Particle–Particle Particle–Tree: A Direct-Tree Hybrid Scheme for Collisional N-Body Simulations

Abstract

Abstract In this paper, we present a new hybrid algorithm for the time integration of collisional $N$-body systems. In this algorithm, the gravitational force between two particles is divided into short- and long-range terms, using a distance-dependent cutoff function. The long-range interaction is calculated using a tree algorithm, and integrated with a constant-timestep leapfrog integrator. The short-range term is calculated directly and integrated with the high-order Hermite scheme. We can reduce the calculation cost per orbital period from $O$($N^2$) to $O$($N$log$N$), without significantly increasing the long-term integration error. The results of our test simulations show that close encounters are integrated accurately. Long-term errors of the total energy show random-walk behaviour, because they are dominated by the error caused by the tree approximation.