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Minist´erio de Ciˆencia y Tecnologia, Observat´orio Nacional/Brasil
Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg/Alemanha
ESPECTROSCOPIA QUANTITATIVA
DE ESTRELAS OB
Hydrogˆenio, H´elio e Carbono
Mar´ıa Fernanda Nieva
Tese para obten¸c˜ao do t´ıtulo de doutor
Programa de orienta¸c˜ao bilateral (co-tutelle)
Orientadores:
Dra. Katia Cunha e Dr. Ulrich Heber
Erlangen - Maio de 2007
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Milhares de livros grátis para download.
A Tese foi apresentada `as 14:30 horas do dia 21 de Maio de 2007, na sala de semi-
narios do departamento de F´ısica do Estado S´olido da Universidade de Erlangen-
N¨urnberg.
A banca examinadora foi composta por:
Dr. Thomas Fauster, presidente (Universit¨at Erlangen-N¨urnberg)
Dr. Ulrich Heber (Universit¨at Erlangen-N¨urnberg)
Dra. Katia Cunha (Observat´orio Nacional)
Dr. Ramiro de la Reza (Observat´orio Nacional)
Dr. Ulrich Katz (Universit¨at Erlangen-N¨urnberg)
Dr. Keith Butler (Ludwig-Maximilians-Universit¨at M¨unchen)
Os referees da Tese foram:
Dr. Ulrich Heber
Dra. Katia Cunha
Dr. Klaus Werner (Universit¨at T¨ubingen)
ads:
Dedicated to my parents, Graciela I. Russo and Felipe J. Nieva,
who gave me the freedom to choose my own path in life.
Dedicado a mis padres, G.I.R y F.J.N., quienes me dieron
la libertad de elegir mi propio camino en la vida.
Resumo
Determina¸c˜oes precisas da composi¸c˜ao qu´ımica de estrelas OB constituem
v´ınculos observacionais fundamentais para a evolu¸c˜ao estelar e evolu¸c˜ao qu´ımica
da Gal´axia. Dentre os elementos leves, o carbono ´e um dos metais mais abun-
dantes no Universo mas a sua an´alise em estrelas jovens apresentou resultados
n˜ao-conclusivos nas ´ultimas d´ecadas. Na vizinhan¸ca solar, as abundˆancias de
carbono obtidas principalmente para estrelas OB n˜ao evolu´ıdas apresentam um
grande espalhamento - com mais de uma ordem de magnitude - e o seu valor
m´edio ´e sistematicamente mais baixo do que aquele obtido para estrelas FG (in-
cluindo o Sol) e regi˜oes H ii. Ambos os resultados n˜ao podem ser explicados em
termos da evolu¸c˜ao estelar e da evolu¸c˜ao qu´ımica da Gal´axia. A an´alise espectral
deste tipo de estrelas tamb´em apresenta abundˆancias discrepantes a partir de
diferentes linhas de C ii e falha ao estabelecer o equil´ıbrio de ioniza¸c˜ao C ii/iii.
Por outro lado, linhas espectrais de hidrogˆenio e h´elio s˜ao ferramentas de
diagn´ostico cruciais para a an´alise quantitativa de estrelas OB, uma vez que elas
s˜ao indicadores prim´arios para a determina¸c˜ao de parˆametros atmosf´ericos funda-
mentais, isto ´e, a temperatura efetiva e a gravidade superficial. A an´alise cuida-
dosa destes parˆametros fornece a base para o estudo posterior de abundˆancias
de metais. C´alculos de forma¸c˜ao de linhas em n˜ao-ETL para estes elementos,
com uma abordagem h´ıbrida, ainda n˜ao foram discutidos completamente at´e o
momento, apesar desta abordagem ser amplamente adotada em an´alises de linhas
met´alicas.
Inicialmente, espectros sint´eticos de hidrogˆenio e h´elio s˜ao calculados com
base na abordagem h´ıbrida n˜ao-ETL a fim de testar a capacidade destes modelos
de reproduzir espectros de alta resolu¸c˜ao e alta raz˜ao sinal/ru´ıdo de estrelas OB
an˜as e gigantes. Modelos atˆomicos e teorias de alargamento de linhas modernos
s˜ao empregados para modelar os espectros de H e He i/ii. Os espectros sint´eticos
h´ıbridos n˜ao-ETL ajustam simultaneamente quase todas as linhas mensur´aveis de
H e He observadas nos espectros de seis estrelas usadas como teste, em um grande
intervalo espectral, do limite da s´erie de Balmer at´e o infravermelho pr´oximo.
Compara¸c˜oes das estruturas atmosf´ericas e dos espectros sint´eticos de H e He
com modelos n˜ao-ETL j´a publicados e que incluem obscurecimento por linhas
demostram que a aproxima¸c˜ao ETL ´e apropriada para modelar a estrutura at-
mosf´erica de estrelas OB an˜as e gigiantes com metalicidades de at´e 1/5 do valor
solar. Esta abordagem evita inconsistˆencias na modelagem dos singletos do He i
encontradas em outros resultados em n˜ao-ETL publicados na literatura. Tamb´em
supera os modelos ETL puros – amplamente aplicados na an´alise de estrelas OB –
em muitos aspectos: o aumento da intensidade das linhas calculadas em n˜ao-ETL
e o uso de dados detalhados de alargamentos de linhas resultam em diferen¸cas
significativas tanto nos perfis de linhas como nas larguras equivalentes das lin-
has de Balmer de do h´elio. Efeitos sistem´aticos na determina¸c˜ao dos parˆametros
estelares s˜ao quantificados. Um procedimento incial confi´avel para estudos de
espectros de metais ´e estabelecido.
Em seguida, um modelo detalhado e robusto para C ii-iv para c´alculos de
forma¸c˜ao de linhas em n˜ao-ETL ´e apresentado. O modelo ´e baseado em dados
atˆomicos selecionados cuidadosamente em uma calibra¸c˜ao emp´ırica. Uma an´alise
espectral quantitativa auto-consistente ´e realizada usando um esquema iterativo
para determinar os parˆamentos atmosf´ericos estelares com grande acur´acia e para
selecionar os dados atˆomicos apropriados para o c´alculo das abundˆancias. O
equil´ıbrio de ioniza¸c˜ao do carbono ´e estabelecido com sucesso com um ´unico
conjunto de dados atˆomicos para todas as estrelas da nossa amostra, que cobre
um grande intervalo de parˆametros. A consistˆencia ´e atingida para um grande
n´umero de linhas de carbono - em um total de 40. Isto inclui, em particular, perfis
mais intensos que s˜ao de grande importˆancia para aplica¸c˜oes extra-gal´acticas.
O duradouro problema de inconsistˆencias na determina¸c˜ao das abundˆancias de
carbono a partir de diferentes linhas e est´agios de ioniza¸c˜ao ´e resolvido.
A an´alise auto-consistente permite definir parˆametros atmosf´ericos e abundˆan-
cias de carbono com acur´acia sem precedentes, com incertezas da ordem de 1%
na temperatura efetiva, 10% na gravidade superficial e 20% na abundˆancia de
carbono, com reduzidos erros sistem´aticos. Isto ´e significativamente melhor do
que os resultados de estudos anteriores, que tipicamente apresentam incertezas
de 5-10%, ∼25% e um fator de ∼2-3, respectivamente.
Al´em disso, uma abundˆancia extremamente homogˆenea de log (C/H) + 12 =
8.32 ±0.04 ´e obtida para as estrelas da nossa amostra. Este resultado estab-
elece a abundˆancia de carbono atual em estrelas da vizinhan¸ca solar como sendo
log (C/H) + 12 8.35±0.05, ap´os pequenos ajustes de < +0.05 dex por estrela,
devido a efeitos evolutivos. Este resultado est´a em acordo com o valor solar
revisto recentemente e com a abundˆancias nebulares na regi˜ao H ii de Orion.
A abordagem apresentada aqui permite que os efeitos de erros sitem´aticos
nos parˆametros fundamentais e nas abundˆancias sejam estabelecidos. Isto sugere
que muitas das dificuldades encontradas em trabalhos anteriores podem estar
relacionadas com grandes efeitos sistem´aticos na an´alise causados por incertezas
nos dados atˆomicos e/ou na determina¸c˜ao dos parˆametros atmosf´ericos.
A obten¸c˜ao de um valor homogˆeneo para a abundˆancia de carbono atual
na vizinhan¸ca solar concorda com an´alises do meio interestelar e tamb´em com
previs˜oes de modelos de evolu¸c˜ao qu´ımica da Gal´axia. A grande acur´acia al-
can¸cada aqui ´e um pr´e-requisito para a determina¸c˜ao do gradiente Gal´actico de
abundˆancias, que ´e da ordem das incertezas atuais (em contraste com um espalha-
mento em abundˆancias da ordem de uma magnitude encontrado em estudos an-
teriores). Modelos de evolu¸c˜ao estelar e de evolu¸c˜ao qu´ımica da Gal´axia podem
assim ser vinculados mais fortemente com abundˆancias de carbono confi´aveis.
Isto pode ser feito em ambientes de diferentes metalicidades (i.e., gal´axias) com
limita¸c˜oes definidas apenas pela qualidade dos espectros estelares.
Zusammenfassung
Unser Verst¨andnis der Entwicklung der Sterne und der Galaxien ruht auf der em-
pirischen Bestimmung der chemischen Zusammensetzung von Sternen und Gas-
nebeln. Um die heute vorherrschenden Elementh¨aufigkeiten zu ermitteln, spielen
die Sterne des Spektraltyps OB eine herausragende Rolle, da sie sehr junge Ob-
jekte sind. Kohlenstoff ist unter den leichten Elementen eines der h¨aufigsten
Metalle. Allerdings liefern die bisherigen Analysen des Kohlenstoffspektrums
widerspr¨uchliche Ergebnisse.
Die Sonnenumgebung ist das am besten untersuchte Himmelsareal. Hier zeigt
sich, dass die Kohlenstoffh¨aufigkeit bei den jungen, unentwickelten OB-Sternen
im Mittel niedriger ist als bei ¨alteren Sternen des Spektraltyps F und G (ein-
schließlich der Sonne) und bei Gasnebeln (H ii-Regionen). Ebenfalls weisen die
Kohlenstoffh¨aufigkeiten von OB-Sternen eine große Streuung von bis zu einer
Gr¨oßenordnung auf. Dies kann weder im Rahmen der Theorie der Sternent-
wicklung noch der Galaxienentwicklung verstanden werden. Bemerkenswert ist
dar¨uberhinaus, dass in vielen Analysen weder die H¨aufigkeiten aus verschiede-
nen Linien des einfach ionisierten Kohlenstoffs, C ii, miteinander in Einklang zu
bringen sind, noch dass das Ionisationsgleichgewicht von C ii/iii eingestellt wer-
den kann.
Wasserstoff und Helium spielen f¨ur die quantitative Spektralanalyse von OB-
Sternen, wie in fast allen astronomischen Objekten, eine wichtige Rolle. Sie wer-
den zur Bestimmung der fundamentalen atmosph¨arischen Parameter, d.h. der
Effektivtemperatur und der Schwerebeschleunigung, genutzt. Ohne die genaue
Kenntnis dieser Parameter k¨onnen die H¨aufigkeiten der Metalle nicht zuverl¨assig
bestimmt werden. Die hier benutzte Analysetechnik, die Spektrumssynthese mit-
tels sogenannter hybriden non-LTE Rechnungen, ist bisher noch nie sorgf¨altig f¨ur
Wasserstoff- und Heliumlinien diskutiert worden, obwohl sie f¨ur Metalllinien weit
verbreitet ist.
Im ersten Arbeitsschritt werden hybride non-LTE Rechnungen f¨ur Wasser-
stoff sowie f¨ur neutrales und einfach ionisiertes Helium durchgef¨uhrt. Die syn-
thetischen Spektren werden anhand von erstklassigem Beobachtungsmaterial,
hochaufgel¨osten, nahezu rauschfreien Spektren von sechs Zwerg- und Riesenster-
nen der Spektraltypen O und B, getestet. Dabei werden die zur Zeit besten
Modellatome und Linienverbreiterungstabellen ber¨ucksichtigt. Die Ergebnisse
sind sehr ¨uberzeugend, denn es gelingt praktisch alle meßbaren Wasserstoff-
und Heliumlinien ¨uber den gesamten Spektralbereich vom Balmersprung bis ins
Nahinfrarote konsistent zu reproduzieren.
Die Verl¨asslichkeit der Methode wird durch Vergleich mit publizierten non-
LTE Modellen verifiziert. Die Annahme des lokalen thermodynamischen Gleich-
gewichts erweist sich als v¨ollig ausreichend, um die Druck- und Temperatur-
schichtung der Atmosph¨are korrekt wiederzugeben. Dies gilt unabh¨angig vom
Metallgehalt bis hinunter zu einem f¨unftel der solaren Metallizit¨at. Es wird
gezeigt, dass der gew¨ahlte Ansatz Inkonsistenzen beim He i Singlettsystem ver-
meidet, die in anderen non-LTE Rechnungen auftreten. Im Vergleich zu LTE-
Analysen, die immer noch weit verbreitet weil einfacher auszuf¨uhren sind, lassen
sich systematische Abweichungen bei der Bestimmung der atmosph¨arischen Pa-
rameter quantifizieren. Ein verl¨asslicher Ausgangspunkt f¨ur die Analyse der Me-
talllinenspektren wurde gefunden.
Im zweiten Arbeitsschritt wird ein umfassendes und robustes Modellatom
f¨ur Kohlenstoff entwickelt, das die Ionisationsstufen C ii–C iv ber¨ucksichtigt.
Es basiert auf sorgf¨altig ¨uberpr¨uften atomaren Daten und wurde anhand von
beobachteten Spektren kalibriert. F¨ur die selbstkonsistente quantitative Spek-
tralanalyse wird ein umfangreiches Iterationsschema ausgearbeitet, um die at-
mosph¨arischen Parameter noch genauer zu bestimmen und geeignete Atomdaten
zu selektieren. Das Ionisationsgleichgewicht des Kohlenstoffs l¨asst sich mit dem
kalibrierten Modellatom zwanglos einstellen. Ebenfalls werden ¨ubereinstimmende
Kohlenstoffh¨aufigkeiten aus insgesamt 40 Spektrallinien abgeleitet. Dabei ist
besonders bemerkenswert, dass auch die st¨arksten Absorptionslinien repro-
duziert werden, die bei extragalaktischen Sternen von großer Bedeutung sind,
da schw¨achere Linien in diesen Objekten nicht meßbar sind. Der gleiche Befund
ergibt sich bei allen sechs Programmsternen, obwohl sie einen weiten Bereich von
Effektivtemperaturen und Schwerebeschleunigungen ¨uberdecken. Damit kann
das alte Problem der widerspr¨uchlichen H¨aufigkeiten aus verschiedenen Linien des
einfach ionisierten Kohlenstoffs und des Ionisations-Ungleichgewichts als gel¨ost
angesehen werden.
Die Fr¨uchte der Bem¨uhungen um gr¨oßtm¨ogliche Genauigkeit werden nach
der Fehleranalyse deutlich. Mit ∼1% bei der Effektivtemperatur und ∼10% bei
der Schwerebeschleunigung sind die Parameter weit genauer bestimmt als ¨ublich
(5-10% bzw. ∼25%). Dies setzt sich bei den Kohlenstoffh¨aufigkeiten fort, bei
denen bisher nie dagewesene ∼20% erreicht werden, w¨ahrend bei den meisten
publizierten Analysen Fehler von Faktoren 2 bis 3 nicht ungew¨ohnlich sind.
Vergleicht man die Programmsterne miteinander, f¨allt die große Homogenit¨at
der Kohlenstoffh¨aufigkeit auf. Die mittlere H¨aufigkeit aller sechs Programm-
sterne ist log (C/H) + 12 = 8.32 mit einer sehr geringen Streuung von 0.04.
Ber¨ucksichtigt man noch kleine Entwicklungseffekte (<0.05 dex) so ergibt sich die
heutige Kohlenstoffh¨aufigkeit in der Sonnenumgebung zu log (C/H) + 12 = 8.35
±0.05 in guter
¨
Ubereinstimmung mit den j¨ungsten Resultaten f¨ur die Sonne
und f¨ur den Orionnebel. Die Ergebnisse der Analyse legen nahe, dass die Un-
gereimtheiten und Widerspr¨uche vieler publizierter Analysen auf untersch¨atzte
systematische Effekte in den Spektralanalysen zur¨uckzuf¨uhren sind, die wiederum
aus Ungenauigkeiten bei den Atomdaten und/oder der atmosph¨arischen Para-
meter herr¨uhren.
Die hier erzielten Analyseergebnisse stehen im Einklang mit den Vorhersagen
galakto-chemischer Entwicklungsmodelle. Geht man ¨uber die Sonnenumgebung
hinaus, sind galaktische H¨aufigkeitsgradienten zu erwarten. Die hohe Genauigkeit
der gegenw¨artigen Analysen ist unabdingbare Voraussetzung, um solche Gradi-
enten zu vermessen, die nach Vorhersagen von Entwicklungsmodellen klein sein
sollten. Die große Streuung alter Resultate erlaubte bisher keine ¨uberzeugenden
Schlußfolgerungen zu treffen.
Mit der hier entwickelte Analysemethodik f¨ur Kohlenstoffh¨aufigkeiten werden
k¨unftig Modelle f¨ur die Sternentwicklung wie auch f¨ur die chemische Entwicklung
von Galaxien sehr viel genauer als bisher ¨uberpr¨uft werden k¨onnen. Dies kann
auf Umgebungen sehr unterschiedlicher Metallizit¨at ausgedehnt werden, z.B. in
anderen Galaxien, beschr¨ankt nur durch die Qualit¨at des Beobachtungsmaterials.
Abstract
Precise determinations of the chemical composition of OB-type stars constitute
fundamental observational constraints to stellar and galactochemical evolution.
Among the light elements, carbon is one of the most abundant metals in the
Universe but analyses in early-type stars showed inconclusive results in the past
decades. In the solar vicinity, carbon abundances derived from mostly unevolved
OB stars indicate a large scatter – by more than one order of magnitude – and
the mean value is systematically lower than those derived from FG-type stars
(including the Sun) and H ii regions. Both results cannot be explained in terms
of stellar evolution and chemical evolution of the Galaxy. Spectral analyses of
this kind of star also give largely discrepant abundances from different C ii lines
and fail to establish the C ii/iii ionization balance.
On the other hand, hydrogen and helium line spectra are crucial diagnostic
features for the quantitative analysis of OB stars as they are primary indicators for
deriving the fundamental atmospheric parameters, i.e. the effective temperature
and the surface gravity. Their careful analysis provides the basis for any further
study of metal abundances. Hybrid non-LTE line-formation calculations for these
elements have not been discussed thoroughly so far, despite a wide use of this
approach for analyses of metal line spectra.
In a first step, synthetic spectra of hydrogen and helium are computed on the
basis of a hybrid non-LTE approach in order to test the ability of these models
to reproduce high-resolution and high-S/N spectra of dwarf and giant OB stars.
State-of-the-art model atoms and line-broadening theories are employed to model
the H and He i/ii spectra. The present hybrid non-LTE synthetic spectra match
simultaneously almost all measurable hydrogen and helium lines observed in six
test stars over a wide spectral range from the Balmer limit to the near-infrared.
A comparison of atmospheric structures and synthetic spectra of H and He
with published line-blanketed non-LTE models validates the suitability of the
LTE approximation for modelling the atmospheric structure of dwarf and giant
OB stars at metallicities down to (at least) 1/5 ×solar. The present approach
avoids inconsistencies in the modelling of the He i singlets found in other published
non-LTE calculations. It improves on pure LTE models – widely applied for
OB star analyses – in many aspects: non-LTE strengthening and the use of
detailed line-broadening data result in significant differences in the line profiles
and equivalent widths of the Balmer and helium lines. Systematic effects on
the stellar parameter determination are quantified. A reliable starting point for
studies of the metal spectra is established.
In a second step, a comprehensive and robust C ii-iv model for non-LTE line-
formation calculations is presented. The model is based on atomic data carefully
selected in an empirical calibration. A self-consistent quantitative spectrum ana-
lysis is performed using an extensive iteration scheme to determine stellar atmos-
pheric parameters with improved accuracy and to select the appropriate atomic
data to be used for the derivation of abundances. The carbon ionization balance
is successfully established with a unique set of input atomic data for all sample
stars, which cover a wide parameter range. Consistency is achieved for a large
number of carbon lines – in total 40. This includes in particular the strongest
features that are of highest importance for extragalactic applications. The long-
standing problem of inconsistencies in the determination of carbon abundances
from different lines and ionization stages is solved.
The self-consistent analysis provides atmospheric parameters and carbon
abundances with unprecedented accuracy, with uncertainties as low as ∼1% in
effective temperature, ∼10% in surface gravity and ∼20% in carbon abundance,
with reduced systematic error. This improves significantly on results from previ-
ous studies, which typically give uncertainties of 5-10%, ∼25% and a factor ∼2-3,
respectively.
Moreover, an extremely homogeneous abundance of log (C/H) + 12 = 8.32
±0.04 is derived from the star sample, This result constrains the present-day stel-
lar carbon abundance in the solar neighbourhood to log (C/H) + 12 8.35±0.05,
after small adjustments by < +0.05 dex per star for evolutionary effects. This is
in agreement with the recently revised solar value and with the gas-phase abun-
dance of the Orion H ii region.
The approach presented here allows the effects of systematic errors on funda-
mental parameters and abundances to be constrained. This suggests that most of
the difficulties found in previous work may be related to large systematic effects
in the analysis caused by inaccuracies in the atomic data and/or the atmospheric
parameter determination.
The finding of a homogeneous present-day carbon abundance in the solar
vicinity conforms with analyses of the interstellar medium and also with predic-
tions of chemical-evolution models for the Galaxy. The high accuracy achieved
here is a prerequisite for the determination of the Galactic abundance gradient,
which is of the order of the present uncertainties (in contrast to an overall abun-
dance scatter of one order of magnitude found in previous studies). Stellar and
galactochemical evolution models can from now on be constrained more tightly
with reliable carbon abundances. This can be done for environments of different
metallicities (i.e. galaxies), with the only remaining limitation being the quality
of the observed stellar spectra.
Contents
1 Introduction (in Portuguese) 1
2 Model Atmospheres 7
2.1 Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Classical Stellar Atmospheres . . . . . . . . . . . . . . . . . . . . 14
2.3 Thermodynamic State: LTE vs. Non-LTE . . . . . . . . . . . . . 15
2.4 Metal Line Blanketing . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Spectral Line Formation . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.1 Line Strength & Broadening Mechanisms . . . . . . . . . . 19
2.5.2 Non-LTE Line Formation . . . . . . . . . . . . . . . . . . 22
3 Atomic Data for Spectral Modelling 27
3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Atomic Structure Calculations . . . . . . . . . . . . . . . . . . . . 28
3.3 Scattering Calculations . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Construction of Model Atoms . . . . . . . . . . . . . . . . . . . . 36
4 Spectroscopic Analysis 39
4.1 Atmospheric Parameters . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Chemical Abundances . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Hybrid Non-LTE Approach for H and He Line Formation 47
5.1 Model Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Observational Data . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Applications to Observations . . . . . . . . . . . . . . . . . . . . . 52
5.3.1 Visual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.2 Near-IR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
I
5.4 Comparison to Other Model Predictions . . . . . . . . . . . . . . 58
5.4.1 Atmospheric Structures, SEDs: LTE vs. non-LTE . . . . . 59
5.4.2 Spectra: Hybrid non-LTE vs. non-LTE and LTE . . . . . . 60
5.4.3 Line Formation: Hybrid non-LTE vs. Full non-LTE . . . . 65
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6 Non-LTE Line Formation for Carbon: Self-Consistent Analysis 71
6.1 The C ii/iii/iv Model Atom . . . . . . . . . . . . . . . . . . . . . 72
6.2 Model Atom Calibration . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.1 Extensive Iteration on Fundamental Variables . . . . . . . 76
6.2.2 Sensitivity of C Lines to Atomic Data . . . . . . . . . . . . 77
6.2.3 Line-Formation Details . . . . . . . . . . . . . . . . . . . . 86
6.2.4 Sensitivity of (C) to Parameter Variations . . . . . . . . . 88
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.4 Comparison with Previous Work . . . . . . . . . . . . . . . . . . . 97
6.4.1 Predictions from Different non-LTE Model Atoms . . . . . 97
6.4.2 Effective Temperatures . . . . . . . . . . . . . . . . . . . . 99
6.5 The Stellar Present-Day C Abundance in the Solar Neighbourhood 101
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 Conclusions 109
A Basis of
´
Echelle Data Reduction 113
B Atomic data for H and He 117
C Linefits to H and He 121
D Linefits to C lines 127
Bibliography 133
Acknowledgements 139
II
Chapter 1
Introduction (in Portuguese)
As estrelas de tipo espectral ’O-late’ e de tipo espectral ’B-early’ pertencem a um
subgrupo que ´e comumente conhecido como estrelas OB (Jaschek & Jashek 1990).
Tais estrelas s˜ao jovens (com idades entre 10
6
e 10
7
anos), massivas (com massas
entre aproximadamente 9 e 20 M
) e tamb´em luminosas (com luminosidades
entre 10
4
e 10
6
L
). Em nossa Gal´axia, a via L´actea, a maioria destas estrelas
jovens encontra-se em associa¸c˜oes OB localizadas nos bra¸cos espirais da Gal´axia.
S˜ao objetos de interesse para este trabalho de tese, a maioria das estrelas OB n˜ao
evolu´ıdas que encontram-se ainda na sequˆencia principal (estrelas an˜as); assim
como estrelas mais evolu´ıdas j´a na fase gigante da evolu¸c˜ao estelar.
As estrelas massivas s˜ao os principais propulsores da evolu¸c˜ao qu´ımica e
dinˆamica do meio interestelar (’MI’), e consequentemente da evolu¸c˜ao das gal´axias
como um todo. Tais estrelas contribuem de modo decisivo para o conte´udo global
de energia e momento do MI, sendo fonte de radia¸c˜ao ionizante no ultra-violeta,
atrav´es de seus ventos estelares e ao explodirem como supernovas em sua fase
final de evolu¸c˜ao. No que concerne a evoluc˜ao qu´ımica gal´actica (modelos de
Hou et al. 2000; Chiappini et al. 2001; 2003), as estrelas OB s˜ao v´ınculos impor-
tantes, por representarem a composi¸c˜ao qu´ımica da Gal´axia no tempo presente.
Adicionalmente, estas estrelas fornecem tamb´em informa¸c˜oes sobre a varia¸c˜ao es-
pacial das abundˆancias dos elementos qu´ımicos na vizinhan¸ca solar, assim como
em regi˜oes gal´aticas mais distantes, o que permite a determina¸c˜ao de gradientes
de metalicidade ao longo do disco da Gal´axia (p.e. Gummersbach et al. 1998;
Rollerston et al. 2000; Daflon & Cunha 2004). As referidas abundˆancias que
s˜ao obtidas para estas estrelas jovens servem tamb´em como um dado comple-
mentar para os resultados de abundˆancia que s˜ao obtidos atrav´es dos estudos
de regioes H ii; dado que ambas as popula¸c˜oes representam o estado presente da
nucleos´ıntese de metais no ciclo c´osmico da mat´eria. Com o advento da gera¸c˜ao
moderna de grandes telesc´opios equipados com espectr´ografos de alta-resolu¸c˜ao
abundˆancias ’primordiais’ podem ser obtidas n˜ao somente para estrelas de tipo
’early-type’ de nossa Gal´axia, assim como tamb´em podem ser estudados objetos
nos ambientes de mais baixa metalicidade das Nuvens de Magalh˜aes (p.e., Korn
1
Chapter 1: Introduction (in Portuguese) 2
et al. 2002; 2005; Rolleston et al. 2003; Hunter et al. 2007). Tais estudos
podem at´e ser estendidos para gal´axias mais distantes do grupo local, e mesmo
para al´em do grupo local, analisando-se espectros de resolu¸c˜ao intermedi´aria de
estrelas supergiantes do tipo ’early-type’ (e.g. Trundle et al. 2002; Urbaneja et
al. 2005 a,b).
Adicionalmente, as abundˆancias dos elementos carbono, nitrogˆenio e oxigˆenio
obtidas atrav´es de an´alises quantitativas de espectros de estrelas massivas de
tipo ’early-type’ servem tamb´em como condi¸c˜oes de contorno observacionais para
modelos de evolu¸c˜ao estelar. Informa¸c˜oes sobre os parˆametros estelares b´asicos
juntamente com as determina¸c˜oes dos padr˜oes de abundˆancia dos elementos
leves resultantes da mistura com material processado nuclearmente, permite uma
avaliac˜ao emp´ırica dos diferentes modelos evolutivos (p.e. Heger & Langer 2000;
Maeder & Meynet 2000).
A precis˜ao atingida nas determina¸c˜oes das abundˆancias dos elementos define
em ´ultima an´alise o grau com que uma teoria pode ser testada. Entretanto,
vale lembrar que a espectroscopia quantitativa baseia-se em v´arias hip´oteses.
Em particular, ´e necess´ario se ter uma comprens˜ao detalhada dos processos de
intera¸c˜ao entre o campo radiativo e o plasma da atmosfera estelar. Qualquer
debilidade no modelo f´ısico adotado acaba limitandoo a relevˆancia de uma an´alise,
mesmo que a observa¸c˜oes sejam de muito alta qualidade e que a redu¸c˜ao dos dados
tenha sido efetuada de forma ideal.
Para que as abundˆancias estelares tenham associadas a elas um verdadeiro sig-
nificado, as determina¸c˜oes dos parˆametros estelares para os objetos individuais
estudados n˜ao podem ser afetadas por erros sistem´aticos significativos. As estre-
las de tipo espectral O ’early’ e O ’medio’ apresentam dificuldades consider´aveis
em suas an´alises de modelos de atmosferas devido, por exemplo, `a efeitos como
esfericidade, perda de massa, al´em de efeitos de bloqueamento / obscurecimento
por linhas em n˜ao-ETL (equilibrio termodinˆamico n˜ao local). Por estas raz˜oes
as estrelas massivas OB de menor luminosidade tem sido os principais objetos
dos estudos de abundˆancias estelares durante longo tempo (p.e. Gies & Lambert
1992; Kilian 1992; Cunha & Lambert 1992; e numerosos estudos similares desde
ent˜ao). De um modo geral, as atmosferas das estrelas an˜as e gigantes do tipo
OB s˜ao supostamente razoavelmente bem descritas por modelos de atmosferas
em ETL (equilibrio termodinˆamico local) em uma dimens˜ao, plano paralelos, ho-
mogˆeneos e hidrost´aticos, com blanketing das linhas e que assumem a validade
do equilibrio radiativo . No entanto, estas simplifica¸c˜oes nao implicam em que as
an´alises quantitativas de estrelas OB an˜as e gigantes sejam triviais.
O elementos hidrogˆenio e h´elio s˜ao de maior interesse neste contexto as-
trof´ısico, j´a que estes constituem praticamente todo o plasma emissor de luz.
As transi¸c˜oes de H e He s˜ao as linhas mais fortes que encontram-se presentes nos
espectros das estrelas OB. Enquanto estes perfis de linhas constituem-se nas prin-
cipais ferramentas para diagn´osticos em an´alises estelares ao longo do diagrama
HR, estas medem tamb´em as condi¸c˜oes f´ısicas do plasma em grandes extens˜oes ao
3
longo das atmosferas estelares, especialemente quando comparadas `as extens˜oes
cobertas por linhas met´alicas.
A maioria das an´alises quantitativas de perfis de linhas do H e do He pub-
licadas na literatura seguem dois tipos b´asicos de metodologia: realiza-se uma
an´alise baseada em tratamentos puramente em ETL (p.e. Rolleston et al. 2000
e referˆencias inclu´ıdas); ou bem, se resolve o problema em n˜ao-ETL restrito
adotando-se modelos de atmosferas em ETL, que incluem o ’blanketing’ das lin-
has. Para as estrelas discutidas neste trabalho, um teste completo dos modelos,
no que concerne a reprodu¸c˜ao de espectros do H e do He (via compara¸c˜oes dire-
tas com observa¸c˜oes no vis´ıvel e infravermelho pr´oximo, e cobrindo um intervalo
grande de parametros) n˜ao encontra-se publicado na literatura (note que estes
existem para as estrelas O pelo menos para alguns grupos de linhas: Bouret et al.
2003; Repolust et al. 2004; 2005; Mokien et al. 2005, 2006; Heap et al. 2006). A
disponibilidade de tais testes ajudaria ao usu´ario de redes de espectros sint´eticos
publicadas na literatura a compreender suas limita¸c˜oes e pontos fortes. Usual-
mente, somente uma ou duas linhas de Balmer de hidrogˆenio e algumas linhas
selecionadas de He, no ´otico azul, s˜ao consideradas em estudos da literatura.
Dentre os metais, o elemento carbono tem um importante papel porque este
´e o elemento principal criado no processo triplo-alfa e assim sendo, servindo
como sementes para a s´ıntese subsequente de todos os elementos mais pesados
(Burbidge et al. 1957; Cameron 1957). O carbono ´e um catalisador para a trans-
forma¸c˜ao de hidrogˆenio em h´elio atrav´es do ciclo CNO em estrelas massivas e de
massa intermedi´aria e este elemento tamb´em constitui a base de toda a qu´ımica
orgˆanica. As abundˆancias de carbono derivadas em estrelas ’early-type’ tem
sido o objeto de in´umeros estudos ao longo das ´ultimas d´ecadas, com crescentes
melhoramentos na qualidade dos dados observacionais, e na complexidade e con-
sistˆencia atingida nos c´alculos dos modelos e nas an´alises espectrais. Um passo
crucial neste trajeto foi o abandono da aproxima¸c˜ao de equilibrio termodinˆamico
local nos c´alculos de forma¸c˜ao das linhas, permitindo-se os desvios do ELT (n˜ao-
ETL). In´umeros modelos atˆomicos foram discutidos na literatura (Lenon 1983;
Eber & Butler 1988; Grigsby et al. 1992; Sigut 1996; Lanz & Hubeny 2003;
2007). Em particular, o modelo atˆomico de Eber & Butler tem sido amplamente
aplicado em trabalhos de determina¸c˜ao de abundˆancias em estrelas O e B, na sua
maioria nao evolu´ıdas e da vizinhan¸ca solar (e.g. Gies & Lambert 1992; Kilian
1992; Cunha & Lambert 1992; Gummersbach et al. 1998; Daflon et al. 1999;
2001a,b).
