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 f¨ur Astronomie T¨urkenschanzstraße 17 A-1180 Wien, Osterreich¨

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 in- evitable, 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 dis- cussions, 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 ii 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 Hüllen- masse verlieren. Idealerweise ist dieser Massenverlust sphärisch symmetrisch, wenn die physikalischen Größen an der Sternoberfläche (z.B. Temperatur) und im umge- benden 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 beziehungs- weise Effektivtemperatur an der Sternphotosphäre, welche sich auf die Kondensation des Staubs und daraus folgend auf die Massenverlustrate auswirkt. Der Staubent- stehungsprozess 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 Ober- fläche von AGB-Sternen vorstellbar. Beobachtungen diesbezüglich sind wegen der hohen Staubproduktion in den AGB-Atmosphären nur schwer zu machen. Ver- schiedene Modellrechnungen und theoretische Uberlegungen¨ unterstützen jedoch diese Theorie. In dieser Arbeit wird ein Modell vorgestellt, das zur Untersuchung des Massenver- lustes über diskreten kühlen Flecken dient. Dazu kam eine strahlungshydrodynam- ische 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 Stern- oberfläche 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 Inho- mogenität im Massenverlust verursachen können. Allerdings müssen die beobachte- ten 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 TheCycleofMatter ...... 3 1.1.1 Interstellar Medium ...... 3 1.1.2 ExchangeofMatter ...... 4 1.2 StellarEvolution ...... 4 1.2.1 StarFormation ...... 4 1.2.2 Constant Light of Hydrogen Fusion - The Main Sequence . . 5 1.2.3 FinalStagesofStars...... 5 1.3 Origin and Composition of Stellar Dust ...... 7 1.3.1 PropertiesofAGBstars ...... 7 1.3.2 Detecting and Measuring Interstellar Dust Grains ...... 9 1.3.3 Dust Formation and Destruction ...... 10 1.4 FromAGBstarstoPNe ...... 11

2 Planetary Nebulae 13 2.1 Morphology and Classiﬁcation ...... 13 2.1.1 ListofProminentPNe...... 14 2.2 Examples ...... 15 2.2.1 Proto-PNe (or Young PNe) ...... 15 2.2.2 RoundandElliptical...... 18 2.2.3 Bipolar and Quadrupolar ...... 25 2.3 GlobalModelstoShapeaPN...... 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 MHDModels ...... 31 2.4 DetailsinPNe ...... 32 2.4.1 Halo...... 32 2.4.2 Jets,LobesandAnsae ...... 33 2.4.3 Knots ...... 33

v vi CONTENTS

II Theoretical Models 35

3 Radiation Hydrodynamics Simulation 37 3.1 BasicEquations...... 37 3.1.1 Conservationform ...... 37 3.1.2 GasComponent ...... 38 3.1.3 Radiation Field ...... 40 3.1.4 Dust...... 41 3.2 Additional Equations and Constitutive Relations ...... 42 3.2.1 GridEquation ...... 42 3.2.2 MassEquation ...... 42 3.2.3 PoissonEquation...... 43 3.2.4 EquationofState(EOS)...... 43 3.2.5 OpacityofGasandDust ...... 44 3.2.6 Source Function of Gas and Dust ...... 45 3.2.7 EddingtonFactor...... 45 3.3 BoundaryConditions...... 46 3.3.1 InnerBoundary...... 46 3.3.2 OuterBoundary ...... 46 3.4 InitialModels...... 47 3.4.1 ModellingMethod ...... 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 NumericalMethods...... 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 AGBstarspots...... 59 4.2 PhysicalModel ...... 60 4.2.1 SpotCoverage ...... 60 4.2.2 Temperature Fluctuations ...... 61 4.2.3 Magnetic Field ...... 61 4.2.4 Dust Formation above Cool Spots ...... 62 4.3 FluxTubeModel...... 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 RewrittenEquations ...... 68

III Results and Discussion 71

5 AGB Stars with Spots 73 5.1 InitialModels...... 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 EﬀectsofChemistry ...... 81 5.3 Dynamic Model Results for Flux Tube Geometry ...... 82 5.3.1 Eﬀects of Geometry ...... 85 5.4 BoundaryConditionsoftheFluxTube...... 88 5.4.1 LateralPressure ...... 88 5.4.2 HeatSourcesandSinks ...... 90 5.5 MassLossthroughaFluxTube...... 91

6 Discussion and Perspectives 95 6.1 MagneticField ...... 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 MassLoss...... 97 6.2.1 MassAcquiration...... 97 6.2.2 Stellar Rotation ...... 98 6.3 Small-scaleStructuresinPNe ...... 99 6.3.1 Instabilities ...... 99 6.3.2 InhomogeneousMassLoss...... 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 PermeableBoundary...... 101 6.5.4 Stellar Pulsations ...... 102 viii CONTENTS

IV Appendices 103

A Discretisation 105 A.1 ComputationalDomain ...... 105 A.2 Rules...... 105 A.3 General ...... 106 A.4 Case1:SphericalGeometry ...... 106 A.4.1 Advection...... 106 A.4.2 Mathematical Operators ...... 107 A.5 Case2:FluxTubeGeometry ...... 107 A.5.1 Advection...... 107 A.5.2 Mathematical Operators ...... 107

B Artiﬁcial Viscosity 109 B.1 General ...... 109 B.1.1 ViscousForce...... 110 B.1.2 Viscous Energy Dissipation ...... 110 B.2 Case1:SphericalGeometry ...... 111 B.2.1 Results ...... 112 B.2.2 Discretisation ...... 112 B.3 Case2:FluxTubeGeometry ...... 113 B.3.1 Results ...... 114 B.3.2 Discretisation ...... 114

C Radiation Transfer 115 C.1 RadiationTransferEquation ...... 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 0th-orderMomentEquation ...... 121 C.2.1 Case 1: Spherical Geometry ...... 121 C.2.2 Case 2: Flux Tube Geometry ...... 122 C.3 1st-orderMomentEquation ...... 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 DustFormation...... 128 D.3.1 C-richChemistry...... 128 D.3.2 Nucleation Theory ...... 131 D.3.3 DustPhysics ...... 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 OperationsandOperators ...... 138 E.2.3 Relations / Vector Identities ...... 141 E.3 Tensors ...... 142 E.3.1 Definitions ...... 142 E.3.2 OperationsandOperators ...... 142 E.3.3 Relations / Tensor Identities ...... 144 E.4 MetricandSymmetries ...... 145 E.4.1 Metric...... 145 E.4.2 CoordinateSystems ...... 147

F Full Set of RHD Equations 153 F.1 DiﬀerentialForm ...... 154 F.2 IntegratedForm ...... 155 F.3 DiscretisedForm ...... 157

G Symbols, Constants and Abbreviations 159 G.1 Symbols...... 159 G.2 Fundamental Physical Constants ...... 160 G.3 AstronomicalConstants ...... 160 G.4 Abbreviations...... 161

List of Tables 163

List of Figures 165

Image Credits 167

Bibliography 169

Part I

Introduction and Motivation

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). Ap- proximately 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 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 F λ 1. Thus dense clouds which are opaque in visual light get transparent λ ≈ − 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 circulation process 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 Jeans mass pressure forces, i.e. the mass concentration rises above the Jeans mass

1 3 M ρ− 2 T 2 (1.1) j ∝ (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 protostellar object and fragmentation may reduce the initial mass. The collapse to a protostellar object needs between 104 and 106 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 available fuel M 2.5 τ M − , (1.2) nuc ∼ burning rate ∼ L ∼ 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 0.6 M and for Neutron Stars ⊙ around 1.4 M . ⊙ 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 0.5 M down to objects with 0.08 M . ⊙ ⊙ Below this mass range a stellar object never gets hot enough to initiate hydrogen fusion in the core. 6 1. EVOLUTION OF STARS

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 ⊙ red giant branch the so-called red giant branch (RGB). While the star itself expands the remaining hydrogen burning hydrogen fusion in a shell around the helium rich centre generates energy for fur- shell ther million years. The stellar core contracts and pressure and temperature increase helium fusion until the helium fusion in the stellar centre begins. Now the evolution proceeds asymptotic giant very rapidly and the star is now located on the asymptotic giant branch (AGB) in branch 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 ⊙ 103 to a few 104L 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 pul- long period sations with long periods the stars are also commonly known as long period variables 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 105 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 hydrogen- and oxygen, which enrich the core by mass with these heavy elements. Due to burning shell 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. 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 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). long period The LPVs are divided into several groups according to the regularity of their light variables 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 κ-mechanism 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 convection zone zone moves inward. This causes a mixing of nuclear processed gas upwards. The mixing 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 convection cell 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. α Orionis Observations e.g. of α Orionis (Beteigeuze) (Gilliland & Dupree 1996 [54]) and three- (Beteigeuze) 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 infrared excess 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 terminal wind material where the terminal wind velocities of typically 10 to 40 km/s have been velocities 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 14 than the mass loss produced by the Sun, i.e. about 10− M /a. The mass loss of 7 5 ⊙ AGB stars lies in the range of 10− to 10− 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 5 of the AGB phase the mass loss raises up to 10− 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 4 bon star IRC+10216 (CW Leo) with a relatively high mass loss of about 10− 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 Repub- lic; 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

Allende meteorite - Fall: February 8, 1969; Location: Chihuahua, Mexico; Allende contains an increased concentration of 26Al 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, Aus- tralia; the meteorite was found to contain a wide variety of organic com- pounds, including many of biological relevance such as amino acids. Zag meteorite - Fall: August 4 or 5, 1998; Location: Western Sahara, Mo- rocco; 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 Sagit- tarius. 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 theoretical on Earth or on a spacecraft or by a theoretical approach. approach The process of dust formation in the circumstellar envelopes of LPVs can be two step process 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. There- fore, the equations of the basic concept of classical homogeneous nucleation theory classical homogeneous are generalised to get a consistent incorporation of random chemical reactions of the nucleation theory gas molecules with the dust clusters. 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 of helium burning episodes in the thin helium shell that surrounds the dormant stars carbon core of an AGB star; the helium burning shell does not generate energy at a 12 1. EVOLUTION OF STARS

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 OH/IR stars & 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- central stars tionary phase which lasts for a very short time (about 1000 to 2000 years). During of PNe this phase the temperature of the central star rises from about 2 000 to 30 000 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 born again again objects, like e.g. Sakurai’s Object, back to the AGB phase and finally ends objects 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 to be constructed. This classification is mainly based on morphology, i.e. the ob- scheme served 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 Pe- imbert 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, Man- chado et al. (1997 [96]) compiled a catalogue of more than 240 PNe of the northern hemisphere published by the Instituto de Astrof´ısica 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 ob- Figure 2.2: Details of PPN NGC 7027 served by the HST. 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 hydrogen or photo-dissociation region and (3) a nearly spherical outer region seen structure spherical outer in dust scattered light is the cool, neutral molecular envelope. The interface region region between the inner shell and the bipolar hourglass is structured and filamentary, structured and suggesting the existence of hydrodynamic instabilities (Latter et al. 2000 [89]). filamentary 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 198 000 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

CRL 2688 - Egg Nebula

Figure 2.3: Halo of PPN CRL 2688 ob- Figure 2.4: Infrared-details of PPN served by the HST. CRL 2688 observed by the HST.

The high resolution image from the HST/WFPC2 instrument in Fig. 2.3 shows dark edge-on disc (1) a remarkable dark edge-on disc obscuring the central star, (2) a pair of radial radial “searchlight “searchlight beam” like features, criss-crossed by (3) a large number (at least 25) of beam” circular arcs roughly circular arcs around the center. The arcs probably represent local peaks in faint radial streaks 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 12 700 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 1 (about 320 km s− ). 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 Cat’s Eye Nebula 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 multiple jet-like speeds along a preferred axis and may even have multiple jet-like outflows. The outflows? 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 features: (1) X-shaped structure, (2) lin- X-shaped structure ear features, which look like the “rungs” linear features of a ladder and (3) dark band passing dark band across the central star. The Red Rect- angle Nebula is associated with a post- 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 Figure 2.5: The PPN HD 44179 observed 322 days, a semi-major axis of a sin i = by the HST/WFPC2. 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, extend- ing 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 embedded within the evolved bipolar nebula evolved bipolar OH 231.8+4.2. This central star pul- nebula sates 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 accel- erated by the radiative pressure, shock Figure 2.6: The PPN OH231.8+4.2 ob- fronts are formed on impact and heat served by the HST/WFPC2. the surrounding gas. It is believed that 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 1 000 years from now the Calabash Nebula will become a fully developed bipolar PN (Bujarrabal et al. 2002 [20]). 18 2. PLANETARY NEBULAE

2.2.2 Round and Elliptical

NGC 6720 - Ring Nebula, M57

Figure 2.7: Halo of the PN NGC 6720 Figure 2.8: The PN NGC 6720 observed observed by the Subaru Telescope. 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 filamentary outer halos and other features: (1) filamentary structure of the inner halo consisting structure of loops and arcs, (2) small-scale structures at the main ring like dense knots and loops and arcs dense knots (3) enhanced bands of emission running across the central cavity. The expansion 1 enhanced bands velocity of the PN of 45 km s implies a expansion age of about 1500 220 years − ± (O’Dell et al. 2002 [110]). 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 petal-like appearance to the inner halo. appearance The outer halo is found to show a limb- limb-brightened knotty structure brightened knotty structure similar to the inner halo, but at much fainter lev- Figure 2.9: Details in the PN NGC 6720. els. However, the typical size of the Subimages taken from Fig. 2.8. 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, which are coincidentally seen from a direction close to its axis of symmetry (Bond axisymmetric et al. 1998 [15]).

Spectroscopic investigations show conclu- sively 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 out- 1 ward with a velocity of about 5 km s− . 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- Figure 2.10: Halo of PN NGC 6720 bright streaks gion. Additional observations by O’Dell et observed by the Spitzer Space Tele- al. (2002 [110]) indicate that the streaks are scope in Infrared Wavelength. formed by material inside of the main ring. 20 2. PLANETARY NEBULAE

NGC 7293 - Helix Nebula

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 filamentary Observatory. It reveals a filamentary struc- structure ture consisting of loops and arcs in the halo loops and arcs comet-like filaments and thousands of comet-like filaments, also “cometary knots” known as cometary knots. The model de- twisted components rived from the image consists of two twisted radial rays components of the main ring, radial rays multiplicity of axes 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 Figure 2.11: Halo of PN NGC 7293 to be located in a planar regime as noted by observed by the Kitt Peak National O’Dell 1998 [109]. Speck et al. (2002 [146]) Observatory and the HST/ACS. determined a lower limit of the PN shell mass of about 1.5 M . ⊙ Fig. 2.12 displays a detailed view of the cometary knots in the Helix Nebula. Cal- culations of the neutral core masses of the cometary knots from the observed extinc- 5 tion indicate masses of about 1.5 10− M for the best observed knots (O’Dell & Han-⊙ dron 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 Figure 2.12: Details of PN NGC 7293 by O’Dell et al. (2002 [110]). It was found observed by the HST. Subimage taken that the knots originate in or close to the from Fig. 2.11. 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 knots are these knots are primordial, i.e. going back to the origin formation of what is now primordial? 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 sub- structures, (2) numerous dense knots and dense knots (3) axisymmetric bands of matter in the cen- axisymmetric bands tral region. The inhomogeneous halo contains various structures, such as radial fila- radial filaments, arcs and arc-like ments, arcs and arc-like features (Papamas- features torakis et al. 1993 [114]). The bright jet- like filaments located at the nebula interior Figure 2.13: Halo of PN NGC 6853 seem to obscure the ionising radiation from observed by Robert Gendler. the central star resulting in a dimming of the halo along their directions. The bow-shaped appearance of the halo is probably related to an interaction of the nebula with the ISM. Perpendicular to the long axis of the elongated PN is a skewed, bright-rimmed bright-rimmed elliptical form possessing several internal structures. This geometry suggests a pro- elliptical form late 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 5 10− M (Meaburn & Lopez 1993 [100]). The ⊙ Figure 2.14: Details of PN NGC 6853 knots are forming at the interface between observed by the HST/WFPC2. the hot, ionised and cool, neutral portion of 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

NGC 2392 - Eskimo Nebula

In this HST image (see Fig. 2.15), one can see that the “parka” is really disc-like structure a disc-like structure (similar to the He- comet-shaped lix Nebula) including a ring of comet- objects shaped objects, with their tails streaming away from the central star. The Es- kimo’s “face” also contains some fasci- nating details. It is composed of two elliptically shaped elliptically shaped lobes of matter stream- lobes ing above and below the dying star. In this photo, one bubble lies in front of the other, obscuring part of the second inner bright shell lobe. The inner bright shell has a pro- late structure, with the major axis oriented very closely along the line of sight, external circular Figure 2.15: The PN NGC 2392 observed while the external circular shell has a shell by the HST/WFPC2. 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 Figure 2.16: Detail of PN NGC 2392 observed collision of slow- and fast-mov- by the HST/WFPC2. Subimages taken from ing material by Rayleigh-Taylor Fig. 2.15. 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 Figure 2.17: The PN NGC 6369 ob- one can see fainter arcs of gas that were fainter arcs served by the HST/WFPC2. lost from the star at the beginning of the ejection process. 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 2 000 ly it is one of the nearer known PNe. The gases are expanding away from the central star with a 1 velocity of about 14.5 km s− . This image clearly shows two stars near the centre of the nebula, a bright white one, and an adja- cent, fainter companion to its upper right. (A third, unrelated star lies near the edge of the nebula.) The faint partner is actually Figure 2.18: The PN NGC 3132 ob- the star that has ejected the nebula. An- served 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

IC 418 - Spirograph Nebula

Fig. 2.19 shows an image of the Spiro- graph 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 2850 L was placed on a stellar ⊙ evolutionary track (see e.g. Iben 1995 [72]) lowest mass among the lowest mass progenitor stars for progenitor star PNe, consistent with the low total nebular low total nebular mass masses of 0.2 to 0.7M assumed for the model ⊙ proposed by Meixner et al. (1996 [101]. Fur- 5 thermore, a mass loss rate of a few 10− M 1 ⊙ yr− over a period of 3000 years followed by Figure 2.19: The PN IC 418 observed an abrupt decrease 2000 years ago when the remarkable by the HST/WFPC2. star presumably left the AGB. The remark- textures able textures seen in the nebula are newly revealed by the Hubble telescope, and their origin is still uncertain.

