Dynamical Atmospheres and Winds of M-Type AGB Stars

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1170

Dynamical atmospheres and winds of M-type AGB stars

SARA BLADH

ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9015-7 UPPSALA urn:nbn:se:uu:diva-230645 2014 Dissertation presented at Uppsala University to be publicly examined in Polhemssalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 10 October 2014 at 10:00 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Patricia Whitelock (South African Astronomical Observatory).

Abstract Bladh, S. 2014. Dynamical atmospheres and winds of M-type AGB stars. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1170. 54 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9015-7.

Mass loss, in the form of slow stellar winds, is a decisive factor for the evolution of cool luminous giants, eventually turning them into white dwarfs. These dense outflows are also a key factor in the enrichment of the interstellar medium with newly produced elements from the interior of these stars. There are strong indications that these winds are accelerated by radiation pressure on dust grains, but the actual grain species responsible for driving the outflows in M- type Asymptotic Giant Branch stars are still a matter of debate. Observations of dust features in the circumstellar environment of these stars suggest that magnesium-iron silicates are possible wind-drivers. However, the optical properties of these silicate grains are strongly influenced by the Fe-content. Fe-bearing condensates heat up strongly when interacting with the radiation field and therefore cannot form close enough to the star to trigger outflows. Fe-free condensates, on the other hand, have a low absorption cross-section at near-IR wavelengths where AGB stars emit most of their flux. To solve this conundrum, it has been suggested that winds of M-type AGB stars may be driven by photon scattering on Fe-free silicate grains with sizes comparable to the wavelength of the flux maximum, rather than by true absorption. In this thesis we investigate dynamical models of M-type AGB stars, using Fe-free silicates as the wind-driving dust species. According to our findings these models produce both dynamic and photometric properties consistent with observations. Especially noteworthy are the large photometric variations in the visual band during a pulsation cycle, seen both in the observed and synthetic fluxes. A closer examination of the models reveals that these variations are caused by changes in the molecular layers, and not by changes in the dust. This is a strong indication that stellar winds of M-type AGB stars are driven by dust materials that are very transparent in the visual and near-infrared wavelength regions, otherwise these molecular effects would not be visible.

Keywords: Late-type stars, AGB stars, stellar winds, atmospheres, mass-loss, outflows, circumstellar matter, dust, hydroynamics, radiative transfer

Sara Bladh, Department of Physics and Astronomy, Box 516, Uppsala University, SE-751 20 Uppsala, Sweden.

ISSN 1651-6214 ISBN 978-91-554-9015-7 urn:nbn:se:uu:diva-230645 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-230645) List of papers

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

I Bladh S. & Höfner S., (2012), "Exploring wind-driving dust species in cool luminous giants. I. Basic criteria and dynamical models of M-type AGB stars", Astronomy & Astrophysics, 546, 76.

II Bladh S., Höfner S., Nowotny W., Aringer B. & Eriksson K., (2013), "Exploring wind-driving dust species in cool luminous giants. II. Constraints from photometry of M-type AGB stars", Astronomy & Astrophysics, 553, 20.

III Bladh S., Höfner S., Nowotny W., Aringer B. & Eriksson K., (2014), "Exploring wind-driving dust species in cool luminous giants. III. Wind models for M-type AGB stars: dynamic and photometric properties", to be submitted to Astronomy & Astrophysics.

IV Sacuto S., Ramstedt S., Höfner S., Olofsson H., Bladh, S., Eriksson K., Aringer B., Klotz D. & Maercker M., (2013), "The wind of the M-type AGB star RT Virginis probed by VLTI/ MIDI", Astronomy & Astrophysics, 551, 72.

Reprints were made with permission from the publishers.

Contents

1 Asymptotic Giant Branch stars ...... 7 1.1 Nucleosynthesis and atmospheric chemistry ...... 7 1.2 Dust species in the circumstellar environment ...... 9 1.3 Pulsation-enhanced dust-driven winds ...... 9 1.4 Wind-driving dust species ...... 11 1.5 Observational constraints ...... 12

2 Dust-driven winds: a qualitative picture ...... 14 2.1 The dust opacity ...... 15 2.2 The dust-formation zone ...... 17 2.2.1 Levitation distances ...... 17 2.2.2 Condensation distances ...... 18 2.3 Criteria for wind-drivers ...... 20

3 Dust-driven winds: radiation-hydrodynamical models ...... 22 3.1 Gas dynamics and radiation ﬁeld ...... 22 3.2 Detailed dust description ...... 24 3.3 A set of wind models for M-type AGB stars ...... 26 3.3.1 Dynamical properties ...... 26 3.3.2 Synthetic spectra and photometry ...... 27 3.3.3 Photometric properties in the visual and near-IR ...... 29 3.3.4 Spectral features in the mid-IR ...... 32 3.3.5 Interferometry of RT Vir ...... 33

4 Exploring the effects of model assumptions ...... 36 4.1 Parameterised dust description ...... 36 4.2 Wind models with a parameterised dust opacity ...... 38 4.2.1 Dynamical properties ...... 39 4.2.2 Photometric properties in the visual and near-IR ...... 39 4.3 Grain temperature and the onset of outﬂow ...... 40

5 Summary and Future plans ...... 44

6 Contributions to the included papers ...... 46

7 Swedish summary - Vindar från svala syrerika jättestjärnor ...... 47 7.1 Svala jättestjärnor ...... 47 7.2 Stoftdrivna vindar ...... 47 7.3 Kriterier för vind-drivande stoft ...... 48 7.4 Vind-drivande stoft ...... 48 7.5 Vind modeller för syrerika jättestjärnor ...... 49

8 Acknowledgements ...... 50

References ...... 52 1. Asymptotic Giant Branch stars

Asymptotic Giant Branch (AGB) stars are evolved stars of low and intermediate mass that have exhausted the hydrogen and helium fuel in their centres. The comparatively low mass of these stars prevents their cores from contract- ing and reaching temperatures that will ignite further fusion processes. In- stead AGB stars produce energy by alternately burning hydrogen and helium in shells surrounding the core and are, as a result of this, very luminous and extended, with typical luminosities of the order 5000 10000 L , effective tem- − " peratures between 2500 3500 K and sizes that reach 200 500 R . However, − − " it is generally not the fusion processes that will end the life of these stars; they lose mass through slow stellar winds at a higher rate than the fuel is exhausted in the shell-burning, eventually turning them into white dwarfs. These dense outﬂows, with typical velocities of 5 30 km/s and mass-loss rates of about 7 5 − 10− 10− M /yr, are also responsible for enriching the interstellar medium − " with newly produced elements from the interior of these stars. According to the most widely accepted scenario these winds are driven by radiation pressure on dust.

1.1 Nucleosynthesis and atmospheric chemistry The compact cores of AGB stars consist of carbon and oxygen, and the energy production is mainly sustained by hydrogen burning in a thin shell outside the dormant core. The products from the hydrogen burning create a helium rich layer under the hydrogen shell. This layer eventually reaches such critical temperatures that it ignites, causing the AGB star to experience intense helium shell burning. These helium shell ﬂashes, or thermal pulses, occur at time in- tervals between 10,000-100,000 years and last up to a few thousand years each time. During the thermal pulses heavier elements, such as carbon and oxygen, are dredged-up from the interior of the star by the large convection cells in the outer layers. These convection cells extend inwards and mix newly processed material (elements produced by the s-process and 3α-process) from the top of the helium burning shell into the surface layers, thereby enriching the atmosphere with carbon. Through this process the C/O-ratio can go from smaller than one to larger than one, leading to so-called carbon stars. But not all AGB stars evolve into carbon stars; the efﬁciency of the carbon dredge up is mass dependent. Massive AGB stars lack a radiative layer between the convective envelope and the hydrogen-burning shell and experience hot-bottom burning

7 Figure 1.1. Schematic picture of the atmospheric chemistry in AGB stars: consecutive dredge-ups lead to increased C abundance in the atmosphere of most AGB stars as they evolve. The least abundant element out of C and O will be mostly bound in the very stable CO-molecule, leaving the excess element to form other molecules and dust grains. Adopted from Höfner (2009).

at the bottom of the convection zone, a process that transforms the dredged up carbon to nitrogen. More massive (M > 4M ) AGB stars will therefore not evolve into carbon stars. For more details∗ on nucleosynthetic" processes in the interior of AGB stars and dredge-ups, see e.g. Lattanzio & Wood (2003a). The evolution from an oxygen-dominated chemistry (M-type AGB star) to a carbon-dominated chemistry (C-type AGB star) will have a signiﬁcant effect on what molecules and dust species form in the extended atmosphere surrounding the AGB star (Russell, 1934). Among the most abundant elements, helium is an inert atom, hydrogen and nitrogen form mainly H2 and N2, and most of the carbon and oxygen will be tied up in the very stable CO-molecule. The element that is more abundant out of carbon and oxygen will be available for forming other molecules and dust species (see Fig. 1.1). If carbon is more abundant than oxygen, carbonaceous molecules (e.g. C2, CN, HCN and C2H2) and dust species like amorphous carbon or silicon carbide will form. If instead oxygen is more abundant, we expect oxygen-bearing molecules (e.g. H2O, SiO and TiO) and dust species like silicates and oxides to form (see, e.g., Gail, 2010).

8 1.2 Dust species in the circumstellar environment The rich dust mineralogy observed in the extended atmospheres of evolved stars is strongly affected by the physical and chemical conditions of the circumstellar environment (e.g. Molster et al., 2010), in particular the C/O ratio, and can be used as a tracer for local conditions. Dust species usually show strong features in the ultraviolet, due to electronic resonances, as well as in the mid to far-infrared range, in this case due to vibrational lattice resonances. Between these ranges of strong resonance, dust materials are quite featureless and optical data is often missing (Zeidler et al., 2011). Both amorphous and crystalline silicates have been observed in oxygen- dominated atmospheres (M-type) and about 10% of the silicates are estimated to be in crystalline form. The features of crystalline silicates are, unlike the features for amorphous silicates, very sensitive to chemical composition, and evidence (see Molster et al., 2002a,b) points to the presence of Fe-free silicates such as MgSiO3 and Mg2SiO4 in the outflows of evolved stars. Amorphous and crystalline silicates have also been observed in some carbon stars, likely originating from previous mass loss episodes when the atmospheric chemistry was more dominated by oxygen. Other dust species identified in the circumstellar environment of oxygen-rich cool stars are corundum (Al2O3)(Stencel et al., 1990) and spinel (MgAl2O4)(Posch et al., 1999). More uncertain identi- fications include quartz (SiO2) and simple oxides ([Mg,Fe]O). In carbon stars the identified dust species in the circumstellar environment include amorphous carbon (amC), silicon carbide (SiC) and magnesium sulphide (MgS). There is no convincing evidence of graphite in evolved stars, even if this dust material is often suggested to explain the 2175 Å feature originating from the interstellar medium. For an overview on dust species in the circumstellar environment of AGB stars, see Molster et al. (2010) and Dorschner (2010).