Em um n´umero grande de estrelas de tipo ’early-type’, principalmente aquelas
em meios extra-gal´aticos e aquelas com altas velocidades projetadas de rota¸c˜ao,
apenas um n´umero pequeno de linhas de carbono no ´otico podem ser utilizadas
em determina¸c˜oes de abundˆancias, devido `a uma maior dificuldade em se atingir
uma boa rela¸c˜ao sinal-ruido (S/N) e ao fato de as linhas serem muito alargadas
devido a rota¸c˜ao. Isto inclui a forte linha de C ii em λλ 6578/82
˚
A e, em particular,
o multipleto em λλ 4267
˚
A, que tem se mostrado tradicionalmente bastante dif´ıcil
Chapter 1: Introduction (in Portuguese) 4
de ser modelado em c´alculos em n˜ao-ETL (e.g. Lambert 1993; Sigut 1996). Estes
dois multipletos usualmente n˜ao reproduzem bem as observa¸c˜oes (e.g. Grigsby
et al. 1992; Hunter et al. 2007), encontrando-se resultados de abundˆancia sis-
tematicamente mais baixas do que aqueles derivados a partir de outras linhas
mais fracas de C ii (Gies & Lambert 1992). As linhas de C iiI tamb´em podem ser
modeladas inadequadamente (Grigsby et al. 1992). Uma complica¸c˜ao adicional
´e a impossibilidade de se estabelecer o equilibrio de ioniza¸c˜ao do C ii / C iii.
Diferen¸cas nas abundˆancias obtidas atrav´es destes dois ions podem chegar a um
fator 5-10 (Daflon et al. 2001b; Hunter et al. 2007). Entretanto, existem in-
consistˆencias adicionais entre as abundˆancias de carbono em estrelas ’early-type’
publicadas na literatura que necessitam de outras explica¸c˜oes, dado que a maioria
destes estudos deriva abundˆancias de carbono a partir de linhas que n˜ao s˜ao t˜ao
sens´ıveis a efeitos n˜ao-ETL.
Uma compara¸c˜ao entre os resultados de abundˆancia para estrelas de tipo
’early-type’ indica que a abundˆancias de carbono na vizinhan¸ca solar s˜ao alta-
mente inomogˆeneas (mesmo para estrelas em um mesmo aglomerado) e forte-
mente sub-solares. Tal resultado constrasta-se com a uniformidade encontrada
nas abundˆancias obtidas para a fase gasosa no meio interestelar dentro de um
limite aproximado de 1.5 kpc do Sol (Sofia & Meyer 2001; e referˆencias citadas
neste trabalho). Este resultado de inomogeneidade n˜ao pode tampoco ser com-
preendido no contexto dos modelos recentes de evolu¸c˜ao qu´ımica da Gal´axia. A
expectativa seria que as estrelas massivas e jovens, que se formam a partir de
uma nuvem molecular numa curta escala de tempo, apresentem uma abundˆancia
de carbono homogˆenea, coincidindo com os resultados obtidos de regi˜oes H ii.
Adicionalmente, o valor de abundˆancia deve ser maior do que aquele observado
em objetos provenientes de gera¸c˜oes estelares anteriores em sua vizinhan¸ca, tal
como estrelas de tipo F e G (< 2 Gyr), e tamb´em maior que o valor solar. Na
realidade, diferen¸cas sistem´aticas significativas existem (ver p.e. Sofia & Meyer
2001; Herrero 2003), levantando d´uvidas sobre a realidade das abundˆancias de
carbono derivadas para as estrelas de tipo B.
O objetivo deste trabalho ´e resolver o problema de abundˆancias de carbono
n˜ao confi´aveis obtidas a partir de estrelas de tipo espectral O e B. Com este
prop´osito, melhoramentos na modelagem e na an´alise quantitativa dos espectros
observados ser˜ao realizados. Em um passo preparat´orio a validade da an´alise
n˜ao-ETL h´ıbrida ser´a testada e o status atual dos c´alculos da forma¸c˜ao das lin-
has de H e He – os absorventes mais abundantes – ser´a avaliado. Um modelo
atˆomico de C ii a C iv ser´a constru´ıdo e calibrado empiricamente utilizando as
observa¸c˜oes de seis estrelas de tipo B ’early’ da vizinhan¸ca solar. Uma ˆenfase
especial ser´a dada a uma avalia¸c˜ao cr´ıtica dos dados atˆomicos de entrada. Os
parˆametros atmosf´ericos fundamentais das estrelas estudadas ser˜ao derivados de
modo auto-consistente a partir de simultˆaneos equilibrios de ioniza¸c˜ao. Este
m´etodo permite obter abundˆancias com acur´acia sem precedentes, resultando
em uma determina¸c˜ao precisa da abundˆancia presente do carbono na vizinhan¸ca
5
solar. Partes dos resultados apresentados neste trabalho de tese e aplica¸c˜oes adi-
cionais foram publicados, um artigo foi submetido e outros est˜ao em prepara¸c˜ao:
Korn et al. (2005); Nieva et al. (2003); Nieva & Przybilla (2006ab; 2007 a-d);
Przybilla et al. 2006ab).
A tese encontra-se organizada da seguinte forma: no cap´ıtulo 2 uma in-
trodu¸c˜ao ao modelo de atmosferas estelares e a an´alise espectral tal como consid-
erada neste trabalho ser´a apresentada. No cap´ıtulo 3 os conceitos b´asicos relativos
aos dados atˆomicos de entrada para a modelagem espectral ser˜ao discutidos. No
cap´ıtulo 4 se apresentar´a uma breve descri¸c˜ao da an´alise espectrosc´opica. No
cap´ıtulo 5 uma an´alise do m´etodo n˜ao-ETL h´ıbrido para a forma¸c˜ao das linhas
de H e He ser´a apresentada. No cap´ıtulo 6 uma calibra¸c˜ao auto-consistente e
simultˆanea do modelo de C ii-iv e a an´alise espectral ser´a descrita. Finalmente,
as conclus˜oes deste trabalho de tese ser˜ao sumarizadas no cap´ıtulo 7.
Chapter 2
Model Atmospheres
Almost all physical information of stars has to be inferred from the radiation they
emit. The photons escape to space from a medium located in the external layers
of the star, called the stellar atmosphere. In order to compute the emergent
flux of the star, it is necessary to specify the physical conditions under which
the radiation is transported. A set of state parameters as function of depth is
termed a model atmosphere. The list of state parameters will depend on the basic
assumptions under which the model is constructed (e.g. temperature, density,
etc.). An important component for the construction of the atmospheric structure
is the radiation, and its transfer through and interaction with the atmospheric
medium is a complex problem to be solved. The emergent spectrum of the star,
and in particular the spectral lines, has to be modelled in detail for quantitative
analyses by comparison with the observed spectra.
From the pioneering work of Uns¨old (1955) enormous progress has been
achieved in the field of stellar atmosphere modelling. A detailed theoretical de-
scription of stellar atmospheres can be found in the textbook of Mihalas (1978)
and a discussion of more recent developments is given in Hubeny (1997). Current
realistic models are built on the basis of a well understood underlying physics and
they allow the analysis of stellar atmospheres to be performed quantitatively.
Overall, the atmospheres of late O and early B-type (OB) main sequence
and giant stars are supposed to be described reasonably well by one-dimensional,
plane-parallel, homogeneous and hydrostatic line-blanketed LTE models in ra-
diative equilibrium. However, this does not imply that quantitative analyses of
the spectra of OB dwarfs and giants are trivial. Even for a simple approach, like
the one employed here, the modelling of the stellar atmosphere is complex and
requires the solution of the atmospheric structure equations coupled to the radia-
tive transfer. As a further complication, small discrepancies in the atmospheric
structure due to different model assumptions or input parameters can produce
large variations of spectral line profiles, and can therefore impact the physical
information derived from the observed spectra notably.
7
Chapter 2: Model Atmospheres 8
Line-blanketed plane-parallel stellar atmospheres in hydrostatic equilibrium
are briefly discussed in the present chapter and they are further investigated in
Chapter 5. Models for expanding atmospheres and the coupling of the atmosphere
with the stellar wind (which describe, e.g. Wolf-Rayet stars or supergiants en-
velopes) will not be discussed. The readers can refer to the review of Kudritzki &
Hummer (1990) and other monographs, e.g. Stellar Atmospheres: Beyond Clas-
sical Models (Crivellari et al. 1990); The Atmospheres of Early-Type Stars (Heber
& Jeffery 1992); Stellar Atmosphere Modeling, (Hubeny, Mihalas & Werner 2003).
On the other hand, a realistic calculation of spectral line formation consider-
ing non-local effects caused by the radiation field is also addressed in the present
chapter. In particular, the restricted non-LTE line-formation calculations em-
ployed in this work are described. Both the model atmospheres and the line-
formation calculation are the basis of any further quantitative spectral analysis
that, in order to be meaningful, should be free of systematic errors.
2.1 Radiative Transfer
The radiation field of early-type stars constitutes an important physical property
of the atmosphere and, at the same time, its interaction with the medium de-
termines the state of the plasma. The propagation of the photons through the
stellar atmosphere is described by the radiative transfer equation, which involves
a number of quantities to be defined for the case of a plane-parallel atmosphere
(see Sect. 2.2) as follows.
The specific intensity I
ν
of radiation with frequency ν is defined as the amount
of energy transported by the radiation in frequency range (ν, ν + dν) per unit
area dS into the solid angle dω within a time interval dt
dE = I
ν
dS cos θ dω dν dt, (2.1)
where θ is the angle between the direction of propagation and the surface normal.
The dimension of I
ν
is erg cm
−2
s
−1
hz
−1
sr
−1
(in cgs units). The specific intensity
provides a macroscopic description of the radiation field.
The interaction of the radiation field with matter is described by two phe-
nomenological quantities: the absorption coefficient χ
ν
and the emission coeffi-
cient η
ν
defined as follows
dE = χ
ν
I
ν
dS ds dω dν dt, (2.2)
dE = η
ν
dS ds dω dν dt. (2.3)
In Eqn. (2.2), dE is the energy absorbed by an element of material of cross-
section dS and length ds, from a beam of specific intensity I
ν
(incident normal
to dS into a solid angle dω). In Eqn. (2.3), dE corresponds to the amount of
9 2.1 Radiative Transfer
energy released by the material in form of radiation. The dimension of χ
ν
is
cm
−1
, consequently 1/χ
ν
measures the characteristic distance a photon travels
between absorption processes: the photon mean-free path. On the other hand, the
dimension of η
ν
is erg cm
−3
s
−1
Hz
−1
sr
−1
. Numerical values for the absorption
and emission coefficients are assigned from a microscopic point of view, and they
follow the form
absorption coefficient = number of absorbers × atomic cross-section,
where the relevant cross-sections are given by atomic physics.
The absorption can be distinguished in two different processes: true absorp-
tion, where a photon is destroyed and its energy thermalized, and scattering,
where the photon is absorbed and immediately re-emitted in a different direction
at a Doppler-shifted frequency (scattering can occur in bound states, but also
free electrons can scatter radiation with the same efficiency at all wavelengths).
Therefore, the total absorption coefficient has two components: χ
ν
= κ
ν
+ σ
ν
,
where κ
ν
is the true absorption and σ
ν
the scattering coefficient.
The total continuous ’true’ absorption coefficient is the sum of an ionization
process where a bound-free transition occurs, and a free-free transition, where
one charged particle is accelerated upon passing close to another charged parti-
cle. The remaining possibility is bound-bound transitions, which produce spectral
lines, also included in κ
ν
. In the case of many overlapped and crowded lines,
their effect on the continuous spectrum is considerable (see Sect. 2.4). Most of
the continuous absorption is due to hydrogen, a direct result of the overwhelming
dominance of hydrogen in the chemical composition. Neutral hydrogen is the ma-
jor source of absorption for the temperatures of OB stars. Both the bound-free
and the free-free absorption of hydrogen are important. The ionization limit for
principal quantum number n = 1 (Lyman limit) is at 912
˚
A, for n = 2 (Balmer
limit) is at 3647
˚
A, etc. In most stars, helium is the next most abundant element
after hydrogen, but its electrons are so tightly bound that its contribution to κ
ν
becomes important only at higher temperatures, and primarily in the ultravio-
let. The excitation energy of the first level is E
1
= 19.72 eV and the ionization
energies of He i and He ii are χ
I1
= 24.59 eV and χ
I2
= 54.42 eV, respectively
(χ
I1
corresponds to 504.19
˚
A and χ
I2
to 227.82
˚
A). Because E
1
is so close to χ
I1
,
by the time electrons start populating the excited levels, helium is beginning to
ionize. A single-electron ion (He ii) behaves similarly to hydrogen, but with en-
ergies scaled up and wavelengths scaled down by a factor of four.
In strict thermodynamic equilibrium (TE), the energy removed by matter
from the radiation field is in detailed balance with the energy emitted, and from
Eqns. (2.2) and (2.3) follows that χ
ν
I
ν
= η
ν
. In TE, the radiation intensity is
Chapter 2: Model Atmospheres 10
described by the Planck function, I
ν
= B
ν
, with
B
ν
=
2hν
3
c
2
1
exp(hν/kT ) − 1
. (2.4)
This function depends only on the frequency ν and temperature T ; h is the Planck
constant, k is the Boltzmann constant and c is the speed of light. A consequence
of these considerations in TE is Kirchoff’s law, η
ν
/χ
ν
= B
ν
.
Based on the previous definitions, the radiative transfer equation can be de-
fined, for the planar 1-D case, as
µ
dI
ν
dz
= η
ν
− I
ν
χ
ν
, (2.5)
where z is the geometrical coordinate and µ is the directional cosine, defined by
µ ≡ cos θ. After division by −χ
ν
, it can be written as
µ
dI
ν
dτ
ν
= I
ν
− S
ν
, (2.6)
with the optical depth τ
ν
defined by
dτ
ν
≡ −χ
ν
dz, (2.7)
and the source function S
ν
defined by
S
ν
≡ η
ν
/χ
ν
. (2.8)
The optical depth corresponds to the integrated absorptivity of the material along
the line of sight and measures the number of photon mean-free paths.
The radiation field can be described in terms of different averages or moments
of the specific intensity. The zero-order moment is the mean intensity J
ν
and
corresponds – except for a numerical factor – to the photon energy density
J
ν
=
1
2
1
−1
I
ν
dµ. (2.9)
The first-order moment is defined as the Eddington flux H
ν
and corresponds to
the astrophysical flux (F
ν
= 4H
ν
)
H
ν
=
1
2
1
−1
I
ν
µ dµ. (2.10)
11 2.1 Radiative Transfer
The second-order moment K
ν
is physically related to the radiation pressure
K
ν
=
1
2
1
−1
I
ν
µ
2
dµ. (2.11)
If the source function is known, the transfer equation (2.6) becomes a linear first-
order differential equation with constant coefficients, solved via an integrating
factor exp (−τ
ν
/µ). The formal solution for the emergent intensity from a semi-
infinite atmosphere seen by an external observer (τ = 0) then is
I
ν
(0, µ, ν) =
∞
0
S
ν
(t) exp(−t/µ) dt/µ, (2.12)
i.e. the source function at every depth point along the line of sight is attenuated
(here, t is the optical depth). The emergent intensity has approximately the value
of the source function at optical depth unity along the line of sight (see Fig 2.1).
z
ντ τ
z max
(z)
ν
z 0
τ ν max (z )=0
0
~ 1τ
0 (z )
Figure 2.1: In a plane-parallel atmosphere the emergent spectrum at frequency
ν is formed in the region around z
0
, where the optical depth is τ
ν
(z
0
) ∼ 1.
For the angle-averaged mean intensity, the formal solution can be expressed as
J
ν
(τ
ν
) =
1
2
∞
0
S
ν
(t) E
1
(|t − τ
ν
|) dt = Λ
τ
ν
[S(t)], (2.13)
where E
1
is the first exponential integral. In the second expression, the mean
intensity is formulated in terms of an operator acting on the source function.
This is the so-called Λ-operator. The calculation of this integral has a high
computational cost due to E
1
. For practical applications, the last equation has to
be replaced by a quadrature sum, with the mean intensity and the source function
discretised for a number of depth points. The Λ-operator is then expressed in
Chapter 2: Model Atmospheres 12
the form of the Λ-matrix, which describes the coupling of the contributions of
the source function from all depth points. In that case, the formal solution can
be written as
J
d
=
D
d
Λ
dd
S
d
, (2.14)
where d denotes the depth index.
The following example illustrates the effect of the Λ matrix on the source
function. For the simple case where all the elements of the source function vector
are zero except for the i-th element, which is taken to be 1, S
d
= δ
di
, then the
formal solution reads
J
1
J
2
.
.
.
J
D
=
Λ
11
Λ
12
··· Λ
1D
Λ
21
Λ
22
··· Λ
2D
.
.
.
.
.
.
.
.
.
.
.
.
Λ
D1
Λ
D2
··· Λ
DD
×
0
.
.
.
1
0
=
Λ
1i
Λ
2i
.
.
.
Λ
Di
, (2.15)
In this case, the i-th column of the Λ matrix is a solution of the transfer equation
with the source function given as a unit pulse function. Physically, the source
function S = S
i
= 1 at depth point i of the atmosphere is re-distributed over all
depth points. In practice, the problem is much more complex, since the source
function is not known a priori, and it also depends non-linearly on the intensity.
An iteration scheme can be used to solve the transfer problem, the so-called
Λ-iteration. Starting from a first estimate of the source function (for instance,
equal to the Planck function), the radiation field is computed by solving the
transfer equation. The radiation field is in turn employed to compute a new
source function, and this procedure can be iterated until some convergence cri-
terion is achieved. Nevertheless, in practical applications the Λ-iteration fails
to converge. Photons emitted in the wings of a line can travel large distances
through the medium before being absorbed and thus can transport information
about inhomogeneities (for example the existence of an outer boundary), whereas
photons in the line core can travel only comparatively very short distances and
are essentially trapped at the place of their formation. Although the latter consti-
tute the majority of all photons in the line, the line source function nevertheless
can respond quickly to inhomogeneities. This is because of the large number of
interactions which redistribute absorbed photons over the line profile and thus
convert a core photon into a wing photon in a short time. By doing a Λ-iteration,
we follow this process from scattering to scattering and finally need as many it-
erations as necessary to give each core photon a chance to be emitted into the
wing at least once (or to be destroyed in a thermalisation process). This involves
a large number of iterations which prevents the applicability of the Λ-procedure.
Therefore other algorithms have to be considered.
13 2.1 Radiative Transfer
Deep in the stellar atmosphere, the source function approaches to the Planck
function: S
ν
→ B
ν
, as practically no photons escape, and thus the medium
is close to TE. Under these conditions, the transfer problem can be expressed
in terms of a diffusion process (see Mihalas 1978). The total radiation flux in
the diffusion approximation is proportional to the temperature gradient and an
averaged opacity, defined by the Rosseland mean opacity
1
χ
R
dB
dT
=
∞
0
1
χ
ν
dB
ν
dT
dν. (2.16)
The importance of the Rosseland opacity lies in the fact that it yields the exact
total radiation flux at large depths and therefore the correct temperature struc-
ture. Integration of χ
R
over the geometrical distance gives the Rosseland optical
depth τ
R
.
The general problem of radiative transfer involves the coupling of physical
variables (depths, frequencies, angles) which numerically means inverting large
matrices. Several methods have been developed in order to solve the radia-
tion transfer problem numerically as can be found in Mihalas (1978) and in the
overview of Hubeny (1997). They are not only aimed at the problem of model
atmospheres but also to the line-formation calculations, with an additional cou-
pling of source functions in different transitions via the statistical equilibrium
condition, when local detailed balance is no longer valid (Sect. 2.5.2).
Currently, the most powerful techniques are the Accelerated Lambda Iteration
(ALI) methods, reviewed by Hubeny (1992). The basic concept is to realise that
some part of the physical coupling in the radiation transfer problem is more im-
portant than others. Cannon (1973) introduced the method of operator splitting
into astrophysical radiative transfer. The idea consists of writing
Λ = Λ
∗
+ (Λ − Λ
∗
), (2.17)
where Λ
∗
is an appropriately chosen approximate lambda operator. The choice of
Λ
∗
is arbitrary. From mathematical principles it can be shown that the diagonal of
the true Λ-operator is the optimum choice for a local Λ
∗
, while when considering
nearest-neighbour interactions the tridiagonal operator can be chosen. The mean
intensity resulting from the i-th iteration is then
J
i
= Λ
∗
S
i
+ (Λ − Λ
∗
)S
i−1
. (2.18)
The action of the Λ-operator splits into two contributions: the approximate Λ
∗
-
operator acts on the source function of the current iteration, describing the (slowly
converging) local absorption and emission processes. The difference between the
exact and the approximate operator Λ − Λ
∗
acts on the previous, known, iterate
Chapter 2: Model Atmospheres 14
of the source function, accelerating the convergence.
The ALI approach was addressed in several studies and implemented in the
numerical computation of the radiative transfer. The ALI approach for solving
the non-LTE line-formation problem, as in this work, was first used by Werner
& Husfeld (1985). The preferred recipe for the realisation of the ALI scheme for
numerical radiative transfer of multilevel atoms was developed in the study of
Rybicki & Hummer (1991). This is the formulation implemented in the numeri-
cal codes used in the course of the present work (See Sect. 2.5.2).
2.2 Classical Stellar Atmospheres
The classical stellar atmosphere problem can be described by a horizontally-
homogeneous, plane-parallel and static atmosphere. This model is applicable
to the so-called stellar photosphere, constrained to a thin layer and not signifi-
cantly affected by outer layers where a stellar wind may play an important rˆole.
Here, the basic assumptions and equations are described.
Plane-Parallel Geometry and Homogeneity. The thickness of the atmo-
sphere is small compared to the radius of the star and it is described well by a
plane-parallel geometry. The atmosphere consists of horizontally homogeneous
layers (without spots, granulations, etc.). Only one variable is therefore needed
to specify a given position in the atmosphere. For an observer the optical depth
is zero at the surface and increases into the atmosphere.
Stationarity. It is assumed that no phenomena depending on time such as pul-
sations or variable magnetic fields are present in the atmosphere. The transfer
equation and the populations of atomic levels have to be constant in time.
Hydrostatic Equilibrium. This condition requires that at each point of the
atmosphere the weight of the overlying layers is supported by the total pressure.
The hydrostatic equilibrium equation can be written as:
dP
dz
= −ρ (g − g
rad
), (2.19)
where P is the total pressure, ρ the mass density, g the surface gravity and g
rad
the radiative acceleration, defined by g
rad
=
4π
c
∞
0
χ
ν
H
ν
dν. The radiative ac-
celeration in the stars of interest here (main sequence and giant stars) is small
compared to the surface gravity.
Radiative Equilibrium. All the energy released by nuclear reactions in the
stellar interior is carried through the stellar atmosphere by radiation. This is
equivalent to the conservation of radiative flux:
∞
0
H
ν
dν = const. =
σ
4π
T
4
eff
, (2.20)
15 2.3 Thermodynamic State: LTE vs. Non-LTE
where σ is the Stefan-Boltzmann constant and T
eff
is the effective temperature.
Applying the radiative transfer equation, the latter can be written as follows:
∞
0
(κ
ν
J
ν
− η
ν
) dν =
∞
0
κ
ν
(J
ν
− S
ν
) dν = 0 (2.21)
Here, only true absorption processes enter the energy balance of the medium (κ
ν
instead of χ
ν
) because scattering contributions to the absorptivity and emissivity
cancel.
Charge conservation. This condition expresses the global electric neutrality
of the medium of a certain elemental composition,
i
n
i
Z
i
− n
e
= 0, (2.22)
where Z
i
is the charge associated to the level i, which equals 0 for levels of neutral
species, 1 for those of a singly-ionized species, etc. The summation extends over
all levels of all ions of all chemical species.
2.3 Thermodynamic State: LTE vs. Non-LTE
Two different assumptions can be considered concerning the thermodynamic state
of the atmospheric medium: local thermodynamic equilibrium (LTE ) or depar-
tures from it (non-LTE ). LTE is very restricted but offers an analytical descrip-
tion of the source function, while non-LTE is more realistic but its numerical
solution might be computationally intensive.
LTE. The statistical physics description of the properties of a medium is enor-
mously simplified when thermodynamic equilibrium is achieved. In such a case,
the particle velocity distributions and the distributions of atoms over ionization
and excitation states are specified by only two thermodynamic variables. In the
stellar atmosphere context these variables are chosen to be the absolute temper-
ature T and the total particle number density N, or the electron number density
n
e
. The assumption of TE cannot be applied to the whole stellar atmosphere, as
the stars emit radiation. However, the standard thermodynamic relations can be
applied locally assuming that each individual volume element of the atmosphere
is in thermodynamic equilibrium (in plane-parallel geometry this assumption is
applied for each layer). This concept is called local thermodynamic equilibrium.
Following this approximation, the equilibrium values of distribution functions
are assigned only to massive particles, but the radiation field is allowed to depart
from a Planckian character. The local kinetic temperature determines both the
velocity distribution of the particles, which is therefore Maxwellian, and the dis-
tribution of the particles through their various ionization and excitation stages
Chapter 2: Model Atmospheres 16
by means of the Saha-Boltzmann equations, defined as follows.
The Maxwellian velocity distribution of particles is given by
f(v) dv = (m/2πkT )
3/2
exp(−mv
2
/2kT ) 4πv
2
dv, (2.23)
where k is the Boltzmann constant, v the velocity and m the particle mass.
The Boltzmann excitation formula reads
(n
j
/n
i
) = (g
j
/g
i
) exp(−(E
j
− E
i
)/kT ), (2.24)
where n
i
is the occupation number density, g
i
the statistical weight, E
i
the energy
of the level i and (E
j
−E
i
) = h ν
ij
measures the frequency of the transition i → j.
The occupation number density is often termed as the atomic level population and
refers to the number density of ions in the excited level i.
The Saha ionization equation reads
N
I
N
I+1
= n
e
U
I
U
I+1
CT
−3/2
exp(χ
I
/kT ), (2.25)
where N
I
is the total number density of the ionization stage I, χ
I
is the ionization
potential of the ion I, U =
i
max
1
g
i
exp(−E
i
/kT ) is the partition function, and
C = (h
2
/2πmk)
3/2
is a constant (= 2.07 × 10
−16
in cgs units).
The last three equations describe the LTE state macroscopically. From a mi-
croscopic point of view, LTE holds if all atomic processes are in detailed balance
(i.e. if every process is exactly balanced by its inverse).
Non-LTE. Any state departing from LTE is denoted as being in non-LTE.
This means that the populations of some energy levels of some atoms/ions may
depart from their LTE values, while the velocity distribution of all particles re-
main Maxwellian, with the same kinetic temperature T . In this case, the Saha-
Boltzmann equations have to be replaced by more general equations accounting
for the detailed atomic processes which populate/depopulate the energy levels:
these are the statistical equilibrium or rate equations,
n
i
j=i
(R
ij
+ C
ij
) =
j=i
n
j
(R
ji
+ C
ji
), (2.26)
where R
ij
and C
ij
are the radiative and collisional rates, respectively, for the
transitions from level i to level j.The left-hand side represents the transitions de-
populating the level i, while the right-hand side describes the processes populating
this level.
Radiative upward rates are given by the following expression,
R
ij
= 4π
α
ij
J
ν
hν
dν, (2.27)
17 2.3 Thermodynamic State: LTE vs. Non-LTE
where α
ij
is an atomic cross-section for radiative bound-bound or bound-free
processes. The downward rate is given by
R
ji
= 4π
n
i
n
j
∗
α
ij
hν
2hν
3
c
2
+ J
ν
exp(−hν/kT ) dν, (2.28)
with the asterix denoting LTE populations. The first term is due to spontaneous
emission, while the second describes stimulated emission. Note that both R
ij
and R
ji
are computed on the basis of the same cross-section value.
The collisional upward rates are given by
C
ij
= n
e
σ
ij
(v)f(v)v dv, (2.29)
where f(v) is the velocity distribution of the colliding particles, which for hot
stars (T
eff
10 000 K) are mainly electrons. The collisional downward rates
result from
C
ji
=
n
i
n
j
∗
C
ij
(2.30)
The set of statistical equations for all levels of an atom form a linearly
dependent system. In order to close the system, one of these equations has
to be replaced by another relation, the total number conservation equation
i
n
i
= N
atoms
, with the summation extending over all levels and ions of a
given species.
Conceptually, the non-LTE effects become important when the radiation field
is strong enough for the radiative rates to dominate the atomic transitions over
the collisional rates. A non-local effect takes place when photons coming from dif-
ferent parts of the atmosphere interact with the local medium. In the numerical
calculations this translates to a coupling of the radiation field at different fre-
quencies with the absorption and emission coefficients at different optical depths.
In addition to the basic equations of the classical stellar atmospheres, the cou-
pling of the radiative transfer and rate equations has to be solved simultaneously,
which requires a greater effort than for the LTE case. This leads to a set of highly
coupled, non-linear system of equations to be solved simultaneously. Due to the
complexity of the problem, the basic structural equations have to be discretised
and solved numerically.
In general terms, non-LTE model atmospheres are required in the analysis
of stars at high temperature (high intensities and radiative rates) and low at-
mospheric densities (low collisional rates). Electron collisions cannot restore TE
locally when the electron mean-free path is too large between collisions and the
radiation field is strong enough to dominate the atomic transitions. The limits in
atmospheric parameters (effective temperature, surface gravity) where the LTE
approach is suitable to describe the atmospheric structure is still under investi-
Chapter 2: Model Atmospheres 18
gation, and one of the purposes of this Thesis is to contribute to this study. The
LTE assumption for the model atmosphere is a good approximation for the kind
of star analysed in this work, in particular for the line-formation region of the
spectral lines of interest. This will be described in Chapter 5, where a comparison
with non-LTE model atmospheres is provided.
2.4 Metal Line Blanketing
The inclusion of thousands to millons of lines in the computations of the at-
mospheric structure and the emergent flux changes the theoretical predictions.
These spectral lines are often computed under the LTE approximation since it
is prohibitive to treat all levels of all ions of all elements in non-LTE. Some
computations are, nevertheless, performed in non-LTE for selected elements, in-
cluding some iron-group members, with simplified model atoms accounting for
superlevels. These superlevels group many individual levels of similar energies
and physical properties. The individual levels are in LTE relative to each other
(see e.g. Hubeny & Lanz 1995 for further details). However, LTE background
line opacities are a good approximation for the computation of the atmospheric
structure and the emergent flux for the stars under study here.
Since spectral lines are opaque to radiation, the energy transport takes place
at other frequencies in order to conserve the total flux. For OB stars, numerous
strong spectral features are located in the UV region and the photons escape at
longer wavelengths. This is denoted as line blocking, which affects the overall
shape of the emergent flux. The restriction of the energy transport by absorption
lines produces a steeper temperature gradient in order to drive the flux through
the atmosphere, resulting in an increase of temperature at deeper layers: the
backwarming effect. On the other hand, in the outer layers the presence of lines
give rise to surface cooling. The combined effects are known as line blanketing.
The implementation of line blanketing is performed following two different
approaches: opacity sampling (OS) and opacity distribution functions (ODF).
Both approaches use simplified line-broadening mechanisms. ODFs resample
the detailed line opacity distribution to form a monotonic function of frequency,
represented by a small number of quadrature points. The detailed calculations are
performed once and the ODFs are tabulated (e.g. as a function of temperature
and pressure) for discrete frequency intervals, at a given chemical composition
and microturbulent velocity. These tables can be used in further applications, as
in the present work. In the OS approach, the spectrum is sampled by choosing a
larger number of frequency points than for the ODFs, allowing for a statistically
approximate computation of the radiation flux. It offers many advantages in
the treatment of line blends or overlaps and it allows non-standard chemical
composition to be handled.
19 2.5 Spectral Line Formation
2.5 Spectral Line Formation
Spectral lines are formed by transitions between bound states of atoms and ions
and they can provide information over a wide range of atmospheric depths, from
the outer layers (where the line core is formed) to the deepest observable points
(the continuum formation region). It is possible to gain deep insight into the
physical state of the stellar atmosphere from the analysis of spectral lines. The
line profile depends on the local conditions of the stellar plasma and on the atomic
properties of the atom or ion under investigation. It is thus important to develop
reliable methods to model spectral lines in order to infer the desired physical
information from comparison to the observed spectra. In this section, the basic
principles of spectral line formation are summarised, giving special emphasis to
calculations accounting for departures from LTE.
2.5.1 Line Strength & Broadening Mechanisms
The strength of a spectral line is basically determined by the number of absorbers
and the line absorption cross-section, given by
α
ij
=
πe
2
mc
f
ij
φ
ν
= B
ij
hν
ij
4π
φ
ν
(2.31)
where e is the electron charge, m the electron mass, f
ij
the oscillator strength,
and φ
ν
is the line absorption profile, which is normalised such that
φ
ν
dν = 1.
