Chaos in Astronomy (eBook)

Preface 6
Contents 8
List of Contributors 13
Nikolaos Voglis (1948–2007) 19
Part I Galactic Dynamics 23
Ordered and Chaotic Orbits in Spiral Galaxies 24
1 Introduction 24
2 Normal Spirals 25
3 Barred Spirals 26
4 Velocity Fields 37
5 Conclusions 39
Appendix: The Families of Short Period Orbits 39
References 42
Order and Chaos in Spiral Galaxies Seen through their Morphology 44
1 Introduction 44
2 Observations vs. Models 45
3 Special Regions and Features 46
4 Interpretation of Morphology 47
5 Pitfalls and Other Issues 48
6 Case Studies: Bars and Spirals 49
7 Conclusions 52
References 53
The Flow through the Arms of Normal and Barred-Spiral Galaxies 54
1 Spiral Flows and Orbital Theory 54
3 Barred Galaxies with Spirals 62
4 Conclusions 64
References 65
Ansae in Barred Galaxies, Observations and Simulations 66
1 Introduction 66
2 Observations 68
3 Simulations of Barred Galaxies and Ansae 70
4 Summary and Future Work 71
Acknowledgements 72
References 72
Orbital Structure in Barred Galaxies and the Role of Chaos 74
1 Initial Conditions and Characteristics of the Bar 74
2 Classification in Ordered and Chaotic Orbits 76
3 Orbital Structure of the Barred Galaxy 77
4 Conclusions 81
References 81
Chaos in Galaxies 83
1 Introduction 83
2 Chaos in Galaxies 86
3 Cases of Sensitive Dependence in Galaxy Models 90
4 Conclusions 95
References 96
Boxy/Peanut Bulges: Formation, Evolution and Properties 97
1 Introduction 97
2 Orbital Structure 98
3 Formation and Evolution of Boxes and Peanuts 98
Acknowledgements 103
References 103
Invariant Manifolds as Building Blocks for the Formation of Spiral Arms and Rings in Barred Galaxies 104
1 Introduction 104
2 Description of the Model and Equations of Motion 106
3 Dynamics Around L1 and L2 107
4 Results 109
Acknowledgement 111
References 111
Secular Instabilities of Stellar Systems: Slow Mode Approach 112
1 Introduction 112
2 Spherical Stellar Clusters Around MBH 114
3 Galactic Disks 119
4 General Conclusions 122
References 123
Stellar Velocity Distribution in Galactic Disks 124
1 Introduction 124
2 The Boltzmann Moment Equations: BEADS-2D 126
3 Test Particle Simulations 126
4 Results 128
5 Summary and Future Plans 130
References 131
Dynamical Study of 2D and 3D Barred Galaxy Models 133
1 Introduction 133
2 Definition of the Smaller Aligment Index (SALI) 134
3 The Model 134
4 Results in the 2D and 3D Ferrers Model 136
5 Conclusions 139
Acknowledgments 139
References 140
Chaos in the Mergers of Galaxies 141
1 Introduction 141
2 A Model of Chaos in the Mergers of Galaxies 142
3 Chaotic Decay of Stationary Oscillations in Galaxies Formed by Mergers: A Conjecture 149
4 Epilogue: Transient Chaos in the Mergers of Galaxies 152
Acknowledgments 152
References 152
Hamiltonian Normal Forms and Galactic Potentials 154
1 Introduction 154
2 The Hamiltonian Normal Form 155
3 Applications 159
4 Comments and Outlook 164
Acknowledgments 165
References 165
Local Phase Space: Shaped by Chaos? 167
1 Introduction 167
2 Local Phase Space 168
3 Simulations 169
4 Results 170
6 The Role of Chaos 173
7 Summary and Future Work 173
References 174
Barred Galaxies: An Observer’s Perspective 175
1 Some Basic Facts 175
2 Observed Properties of Barred Galaxies 176
3 Estimating the Dynamical Ages of Bars 179
4 The Structural Properties of Bars 180
5 Future Work 184
References 185
Invariant Manifolds and the Spiral Arms of Barred Galaxies 189
1 Introduction 189
2 Model 190
3 Results 192
4 Conclusions 197
References 198
Collisional N-Body Simulations and Time-Dependent Orbital Complexity 199
1 Introduction 199
2 Method 199
3 Application to the N = 3 Pythagorean Problem 203
4 Conclusion 206
References 206
Resonances in Galactic and Circumstellar Disks 207
1 Introduction 207
2 Lindblad and Mean Motion Resonances 208
