Numerical simulations of collisions in Keplerian systems

Abstract

Computer simulations which were carried out for Keplerian collisional systems of 250 frictionless particles with a ratio of particle radius to mean semi-major axis of 0.001, confirm the theoretically predicted evolution very well until the thickness of the system is a few times the particle radius and the mass-point approximation becomes invalidated. Before this happens, the collisional contraction of denserregions can be observed. The local dispersions of the perihelia and ascending nodes diminish if the local mean orbit is not too close to a circle with zero inclination. When the mass-point approximation ceases to be valid, the system begins to expand, but with parameter values of our standard system this process is much slower than the simultaneously observed evolution toward grazing collisions which do not affect the orbital elements. Therefore, such systems are not dispersed into the space. If the ratio of particle radius to semi-major axis is larger, the expansion becomes faster and the contraction ceases earlier. In late evolutionary phases the thickness of the system remains essentially constant. At the end of the longest simulation (70 000 impacts) the centres of the particles were in a layer of thickness twice the radius of the particles. The cross-section of the system is often wave-like or irregular and may even include detached parts with their own mean plane. Accordingly the thickness as derived from the root-mean-square inclination of the whole system exceeds the true local thickness. The local dispersion of eccentricities may also be considerably smaller than the root-mean-square eccentricity of the whole system.