The absorption profile is identical to the emission profile when complete redistri-
bution is assumed. The basic atomic quantity that determines the line strength
is f
ij
. This is related to the Einstein coefficient B
ij
, which gives the absorption
probability. The line-centre frequency of the transition is denoted by ν
ij
. The
Einstein coefficients for absorption, stimulated emission (B
ji
) and spontaneous
emission (A
ji
) are connected via the Einstein relations
A
ji
= (2hν
3
/c
2
)B
ji
and g
i
B
ij
= g
j
B
ji
. (2.32)
The Einstein coefficients B
ij
and B
ji
can be obtained from a quantum mechanical
treatment of atoms. On the other hand, the ab-initio derivation of A
ji
requires
to quantize the radiation field, which is much more complicated than deriving
B
ij
. Therefore Eqn. 2.32 is useful for obtaining A
ji
once B
ij
is known.
For an ideal and isolated atom with levels of infinite lifetime, the spectral lines
would be perfectly sharp. However, there are several radiative and collisional
mechanisms which produce smearing of the levels of real atoms in a plasma, re-
sulting in a spectral line broadening.
Chapter 2: Model Atmospheres 20
Natural damping. This describes the line width related to the finite lifetime
of the atomic levels set by the radiative decay. It results from Heisenberg’s
uncertainty principle ∆E∆t ≥ h, where ∆t is the characteristic lifetime for a
decaying state and ∆E refers to the energy spread of the state. Only the ground
states of atoms and ions are stable, and the typical lifetimes of excited levels are
of the order of 10
−8
s. The radiation damping produces a Lorentz profile,
φ
ν
=
γ
rad
/4π
2
(ν − ν
ij
)
2
+ (γ
rad
/4π)
2
, (2.33)
with a full half-intensity width γ
rad
(mathematically equivalent to the full width
at half maximum FWHM). It is the sum of the reciprocal mean lifetimes of the
upper and lower levels, which accounts for all possible radiative decays of both
levels (they are obtained experimentally or as a by-product of oscillator strength
computations),
γ
rad
=
n<i
A
in
+
m<j
A
jm
. (2.34)
Pressure broadening. The atoms embedded in a plasma interact via colli-
sions with other atoms or charged particles, producing a pressure broadening of
the lines. Precise collisional line-broadening data are derived from the quantum
theory of pressure broadening; details can be found in Griem (1964, 1974). For
early-type stars, the most important mechanisms of collisional broadening are
the linear Stark effect for hydrogen lines and the quadratic Stark effect for the
non-hydrogenic atoms and ions. Other broadening mechanisms, such as the Van
der Waals interaction become more relevant at lower temperatures, where the
presence of neutral atoms cannot be neglected (e.g. solar-type stars). When
quantum mechanic data for pressure broadening are not available (in general for
metallic lines), a good approximation formula given by Cowley (1971) can be
employed to estimate Stark-widths,
γ
col
= 4.335 × 10
−7
Z
2
(Rc)
2
(E
−2
u
+ E
−2
l
), (2.35)
where Z is the ionic charge (1 for neutrals, 2 for singly-ionized species, etc.),
R = R
∞
µ/m the Rydberg constant (with reduced mass µ and R
∞
= 109737.315
cm
−1
) and E
u/l
the ionization energy of the upper/lower level (in s
−1
). Pressure
broadening also yields a Lorentz profile, except for the Linear Stark effect at
high densities. The combination of radiative and collisional damping gives a
Lorentzian profile with total width of γ = γ
rad
+ γ
col
, as both processes are
uncorrelated.
Doppler broadening. It is caused by the movement of atoms with a velocity
distribution along the line of sight. The profile of each atom is Doppler shifted
according to its individual line-of-sight velocity and the shifted profiles of an
21 2.5 Spectral Line Formation
ensamble of atoms are superimposed to yield the characteristic Gaussian Doppler
broadening
φ
ν
=
1
√
π∆ν
D
exp(−∆ν/∆ν
D
)
2
(2.36)
where ∆ν = ν − ν
ij
and ∆ν
D
is the Doppler width of the line,
∆ν
D
=
ν
ij
c
2kT
m
A
+ ξ
2
, (2.37)
with m
A
being the mass of an atom of the chemical species under consideration
and ξ is the microturbulent velocity. The first term of the Doppler width corre-
sponds to the thermal motion of the atoms in the medium, while the second is
due to a non-termal component: the microturbulence. It is assumed that non-
thermal motions have a Gaussian distribution around a most probable value ξ
and occur on small scales when compared to a photon mean-free-path. Therefore,
this constitutes an additional source of broadening. The most probable thermal
velocity for atoms and ions is
v
th
=
2kT
m
A
= 0.129
T
m
A
km s
−1
, (2.38)
and it is comparable to the velocity of sound in the stellar atmosphere (e.g. at
30 000 K, the thermal velocity for hydrogen is ∼22 km s
−1
).
The total line profile, accounting for natural, collisional and Doppler broaden-
ing, results from the convolution of a Lorentzian and a Gaussian, termed a Voigt
profile,
φ
Voigt
ν
= φ
Doppler
ν
∗ φ
Lorentz
ν
(2.39)
All the previous broadening mechanisms act on microscopic scales and affect both
equivalent width and line shapes. Macroscopic scales also have to be considered
because resolving the star surface is not possible and only the light integrated
over the stellar disk can be measured.
Rotational broadening. In rotating stars the spectral lines can be strongly
affected by the relative Doppler shifts of the light emerging from different parts
of the stellar disk. Only the line shape is affected by the frequency redistribution
of the photons, while the equivalent width remains unchanged. High values of
projected rotational velocities (along the line-of-sight) may cause a complication
in the analysis of spectral lines as strong blends can be overlooked in this case.
Macroturbulence. Turbulent motions can also occur on a large scale com-
pared to a photon mean-free-path. Individual macroturbulence cells give rise
to Doppler shifts corresponding to the velocity of the cell. The effects on the
line profile are well described with the radial-tangential model for macroturbu-
Chapter 2: Model Atmospheres 22
lence. This requires a further convolution of the line profile with an appropriate
macrobroadening function (Gray 1992), introducing the radial-tangential macro-
turbulent velocity ζ as an additional parameter. Lucy (1976) proposed that
non-radial oscillations of A-type supergiants might produce macroturbulent-like
surface motions, imitating the movement of convective cells in cooler stars. More
recent applications are given by Przybilla (2002) and Przybilla et al. (2006) for
A-type supergiants and Ryans et al. (2002) for the case of B-type supergiants.
Macroturbulence may also be important for giant and main sequence stars. Max-
imum values for the macroturbulence are constrained by twice (both directions
of movement of the cells) the sound speed, in order to avoid cells moving at su-
personic velocities.
Instrumental profile. An additional broadening of the spectral lines is due
to the finite resolution of the spectrograph. This is important in cases of
medium-resolution spectra and for high resolution only in cases where the stars
present a low projected rotational velocity and therefore sharper lines. The in-
strumental profile is assumed to be Gaussian, with a FWHM corresponding to
v = c/2
√
ln 2R, where R = ∆λ/λ is the resolving power of the spectrograph.
2.5.2 Non-LTE Line Formation
Non-LTE line formation is also known as the restricted non-LTE problem. In this
approach, the atmospheric structure (temperature, density, etc.) is assumed to be
known from previous calculations (either LTE or simplified non-LTE), and is kept
fixed, while only radiative transfer and statistical equilibrium for a chosen atom
is solved simultaneously. In the present work, the non-LTE line formation is com-
puted with the codes detail and surface (Giddings 1981; Butler & Giddings
1985; both updated by K. Butler). The implementation of the ALI approach
of Rybicki & Hummer (1991) in detail allows us to solve the coupling of the
radiative transfer and the statistical equilibrium equations for multilevel atoms.
The synthetic spectra are calculated with surface on the basis of the non-LTE
populations computed with detail and refined line-broadening theories.
Hundreds of atomic energy levels and several thousands of transitions can
be accounted for in the computations. The multilevel-line problem is treated
for a single or several atoms (with different ionization stages) in a plane-parallel
medium. The temperature and electron density are assumed to be prescribed
as a function of depth. The continuum opacity and emissivity of H and He are
previously computed with detail. For multilevel atoms, one may express the
coupling of the radiative transfer and statistical equations as
S
ij
= S
ij
(n) and n = n (
¯
J
ij
), (2.40)
where S
ij
is the source function, n is the vector of level populations,
¯
J
ij
is the
frequency-averaged mean intensity and ij indicates the transition i → j.
23 2.5 Spectral Line Formation
With careful preconditioning of the statistical equilibrium equations it is pos-
sible to simplify the coupled problem. Rybicki & Hummer (1991) show some
examples where most of the transfer at frequencies of the ’core’ of the line (de-
scribed by the local part of the lambda operator) can cancel out analytically.
The preconditioned statistical equilibrium equations can become linear in the ion
level populations.
The solution scheme follows this procedure: (i) optical-depth scales and source
functions for all frequencies are calculated from current estimates of the occu-
pation numbers (the first estimate may be LTE-values); (ii) correction terms
∆J
ν
= (Λ − Λ
∗
)S
ν
= ΛS
ν
− Λ
∗
S
ν
are computed for all frequencies and depths
by a formal solution (using for instance the scheme of Feautrier 1964) and by
approximate formal solution using the Λ
∗
-operator; (iii) the approximate radia-
tive transfer and the statistical equilibrium equations are simultaneously solved
depth by depth, starting at the inner boundary (requiring that this boundary lies
deep enough in the atmosphere). These steps are iterated until changes in the
source functions and/or occupation numbers are sufficiently small. The iterative
method proceeds as follows. An initial choice of the old populations is made.
This allows one to setup the equations of statistical equilibrium. These equations
are then solved for the new populations. Regarding these to be the old popu-
lations, another cycle of iteration can be made, continuing until convergence is
obtained.
The coupled problem of radiative transfer and statistical equilibrium can be
described for an idealised case of a two-level atom (lower level:1; upper level:2).
This approximation may seem to be inadequate because real atoms contain many
energy levels, however it provides a good description and an explanation of el-
ementary processes that are crucial to understand the non-LTE line formation.
Under this approximation, the source function can be written in terms of the
¯
J (non-local contribution of the radiation field) and the Planck function (local
contribution)
S = (1 − )
¯
J + B
ν
, (2.41)
where is the destruction probability and can be calculated in terms of the colli-
sional rate C
21
and the Einstein coefficient for spontaneous emission A
21
for cases
where hν kT (for details see Mihalas, 1978)
≈
C
21
C
21
+ A
21
, (2.42)
i.e. is the probability that an absorbed photon is destroyed by a collisional
de-excitation process (C
21
) rather than being re-emitted (A
21
).
Equation (2.41) is fundamental for the problem. The first term on the right
hand side represents the photons in the line created by scattering (emission fol-
Chapter 2: Model Atmospheres 24
lowing a previous absorption of a photon), while the second term represents the
thermal creation of a photon (emission following a previous collisional excitation).
Mathematically, the source function for a two-level atom (2.41) is still a linear
function of the mean intensities.
For a two-level atom the source function is equal to the Planck function deep
in the atmosphere and decreases from a point called the thermalization depth
τ
th
≈ 1/ towards the surface to the value
√
B. This can be explained by the
fact that in a homogeneous medium departures from LTE arise only because
of the presence of the boundary through which the photons escape. At large
optical depths, all microscopic processes are in detailed balance, so the LTE
approximation holds. However, as soon as the photons start to escape from
the medium through the boundary, the photoexcitations are no longer balanced
by radiative de-excitations. Since the absorption rate depends on the number of
photons present, while the spontaneous emission does not (neglecting, stimulated
emission for simplicity), the number of radiative excitations drops below the
number of de-excitations as soon as photons start to escape. The lower level
will start to be overpopulated with respect to LTE, while the upper level will
be underpopulated. Since the source function measures the number of photons
created per unit depth and is therefore proportional to the population of the
upper level, the source function has to drop below the Planck function.
In a multilevel atom the source function contains non-linear terms in the
radiation intensity coupling strongly with the optical depth, the frequency and
the direction. In this case the source function may also increase towards the
surface as it will be shown in Chapters 5 and 6. The non-LTE effects on the
level occupations give rise to departures of the line source function S
l
from the
Planck function B
ν
, which can be defined in terms of the departure coefficients
b
i
= n
i
/n
∗
i
(n
i
, n
∗
i
are the level populations in non-LTE and LTE, respectively)
S
l
=
2hν
3
/c
2
b
i
/b
j
exp (hν/kT ) − 1
, (2.43)
with T being the local temperature.
A schematic description of the non-LTE line-formation calculations is dis-
played in Fig. 2.2. The model atmosphere is computed with atlas9 (Kurucz
1993b). Most of the input data of detail were already discussed in this Chap-
ter, except for the model atoms, which will be described in general terms in
Chapter 3 and for the specific case of carbon in Chapter 6. The atmospheric
parameters and the ODFs are input data for computing the atmospheric struc-
ture (atlas9), which is in turn an input for detail and surface. The classical
atmospheric parameters accounted for are the effective temperature T
eff
, surface
gravity log g and the metallicity [M/H]. The metallicity is the abundance of all
elements heavier than He. The abundance of an element in a star is expressed
25 2.5 Spectral Line Formation
Figure 2.2: Schematic description of the non-LTE line-formation calculations
with the codes detail and surface.
in a logarithmic scale relative to hydrogen, which is normalised in the Sun to a
value of 12, (X) = log(X/H) + 12. Here, X and H are the number densities of
absorbers of the species under study and of hydrogen in the star. The chemical
abundance is one of the main parameters to be determined in this work.
The ODFs account for different values of microturbulence and metallicity,
which should be consistent with those values derived from the analysed star.
The line blocking, discussed in Sect. 2.4, is affected by ξ (broader/stronger lines
result at higher microturbulence) and [M/H] (stronger lines result at high metal-
licity). Consequently a variable backwarming effect is expected to give different
temperature T(τ), pressure P(τ), and electron pressure P
e
(τ) structures – in par-
ticular in the line-formation region. detail accounts indirectly for T(τ), P(τ),
and P
e
(τ) through the temperature, number density of heavy particles and of
Chapter 2: Model Atmospheres 26
electrons as a function of the mass-scale instead of the optical depth.
The ODFs and the microturbulence are also crucial in the solution of the
coupled problem of radiative transfer and statistical equilibrium. In this case the
line blocking affects the radiative rates and therefore the ionization balance. The
ODF is accounted for as an input of detail and the same value of ξ is considered
in both detail and surface. A further input parameter of the model structure
is the helium abundance. The chemical abundance of H, He and the metals under
consideration also enter in the detail computations.
surface computes the formal solution of the radiative transfer (with a known
source function from the level populations of detail) for each depth point, re-
sulting in synthetic non-LTE line profiles. Other essential input data for surface
are abundance, microturbulence, oscillator strength of each line component, colli-
sional and radiative parameters (accounting for broadening mechanisms), energy
levels, statistical weights, wavelengths of the transitions of interest. The atmo-
spheric structure is also necessary to compute the line spectrum, as each line is
formed at different depths in the atmosphere. The final line profile is a Voigt
profile, which is subsequently convolved with a rotational, macroturbulent and
instrumental profile (as explained in Sect. 2.5.1) before being compared with the
observed spectra.
Since none of the atmospheric input data are known a priori, one can solve
the problem by estimating some parameters or by performing an iterative (and
extensive) procedure in order to derive them all self-consistently, depending on
the accuracy one aims to achieve. More details of the spectral analysis based on
atlas9, detail and surface are found in Chapters 5 and 6.
Chapter 3
Atomic Data for Spectral
Modelling
The high precision of experimental and observational facilities impose a continu-
ous demand for a large variety of good quality atomic data in order to interpret
laboratory or astrophysical processes. In the case of astrophysics, the informa-
tion concerning the physical state of the plasma in objects where the assumption
of LTE is no longer valid can be inferred from their spectra only to the extent
that radiative and collisional rates are known. Little of the relevant data can be
determined under controlled laboratory conditions and therefore only few exper-
imental data are available, motivating the development of sophisticated compu-
tational methods. Large collaborations have been established in order to provide
an immense amount of accurate atomic data from ab-initio computations rele-
vant for astrophysical studies: the Opacity Project (OP; Seaton 1987; Seaton et
al. 1994) and the Iron Project (IP; Hummer et al. 1993). Large amounts of semi-
empirical atomic data were also provided by Kurucz with an enormous effort over
two decades (Kurucz 1992, and references therein).
A full description of the computation of atomic data is beyond the scope of
the present work, therefore only the basis of theoretical calculations and some
approximations will be described. Special emphasis will be given to some as-
sumptions and possible sources of systematic uncertainties. A basic knowledge of
the limitations and the different levels of accuracy of the atomic data involved is
highly important in the construction of model atoms for non-LTE line-formation
calculations. These are crucial for the modelling and further analysis of the stellar
spectra. In this Chapter a brief description of the basic concepts of atomic data
needed for spectral modelling, quantum mechanic ab-initio computations and
some approximations is given. These are mainly based on Bautista (2000) and
Berrington et al. (1995). Complementary information is also based on Hummer
et al. (1993), Burke et al. (1971), and Nahar (2003). A summary of the construc-
tion of model atoms for detail is also given, mainly based on the manual by
K. Butler (http://ccp7.dur.ac.uk/Docs/detail.ps) and on Przybilla et al. (2001).
27
Chapter 3: Atomic Data for Spectral Modelling 28
3.1 Basic Concepts
Two general atomic physics problems have to be solved in order to provide atomic
data (energy levels and cross-sections) for spectral modelling: the atomic struc-
ture and scattering processes. The atomic structure problem is concerned with
the computation of energy levels and spontaneous transition probabilities among
bound states of the same ion (A-values) and from autoionizing levels of one ion
to levels of the next ionized species. The scattering problem, relevant for the
formation of the spectra in hot plasmas, is related to all the different processes
that occur after a collision of an ion with a photon or other charged particle (in
this case only electron collisions are considered because the electron velocity is
much higher than the ion velocities).
• A (photon+ion) scattering hν + A
i
→ e
−
+ A
f
can produce photoioniza-
tion/radiative recombination or autoionization/dielectronic recombination
1
(the last two are not explicitly treated in the present computations, but are
only implicit in the detailed resonance structures of the photoionization
cross-sections). A
i
, A
f
refer to an ion in its initial and final state, respec-
tively.
• A (electron+ion) scattering e
−
+ A
i
→ e
−
+ A
f
can produce collisional
excitation/de-excitation or collisional ionization/3-body recombination. A
solution for ionization is more complicated because of the 3-body nature of
the problem. Only limited data are available in the literature and for the re-
maining transitions approximate data are incorporated in the computations
of the theoretical spectra.
Some theoretical methods to compute atomic structure, ionization and exci-
tation of atoms and ions from ab-initio quantum mechanical considerations and
other approximations are summarised in the next sections.
3.2 Atomic Structure Calculations
Quantum mechanics can provide a physical description of the interaction be-
tween electrons and photons with isolated atoms and their ions through the
time-independent Schr¨odinger equation with appropriate boundary conditions
HΨ
i
= E
i
Ψ
i
, (3.1)
where i represents a set of quantum numbers necessary to describe the system,
Ψ
i
are the wavefunctions, where all the information of the system is contained,
E
i
are the eigenvalues and H is the Hamiltonian.
1
The inverse process of dielectronic recombination is photoionization via intermediate
double-excited autoionizing states, i.e. resonances in atomic processes.
29 3.2 Atomic Structure Calculations
For light atoms and ions where the relativistic effects can be neglected (the
spin the of electrons in not considered explicitly), the Hamiltonian can be written
as
H =
N
i=1
p
i
2m
e
−
N
i=1
Ze
2
r
i
+
i=j
e
2
|r
i
− r
j
|
, (3.2)
where N is the number of electrons of the system. The first term is the sum of
kinetic energy of all electrons, the second term is the potential energy due to
the Coulomb attraction of all electrons by the nucleus and the third term is the
energy due to electric repulsion between the electrons. The distance between an
electron and the nucleus is r
i
and the distance between two electrons is |r
i
−r
j
|.
The nucleus is assumed to be infinitely heavy and a point charge.
In the presence of the two-electron operator 1/|r
i
−r
j
| one cannot obtain exact
solutions to the Schr¨odinger equation for the N-electron system. Approximate
methods for solving Eqn. (3.2) replace the two-electron term by one-electron
potentials to give an effective Hamiltonian of the form
H
eff
=
N
i=1
H
eff
i
= −
N
i=1
p
i
2m
e
+
Ze
2
r
i
− V
eff
i
(r
i
)
, (3.3)
For highly ionized atoms the central field potential is a good approximation.
There are several techniques regularly employed in the atomic structure calcu-
lations. The most important are: model potentials, methods based on Hartree-
Fock theory, semiempirical methods, perturbation techniques, and the R-matrix
method in the close coupling formalism. The latter will be described in Sect. 3.3.
Model potentials. An effective potential is proposed for specific cases (e.g.,
central potential for alkali atoms). A generally applicable potential is the Thomas-
Fermi-Dirac type of central potential to generate one-electron orbitals (which de-
pend on the angular momentum of the valence electrons). It is implemented in
the widely used computer program SUPERSTRUCTURE (Eissner et al. 1974,
Eissner 1991), which provides accurate results: ∼1% for energy levels and ∼10%
for oscillator strengths.
Methods based on the Hartree-Fock theory. The Hartree-Fock (HF)
method address the computation of one-electron orbitals in the non-local po-
tential (direct and exchange) from electronic orbitals in a self-consistent way
using the variational principle. A modification of this method for simple systems
is the frozen core FC approximation: wave functions are computed by varying
only valence orbitals while keeping orbitals in the core fixed. In most of the cases
equivalent electrons in the same configuration
2
have to be accounted for. It is
necessary to include the electronic configuration interaction, as in the multicon-
2
Particular distribution of electrons among atomic orbitals.
Chapter 3: Atomic Data for Spectral Modelling 30
figuration Hartree-Fock MCHF method (e.g. Froese Fischer 1977). The method
is potentially very accurate, but computationally lengthy (it involves many iter-
ations on the wavefunctions and normalisation coefficients).
Semiempirical methods. These methods compute the atomic structure of ions
by solving simplified forms of the HF equations. The definition for the atomic
potential is mostly empirically motivated and requires preconceived wavefunc-
tions in order to compute the electron density function. The atomic structure
equations are solved iteratively. The method has the advantage of being very
efficient, but requires an enormous care in the construction of the initial elec-
tron density distribution. It is difficult to estimate the accuracy of any given
calculation except by the observed agreement of a limited sample of data with
experimental values. Kurucz (1988) and Kurucz & Peytremann (1975) based the
computation of millons of energy levels and oscillator strengths for most ions of
astrophysical interest on this method.
Some considerations of particular physical interest can also be taken into ac-
count in the atomic structure computations:
LS coupling. Light atoms and ions are well described by the Russell-Saunders
or LS coupling, where the orbital (L) and the spin (S) angular momenta are
assumed to be separately conserved, as is the parity π. An LS-term is a specific
LSπ target state. LS coupling is characterised by a small separation of the fine-
structure levels when compared to the separation of the terms (in contrast to jj
coupling).
Configuration interaction. This refers to the interaction between different possi-
ble arrangements of the electrons in an atom; the resulting electron distribution,
energy levels, and transitions differ from what would occur in the absence of the
interaction. It can be seen as a correction to the Hartree’s single-electron orbital
approximation representing each electron moving individually in the field of the
nucleus screened by the other electrons. In CI, wavefunctions may be obtained
from linear combinations of single configuration wavefunctions of the same to-
tal angular momentum and spin symmetry (Condon & Shortley 1935). The CI
representation has important effects on atomic quantities like oscillator strengths
and it is a standard capability of codes like SUPERSTRUCTURE, CIV3 (Hib-
bert 1975), and Froese Fischer’s MCHF code.
Relativistic effects. For heavy atoms and ions relativistic effects are impor-
tant for the treatment of forbidden dipole transitions (under LS coupling) and
also for the allowed transitions. The relativistic effects can be treated by the full
Dirac formalism or by the addition of the Breit-Pauli operators to non-relativistic
equations, the latter being widely used for practical applications. There are eight
Breit-Pauli operators, but for low atomic number Z only four are considered
(Eqn. 3.4): (1) the non-relativistic term (Eqn. 3.2); (2) the mass operator gives
31 3.3 Scattering Calculations
the correction due to the relativistic variation of mass with velocity; (3) the
Darwin term is characteristic of the Dirac theory and only applies to s orbitals;
(4) the spin-orbit coupling term describes the interaction between the spin and
the orbital magnetic moment of each electron. It splits the LS-term into fine-
structure levels of symmetry Jπ: J = L+S, J being the total angular momentum.
Other terms considering spin-other-orbit, spin-spin and orbit-orbit coupling are
normally neglected, but can be included if necessary.
H
BP
= H
NR
+ H
mass
+ H
Dar
+ H
SO
(3.4)
Most of the current codes for atomic structure calculations as SUPERSTRUC-
TURE and CIV3 use the Breit-Pauli approximation to account for relativistic
effects.
3.3 Scattering Calculations
The scattering system is composed of an atomic target with N electrons
(Sect.3.2) and an additional incoming (electron impact) or outgoing (photoioniza-
tion/electron impact) electron. Thus, the wavefunctions of the (N + 1)-electron
system Ψ can be expanded in terms of products of wavefunctions of the core φ
i
(target wavefunctions obtained from atomic structure calculations) and those of
the electron θ
i
,
Ψ =
i
φ
i
(x
1
, ..., x
N
) θ
i
(x
N+1
) (3.5)
Substitution of Eqn. (3.5) in Eqn. (3.1), for the non-relativistic case, yields a
system of coupled equations
−(∇
2
+ k
2
i
) θ
i
(x) +
i
V
ii
θ
i
(x) = 0, (3.6)
where V
ii
is the potential energy generated by the interaction of the N electrons
of the target, and k
2
i
is defined by
E = E
i
(N) + k
2
i
(3.7)
These equations have to be solved for every value of energy (k
2
), of total angular
momentum L of the (N +1)-electron system. This last condition motivates the so-
called partial waves expansion, in which all states of definite angular momentum
of the free electron are considered separately.
There are several methods to solve the scattering problem and calculate
the cross-sections necessary for practical applications: the central field approx-
imation, the Gaunt factor and the Coulomb-Born approximation, the R-matrix
method, used for both photoionization and electron impact excitation and ion-
Chapter 3: Atomic Data for Spectral Modelling 32
ization, as well as atomic structure. The latter has been implemented in several
codes and has been widely used by groups like the Opacity Project and the Iron
Project. Here, only the R-matrix and the Gaunt factor methods are briefly de-
scribed.
R-matrix. The R-matrix method is a highly sophisticated and accurate tech-
nique. It takes into account nearly all the physical effects that contribute to
cross-sections for astrophysical applications and it is applicable for all kind of
ions, from neutral to highly ionized species. With the increased complexity of
the calculations the R-matrix can be computationally very intensive.
The R-matrix method proposes a division of the configuration space by a
sphere of radius a centered on the target nucleus (Burke et al. 1971). The aim is
to compute the total wave function in the inner region, and the R-matrix on the
boundary. The R-matrix is obtained for all energies by diagonalizing the Hamil-
tonian of the system only once for each set of conserved quantum numbers. The
derived radial wave function of the scattered electron on the boundary depends
on the R-matrix. The radial wave functions of the inner and outer regions have
to be identical on the boundary. The next step in the calculation is to solve the
electron-target scattering problem in the external region. There, the colliding
electron is outside the atom, and can be considered distinct from the N target
electrons. The wavefunction in the external region is fully determined in terms of
the reactance matrix K. This matrix can be computed in terms of the wavefunc-
tion at r = a, and therefore, in terms of the R-matrix. Scattering observables
may be calculated from the reactance matrix through the scattering matrix S.
In the internal region, r < a (where r is the relative coordinate of the free
electron) electron exchange and correlations between the scattered electron and
the target are important and the (N + 1)-electron collision complex is similar to
a bound state. Consequently, a CI expansion of this complex, analogous to that
used in bound state calculations, is adopted (Eqn. 3.8). The total wavefunction
Ψ in the inner region can be written in terms of the basis states set ψ
k
and the
energy-dependent expansion coefficients A
Ek
as
Ψ =
k
A
Ek
ψ
k
. (3.8)
Based on the assumption of the individual behaviour of both the colliding
and the ionic electrons, the ψ
k
functions for the (N + 1)-electron system can be
expanded as the sum of product wavefunctions: the so-called close-coupling ex-
pansion (CC). The general form of the CC expansion for the radial wavefunction
ψ in terms of a N-electron target basis χ
i
and the scattering electron function θ
i
is
ψ =
∞
i
χ
i
θ
i
+
χ
θ
, (3.9)
33 3.3 Scattering Calculations
where the first term address the bound and the second term the continuum wave-
functions (angular coupling and antisymetrization are assumed). For practical
applications, only a finite close-coupling expansion can be used.
The R-matrix at r = a is defined as
R
ij
(E) =
1
2a
k
ω
ik
(a)ω
jk
(a)
E
k
− E
, (3.10)
where the surface amplitudes ω
ik
(a) and the eigenvalues E
k
are determined by
diagonalizing the Hamiltonian.
This R-matrix is the basic solution of the electron-scattering problem in the
internal region and must be matched to the external region solutions. It allows
one to determine the structure of the (N + 1) system, the collision strengths and
photoionization cross sections. Although the basic computations in this method
are very lengthy, it is a very efficient technique for computing large numbers of
frequency points which allow complex resonance structures in the cross-sections
to be accounted for.
In the external region, r > a, an electron exchange between the free electron
and the target can be neglected if a is large enough to contain the charge distribu-
tion of the target. The internal and external regions are linked by the R-matrix
on the boundary r = a. For the external region the scattered electron moves in
a long-range multipole potential of the target. This potential is local and the
solution can be obtained using a standard method for solving coupled differential
equations with an asymptotic expansion or by using perturbation theory. These
coupled equations for r > a yield the solution for the reactance K-matrix and
the scattering S-matrix, given by
S =
1 + iK
1 − iK
(3.11)
The K-matrix contains all the information needed to derive the observables
associated with electron collisions and the S-matrix elements determine the col-
lision strength Ω
ij
for a transition from an initial state i to a final target state
j:
Ω
ij
=
1
2
ω|S
if
− δ
if
|
2
, (3.12)
where ω = (2L + 1)(2S + 1) or (2J + 1) depending on the coupling scheme. The
collision strength is related to the excitation cross-section σ
ij
,
σ
ij
= Ω
ij
πa
2
0
g
i
E
i
, (3.13)
where a
0
is the Bohr radius, g
i
is the statistical weight of the initial state i and E
i
is the energy of the incident electron (in Rydbergs). In non-LTE computations the
Chapter 3: Atomic Data for Spectral Modelling 34
transition rates C
ij
are of interest (See Sect. 2.3). In this context it is convenient
to define the thermally-averaged effective collision strength Υ
ij
,
Υ
ij
=
∞
0
Ω
ij
exp(−E
j
/kT ) d(E
j
/kT ), (3.14)
with the transition rates being proportional to Υ
ij
.
Radiative transition data are also provided by R-matrix computations. Os-
cillator strengths f
ij
(computed with the R-matrix method applied to atomic
structure) and photoionization cross-sections α
iE
are proportional to the line
strength S(j; i):
f
ij
=
8π
2
mν
ij
he
2
g
i
S(j; i). (3.15)
The relation between line absorption cross-section and the oscillator strength is
given by Eqn. (2.31). Using the dipole length operator D
L
= e
i
r
i
, where the
sum is over all atomic electrons, the line strength is
S
L
(j; i) = | < Ψ
j
|D
L
|Ψ
i
> |
2
. (3.16)
An analogous expression, S
V
(j; i) is found when employing the dipole velocity
operator instead of the dipole length operator. Use of exact wavefunctions would
give identical values of both operators, S
L
= S
V
. The difference between both
line-strengths gives an indication of the accuracy achieved in the computations.
For final states Ψ
E
in the continuum (photoionization) with energy E the
oscillator strength per energy unity can be expressed as
df
iE
dE
=
8π
2
mν
iE
he
2
g
i
S(E; i), (3.17)
and the photoionization cross-section is given by
α
iE
=
πe
2
mc
df
iE
dE
. (3.18)
Radiative transitions are restricted by selection rules, which may depend on
the coupling scheme assumed. Electric dipole transitions require a change of the
parity between the initial and final state, and the total angular momentum J
can change only by ∆J = 0, ± 1 (but the transition 0 ↔ 0 is forbidden). For
LS coupling additional selection rules apply: ∆L = 0, ± 1 and ∆S = 0. Inter-
combinations between two spin-systems are not allowed. Forbidden transitions,
obeying other selection rules, can occur through electric quadrupole or magnetic
dipole transitions, with significantly lower transition probability.
35 3.3 Scattering Calculations
The inverse processes of the aforementioned collisional and radiative transi-
tions are accounted for in the statistical equilibrium equations. The collisional
downward rate is proportional to the collisional upward rate. In the case of ra-
diative transitions, both the upward and the downward rates are functions of the
oscillator-strength (for bound-bound processes) or of the photoionization cross
section (for bound-free processes) defined above (see Sect.2.3).
Similar expressions to Eqn. (3.18) can be derived for free-free transitions. In
hot stars such processes can be dominated by interaction of electrons with the
fields of bare nuclei, as hydrogen is almost fully ionized, and they are referred to
as Coulomb free-free transitions. In the case of electrons moving in the field of
ions containing some bound electrons the free-free process due to other – much
less abundant – ions is negligible but the perturbations affecting the line absorp-
tion process produce line broadening by electron impact (see Sect 2.5.1).
There are important physical effects to be taken into account in the evaluation
of photoionization or collisional excitation/ionization cross-sections.
Configuration interaction. The first requirement in any scattering calculation is
a good representation of the target, i.e. accurate wavefunctions for the target ion.
