Rigidly rotating gravitationally bound systems of point particles, compared to polytropes

Abstract

In order to simulate rigidly rotating polytropes, we have simulated systems of [Formula: see text] point particles, with [Formula: see text] up to 1800. Two particles at a distance [Formula: see text] interact by an attractive potential [Formula: see text] and a repulsive potential [Formula: see text]. The repulsion simulates the pressure in a polytropic gas of polytropic index [Formula: see text]. We take the total angular momentum [Formula: see text] to be conserved, but not the total energy [Formula: see text]. The particles are stationary in the rotating coordinate system. The rotational energy is [Formula: see text] where [Formula: see text] is the moment of inertia. Configurations, where the energy [Formula: see text] has a local minimum, are stable. In the continuum limit [Formula: see text], the particles become more and more tightly packed in a finite volume, with the interparticle distances decreasing as [Formula: see text]. We argue that [Formula: see text] is a good parameter for describing the continuum limit. We argue further that the continuum limit is the polytropic gas of index [Formula: see text]. For example, the density profile of the nonrotating gas approaches that computed from the Lane–Emden equation describing the nonrotating polytropic gas. In the case of maximum rotation, the instability occurs by the loss of particles from the equator, which becomes a sharp edge, as predicted by Jeans in his study of rotating polytropes. We describe the minimum energy nonrotating configurations for a number of small values of [Formula: see text].