Page 48 Introduction Page 49 Nuclear networks Page 50 Mass loss Page 51 The diffusion equations Page 52 Implicit finite elements method Page 53 Application to the solar case Page 54 Transport of angular momentum Page 55 Magnetic fields and internal gravity waves Page 56 Tayler-Spruit dynamo Page 57 References Page 58 Input physics Page 61 Radiative diffusion Page 62 Meridional circulation and rotation-induced mixing Page 63 Evolution of angular momentum Page 64 References Page 65 The available packages Page 67 The kinds of precision Page 68 Choice of variables Page 69 The grids Page 70 Initial PMS model Page 71 Evolution with diffusion Page 72 Burgers's flow equations Page 73 Calculation of mean charges Page 74 Convection Page 75 Calibration of the solar model Page 76 Acknowledgements Page 77 References Page 78 Introduction Page 80 Chemical evolution equations Page 81 Atmospheric layers Page 82 Solving for the structure given the abundances Xk,i Page 83 Examples Page 84 References Page 87 Introduction Page 88 Equation of state Page 89 Nuclear reactions Page 90 Convection Page 91 Structure Page 92 Interpolation in tables Page 93 Calibration of solar models Page 94 References Page 95 Introduction Page 97 Equation of state Page 98 The macrophysics: convection models Page 99 Overshooting Page The macrophysics: atmospheric structure and boundary conditions Page References Page Numerics Page Explicit time integration Page Mass loss Page Nuclear reactions Page Opacities Page Summary Page Formulation of the adiabatic equations Page The Galerkin finite-element method Page Internal stability and accuracy Page Impact of stellar model mesh resolution Page Equilibrium model Page Formulation of the equations Page Numerical scheme Page The shooting method Page Improving the frequency precision Page Computed quantities Page Acknowledgements Page Basic equations for linear perturbations Page The equilibrium model Page At the atmosphere Page Numerical variables Page Accuracy of the results Page Adiabatic case Page Non-adiabatic resolution Page Nonradial oscillations Page Outputs Page Additional tools for the initial model Page Introduction Page Dimensionless variables Page Shooting method Page Numerical examples Page ModelJCA: improved hydrostatic equilibrium Page Oscillation modes Page Difference equations Page Applications Page Oscillation frequencies of a pseudo-rotating model Page The boundary conditions Page filou inputs and outputs Page Numerical tests and results Page Results.
The effect of shellular rotation on adiabatic oscillations Page Methods of studying pulsating stars Page Stellar direct method. Page Conclusions Page Input physics and numerical aspects Page Initial parameters of the models Page Evolutionary tracks and stellar models Page Input physics Page Evolutionary tracks Page Stellar models Page Seismic properties Page Numerical tools Page Presentation of the comparisons and general results Page Low-mass models: Cases 1.
Page Intermediate mass models: Cases 1. Page High mass models: Cases 1. Page Convection regions and ionisation zones Page Solar-like oscillations: Cases 1. Page Cases 1. Page Internal structure Page Solar-type stars with convective cores: Cases 3. Page Convection zones Page Helium surface abundance Page Summary and conclusions Page Comparison between relevant physical quantities Page Comparison between the variables used in the frequency computations Page Equation of State Page Effects on global stellar parameters Page Effects on the stellar structure Page Effects on the frequencies Page Atmosphere Page Numerical aspects Page Abstract Page The equilibrium models Page Oscillation codes and requirements Page Radial modes Page Large separation of radial modes Page Small separations delta Page The influence of the gravitational constant G Page The choice of dependent variables and equations Page The choice of independent variable Page A study case Page Frequency comparisons Page Stellar structures Page Stellar structure Page Evolved stars Page Cover illustration: Schematic illustration of the location of different classes of pulsating stars in the HR diagram, originally published by J.