Such a representation usually requires the inclusion of CI in the atomic structure
model, which affects the calculated energy levels and oscillator strengths of the
ion (See Sect. 3.2). Furthermore, comparisons between computed and experi-
mental energies and oscillator strengths of the target are important indicators of
the quality of the target representation and the overall accuracy of the obtained
cross-sections.
Resonances. The resonance structure is an important part of collisional excita-
tion/ionization and photoionization cross-sections. Physically, resonances occur
when the incoming particle with just the right kinetic energy is trapped into an
autoionizing state
3
of the (N +1)-electron system. Then, as the incoming particle
remains trapped for a time before autoionization occurs, the time delay yields a
phase shift in the wavefunction that manifests itself in sharp peaks in the cross-
sections (resonances). Resonances appear in Rydberg series converging onto the
various excitation thresholds of the target. Resonances in the collision strengths
can produce an enhancement of the excitation rates by up to several factors in
the case of valence electron excitation (up to orders of magnitude for inner-shell
excitation). The resonances also enhance photoionization cross-sections.
Convergence of the partial wave expansion. It is a standard approach to expand
the collision strengths in partial waves from 0 to infinity (in practice taking only
the lower terms) for every possible value of the angular momentum of the free
electron. The convergence of the partial wave expansion is very slow for allowed
transitions and the number of partial waves needed for convergence increases with
3
Autoionizing states are compound states of the (electron+ion) system located above the
ionization potential. They result from the excitation of two or more electrons of the system.
Chapter 3: Atomic Data for Spectral Modelling 36
the energy of the free electron. This convergence is a difficult practical problem
and a possible source of error in the collision strengths for highly ionized systems
for which very high collision energies need to be considered.
Convergence of the close coupling expansion. In the standard R-matrix approach,
the integral term that accounts for the target continuum (Eqn. 3.9) is neglected
or replaced by a discrete sum over bound functions. This approximation is good
when considering valence electron excitation among the lowest energy levels of
the ion. However, for excitations to highly excited levels the convergence of CC
expansion must be looked at carefully. The collision strengths can depend on the
size of the CC expansion.
Relativistic effects. The Breit-Pauli operators (see Sect. 3.2) have been imple-
mented into the RMATRX package of codes by the Iron Project.
The Gaunt Factor approximation. Seaton (1962) and Van Regemorter
(1962) suggested an approximate formula to obtain near-threshold collision rates
for optically allowed transitions. The formula is based on the Bethe approxima-
tion and uses a g factor similar to that introduced by Kramers in the radiative
case (the Gaunt factor). Later, the formula was modified to replace g, which is a
varying function of energy, by an empirical parameter of ¯g = 0.2,
Ω(i, i
) =
8π
3
√
3
S(i; i
) ¯g. (3.19)
where S(i; i
) is the line strength. Seaton’s approximation formula applies to the
collision ionization cross-section, which is proportional to the threshold photoion-
ization cross-section. Van Regemorter’s formula applies to collisional excitation
cross-section with ¯g = 0.2 for principal quantum number n = n
and ¯g = 0.7 oth-
erwise. For forbidden transitions, the collision strength can be estimated from
the Allen (1973) semi-empirical formula. Often a value of Ω ≈ 1 is assumed for
low ions. In practice, Ω can vary by orders of magnitude (see Chapter 5).
3.4 Construction of Model Atoms
The radiative and collisional bound-bound (RBB, CBB) and bound-free (RBF,
CBF) transitions derived from atomic structure and scattering calculations are
collected in order to construct model atoms (as indicated schematically in
Fig. 3.1) for non-LTE calculations. They are input data for the codes detail and
surface (Sect. 2.5.2). Figure 3.2 shows an example of the atomic structure of
C ii (singly-ionized carbon), accounted for non-LTE line-formation computations
by Przybilla et al. (2001). The C ii model atom considers simultaneously both
the doublet and the quartet spin systems. All (dipole-allowed) radiative bound-
bound transitions treated explicitly in non-LTE are displayed in the figure. One
of the aims of the present work is to update this model with new input atomic
37 3.4 Construction of Model Atoms
Atomic Structure
(e.g. code SUPERSTRUCTURE)
(e.g. code RMTRX or approx.)
Scattering Cross−Sections
Radiative & Collisional
Energy Levels
Model
Atom
Figure 3.1: A model atom for non-LTE calculations, as the carbon model of
the present work, can be constructed with input atomic data from the literature
(with inhomogeneous accuracy). Energy levels are mostly adopted from NIST
(National Institute of Standards and Technology) database.
data and the inclusion of other ionization stages to be treated simultaneously.
As explained in Chapter 2, detail solves the coupled radiative transfer and
statistical equilibrium equations. For model atoms of light elements these compu-
tations can be performed under the assumption of LS-coupled terms. When fine-
structure data is available in the literature for light elements the cross-sections
are co-added. Experimental level energies – when possible – are considered. Ra-
diative and collisional transitions from different sources can have a wide range
of accuracy, from a typical 10-20 % for the ab-initio calculations to orders of
magnitude for approximation formulae. detail allows robust model atoms with
hundreds of levels to be implemented explicitly in non-LTE, several thousands of
transitions and a large frequency grid allowing the resonance structure of pho-
toionization cross-sections to be considered.
The following describes briefly the general structure of a model atom (of one
or more ionization stages) for detail.
Frequency grid. The frequency grid can have several thousands of points.
Energy levels and statistical weights. For most of the energy levels exper-
imental energies are available. Data for high-excitation levels can be adopted
from theoretical computations when experiments do not provide them.
RBB. Oscillator strengths for dipole-allowed and in some cases also intercombi-
nation lines are taken from ab-initio calculations.
RBF. Photoionization cross-sections are also based on ab-initio calculations. Au-
toionization and dielectronic recombination are consistently considered through
the detailed resonance structure.
CBB. When available, collisional excitation cross-sections are adopted from ab-
initio calculations (in general for the lowest states). When such calculations
are not available, the Van Regemorter (1962) approximation is employed in the
optically-allowed cases and the semi-empirical formula of Allen (1973) in the for-
bidden cases.
CBF. Collisional ionization rates are evaluated according to the Seaton (1962)
Chapter 3: Atomic Data for Spectral Modelling 38
Figure 3.2: Grotrian diagram for the C ii doublet (left hand) and quartet (right
hand) spin systems from the carbon model atom by Przybilla et al. (2001). Dis-
played are all radiative bound-bound transitions treated explicitly in non-LTE.
approximation adopting threshold photoionization cross-sections from ab-initio
calculations.
Continuous opacity from bound-free and free-free transitions of H and He,
which are the main opacity contributors, are also accounted for. Non-LTE level
populations for these may be computed in a previous step.
In most cases it is mandatory to evaluate the input atomic data by comparison
with observed spectra. The empirical calibration of a model atom is the only
way to choose the optimum set of atomic data able to reproduce observations
independently of the physical conditions of the stellar plasma.
surface computes the synthetic non-LTE line profiles based on the atomic
level populations calculated in detail (it can also compute LTE line spectra).
Experimental wavelengths and fine-structure splitting are considered. Collisional
damping parameters accounting for broadening mechanisms are calculated either
from approximations (for metals) or from refined theories (for hydrogen and he-
lium). Radiative damping constants can be derived from experiments or from
oscillator strengths computations, as explained in Sect. 2.5.1.
Chapter 4
Spectroscopic Analysis
Some physical quantities derived from spectral analyses of stars constitute basic
observational constraints for broad astrophysical fields, like stellar evolution and
chemical evolution of the Milky Way and other galaxies. Hence, any systematic
effect in the interpretation of the stellar spectrum might result in non-reliable
conclusions concerning different fields. In particular, a precise atmospheric pa-
rameter determination is of fundamental importance for the further quantitative
study of the stellar spectrum. This is a non-trivial problem to be solved because
the line spectrum and the observed flux distribution provide the information for
the determination of the atmospheric parameters (i.e., the spectroscopic analysis
depends on basic parameters, which in turn depend on the spectrum).
As we have briefly discussed in the last chapters, the modelling of the stellar
spectrum is complex even for the most simple cases of classical stellar atmo-
spheres. The coupling of the radiation field with radiative and collisional transi-
tions increases the number of variables to be taken into account in the analysis.
The observational constraints derived from a spectral analysis rely on a large
quantity of model assumptions: from the model atmosphere to the input atomic
data of the model atoms for non-LTE line formation. In addition, the observed
spectra suffer a long reduction process (see Appendix A) not free of systematic
effects. The quantitative spectral analysis for the derivation of the fundamental
stellar parameters can be schematised as in Fig. 4.1. The line-profile fitting of
synthetic profiles to observed spectra could result in an inconsistent quantitative
analysis when the underlying physics of the problem is not well solved. It is thus
important to test the consequences of employing different theoretical approaches
on the final synthetic lines and on the derived physical information of the star.
In this chapter, some open issues concerning quantitative analyses of stellar
spectra for OB main sequence and giant stars are considered. They constitute
an important motivation to the present study. In particular, some interesting
results derived from an analysis of stars in the Large Magellanic Cloud (LMC)
are presented.
39
Chapter 4: Spectroscopic Analysis 40
Theory: synthetic lines
Model
Atoms
Atomic
Physics Atmospheres
Stellar
Parameters
Stellar
+
Abundances
Line fits:
Chemical
Observations
Spectra
Figure 4.1: Quantitative Spectroscopy
4.1 Atmospheric Parameters
Among the various parameters that specify the physical conditions of stellar
atmospheres, the most important are the stellar effective temperature and the
surface gravity (to a lesser extent also the chemical composition). They are
called fundamental atmospheric parameters and only these are directly related
to fundamental physical properties of stars like the mass, radius and luminos-
ity. Parameters like micro- and macroturbulence are of secondary importance in
the overall description of stars, but nevertheless should not be ignored in a fine
analysis.
The derivation of basic atmospheric parameters is an important issue of stellar
spectroscopy. Numerous studies have contributed with techniques to determine
them in different kind of star. The motivation of fundamental parameter deter-
mination in early-type stars started from a previous study of main sequence B, A
and F-type stars (Nieva 2002; Adelman et al. 2002). For OB stars the accuracy
of the effective temperature determination can be much lower than for cooler
stars when employing standard methods (for instance, photometric calibrations).
Discrepancies up to ∼25% can be found from different approaches for the T
eff
determination (see Table 4.1 below). This can produce large inconsistencies in
the derived chemical abundances, therefore the problem should be addressed with
care before any further analysis.
Photometric Approach. The fundamental parameter T
eff
can be derived
from an analysis of the variations of strategic regions of the stellar flux. The
area under the flux distribution curve delimited by a filter function can be ex-
pressed as a photometric magnitude. Several photometric systems have been
developed to account for parameter-sensitive features in the shape of the stellar
flux distribution via magnitude (colour) differences. The most commonly used
41 4.1 Atmospheric Parameters
are the broad-band Johnson UBV and the intermediate-band Str¨omgren ubvyβ
systems. The photometric colour indices can be calibrated on the basis of emer-
gent fluxes from model atmospheres calculations and therefore it is possible to
find a link between those indices and atmospheric parameters (see Smalley 1996
for an overview of several calibrations). For hot stars, the Str¨omgren system
provides a temperature-sensitive indicator associated with the Balmer jump, the
gravity sensitive β-index, which measures the strength of H
β
, and even a metallic-
ity sensitive indicator. A useful photometric calibration of T
eff
for early-type stars
based on Str¨omgren indices is provided by Napiwotzki, Sch¨onberner & Wenske
(1993). On the other hand, the three-colour Johnson system is less accurate for
a parameter determination because of its broad-band character, but it is never-
theless useful to estimate temperatures when Str¨omgren indices are not available
(as in the case of many stars belonging to several Galactic OB associations).
The accuracy of photometric calibrations depends on the modelling of stellar at-
mospheres and on the sensitivity of the spectral features to the parameters of
interest, with additional uncertainties from the interstellar reddening
1
. A solu-
tion for the latter is the T
eff
-calibration of the reddening-free Q-parameter defined
by Q = (U − B) −0.72(B −V ) (e.g. Daflon et al. 1999; Lyubimkov et al. 2002),
widely used for analyses of large samples of OB-type stars. These Str¨omgren and
Johnson photometric calibrations are derived for main sequence and giant stars,
and therefore applicable – in principle – to the current work.
Spectroscopic Approach. Another method for the stellar atmospheric pa-
rameter determination relies exclusively on the analysis of the line spectrum, the
so-called spectroscopic approach. The analysis involves profiles and line strengths
of spectral features from hydrogen, helium and metals. This method can only be
performed iteratively, and the final parameters are obtained when the overall
conditions set by different spectral indicators are consistently met.
Assuming the metallicity and the helium abundance to be known, we are
interested to derive from the line spectrum: T
eff
, log g and ξ. For early-type stars,
log g can be determined (when the temperature is known) by fitting the wings of
the Balmer lines, which are strongly affected by the linear Stark effect. Both T
eff
and log g can be derived from ionization equilibria, i.e. from the condition that
the chemical abundance is independent from the ionization stage of the elements
to be analysed. Iterations of an ionization equilibrium with the Balmer-lines
analysis result in a solution for (T
eff
, log g). For a simultaneous analysis of three
ionization stages there is a unique solution, independent from other parameters.
For only two ionization stages the solution depends on the microturbulence. On
the other hand, ξ can be derived by demanding the strong and weak lines to
provide the same value of chemical abundance of a given species. This can be
1
A frequency-dependent extinction of the radiation because of scattering by dust and
molecules in the interstellar medium.
Chapter 4: Spectroscopic Analysis 42
Figure 4.2: Effective temperature determination via Si ionization equilibrium for
the star D15 in the LMC cluster NGC2004 (Nieva et al. 2003). Si abundances
are denoted in logarithmic scale.
done for different elements, obtaining the same value of microturbulent velocity
in the optimum case. When the metallicity and the He abundance are not known,
they should also be determined from an iterative process.
In Fig. 4.2 an example of effective temperature determination from the
Si ii/iii/iv ionization equilibrium is shown (calculations are based on the Si model
of Becker & Butler 1990). For each T
eff
value, a corresponding log g was derived
from Hγ in LTE. The analysed star is LMC NGC2004 D15, and the final solu-
tion based on the Si ii/iii/iv ionization equilibrium, after some iterations on all
atmospheric parameters, give T
eff
= 23 000 K, log g= 3.86 dex and ξ =0 km s
−1
.
The microturbulence was simultaneously derived from Si and O lines.
Note that the final T
eff
is highly dependent on the value of ξ when only two
consecutive ionization stages are considered (as it is usual in practice). When
considering only Si ii/iii, a variation in microturbulence ∆ξ = +7 km s
−1
gives a
different solution for the temperature by ∆T
eff
=∼ −1 000 K, while for Si iii/iv,
∆ξ = +7 km s
−1
gives ∆T
eff
=∼ +1 200 K. The Si iii triplet centered in λ4568
˚
A,
chosen for the analysis, is strong in the studied star, and therefore sensitive to the
microturbulence. The Si ii λλ4128/30
˚
A doublet and the Si iv λ4116
˚
A line are
weak and almost insensitive to the microturbulence, consequently the tempera-
ture derived only from those ionization stages is almost insensitive to ξ (Nieva et
al. 2003).
Figure 4.3 shows the effect of the temperature (21 000 ≤ T
eff
≤ 32 000 K) on
some spectral lines of the stars analysed in the present work (for more details
about the observed spectra see Sect. 5.2). Some lines like from Si ii or Si iv can
be very sensitive to temperature, while Si iii lines remain strong in the selected
stars. Both, Si ii and Si iv are good indicators for T
eff
, as well as He ii for higher
temperatures. Lines of neutral helium change little with T
eff
. The strength of Hδ
increases for lower temperatures.
The success of a self-consistent analysis indicates that the modeled spectra
43 4.1 Atmospheric Parameters
Figure 4.3: Sensitivity of some spectral lines to temperature for the stars analysed
in the present work (τ Sco has the highest and HR 5285 the lowest temperature).
account for the necessary underlying physics to reproduce the observations and
that the derived atmospheric parameters are almost free of systematic errors.
Some inconsistencies like discrepant solutions of (T
eff
, log g) for different species
might be related to problems in the iteration procedure (some parameter is still
not converging to a unique value) or to shortcomings in the spectral line modelling
(e.g. incomplete model atoms for non-LTE line formation and/or inappropriate
input atomic data).
Metal lines, and consequently chemical abundances, can be highly sensitive
to atmospheric parameter variations and therefore the final accuracy can be im-
proved when a consistent spectroscopic analysis is performed. Therefore the spec-
troscopic approach can be much more accurate than the photometric methods.
It also allows us to avoid important systematic effects in fundamental param-
eters, but only when the spectral modelling is realistic. This approach can be
very lengthy, in particular when the starting points are chosen far from the final
solution.
In the last decades, (not well-understood) systematic discrepancies of effec-
tive temperatures derived from photometric and spectroscopic approaches were
reported in the literature. These discrepancies were also found in a previous anal-
ysis of early-B type main sequence and giant stars in the Magellanic Clouds. An
example of solutions for T
eff
following different approaches is shown in Table 4.1.
Discrepancies as large as 4 000 K or more are found for this sample stars. For
the star D15, a difference of temperature by −3 000 K (e.g. those derived from
Chapter 4: Spectroscopic Analysis 44
Table 4.1: Values of T
eff
for stars in the LMC cluster NGC2004. They are de-
rived photometrically from different sources and calibrations, and from a spec-
troscopic approach. For the Q and UVB calibrations the indices from Balona &
Jerzykiewicz (1993) are adopted.
Star T
C
eff
T
Ke
eff
T
D
eff
T
L
eff
T
N
eff
T
K
eff
D15 22000 ··· 19442 18148 19100 22500
C16 ··· 28050 ··· ··· ··· 24600
D3 ··· 26180 25575 24063 ··· 23900
C9 ··· ··· 18933 17377 19100 24000
B30 23000 23550 21344 20319 20251 23450
T
C
eff
from Caloi & Cassatella (1995): UV flux.
T
Ke
eff
from Keller et al. (2000): WFPC2 photometry.
T
D
eff
from calibration of Q-parameter (Daflon et al. 1999).
T
L
eff
from calibration of Q-parameter (Lyubimkov et al. 2002).
T
N
eff
from calibration of U-B index (Napiwotzki et al. 1993). E(B-V) from Balona (1993).
T
K
eff
from Korn et al. (2000, 2002): spectroscopic determination (Si, Becker & Butler 1990).
Daflon et al. 1999 and Korn et al. 2002), implies a subsequent correction for
log g of −0.3 dex from a line fit to Hγ (in LTE) and a higher microturbulence by
10 km s
−1
.
Those discrepancies in the fundamental atmospheric parameters are trans-
lated into systematic differences in the derived chemical abundances. For tem-
peratures derived from Daflon et al. (1999), the Si ionization equilibrium is not
achieved (for a visualisation of this see the variation of Si abundances in Fig. 4.2
for a lower T
eff
by ∼−3 000 K). Therefore, another aim of this work is to investi-
gate the discrepancies of the derived fundamental atmospheric parameters.
The above mentioned silicon model atom has some limitations like a rather
small number of levels treated explicitly in non-LTE, in particular for levels with
high excitation energy, compared to a new model atom by Przybilla & But-
ler (2007). In the present work, silicon is not analysed but instead a self-consistent
method to simultaneously derive the atmospheric parameters and select the op-
timum set of input atomic data for a carbon model atom is proposed. Effective
temperature and log g are derived in the present work, when possible, from si-
multaneous ionization equilibria of He i/ii and C ii/iii/iv and from the Balmer
lines in non-LTE.
45 4.2 Chemical Abundances
4.2 Chemical Abundances
The chemical abundance of a given species can also be derived by fitting synthetic
spectral lines to the observed spectrum
2
. The abundance can be determined only
when all other parameters are known (i.e. effective temperature, surface gravity,
microturbulence, macroturbulence, projected rotational velocity, metallicity).
In the case of fundamental atmospheric parameters derived from photometric
calibrations, the process is shortened and both ξ and (X) can be derived faster.
This procedure does not always allow to obtain ionization balance of the species
under consideration. In such cases it is difficult to identify the sources of errors,
i.e. the parameter calibration or the model atom.
With the spectroscopic approach all parameters and abundances of some
species can be simultaneously derived from multiple ionization equilibria. The
optimum solution for the whole set of atmospheric parameters allows chemical
abundances of the remaining species to be derived (see Chapter 6 for a detailed
description of chemical abundance and atmospheric parameter determination).
2
Another method is based on the equivalent widths of the line profiles and the curve of
growth, nevertheless this is not as accurate as line fitting, as line blends with other species may
be more easily overlooked.
Chapter 5
Hybrid Non-LTE Approach for H
and He Line Formation
Hydrogen and helium are of major interest in the astrophysical context, as they
constitute practically all light-emitting plasma. The lines of H and He are the
strongest spectral features in OB stars. Moreover, they are primary diagnostic
tools for stellar analyses throughout the Hertzsprung-Russell diagram. Because
of their strength they can sample the plasma conditions throughout large parts
of the stellar atmosphere, to a far greater extent than do the metal lines.
The information on the temperature and density structure encoded in the
spectra is interpreted by comparison with synthetic spectra, which requires that
the basic atmospheric structure equations in combination with the radiative
transfer problem be solved (as explained in Chapter 2). The model predictions
may differ, depending on the approximations made and on the atomic data used.
Their quality can be assessed by the ability to reproduce observation. In the
optimum case all observational constraints (continua/spectral energy distribu-
tion, line profiles) should be reproduced simultaneously, indicating the absence
of systematic error (assuming a unique solution). A thorough reproduction of the
hydrogen and helium line spectra should therefore be viewed as a precondition
for all further studies.
Non-LTE effects play a dominant rˆole in the formation of the hydrogen and
helium line spectra in early-type stars, as known since the seminal work by Auer
& Mihalas (1972, 1973). Despite the enormous progress made over the past
40 years, some notorious problems have remained. Observations in the near-
infrared provide one key to improve the situation via extension of the present
observational database to a domain of amplified non-LTE effects (in OB stars).
Some of the problems have been related recently to the remaining inaccuracies
in the atomic data. Thus, the modelling of the hydrogen Paschen, Brackett, and
Pfund series in early-type stars could be improved, resulting in corrections of
equivalent widths by as much as a factor 2–3 (Przybilla & Butler 2004; Repolust
et al. 2005). Also, the observed behaviour of the He i λ10 830
˚
A transition in OB
47
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 48
dwarfs could be reproduced for the first time (Przybilla 2005). In other cases, the
reasons for shortcomings in the non-LTE modelling can be subtler. An example
of this is the He i singlet line problem in early-type stars: computations with
non-LTE model-atmosphere codes reveal discrepancies not only between theory
and observation but also between different theoretical calculations. The overlap
of an He i resonance transition with Fe iv lines results in high sensitivity to the
model assumptions (Najarro et al. 2006).
In this chapter, the status of non-LTE line-formation computations is evalu-
ated for the two most abundant elements in the most common targets of massive
star analyses, OB dwarf and giant stars. This study constitutes the basis for
Chapter 6. The hybrid non-LTE approach (Sect. 5.1) is thoroughly tested on
high-quality spectra of six stars in the solar vicinity (Sects. 5.2 and 5.3). In con-
trast to typical studies from the literature, the entire hydrogen and helium line
spectra is investigated in the optical range, plus some additional near-IR data,
taking advantage of recently improved non-LTE model atoms. After making sure
that excellent agreement between theory and observation can be obtained (i.e.
also avoiding the aforementioned He i singlet line problem), this non-LTE mod-
elling is compared with libraries of synthetic fluxes from the literature (Sect. 5.4).
This is done in order to test their suitability for quantitative analyses of OB dwarfs
and giants. Such libraries are required for (automatised) analyses of large ob-
servational datasets obtained with existing or future multi-object spectrographs
(e.g. the VLT-FLAMES survey of massive stars: Evans et al. 2005; GAIA: Per-
ryman et al. 2001). It is shown that reliable modelling of the line spectra of the
two most basic elements is not straightforward. On the contrary, considerable
systematic errors may result for quantitative analyses of OB stars when applying
these libraries blindly. Finally, a summary of this chapter is provided in Sect. 5.5.
5.1 Model Calculations
The hybrid non-LTE approach solves the restricted non-LTE problem on the basis
of prescribed LTE model atmospheres. The approach is physically less elaborate
than fully self-consistent non-LTE calculations, as more approximations are in-
volved. However, at the same time it is superior to the pure LTE approximation.
In particular, it provides an efficient way to compute realistic synthetic spectra in
all cases where the atmospheric structure is close to LTE (which puts restrictions
on the parameter space coverage). The hybrid non-LTE approach also allows
extensive non-LTE model atoms to be implemented, facilitating a highly detailed
treatment of the atomic processes involved (e.g. account for the resolved reso-
nance structure in photoionizations, avoidance of the – powerful, however also
approximate – superlevel formalism).
As described in Chapter 2, line-blanketed, plane-parallel, homogeneous, and
hydrostatic LTE model atmospheres are computed using the atlas9 code (Ku-
49 5.1 Model Calculations
rucz 1993b). Non-LTE population numbers and synthetic spectra are then ob-
tained with recent versions of detail and surface (Giddings 1981; Butler &
Giddings 1985). The coupled radiative transfer and statistical equilibrium equa-
tions are solved with detail, employing the Accelerated Lambda Iteration (ALI)
scheme of Rybicki & Hummer (1991). Synthetic spectra are calculated with sur-
face, using refined line-broadening theories.
The non-LTE model atoms for hydrogen and He i/ii adopted in the present
work are described in detail by Przybilla & Butler (2004) and Przybilla (2005),
respectively. Use of improved atomic data for electron impact excitations, in par-
ticular from ab-initio computations, allows consistent results from the hydrogen
lines in the visual and near-IR to be derived throughout the entire range of early-
A to O stars . A 15-level model is used for modelling main sequence stars, as
well as a 20-level model for the giants. The He i/ii model atom considers all He i
LS-coupled terms up to the principal quantum number n = 5 individually, and
packed levels up to n = 8 separately for the singlet and triplet spin system. All
levels up to n = 20 are considered in the He ii model. This model atom has been
successfully used to reproduce observed trends of the highly non-LTE-sensitive
He i λ10 830
˚
A transition in early-type main sequence stars (Przybilla 2005). The
He i/ii model was also employed to analyse the visual/near-IR spectra of ex-
treme helium stars (Przybilla et al. 2005) and subluminous B stars (Przybilla et
al. 2006b).
Radiative transitions in detail are treated with simplified line broadening for-
malisms: for transitions between hydrogen levels with n ≤ 7 approximate Stark-
broadening (Griem 1960, following the implementation of Auer & Mihalas 1972,
Appendix) is considered, while for all other transitions, also in He i/ii, depth-
dependent Doppler profiles are assumed. Microturbulence is explicitly accounted
for by including the appropriate term in the Doppler width (see Eqn. 2.37). Both
continuous absorption and line blocking (via LTE Kurucz’ opacity distribution
functions, ODFs, Kurucz 1993a, using the ‘little’ wavelength interval versions),
are accounted for as background opacities in solving the radiation transfer. In
this regard the hybrid non-LTE approach has an advantage over present-day ‘ex-
act’ non-LTE computations: all species responsible for metal line blocking and
blanketing can be considered, though approximately. The ‘exact’ non-LTE meth-
ods, on the other hand, are constrained to several abundant light elements and
typically iron (and sometimes nickel), i.e. only the major line opacity sources are
covered. Note that these ODFs were calculated for solar abundances according
to Anders & Grevesse (1989). The latter have been revised in more recent work
such as Grevesse & Sauval (1998). In particular the reduction of the abundance of
iron (the most important line opacity source) by ∼0.2 dex should be considered.
This is done in this work by adopting the Kurucz (1993a) ODFs for appropriately
reduced metallicity; see Sect. 5.4.1 for further discussion.
The resulting non-LTE populations are then used to compute realistic line
profiles with surface. The same microturbulent velocity as in detail and in
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 50
Figure 5.1: Sensitivity of theoretical He i line profiles to modifications of the
microturbulent velocity in the statistical equilibrium calculations. The microtur-
bulence of the atlas9 model atmosphere structure is held fixed at ξ = 8 km s
−1
.
The test calculations have been done for one of our sample stars (HR 3055). Simi-
lar effects are found for other He i lines, while the He ii and H lines are practically
insensitive to this.
the model structure computations with atlas9 is adopted. In this step of the
calculation detailed Stark-broadening data are employed, as summarised in Ta-
ble B.1. All other important data relevant to line formation are also given there:
wavelengths, lower and upper levels involved in the transition, oscillator strengths
log gf, their accuracies, and sources.
Note that, in typical non-LTE computations for OB stars, microturbulence
is only included in the final profile calculation. The present choice is based on
test calculations that indicate line-profile fits are improved if microturbulence
is also included in computing the level populations. The net effect is a slight
strengthening of the lines (Fig. 5.1). However, the effect is far less pronounced
than described by McErlean et al. (1998), who investigated unblanketed non-
LTE models for B-type supergiants at slightly higher microturbulence; cf. their
Figs. 3 & 4.
The hybrid non-LTE approach involving the codes atlas9, detail, and sur-
face (henceforth abbreviated ADS) is tested here for early B-type dwarfs and gi-
ants, supplemented by LTE computations with atlas9 and surface (AS). This
methodology may be applied to a wider range of atmospheric parameters (i.e.
T
eff
and log g). Line-blanketed, static, and plane-parallel LTE models provide
an even more realistic description of stellar atmospheres at lower temperatures
51 5.2 Observational Data
and higher gravities (excited He ii states should be ignored at lower T
eff
in order
to avoid numerical inconsistencies). Slightly higher temperatures (late O-types)
and lower surface gravities (less-luminous supergiants) may also be covered, un-
til the hybrid non-LTE approach meets its limitations when non-LTE effects on
the atmospheric structure and/or hydrodynamic mass-outflow may no longer be
neglected.
5.2 Observational Data
The proposed analysis technique is tested on six bright Galactic objects in the
entire optical range and also for near-IR lines when available. The programme
stars sample the parameter space in effective temperature and surface gravity
covered by typical applications.
High signal-to-noise (S/N)
´
Echelle spectra of τ Sco (HR 6165), HR 3055,
HR 1861, HR 2928, HR 3468, and HR 5285 were obtained by M. Altmann using
FEROS (Fiberfed Extended Range Optical Spectrograph, Kaufer et al. 1999) on
the ESO 2.2m telescope in La Silla. The data reduction (Appendix A) was per-
formed within the FEROS context in the ESO MIDAS package, using optimum
extraction. The spectra were normalised by fitting a spline function to carefully
selected continuum points. This suffices to retain the line profiles of the Balmer
lines in these early-type stars. Finally, the spectra were brought to the wavelength
rest frame by cross-correlation with an appropriate synthetic spectrum. Of the
entire wavelength range covered by FEROS, only the part between ∼3800 and
8000
˚
A meets the quality criteria of this work for further analysis. The spectra are
compromised by the lower sensitivity of the instrument at shorter wavelengths,
and the reduced stellar fluxes in the far red. FEROS provides a resolving power
R λ/∆λ ≈48 000, with 2.2 pixels per ∆λ resolution element. An S/N of up to
∼800 is achieved in B. With respect to resolution and signal-to-noise ratio, the
spectra available to us are of much higher quality than in typical studies of OB
stars, basically excluding observational uncertainties from the error budget.
A supplementary high-S/N spectrum of HR 1861, also covering the higher
Paschen series, was obtained using FOCES (Fibre Optics Cassegrain Echelle
Spectrograph, Pfeiffer et al. 1998) on the Calar Alto 2.2m telescope. The data
were processed in a standard way, using the data reduction routines described
by Pfeiffer et al. (1998). An R 40 000 (2 pixels per ∆λ resolution element) was
achieved. In addition, a high-S/N spectrum in the K-band of τ Sco taken with
Subaru/IRCS (R 12 000) is available for analysis; see Hanson et al. (2005) for
details on the observations and data reduction. Finally, a high-S/N spectrum
in the λ 2.058 µm region of τ Sco taken with UKIRT/CGS4 (R 19 000) is also
available for analysis (Zaal et al. 1999). Additional IUE fluxes and Johnson and
2MASS magnitudes are also employed for constraining the atmospheric parame-
ters, in particular the effective temperature of the programme stars.
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 52
eff
Stellar atmosphere
Synthetic H, He profiles
H, He I/II lines
all measurable
ξ, ζ, ε( )
, vsin i
M = I ?
no
yes
Final T , log g,
eff
ξ, ζ, ε( )
, vsin i
Initial T , log g,
ξ, ζ, ε( )
, vsin i
He
He
He
Comparison with
observed spectra
I
NLTE H, He populations
M
Modified T , log g,
eff
Parameter verification
Figure 5.2: Scheme of the iterative procedure for obtaining a simultaneous fit to
the hydrogen and helium lines.
5.3 Applications to Observations
Theoretical profiles were fitted to observations in an iterative procedure sum-
marised in Fig. 5.2. The final atmospheric parameters T
eff
and log g, projected
rotational velocities v sin i, and micro- and macroturbulent velocities (ξ, ζ) coin-
cide with those derived from C ii/iii and C ii/iii/iv ionization equilibria (Chap-
ter 6). The parameters are summarised in Table 5.1. They are are also able
to reproduce the IUE fluxes and Johnson and 2MASS magnitudes, displayed in
Fig. 5.3. The observed fluxes are dereddened by the values indicated using a
reddening law according to Howarth (1983) and Seaton (1979) assuming a ratio
R
V
= A
V
/E(B − V ) = 3.1 as typical for the local ISM. They were degraded to
the resolution of the atlas9 model fluxes. The models are normalised to the
observed Johnson V magnitudes and shifted for clarity relative to each other.
The impact of stellar parameter variations on non-LTE profile fits to Hγ,
He i λ4026
˚
A, and He ii λ5411
˚
A in the hot giant HR 3055 is visualised in Fig. 5.4.