1.3 Pulsation-enhanced dust-driven winds Since most AGB stars are variable stars, it has long been suspected that the effects of shock waves, created by stellar pulsations, play an important role in the mass-loss mechanism. Also the infrared spectra observed from these objects indicate that the circumstellar envelopes of AGB stars are major production sites for dust, as discussed above. Early time-dependent dynamical models investigated the possibility of winds driven by shock waves alone or a combination of shock waves and radiation pressure on dust (Wood, 1979; Bowen, 1988). Models based purely on shock waves were found to produce too high mass-loss rates, whereas a combination of pulsations and radiation pressure on dust resulted in reasonable wind characteristics. The mass-loss scenario where stellar winds are driven by a combination of shock waves and radiation pressure on dust, i.e. pulsation-enhanced dust- driven winds, is built on a two-stage process. In the ﬁrst stage, stellar pul-

9 Figure 1.2. Microphysics of dust-driven winds: Dust grains acquire momentum from stellar photons and transfer it to the surrounding gas via collisions. Adopted from Höfner (2011). sations trigger shock waves that propagate through the steep density gradient of the atmosphere. The shock waves transfer kinetic energy, originating in the pulsations, into the atmosphere, making the gas move in almost ballistic trajectories. The result is an extended atmosphere with layers of enhanced density at higher altitudes, and consequently, cooler temperatures: an environment favourable for dust formation. In the second stage of this wind scenario momentum is transferred to newly condensed dust particles by absorption and scattering of the numerous stellar photons reaching the dust formation zone. Collisions between the accelerated dust particles and the surrounding gas then trigger a general outﬂow (see Fig. 1.2). For this scheme to work the two stages of the mass loss process have to connect, i.e. dust species with the right optical properties1 have to form in the density-enhanced layers close to the stellar surface so that sufﬁcient momentum can be transferred to the dust particles and, consequently, to the gas (e.g. see recent reviews by Höfner, 2009, 2011). Dynamical models of pulsation-enhanced dust-driven winds usually simulate the effects of stellar pulsations by prescribing temporal variations of physical quantities just below the photosphere. The time-dependent structure of the atmosphere and wind is then calculated by solving the equations

1Although the characteristic spectral features in the mid-IR provide information about dust species present in the circumstellar environment, it is important to remember that the existence of features does not automatically imply that this material will have an impact on the dynamical structure of the atmosphere. Instead, the dynamics of the atmospheres is mostly inﬂuenced by the optical properties of dust in the near-IR wavelength region since most of the stellar ﬂux of AGB stars is emitted in this region.

10 of hydrodynamics, taking into account the variability of the atmosphere during a pulsation cycle and the formation of shock waves. In addition to the dynamical description, a theory for time-dependent grain growth in cool stellar atmospheres and winds, with equations describing the nucleation, growth and evaporation of the dust material, has been developed (e.g. Gail & Sedl- mayr, 1988; Gauger et al., 1990; Gail & Sedlmayr, 1999). This detailed dust description was first applied to stationary outflows. However, a proper treatment of the wind mechanism, including the effect of pulsations, requires that we consider the time-dependence of both gas dynamics and dust formation, as there is a significant feedback between the two processes. The density enhanced layers created by the pulsation-induced shock waves play a significant role for the efficiency of grain growth and, in turn, the radiation pressure produced by newly formed dust particles will strongly influence the dynamics, and, consequently, the atmospheric structure. Wind models combining time-dependent dynamics with a detailed description for grain growth were first produced for C-type AGB stars (e.g. Fleis- cher et al., 1992; Höfner & Dorfi, 1997). The first time-dependent models with detailed dust description for M-type AGB stars were developed by Jeong et al. (2003), although these models only produce outflows for extreme stellar parameters and using unrealistic dust opacities. In recent years dynamical models have been increasingly based on a wavelength-dependent treatment of the radiative transfer, both for the gas and the dust components (e.g. Höfner et al., 2003; Woitke, 2006). A wavelength-dependent treatment of the radiative transfer, although computationally demanding, is essential for achieving realistic atmospheric temperature and density structures and producing synthetic spectra consistent with observations.

1.4 Wind-driving dust species In the previous section we have described the wind scenario without discussing the relevant dust species responsible for triggering the outflow in dust-driven winds. In the case of C-type AGB stars, models that include a time-dependent description of the growth of amorphous carbon have proven capable of producing outflows comparable with observations, both when it comes to dynamical properties such a mass loss and wind velocities (e.g. Winters et al., 2000; Mattsson et al., 2010), low resolution spectra and photometry (e.g. Gautschy- Loidl et al., 2004; Eriksson et al., 2014), as well as high resolution spectra that probe different regions of the atmosphere (e.g Nowotny et al., 2010). Grains of amorphous carbon have large absorption cross-sections in the near-IR, where AGB stars emit most of their stellar flux. In addition, grains of this material have microphysical properties which allow them to form close to the stellar surface and, provided that carbon is sufficiently more abundant than oxygen, there is plenty of material available for dust formation.

11 For M-type AGB stars (and the intermediate S-type) the picture is not as clear-cut. The dust chemistry is more complex and we expect oxygen-bearing dust species, such oxides and silicates, to form (e.g. Gail & Sedlmayr, 1999). The main purpose of this thesis is to explore the dynamical impact of different dust species and consequences for observables in these stars (Paper I-IV). The dynamical effect of certain materials, like Al2O3 and TiO2, will be limited by the low abundance of the constituent elements, resulting in low radiation pressure. Other dust species, such as silicates, consist of abundant elements but are limited as wind-driver by their optical properties. Fe-bearing silicates have large absorption cross-sections in the visual and near-IR such that they heat up when interacting with the radiation field and cannot exist at distances where the outflows are triggered. Fe-free silicates, on the other hand, can form close to the surface but they are very transparent in the near-IR and will for that reason not generate enough momentum to trigger outflows by absorption of stellar photons. A viable suggestion are outflows driven by photon scattering on Fe-free silicates (Höfner, 2008b); when these grains grow to sizes around 0.1-1.0 µm, the scattering cross-section dominates over the absorption cross-section by several orders of magnitude, providing enough momentum to produce stellar winds. Models including Fe-free silicates, taking into account the grain size when calculating the radiative cross-sections, are capable of driving winds with dynamical properties that are consistent with observations (Höfner, 2008b, and Paper III). In this thesis it is also demonstrated that these models produce synthetic observables in good agreement with observations (Papers II-III). Also, very encouraging, recent observations of a selection of M-type AGB stars suggest the existence of dust particles with radii of 0.3µm in the close circumstellar environments (Norris et al., 2012). ∼

1.5 Observational constraints There are many observational techniques available that can help to reveal the dynamical structure, wind properties and chemical composition of AGB stars, including both high- and low-resolution spectra, broad-band photometry and interferometry, covering the spectral range from ultraviolet to radio wavelengths. High-resolution spectra of molecular lines, e.g. CO-lines in infrared and radio, originating from different regions of the atmosphere, can be used to explore the overall atmospheric structures and velocity ﬁelds (see, e.g., reviews by Olofsson, 2005, 2008a; Marengo, 2009). The presence of individual dust species in the circumstellar envelope can be identiﬁed from low-resolution spectra, observed with space-based telescopes operating at mid to far-infrared wavelengths (e.g. Molster et al., 2010; Dorschner, 2010). Pho- tometric observations can be done with reasonable effort for a wide range of stars, covering one or more pulsation cycles, providing important statistical information concerning pulsation modes and phase-dependent light curves for

12 many targets (e.g. Feast et al., 1989; Whitelock et al., 2000, 2006; Lattanzio & Wood, 2003b). In contrast to most other types of observations, photometry provides a practical way of monitoring long-term temporal variations. The increased angular resolution of the new interferometers, such as ESO’s VLTI and ALMA, makes it possible to probe the inner regions of these extended objects. Interferometric observations can help us gain insight into dynamical processes of the atmosphere, probe gas and dust in the circumstellar envelope and to understand the geometry of these objects (e.g. Marengo, 2009; Olofs- son, 2008b; Wittkowski et al., 2011). In this thesis dynamical models for winds of M-type AGB stars are compared to mass-loss rates and wind velocities derived from observations of CO- lines (Papers II-III), visual and near-IR photometry (Papers II-III), mid-IR spectra (Paper III) and interferometric data (Paper IV) in order to constrain the wind mechanism.

13 2. Dust-driven winds: a qualitative picture

A basic understanding of the physics of dust-driven winds can be obtained by a simple construct that captures the essential parts of how the pulsation-induced shock waves and the radiative acceleration of the dust particles connect. The general reasoning in this chapter is based on Höfner (2008a), and the specific examples were developed in Paper I. Consider the dynamics of a fluid element, starting right after it has been accelerated by a shock wave: the main forces acting on the fluid are the gravitational force and the radiative acceleration on newly formed dust particles within the fluid element. The force due to the thermal pressure gradient in the atmosphere between shock waves is small by comparison and we neglect it in this simple construct. It follows that the acceleration of the fluid element in the radial direction is mainly controlled by the relative values of the gravitational and radiative acceleration, which leads to the following equation of motion,

du R 2 = agrav + arad = agrav (1 Γ)= g ∗ (1 Γ), (2.1) dt − − − − ∗ r − ! " assuming a direct coupling between the gas and the dust components. Here u = dr/dt is the radial velocity of the ﬂuid element at distance r from the stellar center, g = GM /R2 denotes the gravitational acceleration at the stellar radius R and Γ∗is a dimensionless∗ ∗ quantity, dependent on r, that measures the ratio between∗ radiative and gravitational acceleration:

arad κ L Γ(r)= = % & ∗ . (2.2) agrav 4πcGM ∗ Included in the expression for Γ are the gravitational constant G, the speed of light c, the stellar mass and luminosity, M and L , respectively, and the total flux-averaged dust opacity κ (including both∗ absorption∗ and scattering), % & ∞ 0 κacc(λ)Fλ dλ κ = ∞ , (2.3) % & # 0 Fλ dλ # where Fλ is the monochromatic flux at wavelength λ and κacc is the monochromatic grain opacity (see Sect. 2.1). We define a critical value of the flux- averaged opacity when the magnitude of gravitational and radiative acceleration are equal, i.e. when Γ = 1: 4πcGM κ = ∗ (2.4) crit L ∗ 14 7

!!"!#$ 5 !!"!!!# !!"!!!$

3 Radial distance [R_star] R_c = 2.5 R_star

1 0.0 0.5 1.0 1.5 2.0 2.5 Time [yr] Figure 2.1. A simple analytical model for the dynamics of the atmosphere, showing the location of a fluid element as a function of time, based on Eq. (2.1), starting after it has been accelerated by a shock wave. The acceleration of the fluid element is determined by a distance-dependent parameter Γ (ratio of radiative acceleration to gravitational acceleration) that is zero until the fluid element reaches distances where dust can condense (in this example at 2.5R ) and set to a constant value beyond that point. Adopted from Höfner (2009). "

Typical trajectories for different values of Γ are shown in Fig. 2.1. Clearly the ﬂuid element will accelerate outwards if Γ > 1, and a closer examination of this quantity can help us pinpoint the grain properties necessary for driving a wind. The expression for Γ can be factored into two parts; one part that solely depends on stellar parameters and fundamental physical constants, L /4πcGM , and one part, the ﬂux-averaged dust opacity κ , that is grain ∗ ∗ % & material dependent.