NGC 6751

Fig. 2.20 displays an image of the Plan- etary Nebula NGC 6751 obtained with the WFPC2 instrument of the Hubble Space Tele- scope. The nebula shows a very complex morphology. Blue regions mark the hottest circular ring gas, which forms a roughly circular ring a- round the central stellar remnant. Parts in orange and red show the locations of cooler long streamers gas. The cool gas tends to lie in long streamers pointing away from the central star, and irregular ring 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 Figure 2.20: The PN NGC 6751 ob- winds from the hot star at the centre. The served by the HST/WFPC2. star’s surface temperature is estimated to be multiple shell about 140 000 K. Furthermore, observations have indicated the presence of a multiple structure 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 Figure 2.22: The PN NGC 6543 observed observed by the NOT. 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 filamentary A more detailed image of the nebula’s core was taken with Hubble’s Advanced structures Camera for Surveys (ACS) and can be seen in Fig. 2.22. It reveals eleven or even more concentric rings, or shells, around the Cat’s Eye Nebula. These photoionised concentric rings or rings are almost certainly the result of periodic spherical mass pulsations by the shells 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 NGC 7027, and Hubble 5 (Hb 5) show similar sets of multiple concentric rings and Hb 5 (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 of the PN core can be found in Balick (2004 [6]) the core 26 2. PLANETARY NEBULAE

MyCn 18 - Hourglass Nebula

In the HST/WFPC2 image (Fig. 2.23) bright elliptical of the young PN MyCn 18 a bright ellipti- ring 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. intersecting For example, there is a pair of intersecting elliptical rings elliptical rings in the central region which appear to be the rims of a smaller hourglass arc-like structures (Sahai et al. 1995 [130]). The arc-like structures 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. Fur- 1 high speed knots thermore, high speed knots with outflow velocities up to about 630 km s− 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

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 high surface seen against the high surface brightness neb- brightness ula. One of the most interesting features dark lanes 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]) high-velocity has revealed the presence of a high-velocity outflow Figure 2.24: The PN IC 4406 ob- outflow directed along the major axis of the served by the HST/WFPC2. bipolar shape. It seems that the outflow has been collimated by an toroidal density enhancement surrounding the central object. 2.2. Examples 27

NGC 6302 - Bug or Butterﬂy Nebula

Figure 2.25: The PN NGC 6302 observed Figure 2.26: Details of the PN NGC 6302 by Wendel and Flach-Wilken. 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 be observed surrounding the central object. From spectral measurements a mass of lane at least 0.03 M 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 380 000 K (Pottasch et al. 1996 [119]). Furthermore, the bipolar axis shows a distinct change with distance from the distinct change of central object. Also the central dust lane looks slightly deformed. Several of such bipolar axis observed structures are similar to those predicted in the warped-disc model of Icke (2003 [76]). 28 2. PLANETARY NEBULAE

Mz 3 - Ant Nebula

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 investiga- dense ionised core tions of the bipolar PN Mz 3 reveals a dense spherical, bipolar ionised core with almost spherical, bipolar lobes lobes. These are contained within a much filamentary bipolar Figure 2.27: The PN Mz 3 observed by more extensive filamentary bipolar nebula. nebula the HST/WFPC2. The expansion velocity of the inner lobes is 1 measured to be about 50 km s− whereas 1 the walls of the outer bipolar structure expand with about 90 km s− . A pair of hypersonic velocity hypersonic velocity features along the bipolar axis of the nebula reaches velocities features of 500 km s 1 (Redman et al. 2000 [121]). ≃ −

M2-9

The HST/WFPC2 image of the bipolar PN M2-9 shown in Fig. 2.28 depicts sev- inner and outer eral details like inner and outer lobes of the lobes bipolar regions on both sides of the centre, ansae ansae at the main axis of the nebula and dense disc a dense disc obscuring the central region. The central star in M2-9 is known to be close pair of stars one of a very close pair of stars. A model Figure 2.28: The PN M2-9 observed by proposed by Livio & Soker (2001 [92]) as- the HST/WFPC2. suming 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 point- symmetric, 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 acting winds theory (or two wind model), in which the slow wind from the AGB theory 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

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]). stellar companion Another model is based on the existence of a stellar companion, where the departure 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 nonaxisymmetric structures and very pronounced bipolar shapes. Soker & Rappaport (2001 [145]) structure 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 Dynamos in AGB stars can be the origin of magnetic fields. If this field is strong AGB stars 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 magnetic shaping 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. binary and If in a close binary system the magnetic field is coupled to the surrounding disc stellar 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 magnetic fields Nowadays magnetic fields are detected in central stars of PNe (see Jordan et in central stars 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. stellar spots 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 many cases this indicates an interaction of the nebula with the ISM. The expanding central star 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

Garc´ıa-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 sim- ulated 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 Garc´ıa-Segura (1997 [50]) and Garc´ıa-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

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 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 (Bal- ick 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 and several parameters of the nebula and the central star. The ansae expand with ‘Saturn Nebula’ 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

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- cometary knots 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 & Han- dron (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

NGC 3918, K1-2 - Corradi et al. (1999 [28]): Jets, knots and tails in planetary nebulae (NGC 3918, and Wr 17-1 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

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 φ dV = Q S (3.1) dt i − j i j VZ(t) X X 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 ∂φ φdV = dV + φ ~udA~ . (3.2) dt ∂t VZ(t) V (Zt0) A(It0)

37 38 3. RADIATION HYDRODYNAMICS SIMULATION

To rewrite this equation in a compact form we use Gauß’ law

d ∂φ φdV = + ~ (φ ~u) dV . (3.3) dt ∂t ∇ · VZ(t) V (Zt0)

Another notation can be derived by using the substantial derivation ( D := ∂ +~u ~ ) Dt t ·∇ d Dφ φdV = + φ ~ ~u dV . (3.4) dt Dt ∇ · VZ(t) V (Zt0)

For a moving coordinate system the conservation equations take the form

∂ X dV + X urel dA = S dV , (3.5) ∂t ZV (t) Z∂V (t) ZV (t) where X is the physical quantity, S represents the source and sink terms and urel is the relative velocity between the co-moving frame and the numerical grid. Thus, for an Eulerian grid we have urel = u (u is the gas velocity) and for a Lagrangean grid urel = 0. In case of an adaptive grid and spherical symmetry the relative velocity is given by urel = u ugrid , (3.6) − where ugrid is the velocity component of the moving coordinate system in an inertial frame of reference ∂r ugrid = . (3.7) ∂t

3.1.2 Gas Component

Equation of Continuity

This equation describes the conservation of mass within a volume V

dm = 0 . (3.8) dt The physical parameter φ is the density ρ and there are no source and sink terms to include d ρdV = 0 . (3.9) dt VZ(t) Thus we can write the conservation form of the continuity equation as

∂ ρ + ~ (ρ~u) = 0 . (3.10) ∂t ∇ · 3.1. Basic Equations 39

Energy Equation

This equation describes the conservation of the energy dE = E(t) . (3.11) dt i Xi The physical parameter φ is the density multiplied by the internal energy per mass ρ e. But ﬁrst we look at the total energy balance which includes the internal and mechanical energy ∂ 1 1 ρ e + v2 + ~ ρ~u e + v2 = ~ ~q + ~ T ~u + ρ ~u F~ . (3.12) ∂t 2 ∇ · 2 −∇ · ∇ · · · If we multiply the momentum equation by ~u from the left side we get the mechanical energy balance

∂ v2 v2 ρ + ~ ρ~u = ~u ~ P + ~u (~ τ)+ ρ ~u F~ . (3.13) ∂t 2 ∇ · 2 − · ∇ · ∇ · · Now we combine these two relations and derive

∂ (ρe)+ ~ (ρe ~u)= P ~ ~u + τ : ~ ~u ~ ~q, (3.14) ∂t ∇ · − ∇ · ∇ − ∇ · where τ : ~ ~u = τ u (3.15) ∇ ij i;j Xi,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

dI~ = F~ . (3.16) dt i Xi The physical parameter φ is the density multiplied by the gas velocity ρ ~u.

dI~ d = (ρ~u)dV (3.17) dt dt VZ(t)

First we will start with the following approach D~u F~ = ρ = ~ T + ρ f~ , (3.18) tot Dt ∇ i Xi where T = P I + 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 f~i are external forces. With the help of the continuity equation we get the conservation form of the equation of motion

∂ (ρ~u)+ ~ (ρ~u~u T )= ~ T + ρ f~ (3.19) ∂t ∇ · ∇ i Xi or

∂ (ρ~u)+ ~ (ρ~u~u T )= ~ P + ~ τ + ρ f~ . (3.20) ∂t ∇ · −∇ ∇ i Xi

3.1.3 Radiation Field

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

1 ∞ 4π E = I dΩdν = J ... radiation energy density (3.21) c ν c Z0 I ∞ F~ = Iν~ndΩdν = 4πH~ ... radiation energy ﬂux (3.22) Z0 I 1 ∞ 4π P = I ~n~ndΩdν = K . . . radiation pressure (3.23) c ν c Z0 I

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

d E 1 d 1 ∞ ρ + ~u F~ + ~ F~ + P : ~ ~u + ~a F~ = q (ω)dω (3.24) dt ρ c2 dt · ∇ · ∇ c2 · 0 Z0 3.1. Basic Equations 41

First Moment Equation of the RTE

ρ d F~ 1 d 1 1 ∞ + (~u P)+ c ~ P + (~aE)+ F~ ~ ~u = ~q(ω)dω (3.25) c dt ρ ! c dt · ∇ · c c · ∇ Z0

3.1.4 Dust

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) d−1 1 = N d f(N ,t) , (3.26) J ℓ ℓ τ 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 1 of size Nℓ and 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 has to be calculated in a diﬀerent way. J If the thermodynamical conditions allow dust grain formation the transition rate is assumed to be equal to the nucleation rate , i.e. the rate at which stable (su- J∗ percritical) 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) = R (N) R↑(N) R (N)+ R↓(N) , (3.27) dt ↑ − − ↓ 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

∞ j/d Kj = N f(N,t) . (3.28) NX=Nℓ 42 3. RADIATION HYDRODYNAMICS SIMULATION

The following set of moment equations can be derived from the master equation

dK 0 = (3.29) dt J dKj j 1 j/d = Kj 1 + N (1 j d) (3.30) dt d τ − ℓ J ≤ ≤

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

∂ 1 n + (n u)= K + N (1 j d) (3.31) ∂t c ∇ · c τ 2 ℓ J ≤ ≤

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 adaptive grid equations. Therefore, an adaptive grid ensures proper resolution of various features like shock fronts or steep gradients. The grid equation takes the following form

nˆl 1 nˆl − = , (3.32) Rl 1 Rl −

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)= ρ(r) dV ′. (3.33) Z0 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 = 4πGρ. (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 GM ~r ~g = = f~ . (3.36) − r2 r g

3.2.4 Equation of State (EOS)

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:

P (ρ, e) = R ρ e pressure (3.37) µ cV e T (ρ, e) = temperature (3.38) cV or µ P ρ(P, T ) = density (3.39) T R e(P, T ) = cV T energy (3.40)

With the speciﬁc heat at a constant volume as ∂e 1 c := = R (3.41) V ∂T µ γ 1 V − we can also write for the pressure in Eq. (3.37)

P (ρ, e) = (γ 1) ρ e . (3.42) − Other useful quantities as the adiabatic temperature gradient ∂lnT γ 1 := = − , (3.43) ∇ad ∂lnP γ S 44 3. RADIATION HYDRODYNAMICS SIMULATION

the speciﬁc heat at constant pressure

cP = γ cV (3.44)

and the relation ∂lnρ δ := = 1 (3.45) − ∂lnT P can be derived from the EOS of an ideal gas.

3.2.5 Opacity of Gas and Dust

Gas Opacity

For our calculations the mass absorption coeﬃcient of the gas κg is set constant, i.e.

g 4 2 1 κ = 2 10− cm g− . (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 Mie approximation as 3 χ = r0πQext′ (T ) K3 , (3.47)

where r0 is the radius of the monomer, Qext′ = Qext/a denotes the extinction effi- extinction efficiency ciency of the dust grain material which is independent of the grain radius a. For optically thin dust shells the flux average of Qext′ 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′ (T ) Q′ (T ) = 5.9 T (3.48) ext ≈ R rad according to Gail & Sedlmayr (1985 [47]). For our calculations we use

QR′ (T ) = 4.4 Trad (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 κd is defined by χ κd = . (3.50) ρ 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 thermodynamic function can be set equal to the Planck function, i.e. equilibrium σ S = T 4 . (3.51) g π g Similarly, the source function of the dust grains is given by σ S = T 4 , (3.52) d π d where Td denotes the dust temperature. 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.

Td = Trad . (3.53)

3.2.7 Eddington Factor

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 Kν fedd = . (3.54) Jν 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 1 medium, we obtain the Eddington approximation of fedd = 3 (i.e. isotropic radia- Eddington tion field), which is achieved e.g. in the stellar envelope. Whereas the Eddington approximation 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 the radiative transfer in extended stellar atmospheres whereas Yorke (1980 [161]) treatment and Balluch (1988 [12]) used the method of characteristics for the integration of the method of static transfer equation. The latter method is implemented in the RHD code. characteristics 46 3. RADIATION HYDRODYNAMICS SIMULATION

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 rel variables like the integrated mass mr, the velocity u 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 urel = 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 outﬂow-boundary - for the dynamical calculation a simple outﬂow-boundary condition is applied condition at the external boundary by assuming that the velocity gradient vanishes, ∂u i.e. ∂r = 0.

