On the Winds of Carbon Stars and the Origin of Carbon - DocsLib

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List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I L. Mattsson,R.Wahlin&S.Höfner,"Dust Driven Mass Loss from Carbon Stars as Function of Stellar Parameters I. - A Grid of Solar-metallicity Wind Models", submitted to Astronomy & Astrophysics (2009) II L. Mattsson, R. Wahlin, S. Höfner & K. Eriksson, "Intense mass loss from C-rich AGB stars at low metallicity?", published in Astronomy & Astrophysics, 484, L5 (2008) III L. Mattsson &S.Höfner,"Dust Driven Mass Loss from Carbon Stars as Function of Stellar Parameters II. - Effects of Relaxing the Small Particle Approximation", preprint (2009) IV L. Mattsson,S.Höfner&F.Herwig,"Mass Loss Evolution and the For- mation of Detached Shells around TP-AGB Stars", published in Astron- omy & Astrophysics, 470, 339 (2007) V L. Mattsson, F. Herwig, S. Höfner, R. Wahlin, M. Lederer, & B. Paxton, "Effects of Carbon-excess Dependent Mass Loss and Molecular Opaci- ties on Models of C-star Evolution", preprint (2009) VI L. Mattsson,"The Origin of Carbon: Low-mass Stars and an Evolving IMF?", preprint (2009)

Reprints were made with permission from the publishers.

Contents

1 Introduction ...... 9 1.1 SettingtheStage...... 9 1.2 EvolutionofLIMStars...... 10 1.3 WindsofCarbonStars...... 11 1.4 TheCosmicMatterCycle:theOriginofCarbon...... 12 1.5 Carbon-basedLifeintheGalaxy...... 13 2 Modelling of Dynamic Atmospheres and Dust Driven Winds ...... 15 2.1 The Success of Numerical Modelling ...... 15 2.2 TheFullSetofEquations...... 16 2.2.1 RadiationHydrodynamics...... 16 2.2.2 Selfgravity...... 18 2.2.3 Equation of State and other aPrioriAssumptions ...... 18 2.2.4 RadiativeTransfer...... 19 2.2.5 DustFormationandOpacity...... 21 2.3 MethodofComputation...... 23 2.3.1 BoundaryConditions...... 23 2.3.2 InitialConditions...... 24 2.3.3 The Modelling Procedure ...... 25 3 CarbonStarMassLossasFunctionofStellarParameters...... 27 3.1 TheGridofWindModels...... 27 3.1.1 GeneralProperties...... 27 3.1.2 Mass-LossThresholdsandtheCarbonExcess ...... 28 3.2 Metallicity Dependence? ...... 32 4 RelaxingtheSmallParticleApproximation...... 35 4.1 SizeableGrains...... 35 4.2 HowBadistheSPA?...... 36 4.3 SizeMatters!...... 37 5 StellarStructureandEvolution ...... 41 5.1 BasicIngredients ...... 41 5.1.1 Hydrostatic Equilibrium and Energy Generation ...... 41 5.1.2 EnergyTransport...... 42 5.1.3 ConvectionintheMLTPicture...... 43 5.2 RadiativeTransferandtheRôleofGasOpacities ...... 44 5.3 StellarEvolutioninBrief...... 45 5.4 Nucleosynthesis...... 47 5.4.1 TheTriple-alphaProcess...... 47 5.4.2 TheCN-cycle ...... 47 5.5 EffectsofMassLossandMixing...... 49 5.5.1 Carbon Production: Dredge-up and Thermal Pulses . . . . . 50 5.5.2 Carbon Destruction: The on-set of HBB and AGB life times 50 6 Mass-LossEvolutionandtheOriginofDetachedShells...... 51 6.1 AConsequenceofHeliumShellFlashes?...... 51 6.2 Two-windInteraction...... 52 6.3 PulsationsareCritical ...... 53 6.4 SoHowDoesDetachedShellsForm?...... 55 7 ModelsofCarbonStarEvolution...... 57 7.1 Mass-lossEffectsonC-starEvolution...... 57 7.2 TheOriginoftheSuperwind...... 58 7.3 LimitedNucleosynthesis...... 59 8 GalacticChemicalEvolutionModels...... 61 8.1 TheMilkyWayGalaxy...... 61 8.1.1 TheGalacticDisc...... 61 8.1.2 TheStellarHaloandthe"ThickDisc" ...... 62 8.2 OriginoftheElements...... 62 8.3 ModelIngredientsandEquations...... 63 8.3.1 GalaxyFormation...... 64 8.3.2 GalacticDynamics...... 65 8.3.3 StarFormation...... 66 8.3.4 TheInitialMassFunction...... 66 8.3.5 EvolutionoftheISMandElementalRatios...... 67 8.3.6 SupernovaetypeIa...... 68 8.4 StellarYields...... 69 9 AnExtensionofthePadoan-NordlundTheoryoftheIMF ...... 71 9.1 AUniversalIMF?...... 71 9.2 TurbulentFragmentation ...... 71 9.3 EnvironmentalVariationsoftheIMF...... 75 9.4 AlternativeMechanisms ...... 77 10TheOriginofCarbon ...... 79 10.1TheMakingofCarbon...... 79 10.2ATop-heavyIMF?...... 80 10.3LowMassStarsandanEvolvingIMF? ...... 82 11ConcludingRemarksandFutureProspects ...... 85 11.1FutureWork ...... 85 11.1.1 Subsolar Metallicity ...... 85 11.1.2DriftandGrainSizes ...... 85 11.1.3DetachedShells ...... 86 11.1.4GalacticChemicalEvolution...... 86 11.2LifefromaWiderPerspective...... 86 11.2.1AGalacticHabitableZone?...... 86 11.2.2CarbonStarsandCarbon-basedLife...... 88 ContributionstoIncludedPapers...... 91 Sammanfattning...... 93 Acknowledgements ...... 97 Bibliography ...... 99 1. Introduction

"We are all in the gutter, but some of us are looking at the stars."

Oscar Wilde

This is a thesis about stars, and to some extent galaxies. More precisely, it is a thesis about the gas, dust and stars that we ﬁnd in our Galaxy and how it all is evolving. The content may at ﬁrst glance seem as if it consists of sep- arate parts, but that is not the case. This introductory chapter presents the context of this thesis in an attempt to explain how models of the build-up of heavy elements in the Universe actually connect with models of dynamic stellar atmospheres.

1.1 Setting the Stage There exist many fundamental, unanswered questions in the natural sci- ences, not the least in astrophysics. Such questions are the origin and early evolution of the universe, as well as the formation and evolution of galaxies. Another question of the same magnitude is the origin of life in the universe (or at least on Earth). Life, as we know it, requires the existence of carbon, nitrogen, oxygen and a few other elements. However, with the exception of hydrogen, helium and a small amount of lithium, all the atomic nuclei in the universe are created in stars, through nuclear reactions (referred to as nucleosynthesis), where light elements are combined into heavier elements. These elements have then been expelled from the stars, either through relatively slow mass loss (stellar winds), as in the case of low and intermediate mass stars, or explosively, as in the case of high mass stars. Thus, the interstellar medium (ISM) in a galaxy is continuously enriched in heavy elements as new generations of stars form, evolve and expel their newly synthesised elements. This is what is referred to as the chemical evolution of galaxies, which in the strict sense of the word has little to do with chemistry, but rather the build-up of heavy elements over time. These new elements then form molecules and dust in stars and interstellar space and are later incorporated into new generations of stars, planets and eventually life on at least one planet. In this thesis, which is part of a rather ambitious project aiming at a better understanding of the rôle of low- and intermediate-mass (LIM) stars in this "big scheme", results are presented regarding the theoretical modelling of mass-loss from carbon stars and the cosmic origin of atomic carbon as well

9 as carbon dust. Some attention is also given to chemical evolution models (CEMs) and their basic ingredients. The nucleosynthetic input (i.e. stellar yields) in such models is in some cases not known to a sufﬁcient degree of accuracy. For LIM stars, the rate of mass loss during the late stages of evolution is critical for how nucleosythesis proceeds, and therefore the understanding of the mass loss is crucial in order to evaluate the contribution from LIM stars in the chemical evolution of galaxies. In particular, this thesis will deal with the still debated origin of carbon.

1.2 Evolution of LIM Stars Stellar evolution is a process in which the internal structure of stars and the observable characteristics change dramatically over time due to changes in the internal force balance (pressure vs. gravity). Such changes are in- evitable, since stars deplete their initial sources of fuel and then exploit new sources, which affects the rate of energy production and hence changes the internal force balance. Thus, the different evolutionary stages stars go through are mainly due to fuel consumption, initial mass and chemical composition. When the reservoir of hydrogen in their centres is exhausted, stars with an initial mass less than about 8 solar masses become cool giant stars, located on the so-called Red Giant Branch in the so-called Hertzprung-Russel (HR) diagram (see Fig. 1.1) and then later evolve into Asymptotic Giant Branch stars (AGB stars) after completion of the helium-burning in their cores. Characteristic physical features of AGB stars are low effective temperatures that rarely exceed 3500 K and high luminosities, typically several thousand solar luminosities. During their evolution, these stars build a degenerate core of carbon and oxygen, with a helium-burning shell around it, and surrounding that shell, a mantle of hydrogen. It is the hydrogen-burning shell in that mantle which provides the energy needed to maintain the high luminosity. Stellar evolution takes place on long time scales and therefore much of what is known about how stars evolve is due to theoretical models, in combination with observations of stars in stellar clusters, which have similar ages, but appear in different evolutionary stages. The models consist of a number of conservation laws with various sources and sinks and the result can be displayed as evolutionary tracks in the HR diagram (again, see Fig. 1.1 for a few schematic examples) that can be compared with observed stellar populations. If combined with a detailed nuclear-reaction network, these stellar evolution models will also give quantitative results on the syn- thesis of heavy elements in stars, usually referred to as stellar yields, i.e., how much of a given element is produced and expelled from the star during its evolution. This thesis is centred around one key ingredient in stellar evolution models of LIM stars: the mass-loss rate during the ﬁnal stages of evolution and how it depends on various stellar parameters.

10 1.3 Winds of Carbon Stars During the giant phase LIM-stars develop a so-called stellar wind and start to lose some of its matter. The mass-loss rate increases several orders of magnitude as the star enters the AGB, and the matter ejection then leads to the formation of a circumstellar envelope around the star and eventually a planetary nebula with a remnant white dwarf in the centre. While it is rather obvious from observations that these stars lose mass, it is less obvious how their winds are formed. It is believed, however, that the momentum transfer by photons incident on dust grains is the phenomenon driving the strong winds of carbon stars. Dust particles indeed form in the circumstellar envelope at some critical distance from the star, where the temperature and radiation ﬁeld are such that dust formation can take place. In this work, we model this wind formation scenario and try to explore the connections between stellar parameters (e.g. mass, luminosity and temperature) and the properties of these dust driven winds. This requires an understanding of the chemistry that determines the types of dust that forms, as well as the complex dynamics of these stars. The precise nature of the circumstellar dust grains depends on the evolution and physical properties of the star, and the chemical composition of the grains depends on the nature of the wind from the star. In turn the chemical composition of the gas is determined by the nucleosynthesis within the star and to some extent the contribution from previous stellar generations. The dust chemistry is usually either dominated by carbon or by oxygen. If the carbon-to-oxygen ratio (C/O) is less than unity, almost all the carbon atoms will be locked into CO molecules, leaving only the excess oxygen atoms available as basis for the dust chemistry. Similarly, if C/O > 1, there will be an excess of carbon, and thus carbon based dust chemistry. The rare exception to these cases is when C/O = 1 and the carbon and oxygen atoms are equally depleted (assuming equilibrium chemistry), leaving other atoms (e.g. sulphur) to form the majority of the molecules and dust. In this thesis we restrict our study to the case where C/O > 1. What is here called dust are microscopic grains that may form in stellar atmospheres and circumstellar envelopes under certain conditions. The study of meteorites has provided useful information on the nature of dust forming in the circumstellar regions around AGB stars. Isotopic studies have shown that meteorites contain so-called presolar grains that originate from such stars. Therefore, understanding the dust formation process is important in more respects than the formation of dust-driven winds. Dust grains also play an important rôle for star formation and the formation of planetary systems.

11 Figure 1.1: The Hertzsprung-Russell diagram (HR diagram) shows the relationship between luminosity and effective temperature of stars. The evolutionary tracks of stars are shown for several initial masses, and the AGB in particular in the case of a 2M-star. Adopted from Wikipedia Commons.

1.4 The Cosmic Matter Cycle: the Origin of Carbon Today, it is widely accepted that essentially all elements heavier than helium are produced either inside stars or in the high-energy environment associated with supernovae. In the early days of the Big Bang cosmology, one of the major problems was that only hydrogen, helium, tiny amounts of lithium, and perhaps beryllium can be synthesised in the early universe. For heavier nuclei to be formed, other nuclear reactions are required – reactions that could never occur under the physical circumstances in the early universe as implied by the Big Bang theory. The existence of heavy elements is an undeniable fact, so there had to be a piece of the puzzle missing, or the Big Bang idea had to be wrong. During the 1950’s a solution to the problem was given, where the interiors of stars were shown to be the most likely production sites for heavy nuclei (Burbidge, Burbidge, Fowler & Hoyle 1957 [19], referred to as B2FH). A consequence of the B2FH-theory is that the abundances of the elements will change over time, which became evident as the theory of stellar evolution emerged in the decades to follow. However, B2FH provided only the concept for the making of the elements. The actual

12 production sites for individual elements remained to some extent an open question. The stars in a galaxy expel some of their newly synthesised elements into the interstellar medium, as the they "die", i.e. reach their final stages of evolution. According to our present understanding of stellar evolution and nucleosynthesis, massive stars "live fast and die young" as supernovae, while most LIM stars are more long lived (our Sun, for instance, is expected to live more than 10 Gyr) and expel material rather gently by a stellar wind, eventually resulting in a planetary nebula and a white dwarf (remnant star). High mass stars are therefore the first to enrich the ISM in a co-eval stellar population (all stars born at the same time), while LIM stars make a delayed entrance on the scene after typically a few Gyr. Effectively, these differences in stellar lifetime would not make a huge difference if it was not for the fact that some elements are produced in large quantities in high mass stars and the subsequent supernova events, but hardly at all in LIM stars and vice versa. For example, most of the oxygen is produced in high-mass stars and is rather consumed by LIM stars, and most of the carbon probably comes from low-mass stars (stars of masses similar to that of the Sun), since it seems that massive stars could only produce significant amounts of carbon under special circumstances. The ratio of carbon to oxygen in a galaxy is therefore an indicator of the age of the stellar component if the stellar initial mass function1 (IMF) and the star formation history are known, or an indicator of the composition of the stellar component if the age is known. There is just one major problem. Despite the conceptual understanding of nucleosynthesis (dating back to B2FH), the relative contribution of several elements in stars of different masses is not known to very high precision, which is the case for carbon. This has to do with the fact that there are several physical processes, e.g., the rate of mass loss and nuclear reactions, associated with stellar evolution and nucleosythesis that are particularly uncertain in some cases. Constraining the composition and evolution of a stellar population from observed abundance ratios is therefore a non-trivial task that requires that we do our best to minimise these uncertainties.

1.5 Carbon-based Life in the Galaxy Carbon is the element of life, at least here on Earth. And no other types of working biochemistries are actually known to science other than the biochemistry we have on our planet, i.e., it is possible (or even likely) that carbon-based life, operating in a water-based medium, with higher forms metabolising oxygen, is the only possibility. Moreover, carbon seems to be the atom best suited to form the long-chain molecules needed for life. In

1 An empirical, statistical distribution of stellar masses according to which stars are thought to be formed.

13 the light of this, it is interesting to note that our solar system seems to be quite generously endowed with carbon, relative to many other elements that have a stellar origin. The outer parts of the Galaxy appear to be have a carbon deficiency. Due to the much lower star formation rate the outer disc has a low abundance of non-primordial elements in general, but some elements seem rarer than others. Carbon is one such an element, which is due to the delayed production of carbon in low-mass stars. If the peak of star formation has not yet been reached or was reach not long ago, the vast majority of low-mass stars have not yet reached the evolved (carbon star) state where they eject matter back to the ISM. Thus, a decreasing trend with galactocentric radius is expected for carbon relative to elements that are primarily formed in high mass stars, e.g., oxygen. Observational constraints on the carbon abundance for the outermost parts of the disc are scarce. In this thesis it shown that carbon stars may be producing more of the Galactic carbon than previously thought. Such a scenario predicts a rather significant carbon deficiency in the outer disc, which is in agreement with observed abundances, although more data is needed in order to confirm it. Does a carbon deficit decrease the likelihood of carbon-based life arising in the outer parts of the Galaxy? Not necessarily. It is known that our solar system has a natural mechanism for forming pockets of carbon: the comets! Comets contain significant amounts of carbon because they formed far enough from the sun for carbon to solidify rather than evaporate into space, as it mostly did near the Sun. It is quite likely that alot of carbon came to our planet through comet impacts early in its history. But the absolute abundances of carbon that are predicted for the outer disc according to observations of Cepheids (pulsating, massive stars), as well as the chemical evolution models presented in this thesis, are so low that even the comets orbiting stars at these galactocentric distances will not be very carbon-rich. So how likely is it that carbon-based life occurs in the out-skirts of the Milky Way? Lineweaver et al. [103] argued that there ought to be a Galactic Habitable Zone (GHZ), much like the commonly discussed habitable zone inside a planetary system. In central parts of the Galaxy, the rate of supernova explosions will be too high for life-friendly worlds to exist, as consequence of the much higher star formation rate. In the outer Galactic disc, the abundances of elements heavier than hydrogen and helium, which include the elements needed to form earth-like planets as well as living organisms, are to low according to Lineweaver et al. The GHZ picture may be questionable, but the results of this thesis now add an interesting piece of the puzzle: the connection between the winds of carbon stars and the origin of carbon.