3 Chaos Due to Resonance Overlap in Galaxies 216
References 218
Regular and Chaotic Motion in Elliptical Galaxies 219
1 Introduction 219
2 Highly Chaotic Triaxial Stellar Systems 220
3 Partially and Fully Chaotic Orbits 222
4 Figure Rotation in Triaxial Systems 224
5 Musings on Orbital Classi.cation Through Frequency Analysis 224
6 Discussion 228
7 Acknowledgements 229
References 229
Orbital Distributions and Self-Consistency in Elliptical Galaxies 231
1 Introduction 231
2 SCF Code 232
3 Models 233
4 Results 234
5 Conclusions 238
References 239
The Connection Between Orbits and Isophotal Shape in Elliptical Galaxies 241
1 Introduction 241
2 Superposition of Orbit Classes and Isophotal Shape 242
3 Discussion and Conclusion 244
References 244
Gas Orbits in a Spiral Potential 246
1 Introduction 246
2 The Simulations 246
3 The Gaseous Orbits 247
References 249
The Structure of the Phase Space in Galactic Potentials of Three Degrees of Freedom 250
1 Introduction 250
2 4D Surfaces of Section 251
References 253
Regular and Chaotic Orbits in Narrow 2D Bar Models 255
1 Model Description 255
2 Variation of the Bar Axial Ratio a/c 256
3 Surfaces of Section 256
4 A Closer Look at the Propeller Orbit Family 257
5 Conclusions 259
References 259
The Coalescence of Invariant Manifolds in Barred-Spiral Galaxies 260
1 Unstable Periodic Orbits and their Invariant Manifolds 260
2 Conclusions 263
References 263
Part II Celestial Mechanics 265
Chaotic Dynamics in Planetary Systems 266
1 Introduction 266
2 Detection Methods 267
3 The Structure of EPS 269
4 Our Planetary System as Dynamical Model 271
5 Multiplanetary Systems 274
6 Conclusions 277
References 279
Routes to Chaos in Resonant Extrasolar Planetary Systems 280
1 Introduction 280
2 Resonant Extrasolar Planetary Systems 281
3 A Stable System at the 2:1 Resonance. (.2 . .1) = 00 284
4 A Stable System at the 2:1 Resonance. .2 . .1 = 1800 286
5 An Unstable System at the 3:1 Resonance. (.2 . .1) = 00 288
6 A Stable System at the 3:1 Resonance. (.2 . .1) = 00 290
7 Discussion 292
References 293
Prometheus and Pandora, the Champions of Dynamical Chaos 295
Acknowledgments 301
References 301
Planets in Multiple Star Systems: A Symplectic Approach 303
1 Introduction 303
2 Method 305
3 Application and Results 307
4 Conclusions 310
References 310
Stabilized Chaos in the Sitnikov Problem 311
1 Formulation of the Problem 311
2 Stabilization of Chaotic Behavior in the Extended Sitnikov Problem 312
References 314
Part III Fundamental Concepts and Methods 316
Nonextensive Statistical Mechanics – An Approach to Complexity 317
1 Introduction 317
2 Nonextensivity of the Nonadditive Entropy Sq 318
3 Entropy Production per Unit Time 319
4 q-Generalization of the Standard and the L´evy-Gnedenko Central Limit Theorems 320
5 Current Applications of Nonextensive Statistical Mechanics to Astrophysical Systems 322
Acknowledgments 322
References 322
Absolute Versus Relative Motion in Mechanics 327
References 331
Far Fields, from Electrodynamics to Gravitation, and the Dark Matter Problem 333
1 Introduction 333
2 Far Fields in Electrodynamics, and the Wheeler Feynman Identity 334
3 The Virial Theorem in the Theory of Gases and in Astrophysics, and the Role of the External Forces 336
4 Relevance of the Far Matter if Hubble’s Law is Taken into Account 337
5 Estimate of the Gravitational Forces. Role of the Discreteness of the Sources, and of the Fractal Nature of the Universe 339
6 Application to the Virial for a Cluster of Galaxies 341
7 Conclusion 342
Acknowledgement 342
References 342
Distribution Functions for Galaxies using Quadratic Programming 344
1 Introduction 344
2 The Quadratic Programming Modeling Method 344
3 Self-consistent Instabilities 346
4 Conclusions and Future Prospects 353
References 353
Chaos Analysis Using the Patterns Method 354
