Evolution and Seismic Tools for Stellar Astrophysics [1 ed.] 1402094396, 9781402094392

Table of contents :
Introduction......Page 8
ESTA organisation and tools......Page 9
Stellar internal structure and evolution codes......Page 10
TASK 1: basic stellar models......Page 11
TASK 2: oscillation frequencies......Page 12
Initial parameters......Page 13
Input physics......Page 14
Conclusion......Page 16
References......Page 17
Introduction......Page 20
Numerical scheme......Page 21
Equation of state......Page 23
Nuclear reactions......Page 24
Treatment of convection......Page 25
Further developments......Page 26
Acknowledgements......Page 27
Appendix 1: Treatment of diffusion and settling......Page 28
References......Page 29
Integration of the stellar structure......Page 32
Atomic diffusion......Page 34
The Standard Solar Model......Page 35
References......Page 36
Introduction......Page 38
Time steps......Page 39
Diffusion......Page 40
Running YREC......Page 41
Model grids......Page 42
Implemented equations......Page 43
Turbulent velocities......Page 44
Solar model with chi and gamma as independent variables......Page 45
SAL peak shift......Page 46
Acknowledgements......Page 47
References......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 100
The macrophysics: atmospheric structure and boundary conditions......Page 101
References......Page 102
Numerics......Page 103
Explicit time integration......Page 104
Mass loss......Page 105
Nuclear reactions......Page 106
Opacities......Page 107
Summary......Page 108
References......Page 109
Formulation of the adiabatic equations......Page 111
The Galerkin finite-element method......Page 112
Internal stability and accuracy......Page 113
Impact of stellar model mesh resolution......Page 114
References......Page 116
Equilibrium model......Page 117
Formulation of the equations......Page 118
Numerical scheme......Page 119
The shooting method......Page 120
Improving the frequency precision......Page 121
Computed quantities......Page 122
Acknowledgements......Page 123
References......Page 124
Basic equations for linear perturbations......Page 125
The equilibrium model......Page 126
At the atmosphere......Page 127
Numerical variables......Page 128
Accuracy of the results......Page 129
Acknowledgements......Page 130
References......Page 131
Adiabatic case......Page 132
Non-adiabatic resolution......Page 134
References......Page 136
Nonradial oscillations......Page 137
Outputs......Page 138
Additional tools for the initial model......Page 139
References......Page 141
Introduction......Page 143
Dimensionless variables......Page 144
Shooting method......Page 145
l=1 modes......Page 146
Numerical examples......Page 147
ModelJCA: improved hydrostatic equilibrium......Page 148
References......Page 149
Oscillation modes......Page 150
Nonradial oscillations......Page 151
Difference equations......Page 153
Applications......Page 154
References......Page 155
Introduction......Page 156
Oscillation frequencies of a pseudo-rotating model......Page 157
The boundary conditions......Page 158
filou inputs and outputs......Page 159
Numerical tests and results......Page 160
Results. The effect of shellular rotation on adiabatic oscillations......Page 161
References......Page 162
Methods of studying pulsating stars......Page 163
Stellar direct method. The LNAWENR numerical method......Page 164
Conclusions......Page 165
References......Page 166
Input physics and numerical aspects......Page 167
Initial parameters of the models......Page 168
ADIPLS oscillation frequencies......Page 169
References......Page 170
Introduction......Page 172
GRID A: evolution from fully convective spheres......Page 173
GRID B: evolution from a birthline......Page 174
Evolutionary tracks and stellar models......Page 175
References......Page 176
Input physics......Page 178
Evolutionary tracks......Page 179
Stellar models......Page 181
Seismic properties......Page 182
References......Page 183
Introduction......Page 185
Numerical tools......Page 186
Presentation of the comparisons and general results......Page 187
Low-mass models: Cases 1.1, 1.2 and 1.3......Page 190
Intermediate mass models: Cases 1.4 and 1.5......Page 192
High mass models: Cases 1.6 and 1.7......Page 193
Convection regions and ionisation zones......Page 194
Solar-like oscillations: Cases 1.1, 1.2 and 1.3......Page 197
Cases 1.4 and 1.5......Page 198
Cases 1.6 and 1.7......Page 199
Presentation of the comparisons and general results......Page 200
Internal structure......Page 202
Solar-type stars with convective cores: Cases 3.2 and 3.3......Page 203
Convection zones......Page 204
Helium surface abundance......Page 205
Summary and conclusions......Page 206
References......Page 210
Comparison between relevant physical quantities......Page 212
Comparison between the variables used in the frequency computations......Page 214
References......Page 215
Equation of State......Page 216
Opacities......Page 218
Effects on global stellar parameters......Page 219
Effects on the stellar structure......Page 221
Effects on the frequencies......Page 222
Atmosphere......Page 223
Numerical aspects......Page 224
Conclusions......Page 225
References......Page 226
Abstract......Page 227
Introduction......Page 228
The equilibrium models......Page 229
Oscillation codes and requirements......Page 230
Radial modes......Page 231
Non-radial modes with l=2......Page 233
Large separation of radial modes......Page 236
Non-radial modes with l=2......Page 237
Small separations delta02......Page 239
The influence of the gravitational constant G......Page 240
The choice of dependent variables and equations......Page 241
The choice of independent variable......Page 242
Conclusions......Page 243
References......Page 244
Introduction......Page 246
A study case......Page 247
The `data'......Page 248
`Modelling' the proxy star......Page 249
Frequency comparisons......Page 251
Comparison with CLES model: ESTA `calibration'......Page 252
Stellar structures......Page 253
Stellar structure......Page 254
Evolved stars......Page 255
References......Page 256

Citation preview

Mário João P.F.G. Monteiro Editor

Evolution and Seismic Tools for Stellar Astrophysics

Reprinted from Astrophysics and Space Science Volume 316, Nos. 1–4, August 2008

Editor Mário João P.F.G. Monteiro Universidade do Porto Centro de Astrofisica rua das Estrelas s/n 4150-762 Porto Portugal Universidade do Porto Faculdade de Ciencias rua do Campo Alegre 687 4169-007 Porto Portugal

ISBN: 978-1-4020-9439-2

e-ISBN: 978-1-4020-9440-8

Library of Congress Control Number: 2008938913

© Springer Science + Business Media B.V., 2008 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover illustration: Schematic illustration of the location of different classes of pulsating stars in the HR diagram, originally published by J. ChristensenDalsgaard in 1999 (Astrophys. Space Science 261, 1–12), and in an updated form in 2004 (Proc. SOHO 14 - GONG 2004: ‘Helio- and Asteroseismology: Towards a golden future’, ESA SP-559, pp. 1–33). The colour version was produced in 2000 for the Eddington proposal by Lars Peter Rasmussen. Printed on acid-free paper springer.com

Contents

M.J.P.F.G. Monteiro / Foreword The context Y. Lebreton, M.J.P.F.G. Monteiro, J. Montalbán, A. Moya, A. Baglin, J. Christensen-Dalsgaard, M.-J. Goupil, E. Michel, J. Provost, I.W. Roxburgh, R. Scuflaire and ESTA Team / The CoRoT evolution and seismic tools activity

1–12

Evolution codes J. Christensen-Dalsgaard / ASTEC—the Aarhus STellar Evolution Code

13–24

S. Degl’Innocenti, P.G. Prada Moroni, M. Marconi and A. Ruoppo / The FRANEC stellar evolutionary code

25–30

P. Demarque, D.B. Guenther, L.H. Li, A. Mazumdar and C.W. Straka / YREC: the Yale rotating stellar evolution code

31–41

P. Eggenberger, G. Meynet, A. Maeder, R. Hirschi, C. Charbonnel, S. Talon and S. Ekström / The Geneva stellar evolution code

43–54

A. Hui-Bon-Hoa / The Toulouse–Geneva Evolution Code (TGEC)

55–60

P. Morel and Y. Lebreton / CESAM: a free code for stellar evolution calculations

61–73

I.W. Roxburgh / The STAROX stellar evolution code

75–82

R. Scuflaire, S. Théado, J. Montalbán, A. Miglio, P.-O. Bourge, M. Godart, A. Thoul and A. Noels / CLÉS, Code Liégeois d’Évolution Stellaire

83–91

P. Ventura, F. D’Antona and I. Mazzitelli / The ATON 3.1 stellar evolutionary code

93–98

A. Weiss and H. Schlattl / GARSTEC—the Garching Stellar Evolution Code

99–106

Seismic codes P. Brassard and S. Charpinet / PULSE: a finite element code for solving adiabatic nonradial pulsation equations

107–112

J. Christensen-Dalsgaard / ADIPLS—the Aarhus adiabatic oscillation package

113–120

M.J.P.F.G. Monteiro / Porto Oscillation Code (POSC)

121–127

A. Moya and R. Garrido / Granada oscillation code (GraCo)

129–133

J. Provost / NOSC: Nice OScillations Code

135–140

I.W. Roxburgh / The OSCROX stellar oscillaton code

141–147

R. Scuflaire, J. Montalbán, S. Théado, P.-O. Bourge, A. Miglio, M. Godart, A. Thoul and A. Noels / The Liège Oscillation code

149–154

J.C. Suárez and M.J. Goupil / FILOU oscillation code

155–161

M.D. Suran / LNAWENR—linear nonadiabatic nonradial waves

163–166

Grids Y. Lebreton and E. Michel / Reference grids of stellar models and oscillation frequencies

167–171

J.P. Marques, M.J.P.F.G. Monteiro and J.M. Fernandes / Grids of stellar evolution models for asteroseismology (CESAM + POSC)

173–178

J. Montalbán, A. Miglio, A. Noels and R. Scuflaire / Grids of stellar models and frequencies with CLÉS + LOSC

179–185

Comparisons Y. Lebreton, J. Montalbán, J. Christensen-Dalsgaard, I.W. Roxburgh and A. Weiss / CoRoT/ESTA–TASK 1 and TASK 3 comparison of the internal structure and seismic properties of representative stellar models

187–213

M. Marconi, S. Degl’Innocenti, P.G. Prada Moroni and A. Ruoppo / FRANEC versus CESAM predictions for selected CoRoT ESTA task 1 models

215–218

J. Montalbán, Y. Lebreton, A. Miglio, R. Scuflaire, P. Morel and A. Noels / Thorough analysis of input physics in CESAM and CLÉS codes

219–229

A. Moya, J. Christensen-Dalsgaard, S. Charpinet, Y. Lebreton, A. Miglio, J. Montalbán, M.J.P.F.G. Monteiro, J. Provost, I.W. Roxburgh, R. Scuflaire, J.C. Suárez and M. Suran / Inter-comparison of the g-, f- and p-modes calculated using different oscillation codes for a given stellar model

231–249

Balance and future M.J. Goupil / Concluding remarks and perspectives on the evolution and seismic tools activity (ESTA) of the CoRoT community

251–261

Foreword This volume is a collection of original articles published by the Astrophysics and Space Science Journal. These are the result of the work initiated in 2002 with the aim to extensively test, compare and optimize the numerical tools used to calculate stellar models and their oscillation frequencies. This effort took place under the activities of the “CoRoT Evolution and Seismic Tools Activity” (CoRoT/ESTA—http://www.astro.up.pt/esta/) of the Seismology Working Group of the CoRoT Mission (http://corot.oamp.fr/). It has been organized through the participation of the European Space Agency (ESA) in this Frenchled mission, following the appointment by ESA of a Co-Investigator on Stellar Models for the CoRoT Team. 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 2002–2007 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 work presented in this volume has been supported by Centro de Astrofísica da Universidade do Porto (CAUP—http://www.astro.up.pt/) with funds from the European Helio and Asteroseismology Network (HELAS—www.helas-eu.org), a major European collaboration funded by the European Commission under FP6 (contract FP6-2004-Infrastructures-5-026138), and from the Fundação para a Ciência e a Tecnologia (www.fct.mctes.pt) and POCI with support of the Portuguese Government and the European Programme FEDER (under grants POCI/CTE-AST/57610/2004, POCI/V.5/B0094/2005 and PTDC/CTE-AST/66181/2006). 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. Janine Provost at the Observatoire de la Côte d’Azur, Prof. Jørgen Christensen-Dalsgaard at the Danish AsteroSeismology Centre in the University of Aarhus and Dr. Christian Straka at CAUP.

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. We also wish to express special thanks to the PI of the CoRoT mission, Dr. Annie Baglin, and to the PI of the CoRoT Seismology Working Group, Dr. 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. . . . Mário J.P.F.G. Monteiro ([email protected]) Centro de Astrofísica da Universidade do Porto Departamento de Matemática Aplicada, Faculdade de Ciências da Universidade do Porto Porto, 9th September 2008

The context The CoRoT evolution and seismic tools activity Goals and tasks Y. Lebreton · M.J.P.F.G. Monteiro · J. Montalbán · A. Moya · A. Baglin · J. Christensen-Dalsgaard · M.-J. Goupil · E. Michel · J. Provost · I.W. Roxburgh · R. Scuflaire · ESTA Team

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-008-9771-1 © Springer Science+Business Media B.V. 2008

Abstract The forthcoming data expected from space missions such as CoRoT require the capacity of the available tools to provide accurate models whose numerical precision is well above the expected observational errors. 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 CoRoT Evolution and Seismic Tool Activity (ESTA) has been set up with this mission. Y. Lebreton () · ESTA Team Observatoire de Paris, GEPI, CNRS UMR 8111, 5 Place Janssen, 92195 Meudon, France e-mail: [email protected] M.J.P.F.G. Monteiro Centro de Astrofísica da Universidade do Porto and Departamento de Matemática Aplicada da Faculdade de Ciências, Universidade do Porto, Porto, Portugal J. Montalbán · R. Scuflaire Institut d’Astrophysique et Geophysique, Université de Liège, Liège, Belgium A. Moya Instituto de Astrofísica de Andalucía-CSIC, Granada, Spain A. Baglin · M.-J. Goupil · E. Michel · I.W. Roxburgh Observatoire de Paris, LESIA, Paris, France J. Christensen-Dalsgaard Institut for Fysik og Astronomi, Aarhus Universitet, Aarhus, Denmark J. Provost Cassiopée, URA CNRS 1362, Observatoire de la Côte d’Azur, Nice, France I.W. Roxburgh Queen Mary University of London, London, UK

Several groups have been involved. 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. Keywords Stars: interiors · Stars: evolution · Stars: oscillations · Methods: numerical PACS 97.10.Cv · 97.10.Sj · 95.75.Pq

1 Introduction The CoRoT satellite was launched into space on December 27, 2006. This experiment will provide us with stellar oscillation data (frequencies, amplitudes, line widths) for stars of various masses and chemical compositions—mainly main sequence solar type stars, δ Scuti stars and β Cephei stars—with an expected accuracy on the frequencies of a few 10−7 Hz (Baglin et al. 2002; Michel et al. 2006). 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 1999; Mazumdar et al. 2006; Cunha and Metcalfe 2007) or the position of the external convection zone (Audard and Provost 1994; Monteiro et al. 2000; Ballot et al. 2004; Verner et al. 2006) and the envelope helium abundance in low mass solar type stars (Monteiro and Thompson 1998; Basu et al. 2004; Verner et al. 2006). High-quality seismic data also make possible inverse analyses to infer the stellar density (Gough and Kosovichev 1993) and rotation profiles from separated multiplets (Goupil et al. 1996). More generally, it is expected that CoRoT will provide constraints

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_1

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on those aspects of the physics of stellar interiors which are still poorly understood, in particular on convection (Samadi et al. 2005) and on transport mechanisms at work in the radiative zones such as microscopic diffusion and rotationally induced transport (Goupil et al. 2004; Théado et al. 2005). 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 2005; Monteiro et al. 2002). 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. 1995; Gough et al. 1996; Monteiro et al. 2002). 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. In the 1990’s the GONG solar model comparison project has been one of the first attempts to compare thoroughly solar structure models calculated with stellar evolution codes differing both in the numerical procedures and input physics. Extensive comparisons were made which allowed to better understand differences between models and to correct or improve the codes under comparison (ChristensenDalsgaard 1988; Gabriel 1991; Turck-Chièze et al. 1998). In the same spirit, within the CoRoT Seismology Working Group, the ESTA group has been set up (see Sect. 3), with the aim to extensively test, compare and optimise the numerical tools used to calculate stellar models and their oscillation frequencies. 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. 4, we present the three ESTA tasks we have focused on. In Sect. 5 we give the specifications of the tasks concerning the input physics and parameters of the stellar models. Finally in Sect. 6 we present reference grids of stellar models and associated oscillation frequencies calculated by ESTA participants in parallel to the main ESTA

Astrophys Space Sci (2008) 316: 1–12

work. These grids have been used all along the preparation of the CoRoT mission and have been made available on the ESTA web site.

2 The observational seismic program of CoRoT The CoRoT mission has been designed and built to satisfy the scientific specifications as expressed in the document COR-SP-0-ETAN-83-PROJ, written in 2001. At that time more recent developments of the seismology theory as well as discoveries of pulsations were evidently not taken into account. More precisely CoRoT has been required to reach: • a photometric noise of 0.6 ppm in five days in the 0.1 to 10 mHz frequency domain, • a duty cycle of 90%, • a percentage of polluted frequencies less than 5% of the frequency interval. The major goals of the seismology programme were then to detect solar type oscillations in a few bright objects, covering the H–R diagram and to see whether classical pulsators (δ Scuti and β Cephei stars in particular) near the main sequence had also lower amplitude modes. With these goals in mind the seismology program has been defined with two components:  the central program having at least 5 sessions of 150 days (see Fig. 1),  the exploratory program with a few short sessions of 20 days, to complement the H–R diagram coverage. The instrument has two different fields: • the seismology field in which 10 bright stars are observed at the same time (5.5 < mV < 9.5) with a sampling of 32 seconds, • the exoplanet field which observes 12 000 stars at the same time (11 < mV < 16) with a sampling of 512 seconds and in a few cases 32 seconds. In this field the required photometric accuracy is 7 × 10−4 in 1 hour for a 15.5 magnitude star. After one year of operations the mission fulfils all its specifications.

3 ESTA organisation and tools The Evolution and Seismic Tools Activity (ESTA) has been set up in 2002, following the ESA announcement of opportunity for participation in the CoRoT mission. The actual work has started in 2004 at the kick-off meeting that took place during the CoRoT Week 7 in Granada (December

Astrophys Space Sci (2008) 316: 1–12

3

Fig. 1 H–R diagram of the targets of the 5 first long runs of the CoRoT mission, as planned presently. 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 2008)

2004). 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. This site is available at the following URL: http://www.astro.up.pt/corot/ ESTA has been open to participation of the whole community and in particular the involvement of groups with evolution codes or seismic codes has been actively pursued. Up to now there are over 40 team members involved in one or several of the tasks corresponding to more than 16 institutes/groups spread over 9 countries. 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.

3.1 Stellar internal structure and evolution codes Ten stellar evolution codes have participated in ESTA at different levels. The list is given below. Each of them is presented in detail in a specific paper in this volume. • ASTEC—The Aarhus Stellar Evolution Code is a Danish stellar evolution code that has been involved in all the ESTA TASK S. It is presented by Christensen-Dalsgaard (2007b). • ATON—This is an Italian stellar evolution code. Up to now it has participated only marginally in ESTA activities. It is presented by Ventura et al. (2007). • CESAM—The Code d’Évolution Stellaire Adaptatif et Modulaire is a French stellar evolution code. This public code1 has been involved in all the ESTA TASK S as well as in the reference grids calculation. It is presented by Morel and Lebreton (2007). 1 Available

at http://www.obs-nice.fr/cesam

4

Astrophys Space Sci (2008) 316: 1–12

• CLÉS—The Code Liégeois d’Évolution Stellaire is a Belgian stellar evolution code that has been involved in all the ESTA TASK S as well as in the reference grids calculation. It is presented by Scuflaire et al. (2007b). • FRANEC—The Frascati Raphson Newton Evolutionary Code is an Italian stellar evolution code that has participated in ESTA TASK S 1 and 3. It is presented by Degl’Innocenti et al. (2007). • GARSTEC—The Garching Stellar Evolution Code is a German stellar evolution code that has participated in ESTA TASK S 1 and 3. It is presented by Weiss and Schlattl (2007). • GENEC—The Geneva Stellar Evolution Code is a Swiss stellar evolution code that has participated in ESTA TASK 1. It is presented by Eggenberger et al. (2007). • STAROX—This is an English stellar evolution code that has participated in ESTA TASK 1. It is presented by Roxburgh (2007b). • TGEC—The Toulouse-Geneva Evolution Code is a French stellar evolution code that has participated in ESTA TASK S 1 and 3. It is presented by Hui-Bon-Hoa (2007). • YREC—The Yale Rotating Stellar Evolution Code is a US stellar evolution code. Up to now it has participated only marginally in ESTA activities. It is presented by Demarque et al. (2007). 3.2 Stellar oscillation codes Nine stellar oscillation codes have been used for the comparisons of ESTA TASK 2. The list is given below. Each of them is presented in details in a specific paper in this volume. • ADIPLS—The Aarhus Adiabatic Pulsation Package2 contains the ADIPLS stellar oscillation code itself but also many programs to manipulate the stellar model files and the output of the pulsation program (cf. Sect. 5). It is presented by Christensen-Dalsgaard (2007a). • FILOU—This stellar oscillations code developed at ParisMeudon Observatory is presented by Suarez and Goupil (2007). • GRACO—The Granada Oscillation Code is presented by Moya and Garrido (2007). • LOSC—The Liège Oscillations Code is presented by Scuflaire et al. (2007a). • NOSC—The Nice Oscillations Code is presented by Provost (2007). • OSCROX—This English stellar oscillations code is presented by Roxburgh (2007a). • POSC—The Porto Oscillations Code is presented by Monteiro (2008). • PULSE—This Canadian stellar oscillations code is presented by Brassard and Charpinet (2008). 2 Available

at http://astro.phys.au.dk/~jcd/adipack.n

• LNAWENR—This Romanian code for Linear Non-Adiabatic NonRadial Waves is presented by Suran (2007).

4 Presentation of the TASKs To perform the comparison of the participating numerical codes, the ESTA group has focused on specific TASK S. 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. Finally, the third task, TASK 3, still in progress, is similar to TASK 1 but for stellar models including microscopic diffusion of chemical elements. 4.1 TASK 1: basic stellar models TASK 1 has been defined with the aim of comparing the properties of stellar models and evolutionary sequences covering a rather large range of stellar parameters (initial mass, initial chemical composition and evolution stage) corresponding to the bulk of the CoRoT targets. 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 0.9–5.0 M . Table 1 Target models for TASK 1 (see Fig. 2). We have considered 7 cases corresponding to different initial masses, chemical compositions and evolutionary stages. One evolutionary sequence (denoted by “Ov” in the 5th column has been calculated with core overshooting—see text) Case

M/M

Y0

Z0

Specification

Type

1.1

0.9

0.28

0.02

Xc = 0.35

MS

1.2

1.2

0.28

0.02

Xc = 0.69

ZAMS

1.3

1.2

0.26

0.01

McHe = 0.10 M

SGB

= 1.9 × 107

1.4

2.0

0.28

0.02

Tc

1.5

2.0

0.26

0.02

Xc = 0.01, Ov

K

TAMS

1.6

3.0

0.28

0.01

Xc = 0.69

ZAMS

1.7

5.0

0.28

0.02

Xc = 0.35

MS

PMS

Astrophys Space Sci (2008) 316: 1–12

5

Fig. 2 H–R diagram showing the targets for TASK 1 (see Table 1). Red lines correspond to the PMS, black lines to the MS and blue lines to the post-main sequence SGB evolution. The targets are ordered in mass and age along the diagram, from Case 1.1 (bottom-right) up to Case 1.7 (top-left)

For the initial chemical composition, several pairs (Y, Z) have been considered by combining two different values of Z (0.01 and 0.02) and two values of Y (0.26 and 0.28). The corresponding values of (Z/X) are in the range 0.014–0.029. We have adopted the so-called GN93 solar mixture (see Grevesse and Noels 1993) which has (Z/X) = 0.0245. We have therefore metallicities such that [Fe/H] ∈ [−0.25, +0.17] dex (with the standard definition [Fe/H] = log (Z/X) − log (Z/X) ). On each evolutionary sequence, an evolutionary stage has been selected, either on the pre-main sequence (PMS), main sequence (MS) or subgiant branch (SGB). On the PMS we have specified the value of the central temperature of the model (Tc = 1.9 × 107 K). On the MS, we have fixed the value of the central hydrogen content: Xc = 0.69 for the model close to the zero age main sequence (ZAMS), Xc = 0.35 for the model in the middle of the MS and Xc = 0.01 for the model close to the terminal age main sequence (TAMS). On the SGB, a model can be chosen by specifying the value of the mass McHe of the central region of the star where the hydrogen abundance is such that X ≤ 0.01. We chose McHe = 0.10 M . 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. 5 below). The first results of TASK 1 have been presented by Monteiro et al. (2006). Detailed results are presented in this volume (Lebreton et al. 2008; Montalbán et al. 2008a, 2008b; Marconi et al. 2008). 4.2 TASK 2: oscillation frequencies Under TASK 2 we have aimed at evaluating the precision and uncertainties affecting the determination of oscillation frequencies. 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. In particular two steps have been considered in this TASK: Step 1—comparison of the frequencies from different codes for the same stellar model;

6

Astrophys Space Sci (2008) 316: 1–12

Table 2 Target models for TASK 3. 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. Stage C is on the SGB Case

M/M

Y0

Z0

Stage

Xc

McHe

3.1

1.0

0.27

0.017

A

0.35

–

3.2

1.2

0.27

0.017

B

0.01

–

3.3

1.3

0.27

0.017

C

0.00

0.05 M

Step 2—comparison of the frequencies from the same seismic code for different stellar models of the same stellar case. In this exercise stellar models at the middle of the main sequence, produced by ASTEC and CESAM, have been used with a mass of 1.5 M . The models have been provided with different number of mesh points. Preliminary results of this task were reported by Moya (2007) while the final results of Step 1 are discussed in this volume by Moya et al. (2008). Step 2 remains to be done. 4.3 TASK 3: stellar models including microscopic diffusion The goals of TASK 3 are to test, compare and optimise stellar evolution codes which include microscopic diffusion of chemical elements. At this stage we only consider diffusion resulting from pressure, temperature and concentration gradients (see Thoul and Montalbán 2007) while we do not take into account diffusion due to the radiative forces, nor the extra-mixing of chemical elements due to differential rotation or internal gravity waves (see Alecian 2007; Mathis et al. 2007; Zahn 2007). The other physical assumptions proposed as the reference for the comparisons are the same as used for TASK 1 and no overshooting (see Sect. 5). 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.e. M < 1.4 M ) in order to keep in a mass range where radiative accelerations can be omitted. Furthermore, this avoids the problems occurring at higher masses where the use of microscopic diffusion alone produces a very substantial depletion of helium and heavy elements at the surface (and a concomitant increase of the hydrogen content) and in turn requires to invoke other mixing processes to control the gravitational settling (see, for instance Turcotte et al. 1998). For the three cases we have adopted a chemical composition close to the solar one (Z/X = 0.0243). For each case, models at different evolutionary stages have been considered. We focused on three particular evolution stages: middle of the MS, TAMS and SGB (respectively stage A, B

Fig. 3 H–R diagram showing the targets for TASK 3 (see Table 2). Evolutionary tracks correspond to Case 3.1 (bottom-right), Case 3.2 (middle) and Case 3.3 (top-left). 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. Preliminary results of TASK 3 have been presented by Lebreton et al. (2007), Montalbán et al. (2008a), ChristensenDalsgaard (2008). Advanced comparisons are presented in this volume (Lebreton et al. 2008).

5 Specifications for the TASK S In order to reduce the sources of discrepancies in the comparison due to sources not relevant in this context it has been necessary to specify the initial parameters and the physics to be used in the models. 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. Here we briefly describe the common specifications used for all TASK S. Further details can be found in the documentation available at the ESTA web-site.3 5.1 Initial parameters A list of reference values of some astronomical and physical constants have been specified to safeguard consistency in the comparison of the output from different codes. We have also fixed the mixture of heavy elements to be used in the calculations. 3 At

http://www.astro.up.pt/corot/compmod/task1

Astrophys Space Sci (2008) 316: 1–12

7

Table 3 Some of the physical constants necessary for the calculation of a stellar model. The units are cgs, except where otherwise noted Boltzmann’s constant

k = 1.380658 × 10−16 = 1.6605402 × 10−24

Atomic mass unit

mu

Perfect gas constant

R = 8.3145111 × 107

Electron mass

me = 9.1093897 × 10−28

Electron charge

e = 1.602177333 × 10−19 C

Planck’s constant

h = 6.6260755 × 10−27

Speed of light

c = 2.99792458 × 1010

Radiation density constant

a = 7.5659122 × 10−15

Stefan-Boltzmann constant

σ = 5.67051 × 10−5

Electron–Volt

1 eV = 1.60217733 × 10−11

Atomic weight of hydrogen

AH = 1.00782500

Atomic weight of helium

AHe = 4.00260330

Ionisation potential for H

χH = 13.595 eV

1st ionisation potential for He

χHe = 24.580 eV

2nd ionisation potential for He

χHe+ = 54.403 eV

(a) Physical constants: We have specified the values of the physical constants necessary for the calculation of a stellar model according to Cohen and Taylor (1987) and Lide (1994) (see Table 3). 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. (b) Astronomical constants: We have chosen the values to consider for the global parameters of the present Sun. We have adopted R = 6.9599 × 1010 cm, for the solar radius (Allen 1973), L = 3.846 × 1033 erg s−1 , for the solar luminosity (Willson et al. 1986) and M = 1.98919 × 1033 g, for the solar mass. Here M is obtained from (GM ) = 1.32712438 × 1026 cm−3 s−2 (Lide 1994) and G = 6.6716823 × 10−8 cm−3 g−1 s−2 . The value R refers to the radius of the model layer where T = Teff = 5777.54 K. (c) Initial abundances of the elements and heavy elements mixture: All models are calculated with the classical GN93 solar mixture of heavy elements (Grevesse and Noels 1993). As shown in Tables 1 and 2 the mass fractions of hydrogen (X), helium (Y ) and heavy elements (Z) are specified for each model.

5.2 Input physics In order to be able to make basic comparisons between the codes, we chose some reference input physics that have the advantage to be currently implemented in stellar evolution codes. (a) Equation of State: We chose to use the OPAL equation of state (EOS) in its 2001 version (Rogers and Nayfonov 2002) available at the OPAL Web site.4 It consists in a series of tables and in an associated interpolation package. The tables provide a set of thermodynamic quantities, i.e. pressure, internal energy, entropy, specific heat Cv , χρ = (d ln P /d ln ρ)T , χT = (d ln P /d ln T )ρ and the adiabatic indices 1 , 2 /(2 −1) and (3 −1), at given values of the temperature, density, X and Z. 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. In addition, it has been pointed out by Boothroyd and Sackmann (2003) that there are some inconsistencies in the OPAL tables with the result that the tabulated values of the adiabatic indexes 1 , 2 /(2 −1) and (3 −1) may differ by several per cent from the values that can be recalculated from the tabulated values of P , Cv , χρ and χT by means of thermodynamic relations. Furthermore, in the course of the comparisons undertaken by the ESTA group it has been shown by one of us (I.W. Roxburgh) that the tabulated values of the specific heat Cv in OPAL tables are incorrect, and also acknowledged by the OPAL team that this parameter should better be obtained from the other quantities provided by OPAL. For these reasons, some codes only draw P , χρ , χT and 1 from the OPAL EOS tables and apply thermodynamic relations to derive the other thermodynamic quantities. (b) Opacities: We chose to use the 1995 OPAL opacity tables (Iglesias and Rogers 1996) complemented at low temperatures by the Alexander and Ferguson (1994) tables, and not to include the conductive opacity. However slightly different OPAL tables are actually used by the different codes. Pre-calculated OPAL tables have been made available by the OPAL group either prior to publication or later on the OPAL Web site. They have been obtained for a “reduced” GN93 solar mixture, i.e. while the GN93 mixture is given for 23 elements, the OPAL opacity calculation only considers 19 elements but adds the abundances of “lacking” elements to the abundances of nearby elements (having close atomic number). Also, on the OPAL Web site there is the option to calculate tables on-line for any desired 19 elements-mixture and 4 http://phys.llnl.gov/Research/OPAL

8

Astrophys Space Sci (2008) 316: 1–12

to pass or not a smoothing filter to reduce the random numerical errors that affect the OPAL opacity computation. 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. (c) Nuclear reaction rates: No specifications have been given for the choice of the nuclear network. The basic pp chain and CNO cycle reaction networks up to the 17 O (p,α)14 N reaction have generally been used. Some models have been calculated with the option/assumption that either one or all the light elements (3 He, 7 Li, 7 Be, 2 H) are at equilibrium while other models have followed them explicitly. We chose to compute the nuclear reaction rates from the analytical formulae provided by the NACRE compilation (Angulo et al. 1999). Different prescriptions have been used for the screening. Most often, weak screening has been assumed under the Salpeter (1954) formulation. In that case the screening factor is written    ρξ , f = exp Az1 z2 T3 where z1 and z2 are the charges of the interacting nuclei. Some codes have used the expression (4-221) of Clayton  (1968) giving A = 1.88 × 108 and ξ = i zi (1+zi )xi where xi is the abundance per mole of element i. 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 ). (d) Convection and overshooting: We chose to use the classical mixing length treatment of Böhm-Vitense (1958) under the formulation of Henyey et al. (1965) taking into account the optical thickness of the convective bubble. The value of the mixing length parameter has been chosen to be αMLT = 1.6. The onset of convection is determined according to the Schwarzschild criterion (∇ad −∇rad < 0) where ∇ad and ∇rad are respectively the adiabatic and radiative temperature gradient. In models with overshooting, the convective core is extended on a distance lov = αov × min(Hp , Rcc ) where Hp is the pressure scale height and Rcc the radius of the convective core. We have chosen the value of the overshooting parameter to be αov = 0.15. The core is mixed in the region corresponding to the convective and overshooting region. In the overshooting region the temperature gradient is taken to be equal to the adiabatic gradient. Note that some codes have provided models including overshooting but following different parameterizations (e.g. 5 At

http://www.astro.up.pt/corot/compmod/task1

STAROX takes the temperature gradient to be equal to

the radiative gradient in the overshooting region while FRANEC uses a formulation of overshooting depending

on the mass). (e) Atmosphere: Eddington’s grey T (τ ) law has been prescribed for the atmosphere calculation:    1 3 2 4 τ+ , T = Teff 4 3 where τ is the optical depth. The radius of the star is taken to be the bolometric radius, i.e. the radius at the level where the local temperature equals the effective temperature (τ = 2/3 for the Eddington’s law). Codes manage differently the integration of the hydrostatic equation in the atmosphere from a specified upper value of the optical depth τ to the connexion with the envelope. (f) Microscopic diffusion: We only considered the diffusion of helium and heavy elements due to pressure, temperature and concentration gradients and we neglected radiative accelerations that are not yet fully included and tested in the participating codes. We did not impose the formalism to be used. As reviewed by Thoul and Montalbán (2007), two approaches to obtain the diffusion equation from the Boltzmann equation for binary or multiple gas mixtures can be followed: one is based on the Chapman-Enskog theory (Chapman and Cowling 1970, hereafter CC70) and the other on the resolution of the Burgers equations (Burgers 1969, hereafter B69). 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. The TGEC code follows the CC70 approach. The other codes follow the B69 approach but differently: (i) the ASTEC and FRANEC codes use the simplified formalism of Michaud and Proffitt (1993), hereafter MP93, (ii) the CLÉS and GARSTEC codes compute the diffusion coefficients by solving Burgers’ equations according to the formalism of Thoul et al. (1994) and (iii) CESAM provides two options to compute diffusion velocities, one based on the MP69 approximation, the other on Burger’s formalism, with collision integrals derived from Paquette et al. (1986). 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 (2008) 316: 1–12

5.3 Output format Each code had previously designed its own output format. 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. In most exercises we have adopted the file format prescribed by the GONG solar model team which is described in the documentation files in the ADIPLS package and on the ESTA Web site.6 This permits a rather easy treatment of the models which can be read, plotted and compared by means of the programme tools available in the ADIPLS package. In addition, a tool MODCONV and associated documentation have been made available on the ESTA Web-site by M. 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.) as well as a wide range of model variables at each mesh point (distance r to the centre, mass inside the sphere of radius r, pressure, density, temperature, chemical composition, opacity and several other physical parameters including quantities of interest for the computation of adiabatic oscillations like the Brunt-Väisälä frequency). 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.). 5.4 Oscillations Stellar oscillations modelers involved in TASK 2 were asked to provide adiabatic frequencies in the range [20, 2500] μHz and with spherical degrees = 0, 1, 2 and 3. 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). (b) Outer mechanical boundary condition: Set the Lagrangian perturbation to the pressure to zero (δP = 0). (c) Physical constants: Use the same as in TASK 1 and 3. (d) Equations: Use linear adiabatic equations.

9

(a) Set of eigenfunctions: Either the Lagrangian or the Eulerian perturbation to the pressure (δP or P  ) have been used which affects the form of the equations. (b) Order of the integration scheme: Either a second-order or a fourth-order scheme has been used. (c) Richardson extrapolation: Some codes using a secondorder scheme have the possibility to use Richardson extrapolation (Shibahashi and Osaki 1981) to decrease the truncation error. The combination of a 2nd order scheme with Richardson extrapolation leads to errors which scale as N −4 , where N is the number of mesh points (e.g. Christensen-Dalsgaard and Mullan 1994). (d) Integration variable: Either the radius r or the ratio r/P have been used as integration variables. (e) Physical constants: Some codes use a value of the gravitational constant G slightly different than the one specified in the other TASK S. As pointed out by Moya et al. (2008), using different values of G in the stellar structure equations and in the oscillation equations gives rise to inconsistencies. The consequences of these choices are discussed in details in Moya et al. (2008).

6 Reference grids In parallel to the three TASK S, grids of models have been especially calculated with the CESAM and CLÉS codes for masses in the range 0.8−10 M and chemical compositions [Fe/H] = 0.0 and −0.10. These reference grids have been used to locate CoRoT potential targets in the H–R diagram in the process of target selection (see Fig. 1) and to study some δ Scuti candidates for CoRoT (Poretti et al. 2005). In addition, associated oscillation frequencies have been calculated for selected models in the grid either with the ADIPLS, POSC or LOSC codes. This material is described in this volume by Montalbán et al. (2007), Lebreton and Michel (2008), Marques et al. (2008). In addition, a database of models of β Cephei stars and their oscillation frequencies called BetaDat has been designed in the context of ESTA, although the models do not use exactly the same input physics and constants. It is accessible through a Web interface.7

7 Conclusion

However, different methods and assumptions are made to write and to solve numerically the differential equations in the participating oscillation codes. In particular:

In this paper we briefly presented the different activities undertaken by the evolution and seismic tools activity (ESTA, see Monteiro et al. 2006) organised under the responsibilities of the Seismology Working Group (see Michel et al.

6 At

7 http://astrotheor3.astro.ulg.ac.be/

http://www.astro.up.pt/corot/ntools/

10

2006) of the CoRoT mission. A description of the ESTA team and TASK S aimed at improving stellar evolution codes and oscillation codes is presented. 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. Acknowledgements This work is being supported in part by the European Helio- and Asteroseismology Network (HELAS), a major international collaboration funded by the European Commission’s Sixth Framework Programme. M.J.P.F.G.M. is supported in part by FCT and FEDER (POCI2010) through projects POCI/CTE-AST/57610/2004 and POCI/V.5/B0094/2005. A.M. acknowledges financial support of the Spanish PNE under Project number ESP 2004-03855-C03-C01. J.M. has received a financial support from the Federal Science Policy Office (BELSPO) in the frame of the ESA/Prodex program, contract C90199—CoRoT—Preparation to exploitation.

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11 Monteiro, M.J.P.F.G.: POSC—The Porto oscillations code. Astrophys. Space Sci. (2008). doi:10.1007/s10509-008-9802-y Monteiro, M.J.P.F.G., Thompson, M.J.: On the seismic signature of the HeII ionization zone in stellar envelopes. In: New Eyes to See Inside the Sun and Stars. IAU Symposia, vol. 185, p. 317 (1998) Monteiro, M.J.P.F.G., Christensen-Dalsgaard, J., Thompson, M.J.: Seismic study of stellar convective regions: the base of the convective envelope in low-mass stars. Mon. Not. R. Astron. Soc. 316, 165 (2000) Monteiro, M.J.P.F.G., Christensen-Dalsgaard, J., Thompson, M.J.: Asteroseismic inference for solar-type stars. In: Battrick, B., Favata, F., Roxburgh, I.W., Galadi, I.W. (eds.) Stellar Structure and Habitable Planet Finding. ESA SP-485, p. 291. ESA Publication Division, Noordwijk (2002) Monteiro, M.J.P.F.G., Lebreton, Y., Montalbán, J., et al.: Report on the CoRoT evolution and seismic tools activity. In: Fridlund, M., Baglin, A., Lochard, J., Conroy, L. (eds.) The CoRoT Mission. ESA SP-1306, p. 363. ESA Publication Division, Noordwijk (2006) Morel, P., Lebreton, Y.: CESAM: a free code for stellar evolution calculations. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9663-9 Moya, A.: Current status of ESTA-Task 2. In: Straka, C.W., Lebreton, Y., Monteiro, M.J.P.F.G. (eds.) Stellar Evolution and Seismic Tools for Asteroseismology. EAS Publication Series, vol. 26, p. 187. EAD/EDP Science, Les Ulis (2007) Moya, A., Garrido R.: Granada oscillation code (GraCo). Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9694-2 Moya, A., Christensen-Dalsgaard, J., Charpinet, S., et al.: Intercomparison of the g-, f - and p-modes calculated using different oscillation codes for a given stellar model. Astrophys. Space Sci. (2008). doi:10.1007/s10509-007-9717-z Paquette, C., Pelletier, C., Fontaine, G., Michaud, G.: Diffusion coefficients for stellar plasmas. Astrophys. J. Suppl. Ser. 61, 177 (1986) Poretti, E., Alonso, R., Amado, P.J., et al.: Preparing the CoRoT space mission: new variable stars in the galactic anticenter direction. Astron. J. 129, 2461 (2005) Provost, J.: NOSC: Nice oscillations code. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9654-x Reiter, J., Walsh, L., Weiss, A.: Solar models: a comparative study of two stellar evolution codes. Mon. Not. R. Astron. Soc. 274, 899 (1995) Rogers, F.J., Nayfonov, A.: Updated and expanded OPAL equation-ofstate tables: implications for helioseismology. Astrophys. J. 576, 1064 (2002) Roxburgh, I.W., Vorontsov, S.V.: Asteroseismological constraints on stellar convective cores. In: Stellar Structure: Theory and Test of Connective Energy Transport. ASP Conf. Ser., vol. 173, p. 257. Astronomical Society of the Pacific, San Francisco (1999) Roxburgh, I.W.: The OSCROX stellar oscillation code. Astrophys. Space Sci. (2007a). doi:10.1007/s10509-007-9607-4 Roxburgh, I.W.: The STAROX stellar evolution code. Astrophys. Space Sci. (2007b). doi:10.1007/s10509-007-9673-7 Salpeter, E.E.: Electron screening and thermonuclear reactions. Aust. J. Phys. 7, 373 (1954) Samadi, R., Goupil, M.J., Alecian, E., et al.: Excitation of solar-like oscillations: from PMS to MS stellar models. J. Astrophys. Astron. 26, 171 (2005) Scuflaire, R., Montalbán, J., Théado, S., et al.: The Liège oscillation code. Astrophys. Space Sci. (2007a). doi:10.1007/s10509-007-9577-6 Scuflaire, R., Théado, S., Montalbán, J., et al.: CLÉS, Code Liégeois d’Evolution stellaire. Astrophys. Space Sci. (2007b). doi:10.1007/s10509-007-9650-1 Shibahashi, H., Osaki, Y.: Theoretical eigenfrequencies of solar oscillations of low harmonic degree L in 5-MINUTE range. Publ. Astron. Soc. Jpn. 33, 713 (1981)

12 Suarez, J.C., Goupil, M.J.: FILOU oscillation code. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9568-7 Suran, M.: LNAWENR—linear nonadiabatic nonradial waves. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9714-2 Théado, S., Vauclair, S., Castro, M., Charpinet, S., Dolez, N.: Asteroseismic tests of element diffusion in solar type stars. Astron. Astrophys. 437, 553 (2005) Thoul, A.A., Montalbán, J.: Microscopic diffusion in stellar plasmas. In: Straka, C.W., Lebreton, Y., Monteiro, M.J.P.F.G. (eds.) Stellar Evolution and Seismic Tools for Asteroseismology. EAS Publications Series, vol. 26, p. 25. EAD/EDP Science, Les Ulis (2007) Thoul, A.A., Bahcall, J.N., Loeb, A.: Element diffusion in the solar interior. Astrophys. J. 421, 828 (1994) Turck-Chièze, S., Basu, S., Berthomieu, G., et al.: Sensitivity of the sound speed to the physical processes included in the standard solar model. In: Korzennik, S. (ed.) Structure and Dynamics of the Interior of the Sun and Sun-like Stars. ESA SP-418, p. 555. ESA Publication Division, Noordwijk (1998)

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Evolution codes ASTEC—the Aarhus STellar Evolution Code Jørgen Christensen-Dalsgaard

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9675-5 © Springer Science+Business Media B.V. 2007

Abstract The Aarhus code is the result of a long development, starting in 1974, and still ongoing. 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. Keywords Stars · Stellar structure · Stellar evolution PACS 97.10.Cv · 95.75.Pq

1 Introduction What has become ASTEC started its development in Cambridge around 1974. The initial goal was to provide an improved equilibrium model for investigations of solar stability, following earlier work by Christensen-Dalsgaard et al. (1974). However, with the initial evidence for solar oscillations and the prospects for helioseismology (ChristensenDalsgaard and Gough 1976) 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. Christensen-Dalsgaard () Institut for Fysik og Astronomi, og Dansk AsteroSeismisk Center, Bygning 1520, Aarhus Universitet, 8000 Aarhus C, Denmark e-mail: [email protected]

of the solar interior, more emphasis was placed on numerical accuracy than was perhaps common at the time. The code drew some inspiration from the Eggleton stellar evolution code (Eggleton 1971), which had been used previously, but the development was fully independent of that code. An early description of the code was given by ChristensenDalsgaard (1982), with further extensive details provided by Christensen-Dalsgaard (1978); 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. 1993). This led to the so-called Model S of the Sun (Christensen-Dalsgaard et al. 1996) which has found extensive use as reference for helioseismic investigations and which at the time provided reasonably up-to-date representations of the physics of the solar interior. 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.

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_2

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Astrophys Space Sci (2008) 316: 13–24

2 Equations and numerical scheme 2.1 Formulation of the equations The equations of stellar structure and evolution are written with x = log10 q as independent variable, where log10 is logarithm to base 10; here q = m/M is the mass fraction where m is the mass interior to the point considered and M is the photospheric mass of the star, defining the photosphere as the point where the temperature is equal to the effective temperature. The equations are written on the form M q ∂ log10 r = , ∂x 4πρ r 3

(1)

∂ log10 p GM 2 q 2 , =− ∂x 4π r 4 p

(2)

∂ log10 p ∂ log10 T =∇ , ∂x ∂x   ∂ log10 L ∂H 1 ∂p q =M − + , ∂x ∂t ρ ∂t L   ∂ ∂Xk ∂Xk ∂ = Rk + Dk + (Vk Xk ), ∂t ∂m ∂m ∂m k = 1, . . . , K.

(3) (4)

κ Lp 3 , 16πa c˜ T 4 Gm

(5)

(6)

where a is the radiation density constant, c˜ is the speed of light, and κ is the opacity. The factor ψ is discussed following (11) below. The calculation of the temperature gradient in convective regions is discussed in Sect. 4. In (5) the derivatives with respect to m should obviously be expressed in terms of derivatives with respect to x. As discussed below, it is convenient to introduce the diffusion velocity on the form Yk = Dk

∂Xk + Vk X k ; ∂m

(7)

then (5) can be written as ∂Xk MMq = (Yk − Vk Xk ), ∂x Dk   ∂Xk ∂Yk = MMq − Rk . ∂x ∂t

2.2 Boundary conditions The centre, r = 0, is obviously a singular point of (1–4). As discussed in considerable detail by Christensen-Dalsgaard (1982), 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. The treatment of diffusion and settling has yet to be included consistently in this expansion; currently, conditions are imposed that essentially set the Y˜k to zero at the innermost meshpoint. The outer boundary of the solution of the full evolution equations is taken to be at the photosphere, defined as the point where T = Teff , the effective temperature. Consequently an obvious boundary condition is L = 4πr 2 σ T 4 ,

Here r is distance to the centre, ρ is density, p is pressure, G is the gravitational constant, T is temperature, L is the luminosity at r,  is the energy generation rate per unit mass, H is enthalpy per unit mass, Xk is the mass fraction of element k, Rk is the rate of change of Xk due to nuclear reactions, and Dk and Vk are the diffusion and settling coefficients for element k. Also, the value of ∇ = d log T /d log p depends on whether the region is convectively stable or not. In the case of a convectively stable region, ∇ = ∇rad ≡ ψ

Here M = log 10, log being the natural logarithm. (In the code Y˜k = M −1 Yk is used as variable.) For elements where diffusion and settling are ignored the original form, (5), with Dk and Vk set to zero is obviously used.

(10)

where σ is the Stefan-Boltzmann constant. To obtain a second condition, a expression is assumed for the dependence of T on optical depth τ , conveniently written as T = Teff [τ + q(τ )]1/4 .

(11)

Based on this the equation of hydrostatic equilibrium, on the form dp g = , dτ κ

(12)

can be integrated assuming, currently, a constant gravitational acceleration g. The surface boundary condition p=

2τg , κ

(13)

based on an approximate treatment of the outer parts of the atmosphere, is applied at the top of the atmosphere, τ = τmin , typically taken to be 10−4 . The boundary condition is obtained by equating the pressure resulting from integrating (12) to the pressure in the interior solution. The T (τ ) relation and (12) obviously define the temperature gradient ∇ in the atmosphere. To ensure a continuous transition to the interior I follow Henyey et al. (1965) and include the factor ψ = 1 + q  (τ ) in (6).

(8)

2.3 Numerical scheme

(9)

As indicated in (1–5, 8, 9), the dependent variables on the left-hand sides of the equations are expressed in terms of

Astrophys Space Sci (2008) 316: 13–24

15

with a similar notation for n,s i , as well as for quantities at timestep t s+1 . Then (15–17) are replaced by the following difference equations:

the set {yi } = {log10 r, log10 p, log10 T , log10 L, Xk , Y˜k  }, i = 1, . . . , I.

(14)

Here the index k runs over all the abundances considered, whereas the index k  runs over those elements for which diffusion is included. 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. As discussed below, this may involve expressing the thermodynamic state in terms of a different set of variables {zj }, related to the {yi } by a non-singular transformation. Using also the reformulation, (8) and (9), of the diffusion equation, the combined set of stellar-evolution equations consists of equations of one of the following three types: ∂yl = fl (x; zi ; t), ∂x

l = 1, . . . , I1 ,

Type I

(15)

I 

pi (x; zj ; t)

i=1

p = I1 + 1, . . . , I1 + I2 ,

− ypn,s+1 ) + (1 − θp )(ypn+1,s

(22)

− ypn,s )

1 = x n {θp (fpn+1,s+1 + fpn,s+1 ) 2 + (1 − θp )(fpn+1,s + fpn,s ) +

I    n+1,s+1 θp pi + (1 − θp ) n+1,s pi i=1

× (zin+1,s+1 − zin+1,s )/ t s +

I    n,s+1 n,s+1 θp pi + (1 − θp ) n,s − zin,s )/ t s }, pi (zi

n = 1, . . . , N − 1, p = I1 + 1, . . . , I1 + I2 ,   yun,s+1 − yun,s = t s θu fun,s+1 + (1 − θu )fun,s ,

∂yi , ∂t

Type II

∂yu = fu (x; zi ; t), ∂t u = I1 + I2 + 1, . . . , I1 + I2 + I3 .

(16)

Type III

(17)

The equations of Type III are obviously relevant to the evolution of abundances of elements for which diffusion is not taken into account. To this must be added the transformation i = 1, . . . , I.

(18)

In practice, zi = yi for i = 2; the choice of z2 depends on the specific equation of state considered. The equations are solved on the interval [x1 , x2 ], with boundary conditions that can be expressed as gα (x1 ; zi (x1 )) = 0,

α = 1, . . . , KA ,

(19)

gβ (x2 ; zi (x2 )) = 0,

β = KA + 1, . . . , KA + KB .

(20)

The equations are solved by means of the NewtonRaphson-Kantorovich scheme (see Richtmyer 1957; Henrici 1962), in the stellar-evolution context known as the Henyey scheme (e.g., Henyey et al. 1959, 1964). We introduce a mesh x1 = x 1 < x 2 < · · · < x N = x2 in x and consider two timesteps t s and t s+1 , where the solution is assumed to be known at timestep t s . Also, we introduce yin,s = yi (x n , t s ),

θp (ypn+1,s+1

i=1

∂yp = fp (x; zi ; t) + ∂x

yi = yi (x; zj ; t),

1 yln+1,s+1 − yln,s+1 = x n (fln+1,s+1 + fln,s+1 ), 2 n = 1, . . . , N − 1, l = 1, . . . I1 ,

zin,s = zi (x n , t s ),

fin,s = fi (x n ; zjn,s ; t s ),

(21)

n = 1, . . . , N, u = I1 + I2 + 1, . . . , I1 + I2 + I3 .

(23)

(24)

Here x n = x n+1 − x n and t s = t s+1 − t s . Also, the parameters θi allow setting the centralization of the difference scheme in time, θi = 1/2 corresponding to time-centred differences and θi = 1 to a fully implicit scheme. 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.g., for the time derivatives in the energy equation where the characteristic timescale is much shorter than the evolution time and hence short compared with the typical step t in time. The algebraic equations (22–24), together with the boundary conditions, (19) and (20), are solved using NewtonRaphson iteration, to determine the solution {zin,s+1 } at the new time step (see also Christensen-Dalsgaard 1982). Given a trial solution {¯zin,s+1 } the equations are linearized in terms of corrections δzin,s+1 = zin,s+1 − z¯ in,s+1 and the resulting linear equations are solved using forwards elimination and backsubstitution (e.g., Baker et al. 1971). 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. (1975) (see also Eggleton 1971), taking into account the variation of several relevant variables. An early version of the implementation was described by Christensen-Dalsgaard (1982), 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. Having computed model at timestep t s+1 the next timestep is determined from the change in the model between timesteps t s and t s+1 ; this involves a fairly complex algorithm limiting the change in, e.g., log10 p, log10 T and X at fixed m. 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 (100) timesteps are needed to reach the end of central hydrogen burning in models without (with) a convective core, and 100 (200) 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.

3 Microphysics As the code has developed over the years a number of options have been included for the microphysics, although not all of these have been kept up to date or properly verified. 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. It should be noted that the use of a different set {zi } of dependent variables on the right-hand side of the equations yields additional flexibility in the replacement of aspects of the physics. Däppen and Guzik (2000) provided a review of 1 This problem is avoided in the implementation by Eggleton (1971) where the equations are solved using an independent variable that is directly tied to such strong variations in the model.

Astrophys Space Sci (2008) 316: 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 (1991). 3.1 Equation of state The original version of the code used the Eggleton et al. (1973) equation of state and that remains an often used option. In this case z2 = log10 f , where f , introduced by Eggleton et al. and related to the electron-degeneracy parameter, is used as one of the thermodynamic variables; this allows explicit calculation of partial ionization and hence a very efficient evaluation of the required thermodynamic quantities. The formulation also includes a crude but thermodynamically consistent implementation of ‘pressure ionization’ (which actually results mainly as a result of the high density in deep stellar interiors) which provides apparently reasonable results. 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. As an upgrade to the basic EFF formulation ChristensenDalsgaard and Däppen (1992) included Coulomb effects, in the Debye-Hückel formulation, following Mihalas et al. (1988); unlike some earlier treatments of the Coulomb effect this ensured that thermodynamic consistency was maintained; as a result, a substantial effect was found also on the degrees of ionization of hydrogen and helium, resulting from the change in the chemical potential. 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. 1990), based on the chemical picture where the thermodynamic state is obtained through minimization of an expression for the free energy including a number of contributions. 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. 1988). Further updates to the MHD equation of state have been made but they

Astrophys Space Sci (2008) 316: 13–24

have so far not been implemented in ASTEC. An alternative description is provided by the so-called OPAL equation of state (Rogers et al. 1996), based on the physical picture where the thermodynamic state is obtained from an activity expansion. This has been the preferred equation of state for solar modelling with ASTEC, used, e.g., in Model S. 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 1998). This has since been corrected (Gong et al. 2001; Rogers and Nayfonov 2002). Also, the OPAL formulation suffered from inconsistencies between some of the variables provided (e.g. Boothroyd and Sackmann 2003; Scuflaire et al. 2007); effects of the inconsistency on the computation of adiabatic oscillations are discussed by Moya et al. (2007). This has been improved in the latest version of the OPAL equation of state,2 which has also been implemented in ASTEC. Interpolation in the OPAL tables uses quadratic interpolation in ρ, T and X. 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. 3.2 Opacity The early treatment of the opacities used tables from Cox and Stewart (1970) and Cox and Tabor (1976) with an interpolation scheme developed by Cline (1974). A major improvement was the implementation of the OPAL opacity tables (Rogers and Iglesias 1992; Iglesias et al. 1992). 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 (1995). Atmospheric opacities must be supplied separately; here tables by Kurucz (1991), Alexander and Ferguson (1994) or Ferguson et al. (2005) have been used, with a smooth matching to the interior tables at log10 T = 4. Electron conduction may be included based on the tables of Itoh et al. (1983). Interpolation in the OPAL tables is carried out with schemes developed by G. Houdek (see Houdek and Rogl 1993, 1996). The tables are provided as functions of (R, T ), where R = ρ/T 3 . 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 1980, 1983). The second scheme uses birational splines (Späth 1991). In practice the latter scheme has been used for most of the model calculations with ASTEC. Interpolation 2 See

http://www-phys.llnl.gov/Research/OPAL/opal.html

17

in X and log Z is carried out using the univariate scheme of Akima (1970). 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 (1998). 3.3 Nuclear reactions The calculation of the nuclear reaction rates is based on the usual approximations to the reaction integrals (e.g., Clayton 1968) using a variety of coefficients (Bahcall and Pinsonneault 1995; Adelberger et al. 1998; Angulo et al. 1999). Electron screening is computed in the Salpeter (1954) approximation. Electron capture by 7 Be is treated according to Bahcall and Moeller (1969). 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. (1974) but generalized to allow a non-zero initial abundance. A typical value is t3 He = 5 × 107 yr. In the CNO cycle the CN part is assumed to be in nuclear equilibrium and controlled by the rate of the 14 N(1 H, γ )15 O reaction. In addition, the reactions 16

O(1 H, γ )17 F(e+ νe )17 O(1 H, 4 He)14 N

and 15

N(1 H, γ )16 O

are included; these play an important role in ensuring the gradual conversion of 16 O to 14 N and hence increasing the importance of the CN cycle. This part of the cycle is assumed to be fully controlled by the 16 O(1 H, γ )17 F reaction and the branching ratio between the 15 N(1 H, 4 He)12 C and 15 N(1 H, γ )16 O reactions. Helium burning has been included in the code using the expressions of Angulo et al. (1999), and including also the

18

reaction 12 C(4 He, γ )16 O. 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 2.3M where ignition takes place in a relatively quiet manner. Also, as in cases of hydrogen burning (cf. Sect. 4), the treatment of semiconvection that may be required in this phase has not been implemented. 3.4 Diffusion and settling Diffusion and settling are treated in the approximations proposed by Michaud and Proffitt (1993), with revisions kindly provided by Proffitt. 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.g., Richer et al. 1998; Turcotte et al. 1998); there are no current plans to do so in the code. Various approximations to turbulent diffusion can be included, partly inspired by Proffitt and Michaud (1991). 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.

4 Treatment of convection The temperature gradient in convection zones is usually computed using the Vitense (1953) and Böhm-Vitense (1958) version of mixing-length theory; for completeness, the expressions used are provided in Appendix 2. The mixing length is a constant multiple, αML Hp , of the pressure scale height Hp . In addition, emulations of the Canuto and Mazzitelli (1991) formulation, established by Monteiro et al. (1996), can be used. 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. In ASTEC this is used in convective envelopes, ensuring that they are chemically uniform. The treatment of convective cores remains a concern and an area of active development, however. Given

Astrophys Space Sci (2008) 316: 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. The rate of change of these abundances can be written dXk,c ¯ k + 1 dqcc [Xk (xcc ) − Xk,c ] + 1 Y˜k (xcc ), =R dt qcc dt qcc (25) here qcc is the mass fraction in the convective core, xcc = log10 (qcc ), and ¯k = 1 R qcc



qcc

Rk dq

(26)

0

is the reaction rate averaged over the convective core. 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.e., if the core is growing and diffusion is neglected. Finally, the term in Y˜k (xcc ) is obviously only included in cases where diffusion is taken into account. 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. 1). This situation arises in intermediate-mass stars (with masses up to around 1.7M ) where the gradual conversion during evolution of 16 O into 14 N causes an increase in the importance of the CNO cycle in the energy generation (for more massive stars the CNO cycle dominates even with the initial 14 N abundance). Since pressure and temperature are obviously continuous, there is also a discontinuity in density and opacity κ, leading also to a jump in the radiative temperature gradient ∇rad (cf. (6)). Since κ increases with X and ρ, while ρ decreases with increasing X, it is not a priori clear how κ and hence ∇rad react at the discontinuity; in practice the common behaviour is that ∇rad increases going from the value of X in the convective core to the higher value in the radiative region just outside it. 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.g., Merryfield 1995). A different scheme, now under implementation, is discussed in Sect. 6. Various options exist for the treatment of convective overshoot. Simple formulations involve compositionally fully

Astrophys Space Sci (2008) 316: 13–24

mixed overshoot regions from the convective core or convective envelope, over a distance αov Hp below a convective envelope, or αov min(rcc , Hp ) above a convective core, where rcc is the radius of the core. 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. (1994), where various forms of the temperature gradient can be specified, still assuming the overshoot region to be fully mixed. This is being extended to emulate the overshoot simulations by Rempel (2004).

5 Implementation details When computing models of the present Sun it is important to adjust the input parameters such as to obtain a model that precisely matches the observed solar radius, luminosity and ratio Zs /Xs between the abundances of heavy elements and hydrogen, at the present age of the Sun. This is achieved by adjusting the initial hydrogen and heavy-element abundances X0 and Z0 as well as the mixing-length parameter αML (or another parameter characterizing the treatment of convection). 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. In addition to summary output files listing global properties of the models in the evolution sequence, output of detailed models, on the full mesh of the calculation, can be made in three different forms: the so-called emdl files, including just the variables {zi } actually used in the calculation, as well as a complete listing of the input parameters, to provide full documentation of the calculation; the amdl files which provide the variables needed for the Aarhus adiabatic pulsation code (see Christensen-Dalsgaard 2007a); and

19

the gong files, giving an extensive set of variables for use in further investigations of the models or plotting. 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.g., when fitting observed frequencies to determine the properties of the model. 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.g., Metcalfe and Charbonneau 2003), under development by T. 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.

6 Further developments From the preceding presentation it is obvious that there are significant deficiencies in ASTEC, and work is ongoing to correct them. 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 2007b); on the other hand, the case of just helium diffusion can be handled. This problem may be related to issues of semiconvection where convective stability is closely related to the details of the composition profile (Montalbán et al. 2007). In ASTEC I have considered two cases of semiconvection, although the implementation is still under development. One case concerns growing convective cores in nondiffusing models (see also Sect. 4), as illustrated in Fig. 1.

20

Fig. 1 Hydrogen-abundance profile and ∇rad − ∇ad against fractional radius in two 1.5M models of age 1.36 Gyr and with Z = 0.02. 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. In the model illustrated by the solid curves a gradient in X is set up such that ∇rad − ∇ad = 0 in the outer parts of the convective core

The dashed curves illustrate the properties of a model computed with the normal ASTEC implementation, where the hydrogen abundance X is discontinuous at the boundary of the core (see panels a and b). As discussed above, the extent of the convective core is determined by the behaviour of the radiative temperature gradient ∇rad in the radiative region (see panel c),3 leading to a convectively stable region just beneath the boundary which nonetheless is assumed to be fully mixed. A possibly more reasonable treatment, illustrated by the continuous curves, assumes that a hydrogenabundance profile is established such that ∇rad = ∇ad in the outermost parts of the convective core. Since this affects a very small part of the core of the model the effects on its global properties are modest; however, it may have some influence on its pulsational properties, particularly for to slight convergence problems in this calculation, ∇rad − ∇ad is in fact positive at the point identified as the convective boundary. 3 Owing

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g modes or interface modes predominantly trapped near the boundary of the core, possibly offering the potential for asteroseismic diagnostics of this behaviour. Significant sensitivity to the treatment of this region was indeed found by Moya et al. (2007); a similar model, but computed without such treatment of semiconvection (and with somewhat inadequate mesh resolution of the critical region) displayed far larger frequency differences than did the model considered in Fig. 1 between oscillation codes using different numerical techniques, particularly for modes with g-mode like behaviour and partly trapped near the edge of the core. 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. (2001) and discussed in more detail by ChristensenDalsgaard and Di Mauro (2007) the gradient in the heavyelement abundance Z established by settling beneath the convection zone leads to convective instability, e.g., in 1M models of age somewhat higher than the present Sun. Complete mixing of the unstable region removes the gradient and hence the cause of the instability, leading to an uncertain situation, characteristic of semiconvection. As above, a perhaps reasonable assumption might be that partial mixing takes place to establish composition gradients that ensure neutral convective stability in the critical region, with ∇rad = ∇ad . 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.

7 Concluding remarks No stellar evolution code is probably ever finished or fully tested. ASTEC has certainly proved useful in a number of applications, and the results for the Sun, as applied to helioseismology, are perhaps reasonably reliable, at least within the framework of ‘standard solar modelling’. It is obvious, however, that application to the increasingly accurate and detailed asteroseismic data will require further development. The tests provided through the ESTA collaboration and extended through the HELAS Coordination Action are certainly most valuable in this regard. Acknowledgements I am very grateful to D.O. 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. Däppen for contributing the MDI and OPAL equation-of-state packages, G. Houdek for providing the opacity interpolation packages and the OPAL data in

Astrophys Space Sci (2008) 316: 13–24

21

the appropriate form, M.J.P.F.G. Monteiro for providing alternate treatments of the parameterization of convection, C.R. Proffitt for help with installing diffusion and settling in the code and M. Bazot for assistance with updating the treatment of nuclear reactions. M.J.P.F.G. Monteiro is thanked for organizing the ESTA Workshop and Y. Lebreton and J. Montalbán for taking care of the stellar evolution item. This project is being supported by the Danish Natural Science Research Council and by the European Helio- and Asteroseismology Network (HELAS), a major international collaboration funded by the European Commission’s Sixth Framework Programme.

where 1/2

1/2

Ck = X(Ak H Ck H − Ak He Ck He ).

(34)

Here Ck H is the collision integral between element k and hydrogen, which Michaud and Proffitt (1993) approximate, fitting numerically determined values, as Ck H =

1 log[exp(1.2 log k H ) + 1], 1.2

(35)

Appendix 1: Treatment of diffusion and settling where For completeness I reproduce the expressions used to compute the diffusion and settling coefficients. These are largely obtained from Michaud and Proffitt (1993), although with minor modifications. In terms of the variable Y˜k introduced in Sect. 2.1, (8) and (9) become ∂Xk MMq (M Y˜k − Vk Xk ), = ∂x Dk   ∂Xk ∂ Y˜k = Mq − Rk . ∂x ∂t

(27)

4πBT 5/2 log HHe (0.7 + 0.3X)





2mu 5π

1/2

5 9 + ∇ 4 8



1 log[exp(1.2 log k He ) + 1], 1.2

(37)

Ak H = (29)

5/2

kB , e4

(38)

Also,

(30)

mu being the atomic mass unit, kB Boltzmann’s constant and e the electron charge. Also, log HHe is the Coulomb logarithm, approximated by   1 X+3 3 1 + log T . log HHe = −19.95 − log ρ − log 2 2 2 2 (31) In the above expressions, cgs units are used and ‘log’ is natural logarithm. The diffusion coefficient is given by D1 = (4πρr 2 )2

Ck He =

k He = HHe − log Zk .

where 15 16

similarly the collision integral Ck H between element k and helium is evaluated as

where

Gρm (1 − X − Z), × p

B=

(36)

(28)

For hydrogen (k = 1) V1 = −

k H = HHe − log Zk + log 2;

(3 + X) BT 5/2 . ρ log HHe (0.7 + 0.3X) (1 + X)(3 + 5X) (32)

Ak AH Ak + AH

(39)

is the reduced mass in the element k, hydrogen system, AH being the atomic mass of hydrogen; the reduced mass Ak,He involving helium is defined similarly. As presented by Michaud and Proffitt (1993) the diffusion velocity of trace elements depends on the gradient in the hydrogen abundance. To incorporate this in the formalism described by (27) and (28), I express ∂X/∂r in terms of Y˜1 and write the coefficient Vk as (1) (2) Vk = Vk + Vk M Y˜1 .

(40)

Here  (1) Vk

= −4πBT

5/2

ρ

2[1 + Zk − Ak (5X + 3)/4] 1/2

51/2 Zk2 (Ck + Ck He Ak He ) Gm 0.54(4.75X + 2.25) ∇ − log HHe + 5 p

For a trace element k, with charge Zk and atomic mass Ak ,

+

2BT 5/2 1 , 2 1/2 5 ρZk (Ck + Ck He A1/2 k He )

2BT 5/2 V1 (4πr 2 ρ)2 1/2 D1 51/2 ρZk2 (Ck + Ck He Ak He )

×

2Zk + 5(1 + X) X, (1 + X)(5X + 3)

Dk = (4πρr 2 )2

(33)

(41)

22

Astrophys Space Sci (2008) 316: 13–24

and Vk(2) = −

For large and small A asymptotic expressions for Y are easily found. For A  1

   1 (51) A−4 , Y A−1 1 − A−2 + 2 − 3λ

(4πr 2 ρ)2 2BT 5/2 2Zk + 5(1 + X) D1 51/2 ρZk2 (1 + X)(5X + 3)

+

Ck 1/2

X(Ck + Ck He Ak He )

− 0.23.

(42)

Appendix 2: Mixing-length formulation The calculation of ∇ in convective regions is carried out using the Vitense (1953) mixing length theory, in the form given by Gough (1977). This is expressed in terms of R=

gδ4 , (K/ρcp )2 Hp

A = 2η−1 R−1/2 (∇rad − ∇ad )−1/2 ,

(43) (44)

and  1/2 4  λ= √ η . 2 3 2

(45)

Here Hp = p/(ρg) is the pressure scale height, g = Gm/r 2 is the gravitational acceleration, δ = −(∂ log ρ/∂ log T )p , K = 4acT 3 /(3κρ) is the radiative conductivity and cp is the specific heat at constant pressure. Also  is the mixing length, which is taken to be a constant multiple of the pressure scale height,  = αML Hp , and η and  are geometrical quantities, assumed be constant, which are√ related to the aspect ratio of the convective cells; for η = 2/9 and  = 2 we get the Böhm-Vitense (1958) expressions (these values are used in the computation). Then ∇ − ∇ad = Y (Y + A)(∇rad − ∇ad ),

(46)

where Y is the positive root of 1 3 Y + Y 2 + AY − 1 = 0. 3λA The general solution to (47) is 

A x − λ(1 − λ) − λ , Y =A A x

(47)

(48)

where

1/3 x = A γ + [γ 2 + λ3 (1 − λ)3 ]1/2 , and

(49)



  3 3 γ =λ −λ . +λ 2 2A2

(50)

and for A 1

 1 1/3 2/3 Y (3λA) 1 − (3λA) . 3

(52)

Equation (51) is used when A ≥ 15 and (52) when A ≤ 10−5 . At these points the relative differences between the asymptotic values of Y and those found from (48) are less than 5 × 10−7 .

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Astrophys Space Sci (2008) 316: 13–24 Christensen-Dalsgaard, J., Di Mauro, M.P.: Diffusion and helioseismology. In: Straka, C.W., Lebreton, Y., Monteiro, M.J.P.F.G. (eds.) Stellar Evolution and Seismic Tools for Asteroseismology: Diffusive Processes in Stars and Seismic Analysis. EAS Publ. Ser., vol. 26, pp. 3–16. EDP Sciences, Les Ulis (2007) Christensen-Dalsgaard, J., Gough, D.O.: Towards a heliological inverse problem. Nature 259, 89–92 (1976) Christensen-Dalsgaard, J., Dilke, F.W.W., Gough, D.O.: The stability of a solar model to non-radial oscillations. Mon. Not. R. Astron. Soc. 169, 429–445 (1974) Christensen-Dalsgaard, J., Däppen, W., Lebreton, Y.: Solar oscillation frequencies and the equation of state. Nature 336, 634–638 (1988) Christensen-Dalsgaard, J., Proffitt, C.R., Thompson, M.J.: Effects of diffusion on solar models and their oscillation frequencies. Astrophys. J. 403, L75–L78 (1993) Christensen-Dalsgaard, J., Däppen, W., Ajukov, S.V., et al.: The current state of solar modeling. Science 272, 1286–1292 (1996) Clayton, D.D.: Principles of Stellar Evolution and Nucleosynthesis. McGraw-Hill, New York (1968) Cline, A.K.: Scalar- and planar-valued curve fitting using splines under tension. Commun. ACM 17, 218–220 (1974) Cox, A.N., Stewart, J.N.: Rosseland opacity tables for Population I compositions. Astrophys. J. Suppl. Ser. 19, 243–279 (1970) Cox, A.N., Tabor, J.E.: Radiative opacity tables for 40 stellar mixtures. Astrophys. J. Suppl. Ser. 31, 271–312 (1976) Däppen, W., Guzik, J.A.: Astrophysical equation of state and opacity. In: ˙Ibano˘glu, C. (ed.) Variable Stars as Essential Astrophysical Tools, pp. 177–212. Kluwer Academic, Dordrecht (2000) Eggleton, P.P.: The evolution of low mass stars. Mon. Not. R. Astron. Soc. 151, 351–364 (1971) Eggleton, P.P., Faulkner, J., Flannery, B.P.: An approximate equation of state for stellar material. Astron. Astrophys. 23, 325–330 (1973) Elliott, J.R., Kosovichev, A.G.: The adiabatic exponent in the solar core. Astrophys. J. 500, L199–L202 (1998) Ferguson, J.W., Alexander, D.R., Allard, F., Barman, T., Bodnarik, J.G., Hauschildt, P.H., Heffner-Wong, A., Tamanai, A.: Lowtemperature opacities. Astrophys. J. 623, 585–596 (2005) Gong, Z., Däppen, W., Zejda, L.: MHD equation of state with relativistic electrons. Astrophys. J. 546, 1178–1182 (2001) Gough, D.O.: Mixing-length theory for pulsating stars. Astrophys. J. 214, 196–213 (1977) Gough, D.O., Spiegel, E.A., Toomre, J.: Highly stretched meshes as functionals of solutions. In: Richtmyer, R.D. (ed.) Lecture Notes in Physics, vol. 35, pp. 191–196. Springer, Heidelberg (1975) Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York (1962) Henyey, L.G., Wilets, L., Böhm, K.H., LeLevier, R., Levee, R.D.: A method for automatic computation of stellar evolution. Astrophys. J. 129, 628–636 (1959) Henyey, L.G., Forbes, J.E., Gould, N.L.: A new method of automatic computation of stellar evolution. Astrophys. J. 139, 306– 317 (1964) Henyey, L.G., Vardya, M.S., Bodenheimer, P.: Studies in stellar evolution. III. The calculation of model envelopes. Astrophys. J. 142, 841–854 (1965) Houdek, G., Rogl, J.: A new interpolation scheme for opacity tables. In: Weiss, W.W. (ed.) Communications in Asteroseismology, No. 60 (1993) Houdek, G., Rogl, J.: On the accuracy of opacity interpolation schemes. Bull. Astron. Soc. India 24, 317–320 (1996) Iglesias, C.A., Rogers, F.J., Wilson, B.G.: Spin-orbit interaction effects on the Rosseland mean opacity. Astrophys. J. 397, 717–728 (1992) Itoh, N., Mitake, S., Iyetomi, H., Ichimaru, S.: Electrical and thermal conductivities of dense matter in the liquid metal phase. I. Hightemperature results. Astrophys. J. 273, 774–782 (1983)

23 Kurucz, R.L.: New opacity calculations. In: Crivellari, L., Hubeny, I., Hummer, D.G. (eds.) Stellar Atmospheres: Beyond Classical Models. NATO ASI Series, pp. 441–448. Kluwer, Dordrecht (1991) Merryfield, W.J.: Hydrodynamics of semiconvection. Astrophys. J. 444, 318–337 (1995) Metcalfe, T.S., Charbonneau, P.: Stellar structure modeling using a parallel genetic algorithm for objective global optimization. J. Comput. Phys. 185, 176–193 (2003) Michaud, G., Proffitt, C.R.: Particle transport processes. In: Baglin, A., Weiss, W.W. (eds.) Proc. IAU Colloq. 137: Inside the Stars. ASP Conf. Ser., vol. 40, pp. 246–259. Astronomical Society of the Pacific, San Francisco (1993) Mihalas, D., Däppen, W., Hummer, D.G.: The equation of state for stellar envelopes. II. Algorithm and selected results. Astrophys. J. 331, 815–825 (1988) Mihalas, D., Hummer, D.G., Mihalas, B.W., Däppen, W.: The equation of state for stellar envelopes. IV. Thermodynamic quantities and selected ionization fractions for six elemental mixes. Astrophys. J. 350, 300–308 (1990) Montalbán, J., Théado, S., Lebreton, Y.: Comparisons for ESTATASK3: CLES and CESAM. In: Straka, C.W., Lebreton, Y., Monteiro, M.J.P.F.G. (eds.) Stellar Evolution and Seismic Tools for Asteroseismology: Diffusive Processes in Stars and Seismic Analysis. EAS Publ. Ser., vol. 26, pp. 167–176. EDP Sciences, Les Ulis (2007) Monteiro, M.J.P.F.G., Christensen-Dalsgaard, J., Thompson, M.J.: Seismic study of overshoot at the base of the solar convective envelope. Astron. Astrophys. 283, 247–262 (1994) Monteiro, M.J.P.F.G., Christensen-Dalsgaard, J., Thompson, M.J.: Seismic properties of the Sun’s superadiabatic layer. I. Theoretical modelling and parametrization of the uncertainties. Astron. Astrophys. 307, 624–634 (1996) Moya, A., Christensen-Dalsgaard, J., Charpinet, S., Lebreton, Y., Miglio, A., Montalbán, J., Monteiro, M.J.P.F.G., Provost, J., Roxburgh, I., Scuflaire, R., Suárez, J.C., Suran, M.: Inter-comparison of the g-, f- and p-modes calculated using different oscillation codes for a given stellar model. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9717-z Nielson, G.: Minimum norm interpolation in triangles. SIAM J. Numer. Anal. 17, 44–62 (1980) Nielson, G.M.: A method for interpolating scattered data based upon a minimum norm network. Math. Comput. 40, 253–271 (1983) Parker, P.D.M., Rolfs, C.E.: Nuclear energy generation in the solar interior. In: Cox, A.N., Livingston, W.C., Matthews, M. (eds.) Solar Interior and Atmosphere. Space Science Series, pp. 31–50. University of Arizona Press, Tucson (1991) Proffitt, C.R., Michaud, G.: Gravitational settling in solar models. Astrophys. J. 380, 238–250 (1991) Rempel, M.: Overshoot at the base of the solar convection zone: a semianalytical approach. Astrophys. J. 607, 1046–1064 (2004) Richer, J., Michaud, G., Rogers, F., Iglesias, C., Turcotte, S., LeBlanc, F.: Radiative accelerations for evolutionary model calculations. Astrophys. J. 492, 833–842 (1998) Richtmyer, R.D.: Difference Methods for Initial-Value Problems. Interscience, New York (1957) Rogers, F.J., Iglesias, C.A.: Radiative atomic Rosseland mean opacity tables. Astrophys. J. Suppl. Ser. 79, 507–568 (1992) Rogers, F.J., Iglesias, C.A.: The OPAL opacity code: new results. In: Adelman, S.J., Wiese, W.L. (eds.) Astrophysical Applications of Powerful New Databases. ASP Conf. Ser., vol. 78, pp. 31–50. Astronomical Society of the Pacific, San Francisco (1995) Rogers, F.J., Nayfonov, A.: Updated and expanded OPAL equation-ofstate tables: implications for helioseismology. Astrophys. J. 576, 1064–1074 (2002)

24 Rogers, F.J., Swenson, F.J., Iglesias, C.A.: OPAL equation-of-state tables for astrophysical applications. Astrophys. J. 456, 902–908 (1996) Salpeter, E.E.: Electron screening and thermonuclear reactions. Aust. J. Phys. 7, 373–388 (1954) Scuflaire, R., Théado, S., Montalbán, J., Miglio, A., Bourge, P.O., Godart, M., Thoul, A., Noels, A.: CLÉS, Code Liégeois d’Évolution Stellaire. Astrophys. Space Sci. (2007). doi:10.1007/ s10509-007-9577-6 Späth, H.: Zweidimensionale Spline-Interpolations-Algorithmen. Oldenburg, München (1991)

Astrophys Space Sci (2008) 316: 13–24 Tripathy, S.C., Christensen-Dalsgaard, J.: Opacity effects on the solar interior. I. Solar structure. Astron. Astrophys. 337, 579–590 (1998) Turcotte, S., Richer, J., Michaud, G., Iglesias, C.A., Rogers, F.J.: Consistent solar evolution model including diffusion and radiative acceleration effects. Astrophys. J. 504, 539–558 (1998) Vitense, E.: Die Wasserstoffkonvektionszone der Sonne. Z. Astrophys. 32, 135–164 (1953)

The FRANEC stellar evolutionary code S. Degl’Innocenti · P.G. Prada Moroni · M. Marconi · A. Ruoppo

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9560-2 © Springer Science+Business Media B.V. 2007

Abstract We summarize the main physical assumptions and numerical procedures adopted by the FRANEC code to compute stellar models in all the evolutionary phases at hydrostatic and thermal equilibrium. An application to the Standard Solar Model is also briefly presented. Keywords Stars: evolution · Stars: interiors · Sun: evolution PACS 97.10.Cv · 97.10.Sj

(1998), Cariulo et al. (2004), and, for the application to white dwarf (WD) structures, Prada Moroni and Straniero (2002). FRANEC has been used to compute models in all the evolutionary phases, from the pre-main sequence (PMS) to the cooling sequence of WDs, covering a mass range from ≈0.1 to ≈25M for several metallicities and Helium abundances. Our code assumes spherical symmetry for the star and negligible effects of rotation and magnetic fields in the stellar interior. Some of the computed models and the related isochrones are available electronically.1 In Fig. 1 we present selected pre-main sequence (upper panel) and MS and post-MS (lower panel) models.