2.1 The dust opacity Assuming for simplicity that all grains in the ﬂuid element have equal radii agr, their collective opacity per mass of stellar matter can be expressed as

π 2 π Qacc(λ,agr) 3 κacc(λ,agr)= agrQacc(λ,agr)ngr = agrngr, (2.5) ρ ρ agr where ngr is the number density of grains in the fluid element, the efficiency Qacc is defined as the ratio between the radiative and the geometrical cross- section of an individual grain, which can be computed from optical data using

15 Mie theory (e.g. Bohren & Huffman, 1983), and ρ is the total mass density of 3 the ﬂuid element. The factor agrngr is related to the fraction of atmospheric volume occupied by dust particles. This quantity can be expressed in terms of the volume of a monomer, the basic building block of the grain material, according to

3 3 3 Amonmp agrngr = Vmonnmon = fcεlimnH. (2.6) 4π 4π ρgr

Here we have expressed the volume of the monomer Vmon in terms of the atomic weight of the monomer Amon, the proton mass mp and the density of the grain material ρgr. The number of monomers condensed into grains per volume of atmosphere nmon can be expressed by the abundance of the limiting element εlim, the fraction of the limiting element condensed into dust particles fc and the total number density of H atoms nH. Using nH = ρ/(mp(1+4εHe)), where εHe is the helium abundance, we obtain the following expression for the dust opacity

3 Amon Qacc(λ,agr) εlim κacc(λ,agr)= fc. (2.7) 4 ρgr agr s(1 + 4εHe) In this context we define the limiting element as the first element that will be completely consumed due to its relative abundance in the atmosphere, adjusted for the number of atoms s contributing to the monomer (stoichiometric coefficient). For example, in the case of Mg2SiO4, the least abundant element would be Si (for a solar composition) but since 2 Mg atoms are used for building one 1 monomer (s = 2), Mg becomes the limiting element. The factor Amon/ρgr is a material-dependent constant and the optical properties of the dust particles are captured in the efficiency Qacc, which includes contributions from both absorption and scattering, Q = Q +(1 g )Q . (2.8) acc abs − sca sca where gsca is the asymmetry factor describing deviations from isotropic scattering (see, e.g., Krügel, 2003). In the small particle limit, where the grain radius is much smaller than the relevant wavelengths, the absorption and scat- 4 tering efficiencies behave like Qabs ∝ agr and Qsca ∝ agr, according to Mie theory (e.g. Bohren & Huffman, 1983). In this limit absorption dominates over scattering, implying that Q Q and consequently that the efficiency per acc ≈ abs grain radius, Q /a Q /a , becomes independent of grain size. We acc gr ≈ abs gr can therefore express the grain opacity during the early stages of grain formation, or during the whole process if particles remain small, as an essentially

1Note that this is a simple way of estimating the maximum amount of condensable material for a speciﬁc dust species, and consequently, an upper limit for the opacity. This, however, does not necessarily mean that the addition of this element to the grain corresponds to the slowest rate (bottle-neck) in building up a monomer.

16 wavelength-dependent function,

3Amon εlim κacc(λ)= Qacc( (λ) fc (2πagr λ) (2.9) 4ρgr · · s(1 + 4εHe) · )

where Qacc( = Qabs/agr. Looking at each of the factors in this expression in- dividually reveals what conditions need to be satisfied for a dust species to be a potential wind-driver. First, the efficiency per grain radius Qacc( has to be sufficiently large in the wavelength region around the stellar flux maximum, given that κ is calculated by taking the flux-mean of the grain opacity κ . % & acc Furthermore, the abundance εlim is a limiting factor in the dust formation process. The last factor in the expression for the grain opacity, the degree of condensation fc, is a measure of how much of the limiting element has actually condensed into solid material. The degree of condensation will remain zero until the grains start to form and grow, which may only happen at distances where the grains can be thermally stable.

2.2 The dust-formation zone In order to estimate if a specific grain material can start to condense in the density-enhanced outer layers of the atmosphere, we introduce the concepts of levitation distance and condensation distance, and some easy-to-use ap- proximations. In the following, the levitation distance R! is defined as the distance to which pulsation-induced shock waves levitate the gas in the atmosphere, without radiative acceleration on dust. The condensation distance Rc is defined as the closest distance to the star were grains of a specific type can exist, i.e. are thermally stable. For a grain material to be considered a possible wind-driver, the two stages of the mass-loss scheme have to connect. The second stage, i.e. the radiative acceleration, can only be initiated if levitation by shock waves lifts gas beyond the condensation distance.

2.2.1 Levitation distances A simple argument using the complete conversion of pulsation-induced kinetic energy into potential energy, ignoring heat loss and pressure effects, can give us an estimate of how high shock waves can lift gas. If we assume that the gas has an initial velocity u0 at distance R0 from the centre of the star we can

17 derive an expression for the levitation distance in the following way 2 mu0 mM G mM G ∗ = 0 ∗ (2.10) 2 − R0 − R! 2 R R u0 ∗ ∗ = 2 (2.11) R! R0 − uesc 1 R R R u 2 − ! = 0 1 0 0 , (2.12) R R − R uesc ∗ ∗ $ ∗ ! " % 1/2 where the escape velocity at the stellar surface is given by uesc =(2M G/R ) . For stellar parameters typical of an AGB star, i.e. M = 1M , L =∗ 5000∗ L ∗ " ∗ " and Teff = 2800K, the escape velocity at the stellar surface is about 36 km/s. Radial velocities of gas right after the passage of a shock wave, derived from observations of the second overtone CO line (∆v = 3), are of the order of 10-15 km/s. According to dynamical models, these lines are formed in the region from the stellar surface out to about 1.5 stellar radii (Nowotny et al., 2010). Assuming an initial velocity of u0 = 15 km/s and a distance R0 = 1.5R , Eq. (2.12) gives a levitation distance of about 2R . This corresponds approx-∗ imately to where interferometric measurements place∗ the inner edges of dust shells around AGB stars (e.g. Wittkowski et al., 2007; Karovicova et al., 2011; Norris et al., 2012).

2.2.2 Condensation distances

In a strong radiation ﬁeld we assume that the grain temperature Td is determined by the condition of radiative equilibrium, κ J κ S(T )=0. (2.13) abs,J − abs,S d Here J is the mean intensity and S is the source function, both integrated over wavelength, and κabs is the dust absorption coefﬁcient, where the subscripts J and S denote means weighted with Jλ or Sλ (see Sect. 3.1). Assuming an opti- cally thin atmosphere, where the incident intensity on the grains is direct star light, the mean intensity Jλ can be approximated by a geometrically diluted Planck function, Jλ = W(r)Bλ (T ). (2.14) ∗ In this expression T is the effective temperature of the star and W(r) is the geometric dilution factor,∗ 1 W(r)= 1 1 (R /r)2 , (2.15) 2 − − ∗ & ' ( which reduces to W(r)=(R /2r)2 for r R . If we assume that the source ∗ * ∗ 4 function S(Td) is given by B(Td)= Bλ (Td)dλ =(σ/π)Td , then the condition # 18 of radiative equilibrium can be reformulated as

4 4 κabs,B(T )W(r)T = κabs,B(Td)Td , (2.16) ∗ ∗

Now, assuming r R and solving for the grain temperature Td gives the * ∗ following expression

1/2 1/4 R κabs,B(T ) Td(r) T ∗ ∗ . (2.17) ≈ ∗ 2r κ (T ) ! " ! abs,B d " If the absorption coefﬁcient can be approximated with a power law function, κ λ p, in the relevant wavelength region, the factor with the Planck-mean abs ∼ − opacities can be simpliﬁed accordingly,

p κabs,B(T ) T ∗ = ∗ , (2.18) κ (T ) T p ! abs,B d " d resulting in the following dependence of the grain temperature on distance

2 R − 4+p Td(r) T ∗ . (2.19) ≈ ∗ 2r ! "

Introducing the condensation temperature Tc and setting Td = Tc we can solve for the condensation distance Rc = r(Tc),

4+p R 1 T − 2 c = c . (2.20) R 2 T ∗ ! ∗ " For more details see, e.g., Lamers & Cassinelli (1999). In Fig. 2.2 we have plotted curves of constant condensation distances as a function of p and Tc, using this formula. Due to the wavelength-dependence of the absorption coefficient (i.e. the value of p) different condensates will react differently to the stellar radiation field. A grain material with a positive value of p tends to heat up when interacting with the stellar radiation field, being more efficient at absorbing than emitting radiation, and thereby shifting the condensation distance further out. The opposite is true for a grain material with a negative value of p. To summarise, it is the combination of condensation temperature and the slope of the absorption coefficient with wavelength that determines how close to a star a grain type can survive. For example, Fe- bearing silicates have a condensation temperature of about 1100K and p 2 ≈ in the near-IR, which according to this simple formula results in a condensation distance of around 10R for T = 2800 K, well beyond the reach of pulsation-induced shock waves.∗ The corresponding∗ value for Fe-free silicates are Tc 1100K and p 1, resulting in a condensation distance of 2R for ≈ ≈− ∗ T = 2800 K, which fits with the estimated levitation distance in Sect. 2.2.1. ∗ 19 Figure 2.2. Curves of constant condensation distance (Rc/R = 2,4,10) as a function ∗ of the power law coefficient p and the condensation temperature Tc, using Eq. (2.20), with T = 2800K (black) and T = 2500K (grey). The filled red circles mark where ∗ ∗ selected dust species are situated in the p/Tc-plane. The dust species in parenthesis have uncertain or interpolated optical data in the near-IR. Figure from Paper I.

