Decoupling in the line-driven winds of early-type massive stars

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1 Astronomical institute Anton Pannekoek Faculty of Exact Sciences (FNWI) University of Amsterdam Decoupling in the line-driven winds of early-type massive stars Master thesis Astronomy and Astrophysics Lucinda Rasmijn Supervisors: prof.dr. Alex de Koter and drs. Lianne Muijres

2 Cover image: NGC 3603 is a star-forming region located in the Carina spiral arm of the Milky Way, about 20,000 light-years (6.1 kpc) away from our Solar System. This stunning composite image of a young star cluster surrounded by glowing clouds of plasma and gas shows various stages of the life cycle of stars in one single view. The giant gaseous pillars of glowing gas (mainly hydrogen) and dust reveal star formation is still in progress, with newly born stars emerging from their dense, gaseous stellar nurseries. Also shown are Bok globules (dark spots in the upper right corner of the image), which may collapse to form new stars and stellar clusters. The young star cluster consists of bright hot mainly O-type stars whose strong stellar winds and ultraviolet radiation have sweeped out nearby interstellar material. Because of their high mass, they exhaust their nuclear fuel at a fast pace and are expected to end their stellar life relatively soon. The bright blue supergiant star Sher 25 is seen above and left of the cluster, which may be only a few thousand years from ending its life in a supernova explosion. Position (J2000): R.A. 11h 15m 9s.10 Dec Constellation: Carina Distance: Approximately 20,000 light-years (6,100 parsecs) away. Dimensions: This image is roughly 3 arcminutes (17 light-years or 5 parsecs) wide. Exposure date: December 29, 2005 Instrument: ACS/WFC Image credit: NASA, ESA, and the Hubble Heritage (STScI/AURA)-ESA/Hubble Collaboration September 23, 2009

3 abstract Metal rich massive stars launch powerful hypersonic outflows during most of their evolution, emit vast quantities of hard, ionizing radiation, and enrich the space around them with heavy nuclei. The first stars that ever formed are predicted to be very massive and practically metalfree. Due to a lack of metals to drive the outflow, their stellar winds are expected to be very weak. In very weak (low density) winds a decoupling may occur of ions that actively participate in the wind driving (typically metals) from ions that are passive, but that represent the bulk of the plasma (i.e. hydrogen and helium). Depending on the location in the wind where this decoupling occurs, this may affect the mass loss rate, or the terminal velocity of the wind. In this study we explore the parameter space in which decoupling may occur in terms of the stars position in the Hertzsprung-Russell diagram and metal content. In this work we study the occurence of ion-decoupling in a stellar wind for the first time using Monte Carlo simulations, in which the detailed processes at work through which photons deposit radial momentum in a stellar wind are simulated. In our approach we have detailed information about the location where the ion-decoupling occurs and the ion species that decouples. We can also estimate the amount of energy that is used to accelerate the decoupled ions and that is therefore not used to accelerate the bulk. From this we can estimate the impact that ion-decoupling has on the stellar wind properties. i

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5 Contents 1 Theory of line-driven stellar winds Physical processes in line-driven winds The basic equations The equation of motion Properties of the equation of motion The radiative acceleration The continuum radiative force The line force The dynamics of line-driven winds: two methods CAK-theory Monte-Carlo simulations Decoupling Decoupling mechanism Slow-down time Drift time Condition for a drift velocity Semi-Analytical Investigation Semi-analytical condition Estimate of Q and Px Results of semi-analytical investigation Dependence on ion charge Dependence on line strength parameter, x iii

6 CONTENTS 4.3 Dependence on radial distance Dependence on metallicity Discussion of the semi-analytical investigation Numerical investigation Determination of the slow down and drift timescales using the MC code Implementing decoupling in the Monte Carlo code Determination of maximum effect on Ṁ and on v of the bulk plasma Determination of terminal velocity of decoupled ions from L Results of numerical calculations Numerical results on the dependence on ion charge Numerical outcome of the dependence on x Numerical outcome of dependence on radial distance Numerical outcome of dependence on metallicity Maximal effect of ion-decoupling on Ṁ and on v Discussion of numerical investigation Discussion and conclusion 99 A Derivation of the slow-down time 105 A.1 Derivation of deflection angle using two-body approximation A.2 Derivation of slow-down time for distant encounters A.3 Multicomponent hydrodynamical equations Nederlandse samenvatting 123 Bibliography 128 iv

7 Introduction Metal rich massive stars launch powerful hypersonic outflows during most of their evolution, emit vast quantities of hard, ionizing radiation, and enrich the space around them with heavy nuclei. The first stars that ever formed are predicted to be very massive and practically metal-free. Due to a lack of metals to drive the outflow, their stellar winds are expected to be very weak. In very weak (low density) winds a decoupling may occur of ions that actively participate in the wind driving (typically metals) from ions that are passive, but that represent the bulk of the plasma (i.e. hydrogen and helium). Depending on the location in the wind where this decoupling occurs, this may affect the mass loss rate, and/or the terminal velocity of the decoupled ions. In this study the parameter space is explored in which ion-decoupling may occur, notably in terms of the stars position in the Hertzsprung-Russell diagram and metal content. Massive stars Our understanding of the importance of massive stars is expressed in the large number and broad context of studies done on these stars. Massive stars are encountered in studies of star formation, because of the generally young absolute age of the ones observed ( 10 7 yr). Therefore, massive stars are often encountered in regions of active star formation, as they did not have had the time to wander off far from their birthplace. Massive stars are also thought to be able to both trigger and halt star formation via their dynamical impact on their surroundings, both due to the supernova explosions with which they end their life, but also due to the dense, high velocity outflows of matter that they can drive during their lifetime in a so-called stellar wind. Another motivation of interest is their role in the chemical evolution of the interstellar medium (ISM). Heavy elements are formed in massive stars. Through their stellar winds and supernova explosions they contribute efficiently to the progressive chemical enrichment of both the ISM and newly born stars. In the context of cosmology, massive stars play a vital role due to their intrinsic brightness. As a result they can be used as extra-galactic standard candles to determine distances to extra-galactic objects. Since massive stars end their life as a black hole or a neutron star, which by themselves are subjects of great interest, the evolution of massive stars that leads to the formation of the compact object need also be understood if we want to understand the formation and distribution of compact objects throughout the universe. 1

8 CONTENTS Figure 1: Schematic overview of different mass-loss mechanisms across the Hertzsprung-Russel diagram. The typical mass-loss rates (in solar masses per year) encountered in the different stellar wind mechanisms is indicated. Evolutionary tracks taken from Schaller et al. (1992).. Stellar winds In many of the motivations mentioned above stellar winds play an important role. Stellar winds are, in short, the continuous ejection of particles from the stellar surface into the ISM. This causes stars to lose mass over the course of their lifetime in a stellar wind. Depending on the efficiency of their mass-loss mechanism, stars may loose up to 80 to 90 percent of their initial mass in a stellar wind. As a result, whether their final fate will be a black hole or a neutron star, depends for a great deal on the amount of mass that was lost in a stellar wind. The removal of angular momentum in a stellar wind in the case of rotating massive stars plays an important role in the spin-down of these stars. This has consequences for their evolution. The chemical impact of a massive star on its environment is intertwined with the phenomenon of stellar winds. This is because the removal of the stellar envelope in a stellar wind exposes the products of nucleosynthesis at the stellar surface. As the stellar wind proceeds to eject the outer layers of the star, as a consequence, the nuclear burning products are injected into the ISM, which leads to its enrichment. Stellar winds also have an impact on the way we observe the stars. This is because, as the star expells its outer layers, the chemically enriched 2

9 CONTENTS interior layers are exposed at the surface. As a result, observed surface abundances can be explained when taking stellar winds into account. Since an outflowing atmosphere typically causes blue-shifted absorption lines, P Cygni profiles, emission lines and an excess emission at infrared and larger wavelengths, it is also important to know the impact of a stellar wind on its emergent spectrum, since it is the emergent spectrum from which we deduce stellar properties. There are several mechanisms through which stars may loose mass through a continuous outflow. Mechanisms that act most efficiently are exclusively in specific parts of the HR diagram. In coronal winds, which are encountered in solar type stars, for example, stars typically lose mass at a rate on the order of M /yr. This is a very modest mass loss. It takes the sun, for example, 30,000 years to lose a mass equal to that of planet Earth. Though the solar wind may have an impact on life on our planet (e.g., it may disturb or even disrupt satellite communications), compared to what other stars may be capable of, the impact of the solar wind on its environment is very modest. Other mass-loss mechanisms are dust driven winds, which are encountered in luminous cool stars in their asymptotic giant branch (AGB) or supergiant phase, and line-driven winds, which are encountered in hot luminous stars of spectral type O, B and A. Both of these mass-loss mechanisms are generally efficient in removing mass from the star. In dust driven winds the amount of mass lost is typically on the order of M /yr [Lamers and Cassinelli (1999)]. In line driven winds the mass loss rates vary from M /yr for the least massive stars, with masses on the order of 7 8M, to 10 5 M /yr for the most massive stars, with masses on the order of M. In the upper right part of the HR diagram, massive stars are found in their luminous blue variable (LBV) phase, which is a phase characterised by instability during which the star suffers a very high mass-loss rate, of the order of 10 4 M /yr [Lamers and Cassinelli (1999)]. In the intermediate region, where the other mass-loss mechanisms are not efficient in driving an outflow, it is thought that the stellar winds are Alfén wave driven. Very briefly, the outflow is driven by the dissipation of energy and momentum associated with the propagation of transverse Alvén waves, which are generated by magnetic oscillations at the base of the wind. For more information on the subject, the reader is referred to Lamers and Cassinelli (1999). In Fig.[1] an approximate overview is depicted of the different mass-loss mechanisms at work throughout the Hertzsprung-Russel (HR) diagram. The focus of my research is on the line driven winds that operate in hot stars. Line driven winds in massive stars Hot stars produce an intense radiation field. This radiation field is absorbed and scattered by thousands of ultraviolet spectral lines of metal ions leading to an outward acceleration of the metals that is much larger than the inward pull of gravity. As hydrogen and helium are less efficient absorbers of radiation, essentially because their number of effective atomic transitions is much less than that of complex metallic ions, these particles merely get dragged along outwards via Coulomb interactions with the metallic ions. Since hydrogen and helium constitute, for all practical purposes, the bulk of the plasma (as they account for at least 98 3

10 CONTENTS % of the mass), this will therefore be termed the passive plasma, and the metallic ions will be termed the active plasma. The radiative line force is able to drive an outflow of matter in stellar winds because of an effective Coulomb coupling between the metals and the bulk plasma. Because in line driven winds the outflow is primarily due to the transfer of photon momentum to the metallic ions, the mass-loss rate of a line-driven wind depends on the metal content ( metallicity ) of the star. The central question in this study is: what happens to the stellar wind properties if the Coulomb coupling between active ion species and passive plasma becomes ineffective in transferring radial momentum from the active ions to the passive bulk plasma? As will be explained in this thesis, if so called ion-decoupling occurs this may have an impact on the mass-loss rate and/or on the terminal velocity of the wind. Depending on the location in the wind where ion-decoupling occurs, and the contribution of that ion species to the line force, the outflow of the passive bulk plasma in a stellar wind may be slowed down or reduced. One of the unresolved problems in the theory of line-driven winds is the so-called weak wind problem, which is the discrepancy between the observed mass-loss rates of low -luminosity (L L ) stars, and theoretical predictions. At L L the discrepancy between theoretical predictions and observed mass-loss rates may reach a factor 100. Several explanations have been put forward to explain this discrepancy. One such proposal is that the mass-loss diagnostics are not yet fully understood, that is, the discrepancy in the mass-loss rate has an observational origin, and in reality the wind strenghts do not differ that much. Another possible explanation could be that the winds of low-luminosity (L L ) stars are really weaker [Mokiem et al. (2007)]. In this study I will investigate whether the assumption of a well-coupled wind is justified across the Hertzsprung-Russel (HR) diagram, and whether, if ionic decoupling occurs for given stellar and wind parameters, this has a significant impact on the stellar wind properties, such as the mass-loss rate and the terminal velocity of the wind. To this purpose we map where in the parameter space spanned by the HR diagram ions tend to decouple. We will also estimate the maximum effect that ion-decoupling may have on the mass-loss rate and on the terminal velocity. Given these answers, we can address the question of whether ion-decoupling can explain the so called weak wind problem. In this thesis I will give a brief introduction to the basic principles of line-driven stellar winds in Chapter 1. In Chapter 2 the mechanism of decoupling will be discussed and the criterion for ion-decoupling will be introduced. In Chapter 3 and 4 this criterion will be tested in a semi-analytical way, where we have used a mass-loss recipe to calculate the mass-loss rate, and estimated properties of the wind structure and radiation field. The purpose of this is to gain a global understanding as to where in the HR diagram ion-decoupling is expected to occur, which is particularly important in the low metallicity regime, as it is increasingly hard to converge numerical models at low metallicity. Also it serves to explain the numerical 4

11 CONTENTS findings, which are presented in Chapter 5 and 6. Our findings will be discussed and our results will be summarised in Chapter 7. 5

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13 CHAPTER 1 Theory of line-driven stellar winds In this chapter we will introduce some of the fundamental equations which intend to describe the fundamental properties of line-driven stellar winds. The three fundamental equations governing the outflow are the equation of motion, mass continuity, and the energy equation. In the so-called standard model, one often assumes that spherical geometry, homogeneity, a monotonically increasing velocity law, and time-independence holds. We will start our description of line-driven winds with a brief overview of the microphysical processes leading to a radiative-driven outflow, in Sec.(1.1), and from that proceed with a description of the current models that describe the macroscopic outflow mathematically, in Sec.(1.2)-Sect.(1.4). 1.1 Physical processes in line-driven winds The winds of hot stars are thought to be driven mainly by radiation pressure acting on the atmospheric gas via the absorption and scattering of radiation in spectral lines. This idea was pioneered by Lucy and Solomon (1970), who showed that the transfer of momentum from the radiation field to the atmospheric gas via line scatterings in a single strong atomic (resonance) transition can be efficient enough for the radiative acceleration to overcome gravity. The theory was improved by Castor et al. (1975) (hereafter CAK), who showed that the radiative acceleration caused by a large number of lines is far more important for the outflow, than the acceleration due to the few strong resonance lines considered by Lucy and Solomon (1970). However, CAK estimated the line acceleration by using an ensemble of C iii lines only. Abbott (1982) calculated the line acceleration using a list of atomic lines and transition probabilities (gf -values), that is essentially complete from the first to the sixth stages of ionisation for the elements H-Zn. After some more refinements, like dropping the radial streaming 7

14 Chapter 1: Theory of line-driven stellar winds approximation 1 by Pauldrach et al. (1986), it was possible to explain basic observed features of a radiatively driven stellar wind. In this section, we will provide a concise overview of the microphysical processes leading to the line acceleration. The basic mechanism driving the winds of hot massive stars is the transfer of momentum from photons to the atmospheric gas by the scattering of the star s continuum radiation by an ensemble of spectral line transitions. This means that the line force is essentially the result of photons undergoing line-interactions with electrons bound in the ions of the atmospheric plasma. In each line-interaction momentum is exchanged between photons and the ions they interact with. Therefore, the mean velocity increase associated with a line-interaction in case a photon coming in from the radial direction (µ = 1) is absorbed and re-emitted isotropically, is the same as for the case of pure absorption: mv = hν 0 c where m is the mass of the ion under consideration [in g], v is its radial velocity [in cm/s], and ν 0 is the rest frequency of the line transition [in s 1 ]. For a derivation of this equation the reader is referred to Lamers and Cassinelli (1999). The constants h and c have their usual meaning. If we consider, for example, the Ciii resonance line λ977, the typical velocity increase associated with one photon absorption of the Ciii ion is hν 0 /c = 34 cm/s. As was discussed extensively by Lamers and Cassinelli (1999), the transfer of momentum via line-interactions is efficient in maintaining a stellar wind because of the Doppler effect in an accelerating outflow. In a static atmosphere the ions in a given outer layer will receive strongly diminished radiation at the wavelength of a given atomic transition, because the emerging photospheric radiation at that frequency has been absorbed or scattered in the lower layers of the atmosphere. In an outflowing atmosphere, however, there is a flow velocity, v = v(r), which increases from v 0cm/s at the base, to a terminal flow velocity, v, in the outer layers. Because of the velocity gradient, the accelerating ions require a continuum photon (in the rest-frame of the observer), that is progressively red-shifted relative to the rest-frame frequency of the transtition being considered. This allows the ion in principle to absorb unattenuated continuum photons, i.e. radiation that is not diluted by the lower layers, in their atomic transitions. However, not all ions are equally likely to be accelerated via this mechanism. The efficiency with which a given ion can absorb momentum from the radiation field depends on the amount of flux that is offered at the frequency of the atomic transition, and therefore, on the luminosity of the star, and on the number and efficiency of the atomic transitions to absorb photons 1 CAK assumed the radiation flow to be radial. At very large radii, where the star appears as a point source, this approximation is valid. However, in the inner part of the wind (between r R and a few stellar radii from the star), the finite angular size of the stellar core is appreciable, which makes this a very crude approximation. 8

15 1.1 Physical processes in line-driven winds from the radiation field. Therefore, the contribution of a given ion to the total line force depends on the product of elemental abundance times the ionization fraction times the efficiency ( oscillator strength ) with which the ions are receptive to absorb photons in a given atomic transition and on the flux at the frequency of the atomic transition. Since the product of elemental abundance times ionization fraction of neutral hydrogen is approximately equal to that of a given ion in its dominant stage of ionization, it is primarily the number of effective atomic transitions ( driving lines ) that is the most important criterion. The main contributors to the line force are therefore relatively complex abundant ions. Despite hydrogen and helium being by far the most abundant elements in the wind, their acceleration due to line absorptions is only marginal, mainly because their simple atomic structure provides only few effective lines in the region of the stellar flux maximum. Fig.(1.1) shows the fraction of the total line acceleration provided by the lines of several elements. Because of their low contribution to the line force, hydrogen and helium are termed the passive, or, the non-absorbing ions. As was shown by Vink et al. (1999), the main contributors to the line force are complex and relatively abundant elements like C and O, and heavy, minor elements with numerous transitions, such as Fe-group elements, which are accordingly termed the active, or, the absorbing ions (see Table 5 from Abbott (1982)). Because the absorbing ions do not constitute the bulk of the plasma, this would result in practically no outflow of matter in a stellar wind, if it were not for the Coulomb coupling between the absorbing ions and the non-absorbing ions. Generally, the absorbing ions and the bulk plasma (i.e. the hydrogen and helium) are tightly coupled, and therefore constitute a single fluid, because of the many collisions occurring between the absorbing ions and the surrounding particles, that constitute the bulk plasma. In each collision, momentum is exchanged between the colliding partners, and generally this results in a transfer of momentum from the absorbing ions to the non-absorbing, bulk plasma. In this fashion, the bulk plasma is accelerated and dragged along outwards. As a result, all chemical species will become part of the stellar wind. However, in low density winds, the lower collision rate between absorbing ions and the bulk plasma may lead to a decoupling of the absorbing ions from the bulk plasma. Springmann and Pauldrach (1992) proposed that this may lead to a so-called ion runaway. Depending on the location where decoupling occurs, and on the importance of the decoupled ions for the line force, this may have an effect on the terminal flow velocity v, and/or on the mass-loss rate Ṁ of the star. The phenomenon of decoupling was investigated by many authors, notably Springmann and Pauldrach (1992), Babel (1995) and Krtička and Kubát (2000). The consensus is that decoupling has no important effect on the mass-loss rate in hot, massive O-type stars at solar-metallicity, but may be of influence in low density winds of low luminosity stars, e.g. main-sequence A and B-type stars. Because of the importance of metals for the line-driving, it is evident that the metal abundance (represented by the metallicity, Z, of a star) is an important parameter that determines 9

16 Chapter 1: Theory of line-driven stellar winds Figure 1.1: The contribution of elements H and He (dotted line), C,N, and O (dash-dotted), Ne through Ca (dashed), and Cr, Mn, Fe, and Ni, a.k.a, the Fe group elements (solid line), to the total line acceleration. Figure taken from Abbott (1982). the strength of the radiation force. This implies that the amount of mass lost in a stellar wind is expected to depend on the metallicity Z. According to Vink et al. (2001), Ṁ varies with Z according to a power law in Z as Ṁ Z 0.69 for O-type stars, and as Ṁ Z 0.64 for B supergiants, over a restricted range of Z. The aim of this research is to map where in the parameter space spanned by the Hertzsprung- Russel diagram and stellar metallicity, decoupling is expected to occur, and, if it does, to determine whether the occurrence of decoupling affects the stellar mass loss and/or on the terminal velocity of the wind. Now that we have a basic understanding of the physical processes at work in a stellar wind, and as a result, of the subject of this research, we will proceed with a discussion of the theoretical background of line-driven stellar winds. First, we will provide the basic equations governing the outflow, in Sec.(1.2), the radiative acceleration, in Sec.(1.3), and briefly review models that have been developed to deal with the wind dynamics, in Sec.(1.4). The phenomenon of decoupling will subsequently be discussed in Chapter 2. 10

17 1.2 The basic equations 1.2 The basic equations To describe a basic model for a radiation-driven stellar wind, we consider the case of a nonisothermal stationary radiatively driven stellar wind that is homogeneous (i.e. no shocks or clumps), spherically symmetric, and in which no chemical separation exists, the so-called standard model. Magnetic fields and rotation are ignored. In that case there is a single equation of motion for all atomic species that make up the wind, which is governed by the inward directed gravitational force, that is counteracted by the outward directed gas pressure and radiative pressure: v dv dr = GM r 2 1 dp ρ dr + g rad (1.1) where v(r), ρ(r), p(r) are respectively, the velocity [in cm/s], density [in gcm 3 ], and pressure [in g/(s 2 cm 2 )] of the gas at some radius r [in cm] from the star, M is the stellar mass [in g], G is the gravitational constant [in cm 3 /(gs 2 )] and g rad is the radiative acceleration [in cm/s 2 ], which can be provided by continuum events and/or line scatterings, as we will discuss in Sec.(1.3). The left-hand side of Eq(1.1) is the acceleration of a unit of mass, which is produced by the net effect of the forces per unit mass on the right-hand side, that is, the inward directed gravitational acceleration (first term) versus the outward directed pressure gradient (second term, note that the pressure gradient will naturally turn out to be outward directed, as the pressure decreases in the outward direction) and the radiative acceleration (third term). The equation of motion, Eq(1.1), primarily determines the velocity structure v(r) of the medium, and, via the continuity equation, the density structure ρ(r) of the medium. This brings us to the second fundamental equation, the continuity equation. The continuity equation reduces, in the case of a stationary ( ρ/ t = 0) and one-dimensional (v = v r = v) flow, to: Ṁ 4πr 2 ρ(r)v(r) = constant (1.2) where Ṁ dm/dt [often given in units of solar masses per year, M yr 1 ] is the amount of matter flowing through a spherical surface with radius r per unit of time, which we will be referring to as the mass-loss rate. The equation of motion is also related to the third fundamental equation, the energy equation, via the dependence of Eq(1.1) to the pressure gradient, dp/dr, which in turn depends on the temperature gradient, as we will show later. The temperature structure T(r) is primarily determined by the energy equation, and radiative transport. The total energy per unit mass e(r), which is the sum of the kinetic and gravitational energy and the enthalpy, is equal to the initial energy e 0 at the lower boundary r 0 of the wind, plus the energy added into the wind between r 0 and r, in the form of heat deposition Q(r) and/or work done by some force, W(r): e(r) v2 (r) 2 GM r + γ RT γ 1 µ = e 0 + W(r) + Q(r) (1.3) 11

18 Chapter 1: Theory of line-driven stellar winds where γ c p /c v is the heat capacity ratio, which equals 5/3 for a monatomic ideal gas, and R is the ideal gas constant [in units of erg K 1 mol 1 ]. To calculate a wind model, with pre-specified stellar parameters, such as the effective stellar temperature T eff, gravitational acceleration, log g, photospheric radius R, and metallicity Z, the stationary hydrodynamic equations, Eq.(1.1), and Eq.(1.2) need to be solved. As we will see later, in order to calculate the radiative acceleration term in Eq.(1.1), the contribution of a very large amount (on the order of ) of lines need to be calculated. This, in turn, requires the occupation numbers of bound and free states of the atmospheric atoms to be determined by the (time-independent) equations of statistical equilibrium, or rate equations, in which all processes into a given level l are related to all processes away from l are related according to: n m P ml n l P lm = 0 (1.4) m l where P ml [in s 1 ] denotes the total rate, which may contain both radiative and collisional terms, from level m to l, and n l denotes the particle density in state l, which is an abbreviation for the excitation and ionization indices i and j of element k 2. The rate equations are in turn coupled to the hydrodynamical equations via the velocity field (through the dependence of the radiative R ij transition rates), and density structure (through the dependence of the collisional C ij rates). Because the radiation field is coupled to the hydrodynamical equations via the radiative line acceleration, and to the rate equations via the radiative rates, the equation that determines the strength of the radiation field also enters the problem. We provide the time-independent transfer equation in spherical geometry that often suffices to determine the radiation field: m l µ I ν r + 1 r (1 µ2 ) I ν µ = η ν χ ν I ν (1.5) where η ν is the emissivity [in ergs cm 3 sterrad 1 Hz 1 s 1 ], χ ν is the extinction coefficient [in cm 1 ] of a given beam of radiation with specific intensity I ν = I ν (r,θ) [in ers cm 2 s 1 Hz 1 sterrad 1 ], at radial distance from the stellar center r, in direction µ cos θ with respect to the radial outward direction (for which µ = 1), with frequency ν. The solution to this total system of coupled equations yields the hydrodynamical structure of the wind, including the mass-loss rate and terminal velocity The equation of motion The condition for an outflow of material in a stellar wind is that the total outward directed force exceeds the inward directed gravitational force in Eq(1.1). Therefore, let us take a more 2 Note that in writing down this expression for statistical equilibrium, it was implicitly assumed that the net number of particles streaming into or out of a unit volume is negligible ( (n l v) = 0), i.e. the time needed to obtain ionization and excitation equilibrium is shorter than the characteristic flow time v. 12

19 1.2 The basic equations detailed look at the gradients occuring in Eq.(1.1), using the conditions created by Eqn.(1.2) and (1.3). The pressure gradient occurring in Eq.(1.1) follows from the following considerations. First, we assume that the flow behaves locally like a perfect gas: p = k BTρ µm H (1.6) where p = p(r) is again the gas pressure, T = T(r) is the gas temperature, µ is the mean mass per free particle (also sometimes referred to as the mean molecular weight), which is assumed to be a constant throughout the wind, and the constants m H and k B have their usual meaning. The pressure gradient in Eq(1.1) can then be rewritten (for a constant assumed value of µ) as: 1 ρ dp dr = k B dt µm H dr + k BT µm H ρ = k B dt µm H dr + a2 ρ where in the latter equality we have introduced the local speed of sound a = a(r) as: k B T(r) a(r) = µm H dρ dr dρ dr (1.7) The density gradient dρ/dr in Eq(1.7) can be eliminated by means of the equation of mass continuity. Since the mass-loss rate, Ṁ is assumed to be a constant (see Eq(1.2)), we can easily derive the density structure ρ(r) from the velocity structure v(r) by setting the derivative d/dr of Eq(1.2) to zero, and solving for dρ/dr, which results in: 1 ρ dρ dr = 1 dv v dr 2 r (1.8) Plugging Eq(1.8) into Eq(1.7) we obtain for the pressure gradient: 1 ρ dp dr = k B dt µm H dr a2 v dv dr 2a2 r (1.9) The pressure gradient depends on the temperature structure, which, in turn, follows from the simultaneous solution to the transport equation, Eq.(1.5), and the energy equation, Eq(1.3). Therefore, Eq.(1.9) shows that the velocity structure in the wind, Eq.(1.1), depends on the thermal energy deposition, Q(r), in Eq.(1.3), and, as a result, also on the radiation field, through its dependence on dt/dr. Plugging Eq(1.9) back into Eq(1.1) we obtain: v dv dr = GM r 2 k dt µm H dr + a2 v dv dr + 2a2 r + g rad (1.10) 13

