SPHERICALLY SYMMETRIC MODEL ATMOSPHERES FOR LATE-TYPE GIANT STARS By PHILIP DESMOND BENNETT B.Sc. (Mathematics) Simon Fraser University M.Sc. (Astronomy) The University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF GEOPHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1991 © PHILIP DESMOND BENNETT, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date n o*r<e&^ mi. DE-6 (2/88) Abstract The ATHENA computer code has been developed to model the extended atmospheres of late-type giant and supergiant stars. The atmospheres are assumed to be static, spheri-cally symmetric and in radiative and hydrostatic equilibrium. Molecular line blanketing (for now) is handled using the simplifying assumption of mean opacities. The complete linearization method of Auer and Mihalas [7], adapted to spherical geometry, is used to solve the model system. The radiative transfer is solved by using variable Eddington factors to close the system of moment transfer equations, and the entire system of trans-fer equations plus constraints is solved efficiently by arrangement into the Rybicki [83] block matrix form. The variable Eddington factors are calculated from the full angle-dependent formal solution of the radiative transfer problem using the impact parameter method of Hummer, Kunasz and Kunasz [47]. We were guided by the work of Mihalas and Hummer [72] in their development of extended models of 0 stars, but our method differs in the choice of the independent variable. The radius depth scale used by Mihalas and Hummer was found to fail because of the strongly temperature-dependent opacities of late-type atmospheres. Instead, we were able to achieve an exact linearization of the radius. This permitted the use of the numerically well-behaved column mass or opti-cal depth scales. The resulting formulation is analogous to the plane-parallel complete linearization method and reduces to this method in the compact atmosphere limit. Models of M giants were calculated for TfJJ = 3000K and 3500K with opacities of the CN, TiO and H 2 0 molecules included, and the results were in general agreement with other published spherical models. These models were calculated assuming radiative equilibrium. The importance of convective energy transport was estimated by calculating the convective flux that would result from the temperature structure of the models. The ii standard local mixing length theory was used for this purpose. Convection was found to be important only at depths with rRM > 15 for the low gravity models with log g = 0, but significant out to TRM ~ 1 at the most transparent frequencies for the higher gravity models with log g — 2. Thus, the temperature structure of the surface layers and the emergent flux for the log g — 0 models should be accurately modelled but the emergent flux for the logg = 2 models may be in error by up to 5% at the most transparent frequencies. The problem of fitting a series of beautiful internally consistent models to honest-to-goodness real stars that are up there, is horrible A. B. Underhill iii Table of Contents Abstract ii List of Tables viii List of Figures ix Acknowledgement xi 1 The Stellar Atmosphere Problem 1 1.1 Introduction 1 1.2 Classical Model Assumptions 2 1.3 Red Giant Models . 4 1.4 The Physics of Late-Type Atmospheres 6 1.5 Plane-Parallel Models . . 8 1.6 Spherically Symmetric Static Models 14 1.7 Objectives of this Research 23 2 The Spherically Symmetric Model System 25 2.1 Introduction . . 25 2.2 The Model Equations 25 2.3 The Formal Solution . 33 2.4 The Boundary Conditions 38 2.4.1 The Formal Solution 38 2.4.2 The Moment Equations of Transfer 48 2.4.3 Other Boundary Conditions . . 0. . . . . 53 iv 3 The Discrete System and Solution Procedure 56 3.1 The Finite Difference System 56 3.1.1 The Formal Solution: Cubic Spline Difference Formulae 56 3.1.2 Quadrature Weights for the Moment Integrals . 61 3.1.3 Moment Equations: Cubic Spline Difference Formulae 68 3.1.4 The Hydrostatic Equilibrium and Radius Equations 71 3.1.5 The Equation of Radiative Equilibrium 76 3.2 The Linearization of the Radius 78 3.3 The Linearized Model Equations 83 3.4 The Initial Solution 91 3.4.1 Introduction 91 3.4.2 The boundary temperature 92 3.4.3 The boundary pressure 93 3.4.4 The boundary radius 94 4 The Model Computations 95 4.1 Introduction 95 4.2 The Equation of State 95 4.3 The Opacity of the Stellar Gas 105 4.4 Convection 107 4.5 Convergence Criteria and Iterative Control I l l 4.6 The Production of Converged Models 115 4.7 Model Results and Discussion 120 5 Conclusions 169 5.1 Discussion 169 5.1.1 The Complete Linearization Method: An Overview . . 169 v 5.1.2 The Effects of Convection 170 5.1.3 The Treatment of the H 2 0 Opacity 171 5.1.4 The Independent Variable Dilemma 171 5.1.5 Numerical Inconsistencies 173 5.2 Comparison with Other Models 175 5.3 Future Research 178 5.3.1 Short Term Improvements 178 5.3.2 Non-LTE Models 181 5.3.3 Self-Consistent Atmosphere and Wind Models 182 5.3.4 Time-Dependent Dynamical Models 183 References 192 Appendices 199 A The Linearized Model Equations 199 A . l The Model Equations 199. A. 1.1 Transfer Equation . 199 A.1.2 Transfer Equation: Inner Boundary Condition 200 A.1.3 Transfer Equation: Outer Boundary Condition 200 A.1.4 Radiative Equilibrium . . 200 A.1.5 Hydrostatic Equilibrium 201 A.1.6 Hydrostatic Equilibrium: Outer Boundary Condition 201 A.1.7 Radius or Depth Equation 201 A.1.8 Radius Equation: Outer Boundary Condition . 201 A.2 The Linearized Equations 202 A.2.1 Transfer Equation 202 A.2.2 Transfer Equation: Outer Boundary Condition 206 vi A.2.3 Transfer Equation: Inner Boundary Condition 209 A.2.4 Radiative Equilibrium 211 A.2.5 Hydrostatic Equilibrium 212 A.2.6 Hydrostatic Equilibrium: Outer Boundary Condition 214 A.2.7 Radius Equation . 214 A. 2.8 Radius Equation: Outer Boundary Condition 216 B Model Output Tables 217 B . l Glossary of Variables . . 218 B.2 Converged Models: Output Tables 219 B. 2.1 Model 01310191: SS 3000/0.0/-3.0, CN, TiO, H20(sm) 220 B.2.2 Model 02310191: SS 3000/2.0/-3.0, CN, TiO, H20(sm) . . . . . . 223 B.2.3 Model 03310191: SS 3000/0.0/-3.0, CN, TiO, H 20(hm) 226 B.2.4 Model 04310191: SS 3000/1.0/-3.0, CN, TiO, H 20(hm) 229 B.2.5 Model 05310191: SS 3000/2.0/-3.0, CN, TiO, H 20(hm) 232 B.2.6 Model 06310191: SS 3500/1.0/-3.0, CN, TiO, H 20(hm) 235 B.2.7 Model 07310191: SS 3500/2.0/-3.0, CN, TiO, H 20(hm) 238 vii List of Tables 4.1 Species Considered in Model Equation of State 98 4.2 Model Opacities . . 106 4.3 Command Timings on the IBM 3081 122 4.4 Converged Models with Molecular Opacities . 123 4.5 Converged Models: Summary of Parameters . . . 124 viii List of Figures 2.1 Impact parameter coordinate system for formal solution . 35 2.2 Domains of calculation of J„ and Hv: Model 01310191 39 2.3 Domains of calculation of Jv and Ev\ Model 02310191 . . . . . . . . . . 40 4.1 Temperature profiles T(rRM) 135 4.2 Temperature profiles T(r R M ) of the outer layers 136 4.3 Pressure structure log J ^ T ^ ) 137 4.4 Electron pressure log ^ ( T R M ) 138 4.5 Atmospheric extension r(rR M)/.R* — 1 . . ; 139 4.6 Rosseland mean opacity profiles log XRM{TRM) 140 4.7 Ratio of flux-mean opacity to Rosseland mean opacity log XH/XRM • • • • 141 4.8 Ratio of convective to total flux log Hcmv{rRM) / H •'. 142 4.9 Profiles of mean convective velocities log vCONV(TRM) 143 4.10 Profiles of cooling functions log $ c o o j(r R M ) 144 4.11 Normalized radiative equilibrium residuals eRB . 145 4.12 Normalized hydrostatic equilibrium residuals eHB . 146 4.13 Normalized radius (depth) equation residuals eDE 147 4.14 RMS normalized transfer equation residuals eTBrm 148 4.15 Ratio of actual to prescribed model luminosity L/Lt 149 4.16 Emergent flux distributions log Hu 150 4.17 Ratio of photospheric to model radii Rp,(v)/R* 151 4.18 Ratio of the mean intensity to the Planck function log «/„/#„ 152 4.19 Ratio of the monochromatic to the Rosseland mean opacity log XV/XRM • 153 ix 4.20 The Eddington factor fv(rRM) 154 4.21 The distribution of opacity sources in model 01310191 155 4.22 The distribution of opacity sources in model 02310191 . 156 4.23 The ionization structure of model 01310191 157 4.24 The ionization structure of model 02310191 158 4.25 The distribution of hydrogen in model 01310191 159 . 4.26 The distribution of hydrogen in model 02310191 160 4.27 The distribution of carbon in model 01310191 161 4.28 The distribution of carbon in model 021310191 . 162 4.29 The distribution of oxygen in model 01310191 163 4.30 The distribution of oxygen in model 02310191 164 4.31 The distribution of nitrogen in model 01310191 165 4.32 The distribution of nitrogen in model 02310191 166 4.33 The distribution of silicon in model 01310191 167 4.34 The distribution of silicon in model 02310191 168 5.1 Temperature profiles of the surface layers: 3000/0.0 models. 186 5.2 Temperature profiles: 3000/0.0 models 187 5.3 Pressure profiles: 3000/0.0 models 188 5.4 Temperature profiles of the surface layers: 3500/2.0 models. . . . . . . . 189 5.5 Temperature profiles: 3500/2^0 models 190 5.6 Pressure profiles: 3500/2.0 models 191 x Acknowledgement I would like to express my gratitude to my supervisor, Dr. Jason Auman, for introducing me to the study of stellar atmospheres, for his encouragement in undertaking the project described in this thesis, and for his continued support over the years in bringing this work to completion. I am also grateful to A l Irwin, Harvey Richer, Stephenson Yang, and Jeff Brown for advice and suggestions which proved helpful along the way. But I am most indebted to my partner and companion, Lynne Robinson, for her emotional support and sustenance over these past few years, without which this work would likely not have come to fruition. Finally, I wish to dedicate this work to my youngest daughter, Caitlin Agawa, and hope that she too may feel the sense of wonder about the universe that has inspired me over the years. xi Chapter 1 The Stellar Atmosphere Problem 1.1 Introduction Much of stellar astronomy depends upon the observation and interpretation of the elec-tromagnetic radiation radiated by stars. The region near the surface of the star from which photons can escape directly to space without undergoing further absorption or scattering is referred to as the atmosphere. It represents the boundary between the relatively simple physics of the stellar interior, where strict thermodynamic equilibrium accurately describes the state of the material, and the surrounding vacuum of space. The atmosphere thus represents a region of large gradients in the physical variables such as the temperature, gas pressure and the radiation field. It is also where the stellar material becomes optically thin with the result that the radiation field is no longer strongly cou-pled to the local environment. As such, stellar atmospheres are comparatively difficult to model. A meaningful interpretation of stellar spectra, however, is only possible with the assistance of consistent models. Consequently, much effort has been devoted to the construction of such models over the past fifty years. This chapter will review the standard assumptions encountered in modelling stellar atmospheres and, in particular, consider the reasons why these may be of limited appli-cability for red giant models. Also, the extensions of classical theory that are needed to realistically model the atmospheres of these stars are discussed. 1 Chapter 1. TJie Stellar Atmosphere Problem 2 1.2 Classical Model Assumptions Traditionally, the stellar atmosphere problem has been to solve for the run of the phys-ical variables through the outer boundary layers down to depths much larger than the photon mean-free-path. Simplifying assumptions were introduced to render the problem amenable to analytic approaches. Typically, these included the following. 1. Simplified opacities. Early analytic treatments often made the assumption that the opacity \ v of the stellar material is independent of frequency, or \v = X> a n approximation referred to as the gray atmosphere. 2. Plane-parallel geometry. The boundary layers of a star are approximated by a semi-infinite slab with planar stratification. 3. Static atmospheres. The atmosphere is assumed to be in hydrostatic equilibrium. Usually it is also assumed that the total pressure is simply the sum of the gas and radiation pressure. 4. Local thermodynamic equilibrium (LTE). The radiation field, which is in strict thermodynamic equilibrium (TE) at depth in the star, must become increasingly^ anisotropic toward the upper boundary of the atmosphere, since there the intensity over the inward directed hemisphere falls to zero. This approximation assumes that the gas can be treated as though it were still in TE, with well-defined (and equal) kinetic, excitation and ionization temperatures. This implies that the gas particle velocities are specified by the Maxwellian distribution and that their level populations and degree of ionization are given by the Saha-Boltzmann distribution. The radiation field, however, is Chapter 1. The Stellar Atmosphere Problem 3 permitted to deviate from its equilibrium Planck distribution (since otherwise strict TE would still hold). Under the assumption of static LTE models, the variables describing the physical state of the atmosphere are the temperature T and the pressure p of the gas, and the spe-cific intensity Iv of the radiation field. In general, for a system lacking any particular symmetry, T and p will depend upon position (r, 0,$) and the intensity Iu additionally upon the propagation direction described by the angles (0, <f)). In this case, the full trans-fer problem is five-dimensional. The assumption of either a plane-parallel or spherically symmetric geometry reduces the problem to a two-dimensional system. One coordinate is usually described by the optical depth r although this is often replaced by the radial distance r in the spherically symmetric case. The intensity retains an additional depen-dence upon the direction of propagation, described either by the angle 9 to the normal direction or by p = cos 9. These variables are coupled through the equation of radiative transfer, the equation of radiative equilibrium (which expresses conservation of energy in the radiation field), and the equation of hydrostatic equilibrium. These equations form a closed system and can be solved by a variety of numerical methods to yield a self-consistent model atmosphere. In many cases, detailed line transfer calculations are unnecessary when modelling atmospheric structure. The source function can usually be adequately represented as a function of just the mean intensity Jv, the zeroth moment of the radiation field. The use of the moment forms of the equation of transfer further reduces the system to a one-dimensional problem since the remaining angle coordinate has been eliminated. From a computational viewpoint, though, quadratures over frequency remain and effectively constitute another dimension. Thus, the full angle dependent model problem is really three-dimensional, and the moment model, two-dimensional, in terms of timings of solu-tions of finite difference representations and the sizes of computationally feasible grids. Chapter 1. The Stellar Atmosphere Problem 4 Adopting a spherically symmetric geometry, instead of plane-parallel, results in a similar system of model equations in the physical variables T, p, and Iv. The radiative transfer equation now contains an additional term due to the increased complexity of the representation of the divergence in spherical coordinates, but a transformation can be made to remove the offending term. Another major difference is that the spatial coor-dinate r (radius) enters into the transfer equation directly so that the optical depth and radius must both be iterated consistently. While this presents considerable complications and has hampered development of spherical models in the past, we will show that the radius r can be included in the consistent solution of the model equations with little increase in computational effort. With the development of both the opacity distribution function (ODF) and the opac-ity sampling (OS) techniques to handle line-blanketing opacities a decade ago, it then became possible to compute fully consistent static, plane-parallel, LTE model atmo-spheres with realistic opacities. Although molecular line lists are incomplete for some important species (e.g. H2O, HCN, and C2H2), grids of such models have now been calculated for K giants [42], for M giants [56] and for C stars [54]. Existing techniques for including realistic opacities using the ODF method can readily be incorporated into spherical models also. 1.3 Red Giant Models Red giants are difficult to model since a variety of the classical assumptions break down. Some of the particular difficulties encountered arise from the following situations. 1.. The presence of molecules. Stars with T,ff < 4500K are sufficiently cool that molecular association becomes appreciable. The formation of molecules complicates both the equation of state and the radiative transfer in the atmosphere. Simple diatomic and polyatomic Chapter 1. The Stellar Atmosphere Problem 5 molecules can give rise to an absorption spectrum consisting of hundreds of thou-sands to millions of lines, which may overlap and blend into a pseudo-continuous opacity. These atmospheres can not be accurately represented by models using gray opacities, or even by mean opacities. The ODF and OS techniques do allow models to be constructed, at least for static atmospheres, but they remain computationally demanding. 2. Atmospheric extension. The atmospheres of red giant stars are weakly bound due to the low gravity and thus have a large scale height. The extremely nongray opacities characteristic of these stars aggravate this effect since the radius at which the atmosphere becomes optically thin is now wavelength dependent. Both effects combine to increase the effective thickness of the atmosphere. The wavelength dependence of the photo-spheric radius expected of extended atmospheres appears confirmed by the optical interferometry available [14]. When the atmosphere thickness is no longer small compared to the photospheric radius of the star, the curvature of the atmosphere becomes important in the treatment of the radiative transfer. At this point plane-parallel models break down, and more realistic geometries such as spherical sym-metry must be used. 3. Mass loss and stellar winds. All M giants appear to be losing mass, typically on the order of 10 - 6M©/yr, due to an apparently steady wind of velocity about 10 - 20 km/s [82]. Dust grains condensing in the cooling outflowing gas both emit thermal radiation in the mid-infrared (the 'infrared excess') and absorb photospheric radiation in the visible and ultraviolet. Modelling these extended stellar atmospheres necessitates giving up the assumption of static models in hydrostatic equilibrium. Instead, this constraint must be replaced by an equation describing momentum transfer to the outflowing Chapter 1. The Stellar Atmosphere Problem 6 stellar gas. Although wind models lie beyond the scope of the present work, it is essential that fully consistent, robust, static models exist to serve as a base to further efforts in this direction. It is hoped that this work will provide a step toward that goal. 4. Variability. Essentially all red giants are variable stars [37]. While some exhibit only small irregular variations in brightness and can likely be represented by static or steady state models, others, such as the Mira variables, undergo large amplitude, quasi-periodic variations which require fully time dependent, hydrodynamical models. It should be noted that all red giant stars with high mass loss rates appear to be Mira variables or OH/IR stars (all long period variables) [13, 106] so that pulsation seems to be a factor in the poorly understood mass loss mechanism. Variable stars lie beyond the scope of this work and will not be considered further. 1.4 The Physics of Late-Type Atmospheres The solution of the stellar atmosphere problem requires an accurate knowledge of the equation of state. This is fairly straight forward for hotter stars, but the degree of molecular association present in the atmospheres of cool stars complicates the problem substantially. Most recent models base their equation of state on Tsuji's [97] compilation of equilibrium constants for several hundred molecules. This is an extension of an earlier work by Vardya [101] on molecular equilibria in K and M stars. More recently, Irwin [50] published a list of polynomial partition function approximations for a large number of atomic and molecular species of astrophysical interest. Sauval and Tatum [84] published a similar compilation of partition function approximations for diatomic molecules, and Irwin [51] has produced an updated list of partition functions of polyatomic molecules. The general subject of molecules in stars has been reviewed by Tsuji [100]. Chapter 1. The Stellar Atmosphere Problem 7 Bennett [17] developed a computer code to solve the equation of state problem for an arbitrary composition, using an iterative Newton-Raphson technique to solve the chem-ical equilibrium. This procedure is fast and is used in the ATHENA model atmosphere code described in this work. An accurate treatment of molecular bound-bound opacities is also needed. The opac-ity distribution function (ODF) method proposed by Strom and Kurucz [94] was the first to allow realistic molecular opacities to be included in model atmospheres. The idea is that since the structure of a stellar atmosphere depends on integrals of the radiation field over frequency, it may be expected that the detailed small scale frequency structure of the molecular band opacities, consisting of millions of lines arranged in a sequence of bands, is unimportant. It should then be possible to rearrange the actual opacity structure over a fairly large frequency interval (but small compared to variations in BV{T)) to achieve a smoothly varying opacity over this interval. This results in an enormous reduction in the number of frequencies needed to accurately specify the opacity over the interval. Carbon [26] constructed model ODFs for the CN red system, and compared fluxes obtained with straight and harmonic mean models [27]. Kurucz [65] also provided a concise summary of the ODF technique. An alternative approach, developed and applied to atomic line blanketing by Peytremann [79], is the opacity sampling (OS) method. This relies on using a Monte Carlo technique to sample the opacities at randomly selected frequencies and has been described by Johnson and Krupp [57]. Sneden et al. [92] applied the OS method to molecular line blanketing by CN, CO, and C 2 in carbon star models. A review of the opacity modelling field was given by Carbon [28]. The treatment of molecular opacities was questioned by Gustafsson and Olander [43], who noted that computed TiO lines tend to be much stronger than the observed ones. They considered it likely that the true effects of molecular opacities upon atmospheric structure (e.g. backwarming and surface cooling or heating) were smaller than indicated Chapter 1. The Stellar Atmosphere Problem 8 by existing models. This may indicate significant departures from LTE in the electronic transitions of certain molecules, such as TiO [46]. Similar non-LTE effects may also occur in the H 2 0 vibrational transitions. Gustafsson and Olander [43] claimed many molecular lines may be more accurately considered to be formed in scattering, rather than absorp-tion. A full treatment would require non-LTE modelling of the (many) molecular levels involved and, in particular, knowledge of the poorly known cross-sections for collisionally excited electronic transitions. Since detailed monochromatic opacities for H2O were not available, Alexander et al. [3] prepared a statistical representation of the water vapour spectrum suitable for use in opacity sampled (OS) models. Model atmospheres were calculated for Teff = 3000K and 3600K using the ATLAS6 model atmosphere code, as modified at Indiana University. An analysis of the differential effect of the treatment of the H 2 0 opacity (straight mean or OS synthetic spectrum) for these M giant models was presented. J0rgensen [59] has calculated ODFs and line lists for HCN combination bands using newly available laboratory data. The polyatomic molecule HCN is abundant in the outer atmospheres of cool carbon stars, and represents a significant source of opacity. The focus of this study was to determine if HCN has potentially observable spectral features in the visible and near infrared (OAfi-lfi) region. The intent was to find more visible or near IR tracers of the outer atmospheric layers. However, it was clearly shown that HCN can not serve as such a tracer since no bands shortward of 1.5/i should be visible. 1.5 Plane-Parallel Models An extensive literature exists on modelling LTE plane-parallel atmospheres. Much of the early analytic work along, with the first numerical solutions, was reviewed in the compre-hensive summary by Pecker [76]. The technique that remains conceptually the simplest, Chapter 1. The Stellar Atmosphere Problem 9 computationally the fastest, and most amenable to generalization is the complete lin-earization method of Auer and Mihalas [7], as adapted to the second order Feautrier [38] form of the equation of radiative transfer. The use of variable Eddington factors as described by Auer and Mihalas [8] permits an exact closure of the moment equations of transfer and so achieves the elimination of the angle coordinate from the transfer prob-lem. Rybicki [83] showed that the many frequency LTE problem, linearized ultimately in terms of the fundamental variables T and n, can be rearranged into a tridiagonal block matrix that requires much less computational effort to solve as compared to the conventional matrix solution. The overall technique is summarized by Auer [5] in a paper which presents a very good review of the subject. More recent reviews of the technique (for both plane-parallel and spherical models) were presented by Auer [6] and Mihalas [71]. The literature, particularly with regard to modelling early-type stars, is very exten-sive. The review considered here is largely restricted to studies in the area of late-type stars. Plane-parallel model codes of late-type stars are now quite well developed, and several extensive model grids have been published in the last decade. None of the models published to date have used the complete linearization method as detailed by Auer and Mihalas [7], although Nordlund's [75] technique, used to compute the models of the Up-psala group, employs a partial linearization. Most are adaptations of earlier, less robust methods that have been coerced into the difficult task of converging models of red giants. Most recent models have used either the ODF or OS method to handle line blanketing opacities. All of the models described in this section assume molecular lines are formed in pure LTE absorption. Auman [10] calculated a grid of models for late-type giants and dwarfs over a range of 2000K< Tcff < 4000K. The abundances used were approximately solar. The opacities included absorption by H, H~, , He - , H j and the metals, and scattering by H, H 2 and Chapter 1. The Stellar Atmosphere Problem 10 free electrons. The molecular vibration-rotation lines of H2O, important in the later M stars, were included as a harmonic mean opacity. The model atmosphere code used the Avrett-Krook temperature correction method [12] iterated to constant flux. Convection was included in the model and treated using the mixing length theory of Bohm-Vitense [20]. Querci et al. [81] calculated model atmospheres for carbon stars. Opacity sources included absorption and scattering contributions by H, H~, H2, Hj , He - , C - , C, e~ and metals as well as pressure induced absorption due to H 2 - H 2 and H 2-He collisions. Molecular line blanketing by CN, C2 and CO was treated using ODFs to represent the opacities. The temperature-pressure structures of these C star models were very different from those of the M star models of Auman [10]. An expanded grid extending to cooler carbon stars was subsequently produced by Querci and Querci [80] using the same model code. Gustafsson et al. [42] calculated a grid of models for G and K Population II giants, the details of which have been published by Bell et al. [15]. Abundances ranged from solar to [Fe/H] = —3. Bound-free and free-free continuum absorption from H~, H, Hj , H + , and C, Mg, A l , Si, and Rayleigh scattering by neutral H, along with electron scattering, were included. Approximately 50000 atomic lines were considered, along with molecular lines from CH, OH, NH, CN, CO and MgH. The line blanketing was handled using ODFs, and the radiative transfer solved by linearization using, the method of Nordlund [75]. This linearization was only partial in that the implicit dependence of the opacity upon temperature and density was not included in either of the radiative transfer or radiative equilibrium equations. As a result, the convergence of these models was poor, and it was found necessary to adopt a modified Aitken extrapolation to obtain solutions. Auman and Woodrow [11] calculated two sequences of model atmospheres for late-type giants. In one set of models the ionization equlibria were calculated assuming LTE. Chapter 1. The Stellar Atmosphere Problem 11 The other set were calculated using the statistical equhbrium (SE) equations to solve for the level populations and ionization equilibria of the metals that were significant sources of free electrons. The approximation that the bound-bound transitions of these metals were in detailed radiative balance was made. The opacities used were similar to those used by Auman [10] with the addition of the bound-free opacities of K, Na, Ca, A l , Mg, and Si. The H2O line opacity was also included using the harmonic mean approximation. They found a significant shift in the ionization equilibrium in the surface layers of the coolest (Teff = 2000K and 2500K) models, resulting in a increase in the electron pressure of 1-2 orders of magnitude. Kurucz [65] published an extensive grid of models of early type stars, extending in spectral type through the G stars (Teft = 5500K). Solar abundances were adopted, and a modified version of Kurucz's ATLAS5 code [64] was used. The usual continuum opacities for hot stars, along with 106 atomic lines from the list of Kurucz and Peytremann [66], but with no molecules, were included. Kurucz [65] presents a good overview of the ODF method used to handle the atomic line blanketing in these models. Tsuji [99] calculated the first models of M giants to include a reasonably accurate treatment of molecular opacities. Models both with solar abundances and with C/O = 1.05 were considered. The continuum sources included bound-free and free-free absorp-tion by H - , H, Si, Mg and Ca, free-free absorption by Hg and He - , Rayleigh scattering by H, H 2 arid He, and electron scattering. Molecular opacities of CO, OH, CN, CH, TiO, MgH, SiH, CaH, H 2 0 and H 2 (collisionally induced) were handled by the Voigt-analog-Elsasser band model (VAEBM) which, although considerably less accurate than ODF or OS treatments, was an improvement upon previous work using mean opacities. Tsuji's models iterated the temperature until flux constancy was attained. Krupp et al. [62] investigated the role of TiO on the atmospheric structure of late-type giants of solar abundance. A significant warming of up to several hundred degrees was Chapter 1. The Stellar Atmosphere Problem 12 found in the outer layers when the TiO opacity was included. The models were adapted from the ATLAS5 code of Kurucz [64], modified to include atomic and molecular line blanketing using the OS method. Johnson et al. [56] also produced a grid of M giant models. The code used was also based on Kurucz's [64] ATLAS5 model atmosphere program. Solar abundances were assumed, and bound-free, free-free and scattering opacities of H, H~, Hj , H 2 , He, He - , C, Mg, A l and Si were included as described by Kurucz [64]. A list of atomic lines provided by Bell (from the list of Bell and Rodgers [16]) were included, along with the molecular lines of CN, CO, C 2 , TiO, CH, NH, OH and MgH, handled using the OS method. Also, H 2 0 was represented by means of straight mean opacities. A subsequent series of models, based on the above code and with similar opacities were subsequently calculated by Johnson [54] for carbon enriched stars. More recently, Saxner and Gustafsson [85] devised a technique for forming composite ODFs (necessary when more than one opacity source is important in the same spectral region) from single species ODFs. This removed one of the previous objections to the ODF method, that it was necessary to calculate new sets of distribution functions when the composition of the atmosphere was changed. Instead, it is now feasible to calculate ODFs for individual opacity sources and then merge these to construct composite ODFs as needed. Implicit in this is the assumption that the individual opacities are uncorrelated over the frequency interval used to construct the ODFs. In addition, a significant penalty in computing time is required, although much less than needed to recompute a full set of ODFs. Eriksson et al. [36] calculated cool carbon star models that included absorption by HCN. The atmosphere code used was a modified version of the program described by Gustafsson et al. [42], with pressure induced opacities from collisions between H 2 - H 2 and H 2-He added and revisions to the atomic and molecular opacity data. The HCN Chapter 1. The Stellar Atmosphere Problem 13 band strengths were essentially unknown and were modelled by analogy to the more well studied molecules CO2 and N2O. The line blanketing obtained from this was handled using ODFs. The effect of inclusion of this HCN opacity in the models was crucial to the atmospheric structure. Models of cool carbon stars (Teff < 2900K) without HCN are compact, whereas those with HCN are more extended by up to two orders of magnitude (as indicated by the gas pressure at a given temperature). It is quite possible that the inclusion of a realistic H2O opacity would have the same effect in cool M giant stars. A comparison of the structure of model atmospheres obtained using the ODF and OS methods was studied by Ekberg et al. [35]. Significant temperature differences of up to 150K were found for cool carbon star models. This appears to be due to the lack of correlation in the fine structure of the opacity over the range of the atmosphere, in violation of one of the basic assumptions of the ODF method. The problem is worst when the surface layers are dominated by polyatomics, since diatomic absorbers will always dominate deeper in the atmosphere. The status of red giant models was reviewed by Johnson [55] with particular empha-sis on the opacity problem. The consistency of current models with other models and observational constraints was discussed. Brett [21] calculated opacities for the VO A-X, B-X, and TiO e band systems using the Just Overlapping Line Approximation (JOLA) technique [96]. No laboratory analyses exist for any of these band systems, which are potentially important opacity sources for the cooler M stars. The plane-parallel code of Wehrse [103] was used to compute a series of models for 2250K< Teff < 3500K, both with and without these new opacities. Brown et al. [23] computed plane-parallel model atmospheres over the range 3000K< Teff < 4000K, and 0 < log<7 < 2, which included energy transport by both radiation and convection. These models were calculated using the Indiana ATLAS6 code. The opacities included the standard bound-free and free-free continuous opacities of H, He Chapter 1. The Stellar Atmosphere Problem 14 and various metals, and included the molecules CO, H 2 0 , TiO, SiO, CN, C 2 , CH, NH, OH, MgH, and SH as well as atomic lines from the list of Kurucz and Peytremann [66]. The molecular and atomic lines were treated using the opacity sampling method. Water vapour was handled using the statistical spectrum generated by Alexander et al. [3]. 1.6 Spherically Symmetric Static Models When the photon mean free path in the atmosphere becomes a significant fraction of the stellar radius, the curvature of the atmospheric layers becomes important and plane-parallel models become inadequate. One must then resort to spherically symmetric mod-els. The complexity of the transfer equations in spherical coordinates made strictly analytic treatments of the problem impossible except under very special conditions, and so no solutions of a general nature were obtained until after the advent of electronic computers. Kosirev [61], assuming a gray, Kramers opacity of the form np = r _ n , was able to find an approximate temperature gradient and, further assuming a Planckian source function, was able to integrate the formal integral of the transfer equation. Chandrasekhar [31] obtained the formal integral of the transfer equation for an infinite, spherical atmosphere assuming the Eddington approximation and demonstrated the characteristic flattening of the continuum that results from extended, as compared with planar, atmospheres. Chapman [32] examined the formal solution of the spherical transfer equation assum-ing a Kramers opacity law with n = 3. He demonstrated that the Eddington approxima-tion assumed in almost all of the early work on spherical transfer was inaccurate (and totally invalid at small optical depths) due to the strong peaking of the radiation field in the outward radial direction. The modern era of spherical models was ushered in by Cassinelli's [30] numerical models of the central stars of planetary nebula. Bound-free and free-free opacities of Chapter 1. The Stellar Atmosphere Problem 15 H, He, He + , ionized metals and scattering by electrons were included. The radiative transfer problem was attacked by discretizing the two dimensional transfer equation in both radius r and angular variable p using the discrete Sjv method of Carlson [29]. The method is stable only for step sizes of small optical depth. Large variations of the opacity with frequency then force the adoption of a grid spacing set by the most opaque frequency. This was avoided by smoothly fitting diffusion approximation results to the discrete ordinate calculation at large optical depths. The models were converged to radiative equilibrium using Lucy's [67] temperature correction procedure. These models yielded the first accurate continuum fluxes for extended stellar atmospheres. Hummer and Rybicki [48] derived approximate solutions of the spherically symmetric transfer problem with power law opacities and introduced the impact parameter method to obtain an accurate numerical solution of the formal transfer problem with arbitrary opacities. The impact parameter method is described in detail, with computer coding and benchmark tests, by Hummer, Kunasz and Kunasz [47]. Auer's [5] introduction of the sphericality factor transform allowed the reduction of the coupled system of moment equations, closed by the use of the variable Eddington factor method of Auer and Mihalas [8] adapted to spherical geometry, to a single second order differential equation of the Feautrier form involving only the mean intensity Jv. This is then exactly analogous to the plane-parallel problem and can be solved by the Rybicki [83] formulation of the complete linearization method of Auer and Mihalas [7]. The method just described is presented in considerable detail by Mihalas and Hummer [72] and forms the basis of this current work. In their paper, Mihalas and Hummer combined this solution of the radiative transfer problem with a simultaneous solution of the equations of radiative and hydrostatic equilibrium to derive the first self-consistent approach to the solution of the spherical atmosphere problem. The opacities considered were those appropriate for early-type stars. Models (LTE and non-LTE) were calculated for a star with a spectral Chapter 1. The Stellar Atmosphere Problem 16 type near 06. Schmid-Burgk [86] solved the spherical transfer problem, under the assumption that the opacities are arbitrary but known functions of radius, by expressing the source func-tion in terms of a cubic spline (with undetermined coefficients) in the optical depth vari-able. An impact parameter scheme was then considered with the inward and outward directed intensities (I~ and I+ respectively) expressed by means of the formal solution of the transfer equation along each impact parameter ray. The intensities I+ and I~ were then written in terms of the unknown spline coefficients. Quadrature over each spher-ical shell then permitted the moments J and H to also be expressed in terms of these spline coefficients. This allowed the definition of the discrete integral operators A and analogous to the standard plane-parallel operators that yield J and H respectively when operating on the source function S. The constancy of the total flux was used to close this matrix system, which was then iteratively solved for the temperature profile. Another study that considered the solution of the full spherical model atmosphere problem, as opposed to just that of the equation of transfer alone, was done by Hundt et al. [49]. The Schmid-Burgk method was adapted to solve the spherical radiative transfer, and this solution iterated until the equation of hydrostatic equilibrium integrated consistently. Continuous opacities were computed following the methods of the ATLAS code [63] and included contributions from H, H + , H _ , He, He + , He - , A l , Si, S i + , the metals C, N , O, Ne up to the triply ionized species, Rayleigh scattering by H, He and electron scattering. Atomic line blanketing was not included, but molecular absorption by H2O and CN was approximated using mean opacities as suggested by Tsuji [95] and Johnson et al. [58]. Solar abundances were assumed. Apparently no actual cool star models were converged at the time this paper was written since the authors describe only test gray models in their summary. A subsequent paper by Watanabe and Kodaira [102] adapted the method of Hundt Chapter 1. The Stellar Atmosphere Problem 17 et al. [49] for solving the spherical transfer problem to converge realistic nongray atmo-spheres of late-type stars. The intent of this study was to demonstrate the differential effects of atmospheric extension through comparison of spherical models with their plane-parallel counterparts. The iterative procedure used to converge to constant flux was found to oscillate, and it was necessary to impose a damping term to actually converge models. Even then, the flux at the innermost layers (rRM > 5) did not always converge, apparently due to the strong temperature dependence of the opacities. In these cases, the converged T-logTRM relation was extrapolated inward. Bound-free and free-free opacities of H and H~, free-free opacities of H 2 and He", Rayleigh scattering by H and H 2 , and Thomson scattering by electrons were all included in the absorption coefficient. Other continuous metal opacities included were adopted from the ATLAS code of Kurucz [63]. No atomic lines were included. The molecular line opacities of H 2 0 , CO, CN, OH and TiO were included. The H 2 0 opacity used was the mean opacity given by Auman [9], the CO opacity was from Tsuji's [95] JOLA opacity of the fundamental and first overtone of the vibration-rotation bands, the CN opacity was from Johnson et al. [58], and the TiO and OH opacities were taken from Tsuji's [98] V A E B M band model results. Solar composition was assumed. Wehrse [104] published spherically symmetric models of red giant and supergiant stars, solving the equation of radiative transfer using the discrete space theory of Pera-iah and Grant [78]. This technique discretizes the full equation of transfer on the cell [^ d>^ d+i] x [/^ i-i/25^ +1/2] and integrates this over the cell to obtain a tridiagonal block matrix system relating the inward and outward specific intensities (I~ and J + ) to the source function for each spherical shell. One significant advantage of this method over the complete linearization approach of Mihalas and Hummer [72] is that the angle points fii are fixed for all radii rj, and so the calculation of the total flux (which involves a quadrature over //) is riot influenced by arbitrary changes in the /^-discretization with Chapter 1. The Stellar Atmosphere Problem 18 varying radius. In effect, this method achieves a solution of the transfer problem along logarithmic spirals (curves of constant p), rather than along the impact parameter rays of the Mihalas and Hummer approach. Wehrse [104] then used a Newton-Raphson tem-perature correction technique to iterate to constant flux. A recent review of the discrete space theory is presented by Peraiah [77]. This method does have its drawbacks. The aspect ratio of the spherical shells must be kept small for stable convergence, and the maximum stable thickness shrinks with decreasing albedo of the atmosphere. In the limit of pure absorption (zero albedo), the method fails altogether. Finally, the formalism of this method and its associated algebraic complexity obscure the underlying physics making intuitive interpretation of the system that much more difficult. Wehrse [104] adopted mean opacities for the absorption due to the molecules OH, CH, CN, MgH, SiH, TiO, CO and H 2 0 in his models. The molecular bands were assumed to be formed in pure absorption. While it was recognized that scattering processes may play an important role in molecular line formation, it was considered infeasible to handle this in the model calculations (which would require a non-LTE treatment of a molecular absorber). The usual sources of bound-free and free-free absorption were included, along with Rayleigh scattering by H, H 2 and He. Wehrse calculated a series of extended models (and one plane-parallel benchmark), all with Teff = 3000K and logg = 0 for varying He, CNO and metal abundances, and found widely ranging amounts of geometric extension of the atmosphere. Values of the extension d, defined by , r(n.2ti = 1) ranged from d = 0.05, which is nearly compact, to d — 0.30. Included in these models, however, were two with considerably enhanced He abundances (up to [He/H] = +2) which essentially had compact atmospheres. Omitting these He-rich models, the extensions were Chapter 1. The Stellar Atmosphere Problem 19 all in the range d = 0.19 — 0.30. Flux distributions and profiles of selected Fe I lines were also calculated. Overall, the direct effect of sphericity on both the temperature structure of the atmosphere and the line profiles considered was found to be quite small. Indirect effects, such as opacity changes induced by temperature changes, were often substantial but unpredictable without detailed modelling. The treatment of radiative transfer in the spectral lines (whether pure absorption or absorption and scattering assumed) also had a major effect on the emergent spectrum. Schmid-Burgk et al. [87] computed nongray model atmospheres of late M giants and supergiants, for both the plane-parallel and spherically symmetric cases. The radiative transfer problem was solved by both the Schmid-Burgk [86] and Peraiah and Grant [78] methods. Both schemes yielded identical results within the 1% error limit adopted for convergence of the model iterations. The equation of state included 52 particle species plus free electrons, and the chem-ical equilibrium was solved by a Newton-Raphson iteration procedure. The continuous absorbers H, H - , H 2 , , Hj", He - , C, C - , Mg, A l , Si and Fe were included along with Rayleigh scattering by H, H 2 and He, and Thomson scattering by free electrons. Also, the species CH, CN, CO, OH, H 2 0 , TiO, MgH, SiH and CaH were