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 started from the photospheric radius of the star equations

L Rphot = ∗ 4 (3.55) s4πσTeff 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

Envelope Atmosphere

Stellar External Core Medium

RADI Rphot Rext RADE

Figure 3.1: Computational domain of the initial model 48 3. RADIATION HYDRODYNAMICS SIMULATION

static stellar and temperature) the static stellar structure equations are used. Fig. 3.1 depicts structure equations 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 2 m(r) = 4π ρ(r′)r′ dr′ . (3.56) Z0 Radiation Flux 2 From the luminosity L = 4πr Fr with the radiation flux Fr = 4π H we can derive the radiation flux as L 1 H(r)= . (3.57) (4π)2 r2

Radiation Energy Density Assuming local thermodynamic equilibrium (LTE), the radiation energy density is equal to the Planck function and we obtain σ J(r) B(T )= T (r)4 . (3.58) ≃ g π g 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 ndust(r) = (ε ε ) ntot(r) (3.59) C C − O H (cf. Appendix D). 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 K (r) 0 (1 j d) . (3.61) j ≡ ≤ ≤ 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

∂m r = 4πr2ρ . (3.62) ∂r

Hydrostatic Equilibrium 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 Gm ∂2r = r ρ ρ , (3.63) ∂r − r2 − ∂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

1 D = v ℓ (3.65) 3 h i p is determined by the average values of mean velocity v and the mean free path ℓp σ 4 h i of the photons. With J = π T (LTE) we obtain 4π E = J = aT 4 , (3.66) c

4σ 15 3 4 where a = c = 7.57 10− erg cm− K− is the radiation density constant, and 1 in Eq. (3.65) v can be replaced by the velocity of light c and ℓp by ℓph = (κρ)− . Assuming spherical symmetry F~ has only a radial component, i.e. F = F~ = F r | | and ~ E reduces to the derivative in the radial direction, i.e. ∇ ∂E ∂T = 4aT 3 . (3.67) ∂r ∂r 50 3. RADIATION HYDRODYNAMICS SIMULATION

Then Eq. (3.64) and (3.65) give immediately that

4ac T 3 ∂T F = . (3.68) − 3 κρ ∂r

Introducing the local luminosity l(r) = 4πr2F , which can be set constant in the outer stellar envelope (l = const. = L ), the derivation of the temperature yields to ∗

∂T 3 ρ κL 3 ρ κL = ∗ = ∗ . (3.69) ∂r −16πac r2 T 3 −64πσ r2 T 3

The derivation of the temperature can also be expressed as

∂T ∂P ∂T Gm T = = r ρ , (3.70) ∂r ∂r ∂P − r2 P ∇ where ∂ ln T = . (3.71) ∇ ∂ ln P If the energy transport is mainly conducted via radiation then

3 κLP = = . (3.72) ∇ ∇ad 64πσG mT 4

3.4.3 Equations for the Stellar Atmosphere

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

∂m r = 4π r2 ρ(r) . (3.73) ∂r

Hydrostatic Equilibrium The pressure gradient is composed of the gravitational acceleration (ﬁrst term of the RHS) and the contribution of the radiation ﬂux (second term)

∂P m 4π = G r ρ(r)+ ρ(r) κH . (3.74) ∂r − r2 c 3.4. Initial Models 51

Diffusion Equation The 1st-order moment equation of the radiation transfer equation is given in the static limit case, i.e. d 0 and no velocity field by dt ≡ ~ K = ρ (κ + σ ) H~ (3.75) ∇ · ν − ν ν ν (cf. Eq. (C.55) in Appendix C). For spherical symmetry and a nearly isotropic radiation field thus ~ K ~ K , where K is the scalar moment of the radiation ∇ · ν → ∇ ν ν pressure, we get ∂K 3K J ν + ν − ν = ρ (κ + σ ) H . (3.76) ∂r r − ν ν ν We can replace the radiation pressure K by the Eddington factor f = Kν and ν ν Jν obtain ∂(f J ) (3f 1)J ν ν + ν − ν = ρ (κ + σ ) H . (3.77) ∂r r − ν ν ν σ 4 Introducing local thermodynamic equilibrium (LTE), i.e. Jν = π T ∂f ∂T 3f 1 π ν σT 4 + f σ 4T 3 + ν − σT 4 = ρ (κ + σ ) H (3.78) ∂r ν ∂r r −σ ν ν ν we finally derive ∂T π H 1 ∂f 3f 1 1 = ρ (κ + σ ) ν ν + ν − T , (3.79) ∂r − σ ν ν T 3 4 f − ∂r r 4 f ν ν which can be written in a compact form as

∂T ρ(r) κH = A A T , (3.80) ∂r − 1 T 3 − 2 where π 1 A1 = (3.81) σ 4 fedd ∂f 3 f 1 1 A = edd + edd − (3.82) 2 ∂r r 4 f edd Radiation Flux Equation With the help of the relation F = 4πH the luminosity can be written as L(r) = 4π r2 F = 16π2r2H . (3.83) Since the luminosity is constant within the atmosphere, i.e. dL 0, it is obvious dr ≡ from dL ∂H = 16π2 2 rH + r2 , (3.84) dr ∂r that the term in parenthesis is equal to zero thus

∂H 2 H = . (3.85) ∂r − r 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 2 L = 4πr Fr (3.86)

and introducing Fr = 4πH we can derive the radiation ﬂux at the outer boundary L H(Rext)= 2 2 . (3.87) 16π Rext

External Temperature

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 1 to 2 . Assuming local thermodynamic equilibrium (LTE) we obtain σ J B(T )= T 4 (3.89) ≃ π g and with Eq. (3.88) and (3.89) we get the temperature at the outermost radius as

1 π H(R ) 4 T (R )= ext . (3.90) g ext σ µ ′

Total Pressure

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 1 a P = E = T 4 . (3.91) rad 3 3 rad 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. Richt- myer & 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 sion of the Jacoby matrix of the system is done by the Henyey method (Henyey et algorithm 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

BEGIN

INPUT OF CONTROL PARAMETER S

INPUT OF INITIAL MODEL

INITIATE HENYEY -ITERATION

CONVERGENCE ? NO

YES X DIVERGENCES NO REACHED?

CALCULATE NEW MATERIAL - YES GENERATE NEW TIME -STEP FUNCTIONS

GENERATE NEW TIME -STEP FORWARD EXTRAPOLATION

RESTORE NEW DATA

FORWARD EXTRAPOLATION

NO STOP -CONDITION FULFILLED?

YES

END

Figure 3.2: Flowchart of the RHD code 54 3. RADIATION HYDRODYNAMICS SIMULATION

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 monotonic advection scheme (van Leer 1977 [151]). General aspects and rules of the used advection scheme 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- artificial tensor nuter & Winkler (1979 [149]) have been developed a coordinate invariant artificial viscosity 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. Fur- thermore, 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 1 500 G (0.15 T) which is about 2 500 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

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 P m , (4.1) T ≈ P0

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

100 R2 f = Earth = 0.0084 (4.2) ⊙ R2 ⊙

for the Sun, where REarth and R are the radius of the Earth and the Sun, respec- ⊙ tively, 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.

ﬂare stars 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.

RS CVn and Starspots: RS Canum Venaticorum (spectral type F and G) and BY Draconis • BY Dra (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 CVn, FK Com and Ap stars. It exploits the correspondence between wavelength FK Com 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 = T T (4.3) eff,p − eff,s 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) ± σ2 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 disc Arel Fs Teff,s + (1 Arel) Fp Teff,p T¯eff = = − (4.4) R F dA Arel Fs + (1 Arel) Fp disc − R 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 ξ T + T T¯ = eff,s eff,p , (4.5) eff ξ + 1 where A F ξ = rel s . (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

2 A = 4π R = As + Au (4.7) ∗ ∗ and consists of the spot area As = f A (4.8) ∗ and the area of the undisturbed surface

Au = (1 f) A , (4.9) − ∗ where f denotes the surface ﬁlling factor. The global luminosity of the star can be written as 4 L = A σ Teﬀ, . (4.10) ∗ ∗ ∗ We decompose the total luminosity by the luminosity of the spot and the luminosity of the undisturbed surface

4 4 L = A σ (f Teff,s + (1 f) Teff,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

1 4 1 4 1 f 4 Teff,s = Teff, − Teff,u (4.12) f ∗ − f 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

1 4 1 4 f 4 Teff,u = Teff, Teff,s (4.13) 1 f ∗ − 1 f − − 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 Sec- tion 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

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 B2 P spot . (4.15) phot ≃ 8π Assuming the pressure of the photosphere by a simple hydrostatic approach (Kip- penhahn & Weigert 1990 [83]) GM P ρ l (4.16) phot ≃ R2 phot ∗ and the deﬁnition of the photosphere where κ l ρphot = 2/3 where l is the density scale height and κ is the opacity, we get 2 GM 1 P . (4.17) phot ≃ 3 R2 κ ∗ Combining Eq. (4.14), (4.15) and (4.17) the average magnetic intensity required to form AGB stellar spots is 1 1 2 − 1 1 3 M R η − 2 B 4 10− ∗ κ− G , (4.18) av ≥ 1M 300R 104 3 ⊙ ⊙ 3 2 1 where κ3 = κ/(3 10− cm g− ). 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

RCB stars 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 RY Sgr 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 Sakurai’s Object, of three old PNe, these are Sakurai’s Object, V605 Aql and FG Sge (Duerbeck & V605 Aql and FG Sge 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 Sec- tion 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

z2 A(z)= A 1+ (4.19) 0 z2 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 2 border case if z0 is very small the area is proportional to z 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 g = 0 x2a 0 , (4.20) ik   0 0 1   where z2 a(z)=1+ 2 (4.21) z0 (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

stellar x photosphere

z = 0 R phot A0 r0

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 r2 As = A0 2 (4.22) r0 and the radial derivation as

dAs 2r = As′ = A0 2 . (4.23) dr r0 2 If A0 = 4π r0 then Eq. (4.22) is reduced to 2 As = 4π r , (4.24) which denotes the surface of a sphere. For a ﬂux tube

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 z2 r2 and z2 z2 (equal to deﬁnition) → 0 → 0 r2 r2 0 1+ z2 for r r0 a =1+ 2 → 0 → (4.26) z0 for z 0 and r> 0 → ∞ 0 →

2r0 2r z2 for r r0 a′ = → 0 → (4.27) z2 for z 0 and r> 0 0 → ∞ 0 →

Version 2: Substitution of z2 (r r )2 and z2 z2 → − 0 0 → 0 2 (r r0) 1 for r r a =1+ − → → 0 (4.28) z2 for z 0 and r = r 0 → ∞ 0 → 6 0

2(r r0) 0 for r r0 a′ = −2 → → (4.29) z for z0 0 and r = r0 0 → ∞ → 6

Version 3: Substitution of z2 (r r )2 and z2 (r + z )2 → − 0 0 → 0 0 2 (r r0) 1 for r r0 → 2 → a =1+ − 2 (r r0) (4.30) (r0 + z0) 1+ −2 for z0 0 and r = r0 → r0 → 6

2(r r ) 0 for r r0 0 → → a′ = − 2 2(r r0) (4.31) (r0 + z0) −2 for z0 0 and r = r0 → r0 → 6

a′ r r0 0 for r r0 = − → r r0 → (4.32) 2 2 2 − 2 for z0 0 and r = r0 2a (r0 + z0) + (r r0) r +(r r0) − → 0 − → 6

Version 4: Substitution of z2 r2 r2 and z2 (r + z )2 → − 0 0 → 0 0 2 2 r r0 1 for r r0 → 2 → a =1+ − 2 r (4.33) (r0 + z0) 2 for z0 0 and r = r0 → r0 → 6

2r0 2r 2 for r r0 → (r0+z0) → a′ = 2 2r (4.34) (r0 + z0) 2 for z0 0 and r = r0 → r0 → 6 r0 a′ r 2 for r r0 = → (r0+z0) → (4.35) 2a (r + z )2 + r2 r2 1 for z 0 and r = r 0 0 − 0 → r 0 → 6 0 66 4. STELLAR SPOTS

Version 5: Substitution of z2 r2 r2 and z2 r2 + z2 → − 0 0 → 0 0 2 2 r r0 1 for r r0 → 2 → a =1+ 2 − 2 r (4.36) r + z r2 for z0 0 and r = r0 0 0 → 0 → 6

2r0 2r 2 2 for r r0 → r0 +z0 → a′ = 2 2 2r (4.37) r0 + z0 2 for z0 0 and r = r0 → r0 → 6 r0 a′ r r2+z2 for r r0 = → 0 0 → (4.38) 2a r2 + z2 1 for z 0 and r = r 0 → r 0 → 6 0

Version 6: Substitution of z2 (r r )2 and z2 r2 + z2 → − 0 0 → 0 0 2 (r r0) 1 for r r0 → 2 → a =1+ 2− 2 (r r0) (4.39) r0 + z0 1+ −2 for z0 0 and r = r0 → r0 → 6

2(r r ) 0 for r r0 0 → → a′ = 2 − 2 2(r r0 (4.40) r + z −2 for z0 0 and r = r0 0 0 → r0 → 6

a′ r r0 0 for r r0 = 2 2 − → r r0 → (4.41) 2 2 − 2 for z0 0 and r = r0 2a r0 + z0 + (r r0) r +(r r0) − → 0 − → 6

For versions 4 and 5 applies for z 0 that A A , A A and 0 → → s ′ → s′ a 1 ′ . (4.42) 2a → r

a′ Since the expression for 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

2 A0 = f A = f 4πRphot . (4.43) ∗ To avoid large deviations from the spherical surface of the star and the non-spherical 2 area of the ﬂux tube geometry we restrict to a surface ﬁlling factor of f = 10− . 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 = 4πH and the luminosity L = 4π r2 F we can write the internal ﬂux as follows L Hint = 2 2 . (4.44) (4π) r0

As the luminosity is constant, i.e. Lint = Lext, we get for the external ﬂux in the ﬂux tube at the radius Rext

A0 L Hext = Hint = 2 (4.45) Aext 2 2 zext (4π) r0 1+ 2 z0 (see also Fig. 4.2). From Eq. (4.45) we see that for large z0 the radiative ﬂux gets constant at any radius r.

Hext

Hint A A0 ext

Figure 4.2: Radiative ﬂux through a ﬂux tube 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

z2 V = A(z) dz . (4.46)

zZ1 In ﬂux tube geometry we can further write

z2 2 z z2 1 3 z2 V = A0 1+ dz = A0 [z] + z = z2 z1 3z2 z1 Z 0 0 z1 z2 + z z + z2 = A (z z ) 1+ 1 1 2 2 =∆V [z , z ] . (4.47) 0 2 − 1 3 z2 1 2 0 If we adopt the chosen ﬂux tube representation in Section 4.3.2 and substitute z r Eq. (4.47) can be given as 1,2 → 1,2 1 2 2 2 3 (r1 + r1r2 + r2) r0 V = A0 (r2 r1) 1+ 2 2 − . (4.48) − " r0 + z0 # 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 z1 z2

A0 A1 V A2

Figure 4.3: Volume of a ﬂux tube segment 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 1 ua′ J + ′ (J u)= ′ H P ′ u (3K J) c ∂t c ∇z −∇z · − c ∇z · − 2a − ρ (κ J κ S) (4.49) − J − S and the 1st-order moment equation as

1 ∂ 1 ∂K 1 ca′ H + ′ (H u)= H ′ u + (3K J) ρκ H , (4.50) c ∂t c ∇z · − ∂z − c ∇z · 2a − − H 1 ∂(a varx ) where varx = h i . A detailed discussion to derive Eq. (4.49) and ∇z′ · h i a ∂r 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

1 ∂ 3 2 ∂u 1 fi = a 2 ℓ ρ ′ u ′ u a3/2 ∂r ∇z · ∂r − 3∇z · 1 ∂ 3 2 ∂(au) ∂u 1 ∂(au) = a 2 ℓ ρ (4.51) √a ∂V ∂V ∂r − 3 ∂V and the dissipated energy per gram

2 2 2 2 ∂u a′ 3 2 ∂(au) ∂u 1 ∂(au) = ℓ ′ u u = ℓ (4.52) EQ −3 ∇z · ∂r − 2a −2 ∂V ∂r − 3 ∂V is the contribution to the energy equation. A detailed derivation is given in Ap- pendix 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

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 , 4 ⊙ L = 10 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. Adopt- ing 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 104L 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 104L , 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 104L , 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 104L , 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 1.72 1013 247 0.5 C 3.43 1013 493 1 D 6.86 1013 986 2 E 1.72 1014 2465 5 F 3.43 1014 4930 10 G 3.43 1015 49300 100

Table 5.2: Model ﬂux tubes for a luminosity of 104L , 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 104L , 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 104L , 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 104L , 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 104L 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 7 1 5 10− M /a− 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 6 1 the stationary wind and reaches values up to some 10− M /a− . ⊙ 80 5. AGB STARS WITH SPOTS

Figure 5.8: Spatial structure of the stationary wind solution in spherical geometry 7 for star model D (Teﬀ = 2600 K) with v = 10.0 km/s, M˙ = 2.20 10− M /a, ∞ ⊙ log ε = 3.17 and ε /ε = 2.2. O − C O

Figure 5.9: Spatial structure of a stellar wind generated by a dust-induced κ- mechanism in spherical geometry for star model C (Teﬀ = 2500 K) with v = 15.4 6 ∞ km/s, M˙ = 1.54 10− 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 ε /ε . Therefore we decide to fix the amount of oxygen of log ε = 3.17, i.e. the C O O − 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 M˙ ∞ [km/s] [M /a] ⊙ 2.0 breeze - - 7 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− )

Table 5.3: Eﬀects of chemistry for log ε = 3.17 and T = 2600 K. O − eﬀ 82 5. AGB STARS WITH SPOTS

log εO Teﬀ εC /εO Result v M˙ ∞ [K] [km/s] [M /a] ⊙ 7 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− 6 2.0 κ-induced wind (15.7) (1.54 10− ) 7 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− 6 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− )

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. 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 13 tube C (z0 = 3.43 10 cm) with a temperature difference ∆T = 200 K and εC /εO = 4 2.3 located at model D (M = 1M , L = 10 L and Teff = 2600 K). ⊙ ⊙

13 Figure 5.11: Same as Fig. 5.10, but for ﬂux tube D (z0 = 6.86 10 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 13 tube geometry for flux tube B (z0 = 1.72 10 cm) with a temperature difference 4 ∆T = 100 K and εC /εO = 2.2 located at model D (M = 1M , L = 10 L and ⊙ ⊙ Teff = 2600 K).