14 2. Modelling of Dynamic Atmo- spheres and Dust Driven Winds

The models for the atmospheres and winds presented in this thesis are computed using frequency-dependent radiative transfer for the gas and dust, including detailed micro-physics of the dust grains and their formation (described by Höfner et al. [79] and Andersen et al. [4]) in combination with time-dependent hydrodynamics. In this chapter, this complex problem is formulated in some detail, and it is also brieﬂy described how it is solved computationally. The code used in this thesis is essentially that of Höfner et al. [79], with some minor modiﬁcations.

2.1 The Success of Numerical Modelling It has been shown that synthetic opacity sampling spectra for carbon-rich AGB stars based on the numerical models by Höfner et al. [79] reproduce the observed spectral energy distributions of these stars rather well [?]. Perhaps more remarkable is that these models are also able to reproduce the observed time-dependent behaviour of fundamental, first and second overtone rotation-vibration lines of CO, which are formed in the outflow, wind acceleration region, and atmosphere respectively [116, 117]. Such consistent results have not been obtained by any previous modelling. An observed quantity that is quite well reproduced is the outflow velocity, or the terminal velocity of the wind. In the results presented in Paper I one may notice that the range of wind speeds obtained from the models are wider than the observed range of wind speeds for C-stars (see, e.g., the work by Schöier & Olofsson [146]), but the highest wind speeds (models − suggest wind speeds up to 60 km s 1) are associated with very high mass- loss rates, so-called superwinds, and it is therefore unlikely that such stars can be detected, since their existence will come to an end very rapidly once the superwind sets in. The mass-loss rates are more difficult to compare directly with observations. But the observational constraints that do exist for C-stars are reasonably well met by the models [167, 146]. As these results are brought together, it appears that the winds of C-stars are now quite well understood, although some issues still remain. However, there should not be much doubt about the general concept: winds from C- stars in late stages of evolution are dust driven, pulsation enhanced winds. It should be noted, however, that for stars with low abundances of free carbon it seems that the observed winds cannot be driven by dust only.

15 2.2 The Full Set of Equations 2.2.1 Radiation Hydrodynamics The hydrodynamic equations are examples of conservations laws. In gener- alised form, conservation laws for fluids can always be written as Df ∂f ≡ +∇·F = Si , (2.1) Dt ∂t i where f is the density and F the flux of any conserved quantity, and Si is an additional source/sink of f .Inradiation hydrodynamics (RHD) one, addi- tionally, introduces sources and sinks that describe the radiation field and its interactions with matter. The usual hydrodynamic equations are thus modified as follows. The equation of continuity (describing mass conservation) is of course left unchanged, i.e. ∂ρ +∇·(ρv) = 0. (2.2) ∂t

The equation of motion is written as a conservation law for momentum, ∂ρ πρ v 4 gas dust +∇·(ρvv) =−∇P − ρ∇Φ + κ + κH H, (2.3) ∂t c H where κH and χH denote the flux mean opacity for gas and dust, respectively, and H is the radiative flux. The equation of energy becomes ∂ρe gas gas +∇·(ρev) =−P∇·v + 4πρ κ J − κ S , (2.4) ∂t J S where e denotes the specific internal energy, J the mean intensity of the radiation field and S is the source function for the gas. The radiative transfer problem is here treated by using moments of the specific intensity Iν, which is defined as 1 1 c Jν = Iν dμ = Uν (2.5) 2 −1 4π 1 1 1 Hν = Iνμdμ = Fν (2.6) 2 −1 4π 1 1 2 c Kν = Iνμ dμ = Pν (2.7) 2 −1 4π where μ = cosθ and θ is the angle between the radial direction and the line of sight (or light beam). These three moments (sometimes referred to as the Eddington moments) correspond to radiation energy densityUν,fluxFν and radiation pressure Pν. The source terms describing the momentum and energy exchange between gas and radiation are obtained in terms of the frequency integrals of

16 these three moments, i.e. ∞ gas gas gas κJ J − κS S = κν (Jν − Sν)dν (2.8) 0

∞ gas gas κH H = κν Hν dν (2.9) 0

∞ dust dust κH H = κν Hν dν (2.10) 0

gas dust where κν and κν are the frequency-dependent gas and dust opacities, gas gas dust respectively. The frequency integrated opacities κJ , κH and κH are obtained as gas 1 gas κ = κν Xν dν, (2.11) X X dust 1 gas κX = κν Xν dν, (2.12) X where X is either one of the moments J or H, or the source function S. The three moments introduced above are determined by the ﬁrst two (zeroth and ﬁrst) moments of the radiative transfer equation (RTE) together 1 with an Eddington factor fedd = J/K that is needed to close the system. The zeroth moment of the RTE (see Eq. 2.18) is obtained by integration over all angles and becomes 1 ∂J 1 1 v 3K − J gas gas + ∇·(Jv) =−∇·H − K ∇·v + − ρ κ J − κ S (2.13) c ∂t c c c r J S

The ﬁrst moment is obtained accordingly (after multiplication by μ and integration over the same) ∂ − 1 H 1 3K J 1 gas dust + ∇·(Hv) =−∇K − − H∇v − ρ κ + κH H. (2.14) c ∂t c r c H

If seen as a conservation equation, the zeroth moment equation represents the conservation of radiation energy. It can be understood in terms of the energy changes within a volume element dV due to radiation ﬂux through its boundaries, radiative pressure, absorption and emission. For the isotropic case (where J = 3K ) the third term on the RHS vanishes, i.e. the equation contains no longer any explicit geometric dependence. The ﬁrst moment equation represents radiation momentum conservation. It mainly describes the effects of radiation pressure and the momentum losses due to radiative acceleration of matter. Once again, the geometric term vanishes for the isotropic case.

1 The Eddington factor cannot simply be prescribed since our medium is neither optically thin, nor optically thick. Thus, at every time step one must solve the RTE for this factor and apply it to the RHD equations in order to close the system of equations.

17 These equations, (2.13) and (2.14), can be greatly simplified if v c,i.e.if the problem is restricted to the non-relativistic regime (which applies very well to AGB star atmospheres). In that case one can drop all terms of order v/c-factor, which means that the zeroth moment equation contains only the divergence of the radiation flux plus absorption and emission terms, and the first moment equation includes just the divergence of the radiation pressure plus absorption and emission terms.

2.2.2 Selfgravity The coupling between gravity and matter is as usual obtained through Pois- son’s equation, ∇2Φ = 4πG ρ, which in this case (spherical symmetry) can be integrated directly and thus be written π r ∇Φ = 4 G 2 ρ = GMr (r,t) (r,t) 2 s (s,t)ds 2 , (2.15) r 0 r where Φ is the gravitational potential and Mr is the mass contained within a radius r . In the equation of motion (2.3), the gravitational acceleration term therefore does not require that one solves for the gravitational potential. It is sufﬁcient to know the cumulative (spherical) mass distribution.

2.2.3 Equation of State and other a Priori Assumptions In order to close the system of differential equations given above, a few assumptions about the properties of the matter and the radiation ﬁeld have to be made. Pressure and energy density are coupled by introducing an equation of state. Equations of state (EOS) exist in various forms corresponding to different states of matter and corrections due to empirical facts. Here it is simply assumed that a stellar atmosphere is sufﬁciently well described by the EOS for a perfect gas, k P = ρTgas, (2.16) μmp where mp is the proton mass and μ is the mean molecular mass, which is assumed to be μ = 1.26. Another assumption that is made, is that the stellar atmosphere is in local thermal equilibrium (LTE). It is assumed that the source function Sν is given by gas dust ρκν Bν(Tgas) + κν Bν(Tdust) Sν = , (2.17) gas dust ρ(κν + κν ) where Bν(T ) is the Planck function for a temperature T .Departuresfrom LTE may occur if shock waves are present or in the outer extension of the atmosphere (see, e.g., Schirrmacher et al. [142]). Here it is assumed that such

18 effects do not have any significant implications on the structure and dynamics of the atmosphere. The third aprioriassumption being made, is that the gas and dust phases of the atmosphere obey full momentum coupling. i.e. the momentum gained by the dust from the radiation field is directly transferred to the gas. A strong coupling between the dust and the gas phase is not obvious, however. Relaxing this phase coupling approximation, Sandin & Höfner [138] demonstrated that the effects of decoupling of the phases can be quite significant. Dust formation may increase significantly, but this does not seem to necessarily increase the predicted mass loss rates for a given set of stellar parameters. In other words: the efficiency of the momentum transfer may be lower (for a given opacity), but this may be compensated for by the increased dust formation, leaving the net acceleration of the gas almost unchanged by the inclusion of drift. In certain cases, however, the effects on the acceleration can be significant. Nevertheless, one may choose to assume full momentum coupling, because the computational cost for relaxing the phase coupling approximation is simply too large with the present code. This would require that other simplifications are made (e.g., grey radiative transfer) in order to make a grid of models, such as that presented in Paper I, computationally feasible. The benefits from the inclusion of drift are not large enough for that to be given priority over e.g. frequency-dependent radiative transfer (see also the discussion in Paper I and IV). However, parallelisation of the whole code and replacing outdated algorithms and solvers could perhaps do the trick, but that is another thesis.

2.2.4 Radiative Transfer The moments appearing in the RHD equations does not explicitly show what is going on in terms of radiative transfer. In modelling radiative transfer one may start from the Boltzmann equation and let the distribution function f = Iν and assume that the velocity |v|=c,wherec is the speed of light. The radiative transfer equation (RTE) is then simply 1 ∂Iν 1 DIν + n ·∇Iν = = jν − ανIν, (2.18) c ∂t c Dt c where jν = jν(x,t;n,ν) is the emission and α = α(x,t;n,ν) is the absorption (or extinction) coefficient. Eq. (2.18) is applicable to the radiation field of any inertial frame in special relativity and was first explicitly derived by Thomas in 1930 [157]. Hiding in the emission and absorption terms is a huge variety of transitions, scattering and ionisation processes. For a spherically symmetric atmosphere, the opacity (absorption coefficient) κν and the density ρ are functions of radial coordinate r , only. The intensity Iν and the source function Sν will, however, have a directional component in addition to the radial dimension r , which is the angular variable

19 P

r θ Z

Figure 2.1: Illustration of the r −θ projection used for solving the spherically symmetric radiative transfer problem. Adopted from Gustafsson et al. [66].

θ. The radiative transfer equation (RTE) then takes the form 1 ∂Iν ∂Iν sinθ ∂Iν − cosθ + =−κν(r,t)ρ(r,t)[Iν(r,θ,t) − Sν(r,θ,t)]. (2.19) c ∂t ∂r r ∂θ

It is common practise to write the RTE in terms of the cosine factor μ ∈ [−1,1] instead of θ, i.e., on the form

1 ∂Iν ∂Iν μ2 − 1 ∂Iν − μ + = ρ(r )κν(r )[Iν(r,μ) − Sν(r,μ)]. (2.20) c ∂t ∂r r ∂μ

The RTE is implemented in the numerical, dynamic atmosphere model in a slightly different form, however. Introducing the new variable ξ ≡ r μ one obtains the much simpler reduced RTE, which is essentially just the equation for the projected intensity along the normal vector, 1 ∂Iν ∂Iν + = ρ(ξ)κν(ξ)[Iν(ξ) − Sν(ξ)]. (2.21) c ∂t ∂ξ

In the RHD code, this equation is solved using a Feautrier-method [?]and then the de-projected solution is reconstructed. However, the latter is trickier than it may seem, since Eq. (2.21), will render degenerate solutions where every ξ corresponds to inﬁnitely many combinations of r and μ. Hence, a method to de-project the solution in terms of r and μ,ormore precisely, ﬁnd Iν at every grid-point, must be introduced. This is done by introducing an impact parameter p ≡ r sinθ (see Fig 2.1), which obviously

20 Figure 2.2: The structure of an amorphous carbon particle. Adopted from Wikipedia Commons.

yields the relations ∂ξ r 1 1 1 ∂ξ p μp r = = = , = + = . (2.22) ∂r ξ 2 r 2 − p2 μ ∂μ 1 − μ2 (1− μ2)3/2 1 − μ2

Using these two relations, it is now possible to solve the reduced RTE and back-transform to the spherical case.

2.2.5 Dust Formation and Opacity Mass loss from AGB carbon stars is mainly due to momentum transfer from radiation to atmospheric dust grains. As such, a wind model for AGB stars is highly dependent on how dust formation is treated. The models presented in this thesis include a time-dependent description of dust grain growth and evaporation using the moment method by Gail & Sedlmayr [45] and Gauger et al. [46]. The dust component is described in terms of moments K j of the grain size distribution function weighted with a power j of the grain radius. The zeroth moment, K0, is the total number density of grains (sim-

21 ply the integral of the size distribution function over all grain sizes), while K3 is proportional to the average volume of the grains (assuming spherical grains). The equations, which determine the evolution of the dust component, are ∂K0 +∇·(K0 v) = J, (2.23) ∂t ∂ K j j 1 j /3 +∇·(K j v) = K j −1 + N J,1≤ j ≤ 3, (2.24) ∂t 3 τ where J is the net grain formation rate per volume, 1/τ is the net growth- rate of the dust grains, and N is lower size limit of grains contributing to K j . The average grain radius a can be obtained from

K1 〈a〉=r0 , (2.25) K0 where r0 is the monomer radius. The typical grain size may affect the dust opacity and therefore also the efficiency of momentum transfer from radiation to dust, an important issue which is discussed in further detail in Paper IV. It is here assumed that the dust can be described as spherical grains consisting of amorphous carbon. Amorphous carbon grains have a very complicated bonding structure and irregular shape that resemble fur-balls that your cat may spit up... (see Fig. 2.2). Consequently, these "fur-balls" can ab- sorb photons rather well, which is a desired property in the present case. Other types of carbon-based dust may form in the circumstellar envelopes of carbon stars, but amorphous carbon grains are most likely dominant in number and definitely dominating as the source of opacity (see Andersen et al. [4] for details about these assumptions). The nucleation, growth and evaporation of grains is assumed to proceed by reactions involving C, C2,C2HandC2H2. In this description of grain growth, a so called sticking coefficient is used, that enters into the net growth rate of the dust grains. This parameter, αS , is not definitely known unless the exact sequence of chemical reactions responsible for the dust formation is known. However, Gail & Sedlmayr [45] argued that the sticking coefficient must be on the order of unity, mainly because it is expected that neutral rad- ical reactions play a major role in the formation of carbon grains. It should be pointed out as well, that with αS = 1, our models nicely reproduce the expected mass loss properties of TP-AGB stars [146, 147]. In order to calculate how much of the radiative energy and momentum is transferred to dust grains, one needs to know the frequency-dependent opacities of these grains. These can be expressed in terms of the efficiency Qrp, i.e, the the ratio of the radiative cross section to the geometrical cross section of the grains. All models in this thesis are calculated using the refractive index data of Rouleau & Martin [131]. In Paper I & II the so-called small-particle limit was used, i.e., it was assumed that the grains

22 are small compared to the photon wavelengths, and Qrp thus becomes a simple function of the grain radius a. In that limit the wavelength- and grain size-dependence of the opacity can thus be separated into two independent factors, which greatly simpliﬁes the calculations. The absorption coefﬁcient for dust then becomes ∞ ρκdust = ν π 3 ν Qrp( ) a n(a)da (2.26) 0

= where Qrp Qrp/a and n(a) is the size distribution of the dust grains. How- ever, as discussed in Paper I, the grains may grow beyond the regime where they can be regarded as small, and in such a case Eq. (2.26) may not be a very good approximation. In the general case, where a need not be small, ∞ dust 2 ρκν = a πQrp(a,ν)n(a)da, (2.27) 0 which means that the computation of dust opacity requires integration over Qrp and therefore becomes computationally more expensive. However, with modern computers this is no longer a problem, and the small-particle approximation can in principle be relaxed. The effects of relaxing this approximation are explored in Paper III.

2.3 Method of Computation To the system of non-linear partial differential equations describing the dynamics, radiation ﬁeld and dust formation of a C-star atmosphere a so called "grid equation" must be added, i.e., an equation which determines the locations of the grid points according to accuracy considerations [39], and an equation keeping track of the condensible amount of carbon. This gives a total of 12 partial differential equations (PDEs) to solve. The system of PDEs is solved implicitly using a Newton-Raphson scheme. All equations are discretised in a volume-integrated conservation form on a stag- gered mesh. The spatial discretisation of the advection term is a monotonic second-order advection scheme (van Leer, [166]). The same order of numerical precision is used for all PDEs. The details of the numerical method can be found in the paper Dorﬁ & Feuchtinger [40].

2.3.1 Boundary Conditions The inner boundary condition is a so-called "piston boundary" with an amplitude Δvp (variable parameter) and the period P locked to the luminosity of the hydrostatic initial model (see below) by an empirical period-luminosity relation [42]. The outer boundary is treated in two different ways. During the expansion phase (before the system has adjusted itself) the outer boundary is free, i.e. the expansion is followed by

23 the grid. When the outer-most point of the adaptive grid reaches some given radius, the program automatically switches to a fixed outer boundary located at this given radius, allowing outflow over this boundary. There are no specific conditions posed on the location of this outer boundary, except that it must (for obvious reasons) be located at a finite radius, i.e. not actually placed at infinity as in the analytical case, but still at a large enough radius to ensure that the wind speed at the outer boundary is close to the terminal velocity. Hence, the outer boundary is essentially free, although the conditions ∂v = 0, H = μ¯J (2.28) ∂r where μ¯ reflects the angular intensity distribution of the radiation field, are imposed on the outer boundary.