included as molec-ular band absorbers, and treated in the JOLA approximation. Since this mean opacity representation 'smears out' existing flux windows, a 'picket fence' test model was also run in which the opacities were set at double the JOLA values for half of the frequency points, and to zero for the remaining frequencies. Models were also computed in which the molecular opacities (except H 2 0) were treated as pure scattering sources instead of as pure absorbers. The resulting models show temperature differences of about 200K in the outer at-mosphere (logr < — 2) between corresponding spherically symmetric and plane-parallel models. Similar differences exist between models run with the varying test opacities just Chapter 1. The Stellar Atmosphere Problem 20 described. The authors also demonstrate that significant extension is common, with the geometric extension d as denned by equation (1.1) > 0.1 for practically all M stars with masses near 1 M 0 and luminosities > 103Z©. Scholz and Wehrse [90] applied the models of Schmid-Burgk et al. [87] in an attempt to devise an observationally verifiable three-dimensional spectral classification scheme for late M giants and supergiants. While the standard two-dimensional MK system [73] essentially measures Tcff and g, the significant extension expected for some late M giants warrants the addition of a third parameter. Scholz and Wehrse proposed the use of the atmospheric extension d, as denned by equation (1.1). If the extension becomes significant, the photospheric radius R is no longer well defined. Scholz and Wehrse then proposed to arbitrarily define the radius by R = r(ri.2M = 1), and to use the standard relations for Teff and g to define these quantites in terms of this R. Scholz and Wehrse calculated a series of models of solar composition with T e / / of 2750K and 3000K, for a variety of gravities and extensions. Synthetic spectra were calculated, and corresponding narrow band colours derived both on Wing's [107] original eight-colour system, and its extension into the infrared [108]. On the basis of these models, the authors suggest it should be possible to deduce the geometric extension d of the atmosphere from a knowledge of the TiO colour indices, using continuum colour indices and line strengths of ionized metals to infer Teff and g respectively. This is because the TiO band strengths are a sensitive indicator of the extent of the cooler, outer layers of the atmosphere. While in principle this seems feasible, models used for this purpose should include VO as an opacity source (Scholz and Wehrse do not), since VO bands are prominent in the infrared near lfi for giants of spectral type later than M5, and may contaminate supposed 'continuum' windows. Scholz and Tsuji [89] computed a series of spherically extended model atmospheres for Teff = 2500K, 3000K and 3500K, for both M and C star compositions using the method Chapter 1. The Stellar Atmosphere Problem 21 of Schmid-Burgk [86] to solve the spherical transfer problem. A revised version of Tsuji's [99] program was used to solve the equation of state, which included 34 elements and 61 molecular species, and to calculate the absorption and scattering coefficients. Continuous opacities considered included H, H~, HJ , He - , Mg, Si and Ca as absorption sources, and H and H 2 Rayleigh scattering and Thomson scattering by electrons as scattering sources. Molecular absorbers included were H 2 (collision-induced dipole), CO, OH, CN, CH, C 2 , TiO, MgH, SiH, CaH and the polyatomic species H 2 0 , HCN and C 2 H 2 , although contributions from the latter two molecules were incompletely included due to the lack of laboratory data. All molecular line blanketing was represented by straight mean opacities. The most obvious result found was the pronounced lowering of temperatures and pressures in the outer layers of the spherically symmetric models as compared with their plane-parallel counterparts. Some problems are evident with the opacities used in these models. Scholz and Tsuji's [89] opacities for C star models at 3000K, even omitting polyatomics, are more than two orders of magnitude larger than those of Johnson [54], for depths near logrRM = —2. The inclusion of HCN and C 2 H 2 at 3000K [89] results in a further increase in log K R M of about 0.5 at this depth. The models of Eriksson et al. [36], which are plane-parallel C star models incorporating a more detailed analysis of the HCN opacity, show an increase in log KRM « 2 when HCN is included at 2500K at l o g r ^ = —2. Unfortunately, neither Scholz and Tsuji [89] nor Eriksson et al. [36] present comparable opacities used in their respective models, and thus a direct comparison of the models remains difficult. Still, it is evident that the cooler temperatures found in the outer layers of extended models greatly exacerbate the problem of calculating molecular opacities. Scholz [88] used the atmosphere code of Scholz and Tsuji [89] to calculate spheri-cally symmetric models of M giants over the range 2500K< Ttff < 3800K for various values of the gravity and atmospheric extension. The purpose of this study was again Chapter 1. The Stellar Atmosphere Problem 22 to examine the construction of a three-dimensional spectral classification scheme based on (Teff,\ogg, d) and its sensitivity to changes in temperature and composition. Various narrow band colour indices were also considered as indicators of (T e / / , log g, d). While extension effects were shown to be significant and sometimes important enough to dom-inate Ttff and g effects, the implementation of this classification scheme will be severely hampered by the complex behaviour of the molecular band strengths. More accurate and consistent opacities are needed. Preliminary work on modelling extended atmospheres of late M supergiants using the opacity sampling (OS) technique of Johnson et al. [56] has been reported by Brown et ah [22]. The code used was based on Wehrse's [104] implementation of the Peraiah and Grant [78] solution of the spherical transfer problem. The bound-bound opacities of atomic lines, CO, CN, TiO, C 2 , CH, NH, OH and MgH have been included using the OS method, while H 2 0 was represented by a straight mean opacity. Difficulties were encountered since these models did not attain the diffusion limit (as is necessary) at depth, due to limitations of the radiative transfer method [24]. These models remain unpublished, and the current status is unclear. Recently, Bessell et al. [19] have published an extensive grid of spherically symmetric models of M giants. The models were computed using the method described previously by Scholz [88]. This study continued progress toward the development of a quantita-tive, three-dimensional system of stellar classification of M giants and supergiants, based on the parameters Teff, g, and extension d. These models incorporated the additional straight mean opacities calculated by Brett [21] due to the VO A-X, B-X, and TiO e molecular band systems. Models were calculated for a variety of luminosities and metal-licities over the range 3000K< Ttft < 3800K with extensions down to Ttff = 2500K in certain cases. An analysis of narrow band colours, mainly on Wing's [107, 105] 8 colour system, and broad band colours was presented. Particular consideration was given to Chapter 1. The Stellar Atmosphere Problem 23 the determination of gravity-sensitive and extension-sensitive indicators. The authors concluded that a photometric separation of the effects of Teff, gravity and extension is in principle possible. Spherically symmetric models of A stars, with line blanketing, were calculated recently by Fieldus, Lester and Rogers [39]. The model code used was a modified version of ATLAS [64] in which the line blanketing was treated using opacity distribution functions. The radiative transfer in spherical geometry was solved using the half-range moment method of Martin, Rogers and Rybicki [68]. The model was converged using a modification of the flux correction scheme of Simonneau and Crivellari [91] for spherical geometry. This procedure iterates an integral form of the first moment transfer equation to improve the estimate of the mean intensity Jv, and therefore the temperature via the equation of radiative equilibrium. Convergence was slowed in the spherical case, however, since some assumed 'invariants' are no longer invariant. No molecules were included in the equation of state or as sources of opacity since the code was intended to model stars of intermediate spectral type. An analysis of the effect of extension upon the temperature structure, emergent flux, and broad band colours was presented for a grid of stars with Tcff = 10000K for varying gravities and extensions. 1.7 Objectives of this Research This research has two main goals: • To develop a method to solve the spherically symmetric atmosphere problem based on the complete linearization method [7]. • To implement a working model atmosphere code for spherical geometry, with the aim of studying the extended atmospheres of late-type giant and supergiant stars. Chapter 1. The Stellar Atmosphere Problem 24 The method used is broadly based upon that of Mihalas and Hummer [72] as employed in their pioneering study of atmospheric extension in 0 stars. The molecular line opacities are represented by mean opacities to reduce the computational effort involved. We would like to develop a model code which minimizes the additional solution difficulties of the spherical atmosphere as compared with the plane-parallel problem. Ideally, we also seek a spherical code that would reduce directly to the corresponding plane-parallel code in the limiting case of compact atmospheres (in order to aid in differential studies of sphericity). We have achieved success in both endeavours. Our hope is that the ATHENA model code developed here will permit a definitive study of the importance of extension in the atmospheres of late-type giant stars, and serve as a base for further work in the areas of steady-state dynamics (i.e. mass loss), pulsation, and non-LTE. Chapter 2 The Spherically Symmetric Model System 2.1 Introduction This chapter describes the theoretical method used to implement fully self-consistent models of late-type giant stars with static, but extended atmospheres. We have broadly followed the complete linearization method of Mihalas and Hummer (MH) [72] in solving the spherically symmetric model atmosphere problem. In this chapter we develop the moment equation formalism and discuss the numerical difficulties encountered in pursuing the MH method. The formal solution procedure of Hummer, Kunasz, and Kunasz (HKK) [47] is reviewed and the problem of numerical inconsistency between the moment and formal solutions is addressed. We present boundary conditions that account for the thermal emission of gas exterior to the outer boundary of the model. 2.2 The Model Equations The complete linearization technique of Auer and Mihalas [7] using variable Eddington factors [8] to close the system of radiative transfer equations and adapted to spherical geometry as described by Mihalas and Hummer (MH) [72] will be implemented. The equation of transfer, assuming spherical symmetry, for a ray with direction fi = cos 0 crossing a radius r is where Iv,Sv,p,Xv a r e the radiation intensity, the source function, the gas density, and the opacity per unit mass respectively. The source function is denned as the ratio of 25 Chapter 2. The Spherically Symmetric Model System 26 emissivity r\v to opacity Xv The opacity used here is the total extinction coefficient. We shall assume that this opacity can be expressed as the sum of a pure absorption component KV and a pure scattering component o~v so that Xv — Kv -\- o~v. We shall further assume that all of these coefficients are isotropic. The emissivity per unit mass can then be decomposed into a thermal emission coefficient rf^ and a scattering emission coefficient 77*°. For an LTE atmosphere, rf^ — KUBU(T) and rj" — o-vJu. The source function is then M ^ T ^ ) < 2- 2 ) which depends only upon the mean intensity J„ of the radiation field and formally, at least, this dependence is linear. Under these conditions, the transfer of radiation in the atmosphere depends only upon «7„. Knowledge of the full angle dependent intensity Iv(p) is unnecessary. This allows the solution of the radiative transfer to be solved using the moment equation analogues of equation (2.1) and so reduces the dimensionality of the problem from 3 to 2. The linearity of Sv with respect to J„ permits the resulting finite difference system to be efficiently solved by arrangement into the Rybicki [83] block matrix form. Multiplication of equation (2.1) by powers of the angle coordinate p followed by integration over p yields the corresponding moment equations, any set of which always contains one more unknown than equations so that an additional relation among the moments must be supplied to close the system. Since only the zero order moment Jv is actually needed to solve the transfer problem, the number of additional moments evaluated should be kept minimal. At depth in the atmosphere strict thermodynamic equilibrium (TE) is approached, the radiation field must approach the Planck field, and the diffusion approximation becomes valid. Here, Jv ~ Bu, and it can also be shown [70] that the second moment Ku & |i?„ so that the ratio of moments KvjJv = fv ~ | . This ratio /„ is called the Eddington Chapter 2. The Spherically Symmetric Model System 27 factor, and its knowledge can provide the additional relation needed to close the moment equations. The spherically symmetric radiative transfer problem can then be reduced to the determination of the Eddington factors and the solution of the system consisting of the first two moments of the transfer equation (2.1). The zero and first order moment equations are, respectively, ±£(r2Hu)=pXv(Sv-Jv) (2.3) ^ + -{3Kv-J„) = -pXvHv (2.4) ar r where knowledge of the Eddington factor implies Kv = fvJv (2.5) and where Hu is the first moment of the intensity, often referred to the as the Eddington flux. Hv is related to the standard flux by Tv = kirHv. Unfortunately, while fv = | is valid at depth, its value in the outer layers of the atmosphere is not known. To obtain fv throughout the atmosphere, it is necessary to solve the full angle-dependent transfer problem for the specific intensity /„(/*) at each radius, and then perform the appropriate quadratures over the angle coordinate p, to obtain the moments Jv and Kv. The Eddington factor /„ = Kv/Jv is then immediately obtained. In this study the impact parameter method of Hummer, Kunasz, and Kunasz (HKK) [47] was used to obtain the formal solution for the specific intensity given the temperature T and the total pressure p. While reverting to the full angle dependent problem just to find /„ and close the moment equations may seem to be a regressive step, it requires much less computational effort to solve the full formal problem (where T and p are assumed known) and to iterate the two-dimensional moment equation system to full consistency, than to directly iterate the full three-dimensional transfer problem. Chapter 2. The Spherically Symmetric Model System 28 The second term on the left hand side of equation (2.4) numerically destabilizes the system [70]. An integrating factor for the left hand side can be found, namely, the sphericality transform, '3 /„- l \<fr' ' introduced by Auer [5]. This permits the analytic removal of the offending term involving 1/r, yielding spherical moment equations exactly analogous to the plane-parallel form. Using the (radial) monochromatic optical depth drv — —pXudr as the independent vari-able, the moment equations (2.3, 2.4) become ±(r2Hv)=r\Jv-Sv) (2.7) -J-(f»q»r2Jv) = r2Hv (2.8) qu drv Eliminating Hv and introducing the generalized optical depth coordinate dXv — qvdrv = —PlvXvdr, we obtain d2 r 2 , ,v2{Uqvr2Jv) = -(Jv-Sv). (2.9) dX£ qv which is the desired equation of radiative transfer for our spherically symmetric, static models. The method just described is essentially that employed by MH in their study of spherically extended O stars. They assumed a fixed radial grid and expressed the optical depth coordinate drv = —pXvdr in terms of the radius scale. The full system of equa-tions consisting of the moment transfer equations, radiative equilibrium and hydrostatic equilibrium were then linearized in terms of the fundamental physical variables, the tem-perature T, the number density n, and the mean intensity of the radiation field Jv. The various constitutive relations describing the density and opacity of the stellar gas were also linearized in'terms of T and n, assuming P = p(T,n) (2.10) Xu = Xu(T,n). (2.11) Chapter 2. The Spherically Symmetric Model System 29 The Eddington factor /„ was obtained from the formal solution of the full angle dependent transfer problem, where T(r) and n(r) were given. The Eddington factor was held constant during the subsequent linearization step. The linear corrections ST, Sn, and SJV found were used to update their respective quantities. The entire cycle was then repeated and (hopefully) iterated to convergence. In practice the MH method fails completely for late-type stellar atmospheres. While it proved possible to converge test models with constant or only slightly variable opacities, we were unable to converge any model with opacities typical of real late-type stars. These stars have continuous opacities dominated by the bound-free and free-free absorption of the H - ion in addition to significant absorption from a variety of molecular sources in their outer layers. The opacity of the stellar gas under these conditions is very temperature sensitive, increasing by 5-6 orders of magnitude from the outer to inner boundary of the atmosphere. The local power law dependence of opacity on temperature, K(T) OC T V (2.12) typically yields values of 7 ~ 10 in the vicinity of the hydrogen ionization zone (T ~ 104K). The temperature structure of the atmosphere is essentially determined by the con-straint of radiative equilibrium. When K(T) exhibits a strong temperature dependence, as above, a typical temperature correction of ~ 10% needed to recover radiative equilib-rium would result in a corresponding change in K by a factor of l . l 1 0 ~ 2.6! Effectively, the optical depth scale slides back and forth with respect to the fixed radius grid as the temperature is perturbed. However, the response of the linearized system obtained using the MH method is generally not to follow the shifting optical depths at each radius (which would require all of the dependent variables T, n, and Jv to shift in unison) but to attempt to compensate for the change in K(T) by an offsetting change in the density of the gas. In the above example, a change in n by a factor of 2.6 will restore the original Chapter 2. The Spherically Symmetric Model System 30 A r but at the expense of destroying hydrostatic equilibrium. These changes far exceed the typical radius of convergence of a nonlinear system of equations (typically ~ 10% ) so that all hope of convergence is lost. The fundamental problem here is that we are not only solving the transfer equation for JV but for TV as well. This greatly complicates the coupling of an already highly nonlinear system and destroys the convergence properties of the linearized system. The standard solution procedure in plane-parallel model atmosphere work is to use a 'natural' variable tied to the intrinsic 'depth' in the atmosphere, such as column mass or optical depth. The use of column mass makes the equation of hydrostatic equlibrium nearly an exact integral and numerically well-behaved. By contrast, the spherical form of the equation of hydrostatic equilibrium expressed in terms of the radius r yields solutions which grow exponentially inward; a difficult situation to handle numerically. The choice of radius for the independent variable appears to be dictated by the explicit appearance of the term involving 1/r in the original first order moment equation (2.4), or equivalently, by the appearance of the sphericality function q(r) in the transformed moment equation analog (2.8). As noted by MH, if r is not chosen as the independent variable and held fixed during the linearization step, it must also be linearized in terms of the alternate depth variable employed. Their conclusion was that linearization of the radius would lead to more complicated equations and that the effect upon the sphericality function q(r) would be to disastrously destabilize the transfer equations. In fact, this is not the case. We shall show in section 3.2 that it is possible to generalize the standard plane-parallel solution to the spherical case with relatively little increase in complexity over the MH formulation. Either column mass or optical depth may be used. Both were implemented in this work. The column mass scale models converged well (even better than did the optical depth based models) if the opacity was well-behaved and possessed no strong dependencies on the temperature or pressure. However, these Chapter 2. The Spherically Symmetric Model System 31 models faired badly when realistic opacities (which are very temperature-dependent) were added. The most suitable depth scale was found to be the Rosseland mean optical depth. The column mass still suffers from the drawback that the differential optical depth drv = Xvdrn remains quite sensitive to temperature perturbations in regions where x(T) is a strongly varying function of temperature T. The advantage using a mass scale is that the hydrostatic equation remains unaffected by such temperature perturbations, provided the gas pressure p rather than the number density n is used as the fundamental variable. Basically, the gas pressure is insensitive to perturbations in the temperature structure of the atmosphere. We have, therefore, retained p as the fundamental variable throughout this work, in contrast to MH. However, the use of a Rosseland mean optical depth scale automatically provides T(TRM) and JV(TRM) = BV(T) at depth, given simply by T 4 = ^ [ r f i M + ? f i M ] , (2.13) where 9H(T) is the Hopf function appearing in the solution of the plane-parallel grey atmosphere problem. Strictly speaking, this result is true only if the atmospheric scale height h <C R* at depth. In practice, this result remains fairly accurate since extension effects do not generally penetrate too deeply into the atmosphere. Furthermore, we now have drv = Xvdm = (xulXRM)dTRM = XvdrRM, (2.14) where Xv — XV/XRM- The overall temperature dependencies of the opacity, such as the rapid increase in x„(T), are largely cancelled out in the ratio x.v(T). The drawback is that the hydrostatic equation now suffers from column mass variations as a result of temperature perturbations. Since the hydrostatic equation seems better able to cope with such perturbations than do the transfer equations, the decision was made to use the Rosseland mean optical depth as the independent variable throughout this work. Chapter 2. The Spherically Symmetric Model System 32 The final formulation of the model uses rRM as the independent variable, and T, p, r and Jv as the dependent variables in the linearization. The system is closed by the use of the constraint equations. One constraint is provided by the equation of radiative equilibrium which ensures the conservation of energy in the radiation field by requiring that the energy radiated is balanced by the energy absorbed in each layer. Therefore, f Jo (v,-xMdu = 0 (2.15) lo which for the source function specified by equation (2.2) reduces to Kv(Bv - Jv) dv = 0. (2.16) f Jo The form of this equation is important. It proved necessary to make the following rear-rangement nvJv dv - 1 = 0 (2.17) I I OO KUBv dv 0 in order to obtain a numerical solution. This is due to the fact that the left-hand side (LHS) of equation (2.16) is a complex, strongly temperature-dependent function for a cool molecular gas where molecular opacities are important. Most of this variation is cancelled out in the latter equation (2.17), however, leaving the LHS a well-behaved, slowly varying, monotonic function of temperature. As a result, the range of convergence of the corresponding linearized equation is greatly increased. A second constraint is provided by the assumption of hydrostatic equilibrium in the stellar atmosphere. For a spherically symmetric atmosphere, this constraint can be ex-pressed using the expression from Mihalas [70] converted to optical depth variables, dP = GM* _ l£L ( 2 l g ) drRM XRMT2 47rcr2' Chapter 2. The Spherically Symmetric Model System 33 where the gas pressure p is assumed to obey the ideal gas law p = nkT. The second term on the right-hand side of this equation arises from the radiation pressure gradient (V • P ) r , and the parameter 7 is defined, following Mihalas [70] as 7 = — ^ / XuH„dv, (2.19) X-RM-" Jo which describes the efficiency of momentum coupling between the radiation field and the stellar gas. The final constraint, the radius (or depth) equation, drRM PXRM provides the relation between the radius and Rosseland mean optical depth scales. 2.3 The Formal Solution The formal solution of the transfer problem refers to the direct solution of the equation of transfer for Jv for given values of the fundamental physical variables T(rRM), p(rRM) and T^HM). This solution by itself is of little physical significance since the quantities T and p in turn depend upon Jv in a real stellar atmosphere, coupling via the equations of radiative and hydrostatic equilibrium. However, it is of value as a step in an iterative solution procedure. The solution of the model atmosphere described so far relies on the collapse of the inherently two-dimensional transfer problem (depth, angle) to a one-dimensional one (depth) through the use of the moment equation system. This system is closed through the use of variable Eddington factors and is then iterated to a self-consistent solution. The angular distribution of the radiation field is then effectively described by a single value (the Eddington factor) which is not linearized in the moment equation solution. There-fore, although iteration of the linearized moment equations will yield a self-consistent Chapter 2. The Spherically Symmetric Model System 34 solution, this procedure ignores any changes in the angular distribution of the radia-tion field arising during these iterations. Therefore, it is necessary from time to time to re-evaluate the Eddington factors, which is done by a formal solution of the full two-dimensional transfer problem. The method of solution of the formal problem used in this study is the impact param-eter scheme of H K K which is also summarized by Mihalas [70]. The procedure involves solving the one dimensional transfer problem along a series of parallel rays (the 'impact parameters') intersecting concentric spherical shells, as illustrated in Figure 2.1. Each ray pi intersects the radial shell rd at an angle with cosine pdi = fi(rd,pi) = — Jrl-p} = zdi/rd (2.21) rd * and so a solution along each of these rays solves the full two dimensional spherical sym-metric transfer problem, since /„(/*<£) is obtained at all grid radii rd. 4 There are two classes of impact ray in this scheme distinguished by differing inner boundary conditions: (1) those rays tangent or exterior to the inner radius of the grid; and (2) the remaining rays which penetrate the 'core', interior to the inner radius. The equation of transfer along each ray has the standard form ± d ^ P i ) = PMr)- X,(r)£(*,ft)] (2.22) where the + and — signs refer to outward and inward directed radiation respectively. Also, it is assumed r = y/p2 + z 2 . Denning the optical depth along the ray, dtu = —pxvdz, and with Sv = i]u/xv, ^ dI±(Pi,tv) dt I„{Pi,tu) ~ Sv{r) (2.23) V It is now convenient to define the variables uv{pi,U) = \[l+{piX) + K{pi,tv)\ (2.24) vv{pi,U) = \[l+(pi,tv)-l;{pi,tv)] (2.25) Figure 2.1: Impact parameter coordinate system for formal solution Chapter 2. The Spherically Symmetric Model System 36 which behave as 'mean-intensity-like' and 'flux-like' quantities respectively. These obey the relations du„(Pi,tv) = _ (2_26) dt„ dvv(pi,tv) = uu(Pi,tu) - Sv(r) (2.27) dtv from which vv may be eliminated to get the final form of the transfer equation along the ray ^ U v ^ t v ) = u„(Pi,tv) -S„(r) (2.28) The solution of this second order differential equation yields uv(pi,tu), and vv(pi,tu) can then be found by equation (2.26). The moments of the radiation field are obtained by quadrature over the angle coordinate p as follows, Jv(r) = ( uv(r,p)dp (2.29) Jo Hu(r) = I pvu(r,p)dp (2.30) Ku(r) = j p2uv(r, p) dp. (2.31) Jo The updated Eddington factor is then given by ^(r) = Ku(r)/Ju(r). There are two difficulties with using this formal solution to determine /„. One is the potentially large number of impact parameter rays needed for problems with opacities displaying a strong dependence upon frequency. This is since the impact parameters chosen must provide good coverage of p over the radial range where the atmosphere becomes optically thin. For the situation mentioned here the required optical depth range may extend over a considerable radial extent. The second drawback is more fundamental. While the calculation of moments by quadrature of the formal solution intensities is mathematically equivalent to the direct solution of the moment equations, assuming /„ is known exactly, the corresponding discrete representations of these problems are not Chapter 2. The Spherically Symmetric Model System 37 identical. Thus, the impact parameter solution can not be made exactly consistent with the direct solution of the moment equations of transfer [71]. This problem is exacerbated since the moments found by the formal solution method are obtained by quadrature over a varying number of intensities at varying values of the angle coordinate y, for each radius. The lack of a consistent quadrature grid results in the introduction of unavoidable numerical 'noise' in the radial behaviour of the moments so obtained. For example, in the outer layers of the atmosphere, r2Jv approaches a constant value. For sufficiently large r, numerical fluctuations in the value of the quadrature integral will eventually dominate over the actual physical variations. Any attempt to use these values of J„ in the moment transfer equation (2.9), which requires the accurate evaluation of the second radial derivatives of J„ will result in severe problems with the convergence of the linearized system. It appears that this difficulty can be circumvented in the outer layers where r2 Jv is nearly constant by retaining Hv from the formal solution instead and integrating equation (2.8) to obtain Jv. Elsewhere in the atmosphere Ju is returned from the formal solution, and Hv is then found by evaluating the derivative on the left hand side of equation (2.8). In detail, the criterion used was to evaluate Jv directly by quadrature and Hv by differentiation using the moment equation (2.8), unless A(fvqur2Ju)/(fvqur2Ju) for the depth interval [£.,•, i,-+i] was < 1%. In this case, Hu was instead evaluated by quadrature and Jv calculated by the integration formula AJV = -—— f3+1 qvr2Hvdtv (2.32) UW2 Jtj derived from equation (2.8). The relative accuracy of the quadrature integrals is of order 10 - 5 -10 - 6 . The expression AJV/JV provides an estimate of the loss of significance incurred in the numerical differentiation of Ju. When this loss of significance exceeds two decimal places, we choose instead to evaluate Hv directly by quadrature. The domains of these modes of evaluation of J„ are shown in Figures 2.2 and 2.3 for two converged Chapter 2. The Spherically Symmetric Model System 38 models. This procedure ensures that the Jv used in the subsequent iteration of the moment equations differentiates to give the correct flux according to equation (2.8) at the boundaries. This is extremely important since both boundary conditions involve flux constraints, and if these are not accurately imposed, systematic errors will be present throughout the entire atmosphere. 2.4 The Boundary Conditions 2.4.1 The Formal Solution First, we shall consider the boundary conditions needed for the formal solution of the equation of transfer. This involves the solution of a series of one dimensional transfer problems along each of the impact rays. The usual boundary conditions for this sec-ond order differential equation involve the specification of the derivative duv/dtv of the intensity at both the inner and outer boundaries. From equations (2.24) - (2.25) we have vv = uv-I~ =1+ - tt„. (2.33) Since duv = < 2- 3 4> we obtain the following relations which supply the necessary boundary conditions, namely, du dt - = uv-I- (2.35) V at the outer boundary, and duv ( 2 - 3 6 ) which proves to be the more convenient form for the inner boundary, since here vv can be directly evaluated in the diffusion limit. Chapter 2. The Spherically Symmetric Model System 39 •L 60| Figure 2.2: Domains of calculation of Ju and Hu: Model 01310191 Chapter 2. The Spherically Symmetric Model System 40 IAIH -L 60| Figure 2.3: Domains of calculation of Jv and H„: Model 02310191 Chapter 2. The Spherically Symmetric Model System 41 Formal Solution: Inner Boundary Condition To express the actual inner boundary condition, we must distinguish the two classes of impact rays mentioned earlier: those rays intersecting the core interior to the innermost radius of the grid, and those rays tangent to, or exterior to, the core. At the inner boundary, it is assumed that the optical depth is sufficiently great that the diffusion approximation is valid. Then, dB /„(/*) « Bv + fi-r^ (2.37) ' v and vu = Wt - K) « ^ (2-38) The gradient dBv/dTv can be directly evaluated, as follows. The flux Hv is given by ^ldBv_ 1 dBv _ IdT (ldBv\ assuming the diffusion approximation. Then, since the total flux, is known, we must have t00 rr , 1 dT r 1 dBv ' , • Therefore, dT ZpH dr S7xZX{dBvldT)du\ and from equation (2.39) dBv IdT ( 1 dBvs drv p dr \xu dT or, after substitution of equation (2.42), d-^ = ZH drv X^jdBjdT) rx^idB./d^du (2.42) (2.43) (2.44) Chapter 2. The Spherically Symmetric Model System 42 An alternate form can be derived using the value of H from equation (2.40) and the Rosseland mean opacity XRM defined by _ Jo~(dB„/dT)du _ 4.T3 . X R M ~ r X^(dBv/dT) du 7r / " x^(dB,/dT) du ^ Therefore, r ± ^ i v = ^Il, (2.46) Jo Xu dT TTXRM and so an alternate form of dBv/dTv is obtained by substitution of the above expressions for H and / xZ1dBv/dT dv into equation (2.44), yielding dBv 3 i , /dB v \ ^ drv 64ir<rr2T3Xv \ dT J ' where Xu = XV/XRM-For impact rays intersecting the core, then dBv Vu = P~rr, (2.48) while for rays exterior to the core the interior boundary condition is applied, not at the inner radius, but rather at z — 0 along the axis of symmetry. Here p = 0, and therefore, vv = 0, (2.49) as also follows directly from symmetry considerations. The general inner boundary con-dition for an impact ray is thus duv 3J&, dtu ^647r<7r2r3x Formal Solution: Outer Boundary Condition Traditionally, the outer boundary for radiative transfer has been derived under the as-sumption of no incident radiation, or I~ = 0. Then, the boundary condition is just duu uv (2.51) dtv Chapter 2. The Spherically Symmetric Model System 43 as follows from equations (2.33)-(2.34). However, stellar material is present exterior to the outermost radius, as is explicitly assumed by the nonzero values of the gas pressure and optical depth adopted on the outer boundary. In this study, thermal emission from gas exterior to the outer boundary is also included, yielding a nonzero boundary value of I~ also. For a spherically symmetric, static isothermal atmosphere, the mass obtained inte-grating equation (2.18) to infinity is itself infinite, as is the optical depth, and the pressure and density remain finite at infinity [30]. For full consistency, the equation of hydrostatic equilibrium should be replaced by its hydrodynamic generalization, which permits a flow. Lacking this, an arbitrary cutoff procedure must be adopted. MH assumed an exponen-tial, isothermal atmosphere existed exterior to the outer optical depth grid point at T\ and also assumed the Rosseland mean opacity to be independent of pressure in this re-gion. We will generalize this latter assumption and permit a power law dependence of the opacity upon density of the form K(P) OC px, where the index A is then given by p &K , x = irP- <2-52> We still ignore any spectral variations in A and evaluate this quantity assuming K = K R M , the Rosseland mean absorption coefficient. Then, we have the following asymptotic behaviour i T(r) = Ti (2.53) p(r) = P l exp (2.54) p(r) = P l exp ( - ^ ) " • » > Xu(r) = xu>exp(- A ( r~ r i )) (2.56) where h is the scale height of the atmosphere at the outer boundary, and radiation pressure is ignored. Chapter 2. The Spherically Symmetric Model System 44 An incident radiation boundary condition for radiative transfer can now be derived. The formal solution procedure of HKK requires the specification of a boundary value of the incoming radiation along each impact ray making an angle, p, = cos 6, with the outer boundary radius r = T\. The transfer equation for radiation incoming along each ray exterior to the atmosphere is then dl~ v ~K-SV (2.57) dtv or dl7, d z PXv(Iv-SV), (2.58) where the source function is as defined by equation (2.2). Assuming in this region that I~ <C SV for r > r i , equation (2.58) reduces to •' ^ • = -PXUSV. (2.59) For late-type stars we can assume negligible back scattering of JV into I~, SV = KVBV/XV, and so we obtain -^ = -px-{-^r)^-pK^- (2-60) For early-type stars it will, in general, probably be necessary to include scattering con-tributions in a calculation of this type. Integrating inward along the ray, assuming I~(oo) = 0, yields J1"(zi) = 5 l v A p(z)Ku(z)dz, (2.61) Jzi where Z\ — z(ri). We define the extension parameter, d = h/ri, and also £ = z/ri and recall that r = <\J'z2 + p2, where z is the distance along the impact ray and p is the transverse coordinate. The reader is again referred to Figure 2.1 for details of the impact parameter coordinate system. Note that £i = Z\jr\ = p,. Substitution of the asymptotic behaviour of p{r) and Xv{r) given by equations (2.54) and (2.56) into the integrand of Chapter 2. The Spherically Symmetric Model System 45 equation (2.61) yields •KM = Pi*i VB\V i J UL exp ( l + A)( r -n) rid* or hub1) = PiKivB\vr\ I exp -(1 + A) f (2.62) di. (2.63) Further manipulation then gives + ? - P ? - I ) di (2.64) which can be written in the form T- / x PlhKlvBiv ( d (2.65) (2.66) 1 + A V 1 + where the function G(p, d') is defined by where = (1 + A) above. There are two simple special cases. First, note that for the radial ray p = 1, the expression for G reduces to G(l,d) = 1, (2.67) while another reduction is realized in the case of a compact atmosphere (d — 0), where G(p,0) = -. (2.68) The Rosseland mean optical depth T R M may be integrated radially inward from infinity to the outer boundary to obtain PlhXl.RM Tl.RM — 1 + A and the corresponding monochromatic (radial) optical depth is T i l / — T1,RM ( X1 " A \Xl,RM/ RiaXlv — 1 + A " (2.69) (2.70) Chapter 2. The Spherically Symmetric Model System 46 where X L / = XV/XRM- Letting = KU/XV, then and so equation (2.65) simplifies to IrAf1) = Ti,RM^iuXiuBlvG(fi,d') = tabStV(fi)Blu, (2.72) where tabs,u{p) = Tl,RM^luXluG(n,d') (2.73) is the absorption optical depth at the outer boundary along the ray with direction fi. One further generalization is useful. Ideally, the boundary optical depth T i | R M is chosen sufficiently small that T\V <C 1 for all frequencies. However, during the production of the models it was found that some frequencies in the far ultraviolet were optically thick due to strong bound-free absorption from abundant light metals and that the optical depth in the strongest molecular bands approached optical depth unity, even with T R M = 10~6. A considerable improvement in accuracy occurs if the above tab3iV is replaced in equation (2.72) by 1 — exp(—£„&,,„). Then, in the optically thin limit of tab,iV <C 1, the previous result, equation (2.72), is recovered for J^,(/i). In the optically thick limit, tabs,v ^ 15 the correct result IM = Biv (2-74) is obtained as well. The final expression for incident radiation at the outer boundary of the atmosphere thus becomes ^ ( ^ ) = { l - exp [ -T ^ , n M ^ I / Xl .G( / i , c^ , ) ] }^ • (2-75) Given this value of J~(/i), the outer boundary condition of the radiative transfer equation along the impact ray with direction fi is ^ = « * - - t t / 0 , (2-76) Chapter 2. The Spherically Symmetric Model System 47 or, duv dt„ -uv-{l- exp[-TliRMivXuG(p,d')]}Bu (2.77) where the depth subscripts have been suppressed for clarity. Parametrization of the G function The function G(p, d') is defined by the integral The change of variable £' = £ — p results in the equivalent form d^ (2.78) 1 f°° G(p,d') = j,J e x P di' (2.79) which is more convenient for numerical evaluation, since the constant limits of integration permit the use of a fixed (p,d') grid. This representation was used in this study to evaluate G(p,d') over the two dimensional grid {(pi,d'k)}, with Pi e {0,0.001,0.002,0.005,0.01,0.02,0.03,... ,0.99,1} d'k <E {0,10-4,10~3,0.01,0.03,0.1,0.3,1,3,10,30,100} (2.80) (2.81) except for the case d' = 0, where the exact value G(p,0) = 1/p was used. In order to permit the rapid evaluation of this integral in the ATHENA atmosphere code, an analytic parametrization was sought. The limiting values in the previously mentioned special cases suggested a function of the form 1 G(p,d') = (2.82) ad.(p-iy + bd.(p-l) + V with ad> = 0,&d< = 1 for the hmiting case of d' = 0. Then the results G(l,d') = 1 and G([i,0) = 1/p are recovered directly. Chapter 2. The Spherically Symmetric Model System 48 In practice, a reasonable fit to G(p, d') was achieved using the following modification of equation (2i82), •G(M') C(p,d') ad,(p - l ) 2 + bd,{p - 1) + 1' where ((p,d') is a slowly varying correction of order unity given by, M A>\ - ~ 0-17 W>d>- M (C„.-1) + 0.83' and the coefficients, which are functions of the extension parameter d', are: ad> = p{& — log d') exp logd 1 - erf logd' + q wi th Cd' = l + t exp with with p = 0.072697 Q = 0.866726 r = 1.355135 s = 0.419602 = 0.166208 r = 1.083183 t = 0.13289 V - 2.14884 w = 0.83685 (2.83) (2.84) (2.85) (2.86) (2.87) This parametrization, obtained using the OPDATA analysis package [18] yields the value of G(p, d') to within ~ ±2% over the ranges p = [0,1] and d' = [0, oo]. This parametriza-tion of G should be more than adequate for our purposes since the assumption of an exponential atmosphere is itself only approximate in spherical geometry, 2.4.2 The Moment Equations of Transfer This system involves the two low order moments of the transfer equation, averaged over angle and closed by the use of the variable Eddington factor /„. Typically, boundary conditions are imposed assuming knowledge of the flux Hv at the boundaries, leading naturally to the use of the first moment equation d dXv {fvqvr2Jv) = r2Hv (2.88) Chapter 2. The Spherically Symmetric Model System 49 for this purpose. Moment Equations: Inner Boundary Condition The diffusion approximation is assumed to be valid at the inner boundary, giving 1 dBv TJ„ Z~oX = 647r<rr2r3Xp where equation (2.47) has been used to evaluate dBu/drv. The inner boundary condition is then ' i < ' " " ^ > = ( # ) • <2-9°) Moment Equations: Outer Boundary Condition The approach of MH was to supply an outer boundary condition for the moment transfer equation by the introduction of a second Eddington factor gv = Hu/Jv. The boundary flux is Hv = gvJv, and the resulting boundary condition becomes •j^-(fuqvraJv) = r'gvJv (2.91) which involves only the fundamental variables T, p, r and Jv. For the case of no incident radiation, MH assume gv to be invariant, thus allowing the boundary condition to be linearized. However, in the presence of nonzero incident radiation, /^"(/i) > 0, this assumption of the constancy of the ratio of H~u/Ju is false. In this case, both Ju and Hv will now depend on the inward directed intensity, which is a function of the boundary temperature, and gv will possess a temperature dependence and can not be assumed invariant. An appropriate generalization of the approach of MH is to define the outgoing radiation moments J+ = \ [ Iv{p)dp (2.92) Jo H+ = \ ( tilv(u-)dp, (2.93) Chapter 2. The Spherically Symmetric Model System 50 and their ratio gi = H*/J+, and to assume that gi may be held constant during subsequent linearization steps. The boundary condition, however, involves the full angle-averaged moments Jv and JET„, SO that «/+ and must be related to Jv and Hv. For completeness, we define the complementary incoming radiation moments, J- = \ J Iv(n)dfi (2.94) H~ = \ J fil„(n)dn, (2.95) and note that Ju = . J+ + J- , (2.96) Hv = H+ + H^. (2.97) Then, Hv Hj + H~ _ Hj(l + HZ I Hj) 9V~ Jv~ J+ + J," ~ J+(l + J-/J+) • ( 2 , 9 8 ) If we assume that I~ <C /J", J,7 <C «7j~, and JET~ <C H*', we obtain, to first order, ( 1 + § - | ) ^ ( 1 + f - f ) (2-"» since iZi, « if+ and Jv « J+ to this level of approximation. The quantities J~ and ITj" are just the moments of the inward directed radiation and, therefore, K = \ I[ Iu{ti)dti = \ I I-(n)dn = ^TltRM(vXuBv [ G(fi,d')dfi (2.100) J-i Jo Jo and H~ = lf nlv{u)dn = -\ f ftl-(ii)dp = -\TIIRMZVXUBV f /iG(fi,d')dfi, (2.101) J-i Jo Jo where, by definition, Iy(fi) = I^—fi). Chapter 2. The Spherically Symmetric Model System 51 Denning the moment functions, Gj(d) = \ f1 G(»., d) dp (2.102) Jo GH(d) = \ f pG(p,d)dp (2.103) Jo the incident radiation moments can now be written as J~ = +rhRM^X,BvGj(d') (2.104) H~ = -TltRM(vXuBvGH{d') (2.105) where d' = d/(l + A), as before. Also, as in the formal solution, the monochromatic absorption optical depth on the outer boundary Tab,,u = TRM^XU (2.106) has been replaced by 1 — exp(—T0&4II/). This approximation is less valid than in the formal solution case since the moment functions Gj and GH were evaluated under the assumption of optically thin outer layers. These functions will therefore be inaccurate for moderately optically thick boundary layers. This approximation, which we used for our model calculations, gives Ju.= +[1-eM-n,RM£vXv)]BvGj(d') (2.107) H~ = -[1 - exp(-ThRMivXv))BvGH(d') (2.108) In retrospect, a better estimate would probably be K = H[l-eM-2n,RMivXvGj(d'))}Bu (2.109) H~ = - exp(-ATliRMiuxvGH(d'))}Bu (2.110) which yields not only the correct optically thin results above but also the correct optically thick results J~ — Bv/2 and H~ = —Bv/4. This estimate will be tried in future models. Chapter 2. The Spherically Symmetric Model System 52 The outer boundary condition can now be written as -^riUqvT2J„) = r2gvJv = r2gt (jv + —H~ - J~) , (2.111) where gv has been expanded using equation (2.99). Since to first order ^H~ = H-, (2.112) 9v we obtain the result d (f,qyJu) = r2[gUu - {9tJ; ~ H-)) (2.113) dXv With this, our goal of finding a boundary condition involving only the fundamental variables Jv and r, and the invariant ratio g+, has been realized. This expression may now be expanded in terms of the moment functions Gj and Gjj as •^•{UqyJu) = r2{gtJv - [1 - exp(-TRM^Xu)][GH(d') + gtGj{d')}Bv} (2.114) and yields the desired outer boundary condition. Parametrization of the Gj and GH Functions The moment functions Gj(d') = \ f G(fi,d')dfi (2.115) Jo GH(d') = \ f pG(n,d')dfi (2.116) Jo can also be parametrized in a form suitable for repeated, rapid evaluation. Numerical quadrature of the previously obtained values of G(fi,d') over the angle grid {^} yielded values of Gj and GH for values of the extension d' G {d'k}- Again, the OPDATA software package was used to obtain analytic parametrizations to Gj and GH-The function Gj was parametrized by the form Gj(d>) = -j-Tj—r~TZV~h~ <2-117) 7d'(l — tid') + gd'hd' Chapter 2. The Spherically Symmetric Model System 53 where 9d> = ad lb 'IT 1 + tanh 1 +. tanh ^log d' + g and, a = 1.13238 b = 0.213648 q = 0.131120 r = 1.13353 s = 1.56082 t = 1.42257. The function GH was represented by the expression 1 GH(d') = 1 + f l 1 + tanh + q with coefficient values (2.118) (2.119) (2.120) (2.121) (2.122) (2.123) q = 0.020576 r = 0.900053 These parametrizations of Gj and GH are also accurate to ±1%. 