ABSTRACT
Discrete particle molecular dynamics (MD) in 3D is used for numerical simulations of azimuthally complete ring systems.Time is discretized and for each (constant) time-step the equations of motion are integrated. Particles are treated as soft spheres and during a collision between particles i and j, with radii ai and aj , a linear repulsive force and dissipation of kinetic energy in normal direction, nij = rij/rij = (ri − rj)/|ri − rj|, is considered (see the linear spring dashpot model described in (Luding 1998)).The loss of kinetic energy at each collision is expressed by the coefficient of normal restitution r = v′(n)ij /v(n)ij that gives the ratio of the relative velocities vij = |vij| = |vi − vj| in normal direction after (primed) and before (unprimed) the collision.The long-ranged forces are split into the external volume force (each particle interacts with the central potential with mass mc) and self-gravitational force (each particle interacts with all others). To summarize, during a time-step on one ring particle i always long-ranged forces are acting,
and, if rij < ai + aj , short-ranged contact forces as well
Here is ric = |ric| = |ri − rc| the distance of i from the central mass with rc as its position vector, γn a measure for how much kinetic energy is dissipated at each collision and k the spring constant that defines the stiffness of a contact. K = cG defines via c the strength of the self-gravitation of the ring particles, where G denotes the gravitational constant.
