Mass Loss by Inhomogeneous Agb-Winds

Mass Loss by Inhomogeneous
AGB-Winds
Detailed Structures in Planetary Nebulae
Dissertation
eingereicht von
Mag. rer. nat. Ch. Reimers
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Fakultät für Geowissenschaften,
Geographie und Astronomie
der Universität Wien
Institut für Astronomie
Türkenschanzstraße 17
A-1180 Wien, Österreich
Oktober 2005

Preface
On the one hand the distances in the universe as well as the dimensions of astrophysical
objects like galaxies are almost unimaginable. On the other hand the time
scales are either immeasurably long as compared with our human being (e.g. the lifetime
of a typical star like our Sun) or they are even faster than a “human thought”
(e.g. the supernova explosion process, the rotation period of fast rotating pulsars or
the atomic vibrational timescales). Therefore, the fascination to study astrophysical
problems is the possibility to model and solve the “physical world” with the help
of computer technology and specific software. This fascination was also a driving
motivation for the realisation of this thesis.
In order to reconstruct astrophysics here on Earth, computer simulations are inevitable,
which divide the space into small units (in the broadest sense this can be
denoted by spatial resolution) and arrange the time as finite intervals, which are
called time steps. This procedure one calls also discretisation, which was realised
by the development of a program (hereafter RHD code) to simulate radiation hydrodynamic
problems at the Institute for Astronomy of the University of Vienna.
The RHD code is already extensively tested by several calculations to various astronomical
objects, e.g. RR Lyra stars, Cepheids, LBVs, protostellar collapse and
AGB stars. Among other things the work is to be understood as an extension to
this RHD code.
I would like to thank my dissertation advisor, Ernst A. Dorfi, for his support and
encouragement over the previous years. I benefited from his teaching of computer
simulation and he led me to break through problems which certainly resulted in a
timely completion of this thesis.
Also many thanks to the preparatory work of the RHD code done by Susanne
Höfner, Michael U. Feuchtinger as well as Ernst A. Dorfi. Furthermore, a big thank
to Roland Ottensamer for proof-reading of this thesis. Finally, I acknowledge the discussions,
inspirations and patience of all the students and combatants, who worked
in the same computer working room as me.
Vienna, October 2005 Mag. Christian Reimers

Abstract
AGB (Asymptotic Giant Branch) stars generate a massive dust driven stellar wind at
the end of their lives. Thereby they lose a large amount of mass. Ideally, this mass
loss is spherical if the physical conditions are homogeneous at the stellar surface
(e.g. temperature) and the stellar vicinity (e.g. density). Indeed, several physical
processes induce deviations from these ideal conditions. A stellar rotation for example
generates an asphericity of the luminosity or alternatively effective temperature
at the stellar photosphere. This will affect the condensation of dust and therefore
the mass loss rate. The dust formation process depends strongly on the temperature
and density.
Inhomogeneities can also caused by cool spots at the stellar surface. For some
time it is known that spots are common on stars and are much often larger than
spots on our Sun. These inhomogeneities of the temperature are able to emanate
from a magnetic field or a huge convection cell within the stellar envelope. Both
options are possible at the surface of AGB-stars. Due to the massive dust formation
in their atmospheres these physical processes are difficult to observe. But several
theoretical calculations and investigations are able to support such a theory.
This thesis introduces a model for the investigation of the mass loss above cool
spots. For that purpose a radiation hydrodynamic simulation (including a gas, a dust
and a radiation component) has been used and modified for the special purposes of
this problem. A flux tube geometry has been chosen which could have been produced
by a magnetic field in the lower stellar atmosphere. Finally, a discussion has been
carried out about the creation of dense knots in planetary nebula as a result of
cool regions at the stellar surface. A large amount of those dense knots or cometary
structures can be observed in many planetary nebula, like in the Helix or the Eskimo
Nebula.
The result supports the theory that stellar spots generate significant inhomogeneities
of the mass loss. But the formation of dense knots in planetary nebulae
have to be interpreted as a combination of inhomogeneities in the mass loss together
with hydrodynamical instabilities. The model investigated describes the formation
of initial inhomogeneities which can be later amplified by an interaction of the slow
AGB wind with the fast tenuous wind of the hot central star of the planetary nebula.

Zusammenfassung
AGB-Sterne (Asymptotic Giant Branch) produzieren am Ende ihres Lebens einen
ausgeprägten staubgetriebenen Sternwind, bei dem sie einen Großteil ihrer Hüllenmasse
verlieren. Idealerweise ist dieser Massenverlust sphärisch symmetrisch, wenn
die physikalischen Größen an der Sternoberfläche (z.B. Temperatur) und im umgebenden
Medium (z.B. Dichte) homogen sind. Allerdings erzeugen verschiedene
physikalische Prozesse Abweichungen von diesen idealen Bedingungen. Zum Beispiel
bewirkt die Rotation des Sterns eine Aspherizität der Sternleuchtkraft beziehungsweise
Effektivtemperatur an der Sternphotosphäre, welche sich auf die Kondensation
des Staubs und daraus folgend auf die Massenverlustrate auswirkt. Der Staubentstehungsprozess
ist stark von Temperatur und Dichte abhängig.
Inhomogenitäten können auch durch kühle Flecken auf der Sternoberfläche erzeugt
werden. Schon seit einiger Zeit ist bekannt, dass es Sterne mit Flecken gibt, die
mitunter einiges größer sind als Sonnenflecken. Diese Temperaturinhomogenitäten
können von einem Magnetfeld oder aber von großräumigen Konvektionszellen in
einer konvektiven äußeren Hülle stammen. Beide Möglichkeiten sind für die Oberfläche
von AGB-Sternen vorstellbar. Beobachtungen diesbezüglich sind wegen der
hohen Staubproduktion in den AGB-Atmosphären nur schwer zu machen. Verschiedene
Modellrechnungen und theoretische Überlegungen unterstützen jedoch
diese Theorie.
In dieser Arbeit wird ein Modell vorgestellt, das zur Untersuchung des Massenverlustes
über diskreten kühlen Flecken dient. Dazu kam eine strahlungshydrodynamische
Simulation zum Einsatz, die eine Gas-, Staub- und Strahlungs-Komponente
beinhaltet, wobei der Computer-Code für die neue Applikation adaptiert werden
musste. Um den komplexen Sachverhalt zu vereinfachen wurde eine Flussröhren-
Geometrie gewählt, die ein Magnetfeld in der unteren Sternatmosphäre erzeugt.
Eine abschließende Diskussion soll klären, ob diese kühlen Regionen auf der Sternoberfläche
die Existenz von dichten Knoten in Planetarischen Nebeln hervorrufen
kann. In vielen Planetarischen Nebeln sind wir in der Lage eine große Anzahl dichter
Knoten oder “kometenartiger” Strukturen zu beobachten (z.B. im Helix- oder im
Eskimo-Nebel).
Das Ergebnis unterstützt die Theorie, dass Sternflecken eine signifikante Inhomogenität
im Massenverlust verursachen können. Allerdings müssen die beobachteten
dichten Knoten in Planetarischen Nebeln in Verbindung mit hydrodynamischen
Instabilitäten entstanden sein. Das untersuchte Modell erzeugt dabei eine anfängliche
Inhomogenität im stellaren Ausfluss, welche später durch die Wechselwirkung des
langsamen AGB-Windes mit dem schnellen dünnen Wind des heißen Zentralsterns
Planetarischer Nebel verstärkt werden kann.

Contents
I Introduction and Motivation 1
1 Evolution of Stars 3
1.1 The Cycle of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Interstellar Medium . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Exchange of Matter . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Constant Light of Hydrogen Fusion - The Main Sequence . . 5
1.2.3 Final Stages of Stars . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Origin and Composition of Stellar Dust . . . . . . . . . . . . . . . . 7
1.3.1 Properties of AGB stars . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Detecting and Measuring Interstellar Dust Grains . . . . . . 9
1.3.3 Dust Formation and Destruction . . . . . . . . . . . . . . . . 10
1.4 From AGB stars to PNe . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Planetary Nebulae 13
2.1 Morphology and Classification . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 List of Prominent PNe . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Proto-PNe (or Young PNe) . . . . . . . . . . . . . . . . . . . 15
2.2.2 Round and Elliptical . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Bipolar and Quadrupolar . . . . . . . . . . . . . . . . . . . . 25
2.3 Global Models to Shape a PN . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Multiple-Winds Model . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Aspherical Mass Loss of AGB stars . . . . . . . . . . . . . . . 29
2.3.3 The Role of Magnetic Fields . . . . . . . . . . . . . . . . . . 30
2.3.4 Interaction with the ISM . . . . . . . . . . . . . . . . . . . . 31
2.3.5 MHD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Details in PNe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 Halo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.2 Jets, Lobes and Ansae . . . . . . . . . . . . . . . . . . . . . . 33
2.4.3 Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
v

vi CONTENTS
II Theoretical Models 35
3 Radiation Hydrodynamics Simulation 37
3.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Conservation form . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.2 Gas Component . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.3 Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.4 Dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Additional Equations and Constitutive Relations . . . . . . . . . . . 42
3.2.1 Grid Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 Mass Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.3 Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.4 Equation of State (EOS) . . . . . . . . . . . . . . . . . . . . . 43
3.2.5 Opacity of Gas and Dust . . . . . . . . . . . . . . . . . . . . 44
3.2.6 Source Function of Gas and Dust . . . . . . . . . . . . . . . . 45
3.2.7 Eddington Factor . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.1 Inner Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.2 Outer Boundary . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Initial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Modelling Method . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Equations for the Stellar Envelope . . . . . . . . . . . . . . . 49
3.4.3 Equations for the Stellar Atmosphere . . . . . . . . . . . . . 50
3.4.4 Additional Notes . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Stellar Spots 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 Solar Magnetic Activity and Sunspots . . . . . . . . . . . . . 55
4.1.2 Stellar Magnetic Activity . . . . . . . . . . . . . . . . . . . . 56
4.1.3 Observations of Stellar Spots . . . . . . . . . . . . . . . . . . 57
4.1.4 AGB star spots . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Spot Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.2 Temperature Fluctuations . . . . . . . . . . . . . . . . . . . . 61
4.2.3 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.4 Dust Formation above Cool Spots . . . . . . . . . . . . . . . 62
4.3 Flux Tube Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.2 Flux Tube Representations . . . . . . . . . . . . . . . . . . . 63

CONTENTS vii
4.3.3 Specific Declarations and Boundary Conditions . . . . . . . . 67
4.3.4 Rewritten Equations . . . . . . . . . . . . . . . . . . . . . . . 68
III Results and Discussion 71
5 AGB Stars with Spots 73
5.1 Initial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.1 Initial Models for Spherical Geometry . . . . . . . . . . . . . 73
5.1.2 Initial Models for Flux Tube Geometry . . . . . . . . . . . . 76
5.2 Dynamic Model Results for Spherical Geometry . . . . . . . . . . . . 79
5.2.1 Effects of Chemistry . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Dynamic Model Results for Flux Tube Geometry . . . . . . . . . . . 82
5.3.1 Effects of Geometry . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Boundary Conditions of the Flux Tube . . . . . . . . . . . . . . . . . 88
5.4.1 Lateral Pressure . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4.2 Heat Sources and Sinks . . . . . . . . . . . . . . . . . . . . . 90
5.5 Mass Loss through a Flux Tube . . . . . . . . . . . . . . . . . . . . . 91
6 Discussion and Perspectives 95
6.1 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.1.1 Lifetime of Stellar Spots . . . . . . . . . . . . . . . . . . . . . 95
6.1.2 Stellar Activity Cycle . . . . . . . . . . . . . . . . . . . . . . 97
6.1.3 Size and Distribution of Stellar Spots . . . . . . . . . . . . . 97
6.2 Mass Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2.1 Mass Acquiration . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2.2 Stellar Rotation . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Small-scale Structures in PNe . . . . . . . . . . . . . . . . . . . . . . 99
6.3.1 Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3.2 Inhomogeneous Mass Loss . . . . . . . . . . . . . . . . . . . . 99
6.3.3 Radial Filaments . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.5 Assumptions and further Perspectives . . . . . . . . . . . . . . . . . 101
6.5.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.2 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.3 Permeable Boundary . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.4 Stellar Pulsations . . . . . . . . . . . . . . . . . . . . . . . . . 102

viii CONTENTS
IV Appendices 103
A Discretisation 105
A.1 Computational Domain . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.3 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.4 Case 1: Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . 106
A.4.1 Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.4.2 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . 107
A.5 Case 2: Flux Tube Geometry . . . . . . . . . . . . . . . . . . . . . . 107
A.5.1 Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.5.2 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . 107
B Artificial Viscosity 109
B.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B.1.1 Viscous Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.1.2 Viscous Energy Dissipation . . . . . . . . . . . . . . . . . . . 110
B.2 Case 1: Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . 111
B.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.2.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.3 Case 2: Flux Tube Geometry . . . . . . . . . . . . . . . . . . . . . . 113
B.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.3.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
C Radiation Transfer 115
C.1 Radiation Transfer Equation . . . . . . . . . . . . . . . . . . . . . . 115
C.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
C.1.2 RTE in General Geometry . . . . . . . . . . . . . . . . . . . . 115
C.1.3 Variables and Moments . . . . . . . . . . . . . . . . . . . . . 119
C.1.4 Radiation Pressure Tensor Identities . . . . . . . . . . . . . . 119
C.2 0 th -order Moment Equation . . . . . . . . . . . . . . . . . . . . . . . 121
C.2.1 Case 1: Spherical Geometry . . . . . . . . . . . . . . . . . . . 121
C.2.2 Case 2: Flux Tube Geometry . . . . . . . . . . . . . . . . . . 122
C.3 1 st -order Moment Equation . . . . . . . . . . . . . . . . . . . . . . . 123
C.3.1 Case 1: Spherical Geometry . . . . . . . . . . . . . . . . . . . 123
C.3.2 Case 2: Flux Tube Geometry . . . . . . . . . . . . . . . . . . 124
C.4 Derivatives in different geometries . . . . . . . . . . . . . . . . . . . 124
C.5 Summary of Spherical Radiation Equations . . . . . . . . . . . . . . 125
C.5.1 Radiation Energy Equation . . . . . . . . . . . . . . . . . . . 125
C.5.2 Radiation Momentum Equation . . . . . . . . . . . . . . . . . 126

CONTENTS ix
D Dust properties 127
D.1 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
D.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
D.3 Dust Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
D.3.1 C-rich Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 128
D.3.2 Nucleation Theory . . . . . . . . . . . . . . . . . . . . . . . . 131
D.3.3 Dust Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
E Tensor Calculus 135
E.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
E.1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . 135
E.1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
E.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
E.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
E.2.2 Operations and Operators . . . . . . . . . . . . . . . . . . . . 138
E.2.3 Relations / Vector Identities . . . . . . . . . . . . . . . . . . 141
E.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
E.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
E.3.2 Operations and Operators . . . . . . . . . . . . . . . . . . . . 142
E.3.3 Relations / Tensor Identities . . . . . . . . . . . . . . . . . . 144
E.4 Metric and Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 145
E.4.1 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
E.4.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . 147
F Full Set of RHD Equations 153
F.1 Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
F.2 Integrated Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
F.3 Discretised Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
G Symbols, Constants and Abbreviations 159
G.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
G.2 Fundamental Physical Constants . . . . . . . . . . . . . . . . . . . . 160
G.3 Astronomical Constants . . . . . . . . . . . . . . . . . . . . . . . . . 160
G.4 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
List of Tables 163
List of Figures 165
Image Credits 167
Bibliography 169

Part I
Introduction and Motivation
1

Chapter 1
Evolution of Stars
The aim of this thesis is to investigate an inhomogeneous mass loss of asymptotic
giant branch stars (hereafter AGB stars). An effective mechanism of mass loss for
these cool stars is the generation of a dust driven stellar wind where the radiation
pressure accelerates the newly formed dust grains. The first chapter gives a brief
summary of the formation and evolution of stellar objects with special regard to
intermediate mass stars like the AGB’s. In the second chapter we describe in detail
the morphology and classification of planetary nebulae (hereafter PNe) with respect
to the generation of models for the explanation of small-scale structures in PNe as
a result of an interaction from the massive mass loss of the AGB progenitor and
the high velocity outflows of the hot central objects. At first we discuss the cycle of
matter in a galactical context. Therein the interstellar medium plays an important
role as origin of the stellar formation process. Furthermore, an enrichment of heavy
elements by the incorporation of nuclear processed material (e.g. AGB wind) leads
to a chemical evolution of stellar objects.
1.1 The Cycle of Matter
1.1.1 Interstellar Medium
The space between the stars in a galaxy is far from being empty. These regions
are filled with gas, dust, solid bodies (like asteroids or comets), magnetic fields and
charged particles and commonly noted as interstellar medium (hereafter ISM). Approximately
99% of the mass of the ISM is in the gaseous form and the remaining 1%
is composed primarily of dust. The matter of the ISM is not distributed uniformly
but is more or less concentrated in interstellar clouds where complex molecules and interstellar clouds
dust particles can be formed. On the one hand the molecules are the seed for the
dust formation process, on the other hand they are at risk to be destroyed by the
interstellar ultraviolet radiation. But in dense clouds they are shielded against this
destructive radiation.
Apart from molecular cloud cores dust particles are formed in several other
astrophysical environments, ranging from stellar outflows (including red giant at- stellar outflows
mospheres and Wolf-Rayet winds) to interstellar shock fronts and explosive ejecta
(e.g. supernovae). These processes are also responsible for the chemical evolution
3

4 1. EVOLUTION OF STARS
dust component
circulation process
Jeans mass
protostellar object
of the ISM by the enrichment of heavy elements. The dust component of the ISM
becomes detectable as (cf. Savage & Mathis 1979 [134]):
• Interstellar extinction and reddening: It is caused by the absorption of light by
matter between the object and the observer and depends on the wavelength like
Fλ ≈ λ −1 . Thus dense clouds which are opaque in visual light get transparent
for higher λ (e.g. infrared radiation).
• Reflection nebulae: The light from some stars embedded in an interstellar
nebula is scattered by gas and dust particles therein.
• Polarisation: Stellar light can become polarised when passing through a dust
cloud if the particles are small compared to the incident wavelength, if they
are extended in length or if they tend to be orientated in the same direction.
• Infrared emission: Stellar radiation and collisions with atoms also heats up the
dust in the stellar vicinity. The absorbed energy is thermalised and as a result
the dust emits a thermal spectrum predominantly in the infrared wavelengths.
Dust can be studied in situ within our Solar System with several methods (see
therefore Section 1.3.2 on page 9).
1.1.2 Exchange of Matter
The ISM is constantly subject to a circulation process. The gas and dust input
to the ISM is provided by supernova remnants, stellar winds and jets, whereas the
losses are due to star formation and accretion on stellar objects (e.g. white dwarfs,
neutron stars, etc.). The ISM matter is lost forever, when it gets trapped by stellar
or galactic black holes.
1.2 Stellar Evolution
1.2.1 Star Formation
The starting point of star formation is gas and dust concentrated in interstellar
molecular clouds. Dynamical processes like shock waves from energetic events in
the surrounding, e.g. supernova explosions, can trigger the gravitational collapse. If
enough matter is concentrated, the gravitational force dominates the counteracting
pressure forces, i.e. the mass concentration rises above the Jeans mass
Mj ∝ ρ −1
2 T 3
2 (1.1)
(since Jeans first demonstrated the nature of this instability in 1902, it is called
Jeans instability and the involved mass is called Jeans mass), the collapse acts in
and fragmentation may reduce the initial mass. The collapse to a protostellar object
needs between 10 4 and 10 6 years.

1.2. Stellar Evolution 5
Depending on the mass involved, stars with main sequence masses in the
• low (0.08 � M[M⊙] � 2),
• intermediate (2 � M[M⊙] � 8) or
• high (M[M⊙] � 8)
mass range can be formed. After the formation of single or double stars the initial
mass remains mostly constant. But if two or more stars orbit each other closely, the mass transfer
gravitational forces can transfer stellar matter from one star to its companion. This
mass transfer has an impact on the further evolution of each star.
Furthermore, remaining matter from stellar formation generates a disc orbiting
the protostellar object. Matter bound in these stellar accretion discs can be the accretion disc
seed for the formation of huge layered grains, clumps and further for planetesimals.
If the conditions are favourable then asteroids, moons and finally planets emanates
from these building components.
1.2.2 Constant Light of Hydrogen Fusion - The Main Sequence
Single stars with masses less than 1.4 M⊙ remain at the main-sequence (hereafter
MS) stage for a very long period. The MS lifetime of a star can be estimated by the main sequence
nuclear timescale
τnuc ∼
available fuel
burning rate
∼ M
L ∼ M −2.5 , (1.2)
where L is the stellar luminosity and for a MS star L ∝ M 3.5 . Due to fusion hydrogen fusion
hydrogen is converted into helium in the stellar core. During this time the chemical
composition of the star changes and the central temperature slowly rises. For single
stars more massive than the Sun, the nuclear timescale (cf. Eq. (1.2)) decreases and
the MS phase gets shorter.
1.2.3 Final Stages of Stars
The final stages depend on the initial masses of stars and the amount of mass which
is stripped by mass loss due to companion stars or stellar winds. The following mass loss
remnants left over from these stages ordered by the mass at the MS:
• White Dwarfs,
• Neutron Stars and
• Black Holes.
At the end typical masses for White Dwarfs are 0.6M⊙ and for Neutron Stars
around 1.4M⊙.
Low Mass Stars
According to Eq. (1.2) small and relatively cool stars, which are also called red
dwarfs, stay for a long time on the MS compared to stars in the higher mass ranges. red dwarf stars
The masses of red dwarfs are less than about 0.5M⊙ down to objects with 0.08M⊙.
Below this mass range a stellar object never gets hot enough to initiate hydrogen
fusion in the core.

6 1. EVOLUTION OF STARS
red giant branch
hydrogen burning
shell
helium fusion
asymptotic giant
branch
long period
variables
Figure 1.1: Evolutionary tracks in the Hertzsprung-Russell-Diagram for stars with
initial masses of 1M⊙, 5M⊙ and 25M⊙. It shows major phases of the stellar evolution
like the core helium flash, thermal pulses and the ejection of the planetary nebula
(from Iben 1985 [73]).
Intermediate Stars
When the hydrogen fuel is exhausted in the centre of a star within an intermediate
mass range of 1 to 8 M⊙ it leaves the main-sequence phase and evolves towards
the so-called red giant branch (RGB). While the star itself expands the remaining
hydrogen fusion in a shell around the helium rich centre generates energy for fur-
ther million years. The stellar core contracts and pressure and temperature increase
until the helium fusion in the stellar centre begins. Now the evolution proceeds
very rapidly and the star is now located on the asymptotic giant branch (AGB) in
the Hertzsprung-Russell-Diagram or short HRD (cf. Fig. 1.1). AGB stars are very
extended objects with radii of a few hundred R⊙ with high luminosities of about
10 3 to a few 10 4 L⊙ and low effective temperatures of typically < 3500 K. During
their evolution along the AGB they begin to pulsate with large amplitudes, get large
convection zones and drive a massive stellar wind. According to the noticeable pulsations
with long periods the stars are also commonly known as long period variables

1.3. Origin and Composition of Stellar Dust 7
(for a more detailed classification see Sect. 1.3.1). These long period pulsations are
known for a long time. The first observations were made by the discoverer of Mira,
David Fabricius in 1596 and 1609. The Mira stars show variations of their visual Mira stars
light curves with amplitudes of several magnitudes and periods of approximately
one year. The pulsations and the stellar mass loss are an observational evidence
of dynamical processes these stars undergo. Later they reach the post-AGB phase post-AGB
and the repelled outer envelope can be seen for about 10 5 years as PNe whilst the
central object cools to a White Dwarf. More about this type of final stage will be White Dwarf
given in Section 1.3 and Section 1.4.
Massive Stars
These stars can continue generating energy by helium fusion after they have
depleted their hydrogen supplies. Their gravitational potential energy enables them
to build up extremely high pressures and temperatures deep in their interior. These
conditions are able to initiate the fusion of helium and further heavier elements.
After a short red giant phase massive stars mostly end their lives in a gigantic heavy elements
explosion, a supernova, leaving behind a Neutron Star or a Black Hole. Although supernovae
this basic picture is supported by observations, the details of the formation process
of Neutron Stars, e.g. as rapidly rotating pulsars, or even Black Holes, still remains
unclear.
1.3 Origin and Composition of Stellar Dust
AGB stars are known to eject much of their envelope into space and this could be
a significant source of interstellar dust grains (e.g. Nittler et al. 1997 [107]). Such
stars have once been like the Sun but have reached a period in their life-cycle where
they are losing massive amounts of dust and gas preceding their final existence as
White Dwarfs.
1.3.1 Properties of AGB stars
Internal Structure and Nucleosynthesis
The core of an AGB star consists mainly of carbon and oxygen after the central
helium fusion has exhausted. Above this core a helium- and hydrogen-burning shell
converts the atomic binding energies into radiation and heavier elements like carbon helium- and
and oxygen, which enrich the core by mass with these heavy elements. Due to
the highly degenerated electrons, the outward diffusion of the energy by electron
conduction is very efficient. Furthermore, the inner part of the core loses energy
by the production of neutrinos. Consequently, the temperature of the core can not
climb over the temperature where carbon-burning ignites.
hydrogenburning
shell
Theoretical models tell us that the observed peculiarities on their surfaces are
directly connected with the nucleosynthesis in the stellar interior. Newly formed
elements like carbon and oxygen are mixed to the surface by a deep convection zone
(in particular during the so-called the third dredge-up). These mixing processes third dredge-up
occur during the thermal pulsing phase (cf. TP-AGB on page 11) which involves also
the external layers (Iben 1981[74]). Observations show two main types of AGB stars

8 1. EVOLUTION OF STARS
surface composition
long period
variables
κ-mechanism
convection zone
mixing
convection cell
α Orionis
(Beteigeuze)
infrared excess
terminal wind
velocities
concerning their surface composition: oxygen-rich (i.e. stars with surface abundances
of ǫC/ǫO < 1) and carbon-rich (i.e. ǫC/ǫO > 1) AGB stars. Due to the possible
evolution from oxygen-rich stars and the effects of the third dredge-up the formation
to the carbon-rich stars can be explained.
Pulsation and Variability
A large fraction of the AGB stars shows variability with periods of about 80 to
1000 days, which are consequently called long period variables (hereafter LPVs).
The LPVs are divided into several groups according to the regularity of their light
curves:
• Miras showing well defined periods and rather regular shapes,
• semi-regular (SR) with semi-regular light curves and smaller amplitudes
compared to Mira variables (for e.g. classification and evolutionary status of
SR variables see Kerschbaum & Hron 1992 [82]) and
• irregular variables which show no regularity in their light curves.
Light curves of such stars can be found e.g. in Querci & Querci (1986 [120]). The
variability of the LPVs can be explained as a radial pulsation with large amplitudes
caused by a κ-mechanism in the hydrogen- and helium-ionisation zones.
Convection
Convection plays an important role for the transport of energy and momentum
throughout most of the outer parts of the star. During the RGB phase the convection
zone moves inward. This causes a mixing of nuclear processed gas upwards. The
mixing to the surface of the star is called dredge-up and can change the surface
composition (Iben 1985 [73]). Schwarzschild (1975 [138]) has estimated the sizes
of the convective elements (scale of the dominant convection or convection cell) for
Red Giant stars. Only few large convection cells should appear at the photosphere.
Observations e.g. of α Orionis (Beteigeuze) (Gilliland & Dupree 1996 [54]) and three-
dimensional MHD-simulations (e.g. Dorch 2004 [35], Freytag 2003 [45] and Freytag
et al. 2002 [46]) also support the fact of large convection cells.
Circumstellar Envelope and Mass Loss
From the observation of a so-called infrared excess the presence of a circumstellar
envelope (hereafter CSE) around an AGB star can be inferred as done by IRAS
observations (e.g. Likkel et al. 1990 [90]). The infrared excess is explained by the
absorption of photospheric radiation by the CSE, thermalisation and re-emission at
longer wavelengths.
The observation of line profiles in the spectra of CSEs shows also expanding
material where the terminal wind velocities of typically 10 to 40 km/s have been
measured. This observed velocities are relatively small and below the escape velocities
near the stellar photosphere indicating that the mechanism for driving the AGB
wind is different from the solar-type wind. A much larger spatial range has to be
responsible for the acceleration of the AGB wind.

1.3. Origin and Composition of Stellar Dust 9
An important aspect of the AGB phase is the mass loss which is much higher
than the mass loss produced by the Sun, i.e. about 10 −14 M⊙/a. The mass loss of
AGB stars lies in the range of 10 −7 to 10 −5 M⊙/a. It turned out that the mass loss
mechanism is the radiation pressure on dust grains which produces a dust driven
wind (e.g. Höfner & Dorfi 1997 [69]). Due to the increasing luminosity at the end dust driven wind
of the AGB phase the mass loss raises up to 10 −5 M⊙/a denoted by the superwind
phase (e.g. Schröder et al. 1999 [135]). Thermal pulses should drive bursts of su- superwind phase
perwind, which could explain the circumstellar shells found with some PNe. This is circumstellar shells
in agreement with the existence of detached CO shells which can be the result for
carbon stars with episodic mass loss (Olofsson et al. 1996 [113]).
The mechanisms of the heavy mass loss depends essentially on the presence of
dust. A lot of AGB stars (e.g. o Ceti (Mira), IK Tau, NML Cyg, IRC+10216, VY
CMa) show evidence for departure from spherical symmetry and episodes of dust
formation and destruction (Danchi & Townes 2001 [32]). Investigations on the car- asphericity
bon star IRC+10216 (CW Leo) with a relatively high mass loss of about 10 −4 M⊙/a IRC+10216
(Wannier et al. 1980 [157]) show that the aspherical circumstellar shell is due to an
aspherical process produced by the central star. The most likely explanation are
non-radial pulsations or a binary component which has spun up the central star.
Some other stars (e.g. o Ceti (Mira), R Cas and χ Cyg) are binary stars and show o Ceti (Mira)
aspherical circumstellar shells (Groenewegen 1996 [60]).
1.3.2 Detecting and Measuring Interstellar Dust Grains
In the atmospheres of AGB stars a large amount of dust grains can be formed due to
low temperatures and large densities. The dust grains play an important role in the
formation of a stellar wind which transports a lot of matter into the circumstellar
vicinity and beyond.
A number of efforts are made to detect and measure the existence, structure
and composition of such grains to learn how these particles can be created and
how they grow or alternatively are destroyed by radiation or collisions. Below some
observational methods and findings are listed: observational
methods
• IR observations: The infrared satellites IRAS (1983) and SST (2003-now)
from NASA and ISO (1996-1998) from ESA are helpful instruments to detect
and study interstellar dust, particularly observable in the infrared wavelength.
• Study of meteorites: Meteorites contain mostly unprocessed material from
the proto-solar nebula with inclusions of interstellar particles (see e.g. Nittler
et al. 1997 [107]). Some meteorites have become generally known, e.g.
Tieschitz meteorite - Fall: July 15, 1878; Location: Moravia, Czech Republic;
the grain structures are very different as their chemical compositions
are. One is a single-crystal of the most common form of aluminium oxide
Al2O3 (called corundum) while the other does not exhibit a crystalline
structure. The evidence has clarified observations that the production of
the two different forms of aluminium oxide is made in AGB outflows (see
Stroud et al. 2004 [147]).

10 1. EVOLUTION OF STARS
theoretical
approach
two step process
Allende meteorite - Fall: February 8, 1969; Location: Chihuahua, Mexico;
Allende contains an increased concentration of 26 Al decay products, which
can only originate from a supernova explosion in our sun’s neighbourhood.
The shock waves of that explosion may have been the cause of the collapse
of the primordial solar nebula.
Murchison meteorite - Fall: September 28, 1969; Location: Victoria, Australia;
the meteorite was found to contain a wide variety of organic compounds,
including many of biological relevance such as amino acids.
Zag meteorite - Fall: August 4 or 5, 1998; Location: Western Sahara, Morocco;
brecciated chondrite containing extraterrestrial water within blue
halite crystals.
• Dust capture by
satellites in the vicinity of the Earth
LDEF (Long Duration Exposure Facility) orbited Earth from 1984 to 1990
and has been designed to provide long-term data on the space environment
and its effects on space systems and operations,
MPAC (Micro-Particles Capturer) experiment on ISS (attached to the outer
hull of the ISS in Oct. 2001) from the formerly Japanese space agency
NASDA.
space probes in the interplanetary space
Stardust (1999-2006), flew within 236 kilometres of comet Wild 2 (Jan. 2004)
and captured thousands of particles in its aerogel collector for return
on Earth in January 2006. Additionally, the Stardust spacecraft will
bring back samples of interstellar dust, including recently discovered
dust streaming into our Solar System from the direction of Sagittarius.
These materials are believed to consist of ancient pre-solar
interstellar grains that include remnants from the formation of the
Solar System.
1.3.3 Dust Formation and Destruction
How dust grains are created, accumulated and destroyed cannot be investigated in
detail in the vicinity of stellar objects. This can only be done either in a laboratory
on Earth or on a spacecraft or by a theoretical approach.
The process of dust formation in the circumstellar envelopes of LPVs can be
described as a two step process (Sedlmayer 1989 [139]): (1) the condensation of
supercritical nuclei out of the gas phase and (2) the growth of macroscopic grains.
Four processes can change the number density of dust grains
• creation of grains by - growth of smaller dust particles or
- destruction of larger ones
• destruction of grains by - growth of larger dust particles or
- evaporation.