2.3.2 Initial Conditions The dynamical calculations are started from hydrostatic dust-free initial models where the outer boundary is located close to the photosphere (about 1.3 − 1.6R). In these initial models, the fundamental stellar parameters [M,L,Teff,log(C− O)] can be chosen freely as initial conditions for the models. These hydrostatic models are computed by solving the a set of ordinary differential equations. obtained by reducing the full set of equations to the static case using πJ = σT 4,whereJ, H and K are deﬁned as above. Together with the EOS and assuming LTE, this system can be integrated numerically. The equations of mass conservation and hydrostatic equilibrium are

dMr = 4πr 2 ρ, (2.29) dr

dP GMr ρ 4π =− + ρκH H, (2.30) dr r 2 c and remain the same throughout the whole procedure. To obtain the temperature gradient, the ﬁrst step is to obtain a solution for the monocromatic (grey) case by solving dT π ρκH H df 3f − 1 =− − edd + edd T, (2.31) 3 dr 4σ feddT dr r

dH 2 =−H, (2.32) dr r where fedd = K /J. The obtained temperature structure is then iteratively corrected for non-grey effects until a full solution is found. Finally, the adaptive grid must now be made to match the resulting static solution while keeping the physical structure ﬁxed (see Dorﬁ & Drury [39]

24 for details) and in order to ensure that the dynamical model is started from a truly hydrostatic situation, the static solution is also "relaxed" by use of the full dynamical code. For further details about the initial conditions and how grey hydrostatic solutions are obtained, see the description by Dorﬁ & Feuchtinger [40].

2.3.3 The Modelling Procedure When the dust equations are switched on, dust formation starts and the resulting radiative acceleration creates an outward motion of the dust and the gas. In the first computational phase the expansion of the atmospheric layers is followed by the grid to about 20−30R (usually around ∼ 1015 cm). At this radius the location of the outer boundary is fixed, allowing outflow. The outflow model then evolves for typically more than 200 years (or periods). To avoid a significant depletion of mass inside the computational domain, the model calculation is stopped after typically 105 time steps or less. This is important, since one cannot (for computational reasons) allow for any mass flow over the inner boundary. Due to this fact, a large enough mass loss will eventually lead to a depletion of mass in the considered part of the circumstellar envelope.

25

3. Carbon Star Mass Loss as Function of Stellar Parameters

(Paper I & II)

Paper I of this thesis presents a large grid of carbon star wind models based on the theory described in the previous chapter. The main objective of the study was to establish how the mass-loss rate depends on basic stellar parameters. Similar studies do already exist in the literature, but the level of realism in those model grids is not the same (mainly due to grey radiative transfer, see previous chapter) and Paper I presents also a slightly different approach to the construction of a mass-loss prescription, to be used primarily in stellar evolution models. In a related work (Paper II) the effects of initial metallcity, carbon excess and kinetic-energy injection by pulsation were compared. It appears that the mass loss of metal poor carbon stars is just as intense as it is for their more metal rich counter parts, provided that the carbon excess is similar. This aspect of the work is still not ﬁnished, and will continue beyond what is presented here.

3.1 The Grid of Wind Models In Paper I, 900 DMAs are used to determine the mass-loss rate of carbon stars as function of stellar parameters. The aim is to cover as much as possible of the stellar parameter space without preference for combinations of stellar parameters which are more "likely" or expected to produce winds, in order to cover also the mass-loss thresholds for dust-driven winds, i.e., the parameter combinations where winds cannot form. The deﬁnition of the grid is given is Table 3.1.

3.1.1 General Properties The models which do form dust-driven ourﬂows show mass-loss rates that −6 −1 are typically a few times 10 M yr and only one model has a really low − mass-loss rate M˙ < 10 7. Fig. 3.2 shows a histogram of all models for the −1 intermediate piston amplitude Δvp = 4.0 km s that produce a wind. The difﬁculty of producing really low mass-loss rates might be a shortcoming of

27 Table 3.1: Deﬁnition of the grid. Δ denotes the grid step.

−1 M [M]log(L/L) Teff [K] log(C− O) + 12 Δup [km s ] Δ = 200 Δ = 0.30 Δ = 2.0 0.75 3.55, 3.70, 3.85 2400 - 3200 7.90 - 9.10 2.0 - 6.0 1.0 3.70, 3.85, 4.00 2400 - 3200 7.90 - 9.10 2.0 - 6.0 1.5 3.85, 4.00, 4.15 2400 - 3200 7.90 - 9.10 2.0 - 6.0 2.0 3.85, 4.00, 4.15 2400 - 3200 7.90 - 9.10 2.0 - 6.0

the present wind model. At the time of writing, it is not known whether this is due to, e.g., the limited number of frequency points (64) in the radiative transfer of the models, or the phase coupling between gas and dust (i.e., absence of drift). A subset of test models with higher resolution (440 frequency points) has been computed in parallel with the other 900 models, but no dramatic differences were found (see Fig. 3.1, not included in Paper I). Furthermore, it should be noted that the grid is rather coarse. For example, the dependence on the effective temperature is considered in steps of 200 K, and luminosity in steps of Δlog(L/L) = 0.15, which may be too big to trace the trends near the mass-loss thresholds (see below). Winds of carbon stars are known to be rather fast, compared to other − long-period variables. The model wind speeds range between 1−60km s 1, where the fastest winds are found in the most carbon rich cases. The lack −1 of observed carbon stars with very high wind speeds (uout > 40km s )may be a selection effect, since the chance of finding an extremely carbon rich AGB star at the very final stage of its evolution must be quite small. The very −1 low wind speeds seen in a few models (uout < 5km s ), but not in observed carbon stars, is an interesting prediction by the models. Such slow outflows may exist, since a wind can, in theory, be maintained as long as the flow surpasses the escape speed at some point in the velocity profile. In the light of the recent models of M-type stars by Höfner [81] we derived the characteristic grain sizes for models with M = 1M and Δvp = 4.0 −1 −5 km s . The characteristic grain radius in these models is agr ∼ 10 cm, which is large enough to cause more than 10% relative difference between Qrp (see Chapter 2, Sect. 2.2.5) and the small-particle approximation at 1μ. More about this issue is presented in the next chapter and in Paper III.

3.1.2 Mass-Loss Thresholds and the Carbon Excess Despite some possible shortcomings in reproducing low mass-loss rates, Paper I is still a signiﬁcant step forward. It is the ﬁrst systematic study of dust-driven mass loss and other wind properties as functions of stellar parameters that is based on a dynamic atmosphere model including both

28 Figure 3.1: Mass-loss rate for high resolution models (440 frequency points) vs. mass-loss rate for low resolution models (64 frequency points). The dashed line shows a one-to-one correlation and the dotted lines display a 25% variation.

frequency-dependent radiative transfer and detailed, time-dependent dust formation, and is probably the most advanced theoretical constraint on the super-wind of carbon stars available at present. The results demonstrate that strong, dust-driven winds cannot develop under arbitrary physical circumstances, i.e., carbon stars that appear in certain regions of (stellar) parameter space, may not experience any signiﬁcant mass loss. Mattsson et al. [113] demonstrated the necessity of considering thresholds in mass-loss prescriptions, i.e., to include the fact that the mass-loss rate may drop to zero for certain combinations of stellar parameters. In Paper I it was found that for ε˜C < 8.20 and/or Teff > 3200 K a dust driven does not appear to form regardless of the combination of other stellar parameters (see Fig. 3.3). Since the surface gravity scales with Teff,anddust formation is a process highly sensitive to temperature, it is quite expected that the mass-loss rate is highly dependent on Teff, too. But, as can be seen in Fig. 3.3, there is a rather strong dependence on condensible carbon for

29 Figure 3.2: Histogram of mass loss rates for all models in the grid with Δup = 4km −1 s that are producing winds. Note the absence of models with mass-loss rates of a −7 −1 few times 10 Myr , which is due to the selection of parameter combinations in the grid and the fact that the grid does not reﬂect a real stellar population.

both the wind velocity and the mass-loss rate. However, Arndt et al. [5] as well as Wachter et al. [168] found a rather weak dependence on C/O [which is their choice of parameter, in Paper I ε˜C = (C/O − 1)εO is used instead]. Their results stand in sharp contrast to the results presented here, although our ﬁndings here are, qualitatively speaking, hardly a new discovery. Höfner and Dorﬁ [78] and Winters [174] have already pointed out the strong C/O- dependence, especially in the critical wind regime, although this has not been widely recognised. The thresholds that appear when, e.g., the effective temperature becomes too high or the abundance of condensible carbon too low, must be taken into account when constructing a mass-loss prescription to be used with other types of stellar models, such as stellar evolution and nucleosynthesis (see Paper V). Simple mass-loss formulae cannot reproduce all aspects of these thresholds and we have therefore not

30 Figure 3.3: Mass-loss rate (left panels) and wind speed (right panels) as function of effective temperature and carbon excess for a 1M star with log(L/L) = −1 3.70,3.85,4.00 and Δvp = 4.0 km s .

31 derived any such formula. An easy-to-use FORTRAN-code that ﬁnds the corresponding mass-loss rate from interpolated data cubes has therefore been developed as part of this thesis work. The code will be made available to the community.

3.2 Metallicity Dependence? Paper I is restricted to wind models at solar abundances. However, it may seem very likely that the metallicity of an AGB star could affect its wind properties since it may influence the atmospheric structure and possibly the availability of condensible material for dust formation. But, is actually unclear if low metallicity should be associated with lower mass-loss rates in general. Bowen & Willson [18] argued that low-metallicity AGB stars should experience a so-called superwind phase, but proposed that this should occur at a higher luminosity than for a metal-rich case, since at lower metallicities the molecular opacities are smaller, which in turn can reduce the extension of the atmosphere. This may be true for M-stars (C/O > 1), but is true for C-stars as well? The onset of dust formation depends on whether the atmosphere has been able to expand enough for the gas temperature to be as low as is required for dust formation to tale place. It thus appears correct to expect that weaker winds occur at lower metallicities. On the other hand, observational studies of metal-poor carbon stars in the Magellanic Clouds and the Milky Way indicate similar mass-loss rates [165, 64]. Paper II presents a set of models where two combinations of stellar parameters are considered along with two chemical compositions, and various positions of the inner (piston) boundary. For a fixed value of ε˜C and a given set of stellar parameters, it was found that the location of the inner boundary can affect significantly the mass-loss rate if Δvp is kept fixed, and that models with a metal-poor composition (apart from carbon) have similar, or higher, mass-loss rates compared to models of solar composition if the kinetic-energy injections by pulsations are similar. Furthermore, models with a solar composition, show lower mass-loss rates and in several cases also lower wind speeds than models with metal-poor composition for fixed Δvp, if the inner boundary is located at similar optical depth. This is due to the difference in density at the inner boundary (see Fig. 3.4, where q is defined as below), but since the number of models considered in Paper II is quite small, it might be premature to conclude that metal-poor carbon stars have stronger winds. It should be emphasised that Paper II compares models with the same absolute abundance of free carbon, which excludes the possibility that the obtained results are due to the accessibility of raw material for dust formation. The difference in the mass-loss properties appears to be correlated with the input of kinetic energy at the inner boundary. As previously described, the piston boundary condition is sinusoidal with a velocity amplitude Δup, which means that the energy injected into a mass shell of mass

32 = 1 Δ 2 dm at the inner boundary is dEp 2 vp dm. If one considers the quantity ≡ Δ 2 ρ = q vp in(t 0), (3.1) which is proportional to the kinetic-energy injection, and compares with the wind properties, one may note that log(M˙ ) ∝ log(q), (3.2) while the wind velocity uout shows a more complicated dependence on q (see Fig. 3.4). The correlation between mass loss and q is caused by the lev- itation of the outer atmospheric layers and the increase in density, which affects the dust-formation efﬁciency. Based on the numerical results, one may conclude that the mass loss of C-rich AGB-stars, for a given set of stellar parameters and abundance of free carbon, is a function of the energy- injection by pulsations as long as the dust formation has not reached the "saturation limit", i.e., when the degree of dust condensation cannot get any higher. When this limit is reached, the mass-loss rate no longer increases with the energy-injection. The range of realistic, or even possible, pulsation energies is limited, and connected to the stellar parameters. The treatment of the inner boundary and pulsations in the case of a non-saturated wind is crucial. Hence, the im- portance of constraints on the energy injection by pulsations cannot be emphasised enough. A quantitatively correct theory, or reliable observational constraints, of long-period stellar pulsation are required to obtain a complete quantitative understanding of mass loss on the AGB.

33 Figure 3.4: Mass-loss rate as fuction of q for two different stellar parameter conﬁg- urations.

34 4. Relaxing the Small Particle Approximation

(Paper III)

In Paper III the effects of relaxing the small particle approximation (SPA) for dust opacities are explored by considering a few less severe approximations for the radiation pressure efﬁciency, which include the effects of grain sizes and momentum transfer by scattering. The purpose of the study is mainly to establish when the SPA can be applied and to quantify the possible errors that may occur when particles grow beyond this regime. That grains of amorphous carbon dust may not always be that small became clear during the analysis of the results presented in Paper I, but does size matter when it comes to wind properties?

4.1 Sizeable Grains Höfner [81] showed that the size of silicate dust grains is a crucial property in the oxygen rich case, since small, iron free silicates cannot drive a wind [170, 80]. But with sufficiently large grains, scattering becomes important, and micron-sized silicates seem to be a possible solution the M-star wind problem. In Paper I it was shown that in models of carbon-star winds (using the SPA), dust grains tend to be, in certain cases, larger than would be justified using the SPA (see Fig 4.1). But if the grains are not that small, how wrong are the wind properties obtained from models based on the SPA? The dependence of the radiation efficiency factor Qrp on grain size changes character as the grains grow larger (see Fig. 4.2), which suggests that the modelled grain sizes obtained in Paper I could be large enough to affect the radiation pressure significantly. Paper III tries to answer this question by considering a few less severe approximations for the radiation pressure efficiency, which include the effects of grain sizes and scattering on momentum transfer. The main objective is to establish when the SPA can be applied and to quantify the possible errors that may occur in cases where the dust particles have time to grow big enough. If the radiative acceleration falls below the gravitational acceleration, a dust-driven wind may not form, and if it does, the wind speed is often quite

35 low according to the SPA models. If dust is forming in the atmosphere, but the outflow is slow, dust grains will have time to grow bigger, which means that they may be much larger than actually anticipated in the SPA. If the − grains are of "intermediate" size (around 10 5 cm), the total cross-section may be significantly larger than the in the SPA. Momentum transfer from radiation to the dust/gas phase is therefore much more efficient, and the star may thus have a chance to develop a wind that is stronger than suggested by the SPA models. Paper III presents the results of re-computations of a selection of models taken from the model grid presented in Paper I, adopting different re- laxations of the SPA. The most consistent inclusion of grain-size effects in Paper III uses different mean grain radii obtained from the moments of the dust-grain size distribution, but fixed grain radii were also considered for reference. The models selected for re-computation were primarily those that seem to form large dust grains in the SPA and those that we for other reasons expect to negatively affected by the SPA in Paper I.

4.2 How Bad is the SPA? It seems that grain-size dependent opacities are cruical in critical cases where strong dust-driven winds do not form in the SPA models. But are all previous results using the SPA (e.g., the models presented in Paper I) simply incorrect? Models from Paper I which produced strong winds turn out not to be much affected by using the "optimal" grain radius (corresponding to the maximum in Qrp) instead of the small particle limit. Furthermore, one may expect that critical wind models would also be signiﬁcantly affected if so-called drift were included, i.e., if the velocity coupling of the gas and dust phases were relaxed. For a stationary wind, the drift-corrected ratio of radiative and gravitational acceleration Γ can be written [115, 63], κ ˜ rp L uout Γ = , (4.1) 4πcGM uout + udrift

−1 where udrift is the steady-state drift velocity. For slow winds (a few km s ) the drift correction may be quite signiﬁcant, and reduce Γ by more than a factor of two. However, κ˜ rp may increase by a similar factor, since in a slow wind the dust grains have time to grow and the characteristic grain size seem to be such that Qrp is close to its maximum. One could very well picture a scenario where, for slow ouﬂows, the effects of grain-size dependent opacities cancels with the effects of relaxing the gas-dust coupling. Thus, before an exact implementation of grain-size dependent dust opacities is developed, the effects of drift must be studied in combination with the grain-size effects.

