R3 fluids

The current paper is aimed at getting more insight on three main points concerning large-scale astrophysical systems, namely: (i) formulation of tensor virial equations from the standpoint of analytical mechanics; (ii) investigation on the role of systematic and random motions with respect to virial equilibrium configurations; (iii) determination of extent to which systematic and random motions are equivalent in flattening or elongating the shape of a mass distribution. The tensor virial equations are formulated regardless of the nature of the system and its constituents, by generalizing and extending a procedure used for the scalar virial equations in presence of discrete subunits (Landau and Lifchitz 1966). In particular, the self potential-energy tensor is shown to be symmetric with respect to the exchange of the indices, (E_{pot})_{pq}= (E_{pot})_{qp}. Then the results are extended to continuous mass distributions. The role of systematic and random motions in collisionless, ideal, self-gravitating fluids is analysed in detail including radial and tangential velocity dispersion on the equatorial plane, and the related mean angular velocity, overline{Ω}, is conceived as a figure rotation. R3 fluids are defined as ideal, self-gravitating fluids in virial equilibrium, with systematic rotation around a principal axis of inertia, taken to be x_3. The related virial equations are written in terms of the moment of inertia tensor, I_{pq}, the self potential-energy tensor, (E_{pot})_ {pq}, and the generalized anisotropy tensor, ζ_{pq} (Caimmi and Marmo 2005, Caimmi 2006a). Additional effort is devoted to the investigation of the properties of axisymmetric and triaxial configurations. A unified theory of systematic and random motions is developed for R3 fluids, taking into consideration imaginary rotation (Caimmi 1996b, 2006a), and a number of theorems previously stated for homeoidally striated Jacobi ellipsoids (Caimmi 2006a) are extended to the more general case of R3 fluids. The effect of random motion excess is shown to be equivalent to an additional real or imaginary rotation, respectively, inducing flattening (along the equatorial plane) or elongation (along the rotation axis). Then it is realized that a R3 fluid always admits an adjoint configuration with isotropic random velocity distribution. In addition, further constraints are established on the amount of random velocity anisotropy along the principal axes, for triaxial configurations. A necessary condition is formulated for the occurrence of bifurcation points from axisymmetric to triaxial configurations in virial equilibrium, which is independent of the anisotropy parameters. A particularization of general relations is made to the special case of homeoidally striated Jacobi ellipsoid, and some previously known results (Caimmi 2006a) are reproduced.