1.4. From AGB stars to PNe 11
To simplify the complicated process of dust formation, we consider carbon-rich
stars where ǫC/ǫO > 1 and the occurrence of the elements H and C and the molecules carbon-rich stars
H2, C2, C2H and C2H2 which should be in chemical equilibrium. Furthermore, we chemical
assume that the dust component consist of pure amorphous carbon clusters.
equilibrium
The carbon clusters are formed by hetero-molecular nucleation and growth. Therefore,
the equations of the basic concept of classical homogeneous nucleation theory classical
are generalised to get a consistent incorporation of random chemical reactions of the
gas molecules with the dust clusters.
homogeneous
nucleation theory
The growth and destruction of macroscopic grains are done by the temporal
evolution of a few moments, Kj, of the grain size distribution function. This leads
to a set of so-called moment equations which describe the growth and destruction moment equations
process of macroscopic grains. For further details see Gail & Sedlmayr (1988 [49]).
1.4 From AGB stars to PNe
The transition from an AGB star to a PN can be divided in the following evolutionary
scheme, where some phases can overlap each other:
• AGB stars
• Post-AGB stars
• Proto-PNe (or Young PNe)
• PNe with hot central star
• White dwarfs
AGB stars
The AGB evolution itself is divided into two phases:
• The early-AGB (E-AGB) phase is characterised by continuous helium shell E-AGB
burning and terminates when hydrogen is reignited in a thin shell and the
thermal pulses start.
• The thermally pulsating-AGB (TP-AGB) phase the mass of the helium-rich TP-AGB
shell below the hydrogen-burning shell increases and after the accumulation
of a critical mass a thermal pulse is initiated. This thermal pulses can occur
several times.
An review about the AGB evolution is given e.g. by Iben & Renzini (1983 [75]) and
Habing (1990 [63]). The AGB phase is characterised by increasing mass loss. The
outflow from the ageing star deposits a large amount of processed material in the
stellar vicinity and produces circumstellar shells which can easily be observed in
the infrared spectral range. Helium shell flash stars are objects which show a series helium shell flash
stars
of helium burning episodes in the thin helium shell that surrounds the dormant
carbon core of an AGB star; the helium burning shell does not generate energy at a

12 1. EVOLUTION OF STARS
OH/IR stars
central stars
of PNe
born again
objects
constant rate but instead produces energy primarily in short flashes. During a flash,
the region just outside the helium-burning shell becomes unstable to convection and
the resultant mixing probably leads to an upward movement of carbon produced by
helium burning. The overheating from a flash also causes an expansion of the star’s
upper layers, followed by an inward motion, leading to large-scale pulsations.
Post-AGB stars
In the latest AGB phase, the post-AGB phase, the star loses so much material
during a super wind phase, that the star becomes completely invisible at visual
wavelengths due to the surrounding gas and dust. The star then emits almost all
of its radiation in the infrared and can be observed as OH/IR stars (see e.g. Kwok
& Chan 1990 [87]). In this phase the star gets rid of its outer stellar body and its
central part further contracts to a tiny hot central star.
Proto-Planetary Nebulae
The transitional appearance between an AGB star and a PN is called Proto-
Planetary Nebula (hereafter PPN). PPNe are rare because they are in an evolu-
tionary phase which lasts for a very short time (about 1000 to 2000 years). During
this phase the temperature of the central star rises from about 2000 to 30000 K.
However, this phase is essential to learn more about the evolution of a star into and
through the PN stage and its interactions with the ISM. PNe are largely asymmetric,
while their progenitors, AGB winds, are mostly spherically symmetric. This
remains one of the fundamental problems of PNe evolution. Therefore, the PPN
object category is very important in trying to understand, e.g how the symmetry
break between the more or less spherical star and a bipolar shape of the PN can
be explained. Such bipolar shapes are frequently observed (e.g. review by Kwok
2001 [86]).
Planetary Nebulae
When the circumstellar shell expands and the density decreases the intense radiation
of the hot stellar body is able to ionise the gas and we see a glorious PN
(see e.g. Iben 1995 [72]). The glowing PN shell dims out due to thinning of the
circumstellar shell and the decline of ionising radiation flux. The matter repelled
once from the AGB star will then be incorporated in the ISM.
White Dwarfs and PG 1159 stars
Later on the central star of the PN evolves to the appropriate White Dwarf cooling
track where its luminosity and effective temperature decreases. If a helium shell flash
experienced very late by a White Dwarf during its early cooling phase after hydrogen
burning has almost ceased then the star is forced to rapidly evolve as so-called born
again objects, like e.g. Sakurai’s Object, back to the AGB phase and finally ends
as a quiescent helium-burning central star of a PN. The observed examples of this
hydrogen-deficient post-AGB stars are also known as very hot PG 1159 stars. Such
objects are expected to exhibit surface layers that are enriched by the products of
the helium burning, particularly carbon (e.g. Althaus et al. 2005 [1]).

Chapter 2
Planetary Nebulae
To study the detailed structure of a planetary nebula, we will have a look on some
selected objects showing an enormous variety of shapes and small-scale structures.
After the presentation of these objects ranging from young proto-planetary nebulae
to the different objects of evolved ones we summarise the facts with the aim to
generate a detailed model how an AGB star can influence the global shape as well
as the appearance of small-scale structures within the nebulae.
2.1 Morphology and Classification
The term “Planetary Nebula” (hereafter PN) has first been used by Sir William
Herschel. He has defined a PN as a nebula associated with a star looking like a disc
through a telescope. Specifically since they usually glow blue he has thought that
they looked like the planet Uranus he has discovered in 1781.
Since the appearance of a PN is far from uniform a classification scheme had classification
scheme
to be constructed. This classification is mainly based on morphology, i.e. the observed
appearance is the basic criterion to distinguish several classes. Basically, the
following shapes can be deduced shape
• ring-like or circular structures (round to elliptical),
• bipolar (butterfly) or quadrupolar and
• irregular.
Several classification schemes have been developed in the past. The most widely
accepted classification of PNe was devised by Vorontsov-Vel’Yamonov (1934 [156]).
Additionally there have been alternative classifications proposed, such as a system
deduced from the spectra of the PN (see therefore Gurzadyan & Egikyan 1991 [62]). spectra
The search for systematic segregations among PNe of different shapes has started
with the morphological analysis of Greig (1972 [58]). The classification from Peimbert
and collaborators (e.g. Peimbert 1978 [115], Peimbert & Torres-Peimbert
1983 [116]) is based on chemistry. Then Zuckerman & Aller (1986) classified a large chemistry
sample of PNe into many morphological types. Balick (1987 [5]) made a major contribution
to morphological classification, by constructing an empirical evolutionary
13

14 2. PLANETARY NEBULAE
sequence. Chu et al. (1987 [23]) released a catalogue of PNe with more than one shell
(multiple shell PNe). The European Southern Observatory (ESO) has published a
catalogue of more than 250 southern PNe. The images by Schwarz et al. (1992 [136])
were used to group the PNe into classes of an existing morphological classification
and further divided into subclasses by Schwarz et al. (1993 [137]), depending on
the additional features in the inner and outer parts of the nebulae. Finally, Manchado
et al. (1997 [96]) compiled a catalogue of more than 240 PNe of the northern
hemisphere published by the Instituto de Astrofísica de Canarias (IAC).
2.1.1 List of Prominent PNe
The objects listed below can be found on the following pages. They are sorted by
their morphology and/or evolutionary phase. A detailed description of the observational
findings including images (mostly taken from the Hubble Space Telescope
orbiting the Earth) are given for the individual objects. The image credits are presented
in the Appendix on page 167.
Object .................................................................... Page
Proto-PNe (or Young PNe)
• NGC 7027 .............................................................. 15
• CRL 2688 - Egg Nebula .................................................16
• HD 44179 - Red Rectangle Nebula ...................................... 17
• OH231.8+4.2 - Rotten Egg Nebula or Calabash Nebula ................. 17
Round and Elliptical
• NGC 6720 - Ring Nebula, M57 ..........................................18
• NGC 7293 - Helix Nebula ...............................................20
• NGC 6853 - Dumbbell Nebula, M27 .....................................21
• NGC 2392 - Eskimo Nebula .............................................22
• NGC 6369 - Little Ghost Nebula ........................................23
• NGC 3132 - Eight-Burst Nebula ........................................ 23
• IC 418 - Spirograph Nebula ............................................. 24
• NGC 6751 .............................................................. 24
Bipolar and Quadrupolar
• NGC 6543 - Cat’s Eye Nebula .......................................... 25
• MyCn 18 - Hourglass Nebula ............................................26
• IC 4406 - Retina Nebula ................................................ 26
• NGC 6302 - Bug or Butterfly Nebula ....................................27
• Mz 3 - Ant Nebula ......................................................28
• M2-9 ................................................................... 28

2.2. Examples 15
2.2 Examples
2.2.1 Proto-PNe (or Young PNe)
NGC 7027
Figure 2.1: Halo of PPN NGC 7027 observed
by the HST.
Figure 2.2: Details of PPN NGC 7027
observed by the HST.
NGC 7027 is the best studied of the young PNe. The photograph in Fig. 2.1 is
taken by the WFPC2 instrument on-board the HST and shows details which consist
of three distinct components: (1) an ellipsoidal shell depicting the ionised core, (2) a ellipsoidal shell
bipolar hourglass structure outside the ionised core represents the excited molecular bipolar hourglass
structure
hydrogen or photo-dissociation region and (3) a nearly spherical outer region seen
in dust scattered light is the cool, neutral molecular envelope. The interface region
spherical outer
region
between the inner shell and the bipolar hourglass is structured and filamentary, structured and
filamentary
suggesting the existence of hydrodynamic instabilities (Latter et al. 2000 [89]).
When it has been initially at its AGB stage the ejection of the outer star layers
has occurred at a low rate and has been spherical. The HST photo reveals that the
initial ejection events have happened episodically to produce the concentric shells. concentric shells
This evolution culminated in a vigorous ejection of all of the remaining outer layers,
which produced the bright inner regions. At this later stage the ejection have been bright inner regions
non-spherical, and dense clouds of dust condensed from the ejected material. Cox
et al. (2002 [31]) have found a notable series of lobes and openings in the molecular lobes and openings
shell. These features are point symmetric about the centre, which implies recent
activity by collimated outflows with a multiple, bipolar geometry. collimated outflows
Fig. 2.2 depicts a HST/NICMOS and WFPC2 composite image, accentuating the
innermost region of the nebula. The central star is clearly revealed and the stellar
temperature was determined to be about 198000 K. Furthermore, it was found that
the photo-dissociation layer is very thin with a bi-conical shape and lies outside the
ionised gas (Latter et al. 2000 [89]).

16 2. PLANETARY NEBULAE
dark edge-on disc
radial “searchlight
beam”
circular arcs
faint radial streaks
Cat’s Eye Nebula
multiple jet-like
outflows?
CRL 2688 - Egg Nebula
Figure 2.3: Halo of PPN CRL 2688 observed
by the HST.
Figure 2.4: Infrared-details of PPN
CRL 2688 observed by the HST.
The high resolution image from the HST/WFPC2 instrument in Fig. 2.3 shows
(1) a remarkable dark edge-on disc obscuring the central star, (2) a pair of radial
“searchlight beam” like features, criss-crossed by (3) a large number (at least 25) of
roughly circular arcs around the center. The arcs probably represent local peaks in
a quasi-periodic mass ejection process. Very faint radial streaks can be seen within
the “searchlight-beam” structures implying that these are jets of matter (Sahai et
al. 1995 [130]).
The arcs of CRL 2688 illustrate a history of mass ejection of a red giant star for
about 12700 years. They represent dense shells of matter within a smooth cloud, and
show that the rate of mass ejection from the central star has varied on time scales of
about 150 to 450 years throughout its mass loss history and lasting over periods of
75 to 250 years. There exist two models of creating the “searchlight beams”. Either
they are formed as a result of starlight escaping from ring-shaped cavities (Sahai et
al. 1998 [131]). Such cavities may be carved out by a tumbling, high-velocity outflow
(about 320 km s −1 ). Or alternatively, they may result from starlight reflected off
fine jet-like streams of matter being ejected from the central region, and confined
to the walls of a conical region around the symmetry axis (Remark: see also the
appearance of jets in the Cat’s Eye Nebula in section 2.2.3 on page 25).
Fig. 2.4 (Sahai et al. 1998 [128]) shows the inner structure observed with the
HST/NICMOS instrument. It reveals, that the dying star ejects matter at high
speeds along a preferred axis and may even have multiple jet-like outflows. The
torus along the assumed stellar equator or the orbital plane of a binary object is
also visible.

2.2. Examples 17
HD 44179 - Red Rectangle Nebula
The image presented in Fig. 2.5 has
been taken with the HST/WFPC2 instrument
and shows the following fea-
tures: (1) X-shaped structure, (2) lin- X-shaped structure
ear features, which look like the “rungs”
of a ladder and (3) dark band passing dark band
across the central star. The Red Rectangle
Nebula is associated with a post-
Figure 2.5: The PPN HD 44179 observed
by the HST/WFPC2.
linear features
AGB binary system (Cohen et al. 2004 binary system
[27]). It turned out that the star in the
centre is actually a close pair of stars
that orbit each other with a period of
322 days, a semi-major axis of a sini =
0.32 AU and an eccentricity of e = 0.34
(e.g. Men’shchikov et al. 2002 [102]). Interactions between these stars have probably
caused the ejection of the thick dust disc that obscures our view towards the binary.
The “rungs” show a quasi-periodic spacing, suggesting that they have arisen from
discrete episodes of mass loss from the central star, separated by a few hundred
years. Soker (2004 [143]) has argued that the bi-conical shape of the nebula can be
formed by intermittent jets generated by the accreting companion star. intermittent jets?
OH231.8+4.2 - Rotten Egg Nebula
Fig. 2.6 illustrates the HST/WFPC2
image of the Rotten Egg Nebula, also
known as the Calabash Nebula, extending
1.4 light-years in diameter and located
about 5000 light-years from Earth
in the constellation Puppis. A Mira vari- Mira variable star
able star, known as QX Pup, is embed-
ded within the evolved bipolar nebula evolved bipolar
nebula
OH 231.8+4.2. This central star pulsates
with a period of about 700 days,
which is remarkable in the light of its
position at the heart of such an unusual
object (Kastner et al. 1999 [79]). Due to
the high speed of the stellar gas accelerated
by the radiative pressure, shock
fronts are formed on impact and heat
the surrounding gas. It is believed that
Figure 2.6: The PPN OH231.8+4.2 observed
by the HST/WFPC2.
such interactions dominate the formation process in PNe. Much of the gas flow observed
today seems to stem from a sudden acceleration that took place only about
800 years ago. Approximately 1000 years from now the Calabash Nebula will become
a fully developed bipolar PN (Bujarrabal et al. 2002 [20]).

18 2. PLANETARY NEBULAE
filamentary
structure
loops and arcs
dense knots
enhanced bands
petal-like
appearance
limb-brightened
knotty structure
2.2.2 Round and Elliptical
NGC 6720 - Ring Nebula, M57
Figure 2.7: Halo of the PN NGC 6720
observed by the Subaru Telescope.
Figure 2.8: The PN NGC 6720 observed
by the HST/WFPC2.
High-resolution images of the Ring Nebula taken with the Subaru Telescope
(Komiyama et al. 2000 [84], see Fig. 2.7) reveal the fine structure of the inner and
outer halos and other features: (1) filamentary structure of the inner halo consisting
of loops and arcs, (2) small-scale structures at the main ring like dense knots and
(3) enhanced bands of emission running across the central cavity. The expansion
velocity of the PN of 45 km s −1 implies a expansion age of about 1500 ± 220 years
(O’Dell et al. 2002 [110]).
Figure 2.9: Details in the PN NGC 6720.
Subimages taken from Fig. 2.8.
The innermost part of the inner halo
just outside of the main ring of the nebula
shows a filamentary structure consisting
of loops and knots, which gives a
petal-like appearance to the inner halo.
The outer halo is found to show a limb-
brightened knotty structure similar to
the inner halo, but at much fainter levels.
However, the typical size of the
knots is clearly different between the two
halos. The corresponding lifetime, which is estimated from the size divided by the
thermal velocity, is 400 years and 1200 years (see therefore Komiyama et al. 2000 [84]).
The HST image of the Ring Nebula (see Fig. 2.8) displays a host of subarcsecond
dark knots or globules around the periphery of the nebula (see also Fig. 2.9). The
fact that no globules are seen projected against the central region demonstrates that
their distribution is in fact toroidal or cylindrical, rather than spherical. Thus the

2.2. Examples 19
Ring Nebula is in reality a non-spherical, axisymmetric PN (like many other PNe), non-spherical,
axisymmetric
which are coincidentally seen from a direction close to its axis of symmetry (Bond
et al. 1998 [15]).
Spectroscopic investigations show conclusively
that the inner halo cannot have the
form of a radially expanding, spherical shell,
but rather have to be a bipolar appearance
(Bryce et al. 1994 [17]). The very faint outer
halo is probably the remnants of the original
AGB superwind, expanding radially outward
with a velocity of about 5 km s −1 .
Fig. 2.10 gives an excellent view of the
Ring Nebula and its extended halo in in- extended halo
frared wavelength taken by the Spitzer Space
Telescope (SST) and shows several looping looping structures
structures in the outer halo as well as the
two bright streaks crossing the central re- bright streaks
gion. Additional observations by O’Dell et
al. (2002 [110]) indicate that the streaks are
formed by material inside of the main ring.
Figure 2.10: Halo of PN NGC 6720
observed by the Spitzer Space Telescope
in Infrared Wavelength.

20 2. PLANETARY NEBULAE
filamentary
structure
loops and arcs
NGC 7293 - Helix Nebula
Figure 2.11: Halo of PN NGC 7293
observed by the Kitt Peak National
Observatory and the HST/ACS.
The picture shown in Fig. 2.11, is a composite
of images from the ACS/WFC instrument
on-board the HST combined with the
wide view taken at the Kitt Peak National
Observatory. It reveals a filamentary struc-
ture consisting of loops and arcs in the halo
and thousands of comet-like filaments, also
comet-like filaments
“cometary knots” known as cometary knots. The model de-
twisted components
radial rays
multiplicity of axes
knots are
primordial?
Figure 2.12: Details of PN NGC 7293
observed by the HST. Subimage taken
from Fig. 2.11.
rived from the image consists of two twisted
components of the main ring, radial rays
surrounding the main ring and a multiplicity
of axes of the outflow (see therefore O’Dell
et al. 2004 [112]). The filamentary components
including the cometary knots appear
to be located in a planar regime as noted by
O’Dell 1998 [109]. Speck et al. (2002 [146])
determined a lower limit of the PN shell
mass of about 1.5M⊙.
Fig. 2.12 displays a detailed view of the
cometary knots in the Helix Nebula. Calculations
of the neutral core masses of the
cometary knots from the observed extinction
indicate masses of about 1.5 10 −5 M⊙
for the best observed knots (O’Dell & Handron
1996 [111]). 313 of these objects were
detected and project a total number of the
entire nebula of 3500. Investigations show
a lifetime exceeding that of the PN stage.
Spatial motions of the knots were measured
by O’Dell et al. (2002 [110]). It was found
that the knots originate in or close to the
main ionisation front and possibly in the
neutral zone outside of this. Various physical models are advanced enough to explain
the presence of such cometary knots. Rayleigh-Taylor instabilities seem to be
the most likely source. However, the less likely possibility cannot be ruled out that
these knots are primordial, i.e. going back to the origin formation of what is now
the central star. Rayleigh-Taylor instabilities can either result from the original PN
ionisation front or with stellar wind interactions with the inside of the PN.

2.2. Examples 21
NGC 6853 - Dumbbell Nebula, M27
Fig. 2.13 is a composite image that includes
eight hours of exposure through a
Hα-filter, tracing the complex details of the
nebula’s faint outer halo which spans light- faint outer halo
years. Features which can be located on
this detailed image are (1) a halo with substructures,
(2) numerous dense knots and dense knots
(3) axisymmetric bands of matter in the cen- axisymmetric bands
tral region. The inhomogeneous halo con-
tains various structures, such as radial fila- radial filaments,
ments, arcs and arc-like features (Papamastorakis
et al. 1993 [114]). The bright jetlike
filaments located at the nebula interior
seem to obscure the ionising radiation from
the central star resulting in a dimming of
Figure 2.13: Halo of PN NGC 6853
observed by Robert Gendler.
the halo along their directions. The bow-shaped appearance of the halo is probably
related to an interaction of the nebula with the ISM.
arcs and arc-like
features
Perpendicular to the long axis of the elongated PN is a skewed, bright-rimmed bright-rimmed
elliptical form
elliptical form possessing several internal structures. This geometry suggests a prolate
spheroid with abroad equatorial concentration of material that is viewed nearly
in the plane of the equator (O’Dell et al. 2002 [110]).
The close-up image of M27, displayed in
Fig. 2.14, show many dense knots, but their
shapes vary. Some look like fingers pointing
at the central star, located just off the upper
left of the image; others are isolated clouds,
with or without tails. Their typical size is
about 1000 AU’s in diameter and each contains
as much mass as three Earths, about
10 −5 M⊙ (Meaburn & Lopez 1993 [100]). The
knots are forming at the interface between
the hot, ionised and cool, neutral portion of
Figure 2.14: Details of PN NGC 6853
observed by the HST/WFPC2.
the nebula. This area of temperature differentiation moves outward from the central
star as the nebula evolves. In the Dumbbell Nebula we are seeing the knots soon
after this hot gas passed by.
Very few knots have a clear cusp and tail structure of the prototype cometary cusp/tail structure
knots found in the Helix Nebula. Radial tails become more common at larger dis- Helix Nebula
tances from the central star. This indicates that we are seeing intrinsically radial
structures but under a variety of orientations (see O’Dell et al. 2002 [110]).

22 2. PLANETARY NEBULAE
disc-like structure
comet-shaped
objects
elliptically shaped
lobes
inner bright shell
external circular
shell
NGC 2392 - Eskimo Nebula
Figure 2.15: The PN NGC 2392 observed
by the HST/WFPC2.
Figure 2.16: Detail of PN NGC 2392 observed
by the HST/WFPC2. Subimages taken from
Fig. 2.15.
In this HST image (see Fig. 2.15),
one can see that the “parka” is really
a disc-like structure (similar to the He-
lix Nebula) including a ring of comet-
shaped objects, with their tails streaming
away from the central star. The Eskimo’s
“face” also contains some fascinating
details. It is composed of two
elliptically shaped lobes of matter stream-
ing above and below the dying star. In
this photo, one bubble lies in front of
the other, obscuring part of the second
lobe. The inner bright shell has a pro-
late structure, with the major axis oriented
very closely along the line of sight,
while the external circular shell has a
more oblate structure (e.g. Phillips &
Cuesta 1999 [118]).
The lobes are not smooth but
have filaments of denser matter.
Each bubble is about 1 ly
long and about 0.5 ly wide. The
origin of the comet-shaped features
in the “parka” remains uncertain
(see Fig. 2.16). One possible
explanation is that these
filamentary objects formed by a
collision of slow- and fast-moving
material by Rayleigh-Taylor
instabilities or by an interaction
of a collimated outflow with the outer shell, which also could explain the X-ray emissions
measured by the XMM spacecraft (Guerrero et al. 2005 [61]).
Unlike the Helix Nebula, where the tails beyond the knots are nearly linear structures
bounded by radial lines from the central star, the tails related to the Eskimo
Nebula’s knots only sometimes are well bounded. They definitely lie close to radial
lines, but they often deviate within the tail feature and show orientatios up to 8 ◦
from the radial direction. Furthermore, many of the tails are widen faster than the
shadow of the bright knot at their head (O’Dell et al. 2002 [110]).

2.2. Examples 23
NGC 6369 - Little Ghost Nebula
Fig. 2.17 shows an image taken from the
PN NGC 6369. The HST photograph, captured
with the WFPC2 instrument depicts
remarkable details of the ejection process
that are not visible from ground-based telescopes
because of the blurring produced by
the Earth’s atmosphere. The prominent
blue-green ring, approximately 1 ly, marks
the location where the energetic UV radiation
ionises the gas in the PN. Even far-
ther outside the main ring of the nebula, main ring
one can see fainter arcs of gas that were fainter arcs
lost from the star at the beginning of the
ejection process.
Figure 2.17: The PN NGC 6369 observed
by the HST/WFPC2.
Spectroscopic observations from Monteiro et al. (2004 [104]) leads to a mass of
1.8M⊙ for the ionised gas and from evolutionary models a mass of about 0.65M⊙
for the central object was derived. Consequently, this implies an initial mass for the
stellar AGB progenitor of ≃ 3M⊙.
NGC 3132 - Eight-Burst Nebula
NGC 3132 (see HST/WFPC2 image in
Fig. 2.18) is nearly 0.5 ly in diameter, and
at a distance of about 2000 ly it is one of
the nearer known PNe. The gases are expanding
away from the central star with a
velocity of about 14.5 km s −1 . This image
clearly shows two stars near the centre of
the nebula, a bright white one, and an adjacent,
fainter companion to its upper right.
(A third, unrelated star lies near the edge
of the nebula.) The faint partner is actually
the star that has ejected the nebula. An-
Figure 2.18: The PN NGC 3132 observed
by the HST/WFPC2.
other peculiarity of this PN are the twisted twisted dust lanes
dust lanes in the front and behind the ionised central cavity.
In low-resolution images NGC 3132 appears ellipsoidal, but simple shell models
do not explain all of the observed characteristics. Monteiro et al. (2000 [103]) has
proposed a model in which the nebula has an hourglass structure that is being viewed
at 40 ◦ relative to the light of sight.

24 2. PLANETARY NEBULAE
lowest mass
progenitor star
low total nebular
mass
remarkable
textures
circular ring
long streamers
irregular ring
multiple shell
structure
IC 418 - Spirograph Nebula
Figure 2.19: The PN IC 418 observed
by the HST/WFPC2.
Fig. 2.19 shows an image of the Spirograph
Nebula (IC 418) obtained with the
WFPC2 of the Hubble Space Telescope which
disclose a complex morphology and the bright
central star. This central star with a derived
luminosity of 2850L⊙ was placed on a stellar
evolutionary track (see e.g. Iben 1995 [72])
among the lowest mass progenitor stars for
PNe, consistent with the low total nebular
masses of 0.2 to 0.7M⊙ assumed for the model
proposed by Meixner et al. (1996 [101]. Furthermore,
a mass loss rate of a few 10 −5 M⊙
yr −1 over a period of 3000 years followed by
an abrupt decrease 2000 years ago when the
star presumably left the AGB. The remark-
able textures seen in the nebula are newly
revealed by the Hubble telescope, and their origin is still uncertain.
NGC 6751
Figure 2.20: The PN NGC 6751 observed
by the HST/WFPC2.
Fig. 2.20 displays an image of the Planetary
Nebula NGC 6751 obtained with the
WFPC2 instrument of the Hubble Space Telescope.
The nebula shows a very complex
morphology. Blue regions mark the hottest
gas, which forms a roughly circular ring a-
round the central stellar remnant. Parts in
orange and red show the locations of cooler
gas. The cool gas tends to lie in long stream-
ers pointing away from the central star, and
in a surrounding, irregular ring at the outer
edge of the nebula. The origin of these cooler
clouds within the nebula is still uncertain,
but the streamers are clear evidence that their
shapes are affected by radiation and stellar
winds from the hot star at the centre. The
star’s surface temperature is estimated to be
about 140000 K. Furthermore, observations have indicated the presence of a multiple
shell structure in the faint envelope (Chu et al. (1991 [24]).

2.2. Examples 25
2.2.3 Bipolar and Quadrupolar
NGC 6543 - Cat’s Eye Nebula
Figure 2.21: Halo of the PN NGC 6543
observed by the NOT.
Figure 2.22: The PN NGC 6543 observed
by the HST/ACS.
Fig. 2.21 shows the extended halo of the PN NGC 6543 observed by the Nordic
Optical Telescope (NOT). Many non-radial filamentary structures are observable. non-radial
A more detailed image of the nebula’s core was taken with Hubble’s Advanced
Camera for Surveys (ACS) and can be seen in Fig. 2.22. It reveals eleven or even
filamentary
structures
more concentric rings, or shells, around the Cat’s Eye Nebula. These photoionised concentric rings or
shells
rings are almost certainly the result of periodic spherical mass pulsations by the
nucleus before the Cat’s Eye Nebula formed. A good fit is obtained if the bubbles
were ejected with constant mass, thickness, and ejection velocity. The model can be
used to estimate the total mass of the rings, ≈ 0.1M⊙, which lies between that of
the core (≈ 0.05M⊙) and the surrounding halo (≈ 0.5M⊙). Assuming an ejection
speed of 10 km s −1 the inter-pulse period is 1500 ± 300 years, the same as the
expansion age of the core itself. Images taken from HST of other PNe, IC 418, IC 418, NGC 7027
and Hb 5
NGC 7027, and Hubble 5 (Hb 5) show similar sets of multiple concentric rings
(Balick & Wilson 2000 [11]).
Approximately 1,000 years ago the pattern of mass loss suddenly changed, and
the Cat’s Eye Nebula started forming inside the dusty shells. It has been expanding
ever since, as discernible in comparing Hubble images taken in 1994, 1997, 2000, and
2002 (see therefore Balick & Hajian 2004 [8]). A summary of the bipolar structure bipolar structure of
the core
of the PN core can be found in Balick (2004 [6])

26 2. PLANETARY NEBULAE
bright elliptical
ring
intersecting
elliptical rings
arc-like structures
high speed knots
high surface
brightness
dark lanes
high-velocity
outflow
MyCn 18 - Hourglass Nebula
In the HST/WFPC2 image (Fig. 2.23)
of the young PN MyCn 18 a bright ellipti-
cal ring can be seen. The hot central star
within this ring is shown clearly off-centre.
Several other features has been also revealed
which are completely new and unexpected.
For example, there is a pair of intersecting
elliptical rings in the central region which
appear to be the rims of a smaller hourglass
(Sahai et al. 1995 [130]). The arc-like struc-
tures on the hourglass walls could be (1) the
remnants of discrete shells ejected from the
star when it has been younger (e.g. as seen
in the Egg Nebula), (2) flow instabilities, or
Figure 2.23: The PN MyCn 18 ob-
(3) the result from the action of a narrow
served by the HST/WFPC2.
beam of matter intersecting the walls. Furthermore,
high speed knots with outflow velocities up to about 630 km s−1 are
located apparently along the main axis of the bipolar shape. Spectrometric observations
also indicate that some pairs of knots have been ejected in opposing directions
with the same velocity. Among several possible explanations of these hypersonic
knots O’Connor et al. (2000 [108]) preferred the model of a recurrent nova-like ejection
from a central binary star.
IC 4406 - Retina Nebula
Figure 2.24: The PN IC 4406 observed
by the HST/WFPC2.
hancement surrounding the central object.
An image of the seemingly square nebula
IC 4406 can be seen in Fig. 2.24. Like
many other PNe, it exhibits a high degree
of symmetry. The central star is faint and
seen against the high surface brightness neb-
ula. One of the most interesting features
of IC 4406 is the irregular lattice of dark
lanes that criss-cross the centre of the nebula.
These lanes are about 160 AU wide
(see O’Dell et al. 2002 [110]). Data from observations
done by Sahai et al. (1991 [132])
has revealed the presence of a high-velocity
outflow directed along the major axis of the
bipolar shape. It seems that the outflow has
been collimated by an toroidal density en

2.2. Examples 27
NGC 6302 - Bug or Butterfly Nebula
Figure 2.25: The PN NGC 6302 observed
by Wendel and Flach-Wilken.
Figure 2.26: Details of the PN NGC 6302
observed by the HST/WFPC2.
The surrounding of the PN NGC 6302 shows Fig. 2.25 whereas Fig. 2.26 depicts
a more detailed image captured by the HST/WFPC2 instrument. NGC 6302 is
known as the prototypical “butterfly” PN, with the fast wind of the hot central star
channelised into a bipolar shape by a central dense structure.
This butterfly-shaped PN is also known as one of the brightest and most extreme
PNe. A dense equatorial lane, probably a dusty disc obscuring the central star, can dense equatorial
lane
be observed surrounding the central object. From spectral measurements a mass of
at least 0.03M⊙ was derived for this dust disc (Matsuura et al. 2005 [98]). Some
other models suggest a much more disc mass. The innermost region of this massive
disc shows an ionised shell which was produced by the very hot central star. This ionised shell
star is believed to be one of the hottest PN central star known with an temperature
of 380000 K (Pottasch et al. 1996 [119]).
Furthermore, the bipolar axis shows a distinct change with distance from the distinct change of
bipolar axis
central object. Also the central dust lane looks slightly deformed. Several of such
observed structures are similar to those predicted in the warped-disc model of Icke
(2003 [76]).

28 2. PLANETARY NEBULAE
dense ionised core
spherical, bipolar
lobes
filamentary bipolar
nebula
hypersonic velocity
features
inner and outer
lobes
ansae
dense disc
close pair of stars
Mz 3 - Ant Nebula
Figure 2.27: The PN Mz 3 observed by
the HST/WFPC2.
Fig. 2.27 displays a detailed view of the
complex structure of the young bipolar nebula
Mz 3 or Ant Nebula. This image was
combined by pictures of two observations
with the HST/WFPC2 instrument using
slightly different filters. Detailed investigations
of the bipolar PN Mz 3 reveals a dense
ionised core with almost spherical, bipolar
lobes. These are contained within a much
more extensive filamentary bipolar nebula.
The expansion velocity of the inner lobes is
measured to be about 50 km s −1 whereas
the walls of the outer bipolar structure expand with about 90 km s−1 . A pair of
hypersonic velocity features along the bipolar axis of the nebula reaches velocities
of ≃ 500 km s−1 (Redman et al. 2000 [121]).
M2-9
Figure 2.28: The PN M2-9 observed by
the HST/WFPC2.
The HST/WFPC2 image of the bipolar
PN M2-9 shown in Fig. 2.28 depicts several
details like inner and outer lobes of the
bipolar regions on both sides of the centre,
ansae at the main axis of the nebula and
a dense disc obscuring the central region.
The central star in M2-9 is known to be
one of a very close pair of stars. A model
proposed by Livio & Soker (2001 [92]) assuming
that the system consist of an AGB
star or post-AGB star and a White Dwarf
companion with an orbital period of about 120 years. It is even possible that the
stars are engulfed in a common envelope of gas. Another explanation could be that
the gravity of the compact White Dwarf rips weakly bound gas of the AGB star
generating a thin and dense disc.

2.3. Global Models to Shape a PN 29
2.3 Global Models to Shape a PN
The observed morphologies of PNe range from round to elliptical to bipolar to pointsymmetric,
whereas their progenitors, the AGB stars, show spherical symmetry in
their envelopes (e.g. Lucas et al. 1992 [93] Sahai & Bieging 1993 [127], Neri et
al. 1998 [106]). Therefore, it is certain that the interacting-winds process plays a
role in the shaping of the nebula.
2.3.1 Multiple-Winds Model
In general main categories of PN shapes like round, elliptical, bipolar, point-symmetric
and irregular can be observed. To explain such various shapes detailed theoretical
and observational efforts have to be undertaken to combine all essential physical
effects and all possible scenarios together to a complete theory which can interpret
the phenomenon of a PN.
First approaches were made when Kwok et al. (1978 [88]) formulated the inter- interacting wind
theory
acting winds theory (or two wind model), in which the slow wind from the AGB
expansion is swept up by a later-developed fast wind originating from the central
star and forming the dense PN shell. Then it has been studied in more detail
by Kwok (1982 [85]) and a quantitative model has been constructed by Kahn &
West (1985 [78]).
Furthermore, a refined model is needed for point-symmetric and irregular shaped
PNe. Also the observed small-scale-structures like features in the halos, jets, ansae
and knots demand a theoretical model. We also have to take the three-dimensional perspective
structure and the different perspectives into account, i.e. some bipolar nebula appear
elliptical or even round when their major axis is oriented fairly close along the line
of sight.
The multiple-winds model can describe the global shaping of a PN but fail to
explain many small-scale structures. It is possible to modify and refine this model
to get further interesting results. E.g. if we rewrite the boundary conditions of the
multiple-winds problem we obtain new classes of shapes.
2.3.2 Aspherical Mass Loss of AGB stars
In the ideal case of the multiple-winds model a spherical fast wind from the central
star of a PN or later the White Dwarf blows into the also spherical slow and much
denser wind of the progenitor AGB star. The result is an exact spherical shaped PN.
However, if the slow wind of the AGB star is generated by aspherical mass loss then
an aspherically shaped PN will be formed. Now we have to think about a model
which can explain such aspherical mass losses.
One explanation is stellar rotation which generates an asphericity of the luminos- stellar rotation
ity or alternatively effective temperature at the stellar photosphere (see e.g. Dorfi
& Höfner 1996 [39]). Due to the strong dependence of the dust formation process
on the temperature and density these small deviations from spherical symmetry will
affect the condensation of dust and therefore the mass loss rate. The result is a
preferred mass loss in the equatorial plane and the formation of a pole-to-equator

30 2. PLANETARY NEBULAE
stellar companion
nonaxisymmetric
structure
dynamos in
AGB stars
magnetic shaping
binary and
stellar disc
magnetic fields
in central stars
stellar spots
density distribution in the vicinity of the AGB star. When the fast wind from the
central object succeeding the AGB phase sweeps into this aspherical density distribution
it is possible to form elliptical up to less-developed bipolar PNe as shown
e.g. in Reimers (1999 [122]) and Reimers et al. (2000 [123]).
Another model is based on the existence of a stellar companion, where the de-
parture from axisymmetric AGB wind structure can also form elliptical to bipolar
PNe (Livio 1982 [91]). Furthermore, close binaries can produce an accretion disc
and jets, which induces a collimated fast wind. This is leading to nonaxisymmetric
structures and very pronounced bipolar shapes. Soker & Rappaport (2001 [145])
looked into this problem and described the mechanism to form asymmetric PNe by
the variation of the mass loss, i.e. the preferred mass loss in the orbital plane and/or
the occurrence of a jet-like outflow formed at the accretion disc of the companion
star.
2.3.3 The Role of Magnetic Fields
Dynamos in AGB stars can be the origin of magnetic fields. If this field is strong
enough to interact with the matter and shows a more regular structure, then it is
possible to get bipolar outflows like jets or influence the stellar surface quantities
(like temperature or pressure). These processes are able to shape aspherical PNe.
Magnetic fields in the stellar vicinity
Blackman et al. (2001 [13]) investigated the shaping of PNe by magnetic fields
originated from dynamos in AGB stars. They predict that magnetic fields should be
apparent in the winds of AGB stars and Proto-PNe and also that collimated flows
in Proto-PNe should have signatures of ordered magnetic fields. The dynamo model
is based on a rapid rotation of the AGB core.
If in a close binary system the magnetic field is coupled to the surrounding disc
it is possible to get a powerful MHD wind that might help to explain multishell or
multipolar PNe (Blackman et al. 2001 [14]). The precession of the disc can change
the symmetry axis and therefore form the observed multipolar PN-shapes.
Magnetic fields on stellar surfaces
Nowadays magnetic fields are detected in central stars of PNe (see Jordan et
al. 2005 [77]) and in White Dwarfs. The magnetic fields observed in White Dwarfs
can be explained as relics from magnetic fields in the main-sequence progenitors
which are enhanced by magnetic flux conservation during the contraction of the
core. The discovery of magnetic fields in White Dwarfs (e.g. Aznar Cuadrado et
al. 2004 [3]) indicates that a substantial fraction of White Dwarfs have a weak
magnetic field.
Also proposed is the influence of the occurrence of cool stellar spots on the mass
loss rate of the AGB star (Soker 2000 [141]). A stellar magnetic field could produce
such cool spots by a non-linear magnetic dynamo like that in the numerical SHD
simulations presented by Dorch (2004 [35]). The idea to produce an aspherical mass
loss by an inhomogeneous temperature distribution at the the stellar surface will be
picked up in this thesis. The used model is described in detail in Section 4.3.