36 Figure 4.1: Histogram of grain sizes for wind-forming models with M = 1M, −1 Δup = 4kms taken from Paper I.

4.3 Size Matters! In conclusion, Paper III shows that, in critical cases, the effect of grain sizes can be quite signiﬁcant. Mass-loss rates may increase by a factor of two, or more, and wind speeds by as much as an order of magnitude. Furthermore, all models with grain-size dependent radiative transfer that have resultant winds appear to have much lower degrees of dust condensation, compared to corresponding SPA models. Consequently, the "dust-loss rates" are much lower in the new models. However, for well-developed dust-driven winds, where the dust formation has saturated, the effect of grain sizes is almost negligible. The SPA is still a reasonable simpliﬁcation that may be used in models of carbon star mass loss. However, it remains uncertain whether previous results for critical winds near the mass-loss threshold are correct. One may note that introducing drift (decoupling of the gas and dust phases) in the models may cancel with the effects of grain-size dependent radiative trans-

37 Figure 4.2: The radiative pressure efficiency factor Qrp and it components (Qext, Qabs and Qsca), relative to the small particle limit used in the SPA models, as functions of grain radius at λ = 1μm. Data is for amorphous carbon dust, taken from Rouleau & Martin [131] and the Q’s are calculated using Mie theory for spherical particles (program BHMIE from Bohren & Huffman [12], modified by Draine, www.astro.princeton.edu/draine/scattering.html). For further details, see Paper III. fer. Thus, it is quite possible that previous results presented in Paper I (using the SPA) predict wind properties reasonably well, also quantitatively. But for our understanding of dust-driven winds, size definitely matters!

38 Figure 4.3: Mass-loss rates (left panel) and wind speeds (right panel) for Type B models relative to the corresponding SPA models.

Figure 4.4: Mass-loss rates (left panel) and wind speeds (right panel) for Type B models relative to the corresponding models with "optimised" grain radius.

39

5. Stellar Structure and Evolution

In this chapter, the basic theory behind the stellar evolution model used in this thesis (see Paper V) is reviewed. This chapter is also an attempt to give an overview of basic stellar evolution theory and does not have many detailed references. However, most of what is reviewed in this chapter can be found in Clayton’s classical text book: "Principles of Stellar Evolution and Nucleosynthesis"[32] or Cox & Giuli’s "Principles of Stellar Structure"[33].

5.1 Basic Ingredients Many of the computer codes used today, e.g., the EVOL code [69, 70] used in Paper IV and the MESA code [125] used in Paper V build upon the pioneering work by Hofmeister, Kippenhahn & Weigert in 1964 [74]. Modern codes may in various aspects be more advanced, but the basic physical concepts and equations are still very much the same.

5.1.1 Hydrostatic Equilibrium and Energy Generation Similar to a stellar atmosphere model, a model of stellar structure and evolution is based on a set of conservation laws with appropriate source/sink terms. These conservation laws lead to equations for the hydrostatic balance, energy generation and energy transport in stars, which constituate, together, the basic equations of stellar evolution. These are the equations being solved in stellar evolution models. Hydrostatic equilibrium means that the pressure gradient forces and the gravitational force balance each other. This is a good approximation of the situation in a star, although acceleration of matter may occur under certain conditions (note that the MESA code includes an inertia term and therefore it is not strictly a hydrostatic model). Assuming hydrostatic equilibrium, the conservation of thermal energy requires that the amount of net energy emitted per unit time (i.e., energy emitted locally, minus energy ab- sorbed from other layers), at some distance r from the centre of the star, is the sum of the energy generated either by gravitational contraction or by nuclear processes at r . During the slow nuclear burning on the main sequence, the structure of the star is essentially maintained over time, apart from the changing composition in the core. Thus one can in principle neglect the gravitational input to the power generation. However, if the star is contracting the gravita-

41 tional component will be positive and in many cases signiﬁcant. If the star is instead expanding, as it is the case during its transition into a red giant, the gravitational component is negative. In the case of red giant, some of the nuclear energy is going into expanding the envelope of the star as the core contracts, rather than just being radiated away.

5.1.2 Energy Transport In case of radiative energy transport, the radiation pressure gradient is the combined result of the radiative flux and gas opacity κ in the stellar interior. It is straight forward to show that, assuming radiative equilibrium, the radiative transfer problem reduces to an equation for the temperature gradient, dT 3 κρ L = , (5.1) dr 64πσ T 3 r 2 where σ is the Stefan-Boltzmann constant, L is the luminosity and T is the gas temperature. Eq. (5.1) represents the so-called diffusion approximation, valid if the mean-free paths of the photons are short compared to typical spatial scales (typically on the order of 1 cm in stellar interiors). This equation gives the temperature gradient in the case of radiative energy transport only. Under most circumstances, radiative energy transport is needed to release the energy that is being provided by the star, through either its nuclear nuclear burning or its gravitational contraction heating, from its surface. However, inside a star, energy may also be transported via gas dynamics, much like in heated water, just before it begins to boil. When the rate of heating is too high, or when radiative transport is too inefficient due to high opacity, convective energy transport may become the dominant energy transport mechanism. In classical stellar evolution modelling, convection is treated as an adiabatic process [145]. Adiabatic convection is a simplifica- tion of reality, mainly since convection may be less efficient in thin gases (in the upper layers of the star), and it is associated with turbulence and other complex gas dynamics phenomena that are usually referred to as internal mixing. An important consequence of convection is the mixing of material from the stellar interior with the surface layers of the star during certain evolutionary phases, which will be discussed later in this chapter. For evolved low-mass stars, which are the primary objects of study in this thesis, low temperatures in their upper layers lead to rather high opacity and inefficient radiative transport. This is why they develop convective envelopes. As the onset of convection depends on the opacity, the choice of gas opacities in a stellar evolution model may have significant consequences for the results regarding the late evolutionary stages. The lack of a good theory of convection is one of the weakest parts of the theory of stellar structure and evolution, but the effects of it can still be quantified by parameterisation, and the most commonly used recipe is the so-called Mixing Length Theory (MLT).

42 5.1.3 Convection in the MLT Picture MLT describes convective energy transport by comparing temperature and density changes in a rising/sinking matter element to the surrounding average stratiﬁcation. The onset of convection can be described by the Schwarzschild criterion for convection [145]. It is reasonable to assume that the speciﬁc enthalpy difference of a convective element with respect to its surroundings arises from a temperature difference between the internal temperature of the element and the ambient temperature of the surroundings. The mean difference over the the so-called mixing length may be estimated as the ratio of the temperature difference to the difference between the actual and adiabatic temperature gradients. The central idea in MLT is that an unbalanced buoyancy force drives a convective element to move through a distance , before the parcel dissolves into the ambient medium. It is conventional to parameterise the mixing length as a multiple α of the pressure scale height HP ,

= α HP , (5.2)

where 1 1 dP = . (5.3) HP P dr

The "mixing length ratio" α is usually taken to be somewhere between 1 or 2. As a free parameter in stellar models it is often adjusted so that the models fit observational constraints. Of course, this parametric freedom weakens the predictive power of stellar evolution models in general. Some improvements of the mixing length theory are usually employed in stellar evolution models, such as "convective overshoot" regions, where it is assumed that mixing extends beyond the region of convective instability. The model used in Paper IV as well as the models presented in Paper V include what is called exponential diffusive overshoot, the particle spreading in the overshoot region is described as a diffusion process, where the diffusion coefficient is D = τ 2 = − 2r = ov(r ) c vrms(r ) D0 exp , Hv fHP , (5.4) Hv where τc is the convection time scale HV is the velocity scale height of the overshooting convective elements at the convective boundary [69]. The MLT picture of the internal gas dynamics is not a complete physical theory, however. The existence of convection means that part of the stellar interior will have a non-hydrostatic stratification, so it is obviously not physically consistent. This in not the main problem, however. The MLT is based on the idea that convection can be described as "bubbles" of hot gas that rise up through the star, cool off, and then the gas falls back down again. This would certainly release energy, but it is not really how convective energy transport takes place in stellar interiors. MLT parameterisation

43 must therefore be carefully made, so that the effects of convection/mixing are properly included, although the underlying theory is not entirely correct. To some extent detailed simulations of convection alone can give useful information about how to calibrate the MLT in stellar evolution [43, 73], but a better theory of "one-dimensional convection" would still be an im- provement.

5.2 Radiative Transfer and the Rôle of Gas Opacities In the previous sections, the opacity κ has been treated as just an unknown function.Ofcourse,itmustbequantifiedinorderforthemodeltobeself- contained. In stellar structure and evolution modelling, this is achieved by calculating a mean opacity that includes as much as possible of the relevant absorption and scattering processes. The calculation of accurate mean gas opacities is a quite challenging task due to large variations with frequency, temperature and density of the absorption coefficient of numerous molecules, atoms and ions. The important part, however, is that the mean opacity is a representative mean in the context it is being used. Under the assumption of Local Thermal Equilibrium (LTE), the frequency-dependent radiation pressure becomes proportional to the Planck distribution. It can now be shown that under LTE conditions, the diffusion approximation (Eq. 5.1) leads to a heat-flux equation of the form ∞ ∞ −1 =−4ac 3 dT 1 dBν ν dBν ν H T ∗ s d d , (5.5) 3ρ dr 0 κν + κν dT 0 dT

∗ where Bν(T ) is the Planck distribution and κν is the reduced absorption, i.e., ∗ −hν/kT κν = κν(1− e ). (5.6)

Following Rosseland [133], the integral in Eq. (5.5) can be seen as a weighted mean opacity, ∞ ∞ −1 1 ≡ 1 dBν ν dBν ν ∗ s d d , (5.7) κR 0 κν + κν dT 0 dT which is referred to as the Rosseland mean opacity. This deﬁnition of mean opacity is obviously very convenient, given the form of the heat transfer equation and is a good approximation for the interior parts of a star. The Rosseland mean devalues opacity at small ν, while it puts the greatest weight around ν = 4kT/h. Although the Rosseland mean opacity is dominated by transparent spectral regions due to the harmonic nature of the averaging process, the absorption lines are quite important, in particular for cool gases, where molecular and atomic lines are strong and numerous. Missing data for such lines is a problem in several cases.

44 Figure 5.1: A schematic (not drawn to scale) cut-away illustration of an AGB star, showing the core, the hydrogen and the helium burning shells, and the envelope (adopted from NOAO News, August 6, 2003).

The Rosseland mean is defined to fit the diffusion approximation, which is a poor approximation of the radiative transfer in the surface layers of a star. The opacity of the surface layers is important for the onset of convection and the diffusion approximation may need to be relaxed as stellar evolution models become more and more precise. However, currently, the implementation of convection and internal mixing in one-dimensional models is by necessity rather simplistic, which is probably a greater problem. Mixing of material into the envelope during late stages of evolution, i.e., dredge-up events, changes the element abundances in the upper layers. In particular, low and intermediate mass (LIM) stars will dredge-up significant amounts of carbon and can therefore make a transition from oxygen-rich to carbon-rich, i.e., they become carbon stars. Hence, a proper treatment of stellar evolution actually requires that the dependence of κR on, e.g., the fractions of carbon XC and oxygen XO is taken into consideration as in Pa- per V, since changes in the C/O-ratio may have significant effects on the stellar structure (again, see Paper V).

5.3 Stellar Evolution in Brief While stars are on the Main Sequence equilibrium between pressure and gravity is maintained. This situation lasts for about 90% of a stars life: 10 Gyr for the Sun, and even more for smaller stars. The energy is produced by nuclear fusion reactions in the centre of the star. Between the time when T > 10 · 106 K and when the central parts become depleted of hydrogen, helium is produced in the core, by the proton-proton chain. When the core

45 temperature reaches T 20 · 106 K the CNO cycle (see below) becomes a dominant nuclear reaction. After the Main Sequence, LIM-stars develop a hydrogen-burning shell when hydrogen is depleted in the core and a core of helium has formed. What then happens is that the helium core shrinks, and thus the temperature rises around the core, which makes the energy production by hydrogen fusion continue at a faster rate in a shell, and hence the star becomes much brighter. Furthermore, the envelope will expand and thus cool down as the core shrinks and heats up. This process will bring the star up to the sub- giant and red giant branch, from which it then evolves into an Asymptotic Giant Branch star (AGB star) after completion of the helium-burning in the core. During the AGB, the very last stages of stellar evolution, the evolutionary time-scale is dominated by the mass loss: the star essentially "bleeds to death" on a relatively short time scale. The circumstellar "bubble" that develops around the star may be ionised by the star’s UV radiation and may become what is known as a planetary nebula with a slowly fading white dwarf (the remainder of the stellar core) in the middle of it. Stars of masses similar to that of the Sun, or slightly greater, are of particular interest in the context of this thesis, since such stars usually become carbon-rich at some point. For a solar-like star helium burning in the core starts abruptly with a so-called "helium core ﬂash" when T ≈ 108 K. The core then expands and the luminosity decreases, as the star evolves to the horizontal branch in the HR diagram. When helium is depleted, helium burning starts in a shell around the carbon core, which now shrinks even further. Consequently, the temperature rises, the fusion rate increases and the envelope expands: the sun-like star has now become an AGB star. During the late AGB, alternate burning of hydrogen and helium (where the latter is the "ashes" of the former) takes place in a shell around the core producing so-called "helium shell ﬂashes", also called thermal pulses. With each thermal pulse, material (mainly carbon) from the interior is dredged up (mixed) to the surface. Eventually, so much carbon has been dredged up that the upper layers have more carbon than oxygen. A star which has reached this stage is referred to as a carbon star. During the thermal-pulse AGB phase (TP-AGB phase) the star’s envelope is ejected into the surrounding ISM by a very strong stellar wind (superwind). The TP-AGB phase, and the strong wind associated with it, is thought to explain the occurrence of thin, detached shells of gas around carbon stars. This hypothesis is the topic of Paper IV. For really low-mass stars (M ≤ 0.5M) the evolution is expected to end with the star becoming a helium white dwarf, since the central parts never become hot enough for helium burning to take place. Such low-mass stars would have main-sequence life times that exceed the age of the Universe, so the observed helium white dwarfs must have different origin.

46 5.4 Nucleosynthesis Since this thesis is mainly about carbon stars, this section will be restricted to some of the most important nucleosynthesis processes in low-mass stars, which were briefly described in the previous section. In particular, focus will be on the nuclear reactions that lead to the production and destruction of carbon. There are four important sites/types of nucleosynthesis that can be iden- tified in AGB stars: the hydrogen burning shell, the helium burning shell (see Fig. 5.1), sites of extra 13C production, and hot bottom burning. He- lium is mainly consumed through the triple-alpha process, which ends with carbon. Carbon is in turn consumed in the CN-cycle, operating at the bottom of the convective envelope (hot bottom burning). These two reaction paths will be briefly discussed here. More details can easily be found in the extensive literature on the subject (see, e.g., the book by Pagel [124] and references therein).

5.4.1 The Triple-alpha Process The vast majority of all carbon (12C) made in AGB stars is produced by the merging of 4He-atoms via the triple-alpha process, i.e., 4He(αα,γ)12C, (5.8) which is a two-step reaction. In the ﬁrst step two alpha-particles interaction to form 8Be and after a while the production rate of 8Be becomes equal to its decay rate, hence, 4He +4 He ↔8 Be. (5.9)

Then, in a second step, another alpha-particle interacts with the 8Be nu- cleus to form 12Cinthereaction 8Be(α,γ)12C. (5.10)

In AGB stars, the triple-alpha reactions occurs mainly during the thermal instability of the He-shell. The intershell convection zone will grow, which means that new helium is mixed down to the shell from the layers above and newly produced 12C is mixed outward by convection. As a consequence, most of the intershell region is homogenised. During the subsequent third dredge-up events the outer layers of this region are mixed up to the surface of the star and then gradually ejected in to the ISM by the stellar wind [67].

5.4.2 The CN-cycle Because the onset of the proton-proton-chain reaction is slow, another set of reactions is also signiﬁcant (see Iliadis [?], pp. 400-410): the CNO cycles.

47 Figure 5.2: Illustration of the triple-alpha process. Adopted from Wikipedia Com- mons.

There are four different reaction cycles, but only one (the ﬁrst, CNO1,see Fig. 5.3) is important in the present context. The abundances of H and 12C may be regarded as approximately constant on the relevant time scales, which leads to a roughly constant 13N/12C-ratio.Thesameistruefor15N 15 and O. Thus, the CNO1 will reach a steady state on the order of a few minutes. The only actually relevant species are the isotopes of carbon and nitrogen. Hence, the reaction is often referred to as the CN-cycle. In brief, 4 protons are added to 12C, followed by proton to neutron conversions, which leads to the recreation of 12C plus helium, in analogy with the proton-proton reactions. In the reaction above, carbon acts as a catalyst and must be present in the chemical composition of the star. The most difﬁcult step here is the fourth (see Iliadis [82], pp. 397), involving the addition of a proton to nitrogen. The effect is that there is a pile-up of nitrogen, since CN processing converts carbon to nitrogen. Note, however, that the sum of CNO1 abundances are always constant. This is important, since it implies a "trade-off" in the carbon and nitrogen production in stars where CN-cycling takes place, i.e., the ratio of carbon to nitrogen is determined by how much carbon is being con- verted in the CN-cycle.

48 Figure 5.3: Illustration of the CNO-cycle. Adopted from Wikipedia Commons.

5.5 Effects of Mass Loss and Mixing One of the most uncertain ingredients in stellar evolution models for LIM- stars is the rate of mass loss on the AGB. It is well established that AGB-stars lose mass and it is likely that observations of hot white dwarfs surrounded by a planetary nebula are explained by this phenomenon. However, mea- suring the mass-loss rate is complicated and determining its dependence on stellar parameters (and their evolution) by observations is difﬁcult. But in recent years theoretical modelling of dynamic carbon-rich AGB atmospheres has reached a stage where the level of realism is sufﬁcient to actually meet many different observational constraints. Unfortunately, there is no consensus in the use of mass-loss prescriptions in stellar evolution modelling and most existing stellar evolution models are based on either old and outdated theoretical work or empirical formulae with large observational biases and uncertainties. In this thesis a mass-loss prescription for the carbon-star phase based on detailed wind models (Paper I) is combined with a state-of-the-art stellar evolution model (Paper V).