2.3 Criteria for wind-drivers The dust chemistry in the circumstellar environment of AGB stars is very complex and there are many dust species that could potentially be relevant for the dynamics of the atmospheres and winds of these stars. However, just because a dust species is observed in the circumstellar envelope, often identified through mid-IR features, does not mean it will have a substantial impact on the dynamical structure. The dynamics of the atmospheres is mostly influenced by the optical properties of the grains in the near-IR wavelength region, given that most of the stellar flux of AGB stars is emitted in this wavelength range and that the momentum per photon is greater there than at longer wavelengths. The criteria that indicate if a grain material will be important for the dynamical structure are the following:

(a) The distance from the star where the grains are thermally stable, determined by the condensation temperature Tc and the near-IR slope of the absorption coefﬁcient p. (b) The absorption and/or scattering efﬁciency of the grains in the wavelength region where most of the stellar radiation is emitted. (c) The abundance of the limiting element of the grain material, putting an upper limit on the total opacity.

Estimates for speciﬁc grain materials can be obtained in the following way: if

20 Table 2.1. Properties of a few selected dust species in small particle limit for full condensation of the limiting element, assuming stellar parameters M = 1M , 2 " L = 5000L ,T = 2800K, resulting in κcrit = 2.6 cm /g. For amorphous carbon we " ∗ set C/O=1.25. The condensation distance Rc is estimated using Eq. (2.20). The dust species in parenthesis have uncertain or interpolated optical data in the near-IR. For references see Paper I. Material lim. ε /sRκ max Γ lim c % & element [R ] [cm2/g] ∗ Fe Fe 3.24 10 5 11.5 1.5 0.6 · − (Al O ) Al 1.48 10 6 3.6 4 10 2 2 10 2 2 3 · − · − · − TiO Ti 9.77 10 8 3.4 6 10 5 2 10 5 2 · − · − · − (SiO ) Si 3.55 10 5 3.9 6 10 2 2 10 2 2 · − · − · − MgSiO Si 3.55 10 5 2.6 5 10 2 2 10 2 3 · − · − · − Mg SiO Mg 1.90 10 5 2.1 3 10 2 1 10 2 2 4 · − · − · − MgFeSiO Fe 3.24 10 5 9.5 2.9 1.1 4 · − amC C 1.85 10 4 1.8 6.9 2.6 · − the particles can form sufficiently close to the stellar surface, as can be checked by Eq. (2.20) and Eq. (2.12) (condensation distance and levitation distance, respectively), the combined effect of the critical abundance εlim and the efficiency Qacc can be investigated by setting fc = 1 in the expression for the dust opacity κacc (Eq. (2.9)) and assuming a suitable stellar flux distribution. If the resulting flux-averaged dust opacity κ is larger than the critical opacity κ % & crit the grain material is a viable candidate for triggering outflows. Tab. 2.1 shows the flux-averaged dust opacity, as well as the ratio between radiative and gravitational acceleration and the condensation distance, for a few selected dust species, estimated with this approach. As noted in Sect. 2.2.2, Fe-bearing silicates will not condense close enough to the stellar surface to trigger outflows. Dust species like Al2O3 and TiO2 are limited by the abundance of the included elements, and are therefore less likely to produce sufficient momentum transfer to trigger stellar winds. Mg-rich silicates are both abundant and can form close to the stellar surface but the individual radiative cross-section of these grains are too small in the small particle limit to drive stellar winds. However, if Mg-rich silicates grow to sizes comparable to the wavelength of the flux maximum, the contribution to the radiative acceleration from photon scattering can be sufficient to overcome the gravitational pull of the star, as discussed in Höfner (2008b) and Sect. 3.3.

21 3. Dust-driven winds: radiation-hydrodynamical models

The analytical estimates discussed above present a very simplified qualitative picture of the physics in the atmospheres of AGB stars; the dynamical effects of the gas pressure are ignored, the radiation field is described by a geometrically diluted Planck function and Γ is considered as a simple function of the flux-averaged grain opacity. In reality, molecular opacities strongly affect the radiation field, pulsation-induced shock waves influence the structure of the atmosphere and Γ is a time-varying function depending on grain growth. To study the complex interplay between gas dynamics, dust formation and the radiation field we use wavelength-dependent radiation-hydrodynamical models, including a detailed description of the growth of Mg2SiO4 grains, developed in Höfner et al. (2003) and Höfner (2008b). In the following sections the different ingredients of the dynamical models are described.

3.1 Gas dynamics and radiation field The models cover a spherical shell with an inner boundary situated just below the photosphere and an outer boundary in accordance with the dynamical properties of the model. In models that develop winds the outer boundary is fixed at the point where the flow velocity has reached its terminal value, allowing outflow, and in models without winds the outer boundary follows the periodic motions of the upper atmospheric layers. The variable structure of the atmosphere is described by the equations of hydrodynamics (equation of continuity, equation of motion and energy equation) and the pulsations are simulated by sinusoidal variations of velocity and luminosity at the inner boundary. The opacities of molecules and dust forming in the outer cool layers of the atmosphere strongly affect the radiation field and in order to achieve realistic density and temperature structures the models include a wavelength-dependent treatment of radiative transfer (see Höfner et al., 2003, for more details). The conservation of mass, momentum and energy is described by the following equations for the gas component: ∂ (ρ)+∇ (ρu)=0 (3.1) ∂t · ∂ Gm 4πρ (ρu)+∇ (ρuu)= ∇P r ρ + κg + κd H (3.2) ∂t · − g − r2 c H H ∂ ) * (ρe)+∇ (ρeu)= P ∇ u + 4πρ κgJ κgS (3.3) ∂t · − g · J − S g + , 22 where u is the radial velocity, ρ is the gas density, Pg(ρ,e) is the thermal gas pressure, mr is the integrated mass within a sphere with radius r and e is the specific internal energy of the gas. In these equation the radiation field is represented by J and H, denoting the wavelength-integrated moments of the intensity. We assume direct coupling between the motion of the gas and the dust, i.e. the momentum gained by the dust from the radiation field is directly transferred to the gas. Both absorption and scattering on dust may contribute d to the overall momentum gain (see Eq. (2.8)) and both are included in κH. The superscripts g and d correspond to the gas and dust components respectively, and the subscripts indicate averages over different moments of the radiative intensity (see below). This means that κd corresponds to κ as defined by H % & Eq. (2.3), (2.7) and (2.8). The energy budget of the dust component, and therefore the grain temperature, is determined by the condition of radiative equilibrium1

d 1/4 d d κabs,J κabs,JJ κabs,SS(Td)=0 Td = d Tr. (3.4) − −→ -κabs,S .

4 In this formula Tr = Jπ/σ denotes the radiation temperature and κabs the true absorption part of the dust opacity. The wavelength-integrated moments / of the intensity, J, H and K, are determined by solving the zeroth and ﬁrst moment equation of the radiative transfer equation ∇ H + ρ κgJ κgS = 0 (3.5) · J − S g 3K J ∇K + +− + ρ κg,+ κd H = 0 (3.6) r H H ) * simultaneously with Eq. (3.1)-(3.3). Note that the terms containing the dust opacity cancel out in Eq. (3.5) since we have assumed that radiative equilibrium holds for the dust grains (see Eq. (3.4)). The remaining unknown quantities required for closing the system of conservation laws (i.e. the wavelength-averaged opacities, κJ, κH and κS, and the Eddington factor fedd = K/J) are determined in a separate step: by solving the wavelength-dependent radiative transfer equation for the current density- temperature structure after each hydrodynamic time-step we obtain Jλ , Hλ and Kλ , which allows us to compute the averaged opacities, 1 ∞ ∞ κ = κ X dλ where X = X dλ, (3.7) X X λ λ λ 00 00 as well as the Eddington factor, that are used in Eq. (3.2)-(3.6). In the current models we use 319 wavelength points, in contrast to 51 points in Höfner et al.

1Gauger et al. (1990) demonstrated that this assumption holds for grains with high absorption coefﬁcients, e.g amorphous carbon. The validity of this assumption for the relatively transparent Mg2SiO4 grains is discussed in Paper III.

23 (2003), for improved representation of the opacities and the radiation ﬁeld. Assuming LTE, we approximate the source functions with Planck functions, i.e. Sg = B(Tg) and Sd = B(Td), and the corresponding averaged opacities are Planck means.

3.2 Detailed dust description Dust formation can be considered a two step process, starting with the formation of tiny seed nuclei from the gas phase, proceeded by the growth of macroscopic sized grains by condensable material from the gas phase onto the seed nuclei. For the atmospheres of M-type AGB stars there is currently no well-established nucleation theory. For simplicity we assume the existence of seed particles that will start to grow when the thermodynamic conditions are favourable. This will result in a uniform grain size for all dust particles at a given distance from the stellar surface. The models for M-type AGB stars use a time-dependent description for the growth and decomposition of Mg2SiO4 grains which is modelled according to the net reaction

2Mg + SiO + 3H O Mg SiO + 3H , (3.8) 2 −→ 2 4 2 under the assumption that the step determining the total growth rate is the addition of SiO molecules to the grain surface. The equation describing the growth and decomposition of Mg2SiO4 grains follows Gail & Sedlmayr (1999) and is given by2

dagr p , Tg = V Jgr Jdec = V α v n v SiO dt Mg2SiO4 SiO SiO Mg2SiO4 SiO SiO SiO kT T − − g 3 d 1 2 & (3.9)( gr dec where agr is the grain radius, assuming spherical grains. JSiO and JSiO denote the condensation and evaporation rate of SiO molecules per grain surface area,

VMg2SiO4 is the volume of the monomer (the basic building block of the grain material), αSiO is a sticking coefﬁcient, vSiO is the thermal velocity of the SiO molecules, nSiO is the number density of SiO molecules in the gas and pv,SiO is the hypothetical partial pressure of SiO molecules in chemical equilibrium between the gas phase and the solid. The output of the equation describing the grain growth (Eq. (3.9)) is the grain radius agr at a given distance and time. If Mg2SiO4 particles grow to sizes comparable to the wavelength of the ﬂux maximum, the contribution to the radiative acceleration from the scattering cross-section is substantial, dominating over true absorption by several orders of magnitude, as can be

2Note that this equation is given here in the co-moving frame, in contrast to the differential equations of radiation-hydrodynamics in the previous section.

