ISBN: 3-540-64864-X
TITLE: Physics of Planetary Rings
AUTHOR: Fridman, Alexei M.; Gorkavyi, Nikolai N.
TOC:
1. Introduction 1
1.1 Rings as Characteristic Features of Astrophysical Discs 1
1.2 The Planetary Rings as Unique Disc Systems 4
1.3 The Planetary Rings as a Proving Ground for Theorists 5
1.4 Historical Journey 6
2. Observational Data 21
2.1 The Saturnian System 21
2.2 The Uranian System 35
2.3 The Jovian System 40
2.4 The Neptunian System 42
2.5 The Solar System 45
2.6 Accretion Discs 46
2.7 Galactic Discs 49
2.8 Comparative Analysis 51
2.8.1 Primary and Secondary Rings 51
2.8.2 Density Distribution in the Systems
of the Giant Planets 52
2.8.3 Dissipation in a Disc System 54
2.8.4 Table of the Parameters of Disc Systems 57
3. Celestial Mechanics Minimum 59
3.1 Basic Equations 59
3.2 Solution for a Single Point Particle 62
3.3 Main Perturbing Factors 68
3.3.1 Equations for the Osculating Orbital Elements 68
3.3.2 Satellite Orbit in the Field of an Aspherical Planet 74
3.3.3 Effect of Aerodynamic Friction
on the Orbit of a Satellite 76
3.3.4 The PoyntingRobertson Effect 77
3.3.5 Collisions and Particle Orbits 78
4. Elementary Particle Dynamics. I Rigid Body Collisions 81
4.1 Theoretical Models 82
4.1.1 Some Relations from the Theory of the Collisions
of Smooth Spheres 82
4.1.2 Break-Up of Ring Particles (Estimates) 84
4.1.3 Model of Collisions Between Particles
Covered by Regolith 85
4.1.4 Restitution Coefficient of a Smooth Particle 86
4.2 Experimental Data 87
4.2.1 Comparison Between the Smooth Particle Model
and the Experimental Data 87
4.2.2 Restitution Coefficient of Particles
Covered by a Regolith Layer 90
5. Elementary Particle Dynamics. II Ring Cosmogony 95
5.1 Tides or Collisions? 95
5.1.1 Discussion of the Traditional Point of View That
the Region of the Primary Rings Is the Roche Zone 96
5.1.2 Collisional Break-Up of Particles
in Grazing Collisions 98
5.2 Dynamics of Particle Fragments in the Four-Body Problem 100
5.3 Collisional Break-Up of Loose Bodies
as the Cause for the Existence of Planetary Rings 109
5.4 Particle Size Distribution 113
6. Elementary Particle Dynamics. III Wave, Photometric,
and Other Effects 115
6.1 A Satellite in a Differentially Rotating Disc 115
6.2 Two Large Bodies in a Disc of Small Particles 118
6.3 Wanderer Particles in the Four-Body Problem 120
6.4 Azimuthal Brightness Asymmetry of the Saturnian Rings 122
7. Collective Dynamics of Disc Particles. I Formalism 131
7.1 Transport Theories for Macroparticles 131
7.1.1 The Larmor Theorem for a Particle
in a Gravitational Field 134
7.1.2 Derivation of the Moment Equations 135
7.1.3 Integro-differential Equation
for the Non-equilibrium Correction
to the Distribution Function 137
7.1.4 Evaluation of the Vectorial Non-equilibrium
Correction to the Distribution Function.
The Heat Flux Vector 140
7.1.5 Evaluation of the Tensor Non-equilibrium
Correction to the Distribution Function.