20 Chapter 1: Theory of line-driven stellar winds Subtracting the two terms containing the velocity gradient (that is, subtracting the third term on the right-hand side from both sides of Eq(1.10)), and dividing both sides by (v 2 a 2 ), we obtain: 1 dv v dr = { GM r 2 k dt µm H dr + 2a2 r + g } { rad / v 2 a 2} (1.11) This is a general form of the momentum equation of a spherically symmetric radiation-driven stellar wind with thermal energy and momentum deposition Properties of the equation of motion In a stellar wind with thermal energy and momentum deposition, the velocity structure is determined by the momentum equation, Eq(1.11), and -via its dependence on the temperature T(r)- the energy equation, Eq(1.3), together with the apropriate boundary conditions. According to Lamers and Cassinelli (1999), the lower boundary condition of Eq(1.11) is located at a radius r 0, which is in general about the photospheric radius. It is easy to show that energy, either in the form of heat and/or momentum input, should be added to the wind in order to drive one. This can be explained by looking at the energy equation, Eq(1.3). At the lower boundary r 0 of the wind, the energy e 0 is essentially negative because of the gravitational binding of the photosphere to the star, as can be seen from Eq(1.3). However, at a large distance from the star, the energy e(r) has to be positive, if the wind is to escape from the gravitational potential well of the star with velocity v. Therefore, energy has to be added to the gas in order for a stellar wind to exist. This energy can be added in the form of thermal energy or heat deposition, Q(r) (e.g. the dissipation of sound or hydromagnetic waves in a coronal wind), and/or in the form of momentum deposition, W(r) (e.g. radiation pressure in a radiation-driven wind). The momentum deposition, W(r) r r 0 f(r)dr, results from the work done by an outward force, f(r), which, in the present case of a radiatively-driven wind, is due to the radiation pressure p rad. In Eq(1.11), the heat or thermal energy deposition is hidden in the pressure gradient via its dependence on the thermal structure, as can be seen in Eq(1.9). The equation of motion, Eq.(1.11), is a first order ordinary differential equation, which is subject to a singularity at the critical point r c, where the denominator equals zero. Therefore, in order for a smooth solution to exist, the numerator of Eq.(1.11) should equal zero as well at r c. In an isothermal wind where the radiative acceleration term g rad is negligible, or in an isothermal wind where g rad does not depend on the gradient of the velocity, dv/dr (which is the case, for example, when the radiation force is provided by electron scattering or when the driving lines are optically thin, see Lamers and Cassinelli (1999)), this implies that the critical point r c is located at the sonic point r s, i.e. where v(r c ) = a, as can be readily verified in Eq.(1.11). However, when including a radiative acceleration g rad such that g rad = g rad (dv/dr) (which is, according to Lamers and Cassinelli (1999), the case when optically thick lines contribute to the line acceleration), then the critical point r c does not coincide necessarily with the sonic point, but is generally located (slightly) more outwards, r c r s [Pauldrach et al. 14

21 1.3 The radiative acceleration (1986)]. The importance of the critical point r c lies in the fact that the requirement of a smooth single transonic solution to exist, implies that the mass-loss rate Ṁ, when assuming it only depends on L,M and T eff, is fixed, and is influenced by the location of r c. As argued by Lamers and Cassinelli (1999), Ṁ is determined by the conditions (notably: on the heat and/or momentum input) in the region between the lower boundary r 0 and the critical point r c, and is therefore not affected by a force applied in the super critical, c.q. supersonic, region. According to Vink et al. (1999), this can be explained by assuming the vdv/dr-term in Eq.(1.11) in the subsonic region to be negligible compared to the gravitational acceleration. From this it follows that the density structure in the subsonic region may be approximated by that of a static atmosphere, i.e. to be determined by hydrostatic equilibrium. When adding an extra outward force, this results in an increase of the pressure-scaleheight, which yields a slower outward decrease in density. This results, in turn, in a higher density just below the critical point (which is here assumed to coincide with the sonic point), than in case there is no extra force. From the equation of mass continuity, Eq.(1.2), it follows that the corresponding massloss rate Ṁ is higher, than in case there is no extra force in the subsonic region. This provides an explanation why Ṁ is determined by forces applied in the subsonic region, and is not affected by a force applied in the supersonic region. As was argued by Lamers and Cassinelli (1999), a force applied in the supersonic region only affects the terminal flow velocity v, and does not affect Ṁ, because information about this force cannot be communicated down to the subsonic region where Ṁ is fixed. Within this framework, it should be noted that a critical point located in the supersonic region of the wind does not imply that the flow is no longer governed by forces applied around the critical point, because of the alleged inability of communicating information upstream in the wind when in the supersonic part of the wind. This apparent inconsistency was explained by Abbott (1980), who showed, using time-dependent equations of motion, that the physical interpretation of the critical point in radiative flows is completely analogous to that of the sonic point in nonradiative supersonic flows. This is because the presence of the radiation force was found to alter the speed of sound, leading to [the introduction of the term] radiative-acoustic waves. Therefore, analogous to the sonic point in nonradiative flows, the critical point is the point farthest downstream in the wind that can still communicate with all other points in the streamline. For an in-depth discussion on this topic the reader is referred to Abbott (1980). 1.3 The radiative acceleration In the previous section, the equation of motion, Eq.(1.1), was introduced to describe the velocity structure throughout the wind. In order to solve this equation, the gradients occurring in Eq.(1.1) need to be known. So far, we have considered the pressure gradient, Eq.(1.9), which turned out to be linked to the temperature structure (and therefore, to the thermal 15

22 Chapter 1: Theory of line-driven stellar winds energy deposition), and the density structure, which in turn is linked to the velocity structure by assuming mass continuity, Eq.(1.2). In doing so, we arrived at Eq.(1.11). The only term yet to be treated in Eq.(1.11) is the radiative acceleration, g rad. In a radiatively-driven stellar wind, it is this term which provides the critical extra outward force needed to accelerate the material out of the gravitational potential well of the star. It is therefore an essential, yet challenging, task to accurately predict the g rad term in the equation of motion. According to the radiation-driven wind theory, the mass-loss and the acceleration of the wind of an early-type star are controlled by radiation pressure acting on the gas via photon scatterings and absorptions. The condition for a star loosing mass in a radiatively-driven stellar wind is that the total outward directed force, provided mainly by the radiative force, exceeds the inward directed gravitational force. The radiative acceleration g rad resulting from the gradient in the monochromatic radiation pressure p ν [in cm 3 hz 1 ] is: g rad d dm 0 p ν dν (1.12) where dm ρdz is an incremented column mass. This can be rewritten as: g rad = 1 ρc 0 χ ν F ν dν (1.13) where χ ν is the extinction coefficient [in cm 1 ], and F ν = F ν (r) is the monochromatic flux, with dimension [ergs cm 2 s 1 Hz 1 ]. The constants have their usual meaning. The value of the extinction coefficient χ ν i χ ν,i depends on the type of interactions i that take place. These types may be line or continuum interactions. We will therefore first look at the radiative acceleration due to the continuum opacity, g cont and then proceed with a consideration of grad line The continuum radiative force Deep inside the photosphere, where the temperature and density are high, the main contribution to the radiative force comes from continuum events involving free electrons, i.e. electron scattering events, and thermal absorption and emission events. rad, Below, we will shortly describe these processes. 1.) Electron scattering events In the atmospheres of early O and B-type stars, the temperature is high enough for hydrogen, which constitutes the bulk of the plasma, to be mostly ionized. Consequently, there is a large number of free electrons available for photons to scatter upon, in a process called Thomson scattering. The extinction coefficient for Thomson scattering χ e is given by: χ e = 8πe4 3m 2 e c4n e (1.14) 16

23 1.3 The radiative acceleration where n e is the electron density, and the constants have their usual meaning. As Thomson scattering is independent of frequency, χ e χ e (ν), the radiative acceleration caused by the scattering of photons on free electrons, g e, can easily be derived from Eq.(1.13): g e = χ e F ρ e c = σ e c L 4πr 2 (1.15) where σ e (r) σ e is the Thomson extinction coefficient per free particle ([σ e ] =cm 2 g 1 ), ρ e = ρ e (r) is the electron density, F the total stellar flux, L is the total stellar luminosity and c is the speed of light. 2.) Thermal absorption and emission events The thermal absorption and emission events consist of bound-free (i.e. photo-ionizations) and free-free (or bremsstrahlung, due to e.g. electron-ion interactions) absorption and emission processes. The total continuum extinction χ cont ν and free-free extinction coefficients: is the sum of the Thomson scattering and bound-free χ cont ν = χ e + χ bf ν + χ ff ν As the bound-free and free-free extinction coefficients depend on the density squared, χ bf ν ρ 2 and χ ff ν ρ 2, whereas the extinction coefficient for Thomson scattering is linearly proportional to the density (see Eq.(1.14)) χ e n e ρ, deep inside the photosphere, the continuum opacity is therefore dominated by bound-free and free-free processes. Further out in the photosphere, however, the total continuum opacity is dominated by Thomson scattering, since χ e ρ falls off less steeply with density than χ bf ν ρ 2 and χ ff ν ρ 2 do. Therefore, in the part of the envelope where the flow velocity is appreciable, the continuum opacity can be approximated to be provided solely by Thomson scattering. In early-type massive stars (O and B-type stars), generally, the outflow can not be driven exclusively by the continuum radiative force. This is because at the location in the wind where the mass-loss rate Ṁ is set, which is around the so-called critical point r c in the outer part of the photosphere (See Sec.(1.2.2)), the dominating source of continuum extinction is extinction caused by Thomson scattering on free electrons. As Eq.(1.15) shows, the radiative acceleration due to Thomson scattering, g e, displays a 1/r 2 dependence on radius. Since the gravitational acceleration displays an identical radius dependency, this implies that the ratio of the outward force caused by Thomson scattering to the inward gravitational force, Γ e g e /g Newton (1.16) remains a constant throughout the wind, in case the H/He ionization is kept constant throughout the wind. The onset of an outflow requires however, that at some radius, the outward force exceeds the inward gravitational force. Therefore, unless the star is above the Eddington limit 17

24 Chapter 1: Theory of line-driven stellar winds (Γ e 1), or there are other mechanisms at work (like, for example, pulsational instabilities), this implies that the continuum radiative force alone is not able to drive a radiation driven stellar wind. It was therefore first proposed by Lucy and Solomon (1970) that early-type massive stars may loose mass in a stellar wind due to the radiative pressure caused mainly by the transfer of momentum from the stellar radiation field to the atmospheric gas through line interactions. The radiation force caused in this fashion is termed the line force, and will be discussed in the next section The line force In the previous section it was pointed out that inside the photosphere the radiative acceleration is dominated by continuum processes. However, as we recede outwards, at some point the density is low enough to yield a (sudden) drop in opacity, which results in the formation of spectral lines. The radiative acceleration is from then on dominated by line processes. As pointed out in Sect.(1.1), the line force is the result of photons undergoing line-interactions with electrons bound in the ions of the atmospheric plasma. A line-interaction occurs when an ion intercepts photons from the stellar radiation field. If the frequency ν in of the photon in the co-moving frame (CMF) of the ion corresponds (within the range that is spanned by the width of the profile function φ( ν)) to the rest frequency ν 0 of an atomic transition of the ion, and if the lower level of that transition is occupied by an electron, the electron may get excited into the higher excited state of that transition. The excited electron may subsequently get de-excited by emitting a photon in a random direction (in the frame of the ion) spontaneously, or by colliding with surrounding ions (collisional de-excitation) or photons (induced radiative de-excitation). This way, absorption and/or emission lines may form in the stellar spectrum. The radiation force resulting from the line-interactions on one gram of gas is equal to the momentum of the absorbed radiation per second. This can be calculated from Eq(1.13), provided that we know the contribution to the line force of all possible lines. The line extinction coefficient χ line ν χ line ν [in cm 1 ] in a given line is: = k ν ρ = πe2 m e c f lun l { 1 n } u g l φ( ν) (1.17) n l g u where k ν is the absorption coefficient per unit mass [in cm 2 g 1 ], (πe 2 /m e c 2 ) is the cross-section of a classical oscillator, f lu is the quantum mechanical correction factor to the classical crosssection, which is termed the oscillator strength of the transition, n l and n u are the number density of the ion producing the line in lower level l and upper level u respectively [in cm 3 ], and g l and g u are the statistical weights of the respective levels. 18

25 1.3 The radiative acceleration In a rapidly expanding atmosphere the total line force per unit mass, which is the sum of the line forces due to all individual lines, may be represented, using the Sobolev approximation for the optical depth (see below), by: gline tot = F ν c lines ν 0 c dv dr ( 1 e τs(µ=1)) (1.18) where τ S is the Sobolev optical depth (see below), and the sum is carried out over the line forces due to both optically thick (τ s 1) and optically thin (τ s 1) lines. In deriving Eq.(1.18) the Sobolev approximation was used, which states that as long as the geometrical volume in which a line can absorb photons with a fixed frequency (the line interaction region ) is small enough, the physical conditions of the medium within that volume do not change significantly. In a rapidly accelerating outflow, because of the Doppler displacement, the line interaction region becomes small, causing the Sobolev approximation to be generally justified in stellar winds. In this approximation the optical depth at location r S in terms of the rest frequency ν 0 of the absorption line is: τ S (µ) = πe2 f l n l (r S )c m e c ν 0 { 1 n } ug l n l g u r S In the radial (µ = 1) direction, this results in: (r/v) rs 1 + (dln v/dln r 1)µ 2 (1.19) τ S (µ = 1) = (k l ρ) rs c ν 0 ( ) dv 1 (1.20) dr and in the tangential direction (µ = 0), this results in: τ S (µ = 0) = (k l ρ) rs c ( v ) 1 (1.21) ν 0 r where µ cos θ, with θ the direction angle with respect to the radial direction, and k l ρφ( ν) = k ν ρ is the absorption coefficient in a given line at r S [in cm 1 ] (for a derivation of the Sobolev optical depth and the line force the reader is referred to Lamers and Cassinelli (1999)). This simplifies the calculation of the line force greatly, because in the Sobolev approximation line scatterings in a given spectral line can only occur at one specific location, r S, along the photon path (when taking thermal and turbulent motions into account, the profile function φ( ν) is no longer a delta function. It was argued by Sobolev, that if the ion densities n u and n l change linearly along the path dz, the same result for the Sobolev optical depth will be obtained.) To calculate the line acceleration, Eq.(1.18), needed to solve the equation of motion, Eq(1.11), the contributions of all possible lines need to be summed. From Eq.(1.18) it can be seen that this requires both the radiation field and the value of τ S to be known. In turn, τ S requires the occupation numbers n l and n u of all these relevant energy levels, and the oscillator strengths f lu of the atomic transitions to be known. 19

26 Chapter 1: Theory of line-driven stellar winds If the line acceleration, gline tot (r) is known as a function of r, the equation of motion, Eq.(1.11) can be solved numerically. However, from Eq.(1.18) it can be seen that the line acceleration itself depends on the velocity gradient, dv/dr. Note that the explicit dependence of the line force on the velocity gradient entered Eq.(1.18) through the dependence of τ on dv/dr, according to Eq.(1.19). Therefore, because of the presence of the velocity gradient in τ S in gline tot, as a result, it is impossible to determine the value of gtot line a priori. This poses the main difficulty when dealing with the dynamics of line-driven winds, because it requires the line acceleration gline tot to be determined while simultaneously solving the equation of motion. g tot line 1.4 The dynamics of line-driven winds: two methods In Sect.(1.2) the equation of motion was introduced, and in Sect.(1.3) the problems with regard to the radiative acceleration in line-driven winds were outlined. In this section, we will consider two methods developed to deal with the radiative force: the CAK-theory, in which the line force is parameterized, in Sect.(1.4.1), and a Monte Carlo simulation, in Sec.(1.4.2), in which the global radiative acceleration is determined from the loss of radiative momentum, which is linked to the gain of momentum of the outflowing material CAK-theory Because of the problems involved in solving the equation of motion, as pointed out in Sect.(1.3.2), due to the presence of an unknown line force, in CAK theory the basic assumption is that the line force can be parameterized. In basic CAK theory, a radial symmetric, stationary, smooth (i.e. no shocks or clumps) onecomponent fluid flow is assumed, where viscosity and heat conduction are ignored. As pointed out in Sect.(1.1), the original CAK theory was subject to some rather unrealistic assumptions, such as the representation of the line force by C iii lines only, and the radial streaming approximation (see below) In later work by Abbott (1982) and Pauldrach et al. (1986) these assumptions were respectively dropped, by including an extensive line list and by relaxing the radial streaming approximation. The modified CAK theory is sometimes referred to as the MCAK theory, and makes use of the following approximations: 1.) Sobolev approximation In CAK theory the Sobolev approximation (see Sect.(1.3.2)) is applied. As was argued by Pauldrach et al. (1986), this may be a poor approximation in the subsonic region where the continuum is formed, and might only be valid for supersonic velocities and large velocity gradients. 2.) Core-halo approximation 20

27 1.4 The dynamics of line-driven winds: two methods In the core-halo approximation, the hydrostatic photosphere, where the continuum is formed, is assumed to be well separated from the wind, where the line acceleration takes place: δi c δr = 0 (1.22) where I c (r,µ) is the specific continuous intensity at the line frequency at location r, in direction µ. In other words, continuum formation in the wind is neglected 3. As a result of the core-halo approximation, the following assumptions are implicitly made: No feedback of the wind on the photosphere, i.e. no wind line-blanketing (which results in even larger opacity differences in a moving atmosphere (Abbott (1982))) or backwarming, due to radiation being back-scattered from the stellar wind, because in the core-halo approximation the emergent radiation field is not affected by the wind. No diffuse line radiation: CAK assumed that only the direct radiation field contributes to the line acceleration, with no contribution from diffuse radiation. In other words, all the radiation from the photosphere that is absorbed at a given point r in the wind, is assumed to be radially streaming (i.e. µ = 1 in Eq.(1.19)) from the star, and therefore the finite cone angle of the radiating photospheric surface is neglected. As was argued by Pauldrach et al. (1986), far from the star, the assumption of radially streaming photons is justified, as, at very large radii from the star, the star appears as a point source. However, close to the star, where the mass-loss rate is set, the radiation may come in at a large angle with respect to the radial direction. The so-called radial streaming approximation, or point source limit, is therefore a crude approximation close to the star, and may lead to an underestimation of the momentum deposition. This led Pauldrach et al. (1986) to drop the radial streaming approximation by introducing a correction factor, which resulted in a lower mass loss and an increased v. Constant excitation and ionization structure, expressed in the constants k, α and δ, throughout the wind. This is because in CAK theory, the ionization and excitation are only determined by the photosphere and not by local conditions. 3.) No limb-darkening In CAK theory, the effect of limb-darkening, is ignored. The assumption is therefore: δi c δµ = 0 (1.23) Together with Eq.(1.22), this implies that I c (r,µ) = I c (R ), in Eq.(1.13). 4.) No multiple scattering effects 3 Note that according to Abbott (1982), the continuum optical depth is not negligible up until far out in the wind for most Wolf-Rayet stars, Of/Wn and some Of stars. For most OB stars, observations show that the continuum optical depth of the wind is negligible. In the MC simulation, no artificial separation between the photosphere and the wind is assumed. 21

28 Chapter 1: Theory of line-driven stellar winds Each line is assumed to interact only once with unattenuated stellar continuum radiation. This results in an underestimation of the momentum deposition. In the Monte Carlo simulation discussed in the next section, multiple scattering is taken into account, and therefore, we will come back to multiple scattering in the next section. 5.) No rotation In general, massive main sequence stars tend to be rapid rotators, with equatorial rotation velocities in the range v rot = km s 1, with B-type stars rotating closest to their critical break-up speed [Howarth et al. (1997)]. Rotation can have a strong effect on the stellar atmosphere and wind. However, in basic models that describe stellar winds, which include CAK theory, rotation is generally not taken into account. A justification for the neglect of rotation in stellar atmosphere models was provided by Pauldrach et al. (1986), who show that as long as v rot is less than 200 km s 1, the effect on Ṁ and v is only marginal. They conclude that for normal O-type stars, it is therefore not necessary to include rotation into the stellar wind models. 6.) No gas pressure In the simple CAK solution (See Eq.(1.30)-(1.32)), gas pressure is not taken into account. This assumption is justified in the region above the critical point, but is no longer valid in the region around and below the critical point. However, when including the gas pressure in the CAK equation of motion, as argued by Lamers and Cassinelli (1999), this results in a marginal change in Ṁ and v. 7.) Temperature structure given by power law In the basic solution to the equation of motion in CAK theory, the temperature is allowed to depend on radius. CAK assumed that near the singular point T r n, with n between 0 and 0.5. Note that in their simplification of the solution, CAK assumed that the singular radius r c is close enough to the photospheric radius, such that the local (isothermal) speed of sound a = a(r), is small enough compared to the local escape velocity, such that a 2 GM (1 Γ)/r c. Since a 2 T, this implies that the dependency on the temperature gradient, which was hidden in the da 2 /dr-term, drops out of the CAK solution to Ṁ. Taking assumptions 1-7 into account, and using an elaborate line list of the C iii ion, CAK calculated the total line force by summing over all strong and weak lines. Because of the dependence of the Sobolev optical depth, Eq.(1.19), to both the wind (through dv/dr) and ionization (through n l and n u ) structure, CAK introduced a dimensionless optical depth parameter t that depends on the structure of the wind, via: t σ ref e v th ρ(dr/dv) (1.24) where σe ref is a reference value for the electron scattering opacity, for which CAK used the value σe ref = cm 2 g 1, and v th is the mean thermal velocity of the hydrogen ions in a wind with an isothermal temperature, equal to the effective temperature T eff of the star: 2kB T eff v th = (1.25) m H 22

29 1.4 The dynamics of line-driven winds: two methods where the constants have their usual meaning. Assuming the point source approximation, CAK showed that the total radiative acceleration g L due to all spectral lines could be expressed in terms of the radiative force due to continuous absorption by free electrons ge ref, multiplied by a factor M(t) to account for the contribution of the lines: g L ge ref M(t) = σref e L M(t) (1.26) c 4πr2 where ge ref is the radiative acceleration due to electron scattering for the reference value of σe ref, and M(t) is a multiplication factor, termed the force multiplier, which was parameterized in the following way [CAK,Abbott (1982)]: M(t) = kt α ( n e W ) δ (1.27) where n e is the electron density [in cm 3 ], t is the dimensionless optical depth, introduced in Eq(1.24), and W(r) 1/2 (1 ) 1 (R /r) 2 is the geometrical dilution factor. The term (n e /W) δ was introduced by Abbott (1982) to account in a simplified way for the dependence of the line force on the radiative ionization of elements throughout the wind. The parameters k,α and δ are the force multiplier parameters, with k being a measure for the effective number of lines, α a constant describing the distribution of strong to weak lines (α = 1[0] if only strong [weak] lines contribute), and δ a value representing the change of the ionization in the wind. Using the parameterization of the line force, Eq.(1.26), in the equation of motion, Eq.(1.11), results in a first order non-linear differential equation: ( ) v dv dr = GM (1 Γ e ) r 2 1 dp ρ dr + σ ( el 4πr 2 c k ne ) α δ Ṁ dr σ e v th W 4πr 2 (1.28) v dv where the continuity equation, Eq.(1.2), was used to eliminate ρ in Eq.(1.24), and where g rad = g e + g L, where g e is the radiative acceleration due to Thomson scattering, which obeys Eq.(1.16), and the line force g L follows from the CAK parametrization, Eq.(1.26), and Eq.(1.27) and Eq.(1.24). Note that the third term n e /W is not strictly a constant due to its dependence on r and v(r) throughout the wind. However, because δ is generally very small, this term may be assumed to be a constant, according to Lamers and Cassinelli (1999). A simple solution to Eq.(1.28) is obtained if we ignore the contribution of the gas pressure and require Eq.(1.28) to have only one unique solution. In other words, the momentum equation is required to yield only one unique velocity law, and, as a result, one unique value of Ṁ. When taking these simplifying assumptions into account, we obtain, after rearranging some terms: v dv = α GM (1 Γ e ) 1 α r 2 dr (1.29) 23

30 Chapter 1: Theory of line-driven stellar winds Integrating Eq.(1.29) from the photosphere R, where v(r ) is negligible, to some distance r, where v = v(r), a β-type velocity law is obtained, with β = 1/2: { ( α 1 v(r) = 1 α 2GM (1 Γ e ) 1 )} 1/2 R r ( = v 1 R ) β (1.30) r where v = α 1 α v esc (1.31) and v esc = 2GM (1 Γ e )/R is the effective escape velocity. From Eq.(1.30), it can be seen that v reflects the terminal velocity of the wind, as in the limit: r, Eq.(1.30) yields: v( ) v. In the CAK theory, when ignoring the gas pressure, the requirement for a single wind solution yields the mass-loss rate Ṁ in terms of the global and wind parameters, according to: Ṁ = ( σe v ) ( th σe ) ( 1/α 1 α 4π 4π α ( L )1 α α ( αk { ne W } δ ) 1/α c ) 1/α (GM (1 Γ e )) (α 1)/α (1.32) where the constants have their usual meaning. The convenience of standard CAK theory, in which the assumptions 1-7 are made, is that Ṁ and v are simultaneously determined, by solving the equation of motion self-consistently. Note that when taking the gas pressure into account in Eq.(1.28), there is no simple, single analytic solution. However, according to CAK, there is only one unique solution that has the unique property of starting subsonically at r = R, ending supersonically at r =, and going smoothly, i.e. without discontinuities, from subsonic to supersonic velocities. When requiring the solution to be this unique solution, the derived mass-loss rate and terminal velocity turn out to be very similar to the terminal velocity and mass-loss rate obtained when the gas pressure is ignored, Eq.(1.30) and Eq(1.32) [Lamers and Cassinelli (1999)]. Because of the inaccuracy introduced due to the parametrization of the line force, and the inaccuracy because of the assumptions 1-7, an alternative method was developed to deal with the dynamics of radiation-driven winds. This method, in which assumptions 2, 3, 4, 6 and 7 are dropped, utilizes a Monte Carlo technique to simulate a stellar wind model, and will be described in the following section. 24

31 1.4 The dynamics of line-driven winds: two methods Monte-Carlo simulations The Monte Carlo (MC) method was developed to simulate the cumulative effects of photon scattering and absorption processes in a stellar wind. The aim is to converge to the right solution for Ṁ, using an input model for the velocity structure throughout the atmosphere. Essentially, in the MC method the loss of radiative momentum due to photon scatterings and absorptions is registered, and from this the gain of momentum of the outflowing material is deduced. Notably, and this may seem strange, in the MC method the equation of motion is not solved throughout the entire wind. Instead, the mass-loss rate is determined in an iterative proces, using a pre-specified velocity law, that consists of a hydrostatic velocity structure in the lower (photospheric) part of the wind, that is connected smoothly to a β-type velocity law in the high velocity (wind) regime. The parameter that describes the steepness of the velocity law, β, can be given an arbitrary value, and will generally be adjusted such as to agree with observed values of β. This approach is a fairly good alternative to deal with the dynamics of radiationdriven winds, as it does not include the (disputable) assumptions made by CAK, whereas the adopted β-type velocity law, with typically, β = 1 for OB stars (Puls et al. (1996)), provides a fairly good description of the shape of the velocity field throughout the wind. For a given input mass-loss rate, the MC simulation is performed using a non-lte Improved Sobolev Approximation model atmosphere (calculated using the ISA-WIND program) as input. In the ISA-WIND program, the thermal, density and ionization and excitation structure is calculated for an extended and expanding stellar atmosphere model, in which the photosphere and the wind are treated in a unified manner. This means that the core-halo approximation is not used, that is, no (artificial) separation is introduced between the photosphere and the wind. In the photospheric (subsonic) part of the wind, the density structure is derived by solving the equation of motion for an outflow goverened by the gas pressure and the radiative pressure due to Thomson scattering on free electrons, which is the main contributor to the continuum radiative acceleration in the wind (see Sect.(1.3.1)). The line acceleration is ignored in the subsonic regime. The velocity structure in the subsonic wind region is subsequently derived from the density structure, by using the equation of mass continuity, Eq.(1.2). In ISA-WIND the equation of motion is not solved for the supersonic part of the wind, because in the supersonic region, the line acceleration, which is not known a priori, is no longer negligible, but instead dominates the radiative acceleration. Therefore, close to the sonic point, a smooth transition is made to a β-type velocity law, Eq.(1.30), where β and v are input parameters of ISA-WIND. For the temperature structure, an extended grey LTE atmosphere is used, under the assumption of radiative equilibrium. The continuum radiation transfer and rate equations are solved by means of an approximate lambda iteration method. For a detailed description of the model atmosphere code ISA-WIND, the reader is referred to de Koter, Schmutz, & Lamers (1993) and de Koter (1993). The calculated wind models are subsequently used as input into a program called MC-WIND (de Koter et al., 1997), that utilizes a Monte Carlo method, based on the Monte Carlo program of Abbott & Lucy (1985), 25

