Polytropes

Georg P. Horedt

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While it seems possible to present a fairly complete uni?ed theory of undistorted polytropes, as attempted in the previous chapter...

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While it seems possible to present a fairly complete uni?ed theory of undistorted polytropes, as attempted in the previous chapter, the theory of distorted polytropes is much more extended and - phisticated, so that I present merely a brief overview of the theories that seem to me most interesting and important. Basically, the methods proposed to study the hydrostatic equilibrium of a distorted self-gravitating mass can be divided into two major groups (Blinnikov 1975): (i) Analytic or semia- lytic methods using a small parameter connected with the distortion of the polytrope. (ii) More or less accurate numerical methods. Lyapunov and later Carleman (see Jardetzky 1958, p. 13) have demonstrated that a sphere is a unique solution to the problem of hydrostatic equilibrium for a ?uid mass at rest in tridimensional space. The problem complicates enormously if the sphere is rotating rigidly or di?erentially in space round an axis, and/or if it is distorted magnetically or tidally. Even for the simplest case of a uniformly rotating ?uid body with constant density not all possible solutions have been found (Zharkov and Trubitsyn 1978, p. 222). The sphere becomes an oblate ?gure, and we have no a priori knowledge of its strati?cation, boundary shape, planes of symmetry, transfer of angular momentum in di?erentially rotating bodies, etc.
Inhalt
1: Polytropic and Adiabatic Processes. 1.1. Basic Concepts. 1.2. Polytropic and Adiabatic Processes in a Perfect Gas. 1.3. Polytropic Processes for a General Equation of State. 1.4. Adiabatic Processes in a Mixture of Black Body Radiation and Perfect Gas. 1.5. Adiabatic Processes in a Mixture of Electron-Positron Pairs and Black Body Radiation. 1.6. Adiabatic Processes in a Completely Degenerate Electron or Neutron Gas. 1.7. Numerical Survey of Equations of State, Adiabatic Exponents, and Polytropic Indices. 1.8. Emden's Theorem. 2: Undistorted Polytropes. 2.1. General Differential Equations. 2.2. The Homology Theorem and Transformations of the Lane-Emden Equation. 2.3. Exact Analytical Solutions of the Lane-Emden Equation. 2.4. Approximate Analytical Solutions. 2.5. Exact Numerical Solutions. 2.6. Physical Characteristics of Undistorted Polytropes. 2.7. Topology of the Lane-Emden Equation. 2.8. Composite and Other Spherical Polytropes. 3: Distorted Polytropes. 3.1. Introduction. 3.2. Chandrasekhar's First Order Theory of Rotationally Distorted Spheres. 3.3. Chandrasekhar's First Order Theory of Tidally Distorted Polytropes. 3.4. Chandrasekhar's Double Star Problem. 3.5. Second Order Extension of Chandrasekhar's Theory to Differentially Rotating Polytropes. 3.6. Double Approximation Method for Rotationally and Tidally Distorted Polytropic Spheres. 3.7. Second Order Level Surface Theory of Rotationally Distorted Polytropes. 3.8. Numerical and Semmumerical Methods Concernmg Distorted Polytropic Spheres. 3.9. Rotating Polytropic Cylinders and Polytropic Rings. 4: Relativistic Polytropes. 4.1. Undistorted Relativistic Polytropes. 4.2. Rotationally Distorted Relativistic Polytropes. 5: Stability and Oscillations. 5.1. Definitions and General Considerations. 5.2. Basic Equations. 5.3. Radial Oscillations of Polytropic Spheres. 5.4. Instability of Truncated Polytropes. 5.5. Nonradial Oscillations of Polytropic Spheres. 5.6. Stability and Oscillations of Polytropic Cylinders. 5.7. Oscillations and Stability of Rotationally and Tidally Distorted Polytropic Spheres. 5.8. The Virial Method for Rotating Polytropes. 5.9. Stability and Oscillations of Rotating Polytropic Cylinders. 5.10. Stability and Oscillations of Rotating Slabs and Disks. 5.11. Stability and Oscillations of Magnetopolytropes. 5.12. Stability and Oscillations of Relativistic Polytropes. 6: Further Applications to Polytropes. 6.1. Applications to Stars and Stellar Systems. 6.2. Polytropic Atmospheres, Polytropic Clouds and Cores, Embedded Polytropes. 6.3. Polytropic Winds. 6.4. Polytropic Accretion Flows, Accretion Disks and Tori. Acknowledgments. Appendix A. Appendix B. Appendix C. References and Author Index. Subject Index.