13 Figure 5.13: Same as Fig. 5.12, but for ﬂux tube B (z0 = 1.72 10 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. 13 We found that for a flux tube model with ∆T = 0 and about z0 = 1.0 10 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

13 Figure 5.14: Occurrence of a dust-driven wind for flux tube B (z0 = 1.72 10 cm) 4 located at the AGB star model D (M = 1M , L = 10 L and Teff = 2600 K) for ⊙ ⊙ various temperature differences ∆T .

13 Figure 5.15: Same as Fig. 5.14, but for ﬂux tube D (z0 = 6.86 10 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 , 4 ⊙ L = 10 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 R P = ρT (5.3) g µ and the magnetic pressure B2 P = 0 (5.4) m,s 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

PP+ B g,s m,s 0 ρ Pg,n s ρn

Ts Tn Rphot

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

µ B2 ρ ∆T = T T = 0 + 1 n T (5.5) n − s 8πR ρ − ρ n s s 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 P = m,s (5.6) Tn Pg,n which is identical to Eq. (4.1) in Section 4.1.1. Fig. 5.19 displays ∆T as a function 7 3 of B0 as example for the Sun with a photospheric pressure of about 7 10− g cm− . As we mentioned in Section 4.1.1 the magnetic field strength for sunspots reaches up to 1 500 G which corresponds with a maximum ∆T of about 2 000 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 hypothet- 9 3 ical AGB star with a photosphere density of about 7.5 10− g cm− . 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 z 0 and ∆T > 0 then the temperature in the flux tube is always 0 ≃ 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 signifi- cantly. 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 (M˙ A,s) and the mass loss of the spherical model (M˙ A,n) through an area A as M˙ ρ v A,s = ft ft , (5.7) M˙ A,n ρsphvsph 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 M˙ A,s/M˙ A,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 M˙ A,n 28 2∗ 9 [10 cm ] [10− 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 M˙ A,s M˙ A,s/M˙ A,n 28 2∗ 9 Model [10 cm ] [K] [10− M /a] ⊙ 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 r2 B τ s 0 , (6.1) ∝ ν0 Be

95 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 (B 3000 G, B 400 G and ν 1000 km2 s 1) the well-known linear area-to- 0 ≈ e ≈ 0 ≈ − lifetime relation of Gnevyshev (1938 [55])

r 2 τ = s 10 [days] (6.2) [104 km] can be derived and for a typical diameter d 5 104 km of a solar spot we get a s ≈ 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 1 A = πr2 = 4πR2 (6.3) 0 s 100 phot the radius of the stellar spot is

1 r = R (6.4) s 5 phot

and can reach up to 108 km, many orders of magnitude larger than solar spots. Thus the lifetime can reach some 104 to 105 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.

vn . Mn

R phot

pulsations

vs 2 rs Ts . Ms

Figure 6.1: Model of AGB star with a cool spot 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 around PPNe (e.g. IRAS 16594-4656 and IRAS 20028+3910), PNe (e.g. NGC 7027, and 20028+3910 NGC 7027, Hb 5 Hb 5 and NGC 6543) and AGB stars (e.g. IRC+10216). The shells are semi-periodic and NGC 6543 with time intervals between consecutive ejection events of about 200 to 1000 years IRC+10216 (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 is determined by the formation mechanism and the evolution of the magnetic field magnetic field formation and of the star. Frank (1995 [44]) investigated the influence of a big stellar spot and evolution 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. 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

2 A(x)= A0(1 + x ) . (6.6)

The following estimation are based on a star with a mass of 1M , a luminosity of 4 13 ⊙ 10 L and a photospheric radius of Rphot = 3.4 10 cm. For a radius of x = 15, ⊙ 17 3 1 a mean density of ρ = 10− g cm− and a mean velocity of v = 10 kms− the 98 6. DISCUSSION AND PERSPECTIVES

9 mass loss rate is approximately 5.21 10− 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 Dumbbell Nebula (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 1 8 the mean velocity is v = 20 kms− the mass loss rate increases to 1.04 10− 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

2 2 1 J mrv = mr ω = 2π mr P − = 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 r 2 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 18 000 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 Eskimo Nebula. They have only be compressed by the subsequent fast wind of the Helix Nebula PN central star to form dense cometary knots (cf. NGC 7293, the Helix Nebula). break of symmetry 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 been proposed by Capriotti (1973 [21]). These instabilities should be common at instability 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 assumed that Kelvin-Helmholtz instabilities could disrupt the flux tube at larger instability 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 inhomogeneous mass loss process can be the reason even for large clumps and filaments. clumps and Nevertheless as a result of density inhomogeneities in the stellar outflow of the AGB filaments 100 6. DISCUSSION AND PERSPECTIVES

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 2 000 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 Eskimo Nebula 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. Eskimo Nebula (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 contin- uously 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 one- dimensional 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

ρ i r e i i m J i i ui (K ) j i Hi Stellar External Envelope Medium NPT NPT−1 i+2i+1i i−1 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 ∆X X (t) X (t δt) l l − l −

spatial diﬀerence ∆Xl Xl 1 Xl Xl Xl+1 − − − 1 1 spatial mean Xl (Xl 1 + Xl) (Xl + Xl+1) 2 − 2 Xad if urel < 0 Xad if u¯rel < 0 upwind diﬀerencing Xad l 1 l l l l X−ad otherwise Xad otherwise l l+1 g Scheme Discretisation ad donor cell Xl = Xl

(Xl Xl+1)(Xl−1 Xl) ad − − if (Xl Xl+1)(Xl 1 Xl) > 0 van Leer X = X + Xl−1 Xl+1 − l l − ad − − ( Xl otherwise

A.3 General

The volume-integrated conservation equations of the RHD system for a moving coordinate system take the form ∂ X dV + X urel dA = S dV . (A.1) ∂t VZ ∂VI VZ The velocity urel is the relative velocity between the co-moving frame and the numerical grid and is deﬁned as δr urel = u l . (A.2) l l − δt

A.4 Case 1: Spherical Geometry

The volume element in spherical geometry is given by ∆r3 ∆V = l , (A.3) l 3 which is equivalent to a spherical shell between two grid points divided by the factor 4π. Thus the mass element is calculated as 1 ∆m = ρ ∆V . (A.4) 4π l l l

A.4.1 Advection

The discretised form of the l.h.s. in Eq. (A.1) is 1 δ(X ∆V )+∆(r2Xadurel) (A.5) δt l l l l l g A.5. Case 2: Flux Tube Geometry 107 if X is a scalar and 1 δ(X ∆V )+∆(r2Xadurel) (A.6) δt l l l l l if X is a vector. g

A.4.2 Mathematical Operators

Gradients and divergences are discretised in spherical geometry according to

X dV = r2 ∆X (A.7) ∇ ⇒ l l VZ and X dV = ∆(r2 X ) , (A.8) ∇ · ⇒ l l VZ respectively.

A.5 Case 2: Flux Tube Geometry

The volume element in flux tube geometry is given by ∆z3 ∆V = A ∆z + l (A.9) l 0 l 3z3 0 without the consideration of a specific flux tube representation. Thus the mass element within a flux tube is calculated as

∆ml = ρl∆Vl . (A.10)

A.5.1 Advection

The discretised form of the l.h.s. in Eq. (A.1) is 1 δ(X ∆V )+∆(a Xadurel) (A.11) δt l l l l l if X is a scalar and 1 g δ(X ∆V )+∆(a Xadurel) (A.12) δt l l l l l 2 zl if X is a vector, where al =1+ 2 . g z0

A.5.2 Mathematical Operators

Gradients and divergences are discretised in ﬂux tube geometry according to

X dV = a ∆X (A.13) ∇ ⇒ l l VZ and X dV = ∆(a X ) , (A.14) ∇ · ⇒ l l VZ respectively. 108 A. DISCRETISATION Appendix B

Artiﬁcial Viscosity

B.1 General

Shock waves represent a problem for numerical difference methods (finite difference finite difference methods), since they appear with ideal liquids as discontinuities. With the artificial methods 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

Qk = ℓ2ρ div(~u) εk α div(~u) δk [1 θdiv(~u)] . (B.1) i i − i − h i The divergence of the velocity ﬁeld ~u can be derived from the covariant derivation divergence of the contravariant vector uk (see also Appendix E)

k k k λ div(~u)= u;k = u,k +Γkλu . (B.2)

k εi designates the mixed tensor of the symmetrised gradient symmetrised velocity ﬁeld k kl εi = g u(l;i), (B.3)

kl where g represents the contravariant metric tensor and u(l;i) the symmetrised covariant velocity tensor 1 u = (u + u ) . (B.4) (l;i) 2 i;l l;i The covariant derivative of the covariant vector is

u = u Γλ u . (B.5) i;l i,l − li λ 109 110 B. ARTIFICIAL VISCOSITY

k nd Furthermore δi represents the Kronecker tensor of the 2 kind (mixed unity tensor) 1 if i = k δk = i 0 otherwise and θ the Heaviside step function

1 if x> 0 θ(x)= . 0 otherwise The parameter α is selected in such a way, that the pressure tensor disappears at a k homologous contraction, i.e. the trace of Qi has to disappear k Qk = 0 . (B.6) In general α depends on the dimension. In case of a three-dimensional system of coordinates, the following applies 1 α = . (B.7) 3 Finally, we can rewrite the formula for the mixed tensor of the viscous pressure as 1 Qk = l2ρ ul ǫk ul δk , (B.8) i ;l i − 3 ;l i g dyn which have the dimension [ cm s2 ] or [ cm2 ].

B.1.1 Viscous Force

The viscous force, which supplies a contribution in the equation of motion, can be k evaluated by the divergence of Qi f = Qk = Qk +Γk Ql Γl Qk , (B.9) i i;k i,k lk i − ik l g dyn with the dimension [ cm2 s2 ] or [ cm3 ].

B.1.2 Viscous Energy Dissipation

The energy dissipated by the viscosity is a result of the contraction of the pressure tensor with the tensor of the symmetrised velocity ﬁeld 1 = Qi εk . (B.10) EQ −ρ k i

k k If we evaluate the equation above and include the relation u;k = εk (trace of ε) then we get the following equation 1 = l2uk (ε1 ε2)2 + (ε1 ε3)2 + (ε3 ε2)2 + 2 ε2ε1 + ε3ε1 + ε3ε2 , EQ − ;k 3 1 − 2 1 − 3 3 − 2 1 2 1 3 2 3 (B.11) 2 with the dimension [ cm ] or [ erg ]. 0 has to be guaranteed, which yields to a s3 gs EQ ≤ 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- k v w variant vector uk = (u, vr, wr sin θ), and the contravariant vector u = (u, r , r sin θ = Ω), we can determine the following steps: Metric tensor 1 0 0 1 0 0 2 ik 1 gik = 0 r 0 ; g = 0 r2 0  2 2   1  0 0 r sin θ 0 0 r2 sin2 θ Christoﬀel symbols of the second kind   Γ1 = r 22 − Γ1 = r2 sin2 θ 33 − Γ2 = sin θ cos θ 33 − 1 Γ2 =Γ2 =Γ3 =Γ3 = 12 21 13 31 r 3 3 Γ23 =Γ32 = cot θ Divergence ∂u 2u 1 ∂v v ∂Ω ∂(r2u) 1 ∂ ∂Ω uk = + + + cot θ + = 3 + (v sin θ)+ (B.12) ;k ∂r r r ∂θ r ∂φ ∂r3 r sin θ ∂θ ∂φ Mixed tensor of the symmetrised gradient (only terms needed) ∂u ǫ1 = g11u = (B.13) 1 1;1 ∂r u 1 ∂v ǫ2 = g22u = + (B.14) 2 2;2 r r ∂θ u v 1 ∂w ǫ3 = g33u = + cot θ + (B.15) 3 3;3 r r r sin θ ∂φ Viscous pressure (only radial terms)

∂u ∂(r2u) 2 ∂r ∂r3 0 0 ∂(r u) − 2 Qk = 3ℓ2ρ u ∂(r u) (B.16) i 3  0 r ∂r3 0  ∂r − 2 0 0 u ∂(r u)  r − ∂r3  and its divergence   (only needed term) ∂Q1 2 ∂Q2 ∂Q3 Q2 + Q3 Qk = 1 + Q1 + 1 + Q2 cot θ + 1 2 3 1;k ∂r r 1 ∂θ 1 ∂φ − r ∂(r2Q1) 1 ∂ ∂Q3 1 = 3 1 + (Q2 sin θ)+ 1 (Q2 + Q3) ∂r3 sin θ ∂θ 1 ∂φ − r 2 3 Qk=0 ∂(r2Q1) 1 1 ∂ ∂Q3 =k 3 1 + Q1 + (Q2 sin θ)+ 1 ∂r3 r 1 sin θ ∂θ 1 ∂φ 3 ∂(r3Q1) 1 ∂ ∂Q3 = 1 + (Q2 sin θ)+ 1 (B.17) r ∂r3 sin θ ∂θ 1 ∂φ 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:

dyn Viscous force [ cm3 ] (only radial term)

3 ∂ ∂(r2u) ∂u ∂(r2u) 2 ∂ ∂(r2u) ∂u u f = r3l2ρ 3 = r3l2ρ i r ∂r3 ∂r3 ∂r − ∂r3 3r ∂V ∂V ∂r − r (B.18) erg Energy per gram [ gs ] (only radial term)

9 ∂(r2u) ∂u ∂(r2u) 2 2 ∂(r2u) ∂u u 2 = l2 = l2 (B.19) EQ −2 ∂r3 ∂r − ∂r3 −3 ∂V ∂r − r

B.2.2 Discretisation

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.

dyn Viscous force [ cm3 ] (only radial term)

2 1 ∆(r2u ) ∆u u 1 f = ∆ r3l2ρ l l l l (B.20) i 3 r l l ∆V ∆r − r ∆V l l l l l erg Energy [ cm3 s ] (only radial term) 2 ∆(r2u ) ∆u u 2 ρ = l2ρ l l l l (B.21) EQ −3 l ∆V ∆r − r l l l 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), co- k w v variant vector uk = (w√a,vx√a, u), contravariant vector u = ( √a , x√a , u), we can determine the following steps: Metric tensor

1 a 0 0 a 0 0 g = 0 x2a 0 ; gik = 0 1 0 ik    x2a  0 0 1 0 0 1     Christoﬀel symbols of the second kind

Γ1 = x 22 − a Γ3 = ′ 11 − 2 x2a Γ3 = ′ 22 − 2 1 Γ2 =Γ2 = 12 21 x a Γ1 =Γ1 =Γ2 =Γ2 = ′ 13 31 23 32 2a Divergence

1 ∂w w 1 ∂v ∂u a 1 ∂(xw) 1 ∂v 1 ∂(au) uk = + + + + ′ u = + + (B.22) ;k √a ∂x x√a x√a ∂ϕ ∂z a x√a ∂x x√a ∂ϕ a ∂z

Mixed tensor of the symmetrised gradient (only terms needed)

1 ∂w a ǫ1 = g11u = + ′ u (B.23) 1 1;1 √a ∂x 2a 1 ∂v w a ǫ2 = g22u = + + ′ u (B.24) 2 2;2 x√a ∂ϕ x√a 2a ∂u ǫ3 = g33u = (B.25) 3 3;3 ∂z Viscous pressure (only radial terms; z r) → a′ 1 ∂(au) 2a u 3a ∂r 0 0 1 ∂(au) − ′ Qk = ℓ2ρ 0 a u 1 ∂(au) 0 (B.26) i a ∂r  2a − 3a ∂r  0 0 ∂u 1 ∂(au)  ∂r − 3a ∂r    and its divergence (only needed term)

∂Q1 1 ∂Q2 ∂Q3 a a Qk = 3 + Q1 + 3 + 3 + ′ Q3 ′ (Q1 + Q2) 3;k ∂x x 3 ∂ϕ ∂r a 3 − 2a 1 2 114 B. ARTIFICIAL VISCOSITY