32 Chapter 1: Theory of line-driven stellar winds Figure 1.2: The path of a photon in a multiple-scattering process. On its way out through the stellar atmosphere, the photon undergoes multiple scatterings, both line and continuum, thereby depositing both energy and momentum.figure taken from Abbott & Lucy (1985). to keep track of the total momentum transfer from the radiation field to the gas particles that constitute the stellar wind. In the MC-WIND program a fixed number of photons 4, are shot in a random, but outward, initial direction, with a random, but weighted with the probability density function [pdf], frequency. The photons are followed on their way from below the photosphere, where they are emitted, throughout the photosphere and wind. On their way through the atmosphere, which is devided in n discrete concentric shells 5, the net energy passing through each shell is being tracked. Figure (1.2) depicts a photon that undergoes multiple interactions with the particles that make up the wind, thereby depositing energy and momentum in the wind. As the initial energy of the emitted energy packets, E packet, is the same for all packets, it is the different number and kinds of interactions that may occur within each shell, which determine the energy flowing out of each shell and into each subsequent shell. The type of interaction that occurs for a given photon (since an energy packet consists of N identical photons, for the sake of brevity, here we shall refer to the energy packets as photons ), emitted at the first shell with a given (random) frequency, and in a given random direction, depends on the (also randomly) chosen optical depth τ ran it is to traverse. Basically, τ ran is compared to the total optical depth along a photon s flight path, τ tot, which is given by: τ tot = τ cont + τ L (1.33) where τ cont is the continuum optical depth and τ L is the line optical depth. Because of the 4 Rather: energy packets of equal energy representing a number of photons that undergo identical interactions 5 Typically, the wind is divided into 60 concentric shells 26

33 1.4 The dynamics of line-driven winds: two methods Sobolev approximation applied, line events can only occur at specific points (the resonance location s L of a given line) along a photon s flight path. Continuum events, on the other hand, may occur at any distance, s c. The probability of a continuum event occuring increases linearly with distance within a given shell. Given these dependencies, the Monte Carlo procedure determines whether a line or continuum event occurs at τ ran. The energy outflow of a shell is not necessarily the same as the energy input, because the photons may be scattered, absorbed and re-emitted, or eliminated because they are scattered back into the star. Because of the expanding atmosphere, each photon is red-shifted in the frame co-moving with the ions in the wind, the CMF: ν CMF in ν CMF out = ν obs in = ν obs out ( 1 µ ) inv ( c 1 µ ) outv c (1.34) where νin CMF and νout CMF are the CMF frequencies of the incident and emerging photon, νin obs and νout obs are the incident and emergent frequencies for an outside observer, µ in and µ out are the photon s incident and emergent direction cosines at the scattering point, and v = v(r) is the local radial bulk velocity, of which the change in velocity per scattering ( v 10cm s 1 ) is assumed to be negligible compared to the flow velocity (v 10 3 km s 1 ). In the MC method, the lines are described in the Sobolev approximation, which implies that ν CMF in where ν 0 is the rest frequency of the line transition. = ν 0 = ν CMF out (1.35) Therefore, for an outside observer, in each scattering event the photons loose a fraction of their energy, E = hν, due to the change in radial momentum, p = hν/c, when the photon interacts with an ion. Substituting Eq.(1.34) into Eq.(1.35), and multiplying with h/v, yields the change in photon energy in terms of the change in radial momentum of the photon under consideration (for an outside observer): E in E out v = p in µ in p out µ out (1.36) In the Monte Carlo approach, it is used that the radiative energy lost by the photons through the various possible interactions, E = E in E out, is equal to the increase of the radial momentum p = m v of the ions, moving with mass m and velocity v in the shells. Using conservation of ion momentum, where v 1 and v 2 are the radial velocities of the ion just before and after the scattering mv 1 + hνobs in µ in = mv 2 + hνobs out µ out (1.37) c c together with Eq.(1.34) and Eq.(1.35), the increase in the radial momentum of the ion under consideration (for an outside observer) is obtained by rewriting Eq.(1.37): m(v 2 v 1 ) = hν 0 c (µ in µ out ) = hνobs in hνobs out v = E v (1.38) 27

34 Chapter 1: Theory of line-driven stellar winds where it was used that v c. Using that v (v 1 +v 2 )/2, because for each scattering v 2 v 1, and rewriting Eq.(1.38) shows that the gain of kinetic energy of the ions equals the energy loss of the photons: 1 2 m(v2 2 v2 1 ) = hνobs in hνobs out (1.39) Therefore, the total radiative acceleration of the wind g rad (r) per shell can be calculated from the loss of photon energy due to all scatterings within each shell : g rad (r) = 1 m(r) E(r) v(r) t (1.40) where E(r) is the energy lost by the photons, which is equal to the kinetic energy gained by the ions in the shell with radius r, containing a mass m(r). Using that for thin concentric shells m(r) = 4πr 2 ρ(r) r, and that E(r)/ t = L, and that mass continuity, Eq.(1.2), holds, yields for g rad : g rad (r) = 1 L(r) (1.41) Ṁ r We are now in a position to derive an expression that actually links the total energy gained by the wind material to the energy lost by the radiation field. This can be done by simply substituting the derived expression for g rad, Eq.(1.41), into the equation of motion, Eq(1.11), with the contribution of the gas pressure set to zero. This is justified, because in the supersonic region, the acceleration is dominated by the radiative acceleration, rather than the acceleration due to the gas pressure. Integrating the equation of motion from the stellar surface to infinity, yields: 1 2Ṁ(v2 + v2 esc ) = L (1.42) Eq.(1.42) shows that in the MC simulation the total energy gained by the wind material, to lift the material out of the potential well of the star and accelerate it to v, is equal to the energy lost by the radiation field, L. Since v and v esc are input values in our model, this means that the value of Ṁ can be obtained from Eq.(1.42), once L = shells L shell is known from the simulation. The mass-loss rate Ṁ = Ṁout obtained in this fashion, needs to be improved iteratively. Therefore, the unique value of Ṁ is determined by converging until the input value of Ṁ = Ṁin implies an efficiency of wind driving such that Ṁ in = Ṁout. Note that this method need not be locally consistent, because the velocity structure v(r) was not obtained by solving the equation of motion throughout the entire wind, but by adopting a β-type velocity law in the supersonic region, and by solving the equation of motion with no line pressure in the subsonic region. The use of a Monte Carlo simulation to predict mass-loss rates in this way, has the advantages that multiple scatterings are taken into account, and that an artificial separation between the stellar photosphere and wind ( core-halo separation ) is avoided. 28

35 CHAPTER 2 Decoupling In this chapter we examine the mechanism by which absorbing ions may decouple from nonabsorbing ions, and introduce the condition that we will use in this research to test whether a given ionic species is decoupled. In the previous chapter the basic theory of line-driven winds was introduced. It was pointed out that the winds of hot O and B-type stars are driven by the radiation force that acts primarily on minor, notably heavy, metallic ions, such as C,N, and O and Fe-group elements, because of their numerous available atomic transitions. These ions are therefore termed the active ions. If the line force in the expanding atmosphere is to be effective in producing a steady outflow of the entire plasma, the momentum gained by the active particles must be shared with the surrounding field particles (i.e.,protons, helium nuclei and electrons). The momentum exchange between the various species occurs via Coulomb collisions, which causes the passive plasma to be dragged along outwards with the active plasma. In the standard theory of line-driven winds the active ions and the passive plasma are assumed to make up a single fluid, that is, they are assumed to be well-coupled. In this research we verify whether the assumption of a well-coupled wind is justified. One simplyfing assumption in the standard theory of stellar winds is the assumption of a one-component single fluid, despite the fact that the radiative pressure acts primarily on minor metallic ions. In their fundamental paper Lucy and Solomon (1970) argued that the assumption of a single fluid is justified, by showing that the velocity difference between their absorbing C iv ions and the bulk plasma is sufficiently small for the assumption of a single velocity for all particle species throughout the wind, to be valid. A similar conclusion was reached by Lamers and Morton (1976), who showed for ζ Puppis that the momentum transfer between active ions and passive plasma is efficient enough to accelerate the entire plasma outwards. However, their calculations were performed on O-type stars with dense winds. Springmann and Pauldrach (1992) proposed that the wind could be considered as a multi-component gas for stars cooler than O-type stars, essentially, stars with less dense winds. 29

36 Chapter 2: Decoupling Because of the uneven distribution of the line force over the individual ions, the multicomponent nature of stellar winds has received considerable attention by several authors [e.g, Springmann and Pauldrach (1992); Babel (1995); Krtička and Kubát (2000)]. Some multicomponent effects considered include frictional heating. According to Springmann and Pauldrach (1992) this may arise in low-density winds because the collisional momentum transfer between absorbing and non-absorbing ions is accompanied by the production of entropy in the form of frictional heating. Springmann and Pauldrach (1992) proposed that in thin, that is, lowdensity winds, a decoupling of the active ions from the passive bulk plasma may occur.as a result of this decoupling, the momentum transfer from ions to the bulk plasma is terminated, and a lower terminal bulk velocity is obtained. According to Krtička and Kubát (2000) in the case of extremely low-density winds, the absorbing ions may be unable to accelerate the passive component of the wind sufficiently, and the passive component may even fall back onto the stellar surface, resulting in a purely metallic wind. The consensus is that multicomponent effects may only be important in low-density stellar winds. In short, the condition for ion-decoupling is that the time t D required to accelerate an absorbing ion due to momentum gain from photons, if there are no interactions with the passive particles, to a radial drift velocity that is equal in magnitude to the thermal velocity v th,p of the passive plasma, is shorter than the time t S needed to slow down the absorbing ion from w v th,p to zero drift velocity due to Coulomb collisions with the passive particles (without being accelerated by the radiation field in the meantime): t D < t S In the next sections we describe the mechanism of decoupling. 2.1 Decoupling mechanism The momentum transfer between the active ions and the passive bulk plasma occurs via Coulomb collisions. Because of the long range of the electrostatic force between the colliding partners, we must consider distant encounters, each of which produce small deflections, rather than close collisions, which may completely change the path of the particle. This is because at any time, a charged particle interacts electrostatically with all the other plasma particles inside a sphere of radius λ Deb (see below). As was pointed out by Spitzer (1962), because of the large number of particles within a Debye sphere, the cumulative effect of many distant small-angle collisions is often larger than the effect of the few close large-angle collisions that may take place. Because the Coulomb force depends on the distance as 1/r 2, the trajectories of the colliding particles in the center of mass frame are hyperbolae, which can be characterized by an impact parameter. At large distances the charge of the ion is shielded by the natural tendency of electrons to cluster in the neighborhood of the ion and other ions to avoid it. The upper 30

37 2.1 Decoupling mechanism cut-off to the impact parameter is therefore approximately equal to the Debye length: kb T λ Deb = 4πn e e 2 (2.1) which is the typical length scale over which the potential of a given ion decreases with a factor of 1/e due to the screening effects of the surrounding electrons [Spitzer (1962)]. In Eq.(2.1) λ Deb is in units of cm, T is the kinetic temperature of the plasma [in K], n e is the electron density [in cm 3 ], and the constants have their usual meaning. The lower cut-off to the impact parameter b 0 is the distance of closest approach [in cm] in a close encounter that produces a deflection of 90, which follows from Coulomb s law: b 0 = Z iz f e 2 mw 2 (2.2) where Z i is the charge of the active ion under consideration, and Z f denotes the charge of the field particles, that is, mostly protons and electrons, with which the ion interacts. Here m denotes the mass of the active ion [in g] and w is its velocity [in cm/s] in the reference frame of the field particles (i.e., the wind). The factor by which small-angle collisions are more effective than large-angle collisions in producing deflections is termed the Coulomb logarithm ln Λ, with Λ given by [Spitzer (1962)]: Λ = λ Deb b 0 = 3 2Z i Z f e 3 ( k 3 B T 3 ) 1/2 (2.3) πn e Here the parameters are as defined before. Using that mw 2 k B T, it follows that in a typical stellar wind (with n e 10 8 cm 3 and T 10 4 K), the Coulomb logarithm is of the order of 10. In general, in passing through a field of ions a charged test particle will undergo many encounters, with various impact parameters and directions. The cumulative effect can be described as a diffusion of radial momentum of the test particle. The rate at which charged test particles are slowed down in an ionized plasma by encounters with field particles was derived by Spitzer (1962), who used the analogy with Brownian motion, which arises from the inverse square character of the electromagnetic force between the colliding bodies: encounters with small values of the impact parameters (which produce appreciable deflections) are uncommon and encounters with large impact parameters, which occur more frequently, are very ineffective. In analogy with Brownian motion, it is therefore the cumulative effect of a large number of encounters, each of them individually having only a marginal effect, which may produce significant changes in the directions and magnitude of the motions, causing in this way a diffussion of radial momentum. In order to change the velocity of a test particle significantly, a very large number of encounters is needed. However, the presence of the electrical force acting between the particles introduces at the same time one essential difference with respect to the Brownian case, since in the latter case the particles are primarily influenced by the Van der Waals interaction force, which falls off as 1/r 6, between the molecules of the surrounding 31

38 Chapter 2: Decoupling fluid [Chandrasekhar (1949)]. We consider a group of ions with mass m moving with an initial velocity w in the rest frame of the field particles in an ionized plasma. The field particles with mass m f move about with a random thermal speed v th in their reference frame. As a result of successive encounters, the velocity distribution is progressively broadened. According to Spitzer (1962) the corresponding diffusion coefficient w [in cm/s 2 ] is given by: ) w = A D v 2 th (1 + mmf G(w/v th ) (2.4) where v th is the thermal velocity of the field particles [in cm/s], which are in our consideration essentially the hydrogen ions, m f is the mass of the field particles [in g], which is essentially the proton mass, and A D is the diffusion constant [in cm 3 /s 4 ], which is defined by [Spitzer (1962)]: A D = 8πe4 n f Z 2 Z 2 f ln Λ m 2 (2.5) Here n f is the density of the field particles, Z i is the charge of the ion under consideration, Z f is the charge of the field particles, and ln Λ is the Coulomb logarithm, which was defined in Eq.(2.3). The function G(x) is the Chandrasekhar function, which is defined in terms of the error function Φ(x) according to [Chandrasekhar (1942)]: G(x) = Φ(x) xφ (x) 2x 2 (2.6) According to Spitzer s definition of the diffusion coefficient, the argument of the Chandrasekhar function is x = w/v th. Equation (2.4) describes the rate at which moving ions are slowed down in the direction parallel to the original direction of motion, by interactions with the field particles 1. Note that Spitzer (1962) also introduced the quantities ( w ) 2, and ( w ) 2 which describe, respectively, the rate at which ( w) 2 (the energy) increases in the direction parallel and perpendicular to the original direction of motion due to successive encounters. As we are interested in the rate at which the absorbing ions share their momentum with the non-absorbing plasma, we only consider the rate w at which the absorbing ions (our test particles) are slowed down by interactions with the non-absorbing plasma (our field particles). Since diffusion of momentum implies that energy is being transported, the diffusion equa- 1 Eq.(2.4) was first introduced by Chandrasekhar (1942), and is proportional to Chandrasekhar s coefficient of dynamical friction, η [in units of 1/s]. It was introduced to account for the invariance of the Maxwellian velocity distribution of a system of stars to the stochastic variations in the velocity which a given star suffers as a result of successive encounters with field stars. The stochastic process was therefore made conservative, meaning that the Maxwellian velocity distribution of the field stars is left unchanged, by the introduction of dynamical friction. 32

39 2.1 Decoupling mechanism 0.25 Chandrasekhar function G(x) 0.25 Chandrasekhar function G(x) G(x) G(x) factor x/g(x) in slow down time 1e+07 1e+06 factor x/g(x) in slow down time x/g(x) x/g(x) relative drift velocity x relative drift velocity x Figure 2.1: Upper panel: The Chandrasekhar function G(x) as a function of x. In our consideration x w/v th, where w is the radial drift velocity of the active ion, and v th is the thermal velocity of the passive plasma [in cm/s]. When the wind is well-coupled, w v th, and an increase in drift velocity between the active and the passive plasma enhances the collisional drag force, which tends to lower the drift velocity. In case w v th the increase in drift velocity w lowers the drag force, which yields progressively larger velocity differences, if the active ions also absorb radial photon momentum in the meantime. This effect is termed ionrunaway. Lower panel: Behaviour of the factor x/g(x) that occurs in the slow down time t S. Left: linear scale, as G(x) is often encountered in the literature. Right: the Chandrasekhar function on a logarithmic scale. From the lower left panel it is seen that in the regime with w < v th the slow down time is approximately a constant with respect to drift velocity. This means that in this regime the Coulomb friction force is approximately linear as a function of drift velocity. tion is essentially analogous to the heat equation 2. In a stellar wind the diffusion of radial ion momentum due to collisions with the passive plasma is therefore analogous to the randomizing of part of the kinetic energy with which the active ions are moving in the radial direction because of the differential streaming of the particles. This leads to the introduction of the term frictional heating by many authors [e.g., Springmann and Pauldrach (1992)]. The frictional force density R if of species f on species i as defined by Springmann and Pauldrach (1992) is equivalent to the coefficient of dynamical friction w as introduced by Spitzer (1962). Note that the net energy transfer from the active ions to the passive plasma is in the radial outward direction, which causes the passive plasma to undergo a net acceleration in the radial direction, and be dragged along outwards in a stellar wind. The behaviour of the Chandrasekhar function in Eq.(2.4) is illustrated in Fig.(2.1). In the limit of small x, that is, small drift velocity, w v th, G(x) can be approximated in a first 2 More precisely, the diffusion coefficient q 1/η [in cm 2 /s] is the proportionality factor that relates the flux j of the diffusing material to the local density gradient, φ( r, t) (Ficks law). The diffusion equation then simply follows from the continuity equation: where φ( r, t) is the density of the diffusing material at location r. φ + (q φ) = 0 (2.7) t 33

40 Chapter 2: Decoupling order Taylor expansion as: G(x) 2x 3 π (2.8) which implies that the diffusion coefficient, Eq.(2.4) is proportional to the relative drift velocity w/v th. This implies that in the reference frame of the wind the Coulomb force between the active ions and the passive ions (which acts as a frictional force between the different species) increases with increasing radial drift velocity, as long as w < v th. The Chandrasekhar function reaches its maximum value for x 1, which means that the Coulomb coupling between the ions i and the field particles is maximal if the drift velocity w of i approaches the most probable thermal velocity of the field particles. This means that in the regime where the ions move with a radial drift velocity that is equal in magnitude to the random velocity of the surrounding passive particles, the radiative acceleration on the active ions i is maximally balanced by the Coulomb force on i because of interactions with the field particles, f. The field particles, f in return, are maximal accelerated. For drift velocities larger than the thermal velocity, x 1, the velocity distribution functions of i and f no longer overlap in velocity space, and the active ions are said to be decoupled from the passive plasma. In this regime the frictional force G(x) decreases as: lim G(x) 1 x 2x 2 (2.9) In this case the radiative driving force on the active ions i can no longer be balanced by the frictional response of the system. In other words, because of the vanishing Coulomb (frictional) force, the active ions are progressively accelerated by the radiative force. As a result, the time needed to slow down the particles due to Coulomb interactions becomes progressively larger. This is therefore termed the runaway-regime. To summarize, for small drift velocities, w < v th, ion i is coupled to j as an increase in the relative velocity between the two species causes an increase of the momentum exchange between the ions. In the domain where w > v th, an increase in relative velocity causes a reduction of the collisional friction, and the two species decouple. This means that the active ions i can accelerate freely and run away, whereas the bulk plasma is deprived from further acceleration by species i. Therefore, the condition for decoupling is that the time scale needed to accelerate the active ions to a radial drift velocity w equal in magnitude to v th of the passive plasma, is shorter than the time scale needed to slow down the active ions from w = v th to zero drift velocity by Coulomb collisions with the passive particles. 34

41 2.2 Slow-down time 2.2 Slow-down time According to Spitzer (1962) the time t S it takes to slow down the parallel velocity component of a charged test particle in an ionized plasma by encounters with field particles is: t S = w (dw/dt) = w < w > (2.10) where for the slow-down rate dw/dt the diffusion coefficient < w > according to Eq.(2.4) is used. In analogy with Spitzer (1962), the slow-down time is obtained by substituting Eq.(2.4) and Eq.(2.5) in Eq.(2.10): t S = 2k B 8πe 4 T f m 2 n f Z 2 Zf 2 ln Λ m f + m w G(w/v th ) (2.11) Here the slow-down time is in seconds and all the other parameters are as defined before. The left lower panel of Fig. [2.1] shows that the factor w/g(w/v th ) in t S is approximately independent on the velocity w in the velocity range 1cm/s < w < v th. This means that in this velocity range t S (w) t S, that is, in this velocity range the slow-down time is approximately independent on the drift velocity w. Since we are interested in the time needed to slow down particles from a drift velocity equal to the thermal velocity v th of the field particles to zero drift velocity, this means that we may approximate Eq.(2.11) by considering the limiting case w v th of Eq.(2.8) to obtain an approximation for the slow-down time in our velocity domain of interest: t S 3 π 2 (2k B ) 3/2 8πe 4 3/2 Tf m 2 n f Z 2 Zf 2 ln Λ m 1/2 f (m f + m) (2.12) This is the time scale needed to decelerate the active ions from w v th to w = 0 cm/s, without absorbing photons in the meantime. Accordingly, we will define the drift time as the time needed to increase the drift velocity from w = 0 to w = v th without interacting with the field particles in the meantime. We assume that the effect on the radial momentum of the active ions of collisions with the electrons can be neglected. This is because the path of the ion is not expected to be deflected much by encounters with electrons. Also, collisions between active ions of species i with active ions of species j can be neglected, as the density of active ions is much smaller than that of hydrogen and helium. As a result, encounters between active ions will be very rare and therefore negligible for all practical purposes. Thus, we consider only encounters between active ions and passive plasma. 35

42 Chapter 2: Decoupling 2.3 Drift time For the drift time, we use the time it takes the active ion to absorb enough photons to be accelerated to the most probable thermal speed of the field particles. According to Lamers and Cassinelli (1999), per absorption the momentum transfer from a photon to an ion is: dp dt = πe2 m e c f osc F ν0 c (2.13) where p = mv is the momentum of the ions, with mass m and velocity v (in the reference frame of the wind). The cross-section for absorption is (πe 2 /m e c)f osc [in units of cm 2 ], where f osc is the oscillator strength of the atomic transition under consideration [which is dimensionless]. The stellar flux at the frequency ν 0 of the transition is F ν0 = F ν0 (r). Rewriting Eq.(2.13) yields the time needed to accelerate the ion by a given amount dv: ( ) πe 2 1 c dt = dp (2.14) m e c f osc F ν0 In the wind the absorbed photons come from the radial direction, but the re-emission is in a random direction in the frame of the atom. Therefore the average effect when summed over all possible angles in which the photon can be re-emitted, is a mean momentum transfer that is about the same as for the case of pure absorption of the photon [for a derivation, see Lamers and Cassinelli (1999)]: mv = hν 0 (2.15) c The net effect of a radial radiation field is that the absorbing ion gains a radial drift velocity w with respect to the bulk plasma. Substituting Eq.(2.15) into Eq.(2.14) yields the average timescale t A needed to accelerate to w: ( ) πe 2 1 hν 0 t A = (2.16) m e c f osc F ν0 We are interested in the time it takes to accelerate the ion to a radial drift velocity equal to the thermal velocity of the surrounding field particles (i.e., of the protons) by photon absorptions. Therefore we define the time t D as the time needed to accelerate the absorbing ions to a drift velocity equal in magnitude to the most probable thermal speed v th of the bulk plasma by photon absorptions, without interacting with the field particles in the meantime: t D = mv v th hν 0 /c = v th ( hν0 mc = v th ( πe 2 m e c 2 t A ) 1 ( ) πe 2 1 hν 0 m e c ) 1 m f osc F ν0 f osc F ν0 = v th g i (2.17) where v th is, for all practical purposes, the thermal velocity of the protons. Here g i is the net radiative acceleration acting on the absorbing ion that yields mv = hν 0 /c. 36

43 2.3 Drift time Condition for a drift velocity The absorbing ion can obtain a drift velocity with respect to the field particles if the average time between two pure absorptions t A is smaller than the time t S it takes to slow it down to zero drift velocity, and if, while accelerating to a radial drift velocity w, in the meantime, the wind itself has not accelerated by an amount equal to that drift velocity (which would set the drift velocity of the test particle to zero again). Therefore, in order to obtain a drift velocity with respect to the reference frame of the field particles, the following conditions must be satisfied: t A < t S t < t wind (2.18) where t = t (w) is the time it takes for an absorbing ion to be accelerated to a drift velocity w and t wind = t wind (δw) is the time to accelerate the field particles [i.e., the wind] to w. The average time between two absorptions is estimated from Eq.(2.16), and is typically of the order sec, for strong lines, with f osc = 1 and ν 0 = ν max, which is the frequency at the fluxmaximum F ν0 (r) = F νmax (r) of a star with T eff K. If the ion is far out in the wind, at a distance of, say 10 stellar radii, it takes a longer average time ( 10 6 sec) to accelerate the ion in the radial direction via pure absorptions because of the lower flux at large distances. Close to the star, the acceleration is more efficient 3, and therefore it requires less time ( 10 8 sec) to be accelerated per pure absorption. The slow-down time t S is calculated from Eq.(2.12) with T K, n f 10 8 cm 3, ln Λ 15, Z f 1 and Z 3, and is of the order of 10 2 sec. This estimate shows that the criterion that t A < t S is satisfied for a typical O-type star. This means that the ion can get a drift velocity w in the reference frame of the wind. Whether this drift velocity is able to grow to the typical thermal speed v th of the gas, depends not only on the ratio between t S and t D (our decoupling condition), but also on the ratio between the typical time t = t D needed to accelerate the ion to v th, and the time t wind needed to accelerate the radial velocity of the wind by an amount equal to v th. Therefore we estimate whether the same ion as before can be accelerated to the thermal velocity. The second condition in (2.18) is then: t D < t wind (v th ) For a normal O-type star we can estimate the acceleration time t wind of the wind from: t wind (v th ) = v th (dv/dt) wind s (2.19) 3 Note that close to the photosphere the radiation field is more isotropic. Since line scatterings are effective in the wind rather than in the photosphere, we assume a radial radiation field, i.e., with the assumption of pure absorption 37

44 Chapter 2: Decoupling where close to the star the acceleration time is short, and further out it is progressively larger. We estimate the value of dv/dt by assuming a beta-type velocity law and taking the time derivative: dv(r) dt = d dt { (v 1 R }) = v R r r 2 v(r) (2.20) From equation (2.17) we obtain an approximation of the drift time of t D sec, which is long far out in the wind, at 10 stellar radii, and short deep in the wind. Since the time needed to accelerate the wind is much longer than the time needed to obtain a drift velocity w v th, our condition that t D < t wind (v th ) is met and therefore with respect to this condition, the absorbing ion can gain a drift velocity equal to v th. In the meantime the wind is not accelerated by the same amount. Whether the ion will be able to gain a drift velocity equal to the most probable thermal velocity of the bulk plasma before it is slowed down thus depends only on our decoupling criterion, and this will be investigated in the next chapters. 38

