Structure and Morphology of Galaxies: Equilibrium and Stability |

## Participants:
Simonetta Filippi
G.P. Imponente RemoRuffini (ICRA,Phys.Dept. at University "La Sapienza") Luis Alberto Sanchez(ICRA, Universidad Nacional, Medellin, Colombia)
Alonso Sepulveda
Costantino
Sigismondi ## Topics of Research:
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 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) 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 ( |