2.3. Global Models to Shape a PN 31
2.3.4 Interaction with the ISM
Some central stars of PNe are displaced clearly from the center of the nebula. In displacement of
central star
many cases this indicates an interaction of the nebula with the ISM. The expanding
shell of those highly evolved PNe realise the existence of the gas pressure between
the stars. The material of the shells are incorporated into the ISM, thus the central
star migrate out of the geometrical centre according to the stellar proper motion
(see therefore e.g. Villaver et al. 2000 [152] and Kerber et al. 2004 [81]).
Furthermore, the interaction affects also the shape of the PNe which gets more
and more irregular due to inhomogeneities and turbulences in the ISM. This phase irregular shape
is hardly be observed because as a result of the declining rate of ionised radiation
from the central star such evolved PNe are rapidly dimming objects.
2.3.5 MHD Models
García-Segura et al. (1999 [51]) studied the influence of several effects on the formation
of PNe by applying two-dimensional magneto-hydrodynamic (MHD) simulations
of the interaction of two suceeding, time-independent stellar winds. The
rotation of a single AGB-star can lead to an equatorially confined wind resulting in
a typical hourglass shape. Further on, the combination of such a rotating AGB wind
with a magnetic post-AGB wind generates highly collimated bipolar nebulae. For
sufficiently strong magnetic fields ansae and jets are formed in the polar regions of
the nebula. Photo-ionisation can also reproduce irregularities in the shape of simulated
nebulae through instabilities in the ionisation-shock front. In special cases
also cometary knots are formed.
The formation of point-symmetric structures in elliptical and bipolar PNe are
investigated by García-Segura (1997 [50]) and García-Segura and López (2000 [52])
by using 3D MHD simulations. Therefore, these small-scale structures can be described
according to a misalignment of the magnetic collimation axis with respect
to the symmetry axis of the bipolar wind outflow. Distinct jets or ansae-like structures
are evolving from this model depending on the mass loss rate of the collimated
outflow.

32 2. PLANETARY NEBULAE
AGB wind
2.4 Details in PNe
With high-resolution imaging a lot of detailed structures can be found in or beyond
PNe. Some of these common features are described in the following subsections.
2.4.1 Halo
An interesting field of research are halos of PNe originating from the AGB wind
before the PN forms. In the halo we can see the mass loss history of the AGB star.
This is important to understand the kinematics and dynamics of the mass loss in
the latest AGB phase. To study the structure of these halos can provide us with
informations about the
• time-dependency of mass loss,
• inhomogeneity of mass loss and
• interaction process of AGB winds with the ISM.
A comprehensive observational study of halos around PNe has been undertaken
by Corradi et al. (2003 [30]) which divides the halos into the following groups:
• circular or slightly elliptical AGB halos, which contain the signature of the
last thermal pulse on the AGB;
• highly asymmetrical AGB halos;
• candidate recombination halos, i.e. limb-brightened extended shells that are
expected to be produced by recombination during the late post-AGB evolution,
when the luminosity of the central star drops rapidly by a significant factor;
• uncertain cases which deserve further study for a reliable classification;
• non-detections.
Many PNe with faint extended halos have been newly discovered as well as very
expanded ones, e.g. the detection of a giant halo around the PPN NGC 7027 (Navarro
et al. 2002 [105]).
Multiple shells
Several explanations have been proposed to describe multiple shells around PNe
including cycles of magnetic activity somewhat similar to our Sun’s sunspot cycle,
the action of companion stars orbiting around the dying star, and stellar pulsations.
Another possibility is that the material is ejected smoothly from the star, and the
rings are created later on due to formation of waves in the outflowing material. It
will take further observations and more theoretical studies to decide between these
and other possible explanations.

2.4. Details in PNe 33
”FLIERs” (fast low-ionisation emission regions)
FLIERs seem to be most common in elliptical PNe. They are supersonic outflows
on opposite sides in the major axis and at the same distance from the central star
(cf. ansae below). An extensive research about FLIERs and other micro-structures
in PNe has been summarised in a set of four papers by Balick et al. (1993 [10],
1994 [9], 1998 [7]) and Hajian et al. (1997 [64]).
Rings and Arcs
The bright aspherical nebulae are often found to be surrounded by faint, roughly
round halos, which are signatures of the progenitor AGB envelopes produced by
an isotropic mass loss. A number of these halos include numerous concentric arcs,
an evidence for quasi-periodic modulation of the mass loss on time scales of a few
hundred years (see e.g. Hrivnak et al. 2001 [70] and Corradi et al. 2004 [29]).
2.4.2 Jets, Lobes and Ansae
Observations of the PPN NGC 7027 show a remarkable series of holes or cavities in
the molecular hydrogen region. These features are ordered point symmetric about
the central star. The most direct interpretation of these point symmetric holes is
the action of multiple, bipolar outflows or jets from the central star. These are seen
to be increasingly common in PNe, and well studied examples of multiple outflows
showing interactions with the surrounding molecular envelope (CRL 2688 and M1- M1-16
16). In both cases the jets are more prominent than in NGC 7027, but the similarity
is striking (cf. Cox et al. 2002 [31]).
PNe with low excitation characteristics show highly aspherical morphology illustrated
in the existence of multipolar bubbles. In some objects bipolar ansae multipolar bubbles
or collimated radial structures are seen indicating the presence of jets. Sahai and
Trauger (1998 [129]) proposed that the shaping of those PNe is done by high-speed
collimated outflows or jets during the late AGB phase and/or early Post-AGB evolutionary
phase. Three pairs of low-ionisation features, called caps, ansae, and jets,
are detected at or beyond the perimeter of the core of the Cat’s Eye Nebula. The
two thin, radial jets do not lie along the symmetry axis of any other features. Long
term observations are made but no evidence of a lateral change has been seen (Balick
2004 [6] and Balick & Haijan 2004 [8]). One prototype of a PN with ansae is the
Saturn Nebula, NGC 7009. Sabbadin et al. (2004 [126]) analysed the 3D-structure NGC 7009
‘Saturn Nebula’
and several parameters of the nebula and the central star. The ansae expand with
a velocity larger than the rest of the nebula. The ansae material is ejected with
a higher velocity in the PPN phase. The proper motion of the ansae has been
measured by Fernández et al. (2004 [43]) giving a velocity of 114 ± 32 km s −1 and
a dynamic age of about 910 ± 260 years. The age of the PN is determined to be
around 2000 years where the ionisation of the main shell is initiated.
2.4.3 Knots
Dense knots of gas and dust seem to be a natural part of the evolution of PNe.
They form at early stages and their shape changes as the nebula expands. Similar

34 2. PLANETARY NEBULAE
cometary knots
NGC 3918, K1-2
and Wr 17-1
knots have been discovered in other nearby PNe that are all part of the same evolutionary
scheme. High-resolution images taken by the HST display several knots
e.g. in the Ring Nebula, the Eskimo Nebula and the Retina Nebula. The appear-
ance of these knots ranges from cloudy as in the Dumbbell Nebula to cometary or
finger like e.g. in the Ring, Helix and Eskimo Nebula. The most serious criticism
of the cometary knots as primordial knots has been annotated by O’Dell & Handron
(1996 [111]). It is therefore not possible to decide if cometary knots exist as
primordial knots. Due to the extreme conditions at the late stage of the AGB star
evolution gravitationally bound objects could be undetectable until they are revealed
at large distances. Some issues about cometary knots (sometimes called globules)
are discussed in the following papers:
- Capriotti (1973 [21]): Structure and evolution of planetary nebulae (Rayleigh-
Taylor instabilities, e.g. Helix Nebula, NGC 7293)
- Henry et al. (1999 [66]): Morphology and composition of the Helix Nebula
- Corradi et al. (1999 [28]): Jets, knots and tails in planetary nebulae (NGC 3918,
K1-2, and Wr 17-1)
- O’Dell et al. (2002 [110]): Knots in nearby planetary nebulae (filaments in
IC 4406, the Retina Nebula, as possible globule precursors)
- Huggins & Mauron (2002 [71]): Small scale structure in circumstellar envelopes
and the origin of globules in planetary nebulae

Part II
Theoretical Models
35

Chapter 3
Radiation Hydrodynamics
Simulation
To describe a stellar interior and atmosphere we need at first a physical system
which characterises the behaviour of the gas and dust embedded in a radiative
environment. Since the star is in the thermodynamic equilibrium, i.e. the produced
energy is equal to the radiated energy, a certain temperature and luminosity adjusts
itself at the photosphere of the star. Furthermore, the physical conditions depend
on the equation of state and the optical properties of the gas and dust component.
The radiative energy at the photosphere is used to force dynamical processes in the
stellar atmosphere which includes shock waves propagating through the atmosphere
as well as the generation and destruction of molecules and dust grains.
In this chapter the derivation of the used physical system, the boundary conditions
and the generation of initial models are given in the special case of spherical
symmetry.
3.1 Basic Equations
3.1.1 Conservation form
The conservation form of physical equations describes the changes of a physical parameter
φ (e.g. momentum, density, etc.) only due to fluxes through the boundaries
and is conserved otherwise. Thus a certain measurable property of an isolated physical
system does not vary as the system evolves. Therefore, the total derivative of a
physical parameter integrated over the volume dV can be written as
�
d
φdV =
dt
�
Qi − �
Sj
(3.1)
V (t)
where Qi represents the source terms and Sj the sink terms. Because the boundaries
depend on the time t we can apply Leibniz’ rule for the r.h.s.
�
d
dt
φdV =
� �
∂φ
dV +
∂t
φ �u d � A . (3.2)
V (t)
V (t0)
i
37
j
A(t0)

38 3. RADIATION HYDRODYNAMICS SIMULATION
To rewrite this equation in a compact form we use Gauß’ law
�
d
dt
V (t)
φdV =
�
V (t0)
�
∂φ
∂t + � �
∇ · (φ �u) dV . (3.3)
Another notation can be derived by using the substantial derivation ( D
Dt := ∂t+�u· � ∇)
�
d
dt
V (t)
φdV =
�
V (t0)
�
Dφ
Dt + φ � �
∇ · �u dV . (3.4)
For a moving coordinate system the conservation equations take the form
� �
∂
X dV + X u
∂t V (t) ∂V (t)
rel �
dA = S dV , (3.5)
V (t)
where X is the physical quantity, S represents the source and sink terms and u rel is
the relative velocity between the co-moving frame and the numerical grid. Thus, for
an Eulerian grid we have u rel = u (u is the gas velocity) and for a Lagrangean grid
u rel = 0. In case of an adaptive grid and spherical symmetry the relative velocity is
given by
u rel = u − u grid , (3.6)
where ugrid is the velocity component of the moving coordinate system in an inertial
frame of reference
u grid = ∂r
. (3.7)
∂t
3.1.2 Gas Component
Equation of Continuity
This equation describes the conservation of mass within a volume V
dm
dt
= 0 . (3.8)
The physical parameter φ is the density ρ and there are no source and sink terms
to include
�
d
dt
ρdV = 0 . (3.9)
Thus we can write the conservation form of the continuity equation as
V (t)
∂
∂t ρ + � ∇ · (ρ�u) = 0 . (3.10)

3.1. Basic Equations 39
Energy Equation
This equation describes the conservation of the energy
dE
dt
= �
i
E (t)
i . (3.11)
The physical parameter φ is the density multiplied by the internal energy per mass
ρe. But first we look at the total energy balance which includes the internal and
mechanical energy
∂
∂t
�
ρ
�
e + 1
2 v2
��
+ � ∇ ·
� �
ρ�u e + 1
2 v2
��
= −� ∇ · �q + � ∇ · T · �u + ρ �u · � F . (3.12)
If we multiply the momentum equation by �u from the left side we get the mechanical
energy balance
�
∂
ρ
∂t
v2
�
+
2
� �
∇ · ρ�u v2
�
= −�u ·
2
� ∇P + �u · ( � ∇ · τ) + ρ �u · � F . (3.13)
Now we combine these two relations and derive
where
∂
∂t (ρe) + �∇ · (ρe�u) = −P �∇ · �u + τ : �∇�u − �∇ · �q , (3.14)
τ : � ∇�u = �
i,j
τijui;j
denotes the contraction of the viscous stress with the divergence of �u.
Momentum Equation (Equation of Motion)
This equation describes the conservation of the momentum
d � I
dt
(3.15)
�
= �Fi . (3.16)
The physical parameter φ is the density multiplied by the gas velocity ρ�u.
d � I
dt
i
�
d
=
dt
V (t)
First we will start with the following approach
�Ftot = ρ D�u
Dt = � ∇T + ρ �
(ρ�u)dV (3.17)
i
�fi , (3.18)
where T = −PI + τ is the stress tensor which describes the momentum of the
molecular processes (e.g. pressure), P is the thermodynamical pressure, τ is the

40 3. RADIATION HYDRODYNAMICS SIMULATION
viscous stress and � fi are external forces. With the help of the continuity equation
we get the conservation form of the equation of motion
or
3.1.3 Radiation Field
∂
∂t (ρ�u) + � ∇ · (ρ�u�u T ) = � ∇T + ρ �
∂
∂t (ρ�u) + � ∇ · (ρ�u�u T ) = −� ∇P + � ∇τ + ρ �
i
i
�fi
(3.19)
�fi . (3.20)
The radiation transfer is treated in the grey (i.e. frequency integrated) approximation.
The radiation field can be characterised by moments of the specific intensity
Iν defined by
E = 1
c
�∞�
0
∞
��
�F =
0
P = 1
c
�∞�
0
IνdΩdν = 4π
J ... radiation energy density (3.21)
c
Iν�ndΩdν = 4π � H ... radiation energy flux (3.22)
Iν�n�ndΩdν = 4π
K ... radiation pressure (3.23)
c
The starting point of the equations needed is the radiation transfer equation (hereafter
RTE), which describes the propagation of photons through a medium with the
ability of absorption and emission of radiation.
Special cases can be deduced from plane-parallel or spherical geometry. But we
will derive the zeroth and first moment equations of the RTE in general geometry
and for a moving medium. The detailed way of deduction is given in Appendix C
(Radiation Transfer). Here we will only write down the moments as derived from
Buchler 1983 [18] (see also Buchler 1986 [19]).
Zeroth Moment Equation of the RTE
ρ d
dt
� �
E
+
ρ
1
c2 d
�
�u ·
dt
� �
F + � ∇ · � F + P : � ∇�u + 1
c2 �
�a · � �
F =
�∞
0
q0(ω)dω (3.24)

3.1. Basic Equations 41
First Moment Equation of the RTE
ρ
c
� �
d �F
dt ρ
3.1.4 Dust
+ 1
c
d
dt
� �
(�u · P) + c �∇ · P + 1 1
� �
(�aE) + �F · ∇�u � =
c c
�∞
0
�q(ω)dω (3.25)
After the inclusion of gas and radiation we will add dust as third component in the
RHD code. The process of dust formation for a carbon-rich chemistry is described
as a two step process according to Sedlmayr 1989 [139]. The first step describes the
condensation of supercritical nuclei out of the gas phase and the second step depicts
the growth of macroscopic grains.
Net Transition and Growth Rate
The condensation of supercritical nuclei is determined with the net transition rate
for spherical dust particles (i.e. d = 3)
J = N d−1
d
ℓ
f(Nℓ,t) 1
τ
, (3.26)
where Nℓ is the lower limit of the dust grain sizes which may be regarded as macroscopic
in the thermodynamical sense, f(Nℓ,t) denotes the number density of grains
of size Nℓ and 1
τ describes the net growth rate which is calculated in the case of
chemical equilibrium (for more details see Appendix D). However, in general the
number density f(N,t) is not known and J has to be calculated in a different way.
If the thermodynamical conditions allow dust grain formation the transition rate
is assumed to be equal to the nucleation rate J∗, i.e. the rate at which stable (supercritical)
dust clusters are formed out of the gas phase. The implementation of
the nucleation rate can also be found in Appendix D and is based on the classical
nucleation theory (see Feder et al. 1966 [42]).
Moment Equations
The net change of the number density of dust grains containing N monomers is given
by the master equation
df(N,t)
dt
= R↑(N) − R ↑ (N) − R↓(N) + R ↓ (N) , (3.27)
where the terms R↑(N) and R ↓ (N) refer to the addition of i-mers while R ↑ (N) and
R↓(N) accounts for the subtraction of i-mers. To describe the dust component we do
not need to handle the complex system like the master equation because fortunately
the information required for a self-consistent model is contained in a few moments
Kj =
∞�
N j/d f(N,t) . (3.28)
N=Nℓ

42 3. RADIATION HYDRODYNAMICS SIMULATION
adaptive grid
The following set of moment equations can be derived from the master equation
dK0
= J
dt
(3.29)
dKj j 1
=
dt d τ Kj−1 + N j/d
ℓ J (1 ≤ j ≤ d) (3.30)
which describe the formation, growth and evaporation of macroscopic dust grains
(Gauger et al. 1990 [53]) in a comoving frame.
Number Density of Free C-atoms
Additionaly, we need also an equation which describes the number density of all free
C-atoms in the gas phase (excluding the molecule CO) which are able to build up
dust particles. Hence, we can write
∂
∂t nc + ∇ · (nc u) = 1
τ K2 + Nℓ J (1 ≤ j ≤ d) (3.31)
for the amount of condensable material.
3.2 Additional Equations and Constitutive Relations
3.2.1 Grid Equation
The spatial distribution of the grid points is determined by the so-called grid equation
developed by Dorfi & Drury (1987 [36]) which is solved together with the RHD
equations. Therefore, an adaptive grid ensures proper resolution of various features
like shock fronts or steep gradients. The grid equation takes the following form
ˆnl−1
Rl−1
= ˆnl
Rl
, (3.32)
where Rl denotes the desired resolution and nl the point concentration (for more
details see Dorfi & Drury 1987 [36]).
3.2.2 Mass Equation
This equation describes the mass integrated up to a radius r, i.e.
m(r) =
�V
0
ρ(r)dV ′ . (3.33)

3.2. Additional Equations and Constitutive Relations 43
In general the density ρ is a combination of the gas density and the dust density,
but the dust density can be neglected in case of the small amount of condensable
material compared to the amount of hydrogen and helium in the gas phase.
3.2.3 Poisson Equation
The Poisson equation describes the dependence of the gravitational potential ψ on
the density distribution ρ, i.e.
∆ψ = 4π Gρ . (3.34)
To get the gravitational acceleration we need to solve following equation
�g = − � ∇ψ (3.35)
and is implemented in Eq. (3.20) as an additional external force, namely the gravitational
force for a point mass
�g = − GM
r 2
3.2.4 Equation of State (EOS)
�r
r = � fg . (3.36)
The EOS is needed to get relations between the density and temperature of the
material on the one hand and its pressure and internal energy, specific heats, etc.,
on the other hand. Due to the low gas density in stellar atmospheres the ideal gas
approach is a good approximation. ideal gas
The following set of equations describe the relations approximated by an ideal
gas:
or
P(ρ,e) = R
µ cV
T(ρ,e) = e
cV
ρ e pressure (3.37)
temperature (3.38)
ρ(P,T) = µ P
R T
density (3.39)
e(P,T) = cV T energy (3.40)
With the specific heat at a constant volume as
� �
∂e
cV := =
∂T
R
� �
1
µ γ − 1
we can also write for the pressure in Eq. (3.37)
V
(3.41)
P(ρ,e) = (γ − 1) ρ e . (3.42)
Other useful quantities as the adiabatic temperature gradient
� �
∂lnT
∇ad := =
∂lnP
γ − 1
, (3.43)
γ
S

44 3. RADIATION HYDRODYNAMICS SIMULATION
Mie approximation
extinction efficiency
the specific heat at constant pressure
and the relation
δ := −
cP = γ cV
� �
∂lnρ
∂lnT P
can be derived from the EOS of an ideal gas.
3.2.5 Opacity of Gas and Dust
Gas Opacity
(3.44)
= 1 (3.45)
For our calculations the mass absorption coefficient of the gas κ g is set constant, i.e.
κ g = 2 10 −4 cm 2 g −1 . (3.46)
This will reduce computation time and is not an inherent limitation of the RHD
code itself. It seems not likely that a constant gas opacity is the source of major
errors in the hydrodynamical calculations (see Bowen 1988 [16]).
Dust Opacity
If we assume that the radius of all dust particles is small compared to the mean
wavelength of the radiation then the Mie theory can be applied. According to Gail
& Sedlmayr (1987 [48]) the dust opacity χ can be written in the Mie approximation
as
χ = r 3 0π Q ′ ext(T)K3 , (3.47)
where r0 is the radius of the monomer, Q ′ ext = Qext/a denotes the extinction efficiency
of the dust grain material which is independent of the grain radius a. For
optically thin dust shells the flux average of Q ′ ext can be replaced by a Planck average,
for thick dust shells it is replaced by a Rosseland mean opacity which again
can be approximated by a power law and is given by
Q ′ ext(T) ≈ Q ′ R(T) = 5.9 Trad
according to Gail & Sedlmayr (1985 [47]). For our calculations we use
Q ′ R(T) = 4.4 Trad
(3.48)
(3.49)
(cf. Sandin & Höfner 2003 [133]) which is based on the optical constants of Maron
(1990 [97]). Finally, the mass absorption coefficient of the dust κ d is defined by
κ d = χ
ρ
. (3.50)

3.2. Additional Equations and Constitutive Relations 45
3.2.6 Source Function of Gas and Dust
The source function represents the radiation emitted by the gas or dust. Assum-
ing local thermodynamic equilibrium (LTE) with a gas temperature Tg the source local
function can be set equal to the Planck function, i.e.
thermodynamic
equilibrium
Similarly, the source function of the dust grains is given by
where Td denotes the dust temperature.
Sg = σ
π T 4 g . (3.51)
Sd = σ
π T 4 d
, (3.52)
Due to the large opacities the dust is effectively thermally coupled to the radiation
field (energy coupling), which implies that the source function of the dust component energy coupling
is approximately equal to the zeroth moment of the radiation field (radiation energy
density) of the gas. Consequently the dust temperature can be approximated by the
temperature of the radiation field, i.e.
3.2.7 Eddington Factor
Td = Trad . (3.53)
To close the system of moment equations for the radiation field which uses three
moments (J, H and K) we need a further relation between J and K, which is given
by the Eddington factor
fedd = Kν
Jν
. (3.54)
It contains information about the angular dependence of the radiation intensity and
depends on the optical depth. For an isotropic radiation field, i.e. an optically thick
medium, we obtain the Eddington approximation of fedd = 1
3
(i.e. isotropic radia- Eddington
approximation
tion field), which is achieved e.g. in the stellar envelope. Whereas the Eddington
factor goes to fedd = 1 for a distant point source. In the case of stellar atmospheres
neither of these conditions are fulfilled and the Eddington factor has to be determined
by solving the radiation transfer equation. There exist several approaches to
approximate the closure condition of the moment equations. Lucy (1971 [94] and
1976 [95]) has been tried to solve the problem by a semi-analytical treatment of semi-analytical
treatment
the radiative transfer in extended stellar atmospheres whereas Yorke (1980 [161])
and Balluch (1988 [12]) used the method of characteristics for the integration of the method of
characteristics
static transfer equation. The latter method is implemented in the RHD code.

46 3. RADIATION HYDRODYNAMICS SIMULATION
outflow-boundary
condition
3.3 Boundary Conditions
3.3.1 Inner Boundary
At the inner boundary we specify a fixed boundary value for the radius r implying
that there is no pulsation. The density ρ, the internal energy e, the radiation energy
density J, the radiation flux H and the number density of free C-atoms nC are
set to the values given by the initial model program (see Section 3.4). All other
variables like the integrated mass mr, the velocity u rel and the moments of the dust
component (K0, K1, K2, K3) are identically set to zero.
3.3.2 Outer Boundary
At the outer boundary we adopt for the radius r
- that it is set according to fulfil u rel = 0, i.e. the computational domain can
propagate outward or inward in the case of a relaxation of the physical system
or dynamical calculations to the critical point of the outflow or
- is set to a fixed boundary value in the case of dynamical calculations with a
outflow-boundary (see below).
The velocity u is either
- be set to zero in the case of a relaxation or
- for the dynamical calculation a simple outflow-boundary condition is applied
at the external boundary by assuming that the velocity gradient vanishes,
= 0.
i.e. ∂u
∂r
The density ρ, the internal energy e, the radiation energy density J, the radiation
flux H and the number density of free C-atoms nC are set to the values given by
the initial model program (see Section 3.4). The integrated mass mr is given by the
solution of the mass equation for the outermost radius. Assuming that the radiative
flux has only an outward component it is determined by H = µ ′ J at the outer
boundary (cf. Section 3.4.4).

3.4. Initial Models 47
3.4 Initial Models
After the summarisation of the conservation equations and additional physical relations
we have to specify appropriate initial conditions. The initial model describes
a complete set of physical variables which are a consistent solution of the system of
equations of the RHD problem. It also represents the spatial structure of a model
at a specific point of time.
3.4.1 Modelling Method
The initial model is completely determined by the stellar parameters luminosity L∗,
effective temperature Teff and total mass Mtot as well as the elemental abundances of
the species relevant to the dust formation, especially the amount of oxygen (log εO;
cf. Section 5.2.1). The integration of the static radiation hydrodynamic equations is static RHD
equations
started from the photospheric radius of the star
�
Rphot =
L∗
4πσT 4 eff
(3.55)
outwards to get the structure of the atmosphere (mass, pressure, temperature and
radiation flux distribution). The pressure at the photosphere is determined iteratively
to fulfil the outer boundary condition for the radiation field at the external
boundary (cf. Section 3.4.4). To determine the Eddington factor fedd which is required
to solve the diffusion equation we need further iteration of the solution. As a
first step the structure of the atmosphere is calculated using the Eddington approximation
(fedd = 1/3) and afterwards fedd is adjusted iteratively. If the atmospheric
structure has been determined successfully the finally integration inward from the
photosphere to the inner boundary (RADI) is accomplished to calculate the profile of
the stellar envelope. For the determination of the envelope structure (mass, pressure
Stellar
Core
Envelope Atmosphere
RADI Rphot
Rext
RADE
Figure 3.1: Computational domain of the initial model
External
Medium

48 3. RADIATION HYDRODYNAMICS SIMULATION
static stellar
structure equations
and temperature) the static stellar structure equations are used. Fig. 3.1 depicts
the computational domain, i.e. stellar envelope and atmosphere) for the calculation
of the physical quantities of the initial model.
To be usable as starting point of the RHD code some variables have to be generated
from the output variables of the initial model code. The derivation of these
variables and according assumptions are summarised below.
Density and Internal Energy
The relations P(ρ,e) and T(ρ,e) from the equation of state (EOS) are used to
determine the radial profile of the density ρ(r) and the internal energy e(r) (see
Section 3.2.4).
Integrated Mass
The integrated mass denotes the mass contained within the radius r and is given
by
�r
m(r) = 4π
0
ρ(r ′ )r ′2 dr ′ . (3.56)
Radiation Flux
From the luminosity L = 4πr 2 Fr with the radiation flux Fr = 4π H we can
derive the radiation flux as
H(r) = L
(4π) 2
1
. (3.57)
r2 Radiation Energy Density
Assuming local thermodynamic equilibrium (LTE), the radiation energy density
is equal to the Planck function and we obtain
J(r) ≃ B(Tg) = σ
π Tg(r) 4 . (3.58)
Number Density of free C-atoms
The number density of carbon C in the dust phase is determined by the amount
of oxygen and carbon as well as the number density (approximately derived from
the gas density ρ(r)) of the hydrogen like
(cf. Appendix D).
n dust
C (r) = (εC − εO)n tot
H
(r) (3.59)
Velocity
In the static limit case of the radiation hydrodynamic equations all time derivatives
and the gas velocity is identically set to zero, i.e.
u(r) ≡ 0 . (3.60)
Moments of the dust size distribution
Assuming that no dust is present in the initial phase of the time-dependent calculations
the moment equations for the dust are not taken into account
Kj(r) ≡ 0 (1 ≤ j ≤ d) . (3.61)

3.4. Initial Models 49
3.4.2 Equations for the Stellar Envelope
The stellar envelope’s structure is calculated by the following differential equations
in spherical symmetry:
Mass Equation
The derivation of the mass along the radial direction for a spherical symmetry is
Hydrostatic Equilibrium
∂mr
∂r = 4πr2 ρ . (3.62)
The pressure gradient is composed of the gravitational acceleration (first term on
the RHS) and the force produced by acceleration of the material (second term)
∂P
∂r
= −Gmr
r2 ρ − ρ∂2 r
, (3.63)
∂t2 whereas the second term on the RHS can be neglected in the stationary case.
Transport Equation
According to the small mean free path of the photons compared to the stellar radius
the radiative transport in stars can be treated as a diffusion process. Therefore,
the diffusive flux of radiative energy � F is given by
�F = −D � ∇E , (3.64)
where D is the diffusion coefficient and � ∇E describes the gradient of the radiation
energy density. The diffusion coefficient
D = 1
3 〈v〉ℓp
(3.65)
is determined by the average values of mean velocity 〈v〉 and the mean free path ℓp
of the photons. With J = σ
π T 4 (LTE) we obtain
E = 4π
c J = aT 4 , (3.66)
where a = 4σ
c = 7.57 10−15 erg cm −3 K −4 is the radiation density constant, and
in Eq. (3.65) v can be replaced by the velocity of light c and ℓp by ℓph = (κρ) −1 .
Assuming spherical symmetry � F has only a radial component, i.e. Fr = | � F | = F
and � ∇E reduces to the derivative in the radial direction, i.e.
∂E
∂r
= 4aT 3∂T
∂r
. (3.67)

50 3. RADIATION HYDRODYNAMICS SIMULATION
Then Eq. (3.64) and (3.65) give immediately that
F = − 4ac T
3
3 ∂T
κρ ∂r
. (3.68)
Introducing the local luminosity l(r) = 4πr 2 F, which can be set constant in the
outer stellar envelope (l = const. = L∗), the derivation of the temperature yields to
∂T
∂r
= − 3
16πac
ρ
r 2
κL∗ 3
= −
T 3 64πσ
The derivation of the temperature can also be expressed as
∂T
∂r
= ∂P
∂r
∂T
∂P
ρ
r 2
T
= −Gmr ρ
r2 P
κL∗
. (3.69)
T 3
∇ , (3.70)
where
∂ ln T
∇ = .
∂ ln P
(3.71)
If the energy transport is mainly conducted via radiation then
∇ = ∇ad =
3
64πσG
3.4.3 Equations for the Stellar Atmosphere
κLP
. (3.72)
mT 4
Unlike the stellar envelope, in the atmosphere we have to treat additionally with the
interaction of the gas with the radiation field. Thus we need a further equation for
the radiative flux. Therefore, the stellar atmospheres structure is calculated by the
following differential equations:
Mass Equation
The derivation of the mass along the radial direction for a spherical symmetry is
Hydrostatic Equilibrium
∂mr
∂r = 4π r2 ρ(r) . (3.73)
The pressure gradient is composed of the gravitational acceleration (first term of
the RHS) and the contribution of the radiation flux (second term)
∂P
∂r
mr 4π
= −G ρ(r) + ρ(r)κH . (3.74)
r2 c

3.4. Initial Models 51
Diffusion Equation
The 1st-order moment equation of the radiation transfer equation is given in the
≡ 0 and no velocity field by
static limit case, i.e. d
dt
�∇ · Kν = −ρ(κν + σν) � Hν
(3.75)
(cf. Eq. (C.55) in Appendix C). For spherical symmetry and a nearly isotropic
radiation field thus � ∇ · Kν → � ∇Kν, where Kν is the scalar moment of the radiation
pressure, we get
∂Kν
∂r + 3Kν − Jν
= −ρ(κν + σν)Hν . (3.76)
r
We can replace the radiation pressure Kν by the Eddington factor fν = Kν
Jν and
obtain
∂(fνJν)
∂r + (3fν − 1)Jν
= −ρ(κν + σν)Hν . (3.77)
r
Introducing local thermodynamic equilibrium (LTE), i.e. Jν = σ
πT 4
we finally derive
∂T
∂r
∂fν
∂r σT 4 3 ∂T
+ fν σ 4T
∂r + 3fν − 1
σT
r
4 = − π
σ ρ(κν + σν)Hν
= −
�
π
σ ρ(κν + σν) Hν
T 3
� 1
4fν
which can be written in a compact form as
where
∂T
∂r
Radiation Flux Equation
= −A1
�
∂fν
−
∂r + 3fν
�
− 1
T
r
1
4fν
(3.78)
, (3.79)
ρ(r)κH
T 3 − A2 T , (3.80)
A1 = π 1
σ 4fedd
�
∂fedd
A2 =
∂r + 3fedd
�
− 1 1
r
4fedd
With the help of the relation F = 4π H the luminosity can be written as
(3.81)
(3.82)
L(r) = 4π r 2 F = 16π 2 r 2 H . (3.83)
Since the luminosity is constant within the atmosphere, i.e. dL
dr ≡ 0, it is obvious
from
� �
dL
2 ∂H
= 16π2 2r H + r , (3.84)
dr ∂r
that the term in parenthesis is equal to zero thus
∂H
∂r
= −2 H
r
. (3.85)

52 3. RADIATION HYDRODYNAMICS SIMULATION
3.4.4 Additional Notes
External Radiation Flux
To estimate the temperature at the external boundary of the stellar atmosphere we
start with the luminosity
L = 4πr 2 Fr
(3.86)
and introducing Fr = 4πH we can derive the radiation flux at the outer boundary
External Temperature
L
H(Rext) =
16π2 R2 . (3.87)
ext
At the outer boundary we assume the radiative flux to have only an outward component
thus it is given by the relation
H = µ ′ J , (3.88)
where µ ′ denoting a quantity accounting for the geometry of the radiation field.
In the case of a variable Eddington factor µ ′ results from the solution of the grey
radiation transfer equation which is calculated after each time-step to determine the
Eddington factor (cf. Section 3.2.7). For the Eddington approximation µ ′ is equal
. Assuming local thermodynamic equilibrium (LTE) we obtain
to 1
2
J ≃ B(T) = σ
π T 4 g
(3.89)
and with Eq. (3.88) and (3.89) we get the temperature at the outermost radius as
Total Pressure
Tg(Rext) =
� π
σ
H(Rext)
µ ′
�1
4
. (3.90)
Due to a large radiation flux in the stellar interior (especially for luminous objects
like the AGB stars) the photons can contribute considerably to the total pressure.
Assuming the radiation is that of a black body the radiative pressure is given by
Prad = 1 a
E =
3 3 T 4 rad . (3.91)
Then the total pressure is calculated by a combination of the gas pressure and the
radiative pressure
Ptot = Pg + Prad . (3.92)

3.5. Numerical Methods 53
3.5 Numerical Methods
To solve the nonlinear system of partial differential equations (short PDEs) an implicit
numerical scheme, i.e. all variables are represented by their values at the new
point of time, is used in order to obtain sufficiently large time steps during the
dynamical evolution. Whereas explicit codes investigating the same problem suffer
from the very restrictive Courant-Friedrichs-Lewy time step condition (e.g. Richtmyer
& Morton 1967 [125]) The resulting algebraic system of difference equations
for the implicit RHD code is solved using a Newton-Raphson algorithm. The inver- Newton-Raphson
algorithm
sion of the Jacoby matrix of the system is done by the Henyey method (Henyey et
al. 1965 [67]). Furthermore, a fully adaptive grid (Dorfi & Drury 1987 [36]) provides Henyey method
a sufficient spatial resolution in regions of steep gradients. adaptive grid
NO
BEGIN
INPUT OF CONTROL PARAMETERS
INPUT OF INITIAL MODEL
INITIATE HENYEY-ITERATION
CONVERGENCE?
YES
CALCULATE NEW MATERIAL-
FUNCTIONS
GENERATE NEW TIME-STEP
RESTORE NEW DATA
FORWARD EXTRAPOLATION
STOP-CONDITION
FULFILLED?
YES
END
NO
X DIVERGENCES
REACHED?
YES
Figure 3.2: Flowchart of the RHD code
NO
GENERATE NEW TIME-STEP
FORWARD EXTRAPOLATION

54 3. RADIATION HYDRODYNAMICS SIMULATION
monotonic
advection scheme
artificial tensor
viscosity
The used RHD code has been developed at the Institute for Astronomy (e.g. Dorfi
& Feuchtinger 1991 [37], Feuchtinger 1999 (implementation of a nonlinear convective
model for radial stellar pulsations) and Dorfi & Höfner 1991 [38] for the implementation
of dust formation in LPV winds) and a general flowchart of the implicit code
is presented in Fig. 3.2.
Discretisation
The system of equations has been discretised employing a second-order monotonic
advection scheme (van Leer 1977 [151]). General aspects and rules of the used
discretisation are summarised in Appendix A.
Artificial Viscosity
For a proper handling of shock fronts, steep gradients or other discontinuities like
ionisation fronts in nonlinear hydrodynamical calculations an artificial viscosity have
to be implemented. This method introduces an additional pseudo-viscous pressure
which broadens the shock fronts and discontents over a few grid points. Tschar-
nuter & Winkler (1979 [149]) have been developed a coordinate invariant artificial
tensor viscosity and a detailed description of the used artificial viscosity is given in
Appendix B.