24 1

−1

−2

−3

−4

log(Q) or log(g) −5 Q_rp −6 Q_abs Q_sca −7 g_sca

−8 −7.0 −6.5 −6.0 −5.5 −5.0 −4.5 −4.0 −3.5 −3.0 log( grain radius ) [cm]

Figure 3.1. The efficiency Q of Mg2SiO4 grains at wavelength λ = 1 µm (near the stellar flux maximum) as a function of radius, using refractive index data from Jäger et al. (2003): logQacc (acceleration efficiency, black), logQabs (absorption efficiency, red), logQsca (scattering efficiency, blue) and loggsca (asymmetry factor, green). Adopted from Höfner (2009). seen in Fig. 3.1. To include the contribution from photon scattering in the radiative acceleration we calculate the grain-size dependent dust opacity per mass accordingly (see Sect. 2.1 for details)

3 Amon Qacc(λ,agr) εSi κacc(λ,agr)= fc, (3.10) 4 ρgr agr 1 + 4εHe

The efﬁciency Qacc(λ,agr) can be computed from optical data using Mie the- d ory (e.g. Bohren & Huffman, 1983). The degree of condensation fc = nSi/nSi (deﬁned as the fraction of silicon bound in dust compared to the total amount of silicon) can be computed from the grain radius agr (given by (Eq. (3.9)), the volume of the monomer VMg2SiO4 , the abundance of seed particles ngr/nH and the elemental abundance of silicon εSi, 3 4πagr(r,t) 1 ngr 1 fc(r,t)= . (3.11) 3 VMg2SiO4 nH εSi The only free input parameter in this model for grain growth is the abundance of seed particles ngr/nH. Generally, within the range where observable wind properties are reproduced, a lower abundance of seed particles will tend to result in condensates with larger grain radius, due to less competition in accumu- lating the surrounding material, whereas a higher abundance of seed particles will usually produce smaller grains (see Paper III).

25 Table 3.1. The combinations of input parameters for the models of M-type AGB stars (mass, luminosity, effective temperature, period, piston velocity and seed particle abundance). For references see Paper III.

DMA M" L" T" P ∆up logngr/nH [M ][L ] [K] [days] [km/s] " " L50 1 5000 2600–3000 310 3.0, 4.0 -16,-15.5,-15,-14.5 L70 1 7000 2600–3100 395 3.0, 4.0 -16,-15.5,-15,-14.5 L10 1 10000 2600–3200 525 2.0, 3.0 -16,-15.5,-15,-14.5

3.3 A set of wind models for M-type AGB stars In Paper III we present the first extensive set of time-dependent wind models for M-type AGB stars. The aim of this study is to investigate whether outflows can be produced for a range of different stellar parameters and whether winds driven by photon scattering on Fe-free silicates is a viable scenario for M-type AGB stars. We also investigate how the observable properties produced by these models are affected by different model parameters and compare model results with observations, focusing on wind properties, photometry and spectra. Wind properties such as mass-loss rate and wind velocity provide constraints on the density structure of the atmospheres, photometry in visual and near-IR wavelength regions give information about on the driving mechanism of the outflows and mid-IR features provide information about the dust material in the circumstellar environment. The current study includes 139 solar-mass models of M-type AGB stars, with effective temperatures ranging from 2600 K to 3200 K. Three different luminosities have been used: 5000L , 7000L and 10000L , as well as different pulsation amplitudes and seed" particle abundances" (modelled" by the parameters ∆up and ngr/nH). Table 3.1 shows the input parameters of the models. For each fixed luminosity, the effective temperature, piston velocity and seed particle abundance are varied according to the values listed.

3.3.1 Dynamical properties A schematic overview of the dynamical properties of all the available models is shown in Fig. 3.2. The range in effective temperature and luminosity is listed on the x-axis and y-axis, respectively. Each subset consisting of twelve boxes is organised such that the piston velocity is increasing upwards and the seed particle abundance is increasing towards the right. The red boxes represent models that develop a stellar wind, the blue boxes represent models without a wind and the grey boxes represent combinations of parameters not tested or where the models fail for numerical reasons. It is clear from this plot that the dynamical models can produce outﬂows for a wide range of stellar parameters,

26 Figure 3.2. Schematic overview showing the dynamic behaviour of the wind models as a function of input parameters. The red rectangles indicate models with a stellar wind, the blue rectangles indicate models without a wind and the grey rectangles indicate combinations of parameters not tested or where the models fail. For each combination of luminosity and effective temperature the seed particle abundance and piston velocity (in km/s) are varied as indicated by the small box. See Tab. 3.1 for the different values of seed particle abundance (a-d). Figure from Paper III. although generally a high effective temperature and low luminosity makes it more difﬁcult to drive a wind. A comparison between the wind velocities and mass-loss rates of the models and corresponding empirical data for M-type AGB stars, derived from observations of several CO-lines, is shown in Fig. 3.3. The agreement with observations is good, not the least considering that this set of models does not correspond to a stellar population, but rather a grid model where not all combination of input parameters are equally likely. Both high and low mass-loss rates are reproduced, and higher mass-loss rates could probably be reached by increasing the stellar luminosity of the wind models further. The top panel of Fig. 3.3 shows the dynamical properties of wind models colour-coded according to luminosity. The models with different input values for the luminosity form bands where higher luminosity correlates with higher mass-loss rates. If we instead plot the dynamical properties colour-coded by seed particle abundance (bottom panel of Fig. 3.3), the models form bands correlating more with wind velocity.

3.3.2 Synthetic spectra and photometry In order to compare models to observations directly, we perform detailed a posteriori radiative transfer calculations. From this long sequence we select

27 -4 Observations (Olofsson et al. 2002, Delgado et al. 2003) L=5000 LO • L=7000 LO • L=10000 LO • -5 /yr]) sun

-6

log (dM/dt [M -7

-8 0 5 10 15 20 u [km/s]

-4 Observations (Olofsson et al. 2002, Delgado et al. 2003) log ngr/nH=-16.0 log ngr/nH=-15.5 log n /n =-15.0 -5 gr H log ngr/nH=-14.5 /yr]) sun

-6

log (dM/dt [M -7

-8 0 5 10 15 20 u [km/s]

Figure 3.3. Observed mass-loss rates vs. wind velocities of M-type AGB stars (Olof- sson et al., 2002; González Delgado et al., 2003, plus signs) and the corresponding properties for the wind models (squares). The models are colour-coded according to stellar luminosity (upper panel) and seed particle abundance (lower panel). Figure from Paper III.

28 a series of snapshots, equidistant in phase, during three consecutive pulsation periods. For these snapshots we produce spectra and photometry in an a posteriori radiative transfer calculation. The synthetic spectra are computed with opacities from the COMA code (Aringer, 2000), resulting in an opacity sam- pling spectrum between 0.3 µm and 25 µm, and mean synthetic photometry is calculated by phase-averaging over the pulsation cycle. For the comparison with observed data we have chosen to use visual and near-IR colours (J K) − vs. (V K) because they give useful diagnostic for the conditions in the stellar − atmosphere and the wind mechanism.

3.3.3 Photometric properties in the visual and near-IR AGB stars are variable stars, with photometric properties that vary with phase, and realistic dynamical models should be able to reproduce this variation during the pulsation cycle. In Fig. 3.4 we plot the synthetic colours (J K) vs. − (V K) during a pulsation cycle for a few selected models, together with − phase-dependent colours constructed from sine ﬁts of observed light curves (for details see Paper II). The photometric variations lead to loops in the colour-colour diagram, with different position, shapes and tilts. For M-type AGB stars, both the synthetic and observed colour loops are characterised by large variations in (V K) and small variations in (J K) − − (see Fig. 3.4). The colour (V K) reaches its peak value during luminosity − minimum, followed by a trough during luminosity maximum. A closer examination of the synthetic spectra reveals that the large variation in (V K) − is caused by changes in the molecular abundance of TiO during the pulsation cycle, and not by changes in the amount of dust present (see Fig. 3.5 and Paper II). This is a strong indication that the wind-driving dust species in M- type AGB stars are quite transparent in the visual and near-IR, otherwise the variations would not be dominated by molecular features It is clear from Fig. 3.4 that the phase-averaged colours will be affected by both the position of the loops in the colour-colour diagram and the variation in colour during a pulsation cycle. In Fig. 3.6 we compare phase-averaged synthetic colours to observed colours for both M-type (grey and red symbols) and C-type AGB stars (green symbols). The synthetic values reproduce the span of observed colours in both (V K) and (J K) of the ﬁeld M-type − − AGB stars presented in Mendoza (1967) and the bulk of the observed colours from Galactic Bulge miras (Groenewegen & Blommaert, 2005). However, the spread in (J K) for the Bulge miras is more pronounced than what the − models produce, probably partly be due to metallicity effects and the fact that we assume solar abundances of C and O in the models. A higher C/O-ratio should give redder values of (V K) and (J K), judging from hydrostatic − − model atmospheres. Note also that in this colour-colour diagram there is a clean separation between the M-type and C-type AGB stars.

29 5 5 RR Sco R Car R Hya 4 R Oct 4 R Vir T Col T Hor RU Vir (C-type) 0 3 0 3 (J-K) (J-K)

2 2

1 1

4 6 8 10 12 14 4

(V-K)0

1.7 1.7 5 Model A2 Model A3 Model B1 4 Model B2 1.5 Model B3 1.5 Model C1 0 0

0 3 (J-K) (J-K) (J-K) 1.3 1.3 2

1.11 1.1 4 6 8 10 12 4 14 4 6 8 (V-K)10 12 14

(V-K)0

Figure 3.4.1.7 Observed (upper panel) and synthetic (lower panel) photometric variations during a pulsation cycle of M-type miras. The observed variations are derived from sine ﬁts of light-curves (see Paper II for more details). Figure from Paper II.

1.5

30 0 (J-K)

1.3 1038 VI JHKLM

1037

1036

35 [erg/s]

λ 10 L λ 1034

33 10 Original spectrum Spectrum calculated without TiO 1032 0.4 0.6 0.8 1.0 2.0 λ [µm]

Figure 3.5. Spectral energy distribution as a function of wavelength for a model with input parameters M = 1M , L = 5000L , T = 2800K, up = 4km/s and logn /n = 14.5. The original" spectrum is shown" ∗ in black and a spectrum, cal- gr H − culated without TiO opacities, is shown in red. Figure from Paper III.

3.5 Bulge miras (G&B 2005) Field C-type LPVs (Bergeat et al. 2001) 3.0 Field M-type LPVs (Mendoza 1967) Teff=2600-2700 K Teff=2800-2900 K Teff=3000-3200 K 2.5 0

(J-K) 2.0

1.5

1.0

4 6 8 10 12 14

(V-K)0

Figure 3.6. Observed (plus signs) and synthetic (squares) mean colours (J K) vs. − (V K). The observational data is compiled from different sources: Galactic Bulge − miras (Groenewegen & Blommaert, 2005, grey), ﬁeld M-type LPVs (Mendoza, 1967, red), and C-rich giants (Bergeat et al., 2001, green). Note that most of the observational data are single epoch measurements, whereas the synthetic colours are means over the pulsation cycle. Figure from Paper III.