1 Introduction FRANEC (Frascati Raphson Newton Evolutionary Code) is an evolutionary stellar code developed by a group of italian researchers in Frascati about 30 years ago and then modified and updated during the years. An exhaustive description can be found in Chieffi and Straniero (1989), whereas the last updates of the version of the code adopted by the Pisa-Naples group, mostly related to changes in the input physics, are reported by Ciacio et al. (1997), Cassisi et al. S. Degl’Innocenti () · P.G. Prada Moroni Università degli Studi di Pisa, Pisa, Italy e-mail: [email protected] P.G. Prada Moroni e-mail: [email protected] M. Marconi · A. Ruoppo INAF-Osservatorio Astronomico di Capodimonte, Napoli, Italy M. Marconi e-mail: [email protected] A. Ruoppo e-mail: [email protected]

2 Integration of the stellar structure To integrate the four equations of stellar structure and evolution the model is divided into three integration zones: the atmosphere, the sub-atmosphere and the interior, adopting as the independent variable of integration the optical depth, the pressure and the mass, respectively. The variations of the physical quantities in a time step cannot exceed for each point of the stellar structure a prefixed value. For example for the models computed for the CoRoT ESTA Task 1 we adopted the percentage variations reported in Table 1. The first four columns of this table report the percentage variations in radius, luminosity, pressure and temperature, whereas the fifth column refers to hydrogen abundance and the last one to the nuclear burning luminosity variation, that assumes the same value for the three most important reactions, namely pp, CNO and 3α. 1 See the URL http://astro.df.unipi.it/SAA/PELZ0.html and http:// www.mporzio.astro.it/~marco/GIPSY/homegipsy.html.

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_3

25

26

Astrophys Space Sci (2008) 316: 25–30

Fig. 1 Upper panel: Pre-main sequence evolutionary tracks for the labeled stellar masses and solar chemical composition (Z = 0.02, Y = 0.27). Lower panel: evolutionary tracks for the labeled stellar masses and chemical composition from the main sequence (MS) to the first thermal pulse during the asymptotic giant branch (AGB) phase or (for the most massive models) to the onset of central C ignition

The computed evolution can start either from the homogeneous zero age main sequence (ZAMS) structures or from the chemically homogeneous quasi-static pre-main sequence models with central temperature values lower than Tc ≈ 106 K. The equations for the chemical evolution of the stellar matter are solved by means of a classical RaphsonNewton method. In each mesh the chemical abundances

are evolved in time, taking into account microscopic diffusion, if required. The convective zones are assumed to be chemically homogeneous. The chemical abundance in the subatmospheric and atmospheric regions is taken as the one of the most external mesh of the stellar interior. The reference values adopted in the code are shown in Table 2.

Astrophys Space Sci (2008) 316: 25–30

27

Table 1 Maximum allowed variation in a time step for the physical quantities of each mesh, for CoRoT ESTA Task 1 models. The term δLn /Ln refers to the percentage variation in the luminosity associated to nuclear burning reactions, and it is the same for pp, CNO and 3α δR/R

δL/L

δP /P

δT /T

δX/X

δLn /Ln

code checks that the nuclear reaction efficiency is lower than a pre-fixed value; if this occurrence is not verified the code automatically reduces the extension of the subatmospheric region to the level where the nuclear efficiency is the required one.

0.002

0.3

0.1

0.3

0.1

0.1

2.3 The treatment of convection

Table 2 Reference values for the stellar code L

M

R

G

[erg/s]

[g]

[cm]

[dyne cm2 g−2 ]

3.846 · 1033

1.989 · 1033

6.9599 · 1010

6.668 · 10−8

Table 3 Maximum allowed variation of the physical quantities from one mesh point to the next one, for CoRoT ESTA Task 1 models. δR/R

δL/L

δP /P

δT /T

δM/Mtot

0.07

0.03

0.15

0.05

0.1

The extension of the convective zones in the stellar structure is evaluated by means of the classical Schwarzschild criterion and the time scale of convective mixing is always assumed to be much shorter than the nuclear burning time scale. A first order Taylor expansion is used to follow the burnings in the convective regions. During the central He burning phase two additional phenomena, namely the induced core overshooting and the semiconvection, are taken into account. Mechanical overshooting of the stellar core during the central H and He burning phases can be either included or neglected. When it is considered the extension of the affected region is calculated from the boundary of the Schwarzschild convective region through the parameter:

2.1 The stellar interior

αov = Mov /Hpm

In the interior the equations are solved with the Henyey method, taking as boundary conditions in the external part the physical values calculated at the base of the subatmosphere. To reach the convergence of a stellar model the equations are required to be verified at least at the 10−5 level in each mesh. The mass distribution of the mesh points is fixed by the requirement that the physical quantities R, L, P, T, M should not vary by more than some pre-fixed amount from one mesh point to the next one. The requirements for the models computed in the context of CoRoT ESTA Task 1 are reported in Table 3. With these requirements the typical number of meshes in the stellar interior is ≈800–900.

where Hpm = −dM/d ln P . In the external layers the required superadiabaticity is evaluated by following the Cox and Giuli (1968) derivation of the Bohm-Vitense mixing lenght formalism.

2.2 The atmosphere and sub-atmosphere In the atmosphere the independent variable is the optical depth, ranging from about 0 to 2/3 or to the onset of convection. We usually adopt the solar scaled relation between temperature and optical depth by Krishna Swamy (1966); however for the CoRoT ESTA Task 1–Task 3 comparisons the Eddington T (τ ) relation has been used. The radius is fixed to the photospheric radius as obtained from the StephanBoltzmann law. In the sub-atmosphere the independent variable is the pressure, from the end of the atmosphere to a defined value in mass. The integration method is a fourth order Runge-Kutta technique with a variable number of meshes, typically ranging from 300 to 400. In the atmosphere and subatmosphere the variation of the chemical composition due to nuclear reactions is required to be negligible. The

2.4 Atomic diffusion The code has the possibility to include diffusion of He and heavier elements in the interior of the star. In particular for 12 C, 14 N, 16 O, 6 Li, 7 Li,9 B, 11 B, 56 Fe diffusion is explicitly calculated, whereas the other heavy elements are assumed to diffuse as 56 Fe. Electron diffusion is also included. The diffusion coefficients by Thoul et al. (1994) are adopted. These are obtained as the solutions of the Burgers equations for a multi-components fluid, including the gravity (pressure), temperature and composition terms. The rate of change of the mass fractions of the element “m”, due to diffusion, is written in dimensionless form as: 5

1 ∂[r 2 Xm T 2 ξm (r)] ∂Xm =− 2 ∂t ∂r ρr

(1)

where ξm (r) are the so called “diffusion functions” which (for each element m) are calculated as: ∂ ln P ∂ ln T + aT (m) ξ(m) = aP (m) ∂r ∂r  ∂ ln X(n) + aX ([−3pt]m, n) ∂r n −

 n



aX (m, n)

j

Z(j )X(j ) ∂ ln X(j ) A(j ) ∂r  Z(i)X(i) i A(i)

(2)

28

where n runs over chemical elements heavier than helium and the sums over i and j are over all the considered ionic species with atomic weight A and atomic number Z. aP , aT , aX are the gravitational, thermal and chemical diffusion coefficients, respectively. The diffusion velocity is progressively reduced toward the outermost layers to reach a zero value at the stellar surface. Radiative acceleration is not included. The variation of the total metallicity, due to diffusion, is taken into account in the opacity calculation while it is not considered in the equation of state.

Astrophys Space Sci (2008) 316: 25–30

3.3 Opacity

In the following we describe the main physical inputs adopted in the FRANEC code, starting with the three fundamental ingredients of a stellar model, namely the energy generation coefficients, the equation of state and the radiative and conductive opacity.

The adopted opacity tables are the latest OPAL ones,2 with chemical mixtures either from Grevesse and Noels (1993), Grevesse and Sauval (1998) or from Asplund et al. (2005), in the temperature range log T ≥ 4.2. In such a way the EOS is consistent with the adopted opacity tables from the same Livermore group. For lower temperatures the molecular opacity tables from Alexander and Ferguson (1994) are adopted for Task 1–Task 3 models and interpolated with a spline function in Z, with a cubic function in temperature and density and with a linear dependence on the hydrogen abundance. The updated version of the code implements also the Ferguson et al. (2005) tables. A mass-weighted combination of He, C and O is adopted in the H exhausted regions. In case of diffusion the effect of the variation of the global metallicity on the opacity values is taken into account. As for conductive opacity the evaluations by Potekhin (1999) or Itoh et al. (1983) are adopted.

3.1 Energy generation

4 The Standard Solar Model

The adopted reaction rates are taken from the results of the NACRE collaboration (Angulo et al. 1999). Current version of FRANEC takes into account the burning of light elements (D, 7 Li, 9 Be, etc.) in the pre-main sequence phase, with the original Deuterium abundance from Geiss and Gloeckler (1998). During H burning the evolution in time toward the equilibrium abundance of 3 He, 12 C, 14 N, 16 O is explicitly followed. The original 3 He abundance is taken from Geiss and Gloeckler (1998). The code implements weak (Salpeter 1954), weakintermediate and intermediate-strong (Graboske et al. 1973; Dewitt et al. 1973) and strong screening (Itoh et al. 1977, 1979). For the CoRoT ESTA Task 1–Task 3 models, as required, only weak screening has been adopted. For neutrino energy losses the values by Haft et al. (1994) and Itoh et al. (1996) have been adopted.

In recent years helioseismology has added important pieces of information on the solar structure, producing severe tests for Standard Solar Model (SSM) calculations. SSMs thus constitute, for each evolutionary code, an important check of the reliability of the physical inputs and of the efficiency of the microscopic mechanisms adopted for the calculations. Bahcall et al. (1995) already showed that, adopting physical inputs very similar to the ones used for the CoRoT ESTA Task 1–Task 3 models, to reach the agreement with helioseismological observables one needs an accurate treatment of helium and heavy element diffusion. Recent works investigates the effects of the last update of the solar chemical composition (Asplund et al. 2005) on the solar characteristics, pointing out a disagreement between the observed and the predicted sound speed (see e.g. Bahcall et al. 2005; Basu and Antia 2004). However, further investigations are needed and this issue is beyond of the purposes of this contribution. Table 4 shows selected characteristics of our SSM calculated with the physical inputs of Task 3 models, including He and heavy elements diffusion. Figure 2 shows the comparison between the helioseismological squared isothermal sound speed U = P /ρ from Degl’Innocenti et al. (1997) and the result of our SSM. We notice that the sound speed in the solar interior is nicely reproduced but a small discrepancy at the bottom of the convective zone is present. This problem

3 Physical inputs

3.2 Equation of state The most recent version of the code uses the OPAL 2005 tabulations for the appropriate metallicity value, but for CoRoT ESTA Task 1–Task 3 models the OPAL 2001 EOS has been adopted. The OPAL tables are interpolated linearly in temperature and pressure and with a spline in Hydrogen abundance. For values of the temperature and pressure outside the OPAL 2001 domain, we use the equation of state by Prada Moroni and Straniero (private communications, see also Straniero 1988).

2 These are computed by adopting the facilities at the URL: http://www. phys.llnl.gov/Research/OPAL/

Astrophys Space Sci (2008) 316: 25–30

29

Fig. 2 Relative difference between the helioseismological sound speed and the result of our standard solar model with the physical inputs of CoRoT ESTA Task 1–Task 3 models. The region between the blue long dashed lines indicates the allowed region taking into account the uncertainties on the helioseismological results (Degl’Innocenti et al. 1997)

Table 4 Selected characteristics of our SSM with the physical inputs of Task 1, Task 3 models. Original (in), present photospheric (photo), central (c) values and the values at the base of the external convective zone (b) are shown

t [Gyr]

4.57

L [erg]

3.846

R [1010 cm]

6.960

(Z/X)photo

0.0245

Xin

0.712

Yin

0.269

Zin

0.0198

Xphoto

0.744

Yphoto

0.238

Zphoto

0.0182

Rb /R

0.716

Tb [106 K]

2.16

cb [107 cm s−1 ]

2.22

Tc [107 K]

1.569

ρc [100 gr cm− 3]

1.517

Yc

0.63

is well known and discussed in the literature, see e.g. Basu and Antia (1997), Richard et al. (1996).

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Bahcall, J.N., Pinsonneault, M.H., Wasserburg, G.J.: Solar models with helium and heavy-element diffusion. Rev. Mod. Phys. 67, 781– 808 (1995) Bahcall, J.N., Basu, S., Pinsonneault, M., Serenelli, A.M.: Helioseismological implications of recent solar abundance determinations. Astrophys. J. 618, 1049–1056 (2005) Basu, S., Antia, H.M.: Seismic measurement of the depth of the solar convection zone. Mon. Not. Roy. Astron. Soc. 287, 189–198 (1997) Basu, S., Antia, H.M.: Constraining solar abundances using helioseismology. Astrophys. J. 606, L85–L88 (2004) Cariulo, P., Degl’Innocenti, S., Castellani, V.: Calibrated stellar models for metal-poor populations. Astron. Astrophys. 421, 1121–1130 (2004) Cassisi, S., Castellani, V., Degl’Innocenti, S., Weiss, A.: An updated theoretical scenario for globular cluster stars. Astron. Astrophys. Suppl. Ser. 129, 267–279 (1998) Chieffi, A., Straniero, S.: Isochrones for hydrogen-burning globular cluster stars. I—The metallicity range (Fe/H) from −2 to −1. Astrophys. J. Suppl. Ser. 71, 47–87 (1989) Ciacio, F., Degl’Innocenti, S., Ricci, B.: Updating standard solar models. Astron. Astrophys. Suppl. Ser. 123, 449–454 (1997) Cox, J.P., Giuli, R.T.: Principles of stellar structure. Gordon and Breach, New York (1968) Degl’Innocenti, S., Dziembowski, W.A., Fiorentini, G., Ricci, B.: Helioseismology and standard solar models. Astropart. Phys. 7, 77– 97 (1997) Dewitt, H.E., Graboske, H.C., Cooper, M.S.: Screening factors for nuclear reactions. General theory. Astrophys. J. 181, 439–456 (1973) Ferguson, J.W., et al.: Low-temperature opacities. Astrophys. J. 623, 585–596 (2005) Geiss, J., Gloeckler, G.: Abundances of Deuterium and Helium-3 in the protosolar cloud. Space Sci. Rev. 84, 239–250 (1998) Graboske, H.C., Dewitt, H.E., Grossman, A.S., Cooper, M.S.: Screening factors for nuclear reactions. 11. Intermediate screening and astrophysical applications. Astrophys. J. 181, 457–474 (1973) Grevesse, N., Noels, A.: Cosmic Abundances of the elements. In: Prantzos, N., Vangioni-Flam, E., Casse, M. (eds.) Origin and Evolution of the Elements, Paris, 22–25 June 1992, pp. 14. Cambridge University Press, Cambridge (1993)

30 Grevesse, N., Sauval, A.J.: Standard solar composition. Space Sci. Rev. 85, 161–174 (1998) Haft, M., Raffelt, G., Weiss, A.: Standard and nonstandard plasma neutrino emission revisited. Astrophys. J. 425, 222–230 (1994) Itoh, N., Totsuji, H., Ichimaru, S.: Enhancement of thermonuclear reaction rate due to strong screening. Astrophys. J. 218, 477–483 (1977) Itoh, N., Totsuji, H., Ichimaru, S., Dewitt, H.E.: Enhancement of thermonuclear reaction rate due to strong screening. II—Ionic mixtures. Astrophys. J. 234, 1079–1084 (1979) Itoh, N., Mitake, S., Iyetomi, H., Ichimaru, S.: Electrical and thermal conductivities of dense matter in the liquid metal phase. I—Hightemperature results. Astrophys. J. 273, 774–782 (1983) Itoh, N., Hayashi, H., Nishikawa, A., Kohyama, Y.: Neutrino energy loss in stellar interiors. VII. Pair, photo-, plasma, bremsstrahlung, and recombination neutrino processes. Astrophys. J. Suppl. Ser. 102, 411–424 (1996)

Astrophys Space Sci (2008) 316: 25–30 Krishna Swamy, K.S.: Astrophys. J. 145, 174 (1966) Potekhin, A.Y.: Electron conduction in magnetized neutron star envelopes. Astron. Astrophys. 351, 787–797 (1999) Prada Moroni, P.G., Straniero, O.: Calibration of white dwarf cooling sequences: theoretical uncertainty. Astrophys. J. 581, 585–597 (2002) Richard, O., Vauclair, S., Charbonnel, C., Dziembowski, W.A.: New solar models including helioseismological constraints and light element depletion. Astron. Astrophys. 312, 1000–1011 (1996) Salpeter, E.E.: Electrons screening and thermonuclear reaction. Aust. J. Phys. 7, 373–388 (1954) Straniero, O.: A tabulation of thermodynamical properties of fully ionized matter in stellar interiors. Astron. Astrophys. Suppl. Ser. 76, 157–184 (1988) Thoul, A., Bahcall, J., Loeb, A.: Element diffusion in the solar interior. Astrophys. J. 421, 828–842 (1994)

YREC: the Yale rotating stellar evolution code Non-rotating version, seismology applications P. Demarque · D.B. Guenther · L.H. Li · A. Mazumdar · C.W. Straka

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9698-y © Springer Science+Business Media B.V. 2007

Abstract The stellar evolution code YREC is outlined with emphasis on its applications to helio- and asteroseismology. The procedure for calculating calibrated solar and stellar models is described. Other features of the code such as a non-local treatment of convective core overshoot, and the implementation of a parametrized description of turbulence in stellar models, are considered in some detail. The code has been extensively used for other astrophysical applications, some of which are briefly mentioned at the end of the paper. Keywords Methods: numerical · Stars: evolution · Stars: interior · Convection PACS 97.10.Cv96.60.Ly · 92.60.hk

P. Demarque · L.H. Li · A. Mazumdar Department of Astronomy, Yale University, New Haven, CT 06520-8101, USA P. Demarque e-mail: [email protected] L.H. Li e-mail: [email protected] A. Mazumdar e-mail: [email protected] D.B. Guenther Department of Astronomy and Physics, Institute for Computational Astrophysics, Saint Mary’s University, Halifax, Nova Scotia, Canada B3H 3C3 e-mail: [email protected] C.W. Straka () Centro de Astrofísica da Universidade do Porto (CAUP), Rua das Estrelas, 4150-762 Porto, Portugal e-mail: [email protected]

1 Introduction The aim of this paper is to provide an overview of the Yale Rotating Stellar Evolution Code (YREC), as it has been applied in the last few years to research in helio- and asteroseismology. Although YREC contains extensions to model the effects of rotation in an oblate coordinate system, we describe here the “non-rotating” version. In addition to a general description, we shall emphasize three features of the code which have been implemented because of their special relevance to seismology. The first feature is the procedure utilized for the automatic calculation of calibrated solar and stellar models whose pulsational properties are to be investigated. The second feature is the treatment of convective core overshoot. Finally, the third feature is the implementation in stellar models of the effects of turbulence on the structure of the surface layers of stars with a convective envelope. The parametrization of turbulence to one dimension is based on three-dimensional radiative hydrodynamical (3D HRD) simulations of the highly superadiabatic layer (SAL) in the atmosphere. The interaction of turbulent convection and radiation in these thin transition regions is poorly known. Oscillation frequencies are sensitively affected by the structure of transition regions between radiative and convective layers. Seismology thus offers a unique opportunity to explore a long standing problem in stellar physics. Like most stellar evolution codes, YREC is a continuously evolving research tool to which many have contributed. As a result, different versions of YREC are in use at several institutions, which have been applied to a variety of research purposes. Some of the most significant applications of YREC are listed in the text and at the end of this paper (see Sect. 13). The rotating version of YREC, originally developed by Pinsonneault (1988), includes a 1.5D treatment

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_4

31

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of rotation, extending the work of Kippenhahn and Thomas (1970) and Endal and Sofia (1981), and using the formalism of Law (1980). A 2D version of YREC has also recently been implemented, specifically to address some fundamental aspects of solar magnetic activity (Li et al. 2006). Section 2 outlines the numerical scheme adopted to solve the classical differential equations of stellar structure and evolution. The treatment of the boundary conditions, of special importance for seismology, are described in Sect. 3. The constitutive physics, i.e. the equation of state and radiative and conductive opacities, are reviewed in Sect. 4, and the nuclear processes are described in Sect. 6. Stellar physics topics such as superadiabatic convection, element diffusion, convective core overshoot, and turbulence in the outer layers, all of which also have important seismological signatures, are covered in Sects. 3.2, 5, 9 and 10, respectively. The operation of the code is described in Sect. 7, with emphasis on helio- and asteroseismic applications. Seismic diagnostics applications are described in Sect. 11. The role played by YREC in the research on solar neutrinos and helio-seismology is summarized in Sect. 12. Studies of advanced evolutionary phases and applications to stellar population studies are listed in Sect. 13.

2 Henyey code The four first-order simultaneous equations of stellar structure are well-known, and have been frequently discussed in the literature (Schwarzschild 1958). YREC uses mass as the independent variable in the formulation of the equations (Lagrangian formulation). The problem is a two-point boundary value problem, with boundary conditions at the center and at the surface of the model. A relaxation technique, based on a finite difference approximation, is used. The method, first applied to the stellar structure problem by Henyey et al. (1959), is known as the Henyey method. Useful descriptions of the Henyey method are given in the paper by Larson and Demarque (1964) and the book by Kippenhahn and Weigert (1990). Specific details about the numerical procedures in the YREC implementation can be found in the Appendices of Prather (1976), which describe an earlier Henyey code on which YREC is based. Prather (1976) also provides information about the treatment of the constitutive physics, although most of the physics details have been updated since then. In the Henyey method, the model star is divided into n concentric shells by means of n + 1 suitably chosen values of the independent variable (mass), or points, in the interval defined by the innermost point (near the center) and the outermost point, which is specified by the user and located at the base of the envelope integrations. The four differential

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equations are replaced in each shell by approximating difference equations relating the values of the dependent variables at adjacent points. There are four dependent variables in each of the n shells, providing a set of 4(n + 1) linear equations which, together with two boundary conditions at the center and two at the surface, can be solved to determine approximate corrections to the 4(n + 1) dependent variables, starting from a first approximation model. The set of simultaneous equation is solved by iteration until the corrections in each variable satisfy a specified convergence limit. 2.1 Shell redistribution The shells are distributed so as to optimize numerical accuracy and efficiency. In order to follow the evolution from the earliest gravitational contraction phase all the way to the hydrogen and helium shell burning phases, it is necessary to redistribute the shells in the model. This is especially critical during shell burning. After a star exhausts its core supply of hydrogen it begins burning hydrogen in a shell. Initially the shell is almost 0.2M thick for a 1M star but the shell quickly narrows to only 0.001M as the star evolves up the giant branch, thinning to 0.0001M at the point of helium flash. Because of the high temperature dependence of helium burning, helium burning shells are even thinner than hydrogen burning shells. YREC will add or remove shells according to the size of gradients in structure (i.e., pressure, temperature, and composition) and gradients in luminosity, as well as the size of Henyey corrections applied during the iteration procedure. The code keeps track of physically real discontinuities so that they are not smoothed during the rezoning process. Interpolation is linear. Our own testing has shown that using higher order methods such as oscillatory spline interpolation introduces numerical oscillations near the tip of the giant branch. 2.2 Time steps The models are advanced in time through two terms in the energy equation, the nuclear energy term (Sect. 6), and the time rate of change of entropy due to contraction or expansion during evolution. Special care is taken to preserve numerical accuracy for small time steps (Prather 1976). One can either specify the time step or have YREC automatically determine the optimum time step during evolution. When producing accurately calibrated solar models, to maintain numerical consistency it helps to specify the time step interval. In most other situations it is best to let YREC determine the time step based on user specified convergence tolerance criteria. During nuclear burning phases of evolution YREC will guess the time step based on the rate at which hydrogen and helium (if applicable), are being consumed in each shell of the model. During gravitational

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contraction phases of evolution YREC will control the time steps by monitoring the change in temperature, pressure, and luminosity from one model to the next. During helium flash if the model fails to converge during a evolutionary time step, YREC is also able reduce the time step by a user specified factor and redo the evolutionary step. More details regarding the operation of YREC can be found in Sect. 7.

3 Boundary conditions 3.1 Center The two inner boundary conditions constrain the values of the radial distance and luminosity variables at the innermost mass shell. Because of the false singularity at the center, the innermost point is not at the very center, but in a shell chosen close to the center. Note that in order to preserve accuracy, special care must be taken with the position of the innermost shell, especially in pulsation calculations (see Sect. 7.3).

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of the envelope (as determined by the local Schwarzschild criterion), the temperature gradient is evaluated according to the formalism of Stothers and Chin (1995), which is designed to describe superadiabatic convection. It is in this region that the peak of the highly superadiabatic transition layer (SAL) is normally located (as it is in the Sun). The main advantage of the Stothers and Chin (1995) formalism is that by a suitable choice of parameters, it can be made to reproduce either the standard mixing length theory (MLT) (Böhm-Vitense 1958) or the theory of Canuto and Mazzitelli (1992), sometimes called FST. In order to preserve continuity in the convective temperature gradient at the envelopeinterior interface, the Stothers and Chin (1995) formalism is used to calculate the convective gradient both in the envelope and in the interior whenever superadiabaticity exceeds a preset value. Another feature of the envelope integration, described in more detail in Sect. 10, includes a 1D parametrization of the effects of turbulent pressure and turbulent kinetic energy in the outer layers.

3.2 Surface 4 Equation of state and opacities The outer boundary conditions depend on the structure near the surface. Because a model of the outer layers depends on the global properties of the star, i.e. its surface gravity log g and effective temperature Teff , the problem is implicit. In order to specify the surface boundary conditions, which relate the variations of the pressure and temperature variables to the total luminosity and radius of the star, three inward envelope integrations are constructed. These envelope integrations are chosen so as to form a triangle in the theoretical HR-diagram (i.e. the log L/L vs. log Teff plane). The inward envelope integrations consist of two main parts. The outermost layers, starting at optical depth near τ = 10−10 , which are effectively isothermal at the start, are described by a gray radiative atmosphere specified by a T (τ ) relation and integrated to the appropriate value of τ at which the temperature reaches Teff (e.g. τ = 2/3 for the Eddington approximation, τ = 0.312156330 for the Krishna Swamy 1966 atmosphere). This surface in the star is usually defined as the photosphere. As an alternative to the atmosphere integrations, more complex atmospheres from pre-computed libraries can be also used, such as those from Kurucz (1998). Below the photosphere, all variables but the luminosity variable (which is held to be constant in the outer envelope) are integrated to a chosen value of the mass. The integration is carried out using log P as the independent variable, to the value of the mass variable at which the surface boundary conditions for the interior are computed (the base of the envelope). The region which extends from this value of the mass to the innermost shell of the star constitutes the interior of the stellar model. In convectively unstable layers

YREC has been updated regularly so as to incorporate the latest research developments regarding equation of state and opacity in the stellar interior, while maintaining backward compatibility with earlier versions of the same. The current version uses the latest OPAL opacities (Iglesias and Rogers 1996) and OPAL equation of state (Rogers and Nayfonov 2002). At low temperatures (log T < 4.1) opacities are obtained from the tables provided by Ferguson et al. (2005). At each mass shell the EOS is obtained by interpolation from the standard tables. Since the EOS is weakly dependent on Z, we use only one set of tables at a fixed Z, obtained by the Z-interpolation routine provided with the OPAL EOS package. For models with metal diffusion, the value of Z at which the EOS is interpolated is chosen at a suitable intermediate value. The EOS quantities at the desired X, T and ρ are obtained by quadratic interpolation from the tables. The results and the derivatives are smoothed by mixing overlapping quadratics. For opacity, a four-point Lagrangian interpolation scheme is used over a 4-dimensional grid of Z, X, T , and ρ.

5 Diffusion The diffusion of chemical elements by gravitational settling and thermal diffusion is implemented following the prescription of Thoul et al. (1994). Options in the code include no diffusion, helium diffusion only, or both Y and Z diffusion. The analytical fits provided by Thoul et al. (1994) can also be used instead of the tabulated diffusion coefficients, to speed up the computations.

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6 Nuclear reactions

6.1 NACRE

The nuclear reaction rates in conjunction with the corresponding energy release (Q-values) are important for the evolution of chemical species, the energy input from nuclear fusion reactions and for the neutrino fluxes. The reactions explicitly calculated in YREC are the following:

YREC provides the option of using the NACRE1 reaction rates (Angulo et al. 1999). In its present version, the Qvalues from each reaction are not changed and are thus not identical to the values published on the NACRE database2 . All relevant reaction rates that are provided by NACRE are included, i.e. those corresponding to (1)–(3); (5)–(11). The NACRE library lists the rate data in tabulated form and also provides fit-formulas, the latter of which are implemented. The fit-formulas are accurate by 3%–25% compared to the tabulated data, with typical deviations of 10%– 15%. Our tests for a standard solar model have shown that a calibrated standard model is not affected by the NACRE reaction rates. The largest differences are found in the neutrino flux of 8 B which differs by about 9%. This difference is comfortably within other theoretical uncertainties (Bahcall et al. 2004).

1

7

H + 1 H → 2 H + e+ + ν

(1)

3

He + 3 He → 4 He + 2 1 H

(2)

3

He + He → Be + γ

(3)

4

7

Be + e− + 1 H → 2 4 He + ν

(4)

Be + 1 H → 2 4 He + γ + e+ + ν

(5)

12

C + 1 H → 13 C + γ + e+ + ν

(6)

13

C+ H→

(7)

7

1

14

N+γ

14

N + 1 H → 15 N + γ + e+ + ν

15

N + 1 H → 12 C + 4 He

16

(8) (9) +

O+2 H→

14

N + He + γ + e + ν

(10)

3 He →

12

C

(11)

1

4

6.2 Light elements

12

C + 4 He → 16 O + γ

(12)

A switch permits keeping track of nuclear burning of the light elements 2 H, 6 Li, 7 Li and 9 Be at the base of the convection zone in models of sun-like stars (Deliyannis 1990).

13

C + 4 He → 16 O + n

(13)

6.3 Neutrino losses

N + He →

(14)

4

14

4

18

O+γ

2 1 H + e− → 2 H + ν 3

He + 1 H → 4 He + e+ + ν

(15) (16)

The first five equations (1)–(5) contain the three alternative pp branches (pp1,pp2,pp3) all of which start with 3 He. Equations (6)–(9) represent the primary CNO cycle, (10) the secondary cycle. The reaction of helium burning is given in (11), followed by the dominant α capture reactions (12)– (14). The last two reactions are only important for the neutrino problem and can be neglected for the energy generation. As is implicitly shown in the nuclear reactions (1)– (16), all β-decay reactions are treated in the instantaneous approximation. In addition, four branching ratios are defined (Bahcall and Ulrich 1988): the fraction of 7 Be that is burned by electron capture (4), the fraction of 7 Be that is burned by proton capture (5), the fraction of 14 N that is burned via 14 N(p, α)12 C and the fraction of 15 N that is burned via 15 N(p, γ )16 O. The energy generation is calculated by multiplying the rates by the Q-values which are taken from Bahcall and Ulrich (1988, Table 21). The standard reaction rates implemented are identical to the rates published in Bahcall (1989).

Neutrino loss rates are taken from the monograph by Bahcall (1989), updated by subsequent private communications from the author. For advanced stages of stellar evolution, the neutrino rates from photo, pair and plasma sources from Itoh et al. (1989) are included.

7 Running YREC YREC can automatically calculate calibrated solar and stellar models. The user provides a complete set of constraints along with allowable parameter variations and YREC will search within the chosen parameter space for a solution. This is especially convenient since the mixing length parameter and in some cases the helium abundance used to compute stellar models must first be established from calibrated solar models. The calibrated values are sensitive to the choice of opacity tables, the equation of state formulation, the inclusion of diffusion, and the choice of model atmosphere. 1 Nuclear

Astrophysics Compilation of REaction Rates.

2 http://pntpm.ulb.ac.be/Nacre.

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7.1 Calibrated solar models To produce a calibrated solar model the user inputs the age of the Sun and its primordial composition, i.e., mass fraction mixture of hydrogen, helium, and metals on the zero age main-sequence. In addition the user specifies the tolerances for the luminosity and radius. YREC will then vary the initial value for the helium abundance and mixing length parameter until it has produced a model at the age of the Sun that has the observed radius, 6.958 × 1010 cm, and observed luminosity, 3.8515 × 1033 erg/s (Edmonds et al. 1992) within the specified tolerances. When including the effects of metal and helium diffusion, the user has the option of inputting the Z/X at the surface and its allowed tolerance. In this case, YREC will adjust the initial helium abundance, metal abundance, and mixing length until a model at the age of the Sun is produced that matches the Sun’s luminosity, radius, and surface Z/X within the specified tolerances. With 64-bit floating point numbers, YREC can compute a tuned solar model with tolerances of 1 part in 106 for radius and luminosity and 1 part in 104 in Z/X after about 10 to 12 iterations. The actual procedure begins by computing one reference run, one run with slightly changed helium abundance, followed by one run with slightly changed mixing length parameter, and then one run, if chosen, with slightly changed metal abundance. The luminosity, radius, and, if chosen, surface Z/X, of the final models are used to compute the derivative matrix of luminosity, radius and surface Z/X with respect to helium abundance, mixing length parameter, and Z. The first order corrections to each parameter are determined from the derivative matrix and a new model is computed. The process is iterated until the model falls within the specified tolerances. 7.2 Calibrated stellar models The process is slightly different for stars because normally the age of a star is unknown. Only the luminosity and surface temperature are used to constrain the model. In the case of stars, YREC adjusts the mass and either the mixing length parameter or the helium abundance in an attempt to produce a stellar evolutionary track that passes though the tolerance box in the theoretical HR-diagram. To produce a stellar model the user inputs the metal abundance, and either the helium abundance or the mixing length parameter. In addition the user specifies the luminosity, effective temperature, and their corresponding tolerances. The user has the option of allowing the code to adjust either the mixing length parameter or the helium abundance. The code generates tracks varying the mass and the chosen parameter using a derivative matrix to produce a model that passes through the specified location in the HR-diagram. Once the

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optimum parameters are determined, the code computes the track a second time but stops the evolution when the model hits the specified location in the HR-diagram. The tuned model is constrained in mass and age. 7.3 Pulsation models The pulsation output files in YREC are tailored for the JIG non-adiabatic oscillation code of Guenther (Guenther 1994). These files can be saved for specified models in an evolutionary sequence (say for a calibrated solar model, or for a calibrated stellar model), or any model for specified ages along the evolutionary track. One of the first things stellar modelers realized when using their solar models for pulsation analysis is that the optimum distribution of shells within the model for structure and evolutionary calculations is different from the optimum distribution of shells for pulsation analysis. Whereas evolutionary models need to resolve well the nuclear burning regions, pulsation models need to resolve the surface layers (for acoustic modes). For example, for evolutionary models great care is needed to fully resolve the thinning hydrogen burning shell (0.001 M to 0.0001 M ) as the models evolve up the giant branch. For pulsation models it is the low density regions, where the sound waves have the largest amplitudes, that need to be well resolved. Therefore, in order to produce viable models for pulsation analysis, the user increases the resolution of shells in the envelope, atmosphere, and the region below the base of the convection zone. Ultimately, in order to achieve frequency accuracies of the order of 1 part in 104 using a first order numerical pulsation program one needs approximately 600 shells in the interior, 600 shells in the envelope defined as the outer 1–5% of the mass, and 600 shells in the atmosphere. To maximize self consistency all thermodynamic variables and their derivatives are obtained directly from the structure model. Related to shell resolution is the distribution of shells near the core. Stellar evolutionary codes do not locate a shell at the center owing to divide by zero complications but set the innermost shell a small distance away from the center. In order to do accurate pulsation analysis of g-modes or to study the p-mode small spacing parameter, both of which are sensitive to the structure of the deep interior, it is necessary to extend the innermost shell closer to the center than normally required by stellar evolutionary calculations: compare 1.0 × 10−3 radius fraction for stellar evolution to 2.0 × 10−7 for stellar pulsation. A stellar model output for pulsation runs from the “central-most” interior shell to the top of atmosphere computation near optical depth τ = 10−10 . 7.4 Model grids A useful feature of YREC is its ability to carry out extensive model calculations without user input. It is possible to

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generate in a single run, tens of thousands of evolutionary tracks, corresponding to tens of millions of models, covering a wide range of masses, compositions, mixing length parameters, with each track tuned to their own particular numerical and physical variables. This has enabled Guenther and Brown (2004) to compute dense grids of stellar models for pulsation analysis throughout the HR-diagram. For other grids of evolutionary sequences, see Sect. 13. 7.5 Backwards compatibility All new physics (e.g., opacities, nuclear reaction rates) have been implemented along with existing physics so that the user can, at any time, run YREC using older physics.

8 Convection By default the local Schwarzschild criterion is implemented in order to determine if a mass shell is labeled as convective or radiative. The Ledoux stability criterion can also be used when a specific parameter choice is made in the local limit of the non-local convection treatment described below. The abundance of chemical species in convective cores is treated under the assumption of instantaneous mixing.

9 Core overshoot Since the eddy velocity at convective boundaries is nonzero, convective motions will penetrate into the radiative region. Two different forms of penetration are commonly distinguished: (a) inefficient penetration that does not alter the temperature gradient, termed “overmixing” here, and (b) subadiabatic penetration (Zahn 1991), where the convective heat transport is efficient enough to establish a nearly adiabatic temperature gradient. YREC offers a number of different options for treating overshoot (OS). All OS options have in common that mixing of chemical species in the OS region is instantaneous and all chemical species are homogenized within the extended zone. Due to the small characteristic time scale of convection in comparison to the thermal and nuclear timescales during the major burning stages this assumption holds to high accuracy. Among the OS options, two different approaches are distinguished: (a) a parametric treatment where the OS extent is a multiple of the pressure scale height taken at the formal Schwarzschild boundary and (b) a physically motivated treatment where the OS extent is calculated from a non-local convection theory originally developed by Kuhfuß (1986) and later extended by Wuchterl and Feuchtinger (1998). In the latter case, the temperature gradient is calculated directly

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from the additional convection equation which is solved in addition to the canonical stellar structure equations at every time-step. 9.1 Parametric treatment The boundary of the OS zone is determined by adding a fraction αOM of the pressure scale height to the boundary at the radius rS , determined by the Schwarzschild criterion: rnew = rS + αOM HP (rS )

(17)

where the pressure scale height HP (rS ) is taken at the Schwarzschild boundary. The temperature stratification in the OS zone is determined by the two options described above, either (a) the temperature gradient is not altered (overmixing) or (b) the temperature gradient is set to the adiabatic temperature gradient. For a fixed αOM the latter option produces larger convective cores (Zahn 1991). 9.2 Non-local convection As an alternative to the purely parametric treatment of OS, the one-dimensional convection theory developed by Kuhfuß (1986) is implemented (Straka et al. 2005). In the framework of anelastic and diffusion-type approximations of the unknown correlation functions, Kuhfuß derives one equation for the turbulent kinetic energy from spherical averages of the first-order perturbed Navier-Stokes equations. The solution of this equation provides the extent of the convective core region and includes the effects of OS naturally, since the velocity of convective motions is zero where the turbulent kinetic energy vanishes. In addition, this equation also provides the temperature gradient in the OS region. 9.2.1 Implemented equations The new equation for the turbulent kinetic energy ω¯ that is solved in YREC is given by: ∇ad cD 3/2 ∂ Dω¯ = ω¯ (4πr 2 jt ) , jω¯ − − Dt ρHP ∂m jt = −4πr 2 ρ 2 αt ω¯ 1/2

∂ ω¯ ∂m

(18) (19)

where D/Dt is the Lagrangian time derivative, ρ density, r stellar radius, m the Lagrangian mass coordinate. , defined as 1/ = 1/(αMLT HP ) + 1/(βr r) is the geometrically limited mixing length scale with ∇ad being the adiabatic gradient. Note that the limiting of the pressure scale height in the central part influences the total core size within the framework of non-local convection theories. A linear implicit extrapolation method is used in order to calculate the stationary solution of (18), i.e. Dω/Dt ¯ ≡ 0.

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37

The solution yields the turbulent kinetic energy ω¯ at every mass shell. We define shells to be convective, if:

Table 1 Parameters of non-local convection theory Parameter

Canonical value

Description

xω¯ < 0.1

αMLT

1.5

mixing length

αs

0.408

turbulent driving

cD

2.177

dissipation efficiency

αt

0.610 αs

overshoot distance

βr

1.0

geometric mixing length

(20)

where xω¯ =

1 , 1 + F ω¯ 1/2

F=

3αs κ ρ 2 CP 16σ T 3

(21)

with the usual notation for the opacity κ, temperature T , specific heat at constant pressure CP and the StefanBoltzmann constant σ . The boundaries are sharply defined by an extreme falloff of ω¯ which is encountered in interior solutions of (18). Finally, the temperature gradient can be calculated from:   ∂ ω¯ ∇ = ∇ad + xω¯ (∇rad − ∇ad ) + (1 − xω¯ ) G +H (22) ∂n with αt HP G= αs CP T ϕ H = ∇μ δ



4πr 2 ρ m

 ,

(23) (24)

where m is the mass enclosed in one shell and ∇μ = d ln μ/d ln P . In the case of an ideal gas with radiation pressure the dimensionless parameters δ and ϕ take on the values δ = (4 − 3β)/β, β = Pgas /P and ϕ = 1 respectively. In convective core regions, the temperature gradient remains very close to the adiabatic one whenever xω¯ < 0.1. A more detailed discussion of the implemented equations and the numerical techniques employed can be found in Straka et al. (2005). 9.2.2 Non-local parameters The implemented non-local convection theory contains five parameters (Table 1). These parameters must be calibrated, preferably on a well selected set of open clusters, or on selected asteroseismic target stars like Procyon A (Straka et al. 2005). Two of the canonical values given in Table 1, i.e. αs and αt can be derived by matching the Kuhfuß result to mixing length theory (MLT) in the local limit (αt = 0). The mixing-length parameter plays a minor role in the core regions, where superadiabaticity of temperature gradient is tiny. The parameter that controls the OS zone is given by αt and is thus the most important one to calibrate. Kuhfuß derives αt = 0.610 αs from theoretical arguments. In the strictly local limit (αt = 0) the Kuhfuß treatment is equivalent to the MLT equations when based on the Ledoux stability criterion.

10 Turbulence A method to incorporate the effects of turbulence into the outer layers of one-dimensional (1D) stellar models has been implemented in YREC (Li et al. 2002). The method requires a detailed three-dimensional hydrodynamical simulation (3D RHD) of the atmosphere and highly superadiabatic layer of stars (Robinson et al. 2003). The basic idea is to extract from the velocity field of the 3D simulation three important quantities: the turbulent pressure, the turbulent kinetic energy and the anisotropy of the flow. Implementation into a 1D stellar model thus requires two additional parameters, i.e. χ , the specific turbulent kinetic energy, and γ , which reflects the flow anisotropy. These parameters, which modify the hydrostatic equilibrium equation and the internal energy equation, must be introduced in a thermodynamically self-consistent way. As a result, they also change the adiabatic and convective temperature gradients, as well as the energy conservation equation. The next section (Sect. 10.1) describes the calculation of χ and γ from the velocity field in the 3D simulation. The introduction of the parameters χ and γ into the stellar structure equations and YREC is summarized in Sect. 10.2, Sect. 10.3 and Sect. 10.4. The effects on p-mode frequencies in a solar model are illustrated in Sect. 10.5. 10.1 Turbulent velocities The physics (thermodynamics, the equation of state, and opacities) in the 3D simulation is the same as in the 1D stellar models. These simulations follow closely the approach described by Kim and Chan (1998), and are described in more detail in the papers of Robinson et al. (2003, 2004). The full hydrodynamical equations were solved in a thin subsection of the stellar model, i.e. a 3D box located in the vicinity of the photosphere. For the radiative transport, the diffusion approximation was used in the deep region (τ > 103 ) of the simulation, while the 3D Eddington approximation was used (Unno and Spiegel 1966) in the region above. After the simulation had reached a steady state, statistical integrations were performed for each simulation for over 2500 seconds in the case of the solar surface convection.

38

Astrophys Space Sci (2008) 316: 31–41

For the derivation of χ and γ , Li et al. (2002) use a selfconsistent approach introduced by Lydon and Sofia (1995) to include magnetic fields in calculating the convective temperature gradient within the MLT framework, and used successfully by Lydon et al. (1996) to explain the variation of solar p-modes with the solar cycle. Turbulence can be measured by the turbulent Mach number M = v  /vs , where v  is the turbulent velocity, and vs is the sound speed. The MLT is valid when M is sufficiently small. In the outer layers of a star like the sun M can be of order unity (Cox and Giuli 1968), but in the deep convection region M is almost zero. The turbulent velocity is defined by the velocity variance: vi

= (vi 2 − vi )

2 1/2

(25)

,

where the overbar denotes a combined horizontal and temporal average, and vi is the total velocity. Using M, we can define the turbulent kinetic energy per unit mass χ as χ=

1 2 2 M vs . 2

(26)

where   μ = ∂∂lnlnPρT T ,χ,γ  ln ρ ν = − ∂∂ ln χ

PT ,T ,γ

10.3 Solar model with turbulent pressure alone The simplest way to take into account turbulence in solar modeling is to include turbulent pressure (or Reynolds stress) alone. In this case, only the hydrostatic equilibrium equation needs to be modified as follows: ∂P GMr =− (1 + β), ∂Mr 4πr 4

(32)

where P = Pgas + Prad , and β=

Sturb = χ/T ,

PT ,T ,χ

As a result, the stability criterion against convection is modified. For similar reasons, both the convective and adiabatic gradients are also modified by turbulence.



The turbulent contribution to the entropy is

  ln ρ μ = − ∂∂ ln T P ,χ,γ  T  ∂ ln ρ  ν = − ∂ ln γ .

2Pturb ∂Pturb − ρgr ∂P



∂Pturb 1+ ∂P

−1 .

(33)

(27)

where ρχ is the turbulent kinetic energy density. Since Pturb = ρvz 2 , γ can be related to the turbulent velocity as follows:

Here 2Pturb /(ρgr) originates from the spherical coordinate system adopted, representing a kind of geometric effect. The equations that govern the envelope integrations also need to be changed accordingly. One can construct a calibrated nonstandard model in the same way as one obtains the standard solar model, assuming that Pturb , set equal to its value for the present sun, does not change from the ZAMS to the present age of the sun. The p-mode oscillation spectrum of this calibrated solar model (psm) is discussed in Sect. 10.5.

γ = 1 + 2(vz /v  )2 .

10.4 Solar model with χ and γ as independent variables

where T is the gas temperature. Turbulence in the stratified layers of a stellar convection zone is not isotropic. We define the parameter γ to reflect the anisotropy of turbulence, Pturb = (γ − 1)ρχ,

(28)

(29)

when turbulence is isotropic (vz

= vx

= vy );

γ = 5/3 γ =3 or γ = 1 when turbulence is completely anisotropic (vz = v  or vz = 0, respectively). The physical meaning of γ is the specific heat ratio due to turbulence. This affects the distribution of the radial turbulent pressure which is then scaled with the gas pressure, Pgas . The total pressure is defined as PT = Pgas + Prad + Pturb .

(30)

The form of the continuity equation and of the equation of transport of energy by radiation are not affected by turbulence. The hydrostatic equation includes a Reynolds stress term due to turbulence 1 d ∂P GMr = − 2 ρ − 2 (r 2 ρvr vr ), ∂r r r dr

(34)

10.2 Convective temperature gradients with the turbulent velocities

where P = Pgas + Prad . Since the last term can be rewritten as ∂Pturb /∂r + 2(γ − 1)χ/r, this equation becomes

Since the parameters χ and γ now appear in the equation of state, they must be included as independent variables in evaluating the density derivative. We have therefore:

∂PT GMr 2(γ − 1)χ =− − . ∂Mr 4πr 4 4πr 3





dρ/ρ = μdPT /PT − μ dT /T − νdχ/χ − ν dγ /γ ,

(31)

(35)

The last term on the right hand side of (35) also embodies the same spheric geometric effect as 2Pturb /(ρgr) in (33).

Astrophys Space Sci (2008) 316: 31–41

Fig. 1 P -mode frequency difference diagrams. Turbulent model minus standard model (ssm), for the turbulent pressure solar model (psm), and the solar model with the turbulent pressure and turbulent kinetic energy (esm). The difference between psm and ssm is of the order of 1 µHz, while at high frequencies esm and ssm differ by more than 10 µHz. Plotted are the l = 0, 1, 2, 3, 4, 10, 20, . . . , 100 p-modes

39

Fig. 2 P -mode frequency difference diagrams, observation minus model, scaled by the mode mass Qnl , for the standard solar model (ssm), the turbulent pressure solar model (psm), a solar model with fixed turbulent pressure and kinetic energy (esm1), and a solar model with evolutionary turbulent pressure and kinetic energy (esm2, almost overlaps with esm1). Plotted are the l = 0, 1, 2, 3, 4, 10, 20, . . . , 100 p-modes

The energy conservation equation is also modified by turbulence because the first law of thermodynamics must now include the turbulent kinetic energy. We have then:

Li et al. (2002) paper, the p-mode frequencies for two calibrated solar models that include the effects of turbulence are compared to the standard solar model (ssm) p-mode fredST ∂Lr quencies. The psm model is obtained by including turbulent , (36) =−T ∂Mr dt pressure alone in the solar modeling, while the esm models are obtained by introducing the turbulent variables χ and where γ which include both turbulent pressure and kinetic energy.   PT μ ν  μ PT μ ν Contrary to a frequently made assertion, the inclusion of turT dST = cp dT − dPT + 1 + dχ + dγ . bulent pressure in the pressure term has only a small effect ρ ρμχ ρμγ (37) on the calculated p-mode frequencies. On the other hand, the inclusion of turbulent kinetic energy is significant. This The equation of energy transport by convection, is illustrated in Fig. 1 which shows that the frequency differences caused by turbulent kinetic energy are much larger ∂T T GMr in size than those caused by turbulent pressure alone. Fig=− ∇conv , (38) ∂Mr PT 4πr 4 ure 2 indicates that the frequency changes caused by turbulent kinetic energy make the computed model frequencies does not change in form, but the convective temperature gramatch the solar data better than the ssm model. This redient, discussed in a previous section, is different from that sult is consistent with the work of Rosenthal et al. (1999) without turbulence. The equations that govern envelope inwho “patched” a modified 3D RHD simulation by Stein and tegrations also need to be changed accordingly. The oscilNordlund (1989) onto a 1D solar model (see their Figs. 1 lation properties of the calibrated solar model constructed and 6). under this assumption (esm) are discussed in the next section. 10.6 SAL peak shift 10.5 Frequency corrections to solar p-modes

Implementing the effects of turbulence in the outer layers of the stellar model modifies the calculated p-mode frequencies at high frequencies. The magnitude of the frequency correction is illustrated in Figs. 1 and 2 for the case of a solar model, taken from the work of Li et al. (2002). In the

While it is always preferable to extract the γ -χ data from a 3D RHD simulation that corresponds to exactly the same atmospheric conditions (log g, log Teff ) as in the 1D model, it is of interest to estimate the turbulence effects in stellar models where the 3D RHD simulation is not available. In such situations, the γ -χ data cannot be used directly, instead the data must be shifted in order to be applied at the

40

correct depth of the model (Straka et al. 2006). This shifting is motivated by an expected characteristic found in all 3D RHD atmosphere simulations: namely that the SAL peak closely coincides with the turbulent pressure peak. 10.7 Calibration The presented method for including turbulence does not remove MLT and therefore the uncertainties inherent in the mixing length parameter remain. In order to make quantitative predictions, both mass and age of the star must be known to high precision in addition to the luminosity L and effective temperature Teff . Instead of the latter an interferometric radius measurement is preferable, since the measurement is usually more precise. In the case of the Sun, the age is known to high precision and the mixing length parameter and hydrogen mass fraction can be calibrated to the known solar luminosity and radius. As demonstrated in Guenther et al. (2005); Straka et al. (2006), asteroseismic data can be instrumental in other stars, since low order p-mode frequencies anchor the interior model effectively in age and mass. When calibrated to the same luminosity and effective temperature, a differential assessment of turbulence effects can be derived. Another example, in which a proper calibration is possible with more asteroseismic data, is the detached binary system α Centauri: The masses of both components are known to high precision, the radius of the A component is measured with interferometric techniques (Kervella et al. 2003) and the luminosity is also well determined through parallax measurements. With the help of future asteroseismic data of the low order p-mode frequency spectrum, the stellar age of α Centauri can be effectively determined. Under such circumstances the methods described to include turbulence in YREC are fully applicable.

11 Seismic diagnostics Stellar models constructed with YREC have been used to develop seismic diagnostics to explore internal structural properties of stars that could not be observed by any other means. Basu et al. (2004) have used low degree acoustic modes to determine the helium abundance in the envelopes of lowmass main sequence stars with precision. The oscillatory signal in the frequencies caused by the depression in 1 in the second helium ionization zone is used. For frequency errors of one part in 104 , the expected σY in the estimated Y ranges from 0.03 for 0.8M stars to 0.01 for 1.2M main sequence stars. In more evolved stars, this approach is feasible if the relative errors in the frequencies are less than 10−4 . Mazumdar et al. (2006) have explored asteroseismic diagnostics of convective core mass using small frequency

Astrophys Space Sci (2008) 316: 31–41

separations of low-degree p-modes. Small separations can also be combined to derive convective core overshoot diagnostics. It was shown that in stars with convective cores, the mass of the convective core can be estimated to within 5% if the total mass is known, although systematic errors in the total mass could introduce errors of up to 20%. The evolutionary stage of the star, determined in terms of the central hydrogen content is much less sensitive to the mass estimate.

12 Solar neutrinos and helioseismology Different versions of YREC have been used to study the structure of the solar interior, the solar neutrino problem and helioseismology (Guenther and Demarque 1997) and references therein. In particular, the important series of papers by Bahcall et al. (2004, 2005) on solar neutrinos, helioseismology and solar abundances also made use of a dedicated version of YREC.

13 Other YREC applications A variety of applications to stellar structure theory and evolution have been carried out using YREC. In addition of the work on stellar rotation mentioned in the introduction (see also Chaboyer et al. 1995 and Barnes 2003), one notes the pioneer work on stellar collisions and mergers (Sills et al. 1997). Important research in stellar population studies and population synthesis continues to be carried out with the YonseiYale isochrones (YY isochrones) (Yi et al. 2001). Frequently quoted research on helium burning phases of evolution (horizontal-branch) has also been carried out with YREC (Lee et al. 1994; Yi et al. 1997). Acknowledgements A large number of people have contributed to YREC, in ways large and small. In addition to the authors of this writeup, researchers currently at Yale who are doing YREC related research include S. Basu and students, F.J. Robinson, and S. Sofia. There is an on-going collaboration with researchers at other institutions: co-author D.B. Guenther and his students at Saint Mary’s University, Y.-C. Kim, Y.-W. Lee and S.K. Yi and their students at Yonsei University, and coauthor C.W. Straka at CAUP (Porto). Thanks are due to M. Pinsonneault now at Ohio State University, who is the main architect of the original YREC created from previous Henyey codes at Yale. He and his students have developed their own version of YREC. So have A. Sills at McMaster University and B. Chaboyer at Dartmouth College and their students and collaborators. S. Barnes at Lowell Observatory and C. Deliyannis at Indiana University also use YREC in their research. J.-H. Woo has worked on core overshoot. Partial support from NASA Astrophysics Theory grants NAG58406 and NAG5-13299 and from HST-GO-10505.03-A is gratefully acknowledged by members of the Yale group. DBG acknowledges the support of the Natural Sciences and Engineering Council of Canada.

Astrophys Space Sci (2008) 316: 31–41 CWS acknowledges support from the European Helio- and Asteroseismology Network (HELAS) funded by the European Union’s Sixth Framework Program. We would like to thank the anonymous referee for doing a meticulous job that provided very helpful comments to improve the manuscript.

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41 Kim, Y.-C., Chan, K.L.: Astrophys. J. Lett. 496, 121 (1998) Kippenhahn, R., Thomas, H.-C.: In: Slettebak, A. (ed.) Proc. IAU Colloq. 4, p. 20 (1970) Kippenhahn, R., Weigert, A.: Stellar Structure and Evolution. Springer, Berlin (1990) Krishna Swamy, K.S.: Astrophys. J. 145, 174 (1966) Kuhfuß, R.: Astron. Astrophys. 160, 116 (1986) Kurucz, R.L.: http://kurucz.harvard.edu/grids.html (1998) Larson, R.B., Demarque, P.R.: Astrophys. J. 140, 524 (1964) Law, W.Y.: Ph.D. Thesis, Yale University (1980) Lee, Y.-W., Demarque, P., Zinn, R.: Astrophys. J. 423, 248 (1994) Li, L.H., Robinson, F.J., Demarque, P., Sofia, S., Guenther, D.B.: Astrophys. J. 567, 1192 (2002) Li, L.H., Ventura, P., Basu, S., Sofia, S., Demarque, P.: Astrophys. J. Suppl. Ser. 164, 215 (2006) Lydon, T.J., Sofia, S.: Astrophys. J. Suppl. Ser. 101, 357 (1995) Lydon, T.J., Guenther, D.B., Sofia, S.: Astrophys. J. Lett. 456, L127 (1996) Mazumdar, A., Basu, S., Collier, B.L., Demarque, P.: Mon. Not. R. Astron. Soc. 372, 949 (2006) Pinsonneault, M.H.: Ph.D. Thesis, Yale University (1988) Prather, M.J.: Ph.D. Thesis, Yale University (1976) Robinson, F.J., Demarque, P., Li, L.H., et al.: Mon. Not. R. Astron. Soc. 340, 923 (2003) Robinson, F.J., Demarque, P., Li, L.H., et al.: Mon. Not. R. Astron. Soc. 347, 1208 (2004) Rogers, F.J., Nayfonov, A.: Astrophys. J. 576, 1064 (2002) Rosenthal, C.S., Christensen-Dalsgaard, J., Nordlund, Å., Stein, R.F., Trampedach, R.: Astron. Astrophys. 351, 689 (1999) Schwarzschild, M.: Structure and Evolution of the Stars. Princeton University Press, Princeton (1958) Sills, A., Lombardi Jr., J.C., Bailyn, C.D., et al.: Astrophys. J. 487, 290 (1997) Stein, R.F., Nordlund, A.: Astrophys. J. Lett. 342, L95 (1989) Stothers, R.B., Chin, C.-W.: Astrophys. J. 440, 297 (1995) Straka, C.W., Demarque, P., Guenther, D.B.: Astrophys. J. 629, 1075 (2005) Straka, C.W., Demarque, P., Guenther, D.B., Li, L., Robinson, F.J.: Astrophys. J. 636, 1078 (2006) Thoul, A.A., Bahcall, J.N., Loeb, A.: Astrophys. J. 421, 828 (1994) Unno, W., Spiegel, E.A.: Publ. Astron. Soc. Jpn. 18, 85 (1966) Wuchterl, G., Feuchtinger, M.U.: Astron. Astrophys. 340, 419 (1998) Yi, S., Demarque, P., Kim, Y.-C.: Astrophys. J. 482, 677 (1997) Yi, S., Demarque, P., Kim, Y.-C., et al.: Astrophys. J. Suppl. Ser. 136, 417 (2001) Zahn, J.-P.: Astron. Astrophys. 252, 179 (1991)

The Geneva stellar evolution code P. Eggenberger · G. Meynet · A. Maeder · R. Hirschi · C. Charbonnel · S. Talon · S. Ekström

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9511-y © Springer Science+Business Media B.V. 2007

Abstract This paper presents the Geneva stellar evolution code with special emphasis on the modeling of solar-type stars. The basic input physics used in the Geneva code as well as the modeling of atomic diffusion is first discussed. The physical description of rotation is then presented. Finally, the modeling of magnetic instabilities and transport of angular momentum by internal gravity waves is briefly summarized. Keywords Stars: evolution

1 Introduction The Geneva evolution code was mainly developed and used for the computation of models of massive stars. These efP. Eggenberger () · G. Meynet · A. Maeder · C. Charbonnel · S. Ekström Observatoire de Genève, Université de Genève, 51 Ch. des Maillettes, 1290 Sauverny, Switzerland e-mail: [email protected] Present address: P. Eggenberger Institut d’Astrophysique et de Géophysique, l’Université de Liège, Allée du 6 Août 17, 4000 Liège, Belgium R. Hirschi Department of Physics and Astronomy, University of Basel, Klingelbergstr. 82, 4052 Basel, Switzerland C. Charbonnel Laboratoire d’Astrophysique de l’OMP, CNRS UMR 5572, 31400 Toulouse, France S. Talon Departement de Physique, Université de Montréal, Montréal PQ H3C 3J7, Canada

forts resulted in the publication of grids of stellar models computed for a wide range of masses and metallicities which have been extensively used by the astronomical community (Maeder and Meynet 1987; Schaller et al. 1992; Schaerer et al. 1993; Charbonnel et al. 1996; Meynet et al. 1994; Mowlavi et al. 1998). Recently, rotational effects have been included in the Geneva code (see Meynet and Maeder 1997, and other papers of the series). These rotating models are found to successfully reproduce many observational features of massive stars (see for instance Maeder and Meynet 2004a). The Geneva evolution code was also used for the computation of stars at very different evolutionary stages, from pre-sequence evolutionary models including the effects of accretion (Behrend and Maeder 2001) to pre-supernova evolution of rotating massive stars (Hirschi et al. 2004). We also mention the study of rotating stars at very low metallicity (see e.g. Meynet et al. 2006; Ekström et al. 2006). Concerning low mass stars, specific grids of models have been computed with the Geneva code (see Charbonnel et al. 1996, 1999) as well as detailed models of solar-type stars for which p-mode frequencies have been observed (Eggenberger et al. 2004, 2005a, Carrier et al. 2005; Eggenberger and Carrier 2006). In this paper, the input physics introduced in the Geneva evolution code for the modeling of solar-type stars is discussed. Section 2 is dedicated to the description of the basic input physics. The modeling of atomic diffusion is discussed in Sect. 3. The physical description of rotation is presented in Sect. 4. The modeling of the Tayler–Spruit dynamo and the transport of angular momentum by internal gravity waves is summarized in Sect. 5, while the conclusion is given in Sect. 6.

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_5

43

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Astrophys Space Sci (2008) 316: 43–54

2 Basic input physics 2.1 Equations of stellar evolution Four equations describe the evolution of the structure of the star. Since rotational effects are included in the Geneva code, spherical symmetry is no longer valid and the effective gravity (sum of the centrifugal force and gravity) can no longer be derived from a potential. By assuming that there is a strong horizontal (along isobars) turbulence, the angular velocity is then constant on isobars (Zahn 1992). The case is referred to as shellular rotation and is described in (Meynet and Maeder 1997) (see also Sect. 4). In this scheme, the four structure equations are the following: • Hydrostatic equilibrium: GMP ∂P =− fP . ∂MP 4πrP4

• the nuclear reaction rates in order to evaluate nucl and ν (see Sect. 2.2); • the equation of state to determine ρ and the needed thermodynamic quantities (see Sect. 2.4); • the opacities to calculate ∇rad (see Sect. 2.3); • a treatment of convection to compute the convective flux (see Sect. 2.5). On top of that, the equations of the evolution of chemical elements abundances are to be followed. In the Geneva evolution code, these equations are calculated separately from the structure equations according to time splitting method. Moreover, when atomic diffusion is included in the computation of a stellar model, equations describing the variation of the chemical composition due to diffusion are needed (see Sect. 3).

(1) 2.2 Nuclear reactions

• Continuity equation: 2.2.1 Nuclear networks

∂rP 1 . = ∂MP 4πrP2 ρ¯

(2) For hydrogen burning the pp chains and the CNO tri-cycle are calculated in detail and the evolution of the main nuclear species is followed explicitly. For helium burning, we take into account the following reactions:

• Energy conservation: ∂LP = nucl − ν + grav ∂MP = nucl − ν − cP

∂T δ ∂P + . ∂t ρ ∂t

(3)

• Energy transport equation:   fT GMP ∂ ln T =− fP min ∇ad , ∇rad ∂MP fP 4πrP4

(4)

where ∇ = ∇ad = ∇ = ∇rad =

Pδ T ρc ¯ P

in convective zones,

κlP 3 16πacG mT 4

and

in radiative zones,

4πrP4 1 , GMP SP g −1   2 4πrP2 1 fT = . SP gg −1 

fP =

x is x average on an isobaric surface, x is x average in the volume separating two successive isobars and the index P refers to the isobar with a pressure equal to P , while other variables have their usual meaning (see Meynet and Maeder 1997). To solve these equations, the following physical ingredients are required:

• the 3α reaction, • 12 C(α, γ ) 16 O(α, γ ) 20 Ne(α, γ ) 24 Mg, • 13 C(α, n) 16 O, • 14 N(α, γ ) 18 F(β, ν) 18 O(α, γ ) 22 Ne(α, n) 25 Mg, • 17 O(α, n) 20 Ne, • 22 Ne(α, γ ) 26 Mg. The system of nuclear reactions and the abundances variations are then determined for 15 isotopes: H, 3 He, 4 He, 12 C, 13 C, 14 N, 15 N, 16 O, 17 O, 18 O, 20 Ne, 22 Ne, 24 Mg, 25 Mg, and 26 Mg. The values of the following isotopic ratios 3 He/He, C/13 C, 14 N/15 N, O/18 O, 18 O/17 O, 21 Ne/20 Ne, 22 Ne/20 Ne, 25 Mg/24 Mg, and 26 Mg/24 Mg used in the Geneva code are listed in Maeder (1983) who chose, when available, the ratios given by radioastronomical observations of the interstellar material. The code can also account for the Ne–Na and Mg–Al chains in H-burning regions (see e.g. Meynet et al. 1997) and for the neutron capture reactions during Heburning (see e.g. Meynet and Arnould 2000). Note that for the computation of massive PopIII stars, a proper treatment of H-burning has been set up (see Ekström et al. 2006, for more details). The Geneva code is also used to compute stellar models during the advanced stages of evolution (see Hirschi et al. 2004, for more details). The list of elements followed during C-, Ne-, O- and Si-burnings is then: α, 12 C, 16 O, 20 Ne, 24 Mg, 28 Si, 32 S, 36 Ar, 40 Ca, 44 Ti, 48 Cr, 52 Fe and 56 Ni.

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2.2.2 Reaction rates The main inputs of nuclear reaction networks are the thermonuclear reaction rates. In the Geneva code, the NACRE nuclear reaction rates (Angulo et al. 1999) are used. Note that the numerical tables of the NACRE compilation are used and not the analytic fits. Screening factors are included according to the prescription by Graboske et al. (1973). 2.2.3 Numerical scheme The chemical changes due to secular evolution are computed implicitly: we start with a model at time t n for which the correct structure S n (pressure profile, temperature profile, etc.) and chemical composition C n is known. In order to compute the same quantities at time t n+1 = t n + δt, an initial estimate of the chemical composition C0n+1 is made. The approximate solution for the structure is then improved during the first iteration (the Henyey method is used) leading to S1n+1 . These refined profiles of pressure and density are then used to compute a new chemical composition C1n+1 . The same scheme is iterated as long as necessary to obtain the desired accuracy. Since the nuclear reaction network is resolved at each iteration, this procedure guarantees a good consistency between the structure and chemical composition at each iteration. Details regarding the computation of the nuclear network and the associated abundance variations are given in (Maeder 1983). 2.3 Opacities The opacities are needed to calculate the energy transport by the radiative transfer and to determine the radiative gradient ∇rad . Two groups, the OPAL group (Iglesias and Rogers 1996) and the Opacity Project (OP) group (Seaton et al. 1994), obtain very similar results using a different approach for computing opacities. In the Geneva evolution code, we use the opacity tables from the OPAL group complemented at low temperatures with the molecular opacities of Alexander and Ferguson (1994). Note that opacity tables are available for the standard solar abundances of (Grevesse and Noels 1993) and (Grevesse and Sauval 1998) as well as for the new solar abundances of (Asplund et al. 2004).

45

is a general equation of state (see Schaller et al. 1992). Although this general equation of state is perfectly suited for models of massive stars, the computation of reliable models of solar-type stars required a more specific and realistic equation of state. Indeed, in the case of low mass stars, non ideal effects, such as Coulomb interactions become important. Two different equations of state are included in the Geneva evolution code for solar-type stars models: the MHD equation of state (Hummer and Mihalas 1988; Mihalas et al. 1988; Daeppen et al. 1988) and the OPAL equation of state (Rogers et al. 1996; Rogers and Nayfonov 2002). 2.5 Convection and overshooting A prescription for the energy transport by convection is required to calculate the temperature gradient in a convective zone. In the Geneva code, the stability of a given layer is determined by using the Schwarzschild criterion. In the outer convective zone, the standard mixing-length formalism for convection is used (Böhm-Vitense 1958). The basic idea of the mixing-length theory (MLT) is to express the parameters of the non local phenomena of convection in terms of local quantities. The value of the mixing length l, usually expressed as a dimensionless parameter α ≡ l/Hp (with Hp the pressure scale height), is a free parameter of this formalism. Concerning the stellar core, an overshoot of the convective core into the radiative zone on a distance dov ≡ αov min[Hp , rcore ] can be included in the computation. We also note that for red supergiants with luminosities brighter than Mbol > −8.5 (Mini  25 M ), the structure of the outer convective envelope is complex. The acoustic flux seems to be the dominant mode of energy transport. In that case, the treatment of convection in the Geneva code includes turbulent pressure, acoustic flux and a density scale height (see Maeder 1987). In the advanced stages of evolution, we also mention that convective diffusion replaces instantaneous convection from oxygen burning onwards because the mixing timescale becomes longer than the evolution timescale at that point. The numerical method used for this purpose is the method used for rotational diffusive mixing (Meynet et al. 2004). The mixing length theory is then used to derive the corresponding diffusion coefficient.

2.4 Equation of state 2.6 Mass loss The equation of state describes the relation between the three physical parameters (p, T , ρ). For a given chemical composition, it enables the determination of the third physical parameter from the two others and also the determination of the thermodynamic quantities needed for the computation of a stellar model (∇ad , δ, cp , Γ1 , . . . ). In the Geneva evolution code, the equation of state usually used

For low mass stars (M  7 M ) the main sequence evolution is calculated at constant mass. On the red giant branch and on the asymptotic giant branch, the mass loss becomes however non negligible and is taken into account by using the prescription by Reimers (1975): M˙ = 4 × 10−13 ηLR/M (in M yr−1 ) with η = 0.5 (see Maeder and Meynet 1989).

46

Astrophys Space Sci (2008) 316: 43–54

Mass loss plays a key role in the physics and evolution of massive stars (see e.g. Maeder et al. 2005). For these stars, the prescription of Vink et al. (2001) is adopted. Note that the expressions by de Jager et al. (1988) are used when the prescription by Vink et al. does not apply. Mass loss rates follow a scaling relation with the metallicity Z of the type   Z α ˙ ˙ (5) M(Z ) M(Z) = Z ˙ where M(Z) is the mass loss rate when the metallicity is ˙  ) is the mass loss rate for the soequal to Z and M(Z lar metallicity. In the metallicity range from 1/30 to 3.0 times solar, the value of α is between 0.5 and 0.8 according to stellar wind models (see Kudritzki et al. 1987; Leitherer et al. 1992; Vink et al. 2001). 2.6.1 Anisotropic stellar winds The Geneva evolution code also includes anisotropies of the mass loss by stellar winds (Maeder 2002). It is indeed interesting to recall that a rotating star has a non uniform surface brightness, and the polar regions are those which have the most powerful radiative flux. Thus one expects that the star will lose mass preferentially along the rotational axis. This is correct for hot stars, for which the dominant source of opacity is electron scattering. In that case the opacity only depends on the mass fraction of hydrogen and does not depends on other physical quantities such as temperature. Thus rotation induces anisotropies of the winds (Maeder and Desjacques 2001; Dwarkadas and Owocki 2002). The quantity of mass lost through radiatively driven stellar winds is enhanced by rotation. The ratio of the mass loss rate of a star with a surface angular velocity Ω to that of a non-rotating star, of the same initial mass, metallicity and lying at the same position in the HR diagram is given by (see Maeder and Meynet 2000): ˙ M(Ω) (1 − Γ )1/α−1  ˙ M(0) [1 − 49 (v/vcrit,1 )2 − Γ ]1/α−1

(6)

where Γ is the electron scattering opacity for a non-rotating star with the same mass and luminosity and α is a force multiplier (Lamers et al. 1995).

3 Atomic diffusion Atomic diffusion on H, He, C, N, O, Ne and Mg is included in the Geneva evolution code by using the routines developed for the Toulouse–Geneva version of the code (see for example Richard et al. 1996). The chemical changes due to diffusion are computed separately from the changes due to nuclear reactions. For the 15

isotopes included in the nuclear network (see Sect. 2.2), the abundance variations due to the nuclear reactions are first computed. The diffusion equation is then solved separately for all the isotopes except hydrogen. Diffusion due to concentration and thermal gradients is included, but the radiative acceleration is neglected as it is negligible for the structure of the low-mass stellar models with extended convective envelopes (Turcotte et al. 1998). The computation of atomic diffusion is based on the Boltzmann equation for dilute collision-dominated plasmas with the use of the Chapman–Enskog method (Chapman and Cowling 1970) to solve this equation. 3.1 The diffusion equations The diffusion equation describes the evolution of the number concentration c of a given element in the stellar interiors. Its general form in spherical coordinates can be written as:   1 ∂ ∂c ∂c = 2 r 2 ρD − r 2 ρcV − λρc (7) ρ ∂t ∂r r ∂r where D is the diffusion coefficient, V the atomic diffusion velocity and λ the nuclear reaction rate. As mentioned above, the chemical changes due to nuclear reactions are calculated separately from the changes due to diffusion. Thus, λ is only included in the diffusion equations for lithium and beryllium, which are treated separately from the nuclear network. In the formalism of the Chapman–Enskog method used in the Geneva code, the diffusion equations for the different isotopes are written in Lagrangian coordinates as:    2 ∂D1i ∂ci ∂ci  ∂ ci  = D1i + − V 1i 2 ∂t ∂mr ∂mr ∂mr    ∂V1i − + λi c i (8) ∂mr where, as before, ci is the number concentration of element i and λi is the nuclear reaction rate. The diffusion coefficient  is given by D1i  D1i = (4πρr 2 )2 (Dturb + D1i )

(9)

where Dturb corresponds to the effective macroscopic diffusion coefficient (see Sect. 4), while D1i represents the atomic diffusion coefficient of element i relative to hydrogen (element 1). V1i is then given by V1i = (4πρr 2 )V1i , with



V1i = −D1i

1 Zi Ai − − 2 2

+ D1i α1i ∇ ln T ,

(10) 

mH Gmr kT r 2



(11)

Astrophys Space Sci (2008) 316: 43–54

47

where α1i is the thermal diffusion coefficient, Ai the atomic mass number of element i, Zi the charge of element i, and mH is the atomic hydrogen mass.

where the screening length λ is taken as the larger of the Debye length  λD =

3.2 Computation of the diffusion coefficients The atomic diffusion coefficient D1i and the thermal diffusion coefficient α1i are computed with the formalism of Paquette et al. (1986). Using the Chapman–Enskog method, the atomic diffusion coefficient for collisions involving particles s and t can be expressed as:

4πe

kT  2

i

1/2 ni Zi2

or the average interionic distance λi = (3/4πni )1/3 . The value of the collision integrals are given by Paquette et al. in the form of high-accuracy analytic fits. These fits (ij ) are provided for the dimensionless collision integrals Fst defined as (ij )

(ij )

3E Dst = 2nm(1 − )

(12)

where n is the total particle number density and m is the sum of the masses of the particles. The terms E and  contain the (i,j ) values of the collision integrals Ωst according to (9–17) of Paquette et al. (1986). In the same way, the thermal diffusion coefficient αst is given by αst =

5C(cs Ss − ct St ) cs2 Qs + ct2 Qt + cs ct Qs t

(13)

where ci is the number concentration of particles of species i (i = s, t), while the terms C, Ss , St , Qs , Qt , and Qst contain (i,j ) the values of the collision integrals Ωst and are defined in (9–17) of Paquette et al. (1986). In order to compute the diffusion coefficients Dst and αst , one needs to determine the values of the collision integrals. (i,j ) The collision integrals Ωst are given by (Paquette et al. 1986):  (i,j )

Ωst

=

kT 2πmMs Mt

(1/2) 

∞

0

e−g g 2j +3 φst dg, 2

(i)

(14)

with (i) φst



∞

= 2π

(1 − cosi χst )b db

0

∞

b2 (rstmin )2

−



Vst (rstmin ) = 0. g 2 kT

(17)

In the formalism of Paquette et al., the collision integrals are computed for a static screened potential Vst (r) = Zs Zt e2

e−r/λ r

Ωst , st

(19)

with  st = π

Zs Zt e2 2kT

2 

kT 2πmMs Mt

1/2 .

(20)

The value of these integrals depend uniquely upon Zs , Zt , λ, and T . The independent variable for the fits to the dimensionless collision integrals is

(21) ψst = ln ln(1 + γst2 ) , where the dimensionless parameter γst is given by γst =

4kT λ . Zs Zt e2

(22)

Tables 1–8 of Paquette et al. contain the values of the coefficients of the analytic fits and are used to determine the values of the collision integrals. These values are then introduced in the expressions for the diffusion coefficients (see (12, 13)) as well as in (11) for the diffusion velocity. The diffusion equation is then solved by using the numerical method described below.

(15)

with Vst (r) the interaction potential and rstmin the distance of closest approach given by the solution of the equation: 1−

=

3.3 Numerical method and

 1/2 −1 b2 Vst (r) 2 b dr r 1 − 2 − 2 , (16) χst = π − 2 r g kT rstmin 

Fst

(18)

Two different numerical methods are included in the Geneva code to solve the diffusion equations: the Crank–Nicholson finite differences method and the implicit finite elements method. Models of solar-type stars are usually computed by using the implicit finite elements method (see Meynet et al. 2004, for a comparison between these different numerical methods). 3.3.1 Implicit finite elements method The detailed description of this method is presented in (Schatzman et al. 1981) (see the appendix by Glowinsky and Angrand). The basic idea of this method is to decompose the unknown function as a linear combination of well chosen independent functions.

48

The radiative zone of the star is divided in K shells, with the Lagrangian mass coordinate of the ith mesh point being mi (mi is the mass inside the sphere of radius ri ). We introduce K functions vi (mr ) defined by ⎧ m −m r i−1 ⎪ ⎪ ⎪ ⎪ m − m i i−1 ⎨ vi (mr ) = mr − mi+1 ⎪ ⎪ ⎪ mi − mi+1 ⎪ ⎩ 0

if mr ∈ [mi−1 , mi ],

if mr ∈ / [mi−1 , mi+1 ].

(23) (24)

 M2  M2 ∂c ∂c ∂vi ∂vi vi dm + dm − dm D V c ∂t ∂m ∂m ∂m M1 M1 M1  M2 ∂ ∂ + λcvi dm + (cMzc1 ) − (cMzc2 ) = 0 (25) ∂t ∂t M1 M2

where the boundary conditions at the edge of the convective zones M1 and M2 have been used (Mzc is the mass of the convective zone). D  and V  are the diffusion coefficient and velocity as defined in (9 and 10). Finally, the unknown quantity c is expressed as a lin ear combination of the functions vi : c = j Cj vj . Integrals in (25) can then be written as: M2

M1

cvi dm ∼ = =



M2 

M1

M2

M1

 ∂vi dm ∼ Cj = ∂m j

=





M2

V  vj

M1

∂vi dm ∂m

Pj i Cj .

(28)

∂ (Mj,i Cj ) + Nj,i Cj − Pj,i Cj ∂t ∂ ∂ + δ1j (C1 Mzc1 ) − δKj (CK Mzc2 ) = 0. ∂t ∂t

(29)

This linear system of equations is then solved by using the LAPACK routines. Note that the diffusion equation has to be solved simultaneously for all the considered elements. The order of magnitude of the time scales generally implies the computation of many iterations of the diffusion process for a single evolutionary time step.



D

3.4 Application to the solar case Using the input physics described above, a solar calibration was performed with the Geneva stellar evolution code. In order to reproduce the solar luminosity, radius and surface chemical composition of Grevesse and Noels (1993) at the age of the Sun (4.57 Gyr), we obtain an initial helium mass fraction Yi = 0.2735, an initial ratio between the mass fraction of heavy elements and hydrogen (Z/X)i = 0.0274 and a mixing-length parameter α = 1.7998. The surface helium mass fraction of this model is Ys = 0.2426, while the bottom of the convective zone is located at rbzc = 0.712 R . The comparison between the sound speed profile of this model and helioseismological measurements is shown in Fig. 1.