Chapter 4
Stellar Spots
The model of this thesis is based on the assumption that the mass loss of the AGB
star is not homogeneously distributed above the stellar photosphere. These inhomogeneities
emanate from cooler regions which are probably caused by stellar spots.
The lower temperature of these spots should favour the generation of dust grains
and therefore induce a different mass loss rate as well as outflow velocities.
In this chapter we will discuss the existence and occurrence of cool stellar spots.
The starting point are the well studied sunspots on the solar photosphere. Furthermore,
a flux tube model simplifying the complicated geometry is presented and
adopted to the numerical scheme of the RHD code.
4.1 Introduction
4.1.1 Solar Magnetic Activity and Sunspots
The structure of the solar magnetic field is very complex. Field lines are dragged
and twisted by the differential rotation. The theory proposed by Babcock (1961 [4]) differential rotation
describes the concept of a magnetic cycle and explains a number of solar phenomena
by the evolution of subsurface magnetic fields. When magnetic field lines shear, cross
and reconnect a large amount of energy will be released which heats the surrounding
gas creating solar flares. The 11 year activity cycle of the Sun is a cycle of twisting activity cycle
and reorganisation of the overall magnetic field.
“Dark spots” on the solar surface are observed and mentioned for a long time
dating back to a couple of centuries B.C. Since the telescopes become available a
great number of observations has been collected implying that the Sun undergoes
a cycle of activity (sunspot cycle). Sunspots live a few days or weeks and then sunspot cycle
disappear again. Hale (1908 [65]) has been proposed the theory that sunspots are
associated with strong magnetic fields. The magnetic field strength where sunspots
appear reaches up to 1500 G (0.15 T) which is about 2500 times stronger than
Earth’s magnetic field (cf. Earth’s magnetic field strength is typically 0.3 to 0.5 G
near the surface) and much higher than anywhere else on the Sun. If the magnetic
pressure associated with the magnetic fields is comparable to the gas pressure it
inhibits the convection and therefore reduces the amount of energy reaching the
55

56 4. STELLAR SPOTS
flare stars
RS CVn and
BY Dra
solar surface. One result of the energy decrease is the lower temperature of the
sunspots. Weiss (1964 [159]) has been considered a relationship of the observed
magnetic fields to the convection in the Sun. Therefore sunspots appear where
magnetic flux tubes from beneath the solar surface escapes the photosphere and an
upper limit to the temperature drop in this flux tube bundle or flux rope is given by
∆T
T
≈ Pm
P0
, (4.1)
where Pm denotes the magnetic pressure and P0 is the ambient pressure. It can be
shown that the temperature difference reaches up to 2000 K which is in agreement
with observations.
The sunspots are typically about the size of the Earth, and during times of
maximum solar activity, hundreds of sunspots are visible. Therefore, we can estimate
a maximum surface filling factor
f⊙ = 100R2 Earth
R 2 ⊙
= 0.0084 (4.2)
for the Sun, where REarth and R⊙ are the radius of the Earth and the Sun, respectively,
i.e. about 1 percent of the solar photosphere is covered with sunspots.
4.1.2 Stellar Magnetic Activity
The Sun is the only star where we can study stellar spots directly. But if we assume
that the Sun is an “average” star it is reasonable to expect that solar-type magnetic
activity occur in other stars. Similar activities are already detected beyond other
stars:
• Chromospheres: Many young, cool stars (especially M stars) exhibit evidence
of very strong chromospheric activity in the form of emission lines in their
spectra.
• Flares: Flare stars are stars which show eruptions like solar flares, but emit
most of the energy in visible light which cause the luminosity of the star
to increase in this wavelength range and cover a large fraction of the stellar
surface.
• Starspots: RS Canum Venaticorum (spectral type F and G) and BY Draconis
(spectral type K and M) stars show variability according to a stellar rotation
in which starspots cover a significant fraction of the stellar surface.
• Magnetic Fields: Current techniques do not permit detection of magnetic fields
as weak as the solar field in other stars, but there are so-called magnetic stars
which have strong magnetic fields.
If stellar magnetic fields are strong enough they can be measured directly, e.g. by
Zeeman-splitting of spectral lines, and several stars are found with field strength up
to 2 kG (0.2 T). These stars also show photometric variations which are ascribed

4.1. Introduction 57
to stellar spots that may cover up to 60% of their surfaces (Weiss 1994 [160]). Due
to magnetic breaking by stellar winds the rotation period of magnetic active stars
increases as they evolve. Furthermore, a cyclic activity is only detected in slow
rotating stars like the Sun (e.g. Tayler 1997 [148]).
For an extension of the solar dynamo theory to the more general stellar case we
have to expect different stellar activity scenarios as a consequence of different stellar
characteristics, in particular due to a different stellar structure, the efficiency of convection,
the depth of the convection zone, the rate of rotation and the evolutionary
age.
4.1.3 Observations of Stellar Spots
Direct Method
Some investigations are done to reveal structure and features of the stellar sur- stellar surface
face. This is nowadays possible with direct methods like high-resolution imaging for
late-type stars which are preferred due to their angular size. First attempts where
made as the Faint Object Camera (FOC) instrument on-board the Hubble Space
Telescope (HST) observes the red giant star Betelgeuse (or α Orionis) and discovered α Ori, ’Betelgeuse’
a single bright, unresolved area on the stellar disc (Gilliland & Dupree 1996 [54]).
This feature may be a result of magnetic activity, atmospheric convection or global
pulsations and shock structures which heat the stellar atmosphere.
High-precision measurements of cool giant stars (especially Mira, o Ceti) with o Ceti, ’Mira’
the Very Large Telescope Interferometer (VLTI) are clearly showing deviations from
spherical symmetry as well as time-variations (Richichi et al. 2003 [124]). This observational
method is useful for accurate measurements of surface structure parameters
(e.g. diameters, diameter variations, asymmetries, centre-to-limb variations, special
features like hot spots) and of circumstellar envelopes.
Indirect Methods
For the search of stellar spots it is also important to look at those objects, which
show activity like chromospherical flares, convections and strong magnetic fields.
Within spectra these activity indicators (e.g. the chromospheric Ca II K spectral activity indicators
line) can be easily identified.
Doppler imaging denotes another indirect method which is successfully used to Doppler imaging
uncover the thermal structure of the stellar disc. This technique is able to generate
resolved images of the stellar disc of certain rapidly rotating late-type stars, like RS RS CVn and
FK Com
CVn, FK Com and Ap stars. It exploits the correspondence between wavelength
Ap stars
position across a rotationally broadened spectral line and spatial position across the
stellar disc (Vogt & Penrod 1983 [154] and Vogt et al. 1987 [155]).
Table 4.1 shows some examples of stars with detected spots. Most of them are
fast rotators and show starspots at the poles but the spots on MS Ser appear at MS Ser
lattitudes of 23 to 48◦ . An important quantity which is used in further investigations
is the temperature difference
∆T = Teff,p − Teff,s
(4.3)
between the effective temperature of the photosphere (Teff,p) and the spot (Teff,s) and
the surface filling factor f describing the fraction of spots covering the photosphere of

58 4. STELLAR SPOTS
Star f Tp Ts ∆T [K] Ref.
MS Ser 0.21 1300 (1)
LQ Hya 0.25 800 (2)
IM Peg 4450 ± 50 3400 − 3700 (3)
IM Peg 0.32 4666 3920 750 (4)
VY Ari 0.41 4916 4030 890 (4)
HK Lac 0.34 4765 3955 810 (4)
HD 17488 5830 ± 50 500 − 1600 (5)
HD 31993 4500 ± 50 200 (6)
σ 2 CrB 5966 p./5673 s. 2000 both (7)
UZ Lib 4800 300 − 1300 (8)
HII 314 5845 400 − 1400 (9)
= V1038 Tauri
Table 4.1: Data of stars which show stellar spots.
References: (1) Alekseev, Kozlova: A&A 403, 205-215 (2003), (2) Alekseev, Kozlova: Astrophysics
(Astrofizika), v.46, Issue 1, p.28-45 (2003), (3) Ribarik, Olah, Strassmeier: Astron. Nachr./AN
324, No.3, 202-214 (2003), (4) Catalano, Biazzo, Frasca, Marilli: A&A 394, 1009-1021 (2002),
(5) Strassmeier, Pichler, Weber, Granzer: A&A 411, 595-604 (2003), (6) Strassmeier, Kratzwald,
Weber: A&A 408, 1103-1113 (2003), (7) Strassmeier, Rice: A&A 399, 315-327 (2003), (8) Olah,
Strassmeier, Weber: A&A 389, 202-212 (2002), (9) Rice, Strassmeier: A&A 377, 264-272 (2001)
the star. As given in reference (4) in Table 4.1 the hemisphere-averaged temperature
can be expressed as
�
F Teff dA
¯Teff
disc
= �
F dA = Arel Fs Teff,s + (1 − Arel)Fp Teff,p
(4.4)
Arel Fs + (1 − Arel)Fp
disc
where Fs and Fp are the fluxes emitted by the spot and the remaining stellar surface,
respectively. If we rewrite Eq. (4.4) we get
¯Teff = ξ Teff,s + Teff,p
ξ + 1
, (4.5)
where
ξ = Arel Fs
. (4.6)
1 + Arel Fp
In Table 4.1 the area Arel is given as the surface filling factor f with the assumption
that the spots are distributed homogeneously on the whole stellar surface. The
stellar spots of these stars are very large compared to the area of their photospheres
represented in a large value of f. We also have to distinguish between two cases,
either where the stars show few spots with large areas or many spots with small
areas. In both cases f should be the same.
The stars listed in Table 4.1 are more or less “common” stars with spectral types
between F and K and their photospheres are uncovered from an opaque circumstellar
shell as often seen at late type stars (e.g. Wolf-Rayet and AGB stars). We have
some clues, that also extended stars, like the AGB stars, have magnetic fields and
therefore an inhomogeneous temperature distribution on the stellar surface caused
by convection cells and/or stellar spots.

4.1. Introduction 59
4.1.4 AGB star spots
Further questions and problems related to AGB star spots are discussed in various
papers:
a) The lower temperature and the magnetic field above the AGB spot facilitate
dust formation closer to the stellar surface (Soker & Clayton 1999 [144]).
b) The temperature gradient above cool stellar spots without shielding is given
by Frank (1995 [44]).
c) According to the opacity the photosphere inside a cool magnetic stellar spot is
at larger radius as the ambient photosphere (Soker & Clayton 1999 [144]). The
opacity therein increases with decreasing temperature which is the opposite of
the situation in the Sun.
d) Soker & Clayton 1999 [144] have approximated the magnetic pressure gradient
inside the AGB spot as a result of the lateral pressure balance. The magnetic
field lines open-up near and above the photosphere of the spot which implies
a magnetic tension.
e) Due to the slow rotation of AGB stars the lifetime of starspots of a few weeks
to a few months is assumed to be shorter as the rotation period (Soker &
Clayton 1999 [144]).
f) The implication of size and coverage of cool spots on AGB stars are discussed
extensively by Frank (1995 [44]). A model with a large round spot or an
equatorial band have been assumed to describe the asphericities in an AGB
wind.
g) Pulsations cause a variation of the spot temperature (Soker & Clayton 1999 [144])
which should have an effect on the dynamical evolution of the mass loss above
the AGB spot.
h) The mass loss above starspots should be higher especially during the last AGB
phase when the mass loss rate is generally high (Soker 2000 [141]). The newly
formed dust shields the region above it from the stellar radiation. This should
lead to a further dust formation in the shaded region as well as a convergence of
the outflowing stream toward the shaded region resulting in a higher density
flow. Furthermore, as a result of the dust shielding the AGB spot should
have a minimum size to generate a significant higher mass loss rate. Without
shielding the temperature above a cool spot does not fall as steeply as the
surrounding temperature.
i) A weaker radiation above AGB starspots should lead to a slower outflow velocity
(Soker 2000 [141]).
j) Soker (2000 [142]) has proposed magnetic activity cycles for AGB stars of
about 200 to 1000 years. This should be the mechanism behind the formation
of concentric shells found in several PPNe and PNe.

60 4. STELLAR SPOTS
4.2 Physical Model
We will now have a look at the physical model of the phenomenon of stellar spots
(i.e. the reason for their appearance) and the influence on the atmospheres of extended
and cool giant stars.
4.2.1 Spot Coverage
First of all, two parameters are important to know and have to be implemented in the
computational model, namely the temperatures of the starspots and the distribution
of starspots over the surfaces of the stars. The latter will be determined by the
surface filling factor, i.e. the fraction of the stellar surface which is covered with cool
starspots.
The following model is as simple as possible to incorporate in the physical model
into the existing RHD code. The surface of the star is described as
and consists of the spot area
and the area of the undisturbed surface
A∗ = 4π R 2 ∗ = As + Au
As = f A∗
(4.7)
(4.8)
Au = (1 − f)A∗ , (4.9)
where f denotes the surface filling factor. The global luminosity of the star can be
written as
L∗ = A∗ σ T 4 eff,∗ . (4.10)
We decompose the total luminosity by the luminosity of the spot and the luminosity
of the undisturbed surface
L∗ = A∗ σ (f T 4 eff,s + (1 − f)T 4 eff,u ) (4.11)
and the effective temperature of the spot can be written as a function of the filling
factor and the effective temperature of the undisturbed surface
Teff,s =
�
1
f T 4 1 − f
eff,∗ − T
f
4 �1
4
eff,u
(4.12)
Also the effective temperature of the undisturbed surface can be written as a function
of the filling factor and the effective temperature of the spot
Teff,u =
�
1
1 − f T 4 f
eff,∗ −
1 − f T 4 �1
4
eff,s
(4.13)
When f is getting zero then the effective temperature of the undisturbed surface is
equal to the global effective temperature Teff,∗. Typical values of f are about 0.2 up
to 0.5 for magnetic active stars with spectral types of F, G and K. For other types
of stars the surface filling factor can be below this range of values, e.g. for less active
stars like our sun, or above, e.g. for stars with huge convection cells.

4.2. Physical Model 61
4.2.2 Temperature Fluctuations
Convection
The entire disc of the Sun is covered at all times by small, bright features separated
by dark lanes called granules. They have characteristic diameters of 1000 km solar granules
and lifetimes of only several minutes. The brightness variations of the solar granules
result strictly from differences in temperature. The upwelling gas is hotter and
therefore emits more radiation than the cooler, downwelling gas. From the bright
centre of the granule to the darker intergranular region, the brightness variation
corresponds to a temperature difference of about 200 K.
3D stellar convection models of giant stars show the appearance of only few few convection cells
convection cells which can e.g. interpret the interferometric data of the well-known
red supergiant Betelgeuse (α Ori). Due to a spacious convection in the star these α Ori, ’Betelgeuse’
data can be modelled by assuming the presence of up to 3 unresolved hot or cool
spots (see therefore Freytag et al. 2002 [46] and Freytag 2003 [45]). The convective
time scale is in order of a couple of hundred days.
Furthermore, all AGB stars should show such surface patterns with some hotter
or cooler spots. The presence of only few convection cells in the envelope of red
giant stars implies a large zone where the gas cools down and flow downwards. This
will lead to a significant temperature difference compared to the upward moving gas
of up to 1000 K as suggested by Schwarzschild (1975 [138]). Different models of
convection in the envelopes of red giants computed by Antia et al. (1984 [2]) reveal
temperature fluctuations of approximately 300 to 400 K at the stellar surface.
Stellar Spots
The temperature difference between the sunspot and the ambient photosphere
is typically 1000 to 1500 K (cf. Section 4.1.1) whereas for stars with spectral types
between F and K it is about 200 to 2000 K (see therefore e.g. Table 4.1 in Section
4.1.3).
4.2.3 Magnetic Field
The measurement of stellar magnetic fields is very sophisticated apart from those
stellar objects with strong magnetic fields like neutron stars or fast rotating stars
which produce an effective dynamo like Ap stars. Nevertheless, the magnetic field
of AGB stars is not strong enough to be detected directly. About the existence of a
possible magnetic field we can draw conclusion from the detection of maser emissions
in the circumstellar envelopes (CSE) surrounding AGB stars.
The observation of SiO masers toward the Mira variable star TX Cam (see Kem- TX Cam
ball & Diamond 1997 [80], Gray et al. [57] and Diamond & Kemball 2003 [34]) reveals
the dynamical evolution of some SiO components over a full pulsation period. The
polarised maser emission can be used to probe the magnitude and orientation of the
primary magnetic field. This fact indicates the presence of a global magnetic field.
The average magnetic field can be approximated as shown by Soker (1998 [140]).
Thus, we can constitute that the magnetic field strength of a stellar spot (Bspot) is

62 4. STELLAR SPOTS
RCB stars
RY Sgr
Sakurai’s Object,
V605 Aql and
FG Sge
by a factor of η stronger than the average field strength at the photosphere, like
Bspot = ηBav . (4.14)
It is also assumed that the magnetic pressure is of the order of the photospheric
pressure
. (4.15)
8π
Assuming the pressure of the photosphere by a simple hydrostatic approach (Kippenhahn
& Weigert 1990 [83])
Pphot ≃ B2 spot
Pphot ≃ GM
R 2 ∗
ρphot l (4.16)
and the definition of the photosphere where κl ρphot = 2/3 where l is the density
scale height and κ is the opacity, we get
Pphot ≃ 2 GM
3 R2 1
. (4.17)
∗ κ
Combining Eq. (4.14), (4.15) and (4.17) the average magnetic intensity required to
form AGB stellar spots is
Bav ≥ 410 −3
� M
1M⊙
�1
2 � R∗
300R⊙
�−1 �
η
104 �−1 κ −1
2
3 G , (4.18)
where κ3 = κ/(310 −3 cm 2 g −1 ). In combination with a dynamo theory it is therefore
able to generate a magnetic field topology providing the AGB photosphere with
discrete spots.
4.2.4 Dust Formation above Cool Spots
R Corona Borealis (RCB) stars are possibly associated with dust formation above
cool spots. First suggestions to describe the behaviour of RCB stars are done by
Wdowiak (1975 [158]). He has assumed that dust forms over large convection cells
which are cooler than the surrounding photosphere. A magnetic activity cycle similar
to the well known solar cycle could fit in well with the observations of RCB stars
(see therefore Clayton et al. 1993 [26]).
First observations of inhomogeneities in the circumstellar envelope of a RCB
variable star have been made by de Laverny & Mékarnia (2004 [33]). The star RY
Sgr shows some bright and very large dust clouds in various directions at several
hundred stellar radii.
It is discussed that there could be a close relationship between RCB stars and
PNe. The cause is the observational findings on the outbursts of the central stars
of three old PNe, these are Sakurai’s Object, V605 Aql and FG Sge (Duerbeck &
Benneti 1996 [40], Clayton & De Marco 1997 [25] and Gonzalez et al. 1998 [56]).
These outbursts have transformed the hot evolved central stars into cool giant stars
with the spectral properties of a RCB star.
Dust formation above cool magnetic spots in evolved stars, like the AGB stars,
has also been discussed in various papers (e.g. Frank 1995 [44], Soker & Clayton
1999 [144] and Soker 2000 [141]). Some topics therein are listed already in Section
4.1.4.

4.3. Flux Tube Model 63
4.3 Flux Tube Model
To implement the model of a stellar spot in the RHD code we have to derive a
mathematical description of the geometrical topology above the stellar spot. We
prefer the flux tube geometry which offers a set of parameters (like the base area
or the radial distance where the flux tube widens) that can be used to vary the
geometrical appearance of the flux tube. Furthermore, this model simplifies the
complicated geometry and reduces the 2D-problem to a 1D one, which is necessary
for the implementation in the RHD code.
4.3.1 Definition
The definition of a flux tube can be done with the area and is calculated like
�
A(z) = A0 1 + z2
z2 �
(4.19)
0
where A0 is the area at the basis, z the distance from the basis area and z0 the
parameter at which radius the flux tube begins to get less cylindrically. This relation
shows that the flux tube tends to a cylindric symmetry if z0 is large and in the other
border case if z0 is very small the area is proportional to z 2 like it is in the spherical
symmetry. This definition is used to evaluate e.g. the volume or other parameters
and derivations which are needed for the integration of the full set of radiation
hydrodynamical equations.
On the other hand we have to calculate mathematical expressions like the divergence
in flux tube geometry. In this case we define the metric tensor for flux tube
geometry
⎛
a 0 0
gik = ⎝ 0 x2 ⎞
a 0 ⎠ , (4.20)
0 0 1
where
a(z) = 1 + z2
z 2 0
(4.21)
(see also Section B.3 in Appendix B). This definition is not the exact one, because
the off-diagonal elements in the metric are neglected. This means, that the metric
is forced to be orthonormal.
4.3.2 Flux Tube Representations
For the implementation of the flux tube symmetry in the RHD code it is necessary
to develop a more detailed flux tube representation which takes into account the
position of the flux tube, i.e. the base area A0 is not located at r = 0 or the
photosphere r = Rphot but at an specified radius r = r0, and the singularity at
z0 = 0 of the flux tube definition in Eq. (4.19). Fig. 4.1 illustrates the extended flux
tube model discussed above.

64 4. STELLAR SPOTS
stellar limb
z = 0
z
A0
r
x
r
0
stellar
photosphere
R phot
Figure 4.1: Model of a flux tube on a stellar surface
Furthermore, it is advantageous to converge in the limiting case to the spherical
symmetry to test the adaptation. The area in the spherical symmetry can be written
as
and the radial derivation as
r
As = A0
2
r2 0
If A0 = 4π r2 0 then Eq. (4.22) is reduced to
which denotes the surface of a sphere. For a flux tube
(4.22)
dAs
dr = A′ 2r
s = A0
r2 . (4.23)
0
As = 4π r 2 , (4.24)
A(r) = A0 a(r) (4.25)
we can derive the same behaviour for large z (and with the substitution z → r) and
z0 = r0. In the following subsections some flux tube representations are discussed

4.3. Flux Tube Model 65
keeping the aforesaid arguments in mind. The variables z and z0 as introduced in
Eq. (4.19) are modified according to the requirements mentioned above. Note: No
declaration change was done for the variable z0.
Version 1: Substitution of z 2 → r 2 and z 2 0 → z2 0
a = 1 + r2
z 2 0
a ′ = 2r
z 2 0
(equal to definition)
→ 1 + r2 0
z2 for r → r0
0
→ ∞ for z0 → 0 and r > 0
→ 2r0
z2 for r → r0
0
→ ∞ for z0 → 0 and r > 0
Version 2: Substitution of z 2 → (r − r0) 2 and z 2 0 → z2 0
(r − r0) 2
a = 1 +
a ′ =
z 2 0
2(r − r0)
z 2 0
→ 1 for r → r0
→ ∞ for z0 → 0 and r �= r0
→ 0 for r → r0
→ ∞ for z0 → 0 and r �= r0
Version 3: Substitution of z 2 → (r − r0) 2 and z 2 0 → (r0 + z0) 2
a ′
2a =
(r − r0) 2
a = 1 +
(r0 + z0) 2
a ′ =
2(r − r0)
(r0 + z0) 2
r − r0
(r0 + z0) 2 + (r − r0) 2
→ 1 for r → r0
→ 1 + for z0 → 0 and r �= r0
(r−r0) 2
r 2 0
→ 0 for r → r0
for z0 → 0 and r �= r0
→ 2(r−r0)
r 2 0
→ 0 for r → r0
r−r0 →
+(r−r0) 2 for z0 → 0 and r �= r0
Version 4: Substitution of z 2 → r 2 − r 2 0 and z2 0 → (r0 + z0) 2
a ′
2a =
a = 1 + r2 − r 2 0
(r0 + z0) 2
a ′ 2r
=
(r0 + z0) 2
r
(r0 + z0) 2 + r 2 − r 2 0
→
r 2 0
→ 1 for r → r0
for z0 → 0 and r �= r0
→ r2
r 2 0
2r0
(r0+z0) 2
→ 2r
r2 0
→
r0
(r0+z0) 2
→ 1
r
for r → r0
for z0 → 0 and r �= r0
for r → r0
for z0 → 0 and r �= r0
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)

66 4. STELLAR SPOTS
Version 5: Substitution of z 2 → r 2 − r 2 0 and z2 0 → r2 0 + z2 0
a = 1 + r2 − r 2 0
r 2 0 + z2 0
a ′ = 2r
r 2 0 + z2 0
a ′
2a =
r
r 2 + z 2 0
→
→ 1 for r → r0
for z0 → 0 and r �= r0
→ r2
r 2 0
2r0
r2 0 +z2 0
→ 2r
r2 0
→
r0
r2 0 +z2 0
→ 1
r
for r → r0
for z0 → 0 and r �= r0
for r → r0
for z0 → 0 and r �= r0
Version 6: Substitution of z 2 → (r − r0) 2 and z 2 0 → r2 0 + z2 0
a ′
2a =
a = 1 +
(r − r0) 2
r 2 0 + z2 0
a ′ =
2(r − r0)
r 2 0 + z2 0
r − r0
r2 0 + z2 0 + (r − r0) 2
→ 1 for r → r0
→ 1 + for z0 → 0 and r �= r0
(r−r0) 2
r 2 0
→ 0 for r → r0
for z0 → 0 and r �= r0
→ 2(r−r0
r 2 0
→ 0 for r → r0
r−r0 →
+(r−r0) 2 for z0 → 0 and r �= r0
For versions 4 and 5 applies for z0 → 0 that A → As, A ′ → A ′ s and
r 2 0
(4.36)
(4.37)
(4.38)
(4.39)
(4.40)
(4.41)
a ′ 1
→ . (4.42)
2a r
Since the expression for a′
2a is easier in version 5 this flux tube representation has
been implemented.

4.3. Flux Tube Model 67
4.3.3 Specific Declarations and Boundary Conditions
Area of the Flux tube Basis (A0)
The area of the basis is derived from the surface filling factor f as
A0 = f A∗ = f 4πR 2 phot . (4.43)
To avoid large deviations from the spherical surface of the star and the non-spherical
area of the flux tube geometry we restrict to a surface filling factor of f = 10 −2 . In
this case the deviation is less than 1%.
Distance of A0 from the Stellar Centre (R0)
The distance of the basis area A0 from the stellar centre was chosen to be about
99 % of the photosphere Rphot.
Outer Boundary Condition of the Radiative Flux
If we combine the radiative flux F = 4π H and the luminosity L = 4π r 2 F we
can write the internal flux as follows
L
Hint =
(4π) 2 r2 . (4.44)
0
As the luminosity is constant, i.e. Lint = Lext, we get for the external flux in the
flux tube at the radius Rext
Hext = A0
Hint =
Aext
(4π) 2 r 2 0
L
�
1 + z2 ext
z 2 0
� (4.45)
(see also Fig. 4.2). From Eq. (4.45) we see that for large z0 the radiative flux gets
constant at any radius r.
R
*
Hint
A0
Figure 4.2: Radiative flux through a flux tube
Hext
Aext

68 4. STELLAR SPOTS
4.3.4 Rewritten Equations
Due to the large amount of space needed the derivation of the rewritten equations
are not given here. So we give only the results of the primarily modified equations.
The detailed descriptions and derivations can be found in the appendices.
Auxiliary Variables
The area and volume and corresponding derivations in flux tube geometry are needed
for the discretised form of the equations. To get the volume of a flux tube segment
between z1 and z2 we have to integrate over the differential volume dV as follows
V =
�z2
In flux tube geometry we can further write
V = A0
�z2
z1
= A0 (z2 − z1)
z1
A(z)dz . (4.46)
�
1 + z2
z2 � �
dz = A0 [z]
0
z2 1
+ z1 3z2 0
� 3
z � �
z2
=
�
1 + z2 1 + z1z2 + z2 2
3z2 �
= ∆V [z1,z2] . (4.47)
0
If we adopt the chosen flux tube representation in Section 4.3.2 and substitute
z1,2 → r1,2 Eq. (4.47) can be given as
V = A0 (r2 − r1)
�
1 +
z1
1
3 (r2 1 + r1r2 + r2 2 ) − r2 0
r2 0 + z2 �
0
. (4.48)
The graphical illustration of the volume V is displayed in Fig. 4.3. The volume in
Eq. (4.47) is used for the calculation of ∆V in the discretised form of the full set of
radiation hydrodynamical equations as summarised in Appendix F.
z = 0
A0 A1 V A2
z1
Figure 4.3: Volume of a flux tube segment
z2

4.3. Flux Tube Model 69
Radiation Transfer
In the case of flux tube symmetry the 0th-order moment equation can be written as
1 ∂ 1
J +
c ∂t c ∇′ z (J u) = − ∇′ �
1
z · H − P ∇
c
′ �
ua′
z · u − (3K − J)
2a
− ρ(κJJ − κSS) (4.49)
and the 1st-order moment equation as
1 ∂ 1
H +
c ∂t c ∇′ �
1
z · (H u) = −∂K − H ∇
∂z c
′ �
ca′
z · u + (3K − J)
2a
where ∇ ′ z · 〈varx 〉 = 1 ∂(a〈varx 〉)
a ∂r
− ρκHH , (4.50)
. A detailed discussion to derive Eq. (4.49) and
Eq. (4.50) is given in Appendix C (Radiation Transfer) and the discretised equations
can be found in Appendix F (Full Set of RHD Equations).
Artificial Viscosity
The viscous force, which contributes to the moment equation (equation of motion),
is derived for the flux tube geometry as
fi = 1
a3/2 �
∂
a
∂r
3
2 ℓ 2 ρ ∇ ′ �
∂u 1
z · u −
∂r 3 ∇′ ��
z · u
= 1
�
∂
√ a
a ∂V
3
2 ℓ 2 ρ ∂(au)
� ��
∂u 1 ∂(au)
− (4.51)
∂V ∂r 3 ∂V
and the dissipated energy per gram
EQ = − 2
3 ℓ2 ∇ ′ �
∂u a′
z · u −
∂r 2a u
�2 = − 3 ∂(au)
ℓ2
2 ∂V
� ∂u
∂r
− 1
3
�2 ∂(au)
∂V
(4.52)
is the contribution to the energy equation. A detailed derivation is given in Appendix
B (Artificial Viscosity) and the implementation can be found in the discretised
form in Appendix F (Full Set of RHD Equations).

70 4. STELLAR SPOTS

Part III
Results and Discussion
71

Chapter 5
AGB Stars with Spots
To investigate the physical behaviour of the stellar atmosphere above cool spots we
have to implement the mathematical and physical model described in part II into
the RHD code. It is also necessary to rewrite the program for generating initial
models which are needed as starting point, i.e. as initial values for the RHD code.
In this chapter we will give the results of the upgraded initial model program
and the calculations obtained with the RHD code adapted for flux tube geometry.
All calculations with the RHD code start from a hydrostatical and dust-free initial
model generated by a standalone initial model code. Then this model is tested for
stability performing a RHD code calculation by switching off the dust equations.
Afterwards a computation with the full RHD equation system is carried out until a
stationary situation or another predefined break condition is reached.
5.1 Initial Models
Each model is completely determined by specifying four parameters which are the
total mass Mtot, the stellar luminosity L∗, the effective temperature Teff and the
relative abundance of carbon to oxygen εC/εO. They were chosen to be Mtot = 1M⊙,
L∗ = 10 4 L⊙ and an Teff as given in Table 5.1, respectively. The relative abundance
of carbon to oxygen is not necessarily needed for the initial model code but used for
the generation of a profile of the number density of the free condensable C-atoms
(nC) as input for the dust equations implemented in the RHD code.
5.1.1 Initial Models for Spherical Geometry
The initial models are first calculated for AGB stars in spherical symmetry. Adopting
the stationary system of RHD equations the distribution of density ρgas(r),
temperature Tgas(r), radiation energy density J(r), radiation flux H(r) and initial
dust distribution nC(r) are calculated within the stellar atmosphere. The results for
star models A to E are given for the density (Fig. 5.1), for the temperature (Fig. 5.2)
and for the radiation flux (Fig. 5.3). The density declines with increasing radius beginning
from the photosphere at approximately the same value and the temperature
shows a strong decline in the lower atmosphere whereas the spatial profile merges at
73

74 5. AGB STARS WITH SPOTS
Star Teff R∗
Model [K] [R⊙]
A 2300 629
B 2400 578
C 2500 533
D 2600 493
E 2700 457
Table 5.1: Model stars for a luminosity of 10 4 L⊙ and a mass of 1M⊙.
the upper atmosphere for all the model stars at a moderate decline. The same profile
of the radiation flux for all stellar models is a result of the unchanged luminosity of
the star.
Figure 5.1: Density distribution from the initial model program for a luminosity of
10 4 L⊙, a mass of 1M⊙ and various effective temperatures (see Table 5.1).