1 Introduction 354
2 The Patterns Method 356
3 Conclusions 360
References 361
On the Topology of Regions of 3-D Particle Motions in Annular Configurations of n Bodies with a Central Post-Newtonian Potential 363
1 Introduction 363
2 The Three-Dimensional Ring Problem of (n + 1) Bodies and the Normalized Equations of Motion 363
3 Equilibrium Zones and Evolution of the Zero-Velocity Surfaces in 3-D Motion 364
4 Applications 365
References 367
The Average Power-Law Growth of Deviation Vector and Tsallis Entropy 368
1 Introduction 368
2 Q-Entropy and the Growth of Deviation Vectors 368
3 Numerical Examples 370
Acknowledgments 371
References 371
Global Dynamics of Coupled Standard Maps 372
1 Definition and Behavior of GALI 372
2 Dynamical Study of a 4D Standard Map 373
Acknowledgments 376
References 376
Part IV Other Astronomical Systems 377
The Chaotic Light Curves of Accreting Black Holes 378
1 Introduction 378
2 From Light Curves to Dynamics 380
3 The Power Density Spectra 380
4 The Lags 382
5 Modeling the X-ray Light Curves 384
6 Conclusions, Discussion 386
Acknowledgments 388
References 388
3D Accretion Discs Dynamics: Numerical Simulations 389
1 Introduction 389
2 Model 390
3 Shock Waves in Accretion Discs 391
4 The “Precessional” Spiral Density Wave in the Cold Accretion Disc 395
5 The Spiral-Vortex Structure in the Hot Accretion Disc 398
6 Conclusions 399
Acknowledgments 399
References 399
Growth of Density Fluctuations at the Time of Leptogenesis 401
References 403
Fully Developed Turbulence in Accretion Discs of Binary Stars: Turbulent Viscosity Coe.cient and Power Spectrum 405
1 Introduction 405
2 Statement of the Problem, Basic Equations, and Numerical Approach 407
3 Collective Mode m=1 408
4 Over-Re.ection Instability 411
5 Estimation of the Turbulent Viscosity Coe.cient 416
6 Spectrum of Turbulence in the Accretion Disc of IP Peg 417
7 Conclusions 420
Acknowledgments 421
References 422
Solar and Stellar Active Regions A Cosmic Laboratory for the study of Complexity 424
1 Solar Active Regions as Driven Non Linear Systems 424
2 Active Region Formation: A Percolating Driver? 425
3 How Active Regions Evolve by Dissipating Energy: Are AR in a Self Organized Critical State? 426
4 Active Regions as Multi-Scale Physics Laboratories 431
5 Summary 433
Acknowledgement 433
References 433
Chaos and Self-Organization in Solar Flares: A Critical Analysis of the Present Approach 435
1 Introduction 435
2 Solar Flares as Manifestation of Chaos III: “Disruption of the Turbulent Current Sheet of the Flare into Random Resistors Network”‘ 440
Acknowledgement 447
References 447
Charged Particles’ Acceleration through Reconnecting Current Sheets in Solar Flares 449
1 Introduction 449
2 Analytical Formulation of the Problem 450
3 Numerical Results 450
4 Conclusions 452
References 452
The Perturbed Photometric-Magnetic Dynamical Model for the Sunspot Evolution 454
1 Introduction 454
2 The Conservative Perturbed PhMD Model 455
3 The Dissipative Perturbed PhMD Model 457
References 458
The Dynamics of Non-Symmetrically Collapsing Stars 459
1 Introduction 459
2 Equations of Motion and Numerical Results 459
3 The Poincar´e Section 461
4 Conclusions 461
Acknowledgments 463
References 463
Cosmic Order out of Primordial Chaos: A Tribute to Nikos Voglis 464
1 Some Early History 464
2 Angular Momentum Generation: The Tidal Mechanism 465
3 Gravitational Instability 469
4 Tidal Shear 471
5 The Cosmic Web 472
7 Tidal Fields and Void Alignment 474
8 Final Remarks 477
Acknowledgments 478
References 478
Cosmological Inflation: A Personal Perspective 481
1 Introduction 481
2 The Cosmological Problems 482
3 Phase Transitions, Baryogenesis 484
4 Resolving the Horizon Problem 485
5 “Nothing Succeeds like Success” 488
6 Conclusions and Speculations 489
Acknowledgments 492
References 492