1 ∂(xQ1) ∂Q2 1 ∂(aQ3) a = 3 + 3 + 3 ′ (Q1 + Q2) x ∂x ∂ϕ a ∂r − 2a 1 2 Qk=0 1 ∂(xQ1) ∂Q2 1 ∂(aQ3) a k= 3 + 3 + 3 + ′ Q3 x ∂x ∂ϕ a ∂r 2a 3 1 ∂(xQ1) ∂Q2 1 ∂(a3/2Q3) = 3 + 3 + 3 (B.27) x ∂x ∂ϕ a3/2 ∂r

B.3.1 Results

For ﬂux tube geometry we can write down the viscous force and the dissipated energy per gram as:

dyn Viscous force [ cm3 ] (only radial term)

1 ∂ 3/2 2 1 ∂(au) ∂u 1 ∂(au) fi = a ℓ ρ a3/2 ∂r a ∂r ∂r − 3a ∂r 1 ∂ ∂(au) ∂u 1 ∂(au) = a3/2ℓ2ρ (B.28) √a ∂V ∂V ∂r − 3 ∂V erg Energy per gram [ gs ] (only radial term)

2 1 ∂(au) ∂u a 2 3 ∂(au) ∂u 1 ∂(au) 2 = ℓ2 ′ u = ℓ2 (B.29) EQ −3 a ∂r ∂r − 2a −2 ∂V ∂r − 3 ∂V

B.3.2 Discretisation

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.

dyn Viscous force [ cm3 ] (only radial term)

1 ∆(a u ) ∆u 1 ∆(a u ) 1 f = ∆ a3/2ℓ2ρ l l l l l (B.30) i √a l l ∆V ∆r − 3 ∆V ∆V l l l l l erg Energy [ cm3 s ] (only radial term)

3 ∆(a u ) ∆u 1 ∆(a u ) 2 ρ = ℓ2ρ l l l l l (B.31) EQ −2 l ∆V ∆r − 3 ∆V l l l 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 xµ , the photon Boltzmann equation { } is given by

d dxµ ∂ dp a ∂ f = + ∗ f = C [f] , (C.1) dt dt ∂xµ dt ∂p a ∗ where µ = 0, 1, 2, 3 , a = 1, 2, 3 and C [f] is the collision operator. Eq. (C.1) can collision operator { } { } also be expressed as dxµ ∂ dea ∂ + pµ µ f = C [f] . (C.2) dt ∂xµ dt ∂p a ∗ In an inertial frame1: and specialising to FF Eq. (C.2) can be written as ∂ ∂~v ∂ p0 + ~p ~ p0 p0 + ~p ~ ~v f = C [f] (C.3) ∂t · ∇− ∂t · ∇ ∂~p 1 µ dxµ In an inertial frame (absence of external forces) and for p = dt we can write ∂ pµ f = C [f] ∂xµ

115 116 C. RADIATION TRANSFER

ω3 speciﬁc intensity Introducing a speciﬁc intensity I = 2 c2h3 f the Boltzmann equation yields to ∂ ∂~v ∂ 3 dI p0 + ~p ~ p0 p0 + ~p ~ ~v ~p I = p0 (C.4) ∂t · ∇− ∂t · ∇ · ∂~p − p2 dt coll Lorentz For the LF we conduct a Lorentz transformation, where to order (v/c) transformation O ∂ d ∂ = = + ~v ~ ′ , (C.5) ∂t dt ∂t′ · ∇ ~ ~ ~v ∂ = ′ + 2 , (C.6) ∇ ∇ c ∂t′ ~ ~v = ~ ′~v, (C.7) ∇ ∇ d~v ∂~v ∂v ~a = = , (C.8) ≡ dt ∂t ∂t′ with the result

0 d ~v d 0 0 ∂ 3 0 dI p + ~p ~ ′ + ~p p p ~a + ~p ~ ′~v ~p I = p . dt · ∇ · c2 dt − · ∇ · ∂~p − p2 dt coll (C.9) With p0 = p/c and after dividing above equation by p and using ~p = p ~n, we can rewrite Eq. (C.9)

1 d 1 d 1 1 ∂ 3 1 dI + ~n ~ ′ + ~n ~v p~a + ~p ~ ′~v ~n I = . c dt · ∇ c2 · dt − c c · ∇ · ∂~p − p c dt coll (C.10) conservative form Putting terms of the last equation in conservative form, we get ∂ ∂ p~a I = (p~aI) ~a ~nI (C.11) · ∂~p ∂~p · − · and ∂ ∂ ~p ~ ′~v I = ~p ~ ′~vI ~ ′ ~vI (C.12) · ∇ · ∂~p ∂~p · · ∇ − ∇ · with the result 1 d 1 d 1 1 ∂ 1 1 + ~n ~ ′ + ~n ~v + ~a ~n + ~ ′ ~v I p~aI + ~p ~ ′~vI c dt · ∇ c2 · dt c2 · c ∇ · − ∂~p · c2 c · ∇ 3 1 1 dI + ~a ~n + ~n ~ ′~v ~n I = . (C.13) c c · · ∇ · c dt coll local tetrad Introducing spherical coordinates in the local tetrad and using the photon energy ω = pc

∂ 1 1 p~aI + ~p ~ ′~vI = ∂~p · c2 c · ∇ or easily (~p = p ~n and p0 = p/c) and introducing a speciﬁc intensity I

1 ∂ 1 dI + ~n · ∇~ I = . » c ∂t – c „ dt «coll C.1. Radiation Transfer Equation 117

1 1 ∂ 1 ∂ = (p3 ~a ~nI)+ (p~aI) c2 p2 ∂p · p ∂~n 1 1 ∂ 2 1 ∂ + (p ~p ~ ′~v ~nI)+ (~p ~ ′~vI) = c p2 ∂p · ∇ · p ∂~n · ∇ 1 1 ∂I ∂ = p3 ~a ~n + I p3 ~a ~n c2 p2 · ∂p ∂p · 1 1 3 ∂I ∂ 3 1 ∂ 1 + p ~n ~ ′~v ~n + I p ~n ~ ′~v ~n + ~aI + ~n ~ ′~vI = c p2 · ∇ · ∂p ∂p · ∇ · c ∂~n · c · ∇ 1 1 ∂I 3 1 = ~a ~n + ~n ~ ′~v ~n ω + ~a ~n + ~n ~ ′~v ~n I c c · · ∇ · ∂ω c c · · ∇ · 1 ∂ 1 + ~aI + ~n ~ ′~vI = c ∂~n · c · ∇ 1 1 ∂ 2 1 = ~a ~n + ~n ~ ′~v ~n (ωI)+ ~a ~n + ~n ~ ′~v ~n I c c · · ∇ · ∂ω c c · · ∇ · 1 ∂ 1 + ~aI + ~n ~ ′~vI (C.14) c ∂~n · c · ∇ with the result 1 d 1 d 2 1 1 + ~n ~ ′ + ~n ~v + ~a ~n + ~ ′ ~v + ~n ~ ′~v ~n I (C.15) c dt · ∇ c2 · dt c2 · c ∇ · c · ∇ · 1 1 ∂ 1 ∂ 1 1 dI ~a ~n + ~n ~ ′~v ~n (ωI) ~aI + ~n ~ ′~vI = − c c · · ∇ · ∂ω − c ∂~n · c · ∇ c dt coll With continuity equation continuity equation

1 d 1 ρ d I + ~ ′ ~v I = (C.16) c dt c ∇ · c dt ρ and the relation 1 d 1 d 1 v2 ~n ~v I = (~n ~vI) ~a ~nI + (C.17) c2 · dt c2 dt · − c2 · O c2 we can cast the transfer equation into a more compact form

ρ d I 1 d 1 1 ∂ + (~v ~nI)+ ~n ~ ′ I + ~a ~n + ~n ~ ′~v ~n I (ωI) c dt ρ c2 dt · · ∇ c c · · ∇ · − ∂ω 1 ∂ 1 1 dI ~aI + ~n ~ ′~vI = (C.18) − c ∂~n · c · ∇ c dt coll 118 C. RADIATION TRANSFER

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, specific intensity the specific intensity can be written as

[ I(~x +∆~x, t +∆t; ~n, ν) I(~x, t; ~n, ν) ] dA dΩ dν dt = − [ η(~x, t; ~n, ν) χ(~x, t; ~n, ν) I(~x, t; ~n, ν) ] ds dA dΩ 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 d I(~x +∆~x, t +∆t; ~n, ν)= I(~x, t; ~n, ν)+ 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 ∂ ∂ + I(~x, t; ~n, ν)= η(~x, t; ~n, ν) χ(~x, t; ~n, ν) I(~x, t; ~n, ν) . (C.21) c ∂t ∂s − emissivity The emissivity coeﬃcient η is given by coeﬃcient η(~x, t; ~n, ν)= ηt(~x, t; ~n, ν)+ ηs(~x, t; ~n, ν) , (C.22)

extinction where ηt is the thermal and ηs the scattering part, and the extinction coefficient χ coefficient is 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

∂ d~n = ~n ~ + ~ . (C.24) ∂s · ∇ ds · ∇n 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 ) 1 J = J(~x, t; ν)= I(~x, t; ~n, ν) dΩ (C.25) ν 4π ZZ erg Monochromatic radiation energy density [ cm3 Hz ] 4π E = E(~n, t; ν)= J (C.26) ν c ν ~ erg Radiation Flux H [ cm2 s Hz ] 1 H~ = H~ (~n, t; ν)= I(~x, t; ~n, ν) ~ndΩ (C.27) ν 4π ZZ ~ erg Monochromatic radiation ﬂux F (~x, t; ν) [ cm2 s Hz ]

F~ν = F~ (~x, t; ν) = 4πH~ (~x, t; ν) (C.28)

K erg Radiation Pressure [ cm2 s Hz ] 1 K = I(~x, t; ~n, ν) ~n~ndΩ (C.29) ν 4π ZZ Radiation pressure (stress) tensor [ erg ] [ dyn ] cm3 Hz ≡ cm2 Hz 4π P = P(~x, t; ν)= K (C.30) ν c ν

Sometimes the equation of continuity can be taken into account to express the moments of the radiation equation

∂ ρ + ~ (ρ~v) = 0 . (C.31) ∂t ∇ · For a spherical geometry this equation is given as

d 1 ∂r2u ln ρ = u = . (C.32) dt −∇r · −r2 ∂r

C.1.4 Radiation Pressure Tensor Identities 1 P = P(~x, t; ν)= I(~x, t; ~n, ν) ~n~ndΩ (C.33) ν c ZZ 1 ∞ P = P(~x, t)= dν I(~x, t; ~n, ν) ~n~ndΩ (C.34) c Z0 ZZ 120 C. RADIATION TRANSFER

1 ∞ Pij = dν I(~x, t; ~n, ν) ni nj dΩ (C.35) c Z0 ZZ In spherical coordinates where dΩ = sinΘdΘdΦ, n1 = sin Θ cos Φ, n2 = sinΘsinΦ, n3 = cos Θ we get

2π π 1 P11 = cos2 ΦdΦ I sin3 ΘdΘ c Z0 Z0 1 1 2π 2π 1 = Idµ Iµ2dµ = (E P ) (C.36) c − c 2 − Z1 Z1 − − 2π π 1 P22 = sin2 ΦdΦ I sin3 ΘdΘ c Z0 Z0 1 1 2π 2π 1 = Idµ Iµ2dµ = (E P ) (C.37) c − c 2 − Z1 Z1 − − 2π π 1 P33 = dΦ I cos2 ΘsinΘdΘ c Z0 Z0 1 2π = Iµ2dµ = P (C.38) c Z1 − (C.39)

thus 1 (E P ) 0 0 2 − P = 0 1 (E P ) 0  2 −  0 0 P P 0 0 3P E 0 0 1 − = 0 P 0 0 3P E 0 (C.40)   − 2  −  0 0 P 0 0 0     and 1 1 ∂(r2P33) ∂P 3P E (~ P) = (P11 + P22)+ = + − (C.41) ∇ r −r r2 ∂r ∂r r and u ∂u (P : ~ ′~v) = (E P )+ P . (C.42) ∇ r r − ∂r C.2. 0th-order Moment Equation 121

C.2 0th-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])

d E 1 d 1 ∞ ρ + ~v F~ + ~ ′ F~ + P : ~ ′~v + ~a F~ = q (ω)dω (C.43) dt ρ c2 dt · ∇ · ∇ c2 · 0 Z0 1 2 3 4 5

| {z } | {z } | {z } | {z } | {z } Term 1: With the continuity equation we can write this term as d E d ρ = E + E ~ ~v = ∂ E + vk E + E vk . (C.44) dt ρ dt ∇ · t ∇k ;k Term 2: Is of order (v2/c2) and can be neglected. Term 3: This is the divergence of the radiation ﬂux

k ~ ′ F~ = F . (C.45) ∇ · ;k Term 4: This is the contraction of the radiation pressure tensor and the divergence of ~v j i P : ~ ′~v = P v . (C.46) ∇ i ;j Term 5: Is of order (v2/c2) and can be neglected.

C.2.1 Case 1: Spherical Geometry

For the expression in Eq. (C.46), we get ∂u u u (P : ~ ′~v) = P + (E P )= P ′ u + (E 3P ) (C.47) ∇ r ∂r r − ∇r · r − where u is the velocity component in the direction of r, thus Eq. (C.43) is d u E + E ′ u + ′ F + P ′ u + (E 3P )= q . (C.48) dt ∇r · ∇r · ∇r · r − 0 Introducing an other variable set (J, H,~ K) and dividing by 4π, we get 1 ∂ 1 1 u 1 1 J + (u ′ ) J + ′ H (3K J)+ J ′ u+ K ′ u = RHS . (C.49) c ∂t c ·∇r ∇r · − c r − c ∇r · c ∇r · In the case of spherical geometry Eq. (C.43) reduces to (Castor 1972 [22])

1 ∂ 1 1 u J + ′ (J u)= ′ H K ′ u (3K J) c ∂t c ∇r · −∇r · − c ∇r · − r − ρ (κ J κ S) (C.50) − J − S 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:

∂u a′ a′ P : ~ ′~v = P + u (E P )= P ′ u + u (E 3P ) , (C.51) ∇ ∂r 2a − ∇z · 2a − thus Eq. (C.43) is

d ua′ E + E ′ u + ′ F + P ′ u + (E 3P )= q . (C.52) dt ∇z · ∇z · ∇z · 2a − 0

Introducing an other variable set (J, H,~ K) and dividing by 4π, we have

1 ∂ 1 1 ua′ 1 1 J + (u ′ ) J + ′ H (3K J)+ J ′ u+ P ′ u = RHS . (C.53) c ∂t c ·∇z ∇z · − c 2a − c ∇z · c ∇z · In the case of ﬂux tube geometry Eq. (C.43) reduces to

1 ∂ 1 1 ua′ J + ′ (J u)= ′ H P ′ u (3K J) c ∂t c ∇z · −∇z · − c ∇z · − 2a − ρ (κ J κ S) (C.54) − J − S C.3. 1st-order Moment Equation 123

C.3 1st-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~ 1 d 1 1 ∞ + (~v P) + c ~ ′ P + ~aE + F~ ~ ′~v = q(ω)dω (C.55) c dt ρ ! c dt · ∇ · c c · ∇ Z0 1 2 3 4 5

| {z } | {z } | {z } | {z } | {z } Term 1: With the continuity equation we can write this term as

ρ d F~ 1 D 1 = F~ + F~ ~ ~v. (C.56) c dt ρ ! c Dt c · ∇ Term 2: Is of order (v2/c2) and can be neglected. Term 3: This is the ’divergence’ of the radiation pressure tensor

j ~ ′ P = P . (C.57) ∇ · i,i Term 4: Is of order (v2/c2) 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 1 (~ ′ P) = + (3P E) , (C.58) ∇ · r ∂r r −

(F~ ~ ′~v) = F ′ u , (C.59) · ∇ r ∇r · 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 ∂P c 2 F + c + (3P E)+ F ′ u = (~q) . (C.60) c dt ∂r r − c ∇r · r Introducing an other variable set (J, H,~ K) and dividing by 4π, we get 1 ∂ 1 ∂K 1 2 H + (u ′ ) H + + (3K J)+ H ′ u = RHS . (C.61) c ∂t c ·∇r ∂r r − c ∇r · In the case of spherical geometry Eq. (C.55) reduces to (Castor 1972 [22])

1 ∂ 1 ∂K 1 c H + ′ (H u)= H ′ u + (3K J) ρκ H . (C.62) c ∂t c ∇r · − ∂r − c ∇r · r − − H 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:

1 ∂(aP ) a ∂P a (~ P ) = + ′ (P E)= + ′ (3P E) , (C.63) ∇ · z a ∂z 2a − ∂z 2a − thus Eq. (C.55) is

1 d ∂P a′ 2 F + c + c (3P E)+ F ′ u = (~q) . (C.64) c dt ∂z 2a − c ∇z · z

Introducing an other variable set (J, H,~ K) and dividing by 4π, we have

1 ∂ 1 ∂K a′ 2 H + (u ′ ) H + + (3K J)+ H ′ u = RHS . (C.65) c ∂t c ·∇z ∂z 2a − c ∇z · In the case of ﬂux tube geometry Eq. (C.55) reduces to

1 ∂ 1 ∂K 1 ca′ H + ′ (H u)= H ′ u + (3K J) ρκ H . (C.66) c ∂t c ∇z · − ∂z − c ∇z · 2a − − H

C.4 Derivatives in diﬀerent geometries

The following table gives an overview of derivatives in diﬀerent geometries.