The Viscous Stress Tensor 142
7.2 Kinetic Theory of Inelastic Macroparticles 145
8. Collective Dynamics of Disc Particles.
II Stability Analysis 153
8.1 General Dispersion Equation 153
8.1.1 Stability of a Uniformly Rotating Disc 155
8.1.2 A Differentially Rotating Disc of Inelastic Particles 159
8.2 Analysis of the Axisymmetric Oscillations of a Disc;
Instabilities Causing the Small-
and Medium-Scale Structure of the Rings 163
8.2.1 Gravitational Instability 163
8.2.2 Energy (Thermal) Instability 164
8.2.3 Negative Diffusion Instability 165
8.2.4 Analysis of the Dispersion Equation 166
8.2.5 Criteria for the Diffusion and Energy Instabilities
for Non-gravitating Smooth Spheres 168
8.2.6 Energy and Diffusion Instabilities
in a Model of Gravitating Particles 169
8.3 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
8.4 Analysis of Non-axisymmetric Oscillations of a Disc
Ellipse Instability 184
9. Resonance Effects in Planetary Rings. I Spiral Waves 189
9.1 Density Waves 189
9.1.1 Frequency Multiplication in an Aspherical Field 190
9.1.2 Resonance Interaction of a Satellite
with Ring Particles (Two-Dimensional Case) 192
9.1.3 Spiral Waves Taking into Account
the Self-gravitation and Pressure of the Disc
(Two-Dimensional Case) 194
9.2 Bending Waves 196
10. Resonance Effects in Planetary Rings.
II Narrow Ringlets and Satellites 199
10.1 Hypotheses About the Origin of the Uranian Rings 199
10.1.1 The Remarkable Properties of the Uranian Rings 199
10.1.2 Hypotheses About the Connection Between
the Rings and the Five Known Uranian Satellites 200
10.1.3 Hypotheses About Unknown Satellites in the Rings
and "Shepherd" Satellites 201
10.1.4 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
10.1.5 Calculation of the Orbital Radii
of Hypothetical Satellites 202
10.1.6 Detection of New Uranian Satellites 205
10.2 Correlation Between the Uranian Rings
and Satellite Resonances 205
10.2.1 Distribution of the Distances Between the Rings
and the Resonances 205
10.2.2 Correlation Between the Positions of the Rings
and Resonances 207
10.2.3 A Study of the Resonance System of Uranian Rings
Using the Correlation Coefficient 209
11. Formation and Stability of the Uranian Rings 213
11.1 Sequence of the Formation of the Uranian Satellites 216
11.2 Particle Drift in the Uranian Proto-disc 219
11.2.1 Aerodynamic Drift in an Expanding Proto-disc 219
11.2.2 Qualitative Discussion of the Ballistic Drift 222
11.2.3 Estimates of the Ballistic Drift
and of the Aerodynamic Friction 226
11.2.4 Numerical Calculation of the Ballistic Drift
in the Present System of Rings 230
11.3 Formation of the Uranian Rings
in the Inner Lindblad Resonances 233
11.3.1 Elementary Capture Dynamics 234
11.3.2 Numerical Calculation of Particle Capture
in Inner Lindblad Resonances 238
11.4 The Present-Day Uranian Ring System 243
11.4.1 Epoch of Free Drift of the Rings and Its Finale
with the Participation of Cordelia and Ophelia 243
11.4.2 Contemporary Picture of the Drift Equilibrium
in the Rings and the Formation
of the 1986U1R or lambda Ring 245
11.4.3 Dust Structures in the Rings 247
11.4.4 On the Stability of the Sharp Edge
of Non-resonance Elliptical Rings 249
11.4.5 Biographical Information About the Uranian Rings 250
11.5 Conclusions 251
12. Origin, Dynamics, and Stability of the Neptunian Rings 253
12.1 Hypotheses About the Dynamics
of the Incomplete Neptunian Rings (Arcs) 253
12.1.1 Dynamical Models of the Neptunian Arcs
in the Framework of the "Shepherd" Concept 253
12.1.2 Model of Intrinsically Stable Neptunian Arcs
on a Continuous Ring 255
12.1.3 The Voyager-2 Fly-Past near Neptune
in August 1989 256
12.1.4 Connection Between Satellite Resonances
and the Neptunian Rings 258
12.2 Stability of a Separate Epiton 259
12.2.1 Particle Motion in an Epiton 259
12.2.2 Stability of an Epiton of Inelastic Particles 262
12.2.3 Evolution of an Epiton in Resonance
with a Satellite 265
12.3 Formation of Arcs on a Continuous Ring 274
