Figures of equilibrium inside a gravitating ring and the limiting oblateness of elliptical galaxies

Abstract

A new class of figures of equilibrium for a rotating gravitating fluid located inside a gravitating ring or torus is studied. These figures form a family of sequences of generalized oblate spheroids, in which there is for any value of the tidal parameter α in the interval 0 ≤ $0 \leqslant \frac{\alpha }{{\pi G\rho }} \leqslant 0.1867$ ≤ 0.1867 a sequence of spheroids with oblatenesses emin (α) ≤ e ≤ e _max (α). A series of classicalMaclaurin spheroids from a sphere to a flat disk is obtained for α = 0. At intermediate values 0 < α ≤ α _max, there are two limiting non-rotating spheroids in each sequence. When α = α _max, the sequence degenerates into a single non-rotating spheroid with e _cr ≈ 0.9600, corresponding to the maximum oblateness of E7 elliptical galaxies. The second part of the paper considers the influence of rings of dark matter on the dynamics of elliptical galaxies. It is proposed that the equilibrium of an oblate isolated non-rotating galaxy is unstable, and it cannot be supported purely by anisotropy of the stellar velocity dispersion. A ring of dark matter can stabilize a weakly rotating galaxy, supplementing standard dynamical models for such stellar systems. In order for a galaxy to acquire appreciable oblateness, the mass of the ring must be an order of magnitude higher than the mass of the galaxy itself, consistent with the ratios of the masses of dark and baryonic matter in the Universe. The influence of massive external rings could shed light on the existence of galaxies with the critical oblateness E7.