20 dex for the three lowest values (0.9, 1.2 and 1.5) and of the order
of 0.5 dex for the larger values (2.0 and 3.0).
– ∆vmacro and ∆v sin i : The errors on the macroturbulent and rotational profiles are derived from the standard deviation of the values resulting from different line profiles, and
will be different for every object.
4.6 Alternative strategy for a grid-based method
The method described in Section 4.4 does not make full use of the grid, since it will only
check a few tens or hundreds well chosen models within the grid of almost 265,000 models.
The following questions may thus arise: “Is the optimal fit we find with the above described
method, really a global minimum? Or is it just a local optimum and are there other solutions
which might fit equally well, or even better, in another subspace of the grid?”. To answer
these questions, we have additionally considered an alternative approach for an automated
procedure for the line profile fitting procedure.
4.6.1 Description of the alternative method
Instead of working with a method which iteratively updates the fundamental parameters by
selecting only the best values at each iteration, another way to proceed would be the following. One could ‘simply’ compare the EWs of all models in the grid and the accompanying
line profiles with the observed ones. To select then, from this grid, only the best fitting models, we need to introduce a criterion which gives the goodness-of-fit of each model. This
criterion should help to decide which model is most likely the best representation of the true
stellar model.
For this purpose, we have introduced the following loglikelihood function, in analogy with
the one introduced in Eq. (4.1) for the EW:
2 #
nj "
NX
lines X
√
1 yj,i − µj,i
−ln(σj ) − ln( 2 π) −
(4.5)
l≡
2
σj
j=1 i=1
with Nlines the total number of (H, He or Si) lines used, nj the considered number of wavelength points within the line profile, yj,i the observed quantity at point i of line j, µj,i the
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GAUDI B star sample
Table 4.5: (1-α) 100% quantile of the χ2p -distribution, with p degrees of freedom and a
significance level α = 0.05, taken from Decin et al. (2007). We have repeated here only the
values for up to eight free parameters.
p
1
2
3
4
5
6
7
8
χ2p
3.8414588
5.9914645
7.8147279
9.4877290
11.070498
12.591587
14.067140
15.507313
expected (i.e. theoretically predicted) value for this quantity, and σj the error on the observed
quantity. For the basic statistics behind this formula, and the assumptions made, we refer to
Decin et al. (2007). The maximum of this loglikelihood distribution will give us the model
which fits the observations best. However, models that are sufficiently close to this maximum
loglikelihood can still be accepted as good models. Similar to Eq. (4.2), sufficiently close is
quantified as
2 (lmax − lmodel) ≤ χ2p ,
(4.6)
where lmodel is the evaluation of the loglikelihood function for the model, or the free parameters, under consideration, p is the number of free parameters that one wants to fix by
applying this criterion, and χ2p determines the 95% quantile of the chi-square distribution
with p degrees of freedom as listed in Table 4.5 and taken from Decin et al. (2007).
We will use Eq. (4.5) with the corresponding criterion in Eq. (4.6) in two different ways.
First, we will use it to fit the equivalent widths of the lines, to determine, e.g., the effective
temperature. In this case, we use Eq. (4.1) and (4.2). We have also developed an alternative
likelihood function by using line-ratios instead of absolute equivalent widths (with an automated detection of which ionisation stages can be used). Second, we will use it to fit the line
profile shapes. In this case, yj,i represents the flux at wavelength point i in the j-th line profile, and µj,i the flux corresponding to the same wavelength point. Note that, for this reason,
the number and position of the wavelength points should be adjusted to the observations if
necessary, before comparing.
The preparation (e.g. EW determination) remains as it is for the AnalyseBstar approach.
Further, for the alternative method, we work in two big steps:
The first step consists of the simultaneous determination of the effective temperature, the
surface gravity, the microturbulent velocity and the abundances of silicon as well as helium,
by exploitation of our knowledge of the EWs of both ions. We have decided to include both
He and Si lines at the same time. In this way we hope to be independent of the temperature
domain (i.e. early versus late spectral type). For the late type stars the He lines will have
a higher weight where the Si lines almost all disappear (except for Si II), as helium reacts
4.6 Alternative strategy for a grid-based method
145
sensitively to changes in the effective temperature. On the other hand, in the domain where
the He lines become independent from Teff , the Si lines will be given a higher weight. We still
make the additional assumption in the cool regime that the Si abundance is known. Therefore,
we should obtain for all three abundance grids solutions with different Teff /ξ, at comparable
likelihoods. The surface gravity log g has also been included in this first iteration, since it
also has a certain influence on the strength of the Si lines, and on the He lines in the hot
temperature regime. In this way. we will fix in first instance 5 parameters from the equivalent
widths. Therefore the models should satisfy the following criterion:
2 (lmax − lmodel ) ≤ 11.070498.
To be able to apply the loglikelihood formula for the EWs and this criterion, we first need
the synthetic EWs. For this purpose we have created extremely large (IDL) structures which
contain the theoretically predicted EWsyn,j for all lines of all models. Once we have this
information, it is straightforward to compare the observed EWs with all theoretical EWs in
the grid.
After a first selection of models that agree with the observed EW (which was done irrespective
the wind parameters), in a second step, we aim at selecting the best line profile fits, not only
for the Balmer lines, which give us information about the three parameters log g, β and log Q,
but also for the He and Si lines, which give a second constraint for the remaining parameters.
First, we derive the best vmacro (in the same way as described in Section 4.4.4) for each of
the models selected in the first step. Then, we use the Balmer lines to constrain log g and the
wind parameters.
Whereas the two wind parameters are determined from only one line (i.e. Hα) and the gravity
from only two or three lines (depending on the wind strength), the He and Si lines are much
more numerous. Therefore, we approximated the overall loglikelihood by weighting the
separate loglikelihoods with the number of lines, N , used to derive the parameters:
l = llogg .
Nlines
Nlines
+ lwind . Nlines + lSi,He .
,
NH
NSi + NHe
where NH is the amount of hydrogen lines used to derive log g (usually 3), NSi and NHe
are the amount of He and Si lines used to derive Teff , vmicro and the abundances of Si and
He (for the second constraint from the line profiles, and not from the equivalent width), and
Nlines is the total amount of lines used (= NH + NHe + NSi ). llogg , lwind and lSi,He are the
corresponding loglikelihoods, calculated respectively from the Balmer lines for log g, from
Hα for the wind and from all He and Si lines for the remaining parameters. This approach
is motivated by two facts: 1) the l values are additive (by construction), and 2) since the
individual l values are a sum over the considered lines, by dividing by the number of lines
used, we construct in principle for each process an l per line, and add those. Multiplication by
Nlines does not change the shape of the likelihood function and is done as final normalisation.
By assigning these weights, the loglikelihoods will get lower weight, when more lines are
used, and higher weight when only few diagnostic lines are used. On the other hand, all
wavelength points have been given equal weight. In a more sophisticated implementation,
one could think about giving higher weight to, e.g., the forbidden lines in the blue wings of
He I 4471 and He I 4922.
146
GAUDI B star sample
The criterion for the models to be selected as well-fitting models is then
2 (lmax − lmodel ) ≤ 15.507313,
where lmax is now the maximum of the combined loglikelihood function of all line profiles,
as mentioned above. In this way we are, in most cases, left with only one solution. It would
obviously be very interesting to compare this unique solution to the one we found for some
of the stars with AnalyseBstar. In the next section, we show the results of such a comparison
for the B2.5 Ib/II supergiant ι CMa.
For the sake of clarity, in the remainder, we will refer to this alternative method as the full
grid search or loglikelihood method (shortly, the l-method). We stress that we have only
implemented a basic version of this method, and it should be further finetuned if one really
wants to rely on the outcoming results. We have refrained from spending time on this, because
it soon became clear that the method has a few disadvantages which make it an insufficient
method for our purposes. We refer to Section 4.6.3 for a discussion of the usability and
reliability of this method.
4.6.2 Comparison with AnalyseBstar for ι CMa
For ι CMa we applied both AnalyseBstar and the full grid search, coupled to the likelihood
functions in Eqs (4.1) and (4.5), to determine the physical parameters. We compare the outcome in Table 4.6. Most parameters agree fairly well, except for the Si abundance. This could
be the result of forcing an optimal fit to one of the grid points when applying the l-method,
whereas in AnalyseBstar we leave the possibility for the Si abundance to lie in between two
grid points. The value for the Si-abundance, derived from our AnalyseBstar analysis, is
log n(Si)/n(H) = −4.78, which makes ‘-4.79’ the closest grid value, whereas the l-method
indicates log n(Si)/n(H) = −4.49 as the best Si abundance.
In Fig. 4.25, we compare the Si line profiles of both models (we verified that the difference in
H and He is minimal). For the Si III lines, there is only little difference between both models.
However, it appears that we are dealing with the same problem as in Chapter 2, namely that
the different components of the Si II doublets cannot be equally well fitted. Where AnalyseBstar chose Si II 4130 and 5056, the l-method seemed to have given preference to Si II 4128.
The Si II 5041 line lies in the middle between the two grid models, as would be expected
from the interpolated Si abundance.
In the first step of the l-method, 7652 models throughout the whole grid have been selected
as well-fitting models on the basis of the equivalent widths of He and Si. Then, finally, all
synthetic line profiles of these almost 7700 models had to be compared with the observed line
profiles. It took about three hours to finally decide on the best model, whereas AnalyseBstar
only needed half of the time to get the same results. Only 66 different possibilities had to be
checked throughout the procedure.
Note that we find an extremely low He abundance. This could be the result of an extrapolation
147
4.6 Alternative strategy for a grid-based method
Table 4.6: Comparison between the resulting parameters for ι CMa obtained by, on the one
hand, the automatic tool ’AnalyseBstar’ (Section 4.4) and, on the other hand, the alternative
method which makes use of a loglikelihood function, and a full grid search.
fit parameters
Teff (K)
log g (cgs)
log Q (char)
β
n(He)/n(H)
log Si/H
ξ (km/s)
vmacro (km/s)
interpolated
(AnalyseBstar)
17200
0.05
-4.78
16
39 ± 16
best fit
(AnalyseBstar)
17000
2.6
b
1.2
0.10
-4.79
15
39 ± 16
loglikelihood
method
16500
2.6
b
1.5
0.10
-4.49
10
34
Figure 4.24: Contour plot of the best fitting models for ι CMa through the application of
loglikelihood functions. In the first step, the grid is constrained to a subspace of the grid
by comparing the equivalent widths of He and Si lines (black dots). In a second step, we
compare the line profile shapes and we are left with only one model, the one indicated in
white. The rainbow colours represent the third dimension of the loglikelihood distribution
over the subgrid (low l (black) to high l (red)).
towards abundances lower than solar, but, on the other hand, Morel (2007) obtained a similar,
suspiciously low He abundance of 0.045 ± 0.02, which may indicate that it might be real.
148
GAUDI B star sample
Figure 4.25: Comparison of the two different models chosen as the best fitting model for the
observed Si lines for ι CMa (grey) by the l-method (full) and AnalyseBstar (dashed).
4.6.3 Discussion points
Several points can be raised against the actual application of this method. In this section we
address a few points of discussion, which should be kept in mind when applying either of the
two methods.
1. A first important point, which becomes especially of crucial importance when aiming at
the analysis of large samples, is the speed of performance. Comparison of equivalent
widths is straightforward and can be done within a minimum of time. However, the large
number of line profiles that need to be checked, slow down the l-method considerably. For
high-quality data with a high SNR (and thus small errors), the l-method requires at least
one hour of computation time to select the best model, rising exponentially with the error
4.6 Alternative strategy for a grid-based method
149
on the equivalent widths. Indeed, for low SNR spectra, where the errors on the equivalent
widths are much larger, the amount of models that were accepted in the first step is very
large, and, therefore, it can take up to more than 20 hours8 before the final solution is
found. Moreover, in most cases, this final solution is very similar to the one found by
AnalyseBstar in less than a quarter of an hour. For AnalyseBstar, the time needed to
converge to the optimal solution will not only depend on the quality of the data, but also
on the temperature region and on the starting values. The closer the initial values are to
the real values, the faster convergence is reached (typically within a few minutes). When
more than one Si ionisation stage is available (generally this is no problem for effective
temperatures above 15,000 K), the final solution is, on average, found within 15 minutes.
When only one ionisation stage of Si is available, also AnalyseBstar is slower. This can
easily be understood, when one remembers that, in the cool regime, we use a procedure
based on the same principles as this alternative method (i.e. using loglikelihood functions).
Therefore, for cool stars, the computation time can go up to one hour and a half, which is
still much faster than the alternative method.
2. A second disadvantage of the alternative method, is the fact that it is bound by the grid.
Whereas our method is able to search in between the grid points (thanks to the parallel
determination of Teff , ξ and abundance), the alternative method searches the best solution
only among the grid models. This means that it forces the observed line profiles to be
exactly represented by the best model. We know that, e.g., with respect to Si abundance,
the grid is too coarse. A lot of B stars seem to display a depletion in Si (cf. Section 4.5.2),
which means that, likely, almost all abundances lie somewhere between -4.49 and -4.79
(or even lower). The alternative method will force the abundance to either of the two
edges, and, consequently, also the temperature and microturbulence will be affected. For
the microturbulence, a same reasoning can be made, although the effect might be a bit less
worrisome because of the smaller grid steps.
