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Tables of Contents for Physics of Planetary Rings
Chapter/Section Title
Page #
Page Count
Introduction
1
20
Rings as Characteristic Features of Astrophysical Dises
1
3
The Planetary Rings as Unique Disc Systems
4
1
The Planetary Rings as a Proving Ground for Theorists
5
1
Historical Journey
6
15
Observational Data
21
38
The Saturnian System
21
14
The Uranian System
35
5
The Jovain System
40
2
The Neptunian System
42
3
The Solar System
45
1
Accretion Discs
46
3
Galactic Discs
49
2
Comparative Analysis
51
8
Primary and Secondary Rings
51
1
Density Distribution in the Systems of the Giant Planets
52
2
Dissipation in a Disc System
54
3
Table of the Parameters of Disc Systems
57
2
Celestial Mechanics Minimum
59
22
Basic Equations
59
3
Solution for a Single Point Particle
62
6
Main Perturbing Factors
68
13
Equations for the Osculating Orbital Elements
68
6
Satellite Orbit in the Field of an Aspherical Planet
74
2
Effect of Aerodynamic Friction on the Orbit of a Satellite
76
1
The Poynting--Robertson Effect
77
1
Collisions and Particle Orbits
78
3
Elementary Particle Dynamics. I Rigid Body Collisions
81
14
Theoretical Models
82
5
Some Relations from the Theory of the Collisions of Smooth Spheres
82
2
Break-Up of Ring Particles (Estimates)
84
1
Model of Collisions Between Particles Covered by Regolith
85
1
Restitution Coefficient of a Smooth Particle
86
1
Experimental Data
87
8
Comparison Between the Smooth Particle Model and the Experimental Data
87
3
Restitution Coefficient of Particles Covered by a Regolith Layer
90
5
Elementary Particle Dynamics. II Ring Cosmogony
95
20
Tides or Collisons?
95
5
Discussion of the Traditional Point of View That the Region of the Primary Rings Is the Roche Zone
96
2
Collisional Break-Up of Particles in Grazing Collisions
98
2
Dynamics of Particle Fragments in the Four-Body Problem
100
9
Collisional Break-Up of Loose Bodies as the Cause for the Existence of Planetary Rings
109
4
Particle Size Distribution
113
2
Elementary Particle Dynamics. III Wave, Photometric, and Other Effects
115
16
A Satellite in a Differentially Rotating Disc
115
3
Two Large Bodies in a Disc of Small Particles
118
2
Wanderer Particles in the Four-Body Problem
120
2
Azimuthal Brightness Asymmetry of the Saturnian Rings
122
9
Collective Dynamics of Disc Particles. I Formalism
131
22
Transport Theories for Macroparticles
131
14
The Larmor Theorem for a Particle in a Gravitational Field
134
1
Derivation of the Moment Equations
135
2
Integro-differential Equation for the Non-equilibrium Correction to the Distribution Fuction
137
3
Evaluation of the Vectorial Non-equilibrium Correction to the Distribution Function. The Heat Flux Vector
140
2
Evaluation of the Tensor Non-equilibrium Correction to the Distribution Function. The Viscous Stress Tensor
142
3
Kinetic Theory of Inelastic Macroparticles
145
8
Collective Dynamics of Disc Particles. II Stability Analysis
153
36
General Dispersion Equation
153
10
Stability of a Uniformly Rotating Disc
155
4
A Differentially Rotating Disc of Inelastic Particles
159
4
Analysis of the Axisymmetric Oscillations of a Disc; Instabilities Causing the Small-and Medium-Scale Structure of the Rings
163
15
Gravitational Instability
163
1
Energy (Thermal) Instability
164
1
Negative Diffusion Instability
165
1
Analysis of the Dispersion Equation
166
2
Criteria for the Diffusion and Energy Instabilities for Non-gravitating Smooth Spheres
168
1
Energy and Diffusion Instabilities in a Model of Gravitating Particles
169
9