5.1. Initial Models 75
Figure 5.2: Temperature distribution from the initial model program for a luminosity
of 10 4 L⊙, a mass of 1M⊙ and various effective temperatures (see Table 5.1).
Figure 5.3: Radiation flux distribution from the initial model program for a luminosity
of 10 4 L⊙, a mass of 1M⊙ and various effective temperatures (see Table 5.1).

76 5. AGB STARS WITH SPOTS
5.1.2 Initial Models for Flux Tube Geometry
The results of the initial model for seven flux tube models, which are characterised
in Table 5.2 and visualised in Fig. 5.4 for a constant base area A0, are discussed in
this subsection. Fig. 5.5 shows the spatial density distribution. For large values of
z0 the density at the outer edge of the computational domain is slightly increased.
For small values the distribution is similar to the spherical case. The corresponding
temperature distributions are plotted in Fig. 5.6. In the outer region of the stellar
atmosphere we get an isothermal structure for the largest value of z0. And finally,
Fig. 5.7 represents the radiation flux distribution, which is conserved if the area of
the flux tube is not widened as in the case of cylindric forms or flux tubes with
moderate z0.
Figure 5.4: Model flux tubes A to G as described in Table 5.2.

5.1. Initial Models 77
Flux Tube z0
Model [cm] [R⊙] [R∗]
A 0 0 0
B 1.7210 13 247 0.5
C 3.4310 13 493 1
D 6.8610 13 986 2
E 1.7210 14 2465 5
F 3.4310 14 4930 10
G 3.4310 15 49300 100
Table 5.2: Model flux tubes for a luminosity of 10 4 L⊙, a mass of 1M⊙ and an
effective temperature of 2600K (model D in Table 5.1).
Figure 5.5: Density distribution from the initial model program for a luminosity
of 10 4 L⊙, a mass of 1M⊙ and effective temperatures of 2600K for seven flux tube
models with different z0 (see Table 5.2).

78 5. AGB STARS WITH SPOTS
Figure 5.6: Temperature distribution from the initial model program for a luminosity
of 10 4 L⊙, a mass of 1M⊙ and effective temperatures of 2600K for seven flux tube
models with different z0 (see Table 5.2).
Figure 5.7: Radiation flux distribution from the initial model program for a luminosity
of 10 4 L⊙, a mass of 1M⊙ and effective temperatures of 2600K for seven flux
tube models with different z0 (see Table 5.2).

5.2. Dynamic Model Results for Spherical Geometry 79
5.2 Dynamic Model Results for Spherical Geometry
We can use the initial models calculated in the previous sections as input models for
the adaptive RHD code. The program is able to deal with several tuning parameters
as for the numerical part like the artificial viscosity, the grid configuration and the
boundary conditions. The latter ones are needed to characterise the inner and outer
boundaries for all physical equations of the RHD code. Furthermore, it is possible
to variate the inner boundary with time to simulate the pulsation of the star. In
previous calculations these time-dependent models were used to explain the time
dependent mass loss of an AGB star with pulsation (for more details see Höfner S.
1994 [68]). In this work this feature of the RHD code will not be used.
Due to the steep decline of the density in the atmosphere obtained from the
hydrostatic initial model the original outer boundary is located close to the photosphere.
Therefore, the first step is to determine the external radius (Rext) following
the expansion of the atmosphere from the gas velocity at the outer boundary. When
Rext reaches about 15R∗ we switch to a outflow boundary condition keeping Rext
constant. Then the models are developed further in time until a specific atmosphere
structure is established. Basically, the resulting models can be characterised by
four wind scenarios: (1) no wind and therefore no mass loss, if not enough dust
is produced to generate the dust driven wind, e.g. due to physical conditions like
temperature distribution, (2) a stationary wind, which shows a constant mass loss
and a constant velocity at large stellar radii, (3) small shock waves propagating
through the atmosphere, that can be traced to a beginning of (4) a dust-induced
κ-mechanism. The transition between scenario (1) and (2), where a stellar wind is
produced but does not reach escape velocity is usually called a breeze solution. The
amount of mass loss and the terminal velocity depend on the conditions at the point
where the stellar wind reaches supersonic velocity.
Fig. 5.8 shows the velocity, gas temperature, degree of condensation and the
density of a stationary dust driven wind of a star with a mass of 1M⊙, a luminosity
of 10 4 L⊙ and an effective temperature of 2600K (model D in Table 5.1). The
approach of the wind velocity to the terminal velocity is clearly visible in this figure.
The mass loss is calculated as
˙M = ρ(Rext)v(Rext)Aext , (5.1)
where Rext is the outermost radius of the model and Aext the area of sphere at Rext.
In the case of a stationary dust driven wind the mass loss range is between 2 and
5 10 −7 M⊙/a −1 for this kind of star.
Fig. 5.9 depicts the spatial wind structure formed by a dust-induced κ-mechanism
for the same star as in Fig. 5.8 but with more carbon in the atmosphere, (i.e. εC/εO
is larger as compared to the stationary model. The dust-induced κ-mechanism is
clearly pronounced with several shock waves propagating through the atmosphere.
To calculate the terminal velocity and the mass loss we have to average over several
periods. This kind of dust driven wind can generate a higher mass loss compared to
the stationary wind and reaches values up to some 10 −6 M⊙/a −1 .

80 5. AGB STARS WITH SPOTS
Figure 5.8: Spatial structure of the stationary wind solution in spherical geometry
for star model D (Teff = 2600 K) with v∞ = 10.0 km/s, ˙ M = 2.2010 −7 M⊙/a,
log εO = −3.17 and εC/εO = 2.2.
Figure 5.9: Spatial structure of a stellar wind generated by a dust-induced κmechanism
in spherical geometry for star model C (Teff = 2500 K) with v∞ = 15.4
km/s, ˙ M = 1.5410 −6 M⊙/a, log εO = −3.18 and εC/εO = 2.0.

5.2. Dynamic Model Results for Spherical Geometry 81
5.2.1 Effects of Chemistry
If the amount ratio of carbon to oxygen εC/εO increases for a given set of stellar
parameters, the influence of the dust component on the gas temperature becomes
more and more important. The result is the development of an instability, i.e. a
dynamical dust driven wind produced by a dust-induced κ-mechanism, which can
be seen in the spherical as well as in the flux tube geometry. This instability leads to
a more or less periodic formation of dust layers and shock waves propagating through
the stellar atmosphere. Table 5.3 shows the dependence of the chemical composition
on the mass loss and terminal velocity in the case of spherical geometry. For the
stellar model D (cf. Table 5.1) dynamical wind calculations have been done for
several chemical compositions parametrised by εC/εO. It is shown that there exist
stationary wind solutions for this model with εC/εO = 2.2 and 2.3. Furthermore, the
instability starts at slightly higher values of the carbon abundance. For εC/εO = 2.4
we find a transitional scenario, i.e. an irregular variation superposed on a steady
outflow and for εC/εO is increased further the dust-induced κ-mechanism is more
dominant. The dynamical wind solution produced by a dust-induced κ-mechanism is
accompanied by a dramatic increase in the mass loss rate compared to the stationary
wind solutions.
The dependence on the amount of oxygen εO is tested for a wide range of values.
Since εO has a great influence on the generation of stationary or κ-induced dynamical
dust driven winds because it defines the amount of carbon εC according to the ratio
εC/εO. Therefore we decide to fix the amount of oxygen of log εO = −3.17, i.e. the
solar abundance as given by Grevesse & Sauval (1998 [59]).
The total amount of heavy elements like carbon and oxygen strongly influences
the resulting stellar wind as shown in Table 5.4 for spherical geometry. As the
value of the amount of oxygen (log εO) is increased the transition from a breeze
solution to a stationary wind as well as the transition from a stationary wind to a
time-dependent κ-mechanism outflow takes place at lower values of εC/εO.
εC/εO Result v∞
[km/s] [M⊙/a]
2.0 breeze - -
2.1 breeze/stationary wind (6.88) (1.43 10 −7 )
2.2 stationary wind 10.0 2.20 10 −7
2.3 stationary wind 13.3 3.31 10 −7
2.4 stationary/κ-induced wind (17.1) (5.61 10 −7 )
2.5 κ-induced wind (23.8) (8.98 10 −7 )
Table 5.3: Effects of chemistry for log εO = −3.17 and Teff = 2600 K.
˙M

82 5. AGB STARS WITH SPOTS
log εO Teff εC/εO Result v∞
[K] [km/s] [M⊙/a]
−3.18 2600 2.1 breeze (6.04) (1.26 10 −7 )
2.2 stationary wind 9.31 2.03 10 −7
2.3 stationary wind 12.4 2.96 10 −7
2.4 stationary wind 15.3 4.04 10 −7
2500 1.9 stationary wind 8.00 4.57 10 −7
2.0 κ-induced wind (15.7) (1.54 10 −6 )
−3.17 2600 2.1 breeze/stationary wind (6.88) (1.43 10 −7 )
2.2 stationary wind 10.0 2.20 10 −7
2.3 stationary wind 13.3 3.31 10 −7
2.4 stationary/κ-induced wind (17.1) (5.61 10 −7 )
2500 1.9 stationary wind 8.67 4.91 10 −7
2.0 κ-induced wind (15.5) (7.81 10 −7 )
−3.07 2600 2.0 stationary wind 12.0 2.87 10 −7
2.1 stationary wind 16.1 4.48 10 −7
2.2 κ-induced wind (28.3) (2.74 10 −6 )
2.3 κ-induced wind (34.1) (4.72 10 −6 )
2500 1.8 κ-induced wind (23.1) (7.13 10 −6 )
1.9 κ-induced wind (29.2) (8.89 10 −6 )
Table 5.4: Effects of chemistry for εC/εO = 1.9, 2.1, 2.2 and 2.3. Terminal velocity
v∞ and mass loss rate ˙ M are taken at 15R∗. Values in parentheses are mean values
averaged over several periods.
5.3 Dynamic Model Results for Flux Tube Geometry
First of all some calculations are done to test the RHD code with the new adapted
flux tube geometry. Therefore the semi-spherical case have been chosen, i.e. the
flux tube widening parameter z0 is set to zero (flux tube model A in Table 5.2) to
fulfil the spherical criteria as shown in Section 4.3.2. Furthermore, the temperature
difference ∆T is also set to zero in such a way as to compare the results with the
dynamic model results for spherical geometry.
Fig. 5.10 and Fig. 5.11 show the velocity, gas temperature, degree of condensation
and the density of two flux tube models with a stationary dust driven wind. Whereas
Fig. 5.12 and Fig. 5.13 plot the spatial wind structure generated by a dust-induced
κ-mechanism. Results can be found in the following subsection.
For increasing z0 the radiative flux is more and more conserved in the flux tube
in radial direction. This influences the region where the radiative flux accelerates
the newly formed dust grains. Especially for a wind scenario with a dust-induced
κ-mechanism we get a lower degree of condensation for models with larger z0 as
compared to the spherical case. Furthermore, shock fronts propagating through the
stellar atmosphere are not so concisely developed in flux tube geometry compared to
the spherical case which is also clearly seen in the moderate degree of condensation
fcond. It never exceeds a value of about 0.30 in the models displayed in Fig. 5.12
and Fig. 5.13.
˙M

5.3. Dynamic Model Results for Flux Tube Geometry 83
Figure 5.10: Spatial structure of a stationary wind in flux tube geometry for flux
tube C (z0 = 3.43 10 13 cm) with a temperature difference ∆T = 200 K and εC/εO =
2.3 located at model D (M = 1M⊙, L = 10 4 L⊙ and Teff = 2600 K).
Figure 5.11: Same as Fig. 5.10, but for flux tube D (z0 = 6.86 10 13 cm), ∆T = 400 K
and εC/εO = 2.2 .

84 5. AGB STARS WITH SPOTS
Figure 5.12: Spatial structure of a wind from dust-induced κ-mechanism in flux
tube geometry for flux tube B (z0 = 1.72 10 13 cm) with a temperature difference
∆T = 100 K and εC/εO = 2.2 located at model D (M = 1M⊙, L = 10 4 L⊙ and
Teff = 2600 K).
Figure 5.13: Same as Fig. 5.12, but for flux tube B (z0 = 1.72 10 13 cm), ∆T = 200 K
and εC/εO = 1.9.

5.3. Dynamic Model Results for Flux Tube Geometry 85
5.3.1 Effects of Geometry
For the investigation of the impact of the geometry on the resulting wind scenario in
a flux tube several models with different flux tube parameters (like temperature difference
∆T, flux tube widening parameter z0 and chemical composition εC/εO) are
calculated. The boxes illustrated in Fig. 5.14 to Fig. 5.17 corresponds to a temperature
difference ∆T (or a specific flux tube geometry) and the amount ratio of carbon
to oxygen εC/εO and are indicated by the resulting wind scenario. The generation
of a stationary wind is labelled as stat. , whereas a wind generated by a dust-induced
κ-mechanism is labelled as kap. . The brighter grey boxes are calculated but only a
breeze solution have been found. The brightest boxes are models which are either
not calculated or the calculations are stopped as a result of a massive dust formation
that decreases the time step dramatically.
First of all it is shown that the range of εC/εO is very small for the generation of
a stationary wind. Furthermore, this range broadens for flux tubes with large z0 and
large ∆T. Due to the conservation of the radiative flux (no cooling through the flux
tube boundary) it is possible that the temperature decreases too slow to generate
enough dust grains to drive a stellar wind in flux tube geometry (cf. Fig. 5.14 and
Fig. 5.15). This means that for flux tubes with large z0 the stellar spot have to be
cooler to be able to generate a wind.
The reduction of effective temperature of the stellar spot reduces also the radiative
flux and increases therefore the probability of the generation of dust grains (cf. also
Section 5.5) and consequently the formation of a stellar wind (cf. Fig. 5.16 and
Fig. 5.17). But if the flux tube widening parameter z0 is large the radiative flux is
more or less conserved along the flux tube and the formation of dust grains is more
and more inhibited.
We found that for a flux tube model with ∆T = 0 and about z0 = 1.010 13 cm
the behaviour of the stellar wind is similar to the spherical case. The discrepancy
why this accordance do not occur at z0 = 0 can be explained as an effect of the non
negligible deviation of the flux tube geometry from an spherical geometry in terms
of the flux tube area.

86 5. AGB STARS WITH SPOTS
Figure 5.14: Occurrence of a dust-driven wind for flux tube B (z0 = 1.7210 13 cm)
located at the AGB star model D (M = 1M⊙, L = 10 4 L⊙ and Teff = 2600 K) for
various temperature differences ∆T.
Figure 5.15: Same as Fig. 5.14, but for flux tube D (z0 = 6.8610 13 cm).

5.3. Dynamic Model Results for Flux Tube Geometry 87
Figure 5.16: Occurrence of a dust-driven wind for different flux tubes (A to G,
see Table 5.2) with ∆T = 100 K located on the AGB star model D (M = 1M⊙,
L = 10 4 L⊙ and Teff = 2600 K).
Figure 5.17: Same as Fig. 5.16, but for ∆T = 300 K.

88 5. AGB STARS WITH SPOTS
5.4 Boundary Conditions of the Flux Tube
In the current model the boundary between the flux tube and the normal stellar
atmosphere is fixed and cannot be moved due to lateral pressure differences and
is also opaque for radiation. Hence, the temperature can neither be increased by
heat sources nor cooled by sinks. In the following subsections we will investigate
the magnetic field needed to generate the flux tube as well as the occurrence of heat
sources and sinks along the flux tube boundary.
5.4.1 Lateral Pressure
Fig. 5.18 illustrates the the scenario of a vertical flux tube generated by a magnetic
field above a cool spot. For the lateral pressure balance at the stellar photosphere
we can write down
Pg,n = Pg,s + Pm,s , (5.2)
where Pg,n and Pg,s denotes the thermal gas pressure by assuming an ideal gas
and the magnetic pressure
Pg = R
µ ρT (5.3)
(5.4)
8π
above a spot. In our model the magnetic pressure of the normal atmosphere can be
neglected. The terms related to the normal (undisturbed) atmosphere are indicated
B s
B 0
R phot
Pm,s = B2 0
Pg,s+ Pm,s
ρs
ρn
Ts Tn P g,n
Figure 5.18: Magnetic pressure above a stellar spot.

5.4. Boundary Conditions of the Flux Tube 89
Figure 5.19: Dependence of temperature difference ∆T on the magnetic field
strength B0 for the solar photosphere.
by n and for the atmosphere above a spot by s. From Eq. (5.2) the relation
∆T = Tn − Ts = µ B
8πR
2 �
0
+ 1 −
ρs
ρn
�
Tn
ρs
(5.5)
for the temperature difference between the spot and the normal atmosphere is derived.
If ρs = ρn then the second term on the right hand side of Eq. (5.5) vanishes
and the temperature difference depends only on the magnetic field strength B0 and
the density ρs at the photosphere of the star. Eq. (5.5) can also be written as
∆T
Tn
= Pm,s
Pg,n
(5.6)
which is identical to Eq. (4.1) in Section 4.1.1. Fig. 5.19 displays ∆T as a function
of B0 as example for the Sun with a photospheric pressure of about 7 10 −7 g cm −3 .
As we mentioned in Section 4.1.1 the magnetic field strength for sunspots reaches up
to 1500 G which corresponds with a maximum ∆T of about 2000 K in our simple
model. Furthermore, if the density decreases we can derive from the relation given
in Eq. (5.5) that the generation of the same ∆T can be made by a weaker magnetic
field. Or in other words it is easier to produce a significant temperature difference
by a magnetic field (especially weaker than those of the Sun) in stellar atmospheres
with less density like in the case of AGB atmospheres.
Fig. 5.20 illustrates the dependence of ∆T on the magnetic field for an hypothetical
AGB star with a photosphere density of about 7.510 −9 g cm −3 . Therefore, only
less than 100 G are sufficient to generate a temperature difference ∆T of several
100 K.

90 5. AGB STARS WITH SPOTS
Figure 5.20: Same as Fig. 5.19, for an AGB star.
5.4.2 Heat Sources and Sinks
We compare the temperature distribution (or internal energy distribution) of the
spherical atmosphere with the temperature profile in the flux tube above a stellar
spot. If z0 ≃ 0 and ∆T > 0 then the temperature in the flux tube is always
below the temperature in the surrounding atmosphere. Taking a greater z0 then
the temperature stratification above the cooler spot exceeds the temperature of
the spherical atmosphere at a certain radius R. From the photosphere at radius
Rphot to R the flux tube will be heated while it will be cooled due to a cooler
atmosphere outside the flux tube. Furthermore, the radial energy transport through
the radiation energy can also not be negligible at the boundary between the flux tube
interior and the ambient atmosphere. This leads to a heating or cooling according
to geometrical effects.
Fig. 5.21 shows that the region where the flux tube gets more thermal energy lies
below the point where the wind becomes supersonic which is generally the case at
about 2 stellar radii. At the moment it can not be clearly said that this will affect
the mass loss and the terminal outflow velocity of a stationary wind or the behaviour
of a dynamical dust-induced κ-mechanism. But when the cooling and heating are
comparable in the lower atmosphere this won’t change the wind behaviour significantly.

5.5. Mass Loss through a Flux Tube 91
Figure 5.21: Region of heating and cooling of the flux tube. Below the line the flux
tubes are heated while above the line they are cooled. The effective temperature of
the undisturbed photosphere is 2600 K and εC/εO = 2.3.
5.5 Mass Loss through a Flux Tube
Due to the reduced temperature of the stellar spot compared to the ambient atmosphere
the zone of dust condensation moves nearer to the stellar photosphere. This
can be clearly seen in Fig. 5.22 where the degree of condensation (fcond) is plotted
for a spherical model and two flux tube models with different ∆T and different wind
scenarios. For an increasing ∆T the dust condensation occurs at even smaller radii.
To compare the mass loss rate of a flux tube to the overall mass loss of a spherical
model the spherical area have to be reduced according to the area of the flux tube at
the reference radius (15R∗). Therefore we can write the relation between the mass
loss of the flux tube ( ˙ MA,s) and the mass loss of the spherical model ( ˙ MA,n) through
an area A as
˙MA,s
˙MA,n
= ρftvft
ρsphvsph
, (5.7)
where ρft and ρsph denotes the density of the flux tube and spherical model, respectively,
whereas vft and vsph denotes the terminal velocity at the reference radius.
The mass loss of the spherical model corresponding to the reduced area is given
in Table 5.5. The area A is comparable to the area at 15R∗ of several flux tube
models with different widening parameter z0. The chosen spherical model shows an
stationary wind for εC/εO = 2.2 and 2.3 of the amount ratio of carbon to oxygen.
Table 5.6 summarises the relation of the mass loss rates ˙ MA,s/ ˙ MA,n for some flux

92 5. AGB STARS WITH SPOTS
Figure 5.22: Degree of condensation of flux tube models B and D compared to the
spherical star model A.
tube models. In case of flux tube model B the RHD calculations show a dynamical
wind produced by a dust-induced κ-mechanism. Therefore the resulting mass loss
rate is about two times higher compared to the spherical mass loss rate. Whereas a
smaller mass loss rate can be found for flux tube models C and D where a stationary
wind scenario exist. Thus a higher mass loss rate through a flux tube is possible in
case of the occurrence of a dust-induced κ-mechanism in the flux tube. Furthermore,
it is not excluded to find a flux tube model with a high temperature difference ∆T
where the reduced temperature prefers a increased condensation of dust grains and
consequently a larger mass loss rate than in the ambient atmosphere. The problems
are the small range of εC/εO-values for the generation of stationary and dynamical
wind scenarios and the strong temperature and density dependence of the dust
formation process.

5.5. Mass Loss through a Flux Tube 93
A at 15R∗ εC/εO
˙MA,n
[1028 cm2 ] [10−9 M⊙/a]
2.64 2.2 2.33
2.3 3.34
1.65 2.2 1.45
2.3 2.09
0.66 2.2 0.59
2.3 0.84
Table 5.5: Mass loss rate ˙ M through a specific area A for a spherical model. The
values are based on the star model A with Teff = 2600 K and stationary wind
scenarios at εC/εO = 2.2 and 2.3.
Flux Tube A at 15R∗ ∆T εC/εO
˙MA,s
Model [1028 cm2 ] [K] [10−9 M⊙/a]
˙MA,s/ ˙
MA,n
B 2.64 100 2.2 5.57 2.39
2.3 6.84 2.05
C 1.65 200 2.2 1.25 0.86
2.3 1.68 0.81
D 0.66 400 2.2 0.19 0.33
2.3 0.21 0.25
Table 5.6: Mass loss rate ˙ M through different flux tube configurations atop a cool
spot. Flux tube model B shows a dust-driven κ-mechanism wind scenario, whereas
models C and D develops stationary wind scenarios.

94 5. AGB STARS WITH SPOTS

Chapter 6
Discussion and Perspectives
In the previous chapter we have seen how a dust driven wind is generated in a
flux tube geometry. It is obvious that thermal disturbances at the stellar AGB
photosphere can drive winds with different physical properties (e.g. density, pressure
and terminal velocity) compared to the surrounding stellar atmosphere. In this
chapter we will discuss and deduce some impacts on the model of an inhomogeneous
mass loss of AGB stars. Furthermore, we take a look at future improvements of the
model and summarise subsequent perspectives.
At first (Section 6.1) we shall discuss the role and accompanying effects of a stellar
magnetic field. Especially the lifetime of such a field and a possible activity cycle of
the star will be reviewed. If we assume that the flux tube structure is maintained for
a relatively long time according to a stable magnetic field configuration we are able
to take a look at the mass loss produced above a cooler spot in Section 6.2. Along
with this appreciation we have to discuss the timescale of a stellar rotation for the
extended AGB stars. In Section 6.3 the description of possible observed small-scale
structures in PNe are given. A summary of perceptions obtained by this thesis can
be found in Section 6.4. In Section 6.5 we list the assumptions of this theoretical
model and give a summary of further perspectives and future improvements of the
model.
6.1 Magnetic Field
6.1.1 Lifetime of Stellar Spots
First of all it is important to investigate the lifetime of a stellar spot on a AGB
photosphere in detail. If stellar spots exist not long enough, then no significant
effects would be observable on the AGB atmosphere and beyond. But if the lifetime
is long enough, the stellar spot can affect the temperature and density stratification
of the stellar atmosphere above the spot. This will further influence the mass loss
rate and the inhomogeneities of the mass loss.
As shown by Petrovay & Moreno-Insertis (1997 [117]) the lifetime of a magnetic
stellar spot in the limit of strong inhibition of turbulence can be given as
τ ∝ r2 s B0
ν0 Be
95
, (6.1)

96 6. DISCUSSION AND PERSPECTIVES
where rs is the radius of the stellar spot, ν0 is the magnetic diffusivity and Be is
the magnetic field strength, for which ν is reduced by 50%. For solar conditions
(B0 ≈ 3000 G, Be ≈ 400 G and ν0 ≈ 1000 km 2 s −1 ) the well-known linear area-tolifetime
relation of Gnevyshev (1938 [55])
�
τ =
rs
[10 4 km]
� 2
10 [days] (6.2)
can be derived and for a typical diameter ds ≈ 5 10 4 km of a solar spot we get a
lifetime of about 2 to 3 months.
The lifetime for larger stellar spots as expected for AGB stars are essentially
longer due to the larger radii of the spot which enters quadratically in Eq. (6.1).
If we assume that the stellar spot covers about one hundredth of the photospheric
surface
A0 = πr 2 s = 1
100 4πR2 phot
(6.3)
the radius of the stellar spot is
rs = 1
5 Rphot
(6.4)
and can reach up to 10 8 km, many orders of magnitude larger than solar spots. Thus
the lifetime can reach some 10 4 to 10 5 years according to the assumed magnetic field
strength and magnetic diffusivity. Fig. 6.1 displays a sketch of a flux tube above a
cool spot with a radius of rs on a AGB star.
R phot
T n
pulsations
2 r s
Ts
Figure 6.1: Model of AGB star with a cool spot
v n
.
M n
v s
.
M s

6.2. Mass Loss 97
6.1.2 Stellar Activity Cycle
As proposed by Soker (2000 [142]) spherical shells in PN halos could be produced by spherical shells
a stellar activity cycle. The larger amount of cool magnetic spots at the maximum of
the cycle can result in an increased mass loss. If these spots are uniformly distributed
over the whole stellar surface, it is possible to explain the spherical shells around
PPNe or in the halo of PNe. Hrivnak et al. (2001 [70]) compiled a list of objects
which show arcs and rings. These rings are almost concentric and can be found IRAS 16594-4656
and 20028+3910
around PPNe (e.g. IRAS 16594-4656 and IRAS 20028+3910), PNe (e.g. NGC 7027,
Hb 5 and NGC 6543) and AGB stars (e.g. IRC+10216). The shells are semi-periodic
with time intervals between consecutive ejection events of about 200 to 1000 years
(see Hrivnak et al. 2001 [70]). The density enhancement of the shells is by a factor
of ∼ 2 (Hrivnak et al. 2001 [70]) up to a factor of ∼ 10 for IRC+10216 (Mauron &
Huggins 1999 [99]) higher relative to the density in-between.
6.1.3 Size and Distribution of Stellar Spots
The size and spatial and/or temporal distribution of spots on the stellar photosphere
NGC 7027, Hb 5
and NGC 6543
IRC+10216
is determined by the formation mechanism and the evolution of the magnetic field magnetic field
of the star. Frank (1995 [44]) investigated the influence of a big stellar spot and
an equatorial band on the asphericity of the mass loss. He has been shown that a
significant departure from an isotropic wind can be produced by such cool starspots.
formation and
evolution
In the later AGB phase a break of symmetry can be done due to an acceleration break of symmetry
of the stellar rotation (for more details see Section 6.2.2), which influences the appearance
of stellar spots on the stellar surface, i.e. they should be more concentrated
towards the stellar equator. If this occurs during the superwind phase of the AGB
star it is probably possible to generate a dense torus in the equatorial plane. As a
consequence of this torus the fast wind of the stellar successor of the AGB star will
form elliptical or even bipolar PNe.
6.2 Mass Loss
6.2.1 Mass Acquiration
The mass loss rate above an area A can be calculated easily by
˙M ≈ ρvA , (6.5)
where ρ and v are the temporal mean value of the density and the velocity of the
stellar wind at the position of the area A, respectively. If we assume a flux tube
(r0 = Rphot and z0 = 0) we can write for the area at the radius r = xRphot
A(x) = A0(1 + x 2 ). (6.6)
The following estimation are based on a star with a mass of 1M⊙, a luminosity of
10 4 L⊙ and a photospheric radius of Rphot = 3.410 13 cm. For a radius of x = 15,
a mean density of ρ = 10 −17 g cm −3 and a mean velocity of v = 10 km s −1 the

98 6. DISCUSSION AND PERSPECTIVES
Dumbbell Nebula
Eskimo Nebula
Helix Nebula
break of symmetry
mass loss rate is approximately 5.21 10 −9 M⊙ per year or 0.00179 MEarth per year
for a flux tube with the base area A0 covers one hundredth of the photosphere.
To accumulate a mass of about 3 MEarth as a typical mass of a cometary knot
(cf. Section 2.2.2 on page 21 for NGC 6853, the Dumbbell Nebula) the spot and
therefore the flux tube have to be existent for about 1680 years. In this time the
material spans over a distance of 1560 stellar radii or 3540 astronomical units. If
the mean velocity is v = 20 km s −1 the mass loss rate increases to 1.04 10 −8 M⊙
per year or 0.00358 MEarth per year, the time for the mass acquiration decreases to
840 years and the distance is equal as before. The distance decreases only if the
mean density or the area at the outer edge of the flux tube increases for small z0.
Therefore, the mean density should be still more increased for flux tubes with a
larger widening parameter z0. The mass loss increases also with the occurrence of a
dust-induced κ-mechanism or stellar pulsation.
6.2.2 Stellar Rotation
Due to their extended envelopes AGB stars are very slow rotators. Assuming angular
momentum conservation from the main sequence to the AGB phase we can estimate
the rotation period of an AGB star. From the total angular momentum
J = Iω , (6.7)
where I is the moment of inertia, and ω is the angular velocity, we get a relation
between the stellar radius and the rotation period
J ∝ mrv = mr 2 ω = 2π mr 2 P −1 = const. , (6.8)
where m is the stellar mass (approximately equal at main sequence and AGB phase),
r the stellar radius, and P the rotation period. Therefore the rotation period can be
derived from
� �2 r
P = P⊙ . (6.9)
R⊙
For a rotation period of about 27 days for our Sun with 1R⊙ and an AGB radius of
493R⊙ the extended star rotates with a period of 18000 years. Thus a stellar spot is
pointing for a long time in almost the same direction, so the matter from the mass
loss process can be accumulated and produce dense radial filaments. Therefore, it is
possible to accumulate the mass of the knots in radial filaments as shown e.g. in the
Eskimo Nebula. They have only be compressed by the subsequent fast wind of the
PN central star to form dense cometary knots (cf. NGC 7293, the Helix Nebula).
Furthermore, we can draw up a new hypothesis about the break of the symmetry
at the late AGB or even post-AGB phase, where the mass loss changes from spherical
to a more bipolar like structure. Apart from the theory of a system of binary stars
we can draw another scenario which can be a candidate for the transition between
a spherical and an aspherical symmetry. Therefore we postulate the capture of a
big planet or a dwarf star by the extended stellar envelope. Thereby the star spins
up due to an increase of angular momentum. The faster rotation of the star should
initiate a more effective stellar dynamo which consequently intensifies the magnetic

6.3. Small-scale Structures in PNe 99
activity of the star. For this reason the formation of stellar spots will be more
frequently and they are more or less concentrated to the stellar equatorial region
as a result of a winding of the magnetic field like it is the case on our Sun. The
numerous appearance of this cool surface features generates a mass loss enhancement
around the stellar equator. This inhomogeneous mass loss process forms a torus like
density distribution, which consequently influences the shaping of the evolving PN.
6.3 Small-scale Structures in PNe
6.3.1 Instabilities
The first idea to explain the cometary knots due to Rayleigh-Taylor instabilities has Rayleigh-Taylor
instability
been proposed by Capriotti (1973 [21]). These instabilities should be common at
the region where the high-velocity outflow has collided with the denser AGB wind.
However, the expected pattern generated by Rayleigh-Taylor instabilities can not
resemble the features we see in most of the PPNe and young PNe, e.g. the irregular
lanes in IC 4406 (cf. Fig. 2.24 in Section 2 on page 26). IC 4406
Furthermore, Vishniac (1994 [153]) described a new type of instability which acts Vishniac instability
at the intersection of the fast wind from the central star and the slow wind of the
progenitor star with higher density. But it is not clear whether this instability can
explain the presence of the youngest knots near the main ionisation front.
A third possible scenario to generate small-scale structures in the outflow of an
AGB star is the existence of shear flows at the boundary between the flux tube above
a cool stellar spot and the ambient atmosphere. If two fluids of gases with different
densities and different velocities are laterally in contact with each other, instabilities
are inescapable. These shear flows can initiate Kelvin-Helmholtz instabilities. It is Kelvin-Helmholtz
instability
assumed that Kelvin-Helmholtz instabilities could disrupt the flux tube at larger
stellar radii, but it is also possible that the lateral pressure balance will inhibit the
growth of Kelvin-Helmholtz instabilities. This idea has to be investigated in further
studies.
6.3.2 Inhomogeneous Mass Loss
Dyson et al. (1989 [41]) have investigated the clumps in NGC 7293, the Helix Nebula. Helix Nebula
According to their estimates they could have been generated due to inhomogeneities
in the red giant atmosphere at the onset of the superwind phase. Furthermore, it
has been suggested that these inhomogeneities have been ejected from the star itself.
Stellar spots are able to produce those instabilities in the atmospheres of cool
and extended stars. These spots are either made by convection or a magnetic field.
As we have shown in Sect. 5.5, the mass loss above such a temperature anomaly
on the stellar surface is different than for the undisturbed part of the atmosphere.
The lower temperature of the spot as well as the flux tube geometry influences the
behaviour of the dust-driven wind.
Due to the lifetime of the spots and the possible stellar activity cycle the inho-
mogeneous mass loss process can be the reason even for large clumps and filaments. clumps and
filaments
Nevertheless as a result of density inhomogeneities in the stellar outflow of the AGB

100 6. DISCUSSION AND PERSPECTIVES
Eskimo Nebula
Eskimo Nebula
star it is obvious to assume the appearance of instabilities when the fast wind from
the central star of the PN interacts with the slow AGB wind. Additionally the inhomogeneous
circumstellar shell will also influence the propagation of the ionisation
front of the intense radiation field produced by the hot central object of the PN.
6.3.3 Radial Filaments
The generation of dense radial filaments are also a possibility to be a result of a cool
spot in the AGB photosphere. Due to the slow rotation of the star the modified
mass loss above the spot should form a density enhancement along a radial ray. To
evaluate the distance those radial filaments could reach we assume a lifetime of the
temperature anomaly of about 2000 years and a velocity range of the mass loss above
the cool spot of 10 to 20 km/s. Therefore, we get a length of the radial filament
of about 0.07 to 0.13 light years, which is in good agreement with the filaments
observed in the Eskimo Nebula.
6.4 Conclusion
Following perceptions can be obtained from this thesis:
(1) The ability to upgrade the implicit RHD-code for additional geometries has been
demonstrated. Particularly, the successful implementation of the artificial viscosity
(see Appendix B), the radiation transfer (see Appendix C and associated derivatives
(as described in Section 4) are further improvements. Although the modularity of
the present RHD code is not satisfyingly done, so many switches had to be included
to distinguish between different geometrical configurations. This should be taken in
mind for further improvements of the RHD code.
(2) The flux tube approximation as well as the spot size do not reproduce the
multiplicity of cometary knots in the shell observed in many PNe like the Helix
Nebula. Due to the fact that the spatial dimension of the knots and their distances
between themselves can not be argued by our model, these knots cannot be described
by this model with cool spots. Only a global averaged effect is possible to trigger
instabilities at the boundary of the fast wind and the slow AGB wind. With our
spotted wind model we can generate inhomogeneities in the AGB wind which can
support the growth of fluctuation by Rayleigh-Taylor instabilities.
(3) The generation of some radial filaments as seen in the Eskimo Nebula can possibly
be produced by a temperature anomaly at the stellar atmosphere. Due to the slow
rotation of the AGB star and the long lifetime of a cool spot the model is able to
generate radial filaments.