Two values for the parameter variations are adopted, according to the uncer-
tainties of 300 K/0.05 dex in T
eff
/log g and typical values from the more recent
literature (1000 K/0.10 dex). All other hydrogen Balmer and helium lines react
in a similar way. The sensitivity of the hydrogen and He i lines to parameter
variations is low, such that the uncertainties cannot be reduced much below the
53 5.3 Applications to Observations
Table 5.1: Atmospheric parameters of the programme stars
τ Sco HR 3055 HR 1861 HR 2928 HR 5285 HR 3468
T
eff
(K) 32 000 31 200 27 000 26 300 21 500 22 900
± 300 300 400 400 400 400
log g (cgs) 4.30 3.95 4.12 4.15 4.20 3.60
± 0.05 0.05 0.05 0.05 0.05 0.05
ξ (km s
−1
) 5 8 3 3 4 5
± 1 1 1 1 1 1
v sin i (km s
−1
) 4 29 12 14 18 11
± 2 5 1 1 1 3
ζ (km s
−1
) 4 37 ··· 20 ··· 20
± 2 1 ··· 2 ··· 2
y (by number) 0.089 0.080 0.089 0.089 0.089 0.089
± 0.01 0.01 0.01 0.01 0.01 0.01
typical values even for high-S/N observations like the ones presented here. Only
the He ii lines are highly sensitive to changes in T
eff
and, to a lesser degree, in log g.
However, by taking metal ionization equilibria into consideration (e.g. C ii/iii or
C ii/iii/iv), which are even more sensitive than He ii lines, the parameters can
be constrained more accurately.
Projected rotational velocities, micro- and macroturbulence values have also
been verified by fitting the carbon lines. Note that the comparatively high macro-
turbulence in HR 3055 amounts to less than twice the sound speed in the atmo-
spheric plasma. The microturbulence values are typically lower than found in
previous work (e.g. Kilian 1992). Differences in T
eff
and log g are also found.
Solar helium abundances y (by number) are found in all cases.
5.3.1 Visual
Synthetic profiles for a selection of 6 hydrogen Balmer and 18 He i/ii lines in the
visual are compared with observation for the sample stars in Figs. 5.5 and C.1-C.5
(in Appendix C). They constitute the best simultaneous fits to the measurable
H and He lines in the available spectra achieved in this work. Preference for this
selection has been given to (mostly) unblended features with good broadening
data. A summary of all available lines is given in Table B.1, where blending
species are also identified. The hybrid non-LTE approach (ADS) allows us to
reproduce the hydrogen Balmer and He i/ii lines in the visual more precisely, with
few (well-understood) exceptions. The ionization equilibrium of He i/ii puts tight
constraints on T
eff
in the two hotter stars. For the hottest dwarf of the sample,
τ Sco, a very good match between model and observation is achieved (Fig. 5.5),
except for the cores of Hα and He ii λ4686
˚
A. This is because of the neglect of
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 54
Figure 5.3: Best fits of theoretical energy distributions to measurements by IUE
(dotted lines) and Johnson and near-infrared 2MASS photometry (diamonds)
from the UV to the near-IR. For atmospheric parameters see with Table 5.1.
Figure 5.4: Impact of stellar parameter variations on non-LTE profile fits, ex-
emplarily for Hγ, He i λ4026
˚
A, and He ii λ5411
˚
A in HR 3055 (B0 III). Synthetic
spectra for the final parameters (see Table 5.1, thick line) and varied parame-
ters (thin lines for the uncertainty estimates and dashed lines for values typically
found in the literature) are compared to observation.
55 5.3 Applications to Observations
the stellar wind; see Przybilla & Butler (2004) and Mokiem et al. (2005) for
results of hydrodynamic computations. The discrepancies in He i λ4121
˚
A occur
because of blends with metal lines (O ii, C iii, and Fe iii; unaccounted for in
the present computations), which can be nicely resolved at this low v sin i. An
improved fit to He i λ4921
˚
A may be obtained with better broadening data for
the forbidden component.The spectral region around Hα suffers from artifacts
introduced by CCD defects that can only partially be compensated in the data
reduction process.
Line fits to the hot giant HR 3055 are displayed in Fig. C.1. Excellent agree-
ment between theory and observation is also found in this case, with a signifi-
cantly improved fit quality of Hα and He ii λ4686
˚
A, because of an apparently
weaker wind. This star shows a higher v sin i and ζ than τ Sco. Therefore the
He i λ4121
˚
A blend is no longer resolved, leading to an apparently worse fit.
He ii λ4686
˚
A is the only visible (weak) feature of He ii in the intermediate
temperature stars HR 1861 and HR 2928; see Figs. C.2 and C.3. Good fits are
obtained for this line and the features of He i (i.e. establishing the ionization
equilibrium) and hydrogen. Lines of He ii are absent at even lower temperatures,
HR 3468 and HR 5285 in the sample; see Figs. C.4 and C.5.
5.3.2 Near-IR
Additional spectra are available in the near-IR for two stars. An excellent fit to
the higher Paschen series is obtained for HR 1861 in non-LTE, despite the rela-
tively low S/N and the presence of telluric lines
1
, see Fig. 5.6. Good agreement
between the non-LTE spectrum synthesis and observation can also be obtained
for the He i λ 2.058 µm feature and practically perfect agreement for the λ2.11 µm
lines in τ Sco, see Figs. 5.7 and 5.8. Note that the λ 2.058 µm transition is situated
in an atmospheric window with a series of strong telluric lines, therefore its shape
is highly sensitive to the detailed approach in the data reduction process (see e.g.
Zaal et al. 1999). The two different observations in Fig. 5.7 may exemplify the
difficulty of accurate telluric line removal in the data reduction process (or alter-
natively an intrinsic time variability of the feature). Good agreement with the
synthetic spectra is obtained in the case of the higher-resolution UKIRT/CGS4
spectrum. The LTE approach is not even able to reproduce the observation
qualitatively. The line fits in the near-IR are based on the same atmospheric
parameters (Table 5.1) as used for the modelling of the optical spectra.
Non-LTE effects can be amplified in the Rayleigh-Jeans part of the spectral en-
ergy distribution, as demonstrated in this case. See e.g. Przybilla & Butler (2004)
for a discussion and for line fits to Brγ in the K-band and to additional Brackett
and Pfund lines in this star. The case of He i λ10 830
˚
A has been discussed by
Przybilla (2005).
1
Lines or bands in the spectrum of an astronomical object that are due to absorption by
gases such as molecular oxygen, water vapour, or carbon dioxide in Earth’s atmosphere.
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 56
Figure 5.5: Non-LTE line fits to observed hydrogen and helium features in τ Sco
(B0.2 V), based on the atmospheric parameters summarised in Table 5.1.
57 5.3 Applications to Observations
Figure 5.6: Modelling of the Paschen series of HR 1861 with the non-LTE (ADS)
approach. Note the presence of numerous sharp telluric H
2
O lines. All theoretical
spectra in the near-IR have been computed with the same atmospheric parameters
than the models in the visual.
Figure 5.7: Modelling of the He i λ2.058 µm singlet line in τ Sco.
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 58
Figure 5.8: Modelling of the He i λ2.11 µm singlet and triplet feature in τ Sco.
As also shown in Fig. 5.7, these near-IR transitions experience stronger non-LTE
effects than the spectral lines in the visual.
5.4 Comparison to Other Model Predictions
In this section, the hybrid non-LTE computations (ADS) are compared with four
other approaches. Two of them are available grids from the literature, and to
understand their discrepancies to the ADS computations, additional non-LTE
and LTE models are calculated.
i) The atlas9 models are replaced by line-blanketed, plane-parallel, and
hydrostatic non-LTE model atmospheres taken from the publically available os-
tar2002 grid (Lanz & Hubeny 2003, LH03) and the non-LTE line-formation
calculation is performed with detail and surface as described above.
ii) Non-LTE calculations provided in the ostar2002 grid (tlusty and non-
LTE line formation with synspec) and published by LH03 are taken.
iii) LTE spectra based on atlas9 atmospheres and subsequent LTE spec-
trum synthesis with surface (AS) are computed.
iv) The Padova grid (Munari et al. 2005) is considered. These computations
are based on atlas9 atmospheres and LTE spectrum synthesis performed with
the synthe code (Kurucz & Avrett 1981; Kurucz 1993c).
59 5.4 Comparison to Other Model Predictions
Figure 5.9: Upper panel: Comparison of atlas9 and tlusty temperature struc-
tures and electron densities (insets) as function of column mass. The computa-
tions have been performed for giant and dwarf models. Lower panel: Comparison
of spectral energy distributions, the radiation field computed by detail on the
basis of the atlas9 atmospheric structure vs. tlusty.
5.4.1 Atmospheric Structures, SEDs: LTE vs. non-LTE
A comparison of LTE (atlas9) and non-LTE (tlusty) atmospheric structures
and of spectral energy distributions (SEDs) computed with atlas9+detail and
tlusty is made in Fig. 5.9. Models for a hot giant and a dwarf (T
eff
= 32 500 K,
log g = 3.75 and 4.25, respectively) are considered, approximately delineating the
upper temperature boundary of the observations (τ Sco). Reduced non-LTE ef-
fects can be expected for cooler models.
Excellent agreement is found for the temperature and density structures. This
is a basic requirement for successful application of the hybrid non-LTE approach
for spectrum synthesis. The temperature structures deviate by less than 1% in
the inner atmosphere, including the regions where the weaker lines and the wings
of the stronger features are formed (log m −1; see Sect. 5.4.3). At the for-
mation depths of the cores of the stronger H and He lines (−3 log m −1.5)
the differences may increase to 2–3%. Stronger deviations may occur only in
the outermost parts of the atmosphere, outside the line-formation depths. Note
that this good a match is obtained only if the effects of metal line-blanketing
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 60
are correctly accounted for. In particular, the computations should be made for
identical metal abundances. This is complicated by the fact that the ODFs of
Kurucz (1993a) were computed assuming scaled solar abundances from Anders
& Grevesse (1989), while the tlusty computations assume abundances from
Grevesse & Sauval (1998). The most important difference is a downward revision
of the iron abundance by ∼0.2 dex in the later work. Consequently, ODFs with
correspondingly ‘sub-solar’ metallicity are used in order to correct for the dis-
crepancies in the line opacities, with [Fe/H] as a metallicity substitute. See also
Przybilla et al. (2006a) for a discussion of such ‘empirical’ corrections. We should
note that, while the differences are small at (near-)solar abundances, the non-
LTE effects on the atmospheric structure will increase with decreasing metallicity.
Nevertheless, this hybrid non-LTE methodology for OB star analyses should be
applicable down to SMC metallicity, as indicated by an analogous comparison for
models at 1/5 ×solar abundances.
The SEDs computed with atlas9+detail and tlusty show excellent agree-
ment over almost the entire wavelength range. Small differences occur in the
EUV, most notably in the He ii continuum. This is a significant improvement
over the comparison of tlusty with atlas9 model fluxes (not shown here), which
predict much lower fluxes in the Lyman and helium continua. The LTE computa-
tions neglect the non-LTE overionization of the hydrogen and He i ground states.
This overionization reduces the bound-free continuum opacity; see Sect. 5.4.3 for
a more comprehensive discussion.
5.4.2 Spectra: Hybrid non-LTE vs. non-LTE and LTE
Comparisons of synthetic profiles of several strategic lines of hydrogen and He i/ii
are made for three test cases, where models are available from the published grids.
These frame the parameter space studied in the present paper. The test cases
comprise a hot dwarf model (T
eff
= 35 000 K, log g = 4.5), at slightly higher T
eff
than covered by the observations
2
, shown in Fig. 5.11; a hot giant model (32 500,
3.75), with similar temperature to τ Sco, see Fig. 5.12; a cool giant model (20 000,
3.00), with both T
eff
and log g lower than covered by the observations, Fig. 5.13.
In order to be consistent with the published grids, the present computations con-
sider solar metal abundances (Grevesse & Sauval 1998) and solar helium abun-
dance. Our synthetic spectra and those of the ostar2002 grid are degraded to
the highest resolution (R = 20 000) available from the Padova grid. Note that
metal lines are neglected when the emergent spectrum is calculated, but they are
considered indirectly via line blanketing/blocking effects.
i) ADS vs. tlusty-DS. This comparison allows effects caused by differences in
the model atmosphere structures to be disentangled. A practically perfect match
2
This approach is expected to be valid at slightly hotter temperatures, at least in main
sequence stars where the stellar wind does not influence the photospheric layers strongly.
61 5.4 Comparison to Other Model Predictions
Figure 5.10: Comparison of the most discrepant hydrogen and He i/ii line profiles
from the hybrid non-LTE approach (ADS) and a tlusty-DS calculation for a
hot main-sequence model. Practically perfect agreement is obtained, with small
discrepancies occurring only in the wings of He ii λ4686
˚
A.
for the (35 000,4.5) model (see Fig. 5.10) is obtained in the ADS and tlusty-DS
computations, which share the same model atom. This indicates good agreement
of the LTE and non-LTE atmospheric structures at even slightly higher temper-
atures than discussed in Fig. 5.9. The discrepancies are even smaller at lower
temperatures.
ii) ADS vs. tlusty+synspec (LH03). These results are obtained using
two independent methods (model atmospheres, model atoms, numerical solu-
tion). Nonetheless, good agreement is found on the whole for the (35 000,4.5)
and (32 500,3.75) models from an inspection of Figs. 5.11 and 5.12, respectively.
Notable differences between ADS and tlusty+synspec occur in the line cores
of the Balmer lines (the latter filled in by emission) and in the He i singlet lines,
which are systematically weaker in the case of tlusty+synspec, in contrast
to observation (see Figs. 5.5 and C.1). The He i triplet lines derived from both
approaches agree well. Small discrepancies occur in the line wings of the He i
lines because of different broadening data. A good match is also obtained for the
He ii lines, with small differences arising in He ii λ4686
˚
A.
From the comparison of the tlusty+synspec and tlusty-DS results, which
match ADS, we can conclude that the aforementioned discrepancies arise because
of subtle differences in solving the statistical equilibrium and radiative trans-
fer problem. While the present approach uses LTE line opacities averaged over
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 62
the ODF wavelength bins, the tlusty+synspec computations employ a more
sophisticated non-LTE opacity sampling technique. This, however, introduces
a strong dependency on the model assumptions for Fe iv
3
(a highly complex
ion), which has lines overlapping with an He i resonance transition (Najarro et
al. 2006)
4
. The same model atmosphere (tlusty) is used and both model atoms
should be sufficiently robust for modelling the lines in the visual; see Przybilla &
Butler (2004) and Przybilla (2005) for a discussion of this.
iii) ADS vs. atlas9+surface. LTE computations with AS produce nar-
rower Balmer lines for the (35 000,4.5) and (32 500,3.75) models (the differences
reducing progressively from Hα to the higher series members), which leads to
overestimated surface gravities in that case. At the same time, all He i lines are
too shallow in LTE, the trend increasing from blue to red and showing larger dis-
crepancies at lower gravity. On the other hand, rather good agreement is found
for the He ii lines, the LTE predictions being slightly weaker than ADS for the
hot giant. In Fig. 5.13 a comparison of the hybrid non-LTE with our pure LTE
prediction is made for a (20 000,3.0) model with a temperature slightly below
than the lower limit of the programme stars, and at significantly reduced surface
gravity. Here, the wings of the Balmer lines show much better agreement than
at higher temperatures (cf. Fig. 5.11), as well as the He i λλ 4437 and 4713
˚
A
lines. The line cores are also discrepant, increasingly so from Hδ to Hα. Many of
the He i lines experience significant non-LTE strengthening, in particular those
in the red. The line broadening data is the same in ADS and AS, so the He i
wings are very similar. The forbidden components are also accounted for in both
approaches.
iv) ADS vs. atlas9+synthe (Munari et al. 2005). The differences of these
approaches were quantified for the (35 000,4.5) model, when possible. The Balmer
lines from the Padova model present similar characteristics as the LTE AS ap-
proach (in Fig. 5.11 they coincide), resulting in lower equivalent widths by up to
∼30% relative to ADS. When using the Hγ wings as a surface-gravity indicator,
this translates to a systematic error in log g by ∼0.2 dex, implying even larger
errors for fits to the Hβ and Hα wings. The He i lines are generally too weak, by
up to a factor of more than 2 in equivalent width, and the He ii lines too narrow.
For the most part, these discrepancies stem from the neglect of non-LTE effects
on the line-formation process, as the differences in the atmospheric structures are
practically insignificant.
3
In particular on the oscillator strengths of the Fe iv transitions involved. A reduction of
the gf-values may alleviate the discrepancy between the He i singlet and triplet line strengths,
seen, for example, in the ostar2002 grid.
4
As a further test we have calculated synthetic spectra for the (35 000,4.5) and (32 500,3.75)
models on the basis of unified, line-blanketed non-LTE model atmospheres (FASTWIND, Puls
et al. 2005). Excellent agreement with ADS results is found for both the He i singlet and triplet
lines. Note that FASTWIND also uses an approximate treatment of line blocking.
63 5.4 Comparison to Other Model Predictions
Figure 5.11: Comparison of selected H and He i/ii line profiles from the hybrid
non-LTE approach (ADS), non-LTE computations from tlusty+synspec, and
two LTE calculations (atlas9+surface and atlas9+synthe) for a hot main-
sequence model.
Another limiting factor of the atlas9+synthe computations is the use of
insufficient Stark broadening data (Voigt profile with constant Stark damping pa-
rameter). The AS and ADS approaches improve on this, as realistic broadening
data is used (see Table B.1). In the atlas9+synthe approach, it will not be
possible to obtain reasonable agreement for the He i and He ii spectra at the same
time. For the (20 000,3.0) model, the Padova profiles are more similar to the AS
approach. However, the diffuse He i lines still suffer from inappropriate broad-
ening data, in particular the forbidden components are unaccounted for. The
He i lines are affected by non-LTE strengthening, increasing to the red. Only few
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 64
Figure 5.12: Comparison of selected H and He i/ii line profiles from the hybrid
non-LTE approach (ADS), the non-LTE computations with tlusty+synspec
and the LTE approach atlas9+surface for a hot giant model. Here, HR 3055
may act as observational discriminator, indicating our results to be appropriate
(see Fig. C.1).
He i lines are quite similar in the three approaches: λλ4437 and 4713
˚
A match
quite well, as do λλ3867, 4121 (despite blends with metallic lines) and 5015 and
5047
˚
A, not displayed here.
The published libraries of synthetic spectra were computed with different
values of microturbulent velocity (ostar2002: 10 km s
−1
; Padova: 2 km s
−1
).
The ADS and tlusty-DS calculations with ξ = 10 km s
−1
were adopted for the
comparison in Fig. 5.11. Tests with a reduced ξ = 2 km s
−1
were made, resulting in
only small changes in the He ii profiles – the most sensitive to modifications of ξ.
65 5.4 Comparison to Other Model Predictions
Figure 5.13: H and He i profiles for a cool giant model: the hybrid non-LTE
approach (ADS) vs. the LTE (atlas9+surface) and the corresponding Padova
model (atlas9+synthe); see Fig. C.4 for the closest observational analogue.
The differences between the Padova grid and the other approaches are indeed due
to the neglect of non-LTE effects and to additionally insufficient broadening data
and not because of discrepant microturbulences. The ADS and AS computations
in Figs. 5.12 and 5.13 were performed using the ξ of the respective libraries.
5.4.3 Line Formation: Hybrid non-LTE vs. Full non-LTE
The physical reasons for the differences in the non-LTE line profiles of hydrogen
and helium in the last comparisons are studied by a closer study of the under-
lying line-formation processes. For this, non-LTE departure coefficients and line
source functions are investigated for three representative hydrogen and six He i/ii
lines, as derived in atlas9+detail and the tlusty computations. The same
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 66
models as discussed in Fig. 5.9 are chosen. For the (32 500, 3.75) model, a direct
comparison with the resulting line profiles is facilitated by inspection of Fig. 5.12.
For the levels involved in the transitions of interest and the hydrogen and
helium ground states, departure coefficients b
i
(referred to the ground state of
the next higher ion) are displayed in Fig. 5.14. The non-LTE effects on the
level occupations give rise to departures of the line source function S
l
from the
Planck function B
ν
; see Fig. 5.15 for a comparison of S
l
/B
ν
(see Eqn. 2.43) from
the atlas9+detail and tlusty computations. For a given T
eff
the non-LTE
effects are strengthened with decreasing surface gravity, implying lower particle
densities and thus larger mean-free-paths between photon absorptions.
Hydrogen. Three hydrogen lines are studied, Hα, Hβ and Hγ. Departure co-
efficients for levels n > 5 behave similarly to those for n = 5, which already traces
the behaviour of the continuum closely. Consequently, the line source functions
for the higher Balmer lines are similar to that of Hγ. The non-LTE depopu-
lation of the H ground state (Fig. 5.14) reduces the Lyman continuum opacity,
giving rise to higher EUV fluxes than in LTE. The Lyman lines are expected
to experience non-LTE weakening. Note that the tlusty calculations indicate
a slightly stronger non-LTE depopulation of the ground state at continuum for-
mation depths than in our case. Good agreement of the departure coefficients
for n = 2 is found, which is overpopulated at line formation depths. In combi-
nation with the higher H states being close to LTE, this explains the non-LTE
strengthening of the Balmer lines. At the formation depths of the line cores, in
particular for Hα, the tlusty results show a less pronounced overpopulation,
eventually leading to an underpopulation of the n = 2 state in the outer atmo-
sphere. This explains the shallower lines from the tlusty computations relative
to ADS, which is a consequence of the upturn of S
l
/B
ν
(Fig. 5.15). The effect
decreases from Hα to the higher Balmer lines, as the core formation depths shift
to deeper atmospheric layers.
He i Singlets. Two representative features are investigated: λλ4921 and
6678
˚
A. In general, the departure coefficients for most of the levels show rather
good agreement, in qualitative behaviour as well as quantitatively. Notable dif-
ferences in the ground-state overionization occur in the outer atmosphere. More
relevant is the behaviour of the 2p
1
P
o
level, the lower level of practically all He i
singlet transitions in the visual. The non-LTE overpopulation in the tlusty
results is far less-pronounced than in our case (Fig. 5.14), in particular for the
giant model. This gives rise to discrepant line source functions (Fig. 5.15) and
consequently differing line profiles (Fig. 5.12) in the two approaches, with the
ADS computations correctly predicting the non-LTE strengthening and thus re-
producing observation (Sect. 5.3). The states at higher excitation energies (n ≥4)
are in LTE relative to the He ii ground state at line-formation depths.
67 5.4 Comparison to Other Model Predictions
Figure 5.14: Departure coefficients b
i
of some strategic hydrogen and helium
levels as a function of Rosseland optical depth τ
Ross
. The comparison is made for
the giant (left) and dwarf atmospheric models (right column) already discussed
in Fig. 5.9, for the hybrid non-LTE approach (atlas9+detail, thick lines) and
the results of LH03 (tlusty, thin lines). Each level is coded by different line
styles; see the legend in the corresponding panels. Line-formation regions (from
core to wing) corresponding to our calculations are indicated. See the text for
further discussion.
He i Triplets. Two representative lines are studied: λλ4471 and 5875
˚
A.
The departure coefficients from the two approaches differ only slightly at line-
formation depths (Fig. 5.14). As a consequence, the source functions (Fig. 5.15)
are also similar, resulting in negligible differences of the line profiles (Fig. 5.12).
Again, the states with n ≥4 are in detailed balance relative to the He ii ground
state at depths relevant for the line formation.
He ii. Two features are analysed: λλ4541 and 4686
˚
A. The higher He ii levels
are close to LTE with the continuum state at line-formation depths. Only the
Chapter 5: Hybrid Non-LTE Approach for H and He Line Formation 68
Figure 5.15: Ratio of line source function S
l
to Planck function B
ν
at line centre
as a function of τ
Ross
for selected spectral lines of hydrogen and helium. The
comparison is made in analogy to Fig. 5.14, for our approach (atlas9+detail,
thick lines) and LH03 (tlusty, thin lines). The individual spectral lines are
encoded by the different line styles indicating the line-formation depths. See the
text for further discussion.
n = 3 level shows a relevant overpopulation, resulting in a non-LTE strengthening
of the λ4686
˚
A line. The differences in the ADS and tlusty departure coefficients
give slightly shallower profiles for this line in the ostar2002 model (Fig. 5.12).
The source function for λ 4541
˚
A is essentially Planckian in the relevant region.
5.5 Summary
The suitability of the hybrid non-LTE line-formation computations was studied
for quantitative analyses of the hydrogen and helium line spectra of OB dwarf
and giant stars. These computations simultaneously reproduce the line spectra
69 5.5 Summary
throughout the visual and near-IR (where available) at high quality, as well as
the measured spectral energy distributions from the UV to near-IR. The only
exceptions in the observational sample are the cores of Hα and He ii λ4686
˚
A
in τ Sco, because the calculations do not account for stellar winds. For two
He i lines blueward of the traditionally analysed spectral region (∼4000–5000
˚
A),
appropriate Stark broadening data is unavailable at present (see Table B.1).
The comparison of state-of-the-art line-blanketed non-LTE and LTE models
confirms that the atmospheric structure of OB dwarf and giant stars is described
well under the assumption of LTE, but not their spectral energy distribution and
also not their line spectra. For these stars in the range 20 000 K ≤T
eff
≤35 000 K
and 3.0 ≤log g ≤4.5 (far from the Eddington limit), the hybrid non-LTE approach
is equivalent to full hydrostatic non-LTE computations (as partially covered by
the ostar2002 grid, Lanz & Hubeny 2003). It succeeds also in providing syn-
thetic spectra that correctly reproduce the observed He i singlet lines, avoiding
inconsistencies recently reported in the literature.
Computations in LTE from the Padova grid, on the other hand, systematically
predict too shallow and/or too narrow line profiles. In particular, the differences
in the Hγ wings – a common surface gravity indicator – result in systematically
overestimated gravities by up to ∼0.2 dex in LTE (for fixed T
eff
). The differences
in the equivalent widths of the H lines can amount to up to ∼30% and in the
He i/ii lines up to a factor >2 compared to our non-LTE calculations, with the
discrepancies increasing with effective temperature. Nevertheless it is not possible
to quantify the differences in effective temperature determinations from non-LTE
and LTE ionization equilibria of He i/ii, as some of the profiles of the Padova
grid do not reproduce observations even qualitatively.
In terms of parameter range and the underlying physics, the hybrid non-LTE
approach is certainly restricted. It may be of limited use at higher temperatures
(early and mid-O-type stars), lower gravities (early B-type and O-type super-
giants), stars with strong winds, or extremely low metallicities. Nevertheless,
the hybrid non-LTE approach is sufficient for studying normal OB dwarfs and
giants, as it allows the observed line spectra to be reproduced in the visual and
near-IR over a wide range of atmospheric parameters. Here it has advantages
over other more sophisticated non-LTE techniques: I) it allows highly robust and
detailed model atoms to be implemented and to be tested efficiently (i.e. concen-
tration on atomic data while avoiding further complications like stellar winds),
e.g. for metals with hundreds of levels and thousands of transitions; and II)
the model calculations are fast: the computation of one H & He i/ii model with
detail+surface takes only a few minutes on a modern PC (as of 2006).
Chapter 6
Non-LTE Line Formation for
Carbon: Self-Consistent Analysis
Carbon is a primary element created in the fundamental 3α-process and as such it
provides the seed for the subsequent synthesis of all heavier elements (Burbidge et
al. 1957; Cameron 1957). Carbon is an essential catalyst for the nucleosynthesis
of H into He through the CNO cycle in massive and intermediate-mass stars. The
element also constitutes the basis of all organic chemistry.
Carbon abundances derived from early-type stars were the subject of numer-
ous studies over the past decades, with rising improvements in the quality of
observed data and the complexity and consistency of the model calculations and
spectral analyses. A crucial step in this was to abandon the approximation of
local thermodynamic equilibrium (LTE) in line-formation calculations and allow
for deviations (non-LTE). Several non-LTE model atoms have been discussed in
literature (Lennon 1983; Eber & Butler 1988; Grigsby et al. 1992; Sigut 1996;
Lanz & Hubeny 2003, 2007). In particular the model atom of Eber & Butler
found wide application for abundance analyses of mostly unevolved early-type
stars in the solar neighbourhood (e.g. Gies & Lambert 1992; Kilian 1992; Cunha
& Lambert 1994; Gummersbach et al. 1998; Daflon et al. 1999, 2001ab).
Only few optical transitions may be used for abundance determinations of
carbon in many (extragalactic/fast-rotating) early-type stars because of S/N-
constraints or rotational smearing. This includes the strong C ii λλ6578/82
˚
A
and in particular the 4267
˚
A multiplet, which is known to pose a challenge to
non-LTE line-formation calculations (e.g. Lambert 1993; Sigut 1996). These
two multiplets usually fail to reproduce observation consistently (e.g. Grigsby et
al. 1992; Hunter et al. 2007), and they are reported to give systematically lower
abundances than derived from other, weaker C ii lines (Gies & Lambert 1992).
Observed trends for C iii lines may also be poorly matched by model calculations
(Grigsby et al. 1992). A further complication is the failure to establish the C ii/iii
ionization equilibrium. Differences in abundance from the two ions can amount
up to a factor ∼5–10 (Daflon et al. 2001b; Hunter et al. 2007). However, there
71
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 72
are even more inconsistencies with published carbon abundances from early-type
stars in a broader context that require other explanation, as most of the published
studies avoid the lines sensitive to non-LTE effects.
A comparison of the available studies of early B-type stars indicates that
present-day carbon abundances in the solar vicinity are highly inhomogeneous
(even for stars within a single cluster) and largely sub-solar. This is in contrast
to the findings of a uniform abundance in the gas-phase of the interstellar medium
(ISM) within 1.5 kpc of the Sun (Sofia & Meyer 2001, and references therein). It
cannot be understood from current stellar and Galactochemical evolution models
either. Young massive stars that form out of a molecular cloud within a short
timescale can be expected to show a homogeneous carbon abundance, which also
coincides with that of their surrounding H ii nebula. Moreover, this abundance
is expected to be higher than that of objects from previous generations of star
formation in their neighbourhood, like young (≤2 Gyr) F and G stars, and that
of the Sun. In reality, significant systematic differences exist (see e.g. Sofia &
Meyer 2001; Herrero 2003), sheding doubt on the reliability of carbon abundances
derived from early B-type stars.
In this Chapter a robust C ii-iv model atom, empirically calibrated with six
apparently slow-rotating early B-type stars from the solar vicinity is presented.
The focus lies on the critical selection of the appropriate atomic input data and
on a self-consistent derivation of the fundamental atmospheric parameters of the
sample stars from ionization equilibria. This results on derived abundances with
unprecedented accuracy. As a consequence, it is possible to provide a precise
determination of the present-day carbon abundance in the solar neighbourhood.
The Chapter is organised as follows: Section 6.1 describes the model calcu-
lations including a description of the carbon model. Section 6.2 describes the
empirical model calibration via an extensive and self-consistent iteration and the
sensitivity of carbon line-formation calculations to atomic data and atmospheric
parameter variations. Section 6.3 summarises the final results for parameters
and carbon abundances of individual lines for the sample stars. Section 6.4 pro-
vides a comparison of the present results to previous studies. The conclusions on
the present-day carbon abundance in the solar neighbourhood are discussed in
Sect. 6.5. A summary is given in Sect. 6.6.
6.1 The C ii/iii/iv Model Atom
The non-LTE line-formation computations for carbon follow the hybrid non-LTE
approach, as discussed in Chapter 5. The computational efforts can thus be
concentrated on robust non-LTE line-formation calculations.
Non-LTE level populations and model spectra are obtained with Detail and
Surface. Continuous opacities due to hydrogen and helium (for actual abun-
dances) are considered in non-LTE. Line blocking is accounted for in LTE via
73 6.1 The C ii/iii/iv Model Atom
Kurucz’ ODFs. Microturbulence is consistently accounted for in both steps with
Detail and Surface. Non-LTE level populations for hydrogen and He i/ii are
computed as explained in Chapter 4. This is a prerequisite for modelling metal
lines which overlap with the (broad) hydrogen or helium features. In particular,
the C ii λλ6578/82
˚
A doublet is affected in the present case.
A short summary of the input atomic data for the construction of the
C ii/iii/iv model atom is given in this section. A more detailed description
of the motivation for choosing these atomic data will be given in Section 6.2, as
this turned out to be critical for the realistic modelling of the observed spectra.
C ii. This model ion considers LS-coupled terms up to principal quantum
number n = 10 and angular momentum = 9 (66 levels) explicitly in the non-LTE
calculations, with all fine-structure sub-levels combined into one. Additional
levels up to n = 14 are computed in LTE relative to the ground state of C iii.
Level energies are adopted from Moore (1993), Sigut (1996) and Quinet (1998).
The doublet and quartet spin systems are treated simultaneously.
Oscillator strengths (gf -values) from three sources are considered: fine-
structure data from ab-initio computations using the multiconfiguration Hartree-
Fock method in the Breit-Pauli approximation of Froese Fischer & Tachiev (2004,
FFT04), data from application of the Breit-Pauli R-matrix method (Nahar 2002a,
N02a) and results obtained in the Opacity Project (OP) from the R-matrix
method assuming LS-coupling (Yan, Taylor & Seaton 1987). The primary source
of gf -values is FFT04, followed by OP and N02a for the remaining transi-
tions. Intercombination transitions are neglected because of very small oscillator
strengths.
Photoionizations cross-sections are adopted from the OP (Yan & Seaton 1987)
for levels up to n = 9 and = 3 with a correction of the threshold frequencies to
observed values. For the remainder, data from Nahar (2002b: N02b) are used.