3. The l-method assumes that, apart from noise, all observed profiles can be represented
perfectly by the model. However, as far as FASTWIND (or any alternative code) concerns, we know that this is not the case and that there may still be systematic deviations
between observed and theoretical profiles (see, e.g., the discrepancy between the two lines
of Si II 4128-4130, as discussed in Section 2.7). As long as these systematic deviations are
rather similar over the whole grid, their presence might not influence the likelihood ratios
too much. In our case, however, deviations between observed and theoretically predicted
profiles seem to be rather different across the B-type spectral range. Though different,
the deviations are typically small.If we could quantify this ”theoretical” bias as a function
of stellar parameters, we could account for this by introducing carefully chosen weights.
This is exactly what the eye is doing in a fit-by-eye procedure (and, therefore, we were
able to correct for this discrepancy in the analysis of the supergiants in Chapter 2). However, it is not evident to translate this into a code. For O-type stars, the reliability of the
individual lines as a function of parameters is well known, thus allowing Mokiem et al.
(2005) to account for it in their genetic algorithm approach. However, for B-type stars,
8 Even though the efficiency might be improved upon, due to the large number of models to be checked the
method will still remain sufficiently slower than the iterative method (see, e.g., the ana- lysis of the late B stars,
which is partly based on the same approach and already takes sufficiently longer computation time), and therefore
(while keeping the other disadvantages in mind), we have not invested more time in it.
150
GAUDI B star sample
this knowledge can only be built up through the analysis of large samples covering a wide
variety of B-type stars. Hopefully we can provide part of this information after the analysis of the large CoRoT sample. Since almost all GAUDI stars are main sequence stars,
we will only be able to give more information for the dwarfs. Of course this problem of
imperfect line predictions is not inherently connected to the l-method. Also our present
approach (AnalyseBstar) suffers from it.
4. One advantage of the l-method is the fact that it is able to point to all relevant optima
in the possible parameter space, and, in this way, it avoids to end up in a local optimum
instead of the global optimum. However, also with our iterative method, we are sure to
end up in the correct domain of the HRD, since there should be only one maximum, due
to the monotonic behaviour of the ionisation fractions.
For both methods the accurate measurement of the observed EWs is of major importance,
since these will fully determine which models are accepted or rejected.
4.7 Results for the GAUDI stars
In Table 4.7 we list the results of the application of AnalyseBstar to a selected sample of
the B-type stars in the GAUDI database9 . As mentioned in Section 4.2 we have immediately
excluded the known spectroscopic binaries and the Be stars, since our methods are not appropriate to handle these stars, and they have been analysed by other members of the CoRoT
team. The quality of both the SARG and CATANIA spectra was insufficient to perform an
accurate detailed analysis, so these have not been considered either.
For HD 174069, we had both an ELODIE and a FEROS spectrum available. From inspection
of the spectra, it was immediately clear that there are systematic differences in the line profiles, in particular the wings of the Balmer lines are much less pronounced in the ELODIE
spectra (see Fig. 4.26). It is not clear what causes this discrepancy. Also other stars, for which
we had both a FEROS and an ELODIE spectrum available, showed the same discrepancy, so
it seems to concern a systematic, maybe instrumental, effect. We fitted both spectra and came
up with a different set of parameters (see Table 4.7). The difference in log g is large and certainly worrisome. When accounting for the difference in Teff , this is a discrepancy of at least
0.6 dex. Our first thought was that the discrepancy was caused by the shorter wavelength
range of each order in the ELODIE spectra, which affects the normalisation. We have downscaled the FEROS normalisation to exactly the same wavelength range as for the ELODIE
spectra. The problem remained, however, and we did not succeed in getting a hold on it. Due
to the fact that the merging of the ELODIE spectra is in any case less reliable than the one
for the FEROS spectra (cf. Section 4.4.1), we have decided to use the FEROS spectra, and to
leave the ELODIE spectra out of our sample, in order to keep our study as homogeneous as
possible. Of course, this significantly reduces the extent of our sample. With our final aims
9 Based on GAUDI, the data archive and access system of the ground-based asteroseismology programme of the
CoRoT mission. The GAUDI system is maintained at LAEFF, which is part of the Space Science Division of INTA.
151
4.7 Results for the GAUDI stars
Figure 4.26: HD 174069: Significant (and apparently systematic) discrepancies are observed
between the line profiles of the FEROS (black) and the ELODIE (grey) spectrum. This leads
to considerable discrepancies in the derived stellar parameters.
in mind, namely to derive accurate fundamental parameter calibrations, we preferred to base
our analysis on thrustworthy stars, with qualitative spectra.
For the same reason, we have also excluded stars with an very bad SNR and the fast rotators,
because an accurate equivalent width determination is impossible in these cases. In this way,
we were left with 35 reliable targets. In Table 4.7, we give an overview of their derived
properties. Their position in the HRD can be found in Fig. 4.27.
To enlarge the number of stars in the hot temperature side of the spectral range and to allow
for a continuous flow towards the O-type regime, R. Mokiem has been so kind to additionally
analyse 2 of the 11 O-type stars in the GAUDI stars, using his genetic algorithm approach
(Mokiem et al. 2005). Since this approach is very accurate and is based on the same atmosphere code (i.e. FASTWIND), they can be added to our sample, without any objection. We
list the resulting parameters in Table 4.8.
4.7.1 Comparison between observed and synthetic equivalent widths
For all the analysed GAUDI stars, we have compared the observed equivalent widths with
the theoretically predicted equivalent widths of the best fitting model, in order to see if there
exist any systematic differences. This simple exercise provides very useful information, to
improve the theoretical line prediction or to confirm the currently existing predictions. We
have investigated the distribution of
EWobs − EWtheo
EWtheo
152
GAUDI B star sample
Table 4.7: Basic fundamental parameters of a selected sample of GAUDI stars, resulting from the
application of AnalyseBstar. ‘Interpolated values’ as well as the derived closest grid points are given.
Spectral types are taken from SIMBAD. For a summary of the errors, we refer to Section 4.5.3. The
errors on v sin i and vmacro are the standard deviations of the values derived from different Si lines. If
only one line was used for deriving vmacro , we fixed the error at 5 km/s. For most of the latest spectral
types (up to B5), there is no Si III anymore and we had to use a different approach, adopting a value for
the Si abundance. In this case the derivation of the interpolated Teff was impossible, and we could only
derive the interpolated microturbulence.
HD SpT
48434
172488
52382
174069F
174069E
45911
45546
170580
44700
45726
51507
181074
43157
48215
58973
177880
178744
48807
51360
42677
44948
47964
48497
49935
51150
53083
44321
44354
48808
48957
53004
54761
173370
181440
182198
B0III
B0.5V
B1I
B1.5V
B1.5V
B2IV-V
B2V
B2V
B3V
B3
B3V
B3
B5V
B5V
B5
B5V
B5
B7Iab
B7III
B8
B8Vp
B8III
B8
B8
B8
B8
B9
B9
B9
B9
B9
B9
B9V
B9III
B9
interpolated
Teff
(K)
grid
Teff
(K)
log geff
(cgs)
28100
22200
23100
22100
19900
20800
20100
19700
17100
21000
17500
19800
16500
15000
15000
15000
14500
13000
13800
10000
11500
12000
14000
12500
13500
11500
11500
13000
12000
12000
11000
11500
11000
11000
11000
28000
22000
23000
22000
20000
21000
20000
19500
17000
21000
17500
20000
16500
15000
15000
15000
14500
13000
14000
10000
11500
12000
14000
12500
13500
11500
11500
13000
12000
12000
11000
11500
11000
11000
11000
3.10
3.10
2.70
4.20
3.40
4.00
3.80
4.00
3.90
4.00
3.50
3.60
3.60
3.90
3.30
3.90
3.50
2.10
3.30
3.10
3.50
3.10
3.90
3.30
2.80
2.90
3.80
3.90
3.20
3.10
3.90
3.10
3.20
3.50
3.30
interpolated
He
Si
0.10
0.10
0.15
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.20
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.15
0.10
0.10
0.10
0.10
0.10
0.10
0.10
-4.50
-4.36
-4.71
-4.50
-4.81
-4.81
-4.81
-4.86
-4.62
-4.44
-4.51
-4.35
-4.79
-4.49
-4.79
-4.79
-4.49
-4.55
-4.04
-4.79
-4.79
-4.79
-4.49
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
grid
Si
-4.49
-4.49
-4.79
-4.49
-4.79
-4.79
-4.79
-4.79
-4.49
-4.49
-4.49
-4.49
-4.79
-4.49
-4.79
-4.79
-4.49
-4.49
-4.19
-4.79
-4.79
-4.79
-4.49
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
-4.79
interpolated grid
ξ
ξ
(km/s) (km/s)
17.40
13.00
19
3.80
8.50
6.60
4.50
5.90
1.40
5.10
4.40
12.6
1.25
3.94
7.38
9.57
11.80
9.70
2.00
7.59
16.30
3.59
1.63
5.78
4.38
3.39
7.10
9.75
4.00
1.63
7.90
4.81
5.84
2.75
6.83
15
12
20
6
10
6
6
6
3
6
3
12
3
3
6
10
12
10
3
6
15
3
3
6
3
3
6
10
3
3
6
6
6
6
3
v sin i
(km/s)
vmacro
(km/s)
63 ± 5
101 ± 7
61 ± 6
35 ± 5
45 ± 4
23 ± 4
66 ± 10
9±1
12 ± 2
113 ± 10
147 ± 6
133 ± 8
26 ± 3
103 ± 1
95 ± 4
45 ± 4
236 ± 32
25 ± 2
68 ± 8
118 ± 17
74 ± 10
50 ± 4
9±2
71 ± 4
12 ± 2
76 ± 20
98 ± 8
121 ± 0
47 ± 15
24 ± 3
51 ± 7
54 ± 1
262 ± 53
53 ± 7
25 ± 1
49.2 ± 5.0
80.0 ± 9.2
61.7 ± 10.3
34.9 ± 7.8
37.6 ± 5.0
0.0 ± 5.0
57.6 ± 3.6
12.7 ± 5.0
0.0 ± 5.0
55.6 ± 1.0
0.0 ± 5.0
0.0 ± 2.8
20.4 ± 3.2
100.2 ± 5.0
82.9 ± 5.0
52.1 ± 1.9
0.0 ± 5.0
26.0 ± 5.0
53.8 ± 9.5
81.2 ± 5.0
106.9 ± 0.0
52.8 ± 29.1
18.0 ± 5.0
77.4 ± 5.0
39.6 ± 5.2
90.0 ± 5.0
0.0 ± 24.6
72.9 ± 5.0
63.3 ± 10.5
23.1 ± 3.9
54.2 ± 9.4
71.8 ± 5.0
0.0 ± 5.0
0.0 ± 5.0
0.0 ± 5.0
as a function of the effective temperature, surface gravity and microturbulent velocity (Figs
4.28 till 4.30). These diagrams not only teach us about systematic differences between the
observed and synthetic equivalent widths, but also tell us which lines are observable in which
regions, and which lines are reliable to use as indicator for Teff , log g or ξ. For each line,
we have also indicated the linear least squares fit. Positive values in the diagram mean that
the observed EWs are larger than the theoretically predicted ones, negative values mean that
153
4.7 Results for the GAUDI stars
Figure 4.27: Position of the GAUDI stars in the HRD, based on the detailed spectroscopic
analysis performed with AnalyseBstar.
Table 4.8: Fundamental parameters of two GAUDI O-type stars, as derived by R. Mokiem,
using the genetic algorithm of Mokiem et al. (2005), and FASTWIND model atmospheres.
Teff (K)
log g (cm/s2 )
log gc (cm/s2 )
n(He)/n(H)
R∗ /R⊙
L∗ /L⊙
ξ (km/s)
Ṁ (M⊙ /yr)
β
v∞ (km/s)
v sin i (km/s)
M∗ /M⊙
HD 47432 (O9.5 III)
29300
3.14
3.18
0.10
17
1.9e+05
18
6.975e-07
1.64
1600
126
16
HD 52266 (O9 V)
31400
3.64
3.73
0.10
11
1.0e+05
19
6.864e-07
0.66
2100
274
22
FASTWIND predicts too large EWs. As a first overall conclusion, we can say that, generally,
the lines are all very well predicted with only a few exceptions. Note that we have not used
the three He I lines, He I 4010, 4120 and 4140, in our spectral line fitting procedure, because
we knew on beforehand that the broadening functions are very approximative and need to
be updated. We see that all three of them are systematically underestimated. The He I lines
can be observed all over the B-type spectral range, whereas we only have one star for which
He II 4541 could be measured, and only two stars for which He II 4686 is detectable. The
Si II lines are visible up to 22,000 K, the Si III triplet at 4552 from 13,000 up to 28,000 K,
whereas the Si III 4716 and the Si III 4813 triplet are both only visible in a very limited range
154
GAUDI B star sample
Figure 4.28: Behaviour of the relative differences between the observed EW and the theoretically predicted EW of the best fitting model EWobs /EWtheo − 1 as a function of the
effective temperature, surface gravity and microturbulent velocity, for eight He I lines.