 Cj vj vi dm

j



M2

Cj

vj vi dm =

M1

j



M1

V c

Γ

where a is a scalar, v a vector and Γ the surface corresponding to the volume Ω, one obtains (see Appendix B of Talon 1997 for more details):



M2

The values of these integrals can then be calculated for every i and j with 1 ≤ i ≤ K and 1 ≤ j ≤ K (see (B7) of Talon 1997). Using (26–28), the diffusion equation (25) is finally expressed as:

if mr ∈ [mi , mi+1 ],

div(av) = adiv(v) + grad(a) · v,   div(v)dV = v · dS



Astrophys Space Sci (2008) 316: 43–54

j

The function vi is equal to one at mr = mi , is equal to zero at mi+1 and mi−1 and varies linearly as a function of mr inbetween. By multiplying the diffusion equation (8) for a given chemical element by each of the functions vi (mr ), one obtains K equations. Each of these K equations is then integrated over the volume Ω corresponding to the radiative zone of the star. By using the following general relations:

V



j

=

j

Mj i Cj ,

(26)

j

 ∂c ∂vi dm ∼ Cj = ∂m ∂m 





M2

M1

N j i Cj ,

D

∂vj ∂vi dm ∂m ∂m (27)

Fig. 1 Relative sound speed differences between helioseismological results and a standard solar model computed with the Geneva stellar evolution code

Astrophys Space Sci (2008) 316: 43–54

49

4 Modeling of rotation In this section, we briefly summarize the basic physical ingredients of the numerical models of rotating stars. 4.1 Shellular rotation In the radiative interiors of rotating stars, meridional circulation is generated as a result of the thermal imbalance induced by the breaking of the spherical symmetry (Eddington 1925; Vogt 1926). This large scale circulation transports matter and angular momentum. As a result, differential rotation takes place in the radiative zones making the stellar interior highly turbulent. The turbulence is very anisotropic, with a much stronger geostrophic-like transport in the horizontal than in the vertical direction (Zahn 1992). The horizontal turbulent coupling favours an essentially constant angular velocity Ω on the isobars. In the context of this hypothesis of shellular rotation, every quantity depends solely on pressure and can be split into a mean value and its latitudinal perturbation f (P , θ ) = f (P ) + f˜(P )P2 (cos θ )

(30)

where P2 (cos θ ) is the second Legendre polynomial. 4.2 Transport of angular momentum For shellular rotation, the transport of angular momentum obeys an advection–diffusion equation written in Lagrangian coordinates (Zahn 1992; Maeder and Zahn 1998): ρ

 1 ∂  4 d 2 (r Ω)Mr = 2 ρr ΩU (r) dt 5r ∂r   1 ∂ 4 ∂Ω ρDr , + 2 ∂r r ∂r

(31)

r being the radius, ρ the density and Ω(r) the mean angular velocity at level r. It is worthwhile to recall here that meridional circulation is treated as a truly advective process in the Geneva evolution code. The vertical component u(r, θ ) of the velocity of the meridional circulation at a distance r to the center and at a colatitude θ can be written u(r, θ ) = U (r)P2 (cos θ ).

(32)

Only the radial term U (r) appears in (31); its expression is given below in (35). The quantity D is the total diffusion coefficient representing the various instabilities which transport the angular momentum: convection, semiconvection and shear turbulence. In convective regions, a very large diffusion coefficient implies a rotation law which is not far from solid body rotation. In radiative zones, we take D = Dshear , since we consider shear mixing and meridional

circulation as extra-convective mixing. The expression of the coefficient Dshear is given below in (36). The full solution of 31 taking into account U (r) and D gives the non-stationary solution of the problem. The expression of U (r) (35) involves derivatives up to the third order; (31) is thus of the fourth order and implies four boundary conditions. The first boundary conditions impose momentum conservation at convective boundaries (Talon et al. 1997)    R ∂ 1 r 4 ρ dr = − r 4 ρΩU + FΩ for r = rt , Ω ∂t 5 rt   rb  ∂ 1 Ω r 4 ρ dr = r 4 ρΩU for r = rb . ∂t 5 0 The other conditions are determined by requiring the absence of differential rotation at convective boundaries ∂Ω = 0 for r = rt , rb , ∂r

(33)

rt and rb correspond respectively to the top (surface) and bottom (center) of the radiative zone. When the star has no convective core, momentum conservation is then simply equivalent to U = 0. FΩ represents the torque applied at the surface of the star. For solar-type stars, this torque corresponds to the magnetic coupling at the stellar surface. Indeed, these stars are assumed to undergo magnetic braking while arriving on main sequence. In the Geneva code, we adopt the braking law of Kawaler (1988) corresponding to a field geometry intermediate between a dipolar and a radial field ⎧     R 1/2 M −1/2 ⎪ 3 ⎪ (Ω ≤ Ωsat ), ⎪ −KΩ R M dJ ⎨ =     ⎪ dt ⎪ R 1/2 M −1/2 ⎪ ⎩ −KΩΩ 2 sat (Ω > Ωsat ). R M The constant K is related to the magnitude of the magnetic field strength; it is usually calibrated on the Sun and taken to be a constant in all stars (see Bouvier et al. 1997, for instance). Ωsat expresses the fact that magnetic field generation saturates at some critical value (Saar 1996, and references therein). This saturation is required in order to retain a sufficient amount of fast rotators in young clusters, as originally suggested by Stauffer and Hartmann (1987). Ωsat is usually fixed to 14 Ω (see Bouvier et al. 1997). A scaling −1 of Ω in τconv sat (τconv is the global convective time-scale as defined by Kim and Demarque 1996) is also suggested for low mass stars (see Palacios et al. 2003, Sect. 3.3 and references therein). Such a scaling of the value of Ωsat can also be included in the Geneva code. Note that by neglecting the evolution of stellar structure and assuming solid body rotation, this braking law leads to the classical Skumanich law in t −1/2 for the surface velocity.

50

Astrophys Space Sci (2008) 316: 43–54

4.3 Transport of chemical elements

4.6 Horizontal turbulence

The vertical transport of chemicals through the combined action of vertical advection and strong horizontal diffusion can be described as a pure diffusive process (Chaboyer and Zahn 1992). The advective transport is then replaced by a diffusive term, with an effective diffusion coefficient Deff =

|rU (r)|2 30Dh

(34)

where Dh is the diffusion coefficient associated to horizontal turbulence (see Sect. 4.6). The vertical transport of chemical elements then obeys a diffusion equation which, in addition to this macroscopic transport, also accounts for (vertical) turbulent transport, nuclear reactions, and atomic diffusion (see Sect. 3).

The usual expression for the coefficient Dh is, according to Zahn (1992), Dh =

 1  r 2V (r) − αU (r) ch

(37)

where U (r) is the vertical component of the meridional circulation velocity, V (r) the horizontal component, ch a conr 2 Ω¯ stant of order unity and α = 12 d ln d ln r . By expressing the balance between the energy dissipated by the horizontal turbulence and the excess of energy present in the differential rotation on an equipotential that can be dissipated in a dynamical time, Maeder (2003) recently derived a new expression for the diffusion coefficient Dh :  1 ¯ [2V − αU ] 3 , Dh = Ar r Ω(r)V

(38)

with 

3 400nπ

1

4.4 Meridional circulation

A=

The velocity of the meridional circulation in the case of shellular rotation was initially derived by Zahn (1992). The effects of the vertical μ-gradient ∇μ and of the horizontal turbulence on meridional circulation were taken into account by Maeder and Zahn (1998). They found

Mathis et al. (2004) borrowed another prescription for the horizontal turbulence from torque measurements in the classical Couette–Taylor experiment. They find



 P L U (r) = (EΩ + Eμ ) . ρgCP T [∇ad − ∇ + (ϕ/δ)∇μ ] M (35) P is the pressure, CP the specific heat, EΩ and Eμ are terms depending on the Ω- and μ-distributions respectively, up to the third order derivatives and on various thermodynamic quantities (see Maeder and Zahn 1998, for more details). 4.5 Shear turbulence The diffusion by shear instabilities is expressed by a coefficient Dshear , namely Dshear =

4(K + Dh ) ϕ [ δ ∇μ (1 + DKh ) + (∇ad

− ∇rad )]     Hp α d ln Ω 2 fΩ − (∇  − ∇) × gδ 4 d ln r

(36)

where f is a numerical factor equal to 0.8836, K is the thermal diffusivity and (∇  − ∇) expresses the difference between the internal nonadiabatic gradient and the local gradient (Maeder and Meynet 2001).

 Dh =

β 10

1 2

3

.

 2

1 1 2 r|2V − αU | 2 , ¯ r Ω(r)

(39)

(40)

with β ∼ = 1.5 10−5 (Richard and Zahn 1999). These three expressions for the horizontal turbulence are included in the Geneva evolution code. Stellar models are usually computed by using the recent prescription by Maeder.

5 Magnetic fields and internal gravity waves Meridional and rotational turbulent diffusion are not able to account for all observed properties of solar-type stars and in particular for the rotation profile of the radiative interior of the Sun as deduced from helioseismic measurements. Indeed, helioseismological results indicate that the angular velocity Ω(r) is constant as a function of the radius r between about 20% and 70% of the total solar radius (Brown et al. 1989; Kosovichev et al. 1997; Couvidat et al. 2003), while meridional and rotational turbulent diffusion produce an insufficient internal coupling to ensure solid body rotation (Pinsonneault et al. 1989; Chaboyer et al. 1995). This suggests that other effects intervene. In the Geneva evolution code, the effects of the Tayler–Spruit dynamo (Spruit 2002; Maeder and Meynet 2004b) as well as the transport of angular momentum by internal gravity waves (Zahn et al. 1997; Talon et al. 2002; Talon and Charbonnel 2005; Charbonnel and Talon 2005) have been included.

Astrophys Space Sci (2008) 316: 43–54

51

By eliminating the expression of N 2 between (45) and (46), we obtain an expression for the magnetic diffusivity,

5.1 Tayler–Spruit dynamo In this section we briefly summarize the consistent system of equations for the Tayler–Spruit dynamo (Spruit 2002; Maeder and Meynet 2004b). The energy density uB of a magnetic field of intensity B per unit volume is uB =

B2 1 2 2 = ρr ωA 8π 2

with ωA =

B 1

(41)

(4πρ) 2 r

where ωA is the Alfvén frequency in a spherical geometry. If due to magnetic field or rotation, some unstable displacements of vertical amplitude l/2 occur around an average stable position, the restoring buoyancy force produces vertical oscillations around the equilibrium position with a frequency equal to the Brunt–Väisälä frequency N . The restoring oscillations will have an average density of kinetic energy uN  fN ρl 2 N 2 ,

2 1 2 ωA 2fN r N 2 .

If

fN = we have the condition for the vertical amplitude of the instability (Spruit 2002, (6)), 1 2,

ωA N

(43)

where r is the radius. This means that there is a maximum size of the vertical length l of a magnetic instability. In order to not be quickly damped by magnetic diffusivity, the vertical length scale of the instability must satisfy η ηΩ = 2 σB ωA

(44)

where Ω is the angular velocity and σB the characteristic growth-rate of the magnetic field. In a rotating star, 2 /Ω) due to the Coriolis force this growth-rate is σB = (ωA (Spruit 2002; Pitts and Tayler 1985). The combination of the limits given by (43) and (44) gives for the case of marginal stability,  4 ωA N2 η (45) = 2 2 . Ω Ω r Ω The equality of the amplification time of Tayler instability τa = N/(ωA Ωq) with the characteristic frequency σB of the magnetic field leads to the equation (Spruit 2002) Ω ωA =q Ω N

with q = −

(47)

Equations (45) and (46) form a coupled system relating the two unknown quantities η and ωA . The fact that the ratio η/K is very small allows us to bring these coupled equations to a system of degree 4 (Maeder and Meynet 2004b),  r 2Ω  2 r 2Ω 3 3 2 4 N x + N − x + 2Nμ2 x − 2Ω 2 q 2 = 0 (48) μ K q 2K T where x = (ωA /Ω)2 . The solution of this equation, which is easily obtained numerically, provides the Alfvén frequency and by (47) the thermal diffusivity. The azimuthal component of the magnetic field is much stronger that the radial one in the Tayler–Spruit dynamo. We have for these components (Spruit 2002) 1

stability. From this inequality, one obtains l 2


r 2 Ω  ωA 6 . Ω q2

(42)

where fN is a geometrical factor of the order of unity. If the magnetic field produces some instability with a vertical component, one must have uB > uN . Otherwise, the restoring force of gravity which acts at the dynamical timescale would immediately counteract the magnetic in-

l ∇ad at both cell edges, i.e., grid points. If the mixing is treated noninstantaneous (optionally), we treat it as a diffusive process with a typical convective velocity vc estimated from mixinglength theory. The cubic equation of MLT is solved with a Newton-iteration, since the analytical solution is numerically inaccurate for high superadiabaticity. Similarly, convective overshooting is considered to be a diffusive process. We use the description by Freytag et al. (1996), with a diffusion constant D(z) = D0 exp

−2z , f HP

(3)

where f is a free constant (f = 0.016 as the standard value) and HP the usual pressure scale height. z is the radial distance from the formal Schwarzschild border, and D0 sets the scale of diffusive speed and is derived from the MLTconvective velocity. Semiconvection has been implemented, again as a diffusive process, but not been tested nor used so far. 3.1.6 Diffusion Atomic diffusion is applied between two consecutive models, as are all other changes with time. The diffusion coefficients are calculated following the prescription in Thoul et al. (1994); elements considered are either hydrogen and

helium only, or any selection of metals in addition. In the latter case, the proper diffusive speed of each element is considered. No radiative levitation is taken into account. While in the standard treatment of diffusion, all elements are assumed to be fully ionized, an extension of the diffusion treatment (Schlattl 2002) is available which considers the actual ionization stage of each element (if this information is provided by the EoS). Furthermore, more accurate diffusion constants can be computed from improved collision integrals with additional quantum corrections (Schlattl and Salaris 2003). The system of second order differential equations is solved on the same spatial grid as the structure equations with the extended Henyey-solver. As the equations are fairly general, other diffusive effects like time-dependent convective mixing, overshooting, extra-mixing, etc., can be included in the same scheme. 3.2 Microphysics 3.2.1 Nuclear reactions The nuclear reaction rates are either from the NACRE cooperation (Angulo et al. 1999), or from the compilations by Fowler and coworkers (Caughlan et al. 1985; Caughlan and Fowler 1988). In all cases we use the analytical approximations provided. For hydrogen reactions in solar models the program also employs the recommended rates by Adelberger et al. (1998). The crucial reaction rate 12 C(α, γ )16 O is taken from (Kunz et al. 2002). Screening is treated in general in the weak limit, following Salpeter’s classical formula (Salpeter 1954). The nuclear reactions are followed by a small network that treats (as the default) either H-burning or He- and higher burning separately. The nuclei explicitly considered are p,3 He, 4 He, 12 C 13 C, 14 N, 15 N, 16 O, and 17 O, respectively p, n, 4 He, 12 C, 16 O, 20 Ne, 24 Mg, 28 Si, and 56 Ni. The treatment of burning phases beyond He-burning is therefore very rudimentary. If needed, the nuclear network can treat both H- and Heburning simultaneously. Since this is usually connected with very short timescales and violent convection, in this case convective mixing is automatically dealt with in the diffusive, non-instantaneous way. The network is solved by an implicit backward-differencing scheme of the linearized equations. Therefore special care has to be taken of the nuclear timesteps (see Sect. 2.2), which are controlled by keeping abundance changes at a level of 10% per step. For both the purpose of following the evolution of the chemical species and the nuclear energy production during the solution of the spatial problem we use the same nuclear network. However, for the nuclear energy production

Astrophys Space Sci (2008) 316: 99–106

103

the abundance changes are computed with the present abundances, temperature and density, i.e. in an explicit rather than in an implicit way. From the abundance changes the mass changes can be computed using the appropriate atomic masses and finally the energy production rate is obtained. This provides a snapshot of the energy production. A comparison of this value with the average energy output of the following time-step serves as an additional quality parameter for the chemical evolution, which can also be applied as a criterion to limit the evolutionary timestep. Alternatively, one could solve the full nuclear network by the implicit method for the present abundance changes. These are then converted to energy output by computing the net change of binding energies. For the nuclear energy production, the abundance changes are determined from the current composition in the model, using a nuclear timestep of 1% of the last evolutionary one. This method has been used when the nuclear network was designed, but is presently disabled. Both methods to compute the nuclear energy generation do not include energy losses by neutrinos; we add them separately using the numbers given in the sources for the reaction rates. We stress that the reduction in stellar mass due to nuclear reactions is taken into account in our code. 3.2.2 Neutrino losses In addition to neutrino losses during nuclear processes, which are taken care of in the nuclear network, neutrinos produced in the hot and dense plasma serve as an energy sink. We use the fitting formulae by Itoh and coworkers (Munakata et al. 1985), except for plasma neutrinos, where we prefer that by Haft et al. (1994). 3.2.3 Thermal energies A standard way of calculating the gravothermal energies εg in stellar evolution theory is to use the approximative formula by Kippenhahn and Weigert (1990), εg = −T

δ ∂P ∂T ∂s = −cP + . ∂t ∂t ρ ∂t

P ∂ρ ∂u + 2 , ∂t ρ ∂t

Our code uses tables of Rosseland mean opacities κ for mixtures quantified by the mass fractions of hydrogen and metals (both ranging from 0 to 1; the total number of tables is of order 80), and with a temperature and density grid of about 85 and 25 grid points. As the density coordinate we actually use the usual log R = log ρ − 3 log T + 18. The interpolation in this grid is done by the mentioned two-dimensional, bi-rational spline algorithm, and in mixture we use parabolic polynomials first in X (hydrogen) between the three tables closest to the actual value, and then in log Z (metallicity). In practice, and as long as the total metallicity is not changing, e.g. either by advanced nuclear burning phases or metal diffusion, tables for only three metallicity values are sufficient for the whole calculation. For evolutionary stages from core helium burning on, we have special core tables. The tables themselves are the end product of the merging of various data sources. We have four main regions in the T –R domain: • log T < 3.8 , for which we use the Wichita State Alexander & Ferguson molecular opacity tables (Alexander and Ferguson 1994; Ferguson et al. 2005). • 4.1 < log T < 8.7, for which OPAL tables are used (Iglesias and Rogers 1996). • High density: here we employ the results by Itoh et al. (1983) for electron conduction opacities. • log T > 8.7, for which no OPAL data are available; here we use the old Los Alamos Opacity Library (Huebner et al. 1977). In between these regions are transitions. Between 3.8 < log T < 4.1, where both Wichita State and OPAL data are available, we have a linear transition in log κ along with log T from one source to the other. The agreement between both tables is excellent and the transition is very smooth. At the high-density edge we add radiative (κr ) and conductive (κc ) opacities according to 1/κ = 1/κr + 1/κc .

(4)

However, this formula ignores changes in entropy and molecular weight due to composition changes (“mixing entropy”), which may contribute in regions of nuclear energy production or moving convective boundaries. Only in the latter this effect may be needed. The exact formula for εg implemented in the code is εg = −

3.2.4 Opacities

(5)

where u is the specific internal energy. For the higher precision of solar models we always use it instead of (4).

Since with increasing density electron conduction is dominating the transport of energy, the radiative contribution can be omitted once κr > κc . However, in particular at log T < 5 the radiative tables end before this situation is reached. In case the gap between the end of the radiative opacity table and the density from which on κc is already lower than the last radiative value available is only 1–2 dex, we boldly interpolate over this gap (cubic spline). If the gap is too large, the final κ-table has to end here. Should we run out of the table definition range during the stellar model calculations, we use the last table value. This happens at isolated points in some low-mass main-sequence envelopes and cannot be

104

Astrophys Space Sci (2008) 316: 99–106

avoided as long as the input tables do not cover the full ρ–T -plane. All our tables are constructed in this same way in a separate step before they are used by the stellar evolution code. In fact, the calculation of the spline coefficients is the last step in the preparation of the tables, such that the stellar evolution code reads only these coefficients and the appropriate parameter for the generalized spline function. 3.2.5 Equation of state The original EoS in the Kippenhahn-code included an ideal gas with radiation pressure and partial ionization of hydrogen and helium. For higher densities partial and full degeneracy was included according to analytical approximation formulae (Kippenhahn and Thomas 1964). These were improved to higher order accuracy by Wagenhuber (1996). The ionization of carbon was included in Weiss (1987) and the solver scheme for the sets of Saha-equation was changed to a Newton-type one. While this EoS is still implemented, it is only used in case that the more modern tabular equations of state do not cover the parameter range (in T and P , our EoS input variables). We preferentially use the OPAL-EoS (Rogers et al. 1996), where different generations of tables have been obtained from the website http://www-phys.llnl.gov/Research/ OPAL. We prefer the EOSPLUS set, i.e. the denser grid tables, which leads to less convergence problems. The latest tables (EOS_2001) are also available. Interpolation in all tables is done by a smoothed quadratic interpolation on a grid of 4 × 4 table values following the OPAL recommendations. Between mixtures we interpolate parabolically. The MHD EoS (Mihalas et al. 1988) is available, too, and was used mainly for solar model calculations, extending the OPAL EoS, when needed. We recently also implemented Irwin’s EoS (Cassisi et al. 2003)1 in the form of pre-calculated tables. In this case interpolation is done with the same generalized two-dimensional spline method discussed already in Sects. 2 and 3.2.4. Finally, an (unpublished) EoS (Weiss 1999) consisting of the merging of OPAL-tables, the Saumon–Chabrier EoS (Saumon et al. 1995), and that by Pols et al. (1995) has been developed; it is available for pure H/He-mixtures only due to the restriction of the Saumon–Chabrier tables. The choice between all these EoS table sets is done before compilation as a pre-compiler option.

4 The GARSOM Standard Solar Model The Garching Standard Solar Model (GARSOM) has been developed and published by Schlattl et al. (1997, 1999), 1 Available

from http://freeeos.sourceforge.net/

Fig. 1 Sound speed profiles (relative deviations from the seismic sound speed by Basu and Antia 1997) for the models GARSOM4 (solid), GARSOM41-2 (short-dashed), GARSOM5-1 (dash-dotted) and GARSOM5-3 (long-dashed) of Table 1

and was also used in Bahcall et al. (2005b) for comparison with the Bahcall & Pinsonneault models. In the Garching publications, the OPAL & Alexander opacities for the solar mixture of Grevesse and Noels (1993), the corresponding OPAL EoS, diffusion as in Sect. 3.1.6, and standard MLT were used. These choices also comprise the standard set of physics treatment in GARSTEC. For several solar models FST convection was employed, too. In addition, we sometimes use full 2d-hydro atmospheric models as in Schlattl et al. (1997) for the outer boundary condition. In the following we present several of these standard solar models. We have calculated them for the Grevesse and Sauval (1998) as well as for the Asplund et al. (2005) new solar composition. We use standard MLT and FTS convection theory, and employ OPAL and Irwin-EOS. H-, He-, and metal diffusion is always fully accounted for, but with different methods for calculating the diffusive speeds (see Sect. 3.1.6). It can easily be seen from the values in the table and the sound speed profile that the main influence comes from the (Z/X)s calibration, i.e. the total metallicity, while the distribution of metals (GN93, GS98, or AGS04) does not affect the solar structure. This is particularly evident from comparing GARSOM4 with GARSOM41-2; the two sound speed profiles (Fig. 1; solid and short-dashed lines) can be discriminated only for r/R < 0.35 and r/R > 0.85. The latter difference can be explained by the fact that GARSOM4 uses the 2d-hydro atmospheres, which leads to different values of αconv , too.

5 Summary We have introduced GARSTEC, the “Garching Stellar Evolution Code”, and sketched its main features and the treat-

Astrophys Space Sci (2008) 316: 99–106

105 opacity tables used. GARSOM41-1 and -2 models were calibrated to Z/X = 0.0245 (Grevesse and Noels 1993, GN93), although they use (composition and opacity tables) the GS98 metal ratios. The column “Atm.” indicates whether gray Eddington (EG), Krishna-Swamy (KS), or 2-d hydro atmospheres (2d) were used for the outer boundary conditions. Additionally, two reference models from Bahcall et al. (2005a) are added, which should be compared to GARSOM41-1 (BP04) and GARSOM5-3 (BS05)

Table 1 Global quantities of various GARSOM Standard Solar models. All models match the solar log L and log Teff (taken here as 5780 K) constraint to better than 10−4 , and that of the measured (Z/X)s to 1%. αconv is the free parameter of the convection model employed, Rbcz /R the depth of the convective zone. TBL stands for (H/He/metal-) diffusion a la Thoul et al. (1994), and QSchl if the improvements by Schlattl (Schlattl and Salaris 2003; Schlattl 2002) are included. The column “Comp.” gives the source for internal metal ratios in the model composition, while “Opac.” contains that of the Model

Comp.

Conv.

Opac.

EOS

Diff.

Atm.

Yi

Zi

αconv

Rbcz /R

Ys

(Z/X)s

GARSOM4

GN93

FST

GN93

OPAL

TBL

2d

0.2746

0.0199

0.975

0.7133

0.2448

0.02450

GARSOM41-1

GS98

MLT

GS98

OPAL

TBL

EG

0.2753

0.0200

1.741

0.7134

0.2453

0.02450

GARSOM41-2

GS98

FST

GS98

OPAL

TBL

EG

0.2754

0.0200

0.899

0.7133

0.2452

0.02456

GARSOM5-1

GS98

FST

GS98

Irwin

QSchl

2d

0.2695

0.0188

0.898

0.7151

0.2409

0.02300

GARSOM5-2

AGS04

FST

GS98

Irwin

QSchl

2d

0.2447

0.0141

0.840

0.7278

0.2165

0.01650

GARSOM5-3

AGS04

FST

AGS04

Irwin

QSchl

2d

0.2599

0.0139

0.912

0.7302

0.2297

0.01650

BP04

GS98

MLT

GS98

OPAL

TBL

KS

0.2734

0.0188

2.07

0.7147

0.243

0.0228

BS05

AGS04

MLT

AGS04

OPAL

TBL

KS

0.2614

0.0140

1.96

0.7289

0.230

0.0165

ment of all physical aspects. Although the code is a historically grown one, modified by generations of students and scientists at the Max-Planck-Institut für Astrophysik in Garching, it is nevertheless fully up-to-date. Its main advantages are the highly modular program structure, the fully implicit scheme for the spatial problem, and the possibility to solve nuclear burning and mixing processes simultaneously. It can be used for a large mass range and most phases of stellar evolution. Nevertheless, great efforts have been undertaken to ensure high accurateness, such that also the requirements imposed by seismology can be matched.

References Adelberger, E., Austin, S., Bahcall, J., Balantekin, A., Bogaert, G., Buchmann, L.: Solar fusion cross sections. Rev. Mod. Phys. 70, 1265–1291 (1998) Alexander, D.R., Ferguson, J.W.: Low-temperature Rosseland opacities. Astrophys. J. 437, 879–891 (1994) Angulo, C., Arnould, M., Rayet, M., et al.: A compilation of chargedparticle induced thermonuclear reaction rates. Nucl. Phys. A 656, 3–183 (1999) Asplund, M., Grevesse, N., Sauval, A.J.: The solar chemical composition. In: Barnes, T.G. III, Bash, F.N. (eds.) Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis. ASP Conf. Ser., vol. 336, pp. 25–38 (2005) Bahcall, J., Serenelli, A., Basu, S.: New solar opacities, abundances, helioseismology, and neutrino fluxes. Astrophys. J. 621, L85–L88 (2005a) Bahcall, J.N., Basu, S., Pinsonneault, M., Serenelli, A.M.: Helioseismological implications of recent solar abundance determinations. Astrophys. J. 618, 1049–1056 (2005b) Basu, S., Antia, H.M.: Seismic measurement of the depth of the solar convective zone. Mon. Not. Roy. Astron. Soc. 287, 198 (1997) Canuto, V., Mazzitelli, I.: Further improvements of a new model for turbulent convection in stars. Astrophys. J. 389, 724–730 (1992)

Cassisi, S., Salaris, M., Irwin, A.: The initial helium content of galactic globular cluster stars from the R-parameter: comparison with the cosmic microwave background constraint. Astrophys. J. 588, 862–870 (2003) Caughlan, G., Fowler, W.: Thermonuclear reaction rates V. At. Data Nucl. Data Tables 40, 283–334 (1988) Caughlan, G., Fowler, W., Harris, H., Zimmerman, B.: Tables of thermonuclear reaction rates for low-mass nuclei (1 ≤ z ≤ 14). At. Data Nucl. Data Tables 32, 197–233 (1985) Ferguson, J.W., Alexander, D.R., Allard, F., Barman, T., Bodnarik, J.G., Hauschildt, P.H., Heffner-Wong, A., Tamanai, A.: Lowtemperature opacities. Astrophys. J. 623, 585–596 (2005) Freytag, B., Ludwig, H.-G., Steffen, M.: Hydrodynamical models of stellar convection. Astron. Astrophys. 313, 497–516 (1996) Grevesse, N., Noels, A.: Atomic data and the spectrum of the solar photosphere. Phys. Scripta T 47, 133–138 (1993) Grevesse, N., Sauval, A.: Standard solar composition. Space Sci. Rev. 85, 161–174 (1998) Haft, M., Raffelt, G., Weiss, A.: Standard and nonstandard plasma neutrino emission revisited. Astrophys. J. 425, 222–230 (1994). Erratum: 438, 1017 (1995) Henyey, L.G., Forbes, J.E., Gould, N.L.: A new method of automatic computation of stellar evolution. Astrophys. J. 139, 306– 317 (1964) Henyey, L., Vardya, M.S., Bodenheimer, P.: Studies in stellar evolution. III. The calculation of model envelopes.. Astrophys. J. 142, 841–854 (1965) Huebner, W.F., Merts, L., Magee, N.H. Jr., Argo, M.F.: Astrophysical opacity library. Scientifical Report LA-6760-M, Los Alamos Scientific Laboratory (1977) Iglesias, C.A., Rogers, F.J.: Updated OPAL opacities. Astrophys. J. 464, 943–953 (1996) Itoh, N., Mitake, S., Iyetomi, H., Ichimaru, S.: Electrical and thermal conductivities of dense matter in the liquid metal phase. I—High-temperature results. Astrophys. J. 273, 774–782 (1983) Kippenhahn, R., Thomas, H.-C.: Integralapproximationen für die Zustandsgleichung eines entarteten Gases. Z. Astrophys. 60, 19–23 (1964) Kippenhahn, R., Weigert, A.: Stellar Structure and Evolution. A&A Library. Springer, Heidelberg (1990)

106 Kippenhahn, R., Weigert, A., Hofmeister, E.: Methods for calculating stellar evolution. Meth. Comp. Phys. 7, 129–190 (1967) Kunz, R., Fey, M., Jaeger, M., Mayer, A., Hammer, J.W., Staudt, G., Harissopulos, S., Paradellis, T.: Astrophysical reaction rate of 12 C(α, γ )16 O. Astrophys. J. 567, 643–650 (2002) Lucy, L.B.: Mass loss by cool carbon stars. Astrophys. J. 205, 482–491 (1976) Mihalas, D., Däppen, W., Hummer, D.G.: The equation of state for stellar evolution II. Astrophys. J. 331, 815–825 (1988) Munakata, H., Kohyama, Y., Itoh, N.: Neutrino energy loss in stellar interiors. Astrophys. J. 296, 197 (1985). Erratum: 304, 580 (1986) Pols, O.R., Tout, C.A., Eggleton, P.P., Han, Z.: Approximate input physics for stellar modelling. Mon. Not. Roy. Astron. Soc. 274, 964–974 (1995) Rogers, F.J., Swenson, F.J., Iglesias, C.A.: OPAL equation-of-state tables for astrophysical applications. Astrophys. J. 456, 902 (1996) Salpeter, E.E.: Electrons screening and thermonuclear reactions. Aust. J. Phys. 7, 373–388 (1954) Saumon, D., Chabrier, G., van Horn, H.M.: An equation of state for low-mass stars and giant planets. Astrophys. J. Suppl. Ser. 99, 713–741 (1995) Schlattl, H.: The Sun, a laboratory for neutrino- and astrophysics. Ph.D. thesis, Technical University München (1999) Schlattl, H.: Microscopic diffusion of partly ionized metals in the Sun and metal-poor stars. Astron. Astrophys. 395, 85–95 (2002) Schlattl, H., Salaris, M.: Quantum corrections to microscopic diffusion constants. Astron. Astrophys. 402, 29–35 (2003)

Astrophys Space Sci (2008) 316: 99–106 Schlattl, H., Weiss, A., Ludwig, H.-G.: A solar model with improved subatmospheric stratification. Astron. Astrophys. 322, 646–652 (1997) Spaeth, H.: Spline-Algorithmen zur Konstruktion glatter Kurven und Flächen. Oldenburg, München (1973) Swamy, K.S.K.: Profiles of strong lines in K-dwarfs. Astrophys. J. 145, 174–194 (1966) Thomas, H.-C.: Sternentwicklung VIII. Der Helium-Flash bei einem Stern von 1.3 Sonnenmassen. Z. Astrophys. 67, 420–455 (1967) Thoul, A.A., Bahcall, J.N., Loeb, A.: Element diffusion in the solar interior. Astrophys. J. 421, 828–842 (1994) Wagenhuber, J.: Entwicklung von sternen verschiedener massen und metallizitäten auf dem asymptotischen riesenast und danach. Ph.D. thesis, Technical University Munich (1996) Wagenhuber, J., Weiss, A.: Numerical methods for AGB evolution. Astron. Astrophys. 286, 121–135 (1994) Weiss, A.: Evolutionary models for R CrB stars. Astron. Astrophys. 185, 165–177 (1987) Weiss, A.: The progenitor of SN 1987A—uncertain evolution of a 20 solar mass star. Astrophys. J. 339, 365–381 (1989) Weiss, A.: Calculation and application of low-mass stellar models. Habilitationsschrift. Universität München (1999) Weiss, A., Schlattl, H.: Age–luminosity relations for low-mass metalpoor stars. Astron. Astrophys. Suppl. Ser. 144, 487–499 (2000)

Seismic codes PULSE: a finite element code for solving adiabatic nonradial pulsation equations Pierre Brassard · Stéphane Charpinet

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9733-z © Springer Science+Business Media B.V. 2008

Abstract We briefly present the nonradial adiabatic pulsation code PULSE first developped for white dwarf asteroseismology and now used to compute adiabatic oscillation properties for various types of stellar objects. Numerical tests show that the code is able to provide the accuracy (for a given stellar model) required to deal with the precision in frequency expected from the COROT long runs. While the ultimate objective is to compare the output of various pulsation codes (see these proceedings), we already emphasize problems that need to be addressed concerning, in particular, the mesh resolution of the input stellar models and its impact on the accuracy at which frequencies can be computed. Keywords Methods: numerics · Stars: oscillations · Stars: asteroseismology

sequence to the late white dwarf stages. It is extensively used for evolved compact stars, which include the three classes of white dwarf pulsators (the GW Virginis, DBV and DAV stars) showing g-mode pulsations (e.g., Brassard et al. 1992b) and the two classes of Extreme Horizontal Branch or subdwarf B (sdB) pulsators (the EC14026 and “Betsy” stars) showing p- and g-modes (e.g., Charpinet et al. 2000, 2002a, 2002b). The code has also been used for adiabatic calculations in solar-like stars showing high order p-modes (e.g., Théado et al. 2005; Bazot et al. 2005). In this paper, we briefly describe the numerical method implemented in the code to solve accurately and efficiently the nonradial adiabatic linearized oscillation equations. We also present accuracy and stability tests for the code using two representative main sequence models constructed for the COROT/ESTA Task 2/Step 1 comparison exercise.

1 Introduction 2 The numerical method Stellar pulsation is a widely encountered phenomenon occuring in a large variety of stars at all stages of evolution. The pulsation code “PULSE” was originally designed and optimised to compute efficiently adiabatic properties of radial and nonradial oscillation modes in evolved, white dwarf stars. However, this code is of universal use and can deal with stellar models from the early phases on pre-main

PULSE is an updated version of the adiabatic code described in detail in Brassard et al. (1992a); the interested reader should refer to this paper for a complete description of the code. It implements a solver based on finite element methods instead of more widely used finite difference schemes. 2.1 Formulation of the adiabatic equations

P. Brassard Université de Montréal, Montreal, Canada e-mail: [email protected] S. Charpinet () Observatoire Midi-Pyrénées, 14 av. Edouard Belin, Toulouse, 31400, France e-mail: [email protected]

The adiabatic linearized equations for nonradial oscillations are solved in their dimensionless formulation using the variables introduced by Dziembowski (1971) ξr (r) y1 ≡ , r

1 y2 ≡ gr



 σ 2 ξh (r) P  + = , ρ g

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_12

(1)

107

108

Astrophys Space Sci (2008) 316: 107–112

y3 ≡

1  , gr

y4 ≡

d

1 g dr

(2)

where ξr , P  , and  denote the Lagrangian radial displacement and the Eulerian perturbations of the pressure and gravitational potential, respectively. The system of adiabatic equations then takes the form (V + 1)

dy1 = dx



   V ( + 1) V y2 −  − 1 y1 + − 1 1 C1 ω 2

+

(V + 1)

V y3 , 1

(3)

dy3 = (3 −  − U )y3 + y4 , (V + 1) dx

+ (2 − U − )y4

(4)

(5)

(6)

(7)

y3 − y4 = 0|centre ,

(8)

y2 − y3 − y1 = 0|surface ,

(9)

Uy1 + ( + 1)y3 + y4 = 0|surface .

(10)

Various quantities introduced in the above formulation are defined in terms of classical physical quantities as follows

4πr 3 ρ d ln Mr = , dr Mr  r 3  M  C1 = , R Mr

(15)

σ 2R3 GM

(11)

,

V=

N 2 d ln ρ 1 d ln P A=− = − g dr 1 dr ρg χT =− (∇ad − ∇ + B) . P χρ

ρgr d ln P = , dr P

(16)

when using quadratic elements: (17)

with 1 φ1 (ξ ) = ξ(ξ − 1), 2 φ2 (ξ ) = (1 + ξ )(1 − ξ ), 1 φ3 (ξ ) = ξ(ξ + 1); 2

(18) (19) (20)

when using cubic elements:

C1 ω y1 − y2 = 0|centre , 2

1 φ2 (ξ ) = (1 + ξ ); 2

y(j) = cj φ1 (ξ ) + dj φ2 (ξ ) + cj+1 φ3 (ξ )

with boundary conditions expressed as

U=

y(j) = cj φ1 (ξ ) + cj+1 φ2 (ξ )

1 φ1 (ξ ) = (1 − ξ ), 2

  dy4 UV UV y3 = −rAUy1 + (V + 1) y2 + ( + 1) − dx 1 1

ω2 =

The code solves the full set of Ordinary Differential Equations (ODE) and Boundary Value Problem (BVP) iteratively using the Galerkin Finite-Element Method (GFEM) of weighted residuals. The star radial stratification is divided into elements where the solution is approximated by either linear, quadratic, or cubic functions. In the j th element, when using linear elements:

with

dy2 = (C1 ω2 + rA)y1 + (3 −  − U − rA)y2 dx + rAy3 ,

x = ln(r/P ),

2.2 The Galerkin finite-element method

(12)

(13)

y(j) = cj φ1 (ξ ) + dj φ2 (ξ ) + dj φ3 (ξ ) + cj+1 φ4 (ξ )

(21)

with    1 9 1 +ξ − ξ (1 − ξ ), φ1 (ξ ) = − 16 3 3   1 27 − ξ (1 − ξ ), φ2 (ξ ) = (1 + ξ ) 16 3   1 27 + ξ (1 − ξ ), φ3 (ξ ) = (1 + ξ ) 16 3    1 1 9 +ξ −ξ . φ4 (ξ ) = − (1 + ξ ) 16 3 3

(22) (23) (24) (25)

In the above formulation, the isoparametric coordinate in the j th-element is defined as   x − xj ξ =2 − 1, ξ ∈ [−1, +1] (26) xj +1 − xj where x is the integration variable. The system is solved by minimising the so-called weighted residual

(14)

R(x; {cj (, dj , ej )}) = y − F (x)y.

(27)

Astrophys Space Sci (2008) 316: 107–112

109

This is done by orthogonalising this residual to each member of the {φj (x)} set 

+1

−1

R(x; {cj (, dj , ej )})φk (ξ )J (ξ )dξ = 0,

k = 1, 2(, 3, 4).

(28)

The code follows a two-step strategy. It first scans, using linear elements and an explicit scheme, the specified frequency range taking one of the boundary conditions as a discriminant. This step provides a first guess for the eigenfrequencies and eigenfunctions. In the second step, it converges iteratively, using the Newton-Raphson algorithm, each identified eigenmode. During this step, linear, quadratic or cubic elements are employed as specified by the user. When the code is used with linear elements, stable second-order convergence of the eigensolutions is achieved. With quadratic elements, stable fourth-order convergence is reached. With cubic elements, the code offer stable, sixth-order convergence rates. In practice, quadratic elements are most often used, as they offer the best compromise between accuracy of the solutions and computational efficiency.

3 Computed frequencies for Task 2/Step 1 models A foreseen application for the adiabatic pulsation code PULSE concerns the interpretation of seismic data obtained with the satellite COROT. The very long time baseline of the COROT long runs (150 days) will allow to measure pulsation mode frequencies at unprecedent accuracy (∼0.08 µHz). Consequently, independently of any consideration on the accuracy of the physics involved, theoretical frequencies computed from stellar models need to be stable and accurate numerically to the same level, at least. We have investigated the behavior of the code in the context of the pulsation code comparison exercise organised by the COROT/ESTA working group. This exercise is based on two test models representing the same star (i.e., having identical stellar parameters), one with 900 mesh points (M15X04N900) and one with 2000 mesh points (M15X04N2000). Frequency comparisons with other adiabatic codes are shown elsewhere (see the paper from A. Moya in these proceedings). In this paper, we evaluate the capacity of the code to provide accurate frequencies given a stellar model.

(2000 mesh points; but the results are similar with 900 mesh points) in the range 20–4000 µHz for modes with  = 0–3. In this model, the frequency range investigated goes from high-order g-modes (the 20–100 µHz region) to high-order p-modes (the 2500–4000 µHz region). A high-accuracy set of frequencies was created with 15,000 quadratic elements and used as a reference.1 Then, the frequencies were recomputed with different values for the number of elements and compared to the reference set. Figure 1 shows the results of these comparisons in four relevant frequency ranges. These comparisons clearly demonstrate that the code is capable of reaching, internaly, a numerical stability and accuracy appropriate for the high-requirements of the COROT long-runs. The average fluctuations in the computed frequencies can be made much lower than the precision of the COROT long runs in the whole frequency range considered (20–4000 µHz), provided that a sufficient number of quadratic elements is used for the computation (other control parameters of the code have negligible impact on the results). While the minimum number of elements to achieve this accuracy depends somewhat on the frequency range of interest (see Fig. 1), using 5,000 quadratic elements appears to be the minimum appropriate to all situations. We note, however, that the behavior of some particular modes can produce frequency variations that are significantly larger than the average fluctuations. This is illustrated by the upper curves in each panel of Fig. 1 that show the largest (as opposed to the average) variation encountered in this comparison exercise. This problem does not occur in all the frequency ranges considered but is mostly present in the 100– 500 µHz domain. This region corresponds to low-order gand p-modes. A close look to the frequencies indicates that only one or two modes, indeed, have this behavior. These may be due to irregularities in the input stellar model. In particular we point out that these modes penetrate deep inside the star, in particular where the chemical stratification between the H-burning core and the envelope builds up. In this region, physical quantities tend to change rapidly and a possible culprit for this behavior may be the lack of resolution, locally, of the stellar model. If the sharp structures at the edge of the core are not sufficiently well resolved, it can have a significant impact when a mode has a node that falls close to these sharp structures. This may induce significant frequency variations for modes particularly sensitive to such “trapping” effects.

3.1 Internal stability and accuracy In order to test the internal stability of the code and the numerical accuracy at which theoretical frequencies can be obtained given a representative main sequence stellar model, we computed the frequencies for the model M15X04N2000

1 Note that the notion of “element” is different from the notion of “mesh-point”. Elements are defined over several mesh-points depending on their degree: linear, quadratic, and cubic elements cover two, three, and four mesh-points, respectively (see Figs. 1, 2 and 3 of Brassard et al. 1992a, 1992b). Hence, specifying, for instance, N quadratic elements generates a grid of 2N mesh-points.

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Fig. 1 Frequency variations ( ν) as a function of the number of quadratic elements used to compute the eigenfrequencies of the model M15X04N2000. The comparison is done with a reference set of frequencies computed with a large number of quadratic elements (15,000). Each panel focuses on a specific range of frequency: 20– 100 µHz (upper left) corresponds to high-order g-modes (γ -Dor like pulsations), 100–500 µHz (upper right) corresponds to low-order gand p-modes (δ-Scuti type oscillations), 500–2500 µHz (lower left)

corresponds to mid-order p-modes (solar-like oscillations), and 2500– 4000 µHz (lower right) corresponds to high-order p-modes. In each panel, the typical accuracy on frequencies from observations with HARPS (usually 7-day runs), from the COROT short runs (20 days), and from the COROT long runs (150 days) are indicated as horizontal dashed lines. The two curves in each panel gives the average fluctuation (lower black curve) and the largest variation (upper red curve) in frequencies, respectively

3.2 Impact of stellar model mesh resolution

for the determination of accurate pulsation frequencies. For that purpose, the frequencies computed for the two models representative of the same star but having different resolutions of the mesh (900 and 2000 mesh points, respectively)

We present a second set of comparisons aimed at assessing the importance of the mesh resolution in input stellar models

Astrophys Space Sci (2008) 316: 107–112

111

Fig. 2 Frequency variations ( ν) as a function of the number of quadratic elements. The comparison is done between frequencies computed for models M15X04N2000 and M15X04N900, having 2000 and 900 mesh points respectively. Each panel focuses on a specific range of frequency: 20–100 µHz (upper left) corresponds to high-order g-modes (γ -Dor like pulsations), 100–500 µHz (upper right) corresponds to low-order g- and p-modes (δ-Scuti type oscillations), 500–2500 µHz (lower left) corresponds to mid-order

p-modes (solar-like oscillations), and 2500–4000 µHz (lower right) corresponds to high-order p-modes. In each panel, the typical accuracy on frequencies from observations with HARPS (usually 7-day runs), from the COROT short runs (20 days), and from the COROT long runs (150 days) are indicated as horizontal dashed lines. The two curves in each panel gives the average variation (lower black curve) and the largest variation (upper red curve) in frequencies

are compared, assuming various values for the number of quadratic elements used by the adiabatic pulsation code. Figure 2 illustrates, in the four frequency ranges considered previously, the results of these comparisons. Very

clearly, the average differences in frequencies (lower curves in each panel of Fig. 2 induced by changing the stellar model resolution from 900 to 2000 mesh points are, regardless of the number of elements used for the pulsation calcu-

112

lation, significantly larger than the accuracy that we seek to achieve. Notably, the largest variations are seen in the 100–500 µHz range corresponding to the low-order p- and g-modes. Such variations in the computed frequencies clearly indicate that the number of mesh points (900) is insufficient in the M15X04N900 model. At this stage, however, it becomes questionable whether 2000 mesh points in the stellar model are enough to bring numerical fluctuations below the required accuracy for the COROT long runs (less than ∼0.1 µHz). Hence, investigations with models offering higher mesh resolutions are needed to clarify this point.

4 Conclusion The tests presented in this paper confirms that the adiabatic pulsation code PULSE is able to produce sets of theoretical frequencies that have the appropriate internal accuracy to deal with the very high precision seismic data expected from the COROT long runs. Of course, at this stage, this statement does not include possible numerical biases that may exist and which the comparison exercise with other adiabatic pulsation codes may reveal. Beyond the pulsation code itself, we stress that a crucial problem that absolutely needs to be solved is linked to the spatial resolution of the input stellar models. The number of mesh points must be sufficient to ensure that numerical fluctuations in computed frequencies, due to the finite spatial resolution of the stellar models, are made lower than the accuracy at which the frequencies are

Astrophys Space Sci (2008) 316: 107–112

measured with the COROT long runs. The optimal number of mesh points needed to achieve this objective remains to be evaluated.

References Bazot, M., Vauclair, S., Bouchy, F., Santos, N.C.: Seismic analysis of the planet-hosting star μ Arae. Astron. Astrophys. 440, 615–621 (2005) Brassard, P., Pelletier, C., Fontaine, G., Wesemael, F.: Adiabatic properties of pulsating DA white dwarfs. III—a finite-element code for solving nonradial pulsation equations. Astrophys. J. Suppl. Ser. 80, 725–752 (1992a) Brassard, P., Fontaine, G., Wesemael, F., Tassoul, M.: Adiabatic properties of pulsating DA white dwarfs. IV—an extensive survey of the period structure of evolutionary models. Astrophys. J. Suppl. Ser. 81, 747–794 (1992b) Charpinet, S., Fontaine, G., Brassard, P., Dorman, B.: Adiabatic survey of subdwarf B star oscillations. I. Pulsation properties of a representative evolutionary model. Astrophys. J. Suppl. Ser. 131, 223–247 (2000) Charpinet, S., Fontaine, G., Brassard, P., Dorman, B.: Adiabatic survey of subdwarf B star oscillations. II. Effects of model parameters on pulsation modes. Astrophys. J. Suppl. Ser. 139, 487–537 (2002a) Charpinet, S., Fontaine, G., Brassard, P., Dorman, B.: Adiabatic survey of subdwarf B star oscillations. III. Effects of extreme horizontal branch stellar evolution on pulsation modes. Astrophys. J. Suppl. Ser. 140, 469–561 (2002b) Dziembowski, W.A.: Nonradial oscillations of evolved stars. I. Quasiadiabatic approximation. Acta Astron. 21, 289–306 (1971) Théado, S., Vauclair, S., Castro, M., Charpinet, S., Dolez, N.: Asteroseismic tests of element diffusion in solar type stars. Astron. Astrophys. 437, 553–560 (2005)

ADIPLS—the Aarhus adiabatic oscillation package Jørgen Christensen-Dalsgaard

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9689-z © Springer Science+Business Media B.V. 2007

Abstract Development of the Aarhus adiabatic pulsation code started around 1978. Although the main features have been stable for more than a decade, development of the code is continuing, concerning numerical properties and output. The code has been provided as a generally available package and has seen substantial use at a number of installations. Further development of the package, including bringing the documentation closer to being up to date, is planned as part of the HELAS Coordination Action. Keywords Stars: oscillations · Numerical methods · Asteroseismology

Fairly extensive documentation of the code, on which the present paper in part is based, is provided with the distribution package.1 Christensen-Dalsgaard and Berthomieu (1991) provided an extensive review of adiabatic stellar oscillations, emphasizing applications to helioseismology, and discussed many aspects and tests of the Aarhus package, whereas Christensen-Dalsgaard and Mullan (1994) carried out careful tests and comparisons of results on polytropic models; this includes extensive tables of frequencies which can be used for comparison with other codes.

2 Equations and numerical scheme 1 Introduction 2.1 Equilibrium model The goal of the development of the code was to have a simple and efficient tool for the computation of adiabatic oscillation frequencies and eigenfunctions for general stellar models, emphasizing also the accuracy of the results. Not surprisingly, given the long development period, the simplicity is now less evident. However, the code offers considerable flexibility in the choice of integration method as well as ability to determine all frequencies of a given model, in a given range of degree and frequency. The choice of variables describing the equilibrium model and oscillations was to a large extent inspired by Dziembowski (1971). As discussed in Sect. 2.1 the equilibrium model is defined in terms of a minimal set of dimensionless variables, as well as by mass and radius of the model. J. Christensen-Dalsgaard () Institut for Fysik og Astronomi, og Dansk AsteroSeismisk Center, Aarhus Universitet, Bygning 1520, 8000 Aarhus C, Denmark e-mail: [email protected]

The equilibrium model is defined in terms of the following dimensionless variables: x ≡ r/R, A1 ≡ q/x 3 , A2 = Vg ≡ −

where q = m/M, 1 d ln p Gmρ = , Γ1 d ln r Γ1 pr

(1)

A3 ≡ Γ1 , A4 = A ≡

1 d ln p d ln ρ − , Γ1 d ln r d ln r

A5 = U ≡

4πρr 3 . m

1 The package is available at http://astro.phys.au.dk/∼jcd/adipack.n

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_13

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Here r is distance to the centre, m is the mass interior to r, R is the photospheric radius of the model and M is its mass; also, G is the gravitational constant, p is pressure, ρ is density, and Γ1 = (∂ ln p/∂ ln ρ)ad , the derivative being at constant specific entropy. In addition, the model file defines M and R, as well as central pressure and density, in dimensional units, and scaled second derivatives of p and ρ at the centre (required from the expansions in the central boundary condition); finally, for models with vanishing surface pressure, assuming a polytropic relation between p and ρ in the near-surface region, the polytropic index is specified. The following relations between the variables defined here and more “physical” variables are often useful: p=

GM 2 x 2 A21 A5 , 4πR 4 A2 A3

GM 2 dp =− xA2 A5 , dr 4πR 5 1

M A1 A5 . ρ= 4πR 3

(2)

We may also express the characteristic frequencies for adiabatic oscillations in terms of these variables. Thus if N is the buoyancy frequency, Sl is the Lamb frequency at degree l and ωa is the acoustical cut-off frequency for an isothermal atmosphere, we have N2 =

GM ˆ 2 GM N = 3 A1 A4 , R3 R

(3)

Sl2 =

l(l + 1)c2 GM ˆ 2 GM l(l + 1)A1 = 3 Sl = 3 , A2 r2 R R

(4)

ωa2 =

c2 4Hp2

=

GM 2 1 GM ωˆ = A1 A2 A23 , R3 a 4 R3

(5)

where c is the adiabatic sound speed, and Hp = p/(gρ) is the pressure scale height, g being the gravitational acceleration. Finally it may be noted that the squared sound speed is given by c2 =

GM 2 GM 2 A1 cˆ = x . R R A2

(6)

These equations also define the dimensionless characteristic frequencies Nˆ , Sˆl and ωˆ a as well as the dimensionless sound speed c, ˆ which are often useful. 2.2 Formulation of the equations As is well known the displacement vector of nonradial (spheroidal) modes can be written in terms of polar coordinates (r, θ, φ) as   m ∂Yl m δr = Re ξr (r)Yl (θ, φ)ar + ξh (r) aθ ∂θ   1 ∂Ylm (7) aφ exp(−iωt) . + sin θ ∂φ

Here Ylm (θ, φ) = clm Plm (cos θ ) exp(imφ) is a spherical harmonic of degree l and azimuthal order m, θ being co-latitude and φ longitude; Plm (x) is an associated Legendre function, and clm is a suitable normalization constant. Also, ar , aθ , and aφ are unit vectors in the r, θ , and φ directions. Finally, t is time and ω is the angular frequency of the mode. Similarly, e.g., the Eulerian perturbation to pressure may be written2  p  (r, θ, φ, t) = Re p  (r)Ylm (θ, φ) exp(−iωt) . (8) As the oscillations are adiabatic (and only conservative boundary conditions are considered) ω is real, and the amplitude functions ξr (r), ξh (r), p  (r), etc., can be chosen to be real. The equations of adiabatic stellar oscillations, in the nonradial case, are expressed in terms of the following variables3 : ξr , R    l(l + 1) l(l + 1) p + Φ ξh , = y2 = x ρ R ω2 r 2

y1 =

Φ y3 = −x , gr   y3 2 d y4 = x . dx x

(9)

Here Φ  is the perturbation to the gravitational potential. Also, we introduce the dimensionless frequency σ by ω2 =

GM 2 σ , R3

(10)

corresponding to (3–5). These quantities satisfy the following equations:   Vg dy1 = (Vg − 2)y1 + 1 − y2 − Vg y3 , (11) x dx η dy2 = [l(l + 1) − ηA]y1 + (A − 1)y2 + ηAy3 , dx dy3 = y3 + y4 , x dx Vg dy4 = −AUy1 − U y2 + [l(l + 1) + U (A − 2) x dx η x

+ U Vg ]y3 + 2(1 − U )y4 .

(12) (13)

(14)

2I

do not here distinguish between the full perturbation and the radial amplitude function. 3 The

somewhat peculiar choice of y3 , y4 results from the earlier use of an unconventional sign convention for Φ  ; now, as usual, Φ  is defined such that the perturbed Poisson equation has the form ∇ 2 Φ  = 4πGρ  , where ρ  is the Eulerian density perturbation.

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115

Here η = l(l + 1)g/(ω2 r) = l(l + 1)A1 /σ 2 , and the notation is otherwise as defined in (1). In the Cowling (1941) approximation, where the perturbation to the gravitational potential is neglected, the terms in y3 are neglected in (11, 12) and (13, 14) are not used. The dependent variables yi in the nonradial case have been chosen in such a way that for l > 0 they all vary as x l−1 for x → 0. For large l a considerable (and fundamentally unnecessary) computational effort would be needed to represent this variation sufficiently accurately with, e.g., a finite difference technique, if these variables were to be used in the numerical integration. Instead I introduce a new set of dependent variables by yˆi = x −l+1 yi ,

i = 1, 2, 3, 4.

(15)

These variables are then O(1) in x near the centre. They are used in the region where the variation in the yi is dominated by the x l−1 behaviour, for x < xev , say, where xev is determined on the basis of the asymptotic properties of the solution. This transformation permits calculating modes of arbitrarily high degree in a complete model. For radial oscillations only y1 and y2 are used, where y1 is defined as above, and y2 =

p . ω2 R 2 ρ

(16)

Here the equations become dy1 σ 2x2 = (Vg − 2)y1 − Vg y2 , dx q  q dy2

= x − 2 2 (A − U ) y1 + Ay2 . x dx σ x

x

model is typically defined at a suitable point in the stellar atmosphere, with non-zero pressure and density. Here the simple condition of vanishing Lagrangian pressure perturbation is implemented and sometimes used. However, more commonly a condition between pressure perturbation and displacement is established by matching continuously to the solution in an isothermal atmosphere extending continuously from the uppermost point in the model.4 A very similar condition was presented by Unno et al. (1989). In addition, in the full nonradial case a condition is obtained from the continuous match of Φ  and its derivative to the vacuum solution outside the star. In full polytropic models, or other models with vanishing surface pressure, the surface is also a singular point. In this case a boundary condition at the outermost nonsingular point is obtained from a series expansion, assuming a near-surface polytropic behaviour (see ChristensenDalsgaard and Mullan 1994, for details). The code also has the option of considering truncated (e.g., envelope) models although at the moment only in the Cowling approximation or for radial oscillations. In this case the innermost boundary condition is typically the vanishing of the radial displacement ξr although other options are available. 2.4 Numerical scheme The numerical problem can be formulated generally as that of solving

(17)

dyi = aij (x)yj (x), dx I

(18)

The equations are solved on the interval [x1 , xs ] in x. Here, in the most common case involving a complete stellar model x1 = , where is a suitably small number such that the series expansion around x = 0 is sufficiently accurate; however, the code can also deal with envelope models with arbitrary x1 , typically imposing ξr = 0 at the bottom of the envelope. The outermost point is defined by xs = Rs /R where Rs is the surface radius, including the atmosphere; thus, typically, xs > 1. 2.3 Boundary conditions The centre of the star, r = 0, is obviously a singular point of the equations. As discussed, e.g., by Christensen-Dalsgaard et al. (1974) boundary conditions at this point are obtained from a series expansion, in the present code to second significant order. In the general case this defines two conditions at the innermost non-zero point in the model. For radial oscillations, or nonradial oscillations in the Cowling approximation, one condition is obtained. The surface in a realistic

for i = 1, . . . , I,

(19)

j =1

with the boundary conditions I

bij yj (x1 ) = 0,

for i = 1, . . . , I /2,

(20)

cij yj (xs ) = 0,

for i = 1, . . . , I /2.

(21)

j =1 I j =1

Here the order I of the system is 4 for the full nonradial case, and 2 for radial oscillations or nonradial oscillations in the Cowling approximation. This system only allows nontrivial solutions for selected values of σ 2 which is thus an eigenvalue of the problem. The programme permits solving these equations with two basically different techniques, each with some variants. The 4 Note that since the frequency, and other variables, are taken to be real this can only be applied for frequencies below the acoustical cut-off frequency in the isothermal extension.

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first is a shooting method, where solutions satisfying the boundary conditions are integrated separately from the inner and outer boundary, and the eigenvalue is found by matching these solutions at a suitable inner fitting point xf . The second technique is to solve the equations together with a normalization condition and all boundary conditions using a relaxation technique; the eigenvalue is then found by requiring continuity of one of the eigenfunctions at an interior matching point. For simplicity I do not distinguish between yˆi and yi (cf. Sect. 2.2) in this section. It is implicitly understood that the dependent variable (which is denoted yi ) is yˆi for x < xev and yi for x ≥ xev . The numerical treatment of the transition between yˆi and yi has required a little care in the coding. 2.5 The shooting method It is convenient here to distinguish between I = 2 and I = 4. For I = 2 the differential equations (19) have a unique (apart from normalization) solution yi(i) satisfying the inner bound(o) ary conditions (20), and a unique solution yi satisfying the outer boundary conditions (21). These are obtained by numerical integration of the equations. The final solution can (i) (o) then be represented as yj = C (i) yj = C (o) yj . The eigenvalue is obtained by requiring that the solutions agree at a suitable matching point x = xf , say. Thus (i)

(o)

(i)

(o)

C (i) y1 (xf ) = C (o) y1 (xf ),

(22)

C (i) y2 (xf ) = C (o) y2 (xf ).

These equations clearly have a non-trivial solution (C (i) , C (o) ) only when their determinant vanishes, i.e., when (i)

(o)

(i)

(o)

Δ = y1 (xf )y2 (xf ) − y2 (xf )y1 (xf ) = 0.

(23)

Equation (23) is therefore the eigenvalue equation. For I = 4 there are two linearly independent solutions satisfying the inner boundary conditions, and two linearly independent solutions satisfying the outer boundary condi(i,1) (i,2) tions. The former set {yi , yi } is chosen by setting (i,1) y1 (x1 ) = 1, (i,2)

y1

(x1 ) = 1,

(i,1) y3 (x1 ) = 0, (i,2)

y3

(o,1)

and the latter set {yi (o,1)

(xs ) = 1,

y3

(o,2)

(xs ) = 1,

y3

y1 y1

(x1 ) = 1, (o,2)

, yi

(24)

} is chosen by setting

(o,1)

(xs ) = 0,

(o,2)

(xs ) = 1.

(25)

The inner and outer boundary conditions are such that, given y1 and y3 , y2 and y4 may be calculated from them; thus (25) and (26) completely specify the solutions, which are

obtained by integrating from the inner or outer boundary. The final solution can then be represented as (i,1)

yj = C (i,1) yj

(i,2)

+ C (i,2) yj

(o,1)

= C (o,1) yj

(o,2)

+ C (o,2) yj

. (26)

At the fitting point xf continuity of the solution requires that (i,1)

C (i,1) yj

(i,2)

(xf ) + C (i,2) yj (o,1)

= C (o,1) yj

(xf ) (o,2)

(xf ) + C (o,2) yj

(xf ),

j = 1, 2, 3, 4. (27)

This set of equations only has a non-trivial solution if ⎧ (i,1) (i,2) (o,1) (o,2) ⎫ ⎪ y1,f y1,f y1,f y1,f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y (i,1) y (i,2) y (o,1) y (o,2) ⎪ ⎬ 2,f 2,f 2,f 2,f = 0, (28) Δ = det (i,1) (i,2) (o,1) (o,2) ⎪ y3,f y3,f y3,f y3,f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (i,1) (i,2) (o,1) (o,2) ⎪ ⎭ y4,f y4,f y4,f y4,f (i,1)

(i,1)

where, e.g., yj,f ≡ yj (xf ). Thus (28) is the eigenvalue equation in this case. Clearly Δ as defined in either (23) or (28) is a smooth function of σ 2 , and the eigenfrequencies are found as the zeros of this function. This is done in the programme using a standard secant technique. However, the programme also has the option for scanning through a given interval in σ 2 to look for changes of sign of Δ, possibly iterating for the eigenfrequency at each change of sign. Thus it is possible to search a given region of the spectrum completely automatically. The programme allows the use of two different techniques for solving the differential equations. One is the standard second-order centred difference technique, where the differential equations are replaced by the difference equations  yin+1 − yin 1 n n n+1 n+1 a , = y + a y ij j ij j x n+1 − x n 2 I

j =1

i = 1, . . . , I. (29)

Here I have introduced a mesh x1 = x 1 < x 2 < · · · < x N = xs in x, where N is the total number of mesh points; yin ≡ yi (x n ), and aijn ≡ aij (x n ). These equations allow the solution at x = x n+1 to be determined from the solution at x = xn. The second technique was proposed by Gabriel and Noels (1976); here on each mesh interval (x n , x n+1 ) we consider the equations dyi (n) = a¯ ijn yj (x), dx (n)

I

j =1

for i = 1, . . . , I,

(30)

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with constant coefficients, where a¯ ijn = 1/2(aijn + aijn+1 ). These equations may be solved analytically on the mesh intervals, and the complete solution is obtained by continuous matching at the mesh points. This technique clearly permits the computation of solutions varying arbitrarily rapidly, i.e., the calculation of modes of arbitrarily high order. On the other hand solving (30) involves finding the eigenvalues and eigenvectors of the coefficient matrix, and therefore becomes very complex and time consuming for higher-order systems. Thus in practice it has only been implemented for systems of order 2, i.e., radial oscillations or nonradial oscillations in the Cowling approximation. 2.6 The relaxation technique

is generally well-behaved, and the secant iteration may be used without problems. As implemented here the shooting technique is considerably faster than the relaxation technique, and so should be used whenever possible (notice that both techniques may use the difference equations (29) and so they are numerically equivalent, in regions of the spectrum where they both work). For second-order systems the shooting technique can probably always be used; the integrations of the inner and outer solutions should cause no problems, and the matching determinant in (23) is well-behaved. For fourth-order systems, however, this needs not be the case. For modes where the perturbation to the gravitational potential has little effect (i,1) (i,2) on the solution, the two solutions yj and yj , and sim(o,1)

If one of the boundary conditions is dropped, the difference equations, with the remaining boundary condition and a normalization condition, constitute a set of linear equations for the {yjn } which can be solved for any value of σ ; this set may be solved efficiently by forward elimination and backsubstitution (e.g. Baker et al. 1971), with a procedure very similar to the so-called Henyey technique (e.g., Henyey et al. 1959; see also Christensen-Dalsgaard 2007) used in stellar modelling. The eigenvalue is then found by requiring that the remaining boundary condition, which effectively takes the role of Δ(σ ), be satisfied. However, as both boundaries, at least in a complete model, are either singular or very nearly singular, the removal of one of the boundary conditions tends to produce solutions that are somewhat ill-behaved, in particular for modes of high degree. This in turn is reflected in the behaviour of Δ as a function of σ . This problem is avoided in a variant of the relaxation technique where the difference equations are solved separately for x ≤ xf and x ≥ xf , by introducing a double point xf− = x nf = x nf +1 = xf+ in the mesh. The solution is furthermore required to satisfy the boundary conditions (20) and (21), a suitable normalization condition (e.g. y1 (xs ) = 1), and continuity of all but one of the variables at x = xf , e.g., y1 (xf− ) = y1 (xf+ ), y3 (xf− ) = y3 (xf+ ),

(31)

y4 (xf− ) = y4 (xf+ ), (when I = 2 clearly only the first continuity condition is used). We then set Δ = y2 (xf− ) − y2 (xf+ ),

2.7 Improving the frequency precision To make full use of the increasingly accurate observed frequencies the computed frequencies should clearly at the very least match the observational accuracy, for a given model. Only in this way do the frequencies provide a faithful representation of the properties of the model, in comparisons with the observations. However, since the numerical errors in the computed frequencies are typically highly systematic, they may affect the asteroseismic inferences even if they are smaller than the random errors in the observations, and hence more stringent requirements should be imposed on the computations. Also, the fact that solar-like oscillations, and several other types of asteroseismically interesting modes, tend to be of high radial order complicates reaching the required precision. The numerical techniques discussed so far are generally of second order. This yields insufficient precision in the evaluation of the eigenfrequencies, unless a very dense mesh is used in the computation (see also Moya et al. 2007). The code may apply two techniques to improve the precision. One technique (cf. Christensen-Dalsgaard 1982) uses the fact that the frequency approximately satisfies a variational principle (Chandrasekhar 1964).5 The variational expression may formally be written as

(32)

and the eigenvalues are found as the zeros of Δ, regarded as a function of σ 2 . With this definition, Δ may have singularities with discontinuous sign changes that are not associated with an eigenvalue, and hence a little care is required in the search for eigenvalues. However, close to an eigenvalue Δ

(o,2)

and yj , are almost linearly ilarly the two solutions yj dependent, and so the matching determinant nearly vanishes for any value of σ 2 . This is therefore the situation where the relaxation technique may be used with advantage. This applies, in particular, to the calculation of modes of moderate and high degree which are essential to helioseismology.

2 ≡ Σ(ξ )2 = σ 2 = σvar

K(ξ ) , I(ξ )

(33)

5 The variational principle is exact, formally, when the surface Lagrangian pressure perturbation is set to zero, but not when the match to an isothermal atmosphere is used.

118

where K and I are integrals over the equilibrium model depending on the eigenfunction, here represented by ξ . The variational property implies that any error δξ in ξ induces an error in Σ 2 that is O(|δξ |2 ). Thus by substituting the computed eigenfunction into the variational expression a more precise determination of σ 2 should result. This has indeed been confirmed (Christensen-Dalsgaard 1982; Christensen-Dalsgaard and Berthomieu 1991; ChristensenDalsgaard and Mullan 1994). The second technique uses explicitly that the difference scheme (29), which is used by one version of the shooting technique, and the relaxation technique, is of second order. Consequently the truncation errors in the eigenfrequency and eigenfunction scale as N −2 . If σ (N/2) and σ (N ) are the eigenfrequencies obtained from solutions with N/2 and N meshpoints, the leading-order error term therefore cancels in    1 1 N . (34) σRi = 4σ (N) − σ 3 2 This procedure, known as Richardson extrapolation, was used by Shibahashi and Osaki (1981). It provides an estimate of the eigenfrequency that is substantially more accurate than σ (N), although of course at some added computational expense. Indeed, since the error in the representation (29) depends only on even powers of N −1 , the leading term of the error in σRi is O(N −4 ). Even with these techniques the precision of the computed frequencies may be inadequate if the mesh used in stellar-evolution calculations is used also for the computation of the oscillations. The number of meshpoints is typically relatively modest and the distribution may not reflect the requirement to resolve properly the eigenfunctions of the modes. Christensen-Dalsgaard and Berthomieu (1991) discussed techniques to redistribute the mesh in a way that takes into account the asymptotic behaviour of the eigenfunctions; a code to do so, based on four-point Lagrangian interpolation, is included in the ADIPLS distribution package. On the other hand, for computing low-order modes (as are typically relevant for, say, δ Scuti or β Cephei stars), the original mesh of the evolution calculation may be adequate. It is difficult to provide general recommendations concerning the required number of points or the need for redistribution, since this depends strongly on the types of modes and the properties of the stellar model. It is recommended to carry out experiments varying the number and distribution of points to obtain estimates of the intrinsic precision of the computation (e.g., Christensen-Dalsgaard and Berthomieu 1991; Christensen-Dalsgaard and Mullan 1994). In the latter case, considering simple polytropic models, it was found that 4801 points yielded a relative precision substantially better than 10−6 for high-order p-modes, when Richardson extrapolation was used.

Astrophys Space Sci (2008) 316: 113–120

In the discussion of the frequency calculation it is important to distinguish between precision and accuracy, the latter obviously referring to the extent to which the computed frequencies represent what might be considered the ‘true’ frequencies of the model. In particular, the manipulations required to derive (33) and to demonstrate its variational property depend on the equation of hydrostatic support being satisfied. If this is not the case, as might well happen in an insufficiently careful stellar model calculation, the value determined from the variational principle may be quite precise, in the sense of numerically stable, but still unacceptably far from the correct value. Indeed, a comparison between σvar and σRi provides some measure of the reliability of the computed frequencies (e.g. Christensen-Dalsgaard and Berthomieu 1991).

3 Computed quantities The programme finds the order of the mode according to the definition developed by Scuflaire (1974) and Osaki (1975), based on earlier work by Eckart (1960). Specifically, the order is defined by   dy1 n=− + n0 . sign y2 (35) dx xz1 >0

Here the sum is over the zeros {xz1 } in y1 (excluding the centre), and sign is the sign function, sign (z) = 1 if z > 0 and sign (z) = −1 if z < 0. For a complete model that includes the centre n0 = 1 for radial oscillations and n0 = 0 for nonradial oscillations. Thus the lowest-order radial oscillation has order n = 1. Although this is contrary to the commonly used convention of assigning order 0 to the fundamental radial oscillation, the convention used here is in fact the more reasonable, in the sense that it ensures that n is invariant under a continuous variation of l from 0 to 1. With this definition n > 0 for p modes, n = 0 for f modes, and n < 0 for g modes, at least in simple models. It has been found that this procedure has serious problems for dipolar modes in centrally condensed models (e.g., Lee 1985; Guenther 1991; Christensen-Dalsgaard and Mullan 1994). The eigenfunctions y1 are shifted such that nodes disappear or otherwise provide spurious results when (35) is used to determine the mode order. A procedure that does not suffer from this difficulty has recently been developed by Takata (2006b); I discuss it further in Sect. 4. A powerful measure of the characteristics of a mode is provided by the normalized inertia. The code calculates this as  Rs 2 2 2 r1 [ξr + l(l + 1)ξh ]ρr dr ˆ E= M[ξr (Rphot )2 + l(l + 1)ξh (Rphot )2 ]

Astrophys Space Sci (2008) 316: 113–120

=

119

 xs  2 2 x1 y1 + y2 / l(l + 1) qU dx/x

4π[y1 (xphot

)2

+ y2 (xphot

)2 / l(l

+ 1)]

(36)

.

(For radial modes the terms in y2 are not included.) Here r1 = Rx1 and Rs = Rxs are the distance of the innermost mesh point from the centre and the surface radius, respectively, and xphot = Rphot /R = 1 is the fractional photospheric radius. The normalization at the photosphere is to some extent arbitrary, of course, but reflects the fact that many radial-velocity observations use lines formed relatively deep in the atmosphere. A more common definition of the inertia is Mmode , E = 4π Eˆ = M

(37)

where Mmode is the so-called mode mass. The code has the option to output the eigenfunctions, in the form of {yj (x n )}. In addition (or instead) the displacement eigenfunctions can be output in a form indicating the region where the mode predominantly resides, in an energetical sense, as  z1 (x) =

4πr 3 ρ M

1/2

 y1 (x) =

4πr 3 ρ M

1/2 4πr 3 ρ y2 (x) M 1/2   ξh (r) 4πr 3 ρ = l(l + 1) M R

1 z2 (x) = √ l(l + 1)

1/2

ξr (r) , R



(38)

(for radial modes only z1 is found). These are defined in such a way that  xs 2 2 x1 [z1 + z2 ]dx/x . (39) Eˆ = 4π[y1 (xphot )2 + y2 (xphot )2 / l(l + 1)] The form provided by the zi is also convenient, e.g., for computing rotational splittings δωnlm = ωnlm − ωnl0 (e.g., Gough 1981), where ωnlm is the frequency of a mode of radial order n, degree l and azimuthal order m. For slow rotation the splittings are obtained from first-order perturbation analysis as  Rs  π Knlm (r, θ )Ω(r, θ )rdrdθ, (40) δωnlm = m 0

0

characterized by kernels Knlm , where in general the angular velocity Ω depends on both r and θ . The code has built in the option to compute kernels for first-order rotational splitting in the special case where Ω depends only on r.

4 Further developments Several revisions of the code have been implemented in preliminary form or are under development. A substantial im-

provement in the numerical solution of the oscillation equations, particularly for high-order modes, is the installation of a fourth-order integration scheme, based on the algorithm of Cash and Moore (1980). This is essentially operational but has so far not been carefully tested. Comparisons with the results of the variational expression and the use of Richardson extrapolation, of the same formal order, will be particularly interesting. As discussed by Moya et al. (2007) the use of p  (or, as here, ξh ) as one of the integration variables has the disadvantage that the quantity A enters into the oscillation equations. In models with a density discontinuity, such as results if the model has a growing convective core and diffusion is neglected, A has a delta-function singularity at the point of the discontinuity. In the ADIPLS calculations this is dealt with by replacing the discontinuity by a very steep and wellresolved slope. However, it would obviously be an advantage to avoid this problem altogether. This can be achieved by using instead the Lagrangian pressure perturbation δp as one of the variables. Implementing this option would be a relatively straightforward modification to the code and is under consideration. The proper classification of dipolar modes of low order in centrally condensed models has been a long-standing problem in the theory of stellar pulsations, as discussed in Sect. 3. Such a scheme must provide a unique order for each mode, such that the order is invariant under continuous changes of the equilibrium model, e.g., as a result of stellar evolution. As a major breakthrough, Takata in a series of papers has elucidated important properties of these modes and defined a new classification scheme satisfying this requirement (Takata 2005, 2006a, 2006b). A preliminary version of this scheme has been implemented and tested; however, the latest and most convenient form of the Takata classification still needs to be installed. A version of the code has been established which computes the first-order rotational splitting for a given rotation profile Ω(r), in addition to setting up the corresponding kernels. This is being extended by K. Burke, Sheffield, to cover also second-order effects of rotation, based on the formalism of Gough and Thompson (1990). An important motivation for this is the integration, discussed by Christensen-Dalsgaard (2007), of the pulsation calculation with the ASTEC evolution code to allow full calculation of oscillation frequencies for a model of specified parameters (mass, age, initial rotation rate, etc.) as the result of a single subroutine call. Acknowledgements I am very grateful to W. Dziembowski and D.O. Gough for illuminating discussions of the properties of stellar oscillations, and to A. Moya and M.J.P.F.G. Monteiro for organizing the comparisons of stellar oscillation and model calculations within the ESTA collaboration. I thank the referee for useful comments which, I hope, have helped improving the presentation. This project is being supported by the Danish Natural Science Research Council and by the

120 European Helio- and Asteroseismology Network (HELAS), a major international collaboration funded by the European Commission’s Sixth Framework Programme.

References Baker, N.H., Moore, D.W., Spiegel, E.A.: Aperiodic behaviour of a non-linear oscillator. Q. Mech. Appl. Math. 24, 391–422 (1971) Cash, J.R., Moore, D.R.: A high order method for the numerical solution of two-point boundary value problems. BIT 20, 44–52 (1980) Chandrasekhar, S.: A general variational principle governing the radial and the non-radial oscillations of gaseous masses. Astrophys. J. 139, 664–674 (1964) Christensen-Dalsgaard, J.: On solar models and their periods of oscillation. Mon. Not. R. Astron. Soc. 199, 735–761 (1982) Christensen-Dalsgaard, J.: ASTEC—the Aarhus stellar evolution code. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9675-5 Christensen-Dalsgaard, J., Berthomieu, G.: Theory of solar oscillations. In: Cox, A.N., Livingston, W.C., Matthews, M. (eds.) Solar Interior and Atmosphere, Space Science Series, pp. 401–478. University of Arizona Press, Tuscon (1991) Christensen-Dalsgaard, J., Mullan, D.J.: Accurate frequencies of polytropic models. Mon. Not. R. Astron. Soc. 270, 921–935 (1994) Christensen-Dalsgaard, J., Dilke, F.W.W., Gough, D.O.: The stability of a solar model to non-radial oscillations. Mon. Not. R. Astron. Soc. 169, 429–445 (1974) Cowling, T.G.: The non-radial oscillations of polytropic stars. Mon. Not. R. Astron. Soc. 101, 367–375 (1941) Dziembowski, W.: Nonradial oscillations of evolved stars. I. Quasiadiabatic approximation. Acta Astron. 21, 289–306 (1971) Eckart, C.: Hydrodynamics of Oceans and Atmospheres. Pergamon, Elmsford (1960) Gabriel, M., Noels, A.: Stability of a 30 M star towards g + modes of high spherical harmonic values. Astron. Astrophys. 53, 149–157 (1976)

Astrophys Space Sci (2008) 316: 113–120 Gough, D.O.: A new measure of the solar rotation. Mon. Not. R. Astron. Soc. 196, 731–745 (1981) Gough, D.O., Thompson, M.J.: The effect of rotation and a buried magnetic field on stellar oscillations. Mon. Not. R. Astron. Soc. 242, 25–55 (1990) Guenther, D.B.: The p-mode oscillation spectra of an evolving 1M

sun-like star. Astrophys. J. 375, 352–365 (1991) Henyey, L.G., Wilets, L., Böhm, K.H., LeLevier, R., Levee, R.D.: A method for automatic computation of stellar evolution. Astrophys. J. 129, 628–636 (1959) Lee, U.: Stability of the Delta Scuti stars against nonradial oscillations with low degree l. Publ. Astron. Soc. Jpn. 37, 279–291 (1985) Moya, A., Christensen-Dalsgaard, J., Charpinet, S., Lebreton, Y., Miglio, A., Montalbán, J., Monteiro, M.J.P.F.G., Provost, J., Roxburgh, I., Scuflaire, R., Suárez, J.C., Suran, M.: Inter-comparison of the g-, f- and p-modes calculated using different oscillation codes for a given stellar model. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9717-z Osaki, Y.: Nonradial oscillations of a 10 solar mass star in the mainsequence stage. Publ. Astron. Soc. Jpn. 27, 237–258 (1975) Scuflaire, R.: The non radial oscillations of condensed polytropes. Astron. Astrophys. 36, 107–111 (1974) Shibahashi, H., Osaki, Y.: Theoretical eigenfrequencies of solar oscillations of low harmonic degree  in five-minute range. Publ. Astron. Soc. Jpn. 33, 713–719 (1981) Takata, M.: Momentum conservation and model classification of the dipolar oscillations in stars. Publ. Astron. Soc. Jpn. 57, 375–389 (2005) Takata, M.: First integrals of adiabatic stellar oscillations. Publ. Astron. Soc. Jpn. 58, 759–775 (2006a) Takata, M.: Analysis of adiabatic dipolar oscillations of stars. Publ. Astron. Soc. Jpn. 58, 893–908 (2006b) Unno, W., Osaki, Y., Ando, H., Saio, H., Shibahashi, H.: Nonradial Oscillations of Stars, 2nd edn. University of Tokyo Press, Tokyo (1989)

Porto Oscillation Code (POSC) Mário J.P.F.G. Monteiro

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-008-9802-y © Springer Science+Business Media B.V. 2008

Abstract The Porto Oscillation Code (Posc) has been developed in 1995 and improved over the years, with the main goal of calculating linear adiabatic oscillations for models of solar-type stars. It has also been used to estimate the frequencies and eigenfunctions of stars from the pre-main sequence up to the sub-giant phase, having a mass between 0.8 and 4 solar masses. The code solves the linearised perturbation equations of adiabatic pulsations for an equilibrium model using a second order numerical integration method. The possibility of using Richardson extrapolation is implemented. Several options for the surface boundary condition can be used. In this work we briefly review the key ingredients of the calculations, namely the equations, the numerical scheme and the output. Keywords Stars: interiors · Stars: oscillations · Methods: numerical

1 Introduction The Porto Oscillation Code (Posc) was initially developed in 1995 to obtain the frequencies of solar models and envelopes. The first description of the code has been given in Monteiro (1996). M.J.P.F.G. Monteiro () Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal e-mail: [email protected] M.J.P.F.G. Monteiro Departamento de Matemática Aplicada da Faculdade de Ciências, Universidade do Porto, Porto, Portugal

The objective of this paper is to present a summary on how Posc calculates the frequencies of oscillations for stellar models. The paper starts with the basic linear equations describing the oscillations and how these are formulated to be solved numerically. The boundary conditions used and their implementation are also discussed as well as the accuracy of the calculations. We end by listing some of the output values provided by the code and some of the applications where the results of the code have been used.