45 CHAPTER 3 Semi-Analytical Investigation Before we start our numerical search for ion-decoupling across the HR diagram, we scan the parameter space for decoupling, using estimated properties for the radiation field and ion structure. In this chapter we provide a description of how the drift time and slow-down timescale were transformed into a condition that could be used to investigate in a semi-analytical way the occurence of ion-decoupling across the HR diagram. This condition contains the global stellar and wind parameters on the one hand and the estimated local wind and ion structure on the other hand. In Chapter 4, the outcome of our semi-analytical search will be presented. 3.1 Semi-analytical condition To investigate in a semi-analytical way where in parameter space we may expect decoupling of a given active ion (in other words, in what part of parameter space t S > t D occurs), we made some assumptions regarding some of the variables in Eqn(2.11) and Eqn(2.17), which we repeat here for the sake of clarity: Slowing Down time The slowing down time according to Spitzer (1962) is: 1 t S = (1 + m/m f ) vth 2 A D w G(w/v th ) where A D is the diffusion constant, which was defined in Eq.(2.5).The subscript f denotes the field particles that make up the passive plasma. Since the passive plasma consists mainly of hydrogen and a fraction of helium, for the moment we will assume the passive plasma to consist solely of hydrogen. All the other parameters are as defined. Introducing w w/v th, the slow down time can be rewritten as: 1 vth 3 w t S = (1 + m/m f ) A D G(w ) (3.1) (3.2) 39

46 Chapter 3: Semi-Analytical Investigation where in our domain of interest w typically varies between 0 and 1. Drift time The drift time according to Lamers and Cassinelli (1999) is: t D = v th g ion (3.3) where v th is the thermal velocity of the field ions in the plasma and g ion is the radiative acceleration of a given absorbing ion with mass m at location r in the wind: g ion = 1 m where the parameters are as defined in Sec.(2.3). πe 2 m e c 2f oscf ν0 (r) (3.4) Decoupling condition The condition for decoupling is that the slow down time t S exceeds the time t D needed to accelerate a given particle to a velocity typically equal to the thermal velocity v th of the field particles: t S t D > 1 (3.5) Substituting Eq.(2.5), Eq.(3.2), Eq.(3.3) and Eq.(3.4) in Eq.(3.5), results in the following condition for decoupling: t S t D = 1 πe 2 8πe 4 m e c 2 Am amu vth 2 w (1 + A/A f ) n f Z 2 Zf 2 ln Λ G(w ) f oscf ν0 (r) > 1 (3.6) where m = Am amu, with A the atomic mass of the active particle under consideration, and m amu is the atomic mass unit [in units of g]. The free parameters in this condition are the atomic mass A and charge Z of both the active ion and the field particles, the wind temperature T (via the thermal velocity v th ) and the density of the field particles n f, the drift velocity w of the active particle, the oscillator strength f osc of the transition, and the flux F ν0 at the line frequency. The density n f (and therefore, the Coulomb logarithm ln Λ) and flux F ν0 in Eq.(3.6) can be estimated from the following considerations, as listed below. Since the wind consists mainly of hydrogen and helium (i.e., the passive plasma), the density n f of the passive plasma can be derived from the continuity equation: n f n(r) = ρ(r) µ a m amu = 1 µ a m amu Ṁ 4πr 2 v(r) where n(r) is the nucleon density, µ a is the mean atomic weight in amu, Ṁ is the mass loss rate and v(r) is the velocity of the wind. We assume a β-type velocity law: ( v(r) = v 1 R ) β (3.7) r 40

47 3.1 Semi-analytical condition where R is the photospheric radius of the wind-free star, v is the terminal velocity of the wind, and the exponent β(> 0) is a parameter that describes the steepness of the velocity law. The terminal velocity v and (1 R /r) β are kept variable in the criterion for decoupling (see below). The Coulomb logarithm Λ is defined according to Spitzer (1962) (see Sec.(2.2)): Λ = 3 2ZZ f e 3 ( k 3 B T 3 e πn e ) 1/2 where T e is the electron temperature, and n e is the electron number density. We assume the star emits a black body spectrum F ν0 = F ν0 (r) = F ν0 (R ) ( ) 2 R r with F ν0 (R ) = πb ν0 (T eff ), where B ν0 (T eff ) is the Planck function We incorporate these assumptions in Eq.(3.6) to obtain: t S t D = A µ a m 2 amu 1 + A/A f 2m e c 2 e 2 v R 2 Ṁ ( 1 R r πv 2 th Z 2 Zf 2 ln Λ ) β w G(w ) f oscf ν0 (R ) > 1 (3.8) Substituting the thermal velocity of the field particles, with mass m f and thermal energy k B T yields: t S t D = A µ a m amu 1 + A/A f m e c 2 e 2 v R 2 Ṁ ( 1 R r πk B T Z 2 Zf 2 ln Λ ) β w G(w ) f oscf ν0 (R ) > 1 (3.9) We rewrite Eq.(3.9) such that the global stellar and wind parameters v,r and Ṁ are on the left-hand side, and the local wind and ion parameters on the right-hand side: Ṁ v R 2 = t D t S A µ a m amu 1 + A/A f m e c 2 e 2 ) β w ( 1 R r πk B T Z 2 Z 2 f ln Λ G(w ) f oscf ν0 (R ) (3.10) Since the condition for decoupling is that t S > t D, we can also write Eq.(3.10) as Ṁ v R 2 < µ am amu πk B T A m e c 2 e 2 Z 2 Zf 2 ln Λ 1 + A/A f ( 1 R ) β w r G(w ) f oscf ν0 (R ) (3.11) 41

48 Chapter 3: Semi-Analytical Investigation Substituting the values of the constants m amu, m e, c, e and k B, and rewriting in units of solar masses per year and solar radii, yields the condition for decoupling: Ṁ[M /yr] R 2 [10R ]v [10 3 km/s] A T < C µ a 1 + A/A f Z 2 Zf 2 ln Λ ( 1 R ) β w r G(w ) f oscf ν0 (R ) (3.12) where the constant C = Note that the r 2 dependence has dropped out due to the dependence of both n f and F ν0 on r 2. The condition is not independent on r however, because of the r-dependence of the velocity law assumed. However, for our parameter search we will merely focus on layers near the critical point, that is, we keep r fixed at a constant value in Eq.(3.7) (for example v(r) 0.1v ). We do not know the drift velocity w of the active particle a priori. However, our aim is to investigate in which part of the parameter space the assumption of a single fluid is no longer valid. Therefore we need to investigate our condition for decoupling in the velocity domain of the particle when it is still effectively coupled to the passive plasma. This means that the drift velocity satisfies 0 < w < v th, or equivalently 0 < w < 1. In this velocity domain, we can approximate the fraction w /G(w ) as follows: w G(w ) lim w 0 ( w G(w ) ) 3 π 2 (3.13) When for a given particle the condition Eq(3.12) is in favor of decoupling, the drift velocity of the particle will exceed the thermal velocity of the gas, and we can no longer use this approximation for the slowing down time (since the particle in question is already decoupled, and therefore, essentially w 1). When investigating Eq.(3.12) for a wind with a given temperature, chemical composition and ionization, we find that the free parameters are essentially: A parameter Q containing the global stellar and wind parameters v,r and Ṁ, that is proportional to the mass density: Q Ṁ[M /yr] R 2 [10R ]v [10 3 km/s] If the value of Q increases, the medium is denser and therefore can be strongly coupled. A parameter proportional to the oscillator strength times the flux of driving photons (per second and per cm 2 ): f osc F ν0 (R ) If the value of f osc F ν0 (R ) decreases, the medium can be strongly coupled, since for weak line forces active particles are not as easily accelerated relative to the passive medium. 42

49 3.2 Estimate of Q and Px Writing the latter variable as a fraction ǫ of the flux maximum F νmax : F ν0 (R ) = ǫ F νmax (R ) where the flux maximum F νmax (R ) can be found by assuming the stellar spectrum is just the Planck function B ν (T eff ). Using this approximation and writing F νmax (R ) = πb νmax (T eff ) we find for the condition to be implemented: Q < C A T µ a 1 + A/A f Z 2 Zf 2 ln Λ ( 1 R ) β πb νmax(t eff) (ǫ f osc) r where the constant C = Collecting all parameters in the above expression in a single parameter P, the condition for decoupling becomes: Q Px < 1 (3.14) where x ǫ f osc is a free parameter, which typically runs from 10 6 to Estimate of Q and Px Our aim is to determine semi-analytically the position in the HR diagram where decoupling of a given active atomic species occurs on the basis of Eq.(3.14). Therefore, we verified the condition for decoupling, Eq.(3.14), for a set of stellar evolutionary models, taken from Schaller et al. (1992), at solar metallicity. For each initial mass 51 models are given to describe the evolutionary tracks. In this section, we describe how we estimated the values of the stellar and wind parameters required to calculate Q and P. The parameter Q in Eq(3.14) was calculated for each stellar model at each evolutionary stage from: the derived stellar radius, R R = L 4πσT 4 eff where L is the stellar luminosity, T eff is the effective temperature, and the constants have their usual meaning. the calculated terminal velocity, v : v = ξ 2GM (1 Γ e ) R 43

50 Chapter 3: Semi-Analytical Investigation where M is the stellar mass, and ξ v /v esc is the ratio of the terminal flow velocity over the escape velocity. The value of ξ is 2.6 or 1.3, depending on the effective temperature T eff of the model (see Sec.(4.1)). Γ e is the Eddington factor of a given model, at a given evolutionary stage: Γ e = σ el 4πcGM (3.15) where σ e is the electron scattering cross-section per unit mass (in cm 2 g 1 ). The value of σ e depends on the effective temperature T eff and composition, as was described by Lamers and Leitherer (1993), according to: σ e = qǫ 1 + 3ǫ (3.16) The ionization fraction q of He ++ varies as (Lamers and Leitherer): q = 1 if T eff 35,000K q = 1/2 if 30,000K T eff < 35,000K q = 0 if T eff < 30,000K and ǫ He/(H+He) is the ratio of hydrogen to helium in the gas. This ratio can be deduced from the mass abundance of hydrogen and helium, by setting the mass of helium equal to four times the hydrogen mass, which corresponds to taking only the most abundant isotope of helium into account. The mass abundances of hydrogen and helium can be calculated from the usual normalization X +Y +Z = 1, where X, Y and Z are the mass abundances of hydrogen, helium and the metals. The mass abundance of helium at a given stage can be extrapolated from the primordial mass abundance of helium, Y p, and the increase of helium abundance per unit increase in Z, Y/ Z: ( ) Y Y = Y p + Z (3.17) Z The primordial helium abundance is adopted from Peimbert et al. (2007), who determined it to be Y p = ± The slope Y/ Z was determined by (need a reference!) to be Y/ Z = the mass loss rate Ṁ, which was calculated using the mass loss recipe of Vink et al. (2001) The parameter P x in Eq(3.14) was calculated for each stellar model at each evolutionary stage from a set of free parameters describing the line strength, x, the ion charge, Z, and atomic number, A, and the location v/v = (1 R /r) β of the wind. In the semi-analytical search the wind temperature T is assumed to be identical to the effective temperature T eff, which is taken from the evolutionary tracks of Schaller et al. (1992). The value of the Coulomb logarithm ln Λ depends on the net ion charge Z, the charge of 44

51 3.2 Estimate of Q and Px parameter value notes [unit] A f 1 atomic weight field particle Z f 1 charge of field particles n elec /N nucleon 1 fraction of electrons to nucleons T e /T eff 1 fraction of electron temperature relative to Teff β 1 slope of velocity law Table 3.1: Estimated wind parameters used in semi-analytical search for decoupling. the field particles, Z f (which was set to 1), the wind temperature T f T eff and the electron density n elec in the wind. The values of the estimated parameters are all summarized in Table (3.1). To summarize, the parameters that were varied in our semi-analytical search for decoupling throughout the HR diagram are: the ion charge Z, the parameter v/v = (1 R /r) β, the line strength parameter x = ǫf osc, and the metallicity, Z, which determines Ṁ. In the semianalytical study, the dependance of Eq.(3.14) on Z is investigated by varying the metallicity Z from Z = Z (solar metallicity) to Z = 0.01Z. 45

52

53 CHAPTER 4 Results of semi-analytical investigation The semi-analytical condition for decoupling derived in Chapter (3) can be tested for a wide range of wind parameters. In this chapter, we present a parameter study covering the upper part of the Hertzsprung-Russel diagram for different metallicities. We have investigated semi-analytically where in the HR diagram we may expect decoupling of important ions, like carbon ions, to occur as a function of atomic charge, line strength, radial distance and metallicity. We present the grid for which the occurence of decoupling was investigated in Table Dependence on ion charge We start with a brief description of the figures made to show the results. The HR diagram, as shown in Fig.(4.1), contains the evolutionary tracks of Schaller et al. (1992). We overplot the value of log Q (colored dots) calculated for each evolutionary timestep. The color legend indicates the calculated values of log Q. As expected, the most massive stars have the largest value of log Q = Ṁ/(R2 v ). When for a given model decoupling is expected to occur for a given ion, this is indicated with a diamond. The progression of log Px throughout the HR diagram is not shown, as this would not result in a clarifying overview. Therefore Fig.(4.2) shows the progression of log Px (stars) and log Q (blue dots), shown here at the ZAMS, as a function of effective temperature. Models for which the ion under consideration decouples, are those for which log Q < log Px, (i.e., those for which the blue dots are below the stars, see Fig.(4.2)). As the value of log Q was calculated using the mass-loss recipe of Vink et al. (2001), who predict a bi-stability jump in Ṁ around 25,000K, this causes the distinct dip in the progression of our global parameter log Q. Note here, that we have extrapolated the mass-loss regime for which Vink et al. (2001)have derived their mass-loss recipe, since their mass-loss recipe was derived for luminosities in the range log(l /L ) = and masses 47

54 Chapter 4: Results of semi-analytical investigation Figure 4.1: Behaviour of log Q (colored dots) across the HR diagram at solar metallicity. The (open) diamonds indicate models for which Q < Px (decoupling) is expected deep in the wind, at the location in the wind where v/v = 0.5, for strongly accelerated lines, that is, with x = 1, at solar metallicity for C iv (top) and for C iii (bottom). 48

55 4.1 Dependence on ion charge Figure 4.2: Behaviour of log Q (blue dots) and log Px (colored stars) as a function of effective temperature, for models at the ZAMS, at solar metallicity, at the location in the wind where v/v = 0.5, for strong lines, with x = 1, for C ii (black stars), C iii (blue stars) and for C iv (light blue stars). 49

56 Chapter 4: Results of semi-analytical investigation Table 4.1: Overview of semi-analytical results for occurence of decoupling as a function of relevant parameters: ion charge, location v/v in the wind, line strength [parameter] x f osc F ν /F νmax, and metallicity Z /Z. Fig. Ion v/v x Z /Z Decoupling expected for 4.1 top C iv M, 12M 9M MS phase before T jump bottom C iii Same as C iv plus ZAMS of 25M, early MS of 7M, and of 20M 4.2 top C ii MS of 12M 25M before T jump, entire MS of 7 9M, and early MS of 40 85M 4.3 top C iii MS of 9M model before T jump bottom C iii no models 4.5 top C iii MS of 9 15M, before T jump bottom C iii MS of 9 20M before T jump, MS of 7M, and early MS of 25M. 4.7 top C iii MS plus later stadii of 7 9M, 12 60M MS before T jump, and 25M early MS bottom C iii M entire MS, and later evolutionary stages, entire MS before T jump in the range M = 20 60M. The same holds therefore for the predicted bi-stability jump. We briefly discuss the bi-stability jump below. Vink et al. (1999) found that in hot luminous 1 stars with an effective temperature T eff typically above approximately 25,000 K, the dominant ionization stage of Fe in the subsonic part of the wind is Fe iv. In stars in the temperature range typically between 12,500 and 25,000 K the dominant ionization stage was found to be Fe iii, and in stars with T eff 12,500 K Fe iii was found to recombine to Fe ii 2. Vink et al. showed, however, that whether or not Fe iii is the dominant ionization stage in the subsonic region, it is either way the most important Fe line driver. This is because Fe iii has many more efficient atomic transitions than Fe ii and Fe iv. This implies that in models where the effective temperature is above the recombination or jump temperature, T jump, of Fe iv [to Fe iii], the radiative acceleration decreases due to the smaller fraction of Fe iii in the subsonic part of the wind. The same applies for models with T eff 12,500K, where Fe ii becomes the dominant ionization stage. This implies a smaller value of Ṁ when going from effective temperatures where Fe iii is the main ionization stage, to temperatures where Fe iii is not the main ionization stage. Generally, this means that when going from spectral types 1 To be precise: in the range of stars with T eff between 12,500 and 50,00 K, log(l /L ) in the range , and M in the range M. 2 Note that the ionization equilibrium depends on the combination of [effective] temperature and on the [characteristic] density in the wind. However, Vink et al. (1999) defined a typical recombination temperature for Fe because of the linear relation found between the effective temperature and the logarithm of the characteristic wind density, ρ 50

57 4.1 Dependence on ion charge with temperatures above T jump, to temperatures below T jump, a downward bi-stability jump in Ṁ is expected. The temperature T jump where recombination of Fe iv to Fe iii is expected is plotted as the dashed blue line in Fig.(4.1). This temperature was calculated using the derived relation for T jump as a function of the characteristic wind density ρ according to Vink et al. (2001). Fig.(4.2) shows that at the ZAMS the value of Px decreases linearly with decreasing temperature, whereas Q shows a bi-stability jump superimposed on an overall behaviour which decreases with decreasing temperature. This is because Q is proportional to the mass-loss rate. The linear decrease of Px with temperature is because P is proportional to the temperature. Note that in this semi-analytical way of estimating the occurence of decoupling, we can not take the actual variation of the ionization structure and radiation field into account. Therefore, the results obtained using the real local values of t S and t D are expected to differ from our globally defined parameter Q and estimated wind parameter P x. Here our aim is to estimate the occurence of decoupling without doing the actual simulations. Because the decoupling condition, Eq.(3.14), depends on the charge Z of the ion under consideration, we show our semi-analytical prediction for decoupling for three different ionization stages in Fig.(4.1). The condition of decoupling is not a very strong function of atomic number, i.e. it depends on atomic number as A/(A + 1). We choose to show the occurence of decoupling for carbon, since this element is an important wind driver in hot stars Abbott (1982). Since carbon lines are generally strong lines, we set the value of x, which is essentially a representation of the local radiative acceleration, as defined in Eq.(3.4), to unity. In the next section we will investigate what happens when the lines become weak (low oscillator strength) and the number of photons that is offered in the line is decreased (low F ν0 ). Fig.(4.1) and the upper panel of Fig.(4.2) show that for all ionization stages decoupling of carbon is expected to occur predominantly in main sequence models that have a low luminosity, i.e., L L (apart for the M cases in the case of C ii), and that are on the cold side of the bi-stability jump. The reason for this is straightforward. As the luminosity increases, so does the mass-loss rate. This is because the mass-loss rate scales with the luminosity approximately as Ṁ L2 [Vink et al. (2001)]. A higher mass-loss rate implies a higher wind density, and therefore C iv and C iii decoupling is not expected to occur in the most luminous (here: with L L ) part of the HR diagram. The vertical cut-off in the decoupling regime is at the temperature of the bi-stability jump. The reason for this is also straightforward: as the stars evolve redwards across the temperature where Fe iv recombines to Fe iii, the effect is that the predicted mass-loss rate is increased with a factor of about 5-7 [Vink et al. (2001)]. As a result the wind density is increased significantly, which also results in a more efficient slowing down of the ions, and no more decoupling is expected. When comparing the upper panel with the lower panel of Fig(4.1), it is seen that as the charge of the ion under consideration goes down, decoupling occurs in more models. This is 51

58 Chapter 4: Results of semi-analytical investigation due to the inverse proportionality of P with the atomic charge Z in Eq.(3.14). Physically a decreased charge causes a reduced electrical potential, and therefore a reduced deflection χ of the relative orbit per encounter. This implies more encounters are needed to decrease the radial momentum of the absorbing ion by the same amount, and therefore a longer timescale t S to slow down the ion is required. This result is also depicted in the lower panel of Fig.(4.2), in which the progression of log Q and log Px is shown (at the ZAMS) for various ion charges as a function of effective temperature. From this figure, it is clear that with reducing ion charge, it is easier for a given ion to become decoupled. This explains why in the top panel of Fig.(4.1), the 20M star shows no decoupling of C iv, whereas the same star does show decoupling of C iii (see bottom panel Fig.(4.1)). The model with M = 7M does not show decoupling in the case of C iv because it is on the red side of the bi-stability jump, which causes Ṁ to increase. In the other cases, the effect of the reduced charge (and therefore Coulomb interaction that slows the ion down) dominates, and the early MS stages of the 7M model are expected to show decoupling. 4.2 Dependence on line strength parameter, x As was pointed out in Sec.(1.1), important line drivers are elements in which there is an optimal combination of number and strength of lines for the line force. In particular, this is relevant for line with frequencies around the flux maximum, where the photon density is highest. Vink et al. (1999) have shown that for hot luminous stars the mass-loss rate is primarily determined by a large number of mainly weak lines, such as Fe-group lines. The terminal velocity, on the other hand, was found to be determined mainly by the contributions of C,N, and O. Therefore, it is relevant to investigate where in the HR diagram we may expect weak and strong lines to decouple. The efficiency with which lines contribute to the line force depends not only on their oscillator strength, f osc, but also on the flux of the incident radiation field at the CMF frequency ν 0 of the transition, that is, on the ratio F ν0 /F νmax. Since the combination x f osc F ν0 /F νmax is a free parameter in our semi-analytical condition for decoupling, Eq.(3.14), the dependence on x can be investigated in a straightforward manner. The parameters for which we investigated the occurence of decoupling as a function of x are listed in Table 4.1. The result can be seen by comparing the lower panel of Fig.(4.1) with the top and the lower panel of Fig.(4.3), and also by looking at the progression of log Q and log Px as a function of effective temperature, at the ZAMS, for various values of x, as shown in Fig.(4.4). From the figures, it can be seen that with increasing value of x, more stars end up in the decoupled regime. More specifically, Fig.(4.4) shows that for ZAMS models there is a tendency for only strong lines, with the value of x between 0.1 and 1, to decouple in low wind-density models (log Q 8). Ions with very low values of x, that is, ions with weak transitions, with 52

59 4.2 Dependence on line strength parameter, x Figure 4.3: Similar to Fig.(4.1), but now for weak C iii lines, with x = 0.1 (top), and for even weaker C iii lines, with x = 0.01 (bottom), to show the effect of a different x on the occurence of decoupling. The effect of decoupling for strong x = 1 Ciii lines was shown in the lower panel of Fig.(4.1). 53

60 Chapter 4: Results of semi-analytical investigation Figure 4.4: Similar to Fig.(4.2), here to show the effect of a different x. Shown for strong for C iii lines, with x = 1 (black stars), weak lines, with x = 0.1 (gray stars) and weaker lines, with x = 0.01 (lighter gray stars). Decoupling occurs for models where log Q < log Px. frequencies that are not at the flux maximum, are never expected to decouple (at v/v = 0.5). As in the previous section, the model with M = 7M does not show decoupling due to the imposed bi-stability jump, which causes the mass-loss rate, and therefore the wind density to go up. Physically, this effect arises because active particles get accelerated more easily if they have strong transitions, with frequencies close to the flux maximum of the incident radiation field. A stronger accleration implies a shorter drift time, and therefore a better chance to become decoupled. 4.3 Dependence on radial distance Apart from investigating decoupling in the part of the wind where v/v = 0.5, where, according to the theory of stellar winds the terminal velocity of the wind is set, it is also of interest to investigate the occurence of decoupling in the lower part, where, according to stellar wind theories the mass-loss rates is set and in layers further out of the wind. To investigate what happens with the decoupling criterion as we move closer and further 54

61 4.4 Dependence on metallicity out in the wind, we repeated the analysis for C iii using the parameters tabulated in Table 4.1, here also for v/v = 0.2 and v/v = 0.9. When comparing Fig.(4.5) and the lower panel of Fig.(4.1), we see that as we recede further from the star (i.e., from v/v = 0.2 to 0.9), for more models C iii ions tend to decouple. This can also be seen from Fig.(4.6), in which we show the progression of log Q and log Px as a function of T eff for various locations v/v in the wind. It is shown here that the decoupling regime is expected to spread more throughout the HR diagram when considering ions far out in the wind. Physically, this is due to the lower density in the outer layers, which causes a smaller number of collisions between active particles and field particles. This reduces the amount of momentum transferred from the active to the field particles, which implies that the time needed to slow down the particles by the same amount is larger. This creates better conditions for the ions to become decoupled from the plasma. 4.4 Dependence on metallicity From the previous paragraphs, we have seen that we don t expect much decoupling in the subsonic region for solar metallicity stars. Therefore, we investigate what happens to our criterion as we move to lower metallicity. The result is plotted in Figure 4.7, for C iii. The choice of C iii is especially relevant in low metallicity models, because according to Vink et al. (2001), the winds of stars at low metallicity are expected to be driven mainly by CNO. We compare the lower panel of Fig.(4.1), and the two panels of Fig.(4.7). As the metallicity Z goes down, the decoupling regime spreads more throughout the HR diagram. In other words, stars show decoupling, at both later evolutionary stadii, and in a wider mass range. This can easily be explained: as the metallicity Z goes down, the mass loss Ṁ, which scales as a given power of Z, goes down as well, which causes a reduced wind density. A reduced wind density implies that the active particle loses less momentum through collisions with neighboring particles. This can also be seen from Fig.(4.8), which is plotted for Z/Z = 0.1 (left panel), and for Z/Z = 0.01 (right panel). This figure shows that with decreasing metallicity, also weaker lines, that is, lines with low x, are expected to become decoupled in some models. In all our models we have used the evolutionary tracks of Schaller et al. (1992) at solar metallicity. Therefore we changed only Z in our condition, Eq(3.14), keeping the other parameters, including the tracks that were used, unchanged. We have investigated whether the use of solar metallicity tracks instead of low Z-tracks when applying our decoupling criterion for low Z results in large differences with respect to the occurence of decoupling. This was done by comparing the results obtained from low Z-tracks with those obtained from solar metallicity tracks, where only Z was changed in the decoupling criterion. This is done in Fig.(4.9) for Z = 0.2Z, as there are no evolutionary tracks from Schaller et al. (1992) available at Z = 0.1Z or at Z = 0.01Z. The figure shows that the use of solar metallicity tracks with Ṁ calculated for low Z does not result in large differences with respect to the use of low Z 55

62 Chapter 4: Results of semi-analytical investigation Figure 4.5: Dependancy on location, v/v in the wind. The open diamonds indicate models for which Q < Px (decoupling) for strong (x = 1) C iii lines, at solar metallicity, and at location v/v = 0.2 (top) and v/v = 0.9 (bottom). For a comparison with v/v = 0.5, see Fig.(4.1). 56

63 4.4 Dependence on metallicity Figure 4.6: Dependancy on location, v/v in the wind. Symbos: same as Fig.[4.2], here for v/v = 0.2 (black stars), v/v = 0.5 (blue stars), and v/v = 0.9 (light blue stars). 57

64 Chapter 4: Results of semi-analytical investigation Figure 4.7: Similar to previous HR diagrams, here to show the effect of varying the metallicity. Top: Z = 0.1 Z. Bottom: Z = 0.01 Z, for strong (i.e. x = 1) C iii lines at a location where v/v =

65 4.5 Discussion of the semi-analytical investigation tracks, with Ṁ calculated for low Z. Figure 4.8: Dependancy on metallicity, for C iii lines at the location in the wind where v/v = 0.5, at Z = 0.1Z (left), and at Z = 0.01Z (right). 4.5 Discussion of the semi-analytical investigation In this chapter, we presented an overview of where we expect active ions to decouple from the passive plasma semi-analytically, i.e., using estimated properties of the radiation field and ion structure. From the previous sections, we have seen that at solar metallicity decoupling of strong lines with wavelengths around the stellar flux maximum (x 1) in the subsonic (v/v 0.2) region is typically expected to occur only in low luminosity MS models, that are on the red side of T jump, with ZAMS masses in the range 9 12M, and with log Ṁ 9. The decoupling regime is expected to spread more throughout the HR diagram with increasing radial distance r from the star, up into the regime where log Ṁ 7, i.e., for stars with a maximal ZAMS mass of 25M. Decoupling of weak lines (x 1) is typically expected to occur only far out in the wind, for the stars in the ZAMS range of 9 12M 3. As we move towards low-metallicity models, we have seen that decoupling of active ions is expected to occur for both low and high luminosity models, at the ZAMS and at later evolutionary stages. We show here that decoupling of active ions is generally not expected to have any significant effect on the stellar evolution. At low-metallicity, decoupling can potentially prevent the occurence of a stellar wind. However, as we will see, in these models, the winds are expected to be weak, and therefore any modification of the wind properties due to decoupling is not expected to have an important impact on the stellar evolution. As discussed in Chapter 2, decoupling of active ions implies that less momentum is trans- 3 Note that if it were not for the bi-stability jump, lower initial masses (M 7M ) would also be included into the decoupling regime. 59

