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142 Inertial modes in slowly rotating stars : An evolutionary description 4.7 Conclusion This article deals with a new hydrodynamical code based on spectral methods in spherical coordinates. This first version of the code was written to study inertial modes in slowly rotating NS, but it can easily be modified to include fast rotation. As it was explained in the Introduction, the slow rotation limit is a very good approximation for a first step. The present version of the code can be used to study both the Newtonian case and the relativistic case with the so-called strong Cowling approximation. Here, we have given the main algorithms it uses and shown some tests of the linear version. The way we overcame the numerical instability is also explained. Furthermore, in order to work with very different time scales, we have introduced an approximation to deal with mass (or baryonic number) conservation that proved itself quite robust and useful to study inertial modes : the anelastic approximation. Even if this approximation is not a necessary one and can be abandoned in further studies, it is very useful to have an idea of the main properties of the inertial modes in slowly rotating stars. The results presented in this paper were obtained in the linear case using very basic models of NS, such as the divergence-free case with a rigid crust (Vr |r=1= 0) or a γ = 2 polytrope with the anelastic approximation and free surface. Deeper studies in the GR framework with more quantitative results on the effects of EOS, of stratification of the star and of BC will follow. Despite the naive aspect of our studies, some common features appeared. Indeed, whatever the approximation (divergence-free or anelastic) with different BC (rigid crust for the divergence-free case and free surface for the anelastic approximation), we saw the appearance of a polar part in the mode, as soon as the background is Newtonian and differentially rotating, or relativistic and rigidly rotating (in the latter case, this phenomenon is due to the frame dragging). Whether in the case of noisy initial data with a radiation reaction force acting on them, or in the case of the free evolution of an initial linear r-mode, from the very beginning the energy of the polar part is at least 1 % of the energy of the axial part. This result is compatible with the analytical work by Lockitch et al. (2001) who proved, in the relativistic framework, the existence of a nonvanishing polar part of the velocity in a sequence of NS with a barotropic equation of state and decreasing rotation rate. Furthermore, another interesting result that we obtained is the “concentration of the motion” near the surface that quickly appears : after less than 15 periods of the linear r-mode in the calculations done up to now. Adding viscosity has shown that this is a robust and physical feature. Moreover, it seems to neither depend on the physical conditions or on the BC. This phenomenon is very similar to the results found by Karino et al. (2001) with an eigenvectors calculation. But, if this is really a singularity of the derivative of the velocity near the surface, as it seems to be in our time evolutions, it would mean that a mode calculation with finite difference schemes can only give quantitative results completely depending on the number of points. Indeed, with this scheme, there is an intrinsic numerical viscosity that depends on the resolution. Moreover, if this phenomenon is verified in future studies in GR, the stratification of the NS or the physics of the crust could be very important to know if inertial modes are relevant for
GW emission. 4.8 Acknowledgments 4.8 Acknowledgments 143 We would like to thank E. Gourgoulhon and B. Carter for carefully reading this paper. The numerous remarks of the anonymous referee were also very useful for us in trying to improve this article. L. Blanchet and J. Ruoff gave us the opportunity to have friendly and fruitful discussions. We would also like to thank the computer department of the observatory for the technical assistance.