2 Basic equations for linear perturbations Our objective here is to review the necessary equations for non-radial adiabatic oscillations of spherically symmetric non-rotating stars. By following the work by Unno et al. (1989) it is possible to start from the hydrodynamic equations (continuity, Poisson and conservation of momentum equations), in order to obtain a set of equations describing the radial dependence of the amplitude functions for small perturbations. These perturbations correspond to; P for pressure,  for gravitational potential, while ξ is the displacement. The solutions are writen as P (t,r,θ ,φ) = P0 (r) + P˜ (r) Ylm (θ ,φ) eiωt , ˜ (t,r,θ ,φ) = 0 (r) + (r) Ylm (θ ,φ) eiωt ,   ∂Y m ξ (r) ∂Ylm ξ (t,r,θ ,φ) = ξr (r), ξh (r) l , h eiωt , ∂θ sin θ ∂φ

(1)

where the equilibrium configuration of the stars is described by the functions; ρ0 (for density), P0 and 0 . Here t is time, ω the frequency for the oscilating solutions, (θ, φ) the horizontal variables while r is radial distance and Ylm (θ ,φ)

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_14

121

122

Astrophys Space Sci (2008) 316: 121–127

the spherical harmonics characterized by the integer numbers l (mode degree) and m (azimutal order with m = −l, .., 0, .., l). By considering an equation for adiabatic perturbations and after eliminating the dependence on the horizontal coordinates and time, the equations describing the radial amplitude of the small perturbations are obtained in the following form;     S2 Sl2 P˜ 1 2 d ˜ = 0, (r 2 ξr ) + l2  − 2 g0 −c0 1− 2 ρ0 r dr ω ω   ˜ 1 g0 d ˜ − (ω2 −N02 ) ξr − d = 0 , + P (2) ρ0 c02 dr dr   ˜ c02 ρ0 c02 N02 Sl2 d d  ˜ + ξr −  r2 = 0. P˜ + g0 4πG dr 4πGr 2 dr

 rg0 d log ρ0 r dy2 + = − (N02 −ω2 )y1 − dr g0 d log r c02    rg0 d log ρ0 rρ0 − 1 y2 − + + 4πG y3 , g0 d log r c02 (6)   4πGrρ0 dy3 y3 + y4 , = 1− r dr g0

r

r

4πGr 2 ρ0 c02 N02 4πGr 2 ρ0 dy4 =− y − y2 1 dr c02 g02 c02  4πGr 2 ρ0 4πGr 2 ρ0 c02 + l(l+1) − − y4 . y 3 rg0 c02 c02

These form the set of equations we need to solve to obtain the radial behaviour of linear adiabatic oscillations of spherically symmetric stars.

The equilibrium structure in these equations is also characterized by quantities as gravity g0 and sound speed c0 ; g0 = − c02 =

3 The equilibrium model

d0 1 dP0 =− , dr ρ0 dr

1,0 P0 ρ0



with 1 ≡

∂ log P ∂ log ρ



(3) ,

S

where the derivative has been calculated at fixed entropy S. There are also two characteristic frequencies; the Lamb frequency Sl and the buoyancy frequency N0 (also known as the Brunt-Väissälä frequency), corresponding to c2 Sl2 = l(l+1) 02 , r   1 d log P0 d log ρ0 N02 = g0 −

1,0 dr dr   g0 ρ 0 d log ρ0 . = −g0 +

1,0 P0 dr

(4)

If we consider the following dimensionless variables y1 = y2 =

ξr , r ω2 g

ξh =

1 rg





P˜ ˜ , − ρ

˜  y3 = , rg y4 =

˜ 1 d , g dr

the equations can be written as     rg0 dy1 rg0 Sl2 rg0 = r −3 y1 + 2 −1 y2 − 2 y3 , 2 2 dr ω c0 c0 c0

(5)

In order to describe the reference/equilibrium model we consider the following dimensionless functions of the equilibrium structure (as defined by Christensen-Dalsgaard 2008b) x≡

r , R

mr,0 R 3 , r3 M 1 d log P0 rg0 = 2 , a2 ≡ −

1,0 d log r c0

a1 ≡

a3 ≡ 1,0 , a4 ≡

r 2 1 d log P0 d log ρ0 − = N ,

1,0 d log r d log r g0 0

a5 ≡

4πr 3 ρ0 , mr,0

(7)

where M and R are respectively the total mass and radius of the star, while mr is the mass within a sphere of radius r. These 5 functions are the result of an evolution code, being necessary as the input of the oscillation code. The four first order differential equations for small amplitudes can now be written simply as   l(l+1) dy1 a1 − a2 y2 + a2 y3 , x = (a2 −3)y1 + dx σ2   2 σ dy2 = − a4 y1 + (1+a4 −a5 )y2 − a4 y3 , x dx a1 x

dy3 = (1−a5 )y3 + y4 , dx

(8)

Astrophys Space Sci (2008) 316: 121–127

x

123

dy4 = a4 a5 y1 + a2 a5 y2 + [l(l+1) − a2 a5 ] y3 dx − a5 y4 ,

These expressions determine the behaviour of the ai ’s near the centre as follows from the definitions (7); a1 ∼

where we have introduced the reduced frequency σ2 =

(9)

We may write these equations in a vectorial form by defining the matrix, with L2 = l(l+1), a2 −3

⎢ ⎢ σ2 A=⎢ ⎢ a1 −a4 ⎣0 a4 a5

a2 ∼ 0,

a3 ∼ 1c ,

a4 ∼ 0,

a5 ∼ 3, (13)

R3 ω2 . GM

⎡

4π ρ¯c , 3

L2 a −a 1 2 σ2 a4 −a5 +1 0 a2 a5

⎤ a2

0

−a4 1−a5 L2 −a2 a5

0 1 −a5

⎥ ⎥ ⎥. ⎥ ⎦

x (10)

d y  Ac · y , dx

(11)

(14)

with ⎡

The system of differential equations is then simply written as d y x = A · y, dx

where ρ¯c = R 3 ρc /M. If we now replace these in the definition (10) of the matrix A, the problem is reduced to a simple system of differential equations with constant coefficients. This is,

⎢ −3 ⎢ 3 ⎢ σ2 Ac = ⎢ ⎢ 4π ρ¯c ⎣0 0

4π ρ¯c L2 3 σ2 −2 0 0

⎤ 0

0 ⎥ ⎥ 0 0 ⎥ ⎥. ⎥ −2 1 ⎦ L2 −3

(15)

This has a general solution (non-zero) given by

where the vector y has the components (y1 , y2 , y3 , y4 ).

yj = x l−2

∞ 

Yij x 2i ,

j =1, 2, 3, 4,

(16)

and Y04 = lY03 .

(17)

i=0

4 Boundary conditions with To complete the required equations it is also necessary to define four boundary conditions. The solution is to be found by integrating the equations between the centre of the star (r = 0 or x = 0) and the top of the atmosphere (r ≥ R or x ≥ 1). So, in fact we shall be establishing two boundary conditions at x = 0 and other two at the surface. The result is an eigenvalue problem with solutions existing for discrete values of σ . These are the eigenvalues associated to the corresponding eigenfunctions, that must satisfy the boundary conditions. 4.1 At the centre Since we have four dependent variables, the interior boundary conditions correspond to fix the values for two of the dependent variables. The other two are then related to these. The boundary conditions have to guarantee that the solutions are regular in the singular point, x = 0, of the differential equations. So, we start by determining the limiting behaviour of the ai ’s when x → 0. For x ≡ r/R  1 we can write (the subscript “c” stands for the value at x = 0); ρ ∼ ρc ,

mr ∼

4πR 3 3

ρc x 3 ,

and P ∼ Pc .

(12)

Y01 =

4π ρ¯c l Y02 3 σ2

These are the two boundary conditions at the centre: from the values of Y02 and Y03 it is possible to determine Y01 and Y04 . 4.2 At the atmosphere Two more boundary conditions need to be imposed at the top of the atmosphere. In a similar fashion to what has been done for the centre, we now need to established what is the limiting behaviour for the a’s (the subscript “S” represents in the following the value at the top of the atmosphere, located at rS /R ≥ 1) for x → xS . There are different options for imposing a boundary condition at the top of the model (surface). The most commonly used one is to assume an isothermal atmosphere for which we have that, a1 ∼ 1,

a2 ∼ a2S ,

a3 ∼ 1S ,

a4 ∼ a2S ,

a5 ∼ 0. (18)

In such an isothermal atmosphere the density decreases exponentially with radius. This behaviour allows one to approximate the actual value of a5S by zero.

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Astrophys Space Sci (2008) 316: 121–127

The set of equations is now written as, x

d y  AS · y, dx

(19)

where the matrix A, has been approximate using the asymptotic values of ai (see 18): ⎡

a2S −3

⎢ ⎢ 2 AS = ⎢ σ −a4S ⎣0 0

L2 −a 2S σ2 1+a4S 0 0

⎤ a2S −a4S 1 L2

x

0

⎥ 0⎥ ⎥. 1⎦ 0

(20)

By redoing the analysis presented in the previous subsection, and using yj = x −l

∞ 

YS,ij x 2i ,

j = 1, 2, 3, 4 ,

(21)

i=0

it follows that (L2 /σ 2 − a2S )YS,02 + a2S YS,03 , a4S +4−γ 1/2   2 L 2 2 γ = (a2S −a4S −4) + 4(σ −a4S ) −a2S , σ2 YS,01 = 2

(22)

YS,04 = −(l+1) YS,03 . When using the first expression one must be careful since the actual value for γ 1/2 can be imaginary. If it happens the solution will have a propagating component at the boundary, which implies that the wave will be loosing energy at this boundary. This does not correspond to the type of solutions we are looking for (standing waves). Therefore we only consider eigenvalues that are real, corresponding to standing waves, i.e. solutions that are evanescent at the boundaries. Note that this imposes restrictions on the values the frequency σ can have for possible modes of oscillation. Solutions are only calculated for γ ≥ 0. Other options for surface boundary conditions are possible (and have been implemented in Posc). The simplest option is to impose full reflection at the top of the model. Such a condition is achieved by setting δP = 0 at x = xS , giving that YS,01 = YS,02 + YS,03 ,

to solve the set of equations (11) with the boundary conditions (17) and (22) (or one of the other alternatives). In this section we describe briefly how this is done. The actual expressions implemented in the code are extracted from the basic dimensionless system of 4 differential equations;

(23)

d y = A · y, dx

(24)

where the matrix A is given in (10). The functions ai are as listed in (7) and known on a mesh from the equilibrium model obtained from solving the stellar structure equations. We use the dimensionless frequency σ , related to the actual frequency of oscillation (ω). As discussed above, under the selected boundary conditions, solutions exist only for discrete values of ω = ωln . The mode order n is associated with the radial structure of the different eigenfunctions that exist for the same mode degree l (the code considers spherical stars, and so the solutions are independent of the azimutal order m). These values and the corresponding solution y are what we are trying to find. The method we use consists in, given a value of the degree l, to determine the values(s) of σ that give a continuous solution at some meeting point (defined below as xf ). This point is where we stop the integration up from the centre, and the integration down from the atmosphere. In other words we find the value(s) of σ that have a global solution satisfying all our four boundary conditions. So what we do in fact is to iterate in σ in order to find the values that give the zeros of a function measuring the fitting of outer and inner solutions at xf . 5.1 Numerical variables Due to numerical control of errors and precision of the calculation we redefine the variables for different regions of the model. We do so by estimating which regions of the star are evanescent for a given frequency. The two points used here to define these regions are xin and xout . These depend on the model and the values of ω and degree l, corresponding to the roots of the following equation, ω2 (ω2 −ωc2 ) − Sl2 (ω2 −N 2 ) = 0 .

(25)

The acoustic cutoff frequency ωc used here is determined by    d log ρ −1  .  where Hρ ≡  dr 

instead of the first expression in (22).

  c2 dHρ ωc ≡ 0 2 1−2 dr 4Hρ

5 Calculation of the solutions

For the inner (near the centre) evanescent region we redefine the variables according to

In order to calculated the eigenvalues (frequencies of oscillation) of a solar model Posc uses a simple numerical scheme

 yin =

x xin

(26)

2−l y .

(27)

Astrophys Space Sci (2008) 316: 121–127

Here, xin is the transition point separating this inner region from the zone where the default variables, as given in (24), are used. For the outer evanescent region (surface layers), above x = xout , we use instead   x l y . (28) yout = xout The equations are integrated from x = 0 to x = xin determining yin . From there to a fitting point xf (well within the oscillatory region) we calculate the solution using the equations for y. Note that the transition from one region to the other is quite natural considering our definitions yin and yout of y. On the other hand we integrate inward from x = xS to x = xout using instead the equations for yout . From there, down to xf we take again the equations for y. Resulting from these two integrations we have the two sets of values at x = xf which are then continuous (after normalization). Since the system of equations is linear, this is so if and only if the value of σ is an eigenvalue. At this point what we actually do is to iterate on σ to find the zeros of the fitting determinant at xf . 5.2 Method of integration The method implemented to solve numerically the equations considered above is a shooting method using a second-order differences representation of the equations. It consists in writing the differential equations relating the values at two mesh points, xn and xn+1 , as   y d y hn d (n) + (n+1) + O(h3n ) , (29) y(n+1) = y(n) + 2 dx dx where hn = xn+1 − xn and with y(n) ≡ y(xn ). In order to replace the derivatives we use the different sets of differential equations discussed above for the regions 0 ≤ xin ≤ xf ≤ xout ≤ xs . We also have to implement the boundary conditions. It is done by setting the values of y1 and y3 at the boundaries (centre and surface) and to calculate the values of y2 and y4 (at both Boundaries) from the relations constructed in the previous section. Both linearly independent solutions are found by setting the central/atmospheric values of y1 equal to one and y3 alternatively to one and to zero. The actual solution is a linear combination of these two (for the interior solution—up to xf , as well as for the external solution— down to xf ). Since we are using a shooting method, from the values of these two solutions (“in” and “out”) at xf , we construct the matching matrix whose determinant has to be zero if σ is an eigenvalue. So the task of finding an eigenvalue is reduced to finding the zero of the determinant for the fitting conditions at xf .

125

We maximize the efficiency of the search for the eigenvalues (zeros of the determinant) by using the fact that these values are separated approximately by   GM 1/2 2πσ (p-modes), σp2 ∼  R −1 R3 c dr 0 0 (30) 1/2  2πσ 3 GM 2 (g-modes). σg ∼   R N2 R3 l+ 12 0 r0 dr 5.3 Accuracy of the results The actual accuracy of the final values of the frequencies are determined by several aspects of the calculation. As it would be expected, the accuracy of the results (frequencies) depends on the accuracy of the equilibrium model being used. Here we do not address this issue, referring the reader to Monteiro et al. (2006) and Lebreton et al. (2008). But another aspect associated with the equilibrium model, and determining the precision of the calculated eigenvalues, is the mesh on which the equilibrium model is given. To minimise this effect, before calculating the frequencies we produce a re-meshing of the equilibrium model. The actual details of the new mesh depends on the type of model and oscillation modes being calculated. We use a receipt similar to the one discussed by Christensen-Dalsgaard and Berthomieu (1991). This allows us to minimize the errors caused by having too few points where the eigenfunctions are expected to vary more strongly. Other aspect determining the accuracy of the eigenvalues is of course the numerical method used to integrate the four differential equations discussed above. In our case we have a second order scheme for the integration of the system of differential equations. To this we have also added the use of reduced dependent variables in the regions where the amplitudes of the eigenfunctions would be otherwise very small. Further to this the code also uses an extrapolation to improve the accuracy of the determination of each frequency. It is known as Richardson Extrapolation. This uses the fact that our second order integration has an error which varies with the inverse of the squared number of mesh points. Using such a fact it can be written that the actual value of the eigenvalue is  2 α 1 N 2 2 2 σ = σ − σ  with α = , (31) α−1 N α−1 N N where σN is the result found for a mesh of N points and σN  for a mesh of N  points. The code uses, by default, N  ∼ N/2, giving α ∼ 4. This extrapolation requires extra work but improves significantly the accuracy of the numerical frequencies (see Moya et al. 2008). The behaviour of the frequencies with increasing number of mesh points is an internal checkpoint that allows one to

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When l = 0 (radial modes) an additional (+1) cross is considered. To a total negative counting of the crosses corresponds a g-mode while p-modes have positive counting results, with the number corresponding to the mode order. The solutions corresponding to f -mode eigenvalues have a total of zero counts. 6.2 Mode inertia and eigenfunctions The eigenfunctions are provided in different formats (several normalisations and/or combinations) depending on what is required. These are obtained from y and correspond to combinations of the functions, Fig. 1 Comparison of the frequencies obtained using an equilibrium model provided with a different number of mesh points. The reference is for a 10 k mesh. Only differences for frequencies with l = 0, 1, 2, 3 and 100 µHz ≤ ν ≤ 3500 µHz are shown

identify where the actual frequencies are no longer affected by the precision of the integration scheme. Such a comparison is shown in Fig. 1 where the frequencies obtained for a solar model in a mesh of 10 k points is compared with the frequencies obtained using the same model with 4 k and 8 k mesh points. We have also performed a detailed comparison with the results from ADIPLS (Christensen-Dalsgaard 2008a) to have an external check on the computation. Further comparisons of Posc with other codes has been performed recently by Moya et al. (2008). In general (when using a model in a mesh of ∼6 k points) P osc frequencies of oscillation for solar-type stars have an estimated numerical uncertainty below 0.001 µHz. This is below the current observational errors of solar frequencies. Similar values are obtained for oscillation modes of stars of different masses and ages if the mesh is adequately adapted (in terms of number of points and its distribution) to the eigenfunctions being calculated (g or p modes).

6 Output The main output of the code are the values of the frequencies of linear adiabatic oscillations of an equilibrium model of a star. The calculation to be done is defined by an interval in frequency (ωa ≤ ω ≤ ωb ) and in mode degree (la ≤ l ≤ lb ). 6.1 Mode classification The mode order of each eigenvalue is obtained using a method similar to the phase diagram as described by Unno et al. (1989). It consists in counting the number of times the solution crosses the line y1 ≡ 0 in the plane (y1 , y2 ). If the cross is clockwise it counts as (−1) otherwise as (+1).

ξr (r) , ξr (R)

ξh (r) , ξh (R)

P˜ (r) , P˜ (R)

˜ (r) . ˜ (R)

(32)

The equilibrium structure is also used to calculate different normalizations of the eigenfunctions. The code provides, in addition to the mode parameters and frequencies, the mode inertia as given by, R 4π 0 [ξr2 (r) + L ξh2 (r)]r 2 ρ dr Eln ≡ . (33) M ξr2 (R) + L ξh2 (R) 7 Conclusion This work provides a brief description of Posc—the Porto Oscillation Code. This code has been developed mainly for calculating linear adiabatic oscillations of stellar models for stars similar to the Sun (in mass). The code is written in Fortran 77 and is modular. It is prepared to accept input models in the AMDL1 format. Tools are also available to convert almost any available stellar model output to the required format to be used by Posc. The code has been applied to several cases, namely the Sun (Monteiro 1996; Monteiro et al. 1996; Monteiro and Thompson 2005) and other stars (Cunha et al. 2003; Fernandes and Monteiro 2003), including pre-main sequence models (Ruoppo et al. 2007). It has also been used to produce the frequencies of reference grids of stellar evolution models for asteroseismology (Marques et al. 2008). Acknowledgements I want to thank J. Christensen-Dalsgaard for all the data and documentation provided over the last 20 years that have allowed the author to implement and improve this code. This work was supported in part by the European Helio- and Asteroseismology Network (HELAS), a major international collaboration funded by the European Commission (FP6), as well as by FCT and POCI2010 (FEDER) through projects POCI/CTE-AST/57610/2004 and POCI/V.5/B0094/2005. 1 See the description of some file formats for stellar evolution models at http://www.astro.up.pt/corot/ntools/

Astrophys Space Sci (2008) 316: 121–127

References Christensen-Dalsgaard, J.: ASTEC—the Aarhus STellar Evolution Code. Astrophys. Space. Sci. (2008a). doi:10.1007/s10509-007-9675-5 Christensen-Dalsgaard, J.: ADIPLS—the Aarhus adiabatic oscillation package. Astrophys. Space. Sci. (2008b). doi:10.1007/s10509-007-9689-z Christensen-Dalsgaard, J., Berthomieu, G.: Theory of solar oscillations. In: Cox, A.N., Livingston, W.C., Matthews, M.S. (eds.) Solar Interior and Atmosphere, p. 401. University of Arizona Press, Tucson (1991) Cunha, M.S., Fernandes, J.M.M.B., Monteiro, M.J.P.F.G.: Seismic tests of the structure of rapidly oscillating Ap stars: HR1217. Mon. Not. R. Astron. Soc. 343, 831 (2003) Fernandes, J., Monteiro, M.J.P.F.G.: HR diagram and asteroseismic analysis of models for beta Hydri. Astron. Astrophys. 399, 243 (2003) Lebreton, Y., Monteiro, M.J.P.F.G., Montalbán, J., et al.: The CoRoT evolution and seismic tools activity. Astrophys. Space. Sci. (2008) this volume Marques, J.P., Monteiro, M.J.P.F.G., Fernandes, J.: Grids of stellar models and frequencies for asteorseismology (CESAM + POSC). Astrophys. Space. Sci. (2008). doi:10.1007/s10509-008-9786-7

127 Monteiro, M.J.P.F.G.: Seismology of the solar convection zone. Ph.D. Thesis, Queen Mary and Westfield College University of London (1996) Monteiro, M.J.P.F.G., Thompson, M.J.: Mon. Not. R. Astron. Soc. 361, 1187 (2005) Monteiro, M.J.P.F.G., Christensen-Dalsgaard, J., Thompson, M.J.: Astron. Astrophys. 307, 624 (1996) Monteiro, M.J.P.F.G., Lebreton, Y., Montalbán, J., et al.: Report on the CoRoT evolution and seismic tools activity. In: Fridlund, M., Baglin, A., Lochard, J., Conroy, L. (eds.) The CoRoT Mission. ESA SP-1306, p. 363. ESA Publication Division, Noordwijk (2006) Moya, A., Christensen-Dalsgaard, J., Charpinet, S., et al.: Intercomparison of the g-, f- and p-modes calculated using different oscillation codes for a given stellar model. Astrophys. Space. Sci. (2008). doi:10.1007/s10509-007-9717-z Ruoppo, A., Marconi, M., Marques, J.P., et al.: A theoretical approach for the interpretation of pulsating PMS intermediate-mass stars. Astron. Astrophys. 466, 261 (2007) Unno, W., Osaki, Y., Ando, H., Saio, H., Shibahashi, H.: Nonradial Oscillations of Stars, 2nd edn. University of Tokyo Press, Tokyo (1989)

Granada oscillation code (GraCo) A. Moya · R. Garrido

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9694-2 © Springer Science+Business Media B.V. 2007

Abstract Granada oscillation code (GraCo) is a software constructed to compute adiabatic and non-adiabatic oscillation eigenfunctions and eigenvalues. The adiabatic version gives the standard numerical resolution, and also the Richardson extrapolation, different sets of eigenfunctions, different outer mechanical boundary conditions or different integration variables. The non-adiabatic version can include the atmosphere-pulsation interaction. The code has been used for intensive studies of δ Scuti, γ Doradus, β Ceph., SdO and, SdB stars. The non adiabatic observables “phaselag” (the phase between the effective temperature variaeff tions and the radial displacement) and δT Teff (relative surface temperature variation) can help to the modal identification. These quantities together with the energy balance (“growth rate”) provide useful additional information to the adiabatic resolution (eigenfrequencies and eigenfunctions). Keywords Stars · Stellar oscillations · Numerical resolution PACS 97.10.Sj · 97.10.Cv · 97.90.+j

1 Introduction GraCo (Moya et al. 2004) is a software developed to solve non-radial adiabatic and non-adiabatic oscillation equations. It is written in fortran95 language. It can be used for models all over the HR diagram. GraCo is able to work with three A. Moya () · R. Garrido Instituto de Astrofísica de Andalucía (CSIC), Cno. Bajo de Huetor, 50, Granada, Spain e-mail: [email protected]

different sources of equilibrium models: CESAM (Morel 1997), Granada Code (Claret 1999) and JMSTAR (Lawlor and MacDonald 2006). The numerical technique used is the so called Henyey relaxation method as it is described in Unno et al. (1989) (Sect. 18.2). The simple representation of the system of differential equations in terms of second-order centred differences is adopted for the numerical resolution.

2 Adiabatic case The adiabatic system of differential equations is described in Unno et al. (1989) (p. 161). The code has the possibility of choosing between two sets of eigenfunctions. Both sets include the radial displacement (ξr ) and the Eulerian perturbation of the gravitational potential ( g1 φ  ). The sets differ in the use of the Lagrangian or the Eulerian variation of the pressure, and the addition or not of a function of the radial displacement (U ξr /r, with U = d ln m/d ln r, m the mass and r the radius), to the derivative of the Eulerian perturbation of the gravitational potential (dφ  /dr) (Vorontsov et al. 1976). As boundary conditions GraCo uses those prescribed in Unno et al. (1989) (p. 162 ff). The solutions must satisfy regularity conditions at the innermost mesh-point. As a first surface condition the continuity of φ  and its first derivative are imposed. As second surface condition, the mechanical one, the program offers two possibilities: (1) The Lagrangian variation of the pressure vanishes (δp = 0), or (2) makes use of the isothermal reflective wave boundary condition (see Unno et al. 1989, p. 163 ff). Another degree of freedom of GraCo is the variable of integration. The program can solve the system of differential equations as a function of the logarithm of the radius (ln r)

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Fig. 1 Frequency comparison of the different adiabatic resolutions given by GraCo (see text). Only radial modes are shown. The equilibrium model used is a 1.5M and Xc = 0.4 model, with 4042 mesh points

or the ratio between the radius and the pressure ( Pr ). The first one largely weights the inner regions and the second the outer ones. Depending on the physics to be tested, the user can choose the most convenient variable. In order to solve the eigenvalue problem, the outer boundary condition for the gravitational potential is removed. In this case, for each trial eigenfrequency we have a unique solution. But not all the solutions obtained with every trial eigenfrequency fulfill the removed boundary condition. The spectra of the system is the set of eigenfrequencies and eigenfunctions fulfilling this outer boundary condition. As the system of differential equations has been replaced by a system of centred difference equations of second order, the truncation error in the eigenfrequencies and the eigenfunctions are of the order N −2 , with N the number of mesh points of the equilibrium model. To obtain more accurate eigenfrequencies, the code can make use of the so called Richardson extrapolation (Shibahashi and Osaki 1981), a combination of the values obtained with N and N/2 mesh points canceling the leading order error.

Finally, GraCo can compute the first order rotational splitting through the Ledoux coefficient (Ledoux 1951). The eigenfrequencies for radial modes can be obtained in two ways: (1) Using the LAWE second order differential equation, or (2) setting  = 0 in the standard non-radial system of equations. Figure 1 shows the adiabatic eigenfrequency differences between all the possible aforementioned options, only radial modes are depicted. The equilibrium model used is the last of the step1 of Task 2 (Moya et al. 2007) (this volume), with 4042 mesh points, 1.5M and Xc = 0.4. As reference we have calculated eigenfrequencies with the following options: X = ( = 0, no Richardson, p  , δp = 0, ln r). For each comparison we have changed one single degree of freedom, remaining the rest unchanged. We show the differences obtained in the range [200,2500] µHz, that is, from the fundamental radial mode to a frequency slightly larger than the cutoff frequency (around 2250 µHz). Top panel presents the differences obtained when two outer mechanical boundary conditions are used. Reference line is δP = 0, and the

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131

Fig. 2 |δTeff /Teff | (top panel) and φ T (in degrees, bottom panel), as a function of the pulsation constant Q (in days) for different modes with spherical degrees l = 0, 1, 2, 3. A model of a 1.8M is studied with Xc = 0.44, a MLT parameter α = 1 and the CEFF equation of state. Results obtained “with” (+) and “without” atmosphere (×) in the non-adiabatic treatment are compared

comparison is with the use of the isothermal reflective wave outer boundary condition. The differences for large frequencies are of the order of units of µHz. These differences are similar to those obtained by other codes (J.C. Suárez, private communication), but a larger study of these differences is still needed. The use of higher order integration procedures, as the Richardson extrapolation, do not change significantly this differences. In bottom panel, the rest of the comparisons are depicted. The Richardson extrapolation gives differences of the order of tenths of µHz, the use of r/P as integration variable provides small differences always lower than 0.008 µHz. The LAWE differential equations and the use of the Lagrangian variation of the pressure as eigenfunction provide similar differences smaller than 0.05 µHz, but its profile is not constant. For a comprehensive study of these differences see (Moya et al. 2007) (this volume).

3 Non-adiabatic resolution Additional information for asteroseismology is provided by GraCo with the resolution of the non-adiabatic set of differ-

ential equations described in Unno et al. (1989) (p. 261 ff). In the non-adiabatic resolution the eigenfrequencies and the eigenfunctions are no longer real. This makes it possible to obtain the so called non-adiabatic observables: (1) The “phase-lag” (φ T ≡ φ(ξr ) − φ(T )) defined as the phase difference between the Lagrangian variation of the effective temperature and the radial displacement. (2) The relative variation of the Lagrangian variation of the effective tempereff ature ( δT Teff ). And (3) The energy balance of each mode measured by the “growth rate”, directly related with the imaginary part of the modal eigenfrequency. The code uses the adiabatic solutions obtained for a given mode as trial functions for the non-adiabatic relaxation procedure. The inner boundary conditions are those described in Unno et al. (1989) (p. 229). It is in the outer region where GraCo presents some complexity. The code can treat the photosphere as a boundary condition or introduce the atmosphere-pulsation interaction resolution described by Dupret et al. (2002). This interaction is described imposing the atmosphere to be in thermal equilibrium and the diffusion approximation for the radiative flux to be no longer

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Fig. 3 The top panel shows theoretical predictions for two specific Strömgren photometric bands ((b − y) and y) for a given theoretical model using three MLT α parameters in the fundamental radial mode regime (pulsation constant near 0.033 days). The 3rd overtone regime (pulsation constant near 0.017 days) is shown in the bottom panel. The usual observational errors for these quantities from the ground are: ±1◦ for the phase differences and ±0.01 for the amplitude ratio

valid. In this case two different sets of differential equations are solved, one for the stellar core and envelope and another for the atmosphere. A transition layer and outer boundary conditions for the atmosphere must be defined. Figure 2 shows the values of the non-adiabatic observables for a standard δ Scuti model of 1.8M . In this figure we can see how the values in the case labeled “with” (where the atmosphere-pulsation interaction is included) are clearly different from those labeled “without” (the photosphere treated as the outer boundary layer). This illustrates the importance of the inclusion of the atmosphere-pulsation interaction for a better description of the non-adiabatic observables. When the atmosphere is treated as a boundary condition, not all the heat exchanges here are correctly taken into account, therefore the phase-lags obtained are closer to the adiabatic prediction (180◦ ). On the other hand, the

system of equations modeling the atmosphere-pulsation interaction takes into account these non-adiabatic processes in the atmosphere, and the resulting phase-lags are much smaller than 180◦ . This atmosphere-pulsation interaction has not influence upon the growth rate, since it takes place in layers not relevant for the driving of the modes due to their very small density. All of the above-mentioned calculations have a direct influence on the phase difference–amplitude ratio diagrams used to discriminate oscillation modes. phase-lags, as well as relative variations in |δTeff /Teff | and δge /ge (also calculated in the non-adiabatic resolution) can be used to overcome the uncertainties in previous phase-ratio color diagrams. In Garrido et al. (1990) these discrimination diagrams were made using parametrized values for departures from adiabaticity and phase lags. The only remaining de-

Astrophys Space Sci (2008) 316: 129–133

gree of freedom is now the choice of the MLT α parameter in order to describe the convection. Therefore, discrimination diagrams depend only on this parameter, as is shown in Fig. 3 for the same equilibrium model used in Fig. 2. Theoretical predictions are plotted for two specific Strömgren photometric bands ((b − y) and y) using three different MLT α parameters in the fundamental radial mode regime (pulsation constant near 0.033 days) and in the 3rd overtone regime (near 0.017 days). A clear separation between the l-values exists for periods around the fundamental radial. Similar behaviour is found for other modes in the proximity of the 3rd radial overtone, although for these shorter periods some overlapping start to appear at the lowest l-values. They also show the same trend as for the fundamental radial mode: high amplitude ratios for low MLT α and the spherical harmonic l = 3. 4 Conclusions GraCo is a complete software for the resolution of systems of differential equations related with the stellar oscillations. The general integration scheme used in the code is the Henyey relaxation method as it is explained in Unno et al. (1989). In the adiabatic frame, different integration schemes can be used to obtain the eigenfrequencies and eigenfunctions: (1) Two outer mechanical boundary conditions, (2) two choices for the set of eigenfunctions, (3) two choices of the integration variables, (4) the use or not of the Richardson extrapolation and, (5) for radial modes the use of the LAWE second order differential equation or set  = 0 in the standard non-radial differential equations. On the other hand, the main characteristic of the code is that the non-adiabatic set of differential equations can also be solved. Two different treatments of the photosphere can be used. The first considers the atmosphere as a single boundary layer, and the second describes the atmospherepulsation interaction. This makes it possible to obtain more eff accurate non-adiabatic observables (phase-lag and δT Teff ) crucial for modal identification through the multicolor photometry. On the other hand, non-adiabatic studies also allow to

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study the modal energy balance, giving a theoretical range of overstable modes. Finally we want to remark that the analytical expressions of the differential equations, eigenfunctions and boundary conditions, and the numerical procedure followed in GraCo can be found in Unno et al. (1989). Acknowledgements This work was supported by the Spanish PNE number ESP 2004-03855-C03-C01.

References Claret, A.: Studies on stellar rotation. I. The theoretical apsidal motion for evolved rotating stars. Astron. Astrophys. 350, 56–62 (1999) Dupret, M.-A., De Ridder, J., Neuforge, C., Aerts, C., Scuflaire, R.: Influence of non-adiabatic temperature variations on line profile variations of slowly rotating beta Cep stars and SPBs. I. Nonadiabatic eigenfunctions in the atmosphere of a pulsating star. Astron. Astrophys. 385, 563–571 (2002) Garrido, R., Garcia-Lobo, E., Rodriguez, E.: Modal discrimination of pulsating stars by using Stromgren photometry. Astron. Astrophys. 234, 262–268 (1990) Lawlor, T.M., MacDonald, J.: The mass of helium in white dwarf stars and the formation and evolution of hydrogen-deficient post-AGB stars. Mon. Not. R. Astron. Soc. 371, 263–282 (2006) Ledoux, P.: The nonradial oscillations of gaseous stars and the problem of Beta Canis Majoris. Astrophys. J. 114, 373–384 (1951) Morel, P.: CESAM: A code for stellar evolution calculations. Astron. Astrophys. Suppl. Ser. 124, 597–614 (1997) Moya, A., Garrido, R., Dupret, M.A.: Non-adiabatic theoretical observables in δ Scuti stars. Astron. Astrophys. 414, 1081–1090 (2004) Moya, A. et al.: Inter-comparison of the g-, f- and p-modes calculated using different oscillation codes for a given stellar model. Astrophys. Space Sci. (2007), this volume Shibahashi, H., Osaki, Y.: Theoretical eigenfrequencies of solar oscillations of low harmonic degree L in 5-minute range. Publ. Astron. Soc. Jpn. 33, 713–719 (1981) Unno, W., Osaki, Y., Ando, H., Saio, H., Shibahashi, H.: Nonradial Oscillation of Stars. University of Tokyo Press, Tokyo (1989) Vorontsov, S.V., Zharkov, V.N., Lubimov, V.M.: The free oscillations of Jupiter and Saturn. Icarus 27, 109–118 (1976)

NOSC: Nice OScillations Code J. Provost

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9654-x © Springer Science+Business Media B.V. 2007

Abstract A short presentation of the Nice OScillations Code (NOSC) and of its general properties is given here. We described the physics and the various numerical tools, that we have developed to check the validity and the internal consistency of the frequency calculations for a given model. We present different examples of adiabatic calculations, with an estimation of the achieved internal consistency and numerical accuracy.

burned half its initial hydrogen, computed by J. ChristensenDalsgaard for ESTA/Step1b comparisons, called “M4k” (see Moya et al., this book). Section 2 describes how nonradial oscillations are computed by NOSC. The numerical techniques are shortly presented (Sect. 3). In Sect. 4 are given the outputs of the code and the additional tools, which are specific of NOSC, with some examples of application.

Keywords Stars: oscillations · Stars: evolution · Sun: oscillations · Sun: evolution

2 Nonradial oscillations

PACS 97.10.Sj · 97.10.Cv · 96.60.Ly · 96.60.Jw

1 Introduction

Our code compute linear adiabatic oscillations, with a nonadiabatic option with frozen convective flux. The system governing the oscillations can be written after a Fourier transform in time, using the linear operator L (cf. Unno et al. 1989):

Here is given a short description of the Nice Oscillations Code (NOSC) and of its general properties: physics, numerics and outputs. We describe also the strategy that we use and the different tools that allow us to check the validity of our results for a given model: use of different variables (Lagrangian or Eulerian), internal tests of accuracy and internal consistency. Some applications to the Sun and to stellar cases studied in the group Evolution and Seismic Tools (ESTA/Task2—Monteiro et al. 2006; Moya 2007) are presented. Hereafter we will in particular present results for the reference solar model, called “model S” (ChristensenDalsgaard et al. 1996) and for a model of 1.5M which has

where ν is the frequency and ξ the displacement of the oscillations. With boundary conditions at the center and at the surface, the frequency ν satisfy an eigenvalue problem, which can be obtained either from a direct calculation providing the eigen-frequency and the eigen-functions ξ or using the variational expression:  ∗ ξ L(ξ )ρdv (ν var )2 =  ∗ . ξ ξρdv

J. Provost () Laboratoire Cassiopée, UNSA, CNRS, OCA BP 4229, 06304 Nice Cedex 4, France e-mail: [email protected]

The direct calculation of the eigenvalue problem is solved by the integration of a system of 4 linear first order differential equations. The code contains two systems of equations, obtained by projection on spherical harmonics Ym of the

ν 2 ξ = L(ξ ),

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perturbations related to the wave. The first one is the classical one, introduced originally by Y. Osaki, and using the following set of variables: y1 =

ξr , r

y2 =

(p  /ρ + φ  ) , gr

y3 =

φ , gr

y4 =

1 dφ  . g dr

(1)

A second system has been added to compute the stellar oscillations in case of eventual discontinuities in the equilibrium model, like discontinuities of density and sound speed for the Jupiter’s solid core (Provost et al. 1993), or difficulties to obtain an accurate determination of the Brunt–Väisälä frequency, which appears only in the coefficients of the system (1). This second system uses the following variables: y˜1 =

ξr , r

φ y˜3 = , gr

y˜2 =

3 Numerical techniques

δp , p

1 dφ  ξr y˜4 = +U , g dr r

Fig. 1 Comparison of frequencies for modes of degree  = 0, 1, 2, 3, computed for the model M4k (N = 4042), with system (1), and using different independent variables: radius r or log(r/p) (in the sense log(r/p) − r)

3.1 Eigenvalue problem (2)

r, ρ, p and g are respectively the radius density, pressure and local gravity of the equilibrium model assumed to the spherical. The quantity U characterizes its mass distribution Mr along the radius: U = ddlog log r . ξr is the radial component of the displacement ξ , p  and δp are respectively the Eulerian and Lagrangian perturbations of the pressure, and φ  is the Eulerian perturbation of the gravitational potential. The variable y1 related to the radial displacement is normalized to 1 at the surface. For the two systems, the integration is performed with the independent variable log(r/p). This variable has been initially implemented in the code with system (1) by Y. Osaki in order to avoid some rounding error which could occur close to the surface if the radius is the independent variable. Nowadays this effect is very small for models with a large number of mesh points N (see Fig. 1). Boundary conditions: Each system satisfy 2 regularity conditions at the center, respectively on the displacement and on the gravitational potential perturbation, which are developed up to second order in radius. At the surface, the perturbation of gravitational potential is fitted with the outside potential and the mechanical condition can be either a perfect wave reflection condition, i.e. δp = 0, or a fit with an isothermal atmosphere (see Unno et al. 1989). In order to be able to compute high degree oscillations, for which the eigen-functions are concentrated near the surface, the variables yi or y˜i are divided by x (−2) .

We solve the eigenvalue problem following Unno et al. (1989). The four differential equations are discretized with a second order scheme at the N mesh points of the model. We solve the system setting aside the surface mechanical condition (for example δp = 0) with an arbitrary eigenvalue. We search for δpsurface small enough for two such arbitrary values. Then we use a Heney-type relaxation method to converge on eigen-frequency and eigen-functions. 3.2 Richardson extrapolation Our difference scheme of second order induces truncation errors in eigen-frequency and eigen-functions in N −2 . Thus the results are improved by a Richardson extrapolation (Shibahashi and Osaki 1981) using the N mesh points of the model and taking a point over 2 to preserve the mesh distribution: 1 2 νRi (N ) = (4ν 2 (N ) − ν 2 (N/2)). 3

4 Outputs and additional tools 4.1 Outputs (i) For each system, two estimations of eigen-frequency by direct computation and “variational” expression, ν(N ) and ν var (N ), are computed. In the expression of ν var (N ), the derivative of the displacement ξ is not

Astrophys Space Sci (2008) 316: 135–140

computed numerically, but it is replaced by its analytical expression. The difference ν(N ) − ν var (N ) is used as a test of a good integration of the equations for the oscillations and for the equilibrium model. (ii) Radial orders np and ng are obtained respectively for p- and g-modes (respectively ng = 0 and np = 0) from phase diagram (e.g. Unno et al. 1989). The consecutive modes are labeled by n = np − ng , varying from −∞ for low frequency g-modes to ∞ for high frequency pmodes. These radial orders indicate the physical nature of the mode. For example a mixed mode has nonzero np ∗ ng . (iii) Eigenfunctions and the following related quantities are provided by NOSC: – The Mode energy which indicates also the physical nature of the mode. – The rotational coefficient giving the rotational splitting for a constant rotation : νn,,m = νn,,0 + mβn,l . – And in option the rotational kernels and the rotational splittings corresponding to a given law of rotation (for example in Loudagh et al. 1993).

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(ii) We estimate the internal accuracy, using Richardson extrapolation both for the initial model with N mesh points and for the model N/2: νRi (N )−νRi (N/2). Figure 2 and upper panel of Fig. 3 shows that in the solar case (model S), the accuracy is better than 0.01 µHz. A much lower accuracy (0.2 µHz) is obtained using the model of the comparison ESTA/STEP1 computed with CESAM code (Morel 1997), with a small initial number of mesh points (N = 900). Such a case requires to use a 2N model obtained by interpolation (see (ii) in Sect. 4.3). (iii) We compare systematically the frequency ν(N ) and its “variational” expression ν var (N ), because any inconsistency either in the computation of the oscillations frequencies or in that of the equilibrium model will give rise to a nonzero value of ν(N ) − ν var (N ). An example is given in Fig. 4 (upper panel), for the solar model S, showing a rather crude internal consistency of the calculation, both at low-frequency with |ν(N ) − ν var (N )|

4.2 Additional tools for the initial model (i) We check the effect of mesh points number of the equilibrium model by comparing ν(N/2), ν(N ) and νRi (N). Due to our second order numerical scheme, there is a large dependence of the frequency on the mesh points number N , as seen in Fig. 2. This leads us to use almost systematically the Richardson extrapolation of the frequency, which is much less dependent of N (plot (c) of Fig. 2 and Fig. 3).

Fig. 2 Effect of the number of mesh points for oscillations modes of degree  = 0, 1, 2 and 3: frequency differences a ν(1240) − νRi (2480), b ν(2480) − νRi (2480), c νRi (1240) − νRi (2480). Frequencies are computed with system (1) for oscillations modes of degree  = 0 (full point), 1 (open star), 2 (full star) and 3 (triangle), for the solar model S (N = 2480)

Fig. 3 Numerical accuracy measured by the frequency differences νRi (N/2) − νRi (N), for oscillations modes of degree  = 0, 1, 2 and 3, for different models. Same symbols than in Fig. 2. Upper panel: same as the plot (c) in Fig. 2, for solar model S (N = 2480). Lower panel: νRi (N) − νRi (2N) for the model of the comparison ESTA/STEP1 N = 900 computed with CESAM code (Morel 1997). In this case ν(2N) is obtained by interpolation of the model

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Fig. 4 Upper panel: difference ν(N) − ν var (N) between eigen-frequency and its variational expression, obtained with system (1) for oscillations modes of degree  = 0 (full point), 1 (open star), 2 (full star) and 3 (triangle), for the solar model S (N = 2480). Lower panel: same, but adding points to the model to ensure enough points to compute the oscillations in the low frequency range (additional tool 4.3(i))

up to 1 µHz, and at high-frequency specially for  = 0 and 1, with |ν(N ) − ν var (N )| up to respectively 0.5 µHz and 0.2 µHz. The comparison can be done also with Richardson var (N ). It appears extrapolation values νRi (N ) and νRi that, in the case of model M4k (Fig. 5—dots), if we use half the number of mesh points (N = 2022), the internal consistency is of order 0.2 µHz in the g-mode lowfrequency range and in the mixed modes range (around 400 µHz), which is a signature of a bad integration for g- and mixed modes, while for the p-modes, the internal consistency is better than of 0.04 µHz. If we use the initial mesh points number (N = 4042), the integration in the mixed modes and low frequency range is much improved, with an internal consistency better than of 0.04 µHz, in all the frequency range. (iv) The comparison of the frequencies obtained by the direct calculation with the two systems (1) and (2), ν1 (N ) and ν2 (N), is another way to test the quality of the

Astrophys Space Sci (2008) 316: 135–140

Fig. 5 Comparison of the frequencies computed for a given model using 4 different ways: direct calculations with system (1) and (2) and their two associated “variational” expressions, for the model M4k. Here the following differences are shown: νRi2 (N) − νRi1 (N) (stars), var (N) − ν var νRi1 Ri1 (N) (open dots), νRi2 (N) − νRi1 (N) (dots). Upper panel: Richardson values with half the number of the initial model, i.e. N = 2022. Lower panel: same with all the points of the initial model, i.e. N = 4042. For this model, the 4 frequency estimations agree by less than 0.2 µHz if one uses 2022 mesh points, and by less than 0.04 µHz with 4042 mesh points

numerical integration. As the Brunt–Väisälä frequency appears only in the coefficients of the system (1), a nonzero value of ν1 (N ) − ν2 (N ) may also reveal problems in the estimation of this Brunt–Väisälä frequency. The frequency differences with the Richardson extrapolation, νRi1 (N ) − νRi2 (N ) are given for the model M4k (Fig. 5—stars symbols), using either all the mesh points of the model (lower panel, N = 4042) or one point over two (upper panel, N = 2042). The results provided by the two systems are very close with differences less than 0.04 µHz in the p-mode frequency range even using the lowest number of mesh points. However it appears necessary to have the largest mesh points number for the computation of g-modes (in the range below 100 µHz) and mixed modes (around 400 µHz).

Astrophys Space Sci (2008) 316: 135–140

As the number N of mesh points and their distribution along the radius may play an important role, we have developed the following tools described in the next subsection. 4.3 Additional tools changing the mesh distribution (i) NOSC has the possibility to add points in the initial model to ensure enough points along the shortest wavelength of the oscillations in a given range of frequency and degree, according to the mode asymptotic behavior (Gabriel 1995). For that purpose, we estimate the highest asymptotic radial wavenumber at each mesh point i, considering both g- and p-modes, then we compare 2π/k with ri − ri−1 , and we add points by interpolation if necessary, in order to have at least 20 points per wavelength. Note that all the modes are computed with the same model. We find that it is very important to add points close to the center for g-modes and mixed modes. Adding points leads in general to a better internal consistency, as seen for example comparing the upper and lower panels in Fig. 4. It appears that, in the case of model S, the internal consistency of the computation of the oscillations is improved at least by a factor 10 in all the considered frequency range and for all the degrees. (ii) If the initial number of mesh points of the initial model is rather small, we increase this number by interpolation, adding one or two points between two consecutive initial points. Figure 3 (lower panel) shows an estimation of the numerical accuracy for a model with only N = 900, using a model with 2N mesh points, obtained by interpolation. The difference |νRi (N ) − νRi (2N )| is smaller than 0.2 µHz, small but still significant, indicating that for this model the number of mesh points N ∼ 900 is too small to obtain a good accuracy of the integration. The different behavior of  = 0 may have several causes: interpolation, mesh distribution, or problem in the profile of the Brunt–Väisälä frequency. (iii) The redistribution of the N mesh points of the model according to the mode asymptotic behavior, with a different mesh for p- and g-modes, is also implemented in NOSC with the recipe of Christensen-Dalsgaard and Berthomieu (1991).

5 Conclusion In conclusion, NOSC provides the eigen-frequencies of adiabatic oscillations of stars by two different systems and using additional tools to check the quality of the results. NOSC has been successfully compared with other codes (Aarhus & M. Gabriel’s codes for solar g- and p-modes, FILOU & Roxburgh’s codes in COROT context (Milestone 2000

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(e.g. Provost 2000)), ESTA Task2 frequency comparisons (Moya et al., this book). The code NOSC uses a second order scheme and Richardson extrapolation. Specific tools for stellar oscillations and various internal tests of accuracy have been developed, showing that it is very important to check internal consistency of both model and oscillations computation by comparing the eigenfrequency ν and its “variational” expression ν var . These tools have shown that accurate frequency computations can be obtained if the equilibrium model is consistent (hydrostatic equilibrium, expression of Brunt–Väisälä frequency, etc.) and if the number of mesh points of the model is large enough, with a good distribution along the radius, well adapted to the considered frequency range and degree and to the internal stellar structure. It appears that the computation of g-modes and of mixed modes may require a large number of mesh points of the equilibrium model than for the p-mode calculation, specially in the case of a star with a convective core. Many applications have been made. As an example, NOSC has been used for the study of solar oscillations (Berthomieu et al. 1980; Zaatri et al. 2007) and of their properties, specially for gravity modes (Provost et al. 2000), and for the inversion of the internal solar rotation (Corbard et al. 1997). In asteroseismology context, NOSC has been used for some theoretical studies like the study of the effect of the microscopic diffusion on the stellar oscillations (Provost et al. 2005), for the interpretation of ground-based observations of oscillations, like those of α Cen A (Thévenin et al. 2002) and of Procyon A (Provost et al. 2006) and for the preparation of COROT in the frame of the Seismology Working Group (Michel et al. 2006) and of ESTA group (Monteiro et al. 2006). Acknowledgements Many thanks to Yoji Osaki who gave us the initial version of NOSC, to Gabrielle Berthomieu who contributed to improve the code and who made constructive remarks on this paper and to Maurice Gabriel for help in developing some additional tools.

References Berthomieu, G., Cooper, A., Gough, D., Osaki, Y., Provost, J., Rocca, A.: Sensitivity of five minutes eigenfrequencies to the structure of the Sun. In: Hill, H., Dziembowski, W.A. (eds.) Nonradial and Nonlinear Stellar Pulsation, vol. 125, p. 307. Springer, Berlin (1980) Christensen-Dalsgaard, J., Berthomieu, G.: In: Cox, A.N., Livingston, W.C., Mattews, M.S. (eds.) Solar Interior and Atmosphere, pp. 401–478. University of Arizona Press, Tucson (1991) Christensen-Dalsgaard, J., Däppen, W., Antia, H.M., et al.: The current state of solar modeling. Science 272, 1286–1292 (1996) Corbard, T., Berthomieu, G., Morel, P., Provost, J., Schou, J., Tomczyk, S.: The solar rotation rate from LOWL data: A 2D regularized lest-squares inversion using B-splines. Astron. Astrophys. 324, 298–310 (1997) Gabriel, M.: Private communication (1995)

140 Loudagh, S., Provost, J., Berthomieu, G., Ehgamberdiev, S., et al.: A measurement of the l = 1 solar rotational splitting. Astron. Astrophys. 275, L25–L28 (1993) Michel, E., Baglin, A., Auvergne, M., et al.: The seismology programme of CoRoT. In: Fridlund, M., Baglin, A., Lochard, J., Conroy, L. (eds.) The CoRoT Mission—Pre-Launch Status—Stellar Seismology and Planet Finding, ESA Publications Division, ESASP-1306, pp. 39–50 (2006) Monteiro, M.J.P.F.G., Lebreton, Y., Montalban, J., et al.: Report on the CoRoT evolution and seismic tools activity. In: Fridlund, M., Baglin, A., Lochard, J., Conroy, L. (eds.) The CoRoT Mission— Pre-Launch Status—Stellar Seismology and Planet Finding, ESA Publications Division, ESA-SP-1306, p. 363 (2006) Morel, P.: CESAM: A code for stellar evolution calculations. Astron. Astrophys. Suppl. Ser. 124, 597–614 (1997) Moya, A.: Current Status of ESTA-Task2. In: Straka, C.W., Lebreton, Y., Monteiro, M.J.P.F.G. (eds.) Stellar Evolution and Seismic Tools for Asteroseismology—Diffusive Processes in Stars and Seismic Analysis. EAS Publications Series, vol. 26, pp. 187– 194 (2007) Provost, J.: About frequency properties of solar like stars. In: Michel, E., Hui-Bon Hoa, A. (eds.) COROT/SWG/Milestone meeting, Paris, 25–26 September 2000. http://www.lesia.obspm.fr

Astrophys Space Sci (2008) 316: 135–140 Provost, J., Mosser, B., Berthomieu, G.: A new asymptotic formalism for Jovian seismology. Astron. Astrophys. 274, 595–611 (1993) Provost, J., Berthomieu, G., Morel, P.: Low-frequency p- and g-mode solar oscillation. Astron. Astrophys. 353, 775–785 (2000) Provost, J., Berthomieu, G., Bigot, L., Morel, P.: Effect of the microscopic diffusion on asteroseismic properties of intermediate mass stars. Astron. Astrophys. 432, 225–233 (2005) Provost, J., Berthomieu, G., Marti´c, M., Morel, P.: Asteroseismology and evolutionary status of Procyon A. Astron. Astrophys. 460, 759–767 (2006) Shibahashi, H., Osaki, Y.: Theoretical eigenfrequencies of solar oscillations of low harmonic degree l in five-minute range. Publ. Astron. Soc. Jpn. 33, 713–719 (1981) Thévenin, F., Provost, J., Morel, P., Berthomieu, G., Bouchy, F., Carrier, F.: Asteroseismology and calibration of α Cen binary system. Astron. Astrophys. 392, L9 (2002) Unno, W., Osaki, Y., Ando, H., Saio, H., Shibahashi, H.: Nonradial Oscillations of Stars. University of Tokyo Press, Tokyo (1989) Zaatri, A., Provost, J., Berthomieu, G., Morel, P., Corbard, T.: Sensitivity of the low degree oscillations to the change of solar abundances. Astron. Astrophys. 469, 1145 (2007)

The OSCROX stellar oscillaton code Ian W. Roxburgh

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9607-4 © Springer Science+Business Media B.V. 2007

Abstract This paper describes the OSCROX stellar oscillation code for the calculation of the adiabatic oscillations of low degree  of a spherical star. There are two principal versions: one in Lagrangian variables (oscroxL), the second in Eulerian variables (oscroxE). The Lagrangian code does not require values of the Brunt Väisälä frequency or equivalently the density gradient. For  = 1 the oscillation equations have both an exact integral and an exact partial wave solution, and codes oscroxL1 and oscroxE1 incorporate these exact solutions. The difference in the frequencies obtained with the various codes gives some estimate of the uncertainty in the results due both to limited accuracy of hydrostatic support of the stellar model, and the limited accuracy of the integration of the oscillation equations. We compare the results of the different methods by calculating the frequencies in the range 20–2500 µHz of a model of a 1.5 M main-sequence star (ModelJC) kindly provided by J. Christensen-Dalsgaard for the purposes of cross comparison of codes, a modified version of this model (ModelJCA) with improved hydrostatic support, and of a highly accurate n = 3 polytropic model of a star with the same mass and radius. For the polytropic model the frequencies as calculated by all codes agree to within 0.001 µHz, whereas for the 1.5 M main sequence model the frequency differences reach a maximum of 0.04 µHz, due primarily to the limited accuracy of hydrostatic support in the model; this is reduced to 0.01 µHz for ModelJCA. I.W. Roxburgh () Astronomy Unit, Queen Mary, University of London, Mile End Road, London E1 4NS, UK e-mail: [email protected] I.W. Roxburgh LESIA, Observatoire de Paris, Meudon 92195, France

Keywords Stars · Oscillations

1 Introduction The equations governing small amplitude adiabatic oscillations of a spherical star are ∂u + ∇p  + ρ∇ψ  + ρ  ∇ψ = 0, ∂t ∂ρ  + ∇.(ρu) = 0, ∂t

(1)

ρ

(2)

∇ 2 ψ  = 4πGρ  ,

(3)

∂p 

d(δp) ≡ + u.∇p dt ∂t   p d(δρ) p ∂ρ  ≡ Γ1 = Γ1 + u.∇ρ ρ dt ρ ∂t

(4)

where ρ, p, ψ are the density, pressure and gravitational potential in the spherical equilibrium model, ρ  , p  , ψ  their Eulerian perturbations, and δp, δρ their Lagrangian (comoving) perturbations. Γ1 is the adiabatic exponent in the equilibrium model and u the velocity. In spherical polar coordinates (r, θ, φ) we express the velocity u = (ur , u⊥ ), the time dependence as ∝ eiωt , and the spatial dependence in terms of spherical harmonics Y,m (θ, φ): ur = iωξ(r)Y,m eiωt , 



p = p (r)Y,m e

iωt

,

ρ  = ρ  (r)Y,m eiωt , 



ψ = ψ (r)Y,m e

iωt

u⊥ = iωζ (r)∇⊥ Y,m eiωt , δp = δp(r)Y,m e

iωt

(5)

,

(6)

δρ = δρ(r)Y,m eiωt ,

(7)

.

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_17

(8) 141

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Astrophys Space Sci (2008) 316: 141–147

After some manipulation the equations governing the oscillations can be reduced to the Lagrangian form

For the Lagrangian equations we define independent variables yj by

dξ 2 ( + 1) 1 =− ξ + ζ− δp, dr r r Γ1 p

(9)

y0 =

ζ , R

y1 =

ξ , R

(10)

y2 =

δp , p

y3 =

R  ψ, GM

(11)

and for the Eulerian equations independent variables zj by

(12)

z1 =

( + 1) 4gρξ dδp = ω2 ρξ + − ρgζ − ρχ, dr r r dψ  = χ − 4πGρξ, dr dχ 2 ( + 1)  ( + 1) =− χ + 4πGρζ ψ + dr r r r2 where rρω2 ζ = δp + ρξg + ρψ 

(13)

dψ  + 4πGρξ. and χ = dr

(14)

In Eulerian form the equations are     g ( + 1) dξ 2 1 p  ( + 1)  = 2− ξ+ + − ψ, dr r c ω2 r 2 c2 ρ ω2 r 2 (15) 1 dp  g p =− 2 + (ω2 − N 2 + 4πGρ)ξ − χ, ρ dr c ρ

(16)

dψ  = χ − 4πGρξ, (17) dr   4πGρ ( + 1) ( + 1) 2 dχ 1 + ψ − χ = 4πG 2 2 p  + dr r ω r r2 ω2 (18) where χ =

dψ  dr

+ 4πGρξ.

(19)

The acceleration due to gravity, g, sound speed, c, and Brunt Väisäla frequency, N are defined by   GMr g p 1 dρ g= 2 , N 2 = −g + 2 . c2 = Γ1 , ρ ρ dr r c (20) The equations are independent of the azimuthal order m.

2 Dimensionless variables For a star of mass M and radius R we define dimensionless structure variables x, ρ ∗ , c∗ , n∗ , g ∗ , p ∗ , ω∗ by r = Rx, GM 2

ρ=

p∗ , 4πR 4 GM g = 2 g∗, R p=

M ∗ ρ , 4πR 3 GM ∗ 2 c , R GM N 2 = 3 n∗ 2 , R c2 =

(21) ω2 =

GM ∗ 2 ω . R3

z3 =

ξ , R

z2 =

R  ψ, GM

(22) y4 =

R2 χ GM

R p , GM ρ z4 =

R2 GM

(23) χ.

On dropping the asterisks we obtain the Lagrangian equations in dimensionless form as 2 ( + 1) 1 dy1 = − y1 + y0 − y2 , dx x x Γ1

(24)

dy2 ω2 ρy1 4gρy1 ( + 1) ρy4 y2 ρg = + − ρgy0 − + , dx p xp xp p p (25) dy3 = y4 − ρy1 , dx 2 dy4 ( + 1) ( + 1) = − y4 + ρy0 , y3 + 2 dx x x x xρω2 y0 = py2 + ρgy1 + ρy3

(26) (27) (28)

and the equations in Eulerian form as     g ( + 1) dz1 2 1 = − + − z2 z 1 dx c2 x x 2 ω2 c2 +

( + 1) z3 , x 2 ω2

dz2 n2 = (ω2 − n2 + ρ) z1 + z2 − z4 , dx g

(29) (30)

dz3 = −ρ z1 + z4 , (31) dx ρ  dz4 ( + 1)  2 ( + 1)ρ 1 + z3 − z4 . (32) z + = 2 2 2 2 2 dx x x ω x ω We note that the relation between the Eulerian and Lagrangian variables is c2 y2 + gy1 , Γ1

z1 = y1 ,

z2 =

z3 = y3 ,

z4 = y4 .

(33)

The Lagrangian equations do not contain the derivative of the density dρ/dx or equivalently the Brunt Väisäla fre-

Astrophys Space Sci (2008) 316: 141–147

quency, whereas the Eulerian equations do. Since the derivative of the density in stellar models has to be calculated numerically, the error introduced by this numerical differentiation is avoided in the Lagrangian scheme. We also note that in deriving the above equations we have assumed that the unperturbed stellar model is known exactly, e.g. by replacing dp/dx by −gρ. The error in the unperturbed model enters into the two formulations in different ways so a comparison of the results of the two schemes gives some estimate of the accuracy of the results.

3 Boundary conditions On the oscillating surface the gravitational potential and the normal component of its gradient are continuous with a solution of Laplace’s equation, and the Lagrangian pressure perturbation vanishes; on the unperturbed surface (x = xs ) y2 = 0, at x = xs , z2 = gz1 , at x = xs ,

+1 y3 = 0 x Lagrangian,

y4 +

+1 z3 = 0 x Eulerian.

y1 = 0,

y2 = x  ,   p y4 = − x −1 . ρ c

(38)

For  = 0 the leading terms in the two series can be taken as y1 = x,

y2 = −3Γ1c ,

y3 = 0,

y4 = 0

(39)

and y1 = 0,

y2 = 0,

y3 = 1,

y4 = 0.

(40)

The series can be carried to higher order but, since all of ρ, p, g/x, Γ1 can be expanded in a series in x 2 , we increase the mesh resolution near x = 0, interpolate the variables ρ, p, g/x, Γ1 onto this fine mesh, and take the starting point for numerical integration at sufficiently small x. The series for the Eulerian equations is given by replacing the yj by zk as in (33).

4 Numerical techniques (34) 4.1 Shooting method

z4 +

(35)

Since a star does not have a distinct surface, the condition that δp = 0 must be applied sufficiently high in the stellar atmosphere that the resulting frequencies are independent of the exact location in the atmosphere at which the boundary condition is imposed. For large frequencies this requires that the atmospheric model goes to very small optical depths. One way round this that is often used is to impose continuity with the analytical solution for an isothermal atmosphere (Dziembowski 1971; Unno et al. 1989) then  ω 2 x y1 ( + 1)g y2 + −4− g x ω2 x   ( + 1)g y3 = 0 at x = xs . + −−1 gx ω2 x 

(36)

The condition for the Eulerian equations is given by replacing the yj by zk as given in (33). At the centre the solution to the equations must be regular which gives two independent series solutions for x 1. For  = 0 the leading terms of these series can be taken as  ρc   y1 = x −1 , x , y2 = 0, y3 = ω 2 −  3 (37)   ρc −1 x y4 =  ω2 + (3 − ) 3 and

143

  p y3 = − x, ρ c

The oscillation equations are solved by a shooting method as follows. For a given value of  we take a starting value of ω and numerically integrate the two independent solutions from a point close to the centre, where they are given by the series solutions, out to a fitting point x = xf , using a 4th order Runge-Kutta integrator with the values of the coefficients at the mid points of an integration step evaluated either by linear interpolation or, if the coefficients are smoothly varying, by cubic interpolation. This gives the values of the two independent solutions at xf y11 ,

y21 ,

y31 ,

y41 ,

(41)

y12 ,

y22 ,

y32 ,

y42 .

(42)

Given the two surface boundary conditions, then for the same  and ω we again have two independent solutions which satisfy the boundary conditions with surface values y1 = 1,

y2 = 0,

y3 = 0,

y4 = 0,

at x = xs

(43)

and y1 = 0, at x = xs .

y2 = 0,

y3 = 1,

y4 = −

+1 , x (44)

With these boundary conditions we integrate inwards from xs , again using a 4th order Runge-Kutta integrator with

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Astrophys Space Sci (2008) 316: 141–147

interpolated values of the coefficients at the mid points, to give the values of the two independent solutions at xf y13 ,

y23 ,

y33 ,

y43 ,

(45)

y14 ,

y24 ,

y34 ,

y44 .

(46)

The general solution for the inner solution at x = xf is yi int = λ1 yi1 + λ2 yi2 ,

i = 1, 4

(47)

and that for the envelope solution at x = xf is yi env = yi3 + λ3 yi4 ,

i = 1, 4

(48)

where we have taken y1 = 1 at x = xs to set the norm on the solution. For arbitrary ω not all of the yi can be made continuous at x = xf . The condition that they are all continuous is that D(ω) = Det{yij } = 0

(49)

which determines the eigenvalues ωe . Given the value of  and an initial value of ω we gradually increase ω until D changes sign and then converge on to the value of ω that gives D = 0. This is labeled ω1, . Keeping the same  we then increase ω until we find the next change in sign of D and then converge on to the next eigenvalue ω2, . This process is repeated until we have found all the eigenvalues ωk, for a given  within the frequency range of interest, and is then repeated for the  values of interest. Note that at this stage k is purely a label. The eigenfrequencies of the stellar model are then given by   GM 1/2 ωk, Hz. (50) νk, = 2π R3 The eigenfunctions can readily be determined by solving for the λi and combining the two independent inner and envelope solutions. 4.2 Mesh resolution For accurate determination of the eigenvalues it is necessary to have a sufficiently large number of mesh points in each wavelength for all modes of interest and, especially for the Lagrangian code, to ensure stability of the integrations. For calculation with a given stellar model we therefore enhance the mesh in x as necessary (usually 60 points in a wavelength), and take a fine mesh in x in the central core; the values of the coefficients in the oscillation equations on this enhanced mesh being obtained by linear or cubic interpolation, depending on the smoothness of the model. Whilst this will introduce some error, the integration remains of 4th order; to integrate numerically even the simple wave equation y  + ω2 y = 0 we need to have a small enough step length to accurately resolve each wavelength; since the coefficients are constant linear and cubic interpolation is exact.

4.3 Order of the modes, n The k value referred to above is simply a label on the modes not the order of the modes. To calculate the order we determine the number of nodes in the eigenfunction for y1 (∝ ξ ) for the highest p-mode frequency and then count downwards so that g-modes have negative n. For simple stellar models (e.g. a polytrope) this is straight forward but, for realistic evolved stellar models, the concept of the order n can lose meaning since there can be trapped modes in the interior and the number of nodes may no longer be monotonic with frequency. For asteroseismological investigations the order n is of no direct relevance (and cannot be measured); what we seek is a model that fits the observed frequencies or some pattern in the frequencies.

5  = 1 modes For  = 1 modes the equations admit an integral, which follows from conservation of momentum (Takata 2005; Roxburgh 2006), and can be expressed as   g  3ψ  4πG 2ψ   1− 2 − − 2 δp = 0 (51) χ+ r r ω r ω r and an exact partial wave solution of the equations, that is an exact solution which satisfies the central boundary conditions, which is (Roxburgh 2006) ξ = λ,

ζ = λ,

ψ  = (ω2 r − g)λ,

δp  = 0, 

χ = ω2 + 2

g r

(52) λ

where λ = const. These integrals can be used to give an independent check on the accuracy of the solutions of the full 4th order sets. The two codes oscroxL1 and oscroxE1 incorporate the exact integrals reducing the order of the governing equations. We again solve these equations by a 4th order RungeKutta shooting method but this time there is only one solution satisfying the surface boundary conditions which for  = 1 require that y2 = 0, at x = xs , z2 = gz1 , at x = xs ,

y3 = 0,

y4 = 0

Lagrangian, z3 = 0, Eulerian,

(53) z4 = 0 (54)

since in a coordinate system centred on the centre of mass there is no external dipole moment (due to conservation of momentum). In practice we use the integral condition (51)

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to eliminate the variable y3 (Lagrangian) and z3 (Eulerian) and solve the resulting 3rd order system for given , ω. For the two solutions from the centre one is known as the exact partial wave solution (52), and the other is obtained numerically by again eliminating y3 , z3 and using the central boundary condition (38). Continuity of y1 , y2 , y4 then determines the eigenvalues ω and hence νn,1 .

6 Numerical examples We compare the value of frequencies obtained with the various codes using a model of a partially evolved main sequence star of 1.5 M and radius 1.73 R calculated by Christensen Dalsgaard for the purposes of comparison of stellar oscillation codes within the ESTA project (ModelJC). The model has 4042 mesh points. We impose the surface boundary conditions (34) that is δp = 0 at the surface. Figure 1 shows the difference δν = νL − νE between the frequencies calculated for this model using the input mesh and linear interpolation for the mid points; δν is everywhere smaller than 0.041 µHz. If we use cubic interpolation this difference is slightly reduced to 0.037 µHz. The input mesh for ModelJC is relatively coarse; for the highest order g-mode ( = 3, n = 42, ν = 20.2 µHz) the smallest number of mesh points between nodes of the eigenfunction (i.e. within a half-wavelength) is 13, whilst the for the highest order p-mode ( = 0, n = 35, ν = 2564.2 µHz) the smallest number of mesh points in a half-wavelength is 36. We therefore calculated the frequencies with mesh enhancement; increasing the mesh to a minimum of 60 mesh points per half-wavelength. This has little effect on the frequencies: ≈ 0.0005 µHz for the Eulerian code and ≈ 0.0011 µHz for the Lagrangian code. The difference on using linear and cubic interpolation is ≈ 0.0083 µHz for the Eulerian code and ≈ 0.0016 µHz for the Lagrangian code. Figure 2 shows the differences between the  = 1 frequencies calculated with the full 4th order solver oscroxL and the code oscroxL1, each with mesh resolution enhanced so that there are at least 60 points within a half-wavelength and with linear interpolation: the maximum difference is of order 0.032 µHz. Figure 3 shows the differences using oscroxE and oscroxE1 for the same mesh resolution: the maximum difference is 0.019 µHz. The maximum difference between frequencies calculated with oscroxL1 and oscroxE1 is 0.039 µHz. All these differences are hardly changed if we use cubic rather than linear interpolation, increase the mesh resolution, change the fitting point between the inwards and outwards integrations, or change the location of the end of the series solution in the centre. This suggests that the primary cause of the differences is that the model is not in exact hydrostatic equilibrium since this assumption enters in different ways in the 4 codes.

Fig. 1 Difference between frequencies calculated with the Lagrangian and Eulerian codes for a main-sequence model of a star of mass 1.5 M and radius 1.73 R . The maximum difference ≈ 0.041 µHz

Fig. 2 Difference between frequencies of the  = 1 modes calculated with the Lagrangian codes oscroxL and oscroxL1, for an main-sequence model of a star of mass 1.5 M and radius 1.73 R . The maximum difference ≈ 0.032 µHz

Fig. 3 Difference between frequencies of the  = 1 modes calculated with the Eulerian codes oscroxE and oscroxE1 for an main-sequence model of a star of mass 1.5 M and radius 1.73 R . The maximum difference ≈ 0.019 µHz

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Astrophys Space Sci (2008) 316: 141–147

To test this we use a polytropic model of index 3 with mass of 1.5 M and radius 1.73 R , and Γ1 = 5/3 with the same mesh as ModelJC. This model was solved by 4th order Runge-Kutta integration and satisfies hydrostatic equilibrium to high accuracy and the Brunt Väisälä frequency does not require numerical differentiation. Figure 4 shows the difference in frequencies calculated for this model with mesh enhancement and linear interpolation; the Eulerian and Lagrangian codes give the same values to within 0.001 µHz. With cubic interpolation this difference can be reduced to 0.00001 µHz. The difference in frequencies obtained with the full codes and the  = 1 codes is of the same order. Since the equations for the 4 codes oscroxL, oscroxL1, oscroxE, oscroxE1 incorporate the assumption of exact hydrostatic equilibrium (dp/dx=−gρ) in different ways, we conclude that the major limitation on the accuracy of the frequencies for the main sequence model (ModelJC) is due to the departure from exact hydrostatic equilibrium, and that with this model the frequencies can only be calculated to an accuracy of ≈ 0.04 µHz.

Fig. 4 Difference between frequencies calculated with the Lagrangian and Eulerian codes for an n = 3 polytropic model of a star of mass 1.5 M and radius 1.73 R . The maximum difference ≈ 0.0008 µHz

7 ModelJCA: improved hydrostatic equilibrium To further explore the effect of the accuracy of hydrostatic support on the frequencies we took the input distribution of density ρ on the mesh in r from ModelJC and constructed a new model (ModelJCA) by integrating the mass and hydrostatic equations using the algorithms 2π 3 (ρi + ρi+1 )(ri+1 − ri3 ), 3  G Mi ρ i Mi+1 ρi+1 2 + − ri2 ) Pi = Pi+1 + (ri+1 3 4 ri3 ri+1 Mi+1 = Mi +

(55) (56)

Fig. 5 Difference between frequencies calculated with the Lagrangian and Eulerian codes for ModelJCA, that is the hydrostatic model generated ρ(r) from modelJC. With no mesh enhancement and cubic interpolation for the mid points the maximum difference ≈ 0.0065 µHz. For linear interpolation this difference is of order 0.01 µHz

with M0 = 0 and PN taken from ModelJC. Γ1 was taken from ModelJC and the Brunt Väisälä frequency was evaluated by first calculating d log ρ/d log P on the mesh ri using local 3 point quadratic differentiation, and then taking 

GMr N =− r2 2

2

ρ P



1 d log ρ − Γ 1 d log P

 .

(57)

Figure 5 shows the difference between frequencies of ModelJCA calculated with oscroxL and oscroxE with no mesh enhancement and cubic interpolation for mid points. The differences are everywhere < 0.0065 µHz. With linear interpolation this limit is ≈ 0.01 µHz. There is no improvement with enhanced mesh resolution. The maximum differences between frequencies calculated with oscroxL and oscroxL1 and cubic interpolation are shown in Fig. 6, the differences are < 0.0075 µHz. This difference is reduced to 0.005 µHz for linear interpolation. Mesh enhancement has negligible effect. Figure 7 shows the

Fig. 6 Difference between frequencies of the  = 1 modes calculated with the Lagrangian codes oscroxL and oscroxL1, for ModelJCA with no mesh enhancement and cubic interpolation for the mid points. The maximum difference ≈ 0.0075 µHz

Astrophys Space Sci (2008) 316: 141–147

Fig. 7 Difference between frequencies of the  = 1 modes calculated with the Eulerian codes oscroxE and oscroxE1 for ModelJCA with no mesh enhancement and cubic interpolation for the mid points. The maximum difference ≈ 0.003 µHz

differences between oscroxE and oscroxE1, here the maximum difference ≈ 0.003 µHz for both linear and cubic interpolation. Again there is no improvement with increased mesh resolution.

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of hydrostatic support dp/dx=−gρ enters the formalisms in different ways. This gives an estimate of 0.04 µHz for the accuracy of the frequencies derived from ModelJC. On recalculating hydrostatic support (ModelJCA) the accuracy is improved to 0.01 µHz. For the highly (4th order) accurate polytropic model an accuracy of 0.001 µHz can be achieved using linear interpolation, and 0.00001 µHz on using cubic interpolation. It should be noted that the condition that the model satisfies hydrostatic support to high accuracy is not the same as requiring that it satisfies a particular discretisation of the structure equations to high accuracy. For example the model may satisfy the discretisation of (56) to high accuracy but it will not necessarily satisfy the equally valid discretisation using log P as an independent variable to the same accuracy. The accuracy depends on the order of the discretisation and on the mesh resolution. Acknowledgements I would like to thank Sergei Vorontsov for valuable input. This work was supported in part by the UK Particle Physics and Astronomy Research Council under grants PPA/G/S/1997/00338 and PPA/G/S/2003/00137.

References 8 Conclusions The accuracy with which the eigenfrequencies of a model can be determined depends on the accuracy with which the model is in hydrostatic equilibrium, on the mesh resolution and, for the Eulerian codes, on the evaluation of the Brunt Väisälä frequency. The error in the frequencies is estimated by comparing the results obtained with the 4 codes: oscroxL, oscroxE, oscroxL1, oscroxE1, since the assumption

Dziembowski, W.: Nonradial oscillations of stars. I. Quasiadiabatic Approximation. Acta Astron. 21, 289 (1971) Roxburgh, I.W.: An exact integral and exact partial wave solution for dipole ( = 1) oscillations of stars. Available at http://www.maths. qmul.ac.uk/~iwr/oscodes (2006) Takata, M.: Momentum conservation and mode classification of the dipole oscillations of stars. Pub. Astron. Soc. Jpn. 57, 375–389 (2005) Unno, W., Osaki, Y., Ando, H., Saio, H., Shibahashi, H.: Nonradial Oscillations of Stars. University of Tokyo Press, Tokyo (1989)

The Liège Oscillation code R. Scuflaire · J. Montalbán · S. Théado · P.-O. Bourge · A. Miglio · M. Godart · A. Thoul · A. Noels

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9577-6 © Springer Science+Business Media B.V. 2007

Abstract The Liège Oscillation code can be used as a stand-alone program or as a library of subroutines that the user calls from a Fortran main program of his own to compute radial and nonradial adiabatic oscillations of stellar models. We describe the variables and the equations used by the program and the methods used to solve them. A brief account is given of the use and the output of the program. Keywords Stars · Adiabatic oscillations · Stellar pulsations · Asteroseismology PACS 97.10.Sj · 95.75.Pq

1 Introduction The Liège Oscillation code (OSC) has been developed in the early 70 s for computing adiabatic pulsations of spherically symmetric stars (no rotation nor magnetic field). It has gone through minor updates and is still presently in use in the asteroseismology group of the Liège Institute of Astrophysics and Geophysics. Besides the frequencies and eigenfunctions, it produces also the coefficients needed to compute the first order rotational frequency splitting for a rigid rotation and the kernels needed to compute the splitting when a non rigid rotation is considered.

2 Stellar models A stellar model is input to OSC as a table describing a few physical quantities at discrete points of the star, ordered from the centre to the surface. In theory, only two functions are necessary to compute stellar oscillations, for instance ρ(r) and Γ1 (r), the density and the first adiabatic exponent in terms of the radius. However, for OSC, the model file must give at each point the values of the radius r, the mass m in the sphere of radius r, the total pressure P , the density ρ and the first adiabatic exponent Γ1 . It is clear that m and P could have been computed from the other quantities. The program does not require the Brunt–Väisälä frequency, often poorly computed by evolution codes. If the first point is not at the centre, OSC computes the oscillations of the given envelope with a rigid boundary condition at the bottom. Of course, when neglecting the oscillatory behaviour of the core, great attention must be paid to the physical meaning of the output of the program. The outer boundary conditions will be applied at the last point, considered as the surface of the star. Inside the code, the model is described by the following five dimensionless quantities: x = r/R, q/x 3 (with q = m/M), RP /GMρ, 4πR 3 ρ/M and Γ1 , where R and M denote the radius and the total mass of the star.

3 Stellar oscillations 3.1 Oscillation modes

R. Scuflaire () · J. Montalbán · S. Théado · P.-O. Bourge · A. Miglio · M. Godart · A. Thoul · A. Noels Institut d’Astrophysique et de Géophysique, Université de Liège, allée du 6 Août 17, 4000 Liège, Belgium e-mail: [email protected]

The small perturbations of a spherical star without rotation or magnetic field may be described as a superposition of normal modes of oscillation which are the solutions of a linear boundary eigenvalue problem. These normal modes may be

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_18

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indexed by three integers k,  and m. Index k is loosely related to the number of nodes of the radial displacement. Indices  and m are the usual indices of the spherical function Ym (θ, φ) describing the angular dependence. Index  may take any null or positive integer value and m may take 2 + 1 values between − and +. In the following description we use the notation δX and X  for the Lagrangian and Eulerian perturbation of any quantity X and σ for the angular frequency. We often use the dimensionless angular frequency  ω = σ τdyn , where the dynamical time τdyn is defined as R 3 /GM.

When an envelope model is given, the condition at the centre is replaced by a condition at the bottom of the envelope,

3.2 Oscillation equations

Note that if R is the value of r at the last point, x = q/x 3 = 1 at the surface but this is not mandatory. The user can choose to apply the more usual condition

The theory of stellar oscillation has been developed in a number of textbooks. We are the most familiar with the paper of Ledoux and Walraven (1958) and the books of Unno et al. (1979) and Cox (1980). We will just write the needed equations in the form they are implemented in our code. 3.2.1 Radial oscillations In the case of radial oscillations ( = m = 0), the equation of Poisson can be integrated and the perturbation of the gravitational potential eliminated. The differential system is then reduced to order two. We describe a normal mode with two functions Y (x) and Z(x). Disregarding an arbitrary phase, the displacement δr and the Lagrangian perturbation of the pressure δP are written in terms of Y (x) and Z(x) in the following way, √ δr = {a(r)e−iσ t er } = 4π {a(r)Y00 (θ, φ)e−iσ t er }, (1) where er is a unit vector in the radial direction. Near the centre, a(r) ∝ r and may be written a(r)/r = Y (x)

or

a(r)/R = xY (x).