2.4.3 Other Boundary Conditions The remaining equations in the model system consist of the equations of radiative equilib-rium, hydrostatic equilibrium, and the equation defining the radius depth scale. Of these, Chapter 2. The Spherically Symmetric Model System 54 the equation of radiative equilibrium, as formulated here, is not a differential equation and requires no specification of boundary conditions. The other equations are both first order differential equations and require the specification of a single boundary condition for the model to be well-posed. It is convenient and numerically preferable to specify these conditions on the outer boundary. Hydrostatic Equilibrium: Outer Boundary Condition The assumption of an isothermal, exponential atmosphere exterior to the outer boundary with an opacity dependence given by XRM(P) C< px allows the equation of hydrostatic equilibrium to be integrated analytically in this boundary region. Assuming the pressure scale height fe<ri gives P l - (1 + A)— -j-Xl.RM 1^ as the boundary pressure, where L*~/lXl,RM "/» 10.>. 4*cGM,(l + A)J ' = ^ ) • (2.125) Xl.RM V °P / 1 The (1 + A) term in the denominator of the radiation pressure term in equation (2.124) was accidentally omitted from the calculation of Pi in the current series of models. The effect on the models should be very small. Radius Equation: Outer Boundary Condition We have simply set the outer boundary radius ri = R = £R., (2.126) where R is a reference outer boundary radius, determined by the scaling the stellar radius Rt (a fundamental model parameter) by the factor £. The factor £, and consequently, the outer reference radius R, remain unchanged by the linearization step. Chapter 2. The Spherically Symmetric Model System 55 The scale factor £ in our models is determined initially such that r(ro) = R*, where the reference optical depth r 0 = 1. This value of £ could be updated during the formal solution step. This has not been done in this study, however, since this would further slow the overall convergence of the already computationally intensive models. The only consequence of this neglect is that while the converged models produced remain fully self-consistent, the parameter il» specifying the stellar radius will have drifted slightly from its originally prescribed value. The values of the originally specified Af* and Z* remained unchanged. In the domain of Teff, g and Miol, the values of Teff and g will drift slightly from their original specifications. In practice, these effects were found to be small with ATtff < 60K and A log 5 < 0.03. Chapter 3 The Discrete System and Solution Procedure 3.1 The Finite Difference System 3.1.1 The Formal Solution: Cubic Spline Difference Formulae The model equations must be discretized in order to obtain a numerical solution. The second order Feautrier form into which both the transfer equation along an impact ray (for the formal solution) and the moment form of the transfer equation have been cast lends itself naturally to a discrete representation by cubic splines. We shall first consider the formal solution using the (one dimensional) transfer equation along impact rays, following the approach of HKK. Since the formal solution proceeds independently for each frequency of interest, all frequency subscripts will be suppressed in this section for clarity. Consider a function u(t), which is desired to approximate the solution of a second order differential equation. It follows that u(t) must be at least twice differentiable; denote this second derivative by £ = "<i). (3.1) Following the approach of Alhberg [1], a simple candidate for a suitable approximating function is obtained by letting M(t) be the piecewise-linear interpolant of the grid {Afj}, or = XT - *) + M*+i (* " *<*)] f o r [** (3-2) where Atd = tj+x — td. Direct integration of Mj(t), constraining the result to interpolate 56 Chapter 3. The Discrete System and Solution Procedure 57 the grid values {ud}, then yields the cubic spline formula ud(t) = -^-[Md(td+i-t)3 + Md+1(t-tdY (3.3) + (6ud - At2dMd)(td+i - t) + (6« d + 1 - At2dMd+1)(t - td)\ for t e [td, td+1] It is seen that the continuity of u(t), and of M(t) = cPu/dt2, is ensured by the interpo-lation of the respective grid values. However, the first derivative U'd{t)={^) " ^d[ZMd+,(t-tdY-ZMd{td+,-t)2 (3.4) + (6ud+i - At2dMd+x) - (6ud - At2dMd)] for t € [td, td+1] is not automatically continuous across the nodes (grid points), and this requirement must be additionally imposed. By direct evalation for t — td, we have u'd(td) = Ud+1A~ Ud - \\Md+i + \Md)Atd (3.5) and «i_i(*a) = Ud Ud~1 + {\Md + iM<J_1)Atd_1 (3.6) L±td-\ which expresses the value of u'(td) as obtained by interpolation on the adjacent intervals [td, td+i] and [£d_i,i<j] respectively. Continuity of u'(t) then requires u'd(td) = u'd_1(td). (3.7) Rearrangement of this constraint yields the desired spline difference equation, Ud+l Ud ( 1 , 1 \ — 1 ^ \td + Atd-J + \Md+1 (-AtdAtd_1/2 Atd-1/2 \A d Atd-iJ Atd^Atd-i/2 where Atd-1/2 = \{Atd-i + A ^ ) - Then, if the second order differential equation is of the form d2u . , ^ = ^ , 4 (3-9) Chapter 3. The Discrete System and Solution Procedure 58 substitution of the {Md} values directly yields the cubic spline discrete representation of the differential equation (3.9). The Feautrier form of the transfer equation along a ray M(t) = d ^ = u-S(t) (3.10) conforms to this form, and we immediately obtain Ud+l Ud ( 1 , 1 \ «<f-l AtdAtd_1/2 Atd_1/2 \Atd Atd^J Atd^Atd_1/2 + ixr^-(«*Hi - Sd+1) + l(ud - Sd) + I - ^ z L ^ - 5< l_1), (3.11) Finally, the above derivation constrains u'(t) at all interior nodes but leaves this derivative unconstrained on the boundaries. Thus, if d = 1 and d = D refer to the inner and outer boundary nodes of the grid respectively (where D is the number of depths), the corresponding boundary values u'(t\) and u'(tr)) must also be specified for uniqueness. This is very convenient in our case of the transfer equation, since the usual boundary conditions involve the specification of du/dt = u'(t), and the discretized versions of the differential equation boundary conditions immediately supply the necessary spline boundary values. Then, from equations (2.35)-(2.36) we have 1 = ^AJT " (^M2 + *ML)A<1 = UL~ IK~ (3'12) «i>-i(*z>) = " JA"T UD~1 + {\MD + §Afi>_1)AiD_i = vD (3.13) as the corresponding discrete boundary conditions for impact ray k. The solution of this tridiagonal difference system requires knowledge of Md = ud-Sd, (3.14) with the source function S assumed to be given by _ KB' + crJ KB + crJ , „ » ' S = = = iB + (l-i)J 3.15 K + (T X Chapter 3. The Discrete System and Solution Procedure 59 where £ = K/X-The immediate difficulty is that J is not known a priori. However, since Jo dp (3.16) this can be written in the discrete case as Wkd^kd, (3.17) fe=i where the coefficients Wkd are integration weights (to be determined) and the summation is taken over the number of impact rays I that intersect the shell of radius rd. Then, for each frequency the discrete system of transfer equations (3.11) and boundary conditions (3.12)—(3.13), after substitution of the values of S and J given by equations (3.15) and (3.17), can be arranged into the following Rybicki block matrix form, / T i 0 0 T 2 0 0 u 2 K 2 \ 0 / (3.18) 0 0 ••• T 7 U 7 uj \ W i W 2 • • • W j E / \ J / where = (tffei)«fe2j • • • 5 Wfer»)r is the vector of length D representing the depth variation of the intensity Ukd along the impact ray k. Similarly, J = (J x , J 2 , . . . , JD)T describes the behaviour with depth of the mean intensity. The other quantities are coefficient terms derived from rearrangement of the spline equation. Here, the terms T*. and are tridiagonal matrices of dimension D x D, is a D x D diagonal matrix containing the quadrature weights Wkd, E = -I is a D x D diagonal matrix equal to the negative of the identity matrix, and K& are vectors of length D. The first I rows of this matrix system represent the transfer equations along each of the / impact rays while the last row describes the quadrature of the mean intensity J in terms of the individual intensities Uk along each ray. Chapter 3. The Discrete System and Solution Procedure 60 In summary, the following nonzero elements are obtained. The subscript k (labelling each impact ray) has been omitted for clarity from the notation for the optical depth intervals, properly written as Atkd. The frequency subscripts have also been suppressed. = ^ ( A + Afcr) + i.' de[2,Dk-i] (Tk)wi = - A ^ - A - + i (^)' <*€[2,Dfc-l]-(Tk)d,d-i - -At^Ut^x +1 (^Si) ' de[2,Dk-1] (Tk)n = 1 + ^ + (Tk)i2 = - ^ + JA*I (Tk)DkDk = (Tk)DkDh-l = (CM** = - | ( i - d e %Dh -1] (uk)d,d+1 '•= - - j f a x ^ U - • & « ) . <*e[2,Dfe-i] (^k,-i =. -I (7^) (1 - 6-i), i € [2,Dk - 1] (^fc)ll = -§A*i(l"6) (^)l2 = -|At!(l-6) {Uk)DkDk • = - |At B | k_i(l {Uk)DkDk-l = --f AiDj-^l - ^£»fc-l) (tffc)i = Atxd^Bx + ^ O + Jfc (#fc)£>fc = A i u ^ i d ^ ^ + l^-iBjn^-i) + vDk (Wk)dd = wkdi d e [r, Dfc] ^ = -1, . de [l,Dk] (3.19) In the above, Dk refers to the index of the innermost radial shell intersected by the fc-th impact ray. If the ray intersects the core, then Dk = D, the total number of radius points. It should be noted that, although the matrices in the Rybicki construction are of dimension D x D, all elements with an index d > Dk are zero. The solution now proceeds by forward elimination, using the fc-th row to eliminate the corresponding element wkd in the final row. After a total of I such steps, all the Chapter 3. The Discrete System and Solution Procedure 61 elements Wkd are gone, and the remaining single entry in the lower right hand corner is decoupled from the rest of the system. This leaves the matrix equation E'J = P', (3.20) where i E' = - I - ^ W k T ^ U k (3.21) fe=i I P ' = - . ^ W k T ^ K k (3.22) which can be solved to get the mean intensity J. Back substitution into row K of the Rybicki block matrix then yields u k = T ^ K k - T ^ U k J , (3.23) and so all the specific intensities Uk along each impact ray have been found. 3.1.2 Quadrature Weights for the Moment Integrals The remaining problem in the completion of the formal solution is the evaluation of the moment quadratures J , H and K. In general, these integrals are of the form Q n = [ nnF(n)dfi (3.24) Jo where Qo = J , Qi = H, and Q2 = K, and { u if n is e\ v if n is oc . _ — even *V)H . , . (3-25) )dd They can be represented by the corresponding discrete formula 1 1 Q" = S wwFkd = wknFk, (3.26) Chapter 3. The Discrete System and Solution Procedure 62 where I is the number of impact rays and the depth subscripts d have been suppressed for clarity. Initially, cubic splines were used to evaluate these integrals, and quadrature formulae were derived following the approach of HKK. However, splines proved not to be the best choice in this case for two reasons. First, the immediate advantage gained in interpolating a function which can be represented by a second order differential equation is not ap-plicable. Second, and more importantly, a disadvantage of cubic splines became evident in use. Cubic splines, it will be recalled, have first derivatives determined globally and not locally. In practice, this often results in unphysical oscillations of the interpolating function, or 'ringing', in regions of large second derivative, due to the poorly constrained derivatives. The radiation field in an extended, spherically symmetric atmosphere becomes strongly peaked in the radially outward (ft = 1) direction at small optical depths. This led to instances of severe ringing in the spline interpolation of the intensities u(fi) for certain test examples. Therefore, the decision was made to abandon splines in favour of a more well-behaved interpolating function for the evaluation of the moment quadratures. Another cubic polynomial approximation is provided by cubic Hermite interpolation [60].' For a point.a; £ [ajj,JCi+i] the cubic Hermite interpolating polynomial Hz is, for a given set of function values {Fi} and the corresponding values of the first derivatives {Si} on the grid {a^}, defined as H3(x) = Fi<t>i(x) + Fi+1<t>i+1(x) + Si1>i(x) + Si+1tl>i+1(x) (3.27) where the basis functions <f>,ip are 1 - 3£2 + 2£3 (3.28) </>i+i(x) 3£2 - 2£3 (3.29) ipi(x) (3.30) Chapter 3. The Discrete System and Solution Procedure 63 = &Xi(t* ~e) (3-31) and where A and £ = (x — Xi)/Axi. In particular, note that 4i(xi) =1, <f>i(xi+1) =0, tf&Zi) =0, <%{xi+1) = 0 <f>i+i(xt) = 0, <j>i+1(xi+1) = 1, <^-+1(xi) = 0, <t>'i+1(xi+1) = 0 i>i{xi) =0, V;(*i+i) -=0, =1, 'il>i(xi+1) =0 and therefore, H3(x) correctly interpolates the specified function values Fi,Fi+i and derivatives Si,S{+i on the interval [xi,a:i+i]. In the case of the moment quadrature problem, however, the first derivatives are not known a priori and must be estimated numerically. These derivatives can be conveniently estimated to second order by finite differences which, for an non-uniformly spaced grid {xi} of data points give Si = + qiFi + n F i + u (3.32) where AXJ • , Vi = — i TT A x (3.33) Ax,-_1(Aa!^1 + Axi) v ; Axi — Axj_i * 1 (3.35) ri — Ax^Axi-x + Axi) which reduces to the familiar 2-point centered difference formula for a constant grid spacing. The use of this method to supply first derivatives for interpolation using cubic Hermite polynomials is sometimes referred to as cubic Bessel interpolation. It provides a robust, (once) differentiate interpolation that, because of its strong local constraint on first derivatives, resists ringing and provides an interpolation that usually looks quite natural to the human eye. This interpolation, unlike splines, does not possess a continuous second Chapter 3. The Discrete System and Solution Procedure 64 derivative, but that lack is of no consequence for our problem. Consequently, cubic Bessel interpolation was chosen to evaluate the moment quadratures for this study. The boundary derivatives Si and Si present a particular problem since they can not be evaluated by the above three point difference formula. For the moment quadrature problem we have used symmetry considerations where possible to provide these deriva-tives; otherwise, we have simply adopted the 'free-end' approximation of zero second derivative. At the p = 0 boundary, symmetry considerations imply F(0) = 0 for n even (J,K) (3.36) F"(0) = 0 for n odd (H) (3.37) At the fi = 1 boundary, no such constraints apply, and the 'free-end' boundary condition was chosen arbitrarily as being the most natural. This implies F"(l) = 0 for all ra. (3.38) Then, in discrete form, we obtain directly S\• = 0 for n even (J,K) 5 * = f ( ^ T 1 ) - ^ for » odd (H), (3.39) Sr = I l^j£f) ~ ^ all n. With this, all of the first derivatives {Sk},k = 1,...,/ can now be directly estimated from the given {Fk}. The moment quadrature Qk, over the interval [pk,pk+i], can now be evaluated using the cubic Bessel interpolating polynomial to yield an expression of the form Qk = (akFk +bkFk+1)Apk + (ckSk+ dkSk+1)Api (3.40) The values of the coefficients ak, bk, ck and dk depend upon the particular moment evaluated, and are listed below. J(n = 0) : ak .= +| Chapter 3. The Discrete System and Solution Procedure H(n = 1) : K(n = 2) : The total quadrature can therefore be written, after h = + } • ak = +\l*k + ^A / ife h = +\pk + ^A/xfe cfe = -rjiPk + gTj^fe dk = -jtPk - ^Apk Ofc = + -fiVkApk + Ts^l bk = +|4 + ^fcA/Xfe + ^A/z* Cfc = +^ 4 + YiPkApk + ^ A / i | 4 = -n/ 4 * _ idVk^k - jo-Apl 1-1 fc=l e algebraic manipulation, in the form I Chapter 3. The Discrete System and Solution Procedure 66 (3.47) where the weights wk are given by a 2 A ^ 2 + 6'iA/ii for k = 1 a'3Apz + b'2Afi2 + c[Api for k = 2 ™fc = •{ a£.+1 A / i f e + 1 + fe^A/ife + 4_xAfik-i + d'fc_2 A / i f c _ 2 for 3 < A: < J - 2 6j_ 1 A/t /_ i + C J _ 2 A / Z J _ 2 + d'l-zApi-z for fc = I — 1 C j _ 1 A / f / _ i + ( d'i-2Afii-2 for k = I (3.46) and afe = CkPkAfik for 2 < < 7 — 2 = (cj_iP7_i + <Z/_itj)A///_i ?4 = afc + (cfcgfc + 4pfe+i)A/ifc f o r l < A ; < J - l cfc = fefe + (c*r* + dkqk+i)Afik f o r l < A j < J - l d'i = (ci<i + c?ir2)A/ti d'fc = dkrk+iAfik- for 2 < fc < J — 2 The derivatives are estimated by Sh=pkFk.x + qhFk + rhFk+1 f o r 2 < f c < 7 - l , (3.48) where the coefficients p^, 5*. and rk are given by the previously mentioned 3-point differ-ence formula, Pk = -n 7 7 ^ x—r (3-49) 9fe = 7 7 (3.50) ApkApk-i r . = A***-i f 3 5 n fe A / x ^ A / x ^ + A ^ ) ' ; for 2 < k < I — 1. The boundary derivatives Si (for H) and Sx (for J , H, and if) can be put into an analogous form, Si = qiFi + nF2 + tiF3 (3.52) Si = tjF^+PiFj-i + qjFj (3.53) Chapter 3. The Discrete System and Solution Procedure 67 with coefficients given by (3.54) * = + 5 . ( ^ T - * ) ( 3 - 5 5 ) h = - | r , . (3.56) and ti •=• (3.57) = - 2 t e + * - v (3-58) » = A (-7^— -ri-i). (3.59) Again, Fk = uk for the evaluation of J and i f , and Fk = vk = (du/dt)k for the evaluation of H. With the moments J , If and i f in hand, the Eddington factors / = K/J can be directly evaluated for all radius and frequency points. Finally, from the definitions of uk and vk, 1+ = uk + vk (3.60) and the equations for Jk and Hk , where J+ = \fl{p)dp (3.61) Jo H+ = \ f pl{p)dp (3.62) J° can also now be evaluated on the outer boundary, for all frequencies, by taking Fk = l£ with the n = 0 and n = 1 quadrature weights tu*. respectively. This yields the value of the second Eddington factor 4- H+ '' s = 7+- (3-63) on the outer boundary, and the formal solution is complete. Chapter 3. The Discrete System and Solution Procedure 68 3.1.3 Moment Equations: Cubic Spline Difference Formulae The moment form of the equation of transfer in spherical geometry d2 r2 -j^(fuqur2Jv) = —(Ju ~ Sv) dX* qv is also of the second order form dH dt2 M(t) (3.64) (3.65) that can be naturally represented by a cubic spline. Therefore, making the following substitutions «d = fdqdTdJd Atd = AX* Md = -±(Jd-Sd) Qd (3.66) (3.67) (3.68) into the spline difference equation (3.8), we immediately obtain the cubic spline finite difference representation of the moment transfer equation. Writing Sd = UBd + (1 - U)Jd (3.69) then, Jd~ Sd = £d{Jd — Bd) and Upon rearrangement, this yields fd+iqd+ird+1 1 r d + 1 £ d + 1 a A A d A A d o qd+1 /d-i?d-i^d_i 1 rd-ltd-l Md qd (Jd - Bd). A X j _ i 6 qd-i Jd+i — AXj_x fdqdfd Jd-l — (3.70) (3.71) + AXd AXd.J 3 qd + o ^<J-l/2 Jd Chapter 3. The Discrete System and Solution Procedure 69 1 r2d+1U+i 2 r ^ A Y n 1 rd-xJd-x — 7 ^^-d-tfd+i — -_—i\Ad-i/2-t>d — ZXAd-iXJd-i. o <) ? d o qd-i (3.72) Let XRM a n d THAT represent the Rosseland mean opacity and optical depth respectively. Then the generalized monochromatic optical depth variable dXv can be expressed in terms of the independent variable (the Rosseland mean optical depth) since dXv = qv drv = qv ( J drRM = qv\v drRM (3.73) \XRMJ where the notation Xv = XV/XRM has been introduced to represent the normalized mono-chomatic opacity. Then, in discrete form, AXd = \(qdXd + qd+iXd+i)Ard (3.74) where Ar j = Td+i — rd shall be taken to refer to the Rosseland mean optical depth. For the remainder of this section, we shall adopt the convention that x a n d r refer to the respective Rosseland mean quantities unless explicitly specified otherwise. The discrete transfer equation then becomes fd+iqd+i 1 &f+i (idXd + qd+iXd+i) Ard 24 qd+1 - fdqd (qdX.d + qd+ix.d+i)ATd d+l 1 1 + . + .(qdXd + qd+\Xd+\)ATd (qd-xXd-\ + qdX.d)ATd. 1 Cd (I'd \ 2 J^ — [{qdXd + qd+iXd+i)Ard + {qd-iXd-\ + qdX.d)ATd-i] ^—J Jd fd-iqd-i 1 £ d - i .(qd-iXd-i + qdXd)Ard-i 24 qd. 'Hqd-iXd-i + qdXd)ATd-i^ ( ^ n ) Jd-1 id+lf - - \ A D - 7 7 {qdXd + qd+iXd+i)ATd[—-I Bd+i - K^Xd + 9 d + i X « t + i ) ^ + ( G f - i X d - i + qdXd)ATd_x] [ — ) Bd 12, qd \RJ i id 2* qd -(qd-iXd-i + qdXd^r^J^) Bd_u (3.75) -i \ it y Chapter 3. The Discrete System and Solution Procedure 70 where R = r\ is the outer grid radius, and has been introduced as a reference radius for scaling purposes. In an exactly analogous manner, substitution into the spline boundary equations (3.12) - (3.13) for du/dr and M, and expanding the terms and vp appearing there, yields the cubic spline representation of the transfer equation boundary conditions. At the inner boundary, we obtain after some manipulation, / r > - i £ D - i 1 63-1, - . - x A 1 / f \ D - i \ 2 7 : ; ~ rr 777 [QD-IXD-I + ?£)XD)Ar r ,_ 1 I — — ) JD_X .{qD-iXD-i + qDXDj&TD-! 2 4 g D _ i J V R J fDlD , 1 £D, - , - \ * 1 (TD\2 T - 7 — : VA + 77;—{VD-iXD-i + gz>xz>)Arz,_1 ( — J JD = .{qD-iXD-i + 9r>Xu)ArD_i 12 qD J V i? / - ~^{qj>-iXD-i + gr>Xu)Ar£,_1 BD -{QD-IXD-I + qDXDjAro-x J BD-I 24 qD. 128ir<rR2TDXD \dT)D Similarly, at the outer boundary, we have 1 (3.76) (qiXi + 92X2)Ar x + 1 ^ | ( 9 1 ^ + ^ 2 ) A r i + \ g t \ G)2 Ji= .(9lXl + ?2X2)AT! - 2 i | ( 9 i ^ i + ^ 2 ) A r i © 2 j B 2 - ^ | ( ^ i + ? 2 ^ A t i ( i ) 2 j B i ' -^[i-e^(-T^lXl)][GH(d') + gtGj(d')} Q ) 2 ^ i , (3.77) where the extension parameter is ,1 d h , IT* = rT(TTA)' ( 3 7 8 ) the scale height h on the outer boundary (neglecting radiation pressure) is given by kTxr\ h ~ miGM.-L.xi/^cy (3-79) Chapter 3. The Discrete System and Solution Procedure 71 and pi refers to the mean mass per particle on the outer boundary. Finally, as before, <"•> 3.1.4 The Hydrostatic Equilibrium and Radius Equations Discrete Formulae for First Order Differential Equations Both the equation of hydrostatic equilibrium and the equation defining the radius depth scale are first order differential equations of the form and can be represented to first order by the trapezoidal rule, namely ( ^ ) d = l{fd + f d + l ) ( 3 - 8 2 ) or, in an alternate form ttrf+i = ud + ^ (/d + fd+1)Ard (3.83) where Ard = r^+i — rd. Thus, the function / (r) is represented by a straight line on the interval'[Td,Td +i]. A more accurate discretization of the above differential equation may be obtained by approximating / ( r ) by a cubic Hermite polynomial on [ r d , T d + i ] . This representation yields the discrete expression of third order, ud+i =ud + i(/d + fd+1)Ard + ^ -( , d r J d \dTJd+ii Arl (3.84) where the derivatives df/dr may be approximated by the 3-point difference formula fdf\ =Pdfd-i+qdfd + rdfd+i, (3.85) Chapter 3. The Discrete System and Solution Procedure 72 with the coefficients pd, qd and r<f defined by = ^ A ^ T <3-87> P«* = A / A ^ " 1 A (3-88) The Equation of Hydrostatic Equilibrium The equation of hydrostatic equilibrium is written in differential form as dp = ^dT-—^-dr 3.89 Xr* 'VKcr1 = ^ { G M „ dm - ^ dAj , (3.90) where, again, both x a n ( i r refer to the Rosseland mean quantities. Note that both terms are exact differentials, except for the term 7 ( T ) and the factor 1/r2, both of which vary slowly with depth. The right hand side is exact for compact stars, and remains nearly so even for reasonably extended stars. Therefore, the simple trapezoidal rule, equation (3.83), should be sufficiently accurate. Thus, we obtain the following discrete form 2 / r>\ 2' (£)+(£) K-^-). >«) where the reference radius R — r x has been introduced for scaling purposes, and the variables, GM, 9 = -JJT (3-92) dpn _ £. 7(r) dr ~ 47TC.R2' { 6 ^ 6 ) representing the gravity at the outer boundary and the radiation pressure term, respec-tively, have been defined. These variables have been introduced in this form since all three Chapter 3. The Discrete System and Solution Procedure 73 (g, R, and dpR/dr) will remain invariant during the linearization step. The remaining problem now is to express Am in terms of the independent variable A r . Since rd-i-(3.94) (3.95) dm 1 we can write, to third order, from equation(3.84) A m j _ 1 = i"(_L +1) Arj_, _ 1 f J _ (f) _ i (fs) 2 \xd-i XdJ 12 Lxd_! V^/d-i x3 V*v( The higher order terms have been included here since the dependence of %(T) can become very large, e.g. in the hydrogen ionization zone, and the right hand side, 1/x, was found to exhibit significant variation over the grid interval [r^-i,^] in some instances. The first order expression was originally used, following MH, but it proved to be inaccurate. Letting which will also be considered invariant during the linearization step, we then have the final form of Am, Am d _ 1 / 1 + J_\ 2 \Xd-i Xd) Yd-i Yd (3.97) 12 \Xd-i Xd, Substitution of this result in equation (3.91) yields the final discrete form of the equation of hydrostatic equilibrium. The Hydrostatic Equation: Outer Boundary Condition The discrete form of the hydrostatic equilibrium outer boundary condition is immediately obtained from equation (2.124), Pi = ( l + A) T I GM, X\ r\ L*KXi or, equivalently, ri ^ \ \ g T l (R Pi = (1 + A ) — — Xi V i where g = GMt/R2 and R = r\ as before. 2 r 47rcGM»(l + A) 4TTCGM,(1 +A) (3.98) (3.99) Chapter 3. The Discrete System and Solution Procedure 74 The Radius Equation The other first order differential equation in the model system is the radius equation, dr dr -PX, (3.100) which defines the radius grid, where both % and r refer to their respective Rosseland mean quantities. To third order, from equation (3.84), the discrete representation is Ar d_i = Upa-iXd-i + pdXd){rd-i - rd) 12 dr Jd-i \ dr ) t Let Hd = — ( 7T~ ) = reciprocal density scale height Pd \OrJd Yd = Xd \dTj/ Xd / d These will be assumed to remain invariant during the linearization step. Then, d {rd_x - rdf. (3.101) (3.102) (3.103) dr (PX) = pdXdHd - pdx\Yd Furthermore, for an ideal gas, J d p — fin — P-P kT (3.104) (3.105) where p is the mean mass per particle. With the above substitutions for d(px)/dr and p, the radius equation in discrete form becomes 1 /Pd-iPd-iXd-i , PdPdXd\ + (rd-i - rd) 2 V Ti.x 1 fPd-iPd-iXd-iH^x fidpdXdHd /^LiPLiXLi^- i , HdP2dXdYd 12 Td.x Td + ' "\"^v" | (rd_i - rdf kArd-x. (3.106) The values of Hd and Yd are obtained from second order differences using the 3-point formula of equation (3.85) for interior grid points d = 2,..., D — 1. At the inner Chapter 3. The Discrete System and Solution Procedure 75 boundary, the values of (dp/dr)r> and (dx/dr)r) are estimated by extrapolation of the available derivatives at D — 1 and D — 1/2, assuming constant second derivatives. This procedure yields the boundary derivative and (dx/dr)i) is represented by an analogous formula. Then, Hp and YD follow imme-diately from their definitions. The corresponding values at the outer boundary are obtained by application of the isothermal, exponential atmosphere assumed to exist exterior to the outermost radius. The Rosseland mean optical depth at the outer boundary was previously shown to be Mxi ' 1 _ 1 + A ' where =-(¥) • Xi \OpJi (3.108) (3.109) and Hx (dp/dr)i is the density scale height on the outer boundary. Then, is immediately obtained. Also, by writing y = i ^ = i / W n Xdr x \Or/drJ and substituting the expressions for (dx/dr)i and (dr/dr)i derived assuming the isother-mal, exponential boundary atmosphere approximation, Chapter 3. The Discrete System and Solution Procedure 76 is obtained. The integration of dt/dz = — px along impact rays for the formal solution is performed using the same discrete equation (3.101), except that the radial coordinate r is replaced by the distance z along the impact ray. The inner boundary value of [d(px)/dz]i) is obtained in an analogous manner to the radial case, and the outer boundary value is similarly evaluated using the isothermal, exponential boundary approximation to yield where, for this one case only, Xi refers to the monochromatic opacity and p\ to the impact ray angle coordinate. The Radius Equation: Outer Boundary Condition Finally, the radius equation boundary condition is supplied at the outer boundary by where £ is a scaling factor defined such that T(TQ) m We have chosen the reference optical depth To = 1 for this models. 3.1.5 The Equation of Radiative Equilibrium The remaining equations to be cast into discrete form are the various quadratures over frequency, needed both for the equation of radiative equilibrium and the transfer equa-tion inner boundary condition as well as for various auxiliary quantities such as the Rosseland mean opacity. These quadratures were evaluated using the trapezoidal rule, which approximates the integrand by a piecewise-linear function. The opacities of stellar material, particularly at low temperatures, are quite discontinuous in frequency as will be the stellar radiation field present in such a medium. Higher order integration for-mulae should not be used in this case since they require continuity and differentiability (3.114) r i = R = CR, (3.115) Chapter 3. The Discrete System and Solution Procedure 77 properties that the integrand may not possess. The trapezoidal rule yields the following frequency quadrature weights, | ( a : i — x2) for i — 1 Wi = < \{xi^ - xi+1) for 1< i < N (3.116) \(XN-I — XN) for i — N where the integration is performed over the frequency grid {xi},i = 1,..., N and N is the number of frequency points. The equation of radiative equilibrium in discrete form is thus N i=l N = 1, (3.117) where the depth subscript d has been suppressed for clarity. The Rosseland mean opacity XRM is defined as J~(dBJdT)dv ^ 3 XRM — »oo — Too 5 (3.118) / xZ\dBv/dT)dv Tr / X:\dBu/dT)du Jo Jo and this is evaluated in discrete form by N i=l \ / t XRM = ^- — — • (3-119) Wi / dB\ Note that the numerator in the discrete form is evaluated by numerical quadrature and not directly using the exact analytic expression. This is because the analytic result represents the quadrature over the full semi-infinite range from zero to infinity while the analogous numerical result corresponds, by necessity, to a finite range of integration extending from some lower cutoff frequency umin to an upper limit umax. Using the ratio Chapter 3. The Discrete System and Solution Procedure 78 of expressions with differing limits of integration would introduce systematic errors in the Rosseland mean opacity calculated. It is more accurate to evaluate both the numerator and denominator over the same finite range of integration. The truncation errors then largely cancel out in the calculation of the ratio. A similar argument holds for the calculation of the flux-weighted mean opacity XH-H~L x"H"du (3-12°) which is evaluated using the discrete expression ' XH = ^~ (3.121) where the total flux H has been calculated using the discrete sum H = ]Ci=i wiHi a n ( l not the analytic expression H — L*/(167r2r2). 3.2 The Linearization of the Radius We seek to solve the discrete model system, consisting of the moment form of the radiative transfer equations, the equations of hydrostatic and radiative equilibrium, and the radius equation, for T, p, r and Jv as functions of rRM. This system of equations is implicitly nonlinear since the constitutive properties of the stellar material, the density p, the mean mass per particle p, the opacities «„ and Xv> a n ( l thus the ratio £ = «/%, depend (often strongly) upon the local temperature and pressure of the gas. However, the temperature T is constrained to depend upon the mean intensity Jv by the constraint of radiative equilibrium while Jv is in turn determined by p and x through the equation of transfer. To solve such a coupled nonlinear problem, we resort to the complete linearization method (CLM) of Auer and Mihalas [7]. Al l quantities are expressed in terms of the fundamental physical variables describing the state of the atmosphere in our model, T, Chapter 3. The Discrete System and Solution Procedure 79 p, r, and J„, and the resulting system of equations is linearized in terms of the differential changes ST, Sp, Sr and SJV in these variables. For the variables that can not be directly expressed in this manner, the linearization proceeds by expansion of the differential changes in these variables in terms of first-order differences in the fundamental variables. Hence, we can express * = (l)*r+(l)*? (3'i23> and so on. All that matters is that the differential form of the equations be entirely expressed in terms of the differential changes ST, Sp, Sr and SJV. Since all of the discrete model equations involve relations between (at most) the immediately adjacent depth points, the linearized equations can be represented by (at most) tridiagonal matrices of dimension D x D, where D is the number of depth points. The entire linearized system can then be organized into a block matrix form, such as that proposed by Rybicki [83], and solved to yield the first order corrections ST, Sp, Sr and SJU. This will be described in more detail below. However, a serious difficulty is encountered in attempting to apply the C L M to a spherically symmetric geometry. This adaptation requires the linearization of the radius variable r, which does not eiiter into the plane-parallel formulation. This is readily accomplished for the explicit occurrences of the variable r in the model equations, but the presence of the sphericality function qv(r) presents more difficulty. As will be recalled, the function qv(r) allows the collapse of the original, cumbersome first moment equation (2.4).to the more convenient form of equation(2.8), analogous to that of the plane-parallel case. This function is an integral transform over the grid points interior to the present radius r and is defined by '3/„ - 1\ dr1' q v { r ) = (ZJrp) exp / fu (3.124) Chapter 3. The Discrete System and Solution Procedure 80 where rmin is an arbitrary reference radius, here taken to be rmin = rr>, the inner-most grid radius, and fv is the Eddington factor. Unlike the rest of the model system, <lv(rd) depends not just on the adjacent depth points Td-\ and r^+i but on all the points {r<i'}, d' < d interior to the one in question. It would appear that linearization of the transfer equation, which involves qv(r), would therefore destroy the tridiagonality of the complete linearization method. This would render the approach computationally infeasi-ble, due to the large number of full (as opposed to tridiagonal in the plane-parallel case) matrix solutions required for the solution of the entire system. This appears to be the reason behind the choice of a radius scale by MH in their pioneering work on the application of linearization methods to the solution of models with extended atmospheres. The choice of radius as the independent variable obviates the need for linearization of (^(r) since the radius scale is now held fixed. The price to be paid in choosing a radius scale is the destabilization of the equation of hydrostatic equilibrium, as described in section 2.2. In our experience, it proved to be impossible to converge any model resembling that of an actual cool giant star using the MH method with the radius as the independent variable. However, we were able to show that a direct linearization of qv{r) was possible. The discovery of a direct method of linearizing qu(r^) is of central importance in this study. Without this method, it would have been impossible to put models on an optical depth scale and it is unlikely that the convergence of such models would have been achieved. The key to the linearization of q(r) lies in the realization that terms involving q in the discrete form of both the transfer equations and their boundary conditions always occur in the form of a rational expression, with both the numerator and denominator a homogenous linear polynomial in q. Then, defining the ratios, ad = *L = 3~d\ (3.125) g.d-i fa = — = «d+i (3-126) Chapter 3. The Discrete System and Solution Procedure 81 it is clear upon inspection that the transfer equation and boundary conditions can be rewritten entirely in terms of the ratios a and B. When this is carried out, equation (3.75) can be rearranged to obtain the following form of the discrete transfer equation, f a - v A TTjtd+iiPdXd + Xd+i)Ard (PdXd + Xd+i)&TD 24 ffl' + 1 (rj\ jd 1 ( rd \ 2 [7J&[(X<* + a<i+iX<i+i)Ard + (Bd_lXd-i + X<f)Ard_1] ^—J J d 12 + d-i 1 ' ' 1 /^d-lN 2 - - ^7&-i(Xd-i + adXd)^^! (^-g-J Jd-i (Xd-l + OLdXd)A.TD. - 77jtd+i(PdXd + Xd+i)Ard ( — - B, 24: \ R ) d+l 1 / 7 " d \ 2 fiUKXd + ad+1xd+i)ATd + (Bd_lXd-i + Xd)Ar<i_i] ^ —J £ d 1 fTd—l\2 -^£d- i (Xd- i+«<iXd)AT D _ 1 (^ -^-J #<f-i, (3.127) with the inner boundary condition now written in the form fo-i 1 (XD-I + O:DXD)ATD^X 24 £ D - I ( X D - I + <XDXD)ATD-I fu + -T;:(D(8D-\XD-I + XD)ATD_X (PD-IXD-I + X D ) A T £ , _ I 12 1 / ^ D \ 2 - — M f o - i X f l - i +XD)ATD_1 ^—J 5ZJ 1 /I'D—1\2 - ^6>-i(xz?-i + a c X i > ) A r £ - i (^^"J (^)z>' 1287r^il2r3Xi3 and the outer boundary condition as (3.128) 1 LCAxi + ^ A n 24 TTjWiXi + X2)Ari Chapter 3. The Discrete System and Solution Procedure 82 A . ( X i + « 2 X 2 ) A r i - ^6 (Axi + X 2 )Ar 1 g ) 2 £ 2 - ^ i ( X i + «2X2)Ar 1 g ) 2 Bi -^ l - exp( - r 1 £ 1 xO ] [GH (^0 + 5 i + ^ ' ) ] ( ^ ) 2 5 1 . (3.129) Note that q(r) no longer appears exphcitly in any of these transfer equations. Direct evaluation of the ratio Bd yields f*n / 3 / - l \ dr'~ <ld _ (rd+i\' qd+i \ rd J f (3.130) The above integrand varies only slowly with r, and so a discrete representation using the trapezoidal rule was judged to be sufficiently accurate. This procedure yields the discrete analogue 2 rd+i rd and similarly ad )^-H[(^ )+(^ )](—)}• Both of these expressions are bidiagonal, and thus only three radius points are involved in the transfer equation. Therefore, the model system retains its tridiagonal nature. We thus have the explicit functional dependences <xd = otd(rd-urd) (3d = 0d(rd,rd+i), (3.133) (3.134) and thus can linearize the ratios a and B as follows 8ad = ® a d 5rd_! + f T ~ I 8rd drd-x drd m = [ ^ i ) s r d + ( ^ ) S r J + 1 . drB dr, d+i (3.135) (3.136) Chapter 3. The Discrete System and Solution Procedure 83 The values of these partial derivatives are listed below, dctd drd-i dad drd drd d/3d drd+i +<*d Old 2 rd ( rd-i 2 V 3/<i = +3d Td 2 rd 2 rd fdrl 1 + 3/d_! 1 Td-l (Zfd-2 V fdrj /d-i^Li 1 Zfa-x-l fd-irl-x rd+i (3/<i+i — 1 _^ 3/d — 1 fd+ir2 d+l Jd+l fdrl r_d (Zfd+i - 1 1 2 V fd^r2+1 + fdr\ This completes the linearization of the sphericality function q(r). (3.137) (3.138) (3.139) (3.140) 3.3 The Linearized Model Equations The transfer equation (3.75) and boundary conditions (3.76)-(3.77), the equation of hydrostatic equilibrium (3.91) and boundary condition (3.99), the equation of radiative equilibrium (3.117), and the radius equation (3.106) and boundary condition (3.115) form a closed system of equations in the fundamental variables «7i, . . . , J/y, T, p, and r (where N is the number of frequencies). We chose to linearize jn = (r/R)2Jn instead of Jn since ] n is nearly constant for rv <C 1, which reduces the coupling between variables. We now proceed with the solution using the complete linearization method (CLM) as follows. First, define the 'block' vector x = ( j i , . . . , j N , T , p , r ) T , (3.141) whose components j i , . of depths), and ,JN,T,p,r are themselves vectors of dimension D (the number Jn — (jnl, • • • , jnD ) T = (Ti, . . .,TD)T P = ( P I , V , P D ) ~ r = (ri,...,rB)T. (3.142) (3.143) (3.144) (3.145) Chapter 3. The Discrete System and Solution Procedure 84 Here the bold font is used to denote quantities which are vectors of dimension D, or matrices of dimension D x D. The system of model equations can then be compactly written as f(x) = 0 (3.146) where each component of the vector function f, f = ( f T E i l , . . . ^ X E . N j f R E j f H E j f D E ) 2 ^ (3.147) represents a constituent equation of the system. In general, the exact solution x is not known. Instead, we have a trial estimate x(') obtained after i iterations, hopefully with f(x^) « 0. (3.148) An improved estimate x( i + 1) can be obtained assuming — x <C x and expanding f ( x ( i+i)) i n a Taylor series about x ^ . Let A x W = x ( * + i ) _ X W (3.149) Then to first order • / fit \ (*) f(x( i + 1>) = f (x« + A x « ) = f (x«) A x « (3.150) which must vanish if x( t + 1) = x, the exact solution. Therefore, the desired first order correction Ax'* ' is found by solving / f)f\ f ( x « ) + f ^ j AxW = 0. (3.151) Denoting the Jacobian matrix (df/dx) of linearized coefficients by R, and f(xW) by —b, the linearized model system after i iterations is R(0 A x w = b ( i ) . (3.152) Chapter 3. The Discrete System and Solution Procedure 85 The solution of this linear system yields the desired correction vector AxW. Iteration of this method will produce estimates that converge to the exact solution of the model equations, provided the initial estimate x^ 1) is chosen sufficiently close to the exact solu-tion. The linearized coefficients Rk„ are given by the partial derivative of the k-th equation (row) with respect to the component x n of the vector x, R-kn = (^) (3.153) evaluated at the current value of x = x^ '^ . Each of these components are vectors of length D, and thus the coefficients Rkn are matrices of dimension D x D, where D is the number of grid depths.1 The block matrix R is not a full matrix. The transfer equation (3.75) for the n-th frequency, written more concisely as . frE,„(x) = 0 (3.154) depends only upon j n , and T , p, and r. Therefore, fTE,n = fTE,n(jn,T,p,r) (3.155) and the only nonvanishing contributions to the differential 6TTE,II are «~ =:- ( « • + ( " r ) s t + ( " r ) $ p + * • =; T n 6j n + VnST + Vn6p + SnSr. (3.156) Furthermore, since ^frE.n (at depth d) depends only upon variables at depths d — 1, <f, and d+ 1, the coefficients T n - R, nn U n = R n , N + 1 = ( ^ ) (3.157) Chapter 3. The Discrete System and Solution Procedure 86 ( C ^ T E n sn - K „ , N + 3 - y—fc-are themselves tridiagonal matrices of order D. The equation of radiative equilibrium (3.117) has the functional dependence f R E = f R E ( j i , . . . , J N , T , p , r ) (3.158) and, therefore, * - - S ( " ? ) « - + ( " ? ) " + ( ^ ) * + ( " f ) f c N = ~T w n £ j n + C6T + T>8p + ESr. (3.159) n = l In this case, the coefficients (If) w„ = R N + l , n = C = R N + I , N + I D = R N + 1 , N + 2 E = R - N + l . N + 3 C^RE dT (3.160) are all diagonal matrices of order D. The equation of hydrostatic equilibrium has the dependence fHE = fHE(T,p,r) (3.161) and thus ( w ) " + ( * ) * + ( ~ ? ) f c = AST + BSp + FSr (3.162) Chapter 3. The Discrete System and Solution Procedure 87 where the coefficients A = R N + 2 . N + 1 — dT B = R N + 2 L N + 2 = ) F = R N + 2 , N + 3 = (3.163) 0f„ dr are bidiagonal matrices of order D. Similarly, for the radius, or depth, equation we have fDE(T,p,r) and 6fnF. - 6T + = Q£T + HSp + GSr with coefficients Q — R N + S , N + I H = R N + 3 , N + 2 G = R N + 3 , N + 3 \dT J dp m that are bidiagonal matrices of order D. Therefore, the coefficient matrix R has the block matrix form R = 0 ... 0 ' Vi Si \ 0 T 2 • •• 0 u 2 v 2 s2 0 0 • T N u N v N S N Wx W 2 • •• W N c D E 0 0 •• 0 A B F 0 0 . . 0 Q H G / (3.164) (3.165) (3.166) (3.167) Chapter 3. The Discrete System and Solution Procedure 88 The differentials of the constitutive variables are expanded in terms of the fundamen-tal variables; thus « « = ( 3 - 1 6 9 ) « . - ( f ) ^ + ( f ) * P (3.170) where /x refers to the mean particle mass of the stellar gas. The gas density p is not expanded in this manner but rather expressed in terms of p, and the other fundamental variables using the relation p = pp/kT. Since p varies through the atmosphere by only a factor of three while p ranges over several orders of magnitude, this further reduces the coupling between the equations and improves the condition of the numerical problem. The sphericality function qv is expressed in terms of the ratios a and 8, as described previously, and the differentials of these quantities are expanded as *<= (£r)*"+(£K <3-171> ...«*= (©M r^)*- <3-172> The right-hand side vector b in equation (3.152) has the components b = ( K 1 , . . . , K N , M , L , P ) r (3.173) where K n = • - f T B,n(x«) (3.174) M = - f M ( x « ) (3.175) L = -fHE(xW) (3.176) P. = -fDE(x ( i )). (3.177) Chapter 3. The Discrete System and Solution Procedure 89 The linearized system (3.152) can thus be expanded into the block matrix form I T x 0 0 Ui V x Sx \ / Ajx \ / K x \ 0 U 2 V 2 S2 0 T 2 0 0 w, w 2 0 0 0 0 j A j 2 A j N = A T M A p L A r j K 3 T N U N V N S N w . j N K N (3.178) W N C D E A B F yo o . . . 0 Q H G • J \ which is of the Rybicki type. The first N rows represent the radiative transfer equations at the N frequencies considered, and the last three rows represent the equation of radiative equilibrium, the equation of hydrostatic equilibrium, and the depth equation defining the radius scale, respectively. The actual solution procedure used normalized variables so that differences were rel-ative rather than absolute. Define the scaled vectors K~ (Sjm SjnD\ ojn - I - — J \ JnX JnD J (8Ji (Srx Sp Si SpiA '' PD J •M rD ) and the corresponding scaled coefficients R-kn = ~Lk„X n = 5xE (3.179) (3.180) (3.181) (3.182) (3.183) Then, the form of the block matrix equation (3.178) remains unchanged under this scaling transformation. For convenience of notation we will hereafter drop all tildes on the variables and coefficients, but it should be assumed that the above scaling has been applied. The complete definitions of the coefficients of the linearized system appearing in the matrix equation (3.178) are provided for reference in Appendix A. Chapter 3. The Discrete System and Solution Procedure 90 This matrix system is solved using the ra-th row, corresponding to the transfer equa-tion for frequency n, to eliminate the W n term in the row representing radiative equi-librium. After repeated application, all JV quantities W n ,n = 1,. ..,JV, are eHminated from the radiative equilibrium row. The lower right hand 3x3 block is now decoupled from the rest of the system and can be solved directly. This reduced block matrix system has the form D' E' \ / A T \ / M' A B F Ap = L (3.184) u H G / V A r / ) where JV c = C - ^ W n T " 1 ^ n=l (3.185) D' JV = D - ^ W . T ^ V o n=l (3.186) E' JV = E - ^ W ^ S , , n=l (3.187) M' JV = M - ^ W J " 1 ^ . n=l (3.188) Matrix equation (3.184), of order 3D x 3D, can now be solved to obtain the corrections A T , Ap and A r for each depth. Finally, the values of Ajn can be obtained by back substitution into row n of the full matrix equation (3.178) and solution of the resulting tridiagonal system. The updated values of T, p, r and Jv are now used to evaluate the necessary secondary quantities, such as p, fi, and as well as the revised sphericality function qv. With this, the block matrix coefficients in equation (3.178) can be re-evaluated and the updated linear system solved for new corrections A T , Ap, Ar and A«/„. This procedure is iterated until convergence is attained, at which point a self-consistent solution T, p, r and J„ of the nonlinear model system has been found for the assumed value of the Eddington Chapter 3. The Discrete System and Solution Procedure 91 factor /„. Then the impact parameter method of HKK is used again to yield the formal moments J„, Hv and Kv calculated by quadrature over the angle coordinate fi. Finally, a revised Eddington factor fv — Kvj Jv is calculated, and another solution of the finite model system consistent with this value of fv is again found by complete linearization. After several such iterations, the whole procedure should converge to yield values of T, p, r and Jv fully consistent with the Eddington factor fv. This completes the solution of the spherically symmetric model atmosphere. 