Physics of Gravitating Systems I

(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.