(2)

In a similar way, δP /P = {Z(x)e−iσ t } =

√ 4π{Z(x)Y00 (θ, φ)e−iσ t }. (3)

Now, the differential equations read 3 1 dY =− Y − Z, dx x Γ1 x   dZ q GMρ q 2 GMρ = 4 3 +ω xY + xZ. dx RP RP x 3 x

(4) (5)

These equations must be completed by the boundary conditions. At the centre, the regularity of the solution is ensured by the condition 3Γ1 Y + Z = 0 at x = 0.

(6)

Y = 0.

(7)

At the surface, we generally apply a condition deduced from the vanishing of the pressure. In this case, the coefficient GMρ/RP in (5) tends to infinity and the regularity of the solution requires that   q q 2 4 3 + ω Y + 3 Z = 0. (8) x x

δP = 0 or Z = 0.

(9)

3.2.2 Nonradial oscillations In the case of nonradial oscillations ( = 0), the differential system is of order four. We describe a normal mode with four functions Y (x), Z(x), U (x) and V (x). These functions as well as the frequency do not depend upon index m (2 + 1-fold degeneracy). The displacement reads   ∂Ym (θ, φ) − → √ δr = 4π a(r)Ym (θ, φ)er + b(r) eθ ∂θ   1 ∂Ym (θ, φ) −iσ t , (10) eφ e + sin θ ∂φ where er , eθ and eφ form the usual local cartesian basis of spherical coordinates. Near the centre, a(r) and b(r) ∝ r −1 and are written a(r)/R = x −1 Y (x),   RP q x −1 Z(x) + 3 Y (x) . b(r)/R = 2 U (x) + GMρ ω x

(11) (12)

The Lagrangian perturbation of pressure δP and the Eulerian perturbation of the gravitational potential Φ  are given by δP √ = 4π{x  Z(x)Ym (θ, φ)e−iσ t }, P RΦ  √ = 4π{x  U (x)Ym (θ, φ)e−iσ t }, GM    4πR 3 ρ R 2 ∂Φ  √ −1 V (x) − = 4π x Y (x) GM ∂r M  −iσ t . × Ym (θ, φ)e

(13) (14)

(15)

Of course, the solution of a linear √ problem may be multiplied by an arbitrary factor and the 4π factor in the above

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expressions may be dropped. We have put it there for aesthetic reasons because the spherical functions are normalized in such a way that |Ym |2 dΩ = 1. (16)

There are no movement nor density perturbations in the core, where the perturbation of the gravitational potential obeys a Laplace equation and has a simple analytical expression. We thus require a continuous match between Φ  and its gradient at the bottom of the envelope. This is expressed as

4π

√ With this 4π factor, the time-average kinetic energy of a mode is given by 1 2 ρv dV E kin = 2 σ2 = [a 2 + ( + 1)b2 ]4πr 2 ρ dr 4 σ2 (17) = [a 2 + ( + 1)b2 ] dm, 4 without any π factor. With these definitions, the oscillation equations read    +1  q x dY RP = −Y + 2 Z+U − Z, Y+ 3 dx x GMρ Γ1 ω x (18)   dZ q Y V GMρ q ω2 + 4 3 = +x 3Z − dx RP x x x x    RP ( + 1) q q Z + U − Z, Y + − GMρ x xω2 x 3 x 3 (19)   dU 1 4πR 3 ρ = V− Y − U , (20) dx x M dV +1 = (U − V ) dx x

  ( + 1) 4πR 3 ρ q RP Z+U . + Y+ M GMρ xω2 x3

(21)

These equations have been published by Boury et al. (1975), but their equation (9) has been affected by a typo (an extra factor ). The regularity of the solution at the centre imposes two conditions at x = 0,    q RP Y = 2 3Y + Z+U , (22) GMρ ω x V =

4πR 3 ρ Y + U. M

(23)

In the case of an envelope, the bottom of the envelope is supposed to behave as a rigid boundary, Y = 0.

(24)

V = U.

(25)

Two boundary conditions must be imposed at the surface. The first one involves the Lagrangian perturbation of the pressure δP . As for the radial case, the default choice is deduced from the requirement of regularity of the solution when P vanishes at the surface. In our variables, this condition reads     q ( + 1) q 2 Y q ω2 + 4 3 − +x 3Z 2 3 x x ω x x −

( + 1) q V U − = 0. 2 3 x ω x x

(26)

The user can however choose to impose the more usual condition δP = 0 or Z = 0.

(27)

The second boundary condition ensures the matching of Φ  and its gradient with the regular solution of the Laplace equation outside the star, V + ( + 1)U = 0.

(28)

Our choice of variables and the way the differential equations are written call for three remarks: (1) The inclusion of a term in Y in the definition of V (15) ensures that the boundary condition (28) is still valid in the (unphysical) case of a non vanishing density at the surface of the model. (2) The use of the Lagrangian perturbation of the pression results in a better precision in the external layers. (3) We do not use the Brunt–Väisälä frequency n in the coefficients of the equations, often badly computed by stellar evolution codes. However, we use non independent functions ρ(r), m(r) and P (r). Troubles may stem from their possible inconsistencies. Maybe it would have been wiser to let OSC compute m and P from ρ. The solutions computed by OSC are normalized in such a way that [a 2 + ( + 1)b2 ] dm = MR 2 .

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3.3 Mode classification The radial modes owe their existence to the compressibility of the stellar material (acoustic or pressure modes). The mode with the lowest frequency is called the fundamental mode, it has no node in the displacement (except at the origin) and is called p1 . By order of increasing frequency and number of nodes, we have then the first harmonics (one node, p2 ), the second harmonics (two nodes, p3 ), . . . . For nonradial modes, the situation is more complicated. For each , we have a spectrum of p-modes. They are of the same nature as the radial modes, owing their existence to the compressibility of the stellar material. They are numbered p1 , p2 , . . . by order of increasing frequency. If the stellar model has a radiative zone, it has a spectrum of g + -modes. Their frequencies are lower than those of the p-modes and have an accumulation point at zero. They are numbered g1+ , g2+ , . . . by order of decreasing frequency. They owe their existence to the buoyancy force. For each value of  > 1, there exists one f -mode, with its frequency between those of the p-modes and the g-modes. This mode does not disappear when the stellar material is incompressible nor when the buoyancy force is zero. When the star has a convective zone, another spectrum appears, the g − -modes. They have an exponential temporal behaviour (their frequencies are imaginary) and are associated with convection. We neglect them in the following discussion. In OSC, we use an integer to denote the type and order of a computed mode: n for pn , −n for gn+ and 0 for f . In the Cowling’s approximation (the Eulerian perturbation of the gravitational potential is neglected), the order of a given mode can be easily deduced from the behaviour of the vertical and horizontal displacements (Scuflaire 1974b; Gabriel and Scuflaire 1979, 1980). In our case, the mode number obtained in the same way is generally correct for models which are not too evolved. But as the condensation (measured by ¯ increases with the age of the model, it can the ratio ρc /ρ) just be considered as a clue and finally looses any meaning. The algorithm described by Lee (1985) gives a clue to the mode number (also computed by OSC) which keeps its utility a bit longer. But when the condensation of the model is really too high, the only reliable identification method consists in the computation of a large number of contiguous modes, up to the asymptotic domain, where the implemented algorithms continue to give reliable mode numbers. Though the order of the mode cannot be obtained safely, its parity can and is provided by OSC. 3.4 Influence of rotation The rotation of the star removes the degeneracy of the non0 denotes the frequency radial oscillation frequencies. If σk in the absence of rotation, a slow solid rotation with angular

Astrophys Space Sci (2008) 316: 149–154

velocity Ω slightly alters the frequencies in the following way 0 + mβk Ω, σkm = σk

(29)

with

βk = 1 −

(b2 + 2ab) dm . [a 2 + ( + 1)b2 ] dm

(30)

When the angular velocity depends on the radius, the altered frequencies may be written 0 σkm = σk + m Kk (x)Ω(x) dx, (31) where the kernel Kk (x), computed by OSC, is given by Kk (r) =

ρr 2 [a 2 + ( + 1)b2 − 2ab − b2 ]

. ρr 2 [a 2 + ( + 1)b2 ] dr

(32)

3.5 Physical description of the modes For low order modes, specially for evolved models, the physical characteristics of a mode is not tightly linked to its g or p label (Scuflaire 1974a, 1980). OSC outputs different indexes allowing the user a quick analysis of the physical behaviour of a computed mode (gravity or pressure wave, trapped mode, . . .).

4 Technique of solution 4.1 Interpolation of the model The grid of points used for the computation of the model is rarely appropriate for the computation of oscillations, as the eigenfunctions can exhibit rapid spatial oscillations in regions where the variables describing the model are wellbehaved. As the oscillatory behaviour of eigenfunctions is easy to foresee, we interpolate the model before any oscillation computation, increasing the number of points where they will prove necessary. The interpolation method we use preserves the continuity of the first derivatives. 4.2 Difference equations We have adopted a difference equation scheme of the fourth order. That is why we do not need to use Richardson extrapolation method to increase the precision of the eigenfrequency. Our difference scheme rests on the following identity satisfied by any vector function y(x) with continuous derivatives up to the fifth order.

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153

h h2 yi + yi + yi 2 12 h h2 = yi+1 − yi+1 + yi+1 + O(h5 ), 2 12

been used by Keeley (1977) to compute stellar radial nonadiabatic oscillations. The principle of the method is easy to describe. Consider the following eigenvalue problem (33) (A − λB)y = 0.

where yi and yi+1 are the values of y(x) at points xi and xi+1 and h = xi+1 − xi . If y is a solution of the linear differential system dy = A(x)y, dx

(34)

identity (33) may be written   h h2 1 + αi + βi yi 2 12   h h2 = 1 − αi+1 + βi+1 yi+1 + O(h5 ), 2 12

yn · yn+1 → λ − λ0 . yn+1 · yn+1 (35)

(36)

dA β = A2 + . dx

(37)

The difference equations are easily obtained from the above equations, neglecting the term O(h5 ). At the centre, certain coefficients of the matrix A are singular and a slightly different treatment is needed. The matrix can be written as

By = 0 .

(39)

It is worth noticing that the rank of B(0) is lower than its dimension and that equation (39) gives the right number of boundary conditions (1 for the radial case and 2 for the nonradial one). The matrices α and β assume the following different forms at the centre (index 0),

β = (2 − B0 )−1

(43)

(44)

The convergence is quite fast. Practically, the yn must be normalized at each step of the computation to avoid overflow. Moreover, these normalized yn tend to the eigenvector associated with λ. It is true that, in the case of nonradial oscillations, the problem to solve is not exactly in the form of (42). But it is put in the right form if we write λ = λ0 + Δλ

(45)

and linearize with respect to the correction Δλ. The solution is then obtained in a few iterations of this process.

(38)

with all the odd order derivatives of matrix B vanishing at x = 0. It is clear that the regularity of the solution requires, at x = 0,

α = 0,

n = 0, 1, 2, . . . .

Then,

α = A,

1 B(x), x

The method is generally exposed with a unit matrix in place of B. Suppose that we know an approximation λ0 of an eigenvalue λ. Starting with an arbitrary vector y0 , we build the sequence yn+1 = −(A − λ0 B)−1 Byn ,

with

A(x) =

(42)



d 2B dx 2

(40)

 .

(41)

5 Use of the program OSC is written in Fortran and can be used either as a standalone program or as a library of subroutines that the user calls from a main program of his own. The stand-alone program is in fact just a user interface to the library. It accepts instructions from the standard input (or from a command file) and prints all kind of information to the standard output. At the request of the user, the eigenfunctions can be saved to a file. For heavy work, the user had better write his own Fortran main program and call the routines of the library. He gains a better control on the computation and access to results not available otherwise (the rotation kernels for a r-dependent angular velocity Ω, for instance).

0

4.3 Inverse iteration method

6 Applications

After the discretization of the differential equations we are left with an algebraic eigenvalue problem where the eigenvalue is λ = ω2 . OSC uses the inverse iteration method. It is a powerful tool in linear eigenvalue problems. It is described in the book of Wilkinson (1965, Chap. 9, Sect. 47). It has

OSC is routinely used in our group in Liège and by members of the Belgian Asteroseismology Group (BAG) for seismic studies of solar-like pulsators such as, e.g., α Cen A+B (Thoul et al. 2003a; Miglio and Montalbán 2005) and of classical β Cephei variables (Aerts et al. 2003; Thoul et al.

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2003b; Dupret et al. 2004; Ausseloos et al. 2004; Thirion and Thoul 2006; Briquet et al. 2007). It is worth mentioning that in these studies, we obtained indications on the internal rotation of the β Cephei stars HD 129929 and θ Ophiuchi. The adiabatic frequencies computed by OSC are also used as first approximations by the program MAD which computes the non adiabatic oscillations of stellar models. See Dupret (2001) for a description of the code. The Liège oscillation code has also taken part in the work and code comparisons realized within the Corot/ESTA group (Montalbán and Scuflaire 2007; Montalbán et al. 2007).

7 Discussion There is not any general agreement on what the outer boundary condition on the perturbation of pressure should be and different boundary conditions are implemented in the existing codes. In our opinion, the precise choice of the outer boundary condition does not matter so much, as, in any way, the oscillation is far from adiabatic in the very external layers of the star. Moreover, Dzhalilov et al. (2000) have shown that in the Sun, waves in the frequency range ν ≈ 2–10 mHz may reach the chromosphere-corona transition regions by means of a tunneling through the atmospheric barrier. When comparing with observations, a reasonable strategy is thus to use expressions of frequencies which are insensitive to the very external stellar layers (Roxburgh and Vorontsov 2003; Roxburgh 2005). Acknowledgements We acknowledge financial support from the Belgian Science Policy Office (BELSPO) in the frame of the ESA PRODEX 8 program (contract C90199), from the Belgian Interuniversity Attraction Pole (grant P5/36) and from the Fonds National de la Recherche Scientifique (FNRS).

References Aerts, C., Thoul, A., Daszy´nska, J., Scuflaire, R., Waelkens, C., Dupret, M.-A., Niemczura, E., Noels, A.: Asteroseismology of HD 129929: Core overshooting and nonrigid rotation. Science 300, 1926–1928 (2003) Ausseloos, M., Scuflaire, R., Thoul, A., Aerts, C.: Asteroseismology of the β Cephei star ν Eridani: Massive exploration of standard and non-standard stellar models to fit the oscillation data. Mon. Not. Roy. Astron. Soc. 355, 352–358 (2004) Boury, A., Gabriel, M., Noels, A., Scuflaire, R., Ledoux, P.: Vibrational instability of a 1 M star towards nonradial oscillations. Astron. Astrophys. 41, 279–285 (1975) Briquet, M., Morel, T., Thoul, A., Scuflaire, R., Miglio, A., Montalbán, J., Aerts, C.: An asteroseismic study of the β Cephei star θ Ophiuchi: constraints on core overshooting and internal rotation. Mon. Not. Roy. Astron. Soc. (2007, in press)

Astrophys Space Sci (2008) 316: 149–154 Cox, J.P.: Theory of Stellar Pulsation. Princeton University Press, Princeton (1980) Dupret, M.-A.: Nonradial nonadiabatic stellar pulsations: A numerical method and its application to a β Cephei model. Astron. Astrophys. 366, 166–173 (2001) Dupret, M.-A., Thoul, A., Scuflaire, R., Daszy´nska-Daskiewicz, J., Aerts, C., Bourge, P.-O., Waelkens, C., Noels, A.: Asteroseismology of the β Cep star HD 129929. II. Seismic constraints on core overshooting, internal rotation and stellar parameters. Astron. Astrophys. 415, 251–257 (2004) Dzhalilov, N.S., Staude, J., Arlt, K.: Influence of the solar atmosphere on the p-mode eigenoscillations. Astron. Astrophys. 361, 1127–1142 (2000) Gabriel, M., Scuflaire, R.: Properties of nonradial stellar oscillations. Acta Astron. 29, 135–149 (1979) Gabriel, M., Scuflaire, R.: Properties of nonradial stellar oscillations. In: Hill, H.A., Dziembowski, W.A. (eds.) Nonradial and Nonlinear Stellar Pulsation, pp. 478–487. Springer, Berlin (1980) Keeley, D.A.: Linear stability analysis of stellar models by the inverse iteration method. Astrophys. J. 211, 926–933 (1977) Ledoux, P., Walraven, T.: Variables stars. In: Flugge, S. (ed.) Handbuch der Physik, pp. 353–604. Springer, Berlin (1958) Lee, U.: Stability of Delta Scuti stars against nonradial oscillations with low degrees . Publ. Astron. Soc. Jpn. 37, 279–291 (1985) Miglio, A., Montalbán, J.: Constraining fundamental stellar parameters using seismology. Application to α Centauri AB. Astron. Astrophys. 441, 615–629 (2005) Montalbán, J., Scuflaire, R.: Grids of stellar models and frequencies with CLÉS and LOSC. Astrophys. Space Sci. (2007). This volume Montalbán, J., Lebreton, Y., Miglio, A.: Code-to-code comparisons: CLÉS/CESAM. Astrophys. Space Sci. (2007). This volume Roxburgh, I.W.: The ratio of small to large separations of stellar p-modes. Astron. Astrophys. 434, 665–669 (2005) Roxburgh, I.W., Vorontsov, S.V.: The ratio of small to large separations of acoustic oscillations as a diagnostic of the interior of solar-like stars. Astron. Astrophys. 411, 215–220 (2003) Scuflaire, R.: Space oscillations of stellar nonradial eigen-functions. Astron. Astrophys. 34, 449–451 (1974a) Scuflaire, R.: The nonradial oscillations of condensed polytropes. Astron. Astrophys. 36, 107–111 (1974b) Scuflaire, R.: Distribution of energy in stellar nonradial oscillations. Bull. Soc. Roy. Sci. de Liège 49, 164–177 (1980) Thirion, A., Thoul, A.: BetaDat: A β Cephei database. In: Fridlund, M., Baglin, A., Lochard, J., Conroy, L. (eds.) The Corot Mission, PreLaunch Status, Stellar Seismology and Planet Finding, SP-1306, ESA, pp. 373–375 (2006) Thoul, A., Scuflaire, R., Noels, A., Vatovez, B., Briquet, M., Dupret, M.-A., Montalbán, J.: A new seismic analysis of Alpha Centauri. Astron. Astrophys. 402, 293–297 (2003a) Thoul, A., Aerts, C., Dupret, M.-A., Scuflaire, R., Korotin, S.A., Egorova, I.A., Andrievsky, S.M., Lehmann, H., Briquet, M., De Ridder, J., Noels, A.: Seismic modelling of the β Cep star EN (16) Lacertae, 2003. Astron. Astrophys. 406, 287–292 (2003b) Unno, W., Osaki, Y., Ando, H., Shibahashi, H.: Nonradial Oscillations of Stars. University of Tokyo Press, Tokyo (1979) Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, Oxford (1965)

FILOU

oscillation code

J.C. Suárez · M.J. Goupil

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9568-7 © Springer Science+Business Media B.V. 2007

Abstract The present paper provides a description of the oscillation code FILOU, its main features, type of applications it can be used for, and some representative solutions. The code is actively involved in CoRoT/ESTA exercises (this volume) for the preparation for the proper interpretation of space data from the CoRoT mission. Although CoRoT/ESTA exercises have been limited to the oscillations computations for non-rotating models, the main characteristic of FILOU is, however, the computation of radial and non-radial oscillation frequencies in presence of rotation. In particular, FILOU calculates (in a perturbative approach) adiabatic oscillation frequencies corrected for the effects of rotation (up to the second order in the rotation rate) including near degeneracy effects. Furthermore, FILOU works with either a uniform rotation or a radial differential rotation profile (shellular rotation), feature which makes the code singular in the field. Keywords Methods: numerical · Stars: evolution · Stars: general · Stars: interiors · Stars: oscillations (including pulsations) · Stars: rotation · (Stars: variables:) delta Scuti PACS 96.60.Ly · 97.10.Cv · 97.10.Kc · 97.10.Sj · 97.20.Ge · 95.75.Pq · 91.30.Ab J.C. Suárez () Instituto de Astrofísica de Andalucía (CSIC), Camino Bajo de Huétor, 50, CP3004, Granada, Spain e-mail: [email protected] M.J. Goupil LESIA, Observatoire de Paris-Meudon, UMR8109, Meudon, France e-mail: [email protected]

1 Introduction Numerous oscillation codes currently provide oscillation modes for polytropes and 1D models representative of different kind of pulsating stars. The history of the oscillation code FILOU is particularly associated with δ Scuti stars. Originally developed by F. Tran Minh and L. Léon at Observatoire de Paris-Meudon (see Tran Minh and Léon 1995), the code has undergone several modifications and improvements in order to correct the oscillation frequencies for the effects of rotation. In particular, the inclusion of these corrections (up to the second order including near degeneracy effects) to the oscillation code and its numerical tests was part of my PhD. work (Suárez 2002) (hereafter S02). In that work oscillation computations were extended to the case of models including a radial differential rotation profile Ω = Ω(r), i.e., a radial-dependent differential rotation (the so-called shellular rotation). This last characteristic is, by now, unique, and makes this code singular in the field. Although FILOU is currently optimised for the study of the pulsational behaviour of intermediate-mass classical pulsators, namely δ Scuti stars and γ Doradus stars, the code is of universal use. It has been used, for instance, to model individual δ Scuti stars like the well-known Altair (Suárez et al. 2005a), or 29 Cygnus (Casas et al. 2006), as well as to study δ Scuti stars in open clusters (Suárez et al. 2002; Fox Machado et al. 2006; Suárez et al. 2007a); Moreover, it has served to model high-amplitude δ Scuti stars (Poretti et al. 2005) and to analyse the effect of rotation on Petersen diagrams (Suárez et al. 2006a, 2007b). Furthermore, it is worth highlighting the work by Suárez et al. (2006b) (from now on SGM06) which takes advantage of the code’s main feature, i.e., the computation of adiabatic oscillation in presence of shellular rotation, to analyse the effect of such type of rotation on adiabatic oscillation of moderately-fast rotating δ

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_19

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Astrophys Space Sci (2008) 316: 155–161

Scuti stars. Concerning γ Doradus stars, FILOU has participated, additionally to other modelling works, in one of the most recent and promising asteroseismic tool for the modelling of such stars, the Frequency Ratio Method (FRM), developed by Moya et al. (2005) and Suárez et al. (2005b). From the point of view of the numerics, FILOU solves full sets of ODE (Ordinary Differential Equations) in a BVP (Boundary Value Problem), using a combined Galerkine–Bsplines method which enhances the numerical precision with which the oscillation frequencies are calculated. Furthermore, as explained in the following sections, it is possible to easily modify numerous numerical parameters in order to adjust the calculation optimally for the required model, which makes of FILOU a highly versatile code.

2 The adiabatic oscillations equations and boundary conditions FILOU is mainly based on the oscillations equations and their perturbations developed in Dziembowski and Goode (1992) and Soufi et al. (1998). In S02 we describe the second-order perturbation formalism used, which includes the effects of near degeneracy and considers the presence of a radial differential rotation (shellular rotation) profile, as well as its implementation in FILOU. The notation followed is similar to that used by many other oscillation codes, but adapted to the theoretical development considered. Although the code works with different calculations schemes, namely, no rotation, Cowling approximation, and rotation (uniform and differential), in the present document only the most general case is considered, i.e., the presence of shellular rotation. In such a case, oscillations are computed from the so-called pseudo-rotating models, which, as explained in S02, are constructed by modifying the stellar structure equations such as to include the spherical symmetric contribution of the centrifugal acceleration, by means of an effective gravity geff = g − Ac (r) where g and Ac (r) are the local gravity component and the centrifugal acceleration, respectively. The effects of the non-spherical components of the deformation of the star are included through a perturbation in the oscillation equations. For instance, the perturbation of the mean density of a pseudo-rotating model ρ0 is considered of the form ρ2 = p22 (r) P2 (cos θ ), where p22 (r) is defined in SGM06 (15). Furthermore, when near degeneracy is taken into account, the eigenfrequency and the eigenfunction of a neardegenerate mode are then assumed of the form:

ωd = ω¯ 0 + ω˜ 1 + ω˜ 2 ,  αj (ξ0,j + ξ1,j ), ξ= j =a,b

(1) (2)

where ω¯ 0 = (ω0,a + ω0,b )/2 and αj represent the coefficients of the linear combination between the two considered degenerate modes. Subscripts a and b represent whatever two rotationally coupled modes. First- and second-order corrections to the eigenfrequency in presence of near degeneracy are represented by ω˜ 1 and ω˜ 2 respectively; ξ0,j and ξ1,j are the non perturbed and first-order (see definitions and details in S02 and SGM06). The computation of individual ω0,j as well as the corresponding zeroth- and first-order eigenfunctions is described in the next sections. 2.1 Oscillation frequencies of a pseudo-rotating model In order to compute the oscillation frequencies of a pseudorotating model, ω0,j the following dimensionless quantities are used   p ξr 1 y02 = φ + , (3) y01 = , r geff r ρ φ , geff r

y03 =

y04 =

1 dφ , geff dr

(4)

where geff represents the effective gravity defined in the previous section. The quantities p and φ, represent the Eulerian perturbation of the pressure and the gravitational potential (definitions of individual terms can be found in FILOU and SGM06). Considering a differential rotation profile of the form: ¯ + η0 (r)], Ω(r) = Ω[1

(5)

where Ω¯ represents the rotation frequency at the stellar surface, the eigenfrequencies (zeroth order) of a pseudorotating model are calculated from the linearised eigenvalue system: dy01 dx dy02 x dx dy03 x dx dy04 x dx

x

= λ − 3y01 +

Λ y02 , Cr σ02

= (Cr σ02 − A∗ )y01 + (A∗ + 1 − Uχ )y02 − A∗ y03 , = (1 − Uχ )y03 + y04 , =

(6)

U [A∗ y01 + Vg (y02 − y03 )] 1 − σr + Λy03 − Uχ y04 ,

which is solved by FILOU using the dimensionless variable x = r/R and R the stellar radius. In the calculation the frequency is expressed in terms of the dimensionless squared frequency σ02 =

ω2 GM/R 3

(7)

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as well as other adiabatic quantities (see Appendix in SGM06) 1 d ln p d ln ρ A = − , Γ d ln r d ln r V d ln p , Vg = , V =− d ln r Γ1 ∗

(8)

2.2 The boundary conditions The system (6) is solved with the appropriate boundary conditions y02 + y01

3 = 0, Vg

y01 − y02

 = 0, Cr σ02

(9)

d ln Mr . U= d ln r

(10)

3y01 + y04 = 0

( = 0),

y04 − y03 = 0 ( = 0)

(14) (15)

at the centre of the star and, As in SGD98, the following variables are also employed  3 r M C Ac C= , Cr = , σr = , R Mr 1 − σr g   dΩ 2 /Ω¯ 2 Ac U −3+ , U x = U + χ, χ= geff dr λ = Vg (y01 − y02 + y03 ),

Λ = ( + 1),

(11) (12) (13)

where M and mr are the stellar mass and the mass enclosed in the sphere of radius r, respectively.

y01 = 1,

(16)

y04 + ( + 1)y03 = 0,     4 + Cσ02 Λ Λ 1 − − y − y01 1 + 02 V V Cσ02 V Cσ02   +1 =0 + y03 1 + V

(17)

(18)

at the stellar surface. Figure 1 illustrates the solutions for the normalised eigenfunctions y01 , y02 , y03 , and y04 , cor-

Fig. 1 Normalised eigenfunctions y01 , y02 , y03 , and y04 , as a function of the normalised radial distance r/R, corresponding to the oscillation mode (n = 8,  = 1) calculated from a 1.8 M δ Scuti star model, with a rotational velocity of 100 km s−1 at the stellar surface

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responding to a non-radial mixed mode (n = 8,  = 1) obtained from a 1.8 M model with a surface rotational velocity of 100 km s−1 , constructed with 2100 mesh points. 2.3 First-order perturbed eigenfunctions When near-degeneracy effects are considered, first-order corrections to the eigenfunctions are required (see (2)). Considering dimensionless variables equivalent to (3, 4) with first-order perturbed quantities (ξ1,r , φ1 , p1 ), and the zerothorder solutions obtained from (6). FILOU calculates such first-order perturbed eigenfunctions solving the following system: x

dy1 Λ = λ1 − 3y1 + y2 dx Cr σ02 + (y01 + z0 )(1 + η0 ) − (η0 + σ1 )Λz0 ,

x

dy2 = (Cr σ02 − A∗ )y1 + (A∗ + 1 − Uχ )y2 − A∗ y3 dx + (σ1 + η0 )y01 − (1 + η0 )z0 ,

dy3 = (1 − Uχ )y3 + y4 , dx dy4 U x [A∗ y1 + Vg y2 − Vg y3 ] + Λy3 − Uχ y4 , = dx 1 − σr

(19)

x

where λ1 = Vg (y1 − y2 + y3 ) and z0 = y02 /Cσ02 . The horizontal component of ξ 1 can be written as follows:   y2 1 + η0 1 + η0 y01 + − σ1 z0 , z1 = + (20) Λ Λ Cσ02 where σ1 = CL − J0 represents the first-order correction of the corresponding eigenfrequency. 2.4 The first- and second-order frequency corrections Second-order frequency corrections in presence of near degeneracy (1) are coded in FILOU using the following equations (S02, SGM06)    μb + μa (2) ω˜ 2 = ± H2,ab , (21) 2 in the case that δω0 = ω0,a − ω0,b is O(Ω 2 ), and  ω˜ 2 =

νb + νa μb + μa δω02 + + 2 2 8ω¯ 0

 ±



(1) H2,ab ,

(22)

if δω0 is O(Ω). In both cases first-order near degeneracy effects are implicit, which are also calculated by FILOU using ω˜ 1 =

ω1,a + ω1,b  ± H1,ab . 2

(23)

Definitions of all terms involved are given in SGM06. The ν and μ variables contains the corrections (up to the second order) for the effect of rotation on individual eigenfrequencies ω0,j obtained from (6). Such corrections are coded in FILOU using the Saio’s notation, ¯ L − 1 − J0 ) + ωj,2 = ω0,j + Ω(C

Ω¯ 2 (D0 + m2 D1 ), (24) ω0,j

where D0 ≡ X1 + X2 and D1 ≡ Y1 + Y2 . Definitions and details can be found in (SGM06, (8–15)).

3 Structure and computation schemes FILOU is composed by a main program and some modules written in C, and two subroutines written in FORTRAN (77, 95), which read input data from the equilibrium models and calculate the near degeneracy effects for the rotationally coupled modes. Computation of radial and non-radial oscillation frequencies of a given resonant cavity (input equilibrium model) is divided into three sequential steps: first, zeroth-order oscillation frequencies (eigenvalue, ω0 ) are computed as described in Sect. 2.1; then, for each eigenfrequency, the corresponding second-order frequency corrections (without including near degeneracy effects) are calculated (see Sect. 2.4). Finally, the code selects, following certain rules (see S02 or SGM06), the rotationally coupled modes (only pairs of coupled modes are considered) and calculates their corresponding near degeneracy correcting terms (also described in Sect. 2.4).

3.1 FILOU inputs and outputs FILOU inpunts are essentially some physical quantities (see Sect. 2.1) which are read from the equilibrium model and some initial parameters. Currently, the most updated version of the code allows the use of the following input models:

• CESAM-type models: v3.*, v4.*, v5.* and 2k. • GENEVE-type models. The input parameters, which are set by the user in a text file (ASCII), are read by the code when executed. The main parameters are: • The input equilibrium model file. • Type of computation. This option allows the user to force some kind of computing regime (for instance, Cowling approximation, no rotation, uniform rotation, differential rotation). As well, the user can choose the type of output files required, for instance (only the list of frequencies, include or not the corrections for the effect of rotation, near degeneracy effects, or even the eigenfunctions).

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• Frequency domain and spherical degree  range. • Type of boundary conditions (finite/infinite V , see Sect. 2.1). • Type of node assignation (zeros of y01 or JCD method). The basic outputs provided by FILOU are the list of eigenfrequencies and eigenfunctions. However, it is possible to obtain output files containing intermediate calculation data.

4 Numerical techniques The numerical technique followed by FILOU is based on the Galerkin method, together with a finite sequence of B-splines, which is characterised for its flexibility, efficiency and robustness. Although the code was conceived to solve the numerical problem of stellar non-radial oscillations, it actually provides solution to any non-linear system of functional equations, and covers several specific cases, such as the method of finite elements, Lagrangian and/or Hermitian of any order, or even the Crank–Nicholson method of finite differences. Systems (6) and (19) are solved by approximating the eigenfunctions (3, 4) with B-Spline functions. The order of such B-Splines functions can be chosen by the user, although optimum results are obtained typically for an optimal order between 4 and 6. The coefficients for each function are computed by integration following the Galerkin method. Firstly, the code scan the user-specified frequency range to obtain a first guess for the eigenfrequencies. Then, exact eigenfrequencies are searched using either the technique of dichotomy or the Newton–Raphson method. It is worth highlighting the numerical versatility of the code, which can be optimised for the oscillations computation of very different pulsating stars, e.g., solar-like pulsators (implying high-order p modes), g-mode pulsators (γ Doradus stars, white dwarfs), δ Scuti stars, etc. This is so due to the numerous numerical parameters that can be adjusted. To name a few, the precision of the solutions (zeroth- and first-order eigenfrequencies and eigenfunctions) required, the size of the internal frequency interval in which the eigenfrequencies are searched for (this optimises the calculations in the cases of high-order and low-order frequencies), the threshold for valid solutions, etc.

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of mesh points (including Richardson extrapolation), their distribution (rezonning), and the type of boundary conditions used. The current version of FILOU does not take neither Richardson extrapolation nor rezonning into account (included in the next release of the code). Nevertheless we have carried out some numerical tests of such effects, which are illustrated in Figs. 2–4, for a typical main-sequence, δ Scuti star model (∼1.8 M ). Notice that all the effects cause a shift in the oscillation frequency which increases as far as the radial order increases. This behaviour is exponential when varying the number of mesh points (Fig. 2) and when using Richardson extrapolation approximation (Fig. 3). In both cases, for high-order p modes, the effects can reach up 10 and 14 μHz, respectively. For the frequency domain of

Fig. 2 Effect of the number of mesh points on the adiabatic oscillation spectrum of a typical δ Scuti star model (1.8 M ) as a function of the radial order n. Curves represent the frequency differences Δν = |ν2000 − ν600 | obtained with 600 and 2000 mesh points

4.1 Numerical tests and results In this section we report succinctly some of the numerical tests carried out on the code for its optimisation, as well as some of the tests carried out in the framework of the ESTA exercises. As has been shown during the ESTA exercises some of the most limiting aspects, from the numerical point of view, in the calculation of adiabatic oscillations are: the number

Fig. 3 Illustration of the impact of applying Richardson extrapolation on the oscillation frequency spectrum computations for the two models used in Fig. 2. The effect is shown through the frequency differences Δν = |νRE − ν|, where νRE represents the oscillation frequencies obtained applying the Richardson extrapolation

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δ Scuti stars, i.e. n  15, such effects can reach up 1 and 2 μHz which are nonnegligible compared with the highprecision in frequency detection that CoRoT mission is expected to provide.

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4.2 Results. The effect of shellular rotation on adiabatic oscillations

Fig. 4 Illustration of the relative difference in frequency obtained, for a given main-sequence, δ Scuti star model (1.8 M ), when using V → 0 or V → ∞ in the outer boundary conditions (see Sect. 2.1)

One of the most important works carried out using the oscillation code FILOU is the study of the effect of shellular rotation on adiabatic oscillations (SGM06). Indeed, that work takes advantage of the main feature of the code, i.e., the calculation of oscillations in presence of a radial differential rotation. Figure 5 (left panel) illustrates the effect of a shellular rotation on the radial displacement eigenfunction y01 , for a mixed mode obtained for a 1.8 M , δ Scuti star model. Such eigenfunctions can be obtained with FILOU when calculating the oscillation spectra of pseudo-rotating models (see S02) computed assuming local conservation of the angular momentum. Furthermore, the effect of shellular rotation on the oscillation frequencies is also significant for δ Scuti stars. In Fig. 5, such an effect is depicted as a function of the radial order. In that figure Δωd and Δω represent mode-to-mode frequency differences with (bottom panel) and without (top panel) taking the effect of near degeneracy into account. These quantities are calculated, for a given mode, as the difference between the oscillation frequency obtained assuming shellular rotation with the oscillation frequency obtained assuming uniform rotation. The term Cij

Fig. 5 Left panel displays weighted radial displacement eigenfunctions for a mixed mode as a function of the normalised radial distance r/R. Solid and dashed lines represent the f function computed for a shellular-rotating model. Dash-dotted lines represent the f function computed for a uniformly-rotating model. Dotted lines represent the rotation profile (scaled in the figure for clarity) given by the radial func-tion η0 (r). Right panel shows mode-to-mode frequency dif-

ferences between differentially and uniformly-rotating 1.8 M models. Symmetric, solid-line branches represent from top to bottom, differences for m = −1 and m = +1 mode frequencies respectively. For m = 0 modes, differences are represented by a dotted line. The shaded region represents an indicative frontier between the region of g and gp modes (left side) and p modes (right side). Taken from SGM06

Astrophys Space Sci (2008) 316: 155–161

represents the additional effect of near degeneracy on the oscillation frequencies for two degenerate modes i and j (see SGM06 for more details). As shown in SGM06, for g and mixed modes, significant effects (up to 3 μHz) on the oscillation frequencies are predicted. For high-frequency p modes, such effects can reach up to 1 μHz. Such effects are likely to be detectable with CoRoT data, provided numerical eigenfrequencies reach the level of precision required. See SGM06 for a detailed discussion on these results. Acknowledgements JCS acknowledges support at the Instituto de Astrofísica de Andalucía (CSIC) by an I3P contract financed by the European Social Fund and also acknowledges support from the Spanish Plan Nacional del Espacio under project ESP2004-03855-C03-01.

References Casas, R., Suárez, J.C., Moya, A., Garrido, R.: Astron. Astrophys. 455, 1019 (2006) Dziembowski, W.A., Goode, P.R.: Astrophys. J. 394, 670 (1992) Fox Machado, L., Pérez Hernández, F., Suárez, J.C., Michel, E., Lebreton, Y.: Astron. Astrophys. 446, 611 (2006)

161 Moya, A., Suárez, J.C., Amado, P.J., Martín-Ruíz, S., Garrido, R.: Astron. Astrophys. 432, 189 (2005) Poretti, E., Suárez, J.C., Niarchos, P.G., et al.: Astron. Astrophys. 440, 1097 (2005) Soufi, F., Goupil, M.J., Dziembowski, W.A.: Astron. Astrophys. 334, 911 (1998) Suárez, J.C.: Ph.D. Thesis, ISBN 84-689-3851-3, ID, 02/PA07/7178 (2002) Suárez, J.C., et al.: Astron. Astrophys. 390, 523 (2002) Suárez, J.C., Bruntt, H., Buzasi, D.: Astron. Astrophys. 438, 633 (2005a) Suárez, J.C., Moya, A., Amado, P.J., Martín-Ruíz, S., Garrido, R.: Astron. Astrophys. 443, 271 (2005b) Suárez, J.C., Garrido, R., Goupil, M.J.: Astron. Astrophys. 447, 649 (2006a) Suárez, J.C., Goupil, M.J., Morel, P.: Astron. Astrophys. 449, 673 (2006b) Suárez, J.C., Michel, E., Pérez Hernández, F., Houdek, G.: Mon. Not. Roy. Astron. Soc. (2007a, in press). DOI: 10.1111/j.13652966.2007.11927.x, arxiv.org/abs/0705.3626 Suárez, J.C., Garrido, R., Goupil, M.J.: Astron. Astrophys. (2007b, in press) Tran Minh, F., Léon, L.: In: Roxburg, I.W., Maxnou, J.L. (eds.) Physical Processes in Astrophysics. Lecture Notes in Physics, vol. 219. Springer, Berlin (1995)

LNAWENR—linear nonadiabatic nonradial waves Marian Doru Suran

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9714-2 © Springer Science+Business Media B.V. 2007

Abstract This is a presentation of the pulsational model LNAWENR (linear nonadiabatic nonradial), included in the asteroseismological package ROMOSC which was implemented in the Bucharest Observatory to be used in the CoRoT mission. Keywords Pulsating stars · Asteroseismology · Space missions

1 Introduction 1.1 Methods of studying pulsating stars In the literature there are different methods of studying stellar pulsation: – direct methods: P th = P th [S(M)] ; – comparison with observations: P th − P obs = P˜ [S(M)] ; – inverse methods (≡ asteroseismological methods)    M(S) = min P˜ −1 P th − P obs S

where we have: M.D. Suran () Astronomical Institute of the Romanian Academy, Bucharest 040557, Romania e-mail: [email protected]

– space of pulsational equations: λ P = [νnlm , Aλnlm , Φnlm , Δnlm , δnlm ]

where the parameters are respectively ν—frequencies, A, Φ—amplitudes and phases (monochromatic or in photometric colors), Δ, δ—large and small separations and (n, l, m)—spherical wavenumbers; – space of star parameters:   M = (M, t0 ), {X, Y, ZjN k }, αMLT , (i, v0 ), (D N d . . .)   = (M, Te ), {X, Y, ZjN k }, αMLT , (i, v0 ), (D N d . . .) where respectively M—mass, t0 —age, (X, Y, Zi )—chemical composition, αMLT —mixing length parameter, i—inclination, v0 —pulsational velocity, D k —various diffusion coefficients and Te —the effective temperature of the star; – space of physical and pulsational internal solutions: S = [X , Y]  Xi = p, T , r, L, ρ, κ, CV , Cp , Γ1 , Γ3 , Lrad , ∇ad , ∇ad , ∇,  dρ ∂T κ, ∂ρ κ, ∂T , ∂ρ , δ, α, , . . . dr i Yi = [(y1 , . . . , y6 )i , (ω, η, ν, P ), (Aλ , Φ λ )]lmn where p, T , r, L—are the physical parameters on the shell i in the star. We develop the small perturbations in the spherical coordinates: f  (r, θ, φ, t) = f  (r)Ylm (θ, φ)eiσ t or δf (r, θ, φ, t) = δf (r)Ylm (θ, φ)eiσ t , where Ylm (θ, φ) is the spherical harmonic, and where ( ) and δ represent the Eulerian and Lagrangian perturbations. We used as pulsational variables in the shell i: y1 = δr/r, y2 =

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_20

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(1/gr)(p  /ρ +ψ  ), y3 = (1/gr)ψ  , y4 = (1/g)(dψ  )/dr, y5 = δs/Cp , y6 = (δL/L)rad . We use also the pulsational parameters: ω the nondimensional frequency of the star (σ 2 = ω2 GM/R 3 ), η the Dziembowski stabil1 ity parameter of the pulsation (η = W/ 0 |dW/dx|dx ∝ Im(ω)/Re(ω)), ν the frequency, P the period of pulsation. For the thermodynamic variables we used the usual notations (see for example Saio and Cox 1980).

2 ROMOSC seismological method ROMOSC is an asteroseismological method implemented in the Bucharest Observatory which we plan to use in the asteroseismological mission CoRoT. ROMOSC is a software package for which a special hardware was designed. ROMOSC is a completely automated software working with whole evolutive tracks. 2.1 ROMOSC model In order to test and calibrate the ROMOSC software package we used the following model approximations: – neglection of rotation and magnetic splitting: P = [νnl ] ; – a set of parameters to be improved: M = [(M, t){X, Y, Z}] ; – a set of solutions: S = [X , Y].

Astrophys Space Sci (2008) 316: 163–166

– SEISMROM—an inverse method (per grid of tracks) representing our asteroseismological best fit (O − C)2 (between observed and calculated frequencies) for an observed star: χ2 =

N 1 (νC,i − νO,i )2 N σi 2 i=1

where σi represents the uncertainty of the frequency i; – additional software for visualization of the internal structure and modes, and for mode identification.

3 Stellar direct method. The LNAWENR numerical method As a direct method, we used the LNAWENR method implemented in the Bucharest Observatory. This method is now incorporated in the ROMOSC seismological method and software package. In our LNAWENR model, numerical method and code we can choose different sets of linear, nonadiabatic, nonradial equations, and boundary conditions. In the treatment of the radiative energy flux we can use the Eddington approximation (Frad = −(4π/3κρ)∇J , J = (ac/4π)T 4 + (1/4πκ)T (ds/dt), see Saio and Cox 1980) or the form in which the ds/dt term is neglected (see Unno et al. 1989). We use an inert treatment of convection in the form δ(∇ · Fc ) = 0. Our LNAWENR model can also choose between different kinds of approximations in the treatment of boundary conditions (conform to the discussion on this subject in Unno et al. 1989). The set of equations are discussed in Suran (1991). The linearization of the nonadiabatic equations of pulsation (see Suran 1991) gives us a system of the type:

2.2 ROMOSC software package The ROMOSC software package includes: – CESAM2k_V2 evolutionary model (Morel 2005, private communication) which provides evolutive tracks and stellar structures; – COEFNAD—a special interface between the evolutionary and pulsational models in order to automatize the application of the pulsational model on the entire evolutive track (furnishing the global matrix of the coefficients of the linear, nonadiabatic, nonradial pulsating model A(r, t|ω), see the coefficients in the equations A1– A6 in Saio and Cox (1980)); – LNAWENR pulsationary model—a linearized, nonadiabatic and nonradial model; – LNAWENRMIU—a model for the determination of the monochromatic or color magnitude variations (in amplitudes and phases) of the different modes given by LNAWENR;

r

dy = A(r, t0 |ω)y dr

completed by a normalization equation: y1,N = 1. We introduce the normalization equation in the system and extract another surface boundary condition. This boundary condition is used as an equation to determine the nondimensional frequency ω. The complete set of equations is: dy = A(r, t0 |ω)y, dr D(ω, y) = 0. r

We discretize this system using a difference scheme with a centering parameter θ (see Suran 1991):  



1 1 θ Ai+ 2 + αi yi + (1 − θ )Ai+ 2 − αi+1 yi+1 = γi ,

Astrophys Space Sci (2008) 316: 163–166

ωi+2 =

ωi+1 Di (ω, y) − ωi Di+1 (ω, y) . Di (ω, y) − Di+1 (ω, y)

The system in yi becomes bi-diagonal. The discretized system is a nonhomogeneous one and we solve it by a bottomup and top-down method for the inversion of bi-diagonal matrix. In the adiabatic case, a secant method is used for solving the additional equation D(ω, y) = 0. In order to find and isolate the roots of this equation we used a bisection method on a specified frequency interval. This it is the best automatic method for the determination of the entire adiabatic frequencies sequence on this interval. When a root is isolated, in its vicinity the complete set of equations is solved iteratively in two steps. In the first step, using an assumed value of the ω, we solve the nonhomogeneous system in yi . Using these new values of yi we solve the additional equation in ω, D(ω, y) = 0, using the secant method. The new value determined for ω is used for the next iterative step in yi , and so on, until precision in ω is reached. -conditions are imposed both as an accuracy condition to solved the D = 0 equation (|D(ω, y)| < εD ) and as a precision condition for the ω iterative determination (|ωk+1 − ωk+1 | < εω ). Typically, values of the -parameters are in the range 10−8 –10−10 . In the nonadiabatic case, the system in y is also solved using a bottom-up top-down method, but in complex variables. The additional D(ω, y) = 0 equation is solved by a NewtonRaphson method, in complex, with the starting points represented by the adiabatic frequencies.

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a particular evolutive track + the corresponding pulsational track. The LNAWENR numerical code is very fast, with a typical time of ≤ 1 s per nonadiabatic mode calculation. The code is also very robust as convergence, needing typically less than 1000 shells per star in order to converge in the given range of nondimensional frequency. With the approximations: – CESAM2k_V2: – without turbulent diffusion; – αMLT = 1.6; – Opal+Alexander opacities; – OPAL 2001 equations of state. – LNAWENR: – with zero upper boundary condition (δp = 0); – without Richardson extrapolation scheme. We used the ROMOSC software package (including LNAWENR model) for calibrations and tests.We have done this by means of the stars observed by MOST space mission: a solar-like pulsating star—ηBoo and a SPB star— HD163830, both stars with high degree modes, p and respectively g modes. The results were presented in Suran (2007). Also, the LNAWENR model was used to compare the seismic properties of the pre and post-MS stars, using entire evolutive (pre- and pos-MS) tracks (Suran et al. 2001). As an application we investigated the evolutive stage of the star V351 Ori (see Suran and Pricopi 2007).

4 LNAWENR code calibration and results ROMOSC software package is meant to perform two different tasks: – to built of a grid of theoretical tracks; – for each observed star, to use this grid in the asteroseismological best fit (O − C)2 . For the first task the LNAWENR model (included in the ROMOSC package) is used as a direct method. The grid (under construction now) consists of: • a grid of stellar evolutive tracks: (M = 0.8M –6M , ΔM = 0.01M , Z = 0.01–0.04); • a grid of stellar seismic tracks: (M = 0.8M –6M , ΔM = 0.01M , Z = 0.01–0.04, g, f, p modes). For the g, f, p modes determination we use a range in the calculation of the nondimensional frequencies of 0.09 < ω < 50. In order to build the LNAWENR grid, a special hardware was designed. We use a supercomputer with multiprocessors, each processor being used to solve, in automatic mode,

5 Conclusions LNAWENR is a direct pulsational model, method and code. The LNAWENR model could work with different options for pulsational and boundary conditions. The LNAWENR numerical method is based on a bottom-up top-down scheme (in complex) to solve the system of pulsational equations. Based on this numerical method, LNAWENR code is a very fast and robust numerical code. With it, we made different sorts of tests and calibrations using ground based observations and some stars observed by the MOST space mission. The LNAWENR code is supplemented by a code to determine the monochromatic or color variations for different modes (see Suran 1991). The resulting parameters are used to enlarge the space of equations used in the seismic inversion method ROMOSC. There are plans to use the LNAWENR model in the CoRoT and KEPLER missions. As a further development to the LNAWENR model, we are now working to implement a version that includes the differential rotation.

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References Saio, H., Cox, P.J.: Linear, nonadiabatic analysis of nonradial oscillations of massive bear main sequence stars. Astrophys. J. 236, 549–559 (1980) Suran, M.D., Goupil, M.J., Baglin, A., Lebreton, Y., Catala, C.: Comparative seismology of pre- and main sequence stars in the instability strip. Astron. Astrophys. 372, 233–240 (2001) Suran, M.D.: Pulsation properties of a Delta Scuti type star with 2.3M . Rom. Astron. J. 1, 39–50 (1991)

Astrophys Space Sci (2008) 316: 163–166 Suran, M.D.: Inversion methods in stellar polsation. In: Fifty Years of Romanian Astrophysics. AIP Conference Proceedings, vol. 895, pp. 219–223 (2007) Suran, M.D., Pricopi, D.: Asteroseismological results for the star V351 Ori. In: Fifty Years of Romanian Astrophysics. AIP Conference Proceedings, vol. 895, pp. 224–227 (2007) Unno, W., Osaki, Y., Ando, H., Saio, H., Shibahashi, H.: Nonradial Oscillations of Stars, 2nd edn. University of Tokyo Press, Tokyo (1989)

Grids Reference grids of stellar models and oscillation frequencies Data from the CESAM stellar evolution code and ADIPLS oscillation programme Yveline Lebreton · Eric Michel

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-008-9781-z © Springer Science+Business Media B.V. 2008

Abstract We present grids of stellar models and their associated oscillation frequencies that have been used by the CoRoT Seismology Working Group during the scientific preparation of the CoRoT mission. The stellar models have been calculated with the CESAM stellar internal structure and evolution code while the oscillation frequencies have been obtained from the CESAM models by means of the ADIPLS adiabatic oscillation programme. The grids cover a range of masses, chemical compositions and evolutionary stages corresponding to those of the CoRoT primary targets. The stellar models and oscillation frequencies are available on line through the Evolution and Seismic Tools Activity (ESTA) web site. Keywords Stars: evolution · Stars: interiors · Stars: oscillations · Methods: numerical PACS 97.10.Cv · 97.10.Sj · 95.75.Pq

sation of the stellar content in the fields surrounding the potential primary targets have required to locate all considered stars (or stellar systems) in the H–R diagram and eventually to estimate their mass, evolutionary stage and expected oscillation spectra. For that purpose we have calculated the internal structure and evolution of stars in a range of masses, chemical compositions and evolution stages of interest for the modelling of CoRoT targets (Michel and Baglin 2005). We have used the CESAM1 stellar evolution code (Morel 1997; Morel and Lebreton 2007). We have selected several models along each evolution sequence and we have calculated their oscillation frequencies by means of the ADIPLS2 programmes (Christensen-Dalsgaard 1982, 2008). In this paper we briefly present these grids of models and oscillation frequencies. Section 2 presents the CESAM numerical code and the input physics and initial parameters used to calculate the grids while Sect. 3 presents the oscillation data obtained from the ADIPLS adiabatic oscillation code. The present models and associated oscillation frequencies have been made available on the ESTA web site.3

1 Introduction During the scientific preparation of the CoRoT mission, a reference frame in the H–R diagram had to be provided for the CoRoT potential target stars. In particular, the critical selection of the primary targets and the census and characteriY. Lebreton () Observatoire de Paris, GEPI, CNRS UMR 8111, 5 Place Janssen, 92195 Meudon, France e-mail: [email protected] E. Michel Observatoire de Paris, LESIA, CNRS UMR 8109, 5 Place Janssen, 92195 Meudon, France

2 CoRoT/CESAM stellar models 2.1 Input physics and numerical aspects The CESAM stellar evolution code has been written and developed by P. Morel and collaborators since the late 1980s. 1 Code

d’Evolution Stellaire Adaptatif et Modulaire.

2 Aarhus Adiabatic Pulsation Package, available at http://www.phys.au.

dk/ jcd/adipack.n. 3 http://www.astro.up.pt/corot/models/

and also on the Paris Observatory web site at http://wwwusr.obspm.fr/~lebreton/Modeles/CESAM_ COROT.html.

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_21

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Astrophys Space Sci (2008) 316: 167–171

The general description of the code is given by Morel and Lebreton (2007) but see also Morel (1997). CESAM is a public code. The source is available on-line at the web site— http://www.obs-nice.fr/cesam—together with a short guide of directions for use and a comprehensive document giving a complete description of the numerics and of the physics implemented. All the models presented here were calculated in 2004 with the 4.2 version of CESAM written in Fortran77. We point out that the present version of CESAM, CESAM2k, was re-programmed in Fortran95; it has also been updated with new input physics and has been enriched with the inclusion of new physical processes (Morel and Lebreton 2007). The models have be initialised on the zero age main sequence (ZAMS) and the evolution has been followed up to the beginning of the red giant branch. The models have typically 600 mesh points in the interior while the atmosphere is restored on a grid of typically 100 grid points. During hydrogen burning the time step is adjusted according to the relative changes of the hydrogen abundances. In the present calculations it takes from 165 to 420 time steps to evolve from the ZAMS to the ascent of the red giant branch, depending on the mass. All the models have been calculated with the same given set of standard input physics. Here, we have neglected microscopic and turbulent diffusion processes as well as rotation or magnetic fields. The input physics are the following:

in the region corresponding to both the convective and overshooting regions and in the overshooting region the temperature gradient is taken to be equal to the adiabatic gradient. – Atmosphere. Eddington’s grey T (τ ) law is used for the atmosphere calculation. The hydrostatic equation is integrated in the atmosphere, starting at the optical depth τ = 10−4 and the connection with the envelope is made at τ = 10 to ensure the validity of the diffusion approximation (Morel et al. 1994). At this level, we insure the continuity of the variables and of their first derivatives with space. The radius of the star is taken to be the bolometric radius, i.e. the radius at the level where the local temperature equals the effective temperature (τ = 2/3 for the Eddington’s law). – Initial abundances of the elements and heavy elements mixture. All models were calculated with the classical GN93 solar mixture of heavy elements from Grevesse and Noels (1993) corresponding to a solar metals to hydrogen ratio (Z/X) = 0.0245 where Z and X are the abundances in mass fraction of heavy elements and hydrogen respectively. In the nuclear reaction network the initial abundance of each chemical species is split between its isotopes according to the isotopic ratio of nuclides. The initial amount of 2 H is assumed to be converted in 3 He.

– Equation of State. We used the so-called CEFF analytical equation of state (Christensen-Dalsgaard and Däppen 1992) which is an improved version of the EFF (Eggleton, Faulkner, and Flannery 1973) equation of state including the Coulomb corrections to pressure in the so-called Debye-Hückel approximation. – Opacities. We used the OPAL95 opacity tables (Iglesias and Rogers 1996) complemented at low temperatures (T  104 K) by the Alexander and Ferguson (1994) tables. The metal mixture of the opacity tables is fixed to the initial mixture of the model (see below). – Nuclear reaction rates. We used the basic pp and CNO reaction networks up to the 17 O(p, α)14 N reaction. In the present models the CESAM code takes 7 Li, 7 Be and 2 H at equilibrium. The nuclear reaction rates are from Caughlan and Fowler (1988). Weak screening is assumed under Salpeter’s formulation (1954). – Convection and overshooting. We use the classical mixing length treatment of Böhm-Vitense (1958) under the formulation of Henyey, Vardya, and Bodenheimer (1965) taking into account the optical thickness of the convective bubble. To find the location of the onset of convection, we used Schwarzschild’s criterion. In models calculated with overshooting, the convective core is extended on a distance lov = αov × min(Hp , Rcc ) where αov is the overshooting parameter, Hp the pressure scale height and Rcc the radius of the convective core. The core is mixed

2.2 Initial parameters of the models The calibration of a solar model in luminosity and radius with the input physics given above leads to an initial abundance of helium Y = 0.2672, and to a solar mixing-length parameter αMLT, = 1.63. The corresponding initial hydrogen and metal abundances are respectively X = 0.7153 and Z = 0.0175. We have calculated models for two values of the [Fe/H] ratio: [Fe/H] = 0.0 (solar) and −0.1 (metal deficient). The metallicity of the models Z has been obtained from the relation [Fe/H] = log(Z/X) − log(Z/X) . The individual abundances of metals have been scaled on the solar ones. For models with non solar metallicity, we have derived the helium abundance in mass fraction Y according to the relation Y = Yp + Z(Y/Z), where Y/Z is the helium to metal enrichment ratio and Yp is the primordial helium abundance. We took Y/Z = (Y − Yp )/Z ∼ 1.3 and Yp = 0.245 (see Olive and Skillman 2004). All the models were calculated with a mixing-length parameter equal to the solar one. Finally, for solar metallicity we calculated two grids of models, one without overshooting and one including overshooting with the value αov = 0.15. We have calculated models with masses in the range covered by the CoRoT targets: 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 3.0, 4.0, 5.0 and 10.0 M .

Astrophys Space Sci (2008) 316: 167–171

Fig. 1 Grid G1: evolutionary sequences in the H–R diagram for models with solar metallicity [Fe/H] = 0.0 corresponding to Z/X = 0.0245, helium abundance Y = 0.2672, no overshooting

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Fig. 2 Grid G2: evolutionary sequences in the H–R diagram for models with solar metallicity [Fe/H] = 0.0 corresponding to Z/X = 0.0245, helium abundance Y = 0.2672 and overshooting αov = 0.15

2.3 Model grids We present three grids of models. Grid G1 consists in evolutionary sequences for the 12 values of the stellar mass listed above (between 1.0 and 10.0 M ), a solar metallicity and no overshooting. This grid covers the domain of mass of main CoRoT targets, including solar-like oscillators and δ Scuti and β Cephei pulsators. In addition to this standard grid, we have calculated two grids restricted to solar-type and δ Scuti stars, i.e. for the 8 lower values of masses listed above, between 1.0 and 2.4 M . Grid G2 consists in evolutionary sequences with solar metallicity and overshooting (αov = 0.15) and grid G3 consists in evolutionary sequences with [Fe/H] = −0.1 and no overshooting. Figures 1, 2 and 3 display the evolutionary sequences in the H–R diagram for the 3 grids of CESAM models available on the ESTA web site.4 For each evolutionary sequence, we also provide on the web site a file in which the properties of the models are displayed: mass, logarithm of the effective temperature log Teff , bolometric magnitude, logarithm of the luminosity log(L/L ), logarithm of the surface gravity log g and age.

3 ADIPLS oscillation frequencies Using the ADIPLS stellar oscillation program, we have calculated the oscillation frequencies for a selection of models 4 http://www.astro.up.pt/corot/models/

Fig. 3 Grid G3: evolutionary sequences in the H–R diagram for models with sub-solar metallicity [Fe/H] = −0.1 corresponding to Z/X = 0.0194, helium abundance Y = 0.2632 and no overshooting

of the G2 grid (i.e. the grid corresponding to models with solar metallicity [Fe/H] = 0.0, Z/X = 0.0245, helium abundance Y = 0.2672 and overshooting αov = 0.15). In relation with CoRoT targets, we have chosen two values of the mass in the G2 grid: M = 1.2 M which is representative of solar like oscillators and M = 1.8 M which is representative of δ Scuti pulsators. For the M = 1.2 M sequence, we have selected models along the evolutionary track by steps of 100 Myr, except on the hook at the end of the main-sequence where steps are of 10 Myr. For the M = 1.8 M sequence, we have selected models by steps of 50 Myr on the main–

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sequence and by steps of 10 Myr from the red point to the bottom of the red giant branch. For each mass we provide two archives on the ESTA web site, a Oi and a Fi archive, where i = 1 and 2 for M = 1.2 and M = 1.8 M respectively. The Oi archive contains ASCII files, with a .osc extension, each corresponding to a selected model on the evolutionary sequence. The .osc files can be read by means of the programme tools available in the ADIPLS package and described on the ESTA web site.5 They list the global properties of the considered model (name, mass, age, initial composition, luminosity, photospheric radius, etc.) as well as a wide range of model variables at each mesh point (distance r to the centre, mass inside the sphere of radius r, pressure, density, temperature, chemical composition, opacity and several other physical parameters including quantities of interest for the computation of adiabatic oscillations like the Brunt-Väisälä frequency). The Fi archive contains ASCII files, with a .freq extension, each corresponding to a selected model on the evolutionary sequence. The .freq files are the result of the frequency calculation by the ADIPLS program. They list the mass, radius, central pressure and density of the model and the frequency of oscillation, for each mode identified by its degree  and order n. We point out that these files are not corrected for problems of numbering of the modes that occur for modes of  = 1 in the region of the fundamental.

4 Conclusion On the ESTA web site we have provided grids of stellar models and oscillation frequencies that have been used by the CoRoT Seismology Working Group during the scientific preparation of the CoRoT mission. The stellar models are provided for a range of masses between 1.0 and 10.0 M and chemical compositions [Fe/H] = 0.0 and −1.0 dex representative of the main CoRoT targets (solar-type stars, δ Scuti stars and β Cephei stars). Oscillation frequencies are provided all along the main-sequence track for a solartype oscillator (M = 1.2 M ) and a δ Scuti pulsator (M = 1.8 M ). So far, these reference grids have been used to locate CoRoT potential targets in the H–R diagram in the process of target selection (see Fig. 1 in Lebreton et al. 2008a) and to study some δ Scuti candidates for CoRoT (Poretti et al. 2005). On the ESTA web site, other grids of stellar models and oscillation frequencies can be found. These stellar models have been calculated either with the CESAM or the CLÉS code and the oscillation frequencies have been calculated 5 http://www.astro.up.pt/corot/ntools/modconv

Astrophys Space Sci (2008) 316: 167–171

either with the ADIPLS, POSC or LOSC code. All this material (codes and grids) is described in this volume by Lebreton et al. (2008a), Montalbán et al. (2008a), Marques et al. (2008). We point out that the thorough comparisons of the stellar evolutionary codes and oscillations codes performed under ESTA have shown a very good compatability between models calculated by the CESAM and CLÉS codes which gives us confidence in the use of the present grids (see Lebreton et al. 2008b; Montalbán et al. 2008b; Moya et al. 2008).

References Alexander, D.R., Ferguson, J.W.: Low-temperature Rosseland opacities. APJ (Linc. Neb.) 437, 879–891 (1994). doi:10.1086/175039 Böhm-Vitense, E.: Über die Wasserstoffkonvektionszone in Sternen verschiedener Effektivtemperaturen und Leuchtkräfte. Mit 5 Textabbildungen. Z. Astrophys. 46, 108 (1958) Caughlan, G.R., Fowler, W.A.: Thermonuclear reaction rates V. At. Data Nucl. Data Tables 40, 283 (1988) Christensen-Dalsgaard, J.: On solar models and their periods of oscillation. MNRAS 199, 735–761 (1982) Christensen-Dalsgaard, J.: ADIPLS: Aarhus adiabatic pulsation package. Astrophys. Space Sci. (2008). doi:10.1007/ s10509-007-9689-z Christensen-Dalsgaard, J., Däppen, W.: Solar oscillations and the equation of state. Astron. Astrophys. 4, 267–361 (1992) Eggleton, P.P., Faulkner, J., Flannery, B.P.: An approximate equation of state for stellar material. Astron. Astrophys. 23, 325 (1973) Grevesse, N., Noels, A.: Cosmic abundances of the elements. In: Origin and Evolution of the Elements. Cambridge Univ. Press, Cambridge (1993) Henyey, L., Vardya, M.S., Bodenheimer, P.: Studies in stellar evolution. III. The calculation of model envelopes. APJ (Linc. Neb.) 142, 841 (1965) Iglesias, C.A., Rogers, F.J.: Updated opal opacities. APJ (Linc. Neb.) 464, 943 (1996) Lebreton, Y., Monteiro, M.J.P.F.G., Montalbán, J., et al.: The CoRoT evolution and seismic tool activity. Astrophys. Space Sci. (2008a). doi:10.1007/s10509-008-9771-1 Lebreton, Y., Montalbán, J., Christensen-Dalsgaard, J., Roxburgh, I., Weiss, A.: CoRoT/ESTA—TASK 1 and TASK 3 comparison of the internal structure and seismic properties of representative stellar models. Astrophys. Space Sci. (2008b). doi:10.1007/s10509-008-9740-8 Marques, J.P., Monteiro, M.J.P.F.G., Fernandes, J.: Grids of stellar evolution models for asteroseismology (CESAM + POSC). Astrophys. Space Sci. (2008). doi:10.1007/s10509-008-9786-7 Michel, E., Baglin, A.: Space and ground-based seismology. In: Casoli, F., Contini, T., Hameury, J.M., Pagani, L. (eds.) SF2A-2005: Semaine de l’Astrophysique Francaise, p. 283, December 2005 Montalbán, J., Miglio, A., Noels, A., Scuflaire, R.: Grids of stellar models and frequencies with CLES + LOSC. Astrophys. Space Sci. (2008a). doi:10.1007/s10509-007-9718-y Montalbán, J., Lebreton, Y., Miglio, A., Scuflaire, R., Morel, P., Noels, A.: Thorough analysis of input physics in CESAM and CLÉS codes. Astrophys. Space Sci. (2008b). doi:10.1007/s10509-008-9803-x Morel, P.: CESAM: A code for stellar evolution calculations. Astron. Astrophys. Suppl. Ser. 124, 597–614 (1997) Morel, P., Lebreton, Y.: CESAM: Code d’evolution stellaire adaptatif et modulaire. Astrophys. Space Sci. (2007) doi:10.1007/ s10509-007-9663-9

Astrophys Space Sci (2008) 316: 167–171 Morel, P., van’t Veer, C., Provost, J., Berthomieu, G., Castelli, F., Cayrel, R., Goupil, M.J., Lebreton, Y.: Incorporating the atmosphere in stellar structure models: the solar case. Astron. Astrophys. Suppl. Ser. 286, 91–102 (1994) Moya, A., Christensen-Dalsgaard, J., Charpinet, S., et al.: Intercomparison of the g-, f - and p-modes calculated using different oscillation codes for a given stellar model. Astrophys. Space Sci. (2008). doi:10.1007/s10509-007-9717-z Olive, K.A., Skillman, E.D.: A realistic determination of the error on the primordial helium abundance: steps toward nonparametric

171 nebular helium abundances. APJ (Linc. Neb.) 617, 29–49 (2004). doi:10.1086/425170 Poretti, E., Alonso, R., Amado, P.J., et al.: Preparing the COROT space mission: new variable stars in the galactic anticenter direction. Astron. J. 129, 2461 (2005) Salpeter, E.E.: Electrons screening and thermonuclear reactions. Aust. J. Phys. 7, 373 (1954)

Grids of stellar evolution models for asteroseismology (CESAM + POSC) João P. Marques · Mário J.P.F.G. Monteiro · João M. Fernandes

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-008-9786-7 © Springer Science+Business Media B.V. 2008

Abstract In this paper we present a grid of stellar evolution models, computed with an up-to-date physical description of the internal structure, using the Code d’Évolution Stellaire Adaptatif et Modulaire (CESAM). The evolutionary sequences span from the pre-main sequence to the beginning of the Red Giant Branch and cover an interval of mass typical for low and intermediate mass stars. The chemical composition (both helium and metal abundance) is the solar one. The frequencies of oscillation, computed for specific stellar models of the grid using the Porto Oscillations Code (POSC), are also provided. This work was accomplished in order to support the preparation of the CoRoT mission within the Evolution and Seismic Tools Activity (CoRoT/ESTA). On the other hand, the grid can also be used, more generally, to interpret the observational properties of either individual stars or stellar populations. The grids (data and documentation) can be found at http://www.astro.up.pt/corot/models/cesam.

J.P. Marques · M.J.P.F.G. Monteiro () Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal e-mail: [email protected] M.J.P.F.G. Monteiro Departamento de Matemática Aplicada da Faculdade de Ciências, Universidade do Porto, Porto, Portugal J.P. Marques · J.M. Fernandes Grupo de Astrofísica da Universidade de Coimbra, Coimbra, Portugal J.M. Fernandes Departamento de Matemática da Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Coimbra, Portugal

Keywords Stars: evolution · Stars: interiors · Stars: oscillations

1 Introduction The computation of large grids of stellar evolution tracks is particularly useful to study a large number of stars for which photometric and/or spectroscopic observations are available. During the last 20 years, both the improvement of the physical description of stellar interiors (in particularly, the opacities and the equation of state) and the computation facilities allowed many groups to produce a huge amount of stellar models. These cover from very-low mass stars (Chabrier and Baraffe 2000) to higher mass ones (Maeder and Meynet 2000), for different values of metal abundances and ages, including the solar standard models (Bahcall et al. 2006). It is difficult to imagine that the latest progress in stellar population synthesis (Gallart et al. 2005) or on the chemical evolution in the Galaxy would have been possible without the help of those grids (Prantzos 2007). The application of a large number of stellar evolution models to resolve single stars can, also, be very important in order to constrain the models themselves. For example, the comparison between isochrones and Hipparcos Population II stars, in the plan of the HR diagram, pointed out the limitation of those models to reproduce the observations of metal poor stars (Lebreton 2000). With the advent of helio- and astero-seismology new observational constraints for theoretical stellar models have become available. It is very well know how sensitive, for instance, the large and the small frequency separations are to both the global characteristics of stars (mass, radius and chemical composition) and to the theoretical treatment of the stellar internal structure (e.g. Monteiro et al. 2002). On

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174 Table 1 Summary of the input physics adopted to calculate the evolution models in the grids described in this work. Further details are also provided by Lebreton et al. (2008)

Astrophys Space Sci (2008) 316: 173–178 Item of the physics

Selection

Reference(s)

Equation of State

OPAL

Rogers et al. (1996, 2005 Tables)

Opacities

OPAL

Iglesias and Rogers (1996) + Alexander and Ferguson (1994)

Nuclear reaction rates

NACRE

Angulo et al. (1999)

Convection

MLT

Böhm-Vitense (1958) + Henyey et al. (1965)

Overshoot

none

–

Diffusion/settling

none

–

Mixture

solar

Grevesse and Noels (1993)

Atmosphere

gray

–

the other hand, for fixed input physics, the models can be particularly interesting to predict the frequencies of oscillation of stars not (yet!) observed. The prediction can be important to select the stellar targets that will be measured. If the observational program is space based, those predictions are not only important but absolutely crucial to define the final target list (Michel et al. 2006). The grids reported here have already been used to support the seismic study of pre-main sequence stars (Ripepi et al. 2007; Ruoppo et al. 2007). In this work we briefly describe the grids of stellar models and their frequencies of oscillations produced in order to support the preparation of the CoRoT Mission (Fridlund et al. 2006). We start by identifying the physics and parameters that have been used to produce the models and then we discuss the data made available online.

Table 2 Summary of the stellar parameters used to calculate the evolution models in the grids described in this work. Here M is the mass of the sun Stellar parameter

Value

Helium abundance

Y = 0.28

Abundance of metals

Z/X = 0.02857

MLT parameter

α = 1.8

Stellar mass

0.8 ≤ M/M ≤ 8.0

summary). The zero age main sequence (ZAMS) and the terminal age main sequence (TAMS) in the grids are defined as follows: ZAMS: when 99% of all energy is produce by nuclear reactions; TAMS: when the abundance of hydrogen at the centre is Xc = 0.01 ± 0.0001.

2 Parameters of the grids The present grid has been computed using the CESAM code (Morel 1997; Morel and Lebreton 2008), version 2k. In order to produce a grid with different initial conditions (birthline) an extension of the above version of CESAM, developed by Marques (2008), has been used.

3 Initial conditions for stellar evolution Two options have been considered for the initial conditions of the evolution for each stellar mass.

2.1 The input physics

3.1 GRID A: evolution from fully convective spheres

The input physics used to calculate the models is similar to the reference physics adopted by the CoRoT/ESTA group (Lebreton et al. 2008). It is summarised in Table 1. The models consider neither microscopic diffusion of chemical elements nor overshoot of the convective core.

The initial models used for the pre-main sequence (PMS) evolution have tradicionaly been fully convective models with arbitrarily large radii (about 50 R , e.g. Hayashi 1966; Iben 1965). The reason behind this is that it was thought that stars were formed by the homologous contraction of the protostellar cloud; this non-hydrostatic phase was assumed to be so fast that the energy radiated was much lower than the gravitational energy of the cloud. A simple energy budget argument (Hayashi 1966) lead then to this initial large radius. This simple picture of star formation is no longer held to be adequate (Palla and Stahler 1991, 1992). Nevertheless, the high convenience of the so-called “classical” initial mod-

2.2 Stellar parameters We have calculated evolution models with masses ranging from 0.8 to 8 M , from the contraction phase to the beginning of the red giant branch (RGB). All models were computed using the solar abundance of metals and helium and also the solar mixing length parameter (see Table 2 for a

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Fig. 2 Stellar age at the ZAMS and TAMS for different stellar masses, in the two grids. The age at the ZAMS decreases must faster with stellar mass for the grid using the birthline (B) than the grid starting from the classical (politrope) initial condition (A). There is no significant difference for the age at the TAMS in the two grids

Fig. 1 HR diagrams for models of different masses at different ages: top panel—pre-main sequence evolution; bottom panel—main sequence evolution (full lines) and post-main sequence evolution (dotted lines). Only PMS models with an age above 1 Myr are shown. The number indicates the mass of the model in solar masses (M )

els means that they continue to be widely used; besides, the influence of the initial conditions fades as the star evolves on the PMS. The example of the grid that uses this initial condition is illustrated in Fig. 1, in particular in the upper panel where the pre-main sequence evolution from an age of about 1 Myr is shown for models of different stellar masses. 3.2 GRID B: evolution from a birthline It is now known that stars are formed in a very nonhomologous way: an hydrostatic core is formed first, on to which matter from the parental cloud accretes, either directly or through an accretion disk. Consequently, more realistic initial conditions are used for the calculation of this second grid. The most important reason to do so is that the PMS phase becomes shorter with increasing stellar mass, i.e. stars with higher masses are born nearer the ZAMS (see Fig. 2). More realistic initial conditions are not relevant for

stars with masses lower than ∼ 1.5 M , but they become more important for stars with higher masses. In particular, stars with masses higher than ∼ 7 M are born so near the ZAMS that they do not have a proper PMS phase. We started the calculation of the initial models with a 0.5 M model evolved along the Hayashi track until its radius was the same as the radius of an accreting protostar with the same mass in Palla and Stahler (1991). Then accretion is turned on at a constant rate of M˙ = 10−5 M year−1 . The actual mass accretion rates vary with each individual case, and are not in any way constant (Schmeja and Klessen 2004). In particular, the mass accretion rate seams to increase with increasing mass. Given the goal of this grid, it is not necessary to include all the complicated phenomena of stellar birth in the calculation of the initial models; our accretion rate is intended to represent a typical value. The evolutionary track of the accreting protostar on the HR diagram is called the birthline. An initial model for PMS evolution is obtained when the accreting protostar reaches the mass of the model. Both, the birthline and some premain sequence evolutionary tracks, are illustrated in Fig. 3. An important influence on the location of the birthline is the deuterium content of the parental cloud. The mass-radius relation of the protostar is set to a high degree by deuterium burning. In “classical” PMS stars, the structural influence of deuterium burning is small: due to its low abundance, the deuterium burning phase is short. In accreting protostars, however, continuous accretion ensures that fresh deuterium is continuously suplied (see Palla and Stahler 1991). Following (Linsky et al. 2006), we select a lower [D/H] number ratio than typically used; this was [D/H] = 1.5×10−5 .

176

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4 Data The data that has been produced, as part of the grid, includes: • evolutionary sequences (files of type *.dat and *.HR), • stellar models at several ages (files of type *.OSC), • oscillation frequencies (files of type *.freq). All files are available online for download at http://www. astro.up.pt/corot/models/cesam/. Specific information or data about the grids can also be provided on request from the authors. 4.1 Evolutionary tracks and stellar models Fig. 3 HR diagram displaying the birthline (full line) and the pre-main sequence evolution of some models (the number indicates the mass of the model in solar masses M )

Fig. 4 Evolution of the size of the convective regions (shaded area) with time, in models of GRID B, having masses M=1.0 M (top panel) and M=3.0 M (bottom panel). The full evolution from the birthline to the RGB are represented (dotted vertical lines represent the ZAMS and the TAMS)

The evolution of models of different stellar masses are provided using the *.dat files (in same cases de *.HR output files from CESAM are also included). Here the evolution of the global parameters of the star is registered. The panels in Fig. 4 illustrate, as an example, how the size of the convective zones changes with time for models of 1.0 M and

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3.0 M stars. The HR diagrams in Figs. 1–3 are also examples of the data available in these files. For each value of stellar mass, several interior models at different ages can be found in the *.OSC files. For the details on the format and contents of the files please see the documentation available at the website. The files with the data on the internal structure of a star (with a specific mass) at a given time are grouped according to the following phases of evolution:

Acknowledgements This work was supported in part by the European Helio- and Asteroseismology Network (HELAS), a major international collaboration funded by the European Commission’s Sixth Framework Programme. JPM and MJPFGM acknowledge the support by FCT and FEDER (POCI2010) through projects POCI/CTEAST/57610/2004, POCI/V.5/B0094/2005 and PTDC/CTE-AST/ 65971/2006.

prems—pre-main sequence evolution, ms—main sequence evolution, postms—post-main sequence evolution.

Alexander, D.R., Ferguson, J.W.: Low-temperature Rosseland opacities. Astrophys. J. 437, 879 (1994) Angulo, C., Arnould, M., Rayet, M., et al.: A compilation of chargedparticle induced thermonuclear reaction rates. Nucl. Phys. A 656, 3 (1999) Bahcall, J.N., Serenelli, A.M., Basu, S.: 10,000 standard solar models: a Monte Carlo simulation. Astrophys. J. Suppl. Ser. 165, 400 (2006) Böhm-Vitense, E.: Über die Wasserstoffkonvektionszone in Sternen verschiedener Effektivtemperaturen und Leuchtkräfte. Mit 5 Textabbildungen. Z. Astrophys. 46, 108 (1958) Chabrier, G., Baraffe, I.: Theory of low-mass stars and substellar objects. Annu. Rev. Astron. Astrophys. 38, 337 (2000) Fridlund, M., Baglin, A., Lochard, J., Conroy, L. (eds.): The CoRoT Mission, ESA SP-1306. ESA Publications Division, Noordwijk (2006) Gallart, C., Zoccali, M., Aparicio, A.: The adequacy of stellar evolution models for the interpretation of the color-magnitude diagrams of resolved stellar populations. Annu. Rev. Astron. Astrophys. 43, 387 (2005) Grevesse, N., Noels, A.: Cosmic abundances of the elements. In: Origin and Evolution of the Elements (1993) Hayashi, C.: Evolution of protostars. Annu. Rev. Astron. Astrophys. 4, 171 (1966) Henyey, L., Vardya, M.S., Bodenheimer, P.: Studies in stellar evolution. III. The calculation of model envelopes. Astrophys. J. 142, 841 (1965) Iben, I.J.: Stellar evolution. I. The approach to the main sequence. Astrophys. J. 141, 993 (1965) Iglesias, C.A., Rogers, F.J.: Updated opal opacities. Astrophys. J. 464, 943 (1996) Lebreton, Y.: Stellar structure and evolution: deductions from hipparcos. Annu. Rev. Astron. Astrophys. 38, 35 (2000) Lebreton, Y., Monteiro, M.J.P.F.G., Montalbán, J., et al.: The CoRoT evolution and seismic tools activity: goals and tasks. Astrophys. Space Sci. (2008). doi:10.1007/s10509-008-9771-1 Linsky, J.L., Draine, B.T., Moos, H.W., Jenkins, E.B., Wood, B.E., Oliveira, C., Blair, W.P., Friedman, S.D., Gry, C., Knauth, D., Kruk, J.W., Lacour, S., Lehner, N., Redfield, S., Shull, J.M., Sonneborn, G., Williger, G.M.: What is the total deuterium abundance in the local galactic disk? Astrophys. J. 647, 1106 (2006) Maeder, A., Meynet, G.: The evolution of rotating stars. Annu. Rev. Astron. Astrophys. 38, 143 (2000) Marques, J.P.: Pre-main sequence evolution of intermediate mass stars. Ph.D. Thesis, Faculdade de Ciências da Universidade do Porto (2008) Michel, E., Baglin, A., Auvergne, M. et al.: In: Fridlund, M., Baglin, A., Lochard, J., Conroy, L. (eds.) The CoRoT Mission. ESA SP-1306, p. 39 (2006) Monteiro, M.J.P.F.G.: The Porto Oscillations Code (POSC). Astrophys. Space Sci. (2008) doi:10.1007/s10509-008-9802-y Monteiro, M.J.P.F.G., Christensen-Dalsgaard, J., Thompson, M.J.: Asteroseismic inference for solar-type stars. In: Favata, F., Roxburgh, I.W., Galadí-Enríquez, D. (eds.) Stellar Structure and Habitable Planet Finding, ESA SP-485, p. 291 (2002)

Each model has a *.OSC file containing all major quantities describing the internal structure of the star (and used to calculate the seismic properties). 4.2 Frequencies of oscillation For some of the models of the grid the frequencies of oscillation were computed using the POSC code (Monteiro 2008). These are the *.freq files containing the basic mode parameters. Only low degree modes are included (0 ≤ l ≤ 3) with mode orders from n = 1 and with a frequency below the acoustic cutoff frequency of the models. In same cases g-modes are also given (identified with a negative mode order). The boundary condition at the atmosphere used in the computations assumes an isothermal atmosphere at the top. For further details please see Monteiro (2008) and the documentation available at the website.