3.4 The Initial Solution 3.4.1 Introduction The iterative procedure outlined in the previous section requires an initial solution before it can be applied. This is not trivial, since such an estimate must not be too far from the true solution if the iteration is to converge. To start the linearization, initial estimates of T(r), p(r) and r(r) are needed. Then, a formal solution of the equation of transfer will yield values of Jv and /„ for this initial atmospheric structure. The linearization step can then be started. After some experimentation, the following prescription was chosen to obtain an initial temperature profile T(r). First, the optical depth scale {rd} is defined. Next, a trial outer radius r\ is denned, usually by scaling the nominal stellar radius i?» (which is a specified model parameter). Then, assuming a uniform disk model, the angle subtended by the stellar disk in the atmosphere is given by 6m = cos - 1 ft*, where solution of the intensity -T„(/i), now on the updated radius grid {rd}, d = 1,..., D, and the r > R* r < Rt. (3.189) Chapter 3. The Discrete System and Solution Procedure 92 3.4.2 The boundary temperature We assume a boundary temperature of the form Ti •= iT?„(l-P*)C(p.) (3.190) i where the factor 1 — //* represents the dilution of the radiation field due to extension. Two limiting cases are: 1. the compact (plane-parallel) limit. Here /z* « 0, and the classical plane-parallel atmosphere has a boundary tempera-ture of ^ = 1^ 9 ( 0 ) (3.191) where q(r) is the Hopf function. Therefore, we can identify C(0) = q(0) = 1/V~ (3.192) 2. the point-source limit. Now p* 1, and the temperature can be well represented by T? = \T*„(1-p.) (3.193) and so C(l) = f (3.194) A reasonable estimate of a general outer boundary temperature might then be ob-tained by linear interpolation between these two limiting cases. This yields C(p.) = lpt + (l-p.)q(0), (3.195) and a boundary temperature of = 121,(1 - /».)[§**. + «(0)(l" - p.)}. (3.196) Chapter 3. The Discrete System and Solution Procedure 93 At depth, we must recover the grey atmosphere result T{TY=\T*„[T + Q{T)}, (3.197) provided the optical depth scale used is the Rosseland mean. An expression that yields the above temperatures for both limits of r <C 1 and r 1 is then given by T(r)4 = f l j , ( l - p.)[r + lp. + q(r)(l - p.)], (3.198) which has been adopted as the initial temperature profile in this study. 3.4.3 The boundary pressure The boundary pressure is obtained by integration inward from infinity of an exponential, isothermal atmosphere. This yields T.GM. Pi = (1 + A ) - — - j - 1 -4TTCGM*(1 + A) (3.199) xi n where and both x a n d T refer to the Rosseland mean quantities. Since the opacity, Xi = x(~i,pi), (3.201) will generally also be a function of pressure, it is usually necessary to iterate the boundary pressure p\ until a value is found consistent with both constraints. A Newton-Raphson iteration loop is used in the current implementation. Convergence is rapid, normally occurring within three iterations. The initial pressure profile p(r) may then be found by integration of the equation of hydrostatic equilibrium inward through the atmosphere. Chapter 3. The Discrete System and Solution Procedure 94 3.4.4 The boundary radius The remaining integration to be done is the radius equation, % = -PK (3-202) which defines the radius scale {rj}, d — 1,..., D, where D is the number of depth points. We start with the outer boundary radius of r = r1=$R< (3.203) where £ is a scale factor depending upon the extension of the atmosphere. Initially, we arbitrarily set £ = 1.01. The stellar radius i?« is not well defined for stars with extended atmospheres. There-fore, some (arbitrary) criterion is required to link the stellar radius (a fundamental model parameter) to the optical depth scale. Most workers have invoked a relation of the type, r(r 0) = R„ (3.204) where ro is a reference optical depth, to relate these two scales. This is the approach taken by MH, who use r 0 = 2/3. It is the approach that we follow also, although we have adopted ro = 1 instead. The coupled integration of r(r) and p(r) now proceeds inward until r(r 0) is reached. This is compared to the specified radius 2?*, and another iteration of the entire starting sequence is performed if the disagreement exceeds the specified tolerance. If another iteration is taken, the scaling ratio | is updated using a Newton-Raphson technique, a better estimate of r i , T\ and pi is made, and the integration performed again. Eventually, convergence is attained, and then the integration continues inward through the entire atmosphere. This yields the desired initial estimates of T(r), p(r) and r(r). A formal solution is now calculated to obtain Jv(r) and / I / ( T ) , and the linearization step can begin. Chapter 4 The Model Computations 4.1 Introduction The previous chapters developed the spherically symmetric model atmosphere equations and the corresponding discrete system. Before these equations can be solved, the equation of state for a given atmospheric composition P = P(T,P) (4.1) and the opacity of the stellar material must be available. For our purpose, it suffices to determine the total opacity per gram of stellar material Xu = Xv(T,p) (4.2) and the ratio £„ = KV/XV of absorption to total opacity, ZV = UT,P) (4.3) The following, sections summarize the treatment of the equation of state and of the stellar opacity. Model implementation details are then described and the results of cal-culations are presented. 4.2 The Equation of State The equation of state of the stellar gas has been determined in this study using the method of Bennett [17] to solve the chemical equilibrium of the gas. Grain condensation was not considered. 95 Chapter 4. The Model Computations 96 In general, the determination of the equation of state of a cool, molecular gas re-quires the solution of a nonlinear system containing an equation for each element that contributes significantly to either the total gas pressure, the electron pressure, or the opacity. The accurate determination of the equilibrium partial pressure of a species es-sentially requires the determination of all species of greater or comparable abundance. Since some important sources of opacity are of only trace abundance, an accurate solution of the equation of state for cool atmospheres requires the inclusion of a large number of elements. Our treatment of the problem considers 25 elements and 105 chemical species (neutral atoms, ions, and molecules) formed from these elements. A complete list of these species is given in Table 4.1. A direct determination of the equation of state thus requires the iterative solution of a matrix of order 25, which would be quite costly for use in a model atmosphere code. The method of Bennett [17] introduces considerable economy into the solution through the division of the elements into the following three groups: 1. Main elements. These comprise the significant sources of pressure which are also involved in molecule formation. We have included the elements H, C, N , 0, S, and Si in this group. The most serious omission from this group is probably Ti, which depletes 0 by up to 2% through the formation of TiO. 2. Metal elements. These elements are not involved signficantly in the formation of molecules, but do contribute to the total pressure or the electron pressure. The metals are assumed to be present only in the neutral or singly ionized state and not associated into molecules. We included He, Ne, Na, Mg, A l , K, Ca, Fe, and Ni in this group. Doubly ionized C a + + strongly violates this assumption at the very bottom of the atmosphere, but this neglect is of no consequence since the resulting effect on the total electron pressure is negligible. Probably the most significant violation is due to the depletion of Mg (by up to 2%) to form MgH at Chapter 4. The Model Computations 97 low temperatures. 3. Minor elements. These include all the remaining elements which do not con-tribute significantly to either the total pressure or the electron pressure but may be responsible for significant sources of opacity. In this group we included the (somewhat arbitrary) list of elements CI, Sc, Ti , V, Cr, Mn, Co, Sr, Y , and Zr. Two reductions of the solution are now possible. The third group of minor elements are of such low abundance that they exert negligible influence on the abundances of elements in the first two groups, and on molecules formed among elements of these two groups. Therefore, the solution of the equation of state for the first two groups (the main elements and the metals) decouples from the solution of the third group (minor) elements, and thus may be solved independently of this third group. A further reduction of the problem can be achieved by lumping together all of the group (ii) metal elements into a single fictitious element, which we shall denote by the symbol Z. We will show that the equation of state problem can be solved using a reduced matrix system involving only the main group (i) elements, along with the fictitious metal Z, and the electron pressure. We shall first briefly describe the solution procedure for the full matrix of group (i) and (ii) elements, and then shall proceed to demonstrate the reduced solution method mentioned above. We use the following nomenclature, specific to this section. Let the index k refer to one of the elemental species included in the linearization and the index m refer to an arbitrary metal element in the neutral atomic state. Also, let K denote the number of elements included in the full linearization (the full matrix of group (i) and (ii) elements), K' denote the number of elements included in the reduced linearization (the matrix of the group (i) elements and the fictitious metal Z), M denote the number of metal elements, JV denote the total number of chemical species formed entirely from elements of the full set, and JV' denote the total number of chemical species formed Chapter 4. The Model Computations 98 Table 4.1: Species Considered in Model Equation of State Element Z Log Abundance (Number) Species containing this element H 1 12.00 H H+ H " H 2 H+ CH C 2 H 2 NH N H 2 N 3 H 2 0 MgH A1H SiH HS H 2S HCl CaH HCN HCO MgOH AlOH CaOH He 2 10.84 He He+ C 6 8.72 C C+ c 2 C 3 . CH C 2 H 2 CN CO C 0 2 CS SiC SiC 2 HCN HCO N 7 8.98 N N+ N 2 NH N H 2 N H 3 CN NO HCN 0 8 8.96 0 0+ o 2 OH H 2 0 NO CO C 0 2 MgO A10 SiO SO CaO ScO Sc0 2 TiO VO V 0 2 YO Y 0 2 ZrO Zr0 2 HCO MgOH AlOH CaOH Ne 10 7.92 Ne Ne+ Na 11 6.25 Na Na+ NaCl Mg 12 7.42 Mg Mg+ Mg++ MgH MgO MgCl MgOH A l 13 6.39 A l A1+ A1H AlO A1C1 AlOH Si 14 7.52 Si Si+ SiH SiO SiS . SiC SiC 2 S 16 7.20 S S+ HS H 2S SO CS SiS TiS CI 17 5.60 CI ci- HCl NaCl MgCl A1C1 CaCl K 19 4.95 K K+ Ca 20 6.30 Ca Ca+ Ca++ CaH CaO CaCl CaOH Sc 21 3.22 Sc Sc+ ScO Sc0 2 Ti 22 5.13 Ti Ti+ TiO TiS V 23 4.40 V V+ VO V 0 2 Cr 24 5.85 Cr Cr+ Mn 25 5.40 . Mn Mn+ Fe 26 7.60 Fe Fe+ Co 27 5.10 Co Co+ Ni 28 6.30 Ni Ni+ Sr 38 2.85 Sr Sr+ Y 39 1.80 Y Y+ YO Y 0 2 Zr 40 2.50 Zr Zr+ ZrO Zr0 2 Chapter 4. The Model Computations 99 entirely from elements of the reduced set. We shall define the fictitious pressure of an element k (denoted pi) to be the partial pressure that would result if all ions of the element and all atoms of the element bound in molecules were free to contribute to the partial pressure of the elemental species pk. This is sometimes referred to as the pressure of the nuclei since it is basically arrived at by counting the nuclei of element k present in the gas. Similarly, the total fictitious pressure is the gas pressure that would be obtained if all molecules were fully dissociated into their constituent atoms, and the electron pressure was ignored. The derivation, of the abundance equations in the following section make use of the fact that the fractional abundance ctk of element k is just (by definition) the ratio pk/p*. We make the following further definitions: Pe = electron pressure Pk = partial pressure of element in neutral atomic form Pi =. partial pressure of chemical species i Pi = fictitious partial pressure of element k P = total gas pressure * V - fictitious total gas pressure ru = total number of atoms in species i nik = number of atoms of element k in species i li = electronic charge of chemical species i ctk fractional abundance of element k. Note that K 5>* = L (4-4) fc=i We choose the pressures pk,k = and pe as the K + 1 independent variables describing the state of the gas. Then the pressure pi of any other chemical species can Chapter 4. The Model Computations 100 be expressed in terms of these variables via the equations of ionization and dissociation equilibrium as K l l p e h=l Here Ii is the ionization equilibrium constant relating the partial pressure Pi to that of the corresponding neutral species p°, and Ki is the dissociation equilibrium constant relating the partial pressure p° of the neutral (molecular) species to the partial pressures Pk, k = 1,..., K of its neutral atomic constituents. For neutral species Ij = 1 while for single atoms Ki = 1. The important point for our purposes is that Pi has the functional dependence Pi = Pi(Pe,Pl,---,PK)- (4.6) For each of the K elements in groups (i) and (ii) we can write an abundance equation of the form N nikPi ^UiPi *=1 by counting the atoms distributed among the various species. Upon rearrangement we have the abundance equation, N (akni - nik)Pi = 0 for k = 1,..., K, (4.8) i=l of which only K — 1 are linearly independent, due to equation (4.4) above. The charge neutrality of the gas yields another relation, N Y J P ^ - P ^ 0 - (4-9) t = l The sum of all the partial pressures must give the total gas pressure and this constraint yields the final equation of the system, N " ^ ^ + P e = P (4.10) i = l Chapter 4. The Model Computations 101 This set of K + 1 equations in the K + 1 variables pe,pi, • • • ,PK forms a closed system which can be solved for the elemental pressures. We proceed by linearizing the above system of equations, solving the resulting linear system to obtain the first-order corrections 5pe,6px,... ,Spx, and iterating until convergence is attained. The solution timing is fairly slow, however, since the coefficient matrix is of order K X K; in our case K = 15. To economize this solution, we consider only the main elements of group (i) above, and we lump the metal elements of group (ii) together and treat them as the single fictitious metal element Z. This yields the following reduced system of equations. Let the fractional abundance of Z be denoted ctz. Then, the abundance of Z is just the sum of the abundances of the individual metals, or M m=l But we also have a z = p (4.12) where p* = pz + pf and thus M M M ?;=p*]Ca™ = E a " ^ = E ^ - (4-13) m=l m=l m=l The reduced abundance equation for element k becomes ^ nikPi = $ =-sr <«4> »=i or JV' ^(akni - nik)pi + akp*z = 0. (4.15) i=l Chapter 4. The Model Computations 102 Since Pi = Pi(pe,P\i • • • ,PK') this equation can be linearized in terms of the independent variables p e ,Pi , • • • ,PK',P*Z- We adopt p*, instead of pz, as our additional independent variable since this results in simpler equations. 1 The abundance equation for the fictitious metal Z is az A" ~T UiPi + p*z i=l or The total pressure constraint for the reduced system is JV' and therefore By the definition of p^, we have m (4.16) JV' a*~~>W-(l-a.)p;=0 (4.17) ~~>+P* + -« = °> (4-18) i = i and the equation of charge neutrality becomes N' X > « f c + P ^ - P e = 0. ( 4- 1 9) 1=1 We now need to express p+ in terms of our independent variable p*. Let 1+ be the equilibrium constant describing the ionization equilibrium m r=* m+ + e~ (4.20) where 4 = 4^ (4-21) ft. = ^ . . (4-22) P„ (l + ~)pi (4-23) Chapter 4. The Model Computations 103 and thus * = n f e - i f n S ; - <4-24> Therefore, the total fictitious metal ion pressure is M M * T _ L M rf = E r i = E ^ - ^ E F ^ - ( 4 - 2 5 ) m=l m = l J m + P e m=l m + Pe and since PI . _* P. az = -^^p* = ^ (4.26) P* «z we finally obtain the desired expression * M T+ a. ~, ft + V, Substitution of this result into equation (4.19) yields the reduced equation of charge neutrality p*z s-^ amIZ a z m=l m ^ in a form dependent only upon the independent variables pi = Pi(Pe,P\, • • • )PK>) and pe,Pz- We now have a system of i f ' + 1 abundance equations, only K' of which are independent (since YJ f c ak + az = 1), and the two constraint equations (total pressure and charge neutrality) for a total of K' + 2 equations in K' + 2 variables. This reduced system can now be solved by linearization and the coefficient matrix is only of order K' 4- 2, which in our case is 8. The individual metal partial pressures pm can be recovered from pi, once the solution has been found, as follows: A - < ^ - ^ ( £ ) - ( ^ ] r f , . (4-29) where the value of p* from equation (4.26) has been substituted. Then, given Chapter 4. The Model Computations 104 which follows from the definition of 7+ in equation (4.21), we obtain Pm = r + . • (4.31) The final problem concerns the solution of the remaining group (iii) elements. In general, this would require another iterative solution of a similar set of nonlinear equations to those of group (i). However, we adopted the following assumptions which allow us to obtain a direct solution: • Group (iii) elements form molecules only with group (i) elements, and not among themselves. In practice, the heavy elements making up group (iii) almost never associate with each other and this assumption holds quite accurately. • Molecules involving group (iii) elements have only one atom of the respective el-ement present in the molecule. This assumption is largely justified on empirical grounds; molecules violating this assumption may exist but would have very low abundances and are not observed in stellar spectra. The first assumption above decouples all the abundance equations for each group (iii) element. The second assumption ensures that each of these abundance equations is linear in the partial pressure pk of each of the group (iii) elements. Since the values of the partial pressures for the group (i) and (ii) elements are already known, a direct solution of each group (iii) Pk is immediately obtained. v • For example, the equilibrium solution considers the following four species involving Ti: these are Ti, T i + , TiO, and TiS. Then the abundance equation for Ti is given by 'an = = ^z(PTi + Pn + PTio + Pns) (4.32) p p v = I( P R I + ^ l M + | ^ + ^ £ £ ) (4.33) P* \ Pe KTxO KTiS ) . ?5 + » + (4.34) P* V Pe KTiO KTiSJ Chapter 4. The Model Computations 105 and thus, PTi = j • (4.35) This completes the solution of the equation of state. Overall, the order of the linearized equation of state matrix has been.reduced from 25 to 15, and then to 8 by the introduction of the fictitious metal. This corresponds to a factor of approximately (25/8)3 ~ 30 saving in timing. A direct solution of the remaining elements (i.e. those not included explicitly in the linearization) is immediately obtained once the linearization step has been solved. The solution obtained remains exact within the limits of the assumptions made. In practice, errors resulting from violations of these assumptions should be at most of order 1-2%. Test results comparing the full and reduced solutions typically showed much smaller differences. In any case, these errors are negligible compared with the accuracy of Tsuji's [97] polynomial fits to molecular equilibrium constants, which depend upon many poorly determined dissocation energies and, in some cases, uncertain partition functions. 4.3 The Opacity of the Stellar Gas The standard bound-free and free-free opacities are included, in addition to bound-bound opacities for the CN, TiO, and H2O molecules. The continuous opacities are similar to those of Auman and Woodrow [11]. The opacities for CN and TiO are represented by straight means, while the H2O opacity was represented by either a straight or an harmonic mean. All molecular lines are assumed to be formed in pure absorption. The details of the opacities included are summarized in Table 4.2. Straight mean opacities do not accurately describe absorption due to molecular bands but suffice for the purpose of this study, which was to demonstrate the viability of the complete linearization method to cool, extended stellar atmospheres. In general, mean opacities will overestimate the true opacity since frequency intervals with low opacities Chapter 4. The Model Computations Table 4.2: Model Opacities Opacity Reference H bound-free, free-free Mihalas [69] H - bound-free, free-free Gingerich [41] He - free-free polynomial fit of Carbon et al. [25] to the calculation of John [52], as quoted by Kurucz [64] Hj free-free Somerville [93] H^" bound-free, free-free Kurucz [64] C bound-free Henry [45], as quoted by Kurucz [64] Na, K, Mg, Ca, A l , Si . bound-free Auman and Woodrow [11] H Rayleigh scattering Dalgarno [34] He Rayleigh scattering Dalgarno [34] H2 Rayleigh scattering Dalgarno [34] e~ Thomson scattering CN red (straight mean) Johnson et al. [58] TiO (straight mean) Collins [33] H2O (straight mean) Auman [9] H 2 0 (harmonic mean) Auman [9] Chapter 4. The Model Computations 107 that may carry significant flux ('flux windows') are given low weight. An alternative considered in this work was to use an harmonic mean opacity instead. This has the advantage that the flux windows will be given a high weight. Furthermore, the harmonic mean becomes exact at depth. However, in the optically thin limit, the cooling of the gas by emission in molecular bands is more accurately described by a straight mean than by an harmonic opacity. In reality, neither the straight mean nor the harmonic mean properly describe the radiative transfer in the molecular bands. A more accurate treatment of opacities would be provided by the use of opacity distribution functions or opacity sampling methods. This study did attempt to estimate the effect of the treatment of molecular opacities upon atmospheric structure for the H2O molecule, which provides the dominant opacity over much of the infrared region of the spectrum. Some models were calculated which varied only in the representation of the H2O opacity, using either a straight mean or an harmonic mean treatment. 4.4 Convection The model equations described assume radiative.equilibrium, and the models to be dis-cussed in section 4.7 were calculated under this assumption. However, late-type stars have convectively unstable regions in their outer layers. Convective energy transport dominates the radiative flux in the atmospheres of late-type main sequence stars. The efficiency of convective transport is generally much reduced in the atmospheres of late-type giant and supergiant stars due to the lower densities. Convection energy transport only becomes dominant at depth in the atmosphere. As long as the convective flux Hconv is only significant in the deepest parts of the atmosphere with TRM >^ 1, the outer parts of the atmosphere with TRM < 1 and the emergent flux will not be affected by the neglect of convection. Chapter 4. The Model Computations 108 To check the importance of the (neglected) convective transport, the convective flux was calculated using the temperature structure of our radiative models after full conver-gence had been attained. If the convective flux obtained is small, then our neglect of convection in the model calculations is justified. However, regions where the convective flux is significant will be inaccurately described by our radiative models. The convective quantities were calculated using a slightly modified version of the standard local mixing length theory as described by Mihalas [69], and the necessary thermodynamic variables were calculated using the formulae given by Kurucz [64]. In particular, the specific heats were calculated using where E is the internal energy of the gas, and p refers, as it does throughout this work, to the gas pressure. The total pressure ptot, ignoring turbulent pressure, is Ptot = P + Prai (4.38) with the radiation pressure given by pTai = — / K„du. (4.39) c Jo The above formulae for the specific heats follow from Kurucz [64]. Kurucz assumed that (dpTad/dT)p oc T 4 , from which (Sg?) (4.40) follows. However, this result is true only for optically thick regions of the atmosphere where the radiation field is locally controlled and thermodyamic equilibrium is approached. It also applies for homologous transformations of the entire atmosphere. In the optically Chapter 4. The Model Computations 109 thin outer layers, the gas temperature T is almost entirely decoupled from the radia-tion field, and (dprai/dT)p is not well-defined. For localized optically thin perturbations, (dpTad/dT)p « 0. For this work, we adopted the formula dT Jp T from which both of the above limits are recovered. Ignoring radiation pressure contributions, the adiabatic gradient is given by [64] ~ \dmp)s~ dPrad\ (dpX _ (dp dT)p\dp)T \6T (4.42) s P2CP while the radiative gradient V#, which equals the actual gradient V for our radiative models, is This was evaluated numerically using, second order 3-point differences. The atmosphere is unstable against convection when V.R > V ^ , according to the Schwarzschild criterion, in which case the standard mixing length theory is invoked to estimate the convective flux. Following Mihalas [69], we consider the gradient V s of a convective bubble and define a convective efficiency 7 by ' - v T ^ ' ( 4 - 4 4 ) We assume the radius of the (spherical) convective bubbles equals the mixing length I. For optically thick bubbles, we then adopt the estimate _ pCpvconv . . where T R M > B = XRMPI is the Rosseland mean optical thickness of the bubble, vconv is the mean convective velocity of the bubble, and cr is the Stefan-Boltzmann constant. For the case of optically thin bubbles, equation (6-285) of Mihalas [69] has been modified since Chapter 4. The Model Computations 110 a better estimate of the volume emissivity of the bubble is given by AnABupp using the Planck mean opacity « p . This yields pCpvconv 1 7.*. = -J-fT— (4-46) where r P i B = Kppl. Linear interpolation between these limiting cases gives the result, i b _ ^ ( I ± i ^ - ) . ( 4 . 4 7 ) Again following Mihalas [69], define ( ^ T ) = - M ) D - (4-48) Q = l-\omiJp v — / p Energy considerations can be used to estimate the convective velocity. Assume that half of the work done on the convective bubble by buoyancy forces eventually ends up as kinetic energy of the element. Mihalas [69] then obtains v^„=l-r\]\gQh{V-VB) (4.49) where h = ptot/pg is the pressure scale height and g = GM+/r2 is the local gravity. The mixing length parameter aconv = l/h is not determined by the 'theory' and must be additionally supplied. Usually, an arbitrary value near unity is adopted; a value of aconv = 1.6 was used throughout this study. In terms of aconv, TRM.B = PXRM<xconvh (4.50) TP,B = pKpacmvh (4-51) and vcm = acmvy/lgQh(V - V B ) . (4.52) Substitution of the above value for vconv into equation (4.47) gives pCpacmv / ! . ~ ' / l + l r P I B T R M I B \ V - V s 7 * = - S ^ T W ^ M V - v f l) [ — ) = V B ^ V ; ( 4 - 5 3 ) Chapter 4. The Model Computations 111 and thus We obtain where l e v ^ r 3 ( rPtB _ \ = (v - vA) - (v - vB) 1,B / pCpOLconv [ ( V - V ^ ) - ( V - V B ) ] 2 V - VB 16y/2(rT3 2D, I P,B pCpaconv TRM,E This quadratic equation in V — Vjg has the solution V - VB = D + (V - V A ) - i/[D + (V- VA)]2-(V -vAy. (4.54) (4.55) (4.56) (4.57) which determines the convective bubble gradient. Finally, the desired convective flux is given by Hconv = ^-acmvPCpTvconv(V - VB). (4.58) The quantity Hcanv, in analogy to the radiative case, refers to the Eddington flux; the physical convective flux is 47r Hconv. 4.5 Convergence Criteria and Iterative Control An iterative control procedure was used to make the solution of the linearized equations more robust. If a trial solution is close to the exact solution, the first-order correction obtained from the linearized system is quite accurate, and repeated iteration will yield rapid (~ quadratic) convergence to the solution. However, less accurate estimates yield less accurate corrections, and sufficiently poor trial estimates that exceed the radius of convergence of the linearized equations will result in immediate divergence if iterated. The remedy employed in this study was to control the flow of the iteration of the linearized system depending upon the accuracy of the current solution estimate, instead of proceeding to iterate blindly. Clearly, it is desirable to retain estimates close to the exact Chapter 4. The Model Computations 112 solution, while rejecting those further away. The exact solution is not known beforehand, of course, and therefore we found it useful to construct a scalar quantity which provides an estimate of the 'goodness' of a particular trial solution of the linearized system of equations. The choice of this 'goodness' estimator is quite arbitrary, but it should vanish for the exact solution and increase in a continous manner away from the solution. We adopted the magnitude of the maximum component of the normalized residual vector £ of the model equations as our goodness indicator. In the notation of section 3.3, the linearized system of equations is R A x = b (4.59) where the correction vector is A x = ( A j i , . . . , A J N , A T , Ap, Ar). The components Aji , A T , Ap, and A r are themselves vectors of dimension D and contain the depth variation of these variables. Each row of the left-hand side matrix R of equation (4.59) refers to one of the transfer equations for frequencies i — l , . . . , i V , or to one of the constraint equations. The magnitude of the right-hand side vector b, when suitably normalized, provides an estimate of the goodness of the current trial solution x and is zero if, and only if, x is an exact solution of the model equations. Therefore, we define the residual of each (transfer or constraint) equation by the vector - = p b ( 4 - 6 0 ) with norm Then, the scalar value ||e|| =max{|eM | , = 1,..., iV + 3, d = 1,..., £ } (4.62) was chosen as our goodness of solution indicator and used to control the iteration strategy. Chapter 4. The Model Computations 113 Blind iteration of the linearized equations will result in divergence given sufficiently poor solution estimates (i.e. large ||e||) due to the neglect of significant higher-order terms. In this situation, the linear derivatives usually underestimate the actual changes, and thus the linear corrections obtained generally overestimate the required correction. A common strategy is to employ a line search retaining the direction of the linear correction vector while scaring the amplitude to minimize the goodness indicator ||e|| (cf. Fletcher [40]). However, line searching requires a relatively large number of evaluations of ||e||, and is impractical in our case where each such evaluation requires considerable computational effort. We adopted a simpler, but robust, strategy. If the goodness indicator for the i + 1 iteration estimate is approximately less than that of the previous iteration z, < c||e||« (4.63) then we proceed and update the solution estimate x( , + 1) directly. Here c is a constant chosen slightly greater than unity (we used c = 1.05) to prevent possible infinite loops. These could occur if the machine precision is reached before the (too stringent) conver-gence criterion is met, in which A choice of c > 1 then allows an escape from the convergence loop. If instead this iteration yields a worse result, H ( m ) > c | | e | | « (4-64) then the linear correction vector Aa;W is successively multiplied by the scaling factor a, with 0 < a < 1, until I e ( x « + a " A x « ) || < || e ( x « ) | | (4.65) holds after n trials. The procedure just outlined is straight forward, but still somewhat inefficient in terms of overall computational effort. An initially poor trial solution can result in a series of Chapter 4. The Model Computations 114 poor linear corrections, each requiring several of the above sub-iterations to correct. It proved economical to put global limits on the size of the linear corrections as well. For our models, user specified limits ATum, Apum, and Afum were used to define the initial scale factor Q . J ATUm Apum Afum 1 3 = min < —^ , — , — , 1 > (4.66) I L\lmax ^Pmax <-*•! max I where ATmax, Apmax, and Afmax are the components of maximum amplitude of the respective correction vectors A T , Ap , and A f . The limits were set by means of the ATHENA program commands SET DTMAX, SET DPMAX and SET DRMAX, which default to the value of 0.1. Note that it is not necessary to restrict AJn in this manner, since the transfer equations are already linear in J n , and so any discrepancy between the linear correction and the actual nonlinear solution must arise from nonlinearities in T, p, and r. A judicious choice of these limits ensures that the correction vector A x remains small enough to ensure the validity of the linear estimate. In rare cases when the equations are very nonlinear, the scaling algorithm previously discussed may operate to further reduce the value of a. In summary, the sequence of solution estimates x(i+i) = x(») + ^ A x W x(i+i) = x « + a / 5 A x ( 0 x(»+i) = x « + a 2 /3Ax ( i ) (4.67) x(i+i) = x W + a « / 3 A x W continues until n is found for which equation (4.65) is satisfied. We used a = 0.35 to compute the models in this study. This iterative control procedure proved very successful at preventing initial divergence of the linearized equations, and at guiding the iteration into the linear regime where Chapter 4. The Model Computations 115 quadratic convergence is eventually attained. As a final refinement, the value of the i; + 1 iteration estimate was actually updated using the expression xf + aAxf if Ax^ > 0 (4.68) xf exp ( aAxf /xf ) if Axf < 0 for each depth component d of the correction vector. These two variants are equivalent to first-order, but the latter form has the advantage of always providing a positive esti-mate for sd t + 1^. This prevents program aborts due to unphysical negative variables from occurring before the iterative control strategy has a chance to work. The iteration of the linearized equations continues until either ||e|| is less than the specified tolerance, in which case convergence to the solution is assumed, or the number of iterations exceeds the maximum specified on the command. 4.6 The Production of Converged Models The solution of the spherical symmetric model equations is implemented in the ATHENA computer code by means of a command language structure. A summary of the most important commands is given below. INIT This command generates an initial estimate of T(r), p(r) and 7*(T) through the atmosphere. SAVE file This saves the current model parameters, arrays and flags into the specified file for later use. RESTORE file This restores a previously saved model from the specified file. Chapter 4. The Model Computations 116 FORMAL This command specifies that a formal solution of the spherically symmetric radiative transfer problem is to be performed. Given T(r), p(r) and r(r), the moments Ju, Hu, Kv and the Eddington factor fv are generated. MOMENT n Given /„, improved values of Jv(r), T(r), P(T), and r(r) are calculated by the linearization of the full system of moment transfer equations, radiative and hydrostatic equilibrium, and the radius equation. The maximum number of iterations to be attempted is given by ra. CONVERGE m ra This command automates the FORMAL + MOMENT loop, and will perform a maximum of m FORMAL solutions, each allowing a maximum of ra MOMENT iterations. PTCORR ra Given Jv(T), /„, and r(r), improved values of T(r) and P(T) are calculated by linearization of the equations of radiative and hydrostatic equilibrium. The integer n specifies the maximum number of iterations to be attempted. JTCORR n Given P(T), f„, and r(r), improved values of JV(T) and T(r) are calculated by linearization of the moment equations of transfer and the equation Chapter 4. The Model Computations 117 of radiative equilibrium. Again, n specifies the maximum number of iterations to be attempted. SET flag value Sets the value of the specified flag or variable. OUTPUT Produces complete output tables for the current model. The actual solution procedure varies considerably depending upon the model details. The inclusion of molecular opacities was found to substantially increase the amount of effort required to converge models. For example, a series of models were calculated at effective temperatures of 3500K, 4500K, and 5500K for a variety of gravities and extensions. These included all of the usual continuous bound-free and free-free sources of opacity, but no molecular opacities. The command sequence INIT FORMAL CONVERGE 8 8 was sufficient to yield convergence for these models. Typically, convergence occurred within 5 FORMAL iterations, and with a maximum of 6 (much less for the later itera-tions) MOMENT iterations per FORMAL solution loop. Models with molecular opacities proved to be more difficult. In this case, iterations of the completely linearized moment equations often failed to converge, probably due to perturbations in the pressure scale arising from the nonlinear behaviour of the opacity XRM(T) of the molecular gas. Convergence was obtained by the use of a sequential partial linearization of pairs of the variables in turn. First, radiative and hydrostatic equilibrium were enforced by linearizing T and p for a given radiation field «/„ and radial structure r. This step was performed by the PTCORR command and always converged readily. Next, Chapter 4. The Model Computations 118 the transfer equations were solved subject to the constraint of radiative equilibrium by linearizing Jv and T while p and r were held fixed. This system is quite well-behaved since temperature-dependent opacities largely cancel out, provided the Rosseland mean optical depth scale is used as the independent variable for the transfer equations and radiative equilibrium is expressed in the form of equation (2.17). This linearization was implemented in the command JTCORR and proved to be considerably more robust than the complete linearization provided by the MOMENT command. With the above changes, it proved possible to converge all the models which were examined. Typically, the command sequence would be to do a FORMAL solution (to get a new Jv from T, p, r), followed by a PTCORR sequence (to improve the estimate of T, p given Jv, r), followed by one or more JTCORR iterations (to update «/„, T from p, r) and closing the loop with another FORMAL solution. This method of sequentially sweeping through the variables always yielded slow improvement as measured both by the residuals of the equations and by the amplitude of the changes. Eventually, when the solution estimate was sufficiently good, the iteration sequence could be reduced simply to a FORMAL solution followed by a full MOMENT iteration loop. The final solution was then attained using a CONVERGE command, which simply automates the FORMAL + MOMENT cycle. As an actual example, the command sequence to converge a model with parameters Teff/log g/Mbol — 3000/0.0/-3.0, and including the molecular opacities of CN, TiO, and H 2 0 , was as follows: RESTORE previous model PTCORR 10 JTCORR 1 FORMAL PTCORR 10 Chapter 4. The Model Computations 119 JTCORR 1 FORMAL PTCORR 10 JTCORR 1 FORMAL PTCORR 10 JTCORR 1 JTCORR 2 JTCORR5 PTCORR 10 MOMENT 1 MOMENT 5 FORMAL PTCORR 10 MOMENT 6 FORMAL MOMENT 8 CONVERGE 5 .4 Another persistent problem occurred in the very opaque, far ultraviolet continuum which arises from the strong resonant bound-free transitions of the abundant metals. For such frequencies, the values of J„ can range over nearly 20 orders of magnitude through the atmosphere. This results in the cubic spline solution of the transfer equa-tion becoming unstable and yielding oscillatory solutions for J„ in the outer regions of the atmosphere. In terms of direct effects upon the emergent flux or the temperature structure, these oscillations are totally negligible due to the extremely small magnitude Chapter 4. The Model Computations 120 of Jv in the affected regions. However, oscillations of sufficient amplitude eventually re-sult in zero values of J„, a consequence of updating very large negative changes in 8JV in accordance with equation (4.68). This causes the tridiagonal solution of the discrete transfer equation to fail due to the presence of a zero diagonal element. Two modifications were made to circumvent this problem. First, a flag was introduced to allow the user to indicate the choice of solution method in FORMAL. In addition to the previously implemented cubic spline method, the less accurate but more stable method of centered finite differences was provided as an alternative to solve the transfer equation. The latter method, in which second derivatives are simply replaced by their corresponding second order finite difference analogs, is much less apt to yield oscillatory solutions in regions of large gradients. Flagging the problematic frequencies in the ultraviolet then alleviated the oscillatory behaviour of the FORMAL solution. While this approach would also work in principle for the moment equations, an al-ternate solution was sought to avoid the large number of program changes required to replicate the linearized coefficients and implement the alternate finite difference method. The method actually used updated values of Jv, ordered by decreasing radius, and then passed these values through a 5-point median filter for each frequency. This removed the oscillations and zero values of Jv for all but the most extreme situations. In such cases, which typically occurred during initial attempts to converge the more difficult models, the calculation of another FORMAL solution remedied the difficulty. 4.7 Model Results and Discussion A series of trial spherically symmetric model atmospheres were calculated using the method just described for a variety of different opacities. Initially, a large number of models were converged using analytic test opacities. A further series of models were then calculated assuming realistic continuous opacities on a 30 point frequency grid. Finally, Chapter 4. The Model Computations 121 several physically realistic models that included molecular opacities were calculated, and the results of these models are presented in this section. Unless specifically stated other-wise, all further references to models shall refer to those described below and summarised in Table 4.4. These models included the standard continuous bound-free, free-free, and scattering opacity sources, and the molecular bound-bound opacities of CN, TiO and H2O. The opacities due to CN and TiO were included using straight means over wavelength inter-vals. The opacity due to H2O was included in some models using straight means, and in other models using harmonic means. A microturbulent velocity of 2.0 km s - 1 was assumed for the calculation of the H2O opacity. A grid of 106 frequency points was used. This grid was chosen to represent the gross structure of the molecular opacities, and to adequately sample the Planck function at the high frequency end at the inner boundary and the low frequency tail at the outer boundary. Failure to provide sufficient frequency coverage yields errors in the frequency quadratures and thus affects the accuracy of the total luminosity. A Rosseland mean optical depth grid of 80 points spanning the range log rRM == —5 to +2 was initially used, along with a corresponding impact parameter (angle) grid of 30 rays. However, it was found that the more extended models were optically thick in the strongest molecular bands, even at the outer boundary. Therefore, an expanded grid of 87 depths covering log TRM = —6 to +2 was used for the extended models. These models also used an expanded grid of 45 impact parameters. Models were computed using the IBM 3081/63 system at the University of British Columbia Computing Centre. This computer has a floating point speed of about 3 mflops (flop = floating point operation per second). Representative timings for the various commands are given in Table 4.3. The details of the calculated models are summarized in Table 4.4. Each model is characterized by the model parameters Teff/\og g/Mbol (where Chapter 4. The Model Computations 122 Mhol is the stellar luminosity specified as a bolometric magnitude), and a brief description of the molecular opacities included. An equivalent alternate set of stellar parameters would be i * , M*, Rt; the stellar luminosity, mass and radius respectively. All abundances are solar and are taken from Allen [2]. A complete list of the model parameters is given in Table 4.5, including actual model radii and effective temperatures T€ff as denned below in the text. Model output tables are presented in Appendix II. The units of all physical quantities appearing in tables and figures in this section are given in the cgs system unless explicitly stated otherwise. Table 4.3: Command Timings on the IBM 3081 Command CPU time (s) FORMAL 390 PTCORR (per iteration) 65 JTCORR (per iteration) 104 MOMENT (per iteration) 108 Grid size: 87. depths x 106 freqs. x 45 angles Models were iterated until the residual ||£|| denned in section 4.5 was < 2 x 10 - 5 . Average timings for various commands executed on the IBM 3081 are given in Table 4.3. Overall, these models took approximately three hours of CPU time to converge. The stellar radius, defined to be R* = r{r = To = 1), was not constrained to remain at R* (the specified model radius) during convergence'of these models. As a consequence, the final Rm varies slightly from R* and Teff, the actual effective temperature, differs accordingly from the nominal Teff. Values of Teff calculated from Chapter 4. The Model Computations 123 Table 4.4: Converged Models with Molecular Opacities Model ID Model Parameters Ttff/logg/Mbol Molecular Opacities Model Grid Sizes Frequency Depth Angle 01310191 3000/0.0/-3.0 CN, TiOMH 20(sm) 106 87 45 02310191 3000/2.0/-3.0 CN, TiO, H20(sm) 106 87 45 03310191 3000/0.0/-3.0 CN, TiO, H 20(hm) 106 87 45 04310191 3000/1.0/-3.0 CN, TiO, H 20(hm) 106 80 30 05310191 3000/2.0/-3.0 CN, TiO, H 20(hm) 106 80 30 06310191 3500/1.0/-3.0 CN, TiO, H 20(hm) 106 80. 30 07310191 3500/2.0/-3.0 CN, TiO, H 20(hm) 106 80 30 Note: hm= harmonic mean, sm= straight mean are listed in Table 4.5. Similarly, the actual gravity log g differs slightly from the specified log g; these values are also given in Table 4.5 for the current models. Temperature profiles r(r R A f) as a function of Rosseland mean optical depth for a sample of these models are displayed in Figure 4.1. These profiles have been scaled by the factors Teff/Tcff to correct for the drift away from the fiducial stellar parameters. This diagram illustrates the expected lowering of the surface temperature of the extended 3000/0.0/-3.0 model as compared with the compact 3000/2.0/-3.0 model. This can be seen more clearly by reference to Figure 4.2 where the detailed temperature structure of the surface layers is shown on an expanded optical depth scale. The extended model Chapter 4. The Model Computations 124 Table 4.5: Converged Models: Summary of Parameters Model ID M*/MQ R*/RQ R*/Ro ttt logy 01310191 0.612 1225. 129.5 124.8 3055.4 0.03 02310191 61.2 1225. 129.5 129.4 3000.1 2.00 03310191 0.612 1225. 129.5 126.6 3033.5 0.02 04310191 6.12 1225. 129.5 129.4 3000.1 1.00 05310191 61.2 1225. 129.5 129.4 3000.2 2.00 06310191 3.30 1225. 95.1 * 93.6 3529.0 1.01 07310191 33.0 1225. 