12.3.1 Break-Up of a Ring Under the Action
of a Satellite Resonance 274
12.3.2 Interaction Between an Epiton and a Ring 276
12.3.3 Formation of a Stable Chain of Epitons (Arcs) 279
12.3.4 General Scenario for the Origin of the System
of Neptunian Arcs 282
13. Self-organisation of the Solar System 285
13.1 Conditions for the Development of Spatial Structures 285
13.1.1 Self-organisation of Open Systems 286
13.1.2 Gravitational Self-organisation 287
13.2 The Law of the Planetary Distances 287
13.2.1 Tendency of the Solar System
Towards Self-organisation 287
13.2.2 Dissipative Instability and the Law
of the Planetary Distances 290
13.2.3 Proposed Characteristics of the Proto-disc 292
14. Space Studies of the Outer Planets 297
14.1 Space Successes in the Period 19591989 297
14.2 The Voyager Missions 301
14.3 The Cassini Mission 303
14.4 The Chronos Mission 305
14.5 The Infrastructure of Planetary Physics 308
Conclusion 313
Appendices
I. The Possibility of Studying the Dynamics
of Astrophysical Discs in a Two-Dimensional Approach 315
1. Introduction 315
2. Original Equations for the "Volume" Functions 316
2.1 Initial Dynamic Equations 316
2.2 Equation of State 317
3. Derivation of the Basic Equations
for the "Plane" Functions 318
3.1 Order-of-Magnitude Estimates of the Terms
in the Initial Equations 318
3.2 The Two Limiting Cases of Astrophysical Discs 321
3.3 Limitations of the Characteristic Times of Processes
Studied in the Two-Dimensional Approximation 326
3.4 Closed System of Integro-differential Equations
for a Barotropic Disc 328
4. Closed Set of Differential Equations for a Polytropic Disc
in an External Gravitational Field 330
4.1 Derivation of the Two-Dimensional Equations 330
4.2 Special Case of the Potential Phi_0 = Phi_0(r), (Phi_0)' = 0 333
4.3 The Applicability of C = constant 334
5. Closed Set of Differential Equations for a Polytropic
Self-gravitating Disc 335
5.1 Derivation of the Two-Dimensional Equations 335
5.2 Why Does the Gradient of the Plane Pressure
Not Have the Physical Meaning of a Force? 338
6. Conclusion 339
II. Small-Amplitude Waves in a Disc
Which Are Symmetric with Respect to Its z = 0-Plane 341
1. Derivation of a Closed Set of Integro-differential Equations 341
2. Derivation of the Dispersion Equation
Describing the Three-Dimensional Perturbations 345
3. Solution of the Poisson Equation for a Disc
of Half-Thickness h 347
4. Dispersion Relation for Waves in the Plane of the Disc 350
5. The Role of Perturbations Along the Rotation Axis 351
5.1 Condition for Neglecting Mass Transfer
Along the Rotation Axis 352
5.1.1 General Case 352
5.1.2 Isothermal Disc 354
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
6. Conclusion 357
III. Derivation of the Linearised Equations for Oscillations
of a Viscous Disc 359
1. Derivation of the Linearised Equations for Oscillations
of a Viscous Uniformly Rotating Disc 359
2. Derivation of the Linearised Equations for Oscillations
of a Viscous Differentially Rotating Disc of Inelastic
Particles with Account of External Matter Fluxes 361
3. Derivation of the General Dispersion Equation 369
IV. Evaluating the Gravitational Potential
Inside and Outside a Triaxial Ellipsoid 371
1. Potential Inside the Ellipsoid 371
2. Potential Outside the Ellipsoid 375
V. A Drift Mechanism for the Formation
of the Cassini Division 379
1. Introduction 379
2. Statement of the Problem 385
3. Derivation of the Non-linear Momentum
Conservation Equations 388
4. Time-Averaged Non-linear Momentum
Conservation Equations 390
5. Absence of Averaged Radial Mass Flux
in a Dissipationless Disc. Large-Scale Convection 392
6. Radial Mass Transfer in a Viscous Disc 395
7. Evolution of the Surface Density of a Disc 400
8. Conditions for the Formation of Different Types
of Resonant Structures: Gaps or Wavetrains? 401
9. Estimate of the Maximum Width of a Gap Produced
by a Density Wave 405
10. Some Additional Remarks 406
VI. Resonance Structures in Saturn's C Ring 409
References 419
Index 429
END