5 Concluding remarks In this paper we have reviewed the basic aspects of reference grids of stellar models produced with the CESAM code (Morel and Lebreton 2008). Two grids are provided with different initial conditions: A—fully convective sphere (politrope); B—the birthline from Palla and Stahler (1991, 1992). The remaining aspects of the physics are fixed as in Lebreton et al. (2008). The grid also provides frequencies of oscillation for p- and g-modes of some of the models. The major goal of the grids is to support the preparation and exploitation of space missions for asteroseismology and in particular the French CoRoT space mission (Fridlund et al. 2006) launched in December 2006. All data is avaliable online.1 1 http://www.astro.up.pt/corot/models/cesam/.

References

178 Morel, P.: CESAM: a code for stellar evolution calculations. Astron. Astrophys. Suppl. Ser. 124, 597 (1997) Morel, P., Lebreton, Y.: CESAM: code d’evolution stellaire adaptatif et modulaire. Astrophys. Space Sci. (2007) doi:10.1007/s10509-007-9663-9 Palla, F., Stahler, S.W.: The evolution of intermediate-mass protostars. I—Basic results. Astrophys. J. 375, 288 (1991) Palla, F., Stahler, S.W.: The evolution of intermediate-mass protostars. II—Influence of the accretion flow. Astrophys. J. 392, 667 (1992) Prantzos, N.: An introduction to galactic chemical evolution. In: Charbonnel, C., Zahn, J.P. (eds.) Stellar Nucleosynthesis: 50 Years After B2FH. EAS publications Series (2007), arXiv:0709.0833

Astrophys Space Sci (2008) 316: 173–178 Ripepi, V., Bernabei, S., Marconi, M., et al.: Discovery of δ Scuti pulsation in the Herbig Ae star VV Serpentis. Astron. Astrophys. 462, 1023 (2007) Rogers, F.J., Swenson, F.J., Iglesias, C.A.: OPAL equation-of-state tables for astrophysical applications. Astrophys. J. 456, 902 (1996) Ruoppo, A., Marconi, M., Marques, J.P., et al.: A theoretical approach for the interpretation of pulsating PMS intermediate-mass stars. Astron. Astrophys. 466, 261 (2007) Schmeja, S., Klessen, R.S.: Protostellar mass accretion rates from gravoturbulent fragmentation. Astron. Astrophys. 419, 405 (2004)

Grids of stellar models and frequencies with CLÉS + LOSC Josefina Montalbán · Andrea Miglio · Arlette Noels · Richard Scuflaire

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9718-y © Springer Science+Business Media B.V. 2007

Abstract We present a grid of stellar models, obtained with the CLÉS evolutionary code, following the specification of ESTA-Task1, and the corresponding seismic properties, computed with the LOSC code. We provide a complete description of the corresponding files that will be available on the ESTA web-pages.

PACS 97.10.Cv · 97.10.Sj · 95.80+p

as the seismic properties we derived by using the adiabatic oscillation code LOSC (Scuflaire et al. 2007b) for models on the main sequence (MS) or close to it. In Sects. 2 and 3 we describe, respectively, the input physics and the set of stellar parameters we used. The evolutionary tracks, stellar models and oscillation frequency data can be found on the ESTA Web site. The description of all this material is given in Sect. 4 for the evolution and structure models, and in Sect. 5 for the seismic properties of the stellar models. Finally, in Sect. 6 we summarize the type and amount of available files.

1 Introduction

2 Input physics

The preparation of the CoRoT mission brought up the need of a reference grid of stellar models in order to locate the possible CoRoT targets in the Hertzsprung–Russell (HR) diagram, and to allow a first interpretation of the forthcoming CoRoT-data. For this purpose, several grids of stellar models were computed according to the ESTA-Task1 specifications. In this paper we present the grids of stellar models we computed with the code CLÉS (Scuflaire et al. 2007a), as well

For three out of four stellar model grids we adopted the reference input physics, as well as the physical and astronomical constants specified for ESTA-Task1 comparisons (see also Sect. 5 in Morel and Lebreton 2007). We stress that these specifications do not always coincide with those of the standard version of CLÉS (see Scuflaire et al. 2007a).

Keywords Stars: evolution · Stars: interiors · Stars: oscillations

J. Montalbán () · A. Miglio · A. Noels · R. Scuflaire Institut d’Astrophysique et Geophysique, Université de Liège, allée du 6 Août 17, 4000 Liège, Belgium e-mail: [email protected] A. Miglio e-mail: [email protected] A. Noels e-mail: [email protected] R. Scuflaire e-mail: [email protected]

– Equation of State (EoS)—OPAL2001 equation of State (Rogers and Nayfonov 2002) available in the OPAL Web site. The tabulated values of CV have been replaced with the values derived from the tabulated values of P , 1 , χT , χρ . For each ρ, T , X and Z we interpolate in the tables by using our own interpolation routine. – Opacities—OPAL96 (Iglesias and Rogers 1996). The opacity tables used in CLÉS have been calculated online for the standard GN93 (Grevesse and Noels 1993) metal mixture, and using the smoothing routine available in the OPAL Web site to build the final opacity tables. The low temperature opacity tables by Alexander and Ferguson (1994) are also smoothly added (see Scuflaire et

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_23

179

180

–

–

–

– –

al. 2007a), and conductive opacities were not included in these computations. As for EoS, we interpolate in ρ, T , X and Z by using our own interpolation routines. Nuclear network—We choose the same nuclear network as in model comparison, that is, basic pp chain and CNO cycle reactions up to the 17 O(p, α)14 N. We adopted the nuclear reaction rates from the analytical formulae provided by the NACRE compilation (Angulo et al. 1999), including that one for the 14 N(p, γ )15 O, and the weak screening factors from Salpeter (1954). The electronic density, taking part in the screening factor computation, was estimated assuming full ionization of chemical elements. Chemical composition—We adopted the metal distribution provided by the Grevesse and Noels (1993) (thereafter GN93) solar mixture. Only the abundances of light elements (Li, Be and B) were modified to be consistent with those used in CESAM. The isotopic ratios are those of Anders and Grevesse (1989) except for the 2 H/1 H and 3 He/4 He ratios for which we took those of Gautier and Morel (1997). Convection—All the evolutionary tracks were computed with the classical mixing length treatment of convection by Böhm-Vitense (1958) and the formulation of Henyey et al. (1965) for optically thin regions. The mixing length parameter αMLT was fixed at 1.6 for all the grids. Note that the latter is the value chosen for ESTA comparisons, but it does not correspond to the value derived from a solar calibration. Overshooting. All the models, regardless of their mass, were computed without convective core overshooting. Atmosphere. ESTA-Task1 specification requires the surface boundary conditions to be provided by integration of a grey atmosphere following the Eddington’s T (τ ) law (grey models, thereafter). Three of the four stellar model grids were computed following the ESTA specification, while for the fourth one, temperature and density at T = Teff were obtained from Kurucz’s atmosphere models (Kurucz 1998).

3 Stellar parameters We provide evolutionary tracks for masses from 0.8 to 8M , with a mass step M = 0.1, from 0.8 to 1.6M , M = 0.2 up to 4.0M , and M = 0.5 from 4.0 to 8.0M . The initial hydrogen mass fraction is X = 0.70 in all grids. Three different values of the metal mass fraction Z (0.02, 0.01 and 0.006) are available for grids of grey models, while only Z = 0.02 has been considered in the grid computed with Kurucz atmosphere boundary conditions.

Astrophys Space Sci (2008) 316: 179–185

4 Data 4.1 Evolutionary tracks For each stellar parameter (M, X, Z) we follow the PMS evolution from the Hayashi track, and the calculation ends either when the temperature at the stellar center is high enough to burn He, or when the age of the model is larger than 20 Gyr. We recall that the computation of these models does not include conductive opacities and that these ones should be taken into account in modeling low mass stars up to the helium ignition. The grids of these evolutionary tracks are available from the ESTA Web site1 and consist of two HRD-files, whose name contains the values of the stellar parameters for which they were computed. For instance, “m1.00Z0.02X0.70HRD1.txt” and “m1.00Z0.02X0.70-HRD2.txt” contain the evolutionary tracks for a 1.00M star with initial chemical composition X = 0.70 and Z = 0.02. The first four lines in HRD1 and HRD2-files provide information about the stellar parameters and the input physics used to build the model. Since most of them have already been described in previous section, we only mention that fg and ft indicate the factor by which the mesh and time step were multiplied. Both, fg and ft were set equal to 0.5, that is, the number of mesh points and the temporal steps were doubled with respect to the standard value used in CLÉS (Scuflaire et al. 2007b). The quantities for which the temporal evolution were tabulated in the HRDs files are the following: HRD1 files: – Column 1—Stellar mass in units of M . – Column 2—Stellar luminosity (log(L/L )). – Column 3—Decimal logarithm of the effective temperature in K (log Teff ). – Column 4—Stellar radius in units of solar radius (R/R ). – Column 5—Stellar age in Myr. – Column 6—Hydrogen mass fraction at the center. – Column 7—log g, with g, the surface gravity. – Column 8—Number of convective-radiative boundaries. – Column 9—Codification of the of boundary type: “2” for boundary from convective to radiative region; and “1” for boundaries from radiative to convective one, going from the stellar center towards the surface. – Column 10–15—Relative radius of the six first convective-radiative boundaries (from the center outwards). HRD2 files:

– Column 1—Number of model. – Column 2—Stellar age in Myr. – Column 3—Stellar radius in solar radius (R/R ). 1 http://www.astro.up.pt/corot/models

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Fig. 1 Theoretical HR diagram for the evolution phase comprised between the initial homogeneous model and that with a hydrogen mass fraction at the center reduced by 1% with respect to the initial model (that is, Xc = 0.693). Evolutionary tracks are labeled by the corresponding stellar mass. Solid lines correspond to grey models and dashed ones to models with Kurucz boundary conditions

Fig. 2 As Fig. 1 for the evolutionary phase of main sequence and post-main sequence. Open circles indicate the location of models that has burned, at the center, half of the initial hydrogen. Solid circles correspond to models with only 0.1% of hydrogen at the center

– Column 4—Stellar luminosity in units of solar luminosity (L/L ). – Column 5—Effective temperature in K. – Column 6—Temperature at the center in units of 107 K. – Column 7—Central density ρc in g cm−3 . – Column 8—Hydrogen mass fraction at the center (Xc ). – Column 9—Central helium mass fraction (Yc ). – Column 10—Number of convective regions. – Column 11—Relative mass of the convective core (mcc /M∗ ). – Column 12—Relative radius of the base of the outer convective region (Rce /R∗ ).

– Column 13—Mass of the helium core in M . The helium core is defined by the mass of the region with a hydrogen mass fraction smaller than 10−2 . The log Teff –log L/L diagram for all the models with a metal mass fraction Z = 0.02 have been plotted in Fig. 1 for the PMS evolution, and in Fig. 2 for the MS and Post-MS evolutionary phases. For each mass there is a pair of curves, one was computed with Eddington’s law as boundary conditions at the photosphere (solid lines) and the other used Kurucz’s atmosphere models (dashed lines). It is interesting to note that, as it concerns the theoretical HR diagram, the atmosphere type affects only models with effective tempera-

182

Fig. 3 Relative mass of the convective core as a function of the stellar mass for models that have burned the half of their initial H content at the center

ture lower than 6300 K. The effects on the seismic properties will be discussed in Sect. 5. In Fig. 2 we marked the model locations at the middle of their MS (central hydrogen mass fraction Xc = 0.35) by open circles and that of models with a central hydrogen mass fraction Xc  0.001 by filled ones. Defining the beginning of MS by the model with a central mass fraction of hydrogen decreased by 1% with respect to the initial one (that is Xc = 0.693 for the present computations), and its end by that with Xc  0.001, the MS-lifetime varies from 5 × 107 yrs for 8M to more than 13 Gyr for 0.9M . The number of computed stellar models that span the main sequence phase of evolution is of the order of 160 for each (M, X, Z). The dependence on the stellar mass of the convective core mass, for models in the middle of MS is shown in Fig. 3, and that of the convective envelope bottom in Fig. 4. Finally, in Fig. 5 we show the mass of the He core at the end of the MS as a function of the stellar mass.

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Fig. 4 Relative radius of the outer envelope bottom as a function of the stellar mass for the same models as in Fig. 3

Fig. 5 Relative mass of the helium core at the end of the MS phase as a function of the stellar mass

4.2 Stellar models In addition to the evolutionary tracks, a selection of internal structure models for each (M, X, Z) are also available: – ZAMS model—defined as the model for which the ratio of gravitational to nuclear luminosity is lower than 2%. – 1MS model—as the model which has burned 1% of H (Xc  0.693). – MS model—as the model with half of the initial H at the center (Xc  0.35). – TAMS model—as the model with Xc  0.01.

– One every five models spanning the MS phase of evolution: from Xc = 0.693 to Xc = 0.001. That means about 30 models for each set of stellar parameters. An individual file is provided for each stellar model whose name is formed with the stellar parameters, for example: m1.00Z0.02X0.70-####.gong, where “####” can be “ZAMS”, “1MS”, “MS”, “TAMS”, or a number corresponding to the number of the model. The format adopted for these

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183

files is the FGONG2 one with the first twenty-five variables. The number of mesh points through the stellar interior is of the order of 2200, and 100 additional mesh points are used to describe the stellar atmosphere from an optical depth τ = 2/3 outwards, up to τ = 10−3 . Among the physical quantities stored in FGONG files there are the mass fraction of the most important contributors to nuclear energy generation. For illustration, we show in Fig. 6 the chemical profile of key elements such as 3 He, 12 C, and 14 N inside the stellar models in the mid-MS.

5 Seismic properties For each equilibrium model in FGONG format, we also provide a file containing the properties of radial and non radial oscillation modes with spherical degree = 1, 2, and 3. These properties were computed by using the code LOSC (Scuflaire et al. 2007b) with the standard surface boundary condition (regularity of solution when P = 0 at the surface, that is, δP /P + (4 + ω2 )δr/r = 0). The angular frequencies of the computed oscillation modes cover thedomain given by 0.3–50 times the dynamical time (τdym = R 3 /G M). The file name is the same as that of the equilibrium model but with the extension “.freq”; the content is the following: – – – – – – –

Column 1—spherical angular degree . Column 2—radial order n. Column 3—dimensionless frequency ω = σ · τdym . Column 4—angular frequency σ . Column 5—frequency ν = σ/2 π in Hz. Column 6—period P in seconds. Column 7—βn = 1 − Cn gives the frequency shift generated by solid rotation, where Cn is the Ledoux’s constant (Ledoux 1951). – Column 8 — fraction of the kinetic energy associated to the radial component of the motion ev = Ekin,V /Ekin . – Column 9—xm = x is the mean value of x = r/R weighted by the kinetic energy of the oscillation mode, and determines in which  region the mode is trapped. – Column 10— = 2 x 2  − x2 , that takes values between 0 and 1, vanishes for a perfectly trapped mode.

Since discontinuities in the sound speed derivatives in the stellar interior (such as those introduced by the boundaries of the convective regions and by the second helium ionization zone) produce periodic signatures in the frequencies of low degree modes (see e.g. Gough 1990), we decided to provide, as well, files containing the time variations of some quantities that may be of interest in the seismic analysis of stellar models. So, for each stellar parameter set (M, X, Z), 2 Document “Description of the file formats used within ESTA/CoRoT” at http://www.astro.up.pt/corot/ntools/info.html.

Fig. 6 Upper panel: equilibrium abundance of 3 He for models with masses from 0.8 to 8M . Lower panel: equilibrium abundance of 14 N (solid lines) and 12 C (dotted lines) for each of the considered stellar masses. The thicker lines correspond to the chemical profile in a 2M model

but only for the MS-lifetime, we built a file whose name is formed, as for HR diagram ones, with the values of the stellar parameters (for instance “m1.00Z0.02X0.70-ACC.txt”), and whose content in the columns 7–13 is the following: – Cutoff frequency (in µHz) at the photosphere (column 7): νac = c/(4πHp ) where c is the sound speed, and Hp the pressure scale height. R – Acoustic radius in seconds (column 8): rac = 0 1/c dr where R is the stellar radius at the photosphere. – Acoustic depth in seconds of the border of the convective core (column 10) corresponding to the linear relative radius rcc /R given in column 9. – Acoustic depth in seconds of the bottom of the convective envelope (column 12) corresponding to the linear relative radius rce /R given in column 11. – Second He ionization region: linear relative radius (rHeII /R) in column 12, and the corresponding acoustic depth τHeII in column 13. The four first lines, and the columns 1–6 are the same as in HRD2-files.

In principle, only oscillation modes with frequencies lower than the acoustic cutoff one are trapped in the stellar interior and should be observed. As an indication of the order of magnitude of the expected oscillation frequencies we show in Fig. 7 the variation with time of the cutoff frequency for models with masses smaller than 2M .

184

Fig. 7 Evolution of the cutoff frequency along the MS evolution of models with masses from 0.8M (upper curve) to 2M (lower curve)

Astrophys Space Sci (2008) 316: 179–185

Fig. 8 Difference between the oscillation frequencies of 1.8M models computed with Eddington’s grey law (νE ) and with Kurucz’s atmospheres (νK )

5.1 Effect of atmospheric boundary conditions For the chemical composition X = 0.70 and Z = 0.02 there are two types of stellar models depending on whether the boundary conditions at T = Teff are given by Kurucz atmosphere models or by grey ones (with Eddington’s law). We showed in Figs. 1 and 2 that the effect on the HRD location was significant only for Teff  6300 K. The effect on the oscillation frequencies may be, however, important even for higher temperatures. As an example we show in Fig. 8 the difference between oscillation frequencies for 1.8M models in mid-MS. The difference in the global parameters for these Eddington and Kurucz stellar models are R/R = 4 × 10−4 , L/L = 1.4 × 10−5 , and Xc < 10−5 . Nevertheless, the density at T = Teff provided by the two atmosphere models is different enough to affect a significant fraction of the star, leading to an important variation of the oscillation frequencies. Note that for this comparison only the subphotospheric stellar structure was considered in the computation of the oscillation frequencies. The differences at high frequency will be even larger if the atmosphere structure is included in the calculation.

6 Conclusions In order to provide a reference grid of stellar models for the first interpretation of the CoRoT data, we have computed three sets of grey stellar models for 29 different masses from 0.8 to 8M . Each grid is computed with a different chemical composition, and altogether they span metallicity values from [M/H] = −0.45 to +0.07 (taking as reference the

value of Z/X = 0.0245 for the Sun given by GN93). There are 3 × 29 × 3 = 261 files containing data on time evolution of global quantities, around 3050 files with the detailed stellar structure (interior and atmosphere), and other 3050 with the corresponding oscillation frequencies for low degree modes ( = 0, 1, 2, and 3). In addition, a set of stellar models with the Kurucz’s atmospheres as boundary conditions were also computed, for the same values of stellar mass, but for only one chemical composition. That means additional ∼2087 files, 87 for the time evolution, and the rest of them for the stellar structure and the corresponding seismic properties. Acknowledgements We acknowledge financial support from the Belgian Science Policy Office (BELSPO) in the frame of the ESA PRODEX 8 program (contract C90199) and from the Fonds National de la Recherche Scientifique (FNRS).

References Alexander, D.R., Ferguson, J.W.: Low-temperature Rosseland opacities. Astrophys. J. 437, 879–891 (1994). doi:10.1086/175039 Anders, E., Grevesse, N.: Abundances of the elements—meteoritic and solar. Geochim. Cosmochim. Acta 53, 197–214 (1989). doi:10.1016/0016-7037(89)90286-X Angulo, C., Arnould, M., Rayet, M., Descouvemont, P., Baye, D., Leclercq-Willain, C., Coc, A., Barhoumi, S., Aguer, P., Rolfs, C., Kunz, R., Hammer, J.W., Mayer, A., Paradellis, T., Kossionides, S., Chronidou, C., Spyrou, K., degl’Innocenti, S., Fiorentini, G., Ricci, B., Zavatarelli, S., Providencia, C., Wolters, H., Soares, J., Grama, C., Rahighi, J., Shotter, A., Lamehi Rachti, M.: A compilation of charged-particle induced thermonuclear reaction rates. Nucl. Phys. A 656, 3–183 (1999)

Astrophys Space Sci (2008) 316: 179–185 Böhm-Vitense, E.: Über die Wasserstoffkonvektionszone in Sternen verschiedener Effektivtemperaturen und Leuchtkräfte. Mit 5 Textabbildungen. Z. Astrophys. 46, 108 (1958) Gautier, D., Morel, P.: A reestimate of the protosolar (2 H/1 H)p ratio from (3 He/4 He)SW solar wind measurements. Astron. Astrophys. 323, L9–L12 (1997) Grevesse, N., Noels, A.: Cosmic abundances of the elements. In: Origin and Evolution of the Elements. Cambridge University Press, Cambridge (1993) Gough, D.O.: Comments on helioseismic inference. In: Osaki, Y., Shibahashi, H. (eds.) Progress of Seismology of the Sun and Stars. Lecture Notes in Physics, vol. 367, pp. 283–318. Springer, Berlin (1990). doi:10.1007/3-540-53091-6_93 Henyey, L., Vardya, M.S., Bodenheimer, P.: Studies in stellar evolution. III. The calculation of model envelopes. Astrophys. J. 142, 841 (1965) Iglesias, C.A., Rogers, F.J.: Updated opal opacities. Astrophys. J. 464, 943 (1996) Kurucz, R.L.: Grids of model atmospheres. http://kurucz.harvard.edu/ grids.html (1998)

185 Ledoux, P.: The nonradial oscillations of gaseous stars and the problem of beta canis majoris. Astrophys. J. 114, 373 (1951) Morel, P., Lebreton, Y.: CESAM: a free code for stellar evolution calculations. Astrophys. Space Sci. (2007). doi:10.1007/ s10509-007-9663-9 Rogers, F.J., Nayfonov, A.: Updated and expanded OPAL equation-ofstate tables: implications for helioseismology. Astrophys. J. 576, 1064–1074 (2002). doi:10.1086/341894 Salpeter, E.E.: Electrons screening and thermonuclear reactions. Aust. J. Phys. 7, 373 (1954) Scuflaire, R., Théado, S., Montalbán, J., Miglio, A., Bourge, P.-O., Godart, M., Thoul, A., Noels, A.: CLÉS, Code Liégeois d’Évolution Stellaire. Astrophys. Space Sci. (2007a). doi:10.1007/s10509-007-9650-1 Scuflaire, R., Montalbán, J., Théado, S., Bourge, P.O., Miglio, A., Godart, M., Thoul, A., Noels, A.: The Liège OScillations Code. Astrophys. Space Sci. (2007b) doi:10.1007/s10509-007-9577-6

Comparisons CoRoT/ESTA–TASK 1 and TASK 3 comparison of the internal structure and seismic properties of representative stellar models Comparisons between the ASTEC, CESAM, CLES, GARSTEC and STAROX codes Yveline Lebreton · Josefina Montalbán · Jørgen Christensen-Dalsgaard · Ian W. Roxburgh · Achim Weiss Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-008-9740-8 © Springer Science+Business Media B.V. 2008

Abstract We compare stellar models produced by different stellar evolution codes for the CoRoT/ESTA project, comparing their global quantities, their physical structure, and their oscillation properties. We discuss the differences between models and identify the underlying reasons for these differences. The stellar models are representative of potential CoRoT targets. Overall we find very good agreement between the five different codes, but with some significant deviations. We find noticeable discrepancies (though still at the per cent level) that result from the handling of the equation of state, of the opacities and of the convective boundaries. The results of our work will be helpful in interpreting future asteroseismology results from CoRoT.

Y. Lebreton () GEPI, Observatoire de Paris, CNRS, Université Paris Diderot, 5 Place Janssen, 92195 Meudon, France e-mail: [email protected] J. Montalbán Institut d’Astrophysique et Géophysique, Université de Liège, Liège, Belgium e-mail: [email protected] J. Christensen-Dalsgaard Institut for Fysik og Astronomi, Aarhus Universitet, Aarhus, Denmark I.W. Roxburgh Queen Mary University of London, London, England I.W. Roxburgh LESIA, Observatoire de Paris, Meudon, France A. Weiss Max-Planck-Institut für Astrophysik, Garching, Germany

Keywords Stars: evolution · Stars: interiors · Stars: oscillations · Methods: numerical PACS 97.10.Cv · 97.10.Sj · 95.75.Pq

1 Introduction The goals of ESTA-TASKs 1 and 3 are to test the numerical tools used in stellar modelling, with the objective to be ready to interpret safely the asteroseismic data that will come from the CoRoT mission. This consists in quantifying the effects of different numerical implementations of the stellar evolution equations and related input physics on the internal structure, evolution and seismic properties of stellar models. As a result, we aim at improving the stellar evolution codes to get a good agreement between models built with different codes and same input physics. For that purpose, several study cases have been defined that cover a large range of stellar masses and evolutionary stages and stellar models have been calculated without (in TASK 1) or with (in TASK 3) microscopic diffusion (see Monteiro et al. 2006; Lebreton et al. 2007a). In this paper, we present the results of the detailed comparisons of the internal structures and seismic properties of TASKs 1 and 3 target models. The comparisons of the global parameters and evolutionary tracks are discussed in Monteiro et al. (2007). In order to ensure that the differences found are mainly determined by the way each code calculates the evolution and the structure of the model and not by significant differences in the input physics, we decided to use and compare models whose global parameters (age, luminosity, and radius) are very similar. Therefore, we selected models computed by five codes among the ten participating in ESTA: ASTEC (Christensen-Dalsgaard 2007a), CESAM (Morel and Lebreton 2007), CLÉS (Scuflaire et al.

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_24

187

188

2007a), GARSTEC (Weiss and Schlattl 2007), and STAROX (Roxburgh 2007). In Sect. 2 we recall the specifications of TASKs 1 and 3 and present the five codes used in the present paper. We then present the comparisons for TASK 1 in Sect. 3 and for TASK 3 in Sect. 4. For each case in each task, we have computed the relative differences of the physical quantities between pairs of models. We display the variation of the differences between the more relevant quantities inside the star and we provide the average and extreme values of the variations. We then compare the location of the boundaries of the convective regions as well as their evolution with time. For models including microscopic diffusion we examine how helium is depleted at the surface as a function of time. Finally, we analyse the effect of internal structure differences on seismic properties of the model.

2 Presentation of the ESTA-TASKs and tools In the following we briefly recall the specifications and tools of TASK 1 and TASK 3 that have been presented in detail by Lebreton et al. (2007a). 2.1 TASK 1: basic stellar models The specifications for the seven cases that have been considered in TASK 1 are recalled in Table 1. For each case, evolutionary sequences have been calculated for the specified 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) up to the evolutionary stage specified. The masses are in the range 0.9–5.0M . For the initial chemical composition, different (Y, Z) couples have been considered by combining two different values of Z (0.01 and 0.02) and two values of Y (0.26 and 0.28). The evolutionary stages considered are either on the pre main sequence (PMS), the main sequence (MS) or the subgiant branch (SGB). On the PMS the central temperature of the model (Tc = 1.9 × 107 K) has been specified. On the MS, the value of the central hydrogen content has been fixed: Xc = 0.69 for a model close to the zero age main sequence (ZAMS), Xc = 0.35 for a model in the middle of the MS and Xc = 0.01 for a model close to the terminal age main sequence (TAMS). On the SGB, a model is chosen by specifying the value of the mass McHe of the central region of the star where the hydrogen abundance is such that X ≤ 0.01. We chose McHe = 0.10M . 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. Also, reference values of some astronomical and physical constants have been fixed as well as the mixture of heavy elements to be used. These specifications are described in Lebreton et al. (2007a).

Astrophys Space Sci (2008) 316: 187–213 Table 1 Target models for TASK 1. We have considered 7 cases corresponding to different initial masses, chemical compositions and evolutionary stages. One evolutionary sequence (denoted by “OV” in the 5th column has been calculated with core overshooting (see text) Case

M/M

Y0

Z0

Specification

Type

1.1

0.9

0.28

0.02

Xc = 0.35

MS

1.2

1.2

0.28

0.02

Xc = 0.69

ZAMS

1.3

1.2

0.26

0.01

McHe = 0.10M

SGB

1.4

2.0

0.28

0.02

Tc = 1.9×107 K

PMS

1.5

2.0

0.26

0.02

Xc = 0.01, OV

TAMS

1.6

3.0

0.28

0.01

Xc = 0.69

ZAMS

1.7

5.0

0.28

0.02

Xc = 0.35

MS

Table 2 Target models for TASK 3. Left: The 3 cases with corresponding masses and initial chemical composition. Right: The 3 evolutionary stages examined for each case. Stages A and B are respectively in the middle and end of the MS stage. Stage C is on the SGB Case

M M

Y0

Z0

Stage

Xc

3.1

1.0

0.27

0.017

A

0.35

–

3.2

1.2

0.27

0.017

B

0.01

–

3.3

1.3

0.27

0.017

C

0.00

0.05Mstar

McHe

2.2 TASK 3: stellar models including microscopic diffusion TASK 3 is dedicated to the comparisons of stellar models

including microscopic diffusion of chemical elements resulting from pressure, temperature and concentration gradients (see Thoul and Montalbán 2007). The other physical assumptions proposed as the reference for the comparisons of TASK 3 are the same as used for TASK 1, and no overshooting. 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) and to a chemical composition close to the standard solar one (Z/X = 0.0243). For each case, models at different evolutionary stages have been considered. We focused on three particular evolution stages: middle of the MS, TAMS and SGB (respectively stage A, B and C). 2.3 Numerical tools Among the stellar evolution codes considered in the comparisons presented by Monteiro et al. (2007), we have considered 5 codes—listed below—for further more detailed comparisons. These codes have shown a very good agreement in the comparison of the global parameters which ensures that they closely follow the specifications of the tasks in terms of input physics and physical and astronomical constants.

Astrophys Space Sci (2008) 316: 187–213

• ASTEC—Aarhus STellar Evolution Code, described in Christensen-Dalsgaard (2007a). • CESAM—Code d’Évolution Stellaire Adaptatif et Modulaire, see Morel and Lebreton (2007). • CLÉS—Code Liégeois d’Évolution Stellaire, see Scuflaire et al. (2007a). • GARSTEC—Garching Evolution Code, presented in Weiss and Schlattl (2007). • STAROX—Roxburgh’s Evolution Code, see Roxburgh (2007). The oscillation frequencies presented in this paper have been calculated with the LOSC adiabatic oscillation code (Liège Oscillations Code, see Scuflaire et al. 2007b). Part of the comparisons between the models has been performed with programs included in the Aarhus Adiabatic Pulsation Package ADIPLS1 (see Christensen-Dalsgaard 2007b).

3 Comparisons for TASK 1 3.1 Presentation of the comparisons and general results The TASK 1 models span different masses and evolutionary phases. Cases 1.1, 1.2 and 1.3 illustrate the internal structure of solar-like, low-mass 0.9M and 1.2M stars, at the beginning of the main sequence of hydrogen burning (Case 1.2), in the middle of the MS when the hydrogen mass fraction in the centre has been reduced to the half of the initial one (Case 1.1) and in the post-main sequence when the star has already built a He core of 0.1M (Case 1.3). Cases 1.4 and 1.5 correspond to intermediate-mass models (2.0M ), the first one, in a phase prior to the MS when the nuclear reactions have not yet begun to play a relevant role, and the second one, at the end of the MS, when the matter in the centre contains only 1% of hydrogen. Finally, Cases 1.6 (3.0M ) and 1.7 (5.0M ) sample the internal structure of models corresponding to middle and late B-type stars. For these more massive models, the beginning and the middle of their MS are examined. The models provided correspond to a different number of mesh points: the number of mesh points is 1202 in the ASTEC models; it is in the range 2300–3700 in CESAM models, 2200–2400 in CLÉS models, 1500–2100 in models by GARSTEC and 1900–2000 in STAROX models. As explained in the papers devoted to the description of the participating codes (Christensen-Dalsgaard 2007a; Morel and Lebreton 2007; Roxburgh 2007; Scuflaire et al. 2007a; Weiss and Schlattl 2007), the numerical methods used to solve the equations and to interpolate in the tables containing physical inputs are specific to each code and so are the 1 Available

at http://astro.phys.au.dk/~jcd/adipack.n

189 Table 3 TASK 1 models: Global parameter differences given in per cent, between each code and CESAM. For each parameter we give the mean and maximum difference of the complete series of TASK 1 models Code

ASTEC CLÉS GARSTEC STAROX

δR/R

δL/L

δTeff /Teff

Mean

Max

Mean

Max

Mean

Max

0.27

0.83

0.49

1.57

0.02

0.03

0.20

0.43

0.16

0.52

0.01

0.01

0.37

0.59

0.23

0.46

0.03

0.06

0.75

3.29

0.31

0.89

0.03

0.13

possibilities to choose the number and repartition of the mesh points in a model or the time step of the evolution calculation and, more generally, the different levels of precision of the computation. The specifications for the tasks have concerned mainly the physical inputs and the constants to be used (Lebreton et al. 2007a) and have let the modelers free to tune up the numerous numerical parameters involved in their calculation which explains why each code deals with different numbers (and repartition) of mesh points. Table 3 gives a brief summary of the differences in the global parameters of the models by providing the mean and maximum differences in radius, luminosity and effective temperature obtained by each code with respect to CESAM models. The mean difference is obtained by averaging over all the cases calculated (not all cases have been calculated by each code). The differences are very small, i.e. below 0.5 per cent for CLÉS and GARSTEC. They are a bit larger for two ASTEC models (1–2% for Cases 1.2 and 1.7) and for two STAROX models (1–3% for Cases 1.2 and 1.5, but note that for the latter overshooting is treated differently than in other codes as explained in Sect. 3.2.2). For a detailed discussion, see Monteiro et al. (2007). For each model we have computed the local differences in the physical variables with respect to the corresponding model built by CESAM. The physical variables we have considered are the following: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

P : pressure ρ: density Lr : luminosity through the sphere with radius r X: hydrogen mass fraction c: sound speed 1 : adiabatic exponent Cp : specific heat at constant pressure ∇ad : adiabatic temperature gradient κ: radiative opacity ρ 2 A = 11 ddlnlnPr − dd ln ln r = NBV r/g, where NBV is the Brunt-Väisälä frequency and g the local gravity.

To compute the differences we used the grid of a given model (either mass grid or radius grid, see below) and we interpolated the physical variables of the CESAM model on

190

Astrophys Space Sci (2008) 316: 187–213

Table 4 TASK 1 models: Mean quadratic difference in the physical variables between each code and CESAM calculated according to (1). The differences are given in per cent (except for δX) Code

and represent an average over the whole star from centre to photospheric radius. The local differences were computed at fixed relative mass

δ ln c

δ ln P

δ ln ρ

δ ln T

δ ln r

δ ln 1

δ ln ∇ad

δ ln Cp

δ ln κ

δX

δ ln Lr

0.02

0.15

0.16

0.04

0.05

2.69 × 10−4

5.84 × 10−4

7.36 × 10−3

0.44

1.2 × 10−4

0.17

3.14 × 10−4

4.06 × 10−3

2.64 × 10−2

3.4 × 10−4 4.7 × 10−4

0.03 0.33

Case 1.1

ASTEC CLÉS GARSTEC STAROX

0.04 0.06

0.25 0.40

0.22 0.44

0.07 0.07

0.06 0.09

1.46 × 10−1

1.13 × 10−1

1.11 × 10−1

0.20 0.45

0.04

0.28

0.25

0.08

0.09

6.47 × 10−4

–

–

–

3.1 × 10−4

0.08

0.11 0.02

0.72 0.17

0.55 0.14

0.20 0.04

0.20 0.05

1.00 × 10−3 2.60 × 10−4

2.21 × 10−3 3.60 × 10−3

1.25 × 10−2 5.67 × 10−3

0.58 0.19

2.0 × 10−4 9.3 × 10−5

0.94 0.08

0.06

0.23

0.23

0.04

0.07

1.55 × 10−1

1.32 × 10−1

1.36 × 10−1

0.39

7.8 × 10−5

0.24

–

–

–

3.7 × 10−5

0.24

Case 1.2

ASTEC CLÉS GARSTEC STAROX

0.02

0.09

0.09

0.05

0.03

8.63 × 10−4

0.30

2.50

1.96

0.68

0.61

3.62 × 10−3

8.16 × 10−3

2.00 × 10−1

1.32

2.1 × 10−3

1.45

2.35 × 10−3

4.88 × 10−3

2.27 × 10−1

2.0 × 10−3 1.4 × 10−3

5.04 0.91

Case 1.3

ASTEC CLÉS GARSTEC

0.08 0.12

0.51 0.76

0.58 0.69

0.17 0.16

0.20 0.21

1.81 × 10−1

1.81 × 10−1

3.01 × 10−1

0.55 1.01

0.03

0.25

0.22

0.07

0.09

7.40 × 10−4

4.80 × 10−3

5.49 × 10−3

0.27

6.9 × 10−5

1.04

1.71 × 10−1

2.18 × 10−1

1.8 × 10−5 4.3 × 10−5

0.62 3.75

Case 1.4

CLÉS GARSTEC STAROX

0.20 0.12

0.89 0.89

0.66 0.84

0.27 0.24

0.23 0.33

1.71 × 10−1 3.07 × 10−3

–

–

0.67 –

0.33 0.14

1.33 1.03

1.15 0.85

0.44 0.20

0.34 0.23

1.93 × 10−3 1.56 × 10−3

5.12 × 10−3 6.35 × 10−3

5.93 × 10−1 2.34 × 10−1

0.76 0.43

7.1 × 10−3 2.2 × 10−3

0.99 0.85

0.78

7.03

5.90

1.22

1.53

8.94 × 10−3

–

–

–

1.0 × 10−2

1.96

0.06

0.48

0.39

0.10

0.11

6.29 × 10−4

1.66 × 10−3

5.53 × 10−2

0.25

7.2 × 10−4

0.45

0.02

0.19

0.20

0.03

0.04

1.38 × 10−3

4.20 × 10−3

1.81 × 10−2

0.26

2.1 × 10−4

0.20

2.21 × 10−1

3.10 × 10−1

1.17

Case 1.5

ASTEC CLÉS STAROX Case 1.6

ASTEC CLÉS GARSTEC STAROX

0.23

2.58

1.95

0.60

0.64

1.96 × 10−1

0.92

5.9 × 10−4

0.04

0.25

0.19

0.07

0.08

8.51 × 10−4

–

–

–

1.2 × 10−4

0.27

0.27

1.77

1.53

0.26

0.38

8.85 × 10−3

2.72 × 10−2

4.27 × 10−1

0.43

4.9 × 10−3

1.21

2.79 × 10−3

8.51 × 10−3

1.06 × 10−1

1.2 × 10−3 2.4 × 10−3

0.24 1.18

3.3 × 10−3

0.77

Case 1.7

ASTEC CLÉS GARSTEC STAROX

0.05 0.19

0.16 0.73

0.19 0.68

0.03 0.09

0.04 0.17

1.88 × 10−1

2.62 × 10−1

6.02 × 10−1

0.15 0.69

0.14

0.50

0.57

0.08

0.11

6.03 × 10−3

–

–

–

that grid. We used the so-called diff-fgong.d routine in the ADIPLS package. We performed both interpolations at fixed relative radius (r/R) and at fixed relative mass (q = m/M). In both cases, we have computed the local logarithmic differences (δ ln Q) of each physical quantity Q with respect to that of the corresponding CESAM model (except for X where we computed δX). The interpolation is cubic (either in r/R or q = m/M) except for the innermost points where it is linear, either in (r/R)2 or (m/M)2/3 , in order to improve the accuracy of the interpolation of L and m/M.

Since not all the codes provide the atmosphere structure, we have calculated the differences inside the star up to the photospheric radius (R). To provide an estimate of these differences we have defined a kind of “mean-quadratic error”:  δx = 0

M

(xCODE − xCESAM )2 ·

dm M

1/2 (1)

where the differences xCODE − xCESAM are calculated at fixed mass. The values of variations resulting from this computation are collected in Table 4. We note that the “mean-

Astrophys Space Sci (2008) 316: 187–213

191

Fig. 1 TASK 1: Plots of the differences at fixed relative mass between pairs of models (CODE-CESAM) corresponding to Cases 1.1, 1.2 and 1.3. Left panel: logarithmic sound speed differences. Centre left panel:

logarithmic density differences. Centre right panel: hydrogen mass fraction differences. Right panel: logarithmic luminosity differences. Horizontal dotted line represents the reference model (CESAM)

quadratic differences” between the codes generally remain quite low except for a few particular cases. For the unknowns of the stellar structure equations P , T , Lr and for r, ρ and κ, the differences range from 0.1 to at most 7%. Concerning the variation in the thermodynamic quantities we note that while the values of δ1 , δ∇ad , δCp for three of the codes are quite small, the differences are systematically larger than 0.1% in the GARSTEC code. Some differences in the thermodynamic quantities might indeed be expected since each code has its own use of the OPAL equation of state package and variables (Rogers and Nayfonov 2002), see the discussion concerning CLÉS and CESAM in Montalbán et al. (2007a). Similarly, we expect some differences in the opacities derived by the codes even though all codes use the OPAL95 opacities (Iglesias and Rogers 1996) and the AF94 opacities (Alexander and Ferguson 1994) at low temperature. In Fig. 2 we provide the differences, with respect to CESAM, of the opacities calculated by ASTEC, CLÉS and STAROX for two (ρ, T , X, Z) profiles extracted from CESAM models. The larger differences are in the range 2–6% and occur in a narrow zone around log T = 4.0. With GARSTEC dif-

ferences are of the same order of magnitude. Those differences correspond to the joining of OPAL95 and AF94 opacity tables. Each code has its own method to merge the tables: CLÉS, GARSTEC and STAROX interpolate between OPAL95 and AF94 values of log κ on a few temperature points of the domain where the tables overlap, CESAM looks for the temperature value where the difference in opacity is the smallest and ASTEC merges the tables at log T = 4.0. However, in any case the differences obtained between the codes do not exceed the intrinsic differences between OPAL95 and AF94 tables in this zone. In the rest of the star differences in opacities are small and do not exceed 2%. As shown by the detailed comparisons between CESAM and CLÉS codes performed by Montalbán et al. (2007a) differences in opacities at given physical conditions may amount to 2 percents due to the way the OPAL95 data are generated and used (e.g. period when the OPAL95 data were downloaded or obtained from the Livermore team, interpolation programme, mixture of heavy elements). As can be seen in Fig. 2, a major source of difference (well noticeable for the 2.0M , Xc = 0.50 model) is due to the fact that ASTEC, CESAM and STAROX (and also GARSTEC) use early delivered OPAL95

192

Fig. 2 TASK 1: Comparisons of opacities calculated by each code with respect to CESAM for fixed physical conditions corresponding to a model of 0.9M and Xc = 0.35 (top panel) and a model of 2.0M , Xc = 0.50 (bottom panel)

Astrophys Space Sci (2008) 316: 187–213

cases found a worsening of the agreement between the structures. We have derived the maximal relative differences in c, P , ρ, 1 , Lr and X from the relative differences, considering the maximum of the differences calculated at fixed r/R and of those obtained at fixed m/M. Note that for the latter estimate we removed the very external zones (i.e. located at m > 0.9999M) where the differences may be very large. We report these maximal differences in Table 5 together with the location (r/R) where they happen. We note that, except for X and Lr , the largest differences are found in the most external layers and (or) at the boundary of the convection regions. This will be discussed in the following. The effects of doubling the number of (spatial) mesh points and of halving the time step for the computation of evolution have been examined in ASTEC and CLÉS models for Cases 1.1, 1.3 and 1.5 (Christensen-Dalsgaard 2005; Miglio and Montalbán 2005; Montalbán et al. 2007a). Some results are displayed in Figs. 3 for Cases 1.3 and 1.5. For the Case 1.3 model of 1.2M on the SGB, the main differences are obtained when the time step if halved and are seen at the very center (percent level), in the convective envelope and close to the surface (0.5% for the sound speed). In the rest of the star they remain lower than 0.2%. For the Case 1.5 model, which is a 2M model at the end of the MS (with overshooting), differences at the percent level are noticeable in the region where the border of the convective core moved during the MS. Differences may also be at the percent level close to the surface. For the Case 1.1 model of 0.9M on the MS (not plotted), the differences are smaller by a factor 5 to 10 than those obtained for Cases 1.3 and 1.5. Further comparisons of CESAM models with various CLÉS models (doubling either the number of mesh points or halving the time step) have shown different trends: doubling the number of mesh points did not change the results in Case 1.3 but improved the agreement in Case 1.5 for L, ρ, c and the internal X-profile, halving the time step worsened the agreement in both cases. 3.2 Internal structure

tables (hereafter unsmoothed) while CLÉS uses tables that were provided later on the OPAL web site together with a (recommanded) routine to smooth the data. Once this source of difference has been removed (see CLÉS curves with and without smoothed opacities) there remain differences which are probably due to interpolation schemes and to slight differences in the chemical mixture in the opacity tables (see Montalbán et al. 2007a). Finally, when comparing models calculated by different codes (i.e. not simply comparing opacities), it is difficult to disentangle differences in the opacity computation from differences in the structure. As an example, Montalbán et al. (2007a) compared CLÉS and CESAM models based on harmonised opacity data and in some

3.2.1 Low-mass models: Cases 1.1, 1.2 and 1.3 For these models, the differences in c, ρ, X, and Lr as a function of the relative radius are plotted in Fig. 1. Table 4 and Fig. 1 show that the five evolution codes provided quite similar stellar models for Case 1.1—which has an internal structure and evolution stage quite similar to the Sun—and for Case 1.2. We note that the variations found in ASTEC model for Case 1.2 are larger than for Case 1.1 which is probably due to the lack of a PMS evolution in present ASTEC computations. On the other hand, the systematic difference in X observed in CLÉS models, even in

Astrophys Space Sci (2008) 316: 187–213

193

Table 5 TASK 1 models: Maximum variations given in per cent (except for δX) of the physical variables between each code and CESAM and value of the relative radius (r/R) where they happen. The local Code

δ ln c

r/R

δ ln P

r/R

δ ln ρ

differences were computed both at fixed relative mass and fixed relative radius and the maximum of the two values was searched (see Sect. 3.1) δ ln 1

r/R

r/R

δX

r/R

δ ln Lr

r/R

Case 1.1

ASTEC STAROX GARSTEC CLÉS

0.08

0.97540

1.03

0.98493

0.91

0.99114

0.14

0.99986

0.00082

0.10762

0.78

0.02275

0.16

0.69724

1.45

0.98064

1.31

0.98964

0.12

0.99984

0.00148

0.10886

0.55

0.07129

0.28

0.36880

3.12

0.92362

2.72

0.94568

0.55

0.99987

0.00232

0.11734

3.60

0.00319

0.14

0.69663

0.97

0.79942

0.89

0.78455

0.09

0.99987

0.00116

0.11140

0.35

0.00441

0.61

0.83067

4.50

0.84074

4.19

0.86702

0.33

0.99167

0.00093

0.05118

9.76

0.00185

Case 1.2

ASTEC STAROX GARSTEC CLÉS

0.34

0.82990

2.52

0.85099

2.34

0.87925

0.18

0.99223

0.00095

0.04868

0.93

0.04651

0.26

0.77098

2.22

0.86256

1.97

0.91370

0.43

0.98926

0.00102

0.05360

2.26

0.00236

0.08

0.99612

1.04

0.55596

0.94

0.61552

0.08

0.99990

0.00045

0.05124

1.20

0.00455

Case 1.3

ASTEC GARSTEC CLÉS

1.23

0.78937

4.86

0.78160

4.86

0.80785

0.73

0.98620

0.00740

0.04161

189.80

0.00042

0.24

0.02940

1.63

0.00031

1.68

0.02892

0.50

0.99990

0.01018

0.02913

12.21

0.00091

0.49

0.02964

2.91

0.94624

2.62

0.97962

0.18

0.99989

0.00928

0.03105

25.42

0.00284

Case 1.4

STAROX GARSTEC CLÉS

0.82

0.99902

5.04

0.99974

5.82

0.99985

1.05

0.99974

0.00034

0.13025

12.16

0.01056

1.03

0.99988

2.68

0.99986

4.37

0.99989

0.45

0.99987

0.00024

0.13009

2.04

0.00158

0.24

0.99988

0.98

0.38180

1.20

0.99989

0.22

0.99834

0.00032

0.13011

3.31

0.01191

Case 1.5

ASTEC STAROX CLÉS

3.18

0.99689

7.17

0.98780

9.13

0.99602

3.13

0.99705

0.07693

0.06230

4.09

0.00382

10.02

0.99677

15.17

0.23988

21.26

0.99580

9.41

0.99687

0.05763

0.06285

11.55

0.00070

0.51

0.06442

2.29

0.83852

2.40

0.06442

0.19

0.98734

0.01582

0.06442

1.99

0.00584

0.53

0.16440

0.99

0.41038

0.83

0.40804

0.18

0.99567

0.01189

0.16416

1.41

0.09659

Case 1.6

ASTEC STAROX GARSTEC CLÉS

0.35

0.99977

0.52

0.99987

0.47

0.99987

0.57

0.99977

0.00408

0.16359

0.93

0.00230

0.68

0.99407

2.53

0.00171

2.89

0.16320

0.65

0.99846

0.01387

0.16320

5.89

0.00192

0.21

0.99910

0.93

0.40301

0.85

0.43433

0.24

0.99969

0.00684

0.16392

1.10

0.00476

Case 1.7

ASTEC STAROX GARSTEC CLÉS

0.90

0.12212

4.52

0.82037

3.73

0.85521

0.63

0.99536

0.01909

0.12436

6.71

0.00184

0.55

0.11242

2.29

0.85773

1.95

0.87951

0.61

0.99948

0.01517

0.13663

1.55

0.00170

0.69

0.13697

2.38

0.83987

2.65

0.13598

0.45

0.99849

0.02655

0.13647

6.50

0.00148

0.44

0.99317

1.52

0.59622

1.40

0.62262

0.47

0.99309

0.00580

0.13739

0.65

0.00962

the outer layers, results from the detailed calculation of deuterium burning in the early PMS. For the most evolved model (Case 1.3) the differences increase drastically with respect to previous cases. It is worth to mention that at m ∼ 0.1M (r ∼ 0.03R) the variations of sound speed, Brunt-Väisälä frequency (Fig. 4) and hydrogen mass fraction are large. This can be understood as follows. Case 1.3 corresponds to a star of high central density which burns H in a shell. The middle of the H-burning

shell—where the nuclear energy generation is maximum— is located at ∼ 0.1M. Moreover, before reaching that stage, i.e. during a large part of the MS, the star had had a growing convective core (see Sect. 3.4 below) which reached a maximum size of m ∼ 0.05M, when the central H content was X ∼ 0.2, before receding. The large differences seen at m ∼ 0.1M can therefore be linked to the size reached by the convective core during the MS as well as to the features of the composition gradients outside this core.

194

Fig. 3 TASK 1: Effect of halving the time step in the evolution calculation (left and center left) and doubling the number of spatial mesh points in a model (right and center right) for Cases 1.3 and 1.5 models

For Case 1.3 we have looked at the values of the gravitational grav and nuclear energy nuc in the very central regions, i.e. in the He core (from the centre to r/R ∼ 0.02) and in the inner part of the H-burning shell (r/R ∈ [0.02, 0.03]). We find that in the He core, from the border to the centre, CLÉS values of grav are larger by 0–30% than the values obtained by GARSTEC and CESAM. This probably explains the large differences in Lr , i.e. around 20–25 per cent, seen for CLÉS model in the central regions (see Fig. 1). We also find differences in grav of a factor of 2 (CLÉS vs. CESAM) and 3 (GARSTEC vs. CESAM) in a very narrow region in the middle of the H-shell, but these differences appear in a region where nuc is large and therefore are less visible in the luminosity differences. The differences in X seen in the ASTEC model for Case 1.3 in the region where r/R ∈ [0.1, 0.3] are probably due to the nuclear reaction network it uses. ASTEC models have no carbon in their mixture since they assume that the CN part of the CNO cycle is in nuclear equilibrium at all times and include the original 12 C abundance into that of 14 N. That means that the nuclear reactions of the CNO cycle that should take place at r ∼ 0.1R do not occur and hence the hydrogen in that region is less depleted than in models built by the other codes. 3.2.2 Intermediate mass models: Cases 1.4 and 1.5 Case 1.4 illustrates the PMS evolution phase of a 2M star when the 12 C and 14 N abundances become that of equilibrium. We can see in Fig. 6 that the largest differences in

Astrophys Space Sci (2008) 316: 187–213

obtained by ASTEC (top) and CLÉS (bottom). Differences between physical quantities have been calculated at fixed mass and plotted as a function of r/R

2 r/g calculated Fig. 4 TASK 1: Logarithmic differences of A = NBV at fixed m/M between pairs of models (CODE-CESAM) as a function of r/R for Case 1.3

X are indeed found in the region where 12 C is transformed into 14 N (i.e. where r  0.14R, see Fig. 5), that is in the region in-between m ∼ 0.1M (edge of the convective core) and m = 0.2M. In central regions, the contribution of the gravitational contraction to the total energy release is important: the ratio of the gravitational to the total energy grav /(grav + nuc ) varies from ∼6% in the centre up to ∼50% at r/R ∼ 0.1.

Astrophys Space Sci (2008) 316: 187–213

Comparisons between CLÉS, CESAM and GARSTEC show differences in grav and nuc of a few per cent which eventually partially cancel. We note in Fig. 6 a difference in Lr of ∼ 12% for the STAROX model but the data made available for this model do not allow to determine if the difference comes from the nuclear or the gravitational energy generation rate.

Fig. 5 TASK 1, Case 1.4: 12 C and 14 N abundances in the region where they become that of equilibrium for models computed by CESAM (long-dash-dotted lines), CLÉS (dash-dotted lines), GARSTEC (dashed lines), and STAROX (dotted lines)

Fig. 6 TASK 1: Plots of the differences at fixed relative mass between pairs of models (CODE-CESAM) corresponding to Cases 1.4 and 1.5. Left panel: logarithmic sound speed differences. Centre left panel: log-

195

Case 1.5 deals with a 2M model at the end of the MS when the central H content is Xc = 0.01. In this model, the star was evolved with a central mixed region increased by 0.15Hp (Hp being the pressure scale height) with respect to the size of the convective core determined by the Schwarzschild criterion. CESAM, ASTEC and CLÉS assume, as specified, an adiabatic stratification in the overshooting region while STAROX generated the model assuming a radiative stratification in this zone. That smaller temperature gradient, even if it affects only a quite small region, works in practice like an increase in opacity. This leads to an evolution with a larger convective core, and therefore to a higher effective temperature and luminosity in the STAROX model. Therefore in Fig. 6 we only show the differences between the ASTEC and CLÉS models with respect to CESAM. The largest differences in c (as well as in ρ and X) are found in the region in-between r = 0.03R and 0.06R (m/M ∈ [0.07–0.2]). They reflect the differences in the mean molecular weight gradient (∇μ ) left by the inwards displacement of the convective core during the MS evolution. The strong peak at r ∼ 0.06R (m ∼ 0.2M) found in ASTEC-curves, as well as the plateau of δX between r ∼ 0.06R and 0.2R probably result from the treatment of chemical evolution in the ASTEC code, that assumes the CN part of the CNO cycle to be in nuclear equilibrium at all times. 3.2.3 High mass models: Cases 1.6 and 1.7 For the more massive models (Cases 1.6 and 1.7) the agreement between the 5 codes is generally quite good. In the

arithmic density differences. Centre right panel: hydrogen mass fraction differences. Right panel: logarithmic luminosity differences. Horizontal dotted line represents the reference model (CESAM)

196

Astrophys Space Sci (2008) 316: 187–213

Fig. 7 TASK 1: Plots of the differences at fixed relative mass between pairs of models (CODE-CESAM) corresponding to Cases 1.6 and 1.7. Left panel: logarithmic sound speed differences. Centre left panel: log-

arithmic density differences. Centre right panel: hydrogen mass fraction differences. Right panel: logarithmic luminosity differences. Horizontal dotted line represents the reference model (CESAM)

ZAMS model (Case 1.6, Fig. 7) only a spike in δX is found

that affects the sound speed and the Brunt-Väisälä frequency and hence the frequencies of g and p-g mixed modes. For each model considered we looked for the location of the borders of the convective regions by searching the zeros 2 r/g. The results, expressed in relof the quantity A = NBV ative radius and in acoustic depth, are collected in Table 6. Since variations of the adiabatic parameter 1 can also introduce periodic signals in the oscillation frequencies, we also display in Table 6 the values of the relative radius and acoustic depth of the second He-ionisation region that were determined by locating a minimum in 1 . We find a good agreement between the radii at the bottom of the convective envelope obtained with the 5 codes. The dispersion in the values is smaller than 0.01% for Cases 1.4, 1.6 and 1.7. It is of the order of 0.3% for Cases 1.1, 1.2 and 1.5 while the largest dispersion (0.7%) is found for Case 1.3. Concerning the mass of the convective core, the differences between codes increase as the stellar mass decreases: the differences are in the range 0.1–4% for Case 1.7, 0.05–2% for Case 1.6, 2.5–4% for Case 1.5, 2.5–17% for Case 1.4 and 3–30% for Case 1.2. We point out that the convective core mass is larger in the Case 1.5 model provided by STAROX which is due to the fact that STAROX sets the temperature gradient to the radiative one in the overshooting region while the other codes take the adiabatic gradient. We now focus on the models for Cases 1.3, 1.5 and 1.7. They illustrate the situations that can be found for the evolution of a convective core on the MS. Case 1.3 deals with a 1.2M star which has a growing convective core during a large fraction of its MS. Case 1.5 considers a 2M star for which the convective core is shrinking during the MS and

at the convective core boundary (r ∼ 0.16R). Again, we can see a plateau of δX above the convective core boundary in the ASTEC models which probably results from the fact that it assumes the CN part of the CNO cycle to be in nuclear equilibrium at all times. We can also note that the model provided by GARSTEC corresponds to a model slightly more evolved than specified, with Xc = 0.6897 instead of 0.69. In the model in the middle of its MS (Case 1.7), the features seen in the differences are similar to those in Case 1.5 models, the largest differences being concentrated in the ∇μ -region left by the shrinking convective core. 3.3 External layers The variations of c, ρ, and 1 at fixed radius in the most external layers of the models are plotted for selected cases in Fig. 8. As we shall show in Sect. 3.5 these differences play an important role in the p-mode frequency variations. 3.4 Convection regions and ionisation zones The location and evolution with time of the convective regions are essential elements in seismology. Rapid changes in the sound speed, like those arising at the boundary of a convective region, introduce a periodic signature in the oscillation frequencies of low-degree modes (Gough 1990) that in turn can be used to derive the location of convective boundaries. In addition, the location and displacement of the convective core edge leave a chemical composition gradient

Astrophys Space Sci (2008) 316: 187–213

197

Fig. 8 TASK 1. Plots of the logarithmic differences calculated at fixed relative radius between pairs of models (CODE-CESAM) for the outer regions of Case 1.1, 1.3 and 1.5 models. Left panel: sound speed differ-

ences. Central panel: pressure differences. Right panel: adiabatic exponent 1 differences. Horizontal dotted line represents the reference model (CESAM)

which undergoes nuclear reactions inside but also outside this core. Finally for the Case 1.7, which is for a 5M star, the convective core is shrinking on the MS with nuclear reactions concentrated in the central region. In Fig. 9 we show, for the 3 cases, the variation of the relative mass in the convective core (qc = mcc /M) as a function of the central H mass fraction (which decreases with evolution). For the most massive models (Cases 1.5 and 1.7), all the codes provide a similar evolution of the mass of the convective core, and the variations of qc between them are in the range 0.5–5% (corresponding to m/M = 2 × 10−4 – 7 × 10−3 ). We note that ASTEC behaves differently for the 2M model at the beginning of the MS stage when Xc  0.5.

This is probably due to the fact that ASTEC does not include in the total energy the part coming from the nuclear reactions that transform 12 C into 14 N. Case 1.3 is the most problematic one. For a given chemical composition there is a range of stellar masses (typically between 1.1 and 1.6M ) where the convective core grows during a large part of the MS. This generates a discontinuity in the chemical composition at its boundary and leads to the onset of semiconvection (see e.g. Gabriel and Noels 1977; Crowe and Matalas 1982). The crumple profiles of qc in Fig. 9 (left) are the signature of a semiconvection process that has not been adequately treated. In fact, none of the codes participating in this comparison treat the semicon-

198

Astrophys Space Sci (2008) 316: 187–213

Table 6 TASK 1: Features relevant for seismic analysis. Columns 1 and 2: acoustic radius τ0 of the model (at the photosphere) in seconds and acoustical cutoff frequency νac in µHz. Columns 3 to 6: relative radius rcz /R at the bottom of the envelope convection zone(s) and corCode

responding acoustic depths τenv in seconds. Columns 7 to 9: relative mass mcc /M, radius rcc /R and acoustic depth τcc of the convective core. Columns 10 and 11: relative radius of the second He-ionisation region and acoustic depth τHeII

1

2

3

4

5

6

7

8

9

10

11

τ0

νac

rcz /R

τenv

rcz /R

τenv

mcc /M

rcc /R

τcc

rHeII

τHeII

Case 1.1

ASTEC CESAM CLÉS GARSTEC STAROX

3134

5356

–

–

0.6985

1904

–

–

–

0.9808

531

3128

5370

–

–

0.6959

1907

–

–

–

0.9807

531

3124

5379

–

–

0.6959

1905

–

–

–

0.9806

533

3107

5401

–

–

0.6980

1889

–

–

–

0.9806

529

3135

5356

–

–

0.6972

1908

–

–

–

0.9806

533

3995

3993

–

–

0.8307

1832

1.0148 × 10−2

0.0512

3926

0.9839

577

3976

4021

–

–

0.8281

1836

8.4785 × 10−3

0.0484

3910

0.9839

576

3969

4030

–

–

0.8285

1831

8.8180 × 10−3

0.0491

3902

0.9838

576

3960

4028

–

–

0.8283

1829

1.1026 × 10−2

0.0531

3888

0.9840

572

3987

4009

–

–

0.8299

1832

7.6050 × 10−3

0.0465

3924

0.9839

576

9915

1136

–

–

0.7816

5244

–

–

–

0.9726

1868

9922

1134

–

–

0.7844

5211

–

–

–

0.9726

1867

9971

1126

–

–

0.7860

5218

–

–

–

0.9725

1874

9885

1139

–

–

0.7873

5159

–

–

–

0.9728

1850

7012

1798

0.9946

329

0.9916

465

9.4057 × 10−2

0.0982

6809

0.9931

398

7000

1801

0.9946

327

0.9916

465

9.8622 × 10−2

0.1003

6793

0.9931

400

Case 1.2

ASTEC CESAM CLÉS GARSTEC STAROX Case 1.3

ASTEC CESAM CLÉS GARSTEC Case 1.4

CESAM CLÉS GARSTEC STAROX

6990

1788

0.9946

328

0.9916

463

9.1552 × 10−2

0.0972

6791

0.9698

400

6956

1826

0.9947

321

0.9917

457

1.0767 × 10−1

0.1044

6741

0.9932

389

17059

645

–

–

0.9873

1359

7.7371 × 10−2

0.03711

16880

0.9919

1004

17052

644

–

–

0.9879

1305

7.6814 × 10−2

0.03692

16874

0.9919

994

17159

639

–

–

0.9880

1309

7.5622 × 10−2

0.03656

16982

0.9918

1002

17805

611

–

–

0.9855

1575

7.9887 × 10−2

0.03635

17624

0.9911

1128

5848

1690

0.99897

82

0.99392

343

2.1263 × 10−1

0.1631

5548

0.9950

295

5832

1696

0.99897

81

0.99393

342

2.0997 × 10−1

0.1624

5533

0.9950

297

5820

1700

0.99899

79

0.99392

341

2.1162 × 10−1

0.1632

5521

0.9950

296

5878

1673

0.99896

80

0.99386

345

2.0774 × 10−1

0.1618

5579

0.9949

298

5831

1685

0.99898

81

0.99392

342

2.1177 × 10−1

0.1628

5532

0.9950

295

13546

556

0.99963

61

0.99291

807

1.5986 × 10−1

0.1098

13084

0.9944

668

Case 1.5

ASTEC CESAM CLÉS STAROX Case 1.6

ASTEC CESAM CLÉS GARSTEC STAROX Case 1.7

ASTEC CESAM CLÉS GARSTEC STAROX

13383

565

0.99967

54

0.99297

794

1.5673 × 10−1

0.1096

12927

0.9945

659

13419

563

1.00000

0

0.99290

802

1.5642 × 10−1

0.1093

12964

0.9944

665

13297

569

0.99967

51

0.99296

789

1.5286 × 10−1

0.1088

12847

0.9945

650

13454

562

0.99971

46

0.99294

799

1.5966 × 10−1

0.1100

12995

0.9945

654

Astrophys Space Sci (2008) 316: 187–213

199

Fig. 9 TASK 1: Relative mass (qc ) at the border of convective core as a function of the central hydrogen mass fraction for Cases 1.3 (left), 1.5 (middle), and 1.7 (right)

vection instability. The large difference between the ASTEC model curve and the CESAM and CLÉS ones results from the way the codes locate convective borders. While ASTEC searches these boundaries downwards starting from the surface, CLÉS and CESAM search upwards beginning from the centre. We point out that semiconvection also appears below the convective envelope of these stars if microscopic diffusion is included in the modelling (see e.g. Richard et al. 2001, and Sect. 4 below).

Table 7 TASK 1: Solar-like oscillations in Cases 1.1, 1.2 and 1.3: cutoff frequency at the photosphere νac , frequency νmax expected at the maximum of the power spectrum and corresponding radial order kmax , and differences δν ( = 0) in the frequencies between the different codes Case

νac (µHz)

νmax (µHz)

kmax

δν ( = 0) (µHz)

1.1

5400

3500

24

0.2–1

1.2

4000

2660

18

0.2–1

1.3

1100

770

8

0.05–0.2

3.5 Seismic properties Using the adiabatic oscillation code LOSC we computed the oscillation frequencies of p- and g-modes with degree = 0, 1, 2, 3 and for frequencies in the range σ = 0.3–70/τdyn where σ is the angular frequency and τdyn = (R 3 /GM)1/2 is the dynamical time. In these computations we used the standard option in LOSC, that is, regularity of solution when P = 0 at the surface (δP /P + (4 + ω2 )δr/r = 0). The frequencies were computed on the basis of the model structure up to the photosphere (optical depth τ = 2/3). When evaluating differences between different models they were scaled to correct for differences in stellar radius. The frequency differences νCODE − νCESAM are displayed in Figs. 10– 14. 3.5.1 Solar-like oscillations: Cases 1.1, 1.2 and 1.3 On the basis of the Kjeldsen and Bedding (1995) theory, we have estimated the frequency νmax at which we expect the maximum in the power spectrum. This value together with (1) the radial order corresponding to the maximum (kmax ), (2) the acoustical cutoff frequency at the photosphere (νac = c/4πHp ), and (3) the differences in the frequencies between different codes are collected in Table 7. The frequency domains covered are in the range ν ∼ 200–5000 µHz for Case 1.1, 200–4000 µHz for Case 1.2

and 100–2000 µHz for Case 1.3 models. The radial orders are in the range k ∼ 0–5. To explore the effects of the model frequencies in the asymptotic p-mode region we have included modes well above the acoustical cutoff frequency. In addition to the differences δν = νCODE − νCESAM , we have computed the large frequency separation for = 0 and 1 (ν ,k = ν ,k − ν ,k−1 ) (see Figs. 10 and 11), and for Cases 1.1 and 1.2 we derived the frequency-separation ratios defined in Roxburgh and Vorontsov (2003). For these quantities the original model frequencies, without corrections for differences in radius, were used; indeed a substantial part of the visible differences in ν are caused by the radius differences. As shown by Roxburgh and Vorontsov (see also Floranes et al. 2005) the frequency-separation ratios have the advantage to be independent of the physical properties of the outer layers. The almost perfect agreement between the value of these ratios for the models computed with all the codes indicates that the differences observed in the frequencies and in the large frequency separation are only determined by the differences in the surface layers (see also Fig. 8). For the highly condensed Case 1.3 model, the differences in = 0 mode frequencies come from surface differences. On the other hand, the peaks observed in the = 1 mode frequency differences and in the large frequency separation come from variations of Brunt-Väisälä frequency and

200

Fig. 10 TASK 1: Left panel: p-mode frequency differences between models produced by different codes, for Case 1.1 (top row) and 1.2 (bottom). CESAM model is taken as reference, and the frequencies have been scaled to remove the effect of different stellar radii. For each code, we plot two curves corresponding to modes

Astrophys Space Sci (2008) 316: 187–213

with degrees = 0 and = 1. Central panel: Large frequency separations ν( = 0) and ν( = 1) versus the radial order k for Case 1.1 and 1.2 models; these are based on unscaled frequencies. Right panel: Frequency separation ratios as a function of the radial order k

Fig. 11 TASK 1: Left panel: p-mode frequency differences between models produced by different codes, for Case 1.3. CESAM model is taken as reference, and the frequencies have been scaled to remove the effect of different stellar radii. For each code, we plot two curves corresponding to modes with degrees = 0 and = 1. Right panel: Large frequency separations ν( = 0) and ν( = 1) versus the radial order k for Case 1.3, based on unscaled frequencies

from the mixed character of the corresponding modes, see Christensen-Dalsgaard et al. (1995) and references therein. The frequencies of the modes trapped in the μ-gradient region depend not only on the location of this gradient but also on its profile. Differences shown in Fig. 4 reflect the different behaviour of the μ-gradient in ASTEC with respect to CLÉS, GARSTEC, and CESAM which in turn can explain the different behaviour of the ASTEC frequencies seen in Fig. 11.

3.5.2 Cases 1.4 and 1.5 Figure 12 (left panel) displays the differences in the p-mode frequencies for the PMS model of 2M . Two bumps appear in the differences between STAROX and CESAM. The inner one at ν ∼ 300 µHz can be attributed to differences in the sound speed close to the centre as seen in Fig. 6. The outer bump at ν ∼ 1500 µHz results from differences in 1 in the second He-ionisation region.

Astrophys Space Sci (2008) 316: 187–213

201

Fig. 12 TASK 1: p-mode frequency differences between models produced by different codes, for Case 1.4 (left) and 1.5 (right). CESAM model is taken as reference, and the frequencies have been scaled to remove the effect of different stellar radii. For each code, we plot two curves corresponding to modes with degrees = 0 and = 1

can be seen in Fig. 12 the frequencies of the mixed modes are also modified. 3.5.3 Cases 1.6 and 1.7

2 r/g in the deep interior Fig. 13 TASK 1: Run of the quantity A = NBV of Case 1.5 models

Figure 12 (right panel) shows the differences in the pmode frequencies for the evolved Case 1.5 model. As in Case 1.3, the differences mainly result from differences in the surface layers (see Fig. 8). Also, this model is sufficiently evolved to present g-p mixed modes. In fact, the peaks observed at low frequency for = 1 modes correspond to modes trapped in the μ-gradient region. Figure 13 displays the profile of A showing that even though the μ gradient is generated at the same depth in the star, its slope is quite different, and therefore the mixed-mode frequencies also differ. We point out that the smoother decrease of A observed in CESAM models with respect to others at r ∼ 0.065R is due to the scheme used for the integration of the temporal evolution of the chemical composition, i.e. an L-stable implicit Runge-Kutta scheme of order 2 (see Morel and Lebreton 2007). We have checked that when the standard Euler backward scheme is used, the A profile becomes quite similar to what is obtained by other codes (see Fig. 13) and that, as

The frequency differences for p-modes in Case 1.6 and 1.7 are smaller than 0.2 µHz except for the GARSTEC models, for which the differences can be slightly larger than 0.2 µHz for the more massive model, and reach 0.8 µHz for the ZAMS one. We recall, however, that this latter has a central hydrogen content slightly smaller than specified, differing by −3.4 × 10−4 from the specified Xc = 0.69. To investigate the effect of such a small difference in the central H content, the frequencies of two CLÉS models differing by δXc = 3.4 × 10−4 have been calculated: they show differences in the range −0.05 to ∼0.3 µHz that only partially account for the differences found. The stellar parameters of Case 1.7 models match quite well those of a typical SPB star (Slowly Pulsating B type star). This type of pulsators presents high-order g-modes with periods ranging from 0.4 to 3.5 days for modes with low degree ( = 1 and 2) (Dziembowski et al. 1993). We have estimated for Case 1.7 models, the period differences for g-modes with radial order k = −30 to −1. As has been shown in Miglio et al. (2006) the periods of g-modes can present also a oscillatory signal, whose periodicity depends on the location of the μ gradient, and whose amplitude is determined by the slope of the chemical composition gradient. The profile of the quantity A for Case 1.7 is quite similar to that of Case 1.5 (Fig. 13), that is with the profile in CESAM model being smoother than in models obtained by the other codes. The effect on the variation of g-mode periods is shown in Fig. 14 (right) where the periodicity of the signature is related to the location of the μ gradient and the amplitude of the difference is increasing with the steepness of the gradient.

202

Astrophys Space Sci (2008) 316: 187–213

Fig. 14 TASK 1: Left and central panels: p-mode frequency differences between models produced by different codes, for Case 1.6 (left) and 1.7 (centre). CESAM model is taken as reference, and the frequencies have been scaled to remove the effect of different stellar radii. For

each code, we plot two curves corresponding to modes with degrees

= 0 and = 1. Right panel: Plots of the g-mode period differences, between models produced by different codes for Case 1.7 (the CESAM model—horizontal dotted line—is taken as reference)

Table 8 TASK 3 models: Global parameter differences given in per cent, between each code and CESAM-MP. For each parameter we give the mean difference and the maximum difference of the complete series of TASK 3 models (i.e. each case and each phase are included) Code

ASTEC CESAM-B69 CLÉS GARSTEC

δM/M

δR/R

δL/L

δTeff /Teff

δage/age

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

0.01

0.01

0.45

0.85

1.06

1.89

0.04

0.06

–

–

0.00

0.00

0.23

0.54

0.11

0.31

0.10

0.22

0.35

1.02

0.00

0.00

0.23

0.54

0.23

0.45

0.07

0.20

0.47

0.77

0.05

0.08

3.21

26.52

4.30

37.02

0.54

3.92

–

–

4 Comparisons for TASK 3 4.1 Presentation of the comparisons and general results TASK 3 deals with models that include microscopic dif-

fusion of helium and metals due to pressure, temperature and concentration gradients. The codes examined here have adopted different treatments of the diffusion processes. The ASTEC code follows the simplified formalism of Michaud and Proffitt (1993) (hereafter MP93) while the CLÉS and GARSTEC codes compute the diffusion coefficients by solving Burgers’ equations (Burgers 1969 hereafter B69) according to the formalism of Thoul et al. (1994). On the other hand, CESAM provides two approaches to compute diffusion velocities: one, which will be denoted by CESAM-MP is based on the MP93 approximation, the other (hereafter CESAM-B69) is based on Burger’s formalism, with collisions integrals derived from Paquette et al. (1986). We point out that after preliminary comparisons for TASK 3 models presented by Montalbán et al. (2007b) and Lebreton et al. (2007b), we fixed some numerical problems found in the CESAM calculations including diffusion with the B69 approach. Therefore, all the CESAM models presented here (both CESAM-MP and CESAM-B69 ones) are new recalculated models.

Low stellar masses (1.0, 1.2 and 1.3M ) corresponding to solar-type stars, for which diffusion resulting from radiative forces can be safely neglected, have been considered at 3 stages of evolution (middle of the MS when Xc = 0.35, end of the MS when Xc = 0.01 and on SGB when the helium core mass represents 5 per cent of the total mass of the star). The models provided again have a different number of mesh points: the number of mesh points is 1200 in the ASTEC and CESAM models, 2300–2500 in CLÉS models and 1700–2000 in models by GARSTEC. Table 8 gives the mean and maximum differences in the global parameters (mass, radius, luminosity, effective temperature and age) obtained by each code with respect to CESAM-MP models. For each code, the mean difference has been obtained by averaging over the number of cases and phases calculated. The differences are generally very small, i.e. below 0.5 per cent for CESAM-B69, CLÉS and GARSTEC. They are a bit larger for ASTEC evolved models and they are high (25–37%) for one GARSTEC model (the Case 3.2C, subgiant model). We note that there are small differences in mass in ASTEC and GARSTEC models: ASTEC uses a value of the solar mass slightly smaller than the one specified for the comparisons while GARSTEC starts from the specified mass but takes into

Astrophys Space Sci (2008) 316: 187–213

203

Table 9 TASK 3 models: Mean quadratic difference in the physical variables between each code and CESAM calculated according to (1). The differences are given in per cent (except for δX) and represent an Code

average over the whole star from centre to photospheric radius. The local differences were calculated at fixed relative mass

δ ln c

δ ln P

δ ln ρ

δ ln T

δ ln r

δ ln 1

δ ln ∇ad

δ ln Cp

δ ln κ

δX

δ ln Lr

0.03 0.02 0.02 0.04

0.14 0.17 0.18 0.46

0.14 0.17 0.18 0.43

0.06 0.03 0.04 0.07

0.06 0.04 0.04 0.11

4.07 × 10−4 1.55 × 10−4 2.41 × 10−3 1.51 × 10−1

7.42 × 10−4 3.21 × 10−4 5.93 × 10−3 1.23 × 10−1

3.51 × 10−2 3.27 × 10−2 3.84 × 10−2 1.40 × 10−1

1.37 0.16 0.24 0.59

0.00049 0.00042 0.00046 0.00065

0.19 0.04 0.17 0.51

0.15 0.04 0.05 0.06

1.66 0.50 0.53 0.69

1.59 0.48 0.49 0.56

0.53 0.09 0.13 0.16

0.58 0.12 0.16 0.15

1.73 × 10−3 5.10 × 10−4 3.98 × 10−3 1.60 × 10−1

3.87 × 10−3 1.02 × 10−3 9.06 × 10−3 1.42 × 10−1

4.36 × 10−1 7.77 × 10−2 7.45 × 10−2 2.80 × 10−1

2.37 0.29 0.39 0.61

0.00406 0.00085 0.00091 0.00186

1.91 0.35 0.76 0.67

0.15 0.05 0.06 0.10

1.21 0.50 0.40 0.87

1.00 0.46 0.41 0.75

0.41 0.11 0.09 0.23

0.35 0.11 0.11 0.19

1.32 × 10−3 5.56 × 10−4 4.53 × 10−3 1.72 × 10−1

2.55 × 10−3 1.22 × 10−3 1.02 × 10−2 1.62 × 10−1

2.32 × 10−1 8.49 × 10−2 1.34 × 10−1 3.81 × 10−1

2.68 0.44 0.39 0.64

0.00241 0.00106 0.00117 0.00242

5.87 0.75 2.87 1.37

0.02 0.03 0.12

0.15 0.15 0.33

0.15 0.16 0.39

0.02 0.02 0.07

0.03 0.04 0.10

2.26 × 10−4 2.05 × 10−3 1.61 × 10−1

4.75 × 10−4 4.96 × 10−3 1.45 × 10−1

3.76 × 10−2 5.20 × 10−2 2.29 × 10−1

0.12 0.22 0.44

0.00041 0.00054 0.00200

0.07 0.43 0.80

0.21 0.23 0.18

1.64 1.71 0.95

1.38 1.48 0.86

0.30 0.28 0.15

0.35 0.37 0.22

1.55 × 10−3 3.64 × 10−3 1.68 × 10−1

4.05 × 10−3 8.86 × 10−3 1.66 × 10−1

3.23 × 10−1 3.96 × 10−1 2.40 × 10−1

0.56 0.71 0.69

0.00276 0.00339 0.00145

0.59 0.80 0.77

0.06 0.20 0.17

0.22 1.47 1.47

0.24 1.25 1.26

0.04 0.32 0.30

0.06 0.34 0.34

2.95 × 10−4 4.30 × 10−3 1.74 × 10−1

1.02 × 10−3 1.02 × 10−2 1.71 × 10−1

1.26 × 10−1 3.35 × 10−1 5.12 × 10−1

0.28 0.29 0.46

0.00117 0.00294 0.00348

2.90 1.49 3.50

0.18 0.04 0.10

0.39 0.06 0.26

0.50 0.11 0.31

0.06 0.03 0.05

0.09 0.02 0.06

1.29 × 10−3 2.58 × 10−3 1.65 × 10−1

3.42 × 10−3 6.69 × 10−3 1.55 × 10−1

3.61 × 10−1 7.85 × 10−2 2.10 × 10−1

0.32 0.24 0.54

0.00373 0.00088 0.00122

0.41 0.84 0.49

0.18 0.18 0.34

1.09 1.21 2.09

0.92 0.98 1.77

0.21 0.26 0.41

0.25 0.28 0.45

1.07 × 10−3 3.23 × 10−3 1.69 × 10−1

3.08 × 10−3 7.89 × 10−3 1.69 × 10−1

2.90 × 10−1 2.71 × 10−1 7.50 × 10−1

0.30 0.22 0.53

0.00264 0.00248 0.00634

0.42 0.42 0.36

0.09 0.11 0.30

0.48 0.28 1.84

0.43 0.38 1.67

0.11 0.11 0.37

0.11 0.10 0.43

7.23 × 10−4 3.32 × 10−3 1.76 × 10−1

1.64 × 10−3 8.15 × 10−3 1.76 × 10−1

1.45 × 10−1 2.41 × 10−1 7.72 × 10−1

0.27 0.30 0.46

0.00138 0.00211 0.00601

4.40 1.28 3.57

Case 3.1A

ASTEC CESAM-B69 CLÉS GARSTEC Case 3.1B

ASTEC CESAM-B69 CLÉS GARSTEC Case 3.1C

ASTEC CESAM-B69 CLÉS GARSTEC Case 3.2A

CESAM-B69 CLÉS GARSTEC Case 3.2B

CESAM-B69 CLÉS GARSTEC Case 3.2C

CESAM-B69 CLÉS GARSTEC Case 3.3A

CESAM-B69 CLÉS GARSTEC Case 3.3B

CESAM-B69 CLÉS GARSTEC Case 3.3C

CESAM-B69 CLÉS GARSTEC

account the decrease of mass during the evolution which results from the energy lost by radiation. As in TASK 1 we have examined the differences in the physical variables computed by the codes. ASTEC results are only considered for Case 3.1 as further studies are under way for models including convective cores (see ChristensenDalsgaard 2007c). Table 9 provides the “mean quadratic differences” calculated according to (1).