ChristensenDalsgaard in Astrophys. Space Science , 1—12 , and in an updated form in Proc. The colour version was produced in for the Eddington proposal by Lars Peter Rasmussen. Printed on acid-free paper springer. Lebreton, M. Monteiro, J. Moya, A. Baglin, J. Christensen-Dalsgaard, M. Goupil, E. Michel, J. Provost, I. Roxburgh, R. Prada Moroni, M. Marconi and A. Demarque, D. Guenther, L. Li, A. Mazumdar and C. Eggenberger, G. Meynet, A. Maeder, R. Hirschi, C.
Charbonnel, S. Talon and S. Morel and Y. Scuflaire, S. Miglio, P. Bourge, M. Godart, A. Thoul and A. Ventura, F. Weiss and H. Brassard and S. Moya and R. Scuflaire, J. Bourge, A. Miglio, M. Lebreton and E. Marques, M. Monteiro and J. Miglio, A. Noels and R. Lebreton, J. Christensen-Dalsgaard, I. Roxburgh and A. Marconi, S. Prada Moroni and A.
Lebreton, A. Miglio, R. Scuflaire, P. Morel and A. Moya, J. Christensen-Dalsgaard, S. Charpinet, Y. Miglio, J. These are the result of the work initiated in with the aim to extensively test, compare and optimize the numerical tools used to calculate stellar models and their oscillation frequencies.
The volume includes i articles describing most of the evolution and seismic codes currently used in the context of Helio- and Astero-Seismology, ii articles reporting the results of the detailed comparisons of the codes carried out over the period — and iii works discussing grids of stellar evolutionary tracks, models, and frequencies, produced for supporting the CoRoT mission. The first paper of the collection discusses the overall aim and methodology of the work while the last paper discusses some of the open questions that still require further work in the future.
This collection of papers provides a unique reference that covers 10 evolution codes and 9 oscillation codes. Most of these have been used to produce numerous results published over the years, and have never before been fully described in the literature. The comparisons that were carried out over a period of 4 years resulted in a comprehensive study that covered the numerical aspects of the different codes and the implementation of the physics these use, and have provided the basis for further development of the codes.
The same work has also allowed for a detailed characterization of the precision and expected shortfalls of the models produced by these tools. Consequently, this volume is expected to be of great relevance for researchers and research students working on the modeling of stars and on the implementation of seismic test of stellar models.
Moreover, it is expected to have a high impact on the analysis of the data acquired by ongoing and future ground-based instruments and space missions in Helio- and Astero-Seismology. The organization of the workshops of the ESTA Team have been made possible through the contribution of colleagues and the support of their host institutes, namely, Dr.
We are very grateful for the hard work and dedication of all ESTA participants, and in particular to the colleagues that have been more actively involved in the activities as leaders of the tasks and work-packages as well as to the code-builders. Eric Michel, for their continuous support of this effort over the last six years. Finally, we must finish by stressing that only the first step has been done towards the ESTA goals. The work is far from complete, but from here we are confident it will be easier to move forward.
DOI: In order to secure that these tools meet the specifications, a team has been established to test and, when necessary, to improve the codes available in the community. The present paper describes the motivation and the organisation of this activity, providing the context and the basis for the presentation of the results that have been achieved so far.
This is not a finished task as future even better data will continue to demand more precise and complete tools for asteroseismology. Such an accuracy is needed to determine some of the key features of the stellar interiors such as the extent of the convective core of intermediate mass stars Roxburgh and Vorontsov ; Mazumdar et al. High-quality seismic data also make possible inverse analyses to infer the stellar density Gough and Kosovichev and rotation profiles from separated multiplets Goupil et al.
More generally, it is expected that CoRoT will provide constraints M. Monteiro ed. As a result stellar models will be improved by seismological inferences, and if they are combined with high-quality observational data of the global stellar parameters luminosity, effective temperature, radius, abundances , it will allow to improve the determination of those stellar parameters not directly accessible through observation like ages and masses of single stars with valuable returns on the understanding of galactic structure and evolution Lebreton ; Monteiro et al.
However, studies in helioseismology have taught us that in order to probe fine details of the solar internal structure, we need both the constraints of high-quality seismic data and extremely accurate numerical solar models Reiter et al.
