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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SELECTED PUBLICATIONS (1970-2000):

1. F.Pacheco, G.Pucacco, R.Ruffini: Astron.Astrophys., 161, 39 (Paper I), 1986; F.Pacheco, G.Pucacco, R.Ruffini: Proc. of the Fourth Marcel Grossmann Meeting, North Holland, Amsterdam, p.1511 (Paper II), 1986;
2. G.Busarello, S.Filippi, R.Ruffini: "Anisotropic tensor virial models for elliptical galaxies with rotation or vorticity", Astron.Astrophys., 197, 91-104 (Paper III), 1988;
3. F.Pacheco, G.Pucacco, R.Ruffini, G.Sebastiani, Astron.Astrophys., 161, 39 (Paper IV), 1986;
4. G.Busarello, S.Filippi, R.Ruffini: "Anisotropic and inhomogeneous tensor virial models for elliptical galaxies with figure rotation and internal streaming", Astron.Astrophys., 213, 80-88 (Paper V), 1989;
5. S.Filippi, G.Busarello and R.Ruffini: "Equilibrium of triaxial self- gravitating ellipsoids with rotation, vorticity and anisotropic pressure", Proc. of the Fifth Marcel Grossmann Meeting, World Scientific 1989;
6. G.Busarello, S.Filippi and R.Ruffini: "Observable properties of generalized Riemann ellipsoids and their application to elliptical galaxies", Proc. of the Fifth Marcel Grossmann Meeting, World Scientific 1989;
7. G.Busarello, S.Filippi, R.Ruffini: "On b-Type Spheroids", Astron.Astrophys., 227, 30-32 (Paper VI), 1990;
8. S.Filippi, R.Ruffini, A.Sepulveda: "On the generalized Riemann configurations and Dedekind's Theorem", Astron.Astrophys., 231, 30 (Paper VII), 1990;
9. S.Filippi, R.Ruffini and A.Sepulveda: "On the generalization of Dedekind-Riemann sequences to non linear velocities", Europhys. Lett., 12, 735, 1990;
10. S.Filippi, R.Ruffini and A.Sepulveda: "Nonlinear Velocities in Generalized Riemann Ellipsoids", Il Nuovo Cimento , Vol.105 B, n.8-9, 1047, 1990;
11. S.Filippi, R.Ruffini, A.Sepulveda: "Generalized Riemann Ellipsoids. Equilibrium and stability" Astron.Astrophys., 246, 59-70 (Paper VIII), 1991;
12. S.Filippi, A.Sepulveda: "Axisimmetric, rotating masses. Dynamical equilibrium and stability", Proc.of the Sixth Marcel Grossmann Meeting, Kyoto, 1991, World Scientific, 1993;
13. S.Filippi, A.Sepulveda, R.Ruffini: "The implications of the n-th order virial equations on the equilibrium of Jacobi, Dedekind and Riemann ellipsoids", Astrophysical Journal, 460, 762-776, (Paper IX) ,1996;
14. S.Filippi and A.Sepulveda: "Models of Self-Gravitating Rotating Systems", Proc.of the Seventh Marcel Grossmann Meeting, Stanford - U.S.A., World Scientific 1996;
15. S.Filippi and A.Sepulveda, "nth-order Virial Theory", Proc.of the Seventh Marcel Grossmann Meeting, Stanford - U.S.A., World Scientific 1996.
16. S. Filippi, R. Ruffini and A. Sepulveda, " The nth-order Virial Theory. Permitted figures of equilibrium", Il Nuovo Cimento, 112B, 299, 1997.
17. S. Filippi, A. Sepulveda, "Toroidal Solutions to the problem of inhomogeneous Rotating, Gravitating Systems", in Proc. Of VIII Marcell Grossmann Meeting, Jerusalem 1997, World Scientific Publ. 1998

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Publications Jan. 2001- Jan. 2002:

1. S. Filippi, R. Ruffini and A. Sepulveda, (2002). Equilibrium Solutions for Self-Gravitating Polytropic Systems. MM Marcel Grossmann Meeting. 2000. in press.
2. S. Filippi, R. Ruffini and A. Sepulveda, (2002). Functional Method to solving the Euler Equation for Self-Gravitating Systems. MM Marcel Grossmann Meeting. 2000. in press
3. S. Filippi, A. Sepulveda, (2002). A general theory for self Gravitating systems: Shapes of Astronomical Objects. "THE SHAPES OF GALAXIES AND THEIR DARK HALOS". New Haven, Connecticut, USA 28 - 30 May 2001. (pp. 200). Proceedings of the Yale Cosmology Workshop, World Scientific.
4. S. Filippi, R. Ruffini, A. Sepulveda. Functional Approach to the problem of self-gravitating systems: Conditions of integrability, Physical Review D, 65, 044019, (Paper X), 2002

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