As in TASK 1 we note that the “mean-quadratic differences” between the codes generally remain quite low. The differences in P , T , Lr , r, ρ and κ range from 0.1 to 6%. The differences in the thermodynamic quantities (1 , ∇ad , Cp ) are often well below 1 per cent with, as in TASK 1, larger differences in the GARSTEC code which are probably due to a different use of the OPAL equation of state package and variables.

204

Astrophys Space Sci (2008) 316: 187–213

Table 10 TASK 3 models: Maximum variations given in per cent (except for δX) of the physical variables between each code and CESAM and value of the relative radius (r/R) where they happen. The local Code

differences were computed both at fixed relative mass and fixed relative radius and the maximum of the two values was searched (see footnote 2)

δ ln c

r/R

δ ln P

r/R

δ ln ρ

r/R

δ ln 1

r/R

δX

r/R

δ ln Lr

0.13 0.07 0.12 0.23

0.68223 0.73259 0.68285 0.25979

0.94 0.64 1.11 2.23

0.96261 0.42254 0.58835 0.51009

0.85 0.57 1.12 2.07

0.96261 0.44886 0.67438 0.66870

0.01 0.00 0.02 0.23

0.96261 0.96260 0.79926 0.00000

0.00271 0.00192 0.00220 0.00271

0.69608 0.68474 0.67861 0.08724

1.14 0.19 0.63 4.03

0.00155 0.06786 0.06038 0.00215

0.49 0.21 0.19 0.35

0.96005 0.71495 0.75105 0.01139

4.96 1.59 2.07 1.62

0.42170 0.35641 0.49726 0.52826

4.34 1.41 1.91 1.65

0.42410 0.37767 0.63770 0.63443

0.04 0.02 0.03 0.30

0.96005 0.96000 0.95661 0.00000

0.01364 0.00343 0.00442 0.00682

0.05022 0.70558 0.71375 0.06100

10.63 2.59 5.95 7.81

0.00644 0.01962 0.01206 0.00130

0.69 0.44 0.41 0.36

0.66252 0.65858 0.03876 0.65262

2.55 1.37 1.62 2.26

0.40052 0.29892 0.41953 0.86451

3.41 1.98 2.05 2.60

0.66153 0.65858 0.65804 0.65653

0.02 0.03 0.03 0.32

0.96068 0.96042 0.96067 0.00000

0.02401 0.01742 0.01116 0.01186

0.66153 0.65858 0.65804 0.65996

232.00 9.27 16.81 15.42

0.00216 0.02439 0.02402 0.00266

0.44 0.68 2.42

0.04631 0.04632 0.04531

1.43 0.59 1.38

0.88799 0.79763 0.36304

1.12 1.49 5.22

0.88799 0.04632 0.04531

0.01 0.05 0.26

0.88799 0.84435 0.00000

0.00895 0.01382 0.04600

0.04631 0.04632 0.04556

0.85 7.17 7.25

0.04674 0.00173 0.00438

1.22 1.40 0.91

0.04918 0.04903 0.04599

2.96 3.15 2.19

0.66237 0.65759 0.68604

3.08 3.42 2.61

0.74359 0.74393 0.74353

0.10 0.05 0.33

0.88773 0.94536 0.00000

0.02190 0.02501 0.01171

0.04968 0.04942 0.04614

2.98 4.07 6.09

0.00520 0.00378 0.00065

0.40 0.94 0.87

0.03964 0.03764 0.03799

2.18 2.07 1.93

0.72949 0.41034 0.28747

1.93 2.41 2.95

0.72949 0.03764 0.03781

0.10 0.06 0.32

0.73083 0.79075 0.01557

0.00788 0.01781 0.02111

0.04007 0.03881 0.03835

22.20 10.13 26.89

0.02376 0.03093 0.02166

3.00 1.51 2.83

0.06563 0.88954 0.89137

5.22 7.43 14.12

0.88988 0.88881 0.88512

6.15 7.41 14.07

0.06563 0.88954 0.88583

0.02 0.07 0.27

0.86685 0.89961 0.00000

0.06233 0.03196 0.04040

0.06563 0.88954 0.88583

1.75 7.91 3.68

0.00764 0.00159 0.00092

1.37 1.02 2.41

0.83342 0.05241 0.05219

11.12 1.48 2.65

0.83342 0.34861 0.33358

8.25 1.68 4.82

0.83342 0.05198 0.05219

0.12 0.06 0.34

0.83342 0.78878 0.00000

0.01953 0.01949 0.05126

0.05247 0.05241 0.05258

1.65 1.51 1.84

0.03892 0.03895 0.00062

0.68 0.86 2.15

0.03866 0.03902 0.03863

2.14 1.52 2.44

0.73005 0.85187 0.30766

2.08 2.42 5.86

0.03866 0.03902 0.03845

0.11 0.06 0.32

0.73005 0.76734 0.01262

0.01320 0.01759 0.04691

0.03907 0.03902 0.03900

24.23 12.60 26.80

0.01925 0.00123 0.02018

r/R

Case 3.1A

ASTEC CESAM-B69 CLÉS GARSTEC Case 3.1B

ASTEC CESAM-B69 CLÉS GARSTEC Case 3.1C

ASTEC CESAM-B69 CLÉS GARSTEC Case 3.2A

CESAM-B69 CLÉS GARSTEC Case 3.2B

CESAM-B69 CLÉS GARSTEC Case 3.2C

CESAM-B69 CLÉS GARSTEC Case 3.3A

CESAM-B69 CLÉS GARSTEC Case 3.3B

CESAM-B69 CLÉS GARSTEC Case 3.3C

CESAM-B69 CLÉS GARSTEC

The maximal differences2 in c, P , ρ, 1 , Lr and X between codes, and the location (r/R) where they happen are reported in Table 10. Again we note that the largest differences are mainly found in the most exter2 The

method of calculation is the same as in Sect. 3.1.

nal layers and (or) at the boundary of the convection regions. 4.2 Internal structure The variations in X, c, Lr and 1 are displayed in Figs. 15, 16, 17, for Cases 3.1, 3.2 and 3.3 respectively. Here, to spare

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205

Fig. 15 TASK 3: Differences between pairs of models (CODECESAM) corresponding to Cases 3.1, phase A (top), B (middle) and C (bottom) plotted as a function of radius. From left to right: for hydrogen mass fraction, logarithmic sound speed, logarithmic luminosity

and adiabatic exponent 1 . Differences have been calculated at fixed relative mass (for X, c, Lr ) or fixed relative radius (for 1 ). Results are given for ASTEC (continuous line), CESAM-B69 (dotted), CLÉS (dot-dash) and GARSTEC (dashed)

space, we do not plot differences in ρ which are reflected in those in c.

speed differences reflect differences (i) in the stellar radius, (ii) in the chemical composition gradients in the central regions and below the convective envelope (see the features in the region where R ∈ [0.65, 0.9]) and finally (iii) in the location of the convective regions boundaries. They remain quite modest except at the border of the convective envelope and in the zone close to the surface. The differences in 1 , seen in the external regions, reflect differences in the He abundance in the regions of second He ionisation. We also note that large differences in luminosity are found on the SGB (phase C).

4.2.1 Solar models: Case 3.1 The solar model is characterised by a radiative interior and a convective envelope which deepens as evolution proceeds. The differences in the hydrogen abundance X seen in Fig. 15 can be compared to those found in TASK 1, Case 1.1 model (Fig. 1, top-right). In the centre, where there exists an H gradient built by nuclear reactions and where (in the present case) H is drawn outwards by diffusion, the differences are roughly of the same order of magnitude. In the middle-upper radiative zone, where the settling of He and metals leads to an H enrichment, and in the convection zone, much larger differences are found which reflect different diffusion velocities and also depend on the extension and downward progression of the convective envelope. We note that differences grow with evolution from phase A to C. The sound

4.2.2 Solar-type stars with convective cores: Cases 3.2 and 3.3 Those stars of 1.2 and 1.3M have, on the MS, a convective envelope and a convective core. The differences in the hydrogen abundance X seen in the centre in Fig. 16 and 17

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Fig. 16 TASK 3: Differences between pairs of models (CODECESAM) corresponding to Cases 3.2, phase A (top), B (middle) and C (bottom) plotted as a function of radius. From left to right: for hydrogen mass fraction, logarithmic sound speed, logarithmic luminosity

and adiabatic exponent 1 . Differences have been calculated at fixed relative mass (for X, c, Lr ) or fixed relative radius (for 1 ). Results are given for ASTEC (continuous line), CESAM-B69 (dotted), CLÉS (dot-dash) and GARSTEC (dashed)

are rather similar to those found in TASK 1, Case 1.2 and 1.3 models (Fig. 1, centre and bottom, right). As explained in Sect. 3.4, the core mass is growing during a large fraction of the MS. Due to nuclear reactions a helium gradient builds up at the border of the core. In these regions, the diffusion due to the He concentration gradient competes with the He settling term and finally dominates which makes He move outwards from the core regions. As a consequence metals also diffuse outwards preventing the metal settling. Because of the metal enrichment which induces an opacity increase, the zone at the border of the convective core is the seat of semiconvection (Richard et al. 2001). Also, large differences appear at the bottom of and in the convective envelope in the presence of diffusion. In fact, in these models, diffusion makes the metals pile up beneath the convective envelope which induces an increase of opacity in this zone which in turn triggers convective instability in the form of semiconvection (see Bahcall et al. 2001).

4.3 Convection zones Figure 18 shows the evolution of the radius of the convective envelope in the models for Cases 3.1, 3.2 and 3.3 while Fig. 19 displays the evolution of the mass of the convective core in Cases 3.2 and 3.3. The crumpled zones in the rcz /R and mcc /M profiles are the signature of the regions of semiconvection which we find for Cases 3.2 and 3.3 either in the regions above the convective core or beneath the convective envelope. As pointed out by Montalbán et al. (2007b), in the presence of metal diffusion, it is difficult to study the evolution of the boundaries of convective regions. The numerical treatment of those boundaries in the codes is crucial for the determination of the evolution of the unstable layers: it affects the outer convective zone depths and surface abundances as well as the masses of the convective cores and therefore the evolution of the star.

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Fig. 17 TASK 3: Differences between pairs of models (CODECESAM) corresponding to Cases 3.3, phase A (top), B (middle) and C (bottom) plotted as a function of radius. From left to right: for hydrogen mass fraction, logarithmic sound speed, logarithmic luminosity

and adiabatic exponent 1 . Differences have been calculated at fixed relative mass (for X, c, Lr ) or fixed relative radius (for 1 ). Results are given for ASTEC (continuous line), CESAM-B69 (dotted), CLÉS (dot-dash) and GARSTEC (dashed)

Rather small differences in the location of the convective boundaries are seen in Figs. 18 and 19. Table 11 displays the properties of the convective zone boundaries. The radii found at the base of the convective envelope in all cases differ by 0.1–0.7 per cent. On the other hand the mass in the convective cores differs by 2–4 per cent except for Case 3.2A where the mass in GARSTEC model differs from the others by more than 10 per cent.

Case 3.3 (phases B and C) whatever the prescription for the diffusion treatment is. For Case 3.3A which is hotter with a thinner convection envelope, the differences between the MP93 prescription for diffusion used in CESAM-MP models and the complete solution of Burger’s equations B69 are rather large. For CESAM-B69 models this difference is of the order of 16%, and a maximum of about 30% in the diffusion efficiency is found between GARSTEC and CESAMMP. Such differences can indeed be expected and result from the different approaches used to treat microscopic diffusion (MP93 vs. B69) and from the approximations made to calculate the collision integrals. For the solar model, Thoul et al. (1994) found differences of about 15 per cent between their results—based on the solution of Burger’s equations but with approximations made to estimate the collision integrals and the MP93 formalism of Michaud and Proffitt (1993)—where the diffusion equations are simplified but in

4.4 Helium surface abundance Figure 20 displays the helium abundance Ys in the convective envelope for the different cases and phases considered. The evolution of Ys is linked to the efficiency of microscopic diffusion inside the star and to the evolution with time of the internal border of the convective envelope. We note that the surface helium abundance differs by less than 2% for Case 3.1 (any phase) and 3% for Case 3.2 (any phase) and

208

which the collision integrals are obtained according to Paquette et al. (1986). Further tests made by one of us (JM) have shown that for Case 3.2, the use of collision integrals

Astrophys Space Sci (2008) 316: 187–213

of Paquette et al. (1986) within the formalism by Thoul et al. (1994) leads to differences in the surface He abundance that may amount to ∼ 2.5 per cent. Since we can expect that the differences of the diffusion coefficients increase as the depth of the convective zone decreases, it is not surprising that in Case 3.3 models differences in the surface helium abundance are even larger. We also point out that the helium depletion is also sensitive to the numerical treatment of convective borders, in particular in the presence of semiconvection. 4.5 Seismic properties

Fig. 18 TASK 3: Evolution of the radius of the convective envelope for Cases 3.1, 3.2 and 3.3 (respectively 1.0, 1.2 and 1.3M ) in models computed by ASTEC, GARSTEC, CLÉS and CESAM with for the latter two different approaches to treat microscopic diffusion (MP93 and B69)

Fig. 19 TASK 3: Evolution of the mass of the convective core for Cases 3.2 and 3.3 (respectively 1.0, 1.2 and 1.3M ) in models computed by CLÉS, GARSTEC and CESAM with for the latter two different approaches to treat microscopic diffusion (MP93 and B69)

As in Sect. 3.5 we present in Fig. 21 the frequency differences νCODE − νCESAM , where again the frequencies have been scaled to correct for differences in the stellar radius. They can be compared to the results obtained in TASK 1 Cases 1.1, 1.2 and 1.3 (Figs. 10 and 11). Differences increase as evolution proceeds and as the mass increases. We find that the trend of the differences found in MS models (phase A) for the 3 cases (3.1, 3.2, 3.3) is very similar to what has been found for Case 1.1 and 1.2 MS models. Again the similar behaviour of curves with different degree indicates that the frequency differences are due to near-surface effects. Differences between curves corresponding to modes of degree = 0 and 1, which reflect differences in the interior structure, remain small, below 0.1–0.2 µHz (a bit larger for ASTEC). The magnitude of the differences is, on the average, higher in models including microscopic diffusion, due to larger differences in the sound speed in particular in the central regions (or border of the convective core) and at the base of the convective envelope. Two different oscillatory components with a periodicity of ∼2000 and ∼4000 s appear in the frequency differences. The first one which is mainly visible in the GARSTEC models is due to differences in the adiabatic exponent, and its amplitude is related to different helium abundances in the convective envelope. The second one makes the “saw-tooth” profile, and is due to differences at the border of convective envelope. For TAMS models (phase B), in addition to differences observed for MS models, peaks become clearly visible at low frequencies for = 1 modes. As in Case 1.3 they can be attributed to differences in the Brunt-Väisälä frequency in the interior and to the mixed character of the corresponding modes. Any difference in the μ gradient in the region just above the border of the helium core is indeed expected to be seen in the frequency differences. This effect is even larger in SGB models (phase C).

5 Summary and conclusions We have presented detailed comparisons of the internal structures and seismic properties of stellar models in a range

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209

Table 11 TASK 3: values of the fractional radius and mass at the base of the convective envelope (rcz /R, mcz /M) and border of the convective core (rcc /R, mcc /M) and hydrogen abundance in the convection zones Code

Case

rcz /R

mcz /M

Xcz

rcc /R

mcc /M

Xcc

ASTEC CESAM-B69 CESAM-MP CLÉS GARSTEC

3.1A

0.7348

0.9835

0.7436

–

–

–

3.1A

0.7326

0.9831

0.7446

–

–

–

3.1A

0.7321

0.9830

0.7449

–

–

–

3.1A

0.7315

0.9829

0.7470

–

–

–

3.1A

0.7357

0.9835

0.7475

–

–

–

ASTEC CESAM-B69 CESAM-MP CLÉS GARSTEC ASTEC CESAM-B69 CESAM-MP CLÉS GARSTEC

3.1B

0.7202

0.9837

0.7630

–

–

–

3.1B

0.7153

0.9832

0.7657

–

–

–

3.1B

0.7143

0.9830

0.7656

–

–

–

3.1B

0.7151

0.9831

0.7695

–

–

–

3.1B

0.7159

0.9832

0.7705

–

–

–

3.1C

0.6676

0.9720

0.7627

–

–

–

3.1C

0.6644

0.9716

0.7657

–

–

–

3.1C

0.6631

0.9713

0.7661

–

–

–

3.1C

0.6643

0.9715

0.7684

–

–

–

3.1C

0.6643

0.9712

0.7691

–

–

–

3.2A

0.8489

0.9990

0.7754

0.0456

0.0170

0.3500

3.2A

0.8451

0.9999

0.7835

0.0451

0.0169

0.3499

3.2A

0.8443

0.9990

0.7817

0.0451

0.0170

0.3500

3.2A

0.8444

0.9990

0.7888

0.0447

0.0165

0.3504

3.2A

0.8450

0.9990

0.7883

0.0433

0.0149

0.3499

CESAM-B69 CESAM-MP CLÉS GARSTEC

3.2B

0.7957

0.9969

0.7723

0.0376

0.0325

0.0099

3.2B

0.7927

0.9968

0.7723

0.0375

0.0328

0.0100

3.2B

0.7956

0.9969

0.7796

0.0373

0.0319

0.0101

3.2B

0.7953

0.9969

0.7767

0.0372

0.0318

0.0100

CESAM-B69 CESAM-MP CLÉS GARSTEC

3.2C

0.7921

0.9972

0.7768

–

–

–

3.2C

0.7913

0.9972

0.7765

–

–

–

3.2C

0.7903

0.9971

0.7843

–

–

–

3.2C

0.7931

0.9972

0.7814

–

–

–

CESAM-B69 CESAM-MP CLÉS GARSTEC

3.3A

0.8916

0.9999

0.8840

0.0641

0.0575

0.3500

3.3A

0.8893

0.9998

0.8631

0.0641

0.0572

0.3490

3.3A

0.8893

0.9998

0.8947

0.0633

0.0557

0.3503

3.3A

0.8920

0.9999

0.8955

0.0631

0.0554

0.3500

CESAM-B69 CESAM-MP CLÉS GARSTEC

3.3B

0.8336

0.9990

0.7866

0.0389

0.0404

0.0099

3.3B

0.8342

0.9990

0.7886

0.0385

0.0396

0.0100

3.3B

0.8335

0.9990

0.7948

0.0387

0.0399

0.0101

3.3B

0.8334

0.9989

0.7923

0.0384

0.0391

0.0100

CESAM-B69 CESAM-MP CLÉS GARSTEC

3.3C

0.8534

0.9996

0.7902

–

–

–

3.3C

0.8517

0.9995

0.7926

–

–

–

3.3C

0.8526

0.9996

0.7988

–

–

–

3.3C

0.8514

0.9995

0.7963

–

–

–

ASTEC CESAM-B69 CESAM-MP CLÉS GARSTEC

210

Astrophys Space Sci (2008) 316: 187–213

Fig. 20 TASK 3: Helium content Ys in the convection envelope for the different codes and cases considered (A and B are for middle and end of the MS respectively while C is for SGB, see Table 2) for 1.0M (left), 1.2M (centre) and 1.3M (right)

Fig. 21 TASK 3: p-mode frequency differences between models produced by different codes, for Case 3.1 (top row), 3.2 (centre) and 3.3 (bottom). CESAM model is taken as reference, and the frequencies

have been scaled to remove the effect of different stellar radii. For each code, we plot two curves corresponding to modes with degrees = 0 and = 1

Astrophys Space Sci (2008) 316: 187–213

of stellar parameters—mass, chemical composition and evolutionary stage—covering those of the CoRoT targets. The models were calculated by 5 codes (ASTEC, CESAM, CLÉS, STAROX, GARSTEC) which have followed rather closely the specifications for the stellar models (input physics, physical and astronomical constants) that were defined by the ESTA group, although some differences remain, sometimes not fully identified. The oscillation frequencies were calculated by the LOSC code (see Sect. 2.3). In a first step, we have examined ESTA-TASK 1 models, calculated for masses in the range 0.9–5M , with different chemical compositions and evolutionary stages from PMS to SGB. In all these models microscopic diffusion of chemical elements has not been included while one model accounts for overshooting of convective cores. In a second step, we have considered the ESTA-TASK 3 models, in the mass range 1.0–1.3M , solar composition, and evolutionary stages from the middle of the MS to the SGB. In all these models, microscopic diffusion has been taken into account. For both tasks we have discussed the maximum and average differences in the physical quantities from centre to surface (hydrogen abundance X, pressure P , density ρ, luminosity Lr , opacity κ, adiabatic exponent 1 and gradient ∇ad , specific heat at constant pressure Cp and sound speed c). We have found that the average differences are in general small. Differences in P , T , Lr , r, ρ and κ are on the percent level while differences in the thermodynamical quantities are often well below 1%. Concerning the maximal differences, we have found that they are mostly located in the outer layers and in the zones close to the frontiers of the convective zones. As expected, differences generally increase as the evolution proceeds. They are larger in models with convective cores, in particular in models where the convective core increases during a large part of the MS before receding. They are also higher in models including microscopic diffusion or overshooting of the convective core. We have then discussed each case individually and tried to identify the origin of the differences. The way the codes handle the OPAL-EOS tables has an impact on the output thermodynamical properties of the models. In particular, the choice of the thermodynamical quantities to be taken from the tables and of those to be recalculated from others by means of thermodynamic relations is critical because it is known that some of the thermodynamical quantities tabulated in the OPAL tables are inconsistent (Boothroyd and Sackmann 2003). In particular, it has been shown by one of us (IW) during one of the ESTA workshops that it is better not to use the tabulated CV -value. Some further detailed comparisons of CLÉS and CESAM models by Montalbán et al. (2007a), have demonstrated that these inconsistencies lead to differences in the stellar models and their oscillation frequencies substantially dominating the uncertainties resulting from the use of different interpolation tools. Similarly, the differences in the

211

opacity derived by the codes do not come from the different interpolation schemes but mainly from the differences in the opacity tables themselves. Discrepancies depend on stellar mass and for the cases considered in this study the maximum differences in opacities are of the order of 2% for 2M models except in a very narrow zone close to log T  4.0 where there are at a few percents level for all models due to the way the OPAL95 and AF94 tables are combined. In unevolved models, differences have been found that pertain either to the lack of a detailed calculation of the PMS phase or to the simplifications in the nuclear reactions in the CN cycle. In evolved models we have identified differences which are due to the method used to solve the set of equations governing the temporal evolution of the chemical composition. The shape and position of the μ gradient and the numerical handling of the temporal evolution of the border of the convection zones are critical as well. The star keeps the memory of the displacements of the convective core (either growing or receding) through the μ gradient. Models with microscopic diffusion show differences in the He and metals distributions which result from differences in the diffusion velocities and that affect in turn thermodynamic quantities and therefore the oscillation frequencies. The situation is particularly thorny for models that undergo semiconvection, either below the convective zone or at the border of the convective core (for models with diffusion), because none of the codes treats this phenomenon. We found that differences in the radius at the bottom of the convective envelope are small, lower than 0.7%. The differences in the mass of the convective core are sometimes large as in models that undergo semiconvection or in PMS and low-mass ZAMS models (up to 17–30%). In other models, the mass of the convective core differs by 0.5 to 5%. Differences in the surface helium abundance in models including microscopic diffusion are of a few per cent except for the 1.3M model on the MS where they are in the range 15–30% due to the different formalisms used to treat diffusion (see Sect. 4). We have examined the differences in the oscillation frequencies of p- and g-modes of degrees = 0, 1, 2, 3. For solar-type stars we also calculated and examined the large frequency separation for = 0, 1 and the frequency separation ratios defined by Roxburgh and Vorontsov (1999). For solar-type stars on the MS, the differences in the frequencies calculated by the different codes are in the range 0.05–0.1 µHz (for = 0, no microscopic diffusion) and 0.1–0.2 µHz (models with diffusion). For advanced models (TAMS or SGB) differences are larger (up to 1 µHz for

= 0 modes). We find that the frequency separation ratios are in excellent agreement which confirms that the differences found in the frequencies and in the large frequency separation have their origin in near-surface effects. Differences are larger in models including microscopic diffusion

212

where the sound speed differences are larger (in the centre and at the borders of convection zones). In addition, in evolved models, at low frequency for = 1 modes, we found differences of up to 4 µHz that result from differences in the Brunt-Väisälä frequency and from the mixed character of the modes. For stars of 2.0M , frequency differences ( = 0, 1 modes) are lower than 0.5 µHz (PMS) and may reach 1 µHz for the evolved model with overshooting. They are due to structure differences in regions close to the surface, in the region of second He-ionisation and close to the centre. In the evolved model some modes are mixed modes sensitive to the location of and features in the μ gradient. Finally, we found that the frequency differences of the massive models (3 and 5M ) are generally smaller than 0.2 µHz. This thorough comparison work has proven to be very useful in understanding in detail the methods used to handle the calculation of stellar models in different stellar evolution codes. Several bugs and inconsistencies in the codes have been found and corrected. The comparisons have shown that some numerical methods had to be improved and that several simplifications made in the input physics are no longer satisfactory if models of high precision are needed, in particular for asteroseismic applications. We are aware of the weaknesses of the models and therefore of the need for further developments and improvements to bring to them and we are able to give an estimate of the precision they can reach. This gives us confidence on their ability to interpret the asteroseismic observations which are beginning to be delivered by the CoRoT mission where an accuracy of a few 10−7 Hz is expected on the oscillation frequencies (Michel et al. 2006) as well as those that will come from future missions as NASA’s Kepler mission to be launched in 2009 (Christensen-Dalsgaard et al. 2007). Acknowledgements JM thanks A. Noels and A. Miglio for fruitful discussions and acknowledges financial support from the Belgium Science Policy Office (BELSPO) in the frame of the ESA PRODEX 8 program (contract C90199). YL is grateful to P. Morel and B. Pichon for their kind help on the CESAM code. The European Helio and Asteroseismology Network (HELAS) is thanked for financial support.

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Astrophys Space Sci (2008) 316: 187–213 J., Conroy, L. (eds.) The CoRoT Mission. ESA Spec. Publ., vol. 1306, pp. 39–50. ESA Publications Division (2006) Miglio, A., Montalbán, J.: Contribution to the CoRoT ESTA Meeting 4, Aarhus, Denmark. Available at 1 ESTA_Web_Meetings/m4/ (2005) Miglio, A., Montalbán, J., Noels, A.: Effects of extra-mixing processes on the periods of high-order gravity modes in main-sequence stars. Commun. Asteroseismol. 147, 89–92 (2006) Montalbán, J., Lebreton, Y., Miglio, A., Scuflaire, R., Morel, P., Noels, A.: Astrophys. Space Sci. (CoRoT/ESTA Volume) (2007a) Montalbán, J., Théado, S., Lebreton, Y.: Comparisons for ESTATask3: CLES and CESAM. In: Engineering and Science. EAS Publications Series, vol. 26, pp. 167–176 (2007b). doi:10.1051/eas:2007135 Monteiro, M.J.P.F.G., Lebreton, Y., Montalbán, J., ChristensenDalsgaard, J., Castro, M., Degl’Innocenti, S., Moya, A., Roxburgh, I.W., Scuflaire, R., Baglin, A., Cunha, M.S., Eggenberger, P., Fernandes, J., Goupil, M.J., Hui-Bon-Hoa, A., Marconi, M., Marques, J.P., Michel, E., Miglio, A., Morel, P., Pichon, B., Prada Moroni, P.G., Provost, J., Ruoppo, A., Suarez, J.C., Suran, M., Teixeira, T.C.: Report on the CoRoT evolution and seismic tools activity. In: Fridlund, M., Baglin, A., Lochard, J., Conroy, L. (eds.) The CoRoT Mission, ESA Spec. Publ., vol. 1306 p. 363. ESA Publications Division (2006) Monteiro, M.J.P.F.G., et al.: Astrophys. Space Sci. (2007). doi:10.1007/s10509-008-9786-7 Morel, P., Lebreton, Y.: CESAM: Code d’Evolution Stellaire Adaptatif et Modulaire. Astrophys. Space Sci. (2007). doi:10.1007/ s10509-007-9663-9 Paquette, C., Pelletier, C., Fontaine, G., Michaud, G.: Diffusion coefficients for stellar plasmas. Astrophys. J. Suppl. Ser. 61, 177–195 (1986). doi:10.1086/191111

213 Richard, O., Michaud, G., Richer, J.: Iron convection zones in B, A, and F stars. Astrophys. J. 558, 377–391 (2001). doi:10.1086/322264 Rogers, F.J., Nayfonov, A.: Updated and expanded OPAL equation-ofstate tables: implications for helioseismology. Astrophys. J. 576, 1064 (2002) Roxburgh, I.: STAROX: Roxburgh’s Evolution Code. Astrophys. Space Sci. (CoRoT/ESTA Volume) (2007). doi:10.1007/s10509007-9673-7 Roxburgh, I.W., Vorontsov, S.V.: Asteroseismological constraints on stellar convective cores. In: Stellar Structure: Theory and Test of Connective Energy Transport. ASP Conf. Ser., vol. 173, p. 257 (1999) Roxburgh, I.W., Vorontsov, S.V.: The ratio of small to large separations of acoustic oscillations as a diagnostic of the interior of solar-like stars. Astron. Astrophys. 411, 215–220 (2003). doi:10.1051/0004-6361:20031318 Scuflaire, R., Théado, S., Montalbán, J., Miglio, A., Bourge, P.O., Godart, M., Thoul, A., Noels, A.: CLES: Code Liegeois d’Evolution Stellaire. Astrophys. Space Sci. (2007a). doi:10.1007/s10509007-9650-1 Scuflaire, R., Montalbán, J., Théado, S., Bourge, P.O., Miglio, A., Godart, M., Thoul, A., Noels, A.: LOSC: Liege Oscillations Code. Astrophys. Space Sci. (2007b). doi:10.1007/s10509-007-9577-6 Thoul, A., Montalbán, J.: Microscopic diffusion in stellar plasmas. In: Engineering and Science. EAS Publications Series, vol. 26, pp. 25–36 (2007). doi:10.1051/eas:2007123 Thoul, A.A., Bahcall, J.N., Loeb, A.: Element diffusion in the solar interior. Astrophys. J. 421, 828–842 (1994). doi:10.1086/173695 Weiss, A., Schlattl, H.: GARSTEC: the Garching Stellar Evolution Code. Astrophys. Space Sci. (2007). doi:10.1007/s10509007-9606-5

FRANEC versus CESAM predictions for selected CoRoT ESTA task 1 models M. Marconi · S. Degl’Innocenti · P.G. Prada Moroni · A. Ruoppo

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9726-y © Springer Science+Business Media B.V. 2008

Abstract We compare the predictions of the FRANEC and CESAM stellar evolutionary codes for selected physical quantities used as input parameters in current asteroseismological calculations. The agreement is satisfactory but the effect of the remaining small discrepancies on the predicted oscillation frequencies should be carefully evaluated. Keywords Stars: evolution · Stars: interiors PACS 97.10.Cv · 97.10.Sj

1 Introduction Within CoRoT ESTA task 1 project we have computed selected evolutionary models with the FRANEC (Frascati Raphson Newton Evolutionary Code) comparing the results with the ones of other codes adopting the same input physics and similar assumptions on macroscopic phenomena (see Degl’Innocenti et al. in this volume for details about the FRANEC code). The goal is to compare and optimize different stellar evolutionary codes which will be used to calculate

M. Marconi · A. Ruoppo INAF-Osservatorio Astronomico di Capodimonte, Napoli, Italy M. Marconi e-mail: [email protected] A. Ruoppo e-mail: [email protected]

the evolutionary characteristics of the CoRoT target stars. The basic idea is that smaller are the differences among the physical quantities produced by the various codes, smaller is the expected uncertainty on the predicted frequencies, to be compared with future COROT measurements which are expected to have an accuracy of ≈0.1 µHz (Baglin et al. 2002). A general description of the chosen input parameters, of the selected target models and of the comparison among the results of the seven evolutionary codes participating to the ESTA project have been described in Monteiro et al. in this volume. In these comparisons the outputs from the CESAM code (see e.g. Pichon and Morel 2005, and the website: http://www.obs-nice.fr/cesam/) have been used as reference results. Here we will focus on the predictions of physical quantities adopted in the asteroseismological codes for two selected models, whose characteristics are described in Table 1 (case 1.1 and 1.4 of the CoRoT ESTA Task 1).

2 Comparison between relevant physical quantities Table 1 shows the properties of the models 1.1 and 1.4 of the CoRoT ESTA task 1, while Table 2 shows the results of the FRANEC and the CESAM codes for some global parameters: age, radius, luminosity, effective temperature, central temperature, density, H abundance and position of the convection zones (convective cores and external convective Table 1 Properties of the models 1.1 and 1.4 of CoRoT ESTA Task 1

S. Degl’Innocenti () · P.G. Prada Moroni Università degli Studi di Pisa, Pisa, Italy e-mail: [email protected]

Case

M/M

Y

Z

Central X

1.1

0.9

0.28

0.02

0.35

P.G. Prada Moroni e-mail: [email protected]

1.4

2.0

0.28

0.02

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_25

Central T

1.9 × 107

215

216

Astrophys Space Sci (2008) 316: 215–218

Table 2 Global parameters for the models 1.1 and 1.4 of CoRoT ESTA Task 1 calculated using two different stellar evolutionary codes, i.e. FRANEC and CESAM. The standard symbols are used. Age is in Myrs, while mass (M), radius (R) and luminosity (L) are in solar units. The temperature and density are in cgs. The mass of convective cores (Mcor ) and the radii of the external convective zones (Renv ) are in stellar units Code

Age

R/R

L/L

Teff

Tc /107

ρc

Xc

Mcor /M

Renv /R

Case 1.1 CESAM

6826

0.8913

0.6242

5440

1.446

150.2

0.3500

–

0.6958

FRANEC

6839

0.8997

0.6273

5421

1.446

151.0

0.3499

–

0.6956

Case 1.4 CESAM

7.684

1.872

16.03

8449

1.900

50.04

0.6994

0.0944

0.9988

FRANEC

6.893

1.876

16.24

8457

1.897

50.03

0.6993

0.0977

0.9986

Fig. 1 Comparison between the physical quantities predicted by the FRANEC and the CESAM codes for the case 1.1 (left panel) and the case 1.4 (right panel)

envelopes). Among these quantities the radius and the convective zone positions influence in a relevant way the oscillation frequencies. In both the codes the onset of convection is determined according to the Schwarzschildt criterion; in the convective cores the temperature gradient is taken to be equal to the adiabatic gradient while in the external regions the Cox and Giuli (1968) derivation of the Bohm-Vitense mixing length formalism has been adopted. For the 1.1 case the differences for all the quantities are of the order of a few 0.1%, reaching 1% for the age. In contrast, for the 1.4 case the differences are higher, of the order of few percent with the exception of age and convective core extension for which they reach about the 10%. This is probably due to the remaining differences in the implementation of the reference set of physics (e.g. adoption of different EOS table out of the range of the OPAL2001 tables, interpolation method of the opacity tables, etc.). Figure 1 and 2 show the relative differences between the results of FRANEC and CESAM computations for the

squared sound speed, temperature, density and pressure profiles for the cases 1.1 and 1.4, respectively. In both cases, the squared sound speed profile closely follows the temperature profile, as expected since, at least in the complete ionization region, the squared sound speed is proportional to the ratio between temperature and mean molecular weight. One notices that small discrepancies between the FRANEC and the CESAM results are still present, reaching at most the level of few percents. In particular, regarding the case 1.1, the squared sound speed and temperature profiles of FRANEC and CESAM are in excellent agreement except in the outermost 5% in mass, where the relative discrepancy reaches the 1% level. This region corresponds to the external convection zone. The situation of the case 1.4 is different, in fact the discrepancy between the FRANEC and CESAM prediction is larger in the whole structure, reaching 0.5%, with the significative exception of the inner 10% in mass, roughly corresponding to the convective core.

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217

Fig. 2 Relative differences between the results by the FRANEC and the CESAM codes for a5 (upper panels) and a2 (lower panels) functions for cases 1.1 (left) and 1.4 (right)

One should also be aware that, as explained before, there are still differences in the treatment of the physical inputs in the two codes, even if the requirements for the standard input physics to be used for the task 1 comparisons have been fulfilled by both the codes. Relative differences for the sound speed profiles for the Task 1 models among the groups participating to the project are shown in other contributions in this volume.

3 Comparison between the variables used in the frequency computations For the purposes of frequency comparisons the most commonly adopted variables are the following functions:

P where 1 = ( δδ log log ρ )ad

a3 (r) = 1 ,

a4 (r) =

that is related to the Brunt-Väisälä Bouyancy frequency, and a5 (r) =

a2 (r) = −

1 d ln p Gmρ = , 1 d ln r 1 pr

4πρr 3 . m

These are generally used as input quantities for current non-radial pulsation codes (see e.g. Christensen-Dalsgaard 1982). The relations between the variables defined above and more “physical” quantities are often useful:

a1 (r) = q/x 3 , where x = r/R and q = m/M are the fractional values of stellar radius and mass.

1 d ln p d ln ρ − 1 d ln r d ln r

p=

GM 2 x 2 a1 2 a5 , 4πR 4 a2 a3

d2 p GM 2 =− xa1 2 a5 , 2 dr 4πR 5

ρ=

M a1 a5 . 4πR 3

218

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Fig. 3 Comparison between the FRANEC and CESAM Brunt-Väisälä frequency for cases 1.1 (left panel) and 1.4 (right panel)

We may also express the characteristic buoyancy frequency as: GM GM N 2 ≡ 3 N˜ 2 = 3 a1 a4 . R R The comparison between the a2 and a5 functions based on the FRANEC models and the CESAM counterparts shows a very good agreement (see Fig. 2); which slightly worses in the very external regions,where it’s already known that substantial differences between models are present (see e.g. Monteiro et al. in this volume). On the other hand Fig. 3 show the comparison between the Brunt-Väisälä frequencies obtained from the FRANEC and the CESAM model for case 1.1 and 1.4 respectively. The plot is restricted to the outermost stellar region because in the inner layers the differences are always negligible. The residual discrepancy in the stellar atmosphere has to be considered carefully. In fact, given the role of the Brunt-Väisälä

frequency in the computation stellar model periodicities we expect that these discrepancies affect the resulting pulsation frequencies and the comparison with the data in a relevant way. Specific computations aimed at to optimizing the comparison between FRANEC and CESAM results, as well as to reducing the final uncertainty on the predicted frequencies, are in progress.

References Baglin, A., Auvergne, M., Barge, P., et al.: In: Stellar Structure and Habitable Planets Finding. ESA SP-485, p. 17 (2002) Christensen-Dalsgaard, J.: Mon. Not. R. Astron. Soc. 199, 735 (1982) Cox, J.P., Giuli, R.T.: Principles of Stellar Structure. Gordon and Breach, New York (1968) Pichon, B., Morel, P.: In: CoRot/ESTA Meeting 3, Nice, France, 2005

Thorough analysis of input physics in CESAM and CLÉS codes Josefina Montalbán · Yveline Lebreton · Andrea Miglio · Richard Scuflaire · Pierre Morel · Arlette Noels

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-008-9803-x © Springer Science+Business Media B.V. 2008

Abstract This contribution is not about the quality of the agreement between stellar models computed by CESAM and CLÉS codes, but more interesting, on what ESTA-Task 1 run has taught us about these codes and about the input physics they use. We also quantify the effects of different implementations of the same physics on the seismic properties of the stellar models, that in fact is the main aim of ESTA experiments. Keywords Stars: internal structure · Stars: oscillations · Stars: numerical models PACS 97.10.Cv · 97.10.Sj · 95.75.Pq 1 Introduction The goal of ESTA-Task 1 experiment is to check the evolution codes and, if necessary, to improve them. The results of J. Montalbán () · A. Miglio · R. Scuflaire · A. Noels Institut d’Astrophysique et Geophysique, Université de Liège, allée du 6 Août 17, 4000 Liège, Belgium e-mail: [email protected] A. Miglio e-mail: [email protected] R. Scuflaire e-mail: [email protected] Y. Lebreton Observatoire de Paris, GEPI, CNRS UMR 8111, 5 Place Janssen, 92195 Meudon, France e-mail: [email protected] P. Morel Département Cassiopée, CNRS UMR 6202, Observatoire de la Côte d’Azur, Nice, France e-mail: [email protected]

Task 1 comparisons were presented in Monteiro et al. (2006) and Lebreton et al. (2008a) for a set of stellar models representative of potential CoRoT targets. The models calculated for TASK 1 were based on rather simple input physics. Moreover, a great effort was done to reduce at maximum the differences between computations by fixing the values of fundamental constants and the physics to be used (see Lebreton et al. 2008b). In spite of that, some differences among stellar models computed by different codes persist. In CESAM and CLÉS computations we paid attention to adopt, not only the same fundamental constants and metal mixture (Grevesse and Noels 1993, thereafter GN93), but also the same isotopic ratios and atomic mass values. Nevertheless, even if the same metal mixture, opacity tables and equation of state were adopted, there is still some freedom on their implementation. In this paper we analyze these different implementations and estimate the consequent effects on the stellar structure and on the seismic properties of the theoretical models. In Sect. 2 we study the equation of state and in Sect. 3 the differences in the opacity tables. The nuclear reaction rates are discussed in Sect. 4 and the effect of different surface boundary conditions in Sect. 5. Finally, in Sect. 6 we analyze the differences due to different numerical techniques.

2 Equation of State As fixed in ESTA we used the OPAL2001 (Rogers and Nayfonov 2002) equation of state which is provided in a tabular form. In CESAM the quantities: density, ρ, internal energy, E, the compressibilities χT = (∂ ln P /∂ ln T )ρ and χρ = (∂ ln P /∂ ln ρ)T , the adiabatic indices Γ1 , Γ2 /(Γ2 − 1), Γ3 − 1, and the specific heat at constant volume CV , are obtained from the variables P , T , X and Z (respectively pres-

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_26

219

220

Fig. 1 Relative differences in the Γ1 values provided for a given (ρ, T ) structure by CESAM and CLÉS EOS routines. The thick lines correspond to Γ1 values derived from CV tabulated values while the thin ones are derived from the tabulated Γ1 . Solid lines corresponds to a 2 M model with a central hydrogen mass fraction Xc = 0.50, and dashed ones to a 0.9 M star in the middle of the main-sequence

sure, temperature, hydrogen and heavy element mass fraction) using the interpolation package provided on the OPAL web site, and the specific heat at constant pressure (Cp ) is derived from (Γ3 − 1). On the other hand, CLÉS interpolates only CV , P , χρ and χT in the OPAL EOS tables by a method ensuring the continuity of first derivatives at cell boundaries in the four-dimensional space defined by the variables ρ, T , X and Z. The other thermodynamic quantities Γ1 , (Γ3 − 1) and Cp are derived from the values of CV , P , χρ and χT by means of the thermodynamic relations. As a first step we want to disentangle the differences in the thermodynamic quantities from their effects on the stellar structure. We estimate therefore the intrinsic differences between the equation of state used in CESAM and in CLÉS. To this purpose we computed the differences between the thermodynamic quantities from the corresponding EoS routines, for a stellar structure defined by ρ, T , X and Z values. In Fig. 1 (thick lines) we show the result of Γ1 comparison for two different stellar models, a 2 M model with a mass fraction of hydrogen in the center Xc = 0.50 (solid line) and a 0.9 M model with Xc = 0.35 (dashed line). By comparing also the other thermodynamic quantities we found that the largest discrepancies between CESAM and CLÉS EoS occurs for log T < 5 (corresponding to the partial He and H ionization regions), and they are, at maximum, of the order of 2% for Γ1 , and of 5% for ∇ad and Cp . By using the OPAL interpolation routine in CLÉS, we verified that the different interpolation schemes used in CESAM and CLÉS can only account for an uncertainty of 0.05% in P , 0.2%

Astrophys Space Sci (2008) 316: 219–229

in Γ1 , and 0.5% in ∇ad and Cp . These remaining differences are probably explained by the fact that CESAM uses as variables (P , T ) and uses subroutine rhoofp of OPALpackage to transform (P , T ) into (ρ, T ), while CLÉS uses directly (ρ, T ). Nevertheless, those discrepancies are an order of magnitude smaller than the differences between CESAM and CLÉS EoS. As it was already noted by Boothroyd and Sackmann (2003), some inconsistencies existed between thermodynamic quantities tabulated in OPAL EOS: “for the OPAL EOS (Rogers et al. 1996), we found that there were significant inconsistencies when we compared their tabulated values of Γ1 , Γ2 /(Γ2 − 1), and (Γ3 − 1) to the values calculated from their tabulated values of P , CV , χρ , and χT . . . . Preliminary tests indicate that this OPAL2001 EOS has larger but smoother inconsistencies in its tabulated thermodynamic quantities. . .”. As a consequence of these inconsistencies, the choice of the basic thermodynamic quantities is not irrelevant, and it was shown by Roxburgh (2005, private communication), that the choice done in CLÉS was the worst one. A direct comparison with the values of CV computed from the derivative of the internal energy as tabulated in OPAL EOS, showed that the OPAL tabulated CV was affected by a large inaccuracy. The OPAL team acknowledged afterwards the CV -issue and recommended not to use it. The EoS tables used in CLÉS have then been changed by replacing the tabulated CV value by that obtained from the tabulated values of P , χρ , χT , and Γ1 . The remaining discrepancies (∼ 0.2%) between the CESAM Γ1 values and those from the new CLÉS–EoS table (hereafter called CLÉS-EoS-Γ1 to tell it apart from the original one CLÉS-EoS-CV ) are due to the different interpolation routine. As shown in Fig. 1 (thin lines) the discrepancies are much smaller for a solar like than for a 2 M model and they appear mainly in the ionization regions. Concerning the quantities that in CLÉS are obtained from thermodynamic relations and in CESAM from interpolation in OPAL tables, the differences come in part from the interpolation routine and in part from the remaining, even if much smaller, inconsistencies between the tabulated values −1 , and (Γ3 − 1). For instance, the of Γ1 , Γ2 /(Γ2 − 1) = ∇ad values of CV derived from Γ1 may differ by 0.5% from the corresponding value obtained from (Γ3 − 1), and that occurs always in the H and He ionization regions. The problem is that even if the thermodynamic relations to derive the adiabatic indices seem more physical, there is some numerical incoherence. In fact, the derivatives of interpolated (very often polynomial) quantities do not fit in those of the interpolated functions (whose behavior is far from polynomial one). All the CLÉS models involved in Task 1 and Task 3 comparisons (Lebreton et al. 2008a) were recomputed with CLÉS-EoS-Γ1 , but the models used for comparisons presented in Monteiro et al. (2006) were not. In fact, most of

Astrophys Space Sci (2008) 316: 219–229

221

clearly that the oscillation is linked to the Γ1 differences. Moreover, the comparison of Fig. 3 with Fig. 6 in Monteiro et al. (2006) confirms that also for the Case1.1 model (see e.g. Lebreton et al. 2008a), the maximum difference of almost 2 μHz between CESAM and CLÉS models found by Monteiro et al. (2006) was due to the inconsistency between the tabulated CV and adiabatic indices.

3 Opacities ESTA specifications require the use of OPAL96 opacity ta-

Fig. 2 Frequency differences between CESAM and CLÉS 2 M models. The two different curves correspond to CLES models computed by using the two different EoS tables (see text)

Fig. 3 As Fig. 2 for 0.9 M model

the frequency differences found in that paper came from CLÉS-EoS-CV . The effect of EoS differences on the seismic properties are illustrated in Fig. 2 for a 2 M model and in Fig. 3 for the solar like model. In those figures we plot the frequency differences of  = 0 modes for CESAM models and two types of CLÉS ones: those computed with EoS-CV (dashed lines) and those computed with EoS-Γ1 (solid line). The period of the oscillatory signature shown by ν (νCLES − νCESAM ) in Fig. 2 is related to the acoustic depth where models differ. A Fourier transform of ν shows

bles (Iglesias and Rogers 1996) complemented at low temperatures by the Alexander and Ferguson (1994) (thereafter AF94) tables. CESAM uses OPAL tables provided by C. Iglesias, prior to their availability on the web site, and interpolates in the opacity tables by means of a fourpoint Lagrangian interpolation. The OPAL opacity tables used by CLÉS were picked up later on the OPAL web site and smoothed according to the prescription found there (xztrin21.f routine), we will call them thereafter OPAL96-S. Furthermore, the interpolation method in CLÉS opacity routine is the same as that used in EoS table interpolation. In both codes the metal mixture adopted in the opacity tables is the GN93 one. To disentangle the differences in the opacity computations from the differences in the stellar structure, we proceed as in EoS table analysis, that is, we estimate the intrinsic differences in the opacity (κ) by comparing the κ values provided by CESAM and by CLÉS routines for the same stellar structure. The results of these comparisons are shown in Figs. 4 and 5, where we plot for two different stellar models the opacity relative differences (κCESAM − κCLES )/κ as a function of the local temperature and of the relative radius. From comparisons of different models it results that the opacity discrepancies depend on the mass of the stellar model and, for a given mass, on the evolutionary state as well. Moreover, a peaked feature at log T  4 which can reach values of the order of 5%, appears in all the comparisons. This is a consequence of the differences between OPAL and AF94 opacities in the domain [9000 K–12000 K] and of the different method used in CLÉS and CESAM to assemble AF94 and OPAL tables. CLÉS uses the procedure described in Scuflaire et al. (2008b) that ensures a smooth passage between both tables, while CESAM searches for the point of minimum discrepancy between OPAL and AF94. In the interior regions the differences between CESAM and CLÉS opacities do not present the oscillatory behavior that we would expect if these differences resulted from the interpolation schemes. On the contrary, the CESAM opacities are systematically larger (by 1–2%) than the CLÉS ones in the region log T ∈ [5.5, 7] of 2 M model. Even if the metal mixture to be used in opacity computations is fixed (GN93), there may be some uncertainties in

222

Fig. 4 Relative differences in the opacity values provided for a given (ρ, T ) structure by CESAM and CLÉS opacity routines. The thick lines correspond to the κ values derived from the smoothed OPAL tables, while the thin ones were obtained by including in the CLÉS opacity routine the OPAL tables without smoothing. Solid lines corresponds to a 2 M model with Xc = 0.50, and dashed ones to a 0.9 M star in the middle of the main-sequence

Fig. 5 The same differences as in Fig. 4 but plotted as a function of the relative stellar radius

its definition. For instance, OPAL uses atomic masses that do not correspond to the values given by the isotopic ratios in Anders and Grevesse (1989). In particular there is a difference of 0.5% for Neon, and 10% for Argon. Moreover, OPAL opacity tables are computed for 19 elements, while the GN93 mixture contains 23 elements. There are two options: either to ignore the mass fraction of F, Sc, V, and Co, or to allot the abundances of these elements among

Astrophys Space Sci (2008) 316: 219–229

the close neighbors. We have analyzed the effects of these uncertainties on the opacity values, but they turned out to be of the same order of the accuracy in OPAL data (0.1–0.2%). Hence, they cannot account for the discrepancy between CESAM and CLÉS opacities. The other important difference between CESAM and CLÉS opacities is on whether they use the OPAL smoothing routine or not. In fact, the OPAL opacity tables are affected by somewhat random numerical errors of a few percent. To overcome undesirable effects the OPAL web site suggests to pass the original tabular data through a smoothing filter before interpolating for Z, X, log T , and R (with R = ρ/T63 ). A direct comparison between the original and smoothed opacity values have shown a difference larger than 2% for log R = −4 and −3.5 and log T ∈ [5.5, 7]. These differences decrease for larger and smaller values of log R. We have computed new opacity tables for the CLÉS opacity routine without passing through the smoothing filter (CLÉS-OPAL96 instead of CLÉS-OPAL96-S). The comparison between CESAM and new CLÉS opacity computations are also shown in Figs. 4, 5 (thin lines). We note that when both codes use similar OPAL96 tables, the discrepancies in the internal regions almost disappear. The remaining differences are due to the interpolation schemes and to the small differences in GN93 definition. The feature at log T ∼ 4 is still there since the method used in CLÉS to assemble AF94 and OPAL96 tables is the same as in CLÉS-OPAL96-S. At variance with the EOS tables, where an error was detected and acknowledged by the OPAL team, we do not have any argument to prefer the smoothed to the original opacity tables, and we think that the differences between both groups of results must be considered an estimate of the precision of current stellar modeling. Therefore, the Liége group decided to provide for Task 1 and Task 3 comparisons (Lebreton et al. 2008a) the modes computed with the standard tables in CLÉS, that is OPAL96-S. A part of the differences between CESAM and CLÉS models that were reported in Lebreton et al. (2008a) should be hence due to the opacity tables we used. In order to estimate these effects we have recomputed with CLÉS and OPAL96 tables (without smoothing) the models for all the cases in TASK 1 (see Table 1 in Lebreton et al. 2008a). In the next three sections we present the effect of the opacity uncertainty on: the global parameters, the stellar structure, and on the seismic properties. 3.1 Effects on global stellar parameters In general, the change of opacity tables decreases the discrepancies between the stellar global parameters provided by CLÉS and CESAM. In Table 1 we collect the differences (in percent) in radius, luminosity, central density, and central temperature between pairs of models for the Task 1 targets. Columns labeled A give the differences

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Table 1 Global parameter differences, in percent, between Task 1target models computed by CLÉS with OPAL opacity tables in the two versions: smoothed (OPAL96-S) and not smoothed (OPAL96). Stellar radius (R), luminosity (L), central density (ρc ) and Case

M/M

Type

R /R A

C1.1

0.9

C1.2

1.2

C1.3

1.2

C1.4

2.0

C1.5

2.0

C1.6

3.0

C1.7

5.0

MS ZAMS SGB PMS TAMS ZAMS MS

central temperature (Tc ). Columns labeled A give the differences XCLES−OPAL96−S − XCLES−OPAL96 , and columns B and C the differences with CESAM, that is, XCLES−OPAL96−S − XCESAM and XCLES−OPAL96 − XCESAM respectively

L/ L B

ρc / ρc

C

A

B

C

A

Tc / Tc B

C

A

B

C

−0.01

−0.02

−0.007

−0.04

−0.05

−0.01

0.10

−0.08

−0.03

−0.05

−0.02

0.00

−0.17

−0.17

−0.05

−0.12

−0.07

0.04

−0.06

−0.10

−0.001

−0.17

−0.17

0.01

0.33

0.31

0.36

0.90

0.53

−0.30

−2.5

−2.2

−0.16

−0.13

0.04

0.26

0.20

0.04

−0.16

0.47

−0.03

0.43

0.46

0.76

0.82

0.07

−0.20

−0.30

−0.14

−0.14

0.00

0.14

−0.14

0.00

−0.06

0.05

0.18

0.13

0.003

0.25

0.25

−0.005

Fig. 6 Evolutionary tracks for stellar parameters corresponding to the case C1.5 in Task 1, and also without overshooting. Solid thick lines: CLÉS models with the default opacity tables (OPAL96-S); solid thin lines: CESAM models; dotted lines: CLÉS models where the opacity tables have been recomputed without using the opal smoothing filter (OPAL96)

XCLES−OPAL96−S − XCLES−OPAL96 , and columns B and C the differences with CESAM, that is, XCLES−OPAL96−S − XCESAM and XCLES−OPAL96 − XCESAM respectively. We note that for the most evolved models (C1.3 and C1.5), the effect on the radius of changing the CLÉS opacity tables is small, and that the agreement with CESAM gets even worse than with the original tables. There is however a significant decrease of the luminosity discrepancy. For the cases C1.4 and C1.6 (PMS and ZAMS respectively) the change from OPAL96-S to OPAL96 is particularly effective, leading to a decrease of R and L by a factor 4 and 6 respectively for C1.4, and dropping the discrepancy to values lower than 0.003% for C1.6.

0.006 −0.02

0.55

0.57

−0.30

0.00

0.00

0.00

−0.10

0.06

0.05

−0.007

−0.05

0.00

0.01

−0.008

0.01

−0.05

−0.05

0.000

0.02

0.02

Fig. 7 Effect of opacity uncertainties on the stellar radius and luminosity along the main-sequence evolution of a 2 M model. Thin lines correspond to differences between CESAM and CLÉS-OPAL96-S global parameters, and thick ones to differences between CESAM and CLÉS-OPAL96 ones. The dotted lines refer to differences in radius (lower curve) and in luminosity (upper curve) between CLÉS models computed with OPAL96-S and OPAL96 opacity tables

We also studied the effect of opacity tables on the mainsequence evolution of a 2 M star (parameters corresponding to C1.5). As shown in Fig. 6 the HR location of the CLÉS-OPAL96-S evolutionary track is significantly modified by adopting OPAL96 tables, and except for the second gravitational contraction, the new track coincides quite well with the CESAM one. The discrepancies in radius and luminosity along the MS as well as the effect of opacity tables on their values are shown in Fig. 7. These discrepancies are significantly reduced by switching from OPAL96-S to OPAL96, nevertheless the difference in the stellar radius at

224

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Fig. 8 Plots in terms of the relative radius of the differences at fixed relative mass (two left panels) for the internal regions, and at fixed radius (two right panels) for the outer layers, between models computed with different opacity routines for the Cases 1.1, 1.2 and 1.3. Solid lines correspond to the difference between two types of CLÉS models: those obtained with the standard CLÉS version that uses the smoother OPAL opacity tables (opal96-S) and those obtained by using an OPAL opacity

table obtained without smoothing. Dotted lines: differences between the standard CLÉS models and those from CESAM. Dashed lines: differences between CLÉS models computed by using OPAL opacity tables without smoothing and CESAM models. Left panel: logarithmic sound speed differences. Central left panel: logarithmic pressure differences. Central right panel: logarithmic sound speed differences. Right panel: logarithmic adiabatic exponent differences

the end of MS phase (Xc < 0.2) is unchanged. That is not due to the uncertainties in opacity as shown by the dotted lines that correspond to the differences between CLÉS models computed with the two different opacity tables. Neither it is a consequence of the treatment of overshooting since stellar models computed without overshooting for the same stellar parameters show similar discrepancies. The reason is in the treatment of the borders of convective regions in CLÉS that leads to a sort of “numerical diffusion” (Scuflaire et al. 2008a). For the stellar evolution, this diffusion at the border of the convective core works as a slightly larger overshooting. A significant increase of the number of mesh points used for computing the models reduces the numerical diffusion and improves the agreement CLÉS-CESAM.

we computed for each TASK 1 model the local differences in the physical variables at fixed relative mass and at fixed relative radius. To this purpose we use the so-called difffgong.d routine in the ADIPLS package.1 In Figs. 8 and 9 we plot the logarithmic differences of the sound speed, c, and pressure, P , for the stellar interior (two left panels) and of sound speed and the adiabatic exponent Γ1 in the external layers. In each panel there are three different curves that correspond to the comparisons labeled A, B, and C in previous section. So, if the differences shown in Lebreton et al. (2008a) come from the differences in the opacity tables, the solid and dotted lines should be close to each other. We call again the attention to the improvement got for the cases C1.4 and C1.6.

3.2 Effects on the stellar structure To analyze to which extent the differences reported in (Lebreton et al. 2008a) come from the uncertainty in the opacity,

1 http://www.corot.pt/ntools.

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225

Fig. 9 As Fig. 10, for Cases 1.4, 1.5, 1.6 and 1.7

3.3 Effects on the frequencies Since the frequency of p-modes firstly depends on the stellar radius, the improved agreement between CESAM and CLÉS models that we obtained by changing from OPAL96-S to OPAL96 implies also a decrease in the frequency discrepancies. The values of ν (νCLES − νCESAM ) change from 7.25 to 6.35 μHz (at 5300 μHz) for the case C1.1; from 7 to 4.5 μHz (at 4000 μHz) for C1.2; from −5 to −4.5 μHz at 1000 μHz (and from −11 to −10 at 2300 μHz) for C1.3; from 3.7 to −1 μHz at 2200 μHz for C1.4. The improvement is only of 0.3 μHz for the case C1.5, and ν at 1200 μHz is of the order of 7.5 μHz. For C1.6 the initial ν ∼ 3 μHz at 1500 μHz drops to values lower than 0.05 μHz, and for C1.7, ν change from 1.6 μHz to 1.2 μHz (at 600 μHz).

By comparing frequencies that have been scaled to the same radius we remove the effect of R and make appear the differences due to discrepancies in the stellar structure. This was done in Fig. 10 where we plot the frequency differences for  = 0 and 1 and for the cases considered in Task 1. There are two curves in each panel, one corresponding to the difference νCLES−OPAL96−S − νCESAM (that is, that appearing also in Lebreton et al. 2008a), and the second one corresponding to νCLES−OPAL96 − νCESAM . The role of opacity on the oscillation frequencies is a intricate problem since the variations of κ lead to changes of the temperature structure in the star, and therefore also of the value of Γ1 . As can be seen in Fig. 9 for the case 1.5, the differences in the outer layers might even increase for the model computed with similar opacity tables (OPAL96), and as consequence, the frequency differences scaled to the

226

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Fig. 10 p-mode frequency differences between TASK 1 models produced by CLÉS with the two kinds of opacity tables (dash-dotted lines: smoothed OPAL; and dashed lines: without smoothing) and CESAM.

The latter is taken as reference, and the frequencies have been scaled to remove the effect of different stellar radii. For each model we plot two curves corresponding to modes with degrees  = 0 and  = 1

same radius (Fig. 10, lower-left panel) show a discrepancy even larger than with OPAL96-S. The oscillatory behavior is produced by the peak in δ ln c at r/R ∼ 0.997. A comparison between CLÉS-OPAL96 and CLÉS-OPAL96-S clearly shows the same oscillatory behavior, but the absolute frequency difference is only ∼ 0.3 μHz (both models have similar radius). In the same way, CESAM and CLÉS 2 M models at Xc = 0.50 (whose radius differ by less than 3 × 10−4 R∗ once CLÉS adopts OPAL96 tables) show frequency differences of the order of 0.4 μHz at 1200 μHz. After normalizing to the same radius an oscillatory component (amplitude 0.01 μHz) remains in the ν because of the differences in Γ1 in the outer layers.

of the order of 3 × 10−4 –2 × 10−3 (except for nuclear reactions involving 7 Li) if screening factors are included, and of the order or 10−8 if not. In both codes weak screening is assumed under Salpeter (1954)’s formulation. The screening 

4 Nuclear reaction rates We used the basic pp and CNO reaction networks up to the 17 O(p,α)14 N reaction. In the present models the CESAM code takes 7 Li, 7 Be and 2 H at equilibrium while CLÉS follows entirely the combustion of 7 Li and 2 H. The nuclear reaction rates are computed using the analytical formulae provided by the NACRE compilation (Angulo et al. 1999). CESAM uses a pre-computed table2 while CLÉS uses directly the analytical expressions. Comparing the nuclear reaction rates for a 2 M stellar structure and a given chemical composition, we found that the relative differences are 2 By

using pre-computed tables the numerical coherence of derivatives required in the CESAM numerical scheme are guaranteed.

factor is written f = exp(Az1 z2 Tρξ3 ) where z1 and z2 are the charges of the interacting nuclei. CESAM uses the ex8 pression  4-221 of Clayton (1968) where A = 1.88 × 10 , ξ = i zi (1 + zi )xi , and xi is the abundance per mole of element i. The standard  version of CLÉS code takes A = 1.879×108 and ξ = 4i=1 zi (1+zi )xi +Z(1+Z)x(Z) where x(Z) is the abundance of an “average” element containing all the elements different from hydrogen and helium, and Z is the average charge of this element. This approximate estimation of ξ has been changed in CLÉS by assuming full ionization and taking the contribution from each mixture element into account. With this new prescription the differences between CESAM and CLÉS nuclear reaction rates are still of the same order, but with CESAM values larger than CLÉS ones at variance with what was found with the standard CLÉS formulation. All the Task 1 and Task 3 CLÉS-models were computed with the updated version.

5 Atmosphere Eddington’s grey T (τ ) law is used for the atmosphere calcu1 lation: T = Teff [ 34 (τ + 23 )] 4 where τ is the optical depth. CESAM integrates the hydrostatic equation in the atmosphere

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227

These comparisons have allowed to show the discrepancies in frequencies that a “black-box” use of an evolution code might lead to. For illustration, in Fig. 11 we show the outer layers of 2 M models at two different evolutionary states (Xc = 0.50, and Xc = 0.01), with effective temperature Teff = 8337 K and 6706 K respectively. While the atmosphere and sub-phostospheric structure of CLÉS and CESAM models is quite close if P (τmin ) option is used in CESAM, discrepancies that increase with the effective temperature of the model appear when ρ(τmin ) with default values (derived for solar-like models) is adopted. The differences induced in the structure of these outer layers by the use of inappropriate limit values in the atmosphere integration are much larger than those due to opacity differences at log T ∼ 4. These outer structure differences can lead in fact to frequency differences of the order of several μHz.

6 Numerical aspects Fig. 11 Stellar structure of the external layers of 2 M models at two different evolutionary stages (Xc = 0.50 and 0.01). Dashed lines: CLÉS models. Dotted lines: CESAM models with boundary conditions given by P (τmin ). Solid lines: CESAM models with boundary conditions given by ρ(τmin ) and default values (see the text)

starting at the optical depth τ = τmin (τmin = 10−4 for solar like models) and makes the connection with the envelope at τ = 10 where the continuity of the variables and of their first derivatives are assured. The radius of the star is taken to be the bolometric radius, i.e. the radius at the level where the local temperature equals the effective temperature (τ = 2/3 for the Eddington’s law). In the stellar structure integration CLÉS gets the external boundary conditions (the values of density and temperature at a given optical depth τ ) by interpolating a pre-computed table, and the stellar radius is defined as the level where T = Teff . The Eddington atmosphere table, which provides ρ and T at τ = 2/3 (therefore at R = R∗ ), was built by integrating the hydrostatic equilibrium equation in the atmosphere starting at an optical depth that can vary between 10−4 and 10−2 . The atmosphere structure for a given model is computed afterwards, by integrating the same equations for the corresponding values of Teff , log g and chemical composition. While in CLÉS atmosphere computations the condition at the optically thin limit (ρ(τmin )) is determined for each (Teff , log g, X, Z) by a Newton-Raphson iteration algorithm, CESAM allows to integrate the atmosphere by fixing either ρ(τmin ) or P (τmin ). We think it is worth warning here about the relevance of an appropriate choice of ρ(τmin ) in CESAM calculations. As explicitly indicated in the corresponding tutorial, the default values were determined for solar like models, and if much different physical conditions are considered, the boundary conditions in the optically thin limit should be coherently changed.

The different numerical techniques in CLÉS and CESAM lead to different distribution of mesh points in the stellar structure and to different values of the time step between two consecutive models. As pointed out in Sect. 3.1 (see also Lebreton et al. 2008a) the disagreement between CLÉS and CESAM models can be partially reduced in some cases by changing the mesh. Even if both sets of models have a similar total number of mesh points, their distribution inside the star, as shown in Fig. 12 (right column), is quite different. In this section we analyze the effect of doubling the number of mesh points (CLÉS-X2) or the number of time steps (CLÉS-T2) on the differences between CLÉS and CESAM. As shown in Fig. 12 the effects of these changes are not the same in all the considered stellar cases. While for the CASE 1.1 the increase of mesh points makes almost disappear the disagreement between CLÉS and CESAM models (δ ln Var ∼ 10−4 − 5 × 10−5 ), the effect is almost negligible for the CASE 1.3. Doubling the number of mesh points in CASE 1.5 leads to a significant effect in decreasing the luminosity of this TAMS model. As pointed out in Sect. 3.1, a larger number of mesh points near the boundary of the convective core decreases the effect of the sort of “numerical diffusion” that changes the chemical composition gradient at the boundary of the convective core, and that works as a slightly larger overshooting. In fact, the differences of hydrogen abundance in the region of chemical composition gradient (r/R between 0.035 and 0.06, Fig. 12 central-left panel for CASE 1.5) also decrease with respect to those obtained with the standard CLÉS models. As already discussed in Sect. 3.1 a part of the disagreement between CLÉS and CESAM comes from the differences in the opacities used in both codes. In the lower panels of Fig. 12 we have also plotted (dotted lines) the results of comparing the models computed with CLÉS doubling the number of mesh points and

228

Astrophys Space Sci (2008) 316: 219–229

Fig. 12 Plots in terms of the relative radius of the differences at fixed relative mass (three left panels) between three different CLÉS computations and CESAM models. Solid lines correspond to differences between standard CLÉS and CESAM. Dashed lines correspond to differences between CLÉS models computed by doubling the number of time steps and CESAM ones. Dot-dashed lines correspond to differ-

ences between CLÉS models computed with a double number of mesh points. In lower panels there the differences between CESAM and CLÉSX2 models computed using the opacity tables without smoothing (opal96) are plotted by using dotted lines. The right column plots show the distribution of mesh points inside the CESAM models (dotted lines) and CLÉS ones (solid lines)

using the OPAL96 tables without smoothing. A significant decrease of luminosity and hydrogen-profiles differences is obtained when both the number of mesh points and the opacity tables are changed. Decreasing the time step in the evolution models does not lead, in general, to better agreement between CLÉS and CESAM.

simply allowed us to understand the origin of some differences that were reported in the above mentioned paper.

7 Conclusions In addition to the quantitative results of code comparison presented in Lebreton et al. (2008a), the analysis of stellar models computed with the codes CESAM and CLÉS has allowed us to reveal some interesting aspects about the two codes, as well as about the input physics, that only a thorough analysis might bring to light. Some of these evidences have led to changes or correction of bugs in the codes, other

– The inconsistencies among the thermodynamic quantities in OPAL2001 equation of state tables lead to differences in the stellar models and in the oscillation frequencies larger than the uncertainties due to different interpolation tools. Even if the quantity CV as tabulated in OPAL2001 tables is not used, the remaining inconsistencies among the three adiabatic indices lead to differences between the model computed with a code that takes the thermodynamic quantities directly from OPAL tables (such as CESAM) and a model computed with a code whose thermodynamic variables are derived by the thermodynamic relations and a minimum of tabulated quantities (such as CLÉS). Furthermore, the discrepancies will depend on the choice of tabulated quantities. – The precision of the theoretical oscillation frequencies is seriously limited by the uncertainties in the opacity computations.

Astrophys Space Sci (2008) 316: 219–229

– Different approaches used to estimate the electron density in CESAM and CLÉS lead to differences in the screening factors that have no relevant effects on the stellar models. – Even with a simple physics, such as the Eddington’s law for gray atmosphere, the details of numerical tools can have significant consequences on the seismic properties of the models. – The different distribution of mesh points in the models can explain part of the disagreement between CESAM and CLÉS models. An increase of mesh points in the internal regions seems to be required in CLÉS to decrease the differences with CESAM. Apart from the discrepancies in the screening factors which does not significantly affect the oscillation frequencies, the other factors analyzed here can affect the absolute oscillation frequencies by up to several μHz. Acknowledgements J.M., A.M., R.S., and A.N. acknowledge financial support from the Belgium Science Policy Office (BELSPO) in the frame of the ESA PREODEX8 program (contract C90199) and from the Fonds National de la Recherche Scientifique (FNRS). P.M. thanks J.P. Marques (Coimbra University) and L. Piau (Brussels University) for their contribution to the OPAL-EoS implementation in CESAM.

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Inter-comparison of the g-, f- and p-modes calculated using different oscillation codes for a given stellar model A. Moya · J. Christensen-Dalsgaard · S. Charpinet · Y. Lebreton · A. Miglio · J. Montalbán · M.J.P.F.G. Monteiro · J. Provost · I.W. Roxburgh · R. Scuflaire · J.C. Suárez · M. Suran

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-007-9717-z © Springer Science+Business Media B.V. 2007

Abstract In order to make asteroseismology a powerful tool to explore stellar interiors, different numerical codes should give the same oscillation frequencies for the same input physics. Any differences found when comparing the numerical values of the eigenfrequencies will be an impor-

A. Moya () · J.C. Suárez Instituto de Astrofísica de Andalucía—CSIC, Cno. Bajo de Huetor, 50, Granada, Spain e-mail: [email protected] J. Christensen-Dalsgaard Institut for Fysik og Astronomi, og Dansk AsteroSeismisk Center, Aarhus Universitet, Aarhus, Denmark S. Charpinet Observatoire Midi-Pyrénées, Toulouse, France Y. Lebreton Observatoire de Paris, GEPI, CNRS UMR 8111, Meudon, France A. Miglio · J. Montalbán · R. Scuflaire Institut d’Astrophysique et Geophysique, Université de Liège, Liège, Belgium M.J.P.F.G. Monteiro Centro de Astrofísica da Universidade do Porto and Departamento de Matemática Aplicada da Faculdade de Ciências, Universidade do Porto, Porto, Portugal J. Provost Observatoire de la Cote d’Azur, Nice, France I.W. Roxburgh Astronomy Unit, Queen Mary, University of London, London, UK M. Suran Astronomical Institute of the Romanian Academy, Bucharest, Romania

tant piece of information regarding the numerical structure of the code. The ESTA group was created to analyze the nonphysical sources of these differences. The work presented in this report is a part of Task 2 of the ESTA group. Basically the work is devoted to test, compare and, if needed, optimize the seismic codes used to calculate the eigenfrequencies to be finally compared with observations. The first step in this comparison is presented here. The oscillation codes of nine research groups in the field have been used in this study. The same physics has been imposed for all the codes in order to isolate the non-physical dependence of any possible difference. Two equilibrium models with different grids, 2172 and 4042 mesh points, have been used, and the latter model includes an explicit modelling of semiconvection just outside the convective core. Comparing the results for these two models illustrates the effect of the number of mesh points and their distribution in particularly critical parts of the model, such as the steep composition gradient outside the convective core. A comprehensive study of the frequency differences found for the different codes is given as well. These differences are mainly due to the use of different numerical integration schemes. The number of mesh points and their distribution are crucial for interpreting the results. The use of a second-order integration scheme plus a Richardson extrapolation provides similar results to a fourth-order integration scheme. The proper numerical description of the Brunt-Väisälä frequency in the equilibrium model is also critical for some modes. This influence depends on the set of the eigenfunctions used for the solution of the differential equations. An unexpected result of this study is the high sensitivity of the frequency differences to the inconsistent use of values of the gravitational constant (G) in the oscillation codes, within the range of the experimentally determined ones, which differ from the value

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_27

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used to compute the equilibrium model. This effect can provide differences for a given equilibrium model substantially larger than those resulting from the use of different codes or numerical techniques; the actual differences between the values of G used by the different codes account for much of the frequency differences found here. Keywords Stars · Stellar oscillations · Numerical solution Keywords 97.10.Sj · 97.10.Cv · 97.90.+j

1 Introduction Asteroseismology is at present being developed as an efficient instrument in the study of stellar interiors and evolution. Pulsational frequencies are the most important asteroseismic observational inputs. It is evident that a meaningful analysis of the observation, in terms of the basic physics of stellar interiors which is the ultimate target of the investigation, requires reliable computation of oscillation frequencies for specified physics. This is a two-step process, involving first the computation of stellar evolutionary models and secondly the computation of frequencies for the resulting models. Lebreton et al. (this volume) provide an overview of the tests of stellar model calculations. Here we consider the computation of the oscillation frequencies. An evident goal is that the computed frequencies, for a given model, should have errors well below the observational error, which in the case of the CoRoT mission is expected to be below 0.1 µHz (Baglin et al. 2006). For the Kepler mission (e.g., Christensen-Dalsgaard et al. 2007), with expected launch in early 2009, selected stars may be observed continuously for several years and errors as low as 10−3 µHz may be reachable, particularly for modes excited by the heat-engine mechanism. Since errors resulting from numerical problems are typically highly systematic, they may affect the asteroseismic inferences even if they are substantially below the random errors in the observed frequencies. This must be kept in mind in the assessment of the estimates of the numerical errors. During the last decades a lot of codes obtaining numerical solutions of an adiabatic system of differential equations describing stellar oscillations have been developed. In order to ascertain whether any possible difference in the description of the same observational data by different numerical codes is due to physical descriptions or to different numerical integration schemes, the inter-comparison of these oscillation codes in a fixed and homogeneous framework is absolutely necessary. Some effort has been already done in the past but only regarding pairs of codes. Some codes have also developed a lot of internal precision tests. However, there is a lack of inter-comparison of a large enough set of codes.

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We aim in this study try to fix a set of minimum requirements for a code to be sure that any difference found is only due to a different physical assumption. Ideally, for a given model there should be a set of ‘true’ frequencies with which the results of the different codes could be compared. This ideal situation could probably be approximated by considering polytropic models for which it is relatively straightforward to calculate the equilibrium structure with essentially arbitrary accuracy (see also, Christensen-Dalsgaard and Mullan 1994). In practice, the situation for realistic stellar models is more complex. Owing to the complexity of the stellar evolution calculation the models are often available on a numerical mesh which is not obviously adequate for the pulsation calculation. The effect of this on the frequency computation depends on the detailed formulation of the equations in the pulsation codes. These formulations are equivalent only if the equilibrium model satisfies the ‘dynamical’ equations of stellar structure, i.e., the mass equation and the equation of hydrostatic support, and this is obviously not exactly true for a model computed on a finite (possibly even relatively sparse) mesh. One might define a consistent set of frequencies for a given model by interpolating it onto a very dense mesh and resetting it to ensure that the relevant equations of stellar structure are satisfied. The model is fully characterized by the dependence on distance r to the centre of density ρ and the adiabatic exponent Γ1 = (∂ ln P /∂ ln ρ)ad , P being pressure and the derivative being at constant specific entropy. Thus one could interpolate ρ(r) and Γ1 (r) to a fine mesh, and recompute the mass distribution and pressure by integrating the mass equation and the equation of hydrostatic equilibrium. Frequencies of this model should then be essentially independent of the formulation of the oscillation equation and would provide a suitable reference with which to compare other frequency calculations. Such a test may be carried out at a later stage in the ESTA effort. In Task 2, for now, we aim at testing, comparing and optimizing the seismic codes used to calculate the oscillations of existing models of different types of stars. In order to do so we consider steps in the comparison by addressing some of the most relevant items that must be compared regarding the seismic characterization of the models: • Step 1: comparison of the frequencies from different seismic codes for the same model. • Step 2: comparison of the frequencies from the same seismic code for different models of the same stellar case provided by different equilibrium codes. • Step 3: comparison of the frequencies for specific pulsators. The work presented here is mostly focused on step 1. At this step three different equilibrium models have been used. Two of them have been computed using CESAM

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(Morel and Lebreton 2007), with 902 and 2172 mesh points, and a third one with 4042 mesh points provided by ASTEC (Christensen-Dalsgaard 2007b).We present intercomparisons using the two models with the larger numbers of mesh points. The same physics and physical constants (except G) are used for all the oscillation codes. Frequencies in the range of [20, 2500] µHz, belonging to spherical degrees  = 0, 1, 2 and 3 have been calculated, in order to recover most of the possible values we can find in the observational photometric data. We present in Sect. 2 the equilibrium models used and we analyze their main features. Section 3 is devoted to the different oscillation codes used, discussing common requirements and variations in the treatment. In Sect. 4 the direct comparison of frequencies for different frequency ranges and spherical degrees is presented in detail. Sections 5 and 6 analyze the same inter-comparison for the values of the large and small separations, respectively. Section 7 discusses the dominant effects that contribute to the differences in frequencies determined by the different codes. Conclusions are given in Sect. 8.

2 The equilibrium models To ensure that any difference obtained in the inter-comparisons is only due to differences in the numerical schemes, we imposed to all the codes the use of the same equilibrium models. These models were supplied in several formats: OSC, FGONG, SROX, and FAMDL. The first model was required to have 900 mesh points, and it was provided by CESAM. The differences in this case reached unacceptable values of 2–3 µHz when the same integration schemes are compared, and even 10 µHz when the use or not of the Richardson extrapolation are compared. The maximum difference found for large separations with this model is 1 µHz. This showed that either a larger number of mesh points or Richardson extrapolation was necessary.1 The second model was also provided by CESAM (referred from now on as M2k). It uses a grid, with 2172 mesh points, more suitable for asteroseismic purposes. General characteristics of the model are presented in Table 1. These are typical of a γ Doradus star showing oscillations in the asymptotic g-mode regime, and also around the fundamental radial mode. A priori, solar-like pulsations cannot be excluded for this type of star and therefore it can be a good candidate for a global study. In Fig. 1, A∗ (which is a quantity directly related to the Brunt-Väisälä frequency N 2 : A∗ = rg −1 N 2 , where g is the gravitational acceleration) is depicted as a function of the relative radius (x = r/R) in a 1 Detailed results of this investigation can be found at http://www.astro. up.pt/corot/compfreqs/task2/.

233 Table 1 General characteristics of the models used for the intercomparison M/M

log Teff

Age

Xc

R/R

(in My)

Mesh

G

points

[10−8 cgs]

1.5

3.826

1355

0.4

1.731

4042

6.6716823

1.5

3.830

1368

0.4

1.724

2172

6.67232

Fig. 1 A∗ (related to the Brunt-Väisälä frequency) as a function of the relative radius for the two equilibrium models discussed in the text in the μ-gradient zone, close to the convective core. The mesh points provided in the models are indicated

region of steeply varying hydrogen abundance, and hence mean molecular weight μ, just outside the convective core. The model is in a phase of a growing convective core. If diffusion and settling are neglected this leads to a discontinuity in the hydrogen abundance and hence, formally, to a delta function in A∗ ; also, there is a region of ‘semiconvection’ at the edge of the core. In fact, the figure shows that three points in this model display an erratic variation in A∗ just in the transition region between the convective and the radiative zone; also, the mesh resolution of this region of rapid variation seems inadequate. This is emphasized by Fig. 2 which shows the distribution of mesh points of this model along the stellar radius, indicating that there are not enough mesh points in this transition zone. As discussed below, these features of model M2k give rise to frequency differences in the comparison, particularly for those modes for which this inner part is critical for their physical description. The third and last model is a 4042 mesh points model (from now on M4k) provided by ASTEC. General characteristics of the model are presented in Table 1. It has been computed to have overall characteristics similar to the previous CESAM model in order to understand better the differences. However, as discussed by Christensen-Dalsgaard (this volume) particular care has been taken in the treatment of the μ-gradient region; the semiconvective region was replaced by a region with a steep gradient in the hydrogen

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abundance, defined such as to ensure neutral stability of the temperature gradient. As a result, A∗ for this model, also presented in Fig. 1, shows a well-defined and reasonable behaviour. The distribution of the mesh points can be found in Fig. 2. In the central and outer parts of the stellar model, the distribution is similar to the CESAM model M2k. It is in the inner zones, particularly in the boundary region between the convective core and the μ-gradient zone, that the models present different mesh-point distributions, with M4k providing a far superior resolution of this critical region.