Effective collision strengths for electron impact excitation among the lowest
16 LS-states are adopted from R-matrix computations of Wilson, Bell & Hud-
son (2005, 2007). An empirical increase by a factor two was applied to the
3s
2
S–3p
2
P
o
data (see Sect. 6.2.2). Collisional excitation for transitions with-
out data are treated using the Van Regemorter (1962) approximation – in the
optically allowed case – and via the semi-empirical Allen (1973) formula in the
optically forbidden case. Collision strengths Ω varying between 0.01 (∆n ≥ 4)
to 100 (∆n = 0) are employed, as suggested by evaluation of the detailed data
from ab-initio computations of Wilson et al. (2005, 2007).
Collisional ionization rates are evaluated according to the Seaton (1962) ap-
proximation. Threshold photoionization cross-sections are adopted from OP and
N02b, allowing for an empirical correction of one order of magnitude higher for
the 6f
2
F
◦
and 6g
2
G levels – corresponding to the upper levels of the C ii λλ6151
and 6462
˚
A transitions, respectively.
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 74
C iii. This model accounts for LS-terms up to n = 7 and = 7 (70 levels) ex-
plicitly in the statistical equilibrium calculations. In a similar way to C ii, levels
up to n = 14 are computed in LTE relative to the ground state of the next ioniza-
tion stage. The two spin systems (singlet and triplet) are treated simultaneously.
Level energies are taken from the NIST database
1
and energies for 9 of the highest
levels are adopted from N02a. Two sources are considered for oscillator strengths:
N02a and additional values from Eber (1987). Intercombination transitions are
also implemented when their values are non-negligible (f > 10
−4
). Photoioniza-
tion cross-sections are taken from S. Nahar’s webpage
2
. Maxwellian-averaged
collision strengths for electron impact excitation among the lowest 24 terms are
adopted from the R-matrix computations of Mitnik et al. (2003). Collisional
excitation for the remaining transitions and collisional ionization are treated in
analogy to the C ii ion, using appropriate gf -values and threshold photoioniza-
tion cross-sections.
C iv. LS-terms up to n = 10 and = 9 (53 levels) are treated explicitly. Ad-
ditional levels up to n = 14 are computed in LTE relative to the ground state
of C v. Oscillator strengths from ab-initio calculations using the Breit-Pauli R-
matrix method (Nahar 2002c) are adopted. Photoionization cross-sections are
also taken from Nahar’s webpage. Effective collision strengths for electron im-
pact excitation of transitions among the lowest 24 fine-structure levels are taken
from Aggarwal & Keenan (2004) and subsequently co-added. All remaining tran-
sitions, as well as collisional ionization, are treated in analogy to C ii.
The resulting C ii/iii/iv model atom accounts for more than 1300 radiative
and more than 5300 collisional transitions, ∼200 LS-coupled energy levels and
over 20 000 frequency points – the latter allow the detailed resonance structure
of the photoionization cross-sections to be considered. Accuracies of the atomic
data can range from a typical 10-20% for ab-initio computations to orders of
magnitude for approximation formulae. Finally, Voigt profiles are adopted in the
formal solution using Surface. Wavelengths and oscillator strengths of most of
the observed transitions are taken from Wiese et al. (1996). For C ii λλ6151.3/5
and 6461.9
˚
A the line transition data are adopted from Kurucz & Bell (1995).
Radiative damping parameters are calculated from OP lifetimes and coeffi-
cients for collisional broadening by electron impact are adopted from Griem (1974,
for the C ii λ4267
˚
A doublet) or computed according to Cowley (1971). Detailed
tabulations from quantummechanical computations for Stark broadening of sev-
eral C iv transitions (Sch¨oning 1993) are also used. The spectral lines used for
abundance analysis are listed later in Table 6.3.
1
http://physics.nist.gov/PhysRefData/ASD/indices.html
2
http://www-astronomy.mps.ohio-state.edu/∼nahar/px.html
75 6.2 Model Atom Calibration
3.10 < log g < 4.30
eff
21500 K < T < 32000 K
Empirically calibrated C II−IV model atom
Selected atomic data
Accurate (C) for 6 Galactic B starsε
NLTE C populations
Parameter verification
observed spectra
Comparison with
Synthetic H, He, C profiles
Empirical calibration
of C model atom
New set of atomic data
M
I
(grid: 5 (C) x 4 )ε ξ
Modified T , log g
eff
eff
minimisation
2
χ
T , log g: ionization equilibrium
eff
ξ: ε(C) vs. EW
vsin i, : line profile fitsζ
checking observed transitions
verification of RBB, RBF, CBB, CBF
◆
◆
◆
◆
◆
◆
◆
◆
min.?C
no
yes
Step 2
M = I?
&
Accurate Stellar Parameters
Accurate Carbon Abundance
Calibrated C model for 1 Star
up to 40 C spectral lines
Initial T , log g
σ[ε( )]
analysis of atomic structure
Stellar atmosphere
Synthetic H, He profiles
H, He I/II lines
all measurable
ξ, ζ, ε( )
, vsin i
M = I ?
no
yes
ξ, ζ, ε( )
, vsin i
, vsin i
He
He
He
Comparison with
observed spectra
I
NLTE H, He populations
M
Step 1
to Step 2
Parameter verification
Verify Step 1
ξ, ζ, ε( )
Initial T , log g,
eff
Modified T , log g,
eff
Final T , log g,
eff
6 early−B III−V stars
Figure 6.1: Flux diagram of the extensive iterative procedure, as applied to each
programme star. The procedure allows the following to be achieved: i) a simul-
taneous derivation of highly accurate stellar parameters, ii) a critical selection of
input atomic data from different sources, iii) the calibration of the C ii-iv model
atom and iv) a determination of precise C abundances, with highly reduced
systematic errors. RBB/CBB: Radiative/Collisional Bound-Bound, RBF/CBF:
Radiative/Collisional Bound-Free transition data. Step 1 is discussed in Chapter
4. See the text for details.
6.2 Model Atom Calibration
Model atoms can be empirically calibrated for non-LTE calculations by demand-
ing that the model reproduce the observations reliably over the entire parameter
space of relevance. A basic requirement for a proper model atom calibration is
an accurate knowledge of the atmospheric parameters of the test sample stars.
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 76
These should be free of systematic errors in order to prevent one from being mis-
lead when optimising the input atomic data. Unfortunately, the parameters of
stars are a priori not known, they also need to be inferred by interpretation of
observation. Consequently, a simultaneous solution for all stellar parameters and
an optimal set of input atomic data needs to be found. At the same time, those
parameters should also allow the hydrogen and helium spectra to be reproduced.
6.2.1 Extensive Iteration on Fundamental Variables
The atmospheric parameter derivation and selection of input atomic data are
simultaneously performed in an extensive iteration process. When possible, the
effects of the fundamental parameters and the atomic data on the synthetic spec-
tra – the basis for the comparison with observation – are disentangled in order to
achieve a better understanding of the problem. This is facilitated by boundary
conditions, like the ionization balance (all ionization stages of an element are
required to indicate the same abundance) or the rules and regularities of atomic
physics.
The iteration is performed on effective temperature T
eff
and surface gravity
log g, as well as micro-, macroturbulent and projected rotational velocities (ξ,
ζ and v sin i, respectively), helium and carbon abundances (hereafter (He) and
(C), respectively) and different sets of atomic data. Only the metallicity is fixed
to a standard solar value (Grevesse & Sauval 1998), a not too critical assumption
which is furthermore validated a posteriori (Przybilla et al. 2007, in preparation).
This comprises an enormous number of variables (atmospheric parameters and
atomic data) in the iterative scheme summarised as a flux diagram in Fig. 6.1.
The first step concerning the H/He spectrum is solved in Chapter 5. There,
the He i/ii ionization equilibrium is the main indicator for T
eff
(for the hotter
stars), all Balmer lines for log g, the He ii lines for ξ and all He lines for ζ and
v sin i. The second step, involving carbon, is required for a fine tuning of the
atmospheric parameter determination since the metal lines are more sensitive to
parameter variations than the hydrogen and helium lines. Therefore it is possible
to derive them with a better precision than only from hydrogen and helium but
at the same time consistently within the error limits. Effective temperature and
log g are refined by establishing the C ii/iii/iv ionization equilibrium in the hotter
stars and the C ii/iii ionization balance in the cooler stars. The microturbulent
velocity is inferred in the standard way by demanding the carbon abundances of
the individual lines to be independent of equivalent width. Macroturbulent and
projected rotational velocities are determined by detailed fitting of the carbon
line profiles. Line fits are performed on the basis of small grids of synthetic
spectra with different ξ and (C) via χ
2
-minimisation.
Since different sets of atomic data are available, they are intercompared and
their reliability to minimise the uncertainties in the (C)-determination is judged.
Starting from an initial model atom (based on input data that are supposed to
77 6.2 Model Atom Calibration
be the most accurate) a careful analysis of the atomic structure of the model ions
and the reactions of the spectrum synthesis to parameter variations is performed.
Examples of this are given in the next subsections.
The atmospheric parameters derived from C ionization equilibrium in Step 2
are verified by re-iterating Step 1 as a final check for consistency. By application
of the procedure to all programme stars it is possible to calibrate the C ii-iv
model empirically over the entire parameter range (21 500 ≤ T
eff
≤ 32 000 K,
3.10 ≤ log g ≤ 4.30, for dwarfs and giants), resulting in a final reference set of
atomic data. As a consequence, highly accurate carbon abundances are obtained
for the stars of the calibration sample, essentially unbiased by systematic errors.
Note that the high-quality spectra are essential for this success. They allow
the analysis of a wide variety of carbon lines to be performed, many of which
have never been considered before in the study of early B stars. Some weak
lines turned out to be highly sensitive to non-LTE effects and/or to atmospheric
parameter variations, namely the C iv lines and C ii λλ6151 and 6462
˚
A, which
change from absorption to emission at higher temperatures. The reproduction of
the observed trends despite this high sensitivity puts strong constraints on the
robustness of the final model atom. Note also that the presence of lines from three
ionization stages, in hotter stars of the sample, allows T
eff
and log g to be derived
from the ionization equilibrium alone, independent of any other indicators.
To summarise here, the strength of this empirical calibration lies in the simul-
taneous analysis of the large number of C lines of different ionization stages in
stars covering a wide parameter range. In this way it is possible to constrain a fi-
nal set of atomic data independent of any specific stellar atmosphere environment.
This reference model can be used in further applications with a simplified itera-
tive scheme, where the only remaining variables are the atmospheric parameters
and the C abundance.
6.2.2 Sensitivity of C Lines to Atomic Data
As explained in previous chapters, two kinds of processes produce transitions
from one state of an atom to another. Collisions act towards establishing detailed
equilibrium locally, while radiative processes are non-local in character (photons
may travel wide distances before interacting with the plasma). LTE is therefore
established only when either collisions dominate the plasma or the radiation field
is isotropic and Planckian. Such conditions prevail deep in stellar atmospheres,
but the strict validity of LTE can not be assumed for the observable layers.
The line formation in early B-type stars requires such deviations from LTE
to be accounted for in detail. A statistical equilibrium is established, with the
radiation field coupling the plasma conditions at each depth in the atmosphere.
The non-linear interdependency of radiation field and level populations is the
essence of the non-LTE problem. It requires an iterative solution.
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 78
Figure 6.2: Grotrian diagram for the C ii doublet and quartet spin systems. Only
the observed multiplets in the spectra are identified. Levels: those marked in bold
correspond to levels discussed in Figs. 6.3 and 6.6. Multiplet transitions: those
marked by thick lines are highly sensitive to variations of input data for photoion-
ization and collisional cross-sections and for collisional excitation (discussed in
Figs. 6.4, 6.6 and 6.7, respectively). The latter and those marked by thin lines are
considered in the linelist (Table 6.3) for abundance derivation. Those marked by
dotted lines are excluded from the analysis because of contamination with telluric
lines (λλ6257-59, 7231-37
˚
A) or they are too weak even at low T
eff
(λ5889-91
˚
A).
Nevertheless they are accounted for in the calculations of the level populations.
Autoionizing levels above the ionization threshold are not treated explicitly, but
are considered in the statistical equilibrium computations via resonances in the
photoionization cross-sections.
The coupled problem of the radiative transfer (Eqn. 2.5) and the statistical
equilibrium (Eqn. 2.26) equations allows reliable level populations (a prerequisite
for an accurate analysis) to be obtained only under strict conditions. i) The local
temperatures and particle densities are known (i.e. the atmospheric structure)
and ii) the radiation field is realistic and iii) all relevant processes are taken
into account and iv) high-quality atomic data are available (i.e. accurate cross-
79 6.2 Model Atom Calibration
Figure 6.3: Comparison of C ii photoionization cross-sections from the Opacity
Project (Yan & Seaton 1987) and Nahar (2002b) for 2p
2 2
S and 4f
2
F
o
as a
function of wavelength. See the text for a discussion.
sections for the transitions). In particular, i) and ii) require a realistic physical
model of the stellar atmosphere (see Chapter 5 for a discussion of the hybrid
non-LTE approach in the context of this) and an accurate atmospheric param-
eter determination. Items iii) and iv) are related to the model atoms for the
non-LTE calculations. Shortcomings in any of i)–iv) result in increased uncer-
tainties/errors of the analysis.
Because of the interdependency of all transitions (over 6000 in this case)
and the non-local character of the radiation field, even a restricted non-LTE
problem like the one investigated here becomes highly complex. In particular, it
is impossible to quantify a priori the sensitivity of the spectral lines to variations
of some of the atomic input data. Therefore, one of the few remaining reasons for
the large spread of carbon abundances found in literature (see Sect. 6.4) may be
different realisations of model atoms (levels/transitions considered, atomic data,
approximations). Choosing an optimum set of input atomic data is not trivial
and the construction of reliable model atoms for non-LTE calculations requires
an empirical calibration, guided by extensive comparisons with observation. In
the following a summary addressing relevant examples is given.
The actual choice of radiative and collisional data turned out to be a crit-
ical factor for line-formation computations of non-LTE-sensitive transitions in
C ii, which is known to be problematic from the literature (e.g. Lambert 1993;
Sigut 1996). A few comparisons of atomic data available from the literature
and the influence of input atomic data on selected C ii lines is provided. These
transitions are highlighted in the Grotrian diagram of the C ii model (Fig. 6.2),
which will help to illustrate some of the channels leading to the marked non-LTE
sensitivity. The Grotrian diagram also gives an overview of the levels involved in
the formation of the observed transitions.
Photoionization cross-sections. The strength of a spectral line can be
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 80
strongly influenced by photoionizations which may impact the level populations
decisively. On the other hand, photoionization rates depend implicitly on the
level populations, R
ij
= R
ij
(α
ij
, J
ν
[n]), i.e. from the coupling of Eqns. 2.5 & 2.26.
The largest contribution to the integral in R
ij
(Eqn. 2.27) comes from fre-
quencies with large flux and large cross-section. The flux maximum in early B
stars is located longward of the Lyman jump at 912
˚
A. Examples of the behaviour
of photoionization cross-sections are given in Fig. 6.3.
The ionization of C ii is essentially determined by the rates from the highly-
populated ground state and the low-excitation levels. Their relative importance
may be strengthened in cases where the ground state ionization potential coin-
cides with that of a major opacity contributor, He i in the present case. Photoion-
izations from the ground state may then be less efficient because of the reduced
stellar flux shortward of the ionization edge. Recombinations, on the other hand,
are important for the population of high-excitation levels (preferentially at high
, i.e. states with large statistical weight), which couple to the low-lying states
via recombination cascades. For the case of C ii this implies: i) an increased sen-
sitivity of the C ii/iii ionization balance to the exact run of the photoionization
cross-sections of levels at low excitation energies and ii) an increased sensitivity
of transitions like C ii λλ6151, 6462 and in particular 4267
˚
A to non-LTE effects
because of their participation in the recombination cascade.
A comparison of photoionization cross-sections from OP (Yan & Seaton 1987)
and N02b for two levels (marked in the Grotrian diagram, Fig. 6.2) is given in
Fig. 6.3. The first term, 2p
2 2
S, is not directly involved in the formation of the
C ii λλ 4267, 6151 and 6462
˚
A transitions. However, it is populated considerably
and therefore contributes to the C ii/iii ionization balance. Note the wavelength
shifts in the resonance structures of both data. This results in different contri-
butions of the region between the threshold and the Lyman edge to the integral
in R
ij
(Eqn. 2.27), which affect the photoionization rates considerably. On the
other hand, the photoionization cross-sections for 4f
2
F
0
agree well, except for
the resonance structure at shortest wavelengths where the stellar flux becomes
negligible. Consequently, both data (and many others for highly-excited levels)
are exchangable without showing consequences for the spectrum synthesis com-
putations. Note that experimental threshold wavelengths are adopted. On the
other hand, for 2p
2 2
S and other levels at lower excitation energy there is a non-
negligible effect on the λλ4267 and 6151
˚
A transitions when exchanging both data,
as can be seen in Fig. 6.4. In an extreme case, accounting for photoionization
cross-sections from N02b for all levels results in a very strong C ii λλ4267
˚
A line.
A reduction of the C abundance by up to ∼0.8 dex is required to fit the observed
line profile in the calibration stars with such a model atom. Preference to the OP
data over the cross-sections of N02b is given in the final model atom (‘model of
reference’), which help to reproduce observation over the entire parameter range
consistently.
81 6.2 Model Atom Calibration
Figure 6.4: Sensitivity of line profiles of two C ii transitions to variations of pho-
toionization cross-sections. The calculations are made for three different model
atoms using the same model atmosphere (as appropriate for HR 5285): the refe-
rence model atom, with cross-sections for 2p
2 2
S from N02b, and the reference
model atom with all photoionization data replaced by values from N02b. The
profiles are not convolved for effects of rotation or instrumental broadening.
Oscillator strengths. Comparisons between multiplet f-values from three
sources of ab-initio computations are shown in Fig. 6.5: data based on i) FFT04,
ii) N02a and iii) the Opacity Project (Yan et al. 1987). The primary source of
f-values is FFT04, which should be most accurate. The preference for the OP
over the N02a data is motivated by the good agreement of the former with FFT04
(with two exceptions), while oscillator strengths from N02a may show significant
differences for several lines. Data from N02a is therefore adopted only in the
cases where the other sources do not provide information.
Collisional ionization cross-sections. High-excitation levels of C ii can cou-
ple collisionally to the C iii continuum. The line-formation calculations for C ii
λλ4267, 6151 and 6462
˚
A are also highly sensitive to the choice of collisional ion-
ization cross-sections, as shown in Fig. 6.6. The approximation of Seaton (1962)
is used to evaluate the collisional ionization rates because of a lack of any data
from ab-initio computations. Threshold cross-sections for the 6f
2
F
o
and 6g
2
G
levels, that are involved directly in the formation of C ii λλ6151 and 6462
˚
A,
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 82
Figure 6.5: Comparison of oscillator strengths from Nahar (2002a: N02a) and
Froese Fischer & Tachiev (2004: FFT04) vs. values from the Opacity Project
(Yan et al. 1987: OP).
and indirectly of C ii λ4267
˚
A (see Grotrian diagram, Fig. 6.2) are scaled with
different factors, i.e. increasing the collisional rates (Eqn. 2.29). The effects on
abundance analyses can also be drastic, as in the previous example. The em-
pirical calibration indicates an increase of the OP data for these two levels by a
factor 10 to be appropriate for reproducing observation over the entire parameter
range in a consistent way.
Collisional excitation cross-sections. Cross-sections for excitation via elec-
tron collisions, which are proportional to the collision strength Ω
ij
, can show
complex behaviour with impact energy. In practice, the cross-sections will be
weighted by a Maxwell distribution (see Eqn. 2.29), such that data from ab-initio
calculations are often tabulated already in thermally-averaged form as effective
collision strengths
Υ
ij
=
∞
0
Ω
ij
exp(−E
j
/kT ) d(E
j
/kT ) , (6.1)
where E
j
is the energy of the outgoing electron, k the Boltzmann constant and
T the temperature.
Larger collision strengths facilitate a tighter collisional coupling and therefore
promote the establishment of LTE. Consequently, the collisional data used in a
model atom will have an influence on the predicted line profiles.
Accurate data from ab-initio calculations for larger sets of transitions have be-
come available only recently. Effective collision strengths of Wilson et al. (2005,
WBH05) are employed to construct the C ii model. A later revision of part of
the data (Wilson et al. 2007, WBH07) had negligible influence on the predicted
line profiles of almost all observable transitions, except for C ii λλ 3918/20 and
83 6.2 Model Atom Calibration
Figure 6.6: Reactions of three highly non-LTE-sensitive lines to changes of col-
lisional ionization cross-sections. The modifications are made for two energy
levels which are directly involved in the formation of C ii λλ6151 and 6462
˚
A and
indirectly in C ii λ4267
˚
A. The calculations are made for different model atoms
with specific values of the reaction cross-section at threshold and the same set of
atmospheric parameters (as appropriate for τ Sco).
6578/82
˚
A, see Fig. 6.7 for an example. The good agreement of abundances de-
rived from these four with other transitions was broken when using the improved
WBH07 data, requiring abundance adjustments of up to ∼0.3 dex to match ob-
servation. The situation may be improved for the stars of the calibration sample
by increasing the effective collision strength for the 3s
2
S–3p
2
P
o
transition by an
empirical factor of two, see Fig. 6.7. This is larger than the typical uncertainty
of such ab-initio data, which amounts to an estimated 10-20%. However, a closer
inspection of the energy-dependent collision strength for this transition shows
that resonances dominate Ω in the region near threshold. The positions and
strengths of the resonances are sensitive to the details of the atomic data calcu-
lations, in particular to the assumptions made for constructing the target. The
near-threshold region in turn contributes most to the integral in Eqn. 6.1. Con-
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 84
Figure 6.7: Effect of employing different effective collision strengths on the C ii λλ
6578 and 3920
˚
A lines (for HR 1861). Only variations of the data from ab-initio
calculations (Wilson et al. 2005, 2007) are considered.
sequently, more comprehensive ab-initio computations are required to investigate
this in detail. However, these are beyond the scope of the present work.
Collisional data from ab-inito calculations are typically available only for tran-
sitions between relatively low-lying energy levels. In the case of C ii the dataset
is complete for levels up to principal quantum number n = 4. Therefore, for
the bulk of the transitions approximation formulae have to be applied. However,
trends and regularities from the ab-inito data may be used to improve on the
standard approximations made for these, e.g. dropping the assumption of an
energy-independent Ω = 1 for the evaluation of the Allen (1973) formula (see
Sect. 6.1).
In practice, simple approximations are used to describe collisional processes in
most of the model atoms available for non-LTE calculations at present. One can
ask what the effects on synthetic profiles will be from such a simplification. This
is shown in Fig. 6.8 for selected lines in two of the programme stars. Notable
corrections in abundance would be required to reproduce the results from the
reference model, an increase in some of the lines but also a decrease in others.
The overall good agreement with observation (Sect. 6.3) would be destroyed: the
non-LTE sensitive lines and the lines ’in LTE’, which are unaffected by such
modifications, would indicate widely different abundances. Note that the effects
vary from star to star.
85 6.2 Model Atom Calibration
Figure 6.8: Comparison of synthetic C ii/iii line profiles for two of the sample
stars using different model atoms. One calculation uses the reference model atom
and the other adopts standard approximation formulae for evaluating collisional
rates for all transitions.
The discussion on the impact of atomic data on line-formation calculations
is concluded in Fig. 6.9. Here, the differences in abundance derived with the
initial model atom (built from the supposedly ’best’ available homogeneous set
of atomic data, N02ab) and with the model of reference (after the empirical
calibration) are quantified for the entire sample of stars. Abundances from the
strong and non-LTE-sensitive C ii λλ4267 and 6578
˚
A transitions are compared to
abundances derived from the weaker C ii λ5145
˚
A line, which is almost insensitive
to the details of the model atom (i.e. it is ‘in LTE’). The relative abundance is
displayed as a function of effective temperature for each star (see Section 6.3
for final results). Pronounced systematic trends exist when the initial model
atom is used, correlating with the strength of the lines (i.e. stronger lines show
a larger sensitivity to non-LTE effects and therefore to the input atomic data).
These trends and abundance differences from C ii λλ4267 and 6578
˚
A almost
vanish when the model of reference is used. The remaining small differences may
be reduced even further when improved atomic data, in particular for collisions
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 86
Figure 6.9: Abundance differences from the analysis of C ii λλ4267 and 6578
˚
A
(highly sensitive to the input atomic data) and C ii λ5145
˚
A (’in LTE’). Displayed
are results for the six sample stars, as a function of effective temperature. A
comparison of the initial model atom (all radiative transition data for C ii from
N02ab) and the final model atom after the empirical calibration is shown. See
the text for details.
involving high-excitation levels, become available. Note, that the C ii λ6582
˚
A
line (not displayed here) shows a trend similar to that of λ6578
˚
A. The whole
multiplet centered on C ii λ5145
˚
A also behaves consistently. This kind of test
has been made for all C ii-iv lines for every set of input atomic data in the
empirical calibration process of the model atom.
The comparison of observation with model spectra for C iii and C iv, as com-
puted with the initial model ions based on the most sophisticated input data
available, reveals little need for improvement. Both ions are relatively simple,
showing (earth)alkali electron configurations, which pose little challenge to ab-
initio computations. Further empirical testing of the C iv model may be desirable,
involving more transitions. However, this is beyond the scope of the present work,
as markedly hotter plasma environments need to be considered, where the hybrid
non-LTE approach may reach its limitations (see the discussion in Chapter 5).
This detailed study of non-LTE and LTE abundances in Sect. 6.3 helps to
identify the ‘lines in LTE’ for the different atmospheric parameters under analysis.
These may provide good starting points for further analyses when one desires to
avoid non-LTE effects.
6.2.3 Line-Formation Details
A closer study of the underlying line-formation processes allows the nature of
the non-LTE effects to be understood. This is facilitated by Fig. 6.10 for some
representative transitions of C ii/iii/iv in the programme star τ Sco.
Departure coefficients b
i
= n
NLTE
i
/n
LTE
i
are displayed in the left panel of
Fig. 6.10 for the levels involved in the transitions of interest and the ion ground
87 6.2 Model Atom Calibration
Figure 6.10: Departure coefficients b
i
and ratio of line source-function S
l
to Planck
function B
ν
at line centre as a function of τ
Ross
for selected carbon lines in τ Sco.
The spectral lines are encoded by the different line styles indicating the line-
formation depths. Thin grey lines on the right-hand panel correspond to S
l
=B
ν
.
states. Values of b
i
> 1 imply an overpopulation relative to detailed equilibrium
and b
i
< 1 an underpopulation. Non-LTE departures of the level occupations
impact the line source function S
l
. The ratio of S
l
(Eqn. 2.43) to the Planck
function B
ν
is shown for selected transitions in the right panel of Fig. 6.10, and
can be expressed as
S
l
B
ν
=
exp (hν
ij
/kT ) − 1
b
i
/b
j
exp (hν
ij
/kT ) − 1
. (6.2)
As can be seen from Eqn. 6.2, S
l
/B
ν
is determined by the ratio of the departure
coefficients for the lower and upper levels (i, j). An overpopulation of the upper
level relative to the lower (i.e. S
l
/B
ν
> 1) results in non-LTE weakening of the
line and may lead to emission in cases of pronounced overpopulation, while the
inverse gives non-LTE strengthening.
The level populations reach detailed equilibrium values deep in the atmo-
sphere, where large collisional rates and small mean-free paths between photon
absorptions (both because of the high densities of the plasma) enforce this in-
ner boundary condition. Double-ionized carbon is the main ionization stage at
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 88
the temperatures of τ Sco - the C iii ground state is close to LTE. Single-ionized
carbon is overionized at line-formation depths, therefore the levels are underpop-
ulated relative to LTE, and C iv and the C v ground state are overpopulated.
In general, level populations in C ii depart most from detailed equilibrium
in the low-excitation states and approach LTE values gradually with increasing
excitation energy, as collisions facilitate coupling with the C iii ground state.
Therefore, most of the non-LTE-sensitive transitions in C ii have upper levels
that are overpopulated relative to the lower level, so that the lines experience
slight (λ 3920
˚
A) to notable weakening (λ 4267
˚
A) relative to LTE and may even
turn into emission (λλ 6151, 6462
˚
A). The λλ 6578/82
˚
A doublet experiences non-
LTE strengthening for lower effective temperatures, while in hotter objects like
τ Sco the lines are found to be close to LTE. The other observable C ii lines arise
in the quartet spin system. They are weaker (C ii λ 5145
˚
A is the strongest), i.e.
they are formed deeper in the atmosphere, and their formation involves high-
excitation levels which are coupled collisionally at these depths, such that these
lines are essentially in LTE. Note that the behaviour of S
l
/B
ν
for λ 4267
˚
A is in
good agreement with the findings of Sigut (1996). On the other hand, notable
differences exist for the λλ 6578/82
˚
A doublet in particular at higher T
eff
. The
reasons for this will be discussed in the next section (but see also Fig. 6.15).
The C iii transitions can also experience both, non-LTE weakening (like the
strong λ 4187
˚
A) and non-LTE strengthening (like the strong triplet 4647-4651
˚
A).
The C iv doublet λλ 5801/12, which becomes observable only in the hottest stars,
shows a pronounced non-LTE strengthening.
6.2.4 Sensitivity of (C) to Parameter Variations
Spectral lines of carbon, like many other metal lines, can react sensitively to
variations of the stellar atmospheric parameters. This property is a powerful tool
for the atmospheric parameter and abundance determination, using the iteration
scheme (Fig. 6.1). This is a non-trivial and crucial step in the analysis, which
must be performed carefully in order to avoid systematic error. The consequences
of systematically biased atmospheric parameters on carbon line profiles and the
derived abundances are investigated in the following.
The offsets for the parameters (effective temperature ∆ T
eff
=−2000 K, sur-
face gravity ∆ log g= +0.2 dex and microturbulent velocity ∆ ξ= +5 km s
−1
) are
representative for systematic discrepancies between the final values and those
from previous studies (and also in between these studies). Note that they are
much larger than the statistical uncertainties derived in this work. Such discre-
pancies may be caused by several factors, among others: i) photometric effective
temperature calibrations based on model atmospheres with insufficient line blan-
keting
3
, ii) spectroscopic ionization equilibria based on predictions of incomplete
3
An important but underestimated source of systematic error are abundance ‘standards’.
89 6.2 Model Atom Calibration
Figure 6.11: Sensitivity of selected C ii/iii/iv lines to atmospheric parameter
variations in two giants: HR 3055 (hotter, B0 III) and HR 3468 (cooler, B1.5 III).
The present solution corresponds to the final parameters from Table 5.1 and a
constant value of (C) for all lines. The parameter offsets are typical for statistical
and systematic uncertainties from published values. The theoretical spectra are
convolved with a rotational profile for v sin i = 20 km s
−1
. The present solution
establishes the ionization equilibrium. HR 3468 is too cool to show C iv lines.
model atoms, iii) the assumption of LTE for the computation of Balmer line
profiles, which may cause a systematic overestimate of log g by up to 0.2 dex
(Chapter 5).
Figure 6.11 shows the sensitivity of selected C ii/iii/iv lines in two giants of
E.g., the widely used classical Kurucz (1993a) ODFs were computed on the basis of (scaled)
solar abundances according to Anders & Grevesse (1989), which indicated a high abundance for
the most important line opacity source, iron (log Fe/H + 12 = 7.67). A later revision (Grevesse
& Sauval 1998) lead to employ ODFs with appropriately reduced metallicity for the present
model atmosphere calculations (see Chapter 5 for a discussion). This changes the atmospheric
temperature structure notably via a reduced backwarming effect, giving lower temperatures in
the line-formation region by up to ∼500K, i.e. slightly higher than the uncertainties in T
eff
derived in this work.
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 90
Figure 6.12: Sensitivity of carbon abundances to stellar parameter variations:
T
eff
(upper panel), log g (mid panel) and microturbulent velocity (lower panel).
The offsets of the parameters are displayed in the lower left part of each panel.
Note the resulting large spread in abundance from individual lines of the three
ionization stages, in particular for variations of T
eff
(upper panel). Note also the
implications of using only few lines of one ionization stage for abundance deter-
minations in the presence of systematic errors in the atmospheric parameters.
The grey bands correspond to 1σ-uncertainties of the stellar carbon abundance
in the final solution (summarised in Table 6.2).
the sample (HR 3055 and HR 3468) to variations of basic atmospheric parameters
like T
eff
, log g and ξ. The profiles accounting for these variations are compared
with those computed with the final atmospheric parameters and an averaged
carbon abundance. The sensitivity to parameter variations differs from line to
line. For the hotter star, the C iv and – when strong enough – the C ii λλ 6151
and 6462
˚
A lines (the latter not displayed in Figure 6.11) are ideal indicators for
both T
eff
and log g, while the rest of the lines is mostly sensitive to changes in
T
eff
and the strong C iii lines also to changes in ξ. The C ii λλ 6151 and 6462
˚
A
lines are too weak to be measured in this case; however, one can still distinguish
between emission and absorption (in τ Sco they are strong enough to be used
in the analysis). For the cooler star, the C ii λλ 4267, 6151 and 6462
˚
A lines are
highly sensitive to variations of T
eff
and ξ, the rest of the lines are mostly sensitive
to T
eff
and the strong C ii lines also to ξ.
The response to variations in log g can be amplified for C ii λλ 6578/82
˚
A at
higher gravities, because they are formed on the red wing of Hα. The local con-
tinuum and therefore the line-formation depths may change as a consequence,
impacting the non-LTE effects. A correct treatment of the hydrogen Balmer
91 6.2 Model Atom Calibration
Table 6.1: Systematic uncertainties in carbon abundances (in dex, relative to
the final results, Table 6.3) caused by atmospheric parameter variations and the
assumption of LTE for the line-formation calculations.