95.1 95.1 3500.0 2.00 M@ =1.989 x l O 3 3 g, LQ =3.826 x lO 3 3 ergs/cm2/s RQ =6.9599xlO10 cm, from Allen [2] 01310191 (3000/0.0/-3.0) is about 150K cooler on the outer boundary than the corre-sponding compact model 02310191 (3000/2.0/-3.0). The effect of the differing treatment of the H 2 0 opacity on the temperature structure is seen by comparison of the model 03310191 (3000/0.0/-3.0, harmonic mean H 2 0) with model 01310191 (3000/0.0/-3.0, straight mean H 2 0) . Temperature differences of up to ~ 100K appear near optical depth log = —1.5 where the number density of H 2 0 reaches its maximum. The pressure structure of the atmosphere is shown in Figure 4.3, where log P(TRM) is plotted for several models. Similarly, the behaviour of the electron pressure logp e(rH M) is shown in Figure 4.4. Chapter 4. The Model Computations 125 The atmospheric extension, defined by r(TRM)/R+ — 1, where R* is the actual stellar radius, is plotted in Figure 4.5. The total extension of the atmosphere for the extended 3000/0.0/-3.0 models is about 0.25. By contrast, the 3000/2.0/-3.0 models are essentially compact, with an extension < 0.01. The Rosseland mean opacity profiles, XRM{TRM), of selected models is shown in Figure 4.6. The oscillatory behaviour is due to the appearance of molecular opacities at various depths in the atmosphere. The prominent peak in XRM near rRM = 2.5 (log TRM = 0.4) is due to the sharp maximum in number fraction of CN at this depth. The broad peak around logr H A f = —2 is due to the presence of TiO, which dissociates at depths greater than this. The effect of H2O, abundant in the surface layers, is to extend the tail of this peak out to log TRM = —4. The use of straight mean opacities in these models enhances the prominence of these opacity peaks. These peaks would be lower in amplitude if a more accurate treatment of the molecular bound-bound opacities had been employed. An artifact is apparent in Figure 4.6 near log T R M ~ —4 where the profiles of XRM{TRM) for the straight and harmonic mean opacities for the compact models intersect. The harmonic mean opacity should always be less than the corresponding straight mean. The problem is due to the choice of the four frequency points adjacent to the low frequency boundary. The opacity routines (in particular for the calculation of the opacity of H2O) truncate the calculation at wavelengths beyond ~ 10//. As a result, the calculated opacity for these four frequencies is very low. The Rosseland mean gives high weight to points of low opacity and so these four boundary frequencies can at times distort the calculation of the mean. Models with straight mean opacities are particularly susceptible due to the overall high opacity through the infrared region. It appears that the Rosseland mean calculation for the models using a straight mean H 2 0 opacity is affected above log = —3. At the outer boundary the Rosseland mean can be in error by 50%. The net result is to introduce a small distortion of the outer portion of the optical depth scale Chapter 4. The Model Computations 126 in the models using a straight mean H2O opacity. The Rosseland mean calculation for models using the harmonic mean H2O opacity is barely affected (the error is < 1% in XRM), and the optical depth scale remains correct. The ratio of the flux-mean opacity XH to the Rosseland mean opacity is displayed in Figure 4.7. This ratio, 7 = X H / X R M , appears as the multiplier in the radiation pressure term of the equation of hydrostatic equlibrium. This ratio should approach unity at depth, as is confirmed by this figure. We have calculated the convective flux for our (radiative) models in the regions where the atmospheres are unstable against convection according to the Schwarzschild criterion. Figure 4.8 shows the resulting ratio of convective to total flux, Hconv/H, that would result if convection were allowed to develop in the radiative equilibrium (RE) models calculated. The corresponding mean convective velocities are shown in Figure 4.9. Typically, there is a convective region of variable extent present in the outer layers, between rRM ~ 10 - 3 -10 - 1 . The convection is, however, extremely inefficient with Hconv/H ~ 10 - 6 , and the effect on the atmospheric structure is totally negligible. The onset of the primary convective zone occurs around TRM ~ 1. For the low gravity (logy = 0) models, the convection does not become efficient until much deeper, with Hconv/H > 0.01 only for TRM > 15. This corresponds to a monochromatic optical depth of rv « 5 at the most transparent frequency (~ 1.9/t). Convection at this depth will have a negligible effect on the structure of the transparent regions of the atmosphere, and on the emergent flux distributions. In this case, we would expect that both the structure of the outer atmosphere ( T R M > 1) and the emergent flux should be accurately described by our RE models. However, for the high gravity (log g = 2) models, the effect of convection does become (barely) significant. In this case, there is an outer peak with Hconv/H ~ 0.2 for TRM ~ 5. Deeper in the atmosphere, Hconv/H falls again to ~ 0.03 near rRM = 10, and then rises Chapter 4. The Model Computations 127 steeply as in the low gravity case. The most transparent frequency at TRM = 5 occurs at 1.74// for which r„ « 1.3 and exp(—r„) = 0.27. Therefore, the anticipated effect of the neglect of convection could be as large as 5% at the most transparent frequencies. Although, this is overall a small effect, it is clear that convection should be included in the model solution for accurate models in this case. The treatment of convection described previously essentially follows the standard local mixing-length theory as formulated by Mihalas [69]. This assumes that an opti-cally thin, rising convective bubble has radiative losses proportional to KpAB, where Kp is the Planck mean opacity. The overall temperature dependence of the cooling is assumed to be that of AB = (4<r/7r)T3AT. By contrast, the behaviour of the actual cooling function $ c o oj = JQ°° KuBvdu, as illustrated in Figure 4.10, exhibits oscillatory behaviour through most of the atmosphere. The destruction of sources of molecular opacity with increasing optical depth (and temperature) actually results in a decreasing cooling function over much of the atmosphere. However, the corresponding heating func-tion Eheat — J0°° KvJvdu, displays a matching temperature dependence. As a result, the net cooling due to a temperature perturbation A T is still, to first order, given by KpAB, as assumed in the mixing-length model. Our models do not display evidence of the molecular cooling induced instabilities of the kind discussed by Muchmore [74]. The heating function appears to closely follow the cooling function (both are basically determined by the temperature structure of the opacity), so that the net radiative cooling remains a monotonically increasing function of temperature. Thus, our models are stable against such radiative instabilities for small temperature perturbations. The normalized residuals for the equation of radiative equilibrium (eRB), the equation of hydrostatic equilibrium (eHB), and the radius (depth) equation (eDB) are shown in Figures 4.11-4.13 rspectively. The rms normalized transfer equation residual, is defined Chapter 4. The Model Computations 128 by 1/2 (4.70) where the summation is over the grid of N frequencies. The weights Wi used were "i=(V^y/4 (4.7i) with J f B a . = max{J i}^. (4.72) The rms residuals eTBrm, are presented in Figure 4.14. This particular choice of weighting was used to ensure that eTBTm, was representative of frequencies where the radiation had significant intensity. The residuals for all equations in the system were < 10 - 6 throughout the atmosphere for these models, demonstrating full convergence to the exact solution. The ratio of the actual luminosity L of the converged models to the prescribed model luminosity L* is presented in Figure 4.15. This quantity serves as a check on model convergence and overall consistency. In all cases, the ratio L/L+ is within ±1.5% of unity. The error appears to be dominated by quadrature truncation errors resulting from the numerical evaluation of the radiation field integrals on a finite frequency grid, instead of the semi-infinite interval [0, oo]. The 106 point frequency grid employed for the present series of models has its shortward boundary at 10.8/z-1 = 926A, just longward of the Lyman continuum edge. The quadrature error in the evaluation of the total flux H on this grid at the inner boundary of the atmosphere is also approximately 1%, mainly due to the lack of points in the Lyman continuum. The addition of such points, with associated extremely high opacities, would likely exacerbate the transfer equation instabilities noted in section 4.6. Therefore, frequencies in the Lyman continuum were deliberately excluded from the grid. The resulting errors in the evaluation of H impose a limit on the accuracy of the calculation of the luminosity L of about ±1%. Chapter 4. The Model Computations 129 Figure 4.16 shows the emergent flux Hv for several models as a function of wavenum-ber. A striking aspect of these flux distributions is the enhanced strength of the TiO bands for the extended models. Comparison of the flux shortward of the flux peak at 0.6/z-1 for the extended 3000/0.0/-3.0 model and the compact 3000/2.0/-3.0 model shows that the flux emitted by the extended model in this spectral region is similar to that of a compact model with effective temperature about 200K cooler. The 'photospheric radius' RpS evaluated as a function of wavenumber for the extended model 01310191 (3000/0^0/-3.0) is presented in Figure 4.17. Here Rp, is defined as the offset distance for which a ray drawn parallel to the symmetry axis of the star (i.e. an impact parameter as defined by HKK in their formal solution) has a total line-of-sight (monochromatic) optical depth of unity. The quantity RpS is indicative of the size of the resolved stellar disk, as seen by an observer at infinity. It is presented as a ratio to the model radius This quantity provides an area in which direct observational confirmation of atmospheric extension may be possible by means of stellar interferometric studies as reported by Beckers [14]. A series of greyscale representations of the full two-dimensional variation of several variables with optical depth and frequency are presented in Figures 4.18-4.20. These include the ratio of mean radiation intensity to the Planck function, Jv/Bv, the ratio of the monochromatic opacity to the Rosseland mean opacity, XVIXRM, and the value of the Eddington factor /„. These figures permit a qualitative presentation of atmospheric structure in a very compact manner, which provides a useful diagnostic check. For example, it is evident at a glance that the radiation field thermalizes at depth («/„/Bv —> 1 and fv 1/3) for all frequencies as expected. The distribution of the main sources of opacity are shown in Figures 4.21 and 4.22 for the extended model 01310191 (3000/0.0/-3.0) and the compact model 02310191 (3000/2.0/-3.0) respectively. These figures display the log number density of significant Chapter 4. The Model Computations 130 opacity sources as a function of Rosseland mean optical depth. These are presented as number densities since the strength of absorption features is determined by the number of absorbers along the line-of-sight. This can be obtained from these profiles by integration over optical depth. The ionization structure of the atmosphere for the same extended and compact models (01310191 and 02310191 respectively) is illustrated in Figures 4.23 and 4.24. The log number fraction of charged species are shown as a function of optical depth. Finally, the series of Figures 4.25-4.34 show the distribution of the abundant elements H, C, N , 0, and Si with Rosseland mean optical depth for these two models. These profiles are also represented in the form of log number fractions. Chapter 4. The Model Computations 131 Figure Captions Figure 4.1. Temperature profiles T(r R A f) for selected models. Note: hm= harmonic mean, sm= straight mean. Figure 4.2. Temperature profiles T( r R A f ) /T e / / for the outer atmospheres of selected models. Figure 4.3. Pressure structure logp(rH M) for selected models. Units are dynes/cm2. Figure 4.4. Electron pressure logp e(TR M) for selected models. Units are dyne's/cm2. Figure 4.5. Atmospheric extension r(rRM)/Rt — 1 for selected models. Figure 4.6. Rosseland mean opacity profiles log X R M { J R M ) for selected models. Units are cm2/gm. Figure 4.7. Ratio of flux-mean opacity to Rosseland mean opacity, log XH/ XSM, for selected models. Figure 4.8. Ratio of convective to total flux, log HCONV(TRM)/H, for selected models. Figure 4.9. Profiles of mean convective velocities log vconv(rRM) for selected models. Units are cm/s. Figure 4.10. Profiles of cooling functions log §COOI{TRM) = log J0°° KVBv dv for selected models. Units are ergs/gm/s. Figure 4.11. Profiles of the normalized residuals eRB of the equation of radiative equi-librium for selected models. Figure 4.12. Profiles of the normalized residuals eHE of the equation of hydrostatic equilibrium for selected models. Chapter 4. The Model Computations 132 Figure 4.13. Profiles of the normalized residuals eDB of the radius (depth) equation for selected models. Figure 4.14. Profiles of the RMS normalized residuals eTETm, of the equations of radia-tive transfer for selected models. Figure 4.15. Ratio of actual to prescribed model luminosity, L/L„, for selected models. Figure 4.16. Emergent flux distributions log Hv for selected models. Units are ergs/cm 2/s/^ _ 1. Figure 4.17. Ratio of photospheric to model radii, Rp,(u)/Rt, for extended model 01310191. RpS is a measure of the stellar disk radius as seen by a distant observer. See text for details. Figure 4.18. Ratio of the mean radiation intensity to the Planck function, log JvjBu, as a function of optical depth TRM for the extended model 01310191 (3000/0.0/-3.0). On this 10-level greyscale, white (level 1) corresponds to a Jv/Bv value about 0.4, the light grey background (level 4) to about 1, and black (level 10) to about 5. The value of JvjBv has been truncated at 5 to preserve dynamic range. Figure 4.19. Ratio of the monochromatic opacity to the Rosseland mean opacity, log XV/XRM, as a function of optical depth TRM for the extended model 01310191 (3000/0.0/-3.0). On this 10-level greyscale, white (level 1) corresponds to XV/XRM values < 0.05, the light grey dominant background at depth (level 4) to values near 1, and black (level 10) to values > 104. Figure 4.20. The Eddington factor /„ as a function of optical depth TRM for the ex-tended model 01310191 (3000/0.0/-3.0). On this 10-level greyscale, white (level 1) corresponds to fv < 0.34 while black (level 10) corresponds to /„ > 0.65. The increment between levels is approximately 0.03 in fv. Chapter 4. The Model Computations 133 Figure 4 . 2 1 . The distribution of opacity sources in the extended model 01310191. The log number density of each species is plotted as a function of optical depth rRM. Units are c m - 3 . Figure 4 . 2 2 . The distribution of opacity sources in the compact model 02310191. The log number density of each species is plotted as a function of optical depth rRM. Units are c m - 3 . Figure 4 . 2 3 . The ionization structure of extended model 01310191. The log number fraction of each species with respect to the electron density is plotted as a function of optical depth TRM. Figure 4 . 2 4 . The ionization structure of compact model 02310191. The log number fraction of each species with respect to the electron density is plotted as a function of optical depth T R ^ . Figure 4 . 2 5 . The distribution of hydrogen in extended model 01310191. The log num-ber fraction of each species is plotted as a function of optical depth TRM . Figure 4 . 2 6 . The distribution of hydrogen in compact model 02310191. The log number fraction of each species is plotted as a function of optical depth rRM. Figure 4 . 2 7 . The distribution of carbon in extended model 01310191. The log number fraction of each species is plotted as a function of optical depth TRM . Figure 4 . 2 8 . The distribution of carbon in compact model 02310191. The log number fraction of each species is plotted as a function of optical depth TRM. Figure 4 . 2 9 . The distribution of oxygen in extended model 01310191. The log number fraction of each species is plotted as a function of optical depth rRM. Chapter 4. The Model Computations 134 Figure 4.30. The distribution of oxygen in compact model 02310191. The log number fraction of each species is plotted as a function of optical depth r R M . Figure 4.31. The distribution of nitrogen in extended model 01310191. The log number fraction of each species is plotted as a function of optical depth TRM. Figure 4.32. The distribution of nitrogen in compact model 02310191. The log number fraction of each species is plotted as a function of optical depth rRM. Figure 4.33. The distribution of silicon in extended model 01310191. The log number fraction of each species is plotted as a function of optical depth rRM. Figure 4.34. The distribution of silicon in compact model 02310191. The log number fraction of each species is plotted as a function of optical depth TJU^. 8f I-I 0> >-« o ST ioooo 8000 -6000-4000 Models - 3000/0.0/-3.0 sm H 20 - 3000/2.0/-3.0 sm H 20 - 3000/0.0/-3.0 hm H 20 — — — 3500/2.0/-3.0 hm H 20 / / t / 2000 0 -6 -2 log r 0 RM Chapter 4. The Model Computations (D "D O O O O o CM o CM o CM o CM I n: E E E E en CO x : o O O o CO 1 CO CO CO 1 O o O o O o i CD o i ^ » O o O o o o o o o o o ID CO CO CO CO o o LO CO O o o CO O O LO CM O O O CM 136 CM i CO I O i LO i CD i O o LO Figure 4.2: Temperature profiles T ( r R M ) of the outer layers d 6o| Figure 4.3: Pressure structure l o g ^ r ^ ) Figure 4.4: Electron pressure logp e(rR J w r) Chapter 4. The Model Computations I I J _ 139 J/5 "D O O O O CM W o | CM E E E CO CO -C CO CO CO I 1 Q o q d oi d o o o o o o o o o CO CO CO CN O CN CO o I CD i LO o LO o LO o LO CN CN • • o o o b O o O o • o • o Figure 4.5: Atmospheric extension r ( r i t M ) / i ? » — 1 Chapter 4. The Model Computations Figure 4.6: Rosseland mean opacity profiles log XRM{TRM) Chapter 4. The Model Computations 141 _G0 CD "D O o o o o o CM CM CM CM CM X X X X e E E E E CO CO sz sz O O O o o CO 1 CO CO 1 CO CO 1 1 O O o o o d oi d oi oi "•••^ Q o o o o O o o o o O o o o LO CO CO CO CO CO CO CN O M X , H X Bo| CN CN i CO o I CD i Figure 4.7: Ratio of flux-mean opacity to Rosseland mean opacity \o%xnlx Chapter 4. The Model Computations CN in CD "a o o o o o CM CM CM W N X I X I E E E E o o p o CO CO CO CO p o p o o c\i o c\i o o o o o o o o o o o o CO CO CO CO CN I I CO CN CN I I | _ | / A U 0 0 | _ | 60| CD i 00 I Figure 4.8: Ratio of convective to total flux log Hconv(rRM)/H Chapter 4. The Model Computations I L CN CD TD O O O O O CM CM <* CM X X • X X E E E E CO CO si O O O O CO CO CO CO -«!^ 1 o o o o d CM' d oi o o o o o o o o o o o o CO CO CO CO / I \ CN i i CD i CD LO Auooyy 6o| CO CN Figure 4.9: Profiles of mean convective velocities log vcm^{rRM') o o o era (3 S O 3 O $ C I 0 5 ^ c era * —• ? Oss CO CO O •e< O 12.5 12.0 11.5 11.0 10.5-10.0 Models 3000/0.0/-3.0 sm H 20 3000/2.0/-3.0 sm H 20 3000/0.0/-3.0 hm H 20 3000/2.0/-3.0 hm H 20 9.5 -6 -4 -2 log r 0 RM Chapter 4. The Model Computations I I _C0 CO O o o o CM CM CN X X E E E CO CO O O O CO 1 CO CO 1 1 i 1 o o o d CN O "-•^ — o O o o o o o o o CO CO CO CO I 00 I o I 3d CO I 9 6o[ Figure 4.11: Normalized radiative equilibrium residuals eT Chapter 4. The Model Computations \ / X/ CD "D O O O o CM CN CM. X X x E E E CO CO - C q q q CO co cd I I I P o p d oi d o o o o o o o o o CO CO CO CO i 00 I o I 3H CN CO I 9 6o| Figure 4.12: Normalized hydrostatic equilibrium residuals e Chapter 4. The Model Computations I 147 _C0 CD "O O CD i o o O CM CM CM X X X E E E CO CO O O o CO CO CO 1 1 o O o d oi d *">»«, o o o o o o o o o CO CO CO 00 I CN CN I CO O i CD i i CO i 9 60| Figure 4.13: Normalized radius (depth) equation residuals ez Chapter 4. The Model Computations I J in CD "D O CO i o o o CM CM CM x x x E E E CO CO -C p o p CO CO CO I I I p o p d oi d o o o o o o o o o CO CO CO 00 1 O i SUJJ31 CM i i 9 60| CO Figure 4.14: RMS normalized transfer equation residuals ea Chapter 4. The Model Computations I ' in CD o o O o O o CM CM CM CM CM X X X X X E E E E E CO CO si O O o O o CO | CO CO CO CO o o o o o d CN d CN CN o o o o O o o o o o o o o o LO CO CO CO CO CO CO o CN O O O CD CO CO CD Figure 4.15: Ratio of actual to prescribed model luminosity L/L, CO CO CD ID "tf "H xn|j; ;U96J9UU3 6O| Figure 4.16: Emergent flux distributions log Hv y / s dy snipej Ojjai|dsoioi|d psziieuuoN Figure 4.17: Ratio of photospheric to model radii Rp,(v)/R* Figure 4.18: Ratio of the mean intensity to the Planck function log Jvj'Bv Figure 4.19: Ratio of the monochromatic to the Rosseland mean opacity logXiz/x Figure 4.20: The Eddington factor fu(rRM) Chapter 4. The Model Computations CN O CN "si" CD 00 i i i i (Aijsuep UOJJ.09|8 / Aijsuep jsqiunu) 6o| Figure 4.23: The ionization structure of model 01310191 Chapter 4. The Model Computations CD o :5 CO +-• o o l _ o i - J -» * * " ^ CO o o c o CO o "•*-» • • CO 0 N TD O o 2 CN (Aiisuep U O J I O 9 | 8 / Aiisuep jeqiunu) 6o| Figure 4.24: The ionization structure of model 02310191 Chapter 4. The Model Computations 159 CN CC CD O CN i i CD i 00 I S9!09ds J;O uojioejj; J9quunu 6o| Figure 4.25: The distribution of hydrogen in model 01310191 Chapter 4. The Model Computations 1 160 o CN CN CO 03 sepeds jo uoi+yoejj. jeqwnu 6o| Figure 4.26: The distribution of hydrogen in model 02310191 Chapter 4. The Model Computations 161 CM CM i i CD CC CO o 00 I ssjOGds jo uojioejj. J9qtunu 6o| Figure 4.27: The distribution of carbon in model 01310191 Chapter 4. The Model Computations 1 162 o CN CN CO 00 sejoeds jo uojioejj. jequunu 6o| Figure 4.28: The distribution of carbon in model 021310191 Chapter 4. The Model Computations CN CN CO 00 SGjoeds jo uoiioejj jeqiunu 6o| Figure 4.31: The distribution of nitrogen in model 01310191 Chapter 4. The Model Computations co o 5 ^ CN O CM <sr CD 00 i i i i S9!09ds ±o uojjoejj jequunu 6o| Figure 4.32: The distribution of nitrogen in model 02310191 Chapter 4. The Model Computations 167 co O Q ^ CN O CN ^ CO 00 i i i i S9!09ds j.o uojjoejj J9qujnu 6o| Figure 4.33: The distribution of silicon in model 01310191 Chapter 4. The Model Computations 168 co o 5 . 5 CM O CM ^J" CO GO i i i i sapeds jo uojioejj. jeqiunu 6o| Figure 4.34: The distribution of silicon in model 02310191 Chapter 5 Conclusions 5.1 Discussion 5.1.1 The Complete Linearization Method: An Overview The development of the solution procedure and implementation of the ATHENA com-puter code for the modelling of extended atmospheres of cool stars using the complete linearization method (CLM) has been demonstrated. This task actually involved two problems: the successful .application of the CLM to late-type atmospheres where molec-ular opacities are important; and the modification of the plane-parallel CLM to a spher-ically symmetric geometry. Both aspects of this project were significant since this work represents, to the best of our knowledge, the first published application of the CLM to any late-type model atmosphere, either plane-parallel or spherically symmetric. Addi-tionally, few spherically symmetric (SS) models have been produced at all for late-type atmospheres. The use of the C L M brings many advantages. It is fundamentally not just a temper-ature correction procedure, although this does form the core of the work for the current LTE models. However, the generality of the CLM permits a straightforward extension to non-LTE problems where the equations of statistical equilibrium must additionally be solved to obtain level populations. The CLM could also be used to produce fully self-consistent coupled stellar atmosphere and (steady-state) wind models. This would require the addition of a velocity variable to the model system and a replacement of hydrostatic equilibrium by an equation describing momentum transfer in the stellar material. 169 Chapter 5. Conclusions 170 The great power of the CLM derives from its simplicity. The underlying physics remains clear, which is often not the case with more formal methods of solution. Conse-quently, the introduction of extended physics (such as non-LTE) into an existing model proceeds in an obvious manner. In this study, the CLM has been implemented in spherical symmetry using an optical depth scale for the independent variable, as is standard for plane-parallel models. This implementation has been achieved by successfully linearizing the radius variable in a manner which does not destabilize the transfer equation. The spherically symmetric implementation described in this work has the further advantage of reducing to the corresponding plane-parallel model equations in the compact atmosphere limit. In this limit of zero atmospheric extent, the depth equation defining the radius scale becomes irrelevent and decouples from the rest of the system while 17(7*) —» 1 and the other equations reduce exactly to the Auer and Mihalas [7] implementation of the C L M . This greatly simplifies differential studies of extension effects since the same model code can be used to compute both compact and extended model analogs. This eliminates possible artifacts due to model inconsistencies in the different cases. 5.1.2 The Effects of Convection The models in this study were all calculated assuming radiative equilibrium (RE). How-ever, the effect of convection was estimated by calculating the convective flux that would result given the (fixed) temperature structure of these radiative models. For the low gravity log g = 0 models, convection was found to be significant only at depths with TRM > 15. Therefore, the temperature structure of the outer atmosphere ( T R M < 5) and the emergent flux should be accurately described by our current RE models. However, for the higher gravity logg = 2 models, convection does become marginally significant with an expected contribution of up to ~ 5% of the emergent flux at the most transparent Chapter 5. Conclusions 171 frequencies. A consistent treatment of convection should be included in the solution to obtain accurate models in this case. At depth, the incorporation of a consistent treatment of convection would reduce the temperature gradient of the models. As a result, some artifacts occur due to the unrealistically large temperature gradient at depth. The most serious of these is a large radiation pressure gradient at the very bottom of the atmosphere, which for the two extended models calculated exceeds the Eddington limit at depths immediately adjacent to the inner boundary. 5.1.3 The Treatment of the H 2 0 Opacity Our models used the approximation of mean opacities to represent molecular bound-bound absorption. Water vapour absorption was modelled using both straight mean opacities, which more accurately represent thermal emission in the surface layers, and harmonic mean opacities, which are exact at depth. A more detailed treatment of the H 2 0 opacity is needed for accurate models. Differential comparisons of straight mean with harmonic mean H 2 0 models show that the greatest change in temperature structure (~ 120K) occurs near logr H M —1.5, where the number density of H 2 0 is near maximum. The emergent flux shows little overall dependence upon the treatment of the H 2 0 opacity, except in the regions of the strong infrared bands. 5.1.4 The Independent Variable Dilemma Model calculations were attempted using both column mass and Rosseland mean optical depth scales. The column mass models converged well as long as the opacity was not overly temperature sensitive. However, when the opacity was a strong function of tem-perature, column mass based models inevitably suffered wild temperature oscillations at depth when the model solution was iterated. Basically this is because the transfer Chapter 5. Conclusions 172 equation couples the radiation intensity to the optical depth in the gas; for a column mass scale at depth, neither the intensity nor the optical depth are known and therefore any iterative improvement will end up varying both (which is disastrous). Models using a Rosseland mean optical depth scale have the enormous advantage in that the temperature at depth is a known function of the independent variable. Further-more, at depth the transfer of radiation is described by the diffusion equation, and this yields the mean intensity directly in terms of the function BU(T) of the (known) tem-perature. Therefore, the use of the Rosseland mean depth scale fixes the temperature structure and the intensity of the radiation field at the bottom of the atmosphere leaving only the outer boundary to be iterated to convergence. Still, the outer boundary is troublesome. The Rosseland mean depends critically upon details such as the frequency points included in the grid and the treatment of the molecular opacities. This sensitivity of the Rosseland mean is shown clearly by the error introduced by an incorrect opacity calculation for four points at the extreme low-frequency end of the grid of 106 points. Furthermore, since the outer atmosphere is optically thin at most frequencies the radiation field is not coupled to the gas and the optical depth is not a particularly useful variable. The physically meaningful variable in the outer atmosphere is the column mass, which is proportional to the pressure. On the other hand, the Rosseland mean optical depth is a very useful independent variable at depth. The dilemma of the modeller then is to reconcile these opposing constraints. One approach for future models may be to use the Rosseland mean of the continuum opacity sources only as an independent variable. This seems much less likely to cause trouble since the continuum bound-free and free-free opacities are much less temperature sensitive than are the molecular opacities presently included in the Rosseland mean calculation. Another promising idea is to construct a composite variable that approaches the Rosseland mean optical depth deep in the atmosphere, while acting like, a column Chapter 5. Conclusions 173 mass for small optical depths. 5.1.5 Numerical Inconsistencies The stellar luminosity was conserved to within ±1.5% in our models, as shown in Figure 4.15. While this may seem poor compared to other models that conserve flux to better than 0.1%, the flux and luminosity are secondary quantities in our models. Models that accurately conserve flux (cf. Scholz [88]) are generally designed to explicitly enforce flux constancy in their solution method. By contrast, fluxes do not appear in our model equa-tions, with the sole exception of Hv supplied as an inner boundary condition. Instead, our models enforce the more basic equation of radiative equilibrium, which is mathemat-ically equivalent to the conservation of flux (actually r2Hu in the spherically symmetric case), and our solutions yield mean intensities. Fluxes were obtained by differentiation of these mean intensities. The residuals of the transfer equations and radiative equilibrium were very small (~ 10 - 1 °) for our converged models, and it is clear our models accurately solve the discrete systems. The implication is that the errors in luminosity arise from inconsistencies between the discrete and continuous model equations, and also possibly between the different discrete forms of the formal and moment solutions. It should be noted that the models of Scholz [88], while iterated to a flux constancy of 10~4, only obey radiative equilibrium to within ±2%. Thus, the level of internal consistency of our models (~ 1.5%) is entirely comparable to that of the models of Scholz [88] and also of Bessell et al. [19]. There appear to be two effects present. First, examination of Figure 4.15 shows a slow rise in.L/L* of ~ 1% from the inner boundary outward to depths of T R M ~ 1. The gentler rise for the 3500/2.0 model agrees very well with the profile of l/u>, where u> (ideally = 1) estimates the frequency quadrature error and is defined by (5.1) n=l Chapter 5. Conclusions 174 with quadrature weights as defined by equation (3.116). Thus, for the 3500/2.0 model at least, the variation in L/L+ appears to be largely a result of quadrature errors due to the finite frequency grid. Our models used a frequency grid with a high frequency boundary at 10.8/z-1, just longward of the Lyman continuum edge. At the inner boundary, the numerically integrated flux is underestimated by about 1%. The monochromatic fluxes supplied for the inner boundary condition are scaled to yield a total luminosity equal to the specified Moving outward from this boundary, the flux spectral distribution shifts to lower frequencies and better matches our grid so that the numerical quadrature errors become quite small. Therefore, we should expect the integrated flux to increase by ~ 1% over the range of optical depth log T R M = 2 to 0, as confirmed by the 3500/2.0 model. However, the cooler 3000K models show a sharper, greater amplitude rise in L/L+ than can be accounted for by the variation in UJ alone. The second effect evident in Figure 4.15 is the sudden drop in luminosity of up to 2% around log rRM ~ —0.5, terminating the previous rise. The cause here is less obvious. The possibility that molecules are responsible, though, is suggested for two reasons. First, the hotter 3500K model, for which molecular opacities should be less important than in the copier 3000K model, shows a much smaller drop in L/L+. Second, the drop for the higher gravity log 5 = 2 models occurs deeper in the atmosphere, nearer to log T R M = 0. This echoes the shift in the distribution of molecular opacities with gravity seen in Figures 4.21-4.22 and particularly in Figure 4.10, where molecular opacities dominate the cooling function for log rRM < 0.5. A probable cause of this lack of luminosity conservation lies in the finite difference form of the transfer equation. We use Xv Arv = — — A T R M = XV&TRM (5.2) XRM to represent the monochromatic optical depth step. Normally \ v is a slowly varying function of depth. However, in regions where molecular opacities first appear, the spectral distribution of the total opacity can change dramatically with depth, and then Xv may Chapter 5. Conclusions 175 vary significantly over a single depth interval. Under these circumstances, the first order differencing used may become inaccurate, resulting in errors in the mean intensity and flux. It is also possible that rapid changes in the spectral characteristics of the opacity may be responsible for the larger variations in L/L* at depth in the 3000K models. At present this remains speculative; further testing is needed for confirmation. Another artifact affects the values of the Eddington factor fv. This is seen in Figure 4.20. At 2.0/i _ 1, fv initially decreases inward from the outer boundary as expected and approaches the diffusion limit value of 1/3, but then rises again to a plateau of about 0.38 near l o g T R j u = 0 before finally thermalizing at logrRM ~ 1. Again it is likely that the source lies with inaccuracies in the solution of the transfer equation in this, region. The problem occurs in regions where J„ is decreasing very steeply at rates up to 50% per depth point. The finite differencing is almost certainly inaccurate for such large changes over a depth interval. Since a similar finite difference equation is used to solve the transfer equation along impact rays in the formal solution, the values of Iv found here will be equally inaccurate. The result will be systematic errors in the moments Ju, Hv, and Kv calculated by the formal solution and, therefore, corresponding errors in the Eddington factor fv as well. Further study is also required to substantiate this explanation. It seems possible, and certainly desirable, to reduce these internal inconsistencies to 0(1O~3), and this will be given consideration in the immediate future. 5.2 Comparison with Other Models A comparison of our models with the plane-parallel (PP) models of Brown et al. [23], the PP and spherically symmetric (SS) models of Scholz and Tsuji [89], and the PP and SS models of Schmid-Burgk et al. [87] is presented. The PP models of Brown et al. were calculated using a modified version of ATLAS [64] incorporating an opacity-sampled treatment of molecular bands and convective energy transport using the local Chapter 5. Conclusions 176 mixing length theory [64]. The models of Scholz and Tsuji were calculated using the code of Schmid-Burgk [86]. These models were calculated assuming radiative equilibrium and employed mean opacities for the molecular bands. Schmid-Burgk et al. calculated a variety of PP and SS models which included the molecular opacities of CH, CN, CO, OH, H2O, TiO, MgH, SiH and CaH calculated assuming the JOLA approximation of Tsuji [95]. They also calculated models with only the opacity of the H2O molecule included. All of the models of Schmid-Burgk et al. were calculated assuming radiative equilibrium. We present detailed comparisons of the temperature and pressure structure of our models, the models of Brown et al., and the Scholz and Tsuji models for two cases: Tetfl log g =3000/0.0 and 3500/2.0. We also present the 3000/0.0 B0 (PP) and B3 (SS) models of Schmid-Burgk et al. , with all molecular opacities included. The detailed temperature structure of the surface layers as a function of Rosseland mean optical depth is shown for the above 3000/0.0 models in Figure 5.1. The entire atmospheric temperature structure is shown in Figure 5.2 for the same models. The temperature profile of our model has been scaled by the factor Teff/Teff = 3055.4/3000 to compensate for the drift of the model radius J?» off the specified fiducial radius Rm. The models of Schmid-Burgk et al., originally on a 1.2p optical depth scale, have been put onto a Rosseland mean scale by using our T I . ^ T R M ) relation. The corresponding pressure profiles for these models are given in Figure 5.3, where the Scholz and Tsuji models (originally calculated with log g = —0.70) have been shifted by A log p = +0.70 to approximate the gravity of our models. Consequently, the Scholz and Tsuji temperature and pressure profiles should be interpreted with caution, since this scaling procedure will not be correct for pressure dependent opacity and equation of state changes. Our SS model is not directly comparable with that of Scholz and Tsuji; their model has about twice the atmospheric extension (~ 50%) of our 3000/0.0/-3.0 model. A better Chapter 5. Conclusions 177 comparison with our model would be a temperature structure intermediate between the Scholz and Tsuji PP and SS curves. We do not obtain the pronounced drop in temper-ature near log TRM ~ —1.5 seen even in the PP Scholz and Tsuji model. The surface layers of our model are much warmer (by ~ 600K) than the profile given by the average of the PP and SS models of Scholz and Tsuji [89] which we assume represents a model with comparable atmospheric extension (25%) to our model. Furthermore, the models of Scholz and Tsuji were calculated for a lower gravity than our log g = 0 and this makes comparison more ambiguous. However, our temperature profile is quite similar in shape, but ~ 100-200K cooler in the surface layers than the model of Brown et al. Our tem-perature profile is also close to that of the B3 SS model of Schmid-Burgk et al., a model which has a very similar atmospheric extension to our 3000/0.0/-3.0 model. The reduc-tion in the temperature gradient at the very bottom of the atmosphere in the Brown et al. model is due to the onset of convection. The pressure structure of our model is in general agreement with the Brown et al. model, and also in reasonable agreement with the PP B3 model of Schmid-Burgk et al.. However, our model pressure is about 0.5 dex lower than these around logrRM ~ —1.5. The steep drop in pressure above l o g T R M ~ —1.5 notable in the Scholz and Tsuji models does not occur in our model. Again, the interpretation of these Scholz and Tsuji models is difficult because they were calculated for a considerably lower gravity. Analogous figures are presented for the 3500/2.0 models. Figure 5.4 shows the detailed temperature structure of the surface layers for our SS model, the PP model of Brown et al., and the models of Scholz and Tsuji (which were calculated for the substantially lower gravity, log g = —0.43). Schmid-Burgk et al. did not calculate models for a temperature of 3500K. As might be expected, the more compact 3500/2.0 models show less variation in temperature structure, with the Brown et al. and the Scholz and Tsuji PP and SS models deviating from each other by < 100K in the outer atmosphere. The temperature Chapter 5. Conclusions 178 structure of the entire atmosphere is shown for the 3500/2.0 models in Figure 5.5. Our 3500/2.0 SS model has a temperature profile similar in shape to the other models, but consistently warmer by ~ 150K throughout the outer atmosphere ( logr^ > 0). This is probably due to the omission of CO opacity in our model. The CO molecule is the dominant coolant of the surface layers in this temperature regime. Finally, the pressure profiles of our model and the PP model of Brown et al. are shown in Figure 5.6 and are seen to be in reasonable agreement. The models of Scholz and Tsuji were not included in this figure since these were calculated for the much lower gravity of log g = —0.43. 5.3 Future Research 5.3.1 Short Term Improvements One of our first priorities will be the production and publication of a grid of spheri-cally symmetric models. These will probably be pure hydrogen models including only a few continuous opacities (e.g. H~ and neutral H bound-free and free-free absorption, H Rayleigh scattering, and Thomson scattering). The intent is to provide a series of easily reproduced standard models that will facilitate the comparison of different model techniques. , Another area of immediate concern is the equation of state calculation. The solution procedure described in Bennett [17] works well, but it has been implemented using Tsuji's [97] polynomial fits to molecular equilibrium constants. Some of this data is now quite dated, and Tsuji's method does not provide for easy updating. A better approach is given by Irwin [50] who provides polynomial fits to the logarithm of the partition functions as a function of temperature. The temperature variation of the equilibrium constants can then be expressed as a polynomial with coefficients that are explicit algebraic functions of the partition function polynomial coefficients and of the dissociation energy of the molecule. This is a very useful arrangement since it is generally the dissociation energies Chapter 5. Conclusions 179 which are the most poorly determined quantities and, therefore, the most subject to change. The representation of the opacities is another limitation of the current models. We need to include more sources of opacity and to improve our treatment of bound-bound molecular absorption in the solution of the radiative transfer. In particular, the vibra-tional bands of the CO molecule, the atomic lines in the blue and ultraviolet, the CN violet system, the CH G band, MgH bands, and a treatment of VO should be considered before the models can begin to realistically represent emergent fluxes and broad band colours. Straight mean opacities overestimate band strengths. Accurate modelling requires a more realistic treatment of molecular opacities. We anticipate the construction of models using opacity distribution functions (ODFs) in the near future. The computation of such models will require the use of several hundred frequency points, several times the number used in the current models. Since the computational effort scales linearly with the number of frequencies, the computing time for these models will be increased accordingly, probably to around 20 hours of IBM 3081 time per converged model. This estimate is probably pessimistic. Economy can be gained by initial computation of the equation of state and ODF grids once over the entire (T,p) parameter space, for each choice of model abundance and microturbulent velocity. The number of depth points can be reduced by decreasing the depth of the inner boundary ( r R M = 100 in the current models), and perhaps by increasing the optical depth grid spacing in the surface layers where most frequencies are optically thin. Any such reduction yields significant computational.savings since the model timing scales as the cube of the number of depths. Most of the time taken for calculation of the current models is spent not in the linearization solution (which takes only ~ 2 times as long to compute as comparable plane-parallel models) but in the formal solution. This step has a timing ~ 5-10 times Chapter 5. Conclusions 180 worse than the plane-parallel case due to the much larger number (about 45 for spherical symmetry compared to 6-8 for plane-parallel models) of angles, or impact parameters, needed to accurately evaluate the radiation field moment integrals. The formal solution procedure can be economized as well. The sole purpose of this step is to provide a current value of the Eddington factor /„ to close the system of moment equations. In the initial stages of the linearization solution, the dependent variables T, p, r, and J„ are all poorly determined, and it is almost certainly not worthwhile to engage in substantial computational effort merely to compute 'accurate' values of fv, as is presently done. A much faster formal solution, on a coarser grid, yielding a rougher estimate of /„ should suffice until the final convergence of the model equations is at hand. Only then need fv be calculated using the final depth-angle grids. This procedure alone could probably halve the formal solution time. Some work remains to be done on improving the consistency of the numerical results obtained by the formal solution as compared to the moment transfer equations. The present scheme of using the formal solution moment quadratures to evaluate Jv at depth and Hv at the surface, and obtaining the other moment by differentiation or integration using moment equation 2.8 works adequately but is not fully satisfactory. The conver-gence of the formal solution/moment equation loop is slowed by consistency problems in these moments in the outer layers of the atmosphere. One approach that will be ex-amined is to replace the numerical formal solution by an analytic solution obtained by series expansion at depth-angle points for which T „ < C 1 , Since much of the impact pa-rameter grid is optically thin at many frequencies, this could also result in a substantial improvement in timing. Altogether, the above steps could yield a factor ~ 5 improvement in overall com-putational speed. This' would permit spherically symmetric models using ODFs to be computed with timings comparable to our present preliminary models, each of which Chapter 5. Conclusions 181 required about 3 hours of CPU time on an IBM 3081 for full convergence. Another improvement that will be considered in the near future is the incorporation of convective energy transport in the models in a fully consistent manner. This will require the generalization of the equation of radiative equilibrium to include a convective flux gradient in addition to the radiative terms. It should then be possible to linearize the resulting equation of energy balance and proceed directly with the complete linearization as in the present radiative case. One difference is that the new energy balance equation would produce a tridiagonal matrix when linearized, instead of the present diagonal matrix obtained from the radiative equilibrium equation, due to the presence of gradients requiring a 3-point finite difference representation. 