49 5.5.1 Carbon Production: Dredge-up and Thermal Pulses The choice of mass-loss law on the AGB affects the theoretical stellar yields in various ways. It determines the TP-AGB lifetime, which ends when the envelope mass is reduced to some critical value, as well as the actual number of thermal pulses an AGB star goes through. The latter thus represents the maximum possible number of third dredge-up (TDU) episodes, but the mass loss rate also determines when (or even if) the TDU begins and ends. The existence of such dependences is a problem, since no complete picture of how the mass loss of AGB stars depends on basic stellar parameters exists, although it seems that the carbon-rich phase may be understood in the near future (see Paper V).

5.5.2 Carbon Destruction: The on-set of HBB and AGB life times Blöcker [11] studied the effect of mass loss on the occurrence of so-called hot bottom burning1 (HBB), but found that efficient HBB is not prevented by high mass-loss rates as opposed to later results by Frost et al. [44]. Since the mass loss is often not implemented in the same way by different authors (compare e.g. van den Hoek & Groenewegen [162], Marigo [107]), one would expect that at least some of the differences in their results are due to the differences in the mass-loss evolution. As shown by van den Hoek & Groenewegen [162], variations of the mass loss rate (all other model parameters fixed) will affect the total CNO-production significantly. Resulting yields increase with lower mass-loss rates as it results in longer AGB lifetimes. This would also lead to a larger number of thermal pulses (note that one may assume that the amount of dredged-up matter during a thermal pulse is roughly constant), which would affect the carbon yield of low-mass stars in particular. The most important factor is therefore when the so-called super wind sets in during the AGB and how it develops as the stellar parameters change.

1 If the star is sufﬁciently massive, the bottom of the deep convective envelope actually pen- etrates the top of the hydrogen shell. Hence, nucleosynthesis may occur at the bottom of the convective envelope itself.

50 6. Mass-Loss Evolution and the Origin of Detached Shells

(Paper IV)

The origin of the so called ’detached shells’ around AGB stars is not fully understood. However, two popular hypotheses state that these shells form through the interaction of distinct wind phases, or an eruptive phase of mass loss associated with a helium shell ﬂash. Paper IV present a model of the formation of a detached shell around a carbon rich asymptotic giant branch star undergoing thermal pulses (TP-AGB star), based on detailed modelling of mass loss and stellar evolution, leading to a combination of eruptive mass loss and wind interaction.

6.1 A Consequence of Helium Shell Flashes? Paper IV explores aspects of the connection between stellar evolution and mass loss, which may have consequences for the evolution of the circumstellar envelope (CSE) as well. The origin of the so-called ’detached shells’ around carbon rich TP-AGB stars in a late evolutionary stage is not fully understood, although they have been known for about 20 years [118]. The hypothesis that they were connected with He-shell ﬂashes emerged soon after their discovery [119], but the exact formation mechanism remained a matter of debate and this question is still not settled. As an alternative, Wareing et al. [169] have shown that shell-like structures like these can emerge from the interaction of an AGB wind with the interstellar medium (ISM) in which the star is moving. The shell structure could then be explained as a bow shock into the ISM and thus not of internal origin. To understand the mass loss evolution of TP-AGB stars it is important to know how stellar evolution affects the physics of the atmosphere and consequently the formation of a stellar wind. This can be done by feeding the stellar parameters resulting from a stellar evolution model into a dynamic stellar atmosphere model. In Paper IV a radiation-hydrodynamics (RHD) model of a pulsating TP-AGB atmosphere and wind acceleration region (described in Chapter 2) is combined with a detailed stellar evolution model [69, 70] and a simple spherical hydrodynamic model of the CSE to

51 −1 Figure 6.1: Upper left panel: the external luminosity for Δvp = 4.0 kms (full line) −1 and Δvp = 6.0 kms (dashed line) as a function of time. Lower left panel: the mass −1 −1 loss rate for Δvp = 4.0 kms (full line) and Δvp = 6.0 kms (dashed line) as a function of time. The right panels shows a close up (the interval between the vertical dotted lines) demonstrating the effects of the thermal pulse. The vertical dashed line marks the luminosity peak of the He-ﬂash.

demonstrate that the wind properties resulting from the detailed dynamic atmosphere models indeed lead to the formation of a detached shell. The dynamical evolution of the large-scale circumstellar envelope (CSE) on long time scales was followed in order to study the possible formation of a geometrically thin detached shell around the star.

6.2 Two-wind Interaction Previous numerical models of the connection between a He-shell flash and the formation of a detached shell have shown that two-wind interaction (between a fast and a slow wind phase) is a crucial ingredient in the shell formation process, but only as a partial explanation. Steffen & Schönberner [154] confirmed the picture of a detached shell forming from a mass loss eruption and subsequent wind interaction qualitatively, using a spherical hydrodynamic model, although the mass loss evolution is prescribed, not modelled, i.e. a more or less ad hoc assumption about the inner boundary. The detailed wind models that were computed for Paper IV (Δvp = 4.0 − km s 1) show that the wind velocity evolves from a slow phase (∼ 10 km − − s 1) before the He-shell flash, into a rather fast wind (∼ 20 km s 1)during He flash event and then back to a slower wind phase again (see Fig. 6.2). This is basically the same type of evolution as the "modified He-shell flash

52 −1 Figure 6.2: Upper left panel: the wind velocity for Δvp = 4.0 kms (full line) and −1 Δvp = 6.0 kms (dashed line) as a function of time. Lower left panel: the mean −1 −1 degree of dust condensation for Δvp = 4.0 kms (full line) and Δvp = 6.0 kms (dashed line) as a function of time. The right panels shows a close up (the interval between the vertical dotted lines) demonstrating the effects of the thermal pulse. The vertical dashed line marks the luminosity peak of the He-ﬂash.

scenario" used by Steffen & Schönberner [154], although in Paper IV it was obtained using a more sophisticated model, which provides both mass lass rate and wind velocity as functions of stellar parameters. The changes in the mass-loss rate shows a development similar to that of the luminosity (see Fig. 6.1), which makes it qualitatively comparable to the mass-loss evolution assumed by Steffen & Schönberner [154]. In Paper IV it is concluded that both mass loss rate and wind velocity increase during the ﬂash event. This is in qualitative agreement with the scenario proposed by Steffen & Schönberner [154], where detached shells are formed due to a combination of burst-like mass loss during a thermal pulse followed by subsequent wind interaction. It was found that geometrically thin detached shells are neither likely to be formed by simple two-wind interaction nor out of a "mass loss eruption" (with no variations in the wind velocity) alone. In fact, the formation of a slow and a fast wind combined with an eruptive mass loss associated with the fast wind turns out to be very critical.

6.3 Pulsations are Critical The amplitude of the internal pulsations of the star has a great effect on the wind velocities before and after the ﬂash and thus on the strength of the

53 Figure 6.3: Nine instants in the evolution of the CSE after the thermal pulse. The plots show the velocity ﬁeld and the logarithmic density of the CSE for Δvp = 4.0 −1 −1 kms (upper panels) and Δvp = 6.0 kms (lower panels).

wind interaction, determining whether a detached shell is formed or not (see Fig. 6.3). This may actually provide an upper constraint to the strength of the pulsations. The fact that for the stellar parameters used in Paper IV a −1 piston amplitude Δvp = 6.0 km s gives too small variations in the wind velocity to render any wind-wind interaction and therefore no detached shell (see Fig. 6.3), might imply that the pulsations cannot be that strong.

54 6.4 So How Does Detached Shells Form? One of the main results of Paper IV is that both mass loss rate and wind velocity increase during the flash event. This is in qualitative agreement with the "modified He-shell flash scenario" mentioned above, and it is concluded that geometrically thin detached shells are neither likely to be formed by simple two-wind interaction (velocity difference without change in mass-loss rate) nor out of a "mass loss eruption" (with no variations in the wind velocity) alone. In fact, the formation of a slow and a fast wind combined with an eruptive mass loss associated with the fast wind turns out to be very critical. The amplitude of the internal pulsations of the star has a great effect on the wind velocities before and after the flash and thus on the strength of the wind interaction, in the case studied in Paper IV. Further work is needed, but so far it can be concluded that the He-shell flash and the associated "mass loss eruption" in combination with wind interaction gives a satisfying and consistent description of the formation of detached shells around TP-AGB stars.

55

7. Models of Carbon Star Evolution

(Paper V)

In Paper V a model of C-star evolution is presented, where the effects of the changing carbon excess (the amount that the abundance carbon ex- ceeds that of oxygen) are taken into account in as much detail as is currently possible. It is demonstrated that the late-stage evolution is regulated by the carbon excess.

7.1 Mass-loss Effects on C-star Evolution In Paper I, the basis for a new mass-loss prescription suitable for stellar evolution modelling is presented. The choice of mass loss prescription matters very much in stellar evolution modelling, so the way one chooses to pre- scribe the mass loss is as critical as all the other parameters in stellar evolution modelling (e.g., the MLT parameters, dredge-up efﬁciency etc.). Mass loss affects stellar evolution in various ways. For carbon stars the duration of the AGB phase and the number of thermal pulses is basically set by the mass loss rate. Also the internal structure depends on the mass loss rate [11] which in turn affects the fundamental stellar parameters. Accord- ing to our wind models, the mass loss rate depends on these stellar parameters, so there is an interesting feedback mechanism at work here, which means that the mass loss prescription put into a stellar evolution model is critical. Perhaps even more interesting is the effect on nucleosynthesis. It is well-known that varying the mass loss rate can have profound effects on the chemical yields of AGB stars [162]. The low temperatures and intense mass loss associated with carbon-rich AGB stars affects the duration of the AGB phase, and therefore the nucleosynthetic yields (especially the s-process nucleosynthesis) as well as the maximum C/O-ratio that can be obtained. With a high mass loss rate, fewer thermal pulses can occur during the carbon-star phase and consequently less nucleosynthesis takes place. The number of thermal pulses represents also the maximum possible number of third dredge-up (TDU) episodes bringing carbon and other newly-synthesised elements to the stellar surface. Furthermore, the

57 Figure 7.1: Luminosity-temperature diagram (cf. HR-diagram) for Model A-D (see Paper VI for a deﬁnition of the different case). All tracks are plotted starting from the zero-age horizontal branch (ZAHB).

mass loss rate does also determine if and when the TDU begins and ends [86].

7.2 The Origin of the Superwind The grid of wind models in Paper I has resulted in a detailed prescription for the mass-loss during the carbon-star phase. The most important new features of this prescription are that it takes into account the steep dependence on the amount of carbon available for dust formation and the mass-loss thresholds that emerge in certain parts of stellar parameter space. In Paper V this mass-loss prescription is implemented, together with abundance- dependent opacity data, in a new detailed stellar evolution model. The Mass-loss evolution changes dramatically. When the new opacities and the mass-loss prescription from Paper I is combined (Model D), a very pronounced superwind develops as the carbon excess increases. The superwind develops as a consequence of the carbon-excess dependence included in the mass-loss and the lower effective temperature (which also favours mass loss) obtained with the new opacities. The superwind does not arise immediately, since the star has to become sufﬁciently carbon-rich

58 and, at the same time, have a low enough temperature. The mass-loss rate − − grows from M˙ ∼ 10 8 up to M˙ ∼ 10 5, while the carbon excess grows by a factor of four and the temperature drops by almost 1000 K. Within less than 200 Myr from the onset of the superwind, half of the stellar mass is lost in Model D (see Fig. 7.2, lower panel). We note that the AGB will probably be terminated even before the next thermal pulse. Hence the star experiences only 4-5 thermal pulses as carbon star. This is due to a run- away process: with each thermal pulse more carbon is dredged up, which increases the carbon excess and hence also the mass-loss rate, that in turn limits the number of thermal pulses by terminating the AGB. One could say that it is a self-regulating mechanism for the number of thermal pulses. In Model B the superwind has not yet been reach after ﬁve thermal pulses, which is due to a combination of a slower decrease in effective temperature and a steep dependence of mass loss on temperature as according to Paper I. The relatively rapid decline in temperature (see Fig. 7.2, upper panel) causes the star to move more or less horizontally to the left in the HR-diagram (Fig. 7.1) in all cases except Model C. The reason why the temperature does not show a signiﬁcant drop in case C seems to be coupled to a lower mass-loss rate. As the star loses mass, it will start expand rapidly at some critical point and become cooler. In Model C, however, this point has not yet been reached.

7.3 Limited Nucleosynthesis Due to the rapid termination of the AGB, the nucleosynthetic yields are thus probably lower compared to many previous models of C-star evolution. This is limiting the amount of carbon and other nucleosynthesis products of AGB stars, most notably the s-process elements, that can be expelled by a carbon star. With only a few thermal pulses occurring in the carbon-star phase, less carbon will be dredged up. The ﬁnal ejecta (due to the superwind) therefore contains less carbon. Furthermore, what do the results of Paper V mean for really low metallicity? The carbon excess increases due to dredge-up of primary carbon, so the star will become carbon rich at an earlier stage. The question is of course if a superwind would still be formed? These questions need to be answered with further modelling in order to understand nucleosynthesis during the carbon-star phase.

59 Figure 7.2: Evolution of the effective temperature and the stellar mass during the TP-AGB, starting at the peak of the ﬁrst thermal pulse. The different lines have the same meaning as in Fig. 7.1.

60 8. Galactic Chemical Evolution Models

Now this thesis makes a sudden jump from stellar physics and evolution to the evolution of stellar populations in galaxies. The perspective is quite different: here the stars become actors in a complicated play, rather than the theme of the play itself. As a part of the thesis project, a new coputer code for modelling the chemical evolution of the Galaxy, i.e. the enrichment of heavy elements over time, has been developed. In this chapter the underlying theory and its mathematical description is presented.

8.1 The Milky Way Galaxy Our galaxy, the Milky Way, consists of a dense centre known as "the bulge", a disc of gas, dust and stars (forming four distinct arm structures spiralling outward) with a bar-shaped central region, and a diffuse spherical halo of stars and stellar clusters. The baryonic mass distribution within the Galaxy closely resembles that of an SBc galaxy, according to the Hubble classiﬁca- tion scheme [76, 77], which is a spiral galaxy with relatively loosely-wound arms and a quite distinct bar structure at the centre. The Milky Way was not originally thought to be a barred spiral galaxy, but some 25 years ago evidence for the existence of a bar structure started to pile up. It was later conﬁrmed by infrared observations with Spitzer Space Telescope in 2005 [30]. The Milky Way is a rather typical, average-sized disc galaxy with a baryonic mass of about 6 · 1011 solar masses (M). Judging from its mass, our galaxy contains about 200 - 400 billion stars, which would correspond to an integrated absolute visual magnitude of about -20.9. Most of the Galaxy is dominated by the disc and the stellar halo, which are two main components that are usually included in galactic chemical evolution models. Be- low, these components are further described.

8.1.1 The Galactic Disc The galactic disc has a diameter of between 20 and 30 kpc. The the galactocentric distance to the Sun R is about 8 − 9 kpc, according to recent es- timates. The central bar of the Galaxy is thought to be about 8 kpc across, passing through the galactic centre at roughly a 45 degree angle to the line between the Sun and the center of the Galaxy (see Fig. 8.1). The bar is sur-

61 rounded by a ring of the molecular hydrogen, sometimes referred to as the "5 kpc ring". This part of the disc also has the most active star formation. The distribution of neutral and molecular hydrogen in the disc is rather well known [34], while the stellar component, and consequently the total baryonic mass distribution, is not. Since the majority of spiral galaxies similar to the Milky Way are known to have roughly exponential mass distributions, it is thought that our Galaxy is no exception. The scale length rd of the Milky Way disc, however, is still a matter of debate. Existing observational constraints give numbers ranging from rd = 2 kpc up to as much as 6 kpc. In Paper VI it is argued that the observed metallicity gradient as well as the hydrogen distribution favours a scale length close to the lower limit, but no ﬁnal word yet.

8.1.2 The Stellar Halo and the "Thick Disc" The galactic disk is surrounded by a roughly spherical halo, consisting of mostly old stars and globular clusters,. The size of the stellar halo is not known very well but a diameter of at least 60 kpc seems probable. It is also not known exactly how the stars in the halo are distributed. Observational constraints suggest that the stellar density of the halo falls off roughly as a α power-law ρs ∝ r ,whereα = 3 − 4[68]. In addition to the halo, the Milky Way, as well as many other galaxies of similar type, seems to have another disc-like component: the thick disc. This is a "ﬂuffed-up" disc surrounding the ordinary disc (usually referred to as the thin disc), which may share a common origin with the stellar halo. However, the origin of the thick disc is still very much debated, and there are essentially two rivalling hypotheses: (a) the thick disc formed by early accretion together with the halo, or (b) the thick disc is the result of tidal heating of the ordinary (thin) disc due to merger events. In Paper IV it is assumed that scenario (a) is correct, but the purpose of that study is not to explicitly model the thick disc evolution.