31 1038 VI JHKLM

1037

1036

35 [erg/s]

λ 10 L λ 1034

Pure Mg2SiO4 grains Mg SiO core, MgFeSiO mantle (1% of the radius) 1033 2 4 4 Mg2SiO4 core, MgFeSiO4 mantle (5% of the radius) Mg2SiO4 core, MgFeSiO4 mantle (10% of the radius) 1032 1 10 λ [µm]

Figure 3.7. Spectral energy distribution as a function of wavelength during luminosity minimum. The black curve shows the energy distribution using pure Mg2SiO2 grains and the stellar parameters M = 1M , L = 5000L and T = 2800K. The " " ∗ coloured curves show the energy distribution using a core of Mg2SiO2 and a mantel of MgFeSiO4. Figure from Paper III.

3.3.4 Spectral features in the mid-IR An issue with the current wind models is that although they show dynamical properties and visual and near-IR photometry in agreement with observations (see Secs. 3.3.1 and 3.3.3), they do not produce the prominent silicate features at 10 µm and 18 µm that are observed in many oxygen-rich AGB stars. A simple test reveals the probable cause of this problem. Setting a lower limit for the grain temperature of 800 K and 500 K in the a posteriori radiative transfer, thereby making the dust particles warmer at large distances from the star than would be the case for pure Mg2SiO4 grains, results in pronounced mid-IR silicate features (see Fig. 11 of Paper III). This demonstrates that the missing features are not caused by a lack of silicate dust in the models. Rather, the energy balance of pure Mg2SiO4 grains is set mostly in the mid-IR, due to low visual and near-IR absorption cross-sections, resulting in a rapidly falling grain temperature in the wind and therefore too low dust emission. Inclusion of materials with a higher near-IR absorption cross-section into the Mg2SiO4 particles, i.e. ’dirty silicates’, can alter this mid-IR dominance and change the grain temperature. We test this scenario by adding a thin mantel of MgFeSiO4 on top of the pure Mg2SiO2 grains in the a posteriori radiative transfer at a distance from the star when such grains can be thermally stable, as if a small fraction of Fe had condensed onto the surface of pure Mg2SiO2 grains. The result is a spectrum

32 Figure 3.8. Best-fitting synthetic visibility profiles (solid red lines) together with MIDI measurements of RT Vir (black lines), using only Mg2SiO2 grains in the models. Red dotted lines represent the corresponding dust-free visibility profiles. White areas mark where silicate dust dominates. The red bars indicate the phase dispersion of the model. Figure from Paper IV.

with strong silicate features, even with a very thin mantel of MgFeSiO4 (and a small amount of Fe inclusions), as can be seen in Fig. 3.7. This indicates that the silicate grains cool down less with increasing distance from the star than the current models predict and that other materials probably contaminate the pure Mg2SiO2 grains when such inclusions can be thermally stable. In this context it is worth noting that the ﬂux in the near-IR is not changed by the thin mantel of MgFeSiO4 on top of the Mg2SiO2 grains (see Fig. 3.7), i.e. the conclusions from the previous section should still hold.

3.3.5 Interferometry of RT Vir Paper IV is an example of how the dynamical models can be used to interpret observations. We compare spectro-interferometric measurements of the semi- variable star RT Vir, taken with the ESO-VLTI instrument MIDI at 8 13 µm, − to synthetic data from a dynamical model ﬁtted to this star. The comparison

33 Figure 3.9. Best-fitting synthetic visibility profiles (solid red lines) together with MIDI measurements of RT Vir (black lines), using Mg2SiO2 grains from the dynamical models and adding extra Al2O3 in the a posteriori radiative transfer. Red dotted lines show visibility profiles calculated without silicate dust. The red bars indicate the phase dispersion of the model. Figure from Paper IV. shown in Fig. 3.8 confirms the presence of silicate dust in the spatial region between 2 3 stellar radii (corresponding to a baseline of about 60 m, middle − panels), seen as a dip in the visibility profiles around 10 µm. It also shows that the current dynamical models are missing material emitting at longer wavelengths, i.e. the region shaded in grey. Adding alumina condensates in the a posteriori radiative transfer can compensate for this missing opacity, as can be seen in Fig. 3.9. The stellar parameters of RT Vir were fitted by comparing broadband photometry and ISO/SWS spectra of this star to synthetic spectra from hydrostatic MARCS models. These stellar parameters are then used as input parameters in the dynamical models, together with different seed particle abundances and pulsation amplitudes. The result is a set of dynamical models with fixed stellar parameters but a range of dynamical properties. The best dynamical model is picked by again comparing synthetic spectra to broadband photometry and ISO/SWS spectra, but also by comparing observed mass-loss rates and wind

34 velocities to wind properties produced by the dynamical models. However, this might not be the best approach for fitting dynamical models, given the difficulties in this study to achieve a density structure that fits the observed interferometric data at all baselines. AGB stars are variable objects and the structure of the atmosphere can temporarily be influenced by passing shock waves or previous episodes of mass-loss. Furthermore, the photometry and spectra used to constrain the stellar parameters of the star are measured at different pulsation cycles and phases, which introduces uncertainties. There are also uncertainties in the observed mass-loss rates and wind velocities used to constrain the wind models; a change in the dynamical properties will affect the density structure of the atmosphere and the output of interferometric data. As seen in Sect. 3.3.3, observed photometric variations in the near-IR, covering a full pulsation cycle, might provide important constraints concerning the inner atmosphere of the star.

35 4. Exploring the effects of model assumptions

The question arises if the close match between the observed and synthetic photometry seen in Fig. 3.4 and Fig. 3.6 is due to generic properties of the wind models for M-type AGB stars or if it gives constraints on the grain material driving the wind. In order to investigate this we systematically study how the photometry of the dynamical models is inﬂuenced by different chemical and optical properties of the wind-driving dust species. This is done by construct- ing a parameterised description of the dust opacity (see Sect. 4.1 and Paper I), capturing the most essential dust properties for the interaction with the radiation ﬁeld, and testing a range of these dust properties. The resulting models are described in Sect. 4.2 and Papers I-II. Due to the low density of the circumstellar environment and the proximity to a strong radiation source, the dust temperature in the dynamical models is assumed to be determined by radiative equilibrium rather than collisions. In Sect. 4.3 and Paper III we discuss the effects of this assumption by comparing the models presented in Sect. 3 to a set where we instead assume that the grain temperature is equal to the gas temperature.

4.1 Parameterised dust description The wavelength-dependence of the absorption coefﬁcient is of major importance for the grain temperature, as brieﬂy discussed in Sect. 2.2.2. Another important factor is the condensation temperature, a chemical property that indicates below which temperature grains of a certain material can exist. As a consequence, we construct a parameterised dust opacity that allows for different optical wavelength-dependences and condensation temperatures, i.e.,

κ (λ)=κˆ(λ) f (r,t,T ). (4.1) acc · c c Our description is inspired by a formula used by Bowen (1988) in his dust- driven wind models, but is here generalised to allow for wavelength-dependent optical properties. The degree of condensation, fc(r,t,Tc), is designed to increase monoton- ically with falling grain temperature, approaching a value of 1 as the grain temperature drops well below the condensation temperature 1 f (r,t,T )= . (4.2) c c (T (r,t) T )/∆T 1 + e d − c 36 1.2 Tc 1.0 !=0.00 =0.25 0.8 =0.50 =0.75 c f 0.6

0.4

0.2

0.0 1600 1400 1200 1000 800 600 400

Td [K] Figure 4.1. The degree of condensation as a function of grain temperature, using the parameterised dust description in Eqs (4.1)-(4.2) (solid curve), and for a model with a detailed description of Mg2SiO4 grains (model A2 in Paper II, dashed curves). Figure from Paper I.

The grain temperature Td is determined by the condition of radiative equilibrium (Eq. 3.4) and it is a function of both time and distance from the star, due to the varying radiation ﬁeld. The parameter Tc sets where fc = 0.5 and ∆T regulates the width of the dust formation zone (see Fig. 4.1). In the parameterised models, discussed in the following section, Eq. (4.2) replaces Eq. (3.9) describing the grain growth. The wavelength-dependent part of the dust opacity κˆ is modeled as

p λ − κˆ(λ)=κ (4.3) 0 λ ! 0 " where p can be obtained by fitting a power-law function to Qacc/agr data in the wavelength region where most of the stellar flux is emitted (see Fig. 4.2). Regions of low flux will not contribute significantly to the energy balance and the radiative acceleration. In this expression κ0 is a scaling factor such that κˆ(λ0)=κ0. In general the dust opacity would be a function of both wavelength and grain radius. However, since we are interested in the onset of grain formation and in particular the location of the dust formation zone, the adopted parameterisation mimics the small particle limit where Qacc/agr is not a function of grain size (small particles have to exist before they can grow beyond the small particle limit). To distinguish between effects of scattering and absorption of photons on dust grains we further introduce a quantity fabs which sets the fraction of the

37 Figure 4.2. Efficiency per grain radius Qacc/agr in the small particle limit, as a function of wavelength, for a selection of dust species. The dashed lines show the power law fit corresponding to Eq. (4.3) and the shaded area indicates the wavelength region for which the optical data is fitted, coinciding with the region of high stellar flux. For references to the sources of optical data see Paper II.

dust opacity κacc that is to be considered as true absorption

κabs κabs(λ)= fabs κˆ(λ) fc(r,t) where fabs = . (4.4) · · κacc

The dust opacity κacc, which includes contributions from both the absorption and scattering cross-sections, is used to calculate the radiative acceleration in the equation of motion, and the true absorption part κabs is used to determine the grain temperature. This allows us to separate the dynamical and thermal effects of the dust opacity, and the parameter fabs can be adjusted to explore the effects of varying degrees of grain transparency. This parameter can also be used to simulate other potential forces that affect the dynamics of the gas without changing the energy distribution of the radiation ﬁeld.