4.7 Results for the GAUDI stars
Figure 4.29: Same as for Fig. 4.28, but for different lines.
155
156
GAUDI B star sample
Figure 4.30: Same as for Fig. 4.28, but for Si lines.
4.8 Photometric calibrations
157
between 18,000 and 23,000 K. Si IV is detectable only in three cases, which all seem to be
supergiants with large microturbulent velocities. Most other stars have low microturbulent
velocities, as was expected.
We have to be very careful in drawing additional conclusions from these diagrams. Differences between observational and synthetic EWs are not necessarily due to errors in the
theoretical predictions, but can also be due to a misfit of the Gaussian profile to the observed
line, in measuring the observed EW. As mentioned before, for lines such as He I 4471 and
He I 4922, it is difficult to get an accurate value for the EW because of the forbidden component in the left wing and the non-Gaussian distribution as well as due to the many blends in
the right wings. We have looked into the line profile fits to see if there are indeed systematic
differences, or if this is just an artificial effect due to an inaccurate EW measurement. As a
means of illustration, we display in Fig. 4.31 until 4.36, a few examples for different types of
stars: a cool/hot dwarf, a cool/hot giant and a hot supergiant (the only one in our sample).
They show the quality of the GAUDI spectra and the difficulties with which we have to deal.
The figures are discussed in their captions.
4.8 Photometric calibrations
Direct means to determine the physical parameters of stars are available, such as, e.g., cluster main sequence fitting, parallax measurement and analysis of binary motion to determine
respectively the age, luminosity, and mass and radius of a star. The unfortunate thing is that
these methods can only be applied to a small number of targets, and they rapidly loose power
when going to faint objects. One way to overcome this problem, apart from the spectroscopic
analysis, is brought to us through multicolour photometric measurements. Photometric colour
indices reflect, by definition, the difference in emitted energy at different wavelength ranges.
Therefore, they are sensitive to stellar properties, such as gravity and effective temperature,
which alter the energy distribution over the spectral range. To derive information about the
star (e.g., its effective temperature), one has to find the most suited colour indices of a given
photometric system, i.e. those colour indices which are most sensitive to changes in this parameter.
For early type stars, the depth of the Balmer discontinuity is very sensitive to temperature
changes, and, therefore, a good colour index is one that combines filters on either side of the
discontinuity. The gravity, on the other hand, can be determined from the equivalent width
of the Hβ line (in case there is no wind contamination). Thus, a good colour index is one
that combines a broad and a narrow filter, both centred on the Hβ rest wavelength. The broad
filter measures the EW of the line and the narrow filter the line depth, which, combined, give
the FWHM of the line. Once we dispose of the right colour indices, the ‘only’ thing required to derive the fundamental parameters, is an accurate calibration of these parameters as
a function of the observed colour indices. Though this may sound easy, much effort has been
done in order to derive such calibrations. Yet, for the hottest stars, it seems hard to establish. The main reason for this is that, at these hot temperatures, we are on the Rayleigh-Jeans
tail throughout the optical spectral range. Therefore, optical colours are rather insensitive to
changes in the effective temperature for O and early B-type stars.
158
GAUDI B star sample
Figure 4.31: Example of the line profile fits for a cool dwarf : HD 44321. The preparation
was not easy for this spectrum: the EW determination of He I 4387 and He I 4922 is hampered
by strong blends, the normalisation for Si II 4128/4130 is uncertain due to their position in
the broad Stark wings of Hδ and the normalisation of He I 4026 seems to bit a bit offset in
the blue wing. At this temperature, there is also no Si III left anymore, so we need to make
an assumption for the Si abundance. Despite these difficulties, we were still able to obtain a
satisfying fit quality. From the Si II profiles it is immediately clear that this is a fast rotator,
where the macroturbulence is absent. Omitted lines were not present in the spectrum.
4.8 Photometric calibrations
159
Figure 4.32: Example of the line profile fits for a cool giant: HD 181440. Despite the low
quality of the data, and the fact that we have no more Si III left in this temperature region, we
are still able to obtain a satisfying fit quality. Omitted lines were not present in the spectrum.
160
GAUDI B star sample
Figure 4.33: Example of the line profile fits for the only cool supergiant in our sample:
HD 48807. The Balmer lines show some bumps in the wings. It is unclear whether these
bumps are real or not. They also appear in the other supergiant in the GAUDI sample. The
He and Si abundances are both lower than solar, though lying closest to the solar grid point.
This clarifies the small discrepancies in the cores of He as well as Si. Note the excellent fit of
the He I 4471 and 4922 forbidden components.
4.8 Photometric calibrations
161
Figure 4.34: Example of the line profile fits for a hot dwarf : HD 44700. The strong lines
on the blue side of Si III 4716 and Si III 4819 are easily mistaken for the Si lines themselves.
This is the reason why we always indicate the exact position of the transition during the
full preparation process. The misfit of He I 4010 and 4140 is due to incomplete broadening
functions of these lines. Remember that these lines were not used during the fitting procedure,
but only as a double check afterwards. We observe a similar behaviour in several other stars.
All other lines, even the weakest, fit very well. The shape of He I 4471 is even perfect.
162
GAUDI B star sample
Figure 4.35: Example of the line profile fits to the ELODIE spectrum for a hot giant:
HD 48434. Both the He and Si abundance are a bit lower than solar, but for the remainder, this model gives a very good fit quality.
4.8 Photometric calibrations
163
Figure 4.36: Example of the line profile fits for the only hot supergiant in our sample: HD 52382. The peak of Hα is reasonably well represented, despite the fact that
the fitting of the wind is restricted to the grid combinations of log Q and β. The Balmer
lines show bumps in the wings. It is not clear whether this is real and due to the strong
wind, or if this is a spectral artefact. It also arises in the cool supergiant. The interpolated Si abundance (log n(Si)/n(H) = −4.71) is higher than the closest grid abundance
(log n(Si)/n(H) = −4.79), which clarifies the discrepancies in the Si line cores.
164
GAUDI B star sample
We aim at verifying and improving (if necessary) the commonly used Teff and log g calibrations for both the Geneva and the Strömgren photometric system. We attempt to do this
by providing accurate, fundamental parameter estimates (as derived from the application of
AnalyseBstar to high-resolution spectroscopic data) for targets for which also photometric
colour indices are available. These will then be used to check and/or to recalibrate both
photometric systems. The GAUDI sample is ideally suited for this job. As mentioned in
Section 4.1.2, in the framework of CoRoT, an intensive ground-based observing programme
was set up to gather Strömgren colour indices of more than 1500 stars, among which some
320 B-type stars. For 250 of these targets, also high-resolution spectroscopy was available.
However, 60 targets were excluded, as they are spectroscopic binaries or Be stars. On the
other hand, we have 183 B-type stars for which we have Geneva photometry at our disposal,
and for 153 among them we have also spectroscopic information. At present, 25 of these
targets have been spectroscopically analysed. In what follows, we will mainly concentrate on
the Strömgren photometric system, but we still thought it useful to discuss also the Geneva
photometric system, as it allows a comparison between the physical parameters derived from
the fundamental parameter calibrations of both systems.
4.8.1 Geneva photometric system
The Geneva system (Golay 1974, 1980) consists of seven filters (U, B, V, B1, B2, V1 and
G). In this multicolour parameter space, Cramer & Maeder (1979) defined an orthogonal
coordinate system (X, Y, Z):
X = 1.3764 [U − B] − 1.2162 [B1 − B] − 0.8498 [B2 − B]
−0.1554 [V 1 − B] + 0.8450 [G − B] + 0.3788,
Y = 0.3235 [U − B] − 2.3228 [B1 − B] + 2.3363 [B2 − B]
+0.7495 [V 1 − B] − 1.0865 [G − B] − 0.8288,
Z = 0.0255 [U − B] − 0.1740 [B1 − B] + 0.4696 [B2 − B]
−1.1205 [V 1 − B] + 0.7994 [G − B] − 0.4572.
This (X, Y , Z)-representation has brought considerable improvement in the separation of
the effects of gravity and temperature, compared to the original colours. Indeed, in the HRD,
the X-axis is oriented in the direction of the main sequence O and B stars, and is therefore
a good temperature indicator (for log g ≥ 3.5). The Y -axis is perpendicular to X, and lies
in the direction of the luminosity variations. Therefore it is a good gravity indicator. Z is
orthogonal to the (X, Y ) plane, and is a measure of stellar peculiarities (North & Nicolet
1990).
Based on this (X, Y , Z)-representation, Cramer & Maeder (1979) have defined an effective
temperature calibration for dwarfs and giants, based on the temperature estimates of a number
4.8 Photometric calibrations
165
of stars by Code et al. (1976), which were derived from measurements of the angular diameter
and the total absolute flux, integrated over the entire spectrum10:
log Teff = 4.496 − 0.453X + 0.086X 2.
At that time, three stars had not been sufficiently well measured in the Geneva system and
the relation performed bad for Teff ≥ 25, 000 K. Cramer (1984b) improved this effective
temperature calibration by using the latest Geneva colour indices for these stars. Moreover,
he excluded the peculiar stars in the list, and corrected the colours of the binaries to the values
of the primary component. In such way, he came up with the following calibration for O-, Band early A-type stars:
log Teff = 4.586 − 1.038X + 1.094X 2 − 0.646X 3 + 0.139X 4.
However, this calibration did not take into account the slight dependence of X on log g.
Therefore, North & Nicolet (1990) proposed a new calibration for B-type stars, allowing
to predict accurate Teff and log g values, for the measured X and Y parameters, for main
sequence stars with Teff ≥ 10, 000 K, based on Kurucz (1979) atmosphere models and a set
of ‘standard’ stars of Code et al. (1976) and Lanz (1987).
With improved Kurucz models (Kurucz 1993), also this calibration needed revision. This was
done by Kunzli et al. (1997). Instead of comparing the observed colour indices with those interpolated in the “direct” grids of synthetic colours from the known fundamental parameters
(as e.g. Lester et al. (1986) for Strömgren), they preferred to compare the fundamental physical parameters with those interpolated in the inverted grid from the observed colours. With
the aim to obtain the position of the CoRoT target stars in the HRD, C. Aerts interpolated in
the (X, Y ) grids of Kunzli et al. (1997), to obtain the fundamental parameters Teff and log g
of all stars in the eyes of CoRoT for which Geneva photometry is available. Fig. 4.37 shows
the derived Teff and log g estimates.
4.8.2 Strömgren photometric system
The Strömgren or ubvyβ system, is the photometric system for which the most attempts have
been made to establish an accurate calibration in terms of effective temperature and gravity
for early type stars. Generally the respective colour indices c0 and β are used for this. The
index c0 is the dereddened version of c1 = (u - v) - (v - b), which is a measure of the depth
of the Balmer discontinuity, and thus of the effective temperature. As mentioned above, the
equivalent width of Hβ (and therefore the Strömgren parameter β) is a measure of the gravity.
Balona (1984) used Kurucz (1979) model atmospheres to establish a theoretical functional
relationship between effective temperature and gravity on the one hand, and the Strömgren
indices c0 and β on the other hand. He fixed the zero-point of this relationship by using the
effective temperatures of the 32 stars as derived by Code et al. (1976), and found that
log Teff = 3.9036 − 0.4816[c] − 0.5290[β] − 0.1260[c]2 + 0.0924[β][c] − 0.4013[β]2,
10 The angular diameters have been measured with the Sydney University Stellar Interferometer (SUSI) at the
Narrabri Observatory (Brown 1968). The entire spectrum covers the range of UV and visual up to infrared.
166
GAUDI B star sample
Figure 4.37: Effective temperatures and gravities of all stars in the eyes of CoRoT for which
Geneva photometry is available, as a function of the Geneva colours X and Y , as derived
from interpolation in the (X, Y ) grids of Kunzli et al. (1997).
4.8 Photometric calibrations
167
with [c] = log (c0 + 0.20) and [β] = log (β - 2.5). With the update of the theoretical ubvyβ
indices from the published and unpublished grids of Kurucz (1979) by Lester et al. (1986),
and an adjustment of the temperature scale of Code et al. (1976) by Beeckmans (1977) (which
was initially not taken into account, see below), also the calibration of Balona (1984) needed
revision. So, ten years after the original paper, Balona 1994 (hereafter BA94) recalibrated the
effective temperature in terms of Strömgren indices, from the latest data available at that time.
