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Topics of Research:
The
equilibrium configurations of rapidly rotating, selfgravitating 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 nonlinear, 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 IVIII).
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 selfgravitating 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
selfconsistent analytical models. It allows the study of rotating and selfgravitating
polytropic systems with nonlinear 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 selfconsistent way, from an arbitrary law of rotation, of the equilibrium
distribution of mass within a selfgravitating system. Such a distribution in general is
nonaxisymmetric and nonellipsoidal. The nonlinear 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 selfgravitating systems,
which are the fundamental tools of the classical works based on the virial theorem.