66 Chapter 4: Results of semi-analytical investigation Figure 4.9: Similar to previous HR diagrams, at Z = 0.2 Z, for evolutionary tracks at solar metallicity, (top), and for evolutionary tracks at Z = 0.2Z (bottom), for strong (i.e. x = 1) C iii lines at v/v = 0.2 In both cases Z = 0.2Z was used both to calculate the mass-loss rate using the mass-loss recipe of Vink et al. (2001), and in the condition, Eq(3.14) 60

67 4.5 Discussion of the semi-analytical investigation Table 4.2: Estimate of percentage of mass lost in the stellar wind, % M/M init, over the course of the stellar lifetime for several values of metal abundance. M [%M init ] Z = Z Z = 0.1Z Z = 0.01Z M init [M ] ferred from the radiation field to the bulk plasma, as the active ions are assumed to no longer share their momentum with the surrounding particles once they are decoupled. If decoupling of an active ion species occurs in the subsonic region, this may therefore result in a reduced mass-loss rate, Ṁ. For a star with a given metallicity Z, the maximal effect of decoupling on Ṁ can be estimated by assuming all absorbing ions decouple, i.e., total decoupling occurs, in the subsonic region. This means that only the metals escape from the stellar atmosphere, leaving the bulk plasma behind. The mass-loss rate Ṁ is then reduced with a factor Z, that is, the maximal effect of decoupling on Ṁ scales as Ṁ dec Z Ṁ. If decoupling occurs in the supersonic region, this may result in a reduced terminal velocity v of the bulk, and accordingly, the terminal velocity of the decoupled ion species is increased. The impact that decoupling of a given ion species has on Ṁ and/or on v depends on the importance of that ion species for the line driving, i.e. if a given ion species contributes significantly to the line driving by absorbing a lot of momentum from the radiation field, then decoupling of that ion species will result in a noticeable effect on Ṁ and/or v. The impact that decoupling of a given ion species has on the stellar evolution, however, depends not only on the effect mentioned above on the wind, but also on the wind itself. It is therefore of practical interest to investigate whether decoupling occurs in models for which Ṁ is large enough to play an important role in the evolution. This can be done by estimating how much mass is lost in the stellar wind during the stellar lifetime. For models with metal abundance Z = Z,Z = 0.2Z and Z = 0.05Z, the mass lost over the course of the stellar lifetime can be derived directly from the evolutionary tracks calculated by Schaller et al. (1992). Since we are interested in models at Z = 0.1Z, and Z = 0.01Z, we will estimate the mass-loss rate from Z = 0.2Z and Z = 0.05Z, using the scaling relationship between Ṁ and Z, as derived by Vink et al. (2001). According to Vink et al., in the metallicity range between 1/30 and 3Z/Z, the mass-loss rate scales with metallicity as Ṁ Z 0.69 for O-type stars 61

68 Chapter 4: Results of semi-analytical investigation with T eff 25,000K, and as Ṁ Z 0.64 for B-type supergiant stars with T eff 25,000K. Extrapolating their constant power law index down to Z = 0.01Z, we can estimate the amount of mass lost in a stellar wind for our model stars, at Z = 0.1Z and Z = 0.01Z. The result is given in Table 4.2, in which the total mass lost in a stellar wind is given in percentage of the initial (ZAMS) mass of the star. From Table 4.2 it can be seen that solar metallicity stars for which decoupling is expected in the subsonic region (stellar models 9 and 12M ), lose a negligible ( 4%) amount of their initial mass in a stellar wind over the course of their lifetime. For these stars, the mass-loss in a stellar wind is therefore not an important parameter for their evolution. This implies that even total decoupling in the subsonic region, which may result in a maximal reduction of Ṁ with a factor of 0.02, i.e., a pure metallic outflow, is not expected to influence the evolution of the star. For metal-poor stars at Z = 0.1Z decoupling in the subsonic region is expected to occur for stars with a maximal ZAMS mass of 25M. As we go to even lower metallicities, e.g., at Z = 0.01Z, decoupling in the subsonic region is expected to occur at all ZAMS masses throughout the entire main sequence, and (part of) the Hertzsprung-Gap. However, since the mass-loss rate goes down with decreasing metallicity, the total mass lost over the course of the stellar lifetime becomes of the order of a few percent of the initial mass, as can be seen from Table 4.2. This implies that even total decoupling in the subsonic region may have a only a marginal effect on the evolution of low-metallicity models. As a result, if decoupling occurs in the subsonic region, it is not expected to substantially influence the stellar evolution, as the mass-loss rate in these stars is not an important parameter for the stellar evolution. Our semi-analytical investigation has shown that for most solar-metallicity stars, decoupling is more likely to occur in the supersonic wind region, where, at most, it affects the bulk terminal velocity, v. 62

69 CHAPTER 5 Numerical investigation In Chapter 4, we presented the results of our semi-analytical search for decoupling, in which estimated properties for the radiation field and ion structure were used. We are now in a position to verify our results using detailed stellar wind models. In this chapter we provide a concise overview of the method used to determine t S and t D numerically, and the method used to determine the maximum effect of decoupling on Ṁ or v. 5.1 Determination of the slow down and drift timescales using the MC code The simulations are performed by first using the non-lte Improved Sobolev Approximation atmosphere code (ISA-WIND) to calculate the input thermal, density and ionization and excitation structure of a wind model. The calculated ISA-WIND models are subsequently used as input in the Monte Carlo method (MC-WIND), in which the total momentum transfer from the radiation field to the gas particles that constitute the stellar wind, is calculated. The basic mechanism of ISA-WIND and MC-WIND was described in Sec.(1.4.2). Our grid of L and T eff models was set up using the evolutionary tracks of Schaller et al. (1992) for stars with M ZAMS 7M. For these massive stars the evolutionary tracks have been calculated up to the end of the C-burning phase. However, as we have seen in Chapter 4, decoupling of absorbing ions is not expected to occur for evolutionary stadii later than the main sequence. Therefore, we confined our search for decoupling to main sequence models, up to the end of core hydrogen burning. The parameters of the models are listed in Table 6.1 and Table 6.2. The value of Ṁ was calculated in an iterative manner, from the procedure as described in Sec.(1.4.2), using a β-type velocity law with β = 1. The adopted value of the ratio of the terminal velocity to the effective escape velocity at the stellar surface (v /v esc ) depends on the model temperature T eff. Via the characteristic density ρ, it also depends on Γ e, and therefore also on the luminosity and mass. Following Vink et al. (2001), if T eff exceeds the jump temperature T jump,1 involved with the recombination of Fe iv to Fe iii, we 63

70 Chapter 5: Numerical investigation use v/v = 2.6. If T eff is less than T jump,1 but larger than T jump,2, which is the temperature involved with the recombination of Fe iii to Fe ii, than v /v esc = 1.3. Finally, if T eff is less than T jump,2, v /v esc = 0.7 is used. Note that we have extrapolated the luminosity (and temperature) range for which Vink et al. (2001) have derived the linear fit relation between T eff and ρ that determines T jump,i, {i = 1,2}. The validity of this extrapolation is discussed in Sec.(6.6). For each galactic model it suffices to follow photon packages when iterating, which is done in generally 5 iterations, towards the final value of Ṁ. When the model is run for the final converged value of Ṁ, we used photon packages. At Z = 0.1Z the line driving mechanism for intermediate to low mass (depending on the luminosity, for M 15M ) models is not efficient enough to reach convergence. This is because the massloss rate scales with the stellar metallicity Z, which causes a lower Ṁ at low Z. As a result, the photon packages travel through a less dense wind and experience a smaller number of line interactions. Therefore we set the number of photon packages to while converging these models. The models for which convergence could not be reached were models for which the typical amount of line scatterings was below 100 in each supersonic shell. These models were left out of our calculations. In the MC approach we know the local ionization and excitation structure and radiation field that determine the slow down time t S and the drift time t D throughout the wind, or, equivalently, the value of log Px. The value of log Q is globally determined, and is essentially our input value in the MC code. Since we cannot know the actual drift velocity w in Eq.(2.11) a priori, our approach will be to assume a well coupled situation, and from the local conditions that this creates for t S and t D verify whether the assumption of a coupled situation was justified. As pointed out in Sec.(1.4.2), when a photon is emitted in the MC simulation, there are a number of possible interactions that may occur. The photon may undergo a Thomson scattering, a thermal absorption and emission, a line scattering, a continuum absorption in the line resonance region, or a line absorption followed by collisional de-excitation. Since the continuum events mainly involve electrons or the passive plasma, these are not assumed to be influenced by the presence of absorbing ions, which are in a far minority in the wind. Therefore, we assume that the only events which are important for decoupling are the line scatterings. The type of event that takes place for a given photon emitted with random frequency ν is determined using a line list that contains about of the strongest lines of the elements H-Zn in the wavelength region between 25 and 10,000 Å. The line list used was extracted from a line list constructed by Kurucz (1988). If the outcome of the MC procedure is a line event, the parameters needed to determine t S, Eq.(2.11), and t D, Eq.(2.17), are known. The atomic number A, ionization Z, and oscillator strength f osc of the absorbing ion are known from the employed line list. The frequency of the photon is known from the MC simulation, and the 64

71 5.2 Implementing decoupling in the Monte Carlo code local temperature and density of the wind are known from ISA-wind. The field particles are assumed to consist solely of hydrogen. We don t know a priori the velocity of the absorbing ion w in Eq.(2.11). Therefore, we assume a well coupled situation, that is, we assume the velocity w of the ion to be much smaller than the thermal velocity of the field particles, v th. Using this assumption, we can employ Eq.(2.12) for the slow down time, which was derived in the limit w v th from a first order Taylor expansion (see Sec.(2.2)). To verify whether the absorbing ion is decoupled or not, we calculate the momentum loss of the photon in the radial direction, which corresponds to the momentum gain of the ion in the radial direction. The local line force g ion in Eq.(2.17) is then calculated from the momentum change divided by the ion mass. In this way, all line events for which t S > t D are registered. 5.2 Implementing decoupling in the Monte Carlo code In this section we describe a method to implement decoupling into the MC code to estimate the maximum effect of decoupling on Ṁ, or on v. Before doing so, we emphasize that this method only provides us with an estimate of the effect, because the velocity law itself might be affected if decoupling of one or more important ion species occurs. However, in the MC method we do not have a means to account for the effect of decoupling in a self-consistent way. Therefore, we assume the velocity law to remain a beta-type velocity law with β = 1. The maximum effect on Ṁ will be estimated by keeping the velocity law, (i.e., both β and v ) unchanged. The maximum effect on v will be estimated by keeping Ṁ and β unchanged. In the MC method we follow the photons on their path through the wind, which has a fixed input density structure, ρ(r). This means that as a photon undergoes a line scattering that leads to the decoupling of a given ion, this has no effect on the subsequent path of the photon. The only effect in our method is that the photon momentum lost to the decoupled event is no longer assumed to contribute to the gain of radial momentum of the outflowing bulk material. As a result, the energy loss of the photons no longer equals the increase in radial momentum p of the bulk plasma. Essentially, we can no longer assume the wind to consist of a single fluid. However, since we merely test whether the assumption of a wellcoupled wind is justified, this implies that we do this by assuming a well-coupled wind, i.e., a single fluid. We estimate the impact of ion-decoupling on the stellar wind properties Ṁv by subtracting the total luminosity L dec lost by the photons into the decoupled events from the total 65

72 Chapter 5: Numerical investigation energy extracted from the radiation field to accelerate the wind. In Eq.(1.42) we therefore replace L by L L dec, where L dec is the total luminosity lost to the decoupled events and L is the total luminosity used to accelerate the entire wind. The cumulative effect L dec of all decoupled events is calculated in the MC method by registering the momentum transferred to decoupled ions. This is done in a straightforward way, by calculating the cumulative value of νobs in νout obs for each decoupled event, in analogy with the normal (coupled) line events. The amount of photon momentum transferred to the decoupled ions per line scattering event is calculated in the same way in which the photon momentum transfer into a normal line event is calculated. This was explained in Sec.(1.4.2). The total amount of photon momentum lost to the decoupled ions is the sum of all the decoupled events: p = h c N i=1 ( ) ν in,i obs νout,i obs = E dec c (5.1) which is equal to the amount of radiative energy transferred to the decoupled ions E dec /c. Here N is the total number of decoupled events, ν in,i obs and νout,i obs are respectively the frequency of the incoming and the outcoming photon in a decoupled event in the observers frame, and all the other parameters are as defined before. The simulated time interval t [in seconds] by the MC method is calculated from the net radiation energy that is injected into the wind at the inner boundary divided by the stellar luminosity: t = L 1 (E k E j ) (5.2) where E k is the total emitted energy and E j is the total energy which is backscattered into the continuum forming layers, and L is the luminosity of the wind-free star. The rate at which the decoupled ions extract energy from the wind is then: E dec t = L E dec E k E j = L dec (5.3) Determination of maximum effect on Ṁ and on v of the bulk plasma As the total luminosity extracted from the radiation field, L, depends on the product of Ṁ and v, both of which may be affected by decoupling of absorbing ions, we can only provide an estimate of the maximum effect on Ṁ, keeping v constant, and vice-versa. Note that the true effect, if anything significant at all, is obviously expected to lie somewhere between these maxima. The maximum change of Ṁ due to ion-decoupling can be estimated by simply subtracting the value of L dec lost to the decoupled events, Eq(5.3), from the amount of luminosity 66

73 5.2 Implementing decoupling in the Monte Carlo code L used to accelerate the entire plasma outwards. The resulting mass-loss rate Ṁ when taking decoupling into account can be calculated from: 1 2Ṁ (v 2 + v2 esc ) = L L dec (5.4) where v = v is the coupled terminal velocity, that is, the terminal velocity is unchanged with respect to v in a model where no decoupling is taken into account. Similarly, the maximum effect of decoupling on the terminal velocity can be calculated from Eq.(5.4), keeping Ṁ unchanged, i.e., M = Ṁ, and solving for v. It should be noted that are some constraints with respect to the maximum luminosity L dec that can be lost to the decoupled events. This is because the total energy gained by the wind material should always be at least equal to the energy needed to lift the material out of the potential well of the star, 1/2M vesc 2, if the material is to escape from the star (this also holds in the coupled case, naturally). Therefore: 1 2Ṁ v 2 = ( L L dec) 1 2Ṁ v 2 esc 0 (5.5) When rewriting Eq.(5.5) it can be readily seen that the maximal amount of radiative energy that can be lost to the decoupeld events that still results in a wind that escapes from the star is: L dec L Ṁ vesc 2 1 Ṁvesc 2 + Ṁv2 (5.6) If we keep the mass-loss rate unchanged with respect to the coupled mass-loss rate, i.e., Ṁ = Ṁ, this yields: L dec L (v /v esc ) 2 (5.7) Eq.(5.7) shows that in case we keep the mass-loss rate unchanged, i.e., Ṁ = Ṁ, the maximal fraction of the radiative energy that can be lost in the decoupled events, L dec / L, and still result in an outflow, depends on the ratio v /v esc. If v /v esc = 2.6 then L dec / L 0.87, whereas if v /v esc = 2.6 then L dec / L Alternatively, if we keep the terminal velocity unchanged, i.e., v = v, then the constraint on the ratio L dec / L is given by Eq.(5.6). Since Ṁ is allowed to become arbitrarily small, Eq.(5.6) essentially tells us that L dec / L should be smaller or equal to unity (in the latter case, there is a pure metallic outflow). The resulting maximum effects on both Ṁ and v will be presented in Table 6.1 and Determination of terminal velocity of decoupled ions from L Apart from investigating the maximum effect of decoupling on the mass-loss rate Ṁ and the terminal velocity v of the bulk plasma, we can also estimate the terminal velocity with 67

74 Chapter 5: Numerical investigation which the decoupled ions escape from the wind. The terminal velocity of the decoupled ions can be estimated from the amount of luminosity L dec that is transferred to the decoupled ions. This can be estimated by integrating Eq(1.1) without the gas pressure term, from the location where a given ion decouples for the first time, r D, to infinity: v r ( vdv = GM ) r 2 + g rad (r) dr v(r D ) r D where g rad is now the radiation force acting on the decoupled ions, and v(r) is the velocity of the decoupled ions. Plugging in the expression as derived in Eq(1.41) for the radiative acceleration acting on the decoupled ions: ( ) v v(r D ) vdv = r r D GM r Ṁ ij L dec r dr where Ṁij is the mass loss rate of the decoupled ion species i in a given state j, and L dec is the luminosity used to accelerate the decoupled ions. After integrating, we obtain: 1 ( v 2 2 v 2 (r D ) ) = GM GM + 1 L dec (5.8) r r D Ṁ ij Rewriting Eq.(5.8) yields: L dec = Ṁij { 1 ( v 2 2 v 2 (r D ) ) GM r + GM } r D = 1 2Ṁij ( v 2 v 2 (r D ) + v 2 esc (r D) ) (5.9) where in the latter expression we used GM /r D = v 2 esc (r D) and GM /r 0, since r. We assume the mass-loss rate of ion species i in ionization state j to be proportional to the total mass loss rate Ṁ: Ṁ ij = q j X i Ṁ (5.10) where q j = n j /N is the ionization fraction of j and X i is the mass fraction of i. Substituting Eq.(5.10) into Eq.(5.9) yields the luminosity lost to the decoupled ions, L dec : L dec = 1 2 q jx i Ṁ ( v 2 v 2 (r D ) + v 2 esc(r D ) ) (5.11) from which the terminal velocity of ion i in state j can be dertermined by rewriting Eq(5.11) v 2 = 2 L dec q j X i Ṁ + v2 (r D ) v 2 esc (r D) (5.12) It should be noted that the terminal velocity of the decoupled ions obtained in this way is only approximately correct, as the decoupled ions will absorb a progressively redshifted portion of the radiation field as they are accelerated beyond the bulk velocity. This cannot be 68

75 5.2 Implementing decoupling in the Monte Carlo code Figure 5.1: Emerging spectrum with the lines indicated that lead to the decoupling of the involved ion at a given location in the wind of a model with M = 20M. implemented into the MC simulation, because of the single-fluid approximation. Therefore, the real luminosity lost to the decoupled ions is expected to be different, depending on whether the lines are redshifted beyond the absorption edges (most importantly the Balmer jump at 3646 Å). The reason for this is that the flux on the red side of a given absorption edge is reduced, and therefore so is the amount of momentum transferred to the ion. This is illustrated in Fig.(5.1), which shows the emerging spectrum from a given layer in the wind. The events that lead to the decoupling of the ions involved are indicated in the plot. From this figure, it can be seen that P iii ions decouple typically at frequencies close to the Lyman limit, at 912 Å. This implies that as this decoupled ion is accelerated, at some point the ion suddenly receives less momentum from the radiation field, because it absorbs photons on the red side of the absorption edge. The terminal velocity of the ion is therefore expected to be reduced, compared to the case where no absorption edge is traversed. 69

76

77 CHAPTER 6 Results of numerical calculations In this chapter, we present the results of our numerical investigation covering the upper part of the Hertzsprung-Russel diagram for different metallicities. Using the procedure summarized in Sec.(5.1), we have simulated model atmospheres for 10 values of M, in the range between 7 and 120M. For every mass we ran the stellar wind models for the first 13 evolutionary tracks of Schaller et al. (1992). This was done for solar metallicity models and for models at Z = 0.1Z. We show the parameters for a subset of models in Table 6.1 and 6.2, in which the stellar properties (mass M, luminosity L, effective temperature T eff, radius R ) and the wind properties (v /v esc, Ṁ V, Ṁ MC ) are tabulated for the models at the ZAMS, at the evolutionary stage with the lowest T eff on the MS evolution, and at an evolutionary stage between these two stages. Here v /v esc is the ratio of the terminal velocity over the escape velocity, which was determined following Vink et al. (2001), and ṀV is the predicted mass-loss rate according to the mass-loss recipe of Vink et al. (2001). Note that we have extrapolated the mass-loss recipe to include predictions with respect to decoupling in models with L L. The numerically obtained mass-loss rate is indicated as Ṁ MC. The fraction of the energy of the photons that is lost to the decoupled events, (see Sec.(5.2.1)), compared to the total energy used to accelerate the wind, is indicated as L dec / L bulk. The maximum effect on the mass-loss rate when taking decoupling into account and assuming decoupling only affects Ṁ is indicated as Ṁ max (even though it will obviously result in a minimum value for Ṁ), and accordingly, the maximum effect on the terminal velocity, keeping the mass-loss rate fixed, is indicated as v,max (see Sec.(5.2.1)). Note that we tabulated the maximal difference in v. All units used are as defined before. In Table 6.3 we list the decoupled ions for a subset of models at the ZAMS, with the location where decoupling occured for the first time indicated in brackets (in stellar radii). 71

78 Chapter 6: Results of numerical calculations Table 6.1: Parameters for the different wind models at solar metallicity. The last two colums give an estimate of the maximum effect that decoupling of absorbing ions has on log Ṁ and on v. If the difference between log ṀMC and log Ṁmax is smaller than the standard error in M of 0.05, the difference between the two values is assumed negligible and is therefore indicated with a -. An x indicates that the radiation field was unable to accelerate a given density of material characterised by the coupled mass-loss rate Ṁ out of the potential well of the star. time M log L T eff R v vesc v log ṀV log ṀMC L dec L bulk log Ṁmax v,max [M ] [L ) [10 3 K] [R ] [km/s] log[m /yr) log[m /yr) % log[m /yr) [km/s] x

79 Table 6.2: Parameters for the different wind models at Z = 0.1Z. For these models, convergence was reached, but Ṁin Ṁout > This means that consistency between Ṁin and the mass-loss rate Ṁout calculated from the radiative acceleration within about 1% was not obtained. Models that did not converge were left out. As x indicates that the radiation field was unable to accelerate a given density of material characterised by the coupled mass-loss rate Ṁ out of the potential well of the star. time M log L T eff R v vesc v log ṀV log ṀMC L dec L bulk log Ṁmax v,max [M ] [L ) [10 3 K] [R ] [km/s] log[m /yr) log[m /yr) % log[m /yr) [km/s] x

80 Chapter 6: Results of numerical calculations M Decoup λ [at distance R] Table 6.3: Decoupled ions for different models at the ZAMS (M ) ions [Å] [in R ] 7 C ii [6.4],1037.0[6.4],1334.8[9.3],1336.0[5.2] C iii 977.2[1.7],1175.2[15.3] N ii 915.7[15.3],916.8[7.5],1085.8[9.3] Si ii [15.3] Si iii [2.9], [1.8],1113.2[1.2],1206.4[1.2],1298.9[1.8] P iii 918.8[3.7] P iv 950.5[1.4] S ii 946.9[15.3] Fe iii [3.7] 9 C ii [7.8],1336.0[4.4] C iii 977.1[1.5] N ii 916.8[5.3],1084.6[17.7] Al iii [7.7] Si iii [2.8],1113.2[2.0],1206.4[1.2],1298.9[2.0] P iv 950.5[1.3] 12 C iii 977.1[1.2],1175.2[3.2] Si iii [2.1],1109.9[1.5],1113.2[2.4],1206.4[1.2],1294.4[4.2],1296.7[1.6],1298.9[2.5],1301.2[2.2] Si iv [1.7],1128.3[2.1],1393.7[5.6] P iv 950.5[1.2] 15 C iii 977.1[1.3],1175.2[3.4],1247.6[7.8] Si iii [2.2],1109.9[2.9],1206.4[1.4],1113.2[1.9],1294.4[8.0],1296.7[6.4],1298.9[2.4] Si iv [2.0],1128.3[2.7],1393.7[10.1] P iv 950.5[1.2] Cl iii [13.5],1014.9[10.2] Cl iv 984.9[6.4] 25 C iii 977.2[1.6] Si iii [12.0],1113.2[4.8],1206.4[2.8],1298.9[6.8] Si iv [5.3],1128.3[8.0] P iv 950.5[1.3] 60 Si iii [7.8] Ca iii 358.0[3.8] 120 Ca iii 358.0[16.0] 74

81 Figure 6.1: Behaviour of log Q [in units of Ṁ /yr ] (colored dots, for a definition of log Q, please see Sec.(3.1)) across the HR diagram (logarithm of the effective temperature [in K] on the horizontal axis, and logarithm of the stellar luminosity [in solar luminosity] on the vertical axis) at solar metallicity. The open diamonds indicate models for which decoupling of C iv (top) and for C iii (bottom) was found in our numerical simulations, at the location in the wind where v/v = 0.5, at solar metallicity. 75

82 Chapter 6: Results of numerical calculations 6.1 Numerical results on the dependence on ion charge We investigate the occurence of decoupling of carbon ions as a function of their ionization. This is done by investigating whether, and for which, models decoupling of C ii, C iii and C iv occurs at a position between the photosphere and v/v = 0.5. Note that in making these plots no selection on the local line force, as parameterized in x f osc Fν 0 F max was made. Fig.(6.1) and Fig.(6.2) show the numerical analogue of Fig.(4.1) and Fig.(4.2). The upper panel of Fig.(6.2) shows that C ii ions do not decouple for any model, whereas the lower panel of Fig.(6.1) shows that C iii ions do decouple for some low luminosity models, on the high temperature side of the bi-stability jump. The upper panel of Fig.(6.1) shows that C iv is not expected to decouple for any model. Generally decoupling of C iii agrees roughly with our analytical expectations. Decoupling of C ii and C iv ions does not agree. The explanation for this behaviour is discussed below, but first we make some general remarks that essentially hold for all dependancies. (i) Difference in Ṁ First, it should be noted that our numerically obtained mass-loss rates do in general not agree for all models with the mass-loss rates obtained by extrapolating the mass-loss recipe of Vink et al. (2001). In particular, for MS models between 9 and 15 M, the mass-loss rates found were larger, by a factor that ranges between 3 for the 15 M model, and up to 7 for the 9 M model. The reason for this discrepancy will be discussed in the last section of this chapter, here we proceed with the implications this has for decoupling of ions. As a result of the larger mass-loss rates in these models the numerically found wind densities are larger (since we obviously maintain the same value for R and v in each model for both semianalytical predictions and numerical simulations). This means that it is harder for a given ion to decouple, and consequently, progressively stronger accelerated lines are required for decoupling of a given ion at a given location in the wind. This implies that in general, for the largest mass-loss rates, decoupling of a given ion is expected to occur more outwards in the wind, than followed from our analytical predictions. The dependancy on location will be discussed in Sec.(6.3). Here it is pointed out that since a selection was made on which region to include in the making of these plots, obviously we miss those decoupled events for which the occurence was pushed outside of our region of interest due to the larger wind density. This holds for the models for which the mass-loss rate were found to be larger. (ii) Difference in wind structure and radiation field properties It should also be noted that the semi-analytical predictions could only be made by estimating the ionization and excitation structure and the radiation field, whereas from our simulations we have detailed information about the local radiation field and density and ionization structure that essentially determines the local value of the slow down and the drift time. Whereas the semi-analytical predictions merely provide us with an upper (and lower, in the cases studied of low x) limit on where to expect decoupling of ions to occur as a function of the stellar 76

