Physics of Gravitating Systems I : Equilibrium and Stability

It would seem that any specialist in plasma physics studying a medium in which the interaction between particles is as distance-dependent as the inter­ action between stars and other gravitating masses would assert that the role of collective effects in the dynamics of gravitating systems must be decisive. However, among astronomers this point of view has been recog­ nized only very recently. So, comparatively recently, serious consideration has been devoted to theories of galactic spiral structure in which the dominant role is played by the orbital properties of individual stars rather than collec­ tive effects. In this connection we would like to draw the reader's attention to a difference in the scientific traditions of plasma physicists and astrono­ mers, whereby the former have explained the delay of the onset of controlled thermonuclear fusion by the "intrigues" of collective processes in the plasma, while many a generation of astronomers were calculating star motions, solar and lunar eclipses, and a number of other fine effects for many years ahead by making excellent use of only the laws of Newtonian mechanics. Therefore, for an astronomer, it is perhaps not easy to agree with the fact that the evolution of stellar systems is controlled mainly by collective effects, and the habitual methods of theoretical mechanics III astronomy must make way for the method of self-consistent fields.

(Volume I).- § 1. Basic Concepts and Equations of Theory.- § 2. Equilibrium States of Collisionless Gravitating Systems.- § 3. Small Oscillations and Stability.- §4. Jeans Instability of a One―Component Uniform Medium.- §5. Jeans Instability of a Multicomponent Uniform Medium.- 5.1. Basic Theorem (on the Stability of a Multicomponent System with Components at Rest).- 5.2. Four Limiting Cases for a Two―Component Medium.- 5.3. Table of Jeans Instabilities of a Uniform Two―Component Medium.- 5.4. General Case of n Components.- §6. Non―Jeans Instabilities.- § 7. Qualitative Discussion of the Stability of Spherical, Cylindrical (and Disk―Shaped) Systems with Respect to Radial Perturbations.- I Theory.- I Equilibrium and Stability of a Nonrotating Flat Gravitating Layer.- § 1. Equilibrium States of a Collisionless Flat Layer.- § 2. Gravitational (Jeans) Instability of the Layer.- § 3. Anisotropic (Fire―Hose) Instability of a Collisionless Flat Layer.- 3.1. Qualitative Considerations.- 3.2. Derivation of the Dispersion Equation for Bending Perturbations of a Thin Layer.- 3.3. Fire―Hose Instability of a Highly Anisotropic Flat Layer.- 3.4. Analysis of the Dispersion Equation.- 3.5. Additional Remarks.- § 4. Derivation of Integra―Differential Equations for Normal Modes of a Flat Gravitating Layer.- § 5. Symmetrical Perturbations of a Flat Layer with an Isotropic Distribution Function Near the Stability Boundary.- § 6. Perpendicular Oscillations of a Homogeneous Collisionless Layer.- 6.1. Derivation of the Characteristic Equation for Eigenfrequencies.- 6.2. Stability of the Model.- 6.3. Permutational Modes.- 6.4. Time―Independent Perturbations (? = 0).- Problems.- II Equilibrium and Stability of a Collisionless Cylinder.- §1. Equilibrium Cylindrical Configurations.- § 2. Jeans Instability of a Cylinder with Finite Radius.- 2.1. Dispersion Equation for Eigenfrequencies of Axial-Symmetrical Perturbations of a Cylinder with Circular Orbits of Particles.- 2.2. Branches of Axial―Symmetrical Oscillations of a Rotating Cylinder with Maxwellian Distribution of Particles in.- 2.3. Longitudinal Velocities.- 2.4. Oscillative Branches of the Rotating Cylinder with a Jackson Distribution Function (in Longitudinal Velocities).- 2.5. Axial―Symmetrical Perturbations of Cylindrical Models of a More General Type.- § 3. Nonaxial Perturbations of a Collisionless Cylinder.- 3.1. The Long―Wave Fire-Hose Instability.