6.5. Assumptions and further Perspectives 101
6.5 Assumptions and further Perspectives
There are some basic assumptions of the model which are summarised in the following
subsections. Simultaneously we will take a look on further improvements and
perspectives.
6.5.1 Geometry
One assumption on the model is related to geometry where we allow a deviation from
orthogonality for large base areas A0. This can be eliminated by the improvement
of the definition of the metric tensor, consequently also the deviation of the area
surfaces (now to be orthogonal to the boundary surface) to the shells of radii of the
spherical star are minimised.
We further assumed that the base area is perfectly round. In reality this should
not be the case, particularly for extended and fully convective stars. Thus, we can
improve our model by a variable base area. It is also possible to develop a method
to determine global results (i.e. stellar properties like the global luminosity or mass
loss rate) from the model with an inhomogeneous wind generated by some smaller
spots or other surface inhomogeneities.
6.5.2 Magnetic Field
For a more detailed physical description of the problem it is necessary to implement
the equations for a magnetic field, which emerges from the stellar photosphere and
forms the boundary of the flux tube. This expansion of the physical equation system
will also provide us the ability to study the interactions of the flux tube and the
ambient atmosphere. In terms of the further development of a permeable boundary
of the flux tube (see next subsection below) the magnetic field have to be continuously
solved along with the other equation system. Because in such a model the
magnetic field becomes a more or less complex topology compared to the simple flux
tube geometry.
6.5.3 Permeable Boundary
In our present computations the boundary of the flux tube is fixed and not transparent
for matter and radiation. But it is obvious that the radiation heats or cools
the flux tube and the lateral difference of pressure tends to contract or stretch the
flux tube. This can cause a clumpy and irregular wind structure above the stellar
spot.
A solution of such a permeable boundary is not easy to implement in a onedimensional
RHD code, because one has to evaluate all physical quantities for the
flux tube interior as well as the surrounding atmosphere at the same time and at
the same radii. Furthermore, the implementation of such a model has to be done
very carefully, because the geometry terms of the spherical stellar atmosphere and
the flux tube environment are not compatible. This could not be realised with an
one-dimensional RHD code.

102 6. DISCUSSION AND PERSPECTIVES
But a solution of this problem could be approached by the development of a
simplified two-dimensional RHD code. The easiest way is to calculate the solution
of the equation system along two lateral rays in radius simultaneously. One ray
in radius will be used for the determination of the radial distribution of the atmosphere
properties of the undisturbed atmosphere while the other ray will be used for
atmospheric properties of the flux tube interior. At the various radius values it is
now possible to implement a permeable boundary. Thus matter and radiation can
interact at the boundary of the flux tube.
6.5.4 Stellar Pulsations
A pulsation of the AGB star is also not included in the current investigation. Such
pulsations are able to change the surface temperature at relatively short timescales.
This will influence the behaviour of the mass loss generation too. Further calculations
considering a stellar pulsation should include the investigation of the modified
density stratification, the propagation of shock waves through the stellar atmosphere,
the dust formation as well as the dissipation of energy.

Part IV
Appendices
103

Appendix A
Discretisation
A.1 Computational Domain
Stellar
Envelope
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Hi
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External
Medium
NPT NPT−1
i+2 i+1 00000000
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2 1
Figure A.1: Description of the numerical grid for RHD calculations
A.2 Rules
Tensor Type Location Examples
even rank (incl. scalars) between two grid points density ρl
internal energy el
radiation energy Jl
dust moments (Kj)l
odd rank at the grid points velocity ul
radiation flux Hl
mass ml
105

106 A. DISCRETISATION
Operator Symbol Tensor of
even rank odd rank
temporal difference ∆Xl Xl(t) − Xl(t − δt)
spatial difference ∆Xl Xl−1 − Xl Xl − Xl+1
spatial mean Xl
upwind differencing
� X ad
l
Scheme Discretisation
= Xl
donor cell X ad
l
van Leer X ad
l = Xl +
A.3 General
� X ad
1
2 (Xl−1 + Xl)
< 0
otherwise
l−1 if urel l
Xad l
� (Xl−Xl+1)(Xl−1−Xl)
Xl−1−Xl+1
X ad
l
� X ad
1
2 (Xl + Xl+1)
l if ¯
urel l < 0
Xad l+1 otherwise
if (Xl − Xl+1)(Xl−1 − Xl) > 0
otherwise
The volume-integrated conservation equations of the RHD system for a moving
coordinate system take the form
� �
∂
X dV + X u
∂t
rel �
dA = S dV . (A.1)
V
∂V
The velocity u rel is the relative velocity between the co-moving frame and the numerical
grid and is defined as
u rel
l = ul − δrl
δt
A.4 Case 1: Spherical Geometry
The volume element in spherical geometry is given by
∆Vl = ∆r3 l
3
V
. (A.2)
, (A.3)
which is equivalent to a spherical shell between two grid points divided by the factor
4π. Thus the mass element is calculated as
A.4.1 Advection
The discretised form of the l.h.s. in Eq. (A.1) is
1
4π ∆ml = ρl ∆Vl . (A.4)
1
δt δ(Xl∆Vl) + ∆(r 2 l � X ad
l urel
l ) (A.5)

A.5. Case 2: Flux Tube Geometry 107
if X is a scalar and
if X is a vector.
A.4.2 Mathematical Operators
1
δt δ(Xl∆Vl) + ∆(r2 �
l Xad l urel
l ) (A.6)
Gradients and divergences are discretised in spherical geometry according to
�
∇X dV =⇒ r 2 l ∆Xl (A.7)
and �
respectively.
A.5 Case 2: Flux Tube Geometry
V
The volume element in flux tube geometry is given by
� �
V
∇ · X dV =⇒ ∆(r 2 l Xl), (A.8)
∆Vl = A0
∆zl + ∆z3 l
3z 3 0
(A.9)
without the consideration of a specific flux tube representation. Thus the mass
element within a flux tube is calculated as
A.5.1 Advection
The discretised form of the l.h.s. in Eq. (A.1) is
if X is a scalar and
if X is a vector, where al = 1 + z2 l
z2 .
0
A.5.2 Mathematical Operators
∆ml = ρl∆Vl . (A.10)
1
δt δ(Xl∆Vl) + ∆(al � X ad
l urel
l ) (A.11)
1
δt δ(Xl∆Vl) + ∆(al � X ad
l urel
l ) (A.12)
Gradients and divergences are discretised in flux tube geometry according to
�
∇X dV =⇒ al ∆Xl (A.13)
and �
respectively.
V
V
∇ · X dV =⇒ ∆(al Xl), (A.14)

108 A. DISCRETISATION

Appendix B
Artificial Viscosity
B.1 General
Shock waves represent a problem for numerical difference methods (finite difference finite
methods), since they appear with ideal liquids as discontinuities. With the artificial
viscosity an additional pseudo viscous pressure is introduced, which broadens the
shock wave fronts over several grid points. Requirements on the artificial viscosity
are:
• expanding ranges must be free of any artificial viscosity,
• homologous contractions may not be affected by artificial viscosity.
Tscharnuter & Winkler (1979 [149]) derived a general form of the artificial viscosity,
whereby the geometry-independent pressure tensor can be written as
Q k i = ℓ2 �
ρ div(�u) ε k i − α div(�u) δk �
i [1 − θdiv(�u)] . (B.1)
difference
methods
The divergence of the velocity field �u can be derived from the covariant derivation divergence
of the contravariant vector u k (see also Appendix E)
ε k i
div(�u) = u k ;k = uk ,k + Γk kλ uλ . (B.2)
designates the mixed tensor of the symmetrised gradient symmetrised
velocity field
ε k i = gkl u (l;i), (B.3)
where g kl represents the contravariant metric tensor and u (l;i) the symmetrised covariant
velocity tensor
u (l;i) = 1
2 (ui;l + ul;i) . (B.4)
The covariant derivative of the covariant vector is
ui;l = ui,l − Γ λ li uλ . (B.5)
109

110 B. ARTIFICIAL VISCOSITY
Furthermore δ k i represents the Kronecker tensor of the 2nd kind (mixed unity tensor)
δ k i =
and θ the Heaviside step function
θ(x) =
� 1 if i = k
0 otherwise
� 1 if x > 0
0 otherwise
The parameter α is selected in such a way, that the pressure tensor disappears at a
homologous contraction, i.e. the trace of Qk i has to disappear
Q k k
.
= 0 . (B.6)
In general α depends on the dimension. In case of a three-dimensional system of
coordinates, the following applies
α = 1
. (B.7)
3
Finally, we can rewrite the formula for the mixed tensor of the viscous pressure as
Q k i = l2ρu l �
;l ǫ k i
�
, (B.8)
g
which have the dimension [ cm s2] or [ dyn
B.1.1 Viscous Force
cm 2].
− 1
3 ul ;l δk i
The viscous force, which supplies a contribution in the equation of motion, can be
evaluated by the divergence of Q k i
g
with the dimension [ cm2 s2] or [ dyn
fi = Q k i;k = Qk i,k + Γk lk Ql i − Γ l ik Qk l
cm 3].
B.1.2 Viscous Energy Dissipation
, (B.9)
The energy dissipated by the viscosity is a result of the contraction of the pressure
tensor with the tensor of the symmetrised velocity field
EQ = − 1
ρ Qi k εk i
. (B.10)
If we evaluate the equation above and include the relation uk ;k = εk k (trace of ε) then
we get the following equation
EQ = −l 2 u k �
1 � 1
;k (ε1 − ε
3
2 2 )2 + (ε 1 1 − ε33 )2 + (ε 3 3 − ε22 )2� + 2 � ε 2 1ε12 + ε31 ε13 + ε32 ε2 �
3
�
,
(B.11)
with the dimension [ cm2
s3 ] or [ erg
g s ]. EQ ≤ 0 has to be guaranteed, which yields to a
sum of quadratic terms.

B.2. Case 1: Spherical Geometry 111
B.2 Case 1: Spherical Geometry
In spherical geometry with the system of coordinates (x1,x2,x3) = (r,θ,φ), the co-
variant vector uk = (u,vr,wr sin θ), and the contravariant vector uk = (u, v w
r , r sinθ =
Ω), we can determine the following steps:
Metric tensor
⎛
1 0 0
gik = ⎝ 0 r2 0 0
0
r2 sin2 ⎞
⎠ ;
θ
g ik ⎛
1 0 0
= ⎝ 0 1
r2 0
Christoffel symbols of the second kind
Divergence
u k ;k
∂u 2u
= +
∂r r
Γ 1 22 = −r
Γ 1 33 = −r 2 sin 2 θ
Γ 2 33 = − sin θ cos θ
Γ 2 12 = Γ2 21 = Γ3 13 = Γ3 31
Γ 3 23 = Γ3 32
1 ∂v v ∂Ω
+ + cot θ +
r ∂θ r
= cot θ
Mixed tensor of the symmetrised gradient
(only terms needed)
Viscous pressure
(only radial terms)
Q k i = 3ℓ2 ρ ∂(r2 u)
∂r 3
and its divergence
(only needed term)
Q k 1;k = ∂Q1 1
∂r
ǫ 1 1 = g 11 u1;1 = ∂u
∂r
ǫ 2 2 = g22 u2;2 = u
r
= 1
r
0 0 1
r 2 sin 2 θ
∂φ = 3∂(r2 u) 1 ∂ ∂Ω
+ (v sin θ) +
∂r3 r sin θ ∂θ ∂φ (B.12)
+ 1
r
∂v
∂θ
ǫ 3 3 = g33u3;3 = u v 1
+ cot θ +
r r r sin θ
⎛
⎜
⎝
∂u
∂r − ∂(r2u) ∂w
∂φ
∂r3 0 0
u 0 r − ∂(r2u) ∂r3 0
u
0 0 r − ∂(r2u) ∂r3 2
+
r Q11 + ∂Q21 ∂θ + Q21 cot θ + ∂Q31 ∂φ − Q22 + Q33 r
= 3 ∂(r2Q1 1 ) 1
+
∂r3 sin θ
Q k k =0
∂
∂θ (Q21 sin θ) + ∂Q31 ∂φ
⎞
⎞
⎠
(B.13)
(B.14)
(B.15)
⎟
⎠ (B.16)
− 1
r (Q2 2 + Q 3 3)
= 3 ∂(r2Q1 1 ) 1
+
∂r3 r Q11 + 1 ∂
sinθ ∂θ (Q21 sin θ) + ∂Q31 ∂φ
= 3 ∂(r
r
3Q1 1 ) 1 ∂
+
∂r3 sin θ ∂θ (Q21 sin θ) + ∂Q31 ∂φ
(B.17)

112 B. ARTIFICIAL VISCOSITY
B.2.1 Results
Finally, we can write down the viscous force and the dissipated energy per gram, for
the spherical geometry:
Viscous force [ dyn
cm 3]
(only radial term)
fi = 3
r
∂
∂r 3
�
r 3 l 2 ρ 3 ∂(r2u) ∂r3 �
∂u
∂r − ∂(r2u) ∂r3 ��
= 2 ∂
3r ∂V
Energy per gram [ erg
g s ]
(only radial term)
�
r 3 l 2 ρ ∂(r2 �
u) ∂u u
−
∂V ∂r r
EQ = − 9
2 l2∂(r2 u)
∂r3 �
∂u
∂r − ∂(r2u) ∂r3 �2 = − 2
3 l2∂(r2 � �2 u) ∂u u
−
∂V ∂r r
B.2.2 Discretisation
��
(B.18)
(B.19)
In Appendix A the scheme how the equations are discretised is given for the RHD
code. The following equations represent the force and energy terms of the artificial
viscosity which are implemented in the equation of motion and the energy equation,
respectively.
Viscous force [ dyn
cm 3]
(only radial term)
Energy [ erg
cm 3 s ]
(only radial term)
fi = 2 1
3
rl
�
∆ r3 l l2 ∆(r
ρl
2 l ul)
�
∆ul
−
∆Vl ∆rl
ul
��
1
rl ∆Vl
ρ EQ = − 2
3 l2 ∆(r
ρl
2 l ul)
�
∆ul
−
∆Vl ∆rl
ul
�2 rl
(B.20)
(B.21)

B.3. Case 2: Flux Tube Geometry 113
B.3 Case 2: Flux Tube Geometry
In flux tube geometry with the system of coordinates (x1,x2,x3) = (x,ϕ,z), covariant
vector uk = (w √ a,vx √ a,u), contravariant vector uk = ( w √ v ,
a x √ ,u), we can
a
determine the following steps:
Metric tensor
⎛
a 0 0
gik = ⎝ 0 x2 ⎞
a 0 ⎠ ; g
0 0 1
ik ⎛ 1
a 0 0
= ⎝ 1 0 x2 ⎞
a 0 ⎠
0 0 1
Christoffel symbols of the second kind
Divergence
u k ;k
Γ 1 22
= −x
Γ 3 11 = −a′
2
Γ 3 22 = −x2 a ′
2
Γ 2 12 = Γ 2 21 = 1
x
Γ 1 13 = Γ 1 31 = Γ 2 23 = Γ 2 32 = a′
2a
1 ∂w w
= √ +
a ∂x x √ 1
+
a x √ ∂v
a ∂ϕ +∂u
∂z +a′
1
u =
a x √ ∂(xw) 1
+
a ∂x x √ a
Mixed tensor of the symmetrised gradient
(only terms needed)
Viscous pressure
(only radial terms; z → r)
⎛
Q k i = ℓ2ρ 1 ∂(au)
a ∂r
and its divergence
(only needed term)
∂v
∂ϕ +1
a
∂(au)
∂z
(B.22)
ǫ 1 1 = g 11 u1;1 = 1 ∂w a′
√ + u
a ∂x 2a
(B.23)
ǫ 2 2 = g22u2;2 = 1
x √ ∂v w
+
a ∂ϕ x √ a′
+ u
a 2a
(B.24)
ǫ 3 3 = g33u3;3 = ∂u
∂z
(B.25)
⎜
⎝
Q k 3;k = ∂Q1 3
∂x
a ′ 1 ∂(au)
2au − 3a
0
∂r 0 0
a ′ 1 ∂(au)
2au − 3a ∂r 0
0 0 ∂u
∂r
− 1
3a
∂(au)
∂r
1
+
x Q13 + ∂Q23 ∂ϕ + ∂Q33 a′
+
∂r a Q33 − a′
2a (Q11 + Q 2 2)
⎞
⎟
⎠ (B.26)

114 B. ARTIFICIAL VISCOSITY
B.3.1 Results
= 1 ∂(xQ
x
1 3 )
∂x + ∂Q23 1 ∂(aQ
+
∂ϕ a
3 3 )
∂r
∂(xQ1 3 )
∂x + ∂Q23 1 ∂(aQ
+
∂ϕ a
3 3 )
∂r
Q k k =0
= 1
x
− a′
2a (Q1 1 + Q2 2 )
+ a′
2a Q3 3
= 1 ∂(xQ
x
1 3 )
∂x + ∂Q23 1
+
∂ϕ a3/2 ∂(a3/2Q3 3 )
∂r
(B.27)
For flux tube geometry we can write down the viscous force and the dissipated energy
per gram as:
Viscous force [ dyn
cm3] (only radial term)
fi = 1
a3/2 �
∂
a
∂r
3/2 ℓ 2 ρ 1
� ��
∂(au) ∂u 1 ∂(au)
−
a ∂r ∂r 3a ∂r
= 1
�
∂
√ a
a ∂V
3/2 ℓ 2 ρ ∂(au)
� ��
∂u 1 ∂(au)
−
∂V ∂r 3 ∂V
Energy per gram [ erg
g s ]
(only radial term)
EQ = − 2
3
B.3.2 Discretisation
�
1 ∂(au) ∂u a′
ℓ2 −
a ∂r ∂r 2a u
�2 = − 3
2 ℓ2∂(au)
�
∂u
∂V ∂r
− 1
3
�2 ∂(au)
∂V
(B.28)
(B.29)
According to the discretisation scheme given in Appendix A, the following equations
represent the force and energy terms of the artificial viscosity in flux tube geometry.
These terms are implemented in the equation of motion and the energy equation,
respectively.
Viscous force [ dyn
cm 3]
(only radial term)
Energy [ erg
cm 3 s ]
(only radial term)
fi = 1
√ ∆
al
�
a 3/2
l ℓ 2 �
∆(alul) ∆ul
ρl
∆Vl ∆rl
ρ EQ = − 3
2 ℓ2 �
∆(alul) ∆ul
ρl
∆Vl ∆rl
− 1
3
− 1
3
��
∆(alul) 1
∆Vl
∆(alul)
∆Vl
� 2
∆Vl
(B.30)
(B.31)

Appendix C
Radiation Transfer
C.1 Radiation Transfer Equation
C.1.1 General
The derivative of the radiation transfer equation can be done by two approaches:
- Boltzmann Equation
- Local Path
We have to take into account:
- Geometry (coordinates): cartesian (slab), cylindrical, spherical
- Observers view (coordinate frame): Fluid Frame (FF), System or Lab Frame
(SF or LF)
- Motion of the matter: v ≪ c or v ≈ c (relativistic motion)
C.1.2 RTE in General Geometry
The Boltzmann Equation (cf. Buchler 1983 [18] and Buchler 1986 [19])
In terms of some specified coordinate system {x µ }, the photon Boltzmann equation
is given by
�
d dx µ
f =
dt dt
∂ dp∗a
+
∂x µ dt
∂
∂p∗a �
f = C [f] , (C.1)
where µ = {0,1,2,3}, a = {1,2,3} and C [f] is the collision operator. Eq. (C.1) can
also be expressed as
collision operator
�
dx µ ∂
dt ∂x µ + pµdea µ ∂
dt ∂p∗a �
f = C [f] . (C.2)
In an inertial frame1 : and specialising to FF Eq. (C.2) can be written as
�
0 ∂
p
∂t + �p · � ∇ − p 0
�
p 0∂�v
∂t + �p · � � �
∂
∇�v f = C [f] (C.3)
∂�p
1 In an inertial frame (absence of external forces) and for p µ = dx µ
p µ ∂
f = C [f]
∂x µ
115
dt
we can write

116 C. RADIATION TRANSFER
specific intensity
Lorentz
transformation
conservative form
local tetrad
Introducing a specific intensity I = 2 ω3
c 2 h 3f the Boltzmann equation yields to
�
0 ∂
p
∂t + �p · � ∇ − p 0
�
p 0∂�v
∂t + �p · � ∇�v
�
·
� ∂
∂�p
3
− �p
p2 ��
I = p 0
� dI
dt
For the LF we conduct a Lorentz transformation, where to order O(v/c)
with the result
�
0 d
p
dt + �p · � ∇ ′ + �p · �v
c2 �
coll
(C.4)
∂ d ∂
= =
∂t dt ∂t ′ + �v · � ∇ ′ , (C.5)
�∇ = � ∇ ′ + �v
c2 ∂
,
∂t ′ (C.6)
�∇�v = � ∇ ′ �v , (C.7)
�a ≡ d�v
dt
= ∂�v
∂t
d
�
− p0 p
dt 0 �a + �p · � ∇ ′ � �
∂
�v ·
∂�p
∂v
= , (C.8)
∂t ′
��
3
− �p I = p
p2 0
� �
dI
dt
(C.9)
With p 0 = p/c and after dividing above equation by p and using �p = p�n, we can
rewrite Eq. (C.9)
� 1
c
d
dt + �n · � ∇ ′ + 1 d
c2�n · �v
dt
�
1 1
−
c c p�a + �p · � ∇ ′ � �
∂ 3
�v · −
∂�p p �n
��
I = 1
c
Putting terms of the last equation in conservative form, we get
and
� �
dI
dt
coll
coll
.
.
(C.10)
p�a · ∂ ∂
I = · (p�a I) − �a · �n I (C.11)
∂�p ∂�p
�p · � ∇ ′ �v · ∂ ∂
I =
∂�p ∂�p ·
�
�p · � ∇ ′ �
�v I − � ∇ ′ · �v I (C.12)
with the result
�
1 d
c dt + �n · � ∇ ′ + 1 d 1 1
c2�n · �v +
dt c2�a · �n +
c � ∇ ′ �
· �v I − ∂
∂�p ·
�
1 1
p�a I +
c2 c �p · � ∇ ′ �
�v I
� �
3 1
+
c c �a · �n + �n · � ∇ ′ ��
�v · �n I = 1
� �
dI
. (C.13)
c dt
Introducing spherical coordinates in the local tetrad and using the photon energy
ω = pc
∂
∂�p ·
�
1 1
p�a I +
c2 c �p · � ∇ ′ �
�v I =
or easily (�p = p�n and p 0 = p/c) and introducing a specific intensity I
»
1 ∂
c ∂t + �n · � –
∇ I = 1
„ «
dI
.
c dt
coll
coll

C.1. Radiation Transfer Equation 117
= 1
c2 �
1
p2 ∂
∂p (p3�a · �n I) + 1
�
∂
(p�a I)
p ∂�n
+ 1
�
1
c p2 ∂
∂p (p2 �p · � ∇ ′ �v · �n I) + 1 ∂
p ∂�n (�p · � ∇ ′ �
�v I) =
= 1
c2 1
p2 �
p 3 �a · �n ∂I ∂ � � 3
+ I p �a · �n
∂p ∂p
�
+ 1 1
c p2 �
p 3 �n · � ∇ ′ �v · �n ∂I ∂
�
+ I p
∂p ∂p
3 �n · � ∇ ′ �
�v · �n
�
+ 1 ∂
c ∂�n ·
�
1
c �a I + �n · � ∇ ′ �
�v I =
= 1
�
1
c c �a · �n + �n · � ∇ ′ �
�v · �n ω ∂I
�
3 1
+
∂ω c c �a · �n + �n · � ∇ ′ �
�v · �n I
+ 1 ∂
c ∂�n ·
�
1
c �a I + �n · � ∇ ′ �
�v I =
= 1
�
1
c c �a · �n + �n · � ∇ ′ � �
∂ 2 1
�v · �n (ωI) +
∂ω c c �a · �n + �n · � ∇ ′ �
�v · �n I
+ 1 ∂
c ∂�n ·
�
1
c �a I + �n · � ∇ ′ �
�v I
(C.14)
with the result
�
1 d
c dt + �n · � ∇ ′ + 1 d 2 1
�n · �v + �a · �n +
c2 dt c2 c � ∇ ′ · �v + 1
c �n · � ∇ ′ �
�v · �n I (C.15)
− 1
�
1
c c �a · �n + �n · � ∇ ′ �
∂ 1 ∂
�v · �n (ωI) −
∂ω c ∂�n ·
�
1
c �a I + �n · � ∇ ′ �
�v I = 1
� �
dI
c dt
With continuity equation continuity equation
�
1 d 1
+
c dt c � ∇ ′ �
· �v
and the relation
� �
1 d
�n · �v I =
c2 dt
1
c2 d
dt
I = ρ
c
d
dt
� �
I
ρ
1
(�n · �v I) − �a · �n I + O
c2 we can cast the transfer equation into a more compact form
� �
ρ d I
+
c dt ρ
1
c2 d
dt (�v · �n I) + �n · � ∇ ′ I + 1
c
− 1 ∂
c ∂�n ·
�
1
c �a I + �n · � ∇ ′ �
�v I = 1
� �
dI
c dt
� �
v2 c 2
coll
(C.16)
(C.17)
�
1
c �a · �n + �n · � ∇ ′ � �
�v · �n I − ∂
∂ω (ωI)
�
coll
(C.18)

118 C. RADIATION TRANSFER
specific intensity
emissivity
coefficient
extinction
coefficient is
The local Path
We can take a volume of matter around the path of photons from a light source.
Then we look about the effects of interaction of the matter and the photons. The
matter absorbs and emits photons, i.e. reallocates the flux of photons. Generally,
the specific intensity can be written as
[I(�x + ∆�x,t + ∆t;�n,ν) − I(�x,t;�n,ν)]dAdΩ dν dt =
[η(�x,t;�n,ν) − χ(�x,t;�n,ν)I(�x,t;�n,ν)] ds dAdΩ dν dt . (C.19)
Expand the specific intensity I(�x + ∆�x,t+∆t;�n,ν) to a Taylor series around �x and
t and introducing ds = c∆t gives
I(�x + ∆�x,t + ∆t;�n,ν) = I(�x,t;�n,ν) + d
I(�x,t;�n,ν)∆t + ... =
�
dt
�
1 ∂ ∂
I(�x,t;�n,ν) + + I(�x,t;�n,ν)ds . (C.20)
c ∂t ∂s
In general geometry the radiation transfer equation can be written as
� 1
c
∂ ∂
+
∂t ∂s
�
I(�x,t;�n,ν) = η(�x,t;�n,ν) − χ(�x,t;�n,ν)I(�x,t;�n,ν) . (C.21)
The emissivity coefficient η is given by
η(�x,t;�n,ν) = η t (�x,t;�n,ν) + η s (�x,t;�n,ν) , (C.22)
where η t is the thermal and η s the scattering part, and the extinction coefficient χ
χ(�x,t;�n,ν) = κ(�x,t;�n,ν) + σ(�x,t;�n,ν) , (C.23)
where κ denotes the true absorption and σ the scattering coefficient. The derivative
along the radiation propagation path can be derived from
∂
∂s = �n · � ∇ + d�n
ds · � ∇n . (C.24)

C.1. Radiation Transfer Equation 119
C.1.3 Variables and Moments
erg
Specific Intensity I(�x,t;�n,ν) [ cm2 s Hz sr )
erg
Mean Intensity J(�x,t;ν) [ cm2 s Hz )
Jν = J(�x,t;ν) = 1
4π
Monochromatic radiation energy density [ cm3 Hz ]
erg
Radiation Flux � H [ cm2 s Hz ]
��
I(�x,t;�n,ν)dΩ (C.25)
erg
Eν = E(�n,t;ν) = 4π
c Jν
�Hν = � H(�n,t;ν) = 1
4π
Monochromatic radiation flux � F(�x,t;ν) [ cm2 s Hz ]
Radiation Pressure K [ cm2 s Hz ]
��
(C.26)
I(�x,t;�n,ν)�n dΩ (C.27)
erg
�Fν = � F(�x,t;ν) = 4π � H(�x,t;ν) (C.28)
erg
Kν = 1
4π
��
Radiation pressure (stress) tensor [
I(�x,t;�n,ν)�n�n dΩ (C.29)
erg
cm3 Hz
] ≡ [ dyn
cm 2 Hz ]
Pν = P(�x,t;ν) = 4π
c Kν
(C.30)
Sometimes the equation of continuity can be taken into account to express the
moments of the radiation equation
For a spherical geometry this equation is given as
∂
∂t ρ + � ∇ · (ρ�v) = 0. (C.31)
d
dt ln ρ = −∇r · u = − 1
r2 ∂r2u . (C.32)
∂r
C.1.4 Radiation Pressure Tensor Identities
Pν = P(�x,t;ν) = 1
��
I(�x,t;�n,ν)�n�n dΩ (C.33)
c
P = P(�x,t) = 1
c
�∞
0
��
dν I(�x,t;�n,ν)�n�n dΩ (C.34)

120 C. RADIATION TRANSFER
P ij = 1
c
�∞
0
��
dν
I(�x,t;�n,ν)n i n j dΩ (C.35)
In spherical coordinates where dΩ = sinΘdΘdΦ, n 1 = sin Θ cos Φ, n 2 = sin Θ sinΦ,
n 3 = cos Θ we get
thus
and
and
P 11 = 1
c
�
0
= 2π
c
P 22 = 1
c
�
0
= 2π
c
P 33 = 1
c
�
0
= 2π
c
2π
�
cos 2 �
ΦdΦ
1
−1
2π
�
0
Idµ − 2π
c
sin 2 �
ΦdΦ
1
−1
2π
�
0
Idµ − 2π
c
�
dΦ
1
−1
0
π
π
π
I sin 3 ΘdΘ
�1
−1
Iµ 2 dµ = 1
(E − P) (C.36)
2
I sin 3 ΘdΘ
�1
−1
I cos 2 Θ sin ΘdΘ
Iµ 2 dµ = 1
(E − P) (C.37)
2
Iµ 2 dµ = P (C.38)
(C.39)
⎛1
⎞
2 (E − P) 0 0
P = ⎝ 1 0 2 (E − P) 0⎠
0 0 P
⎛ ⎞
P 0 0
= ⎝0
P 0⎠
−
0 0 P
1
⎛
⎞
3P − E 0 0
⎝ 0 3P − E 0⎠
(C.40)
2
0 0 0
( � ∇P)r = − 1
r (P11 + P 22 ) + 1
r2 ∂(r2P33 )
=
∂r
∂P
∂r
(P : � ∇ ′ �v)r = u
r
+ 3P − E
r
(C.41)
∂u
(E − P) + P . (C.42)
∂r

C.2. 0 th -order Moment Equation 121
C.2 0 th -order Moment Equation
The zeroth moment of the radiation transfer equation (total radiation energy equation)
in the fluid frame and independent of the coordinates and of the geometrical
symmetry is derived from Eq. (C.18) by the integration over dΩ and can be written
as (Buchler 1983 [18])
ρ d
� �
E
+
dt ρ
� �� �
1
1
c2 d
�
�v ·
dt
� �
F +
� �� �
2
� ∇ ′ · � F + P :
� �� �
3
� ∇ ′ �v +
� �� �
4
1
c2 �
�a · � �
F =
� �� �
5
Term 1: With the continuity equation we can write this term as
ρ d
� �
E
dt ρ
�∞
0
q0(ω)dω (C.43)
= d
dt E + E � ∇ · �v = ∂tE + v k ∇kE + E v k ;k . (C.44)
Term 2: Is of order (v 2 /c 2 ) and can be neglected.
Term 3: This is the divergence of the radiation flux
�∇ ′ · � F = F k
;k . (C.45)
Term 4: This is the contraction of the radiation pressure tensor and the divergence
of �v
P : � ∇ ′ �v = P j
i vi ;j . (C.46)
Term 5: Is of order (v 2 /c 2 ) and can be neglected.
C.2.1 Case 1: Spherical Geometry
For the expression in Eq. (C.46), we get
(P : � ∇ ′ �v)r = ∂u u
P +
∂r r (E − P) = P ∇′ r
u
· u + (E − 3P) (C.47)
r
where u is the velocity component in the direction of r, thus Eq. (C.43) is
d
dt E + E ∇′ r · u + ∇ ′ r · F + P ∇ ′ r · u + u
r (E − 3P) = q0 . (C.48)
Introducing an other variable set (J, � H,K) and dividing by 4π, we get
1
c
∂ 1
J +
∂t c (u·∇′ r) J +∇ ′ r ·H − 1
c
u 1
(3K −J)+
r c J ∇′ r ·u+ 1
c K ∇′ r ·u = RHS . (C.49)
In the case of spherical geometry Eq. (C.43) reduces to (Castor 1972 [22])
1
c
∂
∂t
1
J +
c ∇′ r · (J u) = − ∇′ 1
r · H −
c
�
K ∇ ′ u
�
r · u − (3K − J)
r
− ρ(κJJ − κSS) (C.50)

122 C. RADIATION TRANSFER
C.2.2 Case 2: Flux Tube Geometry
To calculate the zeroth moment of the radiation transfer equation in flux tube geometry,
we need the following derivatives:
P : � ∇ ′ �v = P ∂u a′
+
∂r 2a u (E − P) = P ∇′ z · u + a′
u (E − 3P) , (C.51)
2a
thus Eq. (C.43) is
d
dt E + E ∇′ z · u + ∇ ′ z · F + P ∇ ′ z · u + ua′
2a (E − 3P) = q0 . (C.52)
Introducing an other variable set (J, �H,K) and dividing by 4π, we have
1
c
∂
∂t
1
J+
c (u·∇′ z ) J+∇′ 1 ua
z ·H −
c
′ 1
(3K − J)+
2a c J ∇′ 1
z ·u+
c P ∇′ z ·u = RHS . (C.53)
In the case of flux tube geometry Eq. (C.43) reduces to
1
c
∂
∂t
1
J +
c ∇′ z · (J u) = − ∇′ 1
z · H −
c
�
P ∇ ′ �
ua′
z · u − (3K − J)
2a
− ρ(κJJ − κSS) (C.54)

C.3. 1 st -order Moment Equation 123
C.3 1 st -order Moment Equation
The first moment of the radiation transfer equation (total radiation flux equation) in
the fluid frame and independent of the coordinates and of the geometrical symmetry
is derived from Eq. (C.18) by the multiplication with �n as well as the integration
over dΩ and can be written as (Buchler 1983 [18])
� �
ρ d �F
+
c dt ρ
� �� �
1
1 d
(�v · P) + c
c dt
� �� �
2
� ∇ ′ · P +
� �� �
3
1
c �aE +
� �� �
4
1
c � F · � ∇ ′ �v =
� �� �
5
�∞
0
q(ω)dω (C.55)
Term 1: With the continuity equation we can write this term as
� �
ρ d �F
=
c dt ρ
1 D
c Dt � F + 1
c � F · � ∇�v . (C.56)
Term 2: Is of order (v 2 /c 2 ) and can be neglected.
Term 3: This is the ’divergence’ of the radiation pressure tensor
�∇ ′ · P = P j
i,i . (C.57)
Term 4: Is of order (v 2 /c 2 ) and can be neglected.
Term 5: Denotes the losses caused by radiative acceleration of the matter.
C.3.1 Case 1: Spherical Geometry
For the expressions in Eq. (C.57) and the second term in Eq. (C.56), we get
( � ∇ ′ · P)r = ∂P
∂r
( � F · � ∇ ′ �v)r = F ∇ ′ r
1
+ (3P − E) , (C.58)
r
· u , (C.59)
respectively, where F is the radiation flux component and u the velocity component
in the direction of r, thus Eq. (C.55) is
1 d
F + c∂P
c dt ∂r
c 2
+ (3P − E) +
r c F ∇′ r · u = (�q)r . (C.60)
Introducing an other variable set (J, � H,K) and dividing by 4π, we get
1
c
∂ 1
H +
∂t c (u · ∇′ ∂K
r ) H +
∂r
1 2
+ (3K − J) +
r c H ∇′ r · u = RHS . (C.61)
In the case of spherical geometry Eq. (C.55) reduces to (Castor 1972 [22])
1 ∂ 1
H +
c ∂t c ∇′ 1
�
r · (H u) = −∂K − H ∇
∂r c
′ c
�
r · u + (3K − J) − ρκHH . (C.62)
r

124 C. RADIATION TRANSFER
C.3.2 Case 2: Flux Tube Geometry
To calculate the zeroth moment of the radiation transfer equation in flux tube geometry,
we need the following derivatives:
thus Eq. (C.55) is
( � ∇ · P)z = 1 ∂(aP) a′ ∂P
+ (P − E) =
a ∂z 2a ∂z
1 d ∂P
F + c
c dt ∂z
+ c a′
2a
a′
+ (3P − E) , (C.63)
2a
(3P − E) + 2
c F ∇′ z · u = (�q) z . (C.64)
Introducing an other variable set (J, �H,K) and dividing by 4π, we have
1 ∂ 1
H +
c ∂t c (u · ∇′ ∂K
z ) H +
∂z
In the case of flux tube geometry Eq. (C.55) reduces to
a′ 2
+ (3K − J) +
2a c H ∇′ z · u = RHS . (C.65)
1 ∂ 1
H +
c ∂t c ∇′ �
1
z · (H u) = −∂K − H ∇
∂z c
′ �
ca′
z · u + (3K − J) − ρκHH .(C.66)
2a
C.4 Derivatives in different geometries
The following table gives an overview of derivatives in different geometries.
Derivative Geometries
Spherical Cylindrical Flux Tube
∇ξ1 u
( � ∇ · � K)ξ1
(K : � ∇u)ξ1
∂K
∂r
1
r2 ∂(r2u) ∂r
1
∂K
+ r (3K − J)
∂u u
∂u
∂r K + r (J − K)
∂u
∂z
∂z
∂z K
∂K
∂z
∂u
∂z
1 ∂(au)
a ∂z
a′ + 2a (3K − J)
ua′ K + 2a (J − K)
Table C.1: Derivatives in the spherical, cylindrical and flux tube geometries.

