Structure and Morphology of Galaxies: Equilibrium and Stability Print E-mail
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The equilibrium configurations of rapidly rotating, self-gravitating fluid systems in the analytical works of Maclaurin, Jacobi, Dedekind and Riemann, as reviewed in Chandrasekhar in the book Ellipsoidal Figures of Equilibrium, Dover, New York, 2d ed.(1987), are uniform in density, have figures of equilibrium described by perfect spheroids or ellipsoids, and have simple internal flow linear in the coordinates. These models cannot represent realistic astrophysical systems for many reasons: the non uniform density of rotating stars or galaxies can be highly centrally condensed, the internal flow of these self gravitating systems is non-linear, and the equilibrium configuration not represented by perfect quadratic surfaces. Nevertheless such simplified analytical models have provided some useful general tools to understand the structure and the global properties of rotating stars and galaxies.

This research has produced an analytic generalization of the theory of ellipsoidal figures of equilibrium, endowed both with rotation and vorticity. The series of papers has followed a variety of tentative approaches, consisting of successive generalizations of known results: looking at more general density distributions, non linear velocity fields, selected forms of the pressure tensor, and finally analysing, following the virial method developed by Chandrasekhar, the constraints imposed by the nth order virial equations (Papers I-VIII).

We give the complete set of virial equations of the nth order generalizing the theory of Chandrasekhar (Paper IX). Our results limit the ranges of possible solution of equilibrium and contain as special cases all the previous classical results.

Actually we focused the attention on the study of equilibrium solutions using the Euler equation, the equation of continuity and the Poisson equation (Paper X). Our present research is an analysis of hydrodynamic equation for self-gravitating systems from the point of view of the functional analysis. We demonstrate that the basic quantities as the general density, the geometric form of the fluid, the pressure, the velocity profile and the vorticity can be expressed as functionals of a velocity potential. We assume the hypothesis of incompressibility of the fluid. We formulate the steady state non linear hydrodynamic equation as a functional equation of a velocity potential and we propose some easy arguments to find the analytic solutions in steady state regime and with a polytropic equation of state. Special models of stars and galaxies based on this theory will be developed.

Our theory represents a generalization of self-consistent analytical models. It allows the study of rotating and self-gravitating polytropic systems with non-linear internal flows and demonstrates the existence of analytical figures of equilibrium. The functional technique, by expressing the relevant hydrodynamic quantities as a functional of the velocity potential, permits the deduction in a self-consistent way, from an arbitrary law of rotation, of the equilibrium distribution of mass within a self-gravitating system. Such a distribution in general is non-axisymmetric and non-ellipsoidal. The non-linear velocity field in the solution matches the rotation curves of astronomical objects. The comparison with the classical models represents an important check for the validity of our theory. On the other hand all the results contained in Chandrasekhar (Ellipsoidal Figures of Equilibrium, Dover, New York, 2d ed. (1987)), can be also obtained in an easy way from our hydrodynamic equation, without considering the integral properties of the self-gravitating systems, which are the fundamental tools of the classical works based on the virial theorem.

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