83 6.1 Numerical results on the dependence on ion charge parameters and estimated wind parameters, from our simulations we know exactly at what position in which wind t S > t D for each interaction. Therefore, whereas our predictions were made for either strongly accelerated (x = 1), or little (x < 1) accelerated lines of a given ion in a given ionization stage in the winds of all models throughout the HR diagram, in reality the result with respect to decoupling of a given ions will be somewhere between our estimated maxima. Note that, as mentioned before, in the models where the wind densities are higher, decoupling of a given ion at a given location in the wind requires a strongly accelerated line. This is also what we find numerically. Since C iii is essentially driven by the absorption of (a relatively few) very strong resonance lines, this implies that if in the right state and subject to the right radiation field, a C iii ion undergoes a large acceleration by the radiation field. As a result, the lines that are found to be decoupled in our simulations are in general those lines which are accelerated most, and therefore the assumption that x 1 holds approximately for the decoupled species. (iii) Variation of slow down and drift time At last, it is pointed out that for ions of the same elements, the slow-down time essentially varies as t S T 3/2 /(n f Z 2 ), and the drift time varies as t D T 1/2 /(f osc F ν0 ). Therefore at a given location in the wind, it is essentially the drift time, and therefore the line acceleration, that determines whether a given ion of the same species will decouple. This is because the charge dependancy contributes at most a factor of 10. However, since we are looking at decoupling in the extended region with v/v 0.5, this means that we have to take the density effect into account as well (the temperature variation contributes little), if we consider decoupling throughout the entire wind. Fig.(6.3) shows the progress of the slow-down time and the drift time for the different ion species as a function of radial distance from the star, for a 15M model at the ZAMS. From this figure, it is clear that C ii and C iv do not tend to decouple, as their drift and slow down time scales differ by a too large fraction. With the points (i)-(iii) in mind we discuss the occurence of decoupling for the carbon ions, starting with C ii. For models with M > 40M only maximal accelerated (x = 1) lines of C ii were expected to decouple. Because of the high densities in these winds ions with higher charges were not expected to decouple, because a higher charge decreases the slowdown time even further. In our simulations C ii is not found to participate to the line driving in models with M > 40M, because the high temperature and density in these models is not in favor of C ii lines. As Fig.(6.4) illustrates, the ionization of C ii is very low for high temperature models. This explains obviously why C ii is not decoupeld in the M > 40M models. In the lower luminosity models, where C ii does participate, some strong lines, e.g., C ii λ1010 in a 25M model, are found to participate primarily in the critical region, where the velocity gradient suddenly jumps upward due to the onset of the radiative line force. As a result of mass continuity, the density declines steeply in this region where the velocity makes an upward jump. A lower density implies that higher ionization stages recombine to lower ionization stages, as can be seen from Fig.(6.4), where the highest ionization stage shows a dip and the lower ionization stages have a peak exactly around the location where 77

84 Chapter 6: Results of numerical calculations Figure 6.2: Dependence on ion charge. Top: same as Fig.(6.1), here for C ii. Bottom: behaviour of log Q (blue dots) and log Px (colored stars) as a function of effective temperature, for models at the ZAMS, at solar metallicity, at the location in the wind where v/v = 0.5, for C ii (black stars), C iii (blue stars) and for C iv (aqua stars). For a definition of log Q and log Px, please see Sec.(3.1). 78

85 6.1 Numerical results on the dependence on ion charge 100 1e time [s] time [s] e-05 C II drift C II slow 1e-06 C III drift C III slow C IV drift C IV slow 1e shell nr 1e-06 C II drift C II slow 1e-08 C III drift C III slow r_sonic C IV drift C IV slow 1e shell nr Figure 6.3: Progress of t S and t r_sonic D throughout a 7 (left) and a 15M (right) model at the ZAMS. The horizonal axis indicates the shell number, as the wind is divided into 60 concentric shells, see Sec Red symbols indicate the timescales for C ii, with plus-signs indicating the drift time of C ii, and x indicating the slow down time. Green symbols indicate the drift time (stars) and the slow-down time (open boxes) for C iii, and blue symbols indicate the drift time (closed boxes) and slow down time (open circles) for C iv. The figure shows that the slow down time is essentially the same (apart from a slight offset due to the difference in charge) for all ions at a given location. When decoupling occurs for a given ionization, this can therefore be attributed to the local line force being larger (smaller drift time). For most lines the drift time is longer than the slow down time, and as a result, no decoupling occurs. Decoupling only occurs relatively far out in the wind, around r R, for C iii 79

86 Chapter 6: Results of numerical calculations velocity [km/s] velocity [km/s] v(r) height [Rstar] v(r) height [Rstar] 1 M=07 Msun IT = 01 1 M=15 Msun IT = 01 ionization fraction of C ionization fraction of C C I C II CIII C IV C V HEIGHT [in RCORE] C I C II CIII C IV C V HEIGHT [in RCORE] 1 M=09 Msun IT = 01 1 M=25 Msun IT = 01 ionization fraction of C ionization fraction of C C I C II CIII C IV C V HEIGHT [in RCORE] C I C II CIII C IV C V HEIGHT [in RCORE] Figure 6.4: Progress of ionization fraction of several carbon lines as a function of radial distance [in R ] from the star, for a 7 (top, left), 9 (bottom,right), 15 (top,left) and 25 (bottom,right) solar mass star, at the ZAMS. We expected decoupling of carbon ions to occur primarily in ZAMS models, where the wind density is the highest. 80

87 6.2 Numerical outcome of the dependence on x the velocity increases steeply. The ions are not found to decouple here, because the density is still too high. In the low luminosity models, with M 12M some C ii lines are also found in the outer layers because of the lower temperature and density in the outward layers. Here they experience a lower line force, and as a result it takes longer to accelerate them to the thermal velocity, as can be seen from Fig.(6.3). As a result, C ii is never found to decouple. In general, it is found that the local line acceleration on the strongest accelerated C iii line, which is the C iii λ977 is approximately one order of magnitude larger than the line acceleration on the strongest accelerated C ii and C iv line. As a result the drift time of C iii is approximately one order of magnitude lower than that of the strongest accelerated C ii and C iv line. Because of the large line force due to the strongest C iii line, as a result, the larger density in the MS models between 7 and 15 M is overcome, and as a result C iii is found to be decoupled. In the later stages of the MS the line force on C iii no longer balances the density effect, because the mass-loss rate increases as the star moves to higher luminosity along the MS. This explains why C iii is not decoupled in the 7th evolutionary stage of a 15M model, for example. In the 20 and 25 M models, the numerically obtained mass-loss rate is roughly a factor of 2 less than predicted from the mass-loss recipe. The slight decrease in wind density in combination with the strength of the C iii λ 977, results in decoupling of C iii. This means that for C iii ions for which decoupling was found x 1 is generally a good approximation to indicate the line force. This is because decoupling occurs for precisely those lines that are strongest accelerated (unless in a very low density region or wind). At last, the C iv line was predicted to decouple in the models in the range between 9 and 15 M. The reason why C iv is not found to decouple in our simulations is primarily due to the lower line force on C iv, as can be seen from Fig.(6.3). Note that the higher charge has only a minor impact on the occurence of decoupling, as pointed out before. In the case of C iii the strength of the line force was able to overcome the larger density in the models in the range between 9 and 15 M. For C iv this is not the case, and as a result, no decoupling of C iv occurs at v/v Numerical outcome of the dependence on x Fig.(6.5) shows the numerical analogue of Fig.(4.3), to illustrate the dependence of decoupling on the line strength parameter, x f osc F ν /F νmax g ion. Essentially we test here the dependence of our decoupling criterion on the local line force. The top panel of Fig.(6.5) was made for 0.05 < x 0.5, and the lower panel was made for x For x in the range between 0.5 and 1 the reader is referred to the lower panel of Fig.(6.1). When comparing the semi-analytical predictions for weakly accelerated (x < 1) ions (see top and lower panel of Fig.(4.3) with the numerical outcome (see top and lower panel of Fig.(6.5) ), we see that the results are generally in agreement with the predictions, i.e., no decoupling is found. 81

88 Chapter 6: Results of numerical calculations This is because at a given location in the wind, the atomic transitions in which a given ion is most likely to absorb a photon is the transition with the largest oscillator strength, and with a frequency near the fluxmaximum. As a result, the local line force due to that line is large, which causes it to be more easily decoupled, if the density of the field particles is not too high. In the MC code it is these strong lines that are the most likely to be selected to absorb momentum from the radiation field. Since decoupling occurs precisely for those lines that are strongest accelerated (when fixing the density), the approximation made that x 1 is reasonable. It should be mentioned here though that some care should be taken, and a subtlety with respect to the line strength in connection with the occurence of decoupling is mentioned in the discussion. The lack of decoupling of C iii at x 0.1 in the 9M model can be attributed to the wind density being higher in our numerical models, as compared to our analytical predicted value, as pointed out in the previous section. Due to the higher wind density this obviously results in less favorable conditions for decoupling to occur, especially at low x values. 6.3 Numerical outcome of dependence on radial distance Fig.(6.7) shows the HR diagram, with the numerical models indicated for which C iii decouples, at a location in the wind where v/v 0.2 (top) and at v/v 0.9 (bottom). For a comparison with decoupling of C iii at v/v = 0.5 the reader is referred to Fig.(6.1). Fig.(6.8) shows the numerical analogue of Fig.(4.6), both made for the ZAMS. Note that we took all decoupled events at a location before the location of interest into account as well, because of the assumption made that once an ion is decoupled it stays decoupled. When comparing the semi-analytical predictions with the numerical outcome, in general agreement is reached. Decoupling tends to occur for more models as the ion under consideration is further out in the wind. Early in the wind, decoupling was expected to occur for essentially the entire MS (before the bi-stability jump) for the models with masses between 9 and 15 M. Our detailed calculations reveal however, that decoupling only occurs at the ZAMS for the 12 and 15 M stars, and not at the early stages of the 9 M star. This can be attributed to the mass-loss rate being larger on the red-side of the jump, as pointed out before. As a result, the global wind density is larger, and it is therefore harder to decouple early in the wind, where the local wind density is even larger. The decoupling that is found in the fourth evolutionary stage of the 9 M model can be explained by the global wind density being smaller. As a result, decoupling of C iii can occur earlier in the wind. Note that it is a only a small effect, as C iii actually does decouple in the earlier MS stages, yet this occurs at a position that is slightly more outwards, to wit, at a radial distance of 1.4R. In all models the C iii line that is decoupled is again the 977Å resonance line. In case we consider decoupling of C iii far out in the wind, at v/v 0.9, it is seen that in 82

89 6.3 Numerical outcome of dependence on radial distance Figure 6.5: Dependence on line strength parameter x. Symbols as in Fig.(6.1). Figure made for weakly accelerated x = 0.1 [top], and even weaker accelerated x = 0.01 [bottom] C iii lines. For strongly accelerated lines, please see the lower panel of Fig.(6.1) 83

90 Chapter 6: Results of numerical calculations Figure 6.6: Dependence on x. Symbols as in Fig.(6.2), here plot made for models at the ZAMS, at Z = Z, at the location in the wind where v/v = 0.2, for strong C iii lines, with x = 1 (black stars), weak lines, with x = 0.1 (blue stars), and for weaker lines, with x = 0.01 (aqua stars). 84

91 6.4 Numerical outcome of dependence on metallicity general, predictions agree with numerical findings. However, for the model with M = 40M decoupling of C iii was not predicted to occur at v/v = 0.9. However, in our numerical models it does occur. This is because the numerically obtained mass-loss rate of the 40M model is smaller than the predicted value of Ṁ. This causes the numerical value of log Q to be smaller than the predicted value of log Q, and in the case of the strong C iii λ977 lines far out in the wind this is lucrative for decoupling. Another effect that should be mentioned when considering decoupling far out in the wind, is that far out in the wind the wind temperature is smaller than deep in the wind. In our analytical calculations we fixed the temperature throughout the wind. This means that the semi-analytical predictions are based on a wind temperature that is generally too high. In the inner wind regions, this assumption is justified, as the wind temperature is approximately equal to that of the star. However when we go further out in the wind the actual wind temperature drops. Note that this is only a minor effect, because the thermal velocity determines both the slow down time and the drift time. The effect on the slow down time is slightly larger, which causes decoupling to occur in regions where the temperature is generally lower. 6.4 Numerical outcome of dependence on metallicity Fig.(6.9) shows the HR diagram, with the numerical models indicated for which C iii decouples in wind models at Z = 0.1Z. Fig.(6.10) shows the numerical analogue of Fig.(4.8). At low metallicity the line driving mechanism becomes less effective because of the lower number of absorbing ions in the wind. This results in a less dense wind (lower Ṁ). Therefore, when converging the Z = 0.1Z models, the amount of photon packages was increased to photons, to have good statistics. Nonetheless, for the lowest stellar luminosity models (M = 7, 9 and 12M ), at most of the MS, the line driving mechanism was too inefficient for the radiative acceleration to be consistent with the input mass-loss rate. In other words, for these models convergence was not obtained. These models were therefore left out of our considerations. Note that as the predicted mass-loss rate of these models is of the order of M /yr, the mass-loss is not an important parameter in these low Z low luminosity models. Therefore, whether or not decoupling occurs is not important for the evolution of these models. We will come back to this point in a later section. When comparing our predictions with the numerical outcome, it can be seen that generally the numerical findings are in agreement with the predictions. However, for the 20 and 25M models, decoupling of C iii early in the wind does not occur at the ZAMS in our numerical models. In fact, in these models C iv tends to decouple early in the wind. In the 20M model C iii decouples further out in the wind, at a radial distance r 3R, whereas it is not found to decouple in the 25M model. This is because in the simulations, in the 25M model C iii lines only occur early in the wind, where the velocity increases steeply and as a consequence C iv recombines to C iii. Despite the steep density decrease due to the velocity increase, the density is still too high for decoupling to occur at this point in the wind. Due 85

92 Chapter 6: Results of numerical calculations Figure 6.7: Dependence on location, as parameterized in v/v, in the wind. Symbols as in Fig.(6.1).At location v/v = 0.2 (top) and v/v = 0.9 (bottom). For comparison with decoupling of C iii at v/v = 0.5, please see the lower panel of Fig.(6.1) 86

93 6.4 Numerical outcome of dependence on metallicity Figure 6.8: Dependence on location. Symbols as in Fig.(6.2), for models at the ZAMS, at Z = Z, for C iii lines at the location in the wind where v/v = 0.2 (black stars), v/v = 0.5 (blue stars), and v/v = 0.9 (aqua stars). 87

94 Chapter 6: Results of numerical calculations to the low mass-loss rate the density in the wind is low. As a result the slow down time is increased significantly, which causes C iv lines, which do occur further out in the wind, to decouple. The left panel of Fig.(6.10) was made for the ZAMS and the right lower panel was made for an intermediate MS stage (i.e., at evolutionary time step equal 7). Figure 6.9: Dependence on stellar metallicity. Similar to previous HR diagrams, symbols as in Fig.(6.1), here to show the effect of varying the metallicity. Figure made for Z = 0.1 Z, for C iii lines at a location where v/v = Maximal effect of ion-decoupling on Ṁ and on v Apart from investigating where in the HR diagram ion-decoupling occurs, we also estimated the effect of ion-decoupling on the stellar wind properties. This was done by the method as described in Sec.(5.2.1). In this method we essentially subtracted the total photon energy lost in decoupled events from the total energy extracted from the radiation field to accelerate the bulk. Note that in the previous sections, we choose to show the occurence of decoupling across the HR diagram for carbon ions, but, as can be seen from Table 6.3, many more ion species were found to decouple at several locations in the winds of the stars studied. One such element that is found to decouple in many models is Si iii, as can be seen in Fig.(6.11). This plot was made at a location where v/v = 0.5, to point out that decoupling of absorbing ions is encountered in many stars. Table 6.1 and 6.2 list our findings for a subset of the models calculated. It is found that in solar metallicity stars, decoupling of ions extracts most energy that would otherwise be used to accelerate the bulk in the 7 and 12M models. In both cases 88

95 6.5 Maximal effect of ion-decoupling on M and on v Figure 6.10: Dependence on stellar metallicity. Symbols as in Fig.(6.2), here for C iii lines at Z = 0.1Z Figure 6.11: Occurence of decoupling for Si iii across HR diagram, at a location in the wind where v/v = 0.5. Symbols as in Fig.(6.1). 89

96 Chapter 6: Results of numerical calculations Table 6.4: Numerically obtained values of D mom, in the case of a coupled wind, and difference log DṀ (in dex) when taking decoupling into account, and attributing the entire effect to a change in Ṁ, and difference D v, when attributing all effects to v. The values listed are the stellar mass, logarithm of luminosity, effective temperature, terminal velocity, mass-loss rate, energy loss to decoupled events, and DṀ and D v. Symbols - and x are similar as in Table 6.1. M log L T eff v log Ṁ L dec L bulk log D mom log DṀ log D v [M ] log[l ) [10 3 K] [R ] [log M /yr] [gcm/s 2 ] [dex] [dex] x 90

97 6.5 Maximal effect of ion-decoupling on M and on v approximately 65 % of the radiative energy used to accelerate the bulk is lost to decoupled ions. In both cases, when ion-decoupling impacts only the mass-loss rate, keeping the terminal velocity constant, the resulting mass-loss rate is a factor 3 lower than in the coupled case. Alternatively, if we keep the mass-loss rate fixed, and assume that ion-decoupling only has an effect on the terminal velocity, than the terminal velocity is reduced by a factor of about 2. In the models at 0.1Z, decoupling has a more significant impact on the stellar wind properties, as may readily be verified from Table 6.1. In the case of a 12M model for example, the energy loss in the decoupled events is larger than the energy needed to accelerate the bulk out of the potential well of the star at a rate equal to Ṁcoupled. In other words, in this model the energy lost to the decoupled event is too large to drive an outflow characterised by the mass-loss rate of the fully coupled situation. This means that in this case, the mass-loss rate must diminish in order for an outflow to occur, and as a result, ion-decoupling is sure to impact the mass-loss rate and not only the terminal velocity of the bulk. In connection with the weak wind problem, we calculated the value of the so-called modified wind momentum D mom, which is related to the luminosity according to the so-called modified wind momentum-luminosity relation (WLR ; e.g.,kudritzki and Puls (2000)): ) log D mom log (Ṁv R/R xlog(l /L ) + log D 0 (6.1) where D mom and D 0 are units of grcm/s 2, the coefficient D 0 and slope x vary as a function of spectral type and metal content, and the mass-loss rate Ṁ, terminal velocity v, and radius R are in the usual cgs units. The WLR essentially expresses that the mechanical momentum flow of a stellar wind Ṁv is a function of the photon momentum rate L /c provided by the stellar photosphere and interior. According to the theory of stellar winds, the reciprocal value of x essentially expresses the the line-strength distribution (in terms of weak and strong lines). Therefore the value of x is expected to change if the temperature or the metal content of the star changes. Vink et al. (2000). We calculated the values of D mom when assuming a coupled wind and when taking decoupling into account. Since we only have information with respect to the maximum effect of ion decoupling on Ṁv, we have calculated the new value of D mom both in case ion-decoupling is assumed to control Ṁ, keeping v unchanged, and in case ion-decoupling is assumed to have an impact on v, keeping Ṁ unchanged. We listed the obtained difference (in dex) with respect to the coupled D mom in Table 6.4. As ion-decoupling is found to have most impact in ZAMS models, we only listed our findings for models at the ZAMS. The table shows that the modified wind momentum D mom is at most changed by a factor of approximately 3 in case decoupling is assumed to have an impact on Ṁ. Apart from investigating the effect of ion-decoupling on the acceleration of the bulk, we also investigated the impact of decoupling on the decoupled ions. After all, after being decoupled from the bulk plasma, the ions are no longer assumed to share their momentum with 91

98 Chapter 6: Results of numerical calculations ion R v,i/c [R ] C iii Si iii Si iv Table 6.5: Estimate of the terminal velocity of the decoupled ions in the case of a 12 M star at the ZAMS (solar metallicity). the rest of the plasma, but they can still absorb photons from the radiation field. This will result in an enormous acceleration of the decoupled ions, as they no longer lose momentum to the bulk plasma. The terminal velocity of a given decoupled ion will depend on the location in the wind where it decoupeld for the first time, but also on the flux at the frequencies of its atomic transitions. The way in which we calculated the maximum terminal velocity with which a decoupeld ion escapes from the wind is described in Sec.(5.2.2). We show the maximum velocity of the decoupled ions for a 12M model in Table 6.5. Note that this result is approximately one order of magnitude larger than the result obtained by Babel (1995), who found that typically v,i 6000km/s in a pure metallic wind. 6.6 Discussion of numerical investigation In this chapter we presented our numerical findings on the occurence of decoupling across the HR diagram as a function of ion charge, location, line acceleration, and metallicity. In general, numerical findings and semi-analytical predictions with respect to the occurence of decoupling across the HR diagram are in agreement. The occasions for which a different result was found, were discussed in the relevant sections. With respect to the differences between semi-analytical method and the numerical calculations, some additional remarks are listed below. Shallow bi-stability jump in low luminosity models Models for which the predictions and numerical findings do not agree are mainly models for which lines of a given ionization stage and at a given location do not occur due to temperature and density effects, and models for which the numerically obtained mass-loss rates are too high, compared with the mass-loss rates obtained from the mass-loss recipe of Vink et al. (2001). The latter cause is found to be the case in models between roughly 15 and 7M. A possible explanation for this discrepancy is provided below. In our semi-analytical predictions we extrapolated the mass-loss recipe of Vink et al. (2001). As the mass-loss recipe includes the bi-stability jump, the decoupling that was found on the high temperature side of the jump occured essentially because of the imposed bi-stability jump of a factor of about 5-7 in Ṁ. Stars on the blue side of T jump were expected to have 92

99 6.6 Discussion of numerical investigation a lower mass-loss rate, because of the lower ionization fraction of Fe iii in the inner wind region, where the mass-loss rate is set. Table 6.2 shows that the mass-loss rates in the models with masses approximately between 15 and 9M are respectively a factor 3 and 7 larger than the predictions based on the extrapolated mass-loss recipe. This results in higher wind densities in these models than the densities, as parameterized in Q, that followed using the mass-loss recipe. The higher mass loss rates on the blue side of the jump result in a bi-stability jump that is smaller than the predicted jump in the mass-loss rate when going from the hot side of the jump to the cool side. Fe iv does recombine to Fe iii in the lower part of the wind around the predicted jump temperature, as can be seen in Fig.(6.12). In the 12M model, the upward jump in the mass-loss rate from the blue side of T jump to the red side is a factor of approximately 7. In the 9M model, the recombination of Fe iv to Fe iii results in an upward jump in Ṁ by a smaller factor, i.e., a factor of about 3-4. This means that in low luminosity stars, the bi-stability jump in Ṁ does occur, but it is not as large as in higher luminosity models. An explanation could be that this is because in the 9M model on the red side of the jump, Fe iii lines are important, but not contributing as significantly to the mass-loss, as they do in the 12M model, where Fe iii lines contribute by far most to the line force in the lower part of the wind, as Fig.(6.13) shows. This might explain why the wind density of the 9M model is not changed very significantly from one side of the jump to the other, and why it is generally larger on the blue side of the jump than as predicted from the extrapolation of the mass-loss recipe. The higher wind density on the blue side of the jump in low luminosity models results generally in less chances for decoupling. With respect to Fig.(6.13) we point out that since we do not actually solve the equation of motion throughout the entire wind in the MC simulation, but only in the hydrostatic part where the contribution of the line force is not taken into account, we do not have any information with respect to the actual critical point that would follow if we had solved the equation of motion throughout the wind. Since we still wanted to know approximately which elements contribute most to the mass-loss driving, the way in which this was implemented in the MC code was by registering the amount of photon momentum transferred in the individual photon interactions to the different ion species up untill the sonic point, which is located at approximately r 1.01R. The results as shown in Fig.(6.13) therefore merely provide an indication of the contribution of the different ion species to the line force around the region where Ṁ is set. Line-strength For most of our analytical predictions we assumed an optimal contribution to the line force, by setting the value of x to unity in all models across the HR diagram. This introduced some differences in stars where the ion under consideration does not participate as strongly (or at all) to the line force at the location under consideration, or because the local density was not 93

100 Chapter 6: Results of numerical calculations velocity [km/s] velocity [km/s] v(r) 1e-04 1e height [Rstar] v(r) 1e-04 1e height [Rstar] 1 M=09 Msun IT = 03 1 M=09 Msun IT = 07 ionization fraction of Fe ionization fraction of Fe Fe I Fe II FeIII Fe IV Fe V HEIGHT [in RCORE] Fe I Fe II FeIII Fe IV Fe V HEIGHT [in RCORE] 1 M=12 Msun IT = 08 1 M=12 Msun IT = 11 ionization fraction of Fe ionization fraction of Fe Fe I Fe II FeIII Fe IV Fe V HEIGHT [in RCORE] Fe I Fe II FeIII Fe IV Fe V HEIGHT [in RCORE] Figure 6.12: Progress of ionization fraction of several Fe lines as a function of radial distance [in stellar radii] from the star, for a 9 solar mass star, at an evolutionary stage where T eff is just above the recombination temperature T jump of Fe iv to Fe iii (left), and at an evolutionary stage where the effective temperature has shifted below T jump (right). This figure shows that the Fe iii fraction generally dominates the Fe ionization fraction in the region of the wind where the velocity gradient is largest, on the red side of T jump, both in the 9M model and in the 12M model. This figure also shows the effect of a decreased density on the ionization structure. In the subsonic region, at the location where the wind accelerates most, as a result, the density shows a downward jump because mass-continuity should hold everywhere in the wind. The effect of a decreased density is that there are less electrons available. Therefore Fe iv reionizes to Fe iii in this part of the wind. 94

101 6.6 Discussion of numerical investigation 0.06 M=9Msun Teff=23kK IT04 total contribution 0.06 M=9Msun Teff=22kK IT06 contribution to line force Contribution L ion / L all lines per element Contribution L ion / L all lines per element III III 0 I III III III IV III IV III III III H C N O Si P S Cl Fe Ni ION 0 IV III I III III III III H C N O Si P S Cl Fe Ni ION 0.06 M=12Msun Teff=23kK IT08 contribution to line force 0.06 M=12Msun Teff=23kK IT09 contribution to line force III Contribution L ion / L all lines per element Contribution L ion / L all lines per element III I III III III H C N O Si P S Cl Fe ION I III III III II II III III III H C N O Si P S Cl Cr Fe Ni ION Figure 6.13: Contribution to the line force by several ions in the lower part of the wind, in a 9M model (top), and in a 12M model (bottom), both on the blue (left) and on the red (right) side of the recombination temperature of Fe iv to Fe iii 95

102 Chapter 6: Results of numerical calculations in favor of decoupling, which was mostly due to a difference in predicted (from extrapolations) mass-loss rates, and numerically obtained mass-loss rates. However, in general, the assumption that x 1 turned out to be a good approximation for the decoupled ions, as they turned out to be mostly strongly accelerated ions. This is because in the MC method the atomic transitions which are most likely to absorb a photon are those transitions for which the line optical depth is largest. Since the line optical depth is essentially proportional to the oscillator strength of the transition f osc the number density n i of the ion that absorbs that line the inverse of the velocity gradient (dv/dr) 1 of the wind (Sobolev approximation), this means that the transitions in which an ion is most likely to absorb a photon are those transitions for which the line optical depth has the largest value. Generally this means that at a given position, the ion is most likely to absorb photons into transitions with the largest value of f osc. With respect to this observation, we point the reader to a subtle effect. This is the fact that, as argued by Castor et al. (1975), the radiative acceleration due to a large ensemble of metallic lines 1, was found to be a factor 100 larger than that obtained when taking only a few strong resonance lines into account, as was done by Lucy and Solomon (1970). The validity of the factor 100 is not discussed here. The point here is that in general, metallic lines are considered to be effective line absorbers not so much because of the strength of their individual atomic transitions but rather because they have a very large number of transitions available for the radiation field to act upon. We defined the drift time as the time needed to accelerate a given ion to a radial drift velocity equal to the thermal velocity of the bulk plasma, i.e., the hydrogen ions, without interacting with the field particles in the meantime. The way in which this time scale was calculated in the MC method was by calculating the line force for each individual interaction on a given ion, and then calculating the drift time from its definition t D v th /g ion. In doing so, it is implicitly assumed that the ion under consideration is accelerated to the thermal velocity by N number of identical photon absorptions, i.e., line absorptions into a single transition only. It is not taken into account that the ion can also be accelerated via other transitions, which may be more or less effective. This was done, because in the MC method we cannot take the effect that an ensemble of lines has on the acceleration of a single ion into account. Essentially, this is because in the MC code the path of the emitted photon is traced through the wind and the momentum loss of the photon is registered per interaction. In this method one only considers a single transtion at a time, and calculates the amount of photon energy lost to that transition. This goes well as long as the acceleration of the individual ions is not the topic of interest. However, if one is interested in the individual ions rather than the bulk acceleration, then in principle one should take into account that an individual ion is accelerated by the ensemble of spectral transitions which are available to the radiation field to act upon at a given location in the wind. This is in particular important for ions which are accelerated by many predominantly 1 Notably their results were obtained by considering many lines of a C iii ion 96