Therefore, to be able to draw valuable information from the future CoRoT data, we have to ensure that we will be able to interpret them with models having reached the optimal level of accuracy. The goals of the ESTA group have been i to be able to produce theoretical seismic predictions by means of different numerical codes and to understand the possible differences between them and ii to bring stellar models at the level of accuracy required to interpret the future CoRoT seismic data.
In this introductory paper, we outline the tools and specifications of the comparisons that will be presented in detail in the different papers in this volume. Section 2 provides a brief overview of the characteristics of the CoRoT seismology targets to be modeled.
Section 3 presents the ESTA group participants, tools, past meetings and publications and briefly introduces the numerical tools stellar evolution codes and seismic codes used in the successive steps of the comparison work. In Sect. Finally in Sect. At that time more recent developments of the seismology theory as well as discoveries of pulsations were evidently not taken into account.
Long Runs are numbered as LRx0Y where x is a or c depending on whether it is a direct run in the center or anticenter direction and Y is the order of the run in the given direction. Standard evolutionary tracks, labelled with their mass in solar units and corresponding to solar metallicity, no overshooting, are also shown see Lebreton and Michel Following this several other meetings and workshops 7 so far have been organised as the activities progressed.
A website has been created to coordinate the activities, allowing for an efficient exchange of data and documentation as well as to share the results of the tasks that have been organised around the key objectives of ESTA. It includes the participation of 10 evolution codes and 9 oscillation codes in the tasks and comparisons that have been developed over the last three years.
As part of the effort several results have been published as papers or proceedings of the workshops or made available through the web-page. The list is given below. Each of them is presented in detail in a specific paper in this volume. It is presented by Christensen-Dalsgaard b. Up to now it has participated only marginally in ESTA activities. It is presented by Ventura et al.
It is presented by Morel and Lebreton It is presented by Scuflaire et al. It is presented by Weiss and Schlattl It is presented by Eggenberger et al. It is presented by Roxburgh b. It is presented by Hui-Bon-Hoa It is presented by Demarque et al. Each of them is presented in details in a specific paper in this volume. It is presented by Christensen-Dalsgaard a. The first task, TASK 1, has consisted in comparing stellar models and evolution sequences produced by eight stellar internal structure and evolution codes.
For that purpose we have fixed some standard input physics and initial parameters for the stellar models to be calculated and we have defined several specific study cases corresponding to stars covering the range of masses, evolution stages and chemical compositions expected for the bulk of CoRoT target stars. The second task, TASK 2, has consisted in testing, comparing and optimising the seismic codes by means of the comparison of the frequencies produced by nine different oscillation codes, again for specific stellar cases.
Seven study cases have been considered, the specifications of which are given in Table 1. The cases are defined as follows. Seven evolutionary sequences have been calculated for different values of the stellar mass and initial chemical composition X, Y, Z where X, Y and Z are respectively the initial hydrogen, helium and metallicity in mass fraction. The masses are in the range 0. We have considered 7 cases corresponding to different initial masses, chemical compositions and evolutionary stages.
The targets are ordered in mass and age along the diagram, from Case 1. Figure 2 displays the location in the H—R diagram of the targets for TASK 1 together with their parent evolutionary track.
All models calculated for TASK 1 are based on rather simple input physics, currently implemented in stellar evolution codes and one model has been calculated with overshooting see Sect. Detailed results are presented in this volume Lebreton et al. In particular we have studied the numerical precision of the frequencies using different meshes and with a wide range of options to solve the equations for linear adiabatic radial and non-radial oscillations.
Left: Three cases with corresponding masses and initial chemical composition. Right: Three evolutionary stages examined for each case. Stages A and B are respectively in the middle and end of the MS stage. The models have been provided with different number of mesh points.
Preliminary results of this task were reported by Moya while the final results of Step 1 are discussed in this volume by Moya et al. Step 2 remains to be done. The other physical assumptions proposed as the reference for the comparisons are the same as used for TASK 1 and no overshooting see Sect.
