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186 Spectral methods and vectorial equations B.2.3 Boundary conditions It was already explained how to impose BC for the divergence-free case. For the system of Equations (B.20) or the Equation (B.24), it is worth distinguishing the two cases : either the enthalpy vanishes or does not vanish at the boundary of the integration domain. Here we will only consider the case of EE. Indeed, for NSE we chose to take a viscosity that vanishes at the surface to avoid the need of further BC. If H0 |r=1 > 0, the solution is quite easy : the Equation (B.24) or the system of Equations (B.20) admit a homogeneous solution that can be used to satisfy one BC. If H0(1) vanishes, the system of Equations (B.20) or the Equation (B.24) are degenerate and therefore no BC can be imposed. But we shall show that, in this case, the correct BC are automatically satisfied. Indeed, consider first the case of the linearized exact system of Equations (B.20). On the surface of the nonperturbed star, we have ∂th + W · ∇H0 which is the correct BC. The surface H0 + h = 0 then defines the profile of the perturbed star. To show that the solution in the anelastic approximation also satisfies the BC, it is more convenient to first examine the nonlinear case. Here, h is not infinitesimal, and the Equation (B.23) must be replaced with the equation Γ(H0 + h) div W + W · ∇(H0 + h) = 0 (B.26) that is satisfied if h is a solution of the nonlinearized Equation (B.24) : Γ (H0 + h) ∆h − div F j+1/2 + ( ∇h − F j+1/2 ) · ∇(H0 + h) = 0. (B.27) Once again, the surface of the star is defined by H0 + h = 0. But from the Equation (B.26) we have the correct BC : W · ∇(H0 + h) |H0+h=0 = 0. The surface of the star is then an unknown quantity that is determined by the relation H0 + h = 0 in solving the Equation (B.27) by iteration. This technique was developed in a different context in the already quoted paper Gourgoulhon et al. (2001). In the linear case, we have Wr = 0 at r = 1 [cf. Equation (B.23)], and the enthalpy does not vanish on the nonperturbed surface of the star. But once again the free surface of the star can be obtained by looking for the position where the total enthalpy H0 + h = 0 vanishes. This surface coincides within first order quantities with the nonperturbed surface r = 1. Note that if γ > 0 the pressure P |r=1 ∝ h γ+1 |r=1 vanishes within terms o(h).
Liste des tableaux Etoiles à neutrons 2.1 Caractéristiques typiques de divers objets astrophysiques. . . . . . . . . . . 33 2.2 Description sommaire des trois types de superfluidité considérés. . . . . . . 47 Inertial modes in slowly rotating stars : An evolutionary description 4.1 Characteristic numbers implied in the dynamics of inertial modes of rotating NS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2 Typical values of the different time scales implied in the dynamics of inertial modes of rotating NS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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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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Ω c /Ω max 1.00 0.98 0.96 0.94
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3.3 Modes inertiels et r-modes 97 F
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P K /P 1.0 0.8 0.6 0.4 0.2 3.3 Mode
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3.3 Modes inertiels et r-modes 101
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Chapitre 4 Inertial modes in slowly
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4.1 Introduction 105 what is done w
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4.2 Equations and numbers 107 obvio
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4.2 Equations and numbers 109 tenso
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4.2 Equations and numbers 111 4.2.2
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or 4.2 Equations and numbers 113 Tv
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4.3 Mass conservation, boundary con
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4.4 Test and calibration of the cod
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4.4 Test and calibration of the cod
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Sp(V θ ) 1 4.4 Test and calibratio
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E(t) / E 0 1.15 1.1 1.05 1 4.4 Test
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Sp(S xx ) S xx 0.1 0 -0.1 4.4 Test
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V r Sp(V r ) 0.4 0.2 0 -0.2 -0.4 -0
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4.5.1 Modifications 4.5 Differentia
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Sp(V r ) V r 4 2 0 -2 4.5 Different
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4.6 Strong Cowling approximation in
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Page 155 and 156: GW emission. 4.8 Acknowledgments 4.
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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
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Page 195 and 196: B.2 Solving Euler or Navier-Stokes
Page 197: The divergence-free approximation B
Page 201 and 202: Table des figures Relativité gén
Page 203 and 204: TABLE DES FIGURES 191 4.14 Time evo
Page 205 and 206: Bibliographie Akmal, A., Pandharipa
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Page 209 and 210: Bibliographie 197 Glendenning, N. K
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Page 213 and 214: Bibliographie 201 Murray, S. S., Sl
Page 215 and 216: Bibliographie 203 Ruoff, J., Stavri
Page 218: Cette étude traite de différents
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