Analysis of the Axisymmetric Oscillations of a Disc with Non-diffusion Fluxes; Accretion Instability -- the Cause of the Large-Scale Structure of the Rings
178
6
Analysis of Non-axisymmetric Oscillations of a Disc -- Ellipse Instability
184
5
Resonance Effects in Planetary Rings. I Spiral Waves
189
10
Density Waves
189
7
Frequency Multiplication in an Aspherical Field
190
2
Resonance Interaction of a Satellite with Ring Particles (Two-Dimensional Case)
192
2
Spiral Waves Taking into Account the Self-gravitation and Pressure of the Disc (Two-Dimensional Case)
194
2
Bending Waves
196
3
Resonance Effects in Planetary Rings. II Narrow Ringlets and Satellites
199
14
Hypotheses About the Origin of the Uranian Rings
199
6
The Remarkable Properties of the Uranian Rings
199
1
Hypotheses About the Connection Between the Rings and the Five Known Uranian Satellities
200
1
Hypotheses About Unknown Satellites in the Rings and ``Shepherd'' Satellites
201
1
Hypothesis About the Resonance Nature of the Uranian Rings and the Existence of a Series of Undiscovered Satellites Beyond the Boundary of the Rings
201
1
Calculation of the Orbital Radii of Hypothetical Satellites
202
3
Detection of New Uranian Satellites
205
1
Correlation Between the Uranian Rings and Satellite Resonances
205
8
Distribution of the Distances Between the Rings and the Resonances
205
2
Correlation Between the Positions of the Rings and Resonances
207
2
A Study of the Resonance System of Uranian Rings Using the Correlation Coefficient
209
4
Formation and Stability of the Uranian Rings
213
40
Sequence of the Formation of the Uranian Satellites
216
3
Particle Drift in the Uranian Proto-disc
219
14
Aerodynamic Drift in an Expanding Proto-disc
219
3
Qualitative Discussion of the Ballistic Drift
222
4
Estimates of the Ballistic Drift and of the Aerodynamic Friction
226
4
Numerical Calculation of the Ballistic Drift in the Present System of Rings
230
3
Formation of the Uranian Rings in the Inner Lindblad Resonances
233
10
Elementary Capture Dynamics
234
4
Numerical Calculation of Particle Capture in Inner Lindblad Resonances
238
5
The Present-Day Uranian Ring System
243
8
Epoch of Free Drift of the Rings and Its Finale with the Participation of Cordelia and Ophelia
243
2
Contemporary Picture of the Drift Equilibrium in the Rings and the Formation of the 1986U1R or λ Ring
245
2
Dust Structures in the Rings
247
2
On the Stability of the Sharp Edge of Non-resonance Elliptical Rings
249
1
Biographical Information About the Uranian Rings
250
1
Conclusions
251
2
Origin, Dynamics, and Stability of the Neptunian Rings
253
32
Hypotheses About the Dynamics of the Incomplete Neptunian Rings (Arcs)
253
6
Dynamical Models of the Neptunian Arcs in the Framework of the ``Shepherd'' Concept
253
2
Model of Intrinsically Stable Neptunian Arcs on a Continuous Ring
255
1
The Voyager-2 Fly-Past near Neptune in August 1989
256
2
Connection Between Satellite Resonances and the Neptunian Rings
258
1
Stability of a Separate Epiton
259
15
Particle Motion in an Epiton
259
3
Stability of an Epiton of Inelastic Particles
262
3
Evolution of an Epiton in Resonance with a Satellite
265
9
Formation of Arcs on a Continuous Ring
274
11
Break-Up of a Ring Under the Action of a Satellite Resonance
274
2
Interaction Between an Epiton and a Ring
276
3
Formation of a Stable Chain of Epitons (Arcs)
279
3
General Scenario for the Origin of the System of Neptunian Arcs
282
3
Self-organisation of the Solar System
285
12
Conditions for the Development of Spatial Structures
285
2
Self-organisation of Open Systems
286
1
Gravitational Self-organisation
287
1
The Law of the Planetary Distances
287
10
Tendency of the Solar System Towards Self-organisation
