Numerical self-consistent distribution function of flattened ring models

Abstract

We provide numerical, self-consistent distribution functions for several flat ring models by simultaneously solving the Fokker-Planck equation and the Poisson equation. In particular, we calculated the distribution function of flat ring systems formed by superposing Kuzmin-Toomre disc solutions and an analytic homogeneous ring solution in terms of complete elliptic integrals. We used these geometrical disc solutions, together with physical parameters, to model more realistic physical systems. Moreover, we defined a cutoff radius to handle the infinite Kuzmin-Toomre disc families numerically. The Fokker-Planck equation is solved by a direct numerical method using the finite difference method, the left conjugate direction algorithm, and a simple boundary condition that allows us to find good results for a large set of physical parameters and for values of the collision term of the same order of magnitude (or larger) when compared with the other terms in the Fokker-Planck equation. The collision term of the Fokker-Planck equation is explicitly calculated by applying the known Rosenbluth potentials for gravitational encounters. Limitations of the method are also discussed.