8.2 Origin of the Elements In the current paradigm (B2FH) essentially all elements heavier than helium (in astronomical terminology often referred to as just metals)arepro- duced by nuclear reactions inside stars or in supernovae. Thus, a model for the build-up of heavy elements within a stellar population can be con- structed as a model of the evolution of that population. In chemical evolution models (CEMs) one considers the evolution of stars and interstellar gas in a speciﬁed region with additional information about how much of new metals is being synthesised in the stars. Simple analytic CEMs where developed in pioneering papers by van den Bergh [163] and Schmidt [144], where the rate of star formation was connected with the build-up rate of metals in the Galaxy. Reﬁned, but simi-

62 Figure 8.1: A map over the Milky Way disc (adopted from Wikipedia Commons).

lar, works by Talbot & Arnett [155, 156], Searle & Sargent [148] and Tins- ley [160] formulated the problem in terms of an evolving stellar population combined with results from stellar nucleosynthesis. The evolution of individual elements is often due to several sources, occurring in different environments, which may locally have very different star formation histories. Nonetheless, the problem can still be described in closed form as a set of coupled ordinary differential equations (ODEs), assuming a homogeneous distribution of matter (stars, gas and metals) and that the gas is well-mixed at all times.

8.3 Model Ingredients and Equations The baryonic matter content of the system can be divided into two main components: stars and gas (including metals/dust). Stars, in this context, include both "living" stars and stellar remnants and gas is the total amount

63 of interstellar gas, i.e. primordial gas as well as gas released by evolving stars. It is therefore quite natural to formulate the problem in terms of functions describing these two components. Throughout this chapter (as well as Paper VI) gas mass, stellar mass and all elemental abundances are expressed in terms of mass surface densities rather than total masses. Furthermore, "dot notation" is used for prescribed rate-functions, like the star formation rate, while partial time-derivatives are used for basic physical quantities, e.g., matter densities (stars, gas and metals).

8.3.1 Galaxy Formation It is usually assumed that the Galaxy was formed through baryonic infall into the dark halo potential, or more precisely, by accretion of pristine gas (hydrogen and helium), where the rate of accretion is considered to follow an exponential decay [92, 159]. It is also rather well established that the halo/thick disc and the thin disc components of the Galaxy were assembled on different time scales and, perhaps, with some separation in time, as suggested by Chiappini et al. [26]. This is known as the two-infall model, which consists of two infall episodes where the disc formation starts after some time τmax (which is where the rate of accretion reaches its maximum), i.e., the total rate of accretion is

Σ˙ h(r,t)ift ≤ τmax Σ˙ inf.(r,t) = (8.1) Σ˙ h(r,t) + Σ˙ d(r,t)ift > τmax where, for the halo/thick disc, −1 Σd(r,t0) t0 t Σ˙ h(r,t) = 1 − exp − exp − , (8.2) τh τh τh and for the (thin) disc, −1 Σd(r,t0) (t0 − τmax) (t − τmax) Σ˙ d(r,t) = 1 − exp − exp − , (8.3) τd(r ) τd(r ) τd(r ) where τ is the infall time scale and t = t0,beingtheageofthegalaxy.To simulate the inside-out formation of a galactic disc, i.e., the inner disc form faster than the outer disc, the infall time scale can be assumed to have a linear radial dependence, r τd(r ) = max 0,τ − τ0 1 − , (8.4) R where τ is the infall time scale in the solar neighbourhood, τ0 is an arbitrary constants and R = 8.5 kpc is the galactocentric distance to the Sun. For the halo/thick-disc phase it is usually assumed that τh is the same at all galactocentric radii.

64 The ﬁnal baryonic (thin) disc is assumed to follow an exponential distribution, −r /rd Σd(r,t0) = Σ0e , (8.5) where Σ0 is the present-day central surface density. Hence, the total disc M = πΣ 2 mass becomes d 2 0rd. The halo/thick-disc surface density is ade- quately modelled by a gaussian surface density distribution, 2 2 r − R Σ (r,t) = Σ (R,t )exp − , (8.6) h h 0 2 rh or, as suggested by Renda et al. [129], a Hubble proﬁle [75], 2 2 −1 R r Σh(r,t) = Σh(R,t0) 1 + 1 + , (8.7) rh rh where rh is the halo scale lenght and R is the galactocentric distance of the Sun.

8.3.2 Galactic Dynamics The rotation of the disc and the internal dynamics of the stars and gas cannot really be decoupled from the evolution of the stellar content and abundances is the ISM. So-called chemodynamic models are dealing with this problem, while CEMs are usually based on simplistic assumptions about the baryonic infall, and possibly galactic outflows, as the only kind of "dynamics" (see above). This may be far from reality, although including detailed N-body dynamics for the stellar component and hydrodynamics for the gas component does not seem to give dramatically different results for a Milky Way-type galaxy (compare, e.g., the results of Samland et al. [136] with those of Chiappini et al. [26]). Including detailed dynamics also has to be done at the expense of including detailed nucleosynthetic information and the possibility to explore free-parameter configurations easily. So, the amount of that dynamics should be included or not, depends on the objective of the study. The only actual disc dynamics that goes into the CEM described here, is the angular frequency of the disc Ω(r,t). As the baryonic component grows, and as matter may be redistributed in the disc, Ω(r,t) must be regarded as function of time. However, to first order, the rotational dynamics of the Galaxy is due the dark matter halo, which had assembled prior to the accretion of baryons and the onset of star formation. Thus, it is quite reasonable to use the observed present-day rotation Ω0(r ) (see Fig. 8.2) of the disc as input. The angular frequency is here assumed to be coupled to the star formation efficiency, which will be described below. Furthermore, one may note that Ω(r,t) is also connected to radial flows induced by shearing in a viscous disc, which has been considered by several authors [31, 151, 152,

65 153, 158]. In the following, such possible radial ﬂows are assumed to be negligible, however.

8.3.3 Star Formation The common choice of star-formation prescription is the so-called Kennicutt-Schmidt law [143, 91], which is essentially a power-law of the form Σ˙ = Σ1+ε (r,t) k gas . (8.8)

In chemical evolution modelling, the total mass density is often included in the prescription, e.g.,

1+ε Σgas(r,t) Σ˙ (r,t) = νΣ(r,t) , (8.9) Σ(r,t)

− where ν is the star formation efficiency expressed in Gyr 1. In many cases it is adequate to just assume that ν is constant. In a galactic disc, however, the star formation efficiency can be related to the angular frequency of the disc, [171, 13], Ω0(r ) ν(r,t) = ν0 , (8.10) Ω0(R) where ν0 is a constant, Ω0 is the present-day (t = t0) angular frequency of the disc. Motivated by the stability criterion for a rotationg disc [90], it is also often assumed that there exist a critical gas density for star formation to be efficient, i.e, a kind of star-formation threshold at Σgas = Σc. The critical density Σc may change along the disc, but for simplicity it may be assumed −2 that Σc has a specific value, e.g., 7.0M pc ,overthewholedisc[90]. Exactly how star formation should be prescribed in CEMs is mostly a matter of taste, due to the absence of a solid theory of star formation. Lynden-Bell [105] argued that the star formation rate may be a function of gas mass density, sound speed, angular frequency of the disc, metallicity, magnetic field strength and several other variables. A prescription of star formation with that many ingredients would of course make CEMs incredibly complicated, and perhaps that is why a simple Kennicutt-Schmidt law is the most common choice among modellers.

8.3.4 The Initial Mass Function The initial mass function (IMF) has ever since the concept was invented by Salpeter [135] been a matter of controversy and debate. One may picture the IMF as a statistical distribution function describing the probability of a star of mass m to be formed in a star forming region. Salpeter suggested a simple time-independent power-law distribution with lower and upper mass cuts. This functional form of the IMF is still the most widely used, although it is

66 Figure 8.2: The angular rotation curve of the Milky Way according to Sofue [150]. The crossing dashed lines marks the solar neighbourhood. now well established that the IMF turns over at low masses and is probably truncated at the high-mass end. Several forms of the IMFs has been suggested and include piece-wise power-laws [139], log-normal distributions [114] and inverse gamma distributions [102]. It is still very much debated whether the IMF is universal, i.e., the same regardless of the star formation environment, or if it may evolve over time, e.g. as a consequence of metallicity and/or gas density dependence [41]. In Paper VI it is suggested that the characteristic stellar mass is related to the Jeans mass [85], which corresponds to an IMF that depends on the gas density (but see also the next Chapter).

8.3.5 Evolution of the ISM and Elemental Ratios Since stellar mass loss mainly occurs during post-main sequence evolution, the rate of mass return to the ISM is related to the number of stars evolving off the main sequence at a given time. If the Galaxy consisted only of a single

67 population of stars (i.e. all stars are formed at t = t0), we would have a gas- ejection rate ∂ mu Σ˙ e(r,t) = Mstars φ(m,t0)R(m)dm (8.11) ∂t mto(t) dmto =−Mstars φ(mto,t0)mtoR(mto) (8.12) dt where R(m)isthegasreturnfractionforastarofstellarmassm, mto is the turn-off mass for a population of age t,andmu is the upper mass limit of the IMF φ(m,t), which is the distribution of masses of the stars formed at a given time t. For a galaxy with continuous or episodic star formation, calculation of the mass ejection rate must not only account for the main sequence life-time of the stars, but also the fact that in such a system we have stars of various ages. Hence, we obtain (making use of Leibniz’ formula) the total amount of mass returned from stars at a time t from t mu ∂ Σ˙ e(r,t) = φ(m,t)Σ˙ (r,t − τm)mR(m)dmdt ∂ t 0 mto(t ) mu = φ(m,t)Σ˙ (r,t − τm)R(m)dm. (8.13) mto(t)

The equation for the density of an element i, requires that we keep track of both the amount of new metals generated by a star during its evolution as well as the amount of metals that are lost from the ISM when a star is formed, but are being re-released as the star evolves and return a certain fraction of its mass back to the ISM. Mathematically, this is usually written

∂Σi Σi (r,t) = Σ˙ inf,i (r,t) − Σ˙ (r,t) ∂t Σ (r,t) gas mu Xi (r,t − τm)φ(m,t)Σ˙ (r,t − τm)dm, (8.14) ml where the deﬁnition of Xi is similar to the "production matrix" formalism introduced by Talbot & Arnett [155, 156].

8.3.6 Supernovae type Ia In order to include the contribution from the second most common type of supernova events, the supernovae type Ia (SNIa), which is assumed to be the result of gas accretion onto a white dwarf from its companion (pre- sumably a red giant) in a binary system, the formalism introduced by Greg- gio & Renzini [47] and Matteucci & Renzini [112] is commonly used. In this formalism, the equation describing the evolution of an element i can be

68 written as ∂Σi Σi (r,t) = Σ˙ i (r,t) − Σ˙ (r,t) + ∂t inf, Σ (r,t) gas mu Xi (r,t − τm)φ(m,t)Σ˙ (r,t − τm)dm+ B mu mB l Xi (r,t − τm)φ(m,t)Σ˙ (r,t − τm)dm+ (8.15) ml B mu (1− η) Xi (r,t − τm)φ(m,t)Σ˙ (r,t − τm)dm+ mB l B mu 1/2 η ϕ(μ)Xi (r,t − τm)φ(m,t)Σ˙ (r,t − τm)dμdm, mB μ l min where τm is the life time of a star of mass m, φ(m,t)istheIMF,Xi is deﬁned as above and η is the fraction of low and intermediate stars undergoing an SNIa event. In the equation above, SNIa events originate from binary sys- B B tems more massive than ml and with a maximum mass of mu. The binary distribution function ϕ(μ) is a function of the ratio of the secondary to the combined mass of the binary system μ, which is normalised on the interval μ ∈ [0,0.5].

8.4 Stellar Yields The last, but not least important, ingredient in CEMs is the input data from stellar nucleosynthesis – the stellar yields. This quantity deserves some at- tentions, since it (together with the IMF) has the greatest impact on CEM results. Stellar yields tell us what elements and how much of them is returned to the ISM when stars die. These yields are computed separately in stellar evolution models and models of supernova explosions, which then provide us with tables of estimated yields for various isotopes. A common definition of the yield due to stellar winds is [106] τm mpi ≡ M˙ (m,t)[Xi (t) − Xi (0)]dt (8.16) 0 where m is the initial mass of the star, M˙ (m,t) is the mass-loss rate and Xi (t) is the surface mass fraction of an element i at time t.Withthisdefi- nition there are only two factors that determine the total ejected mass of a specific element: (1) its initial mass, and (2) its initial chemical composition Xi = Xi (0). As argued in Paper VI, the stellar yields may be the largest source of errors in CEMs and such models must therefore be used with caution. For example, CEMs cannot be used to derive detailed constraints on the evolu-

69 tion of galaxies unless one can be sure that the yields are correct. Computa- tions of the explosive yields of supernovae are based on different assumptions about the supernova energy, the nature of mixing and convection processes, and the amount of mass-loss and fall back. Furthermore, certain assumptions have to be made about the nuclear reaction rates, which provide even more parametric freedom in these models. Thus, different authors often obtain quite different yields from otherwise rather similar models (compare, e.g., the results by Woosley & Weaver [175] with those by Chiefﬁ & Limongi [23]). For AGB stars the situation is not at all different – the uncertainties in the AGB-nucleosynthesis is perhaps even greater. This problem is, to some extent, explored and discussed further in the Chapter 10 and in Paper VI.

70 9. An Extension of the Padoan- Nordlund Theory of the IMF

The origin and functional form of the IMF has been debated ever since the concept was invented by Salpeter [135]. An attractive and simple theory for the origin of the IMF has emerged during the last decade, based on the idea that turbulent fragmentation is the dominating mechanism shap- ing the IMF [123]. In this chapter this quite simple theory for time IMF is brieﬂy outlined and extended to non-isothermal conditions. The purpose is to motivate the choice of functional form of the IMF used in Paper VI.

9.1 A Universal IMF? The hypothesis that the IMF is universal and constant over time is often used to limit the degeneracies of star formation rates (SFRs) obtained from observations. However, the interpretation of data is difﬁcult and depends strongly on such basic assumptions, which leads to diverging results. As an example, Massey [109] presented IMF slopes for young stellar clusters and OB associations in the Galaxy and the Magellanic clouds which seem to cluster around the Salpeter value, α =−2.35. The same data was also considered by Scalo [141] who used it as evidence for variations. Elmegreen [50] demonstrated how variations of the slope of the IMF can be obtained if the populations considered are not large enough, which may seem to favour the conclusion by Massey.

9.2 Turbulent Fragmentation From the theoretical point of view, a universal IMF seems perhaps more likely than it does from observations. Larson [95] suggested that the dominant driving mechanism behind star formation is turbulence. Padoan & Nordlund [123] made simulations of super-sonic, turbulent fragmentation in the ISM, indeed showing that the distribution of collapsing cores almost exactly reproduces the observed properties of the IMF. Based on simple assumptions like the power spectrum of velocities, −β E(k) ∼ k , the Larson relations [96, 97] and that the deﬁnition of the thermal Jeans mass [85] applies, Padoan & Nordlund found an analytical approach towards describing the IMF. The power-law end of the IMF is shown to be dependent only on the spectral index β as x = 3/(4 − β).

71 Simulations and observations both imply a spectral index β ∼ 1.8 and the results by Boldyrev, Nordlund & Padoan [15] speciﬁcally suggest the value β = 1.74, from which the canonical Salpeter slope is recovered. In the low-mass end, the formation of collapsing cores becomes dependent on the statistical distribution of Jeans masses which is assumed to be determined by the probability-density function (PDF) for the gas mass density. The fraction of cores of mass m with gravitational energy in excess of their thermal energy is given by the integral of p(mJ)from0tom.Hence, the IMF can be obtained from −3/(4−β)−1 φ(m) = φ0 P (m)m (9.1) where P (m) is the cumulative probability-density function of Jeans masses, i.e. m P (m) ≡ p(mJ)dmJ. (9.2) 0

The form of p(mJ) is a result of simulations. Isothermal simulations [122, 140, 123] have shown that the PDF for the gas density is almost exactly lognormal, i.e., 1 1 ln(ρ) −〈ln(ρ)〉 2 p(ρ) = exp − , (9.3) 2πσρ 2 σ where σ is the dispersion and p0 is a normalisation constant. Since, for −1/2 isothermal conditions, the Jeans mass mJ ∝ ρ ,Padoan&Nordlund [123] concluded that the distribution of Jeans masses p(mJ)is −2 2 2 mJ 1 −ln(mJ) − A p(mJ) = exp − , (9.4) 2πσ mJ,0 2 σ/2 where A = ln(mJ,0) −〈ln(ρ)〉. (9.5)

This PDF leads to an IMF of the form + σ2 4ln(m) −3/(4−β)−1 φ(m) = φ0 1 + erf m , (9.6) 2 2σ where z 2 −t 2 erf(z) ≡ e dt. (9.7) π 0

So far, this is the theory of Padoan & Nordlund [123]. But could one im- prove upon it? First, one may note that the PDF for the gas mass density is only lognormal for the isothermal case. Scalo et al. [140] showed that a power-law asymptote/tail in the PDF will develop if a polytropic equation

72 5 γ=1.0, 4 M=3

3 log N 2

1 -2 -1 0 1 2

5

4 γ=0.3, M=3

3 log N 2

1 -2 -1 0 1 2 log ρ

Figure 9.1: Mass density PDFs measured in numerical experiments with super- sonic turbulence for an isothermal (upper panel), a polytropic equation of state with γ = 0.3 (lower panel). The dashed lines show a log-normal ﬁt to the isothermal case. Adopted from Scalo et al. [140].

of state, ρ γ ρ γ−1 P = P0 , T = T0 , (9.8) ρ0 ρ0 with γ < 1 is assumed (see the example in Fig. 9.1). Such PDFs are well- described by an inverse gamma distribution, n ρ˜ ρ˜ − +n p(ρ) = exp − ρ (1 ), (9.9) Γ(n) ρ where ∞ z− −t Γ(z) ≡ t 1e dt, ℜ(z) > 0, n > 0 (9.10) 0

73 and ρ˜ is a density-scale parameter. The appropriate values of the parameters n and ρ˜ depend on the choice of the polytropic exponent γ and the Mach number. Higher Mach numbers tend to make n smaller, and vice versa. For n = 1.6 this function is very similar to the PDF obtained by Scalo et al. [140]. 2 −1/2 The Jeans mass is mJ ∝ T P . Hence, in case of a polytropic equation of state, α ρ 3γ− 4 mJ = mJ,0 , α = , (9.11) ρ0 2 which gives the following distribution of Jeans masses, n 1/α −(n/α+1) 1 ρ˜ ρ˜ mJ,0 mJ p(mJ) = exp − , (9.12) Γ(n)|α| ρ0 ρ0 mJ mJ,0

If one for convenience assumes that n = 1, one obtains for a cumulative distribution of Jeans masses which is α ρ˜ m 1/ 4 P (m) = 1 − exp − , γ < . (9.13) ρ0 mJ,0 3

The main result is that the cumulative distribution of Jeans masses rises more slowly compared to the result obtained by Padoan & Nordlund [123]. A realistic star-forming gas would have a polytropic index γ that changes depending on the physical circumstances. In particular, γ should be close to unity for high densities (small mJ) and one would expect γ < 1forthingases [84]. A particularely convenient form of P (m) is the inverse exponential, mc ρ˜ P (m) = exp − , mc ∝ mJ,0 , (9.14) m ρ0 which is similar to the lognormal case for small mJ and has the slowly increasing trend of the cumulative inverse gamma distribution for larger mJ (see Fig. 9.2). Assuming that the inverse exponential is a reasonable com- promise, the corresponding IMF takes the form, −(1+x) mc −(1+x) 3 φ(m) = φ0 P (m)m = φ0 exp − m , x = , (9.15) m 4 − β where ∞ −1 x−1 mc φ0 = m φ(m)dm = . (9.16) 0 Γ(x − 1)

This IMF is identical to the one suggested by Larson [102] and mathematically very convenient.