Three study cases have been considered for the models to be compared. Each case corresponds to a given value of the stellar mass see Table 2. We chose rather low values of the masses i. M Fig. Evolutionary tracks correspond to Case 3. On each track filled circles indicate the stages A, B, C and C. Figure 3 displays the location in the H—R diagram of the targets for TASK 3 together with their parent evolutionary track.
Advanced comparisons are presented in this volume Lebreton et al. Special care has also been given to the exchange of data in order to facilitate the exchange of models and evolutionary tracks and their use in detailed comparisons.
Further details can be found in the documentation available at the ESTA web-site. We have also fixed the mixture of heavy elements to be used in the calculations. Furthermore a reference set of values is provided for the mass of neutron and atomic mass of the elements from 1 H to 17 O.
As shown in Tables 1 and 2 the mass fractions of hydrogen X , helium Y and heavy elements Z are specified for each model. The tables provide a set of thermodynamic quantities, i. However, there are differences in the handling of the tables by the different codes.
Some use the interpolation package provided by the OPAL group, others have developed their own interpolation package. Furthermore, in the course of the comparisons undertaken by the ESTA group it has been shown by one of us I. However slightly different OPAL tables are actually used by the different codes. Different groups used either one of these possibilities. Moreover, some codes use the interpolation routines provided by the OPAL group, others designed their own routines.
We chose to compute the nuclear reaction rates from the analytical formulae provided by the NACRE compilation Angulo et al. Different prescriptions have been used for the screening. Most often, weak screening has been assumed under the Salpeter formulation. Finally, in the nuclear reaction network the initial abundance of each chemical species is split between its isotopes according to the isotopic ratio of nuclides for which values have been given see the ESTA Web site5. The radius of the star is taken to be the bolometric radius, i.
We did not impose the formalism to be used. In both methods, approximations have to be made to derive the various coefficients entering the diffusion equations, in particular the diffusion velocities which are written as a function of the collision integrals.
In the stellar evolution codes which have participated in TASK 3, different treatments of the diffusion processes have been adopted. Furthermore, the number of chemical elements considered in the diffusion process differs in the different codes and some codes consider that all the elements are completely ionised while others calculate the ionisation explicitly. Astrophys Space Sci 1—12 5. But to facilitate the comparisons we have adapted some of the output in order to include all information required for detailed code-to-code comparisons of models.
Monteiro to allow the direct conversion between the different formats available. In all cases ESTA participants have been asked to provide ASCII files ready for comparisons which contain, for the TASK case considered, the global properties of the model name, mass, age, initial composition, luminosity, photospheric radius, etc. Modelers have also provided ASCII files giving the variation with time of some global or internal properties of their evolution sequences luminosity, effective temperature, radius, central hydrogen content, surface helium and heavyelement abundances, radius and mass of the convective core and depth and mass of the convection envelope, etc.
To obtain the solution of the equations, the modelers were asked to adopt the following specifications: a Mesh: Use the mesh provided by the equilibrium model no re-meshing.
Christensen-Dalsgaard and Mullan As pointed out by Moya et al. The consequences of these choices are discussed in details in Moya et al. These reference grids have been used to locate CoRoT potential targets in the H—R diagram in the process of target selection see Fig. It is accessible through a Web interface. In particular: In this paper we briefly presented the different activities undertaken by the evolution and seismic tools activity ESTA, see Monteiro et al.
This provides the context and the reference for the more detailed works published in this volume with the descriptions of the codes and the results of the comparisons. The seismic study of the Sun helioseismology had already pushed the need to calculate solar models to a new level of accuracy due to the necessity to calculate theoretical oscillation frequencies that could match the observed solar values.
As we now approach the same level of precision for other stars in asteroseismology the codes used to model the evolution of stars of different masses have now to reach the same high precision for stellar regimes very different from the Sun. As the physics dominating different regimes in the H—R diagram are different a strong effort towards this is required. In the work developed by ESTA we are pursuing that goal. The results of this exercise so far have shown that we are able to meet the present requirements.
But more work is planned to address some pending aspects of the physics that must be included in the models with sufficient precision to be able to reproduce the observed seismic behaviour.