103 6.6 Discussion of numerical investigation weak lines, such as Fe iii. Note that the ensemble of available transitions is expected to shift with the ionization and excitation structure throughout the wind. However, since the local line force on a given metallic ion after N photons absorptions is always dominated by the atomic transition with the largest oscillator strength, the error introduced due to the neglect of the ensemble of lines is limited, as long as the ion does not change it s excitation structure while accelerating to the drift velocity of the field particles. Note that in the derivation of the average momentum increase per absorption mv, essentially is is already assumed that the ion absorbs only photons with frequency ν 0. One way to deal with this problem could be by looking at all interactions in a thin concentric shell involving a given ion species, that is not necessarily in the same excitation stage, as this may change in principle after one photon absorption, and calculating the cumulative line force due to all interactions involving the ion species under consideration. Care should be taken here not to add the same interaction twice. In doing so, one takes account of the different interactions a given ion may undergo as a result of the ensemble of atomic transitions available at a given location in the wind. 97

104

105 CHAPTER 7 Discussion and conclusion In this study the occurence of decoupling of absorbing active ions from the non-absorbing passive bulk plasma is investigated across the Hertzsprung-Russell (HR) diagram, for stars at solar metallicity and at Z = 0.1Z, and the maximal effect of decoupling on the stellar wind properties (mass-loss rate and terminal velocity) is estimated. We discuss our findings below. Discussion Location of ion-decoupling in the HR diagram Ion decoupling has been studied by many authors [e.g.,springmann and Pauldrach (1992); Babel (1995); Krtička and Kubát (2000)]. Though the approaches to the study of ion-decoupling vary, the consensus is that ion-decoupling is not expected to occur in normal (O-main sequence, O giants, and OB supergiants) O-type stars which feature dense winds, but rather in the low-density winds of B-type stars. In some cases, late O-type stars are also included among the possible candidates for which a one-component description of the flow may fail due to ion-decoupling [notably, Springmann and Pauldrach (1992); Krtička et al. (2008)]. In most of these studies the hydrodynamic structure of a multicomponent wind is calculated using a parameterization of the line force in an isothermal wind [e.g.,krtička and Kubát (2000)], or the effect of ion-decoupling on the temperature structure of the wind is calculated without actually solving the hydrodynamic equations [Springmann and Pauldrach (1992)]. In our approach to the problem of ion-decoupling we utilize a Monte Carlo method in which the cumulative energy losses due to scatterings and absorptions of photons is registered, as they traverse the stellar wind. This way, we have detailed information about the local conditions that determine whether, and which, ions decouple, and at what location in the wind this occurs. 99

106 Chapter 7: Discussion and conclusion Our findings are that decoupling of strongly accelerated ions is encountered in a broad range of stellar winds. For solar metallicity models it is found that ion-decoupling occurs predominantly in low -luminosity (L 10 5 L ), high temperature models on the main sequence, as was discussed in Sec To summarize, the approximate upper limit in terms of luminosity can be attributed to the mass-loss rate, which scales approximately as Ṁ L 2 [Vink et al. (2001)], and is in general sufficiently large in models with L 10 5 L to prevent ions from decoupling of the bulk plasma. In later evolutionary stages decoupling was not found to occur. This is attributed to the fact that as a star evolves redwards across the HR diagram, and passes the temperature limit beyond which Fe iv recombines to Fe iii, its mass-loss rate increases due to the bi-stability jump [Vink et al. (1999)]. As a result, ion-decoupling is unlikely to occur in winds of stars with effective temperatures lower than the typical temperature at the bi-stability jump, T jump K. This study shows that ion-decoupling is typically expected to occur in the winds of stars that reside in the region of the HR diagram that is spanned by L 10 5 L on the horizontal axis and the temperature at the bi-stability jump where Fe iv recombines to Fe iii on the vertical axis. This region is approximately indicated in Fig.[7.2] by the green triangle. Figure 7.1: Approximate indication of the location in the HR diagram where ion-decoupling may typically occur. 100

107 Importance of ion-decoupling on the mass-loss rate and terminal velocity In order to obtain an estimate of the effect that the decoupling of absorbing ions from the bulk plasma has on the stellar wind properties, we calculated the effect of ion-decoupling on the energy that is extracted from the radiation field to accelerate the wind. From this we have calculated the maximum effect that ion-decoupling may have on the mass-loss rate, and on the terminal velocity, according to the procedure as described in Sec This was done for both solar metallicity models, and at Z = 0.1. The results are shown in Table 6.1 and 6.2. Maximum effect at solar metallicity It is found that at solar metallicity the energy lost to the decoupled events is maximal 65 % of the total energy used to accelerate the bulk. This maximal effect was found in a 7 and a 12 M model at the zero-age main sequence (ZAMS). As a result, the maximal effect on the mass-loss rate is a reduction of 65 %. In all other models, this factor is smaller, as can be seen from Table 6.1. As pointed out in the introduction, in low -luminosity stars (L L ), there is the problem that is commonly referred to as the weak wind problem. The problem at hand is that for stars with L L, notably, main-sequence O-type stars, the observed massloss rates, or, more precisely, the wind momenta as parameterized in the variable D mom (see Sec.[6.5]), disagree strongly with theoretical predictions, with a discrepancy of a factor of 100. Several explanations have been put forward to explain this problem. Considering the fact that the similarity between the luminosity range (L L ) below which the mass-loss rates are observed to be two orders of magnitude lower than the predicted values, and our findings with respect to the range in the HR diagram below which ion-decoupling may be expected to occur (see discussion above), it is tempting to consider the possibility that the weaker winds could be explained by ion-decoupling. Therefore we investigate whether the problem can be attributed to the decoupling of absorbing ions in the weak winds of these stars. To this end, we calculated the corresponding value of the modified wind momentum log D mom, following its definition. For an extended discussion on the weak wind problem, the reader is referred to Mokiem et al. (2007). Our findings with respect to the change in log D mom due to decoupling are listed in Table 6.4. We overplotted our findings for both a 40 M model and a 12 M model in the so-called modified wind-momentum -luminosity distribution [Mokiem et al. (2007)], see Fig.(7.2). Note that since the effect of ion-decoupling has most impact on D mom when assuming ion-decoupling controls the mass-loss rate (Ṁ) rather than the terminal velocity, the value of D mom that is plotted is that found when attributing all effects of ion-decoupling to a change in Ṁ. We find that when taking the effects of ion-decoupling on the mass-loss rate maximal into account, this results at most in a reduction of the corresponding value of log D mom by a factor 101

108 Chapter 7: Discussion and conclusion Figure 7.2: Modified wind-momentum -luminosity distribution of observed data [black and grey circles], and an emperical fit [solid and dashed line], by Mokiem et al. (2007), with an overplot of our calculated models, for a coupled wind [green circles], and a decoupled wind [red circles]. 3 in the case of the 12 M star. In all other models, the reduction of log D mom due to iondecoupling was found to be less. Therefore ion-decoupling can not explain the discrepancy in the mass-loss rate of a factor of 100. We conclude that the weak wind problem can not be explained by ion-decoupling, even when taking the maximum effect of ion-decoupling on the mass-loss rate into account. Maximum effect at Z = 0.1Z The role of massive stars at low metallicity has been reviewed extensively by many authors. Massive stars are considered to have played a key role in the evolution of galaxies in the early universe. As the first stars ever formed are thought to have been very massive ( 10 2 M ), short lived stars [Bromm et al. (1999)], these stars are thought to have have been important for the re-ionization history of the intergalactic medium at z 6 because of their intrinsically strong radiation field [Bromm et al. (2001)]. Massive, extreme metal poor stars are also thought to be the progenitors of gamma-ray bursts [Lamb and Reichart (2000)]. Because mass-loss determines the fate of massive stars to a large extend, it is important to study the behaviour of mass-loss at low metallicity. Therefore, we have investigated the 102

109 occurence of ion-decoupling in the winds of metal poor stars. This was done numerically for Z = 0.1Z (see Fig.[6.9]) models, and in a semi-analytical way for models at Z = 0.01Z (see lower panel of Fig.[4.7]). We calculated the maximum impact that decoupling may have on the mass-loss rate and on the terminal velocity of our models at Z = 0.1Z. The results are presented in Table 6.2. We found that at Z = 0.1Z the maximum radiative energy loss to the decoupled events is approximately a factor of 95 %, which was found to occur in a 12 M star at the ZAMS. This implies that the mass-loss rate is reduced at most by a factor of 95 %. Since the resulting radiative energy that is used to accelerate the bulk is less than the energy needed to accelerate the unchanged wind density out of the potential well of the star, this implies that the mass-loss rate has to be reduced, in order to drive an outflow at all. However, as was shown in Table 4.2, at these low metallicities, the maximal effect of a fully coupled stellar wind is that the initial mass is reduced by only 0.3%. This means that in these objects the mass-loss rate is unimportant for the evolution of the star. Therefore the reduction or even shutting down of mass loss in a stellar wind has no important impact on the mass, evolutionary path and supernova progenitor properties of the star. Note that the same holds in the case of very low metallicity stars. Since these stars were very hard, if not impossible, to calculate numerically, we have estimated the occurence of decoupling in a semi-analytical way. From our discussion in Sec.[4.4] in the case of solar metallicity stars and stars at Z = 0.1Z, we have confirmed that the semi-analytical calculations provide a good indication on the occurence of ion-decoupling throughout the HR diagram. Fig.[4.7] shows our analytical predictions for Z = 0.01Z. It turns out that in these models, decoupling is expected to occur for all main sequence stars. However, because of the extreme low mass-loss rate M /yr, it is not expected that ion-decoupling has an important effect on stellar evolution. Conclusion We have investigated the occurrence and impact of ion-decoupling in line-driven winds of early-type massive stars at solar metallicity and at Z = 0.1. Our final conclusions are as follows. Ion-decoupling can not explain the weak-wind problem. For stars in which ions may decouple relatively close to the star, i.e., in the regime where the mass-loss rate is set, the mass-loss rates assuming full coupling are already so low that no substantial impact on the mass, rate of spin-down due to mass-loss, evolutionary path, or supernova progenitor properties is to be expected. 103

110

111 APPENDIX A Derivation of the slow-down time In this appendix, the slow-down time used in our criterium for decoupling will be derived. Because of the similarity with the stellar case, the slow-down time is derived in close analogy to Chandrasekhar (1942), who considers encounters between stars in a galaxy. However in those cases where the physics of the problem of the encounter of ions differs from that discussed by Chandrasekhar this will be pointed out clearly in the text. We consider encounters between charged particles. In analogy to Chandrasekhar (1949), the encounters are governed by an inverse-square force acting between the colliding bodies, i.e. in our case the electrostatic force, which causes the resulting path of the colliding bodies to be altered in a way analogous to Brownian motion: due to the long range of the inversesquare force between the colliding bodies, it is solely the cumulative effect of a large number of distant encounters (which produce only marginal deflections individually), which produce sensible changes in the directions and magnitudes of the motions, just like in the Brownian case. Even though in the charged-particle case the colliding bodies exert an influence on each other via the electrostatic force, while in the Brownian case the (neutral) particles are primarily influenced by the molecules of the surrounding fluid, this is of no importance to the overall motion, since in both cases individual encounters hardly affect the motion of the particle under consideration: after all, in both cases it is the cumulative effect of a large number of distant encounters that are of importance. In accordance with Chandrasekhar (1941a) each encounter will be idealized individually as a two-body problem, taking place in some appropriately chosen fixed frame of reference. The orbit in the orbital plane of each particle with respect to the common center of gravity is a hyperbola. In the following section, we will derive the angular deflection of the relative velocity of a charged test particle due to an encounter with a charged field particle, using a two-body approximation. Because of the long range of the electrostatic force, when compared to the range of the Van der Waals force between neutral atoms, it is the effect of a large number of 105

112 Appendix A: Derivation of the slow-down time distant encounters, each of which individually produce only a marginal deflection, rather than the effect of close encounters, which may completely change the particle velocity at once, that determines the eventual path a charged test particle will follow when travelling through a gas of charged particles. Since it is the time it takes to alter the path of a test particle significantly we are after, essentially, we need to look at the net deflection after many encounters. However, first, we will derive derive the angular deflection π 2ψ of the relative velocity V in the orbital plane using a two-body approximation. A.1 Derivation of deflection angle using two-body approximation Adopting the same notation as Chandrasekhar (1942), we consider the effect of encounters on a test particle of mass m 2 with initial velocity v 2 during its motion through other particles, called field particles, which have mass m 1 and velocity v 1. The equation of motion of a particle m 2 with charge q 2 at position r 2 experiencing a field due to the presence of another particle m 1 with charge, q 1 at position r 1, and that of particle m 1 due to m 2, follows from Coulomb s law: m 1 d 2 r 1 dt 2 = q 1q 2 r 1 r 2 r 1 r 2 3 and m 2 d 2 r 2 dt 2 = q 1q 2 r 2 r 1 r 2 r 1 3 From these equations, the equation for the relative motion of the two particles can be readily obtained. Multiplying both equations by m 2, respectively, m 1, and subtracting the right hand equation from the left hand equation yields: Writing r = r 1 r 2 : d 2 dt 2(r 1 r 2 ) = q 1 q 2 r 1 r 2 r 1 r 2 3 ( ) m1 + m 2 m 1 m 2 d 2 r dt 2 = q 1 q 2 ( m1 + m 2 m 1 m 2 ) r r 3 = λ 1 r (A.1) where λ [in cm 3 s 2 ] is defined as: λ = q 1 q 2 m 1 + m 2 m 1 m 2 (A.2) the value of which is negative for a repulsive force and positive for an attractive force. The motion of the particles relative to each other is thus determined by the following scalar potential function: U = λ r = { < 0 opposite sign > 0 same sign 106

113 A.1 Derivation of deflection angle using two-body approximation which is negative in case we consider a system of opposite charges. That is, over time potential energy is converted into kinetic energy as the particles approach each other. Conversely, if U is positive, kinetic energy is converted into potential energy as the particles approach each other. Taking the cross product of Eq.(A.1) and r, we obtain the angular momentum via: r d2 r dt 2 = d ( r dr ) dt dt = 0 This cross product is equal to zero as both vectors are pointed in the same direction. Hence it follows that r dr h, where h is a constant vector denoting the angular momentum [in dt cm 2 /s], which is indeed independent of the time, as one would expect for a movement subject to a central force. χ r A ψ θ F D Figure A.1: Schematic representation of an encounter between two identical particles. In the two-body approximation, an encounter is depicted as an encounter between a particle at rest in one of the foci F and another particle moving around it at distance r and under an angle θ, with respect to the distance at closest approach, in a hyperbolic orbit. In case one considers an attractive force, the test particle can be imagined to be moving along the right branch of the hyperbola, whereas in case one considers a repulsive force, the test particle is depicted to be moving along the left branch of the hyperbola. When the reference frame is chosen such that the center of gravity A is stationary, each particle can be depicted as moving in a hyperbola relative to A. We adopt polar coordinates (r,θ) in the orbital plane. For a definition of r and θ the reader is referred to Fig.(A.1), where an encounter is shown for the case of a hyperbolic orbit. The 107

114 Appendix A: Derivation of the slow-down time Lagrangian function is similar to the Lagrangian described by Chandrasekhar and is: L = T U = 1 2 (ṙ2 + r 2 θ2 ) + λ r where T = 1/2(ṙ 2 + r 2 θ2 ) denotes the kinetic energy and U = λ/r is the potential energy of the system. The corresponding Lagrangian equations are: r = r θ 2 λ r 2 d dt (r2 θ) = 0 Eq.(A.4) yields the angular momentum integral: h = r 2 θ = constant (A.3) (A.4) (A.5) Substituting Eq.(A.5) into Eq.(A.3) yields: r = h2 r 3 λ r 2 (A.6) Analogue to Chandrasekhar we introduce u = r 1 and rewrite the foregoing equation using Eq.(A.5): r = d ( 1 ) d dt u 2 dt u = d ( 1 ) du dt u 2 dθ θ = d dt ( h du dθ = h 2 u 2d2 u dθ 2 ) = h d2 u dθ 2 θ (A.7) In analogy with Chandrasekhar (see Chandrasekhar, 1942, p ) the equation of (the relative) motion of the two particles in polar coordinates and expressed in units of the inverse radius u = r 1 follows by substituting Eq.(A.6) into Eq.(A.7): d 2 u dθ 2 = u + λ h 2 the general solution of which can be written as: u = u 0 cos(θ + θ 0 ) + λ h 2 (A.8) (A.9) where u 0 and θ 0 are constants of integration, which represent the initial values of u and du/dθ. In accordance with Chandrasekhar, the origin of θ is chosen such as to coincide with the direction of closest approach, i.e. minimum r and maximum u (note that this implies θ 0 = 0). Rewriting Eq.(A.9) we obtain: u = λ (ecos θ + 1) h2 for λ > 0 u = λ (ecos θ 1) h2 for λ < 0 108

115 A.1 Derivation of deflection angle using two-body approximation where e was defined as e u 0 h 2 / λ. Since u = 1/r the equation of the relative motion becomes: r = r = h2 λ h2 λ 1 ecos θ ecos θ 1 for λ > 0 for λ < 0 (A.10) (A.11) This is the equation in polar coordinates for a conic section with the origin in a focal point and eccentricity e. The difference between the λ > 0 and the λ < 0 case essentially arises due to the fact that the eccentricity e should always be greater than or equal to zero. This has however no effect for the shape of the orbit (e > 1 case), but as we shall see below, Eq.(A.11) cannot be used when considering circular, ellipse or parabolic orbits. Since r > 0 at all times, we have the restriction that: 1 + ecos θ > 0 for attractive forces (since then λ > 0), which means that we have for: e = 0 : a circle, 0 < e < 1 : an ellipse, e = 1 : a parabola, with restriction: 1 + ecos θ 0, hence θ π and θ π, e > 1 : a hyperbola, with the following restriction on θ: ( 2π cos 1 1 ) ( < θ < cos 1 1 ) e e ecos θ 1 > 0 for repulsive forces (since then λ < 0), which means that we have for: e = 0 0 < e < 1 Ø : since a repulsive force cannot result in closed orbits. e = 1 e > 1 : a hyperbola, with the following restriction on θ: ) ) 2π cos 1 ( 1 e < θ < cos 1 ( 1 e So far, we have considered the trajectory under an inverse-square central force, which has turned out to be a conic section with the center of force in one focus. The conic section that an orbit exhibits depends on the total energy E of the system, (for E < 0 the orbit is bound and is an ellipse, for E = 0 the orbit is unbound and follows a parabolic shape and for E > 0 the orbit is unbound and follows a hyperbola). Since we are deriving the deflection of the orbit of a particle suffered during an encounter, essentially, this means that we are considering an unbound system, for else, there would not be a deflection in the first place. Therefore, we will from now on only consider the equation of motion for a hyperbolic orbit, in other words, systems for which e > 1. For unbound systems like this, conservation of total energy implies that: T λ r = 1 2 V 2 (A.12) 109

116 Appendix A: Derivation of the slow-down time where T is the kinetic energy as defined before, and V is the relative velocity at infinite separation between the particles. One could argue whether or not E = 0 is an allowed solution as well. However, for the parabolic orbit associated with zero total energy, one can not define a unique deflection angle as is naturally the case for the hyperbolic orbit. Therefore, we do not consider the parabolic solution from this point on. We are now in a position to derive an explicit expression for the eccentricity e as a function of the physical parameters describing an encounter. As the total energy of the system is conserved we may use any distance to obtain an expression for the eccentricity. Hence, in order to avoid any unjustified dependance on θ, we use the distance of closest relative approach, in accordance with Chandrasekhar: r 0 = h2 λ 1 e ± 1 with { + for λ > 0 for λ < 0 (A.13) Note that the distance of closest approach is smaller for attractive forces (λ > 0) and larger for repulsive forces (λ < 0), which intuitively makes sense. This can also be seen in Fig.(A.1), where the relative orbit in case of an attractive force can be interpreted as the right branch of the hyperbola (which is curved around the stationary field particle in F), whereas the orbit in case of a repulsive force can be interpreted as the left branch of the hyperbola, around the stationary field particle in F. Substitution of Eq.(A.13) and Eq.(A.5) in the equation for the total energy, Eq.(A.12), yields: 1 2 V 2 = T λ r 0 = 1 2 r2 0 θ 2 λ r 0 = 1 2r0 2 (r0 2 θ) 2 λ = h2 r 0 2r0 2 = λ2 h 2 ( 1 2 (e ± 1)2 (e ± 1) λ r 0 ) (A.14) where it was used that at the distance of closest approach, r 0, ṙ vanishes. Solving this quadratic equation results in an expression for the eccentricity e: e 2 = 1 + V 2 h 2 λ 2 (A.15) Note that the same eccentricity is obtained both for an attractive and a repulsive force. Substituting Eq.(A.2), yields the following expression for the eccentricity: e 2 = 1 + m 2 1 m2 2 D 2 V 4 q 2 1 q2 2 (m 1 + m 2 ) 2 (A.16) where it was used that the angular momentum integral h can be formally written as h = DV, where D [in cm] is termed the impact parameter, which is the distance of closest approach in the absence of forces. Having derived an expression for the eccentricity, we now return to Fig.(A.1). Generally, for 110

117 A.1 Derivation of deflection angle using two-body approximation a hyperbola described by a given semi-major axis a, and semi-minor axis b, the eccentricity e is connected to a and b via: e 2 = 1 + b2 a 2 = 1 + tan2 ψ (A.17) where ψ is the angle between the semi-major axis and one of the asymptotes of the hyperbola. Rewriting Eq.(A.17), using that tan 2 ψ = 1 + 1/cos 2 ψ, we obtain: cos ψ = 1 e = m 2 1 m2 2 D 2 V 4 q 1 q 2 (m 1 +m 2 ) 2 (A.18) which also follows by considering Fig.(A.1), from which it can be seen that ψ is connected to θ via θ max = π ψ, and using that the angle between the asymptotes of the relative orbit can be derived from Eq.(A.10) and Eq.(A.11) for an attractive respectively repulsive force, by looking at the value of θ in the limiting case that r goes to infinity (for details, see Chandrasekhar (1942), p ), which yields: lim r at ( θ = θ max = cos 1 1 ) e θ = θ max = cos 1 ( 1 e ) for λ > 0 for λ < 0 The difference in the maximum allowed angle can be explained by looking at Fig.(A.1), from the perspective of a stationary field particle in F around which the relative orbit of the test particle is depicted, both in case of an attractive force (right branch of the hyperbola) and in case of a repulsive force (left branch of the hyperbola). In case we re considering a repulsive force, the maximum allowed angle θ max is then the angle ψ, whereas in case we re considering an attractive force, the maximum allowed angle is π ψ. Substituting e according to Eq.(A.16), and using that θ max = π ψ in case of an attractive force and that θ max = ψ in case of a repulsive force, we find: cos ψ = 1 e = m 2 1 m2 2 D 2 V 4 q 1 q 2 (m 1 +m 2 ) 2 (A.19) which relates to the deflection χ of the direction of the relative velocity V in the orbital plane (see Chandrasekhar (1942), p. 51 (eq.[2.301])), via: χ = π 2ψ (A.20) Now that we have derived the angular deflection χ of the relative velocity of a test particle due to an encounter with a field particle, we proceed with considering the effect of a large number of encounters and from that derive an expression for the slow-down time. This will be done in the following section. 111

118 Appendix A: Derivation of the slow-down time A.2 Derivation of slow-down time for distant encounters In the previous section, we have derived the angular deflection of the relative velocity of a test particle due to an encounter with a field particle, using a two-body approximation. Because of the long range of the electrostatic force acting between charged particles, when compared to the range of the force between neutral atoms, it is the effect of a large number of distant encounters, each of which individually produce only a marginal deflection, rather than the effect of close encounters, which may completely change the particle velocity at once, that determines the eventual path a test particle will follow when travelling through a gas of charged particles. Since it is the time it takes to alter the path of a test particle significantly we are interested in, essentially, we need to look at the net deflection after many encounters. Both Chandrasekhar (1942) and Spitzer (1962) have worked on essentially the same problemthat is, both have considered the path traversed by a test particle in an ensemble of field particles, under an inverse-square force. However, there is a subtle difference between Spitzer s and Chandrasekhar s approach. Spitzer was interested in the rate at which the velocity distribution of a group of identical test particles is progressively broadened in velocity space, i.e. the rate at which the perpendicular and parallel velocity components of a group of test particles disperses in a medium of field particles, similar to the diffusion of particles in an ordinary gas. Chandrasekhar, on the other hand, was interested in the rate at which the velocity component parallel to the initial random velocity of a single particle, which is part of a distribution of field particles, changes. It is because of this difference in motivation that Spitzer needed to consider a group of (identical) test particles, whereas in Chandrasekhar s case, it suffices to consider a single test particle. Therefore, Chandrasekhar derives the slow-down time for a single test particle that moves with an initial velocity that is in a random direction in an ensemble of other field particles, that are also moving with a random velocity. The test particle is assumed to be part of the same group of field particles, i.e. they constitute a single system of particles. In his derivation, he assumes an isotropic (Maxwellian) distribution of the velocity of the field particles, which causes the change in the perpendicular velocity component v of the test particle to vanish after many encounters, on symmetry grounds. Spitzer, on the other hand, considers a group of test particles, all of which move with the same initial velocity, in the same initial (z-) direction, in an ensemble of field particles, which move in a random velocity. In his derivation, Spitzer states that one cannot know the change of the perpendicular velocity component of a single test particle after N encounters a priori, but when considering a group of test particles, each moving with the same initial velocity in the same initial direction, the change of the sum of the perpendicular velocity components of the test particles does vanish after N encounters, when one assumes the distribution of velocities within the gas to be isotropic. In the derivation of Spitzer, the direction of motion we are to follow is fixed and therefore, he needs to consider a group of test particles, such that the thermal velocity component is averaged out of the initial velocity. 112

119 A.2 Derivation of slow-down time for distant encounters Both problems amount to the same however, since in Chandrasekhar s case the initial velocity of the test particle is the random velocity, i.e. it may be in any direction. Expression A.19 with A.20 yields the deflection angle χ in the orbital plane for a two body encounter. The ultimate goal of this section is to derive the slowing down time of a particle of mass m 2 and charge q 2, labelled a test particle, penetrating in a medium of particles with typical mass m 1 and charge q 1, labelled the field particles. Such a test particle m 2 will experience a series of small deflections and will eventually be slowed down in its initial direction, once the parallel component of its initial velocity has been reduced to zero. In other words, if we position ourselves in the frame moving along with the bulk velocity of field particles, and refer to the direction of the initial velocity of the test particle as the parallel direction, the question is: how long does it take for this parallel velocity component to reduce to zero as a result of many encounters with field particles? In order to obtain an expression for the slow-down time, we consider the effect of many encounters between the test particle with the field particles. This therefore requires a three-dimensional coordinate description. From Chandrasekhar (1942), p.229 (eq.[5.721]) we adopt for the change in velocity v and v which a test particle with velocity v 2 = v 2 and mass m 2 suffers as the result of an encounter with a field particle with mass m 1 and velocity v 1 : v = 2m 1 m 1 + m 2 [(v 2 v 1 cos θ)cos ψ + v 1 sin θ cos Θ sin ψ] cos ψ v = ± 2m 1 m 1 + m 2 [v v 2 2 2v 2 v 1 cos θ {(v 2 v 1 cos θ)cos ψ + v 1 sinθ cos Θ sinψ} 2 ] 1/2 cos ψ (A.21) (A.22) where θ is the angle between the initial velocities v 1 and v 2, cos ψ satisfies Eq.(A.19), π 2ψ is the angular deflection of the relative velocity V in the orbital plane, which satisfies Eq.(A.19), and Θ is the angle between the orbital plane and the fundamental plane, which contains the vectors v 1 and v 2, as can be seen in Fig.(A.2). Let us consider the effect of a large number of encounters on Eq.(A.21) and Eq.(A.22), that is, we consider v and v following Chandrasekhar (1942), p.54. Let N(v 1,θ,ϕ) be the number of field particles per unit volume with velocities in the range (v 1,v 1 + dv 1 ) and in directions confined to the element of solid angle sin θ dθdϕ, where ϕ is the azimuthal angle in a system of coordinates in which the z-axis coincides with the direction of v 2, and θ is as defined before. The number of encounters which take place during time dt is then given by: N(v 1,θ,ϕ)dv 1 dθdϕ dθ 2π 2πDdD V dt (A.23) The contribution of these encounters to the sum v can then be found by multiplying Eq.(A.21) and Eq.(A.22) with Eq.(A.23) and performing the integrations over the relevant range of each of the variables in Eq.(A.23). 113