3 Oscillation codes and requirements All oscillation codes involved in this task were asked to provide adiabatic frequencies in the range of [20, 2500] µHz and spherical degrees  = 0, 1, 2 and 3. In addition, the solution of the equations must satisfy the following requirements:

Fig. 2 Accumulated number of layers as a function of the relative radius for the two equilibrium models explained in the text

• To use the mesh provided by the equilibrium model, no re-meshing is allowed. • To set the Lagrangian perturbation to the pressure to zero (δP = 0) as outer mechanical boundary condition. • To use the physical constants prescribed in Task 1. • To use linear adiabatic equations. Nevertheless, some other schemes for the numerical solution of the differential equations (from now on called for simplicity “degrees of freedom”) remain open. Nine oscillation codes of different research groups in the field have been used in this inter-comparison exercise. A summary of the participating codes and the different “degrees of freedom” provided by each one is found in Table 2 and include: • Set of eigenfunctions: Use of the Lagrangian or the Eulerian perturbation to the pressure (δP or P  ). This obviously affects the form of the equations; in particular, when using δP the equations do not depend explicitly on A∗ . • Order of the integration scheme: Most of the codes use a second-order scheme, but some of them have implemented a fourth-order scheme. • Richardson extrapolation: Some of the codes using a second-order scheme have the possibility to use Richardson extrapolation (Shibahashi and Osaki 1981) to decrease the truncation error; combining a second-order scheme with Richardson extrapolation yields errors scaling as N −4 , N being the number of mesh points (e.g. ChristensenDalsgaard and Mullan 1994). • Integration variable: Two integration variables are used: (1) the radius (r), or (2) the ratio r/P . The latter variable may reduce the effect of rounding errors in the outer layers (see Sect. 7.3). • Despite the requirement that the physical constants be fixed at the values for Task 1, the codes used slightly different values of the gravitational constant G, as listed in

Table 2 List of participating codes in this inter-comparison and “degrees of freedom” for each code: (a) eigenfunctions, (b) order of the integration scheme (2 or 4), (c) use (y) or not (n) of the Richardson extrapolation, (d) integration variable used, and (e) the choice of the gravitational constant G. The references for each code are also given Code

E.F.

I.S.

Rich.

I.V.

G

Reference

[10−8 cgs] ADIPLS

P

FILOU

P

2

n

r

6.67232

Suarez and Goupil (2007)

G RAC O

P  , δP

2

y, n

r, r/P

6.6716823

Moya and Garrido (2007)

LOSC

δP

4

n

r

6.67232

Scuflaire et al. (2007)

NOSC

P  , δP

2

y, n

r, r/P

6.67259

Provost (2007)

OSCROX

P

4

n

r

6.6716823

Roxburgh (2007)

POSC

P

2

y, n

r

6.6716823

Monteiro (2008)

PULSE

P

4

n

r/P

6.6716823

Brassard and Charpinet (2007)

LNAWENR

P

2

n

r

6.67232

Suran (2007)

2

y, n

r

6.67232

Christensen-Dalsgaard (2007a)

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Table 2. Ideally the equilibrium model should have been computed with the prescribed Task 1 value (6.6716823 × 10−8 cgs) which should then have been used for the oscillation calculations. In practice Model M4k was computed with G = 6.67232 × 10−8 cgs. Using different values of G in the oscillation equations clearly gives rise to inconsistencies, with potential effects on the frequencies, as discussed further in Sect. 7.1. Note that most of the oscillation codes put  = 0 in the general non-radial differential equations to obtain the radial modes, except for LOSC that uses the LAWE differential equation (Linear Adiabatic Wave Equation for radial modes), and for G RAC O for which results will be shown for both sets of equations. It should be noted that the LAWE does not depend on A∗ . All the main characteristics and numerical schemes are presented in previous chapters of this volume. The Nice code (NOSC) has the options of using P  or δP as dependent variables, and r or r/P as independent variable. However, all NOSC results presented here use P  and r. The use of the mesh provided in the equilibrium model, rather than meshes optimized for the different kinds of modes, may result in inadequate resolution of the rapid variation of high-order p- and g-modes and hence larger truncation errors in the solution of the differential equations. In these cases, therefore, the second-order scheme may result in unacceptable errors. Here the use of higher-order integration schemes (a fourth-order scheme or a second-order scheme followed by Richardson extrapolation) is therefore expected to give better results. For low- and intermediateorder modes we expect little effect of the use of higher-order schemes.

4 Frequency inter-comparisons In this section, the results of the direct frequency inter-comparison are presented. We have structured this study splitting the frequency range in three parts for the non-radial case, only one for the radial case, and comparing codes only with similar selections of “degrees of freedom”. In addition, the influence of using different selections in the same code (G RAC O in this case) is shown; for these tests the value of G in the oscillation calculations was the same as was used to compute the equilibrium model. 4.1 Radial modes These are shown in a single frequency range. In Fig. 3 the results obtained using the model M2k are presented. The reference line for all the inter-comparisons is selected to be G RAC O. In this figure the reference frequencies have been

Fig. 3 Frequency comparison (reference line is G RAC O) for modes with  = 0 obtained for the model M2k. ADIPLS and NOSC frequencies have been obtained without using the Richardson extrapolation

obtained using P  , second order, no Richardson extrapolation and r as independent variable (see Table 2). Two sets of codes can be identified in the figure, ADIPLS-NOSCG RAC O, with differences lower than 0.25 µHz, and all with the same “degrees of freedom”, and OSCROX-PULSELOSC-POSC with differences for high frequencies lower than 2 µHz with G RAC O, but with differences among them around 0.5 µHz. This second set of codes differs from the first one in the use of a fourth-order numerical scheme instead of a second-order one. In the present figure, in Fig. 5 and in Figs. 7 and 8, showing inter-comparisons with model M2k, POSC has been chosen as representative of the codes using second order plus Richardson extrapolation (see Table 2). We can see how the use of this integration procedure provides similar results as the fourth-order solutions. These differences, around 0.5 µHz for codes using the same integration scheme, and 2 µHz for codes using different schemes, are larger than the expected precision of the coming observational data. Therefore this effect can change any detailed physical description as interpreted by different oscillation codes. Also, we point out that the differences between the codes using fourth-order schemes and G RAC O results using a second-order scheme indicate that the model has an insufficient number of mesh points for asteroseismic studies. Figure 4 shows the results obtained for the model M4k. In the top panel models with second order, no Richardson extrapolation, r, are depicted. All the differences are lower in magnitude than 0.014 µHz, i.e., two orders of magnitude lower than those obtained for model M2k. Therefore improving the mesh, including a doubling of the number of points, provides a very substantial improvement in the precision, making these values more acceptable for theoretical modeling. The middle panel of Fig. 4 presents the differences obtained with models providing fourth-order integration so-

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Fig. 5 Frequency comparison (reference line is G RAC O) for modes with  = 2 for the model M2k around the fundamental radial mode. ADIPLS and NOSC frequencies have been obtained without using Richardson extrapolation

Fig. 4 Frequency comparison (reference line is G RAC O) for modes with  = 0 as a function of the frequency obtained for the model M4k. In the top panel the models with: second order, no Richardson extrapolation, r, are depicted (NOSC-ADIPLS-G RAC O-FILOU-POSC). The middle panel presents the differences obtained for models providing fourth-order integration solutions or second-order plus Richardson extrapolation (LOSC-OSCROX-NOSC-ADIPLS-G RAC O-POSC). In the bottom panel a comparison of the different “degrees of freedom” only using G RAC O is presented

lutions or second-order plus Richardson extrapolation. The global precision here is similar to the previous case or even slightly lower, with differences lower than 0.02 µHz. It

is interesting to point out that we cannot directly distinguish between a fourth-order integration scheme solution (OSCROX) or a second-order plus Richardson extrapolation (the rest). However, it is noticeable that the G RAC OOSCROX-POSC fall in one group and ADIPLS-NOSC in a second, with a slight difference in the latter case. These two groups are distinguished by the value of G (cf. Table 2), with ADIPLS and NOSC having similar but not identical values. This pattern will be found in other cases also. The LOSC behaviour is discussed in the next paragraph. Finally in the bottom panel of Fig. 4 a comparison between the different “degrees of freedom” using only the G RAC O code are presented. As reference we have used the solution with “degrees of freedom”: X =( = 0, no Richardson extrapolation, P  , r). For each comparison we have changed only one of these “degrees of freedom” at a time, keeping the rest unchanged (solutions X  ). The most prominent effect arises from the use of Richardson extrapolation, which changes the frequencies by nearly 0.8 µHz for the highest-order modes, substantially more than the expected observational accuracies. For model M2k (see Fig. 3), we have similarly found a change of 2 µHz, reflecting the smaller number of mesh points. This clearly shows that second-order schemes are inadequate, even for the mesh in M4k, for the computation of high-order acoustic modes; as expected the effect decreases rapidly with decreasing mode order. The use of r/P as integration variable provides small differences, always lower than 0.008 µHz. These differences are of the order of those obtained in the top and middle panels. The use of the Lagrangian perturbation to the pressure as variable (δP ) or the use of the LAWE differential equation provide very similar differences, lower than 0.05 µHz, but with an oscillatory pattern. This pattern is very similar to that observed for LOSC, which also uses LAWE to obtain the radial modes. As discussed in Sect. 7.2, this oscillatory

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237

Fig. 7 Frequency comparison (reference line is G RAC O) for modes with  = 2 obtained for the model M2k in the low frequency region. ADIPLS and NOSC frequencies have been obtained without using the Richardson extrapolation

Fig. 8 Period separation as a function of the frequency in the asymptotic g-mode region for model M2k

Fig. 6 Frequency inter-comparison (reference line is G RAC O) for modes with  = 2 as a function of the frequency obtained for the model M4k and for modes around the fundamental radial. In top panel models with: second order, no Richardson extrapolation, and r are depicted (NOSC-ADIPLS-G RAC O-FILOU-POSC). Middle panel presents the differences obtained for the models providing fourth-order integration solutions or second-order plus Richardson extrapolation (LOSC-OSCROX-NOSC-ADIPLS-G RAC O-POSC). In bottom panel an inter-comparison for the “different degrees” of freedom only with G RAC O is presented

pattern arises from an inconsistency in the thermodynamics of model M4k which affects A∗ ; solutions of equations

that do not depend on A∗ (i.e., the LAWE or the equations based on δP ) are insensitive to this effect. Therefore, even for model M4k and radial modes, the use of different integration procedures can give different values for the oscillation eigenfrequencies that are non-physical in nature. These non-physical sources of differences are mainly some inconsistencies in the equilibrium models (see Sect. 7.2) and the lack of mesh points. But when the same numerical schemes are used, the different codes provide very similar frequencies. 4.2 Non-radial modes with  = 2 To illustrate the differences appearing in the case of nonradial modes, the spherical degree  = 2 has been chosen. We have divided the frequency spectrum into three regions: (1) Large frequencies ([500, 2500] µHz), (2) frequencies around the fundamental radial mode ([80, 500] µHz), and

238

(3) low-frequency region ([20, 80] µHz). In all cases a study similar to that developed in the radial case has been carried out. For the sake of simplicity the high-frequency differences are not represented since the results are very similar to those presented for the radial case. Only a slightly higher precision is found in this case. The results of LOSC present the same pattern as the radial case, owing to the use of δP as eigenfunction in that code (see Sect. 7.2). Figure 5 shows the results obtained for model M2k when comparing  = 2 frequencies around the fundamental radial mode. The main differences are smaller than in the highfrequency region, corresponding to the low order of the modes and the consequent lesser sensitivity to the number of mesh points. The largest differences are found for two modes of frequency near 345 and 362 µHz showing avoided crossing; these modes have a mixed character, with fairly substantial amplitude in the μ-gradient zone. PULSE and LOSC present differences around 3–4 µHz, POSC around 2 µHz, OSCROX less than 1 µHz, and the rest do not present significant differences for these two modes. That is, the largest differences are found in these codes when using a fourth-order integration scheme or a second-order plus Richardson extrapolation. The values of these differences are larger than the expected precision of the satellite data to come. They clearly reflect the inadequate representation of the μ-gradient zone in M2k, with higher-order schemes being more sensitive to the resulting inconsistency in the model. In the top panel of Fig. 6 the inter-comparisons of the frequencies ( = 2) for model M4k, using a second-order scheme without Richardson extrapolation, are depicted for the same frequency range as in Fig. 5. The main differences are two orders of magnitude lower than those obtained for model M2k, and they are also of the same order of magnitude as those obtained for the high-frequency range with M4k. All these differences remain always lower than 0.025 µHz. G RAC O and POSC are extremely close, while FILOU and ADIPLS provide very similar results, with slightly larger differences for NOSC, by about 0.005 µHz, relative to these codes. Thus again the differences are related directly to the different values of G. There remain small wiggles for the mixed modes near 350 µHz but reduced more than two orders of magnitude relative to the largest differences for model M2k, reflecting the superior resolution of the critical region in model M4k. The middle panel of Fig. 6 shows the same inter-comparison as the top panel for codes using a fourth-order scheme or a second-order plus Richardson extrapolation. The precision is similar to the previous case, with differences of the same order of magnitude. We can distinguish two groups of codes providing very similar results: OSCROX-POSC-G RAC O and LOSC-ADIPLS-NOSC. This

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distribution does not depend on the integration scheme, but again reflects the values of G. The wiggle of the mixed modes is also very similar to that obtained with a secondorder scheme without Richardson extrapolation. Finally, an inter-comparison of different “degrees of freedom” using only G RAC O is presented in order to test the differences obtained for the different choices. This comparison is depicted in the bottom panel of Fig. 6. The use of the Richardson extrapolation is not very important, as expected for these modes of low order, with effects generally smaller than 0.002 µHz, although a larger value is present for the mixed modes. Using δP as variable gives differences larger than the differences among codes with the same “degrees of freedom”. The use of the integration variable r/P does not introduce significant differences. Compared with model M2k, we find a general reduction of the differences for M4k, the main effects being in the region of mixed modes with an improvement reaching up to three orders of magnitude. This is obviously not a simple consequence of the doubling of the number of mesh points. The main reason is likely the inadequate resolution shown by model M2k in the description of the Brunt-Väisälä frequency in the region close to the boundary of the convective core, which is not present in model M4k. The avoided-crossing phenomenon and the behaviour of the mixed modes are very sensitive to the detailed treatment of this region, including the effects of semiconvection. Therefore, an accurate description of N 2 is critical for the oscillation codes in order correctly to obtain eigenfrequencies for modes near an avoided crossing. The direct frequency inter-comparison ends with the study of the low-frequency region; as above we concentrate on modes of degree  = 2. Figure 7 shows the results obtained for model M2k. In this case the differences are lower than obtained at higher frequencies for this model (cf. Figs. 3 and 5). The most surprising behaviour is present in PULSE and LOSC. POSC and OSCROX also present some differences in the same region. To understand the reason for these differences and the region where they appear Fig. 8 shows the period separation ΔΠ between adjacent modes. The first-order asymptotic g-mode theory predicts a constant separation of the periods in this regime for a given . However, when the equations are solved numerically, this period spacing presents several minima, and these minima are directly linked with the mode trapping (Brassard et al. 1992). Figure 8 shows that the position of one of these minima is the same as the position of the largest differences. The modes in this region have somewhat enhanced amplitudes in the region just outside the convective core. In PULSE and LOSC this apparently happens for modes somewhat different from the remaining codes. This is the origin of the frequency differences. As the mode trapping in this region for these stellar models is related to the Brunt-Väisälä frequency in the μ-gradient zone, the previously mentioned inadequate

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treatment of this region in model M2k is likely the reason for these differences. The top panel of Fig. 9 presents inter-comparisons for  = 2 frequencies calculated with a second-order scheme without Richardson extrapolation and for model M4k, for the same frequency range as in Fig. 7. POSC is again extremely close to G RAC O, within 2 × 10−4 µHz. FILOU and ADIPLS present very similar results, with NOSC showing slightly larger differences, ranging from 0.0005 up to 0.0015 µHz relative to this group. The differences decrease globally as far as the frequency decreases until reaching a magnitude of 0.0015 µHz for the smallest frequency studied here. The middle panel of Fig. 9 shows the same comparison for codes using a fourth-order scheme or a secondorder plus Richardson extrapolation. Once again it can be seen that the precision resembles the results in the previous panel. Here POSC and OSCROX provide very similar results to G RAC O. ADIPLS-LOSC-NOSC present small differences among them, with NOSC having a small increasing difference with frequency with respect to the other two. Again the general pattern here, and in the top panel, largely reflects the differences in G. In this case we can distinguish the codes using fourth-order integration scheme with its apparently noisy profiles as compared with a solution using second-order integration plus Richardson extrapolation. In the bottom panel of Fig. 9, an inter-comparison for the different “degrees of freedom” using only G RAC O is presented; as in earlier corresponding plots the same value of G is used as in the computation of the equilibrium model. As expected the Richardson extrapolation has a growing influence as the frequency decreases and the mode order increases, with quite substantial differences, compared with the differences between different codes, for the lowestfrequency modes. Thus, with the mesh provided by the evolution calculation the second-order schemes have inadequate numerical precision. In this case, the differences provided by the use of r/P as integration variable are negligible; The use of the Lagrangian perturbation to the pressure (δP ) gives rise to frequency differences exceeding those obtained between the different codes, at the lowest frequencies; we note, however, that a corresponding comparison between ADIPLS and LOSC does not show this effect which may therefore be particular to the G RAC O implementation. Finally, we want to point out that in the case of model M4k, the large differences appearing in the mode trapping region are not found. Figure 10 presents the same period separation as Fig. 8 but for this model. All the codes give quite similar results. Two mode trapping regions appear with the same frequency domain as in Fig. 8. As this model does not present any numerical imprecision in the Brunt-Väsälä frequency pattern, the obvious conclusion is that, as for the mixed modes in

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Fig. 9 Frequency comparison (reference line is G RAC O) for modes with  = 2 as a function of the frequency obtained for the model M4k in the low-frequency region. In the top panel the models with: second order, no Richardson extrapolation, and r, are depicted (NOSC-ADIPLS-G RAC O-FILOU-POSC). The middle panel presents the differences obtained for the models providing fourth-order integration solutions or second-order plus Richardson extrapolation (LOSC-OSCROX-NOSC-ADIPLS-G RAC O-POSC). In the bottom panel an inter-comparison for the different degrees of freedom only using G RAC O is presented

avoided crossing, any numerical imprecision in the description of N 2 coming from the equilibrium model can give rise

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Fig. 10 Period spacing as a function of the frequency in the asymptotic g-mode region for the model M4k

to large differences in the frequencies calculated by different oscillation codes for the g-modes trapped in the μ-gradient zone.

5 Large separations (LS) This section is devoted to the asymptotic behaviour of pmodes through the use of the so called “large separations”, that is, the difference between two consecutive modes with the same spherical degree  (Δ = ν(n, ) − ν(n − 1, ), n being the radial order of the mode). The structure of the section is similar to the previous one. We will use the same definitions as in the previous section to study all the frequencies ranges. From now on, the results with the M2k model will not be discussed, since no additional information is found from the further inter-comparisons. 5.1 Large separation of radial modes Figure 11 shows the results obtained for model M4k. In the top panel differences for the different codes using second order, no Richardson extrapolation, r, are depicted. POSC and G RAC O are again extremely close. The rest of the codes (ADIPLS-NOSC-FILOU-G RAC O) present differences of up to around 0.0015 µHz, generally sharing an oscillatory pattern, particularly at relatively low frequency; we have no explanation for this behaviour. However, the effect is evidently small. In the middle panel of Fig. 11 the LS differences for the different codes, using a fourth-order integration scheme or a second-order plus Richardson extrapolation, are presented. With the exception of LOSC the global behaviour is similar to that obtained without Richardson extrapolation, the NOSC-ADIPLS-OSCROX-G RAC O differences being always lower than 0.002 µHz. The pattern of the LOSC differences, presenting differences one order of magnitude larger

Fig. 11 Large separation inter-comparison (reference line is G RAC O) for modes with  = 0 as a function of the frequency calculated for the model M4k. In the top panel the models with second order, no Richardson extrapolation, r, are depicted (NOSC-ADIPLS-G RAC O-FILOU-POSC). The middle panel presents the differences obtained for the models using fourth-order integration solutions or second-order plus Richardson extrapolation (LOSC-OSCROX-NOSC-ADIPLS-G RAC O-POSC). In the bottom panel an inter-comparison for the different “degrees of freedom” using only G RAC O is presented

than the rest of the codes, is clearly related to the corresponding oscillatory pattern found in Fig. 4; as before it is explained by the use of the LAWE differential equation.

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The bottom panel of Fig. 11 shows the LS differences for the radial modes obtained with G RAC O and model M4k when different “degrees of freedom” are used. The Richardson extrapolation introduces differences increasing with frequency and mode order, as expected, giving the largest differences, around 0.07 µHz. This again emphasizes the inadequacy of the second-order schemes for the highest-order modes, on the M4k mesh. The integration variable r/P gives differences slightly lower than 5 × 10−4 µHz, i.e., much smaller than that found for different codes using the same numerical techniques, as depicted in the previous panels. The use of the Lagrangian perturbation to the pressure δP and the LAWE differential equation show the same oscillating behaviour and values as those previously observed for the LOSC results. As discussed above, this is related to the inconsistency in A∗ in model M4k (see Sect. 7.2). 5.2 Non-radial modes with  = 2 To illustrate the differences appearing in the case of nonradial modes, the spherical degree  = 2 has been arbitrarily chosen. We have divided the frequency spectrum in three regions, like in the direct frequency inter-comparison: 1) highfrequency region, 2) frequencies around the fundamental radial mode and 3) low-frequency region. In all cases, a study similar to that developed in the radial case has been carried out. In the low-frequency region, the more physical period separation is studied, instead of the frequency separation relevant for acoustic modes. In the first region the results are very similar to those obtained for the radial case; therefore the plots are not presented here. As in the direct frequency inter-comparison case, LOSC also presents an oscillating pattern, owing to the use of δP as eigenfunction (see Sect. 7.2). On the other hand, the only noticeable difference, when compared with the radial case in this region, is that for  = 2 the precision among codes using the same integration procedures is slightly higher. The results obtained with the codes using a secondorder scheme, for the modes around the fundamental radial mode, are depicted in the top panel of Fig. 12. POSC remains very close to G RAC O, with the other set of codes (FILOU-NOSC-ADIPLS) extending the oscillatory pattern in the top panel of Fig. 11, with the largest difference being 0.01 µHz for the mixed modes. This set of codes agrees to within differences around 0.001 µHz (slightly larger for NOSC). The middle panel of Fig. 12 depicts the differences for a fourth-order integration scheme or second-order plus Richardson extrapolation. The precision of the different codes is similar to that given by the second-order scheme without Richardson extrapolation. The maximum difference is lower than 0.08 µHz, and most of the codes present differences around 0.005 µHz. Once again the largest differences

Fig. 12 Large separation inter-comparison (reference line is G RAC O) for modes with  = 2 as a function of the frequency calculated for the model M4k and for modes around the fundamental radial one. In the top panel the models with second order, no Richardson extrapolation, r, are depicted (NOSC-ADIPLS-G RAC O-FILOU-POSC). The middle panel presents the differences obtained for the models using fourth-order integration solutions or second-order plus Richardson extrapolation (LOSC-OSCROX-NOSC-ADIPLS-G RAC O-POSC). In the bottom panel an inter-comparison for different “degrees of freedom” using only G RAC O is presented

are obtained in the mixed modes. No different behaviours depending on the integration scheme are found.

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The bottom panel of Fig. 12 is devoted to the differences obtained with G RAC O when different options for the solution of the differential equations are chosen. In this frequency region the replacement of r by r/P as integration variable introduces the smallest differences, one order of magnitude lower than those obtained using the codes with the same “degree of freedom”. Using the Lagrangian perturbation to the pressure (δP ) as eigenfunction causes some difference indicating sensitivity to whether or not A∗ is used. The Richardson extrapolation introduces differences lower than those found for the different codes, except in the avoided crossing zone of mixed modes, where changes comparable to the largest one are found. Given the rather substantial variations in the bottom panel, and the fact that most codes show the same variation in the top two panels, one might suspect that the dominant source of this variation is in fact in the G RAC O results used as reference. The low-frequency region is studied through the period separation for  = 2 (ΔΠ in seconds), illustrated in Fig. 13. The top panel of this figure shows the differences found using only codes with a second-order integration scheme. ADIPLS and NOSC show similar shifts of around 0.1 s relative to G RAC O. FILOU is similar at the higher frequencies but shows a small oscillating behaviour in the mode-trapping regions. Finally POSC presents a quite noisy pattern, varying around zero. Again the overall grouping of the differences (ADIPLS-NOSC-FILOU and G RAC O-POSC) reflects the different values of G. Results obtained using a fourth-order integration scheme or a second-order plus Richardson extrapolation are compared in the middle panel of Fig. 13. The values of the differences found in this case are of the same order as in the previous inter-comparison. Here we can distinguish codes using a fourth-order scheme or a second-order plus Richardson extrapolation, because of the apparently random pattern they present, with differences one order of magnitude larger than the main values. The frequency differences obtained with the OSCROX and LOSC results are those presenting a noisy behaviour, when comparing with a second-order plus Richardson extrapolation solution (as G RAC O does) as the reference line. POSC also presents some differences in the mode-trapped regions when compared with other codes but using the same integration scheme. The bottom panel of Fig. 13 shows the differences obtained with the same code (G RAC O) and different choices of the “degrees of freedom”. The use of the Richardson extrapolation introduces substantial differences, of the order of seconds, with noticeable wiggles in the two mode-trapping regions. As noted previously this reflects the inadequacy of the second-order schemes for high-order modes. The use of the Lagrangian perturbation to the pressure as variable (δP ) also introduces substantial differences, particularly around the trapped modes near 22 µHz, related to the frequency differences found with G RAC O in this region when δP is used

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Fig. 13 Period separation inter-comparison (reference line is G RAC O) for modes with  = 2 as a function of the frequency calculated for the model M4k in the low-frequency region. In the top panel the models with second order, no Richardson extrapolation, r, are depicted (NOSC-ADIPLS-G RAC O-FILOU-POSC). The middle panel presents the differences obtained for models using fourth-order integration solutions or second-order plus Richardson extrapolation (LOSC-OSCROX-NOSC-ADIPLS-G RAC O-POSC). In the bottom panel an inter-comparison for the different choices of “degrees of freedom” using only G RAC O is presented

(cf. Fig. 9). Using r/P as integration variable gives a small difference when compared with r, one order of magnitude

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lower than the differences found between the codes using the same numerical integration schemes.

6 Small separations (SS) In this section we study the inter-comparisons for the small separations (SS) δ ≡ ν(n, ) − ν(n−1, +2). Therefore, two sets of inter-comparisons can be done, one for the ( = 0 −  = 2) modes and another for the ( = 1 −  = 3) modes. In this case only the high-frequency region is studied, since this is where this quantity has physical meaning. As in the previous section we concentrate on results for model M4k. 6.1 Small separations δ02 The results obtained for the small separation δ02 are presented in Fig. 14. The top panel of this figure shows the differences obtained with the codes solving the set of differential equations with a second-order integration scheme. In the avoided crossing region some wiggles occur related to variations for  = 2, but these wiggles are of the order of magnitude of the differences obtained for the high-frequency region. ADIPLS-POSC-G RAC O present similar values for high frequencies and NOSC-FILOU presents differences around 0.004 µHz, and always lower than 0.01 µHz. These differences are much lower than the expected observational errors for CoRoT. The middle panel shows the same inter-comparison using a fourth-order integration scheme or a second-order plus Richardson extrapolation. Once again we cannot distinguish the integration scheme used. Just LOSC shows an oscillatory pattern for high frequencies due to the use of LAWE or δP (see Sect. 7.2). The order of magnitude of the main differences is the same as those obtained using only a secondorder scheme. The wiggles in the avoided crossing regions are still present, and for high frequencies the differences are all in the range [−0.001, 0.001] µHz. ADIPLS and POSC present an almost constant difference with G RAC O, NOSC and OSCROX show a small noisy behaviour. The bottom panel of Fig. 14 presents the SS differences induced in G RAC O when different choices of “degrees of freedom” are selected. In this case they are all below, or of the same order of magnitude as, the spread between the different codes illustrated in the middle panel. Interestingly, Richardson extrapolation introduces differences for higher frequencies far smaller than found for the large separation (Fig. 11). The use of the Lagrangian perturbation to the pressure (δP ) as variable results in the same oscillatory pattern as seen for LOSC in the middle panel. The integration variable r/P gives increasing differences in the range [0.001, 0.003] µHz, similar to the Richardson extrapolation,

Fig. 14 Small separation inter-comparison (reference line is G RAC O) for modes with  = 0–2 as a function of the frequency calculated for the model M4k in the high-frequency region. In the top panel models with second order, no Richardson extrapolation, r, are depicted (NOSC-ADIPLS-G RAC O-FILOU-POSC). The middle panel presents the differences obtained for models using fourth-order integration schemes or second-order plus Richardson extrapolation (LOSC-OSCROX-NOSC-ADIPLS-G RAC O-POSC). In the bottom panel an inter-comparison for different “degrees of freedom” only using G RAC O is presented

probably reflecting a difference in the sensitivity of radial and nonradial modes to the choice of independent variable

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in the G RAC O code; however, the effect is obviously small. Note that in this case the use of the LAWE for the radial modes, keeping the default “degrees of freedom” (including the use of P  ) for  = 2, is hardly meaningful; thus no results are included for LAWE. 6.2 Small separations δ13 In Fig. 15 the inter-comparison of δ13 is done for model M4k. The top panel shows the differences given by the codes using a second-order integration scheme. For POSCADIPLS-FILOU-G RAC O the differences are between −0.001 and 0.002 µHz and NOSC shows somewhat larger values. In any case the precision is good and the patterns rather smooth. The codes using a fourth-order integration scheme or a second-order plus Richardson extrapolation have produced the results presented in the middle panel of this figure. The precision is very similar as in the previous panel or even a little higher. LOSC shows the oscillatory pattern already obtained in all the previous inter-comparisons. POSC and ADIPLS present smooth difference profiles when compared with G RAC O. On the other hand OSCROX and NOSC give a low noisy profile. Once again the integration scheme used cannot be discriminated. Finally, the bottom panel presents the effect in δ13 of using different numerical integration schemes. Using Richardson extrapolation gives large differences in the highfrequency region, even larger than the precision of the different codes while, as expected, the effect is small for loworder modes. If the differential equations are solved with r/P as integration variable, an almost increased difference is introduced, always lower than 0.001 µHz, i.e., of the order of the precision among most of the codes using the same numerical techniques. The use of the Lagrangian perturbation to the pressure as variable introduces the same oscillatory pattern as was obtained with LOSC.

7 Computational variations 7.1 The influence of the gravitational constant G As indicated in Table 2 different values of G are used by different codes. In many cases the value differs from the value, G = 6.67232 × 10−8 cgs, which was used in the computation of model M4k. As a result, as seen by the pulsation code the equilibrium model is not strictly in hydrostatic equilibrium, thus potentially causing errors in the computed frequencies. In this section we examine the consequence of such inconsistencies. Thus we test the influence of the choice of a particular gravitational constant G, within always the recent experimental values found in the literature. The value

Fig. 15 SS inter-comparison (reference line is G RAC O) for modes with  = 1–3 as a function of the frequency obtained for model M4k in the high-frequency region. In the top panel the models with second order, no Richardson extrapolation, r, are depicted (NOSC-ADIPLS-G RAC O-FILOU-POSC). The middle panel presents the differences calculated for models using fourth-order integration solutions or second-order plus Richardson extrapolation (LOSC-OSCROX-NOSC-ADIPLS-G RAC O-POSC). In the bottom panel an inter-comparison of the different “degrees of freedom” using only G RAC O is presented

of this constant is not very accurately known, and the different values we can find in the literature can have an impact

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and G RAC O in the middle panel of Fig. 4 are consistent with this estimate. In general, as already discussed, it appears that much of the differences between codes using the same numerical scheme results simply from the different values of G. Thus, it is obviously important to ensure that consistent values of G are used in the evolution and oscillation calculations. 7.2 The choice of dependent variables and equations

Fig. 16 Frequency comparison of modes with  = 0 and 2 obtained with G RAC O(p  , δP = 0, no Richardson extrapolation, r) and the model M4k when two different values of the gravitational constant G are used

on the frequency calculation of the same order or even larger than the differences studied here. To illustrate this, two extreme values of G found in the literature have been chosen: G1 = 6.6716823 × 10−8 cgs (as fixed for Task 1), and G2 = 6.693 × 10−8 cgs (Fixler et al. 2007), the most recent one, although with a quoted random error of ±0.027 × 10−8 and a systematic error of ±0.021 × 10−8 cgs it is consistent with the previous value. The present recommended value of the Committee on Data for Science and Technology (CODATA) can be found in the World Wide Web at physics.nist.gov/constants, and it is closer to that fixed in Task 1. The comparison has been carried out with G RAC O at fixed “degrees of freedom” (P  , δP = 0, no Richardson extrapolation, r). The differences obtained using both values of G are shown in Fig. 16 for modes with  = 0 and 2. The equilibrium model used is M4k. Surprisingly, the differences obtained in the region around the fundamental radial mode are much larger than those obtained for the direct frequency inter-comparison for the different codes or for the same code using different integration numerical schemes. As expected for radial modes the largest difference is obtained for the fundamental radial mode. The value of this difference (more than 0.3 µHz in magnitude) is really considerable. In the asymptotic regions the influence of the value of this parameter decreases until reaching values lower than 0.1 µHz. Nevertheless, these differences are larger than those obtained by using or not the Richardson extrapolation. The presence of wiggles for  = 2 for the mixed modes is also remarkable. The relative difference between G1 and G2 is around 3 × 10−3 . In contrast, the relative difference between, say, the value chosen for Task 1 and the ADIPLS value is only 5 × 10−5 . Thus we might expect this difference to cause a frequency difference of order 5 × 10−3 µHz. In fact, both the magnitude and shape of the differences between ADIPLS

The most dramatic difference between the different codes is the oscillatory variation shown by the computations using the LAWE or δP , relative to the reference G RAC O results (e.g., Fig. 4). The oscillatory nature and the ‘period’ of the variation indicated that it reflected a sharp difference in some aspect of the model, located approximately in the region of the second helium ionization zone. After various failed attempts to identify the cause of the variation it transpired that it resulted from a, previously known (e.g., Boothroyd and Sackmann 2003), inconsistency in the OPAL equation-of-state tables (the original tables of Rogers et al. 1996) used in the computation of model M4k with the ASTEC code. To understand this we consider the computation of A∗ =

d ln ρ 1 d ln P − . Γ1 d ln r d ln r

(1)

From the point of view of stellar evolution calculation this is somewhat inconvenient, since the gradient of ρ does not appear directly in the equations of stellar structure. Thus evaluation of A∗ would involve numerical differentiation of ρ. A potentially more convenient formulation is obtained by rewriting the expression in terms of the temperature gradient which is known from the equation of energy transport, given that ρ is a function, known from the equation of state, of pressure, temperature and composition which we characterize solely by the hydrogen abundance X. Thus we obtain   ∂ ln ρ d ln P (∇ − ∇ad ) A∗ = − ∂ ln T P ,X d ln r   d ln X ∂ ln ρ ; (2) + ∂ ln X P ,T d ln r here, as usual, ∇ = d ln T /d ln P and ∇ad is its adiabatic value. This expression still requires numerical differentiation of X,2 but in much of the model X varies little or not at all and hence the term in d ln X/d ln r makes a limited contribution. Furthermore, in the bulk of convective zones, with homogeneous composition, (2) accurately reflects the 2 Except if diffusion is taken into account; in that case the gradient of X appears directly in the equations (see, for example, the description of ASTEC by Christensen-Dalsgaard 2007b).

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7.3 The choice of independent variable

Fig. 17 Differences between A∗ as evaluated from (1) and evaluated from (2) as done in ASTEC, for model M4k, against fractional radius. The differences have been normalized by V = |d ln P /d ln r|

small value of A∗ resulting from the small value of ∇ − ∇ad determined, e.g., from the mixing-length treatment. The transformation in (2) obviously assumes that the thermodynamical quantities used are mutually consistent. As noted by Boothroyd and Sackmann (2003), this is not the case of the original OPAL tables provided by Rogers et al. (1996). The effect of this on A∗ is illustrated in Fig. 17 for model M4k, in terms of the difference between the value of A∗ calculated directly from (1), using numerical differentiation to evaluate d ln ρ/d ln r,3 and the value, A∗ASTEC , evaluated from (2). There are obvious systematic differences,4 concentrated in the ionization zones of hydrogen and helium. These would indeed affect the oscillations as a ‘sharp feature’ and hence cause an oscillatory signature in the frequencies, with the ‘period’ seen in Fig. 4 and elsewhere (e.g., Gough 1990; Monteiro and Thompson 2005; Houdek and Gough 2007). To test further this interpretation of the results we have recomputed frequencies for model M4k, but replacing A∗ASTEC with A∗ computed from (1). For this modified model the differences between ADIPLS and LOSC are smaller by an almost an order of magnitude than the differences illustrated in Fig. 4 and with none of the systematic oscillatory character. We finally note that transforming between the equations using P  and δP also depends on the equation of hydrostatic equilibrium and hence on a consistent choice of G in the oscillation equations (cf. Sect. 7.1). However, within the actual range of G-values used by the different codes this effect is smaller than the effect of the inconsistency in A∗ . 3 At this level of precision it does not matter whether d ln P /d ln r

is calculated from the equation of hydrostatic support or through numerical differentiation.

4 The small-scale rapid variation is probably associated with the interpolation procedure in the equation-of-state tables.

We have found a fairly significant dependence on the G value used by the oscillation codes for the differences of using the integration variables r or r/P . Figure 18 shows this dependence in the case of  = 0. In this figure, the differences between the use of r or r/P with the three different G values presented in Table 2, are depicted. All of them have been obtained using G RAC O and M4K. The differences increase with frequency, and their value depend on the difference Geq − Gosc , where Gx is the G value used to obtain the equilibrium model or the oscillation frequencies, respectively. In fact, the transformation between using r and r/P as independent variables uses the equation of hydrostatic equilibrium and hence the result is obviously sensitive to whether or not a consistent value of G is used. By far the smallest differences are obtained when the same G are used in the equilibrium and oscillation codes. Even when the same value of G is used, we have found a slight effect of using r/P as independent variable, as implemented in a few of the codes (see also Provost 2007), instead of r; in Fig. 18 the maximum difference is 0.008 µHz, as also found in Fig. 4. The origin of choosing r/P seems to be a concern that near the surface, where r varies little between adjacent mesh points, rounding errors in the evaluation of differences in r, when representing the differential equations on finite-difference form, could have a significant effect on the results. Given the rapid variation of P in the near-surface layers this problem is obviously avoided if r/P is used instead. While there might well be such problems with using r when variables are represented in single-precision form (with four-byte reals) it seems unlikely to be a problem when using double-precision variables.5 On the other hand, the present test has been carried out with models transferred in the FGONG format, where variables are given with ten significant digits. Here one cannot exclude that rounding errors might be significant. To test this JC-D computed frequencies with ADIPLS using both the original binary (double-precision) version of M4k and the version resulting from conversion to FGONG format. The frequency differences between these two cases were always less than 10−5 µHz, strongly indicating that the use of the FGONG format is not a concern with the present level of accuracy. It is obvious that the choice of independent variable also affects the truncation error, i.e., the error in the representation of the differential equations on finite-difference form. This can presumably be the reason for the differences found with G RAC O when comparing the use of r and r/P (e.g., Fig. 14), although the effects are small. Further analysis is required to decide which is the more accurate representation. 5 In contrast, the problem would be severe if the interior mass were used as independent variable.

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•

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Fig. 18 Differences between the use of r or r/P as integration variable as a function of the mode frequency, for radial modes. Different G values are used in the G RAC O oscillation code: G (M4K) is the G value used to obtain the M4K model, G (M2K) is the G value used to obtain the M2K model, and G NOSC is the G value used by the NOSC code. M4K has been used as equilibrium model for this comparison

•

8 Conclusions A complete inter-comparison of the frequencies calculated by using a set of oscillation codes has been presented. Nine oscillation codes have been used: ADIPLS, FILOU, NOSC, LOSC, G RAC O, OSCROX, PULSE, POSC and LNAWENR. The only free parameter for each code has been the integration numerical scheme. In order to ensure that the same physics has been imposed to all the codes two equilibrium models of a 1.5M star have been supplied, with 4042 (M4k) and 2172 (M2k) mesh points in their grids, respectively. No re-meshing has been allowed. The model M2k presents inadequate numerical resolution in the BruntVäisälä frequency close to the boundary of the convective core. The model M4k does not suffer from these problems. In the present paper we have used these models only to show the sensitivities of the oscillation codes to possible numerical inaccuracies in the equilibrium models. A complete study of the accuracies of the different evolutionary codes is provided by Lebreton et al. in this volume. Several inter-comparisons have been performed: (1) Direct frequency inter-comparison, (2) large separation (LS), (3) small separation (SS), and (4) different experimental values of the gravitational constant G. The main conclusions can be summarized as: • When codes using the same numerical integration schemes are compared, the general precision obtained for model M4k (0.02 µHz) is higher than the expected precision of the observational data obtained with the CoRoT mission. This precision is always one order of magnitude better or even more when compared with that obtained

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•

•

for model M2k (0.5 µHz). The largest differences are obtained for the high-frequency region. The use of a second-order integration scheme plus Richardson extrapolation or a fourth-order integration scheme does not improve the agreement between the codes when compared with the use of a second-order scheme. In the cases here presented it was not possible to distinguish between the use of a fourth-order scheme or Richardson extrapolation, which always give very acceptable consistency for model M4k. However, the use of a fourth-order integration scheme (or a second-order plus Richardson extrapolation) introduces differences larger than 0.1 µHz (1.5 µHz for M2k and 0.8 µHz for M4k) compared with a second-order scheme, mainly located in the regions of high order (high frequency for p-modes and low frequency for g-modes). This indicates that the second-order schemes have inadequate numerical precision on the mesh provided in M4k and, obviously, even more so on the sparser mesh in M2k. For “large separations” the agreement between different comparable codes using model M4k, with every possible numerical scheme is generally better than the expected precision of the data. However, the equation-of-state inconsistency in model M4k (see Sect. 7.2) introduces a signature that it potentially significant. The use of the Richardson extrapolation together with a second-order integration scheme, or a fourth-order scheme, gives differences for high-order p- and g-modes relative to the use of only a second-order scheme that are comparable to the expected observational accuracies, indicating that the accuracy of the latter scheme is inadequate. The “small separations” present differences for model M4k lower than the expected observational accuracies everywhere, although the inconsistency in M4k has a noticeable effect. Use of Richardson extrapolation has a potentially significant effect, indicating that the precision of second-order schemes may be inadequate. The use of r/P as integration variable instead of r has a small influence when a consistent value of G is used. However, an inconsistency between the G values used to obtain the equilibrium model and the oscillation frequencies can give substantial differences between the use of these integration variables, over the range of G values in Table 2. For model M2k, the mixed modes near the avoided crossing present large differences, reaching even up to 4 µHz. These differences are not present when model M4k is used. The problems with N 2 in M2k are undoubtedly the main source of this difference. The same behaviour is found for the trapped g-modes. Therefore, the correct numerical description of N 2 is critical for the value of the frequencies of these trapped modes or in avoided crossing.

248

• The use of δP or P  as eigenfunction, or the solution of the LAWE, may have a significant effect if the equilibrium model is based on thermodynamic quantities that are not internally consistent, as is the case for the OPAL tables used to calculate model M4k. Further effects arise when there are problems with the Brunt-Väisälä frequency (as for model M2k); here a different choice can change the frequencies of some modes since N 2 does not appear in the differential equations when δP or the LAWE are used and it does when using P  . • The value of the gravitational constant G in the oscillation calculations can introduce non-negligible differences as well, if it is not the same as the value used in the equilibrium model. When the two extreme values found in the literature are used, such inconsistency yields differences in the range [−0.35, −0.08] µHz for model M4k, larger than those obtained when different numerical integration schemes are used. Differences between the values of G actually used by the different codes, although less extreme, account for much of the difference between the computed frequencies. Therefore, for a proper pulsational study, we require that the number of mesh points and their distribution must be such as to yield an equilibrium model that satisfies the dynamical equations with sufficient accuracy also in the regions of the star where the physical quantities present rapid variations (e.g., the outer layers and μ-gradient zones). In addition, the mesh used in the pulsation calculation must properly resolve the eigenfunctions of the highest-order modes considered. In the present case model M4k, with 4042 points, appears to satisfy these conditions although the use of a fourth-order integration scheme, or a second-order scheme and Richardson extrapolation, is still needed in the oscillation calculation; these higher-order schemes give significant improvements compared with the use of a simple second-order scheme. The use of a second-order integration plus Richardson extrapolation scheme is not distinguishable, in accuracy and precision, from the use of a fourth-order integration scheme. A correct physical and numerical description of the Brunt-Väisälä frequency is essential when P  is used as eigenfunction; in particular, inconsistencies in the equation of state can have serious effects on the frequencies. Inconsistency between the value of G used in the oscillation calculation and the value used to compute the equilibrium model, within the range of the different values of G found in the literature, may lead to substantial errors in the computed frequencies. We note that the situation is somewhat different if consistent values of G are used in the evolution and oscillation calculations. Then the effect on the frequencies is approximately given, according to homology arguments, as a scaling by (GM)1/2 . However, since the product GM is

Astrophys Space Sci (2008) 316: 231–249

known extremely precisely from planetary motion in the solar system, any variation in G should be reflected in a corresponding change in the assumed value of M . If this is the case, and if the model is characterized by a given value of M/M (as is typically the case) the effect on the frequencies of changes in G are very small (see also ChristensenDalsgaard et al. 2005). In further tests more care is required to secure the full consistency of the models: a consistent equation of state should be used, and the value of G should obviously be the same in the equilibrium-model and the pulsation calculations; indeed, this strongly argues for including the value of G as one of the parameters in the model file. The main conclusion of this extensive investigation, however, it positive: with a properly resolved equilibrium model the broad range of oscillation codes likely to be involved in the asteroseismic analysis of data from CoRoT and other major upcoming projects generally give consistent results, well within the expected errors of the observations. Thus, although the remaining problems in the calculation evidently require attention, we can be reasonably confident in our ability to compute frequencies of given models and hence in the inferences concerning stellar structure drawn from comparing the computed frequencies with the observations. Acknowledgements This work was supported by the Spanish PNE under Project number ESP 2004-03855-C03-C01, and by the European Helio- and Asteroseismology Network (HELAS), a major international collaboration funded by the European Commission’s Sixth Framework Programme. A.M. and J.M. acknowledge financial support from the Belgian Science Policy Office (BELSPO) in the frame of the ESA PRODEX 8 program (contract C90199). MJPFGM is supported in part by FCT and FEDER (POCI2010) through projects POCI/CTEAST/57610/2004 and POCI/V.5/B0094/2005.

References Baglin, A., Auvergne, M., Barge, P., Deleuil, M., Catala, C., Michel, E., Weiss, W.: COROT team: Scientific objectives for a minisat: COROT. ESA SP-1306, pp. 33–37 (2006) Boothroyd, A.I., Sackmann, I.-J.: Our Sun. IV. The standard model and helioseismology: consequences of uncertainties in input physics and in observed solar parameters. Astrophys. J. 583, 1004–1023 (2003) Brassard, P., Charpinet, S.: PULSE: A finite element code for solving adiabatic nonradial pulsation equations. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9733-z Brassard, P., Fontaine, G., Wesemael, F., Hansen, C.J.: Adiabatic properties of pulsating DA white dwarfs. II—Mode trapping in compositionally stratified models. Astrophys. J. Suppl. Ser. 80, 369– 401 (1992) Christensen-Dalsgaard, J.: ADIPLS—the Aarhus adiabatic oscillation package. Astrophys. Space Sci. (2007a). doi:10.1007/s10509007-9689-z Christensen-Dalsgaard, J.: ASTEC—the Aarhus STellar Evolution Code. Astrophys. Space Sci. (2007b). doi:10.1007/s10509-0079675-5 Christensen-Dalsgaard, J., Mullan, D.J.: Accurate frequencies of polytropic models. Mon. Not. R. Astron. Soc. 270, 921–935 (1994)

Astrophys Space Sci (2008) 316: 231–249 Christensen-Dalsgaard, J., Di Mauro, M.P., Schlattl, H., Weiss, A.: On helioseismic tests of basic physics. Mon. Not. R. Astron. Soc. 356, 587–595 (2005) Christensen-Dalsgaard, J., Arentoft, T., Brown, T.M., Gilliland, R.L., Kjeldsen, H., Borucki, W.J., Koch, D.: Asteroseismology with the Kepler mission. In: Handler, G., Houdek, G. (eds.) Proc. Vienna Workshop on the Future of Asteroseismology. CoAst, vol. 150, pp. 350–356 (2007) Fixler, J.B., Foster, G.T., McGuirk, J.M., Kasevich, M.A.: Atom interferometer measurement of the Newtonian constant of gravity. Science 315, 74–77 (2007) Gough, D.O.: Comments on helioseismic inference. In: Osaki, Y., Shibahashi, H. (eds.) Progress of Seismology of the Sun and Stars. Lecture Notes in Physics, vol. 367, pp. 283–318. Springer, Berlin (1990) Houdek, G., Gough, D.O.: An asteroseismic signature of helium ionization. Mon. Not. R. Astron. Soc. 375, 861–880 (2007) Monteiro, M.J.P.F.G.: POSC—The Porto Oscillations Code. Astrophys. Space Sci. (2008). doi:10.1007/s10509-008-9802-y Monteiro, M.J.P.F.G., Thompson, M.J.: Seismic analysis of the second ionization region of helium in the Sun—I. Sensitivity study and methodology. Mon. Not. R. Astron. Soc. 361, 1187–1196 (2005)

249 Morel, P., Lebreton, Y.: CESAM: a free code for stellar evolution calculations. Astrophys. Space Sci. (2007). doi:10.1007/s10509-0079663-9 Moya, A., Garrido, R.: Granada oscillation code (G RAC O). Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9694-2 Provost, J.: NOSC: Nice oscillations code. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9654-x Rogers, F.J., Swenson, F.J., Iglesias, C.A.: OPAL equation-of-state tables for astrophysical applications. Astrophys. J. 456, 902–908 (1996) Roxburgh, I.W.: The OSCROX stellar oscillation code. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9607-4 Scuflaire, R., Montalbán, J., Théado, S., et al.: CLÉS, Code Liégeois d’Évolution Stellaire. Astrophys. Space Sci. (2007). doi:10.1007/ s10509-007-9650-1 Shibahashi, H., Osaki, Y.: Theoretical eigenfrequencies of solar oscillations of low harmonic degree l in five-minute range. Publ. Astron. Soc. Jpn. 33, 713–719 (1981) Suarez, J.C., Goupil, M.J.: FILOU oscillation code. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9568-7 Suran, M.: LNAWENR—linear nonadiabatic nonradial waves. Astrophys. Space Sci. (2007). doi:10.1007/s10509-007-9714-2

Balance and future Concluding remarks and perspectives on the evolution and seismic tools activity (ESTA) of the CoRoT community Marie Jo Goupil

Originally published in the journal Astrophysics and Space Science, Volume 316, Nos 1–4. DOI: 10.1007/s10509-008-9840-5 © Springer Science+Business Media B.V. 2008

Abstract A first part gives a summary of the main conclusions stemmed from the different comparisons reported in this book. One main conclusion is: with some care in the computations, one is able to obtain frequency differences between different computations which are smaller than required by Corot challenges. This is true for high frequency modes. Small frequency differences are much less easily obtained for low frequencies around the fundamental radial mode frequency. Care here means: having the same physics, same constants of physics, same input stellar parameters, computing models and oscillations with enough self numerical consistent accuracy. The ESTA group has built reference grids of models and associated oscillation frequencies and made them available to the community. In a second part of the present paper, a study case is considered in order to show the need and the use of such reference grids. Finally some perspectives concerning the remaining tasks are suggested. Keywords Stars: evolution · Stars: interiors · Stars: oscillations · Methods: numerical PACS 97.10.Cv · 97.10.Sj · 95.75.Pq

1 Introduction The first seismic results of the Corot mission show that the mission is nominally operating; we can therefore expect measurements of frequencies with an accuracy of 0.1 µHz M.J. Goupil () Observatoire de Paris, LESIA, CNRS UMR 8109, 5 Place Janssen, 92195 Meudon, France e-mail: [email protected]

for individual frequencies for coherent modes and 0.5 µHz for stochastically excited modes—at least over some specific frequency ranges and for the brightest target stars observed over a long run of 150 days. Even if the data do not reach this accuracy, we can remain conservative and take these values as maximum acceptable differences between frequencies obtained by different groups for the ESTA comparisons. Differences between observed and theoretical frequencies can have several origins, some unphysical such as due to numerical problems or real such as due to missing physics. Of course, the last ones are those of interest. Differences larger than 0.1–0.5 µHz due to numerical inaccuracy or physical inconsistencies in the calculations must then be removed, or at least their origins identified. When ESTA started, frequency differences for stellar models other than the Sun and computed using different stellar evolution codes and different oscillation codes were known to be significantly larger than the above figures (Monteiro et al. 2006). The underlying goal of this working group was then to insure that the results obtained from comparisons between theoretical frequencies and observed ones using different codes are compatible and do not lead to different conclusions about the missing physics because they used different input physics. To reach these goals, the main tasks of ESTA had then been – to identify the causes of the differences – to eliminate those which can be eliminated – to quantify those which cannot be eliminated so that these can be taken into account in any further interpretation of the differences between observed data and theoretical calculations

M.J.P.F.G. Monteiro (ed.), Evolution and Seismic Tools for Stellar Astrophysics. DOI: 10.1007/978-1-4020-9440-8_28

251

252

Model and frequency calculations and comparisons have been carried out by several groups and involved a large number of people. This tremendous amount of work has been rewarding as the results are quite informative as indicated by the papers included in this book. Several conclusions of these comparison works have been drawn and are reported in papers of the present book; hence Sect. 2 only presents a brief and synthetic summary of the main conclusions. Section 3 investigates a study case in order to illustrate the need and the use for reference grids of models as those which have been built by the ESTA group and which can be found on the web site: http://www.astro.up.pt/corot/models. Finally Sect. 4 suggests a few additional tasks to be performed.

2 Summary of conclusions Comparisons using the same simplified input physics have been quite useful in that they have indeed led to improvements of all codes: each code identified several inconsistencies and removed them. Despite these corrections and improvements in the physics implementation and efforts of homogenisation (same input stellar global parameters, same values for the constants and same simplified input physics), there is still room for sources of differences as the numerical implementation can be different; this is the case for instance for the type of interpolation and smoothing routines used for computing the opacities as discussed in Montalbán et al. (2008b). However differencies in the output global parameters and the ρ, T , 1 , cs stratifications of the models are quite small, leading to frequency differences well below the CoRoT acceptable limit. This is true for low mass, unevolved (main sequence) stars, even including microscopic diffusion. This indicates that using different numerical and/or interpolation schema for the microphysics do not significantly matter as long as a sufficient intrinsic precision is maintained. The location of the base of the upper convection region for low mass main sequence stars, for instance, using solar like oscillations is correctly determined by the various codes within a few 0.1%. Differences as large or larger than 0.5 to 1 µHz for solar like oscillation frequencies, i.e. larger than acceptable, are nevertheless found in some cases; they are identified as due to surface effects and can be partially removed by an appropriate scaling. On the other hand, significant frequency differencies (larger than acceptable) arise when some physics is missing or poorly modelled. The structure then is very sensitive to the adopted physical description and numerical choices. This happens for the transition between radiative and convective layers and/or in the regions of chemical gradients; this corresponds to low mass evolved models and

Astrophys Space Sci (2008) 316: 251–261

low frequencies (mixed modes and g modes) and results in frequency differences of 0.5 up to 4 µHz. These differences arise from nonhomologeous effects which unfortunately cannot be easily removed by some simple rescaling. Turning to comparisons carried out in Task 2 between results of different oscillation codes for the same input stellar model, it has been confirmed, if necessary, that Richardson extrapolation for the frequency should be used, that the number of shells in the model must be large enough even for unevolved stellar models that-is 700 layers for instance is not sufficient, at least 2000 layers are necessary. It was also suggested that using Lagrangian perturbation variables instead of Eulerian ones for low mass stars can help remove effects of a numerically noisy Väissälä frequency as this one no longer directly appears in the oscillation equations. This must be viewed as a possible help for computing more accurate frequencies; of course that does not remove the problem if the perturbation in the Väissälä frequency comes from a physical inconsistency in the stellar model rather than from numerical noise in the computation. Seismology has shown, here again, its usefulness as the main outcome of comparison of frequencies in Task 2 has been to bring into light several physical inconsistencies in the stellar models. These works have confirmed that differences between results of different evolution and oscillation codes, even reduced as much as possible, are unavoidable and in somes cases are not negligible. Therefore in the process of interpreting seismic CoRoT data with its own tools, it is strongly advisable to establish some sort of calibration, that is to compare its own personal stellar models and associated oscillation frequencies with models stemmed from the ESTA grids of reference. These reference models do not necessarily use the most updated physics, but use a basic physical description which is widely adopted as acceptable and available to most groups. These calibrations could serve as a basis for discussions in case of contradictory interpretations of data by different groups. Use of these reference models is illustrated next with a study case.

3 A study case Except otherwise stated, stellar models in this study case are built using CESAM2k1 stellar evolution code (Morel 1997; Morel and Lebreton 2008) in its V3.2.0 version. The models have been initialised on the zero age main sequence. Whenever possible, they have been evolved with the included ‘co’ (for CoRoT) precision i.e. with typically 1000 mesh points in the interior while the atmosphere is restored on a grid of typically 100 grid points. The resulting spline solutions 1 Code

d’Evolution Stellaire Adaptatif et Modulaire.

Astrophys Space Sci (2008) 316: 251–261

for the models selected for computing oscillation frequencies have then been used to add 2 intermediate points between any two original shells so that the model is given about 3100 layers. Oscillation frequencies for these models are Richardson frequencies computed with the adiabatic oscillation code ADIPLS (Christensen-Dalsgaard 2008). Two types of intrinsic verification are performed on individual models: – Verification of the numerical consistency in the computation of frequencies given a stellar model. This is done by comparing frequencies of one model with increasing number of shells: frequency differences between a model with 2000 layers and a model with 3000 layers are found less than 0.1 µHz over the whole interval (0–2100 mHz) of interest here. – Verification of the numerical consistency of a model with a given oscillation code. One recalls that the frequency differences DRich,var ≡ νRicharson − νvariational (both types of frequency made available by ADIPLS), are good indicators of the numerical consistency of a stellar model as they quantify the precision with which stellar equations are satisfied by the model. 3.1 The procedure First, a model for a proxy star is built which oscillation frequencies will be considered as the data. The subsequent work is done in several steps: One model, hereafter named E1, is computed using the simplified physics as specified in Task 1 ESTA comparisons (Lebreton et al. 2008a, 2008b) but with the present version of Cesam code. This model is chosen so as to match the location of the proxy star in the HR diagram as well as the large frequency separation. Because the ESTA physics is simplified on purpose, frequency differences between the ‘observed’ and theoretical frequencies are likely to exist and one does not expect that the differences D1 = |νproxy − νE1 | be small. Indeed the game is to implement improved physics and see the differences |νproxy − νmodel | decrease. Therefore a second model with a more sophisticated (assumed as an improvement!) input physics is built, hereafter referred to as M1 model. If the above differences D1 happen to be smaller than D = |νproxy − νM1 |, one must conclude that the added physical sophistication is not correct either in the physics or in its implementation. In fact, before being able to attribute the above frequency differences to some missing or incorrect physics, one must check that no contribution is due to some numerical inconsistencies. The minimum requirement is that it be done for the simplified basic input physics. Hence one must perform some sort of calibration, this can be done by comparing

253 Table 1 Global parameters of stellar models used in the comparisons. The columns respectively provide the model name, model mass (in solar units, logarithm of the luminosity (in solar unit) and effective temperature, stellar radius (in solar unit). The last column gives the large separation averaged over the frequency range [0.8–2.1] mHz. The constants held fixed and used in the calculations are GM = 1.32712438 × 1026 , M = 1.98919 × 1033 , G = 6.67168234 × 10−8 ; R = 6.9599 × 1010 , L = 3.846 × 1033 (in cgs units) Model

Mass

log L/L

log Teff

R/R

ν (in µHz)

Proxy

1.35

0.584

3.8165

1.522

82

M1

1.35

0.585

3.8160

1.527

82

E1

1.35

0.586

3.8162

1.529

82

E2

1.30

0.5292

3.8088

1.4826

86

C

1.30

0.5299

3.8087

1.4829

86

the closest reference ESTA model, hereafter C model with a corresponding model, hereafter E2 model, computed as the investigated ones M1 and E1. The reference model, C model, is found on the ESTA web site. This does not mean, of course, that the reference models are better but they can help to understand possible different interpretations of the data and the associated seismic diagnostics obtained by different groups. A criterium can be that D2 = |νE2 − νC | must be smaller than D = |νproxy − νM1 | in order to be able to attribute the differences D to a real default in the physics and not to a bug or imprecision in the numerics of the part due to the simplified physics. 3.2 The ‘data’ The proxy star model is chosen to be a 1.35 M model on the main sequence. Its global stellar parameters can be found in Table 1 which provides mass, logarithm of the luminosity log(L/L ), logarithm of the effective temperature log Teff , the stellar radius and the large separation (1) averaged over the frequency interval [0.8–2.1] mHz. Location of this model in a HR diagram is displayed in Fig. 1. Oscillation frequencies are computed for  = 0, 3. Fig. 2 shows some characteristics of the oscillation frequencies in the range [0.5–2.1] mHz for the proxy star model. Frequencies of the ‘best model’ M1 must reproduce those frequencies as closely as possible. First a diagram echelle is obtained for a large separation of 82±0.5 µHz. Then frequency differences, as defined below, are plotted as a function of the frequency: (a) the large separation ,n = ν,n − ν,n−1

(1)

with , n respectively the degree and the radial order of the mode

254

Astrophys Space Sci (2008) 316: 251–261

Fig. 1 Evolutionary tracks in a HR diagram for models corresponding to the proxy star, M1, E1. Crosses indicate the location of the selected models: black solid line: proxy; red dotted line: M1; blue dashed line: E1

(b) the small separation D,+2 = ν,n − ν+2,n−1 (c) small differences δ01 = ν0,n −

ν1,n + ν1,n−1 2

and δ10 = −ν1,n +

ν0,n + ν0,n−1 2

(d) the second difference δ2 = ν,n − 2ν,n+1 + ν,n+2 and finally the ratios r02 =

ν0,n − ν2,n−1 0,n

Fig. 2 Seismic properties for the proxy star model: top panel: echelle diagram; bottom panels: large (upper left), small (upper right) frequency separations; small (down left) and second (down right) frequency differences; solid line:  = 0; dotted lines:  = 1, 2, 3

r13 =

ν1,n − ν3,n−1 1,n

3.3 ‘Modelling’ the proxy star

The ratios r,+2 have been shown to remove surface effects (Roxburgh and Vorontsov 2003). The ratios r02 and r13 are plotted in Fig. 3 for the three models proxy, E1 and M1. It is not the purpose here to discuss the physical origins of these characteristics (they have indeed been extensively discussed in many papers and reviews over the years), the aim here is rather to evaluate how closely M1 and E1 models can reproduce these features.

First a model satisfying the simplified physics ESTA specification is computed, E1. I next consider as a ‘more’ sophisticated physics core overshooting and convective penetration below an upper convective region (as described in Zahn 1991) and microscopic diffusion in order to build the M1 model. In computing the E1 and M1 models, I simplify the problem and hold fixed the metal relative abundance Z = 0.02 and consider no observational uncertainties

Astrophys Space Sci (2008) 316: 251–261

255 Table 2 Input physics and parameters entering the models investigated here. When non specified, input physics is the same for all models. The columns respectively provide the model name, the convection approach and the associated characteristics length parameter, the core overshoot parameter, the parameter for penetrative convection below a convective region (all these three parameters in units of Hp pressure scale height), inclusion or not of microscopic diffusion, the optical depth at the level matching the interior and the atmosphere Model

Convection

αov sup

αov inf

Mic. diff.

τ

Proxy

CGM

0

0.08

Yes

20

0.15

0

Yes

10

0

0

No

10

0

0

No

10

0

0

No

–

αCGM = 0.7 M1

MLT no optth αMLT = 1.76

E1

MLT αMLT = 1.6

E2

MLT αMLT = 1.6

C Fig. 3 Ratios r02 and r13 (see text for definitions) are plotted versus frequency in mHz for the proxy star model (black lines), model E1 (blue lines) and model M1 (red lines); dashed lines represent r13 and solid lines r02

on the location of the star in the HR diagram. A same location in the HR diagram imposes the same radius. Models M1 and E1 are then calculated such that they are quite close to the proxy star model in the HR diagram. I also assume an exact determination of the large separation, then the mass is indeed imposed (given the chemical composition and the physics). The remaining input free parameters are the initial helium relative abundance Y0 and parameters for the convection. The mass is then held fixed at M = 1.35 M . The initial hydrogen and helium abundances X0 , Y0 had then to be changed for E1 to satisfy the above constrains. For sake of simplicity, all models use the OPAL 2001 equation of state and opacity tables as described in Lebreton et al. (2008a). All models were calculated with the classical GN93 solar mixture of heavy elements from Grevesse and Noels (1993). E1 uses the classical mixing length treatment of Böhm-Vitense (1958) under the formulation of Henyey et al. (1965) taking into account the optical thickness of convective eddies. Eddington’s grey T (τ ) law is used for the atmosphere calculation. Unless otherwise stated, the connection with the envelope is made at the optical depth τ = 10 to ensure the validity of the diffusion approximation (Morel et al. 1994). Although fully unrealistic, this study case is enough for the present illustrative purpose. Taking additional uncertainties into account would make the results even less easily tractable. Differences in physical input between the three models are summarized in Table 2. The proxy and M1 models in-

MLT αMLT = 1.6

clude microscopic diffusion whereas E1 does not. Convection in the proxy star model is computed using the CGM (Canuto et al. 1996) approach for convection whereas the classical MLT is used for M1 and E1. The value of the characteristic length for the CGM approach, αCMT is taken as the value which reproduces the present Sun. E1 model takes into account the optical thickness of the eddies whereas M1 does not. The proxy star model includes convective penetration below the base of the upper convective zone (UCZ) by an amount of αov inf Hp as some penetration in the radiative layers below the UCZ is expected (although perhaps quite small as derived from helioseismology). In order for M1 model to match the same coercion stellar parameters than the proxy star, one possibility is to include core overshoot in the modelling of M1: the convective core is then extended over an overshoot distance αov sup × min(Hp , Rcore ) with an amount set to αov sup = 0.15, Hp the pressure scale height and Rcore the radius of the convective core. Indeed MLT convection with αMLT = 1.76 induces a lower luminosity than CGM convection with αCGM = 0.7, everything else held the same. Figure 1 displays the evolutionary sequences in the H–R diagram corresponding to the proxy star, M1 and E1 models. Stellar parameters which are taken as input constrains for M1 and E1 models are listed in Table 1 and output stellar characteristics are given in Table 3. As the proxy and M1 include microscopic diffusion, relative hydrogen and helium abundances at the surface are given in Table 4 together with their initial values.

256

Astrophys Space Sci (2008) 316: 251–261

Table 3 Stellar parameters for models used in the present comparisons. Columns respectively give the central hydrogen relative abundance, the age of the model in Myr, the radius at the base of the upper convective region, the radius and the mass of the convective core. Radii RbZC , Rcore and the core mass, Mcore , are given in units of total stellar radius and total stellar mass respectively Model

Age

Xc

RbZC

Rcore

Mcore

0.876

7.04 × 10−2

6.54 × 10−2 5.02 × 10−2

(Myr) Proxy

0.384

1743.55

M1

0.471

1728.86

0.894

6.54 × 10−2

E1

0.390

1636.91

0.905

6.92 × 10−2

6.28 × 10−2

E2

0.350

1986.1

0.880

6.54 × 10−2

5.48 × 10−2

C

0.348

2056.8

0.879

6.57 × 10−2

5.57 × 10−2

Table 4 Relative abundances of hydrogen and helium: initial (X0 , Y0 ) and surface values (Xsurf , Ysurf ) Model

X0

Y0

Xsurf

Ysurf

Proxy

0.706

0.274

0.830

0.158

M1

0.706

0.274

0.866

0.125

E1

0.702

0.278

0.702

0.278

E2

0.70

0.28

0.70

0.28

C

0.70

0.28

0.70

0.28

3.4 Seismic diagnostics Figures 3, 4 and 5 show oscillation frequency characteristics in the range [0.5–2.1] mHz in a similar display than for the proxy star model (Fig. 2): an echelle diagram and frequency differences, as defined in (1) and below. That the input physics differ both for E1 and M1 compared with that of the proxy star is not obvious from the echelle diagram nor from the small separation D,+2 . On the other hand, the other seismic indicators do give hints that E1 is quite different from the proxy star model. Features seen for the M1 model appear more similar to those of the proxy star. This tends to indicate that microscopic diffusion is necessary to model the proxy star (which here, as it is a study case, we know is true). It would also tend to indicate that core overshooting is necessary (which we know here is not correct!). Of course, a deeper and quantitative seismic study would be required in real life but is not the objective here. 3.5 Frequency comparisons Figures 6 and 7 show frequency differences D1 = |νproxy − νE1 | and D = |νproxy − νM1 | in µHz respectively between the proxy and E1 models and the proxy and M1 models in function of frequency in mHz. Differences are large amounting up to 20 and 30 µHz at high frequency respectively for

Fig. 4 As Fig. 2 but for E1 model

D and D1. The monotonic increase for D and D1 with frequency tends to indicate that these differences are due to surface effects. In order to remove these surface effects, another indicator is built using some sort of scaling. One indeed defines νproxy,,n νE1,,n − proxy,,n E1,,n

(2)

and νproxy,,n νM1,,n − proxy,,n M1,,n

(3)

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Fig. 6 Top panel: Frequency differences in µHz between the proxy and E1 models in function of the frequency in mHz. Black solid line:  = 0; red solid line:  = 1; blue solid line  = 2; blue dotted  = 3. Bottom panel: Scaled frequency differences. The frequencies ν,n,j for model j (j = 1, 2) are divided by the large separation ,n,j given by (1) and noted j in the plot for shortness

Fig. 5 As Fig. 2 but for M1 model

where ,n is the large separation defined in (1). These quantities are plotted in the bottom panels of Figs. 6 and 7. They tell that differences in deeper layers might exist between the proxy star and both E1 and M1. Differences between the proxy star and M1 models appear to be smaller than between the proxy star and E1 models. This suggests that the structure of M1 is closer to the real (i.e. to the proxy star) structure; this is also indicated by the seismic properties, as mentioned in the previous section. This is indeed confirmed in Sect. 3.7 where the knowledge of the proxy star structure in the present study case is used.

Fig. 7 Same as Fig. 6 but for the proxy and M1 models

3.6 Comparison with CLES model: ESTA ‘calibration’ In order to assess the quality of the present models compared to those which have been involved in the ESTA com-

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Fig. 8 Evolutionary tracks for 1.30 M corresponding to model E2 (red dotted line) and CLES model C (black solid line). The blue cross locates model E2 and the black cross locates model C

parison works, I chose a model in the reference grid of the CLES models as close as possible of the present models: the model is ‘m1.30Z0.02X0.7-MS.gong’ which will from now on be referred to as model C. This is a 1.30 M model on the main sequence (Xc = 0.35). CLES reference models are available on line through the Evolution and Seismic Tools Activity (ESTA) web site and described in Montalbán et al. (2008a). CLES reference model C has been computed with initial relative abundance values Y = 0.28, Z = 0.02 and αMLT, = 1.6 with neither microscopic diffusion nor core overshoot included. Oscillation frequencies of model C have been computed with either ADIPLS or LOSC, assuming a surface boundary condition δp = 0 according to ESTA specification. LOSC frequencies have been taken from the web site together with the model. A model E2 is computed with the same (as much as possible) input physics and stellar parameters with the present version of Cesam code; its frequencies are computed with ADIPLS. Stellar parameters for models E2 and C are given in Tables 1, 3, 4. Their differences in physical content are described in Table 2. Figure 8 displays the evolutionary sequences in the H–R diagram corresponding to the E2 and C models. Only small differences, below 1 µHz, result when frequencies computed with ADIPLS and LOSC for model C are compared for all  as seen in Fig. 9. The largest differences arise for  = 0. Part can perhaps be attributed to a slightly different treatment of the oscillation central boundary conditions between both codes or a different way of computing radial modes. Differences become quite negligible in scaled differences except at low frequencies, near

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Fig. 9 Differences between frequencies computed with ADIPLS and LOSC for C model

the fundamental radial mode. These differences are smaller than the ones expected from CoRoT uncertainties on these quantities which are of the order of 1 µHz (0.1%) for the differences and 0.15 (1%) for the scaled differences (2) and (3). Frequency differences (ν,n,E2,ADIPLS − ν,n,C,LOSC ) between E2 and C models are shown in Fig. 10: they range between 0 and less than 4 µHz. They are less than 0.05 for the absolute values of scaled differences except at low frequency. This insures that E models computed with the present code are close to the ESTA specifications. Although most of the basic physical input is the same, no optimisation (on purpose) has been made here to satisfy exactly the ESTA specifications. Furthermore differences in nuclear reactions, in implementation and smoothing routines can be responsible for the remaining frequency differences as shown in Montalbán et al. (2008b). The unscaled differences (ν,n,E2,ADIPLS − ν,n,C,LOSC ) could have been further decreased by choosing a sligthly more evolved E model. In any case, both types of differences are much less than corresponding differences found between E1 and the proxy star model. This is enough to conclude that seismic diagnostics such as those shown in Figs. 3, 6, 7 are meaningful and further improvement compared with M models are necessary and must go in the sense of decreasing frequency differences. 3.7 Stellar structures In real life, one must not expect to have direct access to precise sound speed, 1 and Väissälä frequency profiles for the proxy star but I show them here for discussion.

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Fig. 10 Frequency differences for model E2 (blue line) and CLES model C (black line). Model E2 frequencies have been computed with ADIPLS whereas frequencies of model C have been computed with LOSC

259

Fig. 11 Comparison of the structures of model M1 (black) and E1 (blue) with that of the proxy star top panel: relative sound speed difference; middle panel: relative adiabatic exponent difference; bottom panel: A difference (see text). The ordinate is the logarithm of temperature

Figure 11 shows the relative differences for the sound speed, cs and the adiabatic exponent 1 between the proxy star and M1 models on one hand and the proxy star and E1 models on the other hand. Also displayed are differences δA where A is related to the Väissälä frequency and is defined as 1 d ln p d ln ρ − 1 d ln r d ln r One can see that globally the structure of M1 is closer than that of E1 to the structure of the proxy star, which confirms conclusions which were drawn from the seismic diagnostics in previous sections. On the other hand, a look at Table 3 shows that conclusions concerning the size of the convective core as well as the location of the base of the upper convective region are not so clear. This is not surprising since M1 includes ‘improved’ physics such as microscopic diffusion compared to E1 which does not; on the other hand M1 includes additional physics with respect to E1 such as core overshooting but which is not ‘correct’ (i.e. not included in the proxy star model). This shows the limitations of a crude seismic analysis as done here. Figure 12 shows differences between E2 and C models which are generally no larger than 0.5% for the sound speed, reach at maximum 0.02% for 1 . Differences in |AE2 − AC | are smaller than 0.05 which is a factor almost 40 smaller than for |AE2 − AC | in Fig. 11.

Fig. 12 Same as Fig. 11 but for the structures of model E2 (blue line) and CLES model C (black line). Note the scales about 1 order of magnitude smaller than in Fig. 11

4 Perspective: remaining issues 4.1 Stellar structure After completion of ESTA tasks 1 to 3, a large number of sources of significant numerical inconsistencies have been

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removed. Attention had been drawn for instance upon the necessity of quite consistent calculations (either tabulated or from analytical formulae) of thermodynamical quantities. Another example is the depth at which different opacity tables are matched, it was indeed found by Lebreton et al. (2008a) and also in CESAM-CLES comparisons by Montalbán et al. (2008b) that different matches of opacity tables can introduce differences up to order of 5% in the sound speed for instance. Another issue with a more subtle effect has arisen which is that one must pay attention to the level at which the integration of hydrostatic equilibrium starts as it does indeed differ from one code to another (Montalbán et al. 2008b). On the other hand, sources of important numerical discrepancies have been quantified but they are not easy to remove such as transitions between radiative and convective regions. As these are poorly modelled, they are sensitive to many features, number of shells, smoothing and so on (Lebreton et al. 2008b). It would nevertheless be certainly valuable if some recipe (even if very crude) could be made available so that it can be used in future model calibrations. This is specially needed for core mixing and/or overshooting. Other sources of differences, although likely of small effects, have not been quantified such as the depth at which the atmosphere is matched to the interior solution. Implementation of the nuclear reactions NACRE has not yet been scrutinized in details and assumed to be the same as well as the electron screening factor. Small also probably are the differences arising from the fact that some codes follow the early nuclear combustion of light elements and some other codes do not. It would be valuable however to quantify the impact of these differences. Although the simplified ESTA physics specified the use of the MLT for convection, there are several parameters entering this formulation which are fixed quite arbitrarily and might not be the same from one code to another. For instance, is the Schwarzschild criterium applied the same way in all codes? Is the optical thickness of the eddies taken into account the same way in all codes? If not, are the consequences of these differences significant for CoRoT seismology? As the MLT is a very crude description of the real process, it is easy to get hidden differences. This is a good example of a multiparameter modelling for which one cannot tell which of different possible choices of parameters must be best taken. At best one can only be aware of the effects of different choices. Altogether however, the ESTA comparison work has shown that with the use of specified simplified input physics, differences in the structure and frequencies of stellar models can be made small enough to cope with the CoRoT measurement accuracy. Important frequency differences (observedcalculated) between different codes are then expected only when new physics is added. Seismological studies with Corot targets, i.e. real life!, will indeed naturally lead us to

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include a more sophisticated physical description that the one included in the grids in order to decrease differences with observed frequencies. This ought to give rise to improved stellar models and consequently one foresees the necessity for the grids of reference stellar models to evolve along. One future objective could then be to build a new generation of grids of reference stellar models including more realistic basic physics but still simplified enough that this remains accessible to most codes. It might then be necessary at some level in the future to compare results of implementation of improved physics such as, to name a few: Surface effects Including a more sophisticated model atmosphere and T (τ ) law; for convection, in order of implementation complexity, switching to a local approach taking into account more properly the energy spectrum of turbulent eddies such as the CGM approach; including turbulent pressure; for a nonlocal description, using what can be learned from 3D convection simulations under the form of patched models for instance. In this last case, comparisons between simulations of the nowadays existing 3D codes and associated patched models would be valuable. Chemical mixing and transport of angular momentum Several codes have now been implemented with transport of angular momentum and rotationally induced transport of chemical elements. Comparing results stemmed from these different codes will be valuable for all involved codes. In some instances, even the underlying physical mechanism is not the same (transport by diffusion or advection). In other codes, the description relies upon the same mechanism but they use different prescriptions for turbulent diffusion coefficients. Note that such comparisons have naturally already started (STAREVOL/GENEVA codes; STAREVOL, CESAM2k codes). With comparisons between models including transport of angular momentum and their rotation profiles, a corollary comparison immediately arises concerning adopted prescriptions and implementations of loss of angular momentum through stellar winds. Comparisons can then be extended to more general mass loss for massive stars such as B stars for instance. Distorsion due to the centrifugal force for a rapid rotator is not yet widely implemented but one can foresee that in some future, comparison of ‘rotating’ models will be an important task. Semiconvection For low mass stars in presence of diffusion although some mixing removes it or for massive stars. Evolved stars It is important to plan comparisons of other types of stars with a more complex evolved structure such as giant stars. Some of them do indeed show promising small amplitude high frequency oscillations (Dziembowski et al.

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2001; Barban et al. 2007). A few giants are observed by CoRoT. This concerns shell burning, conductive opacities, cool atmospheres.

261 Acknowledgements I gratefully thank Yveline Lebreton, Pierre Morel and Josephina Montalban for their help in using Cesam code, the ESTA reference grid models and fruitful discussions. I also thank J. Christensen-Dalsgaard for making his oscillation code ADIPLS available to the community.

4.2 Oscillations For future comparisons of frequency computations, several important issues must be considered: Adapting the mesh distribution, depending which type of modes is computed (p or g modes, high or low frequency). This is important particularly when the modes are sensitive to regions where sharp variations of the structure occur. The largest differences exist for the low frequency part of power spectra. Then frequency differences are quite large and can not be removed by a simple scaling. A possible option at the present time would be to elaborate a standard routine which adds points strategically and is made available to the communauty (so that we all do the same—although incorrectmodelling in the comparisons allowing other sources of differences to be detected!). A routine for smoothing the Väissälä frequency, when using Eulerian perturbation variables, based on a simple recipe ought also to be made available to all codes involved in the comparisons. As pointed out by Moya et al. (2008), the next level of comparison is that of eigenfunctions which generally are less accurately computed than frequencies. It will soon also become necessary to compare frequencies corrected for second (possibly higher) order perturbations due to rotation as different approaches are being developed and implemented in several oscillation codes. In a more longer run, one will have to deal with comparisons of 2D calculations of fast rotating stars and their 2D frequency calculations. As mentioned for the stellar structure above, some attention will have to be paid to the computation of frequencies of evolved stars which is quite delicate particularly because of the central regions of the star. Another important step is the comparison of nonadiabatic codes. Several nonadiabatic oscillation codes have been developed with implementation of different formulations to model time dependent convection and its interaction with pulsation. Comparisons of the resulting damping rates and linear instability regimes, amplitude/phase ratios for mode identification, and intensity/velocity amplitude ratios are already strongly needed. As a final comment, it is worth to stress that one important outcome of the ESTA work is the description of stellar evolutionary and oscillation codes collected together in the same (present) book and therefore easily available to interested readers.

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