This is an open project as the arrival of data from CoRoT and other future projects will require our tools to improve further the precision of the models in order to test the highly accurate data made available for asteroseismology. References Alecian, G. In: Straka, C. Stellar Evolution and Seismic Tools for Asteroseismology. EAS Publications Series, vol. A , 3 Audard, N. Mit 5 Textabbildungen. The observational point of view. The standard model and helioseismology: consequences of uncertainties in input physics and in observed solar parameters.
Space Sci. EAS Publication Series, vol. LNP, vol. Springer, Berlin Gough, D. Science , Goupil, M. In: Favata, F. Stellar Structure and Habitable Planet Finding. ESA SP, p. The calculation of model envelopes.
In: Turon, C. The Three-Dimensional Universe with Gaia. Monteiro, M. In: Weiss, W. IAU Colloq. Astronomical Society of the Pacific Conference Series, vol. In: Fridlund, M. The CoRoT Mission. IAU Symposia, vol.
In: Battrick, B. ASP Conf. In: Korzennik, S. Science , Zahn, J. A novel feature is the integration of the computation of adiabatic oscillations for specified models as part of the code. It offers substantial flexibility in terms of microphysics and has been carefully tested for the computation of solar models. However, considerable development is still required in the treatment of nuclear reactions, diffusion and convective mixing.
The initial goal was to provide an improved equilibrium model for investigations of solar stability, following earlier work by Christensen-Dalsgaard et al. However, with the initial evidence for solar oscillations and the prospects for helioseismology ChristensenDalsgaard and Gough the goals were soon extended to provide models for comparison with the observed frequencies.
Given the expected accuracy of these frequencies, and the need to use them to uncover subtle features J. The code drew some inspiration from the Eggleton stellar evolution code Eggleton , which had been used previously, but the development was fully independent of that code. An early description of the code was given by ChristensenDalsgaard , with further extensive details provided by Christensen-Dalsgaard ; many aspects of this still stand and will only be summarized here. With the increasing quality and extent of the helioseismic data the code was further developed, to allow for more realistic physics; a major improvement was the inclusion of diffusion and settling Christensen-Dalsgaard et al.
In parallel, extensions have been made to the code to consider the evolution of stars other than the Sun; these include the treatment of convective cores and core overshoot, attempts to model red-giant evolution and the inclusion of helium burning.
This development is still very much under way. For use in asteroseismic fitting a version of the code has also been developed in the form of a subroutine with a reasonably simple calling structure which also includes the computation of adiabatic oscillation frequencies as part of the computation. The combined package is available as a single tar file, making the installation relatively straightforward, and the code has been successfully implemented on a variety of platforms.
Nevertheless, it is sufficiently complex that a general release is probably not advisable. The calculation of the temperature gradient in convective regions is discussed in Sect. In 5 the derivatives with respect to m should obviously be expressed in terms of derivatives with respect to x. As discussed in considerable detail by Christensen-Dalsgaard , this is treated by expanding the variables to second significant order in r to obtain the required conditions at the innermost point in the numerical solution.
These are written in the form of expressions of the m and L at the innermost mesh point in terms of the other variables. For elements where diffusion and settling are ignored the original form, 5 , with Dk and Vk set to zero is obviously used. The boundary condition is obtained by equating the pressure resulting from integrating 12 to the pressure in the interior solution. To ensure a continuous transition to the interior I follow Henyey et al. In the solution of the equations, the computation of the right-hand sides is the heaviest part of the calculation and hence needs to be optimized in terms of efficiency.
The former clearly provides higher precision but potentially less stability than the latter; thus time-centred differences are typically used for processes occurring on a slow timescale, such as the change in the hydrogen abundance, whereas the implicit scheme is used, e. This process is repeated with the thus corrected solution as trial, until convergence. The initial ZAMS model is assumed to be static and with a prescribed chemical composition.