5.3.2 Non-LTE Models In the long term, an area of considerable interest is the study of non-LTE effects in extended stellar atmospheres. The effect of atmospheric extension might be expected to amplify departures from LTE, as compared to plane-parallel models, due to the increased amount of stellar material in a low density; radiation-dominated environment. Non-LTE effects may affect the ionization equilibrium of the outer atmosphere [11], and possibly also the molecular equilibrium, as well as the level populations. The necessary equations of statistical equilibrium, and rate equations for determining the molecular abundances, can be readily incorporated into the present linearization scheme. However, two major modifications must be made to implement a non-LTE code. First, changes to the equation of state routines would be required to allow for de-partures of the ionization equilibrium from the LTE Saha distribution. Implementation of this should be routine. More substantial effort would be required if departures from molecular equilibrium, e.g. due to non-LTE photodissociation, were to be included. This would entail the addition of a reaction network and rate equations in order to determine Chapter 5. Conclusions 182 molecular abundances, in addition to the usual statistical equilibrium equations. Second, the Rybicki formulation of the model solution presently used is not suitable for non-LTE work, since the computing time scales as the number of constraint equations. This number is already large even for the relatively simple case of a several multi-level atom non-LTE problem. The problem becomes even larger if reaction rate equations are included, and especially if molecular levels are considered in non-LTE. The direct approaches then become completely infeasible. However, the method of Anderson [4] used to solve a large (~ 100 constraints) non-LTE model of early-type atmospheres appears applicable to late-type atmospheres as well. This method groups transitions with similar physics together into blocks, recognizing that the atmospheric structure depends upon the integral properties of the radiation field and not upon the details of the fine structure (an approach analogous to the replacement of the detailed structure of bound-bound molecular transitions with opacity distribution functions). Because of the difficulties, little work has been done towards non-LTE modelling of late-type atmospheres since the pioneering study by Auman and Woodrow [11]. It is hoped our models can provide a base for future non-LTE studies of late-type atmospheres, and particularly of giant stars with extended atmospheres. However, considerable effort will be required before the state of maturity present in the modelling of early-type at-mospheres is attained. 5.3.3 Self-Consistent Atmosphere and Wind Models The author's main interest lies in the study of atmospheric dynamics. The generalization of the current models to encompass a steady-state, spherically symmetric outflowing wind requires the addition of another fundamental variable, the wind velocity v, and the modification of the equation of hydrostatic equilibrium to account for the transfer of momentum from the radiation field to the stellar gas. Also, the equation of radiative Chapter 5. Conclusions 183 equilibrium must be generalized to describe overall energy conservation in the radiation field and the stellar material, and now must consider not just the energy density of the radiation field, but also the kinetic energy of the gas, and the flux of quantities such as the internal energy of the gas. The system of equations can be closed by the addition of the equation of continuity. It should then.be possible, assuming a radiatively driven wind, to construct combined atmosphere and wind models that include a fully consistent treatment of radiative transfer, energy balance, and momentum transfer to the stellar wind. These models will still require the use of some ad-hoc assumptions in the treatment of grain formation and growth in order to model late-type giant and supergiant winds. Nevertheless, such models should help in elucidating details of the mass loss process from evolved stars that is so ubiquitous, and yet so poorly understood. 5.3.4 Time-Dependent Dynamical Models Giant stars of spectral type later than about M6 are almost always variable. Many of these are categorized as Mira variables on the basis of the amplitude of their photometric variations. The underlying cause of the pulsation responsible for Mira variability remains unknown. However, the determination of the temperature structure of the outer atmo-sphere (rRM < 1) of the (static) spherically symmetric models in this study was very sensitive to numerical perturbations. Scholz and Tsuji [89] report a similar effect that results in a sudden temperature drop over a small optical depth range in the outer layers of their models. Muchmore [74] has proposed that radiative cooling instabilities may be present in the surface layers of K and M giant stars. All of this suggests that the outer atmospheres of late-type giants are only marginally stable at best, and may be radiatively unstable for the cooler stars. This radiative instability could be an underlying cause of Mira variability. An example of just such an instability was given by Woodrow and Auman [109], who Chapter 5. Conclusions 184 evolved time-dependent carbon rich models for several thousand time steps. These models employed a simplified grey treatment of radiative transfer coupled with time-dependent hydrodynamics and grain formation. Woodrow and Auman's 2400K model, after an initial perturbation, eventually relaxed to a periodic solution. In this model the pulsation resulted from a radiative instability driven by the extreme temperature sensitivity of the opacity of the surface layers (due to the condensation of graphite grains). While the condensation of grains in the atmosphere is now considered unlikely for late-type giants, a similar but smaller increase of opacity occurs in the surface layers of these stars due to the formation of polyatomic molecules (mainly H2O in M stars) for Teff < 3000K. It is now feasible to contemplate the development of time-dependent model atmo-spheres of Mira variables, incorporating spherically symmetric hydrodynamics and the complete radiative transfer problem. Realistic time-dependent models are needed to definitively address the importance of the opacity-driven radiative instability hypothesis proposed above. These models should evolve the (one-dimensional) hydrodynamics of an extended, spherically symmetric LTE atmosphere consistent with an accurate treatment of the radiative transfer; The time-independent transfer equation would suffice for this purpose since the thermal and dynamical time scales of fluid elements in the atmosphere are much greater than the fight transit times. However, the presence of an atmospheric velocity field complicates the transfer problem since frequency derivative terms must now be retained in the transfer equations to account for Doppler shifted absorption and emission [70]. Handling this Doppler shifted radiative transfer represents a challenging problem. The time evolution of such models would require considerable computational effort but should be feasible with present-day computing capability. These models would represent the first self-consistent treatment of hydrodynamics and radiative transfer for a late-type stellar atmosphere, and would almost certainly provide fundamental insight into the nature of Mira variability. Chapter 5. Conclusions 185 Figure Captions Figure 5.1. Temperature profiles of the surface layers for various 3000/0.0 models. (PP= plane-parallel, SS= spherically symmetric). Figure 5.2. Temperature profiles for various 3000/0.0 models. Figure 5.3. Pressure profiles for various 3000/0.0 models. The models of Scholz and Tsuji were calculated for logg = —0.70 and have been shifted by Alogp = +0.70. (PP= plane-parallel, SS= spherically symmetric). Figure 5.4. Temperature profiles of the surface layers for various 3500/2.0 models. (PP= plane-parallel, SS= spherically symmetric). Figure 5.5. Temperature profiles for various 3500/2.0 models. Figure 5.6. Pressure profiles for various 3500/2.0 models. 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[104] Wehrse, R. 1981, M.N.R.A.S. 195, 553. [105] White, N.M. and Wing, R.F. 1978 Ap. J. 222, 209. [106] Whitelock, P.A., Pottasch, SR. and Feast, M.W. 1987, in Late Stages of Stellar Evolution, ed. S. Kwok and S.R. Pottasch, D. Reidel, Dordrecht, p.269. [107] Wing, R.F. 1971, in Proceedings of the Conference on Late-Type Stars, ed. G.W. Lockwood and H.M. Dyck (Kitt Peak National Observatory Contribution 554), p.145. References 198 [108] Wing, R.F. 1981, in Physical Processes in Red Giants, ed. I. Iben and A. Renzini, D. Reidel, Dordrecht, p.41. [109] Woodrow, J.E.J, and Auman, J.R. 1982, Ap. J., 257, 247. Appendix A The Linearized Model Equations A . l The Model Equations The spherically symmetric model atmosphere equations are summarised below. These equations have been put into the form f(x) = 0 (A.l) of section 3.3. A. 1.1 Transfer Equation fd+iqd+i 1 .(<ldXd + qd+iXd+i)&Td 24.qd+1 (qdXd + qd+iXd+i)Ard (^t1) J d + 1 - fdqd • - 1 -•• ,]r..'^ .{<ldXd + ?d+iXd+i)Ard (qd_xXd-\ + gdXd)Ard_iJ \RJ - i^~^d^d + 1d+iXd+i)A.Td + (qd.lXd-i + ^dX^Ard.i] {-) Jd , [ fd-iqd-i 1 &f-i/ - . - s A 1 frd-i\2 T + 7 ; ; -^TT- " - 777 Ud-iXd-i + 9dX<j)Ar<i_1 I — - ) J d _i \qd-iXd-\ + qdXd)&Td-i 24gd_i J \ R I + 777 KQdXd + qd+iXd+i)&Td I ——- j Bd+1 24 qd+\ V i t / 1 £d / \ 2 • + 12 g^**** + 9d+1^:<i+1)Ar<i + + 9dXd)Ard_!] —^J £d + ^ —(gd-xXd-i + 9dXd)Ard_1 f ^ ) 2 5d-i = 0 (A.2) 24 <7d_i V K / 199 Appendix A. The Linearized Model Equations A.1.2 Transfer Equation: Inner Boundary Condition fD-iqD-l 1 ' (D-1, . . - x A 1 (rD-l \ 2 t 7 : ; ~ TT {QD-IXD-I + qDXD)ATD^ ( — 5 - ) JD {qD-lXD-l + qDXDj&TD..! 24 (Jrj-l J V -R / fDqD . 1 - , - u I frD\2 T + ^ — (iD-lXD-l + qDXD)ATD_x —^j JD (qD-iXD-i + , 9 D X D ) A T £ , _ I 12? c l fr-24 \ K / L* /dB\ + 128*O-R2TDXD \dT)D~ A.1.3 Transfer Equation: Outer Boundary Condition /2?2 1 6 / - . - u 1 / f 2 \ 2 , - , r r r — ~( fcXi + ftXajAr! I — ) J 2 . ( 9 i X i + 92X2)Ari 24q2 J \-R/ . ( 9 i X i + 9 2 X 2 ) A T I 12 gi 2 J V i i / + ^ | t e + ^ z ) A T l ( i ) 2 ^ + ^ | t o + 92X2)Ar 1 Q)2S: + I [ l_exp(-r 1 6xl)][GH(^ + 5 1 + G ^ , ) ] Q ) V = 0 where the extension parameter d' is JI _ D _ H 1 + A n(l + A)r Ti is the density scale height, and 1 K l \dpj y A . 1.4 Radiative Equilibrium JV ^2/WndindX.ndJnd n = 1 - 1 = 0. JV ^ WndtndXndBnd Appendix A. The Linearized Model Equations 201 A. 1.5 Hydrostatic Equilibrium 2 where and Pd - Pd-i -V+(-V Td-iJ \ r d J gAma-x - ^ A T , ^ ) = 0 (A.8) Amd-i = + -) A r w - ^ - *) Ar^ (A.9) 2 VXd-i Xd/ 12 VXd-i XdJ R 9 dpf GM. R? dr ATTCR2 ' (A.10) (A.ll) (A.12) and Pd \drjl Y,= XRM,d dx* di (A.13) A.1.6 Hydrostatic Equilibrium: Outer Boundary Condition Pi ~ (1 + A) gn (R Xi \ri L*-yxi AircGMt(l + A) (A.14) A.1.7 Radius or Depth Equation 1 /fid-iPd-iXd-i , P-dPdXd 2 V Td^ + Td ^ (rd-i - rd) _\_( fid-iPd-iXd-iSd-i fidPdXdHd Pd-iPd-iXd-iYd-i 12 Td-i , kTl, + ~ rdf ~ kAr^ = 0. (A.15) A.1.8 Radius Equation: Outer Boundary Condition r-i - R = 0 (A.16) where R = and the factor £ is defined such that r(r 0) = i?* Appendix A. The Linearized Model Equations 202 A.2 The Linearized Equations The model equations are linearized in terms of the independent variables T, p, r, and jv = (r/R)2Jv. As shown in section 3.3, the resulting linear system is of the block matrix form / T, n . . . n TT, v , s , \ / Ajx \ / K i \ \ t 0 •• 0 Si 0 T 2 • •• 0 u 2 v 2 s 2 0 0 •• T N u N v N SN W i w 2 ..' .'. W N c D E 0 0 . . 0 A B F 0 o •• 0 Q H G V Aj 2 AT Ap Ar K 2 K N M L \ P J (A.17) Here, and throughout this Appendix, it will be assumed that all references to the correc-tions A j ' i , . . . , AJN, AT, Ap, and Ar refer to the scaled variables as defined in equations (3.179-3.182) and similarly, all linear coefficients are assumed to have been scaled as per equation (3.183). A.2.1 Transfer Equation The linearized equation of radiative transfer for frequency n, at depth d in the atmo-sphere, is given by the <i-th row of the matrix equation T n A j n + U n A T + V n Ap + S nAr = K n . (A.18) Since T n , U„, V n , and Sn are all tridiagonal matrices (of order D, the number of depths) this can be expanded into the scalar equation Tn,d,d-\ Aj„i(f_i + Tn>ddAjnd + Tntd,d+l Ajnid+1 + Un,d,d-\ Ar<i_i + UniddATd + Unidid+iATd+i + Vnidtd-1Apd_1 + VniddApd + Vnidid+1Apd+1 (A.19) + -Sn^d-iAr^-i + Sn>ddArd + Sn>d>d+iArd+1 = Knd Appendix A. The Linearized Model Equations 203 These coefficients have the values, for 2 < d .< D — 1, Ol2 Tn,d,d-1 Tn>dd Tn,d,d+1 Un,d,d-1 Un>dd Unid>d+1 where a n ai2 = a13 Jn,d+1 — (8B\-0-22 (i)/a32(I)/a42(§)JTrf a23(i)d+1+a33(i)d+1+a43(i)d+1J Td-x Ki,d,d-i = n^,dd — Ki,d,d+1 = <Sn,d,d-l = Sn,dd — Sn,d,d+1 = Td+1 Pd-i Pd Pd+i a2\dp)d+i a33\dPJd+i. i r<i_i 5^1 + Ci «52 + Cl -&nd = J>, » • a-t-i \ * • a-t-u JJ^)+cJW±A]rd , 1\drd_1) 4 V^d-i/J ^ 1 , / 93d \ , f d a d + 1 \ ass + c2 1 -r I + c 3 rd+1 \ord+1J V ^ + i / J ' ^ d - i S fd-iqd-i €d-i , _ - \ A ' [ ^ T A 0 , (gd-iXd-i + gdXdjArd-! (qd-iXd-i + ?dXd)Ar<i_1 24gd_i /d+l?d+l £d+l / - , - \ * 7—=—; V A WA VldXd + qd+iXd+i)Ard [<ldXd + qd+iXd+i)&Td 24.qd+1K • 1 -fdqd _ A . 12% + A.20 A.21 A.22 A.23 A.24 A.25 A.26 A.27 ;A.28 A.29 r-d(A,30 A.31 A.32 A.33 A.34 (?d-i%d-i + 9dXd)Ard_1 (%xd + gd+iXd+OAr^ [ (gd-iXd-i + qdXd)^-! + (qdxd + gd+iXd+i)Ard , (A.35) Appendix A. The Linearized Model Equations 204 2^1 a22 a 4 1 «43 «42 qd-1 fdqd(r<i/R)2Jnd - fd-iqd-i{rd-i/R)2Jn,d-i (qd-iXd-i + qdXd)2&Td-i id «23 = qd+1 24,tl^-1 (!X) " •Bn'''",) fdqd{rd/R)2Jnd — fd+iqd+i(rd+i/R)2Jn,d+i (qdXd + qd+iXd+i)2&Td &Td (Jri,d+1 - Bn,d+l) 24<fc 4-1 R 12qd A r d Q ) 2 ( J n d - 5 n d ) qd fdqd{rd/R)2Jnd - fd+iqd+i(rd+i/R)2Jn,d+i + (qdXd + qd+iXd+i)2ATd fdqdirdlR)2 Jnd - fd-iqd-i(rd-i/R)2Jn,d-i (qd-iXd-i + qdX.d)2ATd-i d+l ^ — Ar d_x ( " ^ T ) (Jn,d-1 — Bn>d-l) 2Aqd. id 12qd (Ar d _ a + Ard) (^) ' (Jnd - 5 n d ) «31 = «33 = «32 = d (9dXd + qd+lXd+l) ( - ^ ) (^n,d+l - #n,d+l) 2 4 % + 1 1 12?d (gd-iXd-i + qdXd)Ard-i + + qd+iXd+i)Ard (A.36) (A.37) (A.38) (A.39) (A.40) id 2Aqd M<fc-iX«£-i + qdXd)^rd-x ( - ^ p ) ^ ( ^ d + qd^Xd^Ard^—) ^ (?d-iXd-i + gdXd)ATj_i + (g^d + qd+iXd+i)ATd ( ( ^ ) 2 ( J „ d - 5 n ( i ) . (A.41) (A.42) (A.43) (A.44) Appendix A. The Linearized Model Equations 205 <*51 «52 C 3 C 4 u + (A.45) (A.46) ci = ~Xd +tqd [ + **Xd)Ar<*-i + (?dXd + gd+iXd+i)Ard] 5nd(A.47) (^) (gd-lXd-1 + ?dXd)2 Ar d _ i +^i!d-iAr<i_i(Jn id_i — 5n>d_i) c2 = O H . /d+lgd + l«^n,d+l (QdXd + qd+iXd+i)2 A T * +^ +^1 Ar<i(Jn)<i+i — 5 n id +i) nd •R' L(9dXd + gd+iXd+i)2Ard 1 12 £dATd(J n d — -B„d) +Xd-i g rdy fdqp, nd (qd-iXd-i + gdXd)2Ard_1 — — £d&-Td-l{Jnd — Bnd) b = fdqd(rd/R)2Jrui - fd+iqd+i(rd+i/R)2 Jn,d+1 + (?dXd + qd+iXd+i)A>Td fdqd(rd/R)2Jnd - fd-iqd-i(rd-i/R)2Jn,d-i (qd-iXd-i + gdXd)Ard_1 + 24*qd+l (qd*d + 9d+iXd+i)^rd (^ L) (Jn,d+i ~ Bn,d+i) + 2^^(qd-iXd-i + gdXd)Ard_! ( ^ r ) {Jn,d-i - Bn,d-i) + TTT- \ (<ld-iXd-i + 9dXd)Ar<i_1 + (qdJCd + %+iXd+i)Ard I2qd L (A.48) (A.49) (A.50) (A.51) x ( ^ ) V n d--M. (A.52) The ratios a and 8 are defined (A.53) Appendix A. The Linearized Model Equations 206 a 9 d - i qd+i qd and their derivatives with respect to radius are - l d (A.54) dad qd 2 rd 5rd_i qd-i rd-i 2 dpd-i qd-\ " 2 rd drd-x qd rd-i 2 dad qd '2 rd-i drd qd-i rd 2 dBd qd "2 rd+i drd qd+i rd 2 dad+1 ! qd+i 2 rd+i 9rd qd rd 2 dpd-i , qd-i '2 rd-i drd i qd Td 2 d(3d I q d ' 2 rd drd+1 qd+i .rd+l 2 dad+i qd+i 2 rd drd+1 qd Jd+i 2 fdr2d + fd-ir2d_y nfd-i 3 /d - i - i \ '3fd fdr2d 1 + 3fd-i fd+ir2d + fd-ir2^ 3 / r f - l d+l fdrl Zfd+1 - 1 Zfd - 1 + U d+ird+1 + fdrl fdT2d fd-ir2^ Jd+l'd+l Zfd+i - 1 + Zfd fd+ird+1 fdrd A.2.2 Transfer Equation: Outer Boundary Condition (A.55) (A.56) (A.57) (A.58) (A.59) (A.60) (A.61) (A.62) The linearized outer boundary condition of the transfer equation for frequency n is given by the first row of the matrix equation T n A j n + U n A T + V n A p + S n A r = K n . Expanding these tridiagonal matrices yields, for d = 1, TniiAj n l + T n l 2 A j n 2 + tfnuATi + tf„i2Ar2 + VnllAPl + Vnl2Ap2 + SnuAn + S n l 2 A r 2 = Knl (A.63) (A.64) where Tnn — d\2 {^^j Jnl (A.65) Appendix A. The Linearized Model Equations 207 Tn12 — &13 «^ n2 unll a 2 2 ( j r ) / " 8 1 a23 ( ^ ) 2 + «33 (§i)1 + 042 (If), OB \ 6T)1 . I + 4( dB\ ' T2 where 0-12 (?iXi + 92X2)Ari 12gi + q 2 X 2 ^ T l j~2 Q2 £2 °18 = 7—=—T ^ T A ^ — ( 9 l X l + 92X2)AT! ( ? I X I + 92X2JAT1 24q2 \fiqi(ri/R)2Jni ~ f2q2(r2/R)2Jn2 &22 — 9 i -• 7—;—: \2\ 2^3 + = 92 1 1 I" q X2)L±T\ ^ 9 2 ' fiqi(ri/ )2J i - /2g2(r2y (91X1 + 92X2 ) 2 Ar! |ri <jiexp(-T 16xi) G H ' ^ J - ^ X ' fiqi(ri/R)2Jnl - f2q2(r2/R)2Jn2 ( 9 i X i + 92X2 .) 2Ari 6 . /r 2 \» 5n2) 'nl (A.66) (A.67) . (A.68) (A.69) (A.70) (A.71) (A.72) (A.73) (A.74) (A.75) (A.76) Bni (A.77) Appendix A. The Linearized Model Equations 208 Ct32 «33 «42 C 3 A T ! 12?; i /"^lN -(01X1 + 92X2) (^ J - Bnl) + \nXi exp(-r i c;ixi) A n G H 1 + A + gtGj 1 + A 2 4 ? 2 ( ? l X l + ?2X2) Q ) ( ^ n 2-5 n 2) ^ ( 9 1 X 1 + 92X2) A n g ) 2 d 12q + i [ l - e x p ( - n 6 x i ) ] TH 1 + A + 9tGj 1 + A a4 3 = F24 t o + ? 2^ ) A r iG)' © 2 ^ ( A . 7 8 (A.79 (A.81 «52 = + ^ - ( 9 1 X 1 + 9 2 X 2 ) A n Bnl + (1 - [ttr ( T ^ ) + G , ( ^ ) ] ( £ ) BM (A.82 ^ (?lXl + 9 2 X 2 ) ^ 1 ( ^ ) #n2 <*53 = C 2 -12?2 "Xi ,iE2, hq\Jn2 1 . ( 9 i X i + 9 2 X 2 ) 2 A n 24 + — 6 A n ( J n 2 - £ n 2 ) nl (gixi + 9 2 X 2 ) 2 A n 12 - — 6 A T i ( J n l - Bnl) Co -1 - exp(-r i c;ixi) 2(1 +A) 1 - exp(-n6xi) ( d (A.83 (A.8'4 (A.85 (A.86 2(1 +A) + ff-(m)+^(iTA)lQ)! nl (A.87) b = fMri/R)2Jnl - f2q2(v2/R)2Jn2 (91X1 + 92X2) A n + ^ f 2 ^ 1 * 1 + 92X2)An (Jn2 - 5n2) + i^;(?ixi + 92X2)An g ) 2 (Jm - JB„i) + K g ) ^ . - |[1 - exp(-nc;iXi)] G H d 1 + A g ) 2 B l ( A . 8 8 ) Appendix A. The Linearized Model Equations 209 where the quantities Gj and G'H refer to the first derivatives of the respective functions. A.2.3 Transfer Equation: Inner Boundary Condition The linearized inner boundary condition of the transfer equation for frequency n is given by the D-th row of the matrix equatii— Expanding T n A j n + U n A T + V n A p + S n A r = K n . i for row d = D where these tridiagonal matrices gives,TntD,D-l Ajn,D-l + Tn, DD&jnD * m . ,r ) A T d (A.89) Tn,D,D-Tn<DD i,D,D-l Un,DD Vn,D,D-l Vn,DD Sn,D,D-\ Sn,DD KnD + U„tD,D-lATD-i + UntDD + K.JO .D-lApD-l + VniDDApD + Sn,D,D-lArD-l + Sn>DDArD = KnD (A.90) - \ + °-(§) D + ^ ( § ) D +082 TD. •fdx\ , (dt\ I j(|)D+a32(|)J^ L + C l fcj+*fcj]ri (d2B\ 2{dT~2)D + a 9 \ 0 2 2 0 5 1 + C l Q \orD-x «52 + ^ -r \drD drD-(A.91) (A.92) .1 (A.93) D (A.94) (A.95) (A.96) (A.97) (A.98) (A.99) Appendix A. The Linearized Model Equations 210 where Oil a 1 2 = fp-iqp-i CD-0,22 a 4 i «42 «21 = 9D-1 (?i?-iXr>-i + gJDXr»)Ar£,_i 24gp_i fpqp CD (qD-iXD-i + qpX.p)&TP-i 12qD fpqp(rp/R)2JnP - /D-l?D-l(^£)-l/' Rf Jn,D-l —(qD-iXD-i + g z ? X D ) A T r , _ i (A.100) (qD-iXD-i + qDXD)ArD_t (A.101) (qD-lXD-1 + qDXD)2^TD-\ ^ D — ATD_I ( j ^ 1 ) (^",0-l — Bn,D-\) CD / r„\2 24 ? Z ) 3D 1 2 , ^ - ( i ) fDqD(rp/R)2JnD ~ fp-iqD-\{rp-llR)2 Jn,D-l {qD-lXD-1 + gz>xz?)2Ar£,_i ^ — A r p - i ( ~^ L ) («7n,JD-l - -Sn.D-l) 24 ? r J 3^1 &32 = AT£)_I a 5 i = 5^2 = 8^2 9^2 = ^ S T T C T ^ T ^ X 2 , 24?i? ~ ( 9 Z J - l X £ > - 1 + ^DXD) (-^) (•'n.D-l - £n,Z>-l) ;-J^ i(gi?-iXr>-i + 9z?Xz>) (^ ) (^nu - #„D) cjD-i , . - \ A frp-i\2 I ^ ( f l D - i X i , - i + 9 D X D ) A T 1 , _ x ( ^ ) 2 ^-(?JO-IXU-I + 9z?XD)ArZj_i (^^) -Bn.u-g ( t o - lXD- l + qPX.p)^TP-\ (^) #nZ? 12?JD L, 128*<rB*T*XD 3L* (dB\ l28ir<rR?TDXp \dTJD (A.102) (A.103) (A.104) (A.105) (A.106) (A.107) (A.108) (A.109) (A.110) ( A . l l l ) Appendix A. The Linearized Model Equations 211 C l -XD fD-iqr>-lJn,D-l R ) [{qD-\XD-\ + gDX£>) 2AT£,_i + 24^r»-iAr£»_ 1(J r i | Z3_ 1 — jBni£)_i) (A.112) c4 nfl b = R' [(QD-IXD-I + qDXD)2ATD-1 — ~-t,D&TD_i(JnD — BnD) fDqD{rD/R)2JnD ~ fD-iqD-l{rD-l/R)2Jn,D-l (A.113) (qD-iXD-i + qDXD)&TD_i X * ~ ' — • (A.114) 1 fdB\ A.2.4 Radiative Equilibrium The linearized equation of radiative equilibrium appears as row JV + 1 (third from the bottom) of matrix equation (A.17), JV ^ W n A j n + C A T + D A p + E A r = M (A.115) T»=l All of the matrices W n , C, D, and E are diagonal and so the linearized radiative equi-librium constraint in scalar form is X) wn,dd&jnd + CddATd + DddAPd + EddArd = Md . (A.116) n = l Therefore, letting JV (KB)d = ^ WndtndXndBntl n=l JV (Kj)d = Y wndindXndJnd (A.117) (A.118) n = l Appendix A. The Linearized Model Equations 212 R 2 JV J WndCndXndjnd TdJ ^ - f ' n=l (A.119) the coefficients can be expressed JV C, dd («J) N ^2 Wnd ( r, — 1 L \ Snd ( "57^ J + Xnd dT nd- J, nd nd + dT 6id | ) + Xnd Wind! Bnd \ Td (A.120) Ddd = JKj)d JV £ n=l JV E n=l Wnd Wnd ^ l 5 j n d + ^ U / n d . 'nd 'nd > Pd Edd Wn>dd Md 2{KJ)d WndindXndJnd x _ M)d (A.121) (A.122) (A.123) (A.124) A.2.5 Hydrostatic Equilibrium The linearized equation of hydrostatic equilibrium appears as row N + 2 (second from the bottom) of matrix equation (A.17), A A T + B A p + F A r = L (A.125) The matrices involved here are all bidiagonal, and so the corresponding scalar form of this equation is A d | d _ 1 A T ( i _ 1 + AddATd + B ^ - i A ^ . ! + BddAPd ' + Fd^Atd-r + FddArd = Ld (A.126) Appendix A. The Linearized Model Equations 213 where and Ad,d-\ i-dd 1 "2 1 2 \Td-l, + 'IV JDJ 9 1 Td-x \rd-ij \rdJ J V dTc Bd,d-i Bdd — - 5 f R \Td-l ' R \Td-\. + Fd,d-i Fdd Ld 'R Jd d 9 I - 5 - — dpd-i ( dAm-d-i 9 V fyd • Pd-i 'Pd -it)** Td-lJ \rdy -) Td) _ 1 = Pd-l -Pd + -x gArrid-i - ) Ard-i Arrid-! = lURM,d-i+XRM,d)&Td-i ~ i2(xRM,d-i^d-i — XRM,dXd)Arl_1 RM,d V dr ) d Yd = dpr dr 9 = d Id = X ,I GMt R? U 8ircR2 XH (A.127) (A.128) (A.129) (A.130) (A.131) (A-132) (A.133) (A.134) (A.135) . (A.136) (7d-i+7d) (A.137) ) . (A.138) The quantities Yd, g, dpr/dr, and jd remain fixed during the linearization. The derivatives of Amd-\ above are dAm-d-i dTd-i 1 -2 fdXRM, Ard_i d-i 1 - ^ - x A r , . ! 0 (A.139) Appendix A. The Linearized Model Equations 214 dAm, dTd dAmd-i dpd-i dAmd_i _ 1 2 (9XHM\ 1 2 (dx*M\ 1 + -YdAr^ 0 dpd 2 1 -2 / dXRM , A i -•--y d_ 1Ar d_ 1 o l + - y d A r d _ i 6 A.2.6 Hydrostatic Equilibrium: Outer Boundary Condition On the outer boundary, the scalar equation reduces to the form A n A2\ + -BnApi + FnArx = Lr with, A1X = F11 = R 1 + JT ,(1 + A) 2.(1 +A) ( £ ) 2 n y X r m * \ dp J, Pi 2 r 9XRM,I dpT Lx = 71(1 +A) 2 r 1 + A V dr ) lm 1 (dpr\ ] and, as before, dr 9 = I 7i = i ? 2 7i 4TTCR2 XH XRM / i A.2.7 Radius Equation (A.140) (A.141) (A.142) (A.143) (A.144) (A.145) (A.146). (A.1.47) (A.148) (A.149) (A.150) The linearized radius equation appears as (the last) row iV + 3 of matrix equation (A.17), Q A T + H A p + G A r = P (A.151) Appendix A. The Linearized Model Equations 215 The matrices involved here are all bidiagonal, and so the corresponding scalar form of this equation is Q ^ - i AT d _i + QddATd + Hd^Apd^ + HddAVd + (?<*,<*_! Ar-d.! + GddArd = Pd (A.152) where Qd,d-i Qdd HD>D-\ HM Gd>d-i Gdd Pd 0-21 + O41 &22 + «42 3^1 + 041 FFL32 + &42 ^51^-1 ^52^ b Td-X V dT ) ( 0XRM\ V dp )d_x ( 9XRM\ V dp J + 7^2 dp df dp dP;d-u dp + a72 I - s -d \dp/di Pd-i Pd and p = pkT/p is the mean particle mass. Then, setting Xd = pdpdXRM,d/Td, Xd-i &21 «22 = 2Td_1 Xd (rd-i - rd) «31 «32 = a4i = «42 5^1 = 2Td Xd-i 2pd-i Xd (rd-t - rd) (rd-i - rd) 1 - - <2±t±Yd_x){rd_l _ r d ) l + ^ C ^ - ^ ^ K r - d - i - r , ) 1 - ^ ^ - i ) ^ - ! -rd) 2pd Xd-i 2XRM,d-\ xd (rd-i - rd) (rd-i - rd) l + -6(Hd 2Xn Yd)(rd^ - rd) 2A, d-i k )(?•<£-1 - rd) 2XRM,d Xd-i (rd-i - rd) 1 IX 1 + 6 ^ - -Y^d-i - rd) 1 - ^(Ht-t - ^-Yi-Mrt-i - rd) (A.153) (A.154) (A.155) (A.156) (A.157) (A.158) (A.159) (A.160) (A.161) (A.162) (A.163) (A.164) (A.165) Appendix A. The Linearized Model Equations 216 0 5 2 = a 7 i = a 7 2 = b = 2 Xd-i 2~ _ Ad 2 Ad_i 1 + ^{Hd - ^Yd)(rd-i - rd) 1 - ^(H^ - ^ - Y ^ r ^ - rd) l + liH.-^Y^r^-r,) (ra-i - ra) + 2 ^ ( r d - 1 - r d ) 1 1\ 1 + -(#d - - j -Yd )^ - rd) fcArd-! - ^~(rd-i - rd) - y (r<*-i ~ rd"> (A.166) (A.167) (A.168) (A.169) l-I(JET < l_ 1-^Xi_ 1)(r 4i-i-r (,) 1 + k(H* ~ T^X 7^ - rd) 0 K (A.170) A.2.8 Radius Equation: Outer Boundary Condition On the outer boundary, the scalar equation reduces to the form (A.171) with Q u #11 Pi = 0 - IJL R = 1-R (A.172) (A.173) (A.174) (A.1.75) Appendix B Model Output Tables , This appendix gives condensed listings of the frequency-independent or frequency-averaged quantities of interest that are calculated by the models listed in Table 4.4. A sample of the tabular output generated by the OUTPUT command in the ATHENA model atmo-sphere code is presented on the following pages. For reasons of brevity, only every second depth point has been printed and only the first three pages of output included for each model. A glossary of the variables appearing in this output follows. 217 Appendix B. Model Output Tables 218 B . l Glossary of Variables TauRM Rosseland mean optical depth Tj^. Col Mass Column mass m in units of g/cm2. r/R* - 1 Atmospheric extension r/Rt — 1, where i?» is the stellar radius. T Temperature T in degrees K. n Number density n in c m - 3 . Pe Electron pressure pe in dynes. P.gas Gas pressure p in dynes. P_rad Radiation pressure pTad in dynes. Mu Mean molecular weight p. Rho Density p in g/cm3. ChiRM Rosseland mean (total) absorption coeffcient XRM in units of cm 2/g. r Radius of atmospheric layer in cm. TE res Transfer equation rms residuals eTBrm,. RE res Radiative equilibrium residuals HE res Hydrostatic equilibrium residuals €uB. DE res Depth (radius) equation residuals eDB. EPS rms Total rms residual vector (all equations) erms-g_eff/g Ratio of (gravity - radiative acceleration)/gravity. Chi_H/RM Ratio of flux mean to Rosseland mean opacity XH/XRM-Hconv/H Ratio of convective to total flux Hcmv/H. Lum/L* Ratio of luminosity from total flux H to specified stellar luminosity L*. Cp/Cv Ratio of specific heats Cp/Cv. H_p Pressure scale height h. Qconv Convective quantity Q = — (dmp/dhiT)p. DELrad Radiative gradient V# = (d In T/d In P)R. Appendix B. Model Output Tables DELad Adiabatic gradient V 5 = (dlnT/dlnp)s. DELbub Gradient of convective bubble V B . TauRMb Rosseland mean optical thickness of convective bubble. Vconv Mean convective velocity of bubble in cm/s. Hconv Convective flux Hconv in ergs/cm2 /s. B.2 Converged Models: Output Tables Condensed output of the converged models summarized in Table 4.4 follows. Model 01310191 SS 3000./ 0.00/ -3.00 CN. TiO, H20 (sm) TauRM Col Mass r/R* - 1 T n Pe 1 1 .000E-06 7 .946E-03 1 . 928E -01 1 . 789E+03 2 .254E+10 1 .080E -08 3 1 .585E-06 1 .247E-02 1 . 846E -01 1 .797E+03 3 .531E+10 1 .649E -08 5 2 .512E-06 1 .950E-02 1 . 766E -01 1 .807E+03 5 .525E+10 2 . 539E -08 7 3 .981E-06 3 .029E-02 1 . 688E -01 1 .821E+03 8 .589E+10 3 .931E -08 g 6 .310E-06 4 .648E-02 1 .614E -01 1 .839E+03 1 .317E+11 6 .077E -08 11 1 .000E-05 7 .002E-02 1 . 542E -01 1 .861E+03 1 .981E+11 9 . 312E -08 13 1 .585E-05 1 .032E-01 1 .475E -01 1 .886E+03 2 .912E+11 1 . 407E -07 15 2 .512E-05 1 .488E-01 1 . 411E -01 1 .912E+03 4 . 180E+11 2 .090E -07 17 3 .981E-05 2 .097E-01 1 . 351E -01 1 .940E+03 5 .868E+11 3 .047E -07 19 6 . 310E.-05 2 .899E-01 1 . 294E -01 1 .967E+03 8 .074E+11 4 . 368E -07 21 1 .000E-04 3 .940E-01 1 . 239E -01 1 .995E+03 1 .092E+12 6 . 162E -07 23 1 . 585E-04 5 .279E-01 1 . 186E -01 2 .023E+03 1 .457E+12 8 . 581E -07 25 2 .512E-04 6 .993E-01 1 . 135E -01 2 .050E+03 1 .921E+12 1 . 183E -06 27 3 .981E-04 9 .185E-01 1 .085E -01 2 .078E+03 2 .512E+12 1 . . 620E -06 29 6 . 310E-04 1 . .200E+00 1 .035E -01 2 . 106E + 03 3 .268E+12 2 . 210E -06 31 1 .000E-03 1 . .566E+00 9. . 855E -02 2 . 135E + 03 4/244E+12 3. .018E -06 33 1 .585E-03 2 .050E+00 9 . 349E -02 2 . 166E+03 5 .527E+12 4. . 145E -06 35 2 .512E-03 2, .704E+00 8 . 822E -02 2 . 198E + 03 7 .253E+12 5. . 764E -06 37 3 .981E-03 3 :620E+00 8 . 262E -02 2 . 234E+03 9 .652E+12 8 . 178E -06 39 6 . 310E-03 4 .961E+00 7 .650E -02 2. .275E+03 1 .314E+13 1 . . 197E -05 41 1 .000E-02 7 .040E+00 6 .963E -02 2 . 323E+03 1 .849E+13 1 836E -05 43 1 .585E-02 1 . 051E+01 6 . 164E -02 2 . 382E + 03 2 .731E+13 3. 013E -05 45 2 .512E-02 1 . 694E+01 5. . 195E -02 2. 461E+03 4. .337E+13 5. . 488E -05 47 3 .981E-02 3. 080E+01 3. .950E -02 2. . 574E+03 7. . 715E+13 1 , . 176E -04 49 6 . 310E-02 6 . 580E+01 2. . 317E -02 2. . 725E+03 1 . .604E+14 3, .073E -04 51 1 .000E-01 1 , , 432E+02 5. .975E -03 2, . 880E+03 3. .411E+14 8. 201E -04 53 1 . .585E-01 2. . 687E + 02 -8. . 291E -03 3. 023E+03 6. . 268E+14 1 . 923E -03 55 2 .512E-01 4. . 269E + 02 -1 .904E -02 3. . 166E+03 9, .705E+14 4. 048E -03 57 3 .981E-01 5. . 869E + 02 -2, .666E -02 3, , 309E+03 1.. .295E+15 7 . . 770E -03 59 6 .310E-01 7 . . 294E + 02 -3 . 203E -02 3. . 448E + 03 1 561E+15 1 . , 367E -02 61 1 .000E+00 8. . 502E+02 -3. . 595E -02 3. 607E+03 1 , . 752E+15 2. . 367E -02 63 1 . .585E+00 9. 445E+02 -3. 876E -02 3. 814E+03 1 . 850E+15 4. 16 IE -02 65 2. .512E+00 1 . 025E+03 -4. . 108E -02 4. 094E+03 1 . 877E+15 7 . 098E -02 67 3. .981E+00 1 . 140E+03 -4. 434E -02 4. 433E+03 1 . 938E+15 1 . 066E -01 69 6 . . 310E + 00 1 . 376E+03 -5. 055E -02 4. 826E+03 2. . 170E+15 1 . 719E -01 71 1 . .000E+01 1 . 717E+03 -5. 844E -02 5. 307E+03 2. 494E+15 4 . 984E -01 73 1 . .585E+01 1 . 958E+03 -6. . 354E -02 5. 884E+03 2. 584E+15 . 2. 141E+00 75 2. 512E+01 2. 075E+03 -6. 603E -02 6. 555E+03 2. 465E+15 9 . 406E+00 77 3. 981E+01 2. 126E+03 -6. 720E -02 7 . 324E+03 2. 262E+15 3. 749E+01 79 6. .043E+01 2. 145E+03 -6. 770E -02 8. 108E+03 2. 060E+15 1 . 160E+02 81 7. 805E+01 2. 151E+03 -6. 787E -02 8. 633E+03 1 . 938E+15 2. 150E+02 83 9. 233E+01 2. 153E+03 -6. 795E -02 8. 997E+03 1 . 860E+15 3. 100E+02 85 9. 824E+01 2. 154E+03 -6. 797E -02 9. 136E+03 1 . 831E+15 3. 514E+02 87 1 . 00OE+02 2. 154E+03 -6. 798E -02 9. 176E+03 1 . 823E+15 3. 639E+02 Summary of physica l quant i t ies Page 220 P_gas P_rad Mu Rho ChiRM 5 .568E-03 8 .557E-02 1 . 289E + 00 4 .825E- 14 1 . 281E -04 1 8 .759E-03 8 .638E-02 1 . 307E+00 7 .661E- 14 1 . 304E -04 3 1 . 378E-02 8 . 719E-02 1 .325E+00 1 . 216E- 13 1 . 334E -04 5 2 . 159E-02 8 .800E-02 1 .342E+00 1 .914E- 13 1 . 390E -04 7 3 . 345E-02 8 .882E-02 1 .354E+00 2 .962E- 13 1 . 490E -04 9 5 090E-02 8 .966E-02 1 . 361E + 00 4 .477E- 13 1 .647E -04 1 1 7 .580E-02 9 .052E-02 1 . 363E + 00 6 .588E- 13 1 .877E -04 13 1 . 103E-01 9 . 141E-02 1 . 360E + 00 9 .440E- 13 2 . 196E -04 15 1 .571E-01 9 .233E-02 1 . 355E+00 1 .321E- 12 2 .622E -04 17 2 . 193E-01 9 .329E-02 1 . 349E + 00 1 .809E- 12 3 . 183E -04 19 3 .009E-01 9 .429E-02 1 . 343E + 00 2 .437E- 12 3 .906E -04 21 4 .068E-01 9 .535E-02 1 .337E+00 3 .235E- 12 4 .829E -04 23 5 . 438E-01 9 .647E-02 1 . 332E + 00 4 .247E- 12 5 . 9S5E -04 25 7 .207E-01 9 .767E-02 1 . 326E + 00 5 .531E- 12 7 . 403E -04 27 9 .502E-01 9 .897E-02 1 .320E+00 7 .165E- 12 9 .092E -04 29 1 . .251E+00 1 .004E-01 1 .315E+00 9 .269E- 12 1 . 102E -03 31 1 , .652E+00 1 020E-01 1 .310E+00 1 .202E- 1 1 1 . 309E -03 33 2 .201E+00 1 .039E-01 1 .305E+00 1 .571E- 11 1 .51 1E -03 35 2 .977E+00 1 .061E-01 1 .299E+00 2 .082E- 1 1 1 .678E -03 37 4 .126E+00 1 .088E-01 1 . 293E + 00 2 .822E- 1 1 1 . 773E -03 39 5 .929E+00 1 . 124E-01 1 . 287E+00 3 .952E- 11 1 . 757E -03 41 8. 983E+00 1 . 172E-01 1 . 280E + 00 5 .806E- 1 1 1 , . 601E -03 43 1 . 474E+01 1 . . 242E-01 1 .272E+00 9 . 160E- 11 1 . . 293E -03 45 2. . 741E + 01 1 , . 355E-01 1 . 263E + 00 1 . 617E- 10 8 . 733E -04 47 6. 035E+01 1 , . 534E-01 1 .255E+00 3, . 344E- 10 5 .459E -04 49 1 . . 356E+02 1 . . 768E-01 1 . 252E + 00 7 . 089E- 10 4. . 500E -04 51 2. 616E+02 2. 047E-01 1 . 249E + 00 1 . . 300E- 09 4. .994E -04 53 4. . 243E + 02 2. 407E-01 1 . 246E+00 2. 009E- 09 7. 036E -04 55 5. 918E+02 2. 886E-01 1 .244E+00 2. 676E- 09 1 , . 204E -03 57 7 . . 430E + 02 3. 486E-01 1 . 243E + 00 3, .221E- 09 2. . 193E -03 59 8. . 724E + 02 4 .258E-01 1 . 241E+00 3 .611E- 09 4. . 271E -03 61 9. 740E+02 5. . 369E-01 1 . 241E+00 3. 810E- 09 8. 838E -03 63 1 . 061E+03 7. 090E-01 1 . 240E + 00 3. 864E- 09 1 . 349E -02 65 1 . 186E+03 9. 687E-01 1 . 239E+00 3, 987E- 09 1 . 147E -02 67 1 . 446E+03 1 . . 365E+00 1 . 239E+00 4 . 464E- 09 9. . 166E -03 69 1 . 827E+03 2. 004E+00 1 .239E+00 5. . 129E-09 1 . 455E -02 71 2. 100E+03 3. 034E+00 1 . . 238E + 00 5. .312E- 09 4. 338E -02 73 2. 231E+03 4. 680E+00 1 . .234E+00 5. 051E- 09 1 . 523E -01 75 2. 287E+03 7 . . 295E+00 1 . ,219E+00 4. 577E- 09 5. 757E -01 77 2. 306E+03 1 . 097E+01 1 . .177E+00 4, 025E- 09 2. 044E+00 79 2. 309E+03 1 . 411E+01 1 . , 124E+00 3. 616E- 09 4. 452E+00 81 2. 310E+03 1 . 665E+01 1 , 073E+00 3. 313E- 09 7. 251E+00 83 2. 310E+03 1 . 770E+01 1 . .051E+00 3. 195E- 09 8. 603E+00 85 2. 310E+03 1 . 802E+01 1 , 044E+00 3. 160E- 09 9 . 024E+00 87 Mode 1 01310191 SS 3000./ 0.00/ -3.00 CN, TiO, H20 (sm) TauRM r TE r 1 1 000E-06 1 075E+13 4 661E 3 1 585E-06 1 067E+13 2 641E 5 2 512E-06 1 060E+13 2 934E 7 3 981E-06 1 053E+13 3 375E 9 6 310E-06 1 046E+13 3 16 1E 11 1 000E-05 1 040E+13 2 641E 13 1 585E-05 1 034E+13 1 978E 15 2 512E-05 1 028E+13 1 461E 17 3 981E-05 1 023E+13 9 985E 19 6 310E-05 1 018E+13 6 241E 21 1 000E-04 1 013E+13 4 320E 23 1 585E-04 1 008E+13 4 354E 25 2 512E-04 1 003E+13 4 118E 27 3 981E-04 9 988E+12 2 788E 29 6 310E-04 9 943E+12 2 288E 31 1 000E-03 9 899E+12 2 695E 33 1 585E-03 9 853E+12 3 680E 35 2 512E-03 9 805E+12 5 047E 37 3 981E-03 9 755E+12 8 056E 39 6 310E-03 9 700E+12 1 531E 41 1 000E-02 9 638E+12 3 534E 43 1 585E-02 9 566E+12 9 183E 45 2 512E-02 9 479E+12 2 552E 47 3 981E-02 9 366E+12 6 637E 49 6 310E :02 9 219E+12 8 227E 51 1 000E-01 9 064E+12 6 1 19E 53 1 585E-01 8 936E+12 4 901E 55 2 512E-01 8 839E+12 9 343E 57 3 981E-01 8 770E+12 9 808E 59 6 310E-01 8 722E+12 8 460E 61 1 000E+00 8 687E+12 1 1 12E 63 1 585E+00 8 661E+12 1 227E 65 2 512E+00 8 640E+12 3 852E 67 3 981E+00 8 611E+12 3 452E 69 6 310E+00 8 555E+12 1 361E 71 1 000E+01 8 484E+12 3 509E 73 1 585E+01 8 438E+12 5 526E 75 2 512E+01 8 416E+12 1 941E 77 3 981E+01 8 405E+12 3 128E 79 6 043E+01 8 401E+12 4 061E 81 7 805E+01 8 399E+12 4 356E 83 9 233E+01 8 398E+12 4 357E 85 9 824E+01 8 398E+12 3 89 TE 87 1 000E+02 8 398E+12 2 381E RE res HE res DE res 1 T 2 139E- 12 6 607E- 07 0 OOOE+00 15 1 536E- 12 -2 951E- 09 -5 209E- 1 1 15 9 239E- 13 - 1 038E- 09 - 1 745E- 1 1 15 5 093E- 13 -2 003E- 10 - 1 067E- 1 1 15 2 936E- 13 1 439E- 10 -6 763E- 12 15 1 769E- 13 2 944E- 10 -4 360E- 12 15 1 024E- 13 3 719E- 10 - 1 848E- 12 15 5 025E- 14 3 858E- 10 -2 754E- 13 16 1 526E- 14 3 288E- 10 1 292E- 12 16 -6 393E- 15 2 416E- 10 1 639E- 12 16 - 1 761E- 14 1 401E- 10 1 383E- 12 16 -2 231E- 14 8 883E- 11 1 176E- 12 16 -2 373E- 14 7 843E- 11 9 032E- 13 16 -2 274E- 14 4 682E- 11 2 529E- 13 16 -2 086E- 14 3 349E- 1 1 -6 134E- 13 16 - 1 886E- 14 4 352E- 1 1 -1 554E- 12 16 - 1 686E- 14 6 460E- 11 -2 646E- 12 16 - 1 446E- 14 9 144E- 1 1 -3 787E^ 12 16 - 1 155E- 14 1 028E- 10 -5 289E- 12 15 -7 939E- 15 1 004E- 10 -7 314E- 12 15 - 1 748E- 15 5 360E- 1 1 -1 173E- 11 15 1 224E- 14 -7 794E- 1 1 -2 207E- 11 14 5 001E- 14 -2 520E- 10 -3 257E- 11 14 1 322E- 13 8 975E- 10 1 043E- 10 14 1 417E- 13 3 346E- 09 4 527E- 10 14 6 597E- 14 -2 183E- 09 1 524E- 10 14 3 313E- 14 -3 476E- 09 1 307E- 10 14 9 164E- 15 -5 841E- 09 1 157E- 11 14 -7 650E- 14 - 1 078E- 09 -4 801E- 11 14 -7 735E- 14 3 865E- 10 -2 612E- 11 13 -3 914E- 14 8 199E- 10 -2 498E- 11 13 -1 623E- 14 1 535E- 09 4 623E- 11 14 -1 276E- 14 -6 648E- 1 1 2 101E- 11 14 -1 064E- 14 -2 320E- 09 -1 082E- 10 14 -2 661E- 15 - 1 926E- 09 -6 059E- 11 15 1 827E- 16 3 940E- 09 1 273E- 11 16 -3 839E- 16 1 331E- 10 - 1 266E- 11 15 -1 472E- 15 -9 842E- 11 - 1 149E- 11 15 -2 086E- 15 -2 599E- 11 -8 046E- 12 15 -3 373E- 15 -9 861E- 13 -2 793E- 12 15 -3 724E- 15 2 206E- 13 -4 445E- 13 15 -3 957E- 15 9 027E- 14 - 1 088E- 13 15 -3 936E- 15 2 144E- 15 -2 225E- 15 15 -4 153E- 15 1 228E- 16 -2 204E- 17 Convergence checks Page 221 EPS rms g_ef f /g Ch i_H/RM Hconv/H Lum/L* 6 329E- 08 9 971E -01 1 497E+02 0 000E+00 9 982E-01 1 2 827E- 10 9 963E -01 1 837E+02 0 000E+00 9 981E-01 3 9 945E- 11 9 954E -01 2 271E+02 0 000E+00 9 981E-01 5 1 921E- 11 9 944E -01 2 641E+02 0 000E+00 9 981E-01 7 1 380E- 11 9 936E -01 2 790E+02 0 000E+00 9 981E-01 9 2 820E- 11 9 932E -01 2 683E+02 0 000E+00 9 981E-01 1 1 3 562E- 11 9 931E -01 2 385E+02 0 000E+00 9 981E-01 13 3 695E- 11 9 933E -01 2 006E+02 0 000E+00 9 981E-01 15 3 150E- 11 9 935E -01 1 608E+02 0 000E+00 9 981E-01 17 2 315E- 11 9 939E -01 1 242E+02 0 000E+00 9 981E-01 19 1 342E- 11 9 944E -01 9 320E+01 0 000E+00 9 980E-01 21 8 509E- 12 9 949E -01 6 828E+01 0 000E+00 9 980E-01 23 7 513E- 12 9 955E -01 4 931E+01 0 000E+00 9 980E-01 25 4 485E- 12 9 960E -01 3 542E+01 0 000E+00 9 980E-01 27 3 208E- 12 9 964E -01 2 552E+01 0 000E+00 9 980E-01 29 4 171E- 12 9 969E -01 1 857E+01 0 000E+00 9 980E-01 31 6 192E- 12 9 972E -01 1 374E+01 0 OOOE+00 9 980E-01 33 8 766E- 12 9 976E -01 1 040E+01 0 000E+00 9 980E-01 35 9 864E- 12 9 979E -01 8 122E+00 0 OOOE+00 9 980E-01 37 9 640E- 12 9 982E -01 6 583E+00 0 OOOE+00 9 979E-01 39 5 255E- 12 9 985E -01 5 588E+00 0 OOOE+00 9 979E-01 41 7 759E- 12 9 988E -01 5 019E+00 0 OOOE+00 9 979E-01 43 2 434E- 11 9 990E -01 4 873E+00 0 OOOE+00 9 978E-01 45 8 654E- 11 9 993E -01 5 064E+00 0 OOOE+00 9 977E-01 47 3 234E- 10 9 996E -01 4 742E+00 0 OOOE+00 9 976E-01 49 2 096E- 10 9 998E -01 3 537E+00 0 000E+00 9 977E-01 51 3 332E- 10 9 998E -01 2 687E+00 0 OOOE+00 9 986E-01 53 5 595E- 10 9 998E -01 2 230E+00 0 OOOE+00 1 002E+00 55 1 034E- 10 9 997E -01 1 793E+00 0 OOOE+00 1 010E+00 57 3 710E- 11 9 995E -01 1 374E+00 0 OOOE+00 1 014E+00 59 7 856E- 11 9 992E -01 1 169E+00 4 154E- 11 1 013E+00 61 1 471E- 10 9 985E -01 1 105E+00 1 186E-06 1 012E+00 63 6 678E- 12 9 978E -01 1 089E+00 1 050E-05 1 010E+00 65 2 225E- 10 9 982E -01 1 002E+00 3 194E-07 1 012E+00 67 1 845E- 10 9 986E -01 9 876E-01 0 OOOE+00 1 013E+00 69 3 774E- 10 9 978E -01 9 968E-01 1 655E-06 1 009E+00 71 1 281E- 1 1 9 933E -01 1 001E+00 8 752E-03 1 006E+00 73 9 491E- 12 9 766E -01 1 004E+00 1 301E+00 1 004E+00 ' 75 2 606E- 12 9 115E -01 1 004E+00 2 453E+01 1 002E+00 77 2 837E- 13 6 866E -01 1 001E+00 3 837E+02 1 OOOE+00 79 4 773E- 14 3 179E -01 1 000E+00 6 478E+03 1 000E+00 81 1 421E- 14 - 1 108E -01 1 000E+00 8 883E+06 1 000E+00 83 3 867E- 15 -3 180E -01 1 000E+00 0 OOOE+00 1 OOOE+00 85 2 382E- 15 -3 823E -01 1 000E+00 0 OOOE+00 1 000E+00 87 Model 01310191 SS 3000./ 0.00/ -3.00 CN, TiO, H20 (sm) TauRM Cp/Cv H_p Qconv DELrad DELad 1 1 000E-06 1 .667E+00 2 687E+12 2 060E+00 6 759E-03 1 540E -01 3 1 585E-06 1 648E+00 1 743E+12 2 362E+00 1 066E-02 1 500E -01 5 2 512E-06 1 586E+00 1 150E+12 2 650E+00 1 488E-02 1 392E -01 7 3 981E-06 1 503E+00 7 824E+11 2 879E+00 2 011E-02 1 239E -01 9 6 310E-06 1 422E+00 5 568E+1 1 3 022E+00 2 554E-02 1 079E -01 11 1 000E-05 1 356E+00 4 183'E+11 3 081E+00 3 068E-02 9 399E -02 1.3 1 585E-05 1 306E+00 3 324E+11 3 073E+00 3 514E-02 8 328E -02 15 2 512E-05 1 269E+00 2 783E+11 3 017E+00 3 8B6E-02 7 567E -02 17 3 981E-05 1 243E+00 2 434E+11 2 936E+00 4 145E-02 7 056E -02 19 6 310E-05 1 224E+00 2 204E+11 2 840E+00 4 347E-02 6 735E -02 21 1 000E-04 1 210E+00 2 048E+11 2 742E+00 .4 510E-02 6 549E -02 23 1 585E-04 1 199E+00 1 942E+11 2 644E+00 4 600E-02 6 465E -02 25 2 512E-04 1 191E+00 1 869E+11 2 549E+00 4 723E-02 6 455E -02 27 3 981E-04 1 185E+00 1 818E+11 2 456E+00 4 817E-02 6 504E -02 29 6 310E-04 1 180E+00 1 783E+11 2 365E+00 4 921E-02 6 602E -02 31 1 000E-03 1 177E+00 1 760E+11 2 276E+00 5 025E-02 6 746E -02 33 1 585E-03 1 174E+00 1 745E+11 2 187E+00 5 148E-02 6 936E -02 35 2 512E-03 1 173E+00 1 737E+11 2 097E+00 5 272E-02 7 180E -02 37 3 981E-03 1 172E+00 1 735E+1 1 2 003E+00 5 434E-02 7 492E -02 39 6 310E-03 1 173E+00 1 739E+1 1 1 903E+00 5 625E-02 7 902E -02 41 1 000E-02 1 175E+00 1 749E+1 1 1 794E+00 5 896E-02 8 463E -02 43 1 585E-02 1 180E+00 1 767E+1 1 1 671E+00 6 316E-02 9 294E -02 45 2 512E-02 1 191E+00 1 795E+1 1 1 526E+00 6 868E-02 1 067E -01 47 3 981E-02 1 216E+00 1 841E+1 1 1 365E+00 7 379E-02 1 315E -01 49 6 310E-02 1 260E+00 1 894E+1 1 1 239E+00 6 970E-02 1 677E -01 51 1 000E-01 1 304E+00 1 938E+1 1 1 173E+00 6 949E-02 2 001E -01 53 1 585E-01 . 1 350E+00 1 981E+11 1 129E+00 8 351E-02 2 308E -01 55 2 512E-01 1 409E+00 2 034E+11 1 088E+00 1 145E-01 2 679E -01 57 3 981E-01 1 476E+00 2 096E+11 1 056E+00 1 561E-01 3 063E -01 59 6 310E-01 1 536E+00 2 162E+11 1 035E+00 2 243E-01 3 379E -01 61 1 OOOE+OO 1 590E+00 2 246E+11 1 021E+00 3 843E-01 3 642E -01 63 1 585E+00 1 632E+00 2 363E+11 1 012E+00 6 918E-01 3 833E -01 65 2 512E+00 1 654E+00 2 527E+11 1 008E+00 8 505E-01 3 936E -01 67 3 981E+00 1 662E+00 2 719E+11 1 004E+00 5 695E-01 3 969E -01 69 6 310E+00 1 658E+00 2 922E+11 1 002E+00 3 827E-01 3 955E -01 71 1 000E+01 1 631E+00 3 162E+11 1 004E+00 5 405E-01 3 845E -01 73 1 585E+01 1 551E+00 3 471E+11 1 014E+00 1 246E+00 3 492E -01 75 2 512E+01 1 395E+00 3 862E+11 1 055E+00 3 093E+00 2 679E -01 77 3 981E+01 1 250E+00 4 361E+11 1 195E+00 8 487E+00 1 675E -01 79 6 043E+01 1 196E+00 5 003E+11 1 547E+00 2 731E+01 1 078E -01 81 7 805E+01 1 200E+00 5 583E+11 1 950E+00 1 198E+02 8 882E -02 83 9 233E+01 1 216E+00 6 101E+11 2 299E+00 1 149E+04 8 17 1E -02 85 9 824E+01 1 223E+00 6 329E+11 2 439E+00 -1 381E+02 7 993E -02 87 1 OOOE+02 1 226E+00 6 398E+11 2 480E+00 - 1 160E+02 7 950E -02 Convective quant i t i es Page 222 DELbub TauRMb Vconv Hconv Hconv/H 0 OOOE+OO 2 657E-05 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 1 0 000E+00 2 787E-05 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 3 0 000E+00. 2 983E-05 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 5 0 000E+00 3 330E-05 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 7 0 OOOE+OO 3 930E-05 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 9 0 OOOE+OO 4 935E-05 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 11 0 000E+00 6 575E-05 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 13 0 OOOE+OO 9 229E-05 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 15 0 OOOE+OO 1 349E-04 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 17 0 OOOE+OO 2 031E-04 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 19 0 000E+00 3 120E-04 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 21 0 000E+00 4 854E-04 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 23 0 000E+00 7 601E-04 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 25 0 000E+00 1 191E-03 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 27 0 000E+00 1 859E-03 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 29 0 OOOE+OO 2 876E-03 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 31 0 OOOE+OO 4 394E-03 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 33 0 OOOE+OO 6 598E-03 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 35 0 000E+00 9 702E-03 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 37 0 000E+00 1 392E-02 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 39 0 OOOE+OO 1 943E-02 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 41 0 000E+00 2 627E-02 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 43 0 ODOE+00 3 403E-02 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 45 0 OOOE+OO 4 160E-02 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 47 0 OOOE+OO 5 532E-02 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 49 0 OOOE+OO 9 894E-02 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 51 0 OOOE+OO 2 058E-01 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 53 0 OOOE+OO 4 599E-01 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 55 0 OOOE+OO 1 081E+00 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 57 0 OOOE+OO 2 444E+00 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 59 3 843E-01 5 544E+00 2 022E+02 1 633E-02 4 154E-11 61 6 913E-01 1 273E+01 6 187E+03 4 691E+02 1 186E-06 63 8 486E-01 2 107E+01 1 287E+04 4 173E+03 1 050E-05 65 5 694E-01 1 989E+01 3 995E+03 1 278E+02 3 194E-07 67 0 OOOE+OO 1 913E+01 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 69 5 401E-01 3 776E+01 6 349E+03 6 820E+02 1 655E-06 71 1 158E+00 1 280E+02 1 062E+05 3 647E+06 8 752E-03 73 1 253E+00 4 754E+02 5 244E+05 5 451E+08 1 301E+00 75 5 573E-01 1 839E+03 1 233E+06 1 030E+10 2 453E+01 77 2 295E-01 6 587E+03 2 778E+06 1 613E+1 1 3 837E+02 79 1 716E-01 1 438E+04 6 925E+06 2 725E+12 6 478E+03 81 4 845E-01 2 345E+04 7 704E+07 3 737E+15 8 883E+06 83 0 OOOE+OO 2 783E+04 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 85 0 OOOE+OO 2 919E+04 0 OOOE+OO 0 OOOE+OO 0 OOOE+OO 87 Model 02310191 SS 3000./ 2:00/ -TauRM Col Mass r/R* 1 1 .000E-06 4. .267E-03 1 .476E 3 1 .585E-06 6. .296E-03 1 .433E 5 2 .512E-06 9, .213E-03 1 . 392E 7 3 .981E-06 1 .323E-02 1 .355E 9 6 .310E-06 1 .856E-02 1 . 320E 1 1 1 000E-05 2 .545E-02 1 . 288E 13 1 .585E-05 3. .415E-02 1 .259E 15 2 .512E-05 4. .493E-02 1 .231E 17 3 .981E-05 5 .811E-02 1 .205E 19 6 .310E-05 7 , .404E-02 1 . 180E 21 1 .000E-04 9. .319E-02 1 . 156E 23 1 .585E-04 1 . .161E-01 1 . 132E 25 2 .512E-04 1 .437E-01 1 . 109E 27 3 .981E-04 1 , .771E-01 1 .086E 29 6 .310E-04 2. .178E-01 1 .062E 31 1 .000E-03 2, .685E-01 1 .037E 33 1 .585E-03 3. .326E-01 1 .01 IE 35 2 .512E-03 4. .161E-01 9 .825E 37 3 .981E-03 5. .283E-01 9 .510E 39 6 .310E-03 6, 850E-01 9 . 155E 41 1 , .000E-02 9. .135E-01 8 .746E 43 1 . .585E-02 1 . .262E+00 8 . 266E 45 2. .512E-02 1 . 819E+00 7. .699E 47 3. .981E-02 2. .748E+00 7 .029E 49 6, .310E-02 4. .362E+00 6 . 240E 51 1 . .000E-01 7. .272E+00 5. . 317E 53 1 . .585E-01 1 . ,272E+01 4. . 246E 55 2. .512E-01 2, ,315E+01 3. .024E 57 3. .981E-01 4. .234E+01 1 . . 709E 59 6 , .310E-01 7 , .328E+01 4 . 319E 61 1 . .OOOE+00 1 . .126E+02 -6. .413E 63 1 . .585E+00 1 . ,508E+02 -1 . 423E 65 2. .512E+00 1 . .826E+02 -1 . 972E 67 3, 981E+00 2. 084E+02 -2, .379E 69 6 . 310E+00 2. .354E+02 -2. . 787E 71 1 . 000E+01 2. .768E+02 -3. . 384E 73 1 . .585E+01 3. .326E+02 -4. . 126E 75 2. ,512E+01 3. ,740E+02 -4. 651E 7 7 3, .981E+01 3. 965E+02 -4. 943E 79 6. 043E+01 4. 072E+02 -5, .091E 81 7. 805E+01 4. .111E+02 -5, . 150E 83 9 . ,233E+01 4. ,130E+02 -5. , 178E 85 9 . 824E+01 4. .135E+02 -5. . 188E 87 1 . 000E+02 4. ,137E+02 -5. . 190E 1.00 CN, TiO, H20 (sm) - 1 T n Pe 03 1 .912E+03 1 .612E+12 6 .830E -07 03 1 .909E+03 2 . 381E+12 9 . 321E -07 03 1 .913E+03 3 .478E+12 1 .278E -06 03 1 .920E+03 4 .973E+12 1 . 741E -06 03 1 .930E+03 6 .942E+12 2 . 350E -06 03 1 .943E+03 9 .459E+12 3 . 149E -06 03 1 .958E+03 1 .260E+13 4 . 191E -06 03 1 , .975E+03 1 .643E+13 5 . 547E -06 03 1 .994E+03 2 . 105E+13 7 . 306E -06 03 2 .014E+03 2 .655E+13 9 . 583E -06 03 2 .037E+03 3 . 305E+13 1 . 253E -05 03 2 .061E+03 4 .070E+13 1 .631E -05 03 2 .087E+03 4 .975E+13 2 . 118E -05 03 2 . 114E + 03 6 .051E+13 2 . 745E -05 03 2 . 143E+03 7 .344E+13 3 . 558E -05 03 2. . 174E+03 8 .921E+13 4 . 620E -05 03 2 . 207E+03 1 .089E+14 6 .029E -05 04 2. . 243E+03 1 .340E+14 7 .944E -05 04 2. . 282E+03 1 .673E+14 1 .063E -04 04 2. . 325E+03 2 .130E+14 1 .455E -04 04 2. . 372E+03 2 .783E+14 2 .055E -04 04 2. 426E+03 3. .761E+14 3 .027E -04 04 2. .488E+03 5. .286E+14 4 . 689E -04 04 2. . 560E + 03 7. ,761E+14 7 .694E -04 04 2, .645E+03 1 .193E+15 1 .343E -03 04 2. . 745E+03 1 . 916E+15 2 . 498E -03 04 2. . 865E+03 3. .212E+T5 4 .949E -03 04 3. 005E+03 5. . 574E+15 1 .038E -02 04 3. , 164E+03 9. 686E+15 2 . 251E -02 05 3. . 345E+03 1 , .586E+16 4 .954E : 02 05 3, . 552E+03 2. . 296E+16 1 .090E -01 04 3, . 769E + 03 2. .897E+16 2 . 265E -01 04 4, .016E+03 3. 293E+16 4 . 585E -01 04 4, . 322E+03 3. 492E+16 9. .074E -01 04 4. . 708E+03 3. 621E+16 1 . .651E+00 04 5. . 172E + 03 3, 876E+16 2 . 836E+00 04 5. . 722E+03 4. 210E+16 7, .212E+00 04 6 . , 361E+03 4, 259E+16 2. .752E+01 04 7 . 094E+03 4. 049E+16 1 . .097E+02 04 7. 846E+03 3. 760E+16 3. .572E+02 04 8. , 350E+03 3. 566E+16 7 . 025E+02 04 8. 701E+03 3, 438E+16 1 . 072E+03 04 8. 834E+03 3. . 391E+16 1 .247E+03 04 8. 873E+03 3. . 377E+16 1 . . 301E+03 Summary of physica l quant i t ies P_gas P_rad Mu Rho 4 .253E-01 1 .076E-01 1 . 545E+00 4. .136E- 12 6 .276E-01 1 .077E-01 1 .622E+00 6 , , 412E- 12 9 . 185E-01 1 .078E-01 1 .681E+00 9 . 709E- 12 1 . 318E + 00 1 .078E-01 1 .728E+00 1 .427E- 1 1 1 .850E+00 1 .080E-01 1 .763E+00 2. .032E- 1 1 2 .538E+00 1 .081E-01 1 .787E+00 2 .807E- 11 3 . 405E + 00 1 .083E-01 1 .801E+00 3. .768E- 1 1 4 .480E+00 1 .086E-01 1 . 808E+00 4. 932E- 1 1 5 .794E+00 1 .089E-01 1 . 807E+00 6 . .315E- 1 1 7 . 383E+00 1 .093E-01 1 .800E+00 7 . 934E- 1 1 9 .293E+00 1 .097E-01 1 . 787E + 00 9 . 807E- 1 1 1 . . 158E + 01 1 .103E-01 1 . 770E+00 1 . . 196E-10 .1 . 433E+01 1 . 109E-01 1 .749E+00 1 . . 445E- 10 1 . .766E+01 1 . 1 16E-01 1 .725E+00 1 . . 733E- 10 2 .173E+01 1 . 125E-01 1 .698E+00 2. .071E- 10 2 .678E+01 1 . 135E-01 1 .669E+00 2. 472E- 10 3. . 318E + 01 1 .147E-01 1 . 638E+00 2. 962E- 10 4. .151E+01 1 . 161E-01 1 .607E+00 3. . 576E- 10 5, .271E+01 1 .179E-01 1 .574E+00 4 , 374E- 10 6. .835E+01 1 .201E-01 1 . 543E + 00 5. 455E- 10 9. . 116E+01 1 . 229E-01 1 .511E+00 6. ,984E- 10 1 . . 260E+02 1 .266E-01 1 .481E+00 9 . 247E- 10 1 . 816E+02 1 .316E-01 1 .451E+00 1 . . 274E- 09 2. . 743E+02 1 . 384E-01 1 .422E+00 1 . 