HR 3055 ∆T
eff
∆ log g ∆ ξ LTE
−2000 K +0.2 dex +5 km s
−1
C ii 4267.2 −0.33 −0.11 −0.16 −0.40
5133.3 −0.30 −0.10 0.00 0.00
5143.4 −0.40 −0.05 0.00 0.00
5145.2 −0.32 −0.09 −0.02 0.00
5151.1 −0.30 −0.08 0.00 0.00
5662.5 −0.33 −0.13 0.00 0.00
6578.0 −0.40 −0.15 −0.10 −0.01
6582.9 −0.30 +0.02 +0.05 +0.03
C iii 4056.1 +0.21 +0.06 −0.04 +0.08
4162.9 +0.28 +0.09 −0.03 +0.25
4186.9 +0.35 +0.15 −0.08 +0.07
4663.5 +0.22 +0.07 −0.03 +0.22
4665.9 +0.26 +0.08 −0.08 +0.35
5272.5 +0.16 +0.01 0.00 0.00
C iv 5801.3 +1.06 +0.46 −0.03 +0.39
5811.9 +1.06 +0.46 −0.03 +0.39
line-opacity therefore plays an important rˆole in this context. Non-LTE effects
strengthen the Balmer line wings in particular at higher temperatures (Chap-
ter 5). This might be one of the reasons why Sigut (1996) found problems in
reproducing observed trends for the C ii λλ 6578/82
˚
A doublet at T
eff
> 25 000 K:
Hα was assumed to be formed in LTE in that work.
For HR 3055 the systematic effects exemplified in Fig. 6.11 are quantified
(note that C iii λ4647
˚
A is blended and C ii λ6151
˚
A is too weak in this case)
by deriving non-LTE carbon abundances for the modified values of T
eff
, log g
and ξ and comparing them to the final solution for individual lines. This is
summarised in Table 6.1 and visualised in Fig. 6.12. Table 6.1 also shows sys-
tematic offsets that arise from the assumption of LTE for the line-formation
calculations. This comparison helps to identify the relative importance of atmos-
pheric parameters/non-LTE effects for some key spectral lines. Note that the
C iv lines are extremely sensitive to changes in T
eff
and log g at this temperature
(31 200 K), with discrepancies amounting to up to ∼+1.0 dex in abundance for
∆ T
eff
= −2000 K
4
. The C ii/iii ionization balance is also never established for a
4
Even larger offsets in T
eff
are found with respect to the literature, up to ∆T
eff
−4000 K,
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 92
Table 6.2: Carbon abundances of the programme stars.
τ Sco HR 3055 HR 1861 HR 2928 HR 5285 HR 3468
Sco Cen Field Ori OB1 Field Sco Cen Field
(C) (dex) 8.30 ± 0.12 8.33 ± 0.08 8.33 ± 0.08 8.27 ± 0.07 8.29 ± 0.05 8.37 ± 0.10
# C lines 32 19 30 22 22 19
variation of these parameters (abundance changes down to −0.40 dex for C ii and
up to +0.35 dex for C iii when compared to the present solution). An expected
reduction of the abundances from strong lines is obtained for an increased micro-
turbulence. Note that the systematic variations of carbon abundance with ξ for
some lines are significant considering the high accuracy we are aiming at, despite
smaller effects in general than for T
eff
and log g variations.
The solutions for the modified atmospheric parameters are characterised by
a large scatter of carbon abundances from individual lines (an increase of the
statistical 1-σ uncertainties by up to ∼0.4 dex), despite similar average values.
These similar averages are however accidental. Note that variations of T
eff
show
by far the largest effects.
The use of only a few spectral lines from one ionization stage for carbon abun-
dance determinations – which is common practice in the literature – may have
serious implications on the accuracy of the results. Possible systematic discrepan-
cies, as indicated here, may remain unrecognised. Moreover, the opportunity to
improve on the atmospheric parameter determination by establishing the highly
parameter-sensitive ionization balance may be missed in such cases. This high
sensitivity of the carbon lines to atmospheric parameter variations can be used
as an important tool for precise quantitative spectral analyses of this kind of star
when applying the calibrated model atom in the future.
Some lines are practically unaffected by non-LTE effects, as indicated by Ta-
ble 6.1 (see also Sect. 6.2.2). This implies that they are almost insensitive to any
reasonable choice of model atom. On the other hand, they are highly sensitive
to the choice of atmospheric parameters. This property is highly useful for the
model atom calibration because it helps to disentangle effects due to non-LTE
from those due to inaccuracies in the stellar parameters, facilitating a reduction
of systematic errors. A central conclusion is that typical systematic uncertainties
in the atmospheric parameters can have a similar or even higher impact on the
carbon abundance determination than a neglect of non-LTE effects.
see Sects.4.1 and 6.4.
93 6.3 Results
6.3 Results
Accurate atmospheric parameters and carbon abundances are derived from line-
profile fitting by χ
2
minimisation, which puts tighter constraints than matching
only equivalent widths. The parameters are summarised in Table 5.1 and the
abundances in Table 6.2 for the six programme stars (as obtained from the ite-
rative process, Fig. 6.1). The uncertainties of T
eff
and log g are estimated from
the extremely sensitive carbon ionization equilibrium. They are lower for hotter
stars because of the additional restrictions imposed by the presence of lines from
three ionization stages. For velocities, the 1σ-uncertainties from the analysis of
the entire line ensemble are provided. Projected rotational velocity and (radial-
tangential) macroturbulence (Gray 1992, p. 407ff.) were simultaneously derived
allowing for small line-to-line variations in order to obtain an optimum fit (see
Ryans et al. 2002 for the case of B supergiants).
It is important to emphasise that the final set of atmospheric parameters
as derived from the C ii/iii/iv ionization balance is in agreement with those
from previous quantitative analyses of these stars in Chapter 5. In particular,
a simultaneous match is achieved for i) the H and He lines in the visual and
(where available) in the near-IR, including the He i/ii ionization equilibrium in
the hotter stars and ii) the spectral energy distributions from the UV to the
near IR. A similar degree of consistency is typically not obtained in comparable
studies of early-type stars.
A large quantity of carbon lines is analysed for the first time, giving consis-
tent abundances for all of them. The excellent quality of the observed spectra
combined with the present improved analysis technique allow such precise results.
The 1σ-uncertainties from the line-to-line scatter are typically of the order 0.05–
0.10 dex. The systematic uncertainties due to remaining errors in atmospheric
parameters and atomic data are estimated to be of the order 0.10–0.15 dex, taking
Table 6.1 and the present experiences from Sect. 6.2.2 as a guidline.
Details on the analysis of individual lines can be found in Table 6.3. This
summarises line identifications, level designations, log gf values, excitation po-
tential of the lower level χ
l
, and for each star equivalent widths W
λ
and the
derived non-LTE and LTE abundances. Note that as many lines as possible are
analysed per star, excluding only features with strong blends by other chemi-
cal species. This helps to follow the behaviour of each line and the quality of
the modelling at different temperatures and gravities (see the previous section).
Transitions involving autoionizing states are not accounted for explicitly in the
spectrum synthesis (however, they are considered via resonances in the photoion-
ization cross-sections). These are lines like C ii λλ 4075 (a blend with O ii), 4318
(a blend with S ii), 4374.3, 4375.1, 4376.6, 4411.2/5 and 4627.4
˚
A, which have
sometimes been used for abundance determinations in previous studies.
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 94
Table 6.3: Carbon abundance analysis for the programme stars. bl: blend with other line;. . . : too weak or absent;XX: LTE does not reproduce the emission.
τ Sco HR 3055 HR 1861 HR 2928 HR 5285 HR 3468
Ion λ (
˚
A) l − u log gf χ
l
(eV) W
λ
NLTE LTE W
λ
NLTE LTE W
λ
NLTE LTE W
λ
NLTE LTE W
λ
NLTE LTE W
λ
NLTE LTE
C ii 3919.0 3p
2
P − 4s
2
S −0.53 16.33 bl bl bl bl 90 8.21 8.04 bl
3920.6 ” −0.23 16.33 29 8.23 7.98 35 8.40 8.15 68 8.30 8.04 69 8.30 8.09 97 8.21 8.04 123 8.34 8.34
4267.0/2 3d
2
D − 4f
2
F
0
0.56/0.74 18.04 102 8.45 7.88 113 8.46 8.06 135 8.34 7.64 161 8.25 7.59 220 8.26 7.98 230 8.33 7.92
5133.0/3 (
3
P
0
)3s
4
P
0
− (
3
P
0
)3p
4
P −0.21/−0.18 20.70 19 8.29 8.28 19 8.34 8.34 52 8.41 8.41 42 8.27 8.27 52 8.30 8.30 71 8.35 8.33
5137.3 ” −0.91 20.70 ... ... 9 8.46 8.44 6 8.39 8.29 12 8.44 8.44 ...
5139.2 ” −0.71 20.70 5 8.38 8.34 ... 14 8.46 8.42 10 8.39 8.27 13 8.34 8.34 15 8.34 8.34
5143.4 ” −0.22 20.70 11 8.33 8.31 10 8.44 8.44 27 8.39 8.39 26 8.33 8.30 28 8.34 8.34 35 8.36 8.33
5145.2 ” 0.19 20.71 22 8.27 8.29 19 8.36 8.36 50 8.40 8.40 39 8.29 8.28 43 8.29 8.24 62 8.32 8.31
5151.1 ” −0.18 20.71 14 8.33 8.33 11 8.34 8.34 32 8.36 8.44 22 8.32 8.32 27 8.29 8.30 40 8.36 8.32
5648.1 (
3
P
0
)3s
4
P
0
− (
3
P
0
)3p
4
S −0.42 20.70 6 8.27 8.27 ... 18 8.44 8.41 11 8.39 8.37 19 8.34 8.38 28 8.34 8.34
5662.5 ” −0.25 20.71 7 8.27 8.27 4 8.37 8.37 19 8.38 8.36 16 8.34 8.29 21 8.34 8.36 31 8.34 8.29
6578.0 3s
2
S − 3p
2
P
0
−0.03 14.45 50 8.02 8.01 66 8.24 8.24 138 8.20 8.51 115 8.20 8.37 150 8.27 8.77 230 8.40 8.84
6582.9 ” −0.33 14.45 31 7.94 7.97 37 8.29 8.34 108 8.20 8.34 94 8.20 8.25 125 8.28 8.56 165 8.40 8.74
6779.9 (
3
P
0
)3s
4
P
0
− (
3
P
0
)3p
4
D 0.02 20.70 ... ... 19 8.19 8.26 36 8.12 8.14 33 8.24 8.26 60 8.21 8.24
6780.6 ” −0.38 20.70 ... ... 11 8.19 8.26 bl bl bl
6783.1 ” 0.30 20.71 ... 12 8.22 8.14 41 8.24 8.37 5 8.19 8.31 36 8.24 8.33 65 8.24 8.39
6787.2 ” −0.38 20.70 ... ... 11 8.24 8.34 11 8.19 8.29 12 8.29 8.36 30 8.34 8.39
6791.5 ” −0.27 20.70 ... ... 13 8.24 8.31 18 8.19 8.29 20 8.29 8.34 20 8.26 8.29
6800.7 ” −0.34 20.71 ... ... 10 8.27 8.36 12 8.34 8.36 10 8.24 8.26 ...
6151.3/5 4d
2
D − 6f
2
F
0
−0.15/0.02 20.84 −8 8.29 XX ... ... ... 22 8.34 7.89 18 8.60 7.84
6461.9 4f
2
F
0
− 6g
2
G 0.42 20.95 −18 8.39 XX ... −9 8.34 XX ... 28 8.29 7.81 14 8.60 7.84
C iii 4056.1 4d
1
D
0
− 5f
1
F 0.27 40.20 33 8.44 8.44 45 8.28 8.36 10 8.36 8.36 6 8.27 8.19 ... ...
4152.5 (
3
P
0
)3p
3
D − 5f
3
F
0
−0.11 40.06 28 8.37 8.54 bl 7 8.37 8.37 ... ... ...
4162.9 ” −0.84 40.06 46 8.33 8.57 58 8.29 8.54 bl bl ... ...
4186.9 4f
1
F − 5g
1
G 0.92 40.01 67 8.57 8.41 92 8.34 8.41 20 8.32 8.24 15 8.24 8.20 ... 14 8.49 8.45
4515.8 4p
3
P − 5s
3
S −0.28 39.40 18 8.21 8.24 bl 4 8.40 8.40 ... ... ...
4516.8 ” −0.06 39.40 19 8.23 8.25 bl 6 8.37 8.33 ... ... ...
4647.4 3s
3
S − 3p
3
P 0.07 29.53 188 8.30 ≥9.0 bl 89 8.29 8.64 68 8.26 8.54 9 8.26 8.25 54 8.44 8.84
4650.2 ” −0.15 29.53 153 8.34 ≥9.0 bl 66 8.33 8.59 54 8.28 8.59 9 8.34 8.34 bl
4651.5 ” −0.63 29.53 113 8.44 9.00 bl 43 8.39 8.54 35 8.28 8.54 bl bl
4663.5 (
3
P
0
)3s
3
P
0
− (
3
P
0
)3p
3
P −0.61 38.22 13 8.17 8.44 18 8.27 8.49 5 8.37 8.40 ... ... ...
4665.9 ” −0.03 38.22 37 8.16 8.60 50 8.24 8.59 13 8.38 8.62 ... ... ...
5253.6 (
3
P
0
)3s
3
P
0
− (
3
P
0
)3p
3
S −0.71 38.22 9 8.29 8.39 ... ... ... ... ...
5272.5 ” −0.49 38.23 19 8.32 8.42 14 8.28 8.28 ... ... ... ...
5695.9 3p
1
P − 3d
1
D 0.02 32.10 82 8.22 8.29 bl 10 8.34 8.34 bl ... ...
6731.0 (
3
P
0
)3s
3
P
0
− (
3
P
0
)3p
3
D −0.29 38.22 11 8.37 8.18 ... ... ... ... ...
6744.3 ” −0.02 38.23 11 8.33 8.09 ... ... ... ... ...
8500.3 3s
1
S − 3p
1
P −0.48 30.64 84 8.21 8.59 145 8.40 8.65 bl bl ... ...
C iv 5801.3 3s
2
S − 3s
2
P
0
−0.19 37.55 13 8.34 8.59 53 8.45 8.84 ... ... ... ...
5811.9 3s
2
S − 3s
2
P
0
−0.49 37.55 12 8.34 8.59 34 8.45 8.84 ... ... ... ...
95 6.3 Results
Figure 6.13: Examples of line profile fits (black line) to the observed high-quality
spectrum (grey line) of τ Sco.
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 96
Figure 6.14: Non-LTE and LTE abundances derived from line profile fits to in-
dividual C ii-iv lines in the programme stars as a function of equivalent width.
The ID of the stars and the spectral classification is given in the upper left corner
of each panel. The grey rectangles correspond to 1σ-uncertainties of the stellar
carbon abundance from the line-to-line scatter. Identification of lines with high
sensitivity to non-LTE effects is displayed. Emission lines are marked by crosses
(C ii λλ6151 and 6462
˚
A in τ Sco and λ6462
˚
A in HR 1861): LTE calculations are
not able to reproduce them even qualitatively.
The high quality of the line fits to the observed spectrum of τ Sco is demon-
strated in Fig. 6.13, for almost all the analysed transitions. Similar information
for the other programme stars can be found in Figs. D.1–D.5 in Appendix D.
Of central importance is that the abundances derived from the individual lines
show a small scatter in each sample star. Good fits to individual lines can al-
most always be obtained, however this does not imply consistency in the entire
analysis. An example are the abundances from Fig. 6.12, which were also derived
from high-quality line fits, but which show inconsistencies, expressed as a large
line-to-line scatter, nonetheless. Another example from many other studies are
discrepant lines that are excluded from the analysis in order to reduce the sta-
tistical uncertainties. The present work improves on this because the underlying
physics is solved in a more consistent way.
97 6.4 Comparison with Previous Work
A single exception to the overall good line fits is found: the doublet
C ii λλ4267.0/2
˚
A in τ Sco, where the fine-structure components are resolved
(Fig. 6.13). The synthetic profile is slightly broader than the observed one, even
when neglecting micro- and macroturbulent broadening. Such a detail cannot be
observed in the other stars because of their higher v sin i.
The data from Table 6.3 are visualised in Fig. 6.14, where abundances are
shown as a function of W
λ
. Excellent consistency is found in the case of the non-
LTE analysis while the quality of the results is considerably degraded in LTE.
Note that the sub-set of the strong lines provides non-LTE abundances that are
in good agreement with those derived from the entire sample of lines. This proves
that the reference carbon model is suited well for analyses of low-S/N spectra and
fast-rotating stars, where only the strongest lines are measurable and where such
a consistency check is not feasible. Note also that the non-LTE effects differ from
line to line and also from star to star.
One case requires further discussion: the C ii λλ 6578/82
˚
A doublet in τ Sco.
The remaining problems may be an artifact from the data reduction. The loca-
tion of the red wing of Hα coincides with a bad column of the CCD of FEROS
(the spectral orders are oriented along columns in this spectrograph). A perfect
correction for this cannot be provided. Different wavelengths are affected, de-
pending on the radial velocity shifts of the object. Note that the region around
C ii λλ 6578/82
˚
A doublet has been normalised relative to the local continuum in
Fig. 6.13.
6.4 Comparison with Previous Work
The main sources of systematic uncertainties that can bias abundance analyses,
as identified in Sects. 6.2.2 & 6.2.4: atomic data, i.e. different model atoms, and
stellar parameters, i.e. in particular effective temperature scales are discussed in
this section. A more comprehensive comparison is too complex to be performed
because of the large number of variables involved, which in some cases are not even
documented. This includes numerous factors related to observation (e.g. quality
of the analysed spectra, continuum rectification, equivalent width measurements,
line blends), model atmospheres (e.g. codes, abundance standards/linelists con-
sidered for line blanketing) and line-formation calculations (e.g. treatment of
line blocking, oscillator strengths, line broadening). A comparison of the present
results on carbon abundances in the solar neighbourhood with previous studies
will be made in an extra section, because of the wider implications.
6.4.1 Predictions from Different non-LTE Model Atoms
Non-LTE line profiles based on the model atom of reference and the Eber &
Butler (1988, EB) model have been computed and compared with Sigut’s (1996)
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 98
Figure 6.15: Predicted equivalent widths of some C ii lines from different ap-
proaches as a function of T
eff
. All calculations were performed for log g = 4.0,
ξ =5 km s
−1
and (C) = 8.55, in order to facilitate a comparison with Sigut (1996).
non-LTE data and LTE results in Fig. 6.15. All models are calculated for the
same set of atmospheric parameters and carbon abundances. Note that only one
fundamental difference exists: the present computations account for non-LTE
populations for hydrogen while Sigut assumes LTE. The consequences of this
have been discussed in Sect. 6.2.4.
The model of EB is widely applied in the literature for non-LTE abundance de-
terminations in OB stars. On the other hand, Sigut’s alternative model improves
somewhat on reproducing the observed behaviour of C ii λλ 4267 and 6578/82
˚
A
in OB stars. For this reason it constituted the starting point for other C ii model
atoms, by Korotin et al. (1999), by Przybilla et al. (2001) and the present work.
Consistent abundances from the application of the model atom of reference to
observations are obtained for the C ii λ 4267
˚
A transition, as shown in Sect. 6.3.
99 6.4 Comparison with Previous Work
Consequently, the good agreement with Sigut’s predictions indicates that his
model atom is also highly useful for abundance determinations from this line.
Note that Sigut compares his results (for (C) = 8.55) only qualitatively with ob-
served equivalent widths from different sources, using stellar parameters derived
in one of that studies, or from a photometric T
eff
-calibration. A reduction of car-
bon abundance and an improved determination of atmospheric parameters like
proposed in the present work may bring observation in much better agreement
with his predictions (his Fig. 1). In the region ∼22 000 K ≤T
eff
≤28 000 K good
agreement is also found with the predictions of the EB model atom. However,
this model provides too large equivalent widths and therefore lower abundances
outside this region.
The analysis provided in the present work gives abundances from the λλ 6578/
82
˚
A doublet consistent with that from other lines (τ Sco may be an exception, as
discussed above). The trends predicted with the model atom of reference differ
from the three other model calculations. The two other non-LTE predictions
agree qualitatively up to T
eff
∼22 000 K, Sigut’s model indicating larger and EB
smaller equivalent widths, respectively. However, the present model shows non-
LTE strengthening throughout, approaching LTE at the high T
eff
-limit, while the
two other models imply pronounced non-LTE weakening at T
eff
25 000 K. Note
that this doublet experiences significant non-LTE effects only when the lines are
strong, e.g. for high values of carbon abundance like those discussed in Fig. 6.15.
The non-LTE effects are abundance-dependent, they get reduced with decreasing
abundance.
Finally, the predicted equivalent widths for the widely used C ii multiplet
including λ5145
˚
A are practically independent of the model assumptions. The
lines are close to LTE. Any inconsistencies of carbon abundances based on this
multiplet may be related to other effects, but not by the choice of the model
atom. This multiplet, originated in the quartet spin system, was not analysed by
Sigut.
6.4.2 Effective Temperatures
Abundance determinations can be systematically biased even in the case where a
realistic non-LTE model atom is available. The atmospheric parameters need to
be derived accurately as well. The effects of inappropriately chosen atmospheric
parameters have been quantified in Sect. 6.2.4. Largely discrepant abundances
are found from different ionization stages in particular for inaccurate effective
temperatures. This difficulty had to be faced at the beginning of the present work,
when effective temperatures derived from common photometric calibrations were
adopted. It was not possible to establish the ionization equilibrium of C ii/iii
even for the ‘lines in LTE’ in that case. In order to constrain all variables that are
involved in the analysis simultaneously and in a self-consistent way an iterative
approach was provided.
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 100
Figure 6.16: Comparison of the final T
eff
for the sample stars with values de-
rived from photometric calibrations: T
eff
(Q) (Daflon et al. 1999; Lyubimkov et
al. 2002: LRRL) and T
eff
([u − b]) (Napiwotzki et al. 1993). Photometric in-
dices are computed from SIMBAD data. Spectroscopic temperatures according
to Kilian (1992) are also displayed.
The final values of the present work for T
eff
are compared with other deriva-
tions in Fig. 6.16. The comparison concentrates on widely employed photometric
T
eff
-calibrations, which provide a fast determination of atmospheric parameters
as required for the analysis of larger samples of stars. This includes methods
using broad-band Johnson photometry, like the reddening-free Q index (Daflon
et al. 1999; Lyubimkov et al. 2002) or small-band Str¨omgren photometry, as the
T
eff
([u − b]) calibration of Napiwotzki et al. (1993). The required Johnson and
Str¨omgren magnitudes for the individual stars are adopted from the SIMBAD
database. The discrepancies can be large, amounting up to ∼4000 K. This can
be much larger than the offsets studied in Sect. 6.2.4. The differences in abun-
dance from C ii and C iii can then achieve factors 5, similar to what is found
in previous studies, e.g. Daflon et al. (2001b). The T
eff
-values of this work are
typically higher than those from the photometric estimates. The sample of stars
analysed here is too small to facilitate an improved empirical T
eff
(Q)-calibration
on its own. However, there is an indication that giants have cooler atmospheres
than main sequence stars at the same photometric index. This is qualitatively in
agreement with the findings of Lyubimkov et al. (2002).
Spectroscopic T
eff
-determinations by Kilian (1992) are also shown in Fig. 6.16.
Kilian based her study on line-blanketed LTE model atmospheres of Gold (1984),
grids of non-LTE line profiles for hydrogen (Herrero 1987) and non-LTE ionization
101 6.5 The Stellar Present-Day C Abundance in the Solar Neighbourhood
equilibria of Si ii-iv (using the model atom of Becker & Butler 1990). Therefore,
her approach is conceptually the most similar to the present one despite considera-
ble progress made over the past 15 years, and, as a consequence, gives the closest
agreement with the values derived in this Thesis work (except for one giant).
The available photometric calibrations for T
eff
-determinations in early B-type
stars allow T
eff
values to be estimated. These can be used as starting points for
a further refinement of the atmospheric parameters via well-understood spectro-
scopic indicators. In the optimum case the model should agree with all available
indicators simultaneously: multiple H and He lines and metal ionization equili-
bria. All atmospheric parameters could then be tightly constrained. Such a step
is essential if the 1σ-uncertainties in the abundance analysis are required to be
smaller than a factor ∼2 (∼0.3 dex).
6.5 The Stellar Present-Day C Abundance in
the Solar Neighbourhood
Early-type stars can act as tracers for the present-day chemical composition in the
solar neighbourhood. The small sample of stars analysed here is suited to address
this topic, as the objects are randomly distributed over nearby OB associations
(three stars, see Table 6.2) and the field (the other stars). A highly-homogeneous
carbon abundance of (C) = 8.32±0.04 (1σ statistical uncertainty) is found for
the sample. In continuation of the discussion in the last section this value is
compared with results from previous studies of early B-type stars in LTE and
non-LTE. Mean abundances, statistical 1σ-uncertainties and the number of ana-
lysed stars in the individual studies are summarised in Table 6.4. A visualisation
of the abundance distributions is provided in Fig. 6.17. All stars are located at
distances shorter than ∼1 kpc from the Sun and at galactocentric distances R
g
within up to 500 pc difference with respect to the location of the Sun R
g
.
LTE. The abundances from Kane et al. (1980) are much lower than the present
values. They also show a large spread in abundance (∼1.3 dex), which may be a
consequence of the neglect of non-LTE effects for the line-formation calculations
of C ii λ4267
˚
A (the only line analysed) and of the use of a photometric T
eff
-
scale (from Str¨omgren indices), see Sect. 6.4.2. Two stars of the present sample
are discussed by Kane et al. as well. They derive (C) = 7.36 for τ Sco and
(C) = 7.97 for HR 5285 (χ Cen), in contrast to the results derived in the present
work: (C) = 8.30±0.12 and 8.29±0.05, respectively (see Table 6.2).
Two other LTE studies, not displayed in Fig. 6.17, also provide lower car-
bon abundances from early B-type stars in the solar vicinity. Barnett & Mc-
Keith (1988) base their effective temperatures on photometric calibrations and
derive the carbon abundances from the C ii λλ 6578/82
˚
A multiplet. Note that
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 102
they reject several stars, like τ Sco at (C) = 7.24, for computing their average
value: (C) = 8.2±0.2 (all their stars are considered in Table 6.4). More recent
LTE results by Rolleston et al. (2000) indicate the so far most depleted carbon
abundances in the solar neighbourhood. Their results are obtained from the ana-
lysis of the non-LTE-sensitive C ii λλ 3920 and 6578/82
˚
A lines. Their abundance
of the Cep OB3 association is considered here.
Non-LTE. On the other hand, non-LTE analyses (based on the Eber & Butler
model atom, using different linelists) find systematically higher abundances than
the LTE studies. However, a large spread in abundance is found for each sample.
The present results can be directly compared with those of the purely spec-
troscopic study of Kilian (1992). She derived carbon abundances for stars in
three OB associations (Ori OB1, Sco-Cen, Sgr OB1) and the field from up to 6
unblended C ii transitions involving autoionization levels
5
in the spectral range
between ∼4000 to 5000
˚
A. All the programme stars of the present work are in-
cluded in Kilian’s study, her carbon abundances for these six stars tracing almost
the entire width of her total abundance distribution. A mean (C) = 8.19±0.12 is
derived from the six stars when adopting her results. This comparison is highly
important as it shows that a careful analysis may drastically reduce the sta-
tistical scatter and may also imply a considerable systematic shift of the mean
abundance (towards a higher value in this case). Similar to Kilian, Gummersbach
et al. (1998, not shown in Fig. 6.17) also performed a consistent spectroscopic
analysis of five stars in the solar vicinity as part of their Galactic abundance
gradient study with improved model atmospheres (ATLAS9). Only one C ii line
involving autoionization levels was used for the abundance analysis. Their results
also indicate systematically lower abundances and a considerable spread.
Examples of non-LTE analyses for larger samples of stars based on effec-
tive temperatures derived from photometric calibrations are also displayed in
Fig. 6.17. Gies & Lambert (1992) and Cunha & Lambert (1994) derived T
eff
from
calibrations based on Str¨omgren photometry and non-LTE abundances from the
C ii multiplet around λ5145
˚
A and where possible from C ii λλ5648/62
˚
A. Both
multiplets originate in the quartet spin system and they are almost unaffected
by non-LTE effects. For the comparison five supergiant stars have been ex-
cluded from the sample of Gies & Lambert (1992). These objects may expose
nuclear-processed material at their surface to a much higher degree than main
sequence and giant stars, and they are too far away to be considered members
of the solar neighbourhood. Gies & Lambert (1992) have two stars in com-
mon with the present sample, indicating systematically lower abundances: for
HR 2928 ∆(C) −0.3 dex and for HR 1861 ∆(C) −0.2 dex (from the same
lines). They also consider the doublet C ii λλ6578/82
˚
A for several stars of their
5
Unfortunately these lines are not included explicity in the presents spectrum synthesis,
such that it is not possible to study sources of systematic discrepancies in more detail.
103 6.5 The Stellar Present-Day C Abundance in the Solar Neighbourhood
Table 6.4: Carbon abundances from OB stars in the solar neighbourhood.
Study (C) # stars
non-LTE
present work 8.32±0.04 6
Kilian (1992) 8.23±0.15 20
Kilian (1992): present sample 8.19±0.12 6
Gies & Lambert (1992) 8.20±0.16 31
Cunha & Lambert (1994) 8.40±0.11 15
Gummersbach et al. (1998) 8.20±0.11 5
Andrievsky et al. (1999) 8.21±0.19 10
Daflon et al. (1999, 2001a) 8.22±0.13 9
LTE
Kane at al. (1980) 7.77±0.32 28
Barnett & McKeith (1988) 8.15±0.33 12
Rolleston et al. (2000) 7.63±0.16 4
sample, but obtain even lower abundances compared to the lines from the quar-
tet spin system (by approximately another −0.3 dex). Cunha & Lambert (1994)
have only one star in common with us, HR 1861. They found a similar abun-
dance for this star as Gies & Lambert (1992). More recently, Daflon et al. (1999,
2001a) derived temperatures from their calibration of the Johnson Q-parameter
and non-LTE abundances from the C ii multiplet around λ5145
˚
A. In the present
comparison only stars in the Cep OB2, Cyg OB7 and Lac OB1 associations are
considered, their other objects are too distant. None of the present programme
stars is included in their sample. Finally, Andrievsky et al. (1999) estimated
their atmospheric parameters (T
eff
, log g) from photometry alone. Their carbon
abundances are essentially based on the C ii λλ5122-5151 and the 6578/82
˚
A mul-
tiplets.
All the data on carbon abundances from early-type stars summarised above were
derived from the physical interpretation of observation, which may be affected
by many sources of systematic error (see Sect. 6.2). These data are interpreted
in turn to test models of massive star evolution and the chemical evolution of
the Galaxy. These in turn are anchor points for the interpretation of stellar and
galactochemical evolution. Therefore it is important to analyse the conclusions
drawn from a diagram like Fig. 6.17 in this context.
LTE abundance studies are excluded from the further discussion, as they have
been shown to be systematically biased. Previous non-LTE studies all show a
broad range of carbon abundances in the solar neighbourhood, spanning a factor
of ∼10 in total. A cumulative value of (C) = 8.25±0.16 can be derived from
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 104
Figure 6.17: Comparison of carbon abundances derived from the present sample
of early B-type stars in the solar vicinity with results from the literature for similar
objects (R
g
−R
g
≤ 500 pc, the stars belong to the field and to OB associations)
and with the most recent solar values. The binsize of the histograms is related
to the statistical uncertainty of each sample. The present sample is not large
enough for a statistical comparison, therefore a column with a thickness of the
uncertainty is displayed. The programme stars coincide with six objects from
Kilian (1992): her abundances show a larger spread for these. More objects need
to be analysed in order to improve on the statistics. Note that the systematic
effects on the carbon abundances due to rotational mixing in the course of stellar
evolution and due to the Galactic abundance gradient should be only of the order
of the uncertainty found here, see the text for details.
the 75 measurements discussed in Fig. 6.17, similar to that is derived by the
individual studies.
If such large spread were realistic its physical description would require:
i) mechanisms that alter the atmospheric structure, such as sufficiently strong
magnetic fields or diffusion/radiative levitation processes acting on short timescales
and independent of atmospheric parameters and evolutionary age. This would
imply that the basic assumption of homogeneity of the atmosphere may be no
longer valid and as a consequence the modelling techniques applied so far may
be inadequate; and/or
ii) extremely efficient depletion mechanisms in the course of stellar evolution,
105 6.5 The Stellar Present-Day C Abundance in the Solar Neighbourhood
Table 6.5: Carbon abundances of different objects in the solar vicinity.
Objects (C) Source
B stars (pristine value) 8.35±0.05 present work
B stars 8.25±0.08 Herrero (2003)
Orion H ii (gas) 8.42±0.02 Esteban et al. (2004)
Orion H ii (gas+dust) 8.52±0.02 Esteban et al. (2004)
young F and G stars 8.55±0.10 Sofia & Meyer (2001)
ISM 8.15±0.06 Sofia & Meyer (2001)
Sun 8.39±0.05 Asplund et al. (2005)
Sun 8.52±0.06 Grevesse & Sauval (1998)
by one to two orders of magnitude larger than currently predicted (e.g. Meynet
& Maeder 2003). Already on the Main Sequence mixing with matter from the
stellar core would require a higher efficiency than currently predicted for con-
vection during the first dredge-up. Eventually, also enrichment mechanisms may
be required which contradict nucleosynthesis (the 3α-process is not active in OB
dwarfs and giants, carbon is instead depleted by the CNO cycle); and/or
iii) an enormous chemical inhomogeneity of the present-day interstellar material
in the solar neighbourhood out of which the stars have been formed. This would
require the possibility to change abundances practically instantly even within
single clusters by amounts that are otherwise attributed to the past ∼12 Gyrs of
Galactochemical evolution (e.g. Chiappini et al. 2003).