Thus in this case only 22 are solved. Time evolution is started with a very small t for the initial non-zero timestep. The number N of meshpoints is kept fixed during the evolution, but the distribution of points is varied accord- 16 ing to the change in the structure of the model. The mesh algorithm is based on the first-derivative stretching introduced by Gough et al. An early version of the implementation was described by Christensen-Dalsgaard , but this has subsequently been substantially extended and is still under development.
In particular, a dense distribution of points is set up near the boundaries of convection zones, although so far points are not adjusted so as to be exactly at the edges of the zones. After completing the solution at each timestep the new mesh is determined and the model transferred to this mesh using, in general, third-order interpolation; linear interpolation is used near boundaries of convective regions and in other regions where the variation of the variables is not smooth.
To compensate for the fairly crude treatment of the composition in a possible convective core, the change of X in such a core is given higher weight than the general change in X. The algorithm correctly ensures that very short steps in time are taken in rapid evolutionary phases.
In typical simple calculations, assuming 3 He to be in nuclear equilibrium, roughly 35 timesteps are needed to reach the end of central hydrogen burning in models without with a convective core, and steps to reach the base of the red-giant branch. Calculations with more complex physics or requiring higher numerical precision obviously require a substantially higher number of timesteps.
Evolution up the red giant branch typically requires a large number of timesteps owing to the rapid changes at fixed m in the hydrogen-burning shell1 although the timestep algorithm has options to reduce the weight given to this region. As a general principle, the code has been written in a modular way, so that replacing, for example, routines for equation of state or opacity has been relatively straightforward.
Astrophys Space Sci 13—24 the treatment of the equation of state and opacity in stellar modelling. A review of nuclear reactions in the solar interior, of relevance in the more general stellar case, was provided by Parker and Rolfs The fact that the whole calculation is done explicitly makes it entirely feasible, if somewhat cumbersome, to evaluate analytical derivatives. Up to second derivatives of pressure, density and enthalpy are provided in a fully consistent manner, whereas third derivatives, required for the central boundary condition, assume full ionization.
Partial electron degeneracy is included in the form of expansions that cover all levels of degeneracy and relativistic effects. The computation of the Coulomb effects consequently requires some iteration, even if the EFF variables are used, and hence some increase in computing time. However, the resulting equation of state captures major aspects of the more complex forms discussed below. More realistic descriptions of the equation of state require computations that are currently too complex to be included directly in stellar evolution calculations.
Thus interpolation in pre-computed tables is required. The first such set to be included was the so-called MHD equation of state Mihalas et al.
Application of this formulation to a comparison with observed solar oscillation frequencies showed a very substantial improvement in the agreement between the Sun and the model Christensen-Dalsgaard et al. The early versions of both the MHD and OPAL equations of state suffered from a neglect of relativistic effects in the treatment of the electrons Elliott and Kosovichev This has since been corrected Gong et al.
Also, the OPAL formulation suffered from inconsistencies between some of the variables provided e. Boothroyd and Sackmann ; Scuflaire et al. Typically, a single representative value of Z is used, even in cases with diffusion and settling of heavy elements, although the code has the option of using linear interpolation between two sets of tables with different Z.
With various updates of the tables this has since been the basis for the opacity calculation in the code. The most recent tables, including a variety of mixtures of the heavy elements, are based on the computations by Rogers and Iglesias Atmospheric opacities must be supplied separately; here tables by Kurucz , Alexander and Ferguson or Ferguson et al.
Electron conduction may be included based on the tables of Itoh et al. Houdek see Houdek and Rogl , For interpolation in log R, log T two schemes are available. One uses a minimum-norm algorithm with interpolating function defined in a piecewise fashion over triangles in the log R— log T plane Nielson , Note that the relative composition of Z is assumed to be fixed; thus differential settling or changes in heavy-element composition resulting from nuclear burning are not taken into account.
The code includes flexible ways of modifying the opacity, to allow tests of the sensitivity of the model to such modifications. An extensive survey of the response of solar models to localized opacity changes, specified as functions of log10 T , was made by Tripathy and Christensen-Dalsgaard Electron screening is computed in the Salpeter approximation. Electron capture by 7 Be is treated according to Bahcall and Moeller The nuclear network is relatively limited and is one of the points where the code needs improvement.