833E- 09 4. . 355E+02 1 .480E-01 1 .394E+00 2. . 761E- 09 7. . 261E + 02 1 .616E-01 1 . 366E + 00 4. . 347E- 09 1 . . 270E + 03 1 .810E-01 1 . 340E+00 7. . 146E-09 2. . 313E+03 2 .088E-01 1 . 317E + 00 1 . 219E- 08 4. . 231E+03 2 .480E-01 1 .298E+00 2. 087E- 08 7. . 324E+03 3 .050E-01 1 .280E+00 3. .370E- 08 1 . . 126E + 04 3 .882E-01 1 . 264E + 00 4. .817E- 08 1 . .507E+04 4 .989E-01 1 .253E+00 6. 028E- 08 1 . 826E+04 6 . 506E-01 1 .246E+00 6. 815E- 08 2. .084E+04 8 . 792E-01 1 . 242E + 00 7 . 204E- 08 2. .354E+04 1 . 239E + 00 1 . 240E+00 7 . 458E- 08 2. , 768E + 04 1 . 807E + 00 1 . 239E + 00 7 . 977E- 08 3. . 326E + 04 2 .711E+00 1 .239E+00 8. 662E- 08 3. , 740E + 04 4 .145E+00 1 .238E+00 8. 757E- 08 3. .966E+04 6 .420E+00 1 .236E+00 8. 308E- 08 4. 073E+04 9 .614E+00 1 .228E+00 7 . 668E- 08 4. . 1 11E + 04 1 . . 234E + 01 1 .218E+00 7 . 212E- 08 4. . 130E + 04 1 .456E+01 1 .207E+00 6. 890E- 08 4. . 135E + 04 1 .547E+01 1 .202E+00 6. 766E- 08 4, , 137E + 04 1 .574E+01 1 .200E+00 6. . 7 30E-08 Page 223 ChiRM 2 . 800E -04 1 2 .989E -04 3 3. . 375E -04 5 3 .956E -04 7 4 . 785E -04 9 5 .940E -04 1 1 7 .529E -04 13 9 . 703E -04 15 1 .266E -03 17 1 .664E -03 19 2 . 200E : 03 21 2 . 907E -03 23 3. .825E -03 25 4 .990E -03 27 6 .427E -03 29 8 . 132E -03 31 1 . .005E -02 33 1 . . 207E -02 35 1 . . 398E -02 37 1 . . 556E -02 39 1 . .656E -02 41 1 . . 684E -02 43 1 . .637E -02 45 1 . , 526E -02 47 1 . . 369E -02 49 1 . . 183E -02 51 9 . 863E -03 ' 53 8. . 190E -03 55 7 . . 377E -03 57 7 . .947E -03 59 1 , , 148E -02 61 2. 061E -02 63 4. 094E -02 65 7 , ,470E -02 67 9 . . 156E -02 69 8. . 809E -02 71 1 . . 396E -01 73 3. 771E -01 75 1 . . 155E+00 77 3. . 275E+00 79 6. 271E+00 81 9 , 639E+00 83 1 . . 130E+01 85 1 , , 183E + 01 87 to to CO Model 02310191 SS 3000./ 2.00/ -3.00 CN, TiO, H20 (sm) Convergence checks Page 224 TauRM r TE res RE res HE res DE res EPS rms g_ef f /g Ch i_H/RM Hconv/H Lum/L * 1 1 .OOOE-06 9 .024E+12 2. .317E- 10 -8. .664E- 10 -8 .070E -08 0 .OOOE+OO 7. .733E- 09 9 .9g8E-01 4 . 168E+02 0. .OOOE+OO 9 .970E-01 1 3 1 .585E-06 9 .023E+12 6. .568E- 12 -5. . 305E- 10 -3 .641E -08 -5 .300E- 12 3 .488E- 09 9 .gg8E-oi 4 .507E+02 0 .OOOE+OO 9 .970E-01. 3 5 2 .512E-06 9 .023E+12 4. 812E- 12 -3. .409E- 10 -2 . 574E -09 1 .470E- 12 2, .487E- 10 9 .998E-01 4 . 290E+02 0. .OOOE+OO 9 .970E-01 5 7 3 .981E-06 9 .023E+12 7. . 139E-12 - 1 . .946E- 10 -9 . 172E -09 1 .643E- 12 8 .787E- 10 9 .998E-01 3 .810E+02 0 .OOOE+OO 9 .970E-01 7 9 6 .310E-06 9 .022E+12 7 . .055E- 12 -5. .114E- 1 1 -1 . 881E -08 6 .050E- 13 1 .802E- 09 9 .gg8E-oi 3 .223E+02 0 .OOOE+OO 9 .970E-01 9 1 1 1 .000E-05 9 .022E+12 9. 598E- 12 1 . .047E- 10 -2 .757E -08 -1 .906E- 12 2 .641E- 09 9 .998E-01 2 .628E+02 0 .OOOE+OO 9 ,g70E-01 11 13 1 .585E-05 9 .022E+12 2. . 319E- 11 2. .516E- 10 -2 . 790E -08 -4 .720E- 12 2. .672E- 09 9 .998E-01 2 .083E+02 0 .OOOE+OO 9 .970E-01 13 15 2 .512E-05 9 .022E+12 2, .764E- 11 3 .365E- 10 -1 .467E -08 -6 .582E- 12 1 . 406E- 09 9 .9g8E-01 1 .614E+02 0 .OOOE+OO 9 .g70E-01 15 17 3 .981E-05 9 .021E+12 2. .472E- 11 3. .262E- 10 4 .980E -09 -6 .842E- 12 4 .787E- 10 9 .998E-01 1 . 230E + 02 0 OOOE+OO 9 .970E-01 17 19 6 .310E-05 9 .021E+12 1 .674E- 1 1 2 . 505E- 10 2 .901E -08 -5 .137E- 12 2. . 779E- 09 9 .998E-01 9 .253E+01 0 .OOOE+OO 9 .g69E-01 '19 21 1 .000E-04 9 .021E+12 1 . .577E- 11 1 , .653E- 10 4 .002E -08 -3 .388E- 12 3. .834E- 09 9 .998E-01 6 .885E+01 4 .638E- 1 1 9 .969E-01 21 23 1 .585E-04 9 .021E+12 1 .181E- 11 7 .752E- 1 1 4 .804E -08 -8 .026E- 13 4. .6D2E- 09 9 .998E-01 5 .097E+01 3 .522E-09 9 .969E-01 23 25 2 .512E-04 9 .021E+12 6. .363E- 12 8 .607E- 12 4 . 284E -08 8 .396E- 13 4 . 103E-09 9 .998E-01 3 . 765E+01 1 .972E-08 9 .969E-01 25 27 3 .981E-04 9 .020E+12 3. .673E- 12 -3 .243E- 1 1 3 . 100E -08 1 .652E- 12 2 .970E- 09 9 .998E-01 2 . 784E+01 5 . 729E-08 9 .969E-01 27 29 6 .310E-04 9 .020E+12 3, .553E- 12 -4. . 360E- 1 1 1 . 716E -08 1 . 732E- 12 1 . .643E- 09 9 .9g8E-01 2 .067E+01 1 . 184E-07 9 .96gE-01 2g 31 1 .000E-03 9 .020E+12 2 .376E- 12 -3 .876E- 1 1 4 . 225E -09 1 .231E- 12 4 .047E- 10 9 .998E-01 1 . 546E+01 1 .930E-07 9 .969E-01 31 33 1 .585E-03 9 .020E+12 1 . .474E- 12 -2 .836E- 1 1 -4 .722E -09 5 .366E- 13 4. .523E- 10 9 .998E-01 1 . 168E+01 2 . 575E-07 9 .969E-01 33 35 2 .512E-03 9 .019E+12 ' 1 .027E- 12 -1 .809E- 1 1 -8 .432E -09 6 .837E- 14 8, .076E- 10 9 .998E-01 8 .955E+00 2 .832E-07 9 .968E-01 35 37 3 .981E-03 9 .019E+12 5 .467E- 13 -9 .277E- 12 -6 .273E -09 2 .624E- 13 6 .008E- 10 9 .999E-01 6 .992E+00 2 .544E-07 9 .968E-01 37 39 6 .310E-03 9 .019E+12 4. .354E- 13 -2 .701E- 12 -3 .414E -09 5 .594E- 13 3 . 270E- 10 9 .99gE-01 5 . 580E+00 1 .828E-07 9 .968E-01 3g 41 1 .000E-02 9 .018E+12 3 .724E- 13 -7 .558E- 14 -6 .094E -09 -9 .111E- 14 5. 837E- 10 9 ,9ggE-01 4 . 564E+00 1 .017E-07 9 .968E-01 41 43 1 .585E-02 9 .018E+12 3. .539E- 13 -3. .606E- 13 -9 087E -09 -1 .322E- 12 8, . 704E- 10 9 .gggE-01 3. .827E+00 4 . 163E-08 9 .968E-01 43 45 2 .512E-02 9 .017E+12 3. .599E- 13 -7. .931E- 13 -7 . 106E -09 -2 .068E- 12 6. 806E- 10 9 .gggE-01 3. . 285E + 00 1 .152E-08 9 .968E-01 45 47 3 .981E-02 9 .017E+12 4. .676E- 13 -5. 894E- 13 -4. . 142E -09 -2 .693E- 12 3. 968E- 10 9 ,gg9E-oi 2. .875E+00 1 . .700E-09 9 .967E-01 47 49 6 .310E-02 9 .016E+12 6. .524E- 13 -2. 019E- 13 3. .885E -09 -5 .939E- 13 3. . 721E- 10 9 .gg9E-oi 2. . 554E + 00 3 881E-11 9 .967E-01 49 51 1 .000E-01 9. .015E+12 1 , .595E- 12 4, .790E- 13 2. 524E -08 1 . 322E- 11 2. .417E- 09 1 .OOOE+OO 2. . 297E + 00 0. .OOOE+OO 9 .966E-01 51 53 1 .585E-01 9 .014E+12 1 . .319E- 11 -9. .723E- 13 5. . 310E -08 4 .554E- 11 5. .086E- 09 1 OOOE+OO 2. .077E+00 0. OOOE+OO 9 .966E-01 53 55 2 .512E-01 9 .013E+12 6, 948E- 11 -1 , .674E- 1 1 -2. 807E -08 1 .930E- 11 2. .690E- 09 1 .OOOE+OO 1 . .857E+00 0, , OOOE + OO 9 .965E-01 55 57 3. .981E-01 9 .012E+12 1 . 537E- 10 -6, 430E- 11 -2. .054E -07 - 1 .077E- 10 1 . .967E- 08 1 .OOOE+OO 1 . . 660E+00 0, .OOOE+OO 9 ,96gE-01 57 59 6 .310E-01 9 .011E+12 2. .048E- 10 -8. 570E- 1 1 6. . 563E -08 -8. .703E- 11 6. . 289E- 09 1 .OOOE+OO 1 . ,545E+00 0. .OOOE+OO 9 .994E-01 59 61 1 .OOOE+OO 9 .010E+12 1 . .818E- 10 -4. 370E- 1 1 4. . 565E -07 7 .495E- 11 4. . 372E- 08 1 .OOOE+OO 1 . . 351E + 00 0. OOOE+OO 1 .008E+00 61 63 1 .585E+00 9 .009E+12 3. .283E- 11 -2. 919E- 13 - 1 . , 407E -07 2 .906E- 12 1 , ,348E- 08 1 .OOOE+OO 1 . . 1 12E + 00 2. . 133E-05 1 .011E+00 63 65 2. .512E+00 9 .009E+12 3, .133E- 12 1 , 457E- 12 - 1 . . 349E -08 1 .067E- 12 1 . . 292E- 09 9 ,9ggE-oi 1 . 014E+00 1 . . 379E-02 1 .010E+00 65 67 3 .981E+00 9. .008E+12 5, 467E- 12 7 . .320E- 12 - 1 . . 739E -08 -7 .047E- 12 1 . ,666E- 09 9 .gggE-01 9. 97gE-01 1 , .547E-01 1 .007E+00 67 69 6 .310E+OO 9. .008E+12 1 . 683E- 12 9 . 616E- 13 2. . 219E -08 6 .772E- 12 2. . 125E-09 g .g99E-01 9. . g47E-01 1 . 426E-01 1 .006E+00 69 71 1 . .000E+O1 9. .007E+12 1 . 093E- 13 9, 653E- 14 2 . 138E -09 -8 .277E- 13 2. .048E- 10 9 .g99E-01 9. g64E-01 3 .222E-02 1 .006E+00 71 73 1 . .585E+01 9. .007E+12 1 . 245E- 13 1 . 864E- 13 - 1 . 168E -08 - 1 . .365E- 12 1 . .119E- 09 g .998E-01 1 . OOOE+OO 3. .723E-01 1 003E+00 73 75 2. .512E+01 9. .006E+12 4. . 204E- 14 6. 917E- 14 -6. . 865E - 10 - 1 .811E- 13 6. .575E- 1 1 9 ,9g4E-01 1 . 002E+00 5. ,600E+00 1 .002E+00 75 77 3. .981E+01 9. .006E+12 1 . 554E- 14 2. 834E- 14 2. , 703E - 10 2 .792E- 14 2. 589E- 11 g. .982E-01 1 . .003E+00 3. 607E+01 1 .001E+00 77 79 6. .043E+01 9. .006E+12 6, 421E- 15 1 . 233E- 14 4. . 996E - 1 1 2 .706E- 14 4, . 786E- 12 9 ,g50E-01 1 . 001E+00 1 . 728E+02 1 . .001E+00 7g 81 7 . 805E+01 9. .006E+12 3. 619E- 15 6. 931E- 15 3. . 996E -12 5 .249E- 15 3, 828E- 13 g 904E-01 1 . OOOE+OO 4. , 796E + 02 1 . .OOOE+OO 81 83 9. .233E+01 9 .006E+12 2. 558E- 15 4. 724E- 15 7, . 645E -13 1 .506E- 15 7 . .327E- 14 9 .852E-01 1 . OOOE+OO 9 , .701E+02 1 OOOE+OO 83 85 9, .824E+01 9. .006E+12 2. , 135E-15 4. .129E- 15 5. 552E -13 3 .376E- 16 5, , 322E- 14 g .827E-01 1 . OOOE+OO 1 , , 265E + 03 1 .OOOE+OO 85 87 1 . .OOOE+02 9. .006E+12 1 . 538E- 15 4. 060E- 15 -2, , 176E -14 - 1 . .923E- 17 2. 607E- 15 9. 819E-01 9, 999E-01 1 . , 365E + 03 1 . OOOE+OO 87 225 LO CVJ CM ^ fn i n s o) > - r i P M j ^ n i n s 01 ro r o r u <\J CM ' - n i O N C D r o r o r o r o r o T - r o i n ai s r s r s t s t s r r o i n r>- cn r o i n i * ~ o> r o i n N cn ^ m i n N L O i n i o i O L O c o CD CD t o CD r-~ c o c o c o c o O) X o o O O O o o o o o y~ CO CO CO f-. 1—• r^. r~ CO CO CD •r- o O O O O O LD CVJ y~ T- r o y- o cu C\J rvj r o r o o o o O O o o o o o y~ O o o O O o o O O O o o O i — o O O O O O O O o o O o o o O o O O O > + + + + + + + + + + 1 i i t • i 1 i i • » 1 ( i i + + + + + + i • 1 1 • 1 + + + + + + + c LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU o O O O O O O O O O O CO CM CO CD s r O i n CM s r CO r o CVJ O T— O O O O O O r o CD r^. 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CM CO o i n O CD CD i n CM CM CM «* CM CD en r o LO m o CM •a- "» O) O -— i n i n CO i n o co m CO o CM O CO T - m O r o CD *— CD CD CD CD CM i n r o r o CO CO r~- CM tu 1— r o i n LO o i n i n CM *~ r ~ O r o CO •a- CM i n r o O •sf •a- i n r o o O CD r o o CD r~ *» CO •a- r o r o r o •3- 00 •r- r o r o i n CD CM •<3- y- CM r o i n CO — r o r o i n CO r o r o ^ tD CD *— CM r o ^ CD r o CM r o r o CM N . r o -3" CD h~ r ^ CJ cu O O O O O O O O O o CM r o CM CM CM CM CM CM CM CM CM CO CO CM CM O O O O O O > n O O O O O O O O O o O o O O O o o O o o O o o O O O O O O O o o o o o o o o o o O o o O c D + + + + + + + + + + i i i 1 1 • 1 1 • 1 1 • i + + + + + •f 1 1 1 1 o n LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU _ i O O O O O O O O O O >a- i n CD CD i n i n 00 00 CD CO 1— •sf O O O O O O LO O) O) CM r o CM i— O) CM •a- co CM LU O O O O O O O O O O CD CO CM T— r o CO y- r ^ i n 00 CO O CO i n *a O O O O O O T ~ CD r o «a- CM CO CM oo i n r o o o r-. 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CO CO CO CD r*- O O O O O O CM <a- i n i n i n ^ r o CM CM * - T -o CM CM r o CM CM CM CM CM CM CM r o CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM O O O O O O O O O o O O O O O o o o O O o O O O o o o o O O O o o o o o o o o o o o o o ro LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU _ i 00 r o CD i n CD O O CM CO CD O i n CM i n r-N. r o 1— O l O 1— O CM CD CD «a r o O r o y- i n i — i — co CO O s t o LU r*~ o i n CD r~ CO >- f N o •a- r» CM co r o co co i n CO o r*- CM CO CD CD r o CD r o y- co co m r o a> t— y~ r o i n CD o co Q CM o CO r~- r^. co co co CD CD CD O O <- f— r o r o •a- co O) CM i n O CD •*}• CD T - r o oo s t o i n co CD 00 CO T - r o co i n i n s t LO t o •a- *a •<a- TJ- -a- -<t •st *» i n i n i n i n i n i n i n i n i n LO CO CO h~ 1 -^ CO CD - *- r o r o r o r o r o r o r o r o CM y- r - y- _ CM r o r o CO CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM y-~ T— y— T~ ^_ O O O O y- -- y-X) O O o O O o o o o O O O o o o O O o o o o O O O O O O O o o o o o o o o o O O O O O O O to > • > > i 1 1 1 1 i i i i t 1 • i i 1 1 1 i i t i i < i 1 1 1 1 1 1 1 1 1 + + + + + + + l _ LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU _ J *— >a- 00 *— 1^ CM 1— CD 00 CO >a r^. •sr r o CM O CD 00 CO * - CM >— CO "» CD i n T- CO y- 1— co sa- r^. 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TauRM Col Mass 1 1 . OOOE-06 1 . . 317E-02 1 3 1 . , 585E-06 1 . 892E-02 1 5 2. . 512E-06 2. 736E-02 1 7 3. 981E-06 3. 977E-02 1 9 6 . . 310E-06 5. 794E-02 1 1 1. 1 . 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184E+03 -5 77 3, . 981E+01 2, . 240E+O3 -5 79 6 . 043E+01 2. 261E+03 -5 81 7 . 805E+01 2. . 267E+03 -5 83 9 . , 233E+01 2. , 270E+03 -5 85 9 . 824E+01 2. 271E+03 -5 87 1 . 000E+02 2. 271E+03 -5 0/ - 3.00 CN, TiO, H20 (hm) r/R* - 1 T n 928E -01 1 . . 788E + 03 3. .744E+10 864E -01 1 . , 799E + 03 5. .354E+10 799E -01 1 , 811E+03 7 . . 724E+10 733E -01 1 . .826E+03 1 . .122E+1 1 668E -01 1 . 842E+03 1 . 633E+11 604E -01 1 , , 860E+03 2. .371E+1 1 541E -01 1 . . 880E+03 3, 418E+11 480E -01 1 . 900E+03 4, 876E+11 421E -01 1 . . 922E+03 6. 864E+11 365E -01 1 , .943E+03 9 . 527E+1 1 31 1E -01 1 , 965E+03 1 . .304E+12 259E -01 1 . . 988E+03 1 , ,762E+12 209E -01 2, 010E+03 2, ,354E+12 161E -01 2. .033E+03 3. ,117E+12 1 13E -01 2. 057E+03 4. .100E+12 065E -01 2. 082E+03 5, ,374E+12 017E -01 2. . 108E+03 7 . 040E+12 682E -02 2. . 136E+03 9. .249E+12 174E -02 2. . 166E + 03 1 , .224E+13 637E -02 2 . 200E+03 1 . .639E+13 057E -02 2 . 239E+03 2. .236E+13 41 1E -02 2. . 285E + 03 3. .139E+13 663E -02 2, . 345E+03 4. .599E+13 740E -02 2. . 429E+03 7 . .241E+13 421E -02 2. .573E+03 T. .333E+14 480E -02 2. . 786E + 03 3. .115E+14 097E -03 2. 973E+03 6 , 570E+14 156E -03 3, . 138E+03 1 . 072E+15 297E -02 3. . 292E + 03 1 . .427E+15 816E -02 3. 432E+03 1 . .697E+15 197E -02 3. . 585E+03 1 . .893E+15 47 7E -02 3. . 783E+03 1 . 997E+15 699E -02 4. 054E+03 2, 020E+15 985E -02 4. . 391E+03 2. 052E+15 541E -02 4, , 784E+03 2. 240E+15 318E -02 5, . 262E+03 2. 551E+15 853E -02 5. , 835E + 03 2. 653E+15 1 17E -02 6 . , 502E + 03 2. 536E+15 241E -02 7 . . 265E+03 2. . 329E+15 294E -02 8. 043E+03 2. , 122E+15 312E -02 8. , 564E + 03 1 . 996E+15 321E -02 8. 926E+03 1 . 916E+15 323E -02 9. 064E+03 1 . 886E+15 324E -02 9. . 104E+03 1 . 878E+15 Summary of physica l quant i t ies Page 226 Pe P_gas P_rad Mu Rho ChiRM 1. ,694E'-08 9. . 240E-03 8. .489E-02 1 .319E+00 8. . 202E- 14 9. . 732E -05 1 2, .412E-08 1 , .330E-02 8. 552E-02 1 .333E+00 1 . . 185E-13 1 . .055E -04 3 3. 477E-08 1 . .932E-02 8 616E-02 1 .346E+00 1 , .726E- 13 1 . 137E -04 5 5, 065E-08 2 .827E-02 8 .681E-02 1 .359E+00 2 .531E- 13 1 . 226E -04 7 7 . 426E-08 4. . 152E-02 8, . 747E-02 1 .371E+00 3. .717E- 13 1 . 335E -04 9 1 . .091E-07 6. .089E-02 8 812E-02 1 .381E+00 5. 437E- 13 1 .475E -04 1 1 1 . .598E-07 8 .871E-02 8 .878E-02 1 .389E+00 7 . .884E- 13 1 .659E -04 13 2. .323E-07 1 .279E-01 8, 943E-02 1 . 395E+00 1 . .129E- 12 1 .904E -04 15 3, ,341E-07 1 . .821E-01 9. 007E-02 1 . 399E+00 1 . .594E- 12 2 . 226E -04 17 4. . 747E-07 2 .556E-01 9 073E-02 1 .401E+00 2. .217E- 12 2 .646E -04 19 6. .660E-07 3 .539E-01 9 . 139E-02 1 .403E+00 3. .037E- 12 3 . 188E -04 21 9. . 237E-07 4 .836E-01 9 .207E-02 1 .403E+00 4 .104E- 12 3 .881E -04 23 1 . 268E-06 6 .534E-01 9 .278E-02 1 .401E+00 5 .478E- 12 4 . 752E -04 25 1 729E-06 8 .750E-01 9 . 355E-02 1 .399E+00 7 .242E- 12 5 .822E -04 27 2. .346E-06 1 .165E+00 9 .439E-02 1 . 396E + 00 9 .506E- 12 7 . 11 1E -04 29 3. .178E-06 1 .545E+00 9 .532E-02 1 .392E+00 1 .242E- 1 1 8 .625E -04 31 4 .316E-06 2 .049E+00 9 .639E-02 1 . 387E + 00 1 .622E- 1 1 1 .034E -03 33 5 901E-06 2 .727E+00 9 . 764E-02 1 . 382E + 00 2 .122E- 1 1 1 . 217E -03 35 8 .167E-06 3 .660E+00 9 .914E-02 1 . 37.5E + 00 2 .793E- 1 1 1 .401E -03 37 1 .152E-05 4 .977E+00 1 .010E-01 1 .366E+00 3 .718E- 1 1 1 .561E -03 39 1 .674E-05 6 .912E+00 1 .034E-01 1 .356E+00 5 .035E- 1 1 1 .668E -03 41 2 .544E-05 9 . 903E + 00 1 .067E-01 1 .343E+00 7 .002E- 1 1 1 .679E -03 43 4 .144E-05 1 .489E+01 1 . 113E-01 1 .327E+00 1 .014E- 10 1 . 550E -03 45 7 .649E-05 2 .428E+01 1 . 186E-01 1 .306E+00 1 .570E- 10 1 .218E -03 47 1 .844E-04 4 .736E+01 1 . 325E-01 1 . 278E+00 2 .829E- 10 6 .887E -04 49 6 . 105E-04 1 . 198E + 02 1 .592E-01 1 . 259E + 00 6 .513E- 10 3 .881E -04 51 1 . 742E-03 2 .697E+02 1 .918E-01 1 .253E+00 1 .367E- 09 3 .993E -04 53 3 981E-03 4 .643E+02 2 . 304E-01 1 .248E+00 2 .221E- 09 5 .903E -04 55 7 .857E-03 6 . 488E + 02 2 . 790E-01 1 .245E+00 2 .951E- 09 1 . 115E -03 57 1 . . 374E-02 8 .043E+02 3 . 376E-01 1 . 243E + 00 3 .504E- 09 2 .093E -03 59 2. , 340E-02 9 . 370E + 02 4 . 115E-01 1 .242E+00 3 .903E- 09 3 .998E -03 61 4 . .084E-02 1 .043E+03 5 . 170E-01 1 .241E+00 4 .115E- 09 8 .253E -03 63 7. .056E-02 1 . 130E+03 6 .811E-01 1 . 240E + 00 4 .158E- 09 1 . 374E -02 65 1 .081E-01 1 . 244E+03 9 . 331E-01 1 . 239E + 00 4 .223E- 09 1 . 249E -02 67 1 . .672E-01 1 .479E+03 1 .318E+00 1 .239E+00 4 .609E- 09 9 .526E -03 69 4. .531E-01 1 .853E+03 1 .937E+00 1 . 239E+00 5. .247E- 09 1 . 377E -02 71 1 . .926E+00 2 . 137E+03 2 .933E+00 1 .238E+00 5 .453E- 09 3 .989E -02 73 8. .540E+00 2 .276E+03 4 . 528E + 00 1 .234E+00 5. .197E- 09 1 . 396E -01 75 3. .446E+01 2 . 336E + 03 7 .062E+00 1 .221E+00 4. .720E- 09 5 . 252E -01 77 1 . 079E+02 2 . 356E + 03 1 062E+01 1 . 182E + 00 4. . 165E-09 1 .859E+00 79 2. .020E+02 2 . 360E + 03 1 . 366E+01 1 .133E+00 3 , 755E- 09 4. .058E+00 81 2. 934E+02 2 . 361E + 03 1 613E+01 1 .085E+00 3. .452E- 09 6 .649E+00 83 3. . 337E + 02 2 . 361E + 03 1 . .715E+01 1 .064E+00 3. .333E- 09 7 . .914E+00 85 3. . 459E + 02 2, . 361E + 03 1 .745E+01 1 .058E+00 3. .298E- 09 8 .310E+00 87 Model 03310191 SS 3000./ 0.00/ -3.00 CN, TiO, H20 (hm) TauRM r TE r 1 1 000E-06 1 075E+13 5 599E 3 1 585E-06 1 069E+13 5 357E 5 2 512E-06 1 063E+13 2 147E 7 3 981E-06 1 057E+13 7 798E 9 6 310E-06 1 051E+13 6 660E 11 1 000E-05 1 046E+13 5 02'6E 13 1 585E-05 1 O40E+13 3 152E 15 2 512E-05 1 034E+13 2 197E 17 3 981E-05 1 029E+13 2 013E 19 6 310E-05 1 024E+13 2 098E 21 1 000E-04 1 019E+13 2 375E 23 1 585E-04 1 015E+13 2 534E 25 2 512E-04 1 010E+13 3 367E 27 3 981E-04 1 006E+13 3 429E 29 6 310E-04 1 001E+13 5 1 1 1E 31 1 000E-03 9 970E+12 4 510E 33 1 585E-03 9 927E+12 8 430E 35 2 512E-03 9 883E+12 5 591E 37 3 981E-03 9 837E+12 4 895E 39 6 310E-03 9 789E+12 2 563E 41 1 000E-02 9 736E+12 2 837E 43 1 585E-02 9 678E+12 6 039E 45 2 512E-02 9 61 1E+12 2 354E 47 3 98iE-02 9 528E+12 1 204E 49 6 310E-02 9 409E+12 1 347E 51 1 000E-01 9 234E+12 2 861E 53 1 585E-01 9 074E+12 9 131E 55 2 512E-0T 8 964E+12 2 061E 57 3 981E-01 8 894E+12 1 802E 59 6 310E-01 8 847E+12 1 055E 61 1 OOOE+00 8 813E+12 1 217E 63 1 585E+00 8 787E+12 8 094E 65 2 512E+0O 8 767E+12 5 267E 67 3 981E+00 8 742E+12 2 433E 69 - 6 310E+00 8 691E+12 1 576E 71 1 000E+01 8 621E+12 1 953E 73 1 585E+01 8 573E+12 5 949E 75 2 512E+01 8 549E+12 1 423E 77 3 981E+01 8 538E+12 1 976E 79 6 043E+01 8 534E+12 2 269E 81 7 805E+01 8 532E+12 2 402E 83 9 233E+01 8 531E+12 2 405E 85 9 824E+01 8 531E+12 2 137E 87 1 000E+02 8 531E+12 1 307E RE res HE res DE r e s . 10 1 460E- 12 -2 327E- 07 0 OOOE+00 14 1 023E- 12 -9 863E- 1 1 1 205E- 11 14 6 312E- 13 -9 135E- 10 -6 561E- 12 15 3 358E- 13 -6 286E- 10 -1 158E- 1 1 15 1 538E- 13 2 219E- 11 - 1 043E- 1 1 15 6 223E- 14 4 861E- 10 -6 922E- 12 15 2 430E- 14 5 553E- 10 -4 293E- 12 15 1 066E- 14 3 643E- 10 -3 560E- 12 15 5 212E- 15 1 927E- 10 -3 587E- 12 15 2 570E- 15 1 449E- 10 -3 204E- 12 15 1 525E- 15 1 041E- 10 -2 996E- 12 15 1 564E- 15 8 162E- 1 1 -2 982E- 12 15 2 352E- 15 3 397E- 1 1 -3 669E- 12 15 3 018E- 15 1 166E- 10 -2 612E- 12 15 4 071E- 15 9 377E- 11 -2 950E- 12 15 4 958E- 15 1 233E- 10 -2 330E- 12 15 5 169E- 15 1 701E- 10 -5 002E- 13 15 3 031E- 15 2 492E- 10 4 447E- 12 14 -2 082E- 15 3 913E- 10 1 698E- 1 1 14 -4 382E- 15 3 714E- 10 3 453E- 1 1 13 - 1 026E- 14 9 242E- 10 8 511E- 1 1 13 -9 369E- 14 6 208E- 09 4 355E- 10 11 -6 864E- 13 1 557E- 08 1 262E- 09 11 -2 455E- 12 -7 409E- 08 -4 902E- 09 11 -9 T73E- 14 -7 364E- 08 - 1 871E- 09 11 2 088E- 11 1 194E- 07 1 335E- 08 12 8 784E- 12 -8 118E- 08 1 163E- 09 12 8 762E- 13 -8 123E- 08 - 1 303E- 09 13 -4 245E- 14 1 302E- 08 -7 301E- 1 1 13 -3 787E- 14 3 099E- 09 2 095E- 12 13 -2 934E- 14 7 954E- 10 1 838E- 1 1 14 - 1 371E- 14 - 1 674E- 10 6 700E- 12 14 -6 778E- 15 -4 410E- 1 1 6 989E- 12 14 - 1 304E- 15 4 552E- 10 -6 282E- 12 14 -3 387E- 15 -4 098E- 09 - 1 523E- 10 15 -2 400E- 16 3 470E- 09 -2 170E- 1 1 16 -5 282E- 16 .7 089E- 10 -2 371E- 1 1 15 -6 185E- 16 1 397E- 1 1 - 1 788E- 1 1 15 - 1 090E- 15 1 286E- 11 - 1 059E- 11 15 - 1 559E- 15 6 256E- 12 -3 098E- 12 15 - 1 855E- 15 6 470E- 13 -4 511E- 13 15 -2 150E- 15 8 873E- 14 - 1 069E- 13 15 -2 218E- 15 1 767E- 13 8 578E- 15 15 -2 171E- 15 -1 678E- 15 - 1 029E- 16 Convergence checks Page 227 EPS rms g_ef f /g Chi_H/RM Hconv/H Lum/L* 2 229E- 08 9 983E -01 1 122E+02 0 OOOE+00 9 927E-01 1 9 518E- 12 9 983E -01 1 041E+02 0 OOOE+00 9 927E-01 3 8 750E- 1 1 9 983E -01 9 712E+01 0 OOOE+00 9 927E-01 5 6 022E- 11 9 983E -01 9 015E+01 0 OOOE+00 9 926E-01 7 2 349E- 12 9 983E -01 8 253E+01 0 OOOE+00 9 926E-01 9 4 657E- 11 9 983E -01 7 404E+01 0 OOOE+00 9 926E-01 1 1 5 319E- 11 9 984E -01 6 490E+01 0 OOOE+00 9 926E-01 13 3 489E- 11 9 984E -01 5 560E+O1 0 OOOE+00 9 926E-01 15 1 846E- 11 9 984E -01 4 666E+01 0 OOOE+00 9 926E-01 17 1 389E- 1 1 9 984E -01 3 844E+01 0 OOOE+00 9 926E-01 19 9 979E- 12 9 985E -01 3 120E+01 0 000E+00 9 926E-01 21 7 823E- 12 9 985E -01 2 501E+01 0 OOOE+00 9 926E-01 23 3 273E- 12 9 986E -01 1 987E+01 0 OOOE+00 9 926E-01 25 1 1 17E-11 9 986E -01 1 572E+01 0 OOOE+00 9 925E-01 27 8 986E- 12 9 986E -01 1 243E+01 0 OOOE+00 9 925E-01 29 1 181E- 1 1 9 987E -01 9 860E+00 0 OOOE+00 9 925E-01 31 1 630E- 11 9 988E -01 7 875E+00 0 OOOE+00 9 925E-01 33 2 387E- 1 1 9 988E -01 6 357E+00 0 OOOE+00 9 925E-01 35 3 752E- 1 1 9 989E -01 5 216E+00 0 000E+00 9 925E-01 37 3 573E- 1 1 9 990E -01 4 374E+00 0 OOOE+00 9 925E-01 39 8 890E- 11 9 990E -01 3 780E+00 0 OOOE+00 9 925E-01 41 5 961E- 10 9 991E -01 3 395E+00 0 000E+00 9 924E-01 43 1 496E- 09 9 992E -01 3 246E+00 0 OOOE+00 9 924E-01 45 7 1 12E-09 9 993E -01 3 503E+00 0 OOOE+00 9 923E-01 47 7 056E- 09 9 995E -01 4 702E+00 0 OOOE+00 9 921E-01 49 1 151E- 08 9 997E -01 4 519E+00 0 OOOE+00 9 920E-01 51 7 777E- 09 9 998E -01 3 045E+00 0 000E+00 9 931E-01 53 7 781E- 09 9 998E -01 2 393E+00 0 000E+00 9 976E-01 55 1 248E- 09 9 997E -01 1 822E+00 0 OOOE+00 1 008E+00 57 2 968E- 10 9 996E -01 1 365E+00 0 OOOE+00 1 013E+00 59 7 621E- 11 9 993E -01 1 145E+00 1 179E- 10 1 014E+00 61 1 605E- 1 1 9 986E -01 1 077E+00 2 057E-06 1 013E+00 63 4 277E- 12 9 977E -01 1 077E+00 3 671E-05 1 01 1E+00 65 4 360E- 1 1 9 981E -01 1 005E+00 2 999E-06 1 012E+00 67 3 928E- 10 9 986E -01 9 868E-01 1 548E-09 1 013E+00 69 3 323E- 10 9 979E -01 9 963E-01 1 638E-06 1 010E+00 71 6 793E- 11 9 939E -01 1 001E+00 7 719E-03 1 006E+00 73 2 173E- 12 9 785E -01 1 004E+00 1 188E+00 1 004E+00 75 1 596E- 12 9 192E -01 1 004E+00 2 210E+01 1 002E+00 77 6 687E- 13 7 151E -01 1 001E+00 3 214E+02 1 001E+00 79 7 558E- 14 3 781E -01 1 OOOE+00 3 958E+03 1 OOOE+00 81 1 352E- 14 - 1 859E -02 1 OOOE+00 0 OOOE+00 1 OOOE+00 83 1 708E- 14 -2 124E -01 1 OOOE+00 0 OOOE+00 1 OOOE+00 85 1 316E- 15 -2 729E -01 1 OOOE+00 0 000E+00 1 OOOE+00 87 Model 03310191 SS 3000./ 0.00/ -3.00 CN, TiO, H20 (hm) Convective quant i t i es Page 228 TauRM Cp/Cv H_P Oconv DELrad DELad DELbub TauRMb Vconv Hconv Hconv/H 1 1 000E-06 T .660E+00 1 633E+12 2 580E+00 1 534E-02 1 515E -01 0 OOOE+00 2 086E-05 0 OOOE+OO 0 OOOE+00 0 OOOE+00 1 3 1 585E-06 1 .598E+00 1 174E+12 2 771E+00 1 773E-02 ' 1 411E -01 0 OOOE+00 2 349E-05 0 OOOE+00 0 OOOE+00 0 OOOE+OO 3 5 2 512E-06 1 .524E+00 8 506E+1 1 2 945E+00 1 976E-02 1 278E -01 0 OOOE+00 2 671E-05 0 OOOE+00 0 OOOE+00 0 OOOE+00 5 7 3 981E-06 1 .450E+00 6 259E+1 1 3 097E+00 2 196E-02 1 133E -01 0 OOOE+00 3 109E-05 0 OOOE+00 0 OOOE+00 0 OOOE+00 7 9 6 310E-06 1 .384E+00 4 725E+11 3 218E+00 2 431E-02 9 929E -02 0 OOOE+00 3 750E-05 0 OOOE+OO 0 OOOE+00 0 OOOE+OO ' 9 1 1 1 000E-05 1 . 332E + 00 3 690E+11 3 305E+00 2 667E-02 8 705E -02 0 OOOE+00 4 733E-05 0 OOOE+00 0 OOOE+00 0 OOOE+00 1 1 13 . 1 585E-05 1 . 291E+00 2 998E+11 3 362E+00 2 885E-02 7 724E -02 0 OOOE+00 6 275E-05 0 OOOE+00 0 OOOE+00 0 OOOE+00 13 15 2 512E-05 1 .262E+00 2 536E+1 1 3 392E+00 3 074E-02 6 984E -02 0 OOOE+00 8 724E-05 0 OOOE+00 0 OOOE+00 0 OOOE+00 15 17 3 981E-05 1 .241E+00 2 227E+1 1 3 404E+00 3 238E-02 6 451E -02 0 OOOE+00 1 264E-04 0 OOOE+00 0 OOOE+00 0 OOOE+00 17 19 6 310E-05 1 .227E+00 2 018E+1 1 3 400E+00 3 392E-02 6 081E -02 0 OOOE+00 1 894E-04 0 OOOE+00 0 OOOE+00 0 OOOE+OO 19 21 1 000E-04 1 .216E+00 1 875E+11 3 385E+00 3 538E-02 5 831E -02 0 OOOE+00 2 906E-04 0 OOOE+OO 0 OOOE+00 0 OOOE+00 21 23 1 585E-04 1 209E+00 1 778E+1 1 3 359E+00 3 700E-02 5 672E -02 0 OOOE+OO 4 531E-04 0 OOOE+00 0 OOOE+OO 0 OOOE+OO 23 25 2 512E-04 1 204E+00 1 71 1E+1 1 3 324E+00 3 839E-02 5 579E -02 0 OOOE+00 7 129E-04 0 OOOE+00 0 OOOE+00 0 OOOE+OO 25 27 3 981E-04 1 199E+00 1 666E+11 3 281E+00 4 OOOE-02 5 535E -02 0 OOOE+00 1 124E-03 0 OOOE+OO 0 OOOE+00 0 OOOE+00 27 29 6 310E-04 1 196E+00 1 635E+1 1 3 230E+00 4 150E-02 5 529E -02 0 OOOE+00 1 769E-03 0 OOOE+00 0 OOOE+00 0 OOOE+00 29 31 1 000E-03 1 194E+00 1 616E+1 1 3 171E+00 4 317E-02 5 555E -02 0 OOOE+00 2 771E-03 0 OOOE+00 0 OOOE+00 0 OOOE+00 31 33 1 585E-03 1 192E+00 • 1 605E+11 3 103E+00 4 498E-02 5 609E -02 0 OOOE+00 4 306E-03 0 OOOE+00 0 OOOE+OO 0 OOOE+00 33 35 2 512E-03 1 190E+00 1 602E+11 3 025E+00 4 685E-02 5 692E -02 0 OOOE+00 6 620E-03 0 OOOE+00 0 OOOE+00 0 OOOE+00 35 37 3 981E-03 1 188E+00 1 604E+1 1 2 934E+00 4 908E-02 5 805E -02 0 OOOE+00 1 004E-02 0 OOOE+00 0 OOOE+OO 0 OOOE+00 37 39 6 310E-03 1 186E+00 1 612E+11 2 826E+00 5 164E-02 5 959E -02 0 OOOE+00 1 497E-02 0 OOOE+OO 0 OOOE+00 0 OOOE+00 39 41 1 000E-02 1 184E+00 1 627E+11 2 695E+00 5 514E-02 6 169E -02 0 OOOE+00 2 185E-02 0 OOOE+00 0 OOOE+OO 0 OOOE+00 41 43 1 585E-02 1 182E+00 1 649E+11 2 531E+00 5 958E-02 6 473E -02 0 OOOE+00 3 103E-02 0 OOOE+OO 0 OOOE+00 0 OOOE+00 43 45 2 512E-02 1 180E+00 1 683E+11 2 315E+00 6 642E-02 6 958E -02 0 OOOE+00 4 233E-02 0 OOOE+00 0 OOOE+00 0 OOOE+OO 45 47 3 981E-02 1 180E+00 1 738E+11 2 011E+00 7 906E-02 7 909E -02 0 OOOE+OO 5 316E-02 0 OOOE+OO 0 OOOE+00 0 OOOE+00 47 49 6 310E-02 1 194E+00 1 831E+11 1 593E+00 8 874E-02 1 044E -01 0 OOOE+00 5 707E-02 0 OOOE+00 0 OOOE+00 0 OOOE+00 49 51 1 000E-01 1 247E+00 1 935E+11 1 291E+00 8 114E-02 1 551E -01 0 OOOE+00 7 825E-02 0 OOOE+OO 0 OOOE+00 0 OOOE+OO 51 53 1 585E-01 1 306E+00 2 003E+11 1 181E+00 8 654E-02 1 999E -01 0 OOOE+00 1 749E-01 0 OOOE+00 0 OOOE+00 0 OOOE+00 53 55 2 512E-01 1 375E+00 2 070E+11 1 113E+00 1 227E-01 2 463E -01 0 OOOE+OO 4 342E-01 0 OOOE+00 0 OOOE+00 0 OOOE+00 55 57 3 981E-01 1 453E+00 2 143E+1 1 1 067E+00 1 701E-01 2 934E -01 0 OOOE+00 1 128E+00 0 OOOE+00 0 OOOE+00 0 OOOE+OO 57 59 6 310E-01 1 520E+00 2 214E+1 1 1 041E+00 2 332E-01 3 294E -01 0 OOOE+00 2 598E+00 0 OOOE+OO 0 OOOE+00- 0 OOOE+00 59 61 1 000E+00 1 578E+00 2 297E+11 1 024E+00 3 818E-01 3 582E -01 3 818E-01 5 736E+00 2 748E+02 4 504E-02 1 179E- 10 61 63 i 585E+00 1 624E+00 2 412E+1 1 1 014E+00 6 923E-01 3 800E -01 6 917E-01 1 31 1E+01 7 156E+03 7 905E+02 2 057E-06 63 65 2' 512E+00 1 651E+00 2 575E+11 1 009E+00 9 378E-01 3 923E -01 9 338E-01 2 354E+01 1 886E+04 1 417E+04 3 671E-05 65 67 3 981E+00 T 662E+00 2 775E+11 1 004E+00 6 753E-01 3 967E -01 6 746E-01 2 341E+01 8 184E+03 1 165E+03 2 999E-06 67 69 6 310E+00 1 659E+00 2 990E+11 1 002E+00 4 223E-01 3 960E -01 4 223E-01 2 100E+01 6 397E+02 6 082E-01 1 548E-09 69 71 1 000E+01 1 636E+00 3 237E+11 1 003E+00 5 354E-01 3 863E -01 5 351E-01 3 742E+01 6 223E+03 6 538E+02 1 638E-06 71 73 1 585E+01 1 561E+00 3 553E+11 1 012E+00 1 205E+00 3 540E -01 1 126E+00 1 236E+02 1 004E+05 3 116E+06 7 719E-03 73 75 2 512E+01 1 410E+00 3 951E+11 1 049E+00 2 979E+00 2 764E -01 1 256E+00 4 586E+02 5 038E+05 4 821E+08 1 188E+00 75 77 3 981E+01 1 259E+00 4 456E+11 1 177E+00 8 067E+00 1 748E -01 5 763E-01 1 768E+03 1 183E+06 8 994E+09 2 210E+01 77 79 6 043E+01 1 198E+00 5 096E+11 1 502E+00 2 484E+01 1 116E -01 2 367E-01 6 312E+03 2 592E+06 1 310E+11 3 214E+02 79 81 7 805E+01 1 198E+00 5 667E+11 1 882E+00 8 821E+01 9 097E -02 1 673E-01 1 382E+04 5 789E+06 1 613E+12 3 958E+03 81 83 9 233E+01 1 212E+00 6 173E+11 2 218E+00 - 1 157E+03 8 304E -02 0 OOOE+00 2 267E+04 0 OOOE+OO 0 OOOE+00 0 OOOE+00 83 85 9 824E+01 1 219E+00 6 395E+1 1 2 355E+00 -2 040E+02 8 101E -02 0 OOOE+OO 2 699E+04 0 OOOE+00 0 OOOE+00 0 OOOE+OO 85 87 1 000E+02 1 221E+00 6 462E+1 1 2 395E+00 - 1 573E+02 8 051E -02 0 OOOE+00 2 834E+04 0 OOOE+00 0 OOOE+00 0 OOOE+00 87 Model 04310191 SS 3000./ 1.00/ -3.00 CN, TiO, H20 (hm) TauRM Col Mass 1 1 000E-05 4 467E-02 1 3 1 585E-05 5 907E-02 1 5 2 512E-05 7 809E-02 1 7 3 981E-05 1 031E-01 1 9 6 310E-05 1 360E-01 1 11 1 OOOE-04 1 789E-01 1 13 1 585E-04 2 351E-01 1 15 2 512E-04 3 087E-01 1 17 3 981E-04 4 056E-01 1 19 6 310E-04 5 340E-01 1 21 1 OOOE-03 7 059E-01 1 23 1 585E-03 9 386E-01 1 25 2 512E-03 1 258E+00 1 27 3 981E-03 1 703E+00 1 29 6 310E-03 2 337E+00 9 31 1 OOOE-02 3 257E+00 9 33 1 585E-02 4 630E+00 8 35 2 512E-02 6 747E+00 8 37 3 981E-02 1 016E+01 7 39 6 310E-02 1 599E+01 6 41 1 000E-01 2 704E+01 5 43 1 585E-01 5 042E+01 4 45 2 512E-01 9 988E+01 3 47 3 981E-01 1 840E+02 1 49 6 310E-01 2 823E+02 6 51 1 OOOE+00 3 662E+02 -6 53 1 468E+00 4 249E+02 -4 55 2 154E+00 4 737E+02 -7 57 3 162E+00 5 145E+02 -1 59 4 642E+00 5 574E+02 - 1 61 6 813E+00 6 223E+02 -1 63 1 O00E+O1 7 287E+02 -2 65 1 445E+01 8 418E+02 -2 67 2 089E+01 9 214E+02 -3 69 3 020E+01 9 679E+02 -3 71 4 365E+01 9 937E+02 -3 73 6 310E+01 1 008E+03 -3 75 8 610E+01 1 014E+03 -3 77 9 750E+01 1 016E+03 -3 79 9 950E+01 1 016E+03 -3 r/R* - 1 T n 541E -02 2 025E+03 1 550E+12 501E -02 2 033E+03 2 041E+12 462E -02 2 042E+03 2 686E+12 422E -02 2 053E+03 3 530E+12 383E -02 2 066E+03 4 627E+12 345E -02 2 081E+03 6 049E+12 306E -02 2 098E+03 7 890E+12 268E -02 2 115E + 03 1 028E+13 229E -02 2 135E+03 1 339E+13 190E -02 2 155E+03 1 748E+13 150E -02 2 178E+03 2 288E+13 109E -02 2 202E+03 3 011E+13 066E -02 2 229E+03 3 991E+13 022E -02 2 258E+03 5 338E+13 742E -03 2 290E+03 7 225E+13 235E -03 2 328E+03 9 918E+13 686E -03 2 371E+03 1 386E+14 082E -03 2 421E+03 1 980E+14 404E -03 2 483E+03 2 909E+14 618E -03 2 562E+03 4 447E+14 661E -03 2 668E+03 7 233E+14 446E -03 2 813E+03 1 282E+15 012E -03 2 988E+03 2 397E+15 634E -03 3 180E+03 4 160E+15 016E -04 3 381E+03 6 013E+15 730E -05 3 564E+03 7 408E+15 698E -04 3 721E+03 8 239E+15 776E -04 3 909E+03 8 747E+15 025E -03 4 145E+03 8 963E+15 282E -03 4 437E+03 9 074E+15 661E -03 4 777E+03 9 417E+15 247E -03 5 170E+03 1 020E+16 825E -03 5 605E+03 1 087E+16 217E -03 6 097E+03 1 095E+16 450E -03 6 648E+03 1 055E+16 586E -03 7 261E+03 9 914E+15 665E -03 7 938E+03 9 192E+15 705E -03 8 565E+03 8 571E+15 716E -03 8 830E+03 8 327E+15 718E -03 8 874E+03 8 287E+15 Summary of physica l quant i t ies Page 229 Pe P_gas P_rad Mu Rho ChiRM 9 108E-07 4 331E-01 1 042E-01 1 340E+00 3 449E- 12 3 693E -04 1 1 187E-06 5 728E-01 1 043E-01 1 355E+00 4 594E- 12 4 422E -04 3 1 551E-06 7 574E-01 1 044E-01 1 371E+0O 6 1 14E-12 5 313E -04 5 2 034E-06 1 001E+00 1 045E-01 1 385E+00 8 1 16E-12 6 409E -04 7 2 673E-06 1 320E+00 1 046E-01 1 397E+00 1 073E- 1 1 7 757E -04 9 3 520E-06 1 738E+00 1 047E-01 1 408E+00 1 414E- 11 9 404E -04 ' 1 T 4 644E-06 2 285E+00 1 049E-01 1 416E+00 1 855E- 1 1 1 139E -03 13 6 139E-06 3 002E+00 1 051E-01 1 423E+0O 2 429E- 1 1 1 375E -03 15 8 140E-06 3 947E+00 1 053E-01 1 428E+00 3 176E- 1 1 1 651E -03 17 1 084E-05 5 201E+00 1 057E-01 1 432E+00 4 155E- 1 1 1 966E -03 19 1 453E-05 6 880E+00 1 061E-01 1 434E+00 5 450E- 1 1 2 317E -03 21 1 964E-05 9 154E+00 1 067E-01 1 435E+00 7 177E- 1 1 2 696E -03 23 2 685E-05 1 228E+01 1 074E-01 1 435E+00 9 508E- 1 1 3 090E -03 25 3 722E-05 1 664E+01 1 084E-O1 1 433E+00 1 270E- 10 3 482E -03 27 5 251E-05 2 285E+01 1 098E-01 1 429E+00 1 714E- 10 3 845E -03 29 7 576E-05 3 187E+01 1 117E-01 1 422E+00 2 342E- 10 4 144E -03 31 1 124E-04 4 536E+01 1 143E-01 1 412E+00 3 251E- 10 4 340E -03 33 1 731E-04 6 618E+01 1 181E-01 1 399E+00 4 599E- 10 4 382E -03 35 2 799E-04 9 974E+01 1 236E-01 1 381E+00 6 670E- 10 4 212E -03 37 4 850E-04 1 573E+02 1 320E-O1 1 358E+00 1 002E- 09 3 747E -03 39 9 343E-04 2 664E+02 1 450E-01 1 330E+00 1 597E- 09 2 983E -03 41 2 071E-03 4 979E+02 1 664E-01 1 301E+00 2 770E- 09 2 156E -03 43 5 040E-03 9 888E+02 1 991E-01 1 279E+00 5 091E- 09 1 729E -03 45 1 212E-02 1 827E+03 2 460E-01 1 264E+00 8 734E- 09 1 871E -03 47 2 720E-02 2 807E+03 3 115E-01 1 253E+00 1 251E- 08 3 143E -03 49 5 315E-02 3 645E+03 3 917E-01 1 248E+00 1 535E- 08 6 048E -03 51 8 875E-02 4 233E+03 4 740E-01 1 244E+00 1 703E- 08 1 046E -02 53 1 494E-01 4 721E+03 5 846E-01 1 242E+00 1 804E- 08 1 883E -02 55 2 505E-01 5 129E+03 7 434E-01 1 241E+00 1 847E- 08 3 085E -02 57 3 957E-01 5 559E+03 9 757E-01 1 240E+00 1 868E- 08 3 576E -02 59 5 763E-01 6 210E+03 1 312E+O0 1 239E+00 1 938E- 08 3 096E -02 61 9 770E-01 7 277E+03 1 804E+00 1 239E+00 2 098E- 08 3 122E -02 63 2 443E+00 8 414E+03 2 496E+00 1 239E+00 2 236E- 08 5 357E -02 65 7 558E+00 9 214E+03 3 498E+00 1 238E+00 2 250E- 08 1 263E -01 67 2 385E+01 9 681E+03 4 948E+00 1 236E+00 2 165E- 08 3 245E -01 69 7 158E+01 9 938E+03 7 046E+00 1 230E+00 2 025E- 08 8 602E -01 71 1 993E+02 1 007E+04 1 008E+01 1 214E+00 1 854E- 08 2 353E+00 73 4 413E+02 1 014E+04 1 367E+01 1 185E+00 1 687E- 08 5 617E+00 75 5 940E+02 1 015E+04 1 544E+01 1 167E+O0 1 613E- 08 7 954E+00 77 6 228E+02 1 015E+04 1 575E+01 1 163E+00 1 600E- 08 8 416E+00 79 Model 04310191 SS 3000./ 1.00/ -3.00 CN, TiO, H20 (hm) TauRM. r TE r 1 1 .000E-05 9 . 149E+12 1 . 190E 3 1 .585E-05 9 . 146E+12 1 .054E 5 2 .512E-05 9 . 142E+12 2 . 262E 7 3 .981E-05 9 .139E+12 1 .997E 9 6 .310E-05 9 . 135E+12 1 .849E 11 1 .OOOE-04 9 . 132E+12 2 . 127E 13 1 .585E-04 9 . 128E+12 1 . 316E 15 2 .512E-04 9 .125E+12 4 . 322E 17 3 .981E-04 9 .121E+12 8 .657E 19 6 .310E-04 9 .118E+12 3 .021E 21 1 .000E-03 9 .114E+12 6 . 702E 23 1 .585E-03 9 .110E+12 1 .860E 25 2 .512E-03 9 .107E+12 1 .487E 27 3 .981E-03 9 .103E+12 2 .201E 29 6 .310E-03 9 .098E+12 3. .625E 31 1 . .000E-02 9. .094E+12 1 . .687E 33 1 .585E-02 9. .089E+12 5. .943E 35 2 .512E-02 9 .083E+12 8. . 780E 37 3 .981E-02 9 .077E+12 6 . 132E 39 6 .310E-02 9 .070E+12 3 .075E 41 1 .000E-01 9 .062E+12 1 .005E 43 1 .585E-01 9 .051E+12 5 .744E 45 2 .512E-01 9 .038E+12 6 .597E 47 3. .981E-01 9 . 025E+12 1 . 082E 49 6. .310E-01 9. .016E+12 5, .476E 51 1 . .OOOE+00 9. .010E+12 1 . 491E 53 1 . .468E+00 9. 006E+12 3. . 896E 55 2. .154E+00 9 . 004E+12 6. . 470E 57 3. .162E+00 9. .001E+12 2, , 746E 59 4. .642E+00 8. 999E+12 3. 003E 61 6. .813E+00 8. 996E+12 4, . 301E 63 1 . .000E+01 8. 990E+12 3, . 243E 65 1 . 445E+01 8. 985E+12 1 . 884E 67 2. 089E+01 8. 982E+12 9. 631E 69 3. 020E+01 8. 979E+12 4. 922E 71 4. .365E+01 8. 978E+12 2. 462E 73 6. ,310E+01 8. 978E+12 1 . 048E 75 8. 610E+01 8. 977E+12 5. 159E 77 9. .750E+01 8. 977E+12 1 . 044E 79 9. 950E+01 8. 977E+12 6. 119E RE res HE res DE res 08 -7 .202E- 13 4 .979E- 08 0 .OOOE+00 06 - 1 .137E- 10 2 . 191E-09 9 .529E- 13 07 -4 .106E- 13 1 .876E- 09 1 .869E- 12 13 - 1 .910E- 13 1 .071E- 09 2 .387E- 12 13 -6 .508E- 14 .3 .467E- 10 2 .686E- 12 13 8 .557E- 16 -1 .571E- 10 2 .816E- 12 13 - 1 .349E- 14 -5 .391E- 10 2 .932E- 12 13 -8 .065E- 14 -8 .001E- 10 3 .146E- 12 14 - 1 .803E- 13 - 1 .026E- 09 3 .430E- 12 13 -3 .460E- 13 -1 .284E- 09 3 .499E- 12 14 -5 .919E- 13 -1 .262E- 09 4 .555E- 12 13 -8 .480E- 13 -1 .710E- 09 4 .401E- 12 12 -1 .144E- 12 -1 .830E- 09 4 .657E i 12 13 -1 .538E- 12 - 1 .956E- 09 4 .996E- 12 13 -1 .962E- 12 - 1 . .682E- 09 6. 413E- 12 12 -2 . 386E- 12 - 1 . 896E- 09 7 . 493E- 12 13 -2 .909E- 12 - 1 , .832E- 09 8. .228E- 12 13 -3 .417E- 12 - 1 , .736E- 09 9. .800E- 12 12 -4 .014E- 12 2, , 742E- 09 3. 033E- 11 12 -5 .544E- 12 1 . .914E- 09 4 . ,347E- 1 1 1 1 - 1 .070E- 11 1 . .432E- 08 1 , ,304E- 10 11 -2 .754E- 11 -4 .259E- 08 -2. 679E- 10 11 -4 .922E- 11 -9, 851E- 08 -8. , 136E-10 10 -3. .277E- 11 2. 431E- 07 -3. 276E- 10 11 -3 .501E- 13 1 . .118E- 07 7 . 028E- 10 11 5. .210E- 12 -9 . 861E- 08 -1 . 532E- 10 12 1 . .281E- 12 8. 043E- 09 -2. 587E- 11 13 - 1 . .066E- 14 9. 459E- 09 1 . 010E- 11 13 1 . .662E- 13 -4. 553E- 10 -4. 455E- 12 13 3. .219E- 13 2. 084E- 09 6 . 734E- 12 14 3. .301E- 14 3. 038E- 09 3. . 333E- 12 14 4. .313E- 14 -1 . 234E- 09 -4 . 399E- 13 14 2. ,329E- 14 -3. 511E- 09 - 1 . 616E- 12 15 1 . .151E- 14 -4. 569E- 10 8. 159E- 13 15 5. . 380E- 15 -1 . 671E- 1 1 9 . 322E- 13 15 2. 095E- 15 - 1 . 1 13E-12 6 . 762E- 13 15 6. 331E- 16 -1 . 117E- 1 1 4. 499E- 13 16 0. 000E+00 -2. 993E- 12 8. 368E- 14 15 - 1 . 381E- 15 9. 862E- 12 5. 988E- 14 15 1 . 240E- 14 -7. 580E- 13 -3. 536E- 15 Convergence checks Page 230 EPS rms g_ef f /g Chi_H/RM Hconv/H Lum/L * 1 .267E- 08 9 .997E -01 5 . 288E + 01 0 .OOOE+00 9 .868E-01 1 1 .039E- 06 9 .997E -01 4 .320E+01 0 .OOOE+OO 9 .868E-01 3 2 . 231E- 07 9 .997E -01 3 .520E+01 0 .OOOE+00 9 .868E-01 5 1 .026E- 10 9 .997E -01 2 .862E+01 0 .OOOE+00 9 .868E-01 7 3 .321E- 11 9 .997E -01 2 . 323E + 01 0 .OOOE+OO 9 .868E-01 9 1 . 505E- 1 1 9 .997E -01 1 .883E+01 0. .OOOE+00 9 .868E-01 1 1 5 . 164E-11 9 .997E -01 1 . 528E + 01 0 .OOOE+00 9 .868E-01 13 7 .664E- 11 9 .997E -01 1 . 243E + 01 0 .OOOE+00 9 .868E-01 15 9 .829E- 11 9 .997E -01 1 .015E+01 0 .OOOE+OO 9 .868E-01 17 1 . 230E- 10 9 .997E -01 8 .340E+00 0 .OOOE+00 9 .868E-01 19 1 . 209E- 10 9 .998E -01 6 .905E+00 0 .OOOE+OO 9 .868E-01 21 1 .638E- 10 9 .998E -01 5 .771E+00 0 .OOOE+00 9 .868E-0-1 23 1 .753E- 10 9 .998E -01 4 .877E+00 0 .OOOE+OO 9 .867E-01 25 1 .873E- 10 9 .998E -01 4 . 173E + 00 0 .OOOE+00 9 .867E-01 27 1 .611E- 10 9 .998E -01 3 . 623E+00 0. OOOE+00 9 .867E-01 29 1 . .816E- 10 9 .998E -01 3 . 194E + 00 0. OOOE+00 9 .867E-01 31 1 .755E- 10 9 .998E -01 2 . 862E + 00 0, OOOE+00 9 .867E-01 33 1 . .663E- 10 9 998E -01 2 .614E+00 0. .OOOE+00 9 .867E-01 35 2 .627E- 10 9 .998E -01 2 .442E+00 0. OOOE+00 9 .866E-01 37 1 .834E- 10 9 .999E -01 2 . 359E + 00 9 . •668E-15 9 .866E-01 39 1 .372E- 09 9 .999E -01 2 . 390E + 00 5. .081E-16 9 .865E-01 41 4 .080E- 09 9 .999E -01 2 .451E+00 0. .OOOE+OO 9 .863E-01 43 9 .436E- 09 9 .999E -01 2 .235E+00 0. OOOE+00 9 .865E-01 45 2 . 329E- 08 9. .999E -01 2 .010E+00 0. OOOE+OO 9 .889E-01 47 1 . .071E- 08 9. .999E -01 1 .634E+O0 0. OOOE+00 1 .001E+00 49 9. .445E- 09 9. 999E -01 1 . 223E + 00 0. OOOE+00 1 .010E+00 51 7. , 704E- 10 9 .998E -01 1 .068E+00 9. 943E-07 1 . .011E+00 53 9. .060E- 10 9. .997E -01 1 .015E+00 4 . 275E-04 1 , .010E+00 55 4. . 362E- 1 1 9 .995E -01 1 .015E+00 7 , , 379E-03 1 .008E+00 57 1 996E- 10 9. .994E -01 1 . .004E+00 6 . 437E-03 1 .007E+00 59 2. 910E- 10 9 .995E -01 9 .896E-01 4 . 165E-04 1 .008E+00 61 T . 182E-10 9. 995E -01 9, .955E-01 1 . 217E-04 1 .007E+00 63 3. . 363E- 10 9 . 992E -01 9 996E-01 1 . 822E-02 1 . 005E+00 65 4. 376E- 11 9. 981E -01 1 . 001E+00 7 . 210E-01 1 . 004E+00 67 1 . 603E- 12 9. 950E -01 1 . 002E+00 5. 842E+00 1 , 003E+00 69 1 . 248E- 13 9. 868E -01 1 . 002E+00 3. 150E+01 1 . •002E+00 71 1 . 071E- 12 9 . 639E -01 1 . 002E+00 1 . 774E+02 1 . 001E+00 73 2. 867E- 13 9 . 139E -01 1 . OOOE+00 8. 294E+02 1 . OOOE+00 75 9. 446E- 13 8. . 782E -01 1 . OOOE+00 1 . 579E+03 1 . OOOE+OO 77 7. 287E- 14 8. . 7 11E -01 9. 999E-01 1 . 760E+03 1 . OOOE+00 79 to CO o Model 04310191 SS 3000./ 1.00/ -3.00 CN, TiO, H20 (hm) TauRM 1 1 .000E-05 3 1 .585E-05 5 2 .512E-05 7 3 .981E-05 9 6 .310E-05 1 1 1 .000E-04 13 1 . 585E-04 15 2 . 512E-04 17 3 .981E-04 19 6 .310E-04 21 1 .000E-03 23 1 . 585E-03 25 2 .512E-03 27 3 .981E-03 29 6 . 310E-03 31 1 .OOOE-02 33 1 . 585E-02 35 2 .512E-02 37 3. .981E-02 39 6 . . 310E-02 41 1 . 000E-01 43 1 . .585E-01 45 2, , 512E-01 47 3. 981E-01 49 6 . . 310E-01 51 1 . OOOE+OO 53 1 . . 468E + 00 55 2. .154E+00 57 3. . 162E + 00 59 4. 642E+00 61 6 . 813E+00 63 1 . OOOE+01 65 1 . 445E+01 67 2. 089E+01 69 3. 020E+01 71 4. . 365E + 01 73 6 . 310E + 01 75 8. 610E+01 77 9 . 750E+01 79 9 . 950E+01 Cp/Cv 1.205E+00 1.201E+00 1.199E+00 1.198E+00 1.198E+00 1.199E+00 1.200E+00 1.200E+00 1.201E+00 1.202E+00 1.203E+00 1.204E+00 1.204E+00 1.205E+00 1.206E+00 1.206E+00 1.206E+00 1.205E+00 1.203E+00 1.201E+00 1.199E+00 1.204E+00 1.224E+00 1.265E+00 1.340E+00 1.426E+00 1.500E+00 1.570E+00 1.622E+00 1.649E+00 1.658E+00 1.654E+00 1.631E+00 1.575E+00 1.473E+00 1.347E+00 1.250E+00 1.210E+00 1.205E+00 1.205E+00 H_P 1 .606E+10 1.518E+10 1.451E+10 1.401E+10 1.364E+10 1.339E+10 1.322E+10 1.312E+10 1.308E+10 1.308E+10 1.312E+10 1.319E+10 1.331E+10 1.346E+10 1.365E+10 1.391E+10 1.423E+10 1.465E+10 1.519E+10 1;591E+10 1.688E+10 1.814E+10 1.954E+10 2.098E+10 2.246E+10 2.375E+10 2.484E+10 2.612E+10 2.772E+10 2.969E+10 3.195E+10 3.455E+10 3.742E+10 4.070E+10 4.444E+10 4.876E+10 5.400E+10 5.972E+10 6.256E+10 6.306E+10 Qconv 2.680E+00 2.852E+00 3.005E+00 3.131E+00 3.228E+00 3.298E+00 3.344E+00 3.372E+00 3.386E+00 3.388E+00 3.380E+00 3.361E+00 3.332E+00 3.291E+00 3.235E+00 3.160E+00 3.061E+00 2.928E+00 2.748E+00 2.504E+00 2.185E+00 1.824E+00 1.532E+00 1.322E+00 1.172E+00 1.094E+00 1.056E+00 1.031E+00 1.017E+00 1.010E+00 1.004E+00 1.002E+00 1.004E+00 1.011E+00 1.031E+00 1.086E+00 1.220E+00 1.452E+00 1.590E+00 1.616E+00 DELrad 1 . 316E -02 1 .553E -02 1 .808E -02 2 . 113E -02 2 .433E -02 2 . 731E -02 2 .992E -02 3 . 212E -02 3 .410E -02 3 .597E -02 3 .779E -02 3 .974E -02 4 . 181E -02 4 . 400E -02 4 .673E -02 4 .996E -02 5 . 381E -02 5 .871E -02 6. . 484E -02 7 . 233E -02 8. 078E -02 8, 664E -02 9. . 229E -02 1 . . 203E -01 1 . . 747E -01 2. ,442E -01 3. ,643E -01 5. 809E -01 8. , 178E -01 7 . 887E -01 5. 694E -01 4. 957E -01 7. 192E -01 1 . . 330E+00 2. 551E+00 4. 929E+00 9. 841E+0O 1 . 791E+01 2. 299E+01 2. 400E+01 DELad 6 . 370E-02 6.054E-02 5.814E-02 5.641E-02 5.525E-02 5.452E-02 5.413E-02 5.400E-02 5.406E-02 5.429E-02 5.467E-02 5.519E-02 5.586E-02 5.670E-02 5.774E-02 5.906E-02 6.077E-02 6.308E-02 6.638E-02 7.151E-02 8.041E-02 9.687E-02 1.230E-01 1.616E-01 2.185E-01 2.750E-01 3.174E-01 3.537E-01 3.786E-01 3.913E-01 3.958E-01 3.943E-01 3.850E-01 3.611E-01 3. 115E-01 2. 379E-01 1 .653E-01 1 .220E-01 1 . 100E-01 1 .083E-01 Convective quant i t i es Page 231 DELbub TauRMb Vconv Hconv Hconv/H 0 .OOOE+OO 3 . 274E-05 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 1 0 .OOOE+OO 4 .935E-05 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 3 0 .OOOE+OO 7 .542E-05 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 5 0 .OOOE+OO 1 . 166E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 7 0 .OOOE+OO 1 .817E-04 0 .OOOE+OO 0 OOOE+OO 0 .OOOE+OO 9 0 .OOOE+OO 2 .848E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 1 1 0 .OOOE+OO 4 .470E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 13 0 .OOOE+OO 7 .013E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 15 0 .OOOE+OO 1 .097E-03 0 OOOE+OO 0. .OOOE+OO 0 OOOE+OO 17 0 .OOOE+OO ' 1 . 709E-03 0 OOOE+OO 0 OOOE+OO 0. OOOE+OO 19 0 .OOOE+OO 2 .649E-03 0. OOOE+OO 0. .OOOE+OO 0. OOOE+OO 21 0 .OOOE+OO 4 .083E-O3 0 OOOE+OO 0. .OOOE+OO 0. .OOOE+OO 23 0 .OOOE+OO 6 . 254E-03 0 .OOOE+OO 0 .OOOE+OO 0. OOOE+OO 25 0 .OOOE+OO 9 .522E-03 0. .OOOE+OO 0. OOOE+OO 0, OOOE+OO 27 0 .OOOE+OO 1 . 440E-02 0 .OOOE+OO 0. OOOE+OO 0. OOOE+OO 29 0 .OOOE+OO 2 . 160E-02 0 OOOE+OO 0. OOOE+OO 0. OOOE+OO 31 0 .OOOE+OO 3 . 213E-02 0. .OOOE+OO 0. OOOE+OO 0. OOOE+OO 33 0 .OOOE+OO 4 . 724E-02 0. OOOE+OO 0. OOOE+OO 0. OOOE+OO 35 0. .OOOE+OO 6. .830E-02 0. OOOE+OO 0. OOOE+OO 0. OOOE+OO 37 7. . 233E-02 9. . 563E-02 1. 077E+01 3. 487E-06 9. 668E-15 39 8. 078E-02 1 . . 287E-01 3. 595E+00 1. 836E-07 5. 081E-16 41 0. .OOOE+OO 1 . .733E-01 0. OOOE+OO 0. OOOE+OO 0. OOOE+OO 43 0. OOOE+OO 2, , 753E-01 0. OOOE+OO 0. OOOE+OO 0. OOOE+OO 45 0. OOOE+OO 5. 487E-01 0. OOOE+OO 0. OOOE+OO 0. OOOE+OO 47 0. OOOE+OO 1 . .413E+00 0. OOOE+OO 0. OOOE+OO 0. OOOE+OO 49 0. OOOE+OO 3. . 527E + 00 0. OOOE+OO 0. OOOE+OO 0. OOOE+OO 51 3. .641E-01 7 . 076E+00 3. 242E+03 3. 637E+02 9. 943E-07 53 5. . 737E-01 1 . 421E+01 2. 489E+04 1 . 565E+05 4. 275E-04 55 7. . 706E-01 2. .527E+01 6 . 529E+04 2. 702E+06 7 . 379E-03 57 7. 477E-01 3, . 173E + 01 6 . 284E+04 2. 359E+06 6. 437E-03 59 5. 633E-01 3. 067E+01 2. 502E+04 1 . 527E+05 4. 165E-04 61 4. 934E-01 3. 620E+01 1 . 616E+04 4. 467E+04 1 . 217E-04 63 6. 618E-01 7 . , 173E+01 8. 333E+04 6 . 695E+06 1 . 822E-02 65 7. 490E-01 1 . 851E+02 2. 775E+05 2. 652E+08 7 . 210E-01 67 5. 957E-01 4 . 994E+02 5. 374E+05 2. 150E+09 5. 842E+00 69 3. 865E-01 1 . 359E+03 8. 806E+05 1 . 159E+10 3. 150E+01 71 2. 262E-01 3. 769E+03 1 . 429E+06 6 . 531E+10 1 . 774E+02 73 1 . 471E-01 9 . 053E+03 2. 228E+06 3. 054E+11 8. 294E+02 75 1 . 274E-01 1 . . 284E+04 2. 708E+06 5. 813E+1 1 1 . 579E+03 77 1 . 247E-01 1 . 359E+04 2. 801E+06 6 . 481E+1 1 1 . 760E+03 79 to Model 05310191 SS 3000./ 2.00/ -3.00 CN, TiO, H20 (hm) TauRM Col Mass r/R* - 1 T n 1 1 .OOOE-05 1 , .711E-02 1 . 209E -03 2 .056E+03 6. .015E+12 3 1 .585E-05 2 .292E-02 1 . 171E -03 2 .065E+03 8 .018E+12 5 2 .512E-05 3 .049E-02 1 . 135E -03 2 .076E+03 1 .061E+13 7 3 .981E-05 4. .040E-02 1 .099E -03 2 .087E+03 1 . .399E+13 9 6 .3 TOE-05 5 .343E-02 1 .063E -03 2 .099E+03 1 . .840E+13 1 1 1 .000E-04 7 .061E-02 1 .028E -03 2 . 1 12E+03 2 .416E+13 13 1 .585E-04 9 . 335E-02 9 .934E -04 2 . 127E+03 3 .173E+13 15 2 .512E-04 1 . . 236E-01 9 .585E -04 2 . 142E+03 4. . 170E+13 17 3 .981E-04 1 .640E-01 9 .234E -04 2. . 159E+03 5. .491E+13 19 6 .310E-04 2 .185E-01 8 .878E -04 2. . 178E+03 7. .252E+13 21 1 .000E-03 2, .924E-01 8 .517E -04 2. . 198E+03 9. .617E+13 23 1 .585E-03 3. .936E-01 8 . 146E -04 2 . 220E+03 1 . .282E+14 25 2 . 512E-03 5. .333E-01 7 . 766E -04 2 .244E+03 1 . .719E+14 27 3 .981E-03 7 . 283E-01 7 . 373E -04 2 . 270E+03 2, .320E+14 29 6 .310E-03 1 . .003E+00 6 .965E -04 2 . 300E+03 3. .152E+14 31 1 .OOOE-02 1 . . 393E+00 6 .539E -04 2, . 334E+03 4. . 316E+14 33 1 . 585E-02 1 . .955E+00 6 .092E -04 2. . 374E+03 5. .955E+14 35 2 .512E-02 2. . 773E+00 5. .618E -04 2. .422E+03 8. . 284E+14 37 3 .981E-02 3. .988E+00 5 . 106E -04 2 .479E+03 1 . , 164E+15 39 6 .310E-02 5. . 843E+00 4 . 542E -04 2. . 549E+03 1 , .658E+15 41 1 . .OO0E-O1 8. . 822E+00 3. .896E -04 2. .636E+03 2, .422E+15 43 1 . .585E-01 1 . .390E+01 3. . 130E -04 2. . 743E + 03 3. 666E+15 45 2. .512E-01 2. ,321E+01 2. . 191E -04 2. 878E+03 5. 836E+15 47 3. .981E-01 4, .143E+01 1 . .028E -04 3. .052E+03 9, .827E+15 49 6. .310E-01 7 . 523E+01 -2, .906E -05 3, , 269E + 03 1 . 666E+16 51 1 , .OOOE+00 1 , .203E+02 -1 . 433E -04 3, .517E+03 2. 477E+16 53 1 , .468E+00 1 . .545E+02 -2, .092E -04 3 707E+03 3. 019E+16 55 2. .154E+00 1 . 836E+02 -2. .575E -04 3. 904E+03 3. 406E+16 57 3 . 162E+00 2. 075E+02 -2. .938E -04 4. . 136E + 03 3. 635E+16 59 4, .642E+00 2. .283E+02 -3, . 240E -04 4, 419E+03 3. . 743E+16 61 6, .813E+00 2. .517E+02 -3, , 574E ^ 04 4. , 760E + 03 3. 831E+16 63 1 . .OOOE+01 2. 863E+02 -4. 050E -04 ' 5, . 156E + 03 4. 023E+16 65 1 . .445E+01 3. .306E+02 -4. 624E -04 5, . 592E+03 4. . 283E+16 67 2. .089E+01 3. 689E+02 -5. 098E -04 6. 084E+03 4. ,393E+16 69 3. .020E+01 3. 937E+02 -5. . 405E -04 6 . . 635E+03 4. .299E+16 71 4. . 365E + 01 4. 084E+02 -5. . 594E -04 7 . , 247E+03 4. 083E+16 73 6 , 310E+01 4. ,169E+02 -5. , 711E -04 7 . 924E+03 3. .811E+16 75 8. 610E+01 4. 211E+02 -5. . 774E -04 8. , 550E+03 3. 568E+16 77 9 , 750E+01 4. .223E+02 -5. . 793E -04 8, .8T5E+03 3. 471E+16 79 9 . 950E+01 4. .225E+02 -5. . 795E -04 8. 859E+03 3. 455E+16 Summary of physica l quant i t ies Page 232 Pe P_gas P_rad Mu Rho ChiRM 3. .281E-06 1 .707E+00 1 .067E-01 1 .448E+00 1 . .447E- 1 1 9 .086E -04 1 4. .335E-06 2 . 286E+00 1 .067E-01 1 .473E+00 1 . .961E- 1 1 1 . 105E -03 3 5. 698E-06 3 .041E+00 1 .068E-01 1 .497E+00 2 .638E- 1 1 1 . 340E -03 5 7. . 473E-06 4 .031E+00 1 .068E-01 1 .520E+00 3 .530E- 1 1 1 .619E -03 7 9. . 793E-06 5 . 331E + 00 1 .069E-01 1 .541E+00 4. .707E- 1 1 1 .949E -03 9 1 , . 284E-05 7 .045E+00 1 .070E-01 1 .561E+00 6, .262E- 1 1 2 . 339E -03 1 1 1 . 689E-05 9 . 314E + 00 1 .071E-01 1 . 579E+00 8 .319E- 1 1 2 . 795E -03 13 2. .228E-05 1 .233E+01 1 .072E-01 1 . 596E+00 .1 .105E- 10 3 . 323E -03 15 2. 953E-05 1 .637E+01 1 .074E-01 1 .610E+00 1 . 468E- 10 3 .926E -03 17 3. 937E-05 2 .180E+01 1 .076E-01 1 .623E+00 1 .955E- 10 4 .604E -03 19 5, , 286E-05 2 .918E+01 1 .080E-01 1 .635E+00 2. .610E- 10 5. . 355E -03 21 7 . . 159E-05 3 .928E+01 1 .084E-01 1 .644E+00 3 .499E- 10 6 . 174E -03 23 9, . 793E-05 5 . 324E + 01 1 .091E-01 1 .651E+00 4. .712E- 10 7 .051E -03 25 1 . .356E-04 7 .27OE+01 1 .099E-01 1 .656E+00 6 . .377E- 10 7 . .977E -03 27 1 . 903E-04 1 .001E+02 1 . 111E-01 1 . 656E+00 8. 671E- 10 8 .935E -03 29 2. , 714E-04 1 . 391E+02 1 . 128E-01 1 .652E+00 1 , ,184E- 09 9 .907E -03 31 3, 945E-04 1 .952E+02 1 .152E-01 1 .642E+00 1 . .623E- 09 1 .086E -02 33 5, 871E-04 2 .769E+02 1 .186E-01 1 .622E+00 2. .232E- 09 1 . . 172E -02 35 8. 989E-04 3 .983E+02 1 . 236E-01 1 .593E+00 3. 078E- 09 1 . . 238E -02 37 •1 . 428E-03 5 .836E+02 1 . 310E-01 1 .552E+00 4. . 273E- 09 1 . 259E -02 - 39 2. .390E-03 8 .812E+02 1 .418E-01 1 .502E+00 6 . .038E- 09 1 . . 212E -02 41 4. 282E-03 1 . 388E+03 1 .577E-01 1 .446E+00 8. .804E- 09 1 . 092E -02 43 8. . 372E-03 2. . 319E+03 1 .818E-01 1 . 390E + 00 1 . .347E- 08 9, . 108E -03 45 1 . 816E-02 4 . 140E + 03 2 .193E-01 1 .339E+00 2. .186E- 08 7 . . 322E -03 47 4. .329E-02 7. . 520E+03 2 . 792E-01 1 .298E+00 3. .591E- 08 6 . 874E -03 49 1 . 