4.2 Wind models with a parameterised dust opacity A set of dynamical models based on the parameterised opacity described in the previous section is explored in Papers I-II. All models have the same stellar parameters (M = 1M , L = 5000L , T = 2800K and ∆up = 4km/s) and we investigate the dynamical" effects of" different∗ chemical and optical dust properties by varying the input parameters p and Tc in the formula for the parameterised dust opacity (see Sec. 4.1). In addition to p and Tc, we also vary

38 Figure 4.3. The mass loss rates for a grid of dynamical models with parameterised dust opacity. The grid covers a range of chemical and optical dust properties, represented by the variables Tc and p. In the left panel the grains are opaque ( fabs = 1.0) and in the right panel the grains are semi-transparent ( fabs = 0.5). The over-plotted contours are curves of constant condensation distance (Rc = 2,4,10) using the simple estimate Eq. (2.20). Figure from Paper I. the degree to which the dust opacity is considered true absorption by setting the parameter fabs to 1.0 and 0.5 respectively (100% and 50% true absorption).

4.2.1 Dynamical properties There are clear restrictions concerning what combinations of chemical and optical dust properties that can trigger outﬂows. The coloured area in Fig. 4.3 shows the combinations of Tc and p that produce stellar winds. Dust species with properties that fall outside this region will heat up strongly when interacting with the stellar radiation ﬁeld and will therefore not condense at distances where they matter for the triggering of stellar winds. Fe-bearing condensates all fall within this latter regime.

4.2.2 Photometric properties in the visual and near-IR Photometry from the wind models with parameterised dust opacity indicates that the chemical and optical properties of the wind-driving dust species affect the resulting spectra and photometry strongly. As can be seen in Fig. 4.4, there are distinct differences between the photometric variations produced by models with an opaque wind-driving dust species (red loops) and models with a more transparent wind-driving dust species (blue loops). The photometry from models where the wind is driven by more transparent grains are characterised by large variations in (V K) and small variations in (J K). The same − − behaviour can been seen in the photometric variations from detailed models, using Mg2SiO2 grains as the wind-driving dust species, and the photometric

39 variations derived from observed light curves (see Fig. 3.4). As mentioned in Sect. 3.3.3, this large variation in the visual ﬂux of the models is caused by molecular changes during the pulsation cycle and would not be visible unless the dusty envelope was transparent in the near-IR. It is also clear from the photometric loops produced by these models that the wind-driving dust species in M-type AGB stars can not have an opacity dominating in the near-IR wavelength region (p > 0), as this would lead to too large variations in (J K). −

4.3 Grain temperature and the onset of outﬂow

Due to the optical properties of Mg2SiO4, particles of this material tend to be quite cool in the dust formation zone, with grain temperatures generally a few hundred degrees below the gas temperature. This provides us with an excellent way of exploring the importance of the radial position of the wind acceleration zone: setting the grain temperature equal to the warmer gas temperature results in shifting the position of the dust formation zone slightly outwards. This causes a shift in the radial position of the wind acceleration zone which will affect the observable properties of the dynamical models. Fig. 4.5 shows the mass-loss rates and wind velocities for the models with a dust temperature based on radiative equilibrium (red squares, same as in Sect. 3) and the models where we have assumed that the dust temperature is equal to the gas temperature (blue squares). Both ﬁt observations reasonably well, but the latter models (where the wind acceleration zone is shifted outwards to lower densities) do not reproduce the observations with high values of mass-loss rate and wind velocity, while the stars with low mass loss and slow wind velocities seem to be better reproduced. In contrast to the dynamical properties, photometry provides strong constraints on the position of the wind acceleration zone. The top panel of Fig. 4.6 shows the average colours (J K) and (V K) during a pulsation cycle for − − the two types of models. The photometry of the models where the grain temperature is set equal to the gas temperature (blue squares) is very similar to the photometry for pulsating models without wind (see Fig. 4 in Paper II), whereas the photometry of the models based on radiative equilibrium (red squares) are bluer in (V K). An in-depth comparison is given in the lower − panel of Fig. 4.6, showing the photometric variations during a pulsation cycle. The red and blue loops are from two dynamical models where the dust temperature is set in different ways but which otherwise have the same model parameters. The black loops show observed photometric variations from a set of very regular mira stars (for a more detailed description of the observational data see Paper II). The large variations in (V K) seen in the observed pho- − tometry are not reproduced in the models with a grain temperature set equal to gas temperature.

40 Figure 4.4. Photometric variations during a pulsation cycle for a few selected wind models with parameterised dust opacity. The red loops are from models with opaque grains ( fabs = 1.0) and the blue loops are from models with semi-transparent grains ( fabs = 0.5). The top panel shows models with ﬁxed optical properties (p = 0) and varying condensation temperatures. In the bottom panel the models have varying optical properties but a ﬁxed condensation temperature (Tc = 1600 K). Figure from Paper II.

41 -4 Observations (Olofsson et al. 2002, Delgado et al. 2003) Dust temperature det. by rad. equilibrium Dust temperature equal to gas temperature -5 /yr]) sun

-6

log (dM/dt [M -7

-8 0 5 10 15 20 u [km/s]

Figure 4.5. Observed mass-loss rates vs. wind velocities of M-type AGB stars (Olof- sson et al., 2002; González Delgado et al., 2003, plus signs) and the corresponding properties for the dynamical models where the dust temperature is determined by radiative equilibrium (red squares) or set equal to the gas temperature (blue squares). The ﬁlled squares mark the dynamical models where a more detailed analysis is conducted. Figure from Paper III.

As mentioned in Sect. 3.3.4, the large variation observed in (V K) is a − consequence of a change in the abundance of TiO during a pulsation cycle. The position of the onset of outflow affects the overall density and temperature structure of the gas, and consequently the molecular abundances, by radially ’pulling’ the atmosphere outwards. If the wind acceleration zone is shifted to distances where this effect no longer influences the structure of the inner atmosphere, then the resulting photometry will resemble that of a pulsating atmosphere without a stellar wind. It seems that the observed variations in (V K) do not only reveal that the wind-driving dust species in M-type AGB − stars have to be quite transparent in the visual and near-IR, but they also provide an upper limit for the distance of the onset of outflow. Note that this upper limit probably also holds if a different mechanism than radiation pressure on dust accelerates the wind since the diagnostic presented here is based on gas properties.

42 3.5 Bulge miras (G&B 2005) Field C-type LPVs (Bergeat et al. 2001) 3.0 Field M-type LPVs (Mendoza 1967) Dust temperature det. by rad. equilibrium Dust temperature equal to gas temperature 2.5 0

(J-K) 2.0

1.5

1.0

4 6 8 10 12 14

(V-K)0

1.8 Photometric variations from observed light curves Dust temperature det. by rad. equilibrium Dust temperature equal to gas temperature 1.6 0 1.4 (J-K)

1.2

1.0 4 6 8 10 12 14

(V-K)0

Figure 4.6. Observed and synthetic colours. Synthetic phase-averaged colours from wind models where the dust temperature is determined by radiative equilibrium (red squares) or set equal to the gas temperature (blue squares) is shown in the top panel. The bottom panel show photometric variations derived from sine fits of observed light- curves for a sample set of M-type miras (black loops) and synthetic photometric variations for the models marked with filled squares in the top panel (red and blue loops). The models have stellar parameters M = 1M , L = 5000L , T = 2800K, up = 4 " " ∗ km/s and logn /n = 15. See Paper II for more details concerning the sine-fit of gr H − observed light curves. Figure from Paper III.

43 5. Summary and Future plans

The main goal of this thesis is to explore what dust species are responsible for driving the stellar winds of M-type AGB stars. The main conclusions from this work are the following:

- Dynamical models of M-type AGB stars, with outflows driven by photon scattering on Fe-free silicates of grains sizes comparable to the wavelength of the flux maximum, produce realistic mass-loss rates and wind velocities for a wide range of stellar parameters. In addition, these wind models reproduce well visual and near-IR photometry and interferometric data, showing that outflows driven by photon scattering is a viable scenario for M-type AGB stars. - The dusty envelopes of M-type AGB stars have to be quite transparent in the visual and near-IR wavelength regions, otherwise the molecular changes seen in observed spectra and photometry would not be visible. - The current models do not reproduce the characteristic silicate features at 10 and 18 µm. This is not due to too little dust forming in the models, but rather that the grain temperature of pure Mg2SiO4 grains is falling too rapidly with distance from the star. Inclusion of a small amount of Fe on the grains further out in the circumstellar envelope, where such ’dirty’ grains can exist, will increase the grain temperature and result in spectra with strong silicate features. These dirty silicates will not affect the dynamics or the visual and near-IR photometry of the models significantly. - Fe-bearing silicates heat up strongly when interacting with the radiation field and cannot form close enough to the star to trigger outflows. - Using dynamical models with a parameterised dust description and testing different grain properties we show that the outflows in M-type AGB stars cannot be driven by true absorption of stellar photons on dust grains, as this would produce photometry in the visual and near-IR in- compatible with observed values. - Using wind models with a grain temperature that is set equal to the gas temperature, and thereby shifting the dust formation zone outwards, we find that wind-acceleration has to happen within a well defined range in order to reproduce observed photometry.

44 The mass-loss rates of AGB stars are of major importance for how these stars evolve; everything from nucleosynthesis in the interior of the star to the chemistry of the atmosphere will be affected. It is therefore crucial to provide realistic mass-loss rates as input data to stellar and galactic evolution models. One way to achieve this is to implement mass-loss rates from wind models in evolution models. However, before the existing wind models of M-type AGB stars can be applied to stellar evolution models more work is required. For example, the current grid of wind models only includes solar-mass models and we need to produce models with different stellar masses. The current models reproduce dynamic properties and visual and near-IR photometric observations well. They do not, however, reproduce observations at all wavelengths in a satisfactory way, e.g., the characteristic silicate features at 10 and 18 µm. Preliminary results suggest that inclusion of a small amount of Fe on the grains further out in the circumstellar envelope may solve this problem, but the corresponding micro-physics needs to be implemented. Another future project may be to investigate how water features in the dynamical models compare with observations since this may give additional constraints on the atmospheric structure.

45 6. Contributions to the included papers

Paper I I have performed the analytical estimates and carried out the computations of the dynamical models with parameterised dust description, as well as a major part of the analysis of the results. I wrote most of the text in the paper.

Paper II I computed spectra and photometry for the models with parameterised dust description and the models with detailed dust description, as well as perform- ing a major part of the analysis of the results. I wrote most of the text in the paper, except Sect. 5 on comparative observational data.

Paper III I computed the large set of dynamical models with a detailed dust description presented in this paper, including synthetic spectra and photometry. I performed a major part of the analysis of the numerical results and estimates regarding the grain temperature. I wrote most of the text in the paper.

Paper IV I calculated the spectra of the dynamical models for RT Vir. I also added alumina dust and an extra shell of silicate grains in the a posteriori spectral calculation for the selected radial snapshot.

