Collision detection for N-body Kepler systems

In a Keplerian system, a large number of bodies orbit a central mass. Accretion disks, protoplanetary disks, asteroid belts, and planetary rings are examples. Simulations of these systems require algorithms that are computationally efficient. The inclusion of collisions in the simulations is challenging but important. We intend to calculate the time of collision of two astronomical bodies in intersecting Kepler orbits as a function of the orbital elements. The aim is to use the solution in an analytic propagator ($N$-body simulation) that jumps from one collision event to the next. We outline an algorithm that maintains a list of possible collision pairs ordered chronologically. At each step (the soonest event on the list), only the particles created in the collision can cause new collision possibilities. We estimate the collision rate, the length of the list, and the average change in this length at an event, and study the efficiency of the method used. We find that the collision-time problem is equivalent to finding the grid point between two parallel lines that is closest to the origin. The solution is based on the continued fraction of the ratio of orbital periods. Due to the large jumps in time, the algorithm can beat tree codes (octree and $k$-d tree codes can efficiently detect collisions) for specific systems such as the Solar System with $N<10^8$. However, the gravitational interactions between particles can only be treated as gravitational scattering or as a secular perturbation, at the cost of reducing the time-step or at the cost of accuracy. While simulations of this size with high-fidelity propagators can already span vast timescales, the high efficiency of the collision detection allows many runs from one initial state or a large sample set, so that one can study statistics.