Derivative Geometries Spherical Cylindrical Flux Tube

2 u 1 ∂(r u) ∂u 1 ∂(au) ∇ξ1 r2 ∂r ∂z a ∂z ′ (~ K~ ) ∂K + 1 (3K J) ∂K ∂K + a (3K J) ∇ · ξ1 ∂r r − ∂z ∂z 2a − ′ (K : ~ u) ∂u K + u (J K) ∂u K ∂u K + ua (J K) ∇ ξ1 ∂r r − ∂z ∂z 2a −

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: 3 1 3 1 energy per volume and time [erg cm− s− ] or [J m− s− ] 1 ∂ 1 1 u 3K J J + (Ju) = H K u + − ρ(κ J κ S) (C.67) c ∂t c ∇ · −∇ · − c ∇ · c r − J − S 1 2 3 4 5 6

| {z } | {z } | {z } | {z } | {z } | {z } Integral notation (conservation form) Dimension of terms in the equation: 1 1 energy per time [erg s− ] or [J s− ] 1 ∂ 1 J dV + J u dA = H dV c ∂t c − ∇ · VZ ∂VZ VZ 1 2 3 1 1 3K J K u dV |+ {z −} | u dV{z }ρ (κ| J κ{z S) dV} (C.68) − c ∇ · c r − J − S VZ VZ VZ 4 5 6

Declaration| of{z terms: } | {z } | {z }

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

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: erg energy per volume and time [ cm3 s ] 1 ∂ 1 ∂ 1 3K J H + (Hu) = K H u − (κ ρ + χ )H (C.69) c ∂t c ∇ · −∂r − c ∇ · − r − H H 1 2 3 4 5 6

| {z } | {z } | {z } | {z } | {z } | {z } Integral notation (conservation form) Dimension of terms in the equation: erg energy per time [ s ] 1 ∂ 1 ∂ H dV + H u dA = K dV c ∂t c − ∂r VZ ∂VZ VZ 1 2 3 1 3K J H |u dV {z } |− dV{z }(κ |ρ + χ{z)H dV} (C.70) − c ∇ · − r − H H VZ VZ VZ 4 5 6

Declaration| of terms:{z } | {z } | {z }

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 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 −3 ρcond 2.25 gcm mass density of the condensed phase −2 σd 1400 ergcm 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 −24 mP 1.672610 g protonmass

D.2 Variables

Variable Dimension Description N,N∗ critical cluster size N∗,∞ tot −3 nH cm total number density of H −3 ρG g cm mass density of the gas component r0 cm monomer radius 1/τ s−1 net growth rate −1 −3 , ∗ s cm net transition rate, stationary nucleation rate ZJ J Zel’dovich-factor ΘN K surfacefreeenergy/k Θ∞ K surfacefreeenergy/kfor N 2 → ∞ AN cm surface of a dust grain −3 nc cm number density of all free C-atoms in the gas phase f(N,t) cm−3 number density of N-mers −3 Kj cm moments of the grain size distribution X KP 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. gas monomers H C phase dimers H2 C2 molecules C2H C2H2 CO dust dust grains phase

Table D.1: Chemical composition

Abundances

The chemical abundance of an element X is deﬁned by

n εX = . (D.4) n We can also say εgas = ε εdust (D.5) X X − X that means, that the abundance of an element X in the gas phase is the total tot abundance of X (εX = εX ) without the amount of X bound in dust grains.

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 gas tot ρ nH = , (D.6) (1 + 4 εHe) mP

where mP is the mass of a proton, and includes the free H atoms and the H2 dimers, thus tot nH = n = nH + 2nH2 . (D.7) D.3. Dust Formation 129

The total number density of carbon C is given by

tot gas dust nC = n = εC n = nC + nC , (D.8) where gas (gas) nC = nC + 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

(gas) gas nC = nmon = nC + 2 nC2 + 2 nC2H + 2 nC2H2 . (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

ndust = K = ntot n n(gas) = (ε ε ) n n(gas) . (D.11) C d C − − C C − O − C εC = 1 n “ εO − ” | {z } The fraction of condensable material actually condensed into grains is described by the degree of condensation

K ndust f = d = C . (D.12) cond gas (gas) Kd + nmon dust nC + nC

Partial Pressure

The partial pressure for the hydrogen H is generally given by the equation

tot tot PH = nH k T (D.13) and from Eq. (D.7) we get also

tot PH = PH + 2PH2 = 2 H2 = PH + 2PH KP . (D.14) To express the partial pressure of the atomar hydrogen H we look at the result of the quadratic equation

1 H2 tot PH = 1 1 + 8 K P , (D.15) −4 KH2 ± P H P q where only the negative sign gives a meaningful physical solution. Thus, it yields easily tot 2 PH PH = . (D.16) H2 tot 1+ 1 + 8 KP PH From Eq. (D.10) we get q

(gas) (gas) PC = kTnC = PC + 2PC2 + 2PC2H + 2PC2H2 = 2 C2 C2H 2 C2H2 = PC + 2PC (KP + PH KP + PH KP ) . (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

1 P = 1 1+8(. . .) P (gas) . (D.18) C −4 (. . .) ± C q Thus, it yields easily (as in Eq. (D.16))

(gas) 2 kTnC PC = Pmon = . (D.19) 1+ 1 + 8 kTn(gas) ( C2 + P C2H + P 2 C2H2 ) C KP H KP H KP q For the other molecules we get the partial pressures as follows

2 C2 PC2 = PC KP , (D.20)

2 C2H PC2H = PC PH KP , (D.21)

2 2 C2H2 PC2H2 = PC PH KP . (D.22)

The vapour saturation pressure of C1 is approximated by 86300 P (C ) = exp + 32.89 (D.23) sat 1 − T following Gail & Sedlmayr (1988 [49]). D.3. Dust Formation 131

D.3.2 Nucleation Theory

Transition Rate

The net transition rate represents the number of clusters of size N which are J ℓ 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

ΘN 2 fˆ(N)= n exp (N 1) ln (N 1) 3 (D.24) 1 − S− T − (Feder et al. 1966 [42]) with the ﬁrst

∂fˆ(N) 1 1 2 = ln 1 ΘN′ (N 1) + ΘN (D.25) ∂N S− T (N 1) 3 − 3 − and second derivative

2 ˆ 2 ∂ f(N) 2 2 ΘN′ 1 ΘN = (N 1) 3 (D.26) ∂N 2 −T − Θ − 9 (N 1)2 N − 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

Θ ΘN = ∞ 1 , . (D.27) 1 + [N /(N 1)] 3 h − In the limit N and assuming spherical N-mers Θ approaches a constant →∞ N 2 σd Θ = 4πr0 , (D.28) ∞ k 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

2 ∂ΘN Θ 1 Nh − 3 Nh ∞ =ΘN′ = 1 2 (D.29) ∂N 1 + [N /(N 1)] 3 2 3 N 1 (N 1) { h − } − − or simpliﬁed 1 1 ΘN [Nh/(N 1)] 3 ΘN′ = − 1 (D.30) 3 (N 1) 1 + [N /(N 1)] 3 − h − and 2 2 ∂ ΘN ΘN′ 4 1 =Θ′′ = 2 Θ′ . (D.31) ∂N 2 N Θ − 3 N N 1 N − 132 D. DUST PROPERTIES

The critical cluster size is given as the solution of

∂ ln fˆ = 0 (D.32) ∂N by 1 3 1 2 1 N , Nh 3 Nh 3 N =1+ ∗ ∞ 1+ 1 + 2 2 , (D.33) ∗ 8  " N , # − N ,   ∗ ∞ ∗ ∞  where   2Θ 3 N , = ∞ . (D.34) ∗ ∞ 3T ln S

The stationary nucleation rate is deﬁned by

I ˆ 2 = AN∗ f(N ) Z i vth(i) α(i) neﬀ (i) . (D.35) J∗ ∗ Xi=1 where 2 2 3 AN∗ = 4πr0 N (D.36) ∗ is the surface of a dust grain,

1 1 ∂2 ln fˆ(N) 2 Z = (D.37) 2π ∂N 2 " !N∗ # is the Zel’dovich-factor, k T vth(i)= (D.38) r2πmi denotes the thermal velocities of the corresponding species, α(i) are the sticking coefficients and P (i) n (i)= (D.39) eff k T are the effective number densities. D.3. Dust Formation 133

Net Growth Rate

The net growth rate is deﬁned by

I 1 1 1 = A1 i vth(i) α(i) f(i, t) 1 α (i) (D.40) τ − i b ∗ i=1 S i X′ I Mi c 1 1 c + i Ai vth(i, m) αm(i) ni,m 1 i c α (i, m) , ( − bi,m ∗ ) Xi=1 mX=1 S 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 thermodyna∗ mical and chemical equilibrium, and P = mon (D.41) S Psat 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 1 (T ) T = A i v (i) α(i) f(i, t) 1 Ki d g (D.42) τ 1 th − i (T ) T i=1 ( S Ki g s d ) X′ I Mi r 1 (Tg) (T ) c Ki,m i,m d + i Ai vth(i, m) αm(i) ni,m 1 i r K , ( − i,m(Td) i,m(Tg)) Xi=1 mX=1 S K K where denotes the dissociation constant of the molecule of the growth reaction K and r is the dissociation constant of the molecule involved in the reverse reaction. K If we substitute some quantities and write down all relevant terms of the row in Eq. (D.42), we get

1 2 1 1 1 Tg = 4π r αC PC 1 (D.43) τ 0 T √2πkm A − T g P mon ( S s d ! p 1 (T ) T + √2 α P 1 KC2 d g C2 C2 − 2 (T ) T " S KC2 g s d !

1 αC2H 1 C2H (Td) + PC2H 1 2 K 1 + 1/(2 A ) αC − C H (Tg) mon 2 S K 2 !

p 1 αC2H2 1 H2 (Tg) C2H2 (Td) + PC2H2 1 2 K K . 1 + 2/(2 A ) αC − H (Td) C H (Tg) mon 2 S K 2 K 2 2 !#) p 134 D. DUST PROPERTIES

D.3.3 Dust Physics

0th Moment Dust Equation The zeroth moment of the master dust equation (cf. H¨ofner 1994 [68]) describes the change of the number of dust grains due to creation of particles from gas phase or destruction of dust grains. ∂ K + (K u)= (D.44) ∂t 0 ∇ · 0 J

^ K ad δ([K ] ∆V )+∆( 0 δm )= δt ∆V (D.45) 0 l l − ρ l Jl l l denotes the net transition rate per volume from cluster sizes N < N to N > N . J ℓ ℓ 1st - 3rd Moment Dust Equation These moments of the master dust equation determine the time evolution of some means of the particle radius.

∂ j 1 j/d Kj + (Kj u)= Kj 1 + N (1 j d) (D.46) ∂t ∇ · d τ − ℓ J ≤ ≤

^ad Kj j 1 δ([Kj ]l ∆Vl)+∆( δml)= δt [Kj 1]l ∆Vl − ρ d τ − l l + δt N j/d ∆V (1 j d) (D.47) ℓ J l ≤ ≤ 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. ∂ 1 n + (n u)= K + N (1 j d) (D.48) ∂t c ∇ · c τ 2 ℓ J ≤ ≤

^ n ad 1 δ([n ] ∆V )+∆( c δm )= δt [K ] ∆V c l l − ρ l τ 2 l l l l + δt N ∆V (1 j d) (D.49) ℓ J l ≤ ≤ 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 ac- ceptance with the introduction of Einstein’s theory of general relativity, around 1915. General Relativity is formulated completely in the language of tensors, which Ein- stein 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 nth-rank tensor in m-dimensional space is a mathematical object that has n indices and mn 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 aν . 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 ν tensor a into a covariant tensor aµ (index lowering), use the metric tensor gµν to write ν gµν a = 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 Relation1 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 i i components ~a = a ~gi = ai~g , (E.2) where the connection of the vector space basis raised by the metric tensor

i ij ~g = g ~gj , (E.3) j ~gi = gij~g and (E.4) g = ~g ~g . (E.5) ij i · j The connection of the co- and contravariant metric tensor is

ij i g gjk = δk , (E.6)

ij i where (g ) and (gij) are inverse matrices and δk is the Kronecker delta Kronecker delta 1 for i = j δj = . (E.7) i 0 for i = j 6 The connection between the components of a vector is

j ji a = g ai , (E.8) i aj = gji a . (E.9)

The components of a vector are given by

ai = ~g i ~a or (E.10) · a = ~g ~a. (E.11) i i ·

The covariant physical components of ~a is deﬁned as physical components ⋆ (i) ai = a(i)= h a(i) , (E.12) where 1 h(i) g(i)(i) 2 (E.13) ≡ 1general geometry 138 E. TENSOR CALCULUS

are the scale factors. The length of a vector ~a is deﬁned as

~a 2 = (a(i))2 (E.14) | | Xi and is in R3 expressed with the contravariant components ai

2 2 2 ~a 2 = h a1 + h a2 + h a3 , (E.15) | | 1 2 3 where the associated physical components are

(i) a(i)= h(i)a . (E.16)

3 For the covariant components ai we get in R

3 2 2 2 2 2 1 1 1 ~a = (a(i)) = a1 + a2 + a3 . (E.17) | | h1 h2 h3 Xi=1 According to 1 2 g(i)(i) = (E.18) h (i) the physical components are related to covariant components by the expression 1 a(i)= a(i) . (E.19) h(i) orthonormal basis Special case: orthonormal basis, where

i ~e = ~ei (E.20)

~ei ~ej = δij (E.21)

2 permutation and the permutation symbol symbol +1 if (i, j, k) is an even permutation of (1, 2, 3) ~e (~e ~e )= ε = 1 if (i, j, k) is an odd permutation of (1, 2, 3) . (E.22) i· j × k ijk  −  0 otherwise  E.2.2 Operations and Operators

3 dot product The dot product can be written as

~a ~b ai g bj . (E.23) · ≡ ij 4 cross product The cross product can be written as

2The 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. 3The 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 n ~n “directional” gradient vector operator ∇ ∇ · | | ∂ ~n ~ + d~n ~ derivative along a path ∂s · ∇ ds · ∇n d ∂ ∂ dt ∂t + c ∂s hydrodynamical operator (coordinate system at rest) Table E.2: Summary of some speciﬁc vector operators.