C.5. Summary of Spherical Radiation Equations 125
C.5 Summary of Spherical Radiation Equations
C.5.1 Radiation Energy Equation
Differential notation
Dimension of terms in the equation:
energy per volume and time [erg cm −3 s −1 ] or [J m −3 s −1 ]
1 ∂
c ∂t J +
� �� �
1
1
∇ · (Ju) = −∇ · H −
� c �� � � �� �
2 3
1
K∇ · u+
� c �� �
4
u 3K − J
−ρ(κJJ − κSS)
� c ��r ��
�� �
5
6
Integral notation (conservation form)
Dimension of terms in the equation:
energy per time [erg s −1 ] or [J s −1 ]
�
1 ∂
J dV +
c ∂t
�
V
�� �
1
1
� �
J udA = − ∇ · H dV
c
�
∂V
�� �
V
� �� �
2
3
− 1
�
K ∇ · udV +
c
�
V
�� �
4
1
�
�
3K − J
udV − ρ(κJJ − κSS)dV
c r
�
V
��
V
��
�� �
5
6
Declaration of terms:
Term 1: temporal change of the radiation energy in a certain volume
Term 2: radiation energy flow through the surface (advection)
The radiation energy is changed by
Term 3: the flux of radiation through the surface of a certain volume
(C.67)
(C.68)
Term 4: the work done by the radiation pressure and is analogous to the pressure
term in the gas internal energy equation
Term 5: the work done by the radiation pressure and accounts for the fact that the
radiation pressure is not necessarily isotropic
Term 6: the absorption and emission of the radiation energy by the matter

126 C. RADIATION TRANSFER
C.5.2 Radiation Momentum Equation
Differential notation
Dimension of terms in the equation:
energy per volume and time [ erg
cm 3 s ]
1 ∂
c ∂t H +
� �� �
1
1
∇ · (Hu) = −
� c �� �
2
∂
∂r K −
� �� �
3
1 3K − J
H∇ · u − −(κHρ + χH)H
� c �� ��
��r ��
�� �
4 5
6
Integral notation (conservation form)
Dimension of terms in the equation:
energy per time [ erg
s ]
�
1 ∂
H dV +
c ∂t
V
� �� �
1
1
�
H udA
c
∂V
� �� �
2
− 1
� �
�
3K − J
H ∇ · udV − dV −
c
r
V
� �� �
4
Declaration of terms:
V
� �� �
5
V
�
= −
∂
K dV
∂r
V
� ��
3
�
(κHρ + χH)H dV
� �� �
6
Term 1: temporal change of the radiation flux (momentum) in a certain volume
Term 2: radiation flux flow through the surface (advection)
The radiation flux is changed by
(C.69)
(C.70)
Term 3: the spacial gradient radiation pressure and corresponds to the gas pressure
gradient in the equation of motion
Term 4: the losses caused by radiative acceleration of the matter
Term 5: the anisotropic part of the radiation pressure tensor
Term 6: the absorption of radiation flux by the matter

Appendix D
Dust properties
D.1 Constants
Constant Value Dimension Description
Amon 12.01115 atomic weight of the dust-forming material
ρcond 2.25 g cm −3 mass density of the condensed phase
σd 1400 erg cm −2 surface tension of graphite
Nh 5 particle size for which σd reduces to one half
of σd for the bulk material
αC 0.37 sticking coefficient for Ci (i = 1)
αC2 0.34 sticking coefficient for Ci (i = 2)
αC2H 0.34 sticking coefficient for CiHj (i = 2, j = 1)
αC2H2 0.34 sticking coefficient for CiHj (i = 2, j = 2)
Nℓ 1000 lower limit of the grain sizes to be treated as
macroscopic particles
εHe 0.1 abundance of Helium
mP 1.672610 −24 g proton mass
D.2 Variables
Variable Dimension Description
N, N∗
critical cluster size
N∗,∞
ntot H cm−3 ρG g cm
total number density of H
−3 mass density of the gas component
r0 cm monomer radius
1/τ s−1 net growth rate
J , J∗ s−1cm−3 net transition rate, stationary nucleation rate
Z Zel’dovich-factor
ΘN K surface free energy / k
Θ∞ K surface free energy / k for N → ∞
AN cm2 surface of a dust grain
nc cm−3 number density of all free C-atoms in the gas phase
f(N, t) cm−3 number density of N-mers
Kj cm−3 moments of the grain size distribution
KX P
dissociation constant of element X
127

128 D. DUST PROPERTIES
D.3 Dust Formation
D.3.1 C-rich Chemistry
Chemical Reactions
The chemical reactions for grain growth and chemical sputtering in a carbon-hydrogen
chemistry, where the oxygen is completely bound in the molecule CO, are
C + O ⇋ CO (D.1)
C2H2 + CN ⇋ CN+2 + H2
(D.2)
C2H + CN ⇋ CN+2 + H (D.3)
Table D.1 shows the classification of atoms and molecules, which are taken into
account for our calculations.
Abundances
gas monomers H C
phase dimers H2 C2
molecules C2H
dust dust grains
phase
C2H2
CO
Table D.1: Chemical composition
The chemical abundance of an element X is defined by
We can also say
εX = n
n
ε gas
X = εX − ε dust
X
. (D.4)
(D.5)
that means, that the abundance of an element X in the gas phase is the total
) without the amount of X bound in dust grains.
abundance of X (εX = ε tot
X
Number Densities
The total number density of hydrogen H can approximately be derived from the
density of the gas by the elimination of the amount of helium He
n tot
H =
ρ gas
(1 + 4εHe)mP
, (D.6)
where mP is the mass of a proton, and includes the free H atoms and the H2 dimers,
thus
n tot
H = n = nH + 2nH2 . (D.7)

D.3. Dust Formation 129
The total number density of carbon C is given by
where
n tot
C = n = εC n = n gas
C + ndust C , (D.8)
n gas
C
= n(gas)
C + nCO (D.9)
and the number density of C in free C atoms and bound in the dimer C2 and the
molecules C2H and C2H2 is
n (gas)
C
= ngas mon = nC + 2nC2 + 2nC2H + 2nC2H2 . (D.10)
Oxygen O is completely bound in the molecule CO, i.e. nO = nCO, so the amount
of carbon C in the dust phase is
n dust
C = Kd = n tot
C − n − n (gas)
C
= (εC − εO)n −n
� �� �
(gas)
C . (D.11)
“ ”
εC
= −1 n
εO
The fraction of condensable material actually condensed into grains is described by
the degree of condensation
Partial Pressure
fcond =
Kd
Kd + n gas
mon
=
n dust
C
n dust
C
+ n(gas)
C
. (D.12)
The partial pressure for the hydrogen H is generally given by the equation
and from Eq. (D.7) we get also
P tot
H = n tot
H k T (D.13)
P tot
H = PH + 2PH2 =
= PH + 2P 2 HK H2
P . (D.14)
To express the partial pressure of the atomar hydrogen H we look at the result of
the quadratic equation
PH = − 1
4K H2
� �
1 ± 1 + 8K
P
H2
�
P P tot
H , (D.15)
where only the negative sign gives a meaningful physical solution. Thus, it yields
easily
From Eq. (D.10) we get
P (gas)
C
= k T n (gas)
C
PH =
2P tot
H
�
1 + 1 + 8K H2
P P tot
H
= PC + 2PC2 + 2PC2H + 2PC2H2 =
= PC + 2P 2 C (KC2
P
+ PHK C2H
P
. (D.16)
+ P 2 H KC2H2
P ) . (D.17)

130 D. DUST PROPERTIES
To express the partial pressure of the atomar carbon C we look (as in Eq. (D.15))
at the result of the quadratic equation
PC = − 1
� �
1 ± 1 + 8(...)P
4(...)
(gas)
�
C . (D.18)
Thus, it yields easily (as in Eq. (D.16))
PC = Pmon =
2k T n (gas)
�
C
1 + 1 + 8k T n (gas)
C (K C2 C2H
P + PHK
P + P 2 HKC2H2 P
For the other molecules we get the partial pressures as follows
PC2 = P 2 C KC2
P
. (D.19)
)
, (D.20)
PC2H = P 2 C PH K C2H
P , (D.21)
PC2H2 = P 2 C P 2 H KC2H2
P . (D.22)
The vapour saturation pressure of C1 is approximated by
�
Psat(C1) = exp − 86300
�
+ 32.89
T
following Gail & Sedlmayr (1988 [49]).
(D.23)

D.3. Dust Formation 131
D.3.2 Nucleation Theory
Transition Rate
The net transition rate J represents the number of clusters of size Nℓ which are
created or destroyed per second and volume. Small unstable clusters which form
at random from the gas phase have to grow beyond a certain critical size N∗ until
they are stable against collisions and destruction. This separates the domain of
small unstable clusters from the large thermodynamically stable grains. As soon as
a grain becomes larger than N∗ it will grow if no change of the thermodynamical
conditions occur. The equilibrium size distribution of the clusters is given by
(Feder et al. 1966 [42]) with the first
�
ˆf(N) = n1 exp (N − 1)ln S − ΘN
�
(N − 1)23
T
∂ ˆ f(N)
∂N
and second derivative
= ln S − 1
T
1
(N − 1) 1
3
∂2f(N) ˆ
�
2 Θ ′2
= − (N − 1)23
N
∂N 2 T ΘN
�
Θ ′ 2
N (N − 1) +
3 ΘN
�
− 1
9
ΘN
(N − 1) 2
�
(D.24)
(D.25)
(D.26)
which are needed later to calculate the so-called Zel’dovich-factor (see Eq. D.37). It
is convenient to define the quantity ΘN(T) by
ΘN =
Θ∞
1 + [Nh/(N − 1)] 1
3
, . (D.27)
In the limit N → ∞ and assuming spherical N-mers ΘN approaches a constant
Θ∞ = 4πr 2 0
σd
k
, (D.28)
where σd is the surface free energy per surface area of bulk material. kΘ could be
interpreted as the surface free energy per surface site. For further calculations we
need the first and second derivation of Eq. (D.27), thus
or simplified
and
∂ΘN
∂N = Θ′ N =
Θ∞
{1 + [Nh/(N − 1)] 1
3 } 2
Θ ′ N
= 1
3
ΘN
(N − 1)
∂ 2 ΘN
∂N 2 = Θ′′ N = 2 Θ′2 N
ΘN
1
3
�
Nh
� 2
−3 Nh
N − 1 (N − 1) 2
[Nh/(N − 1)] 1
3
1 + [Nh/(N − 1)] 1
3
− 4
3 Θ′ N
(D.29)
(D.30)
1
. (D.31)
N − 1

132 D. DUST PROPERTIES
The critical cluster size is given as the solution of
by
where
where
N∗ = 1 + N∗,∞
8
⎧
⎨
�
1 +
⎩
1 + 2
N∗,∞ =
∂ ln ˆ f
∂N
� Nh
N∗,∞
= 0 (D.32)
�1
�1
2 �
3 Nh
− 2
N∗,∞
� �3 2Θ∞
3T ln S
The stationary nucleation rate is defined by
is the surface of a dust grain,
is the Zel’dovich-factor,
J∗ = AN∗ ˆ f(N∗)Z
�1
3
⎫
⎬
⎭
3
, (D.33)
. (D.34)
I�
i 2 vth(i)α(i)neff(i) . (D.35)
i=1
AN∗ = 4πr 2 0 N 2
3
∗
� �
1 ∂
Z =
2π
2 ln ˆ f(N)
∂N 2
�
vth(i) =
� k T
2π mi
N∗
�1
2
(D.36)
(D.37)
(D.38)
denotes the thermal velocities of the corresponding species, α(i) are the sticking
coefficients and
neff(i) = P(i)
are the effective number densities.
k T
(D.39)

D.3. Dust Formation 133
Net Growth Rate
The net growth rate is defined by
1
τ
= A1
+
I�
ivth(i)α(i)f(i,t)
i=1
�
iAi
I ′
i=1
Mi �
m=1
�
1 − 1
S i
vth(i,m)α c m(i)ni,m
1
bi
�
α∗(i)
�
1 − 1
S i
1
b c i,m
α c ∗(i,m)
�
,
(D.40)
where f(i) and ni,m are the number densities of the i-mers and the molecules containing
i-mers which contribute to the grain growth, α∗ and b are the departure
coefficients, which describe the departures from thermodynamical and chemical equilibrium,
and
S = Pmon
Psat
(D.41)
is the supersaturation ratio, i.e. the ratio of the actual partial pressure of the
monomers in the gas phase to the vapour saturation pressure.
For chemical equilibrium in the gas phase the net growth rate reduces to
I�
�
1
= A1 ivth(i)α(i)f(i,t) 1 −
τ
i=1
1
Si � �
Ki(Td) Tg
(D.42)
Ki(Tg) Td
I
+
′
Mi � �
iAi vth(i,m)α c �
m(i)ni,m 1 − 1
Si Kr i,m (Tg)
Kr i,m (Td)
�
Ki,m(Td)
,
Ki,m(Tg)
i=1
m=1
where K denotes the dissociation constant of the molecule of the growth reaction
and K r is the dissociation constant of the molecule involved in the reverse reaction.
If we substitute some quantities and write down all relevant terms of the row in
Eq. (D.42), we get
1
τ = 4π r2 0
+
+
+
1
� Tg
√ 2 αC2
� �
1
√
2π k mP Amon
αC PC 1 − 1
S
�
PC2
�
1 − 1
S 2
�
1 αC2H
� PC2H 1 −
1 + 1/(2Amon) αC2
1
S2 1 αC2H2
� PC2H2
1 + 2/(2Amon) αC2
�
1 − 1
S 2
�
KC2 (Td)
KC2 (Tg)
Tg
Td
�
KC2H(Td)
KC2H(Tg)
�
Tg
Td
�
�
KH2 (Tg)
KH2 (Td)
KC2H2 (Td)
KC2H2 (Tg)
���
(D.43)
.

134 D. DUST PROPERTIES
D.3.3 Dust Physics
0 th Moment Dust Equation
The zeroth moment of the master dust equation (cf. Höfner 1994 [68]) describes the
change of the number of dust grains due to creation of particles from gas phase or
destruction of dust grains.
δ([K0]l ∆Vl) + ∆(−
∂
∂t K0 + ∇ · (K0 u) = J (D.44)
��� ad
K0
ρ
l
δml) = δt Jl ∆Vl
(D.45)
J denotes the net transition rate per volume from cluster sizes N < Nℓ to N > Nℓ.
1 st - 3 rd Moment Dust Equation
These moments of the master dust equation determine the time evolution of some
means of the particle radius.
∂
∂t Kj + ∇ · (Kj u) = j
d
δ([Kj]l ∆Vl) + ∆(−
��� ad
Kj
ρ
1
τ Kj−1 + N j/d
ℓ J (1 ≤ j ≤ d) (D.46)
l
δml) = δt j
d
1
τl
[Kj−1]l ∆Vl
+ δt N j/d
ℓ J ∆Vl (1 ≤ j ≤ d) (D.47)
The first term on the right hand side describes the changes caused by growth of
grains with N > Nℓ with the net growth rate 1/τ and the second one for particles
entering or leaving this domain of cluster sizes.
nc Dust Equation
Finally we need also to solve the differential equation of the free C atoms which can
be used for the condensation process as condensable material.
∂
∂t nc + ∇ · (nc u) = 1
τ K2 + Nℓ J (1 ≤ j ≤ d) (D.48)
� �ad nc
�
δ([nc]l ∆Vl) + ∆(− δml) = δt
ρ l
1
[K2]l ∆Vl
τl
+ δt Nℓ J ∆Vl
(1 ≤ j ≤ d) (D.49)

Appendix E
Tensor Calculus
E.1 General
E.1.1 Historical Background
The word tensor was introduced by William Rowan Hamilton in 1846, but he used
the word for what is now called modulus. The word was used in its current meaning
by Woldemar Voigt in 1899.
The notation was developed around 1890 by Gregorio Ricci-Curbastro under the
title absolute differential geometry, and made accessible to many mathematicians by
the publication of Tullio Levi-Civita’s classic text The Absolute Differential Calculus
in 1900 (in Italian; translations followed). The tensor calculus achieved broader acceptance
with the introduction of Einstein’s theory of general relativity, around 1915.
General Relativity is formulated completely in the language of tensors, which Einstein
had learnt from Levi-Civita himself with great difficulty. But tensors are used
also within other fields such as continuum mechanics, for example the strain tensor,
(see linear elasticity). Examples of physical tensors are the energy-momentum
tensor, the inertia tensor and the polarisation tensor.
E.1.2 Definitions
Tensor
A tensor is a certain kind of geometrical entity which generalises the concepts of
scalar, vector (spatial) and linear operator in a way that is independent of any
chosen frame of reference. Tensors are of importance in physics and engineering.
An n th -rank tensor in m-dimensional space is a mathematical object that has n
indices and m n components and obeys certain transformation rules. Each index of
a tensor ranges over the number of dimensions of space. However, the dimension of
the space is largely irrelevant in most tensor equations (with the notable exception
of the contracted Kronecker delta). Tensors are generalisation of scalars (that have
no indices), vectors (that have exactly one index), and matrices (that have exactly
two indices) to an arbitrary number of indices.
135

136 E. TENSOR CALCULUS
Tensors provide a natural and concise mathematical framework for formulating
and solving problems in areas of physics such as elasticity, fluid mechanics, and
general relativity.
Co- and contravariant Tensors
In tensor analysis, a covariant coordinate system is reciprocal to a corresponding
contravariant coordinate system.Roughly speaking, a covariant tensor is a vector
field that defines the topology of a space; it is the base against which one measures.
A contravariant vector is thus a measurement or a displacement on this space.A
covariant tensor, denoted with a lowered index (e.g. aµ) is a tensor having specific
transformation properties that, in general, differ from those of a contravariant tensor.
Contravariant tensors are a type of tensor with differing transformation properties,
denoted a ν . However, in three-dimensional Euclidean space contravariant and
covariant tensors are equivalent. Such tensors are known as Cartesian tensor. The
two types of tensors do differ in higher dimensions, however.To turn a contravariant
tensor a ν into a covariant tensor aµ (index lowering), use the metric tensor gµν to
write
gµνa ν = aµ . (E.1)
Covariant and contravariant indices can be used simultaneously in a mixed tensor.
Tensor Calculus
The set of rules for manipulating and calculating with tensors.
Einstein summation
The convention that repeated indices are implicitly summed over. This can greatly
simplify and shorten equations involving tensors.
Tensor rank
The total number of contravariant and covariant indices of a tensor. (rank 0: Scalar;
rank 1: Vector; rank ≥ 2: Tensor)
Tensor contraction
The contraction of a tensor is obtained by setting unlike indices equal and summing
according to the Einstein summation convention. Contraction reduces the tensor
rank by 2.

E.2. Vectors 137
E.2 Vectors
E.2.1 Definitions
Symbol Relation 1 Description
�g vector basis
�ei (i = 1,2,3) orthonormal vector basis
�n n1�e1 + n2�e2 + n3�e3 unit vector
d�s �nds direction vector element
Table E.1: Definitions of some vector symbols.
The co- and contravariant components of a vector are co-/contravariant
components
�a = a i �gi = ai�g i , (E.2)
where the connection of the vector space basis raised by the metric tensor
�g i = g ij �gj , (E.3)
�gi = gij�g j and (E.4)
gij = �gi · �gj . (E.5)
The connection of the co- and contravariant metric tensor is
g ij gjk = δ i k
where (g ij ) and (gij) are inverse matrices and δ i k
δ j
i =
�
1 for i = j
0 for i �= j
The connection between the components of a vector is
The components of a vector are given by
, (E.6)
is the Kronecker delta Kronecker
delta
. (E.7)
a j = g ji ai , (E.8)
aj = gji a i . (E.9)
a i = �g i ·�a or (E.10)
ai = �gi ·�a . (E.11)
The covariant physical components of �a is defined as physical
components
where
1 general geometry
a ⋆ i = a(i) = h(i) a (i) , (E.12)
h (i) ≡
�
g (i)(i)�12
(E.13)

138 E. TENSOR CALCULUS
orthonormal basis
permutation
symbol
dot product
cross product
are the scale factors. The length of a vector �a is defined as
|�a| 2 = �
(a(i)) 2
and is in R 3 expressed with the contravariant components a i
where the associated physical components are
For the covariant components ai we get in R 3
According to
|�a| 2 =
i
(E.14)
|�a| 2 = � h1a 1� 2 + � h2a 2� 2 + � h3a 3� 2 , (E.15)
3�
(a(i)) 2 �
1
=
i=1
a(i) = h (i)a (i) . (E.16)
h1
a1
� 2
g (i)(i) �
1
=
�
1
+
h2
h (i)
� 2
a2
� 2
�
1
+
h3
a3
� 2
. (E.17)
the physical components are related to covariant components by the expression
Special case: orthonormal basis, where
a(i) = 1
h (i)
�e i = �ei
�ei �ej = δij
(E.18)
a (i) . (E.19)
and the permutation symbol2 ⎧
⎨ +1 if (i,j,k) is an even permutation of (1,2,3)
�ei·(�ej × �ek) = εijk = −1 if (i,j,k) is an odd permutation of (1,2,3)
⎩
0 otherwise
E.2.2 Operations and Operators
The dot product 3 can be written as
The cross product 4 can be written as
(E.20)
(E.21)
. (E.22)
�a · � b ≡ a i gij b j . (E.23)
2 The permutation symbol can also be interpreted as a tensor, in which case it is called the
permutation tensor. This tensor is also called the Levi-Civita tensor or isotropic tensor of rank 3.
3 The dot product is also called the scalar product or inner product. For orthonormal basis we
get
�a · � b = aibi .

E.2. Vectors 139
Symbol Relation1 Description
�∇ see Eq. (E.26) spatial gradient vector operator or Nabla operator
�∇n
“directional” gradient vector operator
∂
∂s
d
dt
�∇ · �n
|�n|
�n · � ∇ + d�n
ds · � ∇n derivative along a path
∂ ∂
∂t + c∂s hydrodynamical operator (coordinate system at rest)
Table E.2: Summary of some specific vector operators.
(�a × � b)i = εijk a j b k , (E.24)
(�a × � b) i = ε ijk ajbk . (E.25)
The Nabla 5 operator is defined by Nabla operator
�∇
∂
= �ei∂i = �ei = �ei
∂ξ(i)
1
h (i)
∂
∂ξ (i)
(E.26)
Introducing spherical coordinates in the local tetrad we can write for the terms
�n � �
�
1 ∂ 1 ∂ 1 ∂
∇x1 = �n�e1 + �n�e2 + �n�e3 x1 =
h1 ∂x1 h2 ∂x2 h3 ∂x3
1
�n�e1
h1
= 1
sinΘcos Φ (E.27)
h1
�n � ∇x2 = 1
sin Θ sin Φ (E.28)
where
h2
�n � ∇x3 = 1
cos Θ (E.29)
h3
�n � ∇Θ = − 1
� � ��
�n �∇(�n�e3)
(E.30)
sinΘ
�n � ∇Φ = − 1
� �
1
�n
sinΦ sinΘ � cos Θ cos Φ
∇(�n�e1) +
sin2 ��
�∇(�n�e3) , (E.31)
Θ
h (i) =
� � ∂x
∂xi
� 2
� �2 � �2 ∂y ∂z
+ +
∂xi ∂xi
(i = 1,2,3) (E.32)
4 The cross product is also known as the vector product or outer product. In R 3 the cross product
becomes
�a × �b = �e1(a2b3 − a3b2) − �e2(a1b3 − a3b1) + �e3(a1b2 − a2b1) = ˛
˛
˛ �e1 �e2 �e3 ˛
˛
˛
�e1(a2b3 − a3b2) + �e2(a3b1 − a1b3) + �e3(a1b2 − a2b1) = ˛ a1 a2 a3 ˛
˛
˛
˛
˛ .
b1 b2 b3
5 The Nabla, also called “del” used to denote the gradient and other vector derivatives. In R 3
we get
�∇ = 1
�e1∂ξ1
h1
1 1
+ �e2∂ξ2 + �e3∂ξ3
h2 h3
.

140 E. TENSOR CALCULUS
vector divergence
Christoffel
symbols
are the scale factors and (x,y,z) represents the rectangular coordinate system (cf. Uesugi
& Tsujita 1969 [150]).
The divergence of a vector6 is the covariant derivative of the contravariant com-
ponent, i.e.
whereas the covariant derivative of a covariant vector is
a k ;i = ak ,i + Γk im am , (E.33)
ak;i = ak,i − Γ m ik am . (E.34)
where Γk im (or Γm ik ) are the connection coefficients (also known as Christoffel symbols
or Ricci rotation coefficients) (see also Section E.4.1 about the metric tensor). With
these coefficients and the relation
we get the components
Γ i ij = −Γ j 1 ∂hi
ii =
hihj ∂ξj
( � ∇ ·�a)11 = a 1 ,1 + Γ 1 12 a 2 + Γ 1 13 a 3 = 1
∂a 1
∂ξ1
h1
( � ∇ ·�a)21 = a 1 ,2 + Γ112 a2 = a 1 ,2 − Γ211 a2 = 1 ∂a
h2
1
∂ξ2
( � ∇ ·�a)31 = a 1 ,3 + Γ 1 13 a 3 = a 1 ,3 − Γ 3 11 a 3 = 1 ∂a
h3
1
∂ξ3
( � ∇ ·�a)12 = a 2 ,1 + Γ122 a1 = a 2 ,1 − Γ221 a1 = 1 ∂a2 ( � ∇ ·�a)22 = a 2 ,2 + Γ 2 21 a 1 + Γ 2 23 a 3 = 1
h1
∂a2 ∂ξ2
+ a2 ∂h1
+
h1h2 ∂ξ2
a3 ∂h1
h1h3 ∂ξ3
∂ξ1
h2
( � ∇ ·�a)32 = a 2 ,3 + Γ322 a3 = a 2 ,3 − Γ223 a3 = 1 ∂a
h3
2
∂ξ3
( � ∇ ·�a)13 = a 3 ,1 + Γ 1 33 a 1 = a 3 ,1 − Γ 3 31 a 1 = 1 ∂a
h1
3
∂ξ1
( � ∇ ·�a)23 = a 3 ,2 + Γ233 a2 = a 3 ,2 − Γ332 a2 = 1 ∂a3 ( � ∇ ·�a)33 = a 3 ,3 + Γ 3 31 a 1 + Γ 3 32 a 2 = 1
h3
h2
∂a3 ∂ξ3
− a2 ∂h2
h1h2 ∂ξ1
− a3 ∂h3
h1h3 ∂ξ1
− a1 ∂h1
h1h2 ∂ξ2
+ a1 ∂h2
+
h1h2 ∂ξ1
a3 ∂h2
h2h3 ∂ξ3
∂ξ2
− a3 ∂h3
h2h3 ∂ξ2
− a1 ∂h1
h1h3 ∂ξ3
− a2 ∂h2
h2h3 ∂ξ3
+ a1 ∂h3
+
h1h3 ∂ξ1
a2 ∂h3
h2h3 ∂ξ2
(E.35)
(E.36)
(E.37)
(E.38)
(E.39)
(E.40)
(E.41)
(E.42)
(E.43)
(E.44)
(E.45)
6 For coordinate systems with constant basis, i.e. no derivatives of the basis exist, e.g cartesian
coordinates
In R 3
�∇ · �a = 1
h1
�∇ · �a = 1
h(i)
∂a 1
1
+
∂ξ1 h2
∂a i
∂ξ (i)
∂a 2
1
+
∂ξ2 h3
∂a 3
.
∂ξ3

E.2. Vectors 141
The divergence of a vector in arbitrary orthogonal curvilinear coordinates can be
calculated by
�
�∇
1 ∂
·�a =
∂ξ1(h2h3 a 1 ) + ∂
∂ξ2(h3h1 a 2 ) + ∂
∂ξ3(h1h2 a 3 �
) . (E.46)
h1h2h3
The curl (only in R 3 ) in arbitrary orthogonal curvilinear coordinates is curl
�∇ × �a =
1
h1h2h3
( � ∇× is also denoted by rot for rotation).
�
� h1�g1 h2�g2 h3�g3
�
� ∂
� ∂ξ
�
1
∂
∂ξ2 ∂
∂ξ3 �
�
�
�
�
� a1 a2 a3
=
�
1 ∂
h2h3 ∂ξ2(h3 a 3 ) − ∂
∂ξ3(h2 a 2 �
) �g1
+ 1
�
∂
h1h3 ∂ξ3(h1 a 1 ) − ∂
∂ξ1(h3 a 3 �
) �g2
+ 1
�
∂
h1h2 ∂ξ1(h2 a 2 ) − ∂
∂ξ2(h1 a 1 �
) �g3
(E.47)
Scalar Triple Product scalar triple
product
[�a, � b,�c] ≡ �a · ( � b × �c) (E.48)
E.2.3 Relations / Vector Identities
�∇(�a · � b) = (�a · � ∇) � b + �a × ( � ∇ × � b) + ( � b · � ∇)�a + � b × ( � ∇ × �a) (E.49)
�∇(�n · �ei) = (�n · � ∇)�ei + (�ei · � ∇)�n + �n × rot �ei + �ei × rot �n
= (�n · � ∇)�ei + �n × rot �ei
(E.50)
rot(�a × � b) = � ∇ × (�a × � b) = ( � b · � ∇)�a − (�a · � ∇) � b + �a( � ∇ � b) − � b( � ∇�a) (E.51)
�a × rot � b = �a × ( � ∇ × � b) = � ∇(�a � b) − (�a · � ∇) � b (E.52)
�a · ( �b × �c) = �b · (�c × �a) = �c · (�a × �b) = det(�a � �
�
�
b�c) = �
�
�
a1 a2 a3
b1 b2 b3
c1 c2 c3
�
�
�
�
�
�
(E.53)

142 E. TENSOR CALCULUS
linear transformation
1 st -rank tensor
2 nd -rank tensor
unity tensor
tensor addition
tensor product
E.3 Tensors
E.3.1 Definitions
The Linear transformation of the base can be written as
where the connection is given by the relation
�g i = a k i �gk (E.54)
�g i = a i k �gk , (E.55)
a k i a j
k = δj
i , (E.56)
i.e. ak i und a i k are inverse matrices.
Tensor of tensor rank one (vector): the co- and contravariant components satisfies
the transformation
ti = a j
i tj
t i = a i j t j
(E.57)
(E.58)
Tensor of tensor rank two (matrix with appropriate transformation behaviour):
the co- and contravariant components satisfies the transformation
tij = a k i al j tkl (E.59)
t ij = a i k
a j
l tkl
In general following relations apply to tensors of tensor rank two
A special case is the unity tensor
(E.60)
t = tij �ei ⊗ �ej
(E.61)
�aT = ai tik �ek (E.62)
t � b = tij bj �ei
E.3.2 Operations and Operators
The tensor addition can be written as
(E.63)
tij = �ei t �ej (E.64)
δ = δij �ei ⊗ �ej = �ei ⊗ �ei . (E.65)
r i jk + si jk = ti jk
The tensor product (dyadic product) can be written as
r ij s k l
= tijk
l
(E.66)
(E.67)

E.3. Tensors 143
The tensor contraction 7 (special case: trace 8 ) can be written as tensor contraction
r ij
j
= ti
(E.68)
Tensor contraction of a tensor product 9
r ij sjk = t i k
(E.69)
Correlation of vectors
This operation is a tensor contraction of a tensor product frequently used in physics. vector correlation
Index Raising
Index Lowering
b i = t ij aj
bi = tij a j
(E.70)
(E.71)
The transformation behaviour of t ij and tij ensures the independence of the correlation
from the coordinate system.
Scalar Product (aka Double Dot Product) scalar product
A : B = AijBji
(E.72)
Divergence of a tensor
covariant derivation
tensor divergence
T ij ij i
; k = T , k + Γlk T lj + Γ j il
lk T (E.73)
for a symmetric tensor
S ij
i
; j = Sij
, j + Γkj Skj + Γ j
kj Sik = g −1 2(g 1
2S ij ), j + Γ i jkSjk for an antisymmetric tensor
A ij
; j = g−12(g
1
2A ij ), j
(E.74)
(E.75)
The physical components of a tensor are defined as physical components
T(i,j) = h (i)h (j)T (i)(j)
T(i,j) =
1
h (i)h (j)
where h (i) are the scale factors (cf. Section E.2.2).
7 in german “Verjüngung”
8 in german “Spur”
9 in german “Überschiebung”
(E.76)
T (i)(j) , (E.77)

