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

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∕r2. 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 L2∕(2I) where I is the moment of inertia. Configurations, where the energy E has a local minimum, are stable. In the continuum limit N→∞, 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.