(~a ~b) = ε ajbk , (E.24) × i ijk (~a ~b)i = εijka b . (E.25) × j k 5 The Nabla operator is deﬁned by Nabla operator ∂ 1 ∂ ~ = ~e ∂ = ~e = ~e (E.26) i i i i (i) ∇ ∂ξ(i) h(i) ∂ξ Introducing spherical coordinates in the local tetrad we can write for the terms 1 ∂ 1 ∂ 1 ∂ 1 ~n~ x = ~n~e + ~n~e + ~n~e x = ~n~e ∇ 1 1 h ∂x 2 h ∂x 3 h ∂x 1 h 1 1 1 2 2 3 3 1 1 = sin Θ cos Φ (E.27) h1 1 ~n~ x2 = sinΘsinΦ (E.28) ∇ h2 1 ~n~ x3 = cos Θ (E.29) ∇ h3 1 ~n~ Θ= ~n ~ (~n~e ) (E.30) ∇ −sinΘ ∇ 3 1 h n 1 oi cos Θ cos Φ ~n~ Φ= ~n ~ (~n~e1)+ ~ (~n~e3) , (E.31) ∇ −sinΦ sinΘ∇ sin2 Θ ∇ where ∂x 2 ∂y 2 ∂z 2 h = + + (i = 1, 2, 3) (E.32) (i) ∂x ∂x ∂x s i i i

4The cross product is also known as the vector product or outer product. In R3 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 ˛ ˛ ˛ ˛ ˛ ˛ ˛ 5The Nabla, also called “del” used to denote the gradient and other vector derivatives. In R3 we get 1 1 1 ∇~ = ~e1∂ξ1 + ~e2∂ξ2 + ~e3∂ξ3 . h1 h2 h3 140 E. TENSOR CALCULUS

are the scale factors and (x,y,z) represents the rectangular coordinate system (cf. Ue- sugi & Tsujita 1969 [150]). 6 vector divergence The divergence of a vector is the covariant derivative of the contravariant component, i.e. k k k m a;i = a,i +Γima , (E.33) whereas the covariant derivative of a covariant vector is

a = a Γma . (E.34) k;i k,i − ik m k m where Γim (or Γik) are the connection coefficients (also known as Christoffel symbols Christoffel or Ricci rotation coefficients) (see also Section E.4.1 about the metric tensor). With symbols these coefficients and the relation

i j 1 ∂hi Γij = Γii = (E.35) − hihj ∂ξj

we get the components

1 2 3 ~ 1 1 2 1 3 1 ∂a a ∂h1 a ∂h1 ( ~a)11 = a , 1 +Γ12 a +Γ13 a = + + (E.36) ∇ · h1 ∂ξ1 h1h2 ∂ξ2 h1h3 ∂ξ3 1 2 ~ 1 1 2 1 2 2 1 ∂a a ∂h2 ( ~a)21 = a , 2 +Γ12 a = a , 2 Γ11 a = (E.37) ∇ · − h2 ∂ξ2 − h1h2 ∂ξ1 1 3 ~ 1 1 3 1 3 3 1 ∂a a ∂h3 ( ~a)31 = a , 3 +Γ13 a = a , 3 Γ11 a = (E.38) ∇ · − h3 ∂ξ3 − h1h3 ∂ξ1 2 1 ~ 2 1 1 2 2 1 1 ∂a a ∂h1 ( ~a)12 = a , 1 +Γ22 a = a , 1 Γ21 a = (E.39) ∇ · − h1 ∂ξ1 − h1h2 ∂ξ2 2 1 3 ~ 2 2 1 2 3 1 ∂a a ∂h2 a ∂h2 ( ~a)22 = a , 2 +Γ21 a +Γ23 a = + + (E.40) ∇ · h2 ∂ξ2 h1h2 ∂ξ1 h2h3 ∂ξ3 2 3 ~ 2 3 3 2 2 3 1 ∂a a ∂h3 ( ~a)32 = a , 3 +Γ22 a = a , 3 Γ23 a = (E.41) ∇ · − h3 ∂ξ3 − h2h3 ∂ξ2 3 1 ~ 3 1 1 3 3 1 1 ∂a a ∂h1 ( ~a)13 = a , 1 +Γ33 a = a , 1 Γ31 a = (E.42) ∇ · − h1 ∂ξ1 − h1h3 ∂ξ3 3 2 ~ 3 2 2 3 3 2 1 ∂a a ∂h2 ( ~a)23 = a , 2 +Γ33 a = a , 2 Γ32 a = (E.43) ∇ · − h2 ∂ξ2 − h2h3 ∂ξ3 3 1 2 ~ 3 3 1 3 2 1 ∂a a ∂h3 a ∂h3 ( ~a)33 = a , 3 +Γ31 a +Γ32 a = + + (E.44) ∇ · h3 ∂ξ3 h1h3 ∂ξ1 h2h3 ∂ξ2 (E.45)

6For coordinate systems with constant basis, i.e. no derivatives of the basis exist, e.g cartesian coordinates

i ∇~ · 1 ∂a ~a = (i) h(i) ∂ξ

In R3

1 2 3 ∇~ · 1 ∂a 1 ∂a 1 ∂a ~a = 1 + 2 + 3 . h1 ∂ξ h2 ∂ξ h3 ∂ξ E.2. Vectors 141

The divergence of a vector in arbitrary orthogonal curvilinear coordinates can be calculated by

1 ∂ ∂ ∂ ~ ~a = (h h a1)+ (h h a2)+ (h h a3) . (E.46) ∇ · h h h ∂ξ1 2 3 ∂ξ2 3 1 ∂ξ3 1 2 1 2 3 3 The curl (only in R ) in arbitrary orthogonal curvilinear coordinates is curl

h1~g1 h2~g2 h3~g3 1 ∂ ∂ ∂ ~ ~a = 1 2 3 = ∇× h h h ∂ξ ∂ξ ∂ξ 1 2 3 a a a 1 2 3

1 ∂ 3 ∂ 2 (h3 a ) (h2 a ) ~g1 h h ∂ξ2 − ∂ξ3 2 3 1 ∂ ∂ + (h a1) (h a3) ~g h h ∂ξ3 1 − ∂ξ1 3 2 1 3 1 ∂ ∂ + (h a2) (h a1) ~g (E.47) h h ∂ξ1 2 − ∂ξ2 1 3 1 2 (~ is also denoted by rot for rotation). ∇× Scalar Triple Product scalar triple [~a,~b,~c] ~a (~b ~c) (E.48) product ≡ · ×

E.2.3 Relations / Vector Identities

~ (~a ~b) = (~a ~ )~b + ~a (~ ~b) + (~b ~ )~a +~b (~ ~a) (E.49) ∇ · · ∇ × ∇× · ∇ × ∇×

~ (~n ~e ) = (~n ~ )~e + (~e ~ )~n + ~n rot ~e + ~e rot ~n ∇ · i · ∇ i i · ∇ × i i × = (~n ~ )~e + ~n rot ~e (E.50) · ∇ i × i

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) × × ∇× ∇ − · ∇

a1 a2 a3 ~a (~b ~c)= ~b (~c ~a)= ~c (~a ~b) = det(~a~b~c)= b b b (E.53) · × · × · × 1 2 3 c c c 1 2 3

142 E. TENSOR CALCULUS

E.3 Tensors

E.3.1 Deﬁnitions linear transformation The Linear transformation of the base can be written as

k ~gi = a i ~gk (E.54) i i k ~g = ak ~g , (E.55) where the connection is given by the relation

k j j a i ak = δi , (E.56) k i i.e. a i und ak are inverse matrices. st 1 -rank tensor Tensor of tensor rank one (vector): the co- and contravariant components satis- ﬁes the transformation

j ti = a i tj (E.57) i i j t = aj t (E.58)

nd 2 -rank tensor Tensor of tensor rank two (matrix with appropriate transformation behaviour): the co- and contravariant components satisﬁes the transformation

k l tij = a i a j tkl (E.59) ij i j kl t = ak al t (E.60) In general following relations apply to tensors of tensor rank two

t = t ~e ~e (E.61) ij i ⊗ j ~aT = ai tik ~ek (E.62)

t~b = tij bj ~ei (E.63)

tij = ~ei t ~ej (E.64)

unity tensor A special case is the unity tensor

δ = δ ~e ~e = ~e ~e . (E.65) ij i ⊗ j i ⊗ i

E.3.2 Operations and Operators

tensor addition The tensor addition can be written as

i i i rjk + sjk = tjk (E.66)

tensor product The tensor product (dyadic product) can be written as

ij k ijk r sl = tl (E.67) E.3. Tensors 143

7 8 The tensor contraction (special case: trace ) can be written as tensor contraction

ij rj = ti (E.68)

Tensor contraction of a tensor product9

ij i r sjk = tk (E.69)

Correlation of vectors This operation is a tensor contraction of a tensor product frequently used in physics. vector correlation Index Raising i ij b = t aj (E.70)

Index Lowering j bi = tij a (E.71)

ij The transformation behaviour of t 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 tensor divergence covariant derivation ij ij i lj j il T ; k = T ,k +Γlk T +Γlk T (E.73) for a symmetric tensor

ij ij j 1 1 i kj ik 2 2 ij i jk S ; j = S ,j +Γkj S +Γkj S = g− (g S ),j +ΓjkS (E.74) for an antisymmetric tensor

ij 1 1 2 2 ij A ; j = g− (g A ),j (E.75)

The physical components of a tensor are deﬁned as physical components

(i)(j) T (i, j)= h(i)h(j)T (E.76) 1 T (i, j)= T(i)(j) , (E.77) h(i)h(j) where h(i) are the scale factors (cf. Section E.2.2).

7in german “Verj¨ungung” 8in german “Spur” 9in german “Uberschiebung”¨ 144 E. TENSOR CALCULUS

E.3.3 Relations / Tensor Identities

T = a b T = a b (E.78) ij i j → ii i i

A = B C B C , B C , B C , B C (E.79) ijkl ij kl → ij il ij kj ij ki ij jl =Aijil =Aijkj =Aijki =Aijjl | {z } | {z } | {z } | {z }

A : ~ ~a = A (~ ~a) + A (~ ~a) + A (~ ~a) + ∇ 11 ∇ 11 21 ∇ 21 31 ∇ 31 A (~ ~a) + A (~ ~a) + A (~ ~a) + 12 ∇ 12 22 ∇ 22 32 ∇ 32 A (~ ~a) + A (~ ~a) + A (~ ~a) (E.80) 13 ∇ 13 23 ∇ 23 33 ∇ 33 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 xi = f i(ξk) . (E.81)

The inﬁnitesimal line elements (or diﬀerential displacements) transforms as follows line element

i i k dx = akdξ , (E.82) where ∂f i ai = . (E.83) k ∂ξk

The metric tensor is determined by metric tensor

i i gkl(ξ)= glk(ξ)= akal (E.84) Xi and the inverse metric tensor can be calculated by the rule

il l gki(ξ)g (ξ)= δk , (E.85)

l where δk 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. 10 The quadratic line element follows from the generalised Pythagorean theorem quadratic line element 2 i k i l k l ds = akdξ aldξ = gkl(ξ)dξ dξ . (E.86) Xi If we assume an orthogonal coordinate system in three-space (R3), the metric is diagonal and the quadratic line element can be written

2 (1) 2 (2) 2 (3) 2 ds = (h1dx ) + (h2dx ) + (h3dx ) , (E.87) where hi is the scale factor. The equation of geodetics (like an equation of motion) of a free particle reads as equation of follows geodetics i i k l u¨ +Γklu˙ u˙ = 0 , (E.88) i where Γkl denotes the so-called Christoffel symbols (or connection coefficients) of Christoffel the first kind, which describe the occurrence of fictitious forces (or pseudo forces), symbols e.g. the Coriolis force. The coefficients are defined as 1 Γi := gim(g + g g ) . (E.89) kl 2 km,l lm,k − kl,m

10in german “Abstandsquadrat” 146 E. TENSOR CALCULUS

divergence The divergence is the covariant derivation of the contravariant component, i.e.

k k k i div(u)= u;k = u,k +Γkiu (E.90)

and accordingly the operator of the divergence can be written as

~ = 1/h2~e ∂ , (E.91) ∇ i i i Xi

symmetrised with the base ~e1,~e2,~e3 . The symmetrised covariant tensor is covariant tensor { } 1 u = (u + u ) , (E.92) (k;l) 2 k;l l;k where u = u Γi u . (E.93) k;l k,l − lk i E.4. Metric and Symmetries 147

E.4.2 Coordinate Systems

Cartesian Coordinates

Orthogonal coordinate system:

(ξ1,ξ2,ξ3) = (x,y,z) (E.94)

Scale factors: (h1, h2, h3) = (1, 1, 1) (E.95) Radius vector: x ~r = y (E.96)   z Local tetrad with spherical coordinates: 

(x,y,z, Θ, Φ) (E.97)

Operators:

~n~ ξ = ~n~ x = sin Θ cos Φ (E.98) ∇ 1 ∇ ~n~ ξ = ~n~ y = sin Θ sin Φ (E.99) ∇ 2 ∇ ~n~ ξ = ~n~ z = cos Θ (E.100) ∇ 3 ∇ ~n~ Θ = 0 (E.101) ∇ ~n~ Φ = 0 (E.102) ∇ ∂ ∂ ∂ ~n~ = sin Θ cos Φ + sinΘsinΦ + cos Θ (E.103) ∇ ∂x ∂y ∂z 148 E. TENSOR CALCULUS

Cylindrical Coordinates

Orthogonal coordinate system:

(ξ1,ξ2,ξ3) = (ρ, φ, z) (E.104)

Scale factors: (h1, h2, h3) = (1, ρ, 1) (E.105) Radius vector: ρ cos φ ~r = ρ sin φ (E.106)   z Local tetrad with spherical coordinates: 

(ρ, φ, z, Θ, Φ) (E.107)

Operators:

~n~ ξ = ~n~ ρ = sin Θ cos Φ (E.108) ∇ 1 ∇ 1 ~n~ ξ = ~n~ φ = sin Θ sin Φ (E.109) ∇ 2 ∇ ρ ~n~ ξ = ~n~ z = cos Θ (E.110) ∇ 3 ∇ ~n~ Θ = 0 (E.111) ∇ 1 ~n~ Φ= sin Θ sin Φ (E.112) ∇ −ρ

∂ 1 ∂ ∂ 1 ∂ ~n~ = sin Θ cos Φ + sinΘsinΦ + cos Θ sinΘsinΦ (E.113) ∇ ∂ρ ρ ∂φ ∂z − ρ ∂Φ

(~ ~v) = 0 (E.114) ∇ · 11 (~ ~v) = 0 (E.115) ∇ · 22 ∂u (~ ~v) = (E.116) ∇ · 33 ∂z (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:

(ξ1,ξ2,ξ3) = (θ, φ, r) (E.118) Scale factors: (h1, h2, h3) = (r, r sinΘ, 1) (E.119) Ricci rotation coeﬃcients:

Γ1 = sin θ cos θ (E.120) 22 − 1 Γ1 =Γ1 = (E.121) 31 13 r 2 2 Γ12 =Γ21 = cot θ (E.122) 1 Γ2 =Γ2 = (E.123) 32 23 r Γ3 = r (E.124) 11 − Γ3 = r sin2 θ (E.125) 22 − Metric:

1 2 g 2 = r sin Θ (E.126) 1 1 g− 2 = (E.127) r2 sin θ Radius Vector: sin θ cos φ ~r = r sin θ sin φ (E.128)   cos θ Local tetrad with spherical coordinates: 

(θ, φ, r, Θ, Φ) (E.129)

Operators: 1 ~n~ ξ = ~n~ θ = sin Θ cos Φ (E.130) ∇ 1 ∇ r sinΘsinΦ ~n~ ξ = ~n~ φ = (E.131) ∇ 2 ∇ r sin θ ~n~ ξ = ~n~ r = cos Θ (E.132) ∇ 3 ∇ 1 ~n~ Θ= sin Θ (E.133) ∇ −r sinΘsinΦ ~n~ Φ= (E.134) ∇ − r tan θ

1 ∂ sinΘsinΦ ∂ ∂ 1 ∂ sinΘsinΦ ∂ ~n~ = sin Θ cos Φ + + cos Θ sinΘ ∇ r ∂θ r sin θ ∂φ ∂r − r ∂Θ − r tan θ ∂Φ (E.135) 150 E. TENSOR CALCULUS

Spherical symmetry: ∂ ∂ ∂ 0 (E.136) ∂θ ≡ ∂φ ≡ ∂Φ ≡ Thus ∂ sinΘ ∂ ~n~ = cos Θ (E.137) ∇ ∂r − r ∂Θ Introducing µ = cos θ ∂ 1 µ2 ∂ ~n~ = µ + − (E.138) ∇ ∂r r ∂Θ Plane parallel geometry, i.e. r →∞ ∂ ~n~ = µ (E.139) ∇ ∂r Components of the divergence of velocity ~v = (0, 0, u): 1 ∂v1 v2 ∂h v3 ∂h u (~ ~v) = + 1 + 1 = (E.140) ∇ · 11 r ∂θ r2 sin θ ∂φ r ∂r r 1 ∂v1 v2 ∂h (~ ~v) = 2 = 0 (E.141) ∇ · 21 r sin θ ∂φ − r2 sin θ ∂θ ∂v1 v3 ∂h (~ ~v) = 3 = 0 (E.142) ∇ · 31 ∂r − r ∂θ 1 ∂v2 v1 ∂h (~ ~v) = 1 = 0 (E.143) ∇ · 12 r ∂θ − r2 sin θ ∂φ 1 ∂v2 v1 ∂h v3 ∂h u (~ ~v) = + 2 + 2 = (E.144) ∇ · 22 r sin θ ∂φ r2 sin θ ∂θ r sin θ ∂r r ∂v2 v3 ∂h (~ ~v) = 3 = 0 (E.145) ∇ · 32 ∂r − r sin θ ∂φ 1 ∂v3 v1 ∂h 1 ∂u (~ ~v) = 1 = (E.146) ∇ · 13 r ∂θ − r ∂r r ∂θ 1 ∂v3 v2 ∂h 1 ∂u (~ ~v) = 2 = (E.147) ∇ · 23 r sin θ ∂φ − r sin θ ∂r r sin θ ∂φ ∂v3 v1 ∂h v2 ∂h ∂u (~ ~v) = + 3 + 3 = (E.148) ∇ · 33 ∂r r ∂θ r sin θ ∂φ ∂r (E.149)