This is to some extent a heritage of its origin as a solar-modelling code, as well as a consequence of the fact that pre-main-sequence evolution is not computed. In the pp chains 2 D, 7 Li and 7 Be are assumed to be in nuclear equilibrium. On the other hand, the code has the option of following the evolution of 3 He, although in many calculations it is sufficient to assume 3 He to be in nuclear equilibrium.
To simulate the evolution of the 3 He abundance X 3 He during the pre-mainsequence phase the initial zero-age main-sequence mode assumes the X 3 He that would have resulted from evolution over a specified period t3 He at constant conditions, as described by Christensen-Dalsgaard et al. Helium burning has been included in the code using the expressions of Angulo et al. However, the code is unable to deal with helium ignition in a helium flash. Thus models with helium burning are restricted to masses in excess of 2.
Also, as in cases of hydrogen burning cf. For completeness I give the complete expressions in Appendix 1.
If included, diffusion of heavy elements assumes that all elements behave as fully ionized 16 O; this is a reasonable approximation in the solar case where the outer convection zone is relatively deep, but becomes increasingly questionable in more massive mainsequence stars.
Here, also, effects of selective radiative levitation should be taken into account e. Various approximations to turbulent diffusion can be included, partly inspired by Proffitt and Michaud In addition, the code has the option to suppress settling in the outer parts of the star, to allow modelling of diffusion and settling in the cores of relatively massive stars where otherwise settling beneath the thin outer convection zone would result in a complete depletion of the surface layers of elements heavier than hydrogen.
At present, diffusion and settling is coupled to nuclear evolution in the consistent manner of 5 only for helium. For the remaining elements taking part in the nuclear network diffusion is neglected.
Correcting this deficiency is an obvious priority. In addition, emulations of the Canuto and Mazzitelli formulation, established by Monteiro et al. Convective regions can obviously, at least in stars that are not extremely evolved, be assumed to have uniform composition. This can in principle be achieved by including a very high diffusion coefficient in such regions. The treatment of convective cores remains a concern and an area of active development, however.
Given Astrophys Space Sci 13—24 the lack of a proper treatment of the diffusion of all elements an explicit calculation of the chemical evolution is required. This is characterized by the assumed homogeneous abundances Xk,c of the elements.
In the second term in 25 Xk xcc is evaluated just outside the core; this term only has an effect if there is a composition discontinuity at the edge of the core, i. In models with a growing convective core, and neglecting diffusion, a discontinuity is set up in the hydrogen abundance at the edge of the core see also Fig. This situation arises in intermediate-mass stars with masses up to around 1.
This raises the question of the definition of convective instability: if the border of the convective core is defined using the composition of the core, the region immediately outside the core is therefore convectively unstable. As a consequence, ASTEC defines the border of the convective core by the abundance in the radiative region, leaving a small convectively stable region just below the border, which nevertheless is assumed to be fully mixed in the standard ASTEC implementation.
This may be regarded as an example of semiconvection, of somewhat uncertain physical consequences e. A different scheme, now under implementation, is discussed in Sect.
Various options exist for the treatment of convective overshoot. The overshoot region may be taken to be either adiabatically or radiatively stratified. A more elaborate treatment of overshoot from a convective envelope has been implemented, following Monteiro et al. This is being extended to emulate the overshoot simulations by Rempel In ASTEC the iteration to determine these parameters is carried out automatically, making the computation of solar models, and the exploration of the consequences of modifications to the input physics, rather convenient.
The ASTEC code has grown over three decades, with a substantial number of different uses along the way. This is clearly reflected in the structure of the code as well as in the large number of input parameters and flags that control its different options. These are provided in an input file, in many cases using simply the defaults provided in the source of the code. Also, several different executables can be produced, reflecting partly the evolution of the code and partly different versions of the physics, in particular the equation of state and the opacity, as well as the option to include diffusion and settling.