46 7. Swedish summary - Vindar från svala syrerika jättestjärnor

7.1 Svala jättestjärnor Asymptotiska jättegrensstjärnor (på engelska kallade asymptotic giant branch stars eller AGB stars) är stjärnor med massor på 1 8M som under mil- − " jarder år har producerat energi genom att omvandla väte och helium i stjär- nans centrum till mestadels kol och syre, och nu påbörjat en fas där de producerar energi genom att omväxlande förbränna väte och helium i skal kring kärnan. Stjärnor som befinner sig i denna fas utmärker sig genom att de är väldigt stora och har hög luminositet, som varierar med perioder på 100 till 1000 dagar. Observationer av dessa stjärnor visar att det finns mycket stoft i deras cirkumstellära höljen, samt att de förlorar en stor del av sin massa via massutflöden med typiska hastigheter av 5 30 km/s. Stjärnvindarna är så − kraftiga att den massförlust de orsakar är det som avgör stjärnans återstående livslängd, snarare än den takt som väte och helium förbränns i skal runt kär- nan. För att kunna förutspå hur stjärnorna utvecklas under denna fas är det därför viktigt att förstå mekanismen bakom massutflödet. En annan orsak till att det är viktigt att kunna modellera dessa utflöden är att de fungerar som ett transportmedel av tyngre ämnen som skapas i stjärnornas inre till det omgivande interstellära mediet. Stjärnvindarna är därmed en viktig komponent för förståelsen av galaxernas kemiska utveckling.

7.2 Stoftdrivna vindar Den mest etablerade förklaringen till de massiva utﬂöden som man har ob- serverat hos AGB stjärnor är att de uppstår i en två-stegs process. Det första steget i denna process orsakas av stjärnans pulsationer, som ger upphov till att chockvågor bildas och fortplantar sig upp genom stjärnans atmosfär. Skikt av komprimerad gas bildas bakom chockvågorna och lyfts till svalare områ- den i höljet runt stjärnan. Detta skapar goda förutsättningar för stoftbildning och förklarar varför man ofta observerar stoft kring dessa stjärnor. I det andra steget accelereras de nybildade stoftkornen genom att rörelsemängd överförs till stoftpartiklarna när de sprider eller absorberar stjärnans ljus. Stoftkornen kolliderar sedan med omgivande gaspartiklar och överför sin rörelsemängd till gasen. Om strålningstrycket från stjärnan är tillräckligt stort för att övervinna dess dragningskraft kan en stjärnvind uppstå.

47 7.3 Kriterier för vind-drivande stoft En nyckelfaktor i denna förklaringsmodell är det stoft som bildas i det cirkum- stellära höljet. Generellt sett delas AGB stjärnor in i olika typer (M, S och C), baserat på den kemiska sammansättningen hos stjärnans atmosfär och vilka molekyler och stoftmaterial som observeras. Hos syrerika stjärnor (typ M) finns det mer syre än kol tillgängligt för stoftbildning och därför är syrehaltiga stoftsorter som silikater och oxider vanliga. Stoftmaterial identifieras ofta via karakteristiska spektralband i det infraröda våglängdsområdet, men eftersom det mesta av energin hos AGB stjärnor strålar i nära infrarött är det de optiska egenskaperna i detta område som avgör vilket strålningstryck olika stoftmaterial bidrar med. Bara för att vi observerar stoftmaterial betyder det inte att de har stor påverkan på dynamiken. Så vad är det då som avgör om ett stoftmaterial kan ha stor inverkan på dynamiken hos dessa stjärnor? I Paper I använder vi analytiska och numeriska uppskattningar för detta. Det första kriteriet som stoftmaterial måste uppfylla för att kunna driva ett massutflöde är att de kondenserar nära stjärnan, där densiteten är tillräckligt hög. Hur nära stjärnan stoftpartiklar av ett visst material kan bildas beror dels på materialets kondensationstemperatur och dels på hur lätt de kan absorbera strålning i det våglängdsintervall där stjärnan avger mest energi. Det krävs också att stoftmaterialet som ska driva stjärnvindarna består av vanligt förekommande grundämnen så att stoftbildningen blir effek- tiv och att de optiska egenskaperna hos materialet är sådana att de individuella stoftkornen kan överföra tillräckligt med rörelsemängd till gasen.

7.4 Vind-drivande stoft I kolrika AGB stjärnor (typ C) är amorft kolstoft en naturlig kandidat till att driva massutflöden. Det finns gott om kol i dessa stjärnor och materialet har en hög kondensationstemperatur och kan bildas nära stjärnan. Kolstoft absorberar också tillräckligt med ljus från stjärnan för att strålningstrycket ska kunna övervinna gravitationen. Modeller som beskriver de dynamiska atmos- färerna och vindarna hos kolstjärnor har reproducerat en rad olika typer av observationer, bl.a. massförlusterna, vindhastighetena, fotometri och spektra. I syrerika AGB stjärnor (typ M) är det fortfarande inte helt klart vilket stoft som driver utflödet. Den starkaste kandidaten är silikatkorn innehål- lande magnesium och järn. Silikatstoft har observerats i överflöd i det cirkum- stellära höljet hos syrerika stjärnor och består av vanligt förekommande grund- ämnen, men det är svårt att observationellt bestämma järnhalten hos dessa stoftpartiklar. Detta har stor betydelse eftersom de optiska egenskaperna hos materialet påverkas starkt av järnhalten. Silikatkorn med järn värms upp så mycket att de inte bildas tillräckligt nära stjärnan för att producera utflöden. Järnfria silikat har lågt absorptionstvärsnitt i de våglängder där AGB-stjärnor utstrålar det mesta av sitt ljus. För att lösa detta problem har det föreslagits

48 att vindarna hos syrerika AGB stjärnor kan drivas av spridning av fotoner på järnfria silikatkorn istället för absorption. Både absorption och spridning av stjärnans ljus bidrar till strålningstrycket. Absorption dominerar när kornen är små, men om kornen växer till storlekar som är jämförbara med våglängden där stjärnan strålar mest så spelar spridningen allt större roll och kan i vissa fall dominera över absorptionen med ﬂera storleksordningar. Detta scenario har framgångsrikt implementerats i ett urval vindmodeller för syrerika AGB stjärnor och modellerna ger vindhastigheter och massförluster i överensstäm- melse med observerade värden.

7.5 Vind modeller för syrerika jättestjärnor Avsikten med denna avhandling är att bättre förstå hur vindarna hos syrerika AGB stjärnor uppstår, samt att vidareutveckla de vindmodeller som ﬁnns för dessa stjärnor. Ett viktigt resultat från detta projekt är att vindmodellerna av syrerika AGB stjärnor, där magnesiumsilikat används som vind-drivande stoft och spridning dominerar strålningstrycket, producerar både dynamiska och fotometriska egenskaper i överensstämmelse med observationer (Papers II- IV). Speciellt utmärkande är att både de observerade och syntetiska färgerna uppvisar stora variationer i visuellt ljus under pulsationscykeln. Denna variation är orsakad av förändringar i halterna hos molekyler i den inre delen av atmosfären. Hur mycket av dessa observationella egenskaper beror på det vind-drivande stoftet? För att undersöka detta har vi konstruerat vindmodeller med en för- enklad stoftbeskrivning, som kan justeras för att härma de viktigaste egenskaperna hos olika stoftmaterial (Papers I-II). En tydlig trend hos dessa modeller är att de optiska egenskaperna av det stoft som driver vinden, speciellt hur genomskinligt materialet är, påverkar fotometrin starkt. En slutsats vi kan dra av detta är att det vind-drivande stoftet i syrerika AGB stjärnor måste vara ganska genomskinligt i det visuella och nära infraröda våglängdsområdet, annars skulle inte de molekylära effekterna vara synliga. Vidare, om nu de stoftrika höljena är transparenta i våglängder där stjärnorna utstrålar sin mesta energi så är troligen de observerade utﬂöden hos dessa stjärnor drivna av spridning av fotoner på stora stoftkorn, snarare än absorption. Målet i framtiden är att använda dessa modeller för att bättre förstå syrerika AGB stjärnor, stjärn- utveckling och den galaktiska kemiska utvecklingen.

49 8. Acknowledgements

A typical day as an astrophysicist often starts with morning tea and a hope that computers and clusters will be in a good mood. We discuss the origin of the universe at lunch and try to do some programming in the afternoon. A coffee break is always appreciated, especially if someone brings cake. First I would like to thank my supervisors, Susanne Höfner and Kjell Eriks- son, for being very supportive and instrumental in bringing this project to the finishing line. Susanne, my fondest memories will be the in-depth (nerdy) discussions we’ve had during these years over the physics of AGB stars, I enjoyed those immensely. Kjell, you’re like a never wavering beam of sun- light that knows his radiative transfer, what would we do without you? I also want to thank our international collaborators, Bernhard Aringer and Walter Nowotny, and the rest of the AGB group at Uppsala University for making science more enjoyable. Special thanks goes to Bengt Gustafsson for interest- ing discussions, both scientifically and otherwise, and for being a living proof that you can be a young rebel even after retirement. I also want to thank Niko- lai Piskunov for being a walking, talking encyclopaedia of numerical methods and programming. His door has always been open, something that I have appreciated very much. There are many of my co-workers and fellow Ph.D students, both old and new, that I would like to thank for creating a work environment where it is easy to thrive and feel welcome. It would be a bit tedious to mention all so let me just do some free-style mind-mapping of key elements; star-bursting galaxies, haro11, turbulence, wiener-schnitzel, c scripts, carbon, spirals, dust, asteroids, computers, inverse problems, rotating galaxies, mechanics, nor- rland, gaia, måndagsfika, astro-cinema, canoeing, primaten, VALD, choco- late fondue, ginger cookie houses, lucia, astro-pub, dark star, milou, crashing hard-disks, solar-twins, monday fruit box, magnetic maps, exo-planets, weird christmas parties, coffee machines and clusters. As everyone knows, Lisa is the best office mate in all Ångström and I am envied by many. She is incorruptible and I am thankful she is guarding our office. I also want to thank Sofia for being a much needed friend during the final years of my Ph.D. She and astro-Bettan proves to me that we’re the best vintage! A huge thank you to the extremely talented Samuel for helping me with the cover of this thesis. Lastly, I want to give a shoutout to my family and friends, especially my parents and siblings. They have helped me when life has seemed overwhelming. Since I don’t excel at multitasking it is a comfort to know that sometimes you can delegate. A lot of people have not just been

50 support, but also a source of joy: my family, my sister’s family, my aikido and yoga families, my study family at KTH and Uppsala, childhood friends and various work families. Finally, I want to give myself a pat on the pack for pulling this off. Good work!

Uppsala, 3 September 2014

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