∂h1 ∂h2 ∂h2 where ∂r = 1, ∂r = sin θ, ∂θ = r cos θ, and all others are zero. Or

3 2 1 ∂ v1 + v 1 ∂ v1 v r cos θ ∂ v1 r θ r r sin θ φ − r2 sin θ r   ~ 1 2 1 2 v3 v1 2 ( ~v)=  r ∂θv r sin θ ∂φv + r sin θ sin θ + r2 sin θ r cos θ ∂rv  = ∇ ·      1 3 v1 1 3 v2 3   ∂θv ∂φv sin θ ∂rv   r − r r sin θ − r sin θ     u  r 0 0  u  (E.150) 0 r 0  1 1   ∂θu ∂φu ∂ru   r r sin θ    E.4. Metric and Symmetries 151

Thus 1 ∂u 1 ∂u 1 ∂(r2u) ~ ~v = + + (E.151) ∇ · r ∂θ r sin θ ∂φ r2 ∂r The unit vector ~n is

2 1/2 nθ sin Θ cos Φ (1 µ ) cos Φ − 2 1/2 ~n(Θ, Φ) = nφ = sinΘsinΦ = (1 µ ) sinΦ (E.152)      −  nr cos Θ µ       were µ = cos Θ, the spatial gradient vector operator 1 1 ~ = ~e ∂ + ~e ∂ + ~e ∂ (E.153) ∇ r r θ r θ φ r sin θ φ the derivation d~n dΘ dΦ = ~e + ~e sin θ (E.154) ds Θ ds Φ ds and the ,,directional” gradient vector operator 1 ~ = ~e ∂ + ~e ∂ . (E.155) ∇n Θ Θ Φ sin θ Φ Or

~ 1 1 1 1 1 = ~e1∂ξ1 + ~e2∂ξ2 + ~e3∂ξ3 = ~eθ∂θ + ~eφ∂φ + ~er∂r (E.156) ∇ h1 h2 h3 r r sin θ 1 1 1 ~ ~a = ∂ (sin θa1)+ ∂ a2 + ∂ (r2 a3) (E.157) ∇ · r sin θ θ r sin θ φ r2 r Divergence of a tensor (radial term): 11 1 T = r2 Tθθ 12 1 T = r2 sin θ Tθφ 22 1 T = (r sin θ)2 Tφφ 31 1 T = r Trθ 32 1 T = r sin θ Trφ 33 T = Trr

3j 3j 3 lj j 3l T ; j = T ,j +ΓljT +ΓlkT = 31 32 33 3 11 3 22 1 33 2 33 2 31 T , 1 + T , 2 + T , 3 +Γ11T +Γ22T +Γ31T +Γ32T +Γ12T = 1 1 1 1 1 1 cot θ ∂ T + ∂ T + ∂ T T T + T + T + T = r θ rθ r sin θ φ rφ r rr − r θθ − r φφ r rr r rr r rθ 1 1 1 1 ∂ (r2 T )+ ∂ (sin θ T )+ ∂ T (T + T )= r2 r rr r sin θ θ rθ r sin θ φ rφ − r θθ φφ 1 1 ∂ (r2 T ) (T + T ) (E.158) r2 r rr − r θθ φφ 1 1 T 1j = ∂ (sin θ T ) cot T (E.159) ; j r2 sin θ θ θθ − r φφ 1 T 2j = ∂ T (E.160) ; j (r sin θ)2 φ φφ 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 Equation1 integrated mass m 3 Continuity Equation density ρ 4 Momentum Equation2 velocity u 5 Energy Equation internal energy e 6 Radiation Energy Equation radiation energy J 7 Radiation Flux Equation radiation ﬂux H th th 8 0 Moment Dust Equation 0 moment, dust density K0 st st 9 1 Moment Dust Equation 1 moment, mean radius of grains K1 nd nd 10 2 Moment Dust Equation 2 moment K2 rd rd 11 3 Moment Dust Equation 3 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.

1also called Poisson Equation 2also 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)

∂ 4π (ρu)+ (ρu u)= P ρ ψ + ρ κ H + f (F.2) ∂t ∇ · −∇ − ∇ c H drag Energy Equation ∂ (ρe)+ (ρe u)= P u + 4π ρ (κ J κ S ) (F.3) ∂t ∇ · − ∇ · J − S g Radiation Energy Equation 1 ∂ 1 1 u 3K J J + (Ju)= H K u+ − ρ (κ J κ S ) (χ J χ S ) c ∂t c ∇· −∇· − c ∇· c r − J − S g − J − S d (F.4) Radiation Flux Equation 1 ∂ 1 ∂ 3K J 1 H + (Hu)= K − H u ρ κ H χ H (F.5) c ∂t c ∇ · −∂r − r − c ∇ · − H − H 0th Moment Dust Equation ∂ K + (K u)= (F.6) ∂t 0 ∇ · 0 J 1st - 3rd Moment Dust Equation

∂ j 1 j/d Kj + (Kj u)= Kj 1 + N (1 j d) (F.7) ∂t ∇ · d τ − ℓ J ≤ ≤ F.2. Integrated Form 155

F.2 Integrated Form

Continuity Equation ∂ ρ dV + ρ u dA = 0 (F.8) ∂t VZ ∂VI

Momentum Equation (Equation of Motion)

∂ ρu dV + ρu u dA = P dV ∂t − ∇ VZ ∂VI VZ ρ ψ dV − ∇ VZ 4π + ρ κ H dV c H VZ

+ fdrag dV (F.9) VZ Energy Equation ∂ ρe dV + ρe u dA = P u dV ∂t − ∇ · VZ ∂VI VZ +4π ρ (κ J κ S ) dV (F.10) J − S g VZ Radiation Energy Equation 1 ∂ 1 J dV + J u dA = H dV c ∂t c − ∇ · VZ ∂VI VZ 1 K u dV − c ∇ · VZ 1 3K J + − u dV c r VZ ρ (κ J κ S) dV (F.11) − J − S VZ Radiation Momentum Equation 1 ∂ 1 ∂ H dV + H u dA = K dV c ∂t c − ∂r VZ ∂VI VZ 1 H u dV − c ∇ · VZ 156 F. FULL SET OF RHD EQUATIONS

3K J − dV − r VZ (κ ρ + χ )H dV (F.12) − H H VZ

0th Moment Dust Equation ∂ K dV + K u dA = dV (F.13) ∂t 0 0 J VZ ∂VI VZ

1st - 3rd Moment Dust Equations ∂ j 1 Kj dV + Kj u dA = Kj 1 dV ∂t d τ − VZ ∂VI VZ + N j/d dV (1 j d) (F.14) ℓ J ≤ ≤ VZ F.3. Discretised Form 157

F.3 Discretised Form

Continuity Equation [g]

2 ad rel δ(ρl ∆Vl)+ δt ∆(rl ρl ul ) = 0 (F.15)

Momentum Equation (Equation of Motion)f [dyn s]

δ(u ρ ∆V )+∆( uad δm ) = δt r2 ∆P l l l − l l − l l ml f δt 4πG 2 ρl∆Vl − rl 4π +δt (κg + κd) H ρ ∆V c l l l l l +δt QMU (F.16)

Energy Equation [erg]

δ(e ρ ∆V )+∆( ead δm ) = δt P ∆(r2 u ) l l l − l l − l l l g f +δt 4π κ ρl (Jl [Sg]l)∆Vl l − +δt QME (F.17)

erg cm Radiation Energy Equation [ s ] ^ J ad δ(J ∆V )+∆( δm ) = c δt ∆(r2 H ) l l − ρ l − l l l δt K ∆(r2 u ) − l l l ul +δt (3Kl Jl) ∆Vl − rl cδtκg ρ (J [S ] )∆V (F.18) − l l l − g l l erg cm Radiation Momentum Equation [ s ]

δ(H ∆V )+ δt ∆(r2 Had urel) = cδtr2 ∆K l l l l l − l l 2 g δt Hl r ∆ul − l 3Kl Jl c δt − ∆Vl − rl c δt (κg + κd) H ρ ∆V (F.19) − l l l l l 0th Moment Dust Equation [1] ^ K ad δ([K ] ∆V )+∆( 0 δm )= δt ∆V (F.20) 0 l l − ρ l Jl l l 158 F. FULL SET OF RHD EQUATIONS

1st - 3rd Moment Dust Equations [1]

^ad Kj j 1 δ([Kj ]l ∆Vl)+∆( δml) = δt [Kj 1]l ∆Vl − ρ d τ − l l +δt N j/d ∆V (1 j d) (F.21) ℓ Jl l ≤ ≤ Dust Density Equation [1] ^ n ad 1 δ([n ] ∆V ) ∆( c δm ) = δt [K ] ∆V c l l − − ρ l τ 2 l l l l +δt N ∆V (F.22) ℓ Jl l Mass Equation (Poisson Equation) [g]

∆m ρ ∆V = 0 (F.23) l − l l

Artiﬁcial Viscosity Contributions

2 ∆u u¯ 2 erg QME = µ ρ l l ∆V [ ] (F.24) −3 Q,l l ∆r − r¯ l s l l 2 ∆u u¯ QMU = ∆ µ ρ l l [dyn] (F.25) −3 r Q,l l ∆r − r¯ l l l where ∆(r2 u ) cm2 µ = ℓ c ℓ2 min l l , 0 [ ] (F.26) Q,l 1 s,l − 2 ∆V s l Appendix G

Symbols, Constants and Abbreviations

G.1 Symbols

Symbol Units Comment cgs SI

mr g kg integrated mass 3 3 ρ g cm− kg m− gas density 1 1 e erg g− J kg− specific internal energy of the gas 1 1 ~v, u cm s− m s− gas velocity 2 1 2 J erg cm− s− W m− zeroth moment of the radiation field 3 3 E erg cm− J m radiation energy density 2 1 2 H~ , H erg cm− s− W m− first moment of the radiation field 2 1 2 F~ , F erg cm− s− W m− radiation energy flux 2 1 2 K, K erg cm− s− W m− second order moment of the radiation field 3 3 P, P erg cm− J m radiation pressure 3 3 Kj cm− m− moments of the grain size distribution

2 2 P dyn cm− N m− gas pressure Tg K K gas temperature Tr K K radiation temperature Td K K dust temperature 2 1 2 Sg erg cm− s− W m− source function of the gas 2 1 2 Sd erg cm− s− W m− source function of the dust g 2 1 2 1 κ cm g− m kg− mass absorption coeﬃcient of the gas d 2 1 2 1 κ cm g− m kg− mass absorption coeﬃcient of the gas fedd Eddington factor 1 1 1 τ s− s− net growth rate of the dust grains 1 3 1 3 s− cm− s− m− net grain formation rate per volume J 3 3 ngr cm− m− 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

10 1 c 2.997 924 58 10 cm s− speed of light in vacuum (exact) 8 1 2.997 924 58 10 ms−

8 2 2 G 6.6742(10) 10− dyn cm g− Newtonian constant of gravitation 11 3 1 2 6.6742(10) 10− m kg− s−

5 2 1 4 σ 5.670 400(40) 10− erg cm− s− K− Stefan-Boltzmann constant 8 2 4 5.670 400(40) 10− Wm− K−

16 1 k 1.380 6505(24) 10− erg K− Boltzmann constant 23 1 1.380 6505(24) 10− JK−

7 1 1 R 8.314 472(15) 10 erg mol− K− molar gas constant 1 1 8.314 472(15) J mol− K−

24 mP 1.672 621 71(29) 10− g proton mass 27 1.672 621 71(29) 10− kg

G.3 Astronomical Constants

M 1.989 1033 g solar mass ⊙ 1.989 1030 kg

33 1 L 3.826 10 erg s− solar luminosity ⊙ 3.826 1026 W

R 6.9598 1010 cm solar radius ⊙ 6.9598 108 m

AU 1.49598 1013 cm astronomical unit 1.49598 1011 m

ly 9.46053 1017 cm light year 9.46053 1015 m

pc 3.08568 1018 cm parsec 3.08568 1016 m

27 MEarth 5.79 10 g mass of Earth 5.79 1024 kg

8 REarth 6.378 137 10 cm equatorial radius of the Earth (IUGG value) 6.378 137 106 m G.4. Abbreviations 161

G.4 Abbreviations

ACS Advanced Camera for Surveys OH OH source AFGL Air Force Geophysical Laboratory AGB Asymptotic Giant Branch PG Palomar-Green catalog AU Astronomical Unit PN Planetary Nebula AURA Association of Universities for Research PPN Proto-Planetary Nebula in Astronomy PRC Public Resources Code

CRL Cambridge Research Laboratory RGB Red Giant Branch CSE CircumStellar Envelope RHD Radiation HydroDynamic RTE Radiation Transfer Equation EOS Equation Of State ESA European Space Agency SIRTF Space InfraRed Telescope Facility ESO European Southern Observatory SST Spitzer Space Telescope ST-ECF Space Telescope - European Coordinating FLIER Fast Low-Ionization Emission Region Facility FOC Faint Object Camera STScI Space Telescope Science Institute

Hb Hubble catalog UMIST University of Manchester Institute of HD H. Draper catalog Science and Technology HEIC Hubble European Space Agency Information Centre VLTI Very Large Telescope Interferometer HR Harvard Revised catalog WD White Dwarf HRD Hertzsprung-Russell Diagram WFC Wide Field Channel HST Hubble Space Telescope WFPC Wide Field Planetary Camera IAC Instituto de Astrof´ısica de Canarias Wr Wray catalog IAU International Astronomical Union XMM X-ray Multi-mirror Mission 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 162 G. SYMBOLS, CONSTANTS AND ABBREVIATIONS List of Tables

4.1 Starswithspots ...... 58

5.1 Modelstars ...... 74 5.2 Modelfluxtubes ...... 77 5.3 Effectsofchemistry(1) ...... 81 5.4 Effectsofchemistry(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 Chemicalcomposition ...... 128

E.1 Deﬁnitions of some vector symbols ...... 137 E.2 Summary of some speciﬁc vector operators ...... 139

F.1 ListofRHDequations ...... 153

163 164 LIST OF TABLES List of Figures

1.1 Evolutionary tracks in the Hertzsprung-Russell-Diagram...... 6

2.1 HaloofPPNNGC7027 ...... 15 2.2 Details of PPN NGC 7027 ...... 15 2.3 Halo of PPN CRL 2688 (Egg Nebula) ...... 16 2.4 Infrared-detailsofPPNCRL2688 ...... 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 ThePNNGC6720...... 18 2.9 Details in the PN NGC 6720 ...... 18 2.10 HaloofPNNGC6720inInfrared...... 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 ThePNNGC2392(EskimoNebula) ...... 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 ThePNIC418(SpirographNebula) ...... 24 2.20ThePNNGC6751...... 24 2.21 Halo of the PN NGC 6543 (Cat’s Eye Nebula) ...... 25 2.22ThePNNGC6543...... 25 2.23 ThePNMyCn18(HourglassNebula) ...... 26 2.24 ThePNIC4406(RetinaNebula) ...... 26 2.25 The PN NGC 6302 (Bug or Butterﬂy Nebula) ...... 27 2.26 Details of the PN NGC 6302 ...... 27 2.27 ThePNMz3(AntNebula) ...... 28 2.28ThePNM2-9...... 28

165 166 LIST OF FIGURES

3.1 Computational domain of the initial model ...... 47 3.2 FlowchartoftheRHDcode ...... 53

4.1 Modelofafluxtubeonastellarsurface ...... 64 4.2 Radiativefluxthroughafluxtube ...... 67 4.3 Volumeofafluxtubesegment ...... 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 Modelfluxtubes ...... 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 κ- mechanisminsphericalgeometry ...... 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 tubegeometry ...... 84 5.13 Spatial structure of a wind from dust-induced κ-mechanism in flux tubegeometry ...... 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 =100K...... 87 5.17 Occurrence of a dust-driven Wind at ∆T =300K ...... 87 5.18 Magneticpressure ...... 88 5.19 Dependence of temperature difference ∆T fortheSun ...... 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 Degreeofcondensation...... 92

6.1 ModelofAGBstarwithacoolspot ...... 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 Observa- tory 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 Bibliography

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