AN O(N LOG N) INTEGRATION SCHEME FOR COLLISIONAL STELLAR-SYSTEMS

We present a new tree-based, high-order integration scheme for the numerical simulation of collisional N-body systems. Unlike previous tree codes, in our implementation the tree is allowed to deform with time, and the properties of each component cell are not calculated afresh at each step but instead are predicted only as needed. Cells are represented by a multipole expansion that can extend as far as the octupole term, depending on the relative costs of the multipole and individual-particle calculations. A block time step algorithm is used to simplify scheduling and to allow full vectorization of the code. Both particle and tree time steps are included within the same block structure. Strongly interacting particles are treated through the use of Kustaanheimo-Stiefel regularization, and only the center of mass of a binary appears in the tree structure, effectively limiting the tree's maximum depth. The integrator conserves energy to a few parts per million per crossing time, and its measured O(N1.5) scaling suggests that it will become competitive with NBODY5, the ''standard'' high-precision integrator for small systems, for N greater than or similar to 10(4). In the present implementation, for N < 10(4), the scaling falls somewhat short of the expected O(N log N) behavior, in large part because the tree-traversal algorithm has not quite reached its asymptotic regime. We have applied the new tree code to the problem of core collapse in a 1024 body equal-mass system, and have found results in excellent agreement with those obtained using NBODY5. We conclude that tree-based schemes can be confidently applied to collisional problems without excessive concern that the tree algorithm will seriously disturb the relaxation processes that drive the evolution.