He used the theoretical indices of Lester et al. (1986) to determine the functional behaviour
θ = 0.7515 + 0.2915[c] + 0.4875[c]2 + 0.1216[c]3 + 1.0306[β] + 0.7958[β]2 − 0.4392[β][c],
with θ = 5040/Teff , [c] = log (c0 + 0.25) and [β] = log (β - 2.5). This theoretical relation was
then compared to empirical determinations of Teff from Beeckmans (1977) and Malagnini
et al. (1986). Beeckmans (1977) revised the effective temperatures of Code et al. (1976),
by applying a correction to the absolute UV fluxes (resulting from different instrumental
calibrations). Malagnini et al. (1986) applied a different approach and derived the effective
temperature, gravity, bolometric correction and angular diameter in a self-consistent way by
comparing the observed energy distribution to the flux taken from model atmospheres. For
temperatures hotter than 20,000 K, there seemed to be a significant discrepancy between
these ‘observed’ values for Teff and the theoretically predicted ones, which BA94 suggested
to be probably due to a problem with the synthetic colours of Lester et al. (1986). On the
other hand, Malagnini et al. (1986) already pointed out larger uncertainties in the hotter temperature range, due to both the inadequacy of the LTE assumption and the incompleteness of
the adopted blanketing. Balona corrected for this discrepancy by comparing the theoretical
calibration with empirically determined temperatures of Malagnini et al. (1986) to fix the
zero-point. In this way he finally came up with the following improved relation:
θ = 0.7951 + 0.3226[c] + 0.5395[c]2 + 0.1346[c]3 + 1.1406[β] + 0.8808[β]2 − 0.4861[β][c].
(4.7)
Since, for early type main sequence and giant stars, the photometric error in the observed β
index outweighs the change in temperature as a function of luminosity class, he suggested to
use the mean β value for these stars as a function of c0 , as soon as c0 < −0.18 or β < 2.60:
β = 2.620 + 0.2517 c0 − 0.1400 c20 + 0.1704 c30 .
(4.8)
For the surface gravity, Balona followed a similar procedure as for the effective temperature.
First, he used the theoretical ubvyβ indices of Lester et al. (1986) to define the functional
relationship for a wide variety of effective temperatures and luminosities. Subsequently, he
fixed the zero-point using data from double-lined eclipsing binaries (from Andersen 1991),
for which the surface gravities can directly be determined from the masses and radii of both
components:
log g = 7.492 − 5.7432[c] − 1.9660[c]2 − 0.8074[c]3 + 5.9449[β] − 1.6611[β][c]. (4.9)
Using the calibrations in Eq. (4.7) and (4.9), we have calculated the effective temperatures
and gravities for all GAUDI B stars, for which the c0 and β indices were obtained during the
CoRoT preparatory campaigns. For the stars with a spectral type B3 or earlier, and for stars
which have c0 < −0.18 or β < 2.60, we have used Eq. (4.8), following the suggestion by
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GAUDI B star sample
BA94 (and private communication). Fig. 4.38 shows the derived values for both the effective
temperature (upper panel) and the surface gravity (lower panel) as a function of the diagnostic
colours. The curved trend visible in the lower panel refers to those objects for which Eq. (4.8)
was applied.
4.8.3 Comparison between the two photometric systems
Effective temperatures
We can now compare in how far the effective temperatures derived from the two different
photometric systems agree. In the upper panel of Fig. 4.39, we compare the colour indices,
used to derive the effective temperatures, for the Strömgren (c0 ) and Geneva system (X). For
the range X > 0.5 or c0 > 0.2, there exists a tight relation between both colour indices.
Below these values, the relation is rather chaotic. In the lower panel, the resulting effective
temperatures are shown. From this figure, we see that for log Teff < 4.10, the Geneva
temperatures are generally a bit higher than the Strömgren temperatures, while from log
Teff = 4.10 on, the opposite is true. We can compare this result with what Heynderickx
(1991) found, since he performed a similar study, but restricted to a sample of β Cephei
stars. For comparison, the ranges of his research are indicated as boxes in Fig. 4.39. He
concluded that the effective temperatures obtained by the Geneva system are systematically
higher than in the Strömgren system, which was attributed to a discontinuity in the Strömgren
Teff -calibration around 20,000 K (log Teff = 4.3). This is exactly the temperature where the
scatter around the one-to-one relation starts to grow (see Fig. 4.39). Also Heynderickx (1991)
noted that the scatter at these temperatures is rather large, which we confirm.
Surface gravities
Cramer (1984a) has linked the Geneva colours to the Strömgren β-index for O, B and early Atype stars, through the definition of a similar β, which is a polynomial of the Geneva colours
X and Y :
β = 2.5909 + 0.0667X + 0.1748X 2 − 0.0612X 3 − 0.6801Y − 2.4676Y 2
+0.4418Y 3 − 0.2559XY + 0.1448XY 2 + 0.2582X 2Y.
Although a linear correlation can be observed between the two photometric β indices, the derived gravities differ appreciably (see Fig. 4.40). For the majority of the stars, the Strömgren
gravities are higher than the Geneva gravities. The Strömgren gravities of the GAUDI stars
also occupy a wider range in log g (extending from 2.9 to 5.5) whereas the Geneva gravities
lie in a range between 3.2 and 4.5. Heynderickx et al. (1994) found exactly the opposite for
their sample of β Cephei stars, namely that the Geneva gravities generally occupy a wider
range, which they ascribed to two factors. Either the uncertainties in the Geneva gravity determination are larger, or the Strömgren index β may not be sensitive enough in the β Cephei
4.8 Photometric calibrations
169
Figure 4.38: Effective temperatures and gravities of the GAUDI B stars as derived from the
observed Strömgren colour indices c0 and β, using the Teff and log g calibrations of Balona
(1994). For early spectral types, Eq. (4.8) was used to determine β. Therefore, at the lowest
β values a curved trend is visible.
170
GAUDI B star sample
instability strip, which would lead to a rather constant log g in this range. However, considering a wide variety of B-type stars of all spectral classes, we find exactly the opposite, namely
that the uncertainty in the Strömgren gravity is larger. Gravities higher than 4.5 for normal
B-type main sequence stars, seem rather implausible.
The main conclusions we can draw from this comparative study is that different photometric
systems do not agree, as far as the prediction of the surface gravities concerns. The effective
temperatures do compare well.
4.8.4 Calibration based on standard stars
Both the Strömgren and Geneva photometric calibrations are based on the fundamental parameter determination of the stars studied in Code et al. (1976). Therefore, it is of crucial
importance that these parameters are accurate. The most recent revision of the fundamental
parameters of these stars was performed by Smalley & Dworetsky (1995). They have reevaluated the values for Teff , as derived by Code et al. (1976), to see if modern observational
data and newer model atmosphere fluxes would have any significant effect.
The result was negative: only small changes in Teff , in agreement with the Code et al. (1976)
values within the derived errors, were found. When scanning through the subsample of B
stars in Code et al. (1976), it is striking how ‘non-standard’ the stars are that were used to
derive the Teff -calibration. At first sight, these targets do not seem the most suited to derive
a reliable calibration. From the 16 B stars, there are 3 Be stars (α Eri, β Ori and ǫ Ori), 3
fast rotators (α Leo at 350 km/s, ǫ Sgr at 240 km/s and α Gru at 250 km/s), 3 spectroscopic
binaries (β Cru, α Vir and α Pav) and 1 with a peculiar spectrum (γ Crv, spectral type B8
IIIp HgMn). From the remaining 6, there are 3 supergiants (ǫ CMa, κ Ori and η CMa) and 3
giants (β CMa, γ Ori and ǫ Cen). Moreover, at least half of these stars undergo pulsations,
which can lead to variations of 1,000 K in Teff during the pulsation cycle. If we exclude all
‘special cases’, only a handful of early B-type stars remains, and thus both the Strömgren and
Geneva calibrations are rather fragile. For three of them, we had a spectrum available and
we derived the fundamental parameters using FASTWIND (see Section 4.5.2). We compare
them to the values derived by Code et al. (1976) and Smalley & Dworetsky (1995). Our
values agree very well with both the original Code et al. (1976), and the revised Smalley &
Dworetsky (1995) determination.
Table 4.9: Comparison of Teff and/or log g between Code et al. (1976), Smalley & Dworetsky
(1995) and this study for the three B stars in common.
Teff
Teff
Teff
(Code)
(Smalley)
(Lefever)
γ Ori 21580 ± 790 20930 ± 950 22000 ± 1000
β CMa 25180 ± 1130 24020 ± 1150 24000 ± 1000
ǫ CMa 20990 ± 760 20210 ± 950 22000 ± 1000
Star
log g
(Smalley)
3.6 ± 0.1
3.6 ± 0.1
3.1 ± 0.1
log g
(Lefever)
3.6 ± 0.10
3.4 ± 0.10
3.2 ± 0.10
4.8 Photometric calibrations
171
Figure 4.39: Comparison between the temperature indicators for both the Strömgren and
Geneva photometric systems: the colour indices c0 and X, respectively (upper panel), and
their resulting effective temperatures (lower panel). For both comparisons we show the linear regression curves in grey. The boxes show the range studied by Heynderickx (1991).
Open symbols in the lower panel refer to objects for which the Geneva effective temperatures
become unreliable due to extrapolation outside the (X, Y ) grid. The one-to-one relation between both derived effective temperatures is indicated as a dotted line. The indicated error on
the Strömgren photometry is an upper limit of 5% for a typical error of 0.01 mag in c0 and β.
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GAUDI B star sample
Figure 4.40: Comparison between the β indices of the Geneva and the Strömgren system,
as well as the derived gravities within each system. Line styles have the same meaning as in
Fig. 4.39.
4.8 Photometric calibrations
173
4.8.5 Spectroscopy versus photometry
Now that we realise that large discrepancies occur among parameters derived from the different photometric systems, especially as far as the surface gravity concerns, we can have a
look if spectroscopy can bring an improvement.
Spectroscopy versus Geneva photometry
In the upper panel of Fig. 4.41, we compare the effective temperatures derived from AnalyseBstar, with those derived from interpolation in the (X, Y ) grid of the Geneva colours.
Generally, the effective temperatures agree fairly very well, within an uncertainty of 1,000 K,
at least for temperatures below 20,000 K. The hotter stars (among which a giant and a supergiant), however, lie considerably below the one-to-one relation, leaving a large discrepancy.
Unfortunately, we have too few objects in the hot temperature regime to decide whether this
trend is real. In the upper panel of Fig. 4.42, we show the same comparison, but now as a
function of the Geneva colours. Obviously, the same trends can be observed. Especially for
X > 1 (i.e. the coolest stars), the agreement between the effective temperatures is perfect.
Altogether, there seems no need to improve the Teff -calibration at this point. Even though
there is a large discrepancy for the hottest stars, we need more hot star data to be able to improve the calibration in this range. The lower panels of both Figs 4.41 and 4.42 show similar
comparisons for the surface gravities. They seem to undergo a large systematic shift of, on
the average, about 0.5 dex in log g, between photometric and spectroscopic values, the latter
being systematically lower.
Spectroscopy versus Strömgren photometry
In the upper panel of Fig. 4.43, we compare the spectroscopic effective temperatures (for all
the analysed GAUDI B-type stars for which Strömgren-indices have been measured) with the
photometric values, as derived from Eq. (4.7). We have applied Eq. (4.8) in the appropriate
regions, i.e. when c0 < −0.18 or β < 2.60, or for spectral types B3 or earlier. In the lower
panel, we compare their values as a function of c0 , which is the main temperature indicator11.
We additionally show the Be stars from the GAUDI sample, analysed by Frémat et al. (2006),
excluding 4 objects for which they state possibly inaccurate parameter determination. For
reasons of consistency, we have also omitted those stars, for which the analysis is based
on ELODIE spectra, as we have done for our own analysis (see discussion in Section 4.7
and Fig. 4.26). Frémat et al. (2006) applied a careful and detailed modeling of the stellar
spectra, taking into account the veiling caused by the envelope, as well as the gravitational
darkening and stellar flattening due to rapid rotation. The plane-parallel atmosphere models
they used for effective temperatures ranging from 15,000 K to 27,000 K were computed in
two consecutive steps.
11 Note that also β will still have some impact on the effective temperature (through Eq. (4.7)), as the effective
temperature and gravity are closely coupled.
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GAUDI B star sample
Figure 4.41: Comparison between the spectroscopically determined Teff and log g, and the
Geneva photometric values (derived from interpolation in the (X, Y ) grid) for the few B-type
stars in GAUDI for which Geneva photometry was available. Diamonds denote supergiants,
with luminosity class (l.c.) I, triangles denote ‘giants’, with l.c. II, II-III, III or III-IV, filled
circles represent the ‘dwarfs’, with l.c. IV, IV-V or V. Open symbols are used for those objects
for which the l.c. is unknown.
4.8 Photometric calibrations
175
Figure 4.42: Position of the GAUDI stars in the calibration window of the Geneva indices.
The photometric (black) and spectroscopic (grey) determinations are connected by a vertical
line segment. Symbols have the same meaning as in Fig. 4.41.
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GAUDI B star sample
To account in the most effective way for line-blanketing, the temperature structure of the atmospheres was computed by using Kurucz (1993) and the ATLAS9 FORTRAN program.
