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 $1/r^2$. 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 $N\to\infty$ 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 $N$.