120 Appendix A: Derivation of the slow-down time v2 Vg v1 v2 π 2Ψ ι Φ Φ θ V π 2ψ φ V φ 2ψ fundamental plane orbital plane Θ Figure A.2: Schematic representation of an encounter. The fundamental plane is set up by the vectors v 1 and v 2, which represent the velocities of the particles before an encounter. The velocity of the center of gravity, denoted by V g remains constant during an encounter. In a coordinate system in which the center of gravity is stationary, the particles move in a hyperbola in their orbital plane, which is, in general, inclined at some angle Θ with respect to the fundamental plane. As a result of the encounter, the relative velocity V is deflected by an angle π 2ψ in the orbital plane, and is then denoted by V. The true deflection suffered by the test particle is determined by the angle π 2Ψ between the initial velocity v 2 and the final velocity v

121 A.2 Derivation of slow-down time for distant encounters According to Chandrasekhar (1943), when considering a large number of encounters, the net increment in the velocity component perpendicular to its original direction of motion, v, vanishes because of the symmetry of the problem. However, this is not the case for the net increase a star suffers in the direction of its original motion during a time t: v = t 0 π dv 1 dθ 0 2π 0 dϕ D0 0 0 dd 2π 0 0 dθ 2π 2πN(v 1,θ,ϕ)V D v (A.24) Rewriting Eq.(A.24) using Eq.(A.19) and performing the integral over the inclination of the orbital plane to the fundamental plane dθ: m π 2π Dmax 1 v = 4π t dv 1 dθ dϕ dd m 1 + m N(v 1,θ,ϕ)V (v 2 v 1 cos θ) D 1 + m2 1 m2 2 (m 1 +m 2 ) 2 D 2 V 4 Z 2 1 Z2 2 e4 The relevant range of the impact parameter D can be found from its relation with the angular deflection χ via the angle ψ in Eq.(A.19). Since it seems natural to have 0 and π/2 as the appropriate limits of ψ, accordingly, one may want to integrate D from 0 to. However, this integral does not converge at D =. According to Chandrasekhar (1942), p.55, this divergence arises physically due to the inappropriate use of the two-body approximation to describe distant encounters for systems of (charged) particles: for the use of Eq.(A.19) for the angle between the two asymptotes assumes the particles to be initially at an infinite distance apart and to separate to an infinite distance after the encounter, whereas, in practice, a second deflection already starts to affect the orbit before the first is completed. In other words, Eq.(A.19) overestimates the actual deflection, because when coming in from infinity, a test particle also experiences encounters with other particles. For close encounters, that is, for encounters for which the distance D is much smaller than the average separation D 0 between the colliding bodies, one may assume the actual deflection to be equal to the one derived from the two-body approximation, Eq.(A.19), since encounters with neighboring particles are barely influencing the encounter under consideration. For distant encounters, where D becomes of the same order as D 0, we must take into account that for some impact parameters D greater than a given amount, the two-body approximation is no longer valid, because of the aforementioned reasons. However, simply ignoring all encounters with impact parameters D greater than a given amount, introduces small errors because we are ignoring the small but finite contributions from these (very) distant encounters. Therefore, we have to choose our upper limit D max carefully. Using that x dx 1 + ax 2 = 1 2a ln ( 1 + ax 2) we perform the integral over D with boundaries extending from 0 to D max : v = 2π m 1 + m π 2π 2 m 1 m 2 Z2Z 2 1e 2 4 t dv 1 dθ dϕ { N(v 1,θ,ϕ) 1 V 3(v 2 v 1 cos θ) ln [ } (A.25) 1 + q 2 V 4] 115

122 Appendix A: Derivation of the slow-down time where q was defined as: q m 1m 2 m 1 + m 2 D max Z 1 Z 2 e 2 (A.26) Proceeding with the integration over the angles θ and ϕ, we need to know the form of N(v 1,θ,ϕ). Following Chandrasekhar (1942), p.59, Eq.[2.335], we assume a spherical distribution of the velocities of the field particles: N(v 1,θ,ϕ) = 1 4π N(v 1) sin θ (A.27) with N(v 1 ) = 4πNf(v 1 )v1 2, where f(v 1) denotes the frequency function defining the distribution of the velocities v 1. Substituting Eq.(A.27) into Eq.(A.25) and performing the integration over the azimuthal angle ϕ we obtain: v = π m 1 + m π 2 m 1 m 2 Z2Z 2 1e 2 4 t dv 1 dθ { N(v 1 ) sin θ V 3 (v 2 v 1 cos θ) ln [ 1 + q 2 V 4] } (A.28) The integral over dθ is performed as in Chandrasekhar (1943); using that V 2 = v v2 2 2v 1v 2 cos θ (A.29) and performing the integral over dv instead of over dθ: V dv = v 1 v 2 sinθdθ v 2 v 1 cos θ = 1 (V 2 + v2 2 v 2v 1) 2 2 Performing the integral in Eq.(A.28) over θ using Eq.(A.30): v = π m 1 + m 2 2 m 1 m 2 Z2Z 2 1e 2 4 t { } 1 2 v2 2 dv 1 N(v 1 ) J 0 v 1 where J was defined by Chandrasekhar (1943), Eq[26], as follows: v1 +v 2 ( J = 1 + v2 2 ) v2 1 V 2 ln(1 + q 2 V 4 ) dv v 1 v 2 (A.30) (A.31) (A.32) where q is as defined in Eq (A.26). In order to perform the integral in Eq.(A.32), we need to know the order of magnitude of q 2 V 4. Since q is a function of the upper limit D max, we need to estimate the value of D max. Following Spitzer (1962), p.127, we use the Debye shielding distance λ deb for the upper limit D max, since this is the distance over which charged particles typically influence each other via the electrostatic force. Therefore, we have: D max λ deb = ( ) kb T 1/2 4πn e e 2 (A.33) 116

123 A.2 Derivation of slow-down time for distant encounters where n e denotes the electron number density and T is the kinetic (electron) temperature. Substituting this value for D max in Eq.(A.32), we see that the term q 2 V 4 is generally much larger than unity: ( ) q 2 V 4 m1 m 2 2 k B T = 1 m 1 + m 2 Z 1 Z 2 e 2 ( m1 m m 1 + m 2 Z 1 Z 2 e 2 4πn e e 2 V 4 ) 2 T V 4 1 n e (A.34) and that, therefore, Eq.(A.32) may be reduced in a similar way as was done by Chandrasekhar (1943), Eq.[28] to Eq.[31], ending up with: { 8v 1 ln ( q u 2) if v 1 < v 2 J = (A.35) 0 if v 1 > v 2 where according to Chandrasekhar, u 2 may be taken to denote the mean square velocity of the field particles. From Eq.(A.35), we see that, in analogy with the stellar case described in Chandrasekhar (1943), only particles with velocities less than the one under consideration contribute to v. Combining Eq.(A.35) and Eq.(A.31) to obtain the equivalent of Chandrasekhar (1943), Eq.[32]: v = 4π m 1 + m 2 m 1 m 2 Z 1 Z 2 e 4 t 2 v2 2 ln ( q u 2) v 2 N(v 1 )dv 1 (A.36) 0 Following Chandrasekhar (1943), we assume the velocities v 1 to be distributed according to Maxwell s law: v2 N(v 1 )dv 1 = 4j3 v2 dv 1 e j2 v1 2 v 2 1 (A.37) 0 π 1/2N where N denotes the number of field particles per unit volume and j is a parameter that measures the dispersion of the velocities in the system (Chandrasekhar (1943), Eq[33]). We express the integral on the right hand side of Eq.(A.37) in terms of the error integral: Substitute x = jv 1 : v2 Hence: v2 0 0 dv 1 e j2 v 2 1 v 2 1 = N(v 1 )dv 1 = 4N π 1/2 Φ(x) = 2 π 1/2 x0 0 x 0 x 0 0 dx e x2 x2 x2 dx e j 3 = 1 j 3 e x2 x 2 dx = 4N π 1/2 x 0 dx e x2 x 2 { 1 4 π1/2 Φ(x 0 ) 1 } 2 e x2 0 x0 = N ( Φ(x 0 ) x 0 Φ (x 0 ) ) (A.38) where it was used that x 0 = jv 2. Substitute Eq.(A.38) in Eq.(A.36) to obtain an expression for the net change in the parallel velocity component of the test particle, analogous to the one obtained for the stellar case in Eq.[35] of Chandrasekhar (1943): v = 4π (m 1 + m 2 ) m 1 m 2 2 Z 2 1 Z2 2 e4 v 2 2 ln(q u 2 ) t N [ Φ(x 0 ) x 0 Φ (x 0 ) ] (A.39) 117

124 Appendix A: Derivation of the slow-down time Since the field particles are assumed to obey a Maxwellian distribution, this implies that the constant j in the distribution function is 1/v th, where v th is the thermal velocity of the system. Then Eq.(A.39) becomes: v = 4π (m 1 + m 2 ) m 1 m 2 2 Z 2 1 Z2 2 e4 v 2 2 ln(q u 2 ) t N [ Φ(v 2 /v th ) v 2 v th Φ (v 2 /v th ) ] Rearranging this expression yields: v t = 4π (m 1 + m 2 ) m 1 m 2 2 = 4π (m 1 + m 2 ) = 8πN m 2 2 m 1 m 2 2 ( 1 + m 2 m 1 = A D ( 1 + m 2 m 1 N Z1 2 Z2 2 e4 ln(q u 2 ) v2 th vth 2 N Z1 2 Z2 2 e4 ln(q u 2 ) 1 ) ) 1 v 2 th v 2 th Z 2 1Z 2 2e 4 ln(q u 2 ) 1 G(v 2 /v th ) = dw dt where we have written, in analogy with Spitzer (1962): v 2 th [ Φ(v2 /v th ) v 2 /v th Φ (v 2 /v th ) ] v 2 2 [ Φ(v2 /v th ) v 2 /v th Φ (v 2 /v th ) ] G(v 2 /v th ) v 2 2 /v2 th (A.40) G(x) = Φ(x) xφ (x) 2x 2 A D = 8πe4 NZ 2 1 Z2 2 ln(q u 2 ) m 2 2 (A.41) where G(x) is termed the Chandrasekhar function, as it was introduced for the first time by Chandrasekhar (1943), and A D is termed the diffusion constant [in cm 3 /s 4 ]. The right hand side of Eq.(A.40) gives the rate dw/dt [in cm/s 2 ] at which moving test particles are slowed down by interactions with the field particles. In case the parallel component of the initial velocity v 2 of the test particle reduces from its initial value v 2 = w to zero after many encounters, we can write v = w and using that dw/dt is equal to the right hand side of Eq.(A.40), the corresponding slow-down time t S = t is: t S = w dw/dt = w v2 th (1 + m 2 /m 1 )A D G(w/v th ) (A.42) This expression obtained for the slowing down time is similar to the slowing down time as provided by Spitzer (1962): t s = v 2 th w (1 + m/m f )A D G(w/v th ) where w = v 2, m f = m 1, m = m 2, Z f = Z 1 and Z = Z 2 in our notation, and A D is: A D = 8πe4 n f Z 2 Z 2 f ln Λ m 2 (A.43) (A.44) where n f = N, and ln Λ is the Coulomb logarithm. As we will see below, at first sight the definition of ln λ according to Spitzer does not coincide with the definition which follows from 118

125 A.3 Multicomponent hydrodynamical equations Chandrasekhar, however, this poses no problem, because of the assumption of a common temperature T for all components of the wind. The logarithmic term appearing in Chandrasekhar s diffusion constant, was defined as: ln(q u 2 µ u ) = ln (λ 2 ) deb Z 1 Z 2 e 2 (A.45) where µ = m 1 m 2 /(m 1 + m 2 ) is the reduced mass of the (two-body) system and u 2 denotes the mean square velocity of all the particles, which constitute a single system, considered in the case of Chandrasekhar. Spitzer used the Coulomb logarithm, which is defined according to: ( ) λdeb mw ln Λ = ln = ln (λ 2 ) deb ZZ f e 2 p 0 (A.46) where p 0 is termed the impact parameter, which is the lower cut-off beyond which the encounters are no longer distant encounters, but rather close encounters, which, as mentioned before, presumably are not important when considering encounters between charged particles [Spitzer (1962)]. However, Spitzer defines p 0 to be the distance associated with a deflection of 90, i.e. tan ψ = 1 in Eq.(A.17), assuming the mass of the field particle to be much larger than the mass of the test particle, such that the reduced mass can be approximated to be the mass of his (lighter) test particle: p 0 = ZZ fe 2 mw 2 (A.47) where m denotes the mass of the test particle, and w is its velocity. However, generally, and in particular in our case, the mass of the test particle is not smaller than the mass of the field particles. Notably, in our case, the mass of the test particle is larger than the mass of the field particles. It is therefore preferable to use the reduced mass, as is done in Eq.(A.45). As a result, the choice of the mass ratio based upon which Spitzer defined p 0, does not represent the general, and in particular not our, case. However, since Spitzer assumes the temperature of the test particles to be equal to the temperature of the field particles, T = T 1 = T 2, the kinetic term in Eq.(A.46) can be replaced by the thermal energy according to mw 2 3k B T. Since in Chandrasekhar s case the Maxwellian speed distribution is assumed, also here the kinetic term can be replaced by the thermal energy, µ u 2 3k B T. Therefore, the two logarithmic terms are equivalent after all: ln(q u 2 µ u ) = ln (λ 2 ) ( ) 3k B T deb Z 1 Z 2 e 2 = ln λ deb ZZ f e 2 = ln Λ A.3 Multicomponent hydrodynamical equations If collisions between particles are sufficiently frequent, a hydrodynamic description, as opposed to a far more detailed description by means of distribution functions, can be utilized to 119

126 Appendix A: Derivation of the slow-down time describe a plasma. When considering a multicomponent plasma, consisting of N different metal ion species i, a bulk hydrogen (and helium) plasma f, and electrons e, then the plasma can be described in terms of its macroscopic (average) quantities, i.e. the mean radial velocity v j of particle species j = {i,f,e}, the number of particles of species j per unit volume, n j, and the mean energy or temperature. In thermal equilibrium, i.e. when the distribution function of particle species j is Maxwellian, the mean kinetic energy per particle m vj 2 /2 can be related simply to the temperature T j. When the thermal coupling between the ions and the electrons is strong, i.e. in a high density gas, the relative temperature difference is small, and one can use that T i = T f = T e = T. Note that the assumption of a Maxwellian distribution function is a good approximation, if the collision rate is high enough, as this causes any arbitrary distribution to become Maxwellian. Equations of motion The possibility that the wind could be considered as a multicomponent flow was considered by many authors, notably, Springmann and Pauldrach (1992) and Krtička and Kubát (2000) (several papers). The general consensus is that a multicomponent flow model may be necessary to describe stellar winds, if the collisional decoupling is strong enough. To that purpose, the usual one-component stationary hydrodynamical equations can be extended to a N + 1-component model, consisting of N metal ion species, which make up N fluids, denoted with the subscript i, and bulk matter (protons and α particles), denoted with subscript f. Following Springmann and Pauldrach (1992), the equations of motion are: v f dv f dr v i dv i dr v e dv e dr = g + ee m f + 1 n f m f R fi = g + Z iee m i = g ee + ge Thom m e + g rad i 1 n i m i R fi (A.48) where v j is the radial velocity of particle species j = {i,f,e} (ions i, passive plasma f and electrons e), m j is the particle mass, and n j is the number density. The gravitational acceleration is denoted by g and E is an electric polarisation field, with Z i e the ionic charge. The radiative acceleration due to absorption of radiation through the atomic transitions is the acceleration due to Thomson scattering, which only acts directly on the electrons. R fi = R if is the frictional force(- density) acting between species f and species i, which arises from a multitude of small-angle Coulomb scattering events. Note that the pressure gradient was neglected, which is only a correct approximation in the supercritical region. However, the error introduced due to this approximation is not important for our purpose here, which is only to show how dynamical friction can be understood in terms of the wind dynamics. The collisions with electrons can be neglected, as will be discussed below. The constants have their usual meaning. Note that when multiplying each of Eqs.(A.48) by their respective densities n j m j, dividing over of the ions of species i is given by g rad i, and g Thom e 120

127 A.3 Multicomponent hydrodynamical equations the total density ρ = n j m j = nm, and summing, the usual one-component equation of motion is obtained. In the one-component equation of motion the internal contributions to the individual equations of motion cancel, because of Newton s third law (R if = R fi ) which causes the frictional force to cancel, and the condition of quasi-neutrality, q j n j = 0, and global conservation of charge (zero-current condition), q j n j v j = 0, which causes the electrical force to cancel (here q j is the charge of species j). The frictional force Because of the low ion density, the frictional force R fi due to the existence of a relative velocity v j v i, is assumed to act only between the active ions and the passive plasma, and not among active ions of species i and active ions of species i. According to Springmann and Pauldrach (1992) the frictional force R fi [in units of g cm 2 s 2 ] acting between the passive plasma and active ions of species i can be written as: R fi = n f n i k fi G(x fi ) (A.49) where n f and n i are number densities of the passive plasma and the active ions of species i. The friction coefficient k fi [in units of g cm 4 s 2 ] is given by: k fi = 4π ln Λq2 f q2 i k B T v f v i v f v i (A.50) where ln Λ is the Coulomb logarithm, which arises because of the small-angle cutoff for Coulomb scattering, due to the Debye screening effects in the plasma. The Chandrasekhar function G(x) is defined in terms of the error function Φ(x) according to: G(x) = 1 2x 2 ( Φ(x) x dφ(x) dx ) (A.51) Following Springmann and Pauldrach (1992), the argument x fi of the Chandrasekhar function is proportional to the ratio of the relative kinetic energy of the particles p,i to the thermal energy of the protons f, via: x fi = v f v i A fi (A.52) v th where A fi = A f A i /(A f +A i ) is the dimensionless reduced atomic mass, and v th is the thermal velocity of the protons. Writing the frictional accleration g fric as: g fric = 1 ρ i R fi (A.53) and realizing that g = v/ t, we can write the average time scale t needed to slow down the constituents of the fluid of species i due to collisional friction as: t = v g fric = v n im i R fi (A.54) 121

128 Appendix A: Derivation of the slow-down time Substituting Eq.(A.49) and Eq.(A.50), we obtain: t = v m i n f k fi G(x fi ) = v m i v f v i v f v i k B T 4π lnλq 2 f q2 i n fg(x fi ) (A.55) Using that we are only interested in the change of the velocity component parallel to the radial direction of the wind, i.e. that v = v, and that therefore, v is in the same direction as the unit vector v f v i /(v f v i ), and using that q j Z j e, Eq.(A.55) becomes: t = 2k B 8πe 4 T n f Z 2 f Z2 i ln Λ m i v G(x fi ) (A.56) In the limit of small x fi, i.e., small drift speed compared to the thermal velocity, G(x fi ) can be approached to be a linear function of its argument: T[G(x fi )] xfi 0 2x fi 3 π (A.57) Eq.(A.57) shows that in the linear case, the friction force R fi depends on the square root of the reduced mass. Therefore, collisions of the ions with the electrons are less effective by a factor of 43 than collisions of the ions with the passive plasma [Springmann and Pauldrach (1992)]. It is also argued by Springmann and Pauldrach (1992) that the collisions of the electrons with the passive plasma can be neglected, because the zero-current condition forces the relative drift speed between these two species to be very small. Therefore the only frictional force that is important in the equations of motion is that due to collisions between active ions and passive plasma. Springmann and Pauldrach (1992) show that the electronic force only plays a role in preventing the electrons from being accelerated more than the ions and the passive plasma due to Thomson scattering on the electrons. When ignoring the collisional coupling to the electrons, it is due to this polarization field that the electron velocity is somewhere between the velocities of the ions and that of the passive plasma (zero-current condition). Substituting Eq.(A.57) into Eq.(A.56), we obtain for small drift speeds: t = 3 π 2 2k B 8πe 4 T n f Z 2 f Z2 i ln Λ m i v th Afi = 3 π 2 1 8πe 4 v 3 th n f Z 2 f Z2 i ln Λ m i m f Afi (A.58) where it was used that v = v f v i. The slow-down time as derived here, is approximately equal to the slow-down time as derived in the previous section, if we assume the reduced mass to be equal to unity. 122

129 Nederlandse samenvatting In mijn scriptie heb ik onderzocht of een fenomeen genaamd ionen ontkoppeling optreedt in de door straling gedreven sterrenwinden van massieve sterren. De vraag waarop ik een antwoord getracht heb te vinden is of, en zo ja in welke mate en in welk soort sterren ionen ontkoppeling optreedt en of het optreedt in sterren waarin het belangrijke gevolgen kan hebben voor de evolutie. Sterren bestaan er in allerlei soorten en maten, van de meest heldere (tot wel miljoenen keer zo lichtkrachtig als onze zon) en zware (tot wel honderd keer zwaarder als onze zon) sterren, tot relatief lichte en lichtzwakke sterren, zoals onze eigen zon. In figuur A.3 ziet u een schematische voorstelling van verschillende typen sterren: een O-type ster is het heetst, het meest lichtkrachtig en het grootst en een G-type ster, zoals onze zon, is relatief koel en klein. Alle sterren stralen omdat de temperatuur en dichtheid in de kern hoog genoeg is voor kernfusie, van bijvoorbeeld waterstof in helium gedurende de zogenaamde hoofdreeks -fase. Vanwege de kernfusie komt energie vrij in de vorm van straling (fotonen), waardoor wij de ster kunnen waarnemen. In zware sterren (lees: met een massa aan het begin van het leven van de ster, die groter is dan ongeveer 9 keer de massa van onze zon) is lichtkracht zo groot dat er door impuls overdracht van de fotonen op het gas materie deeltjes de ruimte in ingeblazen worden. De naar buiten gerichte stralingsdruk van de fotonen overwint dus de naar binnen gerichte zwaartekracht in de buitenste lagen van de ster Als gevolg hiervan heeft de ster een sterrenwind. Aangezien in dit geval fotonen, en niet gasdruk of een ander mechanisme verantwoordelijk is voor het wegblazen van de buitenste lagen, spreken we hier van een stralingsgedreven sterrenwind. De hoeveelheid materie die een ster verliest ten gevolge van een sterrenwind kan bepalend zijn voor de verdere evolutie van de ster. Als een zware ster heel veel massa verliest in een sterrenwind (zware sterren kunnen meer dan de helft van hun initiële massa verliezen in een sterrenwind), dan kan het eindproduct van haar evolutie bijvoorbeeld een neutronenster zijn, in plaats van een zwart gat. Bovendien heeft massaverlies ook gevolgen voor de omgeving van een ster. Zo kan de omgeving chemisch verrijkt worden als een ster op een gegeven moment ook de producten van haar kernfusie de ruimte in blaast, maar ook is er sprake van een dynamische invloed ten gevolge van een sterrenwind. Als het materiaal wat de ster de ruimte in blaast op het gas in de interstellaire ruimte botst, dan kan dit leiden tot schokgolven, welke 123

130 Appendix A: Nederlandse samenvatting Figure A.3: Schematische voorstelling van de classificatie van hoofdreeks sterren volgens Morgan & Keenan. Zware sterren zijn het meest heet, het meest lichtkrachtig en het grootst. opzich weer de formatie van nieuwe sterren kunnen triggeren of juist tot stilstand brengen. In figuur A.4 ziet u een ster in de zogenaamde Wolf-Rayet fase van zijn evolutie, gedurende welke zij zeer veel massa verliest in een sterrenwind. Het moge duidelijk zijn dat de studie van de sterrenwinden belangrijk is als men (de evolutie van) sterren en hun omgeving wil begrijpen. Er blijken verschillende manieren te zijn waarop een sterrenwind tot stand kan komen. In mijn onderzoek richt ik me op dat mechanisme, welke in de meest hete en lichtkrachtige sterren voorkomt. Dit is het regime van de lijngedreven sterrenwinden. In deze sterren vindt de overdracht van foton energie op de gasdeeltjes (ionen) plaats via een select groepje deeltjes, de zogenaamde metalen (merk op dat astronomen alle elementen zwaarder dan waterstof en helium al aanduiden als metalen ). Deze deeltjes zijn namelijk het meest efficient in het absorberen van energie van het stralingsveld, en worden daarom aangeduid als de actieve deeltjes. De rest van de deeltjes waaruit de ster atmosfeer bestaat, i.e., waterstof en helium (zo n 98 % van de massa in sterren met een metaalgehalte zoals dat van onze eigen zon) is veel minder efficient in het absorberen van energie van het stralingsveld en wordt daarom aangeduid als het passieve plasma. Deze deeltjes worden naar buiten toe versneld doordat ze botsen met de actieve deeltjes, waarbij de actieve deeltjes energie overdragen op de passieve deeltjes en deze zo met zich meesleuren. Samengevat: de fotonen dragen energie over op de actieve deeltjes, waardoor deze naar buiten toe versneld worden. Op weg naar buiten toe botsen de actieve deeltjes op de passieve deeltjes, waardoor deze ook naar buiten toe versneld worden. Het resultaat is dat de hele sterlaag naar buiten toe versneld wordt. Stel nu dat de botsingen van de actieve deeltjes op de passieve deeltjes niet efficient zijn in het versnellen van de passieve deeltjes. Dit betekent dat de passieve deeltjes niet langer naar buiten toe versneld worden, maar alleen de actieve deeltjes (slechts zo n 2 % in het geval van een zonsachtig metaalgehalte). Dit wordt aangeduid als ontkoppeling van de actieve deeltjes van de passieve deeltjes. Aangezien de steratmosfeer voor het grootste deel bestaat uit 124

131 Figure A.4: Deze ster is heel heet (met een temperatuur van ongeveer 50,000 K) en lichtkrachtig (ongeveer miljoen keer zo helder als onze zon), en bevindt zich in een stadium ( Wolf- Rayet fase) van zijn evolutie gedurende welke de ster heel veel massa verliest in een sterrenwind. De ster is ongeveer 15,000 lichtjaren van ons verwijdert, en bevindt zich in de constellatie Sagittarius. De opname is gemaakt met Hubble s Wide Field Planetary Camera 2. Credit: Y. Grosdidier (U. Montreal) et al., WFPC2, HST, NASA passieve deeltjes, betekent dit dat het massaverlies in een sterrenwind drastisch gereduceerd wordt. Dit kan dan weer drastische gevolgen hebben voor de massa waarmee de ster haar leven beëindigt en op haar omgeving. In mijn scriptie heb ik onderzocht onder welke omstandigheden en in welk soort sterren ontkoppeling van actieve deeltjes van passieve deeltjes optreedt. De manier waarop de ontkoppeling van actieve deeltjes van passieve deeltjes in kaart is gebracht is door voor verschillende steratmosfeermodellen de dynamische processen die zich in zo n steratmosfeer afspelen te simuleren. De conditie welke is gebruikt om te toetsen of een gegeven actief deeltje ontkoppeld is van het passieve plasma is dat de tijd die nodig is om het actieve deeltje te versnellen tot een bepaalde snelheid ten gevolge van foton absorpties korter moet zijn dan de tijd die nodig is om hetzelfde deeltje af te remmen ten gevolge van botsingen met passieve deeltjes. Als dit het geval is, dan is dat actieve deeltje ontkoppeld van het passieve plasma. Op deze manier was het mogelijk om precies vast te leggen waar in de buitenste lagen van de ster welke actieve deeltjes ontkoppelen, en dus wat het effect ervan is op de dynamica van de sterrenwind. Ontkoppeling van actieve deeltjes van de passieve deeltjes komt vooral voor in die regionen waar de dichtheid van de wind laag is. Lage dichtheden komen voor naarmate je verder 125

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