The present work avoids all these fundamental problems. It implies homoge-
neous abundances after improving the modelling and the analysis methodology,
bringing all model aspects into agreement with high quality observed spectra at
once, which was not achieved so far. It conforms with the finding of a highly uni-
form gas-phase carbon abundance in the interstellar medium (out to distances of
1.5 kpc from the Sun, e.g. Sofia & Meyer 2001, and references therein), despite
a systematic offset of the absolute value because of the dust contribution. The
uncertainties of that indicator (see Table 6.5) are determined by the accuracy to
which the oscillator strengths of the resonance lines in the UV are known, and
should not exceed more than ∼10%.
The present work also agrees with galactochemical evolution models, which
predict homogeneous abundances in the solar neighbourhood (e.g. Chiappini et
al. 2003). A variety of hydrodynamic processes should keep the ISM chemically
well-mixed on small time-scales (Roy & Kunth 1995). The variation due to
the Galactic abundance gradient should amount to up to ∼0.04–0.08 dex kpc
−1
(one kpc is the maximum galactocentric distance sampled in our comparison).
Another aspect concerns the notably sub-solar abundances from early-type stars
found so far (e.g. Herrero 2003, with respect to the old solar standard, Grevesse
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 106
& Sauval 1998 – see Table 6.5). The present findings remedy the situation, in
particular if accounting for the recently revised solar carbon abundance (Asplund
et al. 2005, see also Table 6.5). Galactochemical evolution models predict an
enrichment of the carbon abundance by ∼10-20% (0.04-0.08 dex) over the past
4.5 Gyrs since the birth of the Sun, depending on the model assumptions. The
remaining discrepancies can now be explained much more easily by invoking the
usual assumptions: the birth of the Sun in a slightly more metal-rich region of
the Galaxy and subsequent orbit diffusion or a recent infall of metal-poor gas to
the ISM in the solar neighbourhood.
Finally, rotationally-induced mixing with CN-processed material from the core
may change the atmospheric composition. The effect is expected to increase with
initial angular momentum of a star and with evolutionary age. Theory predicts
a depletion of carbon by ∼0.03 dex for a star of 20 M
with initial rotational
velocity 300 km s
−1
evolving from the zero-age Main Sequence to the end of the
Main Sequence stage, reaching ∼0.15 dex in the supergiant stage (e.g. Meynet &
Maeder 2003). The presence of a magnetic field may amplify rotational mixing
(Maeder & Meynet 2005), however the effects on abundances are not expected
to exceed a factor ∼2. As absolute rotational velocities of stars can be measured
only for a few exceptionally cases, this topic can be addressed comprehensively
only by a statistical approach. The sample of stars analysed in this work is not
large enough for this. However, no significant trend of carbon abundances with
evolutionary age is found for these apparently slow-rotating stars (only τ Sco is
suggested to be a real slow rotator, Donati et al. 2006). The abundances de-
rived may not correspond to the pristine values nonetheless, but it is unlikely
that all the stars are very fast rotators seen pole-on. Small corrections by up to
∼+0.05 dex per star (depending on evolution stage) are predicted by theory for
objects of average rotation. A correction of (C)=+0.03 dex for the average value
of the present sample is derived from considering carbon and nitrogen simultane-
ously (Nieva & Przybilla 2007c). The nitrogen abundances have been obtained
from ∼20–70 transitions per star using the same analysis methodology and the
same atmospheric parameters as derived here, applying an updated model atom
of Przybilla & Butler (2001). The correction to carbon allows us to derive an
average pristine abundance.
This pristine C abundance derived from early B-type stars in the solar neigh-
bourhood is compared with other indicators in Table 6.5. The value is not only
in agreement with the revised solar abundance but also with the gas-phase abun-
dance derived for the Orion nebula. The finding of a uniform abundance matches
with a homogeneous interstellar medium abundance. A disagreement is found
only with abundances from young F & G-type stars. This may be resolved when
hydrodynamic 3D-analyses of late-type stars become routine, as the discrepancy
is similar to the difference between the old and the revised solar standard, which
is the result of such an improved analysis. The conclusion of Sofia & Meyer (2001)
that B-type stars are not reliable proxies for present-day abundances in the solar
107 6.6 Summary
neighbourhood may need a revision in view of the present results.
Despite the small sample size analysed so far, the highly accurate results found
here indicate that the sub-solar average value and the large scatter of carbon
abundances in early-type stars found in previous non-LTE studies could be mostly
a consequence of systematic uncertainties. These may be easily introduced by the
choice of inappropriate atomic data and/or stellar parameters, as it was shown
here. However, a confirmation of these findings from analysis of a larger sample
of stars is required, also for other elements.
6.6 Summary
The motivation of this work was the solution of a long-standing problem in stel-
lar astrophysics: the reliable determination of carbon abundances from early-
type stars. For this purpose a sophisticated C ii/iii/iv model atom for non-LTE
line-formation calculations was constructed, based on input atomic data that
were carefully selected by an empirical calibration process. This was performed
through an extensive iteration scheme that allow the input atomic data and simul-
taneously the atmospheric parameters of the programme stars to be constrained,
which is the basis for all further studies. The calibration sample consists of six
bright and apparently slow-rotating early B-type stars in the solar neighbour-
hood, with high-S/N, high-resolution spectra and a broad wavelength coverage.
The C ii/iii/iv ionization balance gives atmospheric parameters that are
highly consistent with previous spectroscopic analyses: results from a non-LTE
study of hydrogen and helium in the visual and in the near-IR, including the
establishment of the He i/ii ionization equilibrium in the hotter stars, and the
reproduction of the spectral energy distributions from the UV to the near-IR
(see Chapter 5). All carbon lines considered here, from up to three ioniza-
tion stages, indicate similar abundances. The statistical 1σ-uncertainties of the
carbon abundance in each star are of the order ∼0.05-0.10 dex. The system-
atic 1σ-uncertainties are estimated to be ∼0.10-0.15 dex. The resulting aver-
age carbon abundance from C ii/iii/iv in the sample stars is highly uniform,
ε(C) = 8.32±0.04.
These results of unprecedented precision provide important constraints for
stellar evolution and the chemical evolution of the Galaxy, despite the small
sample size: i) They suggest that carbon depletion due to rotational mixing in
the course of stellar evolution is small for stars without excessive initial angular
momentum (by <0.05 dex) up to the giant stage, in agreement with theoretical
predictions. ii) In consequence they suggest that the present-day carbon abun-
dance in the solar neighbourhood is higher and much more homogeneous than
indicated by previous work on early-type stars. This is consistent with the uni-
form abundance found in studies of the ISM and with predictions from models
Chapter 6: Non-LTE Line Formation for Carbon: Self-Consistent Analysis 108
of the chemical evolution of the Galaxy. Moreover,the re-evaluated stellar value
is in agreement with the gas-phase abundance derived for the Orion H ii region
and the recently revised solar abundance.
It was also shown how important is the careful choice of input atomic data for
non-LTE analyses. Not only accurate data for radiative transitions are required
but also for collisional transitions. The carbon abundance analysis also turned out
to react highly sensitively to the choice of atmospheric parameters, in particular
to effective temperatures.
Chapter 7
Conclusions
The motivation of this work was to find a solution of a long-standing problem
in stellar astrophysics: the reliable determination of carbon abundances from
OB-type stars. For this purpose, improvements in the spectral modelling and
in the quantitative analysis of observed spectra are made and further applied
to a sample of six stars randomly distributed in the solar vicinity, covering a
wide parameter range. The spectra have high-S/N, high resolution and broad
wavelength coverage.
In a first step, a hybrid non-LTE approach, allowing for departures from local
thermodynamic equilibrium LTE in the spectrum synthesis, has been thoroughly
tested for H and He line-formation calculations in OB stars for the first time.
Such approach is often employed in the literature for analyses of metal lines. The
present analysis uses recently improved model atoms and is much more exten-
sive than published work, which considered selected lines only. The synthetic
spectra match simultaneously almost all measurable hydrogen and helium lines
in the optical and (where available) also in the near-infrared spectral range of the
programme stars, with only a few well-understood exceptions.
The comparison of state-of-the-art line-blanketed non-LTE (ostar2002 grid
of Lanz & Hubeny 2003) and LTE (Padova grid by Munari et al. 2005) mod-
els confirms that the atmospheric structure of dwarf and giant OB stars is de-
scribed well under the assumption of LTE, but their spectral energy distribution
and, most importantly, their line spectra are not. For these stars in the range
20 000 K ≤T
eff
≤35 000 K and 3.0 ≤log g ≤4.5, the present hybrid non-LTE ap-
proach is equivalent to full hydrostatic non-LTE computations. It succeeds also
in providing synthetic spectra that correctly reproduce the observed He i singlet
lines, avoiding inconsistencies recently reported in the literature. In contrast to
this, pure LTE modelling based on the Padova grid (or equivalent computations
with atlas9+synthe) may give rise to considerable systematic errors in the
atmospheric parameter determination (T
eff
, log g) and to subsequent elemental
abundance studies for the hotter stars in particular.
109
Chapter 7: Conclusions 110
In a second step, a sophisticated C ii/iii/iv model atom for non-LTE line-
formation calculations was constructed, based on input atomic data carefully
selected in an empirical calibration process. This was performed through a self-
consistent quantitative spectrum analysis using an extensive iteration scheme
which facilitated both the input atomic data and the atmospheric parameters
of the calibration stars to be constrained accurately. The C ii/iii/iv ionization
balance is successfully established for all analysed carbon lines of the sample,
which provide similar abundances. The linelist includes 40 transitions suitable
for analysis over a wide wavelength range. In particular, the strongest features, of
highest importance for extragalactic applications, are also consistently modelled.
The statistical 1σ-uncertainties of the carbon abundance in each star are of the
order 0.05-0.10 dex. The systematic 1σ-uncertainties are estimated to be ∼0.10-
0.15 dex. Typically, the systematic uncertainties of the abundance determination
remain mostly undetermined in literature, and statistical uncertainties are of a
factor ∼2–3.
Determinations of carbon abundances react highly sensitive to modifications
of the atmospheric parameters. The importance of this was vastly underestimated
in previous work. A self-consistent analysis provides atmospheric parameters
with unprecedented accuracy and with reduced systematic error: for effective
temperature the uncertainties are as low as ∼1% (literature: 5-10%), for surface
gravity ∼10% (literature: ∼25%). The systematic uncertainties in literature are
however found to be much larger than the statistical ones in the present work:
up to ∼20% for temperatures and ∼50% for surface gravities. The atmospheric
parameters derived from the carbon ionization equilibria are highly consistent
with other indicators investigated in the present work: results from a non-LTE
study of hydrogen and helium in the visual and in the near-IR, including an
establishment of the He i/ii ionization equilibrium in the hotter stars, and the
reproduction of the spectral energy distributions from the ultraviolet to the near-
infrared.
The resulting average carbon abundance for the sample stars is highly uni-
form, ε(C) = 8.32±0.04. This result of unprecedented precision provides impor-
tant constraints for stellar evolution and the chemical evolution of the Galaxy,
despite the small sample size which requires improvement on the statistics.
i) It suggest that carbon depletion due to rotational mixing in the course of stel-
lar evolution is small for stars without excessive initial angular momentum (by
<0.05 dex) up to the giant stage, in agreement with theoretical predictions.
ii) In consequence, this result suggests that the present-day carbon abundance in
the solar neighbourhood is higher and much more homogeneous than indicated by
previous work on early-type stars. This is consistent with the uniform abundance
found in studies of the interstellar medium and with predictions from models of
the chemical evolution of the Galaxy. Moreover, the carbon abundance derived
from OB-stars is now in agreement with the gas-phase abundance of the Orion
H ii region and the recently revised solar abundance.
111
This work shows that one should not underestimate the importance of a care-
ful choice of input atomic data for non-LTE analyses. Not only accurate data for
radiative transitions are required but also for collisional transitions. The carbon
abundance analysis also turned out to react highly sensitive to the choice of at-
mospheric parameters, in particular to effective temperatures. If not accounted
for properly, both factors may result in systematic errors in the interpretation of
observed spectra, which cannot be reduced by statistics, i.e. an increase of the
sample size of stars to be analysed. Photometric calibrations for effective temper-
ature determinations are found to provide only estimates. Those may be used as
starting points in the iteration scheme, but they are not sensitive enough for reli-
able quantitative spectroscopic analyses. Detailed comparisons and calibrations
of models with observed spectra are of highest importance, as these constitute
the only empirical constraints for quantitative analyses. Only then can stellar
and Galactochemical evolution models be verified in a meaningful way, and the
studies be extended to stars in other galaxies.
The present work offers wide perspectives for future studies of chemical abun-
dances from OB-type stars. A straightforward application may be the use of
available sophisticated model atoms (N, O, Mg, Si, etc.) for the analysis of
the programme stars in order to derive the present-day abundances in the solar
vicinity and to verify other ionization equilibria. The calibrated carbon model
atom allows a simplified iteration processes to be performed in other stars, i.e.
concentrating only on the stellar parameter and abundance determination, since
the atomic data are already selected. An analysis of a larger sample of stars
in the solar vicinity can constrain the present findings with higher significance.
Analyses of more distant stars can provide an accurate derivation of the Galactic
abundance gradient. OB-type stars in the Magellanic Clouds and in other nearby
galaxies in the Local Group may also be accurately studied using the present gen-
eration of large telescopes, with the limiting factor defined by the quality of the
observed spectra. The next generation of telescopes, that are currently in the
design phase, will allow even stars in more distant galaxies, of other types than
currently accessible, to be analysed.
Appendix A
Basis of
´
Echelle Data Reduction
´
Echelle means ladder in French. In astrophysics, it refers to a diffraction grating
in which the lines are ruled much further apart than those of an ordinary grating.
Spectrographs using a dispersing element like an ordinary diffraction grating or
prism produce a single spectrum. The wavelength range covered by this type
of instrument is limited by the size the CCD. An
´
Echelle grating produces a
spectrum of very high dispersion, but only over a short wavelength range in each
order, implying that the high orders will overlap. In order to overcome this effect
a cross-dispersing element is used to produce an order separation in the direction
perpendicular to the dispersion. Figure A.1 shows a example of a CCD image
of a 2-D spectrum taken with the FEROS (Fiber-fed Extended Range Optical
Spectrograph) spectrograph at the MPG/ESO-2.2m telescope in La Silla (Chile).
Some characteristic features can be seen in the figure: several absorption lines
corresponding to the stellar spectrum, telluric lines (upper right corner), and
extra signal due to the incidence of cosmic rays (bright dots). Orders in the
upper part have longer wavelengths. For each order the wavelength increases to
the right.
The idea of constructing
´
Echelle spectrographs is to work at high orders with
the highest efficiency. These instruments offer a reasonably high resolution and a
wide wavelength coverage, optimum characteristics for the study of e.g. chemical
abundances in stars, in particular for those which have sharp lines. Unfortunately,
the data reduction process of an
´
Echelle spectrum can be complex.
Different exposures are performed in an observing run: exposures with closed
shutter (for bias elimination), a flat field (for corrections of the CCD pixel sen-
sitivity), a comparison lamp (for wavelength calibration) and finally the stars.
The reduction process of an
´
Echelle spectrum from a set of images follows, in
principle, a similar procedure than that of a single-order spectrum.
• Bias elimination. The bias reflects the detector electronics influence. It is
related with the temperature of the detector.
• Flat fielding. A flat field is the result of illuminating the instrument with
113
Chapter A: Basis of
´
Echelle Data Reduction 114
Figure A.1: Example of a 2-D FEROS spectrum.
a uniform source for determining the relative sensitivity of the pixels. Flat
fielding is the process of dividing the spectra by a normalised flat-field to
remove these sensitivity variations of the system.
• Order detection and extraction. A modelling of the order location on the
2-D spectrum is performed. Then, the flux perpendicular to the dispersion
direction is summed up, giving a (extracted) 1-D spectrum. An optimum
extraction takes a weighted sum such that the S/N of the extracted spec-
trum is optimum, allowing the effects of cosmic rays to be reduced.
• Wavelength calibration. Fitting the dispersion relation from a comparison
of the spectrum of a lamp of well-identified lines (e.g. Th-Ar).
• Normalization of continuum. Fitting of the blaze function which follows
the shape of the 1-D spectrum and normalisation of the continuum to 1.
• Radial velocity correction. This step can be performed by comparison to
an appropriate synthetic spectrum.
For
´
Echelle spectra, additional complexity arises because there are more data
to extract, the orders can have a more complex shape than those from a single-
order instrument, the high dispersion used can make it difficult to distinguish
between true spectral features and cosmic-ray events, flat fielding the data can
be difficult. In some cases adjacent orders overlap slightly in the spatial di-
rection making accurate background subtraction difficult. Several other sources
115
of complexity exist, depending on the instrument. For more details on general
procedures for
´
Echelle spectra reduction the reader can refer to the didactic de-
scription of Churchill (1995) and M. Clayton (www.star.ucl.ac.uk/∼mjc/echelle).
The FEROS spectrograph and the reduction process is described in Kaufer et
al. (1999), Stahl et al. (1999) and Hensberge (1996, 2001, 2004).
Appendix B
Atomic data for H and He
Important atomic data relevant to the hydrogen and helium line formation in
the visual and the near-IR (NIR) are summarised in Table B.1. The data are
composed by wavelengths, lower and upper levels involved in the transition, os-
cillator strengths log gf, their accuracies and sources, also for the detailed Stark-
broadening data and some comments.
117
Chapter B: Atomic data for H and He 118
Table B.1: Atomic data for H and He i/ii line formation in the visual and NIR
λ (
˚
A) Transition log gf Acc. Src. Broad. Comment
H i:
3797.90 2 – 10 −1.511 AA GRC SH
3835.38 2 – 9 −1.362 AA GRC SH
3889.05 2 – 8 −1.192 AA GRC SH
3970.07 2 – 7 −0.993 AA GRC SH blend with interstellar Ca ii H
4101.73 2 – 6 −0.753 AA GRC SH
4340.46 2 – 5 −0.447 AA GRC SH
4861.32 2 – 4 −0.020 AA GRC SH
6562.80 2 – 3 0.710 AA GRC SH
8413.32 3 – 19 −1.823 AA GRC SH
8437.96 3 – 18 −1.748 AA GRC SH
8467.26 3 – 17 −1.670 AA GRC SH
8502.49 3 – 16 −1.586 AA GRC SH
8545.39 3 – 15 −1.495 AA GRC SH
8598.39 3 – 14 −1.398 AA GRC SH
8665.02 3 – 13 −1.292 AA GRC SH
8750.47 3 – 12 −1.175 AA GRC SH
He ii:
3796.34 4 – 20 −1.487 AA GRC G60, G67 blend with Hϑ
3813.50 4 – 19 −1.414 AA GRC G60, G67
3833.81 4 – 18 −1.337 AA GRC G60, G67 blend with Hη
3858.08 4 – 17 −1.255 AA GRC G60, G67
3887.45 4 – 16 −1.166 AA GRC G60, G67 blend with Hζ
3923.49 4 – 15 −1.071 AA GRC SB
a
blend with He i
3968.44 4 – 14 −0.967 AA GRC SB
a
blend with H
4025.61 4 – 13 −0.852 AA GRC SB
a
blend with He i
4100.05 4 – 12 −0.725 AA GRC SB blend with Hδ
4199.84 4 – 11 −0.582 AA GRC SB blend with N iii
4338.67 4 – 10 −0.417 AA GRC SB blend with Hγ
4541.59 4 – 9 −0.223 AA GRC SB
4685.70 3 – 4 1.181 AA GRC SB
4859.32 4 – 8 0.014 AA GRC SB blend with Hβ
5411.52 4 – 7 0.321 AA GRC SB
6560.09 4 – 6 0.759 AA GRC SB blend with Hα
a
Unpublished, priv. comm.;
b
vacuum wavelengths
Accuracy indicators – uncertainties within: AA: 1%; A: 3%
Sources of gf-values: FTS: Fernley et al. (1987); GRC: Green et al. (1957); SPA: Schiff et
al. (1971)
Sources for Stark broadening parameters: BCS69: Barnard et al. (1969); C: Cow-
ley (1971); DSB: Dimitrijevi´c & Sahal-Br´echot (1990); G60: Griem (1960); G64:
Griem (1964); G67: Griem (1967); GBKO: Griem et al. (1962); S: Shamey (1969); SB:
Sch¨oning & Butler (1989); SH: Stehl´e & Hutcheon (1999)
119
Table B.1. continued
λ (
˚
A) Transition log gf Acc. Src. Broad. Comment
He i:
3819.60 2p
3
P
o
– 6d
3
D −0.931 A FTS DSB forb. comp. missing
3819.61 2p
3
P
o
– 6d
3
D −1.153 A FTS DSB . . .
3819.76 2p
3
P
o
– 6d
3
D −1.630 A FTS DSB . . .
3867.47 2p
3
P
o
– 6s
3
S −2.037 A FTS DSB
3867.48 2p
3
P
o
– 6s
3
S −2.260 A FTS DSB
3867.63 2p
3
P
o
– 6s
3
S −2.737 A FTS DSB
3888.60 2s
3
S – 3p
3
P
o
−1.668 AA SPA G64 near core of Hζ
3888.65 2s
3
S – 3p
3
P
o
−0.765 AA SPA G64 . . .
3926.54 2p
1
P
o
– 8d
1
D −1.652 A FTS DSB blends by Si iii & S ii/iii
3935.95 2p
1
P
o
– 8s
1
S −2.772 A FTS DSB
3964.73 2s
1
S – 4p
1
P
o
−1.290 A FTS G64 in wing of H
4009.26 2p
1
P
o
– 7d
1
D −1.449 A FTS DSB
4023.98 2p
1
P
o
– 7s
1
S −2.572 A FTS DSB
4026.18 2p
3
P
o
– 5d
3
D −2.600 A FTS S
4026.19 2p
3
P
o
– 5d
3
D −0.633 A FTS S
4026.20 2p
3
P
o
– 5d
3
D −0.851 A FTS S
4026.36 2p
3
P
o
– 5d
3
D −1.328 A FTS S
4120.81 2p
3
P
o
– 5s
3
S −1.722 A FTS GBKO blends O ii, C iii, & Fe iii
4120.82 2p
3
P
o
– 5s
3
S −1.945 A FTS GBKO . . .
4120.99 2p
3
P
o
– 5s
3
S −2.422 A FTS GBKO . . .
4143.76 2p
1
P
o
– 6d
1
D −1.203 A FTS DSB blends O ii & N ii
4168.97 2p
1
P
o
– 6s
1
S −2.332 A FTS DSB strong blend with O ii
4387.93 2p
1
P
o
– 5d
1
D −0.886 A FTS S
4437.55 2p
1
P
o
– 5s
1
S −2.026 A FTS GBKO cont. diff. interst. band
4471.47 2p
3
P
o
– 4d
3
D −0.210 A FTS BCS69
4471.49 2p
3
P
o
– 4d
3
D −0.432 A FTS BCS69
4471.68 2p
3
P
o
– 4d
3
D −0.909 A FTS BCS69
4713.14 2p
3
P
o
– 4s
3
S −1.276 A FTS G64
4713.16 2p
3
P
o
– 4s
3
S −1.498 A FTS G64
4713.38 2p
3
P
o
– 4s
3
S −1.976 A FTS G64
4921.93 2p
1
P
o
– 4d
1
D −0.442 A FTS BCS69 forb. comp. to be improved
5015.68 2s
1
S – 3p
1
P
o
−0.820 AA SPA G64
5047.74 2p
1
P
o
– 4s
1
S −1.600 A FTS GBKO
5875.60 2p
3
P
o
– 3d
3
D −1.511 A FTS G64
5875.61 2p
3
P
o
– 3d
3
D 0.480 A FTS G64
5875.63 2p
3
P
o
– 3d
3
D −0.338 A FTS G64
5875.64 2p
3
P
o
– 3d
3
D 0.138 A FTS G64
5875.97 2p
3
P
o
– 3d
3
D −0.214 A FTS G64
6678.15 2p
1
P
o
– 3d
1
D 0.328 A FTS G64
7065.18 2p
3
P
o
– 3s
3
S −0.458 A FTS GBKO weak telluric line cont.
7065.22 2p
3
P
o
– 3s
3
S −0.680 A FTS GBKO . . .
7065.71 2p
3
P
o
– 3s
3
S −1.157 A FTS GBKO . . .
7281.35 2p
1
P
o
– 3s
1
S −0.854 A FTS GBKO strong telluric line cont.
10829.09 2s
3
S – 2p
3
P
o
−0.745 AA SPA G64
10830.25 2s
3
S – 2p
3
P
o
−0.268 AA SPA G64
10830.34 2s
3
S – 2p
3
P
o
−0.046 AA SPA G64
20586.92
b
2s
1
S – 2p
1
P
o
−0.424 AA SPA DSB strong telluric line cont.
21125.79
b
3p
3
P
o
– 4s
3
S −0.138 A FTS DSB
21125.89
b
3p
3
P
o
– 4s
3
S −0.360 A FTS DSB
21127.09
b
3p
3
P
o
– 4s
3
S −0.837 A FTS DSB
21137.80
b
3p
1
P
o
– 4s
1
S −0.527 A FTS DSB
Appendix C
Linefits to H and He
Synthetic profiles for a selection of 6 hydrogen Balmer and 18 He i/ii lines in the
visual are compared with observation for HR 3055, HR 1861, HR 2928, HR 3468
and HR 5285 in Figs. C.1-C.5. These are the best simultaneous fits within the
uncertainties of the parameters from Table 5.1, as obtained in the present work.
For a discussion see Sect. 5.3.
121
Chapter C: Linefits to H and He 122
Figure C.1: Non-LTE line fits to observed hydrogen and helium features in
HR 3055 (B0 III). Note that the quality of the line fits for Hα and He ii λ 4686
˚
A
in particular is better than for τ Sco apparently because of a weaker stellar wind,
cf. Fig. 5.5.
123
Figure C.2: Line fits for HR 1861 (B1 IV). For atmospheric parameters see Ta-
ble 5.1, and for further discussion see the text.
Chapter C: Linefits to H and He 124
Figure C.3: As Fig. C.2, but for HR 2928 (B1 IV).
125
Figure C.4: As Fig. C.2, but for HR 3468 (B1.5 III).
Chapter C: Linefits to H and He 126
Figure C.5: As Fig. C.2, but for HR 5285 (B2 V).
Appendix D
Linefits to C lines
Synthetic profiles for a almost all analysed carbon lines in HR 3055, HR 1861,
HR 2928, HR 5285, HR 3468. These are the best simultaneous fits within the
uncertainties of the parameters from Table 5.1. For a discussion see Sect. 6.3.
127
Chapter D: Linefits to C lines 128
Figure D.1: As Fig. 6.13, but for HR 3055. Dotted lines indicate the features not
included in the abundance determination.
129
Figure D.2: As Fig. D.1, but for HR 1861.
Chapter D: Linefits to C lines 130
Figure D.3: As Fig. D.1, but for HR 2928.
131
Figure D.4: As Fig. D.1, but for HR 5285.
Chapter D: Linefits to C lines 132
Figure D.5: As Fig. D.1, but for HR 3468.
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Acknowledgements
”Science knows no country, because knowledge belongs to humanity, and is
the torch which illuminates the world.”
Louis Pasteur
In the beginning of my studies at the National University of Tucum´an (Argentina),
I met an astronomer, Arcadio Poveda (Mexico), who encouraged me to follow my
dream of studying the stars through the rigorous physics formalism (nothing trivial for
a woman in South America). I was the first student who did it at that University, which
follows the motto Pedes in Terra ad Sidera Visus. Several years later, I am concluding
the doctorate in Astrophysics.
This thesis work had enormous support from many persons in different countries.
I am grateful to some of them for their direct participation in the thesis, to others for
contributing to my scientific background. Some of them helped me with administrative
issues, gave me good advice and supported me in difficult moments. This enterprise
would not have been realised without them and without the financial support.
I am infinitely grateful to Norbert Przybilla and Ulrich Heber for accepting me
to work in their team at the Dr. Remeis Sternwarte Bamberg, for providing excellent
working conditions, for very fruitful scientific discussions. Also for their respect of
my ideas, my work and my person, for their permanent support, patience, comments
and suggestions. Vielen Dank an Beide! A special thank to Norbert, for his immense
patience in the long process of constructing reliable model atoms for non-LTE cal-
culations. Thanks for letting me understand part of the wonderful and complicated
physics hidden behind these pretty nice lines called spectra. And for the extra working
hours answering all my questions, correcting errors, suggesting new tests, reading the
manuscripts. Danke sch¨on f¨ur deine Details, sie machen doch den Unterschied! Thanks
also to Martin Altmann for the spectra required to calibrate model atoms. The stay in
Germany was funded by DAAD (German Academic Exchange Service). This allowed
me to finish the Ph.D. Danke H. Jupe und W. Gairing f¨ur die Unterst¨utzung.
The research project was proposed by Katia Cunha and a preparatory phase for
this thesis work was done in Brazil (Observat´orio Nacional, ON) under her supervision
and the collaboration of Simone Daflon. I am grateful to both. Thanks Katia for
encouraging me to do a work of high scientific level, for your support, for the possi-
bility to participate in observing runs and for establishing the contact to the German
supervisors. Thanks Simone for introducing me to the non-LTE computations, for the
interesting discussions and for your questions. I hope I have answered some of them
in this work. Muito obrigada! The stay in Brazil was financed by CNPq (National
Counsel of Technological and Scientific Development) and the observing runs by ON.
Hugo Levato and Stella Malaroda provided letters of reference for the DAAD schol-
arship. Muchas gracias por su permanente apoyo! Many thanks to Herman Hensberge
139
ACKNOWLEDGEMENTS 140
for helping me with the reduction of FEROS spectra, for his patience and his extraordi-
nary experience. Thanks for your interest in my work and thanks to both you and your
wife for the kind reception in Brussels. Thanks to Verne Smith and Dmitriy Bizyaev
(and John Olguin) for the support at McDonald Observatory (Texas, USA). Thanks
to the staff at La Silla (Chile, European Southern Observatory) and at CASLEO (Ar-
gentina) for the technical support in observing runs. I am indebt to Keith Butler for
his patience in reading some manuscripts, for his valuable comments, his support and
interest in my work. Thanks also Joachim Puls for suggestions to some manuscripts.
The participation to conferences, schools and trainings were possible because of the
partial financial support of several institutions: the University of Erlangen-N¨urnberg
(Sternwarte Bamberg and Frauenbeauftragte), DAAD, CERN (European Organization
for Nuclear Research), Royal Observatory of Belgium, CNPq, ON. Several persons also
provided interesting ideas and feedbacks to this work through very nice discussions:
M. Asplund, C. Chiappini, T. Beers, A. Herrero, S. Sim´on-D´ıaz, S. Ekstr¨om, S. M¨ohler,
D. Fabbian, L. Pasquini, U. K¨aufl, A. Seifahrt. Thanks to all! I am also indebt to
D. Baade, A. Maeder, A. Kaufer, M. Kissler-Pattig for their support and interest in
my work and career. Thanks to Ulrich Katz, for giving me excellent advice for planning
the last year of my Ph.D. and further career plans through the ARIADNE project at
the University of Erlangen-N¨urnberg and even now, after the programme. Danke sch¨on!
Thanks to my friends/colleagues/professors at ON for a very nice atmosphere. Di-
ana, Sergio, Cecilia, Flavia, Thais, Bia, Ricardo, Roberta, Vinicius, Wagner, Marcelo,
Vladimir, Evgeni, Flavio, Jucira, Kohl, Dalton, Ramiro, Iara, Antares. Obrigada! Rene
Duffard found a new home for me in Rio. Gracias! Thanks Diana Andrade and Thais
Mothe for the support when I had health problems and no health insurance.
Thanks to my friends/colleagues at the Sternwarte Bamberg for the nice environ-
ment. Stefan and Meike Nesslinger, thanks for your help with German: Ich habe viel
gelernt. I am grateful to Rainer Sterzer for his friendship and his private and techni-
cal support, for helping me with my new home when I arrived to Bamberg. Thanks
to Markus Firnstein, Florian Schiller, Heiko Hirsch, Alfred Tillich, Stephan Geier,
Christian Karl, Simon O’Toole, Heinz Edelmann, Manfred Hanke, J¨orn Wilms, Horst
Drechsel, Irmela Bues, Edith Day for making me feel at home in the institute. A spe-
cial thank to Markus and Florian for reading a draft of my thesis. Thanks to Norberto
Castro Rodr´ıguez for the interesting questions and his comments to my thesis.
Thanks Giovanni Pinz´on Estrada for beautiful moments, for our friendship, for
your point of view of life, for excellent discussions about physics. Muchas gracias por
tu amistad! Thanks Patricia Cˆortes Nogueira for your sincere friendship. Vocˆe ´e minha
irm˜a brasileira. Sinto saudades de vocˆe! Gracias Tity y Silvina por ser mis eternas
amigas de la infancia y por apoyarme siempre. Las extra˜no mucho!
Danke Hedwig und Ernst f¨ur ein wunderbares Zuhause in Deutschland. Ihr seid
meine Deutsche Familie und ich bin sehr gl¨ucklich bei euch. Vielen Dank f¨ur Alles!
A special thank to my partner, for his patience and friendship, for the enormous
support and for respecting my dedication to work. Thanks for our life together and
for your love. Gracias a mi familia por el eterno apoyo, despu´es de viajar tanto para
realizar mis sue˜nos que comenzaron en mi ni˜nez, en las noches despejadas de San Javier,
con ustedes. Gracias por su cari˜no y por estar siempre presente cuando los necesito.
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