Non-LTE level populations were then calculated for each of the atoms considered using
TLUSTY (Hubeny & Lanz 1995) and keeping the temperature and density distributions fixed.
Model atmospheres with effective temperatures lower than 15,000 K were treated assuming
full LTE, while those hotter than 27,000 K were taken from the OSTAR2002 NLTE grid
(Lanz & Hubeny 2003).
Generally, the spectroscopically derived effective temperatures agree very well with those
derived from the photometry (see Fig. 4.43), except for some of the Be stars for which the
Strömgren indices, and particularly β, are probably not so reliable, as narrow-band hydrogenline photometry is strongly affected by emission. Most of the stars for which the temperatures
disagree, lie in the hot temperature regime, above roughly 20,000 K, as of which the scatter
seems to grow without any clear trend. In the same temperature range, we find an O star (open
circle), a giant for which we analysed the ELODIE spectrum and which we consider less
reliable (grey triangle), and an additional star (HD 172488) for which the spectroscopically
derived temperature seems very low with respect to its spectral type (B0.5V). In view of the
very good agreement between the spectroscopic and the photometric values, we can confirm
the Teff calibration and there is no need for improvement.
We have made a similar comparison for the surface gravities. The result can be found in
Fig. 4.44. Whereas the effective temperatures agreed very well, the situation is completely
different for the surface gravity. Only four non-Be stars agree within the ‘allowed’ error
of 0.15 dex. Three objects have a higher spectroscopic gravity, whereas, for the remaining
objects, the gravities derived from the BA94 relation are systematically higher. The transition
point seems to lie around log g ≈ 3.5. For spectroscopic gravities larger than roughly 3.5,
the photometric value is too large, whereas for lower spectroscopic gravities, the photometric
value is too low. Note that this is fully consistent with what Balona (1994) also found on the
basis of a confrontation of his calibration with evolutionary models.
Interestingly, most of the stars for which the photometric value is significantly underestimated, are Be stars. The reason for this probably lies in the fact that these stars have Hβ in
emission, which affects the β colour index. Indeed, when the core of Hβ is slightly refilled
by emission, the line suddenly appears much weaker, implying a lower log g value.
Stars which are classified as giants (4 targets) do have lower gravities (lower panel Fig. 4.44),
which gives confidence in our derived values. However, one has to keep in mind that spectral
classification is very heterogeneous, and for a lot of these stars, spectral types are uncertain.
It seems that, among the sample, there are many giants, which were classified as dwarfs or
had no luminosity class assigned.
The top panel of Fig. 4.45 illustrates the position of our sample stars (black filled circles)
with the position of the eclipsing binaries of Andersen (1991) on which the currently applied
calibration is based. The gravity from these eclipsing binaries is based on their masses and
radii, which can usually be obtained in a very precise and reliable way. However, the colour
indices come from the combined light, using the assumption that most systems have (nearly)
4.8 Photometric calibrations
177
identical components. This implies that the uncertainties in the colours may be large. On
the other hand, the colours for our GAUDI sample should be well determined, if they are
single stars, whereas the uncertainties for the derived gravities may be a bit larger than for
the binaries.
We have also investigated the position of a handful of targets for which an asteroseismic
determination of log g was available (from the derived radius and mass). Such an estimate
results from the construction of a structure model for the stellar interior that fits oscillation
frequencies with an extremely high precision. At present there are only 6 such targets, namely
HD 129929 (Dupret et al. 2004), β CMa (Mazumdar et al. 2006), δ Ceti (Aerts et al. 2006),
12 Lac (Handler et al. 2006), θ Oph (Briquet et al. 2007) and β Cep (C. Aerts, in preparation). For each of these six targets, we obtained the Strömgren β-index from the Galactic
β Cephei catalogue of Stankov & Handler (2005). The position of these stars in the top panel
of Fig. 4.45 (dark grey circles) seem to prolonge the curved trend of the eclipsing binaries
towards lower β-values. Unfortunately, seismic estimates of log g are not yet available for
cooler stars and/or further evolved objects.
A new calibration for log g does seem appropriate, but is hampered by the large scatter in
the data. At first sight, no clear correlation between log g and β is visible in the top panel of
Fig. 4.45. Note, however, that this figure includes a dependency of the surface gravity on the
effective temperature, via the colour index c0 in the calibration Eq. (4.9), as log g is dependent
on both β and c0 : log g(β, c0 ). The 3D representation of this dependence is given in the lower
panel of Fig. 4.45. The height of the bars is representative for log g, while the width reflects
the errors on the observed colour indices. Projecting this figure onto the β-plane (fixing c0 )
brings us back to Fig. 4.44. In Fig. 4.46, we make a similar projection, but now onto the
c0 -plane (fixing β). These figures illustrate that fixing a connection between log g, β and
c0 into one formula, is far from obvious. We have made several attempts to calibrate log g
as a function of the Strömgren photometric indices, going from simple linear calibrations to
complicated higher order polynomials (also in the combined terms), but none of them gave a
satisfying and convincing fit. This can be for two reasons: either there simply does not exist a
single relation which is able to cover all luminosity classes at the same time (from dwarfs up
to supergiants), or we were unable to detect it, due to the fact that we have too few datapoints
for which both the gravity and the observed colour indices are very strictly defined. In the
second case, it would be interesting to see how the trend which is now initiated by the 6
asteroseismic dwarfs, would be continued when giants and cooler dwarfs would be added, as
these provide us with very accurate values of log g. On the other hand, it is worthwhile to
investigate if the colours would change significantly when decoupling the combined light of
the eclipsing binaries, as also for this method, log g is very well defined.
In any case, the message from this investigation should be that one should refrain from using
the calibration outlined by Balona (1994), as long as the discrepancies with the spectroscopic
values are not clarified, as the photometry generally leads to an overestimation of the surface
gravity. We have clearly shown that the photometric colours are too rough a measure to
provide an accurate estimate of the stellar gravity and that they cannot compete with accurate
line fitting through high-resolution spectroscopy.
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GAUDI B star sample
Figure 4.43: Comparison between the spectroscopically determined effective temperatures
and the Strömgren photometric values, derived from the application of the BA94 relation.
Crosses indicate the Be stars from Frémat et al. (2006). The grey symbols denote two stars
for which we have used an ELODIE spectrum, and for which the parameter determination is
less reliable. The dark grey symbol represents one of the two O-type dwarfs analysed by R.
Mokiem, the only one for which the photometric indices were available. Circles, triangles
and diamonds have the same meaning as in Fig. 4.41. The filled symbols in the lower panel
refer to the spectroscopic values for Teff , the open symbols to the photometric values.
4.8 Photometric calibrations
179
Figure 4.44: Comparison between the spectroscopically determined surface gravities and the
Strömgren photometric values, derived from the application of the BA94 relation. Symbols
and colours have the same meaning as in Fig. 4.43. Bottom: stars with unknown luminosity
class are now indicated by squares, to distinguish them from the ‘known’ spectral types (from
SIMBAD): supergiants (diamonds), giants (triangles) and dwarfs (filled circles).
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GAUDI B star sample
Figure 4.45: Top: Position of the 35 analysed GAUDI B stars (filled circles) as compared
with the eclipsing binaries (open circles) of Andersen (1991) on which the log g calibration
of Balona (1994) is based. We additionally show the 6 B-type dwarfs for which we have a
very accurate asteroseismic determination of log g.
Bottom: 3D representation, showing the variation of the spectroscopically derived surface
gravity as a function of the colour indices β and c0 . The height of the shaded area’s gives
the value of the surface gravity and the width is representative for the error on the observed
colour indices.
4.8 Photometric calibrations
181
Figure 4.46: Comparison between the photometric and spectroscopic determinations of the
surface gravity as a function of the colour index c0 . Symbols and colours have the same
meaning as in the bottom panel of Fig. 4.44 and the top panel of Fig. 4.45.
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GAUDI B star sample
Conclusions and future perspectives
This very last part summarises the scientific results of research done in the framework of this
PhD dissertation. At the same time we provide some outlooks for further continuation of this
research.
For the first project, we performed a detailed spectroscopic analysis of 40 B-type supergiants, among which 28 show periodic variations, in order to investigate the mechanism behind the variability. An earlier, photometric study (Waelkens et al. 1998) suggested that there
might be a connection with the pulsations detected in main sequence stars, such as the β
Cephei or Slowly Pulsating B stars, in view of their position in the HRD. Since the accuracy
of the photometric determination of the effective temperature, but especially the gravity, is not
satisfying in the B-type spectral range, a more detailed spectroscopic analysis was required
to re-position the stars and confirm, or reject, this connection. To this end, high-resolution
spectra were gathered and we performed a detailed fit-by-eye line profile fitting procedure to
derive the stellar and wind parameters of this sample. Thanks to the more accurate position
in the HRD, which places the stars at the high-gravity edge of the predicted SPB instability
domain, we were able to propose that non-radial gravity-mode oscillations excited by the
opacity mechanism are likely the cause of the periodic variability, similar to the oscillations
in main sequence B stars. We also suggest nine of the twelve remaining B-type supergiants
to be periodic variables, since they show similar microvariations at the millimagnitude level,
although this was not known before this study. In this way, we propose the occurrence of 37
non-radial pulsators in the upper part of the HRD.
The attribution of the variations to non-radial opacity-driven oscillations is one of the most
important findings of this dissertation. Our sample stars extend the scope of asteroseismic
research towards evolved parts of the HRD, thus allowing finetuning of evolutionary models
(from the ZAMS to the supernova stage) from the seismic sounding of B supergiants. The
next necessary steps to be taken in this direction are to perform long-term and high-precision
time series analyses in spectroscopy, as well as photometry, to detect more frequencies and
to achieve mode identification.
Another important result of this first project is that, even with only a limited number of line
184
Conclusions and future perspectives
profiles, one is able to derive fairly accurate values for the stellar and wind parameters, thanks
to the high predictive power of carefully-chosen line profiles. This made it possible for us to
construct a practical recipe, under the form of a polynomial function, to compute the effective
temperature as a function of spectral type.
As a byproduct of this study, we were able to derive, for the first time, wind momenta for late
B-type supergiants. They seem to follow those of the A supergiants remarkably well. The
wind momentum luminosity relation for mid B-type supergiants was found to be consistent
with theory, although Kudritzki et al. (1999) seemed to find that they all lie significantly
below the theoretical predictions. Unfortunately, and undeserved, the mid and late B-type
supergiants have always been treated a bit ‘inferior’ to the more massive O and early Btype supergiants, so we had only poor means for comparison. Since we needed to make
assumptions for both the radius and the terminal wind velocity, our conclusions concerning
the wind momentum luminosity relation need to be confirmed by further independent studies.
This study did not allow us yet to draw definitive conclusions about a possible connection
between the variability detected in these B-type supergiants and their outflowing winds. We
did find an indication that higher photometric amplitudes may be accompanied by a higher
wind density and a higher relative change in the mass-loss rate, but to assess and confirm these
suggestions, a more refined time-resolved analysis of both the oscillation and wind properties
of a few selected targets is necessary. To initiate this topic, we have already gathered 122
high-resolution CORALIE spectra (1.2m Euler telescope, Chile) for the B3 Ia supergiant
HD 53138, spread over 6 observing runs and covering a time span of 14 months, with the
aim to study the mass loss variability, its connection with pulsations, and its effect on the
fundamental parameters.
Even though we succeeded in deriving fairly accurate estimates for the fundamental parameters of the B-type supergiants, which fixes their position in the HRD, this is still insufficient
information for us to draw firm conclusions about their evolutionary status. Indeed, from
this information alone, we have no idea whether the objects are moving to the red or if they
are already on their way back to the blue side of the HRD. One way to gain further insight
would be to perform a detailed seismic study of each target separately and to fine tune in this
way its evolutionary status from the oscillatory behaviour, which is a strong function of the
chemical stratification inside the star and thus of its age. To do so, requires enough oscillation
frequencies to be able to perform a detailed seismic modelling and to obtain mode identification. Unfortunately, at this point, we do not have enough oscillation frequencies, but this is
certainly one of our main objectives for the near future.
The second project started in the framework of the CoRoT space mission, whose goal it is
to perform high-precision photometry for asteroseismic and (exo) planetary studies. In order
to make an optimal target selection for the satellite mission (which was launched at the end
of 2006), it was (and still is) of crucial importance to select the best-suited targets. To asses
the fundamental properties of potential targets, high-resolution spectra as well as photometric colours were gathered prior to the mission. We were assigned the task to determine the
fundamental parameters of all B-type stars in this catalogue of data (GAUDI). To cope with
the large amount of objects (i.e. almost 200 stars), we have developed the first automated
Conclusions and future perspectives
185
procedure for the spectral line fitting in the B-type range, based on FASTWIND model atmosphere predictions and their emergent spectra. To facilitate future reference, we have baptised
it AnalyseBstar.
