Numerical simulations of collisions between rotating particles

Abstract

Numerical simulations of Keplerian systems consisting of 200 mutually colliding rotating particles indicate that friction and surface irregularity typically reduce the equilibrium velocity dispersion. Friction and irregularity also transfer some of the energy of random velocities to rotational velocities. Simulations confirm the theoretical predictions (H. Salo, 1987, Earth, Moon, Planets, in press) according to which the equilibrium ratio of rotational energy random kinetic energy is 2 β/(14 − 5 β), or ≲0.2, for spherical identical particle and 2(1 + α)/7 ≈ 0.44 for irregular but frictionless identical particles, α and β being the coefficients of resitution and friction. In systems of different sized particles, the dependence of the dispersion of rotational velocities, √Ω, versus particle mass is rather similar to that of random velocity dispersion, √ T. For a power-law distribution extending over mass-interval 2 9, with the power-index 2, the values of √Ω and √T for smallest particles are about five- and fourfold as compared to those of the largest particle. Inclusion of rotation does not change the previously observed Rayleigh distribution of eccentricies and inclinations, while the components of the rotational velocities are found to follow a Gaussian distribution. The z-component of the rotational velocities, σω z , attains a positive residual mean value, proportional to particle radius σ. For mass-point systems the mean of σω z is much smaller than its dispersion, but for near monolayers they are comparable, leading to a clear prograde alignment of spin axes. The application to the rarefied regions of Saturn's rings indicates that friction is able to reduce the equilibrium geometric thickness by about one-half. For the layer of smallest centimeter-sized particles this corresponds to about 15 m while the largest particles are confined to a monolayer. Rotational periods are roughly proportional to particle radius, about one-half of the orbital period for meter-sized particles.