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

摘要

In order to simulate rigidly rotating polytropes, we have simulated systems of N point particles, with N up to 1800. Two particles at a distance r interact by an attractive potential -1/r and a repulsive potential 1/r( 2). The repulsion simulates the pressure in a polytropic gas of polytropic index 3/2. We take the total angular momentum L to be conserved, but not the total energy E. The particles are stationary in the rotating coordinate system. The rotational energy is L-2/(2I) where I is the moment of inertia. Configurations, where the energy E has a local minimum, are stable. In the continuum limit N -> infinity, the particles become more and more tightly packed in a finite volume, with the interparticle distances decreasing as N -1/3. We argue that N(-1/3 )is a good parameter for describing the continuum limit. We argue further that the continuum limit is the polytropic gas of index 3/2. 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 N.