048E-01 1 . 203E+04 3 .699E-01 1 .269E+00 5. .220E- 08 1 . 045E -02 51 1 , 974E-01 1 .545E+04 4 .622E-01 1 .257E+00 6 . 302E- 08 1 . . 778E -02 53 3. 569E-01 1 . .836E+04 5 .767E-01 1 .249E+00 7 . 067E- 08 3. . 128E -02. 55 6 . 421E-01 2 .075E+04 7 .335E-01 1 .245E+00 7 . .512E- 08 5. . 586E -02 57 1 . .127E+00 2 . 283E + 04 9 .606E-01 1 .242E+00 7. . 717E- 08 8. . 553E -02 59 1 . 834E+00 2 . 518E+04 1 .294E+00 1 .240E+00 7. 890E- 08 9 . 481E -02 61 2. 873E+00 2. .864E+04 1 .785E+O0 1 .239E+00 8. .280E- 08 9 . 078E -02 63 5. 690E+00 3 . 307E+04 2 .473E+00 1 . 239E + 00 8. 812E- 08 1 . . 205E -01 65 1 . 570E+01 3. .690E+04 3 .469E+00 1 . 239E + 00 9. 035E- 08 2. 457E -01 67 4 . 803E+01 3 .938E+04 4 .909E+00 1 .238E+00 8. 834E- 08 5. 862E -01 69 1 . 437E+02 4. .085E+04 6 .992E+00 1 . 235E+00 8. 371E- 08 1 . . 452E + 00 71 4. 034E+02 4. . 170E + 04 1 .OOOE+01 1 .227E+00 7 . 766E- 08 3. 658E+00 73 9. 073E+02 4. .212E+04 1 . 357E+01 1 .212E+00 7 . 183E- 08 8. 077E+00 75 1 . 233E+03 4. . 224E + 04 1 .534E+01 1 .203E+00 6 . 933E- 08 1 . . 111E + 01 77 1 . 296E+03 4. , 226E+04 1 .564E+01 1 .201E+00 6. 891E- 08 1 . . 170E + 01 79 Model 05310191 SS 3000./ 2.00/ -3.00 CN, TiO, H20 (hm) TauRM r TE res RE res HE res DE res 1 1 . 000E-05 9 .021E+12 3. . 105E-07 -4. , 102E-12 1 . .150E- 08 0. OOOE+OO 3 1 . 585E-05 9 .021E+12 2. 470E- 07 -3. 087E- 12 2. 068E- 10 3. 658E- 13 5 2. .512E-05 9 .021E+12 2. . 140E-13 -2. , 189E-12 1 , .214E- 09 4, , 382E- 13 7 3. .981E-05 9 .020E+12 2. 703E- 13 -1 . . 304E- 12 9. 055E- 10 4. 256E- 13 9 6. . 310E-05 9 .020E+12 1 . 434E- 13 -5. . 165E-13 - 1 . .654E- 10 3. , 700E- 13 1 1 1 . .000E-04 9 .020E+12 3. . 691E- 13 - 1 . 281E- 13 - 1 , ,609E- 09 2. 807E- 13 13 1 . .585E-04 9 .019E+12 2. 967E- 13 - 1 . 512E- 13 -2. ,360E- 09 2. .255E- 13 15 2. .512E-04 9 .019E+12 3. 631E- 13 -3. 562E- 13 - 1 . .805E- 09 2. .661E- 13 17 3. .981E-04 9 .019E+12 1 . . 133E-13 -5. 759E- 13 -4. .860E- 10 3. 634E- 13 19 6. .310E-04 9 .019E+12 1 . 893E- 13 -9 . 316E- 13 1 , .283E- 10 4, . 179E-13 21 1 . .000E-03 9 .018E+12 1 . . 339E- 13 -1 . .387E- 12 -3. .297E- 10 4, ,078E- 13 23 1 .585E-03 9 .018E+12 1 . .720E- 13 - 1 . 875E- 12 -8. .778E- 10 3, . 984E- 13 25 2. .512E-03 9 .018E+12 2. . 715E- 13 -2, , 330E- 12 - 1 , .198E- 09 3. 849E- 13 27 3 .981E-03 9 .017E+12 2, 428E- 13 -2. . 774E- 12 - 1 . .451E- 09 3 . 256E- 13 29 6 .310E-03 9 .017E+12 2. . 788E- 13 -3. 040E- 12 - 1 . .727E- 09 2, .071E- 13 31 1 . .000E-02 9 .016E+12 3. 034E- 13 -3. 014E- 12 - 1 . .960E- 09 4, .474E- 14 33 1 .585E-02 9 .016E+12 3. 015E- 13 -2, 667E- 12 - 1 , .955E- 09 -1 . 044E- 13 35 2. .512E-02 9 016E+12 3. .162E- 13 -2. . 125E-12 - 1 . .435E- 09 - 1 . .259E- 13 37 3 .981E-02 9 .015E+12 4. .731E- 13 -1 . .617E- 12 4. .807E- 10 3 .718E- 13 39 6 .310E-02 9 .015E+12 1 . •015E- 12 -1 . 459E- 12 6. .470E- 09 2 .654E- 12 41 1 .000E-01 9 .014E+12 4. .414E- 12 -2, •296E- 12 1 . .608E- 08 7 .791E- 12 43 1 .585E-01 9 .013E+12 1 . 892E- 11 -6 . 473E- 12 3. .412E- 08 1 . .982E- 11 45 2. . 512E-01 9 .013E+12 7. 982E- 11 -3, 331E- 1 1 3 .288E- 08 3 .047E- 1 1 47 3 .981E-01 9 .01 1E+12 2. , 439E- 10 -1 . 367E- 10 -2 .320E- 07 - 1 .025E- 10 49 6 .310E-01 9 .010E+12 3. 987E- 10 -2. 008E- 10 -2. .107E- 07 -2 .192E- 10 51 1 .OOOE+00 9 .009E+12 2, .183E- 10 -3. 593E- 11 9 .046E- 07 1 .894E- 10 53 1 .468E+00 9 .009E+12 1 , 687E- 11 -7 . 231E- 13 - 1 . .498E- 07 -9 .289E- 12 55 2 .154E+00 9 .008E+12 1 . .484E- 12 2. .352E- 13 -5 .094E- 09 1 .985E- 12 57 3 . 162E + 00 9 .008E+12 2, .885E- 12 3. .245E- 12 - 1 .320E- 08 -1 .407E- 12 59 4 .642E+00 9 .008E+12 4. 660E- 12 5. .644E- 12 -2 .948E- 09 -2 .454E- 12 61 6 .813E+00 9 .007E+12 7. .892E- 13 4, .315E- 13 1 .300E- 08 3 .951E- 12 63 1 .OOOE+01 9 .007E+12 1 , .077.E- 13 1 . .126E- 13 2 .182E- 09 -4 .414E- 14 65 1 .445E+01 9 .006E+12 1 . .629E- 13 2 .533E- 13 -6 .905E- 09 -4 .415E- 13 67 2 .089E+01 9 .006E+12 6, , 438E- 14 9. . 564E- 14 - 1 . .693E- 09. -4 .608E- 15 69 3 .020E+01 9 .006E+12 3. .093E- 14 4. .832E- 14 -6 .982E- 1 1 5 .604E- 14 71 4 .365E+01 9 .O05E+12 1 . . 461E- 14 2. .458E- 14 6 .539E- 1 1 7 .685E- 14 73 6 .310E+01 9 .005E+12 6. .861E- 15 1 . . 182E-14 1 . .991E- 1 1 6 .241E- 14 75 8 .610E+01 9 .005E+12 3. .765E- 15 6. .243E- 15 -3 .169E- 12 1 .242E- 14 77 9 .750E+01 9 .005E+12 3. .350E- 15 6. 583E- 15 -4 .903E- 12 -2 .625E- 15 79 9 .950E+01 9 .005E+12 2. 093E- 15 7 . .322E- 16 4 . 103E-13 1 .325E- 16 Convergence checks Page 233 EPS rms 3. 062E- 07 2. 436E- 07 1 , . 163E-10 8. 673E- 11 1 . .584E- 11 1 . 542E- 10 2. . 260E- 10 1 , 729E- 10 4. 655E- 11 1 , 229E- 11 3. .158E- 1 1 8. 408E- 11 1 . . 148E-10 1 . ,390E- 10 1 . 654E- 10 1 , 878E- 10 1 . 872E- 10 1 .374E- 10 4. .604E- 11 6, .197E- 10 1 . . 540E- 09 3. .269E- 09 3. ,150E- 09 2. .222E- 08 2 ,019E- 08 8 .665E- 08 1 . .435E- 08 4 .879E- 10 1 .264E- 09 2 .824E- 10 1 . .245E- 09 2 .090E- 10 6 .614E- 10 1 621E- 10 6 687E- 12 6. .263E- 12 1 .907E- 12 3 .036E- 13 4 696E- 13 3 .935E- 14 g_ef f /g 1.OOOE+00 1.OOOE+00 1.OOOE+00 1.OOOE+00 1.OOOE+OO 1.OOOE+00 1 .OOOE+OO 1.OOOE+00 1.OOOE+00 1.OOOE+00 1.OOOE+00 1.OOOE+00 1.OOOE+00 1 .OOOE + 00 1 .OOOE + 00 1.OOOE+00 1 .OOOE + 00 1.OOOE+00 1 .OOOE+00 1.OOOE+00 1.OOOE+00 1 .OOOE + 00 1.OOOE+OO 1.OOOE+00 1.OOOE+00 1.OOOE+00 1.OOOE+00 1.OOOE+00 9.999E-01 9.999E-01 9.999E-01 9 .999E-01 9 .998E-01 9.996E-01 9.991E-01 9.978E-01 9.944E-01 9.876E-01 9.830E-01 9.821E-01 Chi_H/RM 3 . 456E+01 2. .864E+01 2 . 376E+01 1 .970E+01 1 . .632E+01 1 . 352E + 01 1 . .121E+01 9 . 322E + 00 7. . 782E+00 6. .543E+00 5 . 549E+00 4 .752E+00 4. .111E+00 3 . 593E + 00 3 .178E+00 2 .846E+00 2 .578E+00 2 .358E+00 2 .173E+00 2 .011E+00 1 .869E+00 1 .753E+00 1 .692E+00 1 .681E+00 1 .685E+00 1 .437E+00 1 .145E+00 1 .024E+00 9 .889E-01 9 .936E-01 9 .930E-01 9 .961E-01 9 .997E-01 1 .001E+00 1 .001E+00 1 .002E+00 1 .001E+00 1 .OOOE+00 1 .OOOE+00 9 .999E-01 Hconv/H 0. .OOOE+00 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 0 .OOOE+OO 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 0 .OOOE+OO 0 .OOOE+00 0 .OOOE+00 0 .OOOE+OO 0 .OOOE+00 4 .196E-09 2 .843E-07 1 .145E-06 1 .926E-06 2 .195E-06 1 .932E-06 1 .224E-06 2 .702E-06 1 .480E-05 1 .680E-04 8 .838E-03 1 .203E-01 3 .453E-01 2 . 309E-01 6 . 763E-02 2 .296E-01 2 . 401E + 00 1 . 285E + 01 5 .278E+01 2 . 185E + 02 7 .637E+02 1 .284E+03 1 . 402E + 03 Lum/L' 9 .889E-01 9.889E-01 9.889E-01 9.889E-01 9.889E-01 9.889E-01 9.889E-01 9 .889E-01 9 .888E-01 9.888E-01 9.888E-01 9.888E-01 9.888E-01 9.888E-01 9.888E-01 9.888E-01 9.888E-01 9.887E-01 9.887E-01 9.887E-01 9.886E-01 9.886E-01 9.885E-01 9.885E-01 9.903E-01 1 .005E + 00 1 .010E + 00 1.009E+00 1.008E+00 1.006E+00 1.006E+00 1 .005E+00 1.004E+00 1.003E+00 1.002E+00 1.001E+00 1.001E+00 1.OOOE+00 1.OOOE+00 1.OOOE+00 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 ^ co Model 05310191 SS 3000./ 2.00/ -3.00 CN, TiO, H20 (hm) TauRM 1 1 .000E-05 3 1 .585E-05 5 2 .512E-05 7 3 .981E-05 9 6 .310E-05 11 1 .OOOE-04 13 1 .585E-04 15 2 .512E-04 17 3 .981E-04 19 6 .310E-04 21 1 .OOOE-03 23 1 .585E-03 25 2 .512E-03 27 3 .981E-03 29 6 .310E-O3 31 1 .000E-02 33 1 .585E-02 35 2 .512E-02 37 3 .981E-02 39 6 .310E-02 41 1 ;000E-01 43 1 . .585E-01 45 2 .512E-01 47 3 .981E-01 49 6. .310E-01 51 1 . .OOOE+OO 53 1 . .468E+00 55 2. .154E+00 57 3. . 162E + 00 59 4. .642E+00 61 6. 813E+00 63 1 . OOOE+01 65 1 . 445E+01 67 2. 089E+01 69 3. 020E+01 71 4. .365E+01 73 6. ,310E+01 75 8. 610E+01 7 7 9. 750E+01 79 9. 950E+01 Cp/Cv 1 .207E+00 1.210E+00 1.212E+00 1.213E+00 1.214E+00 1.215E+00 1.216E+00 1.217E+00 1.217E+00 1.218E+00 1.218E+00 1.219E+00 1.220E+00 1.221E+00 1.222E+00 1.224E+00 1.227E+00 1.230E+00 1.233E+00 1.235E+00 1.235E+00 1.233E+00 T.227E+00 1.225E+00 1.236E+00 1.282E+00 1.345E+00 1.428E+00 1.522E+00 1.598E+00 1.639E+00 1.653E+00 1.646E+00 1.618E+00 1.553E+00 1.448E+00 1.333E+00 1.261E+00 1.242E+00 1.239E+00 H_P 1.257E+09 1.223E+09 1.196E+09 1.175E+09 1.158E+09 1.144E+09 1.135E+09 1.128E+09 1.124E+09 1.123E+09 1.124E+09 1.127E+09 1.134E+09 1.143E+09 1.157E+09 1.177E+09 1.205E+09 1.243E+09 1.296E+09 1.367E+09 1.461E+09 1.578E+09 1.722E+09 1.895E+09 2.094E+09 2.303E+09 2.451E+09 2.597E+09 2.761E+09 2.957E+09 3.189E+09 3.456E+09 3.749E+09 4.081E+09 4.453E+09 4.875E+09 5.364E+09 5.859E+09 6.088E+09 6.128E+09 Qconv 3.600E+00 3.715E+00 3.798E+00 3.855E+00 3.889E+00 3.904E+00 3.905E+00 3.894E+00 3.874E+00 3.849E+00 3.819E+00 3.787E+0O 3.752E+00 3.718E+00 3.683E+0O 3.649E+00 3.615E+00 3.575E+00 3.517E+00 3.416E+00 .3.241E + 00 2.964E+00 2.579E+00 2.127E+00 1.685E+00 1.348E+00 1.199E+00 1.109E+00 1.055E+00 1.027E+00 1.013E+00 1.005E+00 1.004E+00 1.006E+00 1.016E+00 1.042E+00 1.107E+00 1.224E+00 1.296E+00 1.309E+00 DELrad 1 .485E -02 1 .676E -02 1 . 830E -02 1 .989E -02 2 . 159E -02 2 . 339E -02 2 . 525E -02 2 . 708E -02 2 . 884E -02 3 .059E -02 3 . 231E -02 3 .428E -02 3 .657E -02 3 .933E -02 4 . 280E -02 4 . 725E -02 5 . 304E -02 6 .027E -02 6. 864E -02 7 . . 709E -02 8. 441E -02 9. 079E -02 9 . , 705E -02 1 . 061E -01 1 . . 328E -01 1 . 862E -01 2. 510E -01 3. 821E -01 5. 9 11E -01 7. 621E -01 7. 045E -01 5. 686E -01 6. 360E -01 1 . 037E+0O 1 . 868E+0O 3. 374E+00 6. 095E+00 9. 897E+00 1 . 201E+01 1 . 241E+01 DELad 5 . 237E -02 5 . 152E -02 5 .099E -02 5 .070E -02 5 .060E -02 5 .064E -02 5. .081E -02 5. . 108E -02 5. . 143E -02 5. . 187E -02 5. . 237E -02 5. . 294E -02 5. , 357E -02 5. . 428E -02 5. . 507E -02 5. 597E -02 5. . 700E -02 5. 825E -02 5. 984E -02 6. 200E -02 6. 512E -02 6 . 992E -02 7 . 779E -02 9 . 165E -02 1 . 179E -01 1 . 667E -01 2. 168E -01 2. 727E -01 3. 271E -01 3. 663E -01 3. 863E -01 3. 932E -01 3. 912E -01 3. 795E -01 3. 508E -01 2. 974E -01 2. 266E -01 1 . 707E -01 1 . 523E -01 1 . 496E -01 Convective quant i t ies Page 234 DELbub TauRMb Vconv Hconv Hconv/H 0 .OOOE+OO 2 .643E-05 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 1 0 .OOOE+OO 4 . 242E-05 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 3 0 .OOOE+OO 6 . 764E-05 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 5 0 .OOOE+OO 1 .074E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 7 0 .OOOE+OO 1 .700E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 9 0 .OOOE+OO 2 .682E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 11 0 .OOOE+OO 4 .222E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 13 0 .OOOE+OO 6 .627E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 15 0 .OOOE+OO 1 .037E-03 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 17 0 .OOOE+OO 1 .617E-03 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 19 0 .OOOE+OO 2 .514E-03 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 21 0 .OOOE+OO 3 .897E-03 0. .OOOE+OO 0 .OOOE+OO 0. .OOOE+OO 23 0 .OOOE+OO 6 .027E-03 0. OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 25 0 .OOOE+OO 9 . 306E-03 0 OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 27 0 .OOOE+OO 1 .435E-02 0. OOOE+OO 0. .OOOE+OO 0 .OOOE+OO 29 0 .OOOE+OO 2 .210E-02 0. OOOE+OO 0, .OOOE+OO 0 .OOOE+OO 31 0 .OOOE+OO 3 . 396E-02 0. OOOE+OO 0, .OOOE+OO 0 .OOOE+OO 33 6 .027E-02 5 200E-02 5. . 858E+02 1, .532E+00 4, .196E-09 35 6 860E-02 7. 898E-02 2. . 165E+03 1. 038E+02 2. .843E-07 37 7 702E-02 1 . . 177E-01 3. 124E+03 4. .179E+02 1 . . 145E-06 39 8. 433E-02 1 . , 710E-01 3. 365E+03 7. 031E+02 1 . 926E-06 41 9. 072E-02 2, 428E-01 3. 175E+03 8. 016E+02 2. .195E-06 43 9. 699E-02 3. . 381E-01 2. 736E+03 7. 059E+02 1 . 932E-06 45 1 . 061E-01 4. 852E-01 2. 1 12E + 03 4. 473E+02 1 . 224E-06 47 1 .327E-01 8. 271E-01 2. 535E+03 9. 876E+02 2: 702E-06 49 1 . .861E-01 2. 010E+00 4. 429E+03 5. 410E+03 1 . 480E-05 51 2. 499E-01 4. .395E+00 1 . 020E+04 6. 143E+04 1 . 680E-04 53 3. .650E-01 9. 184E+00 3. 972E+04 3. 231E+06 8. 838E-03 55 4. 866E-01 1 . 854E+01 9. 874E+04 4. 399E+07 1 . 203E-01 57 5. 477E-01 3. 123E+01 1 . 444E+05 1 . 263E+08 3. 453E-01 59 5. 472E-01 3. 817E+01 1 . 276E+05 8. 445E+07 2. 309E-01 61 5. 054E-01 4 . 157E+01 8. 389E+04 2. 474E+07 6 . 763E-02 63 5. 099E-01 6 . 367E+01 1 . 233E+05 8. 399E+07 2. 296E-01 65 5. 043E-01 1 . 449E+02 2. 647E+05 8. 782E+08 2. 401E+00 67 4. 434E-01 3. 690E+02 4. 543E+05 4. 700E+09 1 . 285E+01 69 3. 536E-01 9. 479E+02 7 . 011E + 05 1 . 931E+10 5. 278E+01 71 2. 548E-01 2. 439E+03 1 . 054E+06 7. 995E+10 2. 185E+02 73 1 . 842E-01 5. 438E+03 1 . 494E+06 2. 794E+1 1 7 . 637E+02 75 1 . 621E-01 7 . 500E+03 1 . 731E+06 4. 699E+11 1 . 284E+03 77 1 . 589E-01 7 . 904E+03 1 . 775E+06 5. 128E+1 1 1 . 402E+03 79 Mode"! 06310191 SS 3500./ 1.00/ -3.00 CN, TiO, H20 (hm) TauRM Col Mass r/R* - 1 T n 1 1 000E-05 5 772E-01 1 968E -03 2 670E+03 1 559E+13 3 1 585E-05 7 668E-01 1 196E -03 2 687E+03 2 059E+13 5 2 512E-05 1 024E+00' 4 061E -04 2 708E+03 2 728E+13 7 3 981E-05 1 369E+00 -3 958E -04 2 733E+03 3 620E+13 9 6 310E-05 1 833E+00 - 1 206E -03 2 760E+03 4 802E+13 1 1 1 000E-04 2 452E+00 -2 023E -03 2 790E+03 6 364E+13 13 1 585E-04 3 276E+00 -2 843E -03 2 821E+03 8 418E+13 15 2 512E-04 4 366E+O0 -3 664E -03 2 854E+03 1 111E+14 17 3 981E-04 5 800E+00 -4 485E -03 2 889E+03 1 460E+14 19 6 310E-04 7 672E+00 -5 301E -03 2 924E+03 1 910E+14 21 1 000E-03 1 010E+01 -6 1 1 1E -03 2 960E+03 2 487E+14 23 1 585E-03 1 321E+01 -6 913E -03 2 996E+03 3 220E+14 25 2 512E-03 1 719E+01 -7 706E -03 3 034E+03 4 145E+14 27 3 981E-03 2 223E+01 -8 489E -03 3 071E+03 5 301E+14 29 6 310E-03 2 856E+01 -9 261E -03 3 109E+03 6 738E+14 31 1 000E-02 3 647E+01 -1 002E -02 3 149E+03 8 509E+14 33 1 585E-02 4 628E+01 - 1 077E -02 3 190E+03 1 067E+15 35 2 512E-02 5 835E+01 - 1 151E -02 3 233E+03 1 329E+15 37 3 981E-02 7 303E+01 - 1 223E -02 3 280E+03 1 642E+15 39 6 310E-02 9 067E+01 - 1 294E -02 3 332E+03 2 010E+15 41 1 000E-01 1 1 15E+02 - 1 363E -02 3 391E+03 2 432E+15 43 1 585E-01 1 355E+02 - 1 429E -02 3 461E+03 2 901E+15 45 2 512E-01 1 623E+02 - 1 491E -02 3 547E+03 3 393E+15 47 3 981E-01 1 904E+02 - 1 548E -02 3 660E+03 3 862E+15 49 6 310E-01 2 175E+02 - 1 597E -02 3 811E+03 4 240E+15 51 1 OOOE+00 2 424E+02 - 1 638E -02 4 019E+03 4 484E+15 53 1 468E+00 2 640E+02 - 1 673E -02 4 247E+03 4 624E+15 55 2 154E+00 2 940E+02 - 1 720E -02 4 513E+03 4 849E+15 57 3 162E+00 3 442E+02 - 1 792E -02 4 809E+03 5 332E+15 59 4 642E+00 4 200E+02 - 1 890E -02 5 154E+03 6 082E+15 61 6 813E+00 4 985E+02 - 1 980E -02 5 564E+03 6 695E+15 63 1 OOOE+01 5 532E+02 -2 039E -02 6 041E+03 6 849E+15 65 1 445E+01 5 842E+02 -2 073E -02 6 562E+03 6 660E+15 67 2 089E+01 6 016E+02 -2 092E -02 7 149E+03 6 295E+15 69 3 020E+01 6 110E+02 -2 104E -02 7 802E+03 5 856E+15 71 4 365E+01 6 158E+02 -2 11'OE -02 8 526E+03 5 398E+15 73 6 310E+01 6 182E+02 -2 114E -02 9 327E+03 4 950E+15 75 8 610E+01 6 194E+02 -2 116E -02 1 007E+04 4 590E+15 77 9 750E+01 6 198E+02 -2 1 17E -02 1 038E+04 4 452E+15 79 9 950E+01 6 198E+02 -2 117E -02 1 043E+04 4 430E+15 Summary of physica l quant i t ies Page 235 Pe P_gas P_rad Mu Rho ChiRM 3 984E-05 5 748E+00 2 031E-01 1 242E+00 3 217E - 11 2 841E- 05 1 5 227E-05 7 638E+00 2 035E-01 1 243E+00 4 249E - 11 3 316E- 05 3 6 952E-05 1 020E+01 2 042E-01 1 243E+00 5 631E - 11 3 895E- 05 5 9 330E-05 1 366E+01 2 050E-01 1 243E+00 7 474E - 11 4 594E- 05 7 1 261E-04 1 830E+01 2 061E-01 1 244E+00 9 918E -11 5 438E- 05 9 1 712E-04 2 451E+01 2 075E-01 1 244E+00 1 315E - 10 6 463E- 05 1 1 2 333E-04 3 279E+01 2 091E-01 1 244E+00 1 739E - 10 7 718E- 05 13 3 182E-04 4 376E+01 2 110E-01 1 245E+00 2 295E - 10 9 270E- 05 15 4 340E-04 5 821E+01 2 133E-01 1 245E+00 3 017E - 10 1 121E- 04 17 5 909E-04 7 711E+01 2 160E-01 1 245E+O0 3 949E - 10 1 366E- 04 19 8 024E-04 1 016E+02 2 192E-01 1 245E+00 5 143E - 10 1 677E- 04 21 1 085E-03 1 332E+02 2 228E-01 1 245E+00 6 660E - 10 2 075E- 04 23 1 463E-03 1 736E+02 2 271E-01 1 246E+00 8 573E - 10 2 587E- 04 25 1 962E-03 2 248E+02 2 322E-01 1 246E+00 1 097E -09 3 249E- 04 27 2 622E-03 2 892E+02 2 383E-01 1 246E+00 1 394E -09 4 108E- 04 29 3 496E-03 3 699E+02 2 457E-01 1 246E+00 1 761E -09 5 231E- 04 31 4 656E-03 4 701E+02 2 549E-01 1 246E+00 2 209E -09 6 710E- 04 33 6 208E-03 5 934E+02 2 665E-01 1 246E+00 2 751E -09 8 688E- 04 35 8 309E-03 7 438E+02 2 815E-01 1 246E+00 3 398E -09 1 138E- 03 37 1 120E-02 9 247E+02 3 009E-01 1 246E+00 4 158E -09 . 1 513E- 03 39 1 528E-02 1 139E+03 3 261E-01 1 245E+00 5 029E -09 2 050E- 03 41 2 123E-02 1 386E+03 3 591E-01 1 245E+00 5 995E -09 2 856E- 03 43 3 049E-02 1 662E+03 4 024E-01 1 244E+00 7 008E -09 4 164E- 03 45 4 593E-02 1 951E+03 4 609E-01 1 243E+00 7 969E -09 6 546E- 03 47 7 281E-02 2 231E+03 5 441E-01 1 242E+00 8 741E -09 1 120E- 02 49 1 199E-01 2 488E+03 6 691E-01 1 241E+00 9 236E -09 1 888E- 02 51 1 802E-01 2 71 1E+03 8 245E-01 1 240E+00 9 519E -09 2 334E- 02 53 2 553E-01 3 021E+03 1 042E+00 1 239E+00 9 979E -09 2 179E- 02 55 3 623E-01 3 541E+03 1 343E+00 1 239E+00 1 097E -08 1 898E- 02 57 6 475E-01 4 328E+03 1 779E+00 1 239E+00 1 251E -08 2 135E- 02 59 1 665E+00 5 143E+03 2 422E+00 1 239E+00 1 377E -08 3 829E- 02 61 5 181E+00 5 712E+03 3 370E+00 1 238E+00 1 408E -08 9 150E- 02 63 1 586E+01 6 034E+03 4 698E+00 1 236E+00 1 367E -08 2 310E- 01 65 4 681E+01 6 213E+03 6 621E+00 1 230E+00 1 285E -08 6 086E- 01 67 1 296E+02 6 308E+03 9 401E+00 1 214E+00 1 180E -08 1 674E+0O 69 3 298E+02 6 354E+03 1 342E+01 1 175E+00 1 053E -08 4 767E+00 71 7 514E+02 6 374E+03 1 923E+01 1 093E+00 8 985E -09 1 328E+01 73 1 333E+03 6 380E+03 2 61 1E + 01 9 803E-01 7 472E -09 2 806E+01 75 1 610E+03 6 380E+03 2 952E+01 9 266E-01 6 850E -09 3 567E+01 77 1 656E+03 6 380E+03 3 012E+01 9 177E-01 6 750E -09 3 694E+01 79 to C O Model 06310191 SS 3500./ 1.00/ -3.00 CN, TiO, H20 (hm) TauRM r TE res RE res HE res DE res 1 1 .000E-05 6 .633E+12 6 .072E- 08 - 1 .743E- 09 2 .588E- 06 0 .OOOE+OO 3 1 .585E-05 6 .628E+12 7 .153E- 06 -3 .526E- 07 -6 .563E- 09 -3 .666E- 10 5 2 .512E-05 6 .623E+12 2 .818E- 10 -2 .394E- 10 8 .581E- 08 -1 . 201E- 10 7 3 .981E-05 6 .617E+12 1 .351E- 10 -6 .470E- 11 9 .153E- 08 7 . 852E- 12 9 6 . 310E-05 6 .612E+12 5 .935E- 1 1 - 1 .738E- 11 6 .323E- 08 5 . 107E-11 11 1 .OOOE-04 6 .607E+12 2 .046E- 1 1 -5 .710E- 12 3 .074E- 08 4 .089E- 1 1 13 1 .585E-04 6 .601E+12 8 .620E- 12 -2 .270E- 12 1 .089E- 08 2 .310E- 11 15 2 .512E-04 6 .596E+12 3 .OOOE- 12 - 1 .435E- 12 1 .460E- 09 1 .084E- 1 1 17 3 .981E-04 6 .590E+12 1 .307E- 12 -9 .386E- 13 - 1 .631E- 09 5 .696E- 12 19 6 .310E-04 6 .585E+12 5 .803E- 13 -5 .207E- 13 -2 .424E- 09 3 .259E- 12 21 1 .000E-03 6 .580E+12 2 . 361E- 13 -2 .420E- 13 -2 . 335E- 09 1 .827E- 12 23 1 .585E-03 6 .574E+12 8 .536E- 14 -9 .682E- 14 -1 .922E- 09 9 .529E- 13 25 2 .512E-03 6. .569E+12 2 .607E- 14 -4 .095E- 14 - 1 .425E- 09 4. . 171E-13 27 3 .981E-03 6 , .564E+12 1 .157E- 14 -2 .090E- 14 -9 .919E- 10 1 , .633E- 13 29 6 . 310E-03 .6. .559E+12 1 . .378E- 14 -2 .086E- 14 -6 .630E- 10 4. .759E- 14 31 1 .000E-02 6. .554E+12 1 .543E- 14 -3 .666E- 14 -4 .430E- 10 -3, .298E- 14 33 1 .585E-02 6 . ,549E+12 2. .046E- 14 -6 .838E- 14 -3 .102E- 10 - 1 . 444E- 13 35 2 .512E-02 6, . 544E+12 2. .693E- 14 - 1 . 120E-13 -2 .086E- 10 -2. .604E- 13 37 3 .981E-02 6, .539E+12 3. .322E- 14 - 1 .610E- 13 - 1 .259E- 10 -3, 736E- 13 39 6 . 310E-02 6, . 534E+12 3 .378E- 14 -2 .046E- 13 -5 .218E- 11 -4. .553E- 13 41 1 .000E-01 6 . . 530E+12 2. .586E- 14 -2. .294E- 13 1 . .345E- 11 -5. 027E- 13 43 1 .585E-01 6. .525E+12 1 . .953E- 14 -2 .197E- 13 5 .223E- 11 -6. .148E- 13 45 2 .512E-01 6 . 521E+12 2. 543E- 14 - 1 , .728E- 13 4 .963E- 1 1 - 1 . .121E- 12 47 3 .981E-01 6 . 518E+12 3. .876E- 14 - 1 . 088E- 13 7 .234E- 1 1 -2. 401E- 12 49 6 . 310E-01 6. 514E+12 5. .765E- 14 -5. 483E- 14 6 .655E- 10 - 1 , 063E- 12 51 1 . .OOOE+OO 6 . 512E+12 6. 054E- 14 -3. 880E- 14 1 . 468E- 09 7. 848E- 12 53 1 . . 468E + 00 6 . , 509E+12 3. 576E- 14 -4. 793E- 14 1 , .969E- 11 1 . 771E- 12 55 2, .154E+00 6 . 506E+12 5. . 705E- 14 -4. 717E- 14 -1 . ,826E- 09 - 1 . 028E- 11 57 3. . 162E + 00 6 . 501E+12 4. 576E- 14 -3. 516E- 14 -2. .754E- 09 - 1 . 274E- 11 59 4. , 642E+00 6 . 495E+12 2. 462E- 14 -2. 059E- 14 2. .408E- 09 -3. 826E- 12 61 6 . 813E+00 6. 489E+12 1 . 551E- 14 - 1 . 593E- 14 2. ,854E- 09 1 . 836E- 12 63 1 . .OOOE+01 6 . 485E+12 1 . 257E- 14 - 1 . 219E- 14 2. ,848E- 10 9. ,046E- 13 65 1 . 445E+01 6. 483E+12 1 . 036E- 14 -9. 377E- 15 -6. 529E- 1 1 9. 749E- 15 67 2. 089E+01 6 . 481E+12 8. 787E- 15 -7. 662E- 15 -3. 444E- 1 1 - 1 . 703E- 13 69 3. 020E+01 6. 481E+12 7 . 602E- 15 -6. .497E- 15 -7 . 031E- 12 - 1 . 305E- 13 71 4. . 365E+01 6 . 480E+12 6. 728E- 15 -5. 681E- 15 6. 632E- 13 -2. 254E- 14 73 6. . 310E+01 6. 480E+12 6. 026E- 15 -5. 083E- 15 -1 . 165E- 12 3. 120E- 14 75 8. 610E+01 6. 480E+12 5. 460E- 15 -4. 736E- 15 -9. 079E- 13 5. 839E- 15 77 9. . 750E+01 6. 480E+12 4. 814E- 15 -4. 643E- 15 -1 . 643E- 14 6. 945E- 17 79 9. 950E+01 6. 480E+12 3. 328E- 15 -4. 765E- 15 - 1 . 294E- 16 1 . 792E- 17 Convergence checks Page 236 EPS rms g_ef f /g Ch i_H/RM Hconv/H Lum/L* 2 . 551E- 07 9 .998E -01 2 .452E+02 0 .OOOE+OO 1 .008E+00 1 7 .054E- 06 9 .998E -01 2 .287E+02 0 .OOOE+OO 1 .008E+00 3 8 .224E- 09 9 .998E -01 2 .007E+02 0 .OOOE+OO 1 .008E+00 5 8 .768E- 09 9 .998E -01 1 .682E+02 0 .OOOE+OO 1 .008E+00 7 6 .057E- 09 9 .998E -01 1 .355E+02 0 .OOOE+OO 1 .008E+00 9 2 .944E- 09 9 .998E -01 1 .061E+02 0 .OOOE+OO 1 .008E+00 1 1 1 .043E- 09 9 .998E -01 8 .112E+01 0 .OOOE+OO 1 .008E+00 13 1 . 399E- 10 9 .998E -01 6 .104E+01 0 .OOOE+OO 1 .008E+00 15 1 .562E- 10 9 .999E -01 4 .543E+01 0 .OOOE+OO 1 .008E+00 17 2 .322E- 10 9 .999E -01 3 .359E+01 0 .OOOE+OO 1 .008E+00 19 2 .237E- 10 9 .999E -01 2 .478E+01 0 .OOOE+OO 1 .008E+00 21 1 .841E- 10 9 .999E -01 1 .831E+01 0 .OOOE+OO 1 .008E+00 23 1 .365E- 10 9 .999E -01 1 . 362E+01 0 .OOOE+OO 1 .007E+00 25 9 .501E- 1 1 9 .999E -01 1 .024E+01 0 .OOOE+OO 1 .007E+00 27 6 .351E- 1 1 9 .999E -01 7 . 820E+00 0 .OOOE+OO 1 .007E+00 29 4. .243E- 11 9 .999E -01 6 .071E+00 0 .OOOE+OO 1 .007E+00 31 2. .971E- 1 1 9 . 999E -01 4 .804E+00 0. .OOOE+OO 1 .007E+00 33 1 . .998E- 11 9. . 999E -01 3 .864E+00 0 .OOOE+OO 1 .007E+00 35 1 . 206E- 1 1 9 .999E -01 3 .146E+00 0 .OOOE+OO 1 .007E+00 37 4 .998E- 12 9 999E -01 2 .579E+00 0. OOOE+OO 1 .008E+00 39 1 . .289E- 12 9 .999E -01 2 .117E+00 0. OOOE+OO 1 .008E+00 41 5. .003E- 12 9 .999E -01 1 .741E+00 0. .OOOE+OO 1 .008E+00 43 4. .755E- 12 9 998E -01 1 .456E+00 0. OOOE+OO 1 .008E+00 45 6. 933E- 12 9, 998E -01 1 . 268E + 00 0. OOOE+OO 1 .008E+00 47 6. .375E- 1 1 9 996E -01 1 .164E+00 4, , 280E- 10 1 .007E+00 49 1 . 406E- 10 9 994E -01 1 . . 133E + 00 1 . 023E-06 1 .006E+00 51 1 . 894E- 12 9 . 993E -01 1 , .106E+00 2. 669E-06 1 .006E+00 53 1 . 749E- 10 9 . 994E -01 1 . 031E+00 1 . .109E-07 1 .006E+00 55 2. 637E- 10 9 . 995E -01 9 . 923E-01 0. OOOE+OO 1 .006E+00 57 2. 307E- 10 9 , 994E -01 9 . 904E-01 0. OOOE+OO 1 .005E+00 59 2. 734E- 10 9. 989E -01 9. 967E-01 4. 935E-05 1 .003E+00 61 2. 728E- 11 9 . 974E -01 1 . OOOE+OO 2. 594E-02 1 .002E+00 63 6. 254E- 12 9 . 934E -01 1 . 001E+00 8. 657E-01 1 .001E+00 65 3. 299E- 12 9. 827E -01 1 . 002E+00 7 . 542E+00 9 .997E-01 67 6. 737E- 13 9 . 524E -01 1 . 002E+00 5. 017E+01 9 .986E-01 69 6. 391E- 14 8. 645E -01 1 . 001E+00 3. 813E+02 9 .979E-01 71 1 . 1 18E-13 6 . 229E -01 1 . OOOE+OO 4. 267E+03 9 .984E-01 73 8. 713E- 14 2. 037E -01 1 . OOOE+OO 1 . 104E+05 9. .995E-01 75 5. 021E- 15 - 1 . 223E -02 9 . 999E-01 0. OOOE+OO 9 .999E-01 77 3. 314E- 15 -4. 797E -02 9 . 996E-01 0. OOOE+OO 1 . .OOOE+OO 79 to C O Model 06310191 SS 3500./ 1.00/ -3.00 CN, TiO, H20 (hm) TauRM Cp/Cv H_P Qconv DELrad DELad 1 1 .000E-05 1 .472E+00 1 .857E+10 1 .038E+00 1 .971E-02 2 .881E- 01 3 1 .585E-05 1 .461E+00 1 .850E+10 1 .044E+00 2 .519E-02 2 .868E- 01 5 2 .512E-05 1 .451E+00 1 .849E+10 1 .049E+00 2 .903E-02 2 .848E- 01 7 3 .981E-05 1 .442E+00 1 .853E+10 1 .054E+00 3 .238E-02 2 .824E- 01 9 6 .310E-05 1 .434E+00 1 .861E+10 1 .059E+00 3 .527E-02 2 .801E- 01 11 1 .000E-04 1 .428E+00 1 .872E+10 1 .063E+00 3 . 764E-02 2 .780E- 01 13 1 .585E-04 1 .423E+00 1 .886E+10 1 .066E+00 3 .961E-02 2 .761E- 01 15 2 .512E-04 1 .420E+00 1 .902E+10 1 .069E+00 4 . 121E-02 2 .745E- 01 17 3 .981E-04 1 .417E+00 1 .919E+10 1 .072E+00 4 .258E-02 2 .731E- 01 19 6 .310E-04 1 .414E+00 1 .937E+10 1 .075E+00 4 . 376E-02 2 .720E- 01 21 1 .000E-03 1 .413E+00 1 .956E+10 1 .077E+00 4 .485E-02 2 .710E- 01 23 1 .585E-03 1 .411E+00 1 .976E+10 1 .080E+00 4, . 593E-02 2 .702E- 01 25 2 .512E-03 1 .411E+00 1 .996E+10 1 .082E+00 4. .705E-02 2 .696E- 01 27 3 .981E-03 1 .410E+00 2 .017E+10 1 .083E+00 4, .832E-02 2 .693E- 01 29 6 .310E-03 1 .411E+00 2 .038E+10 1 .085E+00 5, .009E-02 2 .693E- 01 31 1 .000E-02 1 .412E+00 2 .060E+10 1 .085E+00 5. 251E-02 2 .699E- 01 33 1 .585E-02 1 .415E+00 2 .084E+10 1 .085E+00 5 .588E-02 2 .713E- 01 35 2 .512E-02 1 .420E+00 2 .109E+10 1 .084E+00 6 083E-02 2 .739E- 01 37 3 .981E-02 1 .427E+00 2 .136E+10 1 .081E+00 6. . 779E-02 2 .781E- 01 39 6 .310E-02 1 .438E+00 2 .168E+10 1 .076E+00 7 .771E-02 2 .844E- 01 41 1 .000E-01 1 .454E+00 2 .204E+10 1 .069E+00 9. . 253E-02 2, .934E- 01 43 1 .585E-01 1 .477E+00 2 .247E+10 1 .060E+00 1 . . 171E-01 3 057E- 01 45 2 .512E-0T 1 .508E+00 2. .301E+10 1 .048E+00 1 . 618E-01 3 224E- 01 47 3 .981E-01 1 .548E+00 2. .374E+10 1 .035E+00 2. 437E-01 3, .432E- 01 49 6 .310E-01 1 .592E+00 2, .472E+10 1 .022E+00 3, 904E-01 3 .649E- 01 51 1 . OOOE+OO V .630E+00 2. 607E+10 1 .014E+00 5. 919E-01 3. 825E- 01 53 1 . .468E+00 1 .649E+00 2. 755E+10 1 .010E+00 6. 315E-01 3. 916E- 01 55 2. .154E+00 1 .658E+00 2. 926E+10 1 .005E+00 4. 821E-01 3. 957E- 01 57 3. .162E+00 1 .660E+00 3. 114E+10 1 .002E+00 3. 587E-01 3. 966E- 01 59 4. 642E+00 1 .651E+00 3. 331E+10 1 .002E+00 3. 729E-01 3. 933E- 01 61 6. .813E+00 V. .626E+00 3. 590E+10 1 .004E+00 5. 963E-01 3. 828E- 01 63 1 . OOOE+01 1 .565E+00 3. 896E+10 1 .012E+00 1 . 149E+00 3. 565E- 01 65 1 . ,445E+01 1 .462E+00 4. 237E+10 1 .O34E+00 2. 208E+00 3. 053E- 01 67 2. 089E+01 1 .337E+00 4. 638E+10 1 .091E+00 4. 311E+00 2. .312E- 01 69 3. 020E+01 1 .242E+00 5. 130E+10 1 .232E+00 8. 839E+00 1 . 595E- 01 71 4. .365E+01 1 .203E+00 5. 795E+10 1 .540E+00 1 . 995E+01 1 . 1 19E-01 73 6. .310E+01 1 .215E+00 6 . 818E+10 2 .115E+00 5. 819E+01 8. 819E- 02 75 8. 610E+01 1 .254E+00 8. 214E+10 2 .735E+00 3. 482E+02 8. 097E- 02 77 9. 750E+01 1 .271E+00 8. 965E+10 2 .927E+00 -6 . 564E+03 8. 057E- 02 79 9. 950E+01 1 , .273E+00 9. 099E+10 2. .950E+00 - 1 . 199E+03 8. 064E- 02 Convective quant i t ies Page 237 DELbub TauRMb Vconv Hconv Hconv/H 0 .OOOE+OO 2 . 715E-05 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 1 0 .OOOE+OO 4 . 170E-05 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 3 0 .OOOE+OO 6 .489E-05 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 5 0 .OOOE+OO 1 .018E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 7 0 .OOOE+OO 1 .606E-O4 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 9 0 . OOOE + OO 2 .546E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 11 0 .OOOE+OO 4 .051E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 13 0 .OOOE+OO 6 .475E-04 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 15 0 .OOOE+OO 1 .039E-03 0 .OOOE+OO 0 .OOOE+OO 0 :OOOE+OO 17 0 .OOOE+OO 1 .672E-03 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 19 0 .OOOE+OO 2 .700E-O3 0 . OOOE + OO 0 .OOOE+OO 0 .OOOE+OO 21 0 .OOOE+OO 4 . 370E-03 0 .OOOE+OO 0. .OOOE+OO 0 .OOOE+OO 23 0 .OOOE+OO 7 .085E-03 0 .OOOE+OO 0. OOOE+OO 0 .OOOE+OO 25 0 .OOOE+OO 1 . 150E-02 0 .OOOE+OO 0. .OOOE+OO 0 .OOOE+OO 27 0. .OOOE+OO 1 .868E-02 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 29 0 .OOOE+OO 3 .036E-02 0 .OOOE+OO 0. OOOE+OO 0 .OOOE+OO 31 0 .OOOE+OO 4 .941E-02 0 .OOOE+OO 0. OOOE+OO 0 .OOOE+OO 33 0. .OOOE+OO 8 .064E-02 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 35 0 .OOOE+OO 1 .322E-01 0 .OOOE+OO 0 .OOOE+OO 0 .OOOE+OO 37 0 .OOOE+OO 2 .181E-01 0 .OOOE+OO 0 OOOE+OO 0 .OOOE+OO 39 0. .OOOE+OO 3 .635E-01 0 .OOOE+OO 0. OOOE+OO 0 .OOOE+OO 41 0 .OOOE+OO 6 . 155E-01 0 .OOOE+OO 0 OOOE+OO 0 .OOOE+OO 43 0 .OOOE+OO 1 .075E+00 0 .OOOE+OO 0 OOOE+OO 0 .OOOE+OO 45 0" .OOOE+OO 1 .982E+00 0 .OOOE+OO 0. .OOOE+OO 0 .OOOE+OO 47 3. •904E-01 3 . 874E + 00 3 .974E+02 2. .993E-01 4 .280E- 10 49 5. 916E-01 7 .275E+00 5 . 301E + 03 7 . 162E+02 1. 023E-06 51 6 309E-01 9. . 792E + 00 7 . . 281E + 03 1. 869E+03 2. 669E-06 53 4. •820E-01 1 . 018E+01 2. .492E+03 7. 777E+01 1 . .109E-07 55 0. OOOE+OO 1 . 038E+01 0 OOOE+OO 0. OOOE+OO 0. OOOE+OO 57 0. OOOE+OO 1 . :423E+01 0. .OOOE+OO 0. OOOE+OO 0. OOOE+OO 59 5. 939E-01 3, 029E+01 1 . 693E+04 3. 477E+04 4. 935E-05 61 1 . 015E+00 8. 030E+01 1 . . 325E + 05 1 . 830E+07 2. . 594E-02 63 1 . 064E+00 2. , 140E+02 4. 090E+05 6. 112E+08 8. 657E-01 65 6 . 846E-01 5. , 806E+O2 7 . 828E+05 5. 327E+09 7 . . 542E + 00 67 3. 493E-01 1 . 621E+03 1 . . 338E + 06 3. 544E+10 5. 017E+01 69 1 . 792E-01 4. 654E+03 2. .428E+06 2. 694E+11 3. 813E+02 71 1 . 147E-01 1 . . 302E + 04 5. 288E+06 3. 015E+12 4. 267E+03 73 1 . 046E-01 2. . 755E + 04 1 . 616E+07 7 . 799E+13 1 . . 104E + 05 75 0. OOOE+OO 3. . 505E+04 0. OOOE+OO 0. OOOE+OO 0. OOOE+OO 77 0. OOOE+OO 3. 630E+04 0. OOOE+OO 0. OOOE+OO 0. OOOE+OO 79 to co Model 07310191 SS 3500./ 2.00/ -3.00 CN, TiO, H20 (hm) TauRM Col Mass 1 1 000E-05 1 794E-01 1 3 1 585E-05 2 217E-01 1 5 2 512E-05 2 800E-01 1 7 3 981E-05 3 596E-01 1 9 6 310E-05 4 685E-01 1 11 1 OOOE-04 6 180E-01 1 13 1 585E-04 8 241E-01 1 15 2 512E-04 1 108E+00 1 17 3 981E-04 1 501E+00 1 19 6 310E-04 2 040E+00 1 21 1 000E-03 2 778E+00 1 23 1 585E-03 3 780E+00 1 25 2 512E-03 '5 128E+00 1 27 3 981E-03 6 922E+00 9 29 6 310E-03 9 277E+00 8 31 1 000E-02 1 233E+01 7 33 1 585E-02 1 625E+01 6 35 2 512E-02 2 120E+01 5 37 3 981E-02 2 740E+01 5 39 6 310E-02 3 505E+01 4 41 1 000E-01 4 432E+01 3 43 1 585E-01 5 533E+01 2 45 2 512E-01 6 808E+01 1 47 3 981E-01 8 232E+01 1 49. 6 310E-01 9 729E+01 6 51 1 OOOE+00 1 118E+02 9 53 1 468E+00 1 233E+02 -3 55 2 154E+00 1 352E+02 -7 57 3 162E+00 1 506E+02 -1 59 4 642E+00 1 744E+02 -1 61 6 813E+00 2 063E+02 -2 63 1 OOOE+01 2 345E+02 -3 65 1 445E+01 2 523E+02 -4 67 2 089E+01 2 631E+02 -4 69 3 020E+01 2 693E+02 -4 71 4 365E+01 2 728E+02 -4 73 6 310E+01 2 748E+02 -4 75 8 610E+01 2 757E+02 -4 77 9 750E+01 2 760E+02 -4 79 9 950E+01 2 761E+02 -4 r/R* - 1 T n 968E -03 2 605E+03 4 967E + 3 913E -03 2 622E+03 6 099E + 3 851E -03 2 641E+03 7 646E + 3 784E -03 2 664E+03 9 739E + 3 713E -03 2 690E+03 1 257E + 4 638E -03 2 719E+03 1 640E + 4 559E -03 2 751E+03 2 162E + 4 477E -03 2 787E+03 2 871E + 4 391E -03 2 825E+03 3 835E + 4 304E -03 2 866E+03 5 140E + 4 215E -03 2 908E+03 6 898E + 4 124E -03 2 953E+03 9 247E + 4 034E -03 2 998E+03 1 236E + 5 432E -04 3 045E+03 1 642E + 5 534E -04 3 093E+03 2 168E + 15 648E -04 3 141E+03 2 838E + 15 777E -04 3 191E+03 3 681E + 5 922E -04 3 242E+03 4 728E + 5 085E -04 3 297E+03 6 010E + 5 267E -04 3 355E+03 7 554E + 5 472E -04 3 421E+03 9 372E + 5 704E -04 3 494E+03 1 146E + 6 969E -04 3 580E+03 1 376E + 6 277E -04 3 686E+03 1 616E + 16 459E -05 3 824E+03 1 841E + 6 636E -06 4 004E+03 2 021E + 16 069E -05 4 201E+03 2 124E + 16 097E -05 4 450E+03 2 199E + 6 216E -04 4 750E+03 2 296E + 6 948E -04 5 097E+03 2 477E + 6 848E -04 5 504E+03 2 713E+ 6 591E -04 5 977E+03 2 841E + 6 054E -04 6 494E+03 2 814E + 16 340E -04 7 074E+03 2 693E + 6 515E -04 7 721E+03 2 526E + 6 623E -04 8 438E+03 2 341E + 6 689E -04 9 230E+03 2 155E + 6 726E -04 9 963E+03 2 003E + 6 738E -04 1 027E+04 1 944E + 6 740E -04 1 032E+04 1 935E+ 6 Summary of physica l quant i t i es Page 238 Pe P_gas P_rad Mu Rho ChiRM 8 798E-05 1 787E+01 2 OOOE-01 1 252E+00 1 032E- 10 1 283E- 04 1 1 085E-04 2 208E+01 2 004E-01 1 253E+00 1 269E- 10 1 472E- 04 3 1 371E-04 2 788E+01 2 009E-01 1 254E+00 1 592E- 10 1 704E- 04 5 1 765E-04 3 582E+01 2 016E-01 1 255E+00 2..029E- 10 1 977E- 04 7 2 315E-04 4 667E+01 2 026E-O1 1 255E+00 2 620E- 10 . 2 288E- 04 9 3 085E-04 6 157E+01 2 038E-01 1 256E+00 3 422E- 10 2 636E- 04 1 1 4 174E-04 8 21 1E + 01 2 053E-01 1 257E+00 4 512E- 10 3 028E- 04 13 5 717E-04 1 105E+02 2 073E-01 1 258E+00 5 996E- 10 3 477E- 04 15 7 908E-04 1 496E+02 2 096E-01 1 258E+00 8 013E- 10 4 000E- 04 17 1 102E-03 2 034E+02 2 124E-01 1 259E+00 1 074E- 09 4 620E- 04 19 1 543E-03 2 770E+02 2 159E-01 1 259E+00 1 442E- 09 5 369E- 04 21 2 165E-03 3 769E+02 2 199E-01 1 260E+00 1 934E- 09 6 291E- 04 23 3 038E-03 5 1 15E + 02 2 246E-01 1 260E+00 2 585E- 09 7 449E- 04 25 4 253E-03 6 904E+02 2 302E-01 1 260E+00 3 437E- 09 8 928E- 04 27 5 930E-03 9 256E+02 2 368E-01 1 26OE+00 4 537E- 09 1 084E- 03 29 8 226E-03 1 231E+03 2 445E-01 1 261E+00 5 940E- 09 1 333E- 03 31 1 136E-02 1 621E+03 2 538E-01 1 261E+00 7 704E- 09 1 658E- 03 33 1 563E-02 2 1 16E + 03 2 653E-01 1 260E+00 9 894E- 09 2 087E- 03 35 2 150E-02 2 735E+03 2 797E-01 1 260E+00 1 257E- 08 2 661E- 03 37 2 967E-02 3 499E+03 2 982E-01 1 259E+00 1 579E- 08 3 445E- 03 39 4 124E-02 4 426E+03 3 225E-01 1 257E+00 1 957E- 08 4 545E- 03 41 5 803E-02 5 527E+03 3 547E-01 1 255E+00 2 388E- 08 6 135E- 03 43 8 341E-02 6 801E+03 3 973E-01 1 253E+00 2 863E- 08 8 526E- 03 45 1 244E-01 8 224E+03 4 545E-01 1 250E+00 3 354E- 08 1 240E- 02 47 1 956E-01 9 720E+03 5 336E-01 1 247E+00 3 813E- 08 1 944E- 02 49 3 249E-01 1 118E+04 6 479E-01 1 244E+00 4 176E- 08 3 277E- 02 51 5 109E-01 1 232E+04 7 869E-01 1 242E+00 4 381E- 08 4 973E- 02 53 8 008E-01 1 351E+04 9 888E-01 1 241E+00 4 529E- 08 6 376E- 02 55 1 197E+00 1 506E+04 1 280E+00 1 240E+00 4 726E- 08 6 434E- 02 57 1 804E+00 1 743E+04 1 702E+00 1 239E+00 5 098E- 08 6 149E- 02 59 3 547E+00 2 062E+04 2 320E+00 1 239E+00 5 582E- 08 8 119E- 02 61 9 760E+00 2 344E+04 3 230E+00 1 239E+00 5 842E- 08 1 648E- 01 63 2 905E+01 2 523E+04 4 505E+00 1 238E+00 5 782E- 08 3 858E- 01 65 8 601E+01 2 630E+04 6 349E+00 1 235E+00 5 522E- 08 9 485E- 01 67 2 419E+02 2 692E+04 9 016E+00 1 228E+00 5 149E- 08 2 395E+00 69 6 342E+02 2 727E+04 1 287E+01 1 210E+00 4 704E- 08 6 194E+00 71 1 527E+03 2 746E+04 1 845E+01 1 170E+00 4 187E- 08 1 619E+01 73 2 953E+03 2 755E+04 2 505E+01 1 106E+00 3 679E- 08 3 519E+01 75 3 745E+03 2 757E+04 2 832E+01 1 071E+00 3 457E- 08 4 686E+01 77 3 887E+03 2 758E+04 2 889E+01 1 064E+00 3 420E- 08 4 901E+01 79 to CO Model 07310191 SS 3500./ 2.00/ -3.00 CN, TiO, H20 (hm) Convergence checks Page 239 TauRM r TE res RE r< 1 1 .OOOE-05 6 .633E+12 1 .621E- 07 3 .044E 3 1 .585E-05 6 .633E+12 7 .261E- 06 -5 . 803E 5 2 .512E-05 6 .632E+12 1 .479E- 07 5 . 180E 7 3 981E-05 6 .632E+12 1 .666E- 12 1 .811E 9 6 .310E-05 6 .631E+12 8 .982E- 12 -2 .675E 11 1 .000E-04 6 .631E+12 9 .228E- 12 -4 . 543E 13 1 .585E-04* 6 .630E+12 7 .601E- 12 -9 . 498E 15 2 .512E-04 6 .630E+12 1 .234E- 11 -9 . 548E 17 3 .981E-04 6 .629E+12 2 .874E- 12 -4 . 763E 19 6 .310E-04 6 .629E+12 4 .591E- 12 -9 .629E 21 1 .000E-03 6 .628E+12 2 . 146E-12 -6 .656E 23 1 .585E-03 6 .627E+12 8 .219E- 14 -8 .086E 25 2 .512E-03 6 .627E+12 1 .080E- 13 -1 . 429E 27 3 .981E-03 6 .626E+12 1 .422E- 13 1 .625E 29 6 .310E-03 6 .626E+12 2 .808E- 14 5 .574E 31 1 .O00E-O2 6 .625E+12 3 .607E- 14 8. . 538E 33 1 .585E-02 6 .624E+12 1 .391E- 14 5 .062E 35 2 .512E-02 6 .624E+12 4 .162E- 14 5, .213E 37 3 .981E-02 6 .623E+12 8 .851E- 14 -6 , 909E 39 6 .310E-02 6 .623E+12 3 .539E- 13 -3. .939E 41 l'. .000E-01 6 .622E+12 4, .872E- 13 9, . 700E 43 1 . .585E-01 6 .622E+12 1 . .801E- 12 3. , 722E 45 2, .512E-01 6 .621E+12 4. .314E- 12 -1 . 253E 47 3 .981E-01 6 .621E+12 6. .019E- 12 -6 . 602E 49 6, .310E-01 6 .620E+12 3. 641E- 12 - 1 . 820E 51 1 . .OOOE+OO 6 .620E+12 1 . 957E- 13 2. . 319E 53 1 . .468E+00 6 .620E+12 4, ,708E- '14 -1 . 071E 55 2. .154E+00 6 .620E+12 2. 934E- 14 -4. . 395E 57 3 .162E+00 6, .619E+12 2. 600E- 14 -2. 375E 59 4. 642E+00 6. . 619E+ 12 4, . 706E- 14 -2. 181E 61 6. 813E+0O 6 . 618E+12 4. 932E- 15 -1 . 334E 63 1 . OOOE+01 6 . 618E+12 3. 071E- 15 -2. 646E 65 1 . 445E+01 6 . 617E+12 1 . 355E- 15 - 1 . 361E 67 2. 089E+01 6 . 617E+12 7 . 026E- 16 -7. 055E 69 3. 020E+01 6. 617E+12 4. 921E- 16 -5. 028E 71 4. .365E+01 6 . 617E+12 3. 690E- 16 -3. 561E 73 6. .310E+01 6. 617E+12 2. 921E- 16 -2. 620E 75 8. 610E+01 6. 617E+12 2. 348E- 16 - 1 . 626E 77 9. 750E+01 6. 617E+12 2. 871E- 16 -2. 199E 79 9. 950E+01 6. 617E+12 1 . 003E- 16 -5 . 512E HE res DE res EPS rms 11 2 .895E- 07 0 .OOOE+OO 1 .623E- 07 9 08 3 .476E- 08 7 .316E- 12 7 .160E- 06 9 1 1 2 .601E- 08 1 .491E- 1 1 1 .458E- 07 9 11 -2 .214E- 09 1 .470E- 11 2 .121E- 10 9 12 -3 . 215E- 08 4 .899E- 12 3 .080E- 09 9 12 -4 .333E- 08- -8 .276E- 12 4 .150E- 09 9 12 -2 .806E- 08 - 1 .547E- 11 2 .688E- 09 9 12 8 .281E- 10 -1 .371E- 11 8 .026E- 11 9 12 2 .601E- 08 -5 .455E- 12 2 .492E- 09 9 13 3 .411E- 08 3 .318E- 12 3 .268E- 09 1 14 2. .412E- 08 7 .545E- 12 2 .310E- 09 1 15 3 .143E- 09 5 .302E- 12 3 .010E- 10 1 14 - 1 .557E- 08 -4 .840E- 12 1 .491E- 09 1 13 5 .759E- 09 2 .876E- 12 5 .516E- 10 1 14 -4 .953E- 09 -2 . 762E- 13 4 .744E- 10 1 16 -2. .784E- 09 -3 .909E- 13 2 .667E- 10 1 15 -3. .660E- 10 8 .531E- 14 3. .506E- 11 1 16 -2. .709E- 10 6 .518E- 14 2 .594E- 11 1 15 - 1 . 828E- 10 1 .989E- 13 1 . .751E- 11 1 14 2, .191E- 10 5 .828E- 13 2. .098E- 11 1 14 - 1 . 464E- 09 5 .325E- 13 1 . .403E- 10 1 13 -6. .626E- 09 -2. .196E- 12 6, , 346E- 10 1 13 -1 . •613E- 09 -4. .433E- 12 1 . 546E- 10 1 13 2. 076E- 08 4. . 363E- 12 1 . 988E- 09 1 14 -5. . 141E-10 7. ,217E- 12 4. 938E- 11 9 13 - 1 . 882E- 08 -7 . 043E- 12 1 . 802E- 09 9 14 5. 186E- 09 1 , 095E- 12 4. 968E- 10 9 15 2: 563E- 10 1 . . 108E-13 2. 455E- 11 9 15 1 . 658E- 10 1 . 502E- 13 1 . 588E- 11 9 14 -2. 737E- 09 - 1 , ,747E- 12 2. 622E- 10 9 15 2. 048E- 09 1 . 952E- 13 1 . 962E- 10 9 15 1 . 355E- 09 2. 663E- 13 1 . 298E- 10 9 15 -3. 482E- 1 1 -7. 406E- 14 3. 335E- 12 9 16 -4. 028E- 11 -4. 638E- 14 3. 858E- 12 9 16 - 1 . 557E- 11 -3. 151E- 14 1 . 491E- 12 9 16 -5. 186E- 13 - 1 . 930E- 14 4. 971E- 14 9 16 1 . 390E- 12 - 1 . 160E- 14 1 . 332E- 13 9 16 1 . 592E- 13 -2. 265E- 15 1 . 525E- 14 9 16 -2. 591E- 13 -2. 924E- 16 2. 482E- 14 8 17 -7 . 460E- 16 1, 186E- 16 1 . 227E- 16 8 g_ef f /g Ch i_H/RM Hconv/H Lum/L* 999E-01 2 .876E+02 0 .OOOE+OO 1 .007E+00 1 999E-01 2 .438E+02 0 .OOOE+OO 1 .007E+00 3 999E-01 2 .025E+02 0 .OOOE+OO 1 .007E+00 5 999E-0.1 1 .655E+02 0 .OOOE+OO 1 .007E+00 7 999E-01 1 . 333E + 02 0 .OOOE+OO 1 .007E+00 9 999E-01 1 .060E+02 0 .OOOE+OO 1 .007E+00 11 999E-01 8 .331E+01 0 .OOOE+OO 1 .007E+00 13 999E-01 6 .468E+01 0 .OOOE+OO 1 .007E+00 15 999E-01 4 .963E+01 0 .OOOE+OO 1 .007E+00 17 OOOE+OO 3 . 765E + 01 0 .OOOE+OO 1 .007E+00 19 OOOE+OO 2 .827E+01 0. .OOOE+OO 1 .007E+00 21 OOOE+OO 2 . 104E+01 0. .OOOE+OO 1 .007E+00 23 OOOE+OO 1 .558E+01 0. OOOE+OO 1 .007E+00 25 OOOE+OO 1 . 152E + 01 0. OOOE+OO 1 .006E+00 27 OOOE+OO 8 .554E+00 0. .OOOE+OO 1 .006E+00 29 OOOE+OO 6 .432E+00 0. .OOOE+OO 1 .006E+00 31 OOOE+OO 4 . 930E + 00 0. OOOE+OO 1 .006E+00 33 OOOE+OO 3 . 865E + 00 0, , OOOE + OO 1 .006E+00 35 OOOE+OO 3 . 103E + 00 0. OOOE+OO 1 .006E+00 37 OOOE+OO 2 . 542E + 00 0. OOOE+OO 1 .006E+00 39 OOOE+OO 2 .113E+00 0. OOOE+OO 1 .007E+00 41 OOOE+OO 1 . .765E+00 0. OOOE+OO 1 .007E+00 43 OOOE+OO 1 . .480E+00 0. OOOE+OO 1 .007E+00 45 OOOE+OO 1 . .262E+00 0. OOOE+OO 1 .007E+00 47 999E-01 1 . .123E+00 0. OOOE+OO 1 .007E+00 49 999E-01 1 . 049E+00 3. 193E-05 1 .006E+00 51 999E-01 . 1 . 031E+00 1. 179E-03 1 .006E+00 53 998E-01 1 , 024E+00 3. 273E-03 1 .005E+00 55 998E-01 1 . OOOE+OO 7. 554E-04 1 .005E+00 57 998E-01 9 . 918E-01 4. 590E-05 1 .005E+00 59 998E-01 9. 965E-01 1 . 796E-03 1 .003E+00 61 995E-01 1 . OOOE+OO 2. 175E-01 1 .002E+00 63 989E-01 1 . 001E+00 2. 474E+00 1 . .001E+00 65 973E-01 1 . 002E+00 1 . 289E+01 9 .999E-01 67 932E-01 1 . 002E+00 5. 838E+01 9. .992E-01 69 824E-01 1 . 001E+00 2. 842E+02 9. .986E-01 71 540E-01 1 . 001E+00 1 . 531E+03 9 . 987E-01 73 001E-01 1 . OOOE+OO 6 . 151E+03 9. 996E-01 75 670E-01 9. 999E-01 1 . 041E+04 9 . 999E-01 77 609E-01 9. 997E-01 1 . 132E+04 1 . OOOE+OO 79 Model 07310191 SS 3500./ 2.1 DO/ -3.00 CN, TiO, H20 (hm) TauRM Cp/Cv H_P Qconv DELrad DELad 1 1 .000E-05 1 . 275E+00 1 .757E+09 1 .191E+00 2 .954E-02 1 . 790E -01 3 1 .585E-05 1 .269E+00 1 . 763E+09 1 .204E+00 3 .081E-02 1 . 748E -01 5 2 .512E-05 1 .264E+00 1 .771E+09 1 .218E+00 3 . 264E-02 1 . 707E -01 7 3 .981E-05 1 . .259E+00 1 .782E+09 1 .232E+00 3 .506E-02 '1 .670E -01 9 6 . 310E-05 1 . .256E+00 1 .795E+09 1 .244E+00 3 .770E-02 1 .642E -01 11 1 .000E-04 1 . .254E+00 1 .811E+09 1 .255E+00 4 .018E-02 1 .623E -01 13 1 .585E-04 1 . .253E+00 1 .830E+09 1 . 263E + 00 4 . 232E-02 1 . 612E -01 15 2 .512E-04 1 . ,253E+00 1 .851E+09 1 . 269E+00 4 .414E-02 1 .608E -01 17 3 .981E-04 1 . .254E+00 1 .874E+09 1 .274E+00 4 .571E-02 1 .609E -01 19 6 .310E-04 1 . .256E+00 1 .900E+09 1 .278E+00 4 . 713E-02 1 .615E -01 21 1 .000E-03 1 . . 258E+00 1 .926E+09 1 .280E+00 4 .845E-02 1 .625E -01 23 1 .585E-03 1 . .261E+00 1 .954E+09 1 . 281E+00 4 .971E-02 1 .637E -01 25 2 .512E-03 1 . . 264E+00 1 .983E+09 1 . 282E+00 5 .095E-02 1 .652E -01 27 3 .981E-03 1 . . 267E + 00 2 .013E+09 1 .281E+00 5 .223E-02 1 .670E -01 29 6 .310E-03 1; 271E+00 2 .044E+09 1 . 280E+00 5 . 372E-02 1 .690E -01 31 1 .000E-02 1. . 275E+00 2 .075E+09 1 .278E+00 5 .562E-02 1 . 714E -01 33 1 .585E-02 1. 279E+00 2 .108E+09 1 .273E+00 5 .830E-02 1 . 743E -01 35 2 .512E-02 1. 285E+0O 2 .142E+09 1 .266E+00 6 .225E-02 1 . 780E -01 37 3 .981E-02 1. 293E+00 2 .178E+09 1 .256E+00 6. 807E-02 1 .831E -01 39 6 .310E-02 1. 303E+00 2 .218E+09 1 . 240E+00 7 . 646E-02 1 . 902E -01 41 1 . .OOOE-01 1. 316E+00 2 . 264E + 09 1 . 218E+00 8. 829E-02 2 .000E -01 43 1 . .585E-01 1. 335E+00 2 . 316E + 09 1 .191E+00 1 . 055E-01 2 . 136E -01 45 2 .512E-01 1. 363E+00 2 .377E+09 1 .158E+00 1 . .332E-01 2 . 324E -01 47 3. .981E-01 1. 403E+00 2 .453E+09 1 .120E+00 1 . 826E-01 2 .585E -01 49 6. .310E-01 1. 46OE+00 2 .550E+09 1 .081E+00 2. . 702E-01 2 .933E -01 51 1 , .OOOE+00 1. 530E+00 2 .677E+09 1 .048E+00 4. .157E-01 3 . 323E -01 53 1 . ,468E+00 1. 586E+00 2 .812E+09 1 .029E+00 5. . 716E-01 3 .611E -01 55 2, .154E+00 1. 627E+00 2 .982E+09 1 .017E+00 6 . , 379E-01 3 .810E -01 57 3. .162E+00 1. 649E+00 3 .185E+09 1 .009E+00 5. 440E-01 3 .912E -01 59 4. ,642E+00 1. 655E+00 3 .419E+09 1 .004E+00 4. 504E-01 3 .945E -01 61 6 . 813E+00 1. 647E+0O 3 .692E+09 1 .003E+00 5. 244E-01 3 .916E -01 63 1 . OOOE+01 1. 618E+00 4 .010E+09 1 .006E+00 8. 786E-01 3 . 796E -01 65 1 . 445E+01 1. 555E+00 4 .360E+09 1 .015E+00 1 . .589E+00 3 .517E -01 67 2. 089E+01 1. 451E+00 4 .760E+09 1. .040E+00 2. 895E+00 2 .995E -01 69 3. 020E+01 1. 335E+00 5 .225E+09 1 .102E+00 5. 305E+00 2 . 287E -01 71 4. 365E+01 1. 250E+00 5 .794E+09 1 . .245E+00 9. 868E+00 1 . 625E -01 73 6. 310E+01 1. 215E+00 6 .557E+09 1 .540E+00 1 . 867E+01 1 . . 18 IE -01 75 8. 610E+01 1. 223E+00 7 .487E+09 1 . .961E+00 3. 106E+01 9 . . 775E -02 77 9 . 750E+01 1. 234E+00 7 .977E+09 2. .172E+00 3. 766E+01 9 . . 300E -02 79 9 . 950E+01 1. 236E+00 8 .065E+09 2 . 208E + 00 3. 887E+01 9 . 237E -02 Convective quant i t i es Page 240 DELbub TauRMb Vconv Hconv Hconv/H 0 .OOOE+00 3 .724E-05 0 .OOOE+OO 0 .OOOE+00 0 .OOOE+00 1 0 .OOOE+00 5 .267E-05 0 .OOOE+00 0 .OOOE+00 0 .OOOE+OO 3 0 .OOOE+OO 7 .687E-05 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 5 0 .OOOE+00 1 .144E-04 0 .OOOE+OO 0. .OOOE+OO 0 .OOOE+OO 7 0 .OOOE+00 1 .722E-04 0 .OOOE+00 0 . OOOE+00 0 .OOOE+00 9 0 .OOOE+00 2 .614E-04 0 .OOOE+OO 0 .OOOE+00 0 .OOOE+00 1 1 0 .OOOE+OO 4 .001E-04 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 13 0 .OOOE+00 6 .176E-04 0 .OOOE+00 0 .OOOE+00 0 .OOOE+OO 15 0 .OOOE+00 9 .613E-04 0 .OOOE+OO 0 .OOOE+00 0 .OOOE+00 17 0 .OOOE+OO 1 .509E-03 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 19 0. .OOOE+00 2 . 387E-03 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 21 0 OOOE+00 3 .805E-03 0 .OOOE+OO 0 .OOOE+00 0 .OOOE+00 23 0. OOOE+00 6 . 111E-03 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 25 0 OOOE+00 9 .885E-03 0 OOOE+00 0 .OOOE+00 0 .OOOE+OO 27 0 .OOOE+00 1 .608E-02 0 .OOOE+OO 0 .OOOE+00 0 .OOOE+00 29 0 OOOE+00 2 .628E-02 0 .OOOE+00 0 .OOOE+00 0 .OOOE+00 31 0. OOOE+OO 4 . 307E-02 0. .OOOE+OO 0 .OOOE+00 0 .OOOE+OO 33 0, .OOOE+00 7 .076E-02 0 OOOE+00 0 .OOOE+00 0 .OOOE+00 35 0 OOOE+OO 1 . 166E-01 0. OOOE+00 0. OOOE+00 0 OOOE+OO 37 0. OOOE+00 1 . 931E-01 0. OOOE+00 0. OOOE+OO 0. .OOOE+00 39 0. OOOE+00 3 .221E-01 0. .OOOE+00 0. .OOOE+00 0, , OOOE + 00 41 0. OOOE+00 5 .428E-01 0. .OOOE+OO 0, OOOE+OO 0. OOOE+00 43 0. OOOE+00 9 .282E-01 0. .OOOE+00 0, OOOE+00 0, , OOOE + OO 45 0. OOOE+00 1 .633E+00 0. OOOE+OO 0. OOOE+00 0, , OOOE + 00 47 0. OOOE+00 3 .024E+00 0. OOOE+00 0. OOOE+00 0, , OOOE+00 49 4. 147E-01 5 .859E+00 9. 527E+03 2. .162E+04 3. . 193E-05 51 5. 604E-01 9 .803E+00 3. . 210E+04 7 . .984E+05 1. . 179E-03 53 6. 167E-01 1 . . 378E + 01 4 . 542E+04 2. .217E+06 3. . 273E-03 55 5. 366E-01 1 . .550E+01 2. 771E+04 5. .116E+05 7 . 554E-04 57 4. 494E-01 1 . 715E + 01 1 . 065E+04 3. . 109E + 04 4 , . 590E-05 59 5. 141E-01 2 .677E+01 3. 500E+04 1 . .217E+06 1 . . 796E-03 61 6. 580E-01 6 . 179E+01 1 . 688E+05 1 . 473E+08 2. . 175E-01 63 6. 154E-01 1 .556E+02 3. 714E+05 1 . 677E+09 2. 474E+00 65 4. 718E-01 3 .989E+02 6. . 197E + 05 8. 737E+09 1 . . 289E + 01 67 3. 172E-01 1 . 031E+03 9 . 591E+05 3. 956E+10 5. 838E+01 69 1 . 997E-01 2. . 701E + 03 1 . 495E+06 1 . 926E+11 2. 842E+02 71 1 . 319E-01 7 , . 112E + 03 2. 449E+06 1 . 037E+12 1 . 531E+03 73 1 . 037E-01 1 . 551E+04 3. 816E+06 4. 169E+12 6. 151E+03 75 9. 740E-02 2. 068E+04 4. 566E+06 7 . 053E+12 1 . 041E+04 77 9. 658E-02 2 . 163E + 04 4. 702E+06 7 . 673E+12 1 . 132E+04 79 •ft-o
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Spherically symmetric model atmospheres for late-type giant stars Bennett, Philip Desmond 1991
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Title | Spherically symmetric model atmospheres for late-type giant stars |
Creator |
Bennett, Philip Desmond |
Publisher | University of British Columbia |
Date Issued | 1991 |
Description | The ATHENA computer code has been developed to model the extended atmospheres of late-type giant and supergiant stars. The atmospheres are assumed to be static, spherically symmetric and in radiative and hydrostatic equilibrium. Molecular line blanketing (for now) is handled using the simplifying assumption of mean opacities. The complete linearization method of Auer and Mihalas [7], adapted to spherical geometry, is used to solve the model system. The radiative transfer is solved by using variable Eddington factors to close the system of moment transfer equations, and the entire system of transfer equations plus constraints is solved efficiently by arrangement into the Rybicki [83] block matrix form. The variable Eddington factors are calculated from the full angle-dependent formal solution of the radiative transfer problem using the impact parameter method of Hummer, Kunasz and Kunasz [47]. We were guided by the work of Mihalas and Hummer [72] in their development of extended models of O stars, but our method differs in the choice of the independent variable. The radius depth scale used by Mihalas and Hummer was found to fail because of the strongly temperature-dependent opacities of late-type atmospheres. Instead, we were able to achieve an exact linearization of the radius. This permitted the use of the numerically well-behaved column mass or optical depth scales. The resulting formulation is analogous to the plane-parallel complete linearization method and reduces to this method in the compact atmosphere limit. Models of M giants were calculated for T[formula omitted] = 3000K and 3500K with opacities of the CN, TiO and H₂0 molecules included, and the results were in general agreement with other published spherical models. These models were calculated assuming radiative equilibrium. The importance of convective energy transport was estimated by calculating the convective flux that would result from the temperature structure of the models. The standard local mixing length theory was used for this purpose. Convection was found to be important only at depths with Ƭ[formula omitted] > 15 for the low gravity models with log g = 0, but significant out to Ƭ[formula omitted] ~ 1 at the most transparent frequencies for the higher gravity models with log g = 2. Thus, the temperature structure of the surface layers and the emergent flux for the log g = 0 models should be accurately modelled but the emergent flux for the log g = 2 models may be in error by up to 5% at the most transparent frequencies. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-01-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085580 |
URI | http://hdl.handle.net/2429/30908 |
Degree |
Doctor of Philosophy - PhD |
Program |
Astronomy |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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