74 Figure 9.2: Cumulative distribution functions of Jeans masses on a normalised mass scale. The inverse-gamma case with n = 1.6, and the lognormal case correspond to the simulations by Scalo et al. [140] for γ = 0.3 and γ = 1.0, respectively.

9.3 Environmental Variations of the IMF The Larson scaling relations referred to above applies, in principle, only to the high-mass end of the spectrum. It is therefore not clear whether the IMF can depend on environment in low-mass regime. Sirianni et al. [149] presented observations of the central parts of 30 Doradus where a turn-over in the IMF at ∼ 2.7 M were found. That is signiﬁcantly above the value derived for the Milky Way disc. There exist plenty of evidence that the "turn- over mass" is not universal (see the review by Elmegreen [41] and references therein) and it is therefore reasonable to assume that mc depends on local properties of the physical environment of the star forming region. The density-scale parameter used above is determining the shape of the density PDF and is probably depending on physical environment, which suggests that mc may vary. In addition to the turn-over at low stellar masses, there is almost certainly some kind of high-mass truncation in the IMF. Elmegreen [51] suggested a damping factor, η m f (m) = exp , (9.17) mu

75 that will occur due to competition for mass. For simplicity, one may assume that η = 1, i.e., the high mass end has a soft truncation given by an e-folding decay of some mass scale mu. As pointed out by Elmegreen [51], the high- mass truncation should shift towards a higher mass as the Jeans mass becomes larger, which introduces another dependence on the physical environment. This can be understood intuitively, by considering a scenario where the Jeans mass is high due to, e.g., low gas density/pressure or high temperature. In such a case, the high mass stars will win the competition for mass, simply because the most probable mass of collapsing clouds is of order the Jeans mass and once a cloud collapses, gravitational accretion may increase the mass even further. With this exponential damping factor, the IMF suggested in the previous section becomes −(1+x) mc(t) m φ(m,t) = φ0(t)m exp − + , (9.18) m mu(t) where the parameters mc and mu are quite likely related to some characteristic mass scale related to the physical origin of the IMF as discussed above. Larson [100, 101, 102] suggested that the "turn-over mass" might be related to a fundamental mass scale in the star formation process such as the Jeans mass and recent observational evidence for an evolving IMF suggest characteristic stellar masses which are consistent with such a picture [164]. In the previous section it was argued that mc is a function of the density-scale parameter ρ˜, which may also be a quantity that sets the mass scale. In the general case, the normalisation of the IMF φ0,isnowafunctionof time as well. More precisely, x μ (t) φ0(t) = 2 , x ≥ 1, (9.19) Kx−1(μ) where μ is a dimensionless variable deﬁned as mc(t) μ(t) ≡ , (9.20) mu(t) and Kn is the modiﬁed Bessel function of the second kind and order n.The characteristic (or mean) stellar mass of this IMF is

Kx−1(μ) m¯ (t) = mu(t)μ(t) , x ≥ 1. (9.21) Kx(μ)

Obviously, the mean stellar mass may be a function of time too. However, in the special case when mc(t) ∝ mu(t), or μ = const., the situation is some- what simpler. This simplifying assumption was used in Paper VI to restrict the degree of freedom of the CEM and because φ0 becomes constant over time and the IMF will not have to be renormalised at every time step in the CEM.

76 9.4 Alternative Mechanisms It is quite interesting that the mass distribution of unstable/collapsing cores found in MHD simulations is so similar to the Chabrier IMF [22] and in qualitative agreement with mass distributions of pre-stellar cores selected from dust emission or molecular line observations. This could be taken as evidence for that gravitational fragmentation, competitive mass accretion or merging (not included in the turbulence simulations discussed above) may not play a very important rôle in the origin of the IMF. However, the IMF may still be a superposition of several processes. There is most likely an isolated star-formation mode which is dominated by fragmentation near the Jeans mass. In dense clusters some extra accretion and clump coagulation will have to enhance the high-mass component relative to the case of isolated star formation. Recent simulations of competitive accretion, seem to support this scenario [16]. The simple theory for the IMF discussed above, is probably far from a complete theory.

77

10. The Origin of Carbon

(Paper VI)

The build-up of heavy elements in the Milky Way (or any other galaxy) is due to the nucleosynthesis in the stars and the composition and evolution of the stellar population and the ISM. Thus, stellar yields and the form of the IMF are crucial ingredients in models of this process. Paper VI focuses on the origin of carbon, which is most likely linked to carbon stars.

10.1 The Making of Carbon Carbon is one of the most common elements in the Universe, but surprisingly little is known about it. Carbon has a stellar origin, as do almost all other elements, but the nucleosynthesis of carbon seems harder to constrain than that of many other elements. The stellar origin of carbon is mainly due to the Triple-Alpha reaction [134] but this reaction may occur in various types of stars. Already in the work by Burbidge et al. [19] it was suggested that carbon was provided by mass loss from red giants and supergiants. Later Dearborn et al. [36] suggested that low-mass stars may be a significant source of carbon based on observed carbon in planetary nebulae. Recent theoretical work on stellar evolution of low and intermediate mass (LIM) stars [107, 83, 59, 88], support this picture, although the results differ quantitatively. Models of chemical evolution (see Chapter 8) are in good agreement with observed abundances if a delayed carbon release from LIM-stars is assumed. Timmes et al. used the nucleosynthetic yields by Woosley & Weaver [175] and Renzini & Voli [130], which led to an overproduction of carbon due to LIM-stars to the Galactic Disc. However, more recent work (using more recent sets of yields for LIM-stars) have confirmed that the carbon enrichment problem can be solved by a significant contribution from LIM-stars [29, 2, 21]. Maeder [106] and Portinari et al. [126], among others, have argued that radiatively driven massive winds from high-mass (HM) stars should provide significant amounts of carbon. Garnett et al. [57] observed that the C/O-ratio increased with increasing O/H in dwarf irregular galaxies, which they interpreted as to be consistent with carbon being produced in massive

79 stars with metallicity-dependent yields, as in the models by Maeder and Portinari et al. Following that idea, Gustafsson et al. [65] suggested that the rising C/O-trend with metallicity that they found in Galactic-disc stars was the result of carbon being produced in HM-stars rather than LIM-stars. Paper VI presents a series of CEMs with different nucleosynthesis prescriptions – five basic cases labelled A-E. The model tracks displayed in this chapter and in Paper VI are labelled A1-E1 or A2-D2, depending on the type of star-formation law being used (see Sect. 10.3). Case A represents a scenario where the LIM-stars are producing most of the carbon, while Case B represents a case where the carbon is to a large extent produced in HM- stars, similar to the scenario suggested by Gustafsson et al. [65]. Case C is an attempt to explain the observed trend in C/O at low metallicity as obtained by Akerman et al. [2] and Fabbian et al. [54]. The modified LIM-star yields in Case D aims at explaining the super-solar carbon abundances observed in the solar neighbourhood (see Paper VI for further details). This modifica- tion is motivated by the results presented in Paper V and many other recent attempts to constrain the carbon production of LIM-stars. Finally, Model E1 is of particular interest, since it utilises the form of an evolving IMF (see Fig. 10.1) derived in the previous chapter, instead of modifications to the nucleosynthesis prescriptions.

10.2 A Top-heavy IMF? Introducing Non-LTE corrections in the derivation of abundances for low metallicity stars [2, 54] have revealed a declining trend for the C/O ratio in the solar neighbourhood at early times. According to these new results, C/O vs. O/H shows a negative slope until the anticipated onset of disc formation, i.e., during the first billion years of Galactic evolution, which is in disagreement with the predictions of CEMs which do not include any modifications of the carbon and oxygen yields [29, 59]. The C/O discrepancy is obviously connected to the first generations of stars in the Milky Way, and may therefore have common origin with other problems in the early chemical evolution. For instance, the underabundance of essentially metal free LIM-stars in the halo, which is often claimed to be the result of a top-heavy initial mass function (IMF) at early times [1, 161, 89], is one such problem. If the IMF has evolved from being initially top-heavy, to the form that is observed in the solar neighbourhood today, it is possible that this may affect the C/O ratio as well. As demonstrated in Chapter 9, an IMF of the form −(1+x) mc(t) m φ(m,t) = φ0(t)m exp − + , (10.1) m mu(t) may be a reasonable ansatz, where mc and mu are the masses defining the low-mass turn over and the high-mass truncation of the IMF, respectively. With x = 1.80, and some dependence of mc and mu on the physical envi-

80 Figure 10.1: Evolution of the IMF with the gas density in Model E1.

ronment, as discussed in Chapter 9, it becomes the IMF considered in Paper VI, which evolves from a top-heavy form for low gas densities to a "normal" IMF as the gas density increases due to infall. The need for a non-universal IMF was seemingly ruled out by Chiappini et al. [27]. One of the IMFs they tested (the Padoan & Nordlund IMF, see Chapter 9) is in many ways similar to that used in Paper VI, and they also found that it gave good agreement with the observed properties of the solar vicinity. However, since the results they obtained with a non-evolving Scalo IMF were almost the same, and since more drastically changing IMF gave unrealistic metallicity distributions, they concluded that a constant IMF was still the best way to explain the observational constraints in the Milky Way. In Paper VI, on the other hand, it is demonstrated that even a very top-heavy IMF during the early phase of chemical evolution, will not affect the metallicity distribution much at all (see Fig. 10.2) and may be a solution to the problem with the very high C/O-ratios in metal-poor stars (see the grey triple-dot-dash line in Fig. 10.3).

81 Figure 10.2: Predicted metallicity (G-dwarf) distributions for the solar neighbourhood according to Model A1-E1 convolved with a gaussian with σ = 0.15 to simulate observational errors.

10.3 Low Mass Stars and an Evolving IMF? It seems clear that carbon is being released to the ISM on time scales comparable to that of iron in the disc. This suggests, provided that a significant portion of the iron is contributed by SNIa events, that the long-lived LIM-stars are significant carbon producers, unless the HM-stars contribution has a strong metallicity dependence. It is expected that HM-star yields have a metallicity dependence, but that may not be enough to explain how observed C/O seems to increase with metallicity. In Paper VI it is concluded that the major source of carbon in the present-day Universe may very well be the LIM-stars, although HM-stars as major carbon producers cannot be excluded. If the best-fit models are correct, the major source of carbon in the present-day Galaxy is the LIM-stars, providing as much as 80% of the carbon to the ISM.

82 Figure 10.3: Carbon abundance relative to oxygen. Note that Galactic and extra- galactic HII-regions, appear to follow a trend similar to that of the solar neighbourhood. The different lines have the same meaning as in Fig. 10.2, i.e., they correspond to Model A1-E1. The yellow-shaded (bright) area indicates where C stars dominate the carbon enrichment of the ISM according to Model C1.

An evolving IMF,being top-heavy (favouring the formation of HM-stars) during the early stages of Galactic evolution can explain the C/O vs. O/H trend seen in the observed abundances obtained by Fabbian et al. [54], without violating any other observational constraints. Whether an evolving IMF is more likely than a scenario with increased carbon yields at Z = 0 is matter of taste, but several recent studies suggest that in the early Uni- verse, star formation took place according to an IMF that was more top- heavy than at present time [161, 35, 164].

83

11. Concluding Remarks and Future Prospects

"I never think of the future. It comes soon enough."

Albert Einstein

11.1 Future Work Despite the words of Einstein, it seems appropriate to end this thesis with some future prospects. In many respects, the work of this thesis represents ﬁrst steps in several different areas that are connected to the winds and mass loss of carbon stars. A lot could be improved upon and several interesting ideas have emerged during the work on this thesis.

11.1.1 Subsolar Metallicity The grid of models in Paper I contains models of solar metallicity only. It would certainly be interesting to explore lower metallicities, even if the results in Paper II suggest there are no signiﬁcant metallicity effects for mod- erately low metallicities. Really metal-poor carbon stars, on the other hand, are much more compact compared to solar-metallicity carbon stars. Hence, they may have higher surface gravities, which in such a case cannot leave the conditions for dust and wind formation unaffected. A low metallicity grid of models was supposed to be part of this thesis, but for various reasons there was never enough time to realise the plans. But it should be done at some point.

11.1.2 Drift and Grain Sizes Given the results presented in Paper III, a full implementation of grain- size dependent opacities in combination with a model of drift seems to be important as future work. Sandin & Höfner [137, 138] explored the effects of decoupling the gas and dust phases (drift) in combination with non- equilibrium dust formation. Their results suggest that the effects of drift are not negligible, although their Planck-mean models should not be directly compared to the frequency-dependent models presented in this thesis. As argued in Paper II, there are reasons to believe that the effects of drift, to some extent, will cancel the effects of grain-size dependent opacities. The

85 inclusion of drift seems to increase the degree of dust condensation in the atmosphere, while grain-size dependent opacities work in the opposite direction. However, this is speculation. To really know, it has to be tested by numerical experiments.

11.1.3 Detached Shells The next version of the mass-loss module used in Paper V will provide information about the wind speed as well. This means that the work on detached shells in Paper IV can be done even more consistently, since the kinematic properties of the outﬂow and how they change will be known directly from the stellar evolution modelling. If this improved wind evolution is combined with a more sophisticated model for the circumstellar envelope, e.g., by replacing the polytropic equation of state with an energy equation and including radiative transfer, this may tell us more about the formation, stability and sustainability of detached shells around carbon stars. Moreover, it might be possible to tell if a carbon star can produce more than one shell.

11.1.4 Galactic Chemical Evolution If the AGB is terminated after only a few thermal pulses, as suggested by the results of Paper V, the amount of carbon and other nucleosynthesis products that can be expelled by a carbon star will be limited. Computing a set of stellar evolution models with the new mass loss in order to obtain new theoretical constraints on the carbon production may shed some light on the solution to the cabon-enrichment problem studied in Paper VI. This should definitely also be done at some point. The new code used in Paper VI is designed to be coupled with models of time-dependent radial gas flows within the disc of a galaxy. Assuming that stars always stay in circular orbits where they form, as in Paper VI, is not very realistic. An even more severe approximation is that of no radial motions of the gas, which must develop due to viscous forces and shearing. It would certainly be interesting to combine the chemical evolution code with a detailed hydrodynamic model of gas flows in a galactic disc.

11.2 Life from a Wider Perspective 11.2.1 A Galactic Habitable Zone? Irrespective of whether carbon is mainly produced by low- and intermediate-mass stars, or if high-mass stars contribute a large fraction of carbon, the outer parts of the Milky Way disc have low carbon abundances and most of the carbon was released quite recently (during the last Gyr, or so). Assume for a moment that low-mass and intermediate-mass stars dominate the carbon production in the Galaxy. Then it has to be taken

86 Figure 11.1: The GHZ in the Galactic disc based on the metallicity, sufﬁcient time for evolution, and absence of life-extinguishing supernova explosions. The white contours encompass 68% (inner) and 95% (outer) of the origins of stars with the highest potential to be harboring complex life today. Adopted from Lineweaver et al. [103]

into consideration that there has not been enough time for the numerous low-mass stars to evolve into mass-losing carbon stars in the outer Galactic disc, where the peak of star formation was reached quite recently (or has not yet been reached). If instead the contribution from high-mass stars is dominant and increases with metallicity, the time evolution of the carbon abundance is not much different. Are the outer parts of the Galaxy therefore disqualiﬁed as life-forming environments? Gonzalez et al. [61] described a simple model of what has become known as the Galactic Habitable Zone (GHZ), which is essentially the annulus of the Galactic disc which has the right conditions for harbouring stars with life-friendly planets. The inner parts of the Galaxy have a too high supernova rate to be "healthy" for most living organisms, while the outer parts are considered too metal poor to build a habitable terrestrial planet. Lineweaver et al. [103] have reﬁned this model and found that there may be only a small window in time and space where complex life is likely to form (see Fig. 11.1). They assume four criteria for complex life:

87 Figure 11.2: The evolution of the carbon mass fraction at different galactocentric distances according to Model A in Paper VI. The thin dashed lines marks the abundance and formation time of the Sun.

the presence of a suitable host star, a sufficient amount of heavy elements to form terrestrial planets, sufficient time for biological evolution and an environment essentially free of supernovae. Combining these criteria with a model of galactic chemical evolution of the kind described in Chapter 8 (and used in Paper VI) appears to favour a narrow region of the disc which includes the galactocentric position of the Sun. The work by Lineweaver et al. has not been met without critique. Prantzos [127] has pointed out the difficulty in quantifying the four criteria mentioned above. Especially how "sufficient amount of heavy elements to form terrestrial planets" should be interpreted in the GHZ picture is a problem. The delayed and quite moderate carbon enrichment of the outer Galaxy is possibly putting a stronger constraint on the formation of life. The results of Paper VI may therefore lend some support to the idea of a GHZ (see Fig. 11.2). In particular if the formation of important chemical building blocks can be associated with carbon stars.