We have made the conscious choice for a grid-based method, and, consequently, developed
the BSTAR06 grid, presented in Chapter 3. To be able to analyse stars with various properties, in different evolutionary stages and covering the full B-type spectral range, the grid
setup comprises a wide range in physical parameters. Aiming at a high-accuracy parameter
estimation, we have taken the grid steps as small as possible, within a reasonable amount
of computation time. Thanks to the fast performance of the FASTWIND code and dedicated computer infrastructure, we were able to calculate a grid of almost 265,000 atmosphere
models/profile sets within a time span of seven months.
On the basis of this grid, we developed the first automated procedure (AnalyseBstar) to derive the fundamental parameters of B-type stars in general, and those contained in the GAUDI
catalogue in particular. AnalyseBstar uses an iterative scheme to fix the fundamental parameters, which makes it very efficient in converging towards the best values. It first separates
the effects of the temperature, Si abundance and microturbulence from the equivalent widths
of the Si lines. Then, after determining the best macroturbulent velocity from well-chosen Si
lines, the gravity is fixed from the wings of the Balmer lines. Finally also the wind parameters
are fixed from the Hα profile. As long as there is improvement in the derived parameters, the
method continues its iterative search algorithm, until finally the best grid model is found.
We performed formal tests of convergence on synthetic profiles, to which artificial noise was
added to mimic real spectra, to assure that the input parameters are indeed correctly reproduced. Moving on to real spectra, we have first analysed some 30 pulsating stars (β Cephei
and SPB stars), for which we had high-resolution spectra with eminent quality available. We
found excellent agreement with the parameter determination of Morel (2007), who used the
completely independent atmosphere codes DETAIL/SURFACE.
This study revealed a general depletion in Si for B-type dwarfs, with abundances generally
between the solar and depleted grid point. We found a mean abundance of log n(Si)/n(H) =
−4.75 ± 0.06 for the β Cephei stars in our sample. Since we needed to make an assumption
for the Si abundance in the low temperature region, we do not consider the abundance of the
SPBs as a good reference for comparison. The abundance of the non-confirmed pulsators,
on the other hand, leads to a mean value of log n(Si)/n(H) = −4.65 ± 0.16, which is more
consistent with the solar Si abundance of Asplund et al. (2005). The difference with the values
found by Morel (2007) for these objects is in some cases uncomfortably large, and needs to
be further clarified. Such low abundances for B-type dwarfs (in the solar neighbourhood)
have been observed before, while B-type supergiants seem to have Si abundances more or
less consistent with the solar values. Hence, we suggest there to exist a difference in the
Si abundance for different phases of evolution. We propose that diffusion effects in B-type
dwarfs may cause a depletion of Si at the surface, when the silicon sinks and gets trapped in
a region close to the iron opacity bump. When the star further evolves, two convection zones
appear, which are likely to merge and release the trapped Si back to the surface. However,
theoretical computations of gravitational settling of Si to confirm this scenario are not yet
186
Conclusions and future perspectives
available.
Besides the comparison with the results of Morel (2007), we additionally compared our optical values with those determined from a detailed UV study, based on low-resolution spectra,
and again found very good agreement for the few stars in common. This makes us confident
that our method is working sufficiently well.
We acknowledge that there is certainly room for improvement in the implementation of AnalyseBstar. In particular, the following issues might be taken into account in the future:
1) Presently, the vmacro determination solely relies on the line wings of the metallic lines, as
these wings depend less on other contributing parameters than the line cores. We have shown
that this leads to a rather flat loglikelihood distribution near the maximum. Hence, vmacro
remains rather ill-defined. Inclusion of the line core information may decrease the uncertainty region. As far as the present investigation concerns, we find typically large values for
vmacro . Whether the vmacro values are really as large as stated, can be confirmed as soon as
the uncertainty in the determination of the maximum can be reduced. Note, however, that the
derivation of the other physical parameters remains unaffected by this uncertainty, thanks to
the way our method is constructed.
2) the treatment of the wind is rather coarse and restricted to the grid values. For our purposes this was sufficient, since most CoRoT targets are non-evolved objects. However, for
detailed wind analyses, a more sophisticated method needs to be developed which is able to
interpolate within the grid points, to find a better fit. We have not invested effort nor time into
that at this point, mainly because it was not needed for our work.
3) one of the major logical extensions envisioned is the inclusion of magnesium as an explicit
element. In the cool temperature regime, we only had one stage of Si at hand, and therefore
we were forced to make an assumption about its abundance. In this domain, the ionisation
equilibrium of Mg II/I could bring a solution. The ionisation ratio of e.g. Mg II 4482 and
Mg I 5172 or 5183 may be well suited. Mg II 4482 is visible throughout the B-type spectral range. It is really strong for B9 dwarfs, gradually decreasing towards earlier types, and
becomes somewhat blended by Al III 4480 from 20,000 K on. Mg I 5172 and 5183 appear
around 14,000 K (for dwarfs), both quickly becoming stronger towards lower temperatures.
Including Mg in the spectral analysis may considerably improve the accuracy and fasten up
the determination of temperatures lower than roughly 15,000 K.
The thorough testing of AnalyseBstar on the 30 well-understood pulsators was a good transition to the lower quality spectra of the CoRoT B stars sample. From the spectra of the
extended original sample of 190 B stars, we have selected the 35 best quality FEROS spectra
of B stars spanning the whole spectral range, to perform a homogeneous study, and to come
up with improved values for the effective temperature and gravity compared to those derived
from multicolour photometry.We have concentrated on a comparison with the Strömgren calibration, for which colour indices were available for almost every star in the GAUDI database.
The currently widely (and blindly) used Strömgren calibrations for Teff and log g go back to
Balona (1994). In view of the very good agreement between the spectroscopic and the photometric values for the effective temperature, we confirm the appropriatness of the photometric
Teff calibration. On the other hand, for log g, we find systematically lower values than what
is predicted by the log g calibration of Balona (1994), which is in agreement with his own
Conclusions and future perspectives
187
statement, in which he points towards rather large discrepancies with gravities derived from
evolutionary models. We have tried to derive improved calibrations for log g as a function of
β and c0 , but no useful clear trend could be identified. An interesting approach to consider
would be to perform a detailed spectroscopic analysis of the eclipsing binaries of Andersen
(1991) by performing spectral disentangling (Hadrava 2001) and by using the flux ratios over
the spectral range obtained in that way to unravel the photometric colours (without assuming
almost equal components as done so far). On the other hand, the inclusion of more seismic
targets (also cooler and more luminous stars) could also imply a huge step forward towards
a better log g calibration, as asteroseismology (besides eclipsing binaries) provides us with a
method which allows to derive very reliable information about all stellar parameters including
the gravity.
With the development of AnalyseBstar, we provided the community with a tool to process
large amounts of spectroscopic data within a minimum of time, and to derive the fundamental
physical parameters of hot stars with a high accuracy. We offer our methodology to the wide
astronomical community, to be used for a wide variety of purposes. We hope that the efficient
and accurate determination of the fundamental parameters of B-type stars in the HRD may
accelerate our understanding of their evolution and, in this way, provide better insights in
stellar structure and galactic evolution.
188
Conclusions and future perspectives
Samenvatting in het Nederlands:
Fundamentele parameters van
B-type sterren
B-type sterren
Naast onze eigen ster, de Zon, zijn er nog miljarden andere sterren in ons melkwegstelsel.
Hoewel ze er met het blote oog misschien allemaal vrij gelijkaardig uitzien, kunnen ze zich
toch in een heel ander stadium van hun leven bevinden, en vaak hebben ze heel uiteenlopende
eigenschappen. Op basis van die eigenschappen worden sterren ingedeeld in klassen. Zo
worden ze volgens hun effectieve temperatuur12 ingedeeld in ruwweg 7 klassen (spectraaltypes): O, B, A, F, G, K en M, met binnen elke klasse een verdere verfijning in subklassen.
Daarnaast worden sterren ook geclassificeerd volgens hun graviteit, in lichtkrachtklassen: I
(superreuzen), II (heldere reuzen), III (reuzen), IV (subreuzen) en V (dwergen). Een derde
dimensie, de leeftijd, wordt toegevoegd door de ster in het Hertzsprung-Russell Diagram
(HRD) te plaatsen. In dit diagram wordt de lichtkracht of de graviteit uitgezet tegenover het
spectraaltype of de temperatuur van sterren. Dit diagram wordt vaak gebruikt om de verschillende stertypes op een evolutionaire manier met elkaar te verbinden. Om de levenscyclus van
een specifieke ster beter te begrijpen, is het essentieel om op een accurate manier haar positie in het Hertzsprung-Russell Diagram te kennen, zoals we verder zullen aantonen. In dit
onderzoek beperken we ons tot de sterren van spectraaltype B. Het bepalen van hun fysische
eigenschappen vormt de rode draad doorheen deze verhandeling.
B-type sterren zijn hete en zware sterren, met effectieve temperaturen tussen 10 000 en 30 000
Kelvin, en massa’s van 3 tot 20 à 25 keer die van de zon. Met het blote oog kunnen ze herkend worden aan hun blauwe kleur. Het zijn zware sterren, die dankzij hun krachtige stralingsgedreven sterrenwinden en hun spectaculaire dood als supernova explosie, een uiterst
12 De effectieve temperatuur van een ster is de temperatuur die een zwarte straler zou hebben als ze dezelfde
lichtkracht per oppervlakte-eenheid zou uitsturen als de ster.
190
Fundamentele parameters van B-type sterren
belangrijke rol spelen in de verrijking van het universum met chemische elementen zwaarder
dan waterstof en helium. Deze chemische elementen zijn de eindproducten van de nucleaire
reacties die in de kern plaatsvonden. In die zin is een gedetailleerde kennis van de inwendige
fysische processen die zich in zware sterren afspelen onontbeerlijk. Die processen beı̈nvloeden namelijk in sterke mate hun evolutie en daarmee uiteindelijk ook de evolutie van sterrenstelsels, door het chemisch materiaal dat door de sterrenwind en bij de supernova vrijkomt.
Hoewel men aanneemt dat de stellaire evolutietheorie vrij volledig is, zijn er toch nog een
aantal leemtes die dienen opgevuld te worden. Zo blijkt dat de heetste sterren misschien een
significante hoeveelheid minder massa verliezen dan aanvankelijk aangenomen werd, doordat
kleine inhomogeniteiten (zogenaamde ‘clumps’) in de sterrenwind kunnen voorkomen, terwijl men tot recent een homogene wind veronderstelde. Ook de invloed van interne rotatie en
convectie op de evolutie van sterren dient nog verder uitgeklaard te worden. Beide processen
zorgen er namelijk voor dat het chemisch materiaal in het sterinwendige vermengd geraakt.
Eén van de weinige manieren om hieraan tegemoet te komen is via asteroseismologie.
Trillingen in B-type sterren
Asteroseismologie is het domein binnen de stellaire astrofysica dat zich toelegt op het bestuderen van sterpulsaties om op die manier meer informatie af te leiden over de inwendige
sterstructuur. Doordat verschillende pulsatiemodi tot op verschillende dieptes in de ster
binnendringen, geven de frequentieverschillen ons informatie over verschillende lagen in het
sterinwendige.
Algemeen kunnen de oscillaties in twee soorten ingedeeld worden, naargelang de terugroepkracht die dominant in werking is: de p-modi of drukmodi, en de g-modi of graviteitsmodi.
Naargelang de aard van de trillingen worden ze ook wel opgedeeld in radiale en niet-radiale
trillingen. Opdat dergelijke pulsatiemodi waarneembaar zouden zijn aan het steroppervlak,
moet een efficiënt mechanisme werkzaam zijn, dat in staat is om kleine verstoringen (bijvoorbeeld in het dichtheidsevenwicht) in de loop van de tijd te laten groeien tot macroscopische
waarneembare amplitudes. Voor de hoofdreekssterren van spectraaltype B (zoals β Cephei
en traag pulserende B sterren, i.e. SPB sterren) is dit mechanisme al langer gekend als het
opaciteits- of κ mechanisme, werkzaam in de partiële ionisatielagen. De meeste sterlagen
zijn stabiel ten opzichte van kleine storingen: als de ster een klein beetje samentrekt, dan
neemt de temperatuur, en hierdoor de druk, in deze lagen toe, waardoor de contractie tegengewerkt wordt, en zo het (hydrostatisch) evenwicht herstelt. In welbepaalde lagen, namelijk
de partiële ionisatielagen, wordt het evenwicht echter niet hersteld bij kleine storingen. De
vrijgekomen energie wordt namelijk niet gebruikt om de druk te doen stijgen, maar om de
laag zo homogeen mogelijk te maken en de ionisatie verder te zetten. Door de vele elektronen die hierbij vrijkomen, blijft de opaciteit13 hoog, in plaats van te dalen zoals dat in
homogene lagen het geval is. De laag wordt dus opaak en blokkeert in zekere zin de energie die vanuit de kern uitgestuurd wordt. De verhoogde stralingsdruk die hierdoor ontstaat,
13 De opaciteit κ geeft aan in welke mate straling afkomstig van het sterinwendige doorgelaten wordt, en is in die
zin goed te vergelijken met de doorlaatbaarheid van mist.