- 3.2. Nonaxial Perturbations of a Cylinder with Circular Particle Orbits 100§ 4. Stability of a Cylinder with Respect to Flute―like Perturbations.- § 5. Local Analysis of the Stability of Cylinders (Flute―like Perturbations).- 5.1. Dispersion Equation for Model (2), § 1.- 5.2. Maxwellian Distribution Function.- § 6. Comparison with Oscillations of an Incompressible Cylinder.- 6.1. Flute―like Perturbations (kz = 0).- § 7. Flute―like Oscillations of a Nonuniform Cylinder with Circular Orbits of Particles.- Problems.- III Equilibrium and Stability of Collisionless Spherically Symmetrical Systems.- § 1. Equilibrium Distribution Functions.- § 2. Stability of Systems with an Isotropic Particle Velocity Distribution.- 2.1. The General Variational Principle for Gravitating Systems with the Isotropic Distribution of Particles in Velocities (f0 = f0(E), f’0 = df0|dE ? 0).- 2.2. Sufficient Condition of Stability.- 2.3. Other Theorems about Stability. Stability with Respect to Nonradial Perturbations.- 2.4. Variational Principle for Radial Perturbations.- 2.5. Hydrodynamical Analogy.- 2.6. On the Stability of Systems with Distribution Functions That Do Not Satisfy the Condition f’0 (E) ? 0.- § 3. Stability of Systems of Gravitating Particles Moving On Circular Trajectories.- 3.1. Stability of a Uniform Sphere.- 3.2. Stability of a Homogeneous System of Particles with Nearly Circular Orbits.- 3.3. Stability of a Homogeneous Sphere with Finite Angular Momentum.- 3.4. Stability of Inhomogeneous Systems.- § 4. Stability of Systems of Gravitating Particles Moving in Elliptical Orbits.- 4.1. Stability of a Sphere with Arbitrary Elliptical Particle Orbits.- 4.2. Instability of a Rotating Freeman Sphere.- § 5. Stability of Systems with Radial Trajectories of Particles.- 5.1. Linear Stability Theory.- 5.2. Simulation of a Nonlinear Stage of Evolution.- § 6. Stability of Spherically Symmetrical Systems of General Form.- 6.1. Series of the Idlis Distribution Functions.- 6.2. First Series of Camm Distribution Functions (Generalized Poly tropes).- 6.3. Shuster’s Model in the Phase Description.- §7. Discussion of the Results.- Problems.- IV Equilibrium and Stability of Collisionless Ellipsoidal Systems.- § 1. Equilibrium Distribution Functions.- 1.1 Freeman’s Ellipsoidal Models.- 1.2. “Hot” Models of Collisionless Ellipsoids of Revolution.- § 2. Stability of a Three―Axial Ellipsoid and an Elliptical Disk.- 2.1. Stability of a Three-Axial Ellipsoid.- 2.2. Stability of Freeman Elliptical Disks.- § 3. Stability of Two―Axial Collisionless Ellipsoidal Systems.- 3.1. Stability of Freeman’s Spheroids.- 3.2. Peebles―Ostriker Stability Criterion. Stability of Uniform Ellipsoids, “Hot” in the Plane of Rotation.- 3.3. The Fire-Hose Instability of Ellipsoidal Stellar Systems.- 3.4. Secular and Dynamical Instability. Characteristic Equation for Eigenfrequencies of Oscillations of Maclaurin Ellipsoids.- Problems.- V Equilibrium and Stability of Flat Gravitating Systems.- § 1. Equilibrium States of Flat Gaseous and Collisionless Systems.- 1.3. Systems with Circular Particle Orbits.- 1.4. Plasma Systems with a Magnetic Field.- 1.5. Gaseous Systems.- 1.6. “Hot” Collisionless Systems.- § 2. Stability of a “Cold” Rotating Disk.- 2.1. Membrane Oscillations of the Disk.- 2.2. Oscillations in the Plane of the Disk.- § 3. Stability of a Plasma Disk with a Magnetic Field.- 3.1. Qualitative Derivation of the Stability Condition.- 3.2. Variational Principle.- 3.3. Short―Wave Approximation.- 3.4. Numerical Analysis of a Specific Model.- § 4. Stability of a “Hot” Rotating Disk.- 4.1. Oscillations in the Plane of the Disk.- 4.2. Bending Perturbations.- 4.3. Methods of the Stability Investigation of General Collisionless Disk Systems.- 4.4. Exact Spectra of Small Perturbations.- 4.5. Global Instabilities of Gaseous Disks. Comparison of Stability Properties of Gaseous and Stellar Disks.- Problems.- References.- Additional References.