11.2.2 Carbon Stars and Carbon-based Life Organic molecules in dense interstellar clouds may be traced back to frag- ments formed in cicumstellar shells of carbon stars. Carbon stars lose up to

88 − 80% of their original envelopes due to mass loss, with typical rates of 10 7 −4 −1 to 10 M yr . The low temperature of the central star, and the long time scales for mass loss, make it possible for molecules to form in the envelope. Carbon-rich envelopes appear to have the largest variety of chemical compounds. Observed spectra reveal signatures of acetylenic chains, such as HCn NandCnN, and several more unusual radicals. There is also evidence for the presence of signiﬁcant amounts Polycyclic Aromatic Hydrocarbons (PAHs) in the circumstellar material around carbon stars [110]. PAHs may have been vital in the formation of early life on Earth, and may be a crucial for the origin of carbon-based life in general. While PAHs may be abundant in interstellar space, this does not prove that terrestrial life (or even its basic building blocks) has extra-terrestrial origins. However, organic compounds are found in meteorites as well. The high D/H (deuterium-to-hydrogen) ratios in some of these compounds indicate that they originated in cold, interstellar clouds [104]. It has been suggested that much of the carbon, and perhaps many organic compounds on Earth, were brought here via meteorites, comets and interplanetary dust particles [3, 24]. In such a way the Earth may have been provided with material that led to the formation of the ﬁrst living organisms. Quite recently, what is considered to be extraterrestrial nucleobases, were found in the famous Murchison meteorite [108]. These new results suggest that organic compounds, which are components of the genetic code in modern biochemistry, were already present in the early solar system and may have played a key rôle in the origin of life. So, did the building blocks of life come from space? That idea is not new. In fact, a theory of this kind was proposed by Benoit de Maillet in 1743 who suggested that "germs" from space had fallen into the oceans, and over time turned into more advanced forms of life. This so-called "panspermia hypothesis" became popular again when it was advocated by Svante Arrhenius about a century ago [6], but still there exists no actual evidence for alien microbes roaming the solar system. Much more likely is a scenario where the molecular components needed to form primitive organisms are formed in space and brought to the Earth in meteorites. In such a case the carbon stars may be key components in the chain of events leading to occurrence of life on Earth.

The Änd.

89

Contributions to Included Papers

Paper I: Wrote the paper and carried out the computation of the grid as well as analysis of the results. Minor modiﬁcations to the dynamic atmosphere code.

Paper II: Wrote the paper and carried out the computations of the models as well as analysis of the results.

Paper III: Wrote the paper and carried out the computations of the models as well as analysis of the results. Minor modiﬁcations to the dynamic atmosphere code.

Paper IV: Wrote the paper. Computed the atmosphere models and prepared the variable inner boundary conditions for CSE model. Minor modiﬁcations to the dynamic atmosphere code.

Paper V: Wrote the paper and the MESA module for mass loss. Carried out the analysis of the results.

Paper VI: Single-author paper. Wrote the new galactic chemical evolution code from scratch as part of the thesis project.

91

Sammanfattning

Kol kan sägas vara "livets grundämne" här på jorden, och dess ursprung är därför en viktig pusselbit i vår förståelse av livets kosmiska ursprung. Det står klart att allt kol i Universum måste ha bildats i stjärnornas inre, men exakt i vilken sorts stjärnor kolet bildas är inte till fullo känt. Det ﬁnns skäl att tro att stjärnor av solens typ, står för en mycket betydande del av kolanrikningen i Vintergatan och andra galaxer. Merparten av dessa stjärnor blir s.k. kolstjärnor i slutfasen av sin utveckling, d.v.s. deras atmosfärer (ytskikt) får ovanligt höga halter av kol. Detta genom en process där kol från stjärnans inre delar förs upp till ytan i samband med s.k. termiska pulser. Kolet binds till stor del upp i molekyler, främst kolmonoxid, medan det resterande "kolöverskottet" under rätt förutsättningar kan bilda stoftpartiklar. Dessa stoftpartiklar absorberar eller sprider ljuset (fotonerna) som kommer från stjärnan, vilket gör att stoftkornen accelereras utåt av strålningen och drar med sig gasen från stjärnans yttre lager. Detta utﬂöde av gas, till följd av fotoners växelverkan med stoftet, kallas en stoftdriven stjärnvind. Kolstjärnor har mycket kraftiga vindar och kan förlora merparten av sin massa inom loppet av ett par hundratusen år, vilket är en kort tid på en kosmisk tidskala.

Stoftdrivna stjärnvindars egenskaper I denna avhandling studeras både kolstjärnevindarnas egenskaper och kolets ursprung ur ett teoretiskt perspektiv. Bildandet av kolstoft och stoftdrivna vindar har undersökts genom datorsimulering av fysiken bakom dessa fenomen (Artikel I - III). Syftet är att nå en bättre förståelse av hur stjärnvindens "styrka" beror av stjärnans egenskaper, t.ex. ljusstyrka, temperatur och kolhalt, och under vilka förutsättningar kolstoft kan bildas i kolstjärnors atmosfärer. Ett centralt resultat är att dessa kolstoftdrivna stjärnvindar inte uppstår så fort det finns ett kolöverskott att bilda stoft av. Faktum är att det finns ett flertal "trösklar" att komma över. Först och främst måste kolöverskottet vara tillräckligt stort1, men eftersom stoftbildningen påverkas av stjärnans temperatur finns det även en

1 Det ﬁnns observationer av vindar från kolstjärnor med synnerligen litet kolöverskott. Ex- akt hur dessa vindar uppstår är oklart och förmodligen en fråga som räcker för ännu en avhandling. Man bör dock notera att stoftdrivna vindar även förekommer hos de stjärnor där syre dominerar över kol, och inget kolstoft kan bildas. I dessa fall bildas istället s.k. si- likater (stoftkorn av huvudsakligen kisel och syre), som av allt att döma också kan ge upphov till ett gasutﬂöde via samma mekanism som i fallet med kolstjärnorna. I övergången mellan

93 "temperaturtröskel" att komma över. I alltför heta kolstjärneatmosfärer kan nästan inget stoft bildas. Följden blir naturligtvis att inte heller någon stjärnvind uppstår. På liknande sätt finns kritiska ljusstyrkor och stjärnmassor. För lite strålning gör att stoftkornen inte accelereras till de hastigheter som krävs för att övervinna gravitationskraften, som ju i sin tur beror av stjärnans massa. Förutsättningarna för stoftbildning är kanske den viktigaste komponen- ten för att förstå kolstjärnevindarna. Men det finns andra aspekter att ta hänsyn till. Exempelvis har stoftkornens storlek en ganska väsentlig betydelse för deras förmåga att absorbera och sprida fotoner. I Artikel I antas att alla stoftkorn kan betraktas som små, vilket innebär en betydande fören- kling av den fysikaliska modellen som gör den lätt att hantera rent beräkn- ingsmässigt. Men det finns skäl att tro att stoftkornen kanske inte är så små. De kan vara av ungefär mikrometerstorlek, vilket är tillräckligt för att ha betydande effekter rent fysikaliskt. I Artikel III undersöks därför hur mycket stoftkornens storlek skulle kunna påverka stjärnvindens egenskaper. Det visar sig att i de fall där stjärnvinden ligger nära någon av de "trösklar" som nämnts ovan, så har kornstorleken betydelse. Utflödeshastigheterna tycks öka och "trösklarna" förskjuts på ett sätt som gynnar bildandet av stoftdrivna vindar. Att ta hänsyn till kornstorlekens effekter i simuleringar av stoftdrivna stjärnvindar är ännu inte rutinmässigt. Det som gjorts i denna avhandling är bara början.

Kolstjärnors utveckling När en stjärna av solens typ når slutfasen av sin utveckling genomgår den en serie termiska pulser, vilket leder till att många stjärnor får ett kolöver- skott och kraftiga stjärnvindar drivna av kolstoft. Utflödet av gas är till slut så stort att så gott som hela stjärnans yttrehölje av gas försvinner, och kvar blir den "avklädda" centraldelen: en s.k. vit dvärgstjärna. I Artikel IV & V betraktas en kolstjärnas utveckling med tiden, också i detta fall ur ett teoretiskt perspektiv och med hjälp av en datorsimulering. Man har observerat s.k. "frikopplade skal" av gas runt flera kolstjärnor i Vin- tergatan. Dessa skal har föreslagits hänga samman med en kraftigare stjärn- vind kopplad till en s.k. termisk puls från stjärnans inre. Under en sådan puls ökar ljusstyrkan samtidigt som temperaturen sjunker, vilket gynnar bildandet av en stoftdriven stjärnvind. I Artikel IV visas att det inte är "vin- dutbrottet" som sådant som ger upphov till ett gasskal runt stjärnan, utan snarare att gasutflödets hastighet ändras just precis så att ett skal uppstår. Genom att kombinera en datormodell av en solliknande stjärnas utveckling med datormodellen för kolstjärnevindar och en ytterligare datormodell (dock ganska enkel) för gashöljet runt stjärnan som bildas, förklaras denna det "syrerika" och det "kolrika" fallet, är det för närvarande problem att förklara vindarnas uppkomst.

94 process. Gas med högre hastighet strömmar ut och kolliderar med gas som har lägre hastighet, vilken lämnade stjärnan innan den termiska pulsen. Re- sultatet blir att gasen trycks ihop till ett skal, inte helt olikt hur en snöhög bildas framför en snöplog när man plogar en väg på vintern. Den stoftdrivna vinden från en kolstjärna har stor betydelse för vad som händer i slutet av dess utveckling. Datormodeller av stjärnutveckling brukar innehålla väldigt enkla uppskattningar av hur stjärnvinden beror av t.ex. stjärnans ljusstyrka och temperatur. I Artikel V visas att när simuleringarna av stjärnvinden inkluderas i en datormodell av utvecklingen under kolstjärnefasen, blir effekterna dramatiska. Främst av allt bildas en s.k. supervind, d.v.s. ett mycket kraftigt utﬂöde av gas, kort efter att kolstjärnefasen inleds. Denna supervind bildas nu mycket snabbare än i tidigare simuleringar, vilket till stor del beror på den ökande kolhalten. Stjärnan förlorar därmed sitt yttre gashölje väldigt snabbt och hinner inte genomgå särskilt många termiska pulser som kolstjärna, innan den reducerats till en vit dvärg. Resultaten är ännu preliminära, men allt i dessa nya simuleringar tyder på att gashöljet förloras betydligt snabbare än man tidigare trott.

Kolets kosmiska ursprung För att slutligen söka svar på vad som är kolets kosmiska ursprung, d.v.s. vilka stjärnor som bidrar med huvuddelen av allt kol, har även den kosmiska materiecykeln simulerats (Artikel VI). Dagens halter av så gott som alla grundämnen tyngre än helium har byggts upp genom att tyngre grundämnen bildas via kärnreaktioner i stjärnornas inre. Stjärnorna i sig bildas av gas i Vintergatan (liksom i andra galaxer), och delar av gasen och de nybildade tyngre ämnena återförs (via t.ex. stjärnvindar) till det interstellära mediet när stjärnorna når slutfasen av sin utveckling. Eftersom t.ex. nya kolatomer hela tiden bildas i varje ny generation stjärnor, kommer kolhalten i den interstellära gasen och kommande generationer stjärnor hela tiden att öka om inte oanrikad gas tillförs utifrån. Detta är effekten av den kosmiska materiecykeln. Två grundläggande scenarior för kolets ursprung har testats. I det första står de vanligt förekommande och långlivade stjärnorna av solens typ för det mesta av kolanrikningen (via de stoftdrivna stjärnvindarna), och i det andra står de betydligt mer sällsynta, stora, massiva och kortlivade stjärnorna (som även de har en vind, dock inte stoftdriven) för merparten av kolproduktionen. Det sistnämnda scenariot förutsätter att dessa massiva stjärnor ger ifrån sig mer och mer nyproducerat kol ju längre gången den kosmiska materiecykeln är. Inget av dessa två scenarior tycks kunna uteslutas baserat på jämförelse av uppmätta halter av kol och en del andra ämnen (främst kväve, syre och järn) i Vintergatan med simulerade halter. Kolets ursprung är fortfarande ovisst, men viss preferens kan ges åt ett scenario

95 där långlivade stjärnor av soltyp dominerar. I ett sådant scenario beror kolanrikningens ökning med tiden huvudsakligen på stjärnors olika livslängder, och styrs inte anrikningen i någon högre grad av den totala halten tyngre grundämnen. Detta är en något enklare förklaringsmodell, som därför antagligen är att föredra.

Kolbaserat liv i Vintergatan Oavsett vilket scenario man föredrar, så är Vintergatans yttre delar (där merparten av stjärnorna är unga) tämligen kolfattiga och det kol som ändå finns där har tillkommit alldeles nyligen, mätt med en kosmisk tidskala. Att kolet i den interstellära gasen tillkommit nyligen gäller inte minst om det är långlivade stjärnor av soltyp som står för kolproduktionen. Innebär detta att sannolikheten för att kolbaserat liv ska uppstå är låg p.g.a. de låga kolhal- terna, och att något mer avancerat liv inte har hunnit bildas? Man kan nog hålla det för troligt att det är så, vilket ger stöd åt idén om att det väsentli- gen bara är Vintergatsskivans mellersta del (där vi befinner oss!) som har de rätta förutsättningarna för att liv ska kunna bildas och existera i en stabil miljö över en längre tidskala. De centrala delarna av Vintergatan har en för hög frekvens av supernovaexplosioner för att vara en hälsosam miljö2,så kvar blir endast ett ringformat område i mitten av Vintergatsskivan. Detta område i Vintergatan brukar på engelska kallas för "the Galactic Habitable Zone", alltså en galaktisk "livszon". Är det en slump eller nödvändighet att vi befinner oss i denna zon? Den frågan går ännu inte att besvara, men det kan noteras att många kolkedjemolekyler som tros vara viktiga för livets uppkomst återfinns i gashöljena runt kolstjärnor. Om kolstjärnornas vindar levererar både kolet och många av de viktiga molekylerna, så är Vinter- gatans yttre delar (där kolstjärnor bör vara sällsynta, se Artikel VI) verkligen inte den mest sannolika platsen för uppkomsten av liv.

2 En supernovaexplosion har lika stort energiutﬂöde som den sammanlagda utstrålningen från alla stjärnor i en mindre galax, men koncentrerat till en lokal punkt i galaxen. I vår del av Vintergatan är supernovor ganska sällsynta, vilket vi ska vara mycket tacksamma för.

96 Acknowledgements

The first be thanked here, are of course my supervisors Susanne Höfner, Nils Bergvall & Kjell Olofsson. They have all been very patient (answering my numerous questions) and supportive. A very special thanks to Susanne for guiding me through the confusing phase of being a new graduate student without a proper research project. Nils is thanked in particular for planting an idea in my mind, that eventually grew into a research project about the rôle of stars in a greater scheme as well as detailed stellar physics. I hope there will be time in the future to finish all those "side projects" that have developed over the years, but did not make it into this thesis. Falk Herwig is thanked for running the stellar evolution simulations and for interesting discussions on stellar evolution. Kjell Eriksson, Rurik Wahlin, Astrid Wachter, Sara Bladh, and other participants of the "AGB meetings" are thanked for interesting discussions about AGB stars, beer, schnitzels, "fur-balls" and the horrors of "klapprande träskor"... I also wish to thank Bengt Gustafsson for reading all my manuscripts, giv- ing valuable comments and diagnosing me with "intellectual diarrhoea"... always good to know what I am suffering from... I must thank my office neighbour and fellow graduate student, Samuel Regandell, for his "spontaneous lectures" about astrobiology. You have really made me interested in the subject, which has changed my perspectives and to some extent the direction of my thesis work. I could also list the names of all the other graduate students. But you know who you are, right? It has been very nice to have you guys around! All personnel at the department not mentioned above are hereby thanked for contributing to a very nice, social and friendly environment. I have come to understand that it is a rare situation that is not found at many places within the university. Thank you all! Finally, I also thank my wife, Jenny, for all her love and support and for keeping me in touch with the world outside my head. Perhaps you should pull me away from the computer and put me on horseback more often... and last, but not least, my mother, Birgit, for... well, being my mother. Moth- ers are like buttons: they hold things together!

Uppsala, 23 March, 2009

97

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