144 E. TENSOR CALCULUS
E.3.3 Relations / Tensor Identities
Tij = ai bj → Tii = ai bi (E.78)
Aijkl = Bij Ckl → Bij Cil,
Bij Ckj,
Bij Cki,
Bij Cjl
� �� �
=Aijil
� �� �
=Aijkj
� �� �
=Aijki
� �� �
=Aijjl
A : � ∇�a = A11( � ∇�a)11 + A21( � ∇�a)21 + A31( � ∇�a)31 +
A12( � ∇�a)12 + A22( � ∇�a)22 + A32( � ∇�a)32 +
A13( � ∇�a)13 + A23( � ∇�a)23 + A33( � ∇�a)33
(E.79)
(E.80)

E.4. Metric and Symmetries 145
E.4 Metric and Symmetries
E.4.1 Metric
In general the metric is defined as a nonnegative function g(x,y) describing the
“distance” between neighbouring points. At first we write the cartesian coordinates
as a function of curvilinear coordinates curvilinear
coordinates
x i = f i (ξ k ) . (E.81)
The infinitesimal line elements (or differential displacements) transforms as follows line element
where
dx i = a i k dξk , (E.82)
a i k
∂fi
= . (E.83)
∂ξk The metric tensor is determined by metric tensor
gkl(ξ) = glk(ξ) = �
i
a i k ai l
and the inverse metric tensor can be calculated by the rule
gki(ξ)g il (ξ) = δ l k
(E.84)
, (E.85)
where δ l k is the Kronecker delta. Very roughly, the metric tensor gij is a function
which tells how to compute the distance between any two points in a given space.
The quadratic line element 10 follows from the generalised Pythagorean theorem quadratic
line element
ds 2 = �
i
a i k dξk a i l dξl = gkl(ξ)dξ k dξ l . (E.86)
If we assume an orthogonal coordinate system in three-space (R 3 ), the metric is
diagonal and the quadratic line element can be written
where hi is the scale factor.
ds 2 = (h1dx (1) ) 2 + (h2dx (2) ) 2 + (h3dx (3) ) 2 , (E.87)
The equation of geodetics (like an equation of motion) of a free particle reads as equation of
geodetics
follows
ü i + Γ i kl ˙uk ˙u l = 0 , (E.88)
where Γ i kl
denotes the so-called Christoffel symbols (or connection coefficients) of Christoffel
symbols
the first kind, which describe the occurrence of fictitious forces (or pseudo forces),
e.g. the Coriolis force. The coefficients are defined as
Γ i kl
10 in german “Abstandsquadrat”
:= 1
2 gim (gkm,l + glm,k − gkl,m) . (E.89)

146 E. TENSOR CALCULUS
divergence
symmetrised
covariant tensor
The divergence is the covariant derivation of the contravariant component, i.e.
div(u) = u k ;k = uk ,k + Γk ki ui
(E.90)
and accordingly the operator of the divergence can be written as
�∇ = �
1/h 2 i�ei∂i , (E.91)
with the base {�e1,�e2,�e3}. The symmetrised covariant tensor is
where
i
u (k;l) = 1
2 (uk;l + ul;k) , (E.92)
uk;l = uk,l − Γ i lk ui . (E.93)

E.4. Metric and Symmetries 147
E.4.2 Coordinate Systems
Cartesian Coordinates
Orthogonal coordinate system:
Scale factors:
Radius vector:
(ξ1,ξ2,ξ3) = (x,y,z) (E.94)
(h1,h2,h3) = (1,1,1) (E.95)
⎛
�r = ⎝
Local tetrad with spherical coordinates:
Operators:
x
y
z
⎞
⎠ (E.96)
(x,y,z,Θ,Φ) (E.97)
�n � ∇ξ1 = �n � ∇x = sin Θ cos Φ (E.98)
�n � ∇ξ2 = �n � ∇y = sin Θ sin Φ (E.99)
�n � ∇ξ3 = �n � ∇z = cos Θ (E.100)
�n � ∇Θ = 0 (E.101)
�n � ∇Φ = 0 (E.102)
�n � ∇ = sin Θ cos Φ ∂ ∂ ∂
+ sin Θ sinΦ + cos Θ
∂x ∂y ∂z
(E.103)

148 E. TENSOR CALCULUS
Cylindrical Coordinates
Orthogonal coordinate system:
Scale factors:
Radius vector:
(ξ1,ξ2,ξ3) = (ρ,φ,z) (E.104)
(h1,h2,h3) = (1,ρ,1) (E.105)
⎛
�r = ⎝
Local tetrad with spherical coordinates:
Operators:
ρcos φ
ρsin φ
z
⎞
⎠ (E.106)
(ρ,φ,z,Θ,Φ) (E.107)
�n � ∇ξ1 = �n � ∇ρ = sin Θ cos Φ (E.108)
�n � ∇ξ2 = �n � ∇φ = 1
sin Θ sin Φ
ρ
(E.109)
�n � ∇ξ3 = �n � ∇z = cos Θ (E.110)
�n � ∇Θ = 0 (E.111)
�n � ∇Φ = − 1
sin Θ sinΦ
ρ
(E.112)
�n � ∇ = sin Θ cos Φ ∂ 1 ∂ ∂ 1 ∂
+ sinΘsin Φ + cos Θ − sinΘsin Φ
∂ρ ρ ∂φ ∂z ρ ∂Φ
(E.113)
( � ∇ · �v)11 = 0 (E.114)
( � ∇ · �v)22 = 0 (E.115)
( � ∇ · �v)33 = ∂u
∂z
(E.116)
(E.117)

E.4. Metric and Symmetries 149
Spherical Coordinates
In the case of sperical coordinates (r,θ,φ,Θ,Φ) we get:
Orthogonal coordinate system:
Scale factors:
Ricci rotation coefficients:
Metric:
Radius Vector:
(ξ1,ξ2,ξ3) = (θ,φ,r) (E.118)
(h1,h2,h3) = (r,r sin Θ,1) (E.119)
Γ 1 22 = − sinθ cos θ (E.120)
(E.121)
r
Γ 2 12 = Γ 2 21 = cot θ (E.122)
Γ 1 31 = Γ 1 13 = 1
r
(E.123)
Γ 3 11 = −r (E.124)
Γ 2 32 = Γ 2 23 = 1
Γ 3 22 = −r sin 2 θ (E.125)
g 1
2 = r 2 sinΘ (E.126)
g −1
2 =
⎛
�r = r ⎝
Local tetrad with spherical coordinates:
Operators:
�n � ∇ = 1 ∂
sin Θ cos Φ
r ∂θ
1
r 2 sin θ
sin θ cos φ
sin θ sin φ
cos θ
⎞
(E.127)
⎠ (E.128)
(θ,φ,r,Θ,Φ) (E.129)
�n � ∇ξ1 = �n � ∇θ = 1
sinΘcos Φ
r
(E.130)
�n � ∇ξ2 = �n � sin Θ sinΦ
∇φ =
r sin θ
(E.131)
�n � ∇ξ3 = �n � ∇r = cos Θ (E.132)
�n � ∇Θ = − 1
sin Θ
r
(E.133)
�n � sin Θ sinΦ
∇Φ = −
r tan θ
(E.134)
+ sin Θ sinΦ
r sin θ
∂
∂φ
∂ 1 ∂
+ cos Θ − sin Θ
∂r r ∂Θ
sin Θ sinΦ ∂
−
r tan θ ∂Φ
(E.135)

150 E. TENSOR CALCULUS
Spherical symmetry:
Thus
∂ ∂ ∂
≡ ≡ ≡ 0 (E.136)
∂θ ∂φ ∂Φ
�n � ∇ = cos Θ ∂
∂r
Introducing µ = cos θ
�n � ∇ = µ ∂
∂r
Plane parallel geometry, i.e. r → ∞
− sin Θ
r
+ 1 − µ2
r
�n � ∇ = µ ∂
∂r
∂
∂Θ
∂
∂Θ
Components of the divergence of velocity �v = (0,0,u):
where ∂h1
∂r
( � ∇ · �v)11 = 1 ∂v
r
1
v2
+
∂θ r2 sin θ
( � ∇ · �v)21 = 1
r sin θ
∂v 1
∂φ
∂h1
∂φ
+ v3
r
v2
−
r2 ∂h2
sin θ ∂θ
∂h1
∂r
( � ∇ · �v)31 = ∂v1 v3 ∂h3
−
∂r r ∂θ
( � ∇ · �v)12 = 1 ∂v
r
2
v1
−
∂θ r2 ∂h1
sin θ ∂φ
( � ∇ · �v)22 = 1 ∂v
r sin θ
2
v1
+
∂φ r2 ∂h2 v3 ∂h2
+
sin θ ∂θ r sinθ ∂r
( � ∇ · �v)32 = ∂v2 v3 ∂h3
−
∂r r sin θ ∂φ
( � ∇ · �v)13 = 1 ∂v
r
3
v1 ∂h1
−
∂θ r ∂r
( � ∇ · �v)23 = 1 ∂v
r sin θ
3
v2 ∂h2
−
∂φ r sinθ ∂r
( � ∇ · �v)33 = ∂v3
∂r
∂h2 = 1, ∂r
⎛
( � ⎜
∇ · �v) = ⎜
⎝
⎛
⎜
⎝
+ v1
r
= sin θ, ∂h2
∂θ
1
r∂θv 1 + v3
r
∂h3
∂θ
v2 ∂h3
+
r sin θ ∂φ
= u
r
(E.137)
(E.138)
(E.139)
(E.140)
= 0 (E.141)
= 0 (E.142)
= 0 (E.143)
= u
r
(E.144)
= 0 (E.145)
= 1 ∂u
r ∂θ
= 1
r sin θ
= ∂u
∂r
= r cos θ, and all others are zero. Or
∂u
∂φ
1
r sinθ ∂φv1 − v2
r2 sin θr cos θ ∂rv1 1
r∂θv 2 1
r sin θ∂φv2 + v3
v1
r sin θ sinθ + r2 sinθ r cos θ ∂rv2 1
r∂θv 3 − v1
r
u
r 0 0
0 u
r 0
1
r∂θu 1
r sin θ∂φu ∂ru
1
r sinθ ∂φv3 − v2
r sinθ sin θ ∂rv3 ⎞
⎟
⎠
⎞
(E.146)
(E.147)
(E.148)
(E.149)
⎟ =
⎟
⎠
(E.150)

E.4. Metric and Symmetries 151
Thus
The unit vector �n is
⎛
�n(Θ,Φ) = ⎝
nθ
nφ
nr
�∇ · �v = 1 ∂u 1 ∂u 1
+ +
r ∂θ r sinθ ∂φ r2 ∂(r2u) ∂r
⎞
⎛
⎠ = ⎝
sin Θ cos Φ
sin Θ sin Φ
cos Θ
⎞
⎛
⎠ = ⎝
were µ = cos Θ, the spatial gradient vector operator
�∇
1
= �er∂r + �eθ
r ∂θ
1
+ �eφ
r sin θ ∂φ
the derivation
d�n dΘ
= �eΘ
ds ds + �eΦ sin θ dΦ
ds
and the ,,directional” gradient vector operator
Or
�∇ = 1
�e1∂ξ1
h1
(1 − µ 2 ) 1/2 cos Φ
(1 − µ 2 ) 1/2 sin Φ
µ
⎞
(E.151)
⎠ (E.152)
(E.153)
(E.154)
1
�∇n = �eΘ∂Θ + �eΦ
sin θ ∂Φ . (E.155)
1 1 1
+ �e2∂ξ2 + �e3∂ξ3 =
h2 h3 r �eθ∂θ + 1
r sinθ �eφ∂φ + �er∂r
(E.156)
�∇ ·�a = 1
r sin θ ∂θ(sin θa 1 ) + 1
r sinθ ∂φa 2 + 1
r 2∂r(r 2 a 3 ) (E.157)
Divergence of a tensor (radial term):
T 11 = 1
r 2Tθθ
T 12 = 1
r 2 sinθ Tθφ
T 22 = 1
(r sinθ) 2Tφφ
T 31 = 1
r Trθ
T 32 = 1
r sin θ Trφ
T 33 = Trr
T 3j 3j
; j = T , j + Γ3lj T lj + Γ j
lkT 3l =
T 31 ,1 + T 32 ,2 + T 33 ,3 + Γ311 T 11 + Γ 3 22T 22 + Γ 1 31T 33 + Γ 2 32T 33 + Γ 2 12T 31 =
1
r ∂θTrθ + 1
r sinθ ∂φTrφ + ∂rTrr − 1
r Tθθ − 1
r Tφφ + 1
r Trr + 1
r Trr
cot θ
+
r Trθ =
1
r2∂r(r 2 Trr) + 1
r sin θ ∂θ(sin θ Trθ) + 1
r sinθ ∂φTrφ − 1
r (Tθθ + Tφφ) =
1
r2∂r(r 2 Trr) − 1
r (Tθθ + Tφφ) (E.158)
T 1j
; j =
1
r2 sin θ ∂θ(sin θ Tθθ) − 1
cot Tφφ
(E.159)
r
T 2j
; j =
1
(r sin θ) 2∂φTφφ
(E.160)

152 E. TENSOR CALCULUS

Appendix F
Full Set of RHD Equations
The radiation hydrodynamics with dust are calculated with a set of twelve equations
# Equation Name Parameter Symbol
1 Grid Equation radius r
2 Mass Equation 1 integrated mass m
3 Continuity Equation density ρ
4 Momentum Equation 2 velocity u
5 Energy Equation internal energy e
6 Radiation Energy Equation radiation energy J
7 Radiation Flux Equation radiation flux H
8 0 th Moment Dust Equation 0 th moment, dust density K0
9 1 st Moment Dust Equation 1 st moment, mean radius of grains K1
10 2 nd Moment Dust Equation 2 nd moment K2
11 3 rd Moment Dust Equation 3 rd moment K3
12 nc Equation amount density Kn
Table F.1: List of RHD equations
In the following sections the equations of the radiation hydrodynamics code (in
spherical symmetry) are given in differential form, integrated form and discretised
form according to the discretisation form in Appendix A.
1 also called Poisson Equation
2 also called Equation of Motion
153

154 F. FULL SET OF RHD EQUATIONS
F.1 Differential Form
Continuity Equation
∂
ρ + ∇ · (ρu) = 0 (F.1)
∂t
Momentum Equation (Equation of Motion)
Energy Equation
∂
4π
(ρu) + ∇ · (ρu u) = −∇P − ρ∇ψ +
∂t c ρ κHH + fdrag
(F.2)
∂
∂t (ρe) + ∇ · (ρe u) = −P ∇ · u + 4π ρ (κJJ − κSSg) (F.3)
Radiation Energy Equation
1 ∂ 1
1 u 3K − J
J + ∇·(Ju) = −∇·H − K∇·u+ −ρ (κJJ −κSSg)−(χJJ −χSSd)
c ∂t c c c r
(F.4)
Radiation Flux Equation
1 ∂ 1 ∂ 3K − J
H + ∇ · (Hu) = − K −
c ∂t c ∂r r
0 th Moment Dust Equation
1 st - 3 rd Moment Dust Equation
∂
∂t Kj + ∇ · (Kj u) = j
d
− 1
c H∇ · u − ρ κHH − χHH (F.5)
∂
∂t K0 + ∇ · (K0 u) = J (F.6)
1
τ Kj−1 + N j/d
ℓ J (1 ≤ j ≤ d) (F.7)

F.2. Integrated Form 155
F.2 Integrated Form
Continuity Equation
�
∂
∂t
V
�
ρ dV +
Momentum Equation (Equation of Motion)
� �
�
∂
ρu dV + ρu u dA = −
∂t
V ∂V
V
�
−
Energy Equation
� �
∂
ρe dV +
∂t
V
∂V
Radiation Energy Equation
�
1 ∂
J dV +
c ∂t
1
�
c
V
∂V
�
ρe u dA = −
∂V
ρ u dA = 0 (F.8)
V
�
+4π
V
∇P dV
ρ∇ψ dV
+ 4π
�
ρ κHH dV
c
�
V
+ fdrag dV (F.9)
V
P ∇ · u dV
V
�
J u dA = −
V
ρ (κJJ − κSSg) dV (F.10)
∇ · H dV
− 1
�
K ∇ · u dV
c
V
+ 1
�
c
V
�
3K − J
u dV
r
− ρ (κJJ − κSS) dV (F.11)
Radiation Momentum Equation
�
1 ∂
H dV +
c ∂t
V
1
�
�
H u dA = −
c
∂
K dV
∂r
∂V
V
− 1
�
H ∇ · u dV
c
V
V

156 F. FULL SET OF RHD EQUATIONS
0th Moment Dust Equation
� �
∂
K0 dV +
∂t
V
∂V
1 st - 3 rd Moment Dust Equations
∂
∂t
�
V
�
Kj dV +
∂V
�
−
V
V
�
K0 u dA =
Kj u dA = j
�
d
V
�
+
V
3K − J
r
dV
�
− (κHρ + χH)H dV (F.12)
V
1
τ Kj−1 dV
N j/d
ℓ
J dV (F.13)
J dV (1 ≤ j ≤ d) (F.14)

F.3. Discretised Form 157
F.3 Discretised Form
Continuity Equation [g]
δ(ρl ∆Vl) + δt ∆(r 2 l � ρ ad
l urel
l ) = 0 (F.15)
Momentum Equation (Equation of Motion) [dyn s]
δ(ul ρl∆Vl) + ∆(− � u ad
l δml) = − δt r 2 l ∆Pl
Energy Equation [erg]
−δt 4πG ml
r 2 l
ρl∆Vl
+δt 4π
c (κg
l + κd l ) Hl ρl∆Vl
δ(el ρl ∆Vl) + ∆(− � e ad
l δml) = − δt Pl ∆(r 2 l ul)
erg cm
Radiation Energy Equation [ s ]
δ(Jl ∆Vl) + ∆(−
erg cm
Radiation Momentum Equation [ s ]
δ(Hl ∆Vl) + δt ∆(r 2 l
0 th Moment Dust Equation [1]
δ([K0]l ∆Vl) + ∆(−
+δt QMU (F.16)
+δt 4π κ g
l ρl (Jl − [Sg]l) ∆Vl
+δt QME (F.17)
�� �ad J
δml) = − c δt ∆(r
ρ l
2 l Hl)
−δt Kl ∆(r 2 l ul)
+δt (3Kl − Jl) ul
∆Vl
rl
−c δt κ g
l ρl (Jl − [Sg]l) ∆Vl
�
H ad
l urel
l ) = − c δt r 2 l ∆Kl
��� ad
K0
ρ
l
−δt Hl r 2 l ∆ul
−c δt 3Kl − Jl
rl
∆Vl
−c δt (κ g
l + κd l ) Hl ρl∆Vl
δml) = δt Jl ∆Vl
(F.18)
(F.19)
(F.20)

158 F. FULL SET OF RHD EQUATIONS
1 st - 3 rd Moment Dust Equations [1]
δ([Kj]l ∆Vl) + ∆(−
Dust Density Equation [1]
��� ad
Kj
ρ
l
δml) = δt j
d
� �ad nc
l
1
τl
[Kj−1]l ∆Vl
+δt N j/d
ℓ Jl ∆Vl (1 ≤ j ≤ d) (F.21)
�
δ([nc]l ∆Vl) − ∆(− δml) = δt
ρ
1
[K2]l ∆Vl
Mass Equation (Poisson Equation) [g]
Artificial Viscosity Contributions
where
τl
+δt Nℓ Jl ∆Vl
(F.22)
∆ml − ρl ∆Vl = 0 (F.23)
QME = − 2
3 µQ,l ρl
�
∆ul
−
∆rl
ūl
�2 ¯rl
∆Vl [ erg
]
s
(F.24)
QMU = − 2
� �
∆ul
∆ µQ,l ρl −
3rl ∆rl
ūl
��
¯rl
[dyn] (F.25)
µQ,l = ℓ1 cs,l − ℓ 2 2 min
�
∆(r2 �
l ul)
,0
∆Vl
[ cm2
] (F.26)
s

Appendix G
Symbols, Constants and
Abbreviations
G.1 Symbols
Symbol Units Comment
cgs SI
mr g kg integrated mass
ρ g cm −3 kg m −3 gas density
e erg g −1 J kg −1 specific internal energy of the gas
�v, u cm s −1 m s −1 gas velocity
J erg cm −2 s −1 W m −2 zeroth moment of the radiation field
E erg cm −3 J m 3 radiation energy density
�H, H erg cm −2 s −1 W m −2 first moment of the radiation field
�F, F erg cm −2 s −1 W m −2 radiation energy flux
K, K erg cm −2 s −1 W m −2 second order moment of the radiation field
P, P erg cm −3 J m 3 radiation pressure
Kj cm −3 m −3 moments of the grain size distribution
P dyn cm−2 N m−2 gas pressure
Tg K K gas temperature
Tr K K radiation temperature
Td K K dust temperature
Sg erg cm−2 s−1 W m−2 source function of the gas
Sd erg cm−2 s−1 W m−2 source function of the dust
κg cm2 g−1 m2 kg−1 mass absorption coefficient of the gas
κd cm2 g−1 m2 kg−1 mass absorption coefficient of the gas
Eddington factor
fedd
1
τ s−1 s−1 J s
net growth rate of the dust grains
−1 cm−3 s−1 m−3 net grain formation rate per volume
ngr cm−3 m−3 number density of dust grains
rgr cm m grain radius
fcond
degree of condensation
159

160 G. SYMBOLS, CONSTANTS AND ABBREVIATIONS
G.2 Fundamental Physical Constants
c 2.997 924 58 10 10 cm s −1 speed of light in vacuum (exact)
2.997 924 58 10 8 m s −1
G 6.6742(10) 10 −8 dyn cm 2 g −2 Newtonian constant of gravitation
6.6742(10) 10 −11 m 3 kg −1 s −2
σ 5.670 400(40) 10 −5 erg cm −2 s −1 K −4 Stefan-Boltzmann constant
5.670 400(40) 10 −8 W m −2 K −4
k 1.380 6505(24) 10 −16 erg K −1 Boltzmann constant
1.380 6505(24) 10 −23 J K −1
R 8.314 472(15) 10 7 erg mol −1 K −1 molar gas constant
8.314 472(15)J mol −1 K −1
mP 1.672 621 71(29) 10 −24 g proton mass
1.672 621 71(29) 10 −27 kg
G.3 Astronomical Constants
M⊙ 1.989 10 33 g solar mass
1.989 10 30 kg
L⊙ 3.826 10 33 erg s −1 solar luminosity
3.826 10 26 W
R⊙ 6.9598 10 10 cm solar radius
6.9598 10 8 m
AU 1.49598 10 13 cm astronomical unit
1.49598 10 11 m
ly 9.46053 10 17 cm light year
9.46053 10 15 m
pc 3.08568 10 18 cm parsec
3.08568 10 16 m
MEarth 5.79 10 27 g mass of Earth
5.79 10 24 kg
REarth 6.378 137 10 8 cm equatorial radius of the Earth (IUGG value)
6.378 137 10 6 m

G.4. Abbreviations 161
G.4 Abbreviations
ACS Advanced Camera for Surveys
AFGL Air Force Geophysical Laboratory
AGB Asymptotic Giant Branch
AU Astronomical Unit
AURA Association of Universities for Research
in Astronomy
CRL Cambridge Research Laboratory
CSE CircumStellar Envelope
EOS Equation Of State
ESA European Space Agency
ESO European Southern Observatory
FLIER Fast Low-Ionization Emission Region
FOC Faint Object Camera
Hb Hubble catalog
HD H. Draper catalog
HEIC Hubble European Space Agency
Information Centre
HR Harvard Revised catalog
HRD Hertzsprung-Russell Diagram
HST Hubble Space Telescope
IAC Instituto de Astrofísica de Canarias
IAU International Astronomical Union
IC Index Catalog
IR InfraRed
IRC InfraRed Catalog
IRAS InfraRed Astronomical Satellite
ISM InterStellar Medium
ISS International Space Station
ISO Infrared Space Observatory
IUGG International Union of Geodesy and
Geophysics
K Kohoutek catalog
LBV Luminous Blue Variables
LDEF Long Duration Exposure Facility
LPV Long Period Variable
LTE Local Thermodynamical Equilibrium
M Messier catalog
MHD Magneto-HydroDynamic
MPAC Micro-PArticles Capturer
MS Main Sequence
MyCn Mayall+Cannon catalog
Mz Menzel catalog
NASA National Aeronautics and Space
Administration
NASDA National Space Development Agency
NGC New General Catalog
NICMOS Near Infrared Camera and Multi-Object
Spectrometer
NOAO National Optical Astronomy Observatory
NOT Nordic Optical Telescope
NRAO National Radio Astronomy Observatory
OH OH source
PG Palomar-Green catalog
PN Planetary Nebula
PPN Proto-Planetary Nebula
PRC Public Resources Code
RGB Red Giant Branch
RHD Radiation HydroDynamic
RTE Radiation Transfer Equation
SIRTF Space InfraRed Telescope Facility
SST Spitzer Space Telescope
ST-ECF Space Telescope - European Coordinating
Facility
STScI Space Telescope Science Institute
UMIST University of Manchester Institute of
Science and Technology
VLTI Very Large Telescope Interferometer
WD White Dwarf
WFC Wide Field Channel
WFPC Wide Field Planetary Camera
Wr Wray catalog
XMM X-ray Multi-mirror Mission

162 G. SYMBOLS, CONSTANTS AND ABBREVIATIONS

List of Tables
4.1 Stars with spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.1 Model stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Model flux tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Effects of chemistry (1) . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Effects of chemistry (2) . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 Mass loss rate ˙ M through a specific area A for a spherical model . . 93
5.6 Mass loss rate ˙ M through different flux tube configurations . . . . . 93
C.1 Derivatives in the spherical, cylindrical and flux tube geometries. . . 124
D.1 Chemical composition . . . . . . . . . . . . . . . . . . . . . . . . . . 128
E.1 Definitions of some vector symbols . . . . . . . . . . . . . . . . . . . 137
E.2 Summary of some specific vector operators . . . . . . . . . . . . . . . 139
F.1 List of RHD equations . . . . . . . . . . . . . . . . . . . . . . . . . . 153
163

164 LIST OF TABLES

List of Figures
1.1 Evolutionary tracks in the Hertzsprung-Russell-Diagram . . . . . . . 6
2.1 Halo of PPN NGC 7027 . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Details of PPN NGC 7027 . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Halo of PPN CRL 2688 (Egg Nebula) . . . . . . . . . . . . . . . . . 16
2.4 Infrared-details of PPN CRL 2688 . . . . . . . . . . . . . . . . . . . 16
2.5 The PPN HD 44179 (Red Rectangle Nebula) . . . . . . . . . . . . . 17
2.6 The PPN OH231.8+4.2 (Rotten Egg Nebula) . . . . . . . . . . . . . 17
2.7 Halo of the PN NGC 6720 (Ring Nebula) . . . . . . . . . . . . . . . 18
2.8 The PN NGC 6720 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.9 Details in the PN NGC 6720 . . . . . . . . . . . . . . . . . . . . . . 18
2.10 Halo of PN NGC 6720 in Infrared . . . . . . . . . . . . . . . . . . . . 19
2.11 Halo of PN NGC 7293 (Helix Nebula) . . . . . . . . . . . . . . . . . 20
2.12 Details of PN NGC 7293 . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.13 Halo of PN NGC 6853 (Dumbbell Nebula) . . . . . . . . . . . . . . . 21
2.14 Details of PN NGC 6853 . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.15 The PN NGC 2392 (Eskimo Nebula) . . . . . . . . . . . . . . . . . . 22
2.16 Detail of PN NGC 2392 . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.17 The PN NGC 6369 (Little Ghost Nebula) . . . . . . . . . . . . . . . 23
2.18 The PN NGC 3132 (Eight-Burst Nebula) . . . . . . . . . . . . . . . 23
2.19 The PN IC 418 (Spirograph Nebula) . . . . . . . . . . . . . . . . . . 24
2.20 The PN NGC 6751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.21 Halo of the PN NGC 6543 (Cat’s Eye Nebula) . . . . . . . . . . . . 25
2.22 The PN NGC 6543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.23 The PN MyCn 18 (Hourglass Nebula) . . . . . . . . . . . . . . . . . 26
2.24 The PN IC 4406 (Retina Nebula) . . . . . . . . . . . . . . . . . . . . 26
2.25 The PN NGC 6302 (Bug or Butterfly Nebula) . . . . . . . . . . . . . 27
2.26 Details of the PN NGC 6302 . . . . . . . . . . . . . . . . . . . . . . 27
2.27 The PN Mz 3 (Ant Nebula) . . . . . . . . . . . . . . . . . . . . . . . 28
2.28 The PN M2-9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
165

166 LIST OF FIGURES
3.1 Computational domain of the initial model . . . . . . . . . . . . . . 47
3.2 Flowchart of the RHD code . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 Model of a flux tube on a stellar surface . . . . . . . . . . . . . . . . 64
4.2 Radiative flux through a flux tube . . . . . . . . . . . . . . . . . . . 67
4.3 Volume of a flux tube segment . . . . . . . . . . . . . . . . . . . . . 68
5.1 Density distribution from the initial model program . . . . . . . . . 74
5.2 Temperature distribution from the initial model program . . . . . . 75
5.3 Radiation flux distribution from the initial model program . . . . . . 75
5.4 Model flux tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5 Density distribution from the initial model program . . . . . . . . . 77
5.6 Temperature distribution from the initial model program . . . . . . 78
5.7 Radiation flux distribution from the initial model program . . . . . . 78
5.8 Spatial structure of the stationary wind solution in spherical geometry 80
5.9 Spatial structure of a stellar wind generated by a dust-induced κmechanism
in spherical geometry . . . . . . . . . . . . . . . . . . . . 80
5.10 Spatial structure of a stationary wind in flux tube geometry . . . . . 83
5.11 Spatial structure of a stationary wind in flux tube geometry . . . . . 83
5.12 Spatial structure of a wind from dust-induced κ-mechanism in flux
tube geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.13 Spatial structure of a wind from dust-induced κ-mechanism in flux
tube geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.14 Occurrence of a dust-driven wind for flux tube B . . . . . . . . . . . 86
5.15 Occurrence of a dust-driven wind for flux tube D . . . . . . . . . . . 86
5.16 Occurrence of a dust-driven wind at ∆T = 100 K . . . . . . . . . . . 87
5.17 Occurrence of a dust-driven Wind at ∆T = 300 K . . . . . . . . . . 87
5.18 Magnetic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.19 Dependence of temperature difference ∆T for the Sun . . . . . . . . 89
5.20 Dependence of temperature difference ∆T for an AGB star . . . . . 90
5.21 Region of heating and cooling of the flux tube . . . . . . . . . . . . . 91
5.22 Degree of condensation . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.1 Model of AGB star with a cool spot . . . . . . . . . . . . . . . . . . 96
A.1 Description of the numerical grid for RHD calculations . . . . . . . . 105

Image Credits
Fig. 2.1 on page 15: STScI-PRC1996-05 released on January 16, 1996, credit by H.
Bond (STScI) and NASA
Fig. 2.2 on page 15: STScI-PRC1998-11a released on March 12, 1998, credit by W.
B. Latter (SIRTF Science Center/Caltech) and NASA
Fig. 2.3 on page 16: STScI-PRC1996-3 released on January 16, 1996, credit by R.
Sahai and J. Trauger (JPL), the WFPC2 science team, and NASA
Fig. 2.4 on page 16: STSci-PRC1997-11 released on May 12, 1997, credit by R.
Thompson (U. Arizona) et al., NICMOS Instrument Definition Team and NASA
Fig. 2.5 on page 17: STScI-PRC2004-11 released in 2004, credit by NASA/ESA, H.
Van Winckel (Catholic University of Leuven, Belgium) and M. Cohen (University
of California, Berkeley
Fig. 2.6 on page 17: credit by NASA/ESA & Valentin Bujarrabal (Observatorio
Astronomico Nacional, Spain)
Fig. 2.7 on page 18: credit by Subaru Telescope, National Astronomical Observatory
of Japan
Fig. 2.8 on page 18: Hubble Heritage PRC99-01, credit by Hubble Heritage Team
Fig. 2.9 on page 18: same as Fig. 2.8
Fig. 2.10 on page 19: SST Release SSC2005-07a, credit by NASA/JPL-Caltech/J.
Hora (Harvard-Smithsonian CfA)
Fig. 2.11 on page 20: released in 2003 (STScI-PRC2003-11a), credit by: NASA,
NOAO, ESA, the Hubble Helix Nebula Team, M. Meixner (STScI), and T.A. Rector
(NRAO)
Fig. 2.12 on page 20: same as Fig. 2.11
Fig. 2.13 on page 21: credit by Robert Gendler
Fig. 2.14 on page 21: released in 2003 (STScI-PRC2003-06), credit by NASA and
the Hubble Heritage Team (STScI/AURA)
Fig. 2.15 on page 22: released on January 24, 2000 (STScI-PRC2000-07), credits
by NASA, Andrew Fruchter and the ERO Team [Sylvia Baggett (STScI), Richard
Hook (ST-ECF), Zoltan Levay (STScI)]
Fig. 2.16 on page 22: same as Fig. 2.15
Fig. 2.17 on page 23: released in 2002 (STScI-PRC2002-25), credits by NASA and
The Hubble Heritage Team (STScI/AURA)
Fig. 2.18 on page 23: released on November 5, 1998 (STScI-PRC1998-39), credits
by Hubble Heritage Team (STScI/AURA/NASA)

168 IMAGE CREDITS
Fig. 2.19 on page 24: released in 2000 (STScI-PRC2000-28), credits by NASA and
The Hubble Heritage Team (STScI/AURA)
Fig. 2.20 on page 24: released in 2000 (STScI-PRC2000-12), credits by NASA, The
Hubble Heritage Team (STScI/AURA)
Fig. 2.21 on page 25: credit by Nordic Optical Telescope (NOT) and R. Corradi
(Isaac Newton Group of Telescopes, Spain)
Fig. 2.22 on page 25: released in 2004 (STScI-PRC2004-27), credits by NASA,
ESA, HEIC, and The Hubble Heritage Team (STScI/AURA)
Fig. 2.23 on page 26: released on January 16, 1996 (STScI-PRC1996-07), credits by
Raghvendra Sahai and John Trauger (JPL), the WFPC2 science team, and NASA
Fig. 2.24 on page 26: released in 2002 (STScI-PRC2002-14), credit by C. R. O’Dell
(Vanderbilt University) et al., Hubble Heritage Team, NASA
Fig. 2.25 on page 27: credit by Wendel and Flach-Wilken
Fig. 2.26 on page 27: released on May 3, 2004 (STScI-PRC2004-46), credits by
NASA, ESA and A. Zijlstra (UMIST, Manchester, UK)
Fig. 2.27 on page 28: released in 2001 (STScI-PRC2001-05), credits by NASA, ESA
and The Hubble Heritage Team (STScI/AURA)
Fig. 2.28 on page 28: released in 1997 (STScI-PRC1997-38), credits by B. Balick
(University of Washington), V. Icke (Leiden University, The Netherlands), G. Mellema
(Stockholm University), and NASA

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