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128 Inertial modes in slowly rotating stars : An evolutionary description S xx Sp(S xx ) 6 4 2 0 -2 -4 -6 0 50 100 150 t Ω 1.5 1 0.5 0 0 2 4 6 8 10 3ω / Ω Figure 4.13 – Time evolution of one of the two independent components of the Sij[t] tensor corresponding to the same run as the results in Figures 4.11 and 4.12. Here can really be seen the fact that there is only one unstable mode with the frequency of the linear r-mode and that the anelastic approximation does not change this feature. background due to frame dragging. Yet, since the main reason for this difference is the modification of the NSE and the possible appearance of what is called a continuous spectrum [see Ruoff & Kokkotas (2001), (2002) and Beyer & Kokkotas (1999)], the same may append even in the Newtonian case if the star is not rigidly but differentially rotating. And there are several reasons why a NS may not be in rigid rotation : first, the birth conditions of the NS themselves ; second, the nonlinear coupling of modes ; then, a possible drift induced by the existence of a magnetic field. As we are here only dealing with linear hydrodynamics and tests of the code, we will not give more details about those processes [but will send the reader to the following articles for more details : Spruit (1999), Rezzolla et al. (2000), (2001a) and (2001b), and Schenk et al. (2002)]. What we have done is just to take as a given that the background star is differentially rotating with quite an arbitrary law and to look at the influence of this law on the existence of the modes. Nevertheless, instead of asking the question “Is there any r-mode left in a differentially rotating NS ?” that is not well defined, we decided to try to answer to two different and more precise questions : - is there anything growing when an RR force is applied on noise in a differentially rotating background ? - what does happen to the linear r-modes if they are chosen to be the initial data in such kind of background ? Some lights on these questions are in the following subsections, but we shall begin with some words about the modifications implied on the equations by differential rotation.
4.5.1 Modifications 4.5 Differential rotation 129 Assuming differential rotation of the background star involves slight modifications of what we said in the Section 4.2. First, terms coming from the spatial derivatives of the non constant Ω[ξ, ϑ, ϕ] must be added in the NSE. Second, we shall now specify what exactly Ω means in the definition of dimensionless variables [cf. Equation (4.6)]. Concerning Ω, we chose to take for a time unit the inverse of its value at the equator. This is uniquely determinate due to the fact we have always assumed that the rotation law is of the form Ω[r, ϑ, ϕ] = Ω[r sin[ϑ]] or of the form Ω[r, ϑ, ϕ] = Ω[r]. The first case corresponds to what must be this law for the background to be stable with respect to the Newtonian EE, and the second case is in a way inspired by GR even if it is not a solution of the full Newtonian EE. For more details see Section 4.6. In the dimensionless NSE, the modifications induced by differential rotation are quite simple. First, ∂ϕ is replaced with ˜ Ω[r, ϑ]∂ϕ where ˜ Ω[r, ϑ] is the dimensionless profile of rotation. Then, we have to add new terms coming from eϕ r sin[ϑ] V · ∇ ˜Ω[r, ϑ] that are in the spherical orthonormal basis 0 0 Vrr sin[ϑ]∂r ˜ Ω + Vϑ sin[ϑ]∂ϑ ˜ (4.20) Ω. 4.5.2 Noise with huge RR force Once again, our goal was to stay as close as possible to the basic and well understood situation to minimize the number of unknowns. Thus, we took noisy initial data, put it in a differentially rotating background, switched on the RR force and looked for what was to happen. In this way, the question was not to look for the existence of r-modes, but simply to look for the existence of modes driven to instability by the RR force in a non rigidly rotating background. The only difference with the basic study done in the case of rigid rotation was that we had to choose a law for Ω. The first choice was very simple and corresponded to a background stable with respect to the Newtonian EE. It was of the form ˜Ω[r, ϑ] = W 1 + βn r 2 sin[ϑ] 2 (4.21) where βn was a constant depending on the run and W calculated to have ˜ Ω 1, π = 1. 2 As explained above, we also tried a law inspired by GR : ˜Ω[r, ϑ] = W 1 + βgr r 2 (4.22) or laws coming from the already quoted article by Karino et al. (2001) : ˜Ω[r, ϑ] = W βL 2 r 2 sin[ϑ] 2 + βL 2 (4.23)
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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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Page 89 and 90: Chapitre 3 Oscillations stellaires
Page 91 and 92: 3.1 Modes d’une étoile à neutro
Page 93 and 94: 3.1.2 Perturbations 3.1 Modes d’u
Page 95 and 96: Un mode propre vérifie donc 3.1 Mo
Page 97 and 98: 3.1 Modes d’une étoile à neutro
Page 99 and 100: 3.2 Oscillations d’une étoile re
Page 107 and 108: Ω c /Ω max 1.00 0.98 0.96 0.94
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
Page 113 and 114: 3.3 Modes inertiels et r-modes 101
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
Page 131 and 132: 4.4 Test and calibration of the cod
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: V r Sp(V r ) 0.4 0.2 0 -0.2 -0.4 -0
Page 143 and 144: Sp(V r ) V r 4 2 0 -2 4.5 Different
Page 145 and 146: 4.6 Strong Cowling approximation in
Page 147 and 148: V θ Sp(V θ ) 10 5 0 -5 -10 4.6 St
Page 149 and 150: can be written 4.6 Strong Cowling a
Page 151 and 152: E Polar / E Total 0.05 0.04 0.03 0.
Page 153 and 154: S xx Sp(S xx ) 0.2 0.1 0 -0.1 4.6 S
Page 155 and 156: GW emission. 4.8 Acknowledgments 4.
Page 157 and 158: Chapitre 5 Modes inertiels dans des
Page 159 and 160: 5.1 Etoiles relativistes en rotatio
Page 167 and 168: 5.2 Equation d’état et hydrodyna
Page 171 and 172: où l’on a introduit 5.2 Equation
Page 175 and 176: 5.3 Résultats numériques 163 [voi
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
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B.1 Spirit of the spectral methods
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B.1 Spirit of the spectral methods
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B.2 Solving Euler or Navier-Stokes
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The divergence-free approximation B
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Liste des tableaux Etoiles à neutr
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Table des figures Relativité gén
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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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