A Freefall-based Switching Criterion for P ³ T Hybrid N -body Methods in Collisional Stellar Systems

The P 3 T scheme is a hybrid method for simulating gravitational N -body systems. It combines a fast particle-tree (PT) algorithm for long-range forces with a high-accuracy particle–particle (PP; direct N -body) solver for short-range interactions. Preserving both PT efficiency and PP accuracy requires a robust PT–PP switching criterion. We introduce a simple freefall-based switching criterion for general stellar systems alongside the commonly used velocity-dispersion-based ( σ -based) criterion. Using the petar code with the P 3 T scheme and slowdown algorithmic regularization for binaries and higher-order multiples, we perform extensive simulations of star clusters to evaluate how each criterion affects energy conservation and binary evolution. For systems in virial equilibrium, we find that the freefall-based criterion is generally more accurate for low- σ or loose clusters containing binaries, whereas the σ -based criterion is better suited for high- σ systems. Under subvirial or fractal initial conditions, both criteria struggle to maintain high energy conservation; however, the freefall-based criterion improves as the tree time step is reduced, whereas the σ -based criterion degrades due to its low-accuracy treatment of two-body encounters.