287
3
Dissipative Instability and the Law of the Planetary Distances
290
2
Proposed Characteristics of the Proto-disc
292
5
Space Studies of the Outer Planets
297
16
Space Successes in the Period 1959--1989
297
4
The Voyager Missions
301
2
The Cassini Mission
303
2
The Chronos Mission
305
3
The Infrastructure of Planetary Physics
308
5
Conclusion
313
2
Appendices
I. The Possibility of Studying the Dynamics of Astrophysical Discs in a Two-Dimensional Approach
315
26
1. Introduction
315
1
2. Original Equations for the ``Volume'' Functions
316
2
2.1 Initial Dynamic Equations
316
1
2.2 Equation of State
317
1
3. Derivation of the Basic Equations for the ``Plane'' Functions
318
12
3.1 Order-of-Magnitude Estimates of the Terms in the Initial Equations
318
3
3.2 The Two Limiting Cases of Astrophysical Discs
321
5
3.3 Limitations of the Characteristic Times of Processes Studied in the Two-Dimensional Approximation
326
2
3.4 Closed System of Integro-differential Equations for a Barotropic Disc
328
2
4. Closed Set of Differential Equations for a Polytropic Disc in an External Gravitational Field
330
5
4.1 Derivation of the Two-Dimensional Equations
330
3
4.2 Special Case of the Potential Φ0 = Φ0 (r), Φ0 = 0
333
1
4.3 The Applicability of C = constant
334
1
5. Closed Set of Differential Equations for a Polytropic Self-gravitating Disc
335
4
5.1 Derivation of the Two-Dimensional Equations
335
3
5.2 Why Does the Gradient of the Plane Pressure Not Have the Physical Meaning of a Force?
338
1
6. Conclusion
339
2
II. Small-Amplitude Waves in a Disc Which Are Symmetric with Respect to Its z = 0-Plane
341
18
1. Derivation of a Closed Set of Integro-differential Equations
341
4
2. Derivation of the Dispersion Equation Describing the Three-Dimensional Perturbations
345
2
3. Solution of the Poisson Equation for a Disc of Half-Thickness h
347
3
4. Dispersion Relation for Waves in the Plane of the Disc
350
1
5. The Role of Perturbations Along the Rotation Axis
351
6
5.1 Condition for Neglecting Mass Transfer Along the Rotation Axis
352
1
5.1.1 General Case
352
2
5.1.2 Isothermal Disc
354
1
5.2 Condition for Neglecting the Inertial Term in the Equation of Motion in the z-Direction -- Condition for Neglecting Oscillations Along the Rotation Axis
355
2
6. Conclusion
357
2
III. Derivation of the Linearised Equations for Oscillations of a Viscous Disc
359
12
1. Derivation of the Linearised Equations for Oscillations of a Viscous Uniformly Rotating Disc
359
2
2. Derivation of the Linearised Equations for Oscillations of a Viscous Differentially Rotating Disc of Inelastic Particles with Account of External Matter Fluxes
361
8
3. Derivation of the General Dispersion Equation
369
2
IV. Evaluating the Gravitational Potential Inside and Outside a Triaxial Ellipsoid
371
8
1. Potential Inside the Ellipsoid
371
4
2. Potential Outside the Ellipsoid
375
4
V. A Drift Mechanism for the Formation of the Cassini Division
379
30
1. Introduction
379
6
2. Statement of the Problem
385
3
3. Derivation of the Non-linear Momentum Conservation Equations
388
2
4. Time-Averaged Non-linear Momentum Conservation Equations
390
2
5. Absence of Averaged Radial Mass Flux in a Dissipationless Disc. Large-Scale Convection
392
3
6. Radial Mass Transfer in a Viscous Disc
395
5
7. Evolution of the Surface Density of a Disc
400
1
8. Conditions for the Formation of Different Types of Resonant Structures: Gaps or Wavetrains?
401
4
9. Estimate of the Maximum Width of a Gap Produced by a Density Wave
405
1
10. Some Additional Remarks
406
3
VI. Resonance Structures in Saturn's C Ring
409
10
References
419
10
Index
429