Fundamentele parameters van B-type sterren
191
duwt de oppervlaktelagen verder naar boven. Aangezien het stermidden daar transparanter
is, wordt de energie vrijgelaten en de laag valt terug tot zijn oorspronkelijke positie. De
ster reageert met een nog grotere samentrekking, wat tot steeds grotere amplitudes kan leiden. Zo kan een microscopisch kleine verstoring uitgroeien tot een macroscopisch effect, dat
soms de volledige ster aan het trillen kan brengen. Opdat dit zou lukken moet de partiële
ionisatielaag aan bepaalde voorwaarden voldoen die sterk afhangen van de effectieve temperatuur, lichtkracht en chemische samenstelling van de ster. Vandaar dat dergelijke pulserende
sterren enkel voorkomen in welbepaalde zones, de zogenaamde instabiliteitsdomeinen, van
het HRD.
Samenvattend kunnen we dus zeggen dat, om de evolutietheorie van sterren en sterrenstelsels
te verbeteren, het noodzakelijk is om de inwendige processen van de ster ten volle te begrijpen. Dit is mogelijk door het bestuderen van stertrillingen, die op zich dan weer afhankelijk
zijn van de effectieve temperatuur, lichtkracht en chemische samenstelling van de ster. In
die zin is het dan ook eenvoudig te begrijpen dat een nauwkeurige positionering van sterren
in het HRD aan de basis van zulke studie ligt. In deze verhandeling willen wij ons dan ook
toespitsen op een zo accuraat mogelijke bepaling van de fysische parameters voor sterren van
spectraal type B.
Bepaling van de fysische parameters
De enige manier om een ster en haar karakteristieken te leren kennen, is onrechtstreeks, via
het licht dat ze uitstuurt. Met behulp van een spectrograaf wordt het licht, dat zovele lichtjaren
geleden door de ster werd uitgezonden, opgevangen, en via een ingewikkeld reductieproces
omgezet in een ééndimensionaal spectrum. Dit spectrum vertelt ons in principe alles wat we
willen weten over de ster. Alleen is het niet eenvoudig om de exacte informatie te decoderen.
De informatie waar we het over hebben, zijn de fysische parameters van de ster, zoals de
effectieve temperatuur en graviteit, de straal en de massa, de abondanties van de verschillende chemische elementen, de rotatiesnelheid en micro- en macroturbulente snelheden, en
van de wind (beschreven door het massaverlies en het snelheidsverloop). De fysische parameters van sterren worden typisch bepaald door waargenomen spectra te vergelijken met
theoretisch voorspelde spectra, uitgerekend voor verschillende combinaties van parameters,
en dit aan de hand van geavanceerde steratmosfeermodellen. Deze modellen voorspellen
hoe de druk, dichtheid en temperatuur doorheen de steratmosfeer verloopt wanneer de ster
een aantal specifieke eigenschappen bezit, i.e. als ze een welbepaalde temperatuur, graviteit,
enz. heeft. Het verloop van deze globale parameters bepaalt in sterke mate hoe het spectrum
van een dergelijke ster eruit zal zien. Dit synthetisch spectrum wordt dan vergeleken met
het waargenomen spectrum. Omdat elke parameter een specifieke (en door ervaring gekende) invloed op het lijnenspectrum heeft, kunnen we de parameters zodanig aanpassen dat
de voorspelling van het spectrum bij die parameters heel nauwkeurig aansluit bij het geobserveerde spectrum. De set van parameters waarbij het verschil tussen het theoretisch en het
waargenomen spectrum minimaal is, bepaalt dan de definitieve sterparameters. Deze procedure wordt lijnprofielfitting genoemd.
192
Fundamentele parameters van B-type sterren
Doorheen de hele verhandeling hebben wij de NLTE14 atmosfeer- en lijnpredictie code FASTWIND15 aangewend om de synthetische lijnprofielen te berekenen voor een grote hoeveelheid
van parametercombinaties. FASTWIND is speciaal ontwikkeld voor het genereren van synthetische spectra voor zware O, B en A-type sterren. Omdat deze sterren soms aanzienlijke
hoeveelheden massa kunnen verliezen onder de vorm van een wind, is een accurate voorspelling van het verloop van de sterrenwind en de invloed hiervan op het lijnenspectrum,
onontbeerlijk. Hoewel er nog een aantal andere atmosfeercodes in omloop zijn, opteerden
we toch voor FASTWIND, omwille van de hoge snelheid waarmee modellen kunnen berekend worden. Bovendien omvat de code alle “state-of-the-art” beschrijvingen van de relevante
fysica.
Voor onze doelen volstaat de voorspelling van slechts drie lijnen: waterstof, helium en silicium. Enerzijds zijn dit de meest eenvoudige elementen met de minste overgangen. Anderzijds kunnen we aan de hand van enkel deze drie elementen alle informatie die we nodig
hebben uit het spectrum extraheren. Uit de waterstoflijnen kunnen we namelijk enerzijds
alle windparameters en anderzijds de oppervlaktegraviteit bepalen. De effectieve temperatuur halen we dan weer uit de siliciumlijnen, samen met de turbulente snelheden, de rotatie en de siliciumabondantie. De heliumlijnen geven ons dan weer de heliumabondantie,
en in bepaalde omstandigheden leveren ze bovendien een controle op de temperatuur en/of
graviteit. In hoofdstuk 1 gaan we uitgebreid in op de betekenis van elk van deze parameters,
hun invloed op de verschillende lijnprofielen en de manier waarop we deze uit het spectrum
kunnen afleiden.
Periodiek variabele B-type superreuzen
In hoofdstuk 2 presenteren we de resultaten van de analyse van een eerste dergelijk lijnprofielenonderzoek voor een eerste steekproef: 40 B-type superreuzen, waarvan er 28 periodieke
variaties vertonen. Ze variëren met heel kleine amplitudes van de orde van enkele tientallen
millimagnituden en hebben periodes van de orde van enkele dagen. Hoewel er intussen al
heel wat sterren tot deze klasse van superreuzen behoren, bestond er tot nog toe grote onduidelijkheid over het variabiliteitsmechanisme van deze sterren. Een eerste vereiste om dat
variabiliteitsmechanisme beter te begrijpen is, zoals we reeds eerder vermeldden, dat we de
superreuzen op een accurate en betrouwbare manier in het HRD kunnen plaatsen. Dit laat
namelijk toe om de positie te vergelijken met variabelen waarvan we het excitatiemechanisme wel kennen. Een eerste onderzoek op basis van fotometrische data (Waelkens et al.
1998) leek erop te wijzen dat er een verband zou zijn met de pulsaties in hoofdreekssterren, zoals die ook voorkomen in de β Cephei en SPB sterren. Omdat de nauwkeurigheid
14 NLTE staat voor Niet Lokaal Thermodynamisch Evenwicht. In LTE veronderstelt men dat het materiaal in
evenwicht is met de lokale kinetische temperatuur, maar dat de straling mag afwijken van zijn thermodynamische
waarde Bν (T). NLTE duidt dus op afwijkingen van deze aanname, en worden vooral belangrijk in hetere ijle
sterren.
15 FASTWIND staat voor Fast Analysis of STellar Atmospheres with WINDs. Deze atmosfeercode werd ontwikkeld in München onder leiding van Joachim Puls, met de bedoeling om op een snelle manier de sterren en hun
stralingsgedreven sterrenwinden te kunnen bestuderen.
Fundamentele parameters van B-type sterren
193
van de fotometrische bepaling van de graviteit voor hete sterren vaak nogal te wensen overlaat, was de enige manier om deze suggestie al dan niet te bevestigen, aan de hand van een
gedetailleerde spectroscopische analyse. Met dit doel voor ogen, werden hoge resolutie spectroscopische data van deze sterren verzameld. Wij bepaalden de stellaire en windparameters
door een gedetailleerd fitten van de waargenomen lijnprofielen. Dankzij deze analyse konden
we de sterren een betrouwbaardere plaats in het HRD geven. De superreuzen bleken nauw
aan te sluiten bij het voorspelde instabiliteitsdomein van de SPB sterren, waar ze aan de rand
met de hogere graviteiten liggen. Wij suggereren hieruit dat de periodieke variabiliteit in B
type superreuzen -tot voor dit werk nog onvoldoende begrepen- te wijten is aan niet-radiale
graviteitsmodi, geëxciteerd door het opaciteitsmechanisme, en dus inderdaad gelijkaardig
aan de oscillaties in B-type hoofdreekssterren, zoals reeds gesuggereerd werd door Waelkens
et al. (1998). Dit is één van de belangrijkste conclusies van deze verhandeling, omwille
van het feit dat ze toelaat om het onderzoeksgebied van de asteroseismologie verder uit te
breiden naar geëvolueerde sterren in het HRD, daar waar dit tot nu toe beperkt bleef tot
hoofdreekssterren. Dit laat toe om de evolutietheorie verder te verfijnen naar latere fasen
toe, door de studie van stertrillingen in B-type superreuzen. De eerste noodzakelijke stap die
nu in deze richting moet genomen worden is het langdurig opmeten van preciese tijdreeksen
van fotometrische en spectroscopische gegevens, om zoveel mogelijk trillingsfrequenties te
detecteren en te interpreteren.
Een belangrijk bijkomend resultaat van dit onderzoek is dat we voor het eerst de relatie tussen
windimpuls en lichtkracht voor late B-type superreuzen konden vastleggen. We vonden grote
gelijkenissen met de relatie voor A-type superreuzen. De relatie voor mid B-type superreuzen
bleek volledig consistent te zijn met theoretische voorspellingen.
Automatische procedure voor het fitten van lijnprofielen
Het jaar 2006 eindigde als een historisch jaar in het onderzoeksdomein van de asteroseismologie, met de succesvolle lancering van de ruimtemissie CoRoT, een acroniem voor Convectie, Rotatie en planetaire Transits. Zoals de naam al verraadt, is deze ruimtemissie gewijd
aan het onderzoeken van de invloed van inwendige processen zoals convectie en rotatie op
sterevolutie, en dit via asteroseismologie (naast de zoektocht naar exoplaneten). De quasicontinue waarnemingen vanuit de ruimte doen grote verwachtingen rijzen bij asteroseismologen. Ze zouden immers leiden tot een grote hoeveelheid data met een ongeziene precisie,
kwaliteit en meetfrequentie.
Met het oog op een optimale selectie van programmasterren voor CoRoT, is kennis van
hun parameters van cruciaal belang. Daarom heeft men een grootscheepse pre-lancerings
waarneemcampagne opgesteld en zowel fotometrische als spectroscopische data verzameld
van alle relatief heldere sterren in het ‘gezichtsveld’ van CoRoT. Die data werden verzameld in de online catalogus GAUDI. Dit archief bevat spectra van bijna 300 B type sterren.
Omwille van de expertise van het Instituut voor Sterrenkunde van de K.U.Leuven in B-type
sterren, werd het analyseren van deze 300 B-type sterren aan ons toevertrouwd.
194
Fundamentele parameters van B-type sterren
Eén van de grootste struikelblokken om een dergelijke grote steekproef te analyseren, is het
ontbreken van een automatische procedure voor het fitten van de lijnprofielen. Tot op heden
werd het fitten van de lijnprofielen altijd manueel gedaan, als volgt. Op basis van het spectraaltype of literatuurwaarden wordt een eerste afschatting gemaakt van de fundamentele
parameters van de ster. De lijnprofielen bij deze parameters worden vergeleken met het
waargenomen spectrum. Door visuele inspectie wordt bepaald hoe goed de fit is, en of er
aanpassingen in de parameters vereist zijn. Zo ja, dan wordt een nieuw model uitgerekend,
waarbij de desbetreffende parameter aangepast wordt, gebruik makend van kennis (door ervaring) van de invloed van deze parameter op de lijnprofielen. Dit proces gaat verder tot de
kwaliteit van de fit volledig naar de wens van de onderzoeker is. Dit is een subjectief en heel
tijdrovend werk, en heeft tot gevolg dat de bestudeerde steekproeven meestal vrij beperkt in
omvang zijn.
Om hieraan tegemoet te komen, hebben wij, voor het eerst, een automatische procedure ontwikkeld voor het fitten van lijnprofielen voor B-type sterren. Verschillende opties werden in
aanmerking genomen, maar uiteindelijk bleek de, voor onze doelen, efficiëntste methode er
één te zijn die gebaseerd is op een statisch grid. Daarom hebben we een uitgebreid grid van
zo’n 650 000 lijnprofielcombinaties opgezet. Het grid is geschikt voor het analyseren van
sterren met heel uiteenlopende eigenschappen, en in verschillende evolutiestadia, en in die
zin representatief voor het hele B-type domein, van koel tot heet, van dwerg tot superreus, met
weinig tot veel massaverlies. Op basis van dit grid (dat volledig beschreven wordt in hoofdstuk 3), hebben we een iteratieve procedure ontwikkeld, waarbij de hierboven beschreven
manuele fitmethode in een computercode werd omgezet. Hierbij worden eveneens objectieve
criteria vastgelegd op basis waarvan beslist wordt of een model een goede fit oplevert of niet.