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Ecole doctorale de Physique de la r
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Table des matières Résumé v Abst
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TABLE DES MATIÈRES iii 4.5.2 Noise
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Résumé Résumé v Cette étude tr
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Abstract Abstract vii This study de
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Introduction Les sens dont l’a do
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Introduction 3 même, leurs observa
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Chapitre 1 Relativité générale e
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1.1 Relativité générale 1.1.1 Gr
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1.1 Relativité générale 9 La rel
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1.1.3 Tests et prédictions 1.1 Rel
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1.2 Ondes gravitationnelles 1.2.1 E
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y 1.2 Ondes gravitationnelles 15 x
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1.3 Sources d’ondes gravitationne
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1.4 Détection 23 Figure 1.6 - Sens
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Chapitre 2 Etoiles à neutrons Somm
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2.1 Naissance d’une étoile à ne
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2.1 Naissance d’une étoile à ne
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astre Terre Soleil naine blanche é
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2.2 Structure interne 35 Figure 2.4
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2.2 Structure interne 37 extensions
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2.2 Structure interne 39 Figure 2.5
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2.2 Structure interne 41 rigidité
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2.2 Structure interne 43 où = h/2
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2.2 Structure interne 45 à des fer
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2.2 Structure interne 47 Type de su
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2.3 Principes de l’évolution d
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est environ 0.7 fois la masse nue 1
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Log 10 (T/10 9 K) 0 -0.5 -1 -1.5 -2
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2.4 Superfluidité et écarts à l
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aboutit à 2.4 Superfluidité et é
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Résultats 2.4 Superfluidité et é
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Chapitre 3 Oscillations stellaires
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3.1 Modes d’une étoile à neutro
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3.1.2 Perturbations 3.1 Modes d’u
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Un mode propre vérifie donc 3.1 Mo
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3.1 Modes d’une étoile à neutro
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3.2 Oscillations d’une étoile re
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3.2 Oscillations d’une étoile re
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Page 109 and 110: 3.3 Modes inertiels et r-modes 97 F
Page 111 and 112: P K /P 1.0 0.8 0.6 0.4 0.2 3.3 Mode
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Page 115 and 116: Chapitre 4 Inertial modes in slowly
Page 117 and 118: 4.1 Introduction 105 what is done w
Page 119 and 120: 4.2 Equations and numbers 107 obvio
Page 121 and 122: 4.2 Equations and numbers 109 tenso
Page 123 and 124: 4.2 Equations and numbers 111 4.2.2
Page 125 and 126: or 4.2 Equations and numbers 113 Tv
Page 127 and 128: 4.3 Mass conservation, boundary con
Page 129 and 130: 4.4 Test and calibration of the cod
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Page 133 and 134: Sp(V θ ) 1 4.4 Test and calibratio
Page 135 and 136: E(t) / E 0 1.15 1.1 1.05 1 4.4 Test
Page 137 and 138: Sp(S xx ) S xx 0.1 0 -0.1 4.4 Test
Page 139 and 140: V r Sp(V r ) 0.4 0.2 0 -0.2 -0.4 -0
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Page 143 and 144: Sp(V r ) V r 4 2 0 -2 4.5 Different
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Page 147 and 148: V θ Sp(V θ ) 10 5 0 -5 -10 4.6 St
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Page 151 and 152: E Polar / E Total 0.05 0.04 0.03 0.
Page 153: S xx Sp(S xx ) 0.2 0.1 0 -0.1 4.6 S
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Page 159 and 160: 5.1 Etoiles relativistes en rotatio
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Page 177 and 178: 5.3 Résultats numériques 165 Il e
Page 179 and 180: 5.4 Conclusions 167 Spectres de pui
Page 181 and 182: Conclusions et perspectives “Pour
Page 183 and 184: Remerciements 171 La liste est long
Page 185 and 186: Appendice A Géométrie différenti
Page 187 and 188: A.3 Métrique et variétés (pseudo
Page 189 and 190: Appendice B Spectral methods and ve
Page 191 and 192: B.1 Spirit of the spectral methods
Page 193 and 194: B.1 Spirit of the spectral methods
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Page 199 and 200: Liste des tableaux Etoiles à neutr
Page 201 and 202: Table des figures Relativité gén
Page 203 and 204: TABLE DES FIGURES 191 4.14 Time evo
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Bibliographie Akmal, A., Pandharipa
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Bibliographie 195 Bonnor, W. B., Sp
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Bibliographie 197 Glendenning, N. K
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Bibliographie 199 Kokkotas, K. D.,
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Bibliographie 201 Murray, S. S., Sl
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Bibliographie 203 Ruoff, J., Stavri
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Cette étude traite de différents
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