To make the code somewhat more user-friendly, scripts have been made which allow simply to change a few key parameters, such as mass, heavy-element abundance and number of timesteps, by editing templates of the input files. Consequently, the code has been used with success by several users, including students and postdocs at the University of Aarhus.
A convenient way to use the code, without overloading storage systems with the large gong files, is to store the full emdl file and subsequently read in models at selected timesteps, to output the corresponding amdl or gong files.
For asteroseismology it is evidently crucial to compute oscillation frequencies of the computed models. The full calculation of frequencies, for given input parameters to the evolution calculation, often needs to be carried out as part of a larger computation, e.
To facilitate this a version of the code has been made where the evolution calculation and all parts of the adiabatic oscillation calculation is carried out by a single subroutine call, with internal passing of the intermediate products of the calculation.
This subroutine can then be called by, for example, a fitting code. An example of such use is the application of the code in geneticalgorithm fitting e. Metcalfe, High Altitude Observatory. The code has been implemented on a variety of platforms and appears to be relatively robust. To simplify the installation, a complete tar package including all the required files, with a setup script and a full makefile, has been established.
However, the complexity of the code and the lack of adequate documentation makes it unrealistic to release it for general use. A fairly trivial issue is the restricted nuclear network and the failure to include all elements undergoing nuclear reactions in the full diffusive treatment.
Rather more serious problems concern the treatment of the borders of convective regions; even though this obviously also involves open issues of a basic physical nature, the code should at least aim at treating these regions in a numerically consistent, even if perhaps not physically adequate, manner.
A serious problem is the failure of the code for models with convective cores, when diffusion and settling of both helium and heavy elements are included cf. Christensen-Dalsgaard b ; on the other hand, the case of just helium diffusion can be handled. One case concerns growing convective cores in nondiffusing models see also Sect.
The dashed curves illustrate the usual treatment in ASTEC, where the boundary of the convective core is determined by the composition in the radiative region just outside it, and the hydrogen abundance X is discontinuous. Significant sensitivity to the treatment of this region was indeed found by Moya et al. The second aspect of semiconvection concerns the base of the convective envelope in models with diffusion and settling of helium and heavy elements.
As noted by Bahcall et al. Complete mixing of the unstable region removes the gradient and hence the cause of the instability, leading to an uncertain situation, characteristic of semiconvection. Since the opacity depends both on X and Z the mixing must consistently affect both abundances. I have attempted to implement this by including a turbulent diffusivity, obviously common to all elements, determined iteratively as a function of depth beneath the convection zone such that the resulting profiles of X and Z lead to neutral stability.
It is obvious, however, that application to the increasingly accurate and detailed asteroseismic data will require further development. Acknowledgements I am very grateful to D. Gough for his assistance in developing the initial version of the code, including the basic package to solve the equations of stellar evolution. Many people have contributed to the development of ASTEC over the years, and I am very grateful for their contributions.
I thank W. Monteiro for providing alternate treatments of the parameterization of convection, C. Proffitt for help with installing diffusion and settling in the code and M. Bazot for assistance with updating the treatment of nuclear reactions. Lebreton and J. These are largely obtained from Michaud and Proffitt , although with minor modifications.
As presented by Michaud and Proffitt the diffusion velocity of trace elements depends on the gradient in the hydrogen abundance. References Adelberger, E. A , 3— Bahcall, J. Wasserburg Bahcall, J. EAS Publ. Nature , 89—92 Christensen-Dalsgaard, J. Nature , — Christensen-Dalsgaard, J. Science , — Clayton, D. ACM 17, — Cox, A. Variable Stars as Essential Astrophysical Tools, pp. Kluwer Academic, Dordrecht Eggleton, P. In: Richtmyer, R. Lecture Notes in Physics, vol. Springer, Heidelberg Henrici, P.
Wiley, New York Henyey, L. Communications in Asteroseismology, No. India 24, — Iglesias, C. Hightemperature results. In: Crivellari, L. Hardcover ISBN : Edition Number : 1. Number of Pages : VI, Topics : Astrophysics. Skip to main content.
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