De procedure, AnalyseBstar, wordt in detail beschreven in hoofdstuk 4. Fig. 4.3 geeft de
grote lijnen van het programma weer en toont de verschillende stappen die doorlopen moeten
worden.
De hete en koele B-type sterren vereisen een andere aanpak voor het bepalen van de effectieve temperatuur, de siliciumabondantie en de microturbulente snelheid. Deze drie parameters beı̈nvloeden tegelijkertijd de equivalente lijnbreedtes van de siliciumlijnen. Het is
dus noodzakelijk de drie effecten van elkaar te scheiden. Daar waar in de hetere sterren de
verhoudingen van equivalente lijnbreedtes van twee verschillende ionisatieniveau’s van siliciumatomen het effect van de abondantie uitschakelen, en zo toelaten om de drie parameters
vast te leggen, is dit voor de laat type sterren niet meer het geval. Daar is namelijk slechts één
ionisatieniveau van silicium beschikbaar en moeten we noodgedwongen een aanname maken
voor de silicium abondantie. We beschouwen altijd drie verschillende abondanties en kiezen
er uiteindelijk diegene uit die de beste fit geeft. In geval van gelijkwaardige fitkwaliteit geven
we de voorkeur aan het model met een kleine siliciumdepletie, omwille van een algemeen
waargenomen trend in deze richting, althans voor B-type dwergen (zie verder).
Vooraleer de methode concreet toe te passen op echte spectra, hebben we de convergentie van
onze methode eerst uitgebreid getest op gesimuleerde spectra, op basis van synthetische lijnprofielen. De methode convergeerde succesvol naar de goede waarden. Enkel de waarde voor
de macroturbulente snelheid moet met de nodige voorzichtigheid behandeld worden. Daarna
hebben we de methode in eerste instantie getest op sterren waarvan we de fundamentele pa-
Fundamentele parameters van B-type sterren
195
rameters al vrij goed kennen, en waarvan we heel hoge resolutie data ter beschikking hadden.
Het gaat om een selectie van β Cephei en SPB sterren, samen met nog een vijftal sterren,
waarvan de pulsaties nog niet bevestigd werden. De β Cephei en de andere pulserende sterren werden reeds geanalyseerd door Morel (2007), gebruik makend van de atmosfeercodes
DETAIL/SURFACE, die geheel onafhankelijk van FASTWIND gedefinieerd werden. De
analyse van de SPB sterren is nog aan de gang (Briquet et al. 2007). Momenteel zijn er al
twee sterren geanalyseerd.
AnalyseBstar leverde parameters op die nagenoeg volledig consistent waren met de waarden
bepaald met DETAIL/SURFACE. De lijnprofielfits zagen er bovendien meer dan bevredigend
uit, wat erop wijst dat onze methode goed werkt.
Een interessant gegeven kwam naar voor uit de studie van deze steekproef, nl. dat er in Btype dwergen een systematische depletie in silicium lijkt op te treden. Het is niet de eerste
keer dat een dergelijke Si abondantie, lager dan in de Zon, gevonden wordt. Literatuur lijkt
aan te tonen dat de abondanties voor superreuzen wel consistent zijn met solaire waarden.
Aangezien Si een product van interne chemische processen is, lijkt het heel vreemd om een
dergelijke depletie waar te nemen, terwijl we eerder een verrijking zouden verwachten. Dit
dient zeker nog verder uitgeklaard te worden.
De goed bekende pulserende sterren vormen een uitstekende overgang naar de GAUDI steekproef, waar de kwaliteit van de spectra veel minder goed was. De catalogus bevat heel wat
Be sterren en spectroscopische dubbelsterren, die wij met FASTWIND niet op een accurate
manier kunnen analyseren. Na een grondige preselectie, waarbij we de Be sterren, de spectroscopische dubbelsterren, de heel snelle rotators, de ELODIE spectra en de spectra met een
erg lage kwaliteit eruit gelaten hebben, bleven nog zo’n 35 goede kwaliteits FEROS spectra
over, gespreid over het hele spectraalgebied van de B-type sterren. Deze kleine selectie van
GAUDI B-type sterren liet toe om calibraties van de effectieve temperatuur en graviteit als
functie van waargenomen fotometrische indices te verifiëren.
Calibraties voor de effectieve temperatuur en graviteit
Omdat, dankzij de grootscheepse waarnemingscampagne ter voorbereiding van de CoRoT
ruimtemissie, bijna voor alle sterren in de GAUDI steekproef Strömgren indices beschikbaar waren hebben we ons voornamelijk op dit fotometrisch systeem geconcentreerd. De
beste calibratie tot nu toe voor handen en door velen toegepast, is de calibratie van Balona
(1994). We hebben geverifieerd dat de spectroscopisch en fotometrisch bepaalde temperaturen in goede overeenstemming waren. Enkel voor sterren heter dan 20 000 Kelvin waren
de afwijkingen iets groter. Voor de graviteit vonden we echter veel significanter discrepanties,
waarbij de spectroscopische waarden systematisch lager lijken te zijn dan de fotometrische
waarden. Balona (1994) raadde af om zijn calibratie voor de graviteit te gebruiken, omwille
van discrepanties met de waarden afgeleid uit evolutietheorie, waarschijnlijk deels te wijten
aan het verwaarlozen van NLTE effecten. Wij hebben op een gedetailleerde manier een zo
nauwkeurige mogelijke waarde voor de graviteit proberen af te leiden. Onze resultaten worden nu door het CoRoT team gebruikt voor de selectie van de bronnen voor de missie, alsook
196
Fundamentele parameters van B-type sterren
voor de toekomstige seismologische modellering ervan.
Besluit
We kunnen de belangrijkste bijdragen van deze verhandeling als volgt samenvatten:
– De periodieke trillingen in B-type superreuzen zijn hoogstwaarschijnlijk niet-radiale trillingen ten gevolge van het κ mechanisme, net als bij hoofdreeks B-type sterren.
– De Wind-Momentum-Luminositeits Relatie voor late B superreuzen is een logische verlenging van deze voor de A-type superreuzen.
– We vonden een indicatie dat grote fotometrische amplitudes samengaan met hoge dichtheden in de wind en een grotere relatieve verandering in massaverlies.
– We hebben voor het eerst een geautomatiseerde procedure ontwikkeld voor het fitten van
lijnprofielen voor B-type sterren, die toelaat om op een objectieve, robuuste en nauwkeurige
manier grote steekproeven te analyseren in een minimum van tijd.
– We suggereren dat er een systematische depletie in silicium optreedt in B type dwergen,
terwijl de Si-abondanties in superreuzen consistent lijken te zijn met die in de Zon.
We hopen dat we hiermee bijgedragen hebben tot het spectroscopisch en asteroseismologisch
onderzoek van B sterren.
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• De Cat, P.; Briquet, M.; Aerts, C.; Goossens, K.; Saesen, S.; Cuypers, J.; Yakut, K.; Scuflaire, R.; Dupret, M.-A.; Uytterhoeven, K.; van Winckel, H.; Raskin, G.; Davignon, G.;
Le Guillou, L.; van Malderen, R.; Reyniers, M.; Acke, B.; de Meester, W.; Vanautgaerden,
J.; Vandenbussche, B.; Verhoelst, T.; Waelkens, C.; Deroo, P.; Reyniers, K.; Ausseloos,
208
Publication list
M.; Broeders, E.; Daszyska-Daszkiewicz, J.; Debosscher, J.; de Ruyter, S.; Lefever, K.;
Decin, G.; Kolenberg, K.; Mazumdar, A.; van Kerckhoven, C.; de Ridder, J.; Drummond,
R.; Barban, C.; Vanhollebeke, E.; Maas, T.; Decin, L., 2007, Long term photometric monitoring with the Mercator telescope. Frequencies and mode identification of variable O-B
stars, Astronomy and Astrophysics, 463, 243-249.
• Arentoft, T.; de Ridder, J.; Grundahl, F.; Glowienka, L.; Waelkens, C.; Dupret, M.-A.;
Grigahcne, A.; Lefever, K.; Jensen, H. R.; Reyniers, M.; Frandsen, S.; Kjeldsen, H.,
2007, Oscillating blue stragglers, Doradus stars and eclipsing binaries in the open cluster
NGC 2506, Astronomy and Astrophysics, 465, 965-979.
Publications in Conference Proceedings
• Krumpe, M.; Coffey, D.; Egger, G.; Vilardell, F.; Lefever, K.; Liermann, A.; Hoffmann,
A.I.; Steiper, J.; Cherix, M.; Albrecht, S.; and 7 coauthors, 2005, X-RED: a satellite
mission concept to detect early universe gamma ray bursts, SPIE, 5898, 419-432 and 438451
• Lefever, K.; Puls, J.; Aerts, 2006, Study of a sample of periodically variable B-type supergiants, Memorie della Societa Astronomica Italiana, 77, 135.
• Arentoft, T.; Grundahl, F.; De Ridder, J.; Glowienka, L.; Lefever, K.; Jensen, H. R.;
Reyniers, M.; Waelkens, C.; Frandsen, S.; Nielsen, T. B.; Kjeldsen, H., 2006, Pulsating
stars and EBs in clusters: NGC 2506, Memorie della Societa Astronomica Italiana, 77,
99.
• Grundahl, F.; Arentoft, T.; Bruntt, H.; Christensen-Dalsgaard, J.; Clausen, J. V.; Frandsen,
S.; Glowienka, L.; Jensen, H. R.; Kjeldsen, H.; Lefever, K.; and 4 coauthors, 2006,
Pinpointing isochrones in clusters, Memorie della Societa Astronomica Italiana, 77, 433.
• Lefever, K.; Puls, J.; Aerts, 2006, Fundamental Parameters of Massive OBA Stars: Studying the GAUDI Sample in Preparation of COROT, ASP Conf.Ser. 349, Astrophysics of
Variable Stars, 349, 277.
• Lefever, K.; Puls, J.; Aerts, 2007, A Grid of FASTWIND NLTE Model Atmospheres of
Massive Stars, ASP Conf. Ser. 364, The Future of Photometric, Spectrophotometric and
Polarimetric Standardization, 364, 545.
• Lefever, K.; Puls, J.; Aerts, C., 2007, Periodically variable B-type supergiants: Empirical
evidence for non-radial gravity-mode oscillations, to appear in ASP Conf. Ser.
In Preparation
• Lefever, K.; Puls, J.; Morel, Th.; Briquet, M.; Decin. L; Aerts, C., 2007, AnalyseBstar:
an automated tool for the high-accuracy estimation of fundamental parameters for B type
stars - Application to the B-type stars in GAUDI (in preparation, to be submitted to A&A)
Acronyms and abbreviations
AI
Ar
ARES
B.C.
BSC
C
CAK
CAT
cdf
CES
CMF
CoRoT
CTIO
ESA
ESO
EW
Fe
FASTWIND
GA
GAUDI
HIPPARCOS
H
HD
He
HRD
IDL
INT
IRAF
IUE
Artifical Intelligence
argon
Automatic Routine for line Equivalent widths
in stellar Spectra
Bolometric Correction
Bright Star Catalogue
Carbon
Castor, Abbott and Klein
Coudé Auxiliary Telescope
cumulative distribution function
Coudé Echelle Spectrograph
Co-Moving Frame
COnvection ROtation and planetary Transits
Cerro Tololo Inter-american Observatory
European Space Agency
European Southern Observatory
Equivalent Width
iron
Fast Analysis of STellar atmospheres with WINDs
Genetic Algorithm
Ground-based Asteroseismology Uniform
Database Interface
HIgh Precision PARallax COllecting Satellite
hydrogen
Henry Draper
Helium
Hertzsprung-Russell Diagram
Interface Description Language
Isaac Newton Telescope
Image Reduction and Analysis Facility
International Ultraviolet Explorer
99
13
108
20
31
13
9
30
88
30
12
94
45
28
30
14
13
10
99
95
27
13
3
13
4
102
45
105
133
210
Publication list
JKT
LAEFF
l.c.
LDF
Mg
N
Ne
NGC
Ni
NLTE
O
OSN
PLC
SIMBAD
SED
S
Si
SPB
SB2
SNR
SUSI
TAMS
UV
WLR
ZAMS
Jacobus Kapteyn Telescope
Laboratorio de Astrofı́sica Espacial y
Fı́sica Fundamental
Luminosity class
Lennon, Dufton and Fitzsimmons
Magnesium
Nitrogen
Neon
New General Catalogue
Nickel
Non Local Thermodynamic Equilibrium
Oxygen
Observatory Sierra Nevada
Period-Luminosity-Colour
Set of Identifications Measurements and
Bibliography for Astronomical Data
Spectral Energy Distribution
Sulfur
Silicon
Slowly Pulsating B star
Double-lined Spectroscopic Binary
Signal-to-noise ratio
Sydney University Stellar Interferometer
Terminal-Age Main Sequence
Ultraviolet
Wind Momentum Luminosity Relation
Zero-Age Main Sequence
45
95
174
46
13
13
13
39
13
10
13
96
29
34
11
13
13
24
97
42
165
63
9
9
5